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l3lab

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miniCTX

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/- Copyright 2023 Daniel J. Velleman -/ import HTPILib.Chap6 namespace HTPI /- Definitions -/ lemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by have h : n + 1 > 0 := Nat.succ_pos n show a % (n + 1) < n + 1 from Nat.mod_lt a h done def gcd (a b : Nat) : Nat := match b with | 0 => a | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd (n + 1) (a % (n + 1)) termination_by b mutual def gcd_c1 (a b : Nat) : Int := match b with | 0 => 1 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c2 (n + 1) (a % (n + 1)) --Corresponds to s = t' termination_by b def gcd_c2 (a b : Nat) : Int := match b with | 0 => 0 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c1 (n + 1) (a % (n + 1)) - (gcd_c2 (n + 1) (a % (n + 1))) * ↑(a / (n + 1)) --Corresponds to t = s' - t'q termination_by b end def prime (n : Nat) : Prop := 2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n def prime_factor (p n : Nat) : Prop := prime p ∧ p ∣ n def all_prime (l : List Nat) : Prop := ∀ p ∈ l, prime p def nondec (l : List Nat) : Prop := match l with | [] => True --Of course, True is a proposition that is always true | n :: L => (∀ m ∈ L, n ≤ m) ∧ nondec L def nondec_prime_list (l : List Nat) : Prop := all_prime l ∧ nondec l def prod (l : List Nat) : Nat := match l with | [] => 1 | n :: L => n * (prod L) def prime_factorization (n : Nat) (l : List Nat) : Prop := nondec_prime_list l ∧ prod l = n def rel_prime (a b : Nat) : Prop := gcd a b = 1 def congr_mod (m : Nat) (a b : Int) : Prop := (↑m : Int) ∣ (a - b) def cc (m : Nat) (a : Int) : ZMod m := (↑a : ZMod m) notation:50 a " ≡ " b " (MOD " m ")" => congr_mod m a b notation:max "["a"]_"m:max => cc m a def invertible {m : Nat} (X : ZMod m) : Prop := ∃ (Y : ZMod m), X * Y = [1]_m def num_rp_below (m k : Nat) : Nat := match k with | 0 => 0 | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j def phi (m : Nat) : Nat := num_rp_below m m def prod_seq {m : Nat} (j k : Nat) (f : Nat → ZMod m) : ZMod m := match j with | 0 => [1]_m | n + 1 => prod_seq n k f * f (k + n) def maps_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i < n, g i < n def one_one_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i1 < n, ∀ i2 < n, g i1 = g i2 → i1 = i2 def onto_below (n : Nat) (g : Nat → Nat) : Prop := ∀ k < n, ∃ i < n, g i = k def perm_below (n : Nat) (g : Nat → Nat) : Prop := maps_below n g ∧ one_one_below n g ∧ onto_below n g def inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m) def swap (u v i : Nat) : Nat := if i = u then v else if i = v then u else i namespace Euler --For definitions specific to Euler's theorem def F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m def G (m a i : Nat) : Nat := (a * i) % m def Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i end Euler /- Section 7.1 -/ theorem dvd_mod_of_dvd_a_b {a b d : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ (a % b) := by set q : Nat := a / b have h3 : b * q + a % b = a := Nat.div_add_mod a b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 define --Goal : ∃ (c : Nat), a % b = d * c apply Exists.intro (j - k * q) show a % b = d * (j - k * q) from calc a % b _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm _ = a - b * q := by rw [h3] _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc] _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm done theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry #eval gcd 672 161 --Answer: 7 lemma gcd_base (a : Nat) : gcd a 0 = a := by rfl lemma gcd_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd a b = gcd b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1)) rfl done lemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b ≠ 0) : a % b < b := by have h1 : b > 0 := Nat.pos_of_ne_zero h show a % b < b from Nat.mod_lt a h1 done lemma dvd_self (n : Nat) : n ∣ n := by apply Exists.intro 1 ring done theorem gcd_dvd : ∀ (b a : Nat), (gcd a b) ∣ a ∧ (gcd a b) ∣ b := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd a b_1) ∣ a ∧ (gcd a b_1) ∣ b_1 fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_base] --Goal: a ∣ a ∧ a ∣ 0 apply And.intro (dvd_self a) define apply Exists.intro 0 rfl done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_nonzero a h1] --Goal : gcd b (a % b) ∣ a ∧ gcd b (a % b) ∣ b have h2 : a % b < b := mod_nonzero_lt a h1 have h3 : (gcd b (a % b)) ∣ b ∧ (gcd b (a % b)) ∣ (a % b) := ih (a % b) h2 b apply And.intro _ h3.left show (gcd b (a % b)) ∣ a from dvd_a_of_dvd_b_mod h3.left h3.right done done theorem gcd_dvd_left (a b : Nat) : (gcd a b) ∣ a := (gcd_dvd b a).left theorem gcd_dvd_right (a b : Nat) : (gcd a b) ∣ b := (gcd_dvd b a).right lemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl lemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c1 a b = gcd_c2 b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done lemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl lemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * ↑(a / b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done theorem gcd_lin_comb : ∀ (b a : Nat), (gcd_c1 a b) * ↑a + (gcd_c2 a b) * ↑b = ↑(gcd a b) := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd_c1 a b_1) * ↑a + (gcd_c2 a b_1) * ↑b_1 = ↑(gcd a b_1) fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base] --Goal : 1 * ↑a + 0 * ↑0 = ↑a ring done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1] --Goal : gcd_c2 b (a % b) * ↑a + -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * ↑(a / b)) * ↑b = -- ↑(gcd b (a % b)) set r : Nat := a % b set q : Nat := a / b set s : Int := gcd_c1 b r set t : Int := gcd_c2 b r --Goal : t * ↑a + (s - t * ↑q) * ↑b = ↑(gcd b r) have h2 : r < b := mod_nonzero_lt a h1 have h3 : s * ↑b + t * ↑r = ↑(gcd b r) := ih r h2 b have h4 : b * q + r = a := Nat.div_add_mod a b rewrite [←h3, ←h4] rewrite [Nat.cast_add, Nat.cast_mul] --Goal : t * (↑b * ↑q + ↑r) + (s - t * ↑q) * ↑b = s * ↑b + t * ↑r ring done done #eval gcd_c1 672 161 --Answer: 6 #eval gcd_c2 672 161 --Answer: -25 --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161 theorem Theorem_7_1_6 {d a b : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ gcd a b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑d ∣ ↑(gcd a b) set s : Int := gcd_c1 a b set t : Int := gcd_c2 a b have h3 : s * ↑a + t * ↑b = ↑(gcd a b) := gcd_lin_comb b a rewrite [←h3] --Goal : ↑d ∣ s * ↑a + t * ↑b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul] --Goal : ↑d ∣ s * (↑d * ↑j) + t * (↑d * ↑k) define apply Exists.intro (s * ↑j + t * ↑k) ring done /- Section 7.2 -/ theorem dvd_trans {a b c : Nat} (h1 : a ∣ b) (h2 : b ∣ c) : a ∣ c := by define at h1; define at h2; define obtain (m : Nat) (h3 : b = a * m) from h1 obtain (n : Nat) (h4 : c = b * n) from h2 rewrite [h3, mul_assoc] at h4 apply Exists.intro (m * n) show c = a * (m * n) from h4 done lemma exists_prime_factor : ∀ (n : Nat), 2 ≤ n → ∃ (p : Nat), prime_factor p n := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, 2 ≤ n_1 → ∃ (p : Nat), prime_factor p n_1 assume h1 : 2 ≤ n by_cases h2 : prime n · -- Case 1. h2 : prime n apply Exists.intro n define --Goal : prime n ∧ n ∣ n show prime n ∧ n ∣ n from And.intro h2 (dvd_self n) done · -- Case 2. h2 : ¬prime n define at h2 --h2 : ¬(2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n) demorgan at h2 disj_syll h2 h1 obtain (a : Nat) (h3 : ∃ (b : Nat), a * b = n ∧ a < n ∧ b < n) from h2 obtain (b : Nat) (h4 : a * b = n ∧ a < n ∧ b < n) from h3 have h5 : 2 ≤ a := by by_contra h6 have h7 : a ≤ 1 := by linarith have h8 : n ≤ b := calc n _ = a * b := h4.left.symm _ ≤ 1 * b := by rel [h7] _ = b := by ring linarith --n ≤ b contradicts b < n done have h6 : ∃ (p : Nat), prime_factor p a := ih a h4.right.left h5 obtain (p : Nat) (h7 : prime_factor p a) from h6 apply Exists.intro p define --Goal : prime p ∧ p ∣ n define at h7 --h7 : prime p ∧ p ∣ a apply And.intro h7.left have h8 : a ∣ n := by apply Exists.intro b show n = a * b from (h4.left).symm done show p ∣ n from dvd_trans h7.right h8 done done lemma exists_least_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q := by set S : Set Nat := {p : Nat | prime_factor p n} have h2 : ∃ (p : Nat), p ∈ S := exists_prime_factor n h show ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q from well_ord_princ S h2 done lemma all_prime_nil : all_prime [] := by define --Goal : ∀ p ∈ [], prime p fix p : Nat contrapos --Goal : ¬prime p → p ∉ [] assume h1 : ¬prime p show p ∉ [] from List.not_mem_nil p done lemma all_prime_cons (n : Nat) (L : List Nat) : all_prime (n :: L) ↔ prime n ∧ all_prime L := by apply Iff.intro · -- (→) assume h1 : all_prime (n :: L) --Goal : prime n ∧ all_prime L define at h1 --h1 : ∀ p ∈ n :: L, prime p apply And.intro (h1 n (List.mem_cons_self n L)) define --Goal : ∀ p ∈ L, prime p fix p : Nat assume h2 : p ∈ L show prime p from h1 p (List.mem_cons_of_mem n h2) done · -- (←) assume h1 : prime n ∧ all_prime L --Goal : all_prime (n :: l) define : all_prime L at h1 define fix p : Nat assume h2 : p ∈ n :: L rewrite [List.mem_cons] at h2 --h2 : p = n ∨ p ∈ L by_cases on h2 · -- Case 1. h2 : p = n rewrite [h2] show prime n from h1.left done · -- Case 2. h2 : p ∈ L show prime p from h1.right p h2 done done done lemma nondec_nil : nondec [] := by define --Goal : True trivial --trivial proves some obviously true statements, such as True done lemma nondec_cons (n : Nat) (L : List Nat) : nondec (n :: L) ↔ (∀ m ∈ L, n ≤ m) ∧ nondec L := by rfl lemma prod_nil : prod [] = 1 := by rfl lemma prod_cons : prod (n :: L) = n * (prod L) := by rfl lemma exists_cons_of_length_eq_succ {A : Type} {l : List A} {n : Nat} (h : l.length = n + 1) : ∃ (a : A) (L : List A), l = a :: L ∧ L.length = n := by have h1 : ¬l.length = 0 := by linarith rewrite [List.length_eq_zero] at h1 obtain (a : A) (h2 : ∃ (L : List A), l = a :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List A) (h3 : l = a :: L) from h2 apply Exists.intro a apply Exists.intro L apply And.intro h3 have h4 : (a :: L).length = L.length + 1 := List.length_cons a L rewrite [←h3, h] at h4 show L.length = n from (Nat.add_right_cancel h4).symm done lemma list_elt_dvd_prod_by_length (a : Nat) : ∀ (n : Nat), ∀ (l : List Nat), l.length = n → a ∈ l → a ∣ prod l := by by_induc · --Base Case fix l : List Nat assume h1 : l.length = 0 rewrite [List.length_eq_zero] at h1 --h1 : l = [] rewrite [h1] --Goal : a ∈ [] → a ∣ prod [] contrapos assume h2 : ¬a ∣ prod [] show a ∉ [] from List.not_mem_nil a done · -- Induction Step fix n : Nat assume ih : ∀ (l : List Nat), List.length l = n → a ∈ l → a ∣ prod l fix l : List Nat assume h1 : l.length = n + 1 --Goal : a ∈ l → a ∣ prod l obtain (b : Nat) (h2 : ∃ (L : List Nat), l = b :: L ∧ L.length = n) from exists_cons_of_length_eq_succ h1 obtain (L : List Nat) (h3 : l = b :: L ∧ L.length = n) from h2 have h4 : a ∈ L → a ∣ prod L := ih L h3.right assume h5 : a ∈ l rewrite [h3.left, prod_cons] --Goal : a ∣ b * prod L rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b ∨ a ∈ L by_cases on h5 · -- Case 1. h5 : a = b apply Exists.intro (prod L) rewrite [h5] rfl done · -- Case 2. h5 : a ∈ L have h6 : a ∣ prod L := h4 h5 have h7 : prod L ∣ b * prod L := by apply Exists.intro b ring done show a ∣ b * prod L from dvd_trans h6 h7 done done done lemma list_elt_dvd_prod {a : Nat} {l : List Nat} (h : a ∈ l) : a ∣ prod l := by set n : Nat := l.length have h1 : l.length = n := by rfl show a ∣ prod l from list_elt_dvd_prod_by_length a n l h1 h done lemma exists_prime_factorization : ∀ (n : Nat), n ≥ 1 → ∃ (l : List Nat), prime_factorization n l := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, n_1 ≥ 1 → ∃ (l : List Nat), prime_factorization n_1 l assume h1 : n ≥ 1 by_cases h2 : n = 1 · -- Case 1. h2 : n = 1 apply Exists.intro [] define apply And.intro · -- Proof of nondec_prime_list [] define show all_prime [] ∧ nondec [] from And.intro all_prime_nil nondec_nil done · -- Proof of prod [] = n rewrite [prod_nil, h2] rfl done done · -- Case 2. h2 : n ≠ 1 have h3 : n ≥ 2 := lt_of_le_of_ne' h1 h2 obtain (p : Nat) (h4 : prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q) from exists_least_prime_factor h3 have p_prime_factor : prime_factor p n := h4.left define at p_prime_factor have p_prime : prime p := p_prime_factor.left have p_dvd_n : p ∣ n := p_prime_factor.right have p_least : ∀ (q : Nat), prime_factor q n → p ≤ q := h4.right obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n have h5 : m ≠ 0 := by contradict h1 with h6 have h7 : n = 0 := calc n _ = p * m := n_eq_pm _ = p * 0 := by rw [h6] _ = 0 := by ring rewrite [h7] decide done have m_pos : 0 < m := Nat.pos_of_ne_zero h5 have m_lt_n : m < n := by define at p_prime show m < n from calc m _ < m + m := by linarith _ = 2 * m := by ring _ ≤ p * m := by rel [p_prime.left] _ = n := n_eq_pm.symm done obtain (L : List Nat) (h6 : prime_factorization m L) from ih m m_lt_n m_pos define at h6 have ndpl_L : nondec_prime_list L := h6.left define at ndpl_L apply Exists.intro (p :: L) define apply And.intro · -- Proof of nondec_prime_list (p :: L) define apply And.intro · -- Proof of all_prime (p :: L) rewrite [all_prime_cons] show prime p ∧ all_prime L from And.intro p_prime ndpl_L.left done · -- Proof of nondec (p :: L) rewrite [nondec_cons] apply And.intro _ ndpl_L.right fix q : Nat assume q_in_L : q ∈ L have h7 : q ∣ prod L := list_elt_dvd_prod q_in_L rewrite [h6.right] at h7 --h7 : q ∣ m have h8 : m ∣ n := by apply Exists.intro p rewrite [n_eq_pm] ring done have q_dvd_n : q ∣ n := dvd_trans h7 h8 have ap_L : all_prime L := ndpl_L.left define at ap_L have q_prime_factor : prime_factor q n := And.intro (ap_L q q_in_L) q_dvd_n show p ≤ q from p_least q q_prime_factor done done · -- Proof of prod (p :: L) = n rewrite [prod_cons, h6.right, n_eq_pm] rfl done done done theorem Theorem_7_2_2 {a b c : Nat} (h1 : c ∣ a * b) (h2 : rel_prime a c) : c ∣ b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑c ∣ ↑b define at h1; define at h2; define obtain (j : Nat) (h3 : a * b = c * j) from h1 set s : Int := gcd_c1 a c set t : Int := gcd_c2 a c have h4 : s * ↑a + t * ↑c = ↑(gcd a c) := gcd_lin_comb c a rewrite [h2, Nat.cast_one] at h4 --h4 : s * ↑a + t * ↑c = (1 : Int) apply Exists.intro (s * ↑j + t * ↑b) show ↑b = ↑c * (s * ↑j + t * ↑b) from calc ↑b _ = (1 : Int) * ↑b := (one_mul _).symm _ = (s * ↑a + t * ↑c) * ↑b := by rw [h4] _ = s * (↑a * ↑b) + t * ↑c * ↑b := by ring _ = s * (↑c * ↑j) + t * ↑c * ↑b := by rw [←Nat.cast_mul a b, h3, Nat.cast_mul c j] _ = ↑c * (s * ↑j + t * ↑b) := by ring done lemma le_nonzero_prod_left {a b : Nat} (h : a * b ≠ 0) : a ≤ a * b := by have h1 : b ≠ 0 := by contradict h with h1 rewrite [h1] ring done have h2 : 1 ≤ b := Nat.pos_of_ne_zero h1 show a ≤ a * b from calc a = a * 1 := (mul_one a).symm _ ≤ a * b := by rel [h2] done lemma le_nonzero_prod_right {a b : Nat} (h : a * b ≠ 0) : b ≤ a * b := by rewrite [mul_comm] rewrite [mul_comm] at h show b ≤ b * a from le_nonzero_prod_left h done lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry lemma rel_prime_of_prime_not_dvd {a p : Nat} (h1 : prime p) (h2 : ¬p ∣ a) : rel_prime a p := by have h3 : gcd a p ∣ a := gcd_dvd_left a p have h4 : gcd a p ∣ p := gcd_dvd_right a p have h5 : gcd a p = 1 ∨ gcd a p = p := dvd_prime h1 h4 have h6 : gcd a p ≠ p := by contradict h2 with h6 rewrite [h6] at h3 show p ∣ a from h3 done disj_syll h5 h6 show rel_prime a p from h5 done theorem Theorem_7_2_3 {a b p : Nat} (h1 : prime p) (h2 : p ∣ a * b) : p ∣ a ∨ p ∣ b := by or_right with h3 have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3 show p ∣ b from Theorem_7_2_2 h2 h4 done lemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a ≥ 1 := by have h1 : a ≠ 0 := by by_contra h1 rewrite [h1] at h contradict h linarith done show a ≥ 1 from Nat.pos_of_ne_zero h1 done lemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by have h1 : a ≥ 1 := ge_one_of_prod_one h have h2 : a * b ≠ 0 := by linarith have h3 : a ≤ a * b := le_nonzero_prod_left h2 rewrite [h] at h3 show a = 1 from Nat.le_antisymm h3 h1 done lemma eq_one_of_dvd_one {n : Nat} (h : n ∣ 1) : n = 1 := by obtain (j : Nat) (h1 : 1 = n * j) from h show n = 1 from eq_one_of_prod_one h1.symm done lemma prime_not_one {p : Nat} (h : prime p) : p ≠ 1 := by define at h linarith done theorem Theorem_7_2_4 {p : Nat} (h1 : prime p) : ∀ (l : List Nat), p ∣ prod l → ∃ a ∈ l, p ∣ a := by apply List.rec · -- Base Case. Goal : p ∣ prod [] → ∃ a ∈ [], p ∣ a rewrite [prod_nil] assume h2 : p ∣ 1 show ∃ a ∈ [], p ∣ a from absurd (eq_one_of_dvd_one h2) (prime_not_one h1) done · -- Induction Step fix b : Nat fix L : List Nat assume ih : p ∣ prod L → ∃ a ∈ L, p ∣ a --Goal : p ∣ prod (b :: L) → ∃ a ∈ b :: L, p ∣ a assume h2 : p ∣ prod (b :: L) rewrite [prod_cons] at h2 have h3 : p ∣ b ∨ p ∣ prod L := Theorem_7_2_3 h1 h2 by_cases on h3 · -- Case 1. h3 : p ∣ b apply Exists.intro b show b ∈ b :: L ∧ p ∣ b from And.intro (List.mem_cons_self b L) h3 done · -- Case 2. h3 : p ∣ prod L obtain (a : Nat) (h4 : a ∈ L ∧ p ∣ a) from ih h3 apply Exists.intro a show a ∈ b :: L ∧ p ∣ a from And.intro (List.mem_cons_of_mem b h4.left) h4.right done done done lemma prime_in_list {p : Nat} {l : List Nat} (h1 : prime p) (h2 : all_prime l) (h3 : p ∣ prod l) : p ∈ l := by obtain (a : Nat) (h4 : a ∈ l ∧ p ∣ a) from Theorem_7_2_4 h1 l h3 define at h2 have h5 : prime a := h2 a h4.left have h6 : p = 1 ∨ p = a := dvd_prime h5 h4.right disj_syll h6 (prime_not_one h1) rewrite [h6] show a ∈ l from h4.left done lemma first_le_first {p q : Nat} {l m : List Nat} (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m)) (h3 : prod (p :: l) = prod (q :: m)) : p ≤ q := by define at h1; define at h2 have h4 : q ∣ prod (p :: l) := by define apply Exists.intro (prod m) rewrite [←prod_cons] show prod (p :: l) = prod (q :: m) from h3 done have h5 : all_prime (q :: m) := h2.left rewrite [all_prime_cons] at h5 have h6 : q ∈ p :: l := prime_in_list h5.left h1.left h4 have h7 : nondec (p :: l) := h1.right rewrite [nondec_cons] at h7 rewrite [List.mem_cons] at h6 by_cases on h6 · -- Case 1. h6 : q = p linarith done · -- Case 2. h6 : q ∈ l have h8 : ∀ m ∈ l, p ≤ m := h7.left show p ≤ q from h8 q h6 done done lemma nondec_prime_list_tail {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by define at h define rewrite [all_prime_cons, nondec_cons] at h show all_prime l ∧ nondec l from And.intro h.left.right h.right.right done lemma cons_prod_not_one {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : prod (p :: l) ≠ 1 := by define at h have h1 : all_prime (p :: l) := h.left rewrite [all_prime_cons] at h1 rewrite [prod_cons] by_contra h2 show False from (prime_not_one h1.left) (eq_one_of_prod_one h2) done lemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) : l = [] ↔ prod l = 1 := by apply Iff.intro · -- (→) assume h1 : l = [] rewrite [h1] show prod [] = 1 from prod_nil done · -- (←) contrapos assume h1 : ¬l = [] obtain (p : Nat) (h2 : ∃ (L : List Nat), l = p :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List Nat) (h3 : l = p :: L) from h2 rewrite [h3] at h rewrite [h3] show ¬prod (p :: L) = 1 from cons_prod_not_one h done done lemma prime_pos {p : Nat} (h : prime p) : p > 0 := by define at h linarith done theorem Theorem_7_2_5 : ∀ (l1 l2 : List Nat), nondec_prime_list l1 → nondec_prime_list l2 → prod l1 = prod l2 → l1 = l2 := by apply List.rec · -- Base Case. Goal : ∀ (l2 : List Nat), nondec_prime_list [] → -- nondec_prime_list l2 → prod [] = prod l2 → [] = l2 fix l2 : List Nat assume h1 : nondec_prime_list [] assume h2 : nondec_prime_list l2 assume h3 : prod [] = prod l2 rewrite [prod_nil, eq_comm, ←list_nil_iff_prod_one h2] at h3 show [] = l2 from h3.symm done · -- Induction Step fix p : Nat fix L1 : List Nat assume ih : ∀ (L2 : List Nat), nondec_prime_list L1 → nondec_prime_list L2 → prod L1 = prod L2 → L1 = L2 -- Goal : ∀ (l2 : List Nat), nondec_prime_list (p :: L1) → -- nondec_prime_list l2 → prod (p :: L1) = prod l2 → p :: L1 = l2 fix l2 : List Nat assume h1 : nondec_prime_list (p :: L1) assume h2 : nondec_prime_list l2 assume h3 : prod (p :: L1) = prod l2 have h4 : ¬prod (p :: L1) = 1 := cons_prod_not_one h1 rewrite [h3, ←list_nil_iff_prod_one h2] at h4 obtain (q : Nat) (h5 : ∃ (L : List Nat), l2 = q :: L) from List.exists_cons_of_ne_nil h4 obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5 rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2) rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2) have h7 : p ≤ q := first_le_first h1 h2 h3 have h8 : q ≤ p := first_le_first h2 h1 h3.symm have h9 : p = q := by linarith rewrite [h9, prod_cons, prod_cons] at h3 --h3 : q * prod L1 = q * prod L2 have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1 have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2 define at h2 have h12 : all_prime (q :: L2) := h2.left rewrite [all_prime_cons] at h12 have h13 : q > 0 := prime_pos h12.left have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3 have h15 : L1 = L2 := ih L2 h10 h11 h14 rewrite [h6, h9, h15] rfl done done theorem fund_thm_arith (n : Nat) (h : n ≥ 1) : ∃! (l : List Nat), prime_factorization n l := by exists_unique · -- Existence show ∃ (l : List Nat), prime_factorization n l from exists_prime_factorization n h done · -- Uniqueness fix l1 : List Nat; fix l2 : List Nat assume h1 : prime_factorization n l1 assume h2 : prime_factorization n l2 define at h1; define at h2 have h3 : prod l1 = n := h1.right rewrite [←h2.right] at h3 show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3 done done /- Section 7.3 -/ theorem congr_refl (m : Nat) : ∀ (a : Int), a ≡ a (MOD m) := by fix a : Int define --Goal : ∃ (c : Int), a - a = ↑m * c apply Exists.intro 0 ring done theorem congr_symm {m : Nat} : ∀ {a b : Int}, a ≡ b (MOD m) → b ≡ a (MOD m) := by fix a : Int; fix b : Int assume h1 : a ≡ b (MOD m) define at h1 --h1 : ∃ (c : Int), a - b = ↑m * c define --Goal : ∃ (c : Int), b - a = ↑m * c obtain (c : Int) (h2 : a - b = m * c) from h1 apply Exists.intro (-c) show b - a = m * (-c) from calc b - a _ = -(a - b) := by ring _ = -(m * c) := by rw [h2] _ = m * (-c) := by ring done theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry /- Fundamental properties of congruence classes -/ lemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) : X = Y ↔ X.val = Y.val := Fin.ext_iff lemma val_nat_eq_mod (n k : Nat) : ([k]_(n + 1)).val = k % (n + 1) := by rfl lemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl theorem cc_rep {m : Nat} (X : ZMod m) : ∃ (a : Int), X = [a]_m := match m with | 0 => by apply Exists.intro X rfl done | n + 1 => by apply Exists.intro ↑(X.val) have h1 : X.val < n + 1 := Fin.prop X rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1] rfl done theorem add_class (m : Nat) (a b : Int) : [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm theorem mul_class (m : Nat) (a b : Int) : [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm lemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ [a - b]_m = [0]_m := by apply Iff.intro · -- (→) assume h1 : [a]_m = [b]_m have h2 : a - b = a + (-b) := by ring have h3 : b + (-b) = 0 := by ring show [a - b]_m = [0]_m from calc [a - b]_m _ = [a + (-b)]_m := by rw [h2] _ = [a]_m + [-b]_m := by rw [add_class] _ = [b]_m + [-b]_m := by rw [h1] _ = [b + -b]_m := by rw [add_class] _ = [0]_m := by rw [h3] done · -- (←) assume h1 : [a - b]_m = [0]_m have h2 : b + (a - b) = a := by ring have h3 : b + 0 = b := by ring show [a]_m = [b]_m from calc [a]_m _ = [b + (a - b)]_m := by rw [h2] _ = [b]_m + [a - b]_m := by rw [add_class] _ = [b]_m + [0]_m := by rw [h1] _ = [b + 0]_m := by rw [add_class] _ = [b]_m := by rw [h3] done done lemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) : [a]_m = [0]_m → [-a]_m = [0]_m := by assume h1 : [a]_m = [0]_m have h2 : 0 + (-a) = -a := by ring have h3 : a + (-a) = 0 := by ring show [-a]_m = [0]_m from calc [-a]_m _ = [0 + (-a)]_m := by rw [h2] _ = [0]_m + [-a]_m := by rw [add_class] _ = [a]_m + [-a]_m := by rw [h1] _ = [a + (-a)]_m := by rw [add_class] _ = [0]_m := by rw [h3] done lemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) : [-a]_m = [0]_m ↔ [a]_m = [0]_m := by apply Iff.intro _ (cc_neg_zero_of_cc_zero m a) assume h1 : [-a]_m = [0]_m have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1 have h3 : -(-a) = a := by ring rewrite [h3] at h2 show [a]_m = [0]_m from h2 done lemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl lemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m ↔ m ∣ k := match m with | 0 => by have h : (0 : Int) = (↑(0 : Nat) : Int) := by rfl rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj] apply Iff.intro · -- (→) assume h1 : k = 0 rewrite [h1] show 0 ∣ 0 from dvd_self 0 done · -- (←) assume h1 : 0 ∣ k obtain (c : Nat) (h2 : k = 0 * c) from h1 rewrite [h2] ring done done | n + 1 => by rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero] show k % (n + 1) = 0 ↔ n + 1 ∣ k from (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm done lemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m ↔ ↑m ∣ a := by obtain (k : Nat) (h1 : a = ↑k ∨ a = -↑k) from Int.eq_nat_or_neg a by_cases on h1 · -- Case 1. h1: a = ↑k rewrite [h1, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done · -- Case 2. h1: a = -↑k rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done done theorem cc_eq_iff_congr (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ a ≡ b (MOD m) := calc [a]_m = [b]_m _ ↔ [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b _ ↔ ↑m ∣ (a - b) := cc_zero_iff_dvd m (a - b) _ ↔ a ≡ b (MOD m) := by rfl /- End of fundamental properties of congruence classes -/ lemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m := by have h1 : (↑m : Int) ≠ 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m) show 0 ≤ a % m from Int.emod_nonneg a h1 done lemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m) have h2 : (↑m : Int) > 0 := (Nat.cast_pos).rtl h1 show a % m < m from Int.emod_lt_of_pos a h2 done lemma congr_mod_mod (m : Nat) (a : Int) : a ≡ a % m (MOD m) := by define have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m apply Exists.intro (a / m) show a - a % m = m * (a / m) from calc a - (a % m) _ = m * (a / m) + a % m - a % m := by rw [h1] _ = m * (a / m) := by ring done lemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) := And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a)) theorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) : ∃! (r : Int), 0 ≤ r ∧ r < m ∧ a ≡ r (MOD m) := by exists_unique · -- Existence apply Exists.intro (a % m) show 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) from mod_cmpl_res m a done · -- Uniqueness fix r1 : Int; fix r2 : Int assume h1 : 0 ≤ r1 ∧ r1 < m ∧ a ≡ r1 (MOD m) assume h2 : 0 ≤ r2 ∧ r2 < m ∧ a ≡ r2 (MOD m) have h3 : r1 ≡ r2 (MOD m) := congr_trans (congr_symm h1.right.right) h2.right.right obtain (d : Int) (h4 : r1 - r2 = m * d) from h3 have h5 : r1 - r2 < m * 1 := by linarith have h6 : m * (-1) < r1 - r2 := by linarith rewrite [h4] at h5 --h5 : m * d < m * 1 rewrite [h4] at h6 --h6 : m * -1 < m * d have h7 : (↑m : Int) ≥ 0 := Nat.cast_nonneg m have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7 have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7 have h10 : d = 0 := by linarith show r1 = r2 from calc r1 _ = r1 - r2 + r2 := by ring _ = m * 0 + r2 := by rw [h4, h10] _ = r2 := by ring done done lemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m := (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a) theorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y rewrite [h1, h2] have h3 : a + b = b + a := by ring show [a]_m + [b]_m = [b]_m + [a]_m from calc [a]_m + [b]_m _ = [a + b]_m := add_class m a b _ = [b + a]_m := by rw [h3] _ = [b]_m + [a]_m := (add_class m b a).symm done theorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X rewrite [h1] have h2 : a * 1 = a := by ring show [a]_m * [1]_m = [a]_m from calc [a]_m * [1]_m _ = [a * 1]_m := mul_class m a 1 _ = [a]_m := by rw [h2] done theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry lemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) : [a]_m * [gcd_c2 m a]_m = [1]_m := by set s : Int := gcd_c1 m a have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m define at h1 rewrite [h1, Nat.cast_one] at h2 --h2 : s * ↑m + gcd_c2 m a * ↑a = 1 rewrite [mul_class, cc_eq_iff_congr] define --Goal : ∃ (c : Int), ↑a * gcd_c2 m a - 1 = ↑m * c apply Exists.intro (-s) show a * (gcd_c2 m a) - 1 = m * (-s) from calc a * (gcd_c2 m a) - 1 _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring _ = 1 + m * (-s) - 1 := by rw [h2] _ = m * (-s) := by ring done theorem Theorem_7_3_7 (m a : Nat) : invertible [a]_m ↔ rel_prime m a := by apply Iff.intro · -- (→) assume h1 : invertible [a]_m define at h1 obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1 obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y rewrite [h3, mul_class, cc_eq_iff_congr] at h2 define at h2 obtain (c : Int) (h4 : a * b - 1 = m * c) from h2 rewrite [Exercise_7_2_6] --Goal : ∃ (s t : Int), s * ↑m + t * ↑a = 1 apply Exists.intro (-c) apply Exists.intro b show (-c) * m + b * a = 1 from calc (-c) * m + b * a _ = (-c) * m + (a * b - 1) + 1 := by ring _ = (-c) * m + m * c + 1 := by rw [h4] _ = 1 := by ring done · -- (←) assume h1 : rel_prime m a define show ∃ (Y : ZMod m), [a]_m * Y = [1]_m from Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1) done done /- Section 7.4 -/ section Euler open Euler lemma num_rp_below_base {m : Nat} : num_rp_below m 0 = 0 := by rfl lemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) : num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by have h1 : num_rp_below m (j + 1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : gcd m j = 1 rewrite [if_pos h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1 show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1 done lemma num_rp_below_step_not_rp {m j : Nat} (h : ¬rel_prime m j) : num_rp_below m (j + 1) = num_rp_below m j := by have h1 : num_rp_below m (j +1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : ¬gcd m j = 1 rewrite [if_neg h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j show num_rp_below m (j + 1) = num_rp_below m j from h1 done lemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl #eval phi 10 --Answer: 4 lemma prod_inv_iff_inv {m : Nat} {X : ZMod m} (h1 : invertible X) (Y : ZMod m) : invertible (X * Y) ↔ invertible Y := by apply Iff.intro · -- (→) assume h2 : invertible (X * Y) obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2 apply Exists.intro (X * Z) rewrite [←h3] --Goal : Y * (X * Z) = X * Y * Z ring --Note that ring can do algebra in ZMod m done · -- (←) assume h2 : invertible Y obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1 obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2 apply Exists.intro (Xi * Yi) show (X * Y) * (Xi * Yi) = [1]_m from calc X * Y * (Xi * Yi) _ = (X * Xi) * (Y * Yi) := by ring _ = [1]_m * [1]_m := by rw [h3, h4] _ = [1]_m := Theorem_7_3_6_7 [1]_m done done lemma F_rp_def {m i : Nat} (h : rel_prime m i) : F m i = [i]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h --h : gcd m i = 1 rewrite [if_pos h] at h1 show F m i = [i]_m from h1 done lemma F_not_rp_def {m i : Nat} (h : ¬rel_prime m i) : F m i = [1]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h rewrite [h1, if_neg h] rfl done lemma prod_seq_base {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 0 k f = [1]_m := by rfl lemma prod_seq_step {m : Nat} (n k : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl lemma prod_seq_zero_step {m : Nat} (n : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by rewrite [prod_seq_step, zero_add] rfl done lemma prod_one {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 1 k f = f k := by rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7] rfl done lemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl lemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m := calc [G m a i]_m _ = [(a * i) % m]_m := by rfl _ = [a * i]_m := (cc_eq_mod m (a * i)).symm _ = [a]_m * [i]_m := (mul_class m a i).symm lemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) : rel_prime m (G m a i) ↔ rel_prime m i := by have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1 show rel_prime m (G m a i) ↔ rel_prime m i from calc rel_prime m (G m a i) _ ↔ invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm _ ↔ invertible ([a]_m * [i]_m) := by rw [cc_G] _ ↔ invertible [i]_m := prod_inv_iff_inv h2 ([i]_m) _ ↔ rel_prime m i := Theorem_7_3_7 m i done lemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) : F m (G m a i) = [a]_m * F m i := by have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2 show F m (G m a i) = [a]_m * F m i from calc F m (G m a i) _ = [G m a i]_m := F_rp_def h3 _ = [a]_m * [i]_m := cc_G m a i _ = [a]_m * F m i := by rw [F_rp_def h2] done lemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : ¬rel_prime m i) : F m (G m a i) = [1]_m := by rewrite [←G_rp_iff h1 i] at h2 show F m (G m a i) = [1]_m from F_not_rp_def h2 done lemma FG_prod {m a : Nat} (h1 : rel_prime m a) : ∀ (k : Nat), prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by by_induc · -- Base Case show prod_seq 0 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from calc prod_seq 0 0 ((F m) ∘ (G m a)) _ = [1]_m := prod_seq_base _ _ _ = [a]_m ^ 0 * [1]_m := by ring _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by rw [num_rp_below_base, prod_seq_base] done · -- Induction Step fix k : Nat assume ih : prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) by_cases h2 : rel_prime m k · -- Case 1. h2 : rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([a]_m * F m k) := by rw [FG_rp h1 h2] _ = [a]_m ^ ((num_rp_below m k) + 1) * ((prod_seq k 0 (F m)) * F m k) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_rp h2, prod_seq_zero_step] done · -- Case 2. h2 : ¬rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([1]_m) := by rw [FG_not_rp h1 h2] _ = [a]_m ^ (num_rp_below m k) * (prod_seq k 0 (F m) * ([1]_m)) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_not_rp h2, prod_seq_zero_step, F_not_rp_def h2] done done done lemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by define --Goal : ∀ i < m, G m a i < m fix i : Nat assume h1 : i < m rewrite [G_def] --Goal : a * i % m < m show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m) done lemma left_inv_one_one_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g' (g i) = i) : one_one_below n g := sorry lemma right_inv_onto_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g (g' i) = i) (h2 : maps_below n g') : onto_below n g := by define at h2; define fix k : Nat assume h3 : k < n apply Exists.intro (g' k) show g' k < n ∧ g (g' k) = k from And.intro (h2 k h3) (h1 k h3) done lemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m * [inv_mod m a]_m = [1]_m := by have h2 : 0 ≤ (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a) show [a]_m * [inv_mod m a]_m = [1]_m from calc [a]_m * [inv_mod m a]_m _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2] _ = [a]_m * [gcd_c2 m a]_m := by rw [←cc_eq_mod] _ = [1]_m := gcd_c2_inv h1 done lemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m := calc a * (b % m) % m = a % m * (b % m % m) % m := Nat.mul_mod _ _ _ _ = a % m * (b % m) % m := by rw [Nat.mod_mod] _ = a * b % m := (Nat.mul_mod _ _ _).symm lemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm] rfl done theorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] : ↑a ≡ ↑b (MOD m) ↔ a % m = b % m := sorry lemma mul_inv_mod_cancel {m a i : Nat} [NeZero m] (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1 rewrite [mul_class, cc_eq_iff_congr, ←Nat.cast_mul, ←Nat.cast_one, congr_iff_mod_eq_Nat] at h3 show a * inv_mod m a * i % m = i from calc a * (inv_mod m a) * i % m _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = 1 % m * i % m := by rw [h3] _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = i % m := by rw [one_mul] _ = i := Nat.mod_eq_of_lt h2 done

lemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i

HTPI.Ginv_def

htpi/HTPILib/Chap7.lean

htpi

/- Copyright 2023 Daniel J. Velleman -/ import HTPILib.Chap6 namespace HTPI /- Definitions -/ lemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by have h : n + 1 > 0 := Nat.succ_pos n show a % (n + 1) < n + 1 from Nat.mod_lt a h done def gcd (a b : Nat) : Nat := match b with | 0 => a | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd (n + 1) (a % (n + 1)) termination_by b mutual def gcd_c1 (a b : Nat) : Int := match b with | 0 => 1 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c2 (n + 1) (a % (n + 1)) --Corresponds to s = t' termination_by b def gcd_c2 (a b : Nat) : Int := match b with | 0 => 0 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c1 (n + 1) (a % (n + 1)) - (gcd_c2 (n + 1) (a % (n + 1))) * ↑(a / (n + 1)) --Corresponds to t = s' - t'q termination_by b end def prime (n : Nat) : Prop := 2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n def prime_factor (p n : Nat) : Prop := prime p ∧ p ∣ n def all_prime (l : List Nat) : Prop := ∀ p ∈ l, prime p def nondec (l : List Nat) : Prop := match l with | [] => True --Of course, True is a proposition that is always true | n :: L => (∀ m ∈ L, n ≤ m) ∧ nondec L def nondec_prime_list (l : List Nat) : Prop := all_prime l ∧ nondec l def prod (l : List Nat) : Nat := match l with | [] => 1 | n :: L => n * (prod L) def prime_factorization (n : Nat) (l : List Nat) : Prop := nondec_prime_list l ∧ prod l = n def rel_prime (a b : Nat) : Prop := gcd a b = 1 def congr_mod (m : Nat) (a b : Int) : Prop := (↑m : Int) ∣ (a - b) def cc (m : Nat) (a : Int) : ZMod m := (↑a : ZMod m) notation:50 a " ≡ " b " (MOD " m ")" => congr_mod m a b notation:max "["a"]_"m:max => cc m a def invertible {m : Nat} (X : ZMod m) : Prop := ∃ (Y : ZMod m), X * Y = [1]_m def num_rp_below (m k : Nat) : Nat := match k with | 0 => 0 | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j def phi (m : Nat) : Nat := num_rp_below m m def prod_seq {m : Nat} (j k : Nat) (f : Nat → ZMod m) : ZMod m := match j with | 0 => [1]_m | n + 1 => prod_seq n k f * f (k + n) def maps_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i < n, g i < n def one_one_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i1 < n, ∀ i2 < n, g i1 = g i2 → i1 = i2 def onto_below (n : Nat) (g : Nat → Nat) : Prop := ∀ k < n, ∃ i < n, g i = k def perm_below (n : Nat) (g : Nat → Nat) : Prop := maps_below n g ∧ one_one_below n g ∧ onto_below n g def inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m) def swap (u v i : Nat) : Nat := if i = u then v else if i = v then u else i namespace Euler --For definitions specific to Euler's theorem def F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m def G (m a i : Nat) : Nat := (a * i) % m def Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i end Euler /- Section 7.1 -/ theorem dvd_mod_of_dvd_a_b {a b d : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ (a % b) := by set q : Nat := a / b have h3 : b * q + a % b = a := Nat.div_add_mod a b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 define --Goal : ∃ (c : Nat), a % b = d * c apply Exists.intro (j - k * q) show a % b = d * (j - k * q) from calc a % b _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm _ = a - b * q := by rw [h3] _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc] _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm done theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry #eval gcd 672 161 --Answer: 7 lemma gcd_base (a : Nat) : gcd a 0 = a := by rfl lemma gcd_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd a b = gcd b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1)) rfl done lemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b ≠ 0) : a % b < b := by have h1 : b > 0 := Nat.pos_of_ne_zero h show a % b < b from Nat.mod_lt a h1 done lemma dvd_self (n : Nat) : n ∣ n := by apply Exists.intro 1 ring done theorem gcd_dvd : ∀ (b a : Nat), (gcd a b) ∣ a ∧ (gcd a b) ∣ b := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd a b_1) ∣ a ∧ (gcd a b_1) ∣ b_1 fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_base] --Goal: a ∣ a ∧ a ∣ 0 apply And.intro (dvd_self a) define apply Exists.intro 0 rfl done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_nonzero a h1] --Goal : gcd b (a % b) ∣ a ∧ gcd b (a % b) ∣ b have h2 : a % b < b := mod_nonzero_lt a h1 have h3 : (gcd b (a % b)) ∣ b ∧ (gcd b (a % b)) ∣ (a % b) := ih (a % b) h2 b apply And.intro _ h3.left show (gcd b (a % b)) ∣ a from dvd_a_of_dvd_b_mod h3.left h3.right done done theorem gcd_dvd_left (a b : Nat) : (gcd a b) ∣ a := (gcd_dvd b a).left theorem gcd_dvd_right (a b : Nat) : (gcd a b) ∣ b := (gcd_dvd b a).right lemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl lemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c1 a b = gcd_c2 b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done lemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl lemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * ↑(a / b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done theorem gcd_lin_comb : ∀ (b a : Nat), (gcd_c1 a b) * ↑a + (gcd_c2 a b) * ↑b = ↑(gcd a b) := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd_c1 a b_1) * ↑a + (gcd_c2 a b_1) * ↑b_1 = ↑(gcd a b_1) fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base] --Goal : 1 * ↑a + 0 * ↑0 = ↑a ring done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1] --Goal : gcd_c2 b (a % b) * ↑a + -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * ↑(a / b)) * ↑b = -- ↑(gcd b (a % b)) set r : Nat := a % b set q : Nat := a / b set s : Int := gcd_c1 b r set t : Int := gcd_c2 b r --Goal : t * ↑a + (s - t * ↑q) * ↑b = ↑(gcd b r) have h2 : r < b := mod_nonzero_lt a h1 have h3 : s * ↑b + t * ↑r = ↑(gcd b r) := ih r h2 b have h4 : b * q + r = a := Nat.div_add_mod a b rewrite [←h3, ←h4] rewrite [Nat.cast_add, Nat.cast_mul] --Goal : t * (↑b * ↑q + ↑r) + (s - t * ↑q) * ↑b = s * ↑b + t * ↑r ring done done #eval gcd_c1 672 161 --Answer: 6 #eval gcd_c2 672 161 --Answer: -25 --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161 theorem Theorem_7_1_6 {d a b : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ gcd a b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑d ∣ ↑(gcd a b) set s : Int := gcd_c1 a b set t : Int := gcd_c2 a b have h3 : s * ↑a + t * ↑b = ↑(gcd a b) := gcd_lin_comb b a rewrite [←h3] --Goal : ↑d ∣ s * ↑a + t * ↑b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul] --Goal : ↑d ∣ s * (↑d * ↑j) + t * (↑d * ↑k) define apply Exists.intro (s * ↑j + t * ↑k) ring done /- Section 7.2 -/ theorem dvd_trans {a b c : Nat} (h1 : a ∣ b) (h2 : b ∣ c) : a ∣ c := by define at h1; define at h2; define obtain (m : Nat) (h3 : b = a * m) from h1 obtain (n : Nat) (h4 : c = b * n) from h2 rewrite [h3, mul_assoc] at h4 apply Exists.intro (m * n) show c = a * (m * n) from h4 done lemma exists_prime_factor : ∀ (n : Nat), 2 ≤ n → ∃ (p : Nat), prime_factor p n := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, 2 ≤ n_1 → ∃ (p : Nat), prime_factor p n_1 assume h1 : 2 ≤ n by_cases h2 : prime n · -- Case 1. h2 : prime n apply Exists.intro n define --Goal : prime n ∧ n ∣ n show prime n ∧ n ∣ n from And.intro h2 (dvd_self n) done · -- Case 2. h2 : ¬prime n define at h2 --h2 : ¬(2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n) demorgan at h2 disj_syll h2 h1 obtain (a : Nat) (h3 : ∃ (b : Nat), a * b = n ∧ a < n ∧ b < n) from h2 obtain (b : Nat) (h4 : a * b = n ∧ a < n ∧ b < n) from h3 have h5 : 2 ≤ a := by by_contra h6 have h7 : a ≤ 1 := by linarith have h8 : n ≤ b := calc n _ = a * b := h4.left.symm _ ≤ 1 * b := by rel [h7] _ = b := by ring linarith --n ≤ b contradicts b < n done have h6 : ∃ (p : Nat), prime_factor p a := ih a h4.right.left h5 obtain (p : Nat) (h7 : prime_factor p a) from h6 apply Exists.intro p define --Goal : prime p ∧ p ∣ n define at h7 --h7 : prime p ∧ p ∣ a apply And.intro h7.left have h8 : a ∣ n := by apply Exists.intro b show n = a * b from (h4.left).symm done show p ∣ n from dvd_trans h7.right h8 done done lemma exists_least_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q := by set S : Set Nat := {p : Nat | prime_factor p n} have h2 : ∃ (p : Nat), p ∈ S := exists_prime_factor n h show ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q from well_ord_princ S h2 done lemma all_prime_nil : all_prime [] := by define --Goal : ∀ p ∈ [], prime p fix p : Nat contrapos --Goal : ¬prime p → p ∉ [] assume h1 : ¬prime p show p ∉ [] from List.not_mem_nil p done lemma all_prime_cons (n : Nat) (L : List Nat) : all_prime (n :: L) ↔ prime n ∧ all_prime L := by apply Iff.intro · -- (→) assume h1 : all_prime (n :: L) --Goal : prime n ∧ all_prime L define at h1 --h1 : ∀ p ∈ n :: L, prime p apply And.intro (h1 n (List.mem_cons_self n L)) define --Goal : ∀ p ∈ L, prime p fix p : Nat assume h2 : p ∈ L show prime p from h1 p (List.mem_cons_of_mem n h2) done · -- (←) assume h1 : prime n ∧ all_prime L --Goal : all_prime (n :: l) define : all_prime L at h1 define fix p : Nat assume h2 : p ∈ n :: L rewrite [List.mem_cons] at h2 --h2 : p = n ∨ p ∈ L by_cases on h2 · -- Case 1. h2 : p = n rewrite [h2] show prime n from h1.left done · -- Case 2. h2 : p ∈ L show prime p from h1.right p h2 done done done lemma nondec_nil : nondec [] := by define --Goal : True trivial --trivial proves some obviously true statements, such as True done lemma nondec_cons (n : Nat) (L : List Nat) : nondec (n :: L) ↔ (∀ m ∈ L, n ≤ m) ∧ nondec L := by rfl lemma prod_nil : prod [] = 1 := by rfl lemma prod_cons : prod (n :: L) = n * (prod L) := by rfl lemma exists_cons_of_length_eq_succ {A : Type} {l : List A} {n : Nat} (h : l.length = n + 1) : ∃ (a : A) (L : List A), l = a :: L ∧ L.length = n := by have h1 : ¬l.length = 0 := by linarith rewrite [List.length_eq_zero] at h1 obtain (a : A) (h2 : ∃ (L : List A), l = a :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List A) (h3 : l = a :: L) from h2 apply Exists.intro a apply Exists.intro L apply And.intro h3 have h4 : (a :: L).length = L.length + 1 := List.length_cons a L rewrite [←h3, h] at h4 show L.length = n from (Nat.add_right_cancel h4).symm done lemma list_elt_dvd_prod_by_length (a : Nat) : ∀ (n : Nat), ∀ (l : List Nat), l.length = n → a ∈ l → a ∣ prod l := by by_induc · --Base Case fix l : List Nat assume h1 : l.length = 0 rewrite [List.length_eq_zero] at h1 --h1 : l = [] rewrite [h1] --Goal : a ∈ [] → a ∣ prod [] contrapos assume h2 : ¬a ∣ prod [] show a ∉ [] from List.not_mem_nil a done · -- Induction Step fix n : Nat assume ih : ∀ (l : List Nat), List.length l = n → a ∈ l → a ∣ prod l fix l : List Nat assume h1 : l.length = n + 1 --Goal : a ∈ l → a ∣ prod l obtain (b : Nat) (h2 : ∃ (L : List Nat), l = b :: L ∧ L.length = n) from exists_cons_of_length_eq_succ h1 obtain (L : List Nat) (h3 : l = b :: L ∧ L.length = n) from h2 have h4 : a ∈ L → a ∣ prod L := ih L h3.right assume h5 : a ∈ l rewrite [h3.left, prod_cons] --Goal : a ∣ b * prod L rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b ∨ a ∈ L by_cases on h5 · -- Case 1. h5 : a = b apply Exists.intro (prod L) rewrite [h5] rfl done · -- Case 2. h5 : a ∈ L have h6 : a ∣ prod L := h4 h5 have h7 : prod L ∣ b * prod L := by apply Exists.intro b ring done show a ∣ b * prod L from dvd_trans h6 h7 done done done lemma list_elt_dvd_prod {a : Nat} {l : List Nat} (h : a ∈ l) : a ∣ prod l := by set n : Nat := l.length have h1 : l.length = n := by rfl show a ∣ prod l from list_elt_dvd_prod_by_length a n l h1 h done lemma exists_prime_factorization : ∀ (n : Nat), n ≥ 1 → ∃ (l : List Nat), prime_factorization n l := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, n_1 ≥ 1 → ∃ (l : List Nat), prime_factorization n_1 l assume h1 : n ≥ 1 by_cases h2 : n = 1 · -- Case 1. h2 : n = 1 apply Exists.intro [] define apply And.intro · -- Proof of nondec_prime_list [] define show all_prime [] ∧ nondec [] from And.intro all_prime_nil nondec_nil done · -- Proof of prod [] = n rewrite [prod_nil, h2] rfl done done · -- Case 2. h2 : n ≠ 1 have h3 : n ≥ 2 := lt_of_le_of_ne' h1 h2 obtain (p : Nat) (h4 : prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q) from exists_least_prime_factor h3 have p_prime_factor : prime_factor p n := h4.left define at p_prime_factor have p_prime : prime p := p_prime_factor.left have p_dvd_n : p ∣ n := p_prime_factor.right have p_least : ∀ (q : Nat), prime_factor q n → p ≤ q := h4.right obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n have h5 : m ≠ 0 := by contradict h1 with h6 have h7 : n = 0 := calc n _ = p * m := n_eq_pm _ = p * 0 := by rw [h6] _ = 0 := by ring rewrite [h7] decide done have m_pos : 0 < m := Nat.pos_of_ne_zero h5 have m_lt_n : m < n := by define at p_prime show m < n from calc m _ < m + m := by linarith _ = 2 * m := by ring _ ≤ p * m := by rel [p_prime.left] _ = n := n_eq_pm.symm done obtain (L : List Nat) (h6 : prime_factorization m L) from ih m m_lt_n m_pos define at h6 have ndpl_L : nondec_prime_list L := h6.left define at ndpl_L apply Exists.intro (p :: L) define apply And.intro · -- Proof of nondec_prime_list (p :: L) define apply And.intro · -- Proof of all_prime (p :: L) rewrite [all_prime_cons] show prime p ∧ all_prime L from And.intro p_prime ndpl_L.left done · -- Proof of nondec (p :: L) rewrite [nondec_cons] apply And.intro _ ndpl_L.right fix q : Nat assume q_in_L : q ∈ L have h7 : q ∣ prod L := list_elt_dvd_prod q_in_L rewrite [h6.right] at h7 --h7 : q ∣ m have h8 : m ∣ n := by apply Exists.intro p rewrite [n_eq_pm] ring done have q_dvd_n : q ∣ n := dvd_trans h7 h8 have ap_L : all_prime L := ndpl_L.left define at ap_L have q_prime_factor : prime_factor q n := And.intro (ap_L q q_in_L) q_dvd_n show p ≤ q from p_least q q_prime_factor done done · -- Proof of prod (p :: L) = n rewrite [prod_cons, h6.right, n_eq_pm] rfl done done done theorem Theorem_7_2_2 {a b c : Nat} (h1 : c ∣ a * b) (h2 : rel_prime a c) : c ∣ b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑c ∣ ↑b define at h1; define at h2; define obtain (j : Nat) (h3 : a * b = c * j) from h1 set s : Int := gcd_c1 a c set t : Int := gcd_c2 a c have h4 : s * ↑a + t * ↑c = ↑(gcd a c) := gcd_lin_comb c a rewrite [h2, Nat.cast_one] at h4 --h4 : s * ↑a + t * ↑c = (1 : Int) apply Exists.intro (s * ↑j + t * ↑b) show ↑b = ↑c * (s * ↑j + t * ↑b) from calc ↑b _ = (1 : Int) * ↑b := (one_mul _).symm _ = (s * ↑a + t * ↑c) * ↑b := by rw [h4] _ = s * (↑a * ↑b) + t * ↑c * ↑b := by ring _ = s * (↑c * ↑j) + t * ↑c * ↑b := by rw [←Nat.cast_mul a b, h3, Nat.cast_mul c j] _ = ↑c * (s * ↑j + t * ↑b) := by ring done lemma le_nonzero_prod_left {a b : Nat} (h : a * b ≠ 0) : a ≤ a * b := by have h1 : b ≠ 0 := by contradict h with h1 rewrite [h1] ring done have h2 : 1 ≤ b := Nat.pos_of_ne_zero h1 show a ≤ a * b from calc a = a * 1 := (mul_one a).symm _ ≤ a * b := by rel [h2] done lemma le_nonzero_prod_right {a b : Nat} (h : a * b ≠ 0) : b ≤ a * b := by rewrite [mul_comm] rewrite [mul_comm] at h show b ≤ b * a from le_nonzero_prod_left h done lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry lemma rel_prime_of_prime_not_dvd {a p : Nat} (h1 : prime p) (h2 : ¬p ∣ a) : rel_prime a p := by have h3 : gcd a p ∣ a := gcd_dvd_left a p have h4 : gcd a p ∣ p := gcd_dvd_right a p have h5 : gcd a p = 1 ∨ gcd a p = p := dvd_prime h1 h4 have h6 : gcd a p ≠ p := by contradict h2 with h6 rewrite [h6] at h3 show p ∣ a from h3 done disj_syll h5 h6 show rel_prime a p from h5 done theorem Theorem_7_2_3 {a b p : Nat} (h1 : prime p) (h2 : p ∣ a * b) : p ∣ a ∨ p ∣ b := by or_right with h3 have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3 show p ∣ b from Theorem_7_2_2 h2 h4 done lemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a ≥ 1 := by have h1 : a ≠ 0 := by by_contra h1 rewrite [h1] at h contradict h linarith done show a ≥ 1 from Nat.pos_of_ne_zero h1 done lemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by have h1 : a ≥ 1 := ge_one_of_prod_one h have h2 : a * b ≠ 0 := by linarith have h3 : a ≤ a * b := le_nonzero_prod_left h2 rewrite [h] at h3 show a = 1 from Nat.le_antisymm h3 h1 done lemma eq_one_of_dvd_one {n : Nat} (h : n ∣ 1) : n = 1 := by obtain (j : Nat) (h1 : 1 = n * j) from h show n = 1 from eq_one_of_prod_one h1.symm done lemma prime_not_one {p : Nat} (h : prime p) : p ≠ 1 := by define at h linarith done theorem Theorem_7_2_4 {p : Nat} (h1 : prime p) : ∀ (l : List Nat), p ∣ prod l → ∃ a ∈ l, p ∣ a := by apply List.rec · -- Base Case. Goal : p ∣ prod [] → ∃ a ∈ [], p ∣ a rewrite [prod_nil] assume h2 : p ∣ 1 show ∃ a ∈ [], p ∣ a from absurd (eq_one_of_dvd_one h2) (prime_not_one h1) done · -- Induction Step fix b : Nat fix L : List Nat assume ih : p ∣ prod L → ∃ a ∈ L, p ∣ a --Goal : p ∣ prod (b :: L) → ∃ a ∈ b :: L, p ∣ a assume h2 : p ∣ prod (b :: L) rewrite [prod_cons] at h2 have h3 : p ∣ b ∨ p ∣ prod L := Theorem_7_2_3 h1 h2 by_cases on h3 · -- Case 1. h3 : p ∣ b apply Exists.intro b show b ∈ b :: L ∧ p ∣ b from And.intro (List.mem_cons_self b L) h3 done · -- Case 2. h3 : p ∣ prod L obtain (a : Nat) (h4 : a ∈ L ∧ p ∣ a) from ih h3 apply Exists.intro a show a ∈ b :: L ∧ p ∣ a from And.intro (List.mem_cons_of_mem b h4.left) h4.right done done done lemma prime_in_list {p : Nat} {l : List Nat} (h1 : prime p) (h2 : all_prime l) (h3 : p ∣ prod l) : p ∈ l := by obtain (a : Nat) (h4 : a ∈ l ∧ p ∣ a) from Theorem_7_2_4 h1 l h3 define at h2 have h5 : prime a := h2 a h4.left have h6 : p = 1 ∨ p = a := dvd_prime h5 h4.right disj_syll h6 (prime_not_one h1) rewrite [h6] show a ∈ l from h4.left done lemma first_le_first {p q : Nat} {l m : List Nat} (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m)) (h3 : prod (p :: l) = prod (q :: m)) : p ≤ q := by define at h1; define at h2 have h4 : q ∣ prod (p :: l) := by define apply Exists.intro (prod m) rewrite [←prod_cons] show prod (p :: l) = prod (q :: m) from h3 done have h5 : all_prime (q :: m) := h2.left rewrite [all_prime_cons] at h5 have h6 : q ∈ p :: l := prime_in_list h5.left h1.left h4 have h7 : nondec (p :: l) := h1.right rewrite [nondec_cons] at h7 rewrite [List.mem_cons] at h6 by_cases on h6 · -- Case 1. h6 : q = p linarith done · -- Case 2. h6 : q ∈ l have h8 : ∀ m ∈ l, p ≤ m := h7.left show p ≤ q from h8 q h6 done done lemma nondec_prime_list_tail {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by define at h define rewrite [all_prime_cons, nondec_cons] at h show all_prime l ∧ nondec l from And.intro h.left.right h.right.right done lemma cons_prod_not_one {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : prod (p :: l) ≠ 1 := by define at h have h1 : all_prime (p :: l) := h.left rewrite [all_prime_cons] at h1 rewrite [prod_cons] by_contra h2 show False from (prime_not_one h1.left) (eq_one_of_prod_one h2) done lemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) : l = [] ↔ prod l = 1 := by apply Iff.intro · -- (→) assume h1 : l = [] rewrite [h1] show prod [] = 1 from prod_nil done · -- (←) contrapos assume h1 : ¬l = [] obtain (p : Nat) (h2 : ∃ (L : List Nat), l = p :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List Nat) (h3 : l = p :: L) from h2 rewrite [h3] at h rewrite [h3] show ¬prod (p :: L) = 1 from cons_prod_not_one h done done lemma prime_pos {p : Nat} (h : prime p) : p > 0 := by define at h linarith done theorem Theorem_7_2_5 : ∀ (l1 l2 : List Nat), nondec_prime_list l1 → nondec_prime_list l2 → prod l1 = prod l2 → l1 = l2 := by apply List.rec · -- Base Case. Goal : ∀ (l2 : List Nat), nondec_prime_list [] → -- nondec_prime_list l2 → prod [] = prod l2 → [] = l2 fix l2 : List Nat assume h1 : nondec_prime_list [] assume h2 : nondec_prime_list l2 assume h3 : prod [] = prod l2 rewrite [prod_nil, eq_comm, ←list_nil_iff_prod_one h2] at h3 show [] = l2 from h3.symm done · -- Induction Step fix p : Nat fix L1 : List Nat assume ih : ∀ (L2 : List Nat), nondec_prime_list L1 → nondec_prime_list L2 → prod L1 = prod L2 → L1 = L2 -- Goal : ∀ (l2 : List Nat), nondec_prime_list (p :: L1) → -- nondec_prime_list l2 → prod (p :: L1) = prod l2 → p :: L1 = l2 fix l2 : List Nat assume h1 : nondec_prime_list (p :: L1) assume h2 : nondec_prime_list l2 assume h3 : prod (p :: L1) = prod l2 have h4 : ¬prod (p :: L1) = 1 := cons_prod_not_one h1 rewrite [h3, ←list_nil_iff_prod_one h2] at h4 obtain (q : Nat) (h5 : ∃ (L : List Nat), l2 = q :: L) from List.exists_cons_of_ne_nil h4 obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5 rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2) rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2) have h7 : p ≤ q := first_le_first h1 h2 h3 have h8 : q ≤ p := first_le_first h2 h1 h3.symm have h9 : p = q := by linarith rewrite [h9, prod_cons, prod_cons] at h3 --h3 : q * prod L1 = q * prod L2 have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1 have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2 define at h2 have h12 : all_prime (q :: L2) := h2.left rewrite [all_prime_cons] at h12 have h13 : q > 0 := prime_pos h12.left have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3 have h15 : L1 = L2 := ih L2 h10 h11 h14 rewrite [h6, h9, h15] rfl done done theorem fund_thm_arith (n : Nat) (h : n ≥ 1) : ∃! (l : List Nat), prime_factorization n l := by exists_unique · -- Existence show ∃ (l : List Nat), prime_factorization n l from exists_prime_factorization n h done · -- Uniqueness fix l1 : List Nat; fix l2 : List Nat assume h1 : prime_factorization n l1 assume h2 : prime_factorization n l2 define at h1; define at h2 have h3 : prod l1 = n := h1.right rewrite [←h2.right] at h3 show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3 done done /- Section 7.3 -/ theorem congr_refl (m : Nat) : ∀ (a : Int), a ≡ a (MOD m) := by fix a : Int define --Goal : ∃ (c : Int), a - a = ↑m * c apply Exists.intro 0 ring done theorem congr_symm {m : Nat} : ∀ {a b : Int}, a ≡ b (MOD m) → b ≡ a (MOD m) := by fix a : Int; fix b : Int assume h1 : a ≡ b (MOD m) define at h1 --h1 : ∃ (c : Int), a - b = ↑m * c define --Goal : ∃ (c : Int), b - a = ↑m * c obtain (c : Int) (h2 : a - b = m * c) from h1 apply Exists.intro (-c) show b - a = m * (-c) from calc b - a _ = -(a - b) := by ring _ = -(m * c) := by rw [h2] _ = m * (-c) := by ring done theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry /- Fundamental properties of congruence classes -/ lemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) : X = Y ↔ X.val = Y.val := Fin.ext_iff lemma val_nat_eq_mod (n k : Nat) : ([k]_(n + 1)).val = k % (n + 1) := by rfl lemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl theorem cc_rep {m : Nat} (X : ZMod m) : ∃ (a : Int), X = [a]_m := match m with | 0 => by apply Exists.intro X rfl done | n + 1 => by apply Exists.intro ↑(X.val) have h1 : X.val < n + 1 := Fin.prop X rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1] rfl done theorem add_class (m : Nat) (a b : Int) : [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm theorem mul_class (m : Nat) (a b : Int) : [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm lemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ [a - b]_m = [0]_m := by apply Iff.intro · -- (→) assume h1 : [a]_m = [b]_m have h2 : a - b = a + (-b) := by ring have h3 : b + (-b) = 0 := by ring show [a - b]_m = [0]_m from calc [a - b]_m _ = [a + (-b)]_m := by rw [h2] _ = [a]_m + [-b]_m := by rw [add_class] _ = [b]_m + [-b]_m := by rw [h1] _ = [b + -b]_m := by rw [add_class] _ = [0]_m := by rw [h3] done · -- (←) assume h1 : [a - b]_m = [0]_m have h2 : b + (a - b) = a := by ring have h3 : b + 0 = b := by ring show [a]_m = [b]_m from calc [a]_m _ = [b + (a - b)]_m := by rw [h2] _ = [b]_m + [a - b]_m := by rw [add_class] _ = [b]_m + [0]_m := by rw [h1] _ = [b + 0]_m := by rw [add_class] _ = [b]_m := by rw [h3] done done lemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) : [a]_m = [0]_m → [-a]_m = [0]_m := by assume h1 : [a]_m = [0]_m have h2 : 0 + (-a) = -a := by ring have h3 : a + (-a) = 0 := by ring show [-a]_m = [0]_m from calc [-a]_m _ = [0 + (-a)]_m := by rw [h2] _ = [0]_m + [-a]_m := by rw [add_class] _ = [a]_m + [-a]_m := by rw [h1] _ = [a + (-a)]_m := by rw [add_class] _ = [0]_m := by rw [h3] done lemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) : [-a]_m = [0]_m ↔ [a]_m = [0]_m := by apply Iff.intro _ (cc_neg_zero_of_cc_zero m a) assume h1 : [-a]_m = [0]_m have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1 have h3 : -(-a) = a := by ring rewrite [h3] at h2 show [a]_m = [0]_m from h2 done lemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl lemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m ↔ m ∣ k := match m with | 0 => by have h : (0 : Int) = (↑(0 : Nat) : Int) := by rfl rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj] apply Iff.intro · -- (→) assume h1 : k = 0 rewrite [h1] show 0 ∣ 0 from dvd_self 0 done · -- (←) assume h1 : 0 ∣ k obtain (c : Nat) (h2 : k = 0 * c) from h1 rewrite [h2] ring done done | n + 1 => by rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero] show k % (n + 1) = 0 ↔ n + 1 ∣ k from (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm done lemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m ↔ ↑m ∣ a := by obtain (k : Nat) (h1 : a = ↑k ∨ a = -↑k) from Int.eq_nat_or_neg a by_cases on h1 · -- Case 1. h1: a = ↑k rewrite [h1, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done · -- Case 2. h1: a = -↑k rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done done theorem cc_eq_iff_congr (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ a ≡ b (MOD m) := calc [a]_m = [b]_m _ ↔ [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b _ ↔ ↑m ∣ (a - b) := cc_zero_iff_dvd m (a - b) _ ↔ a ≡ b (MOD m) := by rfl /- End of fundamental properties of congruence classes -/ lemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m := by have h1 : (↑m : Int) ≠ 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m) show 0 ≤ a % m from Int.emod_nonneg a h1 done lemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m) have h2 : (↑m : Int) > 0 := (Nat.cast_pos).rtl h1 show a % m < m from Int.emod_lt_of_pos a h2 done lemma congr_mod_mod (m : Nat) (a : Int) : a ≡ a % m (MOD m) := by define have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m apply Exists.intro (a / m) show a - a % m = m * (a / m) from calc a - (a % m) _ = m * (a / m) + a % m - a % m := by rw [h1] _ = m * (a / m) := by ring done lemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) := And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a)) theorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) : ∃! (r : Int), 0 ≤ r ∧ r < m ∧ a ≡ r (MOD m) := by exists_unique · -- Existence apply Exists.intro (a % m) show 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) from mod_cmpl_res m a done · -- Uniqueness fix r1 : Int; fix r2 : Int assume h1 : 0 ≤ r1 ∧ r1 < m ∧ a ≡ r1 (MOD m) assume h2 : 0 ≤ r2 ∧ r2 < m ∧ a ≡ r2 (MOD m) have h3 : r1 ≡ r2 (MOD m) := congr_trans (congr_symm h1.right.right) h2.right.right obtain (d : Int) (h4 : r1 - r2 = m * d) from h3 have h5 : r1 - r2 < m * 1 := by linarith have h6 : m * (-1) < r1 - r2 := by linarith rewrite [h4] at h5 --h5 : m * d < m * 1 rewrite [h4] at h6 --h6 : m * -1 < m * d have h7 : (↑m : Int) ≥ 0 := Nat.cast_nonneg m have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7 have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7 have h10 : d = 0 := by linarith show r1 = r2 from calc r1 _ = r1 - r2 + r2 := by ring _ = m * 0 + r2 := by rw [h4, h10] _ = r2 := by ring done done lemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m := (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a) theorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y rewrite [h1, h2] have h3 : a + b = b + a := by ring show [a]_m + [b]_m = [b]_m + [a]_m from calc [a]_m + [b]_m _ = [a + b]_m := add_class m a b _ = [b + a]_m := by rw [h3] _ = [b]_m + [a]_m := (add_class m b a).symm done theorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X rewrite [h1] have h2 : a * 1 = a := by ring show [a]_m * [1]_m = [a]_m from calc [a]_m * [1]_m _ = [a * 1]_m := mul_class m a 1 _ = [a]_m := by rw [h2] done theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry lemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) : [a]_m * [gcd_c2 m a]_m = [1]_m := by set s : Int := gcd_c1 m a have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m define at h1 rewrite [h1, Nat.cast_one] at h2 --h2 : s * ↑m + gcd_c2 m a * ↑a = 1 rewrite [mul_class, cc_eq_iff_congr] define --Goal : ∃ (c : Int), ↑a * gcd_c2 m a - 1 = ↑m * c apply Exists.intro (-s) show a * (gcd_c2 m a) - 1 = m * (-s) from calc a * (gcd_c2 m a) - 1 _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring _ = 1 + m * (-s) - 1 := by rw [h2] _ = m * (-s) := by ring done theorem Theorem_7_3_7 (m a : Nat) : invertible [a]_m ↔ rel_prime m a := by apply Iff.intro · -- (→) assume h1 : invertible [a]_m define at h1 obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1 obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y rewrite [h3, mul_class, cc_eq_iff_congr] at h2 define at h2 obtain (c : Int) (h4 : a * b - 1 = m * c) from h2 rewrite [Exercise_7_2_6] --Goal : ∃ (s t : Int), s * ↑m + t * ↑a = 1 apply Exists.intro (-c) apply Exists.intro b show (-c) * m + b * a = 1 from calc (-c) * m + b * a _ = (-c) * m + (a * b - 1) + 1 := by ring _ = (-c) * m + m * c + 1 := by rw [h4] _ = 1 := by ring done · -- (←) assume h1 : rel_prime m a define show ∃ (Y : ZMod m), [a]_m * Y = [1]_m from Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1) done done /- Section 7.4 -/ section Euler open Euler lemma num_rp_below_base {m : Nat} : num_rp_below m 0 = 0 := by rfl lemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) : num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by have h1 : num_rp_below m (j + 1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : gcd m j = 1 rewrite [if_pos h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1 show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1 done lemma num_rp_below_step_not_rp {m j : Nat} (h : ¬rel_prime m j) : num_rp_below m (j + 1) = num_rp_below m j := by have h1 : num_rp_below m (j +1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : ¬gcd m j = 1 rewrite [if_neg h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j show num_rp_below m (j + 1) = num_rp_below m j from h1 done lemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl #eval phi 10 --Answer: 4 lemma prod_inv_iff_inv {m : Nat} {X : ZMod m} (h1 : invertible X) (Y : ZMod m) : invertible (X * Y) ↔ invertible Y := by apply Iff.intro · -- (→) assume h2 : invertible (X * Y) obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2 apply Exists.intro (X * Z) rewrite [←h3] --Goal : Y * (X * Z) = X * Y * Z ring --Note that ring can do algebra in ZMod m done · -- (←) assume h2 : invertible Y obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1 obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2 apply Exists.intro (Xi * Yi) show (X * Y) * (Xi * Yi) = [1]_m from calc X * Y * (Xi * Yi) _ = (X * Xi) * (Y * Yi) := by ring _ = [1]_m * [1]_m := by rw [h3, h4] _ = [1]_m := Theorem_7_3_6_7 [1]_m done done lemma F_rp_def {m i : Nat} (h : rel_prime m i) : F m i = [i]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h --h : gcd m i = 1 rewrite [if_pos h] at h1 show F m i = [i]_m from h1 done lemma F_not_rp_def {m i : Nat} (h : ¬rel_prime m i) : F m i = [1]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h rewrite [h1, if_neg h] rfl done lemma prod_seq_base {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 0 k f = [1]_m := by rfl lemma prod_seq_step {m : Nat} (n k : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl lemma prod_seq_zero_step {m : Nat} (n : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by rewrite [prod_seq_step, zero_add] rfl done lemma prod_one {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 1 k f = f k := by rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7] rfl done lemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl lemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m := calc [G m a i]_m _ = [(a * i) % m]_m := by rfl _ = [a * i]_m := (cc_eq_mod m (a * i)).symm _ = [a]_m * [i]_m := (mul_class m a i).symm lemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) : rel_prime m (G m a i) ↔ rel_prime m i := by have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1 show rel_prime m (G m a i) ↔ rel_prime m i from calc rel_prime m (G m a i) _ ↔ invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm _ ↔ invertible ([a]_m * [i]_m) := by rw [cc_G] _ ↔ invertible [i]_m := prod_inv_iff_inv h2 ([i]_m) _ ↔ rel_prime m i := Theorem_7_3_7 m i done lemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) : F m (G m a i) = [a]_m * F m i := by have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2 show F m (G m a i) = [a]_m * F m i from calc F m (G m a i) _ = [G m a i]_m := F_rp_def h3 _ = [a]_m * [i]_m := cc_G m a i _ = [a]_m * F m i := by rw [F_rp_def h2] done lemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : ¬rel_prime m i) : F m (G m a i) = [1]_m := by rewrite [←G_rp_iff h1 i] at h2 show F m (G m a i) = [1]_m from F_not_rp_def h2 done lemma FG_prod {m a : Nat} (h1 : rel_prime m a) : ∀ (k : Nat), prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by by_induc · -- Base Case show prod_seq 0 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from calc prod_seq 0 0 ((F m) ∘ (G m a)) _ = [1]_m := prod_seq_base _ _ _ = [a]_m ^ 0 * [1]_m := by ring _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by rw [num_rp_below_base, prod_seq_base] done · -- Induction Step fix k : Nat assume ih : prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) by_cases h2 : rel_prime m k · -- Case 1. h2 : rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([a]_m * F m k) := by rw [FG_rp h1 h2] _ = [a]_m ^ ((num_rp_below m k) + 1) * ((prod_seq k 0 (F m)) * F m k) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_rp h2, prod_seq_zero_step] done · -- Case 2. h2 : ¬rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([1]_m) := by rw [FG_not_rp h1 h2] _ = [a]_m ^ (num_rp_below m k) * (prod_seq k 0 (F m) * ([1]_m)) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_not_rp h2, prod_seq_zero_step, F_not_rp_def h2] done done done lemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by define --Goal : ∀ i < m, G m a i < m fix i : Nat assume h1 : i < m rewrite [G_def] --Goal : a * i % m < m show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m) done lemma left_inv_one_one_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g' (g i) = i) : one_one_below n g := sorry lemma right_inv_onto_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g (g' i) = i) (h2 : maps_below n g') : onto_below n g := by define at h2; define fix k : Nat assume h3 : k < n apply Exists.intro (g' k) show g' k < n ∧ g (g' k) = k from And.intro (h2 k h3) (h1 k h3) done lemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m * [inv_mod m a]_m = [1]_m := by have h2 : 0 ≤ (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a) show [a]_m * [inv_mod m a]_m = [1]_m from calc [a]_m * [inv_mod m a]_m _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2] _ = [a]_m * [gcd_c2 m a]_m := by rw [←cc_eq_mod] _ = [1]_m := gcd_c2_inv h1 done lemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m := calc a * (b % m) % m = a % m * (b % m % m) % m := Nat.mul_mod _ _ _ _ = a % m * (b % m) % m := by rw [Nat.mod_mod] _ = a * b % m := (Nat.mul_mod _ _ _).symm lemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm] rfl done theorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] : ↑a ≡ ↑b (MOD m) ↔ a % m = b % m := sorry lemma mul_inv_mod_cancel {m a i : Nat} [NeZero m] (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1 rewrite [mul_class, cc_eq_iff_congr, ←Nat.cast_mul, ←Nat.cast_one, congr_iff_mod_eq_Nat] at h3 show a * inv_mod m a * i % m = i from calc a * (inv_mod m a) * i % m _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = 1 % m * i % m := by rw [h3] _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = i % m := by rw [one_mul] _ = i := Nat.mod_eq_of_lt h2 done lemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl

lemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, G m a (Ginv m a i) = i

HTPI.Ginv_right_inv

htpi/HTPILib/Chap7.lean

htpi

/- Copyright 2023 Daniel J. Velleman -/ import HTPILib.Chap6 namespace HTPI /- Definitions -/ lemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by have h : n + 1 > 0 := Nat.succ_pos n show a % (n + 1) < n + 1 from Nat.mod_lt a h done def gcd (a b : Nat) : Nat := match b with | 0 => a | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd (n + 1) (a % (n + 1)) termination_by b mutual def gcd_c1 (a b : Nat) : Int := match b with | 0 => 1 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c2 (n + 1) (a % (n + 1)) --Corresponds to s = t' termination_by b def gcd_c2 (a b : Nat) : Int := match b with | 0 => 0 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c1 (n + 1) (a % (n + 1)) - (gcd_c2 (n + 1) (a % (n + 1))) * ↑(a / (n + 1)) --Corresponds to t = s' - t'q termination_by b end def prime (n : Nat) : Prop := 2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n def prime_factor (p n : Nat) : Prop := prime p ∧ p ∣ n def all_prime (l : List Nat) : Prop := ∀ p ∈ l, prime p def nondec (l : List Nat) : Prop := match l with | [] => True --Of course, True is a proposition that is always true | n :: L => (∀ m ∈ L, n ≤ m) ∧ nondec L def nondec_prime_list (l : List Nat) : Prop := all_prime l ∧ nondec l def prod (l : List Nat) : Nat := match l with | [] => 1 | n :: L => n * (prod L) def prime_factorization (n : Nat) (l : List Nat) : Prop := nondec_prime_list l ∧ prod l = n def rel_prime (a b : Nat) : Prop := gcd a b = 1 def congr_mod (m : Nat) (a b : Int) : Prop := (↑m : Int) ∣ (a - b) def cc (m : Nat) (a : Int) : ZMod m := (↑a : ZMod m) notation:50 a " ≡ " b " (MOD " m ")" => congr_mod m a b notation:max "["a"]_"m:max => cc m a def invertible {m : Nat} (X : ZMod m) : Prop := ∃ (Y : ZMod m), X * Y = [1]_m def num_rp_below (m k : Nat) : Nat := match k with | 0 => 0 | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j def phi (m : Nat) : Nat := num_rp_below m m def prod_seq {m : Nat} (j k : Nat) (f : Nat → ZMod m) : ZMod m := match j with | 0 => [1]_m | n + 1 => prod_seq n k f * f (k + n) def maps_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i < n, g i < n def one_one_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i1 < n, ∀ i2 < n, g i1 = g i2 → i1 = i2 def onto_below (n : Nat) (g : Nat → Nat) : Prop := ∀ k < n, ∃ i < n, g i = k def perm_below (n : Nat) (g : Nat → Nat) : Prop := maps_below n g ∧ one_one_below n g ∧ onto_below n g def inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m) def swap (u v i : Nat) : Nat := if i = u then v else if i = v then u else i namespace Euler --For definitions specific to Euler's theorem def F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m def G (m a i : Nat) : Nat := (a * i) % m def Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i end Euler /- Section 7.1 -/ theorem dvd_mod_of_dvd_a_b {a b d : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ (a % b) := by set q : Nat := a / b have h3 : b * q + a % b = a := Nat.div_add_mod a b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 define --Goal : ∃ (c : Nat), a % b = d * c apply Exists.intro (j - k * q) show a % b = d * (j - k * q) from calc a % b _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm _ = a - b * q := by rw [h3] _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc] _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm done theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry #eval gcd 672 161 --Answer: 7 lemma gcd_base (a : Nat) : gcd a 0 = a := by rfl lemma gcd_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd a b = gcd b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1)) rfl done lemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b ≠ 0) : a % b < b := by have h1 : b > 0 := Nat.pos_of_ne_zero h show a % b < b from Nat.mod_lt a h1 done lemma dvd_self (n : Nat) : n ∣ n := by apply Exists.intro 1 ring done theorem gcd_dvd : ∀ (b a : Nat), (gcd a b) ∣ a ∧ (gcd a b) ∣ b := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd a b_1) ∣ a ∧ (gcd a b_1) ∣ b_1 fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_base] --Goal: a ∣ a ∧ a ∣ 0 apply And.intro (dvd_self a) define apply Exists.intro 0 rfl done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_nonzero a h1] --Goal : gcd b (a % b) ∣ a ∧ gcd b (a % b) ∣ b have h2 : a % b < b := mod_nonzero_lt a h1 have h3 : (gcd b (a % b)) ∣ b ∧ (gcd b (a % b)) ∣ (a % b) := ih (a % b) h2 b apply And.intro _ h3.left show (gcd b (a % b)) ∣ a from dvd_a_of_dvd_b_mod h3.left h3.right done done theorem gcd_dvd_left (a b : Nat) : (gcd a b) ∣ a := (gcd_dvd b a).left theorem gcd_dvd_right (a b : Nat) : (gcd a b) ∣ b := (gcd_dvd b a).right lemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl lemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c1 a b = gcd_c2 b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done lemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl lemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * ↑(a / b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done theorem gcd_lin_comb : ∀ (b a : Nat), (gcd_c1 a b) * ↑a + (gcd_c2 a b) * ↑b = ↑(gcd a b) := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd_c1 a b_1) * ↑a + (gcd_c2 a b_1) * ↑b_1 = ↑(gcd a b_1) fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base] --Goal : 1 * ↑a + 0 * ↑0 = ↑a ring done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1] --Goal : gcd_c2 b (a % b) * ↑a + -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * ↑(a / b)) * ↑b = -- ↑(gcd b (a % b)) set r : Nat := a % b set q : Nat := a / b set s : Int := gcd_c1 b r set t : Int := gcd_c2 b r --Goal : t * ↑a + (s - t * ↑q) * ↑b = ↑(gcd b r) have h2 : r < b := mod_nonzero_lt a h1 have h3 : s * ↑b + t * ↑r = ↑(gcd b r) := ih r h2 b have h4 : b * q + r = a := Nat.div_add_mod a b rewrite [←h3, ←h4] rewrite [Nat.cast_add, Nat.cast_mul] --Goal : t * (↑b * ↑q + ↑r) + (s - t * ↑q) * ↑b = s * ↑b + t * ↑r ring done done #eval gcd_c1 672 161 --Answer: 6 #eval gcd_c2 672 161 --Answer: -25 --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161 theorem Theorem_7_1_6 {d a b : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ gcd a b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑d ∣ ↑(gcd a b) set s : Int := gcd_c1 a b set t : Int := gcd_c2 a b have h3 : s * ↑a + t * ↑b = ↑(gcd a b) := gcd_lin_comb b a rewrite [←h3] --Goal : ↑d ∣ s * ↑a + t * ↑b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul] --Goal : ↑d ∣ s * (↑d * ↑j) + t * (↑d * ↑k) define apply Exists.intro (s * ↑j + t * ↑k) ring done /- Section 7.2 -/ theorem dvd_trans {a b c : Nat} (h1 : a ∣ b) (h2 : b ∣ c) : a ∣ c := by define at h1; define at h2; define obtain (m : Nat) (h3 : b = a * m) from h1 obtain (n : Nat) (h4 : c = b * n) from h2 rewrite [h3, mul_assoc] at h4 apply Exists.intro (m * n) show c = a * (m * n) from h4 done lemma exists_prime_factor : ∀ (n : Nat), 2 ≤ n → ∃ (p : Nat), prime_factor p n := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, 2 ≤ n_1 → ∃ (p : Nat), prime_factor p n_1 assume h1 : 2 ≤ n by_cases h2 : prime n · -- Case 1. h2 : prime n apply Exists.intro n define --Goal : prime n ∧ n ∣ n show prime n ∧ n ∣ n from And.intro h2 (dvd_self n) done · -- Case 2. h2 : ¬prime n define at h2 --h2 : ¬(2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n) demorgan at h2 disj_syll h2 h1 obtain (a : Nat) (h3 : ∃ (b : Nat), a * b = n ∧ a < n ∧ b < n) from h2 obtain (b : Nat) (h4 : a * b = n ∧ a < n ∧ b < n) from h3 have h5 : 2 ≤ a := by by_contra h6 have h7 : a ≤ 1 := by linarith have h8 : n ≤ b := calc n _ = a * b := h4.left.symm _ ≤ 1 * b := by rel [h7] _ = b := by ring linarith --n ≤ b contradicts b < n done have h6 : ∃ (p : Nat), prime_factor p a := ih a h4.right.left h5 obtain (p : Nat) (h7 : prime_factor p a) from h6 apply Exists.intro p define --Goal : prime p ∧ p ∣ n define at h7 --h7 : prime p ∧ p ∣ a apply And.intro h7.left have h8 : a ∣ n := by apply Exists.intro b show n = a * b from (h4.left).symm done show p ∣ n from dvd_trans h7.right h8 done done lemma exists_least_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q := by set S : Set Nat := {p : Nat | prime_factor p n} have h2 : ∃ (p : Nat), p ∈ S := exists_prime_factor n h show ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q from well_ord_princ S h2 done lemma all_prime_nil : all_prime [] := by define --Goal : ∀ p ∈ [], prime p fix p : Nat contrapos --Goal : ¬prime p → p ∉ [] assume h1 : ¬prime p show p ∉ [] from List.not_mem_nil p done lemma all_prime_cons (n : Nat) (L : List Nat) : all_prime (n :: L) ↔ prime n ∧ all_prime L := by apply Iff.intro · -- (→) assume h1 : all_prime (n :: L) --Goal : prime n ∧ all_prime L define at h1 --h1 : ∀ p ∈ n :: L, prime p apply And.intro (h1 n (List.mem_cons_self n L)) define --Goal : ∀ p ∈ L, prime p fix p : Nat assume h2 : p ∈ L show prime p from h1 p (List.mem_cons_of_mem n h2) done · -- (←) assume h1 : prime n ∧ all_prime L --Goal : all_prime (n :: l) define : all_prime L at h1 define fix p : Nat assume h2 : p ∈ n :: L rewrite [List.mem_cons] at h2 --h2 : p = n ∨ p ∈ L by_cases on h2 · -- Case 1. h2 : p = n rewrite [h2] show prime n from h1.left done · -- Case 2. h2 : p ∈ L show prime p from h1.right p h2 done done done lemma nondec_nil : nondec [] := by define --Goal : True trivial --trivial proves some obviously true statements, such as True done lemma nondec_cons (n : Nat) (L : List Nat) : nondec (n :: L) ↔ (∀ m ∈ L, n ≤ m) ∧ nondec L := by rfl lemma prod_nil : prod [] = 1 := by rfl lemma prod_cons : prod (n :: L) = n * (prod L) := by rfl lemma exists_cons_of_length_eq_succ {A : Type} {l : List A} {n : Nat} (h : l.length = n + 1) : ∃ (a : A) (L : List A), l = a :: L ∧ L.length = n := by have h1 : ¬l.length = 0 := by linarith rewrite [List.length_eq_zero] at h1 obtain (a : A) (h2 : ∃ (L : List A), l = a :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List A) (h3 : l = a :: L) from h2 apply Exists.intro a apply Exists.intro L apply And.intro h3 have h4 : (a :: L).length = L.length + 1 := List.length_cons a L rewrite [←h3, h] at h4 show L.length = n from (Nat.add_right_cancel h4).symm done lemma list_elt_dvd_prod_by_length (a : Nat) : ∀ (n : Nat), ∀ (l : List Nat), l.length = n → a ∈ l → a ∣ prod l := by by_induc · --Base Case fix l : List Nat assume h1 : l.length = 0 rewrite [List.length_eq_zero] at h1 --h1 : l = [] rewrite [h1] --Goal : a ∈ [] → a ∣ prod [] contrapos assume h2 : ¬a ∣ prod [] show a ∉ [] from List.not_mem_nil a done · -- Induction Step fix n : Nat assume ih : ∀ (l : List Nat), List.length l = n → a ∈ l → a ∣ prod l fix l : List Nat assume h1 : l.length = n + 1 --Goal : a ∈ l → a ∣ prod l obtain (b : Nat) (h2 : ∃ (L : List Nat), l = b :: L ∧ L.length = n) from exists_cons_of_length_eq_succ h1 obtain (L : List Nat) (h3 : l = b :: L ∧ L.length = n) from h2 have h4 : a ∈ L → a ∣ prod L := ih L h3.right assume h5 : a ∈ l rewrite [h3.left, prod_cons] --Goal : a ∣ b * prod L rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b ∨ a ∈ L by_cases on h5 · -- Case 1. h5 : a = b apply Exists.intro (prod L) rewrite [h5] rfl done · -- Case 2. h5 : a ∈ L have h6 : a ∣ prod L := h4 h5 have h7 : prod L ∣ b * prod L := by apply Exists.intro b ring done show a ∣ b * prod L from dvd_trans h6 h7 done done done lemma list_elt_dvd_prod {a : Nat} {l : List Nat} (h : a ∈ l) : a ∣ prod l := by set n : Nat := l.length have h1 : l.length = n := by rfl show a ∣ prod l from list_elt_dvd_prod_by_length a n l h1 h done lemma exists_prime_factorization : ∀ (n : Nat), n ≥ 1 → ∃ (l : List Nat), prime_factorization n l := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, n_1 ≥ 1 → ∃ (l : List Nat), prime_factorization n_1 l assume h1 : n ≥ 1 by_cases h2 : n = 1 · -- Case 1. h2 : n = 1 apply Exists.intro [] define apply And.intro · -- Proof of nondec_prime_list [] define show all_prime [] ∧ nondec [] from And.intro all_prime_nil nondec_nil done · -- Proof of prod [] = n rewrite [prod_nil, h2] rfl done done · -- Case 2. h2 : n ≠ 1 have h3 : n ≥ 2 := lt_of_le_of_ne' h1 h2 obtain (p : Nat) (h4 : prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q) from exists_least_prime_factor h3 have p_prime_factor : prime_factor p n := h4.left define at p_prime_factor have p_prime : prime p := p_prime_factor.left have p_dvd_n : p ∣ n := p_prime_factor.right have p_least : ∀ (q : Nat), prime_factor q n → p ≤ q := h4.right obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n have h5 : m ≠ 0 := by contradict h1 with h6 have h7 : n = 0 := calc n _ = p * m := n_eq_pm _ = p * 0 := by rw [h6] _ = 0 := by ring rewrite [h7] decide done have m_pos : 0 < m := Nat.pos_of_ne_zero h5 have m_lt_n : m < n := by define at p_prime show m < n from calc m _ < m + m := by linarith _ = 2 * m := by ring _ ≤ p * m := by rel [p_prime.left] _ = n := n_eq_pm.symm done obtain (L : List Nat) (h6 : prime_factorization m L) from ih m m_lt_n m_pos define at h6 have ndpl_L : nondec_prime_list L := h6.left define at ndpl_L apply Exists.intro (p :: L) define apply And.intro · -- Proof of nondec_prime_list (p :: L) define apply And.intro · -- Proof of all_prime (p :: L) rewrite [all_prime_cons] show prime p ∧ all_prime L from And.intro p_prime ndpl_L.left done · -- Proof of nondec (p :: L) rewrite [nondec_cons] apply And.intro _ ndpl_L.right fix q : Nat assume q_in_L : q ∈ L have h7 : q ∣ prod L := list_elt_dvd_prod q_in_L rewrite [h6.right] at h7 --h7 : q ∣ m have h8 : m ∣ n := by apply Exists.intro p rewrite [n_eq_pm] ring done have q_dvd_n : q ∣ n := dvd_trans h7 h8 have ap_L : all_prime L := ndpl_L.left define at ap_L have q_prime_factor : prime_factor q n := And.intro (ap_L q q_in_L) q_dvd_n show p ≤ q from p_least q q_prime_factor done done · -- Proof of prod (p :: L) = n rewrite [prod_cons, h6.right, n_eq_pm] rfl done done done theorem Theorem_7_2_2 {a b c : Nat} (h1 : c ∣ a * b) (h2 : rel_prime a c) : c ∣ b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑c ∣ ↑b define at h1; define at h2; define obtain (j : Nat) (h3 : a * b = c * j) from h1 set s : Int := gcd_c1 a c set t : Int := gcd_c2 a c have h4 : s * ↑a + t * ↑c = ↑(gcd a c) := gcd_lin_comb c a rewrite [h2, Nat.cast_one] at h4 --h4 : s * ↑a + t * ↑c = (1 : Int) apply Exists.intro (s * ↑j + t * ↑b) show ↑b = ↑c * (s * ↑j + t * ↑b) from calc ↑b _ = (1 : Int) * ↑b := (one_mul _).symm _ = (s * ↑a + t * ↑c) * ↑b := by rw [h4] _ = s * (↑a * ↑b) + t * ↑c * ↑b := by ring _ = s * (↑c * ↑j) + t * ↑c * ↑b := by rw [←Nat.cast_mul a b, h3, Nat.cast_mul c j] _ = ↑c * (s * ↑j + t * ↑b) := by ring done lemma le_nonzero_prod_left {a b : Nat} (h : a * b ≠ 0) : a ≤ a * b := by have h1 : b ≠ 0 := by contradict h with h1 rewrite [h1] ring done have h2 : 1 ≤ b := Nat.pos_of_ne_zero h1 show a ≤ a * b from calc a = a * 1 := (mul_one a).symm _ ≤ a * b := by rel [h2] done lemma le_nonzero_prod_right {a b : Nat} (h : a * b ≠ 0) : b ≤ a * b := by rewrite [mul_comm] rewrite [mul_comm] at h show b ≤ b * a from le_nonzero_prod_left h done lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry lemma rel_prime_of_prime_not_dvd {a p : Nat} (h1 : prime p) (h2 : ¬p ∣ a) : rel_prime a p := by have h3 : gcd a p ∣ a := gcd_dvd_left a p have h4 : gcd a p ∣ p := gcd_dvd_right a p have h5 : gcd a p = 1 ∨ gcd a p = p := dvd_prime h1 h4 have h6 : gcd a p ≠ p := by contradict h2 with h6 rewrite [h6] at h3 show p ∣ a from h3 done disj_syll h5 h6 show rel_prime a p from h5 done theorem Theorem_7_2_3 {a b p : Nat} (h1 : prime p) (h2 : p ∣ a * b) : p ∣ a ∨ p ∣ b := by or_right with h3 have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3 show p ∣ b from Theorem_7_2_2 h2 h4 done lemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a ≥ 1 := by have h1 : a ≠ 0 := by by_contra h1 rewrite [h1] at h contradict h linarith done show a ≥ 1 from Nat.pos_of_ne_zero h1 done lemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by have h1 : a ≥ 1 := ge_one_of_prod_one h have h2 : a * b ≠ 0 := by linarith have h3 : a ≤ a * b := le_nonzero_prod_left h2 rewrite [h] at h3 show a = 1 from Nat.le_antisymm h3 h1 done lemma eq_one_of_dvd_one {n : Nat} (h : n ∣ 1) : n = 1 := by obtain (j : Nat) (h1 : 1 = n * j) from h show n = 1 from eq_one_of_prod_one h1.symm done lemma prime_not_one {p : Nat} (h : prime p) : p ≠ 1 := by define at h linarith done theorem Theorem_7_2_4 {p : Nat} (h1 : prime p) : ∀ (l : List Nat), p ∣ prod l → ∃ a ∈ l, p ∣ a := by apply List.rec · -- Base Case. Goal : p ∣ prod [] → ∃ a ∈ [], p ∣ a rewrite [prod_nil] assume h2 : p ∣ 1 show ∃ a ∈ [], p ∣ a from absurd (eq_one_of_dvd_one h2) (prime_not_one h1) done · -- Induction Step fix b : Nat fix L : List Nat assume ih : p ∣ prod L → ∃ a ∈ L, p ∣ a --Goal : p ∣ prod (b :: L) → ∃ a ∈ b :: L, p ∣ a assume h2 : p ∣ prod (b :: L) rewrite [prod_cons] at h2 have h3 : p ∣ b ∨ p ∣ prod L := Theorem_7_2_3 h1 h2 by_cases on h3 · -- Case 1. h3 : p ∣ b apply Exists.intro b show b ∈ b :: L ∧ p ∣ b from And.intro (List.mem_cons_self b L) h3 done · -- Case 2. h3 : p ∣ prod L obtain (a : Nat) (h4 : a ∈ L ∧ p ∣ a) from ih h3 apply Exists.intro a show a ∈ b :: L ∧ p ∣ a from And.intro (List.mem_cons_of_mem b h4.left) h4.right done done done lemma prime_in_list {p : Nat} {l : List Nat} (h1 : prime p) (h2 : all_prime l) (h3 : p ∣ prod l) : p ∈ l := by obtain (a : Nat) (h4 : a ∈ l ∧ p ∣ a) from Theorem_7_2_4 h1 l h3 define at h2 have h5 : prime a := h2 a h4.left have h6 : p = 1 ∨ p = a := dvd_prime h5 h4.right disj_syll h6 (prime_not_one h1) rewrite [h6] show a ∈ l from h4.left done lemma first_le_first {p q : Nat} {l m : List Nat} (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m)) (h3 : prod (p :: l) = prod (q :: m)) : p ≤ q := by define at h1; define at h2 have h4 : q ∣ prod (p :: l) := by define apply Exists.intro (prod m) rewrite [←prod_cons] show prod (p :: l) = prod (q :: m) from h3 done have h5 : all_prime (q :: m) := h2.left rewrite [all_prime_cons] at h5 have h6 : q ∈ p :: l := prime_in_list h5.left h1.left h4 have h7 : nondec (p :: l) := h1.right rewrite [nondec_cons] at h7 rewrite [List.mem_cons] at h6 by_cases on h6 · -- Case 1. h6 : q = p linarith done · -- Case 2. h6 : q ∈ l have h8 : ∀ m ∈ l, p ≤ m := h7.left show p ≤ q from h8 q h6 done done lemma nondec_prime_list_tail {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by define at h define rewrite [all_prime_cons, nondec_cons] at h show all_prime l ∧ nondec l from And.intro h.left.right h.right.right done lemma cons_prod_not_one {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : prod (p :: l) ≠ 1 := by define at h have h1 : all_prime (p :: l) := h.left rewrite [all_prime_cons] at h1 rewrite [prod_cons] by_contra h2 show False from (prime_not_one h1.left) (eq_one_of_prod_one h2) done lemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) : l = [] ↔ prod l = 1 := by apply Iff.intro · -- (→) assume h1 : l = [] rewrite [h1] show prod [] = 1 from prod_nil done · -- (←) contrapos assume h1 : ¬l = [] obtain (p : Nat) (h2 : ∃ (L : List Nat), l = p :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List Nat) (h3 : l = p :: L) from h2 rewrite [h3] at h rewrite [h3] show ¬prod (p :: L) = 1 from cons_prod_not_one h done done lemma prime_pos {p : Nat} (h : prime p) : p > 0 := by define at h linarith done theorem Theorem_7_2_5 : ∀ (l1 l2 : List Nat), nondec_prime_list l1 → nondec_prime_list l2 → prod l1 = prod l2 → l1 = l2 := by apply List.rec · -- Base Case. Goal : ∀ (l2 : List Nat), nondec_prime_list [] → -- nondec_prime_list l2 → prod [] = prod l2 → [] = l2 fix l2 : List Nat assume h1 : nondec_prime_list [] assume h2 : nondec_prime_list l2 assume h3 : prod [] = prod l2 rewrite [prod_nil, eq_comm, ←list_nil_iff_prod_one h2] at h3 show [] = l2 from h3.symm done · -- Induction Step fix p : Nat fix L1 : List Nat assume ih : ∀ (L2 : List Nat), nondec_prime_list L1 → nondec_prime_list L2 → prod L1 = prod L2 → L1 = L2 -- Goal : ∀ (l2 : List Nat), nondec_prime_list (p :: L1) → -- nondec_prime_list l2 → prod (p :: L1) = prod l2 → p :: L1 = l2 fix l2 : List Nat assume h1 : nondec_prime_list (p :: L1) assume h2 : nondec_prime_list l2 assume h3 : prod (p :: L1) = prod l2 have h4 : ¬prod (p :: L1) = 1 := cons_prod_not_one h1 rewrite [h3, ←list_nil_iff_prod_one h2] at h4 obtain (q : Nat) (h5 : ∃ (L : List Nat), l2 = q :: L) from List.exists_cons_of_ne_nil h4 obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5 rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2) rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2) have h7 : p ≤ q := first_le_first h1 h2 h3 have h8 : q ≤ p := first_le_first h2 h1 h3.symm have h9 : p = q := by linarith rewrite [h9, prod_cons, prod_cons] at h3 --h3 : q * prod L1 = q * prod L2 have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1 have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2 define at h2 have h12 : all_prime (q :: L2) := h2.left rewrite [all_prime_cons] at h12 have h13 : q > 0 := prime_pos h12.left have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3 have h15 : L1 = L2 := ih L2 h10 h11 h14 rewrite [h6, h9, h15] rfl done done theorem fund_thm_arith (n : Nat) (h : n ≥ 1) : ∃! (l : List Nat), prime_factorization n l := by exists_unique · -- Existence show ∃ (l : List Nat), prime_factorization n l from exists_prime_factorization n h done · -- Uniqueness fix l1 : List Nat; fix l2 : List Nat assume h1 : prime_factorization n l1 assume h2 : prime_factorization n l2 define at h1; define at h2 have h3 : prod l1 = n := h1.right rewrite [←h2.right] at h3 show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3 done done /- Section 7.3 -/ theorem congr_refl (m : Nat) : ∀ (a : Int), a ≡ a (MOD m) := by fix a : Int define --Goal : ∃ (c : Int), a - a = ↑m * c apply Exists.intro 0 ring done theorem congr_symm {m : Nat} : ∀ {a b : Int}, a ≡ b (MOD m) → b ≡ a (MOD m) := by fix a : Int; fix b : Int assume h1 : a ≡ b (MOD m) define at h1 --h1 : ∃ (c : Int), a - b = ↑m * c define --Goal : ∃ (c : Int), b - a = ↑m * c obtain (c : Int) (h2 : a - b = m * c) from h1 apply Exists.intro (-c) show b - a = m * (-c) from calc b - a _ = -(a - b) := by ring _ = -(m * c) := by rw [h2] _ = m * (-c) := by ring done theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry /- Fundamental properties of congruence classes -/ lemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) : X = Y ↔ X.val = Y.val := Fin.ext_iff lemma val_nat_eq_mod (n k : Nat) : ([k]_(n + 1)).val = k % (n + 1) := by rfl lemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl theorem cc_rep {m : Nat} (X : ZMod m) : ∃ (a : Int), X = [a]_m := match m with | 0 => by apply Exists.intro X rfl done | n + 1 => by apply Exists.intro ↑(X.val) have h1 : X.val < n + 1 := Fin.prop X rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1] rfl done theorem add_class (m : Nat) (a b : Int) : [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm theorem mul_class (m : Nat) (a b : Int) : [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm lemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ [a - b]_m = [0]_m := by apply Iff.intro · -- (→) assume h1 : [a]_m = [b]_m have h2 : a - b = a + (-b) := by ring have h3 : b + (-b) = 0 := by ring show [a - b]_m = [0]_m from calc [a - b]_m _ = [a + (-b)]_m := by rw [h2] _ = [a]_m + [-b]_m := by rw [add_class] _ = [b]_m + [-b]_m := by rw [h1] _ = [b + -b]_m := by rw [add_class] _ = [0]_m := by rw [h3] done · -- (←) assume h1 : [a - b]_m = [0]_m have h2 : b + (a - b) = a := by ring have h3 : b + 0 = b := by ring show [a]_m = [b]_m from calc [a]_m _ = [b + (a - b)]_m := by rw [h2] _ = [b]_m + [a - b]_m := by rw [add_class] _ = [b]_m + [0]_m := by rw [h1] _ = [b + 0]_m := by rw [add_class] _ = [b]_m := by rw [h3] done done lemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) : [a]_m = [0]_m → [-a]_m = [0]_m := by assume h1 : [a]_m = [0]_m have h2 : 0 + (-a) = -a := by ring have h3 : a + (-a) = 0 := by ring show [-a]_m = [0]_m from calc [-a]_m _ = [0 + (-a)]_m := by rw [h2] _ = [0]_m + [-a]_m := by rw [add_class] _ = [a]_m + [-a]_m := by rw [h1] _ = [a + (-a)]_m := by rw [add_class] _ = [0]_m := by rw [h3] done lemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) : [-a]_m = [0]_m ↔ [a]_m = [0]_m := by apply Iff.intro _ (cc_neg_zero_of_cc_zero m a) assume h1 : [-a]_m = [0]_m have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1 have h3 : -(-a) = a := by ring rewrite [h3] at h2 show [a]_m = [0]_m from h2 done lemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl lemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m ↔ m ∣ k := match m with | 0 => by have h : (0 : Int) = (↑(0 : Nat) : Int) := by rfl rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj] apply Iff.intro · -- (→) assume h1 : k = 0 rewrite [h1] show 0 ∣ 0 from dvd_self 0 done · -- (←) assume h1 : 0 ∣ k obtain (c : Nat) (h2 : k = 0 * c) from h1 rewrite [h2] ring done done | n + 1 => by rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero] show k % (n + 1) = 0 ↔ n + 1 ∣ k from (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm done lemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m ↔ ↑m ∣ a := by obtain (k : Nat) (h1 : a = ↑k ∨ a = -↑k) from Int.eq_nat_or_neg a by_cases on h1 · -- Case 1. h1: a = ↑k rewrite [h1, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done · -- Case 2. h1: a = -↑k rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done done theorem cc_eq_iff_congr (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ a ≡ b (MOD m) := calc [a]_m = [b]_m _ ↔ [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b _ ↔ ↑m ∣ (a - b) := cc_zero_iff_dvd m (a - b) _ ↔ a ≡ b (MOD m) := by rfl /- End of fundamental properties of congruence classes -/ lemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m := by have h1 : (↑m : Int) ≠ 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m) show 0 ≤ a % m from Int.emod_nonneg a h1 done lemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m) have h2 : (↑m : Int) > 0 := (Nat.cast_pos).rtl h1 show a % m < m from Int.emod_lt_of_pos a h2 done lemma congr_mod_mod (m : Nat) (a : Int) : a ≡ a % m (MOD m) := by define have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m apply Exists.intro (a / m) show a - a % m = m * (a / m) from calc a - (a % m) _ = m * (a / m) + a % m - a % m := by rw [h1] _ = m * (a / m) := by ring done lemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) := And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a)) theorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) : ∃! (r : Int), 0 ≤ r ∧ r < m ∧ a ≡ r (MOD m) := by exists_unique · -- Existence apply Exists.intro (a % m) show 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) from mod_cmpl_res m a done · -- Uniqueness fix r1 : Int; fix r2 : Int assume h1 : 0 ≤ r1 ∧ r1 < m ∧ a ≡ r1 (MOD m) assume h2 : 0 ≤ r2 ∧ r2 < m ∧ a ≡ r2 (MOD m) have h3 : r1 ≡ r2 (MOD m) := congr_trans (congr_symm h1.right.right) h2.right.right obtain (d : Int) (h4 : r1 - r2 = m * d) from h3 have h5 : r1 - r2 < m * 1 := by linarith have h6 : m * (-1) < r1 - r2 := by linarith rewrite [h4] at h5 --h5 : m * d < m * 1 rewrite [h4] at h6 --h6 : m * -1 < m * d have h7 : (↑m : Int) ≥ 0 := Nat.cast_nonneg m have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7 have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7 have h10 : d = 0 := by linarith show r1 = r2 from calc r1 _ = r1 - r2 + r2 := by ring _ = m * 0 + r2 := by rw [h4, h10] _ = r2 := by ring done done lemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m := (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a) theorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y rewrite [h1, h2] have h3 : a + b = b + a := by ring show [a]_m + [b]_m = [b]_m + [a]_m from calc [a]_m + [b]_m _ = [a + b]_m := add_class m a b _ = [b + a]_m := by rw [h3] _ = [b]_m + [a]_m := (add_class m b a).symm done theorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X rewrite [h1] have h2 : a * 1 = a := by ring show [a]_m * [1]_m = [a]_m from calc [a]_m * [1]_m _ = [a * 1]_m := mul_class m a 1 _ = [a]_m := by rw [h2] done theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry lemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) : [a]_m * [gcd_c2 m a]_m = [1]_m := by set s : Int := gcd_c1 m a have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m define at h1 rewrite [h1, Nat.cast_one] at h2 --h2 : s * ↑m + gcd_c2 m a * ↑a = 1 rewrite [mul_class, cc_eq_iff_congr] define --Goal : ∃ (c : Int), ↑a * gcd_c2 m a - 1 = ↑m * c apply Exists.intro (-s) show a * (gcd_c2 m a) - 1 = m * (-s) from calc a * (gcd_c2 m a) - 1 _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring _ = 1 + m * (-s) - 1 := by rw [h2] _ = m * (-s) := by ring done theorem Theorem_7_3_7 (m a : Nat) : invertible [a]_m ↔ rel_prime m a := by apply Iff.intro · -- (→) assume h1 : invertible [a]_m define at h1 obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1 obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y rewrite [h3, mul_class, cc_eq_iff_congr] at h2 define at h2 obtain (c : Int) (h4 : a * b - 1 = m * c) from h2 rewrite [Exercise_7_2_6] --Goal : ∃ (s t : Int), s * ↑m + t * ↑a = 1 apply Exists.intro (-c) apply Exists.intro b show (-c) * m + b * a = 1 from calc (-c) * m + b * a _ = (-c) * m + (a * b - 1) + 1 := by ring _ = (-c) * m + m * c + 1 := by rw [h4] _ = 1 := by ring done · -- (←) assume h1 : rel_prime m a define show ∃ (Y : ZMod m), [a]_m * Y = [1]_m from Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1) done done /- Section 7.4 -/ section Euler open Euler lemma num_rp_below_base {m : Nat} : num_rp_below m 0 = 0 := by rfl lemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) : num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by have h1 : num_rp_below m (j + 1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : gcd m j = 1 rewrite [if_pos h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1 show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1 done lemma num_rp_below_step_not_rp {m j : Nat} (h : ¬rel_prime m j) : num_rp_below m (j + 1) = num_rp_below m j := by have h1 : num_rp_below m (j +1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : ¬gcd m j = 1 rewrite [if_neg h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j show num_rp_below m (j + 1) = num_rp_below m j from h1 done lemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl #eval phi 10 --Answer: 4 lemma prod_inv_iff_inv {m : Nat} {X : ZMod m} (h1 : invertible X) (Y : ZMod m) : invertible (X * Y) ↔ invertible Y := by apply Iff.intro · -- (→) assume h2 : invertible (X * Y) obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2 apply Exists.intro (X * Z) rewrite [←h3] --Goal : Y * (X * Z) = X * Y * Z ring --Note that ring can do algebra in ZMod m done · -- (←) assume h2 : invertible Y obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1 obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2 apply Exists.intro (Xi * Yi) show (X * Y) * (Xi * Yi) = [1]_m from calc X * Y * (Xi * Yi) _ = (X * Xi) * (Y * Yi) := by ring _ = [1]_m * [1]_m := by rw [h3, h4] _ = [1]_m := Theorem_7_3_6_7 [1]_m done done lemma F_rp_def {m i : Nat} (h : rel_prime m i) : F m i = [i]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h --h : gcd m i = 1 rewrite [if_pos h] at h1 show F m i = [i]_m from h1 done lemma F_not_rp_def {m i : Nat} (h : ¬rel_prime m i) : F m i = [1]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h rewrite [h1, if_neg h] rfl done lemma prod_seq_base {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 0 k f = [1]_m := by rfl lemma prod_seq_step {m : Nat} (n k : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl lemma prod_seq_zero_step {m : Nat} (n : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by rewrite [prod_seq_step, zero_add] rfl done lemma prod_one {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 1 k f = f k := by rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7] rfl done lemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl lemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m := calc [G m a i]_m _ = [(a * i) % m]_m := by rfl _ = [a * i]_m := (cc_eq_mod m (a * i)).symm _ = [a]_m * [i]_m := (mul_class m a i).symm lemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) : rel_prime m (G m a i) ↔ rel_prime m i := by have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1 show rel_prime m (G m a i) ↔ rel_prime m i from calc rel_prime m (G m a i) _ ↔ invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm _ ↔ invertible ([a]_m * [i]_m) := by rw [cc_G] _ ↔ invertible [i]_m := prod_inv_iff_inv h2 ([i]_m) _ ↔ rel_prime m i := Theorem_7_3_7 m i done lemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) : F m (G m a i) = [a]_m * F m i := by have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2 show F m (G m a i) = [a]_m * F m i from calc F m (G m a i) _ = [G m a i]_m := F_rp_def h3 _ = [a]_m * [i]_m := cc_G m a i _ = [a]_m * F m i := by rw [F_rp_def h2] done lemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : ¬rel_prime m i) : F m (G m a i) = [1]_m := by rewrite [←G_rp_iff h1 i] at h2 show F m (G m a i) = [1]_m from F_not_rp_def h2 done lemma FG_prod {m a : Nat} (h1 : rel_prime m a) : ∀ (k : Nat), prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by by_induc · -- Base Case show prod_seq 0 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from calc prod_seq 0 0 ((F m) ∘ (G m a)) _ = [1]_m := prod_seq_base _ _ _ = [a]_m ^ 0 * [1]_m := by ring _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by rw [num_rp_below_base, prod_seq_base] done · -- Induction Step fix k : Nat assume ih : prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) by_cases h2 : rel_prime m k · -- Case 1. h2 : rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([a]_m * F m k) := by rw [FG_rp h1 h2] _ = [a]_m ^ ((num_rp_below m k) + 1) * ((prod_seq k 0 (F m)) * F m k) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_rp h2, prod_seq_zero_step] done · -- Case 2. h2 : ¬rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([1]_m) := by rw [FG_not_rp h1 h2] _ = [a]_m ^ (num_rp_below m k) * (prod_seq k 0 (F m) * ([1]_m)) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_not_rp h2, prod_seq_zero_step, F_not_rp_def h2] done done done lemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by define --Goal : ∀ i < m, G m a i < m fix i : Nat assume h1 : i < m rewrite [G_def] --Goal : a * i % m < m show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m) done lemma left_inv_one_one_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g' (g i) = i) : one_one_below n g := sorry lemma right_inv_onto_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g (g' i) = i) (h2 : maps_below n g') : onto_below n g := by define at h2; define fix k : Nat assume h3 : k < n apply Exists.intro (g' k) show g' k < n ∧ g (g' k) = k from And.intro (h2 k h3) (h1 k h3) done lemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m * [inv_mod m a]_m = [1]_m := by have h2 : 0 ≤ (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a) show [a]_m * [inv_mod m a]_m = [1]_m from calc [a]_m * [inv_mod m a]_m _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2] _ = [a]_m * [gcd_c2 m a]_m := by rw [←cc_eq_mod] _ = [1]_m := gcd_c2_inv h1 done lemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m := calc a * (b % m) % m = a % m * (b % m % m) % m := Nat.mul_mod _ _ _ _ = a % m * (b % m) % m := by rw [Nat.mod_mod] _ = a * b % m := (Nat.mul_mod _ _ _).symm lemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm] rfl done theorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] : ↑a ≡ ↑b (MOD m) ↔ a % m = b % m := sorry lemma mul_inv_mod_cancel {m a i : Nat} [NeZero m] (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1 rewrite [mul_class, cc_eq_iff_congr, ←Nat.cast_mul, ←Nat.cast_one, congr_iff_mod_eq_Nat] at h3 show a * inv_mod m a * i % m = i from calc a * (inv_mod m a) * i % m _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = 1 % m * i % m := by rw [h3] _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = i % m := by rw [one_mul] _ = i := Nat.mod_eq_of_lt h2 done lemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl lemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, G m a (Ginv m a i) = i := by fix i : Nat assume h2 : i < m show G m a (Ginv m a i) = i from calc G m a (Ginv m a i) _ = a * ((inv_mod m a * i) % m) % m := by rfl _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc] _ = i := mul_inv_mod_cancel h1 h2 done

lemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, Ginv m a (G m a i) = i

HTPI.Ginv_left_inv

htpi/HTPILib/Chap7.lean

htpi

/- Copyright 2023 Daniel J. Velleman -/ import HTPILib.Chap6 namespace HTPI /- Definitions -/ lemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by have h : n + 1 > 0 := Nat.succ_pos n show a % (n + 1) < n + 1 from Nat.mod_lt a h done def gcd (a b : Nat) : Nat := match b with | 0 => a | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd (n + 1) (a % (n + 1)) termination_by b mutual def gcd_c1 (a b : Nat) : Int := match b with | 0 => 1 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c2 (n + 1) (a % (n + 1)) --Corresponds to s = t' termination_by b def gcd_c2 (a b : Nat) : Int := match b with | 0 => 0 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c1 (n + 1) (a % (n + 1)) - (gcd_c2 (n + 1) (a % (n + 1))) * ↑(a / (n + 1)) --Corresponds to t = s' - t'q termination_by b end def prime (n : Nat) : Prop := 2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n def prime_factor (p n : Nat) : Prop := prime p ∧ p ∣ n def all_prime (l : List Nat) : Prop := ∀ p ∈ l, prime p def nondec (l : List Nat) : Prop := match l with | [] => True --Of course, True is a proposition that is always true | n :: L => (∀ m ∈ L, n ≤ m) ∧ nondec L def nondec_prime_list (l : List Nat) : Prop := all_prime l ∧ nondec l def prod (l : List Nat) : Nat := match l with | [] => 1 | n :: L => n * (prod L) def prime_factorization (n : Nat) (l : List Nat) : Prop := nondec_prime_list l ∧ prod l = n def rel_prime (a b : Nat) : Prop := gcd a b = 1 def congr_mod (m : Nat) (a b : Int) : Prop := (↑m : Int) ∣ (a - b) def cc (m : Nat) (a : Int) : ZMod m := (↑a : ZMod m) notation:50 a " ≡ " b " (MOD " m ")" => congr_mod m a b notation:max "["a"]_"m:max => cc m a def invertible {m : Nat} (X : ZMod m) : Prop := ∃ (Y : ZMod m), X * Y = [1]_m def num_rp_below (m k : Nat) : Nat := match k with | 0 => 0 | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j def phi (m : Nat) : Nat := num_rp_below m m def prod_seq {m : Nat} (j k : Nat) (f : Nat → ZMod m) : ZMod m := match j with | 0 => [1]_m | n + 1 => prod_seq n k f * f (k + n) def maps_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i < n, g i < n def one_one_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i1 < n, ∀ i2 < n, g i1 = g i2 → i1 = i2 def onto_below (n : Nat) (g : Nat → Nat) : Prop := ∀ k < n, ∃ i < n, g i = k def perm_below (n : Nat) (g : Nat → Nat) : Prop := maps_below n g ∧ one_one_below n g ∧ onto_below n g def inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m) def swap (u v i : Nat) : Nat := if i = u then v else if i = v then u else i namespace Euler --For definitions specific to Euler's theorem def F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m def G (m a i : Nat) : Nat := (a * i) % m def Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i end Euler /- Section 7.1 -/ theorem dvd_mod_of_dvd_a_b {a b d : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ (a % b) := by set q : Nat := a / b have h3 : b * q + a % b = a := Nat.div_add_mod a b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 define --Goal : ∃ (c : Nat), a % b = d * c apply Exists.intro (j - k * q) show a % b = d * (j - k * q) from calc a % b _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm _ = a - b * q := by rw [h3] _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc] _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm done theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry #eval gcd 672 161 --Answer: 7 lemma gcd_base (a : Nat) : gcd a 0 = a := by rfl lemma gcd_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd a b = gcd b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1)) rfl done lemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b ≠ 0) : a % b < b := by have h1 : b > 0 := Nat.pos_of_ne_zero h show a % b < b from Nat.mod_lt a h1 done lemma dvd_self (n : Nat) : n ∣ n := by apply Exists.intro 1 ring done theorem gcd_dvd : ∀ (b a : Nat), (gcd a b) ∣ a ∧ (gcd a b) ∣ b := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd a b_1) ∣ a ∧ (gcd a b_1) ∣ b_1 fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_base] --Goal: a ∣ a ∧ a ∣ 0 apply And.intro (dvd_self a) define apply Exists.intro 0 rfl done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_nonzero a h1] --Goal : gcd b (a % b) ∣ a ∧ gcd b (a % b) ∣ b have h2 : a % b < b := mod_nonzero_lt a h1 have h3 : (gcd b (a % b)) ∣ b ∧ (gcd b (a % b)) ∣ (a % b) := ih (a % b) h2 b apply And.intro _ h3.left show (gcd b (a % b)) ∣ a from dvd_a_of_dvd_b_mod h3.left h3.right done done theorem gcd_dvd_left (a b : Nat) : (gcd a b) ∣ a := (gcd_dvd b a).left theorem gcd_dvd_right (a b : Nat) : (gcd a b) ∣ b := (gcd_dvd b a).right lemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl lemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c1 a b = gcd_c2 b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done lemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl lemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * ↑(a / b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done theorem gcd_lin_comb : ∀ (b a : Nat), (gcd_c1 a b) * ↑a + (gcd_c2 a b) * ↑b = ↑(gcd a b) := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd_c1 a b_1) * ↑a + (gcd_c2 a b_1) * ↑b_1 = ↑(gcd a b_1) fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base] --Goal : 1 * ↑a + 0 * ↑0 = ↑a ring done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1] --Goal : gcd_c2 b (a % b) * ↑a + -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * ↑(a / b)) * ↑b = -- ↑(gcd b (a % b)) set r : Nat := a % b set q : Nat := a / b set s : Int := gcd_c1 b r set t : Int := gcd_c2 b r --Goal : t * ↑a + (s - t * ↑q) * ↑b = ↑(gcd b r) have h2 : r < b := mod_nonzero_lt a h1 have h3 : s * ↑b + t * ↑r = ↑(gcd b r) := ih r h2 b have h4 : b * q + r = a := Nat.div_add_mod a b rewrite [←h3, ←h4] rewrite [Nat.cast_add, Nat.cast_mul] --Goal : t * (↑b * ↑q + ↑r) + (s - t * ↑q) * ↑b = s * ↑b + t * ↑r ring done done #eval gcd_c1 672 161 --Answer: 6 #eval gcd_c2 672 161 --Answer: -25 --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161 theorem Theorem_7_1_6 {d a b : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ gcd a b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑d ∣ ↑(gcd a b) set s : Int := gcd_c1 a b set t : Int := gcd_c2 a b have h3 : s * ↑a + t * ↑b = ↑(gcd a b) := gcd_lin_comb b a rewrite [←h3] --Goal : ↑d ∣ s * ↑a + t * ↑b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul] --Goal : ↑d ∣ s * (↑d * ↑j) + t * (↑d * ↑k) define apply Exists.intro (s * ↑j + t * ↑k) ring done /- Section 7.2 -/ theorem dvd_trans {a b c : Nat} (h1 : a ∣ b) (h2 : b ∣ c) : a ∣ c := by define at h1; define at h2; define obtain (m : Nat) (h3 : b = a * m) from h1 obtain (n : Nat) (h4 : c = b * n) from h2 rewrite [h3, mul_assoc] at h4 apply Exists.intro (m * n) show c = a * (m * n) from h4 done lemma exists_prime_factor : ∀ (n : Nat), 2 ≤ n → ∃ (p : Nat), prime_factor p n := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, 2 ≤ n_1 → ∃ (p : Nat), prime_factor p n_1 assume h1 : 2 ≤ n by_cases h2 : prime n · -- Case 1. h2 : prime n apply Exists.intro n define --Goal : prime n ∧ n ∣ n show prime n ∧ n ∣ n from And.intro h2 (dvd_self n) done · -- Case 2. h2 : ¬prime n define at h2 --h2 : ¬(2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n) demorgan at h2 disj_syll h2 h1 obtain (a : Nat) (h3 : ∃ (b : Nat), a * b = n ∧ a < n ∧ b < n) from h2 obtain (b : Nat) (h4 : a * b = n ∧ a < n ∧ b < n) from h3 have h5 : 2 ≤ a := by by_contra h6 have h7 : a ≤ 1 := by linarith have h8 : n ≤ b := calc n _ = a * b := h4.left.symm _ ≤ 1 * b := by rel [h7] _ = b := by ring linarith --n ≤ b contradicts b < n done have h6 : ∃ (p : Nat), prime_factor p a := ih a h4.right.left h5 obtain (p : Nat) (h7 : prime_factor p a) from h6 apply Exists.intro p define --Goal : prime p ∧ p ∣ n define at h7 --h7 : prime p ∧ p ∣ a apply And.intro h7.left have h8 : a ∣ n := by apply Exists.intro b show n = a * b from (h4.left).symm done show p ∣ n from dvd_trans h7.right h8 done done lemma exists_least_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q := by set S : Set Nat := {p : Nat | prime_factor p n} have h2 : ∃ (p : Nat), p ∈ S := exists_prime_factor n h show ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q from well_ord_princ S h2 done lemma all_prime_nil : all_prime [] := by define --Goal : ∀ p ∈ [], prime p fix p : Nat contrapos --Goal : ¬prime p → p ∉ [] assume h1 : ¬prime p show p ∉ [] from List.not_mem_nil p done lemma all_prime_cons (n : Nat) (L : List Nat) : all_prime (n :: L) ↔ prime n ∧ all_prime L := by apply Iff.intro · -- (→) assume h1 : all_prime (n :: L) --Goal : prime n ∧ all_prime L define at h1 --h1 : ∀ p ∈ n :: L, prime p apply And.intro (h1 n (List.mem_cons_self n L)) define --Goal : ∀ p ∈ L, prime p fix p : Nat assume h2 : p ∈ L show prime p from h1 p (List.mem_cons_of_mem n h2) done · -- (←) assume h1 : prime n ∧ all_prime L --Goal : all_prime (n :: l) define : all_prime L at h1 define fix p : Nat assume h2 : p ∈ n :: L rewrite [List.mem_cons] at h2 --h2 : p = n ∨ p ∈ L by_cases on h2 · -- Case 1. h2 : p = n rewrite [h2] show prime n from h1.left done · -- Case 2. h2 : p ∈ L show prime p from h1.right p h2 done done done lemma nondec_nil : nondec [] := by define --Goal : True trivial --trivial proves some obviously true statements, such as True done lemma nondec_cons (n : Nat) (L : List Nat) : nondec (n :: L) ↔ (∀ m ∈ L, n ≤ m) ∧ nondec L := by rfl lemma prod_nil : prod [] = 1 := by rfl lemma prod_cons : prod (n :: L) = n * (prod L) := by rfl lemma exists_cons_of_length_eq_succ {A : Type} {l : List A} {n : Nat} (h : l.length = n + 1) : ∃ (a : A) (L : List A), l = a :: L ∧ L.length = n := by have h1 : ¬l.length = 0 := by linarith rewrite [List.length_eq_zero] at h1 obtain (a : A) (h2 : ∃ (L : List A), l = a :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List A) (h3 : l = a :: L) from h2 apply Exists.intro a apply Exists.intro L apply And.intro h3 have h4 : (a :: L).length = L.length + 1 := List.length_cons a L rewrite [←h3, h] at h4 show L.length = n from (Nat.add_right_cancel h4).symm done lemma list_elt_dvd_prod_by_length (a : Nat) : ∀ (n : Nat), ∀ (l : List Nat), l.length = n → a ∈ l → a ∣ prod l := by by_induc · --Base Case fix l : List Nat assume h1 : l.length = 0 rewrite [List.length_eq_zero] at h1 --h1 : l = [] rewrite [h1] --Goal : a ∈ [] → a ∣ prod [] contrapos assume h2 : ¬a ∣ prod [] show a ∉ [] from List.not_mem_nil a done · -- Induction Step fix n : Nat assume ih : ∀ (l : List Nat), List.length l = n → a ∈ l → a ∣ prod l fix l : List Nat assume h1 : l.length = n + 1 --Goal : a ∈ l → a ∣ prod l obtain (b : Nat) (h2 : ∃ (L : List Nat), l = b :: L ∧ L.length = n) from exists_cons_of_length_eq_succ h1 obtain (L : List Nat) (h3 : l = b :: L ∧ L.length = n) from h2 have h4 : a ∈ L → a ∣ prod L := ih L h3.right assume h5 : a ∈ l rewrite [h3.left, prod_cons] --Goal : a ∣ b * prod L rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b ∨ a ∈ L by_cases on h5 · -- Case 1. h5 : a = b apply Exists.intro (prod L) rewrite [h5] rfl done · -- Case 2. h5 : a ∈ L have h6 : a ∣ prod L := h4 h5 have h7 : prod L ∣ b * prod L := by apply Exists.intro b ring done show a ∣ b * prod L from dvd_trans h6 h7 done done done lemma list_elt_dvd_prod {a : Nat} {l : List Nat} (h : a ∈ l) : a ∣ prod l := by set n : Nat := l.length have h1 : l.length = n := by rfl show a ∣ prod l from list_elt_dvd_prod_by_length a n l h1 h done lemma exists_prime_factorization : ∀ (n : Nat), n ≥ 1 → ∃ (l : List Nat), prime_factorization n l := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, n_1 ≥ 1 → ∃ (l : List Nat), prime_factorization n_1 l assume h1 : n ≥ 1 by_cases h2 : n = 1 · -- Case 1. h2 : n = 1 apply Exists.intro [] define apply And.intro · -- Proof of nondec_prime_list [] define show all_prime [] ∧ nondec [] from And.intro all_prime_nil nondec_nil done · -- Proof of prod [] = n rewrite [prod_nil, h2] rfl done done · -- Case 2. h2 : n ≠ 1 have h3 : n ≥ 2 := lt_of_le_of_ne' h1 h2 obtain (p : Nat) (h4 : prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q) from exists_least_prime_factor h3 have p_prime_factor : prime_factor p n := h4.left define at p_prime_factor have p_prime : prime p := p_prime_factor.left have p_dvd_n : p ∣ n := p_prime_factor.right have p_least : ∀ (q : Nat), prime_factor q n → p ≤ q := h4.right obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n have h5 : m ≠ 0 := by contradict h1 with h6 have h7 : n = 0 := calc n _ = p * m := n_eq_pm _ = p * 0 := by rw [h6] _ = 0 := by ring rewrite [h7] decide done have m_pos : 0 < m := Nat.pos_of_ne_zero h5 have m_lt_n : m < n := by define at p_prime show m < n from calc m _ < m + m := by linarith _ = 2 * m := by ring _ ≤ p * m := by rel [p_prime.left] _ = n := n_eq_pm.symm done obtain (L : List Nat) (h6 : prime_factorization m L) from ih m m_lt_n m_pos define at h6 have ndpl_L : nondec_prime_list L := h6.left define at ndpl_L apply Exists.intro (p :: L) define apply And.intro · -- Proof of nondec_prime_list (p :: L) define apply And.intro · -- Proof of all_prime (p :: L) rewrite [all_prime_cons] show prime p ∧ all_prime L from And.intro p_prime ndpl_L.left done · -- Proof of nondec (p :: L) rewrite [nondec_cons] apply And.intro _ ndpl_L.right fix q : Nat assume q_in_L : q ∈ L have h7 : q ∣ prod L := list_elt_dvd_prod q_in_L rewrite [h6.right] at h7 --h7 : q ∣ m have h8 : m ∣ n := by apply Exists.intro p rewrite [n_eq_pm] ring done have q_dvd_n : q ∣ n := dvd_trans h7 h8 have ap_L : all_prime L := ndpl_L.left define at ap_L have q_prime_factor : prime_factor q n := And.intro (ap_L q q_in_L) q_dvd_n show p ≤ q from p_least q q_prime_factor done done · -- Proof of prod (p :: L) = n rewrite [prod_cons, h6.right, n_eq_pm] rfl done done done theorem Theorem_7_2_2 {a b c : Nat} (h1 : c ∣ a * b) (h2 : rel_prime a c) : c ∣ b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑c ∣ ↑b define at h1; define at h2; define obtain (j : Nat) (h3 : a * b = c * j) from h1 set s : Int := gcd_c1 a c set t : Int := gcd_c2 a c have h4 : s * ↑a + t * ↑c = ↑(gcd a c) := gcd_lin_comb c a rewrite [h2, Nat.cast_one] at h4 --h4 : s * ↑a + t * ↑c = (1 : Int) apply Exists.intro (s * ↑j + t * ↑b) show ↑b = ↑c * (s * ↑j + t * ↑b) from calc ↑b _ = (1 : Int) * ↑b := (one_mul _).symm _ = (s * ↑a + t * ↑c) * ↑b := by rw [h4] _ = s * (↑a * ↑b) + t * ↑c * ↑b := by ring _ = s * (↑c * ↑j) + t * ↑c * ↑b := by rw [←Nat.cast_mul a b, h3, Nat.cast_mul c j] _ = ↑c * (s * ↑j + t * ↑b) := by ring done lemma le_nonzero_prod_left {a b : Nat} (h : a * b ≠ 0) : a ≤ a * b := by have h1 : b ≠ 0 := by contradict h with h1 rewrite [h1] ring done have h2 : 1 ≤ b := Nat.pos_of_ne_zero h1 show a ≤ a * b from calc a = a * 1 := (mul_one a).symm _ ≤ a * b := by rel [h2] done lemma le_nonzero_prod_right {a b : Nat} (h : a * b ≠ 0) : b ≤ a * b := by rewrite [mul_comm] rewrite [mul_comm] at h show b ≤ b * a from le_nonzero_prod_left h done lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry lemma rel_prime_of_prime_not_dvd {a p : Nat} (h1 : prime p) (h2 : ¬p ∣ a) : rel_prime a p := by have h3 : gcd a p ∣ a := gcd_dvd_left a p have h4 : gcd a p ∣ p := gcd_dvd_right a p have h5 : gcd a p = 1 ∨ gcd a p = p := dvd_prime h1 h4 have h6 : gcd a p ≠ p := by contradict h2 with h6 rewrite [h6] at h3 show p ∣ a from h3 done disj_syll h5 h6 show rel_prime a p from h5 done theorem Theorem_7_2_3 {a b p : Nat} (h1 : prime p) (h2 : p ∣ a * b) : p ∣ a ∨ p ∣ b := by or_right with h3 have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3 show p ∣ b from Theorem_7_2_2 h2 h4 done lemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a ≥ 1 := by have h1 : a ≠ 0 := by by_contra h1 rewrite [h1] at h contradict h linarith done show a ≥ 1 from Nat.pos_of_ne_zero h1 done lemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by have h1 : a ≥ 1 := ge_one_of_prod_one h have h2 : a * b ≠ 0 := by linarith have h3 : a ≤ a * b := le_nonzero_prod_left h2 rewrite [h] at h3 show a = 1 from Nat.le_antisymm h3 h1 done lemma eq_one_of_dvd_one {n : Nat} (h : n ∣ 1) : n = 1 := by obtain (j : Nat) (h1 : 1 = n * j) from h show n = 1 from eq_one_of_prod_one h1.symm done lemma prime_not_one {p : Nat} (h : prime p) : p ≠ 1 := by define at h linarith done theorem Theorem_7_2_4 {p : Nat} (h1 : prime p) : ∀ (l : List Nat), p ∣ prod l → ∃ a ∈ l, p ∣ a := by apply List.rec · -- Base Case. Goal : p ∣ prod [] → ∃ a ∈ [], p ∣ a rewrite [prod_nil] assume h2 : p ∣ 1 show ∃ a ∈ [], p ∣ a from absurd (eq_one_of_dvd_one h2) (prime_not_one h1) done · -- Induction Step fix b : Nat fix L : List Nat assume ih : p ∣ prod L → ∃ a ∈ L, p ∣ a --Goal : p ∣ prod (b :: L) → ∃ a ∈ b :: L, p ∣ a assume h2 : p ∣ prod (b :: L) rewrite [prod_cons] at h2 have h3 : p ∣ b ∨ p ∣ prod L := Theorem_7_2_3 h1 h2 by_cases on h3 · -- Case 1. h3 : p ∣ b apply Exists.intro b show b ∈ b :: L ∧ p ∣ b from And.intro (List.mem_cons_self b L) h3 done · -- Case 2. h3 : p ∣ prod L obtain (a : Nat) (h4 : a ∈ L ∧ p ∣ a) from ih h3 apply Exists.intro a show a ∈ b :: L ∧ p ∣ a from And.intro (List.mem_cons_of_mem b h4.left) h4.right done done done lemma prime_in_list {p : Nat} {l : List Nat} (h1 : prime p) (h2 : all_prime l) (h3 : p ∣ prod l) : p ∈ l := by obtain (a : Nat) (h4 : a ∈ l ∧ p ∣ a) from Theorem_7_2_4 h1 l h3 define at h2 have h5 : prime a := h2 a h4.left have h6 : p = 1 ∨ p = a := dvd_prime h5 h4.right disj_syll h6 (prime_not_one h1) rewrite [h6] show a ∈ l from h4.left done lemma first_le_first {p q : Nat} {l m : List Nat} (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m)) (h3 : prod (p :: l) = prod (q :: m)) : p ≤ q := by define at h1; define at h2 have h4 : q ∣ prod (p :: l) := by define apply Exists.intro (prod m) rewrite [←prod_cons] show prod (p :: l) = prod (q :: m) from h3 done have h5 : all_prime (q :: m) := h2.left rewrite [all_prime_cons] at h5 have h6 : q ∈ p :: l := prime_in_list h5.left h1.left h4 have h7 : nondec (p :: l) := h1.right rewrite [nondec_cons] at h7 rewrite [List.mem_cons] at h6 by_cases on h6 · -- Case 1. h6 : q = p linarith done · -- Case 2. h6 : q ∈ l have h8 : ∀ m ∈ l, p ≤ m := h7.left show p ≤ q from h8 q h6 done done lemma nondec_prime_list_tail {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by define at h define rewrite [all_prime_cons, nondec_cons] at h show all_prime l ∧ nondec l from And.intro h.left.right h.right.right done lemma cons_prod_not_one {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : prod (p :: l) ≠ 1 := by define at h have h1 : all_prime (p :: l) := h.left rewrite [all_prime_cons] at h1 rewrite [prod_cons] by_contra h2 show False from (prime_not_one h1.left) (eq_one_of_prod_one h2) done lemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) : l = [] ↔ prod l = 1 := by apply Iff.intro · -- (→) assume h1 : l = [] rewrite [h1] show prod [] = 1 from prod_nil done · -- (←) contrapos assume h1 : ¬l = [] obtain (p : Nat) (h2 : ∃ (L : List Nat), l = p :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List Nat) (h3 : l = p :: L) from h2 rewrite [h3] at h rewrite [h3] show ¬prod (p :: L) = 1 from cons_prod_not_one h done done lemma prime_pos {p : Nat} (h : prime p) : p > 0 := by define at h linarith done theorem Theorem_7_2_5 : ∀ (l1 l2 : List Nat), nondec_prime_list l1 → nondec_prime_list l2 → prod l1 = prod l2 → l1 = l2 := by apply List.rec · -- Base Case. Goal : ∀ (l2 : List Nat), nondec_prime_list [] → -- nondec_prime_list l2 → prod [] = prod l2 → [] = l2 fix l2 : List Nat assume h1 : nondec_prime_list [] assume h2 : nondec_prime_list l2 assume h3 : prod [] = prod l2 rewrite [prod_nil, eq_comm, ←list_nil_iff_prod_one h2] at h3 show [] = l2 from h3.symm done · -- Induction Step fix p : Nat fix L1 : List Nat assume ih : ∀ (L2 : List Nat), nondec_prime_list L1 → nondec_prime_list L2 → prod L1 = prod L2 → L1 = L2 -- Goal : ∀ (l2 : List Nat), nondec_prime_list (p :: L1) → -- nondec_prime_list l2 → prod (p :: L1) = prod l2 → p :: L1 = l2 fix l2 : List Nat assume h1 : nondec_prime_list (p :: L1) assume h2 : nondec_prime_list l2 assume h3 : prod (p :: L1) = prod l2 have h4 : ¬prod (p :: L1) = 1 := cons_prod_not_one h1 rewrite [h3, ←list_nil_iff_prod_one h2] at h4 obtain (q : Nat) (h5 : ∃ (L : List Nat), l2 = q :: L) from List.exists_cons_of_ne_nil h4 obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5 rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2) rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2) have h7 : p ≤ q := first_le_first h1 h2 h3 have h8 : q ≤ p := first_le_first h2 h1 h3.symm have h9 : p = q := by linarith rewrite [h9, prod_cons, prod_cons] at h3 --h3 : q * prod L1 = q * prod L2 have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1 have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2 define at h2 have h12 : all_prime (q :: L2) := h2.left rewrite [all_prime_cons] at h12 have h13 : q > 0 := prime_pos h12.left have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3 have h15 : L1 = L2 := ih L2 h10 h11 h14 rewrite [h6, h9, h15] rfl done done theorem fund_thm_arith (n : Nat) (h : n ≥ 1) : ∃! (l : List Nat), prime_factorization n l := by exists_unique · -- Existence show ∃ (l : List Nat), prime_factorization n l from exists_prime_factorization n h done · -- Uniqueness fix l1 : List Nat; fix l2 : List Nat assume h1 : prime_factorization n l1 assume h2 : prime_factorization n l2 define at h1; define at h2 have h3 : prod l1 = n := h1.right rewrite [←h2.right] at h3 show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3 done done /- Section 7.3 -/ theorem congr_refl (m : Nat) : ∀ (a : Int), a ≡ a (MOD m) := by fix a : Int define --Goal : ∃ (c : Int), a - a = ↑m * c apply Exists.intro 0 ring done theorem congr_symm {m : Nat} : ∀ {a b : Int}, a ≡ b (MOD m) → b ≡ a (MOD m) := by fix a : Int; fix b : Int assume h1 : a ≡ b (MOD m) define at h1 --h1 : ∃ (c : Int), a - b = ↑m * c define --Goal : ∃ (c : Int), b - a = ↑m * c obtain (c : Int) (h2 : a - b = m * c) from h1 apply Exists.intro (-c) show b - a = m * (-c) from calc b - a _ = -(a - b) := by ring _ = -(m * c) := by rw [h2] _ = m * (-c) := by ring done theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry /- Fundamental properties of congruence classes -/ lemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) : X = Y ↔ X.val = Y.val := Fin.ext_iff lemma val_nat_eq_mod (n k : Nat) : ([k]_(n + 1)).val = k % (n + 1) := by rfl lemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl theorem cc_rep {m : Nat} (X : ZMod m) : ∃ (a : Int), X = [a]_m := match m with | 0 => by apply Exists.intro X rfl done | n + 1 => by apply Exists.intro ↑(X.val) have h1 : X.val < n + 1 := Fin.prop X rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1] rfl done theorem add_class (m : Nat) (a b : Int) : [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm theorem mul_class (m : Nat) (a b : Int) : [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm lemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ [a - b]_m = [0]_m := by apply Iff.intro · -- (→) assume h1 : [a]_m = [b]_m have h2 : a - b = a + (-b) := by ring have h3 : b + (-b) = 0 := by ring show [a - b]_m = [0]_m from calc [a - b]_m _ = [a + (-b)]_m := by rw [h2] _ = [a]_m + [-b]_m := by rw [add_class] _ = [b]_m + [-b]_m := by rw [h1] _ = [b + -b]_m := by rw [add_class] _ = [0]_m := by rw [h3] done · -- (←) assume h1 : [a - b]_m = [0]_m have h2 : b + (a - b) = a := by ring have h3 : b + 0 = b := by ring show [a]_m = [b]_m from calc [a]_m _ = [b + (a - b)]_m := by rw [h2] _ = [b]_m + [a - b]_m := by rw [add_class] _ = [b]_m + [0]_m := by rw [h1] _ = [b + 0]_m := by rw [add_class] _ = [b]_m := by rw [h3] done done lemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) : [a]_m = [0]_m → [-a]_m = [0]_m := by assume h1 : [a]_m = [0]_m have h2 : 0 + (-a) = -a := by ring have h3 : a + (-a) = 0 := by ring show [-a]_m = [0]_m from calc [-a]_m _ = [0 + (-a)]_m := by rw [h2] _ = [0]_m + [-a]_m := by rw [add_class] _ = [a]_m + [-a]_m := by rw [h1] _ = [a + (-a)]_m := by rw [add_class] _ = [0]_m := by rw [h3] done lemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) : [-a]_m = [0]_m ↔ [a]_m = [0]_m := by apply Iff.intro _ (cc_neg_zero_of_cc_zero m a) assume h1 : [-a]_m = [0]_m have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1 have h3 : -(-a) = a := by ring rewrite [h3] at h2 show [a]_m = [0]_m from h2 done lemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl lemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m ↔ m ∣ k := match m with | 0 => by have h : (0 : Int) = (↑(0 : Nat) : Int) := by rfl rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj] apply Iff.intro · -- (→) assume h1 : k = 0 rewrite [h1] show 0 ∣ 0 from dvd_self 0 done · -- (←) assume h1 : 0 ∣ k obtain (c : Nat) (h2 : k = 0 * c) from h1 rewrite [h2] ring done done | n + 1 => by rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero] show k % (n + 1) = 0 ↔ n + 1 ∣ k from (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm done lemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m ↔ ↑m ∣ a := by obtain (k : Nat) (h1 : a = ↑k ∨ a = -↑k) from Int.eq_nat_or_neg a by_cases on h1 · -- Case 1. h1: a = ↑k rewrite [h1, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done · -- Case 2. h1: a = -↑k rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done done theorem cc_eq_iff_congr (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ a ≡ b (MOD m) := calc [a]_m = [b]_m _ ↔ [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b _ ↔ ↑m ∣ (a - b) := cc_zero_iff_dvd m (a - b) _ ↔ a ≡ b (MOD m) := by rfl /- End of fundamental properties of congruence classes -/ lemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m := by have h1 : (↑m : Int) ≠ 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m) show 0 ≤ a % m from Int.emod_nonneg a h1 done lemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m) have h2 : (↑m : Int) > 0 := (Nat.cast_pos).rtl h1 show a % m < m from Int.emod_lt_of_pos a h2 done lemma congr_mod_mod (m : Nat) (a : Int) : a ≡ a % m (MOD m) := by define have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m apply Exists.intro (a / m) show a - a % m = m * (a / m) from calc a - (a % m) _ = m * (a / m) + a % m - a % m := by rw [h1] _ = m * (a / m) := by ring done lemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) := And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a)) theorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) : ∃! (r : Int), 0 ≤ r ∧ r < m ∧ a ≡ r (MOD m) := by exists_unique · -- Existence apply Exists.intro (a % m) show 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) from mod_cmpl_res m a done · -- Uniqueness fix r1 : Int; fix r2 : Int assume h1 : 0 ≤ r1 ∧ r1 < m ∧ a ≡ r1 (MOD m) assume h2 : 0 ≤ r2 ∧ r2 < m ∧ a ≡ r2 (MOD m) have h3 : r1 ≡ r2 (MOD m) := congr_trans (congr_symm h1.right.right) h2.right.right obtain (d : Int) (h4 : r1 - r2 = m * d) from h3 have h5 : r1 - r2 < m * 1 := by linarith have h6 : m * (-1) < r1 - r2 := by linarith rewrite [h4] at h5 --h5 : m * d < m * 1 rewrite [h4] at h6 --h6 : m * -1 < m * d have h7 : (↑m : Int) ≥ 0 := Nat.cast_nonneg m have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7 have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7 have h10 : d = 0 := by linarith show r1 = r2 from calc r1 _ = r1 - r2 + r2 := by ring _ = m * 0 + r2 := by rw [h4, h10] _ = r2 := by ring done done lemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m := (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a) theorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y rewrite [h1, h2] have h3 : a + b = b + a := by ring show [a]_m + [b]_m = [b]_m + [a]_m from calc [a]_m + [b]_m _ = [a + b]_m := add_class m a b _ = [b + a]_m := by rw [h3] _ = [b]_m + [a]_m := (add_class m b a).symm done theorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X rewrite [h1] have h2 : a * 1 = a := by ring show [a]_m * [1]_m = [a]_m from calc [a]_m * [1]_m _ = [a * 1]_m := mul_class m a 1 _ = [a]_m := by rw [h2] done theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry lemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) : [a]_m * [gcd_c2 m a]_m = [1]_m := by set s : Int := gcd_c1 m a have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m define at h1 rewrite [h1, Nat.cast_one] at h2 --h2 : s * ↑m + gcd_c2 m a * ↑a = 1 rewrite [mul_class, cc_eq_iff_congr] define --Goal : ∃ (c : Int), ↑a * gcd_c2 m a - 1 = ↑m * c apply Exists.intro (-s) show a * (gcd_c2 m a) - 1 = m * (-s) from calc a * (gcd_c2 m a) - 1 _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring _ = 1 + m * (-s) - 1 := by rw [h2] _ = m * (-s) := by ring done theorem Theorem_7_3_7 (m a : Nat) : invertible [a]_m ↔ rel_prime m a := by apply Iff.intro · -- (→) assume h1 : invertible [a]_m define at h1 obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1 obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y rewrite [h3, mul_class, cc_eq_iff_congr] at h2 define at h2 obtain (c : Int) (h4 : a * b - 1 = m * c) from h2 rewrite [Exercise_7_2_6] --Goal : ∃ (s t : Int), s * ↑m + t * ↑a = 1 apply Exists.intro (-c) apply Exists.intro b show (-c) * m + b * a = 1 from calc (-c) * m + b * a _ = (-c) * m + (a * b - 1) + 1 := by ring _ = (-c) * m + m * c + 1 := by rw [h4] _ = 1 := by ring done · -- (←) assume h1 : rel_prime m a define show ∃ (Y : ZMod m), [a]_m * Y = [1]_m from Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1) done done /- Section 7.4 -/ section Euler open Euler lemma num_rp_below_base {m : Nat} : num_rp_below m 0 = 0 := by rfl lemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) : num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by have h1 : num_rp_below m (j + 1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : gcd m j = 1 rewrite [if_pos h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1 show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1 done lemma num_rp_below_step_not_rp {m j : Nat} (h : ¬rel_prime m j) : num_rp_below m (j + 1) = num_rp_below m j := by have h1 : num_rp_below m (j +1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : ¬gcd m j = 1 rewrite [if_neg h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j show num_rp_below m (j + 1) = num_rp_below m j from h1 done lemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl #eval phi 10 --Answer: 4 lemma prod_inv_iff_inv {m : Nat} {X : ZMod m} (h1 : invertible X) (Y : ZMod m) : invertible (X * Y) ↔ invertible Y := by apply Iff.intro · -- (→) assume h2 : invertible (X * Y) obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2 apply Exists.intro (X * Z) rewrite [←h3] --Goal : Y * (X * Z) = X * Y * Z ring --Note that ring can do algebra in ZMod m done · -- (←) assume h2 : invertible Y obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1 obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2 apply Exists.intro (Xi * Yi) show (X * Y) * (Xi * Yi) = [1]_m from calc X * Y * (Xi * Yi) _ = (X * Xi) * (Y * Yi) := by ring _ = [1]_m * [1]_m := by rw [h3, h4] _ = [1]_m := Theorem_7_3_6_7 [1]_m done done lemma F_rp_def {m i : Nat} (h : rel_prime m i) : F m i = [i]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h --h : gcd m i = 1 rewrite [if_pos h] at h1 show F m i = [i]_m from h1 done lemma F_not_rp_def {m i : Nat} (h : ¬rel_prime m i) : F m i = [1]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h rewrite [h1, if_neg h] rfl done lemma prod_seq_base {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 0 k f = [1]_m := by rfl lemma prod_seq_step {m : Nat} (n k : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl lemma prod_seq_zero_step {m : Nat} (n : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by rewrite [prod_seq_step, zero_add] rfl done lemma prod_one {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 1 k f = f k := by rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7] rfl done lemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl lemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m := calc [G m a i]_m _ = [(a * i) % m]_m := by rfl _ = [a * i]_m := (cc_eq_mod m (a * i)).symm _ = [a]_m * [i]_m := (mul_class m a i).symm lemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) : rel_prime m (G m a i) ↔ rel_prime m i := by have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1 show rel_prime m (G m a i) ↔ rel_prime m i from calc rel_prime m (G m a i) _ ↔ invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm _ ↔ invertible ([a]_m * [i]_m) := by rw [cc_G] _ ↔ invertible [i]_m := prod_inv_iff_inv h2 ([i]_m) _ ↔ rel_prime m i := Theorem_7_3_7 m i done lemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) : F m (G m a i) = [a]_m * F m i := by have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2 show F m (G m a i) = [a]_m * F m i from calc F m (G m a i) _ = [G m a i]_m := F_rp_def h3 _ = [a]_m * [i]_m := cc_G m a i _ = [a]_m * F m i := by rw [F_rp_def h2] done lemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : ¬rel_prime m i) : F m (G m a i) = [1]_m := by rewrite [←G_rp_iff h1 i] at h2 show F m (G m a i) = [1]_m from F_not_rp_def h2 done lemma FG_prod {m a : Nat} (h1 : rel_prime m a) : ∀ (k : Nat), prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by by_induc · -- Base Case show prod_seq 0 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from calc prod_seq 0 0 ((F m) ∘ (G m a)) _ = [1]_m := prod_seq_base _ _ _ = [a]_m ^ 0 * [1]_m := by ring _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by rw [num_rp_below_base, prod_seq_base] done · -- Induction Step fix k : Nat assume ih : prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) by_cases h2 : rel_prime m k · -- Case 1. h2 : rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([a]_m * F m k) := by rw [FG_rp h1 h2] _ = [a]_m ^ ((num_rp_below m k) + 1) * ((prod_seq k 0 (F m)) * F m k) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_rp h2, prod_seq_zero_step] done · -- Case 2. h2 : ¬rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([1]_m) := by rw [FG_not_rp h1 h2] _ = [a]_m ^ (num_rp_below m k) * (prod_seq k 0 (F m) * ([1]_m)) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_not_rp h2, prod_seq_zero_step, F_not_rp_def h2] done done done lemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by define --Goal : ∀ i < m, G m a i < m fix i : Nat assume h1 : i < m rewrite [G_def] --Goal : a * i % m < m show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m) done lemma left_inv_one_one_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g' (g i) = i) : one_one_below n g := sorry lemma right_inv_onto_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g (g' i) = i) (h2 : maps_below n g') : onto_below n g := by define at h2; define fix k : Nat assume h3 : k < n apply Exists.intro (g' k) show g' k < n ∧ g (g' k) = k from And.intro (h2 k h3) (h1 k h3) done lemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m * [inv_mod m a]_m = [1]_m := by have h2 : 0 ≤ (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a) show [a]_m * [inv_mod m a]_m = [1]_m from calc [a]_m * [inv_mod m a]_m _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2] _ = [a]_m * [gcd_c2 m a]_m := by rw [←cc_eq_mod] _ = [1]_m := gcd_c2_inv h1 done lemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m := calc a * (b % m) % m = a % m * (b % m % m) % m := Nat.mul_mod _ _ _ _ = a % m * (b % m) % m := by rw [Nat.mod_mod] _ = a * b % m := (Nat.mul_mod _ _ _).symm lemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm] rfl done theorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] : ↑a ≡ ↑b (MOD m) ↔ a % m = b % m := sorry lemma mul_inv_mod_cancel {m a i : Nat} [NeZero m] (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1 rewrite [mul_class, cc_eq_iff_congr, ←Nat.cast_mul, ←Nat.cast_one, congr_iff_mod_eq_Nat] at h3 show a * inv_mod m a * i % m = i from calc a * (inv_mod m a) * i % m _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = 1 % m * i % m := by rw [h3] _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = i % m := by rw [one_mul] _ = i := Nat.mod_eq_of_lt h2 done lemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl lemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, G m a (Ginv m a i) = i := by fix i : Nat assume h2 : i < m show G m a (Ginv m a i) = i from calc G m a (Ginv m a i) _ = a * ((inv_mod m a * i) % m) % m := by rfl _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, Ginv m a (G m a i) = i := by fix i : Nat assume h2 : i < m show Ginv m a (G m a i) = i from calc Ginv m a (G m a i) _ = inv_mod m a * ((a * i) % m) % m := by rfl _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc, mul_comm (inv_mod m a)] _ = i := mul_inv_mod_cancel h1 h2 done

lemma Ginv_maps_below (m a : Nat) [NeZero m] : maps_below m (Ginv m a)

HTPI.Ginv_maps_below

htpi/HTPILib/Chap7.lean

htpi

/- Copyright 2023 Daniel J. Velleman -/ import HTPILib.Chap6 namespace HTPI /- Definitions -/ lemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by have h : n + 1 > 0 := Nat.succ_pos n show a % (n + 1) < n + 1 from Nat.mod_lt a h done def gcd (a b : Nat) : Nat := match b with | 0 => a | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd (n + 1) (a % (n + 1)) termination_by b mutual def gcd_c1 (a b : Nat) : Int := match b with | 0 => 1 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c2 (n + 1) (a % (n + 1)) --Corresponds to s = t' termination_by b def gcd_c2 (a b : Nat) : Int := match b with | 0 => 0 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c1 (n + 1) (a % (n + 1)) - (gcd_c2 (n + 1) (a % (n + 1))) * ↑(a / (n + 1)) --Corresponds to t = s' - t'q termination_by b end def prime (n : Nat) : Prop := 2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n def prime_factor (p n : Nat) : Prop := prime p ∧ p ∣ n def all_prime (l : List Nat) : Prop := ∀ p ∈ l, prime p def nondec (l : List Nat) : Prop := match l with | [] => True --Of course, True is a proposition that is always true | n :: L => (∀ m ∈ L, n ≤ m) ∧ nondec L def nondec_prime_list (l : List Nat) : Prop := all_prime l ∧ nondec l def prod (l : List Nat) : Nat := match l with | [] => 1 | n :: L => n * (prod L) def prime_factorization (n : Nat) (l : List Nat) : Prop := nondec_prime_list l ∧ prod l = n def rel_prime (a b : Nat) : Prop := gcd a b = 1 def congr_mod (m : Nat) (a b : Int) : Prop := (↑m : Int) ∣ (a - b) def cc (m : Nat) (a : Int) : ZMod m := (↑a : ZMod m) notation:50 a " ≡ " b " (MOD " m ")" => congr_mod m a b notation:max "["a"]_"m:max => cc m a def invertible {m : Nat} (X : ZMod m) : Prop := ∃ (Y : ZMod m), X * Y = [1]_m def num_rp_below (m k : Nat) : Nat := match k with | 0 => 0 | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j def phi (m : Nat) : Nat := num_rp_below m m def prod_seq {m : Nat} (j k : Nat) (f : Nat → ZMod m) : ZMod m := match j with | 0 => [1]_m | n + 1 => prod_seq n k f * f (k + n) def maps_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i < n, g i < n def one_one_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i1 < n, ∀ i2 < n, g i1 = g i2 → i1 = i2 def onto_below (n : Nat) (g : Nat → Nat) : Prop := ∀ k < n, ∃ i < n, g i = k def perm_below (n : Nat) (g : Nat → Nat) : Prop := maps_below n g ∧ one_one_below n g ∧ onto_below n g def inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m) def swap (u v i : Nat) : Nat := if i = u then v else if i = v then u else i namespace Euler --For definitions specific to Euler's theorem def F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m def G (m a i : Nat) : Nat := (a * i) % m def Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i end Euler /- Section 7.1 -/ theorem dvd_mod_of_dvd_a_b {a b d : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ (a % b) := by set q : Nat := a / b have h3 : b * q + a % b = a := Nat.div_add_mod a b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 define --Goal : ∃ (c : Nat), a % b = d * c apply Exists.intro (j - k * q) show a % b = d * (j - k * q) from calc a % b _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm _ = a - b * q := by rw [h3] _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc] _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm done theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry #eval gcd 672 161 --Answer: 7 lemma gcd_base (a : Nat) : gcd a 0 = a := by rfl lemma gcd_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd a b = gcd b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1)) rfl done lemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b ≠ 0) : a % b < b := by have h1 : b > 0 := Nat.pos_of_ne_zero h show a % b < b from Nat.mod_lt a h1 done lemma dvd_self (n : Nat) : n ∣ n := by apply Exists.intro 1 ring done theorem gcd_dvd : ∀ (b a : Nat), (gcd a b) ∣ a ∧ (gcd a b) ∣ b := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd a b_1) ∣ a ∧ (gcd a b_1) ∣ b_1 fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_base] --Goal: a ∣ a ∧ a ∣ 0 apply And.intro (dvd_self a) define apply Exists.intro 0 rfl done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_nonzero a h1] --Goal : gcd b (a % b) ∣ a ∧ gcd b (a % b) ∣ b have h2 : a % b < b := mod_nonzero_lt a h1 have h3 : (gcd b (a % b)) ∣ b ∧ (gcd b (a % b)) ∣ (a % b) := ih (a % b) h2 b apply And.intro _ h3.left show (gcd b (a % b)) ∣ a from dvd_a_of_dvd_b_mod h3.left h3.right done done theorem gcd_dvd_left (a b : Nat) : (gcd a b) ∣ a := (gcd_dvd b a).left theorem gcd_dvd_right (a b : Nat) : (gcd a b) ∣ b := (gcd_dvd b a).right lemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl lemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c1 a b = gcd_c2 b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done lemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl lemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * ↑(a / b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done theorem gcd_lin_comb : ∀ (b a : Nat), (gcd_c1 a b) * ↑a + (gcd_c2 a b) * ↑b = ↑(gcd a b) := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd_c1 a b_1) * ↑a + (gcd_c2 a b_1) * ↑b_1 = ↑(gcd a b_1) fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base] --Goal : 1 * ↑a + 0 * ↑0 = ↑a ring done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1] --Goal : gcd_c2 b (a % b) * ↑a + -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * ↑(a / b)) * ↑b = -- ↑(gcd b (a % b)) set r : Nat := a % b set q : Nat := a / b set s : Int := gcd_c1 b r set t : Int := gcd_c2 b r --Goal : t * ↑a + (s - t * ↑q) * ↑b = ↑(gcd b r) have h2 : r < b := mod_nonzero_lt a h1 have h3 : s * ↑b + t * ↑r = ↑(gcd b r) := ih r h2 b have h4 : b * q + r = a := Nat.div_add_mod a b rewrite [←h3, ←h4] rewrite [Nat.cast_add, Nat.cast_mul] --Goal : t * (↑b * ↑q + ↑r) + (s - t * ↑q) * ↑b = s * ↑b + t * ↑r ring done done #eval gcd_c1 672 161 --Answer: 6 #eval gcd_c2 672 161 --Answer: -25 --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161 theorem Theorem_7_1_6 {d a b : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ gcd a b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑d ∣ ↑(gcd a b) set s : Int := gcd_c1 a b set t : Int := gcd_c2 a b have h3 : s * ↑a + t * ↑b = ↑(gcd a b) := gcd_lin_comb b a rewrite [←h3] --Goal : ↑d ∣ s * ↑a + t * ↑b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul] --Goal : ↑d ∣ s * (↑d * ↑j) + t * (↑d * ↑k) define apply Exists.intro (s * ↑j + t * ↑k) ring done /- Section 7.2 -/ theorem dvd_trans {a b c : Nat} (h1 : a ∣ b) (h2 : b ∣ c) : a ∣ c := by define at h1; define at h2; define obtain (m : Nat) (h3 : b = a * m) from h1 obtain (n : Nat) (h4 : c = b * n) from h2 rewrite [h3, mul_assoc] at h4 apply Exists.intro (m * n) show c = a * (m * n) from h4 done lemma exists_prime_factor : ∀ (n : Nat), 2 ≤ n → ∃ (p : Nat), prime_factor p n := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, 2 ≤ n_1 → ∃ (p : Nat), prime_factor p n_1 assume h1 : 2 ≤ n by_cases h2 : prime n · -- Case 1. h2 : prime n apply Exists.intro n define --Goal : prime n ∧ n ∣ n show prime n ∧ n ∣ n from And.intro h2 (dvd_self n) done · -- Case 2. h2 : ¬prime n define at h2 --h2 : ¬(2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n) demorgan at h2 disj_syll h2 h1 obtain (a : Nat) (h3 : ∃ (b : Nat), a * b = n ∧ a < n ∧ b < n) from h2 obtain (b : Nat) (h4 : a * b = n ∧ a < n ∧ b < n) from h3 have h5 : 2 ≤ a := by by_contra h6 have h7 : a ≤ 1 := by linarith have h8 : n ≤ b := calc n _ = a * b := h4.left.symm _ ≤ 1 * b := by rel [h7] _ = b := by ring linarith --n ≤ b contradicts b < n done have h6 : ∃ (p : Nat), prime_factor p a := ih a h4.right.left h5 obtain (p : Nat) (h7 : prime_factor p a) from h6 apply Exists.intro p define --Goal : prime p ∧ p ∣ n define at h7 --h7 : prime p ∧ p ∣ a apply And.intro h7.left have h8 : a ∣ n := by apply Exists.intro b show n = a * b from (h4.left).symm done show p ∣ n from dvd_trans h7.right h8 done done lemma exists_least_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q := by set S : Set Nat := {p : Nat | prime_factor p n} have h2 : ∃ (p : Nat), p ∈ S := exists_prime_factor n h show ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q from well_ord_princ S h2 done lemma all_prime_nil : all_prime [] := by define --Goal : ∀ p ∈ [], prime p fix p : Nat contrapos --Goal : ¬prime p → p ∉ [] assume h1 : ¬prime p show p ∉ [] from List.not_mem_nil p done lemma all_prime_cons (n : Nat) (L : List Nat) : all_prime (n :: L) ↔ prime n ∧ all_prime L := by apply Iff.intro · -- (→) assume h1 : all_prime (n :: L) --Goal : prime n ∧ all_prime L define at h1 --h1 : ∀ p ∈ n :: L, prime p apply And.intro (h1 n (List.mem_cons_self n L)) define --Goal : ∀ p ∈ L, prime p fix p : Nat assume h2 : p ∈ L show prime p from h1 p (List.mem_cons_of_mem n h2) done · -- (←) assume h1 : prime n ∧ all_prime L --Goal : all_prime (n :: l) define : all_prime L at h1 define fix p : Nat assume h2 : p ∈ n :: L rewrite [List.mem_cons] at h2 --h2 : p = n ∨ p ∈ L by_cases on h2 · -- Case 1. h2 : p = n rewrite [h2] show prime n from h1.left done · -- Case 2. h2 : p ∈ L show prime p from h1.right p h2 done done done lemma nondec_nil : nondec [] := by define --Goal : True trivial --trivial proves some obviously true statements, such as True done lemma nondec_cons (n : Nat) (L : List Nat) : nondec (n :: L) ↔ (∀ m ∈ L, n ≤ m) ∧ nondec L := by rfl lemma prod_nil : prod [] = 1 := by rfl lemma prod_cons : prod (n :: L) = n * (prod L) := by rfl lemma exists_cons_of_length_eq_succ {A : Type} {l : List A} {n : Nat} (h : l.length = n + 1) : ∃ (a : A) (L : List A), l = a :: L ∧ L.length = n := by have h1 : ¬l.length = 0 := by linarith rewrite [List.length_eq_zero] at h1 obtain (a : A) (h2 : ∃ (L : List A), l = a :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List A) (h3 : l = a :: L) from h2 apply Exists.intro a apply Exists.intro L apply And.intro h3 have h4 : (a :: L).length = L.length + 1 := List.length_cons a L rewrite [←h3, h] at h4 show L.length = n from (Nat.add_right_cancel h4).symm done lemma list_elt_dvd_prod_by_length (a : Nat) : ∀ (n : Nat), ∀ (l : List Nat), l.length = n → a ∈ l → a ∣ prod l := by by_induc · --Base Case fix l : List Nat assume h1 : l.length = 0 rewrite [List.length_eq_zero] at h1 --h1 : l = [] rewrite [h1] --Goal : a ∈ [] → a ∣ prod [] contrapos assume h2 : ¬a ∣ prod [] show a ∉ [] from List.not_mem_nil a done · -- Induction Step fix n : Nat assume ih : ∀ (l : List Nat), List.length l = n → a ∈ l → a ∣ prod l fix l : List Nat assume h1 : l.length = n + 1 --Goal : a ∈ l → a ∣ prod l obtain (b : Nat) (h2 : ∃ (L : List Nat), l = b :: L ∧ L.length = n) from exists_cons_of_length_eq_succ h1 obtain (L : List Nat) (h3 : l = b :: L ∧ L.length = n) from h2 have h4 : a ∈ L → a ∣ prod L := ih L h3.right assume h5 : a ∈ l rewrite [h3.left, prod_cons] --Goal : a ∣ b * prod L rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b ∨ a ∈ L by_cases on h5 · -- Case 1. h5 : a = b apply Exists.intro (prod L) rewrite [h5] rfl done · -- Case 2. h5 : a ∈ L have h6 : a ∣ prod L := h4 h5 have h7 : prod L ∣ b * prod L := by apply Exists.intro b ring done show a ∣ b * prod L from dvd_trans h6 h7 done done done lemma list_elt_dvd_prod {a : Nat} {l : List Nat} (h : a ∈ l) : a ∣ prod l := by set n : Nat := l.length have h1 : l.length = n := by rfl show a ∣ prod l from list_elt_dvd_prod_by_length a n l h1 h done lemma exists_prime_factorization : ∀ (n : Nat), n ≥ 1 → ∃ (l : List Nat), prime_factorization n l := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, n_1 ≥ 1 → ∃ (l : List Nat), prime_factorization n_1 l assume h1 : n ≥ 1 by_cases h2 : n = 1 · -- Case 1. h2 : n = 1 apply Exists.intro [] define apply And.intro · -- Proof of nondec_prime_list [] define show all_prime [] ∧ nondec [] from And.intro all_prime_nil nondec_nil done · -- Proof of prod [] = n rewrite [prod_nil, h2] rfl done done · -- Case 2. h2 : n ≠ 1 have h3 : n ≥ 2 := lt_of_le_of_ne' h1 h2 obtain (p : Nat) (h4 : prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q) from exists_least_prime_factor h3 have p_prime_factor : prime_factor p n := h4.left define at p_prime_factor have p_prime : prime p := p_prime_factor.left have p_dvd_n : p ∣ n := p_prime_factor.right have p_least : ∀ (q : Nat), prime_factor q n → p ≤ q := h4.right obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n have h5 : m ≠ 0 := by contradict h1 with h6 have h7 : n = 0 := calc n _ = p * m := n_eq_pm _ = p * 0 := by rw [h6] _ = 0 := by ring rewrite [h7] decide done have m_pos : 0 < m := Nat.pos_of_ne_zero h5 have m_lt_n : m < n := by define at p_prime show m < n from calc m _ < m + m := by linarith _ = 2 * m := by ring _ ≤ p * m := by rel [p_prime.left] _ = n := n_eq_pm.symm done obtain (L : List Nat) (h6 : prime_factorization m L) from ih m m_lt_n m_pos define at h6 have ndpl_L : nondec_prime_list L := h6.left define at ndpl_L apply Exists.intro (p :: L) define apply And.intro · -- Proof of nondec_prime_list (p :: L) define apply And.intro · -- Proof of all_prime (p :: L) rewrite [all_prime_cons] show prime p ∧ all_prime L from And.intro p_prime ndpl_L.left done · -- Proof of nondec (p :: L) rewrite [nondec_cons] apply And.intro _ ndpl_L.right fix q : Nat assume q_in_L : q ∈ L have h7 : q ∣ prod L := list_elt_dvd_prod q_in_L rewrite [h6.right] at h7 --h7 : q ∣ m have h8 : m ∣ n := by apply Exists.intro p rewrite [n_eq_pm] ring done have q_dvd_n : q ∣ n := dvd_trans h7 h8 have ap_L : all_prime L := ndpl_L.left define at ap_L have q_prime_factor : prime_factor q n := And.intro (ap_L q q_in_L) q_dvd_n show p ≤ q from p_least q q_prime_factor done done · -- Proof of prod (p :: L) = n rewrite [prod_cons, h6.right, n_eq_pm] rfl done done done theorem Theorem_7_2_2 {a b c : Nat} (h1 : c ∣ a * b) (h2 : rel_prime a c) : c ∣ b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑c ∣ ↑b define at h1; define at h2; define obtain (j : Nat) (h3 : a * b = c * j) from h1 set s : Int := gcd_c1 a c set t : Int := gcd_c2 a c have h4 : s * ↑a + t * ↑c = ↑(gcd a c) := gcd_lin_comb c a rewrite [h2, Nat.cast_one] at h4 --h4 : s * ↑a + t * ↑c = (1 : Int) apply Exists.intro (s * ↑j + t * ↑b) show ↑b = ↑c * (s * ↑j + t * ↑b) from calc ↑b _ = (1 : Int) * ↑b := (one_mul _).symm _ = (s * ↑a + t * ↑c) * ↑b := by rw [h4] _ = s * (↑a * ↑b) + t * ↑c * ↑b := by ring _ = s * (↑c * ↑j) + t * ↑c * ↑b := by rw [←Nat.cast_mul a b, h3, Nat.cast_mul c j] _ = ↑c * (s * ↑j + t * ↑b) := by ring done lemma le_nonzero_prod_left {a b : Nat} (h : a * b ≠ 0) : a ≤ a * b := by have h1 : b ≠ 0 := by contradict h with h1 rewrite [h1] ring done have h2 : 1 ≤ b := Nat.pos_of_ne_zero h1 show a ≤ a * b from calc a = a * 1 := (mul_one a).symm _ ≤ a * b := by rel [h2] done lemma le_nonzero_prod_right {a b : Nat} (h : a * b ≠ 0) : b ≤ a * b := by rewrite [mul_comm] rewrite [mul_comm] at h show b ≤ b * a from le_nonzero_prod_left h done lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry lemma rel_prime_of_prime_not_dvd {a p : Nat} (h1 : prime p) (h2 : ¬p ∣ a) : rel_prime a p := by have h3 : gcd a p ∣ a := gcd_dvd_left a p have h4 : gcd a p ∣ p := gcd_dvd_right a p have h5 : gcd a p = 1 ∨ gcd a p = p := dvd_prime h1 h4 have h6 : gcd a p ≠ p := by contradict h2 with h6 rewrite [h6] at h3 show p ∣ a from h3 done disj_syll h5 h6 show rel_prime a p from h5 done theorem Theorem_7_2_3 {a b p : Nat} (h1 : prime p) (h2 : p ∣ a * b) : p ∣ a ∨ p ∣ b := by or_right with h3 have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3 show p ∣ b from Theorem_7_2_2 h2 h4 done lemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a ≥ 1 := by have h1 : a ≠ 0 := by by_contra h1 rewrite [h1] at h contradict h linarith done show a ≥ 1 from Nat.pos_of_ne_zero h1 done lemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by have h1 : a ≥ 1 := ge_one_of_prod_one h have h2 : a * b ≠ 0 := by linarith have h3 : a ≤ a * b := le_nonzero_prod_left h2 rewrite [h] at h3 show a = 1 from Nat.le_antisymm h3 h1 done lemma eq_one_of_dvd_one {n : Nat} (h : n ∣ 1) : n = 1 := by obtain (j : Nat) (h1 : 1 = n * j) from h show n = 1 from eq_one_of_prod_one h1.symm done lemma prime_not_one {p : Nat} (h : prime p) : p ≠ 1 := by define at h linarith done theorem Theorem_7_2_4 {p : Nat} (h1 : prime p) : ∀ (l : List Nat), p ∣ prod l → ∃ a ∈ l, p ∣ a := by apply List.rec · -- Base Case. Goal : p ∣ prod [] → ∃ a ∈ [], p ∣ a rewrite [prod_nil] assume h2 : p ∣ 1 show ∃ a ∈ [], p ∣ a from absurd (eq_one_of_dvd_one h2) (prime_not_one h1) done · -- Induction Step fix b : Nat fix L : List Nat assume ih : p ∣ prod L → ∃ a ∈ L, p ∣ a --Goal : p ∣ prod (b :: L) → ∃ a ∈ b :: L, p ∣ a assume h2 : p ∣ prod (b :: L) rewrite [prod_cons] at h2 have h3 : p ∣ b ∨ p ∣ prod L := Theorem_7_2_3 h1 h2 by_cases on h3 · -- Case 1. h3 : p ∣ b apply Exists.intro b show b ∈ b :: L ∧ p ∣ b from And.intro (List.mem_cons_self b L) h3 done · -- Case 2. h3 : p ∣ prod L obtain (a : Nat) (h4 : a ∈ L ∧ p ∣ a) from ih h3 apply Exists.intro a show a ∈ b :: L ∧ p ∣ a from And.intro (List.mem_cons_of_mem b h4.left) h4.right done done done lemma prime_in_list {p : Nat} {l : List Nat} (h1 : prime p) (h2 : all_prime l) (h3 : p ∣ prod l) : p ∈ l := by obtain (a : Nat) (h4 : a ∈ l ∧ p ∣ a) from Theorem_7_2_4 h1 l h3 define at h2 have h5 : prime a := h2 a h4.left have h6 : p = 1 ∨ p = a := dvd_prime h5 h4.right disj_syll h6 (prime_not_one h1) rewrite [h6] show a ∈ l from h4.left done lemma first_le_first {p q : Nat} {l m : List Nat} (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m)) (h3 : prod (p :: l) = prod (q :: m)) : p ≤ q := by define at h1; define at h2 have h4 : q ∣ prod (p :: l) := by define apply Exists.intro (prod m) rewrite [←prod_cons] show prod (p :: l) = prod (q :: m) from h3 done have h5 : all_prime (q :: m) := h2.left rewrite [all_prime_cons] at h5 have h6 : q ∈ p :: l := prime_in_list h5.left h1.left h4 have h7 : nondec (p :: l) := h1.right rewrite [nondec_cons] at h7 rewrite [List.mem_cons] at h6 by_cases on h6 · -- Case 1. h6 : q = p linarith done · -- Case 2. h6 : q ∈ l have h8 : ∀ m ∈ l, p ≤ m := h7.left show p ≤ q from h8 q h6 done done lemma nondec_prime_list_tail {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by define at h define rewrite [all_prime_cons, nondec_cons] at h show all_prime l ∧ nondec l from And.intro h.left.right h.right.right done lemma cons_prod_not_one {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : prod (p :: l) ≠ 1 := by define at h have h1 : all_prime (p :: l) := h.left rewrite [all_prime_cons] at h1 rewrite [prod_cons] by_contra h2 show False from (prime_not_one h1.left) (eq_one_of_prod_one h2) done lemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) : l = [] ↔ prod l = 1 := by apply Iff.intro · -- (→) assume h1 : l = [] rewrite [h1] show prod [] = 1 from prod_nil done · -- (←) contrapos assume h1 : ¬l = [] obtain (p : Nat) (h2 : ∃ (L : List Nat), l = p :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List Nat) (h3 : l = p :: L) from h2 rewrite [h3] at h rewrite [h3] show ¬prod (p :: L) = 1 from cons_prod_not_one h done done lemma prime_pos {p : Nat} (h : prime p) : p > 0 := by define at h linarith done theorem Theorem_7_2_5 : ∀ (l1 l2 : List Nat), nondec_prime_list l1 → nondec_prime_list l2 → prod l1 = prod l2 → l1 = l2 := by apply List.rec · -- Base Case. Goal : ∀ (l2 : List Nat), nondec_prime_list [] → -- nondec_prime_list l2 → prod [] = prod l2 → [] = l2 fix l2 : List Nat assume h1 : nondec_prime_list [] assume h2 : nondec_prime_list l2 assume h3 : prod [] = prod l2 rewrite [prod_nil, eq_comm, ←list_nil_iff_prod_one h2] at h3 show [] = l2 from h3.symm done · -- Induction Step fix p : Nat fix L1 : List Nat assume ih : ∀ (L2 : List Nat), nondec_prime_list L1 → nondec_prime_list L2 → prod L1 = prod L2 → L1 = L2 -- Goal : ∀ (l2 : List Nat), nondec_prime_list (p :: L1) → -- nondec_prime_list l2 → prod (p :: L1) = prod l2 → p :: L1 = l2 fix l2 : List Nat assume h1 : nondec_prime_list (p :: L1) assume h2 : nondec_prime_list l2 assume h3 : prod (p :: L1) = prod l2 have h4 : ¬prod (p :: L1) = 1 := cons_prod_not_one h1 rewrite [h3, ←list_nil_iff_prod_one h2] at h4 obtain (q : Nat) (h5 : ∃ (L : List Nat), l2 = q :: L) from List.exists_cons_of_ne_nil h4 obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5 rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2) rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2) have h7 : p ≤ q := first_le_first h1 h2 h3 have h8 : q ≤ p := first_le_first h2 h1 h3.symm have h9 : p = q := by linarith rewrite [h9, prod_cons, prod_cons] at h3 --h3 : q * prod L1 = q * prod L2 have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1 have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2 define at h2 have h12 : all_prime (q :: L2) := h2.left rewrite [all_prime_cons] at h12 have h13 : q > 0 := prime_pos h12.left have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3 have h15 : L1 = L2 := ih L2 h10 h11 h14 rewrite [h6, h9, h15] rfl done done theorem fund_thm_arith (n : Nat) (h : n ≥ 1) : ∃! (l : List Nat), prime_factorization n l := by exists_unique · -- Existence show ∃ (l : List Nat), prime_factorization n l from exists_prime_factorization n h done · -- Uniqueness fix l1 : List Nat; fix l2 : List Nat assume h1 : prime_factorization n l1 assume h2 : prime_factorization n l2 define at h1; define at h2 have h3 : prod l1 = n := h1.right rewrite [←h2.right] at h3 show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3 done done /- Section 7.3 -/ theorem congr_refl (m : Nat) : ∀ (a : Int), a ≡ a (MOD m) := by fix a : Int define --Goal : ∃ (c : Int), a - a = ↑m * c apply Exists.intro 0 ring done theorem congr_symm {m : Nat} : ∀ {a b : Int}, a ≡ b (MOD m) → b ≡ a (MOD m) := by fix a : Int; fix b : Int assume h1 : a ≡ b (MOD m) define at h1 --h1 : ∃ (c : Int), a - b = ↑m * c define --Goal : ∃ (c : Int), b - a = ↑m * c obtain (c : Int) (h2 : a - b = m * c) from h1 apply Exists.intro (-c) show b - a = m * (-c) from calc b - a _ = -(a - b) := by ring _ = -(m * c) := by rw [h2] _ = m * (-c) := by ring done theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry /- Fundamental properties of congruence classes -/ lemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) : X = Y ↔ X.val = Y.val := Fin.ext_iff lemma val_nat_eq_mod (n k : Nat) : ([k]_(n + 1)).val = k % (n + 1) := by rfl lemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl theorem cc_rep {m : Nat} (X : ZMod m) : ∃ (a : Int), X = [a]_m := match m with | 0 => by apply Exists.intro X rfl done | n + 1 => by apply Exists.intro ↑(X.val) have h1 : X.val < n + 1 := Fin.prop X rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1] rfl done theorem add_class (m : Nat) (a b : Int) : [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm theorem mul_class (m : Nat) (a b : Int) : [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm lemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ [a - b]_m = [0]_m := by apply Iff.intro · -- (→) assume h1 : [a]_m = [b]_m have h2 : a - b = a + (-b) := by ring have h3 : b + (-b) = 0 := by ring show [a - b]_m = [0]_m from calc [a - b]_m _ = [a + (-b)]_m := by rw [h2] _ = [a]_m + [-b]_m := by rw [add_class] _ = [b]_m + [-b]_m := by rw [h1] _ = [b + -b]_m := by rw [add_class] _ = [0]_m := by rw [h3] done · -- (←) assume h1 : [a - b]_m = [0]_m have h2 : b + (a - b) = a := by ring have h3 : b + 0 = b := by ring show [a]_m = [b]_m from calc [a]_m _ = [b + (a - b)]_m := by rw [h2] _ = [b]_m + [a - b]_m := by rw [add_class] _ = [b]_m + [0]_m := by rw [h1] _ = [b + 0]_m := by rw [add_class] _ = [b]_m := by rw [h3] done done lemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) : [a]_m = [0]_m → [-a]_m = [0]_m := by assume h1 : [a]_m = [0]_m have h2 : 0 + (-a) = -a := by ring have h3 : a + (-a) = 0 := by ring show [-a]_m = [0]_m from calc [-a]_m _ = [0 + (-a)]_m := by rw [h2] _ = [0]_m + [-a]_m := by rw [add_class] _ = [a]_m + [-a]_m := by rw [h1] _ = [a + (-a)]_m := by rw [add_class] _ = [0]_m := by rw [h3] done lemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) : [-a]_m = [0]_m ↔ [a]_m = [0]_m := by apply Iff.intro _ (cc_neg_zero_of_cc_zero m a) assume h1 : [-a]_m = [0]_m have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1 have h3 : -(-a) = a := by ring rewrite [h3] at h2 show [a]_m = [0]_m from h2 done lemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl lemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m ↔ m ∣ k := match m with | 0 => by have h : (0 : Int) = (↑(0 : Nat) : Int) := by rfl rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj] apply Iff.intro · -- (→) assume h1 : k = 0 rewrite [h1] show 0 ∣ 0 from dvd_self 0 done · -- (←) assume h1 : 0 ∣ k obtain (c : Nat) (h2 : k = 0 * c) from h1 rewrite [h2] ring done done | n + 1 => by rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero] show k % (n + 1) = 0 ↔ n + 1 ∣ k from (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm done lemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m ↔ ↑m ∣ a := by obtain (k : Nat) (h1 : a = ↑k ∨ a = -↑k) from Int.eq_nat_or_neg a by_cases on h1 · -- Case 1. h1: a = ↑k rewrite [h1, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done · -- Case 2. h1: a = -↑k rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done done theorem cc_eq_iff_congr (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ a ≡ b (MOD m) := calc [a]_m = [b]_m _ ↔ [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b _ ↔ ↑m ∣ (a - b) := cc_zero_iff_dvd m (a - b) _ ↔ a ≡ b (MOD m) := by rfl /- End of fundamental properties of congruence classes -/ lemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m := by have h1 : (↑m : Int) ≠ 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m) show 0 ≤ a % m from Int.emod_nonneg a h1 done lemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m) have h2 : (↑m : Int) > 0 := (Nat.cast_pos).rtl h1 show a % m < m from Int.emod_lt_of_pos a h2 done lemma congr_mod_mod (m : Nat) (a : Int) : a ≡ a % m (MOD m) := by define have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m apply Exists.intro (a / m) show a - a % m = m * (a / m) from calc a - (a % m) _ = m * (a / m) + a % m - a % m := by rw [h1] _ = m * (a / m) := by ring done lemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) := And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a)) theorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) : ∃! (r : Int), 0 ≤ r ∧ r < m ∧ a ≡ r (MOD m) := by exists_unique · -- Existence apply Exists.intro (a % m) show 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) from mod_cmpl_res m a done · -- Uniqueness fix r1 : Int; fix r2 : Int assume h1 : 0 ≤ r1 ∧ r1 < m ∧ a ≡ r1 (MOD m) assume h2 : 0 ≤ r2 ∧ r2 < m ∧ a ≡ r2 (MOD m) have h3 : r1 ≡ r2 (MOD m) := congr_trans (congr_symm h1.right.right) h2.right.right obtain (d : Int) (h4 : r1 - r2 = m * d) from h3 have h5 : r1 - r2 < m * 1 := by linarith have h6 : m * (-1) < r1 - r2 := by linarith rewrite [h4] at h5 --h5 : m * d < m * 1 rewrite [h4] at h6 --h6 : m * -1 < m * d have h7 : (↑m : Int) ≥ 0 := Nat.cast_nonneg m have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7 have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7 have h10 : d = 0 := by linarith show r1 = r2 from calc r1 _ = r1 - r2 + r2 := by ring _ = m * 0 + r2 := by rw [h4, h10] _ = r2 := by ring done done lemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m := (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a) theorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y rewrite [h1, h2] have h3 : a + b = b + a := by ring show [a]_m + [b]_m = [b]_m + [a]_m from calc [a]_m + [b]_m _ = [a + b]_m := add_class m a b _ = [b + a]_m := by rw [h3] _ = [b]_m + [a]_m := (add_class m b a).symm done theorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X rewrite [h1] have h2 : a * 1 = a := by ring show [a]_m * [1]_m = [a]_m from calc [a]_m * [1]_m _ = [a * 1]_m := mul_class m a 1 _ = [a]_m := by rw [h2] done theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry lemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) : [a]_m * [gcd_c2 m a]_m = [1]_m := by set s : Int := gcd_c1 m a have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m define at h1 rewrite [h1, Nat.cast_one] at h2 --h2 : s * ↑m + gcd_c2 m a * ↑a = 1 rewrite [mul_class, cc_eq_iff_congr] define --Goal : ∃ (c : Int), ↑a * gcd_c2 m a - 1 = ↑m * c apply Exists.intro (-s) show a * (gcd_c2 m a) - 1 = m * (-s) from calc a * (gcd_c2 m a) - 1 _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring _ = 1 + m * (-s) - 1 := by rw [h2] _ = m * (-s) := by ring done theorem Theorem_7_3_7 (m a : Nat) : invertible [a]_m ↔ rel_prime m a := by apply Iff.intro · -- (→) assume h1 : invertible [a]_m define at h1 obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1 obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y rewrite [h3, mul_class, cc_eq_iff_congr] at h2 define at h2 obtain (c : Int) (h4 : a * b - 1 = m * c) from h2 rewrite [Exercise_7_2_6] --Goal : ∃ (s t : Int), s * ↑m + t * ↑a = 1 apply Exists.intro (-c) apply Exists.intro b show (-c) * m + b * a = 1 from calc (-c) * m + b * a _ = (-c) * m + (a * b - 1) + 1 := by ring _ = (-c) * m + m * c + 1 := by rw [h4] _ = 1 := by ring done · -- (←) assume h1 : rel_prime m a define show ∃ (Y : ZMod m), [a]_m * Y = [1]_m from Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1) done done /- Section 7.4 -/ section Euler open Euler lemma num_rp_below_base {m : Nat} : num_rp_below m 0 = 0 := by rfl lemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) : num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by have h1 : num_rp_below m (j + 1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : gcd m j = 1 rewrite [if_pos h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1 show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1 done lemma num_rp_below_step_not_rp {m j : Nat} (h : ¬rel_prime m j) : num_rp_below m (j + 1) = num_rp_below m j := by have h1 : num_rp_below m (j +1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : ¬gcd m j = 1 rewrite [if_neg h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j show num_rp_below m (j + 1) = num_rp_below m j from h1 done lemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl #eval phi 10 --Answer: 4 lemma prod_inv_iff_inv {m : Nat} {X : ZMod m} (h1 : invertible X) (Y : ZMod m) : invertible (X * Y) ↔ invertible Y := by apply Iff.intro · -- (→) assume h2 : invertible (X * Y) obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2 apply Exists.intro (X * Z) rewrite [←h3] --Goal : Y * (X * Z) = X * Y * Z ring --Note that ring can do algebra in ZMod m done · -- (←) assume h2 : invertible Y obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1 obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2 apply Exists.intro (Xi * Yi) show (X * Y) * (Xi * Yi) = [1]_m from calc X * Y * (Xi * Yi) _ = (X * Xi) * (Y * Yi) := by ring _ = [1]_m * [1]_m := by rw [h3, h4] _ = [1]_m := Theorem_7_3_6_7 [1]_m done done lemma F_rp_def {m i : Nat} (h : rel_prime m i) : F m i = [i]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h --h : gcd m i = 1 rewrite [if_pos h] at h1 show F m i = [i]_m from h1 done lemma F_not_rp_def {m i : Nat} (h : ¬rel_prime m i) : F m i = [1]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h rewrite [h1, if_neg h] rfl done lemma prod_seq_base {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 0 k f = [1]_m := by rfl lemma prod_seq_step {m : Nat} (n k : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl lemma prod_seq_zero_step {m : Nat} (n : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by rewrite [prod_seq_step, zero_add] rfl done lemma prod_one {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 1 k f = f k := by rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7] rfl done lemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl lemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m := calc [G m a i]_m _ = [(a * i) % m]_m := by rfl _ = [a * i]_m := (cc_eq_mod m (a * i)).symm _ = [a]_m * [i]_m := (mul_class m a i).symm lemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) : rel_prime m (G m a i) ↔ rel_prime m i := by have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1 show rel_prime m (G m a i) ↔ rel_prime m i from calc rel_prime m (G m a i) _ ↔ invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm _ ↔ invertible ([a]_m * [i]_m) := by rw [cc_G] _ ↔ invertible [i]_m := prod_inv_iff_inv h2 ([i]_m) _ ↔ rel_prime m i := Theorem_7_3_7 m i done lemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) : F m (G m a i) = [a]_m * F m i := by have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2 show F m (G m a i) = [a]_m * F m i from calc F m (G m a i) _ = [G m a i]_m := F_rp_def h3 _ = [a]_m * [i]_m := cc_G m a i _ = [a]_m * F m i := by rw [F_rp_def h2] done lemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : ¬rel_prime m i) : F m (G m a i) = [1]_m := by rewrite [←G_rp_iff h1 i] at h2 show F m (G m a i) = [1]_m from F_not_rp_def h2 done lemma FG_prod {m a : Nat} (h1 : rel_prime m a) : ∀ (k : Nat), prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by by_induc · -- Base Case show prod_seq 0 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from calc prod_seq 0 0 ((F m) ∘ (G m a)) _ = [1]_m := prod_seq_base _ _ _ = [a]_m ^ 0 * [1]_m := by ring _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by rw [num_rp_below_base, prod_seq_base] done · -- Induction Step fix k : Nat assume ih : prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) by_cases h2 : rel_prime m k · -- Case 1. h2 : rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([a]_m * F m k) := by rw [FG_rp h1 h2] _ = [a]_m ^ ((num_rp_below m k) + 1) * ((prod_seq k 0 (F m)) * F m k) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_rp h2, prod_seq_zero_step] done · -- Case 2. h2 : ¬rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([1]_m) := by rw [FG_not_rp h1 h2] _ = [a]_m ^ (num_rp_below m k) * (prod_seq k 0 (F m) * ([1]_m)) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_not_rp h2, prod_seq_zero_step, F_not_rp_def h2] done done done lemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by define --Goal : ∀ i < m, G m a i < m fix i : Nat assume h1 : i < m rewrite [G_def] --Goal : a * i % m < m show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m) done lemma left_inv_one_one_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g' (g i) = i) : one_one_below n g := sorry lemma right_inv_onto_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g (g' i) = i) (h2 : maps_below n g') : onto_below n g := by define at h2; define fix k : Nat assume h3 : k < n apply Exists.intro (g' k) show g' k < n ∧ g (g' k) = k from And.intro (h2 k h3) (h1 k h3) done lemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m * [inv_mod m a]_m = [1]_m := by have h2 : 0 ≤ (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a) show [a]_m * [inv_mod m a]_m = [1]_m from calc [a]_m * [inv_mod m a]_m _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2] _ = [a]_m * [gcd_c2 m a]_m := by rw [←cc_eq_mod] _ = [1]_m := gcd_c2_inv h1 done lemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m := calc a * (b % m) % m = a % m * (b % m % m) % m := Nat.mul_mod _ _ _ _ = a % m * (b % m) % m := by rw [Nat.mod_mod] _ = a * b % m := (Nat.mul_mod _ _ _).symm lemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm] rfl done theorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] : ↑a ≡ ↑b (MOD m) ↔ a % m = b % m := sorry lemma mul_inv_mod_cancel {m a i : Nat} [NeZero m] (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1 rewrite [mul_class, cc_eq_iff_congr, ←Nat.cast_mul, ←Nat.cast_one, congr_iff_mod_eq_Nat] at h3 show a * inv_mod m a * i % m = i from calc a * (inv_mod m a) * i % m _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = 1 % m * i % m := by rw [h3] _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = i % m := by rw [one_mul] _ = i := Nat.mod_eq_of_lt h2 done lemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl lemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, G m a (Ginv m a i) = i := by fix i : Nat assume h2 : i < m show G m a (Ginv m a i) = i from calc G m a (Ginv m a i) _ = a * ((inv_mod m a * i) % m) % m := by rfl _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, Ginv m a (G m a i) = i := by fix i : Nat assume h2 : i < m show Ginv m a (G m a i) = i from calc Ginv m a (G m a i) _ = inv_mod m a * ((a * i) % m) % m := by rfl _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc, mul_comm (inv_mod m a)] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_maps_below (m a : Nat) [NeZero m] : maps_below m (Ginv m a) := G_maps_below m (inv_mod m a)

lemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : one_one_below m (G m a)

HTPI.G_one_one_below

htpi/HTPILib/Chap7.lean

htpi

/- Copyright 2023 Daniel J. Velleman -/ import HTPILib.Chap6 namespace HTPI /- Definitions -/ lemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by have h : n + 1 > 0 := Nat.succ_pos n show a % (n + 1) < n + 1 from Nat.mod_lt a h done def gcd (a b : Nat) : Nat := match b with | 0 => a | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd (n + 1) (a % (n + 1)) termination_by b mutual def gcd_c1 (a b : Nat) : Int := match b with | 0 => 1 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c2 (n + 1) (a % (n + 1)) --Corresponds to s = t' termination_by b def gcd_c2 (a b : Nat) : Int := match b with | 0 => 0 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c1 (n + 1) (a % (n + 1)) - (gcd_c2 (n + 1) (a % (n + 1))) * ↑(a / (n + 1)) --Corresponds to t = s' - t'q termination_by b end def prime (n : Nat) : Prop := 2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n def prime_factor (p n : Nat) : Prop := prime p ∧ p ∣ n def all_prime (l : List Nat) : Prop := ∀ p ∈ l, prime p def nondec (l : List Nat) : Prop := match l with | [] => True --Of course, True is a proposition that is always true | n :: L => (∀ m ∈ L, n ≤ m) ∧ nondec L def nondec_prime_list (l : List Nat) : Prop := all_prime l ∧ nondec l def prod (l : List Nat) : Nat := match l with | [] => 1 | n :: L => n * (prod L) def prime_factorization (n : Nat) (l : List Nat) : Prop := nondec_prime_list l ∧ prod l = n def rel_prime (a b : Nat) : Prop := gcd a b = 1 def congr_mod (m : Nat) (a b : Int) : Prop := (↑m : Int) ∣ (a - b) def cc (m : Nat) (a : Int) : ZMod m := (↑a : ZMod m) notation:50 a " ≡ " b " (MOD " m ")" => congr_mod m a b notation:max "["a"]_"m:max => cc m a def invertible {m : Nat} (X : ZMod m) : Prop := ∃ (Y : ZMod m), X * Y = [1]_m def num_rp_below (m k : Nat) : Nat := match k with | 0 => 0 | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j def phi (m : Nat) : Nat := num_rp_below m m def prod_seq {m : Nat} (j k : Nat) (f : Nat → ZMod m) : ZMod m := match j with | 0 => [1]_m | n + 1 => prod_seq n k f * f (k + n) def maps_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i < n, g i < n def one_one_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i1 < n, ∀ i2 < n, g i1 = g i2 → i1 = i2 def onto_below (n : Nat) (g : Nat → Nat) : Prop := ∀ k < n, ∃ i < n, g i = k def perm_below (n : Nat) (g : Nat → Nat) : Prop := maps_below n g ∧ one_one_below n g ∧ onto_below n g def inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m) def swap (u v i : Nat) : Nat := if i = u then v else if i = v then u else i namespace Euler --For definitions specific to Euler's theorem def F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m def G (m a i : Nat) : Nat := (a * i) % m def Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i end Euler /- Section 7.1 -/ theorem dvd_mod_of_dvd_a_b {a b d : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ (a % b) := by set q : Nat := a / b have h3 : b * q + a % b = a := Nat.div_add_mod a b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 define --Goal : ∃ (c : Nat), a % b = d * c apply Exists.intro (j - k * q) show a % b = d * (j - k * q) from calc a % b _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm _ = a - b * q := by rw [h3] _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc] _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm done theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry #eval gcd 672 161 --Answer: 7 lemma gcd_base (a : Nat) : gcd a 0 = a := by rfl lemma gcd_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd a b = gcd b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1)) rfl done lemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b ≠ 0) : a % b < b := by have h1 : b > 0 := Nat.pos_of_ne_zero h show a % b < b from Nat.mod_lt a h1 done lemma dvd_self (n : Nat) : n ∣ n := by apply Exists.intro 1 ring done theorem gcd_dvd : ∀ (b a : Nat), (gcd a b) ∣ a ∧ (gcd a b) ∣ b := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd a b_1) ∣ a ∧ (gcd a b_1) ∣ b_1 fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_base] --Goal: a ∣ a ∧ a ∣ 0 apply And.intro (dvd_self a) define apply Exists.intro 0 rfl done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_nonzero a h1] --Goal : gcd b (a % b) ∣ a ∧ gcd b (a % b) ∣ b have h2 : a % b < b := mod_nonzero_lt a h1 have h3 : (gcd b (a % b)) ∣ b ∧ (gcd b (a % b)) ∣ (a % b) := ih (a % b) h2 b apply And.intro _ h3.left show (gcd b (a % b)) ∣ a from dvd_a_of_dvd_b_mod h3.left h3.right done done theorem gcd_dvd_left (a b : Nat) : (gcd a b) ∣ a := (gcd_dvd b a).left theorem gcd_dvd_right (a b : Nat) : (gcd a b) ∣ b := (gcd_dvd b a).right lemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl lemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c1 a b = gcd_c2 b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done lemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl lemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * ↑(a / b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done theorem gcd_lin_comb : ∀ (b a : Nat), (gcd_c1 a b) * ↑a + (gcd_c2 a b) * ↑b = ↑(gcd a b) := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd_c1 a b_1) * ↑a + (gcd_c2 a b_1) * ↑b_1 = ↑(gcd a b_1) fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base] --Goal : 1 * ↑a + 0 * ↑0 = ↑a ring done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1] --Goal : gcd_c2 b (a % b) * ↑a + -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * ↑(a / b)) * ↑b = -- ↑(gcd b (a % b)) set r : Nat := a % b set q : Nat := a / b set s : Int := gcd_c1 b r set t : Int := gcd_c2 b r --Goal : t * ↑a + (s - t * ↑q) * ↑b = ↑(gcd b r) have h2 : r < b := mod_nonzero_lt a h1 have h3 : s * ↑b + t * ↑r = ↑(gcd b r) := ih r h2 b have h4 : b * q + r = a := Nat.div_add_mod a b rewrite [←h3, ←h4] rewrite [Nat.cast_add, Nat.cast_mul] --Goal : t * (↑b * ↑q + ↑r) + (s - t * ↑q) * ↑b = s * ↑b + t * ↑r ring done done #eval gcd_c1 672 161 --Answer: 6 #eval gcd_c2 672 161 --Answer: -25 --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161 theorem Theorem_7_1_6 {d a b : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ gcd a b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑d ∣ ↑(gcd a b) set s : Int := gcd_c1 a b set t : Int := gcd_c2 a b have h3 : s * ↑a + t * ↑b = ↑(gcd a b) := gcd_lin_comb b a rewrite [←h3] --Goal : ↑d ∣ s * ↑a + t * ↑b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul] --Goal : ↑d ∣ s * (↑d * ↑j) + t * (↑d * ↑k) define apply Exists.intro (s * ↑j + t * ↑k) ring done /- Section 7.2 -/ theorem dvd_trans {a b c : Nat} (h1 : a ∣ b) (h2 : b ∣ c) : a ∣ c := by define at h1; define at h2; define obtain (m : Nat) (h3 : b = a * m) from h1 obtain (n : Nat) (h4 : c = b * n) from h2 rewrite [h3, mul_assoc] at h4 apply Exists.intro (m * n) show c = a * (m * n) from h4 done lemma exists_prime_factor : ∀ (n : Nat), 2 ≤ n → ∃ (p : Nat), prime_factor p n := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, 2 ≤ n_1 → ∃ (p : Nat), prime_factor p n_1 assume h1 : 2 ≤ n by_cases h2 : prime n · -- Case 1. h2 : prime n apply Exists.intro n define --Goal : prime n ∧ n ∣ n show prime n ∧ n ∣ n from And.intro h2 (dvd_self n) done · -- Case 2. h2 : ¬prime n define at h2 --h2 : ¬(2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n) demorgan at h2 disj_syll h2 h1 obtain (a : Nat) (h3 : ∃ (b : Nat), a * b = n ∧ a < n ∧ b < n) from h2 obtain (b : Nat) (h4 : a * b = n ∧ a < n ∧ b < n) from h3 have h5 : 2 ≤ a := by by_contra h6 have h7 : a ≤ 1 := by linarith have h8 : n ≤ b := calc n _ = a * b := h4.left.symm _ ≤ 1 * b := by rel [h7] _ = b := by ring linarith --n ≤ b contradicts b < n done have h6 : ∃ (p : Nat), prime_factor p a := ih a h4.right.left h5 obtain (p : Nat) (h7 : prime_factor p a) from h6 apply Exists.intro p define --Goal : prime p ∧ p ∣ n define at h7 --h7 : prime p ∧ p ∣ a apply And.intro h7.left have h8 : a ∣ n := by apply Exists.intro b show n = a * b from (h4.left).symm done show p ∣ n from dvd_trans h7.right h8 done done lemma exists_least_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q := by set S : Set Nat := {p : Nat | prime_factor p n} have h2 : ∃ (p : Nat), p ∈ S := exists_prime_factor n h show ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q from well_ord_princ S h2 done lemma all_prime_nil : all_prime [] := by define --Goal : ∀ p ∈ [], prime p fix p : Nat contrapos --Goal : ¬prime p → p ∉ [] assume h1 : ¬prime p show p ∉ [] from List.not_mem_nil p done lemma all_prime_cons (n : Nat) (L : List Nat) : all_prime (n :: L) ↔ prime n ∧ all_prime L := by apply Iff.intro · -- (→) assume h1 : all_prime (n :: L) --Goal : prime n ∧ all_prime L define at h1 --h1 : ∀ p ∈ n :: L, prime p apply And.intro (h1 n (List.mem_cons_self n L)) define --Goal : ∀ p ∈ L, prime p fix p : Nat assume h2 : p ∈ L show prime p from h1 p (List.mem_cons_of_mem n h2) done · -- (←) assume h1 : prime n ∧ all_prime L --Goal : all_prime (n :: l) define : all_prime L at h1 define fix p : Nat assume h2 : p ∈ n :: L rewrite [List.mem_cons] at h2 --h2 : p = n ∨ p ∈ L by_cases on h2 · -- Case 1. h2 : p = n rewrite [h2] show prime n from h1.left done · -- Case 2. h2 : p ∈ L show prime p from h1.right p h2 done done done lemma nondec_nil : nondec [] := by define --Goal : True trivial --trivial proves some obviously true statements, such as True done lemma nondec_cons (n : Nat) (L : List Nat) : nondec (n :: L) ↔ (∀ m ∈ L, n ≤ m) ∧ nondec L := by rfl lemma prod_nil : prod [] = 1 := by rfl lemma prod_cons : prod (n :: L) = n * (prod L) := by rfl lemma exists_cons_of_length_eq_succ {A : Type} {l : List A} {n : Nat} (h : l.length = n + 1) : ∃ (a : A) (L : List A), l = a :: L ∧ L.length = n := by have h1 : ¬l.length = 0 := by linarith rewrite [List.length_eq_zero] at h1 obtain (a : A) (h2 : ∃ (L : List A), l = a :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List A) (h3 : l = a :: L) from h2 apply Exists.intro a apply Exists.intro L apply And.intro h3 have h4 : (a :: L).length = L.length + 1 := List.length_cons a L rewrite [←h3, h] at h4 show L.length = n from (Nat.add_right_cancel h4).symm done lemma list_elt_dvd_prod_by_length (a : Nat) : ∀ (n : Nat), ∀ (l : List Nat), l.length = n → a ∈ l → a ∣ prod l := by by_induc · --Base Case fix l : List Nat assume h1 : l.length = 0 rewrite [List.length_eq_zero] at h1 --h1 : l = [] rewrite [h1] --Goal : a ∈ [] → a ∣ prod [] contrapos assume h2 : ¬a ∣ prod [] show a ∉ [] from List.not_mem_nil a done · -- Induction Step fix n : Nat assume ih : ∀ (l : List Nat), List.length l = n → a ∈ l → a ∣ prod l fix l : List Nat assume h1 : l.length = n + 1 --Goal : a ∈ l → a ∣ prod l obtain (b : Nat) (h2 : ∃ (L : List Nat), l = b :: L ∧ L.length = n) from exists_cons_of_length_eq_succ h1 obtain (L : List Nat) (h3 : l = b :: L ∧ L.length = n) from h2 have h4 : a ∈ L → a ∣ prod L := ih L h3.right assume h5 : a ∈ l rewrite [h3.left, prod_cons] --Goal : a ∣ b * prod L rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b ∨ a ∈ L by_cases on h5 · -- Case 1. h5 : a = b apply Exists.intro (prod L) rewrite [h5] rfl done · -- Case 2. h5 : a ∈ L have h6 : a ∣ prod L := h4 h5 have h7 : prod L ∣ b * prod L := by apply Exists.intro b ring done show a ∣ b * prod L from dvd_trans h6 h7 done done done lemma list_elt_dvd_prod {a : Nat} {l : List Nat} (h : a ∈ l) : a ∣ prod l := by set n : Nat := l.length have h1 : l.length = n := by rfl show a ∣ prod l from list_elt_dvd_prod_by_length a n l h1 h done lemma exists_prime_factorization : ∀ (n : Nat), n ≥ 1 → ∃ (l : List Nat), prime_factorization n l := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, n_1 ≥ 1 → ∃ (l : List Nat), prime_factorization n_1 l assume h1 : n ≥ 1 by_cases h2 : n = 1 · -- Case 1. h2 : n = 1 apply Exists.intro [] define apply And.intro · -- Proof of nondec_prime_list [] define show all_prime [] ∧ nondec [] from And.intro all_prime_nil nondec_nil done · -- Proof of prod [] = n rewrite [prod_nil, h2] rfl done done · -- Case 2. h2 : n ≠ 1 have h3 : n ≥ 2 := lt_of_le_of_ne' h1 h2 obtain (p : Nat) (h4 : prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q) from exists_least_prime_factor h3 have p_prime_factor : prime_factor p n := h4.left define at p_prime_factor have p_prime : prime p := p_prime_factor.left have p_dvd_n : p ∣ n := p_prime_factor.right have p_least : ∀ (q : Nat), prime_factor q n → p ≤ q := h4.right obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n have h5 : m ≠ 0 := by contradict h1 with h6 have h7 : n = 0 := calc n _ = p * m := n_eq_pm _ = p * 0 := by rw [h6] _ = 0 := by ring rewrite [h7] decide done have m_pos : 0 < m := Nat.pos_of_ne_zero h5 have m_lt_n : m < n := by define at p_prime show m < n from calc m _ < m + m := by linarith _ = 2 * m := by ring _ ≤ p * m := by rel [p_prime.left] _ = n := n_eq_pm.symm done obtain (L : List Nat) (h6 : prime_factorization m L) from ih m m_lt_n m_pos define at h6 have ndpl_L : nondec_prime_list L := h6.left define at ndpl_L apply Exists.intro (p :: L) define apply And.intro · -- Proof of nondec_prime_list (p :: L) define apply And.intro · -- Proof of all_prime (p :: L) rewrite [all_prime_cons] show prime p ∧ all_prime L from And.intro p_prime ndpl_L.left done · -- Proof of nondec (p :: L) rewrite [nondec_cons] apply And.intro _ ndpl_L.right fix q : Nat assume q_in_L : q ∈ L have h7 : q ∣ prod L := list_elt_dvd_prod q_in_L rewrite [h6.right] at h7 --h7 : q ∣ m have h8 : m ∣ n := by apply Exists.intro p rewrite [n_eq_pm] ring done have q_dvd_n : q ∣ n := dvd_trans h7 h8 have ap_L : all_prime L := ndpl_L.left define at ap_L have q_prime_factor : prime_factor q n := And.intro (ap_L q q_in_L) q_dvd_n show p ≤ q from p_least q q_prime_factor done done · -- Proof of prod (p :: L) = n rewrite [prod_cons, h6.right, n_eq_pm] rfl done done done theorem Theorem_7_2_2 {a b c : Nat} (h1 : c ∣ a * b) (h2 : rel_prime a c) : c ∣ b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑c ∣ ↑b define at h1; define at h2; define obtain (j : Nat) (h3 : a * b = c * j) from h1 set s : Int := gcd_c1 a c set t : Int := gcd_c2 a c have h4 : s * ↑a + t * ↑c = ↑(gcd a c) := gcd_lin_comb c a rewrite [h2, Nat.cast_one] at h4 --h4 : s * ↑a + t * ↑c = (1 : Int) apply Exists.intro (s * ↑j + t * ↑b) show ↑b = ↑c * (s * ↑j + t * ↑b) from calc ↑b _ = (1 : Int) * ↑b := (one_mul _).symm _ = (s * ↑a + t * ↑c) * ↑b := by rw [h4] _ = s * (↑a * ↑b) + t * ↑c * ↑b := by ring _ = s * (↑c * ↑j) + t * ↑c * ↑b := by rw [←Nat.cast_mul a b, h3, Nat.cast_mul c j] _ = ↑c * (s * ↑j + t * ↑b) := by ring done lemma le_nonzero_prod_left {a b : Nat} (h : a * b ≠ 0) : a ≤ a * b := by have h1 : b ≠ 0 := by contradict h with h1 rewrite [h1] ring done have h2 : 1 ≤ b := Nat.pos_of_ne_zero h1 show a ≤ a * b from calc a = a * 1 := (mul_one a).symm _ ≤ a * b := by rel [h2] done lemma le_nonzero_prod_right {a b : Nat} (h : a * b ≠ 0) : b ≤ a * b := by rewrite [mul_comm] rewrite [mul_comm] at h show b ≤ b * a from le_nonzero_prod_left h done lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry lemma rel_prime_of_prime_not_dvd {a p : Nat} (h1 : prime p) (h2 : ¬p ∣ a) : rel_prime a p := by have h3 : gcd a p ∣ a := gcd_dvd_left a p have h4 : gcd a p ∣ p := gcd_dvd_right a p have h5 : gcd a p = 1 ∨ gcd a p = p := dvd_prime h1 h4 have h6 : gcd a p ≠ p := by contradict h2 with h6 rewrite [h6] at h3 show p ∣ a from h3 done disj_syll h5 h6 show rel_prime a p from h5 done theorem Theorem_7_2_3 {a b p : Nat} (h1 : prime p) (h2 : p ∣ a * b) : p ∣ a ∨ p ∣ b := by or_right with h3 have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3 show p ∣ b from Theorem_7_2_2 h2 h4 done lemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a ≥ 1 := by have h1 : a ≠ 0 := by by_contra h1 rewrite [h1] at h contradict h linarith done show a ≥ 1 from Nat.pos_of_ne_zero h1 done lemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by have h1 : a ≥ 1 := ge_one_of_prod_one h have h2 : a * b ≠ 0 := by linarith have h3 : a ≤ a * b := le_nonzero_prod_left h2 rewrite [h] at h3 show a = 1 from Nat.le_antisymm h3 h1 done lemma eq_one_of_dvd_one {n : Nat} (h : n ∣ 1) : n = 1 := by obtain (j : Nat) (h1 : 1 = n * j) from h show n = 1 from eq_one_of_prod_one h1.symm done lemma prime_not_one {p : Nat} (h : prime p) : p ≠ 1 := by define at h linarith done theorem Theorem_7_2_4 {p : Nat} (h1 : prime p) : ∀ (l : List Nat), p ∣ prod l → ∃ a ∈ l, p ∣ a := by apply List.rec · -- Base Case. Goal : p ∣ prod [] → ∃ a ∈ [], p ∣ a rewrite [prod_nil] assume h2 : p ∣ 1 show ∃ a ∈ [], p ∣ a from absurd (eq_one_of_dvd_one h2) (prime_not_one h1) done · -- Induction Step fix b : Nat fix L : List Nat assume ih : p ∣ prod L → ∃ a ∈ L, p ∣ a --Goal : p ∣ prod (b :: L) → ∃ a ∈ b :: L, p ∣ a assume h2 : p ∣ prod (b :: L) rewrite [prod_cons] at h2 have h3 : p ∣ b ∨ p ∣ prod L := Theorem_7_2_3 h1 h2 by_cases on h3 · -- Case 1. h3 : p ∣ b apply Exists.intro b show b ∈ b :: L ∧ p ∣ b from And.intro (List.mem_cons_self b L) h3 done · -- Case 2. h3 : p ∣ prod L obtain (a : Nat) (h4 : a ∈ L ∧ p ∣ a) from ih h3 apply Exists.intro a show a ∈ b :: L ∧ p ∣ a from And.intro (List.mem_cons_of_mem b h4.left) h4.right done done done lemma prime_in_list {p : Nat} {l : List Nat} (h1 : prime p) (h2 : all_prime l) (h3 : p ∣ prod l) : p ∈ l := by obtain (a : Nat) (h4 : a ∈ l ∧ p ∣ a) from Theorem_7_2_4 h1 l h3 define at h2 have h5 : prime a := h2 a h4.left have h6 : p = 1 ∨ p = a := dvd_prime h5 h4.right disj_syll h6 (prime_not_one h1) rewrite [h6] show a ∈ l from h4.left done lemma first_le_first {p q : Nat} {l m : List Nat} (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m)) (h3 : prod (p :: l) = prod (q :: m)) : p ≤ q := by define at h1; define at h2 have h4 : q ∣ prod (p :: l) := by define apply Exists.intro (prod m) rewrite [←prod_cons] show prod (p :: l) = prod (q :: m) from h3 done have h5 : all_prime (q :: m) := h2.left rewrite [all_prime_cons] at h5 have h6 : q ∈ p :: l := prime_in_list h5.left h1.left h4 have h7 : nondec (p :: l) := h1.right rewrite [nondec_cons] at h7 rewrite [List.mem_cons] at h6 by_cases on h6 · -- Case 1. h6 : q = p linarith done · -- Case 2. h6 : q ∈ l have h8 : ∀ m ∈ l, p ≤ m := h7.left show p ≤ q from h8 q h6 done done lemma nondec_prime_list_tail {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by define at h define rewrite [all_prime_cons, nondec_cons] at h show all_prime l ∧ nondec l from And.intro h.left.right h.right.right done lemma cons_prod_not_one {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : prod (p :: l) ≠ 1 := by define at h have h1 : all_prime (p :: l) := h.left rewrite [all_prime_cons] at h1 rewrite [prod_cons] by_contra h2 show False from (prime_not_one h1.left) (eq_one_of_prod_one h2) done lemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) : l = [] ↔ prod l = 1 := by apply Iff.intro · -- (→) assume h1 : l = [] rewrite [h1] show prod [] = 1 from prod_nil done · -- (←) contrapos assume h1 : ¬l = [] obtain (p : Nat) (h2 : ∃ (L : List Nat), l = p :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List Nat) (h3 : l = p :: L) from h2 rewrite [h3] at h rewrite [h3] show ¬prod (p :: L) = 1 from cons_prod_not_one h done done lemma prime_pos {p : Nat} (h : prime p) : p > 0 := by define at h linarith done theorem Theorem_7_2_5 : ∀ (l1 l2 : List Nat), nondec_prime_list l1 → nondec_prime_list l2 → prod l1 = prod l2 → l1 = l2 := by apply List.rec · -- Base Case. Goal : ∀ (l2 : List Nat), nondec_prime_list [] → -- nondec_prime_list l2 → prod [] = prod l2 → [] = l2 fix l2 : List Nat assume h1 : nondec_prime_list [] assume h2 : nondec_prime_list l2 assume h3 : prod [] = prod l2 rewrite [prod_nil, eq_comm, ←list_nil_iff_prod_one h2] at h3 show [] = l2 from h3.symm done · -- Induction Step fix p : Nat fix L1 : List Nat assume ih : ∀ (L2 : List Nat), nondec_prime_list L1 → nondec_prime_list L2 → prod L1 = prod L2 → L1 = L2 -- Goal : ∀ (l2 : List Nat), nondec_prime_list (p :: L1) → -- nondec_prime_list l2 → prod (p :: L1) = prod l2 → p :: L1 = l2 fix l2 : List Nat assume h1 : nondec_prime_list (p :: L1) assume h2 : nondec_prime_list l2 assume h3 : prod (p :: L1) = prod l2 have h4 : ¬prod (p :: L1) = 1 := cons_prod_not_one h1 rewrite [h3, ←list_nil_iff_prod_one h2] at h4 obtain (q : Nat) (h5 : ∃ (L : List Nat), l2 = q :: L) from List.exists_cons_of_ne_nil h4 obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5 rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2) rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2) have h7 : p ≤ q := first_le_first h1 h2 h3 have h8 : q ≤ p := first_le_first h2 h1 h3.symm have h9 : p = q := by linarith rewrite [h9, prod_cons, prod_cons] at h3 --h3 : q * prod L1 = q * prod L2 have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1 have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2 define at h2 have h12 : all_prime (q :: L2) := h2.left rewrite [all_prime_cons] at h12 have h13 : q > 0 := prime_pos h12.left have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3 have h15 : L1 = L2 := ih L2 h10 h11 h14 rewrite [h6, h9, h15] rfl done done theorem fund_thm_arith (n : Nat) (h : n ≥ 1) : ∃! (l : List Nat), prime_factorization n l := by exists_unique · -- Existence show ∃ (l : List Nat), prime_factorization n l from exists_prime_factorization n h done · -- Uniqueness fix l1 : List Nat; fix l2 : List Nat assume h1 : prime_factorization n l1 assume h2 : prime_factorization n l2 define at h1; define at h2 have h3 : prod l1 = n := h1.right rewrite [←h2.right] at h3 show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3 done done /- Section 7.3 -/ theorem congr_refl (m : Nat) : ∀ (a : Int), a ≡ a (MOD m) := by fix a : Int define --Goal : ∃ (c : Int), a - a = ↑m * c apply Exists.intro 0 ring done theorem congr_symm {m : Nat} : ∀ {a b : Int}, a ≡ b (MOD m) → b ≡ a (MOD m) := by fix a : Int; fix b : Int assume h1 : a ≡ b (MOD m) define at h1 --h1 : ∃ (c : Int), a - b = ↑m * c define --Goal : ∃ (c : Int), b - a = ↑m * c obtain (c : Int) (h2 : a - b = m * c) from h1 apply Exists.intro (-c) show b - a = m * (-c) from calc b - a _ = -(a - b) := by ring _ = -(m * c) := by rw [h2] _ = m * (-c) := by ring done theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry /- Fundamental properties of congruence classes -/ lemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) : X = Y ↔ X.val = Y.val := Fin.ext_iff lemma val_nat_eq_mod (n k : Nat) : ([k]_(n + 1)).val = k % (n + 1) := by rfl lemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl theorem cc_rep {m : Nat} (X : ZMod m) : ∃ (a : Int), X = [a]_m := match m with | 0 => by apply Exists.intro X rfl done | n + 1 => by apply Exists.intro ↑(X.val) have h1 : X.val < n + 1 := Fin.prop X rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1] rfl done theorem add_class (m : Nat) (a b : Int) : [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm theorem mul_class (m : Nat) (a b : Int) : [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm lemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ [a - b]_m = [0]_m := by apply Iff.intro · -- (→) assume h1 : [a]_m = [b]_m have h2 : a - b = a + (-b) := by ring have h3 : b + (-b) = 0 := by ring show [a - b]_m = [0]_m from calc [a - b]_m _ = [a + (-b)]_m := by rw [h2] _ = [a]_m + [-b]_m := by rw [add_class] _ = [b]_m + [-b]_m := by rw [h1] _ = [b + -b]_m := by rw [add_class] _ = [0]_m := by rw [h3] done · -- (←) assume h1 : [a - b]_m = [0]_m have h2 : b + (a - b) = a := by ring have h3 : b + 0 = b := by ring show [a]_m = [b]_m from calc [a]_m _ = [b + (a - b)]_m := by rw [h2] _ = [b]_m + [a - b]_m := by rw [add_class] _ = [b]_m + [0]_m := by rw [h1] _ = [b + 0]_m := by rw [add_class] _ = [b]_m := by rw [h3] done done lemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) : [a]_m = [0]_m → [-a]_m = [0]_m := by assume h1 : [a]_m = [0]_m have h2 : 0 + (-a) = -a := by ring have h3 : a + (-a) = 0 := by ring show [-a]_m = [0]_m from calc [-a]_m _ = [0 + (-a)]_m := by rw [h2] _ = [0]_m + [-a]_m := by rw [add_class] _ = [a]_m + [-a]_m := by rw [h1] _ = [a + (-a)]_m := by rw [add_class] _ = [0]_m := by rw [h3] done lemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) : [-a]_m = [0]_m ↔ [a]_m = [0]_m := by apply Iff.intro _ (cc_neg_zero_of_cc_zero m a) assume h1 : [-a]_m = [0]_m have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1 have h3 : -(-a) = a := by ring rewrite [h3] at h2 show [a]_m = [0]_m from h2 done lemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl lemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m ↔ m ∣ k := match m with | 0 => by have h : (0 : Int) = (↑(0 : Nat) : Int) := by rfl rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj] apply Iff.intro · -- (→) assume h1 : k = 0 rewrite [h1] show 0 ∣ 0 from dvd_self 0 done · -- (←) assume h1 : 0 ∣ k obtain (c : Nat) (h2 : k = 0 * c) from h1 rewrite [h2] ring done done | n + 1 => by rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero] show k % (n + 1) = 0 ↔ n + 1 ∣ k from (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm done lemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m ↔ ↑m ∣ a := by obtain (k : Nat) (h1 : a = ↑k ∨ a = -↑k) from Int.eq_nat_or_neg a by_cases on h1 · -- Case 1. h1: a = ↑k rewrite [h1, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done · -- Case 2. h1: a = -↑k rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done done theorem cc_eq_iff_congr (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ a ≡ b (MOD m) := calc [a]_m = [b]_m _ ↔ [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b _ ↔ ↑m ∣ (a - b) := cc_zero_iff_dvd m (a - b) _ ↔ a ≡ b (MOD m) := by rfl /- End of fundamental properties of congruence classes -/ lemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m := by have h1 : (↑m : Int) ≠ 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m) show 0 ≤ a % m from Int.emod_nonneg a h1 done lemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m) have h2 : (↑m : Int) > 0 := (Nat.cast_pos).rtl h1 show a % m < m from Int.emod_lt_of_pos a h2 done lemma congr_mod_mod (m : Nat) (a : Int) : a ≡ a % m (MOD m) := by define have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m apply Exists.intro (a / m) show a - a % m = m * (a / m) from calc a - (a % m) _ = m * (a / m) + a % m - a % m := by rw [h1] _ = m * (a / m) := by ring done lemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) := And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a)) theorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) : ∃! (r : Int), 0 ≤ r ∧ r < m ∧ a ≡ r (MOD m) := by exists_unique · -- Existence apply Exists.intro (a % m) show 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) from mod_cmpl_res m a done · -- Uniqueness fix r1 : Int; fix r2 : Int assume h1 : 0 ≤ r1 ∧ r1 < m ∧ a ≡ r1 (MOD m) assume h2 : 0 ≤ r2 ∧ r2 < m ∧ a ≡ r2 (MOD m) have h3 : r1 ≡ r2 (MOD m) := congr_trans (congr_symm h1.right.right) h2.right.right obtain (d : Int) (h4 : r1 - r2 = m * d) from h3 have h5 : r1 - r2 < m * 1 := by linarith have h6 : m * (-1) < r1 - r2 := by linarith rewrite [h4] at h5 --h5 : m * d < m * 1 rewrite [h4] at h6 --h6 : m * -1 < m * d have h7 : (↑m : Int) ≥ 0 := Nat.cast_nonneg m have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7 have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7 have h10 : d = 0 := by linarith show r1 = r2 from calc r1 _ = r1 - r2 + r2 := by ring _ = m * 0 + r2 := by rw [h4, h10] _ = r2 := by ring done done lemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m := (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a) theorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y rewrite [h1, h2] have h3 : a + b = b + a := by ring show [a]_m + [b]_m = [b]_m + [a]_m from calc [a]_m + [b]_m _ = [a + b]_m := add_class m a b _ = [b + a]_m := by rw [h3] _ = [b]_m + [a]_m := (add_class m b a).symm done theorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X rewrite [h1] have h2 : a * 1 = a := by ring show [a]_m * [1]_m = [a]_m from calc [a]_m * [1]_m _ = [a * 1]_m := mul_class m a 1 _ = [a]_m := by rw [h2] done theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry lemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) : [a]_m * [gcd_c2 m a]_m = [1]_m := by set s : Int := gcd_c1 m a have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m define at h1 rewrite [h1, Nat.cast_one] at h2 --h2 : s * ↑m + gcd_c2 m a * ↑a = 1 rewrite [mul_class, cc_eq_iff_congr] define --Goal : ∃ (c : Int), ↑a * gcd_c2 m a - 1 = ↑m * c apply Exists.intro (-s) show a * (gcd_c2 m a) - 1 = m * (-s) from calc a * (gcd_c2 m a) - 1 _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring _ = 1 + m * (-s) - 1 := by rw [h2] _ = m * (-s) := by ring done theorem Theorem_7_3_7 (m a : Nat) : invertible [a]_m ↔ rel_prime m a := by apply Iff.intro · -- (→) assume h1 : invertible [a]_m define at h1 obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1 obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y rewrite [h3, mul_class, cc_eq_iff_congr] at h2 define at h2 obtain (c : Int) (h4 : a * b - 1 = m * c) from h2 rewrite [Exercise_7_2_6] --Goal : ∃ (s t : Int), s * ↑m + t * ↑a = 1 apply Exists.intro (-c) apply Exists.intro b show (-c) * m + b * a = 1 from calc (-c) * m + b * a _ = (-c) * m + (a * b - 1) + 1 := by ring _ = (-c) * m + m * c + 1 := by rw [h4] _ = 1 := by ring done · -- (←) assume h1 : rel_prime m a define show ∃ (Y : ZMod m), [a]_m * Y = [1]_m from Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1) done done /- Section 7.4 -/ section Euler open Euler lemma num_rp_below_base {m : Nat} : num_rp_below m 0 = 0 := by rfl lemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) : num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by have h1 : num_rp_below m (j + 1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : gcd m j = 1 rewrite [if_pos h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1 show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1 done lemma num_rp_below_step_not_rp {m j : Nat} (h : ¬rel_prime m j) : num_rp_below m (j + 1) = num_rp_below m j := by have h1 : num_rp_below m (j +1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : ¬gcd m j = 1 rewrite [if_neg h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j show num_rp_below m (j + 1) = num_rp_below m j from h1 done lemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl #eval phi 10 --Answer: 4 lemma prod_inv_iff_inv {m : Nat} {X : ZMod m} (h1 : invertible X) (Y : ZMod m) : invertible (X * Y) ↔ invertible Y := by apply Iff.intro · -- (→) assume h2 : invertible (X * Y) obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2 apply Exists.intro (X * Z) rewrite [←h3] --Goal : Y * (X * Z) = X * Y * Z ring --Note that ring can do algebra in ZMod m done · -- (←) assume h2 : invertible Y obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1 obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2 apply Exists.intro (Xi * Yi) show (X * Y) * (Xi * Yi) = [1]_m from calc X * Y * (Xi * Yi) _ = (X * Xi) * (Y * Yi) := by ring _ = [1]_m * [1]_m := by rw [h3, h4] _ = [1]_m := Theorem_7_3_6_7 [1]_m done done lemma F_rp_def {m i : Nat} (h : rel_prime m i) : F m i = [i]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h --h : gcd m i = 1 rewrite [if_pos h] at h1 show F m i = [i]_m from h1 done lemma F_not_rp_def {m i : Nat} (h : ¬rel_prime m i) : F m i = [1]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h rewrite [h1, if_neg h] rfl done lemma prod_seq_base {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 0 k f = [1]_m := by rfl lemma prod_seq_step {m : Nat} (n k : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl lemma prod_seq_zero_step {m : Nat} (n : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by rewrite [prod_seq_step, zero_add] rfl done lemma prod_one {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 1 k f = f k := by rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7] rfl done lemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl lemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m := calc [G m a i]_m _ = [(a * i) % m]_m := by rfl _ = [a * i]_m := (cc_eq_mod m (a * i)).symm _ = [a]_m * [i]_m := (mul_class m a i).symm lemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) : rel_prime m (G m a i) ↔ rel_prime m i := by have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1 show rel_prime m (G m a i) ↔ rel_prime m i from calc rel_prime m (G m a i) _ ↔ invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm _ ↔ invertible ([a]_m * [i]_m) := by rw [cc_G] _ ↔ invertible [i]_m := prod_inv_iff_inv h2 ([i]_m) _ ↔ rel_prime m i := Theorem_7_3_7 m i done lemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) : F m (G m a i) = [a]_m * F m i := by have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2 show F m (G m a i) = [a]_m * F m i from calc F m (G m a i) _ = [G m a i]_m := F_rp_def h3 _ = [a]_m * [i]_m := cc_G m a i _ = [a]_m * F m i := by rw [F_rp_def h2] done lemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : ¬rel_prime m i) : F m (G m a i) = [1]_m := by rewrite [←G_rp_iff h1 i] at h2 show F m (G m a i) = [1]_m from F_not_rp_def h2 done lemma FG_prod {m a : Nat} (h1 : rel_prime m a) : ∀ (k : Nat), prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by by_induc · -- Base Case show prod_seq 0 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from calc prod_seq 0 0 ((F m) ∘ (G m a)) _ = [1]_m := prod_seq_base _ _ _ = [a]_m ^ 0 * [1]_m := by ring _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by rw [num_rp_below_base, prod_seq_base] done · -- Induction Step fix k : Nat assume ih : prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) by_cases h2 : rel_prime m k · -- Case 1. h2 : rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([a]_m * F m k) := by rw [FG_rp h1 h2] _ = [a]_m ^ ((num_rp_below m k) + 1) * ((prod_seq k 0 (F m)) * F m k) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_rp h2, prod_seq_zero_step] done · -- Case 2. h2 : ¬rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([1]_m) := by rw [FG_not_rp h1 h2] _ = [a]_m ^ (num_rp_below m k) * (prod_seq k 0 (F m) * ([1]_m)) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_not_rp h2, prod_seq_zero_step, F_not_rp_def h2] done done done lemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by define --Goal : ∀ i < m, G m a i < m fix i : Nat assume h1 : i < m rewrite [G_def] --Goal : a * i % m < m show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m) done lemma left_inv_one_one_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g' (g i) = i) : one_one_below n g := sorry lemma right_inv_onto_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g (g' i) = i) (h2 : maps_below n g') : onto_below n g := by define at h2; define fix k : Nat assume h3 : k < n apply Exists.intro (g' k) show g' k < n ∧ g (g' k) = k from And.intro (h2 k h3) (h1 k h3) done lemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m * [inv_mod m a]_m = [1]_m := by have h2 : 0 ≤ (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a) show [a]_m * [inv_mod m a]_m = [1]_m from calc [a]_m * [inv_mod m a]_m _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2] _ = [a]_m * [gcd_c2 m a]_m := by rw [←cc_eq_mod] _ = [1]_m := gcd_c2_inv h1 done lemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m := calc a * (b % m) % m = a % m * (b % m % m) % m := Nat.mul_mod _ _ _ _ = a % m * (b % m) % m := by rw [Nat.mod_mod] _ = a * b % m := (Nat.mul_mod _ _ _).symm lemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm] rfl done theorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] : ↑a ≡ ↑b (MOD m) ↔ a % m = b % m := sorry lemma mul_inv_mod_cancel {m a i : Nat} [NeZero m] (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1 rewrite [mul_class, cc_eq_iff_congr, ←Nat.cast_mul, ←Nat.cast_one, congr_iff_mod_eq_Nat] at h3 show a * inv_mod m a * i % m = i from calc a * (inv_mod m a) * i % m _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = 1 % m * i % m := by rw [h3] _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = i % m := by rw [one_mul] _ = i := Nat.mod_eq_of_lt h2 done lemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl lemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, G m a (Ginv m a i) = i := by fix i : Nat assume h2 : i < m show G m a (Ginv m a i) = i from calc G m a (Ginv m a i) _ = a * ((inv_mod m a * i) % m) % m := by rfl _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, Ginv m a (G m a i) = i := by fix i : Nat assume h2 : i < m show Ginv m a (G m a i) = i from calc Ginv m a (G m a i) _ = inv_mod m a * ((a * i) % m) % m := by rfl _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc, mul_comm (inv_mod m a)] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_maps_below (m a : Nat) [NeZero m] : maps_below m (Ginv m a) := G_maps_below m (inv_mod m a) lemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : one_one_below m (G m a) := left_inv_one_one_below (Ginv_left_inv h1)

lemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : onto_below m (G m a)

HTPI.G_onto_below

htpi/HTPILib/Chap7.lean

htpi

/- Copyright 2023 Daniel J. Velleman -/ import HTPILib.Chap6 namespace HTPI /- Definitions -/ lemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by have h : n + 1 > 0 := Nat.succ_pos n show a % (n + 1) < n + 1 from Nat.mod_lt a h done def gcd (a b : Nat) : Nat := match b with | 0 => a | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd (n + 1) (a % (n + 1)) termination_by b mutual def gcd_c1 (a b : Nat) : Int := match b with | 0 => 1 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c2 (n + 1) (a % (n + 1)) --Corresponds to s = t' termination_by b def gcd_c2 (a b : Nat) : Int := match b with | 0 => 0 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c1 (n + 1) (a % (n + 1)) - (gcd_c2 (n + 1) (a % (n + 1))) * ↑(a / (n + 1)) --Corresponds to t = s' - t'q termination_by b end def prime (n : Nat) : Prop := 2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n def prime_factor (p n : Nat) : Prop := prime p ∧ p ∣ n def all_prime (l : List Nat) : Prop := ∀ p ∈ l, prime p def nondec (l : List Nat) : Prop := match l with | [] => True --Of course, True is a proposition that is always true | n :: L => (∀ m ∈ L, n ≤ m) ∧ nondec L def nondec_prime_list (l : List Nat) : Prop := all_prime l ∧ nondec l def prod (l : List Nat) : Nat := match l with | [] => 1 | n :: L => n * (prod L) def prime_factorization (n : Nat) (l : List Nat) : Prop := nondec_prime_list l ∧ prod l = n def rel_prime (a b : Nat) : Prop := gcd a b = 1 def congr_mod (m : Nat) (a b : Int) : Prop := (↑m : Int) ∣ (a - b) def cc (m : Nat) (a : Int) : ZMod m := (↑a : ZMod m) notation:50 a " ≡ " b " (MOD " m ")" => congr_mod m a b notation:max "["a"]_"m:max => cc m a def invertible {m : Nat} (X : ZMod m) : Prop := ∃ (Y : ZMod m), X * Y = [1]_m def num_rp_below (m k : Nat) : Nat := match k with | 0 => 0 | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j def phi (m : Nat) : Nat := num_rp_below m m def prod_seq {m : Nat} (j k : Nat) (f : Nat → ZMod m) : ZMod m := match j with | 0 => [1]_m | n + 1 => prod_seq n k f * f (k + n) def maps_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i < n, g i < n def one_one_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i1 < n, ∀ i2 < n, g i1 = g i2 → i1 = i2 def onto_below (n : Nat) (g : Nat → Nat) : Prop := ∀ k < n, ∃ i < n, g i = k def perm_below (n : Nat) (g : Nat → Nat) : Prop := maps_below n g ∧ one_one_below n g ∧ onto_below n g def inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m) def swap (u v i : Nat) : Nat := if i = u then v else if i = v then u else i namespace Euler --For definitions specific to Euler's theorem def F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m def G (m a i : Nat) : Nat := (a * i) % m def Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i end Euler /- Section 7.1 -/ theorem dvd_mod_of_dvd_a_b {a b d : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ (a % b) := by set q : Nat := a / b have h3 : b * q + a % b = a := Nat.div_add_mod a b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 define --Goal : ∃ (c : Nat), a % b = d * c apply Exists.intro (j - k * q) show a % b = d * (j - k * q) from calc a % b _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm _ = a - b * q := by rw [h3] _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc] _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm done theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry #eval gcd 672 161 --Answer: 7 lemma gcd_base (a : Nat) : gcd a 0 = a := by rfl lemma gcd_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd a b = gcd b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1)) rfl done lemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b ≠ 0) : a % b < b := by have h1 : b > 0 := Nat.pos_of_ne_zero h show a % b < b from Nat.mod_lt a h1 done lemma dvd_self (n : Nat) : n ∣ n := by apply Exists.intro 1 ring done theorem gcd_dvd : ∀ (b a : Nat), (gcd a b) ∣ a ∧ (gcd a b) ∣ b := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd a b_1) ∣ a ∧ (gcd a b_1) ∣ b_1 fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_base] --Goal: a ∣ a ∧ a ∣ 0 apply And.intro (dvd_self a) define apply Exists.intro 0 rfl done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_nonzero a h1] --Goal : gcd b (a % b) ∣ a ∧ gcd b (a % b) ∣ b have h2 : a % b < b := mod_nonzero_lt a h1 have h3 : (gcd b (a % b)) ∣ b ∧ (gcd b (a % b)) ∣ (a % b) := ih (a % b) h2 b apply And.intro _ h3.left show (gcd b (a % b)) ∣ a from dvd_a_of_dvd_b_mod h3.left h3.right done done theorem gcd_dvd_left (a b : Nat) : (gcd a b) ∣ a := (gcd_dvd b a).left theorem gcd_dvd_right (a b : Nat) : (gcd a b) ∣ b := (gcd_dvd b a).right lemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl lemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c1 a b = gcd_c2 b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done lemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl lemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * ↑(a / b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done theorem gcd_lin_comb : ∀ (b a : Nat), (gcd_c1 a b) * ↑a + (gcd_c2 a b) * ↑b = ↑(gcd a b) := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd_c1 a b_1) * ↑a + (gcd_c2 a b_1) * ↑b_1 = ↑(gcd a b_1) fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base] --Goal : 1 * ↑a + 0 * ↑0 = ↑a ring done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1] --Goal : gcd_c2 b (a % b) * ↑a + -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * ↑(a / b)) * ↑b = -- ↑(gcd b (a % b)) set r : Nat := a % b set q : Nat := a / b set s : Int := gcd_c1 b r set t : Int := gcd_c2 b r --Goal : t * ↑a + (s - t * ↑q) * ↑b = ↑(gcd b r) have h2 : r < b := mod_nonzero_lt a h1 have h3 : s * ↑b + t * ↑r = ↑(gcd b r) := ih r h2 b have h4 : b * q + r = a := Nat.div_add_mod a b rewrite [←h3, ←h4] rewrite [Nat.cast_add, Nat.cast_mul] --Goal : t * (↑b * ↑q + ↑r) + (s - t * ↑q) * ↑b = s * ↑b + t * ↑r ring done done #eval gcd_c1 672 161 --Answer: 6 #eval gcd_c2 672 161 --Answer: -25 --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161 theorem Theorem_7_1_6 {d a b : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ gcd a b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑d ∣ ↑(gcd a b) set s : Int := gcd_c1 a b set t : Int := gcd_c2 a b have h3 : s * ↑a + t * ↑b = ↑(gcd a b) := gcd_lin_comb b a rewrite [←h3] --Goal : ↑d ∣ s * ↑a + t * ↑b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul] --Goal : ↑d ∣ s * (↑d * ↑j) + t * (↑d * ↑k) define apply Exists.intro (s * ↑j + t * ↑k) ring done /- Section 7.2 -/ theorem dvd_trans {a b c : Nat} (h1 : a ∣ b) (h2 : b ∣ c) : a ∣ c := by define at h1; define at h2; define obtain (m : Nat) (h3 : b = a * m) from h1 obtain (n : Nat) (h4 : c = b * n) from h2 rewrite [h3, mul_assoc] at h4 apply Exists.intro (m * n) show c = a * (m * n) from h4 done lemma exists_prime_factor : ∀ (n : Nat), 2 ≤ n → ∃ (p : Nat), prime_factor p n := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, 2 ≤ n_1 → ∃ (p : Nat), prime_factor p n_1 assume h1 : 2 ≤ n by_cases h2 : prime n · -- Case 1. h2 : prime n apply Exists.intro n define --Goal : prime n ∧ n ∣ n show prime n ∧ n ∣ n from And.intro h2 (dvd_self n) done · -- Case 2. h2 : ¬prime n define at h2 --h2 : ¬(2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n) demorgan at h2 disj_syll h2 h1 obtain (a : Nat) (h3 : ∃ (b : Nat), a * b = n ∧ a < n ∧ b < n) from h2 obtain (b : Nat) (h4 : a * b = n ∧ a < n ∧ b < n) from h3 have h5 : 2 ≤ a := by by_contra h6 have h7 : a ≤ 1 := by linarith have h8 : n ≤ b := calc n _ = a * b := h4.left.symm _ ≤ 1 * b := by rel [h7] _ = b := by ring linarith --n ≤ b contradicts b < n done have h6 : ∃ (p : Nat), prime_factor p a := ih a h4.right.left h5 obtain (p : Nat) (h7 : prime_factor p a) from h6 apply Exists.intro p define --Goal : prime p ∧ p ∣ n define at h7 --h7 : prime p ∧ p ∣ a apply And.intro h7.left have h8 : a ∣ n := by apply Exists.intro b show n = a * b from (h4.left).symm done show p ∣ n from dvd_trans h7.right h8 done done lemma exists_least_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q := by set S : Set Nat := {p : Nat | prime_factor p n} have h2 : ∃ (p : Nat), p ∈ S := exists_prime_factor n h show ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q from well_ord_princ S h2 done lemma all_prime_nil : all_prime [] := by define --Goal : ∀ p ∈ [], prime p fix p : Nat contrapos --Goal : ¬prime p → p ∉ [] assume h1 : ¬prime p show p ∉ [] from List.not_mem_nil p done lemma all_prime_cons (n : Nat) (L : List Nat) : all_prime (n :: L) ↔ prime n ∧ all_prime L := by apply Iff.intro · -- (→) assume h1 : all_prime (n :: L) --Goal : prime n ∧ all_prime L define at h1 --h1 : ∀ p ∈ n :: L, prime p apply And.intro (h1 n (List.mem_cons_self n L)) define --Goal : ∀ p ∈ L, prime p fix p : Nat assume h2 : p ∈ L show prime p from h1 p (List.mem_cons_of_mem n h2) done · -- (←) assume h1 : prime n ∧ all_prime L --Goal : all_prime (n :: l) define : all_prime L at h1 define fix p : Nat assume h2 : p ∈ n :: L rewrite [List.mem_cons] at h2 --h2 : p = n ∨ p ∈ L by_cases on h2 · -- Case 1. h2 : p = n rewrite [h2] show prime n from h1.left done · -- Case 2. h2 : p ∈ L show prime p from h1.right p h2 done done done lemma nondec_nil : nondec [] := by define --Goal : True trivial --trivial proves some obviously true statements, such as True done lemma nondec_cons (n : Nat) (L : List Nat) : nondec (n :: L) ↔ (∀ m ∈ L, n ≤ m) ∧ nondec L := by rfl lemma prod_nil : prod [] = 1 := by rfl lemma prod_cons : prod (n :: L) = n * (prod L) := by rfl lemma exists_cons_of_length_eq_succ {A : Type} {l : List A} {n : Nat} (h : l.length = n + 1) : ∃ (a : A) (L : List A), l = a :: L ∧ L.length = n := by have h1 : ¬l.length = 0 := by linarith rewrite [List.length_eq_zero] at h1 obtain (a : A) (h2 : ∃ (L : List A), l = a :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List A) (h3 : l = a :: L) from h2 apply Exists.intro a apply Exists.intro L apply And.intro h3 have h4 : (a :: L).length = L.length + 1 := List.length_cons a L rewrite [←h3, h] at h4 show L.length = n from (Nat.add_right_cancel h4).symm done lemma list_elt_dvd_prod_by_length (a : Nat) : ∀ (n : Nat), ∀ (l : List Nat), l.length = n → a ∈ l → a ∣ prod l := by by_induc · --Base Case fix l : List Nat assume h1 : l.length = 0 rewrite [List.length_eq_zero] at h1 --h1 : l = [] rewrite [h1] --Goal : a ∈ [] → a ∣ prod [] contrapos assume h2 : ¬a ∣ prod [] show a ∉ [] from List.not_mem_nil a done · -- Induction Step fix n : Nat assume ih : ∀ (l : List Nat), List.length l = n → a ∈ l → a ∣ prod l fix l : List Nat assume h1 : l.length = n + 1 --Goal : a ∈ l → a ∣ prod l obtain (b : Nat) (h2 : ∃ (L : List Nat), l = b :: L ∧ L.length = n) from exists_cons_of_length_eq_succ h1 obtain (L : List Nat) (h3 : l = b :: L ∧ L.length = n) from h2 have h4 : a ∈ L → a ∣ prod L := ih L h3.right assume h5 : a ∈ l rewrite [h3.left, prod_cons] --Goal : a ∣ b * prod L rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b ∨ a ∈ L by_cases on h5 · -- Case 1. h5 : a = b apply Exists.intro (prod L) rewrite [h5] rfl done · -- Case 2. h5 : a ∈ L have h6 : a ∣ prod L := h4 h5 have h7 : prod L ∣ b * prod L := by apply Exists.intro b ring done show a ∣ b * prod L from dvd_trans h6 h7 done done done lemma list_elt_dvd_prod {a : Nat} {l : List Nat} (h : a ∈ l) : a ∣ prod l := by set n : Nat := l.length have h1 : l.length = n := by rfl show a ∣ prod l from list_elt_dvd_prod_by_length a n l h1 h done lemma exists_prime_factorization : ∀ (n : Nat), n ≥ 1 → ∃ (l : List Nat), prime_factorization n l := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, n_1 ≥ 1 → ∃ (l : List Nat), prime_factorization n_1 l assume h1 : n ≥ 1 by_cases h2 : n = 1 · -- Case 1. h2 : n = 1 apply Exists.intro [] define apply And.intro · -- Proof of nondec_prime_list [] define show all_prime [] ∧ nondec [] from And.intro all_prime_nil nondec_nil done · -- Proof of prod [] = n rewrite [prod_nil, h2] rfl done done · -- Case 2. h2 : n ≠ 1 have h3 : n ≥ 2 := lt_of_le_of_ne' h1 h2 obtain (p : Nat) (h4 : prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q) from exists_least_prime_factor h3 have p_prime_factor : prime_factor p n := h4.left define at p_prime_factor have p_prime : prime p := p_prime_factor.left have p_dvd_n : p ∣ n := p_prime_factor.right have p_least : ∀ (q : Nat), prime_factor q n → p ≤ q := h4.right obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n have h5 : m ≠ 0 := by contradict h1 with h6 have h7 : n = 0 := calc n _ = p * m := n_eq_pm _ = p * 0 := by rw [h6] _ = 0 := by ring rewrite [h7] decide done have m_pos : 0 < m := Nat.pos_of_ne_zero h5 have m_lt_n : m < n := by define at p_prime show m < n from calc m _ < m + m := by linarith _ = 2 * m := by ring _ ≤ p * m := by rel [p_prime.left] _ = n := n_eq_pm.symm done obtain (L : List Nat) (h6 : prime_factorization m L) from ih m m_lt_n m_pos define at h6 have ndpl_L : nondec_prime_list L := h6.left define at ndpl_L apply Exists.intro (p :: L) define apply And.intro · -- Proof of nondec_prime_list (p :: L) define apply And.intro · -- Proof of all_prime (p :: L) rewrite [all_prime_cons] show prime p ∧ all_prime L from And.intro p_prime ndpl_L.left done · -- Proof of nondec (p :: L) rewrite [nondec_cons] apply And.intro _ ndpl_L.right fix q : Nat assume q_in_L : q ∈ L have h7 : q ∣ prod L := list_elt_dvd_prod q_in_L rewrite [h6.right] at h7 --h7 : q ∣ m have h8 : m ∣ n := by apply Exists.intro p rewrite [n_eq_pm] ring done have q_dvd_n : q ∣ n := dvd_trans h7 h8 have ap_L : all_prime L := ndpl_L.left define at ap_L have q_prime_factor : prime_factor q n := And.intro (ap_L q q_in_L) q_dvd_n show p ≤ q from p_least q q_prime_factor done done · -- Proof of prod (p :: L) = n rewrite [prod_cons, h6.right, n_eq_pm] rfl done done done theorem Theorem_7_2_2 {a b c : Nat} (h1 : c ∣ a * b) (h2 : rel_prime a c) : c ∣ b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑c ∣ ↑b define at h1; define at h2; define obtain (j : Nat) (h3 : a * b = c * j) from h1 set s : Int := gcd_c1 a c set t : Int := gcd_c2 a c have h4 : s * ↑a + t * ↑c = ↑(gcd a c) := gcd_lin_comb c a rewrite [h2, Nat.cast_one] at h4 --h4 : s * ↑a + t * ↑c = (1 : Int) apply Exists.intro (s * ↑j + t * ↑b) show ↑b = ↑c * (s * ↑j + t * ↑b) from calc ↑b _ = (1 : Int) * ↑b := (one_mul _).symm _ = (s * ↑a + t * ↑c) * ↑b := by rw [h4] _ = s * (↑a * ↑b) + t * ↑c * ↑b := by ring _ = s * (↑c * ↑j) + t * ↑c * ↑b := by rw [←Nat.cast_mul a b, h3, Nat.cast_mul c j] _ = ↑c * (s * ↑j + t * ↑b) := by ring done lemma le_nonzero_prod_left {a b : Nat} (h : a * b ≠ 0) : a ≤ a * b := by have h1 : b ≠ 0 := by contradict h with h1 rewrite [h1] ring done have h2 : 1 ≤ b := Nat.pos_of_ne_zero h1 show a ≤ a * b from calc a = a * 1 := (mul_one a).symm _ ≤ a * b := by rel [h2] done lemma le_nonzero_prod_right {a b : Nat} (h : a * b ≠ 0) : b ≤ a * b := by rewrite [mul_comm] rewrite [mul_comm] at h show b ≤ b * a from le_nonzero_prod_left h done lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry lemma rel_prime_of_prime_not_dvd {a p : Nat} (h1 : prime p) (h2 : ¬p ∣ a) : rel_prime a p := by have h3 : gcd a p ∣ a := gcd_dvd_left a p have h4 : gcd a p ∣ p := gcd_dvd_right a p have h5 : gcd a p = 1 ∨ gcd a p = p := dvd_prime h1 h4 have h6 : gcd a p ≠ p := by contradict h2 with h6 rewrite [h6] at h3 show p ∣ a from h3 done disj_syll h5 h6 show rel_prime a p from h5 done theorem Theorem_7_2_3 {a b p : Nat} (h1 : prime p) (h2 : p ∣ a * b) : p ∣ a ∨ p ∣ b := by or_right with h3 have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3 show p ∣ b from Theorem_7_2_2 h2 h4 done lemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a ≥ 1 := by have h1 : a ≠ 0 := by by_contra h1 rewrite [h1] at h contradict h linarith done show a ≥ 1 from Nat.pos_of_ne_zero h1 done lemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by have h1 : a ≥ 1 := ge_one_of_prod_one h have h2 : a * b ≠ 0 := by linarith have h3 : a ≤ a * b := le_nonzero_prod_left h2 rewrite [h] at h3 show a = 1 from Nat.le_antisymm h3 h1 done lemma eq_one_of_dvd_one {n : Nat} (h : n ∣ 1) : n = 1 := by obtain (j : Nat) (h1 : 1 = n * j) from h show n = 1 from eq_one_of_prod_one h1.symm done lemma prime_not_one {p : Nat} (h : prime p) : p ≠ 1 := by define at h linarith done theorem Theorem_7_2_4 {p : Nat} (h1 : prime p) : ∀ (l : List Nat), p ∣ prod l → ∃ a ∈ l, p ∣ a := by apply List.rec · -- Base Case. Goal : p ∣ prod [] → ∃ a ∈ [], p ∣ a rewrite [prod_nil] assume h2 : p ∣ 1 show ∃ a ∈ [], p ∣ a from absurd (eq_one_of_dvd_one h2) (prime_not_one h1) done · -- Induction Step fix b : Nat fix L : List Nat assume ih : p ∣ prod L → ∃ a ∈ L, p ∣ a --Goal : p ∣ prod (b :: L) → ∃ a ∈ b :: L, p ∣ a assume h2 : p ∣ prod (b :: L) rewrite [prod_cons] at h2 have h3 : p ∣ b ∨ p ∣ prod L := Theorem_7_2_3 h1 h2 by_cases on h3 · -- Case 1. h3 : p ∣ b apply Exists.intro b show b ∈ b :: L ∧ p ∣ b from And.intro (List.mem_cons_self b L) h3 done · -- Case 2. h3 : p ∣ prod L obtain (a : Nat) (h4 : a ∈ L ∧ p ∣ a) from ih h3 apply Exists.intro a show a ∈ b :: L ∧ p ∣ a from And.intro (List.mem_cons_of_mem b h4.left) h4.right done done done lemma prime_in_list {p : Nat} {l : List Nat} (h1 : prime p) (h2 : all_prime l) (h3 : p ∣ prod l) : p ∈ l := by obtain (a : Nat) (h4 : a ∈ l ∧ p ∣ a) from Theorem_7_2_4 h1 l h3 define at h2 have h5 : prime a := h2 a h4.left have h6 : p = 1 ∨ p = a := dvd_prime h5 h4.right disj_syll h6 (prime_not_one h1) rewrite [h6] show a ∈ l from h4.left done lemma first_le_first {p q : Nat} {l m : List Nat} (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m)) (h3 : prod (p :: l) = prod (q :: m)) : p ≤ q := by define at h1; define at h2 have h4 : q ∣ prod (p :: l) := by define apply Exists.intro (prod m) rewrite [←prod_cons] show prod (p :: l) = prod (q :: m) from h3 done have h5 : all_prime (q :: m) := h2.left rewrite [all_prime_cons] at h5 have h6 : q ∈ p :: l := prime_in_list h5.left h1.left h4 have h7 : nondec (p :: l) := h1.right rewrite [nondec_cons] at h7 rewrite [List.mem_cons] at h6 by_cases on h6 · -- Case 1. h6 : q = p linarith done · -- Case 2. h6 : q ∈ l have h8 : ∀ m ∈ l, p ≤ m := h7.left show p ≤ q from h8 q h6 done done lemma nondec_prime_list_tail {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by define at h define rewrite [all_prime_cons, nondec_cons] at h show all_prime l ∧ nondec l from And.intro h.left.right h.right.right done lemma cons_prod_not_one {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : prod (p :: l) ≠ 1 := by define at h have h1 : all_prime (p :: l) := h.left rewrite [all_prime_cons] at h1 rewrite [prod_cons] by_contra h2 show False from (prime_not_one h1.left) (eq_one_of_prod_one h2) done lemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) : l = [] ↔ prod l = 1 := by apply Iff.intro · -- (→) assume h1 : l = [] rewrite [h1] show prod [] = 1 from prod_nil done · -- (←) contrapos assume h1 : ¬l = [] obtain (p : Nat) (h2 : ∃ (L : List Nat), l = p :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List Nat) (h3 : l = p :: L) from h2 rewrite [h3] at h rewrite [h3] show ¬prod (p :: L) = 1 from cons_prod_not_one h done done lemma prime_pos {p : Nat} (h : prime p) : p > 0 := by define at h linarith done theorem Theorem_7_2_5 : ∀ (l1 l2 : List Nat), nondec_prime_list l1 → nondec_prime_list l2 → prod l1 = prod l2 → l1 = l2 := by apply List.rec · -- Base Case. Goal : ∀ (l2 : List Nat), nondec_prime_list [] → -- nondec_prime_list l2 → prod [] = prod l2 → [] = l2 fix l2 : List Nat assume h1 : nondec_prime_list [] assume h2 : nondec_prime_list l2 assume h3 : prod [] = prod l2 rewrite [prod_nil, eq_comm, ←list_nil_iff_prod_one h2] at h3 show [] = l2 from h3.symm done · -- Induction Step fix p : Nat fix L1 : List Nat assume ih : ∀ (L2 : List Nat), nondec_prime_list L1 → nondec_prime_list L2 → prod L1 = prod L2 → L1 = L2 -- Goal : ∀ (l2 : List Nat), nondec_prime_list (p :: L1) → -- nondec_prime_list l2 → prod (p :: L1) = prod l2 → p :: L1 = l2 fix l2 : List Nat assume h1 : nondec_prime_list (p :: L1) assume h2 : nondec_prime_list l2 assume h3 : prod (p :: L1) = prod l2 have h4 : ¬prod (p :: L1) = 1 := cons_prod_not_one h1 rewrite [h3, ←list_nil_iff_prod_one h2] at h4 obtain (q : Nat) (h5 : ∃ (L : List Nat), l2 = q :: L) from List.exists_cons_of_ne_nil h4 obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5 rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2) rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2) have h7 : p ≤ q := first_le_first h1 h2 h3 have h8 : q ≤ p := first_le_first h2 h1 h3.symm have h9 : p = q := by linarith rewrite [h9, prod_cons, prod_cons] at h3 --h3 : q * prod L1 = q * prod L2 have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1 have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2 define at h2 have h12 : all_prime (q :: L2) := h2.left rewrite [all_prime_cons] at h12 have h13 : q > 0 := prime_pos h12.left have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3 have h15 : L1 = L2 := ih L2 h10 h11 h14 rewrite [h6, h9, h15] rfl done done theorem fund_thm_arith (n : Nat) (h : n ≥ 1) : ∃! (l : List Nat), prime_factorization n l := by exists_unique · -- Existence show ∃ (l : List Nat), prime_factorization n l from exists_prime_factorization n h done · -- Uniqueness fix l1 : List Nat; fix l2 : List Nat assume h1 : prime_factorization n l1 assume h2 : prime_factorization n l2 define at h1; define at h2 have h3 : prod l1 = n := h1.right rewrite [←h2.right] at h3 show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3 done done /- Section 7.3 -/ theorem congr_refl (m : Nat) : ∀ (a : Int), a ≡ a (MOD m) := by fix a : Int define --Goal : ∃ (c : Int), a - a = ↑m * c apply Exists.intro 0 ring done theorem congr_symm {m : Nat} : ∀ {a b : Int}, a ≡ b (MOD m) → b ≡ a (MOD m) := by fix a : Int; fix b : Int assume h1 : a ≡ b (MOD m) define at h1 --h1 : ∃ (c : Int), a - b = ↑m * c define --Goal : ∃ (c : Int), b - a = ↑m * c obtain (c : Int) (h2 : a - b = m * c) from h1 apply Exists.intro (-c) show b - a = m * (-c) from calc b - a _ = -(a - b) := by ring _ = -(m * c) := by rw [h2] _ = m * (-c) := by ring done theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry /- Fundamental properties of congruence classes -/ lemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) : X = Y ↔ X.val = Y.val := Fin.ext_iff lemma val_nat_eq_mod (n k : Nat) : ([k]_(n + 1)).val = k % (n + 1) := by rfl lemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl theorem cc_rep {m : Nat} (X : ZMod m) : ∃ (a : Int), X = [a]_m := match m with | 0 => by apply Exists.intro X rfl done | n + 1 => by apply Exists.intro ↑(X.val) have h1 : X.val < n + 1 := Fin.prop X rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1] rfl done theorem add_class (m : Nat) (a b : Int) : [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm theorem mul_class (m : Nat) (a b : Int) : [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm lemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ [a - b]_m = [0]_m := by apply Iff.intro · -- (→) assume h1 : [a]_m = [b]_m have h2 : a - b = a + (-b) := by ring have h3 : b + (-b) = 0 := by ring show [a - b]_m = [0]_m from calc [a - b]_m _ = [a + (-b)]_m := by rw [h2] _ = [a]_m + [-b]_m := by rw [add_class] _ = [b]_m + [-b]_m := by rw [h1] _ = [b + -b]_m := by rw [add_class] _ = [0]_m := by rw [h3] done · -- (←) assume h1 : [a - b]_m = [0]_m have h2 : b + (a - b) = a := by ring have h3 : b + 0 = b := by ring show [a]_m = [b]_m from calc [a]_m _ = [b + (a - b)]_m := by rw [h2] _ = [b]_m + [a - b]_m := by rw [add_class] _ = [b]_m + [0]_m := by rw [h1] _ = [b + 0]_m := by rw [add_class] _ = [b]_m := by rw [h3] done done lemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) : [a]_m = [0]_m → [-a]_m = [0]_m := by assume h1 : [a]_m = [0]_m have h2 : 0 + (-a) = -a := by ring have h3 : a + (-a) = 0 := by ring show [-a]_m = [0]_m from calc [-a]_m _ = [0 + (-a)]_m := by rw [h2] _ = [0]_m + [-a]_m := by rw [add_class] _ = [a]_m + [-a]_m := by rw [h1] _ = [a + (-a)]_m := by rw [add_class] _ = [0]_m := by rw [h3] done lemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) : [-a]_m = [0]_m ↔ [a]_m = [0]_m := by apply Iff.intro _ (cc_neg_zero_of_cc_zero m a) assume h1 : [-a]_m = [0]_m have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1 have h3 : -(-a) = a := by ring rewrite [h3] at h2 show [a]_m = [0]_m from h2 done lemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl lemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m ↔ m ∣ k := match m with | 0 => by have h : (0 : Int) = (↑(0 : Nat) : Int) := by rfl rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj] apply Iff.intro · -- (→) assume h1 : k = 0 rewrite [h1] show 0 ∣ 0 from dvd_self 0 done · -- (←) assume h1 : 0 ∣ k obtain (c : Nat) (h2 : k = 0 * c) from h1 rewrite [h2] ring done done | n + 1 => by rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero] show k % (n + 1) = 0 ↔ n + 1 ∣ k from (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm done lemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m ↔ ↑m ∣ a := by obtain (k : Nat) (h1 : a = ↑k ∨ a = -↑k) from Int.eq_nat_or_neg a by_cases on h1 · -- Case 1. h1: a = ↑k rewrite [h1, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done · -- Case 2. h1: a = -↑k rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done done theorem cc_eq_iff_congr (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ a ≡ b (MOD m) := calc [a]_m = [b]_m _ ↔ [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b _ ↔ ↑m ∣ (a - b) := cc_zero_iff_dvd m (a - b) _ ↔ a ≡ b (MOD m) := by rfl /- End of fundamental properties of congruence classes -/ lemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m := by have h1 : (↑m : Int) ≠ 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m) show 0 ≤ a % m from Int.emod_nonneg a h1 done lemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m) have h2 : (↑m : Int) > 0 := (Nat.cast_pos).rtl h1 show a % m < m from Int.emod_lt_of_pos a h2 done lemma congr_mod_mod (m : Nat) (a : Int) : a ≡ a % m (MOD m) := by define have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m apply Exists.intro (a / m) show a - a % m = m * (a / m) from calc a - (a % m) _ = m * (a / m) + a % m - a % m := by rw [h1] _ = m * (a / m) := by ring done lemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) := And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a)) theorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) : ∃! (r : Int), 0 ≤ r ∧ r < m ∧ a ≡ r (MOD m) := by exists_unique · -- Existence apply Exists.intro (a % m) show 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) from mod_cmpl_res m a done · -- Uniqueness fix r1 : Int; fix r2 : Int assume h1 : 0 ≤ r1 ∧ r1 < m ∧ a ≡ r1 (MOD m) assume h2 : 0 ≤ r2 ∧ r2 < m ∧ a ≡ r2 (MOD m) have h3 : r1 ≡ r2 (MOD m) := congr_trans (congr_symm h1.right.right) h2.right.right obtain (d : Int) (h4 : r1 - r2 = m * d) from h3 have h5 : r1 - r2 < m * 1 := by linarith have h6 : m * (-1) < r1 - r2 := by linarith rewrite [h4] at h5 --h5 : m * d < m * 1 rewrite [h4] at h6 --h6 : m * -1 < m * d have h7 : (↑m : Int) ≥ 0 := Nat.cast_nonneg m have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7 have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7 have h10 : d = 0 := by linarith show r1 = r2 from calc r1 _ = r1 - r2 + r2 := by ring _ = m * 0 + r2 := by rw [h4, h10] _ = r2 := by ring done done lemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m := (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a) theorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y rewrite [h1, h2] have h3 : a + b = b + a := by ring show [a]_m + [b]_m = [b]_m + [a]_m from calc [a]_m + [b]_m _ = [a + b]_m := add_class m a b _ = [b + a]_m := by rw [h3] _ = [b]_m + [a]_m := (add_class m b a).symm done theorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X rewrite [h1] have h2 : a * 1 = a := by ring show [a]_m * [1]_m = [a]_m from calc [a]_m * [1]_m _ = [a * 1]_m := mul_class m a 1 _ = [a]_m := by rw [h2] done theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry lemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) : [a]_m * [gcd_c2 m a]_m = [1]_m := by set s : Int := gcd_c1 m a have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m define at h1 rewrite [h1, Nat.cast_one] at h2 --h2 : s * ↑m + gcd_c2 m a * ↑a = 1 rewrite [mul_class, cc_eq_iff_congr] define --Goal : ∃ (c : Int), ↑a * gcd_c2 m a - 1 = ↑m * c apply Exists.intro (-s) show a * (gcd_c2 m a) - 1 = m * (-s) from calc a * (gcd_c2 m a) - 1 _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring _ = 1 + m * (-s) - 1 := by rw [h2] _ = m * (-s) := by ring done theorem Theorem_7_3_7 (m a : Nat) : invertible [a]_m ↔ rel_prime m a := by apply Iff.intro · -- (→) assume h1 : invertible [a]_m define at h1 obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1 obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y rewrite [h3, mul_class, cc_eq_iff_congr] at h2 define at h2 obtain (c : Int) (h4 : a * b - 1 = m * c) from h2 rewrite [Exercise_7_2_6] --Goal : ∃ (s t : Int), s * ↑m + t * ↑a = 1 apply Exists.intro (-c) apply Exists.intro b show (-c) * m + b * a = 1 from calc (-c) * m + b * a _ = (-c) * m + (a * b - 1) + 1 := by ring _ = (-c) * m + m * c + 1 := by rw [h4] _ = 1 := by ring done · -- (←) assume h1 : rel_prime m a define show ∃ (Y : ZMod m), [a]_m * Y = [1]_m from Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1) done done /- Section 7.4 -/ section Euler open Euler lemma num_rp_below_base {m : Nat} : num_rp_below m 0 = 0 := by rfl lemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) : num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by have h1 : num_rp_below m (j + 1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : gcd m j = 1 rewrite [if_pos h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1 show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1 done lemma num_rp_below_step_not_rp {m j : Nat} (h : ¬rel_prime m j) : num_rp_below m (j + 1) = num_rp_below m j := by have h1 : num_rp_below m (j +1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : ¬gcd m j = 1 rewrite [if_neg h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j show num_rp_below m (j + 1) = num_rp_below m j from h1 done lemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl #eval phi 10 --Answer: 4 lemma prod_inv_iff_inv {m : Nat} {X : ZMod m} (h1 : invertible X) (Y : ZMod m) : invertible (X * Y) ↔ invertible Y := by apply Iff.intro · -- (→) assume h2 : invertible (X * Y) obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2 apply Exists.intro (X * Z) rewrite [←h3] --Goal : Y * (X * Z) = X * Y * Z ring --Note that ring can do algebra in ZMod m done · -- (←) assume h2 : invertible Y obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1 obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2 apply Exists.intro (Xi * Yi) show (X * Y) * (Xi * Yi) = [1]_m from calc X * Y * (Xi * Yi) _ = (X * Xi) * (Y * Yi) := by ring _ = [1]_m * [1]_m := by rw [h3, h4] _ = [1]_m := Theorem_7_3_6_7 [1]_m done done lemma F_rp_def {m i : Nat} (h : rel_prime m i) : F m i = [i]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h --h : gcd m i = 1 rewrite [if_pos h] at h1 show F m i = [i]_m from h1 done lemma F_not_rp_def {m i : Nat} (h : ¬rel_prime m i) : F m i = [1]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h rewrite [h1, if_neg h] rfl done lemma prod_seq_base {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 0 k f = [1]_m := by rfl lemma prod_seq_step {m : Nat} (n k : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl lemma prod_seq_zero_step {m : Nat} (n : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by rewrite [prod_seq_step, zero_add] rfl done lemma prod_one {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 1 k f = f k := by rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7] rfl done lemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl lemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m := calc [G m a i]_m _ = [(a * i) % m]_m := by rfl _ = [a * i]_m := (cc_eq_mod m (a * i)).symm _ = [a]_m * [i]_m := (mul_class m a i).symm lemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) : rel_prime m (G m a i) ↔ rel_prime m i := by have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1 show rel_prime m (G m a i) ↔ rel_prime m i from calc rel_prime m (G m a i) _ ↔ invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm _ ↔ invertible ([a]_m * [i]_m) := by rw [cc_G] _ ↔ invertible [i]_m := prod_inv_iff_inv h2 ([i]_m) _ ↔ rel_prime m i := Theorem_7_3_7 m i done lemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) : F m (G m a i) = [a]_m * F m i := by have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2 show F m (G m a i) = [a]_m * F m i from calc F m (G m a i) _ = [G m a i]_m := F_rp_def h3 _ = [a]_m * [i]_m := cc_G m a i _ = [a]_m * F m i := by rw [F_rp_def h2] done lemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : ¬rel_prime m i) : F m (G m a i) = [1]_m := by rewrite [←G_rp_iff h1 i] at h2 show F m (G m a i) = [1]_m from F_not_rp_def h2 done lemma FG_prod {m a : Nat} (h1 : rel_prime m a) : ∀ (k : Nat), prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by by_induc · -- Base Case show prod_seq 0 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from calc prod_seq 0 0 ((F m) ∘ (G m a)) _ = [1]_m := prod_seq_base _ _ _ = [a]_m ^ 0 * [1]_m := by ring _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by rw [num_rp_below_base, prod_seq_base] done · -- Induction Step fix k : Nat assume ih : prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) by_cases h2 : rel_prime m k · -- Case 1. h2 : rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([a]_m * F m k) := by rw [FG_rp h1 h2] _ = [a]_m ^ ((num_rp_below m k) + 1) * ((prod_seq k 0 (F m)) * F m k) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_rp h2, prod_seq_zero_step] done · -- Case 2. h2 : ¬rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([1]_m) := by rw [FG_not_rp h1 h2] _ = [a]_m ^ (num_rp_below m k) * (prod_seq k 0 (F m) * ([1]_m)) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_not_rp h2, prod_seq_zero_step, F_not_rp_def h2] done done done lemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by define --Goal : ∀ i < m, G m a i < m fix i : Nat assume h1 : i < m rewrite [G_def] --Goal : a * i % m < m show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m) done lemma left_inv_one_one_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g' (g i) = i) : one_one_below n g := sorry lemma right_inv_onto_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g (g' i) = i) (h2 : maps_below n g') : onto_below n g := by define at h2; define fix k : Nat assume h3 : k < n apply Exists.intro (g' k) show g' k < n ∧ g (g' k) = k from And.intro (h2 k h3) (h1 k h3) done lemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m * [inv_mod m a]_m = [1]_m := by have h2 : 0 ≤ (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a) show [a]_m * [inv_mod m a]_m = [1]_m from calc [a]_m * [inv_mod m a]_m _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2] _ = [a]_m * [gcd_c2 m a]_m := by rw [←cc_eq_mod] _ = [1]_m := gcd_c2_inv h1 done lemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m := calc a * (b % m) % m = a % m * (b % m % m) % m := Nat.mul_mod _ _ _ _ = a % m * (b % m) % m := by rw [Nat.mod_mod] _ = a * b % m := (Nat.mul_mod _ _ _).symm lemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm] rfl done theorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] : ↑a ≡ ↑b (MOD m) ↔ a % m = b % m := sorry lemma mul_inv_mod_cancel {m a i : Nat} [NeZero m] (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1 rewrite [mul_class, cc_eq_iff_congr, ←Nat.cast_mul, ←Nat.cast_one, congr_iff_mod_eq_Nat] at h3 show a * inv_mod m a * i % m = i from calc a * (inv_mod m a) * i % m _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = 1 % m * i % m := by rw [h3] _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = i % m := by rw [one_mul] _ = i := Nat.mod_eq_of_lt h2 done lemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl lemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, G m a (Ginv m a i) = i := by fix i : Nat assume h2 : i < m show G m a (Ginv m a i) = i from calc G m a (Ginv m a i) _ = a * ((inv_mod m a * i) % m) % m := by rfl _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, Ginv m a (G m a i) = i := by fix i : Nat assume h2 : i < m show Ginv m a (G m a i) = i from calc Ginv m a (G m a i) _ = inv_mod m a * ((a * i) % m) % m := by rfl _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc, mul_comm (inv_mod m a)] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_maps_below (m a : Nat) [NeZero m] : maps_below m (Ginv m a) := G_maps_below m (inv_mod m a) lemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : one_one_below m (G m a) := left_inv_one_one_below (Ginv_left_inv h1) lemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : onto_below m (G m a) := right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a)

lemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : perm_below m (G m a)

HTPI.G_perm_below

htpi/HTPILib/Chap7.lean

htpi

/- Copyright 2023 Daniel J. Velleman -/ import HTPILib.Chap6 namespace HTPI /- Definitions -/ lemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by have h : n + 1 > 0 := Nat.succ_pos n show a % (n + 1) < n + 1 from Nat.mod_lt a h done def gcd (a b : Nat) : Nat := match b with | 0 => a | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd (n + 1) (a % (n + 1)) termination_by b mutual def gcd_c1 (a b : Nat) : Int := match b with | 0 => 1 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c2 (n + 1) (a % (n + 1)) --Corresponds to s = t' termination_by b def gcd_c2 (a b : Nat) : Int := match b with | 0 => 0 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c1 (n + 1) (a % (n + 1)) - (gcd_c2 (n + 1) (a % (n + 1))) * ↑(a / (n + 1)) --Corresponds to t = s' - t'q termination_by b end def prime (n : Nat) : Prop := 2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n def prime_factor (p n : Nat) : Prop := prime p ∧ p ∣ n def all_prime (l : List Nat) : Prop := ∀ p ∈ l, prime p def nondec (l : List Nat) : Prop := match l with | [] => True --Of course, True is a proposition that is always true | n :: L => (∀ m ∈ L, n ≤ m) ∧ nondec L def nondec_prime_list (l : List Nat) : Prop := all_prime l ∧ nondec l def prod (l : List Nat) : Nat := match l with | [] => 1 | n :: L => n * (prod L) def prime_factorization (n : Nat) (l : List Nat) : Prop := nondec_prime_list l ∧ prod l = n def rel_prime (a b : Nat) : Prop := gcd a b = 1 def congr_mod (m : Nat) (a b : Int) : Prop := (↑m : Int) ∣ (a - b) def cc (m : Nat) (a : Int) : ZMod m := (↑a : ZMod m) notation:50 a " ≡ " b " (MOD " m ")" => congr_mod m a b notation:max "["a"]_"m:max => cc m a def invertible {m : Nat} (X : ZMod m) : Prop := ∃ (Y : ZMod m), X * Y = [1]_m def num_rp_below (m k : Nat) : Nat := match k with | 0 => 0 | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j def phi (m : Nat) : Nat := num_rp_below m m def prod_seq {m : Nat} (j k : Nat) (f : Nat → ZMod m) : ZMod m := match j with | 0 => [1]_m | n + 1 => prod_seq n k f * f (k + n) def maps_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i < n, g i < n def one_one_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i1 < n, ∀ i2 < n, g i1 = g i2 → i1 = i2 def onto_below (n : Nat) (g : Nat → Nat) : Prop := ∀ k < n, ∃ i < n, g i = k def perm_below (n : Nat) (g : Nat → Nat) : Prop := maps_below n g ∧ one_one_below n g ∧ onto_below n g def inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m) def swap (u v i : Nat) : Nat := if i = u then v else if i = v then u else i namespace Euler --For definitions specific to Euler's theorem def F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m def G (m a i : Nat) : Nat := (a * i) % m def Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i end Euler /- Section 7.1 -/ theorem dvd_mod_of_dvd_a_b {a b d : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ (a % b) := by set q : Nat := a / b have h3 : b * q + a % b = a := Nat.div_add_mod a b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 define --Goal : ∃ (c : Nat), a % b = d * c apply Exists.intro (j - k * q) show a % b = d * (j - k * q) from calc a % b _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm _ = a - b * q := by rw [h3] _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc] _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm done theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry #eval gcd 672 161 --Answer: 7 lemma gcd_base (a : Nat) : gcd a 0 = a := by rfl lemma gcd_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd a b = gcd b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1)) rfl done lemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b ≠ 0) : a % b < b := by have h1 : b > 0 := Nat.pos_of_ne_zero h show a % b < b from Nat.mod_lt a h1 done lemma dvd_self (n : Nat) : n ∣ n := by apply Exists.intro 1 ring done theorem gcd_dvd : ∀ (b a : Nat), (gcd a b) ∣ a ∧ (gcd a b) ∣ b := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd a b_1) ∣ a ∧ (gcd a b_1) ∣ b_1 fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_base] --Goal: a ∣ a ∧ a ∣ 0 apply And.intro (dvd_self a) define apply Exists.intro 0 rfl done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_nonzero a h1] --Goal : gcd b (a % b) ∣ a ∧ gcd b (a % b) ∣ b have h2 : a % b < b := mod_nonzero_lt a h1 have h3 : (gcd b (a % b)) ∣ b ∧ (gcd b (a % b)) ∣ (a % b) := ih (a % b) h2 b apply And.intro _ h3.left show (gcd b (a % b)) ∣ a from dvd_a_of_dvd_b_mod h3.left h3.right done done theorem gcd_dvd_left (a b : Nat) : (gcd a b) ∣ a := (gcd_dvd b a).left theorem gcd_dvd_right (a b : Nat) : (gcd a b) ∣ b := (gcd_dvd b a).right lemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl lemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c1 a b = gcd_c2 b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done lemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl lemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * ↑(a / b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done theorem gcd_lin_comb : ∀ (b a : Nat), (gcd_c1 a b) * ↑a + (gcd_c2 a b) * ↑b = ↑(gcd a b) := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd_c1 a b_1) * ↑a + (gcd_c2 a b_1) * ↑b_1 = ↑(gcd a b_1) fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base] --Goal : 1 * ↑a + 0 * ↑0 = ↑a ring done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1] --Goal : gcd_c2 b (a % b) * ↑a + -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * ↑(a / b)) * ↑b = -- ↑(gcd b (a % b)) set r : Nat := a % b set q : Nat := a / b set s : Int := gcd_c1 b r set t : Int := gcd_c2 b r --Goal : t * ↑a + (s - t * ↑q) * ↑b = ↑(gcd b r) have h2 : r < b := mod_nonzero_lt a h1 have h3 : s * ↑b + t * ↑r = ↑(gcd b r) := ih r h2 b have h4 : b * q + r = a := Nat.div_add_mod a b rewrite [←h3, ←h4] rewrite [Nat.cast_add, Nat.cast_mul] --Goal : t * (↑b * ↑q + ↑r) + (s - t * ↑q) * ↑b = s * ↑b + t * ↑r ring done done #eval gcd_c1 672 161 --Answer: 6 #eval gcd_c2 672 161 --Answer: -25 --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161 theorem Theorem_7_1_6 {d a b : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ gcd a b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑d ∣ ↑(gcd a b) set s : Int := gcd_c1 a b set t : Int := gcd_c2 a b have h3 : s * ↑a + t * ↑b = ↑(gcd a b) := gcd_lin_comb b a rewrite [←h3] --Goal : ↑d ∣ s * ↑a + t * ↑b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul] --Goal : ↑d ∣ s * (↑d * ↑j) + t * (↑d * ↑k) define apply Exists.intro (s * ↑j + t * ↑k) ring done /- Section 7.2 -/ theorem dvd_trans {a b c : Nat} (h1 : a ∣ b) (h2 : b ∣ c) : a ∣ c := by define at h1; define at h2; define obtain (m : Nat) (h3 : b = a * m) from h1 obtain (n : Nat) (h4 : c = b * n) from h2 rewrite [h3, mul_assoc] at h4 apply Exists.intro (m * n) show c = a * (m * n) from h4 done lemma exists_prime_factor : ∀ (n : Nat), 2 ≤ n → ∃ (p : Nat), prime_factor p n := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, 2 ≤ n_1 → ∃ (p : Nat), prime_factor p n_1 assume h1 : 2 ≤ n by_cases h2 : prime n · -- Case 1. h2 : prime n apply Exists.intro n define --Goal : prime n ∧ n ∣ n show prime n ∧ n ∣ n from And.intro h2 (dvd_self n) done · -- Case 2. h2 : ¬prime n define at h2 --h2 : ¬(2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n) demorgan at h2 disj_syll h2 h1 obtain (a : Nat) (h3 : ∃ (b : Nat), a * b = n ∧ a < n ∧ b < n) from h2 obtain (b : Nat) (h4 : a * b = n ∧ a < n ∧ b < n) from h3 have h5 : 2 ≤ a := by by_contra h6 have h7 : a ≤ 1 := by linarith have h8 : n ≤ b := calc n _ = a * b := h4.left.symm _ ≤ 1 * b := by rel [h7] _ = b := by ring linarith --n ≤ b contradicts b < n done have h6 : ∃ (p : Nat), prime_factor p a := ih a h4.right.left h5 obtain (p : Nat) (h7 : prime_factor p a) from h6 apply Exists.intro p define --Goal : prime p ∧ p ∣ n define at h7 --h7 : prime p ∧ p ∣ a apply And.intro h7.left have h8 : a ∣ n := by apply Exists.intro b show n = a * b from (h4.left).symm done show p ∣ n from dvd_trans h7.right h8 done done lemma exists_least_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q := by set S : Set Nat := {p : Nat | prime_factor p n} have h2 : ∃ (p : Nat), p ∈ S := exists_prime_factor n h show ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q from well_ord_princ S h2 done lemma all_prime_nil : all_prime [] := by define --Goal : ∀ p ∈ [], prime p fix p : Nat contrapos --Goal : ¬prime p → p ∉ [] assume h1 : ¬prime p show p ∉ [] from List.not_mem_nil p done lemma all_prime_cons (n : Nat) (L : List Nat) : all_prime (n :: L) ↔ prime n ∧ all_prime L := by apply Iff.intro · -- (→) assume h1 : all_prime (n :: L) --Goal : prime n ∧ all_prime L define at h1 --h1 : ∀ p ∈ n :: L, prime p apply And.intro (h1 n (List.mem_cons_self n L)) define --Goal : ∀ p ∈ L, prime p fix p : Nat assume h2 : p ∈ L show prime p from h1 p (List.mem_cons_of_mem n h2) done · -- (←) assume h1 : prime n ∧ all_prime L --Goal : all_prime (n :: l) define : all_prime L at h1 define fix p : Nat assume h2 : p ∈ n :: L rewrite [List.mem_cons] at h2 --h2 : p = n ∨ p ∈ L by_cases on h2 · -- Case 1. h2 : p = n rewrite [h2] show prime n from h1.left done · -- Case 2. h2 : p ∈ L show prime p from h1.right p h2 done done done lemma nondec_nil : nondec [] := by define --Goal : True trivial --trivial proves some obviously true statements, such as True done lemma nondec_cons (n : Nat) (L : List Nat) : nondec (n :: L) ↔ (∀ m ∈ L, n ≤ m) ∧ nondec L := by rfl lemma prod_nil : prod [] = 1 := by rfl lemma prod_cons : prod (n :: L) = n * (prod L) := by rfl lemma exists_cons_of_length_eq_succ {A : Type} {l : List A} {n : Nat} (h : l.length = n + 1) : ∃ (a : A) (L : List A), l = a :: L ∧ L.length = n := by have h1 : ¬l.length = 0 := by linarith rewrite [List.length_eq_zero] at h1 obtain (a : A) (h2 : ∃ (L : List A), l = a :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List A) (h3 : l = a :: L) from h2 apply Exists.intro a apply Exists.intro L apply And.intro h3 have h4 : (a :: L).length = L.length + 1 := List.length_cons a L rewrite [←h3, h] at h4 show L.length = n from (Nat.add_right_cancel h4).symm done lemma list_elt_dvd_prod_by_length (a : Nat) : ∀ (n : Nat), ∀ (l : List Nat), l.length = n → a ∈ l → a ∣ prod l := by by_induc · --Base Case fix l : List Nat assume h1 : l.length = 0 rewrite [List.length_eq_zero] at h1 --h1 : l = [] rewrite [h1] --Goal : a ∈ [] → a ∣ prod [] contrapos assume h2 : ¬a ∣ prod [] show a ∉ [] from List.not_mem_nil a done · -- Induction Step fix n : Nat assume ih : ∀ (l : List Nat), List.length l = n → a ∈ l → a ∣ prod l fix l : List Nat assume h1 : l.length = n + 1 --Goal : a ∈ l → a ∣ prod l obtain (b : Nat) (h2 : ∃ (L : List Nat), l = b :: L ∧ L.length = n) from exists_cons_of_length_eq_succ h1 obtain (L : List Nat) (h3 : l = b :: L ∧ L.length = n) from h2 have h4 : a ∈ L → a ∣ prod L := ih L h3.right assume h5 : a ∈ l rewrite [h3.left, prod_cons] --Goal : a ∣ b * prod L rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b ∨ a ∈ L by_cases on h5 · -- Case 1. h5 : a = b apply Exists.intro (prod L) rewrite [h5] rfl done · -- Case 2. h5 : a ∈ L have h6 : a ∣ prod L := h4 h5 have h7 : prod L ∣ b * prod L := by apply Exists.intro b ring done show a ∣ b * prod L from dvd_trans h6 h7 done done done lemma list_elt_dvd_prod {a : Nat} {l : List Nat} (h : a ∈ l) : a ∣ prod l := by set n : Nat := l.length have h1 : l.length = n := by rfl show a ∣ prod l from list_elt_dvd_prod_by_length a n l h1 h done lemma exists_prime_factorization : ∀ (n : Nat), n ≥ 1 → ∃ (l : List Nat), prime_factorization n l := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, n_1 ≥ 1 → ∃ (l : List Nat), prime_factorization n_1 l assume h1 : n ≥ 1 by_cases h2 : n = 1 · -- Case 1. h2 : n = 1 apply Exists.intro [] define apply And.intro · -- Proof of nondec_prime_list [] define show all_prime [] ∧ nondec [] from And.intro all_prime_nil nondec_nil done · -- Proof of prod [] = n rewrite [prod_nil, h2] rfl done done · -- Case 2. h2 : n ≠ 1 have h3 : n ≥ 2 := lt_of_le_of_ne' h1 h2 obtain (p : Nat) (h4 : prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q) from exists_least_prime_factor h3 have p_prime_factor : prime_factor p n := h4.left define at p_prime_factor have p_prime : prime p := p_prime_factor.left have p_dvd_n : p ∣ n := p_prime_factor.right have p_least : ∀ (q : Nat), prime_factor q n → p ≤ q := h4.right obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n have h5 : m ≠ 0 := by contradict h1 with h6 have h7 : n = 0 := calc n _ = p * m := n_eq_pm _ = p * 0 := by rw [h6] _ = 0 := by ring rewrite [h7] decide done have m_pos : 0 < m := Nat.pos_of_ne_zero h5 have m_lt_n : m < n := by define at p_prime show m < n from calc m _ < m + m := by linarith _ = 2 * m := by ring _ ≤ p * m := by rel [p_prime.left] _ = n := n_eq_pm.symm done obtain (L : List Nat) (h6 : prime_factorization m L) from ih m m_lt_n m_pos define at h6 have ndpl_L : nondec_prime_list L := h6.left define at ndpl_L apply Exists.intro (p :: L) define apply And.intro · -- Proof of nondec_prime_list (p :: L) define apply And.intro · -- Proof of all_prime (p :: L) rewrite [all_prime_cons] show prime p ∧ all_prime L from And.intro p_prime ndpl_L.left done · -- Proof of nondec (p :: L) rewrite [nondec_cons] apply And.intro _ ndpl_L.right fix q : Nat assume q_in_L : q ∈ L have h7 : q ∣ prod L := list_elt_dvd_prod q_in_L rewrite [h6.right] at h7 --h7 : q ∣ m have h8 : m ∣ n := by apply Exists.intro p rewrite [n_eq_pm] ring done have q_dvd_n : q ∣ n := dvd_trans h7 h8 have ap_L : all_prime L := ndpl_L.left define at ap_L have q_prime_factor : prime_factor q n := And.intro (ap_L q q_in_L) q_dvd_n show p ≤ q from p_least q q_prime_factor done done · -- Proof of prod (p :: L) = n rewrite [prod_cons, h6.right, n_eq_pm] rfl done done done theorem Theorem_7_2_2 {a b c : Nat} (h1 : c ∣ a * b) (h2 : rel_prime a c) : c ∣ b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑c ∣ ↑b define at h1; define at h2; define obtain (j : Nat) (h3 : a * b = c * j) from h1 set s : Int := gcd_c1 a c set t : Int := gcd_c2 a c have h4 : s * ↑a + t * ↑c = ↑(gcd a c) := gcd_lin_comb c a rewrite [h2, Nat.cast_one] at h4 --h4 : s * ↑a + t * ↑c = (1 : Int) apply Exists.intro (s * ↑j + t * ↑b) show ↑b = ↑c * (s * ↑j + t * ↑b) from calc ↑b _ = (1 : Int) * ↑b := (one_mul _).symm _ = (s * ↑a + t * ↑c) * ↑b := by rw [h4] _ = s * (↑a * ↑b) + t * ↑c * ↑b := by ring _ = s * (↑c * ↑j) + t * ↑c * ↑b := by rw [←Nat.cast_mul a b, h3, Nat.cast_mul c j] _ = ↑c * (s * ↑j + t * ↑b) := by ring done lemma le_nonzero_prod_left {a b : Nat} (h : a * b ≠ 0) : a ≤ a * b := by have h1 : b ≠ 0 := by contradict h with h1 rewrite [h1] ring done have h2 : 1 ≤ b := Nat.pos_of_ne_zero h1 show a ≤ a * b from calc a = a * 1 := (mul_one a).symm _ ≤ a * b := by rel [h2] done lemma le_nonzero_prod_right {a b : Nat} (h : a * b ≠ 0) : b ≤ a * b := by rewrite [mul_comm] rewrite [mul_comm] at h show b ≤ b * a from le_nonzero_prod_left h done lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry lemma rel_prime_of_prime_not_dvd {a p : Nat} (h1 : prime p) (h2 : ¬p ∣ a) : rel_prime a p := by have h3 : gcd a p ∣ a := gcd_dvd_left a p have h4 : gcd a p ∣ p := gcd_dvd_right a p have h5 : gcd a p = 1 ∨ gcd a p = p := dvd_prime h1 h4 have h6 : gcd a p ≠ p := by contradict h2 with h6 rewrite [h6] at h3 show p ∣ a from h3 done disj_syll h5 h6 show rel_prime a p from h5 done theorem Theorem_7_2_3 {a b p : Nat} (h1 : prime p) (h2 : p ∣ a * b) : p ∣ a ∨ p ∣ b := by or_right with h3 have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3 show p ∣ b from Theorem_7_2_2 h2 h4 done lemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a ≥ 1 := by have h1 : a ≠ 0 := by by_contra h1 rewrite [h1] at h contradict h linarith done show a ≥ 1 from Nat.pos_of_ne_zero h1 done lemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by have h1 : a ≥ 1 := ge_one_of_prod_one h have h2 : a * b ≠ 0 := by linarith have h3 : a ≤ a * b := le_nonzero_prod_left h2 rewrite [h] at h3 show a = 1 from Nat.le_antisymm h3 h1 done lemma eq_one_of_dvd_one {n : Nat} (h : n ∣ 1) : n = 1 := by obtain (j : Nat) (h1 : 1 = n * j) from h show n = 1 from eq_one_of_prod_one h1.symm done lemma prime_not_one {p : Nat} (h : prime p) : p ≠ 1 := by define at h linarith done theorem Theorem_7_2_4 {p : Nat} (h1 : prime p) : ∀ (l : List Nat), p ∣ prod l → ∃ a ∈ l, p ∣ a := by apply List.rec · -- Base Case. Goal : p ∣ prod [] → ∃ a ∈ [], p ∣ a rewrite [prod_nil] assume h2 : p ∣ 1 show ∃ a ∈ [], p ∣ a from absurd (eq_one_of_dvd_one h2) (prime_not_one h1) done · -- Induction Step fix b : Nat fix L : List Nat assume ih : p ∣ prod L → ∃ a ∈ L, p ∣ a --Goal : p ∣ prod (b :: L) → ∃ a ∈ b :: L, p ∣ a assume h2 : p ∣ prod (b :: L) rewrite [prod_cons] at h2 have h3 : p ∣ b ∨ p ∣ prod L := Theorem_7_2_3 h1 h2 by_cases on h3 · -- Case 1. h3 : p ∣ b apply Exists.intro b show b ∈ b :: L ∧ p ∣ b from And.intro (List.mem_cons_self b L) h3 done · -- Case 2. h3 : p ∣ prod L obtain (a : Nat) (h4 : a ∈ L ∧ p ∣ a) from ih h3 apply Exists.intro a show a ∈ b :: L ∧ p ∣ a from And.intro (List.mem_cons_of_mem b h4.left) h4.right done done done lemma prime_in_list {p : Nat} {l : List Nat} (h1 : prime p) (h2 : all_prime l) (h3 : p ∣ prod l) : p ∈ l := by obtain (a : Nat) (h4 : a ∈ l ∧ p ∣ a) from Theorem_7_2_4 h1 l h3 define at h2 have h5 : prime a := h2 a h4.left have h6 : p = 1 ∨ p = a := dvd_prime h5 h4.right disj_syll h6 (prime_not_one h1) rewrite [h6] show a ∈ l from h4.left done lemma first_le_first {p q : Nat} {l m : List Nat} (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m)) (h3 : prod (p :: l) = prod (q :: m)) : p ≤ q := by define at h1; define at h2 have h4 : q ∣ prod (p :: l) := by define apply Exists.intro (prod m) rewrite [←prod_cons] show prod (p :: l) = prod (q :: m) from h3 done have h5 : all_prime (q :: m) := h2.left rewrite [all_prime_cons] at h5 have h6 : q ∈ p :: l := prime_in_list h5.left h1.left h4 have h7 : nondec (p :: l) := h1.right rewrite [nondec_cons] at h7 rewrite [List.mem_cons] at h6 by_cases on h6 · -- Case 1. h6 : q = p linarith done · -- Case 2. h6 : q ∈ l have h8 : ∀ m ∈ l, p ≤ m := h7.left show p ≤ q from h8 q h6 done done lemma nondec_prime_list_tail {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by define at h define rewrite [all_prime_cons, nondec_cons] at h show all_prime l ∧ nondec l from And.intro h.left.right h.right.right done lemma cons_prod_not_one {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : prod (p :: l) ≠ 1 := by define at h have h1 : all_prime (p :: l) := h.left rewrite [all_prime_cons] at h1 rewrite [prod_cons] by_contra h2 show False from (prime_not_one h1.left) (eq_one_of_prod_one h2) done lemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) : l = [] ↔ prod l = 1 := by apply Iff.intro · -- (→) assume h1 : l = [] rewrite [h1] show prod [] = 1 from prod_nil done · -- (←) contrapos assume h1 : ¬l = [] obtain (p : Nat) (h2 : ∃ (L : List Nat), l = p :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List Nat) (h3 : l = p :: L) from h2 rewrite [h3] at h rewrite [h3] show ¬prod (p :: L) = 1 from cons_prod_not_one h done done lemma prime_pos {p : Nat} (h : prime p) : p > 0 := by define at h linarith done theorem Theorem_7_2_5 : ∀ (l1 l2 : List Nat), nondec_prime_list l1 → nondec_prime_list l2 → prod l1 = prod l2 → l1 = l2 := by apply List.rec · -- Base Case. Goal : ∀ (l2 : List Nat), nondec_prime_list [] → -- nondec_prime_list l2 → prod [] = prod l2 → [] = l2 fix l2 : List Nat assume h1 : nondec_prime_list [] assume h2 : nondec_prime_list l2 assume h3 : prod [] = prod l2 rewrite [prod_nil, eq_comm, ←list_nil_iff_prod_one h2] at h3 show [] = l2 from h3.symm done · -- Induction Step fix p : Nat fix L1 : List Nat assume ih : ∀ (L2 : List Nat), nondec_prime_list L1 → nondec_prime_list L2 → prod L1 = prod L2 → L1 = L2 -- Goal : ∀ (l2 : List Nat), nondec_prime_list (p :: L1) → -- nondec_prime_list l2 → prod (p :: L1) = prod l2 → p :: L1 = l2 fix l2 : List Nat assume h1 : nondec_prime_list (p :: L1) assume h2 : nondec_prime_list l2 assume h3 : prod (p :: L1) = prod l2 have h4 : ¬prod (p :: L1) = 1 := cons_prod_not_one h1 rewrite [h3, ←list_nil_iff_prod_one h2] at h4 obtain (q : Nat) (h5 : ∃ (L : List Nat), l2 = q :: L) from List.exists_cons_of_ne_nil h4 obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5 rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2) rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2) have h7 : p ≤ q := first_le_first h1 h2 h3 have h8 : q ≤ p := first_le_first h2 h1 h3.symm have h9 : p = q := by linarith rewrite [h9, prod_cons, prod_cons] at h3 --h3 : q * prod L1 = q * prod L2 have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1 have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2 define at h2 have h12 : all_prime (q :: L2) := h2.left rewrite [all_prime_cons] at h12 have h13 : q > 0 := prime_pos h12.left have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3 have h15 : L1 = L2 := ih L2 h10 h11 h14 rewrite [h6, h9, h15] rfl done done theorem fund_thm_arith (n : Nat) (h : n ≥ 1) : ∃! (l : List Nat), prime_factorization n l := by exists_unique · -- Existence show ∃ (l : List Nat), prime_factorization n l from exists_prime_factorization n h done · -- Uniqueness fix l1 : List Nat; fix l2 : List Nat assume h1 : prime_factorization n l1 assume h2 : prime_factorization n l2 define at h1; define at h2 have h3 : prod l1 = n := h1.right rewrite [←h2.right] at h3 show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3 done done /- Section 7.3 -/ theorem congr_refl (m : Nat) : ∀ (a : Int), a ≡ a (MOD m) := by fix a : Int define --Goal : ∃ (c : Int), a - a = ↑m * c apply Exists.intro 0 ring done theorem congr_symm {m : Nat} : ∀ {a b : Int}, a ≡ b (MOD m) → b ≡ a (MOD m) := by fix a : Int; fix b : Int assume h1 : a ≡ b (MOD m) define at h1 --h1 : ∃ (c : Int), a - b = ↑m * c define --Goal : ∃ (c : Int), b - a = ↑m * c obtain (c : Int) (h2 : a - b = m * c) from h1 apply Exists.intro (-c) show b - a = m * (-c) from calc b - a _ = -(a - b) := by ring _ = -(m * c) := by rw [h2] _ = m * (-c) := by ring done theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry /- Fundamental properties of congruence classes -/ lemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) : X = Y ↔ X.val = Y.val := Fin.ext_iff lemma val_nat_eq_mod (n k : Nat) : ([k]_(n + 1)).val = k % (n + 1) := by rfl lemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl theorem cc_rep {m : Nat} (X : ZMod m) : ∃ (a : Int), X = [a]_m := match m with | 0 => by apply Exists.intro X rfl done | n + 1 => by apply Exists.intro ↑(X.val) have h1 : X.val < n + 1 := Fin.prop X rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1] rfl done theorem add_class (m : Nat) (a b : Int) : [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm theorem mul_class (m : Nat) (a b : Int) : [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm lemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ [a - b]_m = [0]_m := by apply Iff.intro · -- (→) assume h1 : [a]_m = [b]_m have h2 : a - b = a + (-b) := by ring have h3 : b + (-b) = 0 := by ring show [a - b]_m = [0]_m from calc [a - b]_m _ = [a + (-b)]_m := by rw [h2] _ = [a]_m + [-b]_m := by rw [add_class] _ = [b]_m + [-b]_m := by rw [h1] _ = [b + -b]_m := by rw [add_class] _ = [0]_m := by rw [h3] done · -- (←) assume h1 : [a - b]_m = [0]_m have h2 : b + (a - b) = a := by ring have h3 : b + 0 = b := by ring show [a]_m = [b]_m from calc [a]_m _ = [b + (a - b)]_m := by rw [h2] _ = [b]_m + [a - b]_m := by rw [add_class] _ = [b]_m + [0]_m := by rw [h1] _ = [b + 0]_m := by rw [add_class] _ = [b]_m := by rw [h3] done done lemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) : [a]_m = [0]_m → [-a]_m = [0]_m := by assume h1 : [a]_m = [0]_m have h2 : 0 + (-a) = -a := by ring have h3 : a + (-a) = 0 := by ring show [-a]_m = [0]_m from calc [-a]_m _ = [0 + (-a)]_m := by rw [h2] _ = [0]_m + [-a]_m := by rw [add_class] _ = [a]_m + [-a]_m := by rw [h1] _ = [a + (-a)]_m := by rw [add_class] _ = [0]_m := by rw [h3] done lemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) : [-a]_m = [0]_m ↔ [a]_m = [0]_m := by apply Iff.intro _ (cc_neg_zero_of_cc_zero m a) assume h1 : [-a]_m = [0]_m have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1 have h3 : -(-a) = a := by ring rewrite [h3] at h2 show [a]_m = [0]_m from h2 done lemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl lemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m ↔ m ∣ k := match m with | 0 => by have h : (0 : Int) = (↑(0 : Nat) : Int) := by rfl rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj] apply Iff.intro · -- (→) assume h1 : k = 0 rewrite [h1] show 0 ∣ 0 from dvd_self 0 done · -- (←) assume h1 : 0 ∣ k obtain (c : Nat) (h2 : k = 0 * c) from h1 rewrite [h2] ring done done | n + 1 => by rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero] show k % (n + 1) = 0 ↔ n + 1 ∣ k from (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm done lemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m ↔ ↑m ∣ a := by obtain (k : Nat) (h1 : a = ↑k ∨ a = -↑k) from Int.eq_nat_or_neg a by_cases on h1 · -- Case 1. h1: a = ↑k rewrite [h1, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done · -- Case 2. h1: a = -↑k rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done done theorem cc_eq_iff_congr (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ a ≡ b (MOD m) := calc [a]_m = [b]_m _ ↔ [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b _ ↔ ↑m ∣ (a - b) := cc_zero_iff_dvd m (a - b) _ ↔ a ≡ b (MOD m) := by rfl /- End of fundamental properties of congruence classes -/ lemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m := by have h1 : (↑m : Int) ≠ 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m) show 0 ≤ a % m from Int.emod_nonneg a h1 done lemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m) have h2 : (↑m : Int) > 0 := (Nat.cast_pos).rtl h1 show a % m < m from Int.emod_lt_of_pos a h2 done lemma congr_mod_mod (m : Nat) (a : Int) : a ≡ a % m (MOD m) := by define have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m apply Exists.intro (a / m) show a - a % m = m * (a / m) from calc a - (a % m) _ = m * (a / m) + a % m - a % m := by rw [h1] _ = m * (a / m) := by ring done lemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) := And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a)) theorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) : ∃! (r : Int), 0 ≤ r ∧ r < m ∧ a ≡ r (MOD m) := by exists_unique · -- Existence apply Exists.intro (a % m) show 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) from mod_cmpl_res m a done · -- Uniqueness fix r1 : Int; fix r2 : Int assume h1 : 0 ≤ r1 ∧ r1 < m ∧ a ≡ r1 (MOD m) assume h2 : 0 ≤ r2 ∧ r2 < m ∧ a ≡ r2 (MOD m) have h3 : r1 ≡ r2 (MOD m) := congr_trans (congr_symm h1.right.right) h2.right.right obtain (d : Int) (h4 : r1 - r2 = m * d) from h3 have h5 : r1 - r2 < m * 1 := by linarith have h6 : m * (-1) < r1 - r2 := by linarith rewrite [h4] at h5 --h5 : m * d < m * 1 rewrite [h4] at h6 --h6 : m * -1 < m * d have h7 : (↑m : Int) ≥ 0 := Nat.cast_nonneg m have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7 have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7 have h10 : d = 0 := by linarith show r1 = r2 from calc r1 _ = r1 - r2 + r2 := by ring _ = m * 0 + r2 := by rw [h4, h10] _ = r2 := by ring done done lemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m := (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a) theorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y rewrite [h1, h2] have h3 : a + b = b + a := by ring show [a]_m + [b]_m = [b]_m + [a]_m from calc [a]_m + [b]_m _ = [a + b]_m := add_class m a b _ = [b + a]_m := by rw [h3] _ = [b]_m + [a]_m := (add_class m b a).symm done theorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X rewrite [h1] have h2 : a * 1 = a := by ring show [a]_m * [1]_m = [a]_m from calc [a]_m * [1]_m _ = [a * 1]_m := mul_class m a 1 _ = [a]_m := by rw [h2] done theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry lemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) : [a]_m * [gcd_c2 m a]_m = [1]_m := by set s : Int := gcd_c1 m a have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m define at h1 rewrite [h1, Nat.cast_one] at h2 --h2 : s * ↑m + gcd_c2 m a * ↑a = 1 rewrite [mul_class, cc_eq_iff_congr] define --Goal : ∃ (c : Int), ↑a * gcd_c2 m a - 1 = ↑m * c apply Exists.intro (-s) show a * (gcd_c2 m a) - 1 = m * (-s) from calc a * (gcd_c2 m a) - 1 _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring _ = 1 + m * (-s) - 1 := by rw [h2] _ = m * (-s) := by ring done theorem Theorem_7_3_7 (m a : Nat) : invertible [a]_m ↔ rel_prime m a := by apply Iff.intro · -- (→) assume h1 : invertible [a]_m define at h1 obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1 obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y rewrite [h3, mul_class, cc_eq_iff_congr] at h2 define at h2 obtain (c : Int) (h4 : a * b - 1 = m * c) from h2 rewrite [Exercise_7_2_6] --Goal : ∃ (s t : Int), s * ↑m + t * ↑a = 1 apply Exists.intro (-c) apply Exists.intro b show (-c) * m + b * a = 1 from calc (-c) * m + b * a _ = (-c) * m + (a * b - 1) + 1 := by ring _ = (-c) * m + m * c + 1 := by rw [h4] _ = 1 := by ring done · -- (←) assume h1 : rel_prime m a define show ∃ (Y : ZMod m), [a]_m * Y = [1]_m from Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1) done done /- Section 7.4 -/ section Euler open Euler lemma num_rp_below_base {m : Nat} : num_rp_below m 0 = 0 := by rfl lemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) : num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by have h1 : num_rp_below m (j + 1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : gcd m j = 1 rewrite [if_pos h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1 show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1 done lemma num_rp_below_step_not_rp {m j : Nat} (h : ¬rel_prime m j) : num_rp_below m (j + 1) = num_rp_below m j := by have h1 : num_rp_below m (j +1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : ¬gcd m j = 1 rewrite [if_neg h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j show num_rp_below m (j + 1) = num_rp_below m j from h1 done lemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl #eval phi 10 --Answer: 4 lemma prod_inv_iff_inv {m : Nat} {X : ZMod m} (h1 : invertible X) (Y : ZMod m) : invertible (X * Y) ↔ invertible Y := by apply Iff.intro · -- (→) assume h2 : invertible (X * Y) obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2 apply Exists.intro (X * Z) rewrite [←h3] --Goal : Y * (X * Z) = X * Y * Z ring --Note that ring can do algebra in ZMod m done · -- (←) assume h2 : invertible Y obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1 obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2 apply Exists.intro (Xi * Yi) show (X * Y) * (Xi * Yi) = [1]_m from calc X * Y * (Xi * Yi) _ = (X * Xi) * (Y * Yi) := by ring _ = [1]_m * [1]_m := by rw [h3, h4] _ = [1]_m := Theorem_7_3_6_7 [1]_m done done lemma F_rp_def {m i : Nat} (h : rel_prime m i) : F m i = [i]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h --h : gcd m i = 1 rewrite [if_pos h] at h1 show F m i = [i]_m from h1 done lemma F_not_rp_def {m i : Nat} (h : ¬rel_prime m i) : F m i = [1]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h rewrite [h1, if_neg h] rfl done lemma prod_seq_base {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 0 k f = [1]_m := by rfl lemma prod_seq_step {m : Nat} (n k : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl lemma prod_seq_zero_step {m : Nat} (n : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by rewrite [prod_seq_step, zero_add] rfl done lemma prod_one {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 1 k f = f k := by rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7] rfl done lemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl lemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m := calc [G m a i]_m _ = [(a * i) % m]_m := by rfl _ = [a * i]_m := (cc_eq_mod m (a * i)).symm _ = [a]_m * [i]_m := (mul_class m a i).symm lemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) : rel_prime m (G m a i) ↔ rel_prime m i := by have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1 show rel_prime m (G m a i) ↔ rel_prime m i from calc rel_prime m (G m a i) _ ↔ invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm _ ↔ invertible ([a]_m * [i]_m) := by rw [cc_G] _ ↔ invertible [i]_m := prod_inv_iff_inv h2 ([i]_m) _ ↔ rel_prime m i := Theorem_7_3_7 m i done lemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) : F m (G m a i) = [a]_m * F m i := by have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2 show F m (G m a i) = [a]_m * F m i from calc F m (G m a i) _ = [G m a i]_m := F_rp_def h3 _ = [a]_m * [i]_m := cc_G m a i _ = [a]_m * F m i := by rw [F_rp_def h2] done lemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : ¬rel_prime m i) : F m (G m a i) = [1]_m := by rewrite [←G_rp_iff h1 i] at h2 show F m (G m a i) = [1]_m from F_not_rp_def h2 done lemma FG_prod {m a : Nat} (h1 : rel_prime m a) : ∀ (k : Nat), prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by by_induc · -- Base Case show prod_seq 0 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from calc prod_seq 0 0 ((F m) ∘ (G m a)) _ = [1]_m := prod_seq_base _ _ _ = [a]_m ^ 0 * [1]_m := by ring _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by rw [num_rp_below_base, prod_seq_base] done · -- Induction Step fix k : Nat assume ih : prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) by_cases h2 : rel_prime m k · -- Case 1. h2 : rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([a]_m * F m k) := by rw [FG_rp h1 h2] _ = [a]_m ^ ((num_rp_below m k) + 1) * ((prod_seq k 0 (F m)) * F m k) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_rp h2, prod_seq_zero_step] done · -- Case 2. h2 : ¬rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([1]_m) := by rw [FG_not_rp h1 h2] _ = [a]_m ^ (num_rp_below m k) * (prod_seq k 0 (F m) * ([1]_m)) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_not_rp h2, prod_seq_zero_step, F_not_rp_def h2] done done done lemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by define --Goal : ∀ i < m, G m a i < m fix i : Nat assume h1 : i < m rewrite [G_def] --Goal : a * i % m < m show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m) done lemma left_inv_one_one_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g' (g i) = i) : one_one_below n g := sorry lemma right_inv_onto_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g (g' i) = i) (h2 : maps_below n g') : onto_below n g := by define at h2; define fix k : Nat assume h3 : k < n apply Exists.intro (g' k) show g' k < n ∧ g (g' k) = k from And.intro (h2 k h3) (h1 k h3) done lemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m * [inv_mod m a]_m = [1]_m := by have h2 : 0 ≤ (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a) show [a]_m * [inv_mod m a]_m = [1]_m from calc [a]_m * [inv_mod m a]_m _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2] _ = [a]_m * [gcd_c2 m a]_m := by rw [←cc_eq_mod] _ = [1]_m := gcd_c2_inv h1 done lemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m := calc a * (b % m) % m = a % m * (b % m % m) % m := Nat.mul_mod _ _ _ _ = a % m * (b % m) % m := by rw [Nat.mod_mod] _ = a * b % m := (Nat.mul_mod _ _ _).symm lemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm] rfl done theorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] : ↑a ≡ ↑b (MOD m) ↔ a % m = b % m := sorry lemma mul_inv_mod_cancel {m a i : Nat} [NeZero m] (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1 rewrite [mul_class, cc_eq_iff_congr, ←Nat.cast_mul, ←Nat.cast_one, congr_iff_mod_eq_Nat] at h3 show a * inv_mod m a * i % m = i from calc a * (inv_mod m a) * i % m _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = 1 % m * i % m := by rw [h3] _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = i % m := by rw [one_mul] _ = i := Nat.mod_eq_of_lt h2 done lemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl lemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, G m a (Ginv m a i) = i := by fix i : Nat assume h2 : i < m show G m a (Ginv m a i) = i from calc G m a (Ginv m a i) _ = a * ((inv_mod m a * i) % m) % m := by rfl _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, Ginv m a (G m a i) = i := by fix i : Nat assume h2 : i < m show Ginv m a (G m a i) = i from calc Ginv m a (G m a i) _ = inv_mod m a * ((a * i) % m) % m := by rfl _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc, mul_comm (inv_mod m a)] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_maps_below (m a : Nat) [NeZero m] : maps_below m (Ginv m a) := G_maps_below m (inv_mod m a) lemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : one_one_below m (G m a) := left_inv_one_one_below (Ginv_left_inv h1) lemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : onto_below m (G m a) := right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a) lemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : perm_below m (G m a) := And.intro (G_maps_below m a) (And.intro (G_one_one_below h1) (G_onto_below h1)) --Permuting a product of congruence classes doesn't change product

lemma swap_fst (u v : Nat) : swap u v u = v

HTPI.swap_fst

htpi/HTPILib/Chap7.lean

htpi

/- Copyright 2023 Daniel J. Velleman -/ import HTPILib.Chap6 namespace HTPI /- Definitions -/ lemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by have h : n + 1 > 0 := Nat.succ_pos n show a % (n + 1) < n + 1 from Nat.mod_lt a h done def gcd (a b : Nat) : Nat := match b with | 0 => a | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd (n + 1) (a % (n + 1)) termination_by b mutual def gcd_c1 (a b : Nat) : Int := match b with | 0 => 1 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c2 (n + 1) (a % (n + 1)) --Corresponds to s = t' termination_by b def gcd_c2 (a b : Nat) : Int := match b with | 0 => 0 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c1 (n + 1) (a % (n + 1)) - (gcd_c2 (n + 1) (a % (n + 1))) * ↑(a / (n + 1)) --Corresponds to t = s' - t'q termination_by b end def prime (n : Nat) : Prop := 2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n def prime_factor (p n : Nat) : Prop := prime p ∧ p ∣ n def all_prime (l : List Nat) : Prop := ∀ p ∈ l, prime p def nondec (l : List Nat) : Prop := match l with | [] => True --Of course, True is a proposition that is always true | n :: L => (∀ m ∈ L, n ≤ m) ∧ nondec L def nondec_prime_list (l : List Nat) : Prop := all_prime l ∧ nondec l def prod (l : List Nat) : Nat := match l with | [] => 1 | n :: L => n * (prod L) def prime_factorization (n : Nat) (l : List Nat) : Prop := nondec_prime_list l ∧ prod l = n def rel_prime (a b : Nat) : Prop := gcd a b = 1 def congr_mod (m : Nat) (a b : Int) : Prop := (↑m : Int) ∣ (a - b) def cc (m : Nat) (a : Int) : ZMod m := (↑a : ZMod m) notation:50 a " ≡ " b " (MOD " m ")" => congr_mod m a b notation:max "["a"]_"m:max => cc m a def invertible {m : Nat} (X : ZMod m) : Prop := ∃ (Y : ZMod m), X * Y = [1]_m def num_rp_below (m k : Nat) : Nat := match k with | 0 => 0 | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j def phi (m : Nat) : Nat := num_rp_below m m def prod_seq {m : Nat} (j k : Nat) (f : Nat → ZMod m) : ZMod m := match j with | 0 => [1]_m | n + 1 => prod_seq n k f * f (k + n) def maps_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i < n, g i < n def one_one_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i1 < n, ∀ i2 < n, g i1 = g i2 → i1 = i2 def onto_below (n : Nat) (g : Nat → Nat) : Prop := ∀ k < n, ∃ i < n, g i = k def perm_below (n : Nat) (g : Nat → Nat) : Prop := maps_below n g ∧ one_one_below n g ∧ onto_below n g def inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m) def swap (u v i : Nat) : Nat := if i = u then v else if i = v then u else i namespace Euler --For definitions specific to Euler's theorem def F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m def G (m a i : Nat) : Nat := (a * i) % m def Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i end Euler /- Section 7.1 -/ theorem dvd_mod_of_dvd_a_b {a b d : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ (a % b) := by set q : Nat := a / b have h3 : b * q + a % b = a := Nat.div_add_mod a b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 define --Goal : ∃ (c : Nat), a % b = d * c apply Exists.intro (j - k * q) show a % b = d * (j - k * q) from calc a % b _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm _ = a - b * q := by rw [h3] _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc] _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm done theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry #eval gcd 672 161 --Answer: 7 lemma gcd_base (a : Nat) : gcd a 0 = a := by rfl lemma gcd_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd a b = gcd b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1)) rfl done lemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b ≠ 0) : a % b < b := by have h1 : b > 0 := Nat.pos_of_ne_zero h show a % b < b from Nat.mod_lt a h1 done lemma dvd_self (n : Nat) : n ∣ n := by apply Exists.intro 1 ring done theorem gcd_dvd : ∀ (b a : Nat), (gcd a b) ∣ a ∧ (gcd a b) ∣ b := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd a b_1) ∣ a ∧ (gcd a b_1) ∣ b_1 fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_base] --Goal: a ∣ a ∧ a ∣ 0 apply And.intro (dvd_self a) define apply Exists.intro 0 rfl done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_nonzero a h1] --Goal : gcd b (a % b) ∣ a ∧ gcd b (a % b) ∣ b have h2 : a % b < b := mod_nonzero_lt a h1 have h3 : (gcd b (a % b)) ∣ b ∧ (gcd b (a % b)) ∣ (a % b) := ih (a % b) h2 b apply And.intro _ h3.left show (gcd b (a % b)) ∣ a from dvd_a_of_dvd_b_mod h3.left h3.right done done theorem gcd_dvd_left (a b : Nat) : (gcd a b) ∣ a := (gcd_dvd b a).left theorem gcd_dvd_right (a b : Nat) : (gcd a b) ∣ b := (gcd_dvd b a).right lemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl lemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c1 a b = gcd_c2 b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done lemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl lemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * ↑(a / b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done theorem gcd_lin_comb : ∀ (b a : Nat), (gcd_c1 a b) * ↑a + (gcd_c2 a b) * ↑b = ↑(gcd a b) := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd_c1 a b_1) * ↑a + (gcd_c2 a b_1) * ↑b_1 = ↑(gcd a b_1) fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base] --Goal : 1 * ↑a + 0 * ↑0 = ↑a ring done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1] --Goal : gcd_c2 b (a % b) * ↑a + -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * ↑(a / b)) * ↑b = -- ↑(gcd b (a % b)) set r : Nat := a % b set q : Nat := a / b set s : Int := gcd_c1 b r set t : Int := gcd_c2 b r --Goal : t * ↑a + (s - t * ↑q) * ↑b = ↑(gcd b r) have h2 : r < b := mod_nonzero_lt a h1 have h3 : s * ↑b + t * ↑r = ↑(gcd b r) := ih r h2 b have h4 : b * q + r = a := Nat.div_add_mod a b rewrite [←h3, ←h4] rewrite [Nat.cast_add, Nat.cast_mul] --Goal : t * (↑b * ↑q + ↑r) + (s - t * ↑q) * ↑b = s * ↑b + t * ↑r ring done done #eval gcd_c1 672 161 --Answer: 6 #eval gcd_c2 672 161 --Answer: -25 --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161 theorem Theorem_7_1_6 {d a b : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ gcd a b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑d ∣ ↑(gcd a b) set s : Int := gcd_c1 a b set t : Int := gcd_c2 a b have h3 : s * ↑a + t * ↑b = ↑(gcd a b) := gcd_lin_comb b a rewrite [←h3] --Goal : ↑d ∣ s * ↑a + t * ↑b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul] --Goal : ↑d ∣ s * (↑d * ↑j) + t * (↑d * ↑k) define apply Exists.intro (s * ↑j + t * ↑k) ring done /- Section 7.2 -/ theorem dvd_trans {a b c : Nat} (h1 : a ∣ b) (h2 : b ∣ c) : a ∣ c := by define at h1; define at h2; define obtain (m : Nat) (h3 : b = a * m) from h1 obtain (n : Nat) (h4 : c = b * n) from h2 rewrite [h3, mul_assoc] at h4 apply Exists.intro (m * n) show c = a * (m * n) from h4 done lemma exists_prime_factor : ∀ (n : Nat), 2 ≤ n → ∃ (p : Nat), prime_factor p n := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, 2 ≤ n_1 → ∃ (p : Nat), prime_factor p n_1 assume h1 : 2 ≤ n by_cases h2 : prime n · -- Case 1. h2 : prime n apply Exists.intro n define --Goal : prime n ∧ n ∣ n show prime n ∧ n ∣ n from And.intro h2 (dvd_self n) done · -- Case 2. h2 : ¬prime n define at h2 --h2 : ¬(2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n) demorgan at h2 disj_syll h2 h1 obtain (a : Nat) (h3 : ∃ (b : Nat), a * b = n ∧ a < n ∧ b < n) from h2 obtain (b : Nat) (h4 : a * b = n ∧ a < n ∧ b < n) from h3 have h5 : 2 ≤ a := by by_contra h6 have h7 : a ≤ 1 := by linarith have h8 : n ≤ b := calc n _ = a * b := h4.left.symm _ ≤ 1 * b := by rel [h7] _ = b := by ring linarith --n ≤ b contradicts b < n done have h6 : ∃ (p : Nat), prime_factor p a := ih a h4.right.left h5 obtain (p : Nat) (h7 : prime_factor p a) from h6 apply Exists.intro p define --Goal : prime p ∧ p ∣ n define at h7 --h7 : prime p ∧ p ∣ a apply And.intro h7.left have h8 : a ∣ n := by apply Exists.intro b show n = a * b from (h4.left).symm done show p ∣ n from dvd_trans h7.right h8 done done lemma exists_least_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q := by set S : Set Nat := {p : Nat | prime_factor p n} have h2 : ∃ (p : Nat), p ∈ S := exists_prime_factor n h show ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q from well_ord_princ S h2 done lemma all_prime_nil : all_prime [] := by define --Goal : ∀ p ∈ [], prime p fix p : Nat contrapos --Goal : ¬prime p → p ∉ [] assume h1 : ¬prime p show p ∉ [] from List.not_mem_nil p done lemma all_prime_cons (n : Nat) (L : List Nat) : all_prime (n :: L) ↔ prime n ∧ all_prime L := by apply Iff.intro · -- (→) assume h1 : all_prime (n :: L) --Goal : prime n ∧ all_prime L define at h1 --h1 : ∀ p ∈ n :: L, prime p apply And.intro (h1 n (List.mem_cons_self n L)) define --Goal : ∀ p ∈ L, prime p fix p : Nat assume h2 : p ∈ L show prime p from h1 p (List.mem_cons_of_mem n h2) done · -- (←) assume h1 : prime n ∧ all_prime L --Goal : all_prime (n :: l) define : all_prime L at h1 define fix p : Nat assume h2 : p ∈ n :: L rewrite [List.mem_cons] at h2 --h2 : p = n ∨ p ∈ L by_cases on h2 · -- Case 1. h2 : p = n rewrite [h2] show prime n from h1.left done · -- Case 2. h2 : p ∈ L show prime p from h1.right p h2 done done done lemma nondec_nil : nondec [] := by define --Goal : True trivial --trivial proves some obviously true statements, such as True done lemma nondec_cons (n : Nat) (L : List Nat) : nondec (n :: L) ↔ (∀ m ∈ L, n ≤ m) ∧ nondec L := by rfl lemma prod_nil : prod [] = 1 := by rfl lemma prod_cons : prod (n :: L) = n * (prod L) := by rfl lemma exists_cons_of_length_eq_succ {A : Type} {l : List A} {n : Nat} (h : l.length = n + 1) : ∃ (a : A) (L : List A), l = a :: L ∧ L.length = n := by have h1 : ¬l.length = 0 := by linarith rewrite [List.length_eq_zero] at h1 obtain (a : A) (h2 : ∃ (L : List A), l = a :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List A) (h3 : l = a :: L) from h2 apply Exists.intro a apply Exists.intro L apply And.intro h3 have h4 : (a :: L).length = L.length + 1 := List.length_cons a L rewrite [←h3, h] at h4 show L.length = n from (Nat.add_right_cancel h4).symm done lemma list_elt_dvd_prod_by_length (a : Nat) : ∀ (n : Nat), ∀ (l : List Nat), l.length = n → a ∈ l → a ∣ prod l := by by_induc · --Base Case fix l : List Nat assume h1 : l.length = 0 rewrite [List.length_eq_zero] at h1 --h1 : l = [] rewrite [h1] --Goal : a ∈ [] → a ∣ prod [] contrapos assume h2 : ¬a ∣ prod [] show a ∉ [] from List.not_mem_nil a done · -- Induction Step fix n : Nat assume ih : ∀ (l : List Nat), List.length l = n → a ∈ l → a ∣ prod l fix l : List Nat assume h1 : l.length = n + 1 --Goal : a ∈ l → a ∣ prod l obtain (b : Nat) (h2 : ∃ (L : List Nat), l = b :: L ∧ L.length = n) from exists_cons_of_length_eq_succ h1 obtain (L : List Nat) (h3 : l = b :: L ∧ L.length = n) from h2 have h4 : a ∈ L → a ∣ prod L := ih L h3.right assume h5 : a ∈ l rewrite [h3.left, prod_cons] --Goal : a ∣ b * prod L rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b ∨ a ∈ L by_cases on h5 · -- Case 1. h5 : a = b apply Exists.intro (prod L) rewrite [h5] rfl done · -- Case 2. h5 : a ∈ L have h6 : a ∣ prod L := h4 h5 have h7 : prod L ∣ b * prod L := by apply Exists.intro b ring done show a ∣ b * prod L from dvd_trans h6 h7 done done done lemma list_elt_dvd_prod {a : Nat} {l : List Nat} (h : a ∈ l) : a ∣ prod l := by set n : Nat := l.length have h1 : l.length = n := by rfl show a ∣ prod l from list_elt_dvd_prod_by_length a n l h1 h done lemma exists_prime_factorization : ∀ (n : Nat), n ≥ 1 → ∃ (l : List Nat), prime_factorization n l := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, n_1 ≥ 1 → ∃ (l : List Nat), prime_factorization n_1 l assume h1 : n ≥ 1 by_cases h2 : n = 1 · -- Case 1. h2 : n = 1 apply Exists.intro [] define apply And.intro · -- Proof of nondec_prime_list [] define show all_prime [] ∧ nondec [] from And.intro all_prime_nil nondec_nil done · -- Proof of prod [] = n rewrite [prod_nil, h2] rfl done done · -- Case 2. h2 : n ≠ 1 have h3 : n ≥ 2 := lt_of_le_of_ne' h1 h2 obtain (p : Nat) (h4 : prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q) from exists_least_prime_factor h3 have p_prime_factor : prime_factor p n := h4.left define at p_prime_factor have p_prime : prime p := p_prime_factor.left have p_dvd_n : p ∣ n := p_prime_factor.right have p_least : ∀ (q : Nat), prime_factor q n → p ≤ q := h4.right obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n have h5 : m ≠ 0 := by contradict h1 with h6 have h7 : n = 0 := calc n _ = p * m := n_eq_pm _ = p * 0 := by rw [h6] _ = 0 := by ring rewrite [h7] decide done have m_pos : 0 < m := Nat.pos_of_ne_zero h5 have m_lt_n : m < n := by define at p_prime show m < n from calc m _ < m + m := by linarith _ = 2 * m := by ring _ ≤ p * m := by rel [p_prime.left] _ = n := n_eq_pm.symm done obtain (L : List Nat) (h6 : prime_factorization m L) from ih m m_lt_n m_pos define at h6 have ndpl_L : nondec_prime_list L := h6.left define at ndpl_L apply Exists.intro (p :: L) define apply And.intro · -- Proof of nondec_prime_list (p :: L) define apply And.intro · -- Proof of all_prime (p :: L) rewrite [all_prime_cons] show prime p ∧ all_prime L from And.intro p_prime ndpl_L.left done · -- Proof of nondec (p :: L) rewrite [nondec_cons] apply And.intro _ ndpl_L.right fix q : Nat assume q_in_L : q ∈ L have h7 : q ∣ prod L := list_elt_dvd_prod q_in_L rewrite [h6.right] at h7 --h7 : q ∣ m have h8 : m ∣ n := by apply Exists.intro p rewrite [n_eq_pm] ring done have q_dvd_n : q ∣ n := dvd_trans h7 h8 have ap_L : all_prime L := ndpl_L.left define at ap_L have q_prime_factor : prime_factor q n := And.intro (ap_L q q_in_L) q_dvd_n show p ≤ q from p_least q q_prime_factor done done · -- Proof of prod (p :: L) = n rewrite [prod_cons, h6.right, n_eq_pm] rfl done done done theorem Theorem_7_2_2 {a b c : Nat} (h1 : c ∣ a * b) (h2 : rel_prime a c) : c ∣ b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑c ∣ ↑b define at h1; define at h2; define obtain (j : Nat) (h3 : a * b = c * j) from h1 set s : Int := gcd_c1 a c set t : Int := gcd_c2 a c have h4 : s * ↑a + t * ↑c = ↑(gcd a c) := gcd_lin_comb c a rewrite [h2, Nat.cast_one] at h4 --h4 : s * ↑a + t * ↑c = (1 : Int) apply Exists.intro (s * ↑j + t * ↑b) show ↑b = ↑c * (s * ↑j + t * ↑b) from calc ↑b _ = (1 : Int) * ↑b := (one_mul _).symm _ = (s * ↑a + t * ↑c) * ↑b := by rw [h4] _ = s * (↑a * ↑b) + t * ↑c * ↑b := by ring _ = s * (↑c * ↑j) + t * ↑c * ↑b := by rw [←Nat.cast_mul a b, h3, Nat.cast_mul c j] _ = ↑c * (s * ↑j + t * ↑b) := by ring done lemma le_nonzero_prod_left {a b : Nat} (h : a * b ≠ 0) : a ≤ a * b := by have h1 : b ≠ 0 := by contradict h with h1 rewrite [h1] ring done have h2 : 1 ≤ b := Nat.pos_of_ne_zero h1 show a ≤ a * b from calc a = a * 1 := (mul_one a).symm _ ≤ a * b := by rel [h2] done lemma le_nonzero_prod_right {a b : Nat} (h : a * b ≠ 0) : b ≤ a * b := by rewrite [mul_comm] rewrite [mul_comm] at h show b ≤ b * a from le_nonzero_prod_left h done lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry lemma rel_prime_of_prime_not_dvd {a p : Nat} (h1 : prime p) (h2 : ¬p ∣ a) : rel_prime a p := by have h3 : gcd a p ∣ a := gcd_dvd_left a p have h4 : gcd a p ∣ p := gcd_dvd_right a p have h5 : gcd a p = 1 ∨ gcd a p = p := dvd_prime h1 h4 have h6 : gcd a p ≠ p := by contradict h2 with h6 rewrite [h6] at h3 show p ∣ a from h3 done disj_syll h5 h6 show rel_prime a p from h5 done theorem Theorem_7_2_3 {a b p : Nat} (h1 : prime p) (h2 : p ∣ a * b) : p ∣ a ∨ p ∣ b := by or_right with h3 have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3 show p ∣ b from Theorem_7_2_2 h2 h4 done lemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a ≥ 1 := by have h1 : a ≠ 0 := by by_contra h1 rewrite [h1] at h contradict h linarith done show a ≥ 1 from Nat.pos_of_ne_zero h1 done lemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by have h1 : a ≥ 1 := ge_one_of_prod_one h have h2 : a * b ≠ 0 := by linarith have h3 : a ≤ a * b := le_nonzero_prod_left h2 rewrite [h] at h3 show a = 1 from Nat.le_antisymm h3 h1 done lemma eq_one_of_dvd_one {n : Nat} (h : n ∣ 1) : n = 1 := by obtain (j : Nat) (h1 : 1 = n * j) from h show n = 1 from eq_one_of_prod_one h1.symm done lemma prime_not_one {p : Nat} (h : prime p) : p ≠ 1 := by define at h linarith done theorem Theorem_7_2_4 {p : Nat} (h1 : prime p) : ∀ (l : List Nat), p ∣ prod l → ∃ a ∈ l, p ∣ a := by apply List.rec · -- Base Case. Goal : p ∣ prod [] → ∃ a ∈ [], p ∣ a rewrite [prod_nil] assume h2 : p ∣ 1 show ∃ a ∈ [], p ∣ a from absurd (eq_one_of_dvd_one h2) (prime_not_one h1) done · -- Induction Step fix b : Nat fix L : List Nat assume ih : p ∣ prod L → ∃ a ∈ L, p ∣ a --Goal : p ∣ prod (b :: L) → ∃ a ∈ b :: L, p ∣ a assume h2 : p ∣ prod (b :: L) rewrite [prod_cons] at h2 have h3 : p ∣ b ∨ p ∣ prod L := Theorem_7_2_3 h1 h2 by_cases on h3 · -- Case 1. h3 : p ∣ b apply Exists.intro b show b ∈ b :: L ∧ p ∣ b from And.intro (List.mem_cons_self b L) h3 done · -- Case 2. h3 : p ∣ prod L obtain (a : Nat) (h4 : a ∈ L ∧ p ∣ a) from ih h3 apply Exists.intro a show a ∈ b :: L ∧ p ∣ a from And.intro (List.mem_cons_of_mem b h4.left) h4.right done done done lemma prime_in_list {p : Nat} {l : List Nat} (h1 : prime p) (h2 : all_prime l) (h3 : p ∣ prod l) : p ∈ l := by obtain (a : Nat) (h4 : a ∈ l ∧ p ∣ a) from Theorem_7_2_4 h1 l h3 define at h2 have h5 : prime a := h2 a h4.left have h6 : p = 1 ∨ p = a := dvd_prime h5 h4.right disj_syll h6 (prime_not_one h1) rewrite [h6] show a ∈ l from h4.left done lemma first_le_first {p q : Nat} {l m : List Nat} (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m)) (h3 : prod (p :: l) = prod (q :: m)) : p ≤ q := by define at h1; define at h2 have h4 : q ∣ prod (p :: l) := by define apply Exists.intro (prod m) rewrite [←prod_cons] show prod (p :: l) = prod (q :: m) from h3 done have h5 : all_prime (q :: m) := h2.left rewrite [all_prime_cons] at h5 have h6 : q ∈ p :: l := prime_in_list h5.left h1.left h4 have h7 : nondec (p :: l) := h1.right rewrite [nondec_cons] at h7 rewrite [List.mem_cons] at h6 by_cases on h6 · -- Case 1. h6 : q = p linarith done · -- Case 2. h6 : q ∈ l have h8 : ∀ m ∈ l, p ≤ m := h7.left show p ≤ q from h8 q h6 done done lemma nondec_prime_list_tail {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by define at h define rewrite [all_prime_cons, nondec_cons] at h show all_prime l ∧ nondec l from And.intro h.left.right h.right.right done lemma cons_prod_not_one {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : prod (p :: l) ≠ 1 := by define at h have h1 : all_prime (p :: l) := h.left rewrite [all_prime_cons] at h1 rewrite [prod_cons] by_contra h2 show False from (prime_not_one h1.left) (eq_one_of_prod_one h2) done lemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) : l = [] ↔ prod l = 1 := by apply Iff.intro · -- (→) assume h1 : l = [] rewrite [h1] show prod [] = 1 from prod_nil done · -- (←) contrapos assume h1 : ¬l = [] obtain (p : Nat) (h2 : ∃ (L : List Nat), l = p :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List Nat) (h3 : l = p :: L) from h2 rewrite [h3] at h rewrite [h3] show ¬prod (p :: L) = 1 from cons_prod_not_one h done done lemma prime_pos {p : Nat} (h : prime p) : p > 0 := by define at h linarith done theorem Theorem_7_2_5 : ∀ (l1 l2 : List Nat), nondec_prime_list l1 → nondec_prime_list l2 → prod l1 = prod l2 → l1 = l2 := by apply List.rec · -- Base Case. Goal : ∀ (l2 : List Nat), nondec_prime_list [] → -- nondec_prime_list l2 → prod [] = prod l2 → [] = l2 fix l2 : List Nat assume h1 : nondec_prime_list [] assume h2 : nondec_prime_list l2 assume h3 : prod [] = prod l2 rewrite [prod_nil, eq_comm, ←list_nil_iff_prod_one h2] at h3 show [] = l2 from h3.symm done · -- Induction Step fix p : Nat fix L1 : List Nat assume ih : ∀ (L2 : List Nat), nondec_prime_list L1 → nondec_prime_list L2 → prod L1 = prod L2 → L1 = L2 -- Goal : ∀ (l2 : List Nat), nondec_prime_list (p :: L1) → -- nondec_prime_list l2 → prod (p :: L1) = prod l2 → p :: L1 = l2 fix l2 : List Nat assume h1 : nondec_prime_list (p :: L1) assume h2 : nondec_prime_list l2 assume h3 : prod (p :: L1) = prod l2 have h4 : ¬prod (p :: L1) = 1 := cons_prod_not_one h1 rewrite [h3, ←list_nil_iff_prod_one h2] at h4 obtain (q : Nat) (h5 : ∃ (L : List Nat), l2 = q :: L) from List.exists_cons_of_ne_nil h4 obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5 rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2) rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2) have h7 : p ≤ q := first_le_first h1 h2 h3 have h8 : q ≤ p := first_le_first h2 h1 h3.symm have h9 : p = q := by linarith rewrite [h9, prod_cons, prod_cons] at h3 --h3 : q * prod L1 = q * prod L2 have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1 have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2 define at h2 have h12 : all_prime (q :: L2) := h2.left rewrite [all_prime_cons] at h12 have h13 : q > 0 := prime_pos h12.left have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3 have h15 : L1 = L2 := ih L2 h10 h11 h14 rewrite [h6, h9, h15] rfl done done theorem fund_thm_arith (n : Nat) (h : n ≥ 1) : ∃! (l : List Nat), prime_factorization n l := by exists_unique · -- Existence show ∃ (l : List Nat), prime_factorization n l from exists_prime_factorization n h done · -- Uniqueness fix l1 : List Nat; fix l2 : List Nat assume h1 : prime_factorization n l1 assume h2 : prime_factorization n l2 define at h1; define at h2 have h3 : prod l1 = n := h1.right rewrite [←h2.right] at h3 show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3 done done /- Section 7.3 -/ theorem congr_refl (m : Nat) : ∀ (a : Int), a ≡ a (MOD m) := by fix a : Int define --Goal : ∃ (c : Int), a - a = ↑m * c apply Exists.intro 0 ring done theorem congr_symm {m : Nat} : ∀ {a b : Int}, a ≡ b (MOD m) → b ≡ a (MOD m) := by fix a : Int; fix b : Int assume h1 : a ≡ b (MOD m) define at h1 --h1 : ∃ (c : Int), a - b = ↑m * c define --Goal : ∃ (c : Int), b - a = ↑m * c obtain (c : Int) (h2 : a - b = m * c) from h1 apply Exists.intro (-c) show b - a = m * (-c) from calc b - a _ = -(a - b) := by ring _ = -(m * c) := by rw [h2] _ = m * (-c) := by ring done theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry /- Fundamental properties of congruence classes -/ lemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) : X = Y ↔ X.val = Y.val := Fin.ext_iff lemma val_nat_eq_mod (n k : Nat) : ([k]_(n + 1)).val = k % (n + 1) := by rfl lemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl theorem cc_rep {m : Nat} (X : ZMod m) : ∃ (a : Int), X = [a]_m := match m with | 0 => by apply Exists.intro X rfl done | n + 1 => by apply Exists.intro ↑(X.val) have h1 : X.val < n + 1 := Fin.prop X rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1] rfl done theorem add_class (m : Nat) (a b : Int) : [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm theorem mul_class (m : Nat) (a b : Int) : [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm lemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ [a - b]_m = [0]_m := by apply Iff.intro · -- (→) assume h1 : [a]_m = [b]_m have h2 : a - b = a + (-b) := by ring have h3 : b + (-b) = 0 := by ring show [a - b]_m = [0]_m from calc [a - b]_m _ = [a + (-b)]_m := by rw [h2] _ = [a]_m + [-b]_m := by rw [add_class] _ = [b]_m + [-b]_m := by rw [h1] _ = [b + -b]_m := by rw [add_class] _ = [0]_m := by rw [h3] done · -- (←) assume h1 : [a - b]_m = [0]_m have h2 : b + (a - b) = a := by ring have h3 : b + 0 = b := by ring show [a]_m = [b]_m from calc [a]_m _ = [b + (a - b)]_m := by rw [h2] _ = [b]_m + [a - b]_m := by rw [add_class] _ = [b]_m + [0]_m := by rw [h1] _ = [b + 0]_m := by rw [add_class] _ = [b]_m := by rw [h3] done done lemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) : [a]_m = [0]_m → [-a]_m = [0]_m := by assume h1 : [a]_m = [0]_m have h2 : 0 + (-a) = -a := by ring have h3 : a + (-a) = 0 := by ring show [-a]_m = [0]_m from calc [-a]_m _ = [0 + (-a)]_m := by rw [h2] _ = [0]_m + [-a]_m := by rw [add_class] _ = [a]_m + [-a]_m := by rw [h1] _ = [a + (-a)]_m := by rw [add_class] _ = [0]_m := by rw [h3] done lemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) : [-a]_m = [0]_m ↔ [a]_m = [0]_m := by apply Iff.intro _ (cc_neg_zero_of_cc_zero m a) assume h1 : [-a]_m = [0]_m have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1 have h3 : -(-a) = a := by ring rewrite [h3] at h2 show [a]_m = [0]_m from h2 done lemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl lemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m ↔ m ∣ k := match m with | 0 => by have h : (0 : Int) = (↑(0 : Nat) : Int) := by rfl rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj] apply Iff.intro · -- (→) assume h1 : k = 0 rewrite [h1] show 0 ∣ 0 from dvd_self 0 done · -- (←) assume h1 : 0 ∣ k obtain (c : Nat) (h2 : k = 0 * c) from h1 rewrite [h2] ring done done | n + 1 => by rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero] show k % (n + 1) = 0 ↔ n + 1 ∣ k from (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm done lemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m ↔ ↑m ∣ a := by obtain (k : Nat) (h1 : a = ↑k ∨ a = -↑k) from Int.eq_nat_or_neg a by_cases on h1 · -- Case 1. h1: a = ↑k rewrite [h1, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done · -- Case 2. h1: a = -↑k rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done done theorem cc_eq_iff_congr (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ a ≡ b (MOD m) := calc [a]_m = [b]_m _ ↔ [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b _ ↔ ↑m ∣ (a - b) := cc_zero_iff_dvd m (a - b) _ ↔ a ≡ b (MOD m) := by rfl /- End of fundamental properties of congruence classes -/ lemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m := by have h1 : (↑m : Int) ≠ 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m) show 0 ≤ a % m from Int.emod_nonneg a h1 done lemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m) have h2 : (↑m : Int) > 0 := (Nat.cast_pos).rtl h1 show a % m < m from Int.emod_lt_of_pos a h2 done lemma congr_mod_mod (m : Nat) (a : Int) : a ≡ a % m (MOD m) := by define have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m apply Exists.intro (a / m) show a - a % m = m * (a / m) from calc a - (a % m) _ = m * (a / m) + a % m - a % m := by rw [h1] _ = m * (a / m) := by ring done lemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) := And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a)) theorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) : ∃! (r : Int), 0 ≤ r ∧ r < m ∧ a ≡ r (MOD m) := by exists_unique · -- Existence apply Exists.intro (a % m) show 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) from mod_cmpl_res m a done · -- Uniqueness fix r1 : Int; fix r2 : Int assume h1 : 0 ≤ r1 ∧ r1 < m ∧ a ≡ r1 (MOD m) assume h2 : 0 ≤ r2 ∧ r2 < m ∧ a ≡ r2 (MOD m) have h3 : r1 ≡ r2 (MOD m) := congr_trans (congr_symm h1.right.right) h2.right.right obtain (d : Int) (h4 : r1 - r2 = m * d) from h3 have h5 : r1 - r2 < m * 1 := by linarith have h6 : m * (-1) < r1 - r2 := by linarith rewrite [h4] at h5 --h5 : m * d < m * 1 rewrite [h4] at h6 --h6 : m * -1 < m * d have h7 : (↑m : Int) ≥ 0 := Nat.cast_nonneg m have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7 have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7 have h10 : d = 0 := by linarith show r1 = r2 from calc r1 _ = r1 - r2 + r2 := by ring _ = m * 0 + r2 := by rw [h4, h10] _ = r2 := by ring done done lemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m := (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a) theorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y rewrite [h1, h2] have h3 : a + b = b + a := by ring show [a]_m + [b]_m = [b]_m + [a]_m from calc [a]_m + [b]_m _ = [a + b]_m := add_class m a b _ = [b + a]_m := by rw [h3] _ = [b]_m + [a]_m := (add_class m b a).symm done theorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X rewrite [h1] have h2 : a * 1 = a := by ring show [a]_m * [1]_m = [a]_m from calc [a]_m * [1]_m _ = [a * 1]_m := mul_class m a 1 _ = [a]_m := by rw [h2] done theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry lemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) : [a]_m * [gcd_c2 m a]_m = [1]_m := by set s : Int := gcd_c1 m a have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m define at h1 rewrite [h1, Nat.cast_one] at h2 --h2 : s * ↑m + gcd_c2 m a * ↑a = 1 rewrite [mul_class, cc_eq_iff_congr] define --Goal : ∃ (c : Int), ↑a * gcd_c2 m a - 1 = ↑m * c apply Exists.intro (-s) show a * (gcd_c2 m a) - 1 = m * (-s) from calc a * (gcd_c2 m a) - 1 _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring _ = 1 + m * (-s) - 1 := by rw [h2] _ = m * (-s) := by ring done theorem Theorem_7_3_7 (m a : Nat) : invertible [a]_m ↔ rel_prime m a := by apply Iff.intro · -- (→) assume h1 : invertible [a]_m define at h1 obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1 obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y rewrite [h3, mul_class, cc_eq_iff_congr] at h2 define at h2 obtain (c : Int) (h4 : a * b - 1 = m * c) from h2 rewrite [Exercise_7_2_6] --Goal : ∃ (s t : Int), s * ↑m + t * ↑a = 1 apply Exists.intro (-c) apply Exists.intro b show (-c) * m + b * a = 1 from calc (-c) * m + b * a _ = (-c) * m + (a * b - 1) + 1 := by ring _ = (-c) * m + m * c + 1 := by rw [h4] _ = 1 := by ring done · -- (←) assume h1 : rel_prime m a define show ∃ (Y : ZMod m), [a]_m * Y = [1]_m from Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1) done done /- Section 7.4 -/ section Euler open Euler lemma num_rp_below_base {m : Nat} : num_rp_below m 0 = 0 := by rfl lemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) : num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by have h1 : num_rp_below m (j + 1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : gcd m j = 1 rewrite [if_pos h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1 show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1 done lemma num_rp_below_step_not_rp {m j : Nat} (h : ¬rel_prime m j) : num_rp_below m (j + 1) = num_rp_below m j := by have h1 : num_rp_below m (j +1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : ¬gcd m j = 1 rewrite [if_neg h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j show num_rp_below m (j + 1) = num_rp_below m j from h1 done lemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl #eval phi 10 --Answer: 4 lemma prod_inv_iff_inv {m : Nat} {X : ZMod m} (h1 : invertible X) (Y : ZMod m) : invertible (X * Y) ↔ invertible Y := by apply Iff.intro · -- (→) assume h2 : invertible (X * Y) obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2 apply Exists.intro (X * Z) rewrite [←h3] --Goal : Y * (X * Z) = X * Y * Z ring --Note that ring can do algebra in ZMod m done · -- (←) assume h2 : invertible Y obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1 obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2 apply Exists.intro (Xi * Yi) show (X * Y) * (Xi * Yi) = [1]_m from calc X * Y * (Xi * Yi) _ = (X * Xi) * (Y * Yi) := by ring _ = [1]_m * [1]_m := by rw [h3, h4] _ = [1]_m := Theorem_7_3_6_7 [1]_m done done lemma F_rp_def {m i : Nat} (h : rel_prime m i) : F m i = [i]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h --h : gcd m i = 1 rewrite [if_pos h] at h1 show F m i = [i]_m from h1 done lemma F_not_rp_def {m i : Nat} (h : ¬rel_prime m i) : F m i = [1]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h rewrite [h1, if_neg h] rfl done lemma prod_seq_base {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 0 k f = [1]_m := by rfl lemma prod_seq_step {m : Nat} (n k : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl lemma prod_seq_zero_step {m : Nat} (n : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by rewrite [prod_seq_step, zero_add] rfl done lemma prod_one {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 1 k f = f k := by rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7] rfl done lemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl lemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m := calc [G m a i]_m _ = [(a * i) % m]_m := by rfl _ = [a * i]_m := (cc_eq_mod m (a * i)).symm _ = [a]_m * [i]_m := (mul_class m a i).symm lemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) : rel_prime m (G m a i) ↔ rel_prime m i := by have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1 show rel_prime m (G m a i) ↔ rel_prime m i from calc rel_prime m (G m a i) _ ↔ invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm _ ↔ invertible ([a]_m * [i]_m) := by rw [cc_G] _ ↔ invertible [i]_m := prod_inv_iff_inv h2 ([i]_m) _ ↔ rel_prime m i := Theorem_7_3_7 m i done lemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) : F m (G m a i) = [a]_m * F m i := by have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2 show F m (G m a i) = [a]_m * F m i from calc F m (G m a i) _ = [G m a i]_m := F_rp_def h3 _ = [a]_m * [i]_m := cc_G m a i _ = [a]_m * F m i := by rw [F_rp_def h2] done lemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : ¬rel_prime m i) : F m (G m a i) = [1]_m := by rewrite [←G_rp_iff h1 i] at h2 show F m (G m a i) = [1]_m from F_not_rp_def h2 done lemma FG_prod {m a : Nat} (h1 : rel_prime m a) : ∀ (k : Nat), prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by by_induc · -- Base Case show prod_seq 0 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from calc prod_seq 0 0 ((F m) ∘ (G m a)) _ = [1]_m := prod_seq_base _ _ _ = [a]_m ^ 0 * [1]_m := by ring _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by rw [num_rp_below_base, prod_seq_base] done · -- Induction Step fix k : Nat assume ih : prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) by_cases h2 : rel_prime m k · -- Case 1. h2 : rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([a]_m * F m k) := by rw [FG_rp h1 h2] _ = [a]_m ^ ((num_rp_below m k) + 1) * ((prod_seq k 0 (F m)) * F m k) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_rp h2, prod_seq_zero_step] done · -- Case 2. h2 : ¬rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([1]_m) := by rw [FG_not_rp h1 h2] _ = [a]_m ^ (num_rp_below m k) * (prod_seq k 0 (F m) * ([1]_m)) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_not_rp h2, prod_seq_zero_step, F_not_rp_def h2] done done done lemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by define --Goal : ∀ i < m, G m a i < m fix i : Nat assume h1 : i < m rewrite [G_def] --Goal : a * i % m < m show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m) done lemma left_inv_one_one_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g' (g i) = i) : one_one_below n g := sorry lemma right_inv_onto_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g (g' i) = i) (h2 : maps_below n g') : onto_below n g := by define at h2; define fix k : Nat assume h3 : k < n apply Exists.intro (g' k) show g' k < n ∧ g (g' k) = k from And.intro (h2 k h3) (h1 k h3) done lemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m * [inv_mod m a]_m = [1]_m := by have h2 : 0 ≤ (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a) show [a]_m * [inv_mod m a]_m = [1]_m from calc [a]_m * [inv_mod m a]_m _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2] _ = [a]_m * [gcd_c2 m a]_m := by rw [←cc_eq_mod] _ = [1]_m := gcd_c2_inv h1 done lemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m := calc a * (b % m) % m = a % m * (b % m % m) % m := Nat.mul_mod _ _ _ _ = a % m * (b % m) % m := by rw [Nat.mod_mod] _ = a * b % m := (Nat.mul_mod _ _ _).symm lemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm] rfl done theorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] : ↑a ≡ ↑b (MOD m) ↔ a % m = b % m := sorry lemma mul_inv_mod_cancel {m a i : Nat} [NeZero m] (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1 rewrite [mul_class, cc_eq_iff_congr, ←Nat.cast_mul, ←Nat.cast_one, congr_iff_mod_eq_Nat] at h3 show a * inv_mod m a * i % m = i from calc a * (inv_mod m a) * i % m _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = 1 % m * i % m := by rw [h3] _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = i % m := by rw [one_mul] _ = i := Nat.mod_eq_of_lt h2 done lemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl lemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, G m a (Ginv m a i) = i := by fix i : Nat assume h2 : i < m show G m a (Ginv m a i) = i from calc G m a (Ginv m a i) _ = a * ((inv_mod m a * i) % m) % m := by rfl _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, Ginv m a (G m a i) = i := by fix i : Nat assume h2 : i < m show Ginv m a (G m a i) = i from calc Ginv m a (G m a i) _ = inv_mod m a * ((a * i) % m) % m := by rfl _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc, mul_comm (inv_mod m a)] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_maps_below (m a : Nat) [NeZero m] : maps_below m (Ginv m a) := G_maps_below m (inv_mod m a) lemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : one_one_below m (G m a) := left_inv_one_one_below (Ginv_left_inv h1) lemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : onto_below m (G m a) := right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a) lemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : perm_below m (G m a) := And.intro (G_maps_below m a) (And.intro (G_one_one_below h1) (G_onto_below h1)) --Permuting a product of congruence classes doesn't change product lemma swap_fst (u v : Nat) : swap u v u = v := by define : swap u v u --Goal : (if u = u then v else if u = v then u else u) = v have h : u = u := by rfl rewrite [if_pos h] rfl done

lemma swap_snd (u v : Nat) : swap u v v = u

HTPI.swap_snd

htpi/HTPILib/Chap7.lean

htpi

/- Copyright 2023 Daniel J. Velleman -/ import HTPILib.Chap6 namespace HTPI /- Definitions -/ lemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by have h : n + 1 > 0 := Nat.succ_pos n show a % (n + 1) < n + 1 from Nat.mod_lt a h done def gcd (a b : Nat) : Nat := match b with | 0 => a | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd (n + 1) (a % (n + 1)) termination_by b mutual def gcd_c1 (a b : Nat) : Int := match b with | 0 => 1 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c2 (n + 1) (a % (n + 1)) --Corresponds to s = t' termination_by b def gcd_c2 (a b : Nat) : Int := match b with | 0 => 0 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c1 (n + 1) (a % (n + 1)) - (gcd_c2 (n + 1) (a % (n + 1))) * ↑(a / (n + 1)) --Corresponds to t = s' - t'q termination_by b end def prime (n : Nat) : Prop := 2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n def prime_factor (p n : Nat) : Prop := prime p ∧ p ∣ n def all_prime (l : List Nat) : Prop := ∀ p ∈ l, prime p def nondec (l : List Nat) : Prop := match l with | [] => True --Of course, True is a proposition that is always true | n :: L => (∀ m ∈ L, n ≤ m) ∧ nondec L def nondec_prime_list (l : List Nat) : Prop := all_prime l ∧ nondec l def prod (l : List Nat) : Nat := match l with | [] => 1 | n :: L => n * (prod L) def prime_factorization (n : Nat) (l : List Nat) : Prop := nondec_prime_list l ∧ prod l = n def rel_prime (a b : Nat) : Prop := gcd a b = 1 def congr_mod (m : Nat) (a b : Int) : Prop := (↑m : Int) ∣ (a - b) def cc (m : Nat) (a : Int) : ZMod m := (↑a : ZMod m) notation:50 a " ≡ " b " (MOD " m ")" => congr_mod m a b notation:max "["a"]_"m:max => cc m a def invertible {m : Nat} (X : ZMod m) : Prop := ∃ (Y : ZMod m), X * Y = [1]_m def num_rp_below (m k : Nat) : Nat := match k with | 0 => 0 | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j def phi (m : Nat) : Nat := num_rp_below m m def prod_seq {m : Nat} (j k : Nat) (f : Nat → ZMod m) : ZMod m := match j with | 0 => [1]_m | n + 1 => prod_seq n k f * f (k + n) def maps_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i < n, g i < n def one_one_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i1 < n, ∀ i2 < n, g i1 = g i2 → i1 = i2 def onto_below (n : Nat) (g : Nat → Nat) : Prop := ∀ k < n, ∃ i < n, g i = k def perm_below (n : Nat) (g : Nat → Nat) : Prop := maps_below n g ∧ one_one_below n g ∧ onto_below n g def inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m) def swap (u v i : Nat) : Nat := if i = u then v else if i = v then u else i namespace Euler --For definitions specific to Euler's theorem def F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m def G (m a i : Nat) : Nat := (a * i) % m def Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i end Euler /- Section 7.1 -/ theorem dvd_mod_of_dvd_a_b {a b d : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ (a % b) := by set q : Nat := a / b have h3 : b * q + a % b = a := Nat.div_add_mod a b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 define --Goal : ∃ (c : Nat), a % b = d * c apply Exists.intro (j - k * q) show a % b = d * (j - k * q) from calc a % b _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm _ = a - b * q := by rw [h3] _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc] _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm done theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry #eval gcd 672 161 --Answer: 7 lemma gcd_base (a : Nat) : gcd a 0 = a := by rfl lemma gcd_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd a b = gcd b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1)) rfl done lemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b ≠ 0) : a % b < b := by have h1 : b > 0 := Nat.pos_of_ne_zero h show a % b < b from Nat.mod_lt a h1 done lemma dvd_self (n : Nat) : n ∣ n := by apply Exists.intro 1 ring done theorem gcd_dvd : ∀ (b a : Nat), (gcd a b) ∣ a ∧ (gcd a b) ∣ b := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd a b_1) ∣ a ∧ (gcd a b_1) ∣ b_1 fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_base] --Goal: a ∣ a ∧ a ∣ 0 apply And.intro (dvd_self a) define apply Exists.intro 0 rfl done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_nonzero a h1] --Goal : gcd b (a % b) ∣ a ∧ gcd b (a % b) ∣ b have h2 : a % b < b := mod_nonzero_lt a h1 have h3 : (gcd b (a % b)) ∣ b ∧ (gcd b (a % b)) ∣ (a % b) := ih (a % b) h2 b apply And.intro _ h3.left show (gcd b (a % b)) ∣ a from dvd_a_of_dvd_b_mod h3.left h3.right done done theorem gcd_dvd_left (a b : Nat) : (gcd a b) ∣ a := (gcd_dvd b a).left theorem gcd_dvd_right (a b : Nat) : (gcd a b) ∣ b := (gcd_dvd b a).right lemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl lemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c1 a b = gcd_c2 b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done lemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl lemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * ↑(a / b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done theorem gcd_lin_comb : ∀ (b a : Nat), (gcd_c1 a b) * ↑a + (gcd_c2 a b) * ↑b = ↑(gcd a b) := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd_c1 a b_1) * ↑a + (gcd_c2 a b_1) * ↑b_1 = ↑(gcd a b_1) fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base] --Goal : 1 * ↑a + 0 * ↑0 = ↑a ring done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1] --Goal : gcd_c2 b (a % b) * ↑a + -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * ↑(a / b)) * ↑b = -- ↑(gcd b (a % b)) set r : Nat := a % b set q : Nat := a / b set s : Int := gcd_c1 b r set t : Int := gcd_c2 b r --Goal : t * ↑a + (s - t * ↑q) * ↑b = ↑(gcd b r) have h2 : r < b := mod_nonzero_lt a h1 have h3 : s * ↑b + t * ↑r = ↑(gcd b r) := ih r h2 b have h4 : b * q + r = a := Nat.div_add_mod a b rewrite [←h3, ←h4] rewrite [Nat.cast_add, Nat.cast_mul] --Goal : t * (↑b * ↑q + ↑r) + (s - t * ↑q) * ↑b = s * ↑b + t * ↑r ring done done #eval gcd_c1 672 161 --Answer: 6 #eval gcd_c2 672 161 --Answer: -25 --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161 theorem Theorem_7_1_6 {d a b : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ gcd a b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑d ∣ ↑(gcd a b) set s : Int := gcd_c1 a b set t : Int := gcd_c2 a b have h3 : s * ↑a + t * ↑b = ↑(gcd a b) := gcd_lin_comb b a rewrite [←h3] --Goal : ↑d ∣ s * ↑a + t * ↑b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul] --Goal : ↑d ∣ s * (↑d * ↑j) + t * (↑d * ↑k) define apply Exists.intro (s * ↑j + t * ↑k) ring done /- Section 7.2 -/ theorem dvd_trans {a b c : Nat} (h1 : a ∣ b) (h2 : b ∣ c) : a ∣ c := by define at h1; define at h2; define obtain (m : Nat) (h3 : b = a * m) from h1 obtain (n : Nat) (h4 : c = b * n) from h2 rewrite [h3, mul_assoc] at h4 apply Exists.intro (m * n) show c = a * (m * n) from h4 done lemma exists_prime_factor : ∀ (n : Nat), 2 ≤ n → ∃ (p : Nat), prime_factor p n := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, 2 ≤ n_1 → ∃ (p : Nat), prime_factor p n_1 assume h1 : 2 ≤ n by_cases h2 : prime n · -- Case 1. h2 : prime n apply Exists.intro n define --Goal : prime n ∧ n ∣ n show prime n ∧ n ∣ n from And.intro h2 (dvd_self n) done · -- Case 2. h2 : ¬prime n define at h2 --h2 : ¬(2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n) demorgan at h2 disj_syll h2 h1 obtain (a : Nat) (h3 : ∃ (b : Nat), a * b = n ∧ a < n ∧ b < n) from h2 obtain (b : Nat) (h4 : a * b = n ∧ a < n ∧ b < n) from h3 have h5 : 2 ≤ a := by by_contra h6 have h7 : a ≤ 1 := by linarith have h8 : n ≤ b := calc n _ = a * b := h4.left.symm _ ≤ 1 * b := by rel [h7] _ = b := by ring linarith --n ≤ b contradicts b < n done have h6 : ∃ (p : Nat), prime_factor p a := ih a h4.right.left h5 obtain (p : Nat) (h7 : prime_factor p a) from h6 apply Exists.intro p define --Goal : prime p ∧ p ∣ n define at h7 --h7 : prime p ∧ p ∣ a apply And.intro h7.left have h8 : a ∣ n := by apply Exists.intro b show n = a * b from (h4.left).symm done show p ∣ n from dvd_trans h7.right h8 done done lemma exists_least_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q := by set S : Set Nat := {p : Nat | prime_factor p n} have h2 : ∃ (p : Nat), p ∈ S := exists_prime_factor n h show ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q from well_ord_princ S h2 done lemma all_prime_nil : all_prime [] := by define --Goal : ∀ p ∈ [], prime p fix p : Nat contrapos --Goal : ¬prime p → p ∉ [] assume h1 : ¬prime p show p ∉ [] from List.not_mem_nil p done lemma all_prime_cons (n : Nat) (L : List Nat) : all_prime (n :: L) ↔ prime n ∧ all_prime L := by apply Iff.intro · -- (→) assume h1 : all_prime (n :: L) --Goal : prime n ∧ all_prime L define at h1 --h1 : ∀ p ∈ n :: L, prime p apply And.intro (h1 n (List.mem_cons_self n L)) define --Goal : ∀ p ∈ L, prime p fix p : Nat assume h2 : p ∈ L show prime p from h1 p (List.mem_cons_of_mem n h2) done · -- (←) assume h1 : prime n ∧ all_prime L --Goal : all_prime (n :: l) define : all_prime L at h1 define fix p : Nat assume h2 : p ∈ n :: L rewrite [List.mem_cons] at h2 --h2 : p = n ∨ p ∈ L by_cases on h2 · -- Case 1. h2 : p = n rewrite [h2] show prime n from h1.left done · -- Case 2. h2 : p ∈ L show prime p from h1.right p h2 done done done lemma nondec_nil : nondec [] := by define --Goal : True trivial --trivial proves some obviously true statements, such as True done lemma nondec_cons (n : Nat) (L : List Nat) : nondec (n :: L) ↔ (∀ m ∈ L, n ≤ m) ∧ nondec L := by rfl lemma prod_nil : prod [] = 1 := by rfl lemma prod_cons : prod (n :: L) = n * (prod L) := by rfl lemma exists_cons_of_length_eq_succ {A : Type} {l : List A} {n : Nat} (h : l.length = n + 1) : ∃ (a : A) (L : List A), l = a :: L ∧ L.length = n := by have h1 : ¬l.length = 0 := by linarith rewrite [List.length_eq_zero] at h1 obtain (a : A) (h2 : ∃ (L : List A), l = a :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List A) (h3 : l = a :: L) from h2 apply Exists.intro a apply Exists.intro L apply And.intro h3 have h4 : (a :: L).length = L.length + 1 := List.length_cons a L rewrite [←h3, h] at h4 show L.length = n from (Nat.add_right_cancel h4).symm done lemma list_elt_dvd_prod_by_length (a : Nat) : ∀ (n : Nat), ∀ (l : List Nat), l.length = n → a ∈ l → a ∣ prod l := by by_induc · --Base Case fix l : List Nat assume h1 : l.length = 0 rewrite [List.length_eq_zero] at h1 --h1 : l = [] rewrite [h1] --Goal : a ∈ [] → a ∣ prod [] contrapos assume h2 : ¬a ∣ prod [] show a ∉ [] from List.not_mem_nil a done · -- Induction Step fix n : Nat assume ih : ∀ (l : List Nat), List.length l = n → a ∈ l → a ∣ prod l fix l : List Nat assume h1 : l.length = n + 1 --Goal : a ∈ l → a ∣ prod l obtain (b : Nat) (h2 : ∃ (L : List Nat), l = b :: L ∧ L.length = n) from exists_cons_of_length_eq_succ h1 obtain (L : List Nat) (h3 : l = b :: L ∧ L.length = n) from h2 have h4 : a ∈ L → a ∣ prod L := ih L h3.right assume h5 : a ∈ l rewrite [h3.left, prod_cons] --Goal : a ∣ b * prod L rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b ∨ a ∈ L by_cases on h5 · -- Case 1. h5 : a = b apply Exists.intro (prod L) rewrite [h5] rfl done · -- Case 2. h5 : a ∈ L have h6 : a ∣ prod L := h4 h5 have h7 : prod L ∣ b * prod L := by apply Exists.intro b ring done show a ∣ b * prod L from dvd_trans h6 h7 done done done lemma list_elt_dvd_prod {a : Nat} {l : List Nat} (h : a ∈ l) : a ∣ prod l := by set n : Nat := l.length have h1 : l.length = n := by rfl show a ∣ prod l from list_elt_dvd_prod_by_length a n l h1 h done lemma exists_prime_factorization : ∀ (n : Nat), n ≥ 1 → ∃ (l : List Nat), prime_factorization n l := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, n_1 ≥ 1 → ∃ (l : List Nat), prime_factorization n_1 l assume h1 : n ≥ 1 by_cases h2 : n = 1 · -- Case 1. h2 : n = 1 apply Exists.intro [] define apply And.intro · -- Proof of nondec_prime_list [] define show all_prime [] ∧ nondec [] from And.intro all_prime_nil nondec_nil done · -- Proof of prod [] = n rewrite [prod_nil, h2] rfl done done · -- Case 2. h2 : n ≠ 1 have h3 : n ≥ 2 := lt_of_le_of_ne' h1 h2 obtain (p : Nat) (h4 : prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q) from exists_least_prime_factor h3 have p_prime_factor : prime_factor p n := h4.left define at p_prime_factor have p_prime : prime p := p_prime_factor.left have p_dvd_n : p ∣ n := p_prime_factor.right have p_least : ∀ (q : Nat), prime_factor q n → p ≤ q := h4.right obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n have h5 : m ≠ 0 := by contradict h1 with h6 have h7 : n = 0 := calc n _ = p * m := n_eq_pm _ = p * 0 := by rw [h6] _ = 0 := by ring rewrite [h7] decide done have m_pos : 0 < m := Nat.pos_of_ne_zero h5 have m_lt_n : m < n := by define at p_prime show m < n from calc m _ < m + m := by linarith _ = 2 * m := by ring _ ≤ p * m := by rel [p_prime.left] _ = n := n_eq_pm.symm done obtain (L : List Nat) (h6 : prime_factorization m L) from ih m m_lt_n m_pos define at h6 have ndpl_L : nondec_prime_list L := h6.left define at ndpl_L apply Exists.intro (p :: L) define apply And.intro · -- Proof of nondec_prime_list (p :: L) define apply And.intro · -- Proof of all_prime (p :: L) rewrite [all_prime_cons] show prime p ∧ all_prime L from And.intro p_prime ndpl_L.left done · -- Proof of nondec (p :: L) rewrite [nondec_cons] apply And.intro _ ndpl_L.right fix q : Nat assume q_in_L : q ∈ L have h7 : q ∣ prod L := list_elt_dvd_prod q_in_L rewrite [h6.right] at h7 --h7 : q ∣ m have h8 : m ∣ n := by apply Exists.intro p rewrite [n_eq_pm] ring done have q_dvd_n : q ∣ n := dvd_trans h7 h8 have ap_L : all_prime L := ndpl_L.left define at ap_L have q_prime_factor : prime_factor q n := And.intro (ap_L q q_in_L) q_dvd_n show p ≤ q from p_least q q_prime_factor done done · -- Proof of prod (p :: L) = n rewrite [prod_cons, h6.right, n_eq_pm] rfl done done done theorem Theorem_7_2_2 {a b c : Nat} (h1 : c ∣ a * b) (h2 : rel_prime a c) : c ∣ b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑c ∣ ↑b define at h1; define at h2; define obtain (j : Nat) (h3 : a * b = c * j) from h1 set s : Int := gcd_c1 a c set t : Int := gcd_c2 a c have h4 : s * ↑a + t * ↑c = ↑(gcd a c) := gcd_lin_comb c a rewrite [h2, Nat.cast_one] at h4 --h4 : s * ↑a + t * ↑c = (1 : Int) apply Exists.intro (s * ↑j + t * ↑b) show ↑b = ↑c * (s * ↑j + t * ↑b) from calc ↑b _ = (1 : Int) * ↑b := (one_mul _).symm _ = (s * ↑a + t * ↑c) * ↑b := by rw [h4] _ = s * (↑a * ↑b) + t * ↑c * ↑b := by ring _ = s * (↑c * ↑j) + t * ↑c * ↑b := by rw [←Nat.cast_mul a b, h3, Nat.cast_mul c j] _ = ↑c * (s * ↑j + t * ↑b) := by ring done lemma le_nonzero_prod_left {a b : Nat} (h : a * b ≠ 0) : a ≤ a * b := by have h1 : b ≠ 0 := by contradict h with h1 rewrite [h1] ring done have h2 : 1 ≤ b := Nat.pos_of_ne_zero h1 show a ≤ a * b from calc a = a * 1 := (mul_one a).symm _ ≤ a * b := by rel [h2] done lemma le_nonzero_prod_right {a b : Nat} (h : a * b ≠ 0) : b ≤ a * b := by rewrite [mul_comm] rewrite [mul_comm] at h show b ≤ b * a from le_nonzero_prod_left h done lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry lemma rel_prime_of_prime_not_dvd {a p : Nat} (h1 : prime p) (h2 : ¬p ∣ a) : rel_prime a p := by have h3 : gcd a p ∣ a := gcd_dvd_left a p have h4 : gcd a p ∣ p := gcd_dvd_right a p have h5 : gcd a p = 1 ∨ gcd a p = p := dvd_prime h1 h4 have h6 : gcd a p ≠ p := by contradict h2 with h6 rewrite [h6] at h3 show p ∣ a from h3 done disj_syll h5 h6 show rel_prime a p from h5 done theorem Theorem_7_2_3 {a b p : Nat} (h1 : prime p) (h2 : p ∣ a * b) : p ∣ a ∨ p ∣ b := by or_right with h3 have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3 show p ∣ b from Theorem_7_2_2 h2 h4 done lemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a ≥ 1 := by have h1 : a ≠ 0 := by by_contra h1 rewrite [h1] at h contradict h linarith done show a ≥ 1 from Nat.pos_of_ne_zero h1 done lemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by have h1 : a ≥ 1 := ge_one_of_prod_one h have h2 : a * b ≠ 0 := by linarith have h3 : a ≤ a * b := le_nonzero_prod_left h2 rewrite [h] at h3 show a = 1 from Nat.le_antisymm h3 h1 done lemma eq_one_of_dvd_one {n : Nat} (h : n ∣ 1) : n = 1 := by obtain (j : Nat) (h1 : 1 = n * j) from h show n = 1 from eq_one_of_prod_one h1.symm done lemma prime_not_one {p : Nat} (h : prime p) : p ≠ 1 := by define at h linarith done theorem Theorem_7_2_4 {p : Nat} (h1 : prime p) : ∀ (l : List Nat), p ∣ prod l → ∃ a ∈ l, p ∣ a := by apply List.rec · -- Base Case. Goal : p ∣ prod [] → ∃ a ∈ [], p ∣ a rewrite [prod_nil] assume h2 : p ∣ 1 show ∃ a ∈ [], p ∣ a from absurd (eq_one_of_dvd_one h2) (prime_not_one h1) done · -- Induction Step fix b : Nat fix L : List Nat assume ih : p ∣ prod L → ∃ a ∈ L, p ∣ a --Goal : p ∣ prod (b :: L) → ∃ a ∈ b :: L, p ∣ a assume h2 : p ∣ prod (b :: L) rewrite [prod_cons] at h2 have h3 : p ∣ b ∨ p ∣ prod L := Theorem_7_2_3 h1 h2 by_cases on h3 · -- Case 1. h3 : p ∣ b apply Exists.intro b show b ∈ b :: L ∧ p ∣ b from And.intro (List.mem_cons_self b L) h3 done · -- Case 2. h3 : p ∣ prod L obtain (a : Nat) (h4 : a ∈ L ∧ p ∣ a) from ih h3 apply Exists.intro a show a ∈ b :: L ∧ p ∣ a from And.intro (List.mem_cons_of_mem b h4.left) h4.right done done done lemma prime_in_list {p : Nat} {l : List Nat} (h1 : prime p) (h2 : all_prime l) (h3 : p ∣ prod l) : p ∈ l := by obtain (a : Nat) (h4 : a ∈ l ∧ p ∣ a) from Theorem_7_2_4 h1 l h3 define at h2 have h5 : prime a := h2 a h4.left have h6 : p = 1 ∨ p = a := dvd_prime h5 h4.right disj_syll h6 (prime_not_one h1) rewrite [h6] show a ∈ l from h4.left done lemma first_le_first {p q : Nat} {l m : List Nat} (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m)) (h3 : prod (p :: l) = prod (q :: m)) : p ≤ q := by define at h1; define at h2 have h4 : q ∣ prod (p :: l) := by define apply Exists.intro (prod m) rewrite [←prod_cons] show prod (p :: l) = prod (q :: m) from h3 done have h5 : all_prime (q :: m) := h2.left rewrite [all_prime_cons] at h5 have h6 : q ∈ p :: l := prime_in_list h5.left h1.left h4 have h7 : nondec (p :: l) := h1.right rewrite [nondec_cons] at h7 rewrite [List.mem_cons] at h6 by_cases on h6 · -- Case 1. h6 : q = p linarith done · -- Case 2. h6 : q ∈ l have h8 : ∀ m ∈ l, p ≤ m := h7.left show p ≤ q from h8 q h6 done done lemma nondec_prime_list_tail {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by define at h define rewrite [all_prime_cons, nondec_cons] at h show all_prime l ∧ nondec l from And.intro h.left.right h.right.right done lemma cons_prod_not_one {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : prod (p :: l) ≠ 1 := by define at h have h1 : all_prime (p :: l) := h.left rewrite [all_prime_cons] at h1 rewrite [prod_cons] by_contra h2 show False from (prime_not_one h1.left) (eq_one_of_prod_one h2) done lemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) : l = [] ↔ prod l = 1 := by apply Iff.intro · -- (→) assume h1 : l = [] rewrite [h1] show prod [] = 1 from prod_nil done · -- (←) contrapos assume h1 : ¬l = [] obtain (p : Nat) (h2 : ∃ (L : List Nat), l = p :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List Nat) (h3 : l = p :: L) from h2 rewrite [h3] at h rewrite [h3] show ¬prod (p :: L) = 1 from cons_prod_not_one h done done lemma prime_pos {p : Nat} (h : prime p) : p > 0 := by define at h linarith done theorem Theorem_7_2_5 : ∀ (l1 l2 : List Nat), nondec_prime_list l1 → nondec_prime_list l2 → prod l1 = prod l2 → l1 = l2 := by apply List.rec · -- Base Case. Goal : ∀ (l2 : List Nat), nondec_prime_list [] → -- nondec_prime_list l2 → prod [] = prod l2 → [] = l2 fix l2 : List Nat assume h1 : nondec_prime_list [] assume h2 : nondec_prime_list l2 assume h3 : prod [] = prod l2 rewrite [prod_nil, eq_comm, ←list_nil_iff_prod_one h2] at h3 show [] = l2 from h3.symm done · -- Induction Step fix p : Nat fix L1 : List Nat assume ih : ∀ (L2 : List Nat), nondec_prime_list L1 → nondec_prime_list L2 → prod L1 = prod L2 → L1 = L2 -- Goal : ∀ (l2 : List Nat), nondec_prime_list (p :: L1) → -- nondec_prime_list l2 → prod (p :: L1) = prod l2 → p :: L1 = l2 fix l2 : List Nat assume h1 : nondec_prime_list (p :: L1) assume h2 : nondec_prime_list l2 assume h3 : prod (p :: L1) = prod l2 have h4 : ¬prod (p :: L1) = 1 := cons_prod_not_one h1 rewrite [h3, ←list_nil_iff_prod_one h2] at h4 obtain (q : Nat) (h5 : ∃ (L : List Nat), l2 = q :: L) from List.exists_cons_of_ne_nil h4 obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5 rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2) rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2) have h7 : p ≤ q := first_le_first h1 h2 h3 have h8 : q ≤ p := first_le_first h2 h1 h3.symm have h9 : p = q := by linarith rewrite [h9, prod_cons, prod_cons] at h3 --h3 : q * prod L1 = q * prod L2 have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1 have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2 define at h2 have h12 : all_prime (q :: L2) := h2.left rewrite [all_prime_cons] at h12 have h13 : q > 0 := prime_pos h12.left have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3 have h15 : L1 = L2 := ih L2 h10 h11 h14 rewrite [h6, h9, h15] rfl done done theorem fund_thm_arith (n : Nat) (h : n ≥ 1) : ∃! (l : List Nat), prime_factorization n l := by exists_unique · -- Existence show ∃ (l : List Nat), prime_factorization n l from exists_prime_factorization n h done · -- Uniqueness fix l1 : List Nat; fix l2 : List Nat assume h1 : prime_factorization n l1 assume h2 : prime_factorization n l2 define at h1; define at h2 have h3 : prod l1 = n := h1.right rewrite [←h2.right] at h3 show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3 done done /- Section 7.3 -/ theorem congr_refl (m : Nat) : ∀ (a : Int), a ≡ a (MOD m) := by fix a : Int define --Goal : ∃ (c : Int), a - a = ↑m * c apply Exists.intro 0 ring done theorem congr_symm {m : Nat} : ∀ {a b : Int}, a ≡ b (MOD m) → b ≡ a (MOD m) := by fix a : Int; fix b : Int assume h1 : a ≡ b (MOD m) define at h1 --h1 : ∃ (c : Int), a - b = ↑m * c define --Goal : ∃ (c : Int), b - a = ↑m * c obtain (c : Int) (h2 : a - b = m * c) from h1 apply Exists.intro (-c) show b - a = m * (-c) from calc b - a _ = -(a - b) := by ring _ = -(m * c) := by rw [h2] _ = m * (-c) := by ring done theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry /- Fundamental properties of congruence classes -/ lemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) : X = Y ↔ X.val = Y.val := Fin.ext_iff lemma val_nat_eq_mod (n k : Nat) : ([k]_(n + 1)).val = k % (n + 1) := by rfl lemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl theorem cc_rep {m : Nat} (X : ZMod m) : ∃ (a : Int), X = [a]_m := match m with | 0 => by apply Exists.intro X rfl done | n + 1 => by apply Exists.intro ↑(X.val) have h1 : X.val < n + 1 := Fin.prop X rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1] rfl done theorem add_class (m : Nat) (a b : Int) : [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm theorem mul_class (m : Nat) (a b : Int) : [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm lemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ [a - b]_m = [0]_m := by apply Iff.intro · -- (→) assume h1 : [a]_m = [b]_m have h2 : a - b = a + (-b) := by ring have h3 : b + (-b) = 0 := by ring show [a - b]_m = [0]_m from calc [a - b]_m _ = [a + (-b)]_m := by rw [h2] _ = [a]_m + [-b]_m := by rw [add_class] _ = [b]_m + [-b]_m := by rw [h1] _ = [b + -b]_m := by rw [add_class] _ = [0]_m := by rw [h3] done · -- (←) assume h1 : [a - b]_m = [0]_m have h2 : b + (a - b) = a := by ring have h3 : b + 0 = b := by ring show [a]_m = [b]_m from calc [a]_m _ = [b + (a - b)]_m := by rw [h2] _ = [b]_m + [a - b]_m := by rw [add_class] _ = [b]_m + [0]_m := by rw [h1] _ = [b + 0]_m := by rw [add_class] _ = [b]_m := by rw [h3] done done lemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) : [a]_m = [0]_m → [-a]_m = [0]_m := by assume h1 : [a]_m = [0]_m have h2 : 0 + (-a) = -a := by ring have h3 : a + (-a) = 0 := by ring show [-a]_m = [0]_m from calc [-a]_m _ = [0 + (-a)]_m := by rw [h2] _ = [0]_m + [-a]_m := by rw [add_class] _ = [a]_m + [-a]_m := by rw [h1] _ = [a + (-a)]_m := by rw [add_class] _ = [0]_m := by rw [h3] done lemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) : [-a]_m = [0]_m ↔ [a]_m = [0]_m := by apply Iff.intro _ (cc_neg_zero_of_cc_zero m a) assume h1 : [-a]_m = [0]_m have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1 have h3 : -(-a) = a := by ring rewrite [h3] at h2 show [a]_m = [0]_m from h2 done lemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl lemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m ↔ m ∣ k := match m with | 0 => by have h : (0 : Int) = (↑(0 : Nat) : Int) := by rfl rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj] apply Iff.intro · -- (→) assume h1 : k = 0 rewrite [h1] show 0 ∣ 0 from dvd_self 0 done · -- (←) assume h1 : 0 ∣ k obtain (c : Nat) (h2 : k = 0 * c) from h1 rewrite [h2] ring done done | n + 1 => by rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero] show k % (n + 1) = 0 ↔ n + 1 ∣ k from (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm done lemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m ↔ ↑m ∣ a := by obtain (k : Nat) (h1 : a = ↑k ∨ a = -↑k) from Int.eq_nat_or_neg a by_cases on h1 · -- Case 1. h1: a = ↑k rewrite [h1, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done · -- Case 2. h1: a = -↑k rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done done theorem cc_eq_iff_congr (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ a ≡ b (MOD m) := calc [a]_m = [b]_m _ ↔ [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b _ ↔ ↑m ∣ (a - b) := cc_zero_iff_dvd m (a - b) _ ↔ a ≡ b (MOD m) := by rfl /- End of fundamental properties of congruence classes -/ lemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m := by have h1 : (↑m : Int) ≠ 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m) show 0 ≤ a % m from Int.emod_nonneg a h1 done lemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m) have h2 : (↑m : Int) > 0 := (Nat.cast_pos).rtl h1 show a % m < m from Int.emod_lt_of_pos a h2 done lemma congr_mod_mod (m : Nat) (a : Int) : a ≡ a % m (MOD m) := by define have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m apply Exists.intro (a / m) show a - a % m = m * (a / m) from calc a - (a % m) _ = m * (a / m) + a % m - a % m := by rw [h1] _ = m * (a / m) := by ring done lemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) := And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a)) theorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) : ∃! (r : Int), 0 ≤ r ∧ r < m ∧ a ≡ r (MOD m) := by exists_unique · -- Existence apply Exists.intro (a % m) show 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) from mod_cmpl_res m a done · -- Uniqueness fix r1 : Int; fix r2 : Int assume h1 : 0 ≤ r1 ∧ r1 < m ∧ a ≡ r1 (MOD m) assume h2 : 0 ≤ r2 ∧ r2 < m ∧ a ≡ r2 (MOD m) have h3 : r1 ≡ r2 (MOD m) := congr_trans (congr_symm h1.right.right) h2.right.right obtain (d : Int) (h4 : r1 - r2 = m * d) from h3 have h5 : r1 - r2 < m * 1 := by linarith have h6 : m * (-1) < r1 - r2 := by linarith rewrite [h4] at h5 --h5 : m * d < m * 1 rewrite [h4] at h6 --h6 : m * -1 < m * d have h7 : (↑m : Int) ≥ 0 := Nat.cast_nonneg m have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7 have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7 have h10 : d = 0 := by linarith show r1 = r2 from calc r1 _ = r1 - r2 + r2 := by ring _ = m * 0 + r2 := by rw [h4, h10] _ = r2 := by ring done done lemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m := (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a) theorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y rewrite [h1, h2] have h3 : a + b = b + a := by ring show [a]_m + [b]_m = [b]_m + [a]_m from calc [a]_m + [b]_m _ = [a + b]_m := add_class m a b _ = [b + a]_m := by rw [h3] _ = [b]_m + [a]_m := (add_class m b a).symm done theorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X rewrite [h1] have h2 : a * 1 = a := by ring show [a]_m * [1]_m = [a]_m from calc [a]_m * [1]_m _ = [a * 1]_m := mul_class m a 1 _ = [a]_m := by rw [h2] done theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry lemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) : [a]_m * [gcd_c2 m a]_m = [1]_m := by set s : Int := gcd_c1 m a have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m define at h1 rewrite [h1, Nat.cast_one] at h2 --h2 : s * ↑m + gcd_c2 m a * ↑a = 1 rewrite [mul_class, cc_eq_iff_congr] define --Goal : ∃ (c : Int), ↑a * gcd_c2 m a - 1 = ↑m * c apply Exists.intro (-s) show a * (gcd_c2 m a) - 1 = m * (-s) from calc a * (gcd_c2 m a) - 1 _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring _ = 1 + m * (-s) - 1 := by rw [h2] _ = m * (-s) := by ring done theorem Theorem_7_3_7 (m a : Nat) : invertible [a]_m ↔ rel_prime m a := by apply Iff.intro · -- (→) assume h1 : invertible [a]_m define at h1 obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1 obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y rewrite [h3, mul_class, cc_eq_iff_congr] at h2 define at h2 obtain (c : Int) (h4 : a * b - 1 = m * c) from h2 rewrite [Exercise_7_2_6] --Goal : ∃ (s t : Int), s * ↑m + t * ↑a = 1 apply Exists.intro (-c) apply Exists.intro b show (-c) * m + b * a = 1 from calc (-c) * m + b * a _ = (-c) * m + (a * b - 1) + 1 := by ring _ = (-c) * m + m * c + 1 := by rw [h4] _ = 1 := by ring done · -- (←) assume h1 : rel_prime m a define show ∃ (Y : ZMod m), [a]_m * Y = [1]_m from Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1) done done /- Section 7.4 -/ section Euler open Euler lemma num_rp_below_base {m : Nat} : num_rp_below m 0 = 0 := by rfl lemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) : num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by have h1 : num_rp_below m (j + 1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : gcd m j = 1 rewrite [if_pos h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1 show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1 done lemma num_rp_below_step_not_rp {m j : Nat} (h : ¬rel_prime m j) : num_rp_below m (j + 1) = num_rp_below m j := by have h1 : num_rp_below m (j +1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : ¬gcd m j = 1 rewrite [if_neg h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j show num_rp_below m (j + 1) = num_rp_below m j from h1 done lemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl #eval phi 10 --Answer: 4 lemma prod_inv_iff_inv {m : Nat} {X : ZMod m} (h1 : invertible X) (Y : ZMod m) : invertible (X * Y) ↔ invertible Y := by apply Iff.intro · -- (→) assume h2 : invertible (X * Y) obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2 apply Exists.intro (X * Z) rewrite [←h3] --Goal : Y * (X * Z) = X * Y * Z ring --Note that ring can do algebra in ZMod m done · -- (←) assume h2 : invertible Y obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1 obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2 apply Exists.intro (Xi * Yi) show (X * Y) * (Xi * Yi) = [1]_m from calc X * Y * (Xi * Yi) _ = (X * Xi) * (Y * Yi) := by ring _ = [1]_m * [1]_m := by rw [h3, h4] _ = [1]_m := Theorem_7_3_6_7 [1]_m done done lemma F_rp_def {m i : Nat} (h : rel_prime m i) : F m i = [i]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h --h : gcd m i = 1 rewrite [if_pos h] at h1 show F m i = [i]_m from h1 done lemma F_not_rp_def {m i : Nat} (h : ¬rel_prime m i) : F m i = [1]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h rewrite [h1, if_neg h] rfl done lemma prod_seq_base {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 0 k f = [1]_m := by rfl lemma prod_seq_step {m : Nat} (n k : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl lemma prod_seq_zero_step {m : Nat} (n : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by rewrite [prod_seq_step, zero_add] rfl done lemma prod_one {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 1 k f = f k := by rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7] rfl done lemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl lemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m := calc [G m a i]_m _ = [(a * i) % m]_m := by rfl _ = [a * i]_m := (cc_eq_mod m (a * i)).symm _ = [a]_m * [i]_m := (mul_class m a i).symm lemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) : rel_prime m (G m a i) ↔ rel_prime m i := by have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1 show rel_prime m (G m a i) ↔ rel_prime m i from calc rel_prime m (G m a i) _ ↔ invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm _ ↔ invertible ([a]_m * [i]_m) := by rw [cc_G] _ ↔ invertible [i]_m := prod_inv_iff_inv h2 ([i]_m) _ ↔ rel_prime m i := Theorem_7_3_7 m i done lemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) : F m (G m a i) = [a]_m * F m i := by have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2 show F m (G m a i) = [a]_m * F m i from calc F m (G m a i) _ = [G m a i]_m := F_rp_def h3 _ = [a]_m * [i]_m := cc_G m a i _ = [a]_m * F m i := by rw [F_rp_def h2] done lemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : ¬rel_prime m i) : F m (G m a i) = [1]_m := by rewrite [←G_rp_iff h1 i] at h2 show F m (G m a i) = [1]_m from F_not_rp_def h2 done lemma FG_prod {m a : Nat} (h1 : rel_prime m a) : ∀ (k : Nat), prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by by_induc · -- Base Case show prod_seq 0 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from calc prod_seq 0 0 ((F m) ∘ (G m a)) _ = [1]_m := prod_seq_base _ _ _ = [a]_m ^ 0 * [1]_m := by ring _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by rw [num_rp_below_base, prod_seq_base] done · -- Induction Step fix k : Nat assume ih : prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) by_cases h2 : rel_prime m k · -- Case 1. h2 : rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([a]_m * F m k) := by rw [FG_rp h1 h2] _ = [a]_m ^ ((num_rp_below m k) + 1) * ((prod_seq k 0 (F m)) * F m k) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_rp h2, prod_seq_zero_step] done · -- Case 2. h2 : ¬rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([1]_m) := by rw [FG_not_rp h1 h2] _ = [a]_m ^ (num_rp_below m k) * (prod_seq k 0 (F m) * ([1]_m)) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_not_rp h2, prod_seq_zero_step, F_not_rp_def h2] done done done lemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by define --Goal : ∀ i < m, G m a i < m fix i : Nat assume h1 : i < m rewrite [G_def] --Goal : a * i % m < m show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m) done lemma left_inv_one_one_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g' (g i) = i) : one_one_below n g := sorry lemma right_inv_onto_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g (g' i) = i) (h2 : maps_below n g') : onto_below n g := by define at h2; define fix k : Nat assume h3 : k < n apply Exists.intro (g' k) show g' k < n ∧ g (g' k) = k from And.intro (h2 k h3) (h1 k h3) done lemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m * [inv_mod m a]_m = [1]_m := by have h2 : 0 ≤ (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a) show [a]_m * [inv_mod m a]_m = [1]_m from calc [a]_m * [inv_mod m a]_m _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2] _ = [a]_m * [gcd_c2 m a]_m := by rw [←cc_eq_mod] _ = [1]_m := gcd_c2_inv h1 done lemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m := calc a * (b % m) % m = a % m * (b % m % m) % m := Nat.mul_mod _ _ _ _ = a % m * (b % m) % m := by rw [Nat.mod_mod] _ = a * b % m := (Nat.mul_mod _ _ _).symm lemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm] rfl done theorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] : ↑a ≡ ↑b (MOD m) ↔ a % m = b % m := sorry lemma mul_inv_mod_cancel {m a i : Nat} [NeZero m] (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1 rewrite [mul_class, cc_eq_iff_congr, ←Nat.cast_mul, ←Nat.cast_one, congr_iff_mod_eq_Nat] at h3 show a * inv_mod m a * i % m = i from calc a * (inv_mod m a) * i % m _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = 1 % m * i % m := by rw [h3] _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = i % m := by rw [one_mul] _ = i := Nat.mod_eq_of_lt h2 done lemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl lemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, G m a (Ginv m a i) = i := by fix i : Nat assume h2 : i < m show G m a (Ginv m a i) = i from calc G m a (Ginv m a i) _ = a * ((inv_mod m a * i) % m) % m := by rfl _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, Ginv m a (G m a i) = i := by fix i : Nat assume h2 : i < m show Ginv m a (G m a i) = i from calc Ginv m a (G m a i) _ = inv_mod m a * ((a * i) % m) % m := by rfl _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc, mul_comm (inv_mod m a)] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_maps_below (m a : Nat) [NeZero m] : maps_below m (Ginv m a) := G_maps_below m (inv_mod m a) lemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : one_one_below m (G m a) := left_inv_one_one_below (Ginv_left_inv h1) lemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : onto_below m (G m a) := right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a) lemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : perm_below m (G m a) := And.intro (G_maps_below m a) (And.intro (G_one_one_below h1) (G_onto_below h1)) --Permuting a product of congruence classes doesn't change product lemma swap_fst (u v : Nat) : swap u v u = v := by define : swap u v u --Goal : (if u = u then v else if u = v then u else u) = v have h : u = u := by rfl rewrite [if_pos h] rfl done lemma swap_snd (u v : Nat) : swap u v v = u := by define : swap u v v by_cases h1 : v = u · -- Case 1. h1 : v = u rewrite [if_pos h1] show v = u from h1 done · -- Case 2. h1 : v ≠ u rewrite [if_neg h1] have h2 : v = v := by rfl rewrite [if_pos h2] rfl done done

lemma swap_other {u v i : Nat} (h1 : i ≠ u) (h2 : i ≠ v) : swap u v i = i

HTPI.swap_other

htpi/HTPILib/Chap7.lean

htpi

/- Copyright 2023 Daniel J. Velleman -/ import HTPILib.Chap6 namespace HTPI /- Definitions -/ lemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by have h : n + 1 > 0 := Nat.succ_pos n show a % (n + 1) < n + 1 from Nat.mod_lt a h done def gcd (a b : Nat) : Nat := match b with | 0 => a | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd (n + 1) (a % (n + 1)) termination_by b mutual def gcd_c1 (a b : Nat) : Int := match b with | 0 => 1 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c2 (n + 1) (a % (n + 1)) --Corresponds to s = t' termination_by b def gcd_c2 (a b : Nat) : Int := match b with | 0 => 0 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c1 (n + 1) (a % (n + 1)) - (gcd_c2 (n + 1) (a % (n + 1))) * ↑(a / (n + 1)) --Corresponds to t = s' - t'q termination_by b end def prime (n : Nat) : Prop := 2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n def prime_factor (p n : Nat) : Prop := prime p ∧ p ∣ n def all_prime (l : List Nat) : Prop := ∀ p ∈ l, prime p def nondec (l : List Nat) : Prop := match l with | [] => True --Of course, True is a proposition that is always true | n :: L => (∀ m ∈ L, n ≤ m) ∧ nondec L def nondec_prime_list (l : List Nat) : Prop := all_prime l ∧ nondec l def prod (l : List Nat) : Nat := match l with | [] => 1 | n :: L => n * (prod L) def prime_factorization (n : Nat) (l : List Nat) : Prop := nondec_prime_list l ∧ prod l = n def rel_prime (a b : Nat) : Prop := gcd a b = 1 def congr_mod (m : Nat) (a b : Int) : Prop := (↑m : Int) ∣ (a - b) def cc (m : Nat) (a : Int) : ZMod m := (↑a : ZMod m) notation:50 a " ≡ " b " (MOD " m ")" => congr_mod m a b notation:max "["a"]_"m:max => cc m a def invertible {m : Nat} (X : ZMod m) : Prop := ∃ (Y : ZMod m), X * Y = [1]_m def num_rp_below (m k : Nat) : Nat := match k with | 0 => 0 | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j def phi (m : Nat) : Nat := num_rp_below m m def prod_seq {m : Nat} (j k : Nat) (f : Nat → ZMod m) : ZMod m := match j with | 0 => [1]_m | n + 1 => prod_seq n k f * f (k + n) def maps_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i < n, g i < n def one_one_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i1 < n, ∀ i2 < n, g i1 = g i2 → i1 = i2 def onto_below (n : Nat) (g : Nat → Nat) : Prop := ∀ k < n, ∃ i < n, g i = k def perm_below (n : Nat) (g : Nat → Nat) : Prop := maps_below n g ∧ one_one_below n g ∧ onto_below n g def inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m) def swap (u v i : Nat) : Nat := if i = u then v else if i = v then u else i namespace Euler --For definitions specific to Euler's theorem def F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m def G (m a i : Nat) : Nat := (a * i) % m def Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i end Euler /- Section 7.1 -/ theorem dvd_mod_of_dvd_a_b {a b d : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ (a % b) := by set q : Nat := a / b have h3 : b * q + a % b = a := Nat.div_add_mod a b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 define --Goal : ∃ (c : Nat), a % b = d * c apply Exists.intro (j - k * q) show a % b = d * (j - k * q) from calc a % b _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm _ = a - b * q := by rw [h3] _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc] _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm done theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry #eval gcd 672 161 --Answer: 7 lemma gcd_base (a : Nat) : gcd a 0 = a := by rfl lemma gcd_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd a b = gcd b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1)) rfl done lemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b ≠ 0) : a % b < b := by have h1 : b > 0 := Nat.pos_of_ne_zero h show a % b < b from Nat.mod_lt a h1 done lemma dvd_self (n : Nat) : n ∣ n := by apply Exists.intro 1 ring done theorem gcd_dvd : ∀ (b a : Nat), (gcd a b) ∣ a ∧ (gcd a b) ∣ b := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd a b_1) ∣ a ∧ (gcd a b_1) ∣ b_1 fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_base] --Goal: a ∣ a ∧ a ∣ 0 apply And.intro (dvd_self a) define apply Exists.intro 0 rfl done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_nonzero a h1] --Goal : gcd b (a % b) ∣ a ∧ gcd b (a % b) ∣ b have h2 : a % b < b := mod_nonzero_lt a h1 have h3 : (gcd b (a % b)) ∣ b ∧ (gcd b (a % b)) ∣ (a % b) := ih (a % b) h2 b apply And.intro _ h3.left show (gcd b (a % b)) ∣ a from dvd_a_of_dvd_b_mod h3.left h3.right done done theorem gcd_dvd_left (a b : Nat) : (gcd a b) ∣ a := (gcd_dvd b a).left theorem gcd_dvd_right (a b : Nat) : (gcd a b) ∣ b := (gcd_dvd b a).right lemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl lemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c1 a b = gcd_c2 b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done lemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl lemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * ↑(a / b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done theorem gcd_lin_comb : ∀ (b a : Nat), (gcd_c1 a b) * ↑a + (gcd_c2 a b) * ↑b = ↑(gcd a b) := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd_c1 a b_1) * ↑a + (gcd_c2 a b_1) * ↑b_1 = ↑(gcd a b_1) fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base] --Goal : 1 * ↑a + 0 * ↑0 = ↑a ring done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1] --Goal : gcd_c2 b (a % b) * ↑a + -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * ↑(a / b)) * ↑b = -- ↑(gcd b (a % b)) set r : Nat := a % b set q : Nat := a / b set s : Int := gcd_c1 b r set t : Int := gcd_c2 b r --Goal : t * ↑a + (s - t * ↑q) * ↑b = ↑(gcd b r) have h2 : r < b := mod_nonzero_lt a h1 have h3 : s * ↑b + t * ↑r = ↑(gcd b r) := ih r h2 b have h4 : b * q + r = a := Nat.div_add_mod a b rewrite [←h3, ←h4] rewrite [Nat.cast_add, Nat.cast_mul] --Goal : t * (↑b * ↑q + ↑r) + (s - t * ↑q) * ↑b = s * ↑b + t * ↑r ring done done #eval gcd_c1 672 161 --Answer: 6 #eval gcd_c2 672 161 --Answer: -25 --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161 theorem Theorem_7_1_6 {d a b : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ gcd a b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑d ∣ ↑(gcd a b) set s : Int := gcd_c1 a b set t : Int := gcd_c2 a b have h3 : s * ↑a + t * ↑b = ↑(gcd a b) := gcd_lin_comb b a rewrite [←h3] --Goal : ↑d ∣ s * ↑a + t * ↑b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul] --Goal : ↑d ∣ s * (↑d * ↑j) + t * (↑d * ↑k) define apply Exists.intro (s * ↑j + t * ↑k) ring done /- Section 7.2 -/ theorem dvd_trans {a b c : Nat} (h1 : a ∣ b) (h2 : b ∣ c) : a ∣ c := by define at h1; define at h2; define obtain (m : Nat) (h3 : b = a * m) from h1 obtain (n : Nat) (h4 : c = b * n) from h2 rewrite [h3, mul_assoc] at h4 apply Exists.intro (m * n) show c = a * (m * n) from h4 done lemma exists_prime_factor : ∀ (n : Nat), 2 ≤ n → ∃ (p : Nat), prime_factor p n := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, 2 ≤ n_1 → ∃ (p : Nat), prime_factor p n_1 assume h1 : 2 ≤ n by_cases h2 : prime n · -- Case 1. h2 : prime n apply Exists.intro n define --Goal : prime n ∧ n ∣ n show prime n ∧ n ∣ n from And.intro h2 (dvd_self n) done · -- Case 2. h2 : ¬prime n define at h2 --h2 : ¬(2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n) demorgan at h2 disj_syll h2 h1 obtain (a : Nat) (h3 : ∃ (b : Nat), a * b = n ∧ a < n ∧ b < n) from h2 obtain (b : Nat) (h4 : a * b = n ∧ a < n ∧ b < n) from h3 have h5 : 2 ≤ a := by by_contra h6 have h7 : a ≤ 1 := by linarith have h8 : n ≤ b := calc n _ = a * b := h4.left.symm _ ≤ 1 * b := by rel [h7] _ = b := by ring linarith --n ≤ b contradicts b < n done have h6 : ∃ (p : Nat), prime_factor p a := ih a h4.right.left h5 obtain (p : Nat) (h7 : prime_factor p a) from h6 apply Exists.intro p define --Goal : prime p ∧ p ∣ n define at h7 --h7 : prime p ∧ p ∣ a apply And.intro h7.left have h8 : a ∣ n := by apply Exists.intro b show n = a * b from (h4.left).symm done show p ∣ n from dvd_trans h7.right h8 done done lemma exists_least_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q := by set S : Set Nat := {p : Nat | prime_factor p n} have h2 : ∃ (p : Nat), p ∈ S := exists_prime_factor n h show ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q from well_ord_princ S h2 done lemma all_prime_nil : all_prime [] := by define --Goal : ∀ p ∈ [], prime p fix p : Nat contrapos --Goal : ¬prime p → p ∉ [] assume h1 : ¬prime p show p ∉ [] from List.not_mem_nil p done lemma all_prime_cons (n : Nat) (L : List Nat) : all_prime (n :: L) ↔ prime n ∧ all_prime L := by apply Iff.intro · -- (→) assume h1 : all_prime (n :: L) --Goal : prime n ∧ all_prime L define at h1 --h1 : ∀ p ∈ n :: L, prime p apply And.intro (h1 n (List.mem_cons_self n L)) define --Goal : ∀ p ∈ L, prime p fix p : Nat assume h2 : p ∈ L show prime p from h1 p (List.mem_cons_of_mem n h2) done · -- (←) assume h1 : prime n ∧ all_prime L --Goal : all_prime (n :: l) define : all_prime L at h1 define fix p : Nat assume h2 : p ∈ n :: L rewrite [List.mem_cons] at h2 --h2 : p = n ∨ p ∈ L by_cases on h2 · -- Case 1. h2 : p = n rewrite [h2] show prime n from h1.left done · -- Case 2. h2 : p ∈ L show prime p from h1.right p h2 done done done lemma nondec_nil : nondec [] := by define --Goal : True trivial --trivial proves some obviously true statements, such as True done lemma nondec_cons (n : Nat) (L : List Nat) : nondec (n :: L) ↔ (∀ m ∈ L, n ≤ m) ∧ nondec L := by rfl lemma prod_nil : prod [] = 1 := by rfl lemma prod_cons : prod (n :: L) = n * (prod L) := by rfl lemma exists_cons_of_length_eq_succ {A : Type} {l : List A} {n : Nat} (h : l.length = n + 1) : ∃ (a : A) (L : List A), l = a :: L ∧ L.length = n := by have h1 : ¬l.length = 0 := by linarith rewrite [List.length_eq_zero] at h1 obtain (a : A) (h2 : ∃ (L : List A), l = a :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List A) (h3 : l = a :: L) from h2 apply Exists.intro a apply Exists.intro L apply And.intro h3 have h4 : (a :: L).length = L.length + 1 := List.length_cons a L rewrite [←h3, h] at h4 show L.length = n from (Nat.add_right_cancel h4).symm done lemma list_elt_dvd_prod_by_length (a : Nat) : ∀ (n : Nat), ∀ (l : List Nat), l.length = n → a ∈ l → a ∣ prod l := by by_induc · --Base Case fix l : List Nat assume h1 : l.length = 0 rewrite [List.length_eq_zero] at h1 --h1 : l = [] rewrite [h1] --Goal : a ∈ [] → a ∣ prod [] contrapos assume h2 : ¬a ∣ prod [] show a ∉ [] from List.not_mem_nil a done · -- Induction Step fix n : Nat assume ih : ∀ (l : List Nat), List.length l = n → a ∈ l → a ∣ prod l fix l : List Nat assume h1 : l.length = n + 1 --Goal : a ∈ l → a ∣ prod l obtain (b : Nat) (h2 : ∃ (L : List Nat), l = b :: L ∧ L.length = n) from exists_cons_of_length_eq_succ h1 obtain (L : List Nat) (h3 : l = b :: L ∧ L.length = n) from h2 have h4 : a ∈ L → a ∣ prod L := ih L h3.right assume h5 : a ∈ l rewrite [h3.left, prod_cons] --Goal : a ∣ b * prod L rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b ∨ a ∈ L by_cases on h5 · -- Case 1. h5 : a = b apply Exists.intro (prod L) rewrite [h5] rfl done · -- Case 2. h5 : a ∈ L have h6 : a ∣ prod L := h4 h5 have h7 : prod L ∣ b * prod L := by apply Exists.intro b ring done show a ∣ b * prod L from dvd_trans h6 h7 done done done lemma list_elt_dvd_prod {a : Nat} {l : List Nat} (h : a ∈ l) : a ∣ prod l := by set n : Nat := l.length have h1 : l.length = n := by rfl show a ∣ prod l from list_elt_dvd_prod_by_length a n l h1 h done lemma exists_prime_factorization : ∀ (n : Nat), n ≥ 1 → ∃ (l : List Nat), prime_factorization n l := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, n_1 ≥ 1 → ∃ (l : List Nat), prime_factorization n_1 l assume h1 : n ≥ 1 by_cases h2 : n = 1 · -- Case 1. h2 : n = 1 apply Exists.intro [] define apply And.intro · -- Proof of nondec_prime_list [] define show all_prime [] ∧ nondec [] from And.intro all_prime_nil nondec_nil done · -- Proof of prod [] = n rewrite [prod_nil, h2] rfl done done · -- Case 2. h2 : n ≠ 1 have h3 : n ≥ 2 := lt_of_le_of_ne' h1 h2 obtain (p : Nat) (h4 : prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q) from exists_least_prime_factor h3 have p_prime_factor : prime_factor p n := h4.left define at p_prime_factor have p_prime : prime p := p_prime_factor.left have p_dvd_n : p ∣ n := p_prime_factor.right have p_least : ∀ (q : Nat), prime_factor q n → p ≤ q := h4.right obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n have h5 : m ≠ 0 := by contradict h1 with h6 have h7 : n = 0 := calc n _ = p * m := n_eq_pm _ = p * 0 := by rw [h6] _ = 0 := by ring rewrite [h7] decide done have m_pos : 0 < m := Nat.pos_of_ne_zero h5 have m_lt_n : m < n := by define at p_prime show m < n from calc m _ < m + m := by linarith _ = 2 * m := by ring _ ≤ p * m := by rel [p_prime.left] _ = n := n_eq_pm.symm done obtain (L : List Nat) (h6 : prime_factorization m L) from ih m m_lt_n m_pos define at h6 have ndpl_L : nondec_prime_list L := h6.left define at ndpl_L apply Exists.intro (p :: L) define apply And.intro · -- Proof of nondec_prime_list (p :: L) define apply And.intro · -- Proof of all_prime (p :: L) rewrite [all_prime_cons] show prime p ∧ all_prime L from And.intro p_prime ndpl_L.left done · -- Proof of nondec (p :: L) rewrite [nondec_cons] apply And.intro _ ndpl_L.right fix q : Nat assume q_in_L : q ∈ L have h7 : q ∣ prod L := list_elt_dvd_prod q_in_L rewrite [h6.right] at h7 --h7 : q ∣ m have h8 : m ∣ n := by apply Exists.intro p rewrite [n_eq_pm] ring done have q_dvd_n : q ∣ n := dvd_trans h7 h8 have ap_L : all_prime L := ndpl_L.left define at ap_L have q_prime_factor : prime_factor q n := And.intro (ap_L q q_in_L) q_dvd_n show p ≤ q from p_least q q_prime_factor done done · -- Proof of prod (p :: L) = n rewrite [prod_cons, h6.right, n_eq_pm] rfl done done done theorem Theorem_7_2_2 {a b c : Nat} (h1 : c ∣ a * b) (h2 : rel_prime a c) : c ∣ b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑c ∣ ↑b define at h1; define at h2; define obtain (j : Nat) (h3 : a * b = c * j) from h1 set s : Int := gcd_c1 a c set t : Int := gcd_c2 a c have h4 : s * ↑a + t * ↑c = ↑(gcd a c) := gcd_lin_comb c a rewrite [h2, Nat.cast_one] at h4 --h4 : s * ↑a + t * ↑c = (1 : Int) apply Exists.intro (s * ↑j + t * ↑b) show ↑b = ↑c * (s * ↑j + t * ↑b) from calc ↑b _ = (1 : Int) * ↑b := (one_mul _).symm _ = (s * ↑a + t * ↑c) * ↑b := by rw [h4] _ = s * (↑a * ↑b) + t * ↑c * ↑b := by ring _ = s * (↑c * ↑j) + t * ↑c * ↑b := by rw [←Nat.cast_mul a b, h3, Nat.cast_mul c j] _ = ↑c * (s * ↑j + t * ↑b) := by ring done lemma le_nonzero_prod_left {a b : Nat} (h : a * b ≠ 0) : a ≤ a * b := by have h1 : b ≠ 0 := by contradict h with h1 rewrite [h1] ring done have h2 : 1 ≤ b := Nat.pos_of_ne_zero h1 show a ≤ a * b from calc a = a * 1 := (mul_one a).symm _ ≤ a * b := by rel [h2] done lemma le_nonzero_prod_right {a b : Nat} (h : a * b ≠ 0) : b ≤ a * b := by rewrite [mul_comm] rewrite [mul_comm] at h show b ≤ b * a from le_nonzero_prod_left h done lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry lemma rel_prime_of_prime_not_dvd {a p : Nat} (h1 : prime p) (h2 : ¬p ∣ a) : rel_prime a p := by have h3 : gcd a p ∣ a := gcd_dvd_left a p have h4 : gcd a p ∣ p := gcd_dvd_right a p have h5 : gcd a p = 1 ∨ gcd a p = p := dvd_prime h1 h4 have h6 : gcd a p ≠ p := by contradict h2 with h6 rewrite [h6] at h3 show p ∣ a from h3 done disj_syll h5 h6 show rel_prime a p from h5 done theorem Theorem_7_2_3 {a b p : Nat} (h1 : prime p) (h2 : p ∣ a * b) : p ∣ a ∨ p ∣ b := by or_right with h3 have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3 show p ∣ b from Theorem_7_2_2 h2 h4 done lemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a ≥ 1 := by have h1 : a ≠ 0 := by by_contra h1 rewrite [h1] at h contradict h linarith done show a ≥ 1 from Nat.pos_of_ne_zero h1 done lemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by have h1 : a ≥ 1 := ge_one_of_prod_one h have h2 : a * b ≠ 0 := by linarith have h3 : a ≤ a * b := le_nonzero_prod_left h2 rewrite [h] at h3 show a = 1 from Nat.le_antisymm h3 h1 done lemma eq_one_of_dvd_one {n : Nat} (h : n ∣ 1) : n = 1 := by obtain (j : Nat) (h1 : 1 = n * j) from h show n = 1 from eq_one_of_prod_one h1.symm done lemma prime_not_one {p : Nat} (h : prime p) : p ≠ 1 := by define at h linarith done theorem Theorem_7_2_4 {p : Nat} (h1 : prime p) : ∀ (l : List Nat), p ∣ prod l → ∃ a ∈ l, p ∣ a := by apply List.rec · -- Base Case. Goal : p ∣ prod [] → ∃ a ∈ [], p ∣ a rewrite [prod_nil] assume h2 : p ∣ 1 show ∃ a ∈ [], p ∣ a from absurd (eq_one_of_dvd_one h2) (prime_not_one h1) done · -- Induction Step fix b : Nat fix L : List Nat assume ih : p ∣ prod L → ∃ a ∈ L, p ∣ a --Goal : p ∣ prod (b :: L) → ∃ a ∈ b :: L, p ∣ a assume h2 : p ∣ prod (b :: L) rewrite [prod_cons] at h2 have h3 : p ∣ b ∨ p ∣ prod L := Theorem_7_2_3 h1 h2 by_cases on h3 · -- Case 1. h3 : p ∣ b apply Exists.intro b show b ∈ b :: L ∧ p ∣ b from And.intro (List.mem_cons_self b L) h3 done · -- Case 2. h3 : p ∣ prod L obtain (a : Nat) (h4 : a ∈ L ∧ p ∣ a) from ih h3 apply Exists.intro a show a ∈ b :: L ∧ p ∣ a from And.intro (List.mem_cons_of_mem b h4.left) h4.right done done done lemma prime_in_list {p : Nat} {l : List Nat} (h1 : prime p) (h2 : all_prime l) (h3 : p ∣ prod l) : p ∈ l := by obtain (a : Nat) (h4 : a ∈ l ∧ p ∣ a) from Theorem_7_2_4 h1 l h3 define at h2 have h5 : prime a := h2 a h4.left have h6 : p = 1 ∨ p = a := dvd_prime h5 h4.right disj_syll h6 (prime_not_one h1) rewrite [h6] show a ∈ l from h4.left done lemma first_le_first {p q : Nat} {l m : List Nat} (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m)) (h3 : prod (p :: l) = prod (q :: m)) : p ≤ q := by define at h1; define at h2 have h4 : q ∣ prod (p :: l) := by define apply Exists.intro (prod m) rewrite [←prod_cons] show prod (p :: l) = prod (q :: m) from h3 done have h5 : all_prime (q :: m) := h2.left rewrite [all_prime_cons] at h5 have h6 : q ∈ p :: l := prime_in_list h5.left h1.left h4 have h7 : nondec (p :: l) := h1.right rewrite [nondec_cons] at h7 rewrite [List.mem_cons] at h6 by_cases on h6 · -- Case 1. h6 : q = p linarith done · -- Case 2. h6 : q ∈ l have h8 : ∀ m ∈ l, p ≤ m := h7.left show p ≤ q from h8 q h6 done done lemma nondec_prime_list_tail {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by define at h define rewrite [all_prime_cons, nondec_cons] at h show all_prime l ∧ nondec l from And.intro h.left.right h.right.right done lemma cons_prod_not_one {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : prod (p :: l) ≠ 1 := by define at h have h1 : all_prime (p :: l) := h.left rewrite [all_prime_cons] at h1 rewrite [prod_cons] by_contra h2 show False from (prime_not_one h1.left) (eq_one_of_prod_one h2) done lemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) : l = [] ↔ prod l = 1 := by apply Iff.intro · -- (→) assume h1 : l = [] rewrite [h1] show prod [] = 1 from prod_nil done · -- (←) contrapos assume h1 : ¬l = [] obtain (p : Nat) (h2 : ∃ (L : List Nat), l = p :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List Nat) (h3 : l = p :: L) from h2 rewrite [h3] at h rewrite [h3] show ¬prod (p :: L) = 1 from cons_prod_not_one h done done lemma prime_pos {p : Nat} (h : prime p) : p > 0 := by define at h linarith done theorem Theorem_7_2_5 : ∀ (l1 l2 : List Nat), nondec_prime_list l1 → nondec_prime_list l2 → prod l1 = prod l2 → l1 = l2 := by apply List.rec · -- Base Case. Goal : ∀ (l2 : List Nat), nondec_prime_list [] → -- nondec_prime_list l2 → prod [] = prod l2 → [] = l2 fix l2 : List Nat assume h1 : nondec_prime_list [] assume h2 : nondec_prime_list l2 assume h3 : prod [] = prod l2 rewrite [prod_nil, eq_comm, ←list_nil_iff_prod_one h2] at h3 show [] = l2 from h3.symm done · -- Induction Step fix p : Nat fix L1 : List Nat assume ih : ∀ (L2 : List Nat), nondec_prime_list L1 → nondec_prime_list L2 → prod L1 = prod L2 → L1 = L2 -- Goal : ∀ (l2 : List Nat), nondec_prime_list (p :: L1) → -- nondec_prime_list l2 → prod (p :: L1) = prod l2 → p :: L1 = l2 fix l2 : List Nat assume h1 : nondec_prime_list (p :: L1) assume h2 : nondec_prime_list l2 assume h3 : prod (p :: L1) = prod l2 have h4 : ¬prod (p :: L1) = 1 := cons_prod_not_one h1 rewrite [h3, ←list_nil_iff_prod_one h2] at h4 obtain (q : Nat) (h5 : ∃ (L : List Nat), l2 = q :: L) from List.exists_cons_of_ne_nil h4 obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5 rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2) rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2) have h7 : p ≤ q := first_le_first h1 h2 h3 have h8 : q ≤ p := first_le_first h2 h1 h3.symm have h9 : p = q := by linarith rewrite [h9, prod_cons, prod_cons] at h3 --h3 : q * prod L1 = q * prod L2 have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1 have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2 define at h2 have h12 : all_prime (q :: L2) := h2.left rewrite [all_prime_cons] at h12 have h13 : q > 0 := prime_pos h12.left have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3 have h15 : L1 = L2 := ih L2 h10 h11 h14 rewrite [h6, h9, h15] rfl done done theorem fund_thm_arith (n : Nat) (h : n ≥ 1) : ∃! (l : List Nat), prime_factorization n l := by exists_unique · -- Existence show ∃ (l : List Nat), prime_factorization n l from exists_prime_factorization n h done · -- Uniqueness fix l1 : List Nat; fix l2 : List Nat assume h1 : prime_factorization n l1 assume h2 : prime_factorization n l2 define at h1; define at h2 have h3 : prod l1 = n := h1.right rewrite [←h2.right] at h3 show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3 done done /- Section 7.3 -/ theorem congr_refl (m : Nat) : ∀ (a : Int), a ≡ a (MOD m) := by fix a : Int define --Goal : ∃ (c : Int), a - a = ↑m * c apply Exists.intro 0 ring done theorem congr_symm {m : Nat} : ∀ {a b : Int}, a ≡ b (MOD m) → b ≡ a (MOD m) := by fix a : Int; fix b : Int assume h1 : a ≡ b (MOD m) define at h1 --h1 : ∃ (c : Int), a - b = ↑m * c define --Goal : ∃ (c : Int), b - a = ↑m * c obtain (c : Int) (h2 : a - b = m * c) from h1 apply Exists.intro (-c) show b - a = m * (-c) from calc b - a _ = -(a - b) := by ring _ = -(m * c) := by rw [h2] _ = m * (-c) := by ring done theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry /- Fundamental properties of congruence classes -/ lemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) : X = Y ↔ X.val = Y.val := Fin.ext_iff lemma val_nat_eq_mod (n k : Nat) : ([k]_(n + 1)).val = k % (n + 1) := by rfl lemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl theorem cc_rep {m : Nat} (X : ZMod m) : ∃ (a : Int), X = [a]_m := match m with | 0 => by apply Exists.intro X rfl done | n + 1 => by apply Exists.intro ↑(X.val) have h1 : X.val < n + 1 := Fin.prop X rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1] rfl done theorem add_class (m : Nat) (a b : Int) : [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm theorem mul_class (m : Nat) (a b : Int) : [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm lemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ [a - b]_m = [0]_m := by apply Iff.intro · -- (→) assume h1 : [a]_m = [b]_m have h2 : a - b = a + (-b) := by ring have h3 : b + (-b) = 0 := by ring show [a - b]_m = [0]_m from calc [a - b]_m _ = [a + (-b)]_m := by rw [h2] _ = [a]_m + [-b]_m := by rw [add_class] _ = [b]_m + [-b]_m := by rw [h1] _ = [b + -b]_m := by rw [add_class] _ = [0]_m := by rw [h3] done · -- (←) assume h1 : [a - b]_m = [0]_m have h2 : b + (a - b) = a := by ring have h3 : b + 0 = b := by ring show [a]_m = [b]_m from calc [a]_m _ = [b + (a - b)]_m := by rw [h2] _ = [b]_m + [a - b]_m := by rw [add_class] _ = [b]_m + [0]_m := by rw [h1] _ = [b + 0]_m := by rw [add_class] _ = [b]_m := by rw [h3] done done lemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) : [a]_m = [0]_m → [-a]_m = [0]_m := by assume h1 : [a]_m = [0]_m have h2 : 0 + (-a) = -a := by ring have h3 : a + (-a) = 0 := by ring show [-a]_m = [0]_m from calc [-a]_m _ = [0 + (-a)]_m := by rw [h2] _ = [0]_m + [-a]_m := by rw [add_class] _ = [a]_m + [-a]_m := by rw [h1] _ = [a + (-a)]_m := by rw [add_class] _ = [0]_m := by rw [h3] done lemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) : [-a]_m = [0]_m ↔ [a]_m = [0]_m := by apply Iff.intro _ (cc_neg_zero_of_cc_zero m a) assume h1 : [-a]_m = [0]_m have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1 have h3 : -(-a) = a := by ring rewrite [h3] at h2 show [a]_m = [0]_m from h2 done lemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl lemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m ↔ m ∣ k := match m with | 0 => by have h : (0 : Int) = (↑(0 : Nat) : Int) := by rfl rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj] apply Iff.intro · -- (→) assume h1 : k = 0 rewrite [h1] show 0 ∣ 0 from dvd_self 0 done · -- (←) assume h1 : 0 ∣ k obtain (c : Nat) (h2 : k = 0 * c) from h1 rewrite [h2] ring done done | n + 1 => by rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero] show k % (n + 1) = 0 ↔ n + 1 ∣ k from (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm done lemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m ↔ ↑m ∣ a := by obtain (k : Nat) (h1 : a = ↑k ∨ a = -↑k) from Int.eq_nat_or_neg a by_cases on h1 · -- Case 1. h1: a = ↑k rewrite [h1, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done · -- Case 2. h1: a = -↑k rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done done theorem cc_eq_iff_congr (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ a ≡ b (MOD m) := calc [a]_m = [b]_m _ ↔ [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b _ ↔ ↑m ∣ (a - b) := cc_zero_iff_dvd m (a - b) _ ↔ a ≡ b (MOD m) := by rfl /- End of fundamental properties of congruence classes -/ lemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m := by have h1 : (↑m : Int) ≠ 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m) show 0 ≤ a % m from Int.emod_nonneg a h1 done lemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m) have h2 : (↑m : Int) > 0 := (Nat.cast_pos).rtl h1 show a % m < m from Int.emod_lt_of_pos a h2 done lemma congr_mod_mod (m : Nat) (a : Int) : a ≡ a % m (MOD m) := by define have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m apply Exists.intro (a / m) show a - a % m = m * (a / m) from calc a - (a % m) _ = m * (a / m) + a % m - a % m := by rw [h1] _ = m * (a / m) := by ring done lemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) := And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a)) theorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) : ∃! (r : Int), 0 ≤ r ∧ r < m ∧ a ≡ r (MOD m) := by exists_unique · -- Existence apply Exists.intro (a % m) show 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) from mod_cmpl_res m a done · -- Uniqueness fix r1 : Int; fix r2 : Int assume h1 : 0 ≤ r1 ∧ r1 < m ∧ a ≡ r1 (MOD m) assume h2 : 0 ≤ r2 ∧ r2 < m ∧ a ≡ r2 (MOD m) have h3 : r1 ≡ r2 (MOD m) := congr_trans (congr_symm h1.right.right) h2.right.right obtain (d : Int) (h4 : r1 - r2 = m * d) from h3 have h5 : r1 - r2 < m * 1 := by linarith have h6 : m * (-1) < r1 - r2 := by linarith rewrite [h4] at h5 --h5 : m * d < m * 1 rewrite [h4] at h6 --h6 : m * -1 < m * d have h7 : (↑m : Int) ≥ 0 := Nat.cast_nonneg m have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7 have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7 have h10 : d = 0 := by linarith show r1 = r2 from calc r1 _ = r1 - r2 + r2 := by ring _ = m * 0 + r2 := by rw [h4, h10] _ = r2 := by ring done done lemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m := (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a) theorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y rewrite [h1, h2] have h3 : a + b = b + a := by ring show [a]_m + [b]_m = [b]_m + [a]_m from calc [a]_m + [b]_m _ = [a + b]_m := add_class m a b _ = [b + a]_m := by rw [h3] _ = [b]_m + [a]_m := (add_class m b a).symm done theorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X rewrite [h1] have h2 : a * 1 = a := by ring show [a]_m * [1]_m = [a]_m from calc [a]_m * [1]_m _ = [a * 1]_m := mul_class m a 1 _ = [a]_m := by rw [h2] done theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry lemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) : [a]_m * [gcd_c2 m a]_m = [1]_m := by set s : Int := gcd_c1 m a have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m define at h1 rewrite [h1, Nat.cast_one] at h2 --h2 : s * ↑m + gcd_c2 m a * ↑a = 1 rewrite [mul_class, cc_eq_iff_congr] define --Goal : ∃ (c : Int), ↑a * gcd_c2 m a - 1 = ↑m * c apply Exists.intro (-s) show a * (gcd_c2 m a) - 1 = m * (-s) from calc a * (gcd_c2 m a) - 1 _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring _ = 1 + m * (-s) - 1 := by rw [h2] _ = m * (-s) := by ring done theorem Theorem_7_3_7 (m a : Nat) : invertible [a]_m ↔ rel_prime m a := by apply Iff.intro · -- (→) assume h1 : invertible [a]_m define at h1 obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1 obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y rewrite [h3, mul_class, cc_eq_iff_congr] at h2 define at h2 obtain (c : Int) (h4 : a * b - 1 = m * c) from h2 rewrite [Exercise_7_2_6] --Goal : ∃ (s t : Int), s * ↑m + t * ↑a = 1 apply Exists.intro (-c) apply Exists.intro b show (-c) * m + b * a = 1 from calc (-c) * m + b * a _ = (-c) * m + (a * b - 1) + 1 := by ring _ = (-c) * m + m * c + 1 := by rw [h4] _ = 1 := by ring done · -- (←) assume h1 : rel_prime m a define show ∃ (Y : ZMod m), [a]_m * Y = [1]_m from Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1) done done /- Section 7.4 -/ section Euler open Euler lemma num_rp_below_base {m : Nat} : num_rp_below m 0 = 0 := by rfl lemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) : num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by have h1 : num_rp_below m (j + 1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : gcd m j = 1 rewrite [if_pos h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1 show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1 done lemma num_rp_below_step_not_rp {m j : Nat} (h : ¬rel_prime m j) : num_rp_below m (j + 1) = num_rp_below m j := by have h1 : num_rp_below m (j +1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : ¬gcd m j = 1 rewrite [if_neg h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j show num_rp_below m (j + 1) = num_rp_below m j from h1 done lemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl #eval phi 10 --Answer: 4 lemma prod_inv_iff_inv {m : Nat} {X : ZMod m} (h1 : invertible X) (Y : ZMod m) : invertible (X * Y) ↔ invertible Y := by apply Iff.intro · -- (→) assume h2 : invertible (X * Y) obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2 apply Exists.intro (X * Z) rewrite [←h3] --Goal : Y * (X * Z) = X * Y * Z ring --Note that ring can do algebra in ZMod m done · -- (←) assume h2 : invertible Y obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1 obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2 apply Exists.intro (Xi * Yi) show (X * Y) * (Xi * Yi) = [1]_m from calc X * Y * (Xi * Yi) _ = (X * Xi) * (Y * Yi) := by ring _ = [1]_m * [1]_m := by rw [h3, h4] _ = [1]_m := Theorem_7_3_6_7 [1]_m done done lemma F_rp_def {m i : Nat} (h : rel_prime m i) : F m i = [i]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h --h : gcd m i = 1 rewrite [if_pos h] at h1 show F m i = [i]_m from h1 done lemma F_not_rp_def {m i : Nat} (h : ¬rel_prime m i) : F m i = [1]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h rewrite [h1, if_neg h] rfl done lemma prod_seq_base {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 0 k f = [1]_m := by rfl lemma prod_seq_step {m : Nat} (n k : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl lemma prod_seq_zero_step {m : Nat} (n : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by rewrite [prod_seq_step, zero_add] rfl done lemma prod_one {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 1 k f = f k := by rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7] rfl done lemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl lemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m := calc [G m a i]_m _ = [(a * i) % m]_m := by rfl _ = [a * i]_m := (cc_eq_mod m (a * i)).symm _ = [a]_m * [i]_m := (mul_class m a i).symm lemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) : rel_prime m (G m a i) ↔ rel_prime m i := by have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1 show rel_prime m (G m a i) ↔ rel_prime m i from calc rel_prime m (G m a i) _ ↔ invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm _ ↔ invertible ([a]_m * [i]_m) := by rw [cc_G] _ ↔ invertible [i]_m := prod_inv_iff_inv h2 ([i]_m) _ ↔ rel_prime m i := Theorem_7_3_7 m i done lemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) : F m (G m a i) = [a]_m * F m i := by have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2 show F m (G m a i) = [a]_m * F m i from calc F m (G m a i) _ = [G m a i]_m := F_rp_def h3 _ = [a]_m * [i]_m := cc_G m a i _ = [a]_m * F m i := by rw [F_rp_def h2] done lemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : ¬rel_prime m i) : F m (G m a i) = [1]_m := by rewrite [←G_rp_iff h1 i] at h2 show F m (G m a i) = [1]_m from F_not_rp_def h2 done lemma FG_prod {m a : Nat} (h1 : rel_prime m a) : ∀ (k : Nat), prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by by_induc · -- Base Case show prod_seq 0 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from calc prod_seq 0 0 ((F m) ∘ (G m a)) _ = [1]_m := prod_seq_base _ _ _ = [a]_m ^ 0 * [1]_m := by ring _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by rw [num_rp_below_base, prod_seq_base] done · -- Induction Step fix k : Nat assume ih : prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) by_cases h2 : rel_prime m k · -- Case 1. h2 : rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([a]_m * F m k) := by rw [FG_rp h1 h2] _ = [a]_m ^ ((num_rp_below m k) + 1) * ((prod_seq k 0 (F m)) * F m k) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_rp h2, prod_seq_zero_step] done · -- Case 2. h2 : ¬rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([1]_m) := by rw [FG_not_rp h1 h2] _ = [a]_m ^ (num_rp_below m k) * (prod_seq k 0 (F m) * ([1]_m)) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_not_rp h2, prod_seq_zero_step, F_not_rp_def h2] done done done lemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by define --Goal : ∀ i < m, G m a i < m fix i : Nat assume h1 : i < m rewrite [G_def] --Goal : a * i % m < m show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m) done lemma left_inv_one_one_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g' (g i) = i) : one_one_below n g := sorry lemma right_inv_onto_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g (g' i) = i) (h2 : maps_below n g') : onto_below n g := by define at h2; define fix k : Nat assume h3 : k < n apply Exists.intro (g' k) show g' k < n ∧ g (g' k) = k from And.intro (h2 k h3) (h1 k h3) done lemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m * [inv_mod m a]_m = [1]_m := by have h2 : 0 ≤ (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a) show [a]_m * [inv_mod m a]_m = [1]_m from calc [a]_m * [inv_mod m a]_m _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2] _ = [a]_m * [gcd_c2 m a]_m := by rw [←cc_eq_mod] _ = [1]_m := gcd_c2_inv h1 done lemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m := calc a * (b % m) % m = a % m * (b % m % m) % m := Nat.mul_mod _ _ _ _ = a % m * (b % m) % m := by rw [Nat.mod_mod] _ = a * b % m := (Nat.mul_mod _ _ _).symm lemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm] rfl done theorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] : ↑a ≡ ↑b (MOD m) ↔ a % m = b % m := sorry lemma mul_inv_mod_cancel {m a i : Nat} [NeZero m] (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1 rewrite [mul_class, cc_eq_iff_congr, ←Nat.cast_mul, ←Nat.cast_one, congr_iff_mod_eq_Nat] at h3 show a * inv_mod m a * i % m = i from calc a * (inv_mod m a) * i % m _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = 1 % m * i % m := by rw [h3] _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = i % m := by rw [one_mul] _ = i := Nat.mod_eq_of_lt h2 done lemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl lemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, G m a (Ginv m a i) = i := by fix i : Nat assume h2 : i < m show G m a (Ginv m a i) = i from calc G m a (Ginv m a i) _ = a * ((inv_mod m a * i) % m) % m := by rfl _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, Ginv m a (G m a i) = i := by fix i : Nat assume h2 : i < m show Ginv m a (G m a i) = i from calc Ginv m a (G m a i) _ = inv_mod m a * ((a * i) % m) % m := by rfl _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc, mul_comm (inv_mod m a)] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_maps_below (m a : Nat) [NeZero m] : maps_below m (Ginv m a) := G_maps_below m (inv_mod m a) lemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : one_one_below m (G m a) := left_inv_one_one_below (Ginv_left_inv h1) lemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : onto_below m (G m a) := right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a) lemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : perm_below m (G m a) := And.intro (G_maps_below m a) (And.intro (G_one_one_below h1) (G_onto_below h1)) --Permuting a product of congruence classes doesn't change product lemma swap_fst (u v : Nat) : swap u v u = v := by define : swap u v u --Goal : (if u = u then v else if u = v then u else u) = v have h : u = u := by rfl rewrite [if_pos h] rfl done lemma swap_snd (u v : Nat) : swap u v v = u := by define : swap u v v by_cases h1 : v = u · -- Case 1. h1 : v = u rewrite [if_pos h1] show v = u from h1 done · -- Case 2. h1 : v ≠ u rewrite [if_neg h1] have h2 : v = v := by rfl rewrite [if_pos h2] rfl done done lemma swap_other {u v i : Nat} (h1 : i ≠ u) (h2 : i ≠ v) : swap u v i = i := by define : swap u v i rewrite [if_neg h1, if_neg h2] rfl done

lemma swap_values (u v i : Nat) : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i

HTPI.swap_values

htpi/HTPILib/Chap7.lean

htpi

/- Copyright 2023 Daniel J. Velleman -/ import HTPILib.Chap6 namespace HTPI /- Definitions -/ lemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by have h : n + 1 > 0 := Nat.succ_pos n show a % (n + 1) < n + 1 from Nat.mod_lt a h done def gcd (a b : Nat) : Nat := match b with | 0 => a | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd (n + 1) (a % (n + 1)) termination_by b mutual def gcd_c1 (a b : Nat) : Int := match b with | 0 => 1 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c2 (n + 1) (a % (n + 1)) --Corresponds to s = t' termination_by b def gcd_c2 (a b : Nat) : Int := match b with | 0 => 0 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c1 (n + 1) (a % (n + 1)) - (gcd_c2 (n + 1) (a % (n + 1))) * ↑(a / (n + 1)) --Corresponds to t = s' - t'q termination_by b end def prime (n : Nat) : Prop := 2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n def prime_factor (p n : Nat) : Prop := prime p ∧ p ∣ n def all_prime (l : List Nat) : Prop := ∀ p ∈ l, prime p def nondec (l : List Nat) : Prop := match l with | [] => True --Of course, True is a proposition that is always true | n :: L => (∀ m ∈ L, n ≤ m) ∧ nondec L def nondec_prime_list (l : List Nat) : Prop := all_prime l ∧ nondec l def prod (l : List Nat) : Nat := match l with | [] => 1 | n :: L => n * (prod L) def prime_factorization (n : Nat) (l : List Nat) : Prop := nondec_prime_list l ∧ prod l = n def rel_prime (a b : Nat) : Prop := gcd a b = 1 def congr_mod (m : Nat) (a b : Int) : Prop := (↑m : Int) ∣ (a - b) def cc (m : Nat) (a : Int) : ZMod m := (↑a : ZMod m) notation:50 a " ≡ " b " (MOD " m ")" => congr_mod m a b notation:max "["a"]_"m:max => cc m a def invertible {m : Nat} (X : ZMod m) : Prop := ∃ (Y : ZMod m), X * Y = [1]_m def num_rp_below (m k : Nat) : Nat := match k with | 0 => 0 | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j def phi (m : Nat) : Nat := num_rp_below m m def prod_seq {m : Nat} (j k : Nat) (f : Nat → ZMod m) : ZMod m := match j with | 0 => [1]_m | n + 1 => prod_seq n k f * f (k + n) def maps_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i < n, g i < n def one_one_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i1 < n, ∀ i2 < n, g i1 = g i2 → i1 = i2 def onto_below (n : Nat) (g : Nat → Nat) : Prop := ∀ k < n, ∃ i < n, g i = k def perm_below (n : Nat) (g : Nat → Nat) : Prop := maps_below n g ∧ one_one_below n g ∧ onto_below n g def inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m) def swap (u v i : Nat) : Nat := if i = u then v else if i = v then u else i namespace Euler --For definitions specific to Euler's theorem def F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m def G (m a i : Nat) : Nat := (a * i) % m def Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i end Euler /- Section 7.1 -/ theorem dvd_mod_of_dvd_a_b {a b d : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ (a % b) := by set q : Nat := a / b have h3 : b * q + a % b = a := Nat.div_add_mod a b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 define --Goal : ∃ (c : Nat), a % b = d * c apply Exists.intro (j - k * q) show a % b = d * (j - k * q) from calc a % b _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm _ = a - b * q := by rw [h3] _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc] _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm done theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry #eval gcd 672 161 --Answer: 7 lemma gcd_base (a : Nat) : gcd a 0 = a := by rfl lemma gcd_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd a b = gcd b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1)) rfl done lemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b ≠ 0) : a % b < b := by have h1 : b > 0 := Nat.pos_of_ne_zero h show a % b < b from Nat.mod_lt a h1 done lemma dvd_self (n : Nat) : n ∣ n := by apply Exists.intro 1 ring done theorem gcd_dvd : ∀ (b a : Nat), (gcd a b) ∣ a ∧ (gcd a b) ∣ b := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd a b_1) ∣ a ∧ (gcd a b_1) ∣ b_1 fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_base] --Goal: a ∣ a ∧ a ∣ 0 apply And.intro (dvd_self a) define apply Exists.intro 0 rfl done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_nonzero a h1] --Goal : gcd b (a % b) ∣ a ∧ gcd b (a % b) ∣ b have h2 : a % b < b := mod_nonzero_lt a h1 have h3 : (gcd b (a % b)) ∣ b ∧ (gcd b (a % b)) ∣ (a % b) := ih (a % b) h2 b apply And.intro _ h3.left show (gcd b (a % b)) ∣ a from dvd_a_of_dvd_b_mod h3.left h3.right done done theorem gcd_dvd_left (a b : Nat) : (gcd a b) ∣ a := (gcd_dvd b a).left theorem gcd_dvd_right (a b : Nat) : (gcd a b) ∣ b := (gcd_dvd b a).right lemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl lemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c1 a b = gcd_c2 b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done lemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl lemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * ↑(a / b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done theorem gcd_lin_comb : ∀ (b a : Nat), (gcd_c1 a b) * ↑a + (gcd_c2 a b) * ↑b = ↑(gcd a b) := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd_c1 a b_1) * ↑a + (gcd_c2 a b_1) * ↑b_1 = ↑(gcd a b_1) fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base] --Goal : 1 * ↑a + 0 * ↑0 = ↑a ring done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1] --Goal : gcd_c2 b (a % b) * ↑a + -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * ↑(a / b)) * ↑b = -- ↑(gcd b (a % b)) set r : Nat := a % b set q : Nat := a / b set s : Int := gcd_c1 b r set t : Int := gcd_c2 b r --Goal : t * ↑a + (s - t * ↑q) * ↑b = ↑(gcd b r) have h2 : r < b := mod_nonzero_lt a h1 have h3 : s * ↑b + t * ↑r = ↑(gcd b r) := ih r h2 b have h4 : b * q + r = a := Nat.div_add_mod a b rewrite [←h3, ←h4] rewrite [Nat.cast_add, Nat.cast_mul] --Goal : t * (↑b * ↑q + ↑r) + (s - t * ↑q) * ↑b = s * ↑b + t * ↑r ring done done #eval gcd_c1 672 161 --Answer: 6 #eval gcd_c2 672 161 --Answer: -25 --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161 theorem Theorem_7_1_6 {d a b : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ gcd a b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑d ∣ ↑(gcd a b) set s : Int := gcd_c1 a b set t : Int := gcd_c2 a b have h3 : s * ↑a + t * ↑b = ↑(gcd a b) := gcd_lin_comb b a rewrite [←h3] --Goal : ↑d ∣ s * ↑a + t * ↑b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul] --Goal : ↑d ∣ s * (↑d * ↑j) + t * (↑d * ↑k) define apply Exists.intro (s * ↑j + t * ↑k) ring done /- Section 7.2 -/ theorem dvd_trans {a b c : Nat} (h1 : a ∣ b) (h2 : b ∣ c) : a ∣ c := by define at h1; define at h2; define obtain (m : Nat) (h3 : b = a * m) from h1 obtain (n : Nat) (h4 : c = b * n) from h2 rewrite [h3, mul_assoc] at h4 apply Exists.intro (m * n) show c = a * (m * n) from h4 done lemma exists_prime_factor : ∀ (n : Nat), 2 ≤ n → ∃ (p : Nat), prime_factor p n := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, 2 ≤ n_1 → ∃ (p : Nat), prime_factor p n_1 assume h1 : 2 ≤ n by_cases h2 : prime n · -- Case 1. h2 : prime n apply Exists.intro n define --Goal : prime n ∧ n ∣ n show prime n ∧ n ∣ n from And.intro h2 (dvd_self n) done · -- Case 2. h2 : ¬prime n define at h2 --h2 : ¬(2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n) demorgan at h2 disj_syll h2 h1 obtain (a : Nat) (h3 : ∃ (b : Nat), a * b = n ∧ a < n ∧ b < n) from h2 obtain (b : Nat) (h4 : a * b = n ∧ a < n ∧ b < n) from h3 have h5 : 2 ≤ a := by by_contra h6 have h7 : a ≤ 1 := by linarith have h8 : n ≤ b := calc n _ = a * b := h4.left.symm _ ≤ 1 * b := by rel [h7] _ = b := by ring linarith --n ≤ b contradicts b < n done have h6 : ∃ (p : Nat), prime_factor p a := ih a h4.right.left h5 obtain (p : Nat) (h7 : prime_factor p a) from h6 apply Exists.intro p define --Goal : prime p ∧ p ∣ n define at h7 --h7 : prime p ∧ p ∣ a apply And.intro h7.left have h8 : a ∣ n := by apply Exists.intro b show n = a * b from (h4.left).symm done show p ∣ n from dvd_trans h7.right h8 done done lemma exists_least_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q := by set S : Set Nat := {p : Nat | prime_factor p n} have h2 : ∃ (p : Nat), p ∈ S := exists_prime_factor n h show ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q from well_ord_princ S h2 done lemma all_prime_nil : all_prime [] := by define --Goal : ∀ p ∈ [], prime p fix p : Nat contrapos --Goal : ¬prime p → p ∉ [] assume h1 : ¬prime p show p ∉ [] from List.not_mem_nil p done lemma all_prime_cons (n : Nat) (L : List Nat) : all_prime (n :: L) ↔ prime n ∧ all_prime L := by apply Iff.intro · -- (→) assume h1 : all_prime (n :: L) --Goal : prime n ∧ all_prime L define at h1 --h1 : ∀ p ∈ n :: L, prime p apply And.intro (h1 n (List.mem_cons_self n L)) define --Goal : ∀ p ∈ L, prime p fix p : Nat assume h2 : p ∈ L show prime p from h1 p (List.mem_cons_of_mem n h2) done · -- (←) assume h1 : prime n ∧ all_prime L --Goal : all_prime (n :: l) define : all_prime L at h1 define fix p : Nat assume h2 : p ∈ n :: L rewrite [List.mem_cons] at h2 --h2 : p = n ∨ p ∈ L by_cases on h2 · -- Case 1. h2 : p = n rewrite [h2] show prime n from h1.left done · -- Case 2. h2 : p ∈ L show prime p from h1.right p h2 done done done lemma nondec_nil : nondec [] := by define --Goal : True trivial --trivial proves some obviously true statements, such as True done lemma nondec_cons (n : Nat) (L : List Nat) : nondec (n :: L) ↔ (∀ m ∈ L, n ≤ m) ∧ nondec L := by rfl lemma prod_nil : prod [] = 1 := by rfl lemma prod_cons : prod (n :: L) = n * (prod L) := by rfl lemma exists_cons_of_length_eq_succ {A : Type} {l : List A} {n : Nat} (h : l.length = n + 1) : ∃ (a : A) (L : List A), l = a :: L ∧ L.length = n := by have h1 : ¬l.length = 0 := by linarith rewrite [List.length_eq_zero] at h1 obtain (a : A) (h2 : ∃ (L : List A), l = a :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List A) (h3 : l = a :: L) from h2 apply Exists.intro a apply Exists.intro L apply And.intro h3 have h4 : (a :: L).length = L.length + 1 := List.length_cons a L rewrite [←h3, h] at h4 show L.length = n from (Nat.add_right_cancel h4).symm done lemma list_elt_dvd_prod_by_length (a : Nat) : ∀ (n : Nat), ∀ (l : List Nat), l.length = n → a ∈ l → a ∣ prod l := by by_induc · --Base Case fix l : List Nat assume h1 : l.length = 0 rewrite [List.length_eq_zero] at h1 --h1 : l = [] rewrite [h1] --Goal : a ∈ [] → a ∣ prod [] contrapos assume h2 : ¬a ∣ prod [] show a ∉ [] from List.not_mem_nil a done · -- Induction Step fix n : Nat assume ih : ∀ (l : List Nat), List.length l = n → a ∈ l → a ∣ prod l fix l : List Nat assume h1 : l.length = n + 1 --Goal : a ∈ l → a ∣ prod l obtain (b : Nat) (h2 : ∃ (L : List Nat), l = b :: L ∧ L.length = n) from exists_cons_of_length_eq_succ h1 obtain (L : List Nat) (h3 : l = b :: L ∧ L.length = n) from h2 have h4 : a ∈ L → a ∣ prod L := ih L h3.right assume h5 : a ∈ l rewrite [h3.left, prod_cons] --Goal : a ∣ b * prod L rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b ∨ a ∈ L by_cases on h5 · -- Case 1. h5 : a = b apply Exists.intro (prod L) rewrite [h5] rfl done · -- Case 2. h5 : a ∈ L have h6 : a ∣ prod L := h4 h5 have h7 : prod L ∣ b * prod L := by apply Exists.intro b ring done show a ∣ b * prod L from dvd_trans h6 h7 done done done lemma list_elt_dvd_prod {a : Nat} {l : List Nat} (h : a ∈ l) : a ∣ prod l := by set n : Nat := l.length have h1 : l.length = n := by rfl show a ∣ prod l from list_elt_dvd_prod_by_length a n l h1 h done lemma exists_prime_factorization : ∀ (n : Nat), n ≥ 1 → ∃ (l : List Nat), prime_factorization n l := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, n_1 ≥ 1 → ∃ (l : List Nat), prime_factorization n_1 l assume h1 : n ≥ 1 by_cases h2 : n = 1 · -- Case 1. h2 : n = 1 apply Exists.intro [] define apply And.intro · -- Proof of nondec_prime_list [] define show all_prime [] ∧ nondec [] from And.intro all_prime_nil nondec_nil done · -- Proof of prod [] = n rewrite [prod_nil, h2] rfl done done · -- Case 2. h2 : n ≠ 1 have h3 : n ≥ 2 := lt_of_le_of_ne' h1 h2 obtain (p : Nat) (h4 : prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q) from exists_least_prime_factor h3 have p_prime_factor : prime_factor p n := h4.left define at p_prime_factor have p_prime : prime p := p_prime_factor.left have p_dvd_n : p ∣ n := p_prime_factor.right have p_least : ∀ (q : Nat), prime_factor q n → p ≤ q := h4.right obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n have h5 : m ≠ 0 := by contradict h1 with h6 have h7 : n = 0 := calc n _ = p * m := n_eq_pm _ = p * 0 := by rw [h6] _ = 0 := by ring rewrite [h7] decide done have m_pos : 0 < m := Nat.pos_of_ne_zero h5 have m_lt_n : m < n := by define at p_prime show m < n from calc m _ < m + m := by linarith _ = 2 * m := by ring _ ≤ p * m := by rel [p_prime.left] _ = n := n_eq_pm.symm done obtain (L : List Nat) (h6 : prime_factorization m L) from ih m m_lt_n m_pos define at h6 have ndpl_L : nondec_prime_list L := h6.left define at ndpl_L apply Exists.intro (p :: L) define apply And.intro · -- Proof of nondec_prime_list (p :: L) define apply And.intro · -- Proof of all_prime (p :: L) rewrite [all_prime_cons] show prime p ∧ all_prime L from And.intro p_prime ndpl_L.left done · -- Proof of nondec (p :: L) rewrite [nondec_cons] apply And.intro _ ndpl_L.right fix q : Nat assume q_in_L : q ∈ L have h7 : q ∣ prod L := list_elt_dvd_prod q_in_L rewrite [h6.right] at h7 --h7 : q ∣ m have h8 : m ∣ n := by apply Exists.intro p rewrite [n_eq_pm] ring done have q_dvd_n : q ∣ n := dvd_trans h7 h8 have ap_L : all_prime L := ndpl_L.left define at ap_L have q_prime_factor : prime_factor q n := And.intro (ap_L q q_in_L) q_dvd_n show p ≤ q from p_least q q_prime_factor done done · -- Proof of prod (p :: L) = n rewrite [prod_cons, h6.right, n_eq_pm] rfl done done done theorem Theorem_7_2_2 {a b c : Nat} (h1 : c ∣ a * b) (h2 : rel_prime a c) : c ∣ b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑c ∣ ↑b define at h1; define at h2; define obtain (j : Nat) (h3 : a * b = c * j) from h1 set s : Int := gcd_c1 a c set t : Int := gcd_c2 a c have h4 : s * ↑a + t * ↑c = ↑(gcd a c) := gcd_lin_comb c a rewrite [h2, Nat.cast_one] at h4 --h4 : s * ↑a + t * ↑c = (1 : Int) apply Exists.intro (s * ↑j + t * ↑b) show ↑b = ↑c * (s * ↑j + t * ↑b) from calc ↑b _ = (1 : Int) * ↑b := (one_mul _).symm _ = (s * ↑a + t * ↑c) * ↑b := by rw [h4] _ = s * (↑a * ↑b) + t * ↑c * ↑b := by ring _ = s * (↑c * ↑j) + t * ↑c * ↑b := by rw [←Nat.cast_mul a b, h3, Nat.cast_mul c j] _ = ↑c * (s * ↑j + t * ↑b) := by ring done lemma le_nonzero_prod_left {a b : Nat} (h : a * b ≠ 0) : a ≤ a * b := by have h1 : b ≠ 0 := by contradict h with h1 rewrite [h1] ring done have h2 : 1 ≤ b := Nat.pos_of_ne_zero h1 show a ≤ a * b from calc a = a * 1 := (mul_one a).symm _ ≤ a * b := by rel [h2] done lemma le_nonzero_prod_right {a b : Nat} (h : a * b ≠ 0) : b ≤ a * b := by rewrite [mul_comm] rewrite [mul_comm] at h show b ≤ b * a from le_nonzero_prod_left h done lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry lemma rel_prime_of_prime_not_dvd {a p : Nat} (h1 : prime p) (h2 : ¬p ∣ a) : rel_prime a p := by have h3 : gcd a p ∣ a := gcd_dvd_left a p have h4 : gcd a p ∣ p := gcd_dvd_right a p have h5 : gcd a p = 1 ∨ gcd a p = p := dvd_prime h1 h4 have h6 : gcd a p ≠ p := by contradict h2 with h6 rewrite [h6] at h3 show p ∣ a from h3 done disj_syll h5 h6 show rel_prime a p from h5 done theorem Theorem_7_2_3 {a b p : Nat} (h1 : prime p) (h2 : p ∣ a * b) : p ∣ a ∨ p ∣ b := by or_right with h3 have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3 show p ∣ b from Theorem_7_2_2 h2 h4 done lemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a ≥ 1 := by have h1 : a ≠ 0 := by by_contra h1 rewrite [h1] at h contradict h linarith done show a ≥ 1 from Nat.pos_of_ne_zero h1 done lemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by have h1 : a ≥ 1 := ge_one_of_prod_one h have h2 : a * b ≠ 0 := by linarith have h3 : a ≤ a * b := le_nonzero_prod_left h2 rewrite [h] at h3 show a = 1 from Nat.le_antisymm h3 h1 done lemma eq_one_of_dvd_one {n : Nat} (h : n ∣ 1) : n = 1 := by obtain (j : Nat) (h1 : 1 = n * j) from h show n = 1 from eq_one_of_prod_one h1.symm done lemma prime_not_one {p : Nat} (h : prime p) : p ≠ 1 := by define at h linarith done theorem Theorem_7_2_4 {p : Nat} (h1 : prime p) : ∀ (l : List Nat), p ∣ prod l → ∃ a ∈ l, p ∣ a := by apply List.rec · -- Base Case. Goal : p ∣ prod [] → ∃ a ∈ [], p ∣ a rewrite [prod_nil] assume h2 : p ∣ 1 show ∃ a ∈ [], p ∣ a from absurd (eq_one_of_dvd_one h2) (prime_not_one h1) done · -- Induction Step fix b : Nat fix L : List Nat assume ih : p ∣ prod L → ∃ a ∈ L, p ∣ a --Goal : p ∣ prod (b :: L) → ∃ a ∈ b :: L, p ∣ a assume h2 : p ∣ prod (b :: L) rewrite [prod_cons] at h2 have h3 : p ∣ b ∨ p ∣ prod L := Theorem_7_2_3 h1 h2 by_cases on h3 · -- Case 1. h3 : p ∣ b apply Exists.intro b show b ∈ b :: L ∧ p ∣ b from And.intro (List.mem_cons_self b L) h3 done · -- Case 2. h3 : p ∣ prod L obtain (a : Nat) (h4 : a ∈ L ∧ p ∣ a) from ih h3 apply Exists.intro a show a ∈ b :: L ∧ p ∣ a from And.intro (List.mem_cons_of_mem b h4.left) h4.right done done done lemma prime_in_list {p : Nat} {l : List Nat} (h1 : prime p) (h2 : all_prime l) (h3 : p ∣ prod l) : p ∈ l := by obtain (a : Nat) (h4 : a ∈ l ∧ p ∣ a) from Theorem_7_2_4 h1 l h3 define at h2 have h5 : prime a := h2 a h4.left have h6 : p = 1 ∨ p = a := dvd_prime h5 h4.right disj_syll h6 (prime_not_one h1) rewrite [h6] show a ∈ l from h4.left done lemma first_le_first {p q : Nat} {l m : List Nat} (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m)) (h3 : prod (p :: l) = prod (q :: m)) : p ≤ q := by define at h1; define at h2 have h4 : q ∣ prod (p :: l) := by define apply Exists.intro (prod m) rewrite [←prod_cons] show prod (p :: l) = prod (q :: m) from h3 done have h5 : all_prime (q :: m) := h2.left rewrite [all_prime_cons] at h5 have h6 : q ∈ p :: l := prime_in_list h5.left h1.left h4 have h7 : nondec (p :: l) := h1.right rewrite [nondec_cons] at h7 rewrite [List.mem_cons] at h6 by_cases on h6 · -- Case 1. h6 : q = p linarith done · -- Case 2. h6 : q ∈ l have h8 : ∀ m ∈ l, p ≤ m := h7.left show p ≤ q from h8 q h6 done done lemma nondec_prime_list_tail {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by define at h define rewrite [all_prime_cons, nondec_cons] at h show all_prime l ∧ nondec l from And.intro h.left.right h.right.right done lemma cons_prod_not_one {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : prod (p :: l) ≠ 1 := by define at h have h1 : all_prime (p :: l) := h.left rewrite [all_prime_cons] at h1 rewrite [prod_cons] by_contra h2 show False from (prime_not_one h1.left) (eq_one_of_prod_one h2) done lemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) : l = [] ↔ prod l = 1 := by apply Iff.intro · -- (→) assume h1 : l = [] rewrite [h1] show prod [] = 1 from prod_nil done · -- (←) contrapos assume h1 : ¬l = [] obtain (p : Nat) (h2 : ∃ (L : List Nat), l = p :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List Nat) (h3 : l = p :: L) from h2 rewrite [h3] at h rewrite [h3] show ¬prod (p :: L) = 1 from cons_prod_not_one h done done lemma prime_pos {p : Nat} (h : prime p) : p > 0 := by define at h linarith done theorem Theorem_7_2_5 : ∀ (l1 l2 : List Nat), nondec_prime_list l1 → nondec_prime_list l2 → prod l1 = prod l2 → l1 = l2 := by apply List.rec · -- Base Case. Goal : ∀ (l2 : List Nat), nondec_prime_list [] → -- nondec_prime_list l2 → prod [] = prod l2 → [] = l2 fix l2 : List Nat assume h1 : nondec_prime_list [] assume h2 : nondec_prime_list l2 assume h3 : prod [] = prod l2 rewrite [prod_nil, eq_comm, ←list_nil_iff_prod_one h2] at h3 show [] = l2 from h3.symm done · -- Induction Step fix p : Nat fix L1 : List Nat assume ih : ∀ (L2 : List Nat), nondec_prime_list L1 → nondec_prime_list L2 → prod L1 = prod L2 → L1 = L2 -- Goal : ∀ (l2 : List Nat), nondec_prime_list (p :: L1) → -- nondec_prime_list l2 → prod (p :: L1) = prod l2 → p :: L1 = l2 fix l2 : List Nat assume h1 : nondec_prime_list (p :: L1) assume h2 : nondec_prime_list l2 assume h3 : prod (p :: L1) = prod l2 have h4 : ¬prod (p :: L1) = 1 := cons_prod_not_one h1 rewrite [h3, ←list_nil_iff_prod_one h2] at h4 obtain (q : Nat) (h5 : ∃ (L : List Nat), l2 = q :: L) from List.exists_cons_of_ne_nil h4 obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5 rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2) rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2) have h7 : p ≤ q := first_le_first h1 h2 h3 have h8 : q ≤ p := first_le_first h2 h1 h3.symm have h9 : p = q := by linarith rewrite [h9, prod_cons, prod_cons] at h3 --h3 : q * prod L1 = q * prod L2 have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1 have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2 define at h2 have h12 : all_prime (q :: L2) := h2.left rewrite [all_prime_cons] at h12 have h13 : q > 0 := prime_pos h12.left have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3 have h15 : L1 = L2 := ih L2 h10 h11 h14 rewrite [h6, h9, h15] rfl done done theorem fund_thm_arith (n : Nat) (h : n ≥ 1) : ∃! (l : List Nat), prime_factorization n l := by exists_unique · -- Existence show ∃ (l : List Nat), prime_factorization n l from exists_prime_factorization n h done · -- Uniqueness fix l1 : List Nat; fix l2 : List Nat assume h1 : prime_factorization n l1 assume h2 : prime_factorization n l2 define at h1; define at h2 have h3 : prod l1 = n := h1.right rewrite [←h2.right] at h3 show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3 done done /- Section 7.3 -/ theorem congr_refl (m : Nat) : ∀ (a : Int), a ≡ a (MOD m) := by fix a : Int define --Goal : ∃ (c : Int), a - a = ↑m * c apply Exists.intro 0 ring done theorem congr_symm {m : Nat} : ∀ {a b : Int}, a ≡ b (MOD m) → b ≡ a (MOD m) := by fix a : Int; fix b : Int assume h1 : a ≡ b (MOD m) define at h1 --h1 : ∃ (c : Int), a - b = ↑m * c define --Goal : ∃ (c : Int), b - a = ↑m * c obtain (c : Int) (h2 : a - b = m * c) from h1 apply Exists.intro (-c) show b - a = m * (-c) from calc b - a _ = -(a - b) := by ring _ = -(m * c) := by rw [h2] _ = m * (-c) := by ring done theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry /- Fundamental properties of congruence classes -/ lemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) : X = Y ↔ X.val = Y.val := Fin.ext_iff lemma val_nat_eq_mod (n k : Nat) : ([k]_(n + 1)).val = k % (n + 1) := by rfl lemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl theorem cc_rep {m : Nat} (X : ZMod m) : ∃ (a : Int), X = [a]_m := match m with | 0 => by apply Exists.intro X rfl done | n + 1 => by apply Exists.intro ↑(X.val) have h1 : X.val < n + 1 := Fin.prop X rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1] rfl done theorem add_class (m : Nat) (a b : Int) : [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm theorem mul_class (m : Nat) (a b : Int) : [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm lemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ [a - b]_m = [0]_m := by apply Iff.intro · -- (→) assume h1 : [a]_m = [b]_m have h2 : a - b = a + (-b) := by ring have h3 : b + (-b) = 0 := by ring show [a - b]_m = [0]_m from calc [a - b]_m _ = [a + (-b)]_m := by rw [h2] _ = [a]_m + [-b]_m := by rw [add_class] _ = [b]_m + [-b]_m := by rw [h1] _ = [b + -b]_m := by rw [add_class] _ = [0]_m := by rw [h3] done · -- (←) assume h1 : [a - b]_m = [0]_m have h2 : b + (a - b) = a := by ring have h3 : b + 0 = b := by ring show [a]_m = [b]_m from calc [a]_m _ = [b + (a - b)]_m := by rw [h2] _ = [b]_m + [a - b]_m := by rw [add_class] _ = [b]_m + [0]_m := by rw [h1] _ = [b + 0]_m := by rw [add_class] _ = [b]_m := by rw [h3] done done lemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) : [a]_m = [0]_m → [-a]_m = [0]_m := by assume h1 : [a]_m = [0]_m have h2 : 0 + (-a) = -a := by ring have h3 : a + (-a) = 0 := by ring show [-a]_m = [0]_m from calc [-a]_m _ = [0 + (-a)]_m := by rw [h2] _ = [0]_m + [-a]_m := by rw [add_class] _ = [a]_m + [-a]_m := by rw [h1] _ = [a + (-a)]_m := by rw [add_class] _ = [0]_m := by rw [h3] done lemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) : [-a]_m = [0]_m ↔ [a]_m = [0]_m := by apply Iff.intro _ (cc_neg_zero_of_cc_zero m a) assume h1 : [-a]_m = [0]_m have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1 have h3 : -(-a) = a := by ring rewrite [h3] at h2 show [a]_m = [0]_m from h2 done lemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl lemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m ↔ m ∣ k := match m with | 0 => by have h : (0 : Int) = (↑(0 : Nat) : Int) := by rfl rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj] apply Iff.intro · -- (→) assume h1 : k = 0 rewrite [h1] show 0 ∣ 0 from dvd_self 0 done · -- (←) assume h1 : 0 ∣ k obtain (c : Nat) (h2 : k = 0 * c) from h1 rewrite [h2] ring done done | n + 1 => by rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero] show k % (n + 1) = 0 ↔ n + 1 ∣ k from (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm done lemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m ↔ ↑m ∣ a := by obtain (k : Nat) (h1 : a = ↑k ∨ a = -↑k) from Int.eq_nat_or_neg a by_cases on h1 · -- Case 1. h1: a = ↑k rewrite [h1, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done · -- Case 2. h1: a = -↑k rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done done theorem cc_eq_iff_congr (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ a ≡ b (MOD m) := calc [a]_m = [b]_m _ ↔ [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b _ ↔ ↑m ∣ (a - b) := cc_zero_iff_dvd m (a - b) _ ↔ a ≡ b (MOD m) := by rfl /- End of fundamental properties of congruence classes -/ lemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m := by have h1 : (↑m : Int) ≠ 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m) show 0 ≤ a % m from Int.emod_nonneg a h1 done lemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m) have h2 : (↑m : Int) > 0 := (Nat.cast_pos).rtl h1 show a % m < m from Int.emod_lt_of_pos a h2 done lemma congr_mod_mod (m : Nat) (a : Int) : a ≡ a % m (MOD m) := by define have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m apply Exists.intro (a / m) show a - a % m = m * (a / m) from calc a - (a % m) _ = m * (a / m) + a % m - a % m := by rw [h1] _ = m * (a / m) := by ring done lemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) := And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a)) theorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) : ∃! (r : Int), 0 ≤ r ∧ r < m ∧ a ≡ r (MOD m) := by exists_unique · -- Existence apply Exists.intro (a % m) show 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) from mod_cmpl_res m a done · -- Uniqueness fix r1 : Int; fix r2 : Int assume h1 : 0 ≤ r1 ∧ r1 < m ∧ a ≡ r1 (MOD m) assume h2 : 0 ≤ r2 ∧ r2 < m ∧ a ≡ r2 (MOD m) have h3 : r1 ≡ r2 (MOD m) := congr_trans (congr_symm h1.right.right) h2.right.right obtain (d : Int) (h4 : r1 - r2 = m * d) from h3 have h5 : r1 - r2 < m * 1 := by linarith have h6 : m * (-1) < r1 - r2 := by linarith rewrite [h4] at h5 --h5 : m * d < m * 1 rewrite [h4] at h6 --h6 : m * -1 < m * d have h7 : (↑m : Int) ≥ 0 := Nat.cast_nonneg m have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7 have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7 have h10 : d = 0 := by linarith show r1 = r2 from calc r1 _ = r1 - r2 + r2 := by ring _ = m * 0 + r2 := by rw [h4, h10] _ = r2 := by ring done done lemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m := (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a) theorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y rewrite [h1, h2] have h3 : a + b = b + a := by ring show [a]_m + [b]_m = [b]_m + [a]_m from calc [a]_m + [b]_m _ = [a + b]_m := add_class m a b _ = [b + a]_m := by rw [h3] _ = [b]_m + [a]_m := (add_class m b a).symm done theorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X rewrite [h1] have h2 : a * 1 = a := by ring show [a]_m * [1]_m = [a]_m from calc [a]_m * [1]_m _ = [a * 1]_m := mul_class m a 1 _ = [a]_m := by rw [h2] done theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry lemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) : [a]_m * [gcd_c2 m a]_m = [1]_m := by set s : Int := gcd_c1 m a have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m define at h1 rewrite [h1, Nat.cast_one] at h2 --h2 : s * ↑m + gcd_c2 m a * ↑a = 1 rewrite [mul_class, cc_eq_iff_congr] define --Goal : ∃ (c : Int), ↑a * gcd_c2 m a - 1 = ↑m * c apply Exists.intro (-s) show a * (gcd_c2 m a) - 1 = m * (-s) from calc a * (gcd_c2 m a) - 1 _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring _ = 1 + m * (-s) - 1 := by rw [h2] _ = m * (-s) := by ring done theorem Theorem_7_3_7 (m a : Nat) : invertible [a]_m ↔ rel_prime m a := by apply Iff.intro · -- (→) assume h1 : invertible [a]_m define at h1 obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1 obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y rewrite [h3, mul_class, cc_eq_iff_congr] at h2 define at h2 obtain (c : Int) (h4 : a * b - 1 = m * c) from h2 rewrite [Exercise_7_2_6] --Goal : ∃ (s t : Int), s * ↑m + t * ↑a = 1 apply Exists.intro (-c) apply Exists.intro b show (-c) * m + b * a = 1 from calc (-c) * m + b * a _ = (-c) * m + (a * b - 1) + 1 := by ring _ = (-c) * m + m * c + 1 := by rw [h4] _ = 1 := by ring done · -- (←) assume h1 : rel_prime m a define show ∃ (Y : ZMod m), [a]_m * Y = [1]_m from Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1) done done /- Section 7.4 -/ section Euler open Euler lemma num_rp_below_base {m : Nat} : num_rp_below m 0 = 0 := by rfl lemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) : num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by have h1 : num_rp_below m (j + 1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : gcd m j = 1 rewrite [if_pos h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1 show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1 done lemma num_rp_below_step_not_rp {m j : Nat} (h : ¬rel_prime m j) : num_rp_below m (j + 1) = num_rp_below m j := by have h1 : num_rp_below m (j +1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : ¬gcd m j = 1 rewrite [if_neg h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j show num_rp_below m (j + 1) = num_rp_below m j from h1 done lemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl #eval phi 10 --Answer: 4 lemma prod_inv_iff_inv {m : Nat} {X : ZMod m} (h1 : invertible X) (Y : ZMod m) : invertible (X * Y) ↔ invertible Y := by apply Iff.intro · -- (→) assume h2 : invertible (X * Y) obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2 apply Exists.intro (X * Z) rewrite [←h3] --Goal : Y * (X * Z) = X * Y * Z ring --Note that ring can do algebra in ZMod m done · -- (←) assume h2 : invertible Y obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1 obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2 apply Exists.intro (Xi * Yi) show (X * Y) * (Xi * Yi) = [1]_m from calc X * Y * (Xi * Yi) _ = (X * Xi) * (Y * Yi) := by ring _ = [1]_m * [1]_m := by rw [h3, h4] _ = [1]_m := Theorem_7_3_6_7 [1]_m done done lemma F_rp_def {m i : Nat} (h : rel_prime m i) : F m i = [i]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h --h : gcd m i = 1 rewrite [if_pos h] at h1 show F m i = [i]_m from h1 done lemma F_not_rp_def {m i : Nat} (h : ¬rel_prime m i) : F m i = [1]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h rewrite [h1, if_neg h] rfl done lemma prod_seq_base {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 0 k f = [1]_m := by rfl lemma prod_seq_step {m : Nat} (n k : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl lemma prod_seq_zero_step {m : Nat} (n : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by rewrite [prod_seq_step, zero_add] rfl done lemma prod_one {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 1 k f = f k := by rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7] rfl done lemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl lemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m := calc [G m a i]_m _ = [(a * i) % m]_m := by rfl _ = [a * i]_m := (cc_eq_mod m (a * i)).symm _ = [a]_m * [i]_m := (mul_class m a i).symm lemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) : rel_prime m (G m a i) ↔ rel_prime m i := by have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1 show rel_prime m (G m a i) ↔ rel_prime m i from calc rel_prime m (G m a i) _ ↔ invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm _ ↔ invertible ([a]_m * [i]_m) := by rw [cc_G] _ ↔ invertible [i]_m := prod_inv_iff_inv h2 ([i]_m) _ ↔ rel_prime m i := Theorem_7_3_7 m i done lemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) : F m (G m a i) = [a]_m * F m i := by have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2 show F m (G m a i) = [a]_m * F m i from calc F m (G m a i) _ = [G m a i]_m := F_rp_def h3 _ = [a]_m * [i]_m := cc_G m a i _ = [a]_m * F m i := by rw [F_rp_def h2] done lemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : ¬rel_prime m i) : F m (G m a i) = [1]_m := by rewrite [←G_rp_iff h1 i] at h2 show F m (G m a i) = [1]_m from F_not_rp_def h2 done lemma FG_prod {m a : Nat} (h1 : rel_prime m a) : ∀ (k : Nat), prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by by_induc · -- Base Case show prod_seq 0 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from calc prod_seq 0 0 ((F m) ∘ (G m a)) _ = [1]_m := prod_seq_base _ _ _ = [a]_m ^ 0 * [1]_m := by ring _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by rw [num_rp_below_base, prod_seq_base] done · -- Induction Step fix k : Nat assume ih : prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) by_cases h2 : rel_prime m k · -- Case 1. h2 : rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([a]_m * F m k) := by rw [FG_rp h1 h2] _ = [a]_m ^ ((num_rp_below m k) + 1) * ((prod_seq k 0 (F m)) * F m k) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_rp h2, prod_seq_zero_step] done · -- Case 2. h2 : ¬rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([1]_m) := by rw [FG_not_rp h1 h2] _ = [a]_m ^ (num_rp_below m k) * (prod_seq k 0 (F m) * ([1]_m)) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_not_rp h2, prod_seq_zero_step, F_not_rp_def h2] done done done lemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by define --Goal : ∀ i < m, G m a i < m fix i : Nat assume h1 : i < m rewrite [G_def] --Goal : a * i % m < m show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m) done lemma left_inv_one_one_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g' (g i) = i) : one_one_below n g := sorry lemma right_inv_onto_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g (g' i) = i) (h2 : maps_below n g') : onto_below n g := by define at h2; define fix k : Nat assume h3 : k < n apply Exists.intro (g' k) show g' k < n ∧ g (g' k) = k from And.intro (h2 k h3) (h1 k h3) done lemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m * [inv_mod m a]_m = [1]_m := by have h2 : 0 ≤ (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a) show [a]_m * [inv_mod m a]_m = [1]_m from calc [a]_m * [inv_mod m a]_m _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2] _ = [a]_m * [gcd_c2 m a]_m := by rw [←cc_eq_mod] _ = [1]_m := gcd_c2_inv h1 done lemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m := calc a * (b % m) % m = a % m * (b % m % m) % m := Nat.mul_mod _ _ _ _ = a % m * (b % m) % m := by rw [Nat.mod_mod] _ = a * b % m := (Nat.mul_mod _ _ _).symm lemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm] rfl done theorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] : ↑a ≡ ↑b (MOD m) ↔ a % m = b % m := sorry lemma mul_inv_mod_cancel {m a i : Nat} [NeZero m] (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1 rewrite [mul_class, cc_eq_iff_congr, ←Nat.cast_mul, ←Nat.cast_one, congr_iff_mod_eq_Nat] at h3 show a * inv_mod m a * i % m = i from calc a * (inv_mod m a) * i % m _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = 1 % m * i % m := by rw [h3] _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = i % m := by rw [one_mul] _ = i := Nat.mod_eq_of_lt h2 done lemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl lemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, G m a (Ginv m a i) = i := by fix i : Nat assume h2 : i < m show G m a (Ginv m a i) = i from calc G m a (Ginv m a i) _ = a * ((inv_mod m a * i) % m) % m := by rfl _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, Ginv m a (G m a i) = i := by fix i : Nat assume h2 : i < m show Ginv m a (G m a i) = i from calc Ginv m a (G m a i) _ = inv_mod m a * ((a * i) % m) % m := by rfl _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc, mul_comm (inv_mod m a)] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_maps_below (m a : Nat) [NeZero m] : maps_below m (Ginv m a) := G_maps_below m (inv_mod m a) lemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : one_one_below m (G m a) := left_inv_one_one_below (Ginv_left_inv h1) lemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : onto_below m (G m a) := right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a) lemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : perm_below m (G m a) := And.intro (G_maps_below m a) (And.intro (G_one_one_below h1) (G_onto_below h1)) --Permuting a product of congruence classes doesn't change product lemma swap_fst (u v : Nat) : swap u v u = v := by define : swap u v u --Goal : (if u = u then v else if u = v then u else u) = v have h : u = u := by rfl rewrite [if_pos h] rfl done lemma swap_snd (u v : Nat) : swap u v v = u := by define : swap u v v by_cases h1 : v = u · -- Case 1. h1 : v = u rewrite [if_pos h1] show v = u from h1 done · -- Case 2. h1 : v ≠ u rewrite [if_neg h1] have h2 : v = v := by rfl rewrite [if_pos h2] rfl done done lemma swap_other {u v i : Nat} (h1 : i ≠ u) (h2 : i ≠ v) : swap u v i = i := by define : swap u v i rewrite [if_neg h1, if_neg h2] rfl done lemma swap_values (u v i : Nat) : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := by by_cases h1 : i = u · -- Case 1. h1 : i = u apply Or.inl rewrite [h1] show swap u v u = v from swap_fst u v done · -- Case 2. h1 : i ≠ u apply Or.inr by_cases h2 : i = v · -- Case 2.1. h2 : i = v apply Or.inl rewrite [h2] show swap u v v = u from swap_snd u v done · -- Case 2.2. h2 : i ≠ v apply Or.inr show swap u v i = i from swap_other h1 h2 done done done

lemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v)

HTPI.swap_maps_below

htpi/HTPILib/Chap7.lean

htpi

/- Copyright 2023 Daniel J. Velleman -/ import HTPILib.Chap6 namespace HTPI /- Definitions -/ lemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by have h : n + 1 > 0 := Nat.succ_pos n show a % (n + 1) < n + 1 from Nat.mod_lt a h done def gcd (a b : Nat) : Nat := match b with | 0 => a | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd (n + 1) (a % (n + 1)) termination_by b mutual def gcd_c1 (a b : Nat) : Int := match b with | 0 => 1 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c2 (n + 1) (a % (n + 1)) --Corresponds to s = t' termination_by b def gcd_c2 (a b : Nat) : Int := match b with | 0 => 0 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c1 (n + 1) (a % (n + 1)) - (gcd_c2 (n + 1) (a % (n + 1))) * ↑(a / (n + 1)) --Corresponds to t = s' - t'q termination_by b end def prime (n : Nat) : Prop := 2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n def prime_factor (p n : Nat) : Prop := prime p ∧ p ∣ n def all_prime (l : List Nat) : Prop := ∀ p ∈ l, prime p def nondec (l : List Nat) : Prop := match l with | [] => True --Of course, True is a proposition that is always true | n :: L => (∀ m ∈ L, n ≤ m) ∧ nondec L def nondec_prime_list (l : List Nat) : Prop := all_prime l ∧ nondec l def prod (l : List Nat) : Nat := match l with | [] => 1 | n :: L => n * (prod L) def prime_factorization (n : Nat) (l : List Nat) : Prop := nondec_prime_list l ∧ prod l = n def rel_prime (a b : Nat) : Prop := gcd a b = 1 def congr_mod (m : Nat) (a b : Int) : Prop := (↑m : Int) ∣ (a - b) def cc (m : Nat) (a : Int) : ZMod m := (↑a : ZMod m) notation:50 a " ≡ " b " (MOD " m ")" => congr_mod m a b notation:max "["a"]_"m:max => cc m a def invertible {m : Nat} (X : ZMod m) : Prop := ∃ (Y : ZMod m), X * Y = [1]_m def num_rp_below (m k : Nat) : Nat := match k with | 0 => 0 | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j def phi (m : Nat) : Nat := num_rp_below m m def prod_seq {m : Nat} (j k : Nat) (f : Nat → ZMod m) : ZMod m := match j with | 0 => [1]_m | n + 1 => prod_seq n k f * f (k + n) def maps_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i < n, g i < n def one_one_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i1 < n, ∀ i2 < n, g i1 = g i2 → i1 = i2 def onto_below (n : Nat) (g : Nat → Nat) : Prop := ∀ k < n, ∃ i < n, g i = k def perm_below (n : Nat) (g : Nat → Nat) : Prop := maps_below n g ∧ one_one_below n g ∧ onto_below n g def inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m) def swap (u v i : Nat) : Nat := if i = u then v else if i = v then u else i namespace Euler --For definitions specific to Euler's theorem def F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m def G (m a i : Nat) : Nat := (a * i) % m def Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i end Euler /- Section 7.1 -/ theorem dvd_mod_of_dvd_a_b {a b d : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ (a % b) := by set q : Nat := a / b have h3 : b * q + a % b = a := Nat.div_add_mod a b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 define --Goal : ∃ (c : Nat), a % b = d * c apply Exists.intro (j - k * q) show a % b = d * (j - k * q) from calc a % b _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm _ = a - b * q := by rw [h3] _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc] _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm done theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry #eval gcd 672 161 --Answer: 7 lemma gcd_base (a : Nat) : gcd a 0 = a := by rfl lemma gcd_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd a b = gcd b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1)) rfl done lemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b ≠ 0) : a % b < b := by have h1 : b > 0 := Nat.pos_of_ne_zero h show a % b < b from Nat.mod_lt a h1 done lemma dvd_self (n : Nat) : n ∣ n := by apply Exists.intro 1 ring done theorem gcd_dvd : ∀ (b a : Nat), (gcd a b) ∣ a ∧ (gcd a b) ∣ b := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd a b_1) ∣ a ∧ (gcd a b_1) ∣ b_1 fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_base] --Goal: a ∣ a ∧ a ∣ 0 apply And.intro (dvd_self a) define apply Exists.intro 0 rfl done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_nonzero a h1] --Goal : gcd b (a % b) ∣ a ∧ gcd b (a % b) ∣ b have h2 : a % b < b := mod_nonzero_lt a h1 have h3 : (gcd b (a % b)) ∣ b ∧ (gcd b (a % b)) ∣ (a % b) := ih (a % b) h2 b apply And.intro _ h3.left show (gcd b (a % b)) ∣ a from dvd_a_of_dvd_b_mod h3.left h3.right done done theorem gcd_dvd_left (a b : Nat) : (gcd a b) ∣ a := (gcd_dvd b a).left theorem gcd_dvd_right (a b : Nat) : (gcd a b) ∣ b := (gcd_dvd b a).right lemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl lemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c1 a b = gcd_c2 b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done lemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl lemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * ↑(a / b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done theorem gcd_lin_comb : ∀ (b a : Nat), (gcd_c1 a b) * ↑a + (gcd_c2 a b) * ↑b = ↑(gcd a b) := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd_c1 a b_1) * ↑a + (gcd_c2 a b_1) * ↑b_1 = ↑(gcd a b_1) fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base] --Goal : 1 * ↑a + 0 * ↑0 = ↑a ring done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1] --Goal : gcd_c2 b (a % b) * ↑a + -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * ↑(a / b)) * ↑b = -- ↑(gcd b (a % b)) set r : Nat := a % b set q : Nat := a / b set s : Int := gcd_c1 b r set t : Int := gcd_c2 b r --Goal : t * ↑a + (s - t * ↑q) * ↑b = ↑(gcd b r) have h2 : r < b := mod_nonzero_lt a h1 have h3 : s * ↑b + t * ↑r = ↑(gcd b r) := ih r h2 b have h4 : b * q + r = a := Nat.div_add_mod a b rewrite [←h3, ←h4] rewrite [Nat.cast_add, Nat.cast_mul] --Goal : t * (↑b * ↑q + ↑r) + (s - t * ↑q) * ↑b = s * ↑b + t * ↑r ring done done #eval gcd_c1 672 161 --Answer: 6 #eval gcd_c2 672 161 --Answer: -25 --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161 theorem Theorem_7_1_6 {d a b : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ gcd a b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑d ∣ ↑(gcd a b) set s : Int := gcd_c1 a b set t : Int := gcd_c2 a b have h3 : s * ↑a + t * ↑b = ↑(gcd a b) := gcd_lin_comb b a rewrite [←h3] --Goal : ↑d ∣ s * ↑a + t * ↑b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul] --Goal : ↑d ∣ s * (↑d * ↑j) + t * (↑d * ↑k) define apply Exists.intro (s * ↑j + t * ↑k) ring done /- Section 7.2 -/ theorem dvd_trans {a b c : Nat} (h1 : a ∣ b) (h2 : b ∣ c) : a ∣ c := by define at h1; define at h2; define obtain (m : Nat) (h3 : b = a * m) from h1 obtain (n : Nat) (h4 : c = b * n) from h2 rewrite [h3, mul_assoc] at h4 apply Exists.intro (m * n) show c = a * (m * n) from h4 done lemma exists_prime_factor : ∀ (n : Nat), 2 ≤ n → ∃ (p : Nat), prime_factor p n := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, 2 ≤ n_1 → ∃ (p : Nat), prime_factor p n_1 assume h1 : 2 ≤ n by_cases h2 : prime n · -- Case 1. h2 : prime n apply Exists.intro n define --Goal : prime n ∧ n ∣ n show prime n ∧ n ∣ n from And.intro h2 (dvd_self n) done · -- Case 2. h2 : ¬prime n define at h2 --h2 : ¬(2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n) demorgan at h2 disj_syll h2 h1 obtain (a : Nat) (h3 : ∃ (b : Nat), a * b = n ∧ a < n ∧ b < n) from h2 obtain (b : Nat) (h4 : a * b = n ∧ a < n ∧ b < n) from h3 have h5 : 2 ≤ a := by by_contra h6 have h7 : a ≤ 1 := by linarith have h8 : n ≤ b := calc n _ = a * b := h4.left.symm _ ≤ 1 * b := by rel [h7] _ = b := by ring linarith --n ≤ b contradicts b < n done have h6 : ∃ (p : Nat), prime_factor p a := ih a h4.right.left h5 obtain (p : Nat) (h7 : prime_factor p a) from h6 apply Exists.intro p define --Goal : prime p ∧ p ∣ n define at h7 --h7 : prime p ∧ p ∣ a apply And.intro h7.left have h8 : a ∣ n := by apply Exists.intro b show n = a * b from (h4.left).symm done show p ∣ n from dvd_trans h7.right h8 done done lemma exists_least_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q := by set S : Set Nat := {p : Nat | prime_factor p n} have h2 : ∃ (p : Nat), p ∈ S := exists_prime_factor n h show ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q from well_ord_princ S h2 done lemma all_prime_nil : all_prime [] := by define --Goal : ∀ p ∈ [], prime p fix p : Nat contrapos --Goal : ¬prime p → p ∉ [] assume h1 : ¬prime p show p ∉ [] from List.not_mem_nil p done lemma all_prime_cons (n : Nat) (L : List Nat) : all_prime (n :: L) ↔ prime n ∧ all_prime L := by apply Iff.intro · -- (→) assume h1 : all_prime (n :: L) --Goal : prime n ∧ all_prime L define at h1 --h1 : ∀ p ∈ n :: L, prime p apply And.intro (h1 n (List.mem_cons_self n L)) define --Goal : ∀ p ∈ L, prime p fix p : Nat assume h2 : p ∈ L show prime p from h1 p (List.mem_cons_of_mem n h2) done · -- (←) assume h1 : prime n ∧ all_prime L --Goal : all_prime (n :: l) define : all_prime L at h1 define fix p : Nat assume h2 : p ∈ n :: L rewrite [List.mem_cons] at h2 --h2 : p = n ∨ p ∈ L by_cases on h2 · -- Case 1. h2 : p = n rewrite [h2] show prime n from h1.left done · -- Case 2. h2 : p ∈ L show prime p from h1.right p h2 done done done lemma nondec_nil : nondec [] := by define --Goal : True trivial --trivial proves some obviously true statements, such as True done lemma nondec_cons (n : Nat) (L : List Nat) : nondec (n :: L) ↔ (∀ m ∈ L, n ≤ m) ∧ nondec L := by rfl lemma prod_nil : prod [] = 1 := by rfl lemma prod_cons : prod (n :: L) = n * (prod L) := by rfl lemma exists_cons_of_length_eq_succ {A : Type} {l : List A} {n : Nat} (h : l.length = n + 1) : ∃ (a : A) (L : List A), l = a :: L ∧ L.length = n := by have h1 : ¬l.length = 0 := by linarith rewrite [List.length_eq_zero] at h1 obtain (a : A) (h2 : ∃ (L : List A), l = a :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List A) (h3 : l = a :: L) from h2 apply Exists.intro a apply Exists.intro L apply And.intro h3 have h4 : (a :: L).length = L.length + 1 := List.length_cons a L rewrite [←h3, h] at h4 show L.length = n from (Nat.add_right_cancel h4).symm done lemma list_elt_dvd_prod_by_length (a : Nat) : ∀ (n : Nat), ∀ (l : List Nat), l.length = n → a ∈ l → a ∣ prod l := by by_induc · --Base Case fix l : List Nat assume h1 : l.length = 0 rewrite [List.length_eq_zero] at h1 --h1 : l = [] rewrite [h1] --Goal : a ∈ [] → a ∣ prod [] contrapos assume h2 : ¬a ∣ prod [] show a ∉ [] from List.not_mem_nil a done · -- Induction Step fix n : Nat assume ih : ∀ (l : List Nat), List.length l = n → a ∈ l → a ∣ prod l fix l : List Nat assume h1 : l.length = n + 1 --Goal : a ∈ l → a ∣ prod l obtain (b : Nat) (h2 : ∃ (L : List Nat), l = b :: L ∧ L.length = n) from exists_cons_of_length_eq_succ h1 obtain (L : List Nat) (h3 : l = b :: L ∧ L.length = n) from h2 have h4 : a ∈ L → a ∣ prod L := ih L h3.right assume h5 : a ∈ l rewrite [h3.left, prod_cons] --Goal : a ∣ b * prod L rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b ∨ a ∈ L by_cases on h5 · -- Case 1. h5 : a = b apply Exists.intro (prod L) rewrite [h5] rfl done · -- Case 2. h5 : a ∈ L have h6 : a ∣ prod L := h4 h5 have h7 : prod L ∣ b * prod L := by apply Exists.intro b ring done show a ∣ b * prod L from dvd_trans h6 h7 done done done lemma list_elt_dvd_prod {a : Nat} {l : List Nat} (h : a ∈ l) : a ∣ prod l := by set n : Nat := l.length have h1 : l.length = n := by rfl show a ∣ prod l from list_elt_dvd_prod_by_length a n l h1 h done lemma exists_prime_factorization : ∀ (n : Nat), n ≥ 1 → ∃ (l : List Nat), prime_factorization n l := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, n_1 ≥ 1 → ∃ (l : List Nat), prime_factorization n_1 l assume h1 : n ≥ 1 by_cases h2 : n = 1 · -- Case 1. h2 : n = 1 apply Exists.intro [] define apply And.intro · -- Proof of nondec_prime_list [] define show all_prime [] ∧ nondec [] from And.intro all_prime_nil nondec_nil done · -- Proof of prod [] = n rewrite [prod_nil, h2] rfl done done · -- Case 2. h2 : n ≠ 1 have h3 : n ≥ 2 := lt_of_le_of_ne' h1 h2 obtain (p : Nat) (h4 : prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q) from exists_least_prime_factor h3 have p_prime_factor : prime_factor p n := h4.left define at p_prime_factor have p_prime : prime p := p_prime_factor.left have p_dvd_n : p ∣ n := p_prime_factor.right have p_least : ∀ (q : Nat), prime_factor q n → p ≤ q := h4.right obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n have h5 : m ≠ 0 := by contradict h1 with h6 have h7 : n = 0 := calc n _ = p * m := n_eq_pm _ = p * 0 := by rw [h6] _ = 0 := by ring rewrite [h7] decide done have m_pos : 0 < m := Nat.pos_of_ne_zero h5 have m_lt_n : m < n := by define at p_prime show m < n from calc m _ < m + m := by linarith _ = 2 * m := by ring _ ≤ p * m := by rel [p_prime.left] _ = n := n_eq_pm.symm done obtain (L : List Nat) (h6 : prime_factorization m L) from ih m m_lt_n m_pos define at h6 have ndpl_L : nondec_prime_list L := h6.left define at ndpl_L apply Exists.intro (p :: L) define apply And.intro · -- Proof of nondec_prime_list (p :: L) define apply And.intro · -- Proof of all_prime (p :: L) rewrite [all_prime_cons] show prime p ∧ all_prime L from And.intro p_prime ndpl_L.left done · -- Proof of nondec (p :: L) rewrite [nondec_cons] apply And.intro _ ndpl_L.right fix q : Nat assume q_in_L : q ∈ L have h7 : q ∣ prod L := list_elt_dvd_prod q_in_L rewrite [h6.right] at h7 --h7 : q ∣ m have h8 : m ∣ n := by apply Exists.intro p rewrite [n_eq_pm] ring done have q_dvd_n : q ∣ n := dvd_trans h7 h8 have ap_L : all_prime L := ndpl_L.left define at ap_L have q_prime_factor : prime_factor q n := And.intro (ap_L q q_in_L) q_dvd_n show p ≤ q from p_least q q_prime_factor done done · -- Proof of prod (p :: L) = n rewrite [prod_cons, h6.right, n_eq_pm] rfl done done done theorem Theorem_7_2_2 {a b c : Nat} (h1 : c ∣ a * b) (h2 : rel_prime a c) : c ∣ b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑c ∣ ↑b define at h1; define at h2; define obtain (j : Nat) (h3 : a * b = c * j) from h1 set s : Int := gcd_c1 a c set t : Int := gcd_c2 a c have h4 : s * ↑a + t * ↑c = ↑(gcd a c) := gcd_lin_comb c a rewrite [h2, Nat.cast_one] at h4 --h4 : s * ↑a + t * ↑c = (1 : Int) apply Exists.intro (s * ↑j + t * ↑b) show ↑b = ↑c * (s * ↑j + t * ↑b) from calc ↑b _ = (1 : Int) * ↑b := (one_mul _).symm _ = (s * ↑a + t * ↑c) * ↑b := by rw [h4] _ = s * (↑a * ↑b) + t * ↑c * ↑b := by ring _ = s * (↑c * ↑j) + t * ↑c * ↑b := by rw [←Nat.cast_mul a b, h3, Nat.cast_mul c j] _ = ↑c * (s * ↑j + t * ↑b) := by ring done lemma le_nonzero_prod_left {a b : Nat} (h : a * b ≠ 0) : a ≤ a * b := by have h1 : b ≠ 0 := by contradict h with h1 rewrite [h1] ring done have h2 : 1 ≤ b := Nat.pos_of_ne_zero h1 show a ≤ a * b from calc a = a * 1 := (mul_one a).symm _ ≤ a * b := by rel [h2] done lemma le_nonzero_prod_right {a b : Nat} (h : a * b ≠ 0) : b ≤ a * b := by rewrite [mul_comm] rewrite [mul_comm] at h show b ≤ b * a from le_nonzero_prod_left h done lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry lemma rel_prime_of_prime_not_dvd {a p : Nat} (h1 : prime p) (h2 : ¬p ∣ a) : rel_prime a p := by have h3 : gcd a p ∣ a := gcd_dvd_left a p have h4 : gcd a p ∣ p := gcd_dvd_right a p have h5 : gcd a p = 1 ∨ gcd a p = p := dvd_prime h1 h4 have h6 : gcd a p ≠ p := by contradict h2 with h6 rewrite [h6] at h3 show p ∣ a from h3 done disj_syll h5 h6 show rel_prime a p from h5 done theorem Theorem_7_2_3 {a b p : Nat} (h1 : prime p) (h2 : p ∣ a * b) : p ∣ a ∨ p ∣ b := by or_right with h3 have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3 show p ∣ b from Theorem_7_2_2 h2 h4 done lemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a ≥ 1 := by have h1 : a ≠ 0 := by by_contra h1 rewrite [h1] at h contradict h linarith done show a ≥ 1 from Nat.pos_of_ne_zero h1 done lemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by have h1 : a ≥ 1 := ge_one_of_prod_one h have h2 : a * b ≠ 0 := by linarith have h3 : a ≤ a * b := le_nonzero_prod_left h2 rewrite [h] at h3 show a = 1 from Nat.le_antisymm h3 h1 done lemma eq_one_of_dvd_one {n : Nat} (h : n ∣ 1) : n = 1 := by obtain (j : Nat) (h1 : 1 = n * j) from h show n = 1 from eq_one_of_prod_one h1.symm done lemma prime_not_one {p : Nat} (h : prime p) : p ≠ 1 := by define at h linarith done theorem Theorem_7_2_4 {p : Nat} (h1 : prime p) : ∀ (l : List Nat), p ∣ prod l → ∃ a ∈ l, p ∣ a := by apply List.rec · -- Base Case. Goal : p ∣ prod [] → ∃ a ∈ [], p ∣ a rewrite [prod_nil] assume h2 : p ∣ 1 show ∃ a ∈ [], p ∣ a from absurd (eq_one_of_dvd_one h2) (prime_not_one h1) done · -- Induction Step fix b : Nat fix L : List Nat assume ih : p ∣ prod L → ∃ a ∈ L, p ∣ a --Goal : p ∣ prod (b :: L) → ∃ a ∈ b :: L, p ∣ a assume h2 : p ∣ prod (b :: L) rewrite [prod_cons] at h2 have h3 : p ∣ b ∨ p ∣ prod L := Theorem_7_2_3 h1 h2 by_cases on h3 · -- Case 1. h3 : p ∣ b apply Exists.intro b show b ∈ b :: L ∧ p ∣ b from And.intro (List.mem_cons_self b L) h3 done · -- Case 2. h3 : p ∣ prod L obtain (a : Nat) (h4 : a ∈ L ∧ p ∣ a) from ih h3 apply Exists.intro a show a ∈ b :: L ∧ p ∣ a from And.intro (List.mem_cons_of_mem b h4.left) h4.right done done done lemma prime_in_list {p : Nat} {l : List Nat} (h1 : prime p) (h2 : all_prime l) (h3 : p ∣ prod l) : p ∈ l := by obtain (a : Nat) (h4 : a ∈ l ∧ p ∣ a) from Theorem_7_2_4 h1 l h3 define at h2 have h5 : prime a := h2 a h4.left have h6 : p = 1 ∨ p = a := dvd_prime h5 h4.right disj_syll h6 (prime_not_one h1) rewrite [h6] show a ∈ l from h4.left done lemma first_le_first {p q : Nat} {l m : List Nat} (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m)) (h3 : prod (p :: l) = prod (q :: m)) : p ≤ q := by define at h1; define at h2 have h4 : q ∣ prod (p :: l) := by define apply Exists.intro (prod m) rewrite [←prod_cons] show prod (p :: l) = prod (q :: m) from h3 done have h5 : all_prime (q :: m) := h2.left rewrite [all_prime_cons] at h5 have h6 : q ∈ p :: l := prime_in_list h5.left h1.left h4 have h7 : nondec (p :: l) := h1.right rewrite [nondec_cons] at h7 rewrite [List.mem_cons] at h6 by_cases on h6 · -- Case 1. h6 : q = p linarith done · -- Case 2. h6 : q ∈ l have h8 : ∀ m ∈ l, p ≤ m := h7.left show p ≤ q from h8 q h6 done done lemma nondec_prime_list_tail {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by define at h define rewrite [all_prime_cons, nondec_cons] at h show all_prime l ∧ nondec l from And.intro h.left.right h.right.right done lemma cons_prod_not_one {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : prod (p :: l) ≠ 1 := by define at h have h1 : all_prime (p :: l) := h.left rewrite [all_prime_cons] at h1 rewrite [prod_cons] by_contra h2 show False from (prime_not_one h1.left) (eq_one_of_prod_one h2) done lemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) : l = [] ↔ prod l = 1 := by apply Iff.intro · -- (→) assume h1 : l = [] rewrite [h1] show prod [] = 1 from prod_nil done · -- (←) contrapos assume h1 : ¬l = [] obtain (p : Nat) (h2 : ∃ (L : List Nat), l = p :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List Nat) (h3 : l = p :: L) from h2 rewrite [h3] at h rewrite [h3] show ¬prod (p :: L) = 1 from cons_prod_not_one h done done lemma prime_pos {p : Nat} (h : prime p) : p > 0 := by define at h linarith done theorem Theorem_7_2_5 : ∀ (l1 l2 : List Nat), nondec_prime_list l1 → nondec_prime_list l2 → prod l1 = prod l2 → l1 = l2 := by apply List.rec · -- Base Case. Goal : ∀ (l2 : List Nat), nondec_prime_list [] → -- nondec_prime_list l2 → prod [] = prod l2 → [] = l2 fix l2 : List Nat assume h1 : nondec_prime_list [] assume h2 : nondec_prime_list l2 assume h3 : prod [] = prod l2 rewrite [prod_nil, eq_comm, ←list_nil_iff_prod_one h2] at h3 show [] = l2 from h3.symm done · -- Induction Step fix p : Nat fix L1 : List Nat assume ih : ∀ (L2 : List Nat), nondec_prime_list L1 → nondec_prime_list L2 → prod L1 = prod L2 → L1 = L2 -- Goal : ∀ (l2 : List Nat), nondec_prime_list (p :: L1) → -- nondec_prime_list l2 → prod (p :: L1) = prod l2 → p :: L1 = l2 fix l2 : List Nat assume h1 : nondec_prime_list (p :: L1) assume h2 : nondec_prime_list l2 assume h3 : prod (p :: L1) = prod l2 have h4 : ¬prod (p :: L1) = 1 := cons_prod_not_one h1 rewrite [h3, ←list_nil_iff_prod_one h2] at h4 obtain (q : Nat) (h5 : ∃ (L : List Nat), l2 = q :: L) from List.exists_cons_of_ne_nil h4 obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5 rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2) rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2) have h7 : p ≤ q := first_le_first h1 h2 h3 have h8 : q ≤ p := first_le_first h2 h1 h3.symm have h9 : p = q := by linarith rewrite [h9, prod_cons, prod_cons] at h3 --h3 : q * prod L1 = q * prod L2 have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1 have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2 define at h2 have h12 : all_prime (q :: L2) := h2.left rewrite [all_prime_cons] at h12 have h13 : q > 0 := prime_pos h12.left have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3 have h15 : L1 = L2 := ih L2 h10 h11 h14 rewrite [h6, h9, h15] rfl done done theorem fund_thm_arith (n : Nat) (h : n ≥ 1) : ∃! (l : List Nat), prime_factorization n l := by exists_unique · -- Existence show ∃ (l : List Nat), prime_factorization n l from exists_prime_factorization n h done · -- Uniqueness fix l1 : List Nat; fix l2 : List Nat assume h1 : prime_factorization n l1 assume h2 : prime_factorization n l2 define at h1; define at h2 have h3 : prod l1 = n := h1.right rewrite [←h2.right] at h3 show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3 done done /- Section 7.3 -/ theorem congr_refl (m : Nat) : ∀ (a : Int), a ≡ a (MOD m) := by fix a : Int define --Goal : ∃ (c : Int), a - a = ↑m * c apply Exists.intro 0 ring done theorem congr_symm {m : Nat} : ∀ {a b : Int}, a ≡ b (MOD m) → b ≡ a (MOD m) := by fix a : Int; fix b : Int assume h1 : a ≡ b (MOD m) define at h1 --h1 : ∃ (c : Int), a - b = ↑m * c define --Goal : ∃ (c : Int), b - a = ↑m * c obtain (c : Int) (h2 : a - b = m * c) from h1 apply Exists.intro (-c) show b - a = m * (-c) from calc b - a _ = -(a - b) := by ring _ = -(m * c) := by rw [h2] _ = m * (-c) := by ring done theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry /- Fundamental properties of congruence classes -/ lemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) : X = Y ↔ X.val = Y.val := Fin.ext_iff lemma val_nat_eq_mod (n k : Nat) : ([k]_(n + 1)).val = k % (n + 1) := by rfl lemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl theorem cc_rep {m : Nat} (X : ZMod m) : ∃ (a : Int), X = [a]_m := match m with | 0 => by apply Exists.intro X rfl done | n + 1 => by apply Exists.intro ↑(X.val) have h1 : X.val < n + 1 := Fin.prop X rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1] rfl done theorem add_class (m : Nat) (a b : Int) : [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm theorem mul_class (m : Nat) (a b : Int) : [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm lemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ [a - b]_m = [0]_m := by apply Iff.intro · -- (→) assume h1 : [a]_m = [b]_m have h2 : a - b = a + (-b) := by ring have h3 : b + (-b) = 0 := by ring show [a - b]_m = [0]_m from calc [a - b]_m _ = [a + (-b)]_m := by rw [h2] _ = [a]_m + [-b]_m := by rw [add_class] _ = [b]_m + [-b]_m := by rw [h1] _ = [b + -b]_m := by rw [add_class] _ = [0]_m := by rw [h3] done · -- (←) assume h1 : [a - b]_m = [0]_m have h2 : b + (a - b) = a := by ring have h3 : b + 0 = b := by ring show [a]_m = [b]_m from calc [a]_m _ = [b + (a - b)]_m := by rw [h2] _ = [b]_m + [a - b]_m := by rw [add_class] _ = [b]_m + [0]_m := by rw [h1] _ = [b + 0]_m := by rw [add_class] _ = [b]_m := by rw [h3] done done lemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) : [a]_m = [0]_m → [-a]_m = [0]_m := by assume h1 : [a]_m = [0]_m have h2 : 0 + (-a) = -a := by ring have h3 : a + (-a) = 0 := by ring show [-a]_m = [0]_m from calc [-a]_m _ = [0 + (-a)]_m := by rw [h2] _ = [0]_m + [-a]_m := by rw [add_class] _ = [a]_m + [-a]_m := by rw [h1] _ = [a + (-a)]_m := by rw [add_class] _ = [0]_m := by rw [h3] done lemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) : [-a]_m = [0]_m ↔ [a]_m = [0]_m := by apply Iff.intro _ (cc_neg_zero_of_cc_zero m a) assume h1 : [-a]_m = [0]_m have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1 have h3 : -(-a) = a := by ring rewrite [h3] at h2 show [a]_m = [0]_m from h2 done lemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl lemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m ↔ m ∣ k := match m with | 0 => by have h : (0 : Int) = (↑(0 : Nat) : Int) := by rfl rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj] apply Iff.intro · -- (→) assume h1 : k = 0 rewrite [h1] show 0 ∣ 0 from dvd_self 0 done · -- (←) assume h1 : 0 ∣ k obtain (c : Nat) (h2 : k = 0 * c) from h1 rewrite [h2] ring done done | n + 1 => by rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero] show k % (n + 1) = 0 ↔ n + 1 ∣ k from (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm done lemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m ↔ ↑m ∣ a := by obtain (k : Nat) (h1 : a = ↑k ∨ a = -↑k) from Int.eq_nat_or_neg a by_cases on h1 · -- Case 1. h1: a = ↑k rewrite [h1, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done · -- Case 2. h1: a = -↑k rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done done theorem cc_eq_iff_congr (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ a ≡ b (MOD m) := calc [a]_m = [b]_m _ ↔ [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b _ ↔ ↑m ∣ (a - b) := cc_zero_iff_dvd m (a - b) _ ↔ a ≡ b (MOD m) := by rfl /- End of fundamental properties of congruence classes -/ lemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m := by have h1 : (↑m : Int) ≠ 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m) show 0 ≤ a % m from Int.emod_nonneg a h1 done lemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m) have h2 : (↑m : Int) > 0 := (Nat.cast_pos).rtl h1 show a % m < m from Int.emod_lt_of_pos a h2 done lemma congr_mod_mod (m : Nat) (a : Int) : a ≡ a % m (MOD m) := by define have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m apply Exists.intro (a / m) show a - a % m = m * (a / m) from calc a - (a % m) _ = m * (a / m) + a % m - a % m := by rw [h1] _ = m * (a / m) := by ring done lemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) := And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a)) theorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) : ∃! (r : Int), 0 ≤ r ∧ r < m ∧ a ≡ r (MOD m) := by exists_unique · -- Existence apply Exists.intro (a % m) show 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) from mod_cmpl_res m a done · -- Uniqueness fix r1 : Int; fix r2 : Int assume h1 : 0 ≤ r1 ∧ r1 < m ∧ a ≡ r1 (MOD m) assume h2 : 0 ≤ r2 ∧ r2 < m ∧ a ≡ r2 (MOD m) have h3 : r1 ≡ r2 (MOD m) := congr_trans (congr_symm h1.right.right) h2.right.right obtain (d : Int) (h4 : r1 - r2 = m * d) from h3 have h5 : r1 - r2 < m * 1 := by linarith have h6 : m * (-1) < r1 - r2 := by linarith rewrite [h4] at h5 --h5 : m * d < m * 1 rewrite [h4] at h6 --h6 : m * -1 < m * d have h7 : (↑m : Int) ≥ 0 := Nat.cast_nonneg m have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7 have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7 have h10 : d = 0 := by linarith show r1 = r2 from calc r1 _ = r1 - r2 + r2 := by ring _ = m * 0 + r2 := by rw [h4, h10] _ = r2 := by ring done done lemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m := (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a) theorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y rewrite [h1, h2] have h3 : a + b = b + a := by ring show [a]_m + [b]_m = [b]_m + [a]_m from calc [a]_m + [b]_m _ = [a + b]_m := add_class m a b _ = [b + a]_m := by rw [h3] _ = [b]_m + [a]_m := (add_class m b a).symm done theorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X rewrite [h1] have h2 : a * 1 = a := by ring show [a]_m * [1]_m = [a]_m from calc [a]_m * [1]_m _ = [a * 1]_m := mul_class m a 1 _ = [a]_m := by rw [h2] done theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry lemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) : [a]_m * [gcd_c2 m a]_m = [1]_m := by set s : Int := gcd_c1 m a have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m define at h1 rewrite [h1, Nat.cast_one] at h2 --h2 : s * ↑m + gcd_c2 m a * ↑a = 1 rewrite [mul_class, cc_eq_iff_congr] define --Goal : ∃ (c : Int), ↑a * gcd_c2 m a - 1 = ↑m * c apply Exists.intro (-s) show a * (gcd_c2 m a) - 1 = m * (-s) from calc a * (gcd_c2 m a) - 1 _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring _ = 1 + m * (-s) - 1 := by rw [h2] _ = m * (-s) := by ring done theorem Theorem_7_3_7 (m a : Nat) : invertible [a]_m ↔ rel_prime m a := by apply Iff.intro · -- (→) assume h1 : invertible [a]_m define at h1 obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1 obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y rewrite [h3, mul_class, cc_eq_iff_congr] at h2 define at h2 obtain (c : Int) (h4 : a * b - 1 = m * c) from h2 rewrite [Exercise_7_2_6] --Goal : ∃ (s t : Int), s * ↑m + t * ↑a = 1 apply Exists.intro (-c) apply Exists.intro b show (-c) * m + b * a = 1 from calc (-c) * m + b * a _ = (-c) * m + (a * b - 1) + 1 := by ring _ = (-c) * m + m * c + 1 := by rw [h4] _ = 1 := by ring done · -- (←) assume h1 : rel_prime m a define show ∃ (Y : ZMod m), [a]_m * Y = [1]_m from Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1) done done /- Section 7.4 -/ section Euler open Euler lemma num_rp_below_base {m : Nat} : num_rp_below m 0 = 0 := by rfl lemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) : num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by have h1 : num_rp_below m (j + 1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : gcd m j = 1 rewrite [if_pos h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1 show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1 done lemma num_rp_below_step_not_rp {m j : Nat} (h : ¬rel_prime m j) : num_rp_below m (j + 1) = num_rp_below m j := by have h1 : num_rp_below m (j +1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : ¬gcd m j = 1 rewrite [if_neg h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j show num_rp_below m (j + 1) = num_rp_below m j from h1 done lemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl #eval phi 10 --Answer: 4 lemma prod_inv_iff_inv {m : Nat} {X : ZMod m} (h1 : invertible X) (Y : ZMod m) : invertible (X * Y) ↔ invertible Y := by apply Iff.intro · -- (→) assume h2 : invertible (X * Y) obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2 apply Exists.intro (X * Z) rewrite [←h3] --Goal : Y * (X * Z) = X * Y * Z ring --Note that ring can do algebra in ZMod m done · -- (←) assume h2 : invertible Y obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1 obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2 apply Exists.intro (Xi * Yi) show (X * Y) * (Xi * Yi) = [1]_m from calc X * Y * (Xi * Yi) _ = (X * Xi) * (Y * Yi) := by ring _ = [1]_m * [1]_m := by rw [h3, h4] _ = [1]_m := Theorem_7_3_6_7 [1]_m done done lemma F_rp_def {m i : Nat} (h : rel_prime m i) : F m i = [i]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h --h : gcd m i = 1 rewrite [if_pos h] at h1 show F m i = [i]_m from h1 done lemma F_not_rp_def {m i : Nat} (h : ¬rel_prime m i) : F m i = [1]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h rewrite [h1, if_neg h] rfl done lemma prod_seq_base {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 0 k f = [1]_m := by rfl lemma prod_seq_step {m : Nat} (n k : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl lemma prod_seq_zero_step {m : Nat} (n : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by rewrite [prod_seq_step, zero_add] rfl done lemma prod_one {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 1 k f = f k := by rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7] rfl done lemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl lemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m := calc [G m a i]_m _ = [(a * i) % m]_m := by rfl _ = [a * i]_m := (cc_eq_mod m (a * i)).symm _ = [a]_m * [i]_m := (mul_class m a i).symm lemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) : rel_prime m (G m a i) ↔ rel_prime m i := by have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1 show rel_prime m (G m a i) ↔ rel_prime m i from calc rel_prime m (G m a i) _ ↔ invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm _ ↔ invertible ([a]_m * [i]_m) := by rw [cc_G] _ ↔ invertible [i]_m := prod_inv_iff_inv h2 ([i]_m) _ ↔ rel_prime m i := Theorem_7_3_7 m i done lemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) : F m (G m a i) = [a]_m * F m i := by have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2 show F m (G m a i) = [a]_m * F m i from calc F m (G m a i) _ = [G m a i]_m := F_rp_def h3 _ = [a]_m * [i]_m := cc_G m a i _ = [a]_m * F m i := by rw [F_rp_def h2] done lemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : ¬rel_prime m i) : F m (G m a i) = [1]_m := by rewrite [←G_rp_iff h1 i] at h2 show F m (G m a i) = [1]_m from F_not_rp_def h2 done lemma FG_prod {m a : Nat} (h1 : rel_prime m a) : ∀ (k : Nat), prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by by_induc · -- Base Case show prod_seq 0 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from calc prod_seq 0 0 ((F m) ∘ (G m a)) _ = [1]_m := prod_seq_base _ _ _ = [a]_m ^ 0 * [1]_m := by ring _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by rw [num_rp_below_base, prod_seq_base] done · -- Induction Step fix k : Nat assume ih : prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) by_cases h2 : rel_prime m k · -- Case 1. h2 : rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([a]_m * F m k) := by rw [FG_rp h1 h2] _ = [a]_m ^ ((num_rp_below m k) + 1) * ((prod_seq k 0 (F m)) * F m k) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_rp h2, prod_seq_zero_step] done · -- Case 2. h2 : ¬rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([1]_m) := by rw [FG_not_rp h1 h2] _ = [a]_m ^ (num_rp_below m k) * (prod_seq k 0 (F m) * ([1]_m)) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_not_rp h2, prod_seq_zero_step, F_not_rp_def h2] done done done lemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by define --Goal : ∀ i < m, G m a i < m fix i : Nat assume h1 : i < m rewrite [G_def] --Goal : a * i % m < m show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m) done lemma left_inv_one_one_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g' (g i) = i) : one_one_below n g := sorry lemma right_inv_onto_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g (g' i) = i) (h2 : maps_below n g') : onto_below n g := by define at h2; define fix k : Nat assume h3 : k < n apply Exists.intro (g' k) show g' k < n ∧ g (g' k) = k from And.intro (h2 k h3) (h1 k h3) done lemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m * [inv_mod m a]_m = [1]_m := by have h2 : 0 ≤ (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a) show [a]_m * [inv_mod m a]_m = [1]_m from calc [a]_m * [inv_mod m a]_m _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2] _ = [a]_m * [gcd_c2 m a]_m := by rw [←cc_eq_mod] _ = [1]_m := gcd_c2_inv h1 done lemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m := calc a * (b % m) % m = a % m * (b % m % m) % m := Nat.mul_mod _ _ _ _ = a % m * (b % m) % m := by rw [Nat.mod_mod] _ = a * b % m := (Nat.mul_mod _ _ _).symm lemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm] rfl done theorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] : ↑a ≡ ↑b (MOD m) ↔ a % m = b % m := sorry lemma mul_inv_mod_cancel {m a i : Nat} [NeZero m] (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1 rewrite [mul_class, cc_eq_iff_congr, ←Nat.cast_mul, ←Nat.cast_one, congr_iff_mod_eq_Nat] at h3 show a * inv_mod m a * i % m = i from calc a * (inv_mod m a) * i % m _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = 1 % m * i % m := by rw [h3] _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = i % m := by rw [one_mul] _ = i := Nat.mod_eq_of_lt h2 done lemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl lemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, G m a (Ginv m a i) = i := by fix i : Nat assume h2 : i < m show G m a (Ginv m a i) = i from calc G m a (Ginv m a i) _ = a * ((inv_mod m a * i) % m) % m := by rfl _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, Ginv m a (G m a i) = i := by fix i : Nat assume h2 : i < m show Ginv m a (G m a i) = i from calc Ginv m a (G m a i) _ = inv_mod m a * ((a * i) % m) % m := by rfl _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc, mul_comm (inv_mod m a)] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_maps_below (m a : Nat) [NeZero m] : maps_below m (Ginv m a) := G_maps_below m (inv_mod m a) lemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : one_one_below m (G m a) := left_inv_one_one_below (Ginv_left_inv h1) lemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : onto_below m (G m a) := right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a) lemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : perm_below m (G m a) := And.intro (G_maps_below m a) (And.intro (G_one_one_below h1) (G_onto_below h1)) --Permuting a product of congruence classes doesn't change product lemma swap_fst (u v : Nat) : swap u v u = v := by define : swap u v u --Goal : (if u = u then v else if u = v then u else u) = v have h : u = u := by rfl rewrite [if_pos h] rfl done lemma swap_snd (u v : Nat) : swap u v v = u := by define : swap u v v by_cases h1 : v = u · -- Case 1. h1 : v = u rewrite [if_pos h1] show v = u from h1 done · -- Case 2. h1 : v ≠ u rewrite [if_neg h1] have h2 : v = v := by rfl rewrite [if_pos h2] rfl done done lemma swap_other {u v i : Nat} (h1 : i ≠ u) (h2 : i ≠ v) : swap u v i = i := by define : swap u v i rewrite [if_neg h1, if_neg h2] rfl done lemma swap_values (u v i : Nat) : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := by by_cases h1 : i = u · -- Case 1. h1 : i = u apply Or.inl rewrite [h1] show swap u v u = v from swap_fst u v done · -- Case 2. h1 : i ≠ u apply Or.inr by_cases h2 : i = v · -- Case 2.1. h2 : i = v apply Or.inl rewrite [h2] show swap u v v = u from swap_snd u v done · -- Case 2.2. h2 : i ≠ v apply Or.inr show swap u v i = i from swap_other h1 h2 done done done lemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by define fix i : Nat assume h3 : i < n have h4 : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := swap_values u v i by_cases on h4 · -- Case 1. h4 : swap u v i = v rewrite [h4] show v < n from h2 done · -- Case 2. by_cases on h4 · -- Case 2.1. h4 : swap u v i = u rewrite [h4] show u < n from h1 done · -- Case 2.2. h4 : swap u v i = i rewrite [h4] show i < n from h3 done done done

lemma swap_swap (u v n : Nat) : ∀ i < n, swap u v (swap u v i) = i

HTPI.swap_swap

htpi/HTPILib/Chap7.lean

htpi

/- Copyright 2023 Daniel J. Velleman -/ import HTPILib.Chap6 namespace HTPI /- Definitions -/ lemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by have h : n + 1 > 0 := Nat.succ_pos n show a % (n + 1) < n + 1 from Nat.mod_lt a h done def gcd (a b : Nat) : Nat := match b with | 0 => a | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd (n + 1) (a % (n + 1)) termination_by b mutual def gcd_c1 (a b : Nat) : Int := match b with | 0 => 1 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c2 (n + 1) (a % (n + 1)) --Corresponds to s = t' termination_by b def gcd_c2 (a b : Nat) : Int := match b with | 0 => 0 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c1 (n + 1) (a % (n + 1)) - (gcd_c2 (n + 1) (a % (n + 1))) * ↑(a / (n + 1)) --Corresponds to t = s' - t'q termination_by b end def prime (n : Nat) : Prop := 2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n def prime_factor (p n : Nat) : Prop := prime p ∧ p ∣ n def all_prime (l : List Nat) : Prop := ∀ p ∈ l, prime p def nondec (l : List Nat) : Prop := match l with | [] => True --Of course, True is a proposition that is always true | n :: L => (∀ m ∈ L, n ≤ m) ∧ nondec L def nondec_prime_list (l : List Nat) : Prop := all_prime l ∧ nondec l def prod (l : List Nat) : Nat := match l with | [] => 1 | n :: L => n * (prod L) def prime_factorization (n : Nat) (l : List Nat) : Prop := nondec_prime_list l ∧ prod l = n def rel_prime (a b : Nat) : Prop := gcd a b = 1 def congr_mod (m : Nat) (a b : Int) : Prop := (↑m : Int) ∣ (a - b) def cc (m : Nat) (a : Int) : ZMod m := (↑a : ZMod m) notation:50 a " ≡ " b " (MOD " m ")" => congr_mod m a b notation:max "["a"]_"m:max => cc m a def invertible {m : Nat} (X : ZMod m) : Prop := ∃ (Y : ZMod m), X * Y = [1]_m def num_rp_below (m k : Nat) : Nat := match k with | 0 => 0 | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j def phi (m : Nat) : Nat := num_rp_below m m def prod_seq {m : Nat} (j k : Nat) (f : Nat → ZMod m) : ZMod m := match j with | 0 => [1]_m | n + 1 => prod_seq n k f * f (k + n) def maps_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i < n, g i < n def one_one_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i1 < n, ∀ i2 < n, g i1 = g i2 → i1 = i2 def onto_below (n : Nat) (g : Nat → Nat) : Prop := ∀ k < n, ∃ i < n, g i = k def perm_below (n : Nat) (g : Nat → Nat) : Prop := maps_below n g ∧ one_one_below n g ∧ onto_below n g def inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m) def swap (u v i : Nat) : Nat := if i = u then v else if i = v then u else i namespace Euler --For definitions specific to Euler's theorem def F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m def G (m a i : Nat) : Nat := (a * i) % m def Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i end Euler /- Section 7.1 -/ theorem dvd_mod_of_dvd_a_b {a b d : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ (a % b) := by set q : Nat := a / b have h3 : b * q + a % b = a := Nat.div_add_mod a b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 define --Goal : ∃ (c : Nat), a % b = d * c apply Exists.intro (j - k * q) show a % b = d * (j - k * q) from calc a % b _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm _ = a - b * q := by rw [h3] _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc] _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm done theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry #eval gcd 672 161 --Answer: 7 lemma gcd_base (a : Nat) : gcd a 0 = a := by rfl lemma gcd_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd a b = gcd b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1)) rfl done lemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b ≠ 0) : a % b < b := by have h1 : b > 0 := Nat.pos_of_ne_zero h show a % b < b from Nat.mod_lt a h1 done lemma dvd_self (n : Nat) : n ∣ n := by apply Exists.intro 1 ring done theorem gcd_dvd : ∀ (b a : Nat), (gcd a b) ∣ a ∧ (gcd a b) ∣ b := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd a b_1) ∣ a ∧ (gcd a b_1) ∣ b_1 fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_base] --Goal: a ∣ a ∧ a ∣ 0 apply And.intro (dvd_self a) define apply Exists.intro 0 rfl done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_nonzero a h1] --Goal : gcd b (a % b) ∣ a ∧ gcd b (a % b) ∣ b have h2 : a % b < b := mod_nonzero_lt a h1 have h3 : (gcd b (a % b)) ∣ b ∧ (gcd b (a % b)) ∣ (a % b) := ih (a % b) h2 b apply And.intro _ h3.left show (gcd b (a % b)) ∣ a from dvd_a_of_dvd_b_mod h3.left h3.right done done theorem gcd_dvd_left (a b : Nat) : (gcd a b) ∣ a := (gcd_dvd b a).left theorem gcd_dvd_right (a b : Nat) : (gcd a b) ∣ b := (gcd_dvd b a).right lemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl lemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c1 a b = gcd_c2 b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done lemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl lemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * ↑(a / b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done theorem gcd_lin_comb : ∀ (b a : Nat), (gcd_c1 a b) * ↑a + (gcd_c2 a b) * ↑b = ↑(gcd a b) := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd_c1 a b_1) * ↑a + (gcd_c2 a b_1) * ↑b_1 = ↑(gcd a b_1) fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base] --Goal : 1 * ↑a + 0 * ↑0 = ↑a ring done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1] --Goal : gcd_c2 b (a % b) * ↑a + -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * ↑(a / b)) * ↑b = -- ↑(gcd b (a % b)) set r : Nat := a % b set q : Nat := a / b set s : Int := gcd_c1 b r set t : Int := gcd_c2 b r --Goal : t * ↑a + (s - t * ↑q) * ↑b = ↑(gcd b r) have h2 : r < b := mod_nonzero_lt a h1 have h3 : s * ↑b + t * ↑r = ↑(gcd b r) := ih r h2 b have h4 : b * q + r = a := Nat.div_add_mod a b rewrite [←h3, ←h4] rewrite [Nat.cast_add, Nat.cast_mul] --Goal : t * (↑b * ↑q + ↑r) + (s - t * ↑q) * ↑b = s * ↑b + t * ↑r ring done done #eval gcd_c1 672 161 --Answer: 6 #eval gcd_c2 672 161 --Answer: -25 --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161 theorem Theorem_7_1_6 {d a b : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ gcd a b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑d ∣ ↑(gcd a b) set s : Int := gcd_c1 a b set t : Int := gcd_c2 a b have h3 : s * ↑a + t * ↑b = ↑(gcd a b) := gcd_lin_comb b a rewrite [←h3] --Goal : ↑d ∣ s * ↑a + t * ↑b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul] --Goal : ↑d ∣ s * (↑d * ↑j) + t * (↑d * ↑k) define apply Exists.intro (s * ↑j + t * ↑k) ring done /- Section 7.2 -/ theorem dvd_trans {a b c : Nat} (h1 : a ∣ b) (h2 : b ∣ c) : a ∣ c := by define at h1; define at h2; define obtain (m : Nat) (h3 : b = a * m) from h1 obtain (n : Nat) (h4 : c = b * n) from h2 rewrite [h3, mul_assoc] at h4 apply Exists.intro (m * n) show c = a * (m * n) from h4 done lemma exists_prime_factor : ∀ (n : Nat), 2 ≤ n → ∃ (p : Nat), prime_factor p n := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, 2 ≤ n_1 → ∃ (p : Nat), prime_factor p n_1 assume h1 : 2 ≤ n by_cases h2 : prime n · -- Case 1. h2 : prime n apply Exists.intro n define --Goal : prime n ∧ n ∣ n show prime n ∧ n ∣ n from And.intro h2 (dvd_self n) done · -- Case 2. h2 : ¬prime n define at h2 --h2 : ¬(2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n) demorgan at h2 disj_syll h2 h1 obtain (a : Nat) (h3 : ∃ (b : Nat), a * b = n ∧ a < n ∧ b < n) from h2 obtain (b : Nat) (h4 : a * b = n ∧ a < n ∧ b < n) from h3 have h5 : 2 ≤ a := by by_contra h6 have h7 : a ≤ 1 := by linarith have h8 : n ≤ b := calc n _ = a * b := h4.left.symm _ ≤ 1 * b := by rel [h7] _ = b := by ring linarith --n ≤ b contradicts b < n done have h6 : ∃ (p : Nat), prime_factor p a := ih a h4.right.left h5 obtain (p : Nat) (h7 : prime_factor p a) from h6 apply Exists.intro p define --Goal : prime p ∧ p ∣ n define at h7 --h7 : prime p ∧ p ∣ a apply And.intro h7.left have h8 : a ∣ n := by apply Exists.intro b show n = a * b from (h4.left).symm done show p ∣ n from dvd_trans h7.right h8 done done lemma exists_least_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q := by set S : Set Nat := {p : Nat | prime_factor p n} have h2 : ∃ (p : Nat), p ∈ S := exists_prime_factor n h show ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q from well_ord_princ S h2 done lemma all_prime_nil : all_prime [] := by define --Goal : ∀ p ∈ [], prime p fix p : Nat contrapos --Goal : ¬prime p → p ∉ [] assume h1 : ¬prime p show p ∉ [] from List.not_mem_nil p done lemma all_prime_cons (n : Nat) (L : List Nat) : all_prime (n :: L) ↔ prime n ∧ all_prime L := by apply Iff.intro · -- (→) assume h1 : all_prime (n :: L) --Goal : prime n ∧ all_prime L define at h1 --h1 : ∀ p ∈ n :: L, prime p apply And.intro (h1 n (List.mem_cons_self n L)) define --Goal : ∀ p ∈ L, prime p fix p : Nat assume h2 : p ∈ L show prime p from h1 p (List.mem_cons_of_mem n h2) done · -- (←) assume h1 : prime n ∧ all_prime L --Goal : all_prime (n :: l) define : all_prime L at h1 define fix p : Nat assume h2 : p ∈ n :: L rewrite [List.mem_cons] at h2 --h2 : p = n ∨ p ∈ L by_cases on h2 · -- Case 1. h2 : p = n rewrite [h2] show prime n from h1.left done · -- Case 2. h2 : p ∈ L show prime p from h1.right p h2 done done done lemma nondec_nil : nondec [] := by define --Goal : True trivial --trivial proves some obviously true statements, such as True done lemma nondec_cons (n : Nat) (L : List Nat) : nondec (n :: L) ↔ (∀ m ∈ L, n ≤ m) ∧ nondec L := by rfl lemma prod_nil : prod [] = 1 := by rfl lemma prod_cons : prod (n :: L) = n * (prod L) := by rfl lemma exists_cons_of_length_eq_succ {A : Type} {l : List A} {n : Nat} (h : l.length = n + 1) : ∃ (a : A) (L : List A), l = a :: L ∧ L.length = n := by have h1 : ¬l.length = 0 := by linarith rewrite [List.length_eq_zero] at h1 obtain (a : A) (h2 : ∃ (L : List A), l = a :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List A) (h3 : l = a :: L) from h2 apply Exists.intro a apply Exists.intro L apply And.intro h3 have h4 : (a :: L).length = L.length + 1 := List.length_cons a L rewrite [←h3, h] at h4 show L.length = n from (Nat.add_right_cancel h4).symm done lemma list_elt_dvd_prod_by_length (a : Nat) : ∀ (n : Nat), ∀ (l : List Nat), l.length = n → a ∈ l → a ∣ prod l := by by_induc · --Base Case fix l : List Nat assume h1 : l.length = 0 rewrite [List.length_eq_zero] at h1 --h1 : l = [] rewrite [h1] --Goal : a ∈ [] → a ∣ prod [] contrapos assume h2 : ¬a ∣ prod [] show a ∉ [] from List.not_mem_nil a done · -- Induction Step fix n : Nat assume ih : ∀ (l : List Nat), List.length l = n → a ∈ l → a ∣ prod l fix l : List Nat assume h1 : l.length = n + 1 --Goal : a ∈ l → a ∣ prod l obtain (b : Nat) (h2 : ∃ (L : List Nat), l = b :: L ∧ L.length = n) from exists_cons_of_length_eq_succ h1 obtain (L : List Nat) (h3 : l = b :: L ∧ L.length = n) from h2 have h4 : a ∈ L → a ∣ prod L := ih L h3.right assume h5 : a ∈ l rewrite [h3.left, prod_cons] --Goal : a ∣ b * prod L rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b ∨ a ∈ L by_cases on h5 · -- Case 1. h5 : a = b apply Exists.intro (prod L) rewrite [h5] rfl done · -- Case 2. h5 : a ∈ L have h6 : a ∣ prod L := h4 h5 have h7 : prod L ∣ b * prod L := by apply Exists.intro b ring done show a ∣ b * prod L from dvd_trans h6 h7 done done done lemma list_elt_dvd_prod {a : Nat} {l : List Nat} (h : a ∈ l) : a ∣ prod l := by set n : Nat := l.length have h1 : l.length = n := by rfl show a ∣ prod l from list_elt_dvd_prod_by_length a n l h1 h done lemma exists_prime_factorization : ∀ (n : Nat), n ≥ 1 → ∃ (l : List Nat), prime_factorization n l := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, n_1 ≥ 1 → ∃ (l : List Nat), prime_factorization n_1 l assume h1 : n ≥ 1 by_cases h2 : n = 1 · -- Case 1. h2 : n = 1 apply Exists.intro [] define apply And.intro · -- Proof of nondec_prime_list [] define show all_prime [] ∧ nondec [] from And.intro all_prime_nil nondec_nil done · -- Proof of prod [] = n rewrite [prod_nil, h2] rfl done done · -- Case 2. h2 : n ≠ 1 have h3 : n ≥ 2 := lt_of_le_of_ne' h1 h2 obtain (p : Nat) (h4 : prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q) from exists_least_prime_factor h3 have p_prime_factor : prime_factor p n := h4.left define at p_prime_factor have p_prime : prime p := p_prime_factor.left have p_dvd_n : p ∣ n := p_prime_factor.right have p_least : ∀ (q : Nat), prime_factor q n → p ≤ q := h4.right obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n have h5 : m ≠ 0 := by contradict h1 with h6 have h7 : n = 0 := calc n _ = p * m := n_eq_pm _ = p * 0 := by rw [h6] _ = 0 := by ring rewrite [h7] decide done have m_pos : 0 < m := Nat.pos_of_ne_zero h5 have m_lt_n : m < n := by define at p_prime show m < n from calc m _ < m + m := by linarith _ = 2 * m := by ring _ ≤ p * m := by rel [p_prime.left] _ = n := n_eq_pm.symm done obtain (L : List Nat) (h6 : prime_factorization m L) from ih m m_lt_n m_pos define at h6 have ndpl_L : nondec_prime_list L := h6.left define at ndpl_L apply Exists.intro (p :: L) define apply And.intro · -- Proof of nondec_prime_list (p :: L) define apply And.intro · -- Proof of all_prime (p :: L) rewrite [all_prime_cons] show prime p ∧ all_prime L from And.intro p_prime ndpl_L.left done · -- Proof of nondec (p :: L) rewrite [nondec_cons] apply And.intro _ ndpl_L.right fix q : Nat assume q_in_L : q ∈ L have h7 : q ∣ prod L := list_elt_dvd_prod q_in_L rewrite [h6.right] at h7 --h7 : q ∣ m have h8 : m ∣ n := by apply Exists.intro p rewrite [n_eq_pm] ring done have q_dvd_n : q ∣ n := dvd_trans h7 h8 have ap_L : all_prime L := ndpl_L.left define at ap_L have q_prime_factor : prime_factor q n := And.intro (ap_L q q_in_L) q_dvd_n show p ≤ q from p_least q q_prime_factor done done · -- Proof of prod (p :: L) = n rewrite [prod_cons, h6.right, n_eq_pm] rfl done done done theorem Theorem_7_2_2 {a b c : Nat} (h1 : c ∣ a * b) (h2 : rel_prime a c) : c ∣ b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑c ∣ ↑b define at h1; define at h2; define obtain (j : Nat) (h3 : a * b = c * j) from h1 set s : Int := gcd_c1 a c set t : Int := gcd_c2 a c have h4 : s * ↑a + t * ↑c = ↑(gcd a c) := gcd_lin_comb c a rewrite [h2, Nat.cast_one] at h4 --h4 : s * ↑a + t * ↑c = (1 : Int) apply Exists.intro (s * ↑j + t * ↑b) show ↑b = ↑c * (s * ↑j + t * ↑b) from calc ↑b _ = (1 : Int) * ↑b := (one_mul _).symm _ = (s * ↑a + t * ↑c) * ↑b := by rw [h4] _ = s * (↑a * ↑b) + t * ↑c * ↑b := by ring _ = s * (↑c * ↑j) + t * ↑c * ↑b := by rw [←Nat.cast_mul a b, h3, Nat.cast_mul c j] _ = ↑c * (s * ↑j + t * ↑b) := by ring done lemma le_nonzero_prod_left {a b : Nat} (h : a * b ≠ 0) : a ≤ a * b := by have h1 : b ≠ 0 := by contradict h with h1 rewrite [h1] ring done have h2 : 1 ≤ b := Nat.pos_of_ne_zero h1 show a ≤ a * b from calc a = a * 1 := (mul_one a).symm _ ≤ a * b := by rel [h2] done lemma le_nonzero_prod_right {a b : Nat} (h : a * b ≠ 0) : b ≤ a * b := by rewrite [mul_comm] rewrite [mul_comm] at h show b ≤ b * a from le_nonzero_prod_left h done lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry lemma rel_prime_of_prime_not_dvd {a p : Nat} (h1 : prime p) (h2 : ¬p ∣ a) : rel_prime a p := by have h3 : gcd a p ∣ a := gcd_dvd_left a p have h4 : gcd a p ∣ p := gcd_dvd_right a p have h5 : gcd a p = 1 ∨ gcd a p = p := dvd_prime h1 h4 have h6 : gcd a p ≠ p := by contradict h2 with h6 rewrite [h6] at h3 show p ∣ a from h3 done disj_syll h5 h6 show rel_prime a p from h5 done theorem Theorem_7_2_3 {a b p : Nat} (h1 : prime p) (h2 : p ∣ a * b) : p ∣ a ∨ p ∣ b := by or_right with h3 have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3 show p ∣ b from Theorem_7_2_2 h2 h4 done lemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a ≥ 1 := by have h1 : a ≠ 0 := by by_contra h1 rewrite [h1] at h contradict h linarith done show a ≥ 1 from Nat.pos_of_ne_zero h1 done lemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by have h1 : a ≥ 1 := ge_one_of_prod_one h have h2 : a * b ≠ 0 := by linarith have h3 : a ≤ a * b := le_nonzero_prod_left h2 rewrite [h] at h3 show a = 1 from Nat.le_antisymm h3 h1 done lemma eq_one_of_dvd_one {n : Nat} (h : n ∣ 1) : n = 1 := by obtain (j : Nat) (h1 : 1 = n * j) from h show n = 1 from eq_one_of_prod_one h1.symm done lemma prime_not_one {p : Nat} (h : prime p) : p ≠ 1 := by define at h linarith done theorem Theorem_7_2_4 {p : Nat} (h1 : prime p) : ∀ (l : List Nat), p ∣ prod l → ∃ a ∈ l, p ∣ a := by apply List.rec · -- Base Case. Goal : p ∣ prod [] → ∃ a ∈ [], p ∣ a rewrite [prod_nil] assume h2 : p ∣ 1 show ∃ a ∈ [], p ∣ a from absurd (eq_one_of_dvd_one h2) (prime_not_one h1) done · -- Induction Step fix b : Nat fix L : List Nat assume ih : p ∣ prod L → ∃ a ∈ L, p ∣ a --Goal : p ∣ prod (b :: L) → ∃ a ∈ b :: L, p ∣ a assume h2 : p ∣ prod (b :: L) rewrite [prod_cons] at h2 have h3 : p ∣ b ∨ p ∣ prod L := Theorem_7_2_3 h1 h2 by_cases on h3 · -- Case 1. h3 : p ∣ b apply Exists.intro b show b ∈ b :: L ∧ p ∣ b from And.intro (List.mem_cons_self b L) h3 done · -- Case 2. h3 : p ∣ prod L obtain (a : Nat) (h4 : a ∈ L ∧ p ∣ a) from ih h3 apply Exists.intro a show a ∈ b :: L ∧ p ∣ a from And.intro (List.mem_cons_of_mem b h4.left) h4.right done done done lemma prime_in_list {p : Nat} {l : List Nat} (h1 : prime p) (h2 : all_prime l) (h3 : p ∣ prod l) : p ∈ l := by obtain (a : Nat) (h4 : a ∈ l ∧ p ∣ a) from Theorem_7_2_4 h1 l h3 define at h2 have h5 : prime a := h2 a h4.left have h6 : p = 1 ∨ p = a := dvd_prime h5 h4.right disj_syll h6 (prime_not_one h1) rewrite [h6] show a ∈ l from h4.left done lemma first_le_first {p q : Nat} {l m : List Nat} (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m)) (h3 : prod (p :: l) = prod (q :: m)) : p ≤ q := by define at h1; define at h2 have h4 : q ∣ prod (p :: l) := by define apply Exists.intro (prod m) rewrite [←prod_cons] show prod (p :: l) = prod (q :: m) from h3 done have h5 : all_prime (q :: m) := h2.left rewrite [all_prime_cons] at h5 have h6 : q ∈ p :: l := prime_in_list h5.left h1.left h4 have h7 : nondec (p :: l) := h1.right rewrite [nondec_cons] at h7 rewrite [List.mem_cons] at h6 by_cases on h6 · -- Case 1. h6 : q = p linarith done · -- Case 2. h6 : q ∈ l have h8 : ∀ m ∈ l, p ≤ m := h7.left show p ≤ q from h8 q h6 done done lemma nondec_prime_list_tail {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by define at h define rewrite [all_prime_cons, nondec_cons] at h show all_prime l ∧ nondec l from And.intro h.left.right h.right.right done lemma cons_prod_not_one {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : prod (p :: l) ≠ 1 := by define at h have h1 : all_prime (p :: l) := h.left rewrite [all_prime_cons] at h1 rewrite [prod_cons] by_contra h2 show False from (prime_not_one h1.left) (eq_one_of_prod_one h2) done lemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) : l = [] ↔ prod l = 1 := by apply Iff.intro · -- (→) assume h1 : l = [] rewrite [h1] show prod [] = 1 from prod_nil done · -- (←) contrapos assume h1 : ¬l = [] obtain (p : Nat) (h2 : ∃ (L : List Nat), l = p :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List Nat) (h3 : l = p :: L) from h2 rewrite [h3] at h rewrite [h3] show ¬prod (p :: L) = 1 from cons_prod_not_one h done done lemma prime_pos {p : Nat} (h : prime p) : p > 0 := by define at h linarith done theorem Theorem_7_2_5 : ∀ (l1 l2 : List Nat), nondec_prime_list l1 → nondec_prime_list l2 → prod l1 = prod l2 → l1 = l2 := by apply List.rec · -- Base Case. Goal : ∀ (l2 : List Nat), nondec_prime_list [] → -- nondec_prime_list l2 → prod [] = prod l2 → [] = l2 fix l2 : List Nat assume h1 : nondec_prime_list [] assume h2 : nondec_prime_list l2 assume h3 : prod [] = prod l2 rewrite [prod_nil, eq_comm, ←list_nil_iff_prod_one h2] at h3 show [] = l2 from h3.symm done · -- Induction Step fix p : Nat fix L1 : List Nat assume ih : ∀ (L2 : List Nat), nondec_prime_list L1 → nondec_prime_list L2 → prod L1 = prod L2 → L1 = L2 -- Goal : ∀ (l2 : List Nat), nondec_prime_list (p :: L1) → -- nondec_prime_list l2 → prod (p :: L1) = prod l2 → p :: L1 = l2 fix l2 : List Nat assume h1 : nondec_prime_list (p :: L1) assume h2 : nondec_prime_list l2 assume h3 : prod (p :: L1) = prod l2 have h4 : ¬prod (p :: L1) = 1 := cons_prod_not_one h1 rewrite [h3, ←list_nil_iff_prod_one h2] at h4 obtain (q : Nat) (h5 : ∃ (L : List Nat), l2 = q :: L) from List.exists_cons_of_ne_nil h4 obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5 rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2) rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2) have h7 : p ≤ q := first_le_first h1 h2 h3 have h8 : q ≤ p := first_le_first h2 h1 h3.symm have h9 : p = q := by linarith rewrite [h9, prod_cons, prod_cons] at h3 --h3 : q * prod L1 = q * prod L2 have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1 have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2 define at h2 have h12 : all_prime (q :: L2) := h2.left rewrite [all_prime_cons] at h12 have h13 : q > 0 := prime_pos h12.left have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3 have h15 : L1 = L2 := ih L2 h10 h11 h14 rewrite [h6, h9, h15] rfl done done theorem fund_thm_arith (n : Nat) (h : n ≥ 1) : ∃! (l : List Nat), prime_factorization n l := by exists_unique · -- Existence show ∃ (l : List Nat), prime_factorization n l from exists_prime_factorization n h done · -- Uniqueness fix l1 : List Nat; fix l2 : List Nat assume h1 : prime_factorization n l1 assume h2 : prime_factorization n l2 define at h1; define at h2 have h3 : prod l1 = n := h1.right rewrite [←h2.right] at h3 show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3 done done /- Section 7.3 -/ theorem congr_refl (m : Nat) : ∀ (a : Int), a ≡ a (MOD m) := by fix a : Int define --Goal : ∃ (c : Int), a - a = ↑m * c apply Exists.intro 0 ring done theorem congr_symm {m : Nat} : ∀ {a b : Int}, a ≡ b (MOD m) → b ≡ a (MOD m) := by fix a : Int; fix b : Int assume h1 : a ≡ b (MOD m) define at h1 --h1 : ∃ (c : Int), a - b = ↑m * c define --Goal : ∃ (c : Int), b - a = ↑m * c obtain (c : Int) (h2 : a - b = m * c) from h1 apply Exists.intro (-c) show b - a = m * (-c) from calc b - a _ = -(a - b) := by ring _ = -(m * c) := by rw [h2] _ = m * (-c) := by ring done theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry /- Fundamental properties of congruence classes -/ lemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) : X = Y ↔ X.val = Y.val := Fin.ext_iff lemma val_nat_eq_mod (n k : Nat) : ([k]_(n + 1)).val = k % (n + 1) := by rfl lemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl theorem cc_rep {m : Nat} (X : ZMod m) : ∃ (a : Int), X = [a]_m := match m with | 0 => by apply Exists.intro X rfl done | n + 1 => by apply Exists.intro ↑(X.val) have h1 : X.val < n + 1 := Fin.prop X rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1] rfl done theorem add_class (m : Nat) (a b : Int) : [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm theorem mul_class (m : Nat) (a b : Int) : [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm lemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ [a - b]_m = [0]_m := by apply Iff.intro · -- (→) assume h1 : [a]_m = [b]_m have h2 : a - b = a + (-b) := by ring have h3 : b + (-b) = 0 := by ring show [a - b]_m = [0]_m from calc [a - b]_m _ = [a + (-b)]_m := by rw [h2] _ = [a]_m + [-b]_m := by rw [add_class] _ = [b]_m + [-b]_m := by rw [h1] _ = [b + -b]_m := by rw [add_class] _ = [0]_m := by rw [h3] done · -- (←) assume h1 : [a - b]_m = [0]_m have h2 : b + (a - b) = a := by ring have h3 : b + 0 = b := by ring show [a]_m = [b]_m from calc [a]_m _ = [b + (a - b)]_m := by rw [h2] _ = [b]_m + [a - b]_m := by rw [add_class] _ = [b]_m + [0]_m := by rw [h1] _ = [b + 0]_m := by rw [add_class] _ = [b]_m := by rw [h3] done done lemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) : [a]_m = [0]_m → [-a]_m = [0]_m := by assume h1 : [a]_m = [0]_m have h2 : 0 + (-a) = -a := by ring have h3 : a + (-a) = 0 := by ring show [-a]_m = [0]_m from calc [-a]_m _ = [0 + (-a)]_m := by rw [h2] _ = [0]_m + [-a]_m := by rw [add_class] _ = [a]_m + [-a]_m := by rw [h1] _ = [a + (-a)]_m := by rw [add_class] _ = [0]_m := by rw [h3] done lemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) : [-a]_m = [0]_m ↔ [a]_m = [0]_m := by apply Iff.intro _ (cc_neg_zero_of_cc_zero m a) assume h1 : [-a]_m = [0]_m have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1 have h3 : -(-a) = a := by ring rewrite [h3] at h2 show [a]_m = [0]_m from h2 done lemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl lemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m ↔ m ∣ k := match m with | 0 => by have h : (0 : Int) = (↑(0 : Nat) : Int) := by rfl rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj] apply Iff.intro · -- (→) assume h1 : k = 0 rewrite [h1] show 0 ∣ 0 from dvd_self 0 done · -- (←) assume h1 : 0 ∣ k obtain (c : Nat) (h2 : k = 0 * c) from h1 rewrite [h2] ring done done | n + 1 => by rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero] show k % (n + 1) = 0 ↔ n + 1 ∣ k from (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm done lemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m ↔ ↑m ∣ a := by obtain (k : Nat) (h1 : a = ↑k ∨ a = -↑k) from Int.eq_nat_or_neg a by_cases on h1 · -- Case 1. h1: a = ↑k rewrite [h1, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done · -- Case 2. h1: a = -↑k rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done done theorem cc_eq_iff_congr (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ a ≡ b (MOD m) := calc [a]_m = [b]_m _ ↔ [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b _ ↔ ↑m ∣ (a - b) := cc_zero_iff_dvd m (a - b) _ ↔ a ≡ b (MOD m) := by rfl /- End of fundamental properties of congruence classes -/ lemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m := by have h1 : (↑m : Int) ≠ 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m) show 0 ≤ a % m from Int.emod_nonneg a h1 done lemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m) have h2 : (↑m : Int) > 0 := (Nat.cast_pos).rtl h1 show a % m < m from Int.emod_lt_of_pos a h2 done lemma congr_mod_mod (m : Nat) (a : Int) : a ≡ a % m (MOD m) := by define have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m apply Exists.intro (a / m) show a - a % m = m * (a / m) from calc a - (a % m) _ = m * (a / m) + a % m - a % m := by rw [h1] _ = m * (a / m) := by ring done lemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) := And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a)) theorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) : ∃! (r : Int), 0 ≤ r ∧ r < m ∧ a ≡ r (MOD m) := by exists_unique · -- Existence apply Exists.intro (a % m) show 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) from mod_cmpl_res m a done · -- Uniqueness fix r1 : Int; fix r2 : Int assume h1 : 0 ≤ r1 ∧ r1 < m ∧ a ≡ r1 (MOD m) assume h2 : 0 ≤ r2 ∧ r2 < m ∧ a ≡ r2 (MOD m) have h3 : r1 ≡ r2 (MOD m) := congr_trans (congr_symm h1.right.right) h2.right.right obtain (d : Int) (h4 : r1 - r2 = m * d) from h3 have h5 : r1 - r2 < m * 1 := by linarith have h6 : m * (-1) < r1 - r2 := by linarith rewrite [h4] at h5 --h5 : m * d < m * 1 rewrite [h4] at h6 --h6 : m * -1 < m * d have h7 : (↑m : Int) ≥ 0 := Nat.cast_nonneg m have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7 have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7 have h10 : d = 0 := by linarith show r1 = r2 from calc r1 _ = r1 - r2 + r2 := by ring _ = m * 0 + r2 := by rw [h4, h10] _ = r2 := by ring done done lemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m := (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a) theorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y rewrite [h1, h2] have h3 : a + b = b + a := by ring show [a]_m + [b]_m = [b]_m + [a]_m from calc [a]_m + [b]_m _ = [a + b]_m := add_class m a b _ = [b + a]_m := by rw [h3] _ = [b]_m + [a]_m := (add_class m b a).symm done theorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X rewrite [h1] have h2 : a * 1 = a := by ring show [a]_m * [1]_m = [a]_m from calc [a]_m * [1]_m _ = [a * 1]_m := mul_class m a 1 _ = [a]_m := by rw [h2] done theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry lemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) : [a]_m * [gcd_c2 m a]_m = [1]_m := by set s : Int := gcd_c1 m a have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m define at h1 rewrite [h1, Nat.cast_one] at h2 --h2 : s * ↑m + gcd_c2 m a * ↑a = 1 rewrite [mul_class, cc_eq_iff_congr] define --Goal : ∃ (c : Int), ↑a * gcd_c2 m a - 1 = ↑m * c apply Exists.intro (-s) show a * (gcd_c2 m a) - 1 = m * (-s) from calc a * (gcd_c2 m a) - 1 _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring _ = 1 + m * (-s) - 1 := by rw [h2] _ = m * (-s) := by ring done theorem Theorem_7_3_7 (m a : Nat) : invertible [a]_m ↔ rel_prime m a := by apply Iff.intro · -- (→) assume h1 : invertible [a]_m define at h1 obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1 obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y rewrite [h3, mul_class, cc_eq_iff_congr] at h2 define at h2 obtain (c : Int) (h4 : a * b - 1 = m * c) from h2 rewrite [Exercise_7_2_6] --Goal : ∃ (s t : Int), s * ↑m + t * ↑a = 1 apply Exists.intro (-c) apply Exists.intro b show (-c) * m + b * a = 1 from calc (-c) * m + b * a _ = (-c) * m + (a * b - 1) + 1 := by ring _ = (-c) * m + m * c + 1 := by rw [h4] _ = 1 := by ring done · -- (←) assume h1 : rel_prime m a define show ∃ (Y : ZMod m), [a]_m * Y = [1]_m from Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1) done done /- Section 7.4 -/ section Euler open Euler lemma num_rp_below_base {m : Nat} : num_rp_below m 0 = 0 := by rfl lemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) : num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by have h1 : num_rp_below m (j + 1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : gcd m j = 1 rewrite [if_pos h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1 show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1 done lemma num_rp_below_step_not_rp {m j : Nat} (h : ¬rel_prime m j) : num_rp_below m (j + 1) = num_rp_below m j := by have h1 : num_rp_below m (j +1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : ¬gcd m j = 1 rewrite [if_neg h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j show num_rp_below m (j + 1) = num_rp_below m j from h1 done lemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl #eval phi 10 --Answer: 4 lemma prod_inv_iff_inv {m : Nat} {X : ZMod m} (h1 : invertible X) (Y : ZMod m) : invertible (X * Y) ↔ invertible Y := by apply Iff.intro · -- (→) assume h2 : invertible (X * Y) obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2 apply Exists.intro (X * Z) rewrite [←h3] --Goal : Y * (X * Z) = X * Y * Z ring --Note that ring can do algebra in ZMod m done · -- (←) assume h2 : invertible Y obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1 obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2 apply Exists.intro (Xi * Yi) show (X * Y) * (Xi * Yi) = [1]_m from calc X * Y * (Xi * Yi) _ = (X * Xi) * (Y * Yi) := by ring _ = [1]_m * [1]_m := by rw [h3, h4] _ = [1]_m := Theorem_7_3_6_7 [1]_m done done lemma F_rp_def {m i : Nat} (h : rel_prime m i) : F m i = [i]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h --h : gcd m i = 1 rewrite [if_pos h] at h1 show F m i = [i]_m from h1 done lemma F_not_rp_def {m i : Nat} (h : ¬rel_prime m i) : F m i = [1]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h rewrite [h1, if_neg h] rfl done lemma prod_seq_base {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 0 k f = [1]_m := by rfl lemma prod_seq_step {m : Nat} (n k : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl lemma prod_seq_zero_step {m : Nat} (n : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by rewrite [prod_seq_step, zero_add] rfl done lemma prod_one {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 1 k f = f k := by rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7] rfl done lemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl lemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m := calc [G m a i]_m _ = [(a * i) % m]_m := by rfl _ = [a * i]_m := (cc_eq_mod m (a * i)).symm _ = [a]_m * [i]_m := (mul_class m a i).symm lemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) : rel_prime m (G m a i) ↔ rel_prime m i := by have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1 show rel_prime m (G m a i) ↔ rel_prime m i from calc rel_prime m (G m a i) _ ↔ invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm _ ↔ invertible ([a]_m * [i]_m) := by rw [cc_G] _ ↔ invertible [i]_m := prod_inv_iff_inv h2 ([i]_m) _ ↔ rel_prime m i := Theorem_7_3_7 m i done lemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) : F m (G m a i) = [a]_m * F m i := by have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2 show F m (G m a i) = [a]_m * F m i from calc F m (G m a i) _ = [G m a i]_m := F_rp_def h3 _ = [a]_m * [i]_m := cc_G m a i _ = [a]_m * F m i := by rw [F_rp_def h2] done lemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : ¬rel_prime m i) : F m (G m a i) = [1]_m := by rewrite [←G_rp_iff h1 i] at h2 show F m (G m a i) = [1]_m from F_not_rp_def h2 done lemma FG_prod {m a : Nat} (h1 : rel_prime m a) : ∀ (k : Nat), prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by by_induc · -- Base Case show prod_seq 0 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from calc prod_seq 0 0 ((F m) ∘ (G m a)) _ = [1]_m := prod_seq_base _ _ _ = [a]_m ^ 0 * [1]_m := by ring _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by rw [num_rp_below_base, prod_seq_base] done · -- Induction Step fix k : Nat assume ih : prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) by_cases h2 : rel_prime m k · -- Case 1. h2 : rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([a]_m * F m k) := by rw [FG_rp h1 h2] _ = [a]_m ^ ((num_rp_below m k) + 1) * ((prod_seq k 0 (F m)) * F m k) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_rp h2, prod_seq_zero_step] done · -- Case 2. h2 : ¬rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([1]_m) := by rw [FG_not_rp h1 h2] _ = [a]_m ^ (num_rp_below m k) * (prod_seq k 0 (F m) * ([1]_m)) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_not_rp h2, prod_seq_zero_step, F_not_rp_def h2] done done done lemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by define --Goal : ∀ i < m, G m a i < m fix i : Nat assume h1 : i < m rewrite [G_def] --Goal : a * i % m < m show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m) done lemma left_inv_one_one_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g' (g i) = i) : one_one_below n g := sorry lemma right_inv_onto_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g (g' i) = i) (h2 : maps_below n g') : onto_below n g := by define at h2; define fix k : Nat assume h3 : k < n apply Exists.intro (g' k) show g' k < n ∧ g (g' k) = k from And.intro (h2 k h3) (h1 k h3) done lemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m * [inv_mod m a]_m = [1]_m := by have h2 : 0 ≤ (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a) show [a]_m * [inv_mod m a]_m = [1]_m from calc [a]_m * [inv_mod m a]_m _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2] _ = [a]_m * [gcd_c2 m a]_m := by rw [←cc_eq_mod] _ = [1]_m := gcd_c2_inv h1 done lemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m := calc a * (b % m) % m = a % m * (b % m % m) % m := Nat.mul_mod _ _ _ _ = a % m * (b % m) % m := by rw [Nat.mod_mod] _ = a * b % m := (Nat.mul_mod _ _ _).symm lemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm] rfl done theorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] : ↑a ≡ ↑b (MOD m) ↔ a % m = b % m := sorry lemma mul_inv_mod_cancel {m a i : Nat} [NeZero m] (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1 rewrite [mul_class, cc_eq_iff_congr, ←Nat.cast_mul, ←Nat.cast_one, congr_iff_mod_eq_Nat] at h3 show a * inv_mod m a * i % m = i from calc a * (inv_mod m a) * i % m _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = 1 % m * i % m := by rw [h3] _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = i % m := by rw [one_mul] _ = i := Nat.mod_eq_of_lt h2 done lemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl lemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, G m a (Ginv m a i) = i := by fix i : Nat assume h2 : i < m show G m a (Ginv m a i) = i from calc G m a (Ginv m a i) _ = a * ((inv_mod m a * i) % m) % m := by rfl _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, Ginv m a (G m a i) = i := by fix i : Nat assume h2 : i < m show Ginv m a (G m a i) = i from calc Ginv m a (G m a i) _ = inv_mod m a * ((a * i) % m) % m := by rfl _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc, mul_comm (inv_mod m a)] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_maps_below (m a : Nat) [NeZero m] : maps_below m (Ginv m a) := G_maps_below m (inv_mod m a) lemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : one_one_below m (G m a) := left_inv_one_one_below (Ginv_left_inv h1) lemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : onto_below m (G m a) := right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a) lemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : perm_below m (G m a) := And.intro (G_maps_below m a) (And.intro (G_one_one_below h1) (G_onto_below h1)) --Permuting a product of congruence classes doesn't change product lemma swap_fst (u v : Nat) : swap u v u = v := by define : swap u v u --Goal : (if u = u then v else if u = v then u else u) = v have h : u = u := by rfl rewrite [if_pos h] rfl done lemma swap_snd (u v : Nat) : swap u v v = u := by define : swap u v v by_cases h1 : v = u · -- Case 1. h1 : v = u rewrite [if_pos h1] show v = u from h1 done · -- Case 2. h1 : v ≠ u rewrite [if_neg h1] have h2 : v = v := by rfl rewrite [if_pos h2] rfl done done lemma swap_other {u v i : Nat} (h1 : i ≠ u) (h2 : i ≠ v) : swap u v i = i := by define : swap u v i rewrite [if_neg h1, if_neg h2] rfl done lemma swap_values (u v i : Nat) : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := by by_cases h1 : i = u · -- Case 1. h1 : i = u apply Or.inl rewrite [h1] show swap u v u = v from swap_fst u v done · -- Case 2. h1 : i ≠ u apply Or.inr by_cases h2 : i = v · -- Case 2.1. h2 : i = v apply Or.inl rewrite [h2] show swap u v v = u from swap_snd u v done · -- Case 2.2. h2 : i ≠ v apply Or.inr show swap u v i = i from swap_other h1 h2 done done done lemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by define fix i : Nat assume h3 : i < n have h4 : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := swap_values u v i by_cases on h4 · -- Case 1. h4 : swap u v i = v rewrite [h4] show v < n from h2 done · -- Case 2. by_cases on h4 · -- Case 2.1. h4 : swap u v i = u rewrite [h4] show u < n from h1 done · -- Case 2.2. h4 : swap u v i = i rewrite [h4] show i < n from h3 done done done lemma swap_swap (u v n : Nat) : ∀ i < n, swap u v (swap u v i) = i := by fix i : Nat assume h : i < n by_cases h1 : i = u · -- Case 1. h1 : i = u rewrite [h1, swap_fst, swap_snd] rfl done · -- Case 2. h1 : i ≠ u by_cases h2 : i = v · -- Case 2.1. h2 : i = v rewrite [h2, swap_snd, swap_fst] rfl done · -- Case 2.2. h2 : i ≠ v rewrite [swap_other h1 h2, swap_other h1 h2] rfl done done done

lemma swap_one_one_below (u v n) : one_one_below n (swap u v)

HTPI.swap_one_one_below

htpi/HTPILib/Chap7.lean

htpi

/- Copyright 2023 Daniel J. Velleman -/ import HTPILib.Chap6 namespace HTPI /- Definitions -/ lemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by have h : n + 1 > 0 := Nat.succ_pos n show a % (n + 1) < n + 1 from Nat.mod_lt a h done def gcd (a b : Nat) : Nat := match b with | 0 => a | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd (n + 1) (a % (n + 1)) termination_by b mutual def gcd_c1 (a b : Nat) : Int := match b with | 0 => 1 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c2 (n + 1) (a % (n + 1)) --Corresponds to s = t' termination_by b def gcd_c2 (a b : Nat) : Int := match b with | 0 => 0 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c1 (n + 1) (a % (n + 1)) - (gcd_c2 (n + 1) (a % (n + 1))) * ↑(a / (n + 1)) --Corresponds to t = s' - t'q termination_by b end def prime (n : Nat) : Prop := 2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n def prime_factor (p n : Nat) : Prop := prime p ∧ p ∣ n def all_prime (l : List Nat) : Prop := ∀ p ∈ l, prime p def nondec (l : List Nat) : Prop := match l with | [] => True --Of course, True is a proposition that is always true | n :: L => (∀ m ∈ L, n ≤ m) ∧ nondec L def nondec_prime_list (l : List Nat) : Prop := all_prime l ∧ nondec l def prod (l : List Nat) : Nat := match l with | [] => 1 | n :: L => n * (prod L) def prime_factorization (n : Nat) (l : List Nat) : Prop := nondec_prime_list l ∧ prod l = n def rel_prime (a b : Nat) : Prop := gcd a b = 1 def congr_mod (m : Nat) (a b : Int) : Prop := (↑m : Int) ∣ (a - b) def cc (m : Nat) (a : Int) : ZMod m := (↑a : ZMod m) notation:50 a " ≡ " b " (MOD " m ")" => congr_mod m a b notation:max "["a"]_"m:max => cc m a def invertible {m : Nat} (X : ZMod m) : Prop := ∃ (Y : ZMod m), X * Y = [1]_m def num_rp_below (m k : Nat) : Nat := match k with | 0 => 0 | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j def phi (m : Nat) : Nat := num_rp_below m m def prod_seq {m : Nat} (j k : Nat) (f : Nat → ZMod m) : ZMod m := match j with | 0 => [1]_m | n + 1 => prod_seq n k f * f (k + n) def maps_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i < n, g i < n def one_one_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i1 < n, ∀ i2 < n, g i1 = g i2 → i1 = i2 def onto_below (n : Nat) (g : Nat → Nat) : Prop := ∀ k < n, ∃ i < n, g i = k def perm_below (n : Nat) (g : Nat → Nat) : Prop := maps_below n g ∧ one_one_below n g ∧ onto_below n g def inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m) def swap (u v i : Nat) : Nat := if i = u then v else if i = v then u else i namespace Euler --For definitions specific to Euler's theorem def F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m def G (m a i : Nat) : Nat := (a * i) % m def Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i end Euler /- Section 7.1 -/ theorem dvd_mod_of_dvd_a_b {a b d : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ (a % b) := by set q : Nat := a / b have h3 : b * q + a % b = a := Nat.div_add_mod a b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 define --Goal : ∃ (c : Nat), a % b = d * c apply Exists.intro (j - k * q) show a % b = d * (j - k * q) from calc a % b _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm _ = a - b * q := by rw [h3] _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc] _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm done theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry #eval gcd 672 161 --Answer: 7 lemma gcd_base (a : Nat) : gcd a 0 = a := by rfl lemma gcd_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd a b = gcd b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1)) rfl done lemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b ≠ 0) : a % b < b := by have h1 : b > 0 := Nat.pos_of_ne_zero h show a % b < b from Nat.mod_lt a h1 done lemma dvd_self (n : Nat) : n ∣ n := by apply Exists.intro 1 ring done theorem gcd_dvd : ∀ (b a : Nat), (gcd a b) ∣ a ∧ (gcd a b) ∣ b := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd a b_1) ∣ a ∧ (gcd a b_1) ∣ b_1 fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_base] --Goal: a ∣ a ∧ a ∣ 0 apply And.intro (dvd_self a) define apply Exists.intro 0 rfl done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_nonzero a h1] --Goal : gcd b (a % b) ∣ a ∧ gcd b (a % b) ∣ b have h2 : a % b < b := mod_nonzero_lt a h1 have h3 : (gcd b (a % b)) ∣ b ∧ (gcd b (a % b)) ∣ (a % b) := ih (a % b) h2 b apply And.intro _ h3.left show (gcd b (a % b)) ∣ a from dvd_a_of_dvd_b_mod h3.left h3.right done done theorem gcd_dvd_left (a b : Nat) : (gcd a b) ∣ a := (gcd_dvd b a).left theorem gcd_dvd_right (a b : Nat) : (gcd a b) ∣ b := (gcd_dvd b a).right lemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl lemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c1 a b = gcd_c2 b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done lemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl lemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * ↑(a / b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done theorem gcd_lin_comb : ∀ (b a : Nat), (gcd_c1 a b) * ↑a + (gcd_c2 a b) * ↑b = ↑(gcd a b) := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd_c1 a b_1) * ↑a + (gcd_c2 a b_1) * ↑b_1 = ↑(gcd a b_1) fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base] --Goal : 1 * ↑a + 0 * ↑0 = ↑a ring done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1] --Goal : gcd_c2 b (a % b) * ↑a + -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * ↑(a / b)) * ↑b = -- ↑(gcd b (a % b)) set r : Nat := a % b set q : Nat := a / b set s : Int := gcd_c1 b r set t : Int := gcd_c2 b r --Goal : t * ↑a + (s - t * ↑q) * ↑b = ↑(gcd b r) have h2 : r < b := mod_nonzero_lt a h1 have h3 : s * ↑b + t * ↑r = ↑(gcd b r) := ih r h2 b have h4 : b * q + r = a := Nat.div_add_mod a b rewrite [←h3, ←h4] rewrite [Nat.cast_add, Nat.cast_mul] --Goal : t * (↑b * ↑q + ↑r) + (s - t * ↑q) * ↑b = s * ↑b + t * ↑r ring done done #eval gcd_c1 672 161 --Answer: 6 #eval gcd_c2 672 161 --Answer: -25 --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161 theorem Theorem_7_1_6 {d a b : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ gcd a b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑d ∣ ↑(gcd a b) set s : Int := gcd_c1 a b set t : Int := gcd_c2 a b have h3 : s * ↑a + t * ↑b = ↑(gcd a b) := gcd_lin_comb b a rewrite [←h3] --Goal : ↑d ∣ s * ↑a + t * ↑b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul] --Goal : ↑d ∣ s * (↑d * ↑j) + t * (↑d * ↑k) define apply Exists.intro (s * ↑j + t * ↑k) ring done /- Section 7.2 -/ theorem dvd_trans {a b c : Nat} (h1 : a ∣ b) (h2 : b ∣ c) : a ∣ c := by define at h1; define at h2; define obtain (m : Nat) (h3 : b = a * m) from h1 obtain (n : Nat) (h4 : c = b * n) from h2 rewrite [h3, mul_assoc] at h4 apply Exists.intro (m * n) show c = a * (m * n) from h4 done lemma exists_prime_factor : ∀ (n : Nat), 2 ≤ n → ∃ (p : Nat), prime_factor p n := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, 2 ≤ n_1 → ∃ (p : Nat), prime_factor p n_1 assume h1 : 2 ≤ n by_cases h2 : prime n · -- Case 1. h2 : prime n apply Exists.intro n define --Goal : prime n ∧ n ∣ n show prime n ∧ n ∣ n from And.intro h2 (dvd_self n) done · -- Case 2. h2 : ¬prime n define at h2 --h2 : ¬(2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n) demorgan at h2 disj_syll h2 h1 obtain (a : Nat) (h3 : ∃ (b : Nat), a * b = n ∧ a < n ∧ b < n) from h2 obtain (b : Nat) (h4 : a * b = n ∧ a < n ∧ b < n) from h3 have h5 : 2 ≤ a := by by_contra h6 have h7 : a ≤ 1 := by linarith have h8 : n ≤ b := calc n _ = a * b := h4.left.symm _ ≤ 1 * b := by rel [h7] _ = b := by ring linarith --n ≤ b contradicts b < n done have h6 : ∃ (p : Nat), prime_factor p a := ih a h4.right.left h5 obtain (p : Nat) (h7 : prime_factor p a) from h6 apply Exists.intro p define --Goal : prime p ∧ p ∣ n define at h7 --h7 : prime p ∧ p ∣ a apply And.intro h7.left have h8 : a ∣ n := by apply Exists.intro b show n = a * b from (h4.left).symm done show p ∣ n from dvd_trans h7.right h8 done done lemma exists_least_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q := by set S : Set Nat := {p : Nat | prime_factor p n} have h2 : ∃ (p : Nat), p ∈ S := exists_prime_factor n h show ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q from well_ord_princ S h2 done lemma all_prime_nil : all_prime [] := by define --Goal : ∀ p ∈ [], prime p fix p : Nat contrapos --Goal : ¬prime p → p ∉ [] assume h1 : ¬prime p show p ∉ [] from List.not_mem_nil p done lemma all_prime_cons (n : Nat) (L : List Nat) : all_prime (n :: L) ↔ prime n ∧ all_prime L := by apply Iff.intro · -- (→) assume h1 : all_prime (n :: L) --Goal : prime n ∧ all_prime L define at h1 --h1 : ∀ p ∈ n :: L, prime p apply And.intro (h1 n (List.mem_cons_self n L)) define --Goal : ∀ p ∈ L, prime p fix p : Nat assume h2 : p ∈ L show prime p from h1 p (List.mem_cons_of_mem n h2) done · -- (←) assume h1 : prime n ∧ all_prime L --Goal : all_prime (n :: l) define : all_prime L at h1 define fix p : Nat assume h2 : p ∈ n :: L rewrite [List.mem_cons] at h2 --h2 : p = n ∨ p ∈ L by_cases on h2 · -- Case 1. h2 : p = n rewrite [h2] show prime n from h1.left done · -- Case 2. h2 : p ∈ L show prime p from h1.right p h2 done done done lemma nondec_nil : nondec [] := by define --Goal : True trivial --trivial proves some obviously true statements, such as True done lemma nondec_cons (n : Nat) (L : List Nat) : nondec (n :: L) ↔ (∀ m ∈ L, n ≤ m) ∧ nondec L := by rfl lemma prod_nil : prod [] = 1 := by rfl lemma prod_cons : prod (n :: L) = n * (prod L) := by rfl lemma exists_cons_of_length_eq_succ {A : Type} {l : List A} {n : Nat} (h : l.length = n + 1) : ∃ (a : A) (L : List A), l = a :: L ∧ L.length = n := by have h1 : ¬l.length = 0 := by linarith rewrite [List.length_eq_zero] at h1 obtain (a : A) (h2 : ∃ (L : List A), l = a :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List A) (h3 : l = a :: L) from h2 apply Exists.intro a apply Exists.intro L apply And.intro h3 have h4 : (a :: L).length = L.length + 1 := List.length_cons a L rewrite [←h3, h] at h4 show L.length = n from (Nat.add_right_cancel h4).symm done lemma list_elt_dvd_prod_by_length (a : Nat) : ∀ (n : Nat), ∀ (l : List Nat), l.length = n → a ∈ l → a ∣ prod l := by by_induc · --Base Case fix l : List Nat assume h1 : l.length = 0 rewrite [List.length_eq_zero] at h1 --h1 : l = [] rewrite [h1] --Goal : a ∈ [] → a ∣ prod [] contrapos assume h2 : ¬a ∣ prod [] show a ∉ [] from List.not_mem_nil a done · -- Induction Step fix n : Nat assume ih : ∀ (l : List Nat), List.length l = n → a ∈ l → a ∣ prod l fix l : List Nat assume h1 : l.length = n + 1 --Goal : a ∈ l → a ∣ prod l obtain (b : Nat) (h2 : ∃ (L : List Nat), l = b :: L ∧ L.length = n) from exists_cons_of_length_eq_succ h1 obtain (L : List Nat) (h3 : l = b :: L ∧ L.length = n) from h2 have h4 : a ∈ L → a ∣ prod L := ih L h3.right assume h5 : a ∈ l rewrite [h3.left, prod_cons] --Goal : a ∣ b * prod L rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b ∨ a ∈ L by_cases on h5 · -- Case 1. h5 : a = b apply Exists.intro (prod L) rewrite [h5] rfl done · -- Case 2. h5 : a ∈ L have h6 : a ∣ prod L := h4 h5 have h7 : prod L ∣ b * prod L := by apply Exists.intro b ring done show a ∣ b * prod L from dvd_trans h6 h7 done done done lemma list_elt_dvd_prod {a : Nat} {l : List Nat} (h : a ∈ l) : a ∣ prod l := by set n : Nat := l.length have h1 : l.length = n := by rfl show a ∣ prod l from list_elt_dvd_prod_by_length a n l h1 h done lemma exists_prime_factorization : ∀ (n : Nat), n ≥ 1 → ∃ (l : List Nat), prime_factorization n l := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, n_1 ≥ 1 → ∃ (l : List Nat), prime_factorization n_1 l assume h1 : n ≥ 1 by_cases h2 : n = 1 · -- Case 1. h2 : n = 1 apply Exists.intro [] define apply And.intro · -- Proof of nondec_prime_list [] define show all_prime [] ∧ nondec [] from And.intro all_prime_nil nondec_nil done · -- Proof of prod [] = n rewrite [prod_nil, h2] rfl done done · -- Case 2. h2 : n ≠ 1 have h3 : n ≥ 2 := lt_of_le_of_ne' h1 h2 obtain (p : Nat) (h4 : prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q) from exists_least_prime_factor h3 have p_prime_factor : prime_factor p n := h4.left define at p_prime_factor have p_prime : prime p := p_prime_factor.left have p_dvd_n : p ∣ n := p_prime_factor.right have p_least : ∀ (q : Nat), prime_factor q n → p ≤ q := h4.right obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n have h5 : m ≠ 0 := by contradict h1 with h6 have h7 : n = 0 := calc n _ = p * m := n_eq_pm _ = p * 0 := by rw [h6] _ = 0 := by ring rewrite [h7] decide done have m_pos : 0 < m := Nat.pos_of_ne_zero h5 have m_lt_n : m < n := by define at p_prime show m < n from calc m _ < m + m := by linarith _ = 2 * m := by ring _ ≤ p * m := by rel [p_prime.left] _ = n := n_eq_pm.symm done obtain (L : List Nat) (h6 : prime_factorization m L) from ih m m_lt_n m_pos define at h6 have ndpl_L : nondec_prime_list L := h6.left define at ndpl_L apply Exists.intro (p :: L) define apply And.intro · -- Proof of nondec_prime_list (p :: L) define apply And.intro · -- Proof of all_prime (p :: L) rewrite [all_prime_cons] show prime p ∧ all_prime L from And.intro p_prime ndpl_L.left done · -- Proof of nondec (p :: L) rewrite [nondec_cons] apply And.intro _ ndpl_L.right fix q : Nat assume q_in_L : q ∈ L have h7 : q ∣ prod L := list_elt_dvd_prod q_in_L rewrite [h6.right] at h7 --h7 : q ∣ m have h8 : m ∣ n := by apply Exists.intro p rewrite [n_eq_pm] ring done have q_dvd_n : q ∣ n := dvd_trans h7 h8 have ap_L : all_prime L := ndpl_L.left define at ap_L have q_prime_factor : prime_factor q n := And.intro (ap_L q q_in_L) q_dvd_n show p ≤ q from p_least q q_prime_factor done done · -- Proof of prod (p :: L) = n rewrite [prod_cons, h6.right, n_eq_pm] rfl done done done theorem Theorem_7_2_2 {a b c : Nat} (h1 : c ∣ a * b) (h2 : rel_prime a c) : c ∣ b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑c ∣ ↑b define at h1; define at h2; define obtain (j : Nat) (h3 : a * b = c * j) from h1 set s : Int := gcd_c1 a c set t : Int := gcd_c2 a c have h4 : s * ↑a + t * ↑c = ↑(gcd a c) := gcd_lin_comb c a rewrite [h2, Nat.cast_one] at h4 --h4 : s * ↑a + t * ↑c = (1 : Int) apply Exists.intro (s * ↑j + t * ↑b) show ↑b = ↑c * (s * ↑j + t * ↑b) from calc ↑b _ = (1 : Int) * ↑b := (one_mul _).symm _ = (s * ↑a + t * ↑c) * ↑b := by rw [h4] _ = s * (↑a * ↑b) + t * ↑c * ↑b := by ring _ = s * (↑c * ↑j) + t * ↑c * ↑b := by rw [←Nat.cast_mul a b, h3, Nat.cast_mul c j] _ = ↑c * (s * ↑j + t * ↑b) := by ring done lemma le_nonzero_prod_left {a b : Nat} (h : a * b ≠ 0) : a ≤ a * b := by have h1 : b ≠ 0 := by contradict h with h1 rewrite [h1] ring done have h2 : 1 ≤ b := Nat.pos_of_ne_zero h1 show a ≤ a * b from calc a = a * 1 := (mul_one a).symm _ ≤ a * b := by rel [h2] done lemma le_nonzero_prod_right {a b : Nat} (h : a * b ≠ 0) : b ≤ a * b := by rewrite [mul_comm] rewrite [mul_comm] at h show b ≤ b * a from le_nonzero_prod_left h done lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry lemma rel_prime_of_prime_not_dvd {a p : Nat} (h1 : prime p) (h2 : ¬p ∣ a) : rel_prime a p := by have h3 : gcd a p ∣ a := gcd_dvd_left a p have h4 : gcd a p ∣ p := gcd_dvd_right a p have h5 : gcd a p = 1 ∨ gcd a p = p := dvd_prime h1 h4 have h6 : gcd a p ≠ p := by contradict h2 with h6 rewrite [h6] at h3 show p ∣ a from h3 done disj_syll h5 h6 show rel_prime a p from h5 done theorem Theorem_7_2_3 {a b p : Nat} (h1 : prime p) (h2 : p ∣ a * b) : p ∣ a ∨ p ∣ b := by or_right with h3 have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3 show p ∣ b from Theorem_7_2_2 h2 h4 done lemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a ≥ 1 := by have h1 : a ≠ 0 := by by_contra h1 rewrite [h1] at h contradict h linarith done show a ≥ 1 from Nat.pos_of_ne_zero h1 done lemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by have h1 : a ≥ 1 := ge_one_of_prod_one h have h2 : a * b ≠ 0 := by linarith have h3 : a ≤ a * b := le_nonzero_prod_left h2 rewrite [h] at h3 show a = 1 from Nat.le_antisymm h3 h1 done lemma eq_one_of_dvd_one {n : Nat} (h : n ∣ 1) : n = 1 := by obtain (j : Nat) (h1 : 1 = n * j) from h show n = 1 from eq_one_of_prod_one h1.symm done lemma prime_not_one {p : Nat} (h : prime p) : p ≠ 1 := by define at h linarith done theorem Theorem_7_2_4 {p : Nat} (h1 : prime p) : ∀ (l : List Nat), p ∣ prod l → ∃ a ∈ l, p ∣ a := by apply List.rec · -- Base Case. Goal : p ∣ prod [] → ∃ a ∈ [], p ∣ a rewrite [prod_nil] assume h2 : p ∣ 1 show ∃ a ∈ [], p ∣ a from absurd (eq_one_of_dvd_one h2) (prime_not_one h1) done · -- Induction Step fix b : Nat fix L : List Nat assume ih : p ∣ prod L → ∃ a ∈ L, p ∣ a --Goal : p ∣ prod (b :: L) → ∃ a ∈ b :: L, p ∣ a assume h2 : p ∣ prod (b :: L) rewrite [prod_cons] at h2 have h3 : p ∣ b ∨ p ∣ prod L := Theorem_7_2_3 h1 h2 by_cases on h3 · -- Case 1. h3 : p ∣ b apply Exists.intro b show b ∈ b :: L ∧ p ∣ b from And.intro (List.mem_cons_self b L) h3 done · -- Case 2. h3 : p ∣ prod L obtain (a : Nat) (h4 : a ∈ L ∧ p ∣ a) from ih h3 apply Exists.intro a show a ∈ b :: L ∧ p ∣ a from And.intro (List.mem_cons_of_mem b h4.left) h4.right done done done lemma prime_in_list {p : Nat} {l : List Nat} (h1 : prime p) (h2 : all_prime l) (h3 : p ∣ prod l) : p ∈ l := by obtain (a : Nat) (h4 : a ∈ l ∧ p ∣ a) from Theorem_7_2_4 h1 l h3 define at h2 have h5 : prime a := h2 a h4.left have h6 : p = 1 ∨ p = a := dvd_prime h5 h4.right disj_syll h6 (prime_not_one h1) rewrite [h6] show a ∈ l from h4.left done lemma first_le_first {p q : Nat} {l m : List Nat} (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m)) (h3 : prod (p :: l) = prod (q :: m)) : p ≤ q := by define at h1; define at h2 have h4 : q ∣ prod (p :: l) := by define apply Exists.intro (prod m) rewrite [←prod_cons] show prod (p :: l) = prod (q :: m) from h3 done have h5 : all_prime (q :: m) := h2.left rewrite [all_prime_cons] at h5 have h6 : q ∈ p :: l := prime_in_list h5.left h1.left h4 have h7 : nondec (p :: l) := h1.right rewrite [nondec_cons] at h7 rewrite [List.mem_cons] at h6 by_cases on h6 · -- Case 1. h6 : q = p linarith done · -- Case 2. h6 : q ∈ l have h8 : ∀ m ∈ l, p ≤ m := h7.left show p ≤ q from h8 q h6 done done lemma nondec_prime_list_tail {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by define at h define rewrite [all_prime_cons, nondec_cons] at h show all_prime l ∧ nondec l from And.intro h.left.right h.right.right done lemma cons_prod_not_one {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : prod (p :: l) ≠ 1 := by define at h have h1 : all_prime (p :: l) := h.left rewrite [all_prime_cons] at h1 rewrite [prod_cons] by_contra h2 show False from (prime_not_one h1.left) (eq_one_of_prod_one h2) done lemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) : l = [] ↔ prod l = 1 := by apply Iff.intro · -- (→) assume h1 : l = [] rewrite [h1] show prod [] = 1 from prod_nil done · -- (←) contrapos assume h1 : ¬l = [] obtain (p : Nat) (h2 : ∃ (L : List Nat), l = p :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List Nat) (h3 : l = p :: L) from h2 rewrite [h3] at h rewrite [h3] show ¬prod (p :: L) = 1 from cons_prod_not_one h done done lemma prime_pos {p : Nat} (h : prime p) : p > 0 := by define at h linarith done theorem Theorem_7_2_5 : ∀ (l1 l2 : List Nat), nondec_prime_list l1 → nondec_prime_list l2 → prod l1 = prod l2 → l1 = l2 := by apply List.rec · -- Base Case. Goal : ∀ (l2 : List Nat), nondec_prime_list [] → -- nondec_prime_list l2 → prod [] = prod l2 → [] = l2 fix l2 : List Nat assume h1 : nondec_prime_list [] assume h2 : nondec_prime_list l2 assume h3 : prod [] = prod l2 rewrite [prod_nil, eq_comm, ←list_nil_iff_prod_one h2] at h3 show [] = l2 from h3.symm done · -- Induction Step fix p : Nat fix L1 : List Nat assume ih : ∀ (L2 : List Nat), nondec_prime_list L1 → nondec_prime_list L2 → prod L1 = prod L2 → L1 = L2 -- Goal : ∀ (l2 : List Nat), nondec_prime_list (p :: L1) → -- nondec_prime_list l2 → prod (p :: L1) = prod l2 → p :: L1 = l2 fix l2 : List Nat assume h1 : nondec_prime_list (p :: L1) assume h2 : nondec_prime_list l2 assume h3 : prod (p :: L1) = prod l2 have h4 : ¬prod (p :: L1) = 1 := cons_prod_not_one h1 rewrite [h3, ←list_nil_iff_prod_one h2] at h4 obtain (q : Nat) (h5 : ∃ (L : List Nat), l2 = q :: L) from List.exists_cons_of_ne_nil h4 obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5 rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2) rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2) have h7 : p ≤ q := first_le_first h1 h2 h3 have h8 : q ≤ p := first_le_first h2 h1 h3.symm have h9 : p = q := by linarith rewrite [h9, prod_cons, prod_cons] at h3 --h3 : q * prod L1 = q * prod L2 have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1 have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2 define at h2 have h12 : all_prime (q :: L2) := h2.left rewrite [all_prime_cons] at h12 have h13 : q > 0 := prime_pos h12.left have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3 have h15 : L1 = L2 := ih L2 h10 h11 h14 rewrite [h6, h9, h15] rfl done done theorem fund_thm_arith (n : Nat) (h : n ≥ 1) : ∃! (l : List Nat), prime_factorization n l := by exists_unique · -- Existence show ∃ (l : List Nat), prime_factorization n l from exists_prime_factorization n h done · -- Uniqueness fix l1 : List Nat; fix l2 : List Nat assume h1 : prime_factorization n l1 assume h2 : prime_factorization n l2 define at h1; define at h2 have h3 : prod l1 = n := h1.right rewrite [←h2.right] at h3 show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3 done done /- Section 7.3 -/ theorem congr_refl (m : Nat) : ∀ (a : Int), a ≡ a (MOD m) := by fix a : Int define --Goal : ∃ (c : Int), a - a = ↑m * c apply Exists.intro 0 ring done theorem congr_symm {m : Nat} : ∀ {a b : Int}, a ≡ b (MOD m) → b ≡ a (MOD m) := by fix a : Int; fix b : Int assume h1 : a ≡ b (MOD m) define at h1 --h1 : ∃ (c : Int), a - b = ↑m * c define --Goal : ∃ (c : Int), b - a = ↑m * c obtain (c : Int) (h2 : a - b = m * c) from h1 apply Exists.intro (-c) show b - a = m * (-c) from calc b - a _ = -(a - b) := by ring _ = -(m * c) := by rw [h2] _ = m * (-c) := by ring done theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry /- Fundamental properties of congruence classes -/ lemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) : X = Y ↔ X.val = Y.val := Fin.ext_iff lemma val_nat_eq_mod (n k : Nat) : ([k]_(n + 1)).val = k % (n + 1) := by rfl lemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl theorem cc_rep {m : Nat} (X : ZMod m) : ∃ (a : Int), X = [a]_m := match m with | 0 => by apply Exists.intro X rfl done | n + 1 => by apply Exists.intro ↑(X.val) have h1 : X.val < n + 1 := Fin.prop X rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1] rfl done theorem add_class (m : Nat) (a b : Int) : [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm theorem mul_class (m : Nat) (a b : Int) : [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm lemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ [a - b]_m = [0]_m := by apply Iff.intro · -- (→) assume h1 : [a]_m = [b]_m have h2 : a - b = a + (-b) := by ring have h3 : b + (-b) = 0 := by ring show [a - b]_m = [0]_m from calc [a - b]_m _ = [a + (-b)]_m := by rw [h2] _ = [a]_m + [-b]_m := by rw [add_class] _ = [b]_m + [-b]_m := by rw [h1] _ = [b + -b]_m := by rw [add_class] _ = [0]_m := by rw [h3] done · -- (←) assume h1 : [a - b]_m = [0]_m have h2 : b + (a - b) = a := by ring have h3 : b + 0 = b := by ring show [a]_m = [b]_m from calc [a]_m _ = [b + (a - b)]_m := by rw [h2] _ = [b]_m + [a - b]_m := by rw [add_class] _ = [b]_m + [0]_m := by rw [h1] _ = [b + 0]_m := by rw [add_class] _ = [b]_m := by rw [h3] done done lemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) : [a]_m = [0]_m → [-a]_m = [0]_m := by assume h1 : [a]_m = [0]_m have h2 : 0 + (-a) = -a := by ring have h3 : a + (-a) = 0 := by ring show [-a]_m = [0]_m from calc [-a]_m _ = [0 + (-a)]_m := by rw [h2] _ = [0]_m + [-a]_m := by rw [add_class] _ = [a]_m + [-a]_m := by rw [h1] _ = [a + (-a)]_m := by rw [add_class] _ = [0]_m := by rw [h3] done lemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) : [-a]_m = [0]_m ↔ [a]_m = [0]_m := by apply Iff.intro _ (cc_neg_zero_of_cc_zero m a) assume h1 : [-a]_m = [0]_m have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1 have h3 : -(-a) = a := by ring rewrite [h3] at h2 show [a]_m = [0]_m from h2 done lemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl lemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m ↔ m ∣ k := match m with | 0 => by have h : (0 : Int) = (↑(0 : Nat) : Int) := by rfl rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj] apply Iff.intro · -- (→) assume h1 : k = 0 rewrite [h1] show 0 ∣ 0 from dvd_self 0 done · -- (←) assume h1 : 0 ∣ k obtain (c : Nat) (h2 : k = 0 * c) from h1 rewrite [h2] ring done done | n + 1 => by rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero] show k % (n + 1) = 0 ↔ n + 1 ∣ k from (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm done lemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m ↔ ↑m ∣ a := by obtain (k : Nat) (h1 : a = ↑k ∨ a = -↑k) from Int.eq_nat_or_neg a by_cases on h1 · -- Case 1. h1: a = ↑k rewrite [h1, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done · -- Case 2. h1: a = -↑k rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done done theorem cc_eq_iff_congr (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ a ≡ b (MOD m) := calc [a]_m = [b]_m _ ↔ [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b _ ↔ ↑m ∣ (a - b) := cc_zero_iff_dvd m (a - b) _ ↔ a ≡ b (MOD m) := by rfl /- End of fundamental properties of congruence classes -/ lemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m := by have h1 : (↑m : Int) ≠ 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m) show 0 ≤ a % m from Int.emod_nonneg a h1 done lemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m) have h2 : (↑m : Int) > 0 := (Nat.cast_pos).rtl h1 show a % m < m from Int.emod_lt_of_pos a h2 done lemma congr_mod_mod (m : Nat) (a : Int) : a ≡ a % m (MOD m) := by define have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m apply Exists.intro (a / m) show a - a % m = m * (a / m) from calc a - (a % m) _ = m * (a / m) + a % m - a % m := by rw [h1] _ = m * (a / m) := by ring done lemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) := And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a)) theorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) : ∃! (r : Int), 0 ≤ r ∧ r < m ∧ a ≡ r (MOD m) := by exists_unique · -- Existence apply Exists.intro (a % m) show 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) from mod_cmpl_res m a done · -- Uniqueness fix r1 : Int; fix r2 : Int assume h1 : 0 ≤ r1 ∧ r1 < m ∧ a ≡ r1 (MOD m) assume h2 : 0 ≤ r2 ∧ r2 < m ∧ a ≡ r2 (MOD m) have h3 : r1 ≡ r2 (MOD m) := congr_trans (congr_symm h1.right.right) h2.right.right obtain (d : Int) (h4 : r1 - r2 = m * d) from h3 have h5 : r1 - r2 < m * 1 := by linarith have h6 : m * (-1) < r1 - r2 := by linarith rewrite [h4] at h5 --h5 : m * d < m * 1 rewrite [h4] at h6 --h6 : m * -1 < m * d have h7 : (↑m : Int) ≥ 0 := Nat.cast_nonneg m have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7 have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7 have h10 : d = 0 := by linarith show r1 = r2 from calc r1 _ = r1 - r2 + r2 := by ring _ = m * 0 + r2 := by rw [h4, h10] _ = r2 := by ring done done lemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m := (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a) theorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y rewrite [h1, h2] have h3 : a + b = b + a := by ring show [a]_m + [b]_m = [b]_m + [a]_m from calc [a]_m + [b]_m _ = [a + b]_m := add_class m a b _ = [b + a]_m := by rw [h3] _ = [b]_m + [a]_m := (add_class m b a).symm done theorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X rewrite [h1] have h2 : a * 1 = a := by ring show [a]_m * [1]_m = [a]_m from calc [a]_m * [1]_m _ = [a * 1]_m := mul_class m a 1 _ = [a]_m := by rw [h2] done theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry lemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) : [a]_m * [gcd_c2 m a]_m = [1]_m := by set s : Int := gcd_c1 m a have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m define at h1 rewrite [h1, Nat.cast_one] at h2 --h2 : s * ↑m + gcd_c2 m a * ↑a = 1 rewrite [mul_class, cc_eq_iff_congr] define --Goal : ∃ (c : Int), ↑a * gcd_c2 m a - 1 = ↑m * c apply Exists.intro (-s) show a * (gcd_c2 m a) - 1 = m * (-s) from calc a * (gcd_c2 m a) - 1 _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring _ = 1 + m * (-s) - 1 := by rw [h2] _ = m * (-s) := by ring done theorem Theorem_7_3_7 (m a : Nat) : invertible [a]_m ↔ rel_prime m a := by apply Iff.intro · -- (→) assume h1 : invertible [a]_m define at h1 obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1 obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y rewrite [h3, mul_class, cc_eq_iff_congr] at h2 define at h2 obtain (c : Int) (h4 : a * b - 1 = m * c) from h2 rewrite [Exercise_7_2_6] --Goal : ∃ (s t : Int), s * ↑m + t * ↑a = 1 apply Exists.intro (-c) apply Exists.intro b show (-c) * m + b * a = 1 from calc (-c) * m + b * a _ = (-c) * m + (a * b - 1) + 1 := by ring _ = (-c) * m + m * c + 1 := by rw [h4] _ = 1 := by ring done · -- (←) assume h1 : rel_prime m a define show ∃ (Y : ZMod m), [a]_m * Y = [1]_m from Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1) done done /- Section 7.4 -/ section Euler open Euler lemma num_rp_below_base {m : Nat} : num_rp_below m 0 = 0 := by rfl lemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) : num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by have h1 : num_rp_below m (j + 1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : gcd m j = 1 rewrite [if_pos h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1 show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1 done lemma num_rp_below_step_not_rp {m j : Nat} (h : ¬rel_prime m j) : num_rp_below m (j + 1) = num_rp_below m j := by have h1 : num_rp_below m (j +1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : ¬gcd m j = 1 rewrite [if_neg h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j show num_rp_below m (j + 1) = num_rp_below m j from h1 done lemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl #eval phi 10 --Answer: 4 lemma prod_inv_iff_inv {m : Nat} {X : ZMod m} (h1 : invertible X) (Y : ZMod m) : invertible (X * Y) ↔ invertible Y := by apply Iff.intro · -- (→) assume h2 : invertible (X * Y) obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2 apply Exists.intro (X * Z) rewrite [←h3] --Goal : Y * (X * Z) = X * Y * Z ring --Note that ring can do algebra in ZMod m done · -- (←) assume h2 : invertible Y obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1 obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2 apply Exists.intro (Xi * Yi) show (X * Y) * (Xi * Yi) = [1]_m from calc X * Y * (Xi * Yi) _ = (X * Xi) * (Y * Yi) := by ring _ = [1]_m * [1]_m := by rw [h3, h4] _ = [1]_m := Theorem_7_3_6_7 [1]_m done done lemma F_rp_def {m i : Nat} (h : rel_prime m i) : F m i = [i]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h --h : gcd m i = 1 rewrite [if_pos h] at h1 show F m i = [i]_m from h1 done lemma F_not_rp_def {m i : Nat} (h : ¬rel_prime m i) : F m i = [1]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h rewrite [h1, if_neg h] rfl done lemma prod_seq_base {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 0 k f = [1]_m := by rfl lemma prod_seq_step {m : Nat} (n k : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl lemma prod_seq_zero_step {m : Nat} (n : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by rewrite [prod_seq_step, zero_add] rfl done lemma prod_one {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 1 k f = f k := by rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7] rfl done lemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl lemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m := calc [G m a i]_m _ = [(a * i) % m]_m := by rfl _ = [a * i]_m := (cc_eq_mod m (a * i)).symm _ = [a]_m * [i]_m := (mul_class m a i).symm lemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) : rel_prime m (G m a i) ↔ rel_prime m i := by have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1 show rel_prime m (G m a i) ↔ rel_prime m i from calc rel_prime m (G m a i) _ ↔ invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm _ ↔ invertible ([a]_m * [i]_m) := by rw [cc_G] _ ↔ invertible [i]_m := prod_inv_iff_inv h2 ([i]_m) _ ↔ rel_prime m i := Theorem_7_3_7 m i done lemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) : F m (G m a i) = [a]_m * F m i := by have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2 show F m (G m a i) = [a]_m * F m i from calc F m (G m a i) _ = [G m a i]_m := F_rp_def h3 _ = [a]_m * [i]_m := cc_G m a i _ = [a]_m * F m i := by rw [F_rp_def h2] done lemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : ¬rel_prime m i) : F m (G m a i) = [1]_m := by rewrite [←G_rp_iff h1 i] at h2 show F m (G m a i) = [1]_m from F_not_rp_def h2 done lemma FG_prod {m a : Nat} (h1 : rel_prime m a) : ∀ (k : Nat), prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by by_induc · -- Base Case show prod_seq 0 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from calc prod_seq 0 0 ((F m) ∘ (G m a)) _ = [1]_m := prod_seq_base _ _ _ = [a]_m ^ 0 * [1]_m := by ring _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by rw [num_rp_below_base, prod_seq_base] done · -- Induction Step fix k : Nat assume ih : prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) by_cases h2 : rel_prime m k · -- Case 1. h2 : rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([a]_m * F m k) := by rw [FG_rp h1 h2] _ = [a]_m ^ ((num_rp_below m k) + 1) * ((prod_seq k 0 (F m)) * F m k) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_rp h2, prod_seq_zero_step] done · -- Case 2. h2 : ¬rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([1]_m) := by rw [FG_not_rp h1 h2] _ = [a]_m ^ (num_rp_below m k) * (prod_seq k 0 (F m) * ([1]_m)) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_not_rp h2, prod_seq_zero_step, F_not_rp_def h2] done done done lemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by define --Goal : ∀ i < m, G m a i < m fix i : Nat assume h1 : i < m rewrite [G_def] --Goal : a * i % m < m show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m) done lemma left_inv_one_one_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g' (g i) = i) : one_one_below n g := sorry lemma right_inv_onto_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g (g' i) = i) (h2 : maps_below n g') : onto_below n g := by define at h2; define fix k : Nat assume h3 : k < n apply Exists.intro (g' k) show g' k < n ∧ g (g' k) = k from And.intro (h2 k h3) (h1 k h3) done lemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m * [inv_mod m a]_m = [1]_m := by have h2 : 0 ≤ (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a) show [a]_m * [inv_mod m a]_m = [1]_m from calc [a]_m * [inv_mod m a]_m _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2] _ = [a]_m * [gcd_c2 m a]_m := by rw [←cc_eq_mod] _ = [1]_m := gcd_c2_inv h1 done lemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m := calc a * (b % m) % m = a % m * (b % m % m) % m := Nat.mul_mod _ _ _ _ = a % m * (b % m) % m := by rw [Nat.mod_mod] _ = a * b % m := (Nat.mul_mod _ _ _).symm lemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm] rfl done theorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] : ↑a ≡ ↑b (MOD m) ↔ a % m = b % m := sorry lemma mul_inv_mod_cancel {m a i : Nat} [NeZero m] (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1 rewrite [mul_class, cc_eq_iff_congr, ←Nat.cast_mul, ←Nat.cast_one, congr_iff_mod_eq_Nat] at h3 show a * inv_mod m a * i % m = i from calc a * (inv_mod m a) * i % m _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = 1 % m * i % m := by rw [h3] _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = i % m := by rw [one_mul] _ = i := Nat.mod_eq_of_lt h2 done lemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl lemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, G m a (Ginv m a i) = i := by fix i : Nat assume h2 : i < m show G m a (Ginv m a i) = i from calc G m a (Ginv m a i) _ = a * ((inv_mod m a * i) % m) % m := by rfl _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, Ginv m a (G m a i) = i := by fix i : Nat assume h2 : i < m show Ginv m a (G m a i) = i from calc Ginv m a (G m a i) _ = inv_mod m a * ((a * i) % m) % m := by rfl _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc, mul_comm (inv_mod m a)] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_maps_below (m a : Nat) [NeZero m] : maps_below m (Ginv m a) := G_maps_below m (inv_mod m a) lemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : one_one_below m (G m a) := left_inv_one_one_below (Ginv_left_inv h1) lemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : onto_below m (G m a) := right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a) lemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : perm_below m (G m a) := And.intro (G_maps_below m a) (And.intro (G_one_one_below h1) (G_onto_below h1)) --Permuting a product of congruence classes doesn't change product lemma swap_fst (u v : Nat) : swap u v u = v := by define : swap u v u --Goal : (if u = u then v else if u = v then u else u) = v have h : u = u := by rfl rewrite [if_pos h] rfl done lemma swap_snd (u v : Nat) : swap u v v = u := by define : swap u v v by_cases h1 : v = u · -- Case 1. h1 : v = u rewrite [if_pos h1] show v = u from h1 done · -- Case 2. h1 : v ≠ u rewrite [if_neg h1] have h2 : v = v := by rfl rewrite [if_pos h2] rfl done done lemma swap_other {u v i : Nat} (h1 : i ≠ u) (h2 : i ≠ v) : swap u v i = i := by define : swap u v i rewrite [if_neg h1, if_neg h2] rfl done lemma swap_values (u v i : Nat) : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := by by_cases h1 : i = u · -- Case 1. h1 : i = u apply Or.inl rewrite [h1] show swap u v u = v from swap_fst u v done · -- Case 2. h1 : i ≠ u apply Or.inr by_cases h2 : i = v · -- Case 2.1. h2 : i = v apply Or.inl rewrite [h2] show swap u v v = u from swap_snd u v done · -- Case 2.2. h2 : i ≠ v apply Or.inr show swap u v i = i from swap_other h1 h2 done done done lemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by define fix i : Nat assume h3 : i < n have h4 : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := swap_values u v i by_cases on h4 · -- Case 1. h4 : swap u v i = v rewrite [h4] show v < n from h2 done · -- Case 2. by_cases on h4 · -- Case 2.1. h4 : swap u v i = u rewrite [h4] show u < n from h1 done · -- Case 2.2. h4 : swap u v i = i rewrite [h4] show i < n from h3 done done done lemma swap_swap (u v n : Nat) : ∀ i < n, swap u v (swap u v i) = i := by fix i : Nat assume h : i < n by_cases h1 : i = u · -- Case 1. h1 : i = u rewrite [h1, swap_fst, swap_snd] rfl done · -- Case 2. h1 : i ≠ u by_cases h2 : i = v · -- Case 2.1. h2 : i = v rewrite [h2, swap_snd, swap_fst] rfl done · -- Case 2.2. h2 : i ≠ v rewrite [swap_other h1 h2, swap_other h1 h2] rfl done done done lemma swap_one_one_below (u v n) : one_one_below n (swap u v) := left_inv_one_one_below (swap_swap u v n)

lemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v)

HTPI.swap_onto_below

htpi/HTPILib/Chap7.lean

htpi

/- Copyright 2023 Daniel J. Velleman -/ import HTPILib.Chap6 namespace HTPI /- Definitions -/ lemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by have h : n + 1 > 0 := Nat.succ_pos n show a % (n + 1) < n + 1 from Nat.mod_lt a h done def gcd (a b : Nat) : Nat := match b with | 0 => a | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd (n + 1) (a % (n + 1)) termination_by b mutual def gcd_c1 (a b : Nat) : Int := match b with | 0 => 1 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c2 (n + 1) (a % (n + 1)) --Corresponds to s = t' termination_by b def gcd_c2 (a b : Nat) : Int := match b with | 0 => 0 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c1 (n + 1) (a % (n + 1)) - (gcd_c2 (n + 1) (a % (n + 1))) * ↑(a / (n + 1)) --Corresponds to t = s' - t'q termination_by b end def prime (n : Nat) : Prop := 2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n def prime_factor (p n : Nat) : Prop := prime p ∧ p ∣ n def all_prime (l : List Nat) : Prop := ∀ p ∈ l, prime p def nondec (l : List Nat) : Prop := match l with | [] => True --Of course, True is a proposition that is always true | n :: L => (∀ m ∈ L, n ≤ m) ∧ nondec L def nondec_prime_list (l : List Nat) : Prop := all_prime l ∧ nondec l def prod (l : List Nat) : Nat := match l with | [] => 1 | n :: L => n * (prod L) def prime_factorization (n : Nat) (l : List Nat) : Prop := nondec_prime_list l ∧ prod l = n def rel_prime (a b : Nat) : Prop := gcd a b = 1 def congr_mod (m : Nat) (a b : Int) : Prop := (↑m : Int) ∣ (a - b) def cc (m : Nat) (a : Int) : ZMod m := (↑a : ZMod m) notation:50 a " ≡ " b " (MOD " m ")" => congr_mod m a b notation:max "["a"]_"m:max => cc m a def invertible {m : Nat} (X : ZMod m) : Prop := ∃ (Y : ZMod m), X * Y = [1]_m def num_rp_below (m k : Nat) : Nat := match k with | 0 => 0 | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j def phi (m : Nat) : Nat := num_rp_below m m def prod_seq {m : Nat} (j k : Nat) (f : Nat → ZMod m) : ZMod m := match j with | 0 => [1]_m | n + 1 => prod_seq n k f * f (k + n) def maps_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i < n, g i < n def one_one_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i1 < n, ∀ i2 < n, g i1 = g i2 → i1 = i2 def onto_below (n : Nat) (g : Nat → Nat) : Prop := ∀ k < n, ∃ i < n, g i = k def perm_below (n : Nat) (g : Nat → Nat) : Prop := maps_below n g ∧ one_one_below n g ∧ onto_below n g def inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m) def swap (u v i : Nat) : Nat := if i = u then v else if i = v then u else i namespace Euler --For definitions specific to Euler's theorem def F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m def G (m a i : Nat) : Nat := (a * i) % m def Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i end Euler /- Section 7.1 -/ theorem dvd_mod_of_dvd_a_b {a b d : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ (a % b) := by set q : Nat := a / b have h3 : b * q + a % b = a := Nat.div_add_mod a b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 define --Goal : ∃ (c : Nat), a % b = d * c apply Exists.intro (j - k * q) show a % b = d * (j - k * q) from calc a % b _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm _ = a - b * q := by rw [h3] _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc] _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm done theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry #eval gcd 672 161 --Answer: 7 lemma gcd_base (a : Nat) : gcd a 0 = a := by rfl lemma gcd_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd a b = gcd b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1)) rfl done lemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b ≠ 0) : a % b < b := by have h1 : b > 0 := Nat.pos_of_ne_zero h show a % b < b from Nat.mod_lt a h1 done lemma dvd_self (n : Nat) : n ∣ n := by apply Exists.intro 1 ring done theorem gcd_dvd : ∀ (b a : Nat), (gcd a b) ∣ a ∧ (gcd a b) ∣ b := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd a b_1) ∣ a ∧ (gcd a b_1) ∣ b_1 fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_base] --Goal: a ∣ a ∧ a ∣ 0 apply And.intro (dvd_self a) define apply Exists.intro 0 rfl done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_nonzero a h1] --Goal : gcd b (a % b) ∣ a ∧ gcd b (a % b) ∣ b have h2 : a % b < b := mod_nonzero_lt a h1 have h3 : (gcd b (a % b)) ∣ b ∧ (gcd b (a % b)) ∣ (a % b) := ih (a % b) h2 b apply And.intro _ h3.left show (gcd b (a % b)) ∣ a from dvd_a_of_dvd_b_mod h3.left h3.right done done theorem gcd_dvd_left (a b : Nat) : (gcd a b) ∣ a := (gcd_dvd b a).left theorem gcd_dvd_right (a b : Nat) : (gcd a b) ∣ b := (gcd_dvd b a).right lemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl lemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c1 a b = gcd_c2 b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done lemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl lemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * ↑(a / b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done theorem gcd_lin_comb : ∀ (b a : Nat), (gcd_c1 a b) * ↑a + (gcd_c2 a b) * ↑b = ↑(gcd a b) := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd_c1 a b_1) * ↑a + (gcd_c2 a b_1) * ↑b_1 = ↑(gcd a b_1) fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base] --Goal : 1 * ↑a + 0 * ↑0 = ↑a ring done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1] --Goal : gcd_c2 b (a % b) * ↑a + -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * ↑(a / b)) * ↑b = -- ↑(gcd b (a % b)) set r : Nat := a % b set q : Nat := a / b set s : Int := gcd_c1 b r set t : Int := gcd_c2 b r --Goal : t * ↑a + (s - t * ↑q) * ↑b = ↑(gcd b r) have h2 : r < b := mod_nonzero_lt a h1 have h3 : s * ↑b + t * ↑r = ↑(gcd b r) := ih r h2 b have h4 : b * q + r = a := Nat.div_add_mod a b rewrite [←h3, ←h4] rewrite [Nat.cast_add, Nat.cast_mul] --Goal : t * (↑b * ↑q + ↑r) + (s - t * ↑q) * ↑b = s * ↑b + t * ↑r ring done done #eval gcd_c1 672 161 --Answer: 6 #eval gcd_c2 672 161 --Answer: -25 --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161 theorem Theorem_7_1_6 {d a b : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ gcd a b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑d ∣ ↑(gcd a b) set s : Int := gcd_c1 a b set t : Int := gcd_c2 a b have h3 : s * ↑a + t * ↑b = ↑(gcd a b) := gcd_lin_comb b a rewrite [←h3] --Goal : ↑d ∣ s * ↑a + t * ↑b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul] --Goal : ↑d ∣ s * (↑d * ↑j) + t * (↑d * ↑k) define apply Exists.intro (s * ↑j + t * ↑k) ring done /- Section 7.2 -/ theorem dvd_trans {a b c : Nat} (h1 : a ∣ b) (h2 : b ∣ c) : a ∣ c := by define at h1; define at h2; define obtain (m : Nat) (h3 : b = a * m) from h1 obtain (n : Nat) (h4 : c = b * n) from h2 rewrite [h3, mul_assoc] at h4 apply Exists.intro (m * n) show c = a * (m * n) from h4 done lemma exists_prime_factor : ∀ (n : Nat), 2 ≤ n → ∃ (p : Nat), prime_factor p n := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, 2 ≤ n_1 → ∃ (p : Nat), prime_factor p n_1 assume h1 : 2 ≤ n by_cases h2 : prime n · -- Case 1. h2 : prime n apply Exists.intro n define --Goal : prime n ∧ n ∣ n show prime n ∧ n ∣ n from And.intro h2 (dvd_self n) done · -- Case 2. h2 : ¬prime n define at h2 --h2 : ¬(2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n) demorgan at h2 disj_syll h2 h1 obtain (a : Nat) (h3 : ∃ (b : Nat), a * b = n ∧ a < n ∧ b < n) from h2 obtain (b : Nat) (h4 : a * b = n ∧ a < n ∧ b < n) from h3 have h5 : 2 ≤ a := by by_contra h6 have h7 : a ≤ 1 := by linarith have h8 : n ≤ b := calc n _ = a * b := h4.left.symm _ ≤ 1 * b := by rel [h7] _ = b := by ring linarith --n ≤ b contradicts b < n done have h6 : ∃ (p : Nat), prime_factor p a := ih a h4.right.left h5 obtain (p : Nat) (h7 : prime_factor p a) from h6 apply Exists.intro p define --Goal : prime p ∧ p ∣ n define at h7 --h7 : prime p ∧ p ∣ a apply And.intro h7.left have h8 : a ∣ n := by apply Exists.intro b show n = a * b from (h4.left).symm done show p ∣ n from dvd_trans h7.right h8 done done lemma exists_least_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q := by set S : Set Nat := {p : Nat | prime_factor p n} have h2 : ∃ (p : Nat), p ∈ S := exists_prime_factor n h show ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q from well_ord_princ S h2 done lemma all_prime_nil : all_prime [] := by define --Goal : ∀ p ∈ [], prime p fix p : Nat contrapos --Goal : ¬prime p → p ∉ [] assume h1 : ¬prime p show p ∉ [] from List.not_mem_nil p done lemma all_prime_cons (n : Nat) (L : List Nat) : all_prime (n :: L) ↔ prime n ∧ all_prime L := by apply Iff.intro · -- (→) assume h1 : all_prime (n :: L) --Goal : prime n ∧ all_prime L define at h1 --h1 : ∀ p ∈ n :: L, prime p apply And.intro (h1 n (List.mem_cons_self n L)) define --Goal : ∀ p ∈ L, prime p fix p : Nat assume h2 : p ∈ L show prime p from h1 p (List.mem_cons_of_mem n h2) done · -- (←) assume h1 : prime n ∧ all_prime L --Goal : all_prime (n :: l) define : all_prime L at h1 define fix p : Nat assume h2 : p ∈ n :: L rewrite [List.mem_cons] at h2 --h2 : p = n ∨ p ∈ L by_cases on h2 · -- Case 1. h2 : p = n rewrite [h2] show prime n from h1.left done · -- Case 2. h2 : p ∈ L show prime p from h1.right p h2 done done done lemma nondec_nil : nondec [] := by define --Goal : True trivial --trivial proves some obviously true statements, such as True done lemma nondec_cons (n : Nat) (L : List Nat) : nondec (n :: L) ↔ (∀ m ∈ L, n ≤ m) ∧ nondec L := by rfl lemma prod_nil : prod [] = 1 := by rfl lemma prod_cons : prod (n :: L) = n * (prod L) := by rfl lemma exists_cons_of_length_eq_succ {A : Type} {l : List A} {n : Nat} (h : l.length = n + 1) : ∃ (a : A) (L : List A), l = a :: L ∧ L.length = n := by have h1 : ¬l.length = 0 := by linarith rewrite [List.length_eq_zero] at h1 obtain (a : A) (h2 : ∃ (L : List A), l = a :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List A) (h3 : l = a :: L) from h2 apply Exists.intro a apply Exists.intro L apply And.intro h3 have h4 : (a :: L).length = L.length + 1 := List.length_cons a L rewrite [←h3, h] at h4 show L.length = n from (Nat.add_right_cancel h4).symm done lemma list_elt_dvd_prod_by_length (a : Nat) : ∀ (n : Nat), ∀ (l : List Nat), l.length = n → a ∈ l → a ∣ prod l := by by_induc · --Base Case fix l : List Nat assume h1 : l.length = 0 rewrite [List.length_eq_zero] at h1 --h1 : l = [] rewrite [h1] --Goal : a ∈ [] → a ∣ prod [] contrapos assume h2 : ¬a ∣ prod [] show a ∉ [] from List.not_mem_nil a done · -- Induction Step fix n : Nat assume ih : ∀ (l : List Nat), List.length l = n → a ∈ l → a ∣ prod l fix l : List Nat assume h1 : l.length = n + 1 --Goal : a ∈ l → a ∣ prod l obtain (b : Nat) (h2 : ∃ (L : List Nat), l = b :: L ∧ L.length = n) from exists_cons_of_length_eq_succ h1 obtain (L : List Nat) (h3 : l = b :: L ∧ L.length = n) from h2 have h4 : a ∈ L → a ∣ prod L := ih L h3.right assume h5 : a ∈ l rewrite [h3.left, prod_cons] --Goal : a ∣ b * prod L rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b ∨ a ∈ L by_cases on h5 · -- Case 1. h5 : a = b apply Exists.intro (prod L) rewrite [h5] rfl done · -- Case 2. h5 : a ∈ L have h6 : a ∣ prod L := h4 h5 have h7 : prod L ∣ b * prod L := by apply Exists.intro b ring done show a ∣ b * prod L from dvd_trans h6 h7 done done done lemma list_elt_dvd_prod {a : Nat} {l : List Nat} (h : a ∈ l) : a ∣ prod l := by set n : Nat := l.length have h1 : l.length = n := by rfl show a ∣ prod l from list_elt_dvd_prod_by_length a n l h1 h done lemma exists_prime_factorization : ∀ (n : Nat), n ≥ 1 → ∃ (l : List Nat), prime_factorization n l := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, n_1 ≥ 1 → ∃ (l : List Nat), prime_factorization n_1 l assume h1 : n ≥ 1 by_cases h2 : n = 1 · -- Case 1. h2 : n = 1 apply Exists.intro [] define apply And.intro · -- Proof of nondec_prime_list [] define show all_prime [] ∧ nondec [] from And.intro all_prime_nil nondec_nil done · -- Proof of prod [] = n rewrite [prod_nil, h2] rfl done done · -- Case 2. h2 : n ≠ 1 have h3 : n ≥ 2 := lt_of_le_of_ne' h1 h2 obtain (p : Nat) (h4 : prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q) from exists_least_prime_factor h3 have p_prime_factor : prime_factor p n := h4.left define at p_prime_factor have p_prime : prime p := p_prime_factor.left have p_dvd_n : p ∣ n := p_prime_factor.right have p_least : ∀ (q : Nat), prime_factor q n → p ≤ q := h4.right obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n have h5 : m ≠ 0 := by contradict h1 with h6 have h7 : n = 0 := calc n _ = p * m := n_eq_pm _ = p * 0 := by rw [h6] _ = 0 := by ring rewrite [h7] decide done have m_pos : 0 < m := Nat.pos_of_ne_zero h5 have m_lt_n : m < n := by define at p_prime show m < n from calc m _ < m + m := by linarith _ = 2 * m := by ring _ ≤ p * m := by rel [p_prime.left] _ = n := n_eq_pm.symm done obtain (L : List Nat) (h6 : prime_factorization m L) from ih m m_lt_n m_pos define at h6 have ndpl_L : nondec_prime_list L := h6.left define at ndpl_L apply Exists.intro (p :: L) define apply And.intro · -- Proof of nondec_prime_list (p :: L) define apply And.intro · -- Proof of all_prime (p :: L) rewrite [all_prime_cons] show prime p ∧ all_prime L from And.intro p_prime ndpl_L.left done · -- Proof of nondec (p :: L) rewrite [nondec_cons] apply And.intro _ ndpl_L.right fix q : Nat assume q_in_L : q ∈ L have h7 : q ∣ prod L := list_elt_dvd_prod q_in_L rewrite [h6.right] at h7 --h7 : q ∣ m have h8 : m ∣ n := by apply Exists.intro p rewrite [n_eq_pm] ring done have q_dvd_n : q ∣ n := dvd_trans h7 h8 have ap_L : all_prime L := ndpl_L.left define at ap_L have q_prime_factor : prime_factor q n := And.intro (ap_L q q_in_L) q_dvd_n show p ≤ q from p_least q q_prime_factor done done · -- Proof of prod (p :: L) = n rewrite [prod_cons, h6.right, n_eq_pm] rfl done done done theorem Theorem_7_2_2 {a b c : Nat} (h1 : c ∣ a * b) (h2 : rel_prime a c) : c ∣ b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑c ∣ ↑b define at h1; define at h2; define obtain (j : Nat) (h3 : a * b = c * j) from h1 set s : Int := gcd_c1 a c set t : Int := gcd_c2 a c have h4 : s * ↑a + t * ↑c = ↑(gcd a c) := gcd_lin_comb c a rewrite [h2, Nat.cast_one] at h4 --h4 : s * ↑a + t * ↑c = (1 : Int) apply Exists.intro (s * ↑j + t * ↑b) show ↑b = ↑c * (s * ↑j + t * ↑b) from calc ↑b _ = (1 : Int) * ↑b := (one_mul _).symm _ = (s * ↑a + t * ↑c) * ↑b := by rw [h4] _ = s * (↑a * ↑b) + t * ↑c * ↑b := by ring _ = s * (↑c * ↑j) + t * ↑c * ↑b := by rw [←Nat.cast_mul a b, h3, Nat.cast_mul c j] _ = ↑c * (s * ↑j + t * ↑b) := by ring done lemma le_nonzero_prod_left {a b : Nat} (h : a * b ≠ 0) : a ≤ a * b := by have h1 : b ≠ 0 := by contradict h with h1 rewrite [h1] ring done have h2 : 1 ≤ b := Nat.pos_of_ne_zero h1 show a ≤ a * b from calc a = a * 1 := (mul_one a).symm _ ≤ a * b := by rel [h2] done lemma le_nonzero_prod_right {a b : Nat} (h : a * b ≠ 0) : b ≤ a * b := by rewrite [mul_comm] rewrite [mul_comm] at h show b ≤ b * a from le_nonzero_prod_left h done lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry lemma rel_prime_of_prime_not_dvd {a p : Nat} (h1 : prime p) (h2 : ¬p ∣ a) : rel_prime a p := by have h3 : gcd a p ∣ a := gcd_dvd_left a p have h4 : gcd a p ∣ p := gcd_dvd_right a p have h5 : gcd a p = 1 ∨ gcd a p = p := dvd_prime h1 h4 have h6 : gcd a p ≠ p := by contradict h2 with h6 rewrite [h6] at h3 show p ∣ a from h3 done disj_syll h5 h6 show rel_prime a p from h5 done theorem Theorem_7_2_3 {a b p : Nat} (h1 : prime p) (h2 : p ∣ a * b) : p ∣ a ∨ p ∣ b := by or_right with h3 have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3 show p ∣ b from Theorem_7_2_2 h2 h4 done lemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a ≥ 1 := by have h1 : a ≠ 0 := by by_contra h1 rewrite [h1] at h contradict h linarith done show a ≥ 1 from Nat.pos_of_ne_zero h1 done lemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by have h1 : a ≥ 1 := ge_one_of_prod_one h have h2 : a * b ≠ 0 := by linarith have h3 : a ≤ a * b := le_nonzero_prod_left h2 rewrite [h] at h3 show a = 1 from Nat.le_antisymm h3 h1 done lemma eq_one_of_dvd_one {n : Nat} (h : n ∣ 1) : n = 1 := by obtain (j : Nat) (h1 : 1 = n * j) from h show n = 1 from eq_one_of_prod_one h1.symm done lemma prime_not_one {p : Nat} (h : prime p) : p ≠ 1 := by define at h linarith done theorem Theorem_7_2_4 {p : Nat} (h1 : prime p) : ∀ (l : List Nat), p ∣ prod l → ∃ a ∈ l, p ∣ a := by apply List.rec · -- Base Case. Goal : p ∣ prod [] → ∃ a ∈ [], p ∣ a rewrite [prod_nil] assume h2 : p ∣ 1 show ∃ a ∈ [], p ∣ a from absurd (eq_one_of_dvd_one h2) (prime_not_one h1) done · -- Induction Step fix b : Nat fix L : List Nat assume ih : p ∣ prod L → ∃ a ∈ L, p ∣ a --Goal : p ∣ prod (b :: L) → ∃ a ∈ b :: L, p ∣ a assume h2 : p ∣ prod (b :: L) rewrite [prod_cons] at h2 have h3 : p ∣ b ∨ p ∣ prod L := Theorem_7_2_3 h1 h2 by_cases on h3 · -- Case 1. h3 : p ∣ b apply Exists.intro b show b ∈ b :: L ∧ p ∣ b from And.intro (List.mem_cons_self b L) h3 done · -- Case 2. h3 : p ∣ prod L obtain (a : Nat) (h4 : a ∈ L ∧ p ∣ a) from ih h3 apply Exists.intro a show a ∈ b :: L ∧ p ∣ a from And.intro (List.mem_cons_of_mem b h4.left) h4.right done done done lemma prime_in_list {p : Nat} {l : List Nat} (h1 : prime p) (h2 : all_prime l) (h3 : p ∣ prod l) : p ∈ l := by obtain (a : Nat) (h4 : a ∈ l ∧ p ∣ a) from Theorem_7_2_4 h1 l h3 define at h2 have h5 : prime a := h2 a h4.left have h6 : p = 1 ∨ p = a := dvd_prime h5 h4.right disj_syll h6 (prime_not_one h1) rewrite [h6] show a ∈ l from h4.left done lemma first_le_first {p q : Nat} {l m : List Nat} (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m)) (h3 : prod (p :: l) = prod (q :: m)) : p ≤ q := by define at h1; define at h2 have h4 : q ∣ prod (p :: l) := by define apply Exists.intro (prod m) rewrite [←prod_cons] show prod (p :: l) = prod (q :: m) from h3 done have h5 : all_prime (q :: m) := h2.left rewrite [all_prime_cons] at h5 have h6 : q ∈ p :: l := prime_in_list h5.left h1.left h4 have h7 : nondec (p :: l) := h1.right rewrite [nondec_cons] at h7 rewrite [List.mem_cons] at h6 by_cases on h6 · -- Case 1. h6 : q = p linarith done · -- Case 2. h6 : q ∈ l have h8 : ∀ m ∈ l, p ≤ m := h7.left show p ≤ q from h8 q h6 done done lemma nondec_prime_list_tail {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by define at h define rewrite [all_prime_cons, nondec_cons] at h show all_prime l ∧ nondec l from And.intro h.left.right h.right.right done lemma cons_prod_not_one {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : prod (p :: l) ≠ 1 := by define at h have h1 : all_prime (p :: l) := h.left rewrite [all_prime_cons] at h1 rewrite [prod_cons] by_contra h2 show False from (prime_not_one h1.left) (eq_one_of_prod_one h2) done lemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) : l = [] ↔ prod l = 1 := by apply Iff.intro · -- (→) assume h1 : l = [] rewrite [h1] show prod [] = 1 from prod_nil done · -- (←) contrapos assume h1 : ¬l = [] obtain (p : Nat) (h2 : ∃ (L : List Nat), l = p :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List Nat) (h3 : l = p :: L) from h2 rewrite [h3] at h rewrite [h3] show ¬prod (p :: L) = 1 from cons_prod_not_one h done done lemma prime_pos {p : Nat} (h : prime p) : p > 0 := by define at h linarith done theorem Theorem_7_2_5 : ∀ (l1 l2 : List Nat), nondec_prime_list l1 → nondec_prime_list l2 → prod l1 = prod l2 → l1 = l2 := by apply List.rec · -- Base Case. Goal : ∀ (l2 : List Nat), nondec_prime_list [] → -- nondec_prime_list l2 → prod [] = prod l2 → [] = l2 fix l2 : List Nat assume h1 : nondec_prime_list [] assume h2 : nondec_prime_list l2 assume h3 : prod [] = prod l2 rewrite [prod_nil, eq_comm, ←list_nil_iff_prod_one h2] at h3 show [] = l2 from h3.symm done · -- Induction Step fix p : Nat fix L1 : List Nat assume ih : ∀ (L2 : List Nat), nondec_prime_list L1 → nondec_prime_list L2 → prod L1 = prod L2 → L1 = L2 -- Goal : ∀ (l2 : List Nat), nondec_prime_list (p :: L1) → -- nondec_prime_list l2 → prod (p :: L1) = prod l2 → p :: L1 = l2 fix l2 : List Nat assume h1 : nondec_prime_list (p :: L1) assume h2 : nondec_prime_list l2 assume h3 : prod (p :: L1) = prod l2 have h4 : ¬prod (p :: L1) = 1 := cons_prod_not_one h1 rewrite [h3, ←list_nil_iff_prod_one h2] at h4 obtain (q : Nat) (h5 : ∃ (L : List Nat), l2 = q :: L) from List.exists_cons_of_ne_nil h4 obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5 rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2) rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2) have h7 : p ≤ q := first_le_first h1 h2 h3 have h8 : q ≤ p := first_le_first h2 h1 h3.symm have h9 : p = q := by linarith rewrite [h9, prod_cons, prod_cons] at h3 --h3 : q * prod L1 = q * prod L2 have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1 have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2 define at h2 have h12 : all_prime (q :: L2) := h2.left rewrite [all_prime_cons] at h12 have h13 : q > 0 := prime_pos h12.left have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3 have h15 : L1 = L2 := ih L2 h10 h11 h14 rewrite [h6, h9, h15] rfl done done theorem fund_thm_arith (n : Nat) (h : n ≥ 1) : ∃! (l : List Nat), prime_factorization n l := by exists_unique · -- Existence show ∃ (l : List Nat), prime_factorization n l from exists_prime_factorization n h done · -- Uniqueness fix l1 : List Nat; fix l2 : List Nat assume h1 : prime_factorization n l1 assume h2 : prime_factorization n l2 define at h1; define at h2 have h3 : prod l1 = n := h1.right rewrite [←h2.right] at h3 show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3 done done /- Section 7.3 -/ theorem congr_refl (m : Nat) : ∀ (a : Int), a ≡ a (MOD m) := by fix a : Int define --Goal : ∃ (c : Int), a - a = ↑m * c apply Exists.intro 0 ring done theorem congr_symm {m : Nat} : ∀ {a b : Int}, a ≡ b (MOD m) → b ≡ a (MOD m) := by fix a : Int; fix b : Int assume h1 : a ≡ b (MOD m) define at h1 --h1 : ∃ (c : Int), a - b = ↑m * c define --Goal : ∃ (c : Int), b - a = ↑m * c obtain (c : Int) (h2 : a - b = m * c) from h1 apply Exists.intro (-c) show b - a = m * (-c) from calc b - a _ = -(a - b) := by ring _ = -(m * c) := by rw [h2] _ = m * (-c) := by ring done theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry /- Fundamental properties of congruence classes -/ lemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) : X = Y ↔ X.val = Y.val := Fin.ext_iff lemma val_nat_eq_mod (n k : Nat) : ([k]_(n + 1)).val = k % (n + 1) := by rfl lemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl theorem cc_rep {m : Nat} (X : ZMod m) : ∃ (a : Int), X = [a]_m := match m with | 0 => by apply Exists.intro X rfl done | n + 1 => by apply Exists.intro ↑(X.val) have h1 : X.val < n + 1 := Fin.prop X rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1] rfl done theorem add_class (m : Nat) (a b : Int) : [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm theorem mul_class (m : Nat) (a b : Int) : [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm lemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ [a - b]_m = [0]_m := by apply Iff.intro · -- (→) assume h1 : [a]_m = [b]_m have h2 : a - b = a + (-b) := by ring have h3 : b + (-b) = 0 := by ring show [a - b]_m = [0]_m from calc [a - b]_m _ = [a + (-b)]_m := by rw [h2] _ = [a]_m + [-b]_m := by rw [add_class] _ = [b]_m + [-b]_m := by rw [h1] _ = [b + -b]_m := by rw [add_class] _ = [0]_m := by rw [h3] done · -- (←) assume h1 : [a - b]_m = [0]_m have h2 : b + (a - b) = a := by ring have h3 : b + 0 = b := by ring show [a]_m = [b]_m from calc [a]_m _ = [b + (a - b)]_m := by rw [h2] _ = [b]_m + [a - b]_m := by rw [add_class] _ = [b]_m + [0]_m := by rw [h1] _ = [b + 0]_m := by rw [add_class] _ = [b]_m := by rw [h3] done done lemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) : [a]_m = [0]_m → [-a]_m = [0]_m := by assume h1 : [a]_m = [0]_m have h2 : 0 + (-a) = -a := by ring have h3 : a + (-a) = 0 := by ring show [-a]_m = [0]_m from calc [-a]_m _ = [0 + (-a)]_m := by rw [h2] _ = [0]_m + [-a]_m := by rw [add_class] _ = [a]_m + [-a]_m := by rw [h1] _ = [a + (-a)]_m := by rw [add_class] _ = [0]_m := by rw [h3] done lemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) : [-a]_m = [0]_m ↔ [a]_m = [0]_m := by apply Iff.intro _ (cc_neg_zero_of_cc_zero m a) assume h1 : [-a]_m = [0]_m have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1 have h3 : -(-a) = a := by ring rewrite [h3] at h2 show [a]_m = [0]_m from h2 done lemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl lemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m ↔ m ∣ k := match m with | 0 => by have h : (0 : Int) = (↑(0 : Nat) : Int) := by rfl rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj] apply Iff.intro · -- (→) assume h1 : k = 0 rewrite [h1] show 0 ∣ 0 from dvd_self 0 done · -- (←) assume h1 : 0 ∣ k obtain (c : Nat) (h2 : k = 0 * c) from h1 rewrite [h2] ring done done | n + 1 => by rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero] show k % (n + 1) = 0 ↔ n + 1 ∣ k from (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm done lemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m ↔ ↑m ∣ a := by obtain (k : Nat) (h1 : a = ↑k ∨ a = -↑k) from Int.eq_nat_or_neg a by_cases on h1 · -- Case 1. h1: a = ↑k rewrite [h1, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done · -- Case 2. h1: a = -↑k rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done done theorem cc_eq_iff_congr (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ a ≡ b (MOD m) := calc [a]_m = [b]_m _ ↔ [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b _ ↔ ↑m ∣ (a - b) := cc_zero_iff_dvd m (a - b) _ ↔ a ≡ b (MOD m) := by rfl /- End of fundamental properties of congruence classes -/ lemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m := by have h1 : (↑m : Int) ≠ 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m) show 0 ≤ a % m from Int.emod_nonneg a h1 done lemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m) have h2 : (↑m : Int) > 0 := (Nat.cast_pos).rtl h1 show a % m < m from Int.emod_lt_of_pos a h2 done lemma congr_mod_mod (m : Nat) (a : Int) : a ≡ a % m (MOD m) := by define have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m apply Exists.intro (a / m) show a - a % m = m * (a / m) from calc a - (a % m) _ = m * (a / m) + a % m - a % m := by rw [h1] _ = m * (a / m) := by ring done lemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) := And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a)) theorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) : ∃! (r : Int), 0 ≤ r ∧ r < m ∧ a ≡ r (MOD m) := by exists_unique · -- Existence apply Exists.intro (a % m) show 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) from mod_cmpl_res m a done · -- Uniqueness fix r1 : Int; fix r2 : Int assume h1 : 0 ≤ r1 ∧ r1 < m ∧ a ≡ r1 (MOD m) assume h2 : 0 ≤ r2 ∧ r2 < m ∧ a ≡ r2 (MOD m) have h3 : r1 ≡ r2 (MOD m) := congr_trans (congr_symm h1.right.right) h2.right.right obtain (d : Int) (h4 : r1 - r2 = m * d) from h3 have h5 : r1 - r2 < m * 1 := by linarith have h6 : m * (-1) < r1 - r2 := by linarith rewrite [h4] at h5 --h5 : m * d < m * 1 rewrite [h4] at h6 --h6 : m * -1 < m * d have h7 : (↑m : Int) ≥ 0 := Nat.cast_nonneg m have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7 have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7 have h10 : d = 0 := by linarith show r1 = r2 from calc r1 _ = r1 - r2 + r2 := by ring _ = m * 0 + r2 := by rw [h4, h10] _ = r2 := by ring done done lemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m := (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a) theorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y rewrite [h1, h2] have h3 : a + b = b + a := by ring show [a]_m + [b]_m = [b]_m + [a]_m from calc [a]_m + [b]_m _ = [a + b]_m := add_class m a b _ = [b + a]_m := by rw [h3] _ = [b]_m + [a]_m := (add_class m b a).symm done theorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X rewrite [h1] have h2 : a * 1 = a := by ring show [a]_m * [1]_m = [a]_m from calc [a]_m * [1]_m _ = [a * 1]_m := mul_class m a 1 _ = [a]_m := by rw [h2] done theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry lemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) : [a]_m * [gcd_c2 m a]_m = [1]_m := by set s : Int := gcd_c1 m a have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m define at h1 rewrite [h1, Nat.cast_one] at h2 --h2 : s * ↑m + gcd_c2 m a * ↑a = 1 rewrite [mul_class, cc_eq_iff_congr] define --Goal : ∃ (c : Int), ↑a * gcd_c2 m a - 1 = ↑m * c apply Exists.intro (-s) show a * (gcd_c2 m a) - 1 = m * (-s) from calc a * (gcd_c2 m a) - 1 _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring _ = 1 + m * (-s) - 1 := by rw [h2] _ = m * (-s) := by ring done theorem Theorem_7_3_7 (m a : Nat) : invertible [a]_m ↔ rel_prime m a := by apply Iff.intro · -- (→) assume h1 : invertible [a]_m define at h1 obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1 obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y rewrite [h3, mul_class, cc_eq_iff_congr] at h2 define at h2 obtain (c : Int) (h4 : a * b - 1 = m * c) from h2 rewrite [Exercise_7_2_6] --Goal : ∃ (s t : Int), s * ↑m + t * ↑a = 1 apply Exists.intro (-c) apply Exists.intro b show (-c) * m + b * a = 1 from calc (-c) * m + b * a _ = (-c) * m + (a * b - 1) + 1 := by ring _ = (-c) * m + m * c + 1 := by rw [h4] _ = 1 := by ring done · -- (←) assume h1 : rel_prime m a define show ∃ (Y : ZMod m), [a]_m * Y = [1]_m from Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1) done done /- Section 7.4 -/ section Euler open Euler lemma num_rp_below_base {m : Nat} : num_rp_below m 0 = 0 := by rfl lemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) : num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by have h1 : num_rp_below m (j + 1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : gcd m j = 1 rewrite [if_pos h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1 show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1 done lemma num_rp_below_step_not_rp {m j : Nat} (h : ¬rel_prime m j) : num_rp_below m (j + 1) = num_rp_below m j := by have h1 : num_rp_below m (j +1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : ¬gcd m j = 1 rewrite [if_neg h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j show num_rp_below m (j + 1) = num_rp_below m j from h1 done lemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl #eval phi 10 --Answer: 4 lemma prod_inv_iff_inv {m : Nat} {X : ZMod m} (h1 : invertible X) (Y : ZMod m) : invertible (X * Y) ↔ invertible Y := by apply Iff.intro · -- (→) assume h2 : invertible (X * Y) obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2 apply Exists.intro (X * Z) rewrite [←h3] --Goal : Y * (X * Z) = X * Y * Z ring --Note that ring can do algebra in ZMod m done · -- (←) assume h2 : invertible Y obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1 obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2 apply Exists.intro (Xi * Yi) show (X * Y) * (Xi * Yi) = [1]_m from calc X * Y * (Xi * Yi) _ = (X * Xi) * (Y * Yi) := by ring _ = [1]_m * [1]_m := by rw [h3, h4] _ = [1]_m := Theorem_7_3_6_7 [1]_m done done lemma F_rp_def {m i : Nat} (h : rel_prime m i) : F m i = [i]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h --h : gcd m i = 1 rewrite [if_pos h] at h1 show F m i = [i]_m from h1 done lemma F_not_rp_def {m i : Nat} (h : ¬rel_prime m i) : F m i = [1]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h rewrite [h1, if_neg h] rfl done lemma prod_seq_base {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 0 k f = [1]_m := by rfl lemma prod_seq_step {m : Nat} (n k : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl lemma prod_seq_zero_step {m : Nat} (n : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by rewrite [prod_seq_step, zero_add] rfl done lemma prod_one {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 1 k f = f k := by rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7] rfl done lemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl lemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m := calc [G m a i]_m _ = [(a * i) % m]_m := by rfl _ = [a * i]_m := (cc_eq_mod m (a * i)).symm _ = [a]_m * [i]_m := (mul_class m a i).symm lemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) : rel_prime m (G m a i) ↔ rel_prime m i := by have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1 show rel_prime m (G m a i) ↔ rel_prime m i from calc rel_prime m (G m a i) _ ↔ invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm _ ↔ invertible ([a]_m * [i]_m) := by rw [cc_G] _ ↔ invertible [i]_m := prod_inv_iff_inv h2 ([i]_m) _ ↔ rel_prime m i := Theorem_7_3_7 m i done lemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) : F m (G m a i) = [a]_m * F m i := by have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2 show F m (G m a i) = [a]_m * F m i from calc F m (G m a i) _ = [G m a i]_m := F_rp_def h3 _ = [a]_m * [i]_m := cc_G m a i _ = [a]_m * F m i := by rw [F_rp_def h2] done lemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : ¬rel_prime m i) : F m (G m a i) = [1]_m := by rewrite [←G_rp_iff h1 i] at h2 show F m (G m a i) = [1]_m from F_not_rp_def h2 done lemma FG_prod {m a : Nat} (h1 : rel_prime m a) : ∀ (k : Nat), prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by by_induc · -- Base Case show prod_seq 0 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from calc prod_seq 0 0 ((F m) ∘ (G m a)) _ = [1]_m := prod_seq_base _ _ _ = [a]_m ^ 0 * [1]_m := by ring _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by rw [num_rp_below_base, prod_seq_base] done · -- Induction Step fix k : Nat assume ih : prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) by_cases h2 : rel_prime m k · -- Case 1. h2 : rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([a]_m * F m k) := by rw [FG_rp h1 h2] _ = [a]_m ^ ((num_rp_below m k) + 1) * ((prod_seq k 0 (F m)) * F m k) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_rp h2, prod_seq_zero_step] done · -- Case 2. h2 : ¬rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([1]_m) := by rw [FG_not_rp h1 h2] _ = [a]_m ^ (num_rp_below m k) * (prod_seq k 0 (F m) * ([1]_m)) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_not_rp h2, prod_seq_zero_step, F_not_rp_def h2] done done done lemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by define --Goal : ∀ i < m, G m a i < m fix i : Nat assume h1 : i < m rewrite [G_def] --Goal : a * i % m < m show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m) done lemma left_inv_one_one_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g' (g i) = i) : one_one_below n g := sorry lemma right_inv_onto_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g (g' i) = i) (h2 : maps_below n g') : onto_below n g := by define at h2; define fix k : Nat assume h3 : k < n apply Exists.intro (g' k) show g' k < n ∧ g (g' k) = k from And.intro (h2 k h3) (h1 k h3) done lemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m * [inv_mod m a]_m = [1]_m := by have h2 : 0 ≤ (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a) show [a]_m * [inv_mod m a]_m = [1]_m from calc [a]_m * [inv_mod m a]_m _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2] _ = [a]_m * [gcd_c2 m a]_m := by rw [←cc_eq_mod] _ = [1]_m := gcd_c2_inv h1 done lemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m := calc a * (b % m) % m = a % m * (b % m % m) % m := Nat.mul_mod _ _ _ _ = a % m * (b % m) % m := by rw [Nat.mod_mod] _ = a * b % m := (Nat.mul_mod _ _ _).symm lemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm] rfl done theorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] : ↑a ≡ ↑b (MOD m) ↔ a % m = b % m := sorry lemma mul_inv_mod_cancel {m a i : Nat} [NeZero m] (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1 rewrite [mul_class, cc_eq_iff_congr, ←Nat.cast_mul, ←Nat.cast_one, congr_iff_mod_eq_Nat] at h3 show a * inv_mod m a * i % m = i from calc a * (inv_mod m a) * i % m _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = 1 % m * i % m := by rw [h3] _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = i % m := by rw [one_mul] _ = i := Nat.mod_eq_of_lt h2 done lemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl lemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, G m a (Ginv m a i) = i := by fix i : Nat assume h2 : i < m show G m a (Ginv m a i) = i from calc G m a (Ginv m a i) _ = a * ((inv_mod m a * i) % m) % m := by rfl _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, Ginv m a (G m a i) = i := by fix i : Nat assume h2 : i < m show Ginv m a (G m a i) = i from calc Ginv m a (G m a i) _ = inv_mod m a * ((a * i) % m) % m := by rfl _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc, mul_comm (inv_mod m a)] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_maps_below (m a : Nat) [NeZero m] : maps_below m (Ginv m a) := G_maps_below m (inv_mod m a) lemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : one_one_below m (G m a) := left_inv_one_one_below (Ginv_left_inv h1) lemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : onto_below m (G m a) := right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a) lemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : perm_below m (G m a) := And.intro (G_maps_below m a) (And.intro (G_one_one_below h1) (G_onto_below h1)) --Permuting a product of congruence classes doesn't change product lemma swap_fst (u v : Nat) : swap u v u = v := by define : swap u v u --Goal : (if u = u then v else if u = v then u else u) = v have h : u = u := by rfl rewrite [if_pos h] rfl done lemma swap_snd (u v : Nat) : swap u v v = u := by define : swap u v v by_cases h1 : v = u · -- Case 1. h1 : v = u rewrite [if_pos h1] show v = u from h1 done · -- Case 2. h1 : v ≠ u rewrite [if_neg h1] have h2 : v = v := by rfl rewrite [if_pos h2] rfl done done lemma swap_other {u v i : Nat} (h1 : i ≠ u) (h2 : i ≠ v) : swap u v i = i := by define : swap u v i rewrite [if_neg h1, if_neg h2] rfl done lemma swap_values (u v i : Nat) : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := by by_cases h1 : i = u · -- Case 1. h1 : i = u apply Or.inl rewrite [h1] show swap u v u = v from swap_fst u v done · -- Case 2. h1 : i ≠ u apply Or.inr by_cases h2 : i = v · -- Case 2.1. h2 : i = v apply Or.inl rewrite [h2] show swap u v v = u from swap_snd u v done · -- Case 2.2. h2 : i ≠ v apply Or.inr show swap u v i = i from swap_other h1 h2 done done done lemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by define fix i : Nat assume h3 : i < n have h4 : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := swap_values u v i by_cases on h4 · -- Case 1. h4 : swap u v i = v rewrite [h4] show v < n from h2 done · -- Case 2. by_cases on h4 · -- Case 2.1. h4 : swap u v i = u rewrite [h4] show u < n from h1 done · -- Case 2.2. h4 : swap u v i = i rewrite [h4] show i < n from h3 done done done lemma swap_swap (u v n : Nat) : ∀ i < n, swap u v (swap u v i) = i := by fix i : Nat assume h : i < n by_cases h1 : i = u · -- Case 1. h1 : i = u rewrite [h1, swap_fst, swap_snd] rfl done · -- Case 2. h1 : i ≠ u by_cases h2 : i = v · -- Case 2.1. h2 : i = v rewrite [h2, swap_snd, swap_fst] rfl done · -- Case 2.2. h2 : i ≠ v rewrite [swap_other h1 h2, swap_other h1 h2] rfl done done done lemma swap_one_one_below (u v n) : one_one_below n (swap u v) := left_inv_one_one_below (swap_swap u v n) lemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) := right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2)

lemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v)

HTPI.swap_perm_below

htpi/HTPILib/Chap7.lean

htpi

/- Copyright 2023 Daniel J. Velleman -/ import HTPILib.Chap6 namespace HTPI /- Definitions -/ lemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by have h : n + 1 > 0 := Nat.succ_pos n show a % (n + 1) < n + 1 from Nat.mod_lt a h done def gcd (a b : Nat) : Nat := match b with | 0 => a | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd (n + 1) (a % (n + 1)) termination_by b mutual def gcd_c1 (a b : Nat) : Int := match b with | 0 => 1 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c2 (n + 1) (a % (n + 1)) --Corresponds to s = t' termination_by b def gcd_c2 (a b : Nat) : Int := match b with | 0 => 0 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c1 (n + 1) (a % (n + 1)) - (gcd_c2 (n + 1) (a % (n + 1))) * ↑(a / (n + 1)) --Corresponds to t = s' - t'q termination_by b end def prime (n : Nat) : Prop := 2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n def prime_factor (p n : Nat) : Prop := prime p ∧ p ∣ n def all_prime (l : List Nat) : Prop := ∀ p ∈ l, prime p def nondec (l : List Nat) : Prop := match l with | [] => True --Of course, True is a proposition that is always true | n :: L => (∀ m ∈ L, n ≤ m) ∧ nondec L def nondec_prime_list (l : List Nat) : Prop := all_prime l ∧ nondec l def prod (l : List Nat) : Nat := match l with | [] => 1 | n :: L => n * (prod L) def prime_factorization (n : Nat) (l : List Nat) : Prop := nondec_prime_list l ∧ prod l = n def rel_prime (a b : Nat) : Prop := gcd a b = 1 def congr_mod (m : Nat) (a b : Int) : Prop := (↑m : Int) ∣ (a - b) def cc (m : Nat) (a : Int) : ZMod m := (↑a : ZMod m) notation:50 a " ≡ " b " (MOD " m ")" => congr_mod m a b notation:max "["a"]_"m:max => cc m a def invertible {m : Nat} (X : ZMod m) : Prop := ∃ (Y : ZMod m), X * Y = [1]_m def num_rp_below (m k : Nat) : Nat := match k with | 0 => 0 | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j def phi (m : Nat) : Nat := num_rp_below m m def prod_seq {m : Nat} (j k : Nat) (f : Nat → ZMod m) : ZMod m := match j with | 0 => [1]_m | n + 1 => prod_seq n k f * f (k + n) def maps_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i < n, g i < n def one_one_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i1 < n, ∀ i2 < n, g i1 = g i2 → i1 = i2 def onto_below (n : Nat) (g : Nat → Nat) : Prop := ∀ k < n, ∃ i < n, g i = k def perm_below (n : Nat) (g : Nat → Nat) : Prop := maps_below n g ∧ one_one_below n g ∧ onto_below n g def inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m) def swap (u v i : Nat) : Nat := if i = u then v else if i = v then u else i namespace Euler --For definitions specific to Euler's theorem def F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m def G (m a i : Nat) : Nat := (a * i) % m def Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i end Euler /- Section 7.1 -/ theorem dvd_mod_of_dvd_a_b {a b d : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ (a % b) := by set q : Nat := a / b have h3 : b * q + a % b = a := Nat.div_add_mod a b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 define --Goal : ∃ (c : Nat), a % b = d * c apply Exists.intro (j - k * q) show a % b = d * (j - k * q) from calc a % b _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm _ = a - b * q := by rw [h3] _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc] _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm done theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry #eval gcd 672 161 --Answer: 7 lemma gcd_base (a : Nat) : gcd a 0 = a := by rfl lemma gcd_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd a b = gcd b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1)) rfl done lemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b ≠ 0) : a % b < b := by have h1 : b > 0 := Nat.pos_of_ne_zero h show a % b < b from Nat.mod_lt a h1 done lemma dvd_self (n : Nat) : n ∣ n := by apply Exists.intro 1 ring done theorem gcd_dvd : ∀ (b a : Nat), (gcd a b) ∣ a ∧ (gcd a b) ∣ b := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd a b_1) ∣ a ∧ (gcd a b_1) ∣ b_1 fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_base] --Goal: a ∣ a ∧ a ∣ 0 apply And.intro (dvd_self a) define apply Exists.intro 0 rfl done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_nonzero a h1] --Goal : gcd b (a % b) ∣ a ∧ gcd b (a % b) ∣ b have h2 : a % b < b := mod_nonzero_lt a h1 have h3 : (gcd b (a % b)) ∣ b ∧ (gcd b (a % b)) ∣ (a % b) := ih (a % b) h2 b apply And.intro _ h3.left show (gcd b (a % b)) ∣ a from dvd_a_of_dvd_b_mod h3.left h3.right done done theorem gcd_dvd_left (a b : Nat) : (gcd a b) ∣ a := (gcd_dvd b a).left theorem gcd_dvd_right (a b : Nat) : (gcd a b) ∣ b := (gcd_dvd b a).right lemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl lemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c1 a b = gcd_c2 b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done lemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl lemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * ↑(a / b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done theorem gcd_lin_comb : ∀ (b a : Nat), (gcd_c1 a b) * ↑a + (gcd_c2 a b) * ↑b = ↑(gcd a b) := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd_c1 a b_1) * ↑a + (gcd_c2 a b_1) * ↑b_1 = ↑(gcd a b_1) fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base] --Goal : 1 * ↑a + 0 * ↑0 = ↑a ring done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1] --Goal : gcd_c2 b (a % b) * ↑a + -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * ↑(a / b)) * ↑b = -- ↑(gcd b (a % b)) set r : Nat := a % b set q : Nat := a / b set s : Int := gcd_c1 b r set t : Int := gcd_c2 b r --Goal : t * ↑a + (s - t * ↑q) * ↑b = ↑(gcd b r) have h2 : r < b := mod_nonzero_lt a h1 have h3 : s * ↑b + t * ↑r = ↑(gcd b r) := ih r h2 b have h4 : b * q + r = a := Nat.div_add_mod a b rewrite [←h3, ←h4] rewrite [Nat.cast_add, Nat.cast_mul] --Goal : t * (↑b * ↑q + ↑r) + (s - t * ↑q) * ↑b = s * ↑b + t * ↑r ring done done #eval gcd_c1 672 161 --Answer: 6 #eval gcd_c2 672 161 --Answer: -25 --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161 theorem Theorem_7_1_6 {d a b : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ gcd a b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑d ∣ ↑(gcd a b) set s : Int := gcd_c1 a b set t : Int := gcd_c2 a b have h3 : s * ↑a + t * ↑b = ↑(gcd a b) := gcd_lin_comb b a rewrite [←h3] --Goal : ↑d ∣ s * ↑a + t * ↑b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul] --Goal : ↑d ∣ s * (↑d * ↑j) + t * (↑d * ↑k) define apply Exists.intro (s * ↑j + t * ↑k) ring done /- Section 7.2 -/ theorem dvd_trans {a b c : Nat} (h1 : a ∣ b) (h2 : b ∣ c) : a ∣ c := by define at h1; define at h2; define obtain (m : Nat) (h3 : b = a * m) from h1 obtain (n : Nat) (h4 : c = b * n) from h2 rewrite [h3, mul_assoc] at h4 apply Exists.intro (m * n) show c = a * (m * n) from h4 done lemma exists_prime_factor : ∀ (n : Nat), 2 ≤ n → ∃ (p : Nat), prime_factor p n := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, 2 ≤ n_1 → ∃ (p : Nat), prime_factor p n_1 assume h1 : 2 ≤ n by_cases h2 : prime n · -- Case 1. h2 : prime n apply Exists.intro n define --Goal : prime n ∧ n ∣ n show prime n ∧ n ∣ n from And.intro h2 (dvd_self n) done · -- Case 2. h2 : ¬prime n define at h2 --h2 : ¬(2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n) demorgan at h2 disj_syll h2 h1 obtain (a : Nat) (h3 : ∃ (b : Nat), a * b = n ∧ a < n ∧ b < n) from h2 obtain (b : Nat) (h4 : a * b = n ∧ a < n ∧ b < n) from h3 have h5 : 2 ≤ a := by by_contra h6 have h7 : a ≤ 1 := by linarith have h8 : n ≤ b := calc n _ = a * b := h4.left.symm _ ≤ 1 * b := by rel [h7] _ = b := by ring linarith --n ≤ b contradicts b < n done have h6 : ∃ (p : Nat), prime_factor p a := ih a h4.right.left h5 obtain (p : Nat) (h7 : prime_factor p a) from h6 apply Exists.intro p define --Goal : prime p ∧ p ∣ n define at h7 --h7 : prime p ∧ p ∣ a apply And.intro h7.left have h8 : a ∣ n := by apply Exists.intro b show n = a * b from (h4.left).symm done show p ∣ n from dvd_trans h7.right h8 done done lemma exists_least_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q := by set S : Set Nat := {p : Nat | prime_factor p n} have h2 : ∃ (p : Nat), p ∈ S := exists_prime_factor n h show ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q from well_ord_princ S h2 done lemma all_prime_nil : all_prime [] := by define --Goal : ∀ p ∈ [], prime p fix p : Nat contrapos --Goal : ¬prime p → p ∉ [] assume h1 : ¬prime p show p ∉ [] from List.not_mem_nil p done lemma all_prime_cons (n : Nat) (L : List Nat) : all_prime (n :: L) ↔ prime n ∧ all_prime L := by apply Iff.intro · -- (→) assume h1 : all_prime (n :: L) --Goal : prime n ∧ all_prime L define at h1 --h1 : ∀ p ∈ n :: L, prime p apply And.intro (h1 n (List.mem_cons_self n L)) define --Goal : ∀ p ∈ L, prime p fix p : Nat assume h2 : p ∈ L show prime p from h1 p (List.mem_cons_of_mem n h2) done · -- (←) assume h1 : prime n ∧ all_prime L --Goal : all_prime (n :: l) define : all_prime L at h1 define fix p : Nat assume h2 : p ∈ n :: L rewrite [List.mem_cons] at h2 --h2 : p = n ∨ p ∈ L by_cases on h2 · -- Case 1. h2 : p = n rewrite [h2] show prime n from h1.left done · -- Case 2. h2 : p ∈ L show prime p from h1.right p h2 done done done lemma nondec_nil : nondec [] := by define --Goal : True trivial --trivial proves some obviously true statements, such as True done lemma nondec_cons (n : Nat) (L : List Nat) : nondec (n :: L) ↔ (∀ m ∈ L, n ≤ m) ∧ nondec L := by rfl lemma prod_nil : prod [] = 1 := by rfl lemma prod_cons : prod (n :: L) = n * (prod L) := by rfl lemma exists_cons_of_length_eq_succ {A : Type} {l : List A} {n : Nat} (h : l.length = n + 1) : ∃ (a : A) (L : List A), l = a :: L ∧ L.length = n := by have h1 : ¬l.length = 0 := by linarith rewrite [List.length_eq_zero] at h1 obtain (a : A) (h2 : ∃ (L : List A), l = a :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List A) (h3 : l = a :: L) from h2 apply Exists.intro a apply Exists.intro L apply And.intro h3 have h4 : (a :: L).length = L.length + 1 := List.length_cons a L rewrite [←h3, h] at h4 show L.length = n from (Nat.add_right_cancel h4).symm done lemma list_elt_dvd_prod_by_length (a : Nat) : ∀ (n : Nat), ∀ (l : List Nat), l.length = n → a ∈ l → a ∣ prod l := by by_induc · --Base Case fix l : List Nat assume h1 : l.length = 0 rewrite [List.length_eq_zero] at h1 --h1 : l = [] rewrite [h1] --Goal : a ∈ [] → a ∣ prod [] contrapos assume h2 : ¬a ∣ prod [] show a ∉ [] from List.not_mem_nil a done · -- Induction Step fix n : Nat assume ih : ∀ (l : List Nat), List.length l = n → a ∈ l → a ∣ prod l fix l : List Nat assume h1 : l.length = n + 1 --Goal : a ∈ l → a ∣ prod l obtain (b : Nat) (h2 : ∃ (L : List Nat), l = b :: L ∧ L.length = n) from exists_cons_of_length_eq_succ h1 obtain (L : List Nat) (h3 : l = b :: L ∧ L.length = n) from h2 have h4 : a ∈ L → a ∣ prod L := ih L h3.right assume h5 : a ∈ l rewrite [h3.left, prod_cons] --Goal : a ∣ b * prod L rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b ∨ a ∈ L by_cases on h5 · -- Case 1. h5 : a = b apply Exists.intro (prod L) rewrite [h5] rfl done · -- Case 2. h5 : a ∈ L have h6 : a ∣ prod L := h4 h5 have h7 : prod L ∣ b * prod L := by apply Exists.intro b ring done show a ∣ b * prod L from dvd_trans h6 h7 done done done lemma list_elt_dvd_prod {a : Nat} {l : List Nat} (h : a ∈ l) : a ∣ prod l := by set n : Nat := l.length have h1 : l.length = n := by rfl show a ∣ prod l from list_elt_dvd_prod_by_length a n l h1 h done lemma exists_prime_factorization : ∀ (n : Nat), n ≥ 1 → ∃ (l : List Nat), prime_factorization n l := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, n_1 ≥ 1 → ∃ (l : List Nat), prime_factorization n_1 l assume h1 : n ≥ 1 by_cases h2 : n = 1 · -- Case 1. h2 : n = 1 apply Exists.intro [] define apply And.intro · -- Proof of nondec_prime_list [] define show all_prime [] ∧ nondec [] from And.intro all_prime_nil nondec_nil done · -- Proof of prod [] = n rewrite [prod_nil, h2] rfl done done · -- Case 2. h2 : n ≠ 1 have h3 : n ≥ 2 := lt_of_le_of_ne' h1 h2 obtain (p : Nat) (h4 : prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q) from exists_least_prime_factor h3 have p_prime_factor : prime_factor p n := h4.left define at p_prime_factor have p_prime : prime p := p_prime_factor.left have p_dvd_n : p ∣ n := p_prime_factor.right have p_least : ∀ (q : Nat), prime_factor q n → p ≤ q := h4.right obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n have h5 : m ≠ 0 := by contradict h1 with h6 have h7 : n = 0 := calc n _ = p * m := n_eq_pm _ = p * 0 := by rw [h6] _ = 0 := by ring rewrite [h7] decide done have m_pos : 0 < m := Nat.pos_of_ne_zero h5 have m_lt_n : m < n := by define at p_prime show m < n from calc m _ < m + m := by linarith _ = 2 * m := by ring _ ≤ p * m := by rel [p_prime.left] _ = n := n_eq_pm.symm done obtain (L : List Nat) (h6 : prime_factorization m L) from ih m m_lt_n m_pos define at h6 have ndpl_L : nondec_prime_list L := h6.left define at ndpl_L apply Exists.intro (p :: L) define apply And.intro · -- Proof of nondec_prime_list (p :: L) define apply And.intro · -- Proof of all_prime (p :: L) rewrite [all_prime_cons] show prime p ∧ all_prime L from And.intro p_prime ndpl_L.left done · -- Proof of nondec (p :: L) rewrite [nondec_cons] apply And.intro _ ndpl_L.right fix q : Nat assume q_in_L : q ∈ L have h7 : q ∣ prod L := list_elt_dvd_prod q_in_L rewrite [h6.right] at h7 --h7 : q ∣ m have h8 : m ∣ n := by apply Exists.intro p rewrite [n_eq_pm] ring done have q_dvd_n : q ∣ n := dvd_trans h7 h8 have ap_L : all_prime L := ndpl_L.left define at ap_L have q_prime_factor : prime_factor q n := And.intro (ap_L q q_in_L) q_dvd_n show p ≤ q from p_least q q_prime_factor done done · -- Proof of prod (p :: L) = n rewrite [prod_cons, h6.right, n_eq_pm] rfl done done done theorem Theorem_7_2_2 {a b c : Nat} (h1 : c ∣ a * b) (h2 : rel_prime a c) : c ∣ b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑c ∣ ↑b define at h1; define at h2; define obtain (j : Nat) (h3 : a * b = c * j) from h1 set s : Int := gcd_c1 a c set t : Int := gcd_c2 a c have h4 : s * ↑a + t * ↑c = ↑(gcd a c) := gcd_lin_comb c a rewrite [h2, Nat.cast_one] at h4 --h4 : s * ↑a + t * ↑c = (1 : Int) apply Exists.intro (s * ↑j + t * ↑b) show ↑b = ↑c * (s * ↑j + t * ↑b) from calc ↑b _ = (1 : Int) * ↑b := (one_mul _).symm _ = (s * ↑a + t * ↑c) * ↑b := by rw [h4] _ = s * (↑a * ↑b) + t * ↑c * ↑b := by ring _ = s * (↑c * ↑j) + t * ↑c * ↑b := by rw [←Nat.cast_mul a b, h3, Nat.cast_mul c j] _ = ↑c * (s * ↑j + t * ↑b) := by ring done lemma le_nonzero_prod_left {a b : Nat} (h : a * b ≠ 0) : a ≤ a * b := by have h1 : b ≠ 0 := by contradict h with h1 rewrite [h1] ring done have h2 : 1 ≤ b := Nat.pos_of_ne_zero h1 show a ≤ a * b from calc a = a * 1 := (mul_one a).symm _ ≤ a * b := by rel [h2] done lemma le_nonzero_prod_right {a b : Nat} (h : a * b ≠ 0) : b ≤ a * b := by rewrite [mul_comm] rewrite [mul_comm] at h show b ≤ b * a from le_nonzero_prod_left h done lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry lemma rel_prime_of_prime_not_dvd {a p : Nat} (h1 : prime p) (h2 : ¬p ∣ a) : rel_prime a p := by have h3 : gcd a p ∣ a := gcd_dvd_left a p have h4 : gcd a p ∣ p := gcd_dvd_right a p have h5 : gcd a p = 1 ∨ gcd a p = p := dvd_prime h1 h4 have h6 : gcd a p ≠ p := by contradict h2 with h6 rewrite [h6] at h3 show p ∣ a from h3 done disj_syll h5 h6 show rel_prime a p from h5 done theorem Theorem_7_2_3 {a b p : Nat} (h1 : prime p) (h2 : p ∣ a * b) : p ∣ a ∨ p ∣ b := by or_right with h3 have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3 show p ∣ b from Theorem_7_2_2 h2 h4 done lemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a ≥ 1 := by have h1 : a ≠ 0 := by by_contra h1 rewrite [h1] at h contradict h linarith done show a ≥ 1 from Nat.pos_of_ne_zero h1 done lemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by have h1 : a ≥ 1 := ge_one_of_prod_one h have h2 : a * b ≠ 0 := by linarith have h3 : a ≤ a * b := le_nonzero_prod_left h2 rewrite [h] at h3 show a = 1 from Nat.le_antisymm h3 h1 done lemma eq_one_of_dvd_one {n : Nat} (h : n ∣ 1) : n = 1 := by obtain (j : Nat) (h1 : 1 = n * j) from h show n = 1 from eq_one_of_prod_one h1.symm done lemma prime_not_one {p : Nat} (h : prime p) : p ≠ 1 := by define at h linarith done theorem Theorem_7_2_4 {p : Nat} (h1 : prime p) : ∀ (l : List Nat), p ∣ prod l → ∃ a ∈ l, p ∣ a := by apply List.rec · -- Base Case. Goal : p ∣ prod [] → ∃ a ∈ [], p ∣ a rewrite [prod_nil] assume h2 : p ∣ 1 show ∃ a ∈ [], p ∣ a from absurd (eq_one_of_dvd_one h2) (prime_not_one h1) done · -- Induction Step fix b : Nat fix L : List Nat assume ih : p ∣ prod L → ∃ a ∈ L, p ∣ a --Goal : p ∣ prod (b :: L) → ∃ a ∈ b :: L, p ∣ a assume h2 : p ∣ prod (b :: L) rewrite [prod_cons] at h2 have h3 : p ∣ b ∨ p ∣ prod L := Theorem_7_2_3 h1 h2 by_cases on h3 · -- Case 1. h3 : p ∣ b apply Exists.intro b show b ∈ b :: L ∧ p ∣ b from And.intro (List.mem_cons_self b L) h3 done · -- Case 2. h3 : p ∣ prod L obtain (a : Nat) (h4 : a ∈ L ∧ p ∣ a) from ih h3 apply Exists.intro a show a ∈ b :: L ∧ p ∣ a from And.intro (List.mem_cons_of_mem b h4.left) h4.right done done done lemma prime_in_list {p : Nat} {l : List Nat} (h1 : prime p) (h2 : all_prime l) (h3 : p ∣ prod l) : p ∈ l := by obtain (a : Nat) (h4 : a ∈ l ∧ p ∣ a) from Theorem_7_2_4 h1 l h3 define at h2 have h5 : prime a := h2 a h4.left have h6 : p = 1 ∨ p = a := dvd_prime h5 h4.right disj_syll h6 (prime_not_one h1) rewrite [h6] show a ∈ l from h4.left done lemma first_le_first {p q : Nat} {l m : List Nat} (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m)) (h3 : prod (p :: l) = prod (q :: m)) : p ≤ q := by define at h1; define at h2 have h4 : q ∣ prod (p :: l) := by define apply Exists.intro (prod m) rewrite [←prod_cons] show prod (p :: l) = prod (q :: m) from h3 done have h5 : all_prime (q :: m) := h2.left rewrite [all_prime_cons] at h5 have h6 : q ∈ p :: l := prime_in_list h5.left h1.left h4 have h7 : nondec (p :: l) := h1.right rewrite [nondec_cons] at h7 rewrite [List.mem_cons] at h6 by_cases on h6 · -- Case 1. h6 : q = p linarith done · -- Case 2. h6 : q ∈ l have h8 : ∀ m ∈ l, p ≤ m := h7.left show p ≤ q from h8 q h6 done done lemma nondec_prime_list_tail {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by define at h define rewrite [all_prime_cons, nondec_cons] at h show all_prime l ∧ nondec l from And.intro h.left.right h.right.right done lemma cons_prod_not_one {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : prod (p :: l) ≠ 1 := by define at h have h1 : all_prime (p :: l) := h.left rewrite [all_prime_cons] at h1 rewrite [prod_cons] by_contra h2 show False from (prime_not_one h1.left) (eq_one_of_prod_one h2) done lemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) : l = [] ↔ prod l = 1 := by apply Iff.intro · -- (→) assume h1 : l = [] rewrite [h1] show prod [] = 1 from prod_nil done · -- (←) contrapos assume h1 : ¬l = [] obtain (p : Nat) (h2 : ∃ (L : List Nat), l = p :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List Nat) (h3 : l = p :: L) from h2 rewrite [h3] at h rewrite [h3] show ¬prod (p :: L) = 1 from cons_prod_not_one h done done lemma prime_pos {p : Nat} (h : prime p) : p > 0 := by define at h linarith done theorem Theorem_7_2_5 : ∀ (l1 l2 : List Nat), nondec_prime_list l1 → nondec_prime_list l2 → prod l1 = prod l2 → l1 = l2 := by apply List.rec · -- Base Case. Goal : ∀ (l2 : List Nat), nondec_prime_list [] → -- nondec_prime_list l2 → prod [] = prod l2 → [] = l2 fix l2 : List Nat assume h1 : nondec_prime_list [] assume h2 : nondec_prime_list l2 assume h3 : prod [] = prod l2 rewrite [prod_nil, eq_comm, ←list_nil_iff_prod_one h2] at h3 show [] = l2 from h3.symm done · -- Induction Step fix p : Nat fix L1 : List Nat assume ih : ∀ (L2 : List Nat), nondec_prime_list L1 → nondec_prime_list L2 → prod L1 = prod L2 → L1 = L2 -- Goal : ∀ (l2 : List Nat), nondec_prime_list (p :: L1) → -- nondec_prime_list l2 → prod (p :: L1) = prod l2 → p :: L1 = l2 fix l2 : List Nat assume h1 : nondec_prime_list (p :: L1) assume h2 : nondec_prime_list l2 assume h3 : prod (p :: L1) = prod l2 have h4 : ¬prod (p :: L1) = 1 := cons_prod_not_one h1 rewrite [h3, ←list_nil_iff_prod_one h2] at h4 obtain (q : Nat) (h5 : ∃ (L : List Nat), l2 = q :: L) from List.exists_cons_of_ne_nil h4 obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5 rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2) rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2) have h7 : p ≤ q := first_le_first h1 h2 h3 have h8 : q ≤ p := first_le_first h2 h1 h3.symm have h9 : p = q := by linarith rewrite [h9, prod_cons, prod_cons] at h3 --h3 : q * prod L1 = q * prod L2 have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1 have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2 define at h2 have h12 : all_prime (q :: L2) := h2.left rewrite [all_prime_cons] at h12 have h13 : q > 0 := prime_pos h12.left have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3 have h15 : L1 = L2 := ih L2 h10 h11 h14 rewrite [h6, h9, h15] rfl done done theorem fund_thm_arith (n : Nat) (h : n ≥ 1) : ∃! (l : List Nat), prime_factorization n l := by exists_unique · -- Existence show ∃ (l : List Nat), prime_factorization n l from exists_prime_factorization n h done · -- Uniqueness fix l1 : List Nat; fix l2 : List Nat assume h1 : prime_factorization n l1 assume h2 : prime_factorization n l2 define at h1; define at h2 have h3 : prod l1 = n := h1.right rewrite [←h2.right] at h3 show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3 done done /- Section 7.3 -/ theorem congr_refl (m : Nat) : ∀ (a : Int), a ≡ a (MOD m) := by fix a : Int define --Goal : ∃ (c : Int), a - a = ↑m * c apply Exists.intro 0 ring done theorem congr_symm {m : Nat} : ∀ {a b : Int}, a ≡ b (MOD m) → b ≡ a (MOD m) := by fix a : Int; fix b : Int assume h1 : a ≡ b (MOD m) define at h1 --h1 : ∃ (c : Int), a - b = ↑m * c define --Goal : ∃ (c : Int), b - a = ↑m * c obtain (c : Int) (h2 : a - b = m * c) from h1 apply Exists.intro (-c) show b - a = m * (-c) from calc b - a _ = -(a - b) := by ring _ = -(m * c) := by rw [h2] _ = m * (-c) := by ring done theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry /- Fundamental properties of congruence classes -/ lemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) : X = Y ↔ X.val = Y.val := Fin.ext_iff lemma val_nat_eq_mod (n k : Nat) : ([k]_(n + 1)).val = k % (n + 1) := by rfl lemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl theorem cc_rep {m : Nat} (X : ZMod m) : ∃ (a : Int), X = [a]_m := match m with | 0 => by apply Exists.intro X rfl done | n + 1 => by apply Exists.intro ↑(X.val) have h1 : X.val < n + 1 := Fin.prop X rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1] rfl done theorem add_class (m : Nat) (a b : Int) : [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm theorem mul_class (m : Nat) (a b : Int) : [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm lemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ [a - b]_m = [0]_m := by apply Iff.intro · -- (→) assume h1 : [a]_m = [b]_m have h2 : a - b = a + (-b) := by ring have h3 : b + (-b) = 0 := by ring show [a - b]_m = [0]_m from calc [a - b]_m _ = [a + (-b)]_m := by rw [h2] _ = [a]_m + [-b]_m := by rw [add_class] _ = [b]_m + [-b]_m := by rw [h1] _ = [b + -b]_m := by rw [add_class] _ = [0]_m := by rw [h3] done · -- (←) assume h1 : [a - b]_m = [0]_m have h2 : b + (a - b) = a := by ring have h3 : b + 0 = b := by ring show [a]_m = [b]_m from calc [a]_m _ = [b + (a - b)]_m := by rw [h2] _ = [b]_m + [a - b]_m := by rw [add_class] _ = [b]_m + [0]_m := by rw [h1] _ = [b + 0]_m := by rw [add_class] _ = [b]_m := by rw [h3] done done lemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) : [a]_m = [0]_m → [-a]_m = [0]_m := by assume h1 : [a]_m = [0]_m have h2 : 0 + (-a) = -a := by ring have h3 : a + (-a) = 0 := by ring show [-a]_m = [0]_m from calc [-a]_m _ = [0 + (-a)]_m := by rw [h2] _ = [0]_m + [-a]_m := by rw [add_class] _ = [a]_m + [-a]_m := by rw [h1] _ = [a + (-a)]_m := by rw [add_class] _ = [0]_m := by rw [h3] done lemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) : [-a]_m = [0]_m ↔ [a]_m = [0]_m := by apply Iff.intro _ (cc_neg_zero_of_cc_zero m a) assume h1 : [-a]_m = [0]_m have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1 have h3 : -(-a) = a := by ring rewrite [h3] at h2 show [a]_m = [0]_m from h2 done lemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl lemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m ↔ m ∣ k := match m with | 0 => by have h : (0 : Int) = (↑(0 : Nat) : Int) := by rfl rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj] apply Iff.intro · -- (→) assume h1 : k = 0 rewrite [h1] show 0 ∣ 0 from dvd_self 0 done · -- (←) assume h1 : 0 ∣ k obtain (c : Nat) (h2 : k = 0 * c) from h1 rewrite [h2] ring done done | n + 1 => by rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero] show k % (n + 1) = 0 ↔ n + 1 ∣ k from (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm done lemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m ↔ ↑m ∣ a := by obtain (k : Nat) (h1 : a = ↑k ∨ a = -↑k) from Int.eq_nat_or_neg a by_cases on h1 · -- Case 1. h1: a = ↑k rewrite [h1, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done · -- Case 2. h1: a = -↑k rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done done theorem cc_eq_iff_congr (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ a ≡ b (MOD m) := calc [a]_m = [b]_m _ ↔ [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b _ ↔ ↑m ∣ (a - b) := cc_zero_iff_dvd m (a - b) _ ↔ a ≡ b (MOD m) := by rfl /- End of fundamental properties of congruence classes -/ lemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m := by have h1 : (↑m : Int) ≠ 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m) show 0 ≤ a % m from Int.emod_nonneg a h1 done lemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m) have h2 : (↑m : Int) > 0 := (Nat.cast_pos).rtl h1 show a % m < m from Int.emod_lt_of_pos a h2 done lemma congr_mod_mod (m : Nat) (a : Int) : a ≡ a % m (MOD m) := by define have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m apply Exists.intro (a / m) show a - a % m = m * (a / m) from calc a - (a % m) _ = m * (a / m) + a % m - a % m := by rw [h1] _ = m * (a / m) := by ring done lemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) := And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a)) theorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) : ∃! (r : Int), 0 ≤ r ∧ r < m ∧ a ≡ r (MOD m) := by exists_unique · -- Existence apply Exists.intro (a % m) show 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) from mod_cmpl_res m a done · -- Uniqueness fix r1 : Int; fix r2 : Int assume h1 : 0 ≤ r1 ∧ r1 < m ∧ a ≡ r1 (MOD m) assume h2 : 0 ≤ r2 ∧ r2 < m ∧ a ≡ r2 (MOD m) have h3 : r1 ≡ r2 (MOD m) := congr_trans (congr_symm h1.right.right) h2.right.right obtain (d : Int) (h4 : r1 - r2 = m * d) from h3 have h5 : r1 - r2 < m * 1 := by linarith have h6 : m * (-1) < r1 - r2 := by linarith rewrite [h4] at h5 --h5 : m * d < m * 1 rewrite [h4] at h6 --h6 : m * -1 < m * d have h7 : (↑m : Int) ≥ 0 := Nat.cast_nonneg m have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7 have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7 have h10 : d = 0 := by linarith show r1 = r2 from calc r1 _ = r1 - r2 + r2 := by ring _ = m * 0 + r2 := by rw [h4, h10] _ = r2 := by ring done done lemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m := (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a) theorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y rewrite [h1, h2] have h3 : a + b = b + a := by ring show [a]_m + [b]_m = [b]_m + [a]_m from calc [a]_m + [b]_m _ = [a + b]_m := add_class m a b _ = [b + a]_m := by rw [h3] _ = [b]_m + [a]_m := (add_class m b a).symm done theorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X rewrite [h1] have h2 : a * 1 = a := by ring show [a]_m * [1]_m = [a]_m from calc [a]_m * [1]_m _ = [a * 1]_m := mul_class m a 1 _ = [a]_m := by rw [h2] done theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry lemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) : [a]_m * [gcd_c2 m a]_m = [1]_m := by set s : Int := gcd_c1 m a have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m define at h1 rewrite [h1, Nat.cast_one] at h2 --h2 : s * ↑m + gcd_c2 m a * ↑a = 1 rewrite [mul_class, cc_eq_iff_congr] define --Goal : ∃ (c : Int), ↑a * gcd_c2 m a - 1 = ↑m * c apply Exists.intro (-s) show a * (gcd_c2 m a) - 1 = m * (-s) from calc a * (gcd_c2 m a) - 1 _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring _ = 1 + m * (-s) - 1 := by rw [h2] _ = m * (-s) := by ring done theorem Theorem_7_3_7 (m a : Nat) : invertible [a]_m ↔ rel_prime m a := by apply Iff.intro · -- (→) assume h1 : invertible [a]_m define at h1 obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1 obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y rewrite [h3, mul_class, cc_eq_iff_congr] at h2 define at h2 obtain (c : Int) (h4 : a * b - 1 = m * c) from h2 rewrite [Exercise_7_2_6] --Goal : ∃ (s t : Int), s * ↑m + t * ↑a = 1 apply Exists.intro (-c) apply Exists.intro b show (-c) * m + b * a = 1 from calc (-c) * m + b * a _ = (-c) * m + (a * b - 1) + 1 := by ring _ = (-c) * m + m * c + 1 := by rw [h4] _ = 1 := by ring done · -- (←) assume h1 : rel_prime m a define show ∃ (Y : ZMod m), [a]_m * Y = [1]_m from Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1) done done /- Section 7.4 -/ section Euler open Euler lemma num_rp_below_base {m : Nat} : num_rp_below m 0 = 0 := by rfl lemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) : num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by have h1 : num_rp_below m (j + 1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : gcd m j = 1 rewrite [if_pos h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1 show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1 done lemma num_rp_below_step_not_rp {m j : Nat} (h : ¬rel_prime m j) : num_rp_below m (j + 1) = num_rp_below m j := by have h1 : num_rp_below m (j +1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : ¬gcd m j = 1 rewrite [if_neg h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j show num_rp_below m (j + 1) = num_rp_below m j from h1 done lemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl #eval phi 10 --Answer: 4 lemma prod_inv_iff_inv {m : Nat} {X : ZMod m} (h1 : invertible X) (Y : ZMod m) : invertible (X * Y) ↔ invertible Y := by apply Iff.intro · -- (→) assume h2 : invertible (X * Y) obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2 apply Exists.intro (X * Z) rewrite [←h3] --Goal : Y * (X * Z) = X * Y * Z ring --Note that ring can do algebra in ZMod m done · -- (←) assume h2 : invertible Y obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1 obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2 apply Exists.intro (Xi * Yi) show (X * Y) * (Xi * Yi) = [1]_m from calc X * Y * (Xi * Yi) _ = (X * Xi) * (Y * Yi) := by ring _ = [1]_m * [1]_m := by rw [h3, h4] _ = [1]_m := Theorem_7_3_6_7 [1]_m done done lemma F_rp_def {m i : Nat} (h : rel_prime m i) : F m i = [i]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h --h : gcd m i = 1 rewrite [if_pos h] at h1 show F m i = [i]_m from h1 done lemma F_not_rp_def {m i : Nat} (h : ¬rel_prime m i) : F m i = [1]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h rewrite [h1, if_neg h] rfl done lemma prod_seq_base {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 0 k f = [1]_m := by rfl lemma prod_seq_step {m : Nat} (n k : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl lemma prod_seq_zero_step {m : Nat} (n : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by rewrite [prod_seq_step, zero_add] rfl done lemma prod_one {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 1 k f = f k := by rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7] rfl done lemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl lemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m := calc [G m a i]_m _ = [(a * i) % m]_m := by rfl _ = [a * i]_m := (cc_eq_mod m (a * i)).symm _ = [a]_m * [i]_m := (mul_class m a i).symm lemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) : rel_prime m (G m a i) ↔ rel_prime m i := by have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1 show rel_prime m (G m a i) ↔ rel_prime m i from calc rel_prime m (G m a i) _ ↔ invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm _ ↔ invertible ([a]_m * [i]_m) := by rw [cc_G] _ ↔ invertible [i]_m := prod_inv_iff_inv h2 ([i]_m) _ ↔ rel_prime m i := Theorem_7_3_7 m i done lemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) : F m (G m a i) = [a]_m * F m i := by have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2 show F m (G m a i) = [a]_m * F m i from calc F m (G m a i) _ = [G m a i]_m := F_rp_def h3 _ = [a]_m * [i]_m := cc_G m a i _ = [a]_m * F m i := by rw [F_rp_def h2] done lemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : ¬rel_prime m i) : F m (G m a i) = [1]_m := by rewrite [←G_rp_iff h1 i] at h2 show F m (G m a i) = [1]_m from F_not_rp_def h2 done lemma FG_prod {m a : Nat} (h1 : rel_prime m a) : ∀ (k : Nat), prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by by_induc · -- Base Case show prod_seq 0 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from calc prod_seq 0 0 ((F m) ∘ (G m a)) _ = [1]_m := prod_seq_base _ _ _ = [a]_m ^ 0 * [1]_m := by ring _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by rw [num_rp_below_base, prod_seq_base] done · -- Induction Step fix k : Nat assume ih : prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) by_cases h2 : rel_prime m k · -- Case 1. h2 : rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([a]_m * F m k) := by rw [FG_rp h1 h2] _ = [a]_m ^ ((num_rp_below m k) + 1) * ((prod_seq k 0 (F m)) * F m k) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_rp h2, prod_seq_zero_step] done · -- Case 2. h2 : ¬rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([1]_m) := by rw [FG_not_rp h1 h2] _ = [a]_m ^ (num_rp_below m k) * (prod_seq k 0 (F m) * ([1]_m)) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_not_rp h2, prod_seq_zero_step, F_not_rp_def h2] done done done lemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by define --Goal : ∀ i < m, G m a i < m fix i : Nat assume h1 : i < m rewrite [G_def] --Goal : a * i % m < m show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m) done lemma left_inv_one_one_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g' (g i) = i) : one_one_below n g := sorry lemma right_inv_onto_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g (g' i) = i) (h2 : maps_below n g') : onto_below n g := by define at h2; define fix k : Nat assume h3 : k < n apply Exists.intro (g' k) show g' k < n ∧ g (g' k) = k from And.intro (h2 k h3) (h1 k h3) done lemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m * [inv_mod m a]_m = [1]_m := by have h2 : 0 ≤ (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a) show [a]_m * [inv_mod m a]_m = [1]_m from calc [a]_m * [inv_mod m a]_m _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2] _ = [a]_m * [gcd_c2 m a]_m := by rw [←cc_eq_mod] _ = [1]_m := gcd_c2_inv h1 done lemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m := calc a * (b % m) % m = a % m * (b % m % m) % m := Nat.mul_mod _ _ _ _ = a % m * (b % m) % m := by rw [Nat.mod_mod] _ = a * b % m := (Nat.mul_mod _ _ _).symm lemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm] rfl done theorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] : ↑a ≡ ↑b (MOD m) ↔ a % m = b % m := sorry lemma mul_inv_mod_cancel {m a i : Nat} [NeZero m] (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1 rewrite [mul_class, cc_eq_iff_congr, ←Nat.cast_mul, ←Nat.cast_one, congr_iff_mod_eq_Nat] at h3 show a * inv_mod m a * i % m = i from calc a * (inv_mod m a) * i % m _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = 1 % m * i % m := by rw [h3] _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = i % m := by rw [one_mul] _ = i := Nat.mod_eq_of_lt h2 done lemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl lemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, G m a (Ginv m a i) = i := by fix i : Nat assume h2 : i < m show G m a (Ginv m a i) = i from calc G m a (Ginv m a i) _ = a * ((inv_mod m a * i) % m) % m := by rfl _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, Ginv m a (G m a i) = i := by fix i : Nat assume h2 : i < m show Ginv m a (G m a i) = i from calc Ginv m a (G m a i) _ = inv_mod m a * ((a * i) % m) % m := by rfl _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc, mul_comm (inv_mod m a)] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_maps_below (m a : Nat) [NeZero m] : maps_below m (Ginv m a) := G_maps_below m (inv_mod m a) lemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : one_one_below m (G m a) := left_inv_one_one_below (Ginv_left_inv h1) lemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : onto_below m (G m a) := right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a) lemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : perm_below m (G m a) := And.intro (G_maps_below m a) (And.intro (G_one_one_below h1) (G_onto_below h1)) --Permuting a product of congruence classes doesn't change product lemma swap_fst (u v : Nat) : swap u v u = v := by define : swap u v u --Goal : (if u = u then v else if u = v then u else u) = v have h : u = u := by rfl rewrite [if_pos h] rfl done lemma swap_snd (u v : Nat) : swap u v v = u := by define : swap u v v by_cases h1 : v = u · -- Case 1. h1 : v = u rewrite [if_pos h1] show v = u from h1 done · -- Case 2. h1 : v ≠ u rewrite [if_neg h1] have h2 : v = v := by rfl rewrite [if_pos h2] rfl done done lemma swap_other {u v i : Nat} (h1 : i ≠ u) (h2 : i ≠ v) : swap u v i = i := by define : swap u v i rewrite [if_neg h1, if_neg h2] rfl done lemma swap_values (u v i : Nat) : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := by by_cases h1 : i = u · -- Case 1. h1 : i = u apply Or.inl rewrite [h1] show swap u v u = v from swap_fst u v done · -- Case 2. h1 : i ≠ u apply Or.inr by_cases h2 : i = v · -- Case 2.1. h2 : i = v apply Or.inl rewrite [h2] show swap u v v = u from swap_snd u v done · -- Case 2.2. h2 : i ≠ v apply Or.inr show swap u v i = i from swap_other h1 h2 done done done lemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by define fix i : Nat assume h3 : i < n have h4 : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := swap_values u v i by_cases on h4 · -- Case 1. h4 : swap u v i = v rewrite [h4] show v < n from h2 done · -- Case 2. by_cases on h4 · -- Case 2.1. h4 : swap u v i = u rewrite [h4] show u < n from h1 done · -- Case 2.2. h4 : swap u v i = i rewrite [h4] show i < n from h3 done done done lemma swap_swap (u v n : Nat) : ∀ i < n, swap u v (swap u v i) = i := by fix i : Nat assume h : i < n by_cases h1 : i = u · -- Case 1. h1 : i = u rewrite [h1, swap_fst, swap_snd] rfl done · -- Case 2. h1 : i ≠ u by_cases h2 : i = v · -- Case 2.1. h2 : i = v rewrite [h2, swap_snd, swap_fst] rfl done · -- Case 2.2. h2 : i ≠ v rewrite [swap_other h1 h2, swap_other h1 h2] rfl done done done lemma swap_one_one_below (u v n) : one_one_below n (swap u v) := left_inv_one_one_below (swap_swap u v n) lemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) := right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2) lemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) := And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2))

lemma comp_perm_below {n : Nat} {f g : Nat → Nat} (h1 : perm_below n f) (h2 : perm_below n g) : perm_below n (f ∘ g)

HTPI.comp_perm_below

htpi/HTPILib/Chap7.lean

htpi

/- Copyright 2023 Daniel J. Velleman -/ import HTPILib.Chap6 namespace HTPI /- Definitions -/ lemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by have h : n + 1 > 0 := Nat.succ_pos n show a % (n + 1) < n + 1 from Nat.mod_lt a h done def gcd (a b : Nat) : Nat := match b with | 0 => a | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd (n + 1) (a % (n + 1)) termination_by b mutual def gcd_c1 (a b : Nat) : Int := match b with | 0 => 1 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c2 (n + 1) (a % (n + 1)) --Corresponds to s = t' termination_by b def gcd_c2 (a b : Nat) : Int := match b with | 0 => 0 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c1 (n + 1) (a % (n + 1)) - (gcd_c2 (n + 1) (a % (n + 1))) * ↑(a / (n + 1)) --Corresponds to t = s' - t'q termination_by b end def prime (n : Nat) : Prop := 2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n def prime_factor (p n : Nat) : Prop := prime p ∧ p ∣ n def all_prime (l : List Nat) : Prop := ∀ p ∈ l, prime p def nondec (l : List Nat) : Prop := match l with | [] => True --Of course, True is a proposition that is always true | n :: L => (∀ m ∈ L, n ≤ m) ∧ nondec L def nondec_prime_list (l : List Nat) : Prop := all_prime l ∧ nondec l def prod (l : List Nat) : Nat := match l with | [] => 1 | n :: L => n * (prod L) def prime_factorization (n : Nat) (l : List Nat) : Prop := nondec_prime_list l ∧ prod l = n def rel_prime (a b : Nat) : Prop := gcd a b = 1 def congr_mod (m : Nat) (a b : Int) : Prop := (↑m : Int) ∣ (a - b) def cc (m : Nat) (a : Int) : ZMod m := (↑a : ZMod m) notation:50 a " ≡ " b " (MOD " m ")" => congr_mod m a b notation:max "["a"]_"m:max => cc m a def invertible {m : Nat} (X : ZMod m) : Prop := ∃ (Y : ZMod m), X * Y = [1]_m def num_rp_below (m k : Nat) : Nat := match k with | 0 => 0 | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j def phi (m : Nat) : Nat := num_rp_below m m def prod_seq {m : Nat} (j k : Nat) (f : Nat → ZMod m) : ZMod m := match j with | 0 => [1]_m | n + 1 => prod_seq n k f * f (k + n) def maps_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i < n, g i < n def one_one_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i1 < n, ∀ i2 < n, g i1 = g i2 → i1 = i2 def onto_below (n : Nat) (g : Nat → Nat) : Prop := ∀ k < n, ∃ i < n, g i = k def perm_below (n : Nat) (g : Nat → Nat) : Prop := maps_below n g ∧ one_one_below n g ∧ onto_below n g def inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m) def swap (u v i : Nat) : Nat := if i = u then v else if i = v then u else i namespace Euler --For definitions specific to Euler's theorem def F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m def G (m a i : Nat) : Nat := (a * i) % m def Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i end Euler /- Section 7.1 -/ theorem dvd_mod_of_dvd_a_b {a b d : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ (a % b) := by set q : Nat := a / b have h3 : b * q + a % b = a := Nat.div_add_mod a b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 define --Goal : ∃ (c : Nat), a % b = d * c apply Exists.intro (j - k * q) show a % b = d * (j - k * q) from calc a % b _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm _ = a - b * q := by rw [h3] _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc] _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm done theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry #eval gcd 672 161 --Answer: 7 lemma gcd_base (a : Nat) : gcd a 0 = a := by rfl lemma gcd_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd a b = gcd b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1)) rfl done lemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b ≠ 0) : a % b < b := by have h1 : b > 0 := Nat.pos_of_ne_zero h show a % b < b from Nat.mod_lt a h1 done lemma dvd_self (n : Nat) : n ∣ n := by apply Exists.intro 1 ring done theorem gcd_dvd : ∀ (b a : Nat), (gcd a b) ∣ a ∧ (gcd a b) ∣ b := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd a b_1) ∣ a ∧ (gcd a b_1) ∣ b_1 fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_base] --Goal: a ∣ a ∧ a ∣ 0 apply And.intro (dvd_self a) define apply Exists.intro 0 rfl done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_nonzero a h1] --Goal : gcd b (a % b) ∣ a ∧ gcd b (a % b) ∣ b have h2 : a % b < b := mod_nonzero_lt a h1 have h3 : (gcd b (a % b)) ∣ b ∧ (gcd b (a % b)) ∣ (a % b) := ih (a % b) h2 b apply And.intro _ h3.left show (gcd b (a % b)) ∣ a from dvd_a_of_dvd_b_mod h3.left h3.right done done theorem gcd_dvd_left (a b : Nat) : (gcd a b) ∣ a := (gcd_dvd b a).left theorem gcd_dvd_right (a b : Nat) : (gcd a b) ∣ b := (gcd_dvd b a).right lemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl lemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c1 a b = gcd_c2 b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done lemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl lemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * ↑(a / b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done theorem gcd_lin_comb : ∀ (b a : Nat), (gcd_c1 a b) * ↑a + (gcd_c2 a b) * ↑b = ↑(gcd a b) := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd_c1 a b_1) * ↑a + (gcd_c2 a b_1) * ↑b_1 = ↑(gcd a b_1) fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base] --Goal : 1 * ↑a + 0 * ↑0 = ↑a ring done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1] --Goal : gcd_c2 b (a % b) * ↑a + -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * ↑(a / b)) * ↑b = -- ↑(gcd b (a % b)) set r : Nat := a % b set q : Nat := a / b set s : Int := gcd_c1 b r set t : Int := gcd_c2 b r --Goal : t * ↑a + (s - t * ↑q) * ↑b = ↑(gcd b r) have h2 : r < b := mod_nonzero_lt a h1 have h3 : s * ↑b + t * ↑r = ↑(gcd b r) := ih r h2 b have h4 : b * q + r = a := Nat.div_add_mod a b rewrite [←h3, ←h4] rewrite [Nat.cast_add, Nat.cast_mul] --Goal : t * (↑b * ↑q + ↑r) + (s - t * ↑q) * ↑b = s * ↑b + t * ↑r ring done done #eval gcd_c1 672 161 --Answer: 6 #eval gcd_c2 672 161 --Answer: -25 --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161 theorem Theorem_7_1_6 {d a b : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ gcd a b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑d ∣ ↑(gcd a b) set s : Int := gcd_c1 a b set t : Int := gcd_c2 a b have h3 : s * ↑a + t * ↑b = ↑(gcd a b) := gcd_lin_comb b a rewrite [←h3] --Goal : ↑d ∣ s * ↑a + t * ↑b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul] --Goal : ↑d ∣ s * (↑d * ↑j) + t * (↑d * ↑k) define apply Exists.intro (s * ↑j + t * ↑k) ring done /- Section 7.2 -/ theorem dvd_trans {a b c : Nat} (h1 : a ∣ b) (h2 : b ∣ c) : a ∣ c := by define at h1; define at h2; define obtain (m : Nat) (h3 : b = a * m) from h1 obtain (n : Nat) (h4 : c = b * n) from h2 rewrite [h3, mul_assoc] at h4 apply Exists.intro (m * n) show c = a * (m * n) from h4 done lemma exists_prime_factor : ∀ (n : Nat), 2 ≤ n → ∃ (p : Nat), prime_factor p n := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, 2 ≤ n_1 → ∃ (p : Nat), prime_factor p n_1 assume h1 : 2 ≤ n by_cases h2 : prime n · -- Case 1. h2 : prime n apply Exists.intro n define --Goal : prime n ∧ n ∣ n show prime n ∧ n ∣ n from And.intro h2 (dvd_self n) done · -- Case 2. h2 : ¬prime n define at h2 --h2 : ¬(2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n) demorgan at h2 disj_syll h2 h1 obtain (a : Nat) (h3 : ∃ (b : Nat), a * b = n ∧ a < n ∧ b < n) from h2 obtain (b : Nat) (h4 : a * b = n ∧ a < n ∧ b < n) from h3 have h5 : 2 ≤ a := by by_contra h6 have h7 : a ≤ 1 := by linarith have h8 : n ≤ b := calc n _ = a * b := h4.left.symm _ ≤ 1 * b := by rel [h7] _ = b := by ring linarith --n ≤ b contradicts b < n done have h6 : ∃ (p : Nat), prime_factor p a := ih a h4.right.left h5 obtain (p : Nat) (h7 : prime_factor p a) from h6 apply Exists.intro p define --Goal : prime p ∧ p ∣ n define at h7 --h7 : prime p ∧ p ∣ a apply And.intro h7.left have h8 : a ∣ n := by apply Exists.intro b show n = a * b from (h4.left).symm done show p ∣ n from dvd_trans h7.right h8 done done lemma exists_least_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q := by set S : Set Nat := {p : Nat | prime_factor p n} have h2 : ∃ (p : Nat), p ∈ S := exists_prime_factor n h show ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q from well_ord_princ S h2 done lemma all_prime_nil : all_prime [] := by define --Goal : ∀ p ∈ [], prime p fix p : Nat contrapos --Goal : ¬prime p → p ∉ [] assume h1 : ¬prime p show p ∉ [] from List.not_mem_nil p done lemma all_prime_cons (n : Nat) (L : List Nat) : all_prime (n :: L) ↔ prime n ∧ all_prime L := by apply Iff.intro · -- (→) assume h1 : all_prime (n :: L) --Goal : prime n ∧ all_prime L define at h1 --h1 : ∀ p ∈ n :: L, prime p apply And.intro (h1 n (List.mem_cons_self n L)) define --Goal : ∀ p ∈ L, prime p fix p : Nat assume h2 : p ∈ L show prime p from h1 p (List.mem_cons_of_mem n h2) done · -- (←) assume h1 : prime n ∧ all_prime L --Goal : all_prime (n :: l) define : all_prime L at h1 define fix p : Nat assume h2 : p ∈ n :: L rewrite [List.mem_cons] at h2 --h2 : p = n ∨ p ∈ L by_cases on h2 · -- Case 1. h2 : p = n rewrite [h2] show prime n from h1.left done · -- Case 2. h2 : p ∈ L show prime p from h1.right p h2 done done done lemma nondec_nil : nondec [] := by define --Goal : True trivial --trivial proves some obviously true statements, such as True done lemma nondec_cons (n : Nat) (L : List Nat) : nondec (n :: L) ↔ (∀ m ∈ L, n ≤ m) ∧ nondec L := by rfl lemma prod_nil : prod [] = 1 := by rfl lemma prod_cons : prod (n :: L) = n * (prod L) := by rfl lemma exists_cons_of_length_eq_succ {A : Type} {l : List A} {n : Nat} (h : l.length = n + 1) : ∃ (a : A) (L : List A), l = a :: L ∧ L.length = n := by have h1 : ¬l.length = 0 := by linarith rewrite [List.length_eq_zero] at h1 obtain (a : A) (h2 : ∃ (L : List A), l = a :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List A) (h3 : l = a :: L) from h2 apply Exists.intro a apply Exists.intro L apply And.intro h3 have h4 : (a :: L).length = L.length + 1 := List.length_cons a L rewrite [←h3, h] at h4 show L.length = n from (Nat.add_right_cancel h4).symm done lemma list_elt_dvd_prod_by_length (a : Nat) : ∀ (n : Nat), ∀ (l : List Nat), l.length = n → a ∈ l → a ∣ prod l := by by_induc · --Base Case fix l : List Nat assume h1 : l.length = 0 rewrite [List.length_eq_zero] at h1 --h1 : l = [] rewrite [h1] --Goal : a ∈ [] → a ∣ prod [] contrapos assume h2 : ¬a ∣ prod [] show a ∉ [] from List.not_mem_nil a done · -- Induction Step fix n : Nat assume ih : ∀ (l : List Nat), List.length l = n → a ∈ l → a ∣ prod l fix l : List Nat assume h1 : l.length = n + 1 --Goal : a ∈ l → a ∣ prod l obtain (b : Nat) (h2 : ∃ (L : List Nat), l = b :: L ∧ L.length = n) from exists_cons_of_length_eq_succ h1 obtain (L : List Nat) (h3 : l = b :: L ∧ L.length = n) from h2 have h4 : a ∈ L → a ∣ prod L := ih L h3.right assume h5 : a ∈ l rewrite [h3.left, prod_cons] --Goal : a ∣ b * prod L rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b ∨ a ∈ L by_cases on h5 · -- Case 1. h5 : a = b apply Exists.intro (prod L) rewrite [h5] rfl done · -- Case 2. h5 : a ∈ L have h6 : a ∣ prod L := h4 h5 have h7 : prod L ∣ b * prod L := by apply Exists.intro b ring done show a ∣ b * prod L from dvd_trans h6 h7 done done done lemma list_elt_dvd_prod {a : Nat} {l : List Nat} (h : a ∈ l) : a ∣ prod l := by set n : Nat := l.length have h1 : l.length = n := by rfl show a ∣ prod l from list_elt_dvd_prod_by_length a n l h1 h done lemma exists_prime_factorization : ∀ (n : Nat), n ≥ 1 → ∃ (l : List Nat), prime_factorization n l := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, n_1 ≥ 1 → ∃ (l : List Nat), prime_factorization n_1 l assume h1 : n ≥ 1 by_cases h2 : n = 1 · -- Case 1. h2 : n = 1 apply Exists.intro [] define apply And.intro · -- Proof of nondec_prime_list [] define show all_prime [] ∧ nondec [] from And.intro all_prime_nil nondec_nil done · -- Proof of prod [] = n rewrite [prod_nil, h2] rfl done done · -- Case 2. h2 : n ≠ 1 have h3 : n ≥ 2 := lt_of_le_of_ne' h1 h2 obtain (p : Nat) (h4 : prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q) from exists_least_prime_factor h3 have p_prime_factor : prime_factor p n := h4.left define at p_prime_factor have p_prime : prime p := p_prime_factor.left have p_dvd_n : p ∣ n := p_prime_factor.right have p_least : ∀ (q : Nat), prime_factor q n → p ≤ q := h4.right obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n have h5 : m ≠ 0 := by contradict h1 with h6 have h7 : n = 0 := calc n _ = p * m := n_eq_pm _ = p * 0 := by rw [h6] _ = 0 := by ring rewrite [h7] decide done have m_pos : 0 < m := Nat.pos_of_ne_zero h5 have m_lt_n : m < n := by define at p_prime show m < n from calc m _ < m + m := by linarith _ = 2 * m := by ring _ ≤ p * m := by rel [p_prime.left] _ = n := n_eq_pm.symm done obtain (L : List Nat) (h6 : prime_factorization m L) from ih m m_lt_n m_pos define at h6 have ndpl_L : nondec_prime_list L := h6.left define at ndpl_L apply Exists.intro (p :: L) define apply And.intro · -- Proof of nondec_prime_list (p :: L) define apply And.intro · -- Proof of all_prime (p :: L) rewrite [all_prime_cons] show prime p ∧ all_prime L from And.intro p_prime ndpl_L.left done · -- Proof of nondec (p :: L) rewrite [nondec_cons] apply And.intro _ ndpl_L.right fix q : Nat assume q_in_L : q ∈ L have h7 : q ∣ prod L := list_elt_dvd_prod q_in_L rewrite [h6.right] at h7 --h7 : q ∣ m have h8 : m ∣ n := by apply Exists.intro p rewrite [n_eq_pm] ring done have q_dvd_n : q ∣ n := dvd_trans h7 h8 have ap_L : all_prime L := ndpl_L.left define at ap_L have q_prime_factor : prime_factor q n := And.intro (ap_L q q_in_L) q_dvd_n show p ≤ q from p_least q q_prime_factor done done · -- Proof of prod (p :: L) = n rewrite [prod_cons, h6.right, n_eq_pm] rfl done done done theorem Theorem_7_2_2 {a b c : Nat} (h1 : c ∣ a * b) (h2 : rel_prime a c) : c ∣ b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑c ∣ ↑b define at h1; define at h2; define obtain (j : Nat) (h3 : a * b = c * j) from h1 set s : Int := gcd_c1 a c set t : Int := gcd_c2 a c have h4 : s * ↑a + t * ↑c = ↑(gcd a c) := gcd_lin_comb c a rewrite [h2, Nat.cast_one] at h4 --h4 : s * ↑a + t * ↑c = (1 : Int) apply Exists.intro (s * ↑j + t * ↑b) show ↑b = ↑c * (s * ↑j + t * ↑b) from calc ↑b _ = (1 : Int) * ↑b := (one_mul _).symm _ = (s * ↑a + t * ↑c) * ↑b := by rw [h4] _ = s * (↑a * ↑b) + t * ↑c * ↑b := by ring _ = s * (↑c * ↑j) + t * ↑c * ↑b := by rw [←Nat.cast_mul a b, h3, Nat.cast_mul c j] _ = ↑c * (s * ↑j + t * ↑b) := by ring done lemma le_nonzero_prod_left {a b : Nat} (h : a * b ≠ 0) : a ≤ a * b := by have h1 : b ≠ 0 := by contradict h with h1 rewrite [h1] ring done have h2 : 1 ≤ b := Nat.pos_of_ne_zero h1 show a ≤ a * b from calc a = a * 1 := (mul_one a).symm _ ≤ a * b := by rel [h2] done lemma le_nonzero_prod_right {a b : Nat} (h : a * b ≠ 0) : b ≤ a * b := by rewrite [mul_comm] rewrite [mul_comm] at h show b ≤ b * a from le_nonzero_prod_left h done lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry lemma rel_prime_of_prime_not_dvd {a p : Nat} (h1 : prime p) (h2 : ¬p ∣ a) : rel_prime a p := by have h3 : gcd a p ∣ a := gcd_dvd_left a p have h4 : gcd a p ∣ p := gcd_dvd_right a p have h5 : gcd a p = 1 ∨ gcd a p = p := dvd_prime h1 h4 have h6 : gcd a p ≠ p := by contradict h2 with h6 rewrite [h6] at h3 show p ∣ a from h3 done disj_syll h5 h6 show rel_prime a p from h5 done theorem Theorem_7_2_3 {a b p : Nat} (h1 : prime p) (h2 : p ∣ a * b) : p ∣ a ∨ p ∣ b := by or_right with h3 have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3 show p ∣ b from Theorem_7_2_2 h2 h4 done lemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a ≥ 1 := by have h1 : a ≠ 0 := by by_contra h1 rewrite [h1] at h contradict h linarith done show a ≥ 1 from Nat.pos_of_ne_zero h1 done lemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by have h1 : a ≥ 1 := ge_one_of_prod_one h have h2 : a * b ≠ 0 := by linarith have h3 : a ≤ a * b := le_nonzero_prod_left h2 rewrite [h] at h3 show a = 1 from Nat.le_antisymm h3 h1 done lemma eq_one_of_dvd_one {n : Nat} (h : n ∣ 1) : n = 1 := by obtain (j : Nat) (h1 : 1 = n * j) from h show n = 1 from eq_one_of_prod_one h1.symm done lemma prime_not_one {p : Nat} (h : prime p) : p ≠ 1 := by define at h linarith done theorem Theorem_7_2_4 {p : Nat} (h1 : prime p) : ∀ (l : List Nat), p ∣ prod l → ∃ a ∈ l, p ∣ a := by apply List.rec · -- Base Case. Goal : p ∣ prod [] → ∃ a ∈ [], p ∣ a rewrite [prod_nil] assume h2 : p ∣ 1 show ∃ a ∈ [], p ∣ a from absurd (eq_one_of_dvd_one h2) (prime_not_one h1) done · -- Induction Step fix b : Nat fix L : List Nat assume ih : p ∣ prod L → ∃ a ∈ L, p ∣ a --Goal : p ∣ prod (b :: L) → ∃ a ∈ b :: L, p ∣ a assume h2 : p ∣ prod (b :: L) rewrite [prod_cons] at h2 have h3 : p ∣ b ∨ p ∣ prod L := Theorem_7_2_3 h1 h2 by_cases on h3 · -- Case 1. h3 : p ∣ b apply Exists.intro b show b ∈ b :: L ∧ p ∣ b from And.intro (List.mem_cons_self b L) h3 done · -- Case 2. h3 : p ∣ prod L obtain (a : Nat) (h4 : a ∈ L ∧ p ∣ a) from ih h3 apply Exists.intro a show a ∈ b :: L ∧ p ∣ a from And.intro (List.mem_cons_of_mem b h4.left) h4.right done done done lemma prime_in_list {p : Nat} {l : List Nat} (h1 : prime p) (h2 : all_prime l) (h3 : p ∣ prod l) : p ∈ l := by obtain (a : Nat) (h4 : a ∈ l ∧ p ∣ a) from Theorem_7_2_4 h1 l h3 define at h2 have h5 : prime a := h2 a h4.left have h6 : p = 1 ∨ p = a := dvd_prime h5 h4.right disj_syll h6 (prime_not_one h1) rewrite [h6] show a ∈ l from h4.left done lemma first_le_first {p q : Nat} {l m : List Nat} (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m)) (h3 : prod (p :: l) = prod (q :: m)) : p ≤ q := by define at h1; define at h2 have h4 : q ∣ prod (p :: l) := by define apply Exists.intro (prod m) rewrite [←prod_cons] show prod (p :: l) = prod (q :: m) from h3 done have h5 : all_prime (q :: m) := h2.left rewrite [all_prime_cons] at h5 have h6 : q ∈ p :: l := prime_in_list h5.left h1.left h4 have h7 : nondec (p :: l) := h1.right rewrite [nondec_cons] at h7 rewrite [List.mem_cons] at h6 by_cases on h6 · -- Case 1. h6 : q = p linarith done · -- Case 2. h6 : q ∈ l have h8 : ∀ m ∈ l, p ≤ m := h7.left show p ≤ q from h8 q h6 done done lemma nondec_prime_list_tail {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by define at h define rewrite [all_prime_cons, nondec_cons] at h show all_prime l ∧ nondec l from And.intro h.left.right h.right.right done lemma cons_prod_not_one {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : prod (p :: l) ≠ 1 := by define at h have h1 : all_prime (p :: l) := h.left rewrite [all_prime_cons] at h1 rewrite [prod_cons] by_contra h2 show False from (prime_not_one h1.left) (eq_one_of_prod_one h2) done lemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) : l = [] ↔ prod l = 1 := by apply Iff.intro · -- (→) assume h1 : l = [] rewrite [h1] show prod [] = 1 from prod_nil done · -- (←) contrapos assume h1 : ¬l = [] obtain (p : Nat) (h2 : ∃ (L : List Nat), l = p :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List Nat) (h3 : l = p :: L) from h2 rewrite [h3] at h rewrite [h3] show ¬prod (p :: L) = 1 from cons_prod_not_one h done done lemma prime_pos {p : Nat} (h : prime p) : p > 0 := by define at h linarith done theorem Theorem_7_2_5 : ∀ (l1 l2 : List Nat), nondec_prime_list l1 → nondec_prime_list l2 → prod l1 = prod l2 → l1 = l2 := by apply List.rec · -- Base Case. Goal : ∀ (l2 : List Nat), nondec_prime_list [] → -- nondec_prime_list l2 → prod [] = prod l2 → [] = l2 fix l2 : List Nat assume h1 : nondec_prime_list [] assume h2 : nondec_prime_list l2 assume h3 : prod [] = prod l2 rewrite [prod_nil, eq_comm, ←list_nil_iff_prod_one h2] at h3 show [] = l2 from h3.symm done · -- Induction Step fix p : Nat fix L1 : List Nat assume ih : ∀ (L2 : List Nat), nondec_prime_list L1 → nondec_prime_list L2 → prod L1 = prod L2 → L1 = L2 -- Goal : ∀ (l2 : List Nat), nondec_prime_list (p :: L1) → -- nondec_prime_list l2 → prod (p :: L1) = prod l2 → p :: L1 = l2 fix l2 : List Nat assume h1 : nondec_prime_list (p :: L1) assume h2 : nondec_prime_list l2 assume h3 : prod (p :: L1) = prod l2 have h4 : ¬prod (p :: L1) = 1 := cons_prod_not_one h1 rewrite [h3, ←list_nil_iff_prod_one h2] at h4 obtain (q : Nat) (h5 : ∃ (L : List Nat), l2 = q :: L) from List.exists_cons_of_ne_nil h4 obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5 rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2) rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2) have h7 : p ≤ q := first_le_first h1 h2 h3 have h8 : q ≤ p := first_le_first h2 h1 h3.symm have h9 : p = q := by linarith rewrite [h9, prod_cons, prod_cons] at h3 --h3 : q * prod L1 = q * prod L2 have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1 have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2 define at h2 have h12 : all_prime (q :: L2) := h2.left rewrite [all_prime_cons] at h12 have h13 : q > 0 := prime_pos h12.left have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3 have h15 : L1 = L2 := ih L2 h10 h11 h14 rewrite [h6, h9, h15] rfl done done theorem fund_thm_arith (n : Nat) (h : n ≥ 1) : ∃! (l : List Nat), prime_factorization n l := by exists_unique · -- Existence show ∃ (l : List Nat), prime_factorization n l from exists_prime_factorization n h done · -- Uniqueness fix l1 : List Nat; fix l2 : List Nat assume h1 : prime_factorization n l1 assume h2 : prime_factorization n l2 define at h1; define at h2 have h3 : prod l1 = n := h1.right rewrite [←h2.right] at h3 show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3 done done /- Section 7.3 -/ theorem congr_refl (m : Nat) : ∀ (a : Int), a ≡ a (MOD m) := by fix a : Int define --Goal : ∃ (c : Int), a - a = ↑m * c apply Exists.intro 0 ring done theorem congr_symm {m : Nat} : ∀ {a b : Int}, a ≡ b (MOD m) → b ≡ a (MOD m) := by fix a : Int; fix b : Int assume h1 : a ≡ b (MOD m) define at h1 --h1 : ∃ (c : Int), a - b = ↑m * c define --Goal : ∃ (c : Int), b - a = ↑m * c obtain (c : Int) (h2 : a - b = m * c) from h1 apply Exists.intro (-c) show b - a = m * (-c) from calc b - a _ = -(a - b) := by ring _ = -(m * c) := by rw [h2] _ = m * (-c) := by ring done theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry /- Fundamental properties of congruence classes -/ lemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) : X = Y ↔ X.val = Y.val := Fin.ext_iff lemma val_nat_eq_mod (n k : Nat) : ([k]_(n + 1)).val = k % (n + 1) := by rfl lemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl theorem cc_rep {m : Nat} (X : ZMod m) : ∃ (a : Int), X = [a]_m := match m with | 0 => by apply Exists.intro X rfl done | n + 1 => by apply Exists.intro ↑(X.val) have h1 : X.val < n + 1 := Fin.prop X rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1] rfl done theorem add_class (m : Nat) (a b : Int) : [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm theorem mul_class (m : Nat) (a b : Int) : [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm lemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ [a - b]_m = [0]_m := by apply Iff.intro · -- (→) assume h1 : [a]_m = [b]_m have h2 : a - b = a + (-b) := by ring have h3 : b + (-b) = 0 := by ring show [a - b]_m = [0]_m from calc [a - b]_m _ = [a + (-b)]_m := by rw [h2] _ = [a]_m + [-b]_m := by rw [add_class] _ = [b]_m + [-b]_m := by rw [h1] _ = [b + -b]_m := by rw [add_class] _ = [0]_m := by rw [h3] done · -- (←) assume h1 : [a - b]_m = [0]_m have h2 : b + (a - b) = a := by ring have h3 : b + 0 = b := by ring show [a]_m = [b]_m from calc [a]_m _ = [b + (a - b)]_m := by rw [h2] _ = [b]_m + [a - b]_m := by rw [add_class] _ = [b]_m + [0]_m := by rw [h1] _ = [b + 0]_m := by rw [add_class] _ = [b]_m := by rw [h3] done done lemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) : [a]_m = [0]_m → [-a]_m = [0]_m := by assume h1 : [a]_m = [0]_m have h2 : 0 + (-a) = -a := by ring have h3 : a + (-a) = 0 := by ring show [-a]_m = [0]_m from calc [-a]_m _ = [0 + (-a)]_m := by rw [h2] _ = [0]_m + [-a]_m := by rw [add_class] _ = [a]_m + [-a]_m := by rw [h1] _ = [a + (-a)]_m := by rw [add_class] _ = [0]_m := by rw [h3] done lemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) : [-a]_m = [0]_m ↔ [a]_m = [0]_m := by apply Iff.intro _ (cc_neg_zero_of_cc_zero m a) assume h1 : [-a]_m = [0]_m have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1 have h3 : -(-a) = a := by ring rewrite [h3] at h2 show [a]_m = [0]_m from h2 done lemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl lemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m ↔ m ∣ k := match m with | 0 => by have h : (0 : Int) = (↑(0 : Nat) : Int) := by rfl rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj] apply Iff.intro · -- (→) assume h1 : k = 0 rewrite [h1] show 0 ∣ 0 from dvd_self 0 done · -- (←) assume h1 : 0 ∣ k obtain (c : Nat) (h2 : k = 0 * c) from h1 rewrite [h2] ring done done | n + 1 => by rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero] show k % (n + 1) = 0 ↔ n + 1 ∣ k from (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm done lemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m ↔ ↑m ∣ a := by obtain (k : Nat) (h1 : a = ↑k ∨ a = -↑k) from Int.eq_nat_or_neg a by_cases on h1 · -- Case 1. h1: a = ↑k rewrite [h1, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done · -- Case 2. h1: a = -↑k rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done done theorem cc_eq_iff_congr (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ a ≡ b (MOD m) := calc [a]_m = [b]_m _ ↔ [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b _ ↔ ↑m ∣ (a - b) := cc_zero_iff_dvd m (a - b) _ ↔ a ≡ b (MOD m) := by rfl /- End of fundamental properties of congruence classes -/ lemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m := by have h1 : (↑m : Int) ≠ 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m) show 0 ≤ a % m from Int.emod_nonneg a h1 done lemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m) have h2 : (↑m : Int) > 0 := (Nat.cast_pos).rtl h1 show a % m < m from Int.emod_lt_of_pos a h2 done lemma congr_mod_mod (m : Nat) (a : Int) : a ≡ a % m (MOD m) := by define have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m apply Exists.intro (a / m) show a - a % m = m * (a / m) from calc a - (a % m) _ = m * (a / m) + a % m - a % m := by rw [h1] _ = m * (a / m) := by ring done lemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) := And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a)) theorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) : ∃! (r : Int), 0 ≤ r ∧ r < m ∧ a ≡ r (MOD m) := by exists_unique · -- Existence apply Exists.intro (a % m) show 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) from mod_cmpl_res m a done · -- Uniqueness fix r1 : Int; fix r2 : Int assume h1 : 0 ≤ r1 ∧ r1 < m ∧ a ≡ r1 (MOD m) assume h2 : 0 ≤ r2 ∧ r2 < m ∧ a ≡ r2 (MOD m) have h3 : r1 ≡ r2 (MOD m) := congr_trans (congr_symm h1.right.right) h2.right.right obtain (d : Int) (h4 : r1 - r2 = m * d) from h3 have h5 : r1 - r2 < m * 1 := by linarith have h6 : m * (-1) < r1 - r2 := by linarith rewrite [h4] at h5 --h5 : m * d < m * 1 rewrite [h4] at h6 --h6 : m * -1 < m * d have h7 : (↑m : Int) ≥ 0 := Nat.cast_nonneg m have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7 have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7 have h10 : d = 0 := by linarith show r1 = r2 from calc r1 _ = r1 - r2 + r2 := by ring _ = m * 0 + r2 := by rw [h4, h10] _ = r2 := by ring done done lemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m := (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a) theorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y rewrite [h1, h2] have h3 : a + b = b + a := by ring show [a]_m + [b]_m = [b]_m + [a]_m from calc [a]_m + [b]_m _ = [a + b]_m := add_class m a b _ = [b + a]_m := by rw [h3] _ = [b]_m + [a]_m := (add_class m b a).symm done theorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X rewrite [h1] have h2 : a * 1 = a := by ring show [a]_m * [1]_m = [a]_m from calc [a]_m * [1]_m _ = [a * 1]_m := mul_class m a 1 _ = [a]_m := by rw [h2] done theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry lemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) : [a]_m * [gcd_c2 m a]_m = [1]_m := by set s : Int := gcd_c1 m a have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m define at h1 rewrite [h1, Nat.cast_one] at h2 --h2 : s * ↑m + gcd_c2 m a * ↑a = 1 rewrite [mul_class, cc_eq_iff_congr] define --Goal : ∃ (c : Int), ↑a * gcd_c2 m a - 1 = ↑m * c apply Exists.intro (-s) show a * (gcd_c2 m a) - 1 = m * (-s) from calc a * (gcd_c2 m a) - 1 _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring _ = 1 + m * (-s) - 1 := by rw [h2] _ = m * (-s) := by ring done theorem Theorem_7_3_7 (m a : Nat) : invertible [a]_m ↔ rel_prime m a := by apply Iff.intro · -- (→) assume h1 : invertible [a]_m define at h1 obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1 obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y rewrite [h3, mul_class, cc_eq_iff_congr] at h2 define at h2 obtain (c : Int) (h4 : a * b - 1 = m * c) from h2 rewrite [Exercise_7_2_6] --Goal : ∃ (s t : Int), s * ↑m + t * ↑a = 1 apply Exists.intro (-c) apply Exists.intro b show (-c) * m + b * a = 1 from calc (-c) * m + b * a _ = (-c) * m + (a * b - 1) + 1 := by ring _ = (-c) * m + m * c + 1 := by rw [h4] _ = 1 := by ring done · -- (←) assume h1 : rel_prime m a define show ∃ (Y : ZMod m), [a]_m * Y = [1]_m from Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1) done done /- Section 7.4 -/ section Euler open Euler lemma num_rp_below_base {m : Nat} : num_rp_below m 0 = 0 := by rfl lemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) : num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by have h1 : num_rp_below m (j + 1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : gcd m j = 1 rewrite [if_pos h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1 show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1 done lemma num_rp_below_step_not_rp {m j : Nat} (h : ¬rel_prime m j) : num_rp_below m (j + 1) = num_rp_below m j := by have h1 : num_rp_below m (j +1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : ¬gcd m j = 1 rewrite [if_neg h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j show num_rp_below m (j + 1) = num_rp_below m j from h1 done lemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl #eval phi 10 --Answer: 4 lemma prod_inv_iff_inv {m : Nat} {X : ZMod m} (h1 : invertible X) (Y : ZMod m) : invertible (X * Y) ↔ invertible Y := by apply Iff.intro · -- (→) assume h2 : invertible (X * Y) obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2 apply Exists.intro (X * Z) rewrite [←h3] --Goal : Y * (X * Z) = X * Y * Z ring --Note that ring can do algebra in ZMod m done · -- (←) assume h2 : invertible Y obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1 obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2 apply Exists.intro (Xi * Yi) show (X * Y) * (Xi * Yi) = [1]_m from calc X * Y * (Xi * Yi) _ = (X * Xi) * (Y * Yi) := by ring _ = [1]_m * [1]_m := by rw [h3, h4] _ = [1]_m := Theorem_7_3_6_7 [1]_m done done lemma F_rp_def {m i : Nat} (h : rel_prime m i) : F m i = [i]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h --h : gcd m i = 1 rewrite [if_pos h] at h1 show F m i = [i]_m from h1 done lemma F_not_rp_def {m i : Nat} (h : ¬rel_prime m i) : F m i = [1]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h rewrite [h1, if_neg h] rfl done lemma prod_seq_base {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 0 k f = [1]_m := by rfl lemma prod_seq_step {m : Nat} (n k : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl lemma prod_seq_zero_step {m : Nat} (n : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by rewrite [prod_seq_step, zero_add] rfl done lemma prod_one {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 1 k f = f k := by rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7] rfl done lemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl lemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m := calc [G m a i]_m _ = [(a * i) % m]_m := by rfl _ = [a * i]_m := (cc_eq_mod m (a * i)).symm _ = [a]_m * [i]_m := (mul_class m a i).symm lemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) : rel_prime m (G m a i) ↔ rel_prime m i := by have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1 show rel_prime m (G m a i) ↔ rel_prime m i from calc rel_prime m (G m a i) _ ↔ invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm _ ↔ invertible ([a]_m * [i]_m) := by rw [cc_G] _ ↔ invertible [i]_m := prod_inv_iff_inv h2 ([i]_m) _ ↔ rel_prime m i := Theorem_7_3_7 m i done lemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) : F m (G m a i) = [a]_m * F m i := by have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2 show F m (G m a i) = [a]_m * F m i from calc F m (G m a i) _ = [G m a i]_m := F_rp_def h3 _ = [a]_m * [i]_m := cc_G m a i _ = [a]_m * F m i := by rw [F_rp_def h2] done lemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : ¬rel_prime m i) : F m (G m a i) = [1]_m := by rewrite [←G_rp_iff h1 i] at h2 show F m (G m a i) = [1]_m from F_not_rp_def h2 done lemma FG_prod {m a : Nat} (h1 : rel_prime m a) : ∀ (k : Nat), prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by by_induc · -- Base Case show prod_seq 0 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from calc prod_seq 0 0 ((F m) ∘ (G m a)) _ = [1]_m := prod_seq_base _ _ _ = [a]_m ^ 0 * [1]_m := by ring _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by rw [num_rp_below_base, prod_seq_base] done · -- Induction Step fix k : Nat assume ih : prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) by_cases h2 : rel_prime m k · -- Case 1. h2 : rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([a]_m * F m k) := by rw [FG_rp h1 h2] _ = [a]_m ^ ((num_rp_below m k) + 1) * ((prod_seq k 0 (F m)) * F m k) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_rp h2, prod_seq_zero_step] done · -- Case 2. h2 : ¬rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([1]_m) := by rw [FG_not_rp h1 h2] _ = [a]_m ^ (num_rp_below m k) * (prod_seq k 0 (F m) * ([1]_m)) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_not_rp h2, prod_seq_zero_step, F_not_rp_def h2] done done done lemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by define --Goal : ∀ i < m, G m a i < m fix i : Nat assume h1 : i < m rewrite [G_def] --Goal : a * i % m < m show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m) done lemma left_inv_one_one_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g' (g i) = i) : one_one_below n g := sorry lemma right_inv_onto_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g (g' i) = i) (h2 : maps_below n g') : onto_below n g := by define at h2; define fix k : Nat assume h3 : k < n apply Exists.intro (g' k) show g' k < n ∧ g (g' k) = k from And.intro (h2 k h3) (h1 k h3) done lemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m * [inv_mod m a]_m = [1]_m := by have h2 : 0 ≤ (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a) show [a]_m * [inv_mod m a]_m = [1]_m from calc [a]_m * [inv_mod m a]_m _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2] _ = [a]_m * [gcd_c2 m a]_m := by rw [←cc_eq_mod] _ = [1]_m := gcd_c2_inv h1 done lemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m := calc a * (b % m) % m = a % m * (b % m % m) % m := Nat.mul_mod _ _ _ _ = a % m * (b % m) % m := by rw [Nat.mod_mod] _ = a * b % m := (Nat.mul_mod _ _ _).symm lemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm] rfl done theorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] : ↑a ≡ ↑b (MOD m) ↔ a % m = b % m := sorry lemma mul_inv_mod_cancel {m a i : Nat} [NeZero m] (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1 rewrite [mul_class, cc_eq_iff_congr, ←Nat.cast_mul, ←Nat.cast_one, congr_iff_mod_eq_Nat] at h3 show a * inv_mod m a * i % m = i from calc a * (inv_mod m a) * i % m _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = 1 % m * i % m := by rw [h3] _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = i % m := by rw [one_mul] _ = i := Nat.mod_eq_of_lt h2 done lemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl lemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, G m a (Ginv m a i) = i := by fix i : Nat assume h2 : i < m show G m a (Ginv m a i) = i from calc G m a (Ginv m a i) _ = a * ((inv_mod m a * i) % m) % m := by rfl _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, Ginv m a (G m a i) = i := by fix i : Nat assume h2 : i < m show Ginv m a (G m a i) = i from calc Ginv m a (G m a i) _ = inv_mod m a * ((a * i) % m) % m := by rfl _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc, mul_comm (inv_mod m a)] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_maps_below (m a : Nat) [NeZero m] : maps_below m (Ginv m a) := G_maps_below m (inv_mod m a) lemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : one_one_below m (G m a) := left_inv_one_one_below (Ginv_left_inv h1) lemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : onto_below m (G m a) := right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a) lemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : perm_below m (G m a) := And.intro (G_maps_below m a) (And.intro (G_one_one_below h1) (G_onto_below h1)) --Permuting a product of congruence classes doesn't change product lemma swap_fst (u v : Nat) : swap u v u = v := by define : swap u v u --Goal : (if u = u then v else if u = v then u else u) = v have h : u = u := by rfl rewrite [if_pos h] rfl done lemma swap_snd (u v : Nat) : swap u v v = u := by define : swap u v v by_cases h1 : v = u · -- Case 1. h1 : v = u rewrite [if_pos h1] show v = u from h1 done · -- Case 2. h1 : v ≠ u rewrite [if_neg h1] have h2 : v = v := by rfl rewrite [if_pos h2] rfl done done lemma swap_other {u v i : Nat} (h1 : i ≠ u) (h2 : i ≠ v) : swap u v i = i := by define : swap u v i rewrite [if_neg h1, if_neg h2] rfl done lemma swap_values (u v i : Nat) : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := by by_cases h1 : i = u · -- Case 1. h1 : i = u apply Or.inl rewrite [h1] show swap u v u = v from swap_fst u v done · -- Case 2. h1 : i ≠ u apply Or.inr by_cases h2 : i = v · -- Case 2.1. h2 : i = v apply Or.inl rewrite [h2] show swap u v v = u from swap_snd u v done · -- Case 2.2. h2 : i ≠ v apply Or.inr show swap u v i = i from swap_other h1 h2 done done done lemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by define fix i : Nat assume h3 : i < n have h4 : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := swap_values u v i by_cases on h4 · -- Case 1. h4 : swap u v i = v rewrite [h4] show v < n from h2 done · -- Case 2. by_cases on h4 · -- Case 2.1. h4 : swap u v i = u rewrite [h4] show u < n from h1 done · -- Case 2.2. h4 : swap u v i = i rewrite [h4] show i < n from h3 done done done lemma swap_swap (u v n : Nat) : ∀ i < n, swap u v (swap u v i) = i := by fix i : Nat assume h : i < n by_cases h1 : i = u · -- Case 1. h1 : i = u rewrite [h1, swap_fst, swap_snd] rfl done · -- Case 2. h1 : i ≠ u by_cases h2 : i = v · -- Case 2.1. h2 : i = v rewrite [h2, swap_snd, swap_fst] rfl done · -- Case 2.2. h2 : i ≠ v rewrite [swap_other h1 h2, swap_other h1 h2] rfl done done done lemma swap_one_one_below (u v n) : one_one_below n (swap u v) := left_inv_one_one_below (swap_swap u v n) lemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) := right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2) lemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) := And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2)) lemma comp_perm_below {n : Nat} {f g : Nat → Nat} (h1 : perm_below n f) (h2 : perm_below n g) : perm_below n (f ∘ g) := sorry

lemma trivial_swap (u : Nat) : swap u u = id

HTPI.trivial_swap

htpi/HTPILib/Chap7.lean

htpi

/- Copyright 2023 Daniel J. Velleman -/ import HTPILib.Chap6 namespace HTPI /- Definitions -/ lemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by have h : n + 1 > 0 := Nat.succ_pos n show a % (n + 1) < n + 1 from Nat.mod_lt a h done def gcd (a b : Nat) : Nat := match b with | 0 => a | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd (n + 1) (a % (n + 1)) termination_by b mutual def gcd_c1 (a b : Nat) : Int := match b with | 0 => 1 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c2 (n + 1) (a % (n + 1)) --Corresponds to s = t' termination_by b def gcd_c2 (a b : Nat) : Int := match b with | 0 => 0 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c1 (n + 1) (a % (n + 1)) - (gcd_c2 (n + 1) (a % (n + 1))) * ↑(a / (n + 1)) --Corresponds to t = s' - t'q termination_by b end def prime (n : Nat) : Prop := 2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n def prime_factor (p n : Nat) : Prop := prime p ∧ p ∣ n def all_prime (l : List Nat) : Prop := ∀ p ∈ l, prime p def nondec (l : List Nat) : Prop := match l with | [] => True --Of course, True is a proposition that is always true | n :: L => (∀ m ∈ L, n ≤ m) ∧ nondec L def nondec_prime_list (l : List Nat) : Prop := all_prime l ∧ nondec l def prod (l : List Nat) : Nat := match l with | [] => 1 | n :: L => n * (prod L) def prime_factorization (n : Nat) (l : List Nat) : Prop := nondec_prime_list l ∧ prod l = n def rel_prime (a b : Nat) : Prop := gcd a b = 1 def congr_mod (m : Nat) (a b : Int) : Prop := (↑m : Int) ∣ (a - b) def cc (m : Nat) (a : Int) : ZMod m := (↑a : ZMod m) notation:50 a " ≡ " b " (MOD " m ")" => congr_mod m a b notation:max "["a"]_"m:max => cc m a def invertible {m : Nat} (X : ZMod m) : Prop := ∃ (Y : ZMod m), X * Y = [1]_m def num_rp_below (m k : Nat) : Nat := match k with | 0 => 0 | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j def phi (m : Nat) : Nat := num_rp_below m m def prod_seq {m : Nat} (j k : Nat) (f : Nat → ZMod m) : ZMod m := match j with | 0 => [1]_m | n + 1 => prod_seq n k f * f (k + n) def maps_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i < n, g i < n def one_one_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i1 < n, ∀ i2 < n, g i1 = g i2 → i1 = i2 def onto_below (n : Nat) (g : Nat → Nat) : Prop := ∀ k < n, ∃ i < n, g i = k def perm_below (n : Nat) (g : Nat → Nat) : Prop := maps_below n g ∧ one_one_below n g ∧ onto_below n g def inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m) def swap (u v i : Nat) : Nat := if i = u then v else if i = v then u else i namespace Euler --For definitions specific to Euler's theorem def F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m def G (m a i : Nat) : Nat := (a * i) % m def Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i end Euler /- Section 7.1 -/ theorem dvd_mod_of_dvd_a_b {a b d : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ (a % b) := by set q : Nat := a / b have h3 : b * q + a % b = a := Nat.div_add_mod a b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 define --Goal : ∃ (c : Nat), a % b = d * c apply Exists.intro (j - k * q) show a % b = d * (j - k * q) from calc a % b _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm _ = a - b * q := by rw [h3] _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc] _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm done theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry #eval gcd 672 161 --Answer: 7 lemma gcd_base (a : Nat) : gcd a 0 = a := by rfl lemma gcd_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd a b = gcd b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1)) rfl done lemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b ≠ 0) : a % b < b := by have h1 : b > 0 := Nat.pos_of_ne_zero h show a % b < b from Nat.mod_lt a h1 done lemma dvd_self (n : Nat) : n ∣ n := by apply Exists.intro 1 ring done theorem gcd_dvd : ∀ (b a : Nat), (gcd a b) ∣ a ∧ (gcd a b) ∣ b := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd a b_1) ∣ a ∧ (gcd a b_1) ∣ b_1 fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_base] --Goal: a ∣ a ∧ a ∣ 0 apply And.intro (dvd_self a) define apply Exists.intro 0 rfl done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_nonzero a h1] --Goal : gcd b (a % b) ∣ a ∧ gcd b (a % b) ∣ b have h2 : a % b < b := mod_nonzero_lt a h1 have h3 : (gcd b (a % b)) ∣ b ∧ (gcd b (a % b)) ∣ (a % b) := ih (a % b) h2 b apply And.intro _ h3.left show (gcd b (a % b)) ∣ a from dvd_a_of_dvd_b_mod h3.left h3.right done done theorem gcd_dvd_left (a b : Nat) : (gcd a b) ∣ a := (gcd_dvd b a).left theorem gcd_dvd_right (a b : Nat) : (gcd a b) ∣ b := (gcd_dvd b a).right lemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl lemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c1 a b = gcd_c2 b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done lemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl lemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * ↑(a / b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done theorem gcd_lin_comb : ∀ (b a : Nat), (gcd_c1 a b) * ↑a + (gcd_c2 a b) * ↑b = ↑(gcd a b) := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd_c1 a b_1) * ↑a + (gcd_c2 a b_1) * ↑b_1 = ↑(gcd a b_1) fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base] --Goal : 1 * ↑a + 0 * ↑0 = ↑a ring done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1] --Goal : gcd_c2 b (a % b) * ↑a + -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * ↑(a / b)) * ↑b = -- ↑(gcd b (a % b)) set r : Nat := a % b set q : Nat := a / b set s : Int := gcd_c1 b r set t : Int := gcd_c2 b r --Goal : t * ↑a + (s - t * ↑q) * ↑b = ↑(gcd b r) have h2 : r < b := mod_nonzero_lt a h1 have h3 : s * ↑b + t * ↑r = ↑(gcd b r) := ih r h2 b have h4 : b * q + r = a := Nat.div_add_mod a b rewrite [←h3, ←h4] rewrite [Nat.cast_add, Nat.cast_mul] --Goal : t * (↑b * ↑q + ↑r) + (s - t * ↑q) * ↑b = s * ↑b + t * ↑r ring done done #eval gcd_c1 672 161 --Answer: 6 #eval gcd_c2 672 161 --Answer: -25 --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161 theorem Theorem_7_1_6 {d a b : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ gcd a b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑d ∣ ↑(gcd a b) set s : Int := gcd_c1 a b set t : Int := gcd_c2 a b have h3 : s * ↑a + t * ↑b = ↑(gcd a b) := gcd_lin_comb b a rewrite [←h3] --Goal : ↑d ∣ s * ↑a + t * ↑b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul] --Goal : ↑d ∣ s * (↑d * ↑j) + t * (↑d * ↑k) define apply Exists.intro (s * ↑j + t * ↑k) ring done /- Section 7.2 -/ theorem dvd_trans {a b c : Nat} (h1 : a ∣ b) (h2 : b ∣ c) : a ∣ c := by define at h1; define at h2; define obtain (m : Nat) (h3 : b = a * m) from h1 obtain (n : Nat) (h4 : c = b * n) from h2 rewrite [h3, mul_assoc] at h4 apply Exists.intro (m * n) show c = a * (m * n) from h4 done lemma exists_prime_factor : ∀ (n : Nat), 2 ≤ n → ∃ (p : Nat), prime_factor p n := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, 2 ≤ n_1 → ∃ (p : Nat), prime_factor p n_1 assume h1 : 2 ≤ n by_cases h2 : prime n · -- Case 1. h2 : prime n apply Exists.intro n define --Goal : prime n ∧ n ∣ n show prime n ∧ n ∣ n from And.intro h2 (dvd_self n) done · -- Case 2. h2 : ¬prime n define at h2 --h2 : ¬(2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n) demorgan at h2 disj_syll h2 h1 obtain (a : Nat) (h3 : ∃ (b : Nat), a * b = n ∧ a < n ∧ b < n) from h2 obtain (b : Nat) (h4 : a * b = n ∧ a < n ∧ b < n) from h3 have h5 : 2 ≤ a := by by_contra h6 have h7 : a ≤ 1 := by linarith have h8 : n ≤ b := calc n _ = a * b := h4.left.symm _ ≤ 1 * b := by rel [h7] _ = b := by ring linarith --n ≤ b contradicts b < n done have h6 : ∃ (p : Nat), prime_factor p a := ih a h4.right.left h5 obtain (p : Nat) (h7 : prime_factor p a) from h6 apply Exists.intro p define --Goal : prime p ∧ p ∣ n define at h7 --h7 : prime p ∧ p ∣ a apply And.intro h7.left have h8 : a ∣ n := by apply Exists.intro b show n = a * b from (h4.left).symm done show p ∣ n from dvd_trans h7.right h8 done done lemma exists_least_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q := by set S : Set Nat := {p : Nat | prime_factor p n} have h2 : ∃ (p : Nat), p ∈ S := exists_prime_factor n h show ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q from well_ord_princ S h2 done lemma all_prime_nil : all_prime [] := by define --Goal : ∀ p ∈ [], prime p fix p : Nat contrapos --Goal : ¬prime p → p ∉ [] assume h1 : ¬prime p show p ∉ [] from List.not_mem_nil p done lemma all_prime_cons (n : Nat) (L : List Nat) : all_prime (n :: L) ↔ prime n ∧ all_prime L := by apply Iff.intro · -- (→) assume h1 : all_prime (n :: L) --Goal : prime n ∧ all_prime L define at h1 --h1 : ∀ p ∈ n :: L, prime p apply And.intro (h1 n (List.mem_cons_self n L)) define --Goal : ∀ p ∈ L, prime p fix p : Nat assume h2 : p ∈ L show prime p from h1 p (List.mem_cons_of_mem n h2) done · -- (←) assume h1 : prime n ∧ all_prime L --Goal : all_prime (n :: l) define : all_prime L at h1 define fix p : Nat assume h2 : p ∈ n :: L rewrite [List.mem_cons] at h2 --h2 : p = n ∨ p ∈ L by_cases on h2 · -- Case 1. h2 : p = n rewrite [h2] show prime n from h1.left done · -- Case 2. h2 : p ∈ L show prime p from h1.right p h2 done done done lemma nondec_nil : nondec [] := by define --Goal : True trivial --trivial proves some obviously true statements, such as True done lemma nondec_cons (n : Nat) (L : List Nat) : nondec (n :: L) ↔ (∀ m ∈ L, n ≤ m) ∧ nondec L := by rfl lemma prod_nil : prod [] = 1 := by rfl lemma prod_cons : prod (n :: L) = n * (prod L) := by rfl lemma exists_cons_of_length_eq_succ {A : Type} {l : List A} {n : Nat} (h : l.length = n + 1) : ∃ (a : A) (L : List A), l = a :: L ∧ L.length = n := by have h1 : ¬l.length = 0 := by linarith rewrite [List.length_eq_zero] at h1 obtain (a : A) (h2 : ∃ (L : List A), l = a :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List A) (h3 : l = a :: L) from h2 apply Exists.intro a apply Exists.intro L apply And.intro h3 have h4 : (a :: L).length = L.length + 1 := List.length_cons a L rewrite [←h3, h] at h4 show L.length = n from (Nat.add_right_cancel h4).symm done lemma list_elt_dvd_prod_by_length (a : Nat) : ∀ (n : Nat), ∀ (l : List Nat), l.length = n → a ∈ l → a ∣ prod l := by by_induc · --Base Case fix l : List Nat assume h1 : l.length = 0 rewrite [List.length_eq_zero] at h1 --h1 : l = [] rewrite [h1] --Goal : a ∈ [] → a ∣ prod [] contrapos assume h2 : ¬a ∣ prod [] show a ∉ [] from List.not_mem_nil a done · -- Induction Step fix n : Nat assume ih : ∀ (l : List Nat), List.length l = n → a ∈ l → a ∣ prod l fix l : List Nat assume h1 : l.length = n + 1 --Goal : a ∈ l → a ∣ prod l obtain (b : Nat) (h2 : ∃ (L : List Nat), l = b :: L ∧ L.length = n) from exists_cons_of_length_eq_succ h1 obtain (L : List Nat) (h3 : l = b :: L ∧ L.length = n) from h2 have h4 : a ∈ L → a ∣ prod L := ih L h3.right assume h5 : a ∈ l rewrite [h3.left, prod_cons] --Goal : a ∣ b * prod L rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b ∨ a ∈ L by_cases on h5 · -- Case 1. h5 : a = b apply Exists.intro (prod L) rewrite [h5] rfl done · -- Case 2. h5 : a ∈ L have h6 : a ∣ prod L := h4 h5 have h7 : prod L ∣ b * prod L := by apply Exists.intro b ring done show a ∣ b * prod L from dvd_trans h6 h7 done done done lemma list_elt_dvd_prod {a : Nat} {l : List Nat} (h : a ∈ l) : a ∣ prod l := by set n : Nat := l.length have h1 : l.length = n := by rfl show a ∣ prod l from list_elt_dvd_prod_by_length a n l h1 h done lemma exists_prime_factorization : ∀ (n : Nat), n ≥ 1 → ∃ (l : List Nat), prime_factorization n l := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, n_1 ≥ 1 → ∃ (l : List Nat), prime_factorization n_1 l assume h1 : n ≥ 1 by_cases h2 : n = 1 · -- Case 1. h2 : n = 1 apply Exists.intro [] define apply And.intro · -- Proof of nondec_prime_list [] define show all_prime [] ∧ nondec [] from And.intro all_prime_nil nondec_nil done · -- Proof of prod [] = n rewrite [prod_nil, h2] rfl done done · -- Case 2. h2 : n ≠ 1 have h3 : n ≥ 2 := lt_of_le_of_ne' h1 h2 obtain (p : Nat) (h4 : prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q) from exists_least_prime_factor h3 have p_prime_factor : prime_factor p n := h4.left define at p_prime_factor have p_prime : prime p := p_prime_factor.left have p_dvd_n : p ∣ n := p_prime_factor.right have p_least : ∀ (q : Nat), prime_factor q n → p ≤ q := h4.right obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n have h5 : m ≠ 0 := by contradict h1 with h6 have h7 : n = 0 := calc n _ = p * m := n_eq_pm _ = p * 0 := by rw [h6] _ = 0 := by ring rewrite [h7] decide done have m_pos : 0 < m := Nat.pos_of_ne_zero h5 have m_lt_n : m < n := by define at p_prime show m < n from calc m _ < m + m := by linarith _ = 2 * m := by ring _ ≤ p * m := by rel [p_prime.left] _ = n := n_eq_pm.symm done obtain (L : List Nat) (h6 : prime_factorization m L) from ih m m_lt_n m_pos define at h6 have ndpl_L : nondec_prime_list L := h6.left define at ndpl_L apply Exists.intro (p :: L) define apply And.intro · -- Proof of nondec_prime_list (p :: L) define apply And.intro · -- Proof of all_prime (p :: L) rewrite [all_prime_cons] show prime p ∧ all_prime L from And.intro p_prime ndpl_L.left done · -- Proof of nondec (p :: L) rewrite [nondec_cons] apply And.intro _ ndpl_L.right fix q : Nat assume q_in_L : q ∈ L have h7 : q ∣ prod L := list_elt_dvd_prod q_in_L rewrite [h6.right] at h7 --h7 : q ∣ m have h8 : m ∣ n := by apply Exists.intro p rewrite [n_eq_pm] ring done have q_dvd_n : q ∣ n := dvd_trans h7 h8 have ap_L : all_prime L := ndpl_L.left define at ap_L have q_prime_factor : prime_factor q n := And.intro (ap_L q q_in_L) q_dvd_n show p ≤ q from p_least q q_prime_factor done done · -- Proof of prod (p :: L) = n rewrite [prod_cons, h6.right, n_eq_pm] rfl done done done theorem Theorem_7_2_2 {a b c : Nat} (h1 : c ∣ a * b) (h2 : rel_prime a c) : c ∣ b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑c ∣ ↑b define at h1; define at h2; define obtain (j : Nat) (h3 : a * b = c * j) from h1 set s : Int := gcd_c1 a c set t : Int := gcd_c2 a c have h4 : s * ↑a + t * ↑c = ↑(gcd a c) := gcd_lin_comb c a rewrite [h2, Nat.cast_one] at h4 --h4 : s * ↑a + t * ↑c = (1 : Int) apply Exists.intro (s * ↑j + t * ↑b) show ↑b = ↑c * (s * ↑j + t * ↑b) from calc ↑b _ = (1 : Int) * ↑b := (one_mul _).symm _ = (s * ↑a + t * ↑c) * ↑b := by rw [h4] _ = s * (↑a * ↑b) + t * ↑c * ↑b := by ring _ = s * (↑c * ↑j) + t * ↑c * ↑b := by rw [←Nat.cast_mul a b, h3, Nat.cast_mul c j] _ = ↑c * (s * ↑j + t * ↑b) := by ring done lemma le_nonzero_prod_left {a b : Nat} (h : a * b ≠ 0) : a ≤ a * b := by have h1 : b ≠ 0 := by contradict h with h1 rewrite [h1] ring done have h2 : 1 ≤ b := Nat.pos_of_ne_zero h1 show a ≤ a * b from calc a = a * 1 := (mul_one a).symm _ ≤ a * b := by rel [h2] done lemma le_nonzero_prod_right {a b : Nat} (h : a * b ≠ 0) : b ≤ a * b := by rewrite [mul_comm] rewrite [mul_comm] at h show b ≤ b * a from le_nonzero_prod_left h done lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry lemma rel_prime_of_prime_not_dvd {a p : Nat} (h1 : prime p) (h2 : ¬p ∣ a) : rel_prime a p := by have h3 : gcd a p ∣ a := gcd_dvd_left a p have h4 : gcd a p ∣ p := gcd_dvd_right a p have h5 : gcd a p = 1 ∨ gcd a p = p := dvd_prime h1 h4 have h6 : gcd a p ≠ p := by contradict h2 with h6 rewrite [h6] at h3 show p ∣ a from h3 done disj_syll h5 h6 show rel_prime a p from h5 done theorem Theorem_7_2_3 {a b p : Nat} (h1 : prime p) (h2 : p ∣ a * b) : p ∣ a ∨ p ∣ b := by or_right with h3 have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3 show p ∣ b from Theorem_7_2_2 h2 h4 done lemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a ≥ 1 := by have h1 : a ≠ 0 := by by_contra h1 rewrite [h1] at h contradict h linarith done show a ≥ 1 from Nat.pos_of_ne_zero h1 done lemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by have h1 : a ≥ 1 := ge_one_of_prod_one h have h2 : a * b ≠ 0 := by linarith have h3 : a ≤ a * b := le_nonzero_prod_left h2 rewrite [h] at h3 show a = 1 from Nat.le_antisymm h3 h1 done lemma eq_one_of_dvd_one {n : Nat} (h : n ∣ 1) : n = 1 := by obtain (j : Nat) (h1 : 1 = n * j) from h show n = 1 from eq_one_of_prod_one h1.symm done lemma prime_not_one {p : Nat} (h : prime p) : p ≠ 1 := by define at h linarith done theorem Theorem_7_2_4 {p : Nat} (h1 : prime p) : ∀ (l : List Nat), p ∣ prod l → ∃ a ∈ l, p ∣ a := by apply List.rec · -- Base Case. Goal : p ∣ prod [] → ∃ a ∈ [], p ∣ a rewrite [prod_nil] assume h2 : p ∣ 1 show ∃ a ∈ [], p ∣ a from absurd (eq_one_of_dvd_one h2) (prime_not_one h1) done · -- Induction Step fix b : Nat fix L : List Nat assume ih : p ∣ prod L → ∃ a ∈ L, p ∣ a --Goal : p ∣ prod (b :: L) → ∃ a ∈ b :: L, p ∣ a assume h2 : p ∣ prod (b :: L) rewrite [prod_cons] at h2 have h3 : p ∣ b ∨ p ∣ prod L := Theorem_7_2_3 h1 h2 by_cases on h3 · -- Case 1. h3 : p ∣ b apply Exists.intro b show b ∈ b :: L ∧ p ∣ b from And.intro (List.mem_cons_self b L) h3 done · -- Case 2. h3 : p ∣ prod L obtain (a : Nat) (h4 : a ∈ L ∧ p ∣ a) from ih h3 apply Exists.intro a show a ∈ b :: L ∧ p ∣ a from And.intro (List.mem_cons_of_mem b h4.left) h4.right done done done lemma prime_in_list {p : Nat} {l : List Nat} (h1 : prime p) (h2 : all_prime l) (h3 : p ∣ prod l) : p ∈ l := by obtain (a : Nat) (h4 : a ∈ l ∧ p ∣ a) from Theorem_7_2_4 h1 l h3 define at h2 have h5 : prime a := h2 a h4.left have h6 : p = 1 ∨ p = a := dvd_prime h5 h4.right disj_syll h6 (prime_not_one h1) rewrite [h6] show a ∈ l from h4.left done lemma first_le_first {p q : Nat} {l m : List Nat} (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m)) (h3 : prod (p :: l) = prod (q :: m)) : p ≤ q := by define at h1; define at h2 have h4 : q ∣ prod (p :: l) := by define apply Exists.intro (prod m) rewrite [←prod_cons] show prod (p :: l) = prod (q :: m) from h3 done have h5 : all_prime (q :: m) := h2.left rewrite [all_prime_cons] at h5 have h6 : q ∈ p :: l := prime_in_list h5.left h1.left h4 have h7 : nondec (p :: l) := h1.right rewrite [nondec_cons] at h7 rewrite [List.mem_cons] at h6 by_cases on h6 · -- Case 1. h6 : q = p linarith done · -- Case 2. h6 : q ∈ l have h8 : ∀ m ∈ l, p ≤ m := h7.left show p ≤ q from h8 q h6 done done lemma nondec_prime_list_tail {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by define at h define rewrite [all_prime_cons, nondec_cons] at h show all_prime l ∧ nondec l from And.intro h.left.right h.right.right done lemma cons_prod_not_one {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : prod (p :: l) ≠ 1 := by define at h have h1 : all_prime (p :: l) := h.left rewrite [all_prime_cons] at h1 rewrite [prod_cons] by_contra h2 show False from (prime_not_one h1.left) (eq_one_of_prod_one h2) done lemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) : l = [] ↔ prod l = 1 := by apply Iff.intro · -- (→) assume h1 : l = [] rewrite [h1] show prod [] = 1 from prod_nil done · -- (←) contrapos assume h1 : ¬l = [] obtain (p : Nat) (h2 : ∃ (L : List Nat), l = p :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List Nat) (h3 : l = p :: L) from h2 rewrite [h3] at h rewrite [h3] show ¬prod (p :: L) = 1 from cons_prod_not_one h done done lemma prime_pos {p : Nat} (h : prime p) : p > 0 := by define at h linarith done theorem Theorem_7_2_5 : ∀ (l1 l2 : List Nat), nondec_prime_list l1 → nondec_prime_list l2 → prod l1 = prod l2 → l1 = l2 := by apply List.rec · -- Base Case. Goal : ∀ (l2 : List Nat), nondec_prime_list [] → -- nondec_prime_list l2 → prod [] = prod l2 → [] = l2 fix l2 : List Nat assume h1 : nondec_prime_list [] assume h2 : nondec_prime_list l2 assume h3 : prod [] = prod l2 rewrite [prod_nil, eq_comm, ←list_nil_iff_prod_one h2] at h3 show [] = l2 from h3.symm done · -- Induction Step fix p : Nat fix L1 : List Nat assume ih : ∀ (L2 : List Nat), nondec_prime_list L1 → nondec_prime_list L2 → prod L1 = prod L2 → L1 = L2 -- Goal : ∀ (l2 : List Nat), nondec_prime_list (p :: L1) → -- nondec_prime_list l2 → prod (p :: L1) = prod l2 → p :: L1 = l2 fix l2 : List Nat assume h1 : nondec_prime_list (p :: L1) assume h2 : nondec_prime_list l2 assume h3 : prod (p :: L1) = prod l2 have h4 : ¬prod (p :: L1) = 1 := cons_prod_not_one h1 rewrite [h3, ←list_nil_iff_prod_one h2] at h4 obtain (q : Nat) (h5 : ∃ (L : List Nat), l2 = q :: L) from List.exists_cons_of_ne_nil h4 obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5 rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2) rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2) have h7 : p ≤ q := first_le_first h1 h2 h3 have h8 : q ≤ p := first_le_first h2 h1 h3.symm have h9 : p = q := by linarith rewrite [h9, prod_cons, prod_cons] at h3 --h3 : q * prod L1 = q * prod L2 have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1 have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2 define at h2 have h12 : all_prime (q :: L2) := h2.left rewrite [all_prime_cons] at h12 have h13 : q > 0 := prime_pos h12.left have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3 have h15 : L1 = L2 := ih L2 h10 h11 h14 rewrite [h6, h9, h15] rfl done done theorem fund_thm_arith (n : Nat) (h : n ≥ 1) : ∃! (l : List Nat), prime_factorization n l := by exists_unique · -- Existence show ∃ (l : List Nat), prime_factorization n l from exists_prime_factorization n h done · -- Uniqueness fix l1 : List Nat; fix l2 : List Nat assume h1 : prime_factorization n l1 assume h2 : prime_factorization n l2 define at h1; define at h2 have h3 : prod l1 = n := h1.right rewrite [←h2.right] at h3 show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3 done done /- Section 7.3 -/ theorem congr_refl (m : Nat) : ∀ (a : Int), a ≡ a (MOD m) := by fix a : Int define --Goal : ∃ (c : Int), a - a = ↑m * c apply Exists.intro 0 ring done theorem congr_symm {m : Nat} : ∀ {a b : Int}, a ≡ b (MOD m) → b ≡ a (MOD m) := by fix a : Int; fix b : Int assume h1 : a ≡ b (MOD m) define at h1 --h1 : ∃ (c : Int), a - b = ↑m * c define --Goal : ∃ (c : Int), b - a = ↑m * c obtain (c : Int) (h2 : a - b = m * c) from h1 apply Exists.intro (-c) show b - a = m * (-c) from calc b - a _ = -(a - b) := by ring _ = -(m * c) := by rw [h2] _ = m * (-c) := by ring done theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry /- Fundamental properties of congruence classes -/ lemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) : X = Y ↔ X.val = Y.val := Fin.ext_iff lemma val_nat_eq_mod (n k : Nat) : ([k]_(n + 1)).val = k % (n + 1) := by rfl lemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl theorem cc_rep {m : Nat} (X : ZMod m) : ∃ (a : Int), X = [a]_m := match m with | 0 => by apply Exists.intro X rfl done | n + 1 => by apply Exists.intro ↑(X.val) have h1 : X.val < n + 1 := Fin.prop X rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1] rfl done theorem add_class (m : Nat) (a b : Int) : [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm theorem mul_class (m : Nat) (a b : Int) : [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm lemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ [a - b]_m = [0]_m := by apply Iff.intro · -- (→) assume h1 : [a]_m = [b]_m have h2 : a - b = a + (-b) := by ring have h3 : b + (-b) = 0 := by ring show [a - b]_m = [0]_m from calc [a - b]_m _ = [a + (-b)]_m := by rw [h2] _ = [a]_m + [-b]_m := by rw [add_class] _ = [b]_m + [-b]_m := by rw [h1] _ = [b + -b]_m := by rw [add_class] _ = [0]_m := by rw [h3] done · -- (←) assume h1 : [a - b]_m = [0]_m have h2 : b + (a - b) = a := by ring have h3 : b + 0 = b := by ring show [a]_m = [b]_m from calc [a]_m _ = [b + (a - b)]_m := by rw [h2] _ = [b]_m + [a - b]_m := by rw [add_class] _ = [b]_m + [0]_m := by rw [h1] _ = [b + 0]_m := by rw [add_class] _ = [b]_m := by rw [h3] done done lemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) : [a]_m = [0]_m → [-a]_m = [0]_m := by assume h1 : [a]_m = [0]_m have h2 : 0 + (-a) = -a := by ring have h3 : a + (-a) = 0 := by ring show [-a]_m = [0]_m from calc [-a]_m _ = [0 + (-a)]_m := by rw [h2] _ = [0]_m + [-a]_m := by rw [add_class] _ = [a]_m + [-a]_m := by rw [h1] _ = [a + (-a)]_m := by rw [add_class] _ = [0]_m := by rw [h3] done lemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) : [-a]_m = [0]_m ↔ [a]_m = [0]_m := by apply Iff.intro _ (cc_neg_zero_of_cc_zero m a) assume h1 : [-a]_m = [0]_m have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1 have h3 : -(-a) = a := by ring rewrite [h3] at h2 show [a]_m = [0]_m from h2 done lemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl lemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m ↔ m ∣ k := match m with | 0 => by have h : (0 : Int) = (↑(0 : Nat) : Int) := by rfl rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj] apply Iff.intro · -- (→) assume h1 : k = 0 rewrite [h1] show 0 ∣ 0 from dvd_self 0 done · -- (←) assume h1 : 0 ∣ k obtain (c : Nat) (h2 : k = 0 * c) from h1 rewrite [h2] ring done done | n + 1 => by rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero] show k % (n + 1) = 0 ↔ n + 1 ∣ k from (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm done lemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m ↔ ↑m ∣ a := by obtain (k : Nat) (h1 : a = ↑k ∨ a = -↑k) from Int.eq_nat_or_neg a by_cases on h1 · -- Case 1. h1: a = ↑k rewrite [h1, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done · -- Case 2. h1: a = -↑k rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done done theorem cc_eq_iff_congr (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ a ≡ b (MOD m) := calc [a]_m = [b]_m _ ↔ [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b _ ↔ ↑m ∣ (a - b) := cc_zero_iff_dvd m (a - b) _ ↔ a ≡ b (MOD m) := by rfl /- End of fundamental properties of congruence classes -/ lemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m := by have h1 : (↑m : Int) ≠ 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m) show 0 ≤ a % m from Int.emod_nonneg a h1 done lemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m) have h2 : (↑m : Int) > 0 := (Nat.cast_pos).rtl h1 show a % m < m from Int.emod_lt_of_pos a h2 done lemma congr_mod_mod (m : Nat) (a : Int) : a ≡ a % m (MOD m) := by define have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m apply Exists.intro (a / m) show a - a % m = m * (a / m) from calc a - (a % m) _ = m * (a / m) + a % m - a % m := by rw [h1] _ = m * (a / m) := by ring done lemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) := And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a)) theorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) : ∃! (r : Int), 0 ≤ r ∧ r < m ∧ a ≡ r (MOD m) := by exists_unique · -- Existence apply Exists.intro (a % m) show 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) from mod_cmpl_res m a done · -- Uniqueness fix r1 : Int; fix r2 : Int assume h1 : 0 ≤ r1 ∧ r1 < m ∧ a ≡ r1 (MOD m) assume h2 : 0 ≤ r2 ∧ r2 < m ∧ a ≡ r2 (MOD m) have h3 : r1 ≡ r2 (MOD m) := congr_trans (congr_symm h1.right.right) h2.right.right obtain (d : Int) (h4 : r1 - r2 = m * d) from h3 have h5 : r1 - r2 < m * 1 := by linarith have h6 : m * (-1) < r1 - r2 := by linarith rewrite [h4] at h5 --h5 : m * d < m * 1 rewrite [h4] at h6 --h6 : m * -1 < m * d have h7 : (↑m : Int) ≥ 0 := Nat.cast_nonneg m have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7 have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7 have h10 : d = 0 := by linarith show r1 = r2 from calc r1 _ = r1 - r2 + r2 := by ring _ = m * 0 + r2 := by rw [h4, h10] _ = r2 := by ring done done lemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m := (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a) theorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y rewrite [h1, h2] have h3 : a + b = b + a := by ring show [a]_m + [b]_m = [b]_m + [a]_m from calc [a]_m + [b]_m _ = [a + b]_m := add_class m a b _ = [b + a]_m := by rw [h3] _ = [b]_m + [a]_m := (add_class m b a).symm done theorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X rewrite [h1] have h2 : a * 1 = a := by ring show [a]_m * [1]_m = [a]_m from calc [a]_m * [1]_m _ = [a * 1]_m := mul_class m a 1 _ = [a]_m := by rw [h2] done theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry lemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) : [a]_m * [gcd_c2 m a]_m = [1]_m := by set s : Int := gcd_c1 m a have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m define at h1 rewrite [h1, Nat.cast_one] at h2 --h2 : s * ↑m + gcd_c2 m a * ↑a = 1 rewrite [mul_class, cc_eq_iff_congr] define --Goal : ∃ (c : Int), ↑a * gcd_c2 m a - 1 = ↑m * c apply Exists.intro (-s) show a * (gcd_c2 m a) - 1 = m * (-s) from calc a * (gcd_c2 m a) - 1 _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring _ = 1 + m * (-s) - 1 := by rw [h2] _ = m * (-s) := by ring done theorem Theorem_7_3_7 (m a : Nat) : invertible [a]_m ↔ rel_prime m a := by apply Iff.intro · -- (→) assume h1 : invertible [a]_m define at h1 obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1 obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y rewrite [h3, mul_class, cc_eq_iff_congr] at h2 define at h2 obtain (c : Int) (h4 : a * b - 1 = m * c) from h2 rewrite [Exercise_7_2_6] --Goal : ∃ (s t : Int), s * ↑m + t * ↑a = 1 apply Exists.intro (-c) apply Exists.intro b show (-c) * m + b * a = 1 from calc (-c) * m + b * a _ = (-c) * m + (a * b - 1) + 1 := by ring _ = (-c) * m + m * c + 1 := by rw [h4] _ = 1 := by ring done · -- (←) assume h1 : rel_prime m a define show ∃ (Y : ZMod m), [a]_m * Y = [1]_m from Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1) done done /- Section 7.4 -/ section Euler open Euler lemma num_rp_below_base {m : Nat} : num_rp_below m 0 = 0 := by rfl lemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) : num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by have h1 : num_rp_below m (j + 1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : gcd m j = 1 rewrite [if_pos h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1 show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1 done lemma num_rp_below_step_not_rp {m j : Nat} (h : ¬rel_prime m j) : num_rp_below m (j + 1) = num_rp_below m j := by have h1 : num_rp_below m (j +1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : ¬gcd m j = 1 rewrite [if_neg h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j show num_rp_below m (j + 1) = num_rp_below m j from h1 done lemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl #eval phi 10 --Answer: 4 lemma prod_inv_iff_inv {m : Nat} {X : ZMod m} (h1 : invertible X) (Y : ZMod m) : invertible (X * Y) ↔ invertible Y := by apply Iff.intro · -- (→) assume h2 : invertible (X * Y) obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2 apply Exists.intro (X * Z) rewrite [←h3] --Goal : Y * (X * Z) = X * Y * Z ring --Note that ring can do algebra in ZMod m done · -- (←) assume h2 : invertible Y obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1 obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2 apply Exists.intro (Xi * Yi) show (X * Y) * (Xi * Yi) = [1]_m from calc X * Y * (Xi * Yi) _ = (X * Xi) * (Y * Yi) := by ring _ = [1]_m * [1]_m := by rw [h3, h4] _ = [1]_m := Theorem_7_3_6_7 [1]_m done done lemma F_rp_def {m i : Nat} (h : rel_prime m i) : F m i = [i]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h --h : gcd m i = 1 rewrite [if_pos h] at h1 show F m i = [i]_m from h1 done lemma F_not_rp_def {m i : Nat} (h : ¬rel_prime m i) : F m i = [1]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h rewrite [h1, if_neg h] rfl done lemma prod_seq_base {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 0 k f = [1]_m := by rfl lemma prod_seq_step {m : Nat} (n k : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl lemma prod_seq_zero_step {m : Nat} (n : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by rewrite [prod_seq_step, zero_add] rfl done lemma prod_one {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 1 k f = f k := by rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7] rfl done lemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl lemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m := calc [G m a i]_m _ = [(a * i) % m]_m := by rfl _ = [a * i]_m := (cc_eq_mod m (a * i)).symm _ = [a]_m * [i]_m := (mul_class m a i).symm lemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) : rel_prime m (G m a i) ↔ rel_prime m i := by have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1 show rel_prime m (G m a i) ↔ rel_prime m i from calc rel_prime m (G m a i) _ ↔ invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm _ ↔ invertible ([a]_m * [i]_m) := by rw [cc_G] _ ↔ invertible [i]_m := prod_inv_iff_inv h2 ([i]_m) _ ↔ rel_prime m i := Theorem_7_3_7 m i done lemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) : F m (G m a i) = [a]_m * F m i := by have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2 show F m (G m a i) = [a]_m * F m i from calc F m (G m a i) _ = [G m a i]_m := F_rp_def h3 _ = [a]_m * [i]_m := cc_G m a i _ = [a]_m * F m i := by rw [F_rp_def h2] done lemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : ¬rel_prime m i) : F m (G m a i) = [1]_m := by rewrite [←G_rp_iff h1 i] at h2 show F m (G m a i) = [1]_m from F_not_rp_def h2 done lemma FG_prod {m a : Nat} (h1 : rel_prime m a) : ∀ (k : Nat), prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by by_induc · -- Base Case show prod_seq 0 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from calc prod_seq 0 0 ((F m) ∘ (G m a)) _ = [1]_m := prod_seq_base _ _ _ = [a]_m ^ 0 * [1]_m := by ring _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by rw [num_rp_below_base, prod_seq_base] done · -- Induction Step fix k : Nat assume ih : prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) by_cases h2 : rel_prime m k · -- Case 1. h2 : rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([a]_m * F m k) := by rw [FG_rp h1 h2] _ = [a]_m ^ ((num_rp_below m k) + 1) * ((prod_seq k 0 (F m)) * F m k) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_rp h2, prod_seq_zero_step] done · -- Case 2. h2 : ¬rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([1]_m) := by rw [FG_not_rp h1 h2] _ = [a]_m ^ (num_rp_below m k) * (prod_seq k 0 (F m) * ([1]_m)) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_not_rp h2, prod_seq_zero_step, F_not_rp_def h2] done done done lemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by define --Goal : ∀ i < m, G m a i < m fix i : Nat assume h1 : i < m rewrite [G_def] --Goal : a * i % m < m show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m) done lemma left_inv_one_one_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g' (g i) = i) : one_one_below n g := sorry lemma right_inv_onto_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g (g' i) = i) (h2 : maps_below n g') : onto_below n g := by define at h2; define fix k : Nat assume h3 : k < n apply Exists.intro (g' k) show g' k < n ∧ g (g' k) = k from And.intro (h2 k h3) (h1 k h3) done lemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m * [inv_mod m a]_m = [1]_m := by have h2 : 0 ≤ (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a) show [a]_m * [inv_mod m a]_m = [1]_m from calc [a]_m * [inv_mod m a]_m _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2] _ = [a]_m * [gcd_c2 m a]_m := by rw [←cc_eq_mod] _ = [1]_m := gcd_c2_inv h1 done lemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m := calc a * (b % m) % m = a % m * (b % m % m) % m := Nat.mul_mod _ _ _ _ = a % m * (b % m) % m := by rw [Nat.mod_mod] _ = a * b % m := (Nat.mul_mod _ _ _).symm lemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm] rfl done theorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] : ↑a ≡ ↑b (MOD m) ↔ a % m = b % m := sorry lemma mul_inv_mod_cancel {m a i : Nat} [NeZero m] (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1 rewrite [mul_class, cc_eq_iff_congr, ←Nat.cast_mul, ←Nat.cast_one, congr_iff_mod_eq_Nat] at h3 show a * inv_mod m a * i % m = i from calc a * (inv_mod m a) * i % m _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = 1 % m * i % m := by rw [h3] _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = i % m := by rw [one_mul] _ = i := Nat.mod_eq_of_lt h2 done lemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl lemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, G m a (Ginv m a i) = i := by fix i : Nat assume h2 : i < m show G m a (Ginv m a i) = i from calc G m a (Ginv m a i) _ = a * ((inv_mod m a * i) % m) % m := by rfl _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, Ginv m a (G m a i) = i := by fix i : Nat assume h2 : i < m show Ginv m a (G m a i) = i from calc Ginv m a (G m a i) _ = inv_mod m a * ((a * i) % m) % m := by rfl _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc, mul_comm (inv_mod m a)] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_maps_below (m a : Nat) [NeZero m] : maps_below m (Ginv m a) := G_maps_below m (inv_mod m a) lemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : one_one_below m (G m a) := left_inv_one_one_below (Ginv_left_inv h1) lemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : onto_below m (G m a) := right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a) lemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : perm_below m (G m a) := And.intro (G_maps_below m a) (And.intro (G_one_one_below h1) (G_onto_below h1)) --Permuting a product of congruence classes doesn't change product lemma swap_fst (u v : Nat) : swap u v u = v := by define : swap u v u --Goal : (if u = u then v else if u = v then u else u) = v have h : u = u := by rfl rewrite [if_pos h] rfl done lemma swap_snd (u v : Nat) : swap u v v = u := by define : swap u v v by_cases h1 : v = u · -- Case 1. h1 : v = u rewrite [if_pos h1] show v = u from h1 done · -- Case 2. h1 : v ≠ u rewrite [if_neg h1] have h2 : v = v := by rfl rewrite [if_pos h2] rfl done done lemma swap_other {u v i : Nat} (h1 : i ≠ u) (h2 : i ≠ v) : swap u v i = i := by define : swap u v i rewrite [if_neg h1, if_neg h2] rfl done lemma swap_values (u v i : Nat) : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := by by_cases h1 : i = u · -- Case 1. h1 : i = u apply Or.inl rewrite [h1] show swap u v u = v from swap_fst u v done · -- Case 2. h1 : i ≠ u apply Or.inr by_cases h2 : i = v · -- Case 2.1. h2 : i = v apply Or.inl rewrite [h2] show swap u v v = u from swap_snd u v done · -- Case 2.2. h2 : i ≠ v apply Or.inr show swap u v i = i from swap_other h1 h2 done done done lemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by define fix i : Nat assume h3 : i < n have h4 : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := swap_values u v i by_cases on h4 · -- Case 1. h4 : swap u v i = v rewrite [h4] show v < n from h2 done · -- Case 2. by_cases on h4 · -- Case 2.1. h4 : swap u v i = u rewrite [h4] show u < n from h1 done · -- Case 2.2. h4 : swap u v i = i rewrite [h4] show i < n from h3 done done done lemma swap_swap (u v n : Nat) : ∀ i < n, swap u v (swap u v i) = i := by fix i : Nat assume h : i < n by_cases h1 : i = u · -- Case 1. h1 : i = u rewrite [h1, swap_fst, swap_snd] rfl done · -- Case 2. h1 : i ≠ u by_cases h2 : i = v · -- Case 2.1. h2 : i = v rewrite [h2, swap_snd, swap_fst] rfl done · -- Case 2.2. h2 : i ≠ v rewrite [swap_other h1 h2, swap_other h1 h2] rfl done done done lemma swap_one_one_below (u v n) : one_one_below n (swap u v) := left_inv_one_one_below (swap_swap u v n) lemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) := right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2) lemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) := And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2)) lemma comp_perm_below {n : Nat} {f g : Nat → Nat} (h1 : perm_below n f) (h2 : perm_below n g) : perm_below n (f ∘ g) := sorry lemma trivial_swap (u : Nat) : swap u u = id := by apply funext fix x : Nat by_cases h1 : x = u · -- Case 1. h1 : x = u rewrite [h1, swap_fst] rfl done · -- Case 2. h1 : x ≠ u rewrite [swap_other h1 h1] rfl done done

lemma prod_eq_fun {m : Nat} (f g : Nat → ZMod m) (k : Nat) : ∀ (n : Nat), (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g

HTPI.prod_eq_fun

htpi/HTPILib/Chap7.lean

htpi

/- Copyright 2023 Daniel J. Velleman -/ import HTPILib.Chap6 namespace HTPI /- Definitions -/ lemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by have h : n + 1 > 0 := Nat.succ_pos n show a % (n + 1) < n + 1 from Nat.mod_lt a h done def gcd (a b : Nat) : Nat := match b with | 0 => a | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd (n + 1) (a % (n + 1)) termination_by b mutual def gcd_c1 (a b : Nat) : Int := match b with | 0 => 1 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c2 (n + 1) (a % (n + 1)) --Corresponds to s = t' termination_by b def gcd_c2 (a b : Nat) : Int := match b with | 0 => 0 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c1 (n + 1) (a % (n + 1)) - (gcd_c2 (n + 1) (a % (n + 1))) * ↑(a / (n + 1)) --Corresponds to t = s' - t'q termination_by b end def prime (n : Nat) : Prop := 2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n def prime_factor (p n : Nat) : Prop := prime p ∧ p ∣ n def all_prime (l : List Nat) : Prop := ∀ p ∈ l, prime p def nondec (l : List Nat) : Prop := match l with | [] => True --Of course, True is a proposition that is always true | n :: L => (∀ m ∈ L, n ≤ m) ∧ nondec L def nondec_prime_list (l : List Nat) : Prop := all_prime l ∧ nondec l def prod (l : List Nat) : Nat := match l with | [] => 1 | n :: L => n * (prod L) def prime_factorization (n : Nat) (l : List Nat) : Prop := nondec_prime_list l ∧ prod l = n def rel_prime (a b : Nat) : Prop := gcd a b = 1 def congr_mod (m : Nat) (a b : Int) : Prop := (↑m : Int) ∣ (a - b) def cc (m : Nat) (a : Int) : ZMod m := (↑a : ZMod m) notation:50 a " ≡ " b " (MOD " m ")" => congr_mod m a b notation:max "["a"]_"m:max => cc m a def invertible {m : Nat} (X : ZMod m) : Prop := ∃ (Y : ZMod m), X * Y = [1]_m def num_rp_below (m k : Nat) : Nat := match k with | 0 => 0 | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j def phi (m : Nat) : Nat := num_rp_below m m def prod_seq {m : Nat} (j k : Nat) (f : Nat → ZMod m) : ZMod m := match j with | 0 => [1]_m | n + 1 => prod_seq n k f * f (k + n) def maps_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i < n, g i < n def one_one_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i1 < n, ∀ i2 < n, g i1 = g i2 → i1 = i2 def onto_below (n : Nat) (g : Nat → Nat) : Prop := ∀ k < n, ∃ i < n, g i = k def perm_below (n : Nat) (g : Nat → Nat) : Prop := maps_below n g ∧ one_one_below n g ∧ onto_below n g def inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m) def swap (u v i : Nat) : Nat := if i = u then v else if i = v then u else i namespace Euler --For definitions specific to Euler's theorem def F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m def G (m a i : Nat) : Nat := (a * i) % m def Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i end Euler /- Section 7.1 -/ theorem dvd_mod_of_dvd_a_b {a b d : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ (a % b) := by set q : Nat := a / b have h3 : b * q + a % b = a := Nat.div_add_mod a b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 define --Goal : ∃ (c : Nat), a % b = d * c apply Exists.intro (j - k * q) show a % b = d * (j - k * q) from calc a % b _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm _ = a - b * q := by rw [h3] _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc] _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm done theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry #eval gcd 672 161 --Answer: 7 lemma gcd_base (a : Nat) : gcd a 0 = a := by rfl lemma gcd_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd a b = gcd b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1)) rfl done lemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b ≠ 0) : a % b < b := by have h1 : b > 0 := Nat.pos_of_ne_zero h show a % b < b from Nat.mod_lt a h1 done lemma dvd_self (n : Nat) : n ∣ n := by apply Exists.intro 1 ring done theorem gcd_dvd : ∀ (b a : Nat), (gcd a b) ∣ a ∧ (gcd a b) ∣ b := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd a b_1) ∣ a ∧ (gcd a b_1) ∣ b_1 fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_base] --Goal: a ∣ a ∧ a ∣ 0 apply And.intro (dvd_self a) define apply Exists.intro 0 rfl done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_nonzero a h1] --Goal : gcd b (a % b) ∣ a ∧ gcd b (a % b) ∣ b have h2 : a % b < b := mod_nonzero_lt a h1 have h3 : (gcd b (a % b)) ∣ b ∧ (gcd b (a % b)) ∣ (a % b) := ih (a % b) h2 b apply And.intro _ h3.left show (gcd b (a % b)) ∣ a from dvd_a_of_dvd_b_mod h3.left h3.right done done theorem gcd_dvd_left (a b : Nat) : (gcd a b) ∣ a := (gcd_dvd b a).left theorem gcd_dvd_right (a b : Nat) : (gcd a b) ∣ b := (gcd_dvd b a).right lemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl lemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c1 a b = gcd_c2 b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done lemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl lemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * ↑(a / b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done theorem gcd_lin_comb : ∀ (b a : Nat), (gcd_c1 a b) * ↑a + (gcd_c2 a b) * ↑b = ↑(gcd a b) := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd_c1 a b_1) * ↑a + (gcd_c2 a b_1) * ↑b_1 = ↑(gcd a b_1) fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base] --Goal : 1 * ↑a + 0 * ↑0 = ↑a ring done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1] --Goal : gcd_c2 b (a % b) * ↑a + -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * ↑(a / b)) * ↑b = -- ↑(gcd b (a % b)) set r : Nat := a % b set q : Nat := a / b set s : Int := gcd_c1 b r set t : Int := gcd_c2 b r --Goal : t * ↑a + (s - t * ↑q) * ↑b = ↑(gcd b r) have h2 : r < b := mod_nonzero_lt a h1 have h3 : s * ↑b + t * ↑r = ↑(gcd b r) := ih r h2 b have h4 : b * q + r = a := Nat.div_add_mod a b rewrite [←h3, ←h4] rewrite [Nat.cast_add, Nat.cast_mul] --Goal : t * (↑b * ↑q + ↑r) + (s - t * ↑q) * ↑b = s * ↑b + t * ↑r ring done done #eval gcd_c1 672 161 --Answer: 6 #eval gcd_c2 672 161 --Answer: -25 --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161 theorem Theorem_7_1_6 {d a b : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ gcd a b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑d ∣ ↑(gcd a b) set s : Int := gcd_c1 a b set t : Int := gcd_c2 a b have h3 : s * ↑a + t * ↑b = ↑(gcd a b) := gcd_lin_comb b a rewrite [←h3] --Goal : ↑d ∣ s * ↑a + t * ↑b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul] --Goal : ↑d ∣ s * (↑d * ↑j) + t * (↑d * ↑k) define apply Exists.intro (s * ↑j + t * ↑k) ring done /- Section 7.2 -/ theorem dvd_trans {a b c : Nat} (h1 : a ∣ b) (h2 : b ∣ c) : a ∣ c := by define at h1; define at h2; define obtain (m : Nat) (h3 : b = a * m) from h1 obtain (n : Nat) (h4 : c = b * n) from h2 rewrite [h3, mul_assoc] at h4 apply Exists.intro (m * n) show c = a * (m * n) from h4 done lemma exists_prime_factor : ∀ (n : Nat), 2 ≤ n → ∃ (p : Nat), prime_factor p n := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, 2 ≤ n_1 → ∃ (p : Nat), prime_factor p n_1 assume h1 : 2 ≤ n by_cases h2 : prime n · -- Case 1. h2 : prime n apply Exists.intro n define --Goal : prime n ∧ n ∣ n show prime n ∧ n ∣ n from And.intro h2 (dvd_self n) done · -- Case 2. h2 : ¬prime n define at h2 --h2 : ¬(2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n) demorgan at h2 disj_syll h2 h1 obtain (a : Nat) (h3 : ∃ (b : Nat), a * b = n ∧ a < n ∧ b < n) from h2 obtain (b : Nat) (h4 : a * b = n ∧ a < n ∧ b < n) from h3 have h5 : 2 ≤ a := by by_contra h6 have h7 : a ≤ 1 := by linarith have h8 : n ≤ b := calc n _ = a * b := h4.left.symm _ ≤ 1 * b := by rel [h7] _ = b := by ring linarith --n ≤ b contradicts b < n done have h6 : ∃ (p : Nat), prime_factor p a := ih a h4.right.left h5 obtain (p : Nat) (h7 : prime_factor p a) from h6 apply Exists.intro p define --Goal : prime p ∧ p ∣ n define at h7 --h7 : prime p ∧ p ∣ a apply And.intro h7.left have h8 : a ∣ n := by apply Exists.intro b show n = a * b from (h4.left).symm done show p ∣ n from dvd_trans h7.right h8 done done lemma exists_least_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q := by set S : Set Nat := {p : Nat | prime_factor p n} have h2 : ∃ (p : Nat), p ∈ S := exists_prime_factor n h show ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q from well_ord_princ S h2 done lemma all_prime_nil : all_prime [] := by define --Goal : ∀ p ∈ [], prime p fix p : Nat contrapos --Goal : ¬prime p → p ∉ [] assume h1 : ¬prime p show p ∉ [] from List.not_mem_nil p done lemma all_prime_cons (n : Nat) (L : List Nat) : all_prime (n :: L) ↔ prime n ∧ all_prime L := by apply Iff.intro · -- (→) assume h1 : all_prime (n :: L) --Goal : prime n ∧ all_prime L define at h1 --h1 : ∀ p ∈ n :: L, prime p apply And.intro (h1 n (List.mem_cons_self n L)) define --Goal : ∀ p ∈ L, prime p fix p : Nat assume h2 : p ∈ L show prime p from h1 p (List.mem_cons_of_mem n h2) done · -- (←) assume h1 : prime n ∧ all_prime L --Goal : all_prime (n :: l) define : all_prime L at h1 define fix p : Nat assume h2 : p ∈ n :: L rewrite [List.mem_cons] at h2 --h2 : p = n ∨ p ∈ L by_cases on h2 · -- Case 1. h2 : p = n rewrite [h2] show prime n from h1.left done · -- Case 2. h2 : p ∈ L show prime p from h1.right p h2 done done done lemma nondec_nil : nondec [] := by define --Goal : True trivial --trivial proves some obviously true statements, such as True done lemma nondec_cons (n : Nat) (L : List Nat) : nondec (n :: L) ↔ (∀ m ∈ L, n ≤ m) ∧ nondec L := by rfl lemma prod_nil : prod [] = 1 := by rfl lemma prod_cons : prod (n :: L) = n * (prod L) := by rfl lemma exists_cons_of_length_eq_succ {A : Type} {l : List A} {n : Nat} (h : l.length = n + 1) : ∃ (a : A) (L : List A), l = a :: L ∧ L.length = n := by have h1 : ¬l.length = 0 := by linarith rewrite [List.length_eq_zero] at h1 obtain (a : A) (h2 : ∃ (L : List A), l = a :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List A) (h3 : l = a :: L) from h2 apply Exists.intro a apply Exists.intro L apply And.intro h3 have h4 : (a :: L).length = L.length + 1 := List.length_cons a L rewrite [←h3, h] at h4 show L.length = n from (Nat.add_right_cancel h4).symm done lemma list_elt_dvd_prod_by_length (a : Nat) : ∀ (n : Nat), ∀ (l : List Nat), l.length = n → a ∈ l → a ∣ prod l := by by_induc · --Base Case fix l : List Nat assume h1 : l.length = 0 rewrite [List.length_eq_zero] at h1 --h1 : l = [] rewrite [h1] --Goal : a ∈ [] → a ∣ prod [] contrapos assume h2 : ¬a ∣ prod [] show a ∉ [] from List.not_mem_nil a done · -- Induction Step fix n : Nat assume ih : ∀ (l : List Nat), List.length l = n → a ∈ l → a ∣ prod l fix l : List Nat assume h1 : l.length = n + 1 --Goal : a ∈ l → a ∣ prod l obtain (b : Nat) (h2 : ∃ (L : List Nat), l = b :: L ∧ L.length = n) from exists_cons_of_length_eq_succ h1 obtain (L : List Nat) (h3 : l = b :: L ∧ L.length = n) from h2 have h4 : a ∈ L → a ∣ prod L := ih L h3.right assume h5 : a ∈ l rewrite [h3.left, prod_cons] --Goal : a ∣ b * prod L rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b ∨ a ∈ L by_cases on h5 · -- Case 1. h5 : a = b apply Exists.intro (prod L) rewrite [h5] rfl done · -- Case 2. h5 : a ∈ L have h6 : a ∣ prod L := h4 h5 have h7 : prod L ∣ b * prod L := by apply Exists.intro b ring done show a ∣ b * prod L from dvd_trans h6 h7 done done done lemma list_elt_dvd_prod {a : Nat} {l : List Nat} (h : a ∈ l) : a ∣ prod l := by set n : Nat := l.length have h1 : l.length = n := by rfl show a ∣ prod l from list_elt_dvd_prod_by_length a n l h1 h done lemma exists_prime_factorization : ∀ (n : Nat), n ≥ 1 → ∃ (l : List Nat), prime_factorization n l := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, n_1 ≥ 1 → ∃ (l : List Nat), prime_factorization n_1 l assume h1 : n ≥ 1 by_cases h2 : n = 1 · -- Case 1. h2 : n = 1 apply Exists.intro [] define apply And.intro · -- Proof of nondec_prime_list [] define show all_prime [] ∧ nondec [] from And.intro all_prime_nil nondec_nil done · -- Proof of prod [] = n rewrite [prod_nil, h2] rfl done done · -- Case 2. h2 : n ≠ 1 have h3 : n ≥ 2 := lt_of_le_of_ne' h1 h2 obtain (p : Nat) (h4 : prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q) from exists_least_prime_factor h3 have p_prime_factor : prime_factor p n := h4.left define at p_prime_factor have p_prime : prime p := p_prime_factor.left have p_dvd_n : p ∣ n := p_prime_factor.right have p_least : ∀ (q : Nat), prime_factor q n → p ≤ q := h4.right obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n have h5 : m ≠ 0 := by contradict h1 with h6 have h7 : n = 0 := calc n _ = p * m := n_eq_pm _ = p * 0 := by rw [h6] _ = 0 := by ring rewrite [h7] decide done have m_pos : 0 < m := Nat.pos_of_ne_zero h5 have m_lt_n : m < n := by define at p_prime show m < n from calc m _ < m + m := by linarith _ = 2 * m := by ring _ ≤ p * m := by rel [p_prime.left] _ = n := n_eq_pm.symm done obtain (L : List Nat) (h6 : prime_factorization m L) from ih m m_lt_n m_pos define at h6 have ndpl_L : nondec_prime_list L := h6.left define at ndpl_L apply Exists.intro (p :: L) define apply And.intro · -- Proof of nondec_prime_list (p :: L) define apply And.intro · -- Proof of all_prime (p :: L) rewrite [all_prime_cons] show prime p ∧ all_prime L from And.intro p_prime ndpl_L.left done · -- Proof of nondec (p :: L) rewrite [nondec_cons] apply And.intro _ ndpl_L.right fix q : Nat assume q_in_L : q ∈ L have h7 : q ∣ prod L := list_elt_dvd_prod q_in_L rewrite [h6.right] at h7 --h7 : q ∣ m have h8 : m ∣ n := by apply Exists.intro p rewrite [n_eq_pm] ring done have q_dvd_n : q ∣ n := dvd_trans h7 h8 have ap_L : all_prime L := ndpl_L.left define at ap_L have q_prime_factor : prime_factor q n := And.intro (ap_L q q_in_L) q_dvd_n show p ≤ q from p_least q q_prime_factor done done · -- Proof of prod (p :: L) = n rewrite [prod_cons, h6.right, n_eq_pm] rfl done done done theorem Theorem_7_2_2 {a b c : Nat} (h1 : c ∣ a * b) (h2 : rel_prime a c) : c ∣ b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑c ∣ ↑b define at h1; define at h2; define obtain (j : Nat) (h3 : a * b = c * j) from h1 set s : Int := gcd_c1 a c set t : Int := gcd_c2 a c have h4 : s * ↑a + t * ↑c = ↑(gcd a c) := gcd_lin_comb c a rewrite [h2, Nat.cast_one] at h4 --h4 : s * ↑a + t * ↑c = (1 : Int) apply Exists.intro (s * ↑j + t * ↑b) show ↑b = ↑c * (s * ↑j + t * ↑b) from calc ↑b _ = (1 : Int) * ↑b := (one_mul _).symm _ = (s * ↑a + t * ↑c) * ↑b := by rw [h4] _ = s * (↑a * ↑b) + t * ↑c * ↑b := by ring _ = s * (↑c * ↑j) + t * ↑c * ↑b := by rw [←Nat.cast_mul a b, h3, Nat.cast_mul c j] _ = ↑c * (s * ↑j + t * ↑b) := by ring done lemma le_nonzero_prod_left {a b : Nat} (h : a * b ≠ 0) : a ≤ a * b := by have h1 : b ≠ 0 := by contradict h with h1 rewrite [h1] ring done have h2 : 1 ≤ b := Nat.pos_of_ne_zero h1 show a ≤ a * b from calc a = a * 1 := (mul_one a).symm _ ≤ a * b := by rel [h2] done lemma le_nonzero_prod_right {a b : Nat} (h : a * b ≠ 0) : b ≤ a * b := by rewrite [mul_comm] rewrite [mul_comm] at h show b ≤ b * a from le_nonzero_prod_left h done lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry lemma rel_prime_of_prime_not_dvd {a p : Nat} (h1 : prime p) (h2 : ¬p ∣ a) : rel_prime a p := by have h3 : gcd a p ∣ a := gcd_dvd_left a p have h4 : gcd a p ∣ p := gcd_dvd_right a p have h5 : gcd a p = 1 ∨ gcd a p = p := dvd_prime h1 h4 have h6 : gcd a p ≠ p := by contradict h2 with h6 rewrite [h6] at h3 show p ∣ a from h3 done disj_syll h5 h6 show rel_prime a p from h5 done theorem Theorem_7_2_3 {a b p : Nat} (h1 : prime p) (h2 : p ∣ a * b) : p ∣ a ∨ p ∣ b := by or_right with h3 have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3 show p ∣ b from Theorem_7_2_2 h2 h4 done lemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a ≥ 1 := by have h1 : a ≠ 0 := by by_contra h1 rewrite [h1] at h contradict h linarith done show a ≥ 1 from Nat.pos_of_ne_zero h1 done lemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by have h1 : a ≥ 1 := ge_one_of_prod_one h have h2 : a * b ≠ 0 := by linarith have h3 : a ≤ a * b := le_nonzero_prod_left h2 rewrite [h] at h3 show a = 1 from Nat.le_antisymm h3 h1 done lemma eq_one_of_dvd_one {n : Nat} (h : n ∣ 1) : n = 1 := by obtain (j : Nat) (h1 : 1 = n * j) from h show n = 1 from eq_one_of_prod_one h1.symm done lemma prime_not_one {p : Nat} (h : prime p) : p ≠ 1 := by define at h linarith done theorem Theorem_7_2_4 {p : Nat} (h1 : prime p) : ∀ (l : List Nat), p ∣ prod l → ∃ a ∈ l, p ∣ a := by apply List.rec · -- Base Case. Goal : p ∣ prod [] → ∃ a ∈ [], p ∣ a rewrite [prod_nil] assume h2 : p ∣ 1 show ∃ a ∈ [], p ∣ a from absurd (eq_one_of_dvd_one h2) (prime_not_one h1) done · -- Induction Step fix b : Nat fix L : List Nat assume ih : p ∣ prod L → ∃ a ∈ L, p ∣ a --Goal : p ∣ prod (b :: L) → ∃ a ∈ b :: L, p ∣ a assume h2 : p ∣ prod (b :: L) rewrite [prod_cons] at h2 have h3 : p ∣ b ∨ p ∣ prod L := Theorem_7_2_3 h1 h2 by_cases on h3 · -- Case 1. h3 : p ∣ b apply Exists.intro b show b ∈ b :: L ∧ p ∣ b from And.intro (List.mem_cons_self b L) h3 done · -- Case 2. h3 : p ∣ prod L obtain (a : Nat) (h4 : a ∈ L ∧ p ∣ a) from ih h3 apply Exists.intro a show a ∈ b :: L ∧ p ∣ a from And.intro (List.mem_cons_of_mem b h4.left) h4.right done done done lemma prime_in_list {p : Nat} {l : List Nat} (h1 : prime p) (h2 : all_prime l) (h3 : p ∣ prod l) : p ∈ l := by obtain (a : Nat) (h4 : a ∈ l ∧ p ∣ a) from Theorem_7_2_4 h1 l h3 define at h2 have h5 : prime a := h2 a h4.left have h6 : p = 1 ∨ p = a := dvd_prime h5 h4.right disj_syll h6 (prime_not_one h1) rewrite [h6] show a ∈ l from h4.left done lemma first_le_first {p q : Nat} {l m : List Nat} (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m)) (h3 : prod (p :: l) = prod (q :: m)) : p ≤ q := by define at h1; define at h2 have h4 : q ∣ prod (p :: l) := by define apply Exists.intro (prod m) rewrite [←prod_cons] show prod (p :: l) = prod (q :: m) from h3 done have h5 : all_prime (q :: m) := h2.left rewrite [all_prime_cons] at h5 have h6 : q ∈ p :: l := prime_in_list h5.left h1.left h4 have h7 : nondec (p :: l) := h1.right rewrite [nondec_cons] at h7 rewrite [List.mem_cons] at h6 by_cases on h6 · -- Case 1. h6 : q = p linarith done · -- Case 2. h6 : q ∈ l have h8 : ∀ m ∈ l, p ≤ m := h7.left show p ≤ q from h8 q h6 done done lemma nondec_prime_list_tail {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by define at h define rewrite [all_prime_cons, nondec_cons] at h show all_prime l ∧ nondec l from And.intro h.left.right h.right.right done lemma cons_prod_not_one {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : prod (p :: l) ≠ 1 := by define at h have h1 : all_prime (p :: l) := h.left rewrite [all_prime_cons] at h1 rewrite [prod_cons] by_contra h2 show False from (prime_not_one h1.left) (eq_one_of_prod_one h2) done lemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) : l = [] ↔ prod l = 1 := by apply Iff.intro · -- (→) assume h1 : l = [] rewrite [h1] show prod [] = 1 from prod_nil done · -- (←) contrapos assume h1 : ¬l = [] obtain (p : Nat) (h2 : ∃ (L : List Nat), l = p :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List Nat) (h3 : l = p :: L) from h2 rewrite [h3] at h rewrite [h3] show ¬prod (p :: L) = 1 from cons_prod_not_one h done done lemma prime_pos {p : Nat} (h : prime p) : p > 0 := by define at h linarith done theorem Theorem_7_2_5 : ∀ (l1 l2 : List Nat), nondec_prime_list l1 → nondec_prime_list l2 → prod l1 = prod l2 → l1 = l2 := by apply List.rec · -- Base Case. Goal : ∀ (l2 : List Nat), nondec_prime_list [] → -- nondec_prime_list l2 → prod [] = prod l2 → [] = l2 fix l2 : List Nat assume h1 : nondec_prime_list [] assume h2 : nondec_prime_list l2 assume h3 : prod [] = prod l2 rewrite [prod_nil, eq_comm, ←list_nil_iff_prod_one h2] at h3 show [] = l2 from h3.symm done · -- Induction Step fix p : Nat fix L1 : List Nat assume ih : ∀ (L2 : List Nat), nondec_prime_list L1 → nondec_prime_list L2 → prod L1 = prod L2 → L1 = L2 -- Goal : ∀ (l2 : List Nat), nondec_prime_list (p :: L1) → -- nondec_prime_list l2 → prod (p :: L1) = prod l2 → p :: L1 = l2 fix l2 : List Nat assume h1 : nondec_prime_list (p :: L1) assume h2 : nondec_prime_list l2 assume h3 : prod (p :: L1) = prod l2 have h4 : ¬prod (p :: L1) = 1 := cons_prod_not_one h1 rewrite [h3, ←list_nil_iff_prod_one h2] at h4 obtain (q : Nat) (h5 : ∃ (L : List Nat), l2 = q :: L) from List.exists_cons_of_ne_nil h4 obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5 rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2) rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2) have h7 : p ≤ q := first_le_first h1 h2 h3 have h8 : q ≤ p := first_le_first h2 h1 h3.symm have h9 : p = q := by linarith rewrite [h9, prod_cons, prod_cons] at h3 --h3 : q * prod L1 = q * prod L2 have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1 have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2 define at h2 have h12 : all_prime (q :: L2) := h2.left rewrite [all_prime_cons] at h12 have h13 : q > 0 := prime_pos h12.left have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3 have h15 : L1 = L2 := ih L2 h10 h11 h14 rewrite [h6, h9, h15] rfl done done theorem fund_thm_arith (n : Nat) (h : n ≥ 1) : ∃! (l : List Nat), prime_factorization n l := by exists_unique · -- Existence show ∃ (l : List Nat), prime_factorization n l from exists_prime_factorization n h done · -- Uniqueness fix l1 : List Nat; fix l2 : List Nat assume h1 : prime_factorization n l1 assume h2 : prime_factorization n l2 define at h1; define at h2 have h3 : prod l1 = n := h1.right rewrite [←h2.right] at h3 show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3 done done /- Section 7.3 -/ theorem congr_refl (m : Nat) : ∀ (a : Int), a ≡ a (MOD m) := by fix a : Int define --Goal : ∃ (c : Int), a - a = ↑m * c apply Exists.intro 0 ring done theorem congr_symm {m : Nat} : ∀ {a b : Int}, a ≡ b (MOD m) → b ≡ a (MOD m) := by fix a : Int; fix b : Int assume h1 : a ≡ b (MOD m) define at h1 --h1 : ∃ (c : Int), a - b = ↑m * c define --Goal : ∃ (c : Int), b - a = ↑m * c obtain (c : Int) (h2 : a - b = m * c) from h1 apply Exists.intro (-c) show b - a = m * (-c) from calc b - a _ = -(a - b) := by ring _ = -(m * c) := by rw [h2] _ = m * (-c) := by ring done theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry /- Fundamental properties of congruence classes -/ lemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) : X = Y ↔ X.val = Y.val := Fin.ext_iff lemma val_nat_eq_mod (n k : Nat) : ([k]_(n + 1)).val = k % (n + 1) := by rfl lemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl theorem cc_rep {m : Nat} (X : ZMod m) : ∃ (a : Int), X = [a]_m := match m with | 0 => by apply Exists.intro X rfl done | n + 1 => by apply Exists.intro ↑(X.val) have h1 : X.val < n + 1 := Fin.prop X rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1] rfl done theorem add_class (m : Nat) (a b : Int) : [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm theorem mul_class (m : Nat) (a b : Int) : [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm lemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ [a - b]_m = [0]_m := by apply Iff.intro · -- (→) assume h1 : [a]_m = [b]_m have h2 : a - b = a + (-b) := by ring have h3 : b + (-b) = 0 := by ring show [a - b]_m = [0]_m from calc [a - b]_m _ = [a + (-b)]_m := by rw [h2] _ = [a]_m + [-b]_m := by rw [add_class] _ = [b]_m + [-b]_m := by rw [h1] _ = [b + -b]_m := by rw [add_class] _ = [0]_m := by rw [h3] done · -- (←) assume h1 : [a - b]_m = [0]_m have h2 : b + (a - b) = a := by ring have h3 : b + 0 = b := by ring show [a]_m = [b]_m from calc [a]_m _ = [b + (a - b)]_m := by rw [h2] _ = [b]_m + [a - b]_m := by rw [add_class] _ = [b]_m + [0]_m := by rw [h1] _ = [b + 0]_m := by rw [add_class] _ = [b]_m := by rw [h3] done done lemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) : [a]_m = [0]_m → [-a]_m = [0]_m := by assume h1 : [a]_m = [0]_m have h2 : 0 + (-a) = -a := by ring have h3 : a + (-a) = 0 := by ring show [-a]_m = [0]_m from calc [-a]_m _ = [0 + (-a)]_m := by rw [h2] _ = [0]_m + [-a]_m := by rw [add_class] _ = [a]_m + [-a]_m := by rw [h1] _ = [a + (-a)]_m := by rw [add_class] _ = [0]_m := by rw [h3] done lemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) : [-a]_m = [0]_m ↔ [a]_m = [0]_m := by apply Iff.intro _ (cc_neg_zero_of_cc_zero m a) assume h1 : [-a]_m = [0]_m have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1 have h3 : -(-a) = a := by ring rewrite [h3] at h2 show [a]_m = [0]_m from h2 done lemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl lemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m ↔ m ∣ k := match m with | 0 => by have h : (0 : Int) = (↑(0 : Nat) : Int) := by rfl rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj] apply Iff.intro · -- (→) assume h1 : k = 0 rewrite [h1] show 0 ∣ 0 from dvd_self 0 done · -- (←) assume h1 : 0 ∣ k obtain (c : Nat) (h2 : k = 0 * c) from h1 rewrite [h2] ring done done | n + 1 => by rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero] show k % (n + 1) = 0 ↔ n + 1 ∣ k from (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm done lemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m ↔ ↑m ∣ a := by obtain (k : Nat) (h1 : a = ↑k ∨ a = -↑k) from Int.eq_nat_or_neg a by_cases on h1 · -- Case 1. h1: a = ↑k rewrite [h1, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done · -- Case 2. h1: a = -↑k rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done done theorem cc_eq_iff_congr (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ a ≡ b (MOD m) := calc [a]_m = [b]_m _ ↔ [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b _ ↔ ↑m ∣ (a - b) := cc_zero_iff_dvd m (a - b) _ ↔ a ≡ b (MOD m) := by rfl /- End of fundamental properties of congruence classes -/ lemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m := by have h1 : (↑m : Int) ≠ 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m) show 0 ≤ a % m from Int.emod_nonneg a h1 done lemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m) have h2 : (↑m : Int) > 0 := (Nat.cast_pos).rtl h1 show a % m < m from Int.emod_lt_of_pos a h2 done lemma congr_mod_mod (m : Nat) (a : Int) : a ≡ a % m (MOD m) := by define have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m apply Exists.intro (a / m) show a - a % m = m * (a / m) from calc a - (a % m) _ = m * (a / m) + a % m - a % m := by rw [h1] _ = m * (a / m) := by ring done lemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) := And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a)) theorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) : ∃! (r : Int), 0 ≤ r ∧ r < m ∧ a ≡ r (MOD m) := by exists_unique · -- Existence apply Exists.intro (a % m) show 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) from mod_cmpl_res m a done · -- Uniqueness fix r1 : Int; fix r2 : Int assume h1 : 0 ≤ r1 ∧ r1 < m ∧ a ≡ r1 (MOD m) assume h2 : 0 ≤ r2 ∧ r2 < m ∧ a ≡ r2 (MOD m) have h3 : r1 ≡ r2 (MOD m) := congr_trans (congr_symm h1.right.right) h2.right.right obtain (d : Int) (h4 : r1 - r2 = m * d) from h3 have h5 : r1 - r2 < m * 1 := by linarith have h6 : m * (-1) < r1 - r2 := by linarith rewrite [h4] at h5 --h5 : m * d < m * 1 rewrite [h4] at h6 --h6 : m * -1 < m * d have h7 : (↑m : Int) ≥ 0 := Nat.cast_nonneg m have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7 have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7 have h10 : d = 0 := by linarith show r1 = r2 from calc r1 _ = r1 - r2 + r2 := by ring _ = m * 0 + r2 := by rw [h4, h10] _ = r2 := by ring done done lemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m := (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a) theorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y rewrite [h1, h2] have h3 : a + b = b + a := by ring show [a]_m + [b]_m = [b]_m + [a]_m from calc [a]_m + [b]_m _ = [a + b]_m := add_class m a b _ = [b + a]_m := by rw [h3] _ = [b]_m + [a]_m := (add_class m b a).symm done theorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X rewrite [h1] have h2 : a * 1 = a := by ring show [a]_m * [1]_m = [a]_m from calc [a]_m * [1]_m _ = [a * 1]_m := mul_class m a 1 _ = [a]_m := by rw [h2] done theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry lemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) : [a]_m * [gcd_c2 m a]_m = [1]_m := by set s : Int := gcd_c1 m a have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m define at h1 rewrite [h1, Nat.cast_one] at h2 --h2 : s * ↑m + gcd_c2 m a * ↑a = 1 rewrite [mul_class, cc_eq_iff_congr] define --Goal : ∃ (c : Int), ↑a * gcd_c2 m a - 1 = ↑m * c apply Exists.intro (-s) show a * (gcd_c2 m a) - 1 = m * (-s) from calc a * (gcd_c2 m a) - 1 _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring _ = 1 + m * (-s) - 1 := by rw [h2] _ = m * (-s) := by ring done theorem Theorem_7_3_7 (m a : Nat) : invertible [a]_m ↔ rel_prime m a := by apply Iff.intro · -- (→) assume h1 : invertible [a]_m define at h1 obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1 obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y rewrite [h3, mul_class, cc_eq_iff_congr] at h2 define at h2 obtain (c : Int) (h4 : a * b - 1 = m * c) from h2 rewrite [Exercise_7_2_6] --Goal : ∃ (s t : Int), s * ↑m + t * ↑a = 1 apply Exists.intro (-c) apply Exists.intro b show (-c) * m + b * a = 1 from calc (-c) * m + b * a _ = (-c) * m + (a * b - 1) + 1 := by ring _ = (-c) * m + m * c + 1 := by rw [h4] _ = 1 := by ring done · -- (←) assume h1 : rel_prime m a define show ∃ (Y : ZMod m), [a]_m * Y = [1]_m from Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1) done done /- Section 7.4 -/ section Euler open Euler lemma num_rp_below_base {m : Nat} : num_rp_below m 0 = 0 := by rfl lemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) : num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by have h1 : num_rp_below m (j + 1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : gcd m j = 1 rewrite [if_pos h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1 show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1 done lemma num_rp_below_step_not_rp {m j : Nat} (h : ¬rel_prime m j) : num_rp_below m (j + 1) = num_rp_below m j := by have h1 : num_rp_below m (j +1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : ¬gcd m j = 1 rewrite [if_neg h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j show num_rp_below m (j + 1) = num_rp_below m j from h1 done lemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl #eval phi 10 --Answer: 4 lemma prod_inv_iff_inv {m : Nat} {X : ZMod m} (h1 : invertible X) (Y : ZMod m) : invertible (X * Y) ↔ invertible Y := by apply Iff.intro · -- (→) assume h2 : invertible (X * Y) obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2 apply Exists.intro (X * Z) rewrite [←h3] --Goal : Y * (X * Z) = X * Y * Z ring --Note that ring can do algebra in ZMod m done · -- (←) assume h2 : invertible Y obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1 obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2 apply Exists.intro (Xi * Yi) show (X * Y) * (Xi * Yi) = [1]_m from calc X * Y * (Xi * Yi) _ = (X * Xi) * (Y * Yi) := by ring _ = [1]_m * [1]_m := by rw [h3, h4] _ = [1]_m := Theorem_7_3_6_7 [1]_m done done lemma F_rp_def {m i : Nat} (h : rel_prime m i) : F m i = [i]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h --h : gcd m i = 1 rewrite [if_pos h] at h1 show F m i = [i]_m from h1 done lemma F_not_rp_def {m i : Nat} (h : ¬rel_prime m i) : F m i = [1]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h rewrite [h1, if_neg h] rfl done lemma prod_seq_base {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 0 k f = [1]_m := by rfl lemma prod_seq_step {m : Nat} (n k : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl lemma prod_seq_zero_step {m : Nat} (n : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by rewrite [prod_seq_step, zero_add] rfl done lemma prod_one {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 1 k f = f k := by rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7] rfl done lemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl lemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m := calc [G m a i]_m _ = [(a * i) % m]_m := by rfl _ = [a * i]_m := (cc_eq_mod m (a * i)).symm _ = [a]_m * [i]_m := (mul_class m a i).symm lemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) : rel_prime m (G m a i) ↔ rel_prime m i := by have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1 show rel_prime m (G m a i) ↔ rel_prime m i from calc rel_prime m (G m a i) _ ↔ invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm _ ↔ invertible ([a]_m * [i]_m) := by rw [cc_G] _ ↔ invertible [i]_m := prod_inv_iff_inv h2 ([i]_m) _ ↔ rel_prime m i := Theorem_7_3_7 m i done lemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) : F m (G m a i) = [a]_m * F m i := by have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2 show F m (G m a i) = [a]_m * F m i from calc F m (G m a i) _ = [G m a i]_m := F_rp_def h3 _ = [a]_m * [i]_m := cc_G m a i _ = [a]_m * F m i := by rw [F_rp_def h2] done lemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : ¬rel_prime m i) : F m (G m a i) = [1]_m := by rewrite [←G_rp_iff h1 i] at h2 show F m (G m a i) = [1]_m from F_not_rp_def h2 done lemma FG_prod {m a : Nat} (h1 : rel_prime m a) : ∀ (k : Nat), prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by by_induc · -- Base Case show prod_seq 0 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from calc prod_seq 0 0 ((F m) ∘ (G m a)) _ = [1]_m := prod_seq_base _ _ _ = [a]_m ^ 0 * [1]_m := by ring _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by rw [num_rp_below_base, prod_seq_base] done · -- Induction Step fix k : Nat assume ih : prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) by_cases h2 : rel_prime m k · -- Case 1. h2 : rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([a]_m * F m k) := by rw [FG_rp h1 h2] _ = [a]_m ^ ((num_rp_below m k) + 1) * ((prod_seq k 0 (F m)) * F m k) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_rp h2, prod_seq_zero_step] done · -- Case 2. h2 : ¬rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([1]_m) := by rw [FG_not_rp h1 h2] _ = [a]_m ^ (num_rp_below m k) * (prod_seq k 0 (F m) * ([1]_m)) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_not_rp h2, prod_seq_zero_step, F_not_rp_def h2] done done done lemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by define --Goal : ∀ i < m, G m a i < m fix i : Nat assume h1 : i < m rewrite [G_def] --Goal : a * i % m < m show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m) done lemma left_inv_one_one_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g' (g i) = i) : one_one_below n g := sorry lemma right_inv_onto_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g (g' i) = i) (h2 : maps_below n g') : onto_below n g := by define at h2; define fix k : Nat assume h3 : k < n apply Exists.intro (g' k) show g' k < n ∧ g (g' k) = k from And.intro (h2 k h3) (h1 k h3) done lemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m * [inv_mod m a]_m = [1]_m := by have h2 : 0 ≤ (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a) show [a]_m * [inv_mod m a]_m = [1]_m from calc [a]_m * [inv_mod m a]_m _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2] _ = [a]_m * [gcd_c2 m a]_m := by rw [←cc_eq_mod] _ = [1]_m := gcd_c2_inv h1 done lemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m := calc a * (b % m) % m = a % m * (b % m % m) % m := Nat.mul_mod _ _ _ _ = a % m * (b % m) % m := by rw [Nat.mod_mod] _ = a * b % m := (Nat.mul_mod _ _ _).symm lemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm] rfl done theorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] : ↑a ≡ ↑b (MOD m) ↔ a % m = b % m := sorry lemma mul_inv_mod_cancel {m a i : Nat} [NeZero m] (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1 rewrite [mul_class, cc_eq_iff_congr, ←Nat.cast_mul, ←Nat.cast_one, congr_iff_mod_eq_Nat] at h3 show a * inv_mod m a * i % m = i from calc a * (inv_mod m a) * i % m _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = 1 % m * i % m := by rw [h3] _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = i % m := by rw [one_mul] _ = i := Nat.mod_eq_of_lt h2 done lemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl lemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, G m a (Ginv m a i) = i := by fix i : Nat assume h2 : i < m show G m a (Ginv m a i) = i from calc G m a (Ginv m a i) _ = a * ((inv_mod m a * i) % m) % m := by rfl _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, Ginv m a (G m a i) = i := by fix i : Nat assume h2 : i < m show Ginv m a (G m a i) = i from calc Ginv m a (G m a i) _ = inv_mod m a * ((a * i) % m) % m := by rfl _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc, mul_comm (inv_mod m a)] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_maps_below (m a : Nat) [NeZero m] : maps_below m (Ginv m a) := G_maps_below m (inv_mod m a) lemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : one_one_below m (G m a) := left_inv_one_one_below (Ginv_left_inv h1) lemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : onto_below m (G m a) := right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a) lemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : perm_below m (G m a) := And.intro (G_maps_below m a) (And.intro (G_one_one_below h1) (G_onto_below h1)) --Permuting a product of congruence classes doesn't change product lemma swap_fst (u v : Nat) : swap u v u = v := by define : swap u v u --Goal : (if u = u then v else if u = v then u else u) = v have h : u = u := by rfl rewrite [if_pos h] rfl done lemma swap_snd (u v : Nat) : swap u v v = u := by define : swap u v v by_cases h1 : v = u · -- Case 1. h1 : v = u rewrite [if_pos h1] show v = u from h1 done · -- Case 2. h1 : v ≠ u rewrite [if_neg h1] have h2 : v = v := by rfl rewrite [if_pos h2] rfl done done lemma swap_other {u v i : Nat} (h1 : i ≠ u) (h2 : i ≠ v) : swap u v i = i := by define : swap u v i rewrite [if_neg h1, if_neg h2] rfl done lemma swap_values (u v i : Nat) : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := by by_cases h1 : i = u · -- Case 1. h1 : i = u apply Or.inl rewrite [h1] show swap u v u = v from swap_fst u v done · -- Case 2. h1 : i ≠ u apply Or.inr by_cases h2 : i = v · -- Case 2.1. h2 : i = v apply Or.inl rewrite [h2] show swap u v v = u from swap_snd u v done · -- Case 2.2. h2 : i ≠ v apply Or.inr show swap u v i = i from swap_other h1 h2 done done done lemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by define fix i : Nat assume h3 : i < n have h4 : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := swap_values u v i by_cases on h4 · -- Case 1. h4 : swap u v i = v rewrite [h4] show v < n from h2 done · -- Case 2. by_cases on h4 · -- Case 2.1. h4 : swap u v i = u rewrite [h4] show u < n from h1 done · -- Case 2.2. h4 : swap u v i = i rewrite [h4] show i < n from h3 done done done lemma swap_swap (u v n : Nat) : ∀ i < n, swap u v (swap u v i) = i := by fix i : Nat assume h : i < n by_cases h1 : i = u · -- Case 1. h1 : i = u rewrite [h1, swap_fst, swap_snd] rfl done · -- Case 2. h1 : i ≠ u by_cases h2 : i = v · -- Case 2.1. h2 : i = v rewrite [h2, swap_snd, swap_fst] rfl done · -- Case 2.2. h2 : i ≠ v rewrite [swap_other h1 h2, swap_other h1 h2] rfl done done done lemma swap_one_one_below (u v n) : one_one_below n (swap u v) := left_inv_one_one_below (swap_swap u v n) lemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) := right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2) lemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) := And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2)) lemma comp_perm_below {n : Nat} {f g : Nat → Nat} (h1 : perm_below n f) (h2 : perm_below n g) : perm_below n (f ∘ g) := sorry lemma trivial_swap (u : Nat) : swap u u = id := by apply funext fix x : Nat by_cases h1 : x = u · -- Case 1. h1 : x = u rewrite [h1, swap_fst] rfl done · -- Case 2. h1 : x ≠ u rewrite [swap_other h1 h1] rfl done done lemma prod_eq_fun {m : Nat} (f g : Nat → ZMod m) (k : Nat) : ∀ (n : Nat), (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g := by by_induc · -- Base Case assume h : (∀ i < 0, f (k + i) = g (k + i)) rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g assume h1 : ∀ i < n + 1, f (k + i) = g (k + i) have h2 : ∀ i < n, f (k + i) = g (k + i) := by fix i : Nat assume h2 : i < n have h3 : i < n + 1 := by linarith show f (k + i) = g (k + i) from h1 i h3 done have h3 : prod_seq n k f = prod_seq n k g := ih h2 have h4 : n < n + 1 := Nat.lt_succ_self n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4] rfl done done

lemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f

HTPI.swap_prod_eq_prod_below

htpi/HTPILib/Chap7.lean

htpi

/- Copyright 2023 Daniel J. Velleman -/ import HTPILib.Chap6 namespace HTPI /- Definitions -/ lemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by have h : n + 1 > 0 := Nat.succ_pos n show a % (n + 1) < n + 1 from Nat.mod_lt a h done def gcd (a b : Nat) : Nat := match b with | 0 => a | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd (n + 1) (a % (n + 1)) termination_by b mutual def gcd_c1 (a b : Nat) : Int := match b with | 0 => 1 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c2 (n + 1) (a % (n + 1)) --Corresponds to s = t' termination_by b def gcd_c2 (a b : Nat) : Int := match b with | 0 => 0 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c1 (n + 1) (a % (n + 1)) - (gcd_c2 (n + 1) (a % (n + 1))) * ↑(a / (n + 1)) --Corresponds to t = s' - t'q termination_by b end def prime (n : Nat) : Prop := 2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n def prime_factor (p n : Nat) : Prop := prime p ∧ p ∣ n def all_prime (l : List Nat) : Prop := ∀ p ∈ l, prime p def nondec (l : List Nat) : Prop := match l with | [] => True --Of course, True is a proposition that is always true | n :: L => (∀ m ∈ L, n ≤ m) ∧ nondec L def nondec_prime_list (l : List Nat) : Prop := all_prime l ∧ nondec l def prod (l : List Nat) : Nat := match l with | [] => 1 | n :: L => n * (prod L) def prime_factorization (n : Nat) (l : List Nat) : Prop := nondec_prime_list l ∧ prod l = n def rel_prime (a b : Nat) : Prop := gcd a b = 1 def congr_mod (m : Nat) (a b : Int) : Prop := (↑m : Int) ∣ (a - b) def cc (m : Nat) (a : Int) : ZMod m := (↑a : ZMod m) notation:50 a " ≡ " b " (MOD " m ")" => congr_mod m a b notation:max "["a"]_"m:max => cc m a def invertible {m : Nat} (X : ZMod m) : Prop := ∃ (Y : ZMod m), X * Y = [1]_m def num_rp_below (m k : Nat) : Nat := match k with | 0 => 0 | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j def phi (m : Nat) : Nat := num_rp_below m m def prod_seq {m : Nat} (j k : Nat) (f : Nat → ZMod m) : ZMod m := match j with | 0 => [1]_m | n + 1 => prod_seq n k f * f (k + n) def maps_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i < n, g i < n def one_one_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i1 < n, ∀ i2 < n, g i1 = g i2 → i1 = i2 def onto_below (n : Nat) (g : Nat → Nat) : Prop := ∀ k < n, ∃ i < n, g i = k def perm_below (n : Nat) (g : Nat → Nat) : Prop := maps_below n g ∧ one_one_below n g ∧ onto_below n g def inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m) def swap (u v i : Nat) : Nat := if i = u then v else if i = v then u else i namespace Euler --For definitions specific to Euler's theorem def F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m def G (m a i : Nat) : Nat := (a * i) % m def Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i end Euler /- Section 7.1 -/ theorem dvd_mod_of_dvd_a_b {a b d : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ (a % b) := by set q : Nat := a / b have h3 : b * q + a % b = a := Nat.div_add_mod a b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 define --Goal : ∃ (c : Nat), a % b = d * c apply Exists.intro (j - k * q) show a % b = d * (j - k * q) from calc a % b _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm _ = a - b * q := by rw [h3] _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc] _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm done theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry #eval gcd 672 161 --Answer: 7 lemma gcd_base (a : Nat) : gcd a 0 = a := by rfl lemma gcd_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd a b = gcd b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1)) rfl done lemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b ≠ 0) : a % b < b := by have h1 : b > 0 := Nat.pos_of_ne_zero h show a % b < b from Nat.mod_lt a h1 done lemma dvd_self (n : Nat) : n ∣ n := by apply Exists.intro 1 ring done theorem gcd_dvd : ∀ (b a : Nat), (gcd a b) ∣ a ∧ (gcd a b) ∣ b := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd a b_1) ∣ a ∧ (gcd a b_1) ∣ b_1 fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_base] --Goal: a ∣ a ∧ a ∣ 0 apply And.intro (dvd_self a) define apply Exists.intro 0 rfl done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_nonzero a h1] --Goal : gcd b (a % b) ∣ a ∧ gcd b (a % b) ∣ b have h2 : a % b < b := mod_nonzero_lt a h1 have h3 : (gcd b (a % b)) ∣ b ∧ (gcd b (a % b)) ∣ (a % b) := ih (a % b) h2 b apply And.intro _ h3.left show (gcd b (a % b)) ∣ a from dvd_a_of_dvd_b_mod h3.left h3.right done done theorem gcd_dvd_left (a b : Nat) : (gcd a b) ∣ a := (gcd_dvd b a).left theorem gcd_dvd_right (a b : Nat) : (gcd a b) ∣ b := (gcd_dvd b a).right lemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl lemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c1 a b = gcd_c2 b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done lemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl lemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * ↑(a / b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done theorem gcd_lin_comb : ∀ (b a : Nat), (gcd_c1 a b) * ↑a + (gcd_c2 a b) * ↑b = ↑(gcd a b) := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd_c1 a b_1) * ↑a + (gcd_c2 a b_1) * ↑b_1 = ↑(gcd a b_1) fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base] --Goal : 1 * ↑a + 0 * ↑0 = ↑a ring done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1] --Goal : gcd_c2 b (a % b) * ↑a + -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * ↑(a / b)) * ↑b = -- ↑(gcd b (a % b)) set r : Nat := a % b set q : Nat := a / b set s : Int := gcd_c1 b r set t : Int := gcd_c2 b r --Goal : t * ↑a + (s - t * ↑q) * ↑b = ↑(gcd b r) have h2 : r < b := mod_nonzero_lt a h1 have h3 : s * ↑b + t * ↑r = ↑(gcd b r) := ih r h2 b have h4 : b * q + r = a := Nat.div_add_mod a b rewrite [←h3, ←h4] rewrite [Nat.cast_add, Nat.cast_mul] --Goal : t * (↑b * ↑q + ↑r) + (s - t * ↑q) * ↑b = s * ↑b + t * ↑r ring done done #eval gcd_c1 672 161 --Answer: 6 #eval gcd_c2 672 161 --Answer: -25 --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161 theorem Theorem_7_1_6 {d a b : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ gcd a b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑d ∣ ↑(gcd a b) set s : Int := gcd_c1 a b set t : Int := gcd_c2 a b have h3 : s * ↑a + t * ↑b = ↑(gcd a b) := gcd_lin_comb b a rewrite [←h3] --Goal : ↑d ∣ s * ↑a + t * ↑b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul] --Goal : ↑d ∣ s * (↑d * ↑j) + t * (↑d * ↑k) define apply Exists.intro (s * ↑j + t * ↑k) ring done /- Section 7.2 -/ theorem dvd_trans {a b c : Nat} (h1 : a ∣ b) (h2 : b ∣ c) : a ∣ c := by define at h1; define at h2; define obtain (m : Nat) (h3 : b = a * m) from h1 obtain (n : Nat) (h4 : c = b * n) from h2 rewrite [h3, mul_assoc] at h4 apply Exists.intro (m * n) show c = a * (m * n) from h4 done lemma exists_prime_factor : ∀ (n : Nat), 2 ≤ n → ∃ (p : Nat), prime_factor p n := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, 2 ≤ n_1 → ∃ (p : Nat), prime_factor p n_1 assume h1 : 2 ≤ n by_cases h2 : prime n · -- Case 1. h2 : prime n apply Exists.intro n define --Goal : prime n ∧ n ∣ n show prime n ∧ n ∣ n from And.intro h2 (dvd_self n) done · -- Case 2. h2 : ¬prime n define at h2 --h2 : ¬(2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n) demorgan at h2 disj_syll h2 h1 obtain (a : Nat) (h3 : ∃ (b : Nat), a * b = n ∧ a < n ∧ b < n) from h2 obtain (b : Nat) (h4 : a * b = n ∧ a < n ∧ b < n) from h3 have h5 : 2 ≤ a := by by_contra h6 have h7 : a ≤ 1 := by linarith have h8 : n ≤ b := calc n _ = a * b := h4.left.symm _ ≤ 1 * b := by rel [h7] _ = b := by ring linarith --n ≤ b contradicts b < n done have h6 : ∃ (p : Nat), prime_factor p a := ih a h4.right.left h5 obtain (p : Nat) (h7 : prime_factor p a) from h6 apply Exists.intro p define --Goal : prime p ∧ p ∣ n define at h7 --h7 : prime p ∧ p ∣ a apply And.intro h7.left have h8 : a ∣ n := by apply Exists.intro b show n = a * b from (h4.left).symm done show p ∣ n from dvd_trans h7.right h8 done done lemma exists_least_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q := by set S : Set Nat := {p : Nat | prime_factor p n} have h2 : ∃ (p : Nat), p ∈ S := exists_prime_factor n h show ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q from well_ord_princ S h2 done lemma all_prime_nil : all_prime [] := by define --Goal : ∀ p ∈ [], prime p fix p : Nat contrapos --Goal : ¬prime p → p ∉ [] assume h1 : ¬prime p show p ∉ [] from List.not_mem_nil p done lemma all_prime_cons (n : Nat) (L : List Nat) : all_prime (n :: L) ↔ prime n ∧ all_prime L := by apply Iff.intro · -- (→) assume h1 : all_prime (n :: L) --Goal : prime n ∧ all_prime L define at h1 --h1 : ∀ p ∈ n :: L, prime p apply And.intro (h1 n (List.mem_cons_self n L)) define --Goal : ∀ p ∈ L, prime p fix p : Nat assume h2 : p ∈ L show prime p from h1 p (List.mem_cons_of_mem n h2) done · -- (←) assume h1 : prime n ∧ all_prime L --Goal : all_prime (n :: l) define : all_prime L at h1 define fix p : Nat assume h2 : p ∈ n :: L rewrite [List.mem_cons] at h2 --h2 : p = n ∨ p ∈ L by_cases on h2 · -- Case 1. h2 : p = n rewrite [h2] show prime n from h1.left done · -- Case 2. h2 : p ∈ L show prime p from h1.right p h2 done done done lemma nondec_nil : nondec [] := by define --Goal : True trivial --trivial proves some obviously true statements, such as True done lemma nondec_cons (n : Nat) (L : List Nat) : nondec (n :: L) ↔ (∀ m ∈ L, n ≤ m) ∧ nondec L := by rfl lemma prod_nil : prod [] = 1 := by rfl lemma prod_cons : prod (n :: L) = n * (prod L) := by rfl lemma exists_cons_of_length_eq_succ {A : Type} {l : List A} {n : Nat} (h : l.length = n + 1) : ∃ (a : A) (L : List A), l = a :: L ∧ L.length = n := by have h1 : ¬l.length = 0 := by linarith rewrite [List.length_eq_zero] at h1 obtain (a : A) (h2 : ∃ (L : List A), l = a :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List A) (h3 : l = a :: L) from h2 apply Exists.intro a apply Exists.intro L apply And.intro h3 have h4 : (a :: L).length = L.length + 1 := List.length_cons a L rewrite [←h3, h] at h4 show L.length = n from (Nat.add_right_cancel h4).symm done lemma list_elt_dvd_prod_by_length (a : Nat) : ∀ (n : Nat), ∀ (l : List Nat), l.length = n → a ∈ l → a ∣ prod l := by by_induc · --Base Case fix l : List Nat assume h1 : l.length = 0 rewrite [List.length_eq_zero] at h1 --h1 : l = [] rewrite [h1] --Goal : a ∈ [] → a ∣ prod [] contrapos assume h2 : ¬a ∣ prod [] show a ∉ [] from List.not_mem_nil a done · -- Induction Step fix n : Nat assume ih : ∀ (l : List Nat), List.length l = n → a ∈ l → a ∣ prod l fix l : List Nat assume h1 : l.length = n + 1 --Goal : a ∈ l → a ∣ prod l obtain (b : Nat) (h2 : ∃ (L : List Nat), l = b :: L ∧ L.length = n) from exists_cons_of_length_eq_succ h1 obtain (L : List Nat) (h3 : l = b :: L ∧ L.length = n) from h2 have h4 : a ∈ L → a ∣ prod L := ih L h3.right assume h5 : a ∈ l rewrite [h3.left, prod_cons] --Goal : a ∣ b * prod L rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b ∨ a ∈ L by_cases on h5 · -- Case 1. h5 : a = b apply Exists.intro (prod L) rewrite [h5] rfl done · -- Case 2. h5 : a ∈ L have h6 : a ∣ prod L := h4 h5 have h7 : prod L ∣ b * prod L := by apply Exists.intro b ring done show a ∣ b * prod L from dvd_trans h6 h7 done done done lemma list_elt_dvd_prod {a : Nat} {l : List Nat} (h : a ∈ l) : a ∣ prod l := by set n : Nat := l.length have h1 : l.length = n := by rfl show a ∣ prod l from list_elt_dvd_prod_by_length a n l h1 h done lemma exists_prime_factorization : ∀ (n : Nat), n ≥ 1 → ∃ (l : List Nat), prime_factorization n l := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, n_1 ≥ 1 → ∃ (l : List Nat), prime_factorization n_1 l assume h1 : n ≥ 1 by_cases h2 : n = 1 · -- Case 1. h2 : n = 1 apply Exists.intro [] define apply And.intro · -- Proof of nondec_prime_list [] define show all_prime [] ∧ nondec [] from And.intro all_prime_nil nondec_nil done · -- Proof of prod [] = n rewrite [prod_nil, h2] rfl done done · -- Case 2. h2 : n ≠ 1 have h3 : n ≥ 2 := lt_of_le_of_ne' h1 h2 obtain (p : Nat) (h4 : prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q) from exists_least_prime_factor h3 have p_prime_factor : prime_factor p n := h4.left define at p_prime_factor have p_prime : prime p := p_prime_factor.left have p_dvd_n : p ∣ n := p_prime_factor.right have p_least : ∀ (q : Nat), prime_factor q n → p ≤ q := h4.right obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n have h5 : m ≠ 0 := by contradict h1 with h6 have h7 : n = 0 := calc n _ = p * m := n_eq_pm _ = p * 0 := by rw [h6] _ = 0 := by ring rewrite [h7] decide done have m_pos : 0 < m := Nat.pos_of_ne_zero h5 have m_lt_n : m < n := by define at p_prime show m < n from calc m _ < m + m := by linarith _ = 2 * m := by ring _ ≤ p * m := by rel [p_prime.left] _ = n := n_eq_pm.symm done obtain (L : List Nat) (h6 : prime_factorization m L) from ih m m_lt_n m_pos define at h6 have ndpl_L : nondec_prime_list L := h6.left define at ndpl_L apply Exists.intro (p :: L) define apply And.intro · -- Proof of nondec_prime_list (p :: L) define apply And.intro · -- Proof of all_prime (p :: L) rewrite [all_prime_cons] show prime p ∧ all_prime L from And.intro p_prime ndpl_L.left done · -- Proof of nondec (p :: L) rewrite [nondec_cons] apply And.intro _ ndpl_L.right fix q : Nat assume q_in_L : q ∈ L have h7 : q ∣ prod L := list_elt_dvd_prod q_in_L rewrite [h6.right] at h7 --h7 : q ∣ m have h8 : m ∣ n := by apply Exists.intro p rewrite [n_eq_pm] ring done have q_dvd_n : q ∣ n := dvd_trans h7 h8 have ap_L : all_prime L := ndpl_L.left define at ap_L have q_prime_factor : prime_factor q n := And.intro (ap_L q q_in_L) q_dvd_n show p ≤ q from p_least q q_prime_factor done done · -- Proof of prod (p :: L) = n rewrite [prod_cons, h6.right, n_eq_pm] rfl done done done theorem Theorem_7_2_2 {a b c : Nat} (h1 : c ∣ a * b) (h2 : rel_prime a c) : c ∣ b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑c ∣ ↑b define at h1; define at h2; define obtain (j : Nat) (h3 : a * b = c * j) from h1 set s : Int := gcd_c1 a c set t : Int := gcd_c2 a c have h4 : s * ↑a + t * ↑c = ↑(gcd a c) := gcd_lin_comb c a rewrite [h2, Nat.cast_one] at h4 --h4 : s * ↑a + t * ↑c = (1 : Int) apply Exists.intro (s * ↑j + t * ↑b) show ↑b = ↑c * (s * ↑j + t * ↑b) from calc ↑b _ = (1 : Int) * ↑b := (one_mul _).symm _ = (s * ↑a + t * ↑c) * ↑b := by rw [h4] _ = s * (↑a * ↑b) + t * ↑c * ↑b := by ring _ = s * (↑c * ↑j) + t * ↑c * ↑b := by rw [←Nat.cast_mul a b, h3, Nat.cast_mul c j] _ = ↑c * (s * ↑j + t * ↑b) := by ring done lemma le_nonzero_prod_left {a b : Nat} (h : a * b ≠ 0) : a ≤ a * b := by have h1 : b ≠ 0 := by contradict h with h1 rewrite [h1] ring done have h2 : 1 ≤ b := Nat.pos_of_ne_zero h1 show a ≤ a * b from calc a = a * 1 := (mul_one a).symm _ ≤ a * b := by rel [h2] done lemma le_nonzero_prod_right {a b : Nat} (h : a * b ≠ 0) : b ≤ a * b := by rewrite [mul_comm] rewrite [mul_comm] at h show b ≤ b * a from le_nonzero_prod_left h done lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry lemma rel_prime_of_prime_not_dvd {a p : Nat} (h1 : prime p) (h2 : ¬p ∣ a) : rel_prime a p := by have h3 : gcd a p ∣ a := gcd_dvd_left a p have h4 : gcd a p ∣ p := gcd_dvd_right a p have h5 : gcd a p = 1 ∨ gcd a p = p := dvd_prime h1 h4 have h6 : gcd a p ≠ p := by contradict h2 with h6 rewrite [h6] at h3 show p ∣ a from h3 done disj_syll h5 h6 show rel_prime a p from h5 done theorem Theorem_7_2_3 {a b p : Nat} (h1 : prime p) (h2 : p ∣ a * b) : p ∣ a ∨ p ∣ b := by or_right with h3 have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3 show p ∣ b from Theorem_7_2_2 h2 h4 done lemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a ≥ 1 := by have h1 : a ≠ 0 := by by_contra h1 rewrite [h1] at h contradict h linarith done show a ≥ 1 from Nat.pos_of_ne_zero h1 done lemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by have h1 : a ≥ 1 := ge_one_of_prod_one h have h2 : a * b ≠ 0 := by linarith have h3 : a ≤ a * b := le_nonzero_prod_left h2 rewrite [h] at h3 show a = 1 from Nat.le_antisymm h3 h1 done lemma eq_one_of_dvd_one {n : Nat} (h : n ∣ 1) : n = 1 := by obtain (j : Nat) (h1 : 1 = n * j) from h show n = 1 from eq_one_of_prod_one h1.symm done lemma prime_not_one {p : Nat} (h : prime p) : p ≠ 1 := by define at h linarith done theorem Theorem_7_2_4 {p : Nat} (h1 : prime p) : ∀ (l : List Nat), p ∣ prod l → ∃ a ∈ l, p ∣ a := by apply List.rec · -- Base Case. Goal : p ∣ prod [] → ∃ a ∈ [], p ∣ a rewrite [prod_nil] assume h2 : p ∣ 1 show ∃ a ∈ [], p ∣ a from absurd (eq_one_of_dvd_one h2) (prime_not_one h1) done · -- Induction Step fix b : Nat fix L : List Nat assume ih : p ∣ prod L → ∃ a ∈ L, p ∣ a --Goal : p ∣ prod (b :: L) → ∃ a ∈ b :: L, p ∣ a assume h2 : p ∣ prod (b :: L) rewrite [prod_cons] at h2 have h3 : p ∣ b ∨ p ∣ prod L := Theorem_7_2_3 h1 h2 by_cases on h3 · -- Case 1. h3 : p ∣ b apply Exists.intro b show b ∈ b :: L ∧ p ∣ b from And.intro (List.mem_cons_self b L) h3 done · -- Case 2. h3 : p ∣ prod L obtain (a : Nat) (h4 : a ∈ L ∧ p ∣ a) from ih h3 apply Exists.intro a show a ∈ b :: L ∧ p ∣ a from And.intro (List.mem_cons_of_mem b h4.left) h4.right done done done lemma prime_in_list {p : Nat} {l : List Nat} (h1 : prime p) (h2 : all_prime l) (h3 : p ∣ prod l) : p ∈ l := by obtain (a : Nat) (h4 : a ∈ l ∧ p ∣ a) from Theorem_7_2_4 h1 l h3 define at h2 have h5 : prime a := h2 a h4.left have h6 : p = 1 ∨ p = a := dvd_prime h5 h4.right disj_syll h6 (prime_not_one h1) rewrite [h6] show a ∈ l from h4.left done lemma first_le_first {p q : Nat} {l m : List Nat} (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m)) (h3 : prod (p :: l) = prod (q :: m)) : p ≤ q := by define at h1; define at h2 have h4 : q ∣ prod (p :: l) := by define apply Exists.intro (prod m) rewrite [←prod_cons] show prod (p :: l) = prod (q :: m) from h3 done have h5 : all_prime (q :: m) := h2.left rewrite [all_prime_cons] at h5 have h6 : q ∈ p :: l := prime_in_list h5.left h1.left h4 have h7 : nondec (p :: l) := h1.right rewrite [nondec_cons] at h7 rewrite [List.mem_cons] at h6 by_cases on h6 · -- Case 1. h6 : q = p linarith done · -- Case 2. h6 : q ∈ l have h8 : ∀ m ∈ l, p ≤ m := h7.left show p ≤ q from h8 q h6 done done lemma nondec_prime_list_tail {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by define at h define rewrite [all_prime_cons, nondec_cons] at h show all_prime l ∧ nondec l from And.intro h.left.right h.right.right done lemma cons_prod_not_one {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : prod (p :: l) ≠ 1 := by define at h have h1 : all_prime (p :: l) := h.left rewrite [all_prime_cons] at h1 rewrite [prod_cons] by_contra h2 show False from (prime_not_one h1.left) (eq_one_of_prod_one h2) done lemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) : l = [] ↔ prod l = 1 := by apply Iff.intro · -- (→) assume h1 : l = [] rewrite [h1] show prod [] = 1 from prod_nil done · -- (←) contrapos assume h1 : ¬l = [] obtain (p : Nat) (h2 : ∃ (L : List Nat), l = p :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List Nat) (h3 : l = p :: L) from h2 rewrite [h3] at h rewrite [h3] show ¬prod (p :: L) = 1 from cons_prod_not_one h done done lemma prime_pos {p : Nat} (h : prime p) : p > 0 := by define at h linarith done theorem Theorem_7_2_5 : ∀ (l1 l2 : List Nat), nondec_prime_list l1 → nondec_prime_list l2 → prod l1 = prod l2 → l1 = l2 := by apply List.rec · -- Base Case. Goal : ∀ (l2 : List Nat), nondec_prime_list [] → -- nondec_prime_list l2 → prod [] = prod l2 → [] = l2 fix l2 : List Nat assume h1 : nondec_prime_list [] assume h2 : nondec_prime_list l2 assume h3 : prod [] = prod l2 rewrite [prod_nil, eq_comm, ←list_nil_iff_prod_one h2] at h3 show [] = l2 from h3.symm done · -- Induction Step fix p : Nat fix L1 : List Nat assume ih : ∀ (L2 : List Nat), nondec_prime_list L1 → nondec_prime_list L2 → prod L1 = prod L2 → L1 = L2 -- Goal : ∀ (l2 : List Nat), nondec_prime_list (p :: L1) → -- nondec_prime_list l2 → prod (p :: L1) = prod l2 → p :: L1 = l2 fix l2 : List Nat assume h1 : nondec_prime_list (p :: L1) assume h2 : nondec_prime_list l2 assume h3 : prod (p :: L1) = prod l2 have h4 : ¬prod (p :: L1) = 1 := cons_prod_not_one h1 rewrite [h3, ←list_nil_iff_prod_one h2] at h4 obtain (q : Nat) (h5 : ∃ (L : List Nat), l2 = q :: L) from List.exists_cons_of_ne_nil h4 obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5 rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2) rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2) have h7 : p ≤ q := first_le_first h1 h2 h3 have h8 : q ≤ p := first_le_first h2 h1 h3.symm have h9 : p = q := by linarith rewrite [h9, prod_cons, prod_cons] at h3 --h3 : q * prod L1 = q * prod L2 have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1 have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2 define at h2 have h12 : all_prime (q :: L2) := h2.left rewrite [all_prime_cons] at h12 have h13 : q > 0 := prime_pos h12.left have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3 have h15 : L1 = L2 := ih L2 h10 h11 h14 rewrite [h6, h9, h15] rfl done done theorem fund_thm_arith (n : Nat) (h : n ≥ 1) : ∃! (l : List Nat), prime_factorization n l := by exists_unique · -- Existence show ∃ (l : List Nat), prime_factorization n l from exists_prime_factorization n h done · -- Uniqueness fix l1 : List Nat; fix l2 : List Nat assume h1 : prime_factorization n l1 assume h2 : prime_factorization n l2 define at h1; define at h2 have h3 : prod l1 = n := h1.right rewrite [←h2.right] at h3 show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3 done done /- Section 7.3 -/ theorem congr_refl (m : Nat) : ∀ (a : Int), a ≡ a (MOD m) := by fix a : Int define --Goal : ∃ (c : Int), a - a = ↑m * c apply Exists.intro 0 ring done theorem congr_symm {m : Nat} : ∀ {a b : Int}, a ≡ b (MOD m) → b ≡ a (MOD m) := by fix a : Int; fix b : Int assume h1 : a ≡ b (MOD m) define at h1 --h1 : ∃ (c : Int), a - b = ↑m * c define --Goal : ∃ (c : Int), b - a = ↑m * c obtain (c : Int) (h2 : a - b = m * c) from h1 apply Exists.intro (-c) show b - a = m * (-c) from calc b - a _ = -(a - b) := by ring _ = -(m * c) := by rw [h2] _ = m * (-c) := by ring done theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry /- Fundamental properties of congruence classes -/ lemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) : X = Y ↔ X.val = Y.val := Fin.ext_iff lemma val_nat_eq_mod (n k : Nat) : ([k]_(n + 1)).val = k % (n + 1) := by rfl lemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl theorem cc_rep {m : Nat} (X : ZMod m) : ∃ (a : Int), X = [a]_m := match m with | 0 => by apply Exists.intro X rfl done | n + 1 => by apply Exists.intro ↑(X.val) have h1 : X.val < n + 1 := Fin.prop X rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1] rfl done theorem add_class (m : Nat) (a b : Int) : [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm theorem mul_class (m : Nat) (a b : Int) : [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm lemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ [a - b]_m = [0]_m := by apply Iff.intro · -- (→) assume h1 : [a]_m = [b]_m have h2 : a - b = a + (-b) := by ring have h3 : b + (-b) = 0 := by ring show [a - b]_m = [0]_m from calc [a - b]_m _ = [a + (-b)]_m := by rw [h2] _ = [a]_m + [-b]_m := by rw [add_class] _ = [b]_m + [-b]_m := by rw [h1] _ = [b + -b]_m := by rw [add_class] _ = [0]_m := by rw [h3] done · -- (←) assume h1 : [a - b]_m = [0]_m have h2 : b + (a - b) = a := by ring have h3 : b + 0 = b := by ring show [a]_m = [b]_m from calc [a]_m _ = [b + (a - b)]_m := by rw [h2] _ = [b]_m + [a - b]_m := by rw [add_class] _ = [b]_m + [0]_m := by rw [h1] _ = [b + 0]_m := by rw [add_class] _ = [b]_m := by rw [h3] done done lemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) : [a]_m = [0]_m → [-a]_m = [0]_m := by assume h1 : [a]_m = [0]_m have h2 : 0 + (-a) = -a := by ring have h3 : a + (-a) = 0 := by ring show [-a]_m = [0]_m from calc [-a]_m _ = [0 + (-a)]_m := by rw [h2] _ = [0]_m + [-a]_m := by rw [add_class] _ = [a]_m + [-a]_m := by rw [h1] _ = [a + (-a)]_m := by rw [add_class] _ = [0]_m := by rw [h3] done lemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) : [-a]_m = [0]_m ↔ [a]_m = [0]_m := by apply Iff.intro _ (cc_neg_zero_of_cc_zero m a) assume h1 : [-a]_m = [0]_m have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1 have h3 : -(-a) = a := by ring rewrite [h3] at h2 show [a]_m = [0]_m from h2 done lemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl lemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m ↔ m ∣ k := match m with | 0 => by have h : (0 : Int) = (↑(0 : Nat) : Int) := by rfl rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj] apply Iff.intro · -- (→) assume h1 : k = 0 rewrite [h1] show 0 ∣ 0 from dvd_self 0 done · -- (←) assume h1 : 0 ∣ k obtain (c : Nat) (h2 : k = 0 * c) from h1 rewrite [h2] ring done done | n + 1 => by rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero] show k % (n + 1) = 0 ↔ n + 1 ∣ k from (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm done lemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m ↔ ↑m ∣ a := by obtain (k : Nat) (h1 : a = ↑k ∨ a = -↑k) from Int.eq_nat_or_neg a by_cases on h1 · -- Case 1. h1: a = ↑k rewrite [h1, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done · -- Case 2. h1: a = -↑k rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done done theorem cc_eq_iff_congr (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ a ≡ b (MOD m) := calc [a]_m = [b]_m _ ↔ [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b _ ↔ ↑m ∣ (a - b) := cc_zero_iff_dvd m (a - b) _ ↔ a ≡ b (MOD m) := by rfl /- End of fundamental properties of congruence classes -/ lemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m := by have h1 : (↑m : Int) ≠ 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m) show 0 ≤ a % m from Int.emod_nonneg a h1 done lemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m) have h2 : (↑m : Int) > 0 := (Nat.cast_pos).rtl h1 show a % m < m from Int.emod_lt_of_pos a h2 done lemma congr_mod_mod (m : Nat) (a : Int) : a ≡ a % m (MOD m) := by define have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m apply Exists.intro (a / m) show a - a % m = m * (a / m) from calc a - (a % m) _ = m * (a / m) + a % m - a % m := by rw [h1] _ = m * (a / m) := by ring done lemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) := And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a)) theorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) : ∃! (r : Int), 0 ≤ r ∧ r < m ∧ a ≡ r (MOD m) := by exists_unique · -- Existence apply Exists.intro (a % m) show 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) from mod_cmpl_res m a done · -- Uniqueness fix r1 : Int; fix r2 : Int assume h1 : 0 ≤ r1 ∧ r1 < m ∧ a ≡ r1 (MOD m) assume h2 : 0 ≤ r2 ∧ r2 < m ∧ a ≡ r2 (MOD m) have h3 : r1 ≡ r2 (MOD m) := congr_trans (congr_symm h1.right.right) h2.right.right obtain (d : Int) (h4 : r1 - r2 = m * d) from h3 have h5 : r1 - r2 < m * 1 := by linarith have h6 : m * (-1) < r1 - r2 := by linarith rewrite [h4] at h5 --h5 : m * d < m * 1 rewrite [h4] at h6 --h6 : m * -1 < m * d have h7 : (↑m : Int) ≥ 0 := Nat.cast_nonneg m have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7 have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7 have h10 : d = 0 := by linarith show r1 = r2 from calc r1 _ = r1 - r2 + r2 := by ring _ = m * 0 + r2 := by rw [h4, h10] _ = r2 := by ring done done lemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m := (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a) theorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y rewrite [h1, h2] have h3 : a + b = b + a := by ring show [a]_m + [b]_m = [b]_m + [a]_m from calc [a]_m + [b]_m _ = [a + b]_m := add_class m a b _ = [b + a]_m := by rw [h3] _ = [b]_m + [a]_m := (add_class m b a).symm done theorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X rewrite [h1] have h2 : a * 1 = a := by ring show [a]_m * [1]_m = [a]_m from calc [a]_m * [1]_m _ = [a * 1]_m := mul_class m a 1 _ = [a]_m := by rw [h2] done theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry lemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) : [a]_m * [gcd_c2 m a]_m = [1]_m := by set s : Int := gcd_c1 m a have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m define at h1 rewrite [h1, Nat.cast_one] at h2 --h2 : s * ↑m + gcd_c2 m a * ↑a = 1 rewrite [mul_class, cc_eq_iff_congr] define --Goal : ∃ (c : Int), ↑a * gcd_c2 m a - 1 = ↑m * c apply Exists.intro (-s) show a * (gcd_c2 m a) - 1 = m * (-s) from calc a * (gcd_c2 m a) - 1 _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring _ = 1 + m * (-s) - 1 := by rw [h2] _ = m * (-s) := by ring done theorem Theorem_7_3_7 (m a : Nat) : invertible [a]_m ↔ rel_prime m a := by apply Iff.intro · -- (→) assume h1 : invertible [a]_m define at h1 obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1 obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y rewrite [h3, mul_class, cc_eq_iff_congr] at h2 define at h2 obtain (c : Int) (h4 : a * b - 1 = m * c) from h2 rewrite [Exercise_7_2_6] --Goal : ∃ (s t : Int), s * ↑m + t * ↑a = 1 apply Exists.intro (-c) apply Exists.intro b show (-c) * m + b * a = 1 from calc (-c) * m + b * a _ = (-c) * m + (a * b - 1) + 1 := by ring _ = (-c) * m + m * c + 1 := by rw [h4] _ = 1 := by ring done · -- (←) assume h1 : rel_prime m a define show ∃ (Y : ZMod m), [a]_m * Y = [1]_m from Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1) done done /- Section 7.4 -/ section Euler open Euler lemma num_rp_below_base {m : Nat} : num_rp_below m 0 = 0 := by rfl lemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) : num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by have h1 : num_rp_below m (j + 1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : gcd m j = 1 rewrite [if_pos h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1 show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1 done lemma num_rp_below_step_not_rp {m j : Nat} (h : ¬rel_prime m j) : num_rp_below m (j + 1) = num_rp_below m j := by have h1 : num_rp_below m (j +1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : ¬gcd m j = 1 rewrite [if_neg h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j show num_rp_below m (j + 1) = num_rp_below m j from h1 done lemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl #eval phi 10 --Answer: 4 lemma prod_inv_iff_inv {m : Nat} {X : ZMod m} (h1 : invertible X) (Y : ZMod m) : invertible (X * Y) ↔ invertible Y := by apply Iff.intro · -- (→) assume h2 : invertible (X * Y) obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2 apply Exists.intro (X * Z) rewrite [←h3] --Goal : Y * (X * Z) = X * Y * Z ring --Note that ring can do algebra in ZMod m done · -- (←) assume h2 : invertible Y obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1 obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2 apply Exists.intro (Xi * Yi) show (X * Y) * (Xi * Yi) = [1]_m from calc X * Y * (Xi * Yi) _ = (X * Xi) * (Y * Yi) := by ring _ = [1]_m * [1]_m := by rw [h3, h4] _ = [1]_m := Theorem_7_3_6_7 [1]_m done done lemma F_rp_def {m i : Nat} (h : rel_prime m i) : F m i = [i]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h --h : gcd m i = 1 rewrite [if_pos h] at h1 show F m i = [i]_m from h1 done lemma F_not_rp_def {m i : Nat} (h : ¬rel_prime m i) : F m i = [1]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h rewrite [h1, if_neg h] rfl done lemma prod_seq_base {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 0 k f = [1]_m := by rfl lemma prod_seq_step {m : Nat} (n k : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl lemma prod_seq_zero_step {m : Nat} (n : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by rewrite [prod_seq_step, zero_add] rfl done lemma prod_one {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 1 k f = f k := by rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7] rfl done lemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl lemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m := calc [G m a i]_m _ = [(a * i) % m]_m := by rfl _ = [a * i]_m := (cc_eq_mod m (a * i)).symm _ = [a]_m * [i]_m := (mul_class m a i).symm lemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) : rel_prime m (G m a i) ↔ rel_prime m i := by have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1 show rel_prime m (G m a i) ↔ rel_prime m i from calc rel_prime m (G m a i) _ ↔ invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm _ ↔ invertible ([a]_m * [i]_m) := by rw [cc_G] _ ↔ invertible [i]_m := prod_inv_iff_inv h2 ([i]_m) _ ↔ rel_prime m i := Theorem_7_3_7 m i done lemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) : F m (G m a i) = [a]_m * F m i := by have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2 show F m (G m a i) = [a]_m * F m i from calc F m (G m a i) _ = [G m a i]_m := F_rp_def h3 _ = [a]_m * [i]_m := cc_G m a i _ = [a]_m * F m i := by rw [F_rp_def h2] done lemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : ¬rel_prime m i) : F m (G m a i) = [1]_m := by rewrite [←G_rp_iff h1 i] at h2 show F m (G m a i) = [1]_m from F_not_rp_def h2 done lemma FG_prod {m a : Nat} (h1 : rel_prime m a) : ∀ (k : Nat), prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by by_induc · -- Base Case show prod_seq 0 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from calc prod_seq 0 0 ((F m) ∘ (G m a)) _ = [1]_m := prod_seq_base _ _ _ = [a]_m ^ 0 * [1]_m := by ring _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by rw [num_rp_below_base, prod_seq_base] done · -- Induction Step fix k : Nat assume ih : prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) by_cases h2 : rel_prime m k · -- Case 1. h2 : rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([a]_m * F m k) := by rw [FG_rp h1 h2] _ = [a]_m ^ ((num_rp_below m k) + 1) * ((prod_seq k 0 (F m)) * F m k) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_rp h2, prod_seq_zero_step] done · -- Case 2. h2 : ¬rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([1]_m) := by rw [FG_not_rp h1 h2] _ = [a]_m ^ (num_rp_below m k) * (prod_seq k 0 (F m) * ([1]_m)) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_not_rp h2, prod_seq_zero_step, F_not_rp_def h2] done done done lemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by define --Goal : ∀ i < m, G m a i < m fix i : Nat assume h1 : i < m rewrite [G_def] --Goal : a * i % m < m show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m) done lemma left_inv_one_one_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g' (g i) = i) : one_one_below n g := sorry lemma right_inv_onto_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g (g' i) = i) (h2 : maps_below n g') : onto_below n g := by define at h2; define fix k : Nat assume h3 : k < n apply Exists.intro (g' k) show g' k < n ∧ g (g' k) = k from And.intro (h2 k h3) (h1 k h3) done lemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m * [inv_mod m a]_m = [1]_m := by have h2 : 0 ≤ (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a) show [a]_m * [inv_mod m a]_m = [1]_m from calc [a]_m * [inv_mod m a]_m _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2] _ = [a]_m * [gcd_c2 m a]_m := by rw [←cc_eq_mod] _ = [1]_m := gcd_c2_inv h1 done lemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m := calc a * (b % m) % m = a % m * (b % m % m) % m := Nat.mul_mod _ _ _ _ = a % m * (b % m) % m := by rw [Nat.mod_mod] _ = a * b % m := (Nat.mul_mod _ _ _).symm lemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm] rfl done theorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] : ↑a ≡ ↑b (MOD m) ↔ a % m = b % m := sorry lemma mul_inv_mod_cancel {m a i : Nat} [NeZero m] (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1 rewrite [mul_class, cc_eq_iff_congr, ←Nat.cast_mul, ←Nat.cast_one, congr_iff_mod_eq_Nat] at h3 show a * inv_mod m a * i % m = i from calc a * (inv_mod m a) * i % m _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = 1 % m * i % m := by rw [h3] _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = i % m := by rw [one_mul] _ = i := Nat.mod_eq_of_lt h2 done lemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl lemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, G m a (Ginv m a i) = i := by fix i : Nat assume h2 : i < m show G m a (Ginv m a i) = i from calc G m a (Ginv m a i) _ = a * ((inv_mod m a * i) % m) % m := by rfl _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, Ginv m a (G m a i) = i := by fix i : Nat assume h2 : i < m show Ginv m a (G m a i) = i from calc Ginv m a (G m a i) _ = inv_mod m a * ((a * i) % m) % m := by rfl _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc, mul_comm (inv_mod m a)] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_maps_below (m a : Nat) [NeZero m] : maps_below m (Ginv m a) := G_maps_below m (inv_mod m a) lemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : one_one_below m (G m a) := left_inv_one_one_below (Ginv_left_inv h1) lemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : onto_below m (G m a) := right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a) lemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : perm_below m (G m a) := And.intro (G_maps_below m a) (And.intro (G_one_one_below h1) (G_onto_below h1)) --Permuting a product of congruence classes doesn't change product lemma swap_fst (u v : Nat) : swap u v u = v := by define : swap u v u --Goal : (if u = u then v else if u = v then u else u) = v have h : u = u := by rfl rewrite [if_pos h] rfl done lemma swap_snd (u v : Nat) : swap u v v = u := by define : swap u v v by_cases h1 : v = u · -- Case 1. h1 : v = u rewrite [if_pos h1] show v = u from h1 done · -- Case 2. h1 : v ≠ u rewrite [if_neg h1] have h2 : v = v := by rfl rewrite [if_pos h2] rfl done done lemma swap_other {u v i : Nat} (h1 : i ≠ u) (h2 : i ≠ v) : swap u v i = i := by define : swap u v i rewrite [if_neg h1, if_neg h2] rfl done lemma swap_values (u v i : Nat) : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := by by_cases h1 : i = u · -- Case 1. h1 : i = u apply Or.inl rewrite [h1] show swap u v u = v from swap_fst u v done · -- Case 2. h1 : i ≠ u apply Or.inr by_cases h2 : i = v · -- Case 2.1. h2 : i = v apply Or.inl rewrite [h2] show swap u v v = u from swap_snd u v done · -- Case 2.2. h2 : i ≠ v apply Or.inr show swap u v i = i from swap_other h1 h2 done done done lemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by define fix i : Nat assume h3 : i < n have h4 : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := swap_values u v i by_cases on h4 · -- Case 1. h4 : swap u v i = v rewrite [h4] show v < n from h2 done · -- Case 2. by_cases on h4 · -- Case 2.1. h4 : swap u v i = u rewrite [h4] show u < n from h1 done · -- Case 2.2. h4 : swap u v i = i rewrite [h4] show i < n from h3 done done done lemma swap_swap (u v n : Nat) : ∀ i < n, swap u v (swap u v i) = i := by fix i : Nat assume h : i < n by_cases h1 : i = u · -- Case 1. h1 : i = u rewrite [h1, swap_fst, swap_snd] rfl done · -- Case 2. h1 : i ≠ u by_cases h2 : i = v · -- Case 2.1. h2 : i = v rewrite [h2, swap_snd, swap_fst] rfl done · -- Case 2.2. h2 : i ≠ v rewrite [swap_other h1 h2, swap_other h1 h2] rfl done done done lemma swap_one_one_below (u v n) : one_one_below n (swap u v) := left_inv_one_one_below (swap_swap u v n) lemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) := right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2) lemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) := And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2)) lemma comp_perm_below {n : Nat} {f g : Nat → Nat} (h1 : perm_below n f) (h2 : perm_below n g) : perm_below n (f ∘ g) := sorry lemma trivial_swap (u : Nat) : swap u u = id := by apply funext fix x : Nat by_cases h1 : x = u · -- Case 1. h1 : x = u rewrite [h1, swap_fst] rfl done · -- Case 2. h1 : x ≠ u rewrite [swap_other h1 h1] rfl done done lemma prod_eq_fun {m : Nat} (f g : Nat → ZMod m) (k : Nat) : ∀ (n : Nat), (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g := by by_induc · -- Base Case assume h : (∀ i < 0, f (k + i) = g (k + i)) rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g assume h1 : ∀ i < n + 1, f (k + i) = g (k + i) have h2 : ∀ i < n, f (k + i) = g (k + i) := by fix i : Nat assume h2 : i < n have h3 : i < n + 1 := by linarith show f (k + i) = g (k + i) from h1 i h3 done have h3 : prod_seq n k f = prod_seq n k g := ih h2 have h4 : n < n + 1 := Nat.lt_succ_self n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4] rfl done done lemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := by have h2 : ∀ (i : Nat), i < u → (f ∘ swap u n) (0 + i) = f (0 + i) := by fix i : Nat assume h2 : i < u have h3 : 0 + i ≠ u := by linarith have h4 : 0 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl done show prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f from prod_eq_fun (f ∘ swap u n) f 0 u h2 done

lemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f

HTPI.swap_prod_eq_prod_between

htpi/HTPILib/Chap7.lean

htpi

/- Copyright 2023 Daniel J. Velleman -/ import HTPILib.Chap6 namespace HTPI /- Definitions -/ lemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by have h : n + 1 > 0 := Nat.succ_pos n show a % (n + 1) < n + 1 from Nat.mod_lt a h done def gcd (a b : Nat) : Nat := match b with | 0 => a | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd (n + 1) (a % (n + 1)) termination_by b mutual def gcd_c1 (a b : Nat) : Int := match b with | 0 => 1 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c2 (n + 1) (a % (n + 1)) --Corresponds to s = t' termination_by b def gcd_c2 (a b : Nat) : Int := match b with | 0 => 0 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c1 (n + 1) (a % (n + 1)) - (gcd_c2 (n + 1) (a % (n + 1))) * ↑(a / (n + 1)) --Corresponds to t = s' - t'q termination_by b end def prime (n : Nat) : Prop := 2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n def prime_factor (p n : Nat) : Prop := prime p ∧ p ∣ n def all_prime (l : List Nat) : Prop := ∀ p ∈ l, prime p def nondec (l : List Nat) : Prop := match l with | [] => True --Of course, True is a proposition that is always true | n :: L => (∀ m ∈ L, n ≤ m) ∧ nondec L def nondec_prime_list (l : List Nat) : Prop := all_prime l ∧ nondec l def prod (l : List Nat) : Nat := match l with | [] => 1 | n :: L => n * (prod L) def prime_factorization (n : Nat) (l : List Nat) : Prop := nondec_prime_list l ∧ prod l = n def rel_prime (a b : Nat) : Prop := gcd a b = 1 def congr_mod (m : Nat) (a b : Int) : Prop := (↑m : Int) ∣ (a - b) def cc (m : Nat) (a : Int) : ZMod m := (↑a : ZMod m) notation:50 a " ≡ " b " (MOD " m ")" => congr_mod m a b notation:max "["a"]_"m:max => cc m a def invertible {m : Nat} (X : ZMod m) : Prop := ∃ (Y : ZMod m), X * Y = [1]_m def num_rp_below (m k : Nat) : Nat := match k with | 0 => 0 | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j def phi (m : Nat) : Nat := num_rp_below m m def prod_seq {m : Nat} (j k : Nat) (f : Nat → ZMod m) : ZMod m := match j with | 0 => [1]_m | n + 1 => prod_seq n k f * f (k + n) def maps_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i < n, g i < n def one_one_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i1 < n, ∀ i2 < n, g i1 = g i2 → i1 = i2 def onto_below (n : Nat) (g : Nat → Nat) : Prop := ∀ k < n, ∃ i < n, g i = k def perm_below (n : Nat) (g : Nat → Nat) : Prop := maps_below n g ∧ one_one_below n g ∧ onto_below n g def inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m) def swap (u v i : Nat) : Nat := if i = u then v else if i = v then u else i namespace Euler --For definitions specific to Euler's theorem def F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m def G (m a i : Nat) : Nat := (a * i) % m def Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i end Euler /- Section 7.1 -/ theorem dvd_mod_of_dvd_a_b {a b d : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ (a % b) := by set q : Nat := a / b have h3 : b * q + a % b = a := Nat.div_add_mod a b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 define --Goal : ∃ (c : Nat), a % b = d * c apply Exists.intro (j - k * q) show a % b = d * (j - k * q) from calc a % b _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm _ = a - b * q := by rw [h3] _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc] _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm done theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry #eval gcd 672 161 --Answer: 7 lemma gcd_base (a : Nat) : gcd a 0 = a := by rfl lemma gcd_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd a b = gcd b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1)) rfl done lemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b ≠ 0) : a % b < b := by have h1 : b > 0 := Nat.pos_of_ne_zero h show a % b < b from Nat.mod_lt a h1 done lemma dvd_self (n : Nat) : n ∣ n := by apply Exists.intro 1 ring done theorem gcd_dvd : ∀ (b a : Nat), (gcd a b) ∣ a ∧ (gcd a b) ∣ b := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd a b_1) ∣ a ∧ (gcd a b_1) ∣ b_1 fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_base] --Goal: a ∣ a ∧ a ∣ 0 apply And.intro (dvd_self a) define apply Exists.intro 0 rfl done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_nonzero a h1] --Goal : gcd b (a % b) ∣ a ∧ gcd b (a % b) ∣ b have h2 : a % b < b := mod_nonzero_lt a h1 have h3 : (gcd b (a % b)) ∣ b ∧ (gcd b (a % b)) ∣ (a % b) := ih (a % b) h2 b apply And.intro _ h3.left show (gcd b (a % b)) ∣ a from dvd_a_of_dvd_b_mod h3.left h3.right done done theorem gcd_dvd_left (a b : Nat) : (gcd a b) ∣ a := (gcd_dvd b a).left theorem gcd_dvd_right (a b : Nat) : (gcd a b) ∣ b := (gcd_dvd b a).right lemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl lemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c1 a b = gcd_c2 b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done lemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl lemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * ↑(a / b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done theorem gcd_lin_comb : ∀ (b a : Nat), (gcd_c1 a b) * ↑a + (gcd_c2 a b) * ↑b = ↑(gcd a b) := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd_c1 a b_1) * ↑a + (gcd_c2 a b_1) * ↑b_1 = ↑(gcd a b_1) fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base] --Goal : 1 * ↑a + 0 * ↑0 = ↑a ring done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1] --Goal : gcd_c2 b (a % b) * ↑a + -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * ↑(a / b)) * ↑b = -- ↑(gcd b (a % b)) set r : Nat := a % b set q : Nat := a / b set s : Int := gcd_c1 b r set t : Int := gcd_c2 b r --Goal : t * ↑a + (s - t * ↑q) * ↑b = ↑(gcd b r) have h2 : r < b := mod_nonzero_lt a h1 have h3 : s * ↑b + t * ↑r = ↑(gcd b r) := ih r h2 b have h4 : b * q + r = a := Nat.div_add_mod a b rewrite [←h3, ←h4] rewrite [Nat.cast_add, Nat.cast_mul] --Goal : t * (↑b * ↑q + ↑r) + (s - t * ↑q) * ↑b = s * ↑b + t * ↑r ring done done #eval gcd_c1 672 161 --Answer: 6 #eval gcd_c2 672 161 --Answer: -25 --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161 theorem Theorem_7_1_6 {d a b : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ gcd a b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑d ∣ ↑(gcd a b) set s : Int := gcd_c1 a b set t : Int := gcd_c2 a b have h3 : s * ↑a + t * ↑b = ↑(gcd a b) := gcd_lin_comb b a rewrite [←h3] --Goal : ↑d ∣ s * ↑a + t * ↑b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul] --Goal : ↑d ∣ s * (↑d * ↑j) + t * (↑d * ↑k) define apply Exists.intro (s * ↑j + t * ↑k) ring done /- Section 7.2 -/ theorem dvd_trans {a b c : Nat} (h1 : a ∣ b) (h2 : b ∣ c) : a ∣ c := by define at h1; define at h2; define obtain (m : Nat) (h3 : b = a * m) from h1 obtain (n : Nat) (h4 : c = b * n) from h2 rewrite [h3, mul_assoc] at h4 apply Exists.intro (m * n) show c = a * (m * n) from h4 done lemma exists_prime_factor : ∀ (n : Nat), 2 ≤ n → ∃ (p : Nat), prime_factor p n := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, 2 ≤ n_1 → ∃ (p : Nat), prime_factor p n_1 assume h1 : 2 ≤ n by_cases h2 : prime n · -- Case 1. h2 : prime n apply Exists.intro n define --Goal : prime n ∧ n ∣ n show prime n ∧ n ∣ n from And.intro h2 (dvd_self n) done · -- Case 2. h2 : ¬prime n define at h2 --h2 : ¬(2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n) demorgan at h2 disj_syll h2 h1 obtain (a : Nat) (h3 : ∃ (b : Nat), a * b = n ∧ a < n ∧ b < n) from h2 obtain (b : Nat) (h4 : a * b = n ∧ a < n ∧ b < n) from h3 have h5 : 2 ≤ a := by by_contra h6 have h7 : a ≤ 1 := by linarith have h8 : n ≤ b := calc n _ = a * b := h4.left.symm _ ≤ 1 * b := by rel [h7] _ = b := by ring linarith --n ≤ b contradicts b < n done have h6 : ∃ (p : Nat), prime_factor p a := ih a h4.right.left h5 obtain (p : Nat) (h7 : prime_factor p a) from h6 apply Exists.intro p define --Goal : prime p ∧ p ∣ n define at h7 --h7 : prime p ∧ p ∣ a apply And.intro h7.left have h8 : a ∣ n := by apply Exists.intro b show n = a * b from (h4.left).symm done show p ∣ n from dvd_trans h7.right h8 done done lemma exists_least_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q := by set S : Set Nat := {p : Nat | prime_factor p n} have h2 : ∃ (p : Nat), p ∈ S := exists_prime_factor n h show ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q from well_ord_princ S h2 done lemma all_prime_nil : all_prime [] := by define --Goal : ∀ p ∈ [], prime p fix p : Nat contrapos --Goal : ¬prime p → p ∉ [] assume h1 : ¬prime p show p ∉ [] from List.not_mem_nil p done lemma all_prime_cons (n : Nat) (L : List Nat) : all_prime (n :: L) ↔ prime n ∧ all_prime L := by apply Iff.intro · -- (→) assume h1 : all_prime (n :: L) --Goal : prime n ∧ all_prime L define at h1 --h1 : ∀ p ∈ n :: L, prime p apply And.intro (h1 n (List.mem_cons_self n L)) define --Goal : ∀ p ∈ L, prime p fix p : Nat assume h2 : p ∈ L show prime p from h1 p (List.mem_cons_of_mem n h2) done · -- (←) assume h1 : prime n ∧ all_prime L --Goal : all_prime (n :: l) define : all_prime L at h1 define fix p : Nat assume h2 : p ∈ n :: L rewrite [List.mem_cons] at h2 --h2 : p = n ∨ p ∈ L by_cases on h2 · -- Case 1. h2 : p = n rewrite [h2] show prime n from h1.left done · -- Case 2. h2 : p ∈ L show prime p from h1.right p h2 done done done lemma nondec_nil : nondec [] := by define --Goal : True trivial --trivial proves some obviously true statements, such as True done lemma nondec_cons (n : Nat) (L : List Nat) : nondec (n :: L) ↔ (∀ m ∈ L, n ≤ m) ∧ nondec L := by rfl lemma prod_nil : prod [] = 1 := by rfl lemma prod_cons : prod (n :: L) = n * (prod L) := by rfl lemma exists_cons_of_length_eq_succ {A : Type} {l : List A} {n : Nat} (h : l.length = n + 1) : ∃ (a : A) (L : List A), l = a :: L ∧ L.length = n := by have h1 : ¬l.length = 0 := by linarith rewrite [List.length_eq_zero] at h1 obtain (a : A) (h2 : ∃ (L : List A), l = a :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List A) (h3 : l = a :: L) from h2 apply Exists.intro a apply Exists.intro L apply And.intro h3 have h4 : (a :: L).length = L.length + 1 := List.length_cons a L rewrite [←h3, h] at h4 show L.length = n from (Nat.add_right_cancel h4).symm done lemma list_elt_dvd_prod_by_length (a : Nat) : ∀ (n : Nat), ∀ (l : List Nat), l.length = n → a ∈ l → a ∣ prod l := by by_induc · --Base Case fix l : List Nat assume h1 : l.length = 0 rewrite [List.length_eq_zero] at h1 --h1 : l = [] rewrite [h1] --Goal : a ∈ [] → a ∣ prod [] contrapos assume h2 : ¬a ∣ prod [] show a ∉ [] from List.not_mem_nil a done · -- Induction Step fix n : Nat assume ih : ∀ (l : List Nat), List.length l = n → a ∈ l → a ∣ prod l fix l : List Nat assume h1 : l.length = n + 1 --Goal : a ∈ l → a ∣ prod l obtain (b : Nat) (h2 : ∃ (L : List Nat), l = b :: L ∧ L.length = n) from exists_cons_of_length_eq_succ h1 obtain (L : List Nat) (h3 : l = b :: L ∧ L.length = n) from h2 have h4 : a ∈ L → a ∣ prod L := ih L h3.right assume h5 : a ∈ l rewrite [h3.left, prod_cons] --Goal : a ∣ b * prod L rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b ∨ a ∈ L by_cases on h5 · -- Case 1. h5 : a = b apply Exists.intro (prod L) rewrite [h5] rfl done · -- Case 2. h5 : a ∈ L have h6 : a ∣ prod L := h4 h5 have h7 : prod L ∣ b * prod L := by apply Exists.intro b ring done show a ∣ b * prod L from dvd_trans h6 h7 done done done lemma list_elt_dvd_prod {a : Nat} {l : List Nat} (h : a ∈ l) : a ∣ prod l := by set n : Nat := l.length have h1 : l.length = n := by rfl show a ∣ prod l from list_elt_dvd_prod_by_length a n l h1 h done lemma exists_prime_factorization : ∀ (n : Nat), n ≥ 1 → ∃ (l : List Nat), prime_factorization n l := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, n_1 ≥ 1 → ∃ (l : List Nat), prime_factorization n_1 l assume h1 : n ≥ 1 by_cases h2 : n = 1 · -- Case 1. h2 : n = 1 apply Exists.intro [] define apply And.intro · -- Proof of nondec_prime_list [] define show all_prime [] ∧ nondec [] from And.intro all_prime_nil nondec_nil done · -- Proof of prod [] = n rewrite [prod_nil, h2] rfl done done · -- Case 2. h2 : n ≠ 1 have h3 : n ≥ 2 := lt_of_le_of_ne' h1 h2 obtain (p : Nat) (h4 : prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q) from exists_least_prime_factor h3 have p_prime_factor : prime_factor p n := h4.left define at p_prime_factor have p_prime : prime p := p_prime_factor.left have p_dvd_n : p ∣ n := p_prime_factor.right have p_least : ∀ (q : Nat), prime_factor q n → p ≤ q := h4.right obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n have h5 : m ≠ 0 := by contradict h1 with h6 have h7 : n = 0 := calc n _ = p * m := n_eq_pm _ = p * 0 := by rw [h6] _ = 0 := by ring rewrite [h7] decide done have m_pos : 0 < m := Nat.pos_of_ne_zero h5 have m_lt_n : m < n := by define at p_prime show m < n from calc m _ < m + m := by linarith _ = 2 * m := by ring _ ≤ p * m := by rel [p_prime.left] _ = n := n_eq_pm.symm done obtain (L : List Nat) (h6 : prime_factorization m L) from ih m m_lt_n m_pos define at h6 have ndpl_L : nondec_prime_list L := h6.left define at ndpl_L apply Exists.intro (p :: L) define apply And.intro · -- Proof of nondec_prime_list (p :: L) define apply And.intro · -- Proof of all_prime (p :: L) rewrite [all_prime_cons] show prime p ∧ all_prime L from And.intro p_prime ndpl_L.left done · -- Proof of nondec (p :: L) rewrite [nondec_cons] apply And.intro _ ndpl_L.right fix q : Nat assume q_in_L : q ∈ L have h7 : q ∣ prod L := list_elt_dvd_prod q_in_L rewrite [h6.right] at h7 --h7 : q ∣ m have h8 : m ∣ n := by apply Exists.intro p rewrite [n_eq_pm] ring done have q_dvd_n : q ∣ n := dvd_trans h7 h8 have ap_L : all_prime L := ndpl_L.left define at ap_L have q_prime_factor : prime_factor q n := And.intro (ap_L q q_in_L) q_dvd_n show p ≤ q from p_least q q_prime_factor done done · -- Proof of prod (p :: L) = n rewrite [prod_cons, h6.right, n_eq_pm] rfl done done done theorem Theorem_7_2_2 {a b c : Nat} (h1 : c ∣ a * b) (h2 : rel_prime a c) : c ∣ b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑c ∣ ↑b define at h1; define at h2; define obtain (j : Nat) (h3 : a * b = c * j) from h1 set s : Int := gcd_c1 a c set t : Int := gcd_c2 a c have h4 : s * ↑a + t * ↑c = ↑(gcd a c) := gcd_lin_comb c a rewrite [h2, Nat.cast_one] at h4 --h4 : s * ↑a + t * ↑c = (1 : Int) apply Exists.intro (s * ↑j + t * ↑b) show ↑b = ↑c * (s * ↑j + t * ↑b) from calc ↑b _ = (1 : Int) * ↑b := (one_mul _).symm _ = (s * ↑a + t * ↑c) * ↑b := by rw [h4] _ = s * (↑a * ↑b) + t * ↑c * ↑b := by ring _ = s * (↑c * ↑j) + t * ↑c * ↑b := by rw [←Nat.cast_mul a b, h3, Nat.cast_mul c j] _ = ↑c * (s * ↑j + t * ↑b) := by ring done lemma le_nonzero_prod_left {a b : Nat} (h : a * b ≠ 0) : a ≤ a * b := by have h1 : b ≠ 0 := by contradict h with h1 rewrite [h1] ring done have h2 : 1 ≤ b := Nat.pos_of_ne_zero h1 show a ≤ a * b from calc a = a * 1 := (mul_one a).symm _ ≤ a * b := by rel [h2] done lemma le_nonzero_prod_right {a b : Nat} (h : a * b ≠ 0) : b ≤ a * b := by rewrite [mul_comm] rewrite [mul_comm] at h show b ≤ b * a from le_nonzero_prod_left h done lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry lemma rel_prime_of_prime_not_dvd {a p : Nat} (h1 : prime p) (h2 : ¬p ∣ a) : rel_prime a p := by have h3 : gcd a p ∣ a := gcd_dvd_left a p have h4 : gcd a p ∣ p := gcd_dvd_right a p have h5 : gcd a p = 1 ∨ gcd a p = p := dvd_prime h1 h4 have h6 : gcd a p ≠ p := by contradict h2 with h6 rewrite [h6] at h3 show p ∣ a from h3 done disj_syll h5 h6 show rel_prime a p from h5 done theorem Theorem_7_2_3 {a b p : Nat} (h1 : prime p) (h2 : p ∣ a * b) : p ∣ a ∨ p ∣ b := by or_right with h3 have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3 show p ∣ b from Theorem_7_2_2 h2 h4 done lemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a ≥ 1 := by have h1 : a ≠ 0 := by by_contra h1 rewrite [h1] at h contradict h linarith done show a ≥ 1 from Nat.pos_of_ne_zero h1 done lemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by have h1 : a ≥ 1 := ge_one_of_prod_one h have h2 : a * b ≠ 0 := by linarith have h3 : a ≤ a * b := le_nonzero_prod_left h2 rewrite [h] at h3 show a = 1 from Nat.le_antisymm h3 h1 done lemma eq_one_of_dvd_one {n : Nat} (h : n ∣ 1) : n = 1 := by obtain (j : Nat) (h1 : 1 = n * j) from h show n = 1 from eq_one_of_prod_one h1.symm done lemma prime_not_one {p : Nat} (h : prime p) : p ≠ 1 := by define at h linarith done theorem Theorem_7_2_4 {p : Nat} (h1 : prime p) : ∀ (l : List Nat), p ∣ prod l → ∃ a ∈ l, p ∣ a := by apply List.rec · -- Base Case. Goal : p ∣ prod [] → ∃ a ∈ [], p ∣ a rewrite [prod_nil] assume h2 : p ∣ 1 show ∃ a ∈ [], p ∣ a from absurd (eq_one_of_dvd_one h2) (prime_not_one h1) done · -- Induction Step fix b : Nat fix L : List Nat assume ih : p ∣ prod L → ∃ a ∈ L, p ∣ a --Goal : p ∣ prod (b :: L) → ∃ a ∈ b :: L, p ∣ a assume h2 : p ∣ prod (b :: L) rewrite [prod_cons] at h2 have h3 : p ∣ b ∨ p ∣ prod L := Theorem_7_2_3 h1 h2 by_cases on h3 · -- Case 1. h3 : p ∣ b apply Exists.intro b show b ∈ b :: L ∧ p ∣ b from And.intro (List.mem_cons_self b L) h3 done · -- Case 2. h3 : p ∣ prod L obtain (a : Nat) (h4 : a ∈ L ∧ p ∣ a) from ih h3 apply Exists.intro a show a ∈ b :: L ∧ p ∣ a from And.intro (List.mem_cons_of_mem b h4.left) h4.right done done done lemma prime_in_list {p : Nat} {l : List Nat} (h1 : prime p) (h2 : all_prime l) (h3 : p ∣ prod l) : p ∈ l := by obtain (a : Nat) (h4 : a ∈ l ∧ p ∣ a) from Theorem_7_2_4 h1 l h3 define at h2 have h5 : prime a := h2 a h4.left have h6 : p = 1 ∨ p = a := dvd_prime h5 h4.right disj_syll h6 (prime_not_one h1) rewrite [h6] show a ∈ l from h4.left done lemma first_le_first {p q : Nat} {l m : List Nat} (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m)) (h3 : prod (p :: l) = prod (q :: m)) : p ≤ q := by define at h1; define at h2 have h4 : q ∣ prod (p :: l) := by define apply Exists.intro (prod m) rewrite [←prod_cons] show prod (p :: l) = prod (q :: m) from h3 done have h5 : all_prime (q :: m) := h2.left rewrite [all_prime_cons] at h5 have h6 : q ∈ p :: l := prime_in_list h5.left h1.left h4 have h7 : nondec (p :: l) := h1.right rewrite [nondec_cons] at h7 rewrite [List.mem_cons] at h6 by_cases on h6 · -- Case 1. h6 : q = p linarith done · -- Case 2. h6 : q ∈ l have h8 : ∀ m ∈ l, p ≤ m := h7.left show p ≤ q from h8 q h6 done done lemma nondec_prime_list_tail {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by define at h define rewrite [all_prime_cons, nondec_cons] at h show all_prime l ∧ nondec l from And.intro h.left.right h.right.right done lemma cons_prod_not_one {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : prod (p :: l) ≠ 1 := by define at h have h1 : all_prime (p :: l) := h.left rewrite [all_prime_cons] at h1 rewrite [prod_cons] by_contra h2 show False from (prime_not_one h1.left) (eq_one_of_prod_one h2) done lemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) : l = [] ↔ prod l = 1 := by apply Iff.intro · -- (→) assume h1 : l = [] rewrite [h1] show prod [] = 1 from prod_nil done · -- (←) contrapos assume h1 : ¬l = [] obtain (p : Nat) (h2 : ∃ (L : List Nat), l = p :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List Nat) (h3 : l = p :: L) from h2 rewrite [h3] at h rewrite [h3] show ¬prod (p :: L) = 1 from cons_prod_not_one h done done lemma prime_pos {p : Nat} (h : prime p) : p > 0 := by define at h linarith done theorem Theorem_7_2_5 : ∀ (l1 l2 : List Nat), nondec_prime_list l1 → nondec_prime_list l2 → prod l1 = prod l2 → l1 = l2 := by apply List.rec · -- Base Case. Goal : ∀ (l2 : List Nat), nondec_prime_list [] → -- nondec_prime_list l2 → prod [] = prod l2 → [] = l2 fix l2 : List Nat assume h1 : nondec_prime_list [] assume h2 : nondec_prime_list l2 assume h3 : prod [] = prod l2 rewrite [prod_nil, eq_comm, ←list_nil_iff_prod_one h2] at h3 show [] = l2 from h3.symm done · -- Induction Step fix p : Nat fix L1 : List Nat assume ih : ∀ (L2 : List Nat), nondec_prime_list L1 → nondec_prime_list L2 → prod L1 = prod L2 → L1 = L2 -- Goal : ∀ (l2 : List Nat), nondec_prime_list (p :: L1) → -- nondec_prime_list l2 → prod (p :: L1) = prod l2 → p :: L1 = l2 fix l2 : List Nat assume h1 : nondec_prime_list (p :: L1) assume h2 : nondec_prime_list l2 assume h3 : prod (p :: L1) = prod l2 have h4 : ¬prod (p :: L1) = 1 := cons_prod_not_one h1 rewrite [h3, ←list_nil_iff_prod_one h2] at h4 obtain (q : Nat) (h5 : ∃ (L : List Nat), l2 = q :: L) from List.exists_cons_of_ne_nil h4 obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5 rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2) rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2) have h7 : p ≤ q := first_le_first h1 h2 h3 have h8 : q ≤ p := first_le_first h2 h1 h3.symm have h9 : p = q := by linarith rewrite [h9, prod_cons, prod_cons] at h3 --h3 : q * prod L1 = q * prod L2 have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1 have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2 define at h2 have h12 : all_prime (q :: L2) := h2.left rewrite [all_prime_cons] at h12 have h13 : q > 0 := prime_pos h12.left have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3 have h15 : L1 = L2 := ih L2 h10 h11 h14 rewrite [h6, h9, h15] rfl done done theorem fund_thm_arith (n : Nat) (h : n ≥ 1) : ∃! (l : List Nat), prime_factorization n l := by exists_unique · -- Existence show ∃ (l : List Nat), prime_factorization n l from exists_prime_factorization n h done · -- Uniqueness fix l1 : List Nat; fix l2 : List Nat assume h1 : prime_factorization n l1 assume h2 : prime_factorization n l2 define at h1; define at h2 have h3 : prod l1 = n := h1.right rewrite [←h2.right] at h3 show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3 done done /- Section 7.3 -/ theorem congr_refl (m : Nat) : ∀ (a : Int), a ≡ a (MOD m) := by fix a : Int define --Goal : ∃ (c : Int), a - a = ↑m * c apply Exists.intro 0 ring done theorem congr_symm {m : Nat} : ∀ {a b : Int}, a ≡ b (MOD m) → b ≡ a (MOD m) := by fix a : Int; fix b : Int assume h1 : a ≡ b (MOD m) define at h1 --h1 : ∃ (c : Int), a - b = ↑m * c define --Goal : ∃ (c : Int), b - a = ↑m * c obtain (c : Int) (h2 : a - b = m * c) from h1 apply Exists.intro (-c) show b - a = m * (-c) from calc b - a _ = -(a - b) := by ring _ = -(m * c) := by rw [h2] _ = m * (-c) := by ring done theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry /- Fundamental properties of congruence classes -/ lemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) : X = Y ↔ X.val = Y.val := Fin.ext_iff lemma val_nat_eq_mod (n k : Nat) : ([k]_(n + 1)).val = k % (n + 1) := by rfl lemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl theorem cc_rep {m : Nat} (X : ZMod m) : ∃ (a : Int), X = [a]_m := match m with | 0 => by apply Exists.intro X rfl done | n + 1 => by apply Exists.intro ↑(X.val) have h1 : X.val < n + 1 := Fin.prop X rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1] rfl done theorem add_class (m : Nat) (a b : Int) : [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm theorem mul_class (m : Nat) (a b : Int) : [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm lemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ [a - b]_m = [0]_m := by apply Iff.intro · -- (→) assume h1 : [a]_m = [b]_m have h2 : a - b = a + (-b) := by ring have h3 : b + (-b) = 0 := by ring show [a - b]_m = [0]_m from calc [a - b]_m _ = [a + (-b)]_m := by rw [h2] _ = [a]_m + [-b]_m := by rw [add_class] _ = [b]_m + [-b]_m := by rw [h1] _ = [b + -b]_m := by rw [add_class] _ = [0]_m := by rw [h3] done · -- (←) assume h1 : [a - b]_m = [0]_m have h2 : b + (a - b) = a := by ring have h3 : b + 0 = b := by ring show [a]_m = [b]_m from calc [a]_m _ = [b + (a - b)]_m := by rw [h2] _ = [b]_m + [a - b]_m := by rw [add_class] _ = [b]_m + [0]_m := by rw [h1] _ = [b + 0]_m := by rw [add_class] _ = [b]_m := by rw [h3] done done lemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) : [a]_m = [0]_m → [-a]_m = [0]_m := by assume h1 : [a]_m = [0]_m have h2 : 0 + (-a) = -a := by ring have h3 : a + (-a) = 0 := by ring show [-a]_m = [0]_m from calc [-a]_m _ = [0 + (-a)]_m := by rw [h2] _ = [0]_m + [-a]_m := by rw [add_class] _ = [a]_m + [-a]_m := by rw [h1] _ = [a + (-a)]_m := by rw [add_class] _ = [0]_m := by rw [h3] done lemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) : [-a]_m = [0]_m ↔ [a]_m = [0]_m := by apply Iff.intro _ (cc_neg_zero_of_cc_zero m a) assume h1 : [-a]_m = [0]_m have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1 have h3 : -(-a) = a := by ring rewrite [h3] at h2 show [a]_m = [0]_m from h2 done lemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl lemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m ↔ m ∣ k := match m with | 0 => by have h : (0 : Int) = (↑(0 : Nat) : Int) := by rfl rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj] apply Iff.intro · -- (→) assume h1 : k = 0 rewrite [h1] show 0 ∣ 0 from dvd_self 0 done · -- (←) assume h1 : 0 ∣ k obtain (c : Nat) (h2 : k = 0 * c) from h1 rewrite [h2] ring done done | n + 1 => by rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero] show k % (n + 1) = 0 ↔ n + 1 ∣ k from (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm done lemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m ↔ ↑m ∣ a := by obtain (k : Nat) (h1 : a = ↑k ∨ a = -↑k) from Int.eq_nat_or_neg a by_cases on h1 · -- Case 1. h1: a = ↑k rewrite [h1, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done · -- Case 2. h1: a = -↑k rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done done theorem cc_eq_iff_congr (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ a ≡ b (MOD m) := calc [a]_m = [b]_m _ ↔ [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b _ ↔ ↑m ∣ (a - b) := cc_zero_iff_dvd m (a - b) _ ↔ a ≡ b (MOD m) := by rfl /- End of fundamental properties of congruence classes -/ lemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m := by have h1 : (↑m : Int) ≠ 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m) show 0 ≤ a % m from Int.emod_nonneg a h1 done lemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m) have h2 : (↑m : Int) > 0 := (Nat.cast_pos).rtl h1 show a % m < m from Int.emod_lt_of_pos a h2 done lemma congr_mod_mod (m : Nat) (a : Int) : a ≡ a % m (MOD m) := by define have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m apply Exists.intro (a / m) show a - a % m = m * (a / m) from calc a - (a % m) _ = m * (a / m) + a % m - a % m := by rw [h1] _ = m * (a / m) := by ring done lemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) := And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a)) theorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) : ∃! (r : Int), 0 ≤ r ∧ r < m ∧ a ≡ r (MOD m) := by exists_unique · -- Existence apply Exists.intro (a % m) show 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) from mod_cmpl_res m a done · -- Uniqueness fix r1 : Int; fix r2 : Int assume h1 : 0 ≤ r1 ∧ r1 < m ∧ a ≡ r1 (MOD m) assume h2 : 0 ≤ r2 ∧ r2 < m ∧ a ≡ r2 (MOD m) have h3 : r1 ≡ r2 (MOD m) := congr_trans (congr_symm h1.right.right) h2.right.right obtain (d : Int) (h4 : r1 - r2 = m * d) from h3 have h5 : r1 - r2 < m * 1 := by linarith have h6 : m * (-1) < r1 - r2 := by linarith rewrite [h4] at h5 --h5 : m * d < m * 1 rewrite [h4] at h6 --h6 : m * -1 < m * d have h7 : (↑m : Int) ≥ 0 := Nat.cast_nonneg m have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7 have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7 have h10 : d = 0 := by linarith show r1 = r2 from calc r1 _ = r1 - r2 + r2 := by ring _ = m * 0 + r2 := by rw [h4, h10] _ = r2 := by ring done done lemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m := (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a) theorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y rewrite [h1, h2] have h3 : a + b = b + a := by ring show [a]_m + [b]_m = [b]_m + [a]_m from calc [a]_m + [b]_m _ = [a + b]_m := add_class m a b _ = [b + a]_m := by rw [h3] _ = [b]_m + [a]_m := (add_class m b a).symm done theorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X rewrite [h1] have h2 : a * 1 = a := by ring show [a]_m * [1]_m = [a]_m from calc [a]_m * [1]_m _ = [a * 1]_m := mul_class m a 1 _ = [a]_m := by rw [h2] done theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry lemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) : [a]_m * [gcd_c2 m a]_m = [1]_m := by set s : Int := gcd_c1 m a have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m define at h1 rewrite [h1, Nat.cast_one] at h2 --h2 : s * ↑m + gcd_c2 m a * ↑a = 1 rewrite [mul_class, cc_eq_iff_congr] define --Goal : ∃ (c : Int), ↑a * gcd_c2 m a - 1 = ↑m * c apply Exists.intro (-s) show a * (gcd_c2 m a) - 1 = m * (-s) from calc a * (gcd_c2 m a) - 1 _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring _ = 1 + m * (-s) - 1 := by rw [h2] _ = m * (-s) := by ring done theorem Theorem_7_3_7 (m a : Nat) : invertible [a]_m ↔ rel_prime m a := by apply Iff.intro · -- (→) assume h1 : invertible [a]_m define at h1 obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1 obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y rewrite [h3, mul_class, cc_eq_iff_congr] at h2 define at h2 obtain (c : Int) (h4 : a * b - 1 = m * c) from h2 rewrite [Exercise_7_2_6] --Goal : ∃ (s t : Int), s * ↑m + t * ↑a = 1 apply Exists.intro (-c) apply Exists.intro b show (-c) * m + b * a = 1 from calc (-c) * m + b * a _ = (-c) * m + (a * b - 1) + 1 := by ring _ = (-c) * m + m * c + 1 := by rw [h4] _ = 1 := by ring done · -- (←) assume h1 : rel_prime m a define show ∃ (Y : ZMod m), [a]_m * Y = [1]_m from Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1) done done /- Section 7.4 -/ section Euler open Euler lemma num_rp_below_base {m : Nat} : num_rp_below m 0 = 0 := by rfl lemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) : num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by have h1 : num_rp_below m (j + 1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : gcd m j = 1 rewrite [if_pos h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1 show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1 done lemma num_rp_below_step_not_rp {m j : Nat} (h : ¬rel_prime m j) : num_rp_below m (j + 1) = num_rp_below m j := by have h1 : num_rp_below m (j +1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : ¬gcd m j = 1 rewrite [if_neg h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j show num_rp_below m (j + 1) = num_rp_below m j from h1 done lemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl #eval phi 10 --Answer: 4 lemma prod_inv_iff_inv {m : Nat} {X : ZMod m} (h1 : invertible X) (Y : ZMod m) : invertible (X * Y) ↔ invertible Y := by apply Iff.intro · -- (→) assume h2 : invertible (X * Y) obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2 apply Exists.intro (X * Z) rewrite [←h3] --Goal : Y * (X * Z) = X * Y * Z ring --Note that ring can do algebra in ZMod m done · -- (←) assume h2 : invertible Y obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1 obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2 apply Exists.intro (Xi * Yi) show (X * Y) * (Xi * Yi) = [1]_m from calc X * Y * (Xi * Yi) _ = (X * Xi) * (Y * Yi) := by ring _ = [1]_m * [1]_m := by rw [h3, h4] _ = [1]_m := Theorem_7_3_6_7 [1]_m done done lemma F_rp_def {m i : Nat} (h : rel_prime m i) : F m i = [i]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h --h : gcd m i = 1 rewrite [if_pos h] at h1 show F m i = [i]_m from h1 done lemma F_not_rp_def {m i : Nat} (h : ¬rel_prime m i) : F m i = [1]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h rewrite [h1, if_neg h] rfl done lemma prod_seq_base {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 0 k f = [1]_m := by rfl lemma prod_seq_step {m : Nat} (n k : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl lemma prod_seq_zero_step {m : Nat} (n : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by rewrite [prod_seq_step, zero_add] rfl done lemma prod_one {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 1 k f = f k := by rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7] rfl done lemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl lemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m := calc [G m a i]_m _ = [(a * i) % m]_m := by rfl _ = [a * i]_m := (cc_eq_mod m (a * i)).symm _ = [a]_m * [i]_m := (mul_class m a i).symm lemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) : rel_prime m (G m a i) ↔ rel_prime m i := by have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1 show rel_prime m (G m a i) ↔ rel_prime m i from calc rel_prime m (G m a i) _ ↔ invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm _ ↔ invertible ([a]_m * [i]_m) := by rw [cc_G] _ ↔ invertible [i]_m := prod_inv_iff_inv h2 ([i]_m) _ ↔ rel_prime m i := Theorem_7_3_7 m i done lemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) : F m (G m a i) = [a]_m * F m i := by have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2 show F m (G m a i) = [a]_m * F m i from calc F m (G m a i) _ = [G m a i]_m := F_rp_def h3 _ = [a]_m * [i]_m := cc_G m a i _ = [a]_m * F m i := by rw [F_rp_def h2] done lemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : ¬rel_prime m i) : F m (G m a i) = [1]_m := by rewrite [←G_rp_iff h1 i] at h2 show F m (G m a i) = [1]_m from F_not_rp_def h2 done lemma FG_prod {m a : Nat} (h1 : rel_prime m a) : ∀ (k : Nat), prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by by_induc · -- Base Case show prod_seq 0 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from calc prod_seq 0 0 ((F m) ∘ (G m a)) _ = [1]_m := prod_seq_base _ _ _ = [a]_m ^ 0 * [1]_m := by ring _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by rw [num_rp_below_base, prod_seq_base] done · -- Induction Step fix k : Nat assume ih : prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) by_cases h2 : rel_prime m k · -- Case 1. h2 : rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([a]_m * F m k) := by rw [FG_rp h1 h2] _ = [a]_m ^ ((num_rp_below m k) + 1) * ((prod_seq k 0 (F m)) * F m k) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_rp h2, prod_seq_zero_step] done · -- Case 2. h2 : ¬rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([1]_m) := by rw [FG_not_rp h1 h2] _ = [a]_m ^ (num_rp_below m k) * (prod_seq k 0 (F m) * ([1]_m)) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_not_rp h2, prod_seq_zero_step, F_not_rp_def h2] done done done lemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by define --Goal : ∀ i < m, G m a i < m fix i : Nat assume h1 : i < m rewrite [G_def] --Goal : a * i % m < m show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m) done lemma left_inv_one_one_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g' (g i) = i) : one_one_below n g := sorry lemma right_inv_onto_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g (g' i) = i) (h2 : maps_below n g') : onto_below n g := by define at h2; define fix k : Nat assume h3 : k < n apply Exists.intro (g' k) show g' k < n ∧ g (g' k) = k from And.intro (h2 k h3) (h1 k h3) done lemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m * [inv_mod m a]_m = [1]_m := by have h2 : 0 ≤ (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a) show [a]_m * [inv_mod m a]_m = [1]_m from calc [a]_m * [inv_mod m a]_m _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2] _ = [a]_m * [gcd_c2 m a]_m := by rw [←cc_eq_mod] _ = [1]_m := gcd_c2_inv h1 done lemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m := calc a * (b % m) % m = a % m * (b % m % m) % m := Nat.mul_mod _ _ _ _ = a % m * (b % m) % m := by rw [Nat.mod_mod] _ = a * b % m := (Nat.mul_mod _ _ _).symm lemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm] rfl done theorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] : ↑a ≡ ↑b (MOD m) ↔ a % m = b % m := sorry lemma mul_inv_mod_cancel {m a i : Nat} [NeZero m] (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1 rewrite [mul_class, cc_eq_iff_congr, ←Nat.cast_mul, ←Nat.cast_one, congr_iff_mod_eq_Nat] at h3 show a * inv_mod m a * i % m = i from calc a * (inv_mod m a) * i % m _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = 1 % m * i % m := by rw [h3] _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = i % m := by rw [one_mul] _ = i := Nat.mod_eq_of_lt h2 done lemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl lemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, G m a (Ginv m a i) = i := by fix i : Nat assume h2 : i < m show G m a (Ginv m a i) = i from calc G m a (Ginv m a i) _ = a * ((inv_mod m a * i) % m) % m := by rfl _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, Ginv m a (G m a i) = i := by fix i : Nat assume h2 : i < m show Ginv m a (G m a i) = i from calc Ginv m a (G m a i) _ = inv_mod m a * ((a * i) % m) % m := by rfl _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc, mul_comm (inv_mod m a)] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_maps_below (m a : Nat) [NeZero m] : maps_below m (Ginv m a) := G_maps_below m (inv_mod m a) lemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : one_one_below m (G m a) := left_inv_one_one_below (Ginv_left_inv h1) lemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : onto_below m (G m a) := right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a) lemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : perm_below m (G m a) := And.intro (G_maps_below m a) (And.intro (G_one_one_below h1) (G_onto_below h1)) --Permuting a product of congruence classes doesn't change product lemma swap_fst (u v : Nat) : swap u v u = v := by define : swap u v u --Goal : (if u = u then v else if u = v then u else u) = v have h : u = u := by rfl rewrite [if_pos h] rfl done lemma swap_snd (u v : Nat) : swap u v v = u := by define : swap u v v by_cases h1 : v = u · -- Case 1. h1 : v = u rewrite [if_pos h1] show v = u from h1 done · -- Case 2. h1 : v ≠ u rewrite [if_neg h1] have h2 : v = v := by rfl rewrite [if_pos h2] rfl done done lemma swap_other {u v i : Nat} (h1 : i ≠ u) (h2 : i ≠ v) : swap u v i = i := by define : swap u v i rewrite [if_neg h1, if_neg h2] rfl done lemma swap_values (u v i : Nat) : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := by by_cases h1 : i = u · -- Case 1. h1 : i = u apply Or.inl rewrite [h1] show swap u v u = v from swap_fst u v done · -- Case 2. h1 : i ≠ u apply Or.inr by_cases h2 : i = v · -- Case 2.1. h2 : i = v apply Or.inl rewrite [h2] show swap u v v = u from swap_snd u v done · -- Case 2.2. h2 : i ≠ v apply Or.inr show swap u v i = i from swap_other h1 h2 done done done lemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by define fix i : Nat assume h3 : i < n have h4 : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := swap_values u v i by_cases on h4 · -- Case 1. h4 : swap u v i = v rewrite [h4] show v < n from h2 done · -- Case 2. by_cases on h4 · -- Case 2.1. h4 : swap u v i = u rewrite [h4] show u < n from h1 done · -- Case 2.2. h4 : swap u v i = i rewrite [h4] show i < n from h3 done done done lemma swap_swap (u v n : Nat) : ∀ i < n, swap u v (swap u v i) = i := by fix i : Nat assume h : i < n by_cases h1 : i = u · -- Case 1. h1 : i = u rewrite [h1, swap_fst, swap_snd] rfl done · -- Case 2. h1 : i ≠ u by_cases h2 : i = v · -- Case 2.1. h2 : i = v rewrite [h2, swap_snd, swap_fst] rfl done · -- Case 2.2. h2 : i ≠ v rewrite [swap_other h1 h2, swap_other h1 h2] rfl done done done lemma swap_one_one_below (u v n) : one_one_below n (swap u v) := left_inv_one_one_below (swap_swap u v n) lemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) := right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2) lemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) := And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2)) lemma comp_perm_below {n : Nat} {f g : Nat → Nat} (h1 : perm_below n f) (h2 : perm_below n g) : perm_below n (f ∘ g) := sorry lemma trivial_swap (u : Nat) : swap u u = id := by apply funext fix x : Nat by_cases h1 : x = u · -- Case 1. h1 : x = u rewrite [h1, swap_fst] rfl done · -- Case 2. h1 : x ≠ u rewrite [swap_other h1 h1] rfl done done lemma prod_eq_fun {m : Nat} (f g : Nat → ZMod m) (k : Nat) : ∀ (n : Nat), (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g := by by_induc · -- Base Case assume h : (∀ i < 0, f (k + i) = g (k + i)) rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g assume h1 : ∀ i < n + 1, f (k + i) = g (k + i) have h2 : ∀ i < n, f (k + i) = g (k + i) := by fix i : Nat assume h2 : i < n have h3 : i < n + 1 := by linarith show f (k + i) = g (k + i) from h1 i h3 done have h3 : prod_seq n k f = prod_seq n k g := ih h2 have h4 : n < n + 1 := Nat.lt_succ_self n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4] rfl done done lemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := by have h2 : ∀ (i : Nat), i < u → (f ∘ swap u n) (0 + i) = f (0 + i) := by fix i : Nat assume h2 : i < u have h3 : 0 + i ≠ u := by linarith have h4 : 0 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl done show prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f from prod_eq_fun (f ∘ swap u n) f 0 u h2 done lemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := by have h2 : ∀ i < j, (f ∘ swap u n) (u + 1 + i) = f (u + 1 + i) := by fix i : Nat assume h2 : i < j have h3 : u + 1 + i ≠ u := by linarith have h4 : u + 1 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl show prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f from prod_eq_fun (f ∘ swap u n) f (u + 1) j h2 done

lemma break_prod {m : Nat} (n : Nat) (f : Nat → ZMod m) : ∀ (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f

HTPI.break_prod

htpi/HTPILib/Chap7.lean

htpi

/- Copyright 2023 Daniel J. Velleman -/ import HTPILib.Chap6 namespace HTPI /- Definitions -/ lemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by have h : n + 1 > 0 := Nat.succ_pos n show a % (n + 1) < n + 1 from Nat.mod_lt a h done def gcd (a b : Nat) : Nat := match b with | 0 => a | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd (n + 1) (a % (n + 1)) termination_by b mutual def gcd_c1 (a b : Nat) : Int := match b with | 0 => 1 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c2 (n + 1) (a % (n + 1)) --Corresponds to s = t' termination_by b def gcd_c2 (a b : Nat) : Int := match b with | 0 => 0 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c1 (n + 1) (a % (n + 1)) - (gcd_c2 (n + 1) (a % (n + 1))) * ↑(a / (n + 1)) --Corresponds to t = s' - t'q termination_by b end def prime (n : Nat) : Prop := 2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n def prime_factor (p n : Nat) : Prop := prime p ∧ p ∣ n def all_prime (l : List Nat) : Prop := ∀ p ∈ l, prime p def nondec (l : List Nat) : Prop := match l with | [] => True --Of course, True is a proposition that is always true | n :: L => (∀ m ∈ L, n ≤ m) ∧ nondec L def nondec_prime_list (l : List Nat) : Prop := all_prime l ∧ nondec l def prod (l : List Nat) : Nat := match l with | [] => 1 | n :: L => n * (prod L) def prime_factorization (n : Nat) (l : List Nat) : Prop := nondec_prime_list l ∧ prod l = n def rel_prime (a b : Nat) : Prop := gcd a b = 1 def congr_mod (m : Nat) (a b : Int) : Prop := (↑m : Int) ∣ (a - b) def cc (m : Nat) (a : Int) : ZMod m := (↑a : ZMod m) notation:50 a " ≡ " b " (MOD " m ")" => congr_mod m a b notation:max "["a"]_"m:max => cc m a def invertible {m : Nat} (X : ZMod m) : Prop := ∃ (Y : ZMod m), X * Y = [1]_m def num_rp_below (m k : Nat) : Nat := match k with | 0 => 0 | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j def phi (m : Nat) : Nat := num_rp_below m m def prod_seq {m : Nat} (j k : Nat) (f : Nat → ZMod m) : ZMod m := match j with | 0 => [1]_m | n + 1 => prod_seq n k f * f (k + n) def maps_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i < n, g i < n def one_one_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i1 < n, ∀ i2 < n, g i1 = g i2 → i1 = i2 def onto_below (n : Nat) (g : Nat → Nat) : Prop := ∀ k < n, ∃ i < n, g i = k def perm_below (n : Nat) (g : Nat → Nat) : Prop := maps_below n g ∧ one_one_below n g ∧ onto_below n g def inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m) def swap (u v i : Nat) : Nat := if i = u then v else if i = v then u else i namespace Euler --For definitions specific to Euler's theorem def F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m def G (m a i : Nat) : Nat := (a * i) % m def Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i end Euler /- Section 7.1 -/ theorem dvd_mod_of_dvd_a_b {a b d : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ (a % b) := by set q : Nat := a / b have h3 : b * q + a % b = a := Nat.div_add_mod a b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 define --Goal : ∃ (c : Nat), a % b = d * c apply Exists.intro (j - k * q) show a % b = d * (j - k * q) from calc a % b _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm _ = a - b * q := by rw [h3] _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc] _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm done theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry #eval gcd 672 161 --Answer: 7 lemma gcd_base (a : Nat) : gcd a 0 = a := by rfl lemma gcd_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd a b = gcd b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1)) rfl done lemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b ≠ 0) : a % b < b := by have h1 : b > 0 := Nat.pos_of_ne_zero h show a % b < b from Nat.mod_lt a h1 done lemma dvd_self (n : Nat) : n ∣ n := by apply Exists.intro 1 ring done theorem gcd_dvd : ∀ (b a : Nat), (gcd a b) ∣ a ∧ (gcd a b) ∣ b := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd a b_1) ∣ a ∧ (gcd a b_1) ∣ b_1 fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_base] --Goal: a ∣ a ∧ a ∣ 0 apply And.intro (dvd_self a) define apply Exists.intro 0 rfl done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_nonzero a h1] --Goal : gcd b (a % b) ∣ a ∧ gcd b (a % b) ∣ b have h2 : a % b < b := mod_nonzero_lt a h1 have h3 : (gcd b (a % b)) ∣ b ∧ (gcd b (a % b)) ∣ (a % b) := ih (a % b) h2 b apply And.intro _ h3.left show (gcd b (a % b)) ∣ a from dvd_a_of_dvd_b_mod h3.left h3.right done done theorem gcd_dvd_left (a b : Nat) : (gcd a b) ∣ a := (gcd_dvd b a).left theorem gcd_dvd_right (a b : Nat) : (gcd a b) ∣ b := (gcd_dvd b a).right lemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl lemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c1 a b = gcd_c2 b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done lemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl lemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * ↑(a / b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done theorem gcd_lin_comb : ∀ (b a : Nat), (gcd_c1 a b) * ↑a + (gcd_c2 a b) * ↑b = ↑(gcd a b) := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd_c1 a b_1) * ↑a + (gcd_c2 a b_1) * ↑b_1 = ↑(gcd a b_1) fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base] --Goal : 1 * ↑a + 0 * ↑0 = ↑a ring done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1] --Goal : gcd_c2 b (a % b) * ↑a + -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * ↑(a / b)) * ↑b = -- ↑(gcd b (a % b)) set r : Nat := a % b set q : Nat := a / b set s : Int := gcd_c1 b r set t : Int := gcd_c2 b r --Goal : t * ↑a + (s - t * ↑q) * ↑b = ↑(gcd b r) have h2 : r < b := mod_nonzero_lt a h1 have h3 : s * ↑b + t * ↑r = ↑(gcd b r) := ih r h2 b have h4 : b * q + r = a := Nat.div_add_mod a b rewrite [←h3, ←h4] rewrite [Nat.cast_add, Nat.cast_mul] --Goal : t * (↑b * ↑q + ↑r) + (s - t * ↑q) * ↑b = s * ↑b + t * ↑r ring done done #eval gcd_c1 672 161 --Answer: 6 #eval gcd_c2 672 161 --Answer: -25 --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161 theorem Theorem_7_1_6 {d a b : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ gcd a b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑d ∣ ↑(gcd a b) set s : Int := gcd_c1 a b set t : Int := gcd_c2 a b have h3 : s * ↑a + t * ↑b = ↑(gcd a b) := gcd_lin_comb b a rewrite [←h3] --Goal : ↑d ∣ s * ↑a + t * ↑b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul] --Goal : ↑d ∣ s * (↑d * ↑j) + t * (↑d * ↑k) define apply Exists.intro (s * ↑j + t * ↑k) ring done /- Section 7.2 -/ theorem dvd_trans {a b c : Nat} (h1 : a ∣ b) (h2 : b ∣ c) : a ∣ c := by define at h1; define at h2; define obtain (m : Nat) (h3 : b = a * m) from h1 obtain (n : Nat) (h4 : c = b * n) from h2 rewrite [h3, mul_assoc] at h4 apply Exists.intro (m * n) show c = a * (m * n) from h4 done lemma exists_prime_factor : ∀ (n : Nat), 2 ≤ n → ∃ (p : Nat), prime_factor p n := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, 2 ≤ n_1 → ∃ (p : Nat), prime_factor p n_1 assume h1 : 2 ≤ n by_cases h2 : prime n · -- Case 1. h2 : prime n apply Exists.intro n define --Goal : prime n ∧ n ∣ n show prime n ∧ n ∣ n from And.intro h2 (dvd_self n) done · -- Case 2. h2 : ¬prime n define at h2 --h2 : ¬(2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n) demorgan at h2 disj_syll h2 h1 obtain (a : Nat) (h3 : ∃ (b : Nat), a * b = n ∧ a < n ∧ b < n) from h2 obtain (b : Nat) (h4 : a * b = n ∧ a < n ∧ b < n) from h3 have h5 : 2 ≤ a := by by_contra h6 have h7 : a ≤ 1 := by linarith have h8 : n ≤ b := calc n _ = a * b := h4.left.symm _ ≤ 1 * b := by rel [h7] _ = b := by ring linarith --n ≤ b contradicts b < n done have h6 : ∃ (p : Nat), prime_factor p a := ih a h4.right.left h5 obtain (p : Nat) (h7 : prime_factor p a) from h6 apply Exists.intro p define --Goal : prime p ∧ p ∣ n define at h7 --h7 : prime p ∧ p ∣ a apply And.intro h7.left have h8 : a ∣ n := by apply Exists.intro b show n = a * b from (h4.left).symm done show p ∣ n from dvd_trans h7.right h8 done done lemma exists_least_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q := by set S : Set Nat := {p : Nat | prime_factor p n} have h2 : ∃ (p : Nat), p ∈ S := exists_prime_factor n h show ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q from well_ord_princ S h2 done lemma all_prime_nil : all_prime [] := by define --Goal : ∀ p ∈ [], prime p fix p : Nat contrapos --Goal : ¬prime p → p ∉ [] assume h1 : ¬prime p show p ∉ [] from List.not_mem_nil p done lemma all_prime_cons (n : Nat) (L : List Nat) : all_prime (n :: L) ↔ prime n ∧ all_prime L := by apply Iff.intro · -- (→) assume h1 : all_prime (n :: L) --Goal : prime n ∧ all_prime L define at h1 --h1 : ∀ p ∈ n :: L, prime p apply And.intro (h1 n (List.mem_cons_self n L)) define --Goal : ∀ p ∈ L, prime p fix p : Nat assume h2 : p ∈ L show prime p from h1 p (List.mem_cons_of_mem n h2) done · -- (←) assume h1 : prime n ∧ all_prime L --Goal : all_prime (n :: l) define : all_prime L at h1 define fix p : Nat assume h2 : p ∈ n :: L rewrite [List.mem_cons] at h2 --h2 : p = n ∨ p ∈ L by_cases on h2 · -- Case 1. h2 : p = n rewrite [h2] show prime n from h1.left done · -- Case 2. h2 : p ∈ L show prime p from h1.right p h2 done done done lemma nondec_nil : nondec [] := by define --Goal : True trivial --trivial proves some obviously true statements, such as True done lemma nondec_cons (n : Nat) (L : List Nat) : nondec (n :: L) ↔ (∀ m ∈ L, n ≤ m) ∧ nondec L := by rfl lemma prod_nil : prod [] = 1 := by rfl lemma prod_cons : prod (n :: L) = n * (prod L) := by rfl lemma exists_cons_of_length_eq_succ {A : Type} {l : List A} {n : Nat} (h : l.length = n + 1) : ∃ (a : A) (L : List A), l = a :: L ∧ L.length = n := by have h1 : ¬l.length = 0 := by linarith rewrite [List.length_eq_zero] at h1 obtain (a : A) (h2 : ∃ (L : List A), l = a :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List A) (h3 : l = a :: L) from h2 apply Exists.intro a apply Exists.intro L apply And.intro h3 have h4 : (a :: L).length = L.length + 1 := List.length_cons a L rewrite [←h3, h] at h4 show L.length = n from (Nat.add_right_cancel h4).symm done lemma list_elt_dvd_prod_by_length (a : Nat) : ∀ (n : Nat), ∀ (l : List Nat), l.length = n → a ∈ l → a ∣ prod l := by by_induc · --Base Case fix l : List Nat assume h1 : l.length = 0 rewrite [List.length_eq_zero] at h1 --h1 : l = [] rewrite [h1] --Goal : a ∈ [] → a ∣ prod [] contrapos assume h2 : ¬a ∣ prod [] show a ∉ [] from List.not_mem_nil a done · -- Induction Step fix n : Nat assume ih : ∀ (l : List Nat), List.length l = n → a ∈ l → a ∣ prod l fix l : List Nat assume h1 : l.length = n + 1 --Goal : a ∈ l → a ∣ prod l obtain (b : Nat) (h2 : ∃ (L : List Nat), l = b :: L ∧ L.length = n) from exists_cons_of_length_eq_succ h1 obtain (L : List Nat) (h3 : l = b :: L ∧ L.length = n) from h2 have h4 : a ∈ L → a ∣ prod L := ih L h3.right assume h5 : a ∈ l rewrite [h3.left, prod_cons] --Goal : a ∣ b * prod L rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b ∨ a ∈ L by_cases on h5 · -- Case 1. h5 : a = b apply Exists.intro (prod L) rewrite [h5] rfl done · -- Case 2. h5 : a ∈ L have h6 : a ∣ prod L := h4 h5 have h7 : prod L ∣ b * prod L := by apply Exists.intro b ring done show a ∣ b * prod L from dvd_trans h6 h7 done done done lemma list_elt_dvd_prod {a : Nat} {l : List Nat} (h : a ∈ l) : a ∣ prod l := by set n : Nat := l.length have h1 : l.length = n := by rfl show a ∣ prod l from list_elt_dvd_prod_by_length a n l h1 h done lemma exists_prime_factorization : ∀ (n : Nat), n ≥ 1 → ∃ (l : List Nat), prime_factorization n l := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, n_1 ≥ 1 → ∃ (l : List Nat), prime_factorization n_1 l assume h1 : n ≥ 1 by_cases h2 : n = 1 · -- Case 1. h2 : n = 1 apply Exists.intro [] define apply And.intro · -- Proof of nondec_prime_list [] define show all_prime [] ∧ nondec [] from And.intro all_prime_nil nondec_nil done · -- Proof of prod [] = n rewrite [prod_nil, h2] rfl done done · -- Case 2. h2 : n ≠ 1 have h3 : n ≥ 2 := lt_of_le_of_ne' h1 h2 obtain (p : Nat) (h4 : prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q) from exists_least_prime_factor h3 have p_prime_factor : prime_factor p n := h4.left define at p_prime_factor have p_prime : prime p := p_prime_factor.left have p_dvd_n : p ∣ n := p_prime_factor.right have p_least : ∀ (q : Nat), prime_factor q n → p ≤ q := h4.right obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n have h5 : m ≠ 0 := by contradict h1 with h6 have h7 : n = 0 := calc n _ = p * m := n_eq_pm _ = p * 0 := by rw [h6] _ = 0 := by ring rewrite [h7] decide done have m_pos : 0 < m := Nat.pos_of_ne_zero h5 have m_lt_n : m < n := by define at p_prime show m < n from calc m _ < m + m := by linarith _ = 2 * m := by ring _ ≤ p * m := by rel [p_prime.left] _ = n := n_eq_pm.symm done obtain (L : List Nat) (h6 : prime_factorization m L) from ih m m_lt_n m_pos define at h6 have ndpl_L : nondec_prime_list L := h6.left define at ndpl_L apply Exists.intro (p :: L) define apply And.intro · -- Proof of nondec_prime_list (p :: L) define apply And.intro · -- Proof of all_prime (p :: L) rewrite [all_prime_cons] show prime p ∧ all_prime L from And.intro p_prime ndpl_L.left done · -- Proof of nondec (p :: L) rewrite [nondec_cons] apply And.intro _ ndpl_L.right fix q : Nat assume q_in_L : q ∈ L have h7 : q ∣ prod L := list_elt_dvd_prod q_in_L rewrite [h6.right] at h7 --h7 : q ∣ m have h8 : m ∣ n := by apply Exists.intro p rewrite [n_eq_pm] ring done have q_dvd_n : q ∣ n := dvd_trans h7 h8 have ap_L : all_prime L := ndpl_L.left define at ap_L have q_prime_factor : prime_factor q n := And.intro (ap_L q q_in_L) q_dvd_n show p ≤ q from p_least q q_prime_factor done done · -- Proof of prod (p :: L) = n rewrite [prod_cons, h6.right, n_eq_pm] rfl done done done theorem Theorem_7_2_2 {a b c : Nat} (h1 : c ∣ a * b) (h2 : rel_prime a c) : c ∣ b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑c ∣ ↑b define at h1; define at h2; define obtain (j : Nat) (h3 : a * b = c * j) from h1 set s : Int := gcd_c1 a c set t : Int := gcd_c2 a c have h4 : s * ↑a + t * ↑c = ↑(gcd a c) := gcd_lin_comb c a rewrite [h2, Nat.cast_one] at h4 --h4 : s * ↑a + t * ↑c = (1 : Int) apply Exists.intro (s * ↑j + t * ↑b) show ↑b = ↑c * (s * ↑j + t * ↑b) from calc ↑b _ = (1 : Int) * ↑b := (one_mul _).symm _ = (s * ↑a + t * ↑c) * ↑b := by rw [h4] _ = s * (↑a * ↑b) + t * ↑c * ↑b := by ring _ = s * (↑c * ↑j) + t * ↑c * ↑b := by rw [←Nat.cast_mul a b, h3, Nat.cast_mul c j] _ = ↑c * (s * ↑j + t * ↑b) := by ring done lemma le_nonzero_prod_left {a b : Nat} (h : a * b ≠ 0) : a ≤ a * b := by have h1 : b ≠ 0 := by contradict h with h1 rewrite [h1] ring done have h2 : 1 ≤ b := Nat.pos_of_ne_zero h1 show a ≤ a * b from calc a = a * 1 := (mul_one a).symm _ ≤ a * b := by rel [h2] done lemma le_nonzero_prod_right {a b : Nat} (h : a * b ≠ 0) : b ≤ a * b := by rewrite [mul_comm] rewrite [mul_comm] at h show b ≤ b * a from le_nonzero_prod_left h done lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry lemma rel_prime_of_prime_not_dvd {a p : Nat} (h1 : prime p) (h2 : ¬p ∣ a) : rel_prime a p := by have h3 : gcd a p ∣ a := gcd_dvd_left a p have h4 : gcd a p ∣ p := gcd_dvd_right a p have h5 : gcd a p = 1 ∨ gcd a p = p := dvd_prime h1 h4 have h6 : gcd a p ≠ p := by contradict h2 with h6 rewrite [h6] at h3 show p ∣ a from h3 done disj_syll h5 h6 show rel_prime a p from h5 done theorem Theorem_7_2_3 {a b p : Nat} (h1 : prime p) (h2 : p ∣ a * b) : p ∣ a ∨ p ∣ b := by or_right with h3 have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3 show p ∣ b from Theorem_7_2_2 h2 h4 done lemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a ≥ 1 := by have h1 : a ≠ 0 := by by_contra h1 rewrite [h1] at h contradict h linarith done show a ≥ 1 from Nat.pos_of_ne_zero h1 done lemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by have h1 : a ≥ 1 := ge_one_of_prod_one h have h2 : a * b ≠ 0 := by linarith have h3 : a ≤ a * b := le_nonzero_prod_left h2 rewrite [h] at h3 show a = 1 from Nat.le_antisymm h3 h1 done lemma eq_one_of_dvd_one {n : Nat} (h : n ∣ 1) : n = 1 := by obtain (j : Nat) (h1 : 1 = n * j) from h show n = 1 from eq_one_of_prod_one h1.symm done lemma prime_not_one {p : Nat} (h : prime p) : p ≠ 1 := by define at h linarith done theorem Theorem_7_2_4 {p : Nat} (h1 : prime p) : ∀ (l : List Nat), p ∣ prod l → ∃ a ∈ l, p ∣ a := by apply List.rec · -- Base Case. Goal : p ∣ prod [] → ∃ a ∈ [], p ∣ a rewrite [prod_nil] assume h2 : p ∣ 1 show ∃ a ∈ [], p ∣ a from absurd (eq_one_of_dvd_one h2) (prime_not_one h1) done · -- Induction Step fix b : Nat fix L : List Nat assume ih : p ∣ prod L → ∃ a ∈ L, p ∣ a --Goal : p ∣ prod (b :: L) → ∃ a ∈ b :: L, p ∣ a assume h2 : p ∣ prod (b :: L) rewrite [prod_cons] at h2 have h3 : p ∣ b ∨ p ∣ prod L := Theorem_7_2_3 h1 h2 by_cases on h3 · -- Case 1. h3 : p ∣ b apply Exists.intro b show b ∈ b :: L ∧ p ∣ b from And.intro (List.mem_cons_self b L) h3 done · -- Case 2. h3 : p ∣ prod L obtain (a : Nat) (h4 : a ∈ L ∧ p ∣ a) from ih h3 apply Exists.intro a show a ∈ b :: L ∧ p ∣ a from And.intro (List.mem_cons_of_mem b h4.left) h4.right done done done lemma prime_in_list {p : Nat} {l : List Nat} (h1 : prime p) (h2 : all_prime l) (h3 : p ∣ prod l) : p ∈ l := by obtain (a : Nat) (h4 : a ∈ l ∧ p ∣ a) from Theorem_7_2_4 h1 l h3 define at h2 have h5 : prime a := h2 a h4.left have h6 : p = 1 ∨ p = a := dvd_prime h5 h4.right disj_syll h6 (prime_not_one h1) rewrite [h6] show a ∈ l from h4.left done lemma first_le_first {p q : Nat} {l m : List Nat} (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m)) (h3 : prod (p :: l) = prod (q :: m)) : p ≤ q := by define at h1; define at h2 have h4 : q ∣ prod (p :: l) := by define apply Exists.intro (prod m) rewrite [←prod_cons] show prod (p :: l) = prod (q :: m) from h3 done have h5 : all_prime (q :: m) := h2.left rewrite [all_prime_cons] at h5 have h6 : q ∈ p :: l := prime_in_list h5.left h1.left h4 have h7 : nondec (p :: l) := h1.right rewrite [nondec_cons] at h7 rewrite [List.mem_cons] at h6 by_cases on h6 · -- Case 1. h6 : q = p linarith done · -- Case 2. h6 : q ∈ l have h8 : ∀ m ∈ l, p ≤ m := h7.left show p ≤ q from h8 q h6 done done lemma nondec_prime_list_tail {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by define at h define rewrite [all_prime_cons, nondec_cons] at h show all_prime l ∧ nondec l from And.intro h.left.right h.right.right done lemma cons_prod_not_one {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : prod (p :: l) ≠ 1 := by define at h have h1 : all_prime (p :: l) := h.left rewrite [all_prime_cons] at h1 rewrite [prod_cons] by_contra h2 show False from (prime_not_one h1.left) (eq_one_of_prod_one h2) done lemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) : l = [] ↔ prod l = 1 := by apply Iff.intro · -- (→) assume h1 : l = [] rewrite [h1] show prod [] = 1 from prod_nil done · -- (←) contrapos assume h1 : ¬l = [] obtain (p : Nat) (h2 : ∃ (L : List Nat), l = p :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List Nat) (h3 : l = p :: L) from h2 rewrite [h3] at h rewrite [h3] show ¬prod (p :: L) = 1 from cons_prod_not_one h done done lemma prime_pos {p : Nat} (h : prime p) : p > 0 := by define at h linarith done theorem Theorem_7_2_5 : ∀ (l1 l2 : List Nat), nondec_prime_list l1 → nondec_prime_list l2 → prod l1 = prod l2 → l1 = l2 := by apply List.rec · -- Base Case. Goal : ∀ (l2 : List Nat), nondec_prime_list [] → -- nondec_prime_list l2 → prod [] = prod l2 → [] = l2 fix l2 : List Nat assume h1 : nondec_prime_list [] assume h2 : nondec_prime_list l2 assume h3 : prod [] = prod l2 rewrite [prod_nil, eq_comm, ←list_nil_iff_prod_one h2] at h3 show [] = l2 from h3.symm done · -- Induction Step fix p : Nat fix L1 : List Nat assume ih : ∀ (L2 : List Nat), nondec_prime_list L1 → nondec_prime_list L2 → prod L1 = prod L2 → L1 = L2 -- Goal : ∀ (l2 : List Nat), nondec_prime_list (p :: L1) → -- nondec_prime_list l2 → prod (p :: L1) = prod l2 → p :: L1 = l2 fix l2 : List Nat assume h1 : nondec_prime_list (p :: L1) assume h2 : nondec_prime_list l2 assume h3 : prod (p :: L1) = prod l2 have h4 : ¬prod (p :: L1) = 1 := cons_prod_not_one h1 rewrite [h3, ←list_nil_iff_prod_one h2] at h4 obtain (q : Nat) (h5 : ∃ (L : List Nat), l2 = q :: L) from List.exists_cons_of_ne_nil h4 obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5 rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2) rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2) have h7 : p ≤ q := first_le_first h1 h2 h3 have h8 : q ≤ p := first_le_first h2 h1 h3.symm have h9 : p = q := by linarith rewrite [h9, prod_cons, prod_cons] at h3 --h3 : q * prod L1 = q * prod L2 have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1 have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2 define at h2 have h12 : all_prime (q :: L2) := h2.left rewrite [all_prime_cons] at h12 have h13 : q > 0 := prime_pos h12.left have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3 have h15 : L1 = L2 := ih L2 h10 h11 h14 rewrite [h6, h9, h15] rfl done done theorem fund_thm_arith (n : Nat) (h : n ≥ 1) : ∃! (l : List Nat), prime_factorization n l := by exists_unique · -- Existence show ∃ (l : List Nat), prime_factorization n l from exists_prime_factorization n h done · -- Uniqueness fix l1 : List Nat; fix l2 : List Nat assume h1 : prime_factorization n l1 assume h2 : prime_factorization n l2 define at h1; define at h2 have h3 : prod l1 = n := h1.right rewrite [←h2.right] at h3 show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3 done done /- Section 7.3 -/ theorem congr_refl (m : Nat) : ∀ (a : Int), a ≡ a (MOD m) := by fix a : Int define --Goal : ∃ (c : Int), a - a = ↑m * c apply Exists.intro 0 ring done theorem congr_symm {m : Nat} : ∀ {a b : Int}, a ≡ b (MOD m) → b ≡ a (MOD m) := by fix a : Int; fix b : Int assume h1 : a ≡ b (MOD m) define at h1 --h1 : ∃ (c : Int), a - b = ↑m * c define --Goal : ∃ (c : Int), b - a = ↑m * c obtain (c : Int) (h2 : a - b = m * c) from h1 apply Exists.intro (-c) show b - a = m * (-c) from calc b - a _ = -(a - b) := by ring _ = -(m * c) := by rw [h2] _ = m * (-c) := by ring done theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry /- Fundamental properties of congruence classes -/ lemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) : X = Y ↔ X.val = Y.val := Fin.ext_iff lemma val_nat_eq_mod (n k : Nat) : ([k]_(n + 1)).val = k % (n + 1) := by rfl lemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl theorem cc_rep {m : Nat} (X : ZMod m) : ∃ (a : Int), X = [a]_m := match m with | 0 => by apply Exists.intro X rfl done | n + 1 => by apply Exists.intro ↑(X.val) have h1 : X.val < n + 1 := Fin.prop X rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1] rfl done theorem add_class (m : Nat) (a b : Int) : [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm theorem mul_class (m : Nat) (a b : Int) : [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm lemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ [a - b]_m = [0]_m := by apply Iff.intro · -- (→) assume h1 : [a]_m = [b]_m have h2 : a - b = a + (-b) := by ring have h3 : b + (-b) = 0 := by ring show [a - b]_m = [0]_m from calc [a - b]_m _ = [a + (-b)]_m := by rw [h2] _ = [a]_m + [-b]_m := by rw [add_class] _ = [b]_m + [-b]_m := by rw [h1] _ = [b + -b]_m := by rw [add_class] _ = [0]_m := by rw [h3] done · -- (←) assume h1 : [a - b]_m = [0]_m have h2 : b + (a - b) = a := by ring have h3 : b + 0 = b := by ring show [a]_m = [b]_m from calc [a]_m _ = [b + (a - b)]_m := by rw [h2] _ = [b]_m + [a - b]_m := by rw [add_class] _ = [b]_m + [0]_m := by rw [h1] _ = [b + 0]_m := by rw [add_class] _ = [b]_m := by rw [h3] done done lemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) : [a]_m = [0]_m → [-a]_m = [0]_m := by assume h1 : [a]_m = [0]_m have h2 : 0 + (-a) = -a := by ring have h3 : a + (-a) = 0 := by ring show [-a]_m = [0]_m from calc [-a]_m _ = [0 + (-a)]_m := by rw [h2] _ = [0]_m + [-a]_m := by rw [add_class] _ = [a]_m + [-a]_m := by rw [h1] _ = [a + (-a)]_m := by rw [add_class] _ = [0]_m := by rw [h3] done lemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) : [-a]_m = [0]_m ↔ [a]_m = [0]_m := by apply Iff.intro _ (cc_neg_zero_of_cc_zero m a) assume h1 : [-a]_m = [0]_m have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1 have h3 : -(-a) = a := by ring rewrite [h3] at h2 show [a]_m = [0]_m from h2 done lemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl lemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m ↔ m ∣ k := match m with | 0 => by have h : (0 : Int) = (↑(0 : Nat) : Int) := by rfl rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj] apply Iff.intro · -- (→) assume h1 : k = 0 rewrite [h1] show 0 ∣ 0 from dvd_self 0 done · -- (←) assume h1 : 0 ∣ k obtain (c : Nat) (h2 : k = 0 * c) from h1 rewrite [h2] ring done done | n + 1 => by rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero] show k % (n + 1) = 0 ↔ n + 1 ∣ k from (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm done lemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m ↔ ↑m ∣ a := by obtain (k : Nat) (h1 : a = ↑k ∨ a = -↑k) from Int.eq_nat_or_neg a by_cases on h1 · -- Case 1. h1: a = ↑k rewrite [h1, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done · -- Case 2. h1: a = -↑k rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done done theorem cc_eq_iff_congr (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ a ≡ b (MOD m) := calc [a]_m = [b]_m _ ↔ [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b _ ↔ ↑m ∣ (a - b) := cc_zero_iff_dvd m (a - b) _ ↔ a ≡ b (MOD m) := by rfl /- End of fundamental properties of congruence classes -/ lemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m := by have h1 : (↑m : Int) ≠ 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m) show 0 ≤ a % m from Int.emod_nonneg a h1 done lemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m) have h2 : (↑m : Int) > 0 := (Nat.cast_pos).rtl h1 show a % m < m from Int.emod_lt_of_pos a h2 done lemma congr_mod_mod (m : Nat) (a : Int) : a ≡ a % m (MOD m) := by define have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m apply Exists.intro (a / m) show a - a % m = m * (a / m) from calc a - (a % m) _ = m * (a / m) + a % m - a % m := by rw [h1] _ = m * (a / m) := by ring done lemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) := And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a)) theorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) : ∃! (r : Int), 0 ≤ r ∧ r < m ∧ a ≡ r (MOD m) := by exists_unique · -- Existence apply Exists.intro (a % m) show 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) from mod_cmpl_res m a done · -- Uniqueness fix r1 : Int; fix r2 : Int assume h1 : 0 ≤ r1 ∧ r1 < m ∧ a ≡ r1 (MOD m) assume h2 : 0 ≤ r2 ∧ r2 < m ∧ a ≡ r2 (MOD m) have h3 : r1 ≡ r2 (MOD m) := congr_trans (congr_symm h1.right.right) h2.right.right obtain (d : Int) (h4 : r1 - r2 = m * d) from h3 have h5 : r1 - r2 < m * 1 := by linarith have h6 : m * (-1) < r1 - r2 := by linarith rewrite [h4] at h5 --h5 : m * d < m * 1 rewrite [h4] at h6 --h6 : m * -1 < m * d have h7 : (↑m : Int) ≥ 0 := Nat.cast_nonneg m have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7 have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7 have h10 : d = 0 := by linarith show r1 = r2 from calc r1 _ = r1 - r2 + r2 := by ring _ = m * 0 + r2 := by rw [h4, h10] _ = r2 := by ring done done lemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m := (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a) theorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y rewrite [h1, h2] have h3 : a + b = b + a := by ring show [a]_m + [b]_m = [b]_m + [a]_m from calc [a]_m + [b]_m _ = [a + b]_m := add_class m a b _ = [b + a]_m := by rw [h3] _ = [b]_m + [a]_m := (add_class m b a).symm done theorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X rewrite [h1] have h2 : a * 1 = a := by ring show [a]_m * [1]_m = [a]_m from calc [a]_m * [1]_m _ = [a * 1]_m := mul_class m a 1 _ = [a]_m := by rw [h2] done theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry lemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) : [a]_m * [gcd_c2 m a]_m = [1]_m := by set s : Int := gcd_c1 m a have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m define at h1 rewrite [h1, Nat.cast_one] at h2 --h2 : s * ↑m + gcd_c2 m a * ↑a = 1 rewrite [mul_class, cc_eq_iff_congr] define --Goal : ∃ (c : Int), ↑a * gcd_c2 m a - 1 = ↑m * c apply Exists.intro (-s) show a * (gcd_c2 m a) - 1 = m * (-s) from calc a * (gcd_c2 m a) - 1 _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring _ = 1 + m * (-s) - 1 := by rw [h2] _ = m * (-s) := by ring done theorem Theorem_7_3_7 (m a : Nat) : invertible [a]_m ↔ rel_prime m a := by apply Iff.intro · -- (→) assume h1 : invertible [a]_m define at h1 obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1 obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y rewrite [h3, mul_class, cc_eq_iff_congr] at h2 define at h2 obtain (c : Int) (h4 : a * b - 1 = m * c) from h2 rewrite [Exercise_7_2_6] --Goal : ∃ (s t : Int), s * ↑m + t * ↑a = 1 apply Exists.intro (-c) apply Exists.intro b show (-c) * m + b * a = 1 from calc (-c) * m + b * a _ = (-c) * m + (a * b - 1) + 1 := by ring _ = (-c) * m + m * c + 1 := by rw [h4] _ = 1 := by ring done · -- (←) assume h1 : rel_prime m a define show ∃ (Y : ZMod m), [a]_m * Y = [1]_m from Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1) done done /- Section 7.4 -/ section Euler open Euler lemma num_rp_below_base {m : Nat} : num_rp_below m 0 = 0 := by rfl lemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) : num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by have h1 : num_rp_below m (j + 1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : gcd m j = 1 rewrite [if_pos h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1 show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1 done lemma num_rp_below_step_not_rp {m j : Nat} (h : ¬rel_prime m j) : num_rp_below m (j + 1) = num_rp_below m j := by have h1 : num_rp_below m (j +1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : ¬gcd m j = 1 rewrite [if_neg h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j show num_rp_below m (j + 1) = num_rp_below m j from h1 done lemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl #eval phi 10 --Answer: 4 lemma prod_inv_iff_inv {m : Nat} {X : ZMod m} (h1 : invertible X) (Y : ZMod m) : invertible (X * Y) ↔ invertible Y := by apply Iff.intro · -- (→) assume h2 : invertible (X * Y) obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2 apply Exists.intro (X * Z) rewrite [←h3] --Goal : Y * (X * Z) = X * Y * Z ring --Note that ring can do algebra in ZMod m done · -- (←) assume h2 : invertible Y obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1 obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2 apply Exists.intro (Xi * Yi) show (X * Y) * (Xi * Yi) = [1]_m from calc X * Y * (Xi * Yi) _ = (X * Xi) * (Y * Yi) := by ring _ = [1]_m * [1]_m := by rw [h3, h4] _ = [1]_m := Theorem_7_3_6_7 [1]_m done done lemma F_rp_def {m i : Nat} (h : rel_prime m i) : F m i = [i]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h --h : gcd m i = 1 rewrite [if_pos h] at h1 show F m i = [i]_m from h1 done lemma F_not_rp_def {m i : Nat} (h : ¬rel_prime m i) : F m i = [1]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h rewrite [h1, if_neg h] rfl done lemma prod_seq_base {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 0 k f = [1]_m := by rfl lemma prod_seq_step {m : Nat} (n k : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl lemma prod_seq_zero_step {m : Nat} (n : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by rewrite [prod_seq_step, zero_add] rfl done lemma prod_one {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 1 k f = f k := by rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7] rfl done lemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl lemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m := calc [G m a i]_m _ = [(a * i) % m]_m := by rfl _ = [a * i]_m := (cc_eq_mod m (a * i)).symm _ = [a]_m * [i]_m := (mul_class m a i).symm lemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) : rel_prime m (G m a i) ↔ rel_prime m i := by have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1 show rel_prime m (G m a i) ↔ rel_prime m i from calc rel_prime m (G m a i) _ ↔ invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm _ ↔ invertible ([a]_m * [i]_m) := by rw [cc_G] _ ↔ invertible [i]_m := prod_inv_iff_inv h2 ([i]_m) _ ↔ rel_prime m i := Theorem_7_3_7 m i done lemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) : F m (G m a i) = [a]_m * F m i := by have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2 show F m (G m a i) = [a]_m * F m i from calc F m (G m a i) _ = [G m a i]_m := F_rp_def h3 _ = [a]_m * [i]_m := cc_G m a i _ = [a]_m * F m i := by rw [F_rp_def h2] done lemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : ¬rel_prime m i) : F m (G m a i) = [1]_m := by rewrite [←G_rp_iff h1 i] at h2 show F m (G m a i) = [1]_m from F_not_rp_def h2 done lemma FG_prod {m a : Nat} (h1 : rel_prime m a) : ∀ (k : Nat), prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by by_induc · -- Base Case show prod_seq 0 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from calc prod_seq 0 0 ((F m) ∘ (G m a)) _ = [1]_m := prod_seq_base _ _ _ = [a]_m ^ 0 * [1]_m := by ring _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by rw [num_rp_below_base, prod_seq_base] done · -- Induction Step fix k : Nat assume ih : prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) by_cases h2 : rel_prime m k · -- Case 1. h2 : rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([a]_m * F m k) := by rw [FG_rp h1 h2] _ = [a]_m ^ ((num_rp_below m k) + 1) * ((prod_seq k 0 (F m)) * F m k) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_rp h2, prod_seq_zero_step] done · -- Case 2. h2 : ¬rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([1]_m) := by rw [FG_not_rp h1 h2] _ = [a]_m ^ (num_rp_below m k) * (prod_seq k 0 (F m) * ([1]_m)) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_not_rp h2, prod_seq_zero_step, F_not_rp_def h2] done done done lemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by define --Goal : ∀ i < m, G m a i < m fix i : Nat assume h1 : i < m rewrite [G_def] --Goal : a * i % m < m show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m) done lemma left_inv_one_one_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g' (g i) = i) : one_one_below n g := sorry lemma right_inv_onto_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g (g' i) = i) (h2 : maps_below n g') : onto_below n g := by define at h2; define fix k : Nat assume h3 : k < n apply Exists.intro (g' k) show g' k < n ∧ g (g' k) = k from And.intro (h2 k h3) (h1 k h3) done lemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m * [inv_mod m a]_m = [1]_m := by have h2 : 0 ≤ (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a) show [a]_m * [inv_mod m a]_m = [1]_m from calc [a]_m * [inv_mod m a]_m _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2] _ = [a]_m * [gcd_c2 m a]_m := by rw [←cc_eq_mod] _ = [1]_m := gcd_c2_inv h1 done lemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m := calc a * (b % m) % m = a % m * (b % m % m) % m := Nat.mul_mod _ _ _ _ = a % m * (b % m) % m := by rw [Nat.mod_mod] _ = a * b % m := (Nat.mul_mod _ _ _).symm lemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm] rfl done theorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] : ↑a ≡ ↑b (MOD m) ↔ a % m = b % m := sorry lemma mul_inv_mod_cancel {m a i : Nat} [NeZero m] (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1 rewrite [mul_class, cc_eq_iff_congr, ←Nat.cast_mul, ←Nat.cast_one, congr_iff_mod_eq_Nat] at h3 show a * inv_mod m a * i % m = i from calc a * (inv_mod m a) * i % m _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = 1 % m * i % m := by rw [h3] _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = i % m := by rw [one_mul] _ = i := Nat.mod_eq_of_lt h2 done lemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl lemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, G m a (Ginv m a i) = i := by fix i : Nat assume h2 : i < m show G m a (Ginv m a i) = i from calc G m a (Ginv m a i) _ = a * ((inv_mod m a * i) % m) % m := by rfl _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, Ginv m a (G m a i) = i := by fix i : Nat assume h2 : i < m show Ginv m a (G m a i) = i from calc Ginv m a (G m a i) _ = inv_mod m a * ((a * i) % m) % m := by rfl _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc, mul_comm (inv_mod m a)] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_maps_below (m a : Nat) [NeZero m] : maps_below m (Ginv m a) := G_maps_below m (inv_mod m a) lemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : one_one_below m (G m a) := left_inv_one_one_below (Ginv_left_inv h1) lemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : onto_below m (G m a) := right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a) lemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : perm_below m (G m a) := And.intro (G_maps_below m a) (And.intro (G_one_one_below h1) (G_onto_below h1)) --Permuting a product of congruence classes doesn't change product lemma swap_fst (u v : Nat) : swap u v u = v := by define : swap u v u --Goal : (if u = u then v else if u = v then u else u) = v have h : u = u := by rfl rewrite [if_pos h] rfl done lemma swap_snd (u v : Nat) : swap u v v = u := by define : swap u v v by_cases h1 : v = u · -- Case 1. h1 : v = u rewrite [if_pos h1] show v = u from h1 done · -- Case 2. h1 : v ≠ u rewrite [if_neg h1] have h2 : v = v := by rfl rewrite [if_pos h2] rfl done done lemma swap_other {u v i : Nat} (h1 : i ≠ u) (h2 : i ≠ v) : swap u v i = i := by define : swap u v i rewrite [if_neg h1, if_neg h2] rfl done lemma swap_values (u v i : Nat) : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := by by_cases h1 : i = u · -- Case 1. h1 : i = u apply Or.inl rewrite [h1] show swap u v u = v from swap_fst u v done · -- Case 2. h1 : i ≠ u apply Or.inr by_cases h2 : i = v · -- Case 2.1. h2 : i = v apply Or.inl rewrite [h2] show swap u v v = u from swap_snd u v done · -- Case 2.2. h2 : i ≠ v apply Or.inr show swap u v i = i from swap_other h1 h2 done done done lemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by define fix i : Nat assume h3 : i < n have h4 : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := swap_values u v i by_cases on h4 · -- Case 1. h4 : swap u v i = v rewrite [h4] show v < n from h2 done · -- Case 2. by_cases on h4 · -- Case 2.1. h4 : swap u v i = u rewrite [h4] show u < n from h1 done · -- Case 2.2. h4 : swap u v i = i rewrite [h4] show i < n from h3 done done done lemma swap_swap (u v n : Nat) : ∀ i < n, swap u v (swap u v i) = i := by fix i : Nat assume h : i < n by_cases h1 : i = u · -- Case 1. h1 : i = u rewrite [h1, swap_fst, swap_snd] rfl done · -- Case 2. h1 : i ≠ u by_cases h2 : i = v · -- Case 2.1. h2 : i = v rewrite [h2, swap_snd, swap_fst] rfl done · -- Case 2.2. h2 : i ≠ v rewrite [swap_other h1 h2, swap_other h1 h2] rfl done done done lemma swap_one_one_below (u v n) : one_one_below n (swap u v) := left_inv_one_one_below (swap_swap u v n) lemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) := right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2) lemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) := And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2)) lemma comp_perm_below {n : Nat} {f g : Nat → Nat} (h1 : perm_below n f) (h2 : perm_below n g) : perm_below n (f ∘ g) := sorry lemma trivial_swap (u : Nat) : swap u u = id := by apply funext fix x : Nat by_cases h1 : x = u · -- Case 1. h1 : x = u rewrite [h1, swap_fst] rfl done · -- Case 2. h1 : x ≠ u rewrite [swap_other h1 h1] rfl done done lemma prod_eq_fun {m : Nat} (f g : Nat → ZMod m) (k : Nat) : ∀ (n : Nat), (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g := by by_induc · -- Base Case assume h : (∀ i < 0, f (k + i) = g (k + i)) rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g assume h1 : ∀ i < n + 1, f (k + i) = g (k + i) have h2 : ∀ i < n, f (k + i) = g (k + i) := by fix i : Nat assume h2 : i < n have h3 : i < n + 1 := by linarith show f (k + i) = g (k + i) from h1 i h3 done have h3 : prod_seq n k f = prod_seq n k g := ih h2 have h4 : n < n + 1 := Nat.lt_succ_self n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4] rfl done done lemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := by have h2 : ∀ (i : Nat), i < u → (f ∘ swap u n) (0 + i) = f (0 + i) := by fix i : Nat assume h2 : i < u have h3 : 0 + i ≠ u := by linarith have h4 : 0 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl done show prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f from prod_eq_fun (f ∘ swap u n) f 0 u h2 done lemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := by have h2 : ∀ i < j, (f ∘ swap u n) (u + 1 + i) = f (u + 1 + i) := by fix i : Nat assume h2 : i < j have h3 : u + 1 + i ≠ u := by linarith have h4 : u + 1 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl show prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f from prod_eq_fun (f ∘ swap u n) f (u + 1) j h2 done lemma break_prod {m : Nat} (n : Nat) (f : Nat → ZMod m) : ∀ (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by by_induc · -- Base Case have h : n + 0 = n := by rfl rewrite [prod_seq_base, h, Theorem_7_3_6_7] rfl done · -- Induction Step fix j : Nat assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f rewrite [←add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc] rfl done done

lemma break_prod_twice {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n

HTPI.break_prod_twice

htpi/HTPILib/Chap7.lean

htpi

/- Copyright 2023 Daniel J. Velleman -/ import HTPILib.Chap6 namespace HTPI /- Definitions -/ lemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by have h : n + 1 > 0 := Nat.succ_pos n show a % (n + 1) < n + 1 from Nat.mod_lt a h done def gcd (a b : Nat) : Nat := match b with | 0 => a | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd (n + 1) (a % (n + 1)) termination_by b mutual def gcd_c1 (a b : Nat) : Int := match b with | 0 => 1 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c2 (n + 1) (a % (n + 1)) --Corresponds to s = t' termination_by b def gcd_c2 (a b : Nat) : Int := match b with | 0 => 0 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c1 (n + 1) (a % (n + 1)) - (gcd_c2 (n + 1) (a % (n + 1))) * ↑(a / (n + 1)) --Corresponds to t = s' - t'q termination_by b end def prime (n : Nat) : Prop := 2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n def prime_factor (p n : Nat) : Prop := prime p ∧ p ∣ n def all_prime (l : List Nat) : Prop := ∀ p ∈ l, prime p def nondec (l : List Nat) : Prop := match l with | [] => True --Of course, True is a proposition that is always true | n :: L => (∀ m ∈ L, n ≤ m) ∧ nondec L def nondec_prime_list (l : List Nat) : Prop := all_prime l ∧ nondec l def prod (l : List Nat) : Nat := match l with | [] => 1 | n :: L => n * (prod L) def prime_factorization (n : Nat) (l : List Nat) : Prop := nondec_prime_list l ∧ prod l = n def rel_prime (a b : Nat) : Prop := gcd a b = 1 def congr_mod (m : Nat) (a b : Int) : Prop := (↑m : Int) ∣ (a - b) def cc (m : Nat) (a : Int) : ZMod m := (↑a : ZMod m) notation:50 a " ≡ " b " (MOD " m ")" => congr_mod m a b notation:max "["a"]_"m:max => cc m a def invertible {m : Nat} (X : ZMod m) : Prop := ∃ (Y : ZMod m), X * Y = [1]_m def num_rp_below (m k : Nat) : Nat := match k with | 0 => 0 | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j def phi (m : Nat) : Nat := num_rp_below m m def prod_seq {m : Nat} (j k : Nat) (f : Nat → ZMod m) : ZMod m := match j with | 0 => [1]_m | n + 1 => prod_seq n k f * f (k + n) def maps_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i < n, g i < n def one_one_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i1 < n, ∀ i2 < n, g i1 = g i2 → i1 = i2 def onto_below (n : Nat) (g : Nat → Nat) : Prop := ∀ k < n, ∃ i < n, g i = k def perm_below (n : Nat) (g : Nat → Nat) : Prop := maps_below n g ∧ one_one_below n g ∧ onto_below n g def inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m) def swap (u v i : Nat) : Nat := if i = u then v else if i = v then u else i namespace Euler --For definitions specific to Euler's theorem def F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m def G (m a i : Nat) : Nat := (a * i) % m def Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i end Euler /- Section 7.1 -/ theorem dvd_mod_of_dvd_a_b {a b d : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ (a % b) := by set q : Nat := a / b have h3 : b * q + a % b = a := Nat.div_add_mod a b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 define --Goal : ∃ (c : Nat), a % b = d * c apply Exists.intro (j - k * q) show a % b = d * (j - k * q) from calc a % b _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm _ = a - b * q := by rw [h3] _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc] _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm done theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry #eval gcd 672 161 --Answer: 7 lemma gcd_base (a : Nat) : gcd a 0 = a := by rfl lemma gcd_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd a b = gcd b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1)) rfl done lemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b ≠ 0) : a % b < b := by have h1 : b > 0 := Nat.pos_of_ne_zero h show a % b < b from Nat.mod_lt a h1 done lemma dvd_self (n : Nat) : n ∣ n := by apply Exists.intro 1 ring done theorem gcd_dvd : ∀ (b a : Nat), (gcd a b) ∣ a ∧ (gcd a b) ∣ b := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd a b_1) ∣ a ∧ (gcd a b_1) ∣ b_1 fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_base] --Goal: a ∣ a ∧ a ∣ 0 apply And.intro (dvd_self a) define apply Exists.intro 0 rfl done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_nonzero a h1] --Goal : gcd b (a % b) ∣ a ∧ gcd b (a % b) ∣ b have h2 : a % b < b := mod_nonzero_lt a h1 have h3 : (gcd b (a % b)) ∣ b ∧ (gcd b (a % b)) ∣ (a % b) := ih (a % b) h2 b apply And.intro _ h3.left show (gcd b (a % b)) ∣ a from dvd_a_of_dvd_b_mod h3.left h3.right done done theorem gcd_dvd_left (a b : Nat) : (gcd a b) ∣ a := (gcd_dvd b a).left theorem gcd_dvd_right (a b : Nat) : (gcd a b) ∣ b := (gcd_dvd b a).right lemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl lemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c1 a b = gcd_c2 b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done lemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl lemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * ↑(a / b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done theorem gcd_lin_comb : ∀ (b a : Nat), (gcd_c1 a b) * ↑a + (gcd_c2 a b) * ↑b = ↑(gcd a b) := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd_c1 a b_1) * ↑a + (gcd_c2 a b_1) * ↑b_1 = ↑(gcd a b_1) fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base] --Goal : 1 * ↑a + 0 * ↑0 = ↑a ring done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1] --Goal : gcd_c2 b (a % b) * ↑a + -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * ↑(a / b)) * ↑b = -- ↑(gcd b (a % b)) set r : Nat := a % b set q : Nat := a / b set s : Int := gcd_c1 b r set t : Int := gcd_c2 b r --Goal : t * ↑a + (s - t * ↑q) * ↑b = ↑(gcd b r) have h2 : r < b := mod_nonzero_lt a h1 have h3 : s * ↑b + t * ↑r = ↑(gcd b r) := ih r h2 b have h4 : b * q + r = a := Nat.div_add_mod a b rewrite [←h3, ←h4] rewrite [Nat.cast_add, Nat.cast_mul] --Goal : t * (↑b * ↑q + ↑r) + (s - t * ↑q) * ↑b = s * ↑b + t * ↑r ring done done #eval gcd_c1 672 161 --Answer: 6 #eval gcd_c2 672 161 --Answer: -25 --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161 theorem Theorem_7_1_6 {d a b : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ gcd a b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑d ∣ ↑(gcd a b) set s : Int := gcd_c1 a b set t : Int := gcd_c2 a b have h3 : s * ↑a + t * ↑b = ↑(gcd a b) := gcd_lin_comb b a rewrite [←h3] --Goal : ↑d ∣ s * ↑a + t * ↑b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul] --Goal : ↑d ∣ s * (↑d * ↑j) + t * (↑d * ↑k) define apply Exists.intro (s * ↑j + t * ↑k) ring done /- Section 7.2 -/ theorem dvd_trans {a b c : Nat} (h1 : a ∣ b) (h2 : b ∣ c) : a ∣ c := by define at h1; define at h2; define obtain (m : Nat) (h3 : b = a * m) from h1 obtain (n : Nat) (h4 : c = b * n) from h2 rewrite [h3, mul_assoc] at h4 apply Exists.intro (m * n) show c = a * (m * n) from h4 done lemma exists_prime_factor : ∀ (n : Nat), 2 ≤ n → ∃ (p : Nat), prime_factor p n := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, 2 ≤ n_1 → ∃ (p : Nat), prime_factor p n_1 assume h1 : 2 ≤ n by_cases h2 : prime n · -- Case 1. h2 : prime n apply Exists.intro n define --Goal : prime n ∧ n ∣ n show prime n ∧ n ∣ n from And.intro h2 (dvd_self n) done · -- Case 2. h2 : ¬prime n define at h2 --h2 : ¬(2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n) demorgan at h2 disj_syll h2 h1 obtain (a : Nat) (h3 : ∃ (b : Nat), a * b = n ∧ a < n ∧ b < n) from h2 obtain (b : Nat) (h4 : a * b = n ∧ a < n ∧ b < n) from h3 have h5 : 2 ≤ a := by by_contra h6 have h7 : a ≤ 1 := by linarith have h8 : n ≤ b := calc n _ = a * b := h4.left.symm _ ≤ 1 * b := by rel [h7] _ = b := by ring linarith --n ≤ b contradicts b < n done have h6 : ∃ (p : Nat), prime_factor p a := ih a h4.right.left h5 obtain (p : Nat) (h7 : prime_factor p a) from h6 apply Exists.intro p define --Goal : prime p ∧ p ∣ n define at h7 --h7 : prime p ∧ p ∣ a apply And.intro h7.left have h8 : a ∣ n := by apply Exists.intro b show n = a * b from (h4.left).symm done show p ∣ n from dvd_trans h7.right h8 done done lemma exists_least_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q := by set S : Set Nat := {p : Nat | prime_factor p n} have h2 : ∃ (p : Nat), p ∈ S := exists_prime_factor n h show ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q from well_ord_princ S h2 done lemma all_prime_nil : all_prime [] := by define --Goal : ∀ p ∈ [], prime p fix p : Nat contrapos --Goal : ¬prime p → p ∉ [] assume h1 : ¬prime p show p ∉ [] from List.not_mem_nil p done lemma all_prime_cons (n : Nat) (L : List Nat) : all_prime (n :: L) ↔ prime n ∧ all_prime L := by apply Iff.intro · -- (→) assume h1 : all_prime (n :: L) --Goal : prime n ∧ all_prime L define at h1 --h1 : ∀ p ∈ n :: L, prime p apply And.intro (h1 n (List.mem_cons_self n L)) define --Goal : ∀ p ∈ L, prime p fix p : Nat assume h2 : p ∈ L show prime p from h1 p (List.mem_cons_of_mem n h2) done · -- (←) assume h1 : prime n ∧ all_prime L --Goal : all_prime (n :: l) define : all_prime L at h1 define fix p : Nat assume h2 : p ∈ n :: L rewrite [List.mem_cons] at h2 --h2 : p = n ∨ p ∈ L by_cases on h2 · -- Case 1. h2 : p = n rewrite [h2] show prime n from h1.left done · -- Case 2. h2 : p ∈ L show prime p from h1.right p h2 done done done lemma nondec_nil : nondec [] := by define --Goal : True trivial --trivial proves some obviously true statements, such as True done lemma nondec_cons (n : Nat) (L : List Nat) : nondec (n :: L) ↔ (∀ m ∈ L, n ≤ m) ∧ nondec L := by rfl lemma prod_nil : prod [] = 1 := by rfl lemma prod_cons : prod (n :: L) = n * (prod L) := by rfl lemma exists_cons_of_length_eq_succ {A : Type} {l : List A} {n : Nat} (h : l.length = n + 1) : ∃ (a : A) (L : List A), l = a :: L ∧ L.length = n := by have h1 : ¬l.length = 0 := by linarith rewrite [List.length_eq_zero] at h1 obtain (a : A) (h2 : ∃ (L : List A), l = a :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List A) (h3 : l = a :: L) from h2 apply Exists.intro a apply Exists.intro L apply And.intro h3 have h4 : (a :: L).length = L.length + 1 := List.length_cons a L rewrite [←h3, h] at h4 show L.length = n from (Nat.add_right_cancel h4).symm done lemma list_elt_dvd_prod_by_length (a : Nat) : ∀ (n : Nat), ∀ (l : List Nat), l.length = n → a ∈ l → a ∣ prod l := by by_induc · --Base Case fix l : List Nat assume h1 : l.length = 0 rewrite [List.length_eq_zero] at h1 --h1 : l = [] rewrite [h1] --Goal : a ∈ [] → a ∣ prod [] contrapos assume h2 : ¬a ∣ prod [] show a ∉ [] from List.not_mem_nil a done · -- Induction Step fix n : Nat assume ih : ∀ (l : List Nat), List.length l = n → a ∈ l → a ∣ prod l fix l : List Nat assume h1 : l.length = n + 1 --Goal : a ∈ l → a ∣ prod l obtain (b : Nat) (h2 : ∃ (L : List Nat), l = b :: L ∧ L.length = n) from exists_cons_of_length_eq_succ h1 obtain (L : List Nat) (h3 : l = b :: L ∧ L.length = n) from h2 have h4 : a ∈ L → a ∣ prod L := ih L h3.right assume h5 : a ∈ l rewrite [h3.left, prod_cons] --Goal : a ∣ b * prod L rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b ∨ a ∈ L by_cases on h5 · -- Case 1. h5 : a = b apply Exists.intro (prod L) rewrite [h5] rfl done · -- Case 2. h5 : a ∈ L have h6 : a ∣ prod L := h4 h5 have h7 : prod L ∣ b * prod L := by apply Exists.intro b ring done show a ∣ b * prod L from dvd_trans h6 h7 done done done lemma list_elt_dvd_prod {a : Nat} {l : List Nat} (h : a ∈ l) : a ∣ prod l := by set n : Nat := l.length have h1 : l.length = n := by rfl show a ∣ prod l from list_elt_dvd_prod_by_length a n l h1 h done lemma exists_prime_factorization : ∀ (n : Nat), n ≥ 1 → ∃ (l : List Nat), prime_factorization n l := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, n_1 ≥ 1 → ∃ (l : List Nat), prime_factorization n_1 l assume h1 : n ≥ 1 by_cases h2 : n = 1 · -- Case 1. h2 : n = 1 apply Exists.intro [] define apply And.intro · -- Proof of nondec_prime_list [] define show all_prime [] ∧ nondec [] from And.intro all_prime_nil nondec_nil done · -- Proof of prod [] = n rewrite [prod_nil, h2] rfl done done · -- Case 2. h2 : n ≠ 1 have h3 : n ≥ 2 := lt_of_le_of_ne' h1 h2 obtain (p : Nat) (h4 : prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q) from exists_least_prime_factor h3 have p_prime_factor : prime_factor p n := h4.left define at p_prime_factor have p_prime : prime p := p_prime_factor.left have p_dvd_n : p ∣ n := p_prime_factor.right have p_least : ∀ (q : Nat), prime_factor q n → p ≤ q := h4.right obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n have h5 : m ≠ 0 := by contradict h1 with h6 have h7 : n = 0 := calc n _ = p * m := n_eq_pm _ = p * 0 := by rw [h6] _ = 0 := by ring rewrite [h7] decide done have m_pos : 0 < m := Nat.pos_of_ne_zero h5 have m_lt_n : m < n := by define at p_prime show m < n from calc m _ < m + m := by linarith _ = 2 * m := by ring _ ≤ p * m := by rel [p_prime.left] _ = n := n_eq_pm.symm done obtain (L : List Nat) (h6 : prime_factorization m L) from ih m m_lt_n m_pos define at h6 have ndpl_L : nondec_prime_list L := h6.left define at ndpl_L apply Exists.intro (p :: L) define apply And.intro · -- Proof of nondec_prime_list (p :: L) define apply And.intro · -- Proof of all_prime (p :: L) rewrite [all_prime_cons] show prime p ∧ all_prime L from And.intro p_prime ndpl_L.left done · -- Proof of nondec (p :: L) rewrite [nondec_cons] apply And.intro _ ndpl_L.right fix q : Nat assume q_in_L : q ∈ L have h7 : q ∣ prod L := list_elt_dvd_prod q_in_L rewrite [h6.right] at h7 --h7 : q ∣ m have h8 : m ∣ n := by apply Exists.intro p rewrite [n_eq_pm] ring done have q_dvd_n : q ∣ n := dvd_trans h7 h8 have ap_L : all_prime L := ndpl_L.left define at ap_L have q_prime_factor : prime_factor q n := And.intro (ap_L q q_in_L) q_dvd_n show p ≤ q from p_least q q_prime_factor done done · -- Proof of prod (p :: L) = n rewrite [prod_cons, h6.right, n_eq_pm] rfl done done done theorem Theorem_7_2_2 {a b c : Nat} (h1 : c ∣ a * b) (h2 : rel_prime a c) : c ∣ b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑c ∣ ↑b define at h1; define at h2; define obtain (j : Nat) (h3 : a * b = c * j) from h1 set s : Int := gcd_c1 a c set t : Int := gcd_c2 a c have h4 : s * ↑a + t * ↑c = ↑(gcd a c) := gcd_lin_comb c a rewrite [h2, Nat.cast_one] at h4 --h4 : s * ↑a + t * ↑c = (1 : Int) apply Exists.intro (s * ↑j + t * ↑b) show ↑b = ↑c * (s * ↑j + t * ↑b) from calc ↑b _ = (1 : Int) * ↑b := (one_mul _).symm _ = (s * ↑a + t * ↑c) * ↑b := by rw [h4] _ = s * (↑a * ↑b) + t * ↑c * ↑b := by ring _ = s * (↑c * ↑j) + t * ↑c * ↑b := by rw [←Nat.cast_mul a b, h3, Nat.cast_mul c j] _ = ↑c * (s * ↑j + t * ↑b) := by ring done lemma le_nonzero_prod_left {a b : Nat} (h : a * b ≠ 0) : a ≤ a * b := by have h1 : b ≠ 0 := by contradict h with h1 rewrite [h1] ring done have h2 : 1 ≤ b := Nat.pos_of_ne_zero h1 show a ≤ a * b from calc a = a * 1 := (mul_one a).symm _ ≤ a * b := by rel [h2] done lemma le_nonzero_prod_right {a b : Nat} (h : a * b ≠ 0) : b ≤ a * b := by rewrite [mul_comm] rewrite [mul_comm] at h show b ≤ b * a from le_nonzero_prod_left h done lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry lemma rel_prime_of_prime_not_dvd {a p : Nat} (h1 : prime p) (h2 : ¬p ∣ a) : rel_prime a p := by have h3 : gcd a p ∣ a := gcd_dvd_left a p have h4 : gcd a p ∣ p := gcd_dvd_right a p have h5 : gcd a p = 1 ∨ gcd a p = p := dvd_prime h1 h4 have h6 : gcd a p ≠ p := by contradict h2 with h6 rewrite [h6] at h3 show p ∣ a from h3 done disj_syll h5 h6 show rel_prime a p from h5 done theorem Theorem_7_2_3 {a b p : Nat} (h1 : prime p) (h2 : p ∣ a * b) : p ∣ a ∨ p ∣ b := by or_right with h3 have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3 show p ∣ b from Theorem_7_2_2 h2 h4 done lemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a ≥ 1 := by have h1 : a ≠ 0 := by by_contra h1 rewrite [h1] at h contradict h linarith done show a ≥ 1 from Nat.pos_of_ne_zero h1 done lemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by have h1 : a ≥ 1 := ge_one_of_prod_one h have h2 : a * b ≠ 0 := by linarith have h3 : a ≤ a * b := le_nonzero_prod_left h2 rewrite [h] at h3 show a = 1 from Nat.le_antisymm h3 h1 done lemma eq_one_of_dvd_one {n : Nat} (h : n ∣ 1) : n = 1 := by obtain (j : Nat) (h1 : 1 = n * j) from h show n = 1 from eq_one_of_prod_one h1.symm done lemma prime_not_one {p : Nat} (h : prime p) : p ≠ 1 := by define at h linarith done theorem Theorem_7_2_4 {p : Nat} (h1 : prime p) : ∀ (l : List Nat), p ∣ prod l → ∃ a ∈ l, p ∣ a := by apply List.rec · -- Base Case. Goal : p ∣ prod [] → ∃ a ∈ [], p ∣ a rewrite [prod_nil] assume h2 : p ∣ 1 show ∃ a ∈ [], p ∣ a from absurd (eq_one_of_dvd_one h2) (prime_not_one h1) done · -- Induction Step fix b : Nat fix L : List Nat assume ih : p ∣ prod L → ∃ a ∈ L, p ∣ a --Goal : p ∣ prod (b :: L) → ∃ a ∈ b :: L, p ∣ a assume h2 : p ∣ prod (b :: L) rewrite [prod_cons] at h2 have h3 : p ∣ b ∨ p ∣ prod L := Theorem_7_2_3 h1 h2 by_cases on h3 · -- Case 1. h3 : p ∣ b apply Exists.intro b show b ∈ b :: L ∧ p ∣ b from And.intro (List.mem_cons_self b L) h3 done · -- Case 2. h3 : p ∣ prod L obtain (a : Nat) (h4 : a ∈ L ∧ p ∣ a) from ih h3 apply Exists.intro a show a ∈ b :: L ∧ p ∣ a from And.intro (List.mem_cons_of_mem b h4.left) h4.right done done done lemma prime_in_list {p : Nat} {l : List Nat} (h1 : prime p) (h2 : all_prime l) (h3 : p ∣ prod l) : p ∈ l := by obtain (a : Nat) (h4 : a ∈ l ∧ p ∣ a) from Theorem_7_2_4 h1 l h3 define at h2 have h5 : prime a := h2 a h4.left have h6 : p = 1 ∨ p = a := dvd_prime h5 h4.right disj_syll h6 (prime_not_one h1) rewrite [h6] show a ∈ l from h4.left done lemma first_le_first {p q : Nat} {l m : List Nat} (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m)) (h3 : prod (p :: l) = prod (q :: m)) : p ≤ q := by define at h1; define at h2 have h4 : q ∣ prod (p :: l) := by define apply Exists.intro (prod m) rewrite [←prod_cons] show prod (p :: l) = prod (q :: m) from h3 done have h5 : all_prime (q :: m) := h2.left rewrite [all_prime_cons] at h5 have h6 : q ∈ p :: l := prime_in_list h5.left h1.left h4 have h7 : nondec (p :: l) := h1.right rewrite [nondec_cons] at h7 rewrite [List.mem_cons] at h6 by_cases on h6 · -- Case 1. h6 : q = p linarith done · -- Case 2. h6 : q ∈ l have h8 : ∀ m ∈ l, p ≤ m := h7.left show p ≤ q from h8 q h6 done done lemma nondec_prime_list_tail {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by define at h define rewrite [all_prime_cons, nondec_cons] at h show all_prime l ∧ nondec l from And.intro h.left.right h.right.right done lemma cons_prod_not_one {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : prod (p :: l) ≠ 1 := by define at h have h1 : all_prime (p :: l) := h.left rewrite [all_prime_cons] at h1 rewrite [prod_cons] by_contra h2 show False from (prime_not_one h1.left) (eq_one_of_prod_one h2) done lemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) : l = [] ↔ prod l = 1 := by apply Iff.intro · -- (→) assume h1 : l = [] rewrite [h1] show prod [] = 1 from prod_nil done · -- (←) contrapos assume h1 : ¬l = [] obtain (p : Nat) (h2 : ∃ (L : List Nat), l = p :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List Nat) (h3 : l = p :: L) from h2 rewrite [h3] at h rewrite [h3] show ¬prod (p :: L) = 1 from cons_prod_not_one h done done lemma prime_pos {p : Nat} (h : prime p) : p > 0 := by define at h linarith done theorem Theorem_7_2_5 : ∀ (l1 l2 : List Nat), nondec_prime_list l1 → nondec_prime_list l2 → prod l1 = prod l2 → l1 = l2 := by apply List.rec · -- Base Case. Goal : ∀ (l2 : List Nat), nondec_prime_list [] → -- nondec_prime_list l2 → prod [] = prod l2 → [] = l2 fix l2 : List Nat assume h1 : nondec_prime_list [] assume h2 : nondec_prime_list l2 assume h3 : prod [] = prod l2 rewrite [prod_nil, eq_comm, ←list_nil_iff_prod_one h2] at h3 show [] = l2 from h3.symm done · -- Induction Step fix p : Nat fix L1 : List Nat assume ih : ∀ (L2 : List Nat), nondec_prime_list L1 → nondec_prime_list L2 → prod L1 = prod L2 → L1 = L2 -- Goal : ∀ (l2 : List Nat), nondec_prime_list (p :: L1) → -- nondec_prime_list l2 → prod (p :: L1) = prod l2 → p :: L1 = l2 fix l2 : List Nat assume h1 : nondec_prime_list (p :: L1) assume h2 : nondec_prime_list l2 assume h3 : prod (p :: L1) = prod l2 have h4 : ¬prod (p :: L1) = 1 := cons_prod_not_one h1 rewrite [h3, ←list_nil_iff_prod_one h2] at h4 obtain (q : Nat) (h5 : ∃ (L : List Nat), l2 = q :: L) from List.exists_cons_of_ne_nil h4 obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5 rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2) rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2) have h7 : p ≤ q := first_le_first h1 h2 h3 have h8 : q ≤ p := first_le_first h2 h1 h3.symm have h9 : p = q := by linarith rewrite [h9, prod_cons, prod_cons] at h3 --h3 : q * prod L1 = q * prod L2 have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1 have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2 define at h2 have h12 : all_prime (q :: L2) := h2.left rewrite [all_prime_cons] at h12 have h13 : q > 0 := prime_pos h12.left have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3 have h15 : L1 = L2 := ih L2 h10 h11 h14 rewrite [h6, h9, h15] rfl done done theorem fund_thm_arith (n : Nat) (h : n ≥ 1) : ∃! (l : List Nat), prime_factorization n l := by exists_unique · -- Existence show ∃ (l : List Nat), prime_factorization n l from exists_prime_factorization n h done · -- Uniqueness fix l1 : List Nat; fix l2 : List Nat assume h1 : prime_factorization n l1 assume h2 : prime_factorization n l2 define at h1; define at h2 have h3 : prod l1 = n := h1.right rewrite [←h2.right] at h3 show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3 done done /- Section 7.3 -/ theorem congr_refl (m : Nat) : ∀ (a : Int), a ≡ a (MOD m) := by fix a : Int define --Goal : ∃ (c : Int), a - a = ↑m * c apply Exists.intro 0 ring done theorem congr_symm {m : Nat} : ∀ {a b : Int}, a ≡ b (MOD m) → b ≡ a (MOD m) := by fix a : Int; fix b : Int assume h1 : a ≡ b (MOD m) define at h1 --h1 : ∃ (c : Int), a - b = ↑m * c define --Goal : ∃ (c : Int), b - a = ↑m * c obtain (c : Int) (h2 : a - b = m * c) from h1 apply Exists.intro (-c) show b - a = m * (-c) from calc b - a _ = -(a - b) := by ring _ = -(m * c) := by rw [h2] _ = m * (-c) := by ring done theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry /- Fundamental properties of congruence classes -/ lemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) : X = Y ↔ X.val = Y.val := Fin.ext_iff lemma val_nat_eq_mod (n k : Nat) : ([k]_(n + 1)).val = k % (n + 1) := by rfl lemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl theorem cc_rep {m : Nat} (X : ZMod m) : ∃ (a : Int), X = [a]_m := match m with | 0 => by apply Exists.intro X rfl done | n + 1 => by apply Exists.intro ↑(X.val) have h1 : X.val < n + 1 := Fin.prop X rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1] rfl done theorem add_class (m : Nat) (a b : Int) : [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm theorem mul_class (m : Nat) (a b : Int) : [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm lemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ [a - b]_m = [0]_m := by apply Iff.intro · -- (→) assume h1 : [a]_m = [b]_m have h2 : a - b = a + (-b) := by ring have h3 : b + (-b) = 0 := by ring show [a - b]_m = [0]_m from calc [a - b]_m _ = [a + (-b)]_m := by rw [h2] _ = [a]_m + [-b]_m := by rw [add_class] _ = [b]_m + [-b]_m := by rw [h1] _ = [b + -b]_m := by rw [add_class] _ = [0]_m := by rw [h3] done · -- (←) assume h1 : [a - b]_m = [0]_m have h2 : b + (a - b) = a := by ring have h3 : b + 0 = b := by ring show [a]_m = [b]_m from calc [a]_m _ = [b + (a - b)]_m := by rw [h2] _ = [b]_m + [a - b]_m := by rw [add_class] _ = [b]_m + [0]_m := by rw [h1] _ = [b + 0]_m := by rw [add_class] _ = [b]_m := by rw [h3] done done lemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) : [a]_m = [0]_m → [-a]_m = [0]_m := by assume h1 : [a]_m = [0]_m have h2 : 0 + (-a) = -a := by ring have h3 : a + (-a) = 0 := by ring show [-a]_m = [0]_m from calc [-a]_m _ = [0 + (-a)]_m := by rw [h2] _ = [0]_m + [-a]_m := by rw [add_class] _ = [a]_m + [-a]_m := by rw [h1] _ = [a + (-a)]_m := by rw [add_class] _ = [0]_m := by rw [h3] done lemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) : [-a]_m = [0]_m ↔ [a]_m = [0]_m := by apply Iff.intro _ (cc_neg_zero_of_cc_zero m a) assume h1 : [-a]_m = [0]_m have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1 have h3 : -(-a) = a := by ring rewrite [h3] at h2 show [a]_m = [0]_m from h2 done lemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl lemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m ↔ m ∣ k := match m with | 0 => by have h : (0 : Int) = (↑(0 : Nat) : Int) := by rfl rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj] apply Iff.intro · -- (→) assume h1 : k = 0 rewrite [h1] show 0 ∣ 0 from dvd_self 0 done · -- (←) assume h1 : 0 ∣ k obtain (c : Nat) (h2 : k = 0 * c) from h1 rewrite [h2] ring done done | n + 1 => by rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero] show k % (n + 1) = 0 ↔ n + 1 ∣ k from (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm done lemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m ↔ ↑m ∣ a := by obtain (k : Nat) (h1 : a = ↑k ∨ a = -↑k) from Int.eq_nat_or_neg a by_cases on h1 · -- Case 1. h1: a = ↑k rewrite [h1, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done · -- Case 2. h1: a = -↑k rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done done theorem cc_eq_iff_congr (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ a ≡ b (MOD m) := calc [a]_m = [b]_m _ ↔ [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b _ ↔ ↑m ∣ (a - b) := cc_zero_iff_dvd m (a - b) _ ↔ a ≡ b (MOD m) := by rfl /- End of fundamental properties of congruence classes -/ lemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m := by have h1 : (↑m : Int) ≠ 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m) show 0 ≤ a % m from Int.emod_nonneg a h1 done lemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m) have h2 : (↑m : Int) > 0 := (Nat.cast_pos).rtl h1 show a % m < m from Int.emod_lt_of_pos a h2 done lemma congr_mod_mod (m : Nat) (a : Int) : a ≡ a % m (MOD m) := by define have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m apply Exists.intro (a / m) show a - a % m = m * (a / m) from calc a - (a % m) _ = m * (a / m) + a % m - a % m := by rw [h1] _ = m * (a / m) := by ring done lemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) := And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a)) theorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) : ∃! (r : Int), 0 ≤ r ∧ r < m ∧ a ≡ r (MOD m) := by exists_unique · -- Existence apply Exists.intro (a % m) show 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) from mod_cmpl_res m a done · -- Uniqueness fix r1 : Int; fix r2 : Int assume h1 : 0 ≤ r1 ∧ r1 < m ∧ a ≡ r1 (MOD m) assume h2 : 0 ≤ r2 ∧ r2 < m ∧ a ≡ r2 (MOD m) have h3 : r1 ≡ r2 (MOD m) := congr_trans (congr_symm h1.right.right) h2.right.right obtain (d : Int) (h4 : r1 - r2 = m * d) from h3 have h5 : r1 - r2 < m * 1 := by linarith have h6 : m * (-1) < r1 - r2 := by linarith rewrite [h4] at h5 --h5 : m * d < m * 1 rewrite [h4] at h6 --h6 : m * -1 < m * d have h7 : (↑m : Int) ≥ 0 := Nat.cast_nonneg m have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7 have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7 have h10 : d = 0 := by linarith show r1 = r2 from calc r1 _ = r1 - r2 + r2 := by ring _ = m * 0 + r2 := by rw [h4, h10] _ = r2 := by ring done done lemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m := (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a) theorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y rewrite [h1, h2] have h3 : a + b = b + a := by ring show [a]_m + [b]_m = [b]_m + [a]_m from calc [a]_m + [b]_m _ = [a + b]_m := add_class m a b _ = [b + a]_m := by rw [h3] _ = [b]_m + [a]_m := (add_class m b a).symm done theorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X rewrite [h1] have h2 : a * 1 = a := by ring show [a]_m * [1]_m = [a]_m from calc [a]_m * [1]_m _ = [a * 1]_m := mul_class m a 1 _ = [a]_m := by rw [h2] done theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry lemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) : [a]_m * [gcd_c2 m a]_m = [1]_m := by set s : Int := gcd_c1 m a have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m define at h1 rewrite [h1, Nat.cast_one] at h2 --h2 : s * ↑m + gcd_c2 m a * ↑a = 1 rewrite [mul_class, cc_eq_iff_congr] define --Goal : ∃ (c : Int), ↑a * gcd_c2 m a - 1 = ↑m * c apply Exists.intro (-s) show a * (gcd_c2 m a) - 1 = m * (-s) from calc a * (gcd_c2 m a) - 1 _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring _ = 1 + m * (-s) - 1 := by rw [h2] _ = m * (-s) := by ring done theorem Theorem_7_3_7 (m a : Nat) : invertible [a]_m ↔ rel_prime m a := by apply Iff.intro · -- (→) assume h1 : invertible [a]_m define at h1 obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1 obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y rewrite [h3, mul_class, cc_eq_iff_congr] at h2 define at h2 obtain (c : Int) (h4 : a * b - 1 = m * c) from h2 rewrite [Exercise_7_2_6] --Goal : ∃ (s t : Int), s * ↑m + t * ↑a = 1 apply Exists.intro (-c) apply Exists.intro b show (-c) * m + b * a = 1 from calc (-c) * m + b * a _ = (-c) * m + (a * b - 1) + 1 := by ring _ = (-c) * m + m * c + 1 := by rw [h4] _ = 1 := by ring done · -- (←) assume h1 : rel_prime m a define show ∃ (Y : ZMod m), [a]_m * Y = [1]_m from Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1) done done /- Section 7.4 -/ section Euler open Euler lemma num_rp_below_base {m : Nat} : num_rp_below m 0 = 0 := by rfl lemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) : num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by have h1 : num_rp_below m (j + 1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : gcd m j = 1 rewrite [if_pos h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1 show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1 done lemma num_rp_below_step_not_rp {m j : Nat} (h : ¬rel_prime m j) : num_rp_below m (j + 1) = num_rp_below m j := by have h1 : num_rp_below m (j +1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : ¬gcd m j = 1 rewrite [if_neg h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j show num_rp_below m (j + 1) = num_rp_below m j from h1 done lemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl #eval phi 10 --Answer: 4 lemma prod_inv_iff_inv {m : Nat} {X : ZMod m} (h1 : invertible X) (Y : ZMod m) : invertible (X * Y) ↔ invertible Y := by apply Iff.intro · -- (→) assume h2 : invertible (X * Y) obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2 apply Exists.intro (X * Z) rewrite [←h3] --Goal : Y * (X * Z) = X * Y * Z ring --Note that ring can do algebra in ZMod m done · -- (←) assume h2 : invertible Y obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1 obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2 apply Exists.intro (Xi * Yi) show (X * Y) * (Xi * Yi) = [1]_m from calc X * Y * (Xi * Yi) _ = (X * Xi) * (Y * Yi) := by ring _ = [1]_m * [1]_m := by rw [h3, h4] _ = [1]_m := Theorem_7_3_6_7 [1]_m done done lemma F_rp_def {m i : Nat} (h : rel_prime m i) : F m i = [i]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h --h : gcd m i = 1 rewrite [if_pos h] at h1 show F m i = [i]_m from h1 done lemma F_not_rp_def {m i : Nat} (h : ¬rel_prime m i) : F m i = [1]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h rewrite [h1, if_neg h] rfl done lemma prod_seq_base {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 0 k f = [1]_m := by rfl lemma prod_seq_step {m : Nat} (n k : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl lemma prod_seq_zero_step {m : Nat} (n : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by rewrite [prod_seq_step, zero_add] rfl done lemma prod_one {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 1 k f = f k := by rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7] rfl done lemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl lemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m := calc [G m a i]_m _ = [(a * i) % m]_m := by rfl _ = [a * i]_m := (cc_eq_mod m (a * i)).symm _ = [a]_m * [i]_m := (mul_class m a i).symm lemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) : rel_prime m (G m a i) ↔ rel_prime m i := by have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1 show rel_prime m (G m a i) ↔ rel_prime m i from calc rel_prime m (G m a i) _ ↔ invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm _ ↔ invertible ([a]_m * [i]_m) := by rw [cc_G] _ ↔ invertible [i]_m := prod_inv_iff_inv h2 ([i]_m) _ ↔ rel_prime m i := Theorem_7_3_7 m i done lemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) : F m (G m a i) = [a]_m * F m i := by have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2 show F m (G m a i) = [a]_m * F m i from calc F m (G m a i) _ = [G m a i]_m := F_rp_def h3 _ = [a]_m * [i]_m := cc_G m a i _ = [a]_m * F m i := by rw [F_rp_def h2] done lemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : ¬rel_prime m i) : F m (G m a i) = [1]_m := by rewrite [←G_rp_iff h1 i] at h2 show F m (G m a i) = [1]_m from F_not_rp_def h2 done lemma FG_prod {m a : Nat} (h1 : rel_prime m a) : ∀ (k : Nat), prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by by_induc · -- Base Case show prod_seq 0 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from calc prod_seq 0 0 ((F m) ∘ (G m a)) _ = [1]_m := prod_seq_base _ _ _ = [a]_m ^ 0 * [1]_m := by ring _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by rw [num_rp_below_base, prod_seq_base] done · -- Induction Step fix k : Nat assume ih : prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) by_cases h2 : rel_prime m k · -- Case 1. h2 : rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([a]_m * F m k) := by rw [FG_rp h1 h2] _ = [a]_m ^ ((num_rp_below m k) + 1) * ((prod_seq k 0 (F m)) * F m k) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_rp h2, prod_seq_zero_step] done · -- Case 2. h2 : ¬rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([1]_m) := by rw [FG_not_rp h1 h2] _ = [a]_m ^ (num_rp_below m k) * (prod_seq k 0 (F m) * ([1]_m)) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_not_rp h2, prod_seq_zero_step, F_not_rp_def h2] done done done lemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by define --Goal : ∀ i < m, G m a i < m fix i : Nat assume h1 : i < m rewrite [G_def] --Goal : a * i % m < m show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m) done lemma left_inv_one_one_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g' (g i) = i) : one_one_below n g := sorry lemma right_inv_onto_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g (g' i) = i) (h2 : maps_below n g') : onto_below n g := by define at h2; define fix k : Nat assume h3 : k < n apply Exists.intro (g' k) show g' k < n ∧ g (g' k) = k from And.intro (h2 k h3) (h1 k h3) done lemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m * [inv_mod m a]_m = [1]_m := by have h2 : 0 ≤ (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a) show [a]_m * [inv_mod m a]_m = [1]_m from calc [a]_m * [inv_mod m a]_m _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2] _ = [a]_m * [gcd_c2 m a]_m := by rw [←cc_eq_mod] _ = [1]_m := gcd_c2_inv h1 done lemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m := calc a * (b % m) % m = a % m * (b % m % m) % m := Nat.mul_mod _ _ _ _ = a % m * (b % m) % m := by rw [Nat.mod_mod] _ = a * b % m := (Nat.mul_mod _ _ _).symm lemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm] rfl done theorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] : ↑a ≡ ↑b (MOD m) ↔ a % m = b % m := sorry lemma mul_inv_mod_cancel {m a i : Nat} [NeZero m] (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1 rewrite [mul_class, cc_eq_iff_congr, ←Nat.cast_mul, ←Nat.cast_one, congr_iff_mod_eq_Nat] at h3 show a * inv_mod m a * i % m = i from calc a * (inv_mod m a) * i % m _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = 1 % m * i % m := by rw [h3] _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = i % m := by rw [one_mul] _ = i := Nat.mod_eq_of_lt h2 done lemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl lemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, G m a (Ginv m a i) = i := by fix i : Nat assume h2 : i < m show G m a (Ginv m a i) = i from calc G m a (Ginv m a i) _ = a * ((inv_mod m a * i) % m) % m := by rfl _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, Ginv m a (G m a i) = i := by fix i : Nat assume h2 : i < m show Ginv m a (G m a i) = i from calc Ginv m a (G m a i) _ = inv_mod m a * ((a * i) % m) % m := by rfl _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc, mul_comm (inv_mod m a)] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_maps_below (m a : Nat) [NeZero m] : maps_below m (Ginv m a) := G_maps_below m (inv_mod m a) lemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : one_one_below m (G m a) := left_inv_one_one_below (Ginv_left_inv h1) lemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : onto_below m (G m a) := right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a) lemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : perm_below m (G m a) := And.intro (G_maps_below m a) (And.intro (G_one_one_below h1) (G_onto_below h1)) --Permuting a product of congruence classes doesn't change product lemma swap_fst (u v : Nat) : swap u v u = v := by define : swap u v u --Goal : (if u = u then v else if u = v then u else u) = v have h : u = u := by rfl rewrite [if_pos h] rfl done lemma swap_snd (u v : Nat) : swap u v v = u := by define : swap u v v by_cases h1 : v = u · -- Case 1. h1 : v = u rewrite [if_pos h1] show v = u from h1 done · -- Case 2. h1 : v ≠ u rewrite [if_neg h1] have h2 : v = v := by rfl rewrite [if_pos h2] rfl done done lemma swap_other {u v i : Nat} (h1 : i ≠ u) (h2 : i ≠ v) : swap u v i = i := by define : swap u v i rewrite [if_neg h1, if_neg h2] rfl done lemma swap_values (u v i : Nat) : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := by by_cases h1 : i = u · -- Case 1. h1 : i = u apply Or.inl rewrite [h1] show swap u v u = v from swap_fst u v done · -- Case 2. h1 : i ≠ u apply Or.inr by_cases h2 : i = v · -- Case 2.1. h2 : i = v apply Or.inl rewrite [h2] show swap u v v = u from swap_snd u v done · -- Case 2.2. h2 : i ≠ v apply Or.inr show swap u v i = i from swap_other h1 h2 done done done lemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by define fix i : Nat assume h3 : i < n have h4 : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := swap_values u v i by_cases on h4 · -- Case 1. h4 : swap u v i = v rewrite [h4] show v < n from h2 done · -- Case 2. by_cases on h4 · -- Case 2.1. h4 : swap u v i = u rewrite [h4] show u < n from h1 done · -- Case 2.2. h4 : swap u v i = i rewrite [h4] show i < n from h3 done done done lemma swap_swap (u v n : Nat) : ∀ i < n, swap u v (swap u v i) = i := by fix i : Nat assume h : i < n by_cases h1 : i = u · -- Case 1. h1 : i = u rewrite [h1, swap_fst, swap_snd] rfl done · -- Case 2. h1 : i ≠ u by_cases h2 : i = v · -- Case 2.1. h2 : i = v rewrite [h2, swap_snd, swap_fst] rfl done · -- Case 2.2. h2 : i ≠ v rewrite [swap_other h1 h2, swap_other h1 h2] rfl done done done lemma swap_one_one_below (u v n) : one_one_below n (swap u v) := left_inv_one_one_below (swap_swap u v n) lemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) := right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2) lemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) := And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2)) lemma comp_perm_below {n : Nat} {f g : Nat → Nat} (h1 : perm_below n f) (h2 : perm_below n g) : perm_below n (f ∘ g) := sorry lemma trivial_swap (u : Nat) : swap u u = id := by apply funext fix x : Nat by_cases h1 : x = u · -- Case 1. h1 : x = u rewrite [h1, swap_fst] rfl done · -- Case 2. h1 : x ≠ u rewrite [swap_other h1 h1] rfl done done lemma prod_eq_fun {m : Nat} (f g : Nat → ZMod m) (k : Nat) : ∀ (n : Nat), (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g := by by_induc · -- Base Case assume h : (∀ i < 0, f (k + i) = g (k + i)) rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g assume h1 : ∀ i < n + 1, f (k + i) = g (k + i) have h2 : ∀ i < n, f (k + i) = g (k + i) := by fix i : Nat assume h2 : i < n have h3 : i < n + 1 := by linarith show f (k + i) = g (k + i) from h1 i h3 done have h3 : prod_seq n k f = prod_seq n k g := ih h2 have h4 : n < n + 1 := Nat.lt_succ_self n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4] rfl done done lemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := by have h2 : ∀ (i : Nat), i < u → (f ∘ swap u n) (0 + i) = f (0 + i) := by fix i : Nat assume h2 : i < u have h3 : 0 + i ≠ u := by linarith have h4 : 0 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl done show prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f from prod_eq_fun (f ∘ swap u n) f 0 u h2 done lemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := by have h2 : ∀ i < j, (f ∘ swap u n) (u + 1 + i) = f (u + 1 + i) := by fix i : Nat assume h2 : i < j have h3 : u + 1 + i ≠ u := by linarith have h4 : u + 1 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl show prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f from prod_eq_fun (f ∘ swap u n) f (u + 1) j h2 done lemma break_prod {m : Nat} (n : Nat) (f : Nat → ZMod m) : ∀ (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by by_induc · -- Base Case have h : n + 0 = n := by rfl rewrite [prod_seq_base, h, Theorem_7_3_6_7] rfl done · -- Induction Step fix j : Nat assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f rewrite [←add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc] rfl done done lemma break_prod_twice {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f := break_prod n f 1 rewrite [prod_one] at h2 have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f := break_prod (u + 1) f j rewrite [←h1] at h3 have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f := break_prod u f 1 rewrite [prod_one] at h4 rewrite [h3, h4] at h2 show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2 done

lemma swap_prod_eq_prod {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f

HTPI.swap_prod_eq_prod

htpi/HTPILib/Chap7.lean

htpi

/- Copyright 2023 Daniel J. Velleman -/ import HTPILib.Chap6 namespace HTPI /- Definitions -/ lemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by have h : n + 1 > 0 := Nat.succ_pos n show a % (n + 1) < n + 1 from Nat.mod_lt a h done def gcd (a b : Nat) : Nat := match b with | 0 => a | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd (n + 1) (a % (n + 1)) termination_by b mutual def gcd_c1 (a b : Nat) : Int := match b with | 0 => 1 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c2 (n + 1) (a % (n + 1)) --Corresponds to s = t' termination_by b def gcd_c2 (a b : Nat) : Int := match b with | 0 => 0 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c1 (n + 1) (a % (n + 1)) - (gcd_c2 (n + 1) (a % (n + 1))) * ↑(a / (n + 1)) --Corresponds to t = s' - t'q termination_by b end def prime (n : Nat) : Prop := 2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n def prime_factor (p n : Nat) : Prop := prime p ∧ p ∣ n def all_prime (l : List Nat) : Prop := ∀ p ∈ l, prime p def nondec (l : List Nat) : Prop := match l with | [] => True --Of course, True is a proposition that is always true | n :: L => (∀ m ∈ L, n ≤ m) ∧ nondec L def nondec_prime_list (l : List Nat) : Prop := all_prime l ∧ nondec l def prod (l : List Nat) : Nat := match l with | [] => 1 | n :: L => n * (prod L) def prime_factorization (n : Nat) (l : List Nat) : Prop := nondec_prime_list l ∧ prod l = n def rel_prime (a b : Nat) : Prop := gcd a b = 1 def congr_mod (m : Nat) (a b : Int) : Prop := (↑m : Int) ∣ (a - b) def cc (m : Nat) (a : Int) : ZMod m := (↑a : ZMod m) notation:50 a " ≡ " b " (MOD " m ")" => congr_mod m a b notation:max "["a"]_"m:max => cc m a def invertible {m : Nat} (X : ZMod m) : Prop := ∃ (Y : ZMod m), X * Y = [1]_m def num_rp_below (m k : Nat) : Nat := match k with | 0 => 0 | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j def phi (m : Nat) : Nat := num_rp_below m m def prod_seq {m : Nat} (j k : Nat) (f : Nat → ZMod m) : ZMod m := match j with | 0 => [1]_m | n + 1 => prod_seq n k f * f (k + n) def maps_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i < n, g i < n def one_one_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i1 < n, ∀ i2 < n, g i1 = g i2 → i1 = i2 def onto_below (n : Nat) (g : Nat → Nat) : Prop := ∀ k < n, ∃ i < n, g i = k def perm_below (n : Nat) (g : Nat → Nat) : Prop := maps_below n g ∧ one_one_below n g ∧ onto_below n g def inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m) def swap (u v i : Nat) : Nat := if i = u then v else if i = v then u else i namespace Euler --For definitions specific to Euler's theorem def F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m def G (m a i : Nat) : Nat := (a * i) % m def Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i end Euler /- Section 7.1 -/ theorem dvd_mod_of_dvd_a_b {a b d : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ (a % b) := by set q : Nat := a / b have h3 : b * q + a % b = a := Nat.div_add_mod a b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 define --Goal : ∃ (c : Nat), a % b = d * c apply Exists.intro (j - k * q) show a % b = d * (j - k * q) from calc a % b _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm _ = a - b * q := by rw [h3] _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc] _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm done theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry #eval gcd 672 161 --Answer: 7 lemma gcd_base (a : Nat) : gcd a 0 = a := by rfl lemma gcd_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd a b = gcd b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1)) rfl done lemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b ≠ 0) : a % b < b := by have h1 : b > 0 := Nat.pos_of_ne_zero h show a % b < b from Nat.mod_lt a h1 done lemma dvd_self (n : Nat) : n ∣ n := by apply Exists.intro 1 ring done theorem gcd_dvd : ∀ (b a : Nat), (gcd a b) ∣ a ∧ (gcd a b) ∣ b := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd a b_1) ∣ a ∧ (gcd a b_1) ∣ b_1 fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_base] --Goal: a ∣ a ∧ a ∣ 0 apply And.intro (dvd_self a) define apply Exists.intro 0 rfl done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_nonzero a h1] --Goal : gcd b (a % b) ∣ a ∧ gcd b (a % b) ∣ b have h2 : a % b < b := mod_nonzero_lt a h1 have h3 : (gcd b (a % b)) ∣ b ∧ (gcd b (a % b)) ∣ (a % b) := ih (a % b) h2 b apply And.intro _ h3.left show (gcd b (a % b)) ∣ a from dvd_a_of_dvd_b_mod h3.left h3.right done done theorem gcd_dvd_left (a b : Nat) : (gcd a b) ∣ a := (gcd_dvd b a).left theorem gcd_dvd_right (a b : Nat) : (gcd a b) ∣ b := (gcd_dvd b a).right lemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl lemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c1 a b = gcd_c2 b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done lemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl lemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * ↑(a / b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done theorem gcd_lin_comb : ∀ (b a : Nat), (gcd_c1 a b) * ↑a + (gcd_c2 a b) * ↑b = ↑(gcd a b) := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd_c1 a b_1) * ↑a + (gcd_c2 a b_1) * ↑b_1 = ↑(gcd a b_1) fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base] --Goal : 1 * ↑a + 0 * ↑0 = ↑a ring done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1] --Goal : gcd_c2 b (a % b) * ↑a + -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * ↑(a / b)) * ↑b = -- ↑(gcd b (a % b)) set r : Nat := a % b set q : Nat := a / b set s : Int := gcd_c1 b r set t : Int := gcd_c2 b r --Goal : t * ↑a + (s - t * ↑q) * ↑b = ↑(gcd b r) have h2 : r < b := mod_nonzero_lt a h1 have h3 : s * ↑b + t * ↑r = ↑(gcd b r) := ih r h2 b have h4 : b * q + r = a := Nat.div_add_mod a b rewrite [←h3, ←h4] rewrite [Nat.cast_add, Nat.cast_mul] --Goal : t * (↑b * ↑q + ↑r) + (s - t * ↑q) * ↑b = s * ↑b + t * ↑r ring done done #eval gcd_c1 672 161 --Answer: 6 #eval gcd_c2 672 161 --Answer: -25 --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161 theorem Theorem_7_1_6 {d a b : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ gcd a b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑d ∣ ↑(gcd a b) set s : Int := gcd_c1 a b set t : Int := gcd_c2 a b have h3 : s * ↑a + t * ↑b = ↑(gcd a b) := gcd_lin_comb b a rewrite [←h3] --Goal : ↑d ∣ s * ↑a + t * ↑b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul] --Goal : ↑d ∣ s * (↑d * ↑j) + t * (↑d * ↑k) define apply Exists.intro (s * ↑j + t * ↑k) ring done /- Section 7.2 -/ theorem dvd_trans {a b c : Nat} (h1 : a ∣ b) (h2 : b ∣ c) : a ∣ c := by define at h1; define at h2; define obtain (m : Nat) (h3 : b = a * m) from h1 obtain (n : Nat) (h4 : c = b * n) from h2 rewrite [h3, mul_assoc] at h4 apply Exists.intro (m * n) show c = a * (m * n) from h4 done lemma exists_prime_factor : ∀ (n : Nat), 2 ≤ n → ∃ (p : Nat), prime_factor p n := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, 2 ≤ n_1 → ∃ (p : Nat), prime_factor p n_1 assume h1 : 2 ≤ n by_cases h2 : prime n · -- Case 1. h2 : prime n apply Exists.intro n define --Goal : prime n ∧ n ∣ n show prime n ∧ n ∣ n from And.intro h2 (dvd_self n) done · -- Case 2. h2 : ¬prime n define at h2 --h2 : ¬(2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n) demorgan at h2 disj_syll h2 h1 obtain (a : Nat) (h3 : ∃ (b : Nat), a * b = n ∧ a < n ∧ b < n) from h2 obtain (b : Nat) (h4 : a * b = n ∧ a < n ∧ b < n) from h3 have h5 : 2 ≤ a := by by_contra h6 have h7 : a ≤ 1 := by linarith have h8 : n ≤ b := calc n _ = a * b := h4.left.symm _ ≤ 1 * b := by rel [h7] _ = b := by ring linarith --n ≤ b contradicts b < n done have h6 : ∃ (p : Nat), prime_factor p a := ih a h4.right.left h5 obtain (p : Nat) (h7 : prime_factor p a) from h6 apply Exists.intro p define --Goal : prime p ∧ p ∣ n define at h7 --h7 : prime p ∧ p ∣ a apply And.intro h7.left have h8 : a ∣ n := by apply Exists.intro b show n = a * b from (h4.left).symm done show p ∣ n from dvd_trans h7.right h8 done done lemma exists_least_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q := by set S : Set Nat := {p : Nat | prime_factor p n} have h2 : ∃ (p : Nat), p ∈ S := exists_prime_factor n h show ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q from well_ord_princ S h2 done lemma all_prime_nil : all_prime [] := by define --Goal : ∀ p ∈ [], prime p fix p : Nat contrapos --Goal : ¬prime p → p ∉ [] assume h1 : ¬prime p show p ∉ [] from List.not_mem_nil p done lemma all_prime_cons (n : Nat) (L : List Nat) : all_prime (n :: L) ↔ prime n ∧ all_prime L := by apply Iff.intro · -- (→) assume h1 : all_prime (n :: L) --Goal : prime n ∧ all_prime L define at h1 --h1 : ∀ p ∈ n :: L, prime p apply And.intro (h1 n (List.mem_cons_self n L)) define --Goal : ∀ p ∈ L, prime p fix p : Nat assume h2 : p ∈ L show prime p from h1 p (List.mem_cons_of_mem n h2) done · -- (←) assume h1 : prime n ∧ all_prime L --Goal : all_prime (n :: l) define : all_prime L at h1 define fix p : Nat assume h2 : p ∈ n :: L rewrite [List.mem_cons] at h2 --h2 : p = n ∨ p ∈ L by_cases on h2 · -- Case 1. h2 : p = n rewrite [h2] show prime n from h1.left done · -- Case 2. h2 : p ∈ L show prime p from h1.right p h2 done done done lemma nondec_nil : nondec [] := by define --Goal : True trivial --trivial proves some obviously true statements, such as True done lemma nondec_cons (n : Nat) (L : List Nat) : nondec (n :: L) ↔ (∀ m ∈ L, n ≤ m) ∧ nondec L := by rfl lemma prod_nil : prod [] = 1 := by rfl lemma prod_cons : prod (n :: L) = n * (prod L) := by rfl lemma exists_cons_of_length_eq_succ {A : Type} {l : List A} {n : Nat} (h : l.length = n + 1) : ∃ (a : A) (L : List A), l = a :: L ∧ L.length = n := by have h1 : ¬l.length = 0 := by linarith rewrite [List.length_eq_zero] at h1 obtain (a : A) (h2 : ∃ (L : List A), l = a :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List A) (h3 : l = a :: L) from h2 apply Exists.intro a apply Exists.intro L apply And.intro h3 have h4 : (a :: L).length = L.length + 1 := List.length_cons a L rewrite [←h3, h] at h4 show L.length = n from (Nat.add_right_cancel h4).symm done lemma list_elt_dvd_prod_by_length (a : Nat) : ∀ (n : Nat), ∀ (l : List Nat), l.length = n → a ∈ l → a ∣ prod l := by by_induc · --Base Case fix l : List Nat assume h1 : l.length = 0 rewrite [List.length_eq_zero] at h1 --h1 : l = [] rewrite [h1] --Goal : a ∈ [] → a ∣ prod [] contrapos assume h2 : ¬a ∣ prod [] show a ∉ [] from List.not_mem_nil a done · -- Induction Step fix n : Nat assume ih : ∀ (l : List Nat), List.length l = n → a ∈ l → a ∣ prod l fix l : List Nat assume h1 : l.length = n + 1 --Goal : a ∈ l → a ∣ prod l obtain (b : Nat) (h2 : ∃ (L : List Nat), l = b :: L ∧ L.length = n) from exists_cons_of_length_eq_succ h1 obtain (L : List Nat) (h3 : l = b :: L ∧ L.length = n) from h2 have h4 : a ∈ L → a ∣ prod L := ih L h3.right assume h5 : a ∈ l rewrite [h3.left, prod_cons] --Goal : a ∣ b * prod L rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b ∨ a ∈ L by_cases on h5 · -- Case 1. h5 : a = b apply Exists.intro (prod L) rewrite [h5] rfl done · -- Case 2. h5 : a ∈ L have h6 : a ∣ prod L := h4 h5 have h7 : prod L ∣ b * prod L := by apply Exists.intro b ring done show a ∣ b * prod L from dvd_trans h6 h7 done done done lemma list_elt_dvd_prod {a : Nat} {l : List Nat} (h : a ∈ l) : a ∣ prod l := by set n : Nat := l.length have h1 : l.length = n := by rfl show a ∣ prod l from list_elt_dvd_prod_by_length a n l h1 h done lemma exists_prime_factorization : ∀ (n : Nat), n ≥ 1 → ∃ (l : List Nat), prime_factorization n l := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, n_1 ≥ 1 → ∃ (l : List Nat), prime_factorization n_1 l assume h1 : n ≥ 1 by_cases h2 : n = 1 · -- Case 1. h2 : n = 1 apply Exists.intro [] define apply And.intro · -- Proof of nondec_prime_list [] define show all_prime [] ∧ nondec [] from And.intro all_prime_nil nondec_nil done · -- Proof of prod [] = n rewrite [prod_nil, h2] rfl done done · -- Case 2. h2 : n ≠ 1 have h3 : n ≥ 2 := lt_of_le_of_ne' h1 h2 obtain (p : Nat) (h4 : prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q) from exists_least_prime_factor h3 have p_prime_factor : prime_factor p n := h4.left define at p_prime_factor have p_prime : prime p := p_prime_factor.left have p_dvd_n : p ∣ n := p_prime_factor.right have p_least : ∀ (q : Nat), prime_factor q n → p ≤ q := h4.right obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n have h5 : m ≠ 0 := by contradict h1 with h6 have h7 : n = 0 := calc n _ = p * m := n_eq_pm _ = p * 0 := by rw [h6] _ = 0 := by ring rewrite [h7] decide done have m_pos : 0 < m := Nat.pos_of_ne_zero h5 have m_lt_n : m < n := by define at p_prime show m < n from calc m _ < m + m := by linarith _ = 2 * m := by ring _ ≤ p * m := by rel [p_prime.left] _ = n := n_eq_pm.symm done obtain (L : List Nat) (h6 : prime_factorization m L) from ih m m_lt_n m_pos define at h6 have ndpl_L : nondec_prime_list L := h6.left define at ndpl_L apply Exists.intro (p :: L) define apply And.intro · -- Proof of nondec_prime_list (p :: L) define apply And.intro · -- Proof of all_prime (p :: L) rewrite [all_prime_cons] show prime p ∧ all_prime L from And.intro p_prime ndpl_L.left done · -- Proof of nondec (p :: L) rewrite [nondec_cons] apply And.intro _ ndpl_L.right fix q : Nat assume q_in_L : q ∈ L have h7 : q ∣ prod L := list_elt_dvd_prod q_in_L rewrite [h6.right] at h7 --h7 : q ∣ m have h8 : m ∣ n := by apply Exists.intro p rewrite [n_eq_pm] ring done have q_dvd_n : q ∣ n := dvd_trans h7 h8 have ap_L : all_prime L := ndpl_L.left define at ap_L have q_prime_factor : prime_factor q n := And.intro (ap_L q q_in_L) q_dvd_n show p ≤ q from p_least q q_prime_factor done done · -- Proof of prod (p :: L) = n rewrite [prod_cons, h6.right, n_eq_pm] rfl done done done theorem Theorem_7_2_2 {a b c : Nat} (h1 : c ∣ a * b) (h2 : rel_prime a c) : c ∣ b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑c ∣ ↑b define at h1; define at h2; define obtain (j : Nat) (h3 : a * b = c * j) from h1 set s : Int := gcd_c1 a c set t : Int := gcd_c2 a c have h4 : s * ↑a + t * ↑c = ↑(gcd a c) := gcd_lin_comb c a rewrite [h2, Nat.cast_one] at h4 --h4 : s * ↑a + t * ↑c = (1 : Int) apply Exists.intro (s * ↑j + t * ↑b) show ↑b = ↑c * (s * ↑j + t * ↑b) from calc ↑b _ = (1 : Int) * ↑b := (one_mul _).symm _ = (s * ↑a + t * ↑c) * ↑b := by rw [h4] _ = s * (↑a * ↑b) + t * ↑c * ↑b := by ring _ = s * (↑c * ↑j) + t * ↑c * ↑b := by rw [←Nat.cast_mul a b, h3, Nat.cast_mul c j] _ = ↑c * (s * ↑j + t * ↑b) := by ring done lemma le_nonzero_prod_left {a b : Nat} (h : a * b ≠ 0) : a ≤ a * b := by have h1 : b ≠ 0 := by contradict h with h1 rewrite [h1] ring done have h2 : 1 ≤ b := Nat.pos_of_ne_zero h1 show a ≤ a * b from calc a = a * 1 := (mul_one a).symm _ ≤ a * b := by rel [h2] done lemma le_nonzero_prod_right {a b : Nat} (h : a * b ≠ 0) : b ≤ a * b := by rewrite [mul_comm] rewrite [mul_comm] at h show b ≤ b * a from le_nonzero_prod_left h done lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry lemma rel_prime_of_prime_not_dvd {a p : Nat} (h1 : prime p) (h2 : ¬p ∣ a) : rel_prime a p := by have h3 : gcd a p ∣ a := gcd_dvd_left a p have h4 : gcd a p ∣ p := gcd_dvd_right a p have h5 : gcd a p = 1 ∨ gcd a p = p := dvd_prime h1 h4 have h6 : gcd a p ≠ p := by contradict h2 with h6 rewrite [h6] at h3 show p ∣ a from h3 done disj_syll h5 h6 show rel_prime a p from h5 done theorem Theorem_7_2_3 {a b p : Nat} (h1 : prime p) (h2 : p ∣ a * b) : p ∣ a ∨ p ∣ b := by or_right with h3 have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3 show p ∣ b from Theorem_7_2_2 h2 h4 done lemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a ≥ 1 := by have h1 : a ≠ 0 := by by_contra h1 rewrite [h1] at h contradict h linarith done show a ≥ 1 from Nat.pos_of_ne_zero h1 done lemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by have h1 : a ≥ 1 := ge_one_of_prod_one h have h2 : a * b ≠ 0 := by linarith have h3 : a ≤ a * b := le_nonzero_prod_left h2 rewrite [h] at h3 show a = 1 from Nat.le_antisymm h3 h1 done lemma eq_one_of_dvd_one {n : Nat} (h : n ∣ 1) : n = 1 := by obtain (j : Nat) (h1 : 1 = n * j) from h show n = 1 from eq_one_of_prod_one h1.symm done lemma prime_not_one {p : Nat} (h : prime p) : p ≠ 1 := by define at h linarith done theorem Theorem_7_2_4 {p : Nat} (h1 : prime p) : ∀ (l : List Nat), p ∣ prod l → ∃ a ∈ l, p ∣ a := by apply List.rec · -- Base Case. Goal : p ∣ prod [] → ∃ a ∈ [], p ∣ a rewrite [prod_nil] assume h2 : p ∣ 1 show ∃ a ∈ [], p ∣ a from absurd (eq_one_of_dvd_one h2) (prime_not_one h1) done · -- Induction Step fix b : Nat fix L : List Nat assume ih : p ∣ prod L → ∃ a ∈ L, p ∣ a --Goal : p ∣ prod (b :: L) → ∃ a ∈ b :: L, p ∣ a assume h2 : p ∣ prod (b :: L) rewrite [prod_cons] at h2 have h3 : p ∣ b ∨ p ∣ prod L := Theorem_7_2_3 h1 h2 by_cases on h3 · -- Case 1. h3 : p ∣ b apply Exists.intro b show b ∈ b :: L ∧ p ∣ b from And.intro (List.mem_cons_self b L) h3 done · -- Case 2. h3 : p ∣ prod L obtain (a : Nat) (h4 : a ∈ L ∧ p ∣ a) from ih h3 apply Exists.intro a show a ∈ b :: L ∧ p ∣ a from And.intro (List.mem_cons_of_mem b h4.left) h4.right done done done lemma prime_in_list {p : Nat} {l : List Nat} (h1 : prime p) (h2 : all_prime l) (h3 : p ∣ prod l) : p ∈ l := by obtain (a : Nat) (h4 : a ∈ l ∧ p ∣ a) from Theorem_7_2_4 h1 l h3 define at h2 have h5 : prime a := h2 a h4.left have h6 : p = 1 ∨ p = a := dvd_prime h5 h4.right disj_syll h6 (prime_not_one h1) rewrite [h6] show a ∈ l from h4.left done lemma first_le_first {p q : Nat} {l m : List Nat} (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m)) (h3 : prod (p :: l) = prod (q :: m)) : p ≤ q := by define at h1; define at h2 have h4 : q ∣ prod (p :: l) := by define apply Exists.intro (prod m) rewrite [←prod_cons] show prod (p :: l) = prod (q :: m) from h3 done have h5 : all_prime (q :: m) := h2.left rewrite [all_prime_cons] at h5 have h6 : q ∈ p :: l := prime_in_list h5.left h1.left h4 have h7 : nondec (p :: l) := h1.right rewrite [nondec_cons] at h7 rewrite [List.mem_cons] at h6 by_cases on h6 · -- Case 1. h6 : q = p linarith done · -- Case 2. h6 : q ∈ l have h8 : ∀ m ∈ l, p ≤ m := h7.left show p ≤ q from h8 q h6 done done lemma nondec_prime_list_tail {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by define at h define rewrite [all_prime_cons, nondec_cons] at h show all_prime l ∧ nondec l from And.intro h.left.right h.right.right done lemma cons_prod_not_one {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : prod (p :: l) ≠ 1 := by define at h have h1 : all_prime (p :: l) := h.left rewrite [all_prime_cons] at h1 rewrite [prod_cons] by_contra h2 show False from (prime_not_one h1.left) (eq_one_of_prod_one h2) done lemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) : l = [] ↔ prod l = 1 := by apply Iff.intro · -- (→) assume h1 : l = [] rewrite [h1] show prod [] = 1 from prod_nil done · -- (←) contrapos assume h1 : ¬l = [] obtain (p : Nat) (h2 : ∃ (L : List Nat), l = p :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List Nat) (h3 : l = p :: L) from h2 rewrite [h3] at h rewrite [h3] show ¬prod (p :: L) = 1 from cons_prod_not_one h done done lemma prime_pos {p : Nat} (h : prime p) : p > 0 := by define at h linarith done theorem Theorem_7_2_5 : ∀ (l1 l2 : List Nat), nondec_prime_list l1 → nondec_prime_list l2 → prod l1 = prod l2 → l1 = l2 := by apply List.rec · -- Base Case. Goal : ∀ (l2 : List Nat), nondec_prime_list [] → -- nondec_prime_list l2 → prod [] = prod l2 → [] = l2 fix l2 : List Nat assume h1 : nondec_prime_list [] assume h2 : nondec_prime_list l2 assume h3 : prod [] = prod l2 rewrite [prod_nil, eq_comm, ←list_nil_iff_prod_one h2] at h3 show [] = l2 from h3.symm done · -- Induction Step fix p : Nat fix L1 : List Nat assume ih : ∀ (L2 : List Nat), nondec_prime_list L1 → nondec_prime_list L2 → prod L1 = prod L2 → L1 = L2 -- Goal : ∀ (l2 : List Nat), nondec_prime_list (p :: L1) → -- nondec_prime_list l2 → prod (p :: L1) = prod l2 → p :: L1 = l2 fix l2 : List Nat assume h1 : nondec_prime_list (p :: L1) assume h2 : nondec_prime_list l2 assume h3 : prod (p :: L1) = prod l2 have h4 : ¬prod (p :: L1) = 1 := cons_prod_not_one h1 rewrite [h3, ←list_nil_iff_prod_one h2] at h4 obtain (q : Nat) (h5 : ∃ (L : List Nat), l2 = q :: L) from List.exists_cons_of_ne_nil h4 obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5 rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2) rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2) have h7 : p ≤ q := first_le_first h1 h2 h3 have h8 : q ≤ p := first_le_first h2 h1 h3.symm have h9 : p = q := by linarith rewrite [h9, prod_cons, prod_cons] at h3 --h3 : q * prod L1 = q * prod L2 have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1 have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2 define at h2 have h12 : all_prime (q :: L2) := h2.left rewrite [all_prime_cons] at h12 have h13 : q > 0 := prime_pos h12.left have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3 have h15 : L1 = L2 := ih L2 h10 h11 h14 rewrite [h6, h9, h15] rfl done done theorem fund_thm_arith (n : Nat) (h : n ≥ 1) : ∃! (l : List Nat), prime_factorization n l := by exists_unique · -- Existence show ∃ (l : List Nat), prime_factorization n l from exists_prime_factorization n h done · -- Uniqueness fix l1 : List Nat; fix l2 : List Nat assume h1 : prime_factorization n l1 assume h2 : prime_factorization n l2 define at h1; define at h2 have h3 : prod l1 = n := h1.right rewrite [←h2.right] at h3 show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3 done done /- Section 7.3 -/ theorem congr_refl (m : Nat) : ∀ (a : Int), a ≡ a (MOD m) := by fix a : Int define --Goal : ∃ (c : Int), a - a = ↑m * c apply Exists.intro 0 ring done theorem congr_symm {m : Nat} : ∀ {a b : Int}, a ≡ b (MOD m) → b ≡ a (MOD m) := by fix a : Int; fix b : Int assume h1 : a ≡ b (MOD m) define at h1 --h1 : ∃ (c : Int), a - b = ↑m * c define --Goal : ∃ (c : Int), b - a = ↑m * c obtain (c : Int) (h2 : a - b = m * c) from h1 apply Exists.intro (-c) show b - a = m * (-c) from calc b - a _ = -(a - b) := by ring _ = -(m * c) := by rw [h2] _ = m * (-c) := by ring done theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry /- Fundamental properties of congruence classes -/ lemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) : X = Y ↔ X.val = Y.val := Fin.ext_iff lemma val_nat_eq_mod (n k : Nat) : ([k]_(n + 1)).val = k % (n + 1) := by rfl lemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl theorem cc_rep {m : Nat} (X : ZMod m) : ∃ (a : Int), X = [a]_m := match m with | 0 => by apply Exists.intro X rfl done | n + 1 => by apply Exists.intro ↑(X.val) have h1 : X.val < n + 1 := Fin.prop X rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1] rfl done theorem add_class (m : Nat) (a b : Int) : [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm theorem mul_class (m : Nat) (a b : Int) : [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm lemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ [a - b]_m = [0]_m := by apply Iff.intro · -- (→) assume h1 : [a]_m = [b]_m have h2 : a - b = a + (-b) := by ring have h3 : b + (-b) = 0 := by ring show [a - b]_m = [0]_m from calc [a - b]_m _ = [a + (-b)]_m := by rw [h2] _ = [a]_m + [-b]_m := by rw [add_class] _ = [b]_m + [-b]_m := by rw [h1] _ = [b + -b]_m := by rw [add_class] _ = [0]_m := by rw [h3] done · -- (←) assume h1 : [a - b]_m = [0]_m have h2 : b + (a - b) = a := by ring have h3 : b + 0 = b := by ring show [a]_m = [b]_m from calc [a]_m _ = [b + (a - b)]_m := by rw [h2] _ = [b]_m + [a - b]_m := by rw [add_class] _ = [b]_m + [0]_m := by rw [h1] _ = [b + 0]_m := by rw [add_class] _ = [b]_m := by rw [h3] done done lemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) : [a]_m = [0]_m → [-a]_m = [0]_m := by assume h1 : [a]_m = [0]_m have h2 : 0 + (-a) = -a := by ring have h3 : a + (-a) = 0 := by ring show [-a]_m = [0]_m from calc [-a]_m _ = [0 + (-a)]_m := by rw [h2] _ = [0]_m + [-a]_m := by rw [add_class] _ = [a]_m + [-a]_m := by rw [h1] _ = [a + (-a)]_m := by rw [add_class] _ = [0]_m := by rw [h3] done lemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) : [-a]_m = [0]_m ↔ [a]_m = [0]_m := by apply Iff.intro _ (cc_neg_zero_of_cc_zero m a) assume h1 : [-a]_m = [0]_m have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1 have h3 : -(-a) = a := by ring rewrite [h3] at h2 show [a]_m = [0]_m from h2 done lemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl lemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m ↔ m ∣ k := match m with | 0 => by have h : (0 : Int) = (↑(0 : Nat) : Int) := by rfl rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj] apply Iff.intro · -- (→) assume h1 : k = 0 rewrite [h1] show 0 ∣ 0 from dvd_self 0 done · -- (←) assume h1 : 0 ∣ k obtain (c : Nat) (h2 : k = 0 * c) from h1 rewrite [h2] ring done done | n + 1 => by rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero] show k % (n + 1) = 0 ↔ n + 1 ∣ k from (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm done lemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m ↔ ↑m ∣ a := by obtain (k : Nat) (h1 : a = ↑k ∨ a = -↑k) from Int.eq_nat_or_neg a by_cases on h1 · -- Case 1. h1: a = ↑k rewrite [h1, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done · -- Case 2. h1: a = -↑k rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done done theorem cc_eq_iff_congr (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ a ≡ b (MOD m) := calc [a]_m = [b]_m _ ↔ [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b _ ↔ ↑m ∣ (a - b) := cc_zero_iff_dvd m (a - b) _ ↔ a ≡ b (MOD m) := by rfl /- End of fundamental properties of congruence classes -/ lemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m := by have h1 : (↑m : Int) ≠ 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m) show 0 ≤ a % m from Int.emod_nonneg a h1 done lemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m) have h2 : (↑m : Int) > 0 := (Nat.cast_pos).rtl h1 show a % m < m from Int.emod_lt_of_pos a h2 done lemma congr_mod_mod (m : Nat) (a : Int) : a ≡ a % m (MOD m) := by define have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m apply Exists.intro (a / m) show a - a % m = m * (a / m) from calc a - (a % m) _ = m * (a / m) + a % m - a % m := by rw [h1] _ = m * (a / m) := by ring done lemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) := And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a)) theorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) : ∃! (r : Int), 0 ≤ r ∧ r < m ∧ a ≡ r (MOD m) := by exists_unique · -- Existence apply Exists.intro (a % m) show 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) from mod_cmpl_res m a done · -- Uniqueness fix r1 : Int; fix r2 : Int assume h1 : 0 ≤ r1 ∧ r1 < m ∧ a ≡ r1 (MOD m) assume h2 : 0 ≤ r2 ∧ r2 < m ∧ a ≡ r2 (MOD m) have h3 : r1 ≡ r2 (MOD m) := congr_trans (congr_symm h1.right.right) h2.right.right obtain (d : Int) (h4 : r1 - r2 = m * d) from h3 have h5 : r1 - r2 < m * 1 := by linarith have h6 : m * (-1) < r1 - r2 := by linarith rewrite [h4] at h5 --h5 : m * d < m * 1 rewrite [h4] at h6 --h6 : m * -1 < m * d have h7 : (↑m : Int) ≥ 0 := Nat.cast_nonneg m have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7 have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7 have h10 : d = 0 := by linarith show r1 = r2 from calc r1 _ = r1 - r2 + r2 := by ring _ = m * 0 + r2 := by rw [h4, h10] _ = r2 := by ring done done lemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m := (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a) theorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y rewrite [h1, h2] have h3 : a + b = b + a := by ring show [a]_m + [b]_m = [b]_m + [a]_m from calc [a]_m + [b]_m _ = [a + b]_m := add_class m a b _ = [b + a]_m := by rw [h3] _ = [b]_m + [a]_m := (add_class m b a).symm done theorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X rewrite [h1] have h2 : a * 1 = a := by ring show [a]_m * [1]_m = [a]_m from calc [a]_m * [1]_m _ = [a * 1]_m := mul_class m a 1 _ = [a]_m := by rw [h2] done theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry lemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) : [a]_m * [gcd_c2 m a]_m = [1]_m := by set s : Int := gcd_c1 m a have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m define at h1 rewrite [h1, Nat.cast_one] at h2 --h2 : s * ↑m + gcd_c2 m a * ↑a = 1 rewrite [mul_class, cc_eq_iff_congr] define --Goal : ∃ (c : Int), ↑a * gcd_c2 m a - 1 = ↑m * c apply Exists.intro (-s) show a * (gcd_c2 m a) - 1 = m * (-s) from calc a * (gcd_c2 m a) - 1 _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring _ = 1 + m * (-s) - 1 := by rw [h2] _ = m * (-s) := by ring done theorem Theorem_7_3_7 (m a : Nat) : invertible [a]_m ↔ rel_prime m a := by apply Iff.intro · -- (→) assume h1 : invertible [a]_m define at h1 obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1 obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y rewrite [h3, mul_class, cc_eq_iff_congr] at h2 define at h2 obtain (c : Int) (h4 : a * b - 1 = m * c) from h2 rewrite [Exercise_7_2_6] --Goal : ∃ (s t : Int), s * ↑m + t * ↑a = 1 apply Exists.intro (-c) apply Exists.intro b show (-c) * m + b * a = 1 from calc (-c) * m + b * a _ = (-c) * m + (a * b - 1) + 1 := by ring _ = (-c) * m + m * c + 1 := by rw [h4] _ = 1 := by ring done · -- (←) assume h1 : rel_prime m a define show ∃ (Y : ZMod m), [a]_m * Y = [1]_m from Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1) done done /- Section 7.4 -/ section Euler open Euler lemma num_rp_below_base {m : Nat} : num_rp_below m 0 = 0 := by rfl lemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) : num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by have h1 : num_rp_below m (j + 1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : gcd m j = 1 rewrite [if_pos h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1 show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1 done lemma num_rp_below_step_not_rp {m j : Nat} (h : ¬rel_prime m j) : num_rp_below m (j + 1) = num_rp_below m j := by have h1 : num_rp_below m (j +1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : ¬gcd m j = 1 rewrite [if_neg h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j show num_rp_below m (j + 1) = num_rp_below m j from h1 done lemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl #eval phi 10 --Answer: 4 lemma prod_inv_iff_inv {m : Nat} {X : ZMod m} (h1 : invertible X) (Y : ZMod m) : invertible (X * Y) ↔ invertible Y := by apply Iff.intro · -- (→) assume h2 : invertible (X * Y) obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2 apply Exists.intro (X * Z) rewrite [←h3] --Goal : Y * (X * Z) = X * Y * Z ring --Note that ring can do algebra in ZMod m done · -- (←) assume h2 : invertible Y obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1 obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2 apply Exists.intro (Xi * Yi) show (X * Y) * (Xi * Yi) = [1]_m from calc X * Y * (Xi * Yi) _ = (X * Xi) * (Y * Yi) := by ring _ = [1]_m * [1]_m := by rw [h3, h4] _ = [1]_m := Theorem_7_3_6_7 [1]_m done done lemma F_rp_def {m i : Nat} (h : rel_prime m i) : F m i = [i]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h --h : gcd m i = 1 rewrite [if_pos h] at h1 show F m i = [i]_m from h1 done lemma F_not_rp_def {m i : Nat} (h : ¬rel_prime m i) : F m i = [1]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h rewrite [h1, if_neg h] rfl done lemma prod_seq_base {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 0 k f = [1]_m := by rfl lemma prod_seq_step {m : Nat} (n k : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl lemma prod_seq_zero_step {m : Nat} (n : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by rewrite [prod_seq_step, zero_add] rfl done lemma prod_one {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 1 k f = f k := by rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7] rfl done lemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl lemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m := calc [G m a i]_m _ = [(a * i) % m]_m := by rfl _ = [a * i]_m := (cc_eq_mod m (a * i)).symm _ = [a]_m * [i]_m := (mul_class m a i).symm lemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) : rel_prime m (G m a i) ↔ rel_prime m i := by have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1 show rel_prime m (G m a i) ↔ rel_prime m i from calc rel_prime m (G m a i) _ ↔ invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm _ ↔ invertible ([a]_m * [i]_m) := by rw [cc_G] _ ↔ invertible [i]_m := prod_inv_iff_inv h2 ([i]_m) _ ↔ rel_prime m i := Theorem_7_3_7 m i done lemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) : F m (G m a i) = [a]_m * F m i := by have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2 show F m (G m a i) = [a]_m * F m i from calc F m (G m a i) _ = [G m a i]_m := F_rp_def h3 _ = [a]_m * [i]_m := cc_G m a i _ = [a]_m * F m i := by rw [F_rp_def h2] done lemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : ¬rel_prime m i) : F m (G m a i) = [1]_m := by rewrite [←G_rp_iff h1 i] at h2 show F m (G m a i) = [1]_m from F_not_rp_def h2 done lemma FG_prod {m a : Nat} (h1 : rel_prime m a) : ∀ (k : Nat), prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by by_induc · -- Base Case show prod_seq 0 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from calc prod_seq 0 0 ((F m) ∘ (G m a)) _ = [1]_m := prod_seq_base _ _ _ = [a]_m ^ 0 * [1]_m := by ring _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by rw [num_rp_below_base, prod_seq_base] done · -- Induction Step fix k : Nat assume ih : prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) by_cases h2 : rel_prime m k · -- Case 1. h2 : rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([a]_m * F m k) := by rw [FG_rp h1 h2] _ = [a]_m ^ ((num_rp_below m k) + 1) * ((prod_seq k 0 (F m)) * F m k) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_rp h2, prod_seq_zero_step] done · -- Case 2. h2 : ¬rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([1]_m) := by rw [FG_not_rp h1 h2] _ = [a]_m ^ (num_rp_below m k) * (prod_seq k 0 (F m) * ([1]_m)) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_not_rp h2, prod_seq_zero_step, F_not_rp_def h2] done done done lemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by define --Goal : ∀ i < m, G m a i < m fix i : Nat assume h1 : i < m rewrite [G_def] --Goal : a * i % m < m show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m) done lemma left_inv_one_one_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g' (g i) = i) : one_one_below n g := sorry lemma right_inv_onto_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g (g' i) = i) (h2 : maps_below n g') : onto_below n g := by define at h2; define fix k : Nat assume h3 : k < n apply Exists.intro (g' k) show g' k < n ∧ g (g' k) = k from And.intro (h2 k h3) (h1 k h3) done lemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m * [inv_mod m a]_m = [1]_m := by have h2 : 0 ≤ (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a) show [a]_m * [inv_mod m a]_m = [1]_m from calc [a]_m * [inv_mod m a]_m _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2] _ = [a]_m * [gcd_c2 m a]_m := by rw [←cc_eq_mod] _ = [1]_m := gcd_c2_inv h1 done lemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m := calc a * (b % m) % m = a % m * (b % m % m) % m := Nat.mul_mod _ _ _ _ = a % m * (b % m) % m := by rw [Nat.mod_mod] _ = a * b % m := (Nat.mul_mod _ _ _).symm lemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm] rfl done theorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] : ↑a ≡ ↑b (MOD m) ↔ a % m = b % m := sorry lemma mul_inv_mod_cancel {m a i : Nat} [NeZero m] (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1 rewrite [mul_class, cc_eq_iff_congr, ←Nat.cast_mul, ←Nat.cast_one, congr_iff_mod_eq_Nat] at h3 show a * inv_mod m a * i % m = i from calc a * (inv_mod m a) * i % m _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = 1 % m * i % m := by rw [h3] _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = i % m := by rw [one_mul] _ = i := Nat.mod_eq_of_lt h2 done lemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl lemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, G m a (Ginv m a i) = i := by fix i : Nat assume h2 : i < m show G m a (Ginv m a i) = i from calc G m a (Ginv m a i) _ = a * ((inv_mod m a * i) % m) % m := by rfl _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, Ginv m a (G m a i) = i := by fix i : Nat assume h2 : i < m show Ginv m a (G m a i) = i from calc Ginv m a (G m a i) _ = inv_mod m a * ((a * i) % m) % m := by rfl _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc, mul_comm (inv_mod m a)] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_maps_below (m a : Nat) [NeZero m] : maps_below m (Ginv m a) := G_maps_below m (inv_mod m a) lemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : one_one_below m (G m a) := left_inv_one_one_below (Ginv_left_inv h1) lemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : onto_below m (G m a) := right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a) lemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : perm_below m (G m a) := And.intro (G_maps_below m a) (And.intro (G_one_one_below h1) (G_onto_below h1)) --Permuting a product of congruence classes doesn't change product lemma swap_fst (u v : Nat) : swap u v u = v := by define : swap u v u --Goal : (if u = u then v else if u = v then u else u) = v have h : u = u := by rfl rewrite [if_pos h] rfl done lemma swap_snd (u v : Nat) : swap u v v = u := by define : swap u v v by_cases h1 : v = u · -- Case 1. h1 : v = u rewrite [if_pos h1] show v = u from h1 done · -- Case 2. h1 : v ≠ u rewrite [if_neg h1] have h2 : v = v := by rfl rewrite [if_pos h2] rfl done done lemma swap_other {u v i : Nat} (h1 : i ≠ u) (h2 : i ≠ v) : swap u v i = i := by define : swap u v i rewrite [if_neg h1, if_neg h2] rfl done lemma swap_values (u v i : Nat) : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := by by_cases h1 : i = u · -- Case 1. h1 : i = u apply Or.inl rewrite [h1] show swap u v u = v from swap_fst u v done · -- Case 2. h1 : i ≠ u apply Or.inr by_cases h2 : i = v · -- Case 2.1. h2 : i = v apply Or.inl rewrite [h2] show swap u v v = u from swap_snd u v done · -- Case 2.2. h2 : i ≠ v apply Or.inr show swap u v i = i from swap_other h1 h2 done done done lemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by define fix i : Nat assume h3 : i < n have h4 : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := swap_values u v i by_cases on h4 · -- Case 1. h4 : swap u v i = v rewrite [h4] show v < n from h2 done · -- Case 2. by_cases on h4 · -- Case 2.1. h4 : swap u v i = u rewrite [h4] show u < n from h1 done · -- Case 2.2. h4 : swap u v i = i rewrite [h4] show i < n from h3 done done done lemma swap_swap (u v n : Nat) : ∀ i < n, swap u v (swap u v i) = i := by fix i : Nat assume h : i < n by_cases h1 : i = u · -- Case 1. h1 : i = u rewrite [h1, swap_fst, swap_snd] rfl done · -- Case 2. h1 : i ≠ u by_cases h2 : i = v · -- Case 2.1. h2 : i = v rewrite [h2, swap_snd, swap_fst] rfl done · -- Case 2.2. h2 : i ≠ v rewrite [swap_other h1 h2, swap_other h1 h2] rfl done done done lemma swap_one_one_below (u v n) : one_one_below n (swap u v) := left_inv_one_one_below (swap_swap u v n) lemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) := right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2) lemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) := And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2)) lemma comp_perm_below {n : Nat} {f g : Nat → Nat} (h1 : perm_below n f) (h2 : perm_below n g) : perm_below n (f ∘ g) := sorry lemma trivial_swap (u : Nat) : swap u u = id := by apply funext fix x : Nat by_cases h1 : x = u · -- Case 1. h1 : x = u rewrite [h1, swap_fst] rfl done · -- Case 2. h1 : x ≠ u rewrite [swap_other h1 h1] rfl done done lemma prod_eq_fun {m : Nat} (f g : Nat → ZMod m) (k : Nat) : ∀ (n : Nat), (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g := by by_induc · -- Base Case assume h : (∀ i < 0, f (k + i) = g (k + i)) rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g assume h1 : ∀ i < n + 1, f (k + i) = g (k + i) have h2 : ∀ i < n, f (k + i) = g (k + i) := by fix i : Nat assume h2 : i < n have h3 : i < n + 1 := by linarith show f (k + i) = g (k + i) from h1 i h3 done have h3 : prod_seq n k f = prod_seq n k g := ih h2 have h4 : n < n + 1 := Nat.lt_succ_self n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4] rfl done done lemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := by have h2 : ∀ (i : Nat), i < u → (f ∘ swap u n) (0 + i) = f (0 + i) := by fix i : Nat assume h2 : i < u have h3 : 0 + i ≠ u := by linarith have h4 : 0 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl done show prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f from prod_eq_fun (f ∘ swap u n) f 0 u h2 done lemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := by have h2 : ∀ i < j, (f ∘ swap u n) (u + 1 + i) = f (u + 1 + i) := by fix i : Nat assume h2 : i < j have h3 : u + 1 + i ≠ u := by linarith have h4 : u + 1 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl show prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f from prod_eq_fun (f ∘ swap u n) f (u + 1) j h2 done lemma break_prod {m : Nat} (n : Nat) (f : Nat → ZMod m) : ∀ (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by by_induc · -- Base Case have h : n + 0 = n := by rfl rewrite [prod_seq_base, h, Theorem_7_3_6_7] rfl done · -- Induction Step fix j : Nat assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f rewrite [←add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc] rfl done done lemma break_prod_twice {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f := break_prod n f 1 rewrite [prod_one] at h2 have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f := break_prod (u + 1) f j rewrite [←h1] at h3 have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f := break_prod u f 1 rewrite [prod_one] at h4 rewrite [h3, h4] at h2 show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2 done lemma swap_prod_eq_prod {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f := by by_cases h2 : u = n · -- Case 1. h2 : u = n rewrite [h2, trivial_swap n] --Goal : prod_seq (n + 1) 0 (f ∘ id) = prod_seq (n + 1) 0 f rfl done · -- Case 2. h2 : ¬u = n have h3 : u + 1 ≤ n := Nat.lt_of_le_of_ne h1 h2 obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3 have break_f : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := break_prod_twice f h4 have break_fs : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_prod_twice (f ∘ swap u n) h4 have f_eq_fs_below : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := swap_prod_eq_prod_below f h1 have f_eq_fs_btwn : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4 show prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f from calc prod_seq (n + 1) 0 (f ∘ swap u n) _ = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_fs _ = prod_seq u 0 f * (f ∘ swap u n) u * prod_seq j (u + 1) f * (f ∘ swap u n) n := by rw [f_eq_fs_below, f_eq_fs_btwn] _ = prod_seq u 0 f * f (swap u n u) * prod_seq j (u + 1) f * f (swap u n n) := by rfl _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by rw [swap_fst, swap_snd] _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring _ = prod_seq (n + 1) 0 f := break_f.symm done done

lemma perm_below_fixed {n : Nat} {g : Nat → Nat} (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g

HTPI.perm_below_fixed

htpi/HTPILib/Chap7.lean

htpi

/- Copyright 2023 Daniel J. Velleman -/ import HTPILib.Chap6 namespace HTPI /- Definitions -/ lemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by have h : n + 1 > 0 := Nat.succ_pos n show a % (n + 1) < n + 1 from Nat.mod_lt a h done def gcd (a b : Nat) : Nat := match b with | 0 => a | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd (n + 1) (a % (n + 1)) termination_by b mutual def gcd_c1 (a b : Nat) : Int := match b with | 0 => 1 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c2 (n + 1) (a % (n + 1)) --Corresponds to s = t' termination_by b def gcd_c2 (a b : Nat) : Int := match b with | 0 => 0 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c1 (n + 1) (a % (n + 1)) - (gcd_c2 (n + 1) (a % (n + 1))) * ↑(a / (n + 1)) --Corresponds to t = s' - t'q termination_by b end def prime (n : Nat) : Prop := 2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n def prime_factor (p n : Nat) : Prop := prime p ∧ p ∣ n def all_prime (l : List Nat) : Prop := ∀ p ∈ l, prime p def nondec (l : List Nat) : Prop := match l with | [] => True --Of course, True is a proposition that is always true | n :: L => (∀ m ∈ L, n ≤ m) ∧ nondec L def nondec_prime_list (l : List Nat) : Prop := all_prime l ∧ nondec l def prod (l : List Nat) : Nat := match l with | [] => 1 | n :: L => n * (prod L) def prime_factorization (n : Nat) (l : List Nat) : Prop := nondec_prime_list l ∧ prod l = n def rel_prime (a b : Nat) : Prop := gcd a b = 1 def congr_mod (m : Nat) (a b : Int) : Prop := (↑m : Int) ∣ (a - b) def cc (m : Nat) (a : Int) : ZMod m := (↑a : ZMod m) notation:50 a " ≡ " b " (MOD " m ")" => congr_mod m a b notation:max "["a"]_"m:max => cc m a def invertible {m : Nat} (X : ZMod m) : Prop := ∃ (Y : ZMod m), X * Y = [1]_m def num_rp_below (m k : Nat) : Nat := match k with | 0 => 0 | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j def phi (m : Nat) : Nat := num_rp_below m m def prod_seq {m : Nat} (j k : Nat) (f : Nat → ZMod m) : ZMod m := match j with | 0 => [1]_m | n + 1 => prod_seq n k f * f (k + n) def maps_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i < n, g i < n def one_one_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i1 < n, ∀ i2 < n, g i1 = g i2 → i1 = i2 def onto_below (n : Nat) (g : Nat → Nat) : Prop := ∀ k < n, ∃ i < n, g i = k def perm_below (n : Nat) (g : Nat → Nat) : Prop := maps_below n g ∧ one_one_below n g ∧ onto_below n g def inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m) def swap (u v i : Nat) : Nat := if i = u then v else if i = v then u else i namespace Euler --For definitions specific to Euler's theorem def F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m def G (m a i : Nat) : Nat := (a * i) % m def Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i end Euler /- Section 7.1 -/ theorem dvd_mod_of_dvd_a_b {a b d : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ (a % b) := by set q : Nat := a / b have h3 : b * q + a % b = a := Nat.div_add_mod a b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 define --Goal : ∃ (c : Nat), a % b = d * c apply Exists.intro (j - k * q) show a % b = d * (j - k * q) from calc a % b _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm _ = a - b * q := by rw [h3] _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc] _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm done theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry #eval gcd 672 161 --Answer: 7 lemma gcd_base (a : Nat) : gcd a 0 = a := by rfl lemma gcd_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd a b = gcd b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1)) rfl done lemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b ≠ 0) : a % b < b := by have h1 : b > 0 := Nat.pos_of_ne_zero h show a % b < b from Nat.mod_lt a h1 done lemma dvd_self (n : Nat) : n ∣ n := by apply Exists.intro 1 ring done theorem gcd_dvd : ∀ (b a : Nat), (gcd a b) ∣ a ∧ (gcd a b) ∣ b := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd a b_1) ∣ a ∧ (gcd a b_1) ∣ b_1 fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_base] --Goal: a ∣ a ∧ a ∣ 0 apply And.intro (dvd_self a) define apply Exists.intro 0 rfl done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_nonzero a h1] --Goal : gcd b (a % b) ∣ a ∧ gcd b (a % b) ∣ b have h2 : a % b < b := mod_nonzero_lt a h1 have h3 : (gcd b (a % b)) ∣ b ∧ (gcd b (a % b)) ∣ (a % b) := ih (a % b) h2 b apply And.intro _ h3.left show (gcd b (a % b)) ∣ a from dvd_a_of_dvd_b_mod h3.left h3.right done done theorem gcd_dvd_left (a b : Nat) : (gcd a b) ∣ a := (gcd_dvd b a).left theorem gcd_dvd_right (a b : Nat) : (gcd a b) ∣ b := (gcd_dvd b a).right lemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl lemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c1 a b = gcd_c2 b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done lemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl lemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * ↑(a / b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done theorem gcd_lin_comb : ∀ (b a : Nat), (gcd_c1 a b) * ↑a + (gcd_c2 a b) * ↑b = ↑(gcd a b) := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd_c1 a b_1) * ↑a + (gcd_c2 a b_1) * ↑b_1 = ↑(gcd a b_1) fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base] --Goal : 1 * ↑a + 0 * ↑0 = ↑a ring done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1] --Goal : gcd_c2 b (a % b) * ↑a + -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * ↑(a / b)) * ↑b = -- ↑(gcd b (a % b)) set r : Nat := a % b set q : Nat := a / b set s : Int := gcd_c1 b r set t : Int := gcd_c2 b r --Goal : t * ↑a + (s - t * ↑q) * ↑b = ↑(gcd b r) have h2 : r < b := mod_nonzero_lt a h1 have h3 : s * ↑b + t * ↑r = ↑(gcd b r) := ih r h2 b have h4 : b * q + r = a := Nat.div_add_mod a b rewrite [←h3, ←h4] rewrite [Nat.cast_add, Nat.cast_mul] --Goal : t * (↑b * ↑q + ↑r) + (s - t * ↑q) * ↑b = s * ↑b + t * ↑r ring done done #eval gcd_c1 672 161 --Answer: 6 #eval gcd_c2 672 161 --Answer: -25 --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161 theorem Theorem_7_1_6 {d a b : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ gcd a b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑d ∣ ↑(gcd a b) set s : Int := gcd_c1 a b set t : Int := gcd_c2 a b have h3 : s * ↑a + t * ↑b = ↑(gcd a b) := gcd_lin_comb b a rewrite [←h3] --Goal : ↑d ∣ s * ↑a + t * ↑b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul] --Goal : ↑d ∣ s * (↑d * ↑j) + t * (↑d * ↑k) define apply Exists.intro (s * ↑j + t * ↑k) ring done /- Section 7.2 -/ theorem dvd_trans {a b c : Nat} (h1 : a ∣ b) (h2 : b ∣ c) : a ∣ c := by define at h1; define at h2; define obtain (m : Nat) (h3 : b = a * m) from h1 obtain (n : Nat) (h4 : c = b * n) from h2 rewrite [h3, mul_assoc] at h4 apply Exists.intro (m * n) show c = a * (m * n) from h4 done lemma exists_prime_factor : ∀ (n : Nat), 2 ≤ n → ∃ (p : Nat), prime_factor p n := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, 2 ≤ n_1 → ∃ (p : Nat), prime_factor p n_1 assume h1 : 2 ≤ n by_cases h2 : prime n · -- Case 1. h2 : prime n apply Exists.intro n define --Goal : prime n ∧ n ∣ n show prime n ∧ n ∣ n from And.intro h2 (dvd_self n) done · -- Case 2. h2 : ¬prime n define at h2 --h2 : ¬(2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n) demorgan at h2 disj_syll h2 h1 obtain (a : Nat) (h3 : ∃ (b : Nat), a * b = n ∧ a < n ∧ b < n) from h2 obtain (b : Nat) (h4 : a * b = n ∧ a < n ∧ b < n) from h3 have h5 : 2 ≤ a := by by_contra h6 have h7 : a ≤ 1 := by linarith have h8 : n ≤ b := calc n _ = a * b := h4.left.symm _ ≤ 1 * b := by rel [h7] _ = b := by ring linarith --n ≤ b contradicts b < n done have h6 : ∃ (p : Nat), prime_factor p a := ih a h4.right.left h5 obtain (p : Nat) (h7 : prime_factor p a) from h6 apply Exists.intro p define --Goal : prime p ∧ p ∣ n define at h7 --h7 : prime p ∧ p ∣ a apply And.intro h7.left have h8 : a ∣ n := by apply Exists.intro b show n = a * b from (h4.left).symm done show p ∣ n from dvd_trans h7.right h8 done done lemma exists_least_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q := by set S : Set Nat := {p : Nat | prime_factor p n} have h2 : ∃ (p : Nat), p ∈ S := exists_prime_factor n h show ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q from well_ord_princ S h2 done lemma all_prime_nil : all_prime [] := by define --Goal : ∀ p ∈ [], prime p fix p : Nat contrapos --Goal : ¬prime p → p ∉ [] assume h1 : ¬prime p show p ∉ [] from List.not_mem_nil p done lemma all_prime_cons (n : Nat) (L : List Nat) : all_prime (n :: L) ↔ prime n ∧ all_prime L := by apply Iff.intro · -- (→) assume h1 : all_prime (n :: L) --Goal : prime n ∧ all_prime L define at h1 --h1 : ∀ p ∈ n :: L, prime p apply And.intro (h1 n (List.mem_cons_self n L)) define --Goal : ∀ p ∈ L, prime p fix p : Nat assume h2 : p ∈ L show prime p from h1 p (List.mem_cons_of_mem n h2) done · -- (←) assume h1 : prime n ∧ all_prime L --Goal : all_prime (n :: l) define : all_prime L at h1 define fix p : Nat assume h2 : p ∈ n :: L rewrite [List.mem_cons] at h2 --h2 : p = n ∨ p ∈ L by_cases on h2 · -- Case 1. h2 : p = n rewrite [h2] show prime n from h1.left done · -- Case 2. h2 : p ∈ L show prime p from h1.right p h2 done done done lemma nondec_nil : nondec [] := by define --Goal : True trivial --trivial proves some obviously true statements, such as True done lemma nondec_cons (n : Nat) (L : List Nat) : nondec (n :: L) ↔ (∀ m ∈ L, n ≤ m) ∧ nondec L := by rfl lemma prod_nil : prod [] = 1 := by rfl lemma prod_cons : prod (n :: L) = n * (prod L) := by rfl lemma exists_cons_of_length_eq_succ {A : Type} {l : List A} {n : Nat} (h : l.length = n + 1) : ∃ (a : A) (L : List A), l = a :: L ∧ L.length = n := by have h1 : ¬l.length = 0 := by linarith rewrite [List.length_eq_zero] at h1 obtain (a : A) (h2 : ∃ (L : List A), l = a :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List A) (h3 : l = a :: L) from h2 apply Exists.intro a apply Exists.intro L apply And.intro h3 have h4 : (a :: L).length = L.length + 1 := List.length_cons a L rewrite [←h3, h] at h4 show L.length = n from (Nat.add_right_cancel h4).symm done lemma list_elt_dvd_prod_by_length (a : Nat) : ∀ (n : Nat), ∀ (l : List Nat), l.length = n → a ∈ l → a ∣ prod l := by by_induc · --Base Case fix l : List Nat assume h1 : l.length = 0 rewrite [List.length_eq_zero] at h1 --h1 : l = [] rewrite [h1] --Goal : a ∈ [] → a ∣ prod [] contrapos assume h2 : ¬a ∣ prod [] show a ∉ [] from List.not_mem_nil a done · -- Induction Step fix n : Nat assume ih : ∀ (l : List Nat), List.length l = n → a ∈ l → a ∣ prod l fix l : List Nat assume h1 : l.length = n + 1 --Goal : a ∈ l → a ∣ prod l obtain (b : Nat) (h2 : ∃ (L : List Nat), l = b :: L ∧ L.length = n) from exists_cons_of_length_eq_succ h1 obtain (L : List Nat) (h3 : l = b :: L ∧ L.length = n) from h2 have h4 : a ∈ L → a ∣ prod L := ih L h3.right assume h5 : a ∈ l rewrite [h3.left, prod_cons] --Goal : a ∣ b * prod L rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b ∨ a ∈ L by_cases on h5 · -- Case 1. h5 : a = b apply Exists.intro (prod L) rewrite [h5] rfl done · -- Case 2. h5 : a ∈ L have h6 : a ∣ prod L := h4 h5 have h7 : prod L ∣ b * prod L := by apply Exists.intro b ring done show a ∣ b * prod L from dvd_trans h6 h7 done done done lemma list_elt_dvd_prod {a : Nat} {l : List Nat} (h : a ∈ l) : a ∣ prod l := by set n : Nat := l.length have h1 : l.length = n := by rfl show a ∣ prod l from list_elt_dvd_prod_by_length a n l h1 h done lemma exists_prime_factorization : ∀ (n : Nat), n ≥ 1 → ∃ (l : List Nat), prime_factorization n l := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, n_1 ≥ 1 → ∃ (l : List Nat), prime_factorization n_1 l assume h1 : n ≥ 1 by_cases h2 : n = 1 · -- Case 1. h2 : n = 1 apply Exists.intro [] define apply And.intro · -- Proof of nondec_prime_list [] define show all_prime [] ∧ nondec [] from And.intro all_prime_nil nondec_nil done · -- Proof of prod [] = n rewrite [prod_nil, h2] rfl done done · -- Case 2. h2 : n ≠ 1 have h3 : n ≥ 2 := lt_of_le_of_ne' h1 h2 obtain (p : Nat) (h4 : prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q) from exists_least_prime_factor h3 have p_prime_factor : prime_factor p n := h4.left define at p_prime_factor have p_prime : prime p := p_prime_factor.left have p_dvd_n : p ∣ n := p_prime_factor.right have p_least : ∀ (q : Nat), prime_factor q n → p ≤ q := h4.right obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n have h5 : m ≠ 0 := by contradict h1 with h6 have h7 : n = 0 := calc n _ = p * m := n_eq_pm _ = p * 0 := by rw [h6] _ = 0 := by ring rewrite [h7] decide done have m_pos : 0 < m := Nat.pos_of_ne_zero h5 have m_lt_n : m < n := by define at p_prime show m < n from calc m _ < m + m := by linarith _ = 2 * m := by ring _ ≤ p * m := by rel [p_prime.left] _ = n := n_eq_pm.symm done obtain (L : List Nat) (h6 : prime_factorization m L) from ih m m_lt_n m_pos define at h6 have ndpl_L : nondec_prime_list L := h6.left define at ndpl_L apply Exists.intro (p :: L) define apply And.intro · -- Proof of nondec_prime_list (p :: L) define apply And.intro · -- Proof of all_prime (p :: L) rewrite [all_prime_cons] show prime p ∧ all_prime L from And.intro p_prime ndpl_L.left done · -- Proof of nondec (p :: L) rewrite [nondec_cons] apply And.intro _ ndpl_L.right fix q : Nat assume q_in_L : q ∈ L have h7 : q ∣ prod L := list_elt_dvd_prod q_in_L rewrite [h6.right] at h7 --h7 : q ∣ m have h8 : m ∣ n := by apply Exists.intro p rewrite [n_eq_pm] ring done have q_dvd_n : q ∣ n := dvd_trans h7 h8 have ap_L : all_prime L := ndpl_L.left define at ap_L have q_prime_factor : prime_factor q n := And.intro (ap_L q q_in_L) q_dvd_n show p ≤ q from p_least q q_prime_factor done done · -- Proof of prod (p :: L) = n rewrite [prod_cons, h6.right, n_eq_pm] rfl done done done theorem Theorem_7_2_2 {a b c : Nat} (h1 : c ∣ a * b) (h2 : rel_prime a c) : c ∣ b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑c ∣ ↑b define at h1; define at h2; define obtain (j : Nat) (h3 : a * b = c * j) from h1 set s : Int := gcd_c1 a c set t : Int := gcd_c2 a c have h4 : s * ↑a + t * ↑c = ↑(gcd a c) := gcd_lin_comb c a rewrite [h2, Nat.cast_one] at h4 --h4 : s * ↑a + t * ↑c = (1 : Int) apply Exists.intro (s * ↑j + t * ↑b) show ↑b = ↑c * (s * ↑j + t * ↑b) from calc ↑b _ = (1 : Int) * ↑b := (one_mul _).symm _ = (s * ↑a + t * ↑c) * ↑b := by rw [h4] _ = s * (↑a * ↑b) + t * ↑c * ↑b := by ring _ = s * (↑c * ↑j) + t * ↑c * ↑b := by rw [←Nat.cast_mul a b, h3, Nat.cast_mul c j] _ = ↑c * (s * ↑j + t * ↑b) := by ring done lemma le_nonzero_prod_left {a b : Nat} (h : a * b ≠ 0) : a ≤ a * b := by have h1 : b ≠ 0 := by contradict h with h1 rewrite [h1] ring done have h2 : 1 ≤ b := Nat.pos_of_ne_zero h1 show a ≤ a * b from calc a = a * 1 := (mul_one a).symm _ ≤ a * b := by rel [h2] done lemma le_nonzero_prod_right {a b : Nat} (h : a * b ≠ 0) : b ≤ a * b := by rewrite [mul_comm] rewrite [mul_comm] at h show b ≤ b * a from le_nonzero_prod_left h done lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry lemma rel_prime_of_prime_not_dvd {a p : Nat} (h1 : prime p) (h2 : ¬p ∣ a) : rel_prime a p := by have h3 : gcd a p ∣ a := gcd_dvd_left a p have h4 : gcd a p ∣ p := gcd_dvd_right a p have h5 : gcd a p = 1 ∨ gcd a p = p := dvd_prime h1 h4 have h6 : gcd a p ≠ p := by contradict h2 with h6 rewrite [h6] at h3 show p ∣ a from h3 done disj_syll h5 h6 show rel_prime a p from h5 done theorem Theorem_7_2_3 {a b p : Nat} (h1 : prime p) (h2 : p ∣ a * b) : p ∣ a ∨ p ∣ b := by or_right with h3 have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3 show p ∣ b from Theorem_7_2_2 h2 h4 done lemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a ≥ 1 := by have h1 : a ≠ 0 := by by_contra h1 rewrite [h1] at h contradict h linarith done show a ≥ 1 from Nat.pos_of_ne_zero h1 done lemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by have h1 : a ≥ 1 := ge_one_of_prod_one h have h2 : a * b ≠ 0 := by linarith have h3 : a ≤ a * b := le_nonzero_prod_left h2 rewrite [h] at h3 show a = 1 from Nat.le_antisymm h3 h1 done lemma eq_one_of_dvd_one {n : Nat} (h : n ∣ 1) : n = 1 := by obtain (j : Nat) (h1 : 1 = n * j) from h show n = 1 from eq_one_of_prod_one h1.symm done lemma prime_not_one {p : Nat} (h : prime p) : p ≠ 1 := by define at h linarith done theorem Theorem_7_2_4 {p : Nat} (h1 : prime p) : ∀ (l : List Nat), p ∣ prod l → ∃ a ∈ l, p ∣ a := by apply List.rec · -- Base Case. Goal : p ∣ prod [] → ∃ a ∈ [], p ∣ a rewrite [prod_nil] assume h2 : p ∣ 1 show ∃ a ∈ [], p ∣ a from absurd (eq_one_of_dvd_one h2) (prime_not_one h1) done · -- Induction Step fix b : Nat fix L : List Nat assume ih : p ∣ prod L → ∃ a ∈ L, p ∣ a --Goal : p ∣ prod (b :: L) → ∃ a ∈ b :: L, p ∣ a assume h2 : p ∣ prod (b :: L) rewrite [prod_cons] at h2 have h3 : p ∣ b ∨ p ∣ prod L := Theorem_7_2_3 h1 h2 by_cases on h3 · -- Case 1. h3 : p ∣ b apply Exists.intro b show b ∈ b :: L ∧ p ∣ b from And.intro (List.mem_cons_self b L) h3 done · -- Case 2. h3 : p ∣ prod L obtain (a : Nat) (h4 : a ∈ L ∧ p ∣ a) from ih h3 apply Exists.intro a show a ∈ b :: L ∧ p ∣ a from And.intro (List.mem_cons_of_mem b h4.left) h4.right done done done lemma prime_in_list {p : Nat} {l : List Nat} (h1 : prime p) (h2 : all_prime l) (h3 : p ∣ prod l) : p ∈ l := by obtain (a : Nat) (h4 : a ∈ l ∧ p ∣ a) from Theorem_7_2_4 h1 l h3 define at h2 have h5 : prime a := h2 a h4.left have h6 : p = 1 ∨ p = a := dvd_prime h5 h4.right disj_syll h6 (prime_not_one h1) rewrite [h6] show a ∈ l from h4.left done lemma first_le_first {p q : Nat} {l m : List Nat} (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m)) (h3 : prod (p :: l) = prod (q :: m)) : p ≤ q := by define at h1; define at h2 have h4 : q ∣ prod (p :: l) := by define apply Exists.intro (prod m) rewrite [←prod_cons] show prod (p :: l) = prod (q :: m) from h3 done have h5 : all_prime (q :: m) := h2.left rewrite [all_prime_cons] at h5 have h6 : q ∈ p :: l := prime_in_list h5.left h1.left h4 have h7 : nondec (p :: l) := h1.right rewrite [nondec_cons] at h7 rewrite [List.mem_cons] at h6 by_cases on h6 · -- Case 1. h6 : q = p linarith done · -- Case 2. h6 : q ∈ l have h8 : ∀ m ∈ l, p ≤ m := h7.left show p ≤ q from h8 q h6 done done lemma nondec_prime_list_tail {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by define at h define rewrite [all_prime_cons, nondec_cons] at h show all_prime l ∧ nondec l from And.intro h.left.right h.right.right done lemma cons_prod_not_one {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : prod (p :: l) ≠ 1 := by define at h have h1 : all_prime (p :: l) := h.left rewrite [all_prime_cons] at h1 rewrite [prod_cons] by_contra h2 show False from (prime_not_one h1.left) (eq_one_of_prod_one h2) done lemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) : l = [] ↔ prod l = 1 := by apply Iff.intro · -- (→) assume h1 : l = [] rewrite [h1] show prod [] = 1 from prod_nil done · -- (←) contrapos assume h1 : ¬l = [] obtain (p : Nat) (h2 : ∃ (L : List Nat), l = p :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List Nat) (h3 : l = p :: L) from h2 rewrite [h3] at h rewrite [h3] show ¬prod (p :: L) = 1 from cons_prod_not_one h done done lemma prime_pos {p : Nat} (h : prime p) : p > 0 := by define at h linarith done theorem Theorem_7_2_5 : ∀ (l1 l2 : List Nat), nondec_prime_list l1 → nondec_prime_list l2 → prod l1 = prod l2 → l1 = l2 := by apply List.rec · -- Base Case. Goal : ∀ (l2 : List Nat), nondec_prime_list [] → -- nondec_prime_list l2 → prod [] = prod l2 → [] = l2 fix l2 : List Nat assume h1 : nondec_prime_list [] assume h2 : nondec_prime_list l2 assume h3 : prod [] = prod l2 rewrite [prod_nil, eq_comm, ←list_nil_iff_prod_one h2] at h3 show [] = l2 from h3.symm done · -- Induction Step fix p : Nat fix L1 : List Nat assume ih : ∀ (L2 : List Nat), nondec_prime_list L1 → nondec_prime_list L2 → prod L1 = prod L2 → L1 = L2 -- Goal : ∀ (l2 : List Nat), nondec_prime_list (p :: L1) → -- nondec_prime_list l2 → prod (p :: L1) = prod l2 → p :: L1 = l2 fix l2 : List Nat assume h1 : nondec_prime_list (p :: L1) assume h2 : nondec_prime_list l2 assume h3 : prod (p :: L1) = prod l2 have h4 : ¬prod (p :: L1) = 1 := cons_prod_not_one h1 rewrite [h3, ←list_nil_iff_prod_one h2] at h4 obtain (q : Nat) (h5 : ∃ (L : List Nat), l2 = q :: L) from List.exists_cons_of_ne_nil h4 obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5 rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2) rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2) have h7 : p ≤ q := first_le_first h1 h2 h3 have h8 : q ≤ p := first_le_first h2 h1 h3.symm have h9 : p = q := by linarith rewrite [h9, prod_cons, prod_cons] at h3 --h3 : q * prod L1 = q * prod L2 have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1 have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2 define at h2 have h12 : all_prime (q :: L2) := h2.left rewrite [all_prime_cons] at h12 have h13 : q > 0 := prime_pos h12.left have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3 have h15 : L1 = L2 := ih L2 h10 h11 h14 rewrite [h6, h9, h15] rfl done done theorem fund_thm_arith (n : Nat) (h : n ≥ 1) : ∃! (l : List Nat), prime_factorization n l := by exists_unique · -- Existence show ∃ (l : List Nat), prime_factorization n l from exists_prime_factorization n h done · -- Uniqueness fix l1 : List Nat; fix l2 : List Nat assume h1 : prime_factorization n l1 assume h2 : prime_factorization n l2 define at h1; define at h2 have h3 : prod l1 = n := h1.right rewrite [←h2.right] at h3 show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3 done done /- Section 7.3 -/ theorem congr_refl (m : Nat) : ∀ (a : Int), a ≡ a (MOD m) := by fix a : Int define --Goal : ∃ (c : Int), a - a = ↑m * c apply Exists.intro 0 ring done theorem congr_symm {m : Nat} : ∀ {a b : Int}, a ≡ b (MOD m) → b ≡ a (MOD m) := by fix a : Int; fix b : Int assume h1 : a ≡ b (MOD m) define at h1 --h1 : ∃ (c : Int), a - b = ↑m * c define --Goal : ∃ (c : Int), b - a = ↑m * c obtain (c : Int) (h2 : a - b = m * c) from h1 apply Exists.intro (-c) show b - a = m * (-c) from calc b - a _ = -(a - b) := by ring _ = -(m * c) := by rw [h2] _ = m * (-c) := by ring done theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry /- Fundamental properties of congruence classes -/ lemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) : X = Y ↔ X.val = Y.val := Fin.ext_iff lemma val_nat_eq_mod (n k : Nat) : ([k]_(n + 1)).val = k % (n + 1) := by rfl lemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl theorem cc_rep {m : Nat} (X : ZMod m) : ∃ (a : Int), X = [a]_m := match m with | 0 => by apply Exists.intro X rfl done | n + 1 => by apply Exists.intro ↑(X.val) have h1 : X.val < n + 1 := Fin.prop X rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1] rfl done theorem add_class (m : Nat) (a b : Int) : [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm theorem mul_class (m : Nat) (a b : Int) : [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm lemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ [a - b]_m = [0]_m := by apply Iff.intro · -- (→) assume h1 : [a]_m = [b]_m have h2 : a - b = a + (-b) := by ring have h3 : b + (-b) = 0 := by ring show [a - b]_m = [0]_m from calc [a - b]_m _ = [a + (-b)]_m := by rw [h2] _ = [a]_m + [-b]_m := by rw [add_class] _ = [b]_m + [-b]_m := by rw [h1] _ = [b + -b]_m := by rw [add_class] _ = [0]_m := by rw [h3] done · -- (←) assume h1 : [a - b]_m = [0]_m have h2 : b + (a - b) = a := by ring have h3 : b + 0 = b := by ring show [a]_m = [b]_m from calc [a]_m _ = [b + (a - b)]_m := by rw [h2] _ = [b]_m + [a - b]_m := by rw [add_class] _ = [b]_m + [0]_m := by rw [h1] _ = [b + 0]_m := by rw [add_class] _ = [b]_m := by rw [h3] done done lemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) : [a]_m = [0]_m → [-a]_m = [0]_m := by assume h1 : [a]_m = [0]_m have h2 : 0 + (-a) = -a := by ring have h3 : a + (-a) = 0 := by ring show [-a]_m = [0]_m from calc [-a]_m _ = [0 + (-a)]_m := by rw [h2] _ = [0]_m + [-a]_m := by rw [add_class] _ = [a]_m + [-a]_m := by rw [h1] _ = [a + (-a)]_m := by rw [add_class] _ = [0]_m := by rw [h3] done lemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) : [-a]_m = [0]_m ↔ [a]_m = [0]_m := by apply Iff.intro _ (cc_neg_zero_of_cc_zero m a) assume h1 : [-a]_m = [0]_m have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1 have h3 : -(-a) = a := by ring rewrite [h3] at h2 show [a]_m = [0]_m from h2 done lemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl lemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m ↔ m ∣ k := match m with | 0 => by have h : (0 : Int) = (↑(0 : Nat) : Int) := by rfl rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj] apply Iff.intro · -- (→) assume h1 : k = 0 rewrite [h1] show 0 ∣ 0 from dvd_self 0 done · -- (←) assume h1 : 0 ∣ k obtain (c : Nat) (h2 : k = 0 * c) from h1 rewrite [h2] ring done done | n + 1 => by rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero] show k % (n + 1) = 0 ↔ n + 1 ∣ k from (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm done lemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m ↔ ↑m ∣ a := by obtain (k : Nat) (h1 : a = ↑k ∨ a = -↑k) from Int.eq_nat_or_neg a by_cases on h1 · -- Case 1. h1: a = ↑k rewrite [h1, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done · -- Case 2. h1: a = -↑k rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done done theorem cc_eq_iff_congr (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ a ≡ b (MOD m) := calc [a]_m = [b]_m _ ↔ [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b _ ↔ ↑m ∣ (a - b) := cc_zero_iff_dvd m (a - b) _ ↔ a ≡ b (MOD m) := by rfl /- End of fundamental properties of congruence classes -/ lemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m := by have h1 : (↑m : Int) ≠ 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m) show 0 ≤ a % m from Int.emod_nonneg a h1 done lemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m) have h2 : (↑m : Int) > 0 := (Nat.cast_pos).rtl h1 show a % m < m from Int.emod_lt_of_pos a h2 done lemma congr_mod_mod (m : Nat) (a : Int) : a ≡ a % m (MOD m) := by define have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m apply Exists.intro (a / m) show a - a % m = m * (a / m) from calc a - (a % m) _ = m * (a / m) + a % m - a % m := by rw [h1] _ = m * (a / m) := by ring done lemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) := And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a)) theorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) : ∃! (r : Int), 0 ≤ r ∧ r < m ∧ a ≡ r (MOD m) := by exists_unique · -- Existence apply Exists.intro (a % m) show 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) from mod_cmpl_res m a done · -- Uniqueness fix r1 : Int; fix r2 : Int assume h1 : 0 ≤ r1 ∧ r1 < m ∧ a ≡ r1 (MOD m) assume h2 : 0 ≤ r2 ∧ r2 < m ∧ a ≡ r2 (MOD m) have h3 : r1 ≡ r2 (MOD m) := congr_trans (congr_symm h1.right.right) h2.right.right obtain (d : Int) (h4 : r1 - r2 = m * d) from h3 have h5 : r1 - r2 < m * 1 := by linarith have h6 : m * (-1) < r1 - r2 := by linarith rewrite [h4] at h5 --h5 : m * d < m * 1 rewrite [h4] at h6 --h6 : m * -1 < m * d have h7 : (↑m : Int) ≥ 0 := Nat.cast_nonneg m have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7 have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7 have h10 : d = 0 := by linarith show r1 = r2 from calc r1 _ = r1 - r2 + r2 := by ring _ = m * 0 + r2 := by rw [h4, h10] _ = r2 := by ring done done lemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m := (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a) theorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y rewrite [h1, h2] have h3 : a + b = b + a := by ring show [a]_m + [b]_m = [b]_m + [a]_m from calc [a]_m + [b]_m _ = [a + b]_m := add_class m a b _ = [b + a]_m := by rw [h3] _ = [b]_m + [a]_m := (add_class m b a).symm done theorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X rewrite [h1] have h2 : a * 1 = a := by ring show [a]_m * [1]_m = [a]_m from calc [a]_m * [1]_m _ = [a * 1]_m := mul_class m a 1 _ = [a]_m := by rw [h2] done theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry lemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) : [a]_m * [gcd_c2 m a]_m = [1]_m := by set s : Int := gcd_c1 m a have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m define at h1 rewrite [h1, Nat.cast_one] at h2 --h2 : s * ↑m + gcd_c2 m a * ↑a = 1 rewrite [mul_class, cc_eq_iff_congr] define --Goal : ∃ (c : Int), ↑a * gcd_c2 m a - 1 = ↑m * c apply Exists.intro (-s) show a * (gcd_c2 m a) - 1 = m * (-s) from calc a * (gcd_c2 m a) - 1 _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring _ = 1 + m * (-s) - 1 := by rw [h2] _ = m * (-s) := by ring done theorem Theorem_7_3_7 (m a : Nat) : invertible [a]_m ↔ rel_prime m a := by apply Iff.intro · -- (→) assume h1 : invertible [a]_m define at h1 obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1 obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y rewrite [h3, mul_class, cc_eq_iff_congr] at h2 define at h2 obtain (c : Int) (h4 : a * b - 1 = m * c) from h2 rewrite [Exercise_7_2_6] --Goal : ∃ (s t : Int), s * ↑m + t * ↑a = 1 apply Exists.intro (-c) apply Exists.intro b show (-c) * m + b * a = 1 from calc (-c) * m + b * a _ = (-c) * m + (a * b - 1) + 1 := by ring _ = (-c) * m + m * c + 1 := by rw [h4] _ = 1 := by ring done · -- (←) assume h1 : rel_prime m a define show ∃ (Y : ZMod m), [a]_m * Y = [1]_m from Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1) done done /- Section 7.4 -/ section Euler open Euler lemma num_rp_below_base {m : Nat} : num_rp_below m 0 = 0 := by rfl lemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) : num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by have h1 : num_rp_below m (j + 1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : gcd m j = 1 rewrite [if_pos h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1 show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1 done lemma num_rp_below_step_not_rp {m j : Nat} (h : ¬rel_prime m j) : num_rp_below m (j + 1) = num_rp_below m j := by have h1 : num_rp_below m (j +1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : ¬gcd m j = 1 rewrite [if_neg h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j show num_rp_below m (j + 1) = num_rp_below m j from h1 done lemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl #eval phi 10 --Answer: 4 lemma prod_inv_iff_inv {m : Nat} {X : ZMod m} (h1 : invertible X) (Y : ZMod m) : invertible (X * Y) ↔ invertible Y := by apply Iff.intro · -- (→) assume h2 : invertible (X * Y) obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2 apply Exists.intro (X * Z) rewrite [←h3] --Goal : Y * (X * Z) = X * Y * Z ring --Note that ring can do algebra in ZMod m done · -- (←) assume h2 : invertible Y obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1 obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2 apply Exists.intro (Xi * Yi) show (X * Y) * (Xi * Yi) = [1]_m from calc X * Y * (Xi * Yi) _ = (X * Xi) * (Y * Yi) := by ring _ = [1]_m * [1]_m := by rw [h3, h4] _ = [1]_m := Theorem_7_3_6_7 [1]_m done done lemma F_rp_def {m i : Nat} (h : rel_prime m i) : F m i = [i]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h --h : gcd m i = 1 rewrite [if_pos h] at h1 show F m i = [i]_m from h1 done lemma F_not_rp_def {m i : Nat} (h : ¬rel_prime m i) : F m i = [1]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h rewrite [h1, if_neg h] rfl done lemma prod_seq_base {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 0 k f = [1]_m := by rfl lemma prod_seq_step {m : Nat} (n k : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl lemma prod_seq_zero_step {m : Nat} (n : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by rewrite [prod_seq_step, zero_add] rfl done lemma prod_one {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 1 k f = f k := by rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7] rfl done lemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl lemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m := calc [G m a i]_m _ = [(a * i) % m]_m := by rfl _ = [a * i]_m := (cc_eq_mod m (a * i)).symm _ = [a]_m * [i]_m := (mul_class m a i).symm lemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) : rel_prime m (G m a i) ↔ rel_prime m i := by have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1 show rel_prime m (G m a i) ↔ rel_prime m i from calc rel_prime m (G m a i) _ ↔ invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm _ ↔ invertible ([a]_m * [i]_m) := by rw [cc_G] _ ↔ invertible [i]_m := prod_inv_iff_inv h2 ([i]_m) _ ↔ rel_prime m i := Theorem_7_3_7 m i done lemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) : F m (G m a i) = [a]_m * F m i := by have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2 show F m (G m a i) = [a]_m * F m i from calc F m (G m a i) _ = [G m a i]_m := F_rp_def h3 _ = [a]_m * [i]_m := cc_G m a i _ = [a]_m * F m i := by rw [F_rp_def h2] done lemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : ¬rel_prime m i) : F m (G m a i) = [1]_m := by rewrite [←G_rp_iff h1 i] at h2 show F m (G m a i) = [1]_m from F_not_rp_def h2 done lemma FG_prod {m a : Nat} (h1 : rel_prime m a) : ∀ (k : Nat), prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by by_induc · -- Base Case show prod_seq 0 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from calc prod_seq 0 0 ((F m) ∘ (G m a)) _ = [1]_m := prod_seq_base _ _ _ = [a]_m ^ 0 * [1]_m := by ring _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by rw [num_rp_below_base, prod_seq_base] done · -- Induction Step fix k : Nat assume ih : prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) by_cases h2 : rel_prime m k · -- Case 1. h2 : rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([a]_m * F m k) := by rw [FG_rp h1 h2] _ = [a]_m ^ ((num_rp_below m k) + 1) * ((prod_seq k 0 (F m)) * F m k) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_rp h2, prod_seq_zero_step] done · -- Case 2. h2 : ¬rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([1]_m) := by rw [FG_not_rp h1 h2] _ = [a]_m ^ (num_rp_below m k) * (prod_seq k 0 (F m) * ([1]_m)) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_not_rp h2, prod_seq_zero_step, F_not_rp_def h2] done done done lemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by define --Goal : ∀ i < m, G m a i < m fix i : Nat assume h1 : i < m rewrite [G_def] --Goal : a * i % m < m show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m) done lemma left_inv_one_one_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g' (g i) = i) : one_one_below n g := sorry lemma right_inv_onto_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g (g' i) = i) (h2 : maps_below n g') : onto_below n g := by define at h2; define fix k : Nat assume h3 : k < n apply Exists.intro (g' k) show g' k < n ∧ g (g' k) = k from And.intro (h2 k h3) (h1 k h3) done lemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m * [inv_mod m a]_m = [1]_m := by have h2 : 0 ≤ (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a) show [a]_m * [inv_mod m a]_m = [1]_m from calc [a]_m * [inv_mod m a]_m _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2] _ = [a]_m * [gcd_c2 m a]_m := by rw [←cc_eq_mod] _ = [1]_m := gcd_c2_inv h1 done lemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m := calc a * (b % m) % m = a % m * (b % m % m) % m := Nat.mul_mod _ _ _ _ = a % m * (b % m) % m := by rw [Nat.mod_mod] _ = a * b % m := (Nat.mul_mod _ _ _).symm lemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm] rfl done theorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] : ↑a ≡ ↑b (MOD m) ↔ a % m = b % m := sorry lemma mul_inv_mod_cancel {m a i : Nat} [NeZero m] (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1 rewrite [mul_class, cc_eq_iff_congr, ←Nat.cast_mul, ←Nat.cast_one, congr_iff_mod_eq_Nat] at h3 show a * inv_mod m a * i % m = i from calc a * (inv_mod m a) * i % m _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = 1 % m * i % m := by rw [h3] _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = i % m := by rw [one_mul] _ = i := Nat.mod_eq_of_lt h2 done lemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl lemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, G m a (Ginv m a i) = i := by fix i : Nat assume h2 : i < m show G m a (Ginv m a i) = i from calc G m a (Ginv m a i) _ = a * ((inv_mod m a * i) % m) % m := by rfl _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, Ginv m a (G m a i) = i := by fix i : Nat assume h2 : i < m show Ginv m a (G m a i) = i from calc Ginv m a (G m a i) _ = inv_mod m a * ((a * i) % m) % m := by rfl _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc, mul_comm (inv_mod m a)] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_maps_below (m a : Nat) [NeZero m] : maps_below m (Ginv m a) := G_maps_below m (inv_mod m a) lemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : one_one_below m (G m a) := left_inv_one_one_below (Ginv_left_inv h1) lemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : onto_below m (G m a) := right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a) lemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : perm_below m (G m a) := And.intro (G_maps_below m a) (And.intro (G_one_one_below h1) (G_onto_below h1)) --Permuting a product of congruence classes doesn't change product lemma swap_fst (u v : Nat) : swap u v u = v := by define : swap u v u --Goal : (if u = u then v else if u = v then u else u) = v have h : u = u := by rfl rewrite [if_pos h] rfl done lemma swap_snd (u v : Nat) : swap u v v = u := by define : swap u v v by_cases h1 : v = u · -- Case 1. h1 : v = u rewrite [if_pos h1] show v = u from h1 done · -- Case 2. h1 : v ≠ u rewrite [if_neg h1] have h2 : v = v := by rfl rewrite [if_pos h2] rfl done done lemma swap_other {u v i : Nat} (h1 : i ≠ u) (h2 : i ≠ v) : swap u v i = i := by define : swap u v i rewrite [if_neg h1, if_neg h2] rfl done lemma swap_values (u v i : Nat) : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := by by_cases h1 : i = u · -- Case 1. h1 : i = u apply Or.inl rewrite [h1] show swap u v u = v from swap_fst u v done · -- Case 2. h1 : i ≠ u apply Or.inr by_cases h2 : i = v · -- Case 2.1. h2 : i = v apply Or.inl rewrite [h2] show swap u v v = u from swap_snd u v done · -- Case 2.2. h2 : i ≠ v apply Or.inr show swap u v i = i from swap_other h1 h2 done done done lemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by define fix i : Nat assume h3 : i < n have h4 : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := swap_values u v i by_cases on h4 · -- Case 1. h4 : swap u v i = v rewrite [h4] show v < n from h2 done · -- Case 2. by_cases on h4 · -- Case 2.1. h4 : swap u v i = u rewrite [h4] show u < n from h1 done · -- Case 2.2. h4 : swap u v i = i rewrite [h4] show i < n from h3 done done done lemma swap_swap (u v n : Nat) : ∀ i < n, swap u v (swap u v i) = i := by fix i : Nat assume h : i < n by_cases h1 : i = u · -- Case 1. h1 : i = u rewrite [h1, swap_fst, swap_snd] rfl done · -- Case 2. h1 : i ≠ u by_cases h2 : i = v · -- Case 2.1. h2 : i = v rewrite [h2, swap_snd, swap_fst] rfl done · -- Case 2.2. h2 : i ≠ v rewrite [swap_other h1 h2, swap_other h1 h2] rfl done done done lemma swap_one_one_below (u v n) : one_one_below n (swap u v) := left_inv_one_one_below (swap_swap u v n) lemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) := right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2) lemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) := And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2)) lemma comp_perm_below {n : Nat} {f g : Nat → Nat} (h1 : perm_below n f) (h2 : perm_below n g) : perm_below n (f ∘ g) := sorry lemma trivial_swap (u : Nat) : swap u u = id := by apply funext fix x : Nat by_cases h1 : x = u · -- Case 1. h1 : x = u rewrite [h1, swap_fst] rfl done · -- Case 2. h1 : x ≠ u rewrite [swap_other h1 h1] rfl done done lemma prod_eq_fun {m : Nat} (f g : Nat → ZMod m) (k : Nat) : ∀ (n : Nat), (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g := by by_induc · -- Base Case assume h : (∀ i < 0, f (k + i) = g (k + i)) rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g assume h1 : ∀ i < n + 1, f (k + i) = g (k + i) have h2 : ∀ i < n, f (k + i) = g (k + i) := by fix i : Nat assume h2 : i < n have h3 : i < n + 1 := by linarith show f (k + i) = g (k + i) from h1 i h3 done have h3 : prod_seq n k f = prod_seq n k g := ih h2 have h4 : n < n + 1 := Nat.lt_succ_self n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4] rfl done done lemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := by have h2 : ∀ (i : Nat), i < u → (f ∘ swap u n) (0 + i) = f (0 + i) := by fix i : Nat assume h2 : i < u have h3 : 0 + i ≠ u := by linarith have h4 : 0 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl done show prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f from prod_eq_fun (f ∘ swap u n) f 0 u h2 done lemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := by have h2 : ∀ i < j, (f ∘ swap u n) (u + 1 + i) = f (u + 1 + i) := by fix i : Nat assume h2 : i < j have h3 : u + 1 + i ≠ u := by linarith have h4 : u + 1 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl show prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f from prod_eq_fun (f ∘ swap u n) f (u + 1) j h2 done lemma break_prod {m : Nat} (n : Nat) (f : Nat → ZMod m) : ∀ (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by by_induc · -- Base Case have h : n + 0 = n := by rfl rewrite [prod_seq_base, h, Theorem_7_3_6_7] rfl done · -- Induction Step fix j : Nat assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f rewrite [←add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc] rfl done done lemma break_prod_twice {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f := break_prod n f 1 rewrite [prod_one] at h2 have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f := break_prod (u + 1) f j rewrite [←h1] at h3 have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f := break_prod u f 1 rewrite [prod_one] at h4 rewrite [h3, h4] at h2 show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2 done lemma swap_prod_eq_prod {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f := by by_cases h2 : u = n · -- Case 1. h2 : u = n rewrite [h2, trivial_swap n] --Goal : prod_seq (n + 1) 0 (f ∘ id) = prod_seq (n + 1) 0 f rfl done · -- Case 2. h2 : ¬u = n have h3 : u + 1 ≤ n := Nat.lt_of_le_of_ne h1 h2 obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3 have break_f : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := break_prod_twice f h4 have break_fs : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_prod_twice (f ∘ swap u n) h4 have f_eq_fs_below : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := swap_prod_eq_prod_below f h1 have f_eq_fs_btwn : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4 show prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f from calc prod_seq (n + 1) 0 (f ∘ swap u n) _ = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_fs _ = prod_seq u 0 f * (f ∘ swap u n) u * prod_seq j (u + 1) f * (f ∘ swap u n) n := by rw [f_eq_fs_below, f_eq_fs_btwn] _ = prod_seq u 0 f * f (swap u n u) * prod_seq j (u + 1) f * f (swap u n n) := by rfl _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by rw [swap_fst, swap_snd] _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring _ = prod_seq (n + 1) 0 f := break_f.symm done done lemma perm_below_fixed {n : Nat} {g : Nat → Nat} (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry

lemma perm_prod {m : Nat} (f : Nat → ZMod m) : ∀ (n : Nat), ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g)

HTPI.perm_prod

htpi/HTPILib/Chap7.lean

htpi

/- Copyright 2023 Daniel J. Velleman -/ import HTPILib.Chap6 namespace HTPI /- Definitions -/ lemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by have h : n + 1 > 0 := Nat.succ_pos n show a % (n + 1) < n + 1 from Nat.mod_lt a h done def gcd (a b : Nat) : Nat := match b with | 0 => a | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd (n + 1) (a % (n + 1)) termination_by b mutual def gcd_c1 (a b : Nat) : Int := match b with | 0 => 1 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c2 (n + 1) (a % (n + 1)) --Corresponds to s = t' termination_by b def gcd_c2 (a b : Nat) : Int := match b with | 0 => 0 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c1 (n + 1) (a % (n + 1)) - (gcd_c2 (n + 1) (a % (n + 1))) * ↑(a / (n + 1)) --Corresponds to t = s' - t'q termination_by b end def prime (n : Nat) : Prop := 2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n def prime_factor (p n : Nat) : Prop := prime p ∧ p ∣ n def all_prime (l : List Nat) : Prop := ∀ p ∈ l, prime p def nondec (l : List Nat) : Prop := match l with | [] => True --Of course, True is a proposition that is always true | n :: L => (∀ m ∈ L, n ≤ m) ∧ nondec L def nondec_prime_list (l : List Nat) : Prop := all_prime l ∧ nondec l def prod (l : List Nat) : Nat := match l with | [] => 1 | n :: L => n * (prod L) def prime_factorization (n : Nat) (l : List Nat) : Prop := nondec_prime_list l ∧ prod l = n def rel_prime (a b : Nat) : Prop := gcd a b = 1 def congr_mod (m : Nat) (a b : Int) : Prop := (↑m : Int) ∣ (a - b) def cc (m : Nat) (a : Int) : ZMod m := (↑a : ZMod m) notation:50 a " ≡ " b " (MOD " m ")" => congr_mod m a b notation:max "["a"]_"m:max => cc m a def invertible {m : Nat} (X : ZMod m) : Prop := ∃ (Y : ZMod m), X * Y = [1]_m def num_rp_below (m k : Nat) : Nat := match k with | 0 => 0 | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j def phi (m : Nat) : Nat := num_rp_below m m def prod_seq {m : Nat} (j k : Nat) (f : Nat → ZMod m) : ZMod m := match j with | 0 => [1]_m | n + 1 => prod_seq n k f * f (k + n) def maps_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i < n, g i < n def one_one_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i1 < n, ∀ i2 < n, g i1 = g i2 → i1 = i2 def onto_below (n : Nat) (g : Nat → Nat) : Prop := ∀ k < n, ∃ i < n, g i = k def perm_below (n : Nat) (g : Nat → Nat) : Prop := maps_below n g ∧ one_one_below n g ∧ onto_below n g def inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m) def swap (u v i : Nat) : Nat := if i = u then v else if i = v then u else i namespace Euler --For definitions specific to Euler's theorem def F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m def G (m a i : Nat) : Nat := (a * i) % m def Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i end Euler /- Section 7.1 -/ theorem dvd_mod_of_dvd_a_b {a b d : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ (a % b) := by set q : Nat := a / b have h3 : b * q + a % b = a := Nat.div_add_mod a b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 define --Goal : ∃ (c : Nat), a % b = d * c apply Exists.intro (j - k * q) show a % b = d * (j - k * q) from calc a % b _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm _ = a - b * q := by rw [h3] _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc] _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm done theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry #eval gcd 672 161 --Answer: 7 lemma gcd_base (a : Nat) : gcd a 0 = a := by rfl lemma gcd_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd a b = gcd b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1)) rfl done lemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b ≠ 0) : a % b < b := by have h1 : b > 0 := Nat.pos_of_ne_zero h show a % b < b from Nat.mod_lt a h1 done lemma dvd_self (n : Nat) : n ∣ n := by apply Exists.intro 1 ring done theorem gcd_dvd : ∀ (b a : Nat), (gcd a b) ∣ a ∧ (gcd a b) ∣ b := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd a b_1) ∣ a ∧ (gcd a b_1) ∣ b_1 fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_base] --Goal: a ∣ a ∧ a ∣ 0 apply And.intro (dvd_self a) define apply Exists.intro 0 rfl done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_nonzero a h1] --Goal : gcd b (a % b) ∣ a ∧ gcd b (a % b) ∣ b have h2 : a % b < b := mod_nonzero_lt a h1 have h3 : (gcd b (a % b)) ∣ b ∧ (gcd b (a % b)) ∣ (a % b) := ih (a % b) h2 b apply And.intro _ h3.left show (gcd b (a % b)) ∣ a from dvd_a_of_dvd_b_mod h3.left h3.right done done theorem gcd_dvd_left (a b : Nat) : (gcd a b) ∣ a := (gcd_dvd b a).left theorem gcd_dvd_right (a b : Nat) : (gcd a b) ∣ b := (gcd_dvd b a).right lemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl lemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c1 a b = gcd_c2 b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done lemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl lemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * ↑(a / b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done theorem gcd_lin_comb : ∀ (b a : Nat), (gcd_c1 a b) * ↑a + (gcd_c2 a b) * ↑b = ↑(gcd a b) := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd_c1 a b_1) * ↑a + (gcd_c2 a b_1) * ↑b_1 = ↑(gcd a b_1) fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base] --Goal : 1 * ↑a + 0 * ↑0 = ↑a ring done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1] --Goal : gcd_c2 b (a % b) * ↑a + -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * ↑(a / b)) * ↑b = -- ↑(gcd b (a % b)) set r : Nat := a % b set q : Nat := a / b set s : Int := gcd_c1 b r set t : Int := gcd_c2 b r --Goal : t * ↑a + (s - t * ↑q) * ↑b = ↑(gcd b r) have h2 : r < b := mod_nonzero_lt a h1 have h3 : s * ↑b + t * ↑r = ↑(gcd b r) := ih r h2 b have h4 : b * q + r = a := Nat.div_add_mod a b rewrite [←h3, ←h4] rewrite [Nat.cast_add, Nat.cast_mul] --Goal : t * (↑b * ↑q + ↑r) + (s - t * ↑q) * ↑b = s * ↑b + t * ↑r ring done done #eval gcd_c1 672 161 --Answer: 6 #eval gcd_c2 672 161 --Answer: -25 --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161 theorem Theorem_7_1_6 {d a b : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ gcd a b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑d ∣ ↑(gcd a b) set s : Int := gcd_c1 a b set t : Int := gcd_c2 a b have h3 : s * ↑a + t * ↑b = ↑(gcd a b) := gcd_lin_comb b a rewrite [←h3] --Goal : ↑d ∣ s * ↑a + t * ↑b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul] --Goal : ↑d ∣ s * (↑d * ↑j) + t * (↑d * ↑k) define apply Exists.intro (s * ↑j + t * ↑k) ring done /- Section 7.2 -/ theorem dvd_trans {a b c : Nat} (h1 : a ∣ b) (h2 : b ∣ c) : a ∣ c := by define at h1; define at h2; define obtain (m : Nat) (h3 : b = a * m) from h1 obtain (n : Nat) (h4 : c = b * n) from h2 rewrite [h3, mul_assoc] at h4 apply Exists.intro (m * n) show c = a * (m * n) from h4 done lemma exists_prime_factor : ∀ (n : Nat), 2 ≤ n → ∃ (p : Nat), prime_factor p n := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, 2 ≤ n_1 → ∃ (p : Nat), prime_factor p n_1 assume h1 : 2 ≤ n by_cases h2 : prime n · -- Case 1. h2 : prime n apply Exists.intro n define --Goal : prime n ∧ n ∣ n show prime n ∧ n ∣ n from And.intro h2 (dvd_self n) done · -- Case 2. h2 : ¬prime n define at h2 --h2 : ¬(2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n) demorgan at h2 disj_syll h2 h1 obtain (a : Nat) (h3 : ∃ (b : Nat), a * b = n ∧ a < n ∧ b < n) from h2 obtain (b : Nat) (h4 : a * b = n ∧ a < n ∧ b < n) from h3 have h5 : 2 ≤ a := by by_contra h6 have h7 : a ≤ 1 := by linarith have h8 : n ≤ b := calc n _ = a * b := h4.left.symm _ ≤ 1 * b := by rel [h7] _ = b := by ring linarith --n ≤ b contradicts b < n done have h6 : ∃ (p : Nat), prime_factor p a := ih a h4.right.left h5 obtain (p : Nat) (h7 : prime_factor p a) from h6 apply Exists.intro p define --Goal : prime p ∧ p ∣ n define at h7 --h7 : prime p ∧ p ∣ a apply And.intro h7.left have h8 : a ∣ n := by apply Exists.intro b show n = a * b from (h4.left).symm done show p ∣ n from dvd_trans h7.right h8 done done lemma exists_least_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q := by set S : Set Nat := {p : Nat | prime_factor p n} have h2 : ∃ (p : Nat), p ∈ S := exists_prime_factor n h show ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q from well_ord_princ S h2 done lemma all_prime_nil : all_prime [] := by define --Goal : ∀ p ∈ [], prime p fix p : Nat contrapos --Goal : ¬prime p → p ∉ [] assume h1 : ¬prime p show p ∉ [] from List.not_mem_nil p done lemma all_prime_cons (n : Nat) (L : List Nat) : all_prime (n :: L) ↔ prime n ∧ all_prime L := by apply Iff.intro · -- (→) assume h1 : all_prime (n :: L) --Goal : prime n ∧ all_prime L define at h1 --h1 : ∀ p ∈ n :: L, prime p apply And.intro (h1 n (List.mem_cons_self n L)) define --Goal : ∀ p ∈ L, prime p fix p : Nat assume h2 : p ∈ L show prime p from h1 p (List.mem_cons_of_mem n h2) done · -- (←) assume h1 : prime n ∧ all_prime L --Goal : all_prime (n :: l) define : all_prime L at h1 define fix p : Nat assume h2 : p ∈ n :: L rewrite [List.mem_cons] at h2 --h2 : p = n ∨ p ∈ L by_cases on h2 · -- Case 1. h2 : p = n rewrite [h2] show prime n from h1.left done · -- Case 2. h2 : p ∈ L show prime p from h1.right p h2 done done done lemma nondec_nil : nondec [] := by define --Goal : True trivial --trivial proves some obviously true statements, such as True done lemma nondec_cons (n : Nat) (L : List Nat) : nondec (n :: L) ↔ (∀ m ∈ L, n ≤ m) ∧ nondec L := by rfl lemma prod_nil : prod [] = 1 := by rfl lemma prod_cons : prod (n :: L) = n * (prod L) := by rfl lemma exists_cons_of_length_eq_succ {A : Type} {l : List A} {n : Nat} (h : l.length = n + 1) : ∃ (a : A) (L : List A), l = a :: L ∧ L.length = n := by have h1 : ¬l.length = 0 := by linarith rewrite [List.length_eq_zero] at h1 obtain (a : A) (h2 : ∃ (L : List A), l = a :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List A) (h3 : l = a :: L) from h2 apply Exists.intro a apply Exists.intro L apply And.intro h3 have h4 : (a :: L).length = L.length + 1 := List.length_cons a L rewrite [←h3, h] at h4 show L.length = n from (Nat.add_right_cancel h4).symm done lemma list_elt_dvd_prod_by_length (a : Nat) : ∀ (n : Nat), ∀ (l : List Nat), l.length = n → a ∈ l → a ∣ prod l := by by_induc · --Base Case fix l : List Nat assume h1 : l.length = 0 rewrite [List.length_eq_zero] at h1 --h1 : l = [] rewrite [h1] --Goal : a ∈ [] → a ∣ prod [] contrapos assume h2 : ¬a ∣ prod [] show a ∉ [] from List.not_mem_nil a done · -- Induction Step fix n : Nat assume ih : ∀ (l : List Nat), List.length l = n → a ∈ l → a ∣ prod l fix l : List Nat assume h1 : l.length = n + 1 --Goal : a ∈ l → a ∣ prod l obtain (b : Nat) (h2 : ∃ (L : List Nat), l = b :: L ∧ L.length = n) from exists_cons_of_length_eq_succ h1 obtain (L : List Nat) (h3 : l = b :: L ∧ L.length = n) from h2 have h4 : a ∈ L → a ∣ prod L := ih L h3.right assume h5 : a ∈ l rewrite [h3.left, prod_cons] --Goal : a ∣ b * prod L rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b ∨ a ∈ L by_cases on h5 · -- Case 1. h5 : a = b apply Exists.intro (prod L) rewrite [h5] rfl done · -- Case 2. h5 : a ∈ L have h6 : a ∣ prod L := h4 h5 have h7 : prod L ∣ b * prod L := by apply Exists.intro b ring done show a ∣ b * prod L from dvd_trans h6 h7 done done done lemma list_elt_dvd_prod {a : Nat} {l : List Nat} (h : a ∈ l) : a ∣ prod l := by set n : Nat := l.length have h1 : l.length = n := by rfl show a ∣ prod l from list_elt_dvd_prod_by_length a n l h1 h done lemma exists_prime_factorization : ∀ (n : Nat), n ≥ 1 → ∃ (l : List Nat), prime_factorization n l := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, n_1 ≥ 1 → ∃ (l : List Nat), prime_factorization n_1 l assume h1 : n ≥ 1 by_cases h2 : n = 1 · -- Case 1. h2 : n = 1 apply Exists.intro [] define apply And.intro · -- Proof of nondec_prime_list [] define show all_prime [] ∧ nondec [] from And.intro all_prime_nil nondec_nil done · -- Proof of prod [] = n rewrite [prod_nil, h2] rfl done done · -- Case 2. h2 : n ≠ 1 have h3 : n ≥ 2 := lt_of_le_of_ne' h1 h2 obtain (p : Nat) (h4 : prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q) from exists_least_prime_factor h3 have p_prime_factor : prime_factor p n := h4.left define at p_prime_factor have p_prime : prime p := p_prime_factor.left have p_dvd_n : p ∣ n := p_prime_factor.right have p_least : ∀ (q : Nat), prime_factor q n → p ≤ q := h4.right obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n have h5 : m ≠ 0 := by contradict h1 with h6 have h7 : n = 0 := calc n _ = p * m := n_eq_pm _ = p * 0 := by rw [h6] _ = 0 := by ring rewrite [h7] decide done have m_pos : 0 < m := Nat.pos_of_ne_zero h5 have m_lt_n : m < n := by define at p_prime show m < n from calc m _ < m + m := by linarith _ = 2 * m := by ring _ ≤ p * m := by rel [p_prime.left] _ = n := n_eq_pm.symm done obtain (L : List Nat) (h6 : prime_factorization m L) from ih m m_lt_n m_pos define at h6 have ndpl_L : nondec_prime_list L := h6.left define at ndpl_L apply Exists.intro (p :: L) define apply And.intro · -- Proof of nondec_prime_list (p :: L) define apply And.intro · -- Proof of all_prime (p :: L) rewrite [all_prime_cons] show prime p ∧ all_prime L from And.intro p_prime ndpl_L.left done · -- Proof of nondec (p :: L) rewrite [nondec_cons] apply And.intro _ ndpl_L.right fix q : Nat assume q_in_L : q ∈ L have h7 : q ∣ prod L := list_elt_dvd_prod q_in_L rewrite [h6.right] at h7 --h7 : q ∣ m have h8 : m ∣ n := by apply Exists.intro p rewrite [n_eq_pm] ring done have q_dvd_n : q ∣ n := dvd_trans h7 h8 have ap_L : all_prime L := ndpl_L.left define at ap_L have q_prime_factor : prime_factor q n := And.intro (ap_L q q_in_L) q_dvd_n show p ≤ q from p_least q q_prime_factor done done · -- Proof of prod (p :: L) = n rewrite [prod_cons, h6.right, n_eq_pm] rfl done done done theorem Theorem_7_2_2 {a b c : Nat} (h1 : c ∣ a * b) (h2 : rel_prime a c) : c ∣ b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑c ∣ ↑b define at h1; define at h2; define obtain (j : Nat) (h3 : a * b = c * j) from h1 set s : Int := gcd_c1 a c set t : Int := gcd_c2 a c have h4 : s * ↑a + t * ↑c = ↑(gcd a c) := gcd_lin_comb c a rewrite [h2, Nat.cast_one] at h4 --h4 : s * ↑a + t * ↑c = (1 : Int) apply Exists.intro (s * ↑j + t * ↑b) show ↑b = ↑c * (s * ↑j + t * ↑b) from calc ↑b _ = (1 : Int) * ↑b := (one_mul _).symm _ = (s * ↑a + t * ↑c) * ↑b := by rw [h4] _ = s * (↑a * ↑b) + t * ↑c * ↑b := by ring _ = s * (↑c * ↑j) + t * ↑c * ↑b := by rw [←Nat.cast_mul a b, h3, Nat.cast_mul c j] _ = ↑c * (s * ↑j + t * ↑b) := by ring done lemma le_nonzero_prod_left {a b : Nat} (h : a * b ≠ 0) : a ≤ a * b := by have h1 : b ≠ 0 := by contradict h with h1 rewrite [h1] ring done have h2 : 1 ≤ b := Nat.pos_of_ne_zero h1 show a ≤ a * b from calc a = a * 1 := (mul_one a).symm _ ≤ a * b := by rel [h2] done lemma le_nonzero_prod_right {a b : Nat} (h : a * b ≠ 0) : b ≤ a * b := by rewrite [mul_comm] rewrite [mul_comm] at h show b ≤ b * a from le_nonzero_prod_left h done lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry lemma rel_prime_of_prime_not_dvd {a p : Nat} (h1 : prime p) (h2 : ¬p ∣ a) : rel_prime a p := by have h3 : gcd a p ∣ a := gcd_dvd_left a p have h4 : gcd a p ∣ p := gcd_dvd_right a p have h5 : gcd a p = 1 ∨ gcd a p = p := dvd_prime h1 h4 have h6 : gcd a p ≠ p := by contradict h2 with h6 rewrite [h6] at h3 show p ∣ a from h3 done disj_syll h5 h6 show rel_prime a p from h5 done theorem Theorem_7_2_3 {a b p : Nat} (h1 : prime p) (h2 : p ∣ a * b) : p ∣ a ∨ p ∣ b := by or_right with h3 have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3 show p ∣ b from Theorem_7_2_2 h2 h4 done lemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a ≥ 1 := by have h1 : a ≠ 0 := by by_contra h1 rewrite [h1] at h contradict h linarith done show a ≥ 1 from Nat.pos_of_ne_zero h1 done lemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by have h1 : a ≥ 1 := ge_one_of_prod_one h have h2 : a * b ≠ 0 := by linarith have h3 : a ≤ a * b := le_nonzero_prod_left h2 rewrite [h] at h3 show a = 1 from Nat.le_antisymm h3 h1 done lemma eq_one_of_dvd_one {n : Nat} (h : n ∣ 1) : n = 1 := by obtain (j : Nat) (h1 : 1 = n * j) from h show n = 1 from eq_one_of_prod_one h1.symm done lemma prime_not_one {p : Nat} (h : prime p) : p ≠ 1 := by define at h linarith done theorem Theorem_7_2_4 {p : Nat} (h1 : prime p) : ∀ (l : List Nat), p ∣ prod l → ∃ a ∈ l, p ∣ a := by apply List.rec · -- Base Case. Goal : p ∣ prod [] → ∃ a ∈ [], p ∣ a rewrite [prod_nil] assume h2 : p ∣ 1 show ∃ a ∈ [], p ∣ a from absurd (eq_one_of_dvd_one h2) (prime_not_one h1) done · -- Induction Step fix b : Nat fix L : List Nat assume ih : p ∣ prod L → ∃ a ∈ L, p ∣ a --Goal : p ∣ prod (b :: L) → ∃ a ∈ b :: L, p ∣ a assume h2 : p ∣ prod (b :: L) rewrite [prod_cons] at h2 have h3 : p ∣ b ∨ p ∣ prod L := Theorem_7_2_3 h1 h2 by_cases on h3 · -- Case 1. h3 : p ∣ b apply Exists.intro b show b ∈ b :: L ∧ p ∣ b from And.intro (List.mem_cons_self b L) h3 done · -- Case 2. h3 : p ∣ prod L obtain (a : Nat) (h4 : a ∈ L ∧ p ∣ a) from ih h3 apply Exists.intro a show a ∈ b :: L ∧ p ∣ a from And.intro (List.mem_cons_of_mem b h4.left) h4.right done done done lemma prime_in_list {p : Nat} {l : List Nat} (h1 : prime p) (h2 : all_prime l) (h3 : p ∣ prod l) : p ∈ l := by obtain (a : Nat) (h4 : a ∈ l ∧ p ∣ a) from Theorem_7_2_4 h1 l h3 define at h2 have h5 : prime a := h2 a h4.left have h6 : p = 1 ∨ p = a := dvd_prime h5 h4.right disj_syll h6 (prime_not_one h1) rewrite [h6] show a ∈ l from h4.left done lemma first_le_first {p q : Nat} {l m : List Nat} (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m)) (h3 : prod (p :: l) = prod (q :: m)) : p ≤ q := by define at h1; define at h2 have h4 : q ∣ prod (p :: l) := by define apply Exists.intro (prod m) rewrite [←prod_cons] show prod (p :: l) = prod (q :: m) from h3 done have h5 : all_prime (q :: m) := h2.left rewrite [all_prime_cons] at h5 have h6 : q ∈ p :: l := prime_in_list h5.left h1.left h4 have h7 : nondec (p :: l) := h1.right rewrite [nondec_cons] at h7 rewrite [List.mem_cons] at h6 by_cases on h6 · -- Case 1. h6 : q = p linarith done · -- Case 2. h6 : q ∈ l have h8 : ∀ m ∈ l, p ≤ m := h7.left show p ≤ q from h8 q h6 done done lemma nondec_prime_list_tail {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by define at h define rewrite [all_prime_cons, nondec_cons] at h show all_prime l ∧ nondec l from And.intro h.left.right h.right.right done lemma cons_prod_not_one {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : prod (p :: l) ≠ 1 := by define at h have h1 : all_prime (p :: l) := h.left rewrite [all_prime_cons] at h1 rewrite [prod_cons] by_contra h2 show False from (prime_not_one h1.left) (eq_one_of_prod_one h2) done lemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) : l = [] ↔ prod l = 1 := by apply Iff.intro · -- (→) assume h1 : l = [] rewrite [h1] show prod [] = 1 from prod_nil done · -- (←) contrapos assume h1 : ¬l = [] obtain (p : Nat) (h2 : ∃ (L : List Nat), l = p :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List Nat) (h3 : l = p :: L) from h2 rewrite [h3] at h rewrite [h3] show ¬prod (p :: L) = 1 from cons_prod_not_one h done done lemma prime_pos {p : Nat} (h : prime p) : p > 0 := by define at h linarith done theorem Theorem_7_2_5 : ∀ (l1 l2 : List Nat), nondec_prime_list l1 → nondec_prime_list l2 → prod l1 = prod l2 → l1 = l2 := by apply List.rec · -- Base Case. Goal : ∀ (l2 : List Nat), nondec_prime_list [] → -- nondec_prime_list l2 → prod [] = prod l2 → [] = l2 fix l2 : List Nat assume h1 : nondec_prime_list [] assume h2 : nondec_prime_list l2 assume h3 : prod [] = prod l2 rewrite [prod_nil, eq_comm, ←list_nil_iff_prod_one h2] at h3 show [] = l2 from h3.symm done · -- Induction Step fix p : Nat fix L1 : List Nat assume ih : ∀ (L2 : List Nat), nondec_prime_list L1 → nondec_prime_list L2 → prod L1 = prod L2 → L1 = L2 -- Goal : ∀ (l2 : List Nat), nondec_prime_list (p :: L1) → -- nondec_prime_list l2 → prod (p :: L1) = prod l2 → p :: L1 = l2 fix l2 : List Nat assume h1 : nondec_prime_list (p :: L1) assume h2 : nondec_prime_list l2 assume h3 : prod (p :: L1) = prod l2 have h4 : ¬prod (p :: L1) = 1 := cons_prod_not_one h1 rewrite [h3, ←list_nil_iff_prod_one h2] at h4 obtain (q : Nat) (h5 : ∃ (L : List Nat), l2 = q :: L) from List.exists_cons_of_ne_nil h4 obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5 rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2) rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2) have h7 : p ≤ q := first_le_first h1 h2 h3 have h8 : q ≤ p := first_le_first h2 h1 h3.symm have h9 : p = q := by linarith rewrite [h9, prod_cons, prod_cons] at h3 --h3 : q * prod L1 = q * prod L2 have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1 have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2 define at h2 have h12 : all_prime (q :: L2) := h2.left rewrite [all_prime_cons] at h12 have h13 : q > 0 := prime_pos h12.left have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3 have h15 : L1 = L2 := ih L2 h10 h11 h14 rewrite [h6, h9, h15] rfl done done theorem fund_thm_arith (n : Nat) (h : n ≥ 1) : ∃! (l : List Nat), prime_factorization n l := by exists_unique · -- Existence show ∃ (l : List Nat), prime_factorization n l from exists_prime_factorization n h done · -- Uniqueness fix l1 : List Nat; fix l2 : List Nat assume h1 : prime_factorization n l1 assume h2 : prime_factorization n l2 define at h1; define at h2 have h3 : prod l1 = n := h1.right rewrite [←h2.right] at h3 show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3 done done /- Section 7.3 -/ theorem congr_refl (m : Nat) : ∀ (a : Int), a ≡ a (MOD m) := by fix a : Int define --Goal : ∃ (c : Int), a - a = ↑m * c apply Exists.intro 0 ring done theorem congr_symm {m : Nat} : ∀ {a b : Int}, a ≡ b (MOD m) → b ≡ a (MOD m) := by fix a : Int; fix b : Int assume h1 : a ≡ b (MOD m) define at h1 --h1 : ∃ (c : Int), a - b = ↑m * c define --Goal : ∃ (c : Int), b - a = ↑m * c obtain (c : Int) (h2 : a - b = m * c) from h1 apply Exists.intro (-c) show b - a = m * (-c) from calc b - a _ = -(a - b) := by ring _ = -(m * c) := by rw [h2] _ = m * (-c) := by ring done theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry /- Fundamental properties of congruence classes -/ lemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) : X = Y ↔ X.val = Y.val := Fin.ext_iff lemma val_nat_eq_mod (n k : Nat) : ([k]_(n + 1)).val = k % (n + 1) := by rfl lemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl theorem cc_rep {m : Nat} (X : ZMod m) : ∃ (a : Int), X = [a]_m := match m with | 0 => by apply Exists.intro X rfl done | n + 1 => by apply Exists.intro ↑(X.val) have h1 : X.val < n + 1 := Fin.prop X rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1] rfl done theorem add_class (m : Nat) (a b : Int) : [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm theorem mul_class (m : Nat) (a b : Int) : [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm lemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ [a - b]_m = [0]_m := by apply Iff.intro · -- (→) assume h1 : [a]_m = [b]_m have h2 : a - b = a + (-b) := by ring have h3 : b + (-b) = 0 := by ring show [a - b]_m = [0]_m from calc [a - b]_m _ = [a + (-b)]_m := by rw [h2] _ = [a]_m + [-b]_m := by rw [add_class] _ = [b]_m + [-b]_m := by rw [h1] _ = [b + -b]_m := by rw [add_class] _ = [0]_m := by rw [h3] done · -- (←) assume h1 : [a - b]_m = [0]_m have h2 : b + (a - b) = a := by ring have h3 : b + 0 = b := by ring show [a]_m = [b]_m from calc [a]_m _ = [b + (a - b)]_m := by rw [h2] _ = [b]_m + [a - b]_m := by rw [add_class] _ = [b]_m + [0]_m := by rw [h1] _ = [b + 0]_m := by rw [add_class] _ = [b]_m := by rw [h3] done done lemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) : [a]_m = [0]_m → [-a]_m = [0]_m := by assume h1 : [a]_m = [0]_m have h2 : 0 + (-a) = -a := by ring have h3 : a + (-a) = 0 := by ring show [-a]_m = [0]_m from calc [-a]_m _ = [0 + (-a)]_m := by rw [h2] _ = [0]_m + [-a]_m := by rw [add_class] _ = [a]_m + [-a]_m := by rw [h1] _ = [a + (-a)]_m := by rw [add_class] _ = [0]_m := by rw [h3] done lemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) : [-a]_m = [0]_m ↔ [a]_m = [0]_m := by apply Iff.intro _ (cc_neg_zero_of_cc_zero m a) assume h1 : [-a]_m = [0]_m have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1 have h3 : -(-a) = a := by ring rewrite [h3] at h2 show [a]_m = [0]_m from h2 done lemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl lemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m ↔ m ∣ k := match m with | 0 => by have h : (0 : Int) = (↑(0 : Nat) : Int) := by rfl rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj] apply Iff.intro · -- (→) assume h1 : k = 0 rewrite [h1] show 0 ∣ 0 from dvd_self 0 done · -- (←) assume h1 : 0 ∣ k obtain (c : Nat) (h2 : k = 0 * c) from h1 rewrite [h2] ring done done | n + 1 => by rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero] show k % (n + 1) = 0 ↔ n + 1 ∣ k from (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm done lemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m ↔ ↑m ∣ a := by obtain (k : Nat) (h1 : a = ↑k ∨ a = -↑k) from Int.eq_nat_or_neg a by_cases on h1 · -- Case 1. h1: a = ↑k rewrite [h1, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done · -- Case 2. h1: a = -↑k rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done done theorem cc_eq_iff_congr (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ a ≡ b (MOD m) := calc [a]_m = [b]_m _ ↔ [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b _ ↔ ↑m ∣ (a - b) := cc_zero_iff_dvd m (a - b) _ ↔ a ≡ b (MOD m) := by rfl /- End of fundamental properties of congruence classes -/ lemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m := by have h1 : (↑m : Int) ≠ 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m) show 0 ≤ a % m from Int.emod_nonneg a h1 done lemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m) have h2 : (↑m : Int) > 0 := (Nat.cast_pos).rtl h1 show a % m < m from Int.emod_lt_of_pos a h2 done lemma congr_mod_mod (m : Nat) (a : Int) : a ≡ a % m (MOD m) := by define have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m apply Exists.intro (a / m) show a - a % m = m * (a / m) from calc a - (a % m) _ = m * (a / m) + a % m - a % m := by rw [h1] _ = m * (a / m) := by ring done lemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) := And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a)) theorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) : ∃! (r : Int), 0 ≤ r ∧ r < m ∧ a ≡ r (MOD m) := by exists_unique · -- Existence apply Exists.intro (a % m) show 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) from mod_cmpl_res m a done · -- Uniqueness fix r1 : Int; fix r2 : Int assume h1 : 0 ≤ r1 ∧ r1 < m ∧ a ≡ r1 (MOD m) assume h2 : 0 ≤ r2 ∧ r2 < m ∧ a ≡ r2 (MOD m) have h3 : r1 ≡ r2 (MOD m) := congr_trans (congr_symm h1.right.right) h2.right.right obtain (d : Int) (h4 : r1 - r2 = m * d) from h3 have h5 : r1 - r2 < m * 1 := by linarith have h6 : m * (-1) < r1 - r2 := by linarith rewrite [h4] at h5 --h5 : m * d < m * 1 rewrite [h4] at h6 --h6 : m * -1 < m * d have h7 : (↑m : Int) ≥ 0 := Nat.cast_nonneg m have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7 have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7 have h10 : d = 0 := by linarith show r1 = r2 from calc r1 _ = r1 - r2 + r2 := by ring _ = m * 0 + r2 := by rw [h4, h10] _ = r2 := by ring done done lemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m := (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a) theorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y rewrite [h1, h2] have h3 : a + b = b + a := by ring show [a]_m + [b]_m = [b]_m + [a]_m from calc [a]_m + [b]_m _ = [a + b]_m := add_class m a b _ = [b + a]_m := by rw [h3] _ = [b]_m + [a]_m := (add_class m b a).symm done theorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X rewrite [h1] have h2 : a * 1 = a := by ring show [a]_m * [1]_m = [a]_m from calc [a]_m * [1]_m _ = [a * 1]_m := mul_class m a 1 _ = [a]_m := by rw [h2] done theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry lemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) : [a]_m * [gcd_c2 m a]_m = [1]_m := by set s : Int := gcd_c1 m a have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m define at h1 rewrite [h1, Nat.cast_one] at h2 --h2 : s * ↑m + gcd_c2 m a * ↑a = 1 rewrite [mul_class, cc_eq_iff_congr] define --Goal : ∃ (c : Int), ↑a * gcd_c2 m a - 1 = ↑m * c apply Exists.intro (-s) show a * (gcd_c2 m a) - 1 = m * (-s) from calc a * (gcd_c2 m a) - 1 _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring _ = 1 + m * (-s) - 1 := by rw [h2] _ = m * (-s) := by ring done theorem Theorem_7_3_7 (m a : Nat) : invertible [a]_m ↔ rel_prime m a := by apply Iff.intro · -- (→) assume h1 : invertible [a]_m define at h1 obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1 obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y rewrite [h3, mul_class, cc_eq_iff_congr] at h2 define at h2 obtain (c : Int) (h4 : a * b - 1 = m * c) from h2 rewrite [Exercise_7_2_6] --Goal : ∃ (s t : Int), s * ↑m + t * ↑a = 1 apply Exists.intro (-c) apply Exists.intro b show (-c) * m + b * a = 1 from calc (-c) * m + b * a _ = (-c) * m + (a * b - 1) + 1 := by ring _ = (-c) * m + m * c + 1 := by rw [h4] _ = 1 := by ring done · -- (←) assume h1 : rel_prime m a define show ∃ (Y : ZMod m), [a]_m * Y = [1]_m from Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1) done done /- Section 7.4 -/ section Euler open Euler lemma num_rp_below_base {m : Nat} : num_rp_below m 0 = 0 := by rfl lemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) : num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by have h1 : num_rp_below m (j + 1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : gcd m j = 1 rewrite [if_pos h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1 show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1 done lemma num_rp_below_step_not_rp {m j : Nat} (h : ¬rel_prime m j) : num_rp_below m (j + 1) = num_rp_below m j := by have h1 : num_rp_below m (j +1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : ¬gcd m j = 1 rewrite [if_neg h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j show num_rp_below m (j + 1) = num_rp_below m j from h1 done lemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl #eval phi 10 --Answer: 4 lemma prod_inv_iff_inv {m : Nat} {X : ZMod m} (h1 : invertible X) (Y : ZMod m) : invertible (X * Y) ↔ invertible Y := by apply Iff.intro · -- (→) assume h2 : invertible (X * Y) obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2 apply Exists.intro (X * Z) rewrite [←h3] --Goal : Y * (X * Z) = X * Y * Z ring --Note that ring can do algebra in ZMod m done · -- (←) assume h2 : invertible Y obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1 obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2 apply Exists.intro (Xi * Yi) show (X * Y) * (Xi * Yi) = [1]_m from calc X * Y * (Xi * Yi) _ = (X * Xi) * (Y * Yi) := by ring _ = [1]_m * [1]_m := by rw [h3, h4] _ = [1]_m := Theorem_7_3_6_7 [1]_m done done lemma F_rp_def {m i : Nat} (h : rel_prime m i) : F m i = [i]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h --h : gcd m i = 1 rewrite [if_pos h] at h1 show F m i = [i]_m from h1 done lemma F_not_rp_def {m i : Nat} (h : ¬rel_prime m i) : F m i = [1]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h rewrite [h1, if_neg h] rfl done lemma prod_seq_base {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 0 k f = [1]_m := by rfl lemma prod_seq_step {m : Nat} (n k : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl lemma prod_seq_zero_step {m : Nat} (n : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by rewrite [prod_seq_step, zero_add] rfl done lemma prod_one {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 1 k f = f k := by rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7] rfl done lemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl lemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m := calc [G m a i]_m _ = [(a * i) % m]_m := by rfl _ = [a * i]_m := (cc_eq_mod m (a * i)).symm _ = [a]_m * [i]_m := (mul_class m a i).symm lemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) : rel_prime m (G m a i) ↔ rel_prime m i := by have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1 show rel_prime m (G m a i) ↔ rel_prime m i from calc rel_prime m (G m a i) _ ↔ invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm _ ↔ invertible ([a]_m * [i]_m) := by rw [cc_G] _ ↔ invertible [i]_m := prod_inv_iff_inv h2 ([i]_m) _ ↔ rel_prime m i := Theorem_7_3_7 m i done lemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) : F m (G m a i) = [a]_m * F m i := by have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2 show F m (G m a i) = [a]_m * F m i from calc F m (G m a i) _ = [G m a i]_m := F_rp_def h3 _ = [a]_m * [i]_m := cc_G m a i _ = [a]_m * F m i := by rw [F_rp_def h2] done lemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : ¬rel_prime m i) : F m (G m a i) = [1]_m := by rewrite [←G_rp_iff h1 i] at h2 show F m (G m a i) = [1]_m from F_not_rp_def h2 done lemma FG_prod {m a : Nat} (h1 : rel_prime m a) : ∀ (k : Nat), prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by by_induc · -- Base Case show prod_seq 0 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from calc prod_seq 0 0 ((F m) ∘ (G m a)) _ = [1]_m := prod_seq_base _ _ _ = [a]_m ^ 0 * [1]_m := by ring _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by rw [num_rp_below_base, prod_seq_base] done · -- Induction Step fix k : Nat assume ih : prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) by_cases h2 : rel_prime m k · -- Case 1. h2 : rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([a]_m * F m k) := by rw [FG_rp h1 h2] _ = [a]_m ^ ((num_rp_below m k) + 1) * ((prod_seq k 0 (F m)) * F m k) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_rp h2, prod_seq_zero_step] done · -- Case 2. h2 : ¬rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([1]_m) := by rw [FG_not_rp h1 h2] _ = [a]_m ^ (num_rp_below m k) * (prod_seq k 0 (F m) * ([1]_m)) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_not_rp h2, prod_seq_zero_step, F_not_rp_def h2] done done done lemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by define --Goal : ∀ i < m, G m a i < m fix i : Nat assume h1 : i < m rewrite [G_def] --Goal : a * i % m < m show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m) done lemma left_inv_one_one_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g' (g i) = i) : one_one_below n g := sorry lemma right_inv_onto_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g (g' i) = i) (h2 : maps_below n g') : onto_below n g := by define at h2; define fix k : Nat assume h3 : k < n apply Exists.intro (g' k) show g' k < n ∧ g (g' k) = k from And.intro (h2 k h3) (h1 k h3) done lemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m * [inv_mod m a]_m = [1]_m := by have h2 : 0 ≤ (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a) show [a]_m * [inv_mod m a]_m = [1]_m from calc [a]_m * [inv_mod m a]_m _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2] _ = [a]_m * [gcd_c2 m a]_m := by rw [←cc_eq_mod] _ = [1]_m := gcd_c2_inv h1 done lemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m := calc a * (b % m) % m = a % m * (b % m % m) % m := Nat.mul_mod _ _ _ _ = a % m * (b % m) % m := by rw [Nat.mod_mod] _ = a * b % m := (Nat.mul_mod _ _ _).symm lemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm] rfl done theorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] : ↑a ≡ ↑b (MOD m) ↔ a % m = b % m := sorry lemma mul_inv_mod_cancel {m a i : Nat} [NeZero m] (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1 rewrite [mul_class, cc_eq_iff_congr, ←Nat.cast_mul, ←Nat.cast_one, congr_iff_mod_eq_Nat] at h3 show a * inv_mod m a * i % m = i from calc a * (inv_mod m a) * i % m _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = 1 % m * i % m := by rw [h3] _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = i % m := by rw [one_mul] _ = i := Nat.mod_eq_of_lt h2 done lemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl lemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, G m a (Ginv m a i) = i := by fix i : Nat assume h2 : i < m show G m a (Ginv m a i) = i from calc G m a (Ginv m a i) _ = a * ((inv_mod m a * i) % m) % m := by rfl _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, Ginv m a (G m a i) = i := by fix i : Nat assume h2 : i < m show Ginv m a (G m a i) = i from calc Ginv m a (G m a i) _ = inv_mod m a * ((a * i) % m) % m := by rfl _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc, mul_comm (inv_mod m a)] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_maps_below (m a : Nat) [NeZero m] : maps_below m (Ginv m a) := G_maps_below m (inv_mod m a) lemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : one_one_below m (G m a) := left_inv_one_one_below (Ginv_left_inv h1) lemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : onto_below m (G m a) := right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a) lemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : perm_below m (G m a) := And.intro (G_maps_below m a) (And.intro (G_one_one_below h1) (G_onto_below h1)) --Permuting a product of congruence classes doesn't change product lemma swap_fst (u v : Nat) : swap u v u = v := by define : swap u v u --Goal : (if u = u then v else if u = v then u else u) = v have h : u = u := by rfl rewrite [if_pos h] rfl done lemma swap_snd (u v : Nat) : swap u v v = u := by define : swap u v v by_cases h1 : v = u · -- Case 1. h1 : v = u rewrite [if_pos h1] show v = u from h1 done · -- Case 2. h1 : v ≠ u rewrite [if_neg h1] have h2 : v = v := by rfl rewrite [if_pos h2] rfl done done lemma swap_other {u v i : Nat} (h1 : i ≠ u) (h2 : i ≠ v) : swap u v i = i := by define : swap u v i rewrite [if_neg h1, if_neg h2] rfl done lemma swap_values (u v i : Nat) : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := by by_cases h1 : i = u · -- Case 1. h1 : i = u apply Or.inl rewrite [h1] show swap u v u = v from swap_fst u v done · -- Case 2. h1 : i ≠ u apply Or.inr by_cases h2 : i = v · -- Case 2.1. h2 : i = v apply Or.inl rewrite [h2] show swap u v v = u from swap_snd u v done · -- Case 2.2. h2 : i ≠ v apply Or.inr show swap u v i = i from swap_other h1 h2 done done done lemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by define fix i : Nat assume h3 : i < n have h4 : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := swap_values u v i by_cases on h4 · -- Case 1. h4 : swap u v i = v rewrite [h4] show v < n from h2 done · -- Case 2. by_cases on h4 · -- Case 2.1. h4 : swap u v i = u rewrite [h4] show u < n from h1 done · -- Case 2.2. h4 : swap u v i = i rewrite [h4] show i < n from h3 done done done lemma swap_swap (u v n : Nat) : ∀ i < n, swap u v (swap u v i) = i := by fix i : Nat assume h : i < n by_cases h1 : i = u · -- Case 1. h1 : i = u rewrite [h1, swap_fst, swap_snd] rfl done · -- Case 2. h1 : i ≠ u by_cases h2 : i = v · -- Case 2.1. h2 : i = v rewrite [h2, swap_snd, swap_fst] rfl done · -- Case 2.2. h2 : i ≠ v rewrite [swap_other h1 h2, swap_other h1 h2] rfl done done done lemma swap_one_one_below (u v n) : one_one_below n (swap u v) := left_inv_one_one_below (swap_swap u v n) lemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) := right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2) lemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) := And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2)) lemma comp_perm_below {n : Nat} {f g : Nat → Nat} (h1 : perm_below n f) (h2 : perm_below n g) : perm_below n (f ∘ g) := sorry lemma trivial_swap (u : Nat) : swap u u = id := by apply funext fix x : Nat by_cases h1 : x = u · -- Case 1. h1 : x = u rewrite [h1, swap_fst] rfl done · -- Case 2. h1 : x ≠ u rewrite [swap_other h1 h1] rfl done done lemma prod_eq_fun {m : Nat} (f g : Nat → ZMod m) (k : Nat) : ∀ (n : Nat), (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g := by by_induc · -- Base Case assume h : (∀ i < 0, f (k + i) = g (k + i)) rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g assume h1 : ∀ i < n + 1, f (k + i) = g (k + i) have h2 : ∀ i < n, f (k + i) = g (k + i) := by fix i : Nat assume h2 : i < n have h3 : i < n + 1 := by linarith show f (k + i) = g (k + i) from h1 i h3 done have h3 : prod_seq n k f = prod_seq n k g := ih h2 have h4 : n < n + 1 := Nat.lt_succ_self n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4] rfl done done lemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := by have h2 : ∀ (i : Nat), i < u → (f ∘ swap u n) (0 + i) = f (0 + i) := by fix i : Nat assume h2 : i < u have h3 : 0 + i ≠ u := by linarith have h4 : 0 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl done show prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f from prod_eq_fun (f ∘ swap u n) f 0 u h2 done lemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := by have h2 : ∀ i < j, (f ∘ swap u n) (u + 1 + i) = f (u + 1 + i) := by fix i : Nat assume h2 : i < j have h3 : u + 1 + i ≠ u := by linarith have h4 : u + 1 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl show prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f from prod_eq_fun (f ∘ swap u n) f (u + 1) j h2 done lemma break_prod {m : Nat} (n : Nat) (f : Nat → ZMod m) : ∀ (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by by_induc · -- Base Case have h : n + 0 = n := by rfl rewrite [prod_seq_base, h, Theorem_7_3_6_7] rfl done · -- Induction Step fix j : Nat assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f rewrite [←add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc] rfl done done lemma break_prod_twice {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f := break_prod n f 1 rewrite [prod_one] at h2 have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f := break_prod (u + 1) f j rewrite [←h1] at h3 have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f := break_prod u f 1 rewrite [prod_one] at h4 rewrite [h3, h4] at h2 show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2 done lemma swap_prod_eq_prod {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f := by by_cases h2 : u = n · -- Case 1. h2 : u = n rewrite [h2, trivial_swap n] --Goal : prod_seq (n + 1) 0 (f ∘ id) = prod_seq (n + 1) 0 f rfl done · -- Case 2. h2 : ¬u = n have h3 : u + 1 ≤ n := Nat.lt_of_le_of_ne h1 h2 obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3 have break_f : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := break_prod_twice f h4 have break_fs : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_prod_twice (f ∘ swap u n) h4 have f_eq_fs_below : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := swap_prod_eq_prod_below f h1 have f_eq_fs_btwn : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4 show prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f from calc prod_seq (n + 1) 0 (f ∘ swap u n) _ = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_fs _ = prod_seq u 0 f * (f ∘ swap u n) u * prod_seq j (u + 1) f * (f ∘ swap u n) n := by rw [f_eq_fs_below, f_eq_fs_btwn] _ = prod_seq u 0 f * f (swap u n u) * prod_seq j (u + 1) f * f (swap u n n) := by rfl _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by rw [swap_fst, swap_snd] _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring _ = prod_seq (n + 1) 0 f := break_f.symm done done lemma perm_below_fixed {n : Nat} {g : Nat → Nat} (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry lemma perm_prod {m : Nat} (f : Nat → ZMod m) : ∀ (n : Nat), ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) := by by_induc · -- Base Case fix g : Nat → Nat assume h1 : perm_below 0 g rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) fix g : Nat → Nat assume g_pb : perm_below (n + 1) g define at g_pb have g_ob : onto_below (n + 1) g := g_pb.right.right define at g_ob have h1 : n < n + 1 := by linarith obtain (u : Nat) (h2 : u < n + 1 ∧ g u = n) from g_ob n h1 have s_pb : perm_below (n + 1) (swap u n) := swap_perm_below h2.left h1 have gs_pb_n1 : perm_below (n + 1) (g ∘ swap u n) := comp_perm_below g_pb s_pb have gs_fix_n : (g ∘ swap u n) n = n := calc (g ∘ swap u n) n _ = g (swap u n n) := by rfl _ = g u := by rw [swap_snd] _ = n := h2.right have gs_pb_n : perm_below n (g ∘ swap u n) := perm_below_fixed gs_pb_n1 gs_fix_n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f ∘ (g ∘ swap u n)) := ih (g ∘ swap u n) gs_pb_n have h3 : u ≤ n := by linarith show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f ∘ g) from calc prod_seq (n + 1) 0 f _ = prod_seq n 0 f * f n := prod_seq_zero_step n f _ = prod_seq n 0 (f ∘ (g ∘ swap u n)) * f ((g ∘ swap u n) n) := by rw [gs_prod, gs_fix_n] _ = prod_seq n 0 (f ∘ g ∘ swap u n) * (f ∘ g ∘ swap u n) n := by rfl _ = prod_seq (n + 1) 0 (f ∘ g ∘ swap u n) := (prod_seq_zero_step n (f ∘ g ∘ swap u n)).symm _ = prod_seq (n + 1) 0 ((f ∘ g) ∘ swap u n) := by rfl _ = prod_seq (n + 1) 0 (f ∘ g) := swap_prod_eq_prod (f ∘ g) h3 done done

lemma F_invertible (m i : Nat) : invertible (F m i)

HTPI.F_invertible

htpi/HTPILib/Chap7.lean

htpi

/- Copyright 2023 Daniel J. Velleman -/ import HTPILib.Chap6 namespace HTPI /- Definitions -/ lemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by have h : n + 1 > 0 := Nat.succ_pos n show a % (n + 1) < n + 1 from Nat.mod_lt a h done def gcd (a b : Nat) : Nat := match b with | 0 => a | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd (n + 1) (a % (n + 1)) termination_by b mutual def gcd_c1 (a b : Nat) : Int := match b with | 0 => 1 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c2 (n + 1) (a % (n + 1)) --Corresponds to s = t' termination_by b def gcd_c2 (a b : Nat) : Int := match b with | 0 => 0 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c1 (n + 1) (a % (n + 1)) - (gcd_c2 (n + 1) (a % (n + 1))) * ↑(a / (n + 1)) --Corresponds to t = s' - t'q termination_by b end def prime (n : Nat) : Prop := 2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n def prime_factor (p n : Nat) : Prop := prime p ∧ p ∣ n def all_prime (l : List Nat) : Prop := ∀ p ∈ l, prime p def nondec (l : List Nat) : Prop := match l with | [] => True --Of course, True is a proposition that is always true | n :: L => (∀ m ∈ L, n ≤ m) ∧ nondec L def nondec_prime_list (l : List Nat) : Prop := all_prime l ∧ nondec l def prod (l : List Nat) : Nat := match l with | [] => 1 | n :: L => n * (prod L) def prime_factorization (n : Nat) (l : List Nat) : Prop := nondec_prime_list l ∧ prod l = n def rel_prime (a b : Nat) : Prop := gcd a b = 1 def congr_mod (m : Nat) (a b : Int) : Prop := (↑m : Int) ∣ (a - b) def cc (m : Nat) (a : Int) : ZMod m := (↑a : ZMod m) notation:50 a " ≡ " b " (MOD " m ")" => congr_mod m a b notation:max "["a"]_"m:max => cc m a def invertible {m : Nat} (X : ZMod m) : Prop := ∃ (Y : ZMod m), X * Y = [1]_m def num_rp_below (m k : Nat) : Nat := match k with | 0 => 0 | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j def phi (m : Nat) : Nat := num_rp_below m m def prod_seq {m : Nat} (j k : Nat) (f : Nat → ZMod m) : ZMod m := match j with | 0 => [1]_m | n + 1 => prod_seq n k f * f (k + n) def maps_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i < n, g i < n def one_one_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i1 < n, ∀ i2 < n, g i1 = g i2 → i1 = i2 def onto_below (n : Nat) (g : Nat → Nat) : Prop := ∀ k < n, ∃ i < n, g i = k def perm_below (n : Nat) (g : Nat → Nat) : Prop := maps_below n g ∧ one_one_below n g ∧ onto_below n g def inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m) def swap (u v i : Nat) : Nat := if i = u then v else if i = v then u else i namespace Euler --For definitions specific to Euler's theorem def F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m def G (m a i : Nat) : Nat := (a * i) % m def Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i end Euler /- Section 7.1 -/ theorem dvd_mod_of_dvd_a_b {a b d : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ (a % b) := by set q : Nat := a / b have h3 : b * q + a % b = a := Nat.div_add_mod a b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 define --Goal : ∃ (c : Nat), a % b = d * c apply Exists.intro (j - k * q) show a % b = d * (j - k * q) from calc a % b _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm _ = a - b * q := by rw [h3] _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc] _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm done theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry #eval gcd 672 161 --Answer: 7 lemma gcd_base (a : Nat) : gcd a 0 = a := by rfl lemma gcd_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd a b = gcd b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1)) rfl done lemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b ≠ 0) : a % b < b := by have h1 : b > 0 := Nat.pos_of_ne_zero h show a % b < b from Nat.mod_lt a h1 done lemma dvd_self (n : Nat) : n ∣ n := by apply Exists.intro 1 ring done theorem gcd_dvd : ∀ (b a : Nat), (gcd a b) ∣ a ∧ (gcd a b) ∣ b := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd a b_1) ∣ a ∧ (gcd a b_1) ∣ b_1 fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_base] --Goal: a ∣ a ∧ a ∣ 0 apply And.intro (dvd_self a) define apply Exists.intro 0 rfl done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_nonzero a h1] --Goal : gcd b (a % b) ∣ a ∧ gcd b (a % b) ∣ b have h2 : a % b < b := mod_nonzero_lt a h1 have h3 : (gcd b (a % b)) ∣ b ∧ (gcd b (a % b)) ∣ (a % b) := ih (a % b) h2 b apply And.intro _ h3.left show (gcd b (a % b)) ∣ a from dvd_a_of_dvd_b_mod h3.left h3.right done done theorem gcd_dvd_left (a b : Nat) : (gcd a b) ∣ a := (gcd_dvd b a).left theorem gcd_dvd_right (a b : Nat) : (gcd a b) ∣ b := (gcd_dvd b a).right lemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl lemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c1 a b = gcd_c2 b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done lemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl lemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * ↑(a / b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done theorem gcd_lin_comb : ∀ (b a : Nat), (gcd_c1 a b) * ↑a + (gcd_c2 a b) * ↑b = ↑(gcd a b) := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd_c1 a b_1) * ↑a + (gcd_c2 a b_1) * ↑b_1 = ↑(gcd a b_1) fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base] --Goal : 1 * ↑a + 0 * ↑0 = ↑a ring done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1] --Goal : gcd_c2 b (a % b) * ↑a + -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * ↑(a / b)) * ↑b = -- ↑(gcd b (a % b)) set r : Nat := a % b set q : Nat := a / b set s : Int := gcd_c1 b r set t : Int := gcd_c2 b r --Goal : t * ↑a + (s - t * ↑q) * ↑b = ↑(gcd b r) have h2 : r < b := mod_nonzero_lt a h1 have h3 : s * ↑b + t * ↑r = ↑(gcd b r) := ih r h2 b have h4 : b * q + r = a := Nat.div_add_mod a b rewrite [←h3, ←h4] rewrite [Nat.cast_add, Nat.cast_mul] --Goal : t * (↑b * ↑q + ↑r) + (s - t * ↑q) * ↑b = s * ↑b + t * ↑r ring done done #eval gcd_c1 672 161 --Answer: 6 #eval gcd_c2 672 161 --Answer: -25 --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161 theorem Theorem_7_1_6 {d a b : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ gcd a b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑d ∣ ↑(gcd a b) set s : Int := gcd_c1 a b set t : Int := gcd_c2 a b have h3 : s * ↑a + t * ↑b = ↑(gcd a b) := gcd_lin_comb b a rewrite [←h3] --Goal : ↑d ∣ s * ↑a + t * ↑b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul] --Goal : ↑d ∣ s * (↑d * ↑j) + t * (↑d * ↑k) define apply Exists.intro (s * ↑j + t * ↑k) ring done /- Section 7.2 -/ theorem dvd_trans {a b c : Nat} (h1 : a ∣ b) (h2 : b ∣ c) : a ∣ c := by define at h1; define at h2; define obtain (m : Nat) (h3 : b = a * m) from h1 obtain (n : Nat) (h4 : c = b * n) from h2 rewrite [h3, mul_assoc] at h4 apply Exists.intro (m * n) show c = a * (m * n) from h4 done lemma exists_prime_factor : ∀ (n : Nat), 2 ≤ n → ∃ (p : Nat), prime_factor p n := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, 2 ≤ n_1 → ∃ (p : Nat), prime_factor p n_1 assume h1 : 2 ≤ n by_cases h2 : prime n · -- Case 1. h2 : prime n apply Exists.intro n define --Goal : prime n ∧ n ∣ n show prime n ∧ n ∣ n from And.intro h2 (dvd_self n) done · -- Case 2. h2 : ¬prime n define at h2 --h2 : ¬(2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n) demorgan at h2 disj_syll h2 h1 obtain (a : Nat) (h3 : ∃ (b : Nat), a * b = n ∧ a < n ∧ b < n) from h2 obtain (b : Nat) (h4 : a * b = n ∧ a < n ∧ b < n) from h3 have h5 : 2 ≤ a := by by_contra h6 have h7 : a ≤ 1 := by linarith have h8 : n ≤ b := calc n _ = a * b := h4.left.symm _ ≤ 1 * b := by rel [h7] _ = b := by ring linarith --n ≤ b contradicts b < n done have h6 : ∃ (p : Nat), prime_factor p a := ih a h4.right.left h5 obtain (p : Nat) (h7 : prime_factor p a) from h6 apply Exists.intro p define --Goal : prime p ∧ p ∣ n define at h7 --h7 : prime p ∧ p ∣ a apply And.intro h7.left have h8 : a ∣ n := by apply Exists.intro b show n = a * b from (h4.left).symm done show p ∣ n from dvd_trans h7.right h8 done done lemma exists_least_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q := by set S : Set Nat := {p : Nat | prime_factor p n} have h2 : ∃ (p : Nat), p ∈ S := exists_prime_factor n h show ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q from well_ord_princ S h2 done lemma all_prime_nil : all_prime [] := by define --Goal : ∀ p ∈ [], prime p fix p : Nat contrapos --Goal : ¬prime p → p ∉ [] assume h1 : ¬prime p show p ∉ [] from List.not_mem_nil p done lemma all_prime_cons (n : Nat) (L : List Nat) : all_prime (n :: L) ↔ prime n ∧ all_prime L := by apply Iff.intro · -- (→) assume h1 : all_prime (n :: L) --Goal : prime n ∧ all_prime L define at h1 --h1 : ∀ p ∈ n :: L, prime p apply And.intro (h1 n (List.mem_cons_self n L)) define --Goal : ∀ p ∈ L, prime p fix p : Nat assume h2 : p ∈ L show prime p from h1 p (List.mem_cons_of_mem n h2) done · -- (←) assume h1 : prime n ∧ all_prime L --Goal : all_prime (n :: l) define : all_prime L at h1 define fix p : Nat assume h2 : p ∈ n :: L rewrite [List.mem_cons] at h2 --h2 : p = n ∨ p ∈ L by_cases on h2 · -- Case 1. h2 : p = n rewrite [h2] show prime n from h1.left done · -- Case 2. h2 : p ∈ L show prime p from h1.right p h2 done done done lemma nondec_nil : nondec [] := by define --Goal : True trivial --trivial proves some obviously true statements, such as True done lemma nondec_cons (n : Nat) (L : List Nat) : nondec (n :: L) ↔ (∀ m ∈ L, n ≤ m) ∧ nondec L := by rfl lemma prod_nil : prod [] = 1 := by rfl lemma prod_cons : prod (n :: L) = n * (prod L) := by rfl lemma exists_cons_of_length_eq_succ {A : Type} {l : List A} {n : Nat} (h : l.length = n + 1) : ∃ (a : A) (L : List A), l = a :: L ∧ L.length = n := by have h1 : ¬l.length = 0 := by linarith rewrite [List.length_eq_zero] at h1 obtain (a : A) (h2 : ∃ (L : List A), l = a :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List A) (h3 : l = a :: L) from h2 apply Exists.intro a apply Exists.intro L apply And.intro h3 have h4 : (a :: L).length = L.length + 1 := List.length_cons a L rewrite [←h3, h] at h4 show L.length = n from (Nat.add_right_cancel h4).symm done lemma list_elt_dvd_prod_by_length (a : Nat) : ∀ (n : Nat), ∀ (l : List Nat), l.length = n → a ∈ l → a ∣ prod l := by by_induc · --Base Case fix l : List Nat assume h1 : l.length = 0 rewrite [List.length_eq_zero] at h1 --h1 : l = [] rewrite [h1] --Goal : a ∈ [] → a ∣ prod [] contrapos assume h2 : ¬a ∣ prod [] show a ∉ [] from List.not_mem_nil a done · -- Induction Step fix n : Nat assume ih : ∀ (l : List Nat), List.length l = n → a ∈ l → a ∣ prod l fix l : List Nat assume h1 : l.length = n + 1 --Goal : a ∈ l → a ∣ prod l obtain (b : Nat) (h2 : ∃ (L : List Nat), l = b :: L ∧ L.length = n) from exists_cons_of_length_eq_succ h1 obtain (L : List Nat) (h3 : l = b :: L ∧ L.length = n) from h2 have h4 : a ∈ L → a ∣ prod L := ih L h3.right assume h5 : a ∈ l rewrite [h3.left, prod_cons] --Goal : a ∣ b * prod L rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b ∨ a ∈ L by_cases on h5 · -- Case 1. h5 : a = b apply Exists.intro (prod L) rewrite [h5] rfl done · -- Case 2. h5 : a ∈ L have h6 : a ∣ prod L := h4 h5 have h7 : prod L ∣ b * prod L := by apply Exists.intro b ring done show a ∣ b * prod L from dvd_trans h6 h7 done done done lemma list_elt_dvd_prod {a : Nat} {l : List Nat} (h : a ∈ l) : a ∣ prod l := by set n : Nat := l.length have h1 : l.length = n := by rfl show a ∣ prod l from list_elt_dvd_prod_by_length a n l h1 h done lemma exists_prime_factorization : ∀ (n : Nat), n ≥ 1 → ∃ (l : List Nat), prime_factorization n l := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, n_1 ≥ 1 → ∃ (l : List Nat), prime_factorization n_1 l assume h1 : n ≥ 1 by_cases h2 : n = 1 · -- Case 1. h2 : n = 1 apply Exists.intro [] define apply And.intro · -- Proof of nondec_prime_list [] define show all_prime [] ∧ nondec [] from And.intro all_prime_nil nondec_nil done · -- Proof of prod [] = n rewrite [prod_nil, h2] rfl done done · -- Case 2. h2 : n ≠ 1 have h3 : n ≥ 2 := lt_of_le_of_ne' h1 h2 obtain (p : Nat) (h4 : prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q) from exists_least_prime_factor h3 have p_prime_factor : prime_factor p n := h4.left define at p_prime_factor have p_prime : prime p := p_prime_factor.left have p_dvd_n : p ∣ n := p_prime_factor.right have p_least : ∀ (q : Nat), prime_factor q n → p ≤ q := h4.right obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n have h5 : m ≠ 0 := by contradict h1 with h6 have h7 : n = 0 := calc n _ = p * m := n_eq_pm _ = p * 0 := by rw [h6] _ = 0 := by ring rewrite [h7] decide done have m_pos : 0 < m := Nat.pos_of_ne_zero h5 have m_lt_n : m < n := by define at p_prime show m < n from calc m _ < m + m := by linarith _ = 2 * m := by ring _ ≤ p * m := by rel [p_prime.left] _ = n := n_eq_pm.symm done obtain (L : List Nat) (h6 : prime_factorization m L) from ih m m_lt_n m_pos define at h6 have ndpl_L : nondec_prime_list L := h6.left define at ndpl_L apply Exists.intro (p :: L) define apply And.intro · -- Proof of nondec_prime_list (p :: L) define apply And.intro · -- Proof of all_prime (p :: L) rewrite [all_prime_cons] show prime p ∧ all_prime L from And.intro p_prime ndpl_L.left done · -- Proof of nondec (p :: L) rewrite [nondec_cons] apply And.intro _ ndpl_L.right fix q : Nat assume q_in_L : q ∈ L have h7 : q ∣ prod L := list_elt_dvd_prod q_in_L rewrite [h6.right] at h7 --h7 : q ∣ m have h8 : m ∣ n := by apply Exists.intro p rewrite [n_eq_pm] ring done have q_dvd_n : q ∣ n := dvd_trans h7 h8 have ap_L : all_prime L := ndpl_L.left define at ap_L have q_prime_factor : prime_factor q n := And.intro (ap_L q q_in_L) q_dvd_n show p ≤ q from p_least q q_prime_factor done done · -- Proof of prod (p :: L) = n rewrite [prod_cons, h6.right, n_eq_pm] rfl done done done theorem Theorem_7_2_2 {a b c : Nat} (h1 : c ∣ a * b) (h2 : rel_prime a c) : c ∣ b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑c ∣ ↑b define at h1; define at h2; define obtain (j : Nat) (h3 : a * b = c * j) from h1 set s : Int := gcd_c1 a c set t : Int := gcd_c2 a c have h4 : s * ↑a + t * ↑c = ↑(gcd a c) := gcd_lin_comb c a rewrite [h2, Nat.cast_one] at h4 --h4 : s * ↑a + t * ↑c = (1 : Int) apply Exists.intro (s * ↑j + t * ↑b) show ↑b = ↑c * (s * ↑j + t * ↑b) from calc ↑b _ = (1 : Int) * ↑b := (one_mul _).symm _ = (s * ↑a + t * ↑c) * ↑b := by rw [h4] _ = s * (↑a * ↑b) + t * ↑c * ↑b := by ring _ = s * (↑c * ↑j) + t * ↑c * ↑b := by rw [←Nat.cast_mul a b, h3, Nat.cast_mul c j] _ = ↑c * (s * ↑j + t * ↑b) := by ring done lemma le_nonzero_prod_left {a b : Nat} (h : a * b ≠ 0) : a ≤ a * b := by have h1 : b ≠ 0 := by contradict h with h1 rewrite [h1] ring done have h2 : 1 ≤ b := Nat.pos_of_ne_zero h1 show a ≤ a * b from calc a = a * 1 := (mul_one a).symm _ ≤ a * b := by rel [h2] done lemma le_nonzero_prod_right {a b : Nat} (h : a * b ≠ 0) : b ≤ a * b := by rewrite [mul_comm] rewrite [mul_comm] at h show b ≤ b * a from le_nonzero_prod_left h done lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry lemma rel_prime_of_prime_not_dvd {a p : Nat} (h1 : prime p) (h2 : ¬p ∣ a) : rel_prime a p := by have h3 : gcd a p ∣ a := gcd_dvd_left a p have h4 : gcd a p ∣ p := gcd_dvd_right a p have h5 : gcd a p = 1 ∨ gcd a p = p := dvd_prime h1 h4 have h6 : gcd a p ≠ p := by contradict h2 with h6 rewrite [h6] at h3 show p ∣ a from h3 done disj_syll h5 h6 show rel_prime a p from h5 done theorem Theorem_7_2_3 {a b p : Nat} (h1 : prime p) (h2 : p ∣ a * b) : p ∣ a ∨ p ∣ b := by or_right with h3 have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3 show p ∣ b from Theorem_7_2_2 h2 h4 done lemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a ≥ 1 := by have h1 : a ≠ 0 := by by_contra h1 rewrite [h1] at h contradict h linarith done show a ≥ 1 from Nat.pos_of_ne_zero h1 done lemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by have h1 : a ≥ 1 := ge_one_of_prod_one h have h2 : a * b ≠ 0 := by linarith have h3 : a ≤ a * b := le_nonzero_prod_left h2 rewrite [h] at h3 show a = 1 from Nat.le_antisymm h3 h1 done lemma eq_one_of_dvd_one {n : Nat} (h : n ∣ 1) : n = 1 := by obtain (j : Nat) (h1 : 1 = n * j) from h show n = 1 from eq_one_of_prod_one h1.symm done lemma prime_not_one {p : Nat} (h : prime p) : p ≠ 1 := by define at h linarith done theorem Theorem_7_2_4 {p : Nat} (h1 : prime p) : ∀ (l : List Nat), p ∣ prod l → ∃ a ∈ l, p ∣ a := by apply List.rec · -- Base Case. Goal : p ∣ prod [] → ∃ a ∈ [], p ∣ a rewrite [prod_nil] assume h2 : p ∣ 1 show ∃ a ∈ [], p ∣ a from absurd (eq_one_of_dvd_one h2) (prime_not_one h1) done · -- Induction Step fix b : Nat fix L : List Nat assume ih : p ∣ prod L → ∃ a ∈ L, p ∣ a --Goal : p ∣ prod (b :: L) → ∃ a ∈ b :: L, p ∣ a assume h2 : p ∣ prod (b :: L) rewrite [prod_cons] at h2 have h3 : p ∣ b ∨ p ∣ prod L := Theorem_7_2_3 h1 h2 by_cases on h3 · -- Case 1. h3 : p ∣ b apply Exists.intro b show b ∈ b :: L ∧ p ∣ b from And.intro (List.mem_cons_self b L) h3 done · -- Case 2. h3 : p ∣ prod L obtain (a : Nat) (h4 : a ∈ L ∧ p ∣ a) from ih h3 apply Exists.intro a show a ∈ b :: L ∧ p ∣ a from And.intro (List.mem_cons_of_mem b h4.left) h4.right done done done lemma prime_in_list {p : Nat} {l : List Nat} (h1 : prime p) (h2 : all_prime l) (h3 : p ∣ prod l) : p ∈ l := by obtain (a : Nat) (h4 : a ∈ l ∧ p ∣ a) from Theorem_7_2_4 h1 l h3 define at h2 have h5 : prime a := h2 a h4.left have h6 : p = 1 ∨ p = a := dvd_prime h5 h4.right disj_syll h6 (prime_not_one h1) rewrite [h6] show a ∈ l from h4.left done lemma first_le_first {p q : Nat} {l m : List Nat} (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m)) (h3 : prod (p :: l) = prod (q :: m)) : p ≤ q := by define at h1; define at h2 have h4 : q ∣ prod (p :: l) := by define apply Exists.intro (prod m) rewrite [←prod_cons] show prod (p :: l) = prod (q :: m) from h3 done have h5 : all_prime (q :: m) := h2.left rewrite [all_prime_cons] at h5 have h6 : q ∈ p :: l := prime_in_list h5.left h1.left h4 have h7 : nondec (p :: l) := h1.right rewrite [nondec_cons] at h7 rewrite [List.mem_cons] at h6 by_cases on h6 · -- Case 1. h6 : q = p linarith done · -- Case 2. h6 : q ∈ l have h8 : ∀ m ∈ l, p ≤ m := h7.left show p ≤ q from h8 q h6 done done lemma nondec_prime_list_tail {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by define at h define rewrite [all_prime_cons, nondec_cons] at h show all_prime l ∧ nondec l from And.intro h.left.right h.right.right done lemma cons_prod_not_one {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : prod (p :: l) ≠ 1 := by define at h have h1 : all_prime (p :: l) := h.left rewrite [all_prime_cons] at h1 rewrite [prod_cons] by_contra h2 show False from (prime_not_one h1.left) (eq_one_of_prod_one h2) done lemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) : l = [] ↔ prod l = 1 := by apply Iff.intro · -- (→) assume h1 : l = [] rewrite [h1] show prod [] = 1 from prod_nil done · -- (←) contrapos assume h1 : ¬l = [] obtain (p : Nat) (h2 : ∃ (L : List Nat), l = p :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List Nat) (h3 : l = p :: L) from h2 rewrite [h3] at h rewrite [h3] show ¬prod (p :: L) = 1 from cons_prod_not_one h done done lemma prime_pos {p : Nat} (h : prime p) : p > 0 := by define at h linarith done theorem Theorem_7_2_5 : ∀ (l1 l2 : List Nat), nondec_prime_list l1 → nondec_prime_list l2 → prod l1 = prod l2 → l1 = l2 := by apply List.rec · -- Base Case. Goal : ∀ (l2 : List Nat), nondec_prime_list [] → -- nondec_prime_list l2 → prod [] = prod l2 → [] = l2 fix l2 : List Nat assume h1 : nondec_prime_list [] assume h2 : nondec_prime_list l2 assume h3 : prod [] = prod l2 rewrite [prod_nil, eq_comm, ←list_nil_iff_prod_one h2] at h3 show [] = l2 from h3.symm done · -- Induction Step fix p : Nat fix L1 : List Nat assume ih : ∀ (L2 : List Nat), nondec_prime_list L1 → nondec_prime_list L2 → prod L1 = prod L2 → L1 = L2 -- Goal : ∀ (l2 : List Nat), nondec_prime_list (p :: L1) → -- nondec_prime_list l2 → prod (p :: L1) = prod l2 → p :: L1 = l2 fix l2 : List Nat assume h1 : nondec_prime_list (p :: L1) assume h2 : nondec_prime_list l2 assume h3 : prod (p :: L1) = prod l2 have h4 : ¬prod (p :: L1) = 1 := cons_prod_not_one h1 rewrite [h3, ←list_nil_iff_prod_one h2] at h4 obtain (q : Nat) (h5 : ∃ (L : List Nat), l2 = q :: L) from List.exists_cons_of_ne_nil h4 obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5 rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2) rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2) have h7 : p ≤ q := first_le_first h1 h2 h3 have h8 : q ≤ p := first_le_first h2 h1 h3.symm have h9 : p = q := by linarith rewrite [h9, prod_cons, prod_cons] at h3 --h3 : q * prod L1 = q * prod L2 have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1 have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2 define at h2 have h12 : all_prime (q :: L2) := h2.left rewrite [all_prime_cons] at h12 have h13 : q > 0 := prime_pos h12.left have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3 have h15 : L1 = L2 := ih L2 h10 h11 h14 rewrite [h6, h9, h15] rfl done done theorem fund_thm_arith (n : Nat) (h : n ≥ 1) : ∃! (l : List Nat), prime_factorization n l := by exists_unique · -- Existence show ∃ (l : List Nat), prime_factorization n l from exists_prime_factorization n h done · -- Uniqueness fix l1 : List Nat; fix l2 : List Nat assume h1 : prime_factorization n l1 assume h2 : prime_factorization n l2 define at h1; define at h2 have h3 : prod l1 = n := h1.right rewrite [←h2.right] at h3 show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3 done done /- Section 7.3 -/ theorem congr_refl (m : Nat) : ∀ (a : Int), a ≡ a (MOD m) := by fix a : Int define --Goal : ∃ (c : Int), a - a = ↑m * c apply Exists.intro 0 ring done theorem congr_symm {m : Nat} : ∀ {a b : Int}, a ≡ b (MOD m) → b ≡ a (MOD m) := by fix a : Int; fix b : Int assume h1 : a ≡ b (MOD m) define at h1 --h1 : ∃ (c : Int), a - b = ↑m * c define --Goal : ∃ (c : Int), b - a = ↑m * c obtain (c : Int) (h2 : a - b = m * c) from h1 apply Exists.intro (-c) show b - a = m * (-c) from calc b - a _ = -(a - b) := by ring _ = -(m * c) := by rw [h2] _ = m * (-c) := by ring done theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry /- Fundamental properties of congruence classes -/ lemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) : X = Y ↔ X.val = Y.val := Fin.ext_iff lemma val_nat_eq_mod (n k : Nat) : ([k]_(n + 1)).val = k % (n + 1) := by rfl lemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl theorem cc_rep {m : Nat} (X : ZMod m) : ∃ (a : Int), X = [a]_m := match m with | 0 => by apply Exists.intro X rfl done | n + 1 => by apply Exists.intro ↑(X.val) have h1 : X.val < n + 1 := Fin.prop X rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1] rfl done theorem add_class (m : Nat) (a b : Int) : [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm theorem mul_class (m : Nat) (a b : Int) : [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm lemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ [a - b]_m = [0]_m := by apply Iff.intro · -- (→) assume h1 : [a]_m = [b]_m have h2 : a - b = a + (-b) := by ring have h3 : b + (-b) = 0 := by ring show [a - b]_m = [0]_m from calc [a - b]_m _ = [a + (-b)]_m := by rw [h2] _ = [a]_m + [-b]_m := by rw [add_class] _ = [b]_m + [-b]_m := by rw [h1] _ = [b + -b]_m := by rw [add_class] _ = [0]_m := by rw [h3] done · -- (←) assume h1 : [a - b]_m = [0]_m have h2 : b + (a - b) = a := by ring have h3 : b + 0 = b := by ring show [a]_m = [b]_m from calc [a]_m _ = [b + (a - b)]_m := by rw [h2] _ = [b]_m + [a - b]_m := by rw [add_class] _ = [b]_m + [0]_m := by rw [h1] _ = [b + 0]_m := by rw [add_class] _ = [b]_m := by rw [h3] done done lemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) : [a]_m = [0]_m → [-a]_m = [0]_m := by assume h1 : [a]_m = [0]_m have h2 : 0 + (-a) = -a := by ring have h3 : a + (-a) = 0 := by ring show [-a]_m = [0]_m from calc [-a]_m _ = [0 + (-a)]_m := by rw [h2] _ = [0]_m + [-a]_m := by rw [add_class] _ = [a]_m + [-a]_m := by rw [h1] _ = [a + (-a)]_m := by rw [add_class] _ = [0]_m := by rw [h3] done lemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) : [-a]_m = [0]_m ↔ [a]_m = [0]_m := by apply Iff.intro _ (cc_neg_zero_of_cc_zero m a) assume h1 : [-a]_m = [0]_m have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1 have h3 : -(-a) = a := by ring rewrite [h3] at h2 show [a]_m = [0]_m from h2 done lemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl lemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m ↔ m ∣ k := match m with | 0 => by have h : (0 : Int) = (↑(0 : Nat) : Int) := by rfl rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj] apply Iff.intro · -- (→) assume h1 : k = 0 rewrite [h1] show 0 ∣ 0 from dvd_self 0 done · -- (←) assume h1 : 0 ∣ k obtain (c : Nat) (h2 : k = 0 * c) from h1 rewrite [h2] ring done done | n + 1 => by rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero] show k % (n + 1) = 0 ↔ n + 1 ∣ k from (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm done lemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m ↔ ↑m ∣ a := by obtain (k : Nat) (h1 : a = ↑k ∨ a = -↑k) from Int.eq_nat_or_neg a by_cases on h1 · -- Case 1. h1: a = ↑k rewrite [h1, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done · -- Case 2. h1: a = -↑k rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done done theorem cc_eq_iff_congr (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ a ≡ b (MOD m) := calc [a]_m = [b]_m _ ↔ [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b _ ↔ ↑m ∣ (a - b) := cc_zero_iff_dvd m (a - b) _ ↔ a ≡ b (MOD m) := by rfl /- End of fundamental properties of congruence classes -/ lemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m := by have h1 : (↑m : Int) ≠ 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m) show 0 ≤ a % m from Int.emod_nonneg a h1 done lemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m) have h2 : (↑m : Int) > 0 := (Nat.cast_pos).rtl h1 show a % m < m from Int.emod_lt_of_pos a h2 done lemma congr_mod_mod (m : Nat) (a : Int) : a ≡ a % m (MOD m) := by define have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m apply Exists.intro (a / m) show a - a % m = m * (a / m) from calc a - (a % m) _ = m * (a / m) + a % m - a % m := by rw [h1] _ = m * (a / m) := by ring done lemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) := And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a)) theorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) : ∃! (r : Int), 0 ≤ r ∧ r < m ∧ a ≡ r (MOD m) := by exists_unique · -- Existence apply Exists.intro (a % m) show 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) from mod_cmpl_res m a done · -- Uniqueness fix r1 : Int; fix r2 : Int assume h1 : 0 ≤ r1 ∧ r1 < m ∧ a ≡ r1 (MOD m) assume h2 : 0 ≤ r2 ∧ r2 < m ∧ a ≡ r2 (MOD m) have h3 : r1 ≡ r2 (MOD m) := congr_trans (congr_symm h1.right.right) h2.right.right obtain (d : Int) (h4 : r1 - r2 = m * d) from h3 have h5 : r1 - r2 < m * 1 := by linarith have h6 : m * (-1) < r1 - r2 := by linarith rewrite [h4] at h5 --h5 : m * d < m * 1 rewrite [h4] at h6 --h6 : m * -1 < m * d have h7 : (↑m : Int) ≥ 0 := Nat.cast_nonneg m have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7 have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7 have h10 : d = 0 := by linarith show r1 = r2 from calc r1 _ = r1 - r2 + r2 := by ring _ = m * 0 + r2 := by rw [h4, h10] _ = r2 := by ring done done lemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m := (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a) theorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y rewrite [h1, h2] have h3 : a + b = b + a := by ring show [a]_m + [b]_m = [b]_m + [a]_m from calc [a]_m + [b]_m _ = [a + b]_m := add_class m a b _ = [b + a]_m := by rw [h3] _ = [b]_m + [a]_m := (add_class m b a).symm done theorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X rewrite [h1] have h2 : a * 1 = a := by ring show [a]_m * [1]_m = [a]_m from calc [a]_m * [1]_m _ = [a * 1]_m := mul_class m a 1 _ = [a]_m := by rw [h2] done theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry lemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) : [a]_m * [gcd_c2 m a]_m = [1]_m := by set s : Int := gcd_c1 m a have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m define at h1 rewrite [h1, Nat.cast_one] at h2 --h2 : s * ↑m + gcd_c2 m a * ↑a = 1 rewrite [mul_class, cc_eq_iff_congr] define --Goal : ∃ (c : Int), ↑a * gcd_c2 m a - 1 = ↑m * c apply Exists.intro (-s) show a * (gcd_c2 m a) - 1 = m * (-s) from calc a * (gcd_c2 m a) - 1 _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring _ = 1 + m * (-s) - 1 := by rw [h2] _ = m * (-s) := by ring done theorem Theorem_7_3_7 (m a : Nat) : invertible [a]_m ↔ rel_prime m a := by apply Iff.intro · -- (→) assume h1 : invertible [a]_m define at h1 obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1 obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y rewrite [h3, mul_class, cc_eq_iff_congr] at h2 define at h2 obtain (c : Int) (h4 : a * b - 1 = m * c) from h2 rewrite [Exercise_7_2_6] --Goal : ∃ (s t : Int), s * ↑m + t * ↑a = 1 apply Exists.intro (-c) apply Exists.intro b show (-c) * m + b * a = 1 from calc (-c) * m + b * a _ = (-c) * m + (a * b - 1) + 1 := by ring _ = (-c) * m + m * c + 1 := by rw [h4] _ = 1 := by ring done · -- (←) assume h1 : rel_prime m a define show ∃ (Y : ZMod m), [a]_m * Y = [1]_m from Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1) done done /- Section 7.4 -/ section Euler open Euler lemma num_rp_below_base {m : Nat} : num_rp_below m 0 = 0 := by rfl lemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) : num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by have h1 : num_rp_below m (j + 1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : gcd m j = 1 rewrite [if_pos h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1 show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1 done lemma num_rp_below_step_not_rp {m j : Nat} (h : ¬rel_prime m j) : num_rp_below m (j + 1) = num_rp_below m j := by have h1 : num_rp_below m (j +1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : ¬gcd m j = 1 rewrite [if_neg h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j show num_rp_below m (j + 1) = num_rp_below m j from h1 done lemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl #eval phi 10 --Answer: 4 lemma prod_inv_iff_inv {m : Nat} {X : ZMod m} (h1 : invertible X) (Y : ZMod m) : invertible (X * Y) ↔ invertible Y := by apply Iff.intro · -- (→) assume h2 : invertible (X * Y) obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2 apply Exists.intro (X * Z) rewrite [←h3] --Goal : Y * (X * Z) = X * Y * Z ring --Note that ring can do algebra in ZMod m done · -- (←) assume h2 : invertible Y obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1 obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2 apply Exists.intro (Xi * Yi) show (X * Y) * (Xi * Yi) = [1]_m from calc X * Y * (Xi * Yi) _ = (X * Xi) * (Y * Yi) := by ring _ = [1]_m * [1]_m := by rw [h3, h4] _ = [1]_m := Theorem_7_3_6_7 [1]_m done done lemma F_rp_def {m i : Nat} (h : rel_prime m i) : F m i = [i]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h --h : gcd m i = 1 rewrite [if_pos h] at h1 show F m i = [i]_m from h1 done lemma F_not_rp_def {m i : Nat} (h : ¬rel_prime m i) : F m i = [1]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h rewrite [h1, if_neg h] rfl done lemma prod_seq_base {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 0 k f = [1]_m := by rfl lemma prod_seq_step {m : Nat} (n k : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl lemma prod_seq_zero_step {m : Nat} (n : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by rewrite [prod_seq_step, zero_add] rfl done lemma prod_one {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 1 k f = f k := by rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7] rfl done lemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl lemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m := calc [G m a i]_m _ = [(a * i) % m]_m := by rfl _ = [a * i]_m := (cc_eq_mod m (a * i)).symm _ = [a]_m * [i]_m := (mul_class m a i).symm lemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) : rel_prime m (G m a i) ↔ rel_prime m i := by have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1 show rel_prime m (G m a i) ↔ rel_prime m i from calc rel_prime m (G m a i) _ ↔ invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm _ ↔ invertible ([a]_m * [i]_m) := by rw [cc_G] _ ↔ invertible [i]_m := prod_inv_iff_inv h2 ([i]_m) _ ↔ rel_prime m i := Theorem_7_3_7 m i done lemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) : F m (G m a i) = [a]_m * F m i := by have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2 show F m (G m a i) = [a]_m * F m i from calc F m (G m a i) _ = [G m a i]_m := F_rp_def h3 _ = [a]_m * [i]_m := cc_G m a i _ = [a]_m * F m i := by rw [F_rp_def h2] done lemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : ¬rel_prime m i) : F m (G m a i) = [1]_m := by rewrite [←G_rp_iff h1 i] at h2 show F m (G m a i) = [1]_m from F_not_rp_def h2 done lemma FG_prod {m a : Nat} (h1 : rel_prime m a) : ∀ (k : Nat), prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by by_induc · -- Base Case show prod_seq 0 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from calc prod_seq 0 0 ((F m) ∘ (G m a)) _ = [1]_m := prod_seq_base _ _ _ = [a]_m ^ 0 * [1]_m := by ring _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by rw [num_rp_below_base, prod_seq_base] done · -- Induction Step fix k : Nat assume ih : prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) by_cases h2 : rel_prime m k · -- Case 1. h2 : rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([a]_m * F m k) := by rw [FG_rp h1 h2] _ = [a]_m ^ ((num_rp_below m k) + 1) * ((prod_seq k 0 (F m)) * F m k) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_rp h2, prod_seq_zero_step] done · -- Case 2. h2 : ¬rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([1]_m) := by rw [FG_not_rp h1 h2] _ = [a]_m ^ (num_rp_below m k) * (prod_seq k 0 (F m) * ([1]_m)) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_not_rp h2, prod_seq_zero_step, F_not_rp_def h2] done done done lemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by define --Goal : ∀ i < m, G m a i < m fix i : Nat assume h1 : i < m rewrite [G_def] --Goal : a * i % m < m show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m) done lemma left_inv_one_one_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g' (g i) = i) : one_one_below n g := sorry lemma right_inv_onto_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g (g' i) = i) (h2 : maps_below n g') : onto_below n g := by define at h2; define fix k : Nat assume h3 : k < n apply Exists.intro (g' k) show g' k < n ∧ g (g' k) = k from And.intro (h2 k h3) (h1 k h3) done lemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m * [inv_mod m a]_m = [1]_m := by have h2 : 0 ≤ (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a) show [a]_m * [inv_mod m a]_m = [1]_m from calc [a]_m * [inv_mod m a]_m _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2] _ = [a]_m * [gcd_c2 m a]_m := by rw [←cc_eq_mod] _ = [1]_m := gcd_c2_inv h1 done lemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m := calc a * (b % m) % m = a % m * (b % m % m) % m := Nat.mul_mod _ _ _ _ = a % m * (b % m) % m := by rw [Nat.mod_mod] _ = a * b % m := (Nat.mul_mod _ _ _).symm lemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm] rfl done theorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] : ↑a ≡ ↑b (MOD m) ↔ a % m = b % m := sorry lemma mul_inv_mod_cancel {m a i : Nat} [NeZero m] (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1 rewrite [mul_class, cc_eq_iff_congr, ←Nat.cast_mul, ←Nat.cast_one, congr_iff_mod_eq_Nat] at h3 show a * inv_mod m a * i % m = i from calc a * (inv_mod m a) * i % m _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = 1 % m * i % m := by rw [h3] _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = i % m := by rw [one_mul] _ = i := Nat.mod_eq_of_lt h2 done lemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl lemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, G m a (Ginv m a i) = i := by fix i : Nat assume h2 : i < m show G m a (Ginv m a i) = i from calc G m a (Ginv m a i) _ = a * ((inv_mod m a * i) % m) % m := by rfl _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, Ginv m a (G m a i) = i := by fix i : Nat assume h2 : i < m show Ginv m a (G m a i) = i from calc Ginv m a (G m a i) _ = inv_mod m a * ((a * i) % m) % m := by rfl _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc, mul_comm (inv_mod m a)] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_maps_below (m a : Nat) [NeZero m] : maps_below m (Ginv m a) := G_maps_below m (inv_mod m a) lemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : one_one_below m (G m a) := left_inv_one_one_below (Ginv_left_inv h1) lemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : onto_below m (G m a) := right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a) lemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : perm_below m (G m a) := And.intro (G_maps_below m a) (And.intro (G_one_one_below h1) (G_onto_below h1)) --Permuting a product of congruence classes doesn't change product lemma swap_fst (u v : Nat) : swap u v u = v := by define : swap u v u --Goal : (if u = u then v else if u = v then u else u) = v have h : u = u := by rfl rewrite [if_pos h] rfl done lemma swap_snd (u v : Nat) : swap u v v = u := by define : swap u v v by_cases h1 : v = u · -- Case 1. h1 : v = u rewrite [if_pos h1] show v = u from h1 done · -- Case 2. h1 : v ≠ u rewrite [if_neg h1] have h2 : v = v := by rfl rewrite [if_pos h2] rfl done done lemma swap_other {u v i : Nat} (h1 : i ≠ u) (h2 : i ≠ v) : swap u v i = i := by define : swap u v i rewrite [if_neg h1, if_neg h2] rfl done lemma swap_values (u v i : Nat) : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := by by_cases h1 : i = u · -- Case 1. h1 : i = u apply Or.inl rewrite [h1] show swap u v u = v from swap_fst u v done · -- Case 2. h1 : i ≠ u apply Or.inr by_cases h2 : i = v · -- Case 2.1. h2 : i = v apply Or.inl rewrite [h2] show swap u v v = u from swap_snd u v done · -- Case 2.2. h2 : i ≠ v apply Or.inr show swap u v i = i from swap_other h1 h2 done done done lemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by define fix i : Nat assume h3 : i < n have h4 : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := swap_values u v i by_cases on h4 · -- Case 1. h4 : swap u v i = v rewrite [h4] show v < n from h2 done · -- Case 2. by_cases on h4 · -- Case 2.1. h4 : swap u v i = u rewrite [h4] show u < n from h1 done · -- Case 2.2. h4 : swap u v i = i rewrite [h4] show i < n from h3 done done done lemma swap_swap (u v n : Nat) : ∀ i < n, swap u v (swap u v i) = i := by fix i : Nat assume h : i < n by_cases h1 : i = u · -- Case 1. h1 : i = u rewrite [h1, swap_fst, swap_snd] rfl done · -- Case 2. h1 : i ≠ u by_cases h2 : i = v · -- Case 2.1. h2 : i = v rewrite [h2, swap_snd, swap_fst] rfl done · -- Case 2.2. h2 : i ≠ v rewrite [swap_other h1 h2, swap_other h1 h2] rfl done done done lemma swap_one_one_below (u v n) : one_one_below n (swap u v) := left_inv_one_one_below (swap_swap u v n) lemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) := right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2) lemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) := And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2)) lemma comp_perm_below {n : Nat} {f g : Nat → Nat} (h1 : perm_below n f) (h2 : perm_below n g) : perm_below n (f ∘ g) := sorry lemma trivial_swap (u : Nat) : swap u u = id := by apply funext fix x : Nat by_cases h1 : x = u · -- Case 1. h1 : x = u rewrite [h1, swap_fst] rfl done · -- Case 2. h1 : x ≠ u rewrite [swap_other h1 h1] rfl done done lemma prod_eq_fun {m : Nat} (f g : Nat → ZMod m) (k : Nat) : ∀ (n : Nat), (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g := by by_induc · -- Base Case assume h : (∀ i < 0, f (k + i) = g (k + i)) rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g assume h1 : ∀ i < n + 1, f (k + i) = g (k + i) have h2 : ∀ i < n, f (k + i) = g (k + i) := by fix i : Nat assume h2 : i < n have h3 : i < n + 1 := by linarith show f (k + i) = g (k + i) from h1 i h3 done have h3 : prod_seq n k f = prod_seq n k g := ih h2 have h4 : n < n + 1 := Nat.lt_succ_self n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4] rfl done done lemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := by have h2 : ∀ (i : Nat), i < u → (f ∘ swap u n) (0 + i) = f (0 + i) := by fix i : Nat assume h2 : i < u have h3 : 0 + i ≠ u := by linarith have h4 : 0 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl done show prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f from prod_eq_fun (f ∘ swap u n) f 0 u h2 done lemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := by have h2 : ∀ i < j, (f ∘ swap u n) (u + 1 + i) = f (u + 1 + i) := by fix i : Nat assume h2 : i < j have h3 : u + 1 + i ≠ u := by linarith have h4 : u + 1 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl show prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f from prod_eq_fun (f ∘ swap u n) f (u + 1) j h2 done lemma break_prod {m : Nat} (n : Nat) (f : Nat → ZMod m) : ∀ (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by by_induc · -- Base Case have h : n + 0 = n := by rfl rewrite [prod_seq_base, h, Theorem_7_3_6_7] rfl done · -- Induction Step fix j : Nat assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f rewrite [←add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc] rfl done done lemma break_prod_twice {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f := break_prod n f 1 rewrite [prod_one] at h2 have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f := break_prod (u + 1) f j rewrite [←h1] at h3 have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f := break_prod u f 1 rewrite [prod_one] at h4 rewrite [h3, h4] at h2 show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2 done lemma swap_prod_eq_prod {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f := by by_cases h2 : u = n · -- Case 1. h2 : u = n rewrite [h2, trivial_swap n] --Goal : prod_seq (n + 1) 0 (f ∘ id) = prod_seq (n + 1) 0 f rfl done · -- Case 2. h2 : ¬u = n have h3 : u + 1 ≤ n := Nat.lt_of_le_of_ne h1 h2 obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3 have break_f : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := break_prod_twice f h4 have break_fs : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_prod_twice (f ∘ swap u n) h4 have f_eq_fs_below : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := swap_prod_eq_prod_below f h1 have f_eq_fs_btwn : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4 show prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f from calc prod_seq (n + 1) 0 (f ∘ swap u n) _ = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_fs _ = prod_seq u 0 f * (f ∘ swap u n) u * prod_seq j (u + 1) f * (f ∘ swap u n) n := by rw [f_eq_fs_below, f_eq_fs_btwn] _ = prod_seq u 0 f * f (swap u n u) * prod_seq j (u + 1) f * f (swap u n n) := by rfl _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by rw [swap_fst, swap_snd] _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring _ = prod_seq (n + 1) 0 f := break_f.symm done done lemma perm_below_fixed {n : Nat} {g : Nat → Nat} (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry lemma perm_prod {m : Nat} (f : Nat → ZMod m) : ∀ (n : Nat), ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) := by by_induc · -- Base Case fix g : Nat → Nat assume h1 : perm_below 0 g rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) fix g : Nat → Nat assume g_pb : perm_below (n + 1) g define at g_pb have g_ob : onto_below (n + 1) g := g_pb.right.right define at g_ob have h1 : n < n + 1 := by linarith obtain (u : Nat) (h2 : u < n + 1 ∧ g u = n) from g_ob n h1 have s_pb : perm_below (n + 1) (swap u n) := swap_perm_below h2.left h1 have gs_pb_n1 : perm_below (n + 1) (g ∘ swap u n) := comp_perm_below g_pb s_pb have gs_fix_n : (g ∘ swap u n) n = n := calc (g ∘ swap u n) n _ = g (swap u n n) := by rfl _ = g u := by rw [swap_snd] _ = n := h2.right have gs_pb_n : perm_below n (g ∘ swap u n) := perm_below_fixed gs_pb_n1 gs_fix_n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f ∘ (g ∘ swap u n)) := ih (g ∘ swap u n) gs_pb_n have h3 : u ≤ n := by linarith show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f ∘ g) from calc prod_seq (n + 1) 0 f _ = prod_seq n 0 f * f n := prod_seq_zero_step n f _ = prod_seq n 0 (f ∘ (g ∘ swap u n)) * f ((g ∘ swap u n) n) := by rw [gs_prod, gs_fix_n] _ = prod_seq n 0 (f ∘ g ∘ swap u n) * (f ∘ g ∘ swap u n) n := by rfl _ = prod_seq (n + 1) 0 (f ∘ g ∘ swap u n) := (prod_seq_zero_step n (f ∘ g ∘ swap u n)).symm _ = prod_seq (n + 1) 0 ((f ∘ g) ∘ swap u n) := by rfl _ = prod_seq (n + 1) 0 (f ∘ g) := swap_prod_eq_prod (f ∘ g) h3 done done lemma F_invertible (m i : Nat) : invertible (F m i) := by by_cases h : rel_prime m i · -- Case 1. h : rel_prime m i rewrite [F_rp_def h] show invertible [i]_m from (Theorem_7_3_7 m i).rtl h done · -- Case 2. h : ¬rel_prime m i rewrite [F_not_rp_def h] apply Exists.intro [1]_m show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m done done

lemma Fprod_invertible (m : Nat) : ∀ (k : Nat), invertible (prod_seq k 0 (F m))

HTPI.Fprod_invertible

htpi/HTPILib/Chap7.lean

htpi

/- Copyright 2023 Daniel J. Velleman -/ import HTPILib.Chap6 namespace HTPI /- Definitions -/ lemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by have h : n + 1 > 0 := Nat.succ_pos n show a % (n + 1) < n + 1 from Nat.mod_lt a h done def gcd (a b : Nat) : Nat := match b with | 0 => a | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd (n + 1) (a % (n + 1)) termination_by b mutual def gcd_c1 (a b : Nat) : Int := match b with | 0 => 1 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c2 (n + 1) (a % (n + 1)) --Corresponds to s = t' termination_by b def gcd_c2 (a b : Nat) : Int := match b with | 0 => 0 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c1 (n + 1) (a % (n + 1)) - (gcd_c2 (n + 1) (a % (n + 1))) * ↑(a / (n + 1)) --Corresponds to t = s' - t'q termination_by b end def prime (n : Nat) : Prop := 2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n def prime_factor (p n : Nat) : Prop := prime p ∧ p ∣ n def all_prime (l : List Nat) : Prop := ∀ p ∈ l, prime p def nondec (l : List Nat) : Prop := match l with | [] => True --Of course, True is a proposition that is always true | n :: L => (∀ m ∈ L, n ≤ m) ∧ nondec L def nondec_prime_list (l : List Nat) : Prop := all_prime l ∧ nondec l def prod (l : List Nat) : Nat := match l with | [] => 1 | n :: L => n * (prod L) def prime_factorization (n : Nat) (l : List Nat) : Prop := nondec_prime_list l ∧ prod l = n def rel_prime (a b : Nat) : Prop := gcd a b = 1 def congr_mod (m : Nat) (a b : Int) : Prop := (↑m : Int) ∣ (a - b) def cc (m : Nat) (a : Int) : ZMod m := (↑a : ZMod m) notation:50 a " ≡ " b " (MOD " m ")" => congr_mod m a b notation:max "["a"]_"m:max => cc m a def invertible {m : Nat} (X : ZMod m) : Prop := ∃ (Y : ZMod m), X * Y = [1]_m def num_rp_below (m k : Nat) : Nat := match k with | 0 => 0 | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j def phi (m : Nat) : Nat := num_rp_below m m def prod_seq {m : Nat} (j k : Nat) (f : Nat → ZMod m) : ZMod m := match j with | 0 => [1]_m | n + 1 => prod_seq n k f * f (k + n) def maps_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i < n, g i < n def one_one_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i1 < n, ∀ i2 < n, g i1 = g i2 → i1 = i2 def onto_below (n : Nat) (g : Nat → Nat) : Prop := ∀ k < n, ∃ i < n, g i = k def perm_below (n : Nat) (g : Nat → Nat) : Prop := maps_below n g ∧ one_one_below n g ∧ onto_below n g def inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m) def swap (u v i : Nat) : Nat := if i = u then v else if i = v then u else i namespace Euler --For definitions specific to Euler's theorem def F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m def G (m a i : Nat) : Nat := (a * i) % m def Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i end Euler /- Section 7.1 -/ theorem dvd_mod_of_dvd_a_b {a b d : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ (a % b) := by set q : Nat := a / b have h3 : b * q + a % b = a := Nat.div_add_mod a b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 define --Goal : ∃ (c : Nat), a % b = d * c apply Exists.intro (j - k * q) show a % b = d * (j - k * q) from calc a % b _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm _ = a - b * q := by rw [h3] _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc] _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm done theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry #eval gcd 672 161 --Answer: 7 lemma gcd_base (a : Nat) : gcd a 0 = a := by rfl lemma gcd_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd a b = gcd b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1)) rfl done lemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b ≠ 0) : a % b < b := by have h1 : b > 0 := Nat.pos_of_ne_zero h show a % b < b from Nat.mod_lt a h1 done lemma dvd_self (n : Nat) : n ∣ n := by apply Exists.intro 1 ring done theorem gcd_dvd : ∀ (b a : Nat), (gcd a b) ∣ a ∧ (gcd a b) ∣ b := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd a b_1) ∣ a ∧ (gcd a b_1) ∣ b_1 fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_base] --Goal: a ∣ a ∧ a ∣ 0 apply And.intro (dvd_self a) define apply Exists.intro 0 rfl done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_nonzero a h1] --Goal : gcd b (a % b) ∣ a ∧ gcd b (a % b) ∣ b have h2 : a % b < b := mod_nonzero_lt a h1 have h3 : (gcd b (a % b)) ∣ b ∧ (gcd b (a % b)) ∣ (a % b) := ih (a % b) h2 b apply And.intro _ h3.left show (gcd b (a % b)) ∣ a from dvd_a_of_dvd_b_mod h3.left h3.right done done theorem gcd_dvd_left (a b : Nat) : (gcd a b) ∣ a := (gcd_dvd b a).left theorem gcd_dvd_right (a b : Nat) : (gcd a b) ∣ b := (gcd_dvd b a).right lemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl lemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c1 a b = gcd_c2 b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done lemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl lemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * ↑(a / b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done theorem gcd_lin_comb : ∀ (b a : Nat), (gcd_c1 a b) * ↑a + (gcd_c2 a b) * ↑b = ↑(gcd a b) := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd_c1 a b_1) * ↑a + (gcd_c2 a b_1) * ↑b_1 = ↑(gcd a b_1) fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base] --Goal : 1 * ↑a + 0 * ↑0 = ↑a ring done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1] --Goal : gcd_c2 b (a % b) * ↑a + -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * ↑(a / b)) * ↑b = -- ↑(gcd b (a % b)) set r : Nat := a % b set q : Nat := a / b set s : Int := gcd_c1 b r set t : Int := gcd_c2 b r --Goal : t * ↑a + (s - t * ↑q) * ↑b = ↑(gcd b r) have h2 : r < b := mod_nonzero_lt a h1 have h3 : s * ↑b + t * ↑r = ↑(gcd b r) := ih r h2 b have h4 : b * q + r = a := Nat.div_add_mod a b rewrite [←h3, ←h4] rewrite [Nat.cast_add, Nat.cast_mul] --Goal : t * (↑b * ↑q + ↑r) + (s - t * ↑q) * ↑b = s * ↑b + t * ↑r ring done done #eval gcd_c1 672 161 --Answer: 6 #eval gcd_c2 672 161 --Answer: -25 --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161 theorem Theorem_7_1_6 {d a b : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ gcd a b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑d ∣ ↑(gcd a b) set s : Int := gcd_c1 a b set t : Int := gcd_c2 a b have h3 : s * ↑a + t * ↑b = ↑(gcd a b) := gcd_lin_comb b a rewrite [←h3] --Goal : ↑d ∣ s * ↑a + t * ↑b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul] --Goal : ↑d ∣ s * (↑d * ↑j) + t * (↑d * ↑k) define apply Exists.intro (s * ↑j + t * ↑k) ring done /- Section 7.2 -/ theorem dvd_trans {a b c : Nat} (h1 : a ∣ b) (h2 : b ∣ c) : a ∣ c := by define at h1; define at h2; define obtain (m : Nat) (h3 : b = a * m) from h1 obtain (n : Nat) (h4 : c = b * n) from h2 rewrite [h3, mul_assoc] at h4 apply Exists.intro (m * n) show c = a * (m * n) from h4 done lemma exists_prime_factor : ∀ (n : Nat), 2 ≤ n → ∃ (p : Nat), prime_factor p n := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, 2 ≤ n_1 → ∃ (p : Nat), prime_factor p n_1 assume h1 : 2 ≤ n by_cases h2 : prime n · -- Case 1. h2 : prime n apply Exists.intro n define --Goal : prime n ∧ n ∣ n show prime n ∧ n ∣ n from And.intro h2 (dvd_self n) done · -- Case 2. h2 : ¬prime n define at h2 --h2 : ¬(2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n) demorgan at h2 disj_syll h2 h1 obtain (a : Nat) (h3 : ∃ (b : Nat), a * b = n ∧ a < n ∧ b < n) from h2 obtain (b : Nat) (h4 : a * b = n ∧ a < n ∧ b < n) from h3 have h5 : 2 ≤ a := by by_contra h6 have h7 : a ≤ 1 := by linarith have h8 : n ≤ b := calc n _ = a * b := h4.left.symm _ ≤ 1 * b := by rel [h7] _ = b := by ring linarith --n ≤ b contradicts b < n done have h6 : ∃ (p : Nat), prime_factor p a := ih a h4.right.left h5 obtain (p : Nat) (h7 : prime_factor p a) from h6 apply Exists.intro p define --Goal : prime p ∧ p ∣ n define at h7 --h7 : prime p ∧ p ∣ a apply And.intro h7.left have h8 : a ∣ n := by apply Exists.intro b show n = a * b from (h4.left).symm done show p ∣ n from dvd_trans h7.right h8 done done lemma exists_least_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q := by set S : Set Nat := {p : Nat | prime_factor p n} have h2 : ∃ (p : Nat), p ∈ S := exists_prime_factor n h show ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q from well_ord_princ S h2 done lemma all_prime_nil : all_prime [] := by define --Goal : ∀ p ∈ [], prime p fix p : Nat contrapos --Goal : ¬prime p → p ∉ [] assume h1 : ¬prime p show p ∉ [] from List.not_mem_nil p done lemma all_prime_cons (n : Nat) (L : List Nat) : all_prime (n :: L) ↔ prime n ∧ all_prime L := by apply Iff.intro · -- (→) assume h1 : all_prime (n :: L) --Goal : prime n ∧ all_prime L define at h1 --h1 : ∀ p ∈ n :: L, prime p apply And.intro (h1 n (List.mem_cons_self n L)) define --Goal : ∀ p ∈ L, prime p fix p : Nat assume h2 : p ∈ L show prime p from h1 p (List.mem_cons_of_mem n h2) done · -- (←) assume h1 : prime n ∧ all_prime L --Goal : all_prime (n :: l) define : all_prime L at h1 define fix p : Nat assume h2 : p ∈ n :: L rewrite [List.mem_cons] at h2 --h2 : p = n ∨ p ∈ L by_cases on h2 · -- Case 1. h2 : p = n rewrite [h2] show prime n from h1.left done · -- Case 2. h2 : p ∈ L show prime p from h1.right p h2 done done done lemma nondec_nil : nondec [] := by define --Goal : True trivial --trivial proves some obviously true statements, such as True done lemma nondec_cons (n : Nat) (L : List Nat) : nondec (n :: L) ↔ (∀ m ∈ L, n ≤ m) ∧ nondec L := by rfl lemma prod_nil : prod [] = 1 := by rfl lemma prod_cons : prod (n :: L) = n * (prod L) := by rfl lemma exists_cons_of_length_eq_succ {A : Type} {l : List A} {n : Nat} (h : l.length = n + 1) : ∃ (a : A) (L : List A), l = a :: L ∧ L.length = n := by have h1 : ¬l.length = 0 := by linarith rewrite [List.length_eq_zero] at h1 obtain (a : A) (h2 : ∃ (L : List A), l = a :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List A) (h3 : l = a :: L) from h2 apply Exists.intro a apply Exists.intro L apply And.intro h3 have h4 : (a :: L).length = L.length + 1 := List.length_cons a L rewrite [←h3, h] at h4 show L.length = n from (Nat.add_right_cancel h4).symm done lemma list_elt_dvd_prod_by_length (a : Nat) : ∀ (n : Nat), ∀ (l : List Nat), l.length = n → a ∈ l → a ∣ prod l := by by_induc · --Base Case fix l : List Nat assume h1 : l.length = 0 rewrite [List.length_eq_zero] at h1 --h1 : l = [] rewrite [h1] --Goal : a ∈ [] → a ∣ prod [] contrapos assume h2 : ¬a ∣ prod [] show a ∉ [] from List.not_mem_nil a done · -- Induction Step fix n : Nat assume ih : ∀ (l : List Nat), List.length l = n → a ∈ l → a ∣ prod l fix l : List Nat assume h1 : l.length = n + 1 --Goal : a ∈ l → a ∣ prod l obtain (b : Nat) (h2 : ∃ (L : List Nat), l = b :: L ∧ L.length = n) from exists_cons_of_length_eq_succ h1 obtain (L : List Nat) (h3 : l = b :: L ∧ L.length = n) from h2 have h4 : a ∈ L → a ∣ prod L := ih L h3.right assume h5 : a ∈ l rewrite [h3.left, prod_cons] --Goal : a ∣ b * prod L rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b ∨ a ∈ L by_cases on h5 · -- Case 1. h5 : a = b apply Exists.intro (prod L) rewrite [h5] rfl done · -- Case 2. h5 : a ∈ L have h6 : a ∣ prod L := h4 h5 have h7 : prod L ∣ b * prod L := by apply Exists.intro b ring done show a ∣ b * prod L from dvd_trans h6 h7 done done done lemma list_elt_dvd_prod {a : Nat} {l : List Nat} (h : a ∈ l) : a ∣ prod l := by set n : Nat := l.length have h1 : l.length = n := by rfl show a ∣ prod l from list_elt_dvd_prod_by_length a n l h1 h done lemma exists_prime_factorization : ∀ (n : Nat), n ≥ 1 → ∃ (l : List Nat), prime_factorization n l := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, n_1 ≥ 1 → ∃ (l : List Nat), prime_factorization n_1 l assume h1 : n ≥ 1 by_cases h2 : n = 1 · -- Case 1. h2 : n = 1 apply Exists.intro [] define apply And.intro · -- Proof of nondec_prime_list [] define show all_prime [] ∧ nondec [] from And.intro all_prime_nil nondec_nil done · -- Proof of prod [] = n rewrite [prod_nil, h2] rfl done done · -- Case 2. h2 : n ≠ 1 have h3 : n ≥ 2 := lt_of_le_of_ne' h1 h2 obtain (p : Nat) (h4 : prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q) from exists_least_prime_factor h3 have p_prime_factor : prime_factor p n := h4.left define at p_prime_factor have p_prime : prime p := p_prime_factor.left have p_dvd_n : p ∣ n := p_prime_factor.right have p_least : ∀ (q : Nat), prime_factor q n → p ≤ q := h4.right obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n have h5 : m ≠ 0 := by contradict h1 with h6 have h7 : n = 0 := calc n _ = p * m := n_eq_pm _ = p * 0 := by rw [h6] _ = 0 := by ring rewrite [h7] decide done have m_pos : 0 < m := Nat.pos_of_ne_zero h5 have m_lt_n : m < n := by define at p_prime show m < n from calc m _ < m + m := by linarith _ = 2 * m := by ring _ ≤ p * m := by rel [p_prime.left] _ = n := n_eq_pm.symm done obtain (L : List Nat) (h6 : prime_factorization m L) from ih m m_lt_n m_pos define at h6 have ndpl_L : nondec_prime_list L := h6.left define at ndpl_L apply Exists.intro (p :: L) define apply And.intro · -- Proof of nondec_prime_list (p :: L) define apply And.intro · -- Proof of all_prime (p :: L) rewrite [all_prime_cons] show prime p ∧ all_prime L from And.intro p_prime ndpl_L.left done · -- Proof of nondec (p :: L) rewrite [nondec_cons] apply And.intro _ ndpl_L.right fix q : Nat assume q_in_L : q ∈ L have h7 : q ∣ prod L := list_elt_dvd_prod q_in_L rewrite [h6.right] at h7 --h7 : q ∣ m have h8 : m ∣ n := by apply Exists.intro p rewrite [n_eq_pm] ring done have q_dvd_n : q ∣ n := dvd_trans h7 h8 have ap_L : all_prime L := ndpl_L.left define at ap_L have q_prime_factor : prime_factor q n := And.intro (ap_L q q_in_L) q_dvd_n show p ≤ q from p_least q q_prime_factor done done · -- Proof of prod (p :: L) = n rewrite [prod_cons, h6.right, n_eq_pm] rfl done done done theorem Theorem_7_2_2 {a b c : Nat} (h1 : c ∣ a * b) (h2 : rel_prime a c) : c ∣ b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑c ∣ ↑b define at h1; define at h2; define obtain (j : Nat) (h3 : a * b = c * j) from h1 set s : Int := gcd_c1 a c set t : Int := gcd_c2 a c have h4 : s * ↑a + t * ↑c = ↑(gcd a c) := gcd_lin_comb c a rewrite [h2, Nat.cast_one] at h4 --h4 : s * ↑a + t * ↑c = (1 : Int) apply Exists.intro (s * ↑j + t * ↑b) show ↑b = ↑c * (s * ↑j + t * ↑b) from calc ↑b _ = (1 : Int) * ↑b := (one_mul _).symm _ = (s * ↑a + t * ↑c) * ↑b := by rw [h4] _ = s * (↑a * ↑b) + t * ↑c * ↑b := by ring _ = s * (↑c * ↑j) + t * ↑c * ↑b := by rw [←Nat.cast_mul a b, h3, Nat.cast_mul c j] _ = ↑c * (s * ↑j + t * ↑b) := by ring done lemma le_nonzero_prod_left {a b : Nat} (h : a * b ≠ 0) : a ≤ a * b := by have h1 : b ≠ 0 := by contradict h with h1 rewrite [h1] ring done have h2 : 1 ≤ b := Nat.pos_of_ne_zero h1 show a ≤ a * b from calc a = a * 1 := (mul_one a).symm _ ≤ a * b := by rel [h2] done lemma le_nonzero_prod_right {a b : Nat} (h : a * b ≠ 0) : b ≤ a * b := by rewrite [mul_comm] rewrite [mul_comm] at h show b ≤ b * a from le_nonzero_prod_left h done lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry lemma rel_prime_of_prime_not_dvd {a p : Nat} (h1 : prime p) (h2 : ¬p ∣ a) : rel_prime a p := by have h3 : gcd a p ∣ a := gcd_dvd_left a p have h4 : gcd a p ∣ p := gcd_dvd_right a p have h5 : gcd a p = 1 ∨ gcd a p = p := dvd_prime h1 h4 have h6 : gcd a p ≠ p := by contradict h2 with h6 rewrite [h6] at h3 show p ∣ a from h3 done disj_syll h5 h6 show rel_prime a p from h5 done theorem Theorem_7_2_3 {a b p : Nat} (h1 : prime p) (h2 : p ∣ a * b) : p ∣ a ∨ p ∣ b := by or_right with h3 have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3 show p ∣ b from Theorem_7_2_2 h2 h4 done lemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a ≥ 1 := by have h1 : a ≠ 0 := by by_contra h1 rewrite [h1] at h contradict h linarith done show a ≥ 1 from Nat.pos_of_ne_zero h1 done lemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by have h1 : a ≥ 1 := ge_one_of_prod_one h have h2 : a * b ≠ 0 := by linarith have h3 : a ≤ a * b := le_nonzero_prod_left h2 rewrite [h] at h3 show a = 1 from Nat.le_antisymm h3 h1 done lemma eq_one_of_dvd_one {n : Nat} (h : n ∣ 1) : n = 1 := by obtain (j : Nat) (h1 : 1 = n * j) from h show n = 1 from eq_one_of_prod_one h1.symm done lemma prime_not_one {p : Nat} (h : prime p) : p ≠ 1 := by define at h linarith done theorem Theorem_7_2_4 {p : Nat} (h1 : prime p) : ∀ (l : List Nat), p ∣ prod l → ∃ a ∈ l, p ∣ a := by apply List.rec · -- Base Case. Goal : p ∣ prod [] → ∃ a ∈ [], p ∣ a rewrite [prod_nil] assume h2 : p ∣ 1 show ∃ a ∈ [], p ∣ a from absurd (eq_one_of_dvd_one h2) (prime_not_one h1) done · -- Induction Step fix b : Nat fix L : List Nat assume ih : p ∣ prod L → ∃ a ∈ L, p ∣ a --Goal : p ∣ prod (b :: L) → ∃ a ∈ b :: L, p ∣ a assume h2 : p ∣ prod (b :: L) rewrite [prod_cons] at h2 have h3 : p ∣ b ∨ p ∣ prod L := Theorem_7_2_3 h1 h2 by_cases on h3 · -- Case 1. h3 : p ∣ b apply Exists.intro b show b ∈ b :: L ∧ p ∣ b from And.intro (List.mem_cons_self b L) h3 done · -- Case 2. h3 : p ∣ prod L obtain (a : Nat) (h4 : a ∈ L ∧ p ∣ a) from ih h3 apply Exists.intro a show a ∈ b :: L ∧ p ∣ a from And.intro (List.mem_cons_of_mem b h4.left) h4.right done done done lemma prime_in_list {p : Nat} {l : List Nat} (h1 : prime p) (h2 : all_prime l) (h3 : p ∣ prod l) : p ∈ l := by obtain (a : Nat) (h4 : a ∈ l ∧ p ∣ a) from Theorem_7_2_4 h1 l h3 define at h2 have h5 : prime a := h2 a h4.left have h6 : p = 1 ∨ p = a := dvd_prime h5 h4.right disj_syll h6 (prime_not_one h1) rewrite [h6] show a ∈ l from h4.left done lemma first_le_first {p q : Nat} {l m : List Nat} (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m)) (h3 : prod (p :: l) = prod (q :: m)) : p ≤ q := by define at h1; define at h2 have h4 : q ∣ prod (p :: l) := by define apply Exists.intro (prod m) rewrite [←prod_cons] show prod (p :: l) = prod (q :: m) from h3 done have h5 : all_prime (q :: m) := h2.left rewrite [all_prime_cons] at h5 have h6 : q ∈ p :: l := prime_in_list h5.left h1.left h4 have h7 : nondec (p :: l) := h1.right rewrite [nondec_cons] at h7 rewrite [List.mem_cons] at h6 by_cases on h6 · -- Case 1. h6 : q = p linarith done · -- Case 2. h6 : q ∈ l have h8 : ∀ m ∈ l, p ≤ m := h7.left show p ≤ q from h8 q h6 done done lemma nondec_prime_list_tail {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by define at h define rewrite [all_prime_cons, nondec_cons] at h show all_prime l ∧ nondec l from And.intro h.left.right h.right.right done lemma cons_prod_not_one {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : prod (p :: l) ≠ 1 := by define at h have h1 : all_prime (p :: l) := h.left rewrite [all_prime_cons] at h1 rewrite [prod_cons] by_contra h2 show False from (prime_not_one h1.left) (eq_one_of_prod_one h2) done lemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) : l = [] ↔ prod l = 1 := by apply Iff.intro · -- (→) assume h1 : l = [] rewrite [h1] show prod [] = 1 from prod_nil done · -- (←) contrapos assume h1 : ¬l = [] obtain (p : Nat) (h2 : ∃ (L : List Nat), l = p :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List Nat) (h3 : l = p :: L) from h2 rewrite [h3] at h rewrite [h3] show ¬prod (p :: L) = 1 from cons_prod_not_one h done done lemma prime_pos {p : Nat} (h : prime p) : p > 0 := by define at h linarith done theorem Theorem_7_2_5 : ∀ (l1 l2 : List Nat), nondec_prime_list l1 → nondec_prime_list l2 → prod l1 = prod l2 → l1 = l2 := by apply List.rec · -- Base Case. Goal : ∀ (l2 : List Nat), nondec_prime_list [] → -- nondec_prime_list l2 → prod [] = prod l2 → [] = l2 fix l2 : List Nat assume h1 : nondec_prime_list [] assume h2 : nondec_prime_list l2 assume h3 : prod [] = prod l2 rewrite [prod_nil, eq_comm, ←list_nil_iff_prod_one h2] at h3 show [] = l2 from h3.symm done · -- Induction Step fix p : Nat fix L1 : List Nat assume ih : ∀ (L2 : List Nat), nondec_prime_list L1 → nondec_prime_list L2 → prod L1 = prod L2 → L1 = L2 -- Goal : ∀ (l2 : List Nat), nondec_prime_list (p :: L1) → -- nondec_prime_list l2 → prod (p :: L1) = prod l2 → p :: L1 = l2 fix l2 : List Nat assume h1 : nondec_prime_list (p :: L1) assume h2 : nondec_prime_list l2 assume h3 : prod (p :: L1) = prod l2 have h4 : ¬prod (p :: L1) = 1 := cons_prod_not_one h1 rewrite [h3, ←list_nil_iff_prod_one h2] at h4 obtain (q : Nat) (h5 : ∃ (L : List Nat), l2 = q :: L) from List.exists_cons_of_ne_nil h4 obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5 rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2) rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2) have h7 : p ≤ q := first_le_first h1 h2 h3 have h8 : q ≤ p := first_le_first h2 h1 h3.symm have h9 : p = q := by linarith rewrite [h9, prod_cons, prod_cons] at h3 --h3 : q * prod L1 = q * prod L2 have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1 have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2 define at h2 have h12 : all_prime (q :: L2) := h2.left rewrite [all_prime_cons] at h12 have h13 : q > 0 := prime_pos h12.left have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3 have h15 : L1 = L2 := ih L2 h10 h11 h14 rewrite [h6, h9, h15] rfl done done theorem fund_thm_arith (n : Nat) (h : n ≥ 1) : ∃! (l : List Nat), prime_factorization n l := by exists_unique · -- Existence show ∃ (l : List Nat), prime_factorization n l from exists_prime_factorization n h done · -- Uniqueness fix l1 : List Nat; fix l2 : List Nat assume h1 : prime_factorization n l1 assume h2 : prime_factorization n l2 define at h1; define at h2 have h3 : prod l1 = n := h1.right rewrite [←h2.right] at h3 show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3 done done /- Section 7.3 -/ theorem congr_refl (m : Nat) : ∀ (a : Int), a ≡ a (MOD m) := by fix a : Int define --Goal : ∃ (c : Int), a - a = ↑m * c apply Exists.intro 0 ring done theorem congr_symm {m : Nat} : ∀ {a b : Int}, a ≡ b (MOD m) → b ≡ a (MOD m) := by fix a : Int; fix b : Int assume h1 : a ≡ b (MOD m) define at h1 --h1 : ∃ (c : Int), a - b = ↑m * c define --Goal : ∃ (c : Int), b - a = ↑m * c obtain (c : Int) (h2 : a - b = m * c) from h1 apply Exists.intro (-c) show b - a = m * (-c) from calc b - a _ = -(a - b) := by ring _ = -(m * c) := by rw [h2] _ = m * (-c) := by ring done theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry /- Fundamental properties of congruence classes -/ lemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) : X = Y ↔ X.val = Y.val := Fin.ext_iff lemma val_nat_eq_mod (n k : Nat) : ([k]_(n + 1)).val = k % (n + 1) := by rfl lemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl theorem cc_rep {m : Nat} (X : ZMod m) : ∃ (a : Int), X = [a]_m := match m with | 0 => by apply Exists.intro X rfl done | n + 1 => by apply Exists.intro ↑(X.val) have h1 : X.val < n + 1 := Fin.prop X rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1] rfl done theorem add_class (m : Nat) (a b : Int) : [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm theorem mul_class (m : Nat) (a b : Int) : [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm lemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ [a - b]_m = [0]_m := by apply Iff.intro · -- (→) assume h1 : [a]_m = [b]_m have h2 : a - b = a + (-b) := by ring have h3 : b + (-b) = 0 := by ring show [a - b]_m = [0]_m from calc [a - b]_m _ = [a + (-b)]_m := by rw [h2] _ = [a]_m + [-b]_m := by rw [add_class] _ = [b]_m + [-b]_m := by rw [h1] _ = [b + -b]_m := by rw [add_class] _ = [0]_m := by rw [h3] done · -- (←) assume h1 : [a - b]_m = [0]_m have h2 : b + (a - b) = a := by ring have h3 : b + 0 = b := by ring show [a]_m = [b]_m from calc [a]_m _ = [b + (a - b)]_m := by rw [h2] _ = [b]_m + [a - b]_m := by rw [add_class] _ = [b]_m + [0]_m := by rw [h1] _ = [b + 0]_m := by rw [add_class] _ = [b]_m := by rw [h3] done done lemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) : [a]_m = [0]_m → [-a]_m = [0]_m := by assume h1 : [a]_m = [0]_m have h2 : 0 + (-a) = -a := by ring have h3 : a + (-a) = 0 := by ring show [-a]_m = [0]_m from calc [-a]_m _ = [0 + (-a)]_m := by rw [h2] _ = [0]_m + [-a]_m := by rw [add_class] _ = [a]_m + [-a]_m := by rw [h1] _ = [a + (-a)]_m := by rw [add_class] _ = [0]_m := by rw [h3] done lemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) : [-a]_m = [0]_m ↔ [a]_m = [0]_m := by apply Iff.intro _ (cc_neg_zero_of_cc_zero m a) assume h1 : [-a]_m = [0]_m have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1 have h3 : -(-a) = a := by ring rewrite [h3] at h2 show [a]_m = [0]_m from h2 done lemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl lemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m ↔ m ∣ k := match m with | 0 => by have h : (0 : Int) = (↑(0 : Nat) : Int) := by rfl rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj] apply Iff.intro · -- (→) assume h1 : k = 0 rewrite [h1] show 0 ∣ 0 from dvd_self 0 done · -- (←) assume h1 : 0 ∣ k obtain (c : Nat) (h2 : k = 0 * c) from h1 rewrite [h2] ring done done | n + 1 => by rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero] show k % (n + 1) = 0 ↔ n + 1 ∣ k from (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm done lemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m ↔ ↑m ∣ a := by obtain (k : Nat) (h1 : a = ↑k ∨ a = -↑k) from Int.eq_nat_or_neg a by_cases on h1 · -- Case 1. h1: a = ↑k rewrite [h1, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done · -- Case 2. h1: a = -↑k rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done done theorem cc_eq_iff_congr (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ a ≡ b (MOD m) := calc [a]_m = [b]_m _ ↔ [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b _ ↔ ↑m ∣ (a - b) := cc_zero_iff_dvd m (a - b) _ ↔ a ≡ b (MOD m) := by rfl /- End of fundamental properties of congruence classes -/ lemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m := by have h1 : (↑m : Int) ≠ 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m) show 0 ≤ a % m from Int.emod_nonneg a h1 done lemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m) have h2 : (↑m : Int) > 0 := (Nat.cast_pos).rtl h1 show a % m < m from Int.emod_lt_of_pos a h2 done lemma congr_mod_mod (m : Nat) (a : Int) : a ≡ a % m (MOD m) := by define have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m apply Exists.intro (a / m) show a - a % m = m * (a / m) from calc a - (a % m) _ = m * (a / m) + a % m - a % m := by rw [h1] _ = m * (a / m) := by ring done lemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) := And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a)) theorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) : ∃! (r : Int), 0 ≤ r ∧ r < m ∧ a ≡ r (MOD m) := by exists_unique · -- Existence apply Exists.intro (a % m) show 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) from mod_cmpl_res m a done · -- Uniqueness fix r1 : Int; fix r2 : Int assume h1 : 0 ≤ r1 ∧ r1 < m ∧ a ≡ r1 (MOD m) assume h2 : 0 ≤ r2 ∧ r2 < m ∧ a ≡ r2 (MOD m) have h3 : r1 ≡ r2 (MOD m) := congr_trans (congr_symm h1.right.right) h2.right.right obtain (d : Int) (h4 : r1 - r2 = m * d) from h3 have h5 : r1 - r2 < m * 1 := by linarith have h6 : m * (-1) < r1 - r2 := by linarith rewrite [h4] at h5 --h5 : m * d < m * 1 rewrite [h4] at h6 --h6 : m * -1 < m * d have h7 : (↑m : Int) ≥ 0 := Nat.cast_nonneg m have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7 have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7 have h10 : d = 0 := by linarith show r1 = r2 from calc r1 _ = r1 - r2 + r2 := by ring _ = m * 0 + r2 := by rw [h4, h10] _ = r2 := by ring done done lemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m := (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a) theorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y rewrite [h1, h2] have h3 : a + b = b + a := by ring show [a]_m + [b]_m = [b]_m + [a]_m from calc [a]_m + [b]_m _ = [a + b]_m := add_class m a b _ = [b + a]_m := by rw [h3] _ = [b]_m + [a]_m := (add_class m b a).symm done theorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X rewrite [h1] have h2 : a * 1 = a := by ring show [a]_m * [1]_m = [a]_m from calc [a]_m * [1]_m _ = [a * 1]_m := mul_class m a 1 _ = [a]_m := by rw [h2] done theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry lemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) : [a]_m * [gcd_c2 m a]_m = [1]_m := by set s : Int := gcd_c1 m a have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m define at h1 rewrite [h1, Nat.cast_one] at h2 --h2 : s * ↑m + gcd_c2 m a * ↑a = 1 rewrite [mul_class, cc_eq_iff_congr] define --Goal : ∃ (c : Int), ↑a * gcd_c2 m a - 1 = ↑m * c apply Exists.intro (-s) show a * (gcd_c2 m a) - 1 = m * (-s) from calc a * (gcd_c2 m a) - 1 _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring _ = 1 + m * (-s) - 1 := by rw [h2] _ = m * (-s) := by ring done theorem Theorem_7_3_7 (m a : Nat) : invertible [a]_m ↔ rel_prime m a := by apply Iff.intro · -- (→) assume h1 : invertible [a]_m define at h1 obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1 obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y rewrite [h3, mul_class, cc_eq_iff_congr] at h2 define at h2 obtain (c : Int) (h4 : a * b - 1 = m * c) from h2 rewrite [Exercise_7_2_6] --Goal : ∃ (s t : Int), s * ↑m + t * ↑a = 1 apply Exists.intro (-c) apply Exists.intro b show (-c) * m + b * a = 1 from calc (-c) * m + b * a _ = (-c) * m + (a * b - 1) + 1 := by ring _ = (-c) * m + m * c + 1 := by rw [h4] _ = 1 := by ring done · -- (←) assume h1 : rel_prime m a define show ∃ (Y : ZMod m), [a]_m * Y = [1]_m from Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1) done done /- Section 7.4 -/ section Euler open Euler lemma num_rp_below_base {m : Nat} : num_rp_below m 0 = 0 := by rfl lemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) : num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by have h1 : num_rp_below m (j + 1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : gcd m j = 1 rewrite [if_pos h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1 show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1 done lemma num_rp_below_step_not_rp {m j : Nat} (h : ¬rel_prime m j) : num_rp_below m (j + 1) = num_rp_below m j := by have h1 : num_rp_below m (j +1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : ¬gcd m j = 1 rewrite [if_neg h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j show num_rp_below m (j + 1) = num_rp_below m j from h1 done lemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl #eval phi 10 --Answer: 4 lemma prod_inv_iff_inv {m : Nat} {X : ZMod m} (h1 : invertible X) (Y : ZMod m) : invertible (X * Y) ↔ invertible Y := by apply Iff.intro · -- (→) assume h2 : invertible (X * Y) obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2 apply Exists.intro (X * Z) rewrite [←h3] --Goal : Y * (X * Z) = X * Y * Z ring --Note that ring can do algebra in ZMod m done · -- (←) assume h2 : invertible Y obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1 obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2 apply Exists.intro (Xi * Yi) show (X * Y) * (Xi * Yi) = [1]_m from calc X * Y * (Xi * Yi) _ = (X * Xi) * (Y * Yi) := by ring _ = [1]_m * [1]_m := by rw [h3, h4] _ = [1]_m := Theorem_7_3_6_7 [1]_m done done lemma F_rp_def {m i : Nat} (h : rel_prime m i) : F m i = [i]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h --h : gcd m i = 1 rewrite [if_pos h] at h1 show F m i = [i]_m from h1 done lemma F_not_rp_def {m i : Nat} (h : ¬rel_prime m i) : F m i = [1]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h rewrite [h1, if_neg h] rfl done lemma prod_seq_base {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 0 k f = [1]_m := by rfl lemma prod_seq_step {m : Nat} (n k : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl lemma prod_seq_zero_step {m : Nat} (n : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by rewrite [prod_seq_step, zero_add] rfl done lemma prod_one {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 1 k f = f k := by rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7] rfl done lemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl lemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m := calc [G m a i]_m _ = [(a * i) % m]_m := by rfl _ = [a * i]_m := (cc_eq_mod m (a * i)).symm _ = [a]_m * [i]_m := (mul_class m a i).symm lemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) : rel_prime m (G m a i) ↔ rel_prime m i := by have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1 show rel_prime m (G m a i) ↔ rel_prime m i from calc rel_prime m (G m a i) _ ↔ invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm _ ↔ invertible ([a]_m * [i]_m) := by rw [cc_G] _ ↔ invertible [i]_m := prod_inv_iff_inv h2 ([i]_m) _ ↔ rel_prime m i := Theorem_7_3_7 m i done lemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) : F m (G m a i) = [a]_m * F m i := by have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2 show F m (G m a i) = [a]_m * F m i from calc F m (G m a i) _ = [G m a i]_m := F_rp_def h3 _ = [a]_m * [i]_m := cc_G m a i _ = [a]_m * F m i := by rw [F_rp_def h2] done lemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : ¬rel_prime m i) : F m (G m a i) = [1]_m := by rewrite [←G_rp_iff h1 i] at h2 show F m (G m a i) = [1]_m from F_not_rp_def h2 done lemma FG_prod {m a : Nat} (h1 : rel_prime m a) : ∀ (k : Nat), prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by by_induc · -- Base Case show prod_seq 0 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from calc prod_seq 0 0 ((F m) ∘ (G m a)) _ = [1]_m := prod_seq_base _ _ _ = [a]_m ^ 0 * [1]_m := by ring _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by rw [num_rp_below_base, prod_seq_base] done · -- Induction Step fix k : Nat assume ih : prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) by_cases h2 : rel_prime m k · -- Case 1. h2 : rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([a]_m * F m k) := by rw [FG_rp h1 h2] _ = [a]_m ^ ((num_rp_below m k) + 1) * ((prod_seq k 0 (F m)) * F m k) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_rp h2, prod_seq_zero_step] done · -- Case 2. h2 : ¬rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([1]_m) := by rw [FG_not_rp h1 h2] _ = [a]_m ^ (num_rp_below m k) * (prod_seq k 0 (F m) * ([1]_m)) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_not_rp h2, prod_seq_zero_step, F_not_rp_def h2] done done done lemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by define --Goal : ∀ i < m, G m a i < m fix i : Nat assume h1 : i < m rewrite [G_def] --Goal : a * i % m < m show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m) done lemma left_inv_one_one_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g' (g i) = i) : one_one_below n g := sorry lemma right_inv_onto_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g (g' i) = i) (h2 : maps_below n g') : onto_below n g := by define at h2; define fix k : Nat assume h3 : k < n apply Exists.intro (g' k) show g' k < n ∧ g (g' k) = k from And.intro (h2 k h3) (h1 k h3) done lemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m * [inv_mod m a]_m = [1]_m := by have h2 : 0 ≤ (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a) show [a]_m * [inv_mod m a]_m = [1]_m from calc [a]_m * [inv_mod m a]_m _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2] _ = [a]_m * [gcd_c2 m a]_m := by rw [←cc_eq_mod] _ = [1]_m := gcd_c2_inv h1 done lemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m := calc a * (b % m) % m = a % m * (b % m % m) % m := Nat.mul_mod _ _ _ _ = a % m * (b % m) % m := by rw [Nat.mod_mod] _ = a * b % m := (Nat.mul_mod _ _ _).symm lemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm] rfl done theorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] : ↑a ≡ ↑b (MOD m) ↔ a % m = b % m := sorry lemma mul_inv_mod_cancel {m a i : Nat} [NeZero m] (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1 rewrite [mul_class, cc_eq_iff_congr, ←Nat.cast_mul, ←Nat.cast_one, congr_iff_mod_eq_Nat] at h3 show a * inv_mod m a * i % m = i from calc a * (inv_mod m a) * i % m _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = 1 % m * i % m := by rw [h3] _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = i % m := by rw [one_mul] _ = i := Nat.mod_eq_of_lt h2 done lemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl lemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, G m a (Ginv m a i) = i := by fix i : Nat assume h2 : i < m show G m a (Ginv m a i) = i from calc G m a (Ginv m a i) _ = a * ((inv_mod m a * i) % m) % m := by rfl _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, Ginv m a (G m a i) = i := by fix i : Nat assume h2 : i < m show Ginv m a (G m a i) = i from calc Ginv m a (G m a i) _ = inv_mod m a * ((a * i) % m) % m := by rfl _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc, mul_comm (inv_mod m a)] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_maps_below (m a : Nat) [NeZero m] : maps_below m (Ginv m a) := G_maps_below m (inv_mod m a) lemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : one_one_below m (G m a) := left_inv_one_one_below (Ginv_left_inv h1) lemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : onto_below m (G m a) := right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a) lemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : perm_below m (G m a) := And.intro (G_maps_below m a) (And.intro (G_one_one_below h1) (G_onto_below h1)) --Permuting a product of congruence classes doesn't change product lemma swap_fst (u v : Nat) : swap u v u = v := by define : swap u v u --Goal : (if u = u then v else if u = v then u else u) = v have h : u = u := by rfl rewrite [if_pos h] rfl done lemma swap_snd (u v : Nat) : swap u v v = u := by define : swap u v v by_cases h1 : v = u · -- Case 1. h1 : v = u rewrite [if_pos h1] show v = u from h1 done · -- Case 2. h1 : v ≠ u rewrite [if_neg h1] have h2 : v = v := by rfl rewrite [if_pos h2] rfl done done lemma swap_other {u v i : Nat} (h1 : i ≠ u) (h2 : i ≠ v) : swap u v i = i := by define : swap u v i rewrite [if_neg h1, if_neg h2] rfl done lemma swap_values (u v i : Nat) : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := by by_cases h1 : i = u · -- Case 1. h1 : i = u apply Or.inl rewrite [h1] show swap u v u = v from swap_fst u v done · -- Case 2. h1 : i ≠ u apply Or.inr by_cases h2 : i = v · -- Case 2.1. h2 : i = v apply Or.inl rewrite [h2] show swap u v v = u from swap_snd u v done · -- Case 2.2. h2 : i ≠ v apply Or.inr show swap u v i = i from swap_other h1 h2 done done done lemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by define fix i : Nat assume h3 : i < n have h4 : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := swap_values u v i by_cases on h4 · -- Case 1. h4 : swap u v i = v rewrite [h4] show v < n from h2 done · -- Case 2. by_cases on h4 · -- Case 2.1. h4 : swap u v i = u rewrite [h4] show u < n from h1 done · -- Case 2.2. h4 : swap u v i = i rewrite [h4] show i < n from h3 done done done lemma swap_swap (u v n : Nat) : ∀ i < n, swap u v (swap u v i) = i := by fix i : Nat assume h : i < n by_cases h1 : i = u · -- Case 1. h1 : i = u rewrite [h1, swap_fst, swap_snd] rfl done · -- Case 2. h1 : i ≠ u by_cases h2 : i = v · -- Case 2.1. h2 : i = v rewrite [h2, swap_snd, swap_fst] rfl done · -- Case 2.2. h2 : i ≠ v rewrite [swap_other h1 h2, swap_other h1 h2] rfl done done done lemma swap_one_one_below (u v n) : one_one_below n (swap u v) := left_inv_one_one_below (swap_swap u v n) lemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) := right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2) lemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) := And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2)) lemma comp_perm_below {n : Nat} {f g : Nat → Nat} (h1 : perm_below n f) (h2 : perm_below n g) : perm_below n (f ∘ g) := sorry lemma trivial_swap (u : Nat) : swap u u = id := by apply funext fix x : Nat by_cases h1 : x = u · -- Case 1. h1 : x = u rewrite [h1, swap_fst] rfl done · -- Case 2. h1 : x ≠ u rewrite [swap_other h1 h1] rfl done done lemma prod_eq_fun {m : Nat} (f g : Nat → ZMod m) (k : Nat) : ∀ (n : Nat), (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g := by by_induc · -- Base Case assume h : (∀ i < 0, f (k + i) = g (k + i)) rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g assume h1 : ∀ i < n + 1, f (k + i) = g (k + i) have h2 : ∀ i < n, f (k + i) = g (k + i) := by fix i : Nat assume h2 : i < n have h3 : i < n + 1 := by linarith show f (k + i) = g (k + i) from h1 i h3 done have h3 : prod_seq n k f = prod_seq n k g := ih h2 have h4 : n < n + 1 := Nat.lt_succ_self n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4] rfl done done lemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := by have h2 : ∀ (i : Nat), i < u → (f ∘ swap u n) (0 + i) = f (0 + i) := by fix i : Nat assume h2 : i < u have h3 : 0 + i ≠ u := by linarith have h4 : 0 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl done show prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f from prod_eq_fun (f ∘ swap u n) f 0 u h2 done lemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := by have h2 : ∀ i < j, (f ∘ swap u n) (u + 1 + i) = f (u + 1 + i) := by fix i : Nat assume h2 : i < j have h3 : u + 1 + i ≠ u := by linarith have h4 : u + 1 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl show prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f from prod_eq_fun (f ∘ swap u n) f (u + 1) j h2 done lemma break_prod {m : Nat} (n : Nat) (f : Nat → ZMod m) : ∀ (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by by_induc · -- Base Case have h : n + 0 = n := by rfl rewrite [prod_seq_base, h, Theorem_7_3_6_7] rfl done · -- Induction Step fix j : Nat assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f rewrite [←add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc] rfl done done lemma break_prod_twice {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f := break_prod n f 1 rewrite [prod_one] at h2 have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f := break_prod (u + 1) f j rewrite [←h1] at h3 have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f := break_prod u f 1 rewrite [prod_one] at h4 rewrite [h3, h4] at h2 show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2 done lemma swap_prod_eq_prod {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f := by by_cases h2 : u = n · -- Case 1. h2 : u = n rewrite [h2, trivial_swap n] --Goal : prod_seq (n + 1) 0 (f ∘ id) = prod_seq (n + 1) 0 f rfl done · -- Case 2. h2 : ¬u = n have h3 : u + 1 ≤ n := Nat.lt_of_le_of_ne h1 h2 obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3 have break_f : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := break_prod_twice f h4 have break_fs : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_prod_twice (f ∘ swap u n) h4 have f_eq_fs_below : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := swap_prod_eq_prod_below f h1 have f_eq_fs_btwn : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4 show prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f from calc prod_seq (n + 1) 0 (f ∘ swap u n) _ = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_fs _ = prod_seq u 0 f * (f ∘ swap u n) u * prod_seq j (u + 1) f * (f ∘ swap u n) n := by rw [f_eq_fs_below, f_eq_fs_btwn] _ = prod_seq u 0 f * f (swap u n u) * prod_seq j (u + 1) f * f (swap u n n) := by rfl _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by rw [swap_fst, swap_snd] _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring _ = prod_seq (n + 1) 0 f := break_f.symm done done lemma perm_below_fixed {n : Nat} {g : Nat → Nat} (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry lemma perm_prod {m : Nat} (f : Nat → ZMod m) : ∀ (n : Nat), ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) := by by_induc · -- Base Case fix g : Nat → Nat assume h1 : perm_below 0 g rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) fix g : Nat → Nat assume g_pb : perm_below (n + 1) g define at g_pb have g_ob : onto_below (n + 1) g := g_pb.right.right define at g_ob have h1 : n < n + 1 := by linarith obtain (u : Nat) (h2 : u < n + 1 ∧ g u = n) from g_ob n h1 have s_pb : perm_below (n + 1) (swap u n) := swap_perm_below h2.left h1 have gs_pb_n1 : perm_below (n + 1) (g ∘ swap u n) := comp_perm_below g_pb s_pb have gs_fix_n : (g ∘ swap u n) n = n := calc (g ∘ swap u n) n _ = g (swap u n n) := by rfl _ = g u := by rw [swap_snd] _ = n := h2.right have gs_pb_n : perm_below n (g ∘ swap u n) := perm_below_fixed gs_pb_n1 gs_fix_n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f ∘ (g ∘ swap u n)) := ih (g ∘ swap u n) gs_pb_n have h3 : u ≤ n := by linarith show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f ∘ g) from calc prod_seq (n + 1) 0 f _ = prod_seq n 0 f * f n := prod_seq_zero_step n f _ = prod_seq n 0 (f ∘ (g ∘ swap u n)) * f ((g ∘ swap u n) n) := by rw [gs_prod, gs_fix_n] _ = prod_seq n 0 (f ∘ g ∘ swap u n) * (f ∘ g ∘ swap u n) n := by rfl _ = prod_seq (n + 1) 0 (f ∘ g ∘ swap u n) := (prod_seq_zero_step n (f ∘ g ∘ swap u n)).symm _ = prod_seq (n + 1) 0 ((f ∘ g) ∘ swap u n) := by rfl _ = prod_seq (n + 1) 0 (f ∘ g) := swap_prod_eq_prod (f ∘ g) h3 done done lemma F_invertible (m i : Nat) : invertible (F m i) := by by_cases h : rel_prime m i · -- Case 1. h : rel_prime m i rewrite [F_rp_def h] show invertible [i]_m from (Theorem_7_3_7 m i).rtl h done · -- Case 2. h : ¬rel_prime m i rewrite [F_not_rp_def h] apply Exists.intro [1]_m show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m done done lemma Fprod_invertible (m : Nat) : ∀ (k : Nat), invertible (prod_seq k 0 (F m)) := by by_induc · -- Base Case apply Exists.intro [1]_m show prod_seq 0 0 (F m) * [1]_m = [1]_m from calc prod_seq 0 0 (F m) * [1]_m _ = [1]_m * [1]_m := by rw [prod_seq_base] _ = [1]_m := Theorem_7_3_6_7 ([1]_m) done · -- Induction Step fix k : Nat assume ih : invertible (prod_seq k 0 (F m)) rewrite [prod_seq_zero_step] show invertible (prod_seq k 0 (F m) * (F m k)) from (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k) done done

theorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m ^ (phi m) = [1]_m

HTPI.Theorem_7_4_2

htpi/HTPILib/Chap7.lean

htpi

/- Copyright 2023 Daniel J. Velleman -/ import HTPILib.Chap6 namespace HTPI /- Definitions -/ lemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by have h : n + 1 > 0 := Nat.succ_pos n show a % (n + 1) < n + 1 from Nat.mod_lt a h done def gcd (a b : Nat) : Nat := match b with | 0 => a | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd (n + 1) (a % (n + 1)) termination_by b mutual def gcd_c1 (a b : Nat) : Int := match b with | 0 => 1 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c2 (n + 1) (a % (n + 1)) --Corresponds to s = t' termination_by b def gcd_c2 (a b : Nat) : Int := match b with | 0 => 0 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c1 (n + 1) (a % (n + 1)) - (gcd_c2 (n + 1) (a % (n + 1))) * ↑(a / (n + 1)) --Corresponds to t = s' - t'q termination_by b end def prime (n : Nat) : Prop := 2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n def prime_factor (p n : Nat) : Prop := prime p ∧ p ∣ n def all_prime (l : List Nat) : Prop := ∀ p ∈ l, prime p def nondec (l : List Nat) : Prop := match l with | [] => True --Of course, True is a proposition that is always true | n :: L => (∀ m ∈ L, n ≤ m) ∧ nondec L def nondec_prime_list (l : List Nat) : Prop := all_prime l ∧ nondec l def prod (l : List Nat) : Nat := match l with | [] => 1 | n :: L => n * (prod L) def prime_factorization (n : Nat) (l : List Nat) : Prop := nondec_prime_list l ∧ prod l = n def rel_prime (a b : Nat) : Prop := gcd a b = 1 def congr_mod (m : Nat) (a b : Int) : Prop := (↑m : Int) ∣ (a - b) def cc (m : Nat) (a : Int) : ZMod m := (↑a : ZMod m) notation:50 a " ≡ " b " (MOD " m ")" => congr_mod m a b notation:max "["a"]_"m:max => cc m a def invertible {m : Nat} (X : ZMod m) : Prop := ∃ (Y : ZMod m), X * Y = [1]_m def num_rp_below (m k : Nat) : Nat := match k with | 0 => 0 | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j def phi (m : Nat) : Nat := num_rp_below m m def prod_seq {m : Nat} (j k : Nat) (f : Nat → ZMod m) : ZMod m := match j with | 0 => [1]_m | n + 1 => prod_seq n k f * f (k + n) def maps_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i < n, g i < n def one_one_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i1 < n, ∀ i2 < n, g i1 = g i2 → i1 = i2 def onto_below (n : Nat) (g : Nat → Nat) : Prop := ∀ k < n, ∃ i < n, g i = k def perm_below (n : Nat) (g : Nat → Nat) : Prop := maps_below n g ∧ one_one_below n g ∧ onto_below n g def inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m) def swap (u v i : Nat) : Nat := if i = u then v else if i = v then u else i namespace Euler --For definitions specific to Euler's theorem def F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m def G (m a i : Nat) : Nat := (a * i) % m def Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i end Euler /- Section 7.1 -/ theorem dvd_mod_of_dvd_a_b {a b d : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ (a % b) := by set q : Nat := a / b have h3 : b * q + a % b = a := Nat.div_add_mod a b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 define --Goal : ∃ (c : Nat), a % b = d * c apply Exists.intro (j - k * q) show a % b = d * (j - k * q) from calc a % b _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm _ = a - b * q := by rw [h3] _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc] _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm done theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry #eval gcd 672 161 --Answer: 7 lemma gcd_base (a : Nat) : gcd a 0 = a := by rfl lemma gcd_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd a b = gcd b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1)) rfl done lemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b ≠ 0) : a % b < b := by have h1 : b > 0 := Nat.pos_of_ne_zero h show a % b < b from Nat.mod_lt a h1 done lemma dvd_self (n : Nat) : n ∣ n := by apply Exists.intro 1 ring done theorem gcd_dvd : ∀ (b a : Nat), (gcd a b) ∣ a ∧ (gcd a b) ∣ b := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd a b_1) ∣ a ∧ (gcd a b_1) ∣ b_1 fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_base] --Goal: a ∣ a ∧ a ∣ 0 apply And.intro (dvd_self a) define apply Exists.intro 0 rfl done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_nonzero a h1] --Goal : gcd b (a % b) ∣ a ∧ gcd b (a % b) ∣ b have h2 : a % b < b := mod_nonzero_lt a h1 have h3 : (gcd b (a % b)) ∣ b ∧ (gcd b (a % b)) ∣ (a % b) := ih (a % b) h2 b apply And.intro _ h3.left show (gcd b (a % b)) ∣ a from dvd_a_of_dvd_b_mod h3.left h3.right done done theorem gcd_dvd_left (a b : Nat) : (gcd a b) ∣ a := (gcd_dvd b a).left theorem gcd_dvd_right (a b : Nat) : (gcd a b) ∣ b := (gcd_dvd b a).right lemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl lemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c1 a b = gcd_c2 b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done lemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl lemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * ↑(a / b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done theorem gcd_lin_comb : ∀ (b a : Nat), (gcd_c1 a b) * ↑a + (gcd_c2 a b) * ↑b = ↑(gcd a b) := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd_c1 a b_1) * ↑a + (gcd_c2 a b_1) * ↑b_1 = ↑(gcd a b_1) fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base] --Goal : 1 * ↑a + 0 * ↑0 = ↑a ring done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1] --Goal : gcd_c2 b (a % b) * ↑a + -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * ↑(a / b)) * ↑b = -- ↑(gcd b (a % b)) set r : Nat := a % b set q : Nat := a / b set s : Int := gcd_c1 b r set t : Int := gcd_c2 b r --Goal : t * ↑a + (s - t * ↑q) * ↑b = ↑(gcd b r) have h2 : r < b := mod_nonzero_lt a h1 have h3 : s * ↑b + t * ↑r = ↑(gcd b r) := ih r h2 b have h4 : b * q + r = a := Nat.div_add_mod a b rewrite [←h3, ←h4] rewrite [Nat.cast_add, Nat.cast_mul] --Goal : t * (↑b * ↑q + ↑r) + (s - t * ↑q) * ↑b = s * ↑b + t * ↑r ring done done #eval gcd_c1 672 161 --Answer: 6 #eval gcd_c2 672 161 --Answer: -25 --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161 theorem Theorem_7_1_6 {d a b : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ gcd a b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑d ∣ ↑(gcd a b) set s : Int := gcd_c1 a b set t : Int := gcd_c2 a b have h3 : s * ↑a + t * ↑b = ↑(gcd a b) := gcd_lin_comb b a rewrite [←h3] --Goal : ↑d ∣ s * ↑a + t * ↑b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul] --Goal : ↑d ∣ s * (↑d * ↑j) + t * (↑d * ↑k) define apply Exists.intro (s * ↑j + t * ↑k) ring done /- Section 7.2 -/ theorem dvd_trans {a b c : Nat} (h1 : a ∣ b) (h2 : b ∣ c) : a ∣ c := by define at h1; define at h2; define obtain (m : Nat) (h3 : b = a * m) from h1 obtain (n : Nat) (h4 : c = b * n) from h2 rewrite [h3, mul_assoc] at h4 apply Exists.intro (m * n) show c = a * (m * n) from h4 done lemma exists_prime_factor : ∀ (n : Nat), 2 ≤ n → ∃ (p : Nat), prime_factor p n := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, 2 ≤ n_1 → ∃ (p : Nat), prime_factor p n_1 assume h1 : 2 ≤ n by_cases h2 : prime n · -- Case 1. h2 : prime n apply Exists.intro n define --Goal : prime n ∧ n ∣ n show prime n ∧ n ∣ n from And.intro h2 (dvd_self n) done · -- Case 2. h2 : ¬prime n define at h2 --h2 : ¬(2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n) demorgan at h2 disj_syll h2 h1 obtain (a : Nat) (h3 : ∃ (b : Nat), a * b = n ∧ a < n ∧ b < n) from h2 obtain (b : Nat) (h4 : a * b = n ∧ a < n ∧ b < n) from h3 have h5 : 2 ≤ a := by by_contra h6 have h7 : a ≤ 1 := by linarith have h8 : n ≤ b := calc n _ = a * b := h4.left.symm _ ≤ 1 * b := by rel [h7] _ = b := by ring linarith --n ≤ b contradicts b < n done have h6 : ∃ (p : Nat), prime_factor p a := ih a h4.right.left h5 obtain (p : Nat) (h7 : prime_factor p a) from h6 apply Exists.intro p define --Goal : prime p ∧ p ∣ n define at h7 --h7 : prime p ∧ p ∣ a apply And.intro h7.left have h8 : a ∣ n := by apply Exists.intro b show n = a * b from (h4.left).symm done show p ∣ n from dvd_trans h7.right h8 done done lemma exists_least_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q := by set S : Set Nat := {p : Nat | prime_factor p n} have h2 : ∃ (p : Nat), p ∈ S := exists_prime_factor n h show ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q from well_ord_princ S h2 done lemma all_prime_nil : all_prime [] := by define --Goal : ∀ p ∈ [], prime p fix p : Nat contrapos --Goal : ¬prime p → p ∉ [] assume h1 : ¬prime p show p ∉ [] from List.not_mem_nil p done lemma all_prime_cons (n : Nat) (L : List Nat) : all_prime (n :: L) ↔ prime n ∧ all_prime L := by apply Iff.intro · -- (→) assume h1 : all_prime (n :: L) --Goal : prime n ∧ all_prime L define at h1 --h1 : ∀ p ∈ n :: L, prime p apply And.intro (h1 n (List.mem_cons_self n L)) define --Goal : ∀ p ∈ L, prime p fix p : Nat assume h2 : p ∈ L show prime p from h1 p (List.mem_cons_of_mem n h2) done · -- (←) assume h1 : prime n ∧ all_prime L --Goal : all_prime (n :: l) define : all_prime L at h1 define fix p : Nat assume h2 : p ∈ n :: L rewrite [List.mem_cons] at h2 --h2 : p = n ∨ p ∈ L by_cases on h2 · -- Case 1. h2 : p = n rewrite [h2] show prime n from h1.left done · -- Case 2. h2 : p ∈ L show prime p from h1.right p h2 done done done lemma nondec_nil : nondec [] := by define --Goal : True trivial --trivial proves some obviously true statements, such as True done lemma nondec_cons (n : Nat) (L : List Nat) : nondec (n :: L) ↔ (∀ m ∈ L, n ≤ m) ∧ nondec L := by rfl lemma prod_nil : prod [] = 1 := by rfl lemma prod_cons : prod (n :: L) = n * (prod L) := by rfl lemma exists_cons_of_length_eq_succ {A : Type} {l : List A} {n : Nat} (h : l.length = n + 1) : ∃ (a : A) (L : List A), l = a :: L ∧ L.length = n := by have h1 : ¬l.length = 0 := by linarith rewrite [List.length_eq_zero] at h1 obtain (a : A) (h2 : ∃ (L : List A), l = a :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List A) (h3 : l = a :: L) from h2 apply Exists.intro a apply Exists.intro L apply And.intro h3 have h4 : (a :: L).length = L.length + 1 := List.length_cons a L rewrite [←h3, h] at h4 show L.length = n from (Nat.add_right_cancel h4).symm done lemma list_elt_dvd_prod_by_length (a : Nat) : ∀ (n : Nat), ∀ (l : List Nat), l.length = n → a ∈ l → a ∣ prod l := by by_induc · --Base Case fix l : List Nat assume h1 : l.length = 0 rewrite [List.length_eq_zero] at h1 --h1 : l = [] rewrite [h1] --Goal : a ∈ [] → a ∣ prod [] contrapos assume h2 : ¬a ∣ prod [] show a ∉ [] from List.not_mem_nil a done · -- Induction Step fix n : Nat assume ih : ∀ (l : List Nat), List.length l = n → a ∈ l → a ∣ prod l fix l : List Nat assume h1 : l.length = n + 1 --Goal : a ∈ l → a ∣ prod l obtain (b : Nat) (h2 : ∃ (L : List Nat), l = b :: L ∧ L.length = n) from exists_cons_of_length_eq_succ h1 obtain (L : List Nat) (h3 : l = b :: L ∧ L.length = n) from h2 have h4 : a ∈ L → a ∣ prod L := ih L h3.right assume h5 : a ∈ l rewrite [h3.left, prod_cons] --Goal : a ∣ b * prod L rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b ∨ a ∈ L by_cases on h5 · -- Case 1. h5 : a = b apply Exists.intro (prod L) rewrite [h5] rfl done · -- Case 2. h5 : a ∈ L have h6 : a ∣ prod L := h4 h5 have h7 : prod L ∣ b * prod L := by apply Exists.intro b ring done show a ∣ b * prod L from dvd_trans h6 h7 done done done lemma list_elt_dvd_prod {a : Nat} {l : List Nat} (h : a ∈ l) : a ∣ prod l := by set n : Nat := l.length have h1 : l.length = n := by rfl show a ∣ prod l from list_elt_dvd_prod_by_length a n l h1 h done lemma exists_prime_factorization : ∀ (n : Nat), n ≥ 1 → ∃ (l : List Nat), prime_factorization n l := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, n_1 ≥ 1 → ∃ (l : List Nat), prime_factorization n_1 l assume h1 : n ≥ 1 by_cases h2 : n = 1 · -- Case 1. h2 : n = 1 apply Exists.intro [] define apply And.intro · -- Proof of nondec_prime_list [] define show all_prime [] ∧ nondec [] from And.intro all_prime_nil nondec_nil done · -- Proof of prod [] = n rewrite [prod_nil, h2] rfl done done · -- Case 2. h2 : n ≠ 1 have h3 : n ≥ 2 := lt_of_le_of_ne' h1 h2 obtain (p : Nat) (h4 : prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q) from exists_least_prime_factor h3 have p_prime_factor : prime_factor p n := h4.left define at p_prime_factor have p_prime : prime p := p_prime_factor.left have p_dvd_n : p ∣ n := p_prime_factor.right have p_least : ∀ (q : Nat), prime_factor q n → p ≤ q := h4.right obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n have h5 : m ≠ 0 := by contradict h1 with h6 have h7 : n = 0 := calc n _ = p * m := n_eq_pm _ = p * 0 := by rw [h6] _ = 0 := by ring rewrite [h7] decide done have m_pos : 0 < m := Nat.pos_of_ne_zero h5 have m_lt_n : m < n := by define at p_prime show m < n from calc m _ < m + m := by linarith _ = 2 * m := by ring _ ≤ p * m := by rel [p_prime.left] _ = n := n_eq_pm.symm done obtain (L : List Nat) (h6 : prime_factorization m L) from ih m m_lt_n m_pos define at h6 have ndpl_L : nondec_prime_list L := h6.left define at ndpl_L apply Exists.intro (p :: L) define apply And.intro · -- Proof of nondec_prime_list (p :: L) define apply And.intro · -- Proof of all_prime (p :: L) rewrite [all_prime_cons] show prime p ∧ all_prime L from And.intro p_prime ndpl_L.left done · -- Proof of nondec (p :: L) rewrite [nondec_cons] apply And.intro _ ndpl_L.right fix q : Nat assume q_in_L : q ∈ L have h7 : q ∣ prod L := list_elt_dvd_prod q_in_L rewrite [h6.right] at h7 --h7 : q ∣ m have h8 : m ∣ n := by apply Exists.intro p rewrite [n_eq_pm] ring done have q_dvd_n : q ∣ n := dvd_trans h7 h8 have ap_L : all_prime L := ndpl_L.left define at ap_L have q_prime_factor : prime_factor q n := And.intro (ap_L q q_in_L) q_dvd_n show p ≤ q from p_least q q_prime_factor done done · -- Proof of prod (p :: L) = n rewrite [prod_cons, h6.right, n_eq_pm] rfl done done done theorem Theorem_7_2_2 {a b c : Nat} (h1 : c ∣ a * b) (h2 : rel_prime a c) : c ∣ b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑c ∣ ↑b define at h1; define at h2; define obtain (j : Nat) (h3 : a * b = c * j) from h1 set s : Int := gcd_c1 a c set t : Int := gcd_c2 a c have h4 : s * ↑a + t * ↑c = ↑(gcd a c) := gcd_lin_comb c a rewrite [h2, Nat.cast_one] at h4 --h4 : s * ↑a + t * ↑c = (1 : Int) apply Exists.intro (s * ↑j + t * ↑b) show ↑b = ↑c * (s * ↑j + t * ↑b) from calc ↑b _ = (1 : Int) * ↑b := (one_mul _).symm _ = (s * ↑a + t * ↑c) * ↑b := by rw [h4] _ = s * (↑a * ↑b) + t * ↑c * ↑b := by ring _ = s * (↑c * ↑j) + t * ↑c * ↑b := by rw [←Nat.cast_mul a b, h3, Nat.cast_mul c j] _ = ↑c * (s * ↑j + t * ↑b) := by ring done lemma le_nonzero_prod_left {a b : Nat} (h : a * b ≠ 0) : a ≤ a * b := by have h1 : b ≠ 0 := by contradict h with h1 rewrite [h1] ring done have h2 : 1 ≤ b := Nat.pos_of_ne_zero h1 show a ≤ a * b from calc a = a * 1 := (mul_one a).symm _ ≤ a * b := by rel [h2] done lemma le_nonzero_prod_right {a b : Nat} (h : a * b ≠ 0) : b ≤ a * b := by rewrite [mul_comm] rewrite [mul_comm] at h show b ≤ b * a from le_nonzero_prod_left h done lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry lemma rel_prime_of_prime_not_dvd {a p : Nat} (h1 : prime p) (h2 : ¬p ∣ a) : rel_prime a p := by have h3 : gcd a p ∣ a := gcd_dvd_left a p have h4 : gcd a p ∣ p := gcd_dvd_right a p have h5 : gcd a p = 1 ∨ gcd a p = p := dvd_prime h1 h4 have h6 : gcd a p ≠ p := by contradict h2 with h6 rewrite [h6] at h3 show p ∣ a from h3 done disj_syll h5 h6 show rel_prime a p from h5 done theorem Theorem_7_2_3 {a b p : Nat} (h1 : prime p) (h2 : p ∣ a * b) : p ∣ a ∨ p ∣ b := by or_right with h3 have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3 show p ∣ b from Theorem_7_2_2 h2 h4 done lemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a ≥ 1 := by have h1 : a ≠ 0 := by by_contra h1 rewrite [h1] at h contradict h linarith done show a ≥ 1 from Nat.pos_of_ne_zero h1 done lemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by have h1 : a ≥ 1 := ge_one_of_prod_one h have h2 : a * b ≠ 0 := by linarith have h3 : a ≤ a * b := le_nonzero_prod_left h2 rewrite [h] at h3 show a = 1 from Nat.le_antisymm h3 h1 done lemma eq_one_of_dvd_one {n : Nat} (h : n ∣ 1) : n = 1 := by obtain (j : Nat) (h1 : 1 = n * j) from h show n = 1 from eq_one_of_prod_one h1.symm done lemma prime_not_one {p : Nat} (h : prime p) : p ≠ 1 := by define at h linarith done theorem Theorem_7_2_4 {p : Nat} (h1 : prime p) : ∀ (l : List Nat), p ∣ prod l → ∃ a ∈ l, p ∣ a := by apply List.rec · -- Base Case. Goal : p ∣ prod [] → ∃ a ∈ [], p ∣ a rewrite [prod_nil] assume h2 : p ∣ 1 show ∃ a ∈ [], p ∣ a from absurd (eq_one_of_dvd_one h2) (prime_not_one h1) done · -- Induction Step fix b : Nat fix L : List Nat assume ih : p ∣ prod L → ∃ a ∈ L, p ∣ a --Goal : p ∣ prod (b :: L) → ∃ a ∈ b :: L, p ∣ a assume h2 : p ∣ prod (b :: L) rewrite [prod_cons] at h2 have h3 : p ∣ b ∨ p ∣ prod L := Theorem_7_2_3 h1 h2 by_cases on h3 · -- Case 1. h3 : p ∣ b apply Exists.intro b show b ∈ b :: L ∧ p ∣ b from And.intro (List.mem_cons_self b L) h3 done · -- Case 2. h3 : p ∣ prod L obtain (a : Nat) (h4 : a ∈ L ∧ p ∣ a) from ih h3 apply Exists.intro a show a ∈ b :: L ∧ p ∣ a from And.intro (List.mem_cons_of_mem b h4.left) h4.right done done done lemma prime_in_list {p : Nat} {l : List Nat} (h1 : prime p) (h2 : all_prime l) (h3 : p ∣ prod l) : p ∈ l := by obtain (a : Nat) (h4 : a ∈ l ∧ p ∣ a) from Theorem_7_2_4 h1 l h3 define at h2 have h5 : prime a := h2 a h4.left have h6 : p = 1 ∨ p = a := dvd_prime h5 h4.right disj_syll h6 (prime_not_one h1) rewrite [h6] show a ∈ l from h4.left done lemma first_le_first {p q : Nat} {l m : List Nat} (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m)) (h3 : prod (p :: l) = prod (q :: m)) : p ≤ q := by define at h1; define at h2 have h4 : q ∣ prod (p :: l) := by define apply Exists.intro (prod m) rewrite [←prod_cons] show prod (p :: l) = prod (q :: m) from h3 done have h5 : all_prime (q :: m) := h2.left rewrite [all_prime_cons] at h5 have h6 : q ∈ p :: l := prime_in_list h5.left h1.left h4 have h7 : nondec (p :: l) := h1.right rewrite [nondec_cons] at h7 rewrite [List.mem_cons] at h6 by_cases on h6 · -- Case 1. h6 : q = p linarith done · -- Case 2. h6 : q ∈ l have h8 : ∀ m ∈ l, p ≤ m := h7.left show p ≤ q from h8 q h6 done done lemma nondec_prime_list_tail {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by define at h define rewrite [all_prime_cons, nondec_cons] at h show all_prime l ∧ nondec l from And.intro h.left.right h.right.right done lemma cons_prod_not_one {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : prod (p :: l) ≠ 1 := by define at h have h1 : all_prime (p :: l) := h.left rewrite [all_prime_cons] at h1 rewrite [prod_cons] by_contra h2 show False from (prime_not_one h1.left) (eq_one_of_prod_one h2) done lemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) : l = [] ↔ prod l = 1 := by apply Iff.intro · -- (→) assume h1 : l = [] rewrite [h1] show prod [] = 1 from prod_nil done · -- (←) contrapos assume h1 : ¬l = [] obtain (p : Nat) (h2 : ∃ (L : List Nat), l = p :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List Nat) (h3 : l = p :: L) from h2 rewrite [h3] at h rewrite [h3] show ¬prod (p :: L) = 1 from cons_prod_not_one h done done lemma prime_pos {p : Nat} (h : prime p) : p > 0 := by define at h linarith done theorem Theorem_7_2_5 : ∀ (l1 l2 : List Nat), nondec_prime_list l1 → nondec_prime_list l2 → prod l1 = prod l2 → l1 = l2 := by apply List.rec · -- Base Case. Goal : ∀ (l2 : List Nat), nondec_prime_list [] → -- nondec_prime_list l2 → prod [] = prod l2 → [] = l2 fix l2 : List Nat assume h1 : nondec_prime_list [] assume h2 : nondec_prime_list l2 assume h3 : prod [] = prod l2 rewrite [prod_nil, eq_comm, ←list_nil_iff_prod_one h2] at h3 show [] = l2 from h3.symm done · -- Induction Step fix p : Nat fix L1 : List Nat assume ih : ∀ (L2 : List Nat), nondec_prime_list L1 → nondec_prime_list L2 → prod L1 = prod L2 → L1 = L2 -- Goal : ∀ (l2 : List Nat), nondec_prime_list (p :: L1) → -- nondec_prime_list l2 → prod (p :: L1) = prod l2 → p :: L1 = l2 fix l2 : List Nat assume h1 : nondec_prime_list (p :: L1) assume h2 : nondec_prime_list l2 assume h3 : prod (p :: L1) = prod l2 have h4 : ¬prod (p :: L1) = 1 := cons_prod_not_one h1 rewrite [h3, ←list_nil_iff_prod_one h2] at h4 obtain (q : Nat) (h5 : ∃ (L : List Nat), l2 = q :: L) from List.exists_cons_of_ne_nil h4 obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5 rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2) rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2) have h7 : p ≤ q := first_le_first h1 h2 h3 have h8 : q ≤ p := first_le_first h2 h1 h3.symm have h9 : p = q := by linarith rewrite [h9, prod_cons, prod_cons] at h3 --h3 : q * prod L1 = q * prod L2 have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1 have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2 define at h2 have h12 : all_prime (q :: L2) := h2.left rewrite [all_prime_cons] at h12 have h13 : q > 0 := prime_pos h12.left have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3 have h15 : L1 = L2 := ih L2 h10 h11 h14 rewrite [h6, h9, h15] rfl done done theorem fund_thm_arith (n : Nat) (h : n ≥ 1) : ∃! (l : List Nat), prime_factorization n l := by exists_unique · -- Existence show ∃ (l : List Nat), prime_factorization n l from exists_prime_factorization n h done · -- Uniqueness fix l1 : List Nat; fix l2 : List Nat assume h1 : prime_factorization n l1 assume h2 : prime_factorization n l2 define at h1; define at h2 have h3 : prod l1 = n := h1.right rewrite [←h2.right] at h3 show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3 done done /- Section 7.3 -/ theorem congr_refl (m : Nat) : ∀ (a : Int), a ≡ a (MOD m) := by fix a : Int define --Goal : ∃ (c : Int), a - a = ↑m * c apply Exists.intro 0 ring done theorem congr_symm {m : Nat} : ∀ {a b : Int}, a ≡ b (MOD m) → b ≡ a (MOD m) := by fix a : Int; fix b : Int assume h1 : a ≡ b (MOD m) define at h1 --h1 : ∃ (c : Int), a - b = ↑m * c define --Goal : ∃ (c : Int), b - a = ↑m * c obtain (c : Int) (h2 : a - b = m * c) from h1 apply Exists.intro (-c) show b - a = m * (-c) from calc b - a _ = -(a - b) := by ring _ = -(m * c) := by rw [h2] _ = m * (-c) := by ring done theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry /- Fundamental properties of congruence classes -/ lemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) : X = Y ↔ X.val = Y.val := Fin.ext_iff lemma val_nat_eq_mod (n k : Nat) : ([k]_(n + 1)).val = k % (n + 1) := by rfl lemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl theorem cc_rep {m : Nat} (X : ZMod m) : ∃ (a : Int), X = [a]_m := match m with | 0 => by apply Exists.intro X rfl done | n + 1 => by apply Exists.intro ↑(X.val) have h1 : X.val < n + 1 := Fin.prop X rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1] rfl done theorem add_class (m : Nat) (a b : Int) : [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm theorem mul_class (m : Nat) (a b : Int) : [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm lemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ [a - b]_m = [0]_m := by apply Iff.intro · -- (→) assume h1 : [a]_m = [b]_m have h2 : a - b = a + (-b) := by ring have h3 : b + (-b) = 0 := by ring show [a - b]_m = [0]_m from calc [a - b]_m _ = [a + (-b)]_m := by rw [h2] _ = [a]_m + [-b]_m := by rw [add_class] _ = [b]_m + [-b]_m := by rw [h1] _ = [b + -b]_m := by rw [add_class] _ = [0]_m := by rw [h3] done · -- (←) assume h1 : [a - b]_m = [0]_m have h2 : b + (a - b) = a := by ring have h3 : b + 0 = b := by ring show [a]_m = [b]_m from calc [a]_m _ = [b + (a - b)]_m := by rw [h2] _ = [b]_m + [a - b]_m := by rw [add_class] _ = [b]_m + [0]_m := by rw [h1] _ = [b + 0]_m := by rw [add_class] _ = [b]_m := by rw [h3] done done lemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) : [a]_m = [0]_m → [-a]_m = [0]_m := by assume h1 : [a]_m = [0]_m have h2 : 0 + (-a) = -a := by ring have h3 : a + (-a) = 0 := by ring show [-a]_m = [0]_m from calc [-a]_m _ = [0 + (-a)]_m := by rw [h2] _ = [0]_m + [-a]_m := by rw [add_class] _ = [a]_m + [-a]_m := by rw [h1] _ = [a + (-a)]_m := by rw [add_class] _ = [0]_m := by rw [h3] done lemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) : [-a]_m = [0]_m ↔ [a]_m = [0]_m := by apply Iff.intro _ (cc_neg_zero_of_cc_zero m a) assume h1 : [-a]_m = [0]_m have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1 have h3 : -(-a) = a := by ring rewrite [h3] at h2 show [a]_m = [0]_m from h2 done lemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl lemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m ↔ m ∣ k := match m with | 0 => by have h : (0 : Int) = (↑(0 : Nat) : Int) := by rfl rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj] apply Iff.intro · -- (→) assume h1 : k = 0 rewrite [h1] show 0 ∣ 0 from dvd_self 0 done · -- (←) assume h1 : 0 ∣ k obtain (c : Nat) (h2 : k = 0 * c) from h1 rewrite [h2] ring done done | n + 1 => by rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero] show k % (n + 1) = 0 ↔ n + 1 ∣ k from (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm done lemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m ↔ ↑m ∣ a := by obtain (k : Nat) (h1 : a = ↑k ∨ a = -↑k) from Int.eq_nat_or_neg a by_cases on h1 · -- Case 1. h1: a = ↑k rewrite [h1, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done · -- Case 2. h1: a = -↑k rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done done theorem cc_eq_iff_congr (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ a ≡ b (MOD m) := calc [a]_m = [b]_m _ ↔ [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b _ ↔ ↑m ∣ (a - b) := cc_zero_iff_dvd m (a - b) _ ↔ a ≡ b (MOD m) := by rfl /- End of fundamental properties of congruence classes -/ lemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m := by have h1 : (↑m : Int) ≠ 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m) show 0 ≤ a % m from Int.emod_nonneg a h1 done lemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m) have h2 : (↑m : Int) > 0 := (Nat.cast_pos).rtl h1 show a % m < m from Int.emod_lt_of_pos a h2 done lemma congr_mod_mod (m : Nat) (a : Int) : a ≡ a % m (MOD m) := by define have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m apply Exists.intro (a / m) show a - a % m = m * (a / m) from calc a - (a % m) _ = m * (a / m) + a % m - a % m := by rw [h1] _ = m * (a / m) := by ring done lemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) := And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a)) theorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) : ∃! (r : Int), 0 ≤ r ∧ r < m ∧ a ≡ r (MOD m) := by exists_unique · -- Existence apply Exists.intro (a % m) show 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) from mod_cmpl_res m a done · -- Uniqueness fix r1 : Int; fix r2 : Int assume h1 : 0 ≤ r1 ∧ r1 < m ∧ a ≡ r1 (MOD m) assume h2 : 0 ≤ r2 ∧ r2 < m ∧ a ≡ r2 (MOD m) have h3 : r1 ≡ r2 (MOD m) := congr_trans (congr_symm h1.right.right) h2.right.right obtain (d : Int) (h4 : r1 - r2 = m * d) from h3 have h5 : r1 - r2 < m * 1 := by linarith have h6 : m * (-1) < r1 - r2 := by linarith rewrite [h4] at h5 --h5 : m * d < m * 1 rewrite [h4] at h6 --h6 : m * -1 < m * d have h7 : (↑m : Int) ≥ 0 := Nat.cast_nonneg m have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7 have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7 have h10 : d = 0 := by linarith show r1 = r2 from calc r1 _ = r1 - r2 + r2 := by ring _ = m * 0 + r2 := by rw [h4, h10] _ = r2 := by ring done done lemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m := (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a) theorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y rewrite [h1, h2] have h3 : a + b = b + a := by ring show [a]_m + [b]_m = [b]_m + [a]_m from calc [a]_m + [b]_m _ = [a + b]_m := add_class m a b _ = [b + a]_m := by rw [h3] _ = [b]_m + [a]_m := (add_class m b a).symm done theorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X rewrite [h1] have h2 : a * 1 = a := by ring show [a]_m * [1]_m = [a]_m from calc [a]_m * [1]_m _ = [a * 1]_m := mul_class m a 1 _ = [a]_m := by rw [h2] done theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry lemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) : [a]_m * [gcd_c2 m a]_m = [1]_m := by set s : Int := gcd_c1 m a have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m define at h1 rewrite [h1, Nat.cast_one] at h2 --h2 : s * ↑m + gcd_c2 m a * ↑a = 1 rewrite [mul_class, cc_eq_iff_congr] define --Goal : ∃ (c : Int), ↑a * gcd_c2 m a - 1 = ↑m * c apply Exists.intro (-s) show a * (gcd_c2 m a) - 1 = m * (-s) from calc a * (gcd_c2 m a) - 1 _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring _ = 1 + m * (-s) - 1 := by rw [h2] _ = m * (-s) := by ring done theorem Theorem_7_3_7 (m a : Nat) : invertible [a]_m ↔ rel_prime m a := by apply Iff.intro · -- (→) assume h1 : invertible [a]_m define at h1 obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1 obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y rewrite [h3, mul_class, cc_eq_iff_congr] at h2 define at h2 obtain (c : Int) (h4 : a * b - 1 = m * c) from h2 rewrite [Exercise_7_2_6] --Goal : ∃ (s t : Int), s * ↑m + t * ↑a = 1 apply Exists.intro (-c) apply Exists.intro b show (-c) * m + b * a = 1 from calc (-c) * m + b * a _ = (-c) * m + (a * b - 1) + 1 := by ring _ = (-c) * m + m * c + 1 := by rw [h4] _ = 1 := by ring done · -- (←) assume h1 : rel_prime m a define show ∃ (Y : ZMod m), [a]_m * Y = [1]_m from Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1) done done /- Section 7.4 -/ section Euler open Euler lemma num_rp_below_base {m : Nat} : num_rp_below m 0 = 0 := by rfl lemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) : num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by have h1 : num_rp_below m (j + 1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : gcd m j = 1 rewrite [if_pos h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1 show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1 done lemma num_rp_below_step_not_rp {m j : Nat} (h : ¬rel_prime m j) : num_rp_below m (j + 1) = num_rp_below m j := by have h1 : num_rp_below m (j +1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : ¬gcd m j = 1 rewrite [if_neg h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j show num_rp_below m (j + 1) = num_rp_below m j from h1 done lemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl #eval phi 10 --Answer: 4 lemma prod_inv_iff_inv {m : Nat} {X : ZMod m} (h1 : invertible X) (Y : ZMod m) : invertible (X * Y) ↔ invertible Y := by apply Iff.intro · -- (→) assume h2 : invertible (X * Y) obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2 apply Exists.intro (X * Z) rewrite [←h3] --Goal : Y * (X * Z) = X * Y * Z ring --Note that ring can do algebra in ZMod m done · -- (←) assume h2 : invertible Y obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1 obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2 apply Exists.intro (Xi * Yi) show (X * Y) * (Xi * Yi) = [1]_m from calc X * Y * (Xi * Yi) _ = (X * Xi) * (Y * Yi) := by ring _ = [1]_m * [1]_m := by rw [h3, h4] _ = [1]_m := Theorem_7_3_6_7 [1]_m done done lemma F_rp_def {m i : Nat} (h : rel_prime m i) : F m i = [i]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h --h : gcd m i = 1 rewrite [if_pos h] at h1 show F m i = [i]_m from h1 done lemma F_not_rp_def {m i : Nat} (h : ¬rel_prime m i) : F m i = [1]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h rewrite [h1, if_neg h] rfl done lemma prod_seq_base {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 0 k f = [1]_m := by rfl lemma prod_seq_step {m : Nat} (n k : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl lemma prod_seq_zero_step {m : Nat} (n : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by rewrite [prod_seq_step, zero_add] rfl done lemma prod_one {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 1 k f = f k := by rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7] rfl done lemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl lemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m := calc [G m a i]_m _ = [(a * i) % m]_m := by rfl _ = [a * i]_m := (cc_eq_mod m (a * i)).symm _ = [a]_m * [i]_m := (mul_class m a i).symm lemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) : rel_prime m (G m a i) ↔ rel_prime m i := by have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1 show rel_prime m (G m a i) ↔ rel_prime m i from calc rel_prime m (G m a i) _ ↔ invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm _ ↔ invertible ([a]_m * [i]_m) := by rw [cc_G] _ ↔ invertible [i]_m := prod_inv_iff_inv h2 ([i]_m) _ ↔ rel_prime m i := Theorem_7_3_7 m i done lemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) : F m (G m a i) = [a]_m * F m i := by have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2 show F m (G m a i) = [a]_m * F m i from calc F m (G m a i) _ = [G m a i]_m := F_rp_def h3 _ = [a]_m * [i]_m := cc_G m a i _ = [a]_m * F m i := by rw [F_rp_def h2] done lemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : ¬rel_prime m i) : F m (G m a i) = [1]_m := by rewrite [←G_rp_iff h1 i] at h2 show F m (G m a i) = [1]_m from F_not_rp_def h2 done lemma FG_prod {m a : Nat} (h1 : rel_prime m a) : ∀ (k : Nat), prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by by_induc · -- Base Case show prod_seq 0 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from calc prod_seq 0 0 ((F m) ∘ (G m a)) _ = [1]_m := prod_seq_base _ _ _ = [a]_m ^ 0 * [1]_m := by ring _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by rw [num_rp_below_base, prod_seq_base] done · -- Induction Step fix k : Nat assume ih : prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) by_cases h2 : rel_prime m k · -- Case 1. h2 : rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([a]_m * F m k) := by rw [FG_rp h1 h2] _ = [a]_m ^ ((num_rp_below m k) + 1) * ((prod_seq k 0 (F m)) * F m k) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_rp h2, prod_seq_zero_step] done · -- Case 2. h2 : ¬rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([1]_m) := by rw [FG_not_rp h1 h2] _ = [a]_m ^ (num_rp_below m k) * (prod_seq k 0 (F m) * ([1]_m)) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_not_rp h2, prod_seq_zero_step, F_not_rp_def h2] done done done lemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by define --Goal : ∀ i < m, G m a i < m fix i : Nat assume h1 : i < m rewrite [G_def] --Goal : a * i % m < m show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m) done lemma left_inv_one_one_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g' (g i) = i) : one_one_below n g := sorry lemma right_inv_onto_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g (g' i) = i) (h2 : maps_below n g') : onto_below n g := by define at h2; define fix k : Nat assume h3 : k < n apply Exists.intro (g' k) show g' k < n ∧ g (g' k) = k from And.intro (h2 k h3) (h1 k h3) done lemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m * [inv_mod m a]_m = [1]_m := by have h2 : 0 ≤ (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a) show [a]_m * [inv_mod m a]_m = [1]_m from calc [a]_m * [inv_mod m a]_m _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2] _ = [a]_m * [gcd_c2 m a]_m := by rw [←cc_eq_mod] _ = [1]_m := gcd_c2_inv h1 done lemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m := calc a * (b % m) % m = a % m * (b % m % m) % m := Nat.mul_mod _ _ _ _ = a % m * (b % m) % m := by rw [Nat.mod_mod] _ = a * b % m := (Nat.mul_mod _ _ _).symm lemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm] rfl done theorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] : ↑a ≡ ↑b (MOD m) ↔ a % m = b % m := sorry lemma mul_inv_mod_cancel {m a i : Nat} [NeZero m] (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1 rewrite [mul_class, cc_eq_iff_congr, ←Nat.cast_mul, ←Nat.cast_one, congr_iff_mod_eq_Nat] at h3 show a * inv_mod m a * i % m = i from calc a * (inv_mod m a) * i % m _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = 1 % m * i % m := by rw [h3] _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = i % m := by rw [one_mul] _ = i := Nat.mod_eq_of_lt h2 done lemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl lemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, G m a (Ginv m a i) = i := by fix i : Nat assume h2 : i < m show G m a (Ginv m a i) = i from calc G m a (Ginv m a i) _ = a * ((inv_mod m a * i) % m) % m := by rfl _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, Ginv m a (G m a i) = i := by fix i : Nat assume h2 : i < m show Ginv m a (G m a i) = i from calc Ginv m a (G m a i) _ = inv_mod m a * ((a * i) % m) % m := by rfl _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc, mul_comm (inv_mod m a)] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_maps_below (m a : Nat) [NeZero m] : maps_below m (Ginv m a) := G_maps_below m (inv_mod m a) lemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : one_one_below m (G m a) := left_inv_one_one_below (Ginv_left_inv h1) lemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : onto_below m (G m a) := right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a) lemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : perm_below m (G m a) := And.intro (G_maps_below m a) (And.intro (G_one_one_below h1) (G_onto_below h1)) --Permuting a product of congruence classes doesn't change product lemma swap_fst (u v : Nat) : swap u v u = v := by define : swap u v u --Goal : (if u = u then v else if u = v then u else u) = v have h : u = u := by rfl rewrite [if_pos h] rfl done lemma swap_snd (u v : Nat) : swap u v v = u := by define : swap u v v by_cases h1 : v = u · -- Case 1. h1 : v = u rewrite [if_pos h1] show v = u from h1 done · -- Case 2. h1 : v ≠ u rewrite [if_neg h1] have h2 : v = v := by rfl rewrite [if_pos h2] rfl done done lemma swap_other {u v i : Nat} (h1 : i ≠ u) (h2 : i ≠ v) : swap u v i = i := by define : swap u v i rewrite [if_neg h1, if_neg h2] rfl done lemma swap_values (u v i : Nat) : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := by by_cases h1 : i = u · -- Case 1. h1 : i = u apply Or.inl rewrite [h1] show swap u v u = v from swap_fst u v done · -- Case 2. h1 : i ≠ u apply Or.inr by_cases h2 : i = v · -- Case 2.1. h2 : i = v apply Or.inl rewrite [h2] show swap u v v = u from swap_snd u v done · -- Case 2.2. h2 : i ≠ v apply Or.inr show swap u v i = i from swap_other h1 h2 done done done lemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by define fix i : Nat assume h3 : i < n have h4 : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := swap_values u v i by_cases on h4 · -- Case 1. h4 : swap u v i = v rewrite [h4] show v < n from h2 done · -- Case 2. by_cases on h4 · -- Case 2.1. h4 : swap u v i = u rewrite [h4] show u < n from h1 done · -- Case 2.2. h4 : swap u v i = i rewrite [h4] show i < n from h3 done done done lemma swap_swap (u v n : Nat) : ∀ i < n, swap u v (swap u v i) = i := by fix i : Nat assume h : i < n by_cases h1 : i = u · -- Case 1. h1 : i = u rewrite [h1, swap_fst, swap_snd] rfl done · -- Case 2. h1 : i ≠ u by_cases h2 : i = v · -- Case 2.1. h2 : i = v rewrite [h2, swap_snd, swap_fst] rfl done · -- Case 2.2. h2 : i ≠ v rewrite [swap_other h1 h2, swap_other h1 h2] rfl done done done lemma swap_one_one_below (u v n) : one_one_below n (swap u v) := left_inv_one_one_below (swap_swap u v n) lemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) := right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2) lemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) := And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2)) lemma comp_perm_below {n : Nat} {f g : Nat → Nat} (h1 : perm_below n f) (h2 : perm_below n g) : perm_below n (f ∘ g) := sorry lemma trivial_swap (u : Nat) : swap u u = id := by apply funext fix x : Nat by_cases h1 : x = u · -- Case 1. h1 : x = u rewrite [h1, swap_fst] rfl done · -- Case 2. h1 : x ≠ u rewrite [swap_other h1 h1] rfl done done lemma prod_eq_fun {m : Nat} (f g : Nat → ZMod m) (k : Nat) : ∀ (n : Nat), (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g := by by_induc · -- Base Case assume h : (∀ i < 0, f (k + i) = g (k + i)) rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g assume h1 : ∀ i < n + 1, f (k + i) = g (k + i) have h2 : ∀ i < n, f (k + i) = g (k + i) := by fix i : Nat assume h2 : i < n have h3 : i < n + 1 := by linarith show f (k + i) = g (k + i) from h1 i h3 done have h3 : prod_seq n k f = prod_seq n k g := ih h2 have h4 : n < n + 1 := Nat.lt_succ_self n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4] rfl done done lemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := by have h2 : ∀ (i : Nat), i < u → (f ∘ swap u n) (0 + i) = f (0 + i) := by fix i : Nat assume h2 : i < u have h3 : 0 + i ≠ u := by linarith have h4 : 0 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl done show prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f from prod_eq_fun (f ∘ swap u n) f 0 u h2 done lemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := by have h2 : ∀ i < j, (f ∘ swap u n) (u + 1 + i) = f (u + 1 + i) := by fix i : Nat assume h2 : i < j have h3 : u + 1 + i ≠ u := by linarith have h4 : u + 1 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl show prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f from prod_eq_fun (f ∘ swap u n) f (u + 1) j h2 done lemma break_prod {m : Nat} (n : Nat) (f : Nat → ZMod m) : ∀ (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by by_induc · -- Base Case have h : n + 0 = n := by rfl rewrite [prod_seq_base, h, Theorem_7_3_6_7] rfl done · -- Induction Step fix j : Nat assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f rewrite [←add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc] rfl done done lemma break_prod_twice {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f := break_prod n f 1 rewrite [prod_one] at h2 have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f := break_prod (u + 1) f j rewrite [←h1] at h3 have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f := break_prod u f 1 rewrite [prod_one] at h4 rewrite [h3, h4] at h2 show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2 done lemma swap_prod_eq_prod {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f := by by_cases h2 : u = n · -- Case 1. h2 : u = n rewrite [h2, trivial_swap n] --Goal : prod_seq (n + 1) 0 (f ∘ id) = prod_seq (n + 1) 0 f rfl done · -- Case 2. h2 : ¬u = n have h3 : u + 1 ≤ n := Nat.lt_of_le_of_ne h1 h2 obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3 have break_f : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := break_prod_twice f h4 have break_fs : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_prod_twice (f ∘ swap u n) h4 have f_eq_fs_below : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := swap_prod_eq_prod_below f h1 have f_eq_fs_btwn : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4 show prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f from calc prod_seq (n + 1) 0 (f ∘ swap u n) _ = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_fs _ = prod_seq u 0 f * (f ∘ swap u n) u * prod_seq j (u + 1) f * (f ∘ swap u n) n := by rw [f_eq_fs_below, f_eq_fs_btwn] _ = prod_seq u 0 f * f (swap u n u) * prod_seq j (u + 1) f * f (swap u n n) := by rfl _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by rw [swap_fst, swap_snd] _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring _ = prod_seq (n + 1) 0 f := break_f.symm done done lemma perm_below_fixed {n : Nat} {g : Nat → Nat} (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry lemma perm_prod {m : Nat} (f : Nat → ZMod m) : ∀ (n : Nat), ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) := by by_induc · -- Base Case fix g : Nat → Nat assume h1 : perm_below 0 g rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) fix g : Nat → Nat assume g_pb : perm_below (n + 1) g define at g_pb have g_ob : onto_below (n + 1) g := g_pb.right.right define at g_ob have h1 : n < n + 1 := by linarith obtain (u : Nat) (h2 : u < n + 1 ∧ g u = n) from g_ob n h1 have s_pb : perm_below (n + 1) (swap u n) := swap_perm_below h2.left h1 have gs_pb_n1 : perm_below (n + 1) (g ∘ swap u n) := comp_perm_below g_pb s_pb have gs_fix_n : (g ∘ swap u n) n = n := calc (g ∘ swap u n) n _ = g (swap u n n) := by rfl _ = g u := by rw [swap_snd] _ = n := h2.right have gs_pb_n : perm_below n (g ∘ swap u n) := perm_below_fixed gs_pb_n1 gs_fix_n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f ∘ (g ∘ swap u n)) := ih (g ∘ swap u n) gs_pb_n have h3 : u ≤ n := by linarith show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f ∘ g) from calc prod_seq (n + 1) 0 f _ = prod_seq n 0 f * f n := prod_seq_zero_step n f _ = prod_seq n 0 (f ∘ (g ∘ swap u n)) * f ((g ∘ swap u n) n) := by rw [gs_prod, gs_fix_n] _ = prod_seq n 0 (f ∘ g ∘ swap u n) * (f ∘ g ∘ swap u n) n := by rfl _ = prod_seq (n + 1) 0 (f ∘ g ∘ swap u n) := (prod_seq_zero_step n (f ∘ g ∘ swap u n)).symm _ = prod_seq (n + 1) 0 ((f ∘ g) ∘ swap u n) := by rfl _ = prod_seq (n + 1) 0 (f ∘ g) := swap_prod_eq_prod (f ∘ g) h3 done done lemma F_invertible (m i : Nat) : invertible (F m i) := by by_cases h : rel_prime m i · -- Case 1. h : rel_prime m i rewrite [F_rp_def h] show invertible [i]_m from (Theorem_7_3_7 m i).rtl h done · -- Case 2. h : ¬rel_prime m i rewrite [F_not_rp_def h] apply Exists.intro [1]_m show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m done done lemma Fprod_invertible (m : Nat) : ∀ (k : Nat), invertible (prod_seq k 0 (F m)) := by by_induc · -- Base Case apply Exists.intro [1]_m show prod_seq 0 0 (F m) * [1]_m = [1]_m from calc prod_seq 0 0 (F m) * [1]_m _ = [1]_m * [1]_m := by rw [prod_seq_base] _ = [1]_m := Theorem_7_3_6_7 ([1]_m) done · -- Induction Step fix k : Nat assume ih : invertible (prod_seq k 0 (F m)) rewrite [prod_seq_zero_step] show invertible (prod_seq k 0 (F m) * (F m k)) from (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k) done done theorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m ^ (phi m) = [1]_m := by have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2 show [a]_m ^ (phi m) = [1]_m from calc [a]_m ^ (phi m) _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3] _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring _ = prod_seq m 0 (F m ∘ G m a) * Y := by rw [FG_prod h1 m, phi_def] _ = prod_seq m 0 (F m) * Y := by rw [perm_prod (F m) m (G m a) (G_perm_below h1)] _ = [1]_m := by rw [h3] done

lemma Exercise_7_4_5_Int (m : Nat) (a : Int) : ∀ (n : Nat), [a]_m ^ n = [a ^ n]_m

HTPI.Exercise_7_4_5_Int

htpi/HTPILib/Chap7.lean

htpi

/- Copyright 2023 Daniel J. Velleman -/ import HTPILib.Chap6 namespace HTPI /- Definitions -/ lemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by have h : n + 1 > 0 := Nat.succ_pos n show a % (n + 1) < n + 1 from Nat.mod_lt a h done def gcd (a b : Nat) : Nat := match b with | 0 => a | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd (n + 1) (a % (n + 1)) termination_by b mutual def gcd_c1 (a b : Nat) : Int := match b with | 0 => 1 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c2 (n + 1) (a % (n + 1)) --Corresponds to s = t' termination_by b def gcd_c2 (a b : Nat) : Int := match b with | 0 => 0 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c1 (n + 1) (a % (n + 1)) - (gcd_c2 (n + 1) (a % (n + 1))) * ↑(a / (n + 1)) --Corresponds to t = s' - t'q termination_by b end def prime (n : Nat) : Prop := 2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n def prime_factor (p n : Nat) : Prop := prime p ∧ p ∣ n def all_prime (l : List Nat) : Prop := ∀ p ∈ l, prime p def nondec (l : List Nat) : Prop := match l with | [] => True --Of course, True is a proposition that is always true | n :: L => (∀ m ∈ L, n ≤ m) ∧ nondec L def nondec_prime_list (l : List Nat) : Prop := all_prime l ∧ nondec l def prod (l : List Nat) : Nat := match l with | [] => 1 | n :: L => n * (prod L) def prime_factorization (n : Nat) (l : List Nat) : Prop := nondec_prime_list l ∧ prod l = n def rel_prime (a b : Nat) : Prop := gcd a b = 1 def congr_mod (m : Nat) (a b : Int) : Prop := (↑m : Int) ∣ (a - b) def cc (m : Nat) (a : Int) : ZMod m := (↑a : ZMod m) notation:50 a " ≡ " b " (MOD " m ")" => congr_mod m a b notation:max "["a"]_"m:max => cc m a def invertible {m : Nat} (X : ZMod m) : Prop := ∃ (Y : ZMod m), X * Y = [1]_m def num_rp_below (m k : Nat) : Nat := match k with | 0 => 0 | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j def phi (m : Nat) : Nat := num_rp_below m m def prod_seq {m : Nat} (j k : Nat) (f : Nat → ZMod m) : ZMod m := match j with | 0 => [1]_m | n + 1 => prod_seq n k f * f (k + n) def maps_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i < n, g i < n def one_one_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i1 < n, ∀ i2 < n, g i1 = g i2 → i1 = i2 def onto_below (n : Nat) (g : Nat → Nat) : Prop := ∀ k < n, ∃ i < n, g i = k def perm_below (n : Nat) (g : Nat → Nat) : Prop := maps_below n g ∧ one_one_below n g ∧ onto_below n g def inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m) def swap (u v i : Nat) : Nat := if i = u then v else if i = v then u else i namespace Euler --For definitions specific to Euler's theorem def F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m def G (m a i : Nat) : Nat := (a * i) % m def Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i end Euler /- Section 7.1 -/ theorem dvd_mod_of_dvd_a_b {a b d : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ (a % b) := by set q : Nat := a / b have h3 : b * q + a % b = a := Nat.div_add_mod a b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 define --Goal : ∃ (c : Nat), a % b = d * c apply Exists.intro (j - k * q) show a % b = d * (j - k * q) from calc a % b _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm _ = a - b * q := by rw [h3] _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc] _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm done theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry #eval gcd 672 161 --Answer: 7 lemma gcd_base (a : Nat) : gcd a 0 = a := by rfl lemma gcd_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd a b = gcd b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1)) rfl done lemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b ≠ 0) : a % b < b := by have h1 : b > 0 := Nat.pos_of_ne_zero h show a % b < b from Nat.mod_lt a h1 done lemma dvd_self (n : Nat) : n ∣ n := by apply Exists.intro 1 ring done theorem gcd_dvd : ∀ (b a : Nat), (gcd a b) ∣ a ∧ (gcd a b) ∣ b := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd a b_1) ∣ a ∧ (gcd a b_1) ∣ b_1 fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_base] --Goal: a ∣ a ∧ a ∣ 0 apply And.intro (dvd_self a) define apply Exists.intro 0 rfl done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_nonzero a h1] --Goal : gcd b (a % b) ∣ a ∧ gcd b (a % b) ∣ b have h2 : a % b < b := mod_nonzero_lt a h1 have h3 : (gcd b (a % b)) ∣ b ∧ (gcd b (a % b)) ∣ (a % b) := ih (a % b) h2 b apply And.intro _ h3.left show (gcd b (a % b)) ∣ a from dvd_a_of_dvd_b_mod h3.left h3.right done done theorem gcd_dvd_left (a b : Nat) : (gcd a b) ∣ a := (gcd_dvd b a).left theorem gcd_dvd_right (a b : Nat) : (gcd a b) ∣ b := (gcd_dvd b a).right lemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl lemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c1 a b = gcd_c2 b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done lemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl lemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * ↑(a / b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done theorem gcd_lin_comb : ∀ (b a : Nat), (gcd_c1 a b) * ↑a + (gcd_c2 a b) * ↑b = ↑(gcd a b) := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd_c1 a b_1) * ↑a + (gcd_c2 a b_1) * ↑b_1 = ↑(gcd a b_1) fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base] --Goal : 1 * ↑a + 0 * ↑0 = ↑a ring done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1] --Goal : gcd_c2 b (a % b) * ↑a + -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * ↑(a / b)) * ↑b = -- ↑(gcd b (a % b)) set r : Nat := a % b set q : Nat := a / b set s : Int := gcd_c1 b r set t : Int := gcd_c2 b r --Goal : t * ↑a + (s - t * ↑q) * ↑b = ↑(gcd b r) have h2 : r < b := mod_nonzero_lt a h1 have h3 : s * ↑b + t * ↑r = ↑(gcd b r) := ih r h2 b have h4 : b * q + r = a := Nat.div_add_mod a b rewrite [←h3, ←h4] rewrite [Nat.cast_add, Nat.cast_mul] --Goal : t * (↑b * ↑q + ↑r) + (s - t * ↑q) * ↑b = s * ↑b + t * ↑r ring done done #eval gcd_c1 672 161 --Answer: 6 #eval gcd_c2 672 161 --Answer: -25 --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161 theorem Theorem_7_1_6 {d a b : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ gcd a b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑d ∣ ↑(gcd a b) set s : Int := gcd_c1 a b set t : Int := gcd_c2 a b have h3 : s * ↑a + t * ↑b = ↑(gcd a b) := gcd_lin_comb b a rewrite [←h3] --Goal : ↑d ∣ s * ↑a + t * ↑b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul] --Goal : ↑d ∣ s * (↑d * ↑j) + t * (↑d * ↑k) define apply Exists.intro (s * ↑j + t * ↑k) ring done /- Section 7.2 -/ theorem dvd_trans {a b c : Nat} (h1 : a ∣ b) (h2 : b ∣ c) : a ∣ c := by define at h1; define at h2; define obtain (m : Nat) (h3 : b = a * m) from h1 obtain (n : Nat) (h4 : c = b * n) from h2 rewrite [h3, mul_assoc] at h4 apply Exists.intro (m * n) show c = a * (m * n) from h4 done lemma exists_prime_factor : ∀ (n : Nat), 2 ≤ n → ∃ (p : Nat), prime_factor p n := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, 2 ≤ n_1 → ∃ (p : Nat), prime_factor p n_1 assume h1 : 2 ≤ n by_cases h2 : prime n · -- Case 1. h2 : prime n apply Exists.intro n define --Goal : prime n ∧ n ∣ n show prime n ∧ n ∣ n from And.intro h2 (dvd_self n) done · -- Case 2. h2 : ¬prime n define at h2 --h2 : ¬(2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n) demorgan at h2 disj_syll h2 h1 obtain (a : Nat) (h3 : ∃ (b : Nat), a * b = n ∧ a < n ∧ b < n) from h2 obtain (b : Nat) (h4 : a * b = n ∧ a < n ∧ b < n) from h3 have h5 : 2 ≤ a := by by_contra h6 have h7 : a ≤ 1 := by linarith have h8 : n ≤ b := calc n _ = a * b := h4.left.symm _ ≤ 1 * b := by rel [h7] _ = b := by ring linarith --n ≤ b contradicts b < n done have h6 : ∃ (p : Nat), prime_factor p a := ih a h4.right.left h5 obtain (p : Nat) (h7 : prime_factor p a) from h6 apply Exists.intro p define --Goal : prime p ∧ p ∣ n define at h7 --h7 : prime p ∧ p ∣ a apply And.intro h7.left have h8 : a ∣ n := by apply Exists.intro b show n = a * b from (h4.left).symm done show p ∣ n from dvd_trans h7.right h8 done done lemma exists_least_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q := by set S : Set Nat := {p : Nat | prime_factor p n} have h2 : ∃ (p : Nat), p ∈ S := exists_prime_factor n h show ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q from well_ord_princ S h2 done lemma all_prime_nil : all_prime [] := by define --Goal : ∀ p ∈ [], prime p fix p : Nat contrapos --Goal : ¬prime p → p ∉ [] assume h1 : ¬prime p show p ∉ [] from List.not_mem_nil p done lemma all_prime_cons (n : Nat) (L : List Nat) : all_prime (n :: L) ↔ prime n ∧ all_prime L := by apply Iff.intro · -- (→) assume h1 : all_prime (n :: L) --Goal : prime n ∧ all_prime L define at h1 --h1 : ∀ p ∈ n :: L, prime p apply And.intro (h1 n (List.mem_cons_self n L)) define --Goal : ∀ p ∈ L, prime p fix p : Nat assume h2 : p ∈ L show prime p from h1 p (List.mem_cons_of_mem n h2) done · -- (←) assume h1 : prime n ∧ all_prime L --Goal : all_prime (n :: l) define : all_prime L at h1 define fix p : Nat assume h2 : p ∈ n :: L rewrite [List.mem_cons] at h2 --h2 : p = n ∨ p ∈ L by_cases on h2 · -- Case 1. h2 : p = n rewrite [h2] show prime n from h1.left done · -- Case 2. h2 : p ∈ L show prime p from h1.right p h2 done done done lemma nondec_nil : nondec [] := by define --Goal : True trivial --trivial proves some obviously true statements, such as True done lemma nondec_cons (n : Nat) (L : List Nat) : nondec (n :: L) ↔ (∀ m ∈ L, n ≤ m) ∧ nondec L := by rfl lemma prod_nil : prod [] = 1 := by rfl lemma prod_cons : prod (n :: L) = n * (prod L) := by rfl lemma exists_cons_of_length_eq_succ {A : Type} {l : List A} {n : Nat} (h : l.length = n + 1) : ∃ (a : A) (L : List A), l = a :: L ∧ L.length = n := by have h1 : ¬l.length = 0 := by linarith rewrite [List.length_eq_zero] at h1 obtain (a : A) (h2 : ∃ (L : List A), l = a :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List A) (h3 : l = a :: L) from h2 apply Exists.intro a apply Exists.intro L apply And.intro h3 have h4 : (a :: L).length = L.length + 1 := List.length_cons a L rewrite [←h3, h] at h4 show L.length = n from (Nat.add_right_cancel h4).symm done lemma list_elt_dvd_prod_by_length (a : Nat) : ∀ (n : Nat), ∀ (l : List Nat), l.length = n → a ∈ l → a ∣ prod l := by by_induc · --Base Case fix l : List Nat assume h1 : l.length = 0 rewrite [List.length_eq_zero] at h1 --h1 : l = [] rewrite [h1] --Goal : a ∈ [] → a ∣ prod [] contrapos assume h2 : ¬a ∣ prod [] show a ∉ [] from List.not_mem_nil a done · -- Induction Step fix n : Nat assume ih : ∀ (l : List Nat), List.length l = n → a ∈ l → a ∣ prod l fix l : List Nat assume h1 : l.length = n + 1 --Goal : a ∈ l → a ∣ prod l obtain (b : Nat) (h2 : ∃ (L : List Nat), l = b :: L ∧ L.length = n) from exists_cons_of_length_eq_succ h1 obtain (L : List Nat) (h3 : l = b :: L ∧ L.length = n) from h2 have h4 : a ∈ L → a ∣ prod L := ih L h3.right assume h5 : a ∈ l rewrite [h3.left, prod_cons] --Goal : a ∣ b * prod L rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b ∨ a ∈ L by_cases on h5 · -- Case 1. h5 : a = b apply Exists.intro (prod L) rewrite [h5] rfl done · -- Case 2. h5 : a ∈ L have h6 : a ∣ prod L := h4 h5 have h7 : prod L ∣ b * prod L := by apply Exists.intro b ring done show a ∣ b * prod L from dvd_trans h6 h7 done done done lemma list_elt_dvd_prod {a : Nat} {l : List Nat} (h : a ∈ l) : a ∣ prod l := by set n : Nat := l.length have h1 : l.length = n := by rfl show a ∣ prod l from list_elt_dvd_prod_by_length a n l h1 h done lemma exists_prime_factorization : ∀ (n : Nat), n ≥ 1 → ∃ (l : List Nat), prime_factorization n l := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, n_1 ≥ 1 → ∃ (l : List Nat), prime_factorization n_1 l assume h1 : n ≥ 1 by_cases h2 : n = 1 · -- Case 1. h2 : n = 1 apply Exists.intro [] define apply And.intro · -- Proof of nondec_prime_list [] define show all_prime [] ∧ nondec [] from And.intro all_prime_nil nondec_nil done · -- Proof of prod [] = n rewrite [prod_nil, h2] rfl done done · -- Case 2. h2 : n ≠ 1 have h3 : n ≥ 2 := lt_of_le_of_ne' h1 h2 obtain (p : Nat) (h4 : prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q) from exists_least_prime_factor h3 have p_prime_factor : prime_factor p n := h4.left define at p_prime_factor have p_prime : prime p := p_prime_factor.left have p_dvd_n : p ∣ n := p_prime_factor.right have p_least : ∀ (q : Nat), prime_factor q n → p ≤ q := h4.right obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n have h5 : m ≠ 0 := by contradict h1 with h6 have h7 : n = 0 := calc n _ = p * m := n_eq_pm _ = p * 0 := by rw [h6] _ = 0 := by ring rewrite [h7] decide done have m_pos : 0 < m := Nat.pos_of_ne_zero h5 have m_lt_n : m < n := by define at p_prime show m < n from calc m _ < m + m := by linarith _ = 2 * m := by ring _ ≤ p * m := by rel [p_prime.left] _ = n := n_eq_pm.symm done obtain (L : List Nat) (h6 : prime_factorization m L) from ih m m_lt_n m_pos define at h6 have ndpl_L : nondec_prime_list L := h6.left define at ndpl_L apply Exists.intro (p :: L) define apply And.intro · -- Proof of nondec_prime_list (p :: L) define apply And.intro · -- Proof of all_prime (p :: L) rewrite [all_prime_cons] show prime p ∧ all_prime L from And.intro p_prime ndpl_L.left done · -- Proof of nondec (p :: L) rewrite [nondec_cons] apply And.intro _ ndpl_L.right fix q : Nat assume q_in_L : q ∈ L have h7 : q ∣ prod L := list_elt_dvd_prod q_in_L rewrite [h6.right] at h7 --h7 : q ∣ m have h8 : m ∣ n := by apply Exists.intro p rewrite [n_eq_pm] ring done have q_dvd_n : q ∣ n := dvd_trans h7 h8 have ap_L : all_prime L := ndpl_L.left define at ap_L have q_prime_factor : prime_factor q n := And.intro (ap_L q q_in_L) q_dvd_n show p ≤ q from p_least q q_prime_factor done done · -- Proof of prod (p :: L) = n rewrite [prod_cons, h6.right, n_eq_pm] rfl done done done theorem Theorem_7_2_2 {a b c : Nat} (h1 : c ∣ a * b) (h2 : rel_prime a c) : c ∣ b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑c ∣ ↑b define at h1; define at h2; define obtain (j : Nat) (h3 : a * b = c * j) from h1 set s : Int := gcd_c1 a c set t : Int := gcd_c2 a c have h4 : s * ↑a + t * ↑c = ↑(gcd a c) := gcd_lin_comb c a rewrite [h2, Nat.cast_one] at h4 --h4 : s * ↑a + t * ↑c = (1 : Int) apply Exists.intro (s * ↑j + t * ↑b) show ↑b = ↑c * (s * ↑j + t * ↑b) from calc ↑b _ = (1 : Int) * ↑b := (one_mul _).symm _ = (s * ↑a + t * ↑c) * ↑b := by rw [h4] _ = s * (↑a * ↑b) + t * ↑c * ↑b := by ring _ = s * (↑c * ↑j) + t * ↑c * ↑b := by rw [←Nat.cast_mul a b, h3, Nat.cast_mul c j] _ = ↑c * (s * ↑j + t * ↑b) := by ring done lemma le_nonzero_prod_left {a b : Nat} (h : a * b ≠ 0) : a ≤ a * b := by have h1 : b ≠ 0 := by contradict h with h1 rewrite [h1] ring done have h2 : 1 ≤ b := Nat.pos_of_ne_zero h1 show a ≤ a * b from calc a = a * 1 := (mul_one a).symm _ ≤ a * b := by rel [h2] done lemma le_nonzero_prod_right {a b : Nat} (h : a * b ≠ 0) : b ≤ a * b := by rewrite [mul_comm] rewrite [mul_comm] at h show b ≤ b * a from le_nonzero_prod_left h done lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry lemma rel_prime_of_prime_not_dvd {a p : Nat} (h1 : prime p) (h2 : ¬p ∣ a) : rel_prime a p := by have h3 : gcd a p ∣ a := gcd_dvd_left a p have h4 : gcd a p ∣ p := gcd_dvd_right a p have h5 : gcd a p = 1 ∨ gcd a p = p := dvd_prime h1 h4 have h6 : gcd a p ≠ p := by contradict h2 with h6 rewrite [h6] at h3 show p ∣ a from h3 done disj_syll h5 h6 show rel_prime a p from h5 done theorem Theorem_7_2_3 {a b p : Nat} (h1 : prime p) (h2 : p ∣ a * b) : p ∣ a ∨ p ∣ b := by or_right with h3 have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3 show p ∣ b from Theorem_7_2_2 h2 h4 done lemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a ≥ 1 := by have h1 : a ≠ 0 := by by_contra h1 rewrite [h1] at h contradict h linarith done show a ≥ 1 from Nat.pos_of_ne_zero h1 done lemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by have h1 : a ≥ 1 := ge_one_of_prod_one h have h2 : a * b ≠ 0 := by linarith have h3 : a ≤ a * b := le_nonzero_prod_left h2 rewrite [h] at h3 show a = 1 from Nat.le_antisymm h3 h1 done lemma eq_one_of_dvd_one {n : Nat} (h : n ∣ 1) : n = 1 := by obtain (j : Nat) (h1 : 1 = n * j) from h show n = 1 from eq_one_of_prod_one h1.symm done lemma prime_not_one {p : Nat} (h : prime p) : p ≠ 1 := by define at h linarith done theorem Theorem_7_2_4 {p : Nat} (h1 : prime p) : ∀ (l : List Nat), p ∣ prod l → ∃ a ∈ l, p ∣ a := by apply List.rec · -- Base Case. Goal : p ∣ prod [] → ∃ a ∈ [], p ∣ a rewrite [prod_nil] assume h2 : p ∣ 1 show ∃ a ∈ [], p ∣ a from absurd (eq_one_of_dvd_one h2) (prime_not_one h1) done · -- Induction Step fix b : Nat fix L : List Nat assume ih : p ∣ prod L → ∃ a ∈ L, p ∣ a --Goal : p ∣ prod (b :: L) → ∃ a ∈ b :: L, p ∣ a assume h2 : p ∣ prod (b :: L) rewrite [prod_cons] at h2 have h3 : p ∣ b ∨ p ∣ prod L := Theorem_7_2_3 h1 h2 by_cases on h3 · -- Case 1. h3 : p ∣ b apply Exists.intro b show b ∈ b :: L ∧ p ∣ b from And.intro (List.mem_cons_self b L) h3 done · -- Case 2. h3 : p ∣ prod L obtain (a : Nat) (h4 : a ∈ L ∧ p ∣ a) from ih h3 apply Exists.intro a show a ∈ b :: L ∧ p ∣ a from And.intro (List.mem_cons_of_mem b h4.left) h4.right done done done lemma prime_in_list {p : Nat} {l : List Nat} (h1 : prime p) (h2 : all_prime l) (h3 : p ∣ prod l) : p ∈ l := by obtain (a : Nat) (h4 : a ∈ l ∧ p ∣ a) from Theorem_7_2_4 h1 l h3 define at h2 have h5 : prime a := h2 a h4.left have h6 : p = 1 ∨ p = a := dvd_prime h5 h4.right disj_syll h6 (prime_not_one h1) rewrite [h6] show a ∈ l from h4.left done lemma first_le_first {p q : Nat} {l m : List Nat} (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m)) (h3 : prod (p :: l) = prod (q :: m)) : p ≤ q := by define at h1; define at h2 have h4 : q ∣ prod (p :: l) := by define apply Exists.intro (prod m) rewrite [←prod_cons] show prod (p :: l) = prod (q :: m) from h3 done have h5 : all_prime (q :: m) := h2.left rewrite [all_prime_cons] at h5 have h6 : q ∈ p :: l := prime_in_list h5.left h1.left h4 have h7 : nondec (p :: l) := h1.right rewrite [nondec_cons] at h7 rewrite [List.mem_cons] at h6 by_cases on h6 · -- Case 1. h6 : q = p linarith done · -- Case 2. h6 : q ∈ l have h8 : ∀ m ∈ l, p ≤ m := h7.left show p ≤ q from h8 q h6 done done lemma nondec_prime_list_tail {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by define at h define rewrite [all_prime_cons, nondec_cons] at h show all_prime l ∧ nondec l from And.intro h.left.right h.right.right done lemma cons_prod_not_one {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : prod (p :: l) ≠ 1 := by define at h have h1 : all_prime (p :: l) := h.left rewrite [all_prime_cons] at h1 rewrite [prod_cons] by_contra h2 show False from (prime_not_one h1.left) (eq_one_of_prod_one h2) done lemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) : l = [] ↔ prod l = 1 := by apply Iff.intro · -- (→) assume h1 : l = [] rewrite [h1] show prod [] = 1 from prod_nil done · -- (←) contrapos assume h1 : ¬l = [] obtain (p : Nat) (h2 : ∃ (L : List Nat), l = p :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List Nat) (h3 : l = p :: L) from h2 rewrite [h3] at h rewrite [h3] show ¬prod (p :: L) = 1 from cons_prod_not_one h done done lemma prime_pos {p : Nat} (h : prime p) : p > 0 := by define at h linarith done theorem Theorem_7_2_5 : ∀ (l1 l2 : List Nat), nondec_prime_list l1 → nondec_prime_list l2 → prod l1 = prod l2 → l1 = l2 := by apply List.rec · -- Base Case. Goal : ∀ (l2 : List Nat), nondec_prime_list [] → -- nondec_prime_list l2 → prod [] = prod l2 → [] = l2 fix l2 : List Nat assume h1 : nondec_prime_list [] assume h2 : nondec_prime_list l2 assume h3 : prod [] = prod l2 rewrite [prod_nil, eq_comm, ←list_nil_iff_prod_one h2] at h3 show [] = l2 from h3.symm done · -- Induction Step fix p : Nat fix L1 : List Nat assume ih : ∀ (L2 : List Nat), nondec_prime_list L1 → nondec_prime_list L2 → prod L1 = prod L2 → L1 = L2 -- Goal : ∀ (l2 : List Nat), nondec_prime_list (p :: L1) → -- nondec_prime_list l2 → prod (p :: L1) = prod l2 → p :: L1 = l2 fix l2 : List Nat assume h1 : nondec_prime_list (p :: L1) assume h2 : nondec_prime_list l2 assume h3 : prod (p :: L1) = prod l2 have h4 : ¬prod (p :: L1) = 1 := cons_prod_not_one h1 rewrite [h3, ←list_nil_iff_prod_one h2] at h4 obtain (q : Nat) (h5 : ∃ (L : List Nat), l2 = q :: L) from List.exists_cons_of_ne_nil h4 obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5 rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2) rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2) have h7 : p ≤ q := first_le_first h1 h2 h3 have h8 : q ≤ p := first_le_first h2 h1 h3.symm have h9 : p = q := by linarith rewrite [h9, prod_cons, prod_cons] at h3 --h3 : q * prod L1 = q * prod L2 have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1 have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2 define at h2 have h12 : all_prime (q :: L2) := h2.left rewrite [all_prime_cons] at h12 have h13 : q > 0 := prime_pos h12.left have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3 have h15 : L1 = L2 := ih L2 h10 h11 h14 rewrite [h6, h9, h15] rfl done done theorem fund_thm_arith (n : Nat) (h : n ≥ 1) : ∃! (l : List Nat), prime_factorization n l := by exists_unique · -- Existence show ∃ (l : List Nat), prime_factorization n l from exists_prime_factorization n h done · -- Uniqueness fix l1 : List Nat; fix l2 : List Nat assume h1 : prime_factorization n l1 assume h2 : prime_factorization n l2 define at h1; define at h2 have h3 : prod l1 = n := h1.right rewrite [←h2.right] at h3 show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3 done done /- Section 7.3 -/ theorem congr_refl (m : Nat) : ∀ (a : Int), a ≡ a (MOD m) := by fix a : Int define --Goal : ∃ (c : Int), a - a = ↑m * c apply Exists.intro 0 ring done theorem congr_symm {m : Nat} : ∀ {a b : Int}, a ≡ b (MOD m) → b ≡ a (MOD m) := by fix a : Int; fix b : Int assume h1 : a ≡ b (MOD m) define at h1 --h1 : ∃ (c : Int), a - b = ↑m * c define --Goal : ∃ (c : Int), b - a = ↑m * c obtain (c : Int) (h2 : a - b = m * c) from h1 apply Exists.intro (-c) show b - a = m * (-c) from calc b - a _ = -(a - b) := by ring _ = -(m * c) := by rw [h2] _ = m * (-c) := by ring done theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry /- Fundamental properties of congruence classes -/ lemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) : X = Y ↔ X.val = Y.val := Fin.ext_iff lemma val_nat_eq_mod (n k : Nat) : ([k]_(n + 1)).val = k % (n + 1) := by rfl lemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl theorem cc_rep {m : Nat} (X : ZMod m) : ∃ (a : Int), X = [a]_m := match m with | 0 => by apply Exists.intro X rfl done | n + 1 => by apply Exists.intro ↑(X.val) have h1 : X.val < n + 1 := Fin.prop X rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1] rfl done theorem add_class (m : Nat) (a b : Int) : [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm theorem mul_class (m : Nat) (a b : Int) : [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm lemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ [a - b]_m = [0]_m := by apply Iff.intro · -- (→) assume h1 : [a]_m = [b]_m have h2 : a - b = a + (-b) := by ring have h3 : b + (-b) = 0 := by ring show [a - b]_m = [0]_m from calc [a - b]_m _ = [a + (-b)]_m := by rw [h2] _ = [a]_m + [-b]_m := by rw [add_class] _ = [b]_m + [-b]_m := by rw [h1] _ = [b + -b]_m := by rw [add_class] _ = [0]_m := by rw [h3] done · -- (←) assume h1 : [a - b]_m = [0]_m have h2 : b + (a - b) = a := by ring have h3 : b + 0 = b := by ring show [a]_m = [b]_m from calc [a]_m _ = [b + (a - b)]_m := by rw [h2] _ = [b]_m + [a - b]_m := by rw [add_class] _ = [b]_m + [0]_m := by rw [h1] _ = [b + 0]_m := by rw [add_class] _ = [b]_m := by rw [h3] done done lemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) : [a]_m = [0]_m → [-a]_m = [0]_m := by assume h1 : [a]_m = [0]_m have h2 : 0 + (-a) = -a := by ring have h3 : a + (-a) = 0 := by ring show [-a]_m = [0]_m from calc [-a]_m _ = [0 + (-a)]_m := by rw [h2] _ = [0]_m + [-a]_m := by rw [add_class] _ = [a]_m + [-a]_m := by rw [h1] _ = [a + (-a)]_m := by rw [add_class] _ = [0]_m := by rw [h3] done lemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) : [-a]_m = [0]_m ↔ [a]_m = [0]_m := by apply Iff.intro _ (cc_neg_zero_of_cc_zero m a) assume h1 : [-a]_m = [0]_m have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1 have h3 : -(-a) = a := by ring rewrite [h3] at h2 show [a]_m = [0]_m from h2 done lemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl lemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m ↔ m ∣ k := match m with | 0 => by have h : (0 : Int) = (↑(0 : Nat) : Int) := by rfl rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj] apply Iff.intro · -- (→) assume h1 : k = 0 rewrite [h1] show 0 ∣ 0 from dvd_self 0 done · -- (←) assume h1 : 0 ∣ k obtain (c : Nat) (h2 : k = 0 * c) from h1 rewrite [h2] ring done done | n + 1 => by rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero] show k % (n + 1) = 0 ↔ n + 1 ∣ k from (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm done lemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m ↔ ↑m ∣ a := by obtain (k : Nat) (h1 : a = ↑k ∨ a = -↑k) from Int.eq_nat_or_neg a by_cases on h1 · -- Case 1. h1: a = ↑k rewrite [h1, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done · -- Case 2. h1: a = -↑k rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done done theorem cc_eq_iff_congr (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ a ≡ b (MOD m) := calc [a]_m = [b]_m _ ↔ [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b _ ↔ ↑m ∣ (a - b) := cc_zero_iff_dvd m (a - b) _ ↔ a ≡ b (MOD m) := by rfl /- End of fundamental properties of congruence classes -/ lemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m := by have h1 : (↑m : Int) ≠ 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m) show 0 ≤ a % m from Int.emod_nonneg a h1 done lemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m) have h2 : (↑m : Int) > 0 := (Nat.cast_pos).rtl h1 show a % m < m from Int.emod_lt_of_pos a h2 done lemma congr_mod_mod (m : Nat) (a : Int) : a ≡ a % m (MOD m) := by define have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m apply Exists.intro (a / m) show a - a % m = m * (a / m) from calc a - (a % m) _ = m * (a / m) + a % m - a % m := by rw [h1] _ = m * (a / m) := by ring done lemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) := And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a)) theorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) : ∃! (r : Int), 0 ≤ r ∧ r < m ∧ a ≡ r (MOD m) := by exists_unique · -- Existence apply Exists.intro (a % m) show 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) from mod_cmpl_res m a done · -- Uniqueness fix r1 : Int; fix r2 : Int assume h1 : 0 ≤ r1 ∧ r1 < m ∧ a ≡ r1 (MOD m) assume h2 : 0 ≤ r2 ∧ r2 < m ∧ a ≡ r2 (MOD m) have h3 : r1 ≡ r2 (MOD m) := congr_trans (congr_symm h1.right.right) h2.right.right obtain (d : Int) (h4 : r1 - r2 = m * d) from h3 have h5 : r1 - r2 < m * 1 := by linarith have h6 : m * (-1) < r1 - r2 := by linarith rewrite [h4] at h5 --h5 : m * d < m * 1 rewrite [h4] at h6 --h6 : m * -1 < m * d have h7 : (↑m : Int) ≥ 0 := Nat.cast_nonneg m have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7 have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7 have h10 : d = 0 := by linarith show r1 = r2 from calc r1 _ = r1 - r2 + r2 := by ring _ = m * 0 + r2 := by rw [h4, h10] _ = r2 := by ring done done lemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m := (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a) theorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y rewrite [h1, h2] have h3 : a + b = b + a := by ring show [a]_m + [b]_m = [b]_m + [a]_m from calc [a]_m + [b]_m _ = [a + b]_m := add_class m a b _ = [b + a]_m := by rw [h3] _ = [b]_m + [a]_m := (add_class m b a).symm done theorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X rewrite [h1] have h2 : a * 1 = a := by ring show [a]_m * [1]_m = [a]_m from calc [a]_m * [1]_m _ = [a * 1]_m := mul_class m a 1 _ = [a]_m := by rw [h2] done theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry lemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) : [a]_m * [gcd_c2 m a]_m = [1]_m := by set s : Int := gcd_c1 m a have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m define at h1 rewrite [h1, Nat.cast_one] at h2 --h2 : s * ↑m + gcd_c2 m a * ↑a = 1 rewrite [mul_class, cc_eq_iff_congr] define --Goal : ∃ (c : Int), ↑a * gcd_c2 m a - 1 = ↑m * c apply Exists.intro (-s) show a * (gcd_c2 m a) - 1 = m * (-s) from calc a * (gcd_c2 m a) - 1 _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring _ = 1 + m * (-s) - 1 := by rw [h2] _ = m * (-s) := by ring done theorem Theorem_7_3_7 (m a : Nat) : invertible [a]_m ↔ rel_prime m a := by apply Iff.intro · -- (→) assume h1 : invertible [a]_m define at h1 obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1 obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y rewrite [h3, mul_class, cc_eq_iff_congr] at h2 define at h2 obtain (c : Int) (h4 : a * b - 1 = m * c) from h2 rewrite [Exercise_7_2_6] --Goal : ∃ (s t : Int), s * ↑m + t * ↑a = 1 apply Exists.intro (-c) apply Exists.intro b show (-c) * m + b * a = 1 from calc (-c) * m + b * a _ = (-c) * m + (a * b - 1) + 1 := by ring _ = (-c) * m + m * c + 1 := by rw [h4] _ = 1 := by ring done · -- (←) assume h1 : rel_prime m a define show ∃ (Y : ZMod m), [a]_m * Y = [1]_m from Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1) done done /- Section 7.4 -/ section Euler open Euler lemma num_rp_below_base {m : Nat} : num_rp_below m 0 = 0 := by rfl lemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) : num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by have h1 : num_rp_below m (j + 1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : gcd m j = 1 rewrite [if_pos h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1 show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1 done lemma num_rp_below_step_not_rp {m j : Nat} (h : ¬rel_prime m j) : num_rp_below m (j + 1) = num_rp_below m j := by have h1 : num_rp_below m (j +1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : ¬gcd m j = 1 rewrite [if_neg h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j show num_rp_below m (j + 1) = num_rp_below m j from h1 done lemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl #eval phi 10 --Answer: 4 lemma prod_inv_iff_inv {m : Nat} {X : ZMod m} (h1 : invertible X) (Y : ZMod m) : invertible (X * Y) ↔ invertible Y := by apply Iff.intro · -- (→) assume h2 : invertible (X * Y) obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2 apply Exists.intro (X * Z) rewrite [←h3] --Goal : Y * (X * Z) = X * Y * Z ring --Note that ring can do algebra in ZMod m done · -- (←) assume h2 : invertible Y obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1 obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2 apply Exists.intro (Xi * Yi) show (X * Y) * (Xi * Yi) = [1]_m from calc X * Y * (Xi * Yi) _ = (X * Xi) * (Y * Yi) := by ring _ = [1]_m * [1]_m := by rw [h3, h4] _ = [1]_m := Theorem_7_3_6_7 [1]_m done done lemma F_rp_def {m i : Nat} (h : rel_prime m i) : F m i = [i]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h --h : gcd m i = 1 rewrite [if_pos h] at h1 show F m i = [i]_m from h1 done lemma F_not_rp_def {m i : Nat} (h : ¬rel_prime m i) : F m i = [1]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h rewrite [h1, if_neg h] rfl done lemma prod_seq_base {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 0 k f = [1]_m := by rfl lemma prod_seq_step {m : Nat} (n k : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl lemma prod_seq_zero_step {m : Nat} (n : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by rewrite [prod_seq_step, zero_add] rfl done lemma prod_one {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 1 k f = f k := by rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7] rfl done lemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl lemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m := calc [G m a i]_m _ = [(a * i) % m]_m := by rfl _ = [a * i]_m := (cc_eq_mod m (a * i)).symm _ = [a]_m * [i]_m := (mul_class m a i).symm lemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) : rel_prime m (G m a i) ↔ rel_prime m i := by have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1 show rel_prime m (G m a i) ↔ rel_prime m i from calc rel_prime m (G m a i) _ ↔ invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm _ ↔ invertible ([a]_m * [i]_m) := by rw [cc_G] _ ↔ invertible [i]_m := prod_inv_iff_inv h2 ([i]_m) _ ↔ rel_prime m i := Theorem_7_3_7 m i done lemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) : F m (G m a i) = [a]_m * F m i := by have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2 show F m (G m a i) = [a]_m * F m i from calc F m (G m a i) _ = [G m a i]_m := F_rp_def h3 _ = [a]_m * [i]_m := cc_G m a i _ = [a]_m * F m i := by rw [F_rp_def h2] done lemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : ¬rel_prime m i) : F m (G m a i) = [1]_m := by rewrite [←G_rp_iff h1 i] at h2 show F m (G m a i) = [1]_m from F_not_rp_def h2 done lemma FG_prod {m a : Nat} (h1 : rel_prime m a) : ∀ (k : Nat), prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by by_induc · -- Base Case show prod_seq 0 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from calc prod_seq 0 0 ((F m) ∘ (G m a)) _ = [1]_m := prod_seq_base _ _ _ = [a]_m ^ 0 * [1]_m := by ring _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by rw [num_rp_below_base, prod_seq_base] done · -- Induction Step fix k : Nat assume ih : prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) by_cases h2 : rel_prime m k · -- Case 1. h2 : rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([a]_m * F m k) := by rw [FG_rp h1 h2] _ = [a]_m ^ ((num_rp_below m k) + 1) * ((prod_seq k 0 (F m)) * F m k) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_rp h2, prod_seq_zero_step] done · -- Case 2. h2 : ¬rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([1]_m) := by rw [FG_not_rp h1 h2] _ = [a]_m ^ (num_rp_below m k) * (prod_seq k 0 (F m) * ([1]_m)) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_not_rp h2, prod_seq_zero_step, F_not_rp_def h2] done done done lemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by define --Goal : ∀ i < m, G m a i < m fix i : Nat assume h1 : i < m rewrite [G_def] --Goal : a * i % m < m show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m) done lemma left_inv_one_one_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g' (g i) = i) : one_one_below n g := sorry lemma right_inv_onto_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g (g' i) = i) (h2 : maps_below n g') : onto_below n g := by define at h2; define fix k : Nat assume h3 : k < n apply Exists.intro (g' k) show g' k < n ∧ g (g' k) = k from And.intro (h2 k h3) (h1 k h3) done lemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m * [inv_mod m a]_m = [1]_m := by have h2 : 0 ≤ (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a) show [a]_m * [inv_mod m a]_m = [1]_m from calc [a]_m * [inv_mod m a]_m _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2] _ = [a]_m * [gcd_c2 m a]_m := by rw [←cc_eq_mod] _ = [1]_m := gcd_c2_inv h1 done lemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m := calc a * (b % m) % m = a % m * (b % m % m) % m := Nat.mul_mod _ _ _ _ = a % m * (b % m) % m := by rw [Nat.mod_mod] _ = a * b % m := (Nat.mul_mod _ _ _).symm lemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm] rfl done theorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] : ↑a ≡ ↑b (MOD m) ↔ a % m = b % m := sorry lemma mul_inv_mod_cancel {m a i : Nat} [NeZero m] (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1 rewrite [mul_class, cc_eq_iff_congr, ←Nat.cast_mul, ←Nat.cast_one, congr_iff_mod_eq_Nat] at h3 show a * inv_mod m a * i % m = i from calc a * (inv_mod m a) * i % m _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = 1 % m * i % m := by rw [h3] _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = i % m := by rw [one_mul] _ = i := Nat.mod_eq_of_lt h2 done lemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl lemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, G m a (Ginv m a i) = i := by fix i : Nat assume h2 : i < m show G m a (Ginv m a i) = i from calc G m a (Ginv m a i) _ = a * ((inv_mod m a * i) % m) % m := by rfl _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, Ginv m a (G m a i) = i := by fix i : Nat assume h2 : i < m show Ginv m a (G m a i) = i from calc Ginv m a (G m a i) _ = inv_mod m a * ((a * i) % m) % m := by rfl _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc, mul_comm (inv_mod m a)] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_maps_below (m a : Nat) [NeZero m] : maps_below m (Ginv m a) := G_maps_below m (inv_mod m a) lemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : one_one_below m (G m a) := left_inv_one_one_below (Ginv_left_inv h1) lemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : onto_below m (G m a) := right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a) lemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : perm_below m (G m a) := And.intro (G_maps_below m a) (And.intro (G_one_one_below h1) (G_onto_below h1)) --Permuting a product of congruence classes doesn't change product lemma swap_fst (u v : Nat) : swap u v u = v := by define : swap u v u --Goal : (if u = u then v else if u = v then u else u) = v have h : u = u := by rfl rewrite [if_pos h] rfl done lemma swap_snd (u v : Nat) : swap u v v = u := by define : swap u v v by_cases h1 : v = u · -- Case 1. h1 : v = u rewrite [if_pos h1] show v = u from h1 done · -- Case 2. h1 : v ≠ u rewrite [if_neg h1] have h2 : v = v := by rfl rewrite [if_pos h2] rfl done done lemma swap_other {u v i : Nat} (h1 : i ≠ u) (h2 : i ≠ v) : swap u v i = i := by define : swap u v i rewrite [if_neg h1, if_neg h2] rfl done lemma swap_values (u v i : Nat) : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := by by_cases h1 : i = u · -- Case 1. h1 : i = u apply Or.inl rewrite [h1] show swap u v u = v from swap_fst u v done · -- Case 2. h1 : i ≠ u apply Or.inr by_cases h2 : i = v · -- Case 2.1. h2 : i = v apply Or.inl rewrite [h2] show swap u v v = u from swap_snd u v done · -- Case 2.2. h2 : i ≠ v apply Or.inr show swap u v i = i from swap_other h1 h2 done done done lemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by define fix i : Nat assume h3 : i < n have h4 : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := swap_values u v i by_cases on h4 · -- Case 1. h4 : swap u v i = v rewrite [h4] show v < n from h2 done · -- Case 2. by_cases on h4 · -- Case 2.1. h4 : swap u v i = u rewrite [h4] show u < n from h1 done · -- Case 2.2. h4 : swap u v i = i rewrite [h4] show i < n from h3 done done done lemma swap_swap (u v n : Nat) : ∀ i < n, swap u v (swap u v i) = i := by fix i : Nat assume h : i < n by_cases h1 : i = u · -- Case 1. h1 : i = u rewrite [h1, swap_fst, swap_snd] rfl done · -- Case 2. h1 : i ≠ u by_cases h2 : i = v · -- Case 2.1. h2 : i = v rewrite [h2, swap_snd, swap_fst] rfl done · -- Case 2.2. h2 : i ≠ v rewrite [swap_other h1 h2, swap_other h1 h2] rfl done done done lemma swap_one_one_below (u v n) : one_one_below n (swap u v) := left_inv_one_one_below (swap_swap u v n) lemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) := right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2) lemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) := And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2)) lemma comp_perm_below {n : Nat} {f g : Nat → Nat} (h1 : perm_below n f) (h2 : perm_below n g) : perm_below n (f ∘ g) := sorry lemma trivial_swap (u : Nat) : swap u u = id := by apply funext fix x : Nat by_cases h1 : x = u · -- Case 1. h1 : x = u rewrite [h1, swap_fst] rfl done · -- Case 2. h1 : x ≠ u rewrite [swap_other h1 h1] rfl done done lemma prod_eq_fun {m : Nat} (f g : Nat → ZMod m) (k : Nat) : ∀ (n : Nat), (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g := by by_induc · -- Base Case assume h : (∀ i < 0, f (k + i) = g (k + i)) rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g assume h1 : ∀ i < n + 1, f (k + i) = g (k + i) have h2 : ∀ i < n, f (k + i) = g (k + i) := by fix i : Nat assume h2 : i < n have h3 : i < n + 1 := by linarith show f (k + i) = g (k + i) from h1 i h3 done have h3 : prod_seq n k f = prod_seq n k g := ih h2 have h4 : n < n + 1 := Nat.lt_succ_self n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4] rfl done done lemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := by have h2 : ∀ (i : Nat), i < u → (f ∘ swap u n) (0 + i) = f (0 + i) := by fix i : Nat assume h2 : i < u have h3 : 0 + i ≠ u := by linarith have h4 : 0 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl done show prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f from prod_eq_fun (f ∘ swap u n) f 0 u h2 done lemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := by have h2 : ∀ i < j, (f ∘ swap u n) (u + 1 + i) = f (u + 1 + i) := by fix i : Nat assume h2 : i < j have h3 : u + 1 + i ≠ u := by linarith have h4 : u + 1 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl show prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f from prod_eq_fun (f ∘ swap u n) f (u + 1) j h2 done lemma break_prod {m : Nat} (n : Nat) (f : Nat → ZMod m) : ∀ (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by by_induc · -- Base Case have h : n + 0 = n := by rfl rewrite [prod_seq_base, h, Theorem_7_3_6_7] rfl done · -- Induction Step fix j : Nat assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f rewrite [←add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc] rfl done done lemma break_prod_twice {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f := break_prod n f 1 rewrite [prod_one] at h2 have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f := break_prod (u + 1) f j rewrite [←h1] at h3 have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f := break_prod u f 1 rewrite [prod_one] at h4 rewrite [h3, h4] at h2 show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2 done lemma swap_prod_eq_prod {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f := by by_cases h2 : u = n · -- Case 1. h2 : u = n rewrite [h2, trivial_swap n] --Goal : prod_seq (n + 1) 0 (f ∘ id) = prod_seq (n + 1) 0 f rfl done · -- Case 2. h2 : ¬u = n have h3 : u + 1 ≤ n := Nat.lt_of_le_of_ne h1 h2 obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3 have break_f : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := break_prod_twice f h4 have break_fs : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_prod_twice (f ∘ swap u n) h4 have f_eq_fs_below : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := swap_prod_eq_prod_below f h1 have f_eq_fs_btwn : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4 show prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f from calc prod_seq (n + 1) 0 (f ∘ swap u n) _ = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_fs _ = prod_seq u 0 f * (f ∘ swap u n) u * prod_seq j (u + 1) f * (f ∘ swap u n) n := by rw [f_eq_fs_below, f_eq_fs_btwn] _ = prod_seq u 0 f * f (swap u n u) * prod_seq j (u + 1) f * f (swap u n n) := by rfl _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by rw [swap_fst, swap_snd] _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring _ = prod_seq (n + 1) 0 f := break_f.symm done done lemma perm_below_fixed {n : Nat} {g : Nat → Nat} (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry lemma perm_prod {m : Nat} (f : Nat → ZMod m) : ∀ (n : Nat), ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) := by by_induc · -- Base Case fix g : Nat → Nat assume h1 : perm_below 0 g rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) fix g : Nat → Nat assume g_pb : perm_below (n + 1) g define at g_pb have g_ob : onto_below (n + 1) g := g_pb.right.right define at g_ob have h1 : n < n + 1 := by linarith obtain (u : Nat) (h2 : u < n + 1 ∧ g u = n) from g_ob n h1 have s_pb : perm_below (n + 1) (swap u n) := swap_perm_below h2.left h1 have gs_pb_n1 : perm_below (n + 1) (g ∘ swap u n) := comp_perm_below g_pb s_pb have gs_fix_n : (g ∘ swap u n) n = n := calc (g ∘ swap u n) n _ = g (swap u n n) := by rfl _ = g u := by rw [swap_snd] _ = n := h2.right have gs_pb_n : perm_below n (g ∘ swap u n) := perm_below_fixed gs_pb_n1 gs_fix_n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f ∘ (g ∘ swap u n)) := ih (g ∘ swap u n) gs_pb_n have h3 : u ≤ n := by linarith show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f ∘ g) from calc prod_seq (n + 1) 0 f _ = prod_seq n 0 f * f n := prod_seq_zero_step n f _ = prod_seq n 0 (f ∘ (g ∘ swap u n)) * f ((g ∘ swap u n) n) := by rw [gs_prod, gs_fix_n] _ = prod_seq n 0 (f ∘ g ∘ swap u n) * (f ∘ g ∘ swap u n) n := by rfl _ = prod_seq (n + 1) 0 (f ∘ g ∘ swap u n) := (prod_seq_zero_step n (f ∘ g ∘ swap u n)).symm _ = prod_seq (n + 1) 0 ((f ∘ g) ∘ swap u n) := by rfl _ = prod_seq (n + 1) 0 (f ∘ g) := swap_prod_eq_prod (f ∘ g) h3 done done lemma F_invertible (m i : Nat) : invertible (F m i) := by by_cases h : rel_prime m i · -- Case 1. h : rel_prime m i rewrite [F_rp_def h] show invertible [i]_m from (Theorem_7_3_7 m i).rtl h done · -- Case 2. h : ¬rel_prime m i rewrite [F_not_rp_def h] apply Exists.intro [1]_m show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m done done lemma Fprod_invertible (m : Nat) : ∀ (k : Nat), invertible (prod_seq k 0 (F m)) := by by_induc · -- Base Case apply Exists.intro [1]_m show prod_seq 0 0 (F m) * [1]_m = [1]_m from calc prod_seq 0 0 (F m) * [1]_m _ = [1]_m * [1]_m := by rw [prod_seq_base] _ = [1]_m := Theorem_7_3_6_7 ([1]_m) done · -- Induction Step fix k : Nat assume ih : invertible (prod_seq k 0 (F m)) rewrite [prod_seq_zero_step] show invertible (prod_seq k 0 (F m) * (F m k)) from (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k) done done theorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m ^ (phi m) = [1]_m := by have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2 show [a]_m ^ (phi m) = [1]_m from calc [a]_m ^ (phi m) _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3] _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring _ = prod_seq m 0 (F m ∘ G m a) * Y := by rw [FG_prod h1 m, phi_def] _ = prod_seq m 0 (F m) * Y := by rw [perm_prod (F m) m (G m a) (G_perm_below h1)] _ = [1]_m := by rw [h3] done lemma Exercise_7_4_5_Int (m : Nat) (a : Int) : ∀ (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry

lemma Exercise_7_4_5_Nat (m a n : Nat) : [a]_m ^ n = [a ^ n]_m

HTPI.Exercise_7_4_5_Nat

htpi/HTPILib/Chap7.lean

htpi

/- Copyright 2023 Daniel J. Velleman -/ import HTPILib.Chap6 namespace HTPI /- Definitions -/ lemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by have h : n + 1 > 0 := Nat.succ_pos n show a % (n + 1) < n + 1 from Nat.mod_lt a h done def gcd (a b : Nat) : Nat := match b with | 0 => a | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd (n + 1) (a % (n + 1)) termination_by b mutual def gcd_c1 (a b : Nat) : Int := match b with | 0 => 1 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c2 (n + 1) (a % (n + 1)) --Corresponds to s = t' termination_by b def gcd_c2 (a b : Nat) : Int := match b with | 0 => 0 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c1 (n + 1) (a % (n + 1)) - (gcd_c2 (n + 1) (a % (n + 1))) * ↑(a / (n + 1)) --Corresponds to t = s' - t'q termination_by b end def prime (n : Nat) : Prop := 2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n def prime_factor (p n : Nat) : Prop := prime p ∧ p ∣ n def all_prime (l : List Nat) : Prop := ∀ p ∈ l, prime p def nondec (l : List Nat) : Prop := match l with | [] => True --Of course, True is a proposition that is always true | n :: L => (∀ m ∈ L, n ≤ m) ∧ nondec L def nondec_prime_list (l : List Nat) : Prop := all_prime l ∧ nondec l def prod (l : List Nat) : Nat := match l with | [] => 1 | n :: L => n * (prod L) def prime_factorization (n : Nat) (l : List Nat) : Prop := nondec_prime_list l ∧ prod l = n def rel_prime (a b : Nat) : Prop := gcd a b = 1 def congr_mod (m : Nat) (a b : Int) : Prop := (↑m : Int) ∣ (a - b) def cc (m : Nat) (a : Int) : ZMod m := (↑a : ZMod m) notation:50 a " ≡ " b " (MOD " m ")" => congr_mod m a b notation:max "["a"]_"m:max => cc m a def invertible {m : Nat} (X : ZMod m) : Prop := ∃ (Y : ZMod m), X * Y = [1]_m def num_rp_below (m k : Nat) : Nat := match k with | 0 => 0 | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j def phi (m : Nat) : Nat := num_rp_below m m def prod_seq {m : Nat} (j k : Nat) (f : Nat → ZMod m) : ZMod m := match j with | 0 => [1]_m | n + 1 => prod_seq n k f * f (k + n) def maps_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i < n, g i < n def one_one_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i1 < n, ∀ i2 < n, g i1 = g i2 → i1 = i2 def onto_below (n : Nat) (g : Nat → Nat) : Prop := ∀ k < n, ∃ i < n, g i = k def perm_below (n : Nat) (g : Nat → Nat) : Prop := maps_below n g ∧ one_one_below n g ∧ onto_below n g def inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m) def swap (u v i : Nat) : Nat := if i = u then v else if i = v then u else i namespace Euler --For definitions specific to Euler's theorem def F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m def G (m a i : Nat) : Nat := (a * i) % m def Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i end Euler /- Section 7.1 -/ theorem dvd_mod_of_dvd_a_b {a b d : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ (a % b) := by set q : Nat := a / b have h3 : b * q + a % b = a := Nat.div_add_mod a b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 define --Goal : ∃ (c : Nat), a % b = d * c apply Exists.intro (j - k * q) show a % b = d * (j - k * q) from calc a % b _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm _ = a - b * q := by rw [h3] _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc] _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm done theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry #eval gcd 672 161 --Answer: 7 lemma gcd_base (a : Nat) : gcd a 0 = a := by rfl lemma gcd_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd a b = gcd b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1)) rfl done lemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b ≠ 0) : a % b < b := by have h1 : b > 0 := Nat.pos_of_ne_zero h show a % b < b from Nat.mod_lt a h1 done lemma dvd_self (n : Nat) : n ∣ n := by apply Exists.intro 1 ring done theorem gcd_dvd : ∀ (b a : Nat), (gcd a b) ∣ a ∧ (gcd a b) ∣ b := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd a b_1) ∣ a ∧ (gcd a b_1) ∣ b_1 fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_base] --Goal: a ∣ a ∧ a ∣ 0 apply And.intro (dvd_self a) define apply Exists.intro 0 rfl done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_nonzero a h1] --Goal : gcd b (a % b) ∣ a ∧ gcd b (a % b) ∣ b have h2 : a % b < b := mod_nonzero_lt a h1 have h3 : (gcd b (a % b)) ∣ b ∧ (gcd b (a % b)) ∣ (a % b) := ih (a % b) h2 b apply And.intro _ h3.left show (gcd b (a % b)) ∣ a from dvd_a_of_dvd_b_mod h3.left h3.right done done theorem gcd_dvd_left (a b : Nat) : (gcd a b) ∣ a := (gcd_dvd b a).left theorem gcd_dvd_right (a b : Nat) : (gcd a b) ∣ b := (gcd_dvd b a).right lemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl lemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c1 a b = gcd_c2 b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done lemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl lemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * ↑(a / b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done theorem gcd_lin_comb : ∀ (b a : Nat), (gcd_c1 a b) * ↑a + (gcd_c2 a b) * ↑b = ↑(gcd a b) := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd_c1 a b_1) * ↑a + (gcd_c2 a b_1) * ↑b_1 = ↑(gcd a b_1) fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base] --Goal : 1 * ↑a + 0 * ↑0 = ↑a ring done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1] --Goal : gcd_c2 b (a % b) * ↑a + -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * ↑(a / b)) * ↑b = -- ↑(gcd b (a % b)) set r : Nat := a % b set q : Nat := a / b set s : Int := gcd_c1 b r set t : Int := gcd_c2 b r --Goal : t * ↑a + (s - t * ↑q) * ↑b = ↑(gcd b r) have h2 : r < b := mod_nonzero_lt a h1 have h3 : s * ↑b + t * ↑r = ↑(gcd b r) := ih r h2 b have h4 : b * q + r = a := Nat.div_add_mod a b rewrite [←h3, ←h4] rewrite [Nat.cast_add, Nat.cast_mul] --Goal : t * (↑b * ↑q + ↑r) + (s - t * ↑q) * ↑b = s * ↑b + t * ↑r ring done done #eval gcd_c1 672 161 --Answer: 6 #eval gcd_c2 672 161 --Answer: -25 --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161 theorem Theorem_7_1_6 {d a b : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ gcd a b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑d ∣ ↑(gcd a b) set s : Int := gcd_c1 a b set t : Int := gcd_c2 a b have h3 : s * ↑a + t * ↑b = ↑(gcd a b) := gcd_lin_comb b a rewrite [←h3] --Goal : ↑d ∣ s * ↑a + t * ↑b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul] --Goal : ↑d ∣ s * (↑d * ↑j) + t * (↑d * ↑k) define apply Exists.intro (s * ↑j + t * ↑k) ring done /- Section 7.2 -/ theorem dvd_trans {a b c : Nat} (h1 : a ∣ b) (h2 : b ∣ c) : a ∣ c := by define at h1; define at h2; define obtain (m : Nat) (h3 : b = a * m) from h1 obtain (n : Nat) (h4 : c = b * n) from h2 rewrite [h3, mul_assoc] at h4 apply Exists.intro (m * n) show c = a * (m * n) from h4 done lemma exists_prime_factor : ∀ (n : Nat), 2 ≤ n → ∃ (p : Nat), prime_factor p n := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, 2 ≤ n_1 → ∃ (p : Nat), prime_factor p n_1 assume h1 : 2 ≤ n by_cases h2 : prime n · -- Case 1. h2 : prime n apply Exists.intro n define --Goal : prime n ∧ n ∣ n show prime n ∧ n ∣ n from And.intro h2 (dvd_self n) done · -- Case 2. h2 : ¬prime n define at h2 --h2 : ¬(2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n) demorgan at h2 disj_syll h2 h1 obtain (a : Nat) (h3 : ∃ (b : Nat), a * b = n ∧ a < n ∧ b < n) from h2 obtain (b : Nat) (h4 : a * b = n ∧ a < n ∧ b < n) from h3 have h5 : 2 ≤ a := by by_contra h6 have h7 : a ≤ 1 := by linarith have h8 : n ≤ b := calc n _ = a * b := h4.left.symm _ ≤ 1 * b := by rel [h7] _ = b := by ring linarith --n ≤ b contradicts b < n done have h6 : ∃ (p : Nat), prime_factor p a := ih a h4.right.left h5 obtain (p : Nat) (h7 : prime_factor p a) from h6 apply Exists.intro p define --Goal : prime p ∧ p ∣ n define at h7 --h7 : prime p ∧ p ∣ a apply And.intro h7.left have h8 : a ∣ n := by apply Exists.intro b show n = a * b from (h4.left).symm done show p ∣ n from dvd_trans h7.right h8 done done lemma exists_least_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q := by set S : Set Nat := {p : Nat | prime_factor p n} have h2 : ∃ (p : Nat), p ∈ S := exists_prime_factor n h show ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q from well_ord_princ S h2 done lemma all_prime_nil : all_prime [] := by define --Goal : ∀ p ∈ [], prime p fix p : Nat contrapos --Goal : ¬prime p → p ∉ [] assume h1 : ¬prime p show p ∉ [] from List.not_mem_nil p done lemma all_prime_cons (n : Nat) (L : List Nat) : all_prime (n :: L) ↔ prime n ∧ all_prime L := by apply Iff.intro · -- (→) assume h1 : all_prime (n :: L) --Goal : prime n ∧ all_prime L define at h1 --h1 : ∀ p ∈ n :: L, prime p apply And.intro (h1 n (List.mem_cons_self n L)) define --Goal : ∀ p ∈ L, prime p fix p : Nat assume h2 : p ∈ L show prime p from h1 p (List.mem_cons_of_mem n h2) done · -- (←) assume h1 : prime n ∧ all_prime L --Goal : all_prime (n :: l) define : all_prime L at h1 define fix p : Nat assume h2 : p ∈ n :: L rewrite [List.mem_cons] at h2 --h2 : p = n ∨ p ∈ L by_cases on h2 · -- Case 1. h2 : p = n rewrite [h2] show prime n from h1.left done · -- Case 2. h2 : p ∈ L show prime p from h1.right p h2 done done done lemma nondec_nil : nondec [] := by define --Goal : True trivial --trivial proves some obviously true statements, such as True done lemma nondec_cons (n : Nat) (L : List Nat) : nondec (n :: L) ↔ (∀ m ∈ L, n ≤ m) ∧ nondec L := by rfl lemma prod_nil : prod [] = 1 := by rfl lemma prod_cons : prod (n :: L) = n * (prod L) := by rfl lemma exists_cons_of_length_eq_succ {A : Type} {l : List A} {n : Nat} (h : l.length = n + 1) : ∃ (a : A) (L : List A), l = a :: L ∧ L.length = n := by have h1 : ¬l.length = 0 := by linarith rewrite [List.length_eq_zero] at h1 obtain (a : A) (h2 : ∃ (L : List A), l = a :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List A) (h3 : l = a :: L) from h2 apply Exists.intro a apply Exists.intro L apply And.intro h3 have h4 : (a :: L).length = L.length + 1 := List.length_cons a L rewrite [←h3, h] at h4 show L.length = n from (Nat.add_right_cancel h4).symm done lemma list_elt_dvd_prod_by_length (a : Nat) : ∀ (n : Nat), ∀ (l : List Nat), l.length = n → a ∈ l → a ∣ prod l := by by_induc · --Base Case fix l : List Nat assume h1 : l.length = 0 rewrite [List.length_eq_zero] at h1 --h1 : l = [] rewrite [h1] --Goal : a ∈ [] → a ∣ prod [] contrapos assume h2 : ¬a ∣ prod [] show a ∉ [] from List.not_mem_nil a done · -- Induction Step fix n : Nat assume ih : ∀ (l : List Nat), List.length l = n → a ∈ l → a ∣ prod l fix l : List Nat assume h1 : l.length = n + 1 --Goal : a ∈ l → a ∣ prod l obtain (b : Nat) (h2 : ∃ (L : List Nat), l = b :: L ∧ L.length = n) from exists_cons_of_length_eq_succ h1 obtain (L : List Nat) (h3 : l = b :: L ∧ L.length = n) from h2 have h4 : a ∈ L → a ∣ prod L := ih L h3.right assume h5 : a ∈ l rewrite [h3.left, prod_cons] --Goal : a ∣ b * prod L rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b ∨ a ∈ L by_cases on h5 · -- Case 1. h5 : a = b apply Exists.intro (prod L) rewrite [h5] rfl done · -- Case 2. h5 : a ∈ L have h6 : a ∣ prod L := h4 h5 have h7 : prod L ∣ b * prod L := by apply Exists.intro b ring done show a ∣ b * prod L from dvd_trans h6 h7 done done done lemma list_elt_dvd_prod {a : Nat} {l : List Nat} (h : a ∈ l) : a ∣ prod l := by set n : Nat := l.length have h1 : l.length = n := by rfl show a ∣ prod l from list_elt_dvd_prod_by_length a n l h1 h done lemma exists_prime_factorization : ∀ (n : Nat), n ≥ 1 → ∃ (l : List Nat), prime_factorization n l := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, n_1 ≥ 1 → ∃ (l : List Nat), prime_factorization n_1 l assume h1 : n ≥ 1 by_cases h2 : n = 1 · -- Case 1. h2 : n = 1 apply Exists.intro [] define apply And.intro · -- Proof of nondec_prime_list [] define show all_prime [] ∧ nondec [] from And.intro all_prime_nil nondec_nil done · -- Proof of prod [] = n rewrite [prod_nil, h2] rfl done done · -- Case 2. h2 : n ≠ 1 have h3 : n ≥ 2 := lt_of_le_of_ne' h1 h2 obtain (p : Nat) (h4 : prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q) from exists_least_prime_factor h3 have p_prime_factor : prime_factor p n := h4.left define at p_prime_factor have p_prime : prime p := p_prime_factor.left have p_dvd_n : p ∣ n := p_prime_factor.right have p_least : ∀ (q : Nat), prime_factor q n → p ≤ q := h4.right obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n have h5 : m ≠ 0 := by contradict h1 with h6 have h7 : n = 0 := calc n _ = p * m := n_eq_pm _ = p * 0 := by rw [h6] _ = 0 := by ring rewrite [h7] decide done have m_pos : 0 < m := Nat.pos_of_ne_zero h5 have m_lt_n : m < n := by define at p_prime show m < n from calc m _ < m + m := by linarith _ = 2 * m := by ring _ ≤ p * m := by rel [p_prime.left] _ = n := n_eq_pm.symm done obtain (L : List Nat) (h6 : prime_factorization m L) from ih m m_lt_n m_pos define at h6 have ndpl_L : nondec_prime_list L := h6.left define at ndpl_L apply Exists.intro (p :: L) define apply And.intro · -- Proof of nondec_prime_list (p :: L) define apply And.intro · -- Proof of all_prime (p :: L) rewrite [all_prime_cons] show prime p ∧ all_prime L from And.intro p_prime ndpl_L.left done · -- Proof of nondec (p :: L) rewrite [nondec_cons] apply And.intro _ ndpl_L.right fix q : Nat assume q_in_L : q ∈ L have h7 : q ∣ prod L := list_elt_dvd_prod q_in_L rewrite [h6.right] at h7 --h7 : q ∣ m have h8 : m ∣ n := by apply Exists.intro p rewrite [n_eq_pm] ring done have q_dvd_n : q ∣ n := dvd_trans h7 h8 have ap_L : all_prime L := ndpl_L.left define at ap_L have q_prime_factor : prime_factor q n := And.intro (ap_L q q_in_L) q_dvd_n show p ≤ q from p_least q q_prime_factor done done · -- Proof of prod (p :: L) = n rewrite [prod_cons, h6.right, n_eq_pm] rfl done done done theorem Theorem_7_2_2 {a b c : Nat} (h1 : c ∣ a * b) (h2 : rel_prime a c) : c ∣ b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑c ∣ ↑b define at h1; define at h2; define obtain (j : Nat) (h3 : a * b = c * j) from h1 set s : Int := gcd_c1 a c set t : Int := gcd_c2 a c have h4 : s * ↑a + t * ↑c = ↑(gcd a c) := gcd_lin_comb c a rewrite [h2, Nat.cast_one] at h4 --h4 : s * ↑a + t * ↑c = (1 : Int) apply Exists.intro (s * ↑j + t * ↑b) show ↑b = ↑c * (s * ↑j + t * ↑b) from calc ↑b _ = (1 : Int) * ↑b := (one_mul _).symm _ = (s * ↑a + t * ↑c) * ↑b := by rw [h4] _ = s * (↑a * ↑b) + t * ↑c * ↑b := by ring _ = s * (↑c * ↑j) + t * ↑c * ↑b := by rw [←Nat.cast_mul a b, h3, Nat.cast_mul c j] _ = ↑c * (s * ↑j + t * ↑b) := by ring done lemma le_nonzero_prod_left {a b : Nat} (h : a * b ≠ 0) : a ≤ a * b := by have h1 : b ≠ 0 := by contradict h with h1 rewrite [h1] ring done have h2 : 1 ≤ b := Nat.pos_of_ne_zero h1 show a ≤ a * b from calc a = a * 1 := (mul_one a).symm _ ≤ a * b := by rel [h2] done lemma le_nonzero_prod_right {a b : Nat} (h : a * b ≠ 0) : b ≤ a * b := by rewrite [mul_comm] rewrite [mul_comm] at h show b ≤ b * a from le_nonzero_prod_left h done lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry lemma rel_prime_of_prime_not_dvd {a p : Nat} (h1 : prime p) (h2 : ¬p ∣ a) : rel_prime a p := by have h3 : gcd a p ∣ a := gcd_dvd_left a p have h4 : gcd a p ∣ p := gcd_dvd_right a p have h5 : gcd a p = 1 ∨ gcd a p = p := dvd_prime h1 h4 have h6 : gcd a p ≠ p := by contradict h2 with h6 rewrite [h6] at h3 show p ∣ a from h3 done disj_syll h5 h6 show rel_prime a p from h5 done theorem Theorem_7_2_3 {a b p : Nat} (h1 : prime p) (h2 : p ∣ a * b) : p ∣ a ∨ p ∣ b := by or_right with h3 have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3 show p ∣ b from Theorem_7_2_2 h2 h4 done lemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a ≥ 1 := by have h1 : a ≠ 0 := by by_contra h1 rewrite [h1] at h contradict h linarith done show a ≥ 1 from Nat.pos_of_ne_zero h1 done lemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by have h1 : a ≥ 1 := ge_one_of_prod_one h have h2 : a * b ≠ 0 := by linarith have h3 : a ≤ a * b := le_nonzero_prod_left h2 rewrite [h] at h3 show a = 1 from Nat.le_antisymm h3 h1 done lemma eq_one_of_dvd_one {n : Nat} (h : n ∣ 1) : n = 1 := by obtain (j : Nat) (h1 : 1 = n * j) from h show n = 1 from eq_one_of_prod_one h1.symm done lemma prime_not_one {p : Nat} (h : prime p) : p ≠ 1 := by define at h linarith done theorem Theorem_7_2_4 {p : Nat} (h1 : prime p) : ∀ (l : List Nat), p ∣ prod l → ∃ a ∈ l, p ∣ a := by apply List.rec · -- Base Case. Goal : p ∣ prod [] → ∃ a ∈ [], p ∣ a rewrite [prod_nil] assume h2 : p ∣ 1 show ∃ a ∈ [], p ∣ a from absurd (eq_one_of_dvd_one h2) (prime_not_one h1) done · -- Induction Step fix b : Nat fix L : List Nat assume ih : p ∣ prod L → ∃ a ∈ L, p ∣ a --Goal : p ∣ prod (b :: L) → ∃ a ∈ b :: L, p ∣ a assume h2 : p ∣ prod (b :: L) rewrite [prod_cons] at h2 have h3 : p ∣ b ∨ p ∣ prod L := Theorem_7_2_3 h1 h2 by_cases on h3 · -- Case 1. h3 : p ∣ b apply Exists.intro b show b ∈ b :: L ∧ p ∣ b from And.intro (List.mem_cons_self b L) h3 done · -- Case 2. h3 : p ∣ prod L obtain (a : Nat) (h4 : a ∈ L ∧ p ∣ a) from ih h3 apply Exists.intro a show a ∈ b :: L ∧ p ∣ a from And.intro (List.mem_cons_of_mem b h4.left) h4.right done done done lemma prime_in_list {p : Nat} {l : List Nat} (h1 : prime p) (h2 : all_prime l) (h3 : p ∣ prod l) : p ∈ l := by obtain (a : Nat) (h4 : a ∈ l ∧ p ∣ a) from Theorem_7_2_4 h1 l h3 define at h2 have h5 : prime a := h2 a h4.left have h6 : p = 1 ∨ p = a := dvd_prime h5 h4.right disj_syll h6 (prime_not_one h1) rewrite [h6] show a ∈ l from h4.left done lemma first_le_first {p q : Nat} {l m : List Nat} (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m)) (h3 : prod (p :: l) = prod (q :: m)) : p ≤ q := by define at h1; define at h2 have h4 : q ∣ prod (p :: l) := by define apply Exists.intro (prod m) rewrite [←prod_cons] show prod (p :: l) = prod (q :: m) from h3 done have h5 : all_prime (q :: m) := h2.left rewrite [all_prime_cons] at h5 have h6 : q ∈ p :: l := prime_in_list h5.left h1.left h4 have h7 : nondec (p :: l) := h1.right rewrite [nondec_cons] at h7 rewrite [List.mem_cons] at h6 by_cases on h6 · -- Case 1. h6 : q = p linarith done · -- Case 2. h6 : q ∈ l have h8 : ∀ m ∈ l, p ≤ m := h7.left show p ≤ q from h8 q h6 done done lemma nondec_prime_list_tail {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by define at h define rewrite [all_prime_cons, nondec_cons] at h show all_prime l ∧ nondec l from And.intro h.left.right h.right.right done lemma cons_prod_not_one {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : prod (p :: l) ≠ 1 := by define at h have h1 : all_prime (p :: l) := h.left rewrite [all_prime_cons] at h1 rewrite [prod_cons] by_contra h2 show False from (prime_not_one h1.left) (eq_one_of_prod_one h2) done lemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) : l = [] ↔ prod l = 1 := by apply Iff.intro · -- (→) assume h1 : l = [] rewrite [h1] show prod [] = 1 from prod_nil done · -- (←) contrapos assume h1 : ¬l = [] obtain (p : Nat) (h2 : ∃ (L : List Nat), l = p :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List Nat) (h3 : l = p :: L) from h2 rewrite [h3] at h rewrite [h3] show ¬prod (p :: L) = 1 from cons_prod_not_one h done done lemma prime_pos {p : Nat} (h : prime p) : p > 0 := by define at h linarith done theorem Theorem_7_2_5 : ∀ (l1 l2 : List Nat), nondec_prime_list l1 → nondec_prime_list l2 → prod l1 = prod l2 → l1 = l2 := by apply List.rec · -- Base Case. Goal : ∀ (l2 : List Nat), nondec_prime_list [] → -- nondec_prime_list l2 → prod [] = prod l2 → [] = l2 fix l2 : List Nat assume h1 : nondec_prime_list [] assume h2 : nondec_prime_list l2 assume h3 : prod [] = prod l2 rewrite [prod_nil, eq_comm, ←list_nil_iff_prod_one h2] at h3 show [] = l2 from h3.symm done · -- Induction Step fix p : Nat fix L1 : List Nat assume ih : ∀ (L2 : List Nat), nondec_prime_list L1 → nondec_prime_list L2 → prod L1 = prod L2 → L1 = L2 -- Goal : ∀ (l2 : List Nat), nondec_prime_list (p :: L1) → -- nondec_prime_list l2 → prod (p :: L1) = prod l2 → p :: L1 = l2 fix l2 : List Nat assume h1 : nondec_prime_list (p :: L1) assume h2 : nondec_prime_list l2 assume h3 : prod (p :: L1) = prod l2 have h4 : ¬prod (p :: L1) = 1 := cons_prod_not_one h1 rewrite [h3, ←list_nil_iff_prod_one h2] at h4 obtain (q : Nat) (h5 : ∃ (L : List Nat), l2 = q :: L) from List.exists_cons_of_ne_nil h4 obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5 rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2) rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2) have h7 : p ≤ q := first_le_first h1 h2 h3 have h8 : q ≤ p := first_le_first h2 h1 h3.symm have h9 : p = q := by linarith rewrite [h9, prod_cons, prod_cons] at h3 --h3 : q * prod L1 = q * prod L2 have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1 have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2 define at h2 have h12 : all_prime (q :: L2) := h2.left rewrite [all_prime_cons] at h12 have h13 : q > 0 := prime_pos h12.left have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3 have h15 : L1 = L2 := ih L2 h10 h11 h14 rewrite [h6, h9, h15] rfl done done theorem fund_thm_arith (n : Nat) (h : n ≥ 1) : ∃! (l : List Nat), prime_factorization n l := by exists_unique · -- Existence show ∃ (l : List Nat), prime_factorization n l from exists_prime_factorization n h done · -- Uniqueness fix l1 : List Nat; fix l2 : List Nat assume h1 : prime_factorization n l1 assume h2 : prime_factorization n l2 define at h1; define at h2 have h3 : prod l1 = n := h1.right rewrite [←h2.right] at h3 show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3 done done /- Section 7.3 -/ theorem congr_refl (m : Nat) : ∀ (a : Int), a ≡ a (MOD m) := by fix a : Int define --Goal : ∃ (c : Int), a - a = ↑m * c apply Exists.intro 0 ring done theorem congr_symm {m : Nat} : ∀ {a b : Int}, a ≡ b (MOD m) → b ≡ a (MOD m) := by fix a : Int; fix b : Int assume h1 : a ≡ b (MOD m) define at h1 --h1 : ∃ (c : Int), a - b = ↑m * c define --Goal : ∃ (c : Int), b - a = ↑m * c obtain (c : Int) (h2 : a - b = m * c) from h1 apply Exists.intro (-c) show b - a = m * (-c) from calc b - a _ = -(a - b) := by ring _ = -(m * c) := by rw [h2] _ = m * (-c) := by ring done theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry /- Fundamental properties of congruence classes -/ lemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) : X = Y ↔ X.val = Y.val := Fin.ext_iff lemma val_nat_eq_mod (n k : Nat) : ([k]_(n + 1)).val = k % (n + 1) := by rfl lemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl theorem cc_rep {m : Nat} (X : ZMod m) : ∃ (a : Int), X = [a]_m := match m with | 0 => by apply Exists.intro X rfl done | n + 1 => by apply Exists.intro ↑(X.val) have h1 : X.val < n + 1 := Fin.prop X rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1] rfl done theorem add_class (m : Nat) (a b : Int) : [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm theorem mul_class (m : Nat) (a b : Int) : [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm lemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ [a - b]_m = [0]_m := by apply Iff.intro · -- (→) assume h1 : [a]_m = [b]_m have h2 : a - b = a + (-b) := by ring have h3 : b + (-b) = 0 := by ring show [a - b]_m = [0]_m from calc [a - b]_m _ = [a + (-b)]_m := by rw [h2] _ = [a]_m + [-b]_m := by rw [add_class] _ = [b]_m + [-b]_m := by rw [h1] _ = [b + -b]_m := by rw [add_class] _ = [0]_m := by rw [h3] done · -- (←) assume h1 : [a - b]_m = [0]_m have h2 : b + (a - b) = a := by ring have h3 : b + 0 = b := by ring show [a]_m = [b]_m from calc [a]_m _ = [b + (a - b)]_m := by rw [h2] _ = [b]_m + [a - b]_m := by rw [add_class] _ = [b]_m + [0]_m := by rw [h1] _ = [b + 0]_m := by rw [add_class] _ = [b]_m := by rw [h3] done done lemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) : [a]_m = [0]_m → [-a]_m = [0]_m := by assume h1 : [a]_m = [0]_m have h2 : 0 + (-a) = -a := by ring have h3 : a + (-a) = 0 := by ring show [-a]_m = [0]_m from calc [-a]_m _ = [0 + (-a)]_m := by rw [h2] _ = [0]_m + [-a]_m := by rw [add_class] _ = [a]_m + [-a]_m := by rw [h1] _ = [a + (-a)]_m := by rw [add_class] _ = [0]_m := by rw [h3] done lemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) : [-a]_m = [0]_m ↔ [a]_m = [0]_m := by apply Iff.intro _ (cc_neg_zero_of_cc_zero m a) assume h1 : [-a]_m = [0]_m have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1 have h3 : -(-a) = a := by ring rewrite [h3] at h2 show [a]_m = [0]_m from h2 done lemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl lemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m ↔ m ∣ k := match m with | 0 => by have h : (0 : Int) = (↑(0 : Nat) : Int) := by rfl rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj] apply Iff.intro · -- (→) assume h1 : k = 0 rewrite [h1] show 0 ∣ 0 from dvd_self 0 done · -- (←) assume h1 : 0 ∣ k obtain (c : Nat) (h2 : k = 0 * c) from h1 rewrite [h2] ring done done | n + 1 => by rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero] show k % (n + 1) = 0 ↔ n + 1 ∣ k from (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm done lemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m ↔ ↑m ∣ a := by obtain (k : Nat) (h1 : a = ↑k ∨ a = -↑k) from Int.eq_nat_or_neg a by_cases on h1 · -- Case 1. h1: a = ↑k rewrite [h1, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done · -- Case 2. h1: a = -↑k rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done done theorem cc_eq_iff_congr (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ a ≡ b (MOD m) := calc [a]_m = [b]_m _ ↔ [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b _ ↔ ↑m ∣ (a - b) := cc_zero_iff_dvd m (a - b) _ ↔ a ≡ b (MOD m) := by rfl /- End of fundamental properties of congruence classes -/ lemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m := by have h1 : (↑m : Int) ≠ 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m) show 0 ≤ a % m from Int.emod_nonneg a h1 done lemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m) have h2 : (↑m : Int) > 0 := (Nat.cast_pos).rtl h1 show a % m < m from Int.emod_lt_of_pos a h2 done lemma congr_mod_mod (m : Nat) (a : Int) : a ≡ a % m (MOD m) := by define have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m apply Exists.intro (a / m) show a - a % m = m * (a / m) from calc a - (a % m) _ = m * (a / m) + a % m - a % m := by rw [h1] _ = m * (a / m) := by ring done lemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) := And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a)) theorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) : ∃! (r : Int), 0 ≤ r ∧ r < m ∧ a ≡ r (MOD m) := by exists_unique · -- Existence apply Exists.intro (a % m) show 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) from mod_cmpl_res m a done · -- Uniqueness fix r1 : Int; fix r2 : Int assume h1 : 0 ≤ r1 ∧ r1 < m ∧ a ≡ r1 (MOD m) assume h2 : 0 ≤ r2 ∧ r2 < m ∧ a ≡ r2 (MOD m) have h3 : r1 ≡ r2 (MOD m) := congr_trans (congr_symm h1.right.right) h2.right.right obtain (d : Int) (h4 : r1 - r2 = m * d) from h3 have h5 : r1 - r2 < m * 1 := by linarith have h6 : m * (-1) < r1 - r2 := by linarith rewrite [h4] at h5 --h5 : m * d < m * 1 rewrite [h4] at h6 --h6 : m * -1 < m * d have h7 : (↑m : Int) ≥ 0 := Nat.cast_nonneg m have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7 have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7 have h10 : d = 0 := by linarith show r1 = r2 from calc r1 _ = r1 - r2 + r2 := by ring _ = m * 0 + r2 := by rw [h4, h10] _ = r2 := by ring done done lemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m := (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a) theorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y rewrite [h1, h2] have h3 : a + b = b + a := by ring show [a]_m + [b]_m = [b]_m + [a]_m from calc [a]_m + [b]_m _ = [a + b]_m := add_class m a b _ = [b + a]_m := by rw [h3] _ = [b]_m + [a]_m := (add_class m b a).symm done theorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X rewrite [h1] have h2 : a * 1 = a := by ring show [a]_m * [1]_m = [a]_m from calc [a]_m * [1]_m _ = [a * 1]_m := mul_class m a 1 _ = [a]_m := by rw [h2] done theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry lemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) : [a]_m * [gcd_c2 m a]_m = [1]_m := by set s : Int := gcd_c1 m a have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m define at h1 rewrite [h1, Nat.cast_one] at h2 --h2 : s * ↑m + gcd_c2 m a * ↑a = 1 rewrite [mul_class, cc_eq_iff_congr] define --Goal : ∃ (c : Int), ↑a * gcd_c2 m a - 1 = ↑m * c apply Exists.intro (-s) show a * (gcd_c2 m a) - 1 = m * (-s) from calc a * (gcd_c2 m a) - 1 _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring _ = 1 + m * (-s) - 1 := by rw [h2] _ = m * (-s) := by ring done theorem Theorem_7_3_7 (m a : Nat) : invertible [a]_m ↔ rel_prime m a := by apply Iff.intro · -- (→) assume h1 : invertible [a]_m define at h1 obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1 obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y rewrite [h3, mul_class, cc_eq_iff_congr] at h2 define at h2 obtain (c : Int) (h4 : a * b - 1 = m * c) from h2 rewrite [Exercise_7_2_6] --Goal : ∃ (s t : Int), s * ↑m + t * ↑a = 1 apply Exists.intro (-c) apply Exists.intro b show (-c) * m + b * a = 1 from calc (-c) * m + b * a _ = (-c) * m + (a * b - 1) + 1 := by ring _ = (-c) * m + m * c + 1 := by rw [h4] _ = 1 := by ring done · -- (←) assume h1 : rel_prime m a define show ∃ (Y : ZMod m), [a]_m * Y = [1]_m from Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1) done done /- Section 7.4 -/ section Euler open Euler lemma num_rp_below_base {m : Nat} : num_rp_below m 0 = 0 := by rfl lemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) : num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by have h1 : num_rp_below m (j + 1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : gcd m j = 1 rewrite [if_pos h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1 show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1 done lemma num_rp_below_step_not_rp {m j : Nat} (h : ¬rel_prime m j) : num_rp_below m (j + 1) = num_rp_below m j := by have h1 : num_rp_below m (j +1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : ¬gcd m j = 1 rewrite [if_neg h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j show num_rp_below m (j + 1) = num_rp_below m j from h1 done lemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl #eval phi 10 --Answer: 4 lemma prod_inv_iff_inv {m : Nat} {X : ZMod m} (h1 : invertible X) (Y : ZMod m) : invertible (X * Y) ↔ invertible Y := by apply Iff.intro · -- (→) assume h2 : invertible (X * Y) obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2 apply Exists.intro (X * Z) rewrite [←h3] --Goal : Y * (X * Z) = X * Y * Z ring --Note that ring can do algebra in ZMod m done · -- (←) assume h2 : invertible Y obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1 obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2 apply Exists.intro (Xi * Yi) show (X * Y) * (Xi * Yi) = [1]_m from calc X * Y * (Xi * Yi) _ = (X * Xi) * (Y * Yi) := by ring _ = [1]_m * [1]_m := by rw [h3, h4] _ = [1]_m := Theorem_7_3_6_7 [1]_m done done lemma F_rp_def {m i : Nat} (h : rel_prime m i) : F m i = [i]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h --h : gcd m i = 1 rewrite [if_pos h] at h1 show F m i = [i]_m from h1 done lemma F_not_rp_def {m i : Nat} (h : ¬rel_prime m i) : F m i = [1]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h rewrite [h1, if_neg h] rfl done lemma prod_seq_base {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 0 k f = [1]_m := by rfl lemma prod_seq_step {m : Nat} (n k : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl lemma prod_seq_zero_step {m : Nat} (n : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by rewrite [prod_seq_step, zero_add] rfl done lemma prod_one {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 1 k f = f k := by rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7] rfl done lemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl lemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m := calc [G m a i]_m _ = [(a * i) % m]_m := by rfl _ = [a * i]_m := (cc_eq_mod m (a * i)).symm _ = [a]_m * [i]_m := (mul_class m a i).symm lemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) : rel_prime m (G m a i) ↔ rel_prime m i := by have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1 show rel_prime m (G m a i) ↔ rel_prime m i from calc rel_prime m (G m a i) _ ↔ invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm _ ↔ invertible ([a]_m * [i]_m) := by rw [cc_G] _ ↔ invertible [i]_m := prod_inv_iff_inv h2 ([i]_m) _ ↔ rel_prime m i := Theorem_7_3_7 m i done lemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) : F m (G m a i) = [a]_m * F m i := by have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2 show F m (G m a i) = [a]_m * F m i from calc F m (G m a i) _ = [G m a i]_m := F_rp_def h3 _ = [a]_m * [i]_m := cc_G m a i _ = [a]_m * F m i := by rw [F_rp_def h2] done lemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : ¬rel_prime m i) : F m (G m a i) = [1]_m := by rewrite [←G_rp_iff h1 i] at h2 show F m (G m a i) = [1]_m from F_not_rp_def h2 done lemma FG_prod {m a : Nat} (h1 : rel_prime m a) : ∀ (k : Nat), prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by by_induc · -- Base Case show prod_seq 0 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from calc prod_seq 0 0 ((F m) ∘ (G m a)) _ = [1]_m := prod_seq_base _ _ _ = [a]_m ^ 0 * [1]_m := by ring _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by rw [num_rp_below_base, prod_seq_base] done · -- Induction Step fix k : Nat assume ih : prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) by_cases h2 : rel_prime m k · -- Case 1. h2 : rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([a]_m * F m k) := by rw [FG_rp h1 h2] _ = [a]_m ^ ((num_rp_below m k) + 1) * ((prod_seq k 0 (F m)) * F m k) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_rp h2, prod_seq_zero_step] done · -- Case 2. h2 : ¬rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([1]_m) := by rw [FG_not_rp h1 h2] _ = [a]_m ^ (num_rp_below m k) * (prod_seq k 0 (F m) * ([1]_m)) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_not_rp h2, prod_seq_zero_step, F_not_rp_def h2] done done done lemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by define --Goal : ∀ i < m, G m a i < m fix i : Nat assume h1 : i < m rewrite [G_def] --Goal : a * i % m < m show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m) done lemma left_inv_one_one_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g' (g i) = i) : one_one_below n g := sorry lemma right_inv_onto_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g (g' i) = i) (h2 : maps_below n g') : onto_below n g := by define at h2; define fix k : Nat assume h3 : k < n apply Exists.intro (g' k) show g' k < n ∧ g (g' k) = k from And.intro (h2 k h3) (h1 k h3) done lemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m * [inv_mod m a]_m = [1]_m := by have h2 : 0 ≤ (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a) show [a]_m * [inv_mod m a]_m = [1]_m from calc [a]_m * [inv_mod m a]_m _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2] _ = [a]_m * [gcd_c2 m a]_m := by rw [←cc_eq_mod] _ = [1]_m := gcd_c2_inv h1 done lemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m := calc a * (b % m) % m = a % m * (b % m % m) % m := Nat.mul_mod _ _ _ _ = a % m * (b % m) % m := by rw [Nat.mod_mod] _ = a * b % m := (Nat.mul_mod _ _ _).symm lemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm] rfl done theorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] : ↑a ≡ ↑b (MOD m) ↔ a % m = b % m := sorry lemma mul_inv_mod_cancel {m a i : Nat} [NeZero m] (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1 rewrite [mul_class, cc_eq_iff_congr, ←Nat.cast_mul, ←Nat.cast_one, congr_iff_mod_eq_Nat] at h3 show a * inv_mod m a * i % m = i from calc a * (inv_mod m a) * i % m _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = 1 % m * i % m := by rw [h3] _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = i % m := by rw [one_mul] _ = i := Nat.mod_eq_of_lt h2 done lemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl lemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, G m a (Ginv m a i) = i := by fix i : Nat assume h2 : i < m show G m a (Ginv m a i) = i from calc G m a (Ginv m a i) _ = a * ((inv_mod m a * i) % m) % m := by rfl _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, Ginv m a (G m a i) = i := by fix i : Nat assume h2 : i < m show Ginv m a (G m a i) = i from calc Ginv m a (G m a i) _ = inv_mod m a * ((a * i) % m) % m := by rfl _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc, mul_comm (inv_mod m a)] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_maps_below (m a : Nat) [NeZero m] : maps_below m (Ginv m a) := G_maps_below m (inv_mod m a) lemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : one_one_below m (G m a) := left_inv_one_one_below (Ginv_left_inv h1) lemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : onto_below m (G m a) := right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a) lemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : perm_below m (G m a) := And.intro (G_maps_below m a) (And.intro (G_one_one_below h1) (G_onto_below h1)) --Permuting a product of congruence classes doesn't change product lemma swap_fst (u v : Nat) : swap u v u = v := by define : swap u v u --Goal : (if u = u then v else if u = v then u else u) = v have h : u = u := by rfl rewrite [if_pos h] rfl done lemma swap_snd (u v : Nat) : swap u v v = u := by define : swap u v v by_cases h1 : v = u · -- Case 1. h1 : v = u rewrite [if_pos h1] show v = u from h1 done · -- Case 2. h1 : v ≠ u rewrite [if_neg h1] have h2 : v = v := by rfl rewrite [if_pos h2] rfl done done lemma swap_other {u v i : Nat} (h1 : i ≠ u) (h2 : i ≠ v) : swap u v i = i := by define : swap u v i rewrite [if_neg h1, if_neg h2] rfl done lemma swap_values (u v i : Nat) : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := by by_cases h1 : i = u · -- Case 1. h1 : i = u apply Or.inl rewrite [h1] show swap u v u = v from swap_fst u v done · -- Case 2. h1 : i ≠ u apply Or.inr by_cases h2 : i = v · -- Case 2.1. h2 : i = v apply Or.inl rewrite [h2] show swap u v v = u from swap_snd u v done · -- Case 2.2. h2 : i ≠ v apply Or.inr show swap u v i = i from swap_other h1 h2 done done done lemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by define fix i : Nat assume h3 : i < n have h4 : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := swap_values u v i by_cases on h4 · -- Case 1. h4 : swap u v i = v rewrite [h4] show v < n from h2 done · -- Case 2. by_cases on h4 · -- Case 2.1. h4 : swap u v i = u rewrite [h4] show u < n from h1 done · -- Case 2.2. h4 : swap u v i = i rewrite [h4] show i < n from h3 done done done lemma swap_swap (u v n : Nat) : ∀ i < n, swap u v (swap u v i) = i := by fix i : Nat assume h : i < n by_cases h1 : i = u · -- Case 1. h1 : i = u rewrite [h1, swap_fst, swap_snd] rfl done · -- Case 2. h1 : i ≠ u by_cases h2 : i = v · -- Case 2.1. h2 : i = v rewrite [h2, swap_snd, swap_fst] rfl done · -- Case 2.2. h2 : i ≠ v rewrite [swap_other h1 h2, swap_other h1 h2] rfl done done done lemma swap_one_one_below (u v n) : one_one_below n (swap u v) := left_inv_one_one_below (swap_swap u v n) lemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) := right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2) lemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) := And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2)) lemma comp_perm_below {n : Nat} {f g : Nat → Nat} (h1 : perm_below n f) (h2 : perm_below n g) : perm_below n (f ∘ g) := sorry lemma trivial_swap (u : Nat) : swap u u = id := by apply funext fix x : Nat by_cases h1 : x = u · -- Case 1. h1 : x = u rewrite [h1, swap_fst] rfl done · -- Case 2. h1 : x ≠ u rewrite [swap_other h1 h1] rfl done done lemma prod_eq_fun {m : Nat} (f g : Nat → ZMod m) (k : Nat) : ∀ (n : Nat), (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g := by by_induc · -- Base Case assume h : (∀ i < 0, f (k + i) = g (k + i)) rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g assume h1 : ∀ i < n + 1, f (k + i) = g (k + i) have h2 : ∀ i < n, f (k + i) = g (k + i) := by fix i : Nat assume h2 : i < n have h3 : i < n + 1 := by linarith show f (k + i) = g (k + i) from h1 i h3 done have h3 : prod_seq n k f = prod_seq n k g := ih h2 have h4 : n < n + 1 := Nat.lt_succ_self n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4] rfl done done lemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := by have h2 : ∀ (i : Nat), i < u → (f ∘ swap u n) (0 + i) = f (0 + i) := by fix i : Nat assume h2 : i < u have h3 : 0 + i ≠ u := by linarith have h4 : 0 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl done show prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f from prod_eq_fun (f ∘ swap u n) f 0 u h2 done lemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := by have h2 : ∀ i < j, (f ∘ swap u n) (u + 1 + i) = f (u + 1 + i) := by fix i : Nat assume h2 : i < j have h3 : u + 1 + i ≠ u := by linarith have h4 : u + 1 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl show prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f from prod_eq_fun (f ∘ swap u n) f (u + 1) j h2 done lemma break_prod {m : Nat} (n : Nat) (f : Nat → ZMod m) : ∀ (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by by_induc · -- Base Case have h : n + 0 = n := by rfl rewrite [prod_seq_base, h, Theorem_7_3_6_7] rfl done · -- Induction Step fix j : Nat assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f rewrite [←add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc] rfl done done lemma break_prod_twice {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f := break_prod n f 1 rewrite [prod_one] at h2 have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f := break_prod (u + 1) f j rewrite [←h1] at h3 have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f := break_prod u f 1 rewrite [prod_one] at h4 rewrite [h3, h4] at h2 show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2 done lemma swap_prod_eq_prod {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f := by by_cases h2 : u = n · -- Case 1. h2 : u = n rewrite [h2, trivial_swap n] --Goal : prod_seq (n + 1) 0 (f ∘ id) = prod_seq (n + 1) 0 f rfl done · -- Case 2. h2 : ¬u = n have h3 : u + 1 ≤ n := Nat.lt_of_le_of_ne h1 h2 obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3 have break_f : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := break_prod_twice f h4 have break_fs : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_prod_twice (f ∘ swap u n) h4 have f_eq_fs_below : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := swap_prod_eq_prod_below f h1 have f_eq_fs_btwn : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4 show prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f from calc prod_seq (n + 1) 0 (f ∘ swap u n) _ = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_fs _ = prod_seq u 0 f * (f ∘ swap u n) u * prod_seq j (u + 1) f * (f ∘ swap u n) n := by rw [f_eq_fs_below, f_eq_fs_btwn] _ = prod_seq u 0 f * f (swap u n u) * prod_seq j (u + 1) f * f (swap u n n) := by rfl _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by rw [swap_fst, swap_snd] _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring _ = prod_seq (n + 1) 0 f := break_f.symm done done lemma perm_below_fixed {n : Nat} {g : Nat → Nat} (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry lemma perm_prod {m : Nat} (f : Nat → ZMod m) : ∀ (n : Nat), ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) := by by_induc · -- Base Case fix g : Nat → Nat assume h1 : perm_below 0 g rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) fix g : Nat → Nat assume g_pb : perm_below (n + 1) g define at g_pb have g_ob : onto_below (n + 1) g := g_pb.right.right define at g_ob have h1 : n < n + 1 := by linarith obtain (u : Nat) (h2 : u < n + 1 ∧ g u = n) from g_ob n h1 have s_pb : perm_below (n + 1) (swap u n) := swap_perm_below h2.left h1 have gs_pb_n1 : perm_below (n + 1) (g ∘ swap u n) := comp_perm_below g_pb s_pb have gs_fix_n : (g ∘ swap u n) n = n := calc (g ∘ swap u n) n _ = g (swap u n n) := by rfl _ = g u := by rw [swap_snd] _ = n := h2.right have gs_pb_n : perm_below n (g ∘ swap u n) := perm_below_fixed gs_pb_n1 gs_fix_n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f ∘ (g ∘ swap u n)) := ih (g ∘ swap u n) gs_pb_n have h3 : u ≤ n := by linarith show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f ∘ g) from calc prod_seq (n + 1) 0 f _ = prod_seq n 0 f * f n := prod_seq_zero_step n f _ = prod_seq n 0 (f ∘ (g ∘ swap u n)) * f ((g ∘ swap u n) n) := by rw [gs_prod, gs_fix_n] _ = prod_seq n 0 (f ∘ g ∘ swap u n) * (f ∘ g ∘ swap u n) n := by rfl _ = prod_seq (n + 1) 0 (f ∘ g ∘ swap u n) := (prod_seq_zero_step n (f ∘ g ∘ swap u n)).symm _ = prod_seq (n + 1) 0 ((f ∘ g) ∘ swap u n) := by rfl _ = prod_seq (n + 1) 0 (f ∘ g) := swap_prod_eq_prod (f ∘ g) h3 done done lemma F_invertible (m i : Nat) : invertible (F m i) := by by_cases h : rel_prime m i · -- Case 1. h : rel_prime m i rewrite [F_rp_def h] show invertible [i]_m from (Theorem_7_3_7 m i).rtl h done · -- Case 2. h : ¬rel_prime m i rewrite [F_not_rp_def h] apply Exists.intro [1]_m show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m done done lemma Fprod_invertible (m : Nat) : ∀ (k : Nat), invertible (prod_seq k 0 (F m)) := by by_induc · -- Base Case apply Exists.intro [1]_m show prod_seq 0 0 (F m) * [1]_m = [1]_m from calc prod_seq 0 0 (F m) * [1]_m _ = [1]_m * [1]_m := by rw [prod_seq_base] _ = [1]_m := Theorem_7_3_6_7 ([1]_m) done · -- Induction Step fix k : Nat assume ih : invertible (prod_seq k 0 (F m)) rewrite [prod_seq_zero_step] show invertible (prod_seq k 0 (F m) * (F m k)) from (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k) done done theorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m ^ (phi m) = [1]_m := by have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2 show [a]_m ^ (phi m) = [1]_m from calc [a]_m ^ (phi m) _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3] _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring _ = prod_seq m 0 (F m ∘ G m a) * Y := by rw [FG_prod h1 m, phi_def] _ = prod_seq m 0 (F m) * Y := by rw [perm_prod (F m) m (G m a) (G_perm_below h1)] _ = [1]_m := by rw [h3] done lemma Exercise_7_4_5_Int (m : Nat) (a : Int) : ∀ (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry lemma Exercise_7_4_5_Nat (m a n : Nat) : [a]_m ^ n = [a ^ n]_m := by rewrite [Exercise_7_4_5_Int] rfl done

theorem Euler's_theorem {m a : Nat} [NeZero m] (h1 : rel_prime m a) : a ^ (phi m) ≡ 1 (MOD m)

HTPI.Euler's_theorem

htpi/HTPILib/Chap7.lean

htpi

/- Copyright 2023 Daniel J. Velleman -/ import HTPILib.Chap6 namespace HTPI /- Definitions -/ lemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by have h : n + 1 > 0 := Nat.succ_pos n show a % (n + 1) < n + 1 from Nat.mod_lt a h done def gcd (a b : Nat) : Nat := match b with | 0 => a | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd (n + 1) (a % (n + 1)) termination_by b mutual def gcd_c1 (a b : Nat) : Int := match b with | 0 => 1 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c2 (n + 1) (a % (n + 1)) --Corresponds to s = t' termination_by b def gcd_c2 (a b : Nat) : Int := match b with | 0 => 0 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c1 (n + 1) (a % (n + 1)) - (gcd_c2 (n + 1) (a % (n + 1))) * ↑(a / (n + 1)) --Corresponds to t = s' - t'q termination_by b end def prime (n : Nat) : Prop := 2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n def prime_factor (p n : Nat) : Prop := prime p ∧ p ∣ n def all_prime (l : List Nat) : Prop := ∀ p ∈ l, prime p def nondec (l : List Nat) : Prop := match l with | [] => True --Of course, True is a proposition that is always true | n :: L => (∀ m ∈ L, n ≤ m) ∧ nondec L def nondec_prime_list (l : List Nat) : Prop := all_prime l ∧ nondec l def prod (l : List Nat) : Nat := match l with | [] => 1 | n :: L => n * (prod L) def prime_factorization (n : Nat) (l : List Nat) : Prop := nondec_prime_list l ∧ prod l = n def rel_prime (a b : Nat) : Prop := gcd a b = 1 def congr_mod (m : Nat) (a b : Int) : Prop := (↑m : Int) ∣ (a - b) def cc (m : Nat) (a : Int) : ZMod m := (↑a : ZMod m) notation:50 a " ≡ " b " (MOD " m ")" => congr_mod m a b notation:max "["a"]_"m:max => cc m a def invertible {m : Nat} (X : ZMod m) : Prop := ∃ (Y : ZMod m), X * Y = [1]_m def num_rp_below (m k : Nat) : Nat := match k with | 0 => 0 | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j def phi (m : Nat) : Nat := num_rp_below m m def prod_seq {m : Nat} (j k : Nat) (f : Nat → ZMod m) : ZMod m := match j with | 0 => [1]_m | n + 1 => prod_seq n k f * f (k + n) def maps_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i < n, g i < n def one_one_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i1 < n, ∀ i2 < n, g i1 = g i2 → i1 = i2 def onto_below (n : Nat) (g : Nat → Nat) : Prop := ∀ k < n, ∃ i < n, g i = k def perm_below (n : Nat) (g : Nat → Nat) : Prop := maps_below n g ∧ one_one_below n g ∧ onto_below n g def inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m) def swap (u v i : Nat) : Nat := if i = u then v else if i = v then u else i namespace Euler --For definitions specific to Euler's theorem def F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m def G (m a i : Nat) : Nat := (a * i) % m def Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i end Euler /- Section 7.1 -/ theorem dvd_mod_of_dvd_a_b {a b d : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ (a % b) := by set q : Nat := a / b have h3 : b * q + a % b = a := Nat.div_add_mod a b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 define --Goal : ∃ (c : Nat), a % b = d * c apply Exists.intro (j - k * q) show a % b = d * (j - k * q) from calc a % b _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm _ = a - b * q := by rw [h3] _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc] _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm done theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry #eval gcd 672 161 --Answer: 7 lemma gcd_base (a : Nat) : gcd a 0 = a := by rfl lemma gcd_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd a b = gcd b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1)) rfl done lemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b ≠ 0) : a % b < b := by have h1 : b > 0 := Nat.pos_of_ne_zero h show a % b < b from Nat.mod_lt a h1 done lemma dvd_self (n : Nat) : n ∣ n := by apply Exists.intro 1 ring done theorem gcd_dvd : ∀ (b a : Nat), (gcd a b) ∣ a ∧ (gcd a b) ∣ b := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd a b_1) ∣ a ∧ (gcd a b_1) ∣ b_1 fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_base] --Goal: a ∣ a ∧ a ∣ 0 apply And.intro (dvd_self a) define apply Exists.intro 0 rfl done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_nonzero a h1] --Goal : gcd b (a % b) ∣ a ∧ gcd b (a % b) ∣ b have h2 : a % b < b := mod_nonzero_lt a h1 have h3 : (gcd b (a % b)) ∣ b ∧ (gcd b (a % b)) ∣ (a % b) := ih (a % b) h2 b apply And.intro _ h3.left show (gcd b (a % b)) ∣ a from dvd_a_of_dvd_b_mod h3.left h3.right done done theorem gcd_dvd_left (a b : Nat) : (gcd a b) ∣ a := (gcd_dvd b a).left theorem gcd_dvd_right (a b : Nat) : (gcd a b) ∣ b := (gcd_dvd b a).right lemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl lemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c1 a b = gcd_c2 b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done lemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl lemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * ↑(a / b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done theorem gcd_lin_comb : ∀ (b a : Nat), (gcd_c1 a b) * ↑a + (gcd_c2 a b) * ↑b = ↑(gcd a b) := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd_c1 a b_1) * ↑a + (gcd_c2 a b_1) * ↑b_1 = ↑(gcd a b_1) fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base] --Goal : 1 * ↑a + 0 * ↑0 = ↑a ring done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1] --Goal : gcd_c2 b (a % b) * ↑a + -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * ↑(a / b)) * ↑b = -- ↑(gcd b (a % b)) set r : Nat := a % b set q : Nat := a / b set s : Int := gcd_c1 b r set t : Int := gcd_c2 b r --Goal : t * ↑a + (s - t * ↑q) * ↑b = ↑(gcd b r) have h2 : r < b := mod_nonzero_lt a h1 have h3 : s * ↑b + t * ↑r = ↑(gcd b r) := ih r h2 b have h4 : b * q + r = a := Nat.div_add_mod a b rewrite [←h3, ←h4] rewrite [Nat.cast_add, Nat.cast_mul] --Goal : t * (↑b * ↑q + ↑r) + (s - t * ↑q) * ↑b = s * ↑b + t * ↑r ring done done #eval gcd_c1 672 161 --Answer: 6 #eval gcd_c2 672 161 --Answer: -25 --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161 theorem Theorem_7_1_6 {d a b : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ gcd a b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑d ∣ ↑(gcd a b) set s : Int := gcd_c1 a b set t : Int := gcd_c2 a b have h3 : s * ↑a + t * ↑b = ↑(gcd a b) := gcd_lin_comb b a rewrite [←h3] --Goal : ↑d ∣ s * ↑a + t * ↑b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul] --Goal : ↑d ∣ s * (↑d * ↑j) + t * (↑d * ↑k) define apply Exists.intro (s * ↑j + t * ↑k) ring done /- Section 7.2 -/ theorem dvd_trans {a b c : Nat} (h1 : a ∣ b) (h2 : b ∣ c) : a ∣ c := by define at h1; define at h2; define obtain (m : Nat) (h3 : b = a * m) from h1 obtain (n : Nat) (h4 : c = b * n) from h2 rewrite [h3, mul_assoc] at h4 apply Exists.intro (m * n) show c = a * (m * n) from h4 done lemma exists_prime_factor : ∀ (n : Nat), 2 ≤ n → ∃ (p : Nat), prime_factor p n := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, 2 ≤ n_1 → ∃ (p : Nat), prime_factor p n_1 assume h1 : 2 ≤ n by_cases h2 : prime n · -- Case 1. h2 : prime n apply Exists.intro n define --Goal : prime n ∧ n ∣ n show prime n ∧ n ∣ n from And.intro h2 (dvd_self n) done · -- Case 2. h2 : ¬prime n define at h2 --h2 : ¬(2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n) demorgan at h2 disj_syll h2 h1 obtain (a : Nat) (h3 : ∃ (b : Nat), a * b = n ∧ a < n ∧ b < n) from h2 obtain (b : Nat) (h4 : a * b = n ∧ a < n ∧ b < n) from h3 have h5 : 2 ≤ a := by by_contra h6 have h7 : a ≤ 1 := by linarith have h8 : n ≤ b := calc n _ = a * b := h4.left.symm _ ≤ 1 * b := by rel [h7] _ = b := by ring linarith --n ≤ b contradicts b < n done have h6 : ∃ (p : Nat), prime_factor p a := ih a h4.right.left h5 obtain (p : Nat) (h7 : prime_factor p a) from h6 apply Exists.intro p define --Goal : prime p ∧ p ∣ n define at h7 --h7 : prime p ∧ p ∣ a apply And.intro h7.left have h8 : a ∣ n := by apply Exists.intro b show n = a * b from (h4.left).symm done show p ∣ n from dvd_trans h7.right h8 done done lemma exists_least_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q := by set S : Set Nat := {p : Nat | prime_factor p n} have h2 : ∃ (p : Nat), p ∈ S := exists_prime_factor n h show ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q from well_ord_princ S h2 done lemma all_prime_nil : all_prime [] := by define --Goal : ∀ p ∈ [], prime p fix p : Nat contrapos --Goal : ¬prime p → p ∉ [] assume h1 : ¬prime p show p ∉ [] from List.not_mem_nil p done lemma all_prime_cons (n : Nat) (L : List Nat) : all_prime (n :: L) ↔ prime n ∧ all_prime L := by apply Iff.intro · -- (→) assume h1 : all_prime (n :: L) --Goal : prime n ∧ all_prime L define at h1 --h1 : ∀ p ∈ n :: L, prime p apply And.intro (h1 n (List.mem_cons_self n L)) define --Goal : ∀ p ∈ L, prime p fix p : Nat assume h2 : p ∈ L show prime p from h1 p (List.mem_cons_of_mem n h2) done · -- (←) assume h1 : prime n ∧ all_prime L --Goal : all_prime (n :: l) define : all_prime L at h1 define fix p : Nat assume h2 : p ∈ n :: L rewrite [List.mem_cons] at h2 --h2 : p = n ∨ p ∈ L by_cases on h2 · -- Case 1. h2 : p = n rewrite [h2] show prime n from h1.left done · -- Case 2. h2 : p ∈ L show prime p from h1.right p h2 done done done lemma nondec_nil : nondec [] := by define --Goal : True trivial --trivial proves some obviously true statements, such as True done lemma nondec_cons (n : Nat) (L : List Nat) : nondec (n :: L) ↔ (∀ m ∈ L, n ≤ m) ∧ nondec L := by rfl lemma prod_nil : prod [] = 1 := by rfl lemma prod_cons : prod (n :: L) = n * (prod L) := by rfl lemma exists_cons_of_length_eq_succ {A : Type} {l : List A} {n : Nat} (h : l.length = n + 1) : ∃ (a : A) (L : List A), l = a :: L ∧ L.length = n := by have h1 : ¬l.length = 0 := by linarith rewrite [List.length_eq_zero] at h1 obtain (a : A) (h2 : ∃ (L : List A), l = a :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List A) (h3 : l = a :: L) from h2 apply Exists.intro a apply Exists.intro L apply And.intro h3 have h4 : (a :: L).length = L.length + 1 := List.length_cons a L rewrite [←h3, h] at h4 show L.length = n from (Nat.add_right_cancel h4).symm done lemma list_elt_dvd_prod_by_length (a : Nat) : ∀ (n : Nat), ∀ (l : List Nat), l.length = n → a ∈ l → a ∣ prod l := by by_induc · --Base Case fix l : List Nat assume h1 : l.length = 0 rewrite [List.length_eq_zero] at h1 --h1 : l = [] rewrite [h1] --Goal : a ∈ [] → a ∣ prod [] contrapos assume h2 : ¬a ∣ prod [] show a ∉ [] from List.not_mem_nil a done · -- Induction Step fix n : Nat assume ih : ∀ (l : List Nat), List.length l = n → a ∈ l → a ∣ prod l fix l : List Nat assume h1 : l.length = n + 1 --Goal : a ∈ l → a ∣ prod l obtain (b : Nat) (h2 : ∃ (L : List Nat), l = b :: L ∧ L.length = n) from exists_cons_of_length_eq_succ h1 obtain (L : List Nat) (h3 : l = b :: L ∧ L.length = n) from h2 have h4 : a ∈ L → a ∣ prod L := ih L h3.right assume h5 : a ∈ l rewrite [h3.left, prod_cons] --Goal : a ∣ b * prod L rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b ∨ a ∈ L by_cases on h5 · -- Case 1. h5 : a = b apply Exists.intro (prod L) rewrite [h5] rfl done · -- Case 2. h5 : a ∈ L have h6 : a ∣ prod L := h4 h5 have h7 : prod L ∣ b * prod L := by apply Exists.intro b ring done show a ∣ b * prod L from dvd_trans h6 h7 done done done lemma list_elt_dvd_prod {a : Nat} {l : List Nat} (h : a ∈ l) : a ∣ prod l := by set n : Nat := l.length have h1 : l.length = n := by rfl show a ∣ prod l from list_elt_dvd_prod_by_length a n l h1 h done lemma exists_prime_factorization : ∀ (n : Nat), n ≥ 1 → ∃ (l : List Nat), prime_factorization n l := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, n_1 ≥ 1 → ∃ (l : List Nat), prime_factorization n_1 l assume h1 : n ≥ 1 by_cases h2 : n = 1 · -- Case 1. h2 : n = 1 apply Exists.intro [] define apply And.intro · -- Proof of nondec_prime_list [] define show all_prime [] ∧ nondec [] from And.intro all_prime_nil nondec_nil done · -- Proof of prod [] = n rewrite [prod_nil, h2] rfl done done · -- Case 2. h2 : n ≠ 1 have h3 : n ≥ 2 := lt_of_le_of_ne' h1 h2 obtain (p : Nat) (h4 : prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q) from exists_least_prime_factor h3 have p_prime_factor : prime_factor p n := h4.left define at p_prime_factor have p_prime : prime p := p_prime_factor.left have p_dvd_n : p ∣ n := p_prime_factor.right have p_least : ∀ (q : Nat), prime_factor q n → p ≤ q := h4.right obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n have h5 : m ≠ 0 := by contradict h1 with h6 have h7 : n = 0 := calc n _ = p * m := n_eq_pm _ = p * 0 := by rw [h6] _ = 0 := by ring rewrite [h7] decide done have m_pos : 0 < m := Nat.pos_of_ne_zero h5 have m_lt_n : m < n := by define at p_prime show m < n from calc m _ < m + m := by linarith _ = 2 * m := by ring _ ≤ p * m := by rel [p_prime.left] _ = n := n_eq_pm.symm done obtain (L : List Nat) (h6 : prime_factorization m L) from ih m m_lt_n m_pos define at h6 have ndpl_L : nondec_prime_list L := h6.left define at ndpl_L apply Exists.intro (p :: L) define apply And.intro · -- Proof of nondec_prime_list (p :: L) define apply And.intro · -- Proof of all_prime (p :: L) rewrite [all_prime_cons] show prime p ∧ all_prime L from And.intro p_prime ndpl_L.left done · -- Proof of nondec (p :: L) rewrite [nondec_cons] apply And.intro _ ndpl_L.right fix q : Nat assume q_in_L : q ∈ L have h7 : q ∣ prod L := list_elt_dvd_prod q_in_L rewrite [h6.right] at h7 --h7 : q ∣ m have h8 : m ∣ n := by apply Exists.intro p rewrite [n_eq_pm] ring done have q_dvd_n : q ∣ n := dvd_trans h7 h8 have ap_L : all_prime L := ndpl_L.left define at ap_L have q_prime_factor : prime_factor q n := And.intro (ap_L q q_in_L) q_dvd_n show p ≤ q from p_least q q_prime_factor done done · -- Proof of prod (p :: L) = n rewrite [prod_cons, h6.right, n_eq_pm] rfl done done done theorem Theorem_7_2_2 {a b c : Nat} (h1 : c ∣ a * b) (h2 : rel_prime a c) : c ∣ b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑c ∣ ↑b define at h1; define at h2; define obtain (j : Nat) (h3 : a * b = c * j) from h1 set s : Int := gcd_c1 a c set t : Int := gcd_c2 a c have h4 : s * ↑a + t * ↑c = ↑(gcd a c) := gcd_lin_comb c a rewrite [h2, Nat.cast_one] at h4 --h4 : s * ↑a + t * ↑c = (1 : Int) apply Exists.intro (s * ↑j + t * ↑b) show ↑b = ↑c * (s * ↑j + t * ↑b) from calc ↑b _ = (1 : Int) * ↑b := (one_mul _).symm _ = (s * ↑a + t * ↑c) * ↑b := by rw [h4] _ = s * (↑a * ↑b) + t * ↑c * ↑b := by ring _ = s * (↑c * ↑j) + t * ↑c * ↑b := by rw [←Nat.cast_mul a b, h3, Nat.cast_mul c j] _ = ↑c * (s * ↑j + t * ↑b) := by ring done lemma le_nonzero_prod_left {a b : Nat} (h : a * b ≠ 0) : a ≤ a * b := by have h1 : b ≠ 0 := by contradict h with h1 rewrite [h1] ring done have h2 : 1 ≤ b := Nat.pos_of_ne_zero h1 show a ≤ a * b from calc a = a * 1 := (mul_one a).symm _ ≤ a * b := by rel [h2] done lemma le_nonzero_prod_right {a b : Nat} (h : a * b ≠ 0) : b ≤ a * b := by rewrite [mul_comm] rewrite [mul_comm] at h show b ≤ b * a from le_nonzero_prod_left h done lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry lemma rel_prime_of_prime_not_dvd {a p : Nat} (h1 : prime p) (h2 : ¬p ∣ a) : rel_prime a p := by have h3 : gcd a p ∣ a := gcd_dvd_left a p have h4 : gcd a p ∣ p := gcd_dvd_right a p have h5 : gcd a p = 1 ∨ gcd a p = p := dvd_prime h1 h4 have h6 : gcd a p ≠ p := by contradict h2 with h6 rewrite [h6] at h3 show p ∣ a from h3 done disj_syll h5 h6 show rel_prime a p from h5 done theorem Theorem_7_2_3 {a b p : Nat} (h1 : prime p) (h2 : p ∣ a * b) : p ∣ a ∨ p ∣ b := by or_right with h3 have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3 show p ∣ b from Theorem_7_2_2 h2 h4 done lemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a ≥ 1 := by have h1 : a ≠ 0 := by by_contra h1 rewrite [h1] at h contradict h linarith done show a ≥ 1 from Nat.pos_of_ne_zero h1 done lemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by have h1 : a ≥ 1 := ge_one_of_prod_one h have h2 : a * b ≠ 0 := by linarith have h3 : a ≤ a * b := le_nonzero_prod_left h2 rewrite [h] at h3 show a = 1 from Nat.le_antisymm h3 h1 done lemma eq_one_of_dvd_one {n : Nat} (h : n ∣ 1) : n = 1 := by obtain (j : Nat) (h1 : 1 = n * j) from h show n = 1 from eq_one_of_prod_one h1.symm done lemma prime_not_one {p : Nat} (h : prime p) : p ≠ 1 := by define at h linarith done theorem Theorem_7_2_4 {p : Nat} (h1 : prime p) : ∀ (l : List Nat), p ∣ prod l → ∃ a ∈ l, p ∣ a := by apply List.rec · -- Base Case. Goal : p ∣ prod [] → ∃ a ∈ [], p ∣ a rewrite [prod_nil] assume h2 : p ∣ 1 show ∃ a ∈ [], p ∣ a from absurd (eq_one_of_dvd_one h2) (prime_not_one h1) done · -- Induction Step fix b : Nat fix L : List Nat assume ih : p ∣ prod L → ∃ a ∈ L, p ∣ a --Goal : p ∣ prod (b :: L) → ∃ a ∈ b :: L, p ∣ a assume h2 : p ∣ prod (b :: L) rewrite [prod_cons] at h2 have h3 : p ∣ b ∨ p ∣ prod L := Theorem_7_2_3 h1 h2 by_cases on h3 · -- Case 1. h3 : p ∣ b apply Exists.intro b show b ∈ b :: L ∧ p ∣ b from And.intro (List.mem_cons_self b L) h3 done · -- Case 2. h3 : p ∣ prod L obtain (a : Nat) (h4 : a ∈ L ∧ p ∣ a) from ih h3 apply Exists.intro a show a ∈ b :: L ∧ p ∣ a from And.intro (List.mem_cons_of_mem b h4.left) h4.right done done done lemma prime_in_list {p : Nat} {l : List Nat} (h1 : prime p) (h2 : all_prime l) (h3 : p ∣ prod l) : p ∈ l := by obtain (a : Nat) (h4 : a ∈ l ∧ p ∣ a) from Theorem_7_2_4 h1 l h3 define at h2 have h5 : prime a := h2 a h4.left have h6 : p = 1 ∨ p = a := dvd_prime h5 h4.right disj_syll h6 (prime_not_one h1) rewrite [h6] show a ∈ l from h4.left done lemma first_le_first {p q : Nat} {l m : List Nat} (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m)) (h3 : prod (p :: l) = prod (q :: m)) : p ≤ q := by define at h1; define at h2 have h4 : q ∣ prod (p :: l) := by define apply Exists.intro (prod m) rewrite [←prod_cons] show prod (p :: l) = prod (q :: m) from h3 done have h5 : all_prime (q :: m) := h2.left rewrite [all_prime_cons] at h5 have h6 : q ∈ p :: l := prime_in_list h5.left h1.left h4 have h7 : nondec (p :: l) := h1.right rewrite [nondec_cons] at h7 rewrite [List.mem_cons] at h6 by_cases on h6 · -- Case 1. h6 : q = p linarith done · -- Case 2. h6 : q ∈ l have h8 : ∀ m ∈ l, p ≤ m := h7.left show p ≤ q from h8 q h6 done done lemma nondec_prime_list_tail {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by define at h define rewrite [all_prime_cons, nondec_cons] at h show all_prime l ∧ nondec l from And.intro h.left.right h.right.right done lemma cons_prod_not_one {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : prod (p :: l) ≠ 1 := by define at h have h1 : all_prime (p :: l) := h.left rewrite [all_prime_cons] at h1 rewrite [prod_cons] by_contra h2 show False from (prime_not_one h1.left) (eq_one_of_prod_one h2) done lemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) : l = [] ↔ prod l = 1 := by apply Iff.intro · -- (→) assume h1 : l = [] rewrite [h1] show prod [] = 1 from prod_nil done · -- (←) contrapos assume h1 : ¬l = [] obtain (p : Nat) (h2 : ∃ (L : List Nat), l = p :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List Nat) (h3 : l = p :: L) from h2 rewrite [h3] at h rewrite [h3] show ¬prod (p :: L) = 1 from cons_prod_not_one h done done lemma prime_pos {p : Nat} (h : prime p) : p > 0 := by define at h linarith done theorem Theorem_7_2_5 : ∀ (l1 l2 : List Nat), nondec_prime_list l1 → nondec_prime_list l2 → prod l1 = prod l2 → l1 = l2 := by apply List.rec · -- Base Case. Goal : ∀ (l2 : List Nat), nondec_prime_list [] → -- nondec_prime_list l2 → prod [] = prod l2 → [] = l2 fix l2 : List Nat assume h1 : nondec_prime_list [] assume h2 : nondec_prime_list l2 assume h3 : prod [] = prod l2 rewrite [prod_nil, eq_comm, ←list_nil_iff_prod_one h2] at h3 show [] = l2 from h3.symm done · -- Induction Step fix p : Nat fix L1 : List Nat assume ih : ∀ (L2 : List Nat), nondec_prime_list L1 → nondec_prime_list L2 → prod L1 = prod L2 → L1 = L2 -- Goal : ∀ (l2 : List Nat), nondec_prime_list (p :: L1) → -- nondec_prime_list l2 → prod (p :: L1) = prod l2 → p :: L1 = l2 fix l2 : List Nat assume h1 : nondec_prime_list (p :: L1) assume h2 : nondec_prime_list l2 assume h3 : prod (p :: L1) = prod l2 have h4 : ¬prod (p :: L1) = 1 := cons_prod_not_one h1 rewrite [h3, ←list_nil_iff_prod_one h2] at h4 obtain (q : Nat) (h5 : ∃ (L : List Nat), l2 = q :: L) from List.exists_cons_of_ne_nil h4 obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5 rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2) rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2) have h7 : p ≤ q := first_le_first h1 h2 h3 have h8 : q ≤ p := first_le_first h2 h1 h3.symm have h9 : p = q := by linarith rewrite [h9, prod_cons, prod_cons] at h3 --h3 : q * prod L1 = q * prod L2 have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1 have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2 define at h2 have h12 : all_prime (q :: L2) := h2.left rewrite [all_prime_cons] at h12 have h13 : q > 0 := prime_pos h12.left have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3 have h15 : L1 = L2 := ih L2 h10 h11 h14 rewrite [h6, h9, h15] rfl done done theorem fund_thm_arith (n : Nat) (h : n ≥ 1) : ∃! (l : List Nat), prime_factorization n l := by exists_unique · -- Existence show ∃ (l : List Nat), prime_factorization n l from exists_prime_factorization n h done · -- Uniqueness fix l1 : List Nat; fix l2 : List Nat assume h1 : prime_factorization n l1 assume h2 : prime_factorization n l2 define at h1; define at h2 have h3 : prod l1 = n := h1.right rewrite [←h2.right] at h3 show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3 done done /- Section 7.3 -/ theorem congr_refl (m : Nat) : ∀ (a : Int), a ≡ a (MOD m) := by fix a : Int define --Goal : ∃ (c : Int), a - a = ↑m * c apply Exists.intro 0 ring done theorem congr_symm {m : Nat} : ∀ {a b : Int}, a ≡ b (MOD m) → b ≡ a (MOD m) := by fix a : Int; fix b : Int assume h1 : a ≡ b (MOD m) define at h1 --h1 : ∃ (c : Int), a - b = ↑m * c define --Goal : ∃ (c : Int), b - a = ↑m * c obtain (c : Int) (h2 : a - b = m * c) from h1 apply Exists.intro (-c) show b - a = m * (-c) from calc b - a _ = -(a - b) := by ring _ = -(m * c) := by rw [h2] _ = m * (-c) := by ring done theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry /- Fundamental properties of congruence classes -/ lemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) : X = Y ↔ X.val = Y.val := Fin.ext_iff lemma val_nat_eq_mod (n k : Nat) : ([k]_(n + 1)).val = k % (n + 1) := by rfl lemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl theorem cc_rep {m : Nat} (X : ZMod m) : ∃ (a : Int), X = [a]_m := match m with | 0 => by apply Exists.intro X rfl done | n + 1 => by apply Exists.intro ↑(X.val) have h1 : X.val < n + 1 := Fin.prop X rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1] rfl done theorem add_class (m : Nat) (a b : Int) : [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm theorem mul_class (m : Nat) (a b : Int) : [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm lemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ [a - b]_m = [0]_m := by apply Iff.intro · -- (→) assume h1 : [a]_m = [b]_m have h2 : a - b = a + (-b) := by ring have h3 : b + (-b) = 0 := by ring show [a - b]_m = [0]_m from calc [a - b]_m _ = [a + (-b)]_m := by rw [h2] _ = [a]_m + [-b]_m := by rw [add_class] _ = [b]_m + [-b]_m := by rw [h1] _ = [b + -b]_m := by rw [add_class] _ = [0]_m := by rw [h3] done · -- (←) assume h1 : [a - b]_m = [0]_m have h2 : b + (a - b) = a := by ring have h3 : b + 0 = b := by ring show [a]_m = [b]_m from calc [a]_m _ = [b + (a - b)]_m := by rw [h2] _ = [b]_m + [a - b]_m := by rw [add_class] _ = [b]_m + [0]_m := by rw [h1] _ = [b + 0]_m := by rw [add_class] _ = [b]_m := by rw [h3] done done lemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) : [a]_m = [0]_m → [-a]_m = [0]_m := by assume h1 : [a]_m = [0]_m have h2 : 0 + (-a) = -a := by ring have h3 : a + (-a) = 0 := by ring show [-a]_m = [0]_m from calc [-a]_m _ = [0 + (-a)]_m := by rw [h2] _ = [0]_m + [-a]_m := by rw [add_class] _ = [a]_m + [-a]_m := by rw [h1] _ = [a + (-a)]_m := by rw [add_class] _ = [0]_m := by rw [h3] done lemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) : [-a]_m = [0]_m ↔ [a]_m = [0]_m := by apply Iff.intro _ (cc_neg_zero_of_cc_zero m a) assume h1 : [-a]_m = [0]_m have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1 have h3 : -(-a) = a := by ring rewrite [h3] at h2 show [a]_m = [0]_m from h2 done lemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl lemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m ↔ m ∣ k := match m with | 0 => by have h : (0 : Int) = (↑(0 : Nat) : Int) := by rfl rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj] apply Iff.intro · -- (→) assume h1 : k = 0 rewrite [h1] show 0 ∣ 0 from dvd_self 0 done · -- (←) assume h1 : 0 ∣ k obtain (c : Nat) (h2 : k = 0 * c) from h1 rewrite [h2] ring done done | n + 1 => by rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero] show k % (n + 1) = 0 ↔ n + 1 ∣ k from (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm done lemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m ↔ ↑m ∣ a := by obtain (k : Nat) (h1 : a = ↑k ∨ a = -↑k) from Int.eq_nat_or_neg a by_cases on h1 · -- Case 1. h1: a = ↑k rewrite [h1, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done · -- Case 2. h1: a = -↑k rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done done theorem cc_eq_iff_congr (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ a ≡ b (MOD m) := calc [a]_m = [b]_m _ ↔ [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b _ ↔ ↑m ∣ (a - b) := cc_zero_iff_dvd m (a - b) _ ↔ a ≡ b (MOD m) := by rfl /- End of fundamental properties of congruence classes -/ lemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m := by have h1 : (↑m : Int) ≠ 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m) show 0 ≤ a % m from Int.emod_nonneg a h1 done lemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m) have h2 : (↑m : Int) > 0 := (Nat.cast_pos).rtl h1 show a % m < m from Int.emod_lt_of_pos a h2 done lemma congr_mod_mod (m : Nat) (a : Int) : a ≡ a % m (MOD m) := by define have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m apply Exists.intro (a / m) show a - a % m = m * (a / m) from calc a - (a % m) _ = m * (a / m) + a % m - a % m := by rw [h1] _ = m * (a / m) := by ring done lemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) := And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a)) theorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) : ∃! (r : Int), 0 ≤ r ∧ r < m ∧ a ≡ r (MOD m) := by exists_unique · -- Existence apply Exists.intro (a % m) show 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) from mod_cmpl_res m a done · -- Uniqueness fix r1 : Int; fix r2 : Int assume h1 : 0 ≤ r1 ∧ r1 < m ∧ a ≡ r1 (MOD m) assume h2 : 0 ≤ r2 ∧ r2 < m ∧ a ≡ r2 (MOD m) have h3 : r1 ≡ r2 (MOD m) := congr_trans (congr_symm h1.right.right) h2.right.right obtain (d : Int) (h4 : r1 - r2 = m * d) from h3 have h5 : r1 - r2 < m * 1 := by linarith have h6 : m * (-1) < r1 - r2 := by linarith rewrite [h4] at h5 --h5 : m * d < m * 1 rewrite [h4] at h6 --h6 : m * -1 < m * d have h7 : (↑m : Int) ≥ 0 := Nat.cast_nonneg m have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7 have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7 have h10 : d = 0 := by linarith show r1 = r2 from calc r1 _ = r1 - r2 + r2 := by ring _ = m * 0 + r2 := by rw [h4, h10] _ = r2 := by ring done done lemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m := (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a) theorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y rewrite [h1, h2] have h3 : a + b = b + a := by ring show [a]_m + [b]_m = [b]_m + [a]_m from calc [a]_m + [b]_m _ = [a + b]_m := add_class m a b _ = [b + a]_m := by rw [h3] _ = [b]_m + [a]_m := (add_class m b a).symm done theorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X rewrite [h1] have h2 : a * 1 = a := by ring show [a]_m * [1]_m = [a]_m from calc [a]_m * [1]_m _ = [a * 1]_m := mul_class m a 1 _ = [a]_m := by rw [h2] done theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry lemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) : [a]_m * [gcd_c2 m a]_m = [1]_m := by set s : Int := gcd_c1 m a have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m define at h1 rewrite [h1, Nat.cast_one] at h2 --h2 : s * ↑m + gcd_c2 m a * ↑a = 1 rewrite [mul_class, cc_eq_iff_congr] define --Goal : ∃ (c : Int), ↑a * gcd_c2 m a - 1 = ↑m * c apply Exists.intro (-s) show a * (gcd_c2 m a) - 1 = m * (-s) from calc a * (gcd_c2 m a) - 1 _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring _ = 1 + m * (-s) - 1 := by rw [h2] _ = m * (-s) := by ring done theorem Theorem_7_3_7 (m a : Nat) : invertible [a]_m ↔ rel_prime m a := by apply Iff.intro · -- (→) assume h1 : invertible [a]_m define at h1 obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1 obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y rewrite [h3, mul_class, cc_eq_iff_congr] at h2 define at h2 obtain (c : Int) (h4 : a * b - 1 = m * c) from h2 rewrite [Exercise_7_2_6] --Goal : ∃ (s t : Int), s * ↑m + t * ↑a = 1 apply Exists.intro (-c) apply Exists.intro b show (-c) * m + b * a = 1 from calc (-c) * m + b * a _ = (-c) * m + (a * b - 1) + 1 := by ring _ = (-c) * m + m * c + 1 := by rw [h4] _ = 1 := by ring done · -- (←) assume h1 : rel_prime m a define show ∃ (Y : ZMod m), [a]_m * Y = [1]_m from Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1) done done /- Section 7.4 -/ section Euler open Euler lemma num_rp_below_base {m : Nat} : num_rp_below m 0 = 0 := by rfl lemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) : num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by have h1 : num_rp_below m (j + 1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : gcd m j = 1 rewrite [if_pos h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1 show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1 done lemma num_rp_below_step_not_rp {m j : Nat} (h : ¬rel_prime m j) : num_rp_below m (j + 1) = num_rp_below m j := by have h1 : num_rp_below m (j +1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : ¬gcd m j = 1 rewrite [if_neg h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j show num_rp_below m (j + 1) = num_rp_below m j from h1 done lemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl #eval phi 10 --Answer: 4 lemma prod_inv_iff_inv {m : Nat} {X : ZMod m} (h1 : invertible X) (Y : ZMod m) : invertible (X * Y) ↔ invertible Y := by apply Iff.intro · -- (→) assume h2 : invertible (X * Y) obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2 apply Exists.intro (X * Z) rewrite [←h3] --Goal : Y * (X * Z) = X * Y * Z ring --Note that ring can do algebra in ZMod m done · -- (←) assume h2 : invertible Y obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1 obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2 apply Exists.intro (Xi * Yi) show (X * Y) * (Xi * Yi) = [1]_m from calc X * Y * (Xi * Yi) _ = (X * Xi) * (Y * Yi) := by ring _ = [1]_m * [1]_m := by rw [h3, h4] _ = [1]_m := Theorem_7_3_6_7 [1]_m done done lemma F_rp_def {m i : Nat} (h : rel_prime m i) : F m i = [i]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h --h : gcd m i = 1 rewrite [if_pos h] at h1 show F m i = [i]_m from h1 done lemma F_not_rp_def {m i : Nat} (h : ¬rel_prime m i) : F m i = [1]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h rewrite [h1, if_neg h] rfl done lemma prod_seq_base {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 0 k f = [1]_m := by rfl lemma prod_seq_step {m : Nat} (n k : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl lemma prod_seq_zero_step {m : Nat} (n : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by rewrite [prod_seq_step, zero_add] rfl done lemma prod_one {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 1 k f = f k := by rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7] rfl done lemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl lemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m := calc [G m a i]_m _ = [(a * i) % m]_m := by rfl _ = [a * i]_m := (cc_eq_mod m (a * i)).symm _ = [a]_m * [i]_m := (mul_class m a i).symm lemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) : rel_prime m (G m a i) ↔ rel_prime m i := by have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1 show rel_prime m (G m a i) ↔ rel_prime m i from calc rel_prime m (G m a i) _ ↔ invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm _ ↔ invertible ([a]_m * [i]_m) := by rw [cc_G] _ ↔ invertible [i]_m := prod_inv_iff_inv h2 ([i]_m) _ ↔ rel_prime m i := Theorem_7_3_7 m i done lemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) : F m (G m a i) = [a]_m * F m i := by have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2 show F m (G m a i) = [a]_m * F m i from calc F m (G m a i) _ = [G m a i]_m := F_rp_def h3 _ = [a]_m * [i]_m := cc_G m a i _ = [a]_m * F m i := by rw [F_rp_def h2] done lemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : ¬rel_prime m i) : F m (G m a i) = [1]_m := by rewrite [←G_rp_iff h1 i] at h2 show F m (G m a i) = [1]_m from F_not_rp_def h2 done lemma FG_prod {m a : Nat} (h1 : rel_prime m a) : ∀ (k : Nat), prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by by_induc · -- Base Case show prod_seq 0 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from calc prod_seq 0 0 ((F m) ∘ (G m a)) _ = [1]_m := prod_seq_base _ _ _ = [a]_m ^ 0 * [1]_m := by ring _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by rw [num_rp_below_base, prod_seq_base] done · -- Induction Step fix k : Nat assume ih : prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) by_cases h2 : rel_prime m k · -- Case 1. h2 : rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([a]_m * F m k) := by rw [FG_rp h1 h2] _ = [a]_m ^ ((num_rp_below m k) + 1) * ((prod_seq k 0 (F m)) * F m k) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_rp h2, prod_seq_zero_step] done · -- Case 2. h2 : ¬rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([1]_m) := by rw [FG_not_rp h1 h2] _ = [a]_m ^ (num_rp_below m k) * (prod_seq k 0 (F m) * ([1]_m)) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_not_rp h2, prod_seq_zero_step, F_not_rp_def h2] done done done lemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by define --Goal : ∀ i < m, G m a i < m fix i : Nat assume h1 : i < m rewrite [G_def] --Goal : a * i % m < m show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m) done lemma left_inv_one_one_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g' (g i) = i) : one_one_below n g := sorry lemma right_inv_onto_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g (g' i) = i) (h2 : maps_below n g') : onto_below n g := by define at h2; define fix k : Nat assume h3 : k < n apply Exists.intro (g' k) show g' k < n ∧ g (g' k) = k from And.intro (h2 k h3) (h1 k h3) done lemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m * [inv_mod m a]_m = [1]_m := by have h2 : 0 ≤ (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a) show [a]_m * [inv_mod m a]_m = [1]_m from calc [a]_m * [inv_mod m a]_m _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2] _ = [a]_m * [gcd_c2 m a]_m := by rw [←cc_eq_mod] _ = [1]_m := gcd_c2_inv h1 done lemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m := calc a * (b % m) % m = a % m * (b % m % m) % m := Nat.mul_mod _ _ _ _ = a % m * (b % m) % m := by rw [Nat.mod_mod] _ = a * b % m := (Nat.mul_mod _ _ _).symm lemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm] rfl done theorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] : ↑a ≡ ↑b (MOD m) ↔ a % m = b % m := sorry lemma mul_inv_mod_cancel {m a i : Nat} [NeZero m] (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1 rewrite [mul_class, cc_eq_iff_congr, ←Nat.cast_mul, ←Nat.cast_one, congr_iff_mod_eq_Nat] at h3 show a * inv_mod m a * i % m = i from calc a * (inv_mod m a) * i % m _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = 1 % m * i % m := by rw [h3] _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = i % m := by rw [one_mul] _ = i := Nat.mod_eq_of_lt h2 done lemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl lemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, G m a (Ginv m a i) = i := by fix i : Nat assume h2 : i < m show G m a (Ginv m a i) = i from calc G m a (Ginv m a i) _ = a * ((inv_mod m a * i) % m) % m := by rfl _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, Ginv m a (G m a i) = i := by fix i : Nat assume h2 : i < m show Ginv m a (G m a i) = i from calc Ginv m a (G m a i) _ = inv_mod m a * ((a * i) % m) % m := by rfl _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc, mul_comm (inv_mod m a)] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_maps_below (m a : Nat) [NeZero m] : maps_below m (Ginv m a) := G_maps_below m (inv_mod m a) lemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : one_one_below m (G m a) := left_inv_one_one_below (Ginv_left_inv h1) lemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : onto_below m (G m a) := right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a) lemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : perm_below m (G m a) := And.intro (G_maps_below m a) (And.intro (G_one_one_below h1) (G_onto_below h1)) --Permuting a product of congruence classes doesn't change product lemma swap_fst (u v : Nat) : swap u v u = v := by define : swap u v u --Goal : (if u = u then v else if u = v then u else u) = v have h : u = u := by rfl rewrite [if_pos h] rfl done lemma swap_snd (u v : Nat) : swap u v v = u := by define : swap u v v by_cases h1 : v = u · -- Case 1. h1 : v = u rewrite [if_pos h1] show v = u from h1 done · -- Case 2. h1 : v ≠ u rewrite [if_neg h1] have h2 : v = v := by rfl rewrite [if_pos h2] rfl done done lemma swap_other {u v i : Nat} (h1 : i ≠ u) (h2 : i ≠ v) : swap u v i = i := by define : swap u v i rewrite [if_neg h1, if_neg h2] rfl done lemma swap_values (u v i : Nat) : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := by by_cases h1 : i = u · -- Case 1. h1 : i = u apply Or.inl rewrite [h1] show swap u v u = v from swap_fst u v done · -- Case 2. h1 : i ≠ u apply Or.inr by_cases h2 : i = v · -- Case 2.1. h2 : i = v apply Or.inl rewrite [h2] show swap u v v = u from swap_snd u v done · -- Case 2.2. h2 : i ≠ v apply Or.inr show swap u v i = i from swap_other h1 h2 done done done lemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by define fix i : Nat assume h3 : i < n have h4 : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := swap_values u v i by_cases on h4 · -- Case 1. h4 : swap u v i = v rewrite [h4] show v < n from h2 done · -- Case 2. by_cases on h4 · -- Case 2.1. h4 : swap u v i = u rewrite [h4] show u < n from h1 done · -- Case 2.2. h4 : swap u v i = i rewrite [h4] show i < n from h3 done done done lemma swap_swap (u v n : Nat) : ∀ i < n, swap u v (swap u v i) = i := by fix i : Nat assume h : i < n by_cases h1 : i = u · -- Case 1. h1 : i = u rewrite [h1, swap_fst, swap_snd] rfl done · -- Case 2. h1 : i ≠ u by_cases h2 : i = v · -- Case 2.1. h2 : i = v rewrite [h2, swap_snd, swap_fst] rfl done · -- Case 2.2. h2 : i ≠ v rewrite [swap_other h1 h2, swap_other h1 h2] rfl done done done lemma swap_one_one_below (u v n) : one_one_below n (swap u v) := left_inv_one_one_below (swap_swap u v n) lemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) := right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2) lemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) := And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2)) lemma comp_perm_below {n : Nat} {f g : Nat → Nat} (h1 : perm_below n f) (h2 : perm_below n g) : perm_below n (f ∘ g) := sorry lemma trivial_swap (u : Nat) : swap u u = id := by apply funext fix x : Nat by_cases h1 : x = u · -- Case 1. h1 : x = u rewrite [h1, swap_fst] rfl done · -- Case 2. h1 : x ≠ u rewrite [swap_other h1 h1] rfl done done lemma prod_eq_fun {m : Nat} (f g : Nat → ZMod m) (k : Nat) : ∀ (n : Nat), (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g := by by_induc · -- Base Case assume h : (∀ i < 0, f (k + i) = g (k + i)) rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g assume h1 : ∀ i < n + 1, f (k + i) = g (k + i) have h2 : ∀ i < n, f (k + i) = g (k + i) := by fix i : Nat assume h2 : i < n have h3 : i < n + 1 := by linarith show f (k + i) = g (k + i) from h1 i h3 done have h3 : prod_seq n k f = prod_seq n k g := ih h2 have h4 : n < n + 1 := Nat.lt_succ_self n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4] rfl done done lemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := by have h2 : ∀ (i : Nat), i < u → (f ∘ swap u n) (0 + i) = f (0 + i) := by fix i : Nat assume h2 : i < u have h3 : 0 + i ≠ u := by linarith have h4 : 0 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl done show prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f from prod_eq_fun (f ∘ swap u n) f 0 u h2 done lemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := by have h2 : ∀ i < j, (f ∘ swap u n) (u + 1 + i) = f (u + 1 + i) := by fix i : Nat assume h2 : i < j have h3 : u + 1 + i ≠ u := by linarith have h4 : u + 1 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl show prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f from prod_eq_fun (f ∘ swap u n) f (u + 1) j h2 done lemma break_prod {m : Nat} (n : Nat) (f : Nat → ZMod m) : ∀ (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by by_induc · -- Base Case have h : n + 0 = n := by rfl rewrite [prod_seq_base, h, Theorem_7_3_6_7] rfl done · -- Induction Step fix j : Nat assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f rewrite [←add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc] rfl done done lemma break_prod_twice {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f := break_prod n f 1 rewrite [prod_one] at h2 have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f := break_prod (u + 1) f j rewrite [←h1] at h3 have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f := break_prod u f 1 rewrite [prod_one] at h4 rewrite [h3, h4] at h2 show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2 done lemma swap_prod_eq_prod {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f := by by_cases h2 : u = n · -- Case 1. h2 : u = n rewrite [h2, trivial_swap n] --Goal : prod_seq (n + 1) 0 (f ∘ id) = prod_seq (n + 1) 0 f rfl done · -- Case 2. h2 : ¬u = n have h3 : u + 1 ≤ n := Nat.lt_of_le_of_ne h1 h2 obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3 have break_f : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := break_prod_twice f h4 have break_fs : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_prod_twice (f ∘ swap u n) h4 have f_eq_fs_below : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := swap_prod_eq_prod_below f h1 have f_eq_fs_btwn : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4 show prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f from calc prod_seq (n + 1) 0 (f ∘ swap u n) _ = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_fs _ = prod_seq u 0 f * (f ∘ swap u n) u * prod_seq j (u + 1) f * (f ∘ swap u n) n := by rw [f_eq_fs_below, f_eq_fs_btwn] _ = prod_seq u 0 f * f (swap u n u) * prod_seq j (u + 1) f * f (swap u n n) := by rfl _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by rw [swap_fst, swap_snd] _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring _ = prod_seq (n + 1) 0 f := break_f.symm done done lemma perm_below_fixed {n : Nat} {g : Nat → Nat} (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry lemma perm_prod {m : Nat} (f : Nat → ZMod m) : ∀ (n : Nat), ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) := by by_induc · -- Base Case fix g : Nat → Nat assume h1 : perm_below 0 g rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) fix g : Nat → Nat assume g_pb : perm_below (n + 1) g define at g_pb have g_ob : onto_below (n + 1) g := g_pb.right.right define at g_ob have h1 : n < n + 1 := by linarith obtain (u : Nat) (h2 : u < n + 1 ∧ g u = n) from g_ob n h1 have s_pb : perm_below (n + 1) (swap u n) := swap_perm_below h2.left h1 have gs_pb_n1 : perm_below (n + 1) (g ∘ swap u n) := comp_perm_below g_pb s_pb have gs_fix_n : (g ∘ swap u n) n = n := calc (g ∘ swap u n) n _ = g (swap u n n) := by rfl _ = g u := by rw [swap_snd] _ = n := h2.right have gs_pb_n : perm_below n (g ∘ swap u n) := perm_below_fixed gs_pb_n1 gs_fix_n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f ∘ (g ∘ swap u n)) := ih (g ∘ swap u n) gs_pb_n have h3 : u ≤ n := by linarith show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f ∘ g) from calc prod_seq (n + 1) 0 f _ = prod_seq n 0 f * f n := prod_seq_zero_step n f _ = prod_seq n 0 (f ∘ (g ∘ swap u n)) * f ((g ∘ swap u n) n) := by rw [gs_prod, gs_fix_n] _ = prod_seq n 0 (f ∘ g ∘ swap u n) * (f ∘ g ∘ swap u n) n := by rfl _ = prod_seq (n + 1) 0 (f ∘ g ∘ swap u n) := (prod_seq_zero_step n (f ∘ g ∘ swap u n)).symm _ = prod_seq (n + 1) 0 ((f ∘ g) ∘ swap u n) := by rfl _ = prod_seq (n + 1) 0 (f ∘ g) := swap_prod_eq_prod (f ∘ g) h3 done done lemma F_invertible (m i : Nat) : invertible (F m i) := by by_cases h : rel_prime m i · -- Case 1. h : rel_prime m i rewrite [F_rp_def h] show invertible [i]_m from (Theorem_7_3_7 m i).rtl h done · -- Case 2. h : ¬rel_prime m i rewrite [F_not_rp_def h] apply Exists.intro [1]_m show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m done done lemma Fprod_invertible (m : Nat) : ∀ (k : Nat), invertible (prod_seq k 0 (F m)) := by by_induc · -- Base Case apply Exists.intro [1]_m show prod_seq 0 0 (F m) * [1]_m = [1]_m from calc prod_seq 0 0 (F m) * [1]_m _ = [1]_m * [1]_m := by rw [prod_seq_base] _ = [1]_m := Theorem_7_3_6_7 ([1]_m) done · -- Induction Step fix k : Nat assume ih : invertible (prod_seq k 0 (F m)) rewrite [prod_seq_zero_step] show invertible (prod_seq k 0 (F m) * (F m k)) from (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k) done done theorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m ^ (phi m) = [1]_m := by have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2 show [a]_m ^ (phi m) = [1]_m from calc [a]_m ^ (phi m) _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3] _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring _ = prod_seq m 0 (F m ∘ G m a) * Y := by rw [FG_prod h1 m, phi_def] _ = prod_seq m 0 (F m) * Y := by rw [perm_prod (F m) m (G m a) (G_perm_below h1)] _ = [1]_m := by rw [h3] done lemma Exercise_7_4_5_Int (m : Nat) (a : Int) : ∀ (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry lemma Exercise_7_4_5_Nat (m a n : Nat) : [a]_m ^ n = [a ^ n]_m := by rewrite [Exercise_7_4_5_Int] rfl done theorem Euler's_theorem {m a : Nat} [NeZero m] (h1 : rel_prime m a) : a ^ (phi m) ≡ 1 (MOD m) := by have h2 : [a]_m ^ (phi m) = [1]_m := Theorem_7_4_2 h1 rewrite [Exercise_7_4_5_Nat m a (phi m)] at h2 --h2 : [a ^ phi m]_m = [1]_m show a ^ (phi m) ≡ 1 (MOD m) from (cc_eq_iff_congr _ _ _).ltr h2 done #eval gcd 10 7 --Answer: 1. So 10 and 7 are relatively prime #eval 7 ^ phi 10 --Answer: 2401, which is congruent to 1 mod 10. end Euler /- Section 7.5 -/

lemma num_rp_prime {p : Nat} (h1 : prime p) : ∀ k < p, num_rp_below p (k + 1) = k

HTPI.num_rp_prime

htpi/HTPILib/Chap7.lean

htpi

/- Copyright 2023 Daniel J. Velleman -/ import HTPILib.Chap6 namespace HTPI /- Definitions -/ lemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by have h : n + 1 > 0 := Nat.succ_pos n show a % (n + 1) < n + 1 from Nat.mod_lt a h done def gcd (a b : Nat) : Nat := match b with | 0 => a | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd (n + 1) (a % (n + 1)) termination_by b mutual def gcd_c1 (a b : Nat) : Int := match b with | 0 => 1 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c2 (n + 1) (a % (n + 1)) --Corresponds to s = t' termination_by b def gcd_c2 (a b : Nat) : Int := match b with | 0 => 0 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c1 (n + 1) (a % (n + 1)) - (gcd_c2 (n + 1) (a % (n + 1))) * ↑(a / (n + 1)) --Corresponds to t = s' - t'q termination_by b end def prime (n : Nat) : Prop := 2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n def prime_factor (p n : Nat) : Prop := prime p ∧ p ∣ n def all_prime (l : List Nat) : Prop := ∀ p ∈ l, prime p def nondec (l : List Nat) : Prop := match l with | [] => True --Of course, True is a proposition that is always true | n :: L => (∀ m ∈ L, n ≤ m) ∧ nondec L def nondec_prime_list (l : List Nat) : Prop := all_prime l ∧ nondec l def prod (l : List Nat) : Nat := match l with | [] => 1 | n :: L => n * (prod L) def prime_factorization (n : Nat) (l : List Nat) : Prop := nondec_prime_list l ∧ prod l = n def rel_prime (a b : Nat) : Prop := gcd a b = 1 def congr_mod (m : Nat) (a b : Int) : Prop := (↑m : Int) ∣ (a - b) def cc (m : Nat) (a : Int) : ZMod m := (↑a : ZMod m) notation:50 a " ≡ " b " (MOD " m ")" => congr_mod m a b notation:max "["a"]_"m:max => cc m a def invertible {m : Nat} (X : ZMod m) : Prop := ∃ (Y : ZMod m), X * Y = [1]_m def num_rp_below (m k : Nat) : Nat := match k with | 0 => 0 | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j def phi (m : Nat) : Nat := num_rp_below m m def prod_seq {m : Nat} (j k : Nat) (f : Nat → ZMod m) : ZMod m := match j with | 0 => [1]_m | n + 1 => prod_seq n k f * f (k + n) def maps_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i < n, g i < n def one_one_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i1 < n, ∀ i2 < n, g i1 = g i2 → i1 = i2 def onto_below (n : Nat) (g : Nat → Nat) : Prop := ∀ k < n, ∃ i < n, g i = k def perm_below (n : Nat) (g : Nat → Nat) : Prop := maps_below n g ∧ one_one_below n g ∧ onto_below n g def inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m) def swap (u v i : Nat) : Nat := if i = u then v else if i = v then u else i namespace Euler --For definitions specific to Euler's theorem def F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m def G (m a i : Nat) : Nat := (a * i) % m def Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i end Euler /- Section 7.1 -/ theorem dvd_mod_of_dvd_a_b {a b d : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ (a % b) := by set q : Nat := a / b have h3 : b * q + a % b = a := Nat.div_add_mod a b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 define --Goal : ∃ (c : Nat), a % b = d * c apply Exists.intro (j - k * q) show a % b = d * (j - k * q) from calc a % b _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm _ = a - b * q := by rw [h3] _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc] _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm done theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry #eval gcd 672 161 --Answer: 7 lemma gcd_base (a : Nat) : gcd a 0 = a := by rfl lemma gcd_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd a b = gcd b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1)) rfl done lemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b ≠ 0) : a % b < b := by have h1 : b > 0 := Nat.pos_of_ne_zero h show a % b < b from Nat.mod_lt a h1 done lemma dvd_self (n : Nat) : n ∣ n := by apply Exists.intro 1 ring done theorem gcd_dvd : ∀ (b a : Nat), (gcd a b) ∣ a ∧ (gcd a b) ∣ b := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd a b_1) ∣ a ∧ (gcd a b_1) ∣ b_1 fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_base] --Goal: a ∣ a ∧ a ∣ 0 apply And.intro (dvd_self a) define apply Exists.intro 0 rfl done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_nonzero a h1] --Goal : gcd b (a % b) ∣ a ∧ gcd b (a % b) ∣ b have h2 : a % b < b := mod_nonzero_lt a h1 have h3 : (gcd b (a % b)) ∣ b ∧ (gcd b (a % b)) ∣ (a % b) := ih (a % b) h2 b apply And.intro _ h3.left show (gcd b (a % b)) ∣ a from dvd_a_of_dvd_b_mod h3.left h3.right done done theorem gcd_dvd_left (a b : Nat) : (gcd a b) ∣ a := (gcd_dvd b a).left theorem gcd_dvd_right (a b : Nat) : (gcd a b) ∣ b := (gcd_dvd b a).right lemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl lemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c1 a b = gcd_c2 b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done lemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl lemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * ↑(a / b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done theorem gcd_lin_comb : ∀ (b a : Nat), (gcd_c1 a b) * ↑a + (gcd_c2 a b) * ↑b = ↑(gcd a b) := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd_c1 a b_1) * ↑a + (gcd_c2 a b_1) * ↑b_1 = ↑(gcd a b_1) fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base] --Goal : 1 * ↑a + 0 * ↑0 = ↑a ring done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1] --Goal : gcd_c2 b (a % b) * ↑a + -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * ↑(a / b)) * ↑b = -- ↑(gcd b (a % b)) set r : Nat := a % b set q : Nat := a / b set s : Int := gcd_c1 b r set t : Int := gcd_c2 b r --Goal : t * ↑a + (s - t * ↑q) * ↑b = ↑(gcd b r) have h2 : r < b := mod_nonzero_lt a h1 have h3 : s * ↑b + t * ↑r = ↑(gcd b r) := ih r h2 b have h4 : b * q + r = a := Nat.div_add_mod a b rewrite [←h3, ←h4] rewrite [Nat.cast_add, Nat.cast_mul] --Goal : t * (↑b * ↑q + ↑r) + (s - t * ↑q) * ↑b = s * ↑b + t * ↑r ring done done #eval gcd_c1 672 161 --Answer: 6 #eval gcd_c2 672 161 --Answer: -25 --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161 theorem Theorem_7_1_6 {d a b : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ gcd a b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑d ∣ ↑(gcd a b) set s : Int := gcd_c1 a b set t : Int := gcd_c2 a b have h3 : s * ↑a + t * ↑b = ↑(gcd a b) := gcd_lin_comb b a rewrite [←h3] --Goal : ↑d ∣ s * ↑a + t * ↑b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul] --Goal : ↑d ∣ s * (↑d * ↑j) + t * (↑d * ↑k) define apply Exists.intro (s * ↑j + t * ↑k) ring done /- Section 7.2 -/ theorem dvd_trans {a b c : Nat} (h1 : a ∣ b) (h2 : b ∣ c) : a ∣ c := by define at h1; define at h2; define obtain (m : Nat) (h3 : b = a * m) from h1 obtain (n : Nat) (h4 : c = b * n) from h2 rewrite [h3, mul_assoc] at h4 apply Exists.intro (m * n) show c = a * (m * n) from h4 done lemma exists_prime_factor : ∀ (n : Nat), 2 ≤ n → ∃ (p : Nat), prime_factor p n := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, 2 ≤ n_1 → ∃ (p : Nat), prime_factor p n_1 assume h1 : 2 ≤ n by_cases h2 : prime n · -- Case 1. h2 : prime n apply Exists.intro n define --Goal : prime n ∧ n ∣ n show prime n ∧ n ∣ n from And.intro h2 (dvd_self n) done · -- Case 2. h2 : ¬prime n define at h2 --h2 : ¬(2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n) demorgan at h2 disj_syll h2 h1 obtain (a : Nat) (h3 : ∃ (b : Nat), a * b = n ∧ a < n ∧ b < n) from h2 obtain (b : Nat) (h4 : a * b = n ∧ a < n ∧ b < n) from h3 have h5 : 2 ≤ a := by by_contra h6 have h7 : a ≤ 1 := by linarith have h8 : n ≤ b := calc n _ = a * b := h4.left.symm _ ≤ 1 * b := by rel [h7] _ = b := by ring linarith --n ≤ b contradicts b < n done have h6 : ∃ (p : Nat), prime_factor p a := ih a h4.right.left h5 obtain (p : Nat) (h7 : prime_factor p a) from h6 apply Exists.intro p define --Goal : prime p ∧ p ∣ n define at h7 --h7 : prime p ∧ p ∣ a apply And.intro h7.left have h8 : a ∣ n := by apply Exists.intro b show n = a * b from (h4.left).symm done show p ∣ n from dvd_trans h7.right h8 done done lemma exists_least_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q := by set S : Set Nat := {p : Nat | prime_factor p n} have h2 : ∃ (p : Nat), p ∈ S := exists_prime_factor n h show ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q from well_ord_princ S h2 done lemma all_prime_nil : all_prime [] := by define --Goal : ∀ p ∈ [], prime p fix p : Nat contrapos --Goal : ¬prime p → p ∉ [] assume h1 : ¬prime p show p ∉ [] from List.not_mem_nil p done lemma all_prime_cons (n : Nat) (L : List Nat) : all_prime (n :: L) ↔ prime n ∧ all_prime L := by apply Iff.intro · -- (→) assume h1 : all_prime (n :: L) --Goal : prime n ∧ all_prime L define at h1 --h1 : ∀ p ∈ n :: L, prime p apply And.intro (h1 n (List.mem_cons_self n L)) define --Goal : ∀ p ∈ L, prime p fix p : Nat assume h2 : p ∈ L show prime p from h1 p (List.mem_cons_of_mem n h2) done · -- (←) assume h1 : prime n ∧ all_prime L --Goal : all_prime (n :: l) define : all_prime L at h1 define fix p : Nat assume h2 : p ∈ n :: L rewrite [List.mem_cons] at h2 --h2 : p = n ∨ p ∈ L by_cases on h2 · -- Case 1. h2 : p = n rewrite [h2] show prime n from h1.left done · -- Case 2. h2 : p ∈ L show prime p from h1.right p h2 done done done lemma nondec_nil : nondec [] := by define --Goal : True trivial --trivial proves some obviously true statements, such as True done lemma nondec_cons (n : Nat) (L : List Nat) : nondec (n :: L) ↔ (∀ m ∈ L, n ≤ m) ∧ nondec L := by rfl lemma prod_nil : prod [] = 1 := by rfl lemma prod_cons : prod (n :: L) = n * (prod L) := by rfl lemma exists_cons_of_length_eq_succ {A : Type} {l : List A} {n : Nat} (h : l.length = n + 1) : ∃ (a : A) (L : List A), l = a :: L ∧ L.length = n := by have h1 : ¬l.length = 0 := by linarith rewrite [List.length_eq_zero] at h1 obtain (a : A) (h2 : ∃ (L : List A), l = a :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List A) (h3 : l = a :: L) from h2 apply Exists.intro a apply Exists.intro L apply And.intro h3 have h4 : (a :: L).length = L.length + 1 := List.length_cons a L rewrite [←h3, h] at h4 show L.length = n from (Nat.add_right_cancel h4).symm done lemma list_elt_dvd_prod_by_length (a : Nat) : ∀ (n : Nat), ∀ (l : List Nat), l.length = n → a ∈ l → a ∣ prod l := by by_induc · --Base Case fix l : List Nat assume h1 : l.length = 0 rewrite [List.length_eq_zero] at h1 --h1 : l = [] rewrite [h1] --Goal : a ∈ [] → a ∣ prod [] contrapos assume h2 : ¬a ∣ prod [] show a ∉ [] from List.not_mem_nil a done · -- Induction Step fix n : Nat assume ih : ∀ (l : List Nat), List.length l = n → a ∈ l → a ∣ prod l fix l : List Nat assume h1 : l.length = n + 1 --Goal : a ∈ l → a ∣ prod l obtain (b : Nat) (h2 : ∃ (L : List Nat), l = b :: L ∧ L.length = n) from exists_cons_of_length_eq_succ h1 obtain (L : List Nat) (h3 : l = b :: L ∧ L.length = n) from h2 have h4 : a ∈ L → a ∣ prod L := ih L h3.right assume h5 : a ∈ l rewrite [h3.left, prod_cons] --Goal : a ∣ b * prod L rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b ∨ a ∈ L by_cases on h5 · -- Case 1. h5 : a = b apply Exists.intro (prod L) rewrite [h5] rfl done · -- Case 2. h5 : a ∈ L have h6 : a ∣ prod L := h4 h5 have h7 : prod L ∣ b * prod L := by apply Exists.intro b ring done show a ∣ b * prod L from dvd_trans h6 h7 done done done lemma list_elt_dvd_prod {a : Nat} {l : List Nat} (h : a ∈ l) : a ∣ prod l := by set n : Nat := l.length have h1 : l.length = n := by rfl show a ∣ prod l from list_elt_dvd_prod_by_length a n l h1 h done lemma exists_prime_factorization : ∀ (n : Nat), n ≥ 1 → ∃ (l : List Nat), prime_factorization n l := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, n_1 ≥ 1 → ∃ (l : List Nat), prime_factorization n_1 l assume h1 : n ≥ 1 by_cases h2 : n = 1 · -- Case 1. h2 : n = 1 apply Exists.intro [] define apply And.intro · -- Proof of nondec_prime_list [] define show all_prime [] ∧ nondec [] from And.intro all_prime_nil nondec_nil done · -- Proof of prod [] = n rewrite [prod_nil, h2] rfl done done · -- Case 2. h2 : n ≠ 1 have h3 : n ≥ 2 := lt_of_le_of_ne' h1 h2 obtain (p : Nat) (h4 : prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q) from exists_least_prime_factor h3 have p_prime_factor : prime_factor p n := h4.left define at p_prime_factor have p_prime : prime p := p_prime_factor.left have p_dvd_n : p ∣ n := p_prime_factor.right have p_least : ∀ (q : Nat), prime_factor q n → p ≤ q := h4.right obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n have h5 : m ≠ 0 := by contradict h1 with h6 have h7 : n = 0 := calc n _ = p * m := n_eq_pm _ = p * 0 := by rw [h6] _ = 0 := by ring rewrite [h7] decide done have m_pos : 0 < m := Nat.pos_of_ne_zero h5 have m_lt_n : m < n := by define at p_prime show m < n from calc m _ < m + m := by linarith _ = 2 * m := by ring _ ≤ p * m := by rel [p_prime.left] _ = n := n_eq_pm.symm done obtain (L : List Nat) (h6 : prime_factorization m L) from ih m m_lt_n m_pos define at h6 have ndpl_L : nondec_prime_list L := h6.left define at ndpl_L apply Exists.intro (p :: L) define apply And.intro · -- Proof of nondec_prime_list (p :: L) define apply And.intro · -- Proof of all_prime (p :: L) rewrite [all_prime_cons] show prime p ∧ all_prime L from And.intro p_prime ndpl_L.left done · -- Proof of nondec (p :: L) rewrite [nondec_cons] apply And.intro _ ndpl_L.right fix q : Nat assume q_in_L : q ∈ L have h7 : q ∣ prod L := list_elt_dvd_prod q_in_L rewrite [h6.right] at h7 --h7 : q ∣ m have h8 : m ∣ n := by apply Exists.intro p rewrite [n_eq_pm] ring done have q_dvd_n : q ∣ n := dvd_trans h7 h8 have ap_L : all_prime L := ndpl_L.left define at ap_L have q_prime_factor : prime_factor q n := And.intro (ap_L q q_in_L) q_dvd_n show p ≤ q from p_least q q_prime_factor done done · -- Proof of prod (p :: L) = n rewrite [prod_cons, h6.right, n_eq_pm] rfl done done done theorem Theorem_7_2_2 {a b c : Nat} (h1 : c ∣ a * b) (h2 : rel_prime a c) : c ∣ b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑c ∣ ↑b define at h1; define at h2; define obtain (j : Nat) (h3 : a * b = c * j) from h1 set s : Int := gcd_c1 a c set t : Int := gcd_c2 a c have h4 : s * ↑a + t * ↑c = ↑(gcd a c) := gcd_lin_comb c a rewrite [h2, Nat.cast_one] at h4 --h4 : s * ↑a + t * ↑c = (1 : Int) apply Exists.intro (s * ↑j + t * ↑b) show ↑b = ↑c * (s * ↑j + t * ↑b) from calc ↑b _ = (1 : Int) * ↑b := (one_mul _).symm _ = (s * ↑a + t * ↑c) * ↑b := by rw [h4] _ = s * (↑a * ↑b) + t * ↑c * ↑b := by ring _ = s * (↑c * ↑j) + t * ↑c * ↑b := by rw [←Nat.cast_mul a b, h3, Nat.cast_mul c j] _ = ↑c * (s * ↑j + t * ↑b) := by ring done lemma le_nonzero_prod_left {a b : Nat} (h : a * b ≠ 0) : a ≤ a * b := by have h1 : b ≠ 0 := by contradict h with h1 rewrite [h1] ring done have h2 : 1 ≤ b := Nat.pos_of_ne_zero h1 show a ≤ a * b from calc a = a * 1 := (mul_one a).symm _ ≤ a * b := by rel [h2] done lemma le_nonzero_prod_right {a b : Nat} (h : a * b ≠ 0) : b ≤ a * b := by rewrite [mul_comm] rewrite [mul_comm] at h show b ≤ b * a from le_nonzero_prod_left h done lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry lemma rel_prime_of_prime_not_dvd {a p : Nat} (h1 : prime p) (h2 : ¬p ∣ a) : rel_prime a p := by have h3 : gcd a p ∣ a := gcd_dvd_left a p have h4 : gcd a p ∣ p := gcd_dvd_right a p have h5 : gcd a p = 1 ∨ gcd a p = p := dvd_prime h1 h4 have h6 : gcd a p ≠ p := by contradict h2 with h6 rewrite [h6] at h3 show p ∣ a from h3 done disj_syll h5 h6 show rel_prime a p from h5 done theorem Theorem_7_2_3 {a b p : Nat} (h1 : prime p) (h2 : p ∣ a * b) : p ∣ a ∨ p ∣ b := by or_right with h3 have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3 show p ∣ b from Theorem_7_2_2 h2 h4 done lemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a ≥ 1 := by have h1 : a ≠ 0 := by by_contra h1 rewrite [h1] at h contradict h linarith done show a ≥ 1 from Nat.pos_of_ne_zero h1 done lemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by have h1 : a ≥ 1 := ge_one_of_prod_one h have h2 : a * b ≠ 0 := by linarith have h3 : a ≤ a * b := le_nonzero_prod_left h2 rewrite [h] at h3 show a = 1 from Nat.le_antisymm h3 h1 done lemma eq_one_of_dvd_one {n : Nat} (h : n ∣ 1) : n = 1 := by obtain (j : Nat) (h1 : 1 = n * j) from h show n = 1 from eq_one_of_prod_one h1.symm done lemma prime_not_one {p : Nat} (h : prime p) : p ≠ 1 := by define at h linarith done theorem Theorem_7_2_4 {p : Nat} (h1 : prime p) : ∀ (l : List Nat), p ∣ prod l → ∃ a ∈ l, p ∣ a := by apply List.rec · -- Base Case. Goal : p ∣ prod [] → ∃ a ∈ [], p ∣ a rewrite [prod_nil] assume h2 : p ∣ 1 show ∃ a ∈ [], p ∣ a from absurd (eq_one_of_dvd_one h2) (prime_not_one h1) done · -- Induction Step fix b : Nat fix L : List Nat assume ih : p ∣ prod L → ∃ a ∈ L, p ∣ a --Goal : p ∣ prod (b :: L) → ∃ a ∈ b :: L, p ∣ a assume h2 : p ∣ prod (b :: L) rewrite [prod_cons] at h2 have h3 : p ∣ b ∨ p ∣ prod L := Theorem_7_2_3 h1 h2 by_cases on h3 · -- Case 1. h3 : p ∣ b apply Exists.intro b show b ∈ b :: L ∧ p ∣ b from And.intro (List.mem_cons_self b L) h3 done · -- Case 2. h3 : p ∣ prod L obtain (a : Nat) (h4 : a ∈ L ∧ p ∣ a) from ih h3 apply Exists.intro a show a ∈ b :: L ∧ p ∣ a from And.intro (List.mem_cons_of_mem b h4.left) h4.right done done done lemma prime_in_list {p : Nat} {l : List Nat} (h1 : prime p) (h2 : all_prime l) (h3 : p ∣ prod l) : p ∈ l := by obtain (a : Nat) (h4 : a ∈ l ∧ p ∣ a) from Theorem_7_2_4 h1 l h3 define at h2 have h5 : prime a := h2 a h4.left have h6 : p = 1 ∨ p = a := dvd_prime h5 h4.right disj_syll h6 (prime_not_one h1) rewrite [h6] show a ∈ l from h4.left done lemma first_le_first {p q : Nat} {l m : List Nat} (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m)) (h3 : prod (p :: l) = prod (q :: m)) : p ≤ q := by define at h1; define at h2 have h4 : q ∣ prod (p :: l) := by define apply Exists.intro (prod m) rewrite [←prod_cons] show prod (p :: l) = prod (q :: m) from h3 done have h5 : all_prime (q :: m) := h2.left rewrite [all_prime_cons] at h5 have h6 : q ∈ p :: l := prime_in_list h5.left h1.left h4 have h7 : nondec (p :: l) := h1.right rewrite [nondec_cons] at h7 rewrite [List.mem_cons] at h6 by_cases on h6 · -- Case 1. h6 : q = p linarith done · -- Case 2. h6 : q ∈ l have h8 : ∀ m ∈ l, p ≤ m := h7.left show p ≤ q from h8 q h6 done done lemma nondec_prime_list_tail {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by define at h define rewrite [all_prime_cons, nondec_cons] at h show all_prime l ∧ nondec l from And.intro h.left.right h.right.right done lemma cons_prod_not_one {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : prod (p :: l) ≠ 1 := by define at h have h1 : all_prime (p :: l) := h.left rewrite [all_prime_cons] at h1 rewrite [prod_cons] by_contra h2 show False from (prime_not_one h1.left) (eq_one_of_prod_one h2) done lemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) : l = [] ↔ prod l = 1 := by apply Iff.intro · -- (→) assume h1 : l = [] rewrite [h1] show prod [] = 1 from prod_nil done · -- (←) contrapos assume h1 : ¬l = [] obtain (p : Nat) (h2 : ∃ (L : List Nat), l = p :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List Nat) (h3 : l = p :: L) from h2 rewrite [h3] at h rewrite [h3] show ¬prod (p :: L) = 1 from cons_prod_not_one h done done lemma prime_pos {p : Nat} (h : prime p) : p > 0 := by define at h linarith done theorem Theorem_7_2_5 : ∀ (l1 l2 : List Nat), nondec_prime_list l1 → nondec_prime_list l2 → prod l1 = prod l2 → l1 = l2 := by apply List.rec · -- Base Case. Goal : ∀ (l2 : List Nat), nondec_prime_list [] → -- nondec_prime_list l2 → prod [] = prod l2 → [] = l2 fix l2 : List Nat assume h1 : nondec_prime_list [] assume h2 : nondec_prime_list l2 assume h3 : prod [] = prod l2 rewrite [prod_nil, eq_comm, ←list_nil_iff_prod_one h2] at h3 show [] = l2 from h3.symm done · -- Induction Step fix p : Nat fix L1 : List Nat assume ih : ∀ (L2 : List Nat), nondec_prime_list L1 → nondec_prime_list L2 → prod L1 = prod L2 → L1 = L2 -- Goal : ∀ (l2 : List Nat), nondec_prime_list (p :: L1) → -- nondec_prime_list l2 → prod (p :: L1) = prod l2 → p :: L1 = l2 fix l2 : List Nat assume h1 : nondec_prime_list (p :: L1) assume h2 : nondec_prime_list l2 assume h3 : prod (p :: L1) = prod l2 have h4 : ¬prod (p :: L1) = 1 := cons_prod_not_one h1 rewrite [h3, ←list_nil_iff_prod_one h2] at h4 obtain (q : Nat) (h5 : ∃ (L : List Nat), l2 = q :: L) from List.exists_cons_of_ne_nil h4 obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5 rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2) rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2) have h7 : p ≤ q := first_le_first h1 h2 h3 have h8 : q ≤ p := first_le_first h2 h1 h3.symm have h9 : p = q := by linarith rewrite [h9, prod_cons, prod_cons] at h3 --h3 : q * prod L1 = q * prod L2 have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1 have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2 define at h2 have h12 : all_prime (q :: L2) := h2.left rewrite [all_prime_cons] at h12 have h13 : q > 0 := prime_pos h12.left have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3 have h15 : L1 = L2 := ih L2 h10 h11 h14 rewrite [h6, h9, h15] rfl done done theorem fund_thm_arith (n : Nat) (h : n ≥ 1) : ∃! (l : List Nat), prime_factorization n l := by exists_unique · -- Existence show ∃ (l : List Nat), prime_factorization n l from exists_prime_factorization n h done · -- Uniqueness fix l1 : List Nat; fix l2 : List Nat assume h1 : prime_factorization n l1 assume h2 : prime_factorization n l2 define at h1; define at h2 have h3 : prod l1 = n := h1.right rewrite [←h2.right] at h3 show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3 done done /- Section 7.3 -/ theorem congr_refl (m : Nat) : ∀ (a : Int), a ≡ a (MOD m) := by fix a : Int define --Goal : ∃ (c : Int), a - a = ↑m * c apply Exists.intro 0 ring done theorem congr_symm {m : Nat} : ∀ {a b : Int}, a ≡ b (MOD m) → b ≡ a (MOD m) := by fix a : Int; fix b : Int assume h1 : a ≡ b (MOD m) define at h1 --h1 : ∃ (c : Int), a - b = ↑m * c define --Goal : ∃ (c : Int), b - a = ↑m * c obtain (c : Int) (h2 : a - b = m * c) from h1 apply Exists.intro (-c) show b - a = m * (-c) from calc b - a _ = -(a - b) := by ring _ = -(m * c) := by rw [h2] _ = m * (-c) := by ring done theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry /- Fundamental properties of congruence classes -/ lemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) : X = Y ↔ X.val = Y.val := Fin.ext_iff lemma val_nat_eq_mod (n k : Nat) : ([k]_(n + 1)).val = k % (n + 1) := by rfl lemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl theorem cc_rep {m : Nat} (X : ZMod m) : ∃ (a : Int), X = [a]_m := match m with | 0 => by apply Exists.intro X rfl done | n + 1 => by apply Exists.intro ↑(X.val) have h1 : X.val < n + 1 := Fin.prop X rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1] rfl done theorem add_class (m : Nat) (a b : Int) : [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm theorem mul_class (m : Nat) (a b : Int) : [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm lemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ [a - b]_m = [0]_m := by apply Iff.intro · -- (→) assume h1 : [a]_m = [b]_m have h2 : a - b = a + (-b) := by ring have h3 : b + (-b) = 0 := by ring show [a - b]_m = [0]_m from calc [a - b]_m _ = [a + (-b)]_m := by rw [h2] _ = [a]_m + [-b]_m := by rw [add_class] _ = [b]_m + [-b]_m := by rw [h1] _ = [b + -b]_m := by rw [add_class] _ = [0]_m := by rw [h3] done · -- (←) assume h1 : [a - b]_m = [0]_m have h2 : b + (a - b) = a := by ring have h3 : b + 0 = b := by ring show [a]_m = [b]_m from calc [a]_m _ = [b + (a - b)]_m := by rw [h2] _ = [b]_m + [a - b]_m := by rw [add_class] _ = [b]_m + [0]_m := by rw [h1] _ = [b + 0]_m := by rw [add_class] _ = [b]_m := by rw [h3] done done lemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) : [a]_m = [0]_m → [-a]_m = [0]_m := by assume h1 : [a]_m = [0]_m have h2 : 0 + (-a) = -a := by ring have h3 : a + (-a) = 0 := by ring show [-a]_m = [0]_m from calc [-a]_m _ = [0 + (-a)]_m := by rw [h2] _ = [0]_m + [-a]_m := by rw [add_class] _ = [a]_m + [-a]_m := by rw [h1] _ = [a + (-a)]_m := by rw [add_class] _ = [0]_m := by rw [h3] done lemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) : [-a]_m = [0]_m ↔ [a]_m = [0]_m := by apply Iff.intro _ (cc_neg_zero_of_cc_zero m a) assume h1 : [-a]_m = [0]_m have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1 have h3 : -(-a) = a := by ring rewrite [h3] at h2 show [a]_m = [0]_m from h2 done lemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl lemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m ↔ m ∣ k := match m with | 0 => by have h : (0 : Int) = (↑(0 : Nat) : Int) := by rfl rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj] apply Iff.intro · -- (→) assume h1 : k = 0 rewrite [h1] show 0 ∣ 0 from dvd_self 0 done · -- (←) assume h1 : 0 ∣ k obtain (c : Nat) (h2 : k = 0 * c) from h1 rewrite [h2] ring done done | n + 1 => by rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero] show k % (n + 1) = 0 ↔ n + 1 ∣ k from (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm done lemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m ↔ ↑m ∣ a := by obtain (k : Nat) (h1 : a = ↑k ∨ a = -↑k) from Int.eq_nat_or_neg a by_cases on h1 · -- Case 1. h1: a = ↑k rewrite [h1, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done · -- Case 2. h1: a = -↑k rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done done theorem cc_eq_iff_congr (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ a ≡ b (MOD m) := calc [a]_m = [b]_m _ ↔ [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b _ ↔ ↑m ∣ (a - b) := cc_zero_iff_dvd m (a - b) _ ↔ a ≡ b (MOD m) := by rfl /- End of fundamental properties of congruence classes -/ lemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m := by have h1 : (↑m : Int) ≠ 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m) show 0 ≤ a % m from Int.emod_nonneg a h1 done lemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m) have h2 : (↑m : Int) > 0 := (Nat.cast_pos).rtl h1 show a % m < m from Int.emod_lt_of_pos a h2 done lemma congr_mod_mod (m : Nat) (a : Int) : a ≡ a % m (MOD m) := by define have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m apply Exists.intro (a / m) show a - a % m = m * (a / m) from calc a - (a % m) _ = m * (a / m) + a % m - a % m := by rw [h1] _ = m * (a / m) := by ring done lemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) := And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a)) theorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) : ∃! (r : Int), 0 ≤ r ∧ r < m ∧ a ≡ r (MOD m) := by exists_unique · -- Existence apply Exists.intro (a % m) show 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) from mod_cmpl_res m a done · -- Uniqueness fix r1 : Int; fix r2 : Int assume h1 : 0 ≤ r1 ∧ r1 < m ∧ a ≡ r1 (MOD m) assume h2 : 0 ≤ r2 ∧ r2 < m ∧ a ≡ r2 (MOD m) have h3 : r1 ≡ r2 (MOD m) := congr_trans (congr_symm h1.right.right) h2.right.right obtain (d : Int) (h4 : r1 - r2 = m * d) from h3 have h5 : r1 - r2 < m * 1 := by linarith have h6 : m * (-1) < r1 - r2 := by linarith rewrite [h4] at h5 --h5 : m * d < m * 1 rewrite [h4] at h6 --h6 : m * -1 < m * d have h7 : (↑m : Int) ≥ 0 := Nat.cast_nonneg m have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7 have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7 have h10 : d = 0 := by linarith show r1 = r2 from calc r1 _ = r1 - r2 + r2 := by ring _ = m * 0 + r2 := by rw [h4, h10] _ = r2 := by ring done done lemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m := (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a) theorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y rewrite [h1, h2] have h3 : a + b = b + a := by ring show [a]_m + [b]_m = [b]_m + [a]_m from calc [a]_m + [b]_m _ = [a + b]_m := add_class m a b _ = [b + a]_m := by rw [h3] _ = [b]_m + [a]_m := (add_class m b a).symm done theorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X rewrite [h1] have h2 : a * 1 = a := by ring show [a]_m * [1]_m = [a]_m from calc [a]_m * [1]_m _ = [a * 1]_m := mul_class m a 1 _ = [a]_m := by rw [h2] done theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry lemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) : [a]_m * [gcd_c2 m a]_m = [1]_m := by set s : Int := gcd_c1 m a have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m define at h1 rewrite [h1, Nat.cast_one] at h2 --h2 : s * ↑m + gcd_c2 m a * ↑a = 1 rewrite [mul_class, cc_eq_iff_congr] define --Goal : ∃ (c : Int), ↑a * gcd_c2 m a - 1 = ↑m * c apply Exists.intro (-s) show a * (gcd_c2 m a) - 1 = m * (-s) from calc a * (gcd_c2 m a) - 1 _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring _ = 1 + m * (-s) - 1 := by rw [h2] _ = m * (-s) := by ring done theorem Theorem_7_3_7 (m a : Nat) : invertible [a]_m ↔ rel_prime m a := by apply Iff.intro · -- (→) assume h1 : invertible [a]_m define at h1 obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1 obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y rewrite [h3, mul_class, cc_eq_iff_congr] at h2 define at h2 obtain (c : Int) (h4 : a * b - 1 = m * c) from h2 rewrite [Exercise_7_2_6] --Goal : ∃ (s t : Int), s * ↑m + t * ↑a = 1 apply Exists.intro (-c) apply Exists.intro b show (-c) * m + b * a = 1 from calc (-c) * m + b * a _ = (-c) * m + (a * b - 1) + 1 := by ring _ = (-c) * m + m * c + 1 := by rw [h4] _ = 1 := by ring done · -- (←) assume h1 : rel_prime m a define show ∃ (Y : ZMod m), [a]_m * Y = [1]_m from Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1) done done /- Section 7.4 -/ section Euler open Euler lemma num_rp_below_base {m : Nat} : num_rp_below m 0 = 0 := by rfl lemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) : num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by have h1 : num_rp_below m (j + 1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : gcd m j = 1 rewrite [if_pos h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1 show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1 done lemma num_rp_below_step_not_rp {m j : Nat} (h : ¬rel_prime m j) : num_rp_below m (j + 1) = num_rp_below m j := by have h1 : num_rp_below m (j +1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : ¬gcd m j = 1 rewrite [if_neg h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j show num_rp_below m (j + 1) = num_rp_below m j from h1 done lemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl #eval phi 10 --Answer: 4 lemma prod_inv_iff_inv {m : Nat} {X : ZMod m} (h1 : invertible X) (Y : ZMod m) : invertible (X * Y) ↔ invertible Y := by apply Iff.intro · -- (→) assume h2 : invertible (X * Y) obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2 apply Exists.intro (X * Z) rewrite [←h3] --Goal : Y * (X * Z) = X * Y * Z ring --Note that ring can do algebra in ZMod m done · -- (←) assume h2 : invertible Y obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1 obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2 apply Exists.intro (Xi * Yi) show (X * Y) * (Xi * Yi) = [1]_m from calc X * Y * (Xi * Yi) _ = (X * Xi) * (Y * Yi) := by ring _ = [1]_m * [1]_m := by rw [h3, h4] _ = [1]_m := Theorem_7_3_6_7 [1]_m done done lemma F_rp_def {m i : Nat} (h : rel_prime m i) : F m i = [i]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h --h : gcd m i = 1 rewrite [if_pos h] at h1 show F m i = [i]_m from h1 done lemma F_not_rp_def {m i : Nat} (h : ¬rel_prime m i) : F m i = [1]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h rewrite [h1, if_neg h] rfl done lemma prod_seq_base {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 0 k f = [1]_m := by rfl lemma prod_seq_step {m : Nat} (n k : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl lemma prod_seq_zero_step {m : Nat} (n : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by rewrite [prod_seq_step, zero_add] rfl done lemma prod_one {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 1 k f = f k := by rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7] rfl done lemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl lemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m := calc [G m a i]_m _ = [(a * i) % m]_m := by rfl _ = [a * i]_m := (cc_eq_mod m (a * i)).symm _ = [a]_m * [i]_m := (mul_class m a i).symm lemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) : rel_prime m (G m a i) ↔ rel_prime m i := by have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1 show rel_prime m (G m a i) ↔ rel_prime m i from calc rel_prime m (G m a i) _ ↔ invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm _ ↔ invertible ([a]_m * [i]_m) := by rw [cc_G] _ ↔ invertible [i]_m := prod_inv_iff_inv h2 ([i]_m) _ ↔ rel_prime m i := Theorem_7_3_7 m i done lemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) : F m (G m a i) = [a]_m * F m i := by have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2 show F m (G m a i) = [a]_m * F m i from calc F m (G m a i) _ = [G m a i]_m := F_rp_def h3 _ = [a]_m * [i]_m := cc_G m a i _ = [a]_m * F m i := by rw [F_rp_def h2] done lemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : ¬rel_prime m i) : F m (G m a i) = [1]_m := by rewrite [←G_rp_iff h1 i] at h2 show F m (G m a i) = [1]_m from F_not_rp_def h2 done lemma FG_prod {m a : Nat} (h1 : rel_prime m a) : ∀ (k : Nat), prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by by_induc · -- Base Case show prod_seq 0 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from calc prod_seq 0 0 ((F m) ∘ (G m a)) _ = [1]_m := prod_seq_base _ _ _ = [a]_m ^ 0 * [1]_m := by ring _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by rw [num_rp_below_base, prod_seq_base] done · -- Induction Step fix k : Nat assume ih : prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) by_cases h2 : rel_prime m k · -- Case 1. h2 : rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([a]_m * F m k) := by rw [FG_rp h1 h2] _ = [a]_m ^ ((num_rp_below m k) + 1) * ((prod_seq k 0 (F m)) * F m k) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_rp h2, prod_seq_zero_step] done · -- Case 2. h2 : ¬rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([1]_m) := by rw [FG_not_rp h1 h2] _ = [a]_m ^ (num_rp_below m k) * (prod_seq k 0 (F m) * ([1]_m)) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_not_rp h2, prod_seq_zero_step, F_not_rp_def h2] done done done lemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by define --Goal : ∀ i < m, G m a i < m fix i : Nat assume h1 : i < m rewrite [G_def] --Goal : a * i % m < m show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m) done lemma left_inv_one_one_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g' (g i) = i) : one_one_below n g := sorry lemma right_inv_onto_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g (g' i) = i) (h2 : maps_below n g') : onto_below n g := by define at h2; define fix k : Nat assume h3 : k < n apply Exists.intro (g' k) show g' k < n ∧ g (g' k) = k from And.intro (h2 k h3) (h1 k h3) done lemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m * [inv_mod m a]_m = [1]_m := by have h2 : 0 ≤ (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a) show [a]_m * [inv_mod m a]_m = [1]_m from calc [a]_m * [inv_mod m a]_m _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2] _ = [a]_m * [gcd_c2 m a]_m := by rw [←cc_eq_mod] _ = [1]_m := gcd_c2_inv h1 done lemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m := calc a * (b % m) % m = a % m * (b % m % m) % m := Nat.mul_mod _ _ _ _ = a % m * (b % m) % m := by rw [Nat.mod_mod] _ = a * b % m := (Nat.mul_mod _ _ _).symm lemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm] rfl done theorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] : ↑a ≡ ↑b (MOD m) ↔ a % m = b % m := sorry lemma mul_inv_mod_cancel {m a i : Nat} [NeZero m] (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1 rewrite [mul_class, cc_eq_iff_congr, ←Nat.cast_mul, ←Nat.cast_one, congr_iff_mod_eq_Nat] at h3 show a * inv_mod m a * i % m = i from calc a * (inv_mod m a) * i % m _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = 1 % m * i % m := by rw [h3] _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = i % m := by rw [one_mul] _ = i := Nat.mod_eq_of_lt h2 done lemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl lemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, G m a (Ginv m a i) = i := by fix i : Nat assume h2 : i < m show G m a (Ginv m a i) = i from calc G m a (Ginv m a i) _ = a * ((inv_mod m a * i) % m) % m := by rfl _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, Ginv m a (G m a i) = i := by fix i : Nat assume h2 : i < m show Ginv m a (G m a i) = i from calc Ginv m a (G m a i) _ = inv_mod m a * ((a * i) % m) % m := by rfl _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc, mul_comm (inv_mod m a)] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_maps_below (m a : Nat) [NeZero m] : maps_below m (Ginv m a) := G_maps_below m (inv_mod m a) lemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : one_one_below m (G m a) := left_inv_one_one_below (Ginv_left_inv h1) lemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : onto_below m (G m a) := right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a) lemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : perm_below m (G m a) := And.intro (G_maps_below m a) (And.intro (G_one_one_below h1) (G_onto_below h1)) --Permuting a product of congruence classes doesn't change product lemma swap_fst (u v : Nat) : swap u v u = v := by define : swap u v u --Goal : (if u = u then v else if u = v then u else u) = v have h : u = u := by rfl rewrite [if_pos h] rfl done lemma swap_snd (u v : Nat) : swap u v v = u := by define : swap u v v by_cases h1 : v = u · -- Case 1. h1 : v = u rewrite [if_pos h1] show v = u from h1 done · -- Case 2. h1 : v ≠ u rewrite [if_neg h1] have h2 : v = v := by rfl rewrite [if_pos h2] rfl done done lemma swap_other {u v i : Nat} (h1 : i ≠ u) (h2 : i ≠ v) : swap u v i = i := by define : swap u v i rewrite [if_neg h1, if_neg h2] rfl done lemma swap_values (u v i : Nat) : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := by by_cases h1 : i = u · -- Case 1. h1 : i = u apply Or.inl rewrite [h1] show swap u v u = v from swap_fst u v done · -- Case 2. h1 : i ≠ u apply Or.inr by_cases h2 : i = v · -- Case 2.1. h2 : i = v apply Or.inl rewrite [h2] show swap u v v = u from swap_snd u v done · -- Case 2.2. h2 : i ≠ v apply Or.inr show swap u v i = i from swap_other h1 h2 done done done lemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by define fix i : Nat assume h3 : i < n have h4 : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := swap_values u v i by_cases on h4 · -- Case 1. h4 : swap u v i = v rewrite [h4] show v < n from h2 done · -- Case 2. by_cases on h4 · -- Case 2.1. h4 : swap u v i = u rewrite [h4] show u < n from h1 done · -- Case 2.2. h4 : swap u v i = i rewrite [h4] show i < n from h3 done done done lemma swap_swap (u v n : Nat) : ∀ i < n, swap u v (swap u v i) = i := by fix i : Nat assume h : i < n by_cases h1 : i = u · -- Case 1. h1 : i = u rewrite [h1, swap_fst, swap_snd] rfl done · -- Case 2. h1 : i ≠ u by_cases h2 : i = v · -- Case 2.1. h2 : i = v rewrite [h2, swap_snd, swap_fst] rfl done · -- Case 2.2. h2 : i ≠ v rewrite [swap_other h1 h2, swap_other h1 h2] rfl done done done lemma swap_one_one_below (u v n) : one_one_below n (swap u v) := left_inv_one_one_below (swap_swap u v n) lemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) := right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2) lemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) := And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2)) lemma comp_perm_below {n : Nat} {f g : Nat → Nat} (h1 : perm_below n f) (h2 : perm_below n g) : perm_below n (f ∘ g) := sorry lemma trivial_swap (u : Nat) : swap u u = id := by apply funext fix x : Nat by_cases h1 : x = u · -- Case 1. h1 : x = u rewrite [h1, swap_fst] rfl done · -- Case 2. h1 : x ≠ u rewrite [swap_other h1 h1] rfl done done lemma prod_eq_fun {m : Nat} (f g : Nat → ZMod m) (k : Nat) : ∀ (n : Nat), (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g := by by_induc · -- Base Case assume h : (∀ i < 0, f (k + i) = g (k + i)) rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g assume h1 : ∀ i < n + 1, f (k + i) = g (k + i) have h2 : ∀ i < n, f (k + i) = g (k + i) := by fix i : Nat assume h2 : i < n have h3 : i < n + 1 := by linarith show f (k + i) = g (k + i) from h1 i h3 done have h3 : prod_seq n k f = prod_seq n k g := ih h2 have h4 : n < n + 1 := Nat.lt_succ_self n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4] rfl done done lemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := by have h2 : ∀ (i : Nat), i < u → (f ∘ swap u n) (0 + i) = f (0 + i) := by fix i : Nat assume h2 : i < u have h3 : 0 + i ≠ u := by linarith have h4 : 0 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl done show prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f from prod_eq_fun (f ∘ swap u n) f 0 u h2 done lemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := by have h2 : ∀ i < j, (f ∘ swap u n) (u + 1 + i) = f (u + 1 + i) := by fix i : Nat assume h2 : i < j have h3 : u + 1 + i ≠ u := by linarith have h4 : u + 1 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl show prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f from prod_eq_fun (f ∘ swap u n) f (u + 1) j h2 done lemma break_prod {m : Nat} (n : Nat) (f : Nat → ZMod m) : ∀ (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by by_induc · -- Base Case have h : n + 0 = n := by rfl rewrite [prod_seq_base, h, Theorem_7_3_6_7] rfl done · -- Induction Step fix j : Nat assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f rewrite [←add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc] rfl done done lemma break_prod_twice {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f := break_prod n f 1 rewrite [prod_one] at h2 have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f := break_prod (u + 1) f j rewrite [←h1] at h3 have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f := break_prod u f 1 rewrite [prod_one] at h4 rewrite [h3, h4] at h2 show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2 done lemma swap_prod_eq_prod {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f := by by_cases h2 : u = n · -- Case 1. h2 : u = n rewrite [h2, trivial_swap n] --Goal : prod_seq (n + 1) 0 (f ∘ id) = prod_seq (n + 1) 0 f rfl done · -- Case 2. h2 : ¬u = n have h3 : u + 1 ≤ n := Nat.lt_of_le_of_ne h1 h2 obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3 have break_f : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := break_prod_twice f h4 have break_fs : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_prod_twice (f ∘ swap u n) h4 have f_eq_fs_below : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := swap_prod_eq_prod_below f h1 have f_eq_fs_btwn : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4 show prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f from calc prod_seq (n + 1) 0 (f ∘ swap u n) _ = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_fs _ = prod_seq u 0 f * (f ∘ swap u n) u * prod_seq j (u + 1) f * (f ∘ swap u n) n := by rw [f_eq_fs_below, f_eq_fs_btwn] _ = prod_seq u 0 f * f (swap u n u) * prod_seq j (u + 1) f * f (swap u n n) := by rfl _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by rw [swap_fst, swap_snd] _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring _ = prod_seq (n + 1) 0 f := break_f.symm done done lemma perm_below_fixed {n : Nat} {g : Nat → Nat} (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry lemma perm_prod {m : Nat} (f : Nat → ZMod m) : ∀ (n : Nat), ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) := by by_induc · -- Base Case fix g : Nat → Nat assume h1 : perm_below 0 g rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) fix g : Nat → Nat assume g_pb : perm_below (n + 1) g define at g_pb have g_ob : onto_below (n + 1) g := g_pb.right.right define at g_ob have h1 : n < n + 1 := by linarith obtain (u : Nat) (h2 : u < n + 1 ∧ g u = n) from g_ob n h1 have s_pb : perm_below (n + 1) (swap u n) := swap_perm_below h2.left h1 have gs_pb_n1 : perm_below (n + 1) (g ∘ swap u n) := comp_perm_below g_pb s_pb have gs_fix_n : (g ∘ swap u n) n = n := calc (g ∘ swap u n) n _ = g (swap u n n) := by rfl _ = g u := by rw [swap_snd] _ = n := h2.right have gs_pb_n : perm_below n (g ∘ swap u n) := perm_below_fixed gs_pb_n1 gs_fix_n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f ∘ (g ∘ swap u n)) := ih (g ∘ swap u n) gs_pb_n have h3 : u ≤ n := by linarith show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f ∘ g) from calc prod_seq (n + 1) 0 f _ = prod_seq n 0 f * f n := prod_seq_zero_step n f _ = prod_seq n 0 (f ∘ (g ∘ swap u n)) * f ((g ∘ swap u n) n) := by rw [gs_prod, gs_fix_n] _ = prod_seq n 0 (f ∘ g ∘ swap u n) * (f ∘ g ∘ swap u n) n := by rfl _ = prod_seq (n + 1) 0 (f ∘ g ∘ swap u n) := (prod_seq_zero_step n (f ∘ g ∘ swap u n)).symm _ = prod_seq (n + 1) 0 ((f ∘ g) ∘ swap u n) := by rfl _ = prod_seq (n + 1) 0 (f ∘ g) := swap_prod_eq_prod (f ∘ g) h3 done done lemma F_invertible (m i : Nat) : invertible (F m i) := by by_cases h : rel_prime m i · -- Case 1. h : rel_prime m i rewrite [F_rp_def h] show invertible [i]_m from (Theorem_7_3_7 m i).rtl h done · -- Case 2. h : ¬rel_prime m i rewrite [F_not_rp_def h] apply Exists.intro [1]_m show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m done done lemma Fprod_invertible (m : Nat) : ∀ (k : Nat), invertible (prod_seq k 0 (F m)) := by by_induc · -- Base Case apply Exists.intro [1]_m show prod_seq 0 0 (F m) * [1]_m = [1]_m from calc prod_seq 0 0 (F m) * [1]_m _ = [1]_m * [1]_m := by rw [prod_seq_base] _ = [1]_m := Theorem_7_3_6_7 ([1]_m) done · -- Induction Step fix k : Nat assume ih : invertible (prod_seq k 0 (F m)) rewrite [prod_seq_zero_step] show invertible (prod_seq k 0 (F m) * (F m k)) from (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k) done done theorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m ^ (phi m) = [1]_m := by have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2 show [a]_m ^ (phi m) = [1]_m from calc [a]_m ^ (phi m) _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3] _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring _ = prod_seq m 0 (F m ∘ G m a) * Y := by rw [FG_prod h1 m, phi_def] _ = prod_seq m 0 (F m) * Y := by rw [perm_prod (F m) m (G m a) (G_perm_below h1)] _ = [1]_m := by rw [h3] done lemma Exercise_7_4_5_Int (m : Nat) (a : Int) : ∀ (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry lemma Exercise_7_4_5_Nat (m a n : Nat) : [a]_m ^ n = [a ^ n]_m := by rewrite [Exercise_7_4_5_Int] rfl done theorem Euler's_theorem {m a : Nat} [NeZero m] (h1 : rel_prime m a) : a ^ (phi m) ≡ 1 (MOD m) := by have h2 : [a]_m ^ (phi m) = [1]_m := Theorem_7_4_2 h1 rewrite [Exercise_7_4_5_Nat m a (phi m)] at h2 --h2 : [a ^ phi m]_m = [1]_m show a ^ (phi m) ≡ 1 (MOD m) from (cc_eq_iff_congr _ _ _).ltr h2 done #eval gcd 10 7 --Answer: 1. So 10 and 7 are relatively prime #eval 7 ^ phi 10 --Answer: 2401, which is congruent to 1 mod 10. end Euler /- Section 7.5 -/ lemma num_rp_prime {p : Nat} (h1 : prime p) : ∀ k < p, num_rp_below p (k + 1) = k := sorry

lemma phi_prime {p : Nat} (h1 : prime p) : phi p = p - 1

HTPI.phi_prime

htpi/HTPILib/Chap7.lean

htpi

/- Copyright 2023 Daniel J. Velleman -/ import HTPILib.Chap6 namespace HTPI /- Definitions -/ lemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by have h : n + 1 > 0 := Nat.succ_pos n show a % (n + 1) < n + 1 from Nat.mod_lt a h done def gcd (a b : Nat) : Nat := match b with | 0 => a | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd (n + 1) (a % (n + 1)) termination_by b mutual def gcd_c1 (a b : Nat) : Int := match b with | 0 => 1 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c2 (n + 1) (a % (n + 1)) --Corresponds to s = t' termination_by b def gcd_c2 (a b : Nat) : Int := match b with | 0 => 0 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c1 (n + 1) (a % (n + 1)) - (gcd_c2 (n + 1) (a % (n + 1))) * ↑(a / (n + 1)) --Corresponds to t = s' - t'q termination_by b end def prime (n : Nat) : Prop := 2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n def prime_factor (p n : Nat) : Prop := prime p ∧ p ∣ n def all_prime (l : List Nat) : Prop := ∀ p ∈ l, prime p def nondec (l : List Nat) : Prop := match l with | [] => True --Of course, True is a proposition that is always true | n :: L => (∀ m ∈ L, n ≤ m) ∧ nondec L def nondec_prime_list (l : List Nat) : Prop := all_prime l ∧ nondec l def prod (l : List Nat) : Nat := match l with | [] => 1 | n :: L => n * (prod L) def prime_factorization (n : Nat) (l : List Nat) : Prop := nondec_prime_list l ∧ prod l = n def rel_prime (a b : Nat) : Prop := gcd a b = 1 def congr_mod (m : Nat) (a b : Int) : Prop := (↑m : Int) ∣ (a - b) def cc (m : Nat) (a : Int) : ZMod m := (↑a : ZMod m) notation:50 a " ≡ " b " (MOD " m ")" => congr_mod m a b notation:max "["a"]_"m:max => cc m a def invertible {m : Nat} (X : ZMod m) : Prop := ∃ (Y : ZMod m), X * Y = [1]_m def num_rp_below (m k : Nat) : Nat := match k with | 0 => 0 | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j def phi (m : Nat) : Nat := num_rp_below m m def prod_seq {m : Nat} (j k : Nat) (f : Nat → ZMod m) : ZMod m := match j with | 0 => [1]_m | n + 1 => prod_seq n k f * f (k + n) def maps_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i < n, g i < n def one_one_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i1 < n, ∀ i2 < n, g i1 = g i2 → i1 = i2 def onto_below (n : Nat) (g : Nat → Nat) : Prop := ∀ k < n, ∃ i < n, g i = k def perm_below (n : Nat) (g : Nat → Nat) : Prop := maps_below n g ∧ one_one_below n g ∧ onto_below n g def inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m) def swap (u v i : Nat) : Nat := if i = u then v else if i = v then u else i namespace Euler --For definitions specific to Euler's theorem def F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m def G (m a i : Nat) : Nat := (a * i) % m def Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i end Euler /- Section 7.1 -/ theorem dvd_mod_of_dvd_a_b {a b d : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ (a % b) := by set q : Nat := a / b have h3 : b * q + a % b = a := Nat.div_add_mod a b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 define --Goal : ∃ (c : Nat), a % b = d * c apply Exists.intro (j - k * q) show a % b = d * (j - k * q) from calc a % b _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm _ = a - b * q := by rw [h3] _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc] _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm done theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry #eval gcd 672 161 --Answer: 7 lemma gcd_base (a : Nat) : gcd a 0 = a := by rfl lemma gcd_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd a b = gcd b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1)) rfl done lemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b ≠ 0) : a % b < b := by have h1 : b > 0 := Nat.pos_of_ne_zero h show a % b < b from Nat.mod_lt a h1 done lemma dvd_self (n : Nat) : n ∣ n := by apply Exists.intro 1 ring done theorem gcd_dvd : ∀ (b a : Nat), (gcd a b) ∣ a ∧ (gcd a b) ∣ b := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd a b_1) ∣ a ∧ (gcd a b_1) ∣ b_1 fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_base] --Goal: a ∣ a ∧ a ∣ 0 apply And.intro (dvd_self a) define apply Exists.intro 0 rfl done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_nonzero a h1] --Goal : gcd b (a % b) ∣ a ∧ gcd b (a % b) ∣ b have h2 : a % b < b := mod_nonzero_lt a h1 have h3 : (gcd b (a % b)) ∣ b ∧ (gcd b (a % b)) ∣ (a % b) := ih (a % b) h2 b apply And.intro _ h3.left show (gcd b (a % b)) ∣ a from dvd_a_of_dvd_b_mod h3.left h3.right done done theorem gcd_dvd_left (a b : Nat) : (gcd a b) ∣ a := (gcd_dvd b a).left theorem gcd_dvd_right (a b : Nat) : (gcd a b) ∣ b := (gcd_dvd b a).right lemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl lemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c1 a b = gcd_c2 b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done lemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl lemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * ↑(a / b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done theorem gcd_lin_comb : ∀ (b a : Nat), (gcd_c1 a b) * ↑a + (gcd_c2 a b) * ↑b = ↑(gcd a b) := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd_c1 a b_1) * ↑a + (gcd_c2 a b_1) * ↑b_1 = ↑(gcd a b_1) fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base] --Goal : 1 * ↑a + 0 * ↑0 = ↑a ring done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1] --Goal : gcd_c2 b (a % b) * ↑a + -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * ↑(a / b)) * ↑b = -- ↑(gcd b (a % b)) set r : Nat := a % b set q : Nat := a / b set s : Int := gcd_c1 b r set t : Int := gcd_c2 b r --Goal : t * ↑a + (s - t * ↑q) * ↑b = ↑(gcd b r) have h2 : r < b := mod_nonzero_lt a h1 have h3 : s * ↑b + t * ↑r = ↑(gcd b r) := ih r h2 b have h4 : b * q + r = a := Nat.div_add_mod a b rewrite [←h3, ←h4] rewrite [Nat.cast_add, Nat.cast_mul] --Goal : t * (↑b * ↑q + ↑r) + (s - t * ↑q) * ↑b = s * ↑b + t * ↑r ring done done #eval gcd_c1 672 161 --Answer: 6 #eval gcd_c2 672 161 --Answer: -25 --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161 theorem Theorem_7_1_6 {d a b : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ gcd a b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑d ∣ ↑(gcd a b) set s : Int := gcd_c1 a b set t : Int := gcd_c2 a b have h3 : s * ↑a + t * ↑b = ↑(gcd a b) := gcd_lin_comb b a rewrite [←h3] --Goal : ↑d ∣ s * ↑a + t * ↑b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul] --Goal : ↑d ∣ s * (↑d * ↑j) + t * (↑d * ↑k) define apply Exists.intro (s * ↑j + t * ↑k) ring done /- Section 7.2 -/ theorem dvd_trans {a b c : Nat} (h1 : a ∣ b) (h2 : b ∣ c) : a ∣ c := by define at h1; define at h2; define obtain (m : Nat) (h3 : b = a * m) from h1 obtain (n : Nat) (h4 : c = b * n) from h2 rewrite [h3, mul_assoc] at h4 apply Exists.intro (m * n) show c = a * (m * n) from h4 done lemma exists_prime_factor : ∀ (n : Nat), 2 ≤ n → ∃ (p : Nat), prime_factor p n := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, 2 ≤ n_1 → ∃ (p : Nat), prime_factor p n_1 assume h1 : 2 ≤ n by_cases h2 : prime n · -- Case 1. h2 : prime n apply Exists.intro n define --Goal : prime n ∧ n ∣ n show prime n ∧ n ∣ n from And.intro h2 (dvd_self n) done · -- Case 2. h2 : ¬prime n define at h2 --h2 : ¬(2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n) demorgan at h2 disj_syll h2 h1 obtain (a : Nat) (h3 : ∃ (b : Nat), a * b = n ∧ a < n ∧ b < n) from h2 obtain (b : Nat) (h4 : a * b = n ∧ a < n ∧ b < n) from h3 have h5 : 2 ≤ a := by by_contra h6 have h7 : a ≤ 1 := by linarith have h8 : n ≤ b := calc n _ = a * b := h4.left.symm _ ≤ 1 * b := by rel [h7] _ = b := by ring linarith --n ≤ b contradicts b < n done have h6 : ∃ (p : Nat), prime_factor p a := ih a h4.right.left h5 obtain (p : Nat) (h7 : prime_factor p a) from h6 apply Exists.intro p define --Goal : prime p ∧ p ∣ n define at h7 --h7 : prime p ∧ p ∣ a apply And.intro h7.left have h8 : a ∣ n := by apply Exists.intro b show n = a * b from (h4.left).symm done show p ∣ n from dvd_trans h7.right h8 done done lemma exists_least_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q := by set S : Set Nat := {p : Nat | prime_factor p n} have h2 : ∃ (p : Nat), p ∈ S := exists_prime_factor n h show ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q from well_ord_princ S h2 done lemma all_prime_nil : all_prime [] := by define --Goal : ∀ p ∈ [], prime p fix p : Nat contrapos --Goal : ¬prime p → p ∉ [] assume h1 : ¬prime p show p ∉ [] from List.not_mem_nil p done lemma all_prime_cons (n : Nat) (L : List Nat) : all_prime (n :: L) ↔ prime n ∧ all_prime L := by apply Iff.intro · -- (→) assume h1 : all_prime (n :: L) --Goal : prime n ∧ all_prime L define at h1 --h1 : ∀ p ∈ n :: L, prime p apply And.intro (h1 n (List.mem_cons_self n L)) define --Goal : ∀ p ∈ L, prime p fix p : Nat assume h2 : p ∈ L show prime p from h1 p (List.mem_cons_of_mem n h2) done · -- (←) assume h1 : prime n ∧ all_prime L --Goal : all_prime (n :: l) define : all_prime L at h1 define fix p : Nat assume h2 : p ∈ n :: L rewrite [List.mem_cons] at h2 --h2 : p = n ∨ p ∈ L by_cases on h2 · -- Case 1. h2 : p = n rewrite [h2] show prime n from h1.left done · -- Case 2. h2 : p ∈ L show prime p from h1.right p h2 done done done lemma nondec_nil : nondec [] := by define --Goal : True trivial --trivial proves some obviously true statements, such as True done lemma nondec_cons (n : Nat) (L : List Nat) : nondec (n :: L) ↔ (∀ m ∈ L, n ≤ m) ∧ nondec L := by rfl lemma prod_nil : prod [] = 1 := by rfl lemma prod_cons : prod (n :: L) = n * (prod L) := by rfl lemma exists_cons_of_length_eq_succ {A : Type} {l : List A} {n : Nat} (h : l.length = n + 1) : ∃ (a : A) (L : List A), l = a :: L ∧ L.length = n := by have h1 : ¬l.length = 0 := by linarith rewrite [List.length_eq_zero] at h1 obtain (a : A) (h2 : ∃ (L : List A), l = a :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List A) (h3 : l = a :: L) from h2 apply Exists.intro a apply Exists.intro L apply And.intro h3 have h4 : (a :: L).length = L.length + 1 := List.length_cons a L rewrite [←h3, h] at h4 show L.length = n from (Nat.add_right_cancel h4).symm done lemma list_elt_dvd_prod_by_length (a : Nat) : ∀ (n : Nat), ∀ (l : List Nat), l.length = n → a ∈ l → a ∣ prod l := by by_induc · --Base Case fix l : List Nat assume h1 : l.length = 0 rewrite [List.length_eq_zero] at h1 --h1 : l = [] rewrite [h1] --Goal : a ∈ [] → a ∣ prod [] contrapos assume h2 : ¬a ∣ prod [] show a ∉ [] from List.not_mem_nil a done · -- Induction Step fix n : Nat assume ih : ∀ (l : List Nat), List.length l = n → a ∈ l → a ∣ prod l fix l : List Nat assume h1 : l.length = n + 1 --Goal : a ∈ l → a ∣ prod l obtain (b : Nat) (h2 : ∃ (L : List Nat), l = b :: L ∧ L.length = n) from exists_cons_of_length_eq_succ h1 obtain (L : List Nat) (h3 : l = b :: L ∧ L.length = n) from h2 have h4 : a ∈ L → a ∣ prod L := ih L h3.right assume h5 : a ∈ l rewrite [h3.left, prod_cons] --Goal : a ∣ b * prod L rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b ∨ a ∈ L by_cases on h5 · -- Case 1. h5 : a = b apply Exists.intro (prod L) rewrite [h5] rfl done · -- Case 2. h5 : a ∈ L have h6 : a ∣ prod L := h4 h5 have h7 : prod L ∣ b * prod L := by apply Exists.intro b ring done show a ∣ b * prod L from dvd_trans h6 h7 done done done lemma list_elt_dvd_prod {a : Nat} {l : List Nat} (h : a ∈ l) : a ∣ prod l := by set n : Nat := l.length have h1 : l.length = n := by rfl show a ∣ prod l from list_elt_dvd_prod_by_length a n l h1 h done lemma exists_prime_factorization : ∀ (n : Nat), n ≥ 1 → ∃ (l : List Nat), prime_factorization n l := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, n_1 ≥ 1 → ∃ (l : List Nat), prime_factorization n_1 l assume h1 : n ≥ 1 by_cases h2 : n = 1 · -- Case 1. h2 : n = 1 apply Exists.intro [] define apply And.intro · -- Proof of nondec_prime_list [] define show all_prime [] ∧ nondec [] from And.intro all_prime_nil nondec_nil done · -- Proof of prod [] = n rewrite [prod_nil, h2] rfl done done · -- Case 2. h2 : n ≠ 1 have h3 : n ≥ 2 := lt_of_le_of_ne' h1 h2 obtain (p : Nat) (h4 : prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q) from exists_least_prime_factor h3 have p_prime_factor : prime_factor p n := h4.left define at p_prime_factor have p_prime : prime p := p_prime_factor.left have p_dvd_n : p ∣ n := p_prime_factor.right have p_least : ∀ (q : Nat), prime_factor q n → p ≤ q := h4.right obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n have h5 : m ≠ 0 := by contradict h1 with h6 have h7 : n = 0 := calc n _ = p * m := n_eq_pm _ = p * 0 := by rw [h6] _ = 0 := by ring rewrite [h7] decide done have m_pos : 0 < m := Nat.pos_of_ne_zero h5 have m_lt_n : m < n := by define at p_prime show m < n from calc m _ < m + m := by linarith _ = 2 * m := by ring _ ≤ p * m := by rel [p_prime.left] _ = n := n_eq_pm.symm done obtain (L : List Nat) (h6 : prime_factorization m L) from ih m m_lt_n m_pos define at h6 have ndpl_L : nondec_prime_list L := h6.left define at ndpl_L apply Exists.intro (p :: L) define apply And.intro · -- Proof of nondec_prime_list (p :: L) define apply And.intro · -- Proof of all_prime (p :: L) rewrite [all_prime_cons] show prime p ∧ all_prime L from And.intro p_prime ndpl_L.left done · -- Proof of nondec (p :: L) rewrite [nondec_cons] apply And.intro _ ndpl_L.right fix q : Nat assume q_in_L : q ∈ L have h7 : q ∣ prod L := list_elt_dvd_prod q_in_L rewrite [h6.right] at h7 --h7 : q ∣ m have h8 : m ∣ n := by apply Exists.intro p rewrite [n_eq_pm] ring done have q_dvd_n : q ∣ n := dvd_trans h7 h8 have ap_L : all_prime L := ndpl_L.left define at ap_L have q_prime_factor : prime_factor q n := And.intro (ap_L q q_in_L) q_dvd_n show p ≤ q from p_least q q_prime_factor done done · -- Proof of prod (p :: L) = n rewrite [prod_cons, h6.right, n_eq_pm] rfl done done done theorem Theorem_7_2_2 {a b c : Nat} (h1 : c ∣ a * b) (h2 : rel_prime a c) : c ∣ b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑c ∣ ↑b define at h1; define at h2; define obtain (j : Nat) (h3 : a * b = c * j) from h1 set s : Int := gcd_c1 a c set t : Int := gcd_c2 a c have h4 : s * ↑a + t * ↑c = ↑(gcd a c) := gcd_lin_comb c a rewrite [h2, Nat.cast_one] at h4 --h4 : s * ↑a + t * ↑c = (1 : Int) apply Exists.intro (s * ↑j + t * ↑b) show ↑b = ↑c * (s * ↑j + t * ↑b) from calc ↑b _ = (1 : Int) * ↑b := (one_mul _).symm _ = (s * ↑a + t * ↑c) * ↑b := by rw [h4] _ = s * (↑a * ↑b) + t * ↑c * ↑b := by ring _ = s * (↑c * ↑j) + t * ↑c * ↑b := by rw [←Nat.cast_mul a b, h3, Nat.cast_mul c j] _ = ↑c * (s * ↑j + t * ↑b) := by ring done lemma le_nonzero_prod_left {a b : Nat} (h : a * b ≠ 0) : a ≤ a * b := by have h1 : b ≠ 0 := by contradict h with h1 rewrite [h1] ring done have h2 : 1 ≤ b := Nat.pos_of_ne_zero h1 show a ≤ a * b from calc a = a * 1 := (mul_one a).symm _ ≤ a * b := by rel [h2] done lemma le_nonzero_prod_right {a b : Nat} (h : a * b ≠ 0) : b ≤ a * b := by rewrite [mul_comm] rewrite [mul_comm] at h show b ≤ b * a from le_nonzero_prod_left h done lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry lemma rel_prime_of_prime_not_dvd {a p : Nat} (h1 : prime p) (h2 : ¬p ∣ a) : rel_prime a p := by have h3 : gcd a p ∣ a := gcd_dvd_left a p have h4 : gcd a p ∣ p := gcd_dvd_right a p have h5 : gcd a p = 1 ∨ gcd a p = p := dvd_prime h1 h4 have h6 : gcd a p ≠ p := by contradict h2 with h6 rewrite [h6] at h3 show p ∣ a from h3 done disj_syll h5 h6 show rel_prime a p from h5 done theorem Theorem_7_2_3 {a b p : Nat} (h1 : prime p) (h2 : p ∣ a * b) : p ∣ a ∨ p ∣ b := by or_right with h3 have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3 show p ∣ b from Theorem_7_2_2 h2 h4 done lemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a ≥ 1 := by have h1 : a ≠ 0 := by by_contra h1 rewrite [h1] at h contradict h linarith done show a ≥ 1 from Nat.pos_of_ne_zero h1 done lemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by have h1 : a ≥ 1 := ge_one_of_prod_one h have h2 : a * b ≠ 0 := by linarith have h3 : a ≤ a * b := le_nonzero_prod_left h2 rewrite [h] at h3 show a = 1 from Nat.le_antisymm h3 h1 done lemma eq_one_of_dvd_one {n : Nat} (h : n ∣ 1) : n = 1 := by obtain (j : Nat) (h1 : 1 = n * j) from h show n = 1 from eq_one_of_prod_one h1.symm done lemma prime_not_one {p : Nat} (h : prime p) : p ≠ 1 := by define at h linarith done theorem Theorem_7_2_4 {p : Nat} (h1 : prime p) : ∀ (l : List Nat), p ∣ prod l → ∃ a ∈ l, p ∣ a := by apply List.rec · -- Base Case. Goal : p ∣ prod [] → ∃ a ∈ [], p ∣ a rewrite [prod_nil] assume h2 : p ∣ 1 show ∃ a ∈ [], p ∣ a from absurd (eq_one_of_dvd_one h2) (prime_not_one h1) done · -- Induction Step fix b : Nat fix L : List Nat assume ih : p ∣ prod L → ∃ a ∈ L, p ∣ a --Goal : p ∣ prod (b :: L) → ∃ a ∈ b :: L, p ∣ a assume h2 : p ∣ prod (b :: L) rewrite [prod_cons] at h2 have h3 : p ∣ b ∨ p ∣ prod L := Theorem_7_2_3 h1 h2 by_cases on h3 · -- Case 1. h3 : p ∣ b apply Exists.intro b show b ∈ b :: L ∧ p ∣ b from And.intro (List.mem_cons_self b L) h3 done · -- Case 2. h3 : p ∣ prod L obtain (a : Nat) (h4 : a ∈ L ∧ p ∣ a) from ih h3 apply Exists.intro a show a ∈ b :: L ∧ p ∣ a from And.intro (List.mem_cons_of_mem b h4.left) h4.right done done done lemma prime_in_list {p : Nat} {l : List Nat} (h1 : prime p) (h2 : all_prime l) (h3 : p ∣ prod l) : p ∈ l := by obtain (a : Nat) (h4 : a ∈ l ∧ p ∣ a) from Theorem_7_2_4 h1 l h3 define at h2 have h5 : prime a := h2 a h4.left have h6 : p = 1 ∨ p = a := dvd_prime h5 h4.right disj_syll h6 (prime_not_one h1) rewrite [h6] show a ∈ l from h4.left done lemma first_le_first {p q : Nat} {l m : List Nat} (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m)) (h3 : prod (p :: l) = prod (q :: m)) : p ≤ q := by define at h1; define at h2 have h4 : q ∣ prod (p :: l) := by define apply Exists.intro (prod m) rewrite [←prod_cons] show prod (p :: l) = prod (q :: m) from h3 done have h5 : all_prime (q :: m) := h2.left rewrite [all_prime_cons] at h5 have h6 : q ∈ p :: l := prime_in_list h5.left h1.left h4 have h7 : nondec (p :: l) := h1.right rewrite [nondec_cons] at h7 rewrite [List.mem_cons] at h6 by_cases on h6 · -- Case 1. h6 : q = p linarith done · -- Case 2. h6 : q ∈ l have h8 : ∀ m ∈ l, p ≤ m := h7.left show p ≤ q from h8 q h6 done done lemma nondec_prime_list_tail {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by define at h define rewrite [all_prime_cons, nondec_cons] at h show all_prime l ∧ nondec l from And.intro h.left.right h.right.right done lemma cons_prod_not_one {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : prod (p :: l) ≠ 1 := by define at h have h1 : all_prime (p :: l) := h.left rewrite [all_prime_cons] at h1 rewrite [prod_cons] by_contra h2 show False from (prime_not_one h1.left) (eq_one_of_prod_one h2) done lemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) : l = [] ↔ prod l = 1 := by apply Iff.intro · -- (→) assume h1 : l = [] rewrite [h1] show prod [] = 1 from prod_nil done · -- (←) contrapos assume h1 : ¬l = [] obtain (p : Nat) (h2 : ∃ (L : List Nat), l = p :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List Nat) (h3 : l = p :: L) from h2 rewrite [h3] at h rewrite [h3] show ¬prod (p :: L) = 1 from cons_prod_not_one h done done lemma prime_pos {p : Nat} (h : prime p) : p > 0 := by define at h linarith done theorem Theorem_7_2_5 : ∀ (l1 l2 : List Nat), nondec_prime_list l1 → nondec_prime_list l2 → prod l1 = prod l2 → l1 = l2 := by apply List.rec · -- Base Case. Goal : ∀ (l2 : List Nat), nondec_prime_list [] → -- nondec_prime_list l2 → prod [] = prod l2 → [] = l2 fix l2 : List Nat assume h1 : nondec_prime_list [] assume h2 : nondec_prime_list l2 assume h3 : prod [] = prod l2 rewrite [prod_nil, eq_comm, ←list_nil_iff_prod_one h2] at h3 show [] = l2 from h3.symm done · -- Induction Step fix p : Nat fix L1 : List Nat assume ih : ∀ (L2 : List Nat), nondec_prime_list L1 → nondec_prime_list L2 → prod L1 = prod L2 → L1 = L2 -- Goal : ∀ (l2 : List Nat), nondec_prime_list (p :: L1) → -- nondec_prime_list l2 → prod (p :: L1) = prod l2 → p :: L1 = l2 fix l2 : List Nat assume h1 : nondec_prime_list (p :: L1) assume h2 : nondec_prime_list l2 assume h3 : prod (p :: L1) = prod l2 have h4 : ¬prod (p :: L1) = 1 := cons_prod_not_one h1 rewrite [h3, ←list_nil_iff_prod_one h2] at h4 obtain (q : Nat) (h5 : ∃ (L : List Nat), l2 = q :: L) from List.exists_cons_of_ne_nil h4 obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5 rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2) rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2) have h7 : p ≤ q := first_le_first h1 h2 h3 have h8 : q ≤ p := first_le_first h2 h1 h3.symm have h9 : p = q := by linarith rewrite [h9, prod_cons, prod_cons] at h3 --h3 : q * prod L1 = q * prod L2 have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1 have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2 define at h2 have h12 : all_prime (q :: L2) := h2.left rewrite [all_prime_cons] at h12 have h13 : q > 0 := prime_pos h12.left have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3 have h15 : L1 = L2 := ih L2 h10 h11 h14 rewrite [h6, h9, h15] rfl done done theorem fund_thm_arith (n : Nat) (h : n ≥ 1) : ∃! (l : List Nat), prime_factorization n l := by exists_unique · -- Existence show ∃ (l : List Nat), prime_factorization n l from exists_prime_factorization n h done · -- Uniqueness fix l1 : List Nat; fix l2 : List Nat assume h1 : prime_factorization n l1 assume h2 : prime_factorization n l2 define at h1; define at h2 have h3 : prod l1 = n := h1.right rewrite [←h2.right] at h3 show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3 done done /- Section 7.3 -/ theorem congr_refl (m : Nat) : ∀ (a : Int), a ≡ a (MOD m) := by fix a : Int define --Goal : ∃ (c : Int), a - a = ↑m * c apply Exists.intro 0 ring done theorem congr_symm {m : Nat} : ∀ {a b : Int}, a ≡ b (MOD m) → b ≡ a (MOD m) := by fix a : Int; fix b : Int assume h1 : a ≡ b (MOD m) define at h1 --h1 : ∃ (c : Int), a - b = ↑m * c define --Goal : ∃ (c : Int), b - a = ↑m * c obtain (c : Int) (h2 : a - b = m * c) from h1 apply Exists.intro (-c) show b - a = m * (-c) from calc b - a _ = -(a - b) := by ring _ = -(m * c) := by rw [h2] _ = m * (-c) := by ring done theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry /- Fundamental properties of congruence classes -/ lemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) : X = Y ↔ X.val = Y.val := Fin.ext_iff lemma val_nat_eq_mod (n k : Nat) : ([k]_(n + 1)).val = k % (n + 1) := by rfl lemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl theorem cc_rep {m : Nat} (X : ZMod m) : ∃ (a : Int), X = [a]_m := match m with | 0 => by apply Exists.intro X rfl done | n + 1 => by apply Exists.intro ↑(X.val) have h1 : X.val < n + 1 := Fin.prop X rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1] rfl done theorem add_class (m : Nat) (a b : Int) : [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm theorem mul_class (m : Nat) (a b : Int) : [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm lemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ [a - b]_m = [0]_m := by apply Iff.intro · -- (→) assume h1 : [a]_m = [b]_m have h2 : a - b = a + (-b) := by ring have h3 : b + (-b) = 0 := by ring show [a - b]_m = [0]_m from calc [a - b]_m _ = [a + (-b)]_m := by rw [h2] _ = [a]_m + [-b]_m := by rw [add_class] _ = [b]_m + [-b]_m := by rw [h1] _ = [b + -b]_m := by rw [add_class] _ = [0]_m := by rw [h3] done · -- (←) assume h1 : [a - b]_m = [0]_m have h2 : b + (a - b) = a := by ring have h3 : b + 0 = b := by ring show [a]_m = [b]_m from calc [a]_m _ = [b + (a - b)]_m := by rw [h2] _ = [b]_m + [a - b]_m := by rw [add_class] _ = [b]_m + [0]_m := by rw [h1] _ = [b + 0]_m := by rw [add_class] _ = [b]_m := by rw [h3] done done lemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) : [a]_m = [0]_m → [-a]_m = [0]_m := by assume h1 : [a]_m = [0]_m have h2 : 0 + (-a) = -a := by ring have h3 : a + (-a) = 0 := by ring show [-a]_m = [0]_m from calc [-a]_m _ = [0 + (-a)]_m := by rw [h2] _ = [0]_m + [-a]_m := by rw [add_class] _ = [a]_m + [-a]_m := by rw [h1] _ = [a + (-a)]_m := by rw [add_class] _ = [0]_m := by rw [h3] done lemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) : [-a]_m = [0]_m ↔ [a]_m = [0]_m := by apply Iff.intro _ (cc_neg_zero_of_cc_zero m a) assume h1 : [-a]_m = [0]_m have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1 have h3 : -(-a) = a := by ring rewrite [h3] at h2 show [a]_m = [0]_m from h2 done lemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl lemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m ↔ m ∣ k := match m with | 0 => by have h : (0 : Int) = (↑(0 : Nat) : Int) := by rfl rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj] apply Iff.intro · -- (→) assume h1 : k = 0 rewrite [h1] show 0 ∣ 0 from dvd_self 0 done · -- (←) assume h1 : 0 ∣ k obtain (c : Nat) (h2 : k = 0 * c) from h1 rewrite [h2] ring done done | n + 1 => by rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero] show k % (n + 1) = 0 ↔ n + 1 ∣ k from (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm done lemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m ↔ ↑m ∣ a := by obtain (k : Nat) (h1 : a = ↑k ∨ a = -↑k) from Int.eq_nat_or_neg a by_cases on h1 · -- Case 1. h1: a = ↑k rewrite [h1, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done · -- Case 2. h1: a = -↑k rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done done theorem cc_eq_iff_congr (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ a ≡ b (MOD m) := calc [a]_m = [b]_m _ ↔ [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b _ ↔ ↑m ∣ (a - b) := cc_zero_iff_dvd m (a - b) _ ↔ a ≡ b (MOD m) := by rfl /- End of fundamental properties of congruence classes -/ lemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m := by have h1 : (↑m : Int) ≠ 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m) show 0 ≤ a % m from Int.emod_nonneg a h1 done lemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m) have h2 : (↑m : Int) > 0 := (Nat.cast_pos).rtl h1 show a % m < m from Int.emod_lt_of_pos a h2 done lemma congr_mod_mod (m : Nat) (a : Int) : a ≡ a % m (MOD m) := by define have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m apply Exists.intro (a / m) show a - a % m = m * (a / m) from calc a - (a % m) _ = m * (a / m) + a % m - a % m := by rw [h1] _ = m * (a / m) := by ring done lemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) := And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a)) theorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) : ∃! (r : Int), 0 ≤ r ∧ r < m ∧ a ≡ r (MOD m) := by exists_unique · -- Existence apply Exists.intro (a % m) show 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) from mod_cmpl_res m a done · -- Uniqueness fix r1 : Int; fix r2 : Int assume h1 : 0 ≤ r1 ∧ r1 < m ∧ a ≡ r1 (MOD m) assume h2 : 0 ≤ r2 ∧ r2 < m ∧ a ≡ r2 (MOD m) have h3 : r1 ≡ r2 (MOD m) := congr_trans (congr_symm h1.right.right) h2.right.right obtain (d : Int) (h4 : r1 - r2 = m * d) from h3 have h5 : r1 - r2 < m * 1 := by linarith have h6 : m * (-1) < r1 - r2 := by linarith rewrite [h4] at h5 --h5 : m * d < m * 1 rewrite [h4] at h6 --h6 : m * -1 < m * d have h7 : (↑m : Int) ≥ 0 := Nat.cast_nonneg m have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7 have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7 have h10 : d = 0 := by linarith show r1 = r2 from calc r1 _ = r1 - r2 + r2 := by ring _ = m * 0 + r2 := by rw [h4, h10] _ = r2 := by ring done done lemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m := (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a) theorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y rewrite [h1, h2] have h3 : a + b = b + a := by ring show [a]_m + [b]_m = [b]_m + [a]_m from calc [a]_m + [b]_m _ = [a + b]_m := add_class m a b _ = [b + a]_m := by rw [h3] _ = [b]_m + [a]_m := (add_class m b a).symm done theorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X rewrite [h1] have h2 : a * 1 = a := by ring show [a]_m * [1]_m = [a]_m from calc [a]_m * [1]_m _ = [a * 1]_m := mul_class m a 1 _ = [a]_m := by rw [h2] done theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry lemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) : [a]_m * [gcd_c2 m a]_m = [1]_m := by set s : Int := gcd_c1 m a have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m define at h1 rewrite [h1, Nat.cast_one] at h2 --h2 : s * ↑m + gcd_c2 m a * ↑a = 1 rewrite [mul_class, cc_eq_iff_congr] define --Goal : ∃ (c : Int), ↑a * gcd_c2 m a - 1 = ↑m * c apply Exists.intro (-s) show a * (gcd_c2 m a) - 1 = m * (-s) from calc a * (gcd_c2 m a) - 1 _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring _ = 1 + m * (-s) - 1 := by rw [h2] _ = m * (-s) := by ring done theorem Theorem_7_3_7 (m a : Nat) : invertible [a]_m ↔ rel_prime m a := by apply Iff.intro · -- (→) assume h1 : invertible [a]_m define at h1 obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1 obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y rewrite [h3, mul_class, cc_eq_iff_congr] at h2 define at h2 obtain (c : Int) (h4 : a * b - 1 = m * c) from h2 rewrite [Exercise_7_2_6] --Goal : ∃ (s t : Int), s * ↑m + t * ↑a = 1 apply Exists.intro (-c) apply Exists.intro b show (-c) * m + b * a = 1 from calc (-c) * m + b * a _ = (-c) * m + (a * b - 1) + 1 := by ring _ = (-c) * m + m * c + 1 := by rw [h4] _ = 1 := by ring done · -- (←) assume h1 : rel_prime m a define show ∃ (Y : ZMod m), [a]_m * Y = [1]_m from Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1) done done /- Section 7.4 -/ section Euler open Euler lemma num_rp_below_base {m : Nat} : num_rp_below m 0 = 0 := by rfl lemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) : num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by have h1 : num_rp_below m (j + 1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : gcd m j = 1 rewrite [if_pos h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1 show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1 done lemma num_rp_below_step_not_rp {m j : Nat} (h : ¬rel_prime m j) : num_rp_below m (j + 1) = num_rp_below m j := by have h1 : num_rp_below m (j +1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : ¬gcd m j = 1 rewrite [if_neg h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j show num_rp_below m (j + 1) = num_rp_below m j from h1 done lemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl #eval phi 10 --Answer: 4 lemma prod_inv_iff_inv {m : Nat} {X : ZMod m} (h1 : invertible X) (Y : ZMod m) : invertible (X * Y) ↔ invertible Y := by apply Iff.intro · -- (→) assume h2 : invertible (X * Y) obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2 apply Exists.intro (X * Z) rewrite [←h3] --Goal : Y * (X * Z) = X * Y * Z ring --Note that ring can do algebra in ZMod m done · -- (←) assume h2 : invertible Y obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1 obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2 apply Exists.intro (Xi * Yi) show (X * Y) * (Xi * Yi) = [1]_m from calc X * Y * (Xi * Yi) _ = (X * Xi) * (Y * Yi) := by ring _ = [1]_m * [1]_m := by rw [h3, h4] _ = [1]_m := Theorem_7_3_6_7 [1]_m done done lemma F_rp_def {m i : Nat} (h : rel_prime m i) : F m i = [i]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h --h : gcd m i = 1 rewrite [if_pos h] at h1 show F m i = [i]_m from h1 done lemma F_not_rp_def {m i : Nat} (h : ¬rel_prime m i) : F m i = [1]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h rewrite [h1, if_neg h] rfl done lemma prod_seq_base {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 0 k f = [1]_m := by rfl lemma prod_seq_step {m : Nat} (n k : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl lemma prod_seq_zero_step {m : Nat} (n : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by rewrite [prod_seq_step, zero_add] rfl done lemma prod_one {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 1 k f = f k := by rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7] rfl done lemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl lemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m := calc [G m a i]_m _ = [(a * i) % m]_m := by rfl _ = [a * i]_m := (cc_eq_mod m (a * i)).symm _ = [a]_m * [i]_m := (mul_class m a i).symm lemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) : rel_prime m (G m a i) ↔ rel_prime m i := by have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1 show rel_prime m (G m a i) ↔ rel_prime m i from calc rel_prime m (G m a i) _ ↔ invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm _ ↔ invertible ([a]_m * [i]_m) := by rw [cc_G] _ ↔ invertible [i]_m := prod_inv_iff_inv h2 ([i]_m) _ ↔ rel_prime m i := Theorem_7_3_7 m i done lemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) : F m (G m a i) = [a]_m * F m i := by have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2 show F m (G m a i) = [a]_m * F m i from calc F m (G m a i) _ = [G m a i]_m := F_rp_def h3 _ = [a]_m * [i]_m := cc_G m a i _ = [a]_m * F m i := by rw [F_rp_def h2] done lemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : ¬rel_prime m i) : F m (G m a i) = [1]_m := by rewrite [←G_rp_iff h1 i] at h2 show F m (G m a i) = [1]_m from F_not_rp_def h2 done lemma FG_prod {m a : Nat} (h1 : rel_prime m a) : ∀ (k : Nat), prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by by_induc · -- Base Case show prod_seq 0 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from calc prod_seq 0 0 ((F m) ∘ (G m a)) _ = [1]_m := prod_seq_base _ _ _ = [a]_m ^ 0 * [1]_m := by ring _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by rw [num_rp_below_base, prod_seq_base] done · -- Induction Step fix k : Nat assume ih : prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) by_cases h2 : rel_prime m k · -- Case 1. h2 : rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([a]_m * F m k) := by rw [FG_rp h1 h2] _ = [a]_m ^ ((num_rp_below m k) + 1) * ((prod_seq k 0 (F m)) * F m k) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_rp h2, prod_seq_zero_step] done · -- Case 2. h2 : ¬rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([1]_m) := by rw [FG_not_rp h1 h2] _ = [a]_m ^ (num_rp_below m k) * (prod_seq k 0 (F m) * ([1]_m)) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_not_rp h2, prod_seq_zero_step, F_not_rp_def h2] done done done lemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by define --Goal : ∀ i < m, G m a i < m fix i : Nat assume h1 : i < m rewrite [G_def] --Goal : a * i % m < m show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m) done lemma left_inv_one_one_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g' (g i) = i) : one_one_below n g := sorry lemma right_inv_onto_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g (g' i) = i) (h2 : maps_below n g') : onto_below n g := by define at h2; define fix k : Nat assume h3 : k < n apply Exists.intro (g' k) show g' k < n ∧ g (g' k) = k from And.intro (h2 k h3) (h1 k h3) done lemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m * [inv_mod m a]_m = [1]_m := by have h2 : 0 ≤ (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a) show [a]_m * [inv_mod m a]_m = [1]_m from calc [a]_m * [inv_mod m a]_m _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2] _ = [a]_m * [gcd_c2 m a]_m := by rw [←cc_eq_mod] _ = [1]_m := gcd_c2_inv h1 done lemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m := calc a * (b % m) % m = a % m * (b % m % m) % m := Nat.mul_mod _ _ _ _ = a % m * (b % m) % m := by rw [Nat.mod_mod] _ = a * b % m := (Nat.mul_mod _ _ _).symm lemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm] rfl done theorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] : ↑a ≡ ↑b (MOD m) ↔ a % m = b % m := sorry lemma mul_inv_mod_cancel {m a i : Nat} [NeZero m] (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1 rewrite [mul_class, cc_eq_iff_congr, ←Nat.cast_mul, ←Nat.cast_one, congr_iff_mod_eq_Nat] at h3 show a * inv_mod m a * i % m = i from calc a * (inv_mod m a) * i % m _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = 1 % m * i % m := by rw [h3] _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = i % m := by rw [one_mul] _ = i := Nat.mod_eq_of_lt h2 done lemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl lemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, G m a (Ginv m a i) = i := by fix i : Nat assume h2 : i < m show G m a (Ginv m a i) = i from calc G m a (Ginv m a i) _ = a * ((inv_mod m a * i) % m) % m := by rfl _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, Ginv m a (G m a i) = i := by fix i : Nat assume h2 : i < m show Ginv m a (G m a i) = i from calc Ginv m a (G m a i) _ = inv_mod m a * ((a * i) % m) % m := by rfl _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc, mul_comm (inv_mod m a)] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_maps_below (m a : Nat) [NeZero m] : maps_below m (Ginv m a) := G_maps_below m (inv_mod m a) lemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : one_one_below m (G m a) := left_inv_one_one_below (Ginv_left_inv h1) lemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : onto_below m (G m a) := right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a) lemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : perm_below m (G m a) := And.intro (G_maps_below m a) (And.intro (G_one_one_below h1) (G_onto_below h1)) --Permuting a product of congruence classes doesn't change product lemma swap_fst (u v : Nat) : swap u v u = v := by define : swap u v u --Goal : (if u = u then v else if u = v then u else u) = v have h : u = u := by rfl rewrite [if_pos h] rfl done lemma swap_snd (u v : Nat) : swap u v v = u := by define : swap u v v by_cases h1 : v = u · -- Case 1. h1 : v = u rewrite [if_pos h1] show v = u from h1 done · -- Case 2. h1 : v ≠ u rewrite [if_neg h1] have h2 : v = v := by rfl rewrite [if_pos h2] rfl done done lemma swap_other {u v i : Nat} (h1 : i ≠ u) (h2 : i ≠ v) : swap u v i = i := by define : swap u v i rewrite [if_neg h1, if_neg h2] rfl done lemma swap_values (u v i : Nat) : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := by by_cases h1 : i = u · -- Case 1. h1 : i = u apply Or.inl rewrite [h1] show swap u v u = v from swap_fst u v done · -- Case 2. h1 : i ≠ u apply Or.inr by_cases h2 : i = v · -- Case 2.1. h2 : i = v apply Or.inl rewrite [h2] show swap u v v = u from swap_snd u v done · -- Case 2.2. h2 : i ≠ v apply Or.inr show swap u v i = i from swap_other h1 h2 done done done lemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by define fix i : Nat assume h3 : i < n have h4 : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := swap_values u v i by_cases on h4 · -- Case 1. h4 : swap u v i = v rewrite [h4] show v < n from h2 done · -- Case 2. by_cases on h4 · -- Case 2.1. h4 : swap u v i = u rewrite [h4] show u < n from h1 done · -- Case 2.2. h4 : swap u v i = i rewrite [h4] show i < n from h3 done done done lemma swap_swap (u v n : Nat) : ∀ i < n, swap u v (swap u v i) = i := by fix i : Nat assume h : i < n by_cases h1 : i = u · -- Case 1. h1 : i = u rewrite [h1, swap_fst, swap_snd] rfl done · -- Case 2. h1 : i ≠ u by_cases h2 : i = v · -- Case 2.1. h2 : i = v rewrite [h2, swap_snd, swap_fst] rfl done · -- Case 2.2. h2 : i ≠ v rewrite [swap_other h1 h2, swap_other h1 h2] rfl done done done lemma swap_one_one_below (u v n) : one_one_below n (swap u v) := left_inv_one_one_below (swap_swap u v n) lemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) := right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2) lemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) := And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2)) lemma comp_perm_below {n : Nat} {f g : Nat → Nat} (h1 : perm_below n f) (h2 : perm_below n g) : perm_below n (f ∘ g) := sorry lemma trivial_swap (u : Nat) : swap u u = id := by apply funext fix x : Nat by_cases h1 : x = u · -- Case 1. h1 : x = u rewrite [h1, swap_fst] rfl done · -- Case 2. h1 : x ≠ u rewrite [swap_other h1 h1] rfl done done lemma prod_eq_fun {m : Nat} (f g : Nat → ZMod m) (k : Nat) : ∀ (n : Nat), (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g := by by_induc · -- Base Case assume h : (∀ i < 0, f (k + i) = g (k + i)) rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g assume h1 : ∀ i < n + 1, f (k + i) = g (k + i) have h2 : ∀ i < n, f (k + i) = g (k + i) := by fix i : Nat assume h2 : i < n have h3 : i < n + 1 := by linarith show f (k + i) = g (k + i) from h1 i h3 done have h3 : prod_seq n k f = prod_seq n k g := ih h2 have h4 : n < n + 1 := Nat.lt_succ_self n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4] rfl done done lemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := by have h2 : ∀ (i : Nat), i < u → (f ∘ swap u n) (0 + i) = f (0 + i) := by fix i : Nat assume h2 : i < u have h3 : 0 + i ≠ u := by linarith have h4 : 0 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl done show prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f from prod_eq_fun (f ∘ swap u n) f 0 u h2 done lemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := by have h2 : ∀ i < j, (f ∘ swap u n) (u + 1 + i) = f (u + 1 + i) := by fix i : Nat assume h2 : i < j have h3 : u + 1 + i ≠ u := by linarith have h4 : u + 1 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl show prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f from prod_eq_fun (f ∘ swap u n) f (u + 1) j h2 done lemma break_prod {m : Nat} (n : Nat) (f : Nat → ZMod m) : ∀ (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by by_induc · -- Base Case have h : n + 0 = n := by rfl rewrite [prod_seq_base, h, Theorem_7_3_6_7] rfl done · -- Induction Step fix j : Nat assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f rewrite [←add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc] rfl done done lemma break_prod_twice {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f := break_prod n f 1 rewrite [prod_one] at h2 have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f := break_prod (u + 1) f j rewrite [←h1] at h3 have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f := break_prod u f 1 rewrite [prod_one] at h4 rewrite [h3, h4] at h2 show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2 done lemma swap_prod_eq_prod {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f := by by_cases h2 : u = n · -- Case 1. h2 : u = n rewrite [h2, trivial_swap n] --Goal : prod_seq (n + 1) 0 (f ∘ id) = prod_seq (n + 1) 0 f rfl done · -- Case 2. h2 : ¬u = n have h3 : u + 1 ≤ n := Nat.lt_of_le_of_ne h1 h2 obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3 have break_f : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := break_prod_twice f h4 have break_fs : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_prod_twice (f ∘ swap u n) h4 have f_eq_fs_below : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := swap_prod_eq_prod_below f h1 have f_eq_fs_btwn : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4 show prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f from calc prod_seq (n + 1) 0 (f ∘ swap u n) _ = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_fs _ = prod_seq u 0 f * (f ∘ swap u n) u * prod_seq j (u + 1) f * (f ∘ swap u n) n := by rw [f_eq_fs_below, f_eq_fs_btwn] _ = prod_seq u 0 f * f (swap u n u) * prod_seq j (u + 1) f * f (swap u n n) := by rfl _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by rw [swap_fst, swap_snd] _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring _ = prod_seq (n + 1) 0 f := break_f.symm done done lemma perm_below_fixed {n : Nat} {g : Nat → Nat} (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry lemma perm_prod {m : Nat} (f : Nat → ZMod m) : ∀ (n : Nat), ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) := by by_induc · -- Base Case fix g : Nat → Nat assume h1 : perm_below 0 g rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) fix g : Nat → Nat assume g_pb : perm_below (n + 1) g define at g_pb have g_ob : onto_below (n + 1) g := g_pb.right.right define at g_ob have h1 : n < n + 1 := by linarith obtain (u : Nat) (h2 : u < n + 1 ∧ g u = n) from g_ob n h1 have s_pb : perm_below (n + 1) (swap u n) := swap_perm_below h2.left h1 have gs_pb_n1 : perm_below (n + 1) (g ∘ swap u n) := comp_perm_below g_pb s_pb have gs_fix_n : (g ∘ swap u n) n = n := calc (g ∘ swap u n) n _ = g (swap u n n) := by rfl _ = g u := by rw [swap_snd] _ = n := h2.right have gs_pb_n : perm_below n (g ∘ swap u n) := perm_below_fixed gs_pb_n1 gs_fix_n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f ∘ (g ∘ swap u n)) := ih (g ∘ swap u n) gs_pb_n have h3 : u ≤ n := by linarith show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f ∘ g) from calc prod_seq (n + 1) 0 f _ = prod_seq n 0 f * f n := prod_seq_zero_step n f _ = prod_seq n 0 (f ∘ (g ∘ swap u n)) * f ((g ∘ swap u n) n) := by rw [gs_prod, gs_fix_n] _ = prod_seq n 0 (f ∘ g ∘ swap u n) * (f ∘ g ∘ swap u n) n := by rfl _ = prod_seq (n + 1) 0 (f ∘ g ∘ swap u n) := (prod_seq_zero_step n (f ∘ g ∘ swap u n)).symm _ = prod_seq (n + 1) 0 ((f ∘ g) ∘ swap u n) := by rfl _ = prod_seq (n + 1) 0 (f ∘ g) := swap_prod_eq_prod (f ∘ g) h3 done done lemma F_invertible (m i : Nat) : invertible (F m i) := by by_cases h : rel_prime m i · -- Case 1. h : rel_prime m i rewrite [F_rp_def h] show invertible [i]_m from (Theorem_7_3_7 m i).rtl h done · -- Case 2. h : ¬rel_prime m i rewrite [F_not_rp_def h] apply Exists.intro [1]_m show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m done done lemma Fprod_invertible (m : Nat) : ∀ (k : Nat), invertible (prod_seq k 0 (F m)) := by by_induc · -- Base Case apply Exists.intro [1]_m show prod_seq 0 0 (F m) * [1]_m = [1]_m from calc prod_seq 0 0 (F m) * [1]_m _ = [1]_m * [1]_m := by rw [prod_seq_base] _ = [1]_m := Theorem_7_3_6_7 ([1]_m) done · -- Induction Step fix k : Nat assume ih : invertible (prod_seq k 0 (F m)) rewrite [prod_seq_zero_step] show invertible (prod_seq k 0 (F m) * (F m k)) from (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k) done done theorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m ^ (phi m) = [1]_m := by have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2 show [a]_m ^ (phi m) = [1]_m from calc [a]_m ^ (phi m) _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3] _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring _ = prod_seq m 0 (F m ∘ G m a) * Y := by rw [FG_prod h1 m, phi_def] _ = prod_seq m 0 (F m) * Y := by rw [perm_prod (F m) m (G m a) (G_perm_below h1)] _ = [1]_m := by rw [h3] done lemma Exercise_7_4_5_Int (m : Nat) (a : Int) : ∀ (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry lemma Exercise_7_4_5_Nat (m a n : Nat) : [a]_m ^ n = [a ^ n]_m := by rewrite [Exercise_7_4_5_Int] rfl done theorem Euler's_theorem {m a : Nat} [NeZero m] (h1 : rel_prime m a) : a ^ (phi m) ≡ 1 (MOD m) := by have h2 : [a]_m ^ (phi m) = [1]_m := Theorem_7_4_2 h1 rewrite [Exercise_7_4_5_Nat m a (phi m)] at h2 --h2 : [a ^ phi m]_m = [1]_m show a ^ (phi m) ≡ 1 (MOD m) from (cc_eq_iff_congr _ _ _).ltr h2 done #eval gcd 10 7 --Answer: 1. So 10 and 7 are relatively prime #eval 7 ^ phi 10 --Answer: 2401, which is congruent to 1 mod 10. end Euler /- Section 7.5 -/ lemma num_rp_prime {p : Nat} (h1 : prime p) : ∀ k < p, num_rp_below p (k + 1) = k := sorry lemma phi_prime {p : Nat} (h1 : prime p) : phi p = p - 1 := by have h2 : 1 ≤ p := prime_pos h1 have h3 : p - 1 + 1 = p := Nat.sub_add_cancel h2 have h4 : p - 1 < p := by linarith have h5 : num_rp_below p (p - 1 + 1) = p - 1 := num_rp_prime h1 (p - 1) h4 rewrite [h3] at h5 show phi p = p - 1 from h5 done

theorem Theorem_7_2_2_Int {a c : Nat} {b : Int} (h1 : ↑c ∣ ↑a * b) (h2 : rel_prime a c) : ↑c ∣ b

HTPI.Theorem_7_2_2_Int

htpi/HTPILib/Chap7.lean

htpi

/- Copyright 2023 Daniel J. Velleman -/ import HTPILib.Chap6 namespace HTPI /- Definitions -/ lemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by have h : n + 1 > 0 := Nat.succ_pos n show a % (n + 1) < n + 1 from Nat.mod_lt a h done def gcd (a b : Nat) : Nat := match b with | 0 => a | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd (n + 1) (a % (n + 1)) termination_by b mutual def gcd_c1 (a b : Nat) : Int := match b with | 0 => 1 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c2 (n + 1) (a % (n + 1)) --Corresponds to s = t' termination_by b def gcd_c2 (a b : Nat) : Int := match b with | 0 => 0 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c1 (n + 1) (a % (n + 1)) - (gcd_c2 (n + 1) (a % (n + 1))) * ↑(a / (n + 1)) --Corresponds to t = s' - t'q termination_by b end def prime (n : Nat) : Prop := 2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n def prime_factor (p n : Nat) : Prop := prime p ∧ p ∣ n def all_prime (l : List Nat) : Prop := ∀ p ∈ l, prime p def nondec (l : List Nat) : Prop := match l with | [] => True --Of course, True is a proposition that is always true | n :: L => (∀ m ∈ L, n ≤ m) ∧ nondec L def nondec_prime_list (l : List Nat) : Prop := all_prime l ∧ nondec l def prod (l : List Nat) : Nat := match l with | [] => 1 | n :: L => n * (prod L) def prime_factorization (n : Nat) (l : List Nat) : Prop := nondec_prime_list l ∧ prod l = n def rel_prime (a b : Nat) : Prop := gcd a b = 1 def congr_mod (m : Nat) (a b : Int) : Prop := (↑m : Int) ∣ (a - b) def cc (m : Nat) (a : Int) : ZMod m := (↑a : ZMod m) notation:50 a " ≡ " b " (MOD " m ")" => congr_mod m a b notation:max "["a"]_"m:max => cc m a def invertible {m : Nat} (X : ZMod m) : Prop := ∃ (Y : ZMod m), X * Y = [1]_m def num_rp_below (m k : Nat) : Nat := match k with | 0 => 0 | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j def phi (m : Nat) : Nat := num_rp_below m m def prod_seq {m : Nat} (j k : Nat) (f : Nat → ZMod m) : ZMod m := match j with | 0 => [1]_m | n + 1 => prod_seq n k f * f (k + n) def maps_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i < n, g i < n def one_one_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i1 < n, ∀ i2 < n, g i1 = g i2 → i1 = i2 def onto_below (n : Nat) (g : Nat → Nat) : Prop := ∀ k < n, ∃ i < n, g i = k def perm_below (n : Nat) (g : Nat → Nat) : Prop := maps_below n g ∧ one_one_below n g ∧ onto_below n g def inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m) def swap (u v i : Nat) : Nat := if i = u then v else if i = v then u else i namespace Euler --For definitions specific to Euler's theorem def F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m def G (m a i : Nat) : Nat := (a * i) % m def Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i end Euler /- Section 7.1 -/ theorem dvd_mod_of_dvd_a_b {a b d : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ (a % b) := by set q : Nat := a / b have h3 : b * q + a % b = a := Nat.div_add_mod a b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 define --Goal : ∃ (c : Nat), a % b = d * c apply Exists.intro (j - k * q) show a % b = d * (j - k * q) from calc a % b _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm _ = a - b * q := by rw [h3] _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc] _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm done theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry #eval gcd 672 161 --Answer: 7 lemma gcd_base (a : Nat) : gcd a 0 = a := by rfl lemma gcd_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd a b = gcd b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1)) rfl done lemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b ≠ 0) : a % b < b := by have h1 : b > 0 := Nat.pos_of_ne_zero h show a % b < b from Nat.mod_lt a h1 done lemma dvd_self (n : Nat) : n ∣ n := by apply Exists.intro 1 ring done theorem gcd_dvd : ∀ (b a : Nat), (gcd a b) ∣ a ∧ (gcd a b) ∣ b := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd a b_1) ∣ a ∧ (gcd a b_1) ∣ b_1 fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_base] --Goal: a ∣ a ∧ a ∣ 0 apply And.intro (dvd_self a) define apply Exists.intro 0 rfl done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_nonzero a h1] --Goal : gcd b (a % b) ∣ a ∧ gcd b (a % b) ∣ b have h2 : a % b < b := mod_nonzero_lt a h1 have h3 : (gcd b (a % b)) ∣ b ∧ (gcd b (a % b)) ∣ (a % b) := ih (a % b) h2 b apply And.intro _ h3.left show (gcd b (a % b)) ∣ a from dvd_a_of_dvd_b_mod h3.left h3.right done done theorem gcd_dvd_left (a b : Nat) : (gcd a b) ∣ a := (gcd_dvd b a).left theorem gcd_dvd_right (a b : Nat) : (gcd a b) ∣ b := (gcd_dvd b a).right lemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl lemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c1 a b = gcd_c2 b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done lemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl lemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * ↑(a / b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done theorem gcd_lin_comb : ∀ (b a : Nat), (gcd_c1 a b) * ↑a + (gcd_c2 a b) * ↑b = ↑(gcd a b) := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd_c1 a b_1) * ↑a + (gcd_c2 a b_1) * ↑b_1 = ↑(gcd a b_1) fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base] --Goal : 1 * ↑a + 0 * ↑0 = ↑a ring done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1] --Goal : gcd_c2 b (a % b) * ↑a + -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * ↑(a / b)) * ↑b = -- ↑(gcd b (a % b)) set r : Nat := a % b set q : Nat := a / b set s : Int := gcd_c1 b r set t : Int := gcd_c2 b r --Goal : t * ↑a + (s - t * ↑q) * ↑b = ↑(gcd b r) have h2 : r < b := mod_nonzero_lt a h1 have h3 : s * ↑b + t * ↑r = ↑(gcd b r) := ih r h2 b have h4 : b * q + r = a := Nat.div_add_mod a b rewrite [←h3, ←h4] rewrite [Nat.cast_add, Nat.cast_mul] --Goal : t * (↑b * ↑q + ↑r) + (s - t * ↑q) * ↑b = s * ↑b + t * ↑r ring done done #eval gcd_c1 672 161 --Answer: 6 #eval gcd_c2 672 161 --Answer: -25 --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161 theorem Theorem_7_1_6 {d a b : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ gcd a b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑d ∣ ↑(gcd a b) set s : Int := gcd_c1 a b set t : Int := gcd_c2 a b have h3 : s * ↑a + t * ↑b = ↑(gcd a b) := gcd_lin_comb b a rewrite [←h3] --Goal : ↑d ∣ s * ↑a + t * ↑b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul] --Goal : ↑d ∣ s * (↑d * ↑j) + t * (↑d * ↑k) define apply Exists.intro (s * ↑j + t * ↑k) ring done /- Section 7.2 -/ theorem dvd_trans {a b c : Nat} (h1 : a ∣ b) (h2 : b ∣ c) : a ∣ c := by define at h1; define at h2; define obtain (m : Nat) (h3 : b = a * m) from h1 obtain (n : Nat) (h4 : c = b * n) from h2 rewrite [h3, mul_assoc] at h4 apply Exists.intro (m * n) show c = a * (m * n) from h4 done lemma exists_prime_factor : ∀ (n : Nat), 2 ≤ n → ∃ (p : Nat), prime_factor p n := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, 2 ≤ n_1 → ∃ (p : Nat), prime_factor p n_1 assume h1 : 2 ≤ n by_cases h2 : prime n · -- Case 1. h2 : prime n apply Exists.intro n define --Goal : prime n ∧ n ∣ n show prime n ∧ n ∣ n from And.intro h2 (dvd_self n) done · -- Case 2. h2 : ¬prime n define at h2 --h2 : ¬(2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n) demorgan at h2 disj_syll h2 h1 obtain (a : Nat) (h3 : ∃ (b : Nat), a * b = n ∧ a < n ∧ b < n) from h2 obtain (b : Nat) (h4 : a * b = n ∧ a < n ∧ b < n) from h3 have h5 : 2 ≤ a := by by_contra h6 have h7 : a ≤ 1 := by linarith have h8 : n ≤ b := calc n _ = a * b := h4.left.symm _ ≤ 1 * b := by rel [h7] _ = b := by ring linarith --n ≤ b contradicts b < n done have h6 : ∃ (p : Nat), prime_factor p a := ih a h4.right.left h5 obtain (p : Nat) (h7 : prime_factor p a) from h6 apply Exists.intro p define --Goal : prime p ∧ p ∣ n define at h7 --h7 : prime p ∧ p ∣ a apply And.intro h7.left have h8 : a ∣ n := by apply Exists.intro b show n = a * b from (h4.left).symm done show p ∣ n from dvd_trans h7.right h8 done done lemma exists_least_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q := by set S : Set Nat := {p : Nat | prime_factor p n} have h2 : ∃ (p : Nat), p ∈ S := exists_prime_factor n h show ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q from well_ord_princ S h2 done lemma all_prime_nil : all_prime [] := by define --Goal : ∀ p ∈ [], prime p fix p : Nat contrapos --Goal : ¬prime p → p ∉ [] assume h1 : ¬prime p show p ∉ [] from List.not_mem_nil p done lemma all_prime_cons (n : Nat) (L : List Nat) : all_prime (n :: L) ↔ prime n ∧ all_prime L := by apply Iff.intro · -- (→) assume h1 : all_prime (n :: L) --Goal : prime n ∧ all_prime L define at h1 --h1 : ∀ p ∈ n :: L, prime p apply And.intro (h1 n (List.mem_cons_self n L)) define --Goal : ∀ p ∈ L, prime p fix p : Nat assume h2 : p ∈ L show prime p from h1 p (List.mem_cons_of_mem n h2) done · -- (←) assume h1 : prime n ∧ all_prime L --Goal : all_prime (n :: l) define : all_prime L at h1 define fix p : Nat assume h2 : p ∈ n :: L rewrite [List.mem_cons] at h2 --h2 : p = n ∨ p ∈ L by_cases on h2 · -- Case 1. h2 : p = n rewrite [h2] show prime n from h1.left done · -- Case 2. h2 : p ∈ L show prime p from h1.right p h2 done done done lemma nondec_nil : nondec [] := by define --Goal : True trivial --trivial proves some obviously true statements, such as True done lemma nondec_cons (n : Nat) (L : List Nat) : nondec (n :: L) ↔ (∀ m ∈ L, n ≤ m) ∧ nondec L := by rfl lemma prod_nil : prod [] = 1 := by rfl lemma prod_cons : prod (n :: L) = n * (prod L) := by rfl lemma exists_cons_of_length_eq_succ {A : Type} {l : List A} {n : Nat} (h : l.length = n + 1) : ∃ (a : A) (L : List A), l = a :: L ∧ L.length = n := by have h1 : ¬l.length = 0 := by linarith rewrite [List.length_eq_zero] at h1 obtain (a : A) (h2 : ∃ (L : List A), l = a :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List A) (h3 : l = a :: L) from h2 apply Exists.intro a apply Exists.intro L apply And.intro h3 have h4 : (a :: L).length = L.length + 1 := List.length_cons a L rewrite [←h3, h] at h4 show L.length = n from (Nat.add_right_cancel h4).symm done lemma list_elt_dvd_prod_by_length (a : Nat) : ∀ (n : Nat), ∀ (l : List Nat), l.length = n → a ∈ l → a ∣ prod l := by by_induc · --Base Case fix l : List Nat assume h1 : l.length = 0 rewrite [List.length_eq_zero] at h1 --h1 : l = [] rewrite [h1] --Goal : a ∈ [] → a ∣ prod [] contrapos assume h2 : ¬a ∣ prod [] show a ∉ [] from List.not_mem_nil a done · -- Induction Step fix n : Nat assume ih : ∀ (l : List Nat), List.length l = n → a ∈ l → a ∣ prod l fix l : List Nat assume h1 : l.length = n + 1 --Goal : a ∈ l → a ∣ prod l obtain (b : Nat) (h2 : ∃ (L : List Nat), l = b :: L ∧ L.length = n) from exists_cons_of_length_eq_succ h1 obtain (L : List Nat) (h3 : l = b :: L ∧ L.length = n) from h2 have h4 : a ∈ L → a ∣ prod L := ih L h3.right assume h5 : a ∈ l rewrite [h3.left, prod_cons] --Goal : a ∣ b * prod L rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b ∨ a ∈ L by_cases on h5 · -- Case 1. h5 : a = b apply Exists.intro (prod L) rewrite [h5] rfl done · -- Case 2. h5 : a ∈ L have h6 : a ∣ prod L := h4 h5 have h7 : prod L ∣ b * prod L := by apply Exists.intro b ring done show a ∣ b * prod L from dvd_trans h6 h7 done done done lemma list_elt_dvd_prod {a : Nat} {l : List Nat} (h : a ∈ l) : a ∣ prod l := by set n : Nat := l.length have h1 : l.length = n := by rfl show a ∣ prod l from list_elt_dvd_prod_by_length a n l h1 h done lemma exists_prime_factorization : ∀ (n : Nat), n ≥ 1 → ∃ (l : List Nat), prime_factorization n l := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, n_1 ≥ 1 → ∃ (l : List Nat), prime_factorization n_1 l assume h1 : n ≥ 1 by_cases h2 : n = 1 · -- Case 1. h2 : n = 1 apply Exists.intro [] define apply And.intro · -- Proof of nondec_prime_list [] define show all_prime [] ∧ nondec [] from And.intro all_prime_nil nondec_nil done · -- Proof of prod [] = n rewrite [prod_nil, h2] rfl done done · -- Case 2. h2 : n ≠ 1 have h3 : n ≥ 2 := lt_of_le_of_ne' h1 h2 obtain (p : Nat) (h4 : prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q) from exists_least_prime_factor h3 have p_prime_factor : prime_factor p n := h4.left define at p_prime_factor have p_prime : prime p := p_prime_factor.left have p_dvd_n : p ∣ n := p_prime_factor.right have p_least : ∀ (q : Nat), prime_factor q n → p ≤ q := h4.right obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n have h5 : m ≠ 0 := by contradict h1 with h6 have h7 : n = 0 := calc n _ = p * m := n_eq_pm _ = p * 0 := by rw [h6] _ = 0 := by ring rewrite [h7] decide done have m_pos : 0 < m := Nat.pos_of_ne_zero h5 have m_lt_n : m < n := by define at p_prime show m < n from calc m _ < m + m := by linarith _ = 2 * m := by ring _ ≤ p * m := by rel [p_prime.left] _ = n := n_eq_pm.symm done obtain (L : List Nat) (h6 : prime_factorization m L) from ih m m_lt_n m_pos define at h6 have ndpl_L : nondec_prime_list L := h6.left define at ndpl_L apply Exists.intro (p :: L) define apply And.intro · -- Proof of nondec_prime_list (p :: L) define apply And.intro · -- Proof of all_prime (p :: L) rewrite [all_prime_cons] show prime p ∧ all_prime L from And.intro p_prime ndpl_L.left done · -- Proof of nondec (p :: L) rewrite [nondec_cons] apply And.intro _ ndpl_L.right fix q : Nat assume q_in_L : q ∈ L have h7 : q ∣ prod L := list_elt_dvd_prod q_in_L rewrite [h6.right] at h7 --h7 : q ∣ m have h8 : m ∣ n := by apply Exists.intro p rewrite [n_eq_pm] ring done have q_dvd_n : q ∣ n := dvd_trans h7 h8 have ap_L : all_prime L := ndpl_L.left define at ap_L have q_prime_factor : prime_factor q n := And.intro (ap_L q q_in_L) q_dvd_n show p ≤ q from p_least q q_prime_factor done done · -- Proof of prod (p :: L) = n rewrite [prod_cons, h6.right, n_eq_pm] rfl done done done theorem Theorem_7_2_2 {a b c : Nat} (h1 : c ∣ a * b) (h2 : rel_prime a c) : c ∣ b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑c ∣ ↑b define at h1; define at h2; define obtain (j : Nat) (h3 : a * b = c * j) from h1 set s : Int := gcd_c1 a c set t : Int := gcd_c2 a c have h4 : s * ↑a + t * ↑c = ↑(gcd a c) := gcd_lin_comb c a rewrite [h2, Nat.cast_one] at h4 --h4 : s * ↑a + t * ↑c = (1 : Int) apply Exists.intro (s * ↑j + t * ↑b) show ↑b = ↑c * (s * ↑j + t * ↑b) from calc ↑b _ = (1 : Int) * ↑b := (one_mul _).symm _ = (s * ↑a + t * ↑c) * ↑b := by rw [h4] _ = s * (↑a * ↑b) + t * ↑c * ↑b := by ring _ = s * (↑c * ↑j) + t * ↑c * ↑b := by rw [←Nat.cast_mul a b, h3, Nat.cast_mul c j] _ = ↑c * (s * ↑j + t * ↑b) := by ring done lemma le_nonzero_prod_left {a b : Nat} (h : a * b ≠ 0) : a ≤ a * b := by have h1 : b ≠ 0 := by contradict h with h1 rewrite [h1] ring done have h2 : 1 ≤ b := Nat.pos_of_ne_zero h1 show a ≤ a * b from calc a = a * 1 := (mul_one a).symm _ ≤ a * b := by rel [h2] done lemma le_nonzero_prod_right {a b : Nat} (h : a * b ≠ 0) : b ≤ a * b := by rewrite [mul_comm] rewrite [mul_comm] at h show b ≤ b * a from le_nonzero_prod_left h done lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry lemma rel_prime_of_prime_not_dvd {a p : Nat} (h1 : prime p) (h2 : ¬p ∣ a) : rel_prime a p := by have h3 : gcd a p ∣ a := gcd_dvd_left a p have h4 : gcd a p ∣ p := gcd_dvd_right a p have h5 : gcd a p = 1 ∨ gcd a p = p := dvd_prime h1 h4 have h6 : gcd a p ≠ p := by contradict h2 with h6 rewrite [h6] at h3 show p ∣ a from h3 done disj_syll h5 h6 show rel_prime a p from h5 done theorem Theorem_7_2_3 {a b p : Nat} (h1 : prime p) (h2 : p ∣ a * b) : p ∣ a ∨ p ∣ b := by or_right with h3 have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3 show p ∣ b from Theorem_7_2_2 h2 h4 done lemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a ≥ 1 := by have h1 : a ≠ 0 := by by_contra h1 rewrite [h1] at h contradict h linarith done show a ≥ 1 from Nat.pos_of_ne_zero h1 done lemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by have h1 : a ≥ 1 := ge_one_of_prod_one h have h2 : a * b ≠ 0 := by linarith have h3 : a ≤ a * b := le_nonzero_prod_left h2 rewrite [h] at h3 show a = 1 from Nat.le_antisymm h3 h1 done lemma eq_one_of_dvd_one {n : Nat} (h : n ∣ 1) : n = 1 := by obtain (j : Nat) (h1 : 1 = n * j) from h show n = 1 from eq_one_of_prod_one h1.symm done lemma prime_not_one {p : Nat} (h : prime p) : p ≠ 1 := by define at h linarith done theorem Theorem_7_2_4 {p : Nat} (h1 : prime p) : ∀ (l : List Nat), p ∣ prod l → ∃ a ∈ l, p ∣ a := by apply List.rec · -- Base Case. Goal : p ∣ prod [] → ∃ a ∈ [], p ∣ a rewrite [prod_nil] assume h2 : p ∣ 1 show ∃ a ∈ [], p ∣ a from absurd (eq_one_of_dvd_one h2) (prime_not_one h1) done · -- Induction Step fix b : Nat fix L : List Nat assume ih : p ∣ prod L → ∃ a ∈ L, p ∣ a --Goal : p ∣ prod (b :: L) → ∃ a ∈ b :: L, p ∣ a assume h2 : p ∣ prod (b :: L) rewrite [prod_cons] at h2 have h3 : p ∣ b ∨ p ∣ prod L := Theorem_7_2_3 h1 h2 by_cases on h3 · -- Case 1. h3 : p ∣ b apply Exists.intro b show b ∈ b :: L ∧ p ∣ b from And.intro (List.mem_cons_self b L) h3 done · -- Case 2. h3 : p ∣ prod L obtain (a : Nat) (h4 : a ∈ L ∧ p ∣ a) from ih h3 apply Exists.intro a show a ∈ b :: L ∧ p ∣ a from And.intro (List.mem_cons_of_mem b h4.left) h4.right done done done lemma prime_in_list {p : Nat} {l : List Nat} (h1 : prime p) (h2 : all_prime l) (h3 : p ∣ prod l) : p ∈ l := by obtain (a : Nat) (h4 : a ∈ l ∧ p ∣ a) from Theorem_7_2_4 h1 l h3 define at h2 have h5 : prime a := h2 a h4.left have h6 : p = 1 ∨ p = a := dvd_prime h5 h4.right disj_syll h6 (prime_not_one h1) rewrite [h6] show a ∈ l from h4.left done lemma first_le_first {p q : Nat} {l m : List Nat} (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m)) (h3 : prod (p :: l) = prod (q :: m)) : p ≤ q := by define at h1; define at h2 have h4 : q ∣ prod (p :: l) := by define apply Exists.intro (prod m) rewrite [←prod_cons] show prod (p :: l) = prod (q :: m) from h3 done have h5 : all_prime (q :: m) := h2.left rewrite [all_prime_cons] at h5 have h6 : q ∈ p :: l := prime_in_list h5.left h1.left h4 have h7 : nondec (p :: l) := h1.right rewrite [nondec_cons] at h7 rewrite [List.mem_cons] at h6 by_cases on h6 · -- Case 1. h6 : q = p linarith done · -- Case 2. h6 : q ∈ l have h8 : ∀ m ∈ l, p ≤ m := h7.left show p ≤ q from h8 q h6 done done lemma nondec_prime_list_tail {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by define at h define rewrite [all_prime_cons, nondec_cons] at h show all_prime l ∧ nondec l from And.intro h.left.right h.right.right done lemma cons_prod_not_one {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : prod (p :: l) ≠ 1 := by define at h have h1 : all_prime (p :: l) := h.left rewrite [all_prime_cons] at h1 rewrite [prod_cons] by_contra h2 show False from (prime_not_one h1.left) (eq_one_of_prod_one h2) done lemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) : l = [] ↔ prod l = 1 := by apply Iff.intro · -- (→) assume h1 : l = [] rewrite [h1] show prod [] = 1 from prod_nil done · -- (←) contrapos assume h1 : ¬l = [] obtain (p : Nat) (h2 : ∃ (L : List Nat), l = p :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List Nat) (h3 : l = p :: L) from h2 rewrite [h3] at h rewrite [h3] show ¬prod (p :: L) = 1 from cons_prod_not_one h done done lemma prime_pos {p : Nat} (h : prime p) : p > 0 := by define at h linarith done theorem Theorem_7_2_5 : ∀ (l1 l2 : List Nat), nondec_prime_list l1 → nondec_prime_list l2 → prod l1 = prod l2 → l1 = l2 := by apply List.rec · -- Base Case. Goal : ∀ (l2 : List Nat), nondec_prime_list [] → -- nondec_prime_list l2 → prod [] = prod l2 → [] = l2 fix l2 : List Nat assume h1 : nondec_prime_list [] assume h2 : nondec_prime_list l2 assume h3 : prod [] = prod l2 rewrite [prod_nil, eq_comm, ←list_nil_iff_prod_one h2] at h3 show [] = l2 from h3.symm done · -- Induction Step fix p : Nat fix L1 : List Nat assume ih : ∀ (L2 : List Nat), nondec_prime_list L1 → nondec_prime_list L2 → prod L1 = prod L2 → L1 = L2 -- Goal : ∀ (l2 : List Nat), nondec_prime_list (p :: L1) → -- nondec_prime_list l2 → prod (p :: L1) = prod l2 → p :: L1 = l2 fix l2 : List Nat assume h1 : nondec_prime_list (p :: L1) assume h2 : nondec_prime_list l2 assume h3 : prod (p :: L1) = prod l2 have h4 : ¬prod (p :: L1) = 1 := cons_prod_not_one h1 rewrite [h3, ←list_nil_iff_prod_one h2] at h4 obtain (q : Nat) (h5 : ∃ (L : List Nat), l2 = q :: L) from List.exists_cons_of_ne_nil h4 obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5 rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2) rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2) have h7 : p ≤ q := first_le_first h1 h2 h3 have h8 : q ≤ p := first_le_first h2 h1 h3.symm have h9 : p = q := by linarith rewrite [h9, prod_cons, prod_cons] at h3 --h3 : q * prod L1 = q * prod L2 have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1 have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2 define at h2 have h12 : all_prime (q :: L2) := h2.left rewrite [all_prime_cons] at h12 have h13 : q > 0 := prime_pos h12.left have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3 have h15 : L1 = L2 := ih L2 h10 h11 h14 rewrite [h6, h9, h15] rfl done done theorem fund_thm_arith (n : Nat) (h : n ≥ 1) : ∃! (l : List Nat), prime_factorization n l := by exists_unique · -- Existence show ∃ (l : List Nat), prime_factorization n l from exists_prime_factorization n h done · -- Uniqueness fix l1 : List Nat; fix l2 : List Nat assume h1 : prime_factorization n l1 assume h2 : prime_factorization n l2 define at h1; define at h2 have h3 : prod l1 = n := h1.right rewrite [←h2.right] at h3 show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3 done done /- Section 7.3 -/ theorem congr_refl (m : Nat) : ∀ (a : Int), a ≡ a (MOD m) := by fix a : Int define --Goal : ∃ (c : Int), a - a = ↑m * c apply Exists.intro 0 ring done theorem congr_symm {m : Nat} : ∀ {a b : Int}, a ≡ b (MOD m) → b ≡ a (MOD m) := by fix a : Int; fix b : Int assume h1 : a ≡ b (MOD m) define at h1 --h1 : ∃ (c : Int), a - b = ↑m * c define --Goal : ∃ (c : Int), b - a = ↑m * c obtain (c : Int) (h2 : a - b = m * c) from h1 apply Exists.intro (-c) show b - a = m * (-c) from calc b - a _ = -(a - b) := by ring _ = -(m * c) := by rw [h2] _ = m * (-c) := by ring done theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry /- Fundamental properties of congruence classes -/ lemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) : X = Y ↔ X.val = Y.val := Fin.ext_iff lemma val_nat_eq_mod (n k : Nat) : ([k]_(n + 1)).val = k % (n + 1) := by rfl lemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl theorem cc_rep {m : Nat} (X : ZMod m) : ∃ (a : Int), X = [a]_m := match m with | 0 => by apply Exists.intro X rfl done | n + 1 => by apply Exists.intro ↑(X.val) have h1 : X.val < n + 1 := Fin.prop X rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1] rfl done theorem add_class (m : Nat) (a b : Int) : [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm theorem mul_class (m : Nat) (a b : Int) : [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm lemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ [a - b]_m = [0]_m := by apply Iff.intro · -- (→) assume h1 : [a]_m = [b]_m have h2 : a - b = a + (-b) := by ring have h3 : b + (-b) = 0 := by ring show [a - b]_m = [0]_m from calc [a - b]_m _ = [a + (-b)]_m := by rw [h2] _ = [a]_m + [-b]_m := by rw [add_class] _ = [b]_m + [-b]_m := by rw [h1] _ = [b + -b]_m := by rw [add_class] _ = [0]_m := by rw [h3] done · -- (←) assume h1 : [a - b]_m = [0]_m have h2 : b + (a - b) = a := by ring have h3 : b + 0 = b := by ring show [a]_m = [b]_m from calc [a]_m _ = [b + (a - b)]_m := by rw [h2] _ = [b]_m + [a - b]_m := by rw [add_class] _ = [b]_m + [0]_m := by rw [h1] _ = [b + 0]_m := by rw [add_class] _ = [b]_m := by rw [h3] done done lemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) : [a]_m = [0]_m → [-a]_m = [0]_m := by assume h1 : [a]_m = [0]_m have h2 : 0 + (-a) = -a := by ring have h3 : a + (-a) = 0 := by ring show [-a]_m = [0]_m from calc [-a]_m _ = [0 + (-a)]_m := by rw [h2] _ = [0]_m + [-a]_m := by rw [add_class] _ = [a]_m + [-a]_m := by rw [h1] _ = [a + (-a)]_m := by rw [add_class] _ = [0]_m := by rw [h3] done lemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) : [-a]_m = [0]_m ↔ [a]_m = [0]_m := by apply Iff.intro _ (cc_neg_zero_of_cc_zero m a) assume h1 : [-a]_m = [0]_m have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1 have h3 : -(-a) = a := by ring rewrite [h3] at h2 show [a]_m = [0]_m from h2 done lemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl lemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m ↔ m ∣ k := match m with | 0 => by have h : (0 : Int) = (↑(0 : Nat) : Int) := by rfl rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj] apply Iff.intro · -- (→) assume h1 : k = 0 rewrite [h1] show 0 ∣ 0 from dvd_self 0 done · -- (←) assume h1 : 0 ∣ k obtain (c : Nat) (h2 : k = 0 * c) from h1 rewrite [h2] ring done done | n + 1 => by rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero] show k % (n + 1) = 0 ↔ n + 1 ∣ k from (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm done lemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m ↔ ↑m ∣ a := by obtain (k : Nat) (h1 : a = ↑k ∨ a = -↑k) from Int.eq_nat_or_neg a by_cases on h1 · -- Case 1. h1: a = ↑k rewrite [h1, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done · -- Case 2. h1: a = -↑k rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done done theorem cc_eq_iff_congr (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ a ≡ b (MOD m) := calc [a]_m = [b]_m _ ↔ [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b _ ↔ ↑m ∣ (a - b) := cc_zero_iff_dvd m (a - b) _ ↔ a ≡ b (MOD m) := by rfl /- End of fundamental properties of congruence classes -/ lemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m := by have h1 : (↑m : Int) ≠ 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m) show 0 ≤ a % m from Int.emod_nonneg a h1 done lemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m) have h2 : (↑m : Int) > 0 := (Nat.cast_pos).rtl h1 show a % m < m from Int.emod_lt_of_pos a h2 done lemma congr_mod_mod (m : Nat) (a : Int) : a ≡ a % m (MOD m) := by define have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m apply Exists.intro (a / m) show a - a % m = m * (a / m) from calc a - (a % m) _ = m * (a / m) + a % m - a % m := by rw [h1] _ = m * (a / m) := by ring done lemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) := And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a)) theorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) : ∃! (r : Int), 0 ≤ r ∧ r < m ∧ a ≡ r (MOD m) := by exists_unique · -- Existence apply Exists.intro (a % m) show 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) from mod_cmpl_res m a done · -- Uniqueness fix r1 : Int; fix r2 : Int assume h1 : 0 ≤ r1 ∧ r1 < m ∧ a ≡ r1 (MOD m) assume h2 : 0 ≤ r2 ∧ r2 < m ∧ a ≡ r2 (MOD m) have h3 : r1 ≡ r2 (MOD m) := congr_trans (congr_symm h1.right.right) h2.right.right obtain (d : Int) (h4 : r1 - r2 = m * d) from h3 have h5 : r1 - r2 < m * 1 := by linarith have h6 : m * (-1) < r1 - r2 := by linarith rewrite [h4] at h5 --h5 : m * d < m * 1 rewrite [h4] at h6 --h6 : m * -1 < m * d have h7 : (↑m : Int) ≥ 0 := Nat.cast_nonneg m have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7 have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7 have h10 : d = 0 := by linarith show r1 = r2 from calc r1 _ = r1 - r2 + r2 := by ring _ = m * 0 + r2 := by rw [h4, h10] _ = r2 := by ring done done lemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m := (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a) theorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y rewrite [h1, h2] have h3 : a + b = b + a := by ring show [a]_m + [b]_m = [b]_m + [a]_m from calc [a]_m + [b]_m _ = [a + b]_m := add_class m a b _ = [b + a]_m := by rw [h3] _ = [b]_m + [a]_m := (add_class m b a).symm done theorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X rewrite [h1] have h2 : a * 1 = a := by ring show [a]_m * [1]_m = [a]_m from calc [a]_m * [1]_m _ = [a * 1]_m := mul_class m a 1 _ = [a]_m := by rw [h2] done theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry lemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) : [a]_m * [gcd_c2 m a]_m = [1]_m := by set s : Int := gcd_c1 m a have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m define at h1 rewrite [h1, Nat.cast_one] at h2 --h2 : s * ↑m + gcd_c2 m a * ↑a = 1 rewrite [mul_class, cc_eq_iff_congr] define --Goal : ∃ (c : Int), ↑a * gcd_c2 m a - 1 = ↑m * c apply Exists.intro (-s) show a * (gcd_c2 m a) - 1 = m * (-s) from calc a * (gcd_c2 m a) - 1 _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring _ = 1 + m * (-s) - 1 := by rw [h2] _ = m * (-s) := by ring done theorem Theorem_7_3_7 (m a : Nat) : invertible [a]_m ↔ rel_prime m a := by apply Iff.intro · -- (→) assume h1 : invertible [a]_m define at h1 obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1 obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y rewrite [h3, mul_class, cc_eq_iff_congr] at h2 define at h2 obtain (c : Int) (h4 : a * b - 1 = m * c) from h2 rewrite [Exercise_7_2_6] --Goal : ∃ (s t : Int), s * ↑m + t * ↑a = 1 apply Exists.intro (-c) apply Exists.intro b show (-c) * m + b * a = 1 from calc (-c) * m + b * a _ = (-c) * m + (a * b - 1) + 1 := by ring _ = (-c) * m + m * c + 1 := by rw [h4] _ = 1 := by ring done · -- (←) assume h1 : rel_prime m a define show ∃ (Y : ZMod m), [a]_m * Y = [1]_m from Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1) done done /- Section 7.4 -/ section Euler open Euler lemma num_rp_below_base {m : Nat} : num_rp_below m 0 = 0 := by rfl lemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) : num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by have h1 : num_rp_below m (j + 1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : gcd m j = 1 rewrite [if_pos h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1 show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1 done lemma num_rp_below_step_not_rp {m j : Nat} (h : ¬rel_prime m j) : num_rp_below m (j + 1) = num_rp_below m j := by have h1 : num_rp_below m (j +1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : ¬gcd m j = 1 rewrite [if_neg h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j show num_rp_below m (j + 1) = num_rp_below m j from h1 done lemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl #eval phi 10 --Answer: 4 lemma prod_inv_iff_inv {m : Nat} {X : ZMod m} (h1 : invertible X) (Y : ZMod m) : invertible (X * Y) ↔ invertible Y := by apply Iff.intro · -- (→) assume h2 : invertible (X * Y) obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2 apply Exists.intro (X * Z) rewrite [←h3] --Goal : Y * (X * Z) = X * Y * Z ring --Note that ring can do algebra in ZMod m done · -- (←) assume h2 : invertible Y obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1 obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2 apply Exists.intro (Xi * Yi) show (X * Y) * (Xi * Yi) = [1]_m from calc X * Y * (Xi * Yi) _ = (X * Xi) * (Y * Yi) := by ring _ = [1]_m * [1]_m := by rw [h3, h4] _ = [1]_m := Theorem_7_3_6_7 [1]_m done done lemma F_rp_def {m i : Nat} (h : rel_prime m i) : F m i = [i]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h --h : gcd m i = 1 rewrite [if_pos h] at h1 show F m i = [i]_m from h1 done lemma F_not_rp_def {m i : Nat} (h : ¬rel_prime m i) : F m i = [1]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h rewrite [h1, if_neg h] rfl done lemma prod_seq_base {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 0 k f = [1]_m := by rfl lemma prod_seq_step {m : Nat} (n k : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl lemma prod_seq_zero_step {m : Nat} (n : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by rewrite [prod_seq_step, zero_add] rfl done lemma prod_one {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 1 k f = f k := by rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7] rfl done lemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl lemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m := calc [G m a i]_m _ = [(a * i) % m]_m := by rfl _ = [a * i]_m := (cc_eq_mod m (a * i)).symm _ = [a]_m * [i]_m := (mul_class m a i).symm lemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) : rel_prime m (G m a i) ↔ rel_prime m i := by have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1 show rel_prime m (G m a i) ↔ rel_prime m i from calc rel_prime m (G m a i) _ ↔ invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm _ ↔ invertible ([a]_m * [i]_m) := by rw [cc_G] _ ↔ invertible [i]_m := prod_inv_iff_inv h2 ([i]_m) _ ↔ rel_prime m i := Theorem_7_3_7 m i done lemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) : F m (G m a i) = [a]_m * F m i := by have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2 show F m (G m a i) = [a]_m * F m i from calc F m (G m a i) _ = [G m a i]_m := F_rp_def h3 _ = [a]_m * [i]_m := cc_G m a i _ = [a]_m * F m i := by rw [F_rp_def h2] done lemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : ¬rel_prime m i) : F m (G m a i) = [1]_m := by rewrite [←G_rp_iff h1 i] at h2 show F m (G m a i) = [1]_m from F_not_rp_def h2 done lemma FG_prod {m a : Nat} (h1 : rel_prime m a) : ∀ (k : Nat), prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by by_induc · -- Base Case show prod_seq 0 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from calc prod_seq 0 0 ((F m) ∘ (G m a)) _ = [1]_m := prod_seq_base _ _ _ = [a]_m ^ 0 * [1]_m := by ring _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by rw [num_rp_below_base, prod_seq_base] done · -- Induction Step fix k : Nat assume ih : prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) by_cases h2 : rel_prime m k · -- Case 1. h2 : rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([a]_m * F m k) := by rw [FG_rp h1 h2] _ = [a]_m ^ ((num_rp_below m k) + 1) * ((prod_seq k 0 (F m)) * F m k) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_rp h2, prod_seq_zero_step] done · -- Case 2. h2 : ¬rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([1]_m) := by rw [FG_not_rp h1 h2] _ = [a]_m ^ (num_rp_below m k) * (prod_seq k 0 (F m) * ([1]_m)) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_not_rp h2, prod_seq_zero_step, F_not_rp_def h2] done done done lemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by define --Goal : ∀ i < m, G m a i < m fix i : Nat assume h1 : i < m rewrite [G_def] --Goal : a * i % m < m show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m) done lemma left_inv_one_one_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g' (g i) = i) : one_one_below n g := sorry lemma right_inv_onto_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g (g' i) = i) (h2 : maps_below n g') : onto_below n g := by define at h2; define fix k : Nat assume h3 : k < n apply Exists.intro (g' k) show g' k < n ∧ g (g' k) = k from And.intro (h2 k h3) (h1 k h3) done lemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m * [inv_mod m a]_m = [1]_m := by have h2 : 0 ≤ (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a) show [a]_m * [inv_mod m a]_m = [1]_m from calc [a]_m * [inv_mod m a]_m _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2] _ = [a]_m * [gcd_c2 m a]_m := by rw [←cc_eq_mod] _ = [1]_m := gcd_c2_inv h1 done lemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m := calc a * (b % m) % m = a % m * (b % m % m) % m := Nat.mul_mod _ _ _ _ = a % m * (b % m) % m := by rw [Nat.mod_mod] _ = a * b % m := (Nat.mul_mod _ _ _).symm lemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm] rfl done theorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] : ↑a ≡ ↑b (MOD m) ↔ a % m = b % m := sorry lemma mul_inv_mod_cancel {m a i : Nat} [NeZero m] (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1 rewrite [mul_class, cc_eq_iff_congr, ←Nat.cast_mul, ←Nat.cast_one, congr_iff_mod_eq_Nat] at h3 show a * inv_mod m a * i % m = i from calc a * (inv_mod m a) * i % m _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = 1 % m * i % m := by rw [h3] _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = i % m := by rw [one_mul] _ = i := Nat.mod_eq_of_lt h2 done lemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl lemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, G m a (Ginv m a i) = i := by fix i : Nat assume h2 : i < m show G m a (Ginv m a i) = i from calc G m a (Ginv m a i) _ = a * ((inv_mod m a * i) % m) % m := by rfl _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, Ginv m a (G m a i) = i := by fix i : Nat assume h2 : i < m show Ginv m a (G m a i) = i from calc Ginv m a (G m a i) _ = inv_mod m a * ((a * i) % m) % m := by rfl _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc, mul_comm (inv_mod m a)] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_maps_below (m a : Nat) [NeZero m] : maps_below m (Ginv m a) := G_maps_below m (inv_mod m a) lemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : one_one_below m (G m a) := left_inv_one_one_below (Ginv_left_inv h1) lemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : onto_below m (G m a) := right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a) lemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : perm_below m (G m a) := And.intro (G_maps_below m a) (And.intro (G_one_one_below h1) (G_onto_below h1)) --Permuting a product of congruence classes doesn't change product lemma swap_fst (u v : Nat) : swap u v u = v := by define : swap u v u --Goal : (if u = u then v else if u = v then u else u) = v have h : u = u := by rfl rewrite [if_pos h] rfl done lemma swap_snd (u v : Nat) : swap u v v = u := by define : swap u v v by_cases h1 : v = u · -- Case 1. h1 : v = u rewrite [if_pos h1] show v = u from h1 done · -- Case 2. h1 : v ≠ u rewrite [if_neg h1] have h2 : v = v := by rfl rewrite [if_pos h2] rfl done done lemma swap_other {u v i : Nat} (h1 : i ≠ u) (h2 : i ≠ v) : swap u v i = i := by define : swap u v i rewrite [if_neg h1, if_neg h2] rfl done lemma swap_values (u v i : Nat) : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := by by_cases h1 : i = u · -- Case 1. h1 : i = u apply Or.inl rewrite [h1] show swap u v u = v from swap_fst u v done · -- Case 2. h1 : i ≠ u apply Or.inr by_cases h2 : i = v · -- Case 2.1. h2 : i = v apply Or.inl rewrite [h2] show swap u v v = u from swap_snd u v done · -- Case 2.2. h2 : i ≠ v apply Or.inr show swap u v i = i from swap_other h1 h2 done done done lemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by define fix i : Nat assume h3 : i < n have h4 : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := swap_values u v i by_cases on h4 · -- Case 1. h4 : swap u v i = v rewrite [h4] show v < n from h2 done · -- Case 2. by_cases on h4 · -- Case 2.1. h4 : swap u v i = u rewrite [h4] show u < n from h1 done · -- Case 2.2. h4 : swap u v i = i rewrite [h4] show i < n from h3 done done done lemma swap_swap (u v n : Nat) : ∀ i < n, swap u v (swap u v i) = i := by fix i : Nat assume h : i < n by_cases h1 : i = u · -- Case 1. h1 : i = u rewrite [h1, swap_fst, swap_snd] rfl done · -- Case 2. h1 : i ≠ u by_cases h2 : i = v · -- Case 2.1. h2 : i = v rewrite [h2, swap_snd, swap_fst] rfl done · -- Case 2.2. h2 : i ≠ v rewrite [swap_other h1 h2, swap_other h1 h2] rfl done done done lemma swap_one_one_below (u v n) : one_one_below n (swap u v) := left_inv_one_one_below (swap_swap u v n) lemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) := right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2) lemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) := And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2)) lemma comp_perm_below {n : Nat} {f g : Nat → Nat} (h1 : perm_below n f) (h2 : perm_below n g) : perm_below n (f ∘ g) := sorry lemma trivial_swap (u : Nat) : swap u u = id := by apply funext fix x : Nat by_cases h1 : x = u · -- Case 1. h1 : x = u rewrite [h1, swap_fst] rfl done · -- Case 2. h1 : x ≠ u rewrite [swap_other h1 h1] rfl done done lemma prod_eq_fun {m : Nat} (f g : Nat → ZMod m) (k : Nat) : ∀ (n : Nat), (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g := by by_induc · -- Base Case assume h : (∀ i < 0, f (k + i) = g (k + i)) rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g assume h1 : ∀ i < n + 1, f (k + i) = g (k + i) have h2 : ∀ i < n, f (k + i) = g (k + i) := by fix i : Nat assume h2 : i < n have h3 : i < n + 1 := by linarith show f (k + i) = g (k + i) from h1 i h3 done have h3 : prod_seq n k f = prod_seq n k g := ih h2 have h4 : n < n + 1 := Nat.lt_succ_self n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4] rfl done done lemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := by have h2 : ∀ (i : Nat), i < u → (f ∘ swap u n) (0 + i) = f (0 + i) := by fix i : Nat assume h2 : i < u have h3 : 0 + i ≠ u := by linarith have h4 : 0 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl done show prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f from prod_eq_fun (f ∘ swap u n) f 0 u h2 done lemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := by have h2 : ∀ i < j, (f ∘ swap u n) (u + 1 + i) = f (u + 1 + i) := by fix i : Nat assume h2 : i < j have h3 : u + 1 + i ≠ u := by linarith have h4 : u + 1 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl show prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f from prod_eq_fun (f ∘ swap u n) f (u + 1) j h2 done lemma break_prod {m : Nat} (n : Nat) (f : Nat → ZMod m) : ∀ (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by by_induc · -- Base Case have h : n + 0 = n := by rfl rewrite [prod_seq_base, h, Theorem_7_3_6_7] rfl done · -- Induction Step fix j : Nat assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f rewrite [←add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc] rfl done done lemma break_prod_twice {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f := break_prod n f 1 rewrite [prod_one] at h2 have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f := break_prod (u + 1) f j rewrite [←h1] at h3 have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f := break_prod u f 1 rewrite [prod_one] at h4 rewrite [h3, h4] at h2 show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2 done lemma swap_prod_eq_prod {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f := by by_cases h2 : u = n · -- Case 1. h2 : u = n rewrite [h2, trivial_swap n] --Goal : prod_seq (n + 1) 0 (f ∘ id) = prod_seq (n + 1) 0 f rfl done · -- Case 2. h2 : ¬u = n have h3 : u + 1 ≤ n := Nat.lt_of_le_of_ne h1 h2 obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3 have break_f : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := break_prod_twice f h4 have break_fs : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_prod_twice (f ∘ swap u n) h4 have f_eq_fs_below : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := swap_prod_eq_prod_below f h1 have f_eq_fs_btwn : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4 show prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f from calc prod_seq (n + 1) 0 (f ∘ swap u n) _ = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_fs _ = prod_seq u 0 f * (f ∘ swap u n) u * prod_seq j (u + 1) f * (f ∘ swap u n) n := by rw [f_eq_fs_below, f_eq_fs_btwn] _ = prod_seq u 0 f * f (swap u n u) * prod_seq j (u + 1) f * f (swap u n n) := by rfl _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by rw [swap_fst, swap_snd] _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring _ = prod_seq (n + 1) 0 f := break_f.symm done done lemma perm_below_fixed {n : Nat} {g : Nat → Nat} (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry lemma perm_prod {m : Nat} (f : Nat → ZMod m) : ∀ (n : Nat), ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) := by by_induc · -- Base Case fix g : Nat → Nat assume h1 : perm_below 0 g rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) fix g : Nat → Nat assume g_pb : perm_below (n + 1) g define at g_pb have g_ob : onto_below (n + 1) g := g_pb.right.right define at g_ob have h1 : n < n + 1 := by linarith obtain (u : Nat) (h2 : u < n + 1 ∧ g u = n) from g_ob n h1 have s_pb : perm_below (n + 1) (swap u n) := swap_perm_below h2.left h1 have gs_pb_n1 : perm_below (n + 1) (g ∘ swap u n) := comp_perm_below g_pb s_pb have gs_fix_n : (g ∘ swap u n) n = n := calc (g ∘ swap u n) n _ = g (swap u n n) := by rfl _ = g u := by rw [swap_snd] _ = n := h2.right have gs_pb_n : perm_below n (g ∘ swap u n) := perm_below_fixed gs_pb_n1 gs_fix_n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f ∘ (g ∘ swap u n)) := ih (g ∘ swap u n) gs_pb_n have h3 : u ≤ n := by linarith show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f ∘ g) from calc prod_seq (n + 1) 0 f _ = prod_seq n 0 f * f n := prod_seq_zero_step n f _ = prod_seq n 0 (f ∘ (g ∘ swap u n)) * f ((g ∘ swap u n) n) := by rw [gs_prod, gs_fix_n] _ = prod_seq n 0 (f ∘ g ∘ swap u n) * (f ∘ g ∘ swap u n) n := by rfl _ = prod_seq (n + 1) 0 (f ∘ g ∘ swap u n) := (prod_seq_zero_step n (f ∘ g ∘ swap u n)).symm _ = prod_seq (n + 1) 0 ((f ∘ g) ∘ swap u n) := by rfl _ = prod_seq (n + 1) 0 (f ∘ g) := swap_prod_eq_prod (f ∘ g) h3 done done lemma F_invertible (m i : Nat) : invertible (F m i) := by by_cases h : rel_prime m i · -- Case 1. h : rel_prime m i rewrite [F_rp_def h] show invertible [i]_m from (Theorem_7_3_7 m i).rtl h done · -- Case 2. h : ¬rel_prime m i rewrite [F_not_rp_def h] apply Exists.intro [1]_m show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m done done lemma Fprod_invertible (m : Nat) : ∀ (k : Nat), invertible (prod_seq k 0 (F m)) := by by_induc · -- Base Case apply Exists.intro [1]_m show prod_seq 0 0 (F m) * [1]_m = [1]_m from calc prod_seq 0 0 (F m) * [1]_m _ = [1]_m * [1]_m := by rw [prod_seq_base] _ = [1]_m := Theorem_7_3_6_7 ([1]_m) done · -- Induction Step fix k : Nat assume ih : invertible (prod_seq k 0 (F m)) rewrite [prod_seq_zero_step] show invertible (prod_seq k 0 (F m) * (F m k)) from (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k) done done theorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m ^ (phi m) = [1]_m := by have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2 show [a]_m ^ (phi m) = [1]_m from calc [a]_m ^ (phi m) _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3] _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring _ = prod_seq m 0 (F m ∘ G m a) * Y := by rw [FG_prod h1 m, phi_def] _ = prod_seq m 0 (F m) * Y := by rw [perm_prod (F m) m (G m a) (G_perm_below h1)] _ = [1]_m := by rw [h3] done lemma Exercise_7_4_5_Int (m : Nat) (a : Int) : ∀ (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry lemma Exercise_7_4_5_Nat (m a n : Nat) : [a]_m ^ n = [a ^ n]_m := by rewrite [Exercise_7_4_5_Int] rfl done theorem Euler's_theorem {m a : Nat} [NeZero m] (h1 : rel_prime m a) : a ^ (phi m) ≡ 1 (MOD m) := by have h2 : [a]_m ^ (phi m) = [1]_m := Theorem_7_4_2 h1 rewrite [Exercise_7_4_5_Nat m a (phi m)] at h2 --h2 : [a ^ phi m]_m = [1]_m show a ^ (phi m) ≡ 1 (MOD m) from (cc_eq_iff_congr _ _ _).ltr h2 done #eval gcd 10 7 --Answer: 1. So 10 and 7 are relatively prime #eval 7 ^ phi 10 --Answer: 2401, which is congruent to 1 mod 10. end Euler /- Section 7.5 -/ lemma num_rp_prime {p : Nat} (h1 : prime p) : ∀ k < p, num_rp_below p (k + 1) = k := sorry lemma phi_prime {p : Nat} (h1 : prime p) : phi p = p - 1 := by have h2 : 1 ≤ p := prime_pos h1 have h3 : p - 1 + 1 = p := Nat.sub_add_cancel h2 have h4 : p - 1 < p := by linarith have h5 : num_rp_below p (p - 1 + 1) = p - 1 := num_rp_prime h1 (p - 1) h4 rewrite [h3] at h5 show phi p = p - 1 from h5 done theorem Theorem_7_2_2_Int {a c : Nat} {b : Int} (h1 : ↑c ∣ ↑a * b) (h2 : rel_prime a c) : ↑c ∣ b := by rewrite [Int.natCast_dvd, Int.natAbs_mul, Int.natAbs_ofNat] at h1 --h1 : c ∣ a * Int.natAbs b rewrite [Int.natCast_dvd] --Goal : c ∣ Int.natAbs b show c ∣ Int.natAbs b from Theorem_7_2_2 h1 h2 done

lemma Lemma_7_4_5 {m n : Nat} (a b : Int) (h1 : rel_prime m n) : a ≡ b (MOD m * n) ↔ a ≡ b (MOD m) ∧ a ≡ b (MOD n)

HTPI.Lemma_7_4_5

htpi/HTPILib/Chap7.lean

htpi

/- Copyright 2023 Daniel J. Velleman -/ import HTPILib.Chap6 namespace HTPI /- Definitions -/ lemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by have h : n + 1 > 0 := Nat.succ_pos n show a % (n + 1) < n + 1 from Nat.mod_lt a h done def gcd (a b : Nat) : Nat := match b with | 0 => a | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd (n + 1) (a % (n + 1)) termination_by b mutual def gcd_c1 (a b : Nat) : Int := match b with | 0 => 1 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c2 (n + 1) (a % (n + 1)) --Corresponds to s = t' termination_by b def gcd_c2 (a b : Nat) : Int := match b with | 0 => 0 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c1 (n + 1) (a % (n + 1)) - (gcd_c2 (n + 1) (a % (n + 1))) * ↑(a / (n + 1)) --Corresponds to t = s' - t'q termination_by b end def prime (n : Nat) : Prop := 2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n def prime_factor (p n : Nat) : Prop := prime p ∧ p ∣ n def all_prime (l : List Nat) : Prop := ∀ p ∈ l, prime p def nondec (l : List Nat) : Prop := match l with | [] => True --Of course, True is a proposition that is always true | n :: L => (∀ m ∈ L, n ≤ m) ∧ nondec L def nondec_prime_list (l : List Nat) : Prop := all_prime l ∧ nondec l def prod (l : List Nat) : Nat := match l with | [] => 1 | n :: L => n * (prod L) def prime_factorization (n : Nat) (l : List Nat) : Prop := nondec_prime_list l ∧ prod l = n def rel_prime (a b : Nat) : Prop := gcd a b = 1 def congr_mod (m : Nat) (a b : Int) : Prop := (↑m : Int) ∣ (a - b) def cc (m : Nat) (a : Int) : ZMod m := (↑a : ZMod m) notation:50 a " ≡ " b " (MOD " m ")" => congr_mod m a b notation:max "["a"]_"m:max => cc m a def invertible {m : Nat} (X : ZMod m) : Prop := ∃ (Y : ZMod m), X * Y = [1]_m def num_rp_below (m k : Nat) : Nat := match k with | 0 => 0 | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j def phi (m : Nat) : Nat := num_rp_below m m def prod_seq {m : Nat} (j k : Nat) (f : Nat → ZMod m) : ZMod m := match j with | 0 => [1]_m | n + 1 => prod_seq n k f * f (k + n) def maps_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i < n, g i < n def one_one_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i1 < n, ∀ i2 < n, g i1 = g i2 → i1 = i2 def onto_below (n : Nat) (g : Nat → Nat) : Prop := ∀ k < n, ∃ i < n, g i = k def perm_below (n : Nat) (g : Nat → Nat) : Prop := maps_below n g ∧ one_one_below n g ∧ onto_below n g def inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m) def swap (u v i : Nat) : Nat := if i = u then v else if i = v then u else i namespace Euler --For definitions specific to Euler's theorem def F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m def G (m a i : Nat) : Nat := (a * i) % m def Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i end Euler /- Section 7.1 -/ theorem dvd_mod_of_dvd_a_b {a b d : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ (a % b) := by set q : Nat := a / b have h3 : b * q + a % b = a := Nat.div_add_mod a b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 define --Goal : ∃ (c : Nat), a % b = d * c apply Exists.intro (j - k * q) show a % b = d * (j - k * q) from calc a % b _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm _ = a - b * q := by rw [h3] _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc] _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm done theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry #eval gcd 672 161 --Answer: 7 lemma gcd_base (a : Nat) : gcd a 0 = a := by rfl lemma gcd_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd a b = gcd b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1)) rfl done lemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b ≠ 0) : a % b < b := by have h1 : b > 0 := Nat.pos_of_ne_zero h show a % b < b from Nat.mod_lt a h1 done lemma dvd_self (n : Nat) : n ∣ n := by apply Exists.intro 1 ring done theorem gcd_dvd : ∀ (b a : Nat), (gcd a b) ∣ a ∧ (gcd a b) ∣ b := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd a b_1) ∣ a ∧ (gcd a b_1) ∣ b_1 fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_base] --Goal: a ∣ a ∧ a ∣ 0 apply And.intro (dvd_self a) define apply Exists.intro 0 rfl done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_nonzero a h1] --Goal : gcd b (a % b) ∣ a ∧ gcd b (a % b) ∣ b have h2 : a % b < b := mod_nonzero_lt a h1 have h3 : (gcd b (a % b)) ∣ b ∧ (gcd b (a % b)) ∣ (a % b) := ih (a % b) h2 b apply And.intro _ h3.left show (gcd b (a % b)) ∣ a from dvd_a_of_dvd_b_mod h3.left h3.right done done theorem gcd_dvd_left (a b : Nat) : (gcd a b) ∣ a := (gcd_dvd b a).left theorem gcd_dvd_right (a b : Nat) : (gcd a b) ∣ b := (gcd_dvd b a).right lemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl lemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c1 a b = gcd_c2 b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done lemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl lemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * ↑(a / b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done theorem gcd_lin_comb : ∀ (b a : Nat), (gcd_c1 a b) * ↑a + (gcd_c2 a b) * ↑b = ↑(gcd a b) := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd_c1 a b_1) * ↑a + (gcd_c2 a b_1) * ↑b_1 = ↑(gcd a b_1) fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base] --Goal : 1 * ↑a + 0 * ↑0 = ↑a ring done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1] --Goal : gcd_c2 b (a % b) * ↑a + -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * ↑(a / b)) * ↑b = -- ↑(gcd b (a % b)) set r : Nat := a % b set q : Nat := a / b set s : Int := gcd_c1 b r set t : Int := gcd_c2 b r --Goal : t * ↑a + (s - t * ↑q) * ↑b = ↑(gcd b r) have h2 : r < b := mod_nonzero_lt a h1 have h3 : s * ↑b + t * ↑r = ↑(gcd b r) := ih r h2 b have h4 : b * q + r = a := Nat.div_add_mod a b rewrite [←h3, ←h4] rewrite [Nat.cast_add, Nat.cast_mul] --Goal : t * (↑b * ↑q + ↑r) + (s - t * ↑q) * ↑b = s * ↑b + t * ↑r ring done done #eval gcd_c1 672 161 --Answer: 6 #eval gcd_c2 672 161 --Answer: -25 --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161 theorem Theorem_7_1_6 {d a b : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ gcd a b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑d ∣ ↑(gcd a b) set s : Int := gcd_c1 a b set t : Int := gcd_c2 a b have h3 : s * ↑a + t * ↑b = ↑(gcd a b) := gcd_lin_comb b a rewrite [←h3] --Goal : ↑d ∣ s * ↑a + t * ↑b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul] --Goal : ↑d ∣ s * (↑d * ↑j) + t * (↑d * ↑k) define apply Exists.intro (s * ↑j + t * ↑k) ring done /- Section 7.2 -/ theorem dvd_trans {a b c : Nat} (h1 : a ∣ b) (h2 : b ∣ c) : a ∣ c := by define at h1; define at h2; define obtain (m : Nat) (h3 : b = a * m) from h1 obtain (n : Nat) (h4 : c = b * n) from h2 rewrite [h3, mul_assoc] at h4 apply Exists.intro (m * n) show c = a * (m * n) from h4 done lemma exists_prime_factor : ∀ (n : Nat), 2 ≤ n → ∃ (p : Nat), prime_factor p n := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, 2 ≤ n_1 → ∃ (p : Nat), prime_factor p n_1 assume h1 : 2 ≤ n by_cases h2 : prime n · -- Case 1. h2 : prime n apply Exists.intro n define --Goal : prime n ∧ n ∣ n show prime n ∧ n ∣ n from And.intro h2 (dvd_self n) done · -- Case 2. h2 : ¬prime n define at h2 --h2 : ¬(2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n) demorgan at h2 disj_syll h2 h1 obtain (a : Nat) (h3 : ∃ (b : Nat), a * b = n ∧ a < n ∧ b < n) from h2 obtain (b : Nat) (h4 : a * b = n ∧ a < n ∧ b < n) from h3 have h5 : 2 ≤ a := by by_contra h6 have h7 : a ≤ 1 := by linarith have h8 : n ≤ b := calc n _ = a * b := h4.left.symm _ ≤ 1 * b := by rel [h7] _ = b := by ring linarith --n ≤ b contradicts b < n done have h6 : ∃ (p : Nat), prime_factor p a := ih a h4.right.left h5 obtain (p : Nat) (h7 : prime_factor p a) from h6 apply Exists.intro p define --Goal : prime p ∧ p ∣ n define at h7 --h7 : prime p ∧ p ∣ a apply And.intro h7.left have h8 : a ∣ n := by apply Exists.intro b show n = a * b from (h4.left).symm done show p ∣ n from dvd_trans h7.right h8 done done lemma exists_least_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q := by set S : Set Nat := {p : Nat | prime_factor p n} have h2 : ∃ (p : Nat), p ∈ S := exists_prime_factor n h show ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q from well_ord_princ S h2 done lemma all_prime_nil : all_prime [] := by define --Goal : ∀ p ∈ [], prime p fix p : Nat contrapos --Goal : ¬prime p → p ∉ [] assume h1 : ¬prime p show p ∉ [] from List.not_mem_nil p done lemma all_prime_cons (n : Nat) (L : List Nat) : all_prime (n :: L) ↔ prime n ∧ all_prime L := by apply Iff.intro · -- (→) assume h1 : all_prime (n :: L) --Goal : prime n ∧ all_prime L define at h1 --h1 : ∀ p ∈ n :: L, prime p apply And.intro (h1 n (List.mem_cons_self n L)) define --Goal : ∀ p ∈ L, prime p fix p : Nat assume h2 : p ∈ L show prime p from h1 p (List.mem_cons_of_mem n h2) done · -- (←) assume h1 : prime n ∧ all_prime L --Goal : all_prime (n :: l) define : all_prime L at h1 define fix p : Nat assume h2 : p ∈ n :: L rewrite [List.mem_cons] at h2 --h2 : p = n ∨ p ∈ L by_cases on h2 · -- Case 1. h2 : p = n rewrite [h2] show prime n from h1.left done · -- Case 2. h2 : p ∈ L show prime p from h1.right p h2 done done done lemma nondec_nil : nondec [] := by define --Goal : True trivial --trivial proves some obviously true statements, such as True done lemma nondec_cons (n : Nat) (L : List Nat) : nondec (n :: L) ↔ (∀ m ∈ L, n ≤ m) ∧ nondec L := by rfl lemma prod_nil : prod [] = 1 := by rfl lemma prod_cons : prod (n :: L) = n * (prod L) := by rfl lemma exists_cons_of_length_eq_succ {A : Type} {l : List A} {n : Nat} (h : l.length = n + 1) : ∃ (a : A) (L : List A), l = a :: L ∧ L.length = n := by have h1 : ¬l.length = 0 := by linarith rewrite [List.length_eq_zero] at h1 obtain (a : A) (h2 : ∃ (L : List A), l = a :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List A) (h3 : l = a :: L) from h2 apply Exists.intro a apply Exists.intro L apply And.intro h3 have h4 : (a :: L).length = L.length + 1 := List.length_cons a L rewrite [←h3, h] at h4 show L.length = n from (Nat.add_right_cancel h4).symm done lemma list_elt_dvd_prod_by_length (a : Nat) : ∀ (n : Nat), ∀ (l : List Nat), l.length = n → a ∈ l → a ∣ prod l := by by_induc · --Base Case fix l : List Nat assume h1 : l.length = 0 rewrite [List.length_eq_zero] at h1 --h1 : l = [] rewrite [h1] --Goal : a ∈ [] → a ∣ prod [] contrapos assume h2 : ¬a ∣ prod [] show a ∉ [] from List.not_mem_nil a done · -- Induction Step fix n : Nat assume ih : ∀ (l : List Nat), List.length l = n → a ∈ l → a ∣ prod l fix l : List Nat assume h1 : l.length = n + 1 --Goal : a ∈ l → a ∣ prod l obtain (b : Nat) (h2 : ∃ (L : List Nat), l = b :: L ∧ L.length = n) from exists_cons_of_length_eq_succ h1 obtain (L : List Nat) (h3 : l = b :: L ∧ L.length = n) from h2 have h4 : a ∈ L → a ∣ prod L := ih L h3.right assume h5 : a ∈ l rewrite [h3.left, prod_cons] --Goal : a ∣ b * prod L rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b ∨ a ∈ L by_cases on h5 · -- Case 1. h5 : a = b apply Exists.intro (prod L) rewrite [h5] rfl done · -- Case 2. h5 : a ∈ L have h6 : a ∣ prod L := h4 h5 have h7 : prod L ∣ b * prod L := by apply Exists.intro b ring done show a ∣ b * prod L from dvd_trans h6 h7 done done done lemma list_elt_dvd_prod {a : Nat} {l : List Nat} (h : a ∈ l) : a ∣ prod l := by set n : Nat := l.length have h1 : l.length = n := by rfl show a ∣ prod l from list_elt_dvd_prod_by_length a n l h1 h done lemma exists_prime_factorization : ∀ (n : Nat), n ≥ 1 → ∃ (l : List Nat), prime_factorization n l := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, n_1 ≥ 1 → ∃ (l : List Nat), prime_factorization n_1 l assume h1 : n ≥ 1 by_cases h2 : n = 1 · -- Case 1. h2 : n = 1 apply Exists.intro [] define apply And.intro · -- Proof of nondec_prime_list [] define show all_prime [] ∧ nondec [] from And.intro all_prime_nil nondec_nil done · -- Proof of prod [] = n rewrite [prod_nil, h2] rfl done done · -- Case 2. h2 : n ≠ 1 have h3 : n ≥ 2 := lt_of_le_of_ne' h1 h2 obtain (p : Nat) (h4 : prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q) from exists_least_prime_factor h3 have p_prime_factor : prime_factor p n := h4.left define at p_prime_factor have p_prime : prime p := p_prime_factor.left have p_dvd_n : p ∣ n := p_prime_factor.right have p_least : ∀ (q : Nat), prime_factor q n → p ≤ q := h4.right obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n have h5 : m ≠ 0 := by contradict h1 with h6 have h7 : n = 0 := calc n _ = p * m := n_eq_pm _ = p * 0 := by rw [h6] _ = 0 := by ring rewrite [h7] decide done have m_pos : 0 < m := Nat.pos_of_ne_zero h5 have m_lt_n : m < n := by define at p_prime show m < n from calc m _ < m + m := by linarith _ = 2 * m := by ring _ ≤ p * m := by rel [p_prime.left] _ = n := n_eq_pm.symm done obtain (L : List Nat) (h6 : prime_factorization m L) from ih m m_lt_n m_pos define at h6 have ndpl_L : nondec_prime_list L := h6.left define at ndpl_L apply Exists.intro (p :: L) define apply And.intro · -- Proof of nondec_prime_list (p :: L) define apply And.intro · -- Proof of all_prime (p :: L) rewrite [all_prime_cons] show prime p ∧ all_prime L from And.intro p_prime ndpl_L.left done · -- Proof of nondec (p :: L) rewrite [nondec_cons] apply And.intro _ ndpl_L.right fix q : Nat assume q_in_L : q ∈ L have h7 : q ∣ prod L := list_elt_dvd_prod q_in_L rewrite [h6.right] at h7 --h7 : q ∣ m have h8 : m ∣ n := by apply Exists.intro p rewrite [n_eq_pm] ring done have q_dvd_n : q ∣ n := dvd_trans h7 h8 have ap_L : all_prime L := ndpl_L.left define at ap_L have q_prime_factor : prime_factor q n := And.intro (ap_L q q_in_L) q_dvd_n show p ≤ q from p_least q q_prime_factor done done · -- Proof of prod (p :: L) = n rewrite [prod_cons, h6.right, n_eq_pm] rfl done done done theorem Theorem_7_2_2 {a b c : Nat} (h1 : c ∣ a * b) (h2 : rel_prime a c) : c ∣ b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑c ∣ ↑b define at h1; define at h2; define obtain (j : Nat) (h3 : a * b = c * j) from h1 set s : Int := gcd_c1 a c set t : Int := gcd_c2 a c have h4 : s * ↑a + t * ↑c = ↑(gcd a c) := gcd_lin_comb c a rewrite [h2, Nat.cast_one] at h4 --h4 : s * ↑a + t * ↑c = (1 : Int) apply Exists.intro (s * ↑j + t * ↑b) show ↑b = ↑c * (s * ↑j + t * ↑b) from calc ↑b _ = (1 : Int) * ↑b := (one_mul _).symm _ = (s * ↑a + t * ↑c) * ↑b := by rw [h4] _ = s * (↑a * ↑b) + t * ↑c * ↑b := by ring _ = s * (↑c * ↑j) + t * ↑c * ↑b := by rw [←Nat.cast_mul a b, h3, Nat.cast_mul c j] _ = ↑c * (s * ↑j + t * ↑b) := by ring done lemma le_nonzero_prod_left {a b : Nat} (h : a * b ≠ 0) : a ≤ a * b := by have h1 : b ≠ 0 := by contradict h with h1 rewrite [h1] ring done have h2 : 1 ≤ b := Nat.pos_of_ne_zero h1 show a ≤ a * b from calc a = a * 1 := (mul_one a).symm _ ≤ a * b := by rel [h2] done lemma le_nonzero_prod_right {a b : Nat} (h : a * b ≠ 0) : b ≤ a * b := by rewrite [mul_comm] rewrite [mul_comm] at h show b ≤ b * a from le_nonzero_prod_left h done lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry lemma rel_prime_of_prime_not_dvd {a p : Nat} (h1 : prime p) (h2 : ¬p ∣ a) : rel_prime a p := by have h3 : gcd a p ∣ a := gcd_dvd_left a p have h4 : gcd a p ∣ p := gcd_dvd_right a p have h5 : gcd a p = 1 ∨ gcd a p = p := dvd_prime h1 h4 have h6 : gcd a p ≠ p := by contradict h2 with h6 rewrite [h6] at h3 show p ∣ a from h3 done disj_syll h5 h6 show rel_prime a p from h5 done theorem Theorem_7_2_3 {a b p : Nat} (h1 : prime p) (h2 : p ∣ a * b) : p ∣ a ∨ p ∣ b := by or_right with h3 have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3 show p ∣ b from Theorem_7_2_2 h2 h4 done lemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a ≥ 1 := by have h1 : a ≠ 0 := by by_contra h1 rewrite [h1] at h contradict h linarith done show a ≥ 1 from Nat.pos_of_ne_zero h1 done lemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by have h1 : a ≥ 1 := ge_one_of_prod_one h have h2 : a * b ≠ 0 := by linarith have h3 : a ≤ a * b := le_nonzero_prod_left h2 rewrite [h] at h3 show a = 1 from Nat.le_antisymm h3 h1 done lemma eq_one_of_dvd_one {n : Nat} (h : n ∣ 1) : n = 1 := by obtain (j : Nat) (h1 : 1 = n * j) from h show n = 1 from eq_one_of_prod_one h1.symm done lemma prime_not_one {p : Nat} (h : prime p) : p ≠ 1 := by define at h linarith done theorem Theorem_7_2_4 {p : Nat} (h1 : prime p) : ∀ (l : List Nat), p ∣ prod l → ∃ a ∈ l, p ∣ a := by apply List.rec · -- Base Case. Goal : p ∣ prod [] → ∃ a ∈ [], p ∣ a rewrite [prod_nil] assume h2 : p ∣ 1 show ∃ a ∈ [], p ∣ a from absurd (eq_one_of_dvd_one h2) (prime_not_one h1) done · -- Induction Step fix b : Nat fix L : List Nat assume ih : p ∣ prod L → ∃ a ∈ L, p ∣ a --Goal : p ∣ prod (b :: L) → ∃ a ∈ b :: L, p ∣ a assume h2 : p ∣ prod (b :: L) rewrite [prod_cons] at h2 have h3 : p ∣ b ∨ p ∣ prod L := Theorem_7_2_3 h1 h2 by_cases on h3 · -- Case 1. h3 : p ∣ b apply Exists.intro b show b ∈ b :: L ∧ p ∣ b from And.intro (List.mem_cons_self b L) h3 done · -- Case 2. h3 : p ∣ prod L obtain (a : Nat) (h4 : a ∈ L ∧ p ∣ a) from ih h3 apply Exists.intro a show a ∈ b :: L ∧ p ∣ a from And.intro (List.mem_cons_of_mem b h4.left) h4.right done done done lemma prime_in_list {p : Nat} {l : List Nat} (h1 : prime p) (h2 : all_prime l) (h3 : p ∣ prod l) : p ∈ l := by obtain (a : Nat) (h4 : a ∈ l ∧ p ∣ a) from Theorem_7_2_4 h1 l h3 define at h2 have h5 : prime a := h2 a h4.left have h6 : p = 1 ∨ p = a := dvd_prime h5 h4.right disj_syll h6 (prime_not_one h1) rewrite [h6] show a ∈ l from h4.left done lemma first_le_first {p q : Nat} {l m : List Nat} (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m)) (h3 : prod (p :: l) = prod (q :: m)) : p ≤ q := by define at h1; define at h2 have h4 : q ∣ prod (p :: l) := by define apply Exists.intro (prod m) rewrite [←prod_cons] show prod (p :: l) = prod (q :: m) from h3 done have h5 : all_prime (q :: m) := h2.left rewrite [all_prime_cons] at h5 have h6 : q ∈ p :: l := prime_in_list h5.left h1.left h4 have h7 : nondec (p :: l) := h1.right rewrite [nondec_cons] at h7 rewrite [List.mem_cons] at h6 by_cases on h6 · -- Case 1. h6 : q = p linarith done · -- Case 2. h6 : q ∈ l have h8 : ∀ m ∈ l, p ≤ m := h7.left show p ≤ q from h8 q h6 done done lemma nondec_prime_list_tail {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by define at h define rewrite [all_prime_cons, nondec_cons] at h show all_prime l ∧ nondec l from And.intro h.left.right h.right.right done lemma cons_prod_not_one {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : prod (p :: l) ≠ 1 := by define at h have h1 : all_prime (p :: l) := h.left rewrite [all_prime_cons] at h1 rewrite [prod_cons] by_contra h2 show False from (prime_not_one h1.left) (eq_one_of_prod_one h2) done lemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) : l = [] ↔ prod l = 1 := by apply Iff.intro · -- (→) assume h1 : l = [] rewrite [h1] show prod [] = 1 from prod_nil done · -- (←) contrapos assume h1 : ¬l = [] obtain (p : Nat) (h2 : ∃ (L : List Nat), l = p :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List Nat) (h3 : l = p :: L) from h2 rewrite [h3] at h rewrite [h3] show ¬prod (p :: L) = 1 from cons_prod_not_one h done done lemma prime_pos {p : Nat} (h : prime p) : p > 0 := by define at h linarith done theorem Theorem_7_2_5 : ∀ (l1 l2 : List Nat), nondec_prime_list l1 → nondec_prime_list l2 → prod l1 = prod l2 → l1 = l2 := by apply List.rec · -- Base Case. Goal : ∀ (l2 : List Nat), nondec_prime_list [] → -- nondec_prime_list l2 → prod [] = prod l2 → [] = l2 fix l2 : List Nat assume h1 : nondec_prime_list [] assume h2 : nondec_prime_list l2 assume h3 : prod [] = prod l2 rewrite [prod_nil, eq_comm, ←list_nil_iff_prod_one h2] at h3 show [] = l2 from h3.symm done · -- Induction Step fix p : Nat fix L1 : List Nat assume ih : ∀ (L2 : List Nat), nondec_prime_list L1 → nondec_prime_list L2 → prod L1 = prod L2 → L1 = L2 -- Goal : ∀ (l2 : List Nat), nondec_prime_list (p :: L1) → -- nondec_prime_list l2 → prod (p :: L1) = prod l2 → p :: L1 = l2 fix l2 : List Nat assume h1 : nondec_prime_list (p :: L1) assume h2 : nondec_prime_list l2 assume h3 : prod (p :: L1) = prod l2 have h4 : ¬prod (p :: L1) = 1 := cons_prod_not_one h1 rewrite [h3, ←list_nil_iff_prod_one h2] at h4 obtain (q : Nat) (h5 : ∃ (L : List Nat), l2 = q :: L) from List.exists_cons_of_ne_nil h4 obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5 rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2) rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2) have h7 : p ≤ q := first_le_first h1 h2 h3 have h8 : q ≤ p := first_le_first h2 h1 h3.symm have h9 : p = q := by linarith rewrite [h9, prod_cons, prod_cons] at h3 --h3 : q * prod L1 = q * prod L2 have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1 have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2 define at h2 have h12 : all_prime (q :: L2) := h2.left rewrite [all_prime_cons] at h12 have h13 : q > 0 := prime_pos h12.left have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3 have h15 : L1 = L2 := ih L2 h10 h11 h14 rewrite [h6, h9, h15] rfl done done theorem fund_thm_arith (n : Nat) (h : n ≥ 1) : ∃! (l : List Nat), prime_factorization n l := by exists_unique · -- Existence show ∃ (l : List Nat), prime_factorization n l from exists_prime_factorization n h done · -- Uniqueness fix l1 : List Nat; fix l2 : List Nat assume h1 : prime_factorization n l1 assume h2 : prime_factorization n l2 define at h1; define at h2 have h3 : prod l1 = n := h1.right rewrite [←h2.right] at h3 show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3 done done /- Section 7.3 -/ theorem congr_refl (m : Nat) : ∀ (a : Int), a ≡ a (MOD m) := by fix a : Int define --Goal : ∃ (c : Int), a - a = ↑m * c apply Exists.intro 0 ring done theorem congr_symm {m : Nat} : ∀ {a b : Int}, a ≡ b (MOD m) → b ≡ a (MOD m) := by fix a : Int; fix b : Int assume h1 : a ≡ b (MOD m) define at h1 --h1 : ∃ (c : Int), a - b = ↑m * c define --Goal : ∃ (c : Int), b - a = ↑m * c obtain (c : Int) (h2 : a - b = m * c) from h1 apply Exists.intro (-c) show b - a = m * (-c) from calc b - a _ = -(a - b) := by ring _ = -(m * c) := by rw [h2] _ = m * (-c) := by ring done theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry /- Fundamental properties of congruence classes -/ lemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) : X = Y ↔ X.val = Y.val := Fin.ext_iff lemma val_nat_eq_mod (n k : Nat) : ([k]_(n + 1)).val = k % (n + 1) := by rfl lemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl theorem cc_rep {m : Nat} (X : ZMod m) : ∃ (a : Int), X = [a]_m := match m with | 0 => by apply Exists.intro X rfl done | n + 1 => by apply Exists.intro ↑(X.val) have h1 : X.val < n + 1 := Fin.prop X rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1] rfl done theorem add_class (m : Nat) (a b : Int) : [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm theorem mul_class (m : Nat) (a b : Int) : [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm lemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ [a - b]_m = [0]_m := by apply Iff.intro · -- (→) assume h1 : [a]_m = [b]_m have h2 : a - b = a + (-b) := by ring have h3 : b + (-b) = 0 := by ring show [a - b]_m = [0]_m from calc [a - b]_m _ = [a + (-b)]_m := by rw [h2] _ = [a]_m + [-b]_m := by rw [add_class] _ = [b]_m + [-b]_m := by rw [h1] _ = [b + -b]_m := by rw [add_class] _ = [0]_m := by rw [h3] done · -- (←) assume h1 : [a - b]_m = [0]_m have h2 : b + (a - b) = a := by ring have h3 : b + 0 = b := by ring show [a]_m = [b]_m from calc [a]_m _ = [b + (a - b)]_m := by rw [h2] _ = [b]_m + [a - b]_m := by rw [add_class] _ = [b]_m + [0]_m := by rw [h1] _ = [b + 0]_m := by rw [add_class] _ = [b]_m := by rw [h3] done done lemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) : [a]_m = [0]_m → [-a]_m = [0]_m := by assume h1 : [a]_m = [0]_m have h2 : 0 + (-a) = -a := by ring have h3 : a + (-a) = 0 := by ring show [-a]_m = [0]_m from calc [-a]_m _ = [0 + (-a)]_m := by rw [h2] _ = [0]_m + [-a]_m := by rw [add_class] _ = [a]_m + [-a]_m := by rw [h1] _ = [a + (-a)]_m := by rw [add_class] _ = [0]_m := by rw [h3] done lemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) : [-a]_m = [0]_m ↔ [a]_m = [0]_m := by apply Iff.intro _ (cc_neg_zero_of_cc_zero m a) assume h1 : [-a]_m = [0]_m have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1 have h3 : -(-a) = a := by ring rewrite [h3] at h2 show [a]_m = [0]_m from h2 done lemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl lemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m ↔ m ∣ k := match m with | 0 => by have h : (0 : Int) = (↑(0 : Nat) : Int) := by rfl rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj] apply Iff.intro · -- (→) assume h1 : k = 0 rewrite [h1] show 0 ∣ 0 from dvd_self 0 done · -- (←) assume h1 : 0 ∣ k obtain (c : Nat) (h2 : k = 0 * c) from h1 rewrite [h2] ring done done | n + 1 => by rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero] show k % (n + 1) = 0 ↔ n + 1 ∣ k from (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm done lemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m ↔ ↑m ∣ a := by obtain (k : Nat) (h1 : a = ↑k ∨ a = -↑k) from Int.eq_nat_or_neg a by_cases on h1 · -- Case 1. h1: a = ↑k rewrite [h1, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done · -- Case 2. h1: a = -↑k rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done done theorem cc_eq_iff_congr (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ a ≡ b (MOD m) := calc [a]_m = [b]_m _ ↔ [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b _ ↔ ↑m ∣ (a - b) := cc_zero_iff_dvd m (a - b) _ ↔ a ≡ b (MOD m) := by rfl /- End of fundamental properties of congruence classes -/ lemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m := by have h1 : (↑m : Int) ≠ 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m) show 0 ≤ a % m from Int.emod_nonneg a h1 done lemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m) have h2 : (↑m : Int) > 0 := (Nat.cast_pos).rtl h1 show a % m < m from Int.emod_lt_of_pos a h2 done lemma congr_mod_mod (m : Nat) (a : Int) : a ≡ a % m (MOD m) := by define have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m apply Exists.intro (a / m) show a - a % m = m * (a / m) from calc a - (a % m) _ = m * (a / m) + a % m - a % m := by rw [h1] _ = m * (a / m) := by ring done lemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) := And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a)) theorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) : ∃! (r : Int), 0 ≤ r ∧ r < m ∧ a ≡ r (MOD m) := by exists_unique · -- Existence apply Exists.intro (a % m) show 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) from mod_cmpl_res m a done · -- Uniqueness fix r1 : Int; fix r2 : Int assume h1 : 0 ≤ r1 ∧ r1 < m ∧ a ≡ r1 (MOD m) assume h2 : 0 ≤ r2 ∧ r2 < m ∧ a ≡ r2 (MOD m) have h3 : r1 ≡ r2 (MOD m) := congr_trans (congr_symm h1.right.right) h2.right.right obtain (d : Int) (h4 : r1 - r2 = m * d) from h3 have h5 : r1 - r2 < m * 1 := by linarith have h6 : m * (-1) < r1 - r2 := by linarith rewrite [h4] at h5 --h5 : m * d < m * 1 rewrite [h4] at h6 --h6 : m * -1 < m * d have h7 : (↑m : Int) ≥ 0 := Nat.cast_nonneg m have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7 have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7 have h10 : d = 0 := by linarith show r1 = r2 from calc r1 _ = r1 - r2 + r2 := by ring _ = m * 0 + r2 := by rw [h4, h10] _ = r2 := by ring done done lemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m := (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a) theorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y rewrite [h1, h2] have h3 : a + b = b + a := by ring show [a]_m + [b]_m = [b]_m + [a]_m from calc [a]_m + [b]_m _ = [a + b]_m := add_class m a b _ = [b + a]_m := by rw [h3] _ = [b]_m + [a]_m := (add_class m b a).symm done theorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X rewrite [h1] have h2 : a * 1 = a := by ring show [a]_m * [1]_m = [a]_m from calc [a]_m * [1]_m _ = [a * 1]_m := mul_class m a 1 _ = [a]_m := by rw [h2] done theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry lemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) : [a]_m * [gcd_c2 m a]_m = [1]_m := by set s : Int := gcd_c1 m a have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m define at h1 rewrite [h1, Nat.cast_one] at h2 --h2 : s * ↑m + gcd_c2 m a * ↑a = 1 rewrite [mul_class, cc_eq_iff_congr] define --Goal : ∃ (c : Int), ↑a * gcd_c2 m a - 1 = ↑m * c apply Exists.intro (-s) show a * (gcd_c2 m a) - 1 = m * (-s) from calc a * (gcd_c2 m a) - 1 _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring _ = 1 + m * (-s) - 1 := by rw [h2] _ = m * (-s) := by ring done theorem Theorem_7_3_7 (m a : Nat) : invertible [a]_m ↔ rel_prime m a := by apply Iff.intro · -- (→) assume h1 : invertible [a]_m define at h1 obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1 obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y rewrite [h3, mul_class, cc_eq_iff_congr] at h2 define at h2 obtain (c : Int) (h4 : a * b - 1 = m * c) from h2 rewrite [Exercise_7_2_6] --Goal : ∃ (s t : Int), s * ↑m + t * ↑a = 1 apply Exists.intro (-c) apply Exists.intro b show (-c) * m + b * a = 1 from calc (-c) * m + b * a _ = (-c) * m + (a * b - 1) + 1 := by ring _ = (-c) * m + m * c + 1 := by rw [h4] _ = 1 := by ring done · -- (←) assume h1 : rel_prime m a define show ∃ (Y : ZMod m), [a]_m * Y = [1]_m from Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1) done done /- Section 7.4 -/ section Euler open Euler lemma num_rp_below_base {m : Nat} : num_rp_below m 0 = 0 := by rfl lemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) : num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by have h1 : num_rp_below m (j + 1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : gcd m j = 1 rewrite [if_pos h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1 show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1 done lemma num_rp_below_step_not_rp {m j : Nat} (h : ¬rel_prime m j) : num_rp_below m (j + 1) = num_rp_below m j := by have h1 : num_rp_below m (j +1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : ¬gcd m j = 1 rewrite [if_neg h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j show num_rp_below m (j + 1) = num_rp_below m j from h1 done lemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl #eval phi 10 --Answer: 4 lemma prod_inv_iff_inv {m : Nat} {X : ZMod m} (h1 : invertible X) (Y : ZMod m) : invertible (X * Y) ↔ invertible Y := by apply Iff.intro · -- (→) assume h2 : invertible (X * Y) obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2 apply Exists.intro (X * Z) rewrite [←h3] --Goal : Y * (X * Z) = X * Y * Z ring --Note that ring can do algebra in ZMod m done · -- (←) assume h2 : invertible Y obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1 obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2 apply Exists.intro (Xi * Yi) show (X * Y) * (Xi * Yi) = [1]_m from calc X * Y * (Xi * Yi) _ = (X * Xi) * (Y * Yi) := by ring _ = [1]_m * [1]_m := by rw [h3, h4] _ = [1]_m := Theorem_7_3_6_7 [1]_m done done lemma F_rp_def {m i : Nat} (h : rel_prime m i) : F m i = [i]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h --h : gcd m i = 1 rewrite [if_pos h] at h1 show F m i = [i]_m from h1 done lemma F_not_rp_def {m i : Nat} (h : ¬rel_prime m i) : F m i = [1]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h rewrite [h1, if_neg h] rfl done lemma prod_seq_base {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 0 k f = [1]_m := by rfl lemma prod_seq_step {m : Nat} (n k : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl lemma prod_seq_zero_step {m : Nat} (n : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by rewrite [prod_seq_step, zero_add] rfl done lemma prod_one {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 1 k f = f k := by rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7] rfl done lemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl lemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m := calc [G m a i]_m _ = [(a * i) % m]_m := by rfl _ = [a * i]_m := (cc_eq_mod m (a * i)).symm _ = [a]_m * [i]_m := (mul_class m a i).symm lemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) : rel_prime m (G m a i) ↔ rel_prime m i := by have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1 show rel_prime m (G m a i) ↔ rel_prime m i from calc rel_prime m (G m a i) _ ↔ invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm _ ↔ invertible ([a]_m * [i]_m) := by rw [cc_G] _ ↔ invertible [i]_m := prod_inv_iff_inv h2 ([i]_m) _ ↔ rel_prime m i := Theorem_7_3_7 m i done lemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) : F m (G m a i) = [a]_m * F m i := by have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2 show F m (G m a i) = [a]_m * F m i from calc F m (G m a i) _ = [G m a i]_m := F_rp_def h3 _ = [a]_m * [i]_m := cc_G m a i _ = [a]_m * F m i := by rw [F_rp_def h2] done lemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : ¬rel_prime m i) : F m (G m a i) = [1]_m := by rewrite [←G_rp_iff h1 i] at h2 show F m (G m a i) = [1]_m from F_not_rp_def h2 done lemma FG_prod {m a : Nat} (h1 : rel_prime m a) : ∀ (k : Nat), prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by by_induc · -- Base Case show prod_seq 0 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from calc prod_seq 0 0 ((F m) ∘ (G m a)) _ = [1]_m := prod_seq_base _ _ _ = [a]_m ^ 0 * [1]_m := by ring _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by rw [num_rp_below_base, prod_seq_base] done · -- Induction Step fix k : Nat assume ih : prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) by_cases h2 : rel_prime m k · -- Case 1. h2 : rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([a]_m * F m k) := by rw [FG_rp h1 h2] _ = [a]_m ^ ((num_rp_below m k) + 1) * ((prod_seq k 0 (F m)) * F m k) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_rp h2, prod_seq_zero_step] done · -- Case 2. h2 : ¬rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([1]_m) := by rw [FG_not_rp h1 h2] _ = [a]_m ^ (num_rp_below m k) * (prod_seq k 0 (F m) * ([1]_m)) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_not_rp h2, prod_seq_zero_step, F_not_rp_def h2] done done done lemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by define --Goal : ∀ i < m, G m a i < m fix i : Nat assume h1 : i < m rewrite [G_def] --Goal : a * i % m < m show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m) done lemma left_inv_one_one_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g' (g i) = i) : one_one_below n g := sorry lemma right_inv_onto_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g (g' i) = i) (h2 : maps_below n g') : onto_below n g := by define at h2; define fix k : Nat assume h3 : k < n apply Exists.intro (g' k) show g' k < n ∧ g (g' k) = k from And.intro (h2 k h3) (h1 k h3) done lemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m * [inv_mod m a]_m = [1]_m := by have h2 : 0 ≤ (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a) show [a]_m * [inv_mod m a]_m = [1]_m from calc [a]_m * [inv_mod m a]_m _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2] _ = [a]_m * [gcd_c2 m a]_m := by rw [←cc_eq_mod] _ = [1]_m := gcd_c2_inv h1 done lemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m := calc a * (b % m) % m = a % m * (b % m % m) % m := Nat.mul_mod _ _ _ _ = a % m * (b % m) % m := by rw [Nat.mod_mod] _ = a * b % m := (Nat.mul_mod _ _ _).symm lemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm] rfl done theorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] : ↑a ≡ ↑b (MOD m) ↔ a % m = b % m := sorry lemma mul_inv_mod_cancel {m a i : Nat} [NeZero m] (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1 rewrite [mul_class, cc_eq_iff_congr, ←Nat.cast_mul, ←Nat.cast_one, congr_iff_mod_eq_Nat] at h3 show a * inv_mod m a * i % m = i from calc a * (inv_mod m a) * i % m _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = 1 % m * i % m := by rw [h3] _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = i % m := by rw [one_mul] _ = i := Nat.mod_eq_of_lt h2 done lemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl lemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, G m a (Ginv m a i) = i := by fix i : Nat assume h2 : i < m show G m a (Ginv m a i) = i from calc G m a (Ginv m a i) _ = a * ((inv_mod m a * i) % m) % m := by rfl _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, Ginv m a (G m a i) = i := by fix i : Nat assume h2 : i < m show Ginv m a (G m a i) = i from calc Ginv m a (G m a i) _ = inv_mod m a * ((a * i) % m) % m := by rfl _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc, mul_comm (inv_mod m a)] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_maps_below (m a : Nat) [NeZero m] : maps_below m (Ginv m a) := G_maps_below m (inv_mod m a) lemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : one_one_below m (G m a) := left_inv_one_one_below (Ginv_left_inv h1) lemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : onto_below m (G m a) := right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a) lemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : perm_below m (G m a) := And.intro (G_maps_below m a) (And.intro (G_one_one_below h1) (G_onto_below h1)) --Permuting a product of congruence classes doesn't change product lemma swap_fst (u v : Nat) : swap u v u = v := by define : swap u v u --Goal : (if u = u then v else if u = v then u else u) = v have h : u = u := by rfl rewrite [if_pos h] rfl done lemma swap_snd (u v : Nat) : swap u v v = u := by define : swap u v v by_cases h1 : v = u · -- Case 1. h1 : v = u rewrite [if_pos h1] show v = u from h1 done · -- Case 2. h1 : v ≠ u rewrite [if_neg h1] have h2 : v = v := by rfl rewrite [if_pos h2] rfl done done lemma swap_other {u v i : Nat} (h1 : i ≠ u) (h2 : i ≠ v) : swap u v i = i := by define : swap u v i rewrite [if_neg h1, if_neg h2] rfl done lemma swap_values (u v i : Nat) : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := by by_cases h1 : i = u · -- Case 1. h1 : i = u apply Or.inl rewrite [h1] show swap u v u = v from swap_fst u v done · -- Case 2. h1 : i ≠ u apply Or.inr by_cases h2 : i = v · -- Case 2.1. h2 : i = v apply Or.inl rewrite [h2] show swap u v v = u from swap_snd u v done · -- Case 2.2. h2 : i ≠ v apply Or.inr show swap u v i = i from swap_other h1 h2 done done done lemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by define fix i : Nat assume h3 : i < n have h4 : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := swap_values u v i by_cases on h4 · -- Case 1. h4 : swap u v i = v rewrite [h4] show v < n from h2 done · -- Case 2. by_cases on h4 · -- Case 2.1. h4 : swap u v i = u rewrite [h4] show u < n from h1 done · -- Case 2.2. h4 : swap u v i = i rewrite [h4] show i < n from h3 done done done lemma swap_swap (u v n : Nat) : ∀ i < n, swap u v (swap u v i) = i := by fix i : Nat assume h : i < n by_cases h1 : i = u · -- Case 1. h1 : i = u rewrite [h1, swap_fst, swap_snd] rfl done · -- Case 2. h1 : i ≠ u by_cases h2 : i = v · -- Case 2.1. h2 : i = v rewrite [h2, swap_snd, swap_fst] rfl done · -- Case 2.2. h2 : i ≠ v rewrite [swap_other h1 h2, swap_other h1 h2] rfl done done done lemma swap_one_one_below (u v n) : one_one_below n (swap u v) := left_inv_one_one_below (swap_swap u v n) lemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) := right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2) lemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) := And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2)) lemma comp_perm_below {n : Nat} {f g : Nat → Nat} (h1 : perm_below n f) (h2 : perm_below n g) : perm_below n (f ∘ g) := sorry lemma trivial_swap (u : Nat) : swap u u = id := by apply funext fix x : Nat by_cases h1 : x = u · -- Case 1. h1 : x = u rewrite [h1, swap_fst] rfl done · -- Case 2. h1 : x ≠ u rewrite [swap_other h1 h1] rfl done done lemma prod_eq_fun {m : Nat} (f g : Nat → ZMod m) (k : Nat) : ∀ (n : Nat), (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g := by by_induc · -- Base Case assume h : (∀ i < 0, f (k + i) = g (k + i)) rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g assume h1 : ∀ i < n + 1, f (k + i) = g (k + i) have h2 : ∀ i < n, f (k + i) = g (k + i) := by fix i : Nat assume h2 : i < n have h3 : i < n + 1 := by linarith show f (k + i) = g (k + i) from h1 i h3 done have h3 : prod_seq n k f = prod_seq n k g := ih h2 have h4 : n < n + 1 := Nat.lt_succ_self n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4] rfl done done lemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := by have h2 : ∀ (i : Nat), i < u → (f ∘ swap u n) (0 + i) = f (0 + i) := by fix i : Nat assume h2 : i < u have h3 : 0 + i ≠ u := by linarith have h4 : 0 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl done show prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f from prod_eq_fun (f ∘ swap u n) f 0 u h2 done lemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := by have h2 : ∀ i < j, (f ∘ swap u n) (u + 1 + i) = f (u + 1 + i) := by fix i : Nat assume h2 : i < j have h3 : u + 1 + i ≠ u := by linarith have h4 : u + 1 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl show prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f from prod_eq_fun (f ∘ swap u n) f (u + 1) j h2 done lemma break_prod {m : Nat} (n : Nat) (f : Nat → ZMod m) : ∀ (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by by_induc · -- Base Case have h : n + 0 = n := by rfl rewrite [prod_seq_base, h, Theorem_7_3_6_7] rfl done · -- Induction Step fix j : Nat assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f rewrite [←add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc] rfl done done lemma break_prod_twice {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f := break_prod n f 1 rewrite [prod_one] at h2 have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f := break_prod (u + 1) f j rewrite [←h1] at h3 have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f := break_prod u f 1 rewrite [prod_one] at h4 rewrite [h3, h4] at h2 show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2 done lemma swap_prod_eq_prod {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f := by by_cases h2 : u = n · -- Case 1. h2 : u = n rewrite [h2, trivial_swap n] --Goal : prod_seq (n + 1) 0 (f ∘ id) = prod_seq (n + 1) 0 f rfl done · -- Case 2. h2 : ¬u = n have h3 : u + 1 ≤ n := Nat.lt_of_le_of_ne h1 h2 obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3 have break_f : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := break_prod_twice f h4 have break_fs : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_prod_twice (f ∘ swap u n) h4 have f_eq_fs_below : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := swap_prod_eq_prod_below f h1 have f_eq_fs_btwn : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4 show prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f from calc prod_seq (n + 1) 0 (f ∘ swap u n) _ = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_fs _ = prod_seq u 0 f * (f ∘ swap u n) u * prod_seq j (u + 1) f * (f ∘ swap u n) n := by rw [f_eq_fs_below, f_eq_fs_btwn] _ = prod_seq u 0 f * f (swap u n u) * prod_seq j (u + 1) f * f (swap u n n) := by rfl _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by rw [swap_fst, swap_snd] _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring _ = prod_seq (n + 1) 0 f := break_f.symm done done lemma perm_below_fixed {n : Nat} {g : Nat → Nat} (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry lemma perm_prod {m : Nat} (f : Nat → ZMod m) : ∀ (n : Nat), ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) := by by_induc · -- Base Case fix g : Nat → Nat assume h1 : perm_below 0 g rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) fix g : Nat → Nat assume g_pb : perm_below (n + 1) g define at g_pb have g_ob : onto_below (n + 1) g := g_pb.right.right define at g_ob have h1 : n < n + 1 := by linarith obtain (u : Nat) (h2 : u < n + 1 ∧ g u = n) from g_ob n h1 have s_pb : perm_below (n + 1) (swap u n) := swap_perm_below h2.left h1 have gs_pb_n1 : perm_below (n + 1) (g ∘ swap u n) := comp_perm_below g_pb s_pb have gs_fix_n : (g ∘ swap u n) n = n := calc (g ∘ swap u n) n _ = g (swap u n n) := by rfl _ = g u := by rw [swap_snd] _ = n := h2.right have gs_pb_n : perm_below n (g ∘ swap u n) := perm_below_fixed gs_pb_n1 gs_fix_n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f ∘ (g ∘ swap u n)) := ih (g ∘ swap u n) gs_pb_n have h3 : u ≤ n := by linarith show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f ∘ g) from calc prod_seq (n + 1) 0 f _ = prod_seq n 0 f * f n := prod_seq_zero_step n f _ = prod_seq n 0 (f ∘ (g ∘ swap u n)) * f ((g ∘ swap u n) n) := by rw [gs_prod, gs_fix_n] _ = prod_seq n 0 (f ∘ g ∘ swap u n) * (f ∘ g ∘ swap u n) n := by rfl _ = prod_seq (n + 1) 0 (f ∘ g ∘ swap u n) := (prod_seq_zero_step n (f ∘ g ∘ swap u n)).symm _ = prod_seq (n + 1) 0 ((f ∘ g) ∘ swap u n) := by rfl _ = prod_seq (n + 1) 0 (f ∘ g) := swap_prod_eq_prod (f ∘ g) h3 done done lemma F_invertible (m i : Nat) : invertible (F m i) := by by_cases h : rel_prime m i · -- Case 1. h : rel_prime m i rewrite [F_rp_def h] show invertible [i]_m from (Theorem_7_3_7 m i).rtl h done · -- Case 2. h : ¬rel_prime m i rewrite [F_not_rp_def h] apply Exists.intro [1]_m show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m done done lemma Fprod_invertible (m : Nat) : ∀ (k : Nat), invertible (prod_seq k 0 (F m)) := by by_induc · -- Base Case apply Exists.intro [1]_m show prod_seq 0 0 (F m) * [1]_m = [1]_m from calc prod_seq 0 0 (F m) * [1]_m _ = [1]_m * [1]_m := by rw [prod_seq_base] _ = [1]_m := Theorem_7_3_6_7 ([1]_m) done · -- Induction Step fix k : Nat assume ih : invertible (prod_seq k 0 (F m)) rewrite [prod_seq_zero_step] show invertible (prod_seq k 0 (F m) * (F m k)) from (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k) done done theorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m ^ (phi m) = [1]_m := by have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2 show [a]_m ^ (phi m) = [1]_m from calc [a]_m ^ (phi m) _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3] _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring _ = prod_seq m 0 (F m ∘ G m a) * Y := by rw [FG_prod h1 m, phi_def] _ = prod_seq m 0 (F m) * Y := by rw [perm_prod (F m) m (G m a) (G_perm_below h1)] _ = [1]_m := by rw [h3] done lemma Exercise_7_4_5_Int (m : Nat) (a : Int) : ∀ (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry lemma Exercise_7_4_5_Nat (m a n : Nat) : [a]_m ^ n = [a ^ n]_m := by rewrite [Exercise_7_4_5_Int] rfl done theorem Euler's_theorem {m a : Nat} [NeZero m] (h1 : rel_prime m a) : a ^ (phi m) ≡ 1 (MOD m) := by have h2 : [a]_m ^ (phi m) = [1]_m := Theorem_7_4_2 h1 rewrite [Exercise_7_4_5_Nat m a (phi m)] at h2 --h2 : [a ^ phi m]_m = [1]_m show a ^ (phi m) ≡ 1 (MOD m) from (cc_eq_iff_congr _ _ _).ltr h2 done #eval gcd 10 7 --Answer: 1. So 10 and 7 are relatively prime #eval 7 ^ phi 10 --Answer: 2401, which is congruent to 1 mod 10. end Euler /- Section 7.5 -/ lemma num_rp_prime {p : Nat} (h1 : prime p) : ∀ k < p, num_rp_below p (k + 1) = k := sorry lemma phi_prime {p : Nat} (h1 : prime p) : phi p = p - 1 := by have h2 : 1 ≤ p := prime_pos h1 have h3 : p - 1 + 1 = p := Nat.sub_add_cancel h2 have h4 : p - 1 < p := by linarith have h5 : num_rp_below p (p - 1 + 1) = p - 1 := num_rp_prime h1 (p - 1) h4 rewrite [h3] at h5 show phi p = p - 1 from h5 done theorem Theorem_7_2_2_Int {a c : Nat} {b : Int} (h1 : ↑c ∣ ↑a * b) (h2 : rel_prime a c) : ↑c ∣ b := by rewrite [Int.natCast_dvd, Int.natAbs_mul, Int.natAbs_ofNat] at h1 --h1 : c ∣ a * Int.natAbs b rewrite [Int.natCast_dvd] --Goal : c ∣ Int.natAbs b show c ∣ Int.natAbs b from Theorem_7_2_2 h1 h2 done lemma Lemma_7_4_5 {m n : Nat} (a b : Int) (h1 : rel_prime m n) : a ≡ b (MOD m * n) ↔ a ≡ b (MOD m) ∧ a ≡ b (MOD n) := by apply Iff.intro · -- (→) assume h2 : a ≡ b (MOD m * n) obtain (j : Int) (h3 : a - b = (m * n) * j) from h2 apply And.intro · -- Proof of a ≡ b (MOD m) apply Exists.intro (n * j) show a - b = m * (n * j) from calc a - b _ = m * n * j := h3 _ = m * (n * j) := by ring done · -- Proof of a ≡ b (MOD n) apply Exists.intro (m * j) show a - b = n * (m * j) from calc a - b _ = m * n * j := h3 _ = n * (m * j) := by ring done done · -- (←) assume h2 : a ≡ b (MOD m) ∧ a ≡ b (MOD n) obtain (j : Int) (h3 : a - b = m * j) from h2.left have h4 : (↑n : Int) ∣ a - b := h2.right rewrite [h3] at h4 --h4 : ↑n ∣ ↑m * j have h5 : ↑n ∣ j := Theorem_7_2_2_Int h4 h1 obtain (k : Int) (h6 : j = n * k) from h5 apply Exists.intro k --Goal : a - b = ↑(m * n) * k rewrite [Nat.cast_mul] --Goal : a - b = ↑m * ↑n * k show a - b = (m * n) * k from calc a - b _ = m * j := h3 _ = m * (n * k) := by rw [h6] _ = (m * n) * k := by ring done done --From exercises of Section 7.2

theorem rel_prime_symm {a b : Nat} (h : rel_prime a b) : rel_prime b a

HTPI.rel_prime_symm

htpi/HTPILib/Chap7.lean

htpi

/- Copyright 2023 Daniel J. Velleman -/ import HTPILib.Chap6 namespace HTPI /- Definitions -/ lemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by have h : n + 1 > 0 := Nat.succ_pos n show a % (n + 1) < n + 1 from Nat.mod_lt a h done def gcd (a b : Nat) : Nat := match b with | 0 => a | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd (n + 1) (a % (n + 1)) termination_by b mutual def gcd_c1 (a b : Nat) : Int := match b with | 0 => 1 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c2 (n + 1) (a % (n + 1)) --Corresponds to s = t' termination_by b def gcd_c2 (a b : Nat) : Int := match b with | 0 => 0 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c1 (n + 1) (a % (n + 1)) - (gcd_c2 (n + 1) (a % (n + 1))) * ↑(a / (n + 1)) --Corresponds to t = s' - t'q termination_by b end def prime (n : Nat) : Prop := 2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n def prime_factor (p n : Nat) : Prop := prime p ∧ p ∣ n def all_prime (l : List Nat) : Prop := ∀ p ∈ l, prime p def nondec (l : List Nat) : Prop := match l with | [] => True --Of course, True is a proposition that is always true | n :: L => (∀ m ∈ L, n ≤ m) ∧ nondec L def nondec_prime_list (l : List Nat) : Prop := all_prime l ∧ nondec l def prod (l : List Nat) : Nat := match l with | [] => 1 | n :: L => n * (prod L) def prime_factorization (n : Nat) (l : List Nat) : Prop := nondec_prime_list l ∧ prod l = n def rel_prime (a b : Nat) : Prop := gcd a b = 1 def congr_mod (m : Nat) (a b : Int) : Prop := (↑m : Int) ∣ (a - b) def cc (m : Nat) (a : Int) : ZMod m := (↑a : ZMod m) notation:50 a " ≡ " b " (MOD " m ")" => congr_mod m a b notation:max "["a"]_"m:max => cc m a def invertible {m : Nat} (X : ZMod m) : Prop := ∃ (Y : ZMod m), X * Y = [1]_m def num_rp_below (m k : Nat) : Nat := match k with | 0 => 0 | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j def phi (m : Nat) : Nat := num_rp_below m m def prod_seq {m : Nat} (j k : Nat) (f : Nat → ZMod m) : ZMod m := match j with | 0 => [1]_m | n + 1 => prod_seq n k f * f (k + n) def maps_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i < n, g i < n def one_one_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i1 < n, ∀ i2 < n, g i1 = g i2 → i1 = i2 def onto_below (n : Nat) (g : Nat → Nat) : Prop := ∀ k < n, ∃ i < n, g i = k def perm_below (n : Nat) (g : Nat → Nat) : Prop := maps_below n g ∧ one_one_below n g ∧ onto_below n g def inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m) def swap (u v i : Nat) : Nat := if i = u then v else if i = v then u else i namespace Euler --For definitions specific to Euler's theorem def F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m def G (m a i : Nat) : Nat := (a * i) % m def Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i end Euler /- Section 7.1 -/ theorem dvd_mod_of_dvd_a_b {a b d : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ (a % b) := by set q : Nat := a / b have h3 : b * q + a % b = a := Nat.div_add_mod a b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 define --Goal : ∃ (c : Nat), a % b = d * c apply Exists.intro (j - k * q) show a % b = d * (j - k * q) from calc a % b _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm _ = a - b * q := by rw [h3] _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc] _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm done theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry #eval gcd 672 161 --Answer: 7 lemma gcd_base (a : Nat) : gcd a 0 = a := by rfl lemma gcd_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd a b = gcd b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1)) rfl done lemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b ≠ 0) : a % b < b := by have h1 : b > 0 := Nat.pos_of_ne_zero h show a % b < b from Nat.mod_lt a h1 done lemma dvd_self (n : Nat) : n ∣ n := by apply Exists.intro 1 ring done theorem gcd_dvd : ∀ (b a : Nat), (gcd a b) ∣ a ∧ (gcd a b) ∣ b := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd a b_1) ∣ a ∧ (gcd a b_1) ∣ b_1 fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_base] --Goal: a ∣ a ∧ a ∣ 0 apply And.intro (dvd_self a) define apply Exists.intro 0 rfl done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_nonzero a h1] --Goal : gcd b (a % b) ∣ a ∧ gcd b (a % b) ∣ b have h2 : a % b < b := mod_nonzero_lt a h1 have h3 : (gcd b (a % b)) ∣ b ∧ (gcd b (a % b)) ∣ (a % b) := ih (a % b) h2 b apply And.intro _ h3.left show (gcd b (a % b)) ∣ a from dvd_a_of_dvd_b_mod h3.left h3.right done done theorem gcd_dvd_left (a b : Nat) : (gcd a b) ∣ a := (gcd_dvd b a).left theorem gcd_dvd_right (a b : Nat) : (gcd a b) ∣ b := (gcd_dvd b a).right lemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl lemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c1 a b = gcd_c2 b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done lemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl lemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * ↑(a / b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done theorem gcd_lin_comb : ∀ (b a : Nat), (gcd_c1 a b) * ↑a + (gcd_c2 a b) * ↑b = ↑(gcd a b) := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd_c1 a b_1) * ↑a + (gcd_c2 a b_1) * ↑b_1 = ↑(gcd a b_1) fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base] --Goal : 1 * ↑a + 0 * ↑0 = ↑a ring done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1] --Goal : gcd_c2 b (a % b) * ↑a + -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * ↑(a / b)) * ↑b = -- ↑(gcd b (a % b)) set r : Nat := a % b set q : Nat := a / b set s : Int := gcd_c1 b r set t : Int := gcd_c2 b r --Goal : t * ↑a + (s - t * ↑q) * ↑b = ↑(gcd b r) have h2 : r < b := mod_nonzero_lt a h1 have h3 : s * ↑b + t * ↑r = ↑(gcd b r) := ih r h2 b have h4 : b * q + r = a := Nat.div_add_mod a b rewrite [←h3, ←h4] rewrite [Nat.cast_add, Nat.cast_mul] --Goal : t * (↑b * ↑q + ↑r) + (s - t * ↑q) * ↑b = s * ↑b + t * ↑r ring done done #eval gcd_c1 672 161 --Answer: 6 #eval gcd_c2 672 161 --Answer: -25 --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161 theorem Theorem_7_1_6 {d a b : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ gcd a b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑d ∣ ↑(gcd a b) set s : Int := gcd_c1 a b set t : Int := gcd_c2 a b have h3 : s * ↑a + t * ↑b = ↑(gcd a b) := gcd_lin_comb b a rewrite [←h3] --Goal : ↑d ∣ s * ↑a + t * ↑b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul] --Goal : ↑d ∣ s * (↑d * ↑j) + t * (↑d * ↑k) define apply Exists.intro (s * ↑j + t * ↑k) ring done /- Section 7.2 -/ theorem dvd_trans {a b c : Nat} (h1 : a ∣ b) (h2 : b ∣ c) : a ∣ c := by define at h1; define at h2; define obtain (m : Nat) (h3 : b = a * m) from h1 obtain (n : Nat) (h4 : c = b * n) from h2 rewrite [h3, mul_assoc] at h4 apply Exists.intro (m * n) show c = a * (m * n) from h4 done lemma exists_prime_factor : ∀ (n : Nat), 2 ≤ n → ∃ (p : Nat), prime_factor p n := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, 2 ≤ n_1 → ∃ (p : Nat), prime_factor p n_1 assume h1 : 2 ≤ n by_cases h2 : prime n · -- Case 1. h2 : prime n apply Exists.intro n define --Goal : prime n ∧ n ∣ n show prime n ∧ n ∣ n from And.intro h2 (dvd_self n) done · -- Case 2. h2 : ¬prime n define at h2 --h2 : ¬(2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n) demorgan at h2 disj_syll h2 h1 obtain (a : Nat) (h3 : ∃ (b : Nat), a * b = n ∧ a < n ∧ b < n) from h2 obtain (b : Nat) (h4 : a * b = n ∧ a < n ∧ b < n) from h3 have h5 : 2 ≤ a := by by_contra h6 have h7 : a ≤ 1 := by linarith have h8 : n ≤ b := calc n _ = a * b := h4.left.symm _ ≤ 1 * b := by rel [h7] _ = b := by ring linarith --n ≤ b contradicts b < n done have h6 : ∃ (p : Nat), prime_factor p a := ih a h4.right.left h5 obtain (p : Nat) (h7 : prime_factor p a) from h6 apply Exists.intro p define --Goal : prime p ∧ p ∣ n define at h7 --h7 : prime p ∧ p ∣ a apply And.intro h7.left have h8 : a ∣ n := by apply Exists.intro b show n = a * b from (h4.left).symm done show p ∣ n from dvd_trans h7.right h8 done done lemma exists_least_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q := by set S : Set Nat := {p : Nat | prime_factor p n} have h2 : ∃ (p : Nat), p ∈ S := exists_prime_factor n h show ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q from well_ord_princ S h2 done lemma all_prime_nil : all_prime [] := by define --Goal : ∀ p ∈ [], prime p fix p : Nat contrapos --Goal : ¬prime p → p ∉ [] assume h1 : ¬prime p show p ∉ [] from List.not_mem_nil p done lemma all_prime_cons (n : Nat) (L : List Nat) : all_prime (n :: L) ↔ prime n ∧ all_prime L := by apply Iff.intro · -- (→) assume h1 : all_prime (n :: L) --Goal : prime n ∧ all_prime L define at h1 --h1 : ∀ p ∈ n :: L, prime p apply And.intro (h1 n (List.mem_cons_self n L)) define --Goal : ∀ p ∈ L, prime p fix p : Nat assume h2 : p ∈ L show prime p from h1 p (List.mem_cons_of_mem n h2) done · -- (←) assume h1 : prime n ∧ all_prime L --Goal : all_prime (n :: l) define : all_prime L at h1 define fix p : Nat assume h2 : p ∈ n :: L rewrite [List.mem_cons] at h2 --h2 : p = n ∨ p ∈ L by_cases on h2 · -- Case 1. h2 : p = n rewrite [h2] show prime n from h1.left done · -- Case 2. h2 : p ∈ L show prime p from h1.right p h2 done done done lemma nondec_nil : nondec [] := by define --Goal : True trivial --trivial proves some obviously true statements, such as True done lemma nondec_cons (n : Nat) (L : List Nat) : nondec (n :: L) ↔ (∀ m ∈ L, n ≤ m) ∧ nondec L := by rfl lemma prod_nil : prod [] = 1 := by rfl lemma prod_cons : prod (n :: L) = n * (prod L) := by rfl lemma exists_cons_of_length_eq_succ {A : Type} {l : List A} {n : Nat} (h : l.length = n + 1) : ∃ (a : A) (L : List A), l = a :: L ∧ L.length = n := by have h1 : ¬l.length = 0 := by linarith rewrite [List.length_eq_zero] at h1 obtain (a : A) (h2 : ∃ (L : List A), l = a :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List A) (h3 : l = a :: L) from h2 apply Exists.intro a apply Exists.intro L apply And.intro h3 have h4 : (a :: L).length = L.length + 1 := List.length_cons a L rewrite [←h3, h] at h4 show L.length = n from (Nat.add_right_cancel h4).symm done lemma list_elt_dvd_prod_by_length (a : Nat) : ∀ (n : Nat), ∀ (l : List Nat), l.length = n → a ∈ l → a ∣ prod l := by by_induc · --Base Case fix l : List Nat assume h1 : l.length = 0 rewrite [List.length_eq_zero] at h1 --h1 : l = [] rewrite [h1] --Goal : a ∈ [] → a ∣ prod [] contrapos assume h2 : ¬a ∣ prod [] show a ∉ [] from List.not_mem_nil a done · -- Induction Step fix n : Nat assume ih : ∀ (l : List Nat), List.length l = n → a ∈ l → a ∣ prod l fix l : List Nat assume h1 : l.length = n + 1 --Goal : a ∈ l → a ∣ prod l obtain (b : Nat) (h2 : ∃ (L : List Nat), l = b :: L ∧ L.length = n) from exists_cons_of_length_eq_succ h1 obtain (L : List Nat) (h3 : l = b :: L ∧ L.length = n) from h2 have h4 : a ∈ L → a ∣ prod L := ih L h3.right assume h5 : a ∈ l rewrite [h3.left, prod_cons] --Goal : a ∣ b * prod L rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b ∨ a ∈ L by_cases on h5 · -- Case 1. h5 : a = b apply Exists.intro (prod L) rewrite [h5] rfl done · -- Case 2. h5 : a ∈ L have h6 : a ∣ prod L := h4 h5 have h7 : prod L ∣ b * prod L := by apply Exists.intro b ring done show a ∣ b * prod L from dvd_trans h6 h7 done done done lemma list_elt_dvd_prod {a : Nat} {l : List Nat} (h : a ∈ l) : a ∣ prod l := by set n : Nat := l.length have h1 : l.length = n := by rfl show a ∣ prod l from list_elt_dvd_prod_by_length a n l h1 h done lemma exists_prime_factorization : ∀ (n : Nat), n ≥ 1 → ∃ (l : List Nat), prime_factorization n l := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, n_1 ≥ 1 → ∃ (l : List Nat), prime_factorization n_1 l assume h1 : n ≥ 1 by_cases h2 : n = 1 · -- Case 1. h2 : n = 1 apply Exists.intro [] define apply And.intro · -- Proof of nondec_prime_list [] define show all_prime [] ∧ nondec [] from And.intro all_prime_nil nondec_nil done · -- Proof of prod [] = n rewrite [prod_nil, h2] rfl done done · -- Case 2. h2 : n ≠ 1 have h3 : n ≥ 2 := lt_of_le_of_ne' h1 h2 obtain (p : Nat) (h4 : prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q) from exists_least_prime_factor h3 have p_prime_factor : prime_factor p n := h4.left define at p_prime_factor have p_prime : prime p := p_prime_factor.left have p_dvd_n : p ∣ n := p_prime_factor.right have p_least : ∀ (q : Nat), prime_factor q n → p ≤ q := h4.right obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n have h5 : m ≠ 0 := by contradict h1 with h6 have h7 : n = 0 := calc n _ = p * m := n_eq_pm _ = p * 0 := by rw [h6] _ = 0 := by ring rewrite [h7] decide done have m_pos : 0 < m := Nat.pos_of_ne_zero h5 have m_lt_n : m < n := by define at p_prime show m < n from calc m _ < m + m := by linarith _ = 2 * m := by ring _ ≤ p * m := by rel [p_prime.left] _ = n := n_eq_pm.symm done obtain (L : List Nat) (h6 : prime_factorization m L) from ih m m_lt_n m_pos define at h6 have ndpl_L : nondec_prime_list L := h6.left define at ndpl_L apply Exists.intro (p :: L) define apply And.intro · -- Proof of nondec_prime_list (p :: L) define apply And.intro · -- Proof of all_prime (p :: L) rewrite [all_prime_cons] show prime p ∧ all_prime L from And.intro p_prime ndpl_L.left done · -- Proof of nondec (p :: L) rewrite [nondec_cons] apply And.intro _ ndpl_L.right fix q : Nat assume q_in_L : q ∈ L have h7 : q ∣ prod L := list_elt_dvd_prod q_in_L rewrite [h6.right] at h7 --h7 : q ∣ m have h8 : m ∣ n := by apply Exists.intro p rewrite [n_eq_pm] ring done have q_dvd_n : q ∣ n := dvd_trans h7 h8 have ap_L : all_prime L := ndpl_L.left define at ap_L have q_prime_factor : prime_factor q n := And.intro (ap_L q q_in_L) q_dvd_n show p ≤ q from p_least q q_prime_factor done done · -- Proof of prod (p :: L) = n rewrite [prod_cons, h6.right, n_eq_pm] rfl done done done theorem Theorem_7_2_2 {a b c : Nat} (h1 : c ∣ a * b) (h2 : rel_prime a c) : c ∣ b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑c ∣ ↑b define at h1; define at h2; define obtain (j : Nat) (h3 : a * b = c * j) from h1 set s : Int := gcd_c1 a c set t : Int := gcd_c2 a c have h4 : s * ↑a + t * ↑c = ↑(gcd a c) := gcd_lin_comb c a rewrite [h2, Nat.cast_one] at h4 --h4 : s * ↑a + t * ↑c = (1 : Int) apply Exists.intro (s * ↑j + t * ↑b) show ↑b = ↑c * (s * ↑j + t * ↑b) from calc ↑b _ = (1 : Int) * ↑b := (one_mul _).symm _ = (s * ↑a + t * ↑c) * ↑b := by rw [h4] _ = s * (↑a * ↑b) + t * ↑c * ↑b := by ring _ = s * (↑c * ↑j) + t * ↑c * ↑b := by rw [←Nat.cast_mul a b, h3, Nat.cast_mul c j] _ = ↑c * (s * ↑j + t * ↑b) := by ring done lemma le_nonzero_prod_left {a b : Nat} (h : a * b ≠ 0) : a ≤ a * b := by have h1 : b ≠ 0 := by contradict h with h1 rewrite [h1] ring done have h2 : 1 ≤ b := Nat.pos_of_ne_zero h1 show a ≤ a * b from calc a = a * 1 := (mul_one a).symm _ ≤ a * b := by rel [h2] done lemma le_nonzero_prod_right {a b : Nat} (h : a * b ≠ 0) : b ≤ a * b := by rewrite [mul_comm] rewrite [mul_comm] at h show b ≤ b * a from le_nonzero_prod_left h done lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry lemma rel_prime_of_prime_not_dvd {a p : Nat} (h1 : prime p) (h2 : ¬p ∣ a) : rel_prime a p := by have h3 : gcd a p ∣ a := gcd_dvd_left a p have h4 : gcd a p ∣ p := gcd_dvd_right a p have h5 : gcd a p = 1 ∨ gcd a p = p := dvd_prime h1 h4 have h6 : gcd a p ≠ p := by contradict h2 with h6 rewrite [h6] at h3 show p ∣ a from h3 done disj_syll h5 h6 show rel_prime a p from h5 done theorem Theorem_7_2_3 {a b p : Nat} (h1 : prime p) (h2 : p ∣ a * b) : p ∣ a ∨ p ∣ b := by or_right with h3 have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3 show p ∣ b from Theorem_7_2_2 h2 h4 done lemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a ≥ 1 := by have h1 : a ≠ 0 := by by_contra h1 rewrite [h1] at h contradict h linarith done show a ≥ 1 from Nat.pos_of_ne_zero h1 done lemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by have h1 : a ≥ 1 := ge_one_of_prod_one h have h2 : a * b ≠ 0 := by linarith have h3 : a ≤ a * b := le_nonzero_prod_left h2 rewrite [h] at h3 show a = 1 from Nat.le_antisymm h3 h1 done lemma eq_one_of_dvd_one {n : Nat} (h : n ∣ 1) : n = 1 := by obtain (j : Nat) (h1 : 1 = n * j) from h show n = 1 from eq_one_of_prod_one h1.symm done lemma prime_not_one {p : Nat} (h : prime p) : p ≠ 1 := by define at h linarith done theorem Theorem_7_2_4 {p : Nat} (h1 : prime p) : ∀ (l : List Nat), p ∣ prod l → ∃ a ∈ l, p ∣ a := by apply List.rec · -- Base Case. Goal : p ∣ prod [] → ∃ a ∈ [], p ∣ a rewrite [prod_nil] assume h2 : p ∣ 1 show ∃ a ∈ [], p ∣ a from absurd (eq_one_of_dvd_one h2) (prime_not_one h1) done · -- Induction Step fix b : Nat fix L : List Nat assume ih : p ∣ prod L → ∃ a ∈ L, p ∣ a --Goal : p ∣ prod (b :: L) → ∃ a ∈ b :: L, p ∣ a assume h2 : p ∣ prod (b :: L) rewrite [prod_cons] at h2 have h3 : p ∣ b ∨ p ∣ prod L := Theorem_7_2_3 h1 h2 by_cases on h3 · -- Case 1. h3 : p ∣ b apply Exists.intro b show b ∈ b :: L ∧ p ∣ b from And.intro (List.mem_cons_self b L) h3 done · -- Case 2. h3 : p ∣ prod L obtain (a : Nat) (h4 : a ∈ L ∧ p ∣ a) from ih h3 apply Exists.intro a show a ∈ b :: L ∧ p ∣ a from And.intro (List.mem_cons_of_mem b h4.left) h4.right done done done lemma prime_in_list {p : Nat} {l : List Nat} (h1 : prime p) (h2 : all_prime l) (h3 : p ∣ prod l) : p ∈ l := by obtain (a : Nat) (h4 : a ∈ l ∧ p ∣ a) from Theorem_7_2_4 h1 l h3 define at h2 have h5 : prime a := h2 a h4.left have h6 : p = 1 ∨ p = a := dvd_prime h5 h4.right disj_syll h6 (prime_not_one h1) rewrite [h6] show a ∈ l from h4.left done lemma first_le_first {p q : Nat} {l m : List Nat} (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m)) (h3 : prod (p :: l) = prod (q :: m)) : p ≤ q := by define at h1; define at h2 have h4 : q ∣ prod (p :: l) := by define apply Exists.intro (prod m) rewrite [←prod_cons] show prod (p :: l) = prod (q :: m) from h3 done have h5 : all_prime (q :: m) := h2.left rewrite [all_prime_cons] at h5 have h6 : q ∈ p :: l := prime_in_list h5.left h1.left h4 have h7 : nondec (p :: l) := h1.right rewrite [nondec_cons] at h7 rewrite [List.mem_cons] at h6 by_cases on h6 · -- Case 1. h6 : q = p linarith done · -- Case 2. h6 : q ∈ l have h8 : ∀ m ∈ l, p ≤ m := h7.left show p ≤ q from h8 q h6 done done lemma nondec_prime_list_tail {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by define at h define rewrite [all_prime_cons, nondec_cons] at h show all_prime l ∧ nondec l from And.intro h.left.right h.right.right done lemma cons_prod_not_one {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : prod (p :: l) ≠ 1 := by define at h have h1 : all_prime (p :: l) := h.left rewrite [all_prime_cons] at h1 rewrite [prod_cons] by_contra h2 show False from (prime_not_one h1.left) (eq_one_of_prod_one h2) done lemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) : l = [] ↔ prod l = 1 := by apply Iff.intro · -- (→) assume h1 : l = [] rewrite [h1] show prod [] = 1 from prod_nil done · -- (←) contrapos assume h1 : ¬l = [] obtain (p : Nat) (h2 : ∃ (L : List Nat), l = p :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List Nat) (h3 : l = p :: L) from h2 rewrite [h3] at h rewrite [h3] show ¬prod (p :: L) = 1 from cons_prod_not_one h done done lemma prime_pos {p : Nat} (h : prime p) : p > 0 := by define at h linarith done theorem Theorem_7_2_5 : ∀ (l1 l2 : List Nat), nondec_prime_list l1 → nondec_prime_list l2 → prod l1 = prod l2 → l1 = l2 := by apply List.rec · -- Base Case. Goal : ∀ (l2 : List Nat), nondec_prime_list [] → -- nondec_prime_list l2 → prod [] = prod l2 → [] = l2 fix l2 : List Nat assume h1 : nondec_prime_list [] assume h2 : nondec_prime_list l2 assume h3 : prod [] = prod l2 rewrite [prod_nil, eq_comm, ←list_nil_iff_prod_one h2] at h3 show [] = l2 from h3.symm done · -- Induction Step fix p : Nat fix L1 : List Nat assume ih : ∀ (L2 : List Nat), nondec_prime_list L1 → nondec_prime_list L2 → prod L1 = prod L2 → L1 = L2 -- Goal : ∀ (l2 : List Nat), nondec_prime_list (p :: L1) → -- nondec_prime_list l2 → prod (p :: L1) = prod l2 → p :: L1 = l2 fix l2 : List Nat assume h1 : nondec_prime_list (p :: L1) assume h2 : nondec_prime_list l2 assume h3 : prod (p :: L1) = prod l2 have h4 : ¬prod (p :: L1) = 1 := cons_prod_not_one h1 rewrite [h3, ←list_nil_iff_prod_one h2] at h4 obtain (q : Nat) (h5 : ∃ (L : List Nat), l2 = q :: L) from List.exists_cons_of_ne_nil h4 obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5 rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2) rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2) have h7 : p ≤ q := first_le_first h1 h2 h3 have h8 : q ≤ p := first_le_first h2 h1 h3.symm have h9 : p = q := by linarith rewrite [h9, prod_cons, prod_cons] at h3 --h3 : q * prod L1 = q * prod L2 have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1 have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2 define at h2 have h12 : all_prime (q :: L2) := h2.left rewrite [all_prime_cons] at h12 have h13 : q > 0 := prime_pos h12.left have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3 have h15 : L1 = L2 := ih L2 h10 h11 h14 rewrite [h6, h9, h15] rfl done done theorem fund_thm_arith (n : Nat) (h : n ≥ 1) : ∃! (l : List Nat), prime_factorization n l := by exists_unique · -- Existence show ∃ (l : List Nat), prime_factorization n l from exists_prime_factorization n h done · -- Uniqueness fix l1 : List Nat; fix l2 : List Nat assume h1 : prime_factorization n l1 assume h2 : prime_factorization n l2 define at h1; define at h2 have h3 : prod l1 = n := h1.right rewrite [←h2.right] at h3 show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3 done done /- Section 7.3 -/ theorem congr_refl (m : Nat) : ∀ (a : Int), a ≡ a (MOD m) := by fix a : Int define --Goal : ∃ (c : Int), a - a = ↑m * c apply Exists.intro 0 ring done theorem congr_symm {m : Nat} : ∀ {a b : Int}, a ≡ b (MOD m) → b ≡ a (MOD m) := by fix a : Int; fix b : Int assume h1 : a ≡ b (MOD m) define at h1 --h1 : ∃ (c : Int), a - b = ↑m * c define --Goal : ∃ (c : Int), b - a = ↑m * c obtain (c : Int) (h2 : a - b = m * c) from h1 apply Exists.intro (-c) show b - a = m * (-c) from calc b - a _ = -(a - b) := by ring _ = -(m * c) := by rw [h2] _ = m * (-c) := by ring done theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry /- Fundamental properties of congruence classes -/ lemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) : X = Y ↔ X.val = Y.val := Fin.ext_iff lemma val_nat_eq_mod (n k : Nat) : ([k]_(n + 1)).val = k % (n + 1) := by rfl lemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl theorem cc_rep {m : Nat} (X : ZMod m) : ∃ (a : Int), X = [a]_m := match m with | 0 => by apply Exists.intro X rfl done | n + 1 => by apply Exists.intro ↑(X.val) have h1 : X.val < n + 1 := Fin.prop X rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1] rfl done theorem add_class (m : Nat) (a b : Int) : [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm theorem mul_class (m : Nat) (a b : Int) : [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm lemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ [a - b]_m = [0]_m := by apply Iff.intro · -- (→) assume h1 : [a]_m = [b]_m have h2 : a - b = a + (-b) := by ring have h3 : b + (-b) = 0 := by ring show [a - b]_m = [0]_m from calc [a - b]_m _ = [a + (-b)]_m := by rw [h2] _ = [a]_m + [-b]_m := by rw [add_class] _ = [b]_m + [-b]_m := by rw [h1] _ = [b + -b]_m := by rw [add_class] _ = [0]_m := by rw [h3] done · -- (←) assume h1 : [a - b]_m = [0]_m have h2 : b + (a - b) = a := by ring have h3 : b + 0 = b := by ring show [a]_m = [b]_m from calc [a]_m _ = [b + (a - b)]_m := by rw [h2] _ = [b]_m + [a - b]_m := by rw [add_class] _ = [b]_m + [0]_m := by rw [h1] _ = [b + 0]_m := by rw [add_class] _ = [b]_m := by rw [h3] done done lemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) : [a]_m = [0]_m → [-a]_m = [0]_m := by assume h1 : [a]_m = [0]_m have h2 : 0 + (-a) = -a := by ring have h3 : a + (-a) = 0 := by ring show [-a]_m = [0]_m from calc [-a]_m _ = [0 + (-a)]_m := by rw [h2] _ = [0]_m + [-a]_m := by rw [add_class] _ = [a]_m + [-a]_m := by rw [h1] _ = [a + (-a)]_m := by rw [add_class] _ = [0]_m := by rw [h3] done lemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) : [-a]_m = [0]_m ↔ [a]_m = [0]_m := by apply Iff.intro _ (cc_neg_zero_of_cc_zero m a) assume h1 : [-a]_m = [0]_m have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1 have h3 : -(-a) = a := by ring rewrite [h3] at h2 show [a]_m = [0]_m from h2 done lemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl lemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m ↔ m ∣ k := match m with | 0 => by have h : (0 : Int) = (↑(0 : Nat) : Int) := by rfl rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj] apply Iff.intro · -- (→) assume h1 : k = 0 rewrite [h1] show 0 ∣ 0 from dvd_self 0 done · -- (←) assume h1 : 0 ∣ k obtain (c : Nat) (h2 : k = 0 * c) from h1 rewrite [h2] ring done done | n + 1 => by rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero] show k % (n + 1) = 0 ↔ n + 1 ∣ k from (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm done lemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m ↔ ↑m ∣ a := by obtain (k : Nat) (h1 : a = ↑k ∨ a = -↑k) from Int.eq_nat_or_neg a by_cases on h1 · -- Case 1. h1: a = ↑k rewrite [h1, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done · -- Case 2. h1: a = -↑k rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done done theorem cc_eq_iff_congr (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ a ≡ b (MOD m) := calc [a]_m = [b]_m _ ↔ [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b _ ↔ ↑m ∣ (a - b) := cc_zero_iff_dvd m (a - b) _ ↔ a ≡ b (MOD m) := by rfl /- End of fundamental properties of congruence classes -/ lemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m := by have h1 : (↑m : Int) ≠ 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m) show 0 ≤ a % m from Int.emod_nonneg a h1 done lemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m) have h2 : (↑m : Int) > 0 := (Nat.cast_pos).rtl h1 show a % m < m from Int.emod_lt_of_pos a h2 done lemma congr_mod_mod (m : Nat) (a : Int) : a ≡ a % m (MOD m) := by define have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m apply Exists.intro (a / m) show a - a % m = m * (a / m) from calc a - (a % m) _ = m * (a / m) + a % m - a % m := by rw [h1] _ = m * (a / m) := by ring done lemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) := And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a)) theorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) : ∃! (r : Int), 0 ≤ r ∧ r < m ∧ a ≡ r (MOD m) := by exists_unique · -- Existence apply Exists.intro (a % m) show 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) from mod_cmpl_res m a done · -- Uniqueness fix r1 : Int; fix r2 : Int assume h1 : 0 ≤ r1 ∧ r1 < m ∧ a ≡ r1 (MOD m) assume h2 : 0 ≤ r2 ∧ r2 < m ∧ a ≡ r2 (MOD m) have h3 : r1 ≡ r2 (MOD m) := congr_trans (congr_symm h1.right.right) h2.right.right obtain (d : Int) (h4 : r1 - r2 = m * d) from h3 have h5 : r1 - r2 < m * 1 := by linarith have h6 : m * (-1) < r1 - r2 := by linarith rewrite [h4] at h5 --h5 : m * d < m * 1 rewrite [h4] at h6 --h6 : m * -1 < m * d have h7 : (↑m : Int) ≥ 0 := Nat.cast_nonneg m have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7 have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7 have h10 : d = 0 := by linarith show r1 = r2 from calc r1 _ = r1 - r2 + r2 := by ring _ = m * 0 + r2 := by rw [h4, h10] _ = r2 := by ring done done lemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m := (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a) theorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y rewrite [h1, h2] have h3 : a + b = b + a := by ring show [a]_m + [b]_m = [b]_m + [a]_m from calc [a]_m + [b]_m _ = [a + b]_m := add_class m a b _ = [b + a]_m := by rw [h3] _ = [b]_m + [a]_m := (add_class m b a).symm done theorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X rewrite [h1] have h2 : a * 1 = a := by ring show [a]_m * [1]_m = [a]_m from calc [a]_m * [1]_m _ = [a * 1]_m := mul_class m a 1 _ = [a]_m := by rw [h2] done theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry lemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) : [a]_m * [gcd_c2 m a]_m = [1]_m := by set s : Int := gcd_c1 m a have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m define at h1 rewrite [h1, Nat.cast_one] at h2 --h2 : s * ↑m + gcd_c2 m a * ↑a = 1 rewrite [mul_class, cc_eq_iff_congr] define --Goal : ∃ (c : Int), ↑a * gcd_c2 m a - 1 = ↑m * c apply Exists.intro (-s) show a * (gcd_c2 m a) - 1 = m * (-s) from calc a * (gcd_c2 m a) - 1 _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring _ = 1 + m * (-s) - 1 := by rw [h2] _ = m * (-s) := by ring done theorem Theorem_7_3_7 (m a : Nat) : invertible [a]_m ↔ rel_prime m a := by apply Iff.intro · -- (→) assume h1 : invertible [a]_m define at h1 obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1 obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y rewrite [h3, mul_class, cc_eq_iff_congr] at h2 define at h2 obtain (c : Int) (h4 : a * b - 1 = m * c) from h2 rewrite [Exercise_7_2_6] --Goal : ∃ (s t : Int), s * ↑m + t * ↑a = 1 apply Exists.intro (-c) apply Exists.intro b show (-c) * m + b * a = 1 from calc (-c) * m + b * a _ = (-c) * m + (a * b - 1) + 1 := by ring _ = (-c) * m + m * c + 1 := by rw [h4] _ = 1 := by ring done · -- (←) assume h1 : rel_prime m a define show ∃ (Y : ZMod m), [a]_m * Y = [1]_m from Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1) done done /- Section 7.4 -/ section Euler open Euler lemma num_rp_below_base {m : Nat} : num_rp_below m 0 = 0 := by rfl lemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) : num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by have h1 : num_rp_below m (j + 1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : gcd m j = 1 rewrite [if_pos h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1 show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1 done lemma num_rp_below_step_not_rp {m j : Nat} (h : ¬rel_prime m j) : num_rp_below m (j + 1) = num_rp_below m j := by have h1 : num_rp_below m (j +1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : ¬gcd m j = 1 rewrite [if_neg h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j show num_rp_below m (j + 1) = num_rp_below m j from h1 done lemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl #eval phi 10 --Answer: 4 lemma prod_inv_iff_inv {m : Nat} {X : ZMod m} (h1 : invertible X) (Y : ZMod m) : invertible (X * Y) ↔ invertible Y := by apply Iff.intro · -- (→) assume h2 : invertible (X * Y) obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2 apply Exists.intro (X * Z) rewrite [←h3] --Goal : Y * (X * Z) = X * Y * Z ring --Note that ring can do algebra in ZMod m done · -- (←) assume h2 : invertible Y obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1 obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2 apply Exists.intro (Xi * Yi) show (X * Y) * (Xi * Yi) = [1]_m from calc X * Y * (Xi * Yi) _ = (X * Xi) * (Y * Yi) := by ring _ = [1]_m * [1]_m := by rw [h3, h4] _ = [1]_m := Theorem_7_3_6_7 [1]_m done done lemma F_rp_def {m i : Nat} (h : rel_prime m i) : F m i = [i]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h --h : gcd m i = 1 rewrite [if_pos h] at h1 show F m i = [i]_m from h1 done lemma F_not_rp_def {m i : Nat} (h : ¬rel_prime m i) : F m i = [1]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h rewrite [h1, if_neg h] rfl done lemma prod_seq_base {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 0 k f = [1]_m := by rfl lemma prod_seq_step {m : Nat} (n k : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl lemma prod_seq_zero_step {m : Nat} (n : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by rewrite [prod_seq_step, zero_add] rfl done lemma prod_one {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 1 k f = f k := by rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7] rfl done lemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl lemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m := calc [G m a i]_m _ = [(a * i) % m]_m := by rfl _ = [a * i]_m := (cc_eq_mod m (a * i)).symm _ = [a]_m * [i]_m := (mul_class m a i).symm lemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) : rel_prime m (G m a i) ↔ rel_prime m i := by have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1 show rel_prime m (G m a i) ↔ rel_prime m i from calc rel_prime m (G m a i) _ ↔ invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm _ ↔ invertible ([a]_m * [i]_m) := by rw [cc_G] _ ↔ invertible [i]_m := prod_inv_iff_inv h2 ([i]_m) _ ↔ rel_prime m i := Theorem_7_3_7 m i done lemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) : F m (G m a i) = [a]_m * F m i := by have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2 show F m (G m a i) = [a]_m * F m i from calc F m (G m a i) _ = [G m a i]_m := F_rp_def h3 _ = [a]_m * [i]_m := cc_G m a i _ = [a]_m * F m i := by rw [F_rp_def h2] done lemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : ¬rel_prime m i) : F m (G m a i) = [1]_m := by rewrite [←G_rp_iff h1 i] at h2 show F m (G m a i) = [1]_m from F_not_rp_def h2 done lemma FG_prod {m a : Nat} (h1 : rel_prime m a) : ∀ (k : Nat), prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by by_induc · -- Base Case show prod_seq 0 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from calc prod_seq 0 0 ((F m) ∘ (G m a)) _ = [1]_m := prod_seq_base _ _ _ = [a]_m ^ 0 * [1]_m := by ring _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by rw [num_rp_below_base, prod_seq_base] done · -- Induction Step fix k : Nat assume ih : prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) by_cases h2 : rel_prime m k · -- Case 1. h2 : rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([a]_m * F m k) := by rw [FG_rp h1 h2] _ = [a]_m ^ ((num_rp_below m k) + 1) * ((prod_seq k 0 (F m)) * F m k) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_rp h2, prod_seq_zero_step] done · -- Case 2. h2 : ¬rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([1]_m) := by rw [FG_not_rp h1 h2] _ = [a]_m ^ (num_rp_below m k) * (prod_seq k 0 (F m) * ([1]_m)) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_not_rp h2, prod_seq_zero_step, F_not_rp_def h2] done done done lemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by define --Goal : ∀ i < m, G m a i < m fix i : Nat assume h1 : i < m rewrite [G_def] --Goal : a * i % m < m show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m) done lemma left_inv_one_one_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g' (g i) = i) : one_one_below n g := sorry lemma right_inv_onto_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g (g' i) = i) (h2 : maps_below n g') : onto_below n g := by define at h2; define fix k : Nat assume h3 : k < n apply Exists.intro (g' k) show g' k < n ∧ g (g' k) = k from And.intro (h2 k h3) (h1 k h3) done lemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m * [inv_mod m a]_m = [1]_m := by have h2 : 0 ≤ (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a) show [a]_m * [inv_mod m a]_m = [1]_m from calc [a]_m * [inv_mod m a]_m _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2] _ = [a]_m * [gcd_c2 m a]_m := by rw [←cc_eq_mod] _ = [1]_m := gcd_c2_inv h1 done lemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m := calc a * (b % m) % m = a % m * (b % m % m) % m := Nat.mul_mod _ _ _ _ = a % m * (b % m) % m := by rw [Nat.mod_mod] _ = a * b % m := (Nat.mul_mod _ _ _).symm lemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm] rfl done theorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] : ↑a ≡ ↑b (MOD m) ↔ a % m = b % m := sorry lemma mul_inv_mod_cancel {m a i : Nat} [NeZero m] (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1 rewrite [mul_class, cc_eq_iff_congr, ←Nat.cast_mul, ←Nat.cast_one, congr_iff_mod_eq_Nat] at h3 show a * inv_mod m a * i % m = i from calc a * (inv_mod m a) * i % m _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = 1 % m * i % m := by rw [h3] _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = i % m := by rw [one_mul] _ = i := Nat.mod_eq_of_lt h2 done lemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl lemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, G m a (Ginv m a i) = i := by fix i : Nat assume h2 : i < m show G m a (Ginv m a i) = i from calc G m a (Ginv m a i) _ = a * ((inv_mod m a * i) % m) % m := by rfl _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, Ginv m a (G m a i) = i := by fix i : Nat assume h2 : i < m show Ginv m a (G m a i) = i from calc Ginv m a (G m a i) _ = inv_mod m a * ((a * i) % m) % m := by rfl _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc, mul_comm (inv_mod m a)] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_maps_below (m a : Nat) [NeZero m] : maps_below m (Ginv m a) := G_maps_below m (inv_mod m a) lemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : one_one_below m (G m a) := left_inv_one_one_below (Ginv_left_inv h1) lemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : onto_below m (G m a) := right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a) lemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : perm_below m (G m a) := And.intro (G_maps_below m a) (And.intro (G_one_one_below h1) (G_onto_below h1)) --Permuting a product of congruence classes doesn't change product lemma swap_fst (u v : Nat) : swap u v u = v := by define : swap u v u --Goal : (if u = u then v else if u = v then u else u) = v have h : u = u := by rfl rewrite [if_pos h] rfl done lemma swap_snd (u v : Nat) : swap u v v = u := by define : swap u v v by_cases h1 : v = u · -- Case 1. h1 : v = u rewrite [if_pos h1] show v = u from h1 done · -- Case 2. h1 : v ≠ u rewrite [if_neg h1] have h2 : v = v := by rfl rewrite [if_pos h2] rfl done done lemma swap_other {u v i : Nat} (h1 : i ≠ u) (h2 : i ≠ v) : swap u v i = i := by define : swap u v i rewrite [if_neg h1, if_neg h2] rfl done lemma swap_values (u v i : Nat) : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := by by_cases h1 : i = u · -- Case 1. h1 : i = u apply Or.inl rewrite [h1] show swap u v u = v from swap_fst u v done · -- Case 2. h1 : i ≠ u apply Or.inr by_cases h2 : i = v · -- Case 2.1. h2 : i = v apply Or.inl rewrite [h2] show swap u v v = u from swap_snd u v done · -- Case 2.2. h2 : i ≠ v apply Or.inr show swap u v i = i from swap_other h1 h2 done done done lemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by define fix i : Nat assume h3 : i < n have h4 : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := swap_values u v i by_cases on h4 · -- Case 1. h4 : swap u v i = v rewrite [h4] show v < n from h2 done · -- Case 2. by_cases on h4 · -- Case 2.1. h4 : swap u v i = u rewrite [h4] show u < n from h1 done · -- Case 2.2. h4 : swap u v i = i rewrite [h4] show i < n from h3 done done done lemma swap_swap (u v n : Nat) : ∀ i < n, swap u v (swap u v i) = i := by fix i : Nat assume h : i < n by_cases h1 : i = u · -- Case 1. h1 : i = u rewrite [h1, swap_fst, swap_snd] rfl done · -- Case 2. h1 : i ≠ u by_cases h2 : i = v · -- Case 2.1. h2 : i = v rewrite [h2, swap_snd, swap_fst] rfl done · -- Case 2.2. h2 : i ≠ v rewrite [swap_other h1 h2, swap_other h1 h2] rfl done done done lemma swap_one_one_below (u v n) : one_one_below n (swap u v) := left_inv_one_one_below (swap_swap u v n) lemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) := right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2) lemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) := And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2)) lemma comp_perm_below {n : Nat} {f g : Nat → Nat} (h1 : perm_below n f) (h2 : perm_below n g) : perm_below n (f ∘ g) := sorry lemma trivial_swap (u : Nat) : swap u u = id := by apply funext fix x : Nat by_cases h1 : x = u · -- Case 1. h1 : x = u rewrite [h1, swap_fst] rfl done · -- Case 2. h1 : x ≠ u rewrite [swap_other h1 h1] rfl done done lemma prod_eq_fun {m : Nat} (f g : Nat → ZMod m) (k : Nat) : ∀ (n : Nat), (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g := by by_induc · -- Base Case assume h : (∀ i < 0, f (k + i) = g (k + i)) rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g assume h1 : ∀ i < n + 1, f (k + i) = g (k + i) have h2 : ∀ i < n, f (k + i) = g (k + i) := by fix i : Nat assume h2 : i < n have h3 : i < n + 1 := by linarith show f (k + i) = g (k + i) from h1 i h3 done have h3 : prod_seq n k f = prod_seq n k g := ih h2 have h4 : n < n + 1 := Nat.lt_succ_self n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4] rfl done done lemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := by have h2 : ∀ (i : Nat), i < u → (f ∘ swap u n) (0 + i) = f (0 + i) := by fix i : Nat assume h2 : i < u have h3 : 0 + i ≠ u := by linarith have h4 : 0 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl done show prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f from prod_eq_fun (f ∘ swap u n) f 0 u h2 done lemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := by have h2 : ∀ i < j, (f ∘ swap u n) (u + 1 + i) = f (u + 1 + i) := by fix i : Nat assume h2 : i < j have h3 : u + 1 + i ≠ u := by linarith have h4 : u + 1 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl show prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f from prod_eq_fun (f ∘ swap u n) f (u + 1) j h2 done lemma break_prod {m : Nat} (n : Nat) (f : Nat → ZMod m) : ∀ (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by by_induc · -- Base Case have h : n + 0 = n := by rfl rewrite [prod_seq_base, h, Theorem_7_3_6_7] rfl done · -- Induction Step fix j : Nat assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f rewrite [←add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc] rfl done done lemma break_prod_twice {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f := break_prod n f 1 rewrite [prod_one] at h2 have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f := break_prod (u + 1) f j rewrite [←h1] at h3 have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f := break_prod u f 1 rewrite [prod_one] at h4 rewrite [h3, h4] at h2 show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2 done lemma swap_prod_eq_prod {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f := by by_cases h2 : u = n · -- Case 1. h2 : u = n rewrite [h2, trivial_swap n] --Goal : prod_seq (n + 1) 0 (f ∘ id) = prod_seq (n + 1) 0 f rfl done · -- Case 2. h2 : ¬u = n have h3 : u + 1 ≤ n := Nat.lt_of_le_of_ne h1 h2 obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3 have break_f : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := break_prod_twice f h4 have break_fs : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_prod_twice (f ∘ swap u n) h4 have f_eq_fs_below : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := swap_prod_eq_prod_below f h1 have f_eq_fs_btwn : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4 show prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f from calc prod_seq (n + 1) 0 (f ∘ swap u n) _ = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_fs _ = prod_seq u 0 f * (f ∘ swap u n) u * prod_seq j (u + 1) f * (f ∘ swap u n) n := by rw [f_eq_fs_below, f_eq_fs_btwn] _ = prod_seq u 0 f * f (swap u n u) * prod_seq j (u + 1) f * f (swap u n n) := by rfl _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by rw [swap_fst, swap_snd] _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring _ = prod_seq (n + 1) 0 f := break_f.symm done done lemma perm_below_fixed {n : Nat} {g : Nat → Nat} (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry lemma perm_prod {m : Nat} (f : Nat → ZMod m) : ∀ (n : Nat), ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) := by by_induc · -- Base Case fix g : Nat → Nat assume h1 : perm_below 0 g rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) fix g : Nat → Nat assume g_pb : perm_below (n + 1) g define at g_pb have g_ob : onto_below (n + 1) g := g_pb.right.right define at g_ob have h1 : n < n + 1 := by linarith obtain (u : Nat) (h2 : u < n + 1 ∧ g u = n) from g_ob n h1 have s_pb : perm_below (n + 1) (swap u n) := swap_perm_below h2.left h1 have gs_pb_n1 : perm_below (n + 1) (g ∘ swap u n) := comp_perm_below g_pb s_pb have gs_fix_n : (g ∘ swap u n) n = n := calc (g ∘ swap u n) n _ = g (swap u n n) := by rfl _ = g u := by rw [swap_snd] _ = n := h2.right have gs_pb_n : perm_below n (g ∘ swap u n) := perm_below_fixed gs_pb_n1 gs_fix_n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f ∘ (g ∘ swap u n)) := ih (g ∘ swap u n) gs_pb_n have h3 : u ≤ n := by linarith show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f ∘ g) from calc prod_seq (n + 1) 0 f _ = prod_seq n 0 f * f n := prod_seq_zero_step n f _ = prod_seq n 0 (f ∘ (g ∘ swap u n)) * f ((g ∘ swap u n) n) := by rw [gs_prod, gs_fix_n] _ = prod_seq n 0 (f ∘ g ∘ swap u n) * (f ∘ g ∘ swap u n) n := by rfl _ = prod_seq (n + 1) 0 (f ∘ g ∘ swap u n) := (prod_seq_zero_step n (f ∘ g ∘ swap u n)).symm _ = prod_seq (n + 1) 0 ((f ∘ g) ∘ swap u n) := by rfl _ = prod_seq (n + 1) 0 (f ∘ g) := swap_prod_eq_prod (f ∘ g) h3 done done lemma F_invertible (m i : Nat) : invertible (F m i) := by by_cases h : rel_prime m i · -- Case 1. h : rel_prime m i rewrite [F_rp_def h] show invertible [i]_m from (Theorem_7_3_7 m i).rtl h done · -- Case 2. h : ¬rel_prime m i rewrite [F_not_rp_def h] apply Exists.intro [1]_m show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m done done lemma Fprod_invertible (m : Nat) : ∀ (k : Nat), invertible (prod_seq k 0 (F m)) := by by_induc · -- Base Case apply Exists.intro [1]_m show prod_seq 0 0 (F m) * [1]_m = [1]_m from calc prod_seq 0 0 (F m) * [1]_m _ = [1]_m * [1]_m := by rw [prod_seq_base] _ = [1]_m := Theorem_7_3_6_7 ([1]_m) done · -- Induction Step fix k : Nat assume ih : invertible (prod_seq k 0 (F m)) rewrite [prod_seq_zero_step] show invertible (prod_seq k 0 (F m) * (F m k)) from (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k) done done theorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m ^ (phi m) = [1]_m := by have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2 show [a]_m ^ (phi m) = [1]_m from calc [a]_m ^ (phi m) _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3] _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring _ = prod_seq m 0 (F m ∘ G m a) * Y := by rw [FG_prod h1 m, phi_def] _ = prod_seq m 0 (F m) * Y := by rw [perm_prod (F m) m (G m a) (G_perm_below h1)] _ = [1]_m := by rw [h3] done lemma Exercise_7_4_5_Int (m : Nat) (a : Int) : ∀ (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry lemma Exercise_7_4_5_Nat (m a n : Nat) : [a]_m ^ n = [a ^ n]_m := by rewrite [Exercise_7_4_5_Int] rfl done theorem Euler's_theorem {m a : Nat} [NeZero m] (h1 : rel_prime m a) : a ^ (phi m) ≡ 1 (MOD m) := by have h2 : [a]_m ^ (phi m) = [1]_m := Theorem_7_4_2 h1 rewrite [Exercise_7_4_5_Nat m a (phi m)] at h2 --h2 : [a ^ phi m]_m = [1]_m show a ^ (phi m) ≡ 1 (MOD m) from (cc_eq_iff_congr _ _ _).ltr h2 done #eval gcd 10 7 --Answer: 1. So 10 and 7 are relatively prime #eval 7 ^ phi 10 --Answer: 2401, which is congruent to 1 mod 10. end Euler /- Section 7.5 -/ lemma num_rp_prime {p : Nat} (h1 : prime p) : ∀ k < p, num_rp_below p (k + 1) = k := sorry lemma phi_prime {p : Nat} (h1 : prime p) : phi p = p - 1 := by have h2 : 1 ≤ p := prime_pos h1 have h3 : p - 1 + 1 = p := Nat.sub_add_cancel h2 have h4 : p - 1 < p := by linarith have h5 : num_rp_below p (p - 1 + 1) = p - 1 := num_rp_prime h1 (p - 1) h4 rewrite [h3] at h5 show phi p = p - 1 from h5 done theorem Theorem_7_2_2_Int {a c : Nat} {b : Int} (h1 : ↑c ∣ ↑a * b) (h2 : rel_prime a c) : ↑c ∣ b := by rewrite [Int.natCast_dvd, Int.natAbs_mul, Int.natAbs_ofNat] at h1 --h1 : c ∣ a * Int.natAbs b rewrite [Int.natCast_dvd] --Goal : c ∣ Int.natAbs b show c ∣ Int.natAbs b from Theorem_7_2_2 h1 h2 done lemma Lemma_7_4_5 {m n : Nat} (a b : Int) (h1 : rel_prime m n) : a ≡ b (MOD m * n) ↔ a ≡ b (MOD m) ∧ a ≡ b (MOD n) := by apply Iff.intro · -- (→) assume h2 : a ≡ b (MOD m * n) obtain (j : Int) (h3 : a - b = (m * n) * j) from h2 apply And.intro · -- Proof of a ≡ b (MOD m) apply Exists.intro (n * j) show a - b = m * (n * j) from calc a - b _ = m * n * j := h3 _ = m * (n * j) := by ring done · -- Proof of a ≡ b (MOD n) apply Exists.intro (m * j) show a - b = n * (m * j) from calc a - b _ = m * n * j := h3 _ = n * (m * j) := by ring done done · -- (←) assume h2 : a ≡ b (MOD m) ∧ a ≡ b (MOD n) obtain (j : Int) (h3 : a - b = m * j) from h2.left have h4 : (↑n : Int) ∣ a - b := h2.right rewrite [h3] at h4 --h4 : ↑n ∣ ↑m * j have h5 : ↑n ∣ j := Theorem_7_2_2_Int h4 h1 obtain (k : Int) (h6 : j = n * k) from h5 apply Exists.intro k --Goal : a - b = ↑(m * n) * k rewrite [Nat.cast_mul] --Goal : a - b = ↑m * ↑n * k show a - b = (m * n) * k from calc a - b _ = m * j := h3 _ = m * (n * k) := by rw [h6] _ = (m * n) * k := by ring done done --From exercises of Section 7.2 theorem rel_prime_symm {a b : Nat} (h : rel_prime a b) : rel_prime b a := sorry

lemma prime_NeZero {p : Nat} (h : prime p) : NeZero p

HTPI.prime_NeZero

htpi/HTPILib/Chap7.lean

htpi

/- Copyright 2023 Daniel J. Velleman -/ import HTPILib.Chap6 namespace HTPI /- Definitions -/ lemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by have h : n + 1 > 0 := Nat.succ_pos n show a % (n + 1) < n + 1 from Nat.mod_lt a h done def gcd (a b : Nat) : Nat := match b with | 0 => a | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd (n + 1) (a % (n + 1)) termination_by b mutual def gcd_c1 (a b : Nat) : Int := match b with | 0 => 1 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c2 (n + 1) (a % (n + 1)) --Corresponds to s = t' termination_by b def gcd_c2 (a b : Nat) : Int := match b with | 0 => 0 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c1 (n + 1) (a % (n + 1)) - (gcd_c2 (n + 1) (a % (n + 1))) * ↑(a / (n + 1)) --Corresponds to t = s' - t'q termination_by b end def prime (n : Nat) : Prop := 2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n def prime_factor (p n : Nat) : Prop := prime p ∧ p ∣ n def all_prime (l : List Nat) : Prop := ∀ p ∈ l, prime p def nondec (l : List Nat) : Prop := match l with | [] => True --Of course, True is a proposition that is always true | n :: L => (∀ m ∈ L, n ≤ m) ∧ nondec L def nondec_prime_list (l : List Nat) : Prop := all_prime l ∧ nondec l def prod (l : List Nat) : Nat := match l with | [] => 1 | n :: L => n * (prod L) def prime_factorization (n : Nat) (l : List Nat) : Prop := nondec_prime_list l ∧ prod l = n def rel_prime (a b : Nat) : Prop := gcd a b = 1 def congr_mod (m : Nat) (a b : Int) : Prop := (↑m : Int) ∣ (a - b) def cc (m : Nat) (a : Int) : ZMod m := (↑a : ZMod m) notation:50 a " ≡ " b " (MOD " m ")" => congr_mod m a b notation:max "["a"]_"m:max => cc m a def invertible {m : Nat} (X : ZMod m) : Prop := ∃ (Y : ZMod m), X * Y = [1]_m def num_rp_below (m k : Nat) : Nat := match k with | 0 => 0 | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j def phi (m : Nat) : Nat := num_rp_below m m def prod_seq {m : Nat} (j k : Nat) (f : Nat → ZMod m) : ZMod m := match j with | 0 => [1]_m | n + 1 => prod_seq n k f * f (k + n) def maps_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i < n, g i < n def one_one_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i1 < n, ∀ i2 < n, g i1 = g i2 → i1 = i2 def onto_below (n : Nat) (g : Nat → Nat) : Prop := ∀ k < n, ∃ i < n, g i = k def perm_below (n : Nat) (g : Nat → Nat) : Prop := maps_below n g ∧ one_one_below n g ∧ onto_below n g def inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m) def swap (u v i : Nat) : Nat := if i = u then v else if i = v then u else i namespace Euler --For definitions specific to Euler's theorem def F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m def G (m a i : Nat) : Nat := (a * i) % m def Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i end Euler /- Section 7.1 -/ theorem dvd_mod_of_dvd_a_b {a b d : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ (a % b) := by set q : Nat := a / b have h3 : b * q + a % b = a := Nat.div_add_mod a b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 define --Goal : ∃ (c : Nat), a % b = d * c apply Exists.intro (j - k * q) show a % b = d * (j - k * q) from calc a % b _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm _ = a - b * q := by rw [h3] _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc] _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm done theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry #eval gcd 672 161 --Answer: 7 lemma gcd_base (a : Nat) : gcd a 0 = a := by rfl lemma gcd_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd a b = gcd b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1)) rfl done lemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b ≠ 0) : a % b < b := by have h1 : b > 0 := Nat.pos_of_ne_zero h show a % b < b from Nat.mod_lt a h1 done lemma dvd_self (n : Nat) : n ∣ n := by apply Exists.intro 1 ring done theorem gcd_dvd : ∀ (b a : Nat), (gcd a b) ∣ a ∧ (gcd a b) ∣ b := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd a b_1) ∣ a ∧ (gcd a b_1) ∣ b_1 fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_base] --Goal: a ∣ a ∧ a ∣ 0 apply And.intro (dvd_self a) define apply Exists.intro 0 rfl done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_nonzero a h1] --Goal : gcd b (a % b) ∣ a ∧ gcd b (a % b) ∣ b have h2 : a % b < b := mod_nonzero_lt a h1 have h3 : (gcd b (a % b)) ∣ b ∧ (gcd b (a % b)) ∣ (a % b) := ih (a % b) h2 b apply And.intro _ h3.left show (gcd b (a % b)) ∣ a from dvd_a_of_dvd_b_mod h3.left h3.right done done theorem gcd_dvd_left (a b : Nat) : (gcd a b) ∣ a := (gcd_dvd b a).left theorem gcd_dvd_right (a b : Nat) : (gcd a b) ∣ b := (gcd_dvd b a).right lemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl lemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c1 a b = gcd_c2 b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done lemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl lemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * ↑(a / b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done theorem gcd_lin_comb : ∀ (b a : Nat), (gcd_c1 a b) * ↑a + (gcd_c2 a b) * ↑b = ↑(gcd a b) := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd_c1 a b_1) * ↑a + (gcd_c2 a b_1) * ↑b_1 = ↑(gcd a b_1) fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base] --Goal : 1 * ↑a + 0 * ↑0 = ↑a ring done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1] --Goal : gcd_c2 b (a % b) * ↑a + -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * ↑(a / b)) * ↑b = -- ↑(gcd b (a % b)) set r : Nat := a % b set q : Nat := a / b set s : Int := gcd_c1 b r set t : Int := gcd_c2 b r --Goal : t * ↑a + (s - t * ↑q) * ↑b = ↑(gcd b r) have h2 : r < b := mod_nonzero_lt a h1 have h3 : s * ↑b + t * ↑r = ↑(gcd b r) := ih r h2 b have h4 : b * q + r = a := Nat.div_add_mod a b rewrite [←h3, ←h4] rewrite [Nat.cast_add, Nat.cast_mul] --Goal : t * (↑b * ↑q + ↑r) + (s - t * ↑q) * ↑b = s * ↑b + t * ↑r ring done done #eval gcd_c1 672 161 --Answer: 6 #eval gcd_c2 672 161 --Answer: -25 --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161 theorem Theorem_7_1_6 {d a b : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ gcd a b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑d ∣ ↑(gcd a b) set s : Int := gcd_c1 a b set t : Int := gcd_c2 a b have h3 : s * ↑a + t * ↑b = ↑(gcd a b) := gcd_lin_comb b a rewrite [←h3] --Goal : ↑d ∣ s * ↑a + t * ↑b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul] --Goal : ↑d ∣ s * (↑d * ↑j) + t * (↑d * ↑k) define apply Exists.intro (s * ↑j + t * ↑k) ring done /- Section 7.2 -/ theorem dvd_trans {a b c : Nat} (h1 : a ∣ b) (h2 : b ∣ c) : a ∣ c := by define at h1; define at h2; define obtain (m : Nat) (h3 : b = a * m) from h1 obtain (n : Nat) (h4 : c = b * n) from h2 rewrite [h3, mul_assoc] at h4 apply Exists.intro (m * n) show c = a * (m * n) from h4 done lemma exists_prime_factor : ∀ (n : Nat), 2 ≤ n → ∃ (p : Nat), prime_factor p n := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, 2 ≤ n_1 → ∃ (p : Nat), prime_factor p n_1 assume h1 : 2 ≤ n by_cases h2 : prime n · -- Case 1. h2 : prime n apply Exists.intro n define --Goal : prime n ∧ n ∣ n show prime n ∧ n ∣ n from And.intro h2 (dvd_self n) done · -- Case 2. h2 : ¬prime n define at h2 --h2 : ¬(2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n) demorgan at h2 disj_syll h2 h1 obtain (a : Nat) (h3 : ∃ (b : Nat), a * b = n ∧ a < n ∧ b < n) from h2 obtain (b : Nat) (h4 : a * b = n ∧ a < n ∧ b < n) from h3 have h5 : 2 ≤ a := by by_contra h6 have h7 : a ≤ 1 := by linarith have h8 : n ≤ b := calc n _ = a * b := h4.left.symm _ ≤ 1 * b := by rel [h7] _ = b := by ring linarith --n ≤ b contradicts b < n done have h6 : ∃ (p : Nat), prime_factor p a := ih a h4.right.left h5 obtain (p : Nat) (h7 : prime_factor p a) from h6 apply Exists.intro p define --Goal : prime p ∧ p ∣ n define at h7 --h7 : prime p ∧ p ∣ a apply And.intro h7.left have h8 : a ∣ n := by apply Exists.intro b show n = a * b from (h4.left).symm done show p ∣ n from dvd_trans h7.right h8 done done lemma exists_least_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q := by set S : Set Nat := {p : Nat | prime_factor p n} have h2 : ∃ (p : Nat), p ∈ S := exists_prime_factor n h show ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q from well_ord_princ S h2 done lemma all_prime_nil : all_prime [] := by define --Goal : ∀ p ∈ [], prime p fix p : Nat contrapos --Goal : ¬prime p → p ∉ [] assume h1 : ¬prime p show p ∉ [] from List.not_mem_nil p done lemma all_prime_cons (n : Nat) (L : List Nat) : all_prime (n :: L) ↔ prime n ∧ all_prime L := by apply Iff.intro · -- (→) assume h1 : all_prime (n :: L) --Goal : prime n ∧ all_prime L define at h1 --h1 : ∀ p ∈ n :: L, prime p apply And.intro (h1 n (List.mem_cons_self n L)) define --Goal : ∀ p ∈ L, prime p fix p : Nat assume h2 : p ∈ L show prime p from h1 p (List.mem_cons_of_mem n h2) done · -- (←) assume h1 : prime n ∧ all_prime L --Goal : all_prime (n :: l) define : all_prime L at h1 define fix p : Nat assume h2 : p ∈ n :: L rewrite [List.mem_cons] at h2 --h2 : p = n ∨ p ∈ L by_cases on h2 · -- Case 1. h2 : p = n rewrite [h2] show prime n from h1.left done · -- Case 2. h2 : p ∈ L show prime p from h1.right p h2 done done done lemma nondec_nil : nondec [] := by define --Goal : True trivial --trivial proves some obviously true statements, such as True done lemma nondec_cons (n : Nat) (L : List Nat) : nondec (n :: L) ↔ (∀ m ∈ L, n ≤ m) ∧ nondec L := by rfl lemma prod_nil : prod [] = 1 := by rfl lemma prod_cons : prod (n :: L) = n * (prod L) := by rfl lemma exists_cons_of_length_eq_succ {A : Type} {l : List A} {n : Nat} (h : l.length = n + 1) : ∃ (a : A) (L : List A), l = a :: L ∧ L.length = n := by have h1 : ¬l.length = 0 := by linarith rewrite [List.length_eq_zero] at h1 obtain (a : A) (h2 : ∃ (L : List A), l = a :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List A) (h3 : l = a :: L) from h2 apply Exists.intro a apply Exists.intro L apply And.intro h3 have h4 : (a :: L).length = L.length + 1 := List.length_cons a L rewrite [←h3, h] at h4 show L.length = n from (Nat.add_right_cancel h4).symm done lemma list_elt_dvd_prod_by_length (a : Nat) : ∀ (n : Nat), ∀ (l : List Nat), l.length = n → a ∈ l → a ∣ prod l := by by_induc · --Base Case fix l : List Nat assume h1 : l.length = 0 rewrite [List.length_eq_zero] at h1 --h1 : l = [] rewrite [h1] --Goal : a ∈ [] → a ∣ prod [] contrapos assume h2 : ¬a ∣ prod [] show a ∉ [] from List.not_mem_nil a done · -- Induction Step fix n : Nat assume ih : ∀ (l : List Nat), List.length l = n → a ∈ l → a ∣ prod l fix l : List Nat assume h1 : l.length = n + 1 --Goal : a ∈ l → a ∣ prod l obtain (b : Nat) (h2 : ∃ (L : List Nat), l = b :: L ∧ L.length = n) from exists_cons_of_length_eq_succ h1 obtain (L : List Nat) (h3 : l = b :: L ∧ L.length = n) from h2 have h4 : a ∈ L → a ∣ prod L := ih L h3.right assume h5 : a ∈ l rewrite [h3.left, prod_cons] --Goal : a ∣ b * prod L rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b ∨ a ∈ L by_cases on h5 · -- Case 1. h5 : a = b apply Exists.intro (prod L) rewrite [h5] rfl done · -- Case 2. h5 : a ∈ L have h6 : a ∣ prod L := h4 h5 have h7 : prod L ∣ b * prod L := by apply Exists.intro b ring done show a ∣ b * prod L from dvd_trans h6 h7 done done done lemma list_elt_dvd_prod {a : Nat} {l : List Nat} (h : a ∈ l) : a ∣ prod l := by set n : Nat := l.length have h1 : l.length = n := by rfl show a ∣ prod l from list_elt_dvd_prod_by_length a n l h1 h done lemma exists_prime_factorization : ∀ (n : Nat), n ≥ 1 → ∃ (l : List Nat), prime_factorization n l := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, n_1 ≥ 1 → ∃ (l : List Nat), prime_factorization n_1 l assume h1 : n ≥ 1 by_cases h2 : n = 1 · -- Case 1. h2 : n = 1 apply Exists.intro [] define apply And.intro · -- Proof of nondec_prime_list [] define show all_prime [] ∧ nondec [] from And.intro all_prime_nil nondec_nil done · -- Proof of prod [] = n rewrite [prod_nil, h2] rfl done done · -- Case 2. h2 : n ≠ 1 have h3 : n ≥ 2 := lt_of_le_of_ne' h1 h2 obtain (p : Nat) (h4 : prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q) from exists_least_prime_factor h3 have p_prime_factor : prime_factor p n := h4.left define at p_prime_factor have p_prime : prime p := p_prime_factor.left have p_dvd_n : p ∣ n := p_prime_factor.right have p_least : ∀ (q : Nat), prime_factor q n → p ≤ q := h4.right obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n have h5 : m ≠ 0 := by contradict h1 with h6 have h7 : n = 0 := calc n _ = p * m := n_eq_pm _ = p * 0 := by rw [h6] _ = 0 := by ring rewrite [h7] decide done have m_pos : 0 < m := Nat.pos_of_ne_zero h5 have m_lt_n : m < n := by define at p_prime show m < n from calc m _ < m + m := by linarith _ = 2 * m := by ring _ ≤ p * m := by rel [p_prime.left] _ = n := n_eq_pm.symm done obtain (L : List Nat) (h6 : prime_factorization m L) from ih m m_lt_n m_pos define at h6 have ndpl_L : nondec_prime_list L := h6.left define at ndpl_L apply Exists.intro (p :: L) define apply And.intro · -- Proof of nondec_prime_list (p :: L) define apply And.intro · -- Proof of all_prime (p :: L) rewrite [all_prime_cons] show prime p ∧ all_prime L from And.intro p_prime ndpl_L.left done · -- Proof of nondec (p :: L) rewrite [nondec_cons] apply And.intro _ ndpl_L.right fix q : Nat assume q_in_L : q ∈ L have h7 : q ∣ prod L := list_elt_dvd_prod q_in_L rewrite [h6.right] at h7 --h7 : q ∣ m have h8 : m ∣ n := by apply Exists.intro p rewrite [n_eq_pm] ring done have q_dvd_n : q ∣ n := dvd_trans h7 h8 have ap_L : all_prime L := ndpl_L.left define at ap_L have q_prime_factor : prime_factor q n := And.intro (ap_L q q_in_L) q_dvd_n show p ≤ q from p_least q q_prime_factor done done · -- Proof of prod (p :: L) = n rewrite [prod_cons, h6.right, n_eq_pm] rfl done done done theorem Theorem_7_2_2 {a b c : Nat} (h1 : c ∣ a * b) (h2 : rel_prime a c) : c ∣ b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑c ∣ ↑b define at h1; define at h2; define obtain (j : Nat) (h3 : a * b = c * j) from h1 set s : Int := gcd_c1 a c set t : Int := gcd_c2 a c have h4 : s * ↑a + t * ↑c = ↑(gcd a c) := gcd_lin_comb c a rewrite [h2, Nat.cast_one] at h4 --h4 : s * ↑a + t * ↑c = (1 : Int) apply Exists.intro (s * ↑j + t * ↑b) show ↑b = ↑c * (s * ↑j + t * ↑b) from calc ↑b _ = (1 : Int) * ↑b := (one_mul _).symm _ = (s * ↑a + t * ↑c) * ↑b := by rw [h4] _ = s * (↑a * ↑b) + t * ↑c * ↑b := by ring _ = s * (↑c * ↑j) + t * ↑c * ↑b := by rw [←Nat.cast_mul a b, h3, Nat.cast_mul c j] _ = ↑c * (s * ↑j + t * ↑b) := by ring done lemma le_nonzero_prod_left {a b : Nat} (h : a * b ≠ 0) : a ≤ a * b := by have h1 : b ≠ 0 := by contradict h with h1 rewrite [h1] ring done have h2 : 1 ≤ b := Nat.pos_of_ne_zero h1 show a ≤ a * b from calc a = a * 1 := (mul_one a).symm _ ≤ a * b := by rel [h2] done lemma le_nonzero_prod_right {a b : Nat} (h : a * b ≠ 0) : b ≤ a * b := by rewrite [mul_comm] rewrite [mul_comm] at h show b ≤ b * a from le_nonzero_prod_left h done lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry lemma rel_prime_of_prime_not_dvd {a p : Nat} (h1 : prime p) (h2 : ¬p ∣ a) : rel_prime a p := by have h3 : gcd a p ∣ a := gcd_dvd_left a p have h4 : gcd a p ∣ p := gcd_dvd_right a p have h5 : gcd a p = 1 ∨ gcd a p = p := dvd_prime h1 h4 have h6 : gcd a p ≠ p := by contradict h2 with h6 rewrite [h6] at h3 show p ∣ a from h3 done disj_syll h5 h6 show rel_prime a p from h5 done theorem Theorem_7_2_3 {a b p : Nat} (h1 : prime p) (h2 : p ∣ a * b) : p ∣ a ∨ p ∣ b := by or_right with h3 have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3 show p ∣ b from Theorem_7_2_2 h2 h4 done lemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a ≥ 1 := by have h1 : a ≠ 0 := by by_contra h1 rewrite [h1] at h contradict h linarith done show a ≥ 1 from Nat.pos_of_ne_zero h1 done lemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by have h1 : a ≥ 1 := ge_one_of_prod_one h have h2 : a * b ≠ 0 := by linarith have h3 : a ≤ a * b := le_nonzero_prod_left h2 rewrite [h] at h3 show a = 1 from Nat.le_antisymm h3 h1 done lemma eq_one_of_dvd_one {n : Nat} (h : n ∣ 1) : n = 1 := by obtain (j : Nat) (h1 : 1 = n * j) from h show n = 1 from eq_one_of_prod_one h1.symm done lemma prime_not_one {p : Nat} (h : prime p) : p ≠ 1 := by define at h linarith done theorem Theorem_7_2_4 {p : Nat} (h1 : prime p) : ∀ (l : List Nat), p ∣ prod l → ∃ a ∈ l, p ∣ a := by apply List.rec · -- Base Case. Goal : p ∣ prod [] → ∃ a ∈ [], p ∣ a rewrite [prod_nil] assume h2 : p ∣ 1 show ∃ a ∈ [], p ∣ a from absurd (eq_one_of_dvd_one h2) (prime_not_one h1) done · -- Induction Step fix b : Nat fix L : List Nat assume ih : p ∣ prod L → ∃ a ∈ L, p ∣ a --Goal : p ∣ prod (b :: L) → ∃ a ∈ b :: L, p ∣ a assume h2 : p ∣ prod (b :: L) rewrite [prod_cons] at h2 have h3 : p ∣ b ∨ p ∣ prod L := Theorem_7_2_3 h1 h2 by_cases on h3 · -- Case 1. h3 : p ∣ b apply Exists.intro b show b ∈ b :: L ∧ p ∣ b from And.intro (List.mem_cons_self b L) h3 done · -- Case 2. h3 : p ∣ prod L obtain (a : Nat) (h4 : a ∈ L ∧ p ∣ a) from ih h3 apply Exists.intro a show a ∈ b :: L ∧ p ∣ a from And.intro (List.mem_cons_of_mem b h4.left) h4.right done done done lemma prime_in_list {p : Nat} {l : List Nat} (h1 : prime p) (h2 : all_prime l) (h3 : p ∣ prod l) : p ∈ l := by obtain (a : Nat) (h4 : a ∈ l ∧ p ∣ a) from Theorem_7_2_4 h1 l h3 define at h2 have h5 : prime a := h2 a h4.left have h6 : p = 1 ∨ p = a := dvd_prime h5 h4.right disj_syll h6 (prime_not_one h1) rewrite [h6] show a ∈ l from h4.left done lemma first_le_first {p q : Nat} {l m : List Nat} (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m)) (h3 : prod (p :: l) = prod (q :: m)) : p ≤ q := by define at h1; define at h2 have h4 : q ∣ prod (p :: l) := by define apply Exists.intro (prod m) rewrite [←prod_cons] show prod (p :: l) = prod (q :: m) from h3 done have h5 : all_prime (q :: m) := h2.left rewrite [all_prime_cons] at h5 have h6 : q ∈ p :: l := prime_in_list h5.left h1.left h4 have h7 : nondec (p :: l) := h1.right rewrite [nondec_cons] at h7 rewrite [List.mem_cons] at h6 by_cases on h6 · -- Case 1. h6 : q = p linarith done · -- Case 2. h6 : q ∈ l have h8 : ∀ m ∈ l, p ≤ m := h7.left show p ≤ q from h8 q h6 done done lemma nondec_prime_list_tail {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by define at h define rewrite [all_prime_cons, nondec_cons] at h show all_prime l ∧ nondec l from And.intro h.left.right h.right.right done lemma cons_prod_not_one {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : prod (p :: l) ≠ 1 := by define at h have h1 : all_prime (p :: l) := h.left rewrite [all_prime_cons] at h1 rewrite [prod_cons] by_contra h2 show False from (prime_not_one h1.left) (eq_one_of_prod_one h2) done lemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) : l = [] ↔ prod l = 1 := by apply Iff.intro · -- (→) assume h1 : l = [] rewrite [h1] show prod [] = 1 from prod_nil done · -- (←) contrapos assume h1 : ¬l = [] obtain (p : Nat) (h2 : ∃ (L : List Nat), l = p :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List Nat) (h3 : l = p :: L) from h2 rewrite [h3] at h rewrite [h3] show ¬prod (p :: L) = 1 from cons_prod_not_one h done done lemma prime_pos {p : Nat} (h : prime p) : p > 0 := by define at h linarith done theorem Theorem_7_2_5 : ∀ (l1 l2 : List Nat), nondec_prime_list l1 → nondec_prime_list l2 → prod l1 = prod l2 → l1 = l2 := by apply List.rec · -- Base Case. Goal : ∀ (l2 : List Nat), nondec_prime_list [] → -- nondec_prime_list l2 → prod [] = prod l2 → [] = l2 fix l2 : List Nat assume h1 : nondec_prime_list [] assume h2 : nondec_prime_list l2 assume h3 : prod [] = prod l2 rewrite [prod_nil, eq_comm, ←list_nil_iff_prod_one h2] at h3 show [] = l2 from h3.symm done · -- Induction Step fix p : Nat fix L1 : List Nat assume ih : ∀ (L2 : List Nat), nondec_prime_list L1 → nondec_prime_list L2 → prod L1 = prod L2 → L1 = L2 -- Goal : ∀ (l2 : List Nat), nondec_prime_list (p :: L1) → -- nondec_prime_list l2 → prod (p :: L1) = prod l2 → p :: L1 = l2 fix l2 : List Nat assume h1 : nondec_prime_list (p :: L1) assume h2 : nondec_prime_list l2 assume h3 : prod (p :: L1) = prod l2 have h4 : ¬prod (p :: L1) = 1 := cons_prod_not_one h1 rewrite [h3, ←list_nil_iff_prod_one h2] at h4 obtain (q : Nat) (h5 : ∃ (L : List Nat), l2 = q :: L) from List.exists_cons_of_ne_nil h4 obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5 rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2) rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2) have h7 : p ≤ q := first_le_first h1 h2 h3 have h8 : q ≤ p := first_le_first h2 h1 h3.symm have h9 : p = q := by linarith rewrite [h9, prod_cons, prod_cons] at h3 --h3 : q * prod L1 = q * prod L2 have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1 have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2 define at h2 have h12 : all_prime (q :: L2) := h2.left rewrite [all_prime_cons] at h12 have h13 : q > 0 := prime_pos h12.left have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3 have h15 : L1 = L2 := ih L2 h10 h11 h14 rewrite [h6, h9, h15] rfl done done theorem fund_thm_arith (n : Nat) (h : n ≥ 1) : ∃! (l : List Nat), prime_factorization n l := by exists_unique · -- Existence show ∃ (l : List Nat), prime_factorization n l from exists_prime_factorization n h done · -- Uniqueness fix l1 : List Nat; fix l2 : List Nat assume h1 : prime_factorization n l1 assume h2 : prime_factorization n l2 define at h1; define at h2 have h3 : prod l1 = n := h1.right rewrite [←h2.right] at h3 show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3 done done /- Section 7.3 -/ theorem congr_refl (m : Nat) : ∀ (a : Int), a ≡ a (MOD m) := by fix a : Int define --Goal : ∃ (c : Int), a - a = ↑m * c apply Exists.intro 0 ring done theorem congr_symm {m : Nat} : ∀ {a b : Int}, a ≡ b (MOD m) → b ≡ a (MOD m) := by fix a : Int; fix b : Int assume h1 : a ≡ b (MOD m) define at h1 --h1 : ∃ (c : Int), a - b = ↑m * c define --Goal : ∃ (c : Int), b - a = ↑m * c obtain (c : Int) (h2 : a - b = m * c) from h1 apply Exists.intro (-c) show b - a = m * (-c) from calc b - a _ = -(a - b) := by ring _ = -(m * c) := by rw [h2] _ = m * (-c) := by ring done theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry /- Fundamental properties of congruence classes -/ lemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) : X = Y ↔ X.val = Y.val := Fin.ext_iff lemma val_nat_eq_mod (n k : Nat) : ([k]_(n + 1)).val = k % (n + 1) := by rfl lemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl theorem cc_rep {m : Nat} (X : ZMod m) : ∃ (a : Int), X = [a]_m := match m with | 0 => by apply Exists.intro X rfl done | n + 1 => by apply Exists.intro ↑(X.val) have h1 : X.val < n + 1 := Fin.prop X rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1] rfl done theorem add_class (m : Nat) (a b : Int) : [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm theorem mul_class (m : Nat) (a b : Int) : [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm lemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ [a - b]_m = [0]_m := by apply Iff.intro · -- (→) assume h1 : [a]_m = [b]_m have h2 : a - b = a + (-b) := by ring have h3 : b + (-b) = 0 := by ring show [a - b]_m = [0]_m from calc [a - b]_m _ = [a + (-b)]_m := by rw [h2] _ = [a]_m + [-b]_m := by rw [add_class] _ = [b]_m + [-b]_m := by rw [h1] _ = [b + -b]_m := by rw [add_class] _ = [0]_m := by rw [h3] done · -- (←) assume h1 : [a - b]_m = [0]_m have h2 : b + (a - b) = a := by ring have h3 : b + 0 = b := by ring show [a]_m = [b]_m from calc [a]_m _ = [b + (a - b)]_m := by rw [h2] _ = [b]_m + [a - b]_m := by rw [add_class] _ = [b]_m + [0]_m := by rw [h1] _ = [b + 0]_m := by rw [add_class] _ = [b]_m := by rw [h3] done done lemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) : [a]_m = [0]_m → [-a]_m = [0]_m := by assume h1 : [a]_m = [0]_m have h2 : 0 + (-a) = -a := by ring have h3 : a + (-a) = 0 := by ring show [-a]_m = [0]_m from calc [-a]_m _ = [0 + (-a)]_m := by rw [h2] _ = [0]_m + [-a]_m := by rw [add_class] _ = [a]_m + [-a]_m := by rw [h1] _ = [a + (-a)]_m := by rw [add_class] _ = [0]_m := by rw [h3] done lemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) : [-a]_m = [0]_m ↔ [a]_m = [0]_m := by apply Iff.intro _ (cc_neg_zero_of_cc_zero m a) assume h1 : [-a]_m = [0]_m have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1 have h3 : -(-a) = a := by ring rewrite [h3] at h2 show [a]_m = [0]_m from h2 done lemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl lemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m ↔ m ∣ k := match m with | 0 => by have h : (0 : Int) = (↑(0 : Nat) : Int) := by rfl rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj] apply Iff.intro · -- (→) assume h1 : k = 0 rewrite [h1] show 0 ∣ 0 from dvd_self 0 done · -- (←) assume h1 : 0 ∣ k obtain (c : Nat) (h2 : k = 0 * c) from h1 rewrite [h2] ring done done | n + 1 => by rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero] show k % (n + 1) = 0 ↔ n + 1 ∣ k from (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm done lemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m ↔ ↑m ∣ a := by obtain (k : Nat) (h1 : a = ↑k ∨ a = -↑k) from Int.eq_nat_or_neg a by_cases on h1 · -- Case 1. h1: a = ↑k rewrite [h1, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done · -- Case 2. h1: a = -↑k rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done done theorem cc_eq_iff_congr (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ a ≡ b (MOD m) := calc [a]_m = [b]_m _ ↔ [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b _ ↔ ↑m ∣ (a - b) := cc_zero_iff_dvd m (a - b) _ ↔ a ≡ b (MOD m) := by rfl /- End of fundamental properties of congruence classes -/ lemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m := by have h1 : (↑m : Int) ≠ 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m) show 0 ≤ a % m from Int.emod_nonneg a h1 done lemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m) have h2 : (↑m : Int) > 0 := (Nat.cast_pos).rtl h1 show a % m < m from Int.emod_lt_of_pos a h2 done lemma congr_mod_mod (m : Nat) (a : Int) : a ≡ a % m (MOD m) := by define have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m apply Exists.intro (a / m) show a - a % m = m * (a / m) from calc a - (a % m) _ = m * (a / m) + a % m - a % m := by rw [h1] _ = m * (a / m) := by ring done lemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) := And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a)) theorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) : ∃! (r : Int), 0 ≤ r ∧ r < m ∧ a ≡ r (MOD m) := by exists_unique · -- Existence apply Exists.intro (a % m) show 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) from mod_cmpl_res m a done · -- Uniqueness fix r1 : Int; fix r2 : Int assume h1 : 0 ≤ r1 ∧ r1 < m ∧ a ≡ r1 (MOD m) assume h2 : 0 ≤ r2 ∧ r2 < m ∧ a ≡ r2 (MOD m) have h3 : r1 ≡ r2 (MOD m) := congr_trans (congr_symm h1.right.right) h2.right.right obtain (d : Int) (h4 : r1 - r2 = m * d) from h3 have h5 : r1 - r2 < m * 1 := by linarith have h6 : m * (-1) < r1 - r2 := by linarith rewrite [h4] at h5 --h5 : m * d < m * 1 rewrite [h4] at h6 --h6 : m * -1 < m * d have h7 : (↑m : Int) ≥ 0 := Nat.cast_nonneg m have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7 have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7 have h10 : d = 0 := by linarith show r1 = r2 from calc r1 _ = r1 - r2 + r2 := by ring _ = m * 0 + r2 := by rw [h4, h10] _ = r2 := by ring done done lemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m := (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a) theorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y rewrite [h1, h2] have h3 : a + b = b + a := by ring show [a]_m + [b]_m = [b]_m + [a]_m from calc [a]_m + [b]_m _ = [a + b]_m := add_class m a b _ = [b + a]_m := by rw [h3] _ = [b]_m + [a]_m := (add_class m b a).symm done theorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X rewrite [h1] have h2 : a * 1 = a := by ring show [a]_m * [1]_m = [a]_m from calc [a]_m * [1]_m _ = [a * 1]_m := mul_class m a 1 _ = [a]_m := by rw [h2] done theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry lemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) : [a]_m * [gcd_c2 m a]_m = [1]_m := by set s : Int := gcd_c1 m a have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m define at h1 rewrite [h1, Nat.cast_one] at h2 --h2 : s * ↑m + gcd_c2 m a * ↑a = 1 rewrite [mul_class, cc_eq_iff_congr] define --Goal : ∃ (c : Int), ↑a * gcd_c2 m a - 1 = ↑m * c apply Exists.intro (-s) show a * (gcd_c2 m a) - 1 = m * (-s) from calc a * (gcd_c2 m a) - 1 _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring _ = 1 + m * (-s) - 1 := by rw [h2] _ = m * (-s) := by ring done theorem Theorem_7_3_7 (m a : Nat) : invertible [a]_m ↔ rel_prime m a := by apply Iff.intro · -- (→) assume h1 : invertible [a]_m define at h1 obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1 obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y rewrite [h3, mul_class, cc_eq_iff_congr] at h2 define at h2 obtain (c : Int) (h4 : a * b - 1 = m * c) from h2 rewrite [Exercise_7_2_6] --Goal : ∃ (s t : Int), s * ↑m + t * ↑a = 1 apply Exists.intro (-c) apply Exists.intro b show (-c) * m + b * a = 1 from calc (-c) * m + b * a _ = (-c) * m + (a * b - 1) + 1 := by ring _ = (-c) * m + m * c + 1 := by rw [h4] _ = 1 := by ring done · -- (←) assume h1 : rel_prime m a define show ∃ (Y : ZMod m), [a]_m * Y = [1]_m from Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1) done done /- Section 7.4 -/ section Euler open Euler lemma num_rp_below_base {m : Nat} : num_rp_below m 0 = 0 := by rfl lemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) : num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by have h1 : num_rp_below m (j + 1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : gcd m j = 1 rewrite [if_pos h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1 show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1 done lemma num_rp_below_step_not_rp {m j : Nat} (h : ¬rel_prime m j) : num_rp_below m (j + 1) = num_rp_below m j := by have h1 : num_rp_below m (j +1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : ¬gcd m j = 1 rewrite [if_neg h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j show num_rp_below m (j + 1) = num_rp_below m j from h1 done lemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl #eval phi 10 --Answer: 4 lemma prod_inv_iff_inv {m : Nat} {X : ZMod m} (h1 : invertible X) (Y : ZMod m) : invertible (X * Y) ↔ invertible Y := by apply Iff.intro · -- (→) assume h2 : invertible (X * Y) obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2 apply Exists.intro (X * Z) rewrite [←h3] --Goal : Y * (X * Z) = X * Y * Z ring --Note that ring can do algebra in ZMod m done · -- (←) assume h2 : invertible Y obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1 obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2 apply Exists.intro (Xi * Yi) show (X * Y) * (Xi * Yi) = [1]_m from calc X * Y * (Xi * Yi) _ = (X * Xi) * (Y * Yi) := by ring _ = [1]_m * [1]_m := by rw [h3, h4] _ = [1]_m := Theorem_7_3_6_7 [1]_m done done lemma F_rp_def {m i : Nat} (h : rel_prime m i) : F m i = [i]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h --h : gcd m i = 1 rewrite [if_pos h] at h1 show F m i = [i]_m from h1 done lemma F_not_rp_def {m i : Nat} (h : ¬rel_prime m i) : F m i = [1]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h rewrite [h1, if_neg h] rfl done lemma prod_seq_base {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 0 k f = [1]_m := by rfl lemma prod_seq_step {m : Nat} (n k : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl lemma prod_seq_zero_step {m : Nat} (n : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by rewrite [prod_seq_step, zero_add] rfl done lemma prod_one {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 1 k f = f k := by rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7] rfl done lemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl lemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m := calc [G m a i]_m _ = [(a * i) % m]_m := by rfl _ = [a * i]_m := (cc_eq_mod m (a * i)).symm _ = [a]_m * [i]_m := (mul_class m a i).symm lemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) : rel_prime m (G m a i) ↔ rel_prime m i := by have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1 show rel_prime m (G m a i) ↔ rel_prime m i from calc rel_prime m (G m a i) _ ↔ invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm _ ↔ invertible ([a]_m * [i]_m) := by rw [cc_G] _ ↔ invertible [i]_m := prod_inv_iff_inv h2 ([i]_m) _ ↔ rel_prime m i := Theorem_7_3_7 m i done lemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) : F m (G m a i) = [a]_m * F m i := by have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2 show F m (G m a i) = [a]_m * F m i from calc F m (G m a i) _ = [G m a i]_m := F_rp_def h3 _ = [a]_m * [i]_m := cc_G m a i _ = [a]_m * F m i := by rw [F_rp_def h2] done lemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : ¬rel_prime m i) : F m (G m a i) = [1]_m := by rewrite [←G_rp_iff h1 i] at h2 show F m (G m a i) = [1]_m from F_not_rp_def h2 done lemma FG_prod {m a : Nat} (h1 : rel_prime m a) : ∀ (k : Nat), prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by by_induc · -- Base Case show prod_seq 0 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from calc prod_seq 0 0 ((F m) ∘ (G m a)) _ = [1]_m := prod_seq_base _ _ _ = [a]_m ^ 0 * [1]_m := by ring _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by rw [num_rp_below_base, prod_seq_base] done · -- Induction Step fix k : Nat assume ih : prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) by_cases h2 : rel_prime m k · -- Case 1. h2 : rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([a]_m * F m k) := by rw [FG_rp h1 h2] _ = [a]_m ^ ((num_rp_below m k) + 1) * ((prod_seq k 0 (F m)) * F m k) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_rp h2, prod_seq_zero_step] done · -- Case 2. h2 : ¬rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([1]_m) := by rw [FG_not_rp h1 h2] _ = [a]_m ^ (num_rp_below m k) * (prod_seq k 0 (F m) * ([1]_m)) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_not_rp h2, prod_seq_zero_step, F_not_rp_def h2] done done done lemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by define --Goal : ∀ i < m, G m a i < m fix i : Nat assume h1 : i < m rewrite [G_def] --Goal : a * i % m < m show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m) done lemma left_inv_one_one_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g' (g i) = i) : one_one_below n g := sorry lemma right_inv_onto_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g (g' i) = i) (h2 : maps_below n g') : onto_below n g := by define at h2; define fix k : Nat assume h3 : k < n apply Exists.intro (g' k) show g' k < n ∧ g (g' k) = k from And.intro (h2 k h3) (h1 k h3) done lemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m * [inv_mod m a]_m = [1]_m := by have h2 : 0 ≤ (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a) show [a]_m * [inv_mod m a]_m = [1]_m from calc [a]_m * [inv_mod m a]_m _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2] _ = [a]_m * [gcd_c2 m a]_m := by rw [←cc_eq_mod] _ = [1]_m := gcd_c2_inv h1 done lemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m := calc a * (b % m) % m = a % m * (b % m % m) % m := Nat.mul_mod _ _ _ _ = a % m * (b % m) % m := by rw [Nat.mod_mod] _ = a * b % m := (Nat.mul_mod _ _ _).symm lemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm] rfl done theorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] : ↑a ≡ ↑b (MOD m) ↔ a % m = b % m := sorry lemma mul_inv_mod_cancel {m a i : Nat} [NeZero m] (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1 rewrite [mul_class, cc_eq_iff_congr, ←Nat.cast_mul, ←Nat.cast_one, congr_iff_mod_eq_Nat] at h3 show a * inv_mod m a * i % m = i from calc a * (inv_mod m a) * i % m _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = 1 % m * i % m := by rw [h3] _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = i % m := by rw [one_mul] _ = i := Nat.mod_eq_of_lt h2 done lemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl lemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, G m a (Ginv m a i) = i := by fix i : Nat assume h2 : i < m show G m a (Ginv m a i) = i from calc G m a (Ginv m a i) _ = a * ((inv_mod m a * i) % m) % m := by rfl _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, Ginv m a (G m a i) = i := by fix i : Nat assume h2 : i < m show Ginv m a (G m a i) = i from calc Ginv m a (G m a i) _ = inv_mod m a * ((a * i) % m) % m := by rfl _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc, mul_comm (inv_mod m a)] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_maps_below (m a : Nat) [NeZero m] : maps_below m (Ginv m a) := G_maps_below m (inv_mod m a) lemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : one_one_below m (G m a) := left_inv_one_one_below (Ginv_left_inv h1) lemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : onto_below m (G m a) := right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a) lemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : perm_below m (G m a) := And.intro (G_maps_below m a) (And.intro (G_one_one_below h1) (G_onto_below h1)) --Permuting a product of congruence classes doesn't change product lemma swap_fst (u v : Nat) : swap u v u = v := by define : swap u v u --Goal : (if u = u then v else if u = v then u else u) = v have h : u = u := by rfl rewrite [if_pos h] rfl done lemma swap_snd (u v : Nat) : swap u v v = u := by define : swap u v v by_cases h1 : v = u · -- Case 1. h1 : v = u rewrite [if_pos h1] show v = u from h1 done · -- Case 2. h1 : v ≠ u rewrite [if_neg h1] have h2 : v = v := by rfl rewrite [if_pos h2] rfl done done lemma swap_other {u v i : Nat} (h1 : i ≠ u) (h2 : i ≠ v) : swap u v i = i := by define : swap u v i rewrite [if_neg h1, if_neg h2] rfl done lemma swap_values (u v i : Nat) : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := by by_cases h1 : i = u · -- Case 1. h1 : i = u apply Or.inl rewrite [h1] show swap u v u = v from swap_fst u v done · -- Case 2. h1 : i ≠ u apply Or.inr by_cases h2 : i = v · -- Case 2.1. h2 : i = v apply Or.inl rewrite [h2] show swap u v v = u from swap_snd u v done · -- Case 2.2. h2 : i ≠ v apply Or.inr show swap u v i = i from swap_other h1 h2 done done done lemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by define fix i : Nat assume h3 : i < n have h4 : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := swap_values u v i by_cases on h4 · -- Case 1. h4 : swap u v i = v rewrite [h4] show v < n from h2 done · -- Case 2. by_cases on h4 · -- Case 2.1. h4 : swap u v i = u rewrite [h4] show u < n from h1 done · -- Case 2.2. h4 : swap u v i = i rewrite [h4] show i < n from h3 done done done lemma swap_swap (u v n : Nat) : ∀ i < n, swap u v (swap u v i) = i := by fix i : Nat assume h : i < n by_cases h1 : i = u · -- Case 1. h1 : i = u rewrite [h1, swap_fst, swap_snd] rfl done · -- Case 2. h1 : i ≠ u by_cases h2 : i = v · -- Case 2.1. h2 : i = v rewrite [h2, swap_snd, swap_fst] rfl done · -- Case 2.2. h2 : i ≠ v rewrite [swap_other h1 h2, swap_other h1 h2] rfl done done done lemma swap_one_one_below (u v n) : one_one_below n (swap u v) := left_inv_one_one_below (swap_swap u v n) lemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) := right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2) lemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) := And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2)) lemma comp_perm_below {n : Nat} {f g : Nat → Nat} (h1 : perm_below n f) (h2 : perm_below n g) : perm_below n (f ∘ g) := sorry lemma trivial_swap (u : Nat) : swap u u = id := by apply funext fix x : Nat by_cases h1 : x = u · -- Case 1. h1 : x = u rewrite [h1, swap_fst] rfl done · -- Case 2. h1 : x ≠ u rewrite [swap_other h1 h1] rfl done done lemma prod_eq_fun {m : Nat} (f g : Nat → ZMod m) (k : Nat) : ∀ (n : Nat), (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g := by by_induc · -- Base Case assume h : (∀ i < 0, f (k + i) = g (k + i)) rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g assume h1 : ∀ i < n + 1, f (k + i) = g (k + i) have h2 : ∀ i < n, f (k + i) = g (k + i) := by fix i : Nat assume h2 : i < n have h3 : i < n + 1 := by linarith show f (k + i) = g (k + i) from h1 i h3 done have h3 : prod_seq n k f = prod_seq n k g := ih h2 have h4 : n < n + 1 := Nat.lt_succ_self n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4] rfl done done lemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := by have h2 : ∀ (i : Nat), i < u → (f ∘ swap u n) (0 + i) = f (0 + i) := by fix i : Nat assume h2 : i < u have h3 : 0 + i ≠ u := by linarith have h4 : 0 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl done show prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f from prod_eq_fun (f ∘ swap u n) f 0 u h2 done lemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := by have h2 : ∀ i < j, (f ∘ swap u n) (u + 1 + i) = f (u + 1 + i) := by fix i : Nat assume h2 : i < j have h3 : u + 1 + i ≠ u := by linarith have h4 : u + 1 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl show prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f from prod_eq_fun (f ∘ swap u n) f (u + 1) j h2 done lemma break_prod {m : Nat} (n : Nat) (f : Nat → ZMod m) : ∀ (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by by_induc · -- Base Case have h : n + 0 = n := by rfl rewrite [prod_seq_base, h, Theorem_7_3_6_7] rfl done · -- Induction Step fix j : Nat assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f rewrite [←add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc] rfl done done lemma break_prod_twice {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f := break_prod n f 1 rewrite [prod_one] at h2 have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f := break_prod (u + 1) f j rewrite [←h1] at h3 have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f := break_prod u f 1 rewrite [prod_one] at h4 rewrite [h3, h4] at h2 show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2 done lemma swap_prod_eq_prod {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f := by by_cases h2 : u = n · -- Case 1. h2 : u = n rewrite [h2, trivial_swap n] --Goal : prod_seq (n + 1) 0 (f ∘ id) = prod_seq (n + 1) 0 f rfl done · -- Case 2. h2 : ¬u = n have h3 : u + 1 ≤ n := Nat.lt_of_le_of_ne h1 h2 obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3 have break_f : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := break_prod_twice f h4 have break_fs : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_prod_twice (f ∘ swap u n) h4 have f_eq_fs_below : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := swap_prod_eq_prod_below f h1 have f_eq_fs_btwn : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4 show prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f from calc prod_seq (n + 1) 0 (f ∘ swap u n) _ = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_fs _ = prod_seq u 0 f * (f ∘ swap u n) u * prod_seq j (u + 1) f * (f ∘ swap u n) n := by rw [f_eq_fs_below, f_eq_fs_btwn] _ = prod_seq u 0 f * f (swap u n u) * prod_seq j (u + 1) f * f (swap u n n) := by rfl _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by rw [swap_fst, swap_snd] _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring _ = prod_seq (n + 1) 0 f := break_f.symm done done lemma perm_below_fixed {n : Nat} {g : Nat → Nat} (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry lemma perm_prod {m : Nat} (f : Nat → ZMod m) : ∀ (n : Nat), ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) := by by_induc · -- Base Case fix g : Nat → Nat assume h1 : perm_below 0 g rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) fix g : Nat → Nat assume g_pb : perm_below (n + 1) g define at g_pb have g_ob : onto_below (n + 1) g := g_pb.right.right define at g_ob have h1 : n < n + 1 := by linarith obtain (u : Nat) (h2 : u < n + 1 ∧ g u = n) from g_ob n h1 have s_pb : perm_below (n + 1) (swap u n) := swap_perm_below h2.left h1 have gs_pb_n1 : perm_below (n + 1) (g ∘ swap u n) := comp_perm_below g_pb s_pb have gs_fix_n : (g ∘ swap u n) n = n := calc (g ∘ swap u n) n _ = g (swap u n n) := by rfl _ = g u := by rw [swap_snd] _ = n := h2.right have gs_pb_n : perm_below n (g ∘ swap u n) := perm_below_fixed gs_pb_n1 gs_fix_n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f ∘ (g ∘ swap u n)) := ih (g ∘ swap u n) gs_pb_n have h3 : u ≤ n := by linarith show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f ∘ g) from calc prod_seq (n + 1) 0 f _ = prod_seq n 0 f * f n := prod_seq_zero_step n f _ = prod_seq n 0 (f ∘ (g ∘ swap u n)) * f ((g ∘ swap u n) n) := by rw [gs_prod, gs_fix_n] _ = prod_seq n 0 (f ∘ g ∘ swap u n) * (f ∘ g ∘ swap u n) n := by rfl _ = prod_seq (n + 1) 0 (f ∘ g ∘ swap u n) := (prod_seq_zero_step n (f ∘ g ∘ swap u n)).symm _ = prod_seq (n + 1) 0 ((f ∘ g) ∘ swap u n) := by rfl _ = prod_seq (n + 1) 0 (f ∘ g) := swap_prod_eq_prod (f ∘ g) h3 done done lemma F_invertible (m i : Nat) : invertible (F m i) := by by_cases h : rel_prime m i · -- Case 1. h : rel_prime m i rewrite [F_rp_def h] show invertible [i]_m from (Theorem_7_3_7 m i).rtl h done · -- Case 2. h : ¬rel_prime m i rewrite [F_not_rp_def h] apply Exists.intro [1]_m show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m done done lemma Fprod_invertible (m : Nat) : ∀ (k : Nat), invertible (prod_seq k 0 (F m)) := by by_induc · -- Base Case apply Exists.intro [1]_m show prod_seq 0 0 (F m) * [1]_m = [1]_m from calc prod_seq 0 0 (F m) * [1]_m _ = [1]_m * [1]_m := by rw [prod_seq_base] _ = [1]_m := Theorem_7_3_6_7 ([1]_m) done · -- Induction Step fix k : Nat assume ih : invertible (prod_seq k 0 (F m)) rewrite [prod_seq_zero_step] show invertible (prod_seq k 0 (F m) * (F m k)) from (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k) done done theorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m ^ (phi m) = [1]_m := by have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2 show [a]_m ^ (phi m) = [1]_m from calc [a]_m ^ (phi m) _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3] _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring _ = prod_seq m 0 (F m ∘ G m a) * Y := by rw [FG_prod h1 m, phi_def] _ = prod_seq m 0 (F m) * Y := by rw [perm_prod (F m) m (G m a) (G_perm_below h1)] _ = [1]_m := by rw [h3] done lemma Exercise_7_4_5_Int (m : Nat) (a : Int) : ∀ (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry lemma Exercise_7_4_5_Nat (m a n : Nat) : [a]_m ^ n = [a ^ n]_m := by rewrite [Exercise_7_4_5_Int] rfl done theorem Euler's_theorem {m a : Nat} [NeZero m] (h1 : rel_prime m a) : a ^ (phi m) ≡ 1 (MOD m) := by have h2 : [a]_m ^ (phi m) = [1]_m := Theorem_7_4_2 h1 rewrite [Exercise_7_4_5_Nat m a (phi m)] at h2 --h2 : [a ^ phi m]_m = [1]_m show a ^ (phi m) ≡ 1 (MOD m) from (cc_eq_iff_congr _ _ _).ltr h2 done #eval gcd 10 7 --Answer: 1. So 10 and 7 are relatively prime #eval 7 ^ phi 10 --Answer: 2401, which is congruent to 1 mod 10. end Euler /- Section 7.5 -/ lemma num_rp_prime {p : Nat} (h1 : prime p) : ∀ k < p, num_rp_below p (k + 1) = k := sorry lemma phi_prime {p : Nat} (h1 : prime p) : phi p = p - 1 := by have h2 : 1 ≤ p := prime_pos h1 have h3 : p - 1 + 1 = p := Nat.sub_add_cancel h2 have h4 : p - 1 < p := by linarith have h5 : num_rp_below p (p - 1 + 1) = p - 1 := num_rp_prime h1 (p - 1) h4 rewrite [h3] at h5 show phi p = p - 1 from h5 done theorem Theorem_7_2_2_Int {a c : Nat} {b : Int} (h1 : ↑c ∣ ↑a * b) (h2 : rel_prime a c) : ↑c ∣ b := by rewrite [Int.natCast_dvd, Int.natAbs_mul, Int.natAbs_ofNat] at h1 --h1 : c ∣ a * Int.natAbs b rewrite [Int.natCast_dvd] --Goal : c ∣ Int.natAbs b show c ∣ Int.natAbs b from Theorem_7_2_2 h1 h2 done lemma Lemma_7_4_5 {m n : Nat} (a b : Int) (h1 : rel_prime m n) : a ≡ b (MOD m * n) ↔ a ≡ b (MOD m) ∧ a ≡ b (MOD n) := by apply Iff.intro · -- (→) assume h2 : a ≡ b (MOD m * n) obtain (j : Int) (h3 : a - b = (m * n) * j) from h2 apply And.intro · -- Proof of a ≡ b (MOD m) apply Exists.intro (n * j) show a - b = m * (n * j) from calc a - b _ = m * n * j := h3 _ = m * (n * j) := by ring done · -- Proof of a ≡ b (MOD n) apply Exists.intro (m * j) show a - b = n * (m * j) from calc a - b _ = m * n * j := h3 _ = n * (m * j) := by ring done done · -- (←) assume h2 : a ≡ b (MOD m) ∧ a ≡ b (MOD n) obtain (j : Int) (h3 : a - b = m * j) from h2.left have h4 : (↑n : Int) ∣ a - b := h2.right rewrite [h3] at h4 --h4 : ↑n ∣ ↑m * j have h5 : ↑n ∣ j := Theorem_7_2_2_Int h4 h1 obtain (k : Int) (h6 : j = n * k) from h5 apply Exists.intro k --Goal : a - b = ↑(m * n) * k rewrite [Nat.cast_mul] --Goal : a - b = ↑m * ↑n * k show a - b = (m * n) * k from calc a - b _ = m * j := h3 _ = m * (n * k) := by rw [h6] _ = (m * n) * k := by ring done done --From exercises of Section 7.2 theorem rel_prime_symm {a b : Nat} (h : rel_prime a b) : rel_prime b a := sorry lemma prime_NeZero {p : Nat} (h : prime p) : NeZero p := by rewrite [neZero_iff] --Goal : p ≠ 0 define at h linarith done

lemma Lemma_7_5_1 {p e d m c s : Nat} {t : Int} (h1 : prime p) (h2 : e * d = (p - 1) * s + 1) (h3 : m ^ e - c = p * t) : c ^ d ≡ m (MOD p)

HTPI.Lemma_7_5_1

htpi/HTPILib/Chap7.lean

htpi

/- Copyright 2023 Daniel J. Velleman -/ import HTPILib.Chap6 namespace HTPI /- Definitions -/ lemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by have h : n + 1 > 0 := Nat.succ_pos n show a % (n + 1) < n + 1 from Nat.mod_lt a h done def gcd (a b : Nat) : Nat := match b with | 0 => a | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd (n + 1) (a % (n + 1)) termination_by b mutual def gcd_c1 (a b : Nat) : Int := match b with | 0 => 1 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c2 (n + 1) (a % (n + 1)) --Corresponds to s = t' termination_by b def gcd_c2 (a b : Nat) : Int := match b with | 0 => 0 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c1 (n + 1) (a % (n + 1)) - (gcd_c2 (n + 1) (a % (n + 1))) * ↑(a / (n + 1)) --Corresponds to t = s' - t'q termination_by b end def prime (n : Nat) : Prop := 2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n def prime_factor (p n : Nat) : Prop := prime p ∧ p ∣ n def all_prime (l : List Nat) : Prop := ∀ p ∈ l, prime p def nondec (l : List Nat) : Prop := match l with | [] => True --Of course, True is a proposition that is always true | n :: L => (∀ m ∈ L, n ≤ m) ∧ nondec L def nondec_prime_list (l : List Nat) : Prop := all_prime l ∧ nondec l def prod (l : List Nat) : Nat := match l with | [] => 1 | n :: L => n * (prod L) def prime_factorization (n : Nat) (l : List Nat) : Prop := nondec_prime_list l ∧ prod l = n def rel_prime (a b : Nat) : Prop := gcd a b = 1 def congr_mod (m : Nat) (a b : Int) : Prop := (↑m : Int) ∣ (a - b) def cc (m : Nat) (a : Int) : ZMod m := (↑a : ZMod m) notation:50 a " ≡ " b " (MOD " m ")" => congr_mod m a b notation:max "["a"]_"m:max => cc m a def invertible {m : Nat} (X : ZMod m) : Prop := ∃ (Y : ZMod m), X * Y = [1]_m def num_rp_below (m k : Nat) : Nat := match k with | 0 => 0 | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j def phi (m : Nat) : Nat := num_rp_below m m def prod_seq {m : Nat} (j k : Nat) (f : Nat → ZMod m) : ZMod m := match j with | 0 => [1]_m | n + 1 => prod_seq n k f * f (k + n) def maps_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i < n, g i < n def one_one_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i1 < n, ∀ i2 < n, g i1 = g i2 → i1 = i2 def onto_below (n : Nat) (g : Nat → Nat) : Prop := ∀ k < n, ∃ i < n, g i = k def perm_below (n : Nat) (g : Nat → Nat) : Prop := maps_below n g ∧ one_one_below n g ∧ onto_below n g def inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m) def swap (u v i : Nat) : Nat := if i = u then v else if i = v then u else i namespace Euler --For definitions specific to Euler's theorem def F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m def G (m a i : Nat) : Nat := (a * i) % m def Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i end Euler /- Section 7.1 -/ theorem dvd_mod_of_dvd_a_b {a b d : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ (a % b) := by set q : Nat := a / b have h3 : b * q + a % b = a := Nat.div_add_mod a b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 define --Goal : ∃ (c : Nat), a % b = d * c apply Exists.intro (j - k * q) show a % b = d * (j - k * q) from calc a % b _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm _ = a - b * q := by rw [h3] _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc] _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm done theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry #eval gcd 672 161 --Answer: 7 lemma gcd_base (a : Nat) : gcd a 0 = a := by rfl lemma gcd_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd a b = gcd b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1)) rfl done lemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b ≠ 0) : a % b < b := by have h1 : b > 0 := Nat.pos_of_ne_zero h show a % b < b from Nat.mod_lt a h1 done lemma dvd_self (n : Nat) : n ∣ n := by apply Exists.intro 1 ring done theorem gcd_dvd : ∀ (b a : Nat), (gcd a b) ∣ a ∧ (gcd a b) ∣ b := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd a b_1) ∣ a ∧ (gcd a b_1) ∣ b_1 fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_base] --Goal: a ∣ a ∧ a ∣ 0 apply And.intro (dvd_self a) define apply Exists.intro 0 rfl done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_nonzero a h1] --Goal : gcd b (a % b) ∣ a ∧ gcd b (a % b) ∣ b have h2 : a % b < b := mod_nonzero_lt a h1 have h3 : (gcd b (a % b)) ∣ b ∧ (gcd b (a % b)) ∣ (a % b) := ih (a % b) h2 b apply And.intro _ h3.left show (gcd b (a % b)) ∣ a from dvd_a_of_dvd_b_mod h3.left h3.right done done theorem gcd_dvd_left (a b : Nat) : (gcd a b) ∣ a := (gcd_dvd b a).left theorem gcd_dvd_right (a b : Nat) : (gcd a b) ∣ b := (gcd_dvd b a).right lemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl lemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c1 a b = gcd_c2 b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done lemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl lemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * ↑(a / b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done theorem gcd_lin_comb : ∀ (b a : Nat), (gcd_c1 a b) * ↑a + (gcd_c2 a b) * ↑b = ↑(gcd a b) := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd_c1 a b_1) * ↑a + (gcd_c2 a b_1) * ↑b_1 = ↑(gcd a b_1) fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base] --Goal : 1 * ↑a + 0 * ↑0 = ↑a ring done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1] --Goal : gcd_c2 b (a % b) * ↑a + -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * ↑(a / b)) * ↑b = -- ↑(gcd b (a % b)) set r : Nat := a % b set q : Nat := a / b set s : Int := gcd_c1 b r set t : Int := gcd_c2 b r --Goal : t * ↑a + (s - t * ↑q) * ↑b = ↑(gcd b r) have h2 : r < b := mod_nonzero_lt a h1 have h3 : s * ↑b + t * ↑r = ↑(gcd b r) := ih r h2 b have h4 : b * q + r = a := Nat.div_add_mod a b rewrite [←h3, ←h4] rewrite [Nat.cast_add, Nat.cast_mul] --Goal : t * (↑b * ↑q + ↑r) + (s - t * ↑q) * ↑b = s * ↑b + t * ↑r ring done done #eval gcd_c1 672 161 --Answer: 6 #eval gcd_c2 672 161 --Answer: -25 --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161 theorem Theorem_7_1_6 {d a b : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ gcd a b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑d ∣ ↑(gcd a b) set s : Int := gcd_c1 a b set t : Int := gcd_c2 a b have h3 : s * ↑a + t * ↑b = ↑(gcd a b) := gcd_lin_comb b a rewrite [←h3] --Goal : ↑d ∣ s * ↑a + t * ↑b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul] --Goal : ↑d ∣ s * (↑d * ↑j) + t * (↑d * ↑k) define apply Exists.intro (s * ↑j + t * ↑k) ring done /- Section 7.2 -/ theorem dvd_trans {a b c : Nat} (h1 : a ∣ b) (h2 : b ∣ c) : a ∣ c := by define at h1; define at h2; define obtain (m : Nat) (h3 : b = a * m) from h1 obtain (n : Nat) (h4 : c = b * n) from h2 rewrite [h3, mul_assoc] at h4 apply Exists.intro (m * n) show c = a * (m * n) from h4 done lemma exists_prime_factor : ∀ (n : Nat), 2 ≤ n → ∃ (p : Nat), prime_factor p n := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, 2 ≤ n_1 → ∃ (p : Nat), prime_factor p n_1 assume h1 : 2 ≤ n by_cases h2 : prime n · -- Case 1. h2 : prime n apply Exists.intro n define --Goal : prime n ∧ n ∣ n show prime n ∧ n ∣ n from And.intro h2 (dvd_self n) done · -- Case 2. h2 : ¬prime n define at h2 --h2 : ¬(2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n) demorgan at h2 disj_syll h2 h1 obtain (a : Nat) (h3 : ∃ (b : Nat), a * b = n ∧ a < n ∧ b < n) from h2 obtain (b : Nat) (h4 : a * b = n ∧ a < n ∧ b < n) from h3 have h5 : 2 ≤ a := by by_contra h6 have h7 : a ≤ 1 := by linarith have h8 : n ≤ b := calc n _ = a * b := h4.left.symm _ ≤ 1 * b := by rel [h7] _ = b := by ring linarith --n ≤ b contradicts b < n done have h6 : ∃ (p : Nat), prime_factor p a := ih a h4.right.left h5 obtain (p : Nat) (h7 : prime_factor p a) from h6 apply Exists.intro p define --Goal : prime p ∧ p ∣ n define at h7 --h7 : prime p ∧ p ∣ a apply And.intro h7.left have h8 : a ∣ n := by apply Exists.intro b show n = a * b from (h4.left).symm done show p ∣ n from dvd_trans h7.right h8 done done lemma exists_least_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q := by set S : Set Nat := {p : Nat | prime_factor p n} have h2 : ∃ (p : Nat), p ∈ S := exists_prime_factor n h show ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q from well_ord_princ S h2 done lemma all_prime_nil : all_prime [] := by define --Goal : ∀ p ∈ [], prime p fix p : Nat contrapos --Goal : ¬prime p → p ∉ [] assume h1 : ¬prime p show p ∉ [] from List.not_mem_nil p done lemma all_prime_cons (n : Nat) (L : List Nat) : all_prime (n :: L) ↔ prime n ∧ all_prime L := by apply Iff.intro · -- (→) assume h1 : all_prime (n :: L) --Goal : prime n ∧ all_prime L define at h1 --h1 : ∀ p ∈ n :: L, prime p apply And.intro (h1 n (List.mem_cons_self n L)) define --Goal : ∀ p ∈ L, prime p fix p : Nat assume h2 : p ∈ L show prime p from h1 p (List.mem_cons_of_mem n h2) done · -- (←) assume h1 : prime n ∧ all_prime L --Goal : all_prime (n :: l) define : all_prime L at h1 define fix p : Nat assume h2 : p ∈ n :: L rewrite [List.mem_cons] at h2 --h2 : p = n ∨ p ∈ L by_cases on h2 · -- Case 1. h2 : p = n rewrite [h2] show prime n from h1.left done · -- Case 2. h2 : p ∈ L show prime p from h1.right p h2 done done done lemma nondec_nil : nondec [] := by define --Goal : True trivial --trivial proves some obviously true statements, such as True done lemma nondec_cons (n : Nat) (L : List Nat) : nondec (n :: L) ↔ (∀ m ∈ L, n ≤ m) ∧ nondec L := by rfl lemma prod_nil : prod [] = 1 := by rfl lemma prod_cons : prod (n :: L) = n * (prod L) := by rfl lemma exists_cons_of_length_eq_succ {A : Type} {l : List A} {n : Nat} (h : l.length = n + 1) : ∃ (a : A) (L : List A), l = a :: L ∧ L.length = n := by have h1 : ¬l.length = 0 := by linarith rewrite [List.length_eq_zero] at h1 obtain (a : A) (h2 : ∃ (L : List A), l = a :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List A) (h3 : l = a :: L) from h2 apply Exists.intro a apply Exists.intro L apply And.intro h3 have h4 : (a :: L).length = L.length + 1 := List.length_cons a L rewrite [←h3, h] at h4 show L.length = n from (Nat.add_right_cancel h4).symm done lemma list_elt_dvd_prod_by_length (a : Nat) : ∀ (n : Nat), ∀ (l : List Nat), l.length = n → a ∈ l → a ∣ prod l := by by_induc · --Base Case fix l : List Nat assume h1 : l.length = 0 rewrite [List.length_eq_zero] at h1 --h1 : l = [] rewrite [h1] --Goal : a ∈ [] → a ∣ prod [] contrapos assume h2 : ¬a ∣ prod [] show a ∉ [] from List.not_mem_nil a done · -- Induction Step fix n : Nat assume ih : ∀ (l : List Nat), List.length l = n → a ∈ l → a ∣ prod l fix l : List Nat assume h1 : l.length = n + 1 --Goal : a ∈ l → a ∣ prod l obtain (b : Nat) (h2 : ∃ (L : List Nat), l = b :: L ∧ L.length = n) from exists_cons_of_length_eq_succ h1 obtain (L : List Nat) (h3 : l = b :: L ∧ L.length = n) from h2 have h4 : a ∈ L → a ∣ prod L := ih L h3.right assume h5 : a ∈ l rewrite [h3.left, prod_cons] --Goal : a ∣ b * prod L rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b ∨ a ∈ L by_cases on h5 · -- Case 1. h5 : a = b apply Exists.intro (prod L) rewrite [h5] rfl done · -- Case 2. h5 : a ∈ L have h6 : a ∣ prod L := h4 h5 have h7 : prod L ∣ b * prod L := by apply Exists.intro b ring done show a ∣ b * prod L from dvd_trans h6 h7 done done done lemma list_elt_dvd_prod {a : Nat} {l : List Nat} (h : a ∈ l) : a ∣ prod l := by set n : Nat := l.length have h1 : l.length = n := by rfl show a ∣ prod l from list_elt_dvd_prod_by_length a n l h1 h done lemma exists_prime_factorization : ∀ (n : Nat), n ≥ 1 → ∃ (l : List Nat), prime_factorization n l := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, n_1 ≥ 1 → ∃ (l : List Nat), prime_factorization n_1 l assume h1 : n ≥ 1 by_cases h2 : n = 1 · -- Case 1. h2 : n = 1 apply Exists.intro [] define apply And.intro · -- Proof of nondec_prime_list [] define show all_prime [] ∧ nondec [] from And.intro all_prime_nil nondec_nil done · -- Proof of prod [] = n rewrite [prod_nil, h2] rfl done done · -- Case 2. h2 : n ≠ 1 have h3 : n ≥ 2 := lt_of_le_of_ne' h1 h2 obtain (p : Nat) (h4 : prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q) from exists_least_prime_factor h3 have p_prime_factor : prime_factor p n := h4.left define at p_prime_factor have p_prime : prime p := p_prime_factor.left have p_dvd_n : p ∣ n := p_prime_factor.right have p_least : ∀ (q : Nat), prime_factor q n → p ≤ q := h4.right obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n have h5 : m ≠ 0 := by contradict h1 with h6 have h7 : n = 0 := calc n _ = p * m := n_eq_pm _ = p * 0 := by rw [h6] _ = 0 := by ring rewrite [h7] decide done have m_pos : 0 < m := Nat.pos_of_ne_zero h5 have m_lt_n : m < n := by define at p_prime show m < n from calc m _ < m + m := by linarith _ = 2 * m := by ring _ ≤ p * m := by rel [p_prime.left] _ = n := n_eq_pm.symm done obtain (L : List Nat) (h6 : prime_factorization m L) from ih m m_lt_n m_pos define at h6 have ndpl_L : nondec_prime_list L := h6.left define at ndpl_L apply Exists.intro (p :: L) define apply And.intro · -- Proof of nondec_prime_list (p :: L) define apply And.intro · -- Proof of all_prime (p :: L) rewrite [all_prime_cons] show prime p ∧ all_prime L from And.intro p_prime ndpl_L.left done · -- Proof of nondec (p :: L) rewrite [nondec_cons] apply And.intro _ ndpl_L.right fix q : Nat assume q_in_L : q ∈ L have h7 : q ∣ prod L := list_elt_dvd_prod q_in_L rewrite [h6.right] at h7 --h7 : q ∣ m have h8 : m ∣ n := by apply Exists.intro p rewrite [n_eq_pm] ring done have q_dvd_n : q ∣ n := dvd_trans h7 h8 have ap_L : all_prime L := ndpl_L.left define at ap_L have q_prime_factor : prime_factor q n := And.intro (ap_L q q_in_L) q_dvd_n show p ≤ q from p_least q q_prime_factor done done · -- Proof of prod (p :: L) = n rewrite [prod_cons, h6.right, n_eq_pm] rfl done done done theorem Theorem_7_2_2 {a b c : Nat} (h1 : c ∣ a * b) (h2 : rel_prime a c) : c ∣ b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑c ∣ ↑b define at h1; define at h2; define obtain (j : Nat) (h3 : a * b = c * j) from h1 set s : Int := gcd_c1 a c set t : Int := gcd_c2 a c have h4 : s * ↑a + t * ↑c = ↑(gcd a c) := gcd_lin_comb c a rewrite [h2, Nat.cast_one] at h4 --h4 : s * ↑a + t * ↑c = (1 : Int) apply Exists.intro (s * ↑j + t * ↑b) show ↑b = ↑c * (s * ↑j + t * ↑b) from calc ↑b _ = (1 : Int) * ↑b := (one_mul _).symm _ = (s * ↑a + t * ↑c) * ↑b := by rw [h4] _ = s * (↑a * ↑b) + t * ↑c * ↑b := by ring _ = s * (↑c * ↑j) + t * ↑c * ↑b := by rw [←Nat.cast_mul a b, h3, Nat.cast_mul c j] _ = ↑c * (s * ↑j + t * ↑b) := by ring done lemma le_nonzero_prod_left {a b : Nat} (h : a * b ≠ 0) : a ≤ a * b := by have h1 : b ≠ 0 := by contradict h with h1 rewrite [h1] ring done have h2 : 1 ≤ b := Nat.pos_of_ne_zero h1 show a ≤ a * b from calc a = a * 1 := (mul_one a).symm _ ≤ a * b := by rel [h2] done lemma le_nonzero_prod_right {a b : Nat} (h : a * b ≠ 0) : b ≤ a * b := by rewrite [mul_comm] rewrite [mul_comm] at h show b ≤ b * a from le_nonzero_prod_left h done lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry lemma rel_prime_of_prime_not_dvd {a p : Nat} (h1 : prime p) (h2 : ¬p ∣ a) : rel_prime a p := by have h3 : gcd a p ∣ a := gcd_dvd_left a p have h4 : gcd a p ∣ p := gcd_dvd_right a p have h5 : gcd a p = 1 ∨ gcd a p = p := dvd_prime h1 h4 have h6 : gcd a p ≠ p := by contradict h2 with h6 rewrite [h6] at h3 show p ∣ a from h3 done disj_syll h5 h6 show rel_prime a p from h5 done theorem Theorem_7_2_3 {a b p : Nat} (h1 : prime p) (h2 : p ∣ a * b) : p ∣ a ∨ p ∣ b := by or_right with h3 have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3 show p ∣ b from Theorem_7_2_2 h2 h4 done lemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a ≥ 1 := by have h1 : a ≠ 0 := by by_contra h1 rewrite [h1] at h contradict h linarith done show a ≥ 1 from Nat.pos_of_ne_zero h1 done lemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by have h1 : a ≥ 1 := ge_one_of_prod_one h have h2 : a * b ≠ 0 := by linarith have h3 : a ≤ a * b := le_nonzero_prod_left h2 rewrite [h] at h3 show a = 1 from Nat.le_antisymm h3 h1 done lemma eq_one_of_dvd_one {n : Nat} (h : n ∣ 1) : n = 1 := by obtain (j : Nat) (h1 : 1 = n * j) from h show n = 1 from eq_one_of_prod_one h1.symm done lemma prime_not_one {p : Nat} (h : prime p) : p ≠ 1 := by define at h linarith done theorem Theorem_7_2_4 {p : Nat} (h1 : prime p) : ∀ (l : List Nat), p ∣ prod l → ∃ a ∈ l, p ∣ a := by apply List.rec · -- Base Case. Goal : p ∣ prod [] → ∃ a ∈ [], p ∣ a rewrite [prod_nil] assume h2 : p ∣ 1 show ∃ a ∈ [], p ∣ a from absurd (eq_one_of_dvd_one h2) (prime_not_one h1) done · -- Induction Step fix b : Nat fix L : List Nat assume ih : p ∣ prod L → ∃ a ∈ L, p ∣ a --Goal : p ∣ prod (b :: L) → ∃ a ∈ b :: L, p ∣ a assume h2 : p ∣ prod (b :: L) rewrite [prod_cons] at h2 have h3 : p ∣ b ∨ p ∣ prod L := Theorem_7_2_3 h1 h2 by_cases on h3 · -- Case 1. h3 : p ∣ b apply Exists.intro b show b ∈ b :: L ∧ p ∣ b from And.intro (List.mem_cons_self b L) h3 done · -- Case 2. h3 : p ∣ prod L obtain (a : Nat) (h4 : a ∈ L ∧ p ∣ a) from ih h3 apply Exists.intro a show a ∈ b :: L ∧ p ∣ a from And.intro (List.mem_cons_of_mem b h4.left) h4.right done done done lemma prime_in_list {p : Nat} {l : List Nat} (h1 : prime p) (h2 : all_prime l) (h3 : p ∣ prod l) : p ∈ l := by obtain (a : Nat) (h4 : a ∈ l ∧ p ∣ a) from Theorem_7_2_4 h1 l h3 define at h2 have h5 : prime a := h2 a h4.left have h6 : p = 1 ∨ p = a := dvd_prime h5 h4.right disj_syll h6 (prime_not_one h1) rewrite [h6] show a ∈ l from h4.left done lemma first_le_first {p q : Nat} {l m : List Nat} (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m)) (h3 : prod (p :: l) = prod (q :: m)) : p ≤ q := by define at h1; define at h2 have h4 : q ∣ prod (p :: l) := by define apply Exists.intro (prod m) rewrite [←prod_cons] show prod (p :: l) = prod (q :: m) from h3 done have h5 : all_prime (q :: m) := h2.left rewrite [all_prime_cons] at h5 have h6 : q ∈ p :: l := prime_in_list h5.left h1.left h4 have h7 : nondec (p :: l) := h1.right rewrite [nondec_cons] at h7 rewrite [List.mem_cons] at h6 by_cases on h6 · -- Case 1. h6 : q = p linarith done · -- Case 2. h6 : q ∈ l have h8 : ∀ m ∈ l, p ≤ m := h7.left show p ≤ q from h8 q h6 done done lemma nondec_prime_list_tail {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by define at h define rewrite [all_prime_cons, nondec_cons] at h show all_prime l ∧ nondec l from And.intro h.left.right h.right.right done lemma cons_prod_not_one {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : prod (p :: l) ≠ 1 := by define at h have h1 : all_prime (p :: l) := h.left rewrite [all_prime_cons] at h1 rewrite [prod_cons] by_contra h2 show False from (prime_not_one h1.left) (eq_one_of_prod_one h2) done lemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) : l = [] ↔ prod l = 1 := by apply Iff.intro · -- (→) assume h1 : l = [] rewrite [h1] show prod [] = 1 from prod_nil done · -- (←) contrapos assume h1 : ¬l = [] obtain (p : Nat) (h2 : ∃ (L : List Nat), l = p :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List Nat) (h3 : l = p :: L) from h2 rewrite [h3] at h rewrite [h3] show ¬prod (p :: L) = 1 from cons_prod_not_one h done done lemma prime_pos {p : Nat} (h : prime p) : p > 0 := by define at h linarith done theorem Theorem_7_2_5 : ∀ (l1 l2 : List Nat), nondec_prime_list l1 → nondec_prime_list l2 → prod l1 = prod l2 → l1 = l2 := by apply List.rec · -- Base Case. Goal : ∀ (l2 : List Nat), nondec_prime_list [] → -- nondec_prime_list l2 → prod [] = prod l2 → [] = l2 fix l2 : List Nat assume h1 : nondec_prime_list [] assume h2 : nondec_prime_list l2 assume h3 : prod [] = prod l2 rewrite [prod_nil, eq_comm, ←list_nil_iff_prod_one h2] at h3 show [] = l2 from h3.symm done · -- Induction Step fix p : Nat fix L1 : List Nat assume ih : ∀ (L2 : List Nat), nondec_prime_list L1 → nondec_prime_list L2 → prod L1 = prod L2 → L1 = L2 -- Goal : ∀ (l2 : List Nat), nondec_prime_list (p :: L1) → -- nondec_prime_list l2 → prod (p :: L1) = prod l2 → p :: L1 = l2 fix l2 : List Nat assume h1 : nondec_prime_list (p :: L1) assume h2 : nondec_prime_list l2 assume h3 : prod (p :: L1) = prod l2 have h4 : ¬prod (p :: L1) = 1 := cons_prod_not_one h1 rewrite [h3, ←list_nil_iff_prod_one h2] at h4 obtain (q : Nat) (h5 : ∃ (L : List Nat), l2 = q :: L) from List.exists_cons_of_ne_nil h4 obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5 rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2) rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2) have h7 : p ≤ q := first_le_first h1 h2 h3 have h8 : q ≤ p := first_le_first h2 h1 h3.symm have h9 : p = q := by linarith rewrite [h9, prod_cons, prod_cons] at h3 --h3 : q * prod L1 = q * prod L2 have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1 have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2 define at h2 have h12 : all_prime (q :: L2) := h2.left rewrite [all_prime_cons] at h12 have h13 : q > 0 := prime_pos h12.left have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3 have h15 : L1 = L2 := ih L2 h10 h11 h14 rewrite [h6, h9, h15] rfl done done theorem fund_thm_arith (n : Nat) (h : n ≥ 1) : ∃! (l : List Nat), prime_factorization n l := by exists_unique · -- Existence show ∃ (l : List Nat), prime_factorization n l from exists_prime_factorization n h done · -- Uniqueness fix l1 : List Nat; fix l2 : List Nat assume h1 : prime_factorization n l1 assume h2 : prime_factorization n l2 define at h1; define at h2 have h3 : prod l1 = n := h1.right rewrite [←h2.right] at h3 show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3 done done /- Section 7.3 -/ theorem congr_refl (m : Nat) : ∀ (a : Int), a ≡ a (MOD m) := by fix a : Int define --Goal : ∃ (c : Int), a - a = ↑m * c apply Exists.intro 0 ring done theorem congr_symm {m : Nat} : ∀ {a b : Int}, a ≡ b (MOD m) → b ≡ a (MOD m) := by fix a : Int; fix b : Int assume h1 : a ≡ b (MOD m) define at h1 --h1 : ∃ (c : Int), a - b = ↑m * c define --Goal : ∃ (c : Int), b - a = ↑m * c obtain (c : Int) (h2 : a - b = m * c) from h1 apply Exists.intro (-c) show b - a = m * (-c) from calc b - a _ = -(a - b) := by ring _ = -(m * c) := by rw [h2] _ = m * (-c) := by ring done theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry /- Fundamental properties of congruence classes -/ lemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) : X = Y ↔ X.val = Y.val := Fin.ext_iff lemma val_nat_eq_mod (n k : Nat) : ([k]_(n + 1)).val = k % (n + 1) := by rfl lemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl theorem cc_rep {m : Nat} (X : ZMod m) : ∃ (a : Int), X = [a]_m := match m with | 0 => by apply Exists.intro X rfl done | n + 1 => by apply Exists.intro ↑(X.val) have h1 : X.val < n + 1 := Fin.prop X rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1] rfl done theorem add_class (m : Nat) (a b : Int) : [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm theorem mul_class (m : Nat) (a b : Int) : [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm lemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ [a - b]_m = [0]_m := by apply Iff.intro · -- (→) assume h1 : [a]_m = [b]_m have h2 : a - b = a + (-b) := by ring have h3 : b + (-b) = 0 := by ring show [a - b]_m = [0]_m from calc [a - b]_m _ = [a + (-b)]_m := by rw [h2] _ = [a]_m + [-b]_m := by rw [add_class] _ = [b]_m + [-b]_m := by rw [h1] _ = [b + -b]_m := by rw [add_class] _ = [0]_m := by rw [h3] done · -- (←) assume h1 : [a - b]_m = [0]_m have h2 : b + (a - b) = a := by ring have h3 : b + 0 = b := by ring show [a]_m = [b]_m from calc [a]_m _ = [b + (a - b)]_m := by rw [h2] _ = [b]_m + [a - b]_m := by rw [add_class] _ = [b]_m + [0]_m := by rw [h1] _ = [b + 0]_m := by rw [add_class] _ = [b]_m := by rw [h3] done done lemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) : [a]_m = [0]_m → [-a]_m = [0]_m := by assume h1 : [a]_m = [0]_m have h2 : 0 + (-a) = -a := by ring have h3 : a + (-a) = 0 := by ring show [-a]_m = [0]_m from calc [-a]_m _ = [0 + (-a)]_m := by rw [h2] _ = [0]_m + [-a]_m := by rw [add_class] _ = [a]_m + [-a]_m := by rw [h1] _ = [a + (-a)]_m := by rw [add_class] _ = [0]_m := by rw [h3] done lemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) : [-a]_m = [0]_m ↔ [a]_m = [0]_m := by apply Iff.intro _ (cc_neg_zero_of_cc_zero m a) assume h1 : [-a]_m = [0]_m have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1 have h3 : -(-a) = a := by ring rewrite [h3] at h2 show [a]_m = [0]_m from h2 done lemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl lemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m ↔ m ∣ k := match m with | 0 => by have h : (0 : Int) = (↑(0 : Nat) : Int) := by rfl rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj] apply Iff.intro · -- (→) assume h1 : k = 0 rewrite [h1] show 0 ∣ 0 from dvd_self 0 done · -- (←) assume h1 : 0 ∣ k obtain (c : Nat) (h2 : k = 0 * c) from h1 rewrite [h2] ring done done | n + 1 => by rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero] show k % (n + 1) = 0 ↔ n + 1 ∣ k from (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm done lemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m ↔ ↑m ∣ a := by obtain (k : Nat) (h1 : a = ↑k ∨ a = -↑k) from Int.eq_nat_or_neg a by_cases on h1 · -- Case 1. h1: a = ↑k rewrite [h1, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done · -- Case 2. h1: a = -↑k rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done done theorem cc_eq_iff_congr (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ a ≡ b (MOD m) := calc [a]_m = [b]_m _ ↔ [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b _ ↔ ↑m ∣ (a - b) := cc_zero_iff_dvd m (a - b) _ ↔ a ≡ b (MOD m) := by rfl /- End of fundamental properties of congruence classes -/ lemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m := by have h1 : (↑m : Int) ≠ 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m) show 0 ≤ a % m from Int.emod_nonneg a h1 done lemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m) have h2 : (↑m : Int) > 0 := (Nat.cast_pos).rtl h1 show a % m < m from Int.emod_lt_of_pos a h2 done lemma congr_mod_mod (m : Nat) (a : Int) : a ≡ a % m (MOD m) := by define have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m apply Exists.intro (a / m) show a - a % m = m * (a / m) from calc a - (a % m) _ = m * (a / m) + a % m - a % m := by rw [h1] _ = m * (a / m) := by ring done lemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) := And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a)) theorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) : ∃! (r : Int), 0 ≤ r ∧ r < m ∧ a ≡ r (MOD m) := by exists_unique · -- Existence apply Exists.intro (a % m) show 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) from mod_cmpl_res m a done · -- Uniqueness fix r1 : Int; fix r2 : Int assume h1 : 0 ≤ r1 ∧ r1 < m ∧ a ≡ r1 (MOD m) assume h2 : 0 ≤ r2 ∧ r2 < m ∧ a ≡ r2 (MOD m) have h3 : r1 ≡ r2 (MOD m) := congr_trans (congr_symm h1.right.right) h2.right.right obtain (d : Int) (h4 : r1 - r2 = m * d) from h3 have h5 : r1 - r2 < m * 1 := by linarith have h6 : m * (-1) < r1 - r2 := by linarith rewrite [h4] at h5 --h5 : m * d < m * 1 rewrite [h4] at h6 --h6 : m * -1 < m * d have h7 : (↑m : Int) ≥ 0 := Nat.cast_nonneg m have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7 have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7 have h10 : d = 0 := by linarith show r1 = r2 from calc r1 _ = r1 - r2 + r2 := by ring _ = m * 0 + r2 := by rw [h4, h10] _ = r2 := by ring done done lemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m := (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a) theorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y rewrite [h1, h2] have h3 : a + b = b + a := by ring show [a]_m + [b]_m = [b]_m + [a]_m from calc [a]_m + [b]_m _ = [a + b]_m := add_class m a b _ = [b + a]_m := by rw [h3] _ = [b]_m + [a]_m := (add_class m b a).symm done theorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X rewrite [h1] have h2 : a * 1 = a := by ring show [a]_m * [1]_m = [a]_m from calc [a]_m * [1]_m _ = [a * 1]_m := mul_class m a 1 _ = [a]_m := by rw [h2] done theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry lemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) : [a]_m * [gcd_c2 m a]_m = [1]_m := by set s : Int := gcd_c1 m a have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m define at h1 rewrite [h1, Nat.cast_one] at h2 --h2 : s * ↑m + gcd_c2 m a * ↑a = 1 rewrite [mul_class, cc_eq_iff_congr] define --Goal : ∃ (c : Int), ↑a * gcd_c2 m a - 1 = ↑m * c apply Exists.intro (-s) show a * (gcd_c2 m a) - 1 = m * (-s) from calc a * (gcd_c2 m a) - 1 _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring _ = 1 + m * (-s) - 1 := by rw [h2] _ = m * (-s) := by ring done theorem Theorem_7_3_7 (m a : Nat) : invertible [a]_m ↔ rel_prime m a := by apply Iff.intro · -- (→) assume h1 : invertible [a]_m define at h1 obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1 obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y rewrite [h3, mul_class, cc_eq_iff_congr] at h2 define at h2 obtain (c : Int) (h4 : a * b - 1 = m * c) from h2 rewrite [Exercise_7_2_6] --Goal : ∃ (s t : Int), s * ↑m + t * ↑a = 1 apply Exists.intro (-c) apply Exists.intro b show (-c) * m + b * a = 1 from calc (-c) * m + b * a _ = (-c) * m + (a * b - 1) + 1 := by ring _ = (-c) * m + m * c + 1 := by rw [h4] _ = 1 := by ring done · -- (←) assume h1 : rel_prime m a define show ∃ (Y : ZMod m), [a]_m * Y = [1]_m from Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1) done done /- Section 7.4 -/ section Euler open Euler lemma num_rp_below_base {m : Nat} : num_rp_below m 0 = 0 := by rfl lemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) : num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by have h1 : num_rp_below m (j + 1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : gcd m j = 1 rewrite [if_pos h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1 show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1 done lemma num_rp_below_step_not_rp {m j : Nat} (h : ¬rel_prime m j) : num_rp_below m (j + 1) = num_rp_below m j := by have h1 : num_rp_below m (j +1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : ¬gcd m j = 1 rewrite [if_neg h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j show num_rp_below m (j + 1) = num_rp_below m j from h1 done lemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl #eval phi 10 --Answer: 4 lemma prod_inv_iff_inv {m : Nat} {X : ZMod m} (h1 : invertible X) (Y : ZMod m) : invertible (X * Y) ↔ invertible Y := by apply Iff.intro · -- (→) assume h2 : invertible (X * Y) obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2 apply Exists.intro (X * Z) rewrite [←h3] --Goal : Y * (X * Z) = X * Y * Z ring --Note that ring can do algebra in ZMod m done · -- (←) assume h2 : invertible Y obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1 obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2 apply Exists.intro (Xi * Yi) show (X * Y) * (Xi * Yi) = [1]_m from calc X * Y * (Xi * Yi) _ = (X * Xi) * (Y * Yi) := by ring _ = [1]_m * [1]_m := by rw [h3, h4] _ = [1]_m := Theorem_7_3_6_7 [1]_m done done lemma F_rp_def {m i : Nat} (h : rel_prime m i) : F m i = [i]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h --h : gcd m i = 1 rewrite [if_pos h] at h1 show F m i = [i]_m from h1 done lemma F_not_rp_def {m i : Nat} (h : ¬rel_prime m i) : F m i = [1]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h rewrite [h1, if_neg h] rfl done lemma prod_seq_base {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 0 k f = [1]_m := by rfl lemma prod_seq_step {m : Nat} (n k : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl lemma prod_seq_zero_step {m : Nat} (n : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by rewrite [prod_seq_step, zero_add] rfl done lemma prod_one {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 1 k f = f k := by rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7] rfl done lemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl lemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m := calc [G m a i]_m _ = [(a * i) % m]_m := by rfl _ = [a * i]_m := (cc_eq_mod m (a * i)).symm _ = [a]_m * [i]_m := (mul_class m a i).symm lemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) : rel_prime m (G m a i) ↔ rel_prime m i := by have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1 show rel_prime m (G m a i) ↔ rel_prime m i from calc rel_prime m (G m a i) _ ↔ invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm _ ↔ invertible ([a]_m * [i]_m) := by rw [cc_G] _ ↔ invertible [i]_m := prod_inv_iff_inv h2 ([i]_m) _ ↔ rel_prime m i := Theorem_7_3_7 m i done lemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) : F m (G m a i) = [a]_m * F m i := by have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2 show F m (G m a i) = [a]_m * F m i from calc F m (G m a i) _ = [G m a i]_m := F_rp_def h3 _ = [a]_m * [i]_m := cc_G m a i _ = [a]_m * F m i := by rw [F_rp_def h2] done lemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : ¬rel_prime m i) : F m (G m a i) = [1]_m := by rewrite [←G_rp_iff h1 i] at h2 show F m (G m a i) = [1]_m from F_not_rp_def h2 done lemma FG_prod {m a : Nat} (h1 : rel_prime m a) : ∀ (k : Nat), prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by by_induc · -- Base Case show prod_seq 0 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from calc prod_seq 0 0 ((F m) ∘ (G m a)) _ = [1]_m := prod_seq_base _ _ _ = [a]_m ^ 0 * [1]_m := by ring _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by rw [num_rp_below_base, prod_seq_base] done · -- Induction Step fix k : Nat assume ih : prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) by_cases h2 : rel_prime m k · -- Case 1. h2 : rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([a]_m * F m k) := by rw [FG_rp h1 h2] _ = [a]_m ^ ((num_rp_below m k) + 1) * ((prod_seq k 0 (F m)) * F m k) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_rp h2, prod_seq_zero_step] done · -- Case 2. h2 : ¬rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([1]_m) := by rw [FG_not_rp h1 h2] _ = [a]_m ^ (num_rp_below m k) * (prod_seq k 0 (F m) * ([1]_m)) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_not_rp h2, prod_seq_zero_step, F_not_rp_def h2] done done done lemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by define --Goal : ∀ i < m, G m a i < m fix i : Nat assume h1 : i < m rewrite [G_def] --Goal : a * i % m < m show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m) done lemma left_inv_one_one_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g' (g i) = i) : one_one_below n g := sorry lemma right_inv_onto_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g (g' i) = i) (h2 : maps_below n g') : onto_below n g := by define at h2; define fix k : Nat assume h3 : k < n apply Exists.intro (g' k) show g' k < n ∧ g (g' k) = k from And.intro (h2 k h3) (h1 k h3) done lemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m * [inv_mod m a]_m = [1]_m := by have h2 : 0 ≤ (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a) show [a]_m * [inv_mod m a]_m = [1]_m from calc [a]_m * [inv_mod m a]_m _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2] _ = [a]_m * [gcd_c2 m a]_m := by rw [←cc_eq_mod] _ = [1]_m := gcd_c2_inv h1 done lemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m := calc a * (b % m) % m = a % m * (b % m % m) % m := Nat.mul_mod _ _ _ _ = a % m * (b % m) % m := by rw [Nat.mod_mod] _ = a * b % m := (Nat.mul_mod _ _ _).symm lemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm] rfl done theorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] : ↑a ≡ ↑b (MOD m) ↔ a % m = b % m := sorry lemma mul_inv_mod_cancel {m a i : Nat} [NeZero m] (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1 rewrite [mul_class, cc_eq_iff_congr, ←Nat.cast_mul, ←Nat.cast_one, congr_iff_mod_eq_Nat] at h3 show a * inv_mod m a * i % m = i from calc a * (inv_mod m a) * i % m _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = 1 % m * i % m := by rw [h3] _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = i % m := by rw [one_mul] _ = i := Nat.mod_eq_of_lt h2 done lemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl lemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, G m a (Ginv m a i) = i := by fix i : Nat assume h2 : i < m show G m a (Ginv m a i) = i from calc G m a (Ginv m a i) _ = a * ((inv_mod m a * i) % m) % m := by rfl _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, Ginv m a (G m a i) = i := by fix i : Nat assume h2 : i < m show Ginv m a (G m a i) = i from calc Ginv m a (G m a i) _ = inv_mod m a * ((a * i) % m) % m := by rfl _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc, mul_comm (inv_mod m a)] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_maps_below (m a : Nat) [NeZero m] : maps_below m (Ginv m a) := G_maps_below m (inv_mod m a) lemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : one_one_below m (G m a) := left_inv_one_one_below (Ginv_left_inv h1) lemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : onto_below m (G m a) := right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a) lemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : perm_below m (G m a) := And.intro (G_maps_below m a) (And.intro (G_one_one_below h1) (G_onto_below h1)) --Permuting a product of congruence classes doesn't change product lemma swap_fst (u v : Nat) : swap u v u = v := by define : swap u v u --Goal : (if u = u then v else if u = v then u else u) = v have h : u = u := by rfl rewrite [if_pos h] rfl done lemma swap_snd (u v : Nat) : swap u v v = u := by define : swap u v v by_cases h1 : v = u · -- Case 1. h1 : v = u rewrite [if_pos h1] show v = u from h1 done · -- Case 2. h1 : v ≠ u rewrite [if_neg h1] have h2 : v = v := by rfl rewrite [if_pos h2] rfl done done lemma swap_other {u v i : Nat} (h1 : i ≠ u) (h2 : i ≠ v) : swap u v i = i := by define : swap u v i rewrite [if_neg h1, if_neg h2] rfl done lemma swap_values (u v i : Nat) : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := by by_cases h1 : i = u · -- Case 1. h1 : i = u apply Or.inl rewrite [h1] show swap u v u = v from swap_fst u v done · -- Case 2. h1 : i ≠ u apply Or.inr by_cases h2 : i = v · -- Case 2.1. h2 : i = v apply Or.inl rewrite [h2] show swap u v v = u from swap_snd u v done · -- Case 2.2. h2 : i ≠ v apply Or.inr show swap u v i = i from swap_other h1 h2 done done done lemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by define fix i : Nat assume h3 : i < n have h4 : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := swap_values u v i by_cases on h4 · -- Case 1. h4 : swap u v i = v rewrite [h4] show v < n from h2 done · -- Case 2. by_cases on h4 · -- Case 2.1. h4 : swap u v i = u rewrite [h4] show u < n from h1 done · -- Case 2.2. h4 : swap u v i = i rewrite [h4] show i < n from h3 done done done lemma swap_swap (u v n : Nat) : ∀ i < n, swap u v (swap u v i) = i := by fix i : Nat assume h : i < n by_cases h1 : i = u · -- Case 1. h1 : i = u rewrite [h1, swap_fst, swap_snd] rfl done · -- Case 2. h1 : i ≠ u by_cases h2 : i = v · -- Case 2.1. h2 : i = v rewrite [h2, swap_snd, swap_fst] rfl done · -- Case 2.2. h2 : i ≠ v rewrite [swap_other h1 h2, swap_other h1 h2] rfl done done done lemma swap_one_one_below (u v n) : one_one_below n (swap u v) := left_inv_one_one_below (swap_swap u v n) lemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) := right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2) lemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) := And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2)) lemma comp_perm_below {n : Nat} {f g : Nat → Nat} (h1 : perm_below n f) (h2 : perm_below n g) : perm_below n (f ∘ g) := sorry lemma trivial_swap (u : Nat) : swap u u = id := by apply funext fix x : Nat by_cases h1 : x = u · -- Case 1. h1 : x = u rewrite [h1, swap_fst] rfl done · -- Case 2. h1 : x ≠ u rewrite [swap_other h1 h1] rfl done done lemma prod_eq_fun {m : Nat} (f g : Nat → ZMod m) (k : Nat) : ∀ (n : Nat), (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g := by by_induc · -- Base Case assume h : (∀ i < 0, f (k + i) = g (k + i)) rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g assume h1 : ∀ i < n + 1, f (k + i) = g (k + i) have h2 : ∀ i < n, f (k + i) = g (k + i) := by fix i : Nat assume h2 : i < n have h3 : i < n + 1 := by linarith show f (k + i) = g (k + i) from h1 i h3 done have h3 : prod_seq n k f = prod_seq n k g := ih h2 have h4 : n < n + 1 := Nat.lt_succ_self n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4] rfl done done lemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := by have h2 : ∀ (i : Nat), i < u → (f ∘ swap u n) (0 + i) = f (0 + i) := by fix i : Nat assume h2 : i < u have h3 : 0 + i ≠ u := by linarith have h4 : 0 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl done show prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f from prod_eq_fun (f ∘ swap u n) f 0 u h2 done lemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := by have h2 : ∀ i < j, (f ∘ swap u n) (u + 1 + i) = f (u + 1 + i) := by fix i : Nat assume h2 : i < j have h3 : u + 1 + i ≠ u := by linarith have h4 : u + 1 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl show prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f from prod_eq_fun (f ∘ swap u n) f (u + 1) j h2 done lemma break_prod {m : Nat} (n : Nat) (f : Nat → ZMod m) : ∀ (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by by_induc · -- Base Case have h : n + 0 = n := by rfl rewrite [prod_seq_base, h, Theorem_7_3_6_7] rfl done · -- Induction Step fix j : Nat assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f rewrite [←add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc] rfl done done lemma break_prod_twice {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f := break_prod n f 1 rewrite [prod_one] at h2 have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f := break_prod (u + 1) f j rewrite [←h1] at h3 have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f := break_prod u f 1 rewrite [prod_one] at h4 rewrite [h3, h4] at h2 show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2 done lemma swap_prod_eq_prod {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f := by by_cases h2 : u = n · -- Case 1. h2 : u = n rewrite [h2, trivial_swap n] --Goal : prod_seq (n + 1) 0 (f ∘ id) = prod_seq (n + 1) 0 f rfl done · -- Case 2. h2 : ¬u = n have h3 : u + 1 ≤ n := Nat.lt_of_le_of_ne h1 h2 obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3 have break_f : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := break_prod_twice f h4 have break_fs : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_prod_twice (f ∘ swap u n) h4 have f_eq_fs_below : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := swap_prod_eq_prod_below f h1 have f_eq_fs_btwn : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4 show prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f from calc prod_seq (n + 1) 0 (f ∘ swap u n) _ = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_fs _ = prod_seq u 0 f * (f ∘ swap u n) u * prod_seq j (u + 1) f * (f ∘ swap u n) n := by rw [f_eq_fs_below, f_eq_fs_btwn] _ = prod_seq u 0 f * f (swap u n u) * prod_seq j (u + 1) f * f (swap u n n) := by rfl _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by rw [swap_fst, swap_snd] _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring _ = prod_seq (n + 1) 0 f := break_f.symm done done lemma perm_below_fixed {n : Nat} {g : Nat → Nat} (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry lemma perm_prod {m : Nat} (f : Nat → ZMod m) : ∀ (n : Nat), ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) := by by_induc · -- Base Case fix g : Nat → Nat assume h1 : perm_below 0 g rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) fix g : Nat → Nat assume g_pb : perm_below (n + 1) g define at g_pb have g_ob : onto_below (n + 1) g := g_pb.right.right define at g_ob have h1 : n < n + 1 := by linarith obtain (u : Nat) (h2 : u < n + 1 ∧ g u = n) from g_ob n h1 have s_pb : perm_below (n + 1) (swap u n) := swap_perm_below h2.left h1 have gs_pb_n1 : perm_below (n + 1) (g ∘ swap u n) := comp_perm_below g_pb s_pb have gs_fix_n : (g ∘ swap u n) n = n := calc (g ∘ swap u n) n _ = g (swap u n n) := by rfl _ = g u := by rw [swap_snd] _ = n := h2.right have gs_pb_n : perm_below n (g ∘ swap u n) := perm_below_fixed gs_pb_n1 gs_fix_n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f ∘ (g ∘ swap u n)) := ih (g ∘ swap u n) gs_pb_n have h3 : u ≤ n := by linarith show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f ∘ g) from calc prod_seq (n + 1) 0 f _ = prod_seq n 0 f * f n := prod_seq_zero_step n f _ = prod_seq n 0 (f ∘ (g ∘ swap u n)) * f ((g ∘ swap u n) n) := by rw [gs_prod, gs_fix_n] _ = prod_seq n 0 (f ∘ g ∘ swap u n) * (f ∘ g ∘ swap u n) n := by rfl _ = prod_seq (n + 1) 0 (f ∘ g ∘ swap u n) := (prod_seq_zero_step n (f ∘ g ∘ swap u n)).symm _ = prod_seq (n + 1) 0 ((f ∘ g) ∘ swap u n) := by rfl _ = prod_seq (n + 1) 0 (f ∘ g) := swap_prod_eq_prod (f ∘ g) h3 done done lemma F_invertible (m i : Nat) : invertible (F m i) := by by_cases h : rel_prime m i · -- Case 1. h : rel_prime m i rewrite [F_rp_def h] show invertible [i]_m from (Theorem_7_3_7 m i).rtl h done · -- Case 2. h : ¬rel_prime m i rewrite [F_not_rp_def h] apply Exists.intro [1]_m show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m done done lemma Fprod_invertible (m : Nat) : ∀ (k : Nat), invertible (prod_seq k 0 (F m)) := by by_induc · -- Base Case apply Exists.intro [1]_m show prod_seq 0 0 (F m) * [1]_m = [1]_m from calc prod_seq 0 0 (F m) * [1]_m _ = [1]_m * [1]_m := by rw [prod_seq_base] _ = [1]_m := Theorem_7_3_6_7 ([1]_m) done · -- Induction Step fix k : Nat assume ih : invertible (prod_seq k 0 (F m)) rewrite [prod_seq_zero_step] show invertible (prod_seq k 0 (F m) * (F m k)) from (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k) done done theorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m ^ (phi m) = [1]_m := by have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2 show [a]_m ^ (phi m) = [1]_m from calc [a]_m ^ (phi m) _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3] _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring _ = prod_seq m 0 (F m ∘ G m a) * Y := by rw [FG_prod h1 m, phi_def] _ = prod_seq m 0 (F m) * Y := by rw [perm_prod (F m) m (G m a) (G_perm_below h1)] _ = [1]_m := by rw [h3] done lemma Exercise_7_4_5_Int (m : Nat) (a : Int) : ∀ (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry lemma Exercise_7_4_5_Nat (m a n : Nat) : [a]_m ^ n = [a ^ n]_m := by rewrite [Exercise_7_4_5_Int] rfl done theorem Euler's_theorem {m a : Nat} [NeZero m] (h1 : rel_prime m a) : a ^ (phi m) ≡ 1 (MOD m) := by have h2 : [a]_m ^ (phi m) = [1]_m := Theorem_7_4_2 h1 rewrite [Exercise_7_4_5_Nat m a (phi m)] at h2 --h2 : [a ^ phi m]_m = [1]_m show a ^ (phi m) ≡ 1 (MOD m) from (cc_eq_iff_congr _ _ _).ltr h2 done #eval gcd 10 7 --Answer: 1. So 10 and 7 are relatively prime #eval 7 ^ phi 10 --Answer: 2401, which is congruent to 1 mod 10. end Euler /- Section 7.5 -/ lemma num_rp_prime {p : Nat} (h1 : prime p) : ∀ k < p, num_rp_below p (k + 1) = k := sorry lemma phi_prime {p : Nat} (h1 : prime p) : phi p = p - 1 := by have h2 : 1 ≤ p := prime_pos h1 have h3 : p - 1 + 1 = p := Nat.sub_add_cancel h2 have h4 : p - 1 < p := by linarith have h5 : num_rp_below p (p - 1 + 1) = p - 1 := num_rp_prime h1 (p - 1) h4 rewrite [h3] at h5 show phi p = p - 1 from h5 done theorem Theorem_7_2_2_Int {a c : Nat} {b : Int} (h1 : ↑c ∣ ↑a * b) (h2 : rel_prime a c) : ↑c ∣ b := by rewrite [Int.natCast_dvd, Int.natAbs_mul, Int.natAbs_ofNat] at h1 --h1 : c ∣ a * Int.natAbs b rewrite [Int.natCast_dvd] --Goal : c ∣ Int.natAbs b show c ∣ Int.natAbs b from Theorem_7_2_2 h1 h2 done lemma Lemma_7_4_5 {m n : Nat} (a b : Int) (h1 : rel_prime m n) : a ≡ b (MOD m * n) ↔ a ≡ b (MOD m) ∧ a ≡ b (MOD n) := by apply Iff.intro · -- (→) assume h2 : a ≡ b (MOD m * n) obtain (j : Int) (h3 : a - b = (m * n) * j) from h2 apply And.intro · -- Proof of a ≡ b (MOD m) apply Exists.intro (n * j) show a - b = m * (n * j) from calc a - b _ = m * n * j := h3 _ = m * (n * j) := by ring done · -- Proof of a ≡ b (MOD n) apply Exists.intro (m * j) show a - b = n * (m * j) from calc a - b _ = m * n * j := h3 _ = n * (m * j) := by ring done done · -- (←) assume h2 : a ≡ b (MOD m) ∧ a ≡ b (MOD n) obtain (j : Int) (h3 : a - b = m * j) from h2.left have h4 : (↑n : Int) ∣ a - b := h2.right rewrite [h3] at h4 --h4 : ↑n ∣ ↑m * j have h5 : ↑n ∣ j := Theorem_7_2_2_Int h4 h1 obtain (k : Int) (h6 : j = n * k) from h5 apply Exists.intro k --Goal : a - b = ↑(m * n) * k rewrite [Nat.cast_mul] --Goal : a - b = ↑m * ↑n * k show a - b = (m * n) * k from calc a - b _ = m * j := h3 _ = m * (n * k) := by rw [h6] _ = (m * n) * k := by ring done done --From exercises of Section 7.2 theorem rel_prime_symm {a b : Nat} (h : rel_prime a b) : rel_prime b a := sorry lemma prime_NeZero {p : Nat} (h : prime p) : NeZero p := by rewrite [neZero_iff] --Goal : p ≠ 0 define at h linarith done lemma Lemma_7_5_1 {p e d m c s : Nat} {t : Int} (h1 : prime p) (h2 : e * d = (p - 1) * s + 1) (h3 : m ^ e - c = p * t) : c ^ d ≡ m (MOD p) := by have h4 : m ^ e ≡ c (MOD p) := Exists.intro t h3 have h5 : [m ^ e]_p = [c]_p := (cc_eq_iff_congr _ _ _).rtl h4 rewrite [←Exercise_7_4_5_Nat] at h5 --h5 : [m]_p ^ e = [c]_p by_cases h6 : p ∣ m · -- Case 1. h6 : p ∣ m have h7 : m ≡ 0 (MOD p) := by obtain (j : Nat) (h8 : m = p * j) from h6 apply Exists.intro (↑j : Int) --Goal : ↑m - 0 = ↑p * ↑j rewrite [h8, Nat.cast_mul] ring done have h8 : [m]_p = [0]_p := (cc_eq_iff_congr _ _ _).rtl h7 have h9 : e * d ≠ 0 := by rewrite [h2] show (p - 1) * s + 1 ≠ 0 from Nat.add_one_ne_zero _ done have h10 : (0 : Int) ^ (e * d) = 0 := zero_pow h9 have h11 : [c ^ d]_p = [m]_p := calc [c ^ d]_p _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat] _ = ([m]_p ^ e) ^ d := by rw [h5] _ = [m]_p ^ (e * d) := by ring _ = [0]_p ^ (e * d) := by rw [h8] _ = [0 ^ (e * d)]_p := Exercise_7_4_5_Int _ _ _ _ = [0]_p := by rw [h10] _ = [m]_p := by rw [h8] show c ^ d ≡ m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11 done · -- Case 2. h6 : ¬p ∣ m have h7 : rel_prime m p := rel_prime_of_prime_not_dvd h1 h6 have h8 : rel_prime p m := rel_prime_symm h7 have h9 : NeZero p := prime_NeZero h1 have h10 : (1 : Int) ^ s = 1 := by ring have h11 : [c ^ d]_p = [m]_p := calc [c ^ d]_p _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat] _ = ([m]_p ^ e) ^ d := by rw [h5] _ = [m]_p ^ (e * d) := by ring _ = [m]_p ^ ((p - 1) * s + 1) := by rw [h2] _ = ([m]_p ^ (p - 1)) ^ s * [m]_p := by ring _ = ([m]_p ^ (phi p)) ^ s * [m]_p := by rw [phi_prime h1] _ = [1]_p ^ s * [m]_p := by rw [Theorem_7_4_2 h8] _ = [1 ^ s]_p * [m]_p := by rw [Exercise_7_4_5_Int] _ = [1]_p * [m]_p := by rw [h10] _ = [m]_p * [1]_p := by ring _ = [m]_p := Theorem_7_3_6_7 _ show c ^ d ≡ m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11 done done

theorem Theorem_7_5_1 (p q n e d k m c : Nat) (p_prime : prime p) (q_prime : prime q) (p_ne_q : p ≠ q) (n_pq : n = p * q) (ed_congr_1 : e * d = k * (p - 1) * (q - 1) + 1) (h1 : [m]_n ^ e = [c]_n) : [c]_n ^ d = [m]_n

HTPI.Theorem_7_5_1

htpi/HTPILib/Chap7.lean

htpi

/- Copyright 2023 Daniel J. Velleman -/ import HTPILib.Chap6 namespace HTPI /- Definitions -/ lemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by have h : n + 1 > 0 := Nat.succ_pos n show a % (n + 1) < n + 1 from Nat.mod_lt a h done def gcd (a b : Nat) : Nat := match b with | 0 => a | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd (n + 1) (a % (n + 1)) termination_by b mutual def gcd_c1 (a b : Nat) : Int := match b with | 0 => 1 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c2 (n + 1) (a % (n + 1)) --Corresponds to s = t' termination_by b def gcd_c2 (a b : Nat) : Int := match b with | 0 => 0 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c1 (n + 1) (a % (n + 1)) - (gcd_c2 (n + 1) (a % (n + 1))) * ↑(a / (n + 1)) --Corresponds to t = s' - t'q termination_by b end def prime (n : Nat) : Prop := 2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n def prime_factor (p n : Nat) : Prop := prime p ∧ p ∣ n def all_prime (l : List Nat) : Prop := ∀ p ∈ l, prime p def nondec (l : List Nat) : Prop := match l with | [] => True --Of course, True is a proposition that is always true | n :: L => (∀ m ∈ L, n ≤ m) ∧ nondec L def nondec_prime_list (l : List Nat) : Prop := all_prime l ∧ nondec l def prod (l : List Nat) : Nat := match l with | [] => 1 | n :: L => n * (prod L) def prime_factorization (n : Nat) (l : List Nat) : Prop := nondec_prime_list l ∧ prod l = n def rel_prime (a b : Nat) : Prop := gcd a b = 1 def congr_mod (m : Nat) (a b : Int) : Prop := (↑m : Int) ∣ (a - b) def cc (m : Nat) (a : Int) : ZMod m := (↑a : ZMod m) notation:50 a " ≡ " b " (MOD " m ")" => congr_mod m a b notation:max "["a"]_"m:max => cc m a def invertible {m : Nat} (X : ZMod m) : Prop := ∃ (Y : ZMod m), X * Y = [1]_m def num_rp_below (m k : Nat) : Nat := match k with | 0 => 0 | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j def phi (m : Nat) : Nat := num_rp_below m m def prod_seq {m : Nat} (j k : Nat) (f : Nat → ZMod m) : ZMod m := match j with | 0 => [1]_m | n + 1 => prod_seq n k f * f (k + n) def maps_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i < n, g i < n def one_one_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i1 < n, ∀ i2 < n, g i1 = g i2 → i1 = i2 def onto_below (n : Nat) (g : Nat → Nat) : Prop := ∀ k < n, ∃ i < n, g i = k def perm_below (n : Nat) (g : Nat → Nat) : Prop := maps_below n g ∧ one_one_below n g ∧ onto_below n g def inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m) def swap (u v i : Nat) : Nat := if i = u then v else if i = v then u else i namespace Euler --For definitions specific to Euler's theorem def F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m def G (m a i : Nat) : Nat := (a * i) % m def Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i end Euler /- Section 7.1 -/ theorem dvd_mod_of_dvd_a_b {a b d : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ (a % b) := by set q : Nat := a / b have h3 : b * q + a % b = a := Nat.div_add_mod a b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 define --Goal : ∃ (c : Nat), a % b = d * c apply Exists.intro (j - k * q) show a % b = d * (j - k * q) from calc a % b _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm _ = a - b * q := by rw [h3] _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc] _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm done theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry #eval gcd 672 161 --Answer: 7 lemma gcd_base (a : Nat) : gcd a 0 = a := by rfl lemma gcd_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd a b = gcd b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1)) rfl done lemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b ≠ 0) : a % b < b := by have h1 : b > 0 := Nat.pos_of_ne_zero h show a % b < b from Nat.mod_lt a h1 done lemma dvd_self (n : Nat) : n ∣ n := by apply Exists.intro 1 ring done theorem gcd_dvd : ∀ (b a : Nat), (gcd a b) ∣ a ∧ (gcd a b) ∣ b := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd a b_1) ∣ a ∧ (gcd a b_1) ∣ b_1 fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_base] --Goal: a ∣ a ∧ a ∣ 0 apply And.intro (dvd_self a) define apply Exists.intro 0 rfl done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_nonzero a h1] --Goal : gcd b (a % b) ∣ a ∧ gcd b (a % b) ∣ b have h2 : a % b < b := mod_nonzero_lt a h1 have h3 : (gcd b (a % b)) ∣ b ∧ (gcd b (a % b)) ∣ (a % b) := ih (a % b) h2 b apply And.intro _ h3.left show (gcd b (a % b)) ∣ a from dvd_a_of_dvd_b_mod h3.left h3.right done done theorem gcd_dvd_left (a b : Nat) : (gcd a b) ∣ a := (gcd_dvd b a).left theorem gcd_dvd_right (a b : Nat) : (gcd a b) ∣ b := (gcd_dvd b a).right lemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl lemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c1 a b = gcd_c2 b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done lemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl lemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * ↑(a / b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done theorem gcd_lin_comb : ∀ (b a : Nat), (gcd_c1 a b) * ↑a + (gcd_c2 a b) * ↑b = ↑(gcd a b) := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd_c1 a b_1) * ↑a + (gcd_c2 a b_1) * ↑b_1 = ↑(gcd a b_1) fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base] --Goal : 1 * ↑a + 0 * ↑0 = ↑a ring done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1] --Goal : gcd_c2 b (a % b) * ↑a + -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * ↑(a / b)) * ↑b = -- ↑(gcd b (a % b)) set r : Nat := a % b set q : Nat := a / b set s : Int := gcd_c1 b r set t : Int := gcd_c2 b r --Goal : t * ↑a + (s - t * ↑q) * ↑b = ↑(gcd b r) have h2 : r < b := mod_nonzero_lt a h1 have h3 : s * ↑b + t * ↑r = ↑(gcd b r) := ih r h2 b have h4 : b * q + r = a := Nat.div_add_mod a b rewrite [←h3, ←h4] rewrite [Nat.cast_add, Nat.cast_mul] --Goal : t * (↑b * ↑q + ↑r) + (s - t * ↑q) * ↑b = s * ↑b + t * ↑r ring done done #eval gcd_c1 672 161 --Answer: 6 #eval gcd_c2 672 161 --Answer: -25 --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161 theorem Theorem_7_1_6 {d a b : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ gcd a b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑d ∣ ↑(gcd a b) set s : Int := gcd_c1 a b set t : Int := gcd_c2 a b have h3 : s * ↑a + t * ↑b = ↑(gcd a b) := gcd_lin_comb b a rewrite [←h3] --Goal : ↑d ∣ s * ↑a + t * ↑b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul] --Goal : ↑d ∣ s * (↑d * ↑j) + t * (↑d * ↑k) define apply Exists.intro (s * ↑j + t * ↑k) ring done /- Section 7.2 -/ theorem dvd_trans {a b c : Nat} (h1 : a ∣ b) (h2 : b ∣ c) : a ∣ c := by define at h1; define at h2; define obtain (m : Nat) (h3 : b = a * m) from h1 obtain (n : Nat) (h4 : c = b * n) from h2 rewrite [h3, mul_assoc] at h4 apply Exists.intro (m * n) show c = a * (m * n) from h4 done lemma exists_prime_factor : ∀ (n : Nat), 2 ≤ n → ∃ (p : Nat), prime_factor p n := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, 2 ≤ n_1 → ∃ (p : Nat), prime_factor p n_1 assume h1 : 2 ≤ n by_cases h2 : prime n · -- Case 1. h2 : prime n apply Exists.intro n define --Goal : prime n ∧ n ∣ n show prime n ∧ n ∣ n from And.intro h2 (dvd_self n) done · -- Case 2. h2 : ¬prime n define at h2 --h2 : ¬(2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n) demorgan at h2 disj_syll h2 h1 obtain (a : Nat) (h3 : ∃ (b : Nat), a * b = n ∧ a < n ∧ b < n) from h2 obtain (b : Nat) (h4 : a * b = n ∧ a < n ∧ b < n) from h3 have h5 : 2 ≤ a := by by_contra h6 have h7 : a ≤ 1 := by linarith have h8 : n ≤ b := calc n _ = a * b := h4.left.symm _ ≤ 1 * b := by rel [h7] _ = b := by ring linarith --n ≤ b contradicts b < n done have h6 : ∃ (p : Nat), prime_factor p a := ih a h4.right.left h5 obtain (p : Nat) (h7 : prime_factor p a) from h6 apply Exists.intro p define --Goal : prime p ∧ p ∣ n define at h7 --h7 : prime p ∧ p ∣ a apply And.intro h7.left have h8 : a ∣ n := by apply Exists.intro b show n = a * b from (h4.left).symm done show p ∣ n from dvd_trans h7.right h8 done done lemma exists_least_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q := by set S : Set Nat := {p : Nat | prime_factor p n} have h2 : ∃ (p : Nat), p ∈ S := exists_prime_factor n h show ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q from well_ord_princ S h2 done lemma all_prime_nil : all_prime [] := by define --Goal : ∀ p ∈ [], prime p fix p : Nat contrapos --Goal : ¬prime p → p ∉ [] assume h1 : ¬prime p show p ∉ [] from List.not_mem_nil p done lemma all_prime_cons (n : Nat) (L : List Nat) : all_prime (n :: L) ↔ prime n ∧ all_prime L := by apply Iff.intro · -- (→) assume h1 : all_prime (n :: L) --Goal : prime n ∧ all_prime L define at h1 --h1 : ∀ p ∈ n :: L, prime p apply And.intro (h1 n (List.mem_cons_self n L)) define --Goal : ∀ p ∈ L, prime p fix p : Nat assume h2 : p ∈ L show prime p from h1 p (List.mem_cons_of_mem n h2) done · -- (←) assume h1 : prime n ∧ all_prime L --Goal : all_prime (n :: l) define : all_prime L at h1 define fix p : Nat assume h2 : p ∈ n :: L rewrite [List.mem_cons] at h2 --h2 : p = n ∨ p ∈ L by_cases on h2 · -- Case 1. h2 : p = n rewrite [h2] show prime n from h1.left done · -- Case 2. h2 : p ∈ L show prime p from h1.right p h2 done done done lemma nondec_nil : nondec [] := by define --Goal : True trivial --trivial proves some obviously true statements, such as True done lemma nondec_cons (n : Nat) (L : List Nat) : nondec (n :: L) ↔ (∀ m ∈ L, n ≤ m) ∧ nondec L := by rfl lemma prod_nil : prod [] = 1 := by rfl lemma prod_cons : prod (n :: L) = n * (prod L) := by rfl lemma exists_cons_of_length_eq_succ {A : Type} {l : List A} {n : Nat} (h : l.length = n + 1) : ∃ (a : A) (L : List A), l = a :: L ∧ L.length = n := by have h1 : ¬l.length = 0 := by linarith rewrite [List.length_eq_zero] at h1 obtain (a : A) (h2 : ∃ (L : List A), l = a :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List A) (h3 : l = a :: L) from h2 apply Exists.intro a apply Exists.intro L apply And.intro h3 have h4 : (a :: L).length = L.length + 1 := List.length_cons a L rewrite [←h3, h] at h4 show L.length = n from (Nat.add_right_cancel h4).symm done lemma list_elt_dvd_prod_by_length (a : Nat) : ∀ (n : Nat), ∀ (l : List Nat), l.length = n → a ∈ l → a ∣ prod l := by by_induc · --Base Case fix l : List Nat assume h1 : l.length = 0 rewrite [List.length_eq_zero] at h1 --h1 : l = [] rewrite [h1] --Goal : a ∈ [] → a ∣ prod [] contrapos assume h2 : ¬a ∣ prod [] show a ∉ [] from List.not_mem_nil a done · -- Induction Step fix n : Nat assume ih : ∀ (l : List Nat), List.length l = n → a ∈ l → a ∣ prod l fix l : List Nat assume h1 : l.length = n + 1 --Goal : a ∈ l → a ∣ prod l obtain (b : Nat) (h2 : ∃ (L : List Nat), l = b :: L ∧ L.length = n) from exists_cons_of_length_eq_succ h1 obtain (L : List Nat) (h3 : l = b :: L ∧ L.length = n) from h2 have h4 : a ∈ L → a ∣ prod L := ih L h3.right assume h5 : a ∈ l rewrite [h3.left, prod_cons] --Goal : a ∣ b * prod L rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b ∨ a ∈ L by_cases on h5 · -- Case 1. h5 : a = b apply Exists.intro (prod L) rewrite [h5] rfl done · -- Case 2. h5 : a ∈ L have h6 : a ∣ prod L := h4 h5 have h7 : prod L ∣ b * prod L := by apply Exists.intro b ring done show a ∣ b * prod L from dvd_trans h6 h7 done done done lemma list_elt_dvd_prod {a : Nat} {l : List Nat} (h : a ∈ l) : a ∣ prod l := by set n : Nat := l.length have h1 : l.length = n := by rfl show a ∣ prod l from list_elt_dvd_prod_by_length a n l h1 h done lemma exists_prime_factorization : ∀ (n : Nat), n ≥ 1 → ∃ (l : List Nat), prime_factorization n l := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, n_1 ≥ 1 → ∃ (l : List Nat), prime_factorization n_1 l assume h1 : n ≥ 1 by_cases h2 : n = 1 · -- Case 1. h2 : n = 1 apply Exists.intro [] define apply And.intro · -- Proof of nondec_prime_list [] define show all_prime [] ∧ nondec [] from And.intro all_prime_nil nondec_nil done · -- Proof of prod [] = n rewrite [prod_nil, h2] rfl done done · -- Case 2. h2 : n ≠ 1 have h3 : n ≥ 2 := lt_of_le_of_ne' h1 h2 obtain (p : Nat) (h4 : prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q) from exists_least_prime_factor h3 have p_prime_factor : prime_factor p n := h4.left define at p_prime_factor have p_prime : prime p := p_prime_factor.left have p_dvd_n : p ∣ n := p_prime_factor.right have p_least : ∀ (q : Nat), prime_factor q n → p ≤ q := h4.right obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n have h5 : m ≠ 0 := by contradict h1 with h6 have h7 : n = 0 := calc n _ = p * m := n_eq_pm _ = p * 0 := by rw [h6] _ = 0 := by ring rewrite [h7] decide done have m_pos : 0 < m := Nat.pos_of_ne_zero h5 have m_lt_n : m < n := by define at p_prime show m < n from calc m _ < m + m := by linarith _ = 2 * m := by ring _ ≤ p * m := by rel [p_prime.left] _ = n := n_eq_pm.symm done obtain (L : List Nat) (h6 : prime_factorization m L) from ih m m_lt_n m_pos define at h6 have ndpl_L : nondec_prime_list L := h6.left define at ndpl_L apply Exists.intro (p :: L) define apply And.intro · -- Proof of nondec_prime_list (p :: L) define apply And.intro · -- Proof of all_prime (p :: L) rewrite [all_prime_cons] show prime p ∧ all_prime L from And.intro p_prime ndpl_L.left done · -- Proof of nondec (p :: L) rewrite [nondec_cons] apply And.intro _ ndpl_L.right fix q : Nat assume q_in_L : q ∈ L have h7 : q ∣ prod L := list_elt_dvd_prod q_in_L rewrite [h6.right] at h7 --h7 : q ∣ m have h8 : m ∣ n := by apply Exists.intro p rewrite [n_eq_pm] ring done have q_dvd_n : q ∣ n := dvd_trans h7 h8 have ap_L : all_prime L := ndpl_L.left define at ap_L have q_prime_factor : prime_factor q n := And.intro (ap_L q q_in_L) q_dvd_n show p ≤ q from p_least q q_prime_factor done done · -- Proof of prod (p :: L) = n rewrite [prod_cons, h6.right, n_eq_pm] rfl done done done theorem Theorem_7_2_2 {a b c : Nat} (h1 : c ∣ a * b) (h2 : rel_prime a c) : c ∣ b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑c ∣ ↑b define at h1; define at h2; define obtain (j : Nat) (h3 : a * b = c * j) from h1 set s : Int := gcd_c1 a c set t : Int := gcd_c2 a c have h4 : s * ↑a + t * ↑c = ↑(gcd a c) := gcd_lin_comb c a rewrite [h2, Nat.cast_one] at h4 --h4 : s * ↑a + t * ↑c = (1 : Int) apply Exists.intro (s * ↑j + t * ↑b) show ↑b = ↑c * (s * ↑j + t * ↑b) from calc ↑b _ = (1 : Int) * ↑b := (one_mul _).symm _ = (s * ↑a + t * ↑c) * ↑b := by rw [h4] _ = s * (↑a * ↑b) + t * ↑c * ↑b := by ring _ = s * (↑c * ↑j) + t * ↑c * ↑b := by rw [←Nat.cast_mul a b, h3, Nat.cast_mul c j] _ = ↑c * (s * ↑j + t * ↑b) := by ring done lemma le_nonzero_prod_left {a b : Nat} (h : a * b ≠ 0) : a ≤ a * b := by have h1 : b ≠ 0 := by contradict h with h1 rewrite [h1] ring done have h2 : 1 ≤ b := Nat.pos_of_ne_zero h1 show a ≤ a * b from calc a = a * 1 := (mul_one a).symm _ ≤ a * b := by rel [h2] done lemma le_nonzero_prod_right {a b : Nat} (h : a * b ≠ 0) : b ≤ a * b := by rewrite [mul_comm] rewrite [mul_comm] at h show b ≤ b * a from le_nonzero_prod_left h done lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry lemma rel_prime_of_prime_not_dvd {a p : Nat} (h1 : prime p) (h2 : ¬p ∣ a) : rel_prime a p := by have h3 : gcd a p ∣ a := gcd_dvd_left a p have h4 : gcd a p ∣ p := gcd_dvd_right a p have h5 : gcd a p = 1 ∨ gcd a p = p := dvd_prime h1 h4 have h6 : gcd a p ≠ p := by contradict h2 with h6 rewrite [h6] at h3 show p ∣ a from h3 done disj_syll h5 h6 show rel_prime a p from h5 done theorem Theorem_7_2_3 {a b p : Nat} (h1 : prime p) (h2 : p ∣ a * b) : p ∣ a ∨ p ∣ b := by or_right with h3 have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3 show p ∣ b from Theorem_7_2_2 h2 h4 done lemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a ≥ 1 := by have h1 : a ≠ 0 := by by_contra h1 rewrite [h1] at h contradict h linarith done show a ≥ 1 from Nat.pos_of_ne_zero h1 done lemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by have h1 : a ≥ 1 := ge_one_of_prod_one h have h2 : a * b ≠ 0 := by linarith have h3 : a ≤ a * b := le_nonzero_prod_left h2 rewrite [h] at h3 show a = 1 from Nat.le_antisymm h3 h1 done lemma eq_one_of_dvd_one {n : Nat} (h : n ∣ 1) : n = 1 := by obtain (j : Nat) (h1 : 1 = n * j) from h show n = 1 from eq_one_of_prod_one h1.symm done lemma prime_not_one {p : Nat} (h : prime p) : p ≠ 1 := by define at h linarith done theorem Theorem_7_2_4 {p : Nat} (h1 : prime p) : ∀ (l : List Nat), p ∣ prod l → ∃ a ∈ l, p ∣ a := by apply List.rec · -- Base Case. Goal : p ∣ prod [] → ∃ a ∈ [], p ∣ a rewrite [prod_nil] assume h2 : p ∣ 1 show ∃ a ∈ [], p ∣ a from absurd (eq_one_of_dvd_one h2) (prime_not_one h1) done · -- Induction Step fix b : Nat fix L : List Nat assume ih : p ∣ prod L → ∃ a ∈ L, p ∣ a --Goal : p ∣ prod (b :: L) → ∃ a ∈ b :: L, p ∣ a assume h2 : p ∣ prod (b :: L) rewrite [prod_cons] at h2 have h3 : p ∣ b ∨ p ∣ prod L := Theorem_7_2_3 h1 h2 by_cases on h3 · -- Case 1. h3 : p ∣ b apply Exists.intro b show b ∈ b :: L ∧ p ∣ b from And.intro (List.mem_cons_self b L) h3 done · -- Case 2. h3 : p ∣ prod L obtain (a : Nat) (h4 : a ∈ L ∧ p ∣ a) from ih h3 apply Exists.intro a show a ∈ b :: L ∧ p ∣ a from And.intro (List.mem_cons_of_mem b h4.left) h4.right done done done lemma prime_in_list {p : Nat} {l : List Nat} (h1 : prime p) (h2 : all_prime l) (h3 : p ∣ prod l) : p ∈ l := by obtain (a : Nat) (h4 : a ∈ l ∧ p ∣ a) from Theorem_7_2_4 h1 l h3 define at h2 have h5 : prime a := h2 a h4.left have h6 : p = 1 ∨ p = a := dvd_prime h5 h4.right disj_syll h6 (prime_not_one h1) rewrite [h6] show a ∈ l from h4.left done lemma first_le_first {p q : Nat} {l m : List Nat} (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m)) (h3 : prod (p :: l) = prod (q :: m)) : p ≤ q := by define at h1; define at h2 have h4 : q ∣ prod (p :: l) := by define apply Exists.intro (prod m) rewrite [←prod_cons] show prod (p :: l) = prod (q :: m) from h3 done have h5 : all_prime (q :: m) := h2.left rewrite [all_prime_cons] at h5 have h6 : q ∈ p :: l := prime_in_list h5.left h1.left h4 have h7 : nondec (p :: l) := h1.right rewrite [nondec_cons] at h7 rewrite [List.mem_cons] at h6 by_cases on h6 · -- Case 1. h6 : q = p linarith done · -- Case 2. h6 : q ∈ l have h8 : ∀ m ∈ l, p ≤ m := h7.left show p ≤ q from h8 q h6 done done lemma nondec_prime_list_tail {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by define at h define rewrite [all_prime_cons, nondec_cons] at h show all_prime l ∧ nondec l from And.intro h.left.right h.right.right done lemma cons_prod_not_one {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : prod (p :: l) ≠ 1 := by define at h have h1 : all_prime (p :: l) := h.left rewrite [all_prime_cons] at h1 rewrite [prod_cons] by_contra h2 show False from (prime_not_one h1.left) (eq_one_of_prod_one h2) done lemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) : l = [] ↔ prod l = 1 := by apply Iff.intro · -- (→) assume h1 : l = [] rewrite [h1] show prod [] = 1 from prod_nil done · -- (←) contrapos assume h1 : ¬l = [] obtain (p : Nat) (h2 : ∃ (L : List Nat), l = p :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List Nat) (h3 : l = p :: L) from h2 rewrite [h3] at h rewrite [h3] show ¬prod (p :: L) = 1 from cons_prod_not_one h done done lemma prime_pos {p : Nat} (h : prime p) : p > 0 := by define at h linarith done theorem Theorem_7_2_5 : ∀ (l1 l2 : List Nat), nondec_prime_list l1 → nondec_prime_list l2 → prod l1 = prod l2 → l1 = l2 := by apply List.rec · -- Base Case. Goal : ∀ (l2 : List Nat), nondec_prime_list [] → -- nondec_prime_list l2 → prod [] = prod l2 → [] = l2 fix l2 : List Nat assume h1 : nondec_prime_list [] assume h2 : nondec_prime_list l2 assume h3 : prod [] = prod l2 rewrite [prod_nil, eq_comm, ←list_nil_iff_prod_one h2] at h3 show [] = l2 from h3.symm done · -- Induction Step fix p : Nat fix L1 : List Nat assume ih : ∀ (L2 : List Nat), nondec_prime_list L1 → nondec_prime_list L2 → prod L1 = prod L2 → L1 = L2 -- Goal : ∀ (l2 : List Nat), nondec_prime_list (p :: L1) → -- nondec_prime_list l2 → prod (p :: L1) = prod l2 → p :: L1 = l2 fix l2 : List Nat assume h1 : nondec_prime_list (p :: L1) assume h2 : nondec_prime_list l2 assume h3 : prod (p :: L1) = prod l2 have h4 : ¬prod (p :: L1) = 1 := cons_prod_not_one h1 rewrite [h3, ←list_nil_iff_prod_one h2] at h4 obtain (q : Nat) (h5 : ∃ (L : List Nat), l2 = q :: L) from List.exists_cons_of_ne_nil h4 obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5 rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2) rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2) have h7 : p ≤ q := first_le_first h1 h2 h3 have h8 : q ≤ p := first_le_first h2 h1 h3.symm have h9 : p = q := by linarith rewrite [h9, prod_cons, prod_cons] at h3 --h3 : q * prod L1 = q * prod L2 have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1 have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2 define at h2 have h12 : all_prime (q :: L2) := h2.left rewrite [all_prime_cons] at h12 have h13 : q > 0 := prime_pos h12.left have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3 have h15 : L1 = L2 := ih L2 h10 h11 h14 rewrite [h6, h9, h15] rfl done done theorem fund_thm_arith (n : Nat) (h : n ≥ 1) : ∃! (l : List Nat), prime_factorization n l := by exists_unique · -- Existence show ∃ (l : List Nat), prime_factorization n l from exists_prime_factorization n h done · -- Uniqueness fix l1 : List Nat; fix l2 : List Nat assume h1 : prime_factorization n l1 assume h2 : prime_factorization n l2 define at h1; define at h2 have h3 : prod l1 = n := h1.right rewrite [←h2.right] at h3 show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3 done done /- Section 7.3 -/ theorem congr_refl (m : Nat) : ∀ (a : Int), a ≡ a (MOD m) := by fix a : Int define --Goal : ∃ (c : Int), a - a = ↑m * c apply Exists.intro 0 ring done theorem congr_symm {m : Nat} : ∀ {a b : Int}, a ≡ b (MOD m) → b ≡ a (MOD m) := by fix a : Int; fix b : Int assume h1 : a ≡ b (MOD m) define at h1 --h1 : ∃ (c : Int), a - b = ↑m * c define --Goal : ∃ (c : Int), b - a = ↑m * c obtain (c : Int) (h2 : a - b = m * c) from h1 apply Exists.intro (-c) show b - a = m * (-c) from calc b - a _ = -(a - b) := by ring _ = -(m * c) := by rw [h2] _ = m * (-c) := by ring done theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry /- Fundamental properties of congruence classes -/ lemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) : X = Y ↔ X.val = Y.val := Fin.ext_iff lemma val_nat_eq_mod (n k : Nat) : ([k]_(n + 1)).val = k % (n + 1) := by rfl lemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl theorem cc_rep {m : Nat} (X : ZMod m) : ∃ (a : Int), X = [a]_m := match m with | 0 => by apply Exists.intro X rfl done | n + 1 => by apply Exists.intro ↑(X.val) have h1 : X.val < n + 1 := Fin.prop X rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1] rfl done theorem add_class (m : Nat) (a b : Int) : [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm theorem mul_class (m : Nat) (a b : Int) : [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm lemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ [a - b]_m = [0]_m := by apply Iff.intro · -- (→) assume h1 : [a]_m = [b]_m have h2 : a - b = a + (-b) := by ring have h3 : b + (-b) = 0 := by ring show [a - b]_m = [0]_m from calc [a - b]_m _ = [a + (-b)]_m := by rw [h2] _ = [a]_m + [-b]_m := by rw [add_class] _ = [b]_m + [-b]_m := by rw [h1] _ = [b + -b]_m := by rw [add_class] _ = [0]_m := by rw [h3] done · -- (←) assume h1 : [a - b]_m = [0]_m have h2 : b + (a - b) = a := by ring have h3 : b + 0 = b := by ring show [a]_m = [b]_m from calc [a]_m _ = [b + (a - b)]_m := by rw [h2] _ = [b]_m + [a - b]_m := by rw [add_class] _ = [b]_m + [0]_m := by rw [h1] _ = [b + 0]_m := by rw [add_class] _ = [b]_m := by rw [h3] done done lemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) : [a]_m = [0]_m → [-a]_m = [0]_m := by assume h1 : [a]_m = [0]_m have h2 : 0 + (-a) = -a := by ring have h3 : a + (-a) = 0 := by ring show [-a]_m = [0]_m from calc [-a]_m _ = [0 + (-a)]_m := by rw [h2] _ = [0]_m + [-a]_m := by rw [add_class] _ = [a]_m + [-a]_m := by rw [h1] _ = [a + (-a)]_m := by rw [add_class] _ = [0]_m := by rw [h3] done lemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) : [-a]_m = [0]_m ↔ [a]_m = [0]_m := by apply Iff.intro _ (cc_neg_zero_of_cc_zero m a) assume h1 : [-a]_m = [0]_m have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1 have h3 : -(-a) = a := by ring rewrite [h3] at h2 show [a]_m = [0]_m from h2 done lemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl lemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m ↔ m ∣ k := match m with | 0 => by have h : (0 : Int) = (↑(0 : Nat) : Int) := by rfl rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj] apply Iff.intro · -- (→) assume h1 : k = 0 rewrite [h1] show 0 ∣ 0 from dvd_self 0 done · -- (←) assume h1 : 0 ∣ k obtain (c : Nat) (h2 : k = 0 * c) from h1 rewrite [h2] ring done done | n + 1 => by rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero] show k % (n + 1) = 0 ↔ n + 1 ∣ k from (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm done lemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m ↔ ↑m ∣ a := by obtain (k : Nat) (h1 : a = ↑k ∨ a = -↑k) from Int.eq_nat_or_neg a by_cases on h1 · -- Case 1. h1: a = ↑k rewrite [h1, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done · -- Case 2. h1: a = -↑k rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done done theorem cc_eq_iff_congr (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ a ≡ b (MOD m) := calc [a]_m = [b]_m _ ↔ [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b _ ↔ ↑m ∣ (a - b) := cc_zero_iff_dvd m (a - b) _ ↔ a ≡ b (MOD m) := by rfl /- End of fundamental properties of congruence classes -/ lemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m := by have h1 : (↑m : Int) ≠ 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m) show 0 ≤ a % m from Int.emod_nonneg a h1 done lemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m) have h2 : (↑m : Int) > 0 := (Nat.cast_pos).rtl h1 show a % m < m from Int.emod_lt_of_pos a h2 done lemma congr_mod_mod (m : Nat) (a : Int) : a ≡ a % m (MOD m) := by define have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m apply Exists.intro (a / m) show a - a % m = m * (a / m) from calc a - (a % m) _ = m * (a / m) + a % m - a % m := by rw [h1] _ = m * (a / m) := by ring done lemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) := And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a)) theorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) : ∃! (r : Int), 0 ≤ r ∧ r < m ∧ a ≡ r (MOD m) := by exists_unique · -- Existence apply Exists.intro (a % m) show 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) from mod_cmpl_res m a done · -- Uniqueness fix r1 : Int; fix r2 : Int assume h1 : 0 ≤ r1 ∧ r1 < m ∧ a ≡ r1 (MOD m) assume h2 : 0 ≤ r2 ∧ r2 < m ∧ a ≡ r2 (MOD m) have h3 : r1 ≡ r2 (MOD m) := congr_trans (congr_symm h1.right.right) h2.right.right obtain (d : Int) (h4 : r1 - r2 = m * d) from h3 have h5 : r1 - r2 < m * 1 := by linarith have h6 : m * (-1) < r1 - r2 := by linarith rewrite [h4] at h5 --h5 : m * d < m * 1 rewrite [h4] at h6 --h6 : m * -1 < m * d have h7 : (↑m : Int) ≥ 0 := Nat.cast_nonneg m have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7 have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7 have h10 : d = 0 := by linarith show r1 = r2 from calc r1 _ = r1 - r2 + r2 := by ring _ = m * 0 + r2 := by rw [h4, h10] _ = r2 := by ring done done lemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m := (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a) theorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y rewrite [h1, h2] have h3 : a + b = b + a := by ring show [a]_m + [b]_m = [b]_m + [a]_m from calc [a]_m + [b]_m _ = [a + b]_m := add_class m a b _ = [b + a]_m := by rw [h3] _ = [b]_m + [a]_m := (add_class m b a).symm done theorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X rewrite [h1] have h2 : a * 1 = a := by ring show [a]_m * [1]_m = [a]_m from calc [a]_m * [1]_m _ = [a * 1]_m := mul_class m a 1 _ = [a]_m := by rw [h2] done theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry lemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) : [a]_m * [gcd_c2 m a]_m = [1]_m := by set s : Int := gcd_c1 m a have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m define at h1 rewrite [h1, Nat.cast_one] at h2 --h2 : s * ↑m + gcd_c2 m a * ↑a = 1 rewrite [mul_class, cc_eq_iff_congr] define --Goal : ∃ (c : Int), ↑a * gcd_c2 m a - 1 = ↑m * c apply Exists.intro (-s) show a * (gcd_c2 m a) - 1 = m * (-s) from calc a * (gcd_c2 m a) - 1 _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring _ = 1 + m * (-s) - 1 := by rw [h2] _ = m * (-s) := by ring done theorem Theorem_7_3_7 (m a : Nat) : invertible [a]_m ↔ rel_prime m a := by apply Iff.intro · -- (→) assume h1 : invertible [a]_m define at h1 obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1 obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y rewrite [h3, mul_class, cc_eq_iff_congr] at h2 define at h2 obtain (c : Int) (h4 : a * b - 1 = m * c) from h2 rewrite [Exercise_7_2_6] --Goal : ∃ (s t : Int), s * ↑m + t * ↑a = 1 apply Exists.intro (-c) apply Exists.intro b show (-c) * m + b * a = 1 from calc (-c) * m + b * a _ = (-c) * m + (a * b - 1) + 1 := by ring _ = (-c) * m + m * c + 1 := by rw [h4] _ = 1 := by ring done · -- (←) assume h1 : rel_prime m a define show ∃ (Y : ZMod m), [a]_m * Y = [1]_m from Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1) done done /- Section 7.4 -/ section Euler open Euler lemma num_rp_below_base {m : Nat} : num_rp_below m 0 = 0 := by rfl lemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) : num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by have h1 : num_rp_below m (j + 1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : gcd m j = 1 rewrite [if_pos h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1 show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1 done lemma num_rp_below_step_not_rp {m j : Nat} (h : ¬rel_prime m j) : num_rp_below m (j + 1) = num_rp_below m j := by have h1 : num_rp_below m (j +1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : ¬gcd m j = 1 rewrite [if_neg h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j show num_rp_below m (j + 1) = num_rp_below m j from h1 done lemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl #eval phi 10 --Answer: 4 lemma prod_inv_iff_inv {m : Nat} {X : ZMod m} (h1 : invertible X) (Y : ZMod m) : invertible (X * Y) ↔ invertible Y := by apply Iff.intro · -- (→) assume h2 : invertible (X * Y) obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2 apply Exists.intro (X * Z) rewrite [←h3] --Goal : Y * (X * Z) = X * Y * Z ring --Note that ring can do algebra in ZMod m done · -- (←) assume h2 : invertible Y obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1 obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2 apply Exists.intro (Xi * Yi) show (X * Y) * (Xi * Yi) = [1]_m from calc X * Y * (Xi * Yi) _ = (X * Xi) * (Y * Yi) := by ring _ = [1]_m * [1]_m := by rw [h3, h4] _ = [1]_m := Theorem_7_3_6_7 [1]_m done done lemma F_rp_def {m i : Nat} (h : rel_prime m i) : F m i = [i]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h --h : gcd m i = 1 rewrite [if_pos h] at h1 show F m i = [i]_m from h1 done lemma F_not_rp_def {m i : Nat} (h : ¬rel_prime m i) : F m i = [1]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h rewrite [h1, if_neg h] rfl done lemma prod_seq_base {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 0 k f = [1]_m := by rfl lemma prod_seq_step {m : Nat} (n k : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl lemma prod_seq_zero_step {m : Nat} (n : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by rewrite [prod_seq_step, zero_add] rfl done lemma prod_one {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 1 k f = f k := by rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7] rfl done lemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl lemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m := calc [G m a i]_m _ = [(a * i) % m]_m := by rfl _ = [a * i]_m := (cc_eq_mod m (a * i)).symm _ = [a]_m * [i]_m := (mul_class m a i).symm lemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) : rel_prime m (G m a i) ↔ rel_prime m i := by have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1 show rel_prime m (G m a i) ↔ rel_prime m i from calc rel_prime m (G m a i) _ ↔ invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm _ ↔ invertible ([a]_m * [i]_m) := by rw [cc_G] _ ↔ invertible [i]_m := prod_inv_iff_inv h2 ([i]_m) _ ↔ rel_prime m i := Theorem_7_3_7 m i done lemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) : F m (G m a i) = [a]_m * F m i := by have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2 show F m (G m a i) = [a]_m * F m i from calc F m (G m a i) _ = [G m a i]_m := F_rp_def h3 _ = [a]_m * [i]_m := cc_G m a i _ = [a]_m * F m i := by rw [F_rp_def h2] done lemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : ¬rel_prime m i) : F m (G m a i) = [1]_m := by rewrite [←G_rp_iff h1 i] at h2 show F m (G m a i) = [1]_m from F_not_rp_def h2 done lemma FG_prod {m a : Nat} (h1 : rel_prime m a) : ∀ (k : Nat), prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by by_induc · -- Base Case show prod_seq 0 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from calc prod_seq 0 0 ((F m) ∘ (G m a)) _ = [1]_m := prod_seq_base _ _ _ = [a]_m ^ 0 * [1]_m := by ring _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by rw [num_rp_below_base, prod_seq_base] done · -- Induction Step fix k : Nat assume ih : prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) by_cases h2 : rel_prime m k · -- Case 1. h2 : rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([a]_m * F m k) := by rw [FG_rp h1 h2] _ = [a]_m ^ ((num_rp_below m k) + 1) * ((prod_seq k 0 (F m)) * F m k) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_rp h2, prod_seq_zero_step] done · -- Case 2. h2 : ¬rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([1]_m) := by rw [FG_not_rp h1 h2] _ = [a]_m ^ (num_rp_below m k) * (prod_seq k 0 (F m) * ([1]_m)) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_not_rp h2, prod_seq_zero_step, F_not_rp_def h2] done done done lemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by define --Goal : ∀ i < m, G m a i < m fix i : Nat assume h1 : i < m rewrite [G_def] --Goal : a * i % m < m show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m) done lemma left_inv_one_one_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g' (g i) = i) : one_one_below n g := sorry lemma right_inv_onto_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g (g' i) = i) (h2 : maps_below n g') : onto_below n g := by define at h2; define fix k : Nat assume h3 : k < n apply Exists.intro (g' k) show g' k < n ∧ g (g' k) = k from And.intro (h2 k h3) (h1 k h3) done lemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m * [inv_mod m a]_m = [1]_m := by have h2 : 0 ≤ (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a) show [a]_m * [inv_mod m a]_m = [1]_m from calc [a]_m * [inv_mod m a]_m _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2] _ = [a]_m * [gcd_c2 m a]_m := by rw [←cc_eq_mod] _ = [1]_m := gcd_c2_inv h1 done lemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m := calc a * (b % m) % m = a % m * (b % m % m) % m := Nat.mul_mod _ _ _ _ = a % m * (b % m) % m := by rw [Nat.mod_mod] _ = a * b % m := (Nat.mul_mod _ _ _).symm lemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm] rfl done theorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] : ↑a ≡ ↑b (MOD m) ↔ a % m = b % m := sorry lemma mul_inv_mod_cancel {m a i : Nat} [NeZero m] (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1 rewrite [mul_class, cc_eq_iff_congr, ←Nat.cast_mul, ←Nat.cast_one, congr_iff_mod_eq_Nat] at h3 show a * inv_mod m a * i % m = i from calc a * (inv_mod m a) * i % m _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = 1 % m * i % m := by rw [h3] _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = i % m := by rw [one_mul] _ = i := Nat.mod_eq_of_lt h2 done lemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl lemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, G m a (Ginv m a i) = i := by fix i : Nat assume h2 : i < m show G m a (Ginv m a i) = i from calc G m a (Ginv m a i) _ = a * ((inv_mod m a * i) % m) % m := by rfl _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, Ginv m a (G m a i) = i := by fix i : Nat assume h2 : i < m show Ginv m a (G m a i) = i from calc Ginv m a (G m a i) _ = inv_mod m a * ((a * i) % m) % m := by rfl _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc, mul_comm (inv_mod m a)] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_maps_below (m a : Nat) [NeZero m] : maps_below m (Ginv m a) := G_maps_below m (inv_mod m a) lemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : one_one_below m (G m a) := left_inv_one_one_below (Ginv_left_inv h1) lemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : onto_below m (G m a) := right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a) lemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : perm_below m (G m a) := And.intro (G_maps_below m a) (And.intro (G_one_one_below h1) (G_onto_below h1)) --Permuting a product of congruence classes doesn't change product lemma swap_fst (u v : Nat) : swap u v u = v := by define : swap u v u --Goal : (if u = u then v else if u = v then u else u) = v have h : u = u := by rfl rewrite [if_pos h] rfl done lemma swap_snd (u v : Nat) : swap u v v = u := by define : swap u v v by_cases h1 : v = u · -- Case 1. h1 : v = u rewrite [if_pos h1] show v = u from h1 done · -- Case 2. h1 : v ≠ u rewrite [if_neg h1] have h2 : v = v := by rfl rewrite [if_pos h2] rfl done done lemma swap_other {u v i : Nat} (h1 : i ≠ u) (h2 : i ≠ v) : swap u v i = i := by define : swap u v i rewrite [if_neg h1, if_neg h2] rfl done lemma swap_values (u v i : Nat) : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := by by_cases h1 : i = u · -- Case 1. h1 : i = u apply Or.inl rewrite [h1] show swap u v u = v from swap_fst u v done · -- Case 2. h1 : i ≠ u apply Or.inr by_cases h2 : i = v · -- Case 2.1. h2 : i = v apply Or.inl rewrite [h2] show swap u v v = u from swap_snd u v done · -- Case 2.2. h2 : i ≠ v apply Or.inr show swap u v i = i from swap_other h1 h2 done done done lemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by define fix i : Nat assume h3 : i < n have h4 : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := swap_values u v i by_cases on h4 · -- Case 1. h4 : swap u v i = v rewrite [h4] show v < n from h2 done · -- Case 2. by_cases on h4 · -- Case 2.1. h4 : swap u v i = u rewrite [h4] show u < n from h1 done · -- Case 2.2. h4 : swap u v i = i rewrite [h4] show i < n from h3 done done done lemma swap_swap (u v n : Nat) : ∀ i < n, swap u v (swap u v i) = i := by fix i : Nat assume h : i < n by_cases h1 : i = u · -- Case 1. h1 : i = u rewrite [h1, swap_fst, swap_snd] rfl done · -- Case 2. h1 : i ≠ u by_cases h2 : i = v · -- Case 2.1. h2 : i = v rewrite [h2, swap_snd, swap_fst] rfl done · -- Case 2.2. h2 : i ≠ v rewrite [swap_other h1 h2, swap_other h1 h2] rfl done done done lemma swap_one_one_below (u v n) : one_one_below n (swap u v) := left_inv_one_one_below (swap_swap u v n) lemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) := right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2) lemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) := And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2)) lemma comp_perm_below {n : Nat} {f g : Nat → Nat} (h1 : perm_below n f) (h2 : perm_below n g) : perm_below n (f ∘ g) := sorry lemma trivial_swap (u : Nat) : swap u u = id := by apply funext fix x : Nat by_cases h1 : x = u · -- Case 1. h1 : x = u rewrite [h1, swap_fst] rfl done · -- Case 2. h1 : x ≠ u rewrite [swap_other h1 h1] rfl done done lemma prod_eq_fun {m : Nat} (f g : Nat → ZMod m) (k : Nat) : ∀ (n : Nat), (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g := by by_induc · -- Base Case assume h : (∀ i < 0, f (k + i) = g (k + i)) rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g assume h1 : ∀ i < n + 1, f (k + i) = g (k + i) have h2 : ∀ i < n, f (k + i) = g (k + i) := by fix i : Nat assume h2 : i < n have h3 : i < n + 1 := by linarith show f (k + i) = g (k + i) from h1 i h3 done have h3 : prod_seq n k f = prod_seq n k g := ih h2 have h4 : n < n + 1 := Nat.lt_succ_self n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4] rfl done done lemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := by have h2 : ∀ (i : Nat), i < u → (f ∘ swap u n) (0 + i) = f (0 + i) := by fix i : Nat assume h2 : i < u have h3 : 0 + i ≠ u := by linarith have h4 : 0 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl done show prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f from prod_eq_fun (f ∘ swap u n) f 0 u h2 done lemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := by have h2 : ∀ i < j, (f ∘ swap u n) (u + 1 + i) = f (u + 1 + i) := by fix i : Nat assume h2 : i < j have h3 : u + 1 + i ≠ u := by linarith have h4 : u + 1 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl show prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f from prod_eq_fun (f ∘ swap u n) f (u + 1) j h2 done lemma break_prod {m : Nat} (n : Nat) (f : Nat → ZMod m) : ∀ (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by by_induc · -- Base Case have h : n + 0 = n := by rfl rewrite [prod_seq_base, h, Theorem_7_3_6_7] rfl done · -- Induction Step fix j : Nat assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f rewrite [←add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc] rfl done done lemma break_prod_twice {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f := break_prod n f 1 rewrite [prod_one] at h2 have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f := break_prod (u + 1) f j rewrite [←h1] at h3 have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f := break_prod u f 1 rewrite [prod_one] at h4 rewrite [h3, h4] at h2 show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2 done lemma swap_prod_eq_prod {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f := by by_cases h2 : u = n · -- Case 1. h2 : u = n rewrite [h2, trivial_swap n] --Goal : prod_seq (n + 1) 0 (f ∘ id) = prod_seq (n + 1) 0 f rfl done · -- Case 2. h2 : ¬u = n have h3 : u + 1 ≤ n := Nat.lt_of_le_of_ne h1 h2 obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3 have break_f : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := break_prod_twice f h4 have break_fs : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_prod_twice (f ∘ swap u n) h4 have f_eq_fs_below : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := swap_prod_eq_prod_below f h1 have f_eq_fs_btwn : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4 show prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f from calc prod_seq (n + 1) 0 (f ∘ swap u n) _ = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_fs _ = prod_seq u 0 f * (f ∘ swap u n) u * prod_seq j (u + 1) f * (f ∘ swap u n) n := by rw [f_eq_fs_below, f_eq_fs_btwn] _ = prod_seq u 0 f * f (swap u n u) * prod_seq j (u + 1) f * f (swap u n n) := by rfl _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by rw [swap_fst, swap_snd] _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring _ = prod_seq (n + 1) 0 f := break_f.symm done done lemma perm_below_fixed {n : Nat} {g : Nat → Nat} (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry lemma perm_prod {m : Nat} (f : Nat → ZMod m) : ∀ (n : Nat), ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) := by by_induc · -- Base Case fix g : Nat → Nat assume h1 : perm_below 0 g rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) fix g : Nat → Nat assume g_pb : perm_below (n + 1) g define at g_pb have g_ob : onto_below (n + 1) g := g_pb.right.right define at g_ob have h1 : n < n + 1 := by linarith obtain (u : Nat) (h2 : u < n + 1 ∧ g u = n) from g_ob n h1 have s_pb : perm_below (n + 1) (swap u n) := swap_perm_below h2.left h1 have gs_pb_n1 : perm_below (n + 1) (g ∘ swap u n) := comp_perm_below g_pb s_pb have gs_fix_n : (g ∘ swap u n) n = n := calc (g ∘ swap u n) n _ = g (swap u n n) := by rfl _ = g u := by rw [swap_snd] _ = n := h2.right have gs_pb_n : perm_below n (g ∘ swap u n) := perm_below_fixed gs_pb_n1 gs_fix_n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f ∘ (g ∘ swap u n)) := ih (g ∘ swap u n) gs_pb_n have h3 : u ≤ n := by linarith show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f ∘ g) from calc prod_seq (n + 1) 0 f _ = prod_seq n 0 f * f n := prod_seq_zero_step n f _ = prod_seq n 0 (f ∘ (g ∘ swap u n)) * f ((g ∘ swap u n) n) := by rw [gs_prod, gs_fix_n] _ = prod_seq n 0 (f ∘ g ∘ swap u n) * (f ∘ g ∘ swap u n) n := by rfl _ = prod_seq (n + 1) 0 (f ∘ g ∘ swap u n) := (prod_seq_zero_step n (f ∘ g ∘ swap u n)).symm _ = prod_seq (n + 1) 0 ((f ∘ g) ∘ swap u n) := by rfl _ = prod_seq (n + 1) 0 (f ∘ g) := swap_prod_eq_prod (f ∘ g) h3 done done lemma F_invertible (m i : Nat) : invertible (F m i) := by by_cases h : rel_prime m i · -- Case 1. h : rel_prime m i rewrite [F_rp_def h] show invertible [i]_m from (Theorem_7_3_7 m i).rtl h done · -- Case 2. h : ¬rel_prime m i rewrite [F_not_rp_def h] apply Exists.intro [1]_m show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m done done lemma Fprod_invertible (m : Nat) : ∀ (k : Nat), invertible (prod_seq k 0 (F m)) := by by_induc · -- Base Case apply Exists.intro [1]_m show prod_seq 0 0 (F m) * [1]_m = [1]_m from calc prod_seq 0 0 (F m) * [1]_m _ = [1]_m * [1]_m := by rw [prod_seq_base] _ = [1]_m := Theorem_7_3_6_7 ([1]_m) done · -- Induction Step fix k : Nat assume ih : invertible (prod_seq k 0 (F m)) rewrite [prod_seq_zero_step] show invertible (prod_seq k 0 (F m) * (F m k)) from (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k) done done theorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m ^ (phi m) = [1]_m := by have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2 show [a]_m ^ (phi m) = [1]_m from calc [a]_m ^ (phi m) _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3] _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring _ = prod_seq m 0 (F m ∘ G m a) * Y := by rw [FG_prod h1 m, phi_def] _ = prod_seq m 0 (F m) * Y := by rw [perm_prod (F m) m (G m a) (G_perm_below h1)] _ = [1]_m := by rw [h3] done lemma Exercise_7_4_5_Int (m : Nat) (a : Int) : ∀ (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry lemma Exercise_7_4_5_Nat (m a n : Nat) : [a]_m ^ n = [a ^ n]_m := by rewrite [Exercise_7_4_5_Int] rfl done theorem Euler's_theorem {m a : Nat} [NeZero m] (h1 : rel_prime m a) : a ^ (phi m) ≡ 1 (MOD m) := by have h2 : [a]_m ^ (phi m) = [1]_m := Theorem_7_4_2 h1 rewrite [Exercise_7_4_5_Nat m a (phi m)] at h2 --h2 : [a ^ phi m]_m = [1]_m show a ^ (phi m) ≡ 1 (MOD m) from (cc_eq_iff_congr _ _ _).ltr h2 done #eval gcd 10 7 --Answer: 1. So 10 and 7 are relatively prime #eval 7 ^ phi 10 --Answer: 2401, which is congruent to 1 mod 10. end Euler /- Section 7.5 -/ lemma num_rp_prime {p : Nat} (h1 : prime p) : ∀ k < p, num_rp_below p (k + 1) = k := sorry lemma phi_prime {p : Nat} (h1 : prime p) : phi p = p - 1 := by have h2 : 1 ≤ p := prime_pos h1 have h3 : p - 1 + 1 = p := Nat.sub_add_cancel h2 have h4 : p - 1 < p := by linarith have h5 : num_rp_below p (p - 1 + 1) = p - 1 := num_rp_prime h1 (p - 1) h4 rewrite [h3] at h5 show phi p = p - 1 from h5 done theorem Theorem_7_2_2_Int {a c : Nat} {b : Int} (h1 : ↑c ∣ ↑a * b) (h2 : rel_prime a c) : ↑c ∣ b := by rewrite [Int.natCast_dvd, Int.natAbs_mul, Int.natAbs_ofNat] at h1 --h1 : c ∣ a * Int.natAbs b rewrite [Int.natCast_dvd] --Goal : c ∣ Int.natAbs b show c ∣ Int.natAbs b from Theorem_7_2_2 h1 h2 done lemma Lemma_7_4_5 {m n : Nat} (a b : Int) (h1 : rel_prime m n) : a ≡ b (MOD m * n) ↔ a ≡ b (MOD m) ∧ a ≡ b (MOD n) := by apply Iff.intro · -- (→) assume h2 : a ≡ b (MOD m * n) obtain (j : Int) (h3 : a - b = (m * n) * j) from h2 apply And.intro · -- Proof of a ≡ b (MOD m) apply Exists.intro (n * j) show a - b = m * (n * j) from calc a - b _ = m * n * j := h3 _ = m * (n * j) := by ring done · -- Proof of a ≡ b (MOD n) apply Exists.intro (m * j) show a - b = n * (m * j) from calc a - b _ = m * n * j := h3 _ = n * (m * j) := by ring done done · -- (←) assume h2 : a ≡ b (MOD m) ∧ a ≡ b (MOD n) obtain (j : Int) (h3 : a - b = m * j) from h2.left have h4 : (↑n : Int) ∣ a - b := h2.right rewrite [h3] at h4 --h4 : ↑n ∣ ↑m * j have h5 : ↑n ∣ j := Theorem_7_2_2_Int h4 h1 obtain (k : Int) (h6 : j = n * k) from h5 apply Exists.intro k --Goal : a - b = ↑(m * n) * k rewrite [Nat.cast_mul] --Goal : a - b = ↑m * ↑n * k show a - b = (m * n) * k from calc a - b _ = m * j := h3 _ = m * (n * k) := by rw [h6] _ = (m * n) * k := by ring done done --From exercises of Section 7.2 theorem rel_prime_symm {a b : Nat} (h : rel_prime a b) : rel_prime b a := sorry lemma prime_NeZero {p : Nat} (h : prime p) : NeZero p := by rewrite [neZero_iff] --Goal : p ≠ 0 define at h linarith done lemma Lemma_7_5_1 {p e d m c s : Nat} {t : Int} (h1 : prime p) (h2 : e * d = (p - 1) * s + 1) (h3 : m ^ e - c = p * t) : c ^ d ≡ m (MOD p) := by have h4 : m ^ e ≡ c (MOD p) := Exists.intro t h3 have h5 : [m ^ e]_p = [c]_p := (cc_eq_iff_congr _ _ _).rtl h4 rewrite [←Exercise_7_4_5_Nat] at h5 --h5 : [m]_p ^ e = [c]_p by_cases h6 : p ∣ m · -- Case 1. h6 : p ∣ m have h7 : m ≡ 0 (MOD p) := by obtain (j : Nat) (h8 : m = p * j) from h6 apply Exists.intro (↑j : Int) --Goal : ↑m - 0 = ↑p * ↑j rewrite [h8, Nat.cast_mul] ring done have h8 : [m]_p = [0]_p := (cc_eq_iff_congr _ _ _).rtl h7 have h9 : e * d ≠ 0 := by rewrite [h2] show (p - 1) * s + 1 ≠ 0 from Nat.add_one_ne_zero _ done have h10 : (0 : Int) ^ (e * d) = 0 := zero_pow h9 have h11 : [c ^ d]_p = [m]_p := calc [c ^ d]_p _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat] _ = ([m]_p ^ e) ^ d := by rw [h5] _ = [m]_p ^ (e * d) := by ring _ = [0]_p ^ (e * d) := by rw [h8] _ = [0 ^ (e * d)]_p := Exercise_7_4_5_Int _ _ _ _ = [0]_p := by rw [h10] _ = [m]_p := by rw [h8] show c ^ d ≡ m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11 done · -- Case 2. h6 : ¬p ∣ m have h7 : rel_prime m p := rel_prime_of_prime_not_dvd h1 h6 have h8 : rel_prime p m := rel_prime_symm h7 have h9 : NeZero p := prime_NeZero h1 have h10 : (1 : Int) ^ s = 1 := by ring have h11 : [c ^ d]_p = [m]_p := calc [c ^ d]_p _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat] _ = ([m]_p ^ e) ^ d := by rw [h5] _ = [m]_p ^ (e * d) := by ring _ = [m]_p ^ ((p - 1) * s + 1) := by rw [h2] _ = ([m]_p ^ (p - 1)) ^ s * [m]_p := by ring _ = ([m]_p ^ (phi p)) ^ s * [m]_p := by rw [phi_prime h1] _ = [1]_p ^ s * [m]_p := by rw [Theorem_7_4_2 h8] _ = [1 ^ s]_p * [m]_p := by rw [Exercise_7_4_5_Int] _ = [1]_p * [m]_p := by rw [h10] _ = [m]_p * [1]_p := by ring _ = [m]_p := Theorem_7_3_6_7 _ show c ^ d ≡ m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11 done done theorem Theorem_7_5_1 (p q n e d k m c : Nat) (p_prime : prime p) (q_prime : prime q) (p_ne_q : p ≠ q) (n_pq : n = p * q) (ed_congr_1 : e * d = k * (p - 1) * (q - 1) + 1) (h1 : [m]_n ^ e = [c]_n) : [c]_n ^ d = [m]_n := by rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] at h1 --h1 : m ^ e ≡ c (MOD n) rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] --Goal : c ^ d ≡ m (MOD n) obtain (j : Int) (h2 : m ^ e - c = n * j) from h1 rewrite [n_pq, Nat.cast_mul] at h2 --h2 : m ^ e - c = p * q * j have h3 : e * d = (p - 1) * (k * (q - 1)) + 1 := by rewrite [ed_congr_1] ring done have h4 : m ^ e - c = p * (q * j) := by rewrite [h2] ring done have congr_p : c ^ d ≡ m (MOD p) := Lemma_7_5_1 p_prime h3 h4 have h5 : e * d = (q - 1) * (k * (p - 1)) + 1 := by rewrite [ed_congr_1] ring done have h6 : m ^ e - c = q * (p * j) := by rewrite [h2] ring done have congr_q : c ^ d ≡ m (MOD q) := Lemma_7_5_1 q_prime h5 h6 have h7 : ¬q ∣ p := by by_contra h8 have h9 : q = 1 ∨ q = p := dvd_prime p_prime h8 disj_syll h9 (prime_not_one q_prime) show False from p_ne_q h9.symm done have h8 : rel_prime p q := rel_prime_of_prime_not_dvd q_prime h7 rewrite [n_pq, Lemma_7_4_5 _ _ h8] show c ^ d ≡ m (MOD p) ∧ c ^ d ≡ m (MOD q) from And.intro congr_p congr_q done /- BEGIN EXERCISES -/ namespace Exercises /- Section 7.1 -/ -- 1.

theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a

HTPI.Exercises.dvd_a_of_dvd_b_mod

htpi/HTPILib/Chap7.lean

htpi

/- Copyright 2023 Daniel J. Velleman -/ import HTPILib.Chap6 namespace HTPI /- Definitions -/ lemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by have h : n + 1 > 0 := Nat.succ_pos n show a % (n + 1) < n + 1 from Nat.mod_lt a h done def gcd (a b : Nat) : Nat := match b with | 0 => a | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd (n + 1) (a % (n + 1)) termination_by b mutual def gcd_c1 (a b : Nat) : Int := match b with | 0 => 1 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c2 (n + 1) (a % (n + 1)) --Corresponds to s = t' termination_by b def gcd_c2 (a b : Nat) : Int := match b with | 0 => 0 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c1 (n + 1) (a % (n + 1)) - (gcd_c2 (n + 1) (a % (n + 1))) * ↑(a / (n + 1)) --Corresponds to t = s' - t'q termination_by b end def prime (n : Nat) : Prop := 2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n def prime_factor (p n : Nat) : Prop := prime p ∧ p ∣ n def all_prime (l : List Nat) : Prop := ∀ p ∈ l, prime p def nondec (l : List Nat) : Prop := match l with | [] => True --Of course, True is a proposition that is always true | n :: L => (∀ m ∈ L, n ≤ m) ∧ nondec L def nondec_prime_list (l : List Nat) : Prop := all_prime l ∧ nondec l def prod (l : List Nat) : Nat := match l with | [] => 1 | n :: L => n * (prod L) def prime_factorization (n : Nat) (l : List Nat) : Prop := nondec_prime_list l ∧ prod l = n def rel_prime (a b : Nat) : Prop := gcd a b = 1 def congr_mod (m : Nat) (a b : Int) : Prop := (↑m : Int) ∣ (a - b) def cc (m : Nat) (a : Int) : ZMod m := (↑a : ZMod m) notation:50 a " ≡ " b " (MOD " m ")" => congr_mod m a b notation:max "["a"]_"m:max => cc m a def invertible {m : Nat} (X : ZMod m) : Prop := ∃ (Y : ZMod m), X * Y = [1]_m def num_rp_below (m k : Nat) : Nat := match k with | 0 => 0 | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j def phi (m : Nat) : Nat := num_rp_below m m def prod_seq {m : Nat} (j k : Nat) (f : Nat → ZMod m) : ZMod m := match j with | 0 => [1]_m | n + 1 => prod_seq n k f * f (k + n) def maps_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i < n, g i < n def one_one_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i1 < n, ∀ i2 < n, g i1 = g i2 → i1 = i2 def onto_below (n : Nat) (g : Nat → Nat) : Prop := ∀ k < n, ∃ i < n, g i = k def perm_below (n : Nat) (g : Nat → Nat) : Prop := maps_below n g ∧ one_one_below n g ∧ onto_below n g def inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m) def swap (u v i : Nat) : Nat := if i = u then v else if i = v then u else i namespace Euler --For definitions specific to Euler's theorem def F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m def G (m a i : Nat) : Nat := (a * i) % m def Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i end Euler /- Section 7.1 -/ theorem dvd_mod_of_dvd_a_b {a b d : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ (a % b) := by set q : Nat := a / b have h3 : b * q + a % b = a := Nat.div_add_mod a b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 define --Goal : ∃ (c : Nat), a % b = d * c apply Exists.intro (j - k * q) show a % b = d * (j - k * q) from calc a % b _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm _ = a - b * q := by rw [h3] _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc] _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm done theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry #eval gcd 672 161 --Answer: 7 lemma gcd_base (a : Nat) : gcd a 0 = a := by rfl lemma gcd_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd a b = gcd b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1)) rfl done lemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b ≠ 0) : a % b < b := by have h1 : b > 0 := Nat.pos_of_ne_zero h show a % b < b from Nat.mod_lt a h1 done lemma dvd_self (n : Nat) : n ∣ n := by apply Exists.intro 1 ring done theorem gcd_dvd : ∀ (b a : Nat), (gcd a b) ∣ a ∧ (gcd a b) ∣ b := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd a b_1) ∣ a ∧ (gcd a b_1) ∣ b_1 fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_base] --Goal: a ∣ a ∧ a ∣ 0 apply And.intro (dvd_self a) define apply Exists.intro 0 rfl done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_nonzero a h1] --Goal : gcd b (a % b) ∣ a ∧ gcd b (a % b) ∣ b have h2 : a % b < b := mod_nonzero_lt a h1 have h3 : (gcd b (a % b)) ∣ b ∧ (gcd b (a % b)) ∣ (a % b) := ih (a % b) h2 b apply And.intro _ h3.left show (gcd b (a % b)) ∣ a from dvd_a_of_dvd_b_mod h3.left h3.right done done theorem gcd_dvd_left (a b : Nat) : (gcd a b) ∣ a := (gcd_dvd b a).left theorem gcd_dvd_right (a b : Nat) : (gcd a b) ∣ b := (gcd_dvd b a).right lemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl lemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c1 a b = gcd_c2 b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done lemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl lemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * ↑(a / b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done theorem gcd_lin_comb : ∀ (b a : Nat), (gcd_c1 a b) * ↑a + (gcd_c2 a b) * ↑b = ↑(gcd a b) := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd_c1 a b_1) * ↑a + (gcd_c2 a b_1) * ↑b_1 = ↑(gcd a b_1) fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base] --Goal : 1 * ↑a + 0 * ↑0 = ↑a ring done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1] --Goal : gcd_c2 b (a % b) * ↑a + -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * ↑(a / b)) * ↑b = -- ↑(gcd b (a % b)) set r : Nat := a % b set q : Nat := a / b set s : Int := gcd_c1 b r set t : Int := gcd_c2 b r --Goal : t * ↑a + (s - t * ↑q) * ↑b = ↑(gcd b r) have h2 : r < b := mod_nonzero_lt a h1 have h3 : s * ↑b + t * ↑r = ↑(gcd b r) := ih r h2 b have h4 : b * q + r = a := Nat.div_add_mod a b rewrite [←h3, ←h4] rewrite [Nat.cast_add, Nat.cast_mul] --Goal : t * (↑b * ↑q + ↑r) + (s - t * ↑q) * ↑b = s * ↑b + t * ↑r ring done done #eval gcd_c1 672 161 --Answer: 6 #eval gcd_c2 672 161 --Answer: -25 --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161 theorem Theorem_7_1_6 {d a b : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ gcd a b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑d ∣ ↑(gcd a b) set s : Int := gcd_c1 a b set t : Int := gcd_c2 a b have h3 : s * ↑a + t * ↑b = ↑(gcd a b) := gcd_lin_comb b a rewrite [←h3] --Goal : ↑d ∣ s * ↑a + t * ↑b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul] --Goal : ↑d ∣ s * (↑d * ↑j) + t * (↑d * ↑k) define apply Exists.intro (s * ↑j + t * ↑k) ring done /- Section 7.2 -/ theorem dvd_trans {a b c : Nat} (h1 : a ∣ b) (h2 : b ∣ c) : a ∣ c := by define at h1; define at h2; define obtain (m : Nat) (h3 : b = a * m) from h1 obtain (n : Nat) (h4 : c = b * n) from h2 rewrite [h3, mul_assoc] at h4 apply Exists.intro (m * n) show c = a * (m * n) from h4 done lemma exists_prime_factor : ∀ (n : Nat), 2 ≤ n → ∃ (p : Nat), prime_factor p n := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, 2 ≤ n_1 → ∃ (p : Nat), prime_factor p n_1 assume h1 : 2 ≤ n by_cases h2 : prime n · -- Case 1. h2 : prime n apply Exists.intro n define --Goal : prime n ∧ n ∣ n show prime n ∧ n ∣ n from And.intro h2 (dvd_self n) done · -- Case 2. h2 : ¬prime n define at h2 --h2 : ¬(2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n) demorgan at h2 disj_syll h2 h1 obtain (a : Nat) (h3 : ∃ (b : Nat), a * b = n ∧ a < n ∧ b < n) from h2 obtain (b : Nat) (h4 : a * b = n ∧ a < n ∧ b < n) from h3 have h5 : 2 ≤ a := by by_contra h6 have h7 : a ≤ 1 := by linarith have h8 : n ≤ b := calc n _ = a * b := h4.left.symm _ ≤ 1 * b := by rel [h7] _ = b := by ring linarith --n ≤ b contradicts b < n done have h6 : ∃ (p : Nat), prime_factor p a := ih a h4.right.left h5 obtain (p : Nat) (h7 : prime_factor p a) from h6 apply Exists.intro p define --Goal : prime p ∧ p ∣ n define at h7 --h7 : prime p ∧ p ∣ a apply And.intro h7.left have h8 : a ∣ n := by apply Exists.intro b show n = a * b from (h4.left).symm done show p ∣ n from dvd_trans h7.right h8 done done lemma exists_least_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q := by set S : Set Nat := {p : Nat | prime_factor p n} have h2 : ∃ (p : Nat), p ∈ S := exists_prime_factor n h show ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q from well_ord_princ S h2 done lemma all_prime_nil : all_prime [] := by define --Goal : ∀ p ∈ [], prime p fix p : Nat contrapos --Goal : ¬prime p → p ∉ [] assume h1 : ¬prime p show p ∉ [] from List.not_mem_nil p done lemma all_prime_cons (n : Nat) (L : List Nat) : all_prime (n :: L) ↔ prime n ∧ all_prime L := by apply Iff.intro · -- (→) assume h1 : all_prime (n :: L) --Goal : prime n ∧ all_prime L define at h1 --h1 : ∀ p ∈ n :: L, prime p apply And.intro (h1 n (List.mem_cons_self n L)) define --Goal : ∀ p ∈ L, prime p fix p : Nat assume h2 : p ∈ L show prime p from h1 p (List.mem_cons_of_mem n h2) done · -- (←) assume h1 : prime n ∧ all_prime L --Goal : all_prime (n :: l) define : all_prime L at h1 define fix p : Nat assume h2 : p ∈ n :: L rewrite [List.mem_cons] at h2 --h2 : p = n ∨ p ∈ L by_cases on h2 · -- Case 1. h2 : p = n rewrite [h2] show prime n from h1.left done · -- Case 2. h2 : p ∈ L show prime p from h1.right p h2 done done done lemma nondec_nil : nondec [] := by define --Goal : True trivial --trivial proves some obviously true statements, such as True done lemma nondec_cons (n : Nat) (L : List Nat) : nondec (n :: L) ↔ (∀ m ∈ L, n ≤ m) ∧ nondec L := by rfl lemma prod_nil : prod [] = 1 := by rfl lemma prod_cons : prod (n :: L) = n * (prod L) := by rfl lemma exists_cons_of_length_eq_succ {A : Type} {l : List A} {n : Nat} (h : l.length = n + 1) : ∃ (a : A) (L : List A), l = a :: L ∧ L.length = n := by have h1 : ¬l.length = 0 := by linarith rewrite [List.length_eq_zero] at h1 obtain (a : A) (h2 : ∃ (L : List A), l = a :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List A) (h3 : l = a :: L) from h2 apply Exists.intro a apply Exists.intro L apply And.intro h3 have h4 : (a :: L).length = L.length + 1 := List.length_cons a L rewrite [←h3, h] at h4 show L.length = n from (Nat.add_right_cancel h4).symm done lemma list_elt_dvd_prod_by_length (a : Nat) : ∀ (n : Nat), ∀ (l : List Nat), l.length = n → a ∈ l → a ∣ prod l := by by_induc · --Base Case fix l : List Nat assume h1 : l.length = 0 rewrite [List.length_eq_zero] at h1 --h1 : l = [] rewrite [h1] --Goal : a ∈ [] → a ∣ prod [] contrapos assume h2 : ¬a ∣ prod [] show a ∉ [] from List.not_mem_nil a done · -- Induction Step fix n : Nat assume ih : ∀ (l : List Nat), List.length l = n → a ∈ l → a ∣ prod l fix l : List Nat assume h1 : l.length = n + 1 --Goal : a ∈ l → a ∣ prod l obtain (b : Nat) (h2 : ∃ (L : List Nat), l = b :: L ∧ L.length = n) from exists_cons_of_length_eq_succ h1 obtain (L : List Nat) (h3 : l = b :: L ∧ L.length = n) from h2 have h4 : a ∈ L → a ∣ prod L := ih L h3.right assume h5 : a ∈ l rewrite [h3.left, prod_cons] --Goal : a ∣ b * prod L rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b ∨ a ∈ L by_cases on h5 · -- Case 1. h5 : a = b apply Exists.intro (prod L) rewrite [h5] rfl done · -- Case 2. h5 : a ∈ L have h6 : a ∣ prod L := h4 h5 have h7 : prod L ∣ b * prod L := by apply Exists.intro b ring done show a ∣ b * prod L from dvd_trans h6 h7 done done done lemma list_elt_dvd_prod {a : Nat} {l : List Nat} (h : a ∈ l) : a ∣ prod l := by set n : Nat := l.length have h1 : l.length = n := by rfl show a ∣ prod l from list_elt_dvd_prod_by_length a n l h1 h done lemma exists_prime_factorization : ∀ (n : Nat), n ≥ 1 → ∃ (l : List Nat), prime_factorization n l := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, n_1 ≥ 1 → ∃ (l : List Nat), prime_factorization n_1 l assume h1 : n ≥ 1 by_cases h2 : n = 1 · -- Case 1. h2 : n = 1 apply Exists.intro [] define apply And.intro · -- Proof of nondec_prime_list [] define show all_prime [] ∧ nondec [] from And.intro all_prime_nil nondec_nil done · -- Proof of prod [] = n rewrite [prod_nil, h2] rfl done done · -- Case 2. h2 : n ≠ 1 have h3 : n ≥ 2 := lt_of_le_of_ne' h1 h2 obtain (p : Nat) (h4 : prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q) from exists_least_prime_factor h3 have p_prime_factor : prime_factor p n := h4.left define at p_prime_factor have p_prime : prime p := p_prime_factor.left have p_dvd_n : p ∣ n := p_prime_factor.right have p_least : ∀ (q : Nat), prime_factor q n → p ≤ q := h4.right obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n have h5 : m ≠ 0 := by contradict h1 with h6 have h7 : n = 0 := calc n _ = p * m := n_eq_pm _ = p * 0 := by rw [h6] _ = 0 := by ring rewrite [h7] decide done have m_pos : 0 < m := Nat.pos_of_ne_zero h5 have m_lt_n : m < n := by define at p_prime show m < n from calc m _ < m + m := by linarith _ = 2 * m := by ring _ ≤ p * m := by rel [p_prime.left] _ = n := n_eq_pm.symm done obtain (L : List Nat) (h6 : prime_factorization m L) from ih m m_lt_n m_pos define at h6 have ndpl_L : nondec_prime_list L := h6.left define at ndpl_L apply Exists.intro (p :: L) define apply And.intro · -- Proof of nondec_prime_list (p :: L) define apply And.intro · -- Proof of all_prime (p :: L) rewrite [all_prime_cons] show prime p ∧ all_prime L from And.intro p_prime ndpl_L.left done · -- Proof of nondec (p :: L) rewrite [nondec_cons] apply And.intro _ ndpl_L.right fix q : Nat assume q_in_L : q ∈ L have h7 : q ∣ prod L := list_elt_dvd_prod q_in_L rewrite [h6.right] at h7 --h7 : q ∣ m have h8 : m ∣ n := by apply Exists.intro p rewrite [n_eq_pm] ring done have q_dvd_n : q ∣ n := dvd_trans h7 h8 have ap_L : all_prime L := ndpl_L.left define at ap_L have q_prime_factor : prime_factor q n := And.intro (ap_L q q_in_L) q_dvd_n show p ≤ q from p_least q q_prime_factor done done · -- Proof of prod (p :: L) = n rewrite [prod_cons, h6.right, n_eq_pm] rfl done done done theorem Theorem_7_2_2 {a b c : Nat} (h1 : c ∣ a * b) (h2 : rel_prime a c) : c ∣ b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑c ∣ ↑b define at h1; define at h2; define obtain (j : Nat) (h3 : a * b = c * j) from h1 set s : Int := gcd_c1 a c set t : Int := gcd_c2 a c have h4 : s * ↑a + t * ↑c = ↑(gcd a c) := gcd_lin_comb c a rewrite [h2, Nat.cast_one] at h4 --h4 : s * ↑a + t * ↑c = (1 : Int) apply Exists.intro (s * ↑j + t * ↑b) show ↑b = ↑c * (s * ↑j + t * ↑b) from calc ↑b _ = (1 : Int) * ↑b := (one_mul _).symm _ = (s * ↑a + t * ↑c) * ↑b := by rw [h4] _ = s * (↑a * ↑b) + t * ↑c * ↑b := by ring _ = s * (↑c * ↑j) + t * ↑c * ↑b := by rw [←Nat.cast_mul a b, h3, Nat.cast_mul c j] _ = ↑c * (s * ↑j + t * ↑b) := by ring done lemma le_nonzero_prod_left {a b : Nat} (h : a * b ≠ 0) : a ≤ a * b := by have h1 : b ≠ 0 := by contradict h with h1 rewrite [h1] ring done have h2 : 1 ≤ b := Nat.pos_of_ne_zero h1 show a ≤ a * b from calc a = a * 1 := (mul_one a).symm _ ≤ a * b := by rel [h2] done lemma le_nonzero_prod_right {a b : Nat} (h : a * b ≠ 0) : b ≤ a * b := by rewrite [mul_comm] rewrite [mul_comm] at h show b ≤ b * a from le_nonzero_prod_left h done lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry lemma rel_prime_of_prime_not_dvd {a p : Nat} (h1 : prime p) (h2 : ¬p ∣ a) : rel_prime a p := by have h3 : gcd a p ∣ a := gcd_dvd_left a p have h4 : gcd a p ∣ p := gcd_dvd_right a p have h5 : gcd a p = 1 ∨ gcd a p = p := dvd_prime h1 h4 have h6 : gcd a p ≠ p := by contradict h2 with h6 rewrite [h6] at h3 show p ∣ a from h3 done disj_syll h5 h6 show rel_prime a p from h5 done theorem Theorem_7_2_3 {a b p : Nat} (h1 : prime p) (h2 : p ∣ a * b) : p ∣ a ∨ p ∣ b := by or_right with h3 have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3 show p ∣ b from Theorem_7_2_2 h2 h4 done lemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a ≥ 1 := by have h1 : a ≠ 0 := by by_contra h1 rewrite [h1] at h contradict h linarith done show a ≥ 1 from Nat.pos_of_ne_zero h1 done lemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by have h1 : a ≥ 1 := ge_one_of_prod_one h have h2 : a * b ≠ 0 := by linarith have h3 : a ≤ a * b := le_nonzero_prod_left h2 rewrite [h] at h3 show a = 1 from Nat.le_antisymm h3 h1 done lemma eq_one_of_dvd_one {n : Nat} (h : n ∣ 1) : n = 1 := by obtain (j : Nat) (h1 : 1 = n * j) from h show n = 1 from eq_one_of_prod_one h1.symm done lemma prime_not_one {p : Nat} (h : prime p) : p ≠ 1 := by define at h linarith done theorem Theorem_7_2_4 {p : Nat} (h1 : prime p) : ∀ (l : List Nat), p ∣ prod l → ∃ a ∈ l, p ∣ a := by apply List.rec · -- Base Case. Goal : p ∣ prod [] → ∃ a ∈ [], p ∣ a rewrite [prod_nil] assume h2 : p ∣ 1 show ∃ a ∈ [], p ∣ a from absurd (eq_one_of_dvd_one h2) (prime_not_one h1) done · -- Induction Step fix b : Nat fix L : List Nat assume ih : p ∣ prod L → ∃ a ∈ L, p ∣ a --Goal : p ∣ prod (b :: L) → ∃ a ∈ b :: L, p ∣ a assume h2 : p ∣ prod (b :: L) rewrite [prod_cons] at h2 have h3 : p ∣ b ∨ p ∣ prod L := Theorem_7_2_3 h1 h2 by_cases on h3 · -- Case 1. h3 : p ∣ b apply Exists.intro b show b ∈ b :: L ∧ p ∣ b from And.intro (List.mem_cons_self b L) h3 done · -- Case 2. h3 : p ∣ prod L obtain (a : Nat) (h4 : a ∈ L ∧ p ∣ a) from ih h3 apply Exists.intro a show a ∈ b :: L ∧ p ∣ a from And.intro (List.mem_cons_of_mem b h4.left) h4.right done done done lemma prime_in_list {p : Nat} {l : List Nat} (h1 : prime p) (h2 : all_prime l) (h3 : p ∣ prod l) : p ∈ l := by obtain (a : Nat) (h4 : a ∈ l ∧ p ∣ a) from Theorem_7_2_4 h1 l h3 define at h2 have h5 : prime a := h2 a h4.left have h6 : p = 1 ∨ p = a := dvd_prime h5 h4.right disj_syll h6 (prime_not_one h1) rewrite [h6] show a ∈ l from h4.left done lemma first_le_first {p q : Nat} {l m : List Nat} (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m)) (h3 : prod (p :: l) = prod (q :: m)) : p ≤ q := by define at h1; define at h2 have h4 : q ∣ prod (p :: l) := by define apply Exists.intro (prod m) rewrite [←prod_cons] show prod (p :: l) = prod (q :: m) from h3 done have h5 : all_prime (q :: m) := h2.left rewrite [all_prime_cons] at h5 have h6 : q ∈ p :: l := prime_in_list h5.left h1.left h4 have h7 : nondec (p :: l) := h1.right rewrite [nondec_cons] at h7 rewrite [List.mem_cons] at h6 by_cases on h6 · -- Case 1. h6 : q = p linarith done · -- Case 2. h6 : q ∈ l have h8 : ∀ m ∈ l, p ≤ m := h7.left show p ≤ q from h8 q h6 done done lemma nondec_prime_list_tail {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by define at h define rewrite [all_prime_cons, nondec_cons] at h show all_prime l ∧ nondec l from And.intro h.left.right h.right.right done lemma cons_prod_not_one {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : prod (p :: l) ≠ 1 := by define at h have h1 : all_prime (p :: l) := h.left rewrite [all_prime_cons] at h1 rewrite [prod_cons] by_contra h2 show False from (prime_not_one h1.left) (eq_one_of_prod_one h2) done lemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) : l = [] ↔ prod l = 1 := by apply Iff.intro · -- (→) assume h1 : l = [] rewrite [h1] show prod [] = 1 from prod_nil done · -- (←) contrapos assume h1 : ¬l = [] obtain (p : Nat) (h2 : ∃ (L : List Nat), l = p :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List Nat) (h3 : l = p :: L) from h2 rewrite [h3] at h rewrite [h3] show ¬prod (p :: L) = 1 from cons_prod_not_one h done done lemma prime_pos {p : Nat} (h : prime p) : p > 0 := by define at h linarith done theorem Theorem_7_2_5 : ∀ (l1 l2 : List Nat), nondec_prime_list l1 → nondec_prime_list l2 → prod l1 = prod l2 → l1 = l2 := by apply List.rec · -- Base Case. Goal : ∀ (l2 : List Nat), nondec_prime_list [] → -- nondec_prime_list l2 → prod [] = prod l2 → [] = l2 fix l2 : List Nat assume h1 : nondec_prime_list [] assume h2 : nondec_prime_list l2 assume h3 : prod [] = prod l2 rewrite [prod_nil, eq_comm, ←list_nil_iff_prod_one h2] at h3 show [] = l2 from h3.symm done · -- Induction Step fix p : Nat fix L1 : List Nat assume ih : ∀ (L2 : List Nat), nondec_prime_list L1 → nondec_prime_list L2 → prod L1 = prod L2 → L1 = L2 -- Goal : ∀ (l2 : List Nat), nondec_prime_list (p :: L1) → -- nondec_prime_list l2 → prod (p :: L1) = prod l2 → p :: L1 = l2 fix l2 : List Nat assume h1 : nondec_prime_list (p :: L1) assume h2 : nondec_prime_list l2 assume h3 : prod (p :: L1) = prod l2 have h4 : ¬prod (p :: L1) = 1 := cons_prod_not_one h1 rewrite [h3, ←list_nil_iff_prod_one h2] at h4 obtain (q : Nat) (h5 : ∃ (L : List Nat), l2 = q :: L) from List.exists_cons_of_ne_nil h4 obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5 rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2) rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2) have h7 : p ≤ q := first_le_first h1 h2 h3 have h8 : q ≤ p := first_le_first h2 h1 h3.symm have h9 : p = q := by linarith rewrite [h9, prod_cons, prod_cons] at h3 --h3 : q * prod L1 = q * prod L2 have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1 have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2 define at h2 have h12 : all_prime (q :: L2) := h2.left rewrite [all_prime_cons] at h12 have h13 : q > 0 := prime_pos h12.left have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3 have h15 : L1 = L2 := ih L2 h10 h11 h14 rewrite [h6, h9, h15] rfl done done theorem fund_thm_arith (n : Nat) (h : n ≥ 1) : ∃! (l : List Nat), prime_factorization n l := by exists_unique · -- Existence show ∃ (l : List Nat), prime_factorization n l from exists_prime_factorization n h done · -- Uniqueness fix l1 : List Nat; fix l2 : List Nat assume h1 : prime_factorization n l1 assume h2 : prime_factorization n l2 define at h1; define at h2 have h3 : prod l1 = n := h1.right rewrite [←h2.right] at h3 show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3 done done /- Section 7.3 -/ theorem congr_refl (m : Nat) : ∀ (a : Int), a ≡ a (MOD m) := by fix a : Int define --Goal : ∃ (c : Int), a - a = ↑m * c apply Exists.intro 0 ring done theorem congr_symm {m : Nat} : ∀ {a b : Int}, a ≡ b (MOD m) → b ≡ a (MOD m) := by fix a : Int; fix b : Int assume h1 : a ≡ b (MOD m) define at h1 --h1 : ∃ (c : Int), a - b = ↑m * c define --Goal : ∃ (c : Int), b - a = ↑m * c obtain (c : Int) (h2 : a - b = m * c) from h1 apply Exists.intro (-c) show b - a = m * (-c) from calc b - a _ = -(a - b) := by ring _ = -(m * c) := by rw [h2] _ = m * (-c) := by ring done theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry /- Fundamental properties of congruence classes -/ lemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) : X = Y ↔ X.val = Y.val := Fin.ext_iff lemma val_nat_eq_mod (n k : Nat) : ([k]_(n + 1)).val = k % (n + 1) := by rfl lemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl theorem cc_rep {m : Nat} (X : ZMod m) : ∃ (a : Int), X = [a]_m := match m with | 0 => by apply Exists.intro X rfl done | n + 1 => by apply Exists.intro ↑(X.val) have h1 : X.val < n + 1 := Fin.prop X rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1] rfl done theorem add_class (m : Nat) (a b : Int) : [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm theorem mul_class (m : Nat) (a b : Int) : [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm lemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ [a - b]_m = [0]_m := by apply Iff.intro · -- (→) assume h1 : [a]_m = [b]_m have h2 : a - b = a + (-b) := by ring have h3 : b + (-b) = 0 := by ring show [a - b]_m = [0]_m from calc [a - b]_m _ = [a + (-b)]_m := by rw [h2] _ = [a]_m + [-b]_m := by rw [add_class] _ = [b]_m + [-b]_m := by rw [h1] _ = [b + -b]_m := by rw [add_class] _ = [0]_m := by rw [h3] done · -- (←) assume h1 : [a - b]_m = [0]_m have h2 : b + (a - b) = a := by ring have h3 : b + 0 = b := by ring show [a]_m = [b]_m from calc [a]_m _ = [b + (a - b)]_m := by rw [h2] _ = [b]_m + [a - b]_m := by rw [add_class] _ = [b]_m + [0]_m := by rw [h1] _ = [b + 0]_m := by rw [add_class] _ = [b]_m := by rw [h3] done done lemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) : [a]_m = [0]_m → [-a]_m = [0]_m := by assume h1 : [a]_m = [0]_m have h2 : 0 + (-a) = -a := by ring have h3 : a + (-a) = 0 := by ring show [-a]_m = [0]_m from calc [-a]_m _ = [0 + (-a)]_m := by rw [h2] _ = [0]_m + [-a]_m := by rw [add_class] _ = [a]_m + [-a]_m := by rw [h1] _ = [a + (-a)]_m := by rw [add_class] _ = [0]_m := by rw [h3] done lemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) : [-a]_m = [0]_m ↔ [a]_m = [0]_m := by apply Iff.intro _ (cc_neg_zero_of_cc_zero m a) assume h1 : [-a]_m = [0]_m have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1 have h3 : -(-a) = a := by ring rewrite [h3] at h2 show [a]_m = [0]_m from h2 done lemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl lemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m ↔ m ∣ k := match m with | 0 => by have h : (0 : Int) = (↑(0 : Nat) : Int) := by rfl rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj] apply Iff.intro · -- (→) assume h1 : k = 0 rewrite [h1] show 0 ∣ 0 from dvd_self 0 done · -- (←) assume h1 : 0 ∣ k obtain (c : Nat) (h2 : k = 0 * c) from h1 rewrite [h2] ring done done | n + 1 => by rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero] show k % (n + 1) = 0 ↔ n + 1 ∣ k from (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm done lemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m ↔ ↑m ∣ a := by obtain (k : Nat) (h1 : a = ↑k ∨ a = -↑k) from Int.eq_nat_or_neg a by_cases on h1 · -- Case 1. h1: a = ↑k rewrite [h1, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done · -- Case 2. h1: a = -↑k rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done done theorem cc_eq_iff_congr (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ a ≡ b (MOD m) := calc [a]_m = [b]_m _ ↔ [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b _ ↔ ↑m ∣ (a - b) := cc_zero_iff_dvd m (a - b) _ ↔ a ≡ b (MOD m) := by rfl /- End of fundamental properties of congruence classes -/ lemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m := by have h1 : (↑m : Int) ≠ 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m) show 0 ≤ a % m from Int.emod_nonneg a h1 done lemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m) have h2 : (↑m : Int) > 0 := (Nat.cast_pos).rtl h1 show a % m < m from Int.emod_lt_of_pos a h2 done lemma congr_mod_mod (m : Nat) (a : Int) : a ≡ a % m (MOD m) := by define have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m apply Exists.intro (a / m) show a - a % m = m * (a / m) from calc a - (a % m) _ = m * (a / m) + a % m - a % m := by rw [h1] _ = m * (a / m) := by ring done lemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) := And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a)) theorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) : ∃! (r : Int), 0 ≤ r ∧ r < m ∧ a ≡ r (MOD m) := by exists_unique · -- Existence apply Exists.intro (a % m) show 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) from mod_cmpl_res m a done · -- Uniqueness fix r1 : Int; fix r2 : Int assume h1 : 0 ≤ r1 ∧ r1 < m ∧ a ≡ r1 (MOD m) assume h2 : 0 ≤ r2 ∧ r2 < m ∧ a ≡ r2 (MOD m) have h3 : r1 ≡ r2 (MOD m) := congr_trans (congr_symm h1.right.right) h2.right.right obtain (d : Int) (h4 : r1 - r2 = m * d) from h3 have h5 : r1 - r2 < m * 1 := by linarith have h6 : m * (-1) < r1 - r2 := by linarith rewrite [h4] at h5 --h5 : m * d < m * 1 rewrite [h4] at h6 --h6 : m * -1 < m * d have h7 : (↑m : Int) ≥ 0 := Nat.cast_nonneg m have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7 have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7 have h10 : d = 0 := by linarith show r1 = r2 from calc r1 _ = r1 - r2 + r2 := by ring _ = m * 0 + r2 := by rw [h4, h10] _ = r2 := by ring done done lemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m := (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a) theorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y rewrite [h1, h2] have h3 : a + b = b + a := by ring show [a]_m + [b]_m = [b]_m + [a]_m from calc [a]_m + [b]_m _ = [a + b]_m := add_class m a b _ = [b + a]_m := by rw [h3] _ = [b]_m + [a]_m := (add_class m b a).symm done theorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X rewrite [h1] have h2 : a * 1 = a := by ring show [a]_m * [1]_m = [a]_m from calc [a]_m * [1]_m _ = [a * 1]_m := mul_class m a 1 _ = [a]_m := by rw [h2] done theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry lemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) : [a]_m * [gcd_c2 m a]_m = [1]_m := by set s : Int := gcd_c1 m a have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m define at h1 rewrite [h1, Nat.cast_one] at h2 --h2 : s * ↑m + gcd_c2 m a * ↑a = 1 rewrite [mul_class, cc_eq_iff_congr] define --Goal : ∃ (c : Int), ↑a * gcd_c2 m a - 1 = ↑m * c apply Exists.intro (-s) show a * (gcd_c2 m a) - 1 = m * (-s) from calc a * (gcd_c2 m a) - 1 _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring _ = 1 + m * (-s) - 1 := by rw [h2] _ = m * (-s) := by ring done theorem Theorem_7_3_7 (m a : Nat) : invertible [a]_m ↔ rel_prime m a := by apply Iff.intro · -- (→) assume h1 : invertible [a]_m define at h1 obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1 obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y rewrite [h3, mul_class, cc_eq_iff_congr] at h2 define at h2 obtain (c : Int) (h4 : a * b - 1 = m * c) from h2 rewrite [Exercise_7_2_6] --Goal : ∃ (s t : Int), s * ↑m + t * ↑a = 1 apply Exists.intro (-c) apply Exists.intro b show (-c) * m + b * a = 1 from calc (-c) * m + b * a _ = (-c) * m + (a * b - 1) + 1 := by ring _ = (-c) * m + m * c + 1 := by rw [h4] _ = 1 := by ring done · -- (←) assume h1 : rel_prime m a define show ∃ (Y : ZMod m), [a]_m * Y = [1]_m from Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1) done done /- Section 7.4 -/ section Euler open Euler lemma num_rp_below_base {m : Nat} : num_rp_below m 0 = 0 := by rfl lemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) : num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by have h1 : num_rp_below m (j + 1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : gcd m j = 1 rewrite [if_pos h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1 show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1 done lemma num_rp_below_step_not_rp {m j : Nat} (h : ¬rel_prime m j) : num_rp_below m (j + 1) = num_rp_below m j := by have h1 : num_rp_below m (j +1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : ¬gcd m j = 1 rewrite [if_neg h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j show num_rp_below m (j + 1) = num_rp_below m j from h1 done lemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl #eval phi 10 --Answer: 4 lemma prod_inv_iff_inv {m : Nat} {X : ZMod m} (h1 : invertible X) (Y : ZMod m) : invertible (X * Y) ↔ invertible Y := by apply Iff.intro · -- (→) assume h2 : invertible (X * Y) obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2 apply Exists.intro (X * Z) rewrite [←h3] --Goal : Y * (X * Z) = X * Y * Z ring --Note that ring can do algebra in ZMod m done · -- (←) assume h2 : invertible Y obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1 obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2 apply Exists.intro (Xi * Yi) show (X * Y) * (Xi * Yi) = [1]_m from calc X * Y * (Xi * Yi) _ = (X * Xi) * (Y * Yi) := by ring _ = [1]_m * [1]_m := by rw [h3, h4] _ = [1]_m := Theorem_7_3_6_7 [1]_m done done lemma F_rp_def {m i : Nat} (h : rel_prime m i) : F m i = [i]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h --h : gcd m i = 1 rewrite [if_pos h] at h1 show F m i = [i]_m from h1 done lemma F_not_rp_def {m i : Nat} (h : ¬rel_prime m i) : F m i = [1]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h rewrite [h1, if_neg h] rfl done lemma prod_seq_base {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 0 k f = [1]_m := by rfl lemma prod_seq_step {m : Nat} (n k : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl lemma prod_seq_zero_step {m : Nat} (n : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by rewrite [prod_seq_step, zero_add] rfl done lemma prod_one {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 1 k f = f k := by rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7] rfl done lemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl lemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m := calc [G m a i]_m _ = [(a * i) % m]_m := by rfl _ = [a * i]_m := (cc_eq_mod m (a * i)).symm _ = [a]_m * [i]_m := (mul_class m a i).symm lemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) : rel_prime m (G m a i) ↔ rel_prime m i := by have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1 show rel_prime m (G m a i) ↔ rel_prime m i from calc rel_prime m (G m a i) _ ↔ invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm _ ↔ invertible ([a]_m * [i]_m) := by rw [cc_G] _ ↔ invertible [i]_m := prod_inv_iff_inv h2 ([i]_m) _ ↔ rel_prime m i := Theorem_7_3_7 m i done lemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) : F m (G m a i) = [a]_m * F m i := by have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2 show F m (G m a i) = [a]_m * F m i from calc F m (G m a i) _ = [G m a i]_m := F_rp_def h3 _ = [a]_m * [i]_m := cc_G m a i _ = [a]_m * F m i := by rw [F_rp_def h2] done lemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : ¬rel_prime m i) : F m (G m a i) = [1]_m := by rewrite [←G_rp_iff h1 i] at h2 show F m (G m a i) = [1]_m from F_not_rp_def h2 done lemma FG_prod {m a : Nat} (h1 : rel_prime m a) : ∀ (k : Nat), prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by by_induc · -- Base Case show prod_seq 0 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from calc prod_seq 0 0 ((F m) ∘ (G m a)) _ = [1]_m := prod_seq_base _ _ _ = [a]_m ^ 0 * [1]_m := by ring _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by rw [num_rp_below_base, prod_seq_base] done · -- Induction Step fix k : Nat assume ih : prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) by_cases h2 : rel_prime m k · -- Case 1. h2 : rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([a]_m * F m k) := by rw [FG_rp h1 h2] _ = [a]_m ^ ((num_rp_below m k) + 1) * ((prod_seq k 0 (F m)) * F m k) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_rp h2, prod_seq_zero_step] done · -- Case 2. h2 : ¬rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([1]_m) := by rw [FG_not_rp h1 h2] _ = [a]_m ^ (num_rp_below m k) * (prod_seq k 0 (F m) * ([1]_m)) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_not_rp h2, prod_seq_zero_step, F_not_rp_def h2] done done done lemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by define --Goal : ∀ i < m, G m a i < m fix i : Nat assume h1 : i < m rewrite [G_def] --Goal : a * i % m < m show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m) done lemma left_inv_one_one_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g' (g i) = i) : one_one_below n g := sorry lemma right_inv_onto_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g (g' i) = i) (h2 : maps_below n g') : onto_below n g := by define at h2; define fix k : Nat assume h3 : k < n apply Exists.intro (g' k) show g' k < n ∧ g (g' k) = k from And.intro (h2 k h3) (h1 k h3) done lemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m * [inv_mod m a]_m = [1]_m := by have h2 : 0 ≤ (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a) show [a]_m * [inv_mod m a]_m = [1]_m from calc [a]_m * [inv_mod m a]_m _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2] _ = [a]_m * [gcd_c2 m a]_m := by rw [←cc_eq_mod] _ = [1]_m := gcd_c2_inv h1 done lemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m := calc a * (b % m) % m = a % m * (b % m % m) % m := Nat.mul_mod _ _ _ _ = a % m * (b % m) % m := by rw [Nat.mod_mod] _ = a * b % m := (Nat.mul_mod _ _ _).symm lemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm] rfl done theorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] : ↑a ≡ ↑b (MOD m) ↔ a % m = b % m := sorry lemma mul_inv_mod_cancel {m a i : Nat} [NeZero m] (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1 rewrite [mul_class, cc_eq_iff_congr, ←Nat.cast_mul, ←Nat.cast_one, congr_iff_mod_eq_Nat] at h3 show a * inv_mod m a * i % m = i from calc a * (inv_mod m a) * i % m _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = 1 % m * i % m := by rw [h3] _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = i % m := by rw [one_mul] _ = i := Nat.mod_eq_of_lt h2 done lemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl lemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, G m a (Ginv m a i) = i := by fix i : Nat assume h2 : i < m show G m a (Ginv m a i) = i from calc G m a (Ginv m a i) _ = a * ((inv_mod m a * i) % m) % m := by rfl _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, Ginv m a (G m a i) = i := by fix i : Nat assume h2 : i < m show Ginv m a (G m a i) = i from calc Ginv m a (G m a i) _ = inv_mod m a * ((a * i) % m) % m := by rfl _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc, mul_comm (inv_mod m a)] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_maps_below (m a : Nat) [NeZero m] : maps_below m (Ginv m a) := G_maps_below m (inv_mod m a) lemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : one_one_below m (G m a) := left_inv_one_one_below (Ginv_left_inv h1) lemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : onto_below m (G m a) := right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a) lemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : perm_below m (G m a) := And.intro (G_maps_below m a) (And.intro (G_one_one_below h1) (G_onto_below h1)) --Permuting a product of congruence classes doesn't change product lemma swap_fst (u v : Nat) : swap u v u = v := by define : swap u v u --Goal : (if u = u then v else if u = v then u else u) = v have h : u = u := by rfl rewrite [if_pos h] rfl done lemma swap_snd (u v : Nat) : swap u v v = u := by define : swap u v v by_cases h1 : v = u · -- Case 1. h1 : v = u rewrite [if_pos h1] show v = u from h1 done · -- Case 2. h1 : v ≠ u rewrite [if_neg h1] have h2 : v = v := by rfl rewrite [if_pos h2] rfl done done lemma swap_other {u v i : Nat} (h1 : i ≠ u) (h2 : i ≠ v) : swap u v i = i := by define : swap u v i rewrite [if_neg h1, if_neg h2] rfl done lemma swap_values (u v i : Nat) : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := by by_cases h1 : i = u · -- Case 1. h1 : i = u apply Or.inl rewrite [h1] show swap u v u = v from swap_fst u v done · -- Case 2. h1 : i ≠ u apply Or.inr by_cases h2 : i = v · -- Case 2.1. h2 : i = v apply Or.inl rewrite [h2] show swap u v v = u from swap_snd u v done · -- Case 2.2. h2 : i ≠ v apply Or.inr show swap u v i = i from swap_other h1 h2 done done done lemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by define fix i : Nat assume h3 : i < n have h4 : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := swap_values u v i by_cases on h4 · -- Case 1. h4 : swap u v i = v rewrite [h4] show v < n from h2 done · -- Case 2. by_cases on h4 · -- Case 2.1. h4 : swap u v i = u rewrite [h4] show u < n from h1 done · -- Case 2.2. h4 : swap u v i = i rewrite [h4] show i < n from h3 done done done lemma swap_swap (u v n : Nat) : ∀ i < n, swap u v (swap u v i) = i := by fix i : Nat assume h : i < n by_cases h1 : i = u · -- Case 1. h1 : i = u rewrite [h1, swap_fst, swap_snd] rfl done · -- Case 2. h1 : i ≠ u by_cases h2 : i = v · -- Case 2.1. h2 : i = v rewrite [h2, swap_snd, swap_fst] rfl done · -- Case 2.2. h2 : i ≠ v rewrite [swap_other h1 h2, swap_other h1 h2] rfl done done done lemma swap_one_one_below (u v n) : one_one_below n (swap u v) := left_inv_one_one_below (swap_swap u v n) lemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) := right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2) lemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) := And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2)) lemma comp_perm_below {n : Nat} {f g : Nat → Nat} (h1 : perm_below n f) (h2 : perm_below n g) : perm_below n (f ∘ g) := sorry lemma trivial_swap (u : Nat) : swap u u = id := by apply funext fix x : Nat by_cases h1 : x = u · -- Case 1. h1 : x = u rewrite [h1, swap_fst] rfl done · -- Case 2. h1 : x ≠ u rewrite [swap_other h1 h1] rfl done done lemma prod_eq_fun {m : Nat} (f g : Nat → ZMod m) (k : Nat) : ∀ (n : Nat), (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g := by by_induc · -- Base Case assume h : (∀ i < 0, f (k + i) = g (k + i)) rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g assume h1 : ∀ i < n + 1, f (k + i) = g (k + i) have h2 : ∀ i < n, f (k + i) = g (k + i) := by fix i : Nat assume h2 : i < n have h3 : i < n + 1 := by linarith show f (k + i) = g (k + i) from h1 i h3 done have h3 : prod_seq n k f = prod_seq n k g := ih h2 have h4 : n < n + 1 := Nat.lt_succ_self n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4] rfl done done lemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := by have h2 : ∀ (i : Nat), i < u → (f ∘ swap u n) (0 + i) = f (0 + i) := by fix i : Nat assume h2 : i < u have h3 : 0 + i ≠ u := by linarith have h4 : 0 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl done show prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f from prod_eq_fun (f ∘ swap u n) f 0 u h2 done lemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := by have h2 : ∀ i < j, (f ∘ swap u n) (u + 1 + i) = f (u + 1 + i) := by fix i : Nat assume h2 : i < j have h3 : u + 1 + i ≠ u := by linarith have h4 : u + 1 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl show prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f from prod_eq_fun (f ∘ swap u n) f (u + 1) j h2 done lemma break_prod {m : Nat} (n : Nat) (f : Nat → ZMod m) : ∀ (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by by_induc · -- Base Case have h : n + 0 = n := by rfl rewrite [prod_seq_base, h, Theorem_7_3_6_7] rfl done · -- Induction Step fix j : Nat assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f rewrite [←add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc] rfl done done lemma break_prod_twice {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f := break_prod n f 1 rewrite [prod_one] at h2 have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f := break_prod (u + 1) f j rewrite [←h1] at h3 have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f := break_prod u f 1 rewrite [prod_one] at h4 rewrite [h3, h4] at h2 show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2 done lemma swap_prod_eq_prod {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f := by by_cases h2 : u = n · -- Case 1. h2 : u = n rewrite [h2, trivial_swap n] --Goal : prod_seq (n + 1) 0 (f ∘ id) = prod_seq (n + 1) 0 f rfl done · -- Case 2. h2 : ¬u = n have h3 : u + 1 ≤ n := Nat.lt_of_le_of_ne h1 h2 obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3 have break_f : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := break_prod_twice f h4 have break_fs : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_prod_twice (f ∘ swap u n) h4 have f_eq_fs_below : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := swap_prod_eq_prod_below f h1 have f_eq_fs_btwn : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4 show prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f from calc prod_seq (n + 1) 0 (f ∘ swap u n) _ = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_fs _ = prod_seq u 0 f * (f ∘ swap u n) u * prod_seq j (u + 1) f * (f ∘ swap u n) n := by rw [f_eq_fs_below, f_eq_fs_btwn] _ = prod_seq u 0 f * f (swap u n u) * prod_seq j (u + 1) f * f (swap u n n) := by rfl _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by rw [swap_fst, swap_snd] _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring _ = prod_seq (n + 1) 0 f := break_f.symm done done lemma perm_below_fixed {n : Nat} {g : Nat → Nat} (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry lemma perm_prod {m : Nat} (f : Nat → ZMod m) : ∀ (n : Nat), ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) := by by_induc · -- Base Case fix g : Nat → Nat assume h1 : perm_below 0 g rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) fix g : Nat → Nat assume g_pb : perm_below (n + 1) g define at g_pb have g_ob : onto_below (n + 1) g := g_pb.right.right define at g_ob have h1 : n < n + 1 := by linarith obtain (u : Nat) (h2 : u < n + 1 ∧ g u = n) from g_ob n h1 have s_pb : perm_below (n + 1) (swap u n) := swap_perm_below h2.left h1 have gs_pb_n1 : perm_below (n + 1) (g ∘ swap u n) := comp_perm_below g_pb s_pb have gs_fix_n : (g ∘ swap u n) n = n := calc (g ∘ swap u n) n _ = g (swap u n n) := by rfl _ = g u := by rw [swap_snd] _ = n := h2.right have gs_pb_n : perm_below n (g ∘ swap u n) := perm_below_fixed gs_pb_n1 gs_fix_n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f ∘ (g ∘ swap u n)) := ih (g ∘ swap u n) gs_pb_n have h3 : u ≤ n := by linarith show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f ∘ g) from calc prod_seq (n + 1) 0 f _ = prod_seq n 0 f * f n := prod_seq_zero_step n f _ = prod_seq n 0 (f ∘ (g ∘ swap u n)) * f ((g ∘ swap u n) n) := by rw [gs_prod, gs_fix_n] _ = prod_seq n 0 (f ∘ g ∘ swap u n) * (f ∘ g ∘ swap u n) n := by rfl _ = prod_seq (n + 1) 0 (f ∘ g ∘ swap u n) := (prod_seq_zero_step n (f ∘ g ∘ swap u n)).symm _ = prod_seq (n + 1) 0 ((f ∘ g) ∘ swap u n) := by rfl _ = prod_seq (n + 1) 0 (f ∘ g) := swap_prod_eq_prod (f ∘ g) h3 done done lemma F_invertible (m i : Nat) : invertible (F m i) := by by_cases h : rel_prime m i · -- Case 1. h : rel_prime m i rewrite [F_rp_def h] show invertible [i]_m from (Theorem_7_3_7 m i).rtl h done · -- Case 2. h : ¬rel_prime m i rewrite [F_not_rp_def h] apply Exists.intro [1]_m show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m done done lemma Fprod_invertible (m : Nat) : ∀ (k : Nat), invertible (prod_seq k 0 (F m)) := by by_induc · -- Base Case apply Exists.intro [1]_m show prod_seq 0 0 (F m) * [1]_m = [1]_m from calc prod_seq 0 0 (F m) * [1]_m _ = [1]_m * [1]_m := by rw [prod_seq_base] _ = [1]_m := Theorem_7_3_6_7 ([1]_m) done · -- Induction Step fix k : Nat assume ih : invertible (prod_seq k 0 (F m)) rewrite [prod_seq_zero_step] show invertible (prod_seq k 0 (F m) * (F m k)) from (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k) done done theorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m ^ (phi m) = [1]_m := by have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2 show [a]_m ^ (phi m) = [1]_m from calc [a]_m ^ (phi m) _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3] _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring _ = prod_seq m 0 (F m ∘ G m a) * Y := by rw [FG_prod h1 m, phi_def] _ = prod_seq m 0 (F m) * Y := by rw [perm_prod (F m) m (G m a) (G_perm_below h1)] _ = [1]_m := by rw [h3] done lemma Exercise_7_4_5_Int (m : Nat) (a : Int) : ∀ (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry lemma Exercise_7_4_5_Nat (m a n : Nat) : [a]_m ^ n = [a ^ n]_m := by rewrite [Exercise_7_4_5_Int] rfl done theorem Euler's_theorem {m a : Nat} [NeZero m] (h1 : rel_prime m a) : a ^ (phi m) ≡ 1 (MOD m) := by have h2 : [a]_m ^ (phi m) = [1]_m := Theorem_7_4_2 h1 rewrite [Exercise_7_4_5_Nat m a (phi m)] at h2 --h2 : [a ^ phi m]_m = [1]_m show a ^ (phi m) ≡ 1 (MOD m) from (cc_eq_iff_congr _ _ _).ltr h2 done #eval gcd 10 7 --Answer: 1. So 10 and 7 are relatively prime #eval 7 ^ phi 10 --Answer: 2401, which is congruent to 1 mod 10. end Euler /- Section 7.5 -/ lemma num_rp_prime {p : Nat} (h1 : prime p) : ∀ k < p, num_rp_below p (k + 1) = k := sorry lemma phi_prime {p : Nat} (h1 : prime p) : phi p = p - 1 := by have h2 : 1 ≤ p := prime_pos h1 have h3 : p - 1 + 1 = p := Nat.sub_add_cancel h2 have h4 : p - 1 < p := by linarith have h5 : num_rp_below p (p - 1 + 1) = p - 1 := num_rp_prime h1 (p - 1) h4 rewrite [h3] at h5 show phi p = p - 1 from h5 done theorem Theorem_7_2_2_Int {a c : Nat} {b : Int} (h1 : ↑c ∣ ↑a * b) (h2 : rel_prime a c) : ↑c ∣ b := by rewrite [Int.natCast_dvd, Int.natAbs_mul, Int.natAbs_ofNat] at h1 --h1 : c ∣ a * Int.natAbs b rewrite [Int.natCast_dvd] --Goal : c ∣ Int.natAbs b show c ∣ Int.natAbs b from Theorem_7_2_2 h1 h2 done lemma Lemma_7_4_5 {m n : Nat} (a b : Int) (h1 : rel_prime m n) : a ≡ b (MOD m * n) ↔ a ≡ b (MOD m) ∧ a ≡ b (MOD n) := by apply Iff.intro · -- (→) assume h2 : a ≡ b (MOD m * n) obtain (j : Int) (h3 : a - b = (m * n) * j) from h2 apply And.intro · -- Proof of a ≡ b (MOD m) apply Exists.intro (n * j) show a - b = m * (n * j) from calc a - b _ = m * n * j := h3 _ = m * (n * j) := by ring done · -- Proof of a ≡ b (MOD n) apply Exists.intro (m * j) show a - b = n * (m * j) from calc a - b _ = m * n * j := h3 _ = n * (m * j) := by ring done done · -- (←) assume h2 : a ≡ b (MOD m) ∧ a ≡ b (MOD n) obtain (j : Int) (h3 : a - b = m * j) from h2.left have h4 : (↑n : Int) ∣ a - b := h2.right rewrite [h3] at h4 --h4 : ↑n ∣ ↑m * j have h5 : ↑n ∣ j := Theorem_7_2_2_Int h4 h1 obtain (k : Int) (h6 : j = n * k) from h5 apply Exists.intro k --Goal : a - b = ↑(m * n) * k rewrite [Nat.cast_mul] --Goal : a - b = ↑m * ↑n * k show a - b = (m * n) * k from calc a - b _ = m * j := h3 _ = m * (n * k) := by rw [h6] _ = (m * n) * k := by ring done done --From exercises of Section 7.2 theorem rel_prime_symm {a b : Nat} (h : rel_prime a b) : rel_prime b a := sorry lemma prime_NeZero {p : Nat} (h : prime p) : NeZero p := by rewrite [neZero_iff] --Goal : p ≠ 0 define at h linarith done lemma Lemma_7_5_1 {p e d m c s : Nat} {t : Int} (h1 : prime p) (h2 : e * d = (p - 1) * s + 1) (h3 : m ^ e - c = p * t) : c ^ d ≡ m (MOD p) := by have h4 : m ^ e ≡ c (MOD p) := Exists.intro t h3 have h5 : [m ^ e]_p = [c]_p := (cc_eq_iff_congr _ _ _).rtl h4 rewrite [←Exercise_7_4_5_Nat] at h5 --h5 : [m]_p ^ e = [c]_p by_cases h6 : p ∣ m · -- Case 1. h6 : p ∣ m have h7 : m ≡ 0 (MOD p) := by obtain (j : Nat) (h8 : m = p * j) from h6 apply Exists.intro (↑j : Int) --Goal : ↑m - 0 = ↑p * ↑j rewrite [h8, Nat.cast_mul] ring done have h8 : [m]_p = [0]_p := (cc_eq_iff_congr _ _ _).rtl h7 have h9 : e * d ≠ 0 := by rewrite [h2] show (p - 1) * s + 1 ≠ 0 from Nat.add_one_ne_zero _ done have h10 : (0 : Int) ^ (e * d) = 0 := zero_pow h9 have h11 : [c ^ d]_p = [m]_p := calc [c ^ d]_p _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat] _ = ([m]_p ^ e) ^ d := by rw [h5] _ = [m]_p ^ (e * d) := by ring _ = [0]_p ^ (e * d) := by rw [h8] _ = [0 ^ (e * d)]_p := Exercise_7_4_5_Int _ _ _ _ = [0]_p := by rw [h10] _ = [m]_p := by rw [h8] show c ^ d ≡ m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11 done · -- Case 2. h6 : ¬p ∣ m have h7 : rel_prime m p := rel_prime_of_prime_not_dvd h1 h6 have h8 : rel_prime p m := rel_prime_symm h7 have h9 : NeZero p := prime_NeZero h1 have h10 : (1 : Int) ^ s = 1 := by ring have h11 : [c ^ d]_p = [m]_p := calc [c ^ d]_p _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat] _ = ([m]_p ^ e) ^ d := by rw [h5] _ = [m]_p ^ (e * d) := by ring _ = [m]_p ^ ((p - 1) * s + 1) := by rw [h2] _ = ([m]_p ^ (p - 1)) ^ s * [m]_p := by ring _ = ([m]_p ^ (phi p)) ^ s * [m]_p := by rw [phi_prime h1] _ = [1]_p ^ s * [m]_p := by rw [Theorem_7_4_2 h8] _ = [1 ^ s]_p * [m]_p := by rw [Exercise_7_4_5_Int] _ = [1]_p * [m]_p := by rw [h10] _ = [m]_p * [1]_p := by ring _ = [m]_p := Theorem_7_3_6_7 _ show c ^ d ≡ m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11 done done theorem Theorem_7_5_1 (p q n e d k m c : Nat) (p_prime : prime p) (q_prime : prime q) (p_ne_q : p ≠ q) (n_pq : n = p * q) (ed_congr_1 : e * d = k * (p - 1) * (q - 1) + 1) (h1 : [m]_n ^ e = [c]_n) : [c]_n ^ d = [m]_n := by rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] at h1 --h1 : m ^ e ≡ c (MOD n) rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] --Goal : c ^ d ≡ m (MOD n) obtain (j : Int) (h2 : m ^ e - c = n * j) from h1 rewrite [n_pq, Nat.cast_mul] at h2 --h2 : m ^ e - c = p * q * j have h3 : e * d = (p - 1) * (k * (q - 1)) + 1 := by rewrite [ed_congr_1] ring done have h4 : m ^ e - c = p * (q * j) := by rewrite [h2] ring done have congr_p : c ^ d ≡ m (MOD p) := Lemma_7_5_1 p_prime h3 h4 have h5 : e * d = (q - 1) * (k * (p - 1)) + 1 := by rewrite [ed_congr_1] ring done have h6 : m ^ e - c = q * (p * j) := by rewrite [h2] ring done have congr_q : c ^ d ≡ m (MOD q) := Lemma_7_5_1 q_prime h5 h6 have h7 : ¬q ∣ p := by by_contra h8 have h9 : q = 1 ∨ q = p := dvd_prime p_prime h8 disj_syll h9 (prime_not_one q_prime) show False from p_ne_q h9.symm done have h8 : rel_prime p q := rel_prime_of_prime_not_dvd q_prime h7 rewrite [n_pq, Lemma_7_4_5 _ _ h8] show c ^ d ≡ m (MOD p) ∧ c ^ d ≡ m (MOD q) from And.intro congr_p congr_q done /- BEGIN EXERCISES -/ namespace Exercises /- Section 7.1 -/ -- 1. theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry -- 2.

lemma gcd_comm_lt {a b : Nat} (h : a < b) : gcd a b = gcd b a

HTPI.Exercises.gcd_comm_lt

htpi/HTPILib/Chap7.lean

htpi

/- Copyright 2023 Daniel J. Velleman -/ import HTPILib.Chap6 namespace HTPI /- Definitions -/ lemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by have h : n + 1 > 0 := Nat.succ_pos n show a % (n + 1) < n + 1 from Nat.mod_lt a h done def gcd (a b : Nat) : Nat := match b with | 0 => a | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd (n + 1) (a % (n + 1)) termination_by b mutual def gcd_c1 (a b : Nat) : Int := match b with | 0 => 1 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c2 (n + 1) (a % (n + 1)) --Corresponds to s = t' termination_by b def gcd_c2 (a b : Nat) : Int := match b with | 0 => 0 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c1 (n + 1) (a % (n + 1)) - (gcd_c2 (n + 1) (a % (n + 1))) * ↑(a / (n + 1)) --Corresponds to t = s' - t'q termination_by b end def prime (n : Nat) : Prop := 2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n def prime_factor (p n : Nat) : Prop := prime p ∧ p ∣ n def all_prime (l : List Nat) : Prop := ∀ p ∈ l, prime p def nondec (l : List Nat) : Prop := match l with | [] => True --Of course, True is a proposition that is always true | n :: L => (∀ m ∈ L, n ≤ m) ∧ nondec L def nondec_prime_list (l : List Nat) : Prop := all_prime l ∧ nondec l def prod (l : List Nat) : Nat := match l with | [] => 1 | n :: L => n * (prod L) def prime_factorization (n : Nat) (l : List Nat) : Prop := nondec_prime_list l ∧ prod l = n def rel_prime (a b : Nat) : Prop := gcd a b = 1 def congr_mod (m : Nat) (a b : Int) : Prop := (↑m : Int) ∣ (a - b) def cc (m : Nat) (a : Int) : ZMod m := (↑a : ZMod m) notation:50 a " ≡ " b " (MOD " m ")" => congr_mod m a b notation:max "["a"]_"m:max => cc m a def invertible {m : Nat} (X : ZMod m) : Prop := ∃ (Y : ZMod m), X * Y = [1]_m def num_rp_below (m k : Nat) : Nat := match k with | 0 => 0 | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j def phi (m : Nat) : Nat := num_rp_below m m def prod_seq {m : Nat} (j k : Nat) (f : Nat → ZMod m) : ZMod m := match j with | 0 => [1]_m | n + 1 => prod_seq n k f * f (k + n) def maps_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i < n, g i < n def one_one_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i1 < n, ∀ i2 < n, g i1 = g i2 → i1 = i2 def onto_below (n : Nat) (g : Nat → Nat) : Prop := ∀ k < n, ∃ i < n, g i = k def perm_below (n : Nat) (g : Nat → Nat) : Prop := maps_below n g ∧ one_one_below n g ∧ onto_below n g def inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m) def swap (u v i : Nat) : Nat := if i = u then v else if i = v then u else i namespace Euler --For definitions specific to Euler's theorem def F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m def G (m a i : Nat) : Nat := (a * i) % m def Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i end Euler /- Section 7.1 -/ theorem dvd_mod_of_dvd_a_b {a b d : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ (a % b) := by set q : Nat := a / b have h3 : b * q + a % b = a := Nat.div_add_mod a b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 define --Goal : ∃ (c : Nat), a % b = d * c apply Exists.intro (j - k * q) show a % b = d * (j - k * q) from calc a % b _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm _ = a - b * q := by rw [h3] _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc] _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm done theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry #eval gcd 672 161 --Answer: 7 lemma gcd_base (a : Nat) : gcd a 0 = a := by rfl lemma gcd_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd a b = gcd b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1)) rfl done lemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b ≠ 0) : a % b < b := by have h1 : b > 0 := Nat.pos_of_ne_zero h show a % b < b from Nat.mod_lt a h1 done lemma dvd_self (n : Nat) : n ∣ n := by apply Exists.intro 1 ring done theorem gcd_dvd : ∀ (b a : Nat), (gcd a b) ∣ a ∧ (gcd a b) ∣ b := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd a b_1) ∣ a ∧ (gcd a b_1) ∣ b_1 fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_base] --Goal: a ∣ a ∧ a ∣ 0 apply And.intro (dvd_self a) define apply Exists.intro 0 rfl done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_nonzero a h1] --Goal : gcd b (a % b) ∣ a ∧ gcd b (a % b) ∣ b have h2 : a % b < b := mod_nonzero_lt a h1 have h3 : (gcd b (a % b)) ∣ b ∧ (gcd b (a % b)) ∣ (a % b) := ih (a % b) h2 b apply And.intro _ h3.left show (gcd b (a % b)) ∣ a from dvd_a_of_dvd_b_mod h3.left h3.right done done theorem gcd_dvd_left (a b : Nat) : (gcd a b) ∣ a := (gcd_dvd b a).left theorem gcd_dvd_right (a b : Nat) : (gcd a b) ∣ b := (gcd_dvd b a).right lemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl lemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c1 a b = gcd_c2 b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done lemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl lemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * ↑(a / b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done theorem gcd_lin_comb : ∀ (b a : Nat), (gcd_c1 a b) * ↑a + (gcd_c2 a b) * ↑b = ↑(gcd a b) := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd_c1 a b_1) * ↑a + (gcd_c2 a b_1) * ↑b_1 = ↑(gcd a b_1) fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base] --Goal : 1 * ↑a + 0 * ↑0 = ↑a ring done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1] --Goal : gcd_c2 b (a % b) * ↑a + -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * ↑(a / b)) * ↑b = -- ↑(gcd b (a % b)) set r : Nat := a % b set q : Nat := a / b set s : Int := gcd_c1 b r set t : Int := gcd_c2 b r --Goal : t * ↑a + (s - t * ↑q) * ↑b = ↑(gcd b r) have h2 : r < b := mod_nonzero_lt a h1 have h3 : s * ↑b + t * ↑r = ↑(gcd b r) := ih r h2 b have h4 : b * q + r = a := Nat.div_add_mod a b rewrite [←h3, ←h4] rewrite [Nat.cast_add, Nat.cast_mul] --Goal : t * (↑b * ↑q + ↑r) + (s - t * ↑q) * ↑b = s * ↑b + t * ↑r ring done done #eval gcd_c1 672 161 --Answer: 6 #eval gcd_c2 672 161 --Answer: -25 --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161 theorem Theorem_7_1_6 {d a b : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ gcd a b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑d ∣ ↑(gcd a b) set s : Int := gcd_c1 a b set t : Int := gcd_c2 a b have h3 : s * ↑a + t * ↑b = ↑(gcd a b) := gcd_lin_comb b a rewrite [←h3] --Goal : ↑d ∣ s * ↑a + t * ↑b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul] --Goal : ↑d ∣ s * (↑d * ↑j) + t * (↑d * ↑k) define apply Exists.intro (s * ↑j + t * ↑k) ring done /- Section 7.2 -/ theorem dvd_trans {a b c : Nat} (h1 : a ∣ b) (h2 : b ∣ c) : a ∣ c := by define at h1; define at h2; define obtain (m : Nat) (h3 : b = a * m) from h1 obtain (n : Nat) (h4 : c = b * n) from h2 rewrite [h3, mul_assoc] at h4 apply Exists.intro (m * n) show c = a * (m * n) from h4 done lemma exists_prime_factor : ∀ (n : Nat), 2 ≤ n → ∃ (p : Nat), prime_factor p n := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, 2 ≤ n_1 → ∃ (p : Nat), prime_factor p n_1 assume h1 : 2 ≤ n by_cases h2 : prime n · -- Case 1. h2 : prime n apply Exists.intro n define --Goal : prime n ∧ n ∣ n show prime n ∧ n ∣ n from And.intro h2 (dvd_self n) done · -- Case 2. h2 : ¬prime n define at h2 --h2 : ¬(2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n) demorgan at h2 disj_syll h2 h1 obtain (a : Nat) (h3 : ∃ (b : Nat), a * b = n ∧ a < n ∧ b < n) from h2 obtain (b : Nat) (h4 : a * b = n ∧ a < n ∧ b < n) from h3 have h5 : 2 ≤ a := by by_contra h6 have h7 : a ≤ 1 := by linarith have h8 : n ≤ b := calc n _ = a * b := h4.left.symm _ ≤ 1 * b := by rel [h7] _ = b := by ring linarith --n ≤ b contradicts b < n done have h6 : ∃ (p : Nat), prime_factor p a := ih a h4.right.left h5 obtain (p : Nat) (h7 : prime_factor p a) from h6 apply Exists.intro p define --Goal : prime p ∧ p ∣ n define at h7 --h7 : prime p ∧ p ∣ a apply And.intro h7.left have h8 : a ∣ n := by apply Exists.intro b show n = a * b from (h4.left).symm done show p ∣ n from dvd_trans h7.right h8 done done lemma exists_least_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q := by set S : Set Nat := {p : Nat | prime_factor p n} have h2 : ∃ (p : Nat), p ∈ S := exists_prime_factor n h show ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q from well_ord_princ S h2 done lemma all_prime_nil : all_prime [] := by define --Goal : ∀ p ∈ [], prime p fix p : Nat contrapos --Goal : ¬prime p → p ∉ [] assume h1 : ¬prime p show p ∉ [] from List.not_mem_nil p done lemma all_prime_cons (n : Nat) (L : List Nat) : all_prime (n :: L) ↔ prime n ∧ all_prime L := by apply Iff.intro · -- (→) assume h1 : all_prime (n :: L) --Goal : prime n ∧ all_prime L define at h1 --h1 : ∀ p ∈ n :: L, prime p apply And.intro (h1 n (List.mem_cons_self n L)) define --Goal : ∀ p ∈ L, prime p fix p : Nat assume h2 : p ∈ L show prime p from h1 p (List.mem_cons_of_mem n h2) done · -- (←) assume h1 : prime n ∧ all_prime L --Goal : all_prime (n :: l) define : all_prime L at h1 define fix p : Nat assume h2 : p ∈ n :: L rewrite [List.mem_cons] at h2 --h2 : p = n ∨ p ∈ L by_cases on h2 · -- Case 1. h2 : p = n rewrite [h2] show prime n from h1.left done · -- Case 2. h2 : p ∈ L show prime p from h1.right p h2 done done done lemma nondec_nil : nondec [] := by define --Goal : True trivial --trivial proves some obviously true statements, such as True done lemma nondec_cons (n : Nat) (L : List Nat) : nondec (n :: L) ↔ (∀ m ∈ L, n ≤ m) ∧ nondec L := by rfl lemma prod_nil : prod [] = 1 := by rfl lemma prod_cons : prod (n :: L) = n * (prod L) := by rfl lemma exists_cons_of_length_eq_succ {A : Type} {l : List A} {n : Nat} (h : l.length = n + 1) : ∃ (a : A) (L : List A), l = a :: L ∧ L.length = n := by have h1 : ¬l.length = 0 := by linarith rewrite [List.length_eq_zero] at h1 obtain (a : A) (h2 : ∃ (L : List A), l = a :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List A) (h3 : l = a :: L) from h2 apply Exists.intro a apply Exists.intro L apply And.intro h3 have h4 : (a :: L).length = L.length + 1 := List.length_cons a L rewrite [←h3, h] at h4 show L.length = n from (Nat.add_right_cancel h4).symm done lemma list_elt_dvd_prod_by_length (a : Nat) : ∀ (n : Nat), ∀ (l : List Nat), l.length = n → a ∈ l → a ∣ prod l := by by_induc · --Base Case fix l : List Nat assume h1 : l.length = 0 rewrite [List.length_eq_zero] at h1 --h1 : l = [] rewrite [h1] --Goal : a ∈ [] → a ∣ prod [] contrapos assume h2 : ¬a ∣ prod [] show a ∉ [] from List.not_mem_nil a done · -- Induction Step fix n : Nat assume ih : ∀ (l : List Nat), List.length l = n → a ∈ l → a ∣ prod l fix l : List Nat assume h1 : l.length = n + 1 --Goal : a ∈ l → a ∣ prod l obtain (b : Nat) (h2 : ∃ (L : List Nat), l = b :: L ∧ L.length = n) from exists_cons_of_length_eq_succ h1 obtain (L : List Nat) (h3 : l = b :: L ∧ L.length = n) from h2 have h4 : a ∈ L → a ∣ prod L := ih L h3.right assume h5 : a ∈ l rewrite [h3.left, prod_cons] --Goal : a ∣ b * prod L rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b ∨ a ∈ L by_cases on h5 · -- Case 1. h5 : a = b apply Exists.intro (prod L) rewrite [h5] rfl done · -- Case 2. h5 : a ∈ L have h6 : a ∣ prod L := h4 h5 have h7 : prod L ∣ b * prod L := by apply Exists.intro b ring done show a ∣ b * prod L from dvd_trans h6 h7 done done done lemma list_elt_dvd_prod {a : Nat} {l : List Nat} (h : a ∈ l) : a ∣ prod l := by set n : Nat := l.length have h1 : l.length = n := by rfl show a ∣ prod l from list_elt_dvd_prod_by_length a n l h1 h done lemma exists_prime_factorization : ∀ (n : Nat), n ≥ 1 → ∃ (l : List Nat), prime_factorization n l := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, n_1 ≥ 1 → ∃ (l : List Nat), prime_factorization n_1 l assume h1 : n ≥ 1 by_cases h2 : n = 1 · -- Case 1. h2 : n = 1 apply Exists.intro [] define apply And.intro · -- Proof of nondec_prime_list [] define show all_prime [] ∧ nondec [] from And.intro all_prime_nil nondec_nil done · -- Proof of prod [] = n rewrite [prod_nil, h2] rfl done done · -- Case 2. h2 : n ≠ 1 have h3 : n ≥ 2 := lt_of_le_of_ne' h1 h2 obtain (p : Nat) (h4 : prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q) from exists_least_prime_factor h3 have p_prime_factor : prime_factor p n := h4.left define at p_prime_factor have p_prime : prime p := p_prime_factor.left have p_dvd_n : p ∣ n := p_prime_factor.right have p_least : ∀ (q : Nat), prime_factor q n → p ≤ q := h4.right obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n have h5 : m ≠ 0 := by contradict h1 with h6 have h7 : n = 0 := calc n _ = p * m := n_eq_pm _ = p * 0 := by rw [h6] _ = 0 := by ring rewrite [h7] decide done have m_pos : 0 < m := Nat.pos_of_ne_zero h5 have m_lt_n : m < n := by define at p_prime show m < n from calc m _ < m + m := by linarith _ = 2 * m := by ring _ ≤ p * m := by rel [p_prime.left] _ = n := n_eq_pm.symm done obtain (L : List Nat) (h6 : prime_factorization m L) from ih m m_lt_n m_pos define at h6 have ndpl_L : nondec_prime_list L := h6.left define at ndpl_L apply Exists.intro (p :: L) define apply And.intro · -- Proof of nondec_prime_list (p :: L) define apply And.intro · -- Proof of all_prime (p :: L) rewrite [all_prime_cons] show prime p ∧ all_prime L from And.intro p_prime ndpl_L.left done · -- Proof of nondec (p :: L) rewrite [nondec_cons] apply And.intro _ ndpl_L.right fix q : Nat assume q_in_L : q ∈ L have h7 : q ∣ prod L := list_elt_dvd_prod q_in_L rewrite [h6.right] at h7 --h7 : q ∣ m have h8 : m ∣ n := by apply Exists.intro p rewrite [n_eq_pm] ring done have q_dvd_n : q ∣ n := dvd_trans h7 h8 have ap_L : all_prime L := ndpl_L.left define at ap_L have q_prime_factor : prime_factor q n := And.intro (ap_L q q_in_L) q_dvd_n show p ≤ q from p_least q q_prime_factor done done · -- Proof of prod (p :: L) = n rewrite [prod_cons, h6.right, n_eq_pm] rfl done done done theorem Theorem_7_2_2 {a b c : Nat} (h1 : c ∣ a * b) (h2 : rel_prime a c) : c ∣ b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑c ∣ ↑b define at h1; define at h2; define obtain (j : Nat) (h3 : a * b = c * j) from h1 set s : Int := gcd_c1 a c set t : Int := gcd_c2 a c have h4 : s * ↑a + t * ↑c = ↑(gcd a c) := gcd_lin_comb c a rewrite [h2, Nat.cast_one] at h4 --h4 : s * ↑a + t * ↑c = (1 : Int) apply Exists.intro (s * ↑j + t * ↑b) show ↑b = ↑c * (s * ↑j + t * ↑b) from calc ↑b _ = (1 : Int) * ↑b := (one_mul _).symm _ = (s * ↑a + t * ↑c) * ↑b := by rw [h4] _ = s * (↑a * ↑b) + t * ↑c * ↑b := by ring _ = s * (↑c * ↑j) + t * ↑c * ↑b := by rw [←Nat.cast_mul a b, h3, Nat.cast_mul c j] _ = ↑c * (s * ↑j + t * ↑b) := by ring done lemma le_nonzero_prod_left {a b : Nat} (h : a * b ≠ 0) : a ≤ a * b := by have h1 : b ≠ 0 := by contradict h with h1 rewrite [h1] ring done have h2 : 1 ≤ b := Nat.pos_of_ne_zero h1 show a ≤ a * b from calc a = a * 1 := (mul_one a).symm _ ≤ a * b := by rel [h2] done lemma le_nonzero_prod_right {a b : Nat} (h : a * b ≠ 0) : b ≤ a * b := by rewrite [mul_comm] rewrite [mul_comm] at h show b ≤ b * a from le_nonzero_prod_left h done lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry lemma rel_prime_of_prime_not_dvd {a p : Nat} (h1 : prime p) (h2 : ¬p ∣ a) : rel_prime a p := by have h3 : gcd a p ∣ a := gcd_dvd_left a p have h4 : gcd a p ∣ p := gcd_dvd_right a p have h5 : gcd a p = 1 ∨ gcd a p = p := dvd_prime h1 h4 have h6 : gcd a p ≠ p := by contradict h2 with h6 rewrite [h6] at h3 show p ∣ a from h3 done disj_syll h5 h6 show rel_prime a p from h5 done theorem Theorem_7_2_3 {a b p : Nat} (h1 : prime p) (h2 : p ∣ a * b) : p ∣ a ∨ p ∣ b := by or_right with h3 have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3 show p ∣ b from Theorem_7_2_2 h2 h4 done lemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a ≥ 1 := by have h1 : a ≠ 0 := by by_contra h1 rewrite [h1] at h contradict h linarith done show a ≥ 1 from Nat.pos_of_ne_zero h1 done lemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by have h1 : a ≥ 1 := ge_one_of_prod_one h have h2 : a * b ≠ 0 := by linarith have h3 : a ≤ a * b := le_nonzero_prod_left h2 rewrite [h] at h3 show a = 1 from Nat.le_antisymm h3 h1 done lemma eq_one_of_dvd_one {n : Nat} (h : n ∣ 1) : n = 1 := by obtain (j : Nat) (h1 : 1 = n * j) from h show n = 1 from eq_one_of_prod_one h1.symm done lemma prime_not_one {p : Nat} (h : prime p) : p ≠ 1 := by define at h linarith done theorem Theorem_7_2_4 {p : Nat} (h1 : prime p) : ∀ (l : List Nat), p ∣ prod l → ∃ a ∈ l, p ∣ a := by apply List.rec · -- Base Case. Goal : p ∣ prod [] → ∃ a ∈ [], p ∣ a rewrite [prod_nil] assume h2 : p ∣ 1 show ∃ a ∈ [], p ∣ a from absurd (eq_one_of_dvd_one h2) (prime_not_one h1) done · -- Induction Step fix b : Nat fix L : List Nat assume ih : p ∣ prod L → ∃ a ∈ L, p ∣ a --Goal : p ∣ prod (b :: L) → ∃ a ∈ b :: L, p ∣ a assume h2 : p ∣ prod (b :: L) rewrite [prod_cons] at h2 have h3 : p ∣ b ∨ p ∣ prod L := Theorem_7_2_3 h1 h2 by_cases on h3 · -- Case 1. h3 : p ∣ b apply Exists.intro b show b ∈ b :: L ∧ p ∣ b from And.intro (List.mem_cons_self b L) h3 done · -- Case 2. h3 : p ∣ prod L obtain (a : Nat) (h4 : a ∈ L ∧ p ∣ a) from ih h3 apply Exists.intro a show a ∈ b :: L ∧ p ∣ a from And.intro (List.mem_cons_of_mem b h4.left) h4.right done done done lemma prime_in_list {p : Nat} {l : List Nat} (h1 : prime p) (h2 : all_prime l) (h3 : p ∣ prod l) : p ∈ l := by obtain (a : Nat) (h4 : a ∈ l ∧ p ∣ a) from Theorem_7_2_4 h1 l h3 define at h2 have h5 : prime a := h2 a h4.left have h6 : p = 1 ∨ p = a := dvd_prime h5 h4.right disj_syll h6 (prime_not_one h1) rewrite [h6] show a ∈ l from h4.left done lemma first_le_first {p q : Nat} {l m : List Nat} (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m)) (h3 : prod (p :: l) = prod (q :: m)) : p ≤ q := by define at h1; define at h2 have h4 : q ∣ prod (p :: l) := by define apply Exists.intro (prod m) rewrite [←prod_cons] show prod (p :: l) = prod (q :: m) from h3 done have h5 : all_prime (q :: m) := h2.left rewrite [all_prime_cons] at h5 have h6 : q ∈ p :: l := prime_in_list h5.left h1.left h4 have h7 : nondec (p :: l) := h1.right rewrite [nondec_cons] at h7 rewrite [List.mem_cons] at h6 by_cases on h6 · -- Case 1. h6 : q = p linarith done · -- Case 2. h6 : q ∈ l have h8 : ∀ m ∈ l, p ≤ m := h7.left show p ≤ q from h8 q h6 done done lemma nondec_prime_list_tail {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by define at h define rewrite [all_prime_cons, nondec_cons] at h show all_prime l ∧ nondec l from And.intro h.left.right h.right.right done lemma cons_prod_not_one {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : prod (p :: l) ≠ 1 := by define at h have h1 : all_prime (p :: l) := h.left rewrite [all_prime_cons] at h1 rewrite [prod_cons] by_contra h2 show False from (prime_not_one h1.left) (eq_one_of_prod_one h2) done lemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) : l = [] ↔ prod l = 1 := by apply Iff.intro · -- (→) assume h1 : l = [] rewrite [h1] show prod [] = 1 from prod_nil done · -- (←) contrapos assume h1 : ¬l = [] obtain (p : Nat) (h2 : ∃ (L : List Nat), l = p :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List Nat) (h3 : l = p :: L) from h2 rewrite [h3] at h rewrite [h3] show ¬prod (p :: L) = 1 from cons_prod_not_one h done done lemma prime_pos {p : Nat} (h : prime p) : p > 0 := by define at h linarith done theorem Theorem_7_2_5 : ∀ (l1 l2 : List Nat), nondec_prime_list l1 → nondec_prime_list l2 → prod l1 = prod l2 → l1 = l2 := by apply List.rec · -- Base Case. Goal : ∀ (l2 : List Nat), nondec_prime_list [] → -- nondec_prime_list l2 → prod [] = prod l2 → [] = l2 fix l2 : List Nat assume h1 : nondec_prime_list [] assume h2 : nondec_prime_list l2 assume h3 : prod [] = prod l2 rewrite [prod_nil, eq_comm, ←list_nil_iff_prod_one h2] at h3 show [] = l2 from h3.symm done · -- Induction Step fix p : Nat fix L1 : List Nat assume ih : ∀ (L2 : List Nat), nondec_prime_list L1 → nondec_prime_list L2 → prod L1 = prod L2 → L1 = L2 -- Goal : ∀ (l2 : List Nat), nondec_prime_list (p :: L1) → -- nondec_prime_list l2 → prod (p :: L1) = prod l2 → p :: L1 = l2 fix l2 : List Nat assume h1 : nondec_prime_list (p :: L1) assume h2 : nondec_prime_list l2 assume h3 : prod (p :: L1) = prod l2 have h4 : ¬prod (p :: L1) = 1 := cons_prod_not_one h1 rewrite [h3, ←list_nil_iff_prod_one h2] at h4 obtain (q : Nat) (h5 : ∃ (L : List Nat), l2 = q :: L) from List.exists_cons_of_ne_nil h4 obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5 rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2) rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2) have h7 : p ≤ q := first_le_first h1 h2 h3 have h8 : q ≤ p := first_le_first h2 h1 h3.symm have h9 : p = q := by linarith rewrite [h9, prod_cons, prod_cons] at h3 --h3 : q * prod L1 = q * prod L2 have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1 have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2 define at h2 have h12 : all_prime (q :: L2) := h2.left rewrite [all_prime_cons] at h12 have h13 : q > 0 := prime_pos h12.left have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3 have h15 : L1 = L2 := ih L2 h10 h11 h14 rewrite [h6, h9, h15] rfl done done theorem fund_thm_arith (n : Nat) (h : n ≥ 1) : ∃! (l : List Nat), prime_factorization n l := by exists_unique · -- Existence show ∃ (l : List Nat), prime_factorization n l from exists_prime_factorization n h done · -- Uniqueness fix l1 : List Nat; fix l2 : List Nat assume h1 : prime_factorization n l1 assume h2 : prime_factorization n l2 define at h1; define at h2 have h3 : prod l1 = n := h1.right rewrite [←h2.right] at h3 show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3 done done /- Section 7.3 -/ theorem congr_refl (m : Nat) : ∀ (a : Int), a ≡ a (MOD m) := by fix a : Int define --Goal : ∃ (c : Int), a - a = ↑m * c apply Exists.intro 0 ring done theorem congr_symm {m : Nat} : ∀ {a b : Int}, a ≡ b (MOD m) → b ≡ a (MOD m) := by fix a : Int; fix b : Int assume h1 : a ≡ b (MOD m) define at h1 --h1 : ∃ (c : Int), a - b = ↑m * c define --Goal : ∃ (c : Int), b - a = ↑m * c obtain (c : Int) (h2 : a - b = m * c) from h1 apply Exists.intro (-c) show b - a = m * (-c) from calc b - a _ = -(a - b) := by ring _ = -(m * c) := by rw [h2] _ = m * (-c) := by ring done theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry /- Fundamental properties of congruence classes -/ lemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) : X = Y ↔ X.val = Y.val := Fin.ext_iff lemma val_nat_eq_mod (n k : Nat) : ([k]_(n + 1)).val = k % (n + 1) := by rfl lemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl theorem cc_rep {m : Nat} (X : ZMod m) : ∃ (a : Int), X = [a]_m := match m with | 0 => by apply Exists.intro X rfl done | n + 1 => by apply Exists.intro ↑(X.val) have h1 : X.val < n + 1 := Fin.prop X rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1] rfl done theorem add_class (m : Nat) (a b : Int) : [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm theorem mul_class (m : Nat) (a b : Int) : [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm lemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ [a - b]_m = [0]_m := by apply Iff.intro · -- (→) assume h1 : [a]_m = [b]_m have h2 : a - b = a + (-b) := by ring have h3 : b + (-b) = 0 := by ring show [a - b]_m = [0]_m from calc [a - b]_m _ = [a + (-b)]_m := by rw [h2] _ = [a]_m + [-b]_m := by rw [add_class] _ = [b]_m + [-b]_m := by rw [h1] _ = [b + -b]_m := by rw [add_class] _ = [0]_m := by rw [h3] done · -- (←) assume h1 : [a - b]_m = [0]_m have h2 : b + (a - b) = a := by ring have h3 : b + 0 = b := by ring show [a]_m = [b]_m from calc [a]_m _ = [b + (a - b)]_m := by rw [h2] _ = [b]_m + [a - b]_m := by rw [add_class] _ = [b]_m + [0]_m := by rw [h1] _ = [b + 0]_m := by rw [add_class] _ = [b]_m := by rw [h3] done done lemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) : [a]_m = [0]_m → [-a]_m = [0]_m := by assume h1 : [a]_m = [0]_m have h2 : 0 + (-a) = -a := by ring have h3 : a + (-a) = 0 := by ring show [-a]_m = [0]_m from calc [-a]_m _ = [0 + (-a)]_m := by rw [h2] _ = [0]_m + [-a]_m := by rw [add_class] _ = [a]_m + [-a]_m := by rw [h1] _ = [a + (-a)]_m := by rw [add_class] _ = [0]_m := by rw [h3] done lemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) : [-a]_m = [0]_m ↔ [a]_m = [0]_m := by apply Iff.intro _ (cc_neg_zero_of_cc_zero m a) assume h1 : [-a]_m = [0]_m have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1 have h3 : -(-a) = a := by ring rewrite [h3] at h2 show [a]_m = [0]_m from h2 done lemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl lemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m ↔ m ∣ k := match m with | 0 => by have h : (0 : Int) = (↑(0 : Nat) : Int) := by rfl rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj] apply Iff.intro · -- (→) assume h1 : k = 0 rewrite [h1] show 0 ∣ 0 from dvd_self 0 done · -- (←) assume h1 : 0 ∣ k obtain (c : Nat) (h2 : k = 0 * c) from h1 rewrite [h2] ring done done | n + 1 => by rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero] show k % (n + 1) = 0 ↔ n + 1 ∣ k from (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm done lemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m ↔ ↑m ∣ a := by obtain (k : Nat) (h1 : a = ↑k ∨ a = -↑k) from Int.eq_nat_or_neg a by_cases on h1 · -- Case 1. h1: a = ↑k rewrite [h1, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done · -- Case 2. h1: a = -↑k rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done done theorem cc_eq_iff_congr (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ a ≡ b (MOD m) := calc [a]_m = [b]_m _ ↔ [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b _ ↔ ↑m ∣ (a - b) := cc_zero_iff_dvd m (a - b) _ ↔ a ≡ b (MOD m) := by rfl /- End of fundamental properties of congruence classes -/ lemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m := by have h1 : (↑m : Int) ≠ 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m) show 0 ≤ a % m from Int.emod_nonneg a h1 done lemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m) have h2 : (↑m : Int) > 0 := (Nat.cast_pos).rtl h1 show a % m < m from Int.emod_lt_of_pos a h2 done lemma congr_mod_mod (m : Nat) (a : Int) : a ≡ a % m (MOD m) := by define have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m apply Exists.intro (a / m) show a - a % m = m * (a / m) from calc a - (a % m) _ = m * (a / m) + a % m - a % m := by rw [h1] _ = m * (a / m) := by ring done lemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) := And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a)) theorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) : ∃! (r : Int), 0 ≤ r ∧ r < m ∧ a ≡ r (MOD m) := by exists_unique · -- Existence apply Exists.intro (a % m) show 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) from mod_cmpl_res m a done · -- Uniqueness fix r1 : Int; fix r2 : Int assume h1 : 0 ≤ r1 ∧ r1 < m ∧ a ≡ r1 (MOD m) assume h2 : 0 ≤ r2 ∧ r2 < m ∧ a ≡ r2 (MOD m) have h3 : r1 ≡ r2 (MOD m) := congr_trans (congr_symm h1.right.right) h2.right.right obtain (d : Int) (h4 : r1 - r2 = m * d) from h3 have h5 : r1 - r2 < m * 1 := by linarith have h6 : m * (-1) < r1 - r2 := by linarith rewrite [h4] at h5 --h5 : m * d < m * 1 rewrite [h4] at h6 --h6 : m * -1 < m * d have h7 : (↑m : Int) ≥ 0 := Nat.cast_nonneg m have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7 have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7 have h10 : d = 0 := by linarith show r1 = r2 from calc r1 _ = r1 - r2 + r2 := by ring _ = m * 0 + r2 := by rw [h4, h10] _ = r2 := by ring done done lemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m := (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a) theorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y rewrite [h1, h2] have h3 : a + b = b + a := by ring show [a]_m + [b]_m = [b]_m + [a]_m from calc [a]_m + [b]_m _ = [a + b]_m := add_class m a b _ = [b + a]_m := by rw [h3] _ = [b]_m + [a]_m := (add_class m b a).symm done theorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X rewrite [h1] have h2 : a * 1 = a := by ring show [a]_m * [1]_m = [a]_m from calc [a]_m * [1]_m _ = [a * 1]_m := mul_class m a 1 _ = [a]_m := by rw [h2] done theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry lemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) : [a]_m * [gcd_c2 m a]_m = [1]_m := by set s : Int := gcd_c1 m a have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m define at h1 rewrite [h1, Nat.cast_one] at h2 --h2 : s * ↑m + gcd_c2 m a * ↑a = 1 rewrite [mul_class, cc_eq_iff_congr] define --Goal : ∃ (c : Int), ↑a * gcd_c2 m a - 1 = ↑m * c apply Exists.intro (-s) show a * (gcd_c2 m a) - 1 = m * (-s) from calc a * (gcd_c2 m a) - 1 _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring _ = 1 + m * (-s) - 1 := by rw [h2] _ = m * (-s) := by ring done theorem Theorem_7_3_7 (m a : Nat) : invertible [a]_m ↔ rel_prime m a := by apply Iff.intro · -- (→) assume h1 : invertible [a]_m define at h1 obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1 obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y rewrite [h3, mul_class, cc_eq_iff_congr] at h2 define at h2 obtain (c : Int) (h4 : a * b - 1 = m * c) from h2 rewrite [Exercise_7_2_6] --Goal : ∃ (s t : Int), s * ↑m + t * ↑a = 1 apply Exists.intro (-c) apply Exists.intro b show (-c) * m + b * a = 1 from calc (-c) * m + b * a _ = (-c) * m + (a * b - 1) + 1 := by ring _ = (-c) * m + m * c + 1 := by rw [h4] _ = 1 := by ring done · -- (←) assume h1 : rel_prime m a define show ∃ (Y : ZMod m), [a]_m * Y = [1]_m from Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1) done done /- Section 7.4 -/ section Euler open Euler lemma num_rp_below_base {m : Nat} : num_rp_below m 0 = 0 := by rfl lemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) : num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by have h1 : num_rp_below m (j + 1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : gcd m j = 1 rewrite [if_pos h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1 show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1 done lemma num_rp_below_step_not_rp {m j : Nat} (h : ¬rel_prime m j) : num_rp_below m (j + 1) = num_rp_below m j := by have h1 : num_rp_below m (j +1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : ¬gcd m j = 1 rewrite [if_neg h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j show num_rp_below m (j + 1) = num_rp_below m j from h1 done lemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl #eval phi 10 --Answer: 4 lemma prod_inv_iff_inv {m : Nat} {X : ZMod m} (h1 : invertible X) (Y : ZMod m) : invertible (X * Y) ↔ invertible Y := by apply Iff.intro · -- (→) assume h2 : invertible (X * Y) obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2 apply Exists.intro (X * Z) rewrite [←h3] --Goal : Y * (X * Z) = X * Y * Z ring --Note that ring can do algebra in ZMod m done · -- (←) assume h2 : invertible Y obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1 obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2 apply Exists.intro (Xi * Yi) show (X * Y) * (Xi * Yi) = [1]_m from calc X * Y * (Xi * Yi) _ = (X * Xi) * (Y * Yi) := by ring _ = [1]_m * [1]_m := by rw [h3, h4] _ = [1]_m := Theorem_7_3_6_7 [1]_m done done lemma F_rp_def {m i : Nat} (h : rel_prime m i) : F m i = [i]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h --h : gcd m i = 1 rewrite [if_pos h] at h1 show F m i = [i]_m from h1 done lemma F_not_rp_def {m i : Nat} (h : ¬rel_prime m i) : F m i = [1]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h rewrite [h1, if_neg h] rfl done lemma prod_seq_base {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 0 k f = [1]_m := by rfl lemma prod_seq_step {m : Nat} (n k : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl lemma prod_seq_zero_step {m : Nat} (n : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by rewrite [prod_seq_step, zero_add] rfl done lemma prod_one {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 1 k f = f k := by rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7] rfl done lemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl lemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m := calc [G m a i]_m _ = [(a * i) % m]_m := by rfl _ = [a * i]_m := (cc_eq_mod m (a * i)).symm _ = [a]_m * [i]_m := (mul_class m a i).symm lemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) : rel_prime m (G m a i) ↔ rel_prime m i := by have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1 show rel_prime m (G m a i) ↔ rel_prime m i from calc rel_prime m (G m a i) _ ↔ invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm _ ↔ invertible ([a]_m * [i]_m) := by rw [cc_G] _ ↔ invertible [i]_m := prod_inv_iff_inv h2 ([i]_m) _ ↔ rel_prime m i := Theorem_7_3_7 m i done lemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) : F m (G m a i) = [a]_m * F m i := by have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2 show F m (G m a i) = [a]_m * F m i from calc F m (G m a i) _ = [G m a i]_m := F_rp_def h3 _ = [a]_m * [i]_m := cc_G m a i _ = [a]_m * F m i := by rw [F_rp_def h2] done lemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : ¬rel_prime m i) : F m (G m a i) = [1]_m := by rewrite [←G_rp_iff h1 i] at h2 show F m (G m a i) = [1]_m from F_not_rp_def h2 done lemma FG_prod {m a : Nat} (h1 : rel_prime m a) : ∀ (k : Nat), prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by by_induc · -- Base Case show prod_seq 0 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from calc prod_seq 0 0 ((F m) ∘ (G m a)) _ = [1]_m := prod_seq_base _ _ _ = [a]_m ^ 0 * [1]_m := by ring _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by rw [num_rp_below_base, prod_seq_base] done · -- Induction Step fix k : Nat assume ih : prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) by_cases h2 : rel_prime m k · -- Case 1. h2 : rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([a]_m * F m k) := by rw [FG_rp h1 h2] _ = [a]_m ^ ((num_rp_below m k) + 1) * ((prod_seq k 0 (F m)) * F m k) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_rp h2, prod_seq_zero_step] done · -- Case 2. h2 : ¬rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([1]_m) := by rw [FG_not_rp h1 h2] _ = [a]_m ^ (num_rp_below m k) * (prod_seq k 0 (F m) * ([1]_m)) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_not_rp h2, prod_seq_zero_step, F_not_rp_def h2] done done done lemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by define --Goal : ∀ i < m, G m a i < m fix i : Nat assume h1 : i < m rewrite [G_def] --Goal : a * i % m < m show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m) done lemma left_inv_one_one_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g' (g i) = i) : one_one_below n g := sorry lemma right_inv_onto_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g (g' i) = i) (h2 : maps_below n g') : onto_below n g := by define at h2; define fix k : Nat assume h3 : k < n apply Exists.intro (g' k) show g' k < n ∧ g (g' k) = k from And.intro (h2 k h3) (h1 k h3) done lemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m * [inv_mod m a]_m = [1]_m := by have h2 : 0 ≤ (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a) show [a]_m * [inv_mod m a]_m = [1]_m from calc [a]_m * [inv_mod m a]_m _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2] _ = [a]_m * [gcd_c2 m a]_m := by rw [←cc_eq_mod] _ = [1]_m := gcd_c2_inv h1 done lemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m := calc a * (b % m) % m = a % m * (b % m % m) % m := Nat.mul_mod _ _ _ _ = a % m * (b % m) % m := by rw [Nat.mod_mod] _ = a * b % m := (Nat.mul_mod _ _ _).symm lemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm] rfl done theorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] : ↑a ≡ ↑b (MOD m) ↔ a % m = b % m := sorry lemma mul_inv_mod_cancel {m a i : Nat} [NeZero m] (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1 rewrite [mul_class, cc_eq_iff_congr, ←Nat.cast_mul, ←Nat.cast_one, congr_iff_mod_eq_Nat] at h3 show a * inv_mod m a * i % m = i from calc a * (inv_mod m a) * i % m _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = 1 % m * i % m := by rw [h3] _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = i % m := by rw [one_mul] _ = i := Nat.mod_eq_of_lt h2 done lemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl lemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, G m a (Ginv m a i) = i := by fix i : Nat assume h2 : i < m show G m a (Ginv m a i) = i from calc G m a (Ginv m a i) _ = a * ((inv_mod m a * i) % m) % m := by rfl _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, Ginv m a (G m a i) = i := by fix i : Nat assume h2 : i < m show Ginv m a (G m a i) = i from calc Ginv m a (G m a i) _ = inv_mod m a * ((a * i) % m) % m := by rfl _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc, mul_comm (inv_mod m a)] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_maps_below (m a : Nat) [NeZero m] : maps_below m (Ginv m a) := G_maps_below m (inv_mod m a) lemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : one_one_below m (G m a) := left_inv_one_one_below (Ginv_left_inv h1) lemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : onto_below m (G m a) := right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a) lemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : perm_below m (G m a) := And.intro (G_maps_below m a) (And.intro (G_one_one_below h1) (G_onto_below h1)) --Permuting a product of congruence classes doesn't change product lemma swap_fst (u v : Nat) : swap u v u = v := by define : swap u v u --Goal : (if u = u then v else if u = v then u else u) = v have h : u = u := by rfl rewrite [if_pos h] rfl done lemma swap_snd (u v : Nat) : swap u v v = u := by define : swap u v v by_cases h1 : v = u · -- Case 1. h1 : v = u rewrite [if_pos h1] show v = u from h1 done · -- Case 2. h1 : v ≠ u rewrite [if_neg h1] have h2 : v = v := by rfl rewrite [if_pos h2] rfl done done lemma swap_other {u v i : Nat} (h1 : i ≠ u) (h2 : i ≠ v) : swap u v i = i := by define : swap u v i rewrite [if_neg h1, if_neg h2] rfl done lemma swap_values (u v i : Nat) : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := by by_cases h1 : i = u · -- Case 1. h1 : i = u apply Or.inl rewrite [h1] show swap u v u = v from swap_fst u v done · -- Case 2. h1 : i ≠ u apply Or.inr by_cases h2 : i = v · -- Case 2.1. h2 : i = v apply Or.inl rewrite [h2] show swap u v v = u from swap_snd u v done · -- Case 2.2. h2 : i ≠ v apply Or.inr show swap u v i = i from swap_other h1 h2 done done done lemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by define fix i : Nat assume h3 : i < n have h4 : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := swap_values u v i by_cases on h4 · -- Case 1. h4 : swap u v i = v rewrite [h4] show v < n from h2 done · -- Case 2. by_cases on h4 · -- Case 2.1. h4 : swap u v i = u rewrite [h4] show u < n from h1 done · -- Case 2.2. h4 : swap u v i = i rewrite [h4] show i < n from h3 done done done lemma swap_swap (u v n : Nat) : ∀ i < n, swap u v (swap u v i) = i := by fix i : Nat assume h : i < n by_cases h1 : i = u · -- Case 1. h1 : i = u rewrite [h1, swap_fst, swap_snd] rfl done · -- Case 2. h1 : i ≠ u by_cases h2 : i = v · -- Case 2.1. h2 : i = v rewrite [h2, swap_snd, swap_fst] rfl done · -- Case 2.2. h2 : i ≠ v rewrite [swap_other h1 h2, swap_other h1 h2] rfl done done done lemma swap_one_one_below (u v n) : one_one_below n (swap u v) := left_inv_one_one_below (swap_swap u v n) lemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) := right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2) lemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) := And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2)) lemma comp_perm_below {n : Nat} {f g : Nat → Nat} (h1 : perm_below n f) (h2 : perm_below n g) : perm_below n (f ∘ g) := sorry lemma trivial_swap (u : Nat) : swap u u = id := by apply funext fix x : Nat by_cases h1 : x = u · -- Case 1. h1 : x = u rewrite [h1, swap_fst] rfl done · -- Case 2. h1 : x ≠ u rewrite [swap_other h1 h1] rfl done done lemma prod_eq_fun {m : Nat} (f g : Nat → ZMod m) (k : Nat) : ∀ (n : Nat), (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g := by by_induc · -- Base Case assume h : (∀ i < 0, f (k + i) = g (k + i)) rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g assume h1 : ∀ i < n + 1, f (k + i) = g (k + i) have h2 : ∀ i < n, f (k + i) = g (k + i) := by fix i : Nat assume h2 : i < n have h3 : i < n + 1 := by linarith show f (k + i) = g (k + i) from h1 i h3 done have h3 : prod_seq n k f = prod_seq n k g := ih h2 have h4 : n < n + 1 := Nat.lt_succ_self n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4] rfl done done lemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := by have h2 : ∀ (i : Nat), i < u → (f ∘ swap u n) (0 + i) = f (0 + i) := by fix i : Nat assume h2 : i < u have h3 : 0 + i ≠ u := by linarith have h4 : 0 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl done show prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f from prod_eq_fun (f ∘ swap u n) f 0 u h2 done lemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := by have h2 : ∀ i < j, (f ∘ swap u n) (u + 1 + i) = f (u + 1 + i) := by fix i : Nat assume h2 : i < j have h3 : u + 1 + i ≠ u := by linarith have h4 : u + 1 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl show prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f from prod_eq_fun (f ∘ swap u n) f (u + 1) j h2 done lemma break_prod {m : Nat} (n : Nat) (f : Nat → ZMod m) : ∀ (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by by_induc · -- Base Case have h : n + 0 = n := by rfl rewrite [prod_seq_base, h, Theorem_7_3_6_7] rfl done · -- Induction Step fix j : Nat assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f rewrite [←add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc] rfl done done lemma break_prod_twice {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f := break_prod n f 1 rewrite [prod_one] at h2 have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f := break_prod (u + 1) f j rewrite [←h1] at h3 have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f := break_prod u f 1 rewrite [prod_one] at h4 rewrite [h3, h4] at h2 show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2 done lemma swap_prod_eq_prod {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f := by by_cases h2 : u = n · -- Case 1. h2 : u = n rewrite [h2, trivial_swap n] --Goal : prod_seq (n + 1) 0 (f ∘ id) = prod_seq (n + 1) 0 f rfl done · -- Case 2. h2 : ¬u = n have h3 : u + 1 ≤ n := Nat.lt_of_le_of_ne h1 h2 obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3 have break_f : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := break_prod_twice f h4 have break_fs : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_prod_twice (f ∘ swap u n) h4 have f_eq_fs_below : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := swap_prod_eq_prod_below f h1 have f_eq_fs_btwn : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4 show prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f from calc prod_seq (n + 1) 0 (f ∘ swap u n) _ = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_fs _ = prod_seq u 0 f * (f ∘ swap u n) u * prod_seq j (u + 1) f * (f ∘ swap u n) n := by rw [f_eq_fs_below, f_eq_fs_btwn] _ = prod_seq u 0 f * f (swap u n u) * prod_seq j (u + 1) f * f (swap u n n) := by rfl _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by rw [swap_fst, swap_snd] _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring _ = prod_seq (n + 1) 0 f := break_f.symm done done lemma perm_below_fixed {n : Nat} {g : Nat → Nat} (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry lemma perm_prod {m : Nat} (f : Nat → ZMod m) : ∀ (n : Nat), ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) := by by_induc · -- Base Case fix g : Nat → Nat assume h1 : perm_below 0 g rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) fix g : Nat → Nat assume g_pb : perm_below (n + 1) g define at g_pb have g_ob : onto_below (n + 1) g := g_pb.right.right define at g_ob have h1 : n < n + 1 := by linarith obtain (u : Nat) (h2 : u < n + 1 ∧ g u = n) from g_ob n h1 have s_pb : perm_below (n + 1) (swap u n) := swap_perm_below h2.left h1 have gs_pb_n1 : perm_below (n + 1) (g ∘ swap u n) := comp_perm_below g_pb s_pb have gs_fix_n : (g ∘ swap u n) n = n := calc (g ∘ swap u n) n _ = g (swap u n n) := by rfl _ = g u := by rw [swap_snd] _ = n := h2.right have gs_pb_n : perm_below n (g ∘ swap u n) := perm_below_fixed gs_pb_n1 gs_fix_n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f ∘ (g ∘ swap u n)) := ih (g ∘ swap u n) gs_pb_n have h3 : u ≤ n := by linarith show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f ∘ g) from calc prod_seq (n + 1) 0 f _ = prod_seq n 0 f * f n := prod_seq_zero_step n f _ = prod_seq n 0 (f ∘ (g ∘ swap u n)) * f ((g ∘ swap u n) n) := by rw [gs_prod, gs_fix_n] _ = prod_seq n 0 (f ∘ g ∘ swap u n) * (f ∘ g ∘ swap u n) n := by rfl _ = prod_seq (n + 1) 0 (f ∘ g ∘ swap u n) := (prod_seq_zero_step n (f ∘ g ∘ swap u n)).symm _ = prod_seq (n + 1) 0 ((f ∘ g) ∘ swap u n) := by rfl _ = prod_seq (n + 1) 0 (f ∘ g) := swap_prod_eq_prod (f ∘ g) h3 done done lemma F_invertible (m i : Nat) : invertible (F m i) := by by_cases h : rel_prime m i · -- Case 1. h : rel_prime m i rewrite [F_rp_def h] show invertible [i]_m from (Theorem_7_3_7 m i).rtl h done · -- Case 2. h : ¬rel_prime m i rewrite [F_not_rp_def h] apply Exists.intro [1]_m show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m done done lemma Fprod_invertible (m : Nat) : ∀ (k : Nat), invertible (prod_seq k 0 (F m)) := by by_induc · -- Base Case apply Exists.intro [1]_m show prod_seq 0 0 (F m) * [1]_m = [1]_m from calc prod_seq 0 0 (F m) * [1]_m _ = [1]_m * [1]_m := by rw [prod_seq_base] _ = [1]_m := Theorem_7_3_6_7 ([1]_m) done · -- Induction Step fix k : Nat assume ih : invertible (prod_seq k 0 (F m)) rewrite [prod_seq_zero_step] show invertible (prod_seq k 0 (F m) * (F m k)) from (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k) done done theorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m ^ (phi m) = [1]_m := by have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2 show [a]_m ^ (phi m) = [1]_m from calc [a]_m ^ (phi m) _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3] _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring _ = prod_seq m 0 (F m ∘ G m a) * Y := by rw [FG_prod h1 m, phi_def] _ = prod_seq m 0 (F m) * Y := by rw [perm_prod (F m) m (G m a) (G_perm_below h1)] _ = [1]_m := by rw [h3] done lemma Exercise_7_4_5_Int (m : Nat) (a : Int) : ∀ (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry lemma Exercise_7_4_5_Nat (m a n : Nat) : [a]_m ^ n = [a ^ n]_m := by rewrite [Exercise_7_4_5_Int] rfl done theorem Euler's_theorem {m a : Nat} [NeZero m] (h1 : rel_prime m a) : a ^ (phi m) ≡ 1 (MOD m) := by have h2 : [a]_m ^ (phi m) = [1]_m := Theorem_7_4_2 h1 rewrite [Exercise_7_4_5_Nat m a (phi m)] at h2 --h2 : [a ^ phi m]_m = [1]_m show a ^ (phi m) ≡ 1 (MOD m) from (cc_eq_iff_congr _ _ _).ltr h2 done #eval gcd 10 7 --Answer: 1. So 10 and 7 are relatively prime #eval 7 ^ phi 10 --Answer: 2401, which is congruent to 1 mod 10. end Euler /- Section 7.5 -/ lemma num_rp_prime {p : Nat} (h1 : prime p) : ∀ k < p, num_rp_below p (k + 1) = k := sorry lemma phi_prime {p : Nat} (h1 : prime p) : phi p = p - 1 := by have h2 : 1 ≤ p := prime_pos h1 have h3 : p - 1 + 1 = p := Nat.sub_add_cancel h2 have h4 : p - 1 < p := by linarith have h5 : num_rp_below p (p - 1 + 1) = p - 1 := num_rp_prime h1 (p - 1) h4 rewrite [h3] at h5 show phi p = p - 1 from h5 done theorem Theorem_7_2_2_Int {a c : Nat} {b : Int} (h1 : ↑c ∣ ↑a * b) (h2 : rel_prime a c) : ↑c ∣ b := by rewrite [Int.natCast_dvd, Int.natAbs_mul, Int.natAbs_ofNat] at h1 --h1 : c ∣ a * Int.natAbs b rewrite [Int.natCast_dvd] --Goal : c ∣ Int.natAbs b show c ∣ Int.natAbs b from Theorem_7_2_2 h1 h2 done lemma Lemma_7_4_5 {m n : Nat} (a b : Int) (h1 : rel_prime m n) : a ≡ b (MOD m * n) ↔ a ≡ b (MOD m) ∧ a ≡ b (MOD n) := by apply Iff.intro · -- (→) assume h2 : a ≡ b (MOD m * n) obtain (j : Int) (h3 : a - b = (m * n) * j) from h2 apply And.intro · -- Proof of a ≡ b (MOD m) apply Exists.intro (n * j) show a - b = m * (n * j) from calc a - b _ = m * n * j := h3 _ = m * (n * j) := by ring done · -- Proof of a ≡ b (MOD n) apply Exists.intro (m * j) show a - b = n * (m * j) from calc a - b _ = m * n * j := h3 _ = n * (m * j) := by ring done done · -- (←) assume h2 : a ≡ b (MOD m) ∧ a ≡ b (MOD n) obtain (j : Int) (h3 : a - b = m * j) from h2.left have h4 : (↑n : Int) ∣ a - b := h2.right rewrite [h3] at h4 --h4 : ↑n ∣ ↑m * j have h5 : ↑n ∣ j := Theorem_7_2_2_Int h4 h1 obtain (k : Int) (h6 : j = n * k) from h5 apply Exists.intro k --Goal : a - b = ↑(m * n) * k rewrite [Nat.cast_mul] --Goal : a - b = ↑m * ↑n * k show a - b = (m * n) * k from calc a - b _ = m * j := h3 _ = m * (n * k) := by rw [h6] _ = (m * n) * k := by ring done done --From exercises of Section 7.2 theorem rel_prime_symm {a b : Nat} (h : rel_prime a b) : rel_prime b a := sorry lemma prime_NeZero {p : Nat} (h : prime p) : NeZero p := by rewrite [neZero_iff] --Goal : p ≠ 0 define at h linarith done lemma Lemma_7_5_1 {p e d m c s : Nat} {t : Int} (h1 : prime p) (h2 : e * d = (p - 1) * s + 1) (h3 : m ^ e - c = p * t) : c ^ d ≡ m (MOD p) := by have h4 : m ^ e ≡ c (MOD p) := Exists.intro t h3 have h5 : [m ^ e]_p = [c]_p := (cc_eq_iff_congr _ _ _).rtl h4 rewrite [←Exercise_7_4_5_Nat] at h5 --h5 : [m]_p ^ e = [c]_p by_cases h6 : p ∣ m · -- Case 1. h6 : p ∣ m have h7 : m ≡ 0 (MOD p) := by obtain (j : Nat) (h8 : m = p * j) from h6 apply Exists.intro (↑j : Int) --Goal : ↑m - 0 = ↑p * ↑j rewrite [h8, Nat.cast_mul] ring done have h8 : [m]_p = [0]_p := (cc_eq_iff_congr _ _ _).rtl h7 have h9 : e * d ≠ 0 := by rewrite [h2] show (p - 1) * s + 1 ≠ 0 from Nat.add_one_ne_zero _ done have h10 : (0 : Int) ^ (e * d) = 0 := zero_pow h9 have h11 : [c ^ d]_p = [m]_p := calc [c ^ d]_p _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat] _ = ([m]_p ^ e) ^ d := by rw [h5] _ = [m]_p ^ (e * d) := by ring _ = [0]_p ^ (e * d) := by rw [h8] _ = [0 ^ (e * d)]_p := Exercise_7_4_5_Int _ _ _ _ = [0]_p := by rw [h10] _ = [m]_p := by rw [h8] show c ^ d ≡ m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11 done · -- Case 2. h6 : ¬p ∣ m have h7 : rel_prime m p := rel_prime_of_prime_not_dvd h1 h6 have h8 : rel_prime p m := rel_prime_symm h7 have h9 : NeZero p := prime_NeZero h1 have h10 : (1 : Int) ^ s = 1 := by ring have h11 : [c ^ d]_p = [m]_p := calc [c ^ d]_p _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat] _ = ([m]_p ^ e) ^ d := by rw [h5] _ = [m]_p ^ (e * d) := by ring _ = [m]_p ^ ((p - 1) * s + 1) := by rw [h2] _ = ([m]_p ^ (p - 1)) ^ s * [m]_p := by ring _ = ([m]_p ^ (phi p)) ^ s * [m]_p := by rw [phi_prime h1] _ = [1]_p ^ s * [m]_p := by rw [Theorem_7_4_2 h8] _ = [1 ^ s]_p * [m]_p := by rw [Exercise_7_4_5_Int] _ = [1]_p * [m]_p := by rw [h10] _ = [m]_p * [1]_p := by ring _ = [m]_p := Theorem_7_3_6_7 _ show c ^ d ≡ m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11 done done theorem Theorem_7_5_1 (p q n e d k m c : Nat) (p_prime : prime p) (q_prime : prime q) (p_ne_q : p ≠ q) (n_pq : n = p * q) (ed_congr_1 : e * d = k * (p - 1) * (q - 1) + 1) (h1 : [m]_n ^ e = [c]_n) : [c]_n ^ d = [m]_n := by rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] at h1 --h1 : m ^ e ≡ c (MOD n) rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] --Goal : c ^ d ≡ m (MOD n) obtain (j : Int) (h2 : m ^ e - c = n * j) from h1 rewrite [n_pq, Nat.cast_mul] at h2 --h2 : m ^ e - c = p * q * j have h3 : e * d = (p - 1) * (k * (q - 1)) + 1 := by rewrite [ed_congr_1] ring done have h4 : m ^ e - c = p * (q * j) := by rewrite [h2] ring done have congr_p : c ^ d ≡ m (MOD p) := Lemma_7_5_1 p_prime h3 h4 have h5 : e * d = (q - 1) * (k * (p - 1)) + 1 := by rewrite [ed_congr_1] ring done have h6 : m ^ e - c = q * (p * j) := by rewrite [h2] ring done have congr_q : c ^ d ≡ m (MOD q) := Lemma_7_5_1 q_prime h5 h6 have h7 : ¬q ∣ p := by by_contra h8 have h9 : q = 1 ∨ q = p := dvd_prime p_prime h8 disj_syll h9 (prime_not_one q_prime) show False from p_ne_q h9.symm done have h8 : rel_prime p q := rel_prime_of_prime_not_dvd q_prime h7 rewrite [n_pq, Lemma_7_4_5 _ _ h8] show c ^ d ≡ m (MOD p) ∧ c ^ d ≡ m (MOD q) from And.intro congr_p congr_q done /- BEGIN EXERCISES -/ namespace Exercises /- Section 7.1 -/ -- 1. theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry -- 2. lemma gcd_comm_lt {a b : Nat} (h : a < b) : gcd a b = gcd b a := sorry

theorem gcd_comm (a b : Nat) : gcd a b = gcd b a

HTPI.Exercises.gcd_comm

htpi/HTPILib/Chap7.lean

htpi

/- Copyright 2023 Daniel J. Velleman -/ import HTPILib.Chap6 namespace HTPI /- Definitions -/ lemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by have h : n + 1 > 0 := Nat.succ_pos n show a % (n + 1) < n + 1 from Nat.mod_lt a h done def gcd (a b : Nat) : Nat := match b with | 0 => a | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd (n + 1) (a % (n + 1)) termination_by b mutual def gcd_c1 (a b : Nat) : Int := match b with | 0 => 1 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c2 (n + 1) (a % (n + 1)) --Corresponds to s = t' termination_by b def gcd_c2 (a b : Nat) : Int := match b with | 0 => 0 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c1 (n + 1) (a % (n + 1)) - (gcd_c2 (n + 1) (a % (n + 1))) * ↑(a / (n + 1)) --Corresponds to t = s' - t'q termination_by b end def prime (n : Nat) : Prop := 2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n def prime_factor (p n : Nat) : Prop := prime p ∧ p ∣ n def all_prime (l : List Nat) : Prop := ∀ p ∈ l, prime p def nondec (l : List Nat) : Prop := match l with | [] => True --Of course, True is a proposition that is always true | n :: L => (∀ m ∈ L, n ≤ m) ∧ nondec L def nondec_prime_list (l : List Nat) : Prop := all_prime l ∧ nondec l def prod (l : List Nat) : Nat := match l with | [] => 1 | n :: L => n * (prod L) def prime_factorization (n : Nat) (l : List Nat) : Prop := nondec_prime_list l ∧ prod l = n def rel_prime (a b : Nat) : Prop := gcd a b = 1 def congr_mod (m : Nat) (a b : Int) : Prop := (↑m : Int) ∣ (a - b) def cc (m : Nat) (a : Int) : ZMod m := (↑a : ZMod m) notation:50 a " ≡ " b " (MOD " m ")" => congr_mod m a b notation:max "["a"]_"m:max => cc m a def invertible {m : Nat} (X : ZMod m) : Prop := ∃ (Y : ZMod m), X * Y = [1]_m def num_rp_below (m k : Nat) : Nat := match k with | 0 => 0 | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j def phi (m : Nat) : Nat := num_rp_below m m def prod_seq {m : Nat} (j k : Nat) (f : Nat → ZMod m) : ZMod m := match j with | 0 => [1]_m | n + 1 => prod_seq n k f * f (k + n) def maps_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i < n, g i < n def one_one_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i1 < n, ∀ i2 < n, g i1 = g i2 → i1 = i2 def onto_below (n : Nat) (g : Nat → Nat) : Prop := ∀ k < n, ∃ i < n, g i = k def perm_below (n : Nat) (g : Nat → Nat) : Prop := maps_below n g ∧ one_one_below n g ∧ onto_below n g def inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m) def swap (u v i : Nat) : Nat := if i = u then v else if i = v then u else i namespace Euler --For definitions specific to Euler's theorem def F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m def G (m a i : Nat) : Nat := (a * i) % m def Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i end Euler /- Section 7.1 -/ theorem dvd_mod_of_dvd_a_b {a b d : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ (a % b) := by set q : Nat := a / b have h3 : b * q + a % b = a := Nat.div_add_mod a b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 define --Goal : ∃ (c : Nat), a % b = d * c apply Exists.intro (j - k * q) show a % b = d * (j - k * q) from calc a % b _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm _ = a - b * q := by rw [h3] _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc] _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm done theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry #eval gcd 672 161 --Answer: 7 lemma gcd_base (a : Nat) : gcd a 0 = a := by rfl lemma gcd_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd a b = gcd b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1)) rfl done lemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b ≠ 0) : a % b < b := by have h1 : b > 0 := Nat.pos_of_ne_zero h show a % b < b from Nat.mod_lt a h1 done lemma dvd_self (n : Nat) : n ∣ n := by apply Exists.intro 1 ring done theorem gcd_dvd : ∀ (b a : Nat), (gcd a b) ∣ a ∧ (gcd a b) ∣ b := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd a b_1) ∣ a ∧ (gcd a b_1) ∣ b_1 fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_base] --Goal: a ∣ a ∧ a ∣ 0 apply And.intro (dvd_self a) define apply Exists.intro 0 rfl done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_nonzero a h1] --Goal : gcd b (a % b) ∣ a ∧ gcd b (a % b) ∣ b have h2 : a % b < b := mod_nonzero_lt a h1 have h3 : (gcd b (a % b)) ∣ b ∧ (gcd b (a % b)) ∣ (a % b) := ih (a % b) h2 b apply And.intro _ h3.left show (gcd b (a % b)) ∣ a from dvd_a_of_dvd_b_mod h3.left h3.right done done theorem gcd_dvd_left (a b : Nat) : (gcd a b) ∣ a := (gcd_dvd b a).left theorem gcd_dvd_right (a b : Nat) : (gcd a b) ∣ b := (gcd_dvd b a).right lemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl lemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c1 a b = gcd_c2 b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done lemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl lemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * ↑(a / b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done theorem gcd_lin_comb : ∀ (b a : Nat), (gcd_c1 a b) * ↑a + (gcd_c2 a b) * ↑b = ↑(gcd a b) := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd_c1 a b_1) * ↑a + (gcd_c2 a b_1) * ↑b_1 = ↑(gcd a b_1) fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base] --Goal : 1 * ↑a + 0 * ↑0 = ↑a ring done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1] --Goal : gcd_c2 b (a % b) * ↑a + -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * ↑(a / b)) * ↑b = -- ↑(gcd b (a % b)) set r : Nat := a % b set q : Nat := a / b set s : Int := gcd_c1 b r set t : Int := gcd_c2 b r --Goal : t * ↑a + (s - t * ↑q) * ↑b = ↑(gcd b r) have h2 : r < b := mod_nonzero_lt a h1 have h3 : s * ↑b + t * ↑r = ↑(gcd b r) := ih r h2 b have h4 : b * q + r = a := Nat.div_add_mod a b rewrite [←h3, ←h4] rewrite [Nat.cast_add, Nat.cast_mul] --Goal : t * (↑b * ↑q + ↑r) + (s - t * ↑q) * ↑b = s * ↑b + t * ↑r ring done done #eval gcd_c1 672 161 --Answer: 6 #eval gcd_c2 672 161 --Answer: -25 --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161 theorem Theorem_7_1_6 {d a b : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ gcd a b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑d ∣ ↑(gcd a b) set s : Int := gcd_c1 a b set t : Int := gcd_c2 a b have h3 : s * ↑a + t * ↑b = ↑(gcd a b) := gcd_lin_comb b a rewrite [←h3] --Goal : ↑d ∣ s * ↑a + t * ↑b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul] --Goal : ↑d ∣ s * (↑d * ↑j) + t * (↑d * ↑k) define apply Exists.intro (s * ↑j + t * ↑k) ring done /- Section 7.2 -/ theorem dvd_trans {a b c : Nat} (h1 : a ∣ b) (h2 : b ∣ c) : a ∣ c := by define at h1; define at h2; define obtain (m : Nat) (h3 : b = a * m) from h1 obtain (n : Nat) (h4 : c = b * n) from h2 rewrite [h3, mul_assoc] at h4 apply Exists.intro (m * n) show c = a * (m * n) from h4 done lemma exists_prime_factor : ∀ (n : Nat), 2 ≤ n → ∃ (p : Nat), prime_factor p n := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, 2 ≤ n_1 → ∃ (p : Nat), prime_factor p n_1 assume h1 : 2 ≤ n by_cases h2 : prime n · -- Case 1. h2 : prime n apply Exists.intro n define --Goal : prime n ∧ n ∣ n show prime n ∧ n ∣ n from And.intro h2 (dvd_self n) done · -- Case 2. h2 : ¬prime n define at h2 --h2 : ¬(2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n) demorgan at h2 disj_syll h2 h1 obtain (a : Nat) (h3 : ∃ (b : Nat), a * b = n ∧ a < n ∧ b < n) from h2 obtain (b : Nat) (h4 : a * b = n ∧ a < n ∧ b < n) from h3 have h5 : 2 ≤ a := by by_contra h6 have h7 : a ≤ 1 := by linarith have h8 : n ≤ b := calc n _ = a * b := h4.left.symm _ ≤ 1 * b := by rel [h7] _ = b := by ring linarith --n ≤ b contradicts b < n done have h6 : ∃ (p : Nat), prime_factor p a := ih a h4.right.left h5 obtain (p : Nat) (h7 : prime_factor p a) from h6 apply Exists.intro p define --Goal : prime p ∧ p ∣ n define at h7 --h7 : prime p ∧ p ∣ a apply And.intro h7.left have h8 : a ∣ n := by apply Exists.intro b show n = a * b from (h4.left).symm done show p ∣ n from dvd_trans h7.right h8 done done lemma exists_least_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q := by set S : Set Nat := {p : Nat | prime_factor p n} have h2 : ∃ (p : Nat), p ∈ S := exists_prime_factor n h show ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q from well_ord_princ S h2 done lemma all_prime_nil : all_prime [] := by define --Goal : ∀ p ∈ [], prime p fix p : Nat contrapos --Goal : ¬prime p → p ∉ [] assume h1 : ¬prime p show p ∉ [] from List.not_mem_nil p done lemma all_prime_cons (n : Nat) (L : List Nat) : all_prime (n :: L) ↔ prime n ∧ all_prime L := by apply Iff.intro · -- (→) assume h1 : all_prime (n :: L) --Goal : prime n ∧ all_prime L define at h1 --h1 : ∀ p ∈ n :: L, prime p apply And.intro (h1 n (List.mem_cons_self n L)) define --Goal : ∀ p ∈ L, prime p fix p : Nat assume h2 : p ∈ L show prime p from h1 p (List.mem_cons_of_mem n h2) done · -- (←) assume h1 : prime n ∧ all_prime L --Goal : all_prime (n :: l) define : all_prime L at h1 define fix p : Nat assume h2 : p ∈ n :: L rewrite [List.mem_cons] at h2 --h2 : p = n ∨ p ∈ L by_cases on h2 · -- Case 1. h2 : p = n rewrite [h2] show prime n from h1.left done · -- Case 2. h2 : p ∈ L show prime p from h1.right p h2 done done done lemma nondec_nil : nondec [] := by define --Goal : True trivial --trivial proves some obviously true statements, such as True done lemma nondec_cons (n : Nat) (L : List Nat) : nondec (n :: L) ↔ (∀ m ∈ L, n ≤ m) ∧ nondec L := by rfl lemma prod_nil : prod [] = 1 := by rfl lemma prod_cons : prod (n :: L) = n * (prod L) := by rfl lemma exists_cons_of_length_eq_succ {A : Type} {l : List A} {n : Nat} (h : l.length = n + 1) : ∃ (a : A) (L : List A), l = a :: L ∧ L.length = n := by have h1 : ¬l.length = 0 := by linarith rewrite [List.length_eq_zero] at h1 obtain (a : A) (h2 : ∃ (L : List A), l = a :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List A) (h3 : l = a :: L) from h2 apply Exists.intro a apply Exists.intro L apply And.intro h3 have h4 : (a :: L).length = L.length + 1 := List.length_cons a L rewrite [←h3, h] at h4 show L.length = n from (Nat.add_right_cancel h4).symm done lemma list_elt_dvd_prod_by_length (a : Nat) : ∀ (n : Nat), ∀ (l : List Nat), l.length = n → a ∈ l → a ∣ prod l := by by_induc · --Base Case fix l : List Nat assume h1 : l.length = 0 rewrite [List.length_eq_zero] at h1 --h1 : l = [] rewrite [h1] --Goal : a ∈ [] → a ∣ prod [] contrapos assume h2 : ¬a ∣ prod [] show a ∉ [] from List.not_mem_nil a done · -- Induction Step fix n : Nat assume ih : ∀ (l : List Nat), List.length l = n → a ∈ l → a ∣ prod l fix l : List Nat assume h1 : l.length = n + 1 --Goal : a ∈ l → a ∣ prod l obtain (b : Nat) (h2 : ∃ (L : List Nat), l = b :: L ∧ L.length = n) from exists_cons_of_length_eq_succ h1 obtain (L : List Nat) (h3 : l = b :: L ∧ L.length = n) from h2 have h4 : a ∈ L → a ∣ prod L := ih L h3.right assume h5 : a ∈ l rewrite [h3.left, prod_cons] --Goal : a ∣ b * prod L rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b ∨ a ∈ L by_cases on h5 · -- Case 1. h5 : a = b apply Exists.intro (prod L) rewrite [h5] rfl done · -- Case 2. h5 : a ∈ L have h6 : a ∣ prod L := h4 h5 have h7 : prod L ∣ b * prod L := by apply Exists.intro b ring done show a ∣ b * prod L from dvd_trans h6 h7 done done done lemma list_elt_dvd_prod {a : Nat} {l : List Nat} (h : a ∈ l) : a ∣ prod l := by set n : Nat := l.length have h1 : l.length = n := by rfl show a ∣ prod l from list_elt_dvd_prod_by_length a n l h1 h done lemma exists_prime_factorization : ∀ (n : Nat), n ≥ 1 → ∃ (l : List Nat), prime_factorization n l := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, n_1 ≥ 1 → ∃ (l : List Nat), prime_factorization n_1 l assume h1 : n ≥ 1 by_cases h2 : n = 1 · -- Case 1. h2 : n = 1 apply Exists.intro [] define apply And.intro · -- Proof of nondec_prime_list [] define show all_prime [] ∧ nondec [] from And.intro all_prime_nil nondec_nil done · -- Proof of prod [] = n rewrite [prod_nil, h2] rfl done done · -- Case 2. h2 : n ≠ 1 have h3 : n ≥ 2 := lt_of_le_of_ne' h1 h2 obtain (p : Nat) (h4 : prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q) from exists_least_prime_factor h3 have p_prime_factor : prime_factor p n := h4.left define at p_prime_factor have p_prime : prime p := p_prime_factor.left have p_dvd_n : p ∣ n := p_prime_factor.right have p_least : ∀ (q : Nat), prime_factor q n → p ≤ q := h4.right obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n have h5 : m ≠ 0 := by contradict h1 with h6 have h7 : n = 0 := calc n _ = p * m := n_eq_pm _ = p * 0 := by rw [h6] _ = 0 := by ring rewrite [h7] decide done have m_pos : 0 < m := Nat.pos_of_ne_zero h5 have m_lt_n : m < n := by define at p_prime show m < n from calc m _ < m + m := by linarith _ = 2 * m := by ring _ ≤ p * m := by rel [p_prime.left] _ = n := n_eq_pm.symm done obtain (L : List Nat) (h6 : prime_factorization m L) from ih m m_lt_n m_pos define at h6 have ndpl_L : nondec_prime_list L := h6.left define at ndpl_L apply Exists.intro (p :: L) define apply And.intro · -- Proof of nondec_prime_list (p :: L) define apply And.intro · -- Proof of all_prime (p :: L) rewrite [all_prime_cons] show prime p ∧ all_prime L from And.intro p_prime ndpl_L.left done · -- Proof of nondec (p :: L) rewrite [nondec_cons] apply And.intro _ ndpl_L.right fix q : Nat assume q_in_L : q ∈ L have h7 : q ∣ prod L := list_elt_dvd_prod q_in_L rewrite [h6.right] at h7 --h7 : q ∣ m have h8 : m ∣ n := by apply Exists.intro p rewrite [n_eq_pm] ring done have q_dvd_n : q ∣ n := dvd_trans h7 h8 have ap_L : all_prime L := ndpl_L.left define at ap_L have q_prime_factor : prime_factor q n := And.intro (ap_L q q_in_L) q_dvd_n show p ≤ q from p_least q q_prime_factor done done · -- Proof of prod (p :: L) = n rewrite [prod_cons, h6.right, n_eq_pm] rfl done done done theorem Theorem_7_2_2 {a b c : Nat} (h1 : c ∣ a * b) (h2 : rel_prime a c) : c ∣ b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑c ∣ ↑b define at h1; define at h2; define obtain (j : Nat) (h3 : a * b = c * j) from h1 set s : Int := gcd_c1 a c set t : Int := gcd_c2 a c have h4 : s * ↑a + t * ↑c = ↑(gcd a c) := gcd_lin_comb c a rewrite [h2, Nat.cast_one] at h4 --h4 : s * ↑a + t * ↑c = (1 : Int) apply Exists.intro (s * ↑j + t * ↑b) show ↑b = ↑c * (s * ↑j + t * ↑b) from calc ↑b _ = (1 : Int) * ↑b := (one_mul _).symm _ = (s * ↑a + t * ↑c) * ↑b := by rw [h4] _ = s * (↑a * ↑b) + t * ↑c * ↑b := by ring _ = s * (↑c * ↑j) + t * ↑c * ↑b := by rw [←Nat.cast_mul a b, h3, Nat.cast_mul c j] _ = ↑c * (s * ↑j + t * ↑b) := by ring done lemma le_nonzero_prod_left {a b : Nat} (h : a * b ≠ 0) : a ≤ a * b := by have h1 : b ≠ 0 := by contradict h with h1 rewrite [h1] ring done have h2 : 1 ≤ b := Nat.pos_of_ne_zero h1 show a ≤ a * b from calc a = a * 1 := (mul_one a).symm _ ≤ a * b := by rel [h2] done lemma le_nonzero_prod_right {a b : Nat} (h : a * b ≠ 0) : b ≤ a * b := by rewrite [mul_comm] rewrite [mul_comm] at h show b ≤ b * a from le_nonzero_prod_left h done lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry lemma rel_prime_of_prime_not_dvd {a p : Nat} (h1 : prime p) (h2 : ¬p ∣ a) : rel_prime a p := by have h3 : gcd a p ∣ a := gcd_dvd_left a p have h4 : gcd a p ∣ p := gcd_dvd_right a p have h5 : gcd a p = 1 ∨ gcd a p = p := dvd_prime h1 h4 have h6 : gcd a p ≠ p := by contradict h2 with h6 rewrite [h6] at h3 show p ∣ a from h3 done disj_syll h5 h6 show rel_prime a p from h5 done theorem Theorem_7_2_3 {a b p : Nat} (h1 : prime p) (h2 : p ∣ a * b) : p ∣ a ∨ p ∣ b := by or_right with h3 have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3 show p ∣ b from Theorem_7_2_2 h2 h4 done lemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a ≥ 1 := by have h1 : a ≠ 0 := by by_contra h1 rewrite [h1] at h contradict h linarith done show a ≥ 1 from Nat.pos_of_ne_zero h1 done lemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by have h1 : a ≥ 1 := ge_one_of_prod_one h have h2 : a * b ≠ 0 := by linarith have h3 : a ≤ a * b := le_nonzero_prod_left h2 rewrite [h] at h3 show a = 1 from Nat.le_antisymm h3 h1 done lemma eq_one_of_dvd_one {n : Nat} (h : n ∣ 1) : n = 1 := by obtain (j : Nat) (h1 : 1 = n * j) from h show n = 1 from eq_one_of_prod_one h1.symm done lemma prime_not_one {p : Nat} (h : prime p) : p ≠ 1 := by define at h linarith done theorem Theorem_7_2_4 {p : Nat} (h1 : prime p) : ∀ (l : List Nat), p ∣ prod l → ∃ a ∈ l, p ∣ a := by apply List.rec · -- Base Case. Goal : p ∣ prod [] → ∃ a ∈ [], p ∣ a rewrite [prod_nil] assume h2 : p ∣ 1 show ∃ a ∈ [], p ∣ a from absurd (eq_one_of_dvd_one h2) (prime_not_one h1) done · -- Induction Step fix b : Nat fix L : List Nat assume ih : p ∣ prod L → ∃ a ∈ L, p ∣ a --Goal : p ∣ prod (b :: L) → ∃ a ∈ b :: L, p ∣ a assume h2 : p ∣ prod (b :: L) rewrite [prod_cons] at h2 have h3 : p ∣ b ∨ p ∣ prod L := Theorem_7_2_3 h1 h2 by_cases on h3 · -- Case 1. h3 : p ∣ b apply Exists.intro b show b ∈ b :: L ∧ p ∣ b from And.intro (List.mem_cons_self b L) h3 done · -- Case 2. h3 : p ∣ prod L obtain (a : Nat) (h4 : a ∈ L ∧ p ∣ a) from ih h3 apply Exists.intro a show a ∈ b :: L ∧ p ∣ a from And.intro (List.mem_cons_of_mem b h4.left) h4.right done done done lemma prime_in_list {p : Nat} {l : List Nat} (h1 : prime p) (h2 : all_prime l) (h3 : p ∣ prod l) : p ∈ l := by obtain (a : Nat) (h4 : a ∈ l ∧ p ∣ a) from Theorem_7_2_4 h1 l h3 define at h2 have h5 : prime a := h2 a h4.left have h6 : p = 1 ∨ p = a := dvd_prime h5 h4.right disj_syll h6 (prime_not_one h1) rewrite [h6] show a ∈ l from h4.left done lemma first_le_first {p q : Nat} {l m : List Nat} (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m)) (h3 : prod (p :: l) = prod (q :: m)) : p ≤ q := by define at h1; define at h2 have h4 : q ∣ prod (p :: l) := by define apply Exists.intro (prod m) rewrite [←prod_cons] show prod (p :: l) = prod (q :: m) from h3 done have h5 : all_prime (q :: m) := h2.left rewrite [all_prime_cons] at h5 have h6 : q ∈ p :: l := prime_in_list h5.left h1.left h4 have h7 : nondec (p :: l) := h1.right rewrite [nondec_cons] at h7 rewrite [List.mem_cons] at h6 by_cases on h6 · -- Case 1. h6 : q = p linarith done · -- Case 2. h6 : q ∈ l have h8 : ∀ m ∈ l, p ≤ m := h7.left show p ≤ q from h8 q h6 done done lemma nondec_prime_list_tail {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by define at h define rewrite [all_prime_cons, nondec_cons] at h show all_prime l ∧ nondec l from And.intro h.left.right h.right.right done lemma cons_prod_not_one {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : prod (p :: l) ≠ 1 := by define at h have h1 : all_prime (p :: l) := h.left rewrite [all_prime_cons] at h1 rewrite [prod_cons] by_contra h2 show False from (prime_not_one h1.left) (eq_one_of_prod_one h2) done lemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) : l = [] ↔ prod l = 1 := by apply Iff.intro · -- (→) assume h1 : l = [] rewrite [h1] show prod [] = 1 from prod_nil done · -- (←) contrapos assume h1 : ¬l = [] obtain (p : Nat) (h2 : ∃ (L : List Nat), l = p :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List Nat) (h3 : l = p :: L) from h2 rewrite [h3] at h rewrite [h3] show ¬prod (p :: L) = 1 from cons_prod_not_one h done done lemma prime_pos {p : Nat} (h : prime p) : p > 0 := by define at h linarith done theorem Theorem_7_2_5 : ∀ (l1 l2 : List Nat), nondec_prime_list l1 → nondec_prime_list l2 → prod l1 = prod l2 → l1 = l2 := by apply List.rec · -- Base Case. Goal : ∀ (l2 : List Nat), nondec_prime_list [] → -- nondec_prime_list l2 → prod [] = prod l2 → [] = l2 fix l2 : List Nat assume h1 : nondec_prime_list [] assume h2 : nondec_prime_list l2 assume h3 : prod [] = prod l2 rewrite [prod_nil, eq_comm, ←list_nil_iff_prod_one h2] at h3 show [] = l2 from h3.symm done · -- Induction Step fix p : Nat fix L1 : List Nat assume ih : ∀ (L2 : List Nat), nondec_prime_list L1 → nondec_prime_list L2 → prod L1 = prod L2 → L1 = L2 -- Goal : ∀ (l2 : List Nat), nondec_prime_list (p :: L1) → -- nondec_prime_list l2 → prod (p :: L1) = prod l2 → p :: L1 = l2 fix l2 : List Nat assume h1 : nondec_prime_list (p :: L1) assume h2 : nondec_prime_list l2 assume h3 : prod (p :: L1) = prod l2 have h4 : ¬prod (p :: L1) = 1 := cons_prod_not_one h1 rewrite [h3, ←list_nil_iff_prod_one h2] at h4 obtain (q : Nat) (h5 : ∃ (L : List Nat), l2 = q :: L) from List.exists_cons_of_ne_nil h4 obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5 rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2) rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2) have h7 : p ≤ q := first_le_first h1 h2 h3 have h8 : q ≤ p := first_le_first h2 h1 h3.symm have h9 : p = q := by linarith rewrite [h9, prod_cons, prod_cons] at h3 --h3 : q * prod L1 = q * prod L2 have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1 have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2 define at h2 have h12 : all_prime (q :: L2) := h2.left rewrite [all_prime_cons] at h12 have h13 : q > 0 := prime_pos h12.left have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3 have h15 : L1 = L2 := ih L2 h10 h11 h14 rewrite [h6, h9, h15] rfl done done theorem fund_thm_arith (n : Nat) (h : n ≥ 1) : ∃! (l : List Nat), prime_factorization n l := by exists_unique · -- Existence show ∃ (l : List Nat), prime_factorization n l from exists_prime_factorization n h done · -- Uniqueness fix l1 : List Nat; fix l2 : List Nat assume h1 : prime_factorization n l1 assume h2 : prime_factorization n l2 define at h1; define at h2 have h3 : prod l1 = n := h1.right rewrite [←h2.right] at h3 show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3 done done /- Section 7.3 -/ theorem congr_refl (m : Nat) : ∀ (a : Int), a ≡ a (MOD m) := by fix a : Int define --Goal : ∃ (c : Int), a - a = ↑m * c apply Exists.intro 0 ring done theorem congr_symm {m : Nat} : ∀ {a b : Int}, a ≡ b (MOD m) → b ≡ a (MOD m) := by fix a : Int; fix b : Int assume h1 : a ≡ b (MOD m) define at h1 --h1 : ∃ (c : Int), a - b = ↑m * c define --Goal : ∃ (c : Int), b - a = ↑m * c obtain (c : Int) (h2 : a - b = m * c) from h1 apply Exists.intro (-c) show b - a = m * (-c) from calc b - a _ = -(a - b) := by ring _ = -(m * c) := by rw [h2] _ = m * (-c) := by ring done theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry /- Fundamental properties of congruence classes -/ lemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) : X = Y ↔ X.val = Y.val := Fin.ext_iff lemma val_nat_eq_mod (n k : Nat) : ([k]_(n + 1)).val = k % (n + 1) := by rfl lemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl theorem cc_rep {m : Nat} (X : ZMod m) : ∃ (a : Int), X = [a]_m := match m with | 0 => by apply Exists.intro X rfl done | n + 1 => by apply Exists.intro ↑(X.val) have h1 : X.val < n + 1 := Fin.prop X rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1] rfl done theorem add_class (m : Nat) (a b : Int) : [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm theorem mul_class (m : Nat) (a b : Int) : [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm lemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ [a - b]_m = [0]_m := by apply Iff.intro · -- (→) assume h1 : [a]_m = [b]_m have h2 : a - b = a + (-b) := by ring have h3 : b + (-b) = 0 := by ring show [a - b]_m = [0]_m from calc [a - b]_m _ = [a + (-b)]_m := by rw [h2] _ = [a]_m + [-b]_m := by rw [add_class] _ = [b]_m + [-b]_m := by rw [h1] _ = [b + -b]_m := by rw [add_class] _ = [0]_m := by rw [h3] done · -- (←) assume h1 : [a - b]_m = [0]_m have h2 : b + (a - b) = a := by ring have h3 : b + 0 = b := by ring show [a]_m = [b]_m from calc [a]_m _ = [b + (a - b)]_m := by rw [h2] _ = [b]_m + [a - b]_m := by rw [add_class] _ = [b]_m + [0]_m := by rw [h1] _ = [b + 0]_m := by rw [add_class] _ = [b]_m := by rw [h3] done done lemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) : [a]_m = [0]_m → [-a]_m = [0]_m := by assume h1 : [a]_m = [0]_m have h2 : 0 + (-a) = -a := by ring have h3 : a + (-a) = 0 := by ring show [-a]_m = [0]_m from calc [-a]_m _ = [0 + (-a)]_m := by rw [h2] _ = [0]_m + [-a]_m := by rw [add_class] _ = [a]_m + [-a]_m := by rw [h1] _ = [a + (-a)]_m := by rw [add_class] _ = [0]_m := by rw [h3] done lemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) : [-a]_m = [0]_m ↔ [a]_m = [0]_m := by apply Iff.intro _ (cc_neg_zero_of_cc_zero m a) assume h1 : [-a]_m = [0]_m have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1 have h3 : -(-a) = a := by ring rewrite [h3] at h2 show [a]_m = [0]_m from h2 done lemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl lemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m ↔ m ∣ k := match m with | 0 => by have h : (0 : Int) = (↑(0 : Nat) : Int) := by rfl rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj] apply Iff.intro · -- (→) assume h1 : k = 0 rewrite [h1] show 0 ∣ 0 from dvd_self 0 done · -- (←) assume h1 : 0 ∣ k obtain (c : Nat) (h2 : k = 0 * c) from h1 rewrite [h2] ring done done | n + 1 => by rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero] show k % (n + 1) = 0 ↔ n + 1 ∣ k from (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm done lemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m ↔ ↑m ∣ a := by obtain (k : Nat) (h1 : a = ↑k ∨ a = -↑k) from Int.eq_nat_or_neg a by_cases on h1 · -- Case 1. h1: a = ↑k rewrite [h1, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done · -- Case 2. h1: a = -↑k rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done done theorem cc_eq_iff_congr (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ a ≡ b (MOD m) := calc [a]_m = [b]_m _ ↔ [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b _ ↔ ↑m ∣ (a - b) := cc_zero_iff_dvd m (a - b) _ ↔ a ≡ b (MOD m) := by rfl /- End of fundamental properties of congruence classes -/ lemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m := by have h1 : (↑m : Int) ≠ 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m) show 0 ≤ a % m from Int.emod_nonneg a h1 done lemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m) have h2 : (↑m : Int) > 0 := (Nat.cast_pos).rtl h1 show a % m < m from Int.emod_lt_of_pos a h2 done lemma congr_mod_mod (m : Nat) (a : Int) : a ≡ a % m (MOD m) := by define have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m apply Exists.intro (a / m) show a - a % m = m * (a / m) from calc a - (a % m) _ = m * (a / m) + a % m - a % m := by rw [h1] _ = m * (a / m) := by ring done lemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) := And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a)) theorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) : ∃! (r : Int), 0 ≤ r ∧ r < m ∧ a ≡ r (MOD m) := by exists_unique · -- Existence apply Exists.intro (a % m) show 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) from mod_cmpl_res m a done · -- Uniqueness fix r1 : Int; fix r2 : Int assume h1 : 0 ≤ r1 ∧ r1 < m ∧ a ≡ r1 (MOD m) assume h2 : 0 ≤ r2 ∧ r2 < m ∧ a ≡ r2 (MOD m) have h3 : r1 ≡ r2 (MOD m) := congr_trans (congr_symm h1.right.right) h2.right.right obtain (d : Int) (h4 : r1 - r2 = m * d) from h3 have h5 : r1 - r2 < m * 1 := by linarith have h6 : m * (-1) < r1 - r2 := by linarith rewrite [h4] at h5 --h5 : m * d < m * 1 rewrite [h4] at h6 --h6 : m * -1 < m * d have h7 : (↑m : Int) ≥ 0 := Nat.cast_nonneg m have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7 have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7 have h10 : d = 0 := by linarith show r1 = r2 from calc r1 _ = r1 - r2 + r2 := by ring _ = m * 0 + r2 := by rw [h4, h10] _ = r2 := by ring done done lemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m := (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a) theorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y rewrite [h1, h2] have h3 : a + b = b + a := by ring show [a]_m + [b]_m = [b]_m + [a]_m from calc [a]_m + [b]_m _ = [a + b]_m := add_class m a b _ = [b + a]_m := by rw [h3] _ = [b]_m + [a]_m := (add_class m b a).symm done theorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X rewrite [h1] have h2 : a * 1 = a := by ring show [a]_m * [1]_m = [a]_m from calc [a]_m * [1]_m _ = [a * 1]_m := mul_class m a 1 _ = [a]_m := by rw [h2] done theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry lemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) : [a]_m * [gcd_c2 m a]_m = [1]_m := by set s : Int := gcd_c1 m a have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m define at h1 rewrite [h1, Nat.cast_one] at h2 --h2 : s * ↑m + gcd_c2 m a * ↑a = 1 rewrite [mul_class, cc_eq_iff_congr] define --Goal : ∃ (c : Int), ↑a * gcd_c2 m a - 1 = ↑m * c apply Exists.intro (-s) show a * (gcd_c2 m a) - 1 = m * (-s) from calc a * (gcd_c2 m a) - 1 _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring _ = 1 + m * (-s) - 1 := by rw [h2] _ = m * (-s) := by ring done theorem Theorem_7_3_7 (m a : Nat) : invertible [a]_m ↔ rel_prime m a := by apply Iff.intro · -- (→) assume h1 : invertible [a]_m define at h1 obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1 obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y rewrite [h3, mul_class, cc_eq_iff_congr] at h2 define at h2 obtain (c : Int) (h4 : a * b - 1 = m * c) from h2 rewrite [Exercise_7_2_6] --Goal : ∃ (s t : Int), s * ↑m + t * ↑a = 1 apply Exists.intro (-c) apply Exists.intro b show (-c) * m + b * a = 1 from calc (-c) * m + b * a _ = (-c) * m + (a * b - 1) + 1 := by ring _ = (-c) * m + m * c + 1 := by rw [h4] _ = 1 := by ring done · -- (←) assume h1 : rel_prime m a define show ∃ (Y : ZMod m), [a]_m * Y = [1]_m from Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1) done done /- Section 7.4 -/ section Euler open Euler lemma num_rp_below_base {m : Nat} : num_rp_below m 0 = 0 := by rfl lemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) : num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by have h1 : num_rp_below m (j + 1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : gcd m j = 1 rewrite [if_pos h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1 show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1 done lemma num_rp_below_step_not_rp {m j : Nat} (h : ¬rel_prime m j) : num_rp_below m (j + 1) = num_rp_below m j := by have h1 : num_rp_below m (j +1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : ¬gcd m j = 1 rewrite [if_neg h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j show num_rp_below m (j + 1) = num_rp_below m j from h1 done lemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl #eval phi 10 --Answer: 4 lemma prod_inv_iff_inv {m : Nat} {X : ZMod m} (h1 : invertible X) (Y : ZMod m) : invertible (X * Y) ↔ invertible Y := by apply Iff.intro · -- (→) assume h2 : invertible (X * Y) obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2 apply Exists.intro (X * Z) rewrite [←h3] --Goal : Y * (X * Z) = X * Y * Z ring --Note that ring can do algebra in ZMod m done · -- (←) assume h2 : invertible Y obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1 obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2 apply Exists.intro (Xi * Yi) show (X * Y) * (Xi * Yi) = [1]_m from calc X * Y * (Xi * Yi) _ = (X * Xi) * (Y * Yi) := by ring _ = [1]_m * [1]_m := by rw [h3, h4] _ = [1]_m := Theorem_7_3_6_7 [1]_m done done lemma F_rp_def {m i : Nat} (h : rel_prime m i) : F m i = [i]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h --h : gcd m i = 1 rewrite [if_pos h] at h1 show F m i = [i]_m from h1 done lemma F_not_rp_def {m i : Nat} (h : ¬rel_prime m i) : F m i = [1]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h rewrite [h1, if_neg h] rfl done lemma prod_seq_base {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 0 k f = [1]_m := by rfl lemma prod_seq_step {m : Nat} (n k : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl lemma prod_seq_zero_step {m : Nat} (n : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by rewrite [prod_seq_step, zero_add] rfl done lemma prod_one {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 1 k f = f k := by rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7] rfl done lemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl lemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m := calc [G m a i]_m _ = [(a * i) % m]_m := by rfl _ = [a * i]_m := (cc_eq_mod m (a * i)).symm _ = [a]_m * [i]_m := (mul_class m a i).symm lemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) : rel_prime m (G m a i) ↔ rel_prime m i := by have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1 show rel_prime m (G m a i) ↔ rel_prime m i from calc rel_prime m (G m a i) _ ↔ invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm _ ↔ invertible ([a]_m * [i]_m) := by rw [cc_G] _ ↔ invertible [i]_m := prod_inv_iff_inv h2 ([i]_m) _ ↔ rel_prime m i := Theorem_7_3_7 m i done lemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) : F m (G m a i) = [a]_m * F m i := by have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2 show F m (G m a i) = [a]_m * F m i from calc F m (G m a i) _ = [G m a i]_m := F_rp_def h3 _ = [a]_m * [i]_m := cc_G m a i _ = [a]_m * F m i := by rw [F_rp_def h2] done lemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : ¬rel_prime m i) : F m (G m a i) = [1]_m := by rewrite [←G_rp_iff h1 i] at h2 show F m (G m a i) = [1]_m from F_not_rp_def h2 done lemma FG_prod {m a : Nat} (h1 : rel_prime m a) : ∀ (k : Nat), prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by by_induc · -- Base Case show prod_seq 0 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from calc prod_seq 0 0 ((F m) ∘ (G m a)) _ = [1]_m := prod_seq_base _ _ _ = [a]_m ^ 0 * [1]_m := by ring _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by rw [num_rp_below_base, prod_seq_base] done · -- Induction Step fix k : Nat assume ih : prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) by_cases h2 : rel_prime m k · -- Case 1. h2 : rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([a]_m * F m k) := by rw [FG_rp h1 h2] _ = [a]_m ^ ((num_rp_below m k) + 1) * ((prod_seq k 0 (F m)) * F m k) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_rp h2, prod_seq_zero_step] done · -- Case 2. h2 : ¬rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([1]_m) := by rw [FG_not_rp h1 h2] _ = [a]_m ^ (num_rp_below m k) * (prod_seq k 0 (F m) * ([1]_m)) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_not_rp h2, prod_seq_zero_step, F_not_rp_def h2] done done done lemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by define --Goal : ∀ i < m, G m a i < m fix i : Nat assume h1 : i < m rewrite [G_def] --Goal : a * i % m < m show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m) done lemma left_inv_one_one_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g' (g i) = i) : one_one_below n g := sorry lemma right_inv_onto_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g (g' i) = i) (h2 : maps_below n g') : onto_below n g := by define at h2; define fix k : Nat assume h3 : k < n apply Exists.intro (g' k) show g' k < n ∧ g (g' k) = k from And.intro (h2 k h3) (h1 k h3) done lemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m * [inv_mod m a]_m = [1]_m := by have h2 : 0 ≤ (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a) show [a]_m * [inv_mod m a]_m = [1]_m from calc [a]_m * [inv_mod m a]_m _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2] _ = [a]_m * [gcd_c2 m a]_m := by rw [←cc_eq_mod] _ = [1]_m := gcd_c2_inv h1 done lemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m := calc a * (b % m) % m = a % m * (b % m % m) % m := Nat.mul_mod _ _ _ _ = a % m * (b % m) % m := by rw [Nat.mod_mod] _ = a * b % m := (Nat.mul_mod _ _ _).symm lemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm] rfl done theorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] : ↑a ≡ ↑b (MOD m) ↔ a % m = b % m := sorry lemma mul_inv_mod_cancel {m a i : Nat} [NeZero m] (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1 rewrite [mul_class, cc_eq_iff_congr, ←Nat.cast_mul, ←Nat.cast_one, congr_iff_mod_eq_Nat] at h3 show a * inv_mod m a * i % m = i from calc a * (inv_mod m a) * i % m _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = 1 % m * i % m := by rw [h3] _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = i % m := by rw [one_mul] _ = i := Nat.mod_eq_of_lt h2 done lemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl lemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, G m a (Ginv m a i) = i := by fix i : Nat assume h2 : i < m show G m a (Ginv m a i) = i from calc G m a (Ginv m a i) _ = a * ((inv_mod m a * i) % m) % m := by rfl _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, Ginv m a (G m a i) = i := by fix i : Nat assume h2 : i < m show Ginv m a (G m a i) = i from calc Ginv m a (G m a i) _ = inv_mod m a * ((a * i) % m) % m := by rfl _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc, mul_comm (inv_mod m a)] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_maps_below (m a : Nat) [NeZero m] : maps_below m (Ginv m a) := G_maps_below m (inv_mod m a) lemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : one_one_below m (G m a) := left_inv_one_one_below (Ginv_left_inv h1) lemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : onto_below m (G m a) := right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a) lemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : perm_below m (G m a) := And.intro (G_maps_below m a) (And.intro (G_one_one_below h1) (G_onto_below h1)) --Permuting a product of congruence classes doesn't change product lemma swap_fst (u v : Nat) : swap u v u = v := by define : swap u v u --Goal : (if u = u then v else if u = v then u else u) = v have h : u = u := by rfl rewrite [if_pos h] rfl done lemma swap_snd (u v : Nat) : swap u v v = u := by define : swap u v v by_cases h1 : v = u · -- Case 1. h1 : v = u rewrite [if_pos h1] show v = u from h1 done · -- Case 2. h1 : v ≠ u rewrite [if_neg h1] have h2 : v = v := by rfl rewrite [if_pos h2] rfl done done lemma swap_other {u v i : Nat} (h1 : i ≠ u) (h2 : i ≠ v) : swap u v i = i := by define : swap u v i rewrite [if_neg h1, if_neg h2] rfl done lemma swap_values (u v i : Nat) : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := by by_cases h1 : i = u · -- Case 1. h1 : i = u apply Or.inl rewrite [h1] show swap u v u = v from swap_fst u v done · -- Case 2. h1 : i ≠ u apply Or.inr by_cases h2 : i = v · -- Case 2.1. h2 : i = v apply Or.inl rewrite [h2] show swap u v v = u from swap_snd u v done · -- Case 2.2. h2 : i ≠ v apply Or.inr show swap u v i = i from swap_other h1 h2 done done done lemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by define fix i : Nat assume h3 : i < n have h4 : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := swap_values u v i by_cases on h4 · -- Case 1. h4 : swap u v i = v rewrite [h4] show v < n from h2 done · -- Case 2. by_cases on h4 · -- Case 2.1. h4 : swap u v i = u rewrite [h4] show u < n from h1 done · -- Case 2.2. h4 : swap u v i = i rewrite [h4] show i < n from h3 done done done lemma swap_swap (u v n : Nat) : ∀ i < n, swap u v (swap u v i) = i := by fix i : Nat assume h : i < n by_cases h1 : i = u · -- Case 1. h1 : i = u rewrite [h1, swap_fst, swap_snd] rfl done · -- Case 2. h1 : i ≠ u by_cases h2 : i = v · -- Case 2.1. h2 : i = v rewrite [h2, swap_snd, swap_fst] rfl done · -- Case 2.2. h2 : i ≠ v rewrite [swap_other h1 h2, swap_other h1 h2] rfl done done done lemma swap_one_one_below (u v n) : one_one_below n (swap u v) := left_inv_one_one_below (swap_swap u v n) lemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) := right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2) lemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) := And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2)) lemma comp_perm_below {n : Nat} {f g : Nat → Nat} (h1 : perm_below n f) (h2 : perm_below n g) : perm_below n (f ∘ g) := sorry lemma trivial_swap (u : Nat) : swap u u = id := by apply funext fix x : Nat by_cases h1 : x = u · -- Case 1. h1 : x = u rewrite [h1, swap_fst] rfl done · -- Case 2. h1 : x ≠ u rewrite [swap_other h1 h1] rfl done done lemma prod_eq_fun {m : Nat} (f g : Nat → ZMod m) (k : Nat) : ∀ (n : Nat), (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g := by by_induc · -- Base Case assume h : (∀ i < 0, f (k + i) = g (k + i)) rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g assume h1 : ∀ i < n + 1, f (k + i) = g (k + i) have h2 : ∀ i < n, f (k + i) = g (k + i) := by fix i : Nat assume h2 : i < n have h3 : i < n + 1 := by linarith show f (k + i) = g (k + i) from h1 i h3 done have h3 : prod_seq n k f = prod_seq n k g := ih h2 have h4 : n < n + 1 := Nat.lt_succ_self n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4] rfl done done lemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := by have h2 : ∀ (i : Nat), i < u → (f ∘ swap u n) (0 + i) = f (0 + i) := by fix i : Nat assume h2 : i < u have h3 : 0 + i ≠ u := by linarith have h4 : 0 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl done show prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f from prod_eq_fun (f ∘ swap u n) f 0 u h2 done lemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := by have h2 : ∀ i < j, (f ∘ swap u n) (u + 1 + i) = f (u + 1 + i) := by fix i : Nat assume h2 : i < j have h3 : u + 1 + i ≠ u := by linarith have h4 : u + 1 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl show prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f from prod_eq_fun (f ∘ swap u n) f (u + 1) j h2 done lemma break_prod {m : Nat} (n : Nat) (f : Nat → ZMod m) : ∀ (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by by_induc · -- Base Case have h : n + 0 = n := by rfl rewrite [prod_seq_base, h, Theorem_7_3_6_7] rfl done · -- Induction Step fix j : Nat assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f rewrite [←add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc] rfl done done lemma break_prod_twice {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f := break_prod n f 1 rewrite [prod_one] at h2 have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f := break_prod (u + 1) f j rewrite [←h1] at h3 have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f := break_prod u f 1 rewrite [prod_one] at h4 rewrite [h3, h4] at h2 show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2 done lemma swap_prod_eq_prod {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f := by by_cases h2 : u = n · -- Case 1. h2 : u = n rewrite [h2, trivial_swap n] --Goal : prod_seq (n + 1) 0 (f ∘ id) = prod_seq (n + 1) 0 f rfl done · -- Case 2. h2 : ¬u = n have h3 : u + 1 ≤ n := Nat.lt_of_le_of_ne h1 h2 obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3 have break_f : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := break_prod_twice f h4 have break_fs : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_prod_twice (f ∘ swap u n) h4 have f_eq_fs_below : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := swap_prod_eq_prod_below f h1 have f_eq_fs_btwn : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4 show prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f from calc prod_seq (n + 1) 0 (f ∘ swap u n) _ = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_fs _ = prod_seq u 0 f * (f ∘ swap u n) u * prod_seq j (u + 1) f * (f ∘ swap u n) n := by rw [f_eq_fs_below, f_eq_fs_btwn] _ = prod_seq u 0 f * f (swap u n u) * prod_seq j (u + 1) f * f (swap u n n) := by rfl _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by rw [swap_fst, swap_snd] _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring _ = prod_seq (n + 1) 0 f := break_f.symm done done lemma perm_below_fixed {n : Nat} {g : Nat → Nat} (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry lemma perm_prod {m : Nat} (f : Nat → ZMod m) : ∀ (n : Nat), ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) := by by_induc · -- Base Case fix g : Nat → Nat assume h1 : perm_below 0 g rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) fix g : Nat → Nat assume g_pb : perm_below (n + 1) g define at g_pb have g_ob : onto_below (n + 1) g := g_pb.right.right define at g_ob have h1 : n < n + 1 := by linarith obtain (u : Nat) (h2 : u < n + 1 ∧ g u = n) from g_ob n h1 have s_pb : perm_below (n + 1) (swap u n) := swap_perm_below h2.left h1 have gs_pb_n1 : perm_below (n + 1) (g ∘ swap u n) := comp_perm_below g_pb s_pb have gs_fix_n : (g ∘ swap u n) n = n := calc (g ∘ swap u n) n _ = g (swap u n n) := by rfl _ = g u := by rw [swap_snd] _ = n := h2.right have gs_pb_n : perm_below n (g ∘ swap u n) := perm_below_fixed gs_pb_n1 gs_fix_n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f ∘ (g ∘ swap u n)) := ih (g ∘ swap u n) gs_pb_n have h3 : u ≤ n := by linarith show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f ∘ g) from calc prod_seq (n + 1) 0 f _ = prod_seq n 0 f * f n := prod_seq_zero_step n f _ = prod_seq n 0 (f ∘ (g ∘ swap u n)) * f ((g ∘ swap u n) n) := by rw [gs_prod, gs_fix_n] _ = prod_seq n 0 (f ∘ g ∘ swap u n) * (f ∘ g ∘ swap u n) n := by rfl _ = prod_seq (n + 1) 0 (f ∘ g ∘ swap u n) := (prod_seq_zero_step n (f ∘ g ∘ swap u n)).symm _ = prod_seq (n + 1) 0 ((f ∘ g) ∘ swap u n) := by rfl _ = prod_seq (n + 1) 0 (f ∘ g) := swap_prod_eq_prod (f ∘ g) h3 done done lemma F_invertible (m i : Nat) : invertible (F m i) := by by_cases h : rel_prime m i · -- Case 1. h : rel_prime m i rewrite [F_rp_def h] show invertible [i]_m from (Theorem_7_3_7 m i).rtl h done · -- Case 2. h : ¬rel_prime m i rewrite [F_not_rp_def h] apply Exists.intro [1]_m show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m done done lemma Fprod_invertible (m : Nat) : ∀ (k : Nat), invertible (prod_seq k 0 (F m)) := by by_induc · -- Base Case apply Exists.intro [1]_m show prod_seq 0 0 (F m) * [1]_m = [1]_m from calc prod_seq 0 0 (F m) * [1]_m _ = [1]_m * [1]_m := by rw [prod_seq_base] _ = [1]_m := Theorem_7_3_6_7 ([1]_m) done · -- Induction Step fix k : Nat assume ih : invertible (prod_seq k 0 (F m)) rewrite [prod_seq_zero_step] show invertible (prod_seq k 0 (F m) * (F m k)) from (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k) done done theorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m ^ (phi m) = [1]_m := by have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2 show [a]_m ^ (phi m) = [1]_m from calc [a]_m ^ (phi m) _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3] _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring _ = prod_seq m 0 (F m ∘ G m a) * Y := by rw [FG_prod h1 m, phi_def] _ = prod_seq m 0 (F m) * Y := by rw [perm_prod (F m) m (G m a) (G_perm_below h1)] _ = [1]_m := by rw [h3] done lemma Exercise_7_4_5_Int (m : Nat) (a : Int) : ∀ (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry lemma Exercise_7_4_5_Nat (m a n : Nat) : [a]_m ^ n = [a ^ n]_m := by rewrite [Exercise_7_4_5_Int] rfl done theorem Euler's_theorem {m a : Nat} [NeZero m] (h1 : rel_prime m a) : a ^ (phi m) ≡ 1 (MOD m) := by have h2 : [a]_m ^ (phi m) = [1]_m := Theorem_7_4_2 h1 rewrite [Exercise_7_4_5_Nat m a (phi m)] at h2 --h2 : [a ^ phi m]_m = [1]_m show a ^ (phi m) ≡ 1 (MOD m) from (cc_eq_iff_congr _ _ _).ltr h2 done #eval gcd 10 7 --Answer: 1. So 10 and 7 are relatively prime #eval 7 ^ phi 10 --Answer: 2401, which is congruent to 1 mod 10. end Euler /- Section 7.5 -/ lemma num_rp_prime {p : Nat} (h1 : prime p) : ∀ k < p, num_rp_below p (k + 1) = k := sorry lemma phi_prime {p : Nat} (h1 : prime p) : phi p = p - 1 := by have h2 : 1 ≤ p := prime_pos h1 have h3 : p - 1 + 1 = p := Nat.sub_add_cancel h2 have h4 : p - 1 < p := by linarith have h5 : num_rp_below p (p - 1 + 1) = p - 1 := num_rp_prime h1 (p - 1) h4 rewrite [h3] at h5 show phi p = p - 1 from h5 done theorem Theorem_7_2_2_Int {a c : Nat} {b : Int} (h1 : ↑c ∣ ↑a * b) (h2 : rel_prime a c) : ↑c ∣ b := by rewrite [Int.natCast_dvd, Int.natAbs_mul, Int.natAbs_ofNat] at h1 --h1 : c ∣ a * Int.natAbs b rewrite [Int.natCast_dvd] --Goal : c ∣ Int.natAbs b show c ∣ Int.natAbs b from Theorem_7_2_2 h1 h2 done lemma Lemma_7_4_5 {m n : Nat} (a b : Int) (h1 : rel_prime m n) : a ≡ b (MOD m * n) ↔ a ≡ b (MOD m) ∧ a ≡ b (MOD n) := by apply Iff.intro · -- (→) assume h2 : a ≡ b (MOD m * n) obtain (j : Int) (h3 : a - b = (m * n) * j) from h2 apply And.intro · -- Proof of a ≡ b (MOD m) apply Exists.intro (n * j) show a - b = m * (n * j) from calc a - b _ = m * n * j := h3 _ = m * (n * j) := by ring done · -- Proof of a ≡ b (MOD n) apply Exists.intro (m * j) show a - b = n * (m * j) from calc a - b _ = m * n * j := h3 _ = n * (m * j) := by ring done done · -- (←) assume h2 : a ≡ b (MOD m) ∧ a ≡ b (MOD n) obtain (j : Int) (h3 : a - b = m * j) from h2.left have h4 : (↑n : Int) ∣ a - b := h2.right rewrite [h3] at h4 --h4 : ↑n ∣ ↑m * j have h5 : ↑n ∣ j := Theorem_7_2_2_Int h4 h1 obtain (k : Int) (h6 : j = n * k) from h5 apply Exists.intro k --Goal : a - b = ↑(m * n) * k rewrite [Nat.cast_mul] --Goal : a - b = ↑m * ↑n * k show a - b = (m * n) * k from calc a - b _ = m * j := h3 _ = m * (n * k) := by rw [h6] _ = (m * n) * k := by ring done done --From exercises of Section 7.2 theorem rel_prime_symm {a b : Nat} (h : rel_prime a b) : rel_prime b a := sorry lemma prime_NeZero {p : Nat} (h : prime p) : NeZero p := by rewrite [neZero_iff] --Goal : p ≠ 0 define at h linarith done lemma Lemma_7_5_1 {p e d m c s : Nat} {t : Int} (h1 : prime p) (h2 : e * d = (p - 1) * s + 1) (h3 : m ^ e - c = p * t) : c ^ d ≡ m (MOD p) := by have h4 : m ^ e ≡ c (MOD p) := Exists.intro t h3 have h5 : [m ^ e]_p = [c]_p := (cc_eq_iff_congr _ _ _).rtl h4 rewrite [←Exercise_7_4_5_Nat] at h5 --h5 : [m]_p ^ e = [c]_p by_cases h6 : p ∣ m · -- Case 1. h6 : p ∣ m have h7 : m ≡ 0 (MOD p) := by obtain (j : Nat) (h8 : m = p * j) from h6 apply Exists.intro (↑j : Int) --Goal : ↑m - 0 = ↑p * ↑j rewrite [h8, Nat.cast_mul] ring done have h8 : [m]_p = [0]_p := (cc_eq_iff_congr _ _ _).rtl h7 have h9 : e * d ≠ 0 := by rewrite [h2] show (p - 1) * s + 1 ≠ 0 from Nat.add_one_ne_zero _ done have h10 : (0 : Int) ^ (e * d) = 0 := zero_pow h9 have h11 : [c ^ d]_p = [m]_p := calc [c ^ d]_p _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat] _ = ([m]_p ^ e) ^ d := by rw [h5] _ = [m]_p ^ (e * d) := by ring _ = [0]_p ^ (e * d) := by rw [h8] _ = [0 ^ (e * d)]_p := Exercise_7_4_5_Int _ _ _ _ = [0]_p := by rw [h10] _ = [m]_p := by rw [h8] show c ^ d ≡ m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11 done · -- Case 2. h6 : ¬p ∣ m have h7 : rel_prime m p := rel_prime_of_prime_not_dvd h1 h6 have h8 : rel_prime p m := rel_prime_symm h7 have h9 : NeZero p := prime_NeZero h1 have h10 : (1 : Int) ^ s = 1 := by ring have h11 : [c ^ d]_p = [m]_p := calc [c ^ d]_p _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat] _ = ([m]_p ^ e) ^ d := by rw [h5] _ = [m]_p ^ (e * d) := by ring _ = [m]_p ^ ((p - 1) * s + 1) := by rw [h2] _ = ([m]_p ^ (p - 1)) ^ s * [m]_p := by ring _ = ([m]_p ^ (phi p)) ^ s * [m]_p := by rw [phi_prime h1] _ = [1]_p ^ s * [m]_p := by rw [Theorem_7_4_2 h8] _ = [1 ^ s]_p * [m]_p := by rw [Exercise_7_4_5_Int] _ = [1]_p * [m]_p := by rw [h10] _ = [m]_p * [1]_p := by ring _ = [m]_p := Theorem_7_3_6_7 _ show c ^ d ≡ m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11 done done theorem Theorem_7_5_1 (p q n e d k m c : Nat) (p_prime : prime p) (q_prime : prime q) (p_ne_q : p ≠ q) (n_pq : n = p * q) (ed_congr_1 : e * d = k * (p - 1) * (q - 1) + 1) (h1 : [m]_n ^ e = [c]_n) : [c]_n ^ d = [m]_n := by rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] at h1 --h1 : m ^ e ≡ c (MOD n) rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] --Goal : c ^ d ≡ m (MOD n) obtain (j : Int) (h2 : m ^ e - c = n * j) from h1 rewrite [n_pq, Nat.cast_mul] at h2 --h2 : m ^ e - c = p * q * j have h3 : e * d = (p - 1) * (k * (q - 1)) + 1 := by rewrite [ed_congr_1] ring done have h4 : m ^ e - c = p * (q * j) := by rewrite [h2] ring done have congr_p : c ^ d ≡ m (MOD p) := Lemma_7_5_1 p_prime h3 h4 have h5 : e * d = (q - 1) * (k * (p - 1)) + 1 := by rewrite [ed_congr_1] ring done have h6 : m ^ e - c = q * (p * j) := by rewrite [h2] ring done have congr_q : c ^ d ≡ m (MOD q) := Lemma_7_5_1 q_prime h5 h6 have h7 : ¬q ∣ p := by by_contra h8 have h9 : q = 1 ∨ q = p := dvd_prime p_prime h8 disj_syll h9 (prime_not_one q_prime) show False from p_ne_q h9.symm done have h8 : rel_prime p q := rel_prime_of_prime_not_dvd q_prime h7 rewrite [n_pq, Lemma_7_4_5 _ _ h8] show c ^ d ≡ m (MOD p) ∧ c ^ d ≡ m (MOD q) from And.intro congr_p congr_q done /- BEGIN EXERCISES -/ namespace Exercises /- Section 7.1 -/ -- 1. theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry -- 2. lemma gcd_comm_lt {a b : Nat} (h : a < b) : gcd a b = gcd b a := sorry theorem gcd_comm (a b : Nat) : gcd a b = gcd b a := sorry -- 3.

theorem Exercise_7_1_5 (a b : Nat) (n : Int) : (∃ (s t : Int), s * a + t * b = n) ↔ (↑(gcd a b) : Int) ∣ n

HTPI.Exercises.Exercise_7_1_5

htpi/HTPILib/Chap7.lean

htpi

/- Copyright 2023 Daniel J. Velleman -/ import HTPILib.Chap6 namespace HTPI /- Definitions -/ lemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by have h : n + 1 > 0 := Nat.succ_pos n show a % (n + 1) < n + 1 from Nat.mod_lt a h done def gcd (a b : Nat) : Nat := match b with | 0 => a | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd (n + 1) (a % (n + 1)) termination_by b mutual def gcd_c1 (a b : Nat) : Int := match b with | 0 => 1 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c2 (n + 1) (a % (n + 1)) --Corresponds to s = t' termination_by b def gcd_c2 (a b : Nat) : Int := match b with | 0 => 0 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c1 (n + 1) (a % (n + 1)) - (gcd_c2 (n + 1) (a % (n + 1))) * ↑(a / (n + 1)) --Corresponds to t = s' - t'q termination_by b end def prime (n : Nat) : Prop := 2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n def prime_factor (p n : Nat) : Prop := prime p ∧ p ∣ n def all_prime (l : List Nat) : Prop := ∀ p ∈ l, prime p def nondec (l : List Nat) : Prop := match l with | [] => True --Of course, True is a proposition that is always true | n :: L => (∀ m ∈ L, n ≤ m) ∧ nondec L def nondec_prime_list (l : List Nat) : Prop := all_prime l ∧ nondec l def prod (l : List Nat) : Nat := match l with | [] => 1 | n :: L => n * (prod L) def prime_factorization (n : Nat) (l : List Nat) : Prop := nondec_prime_list l ∧ prod l = n def rel_prime (a b : Nat) : Prop := gcd a b = 1 def congr_mod (m : Nat) (a b : Int) : Prop := (↑m : Int) ∣ (a - b) def cc (m : Nat) (a : Int) : ZMod m := (↑a : ZMod m) notation:50 a " ≡ " b " (MOD " m ")" => congr_mod m a b notation:max "["a"]_"m:max => cc m a def invertible {m : Nat} (X : ZMod m) : Prop := ∃ (Y : ZMod m), X * Y = [1]_m def num_rp_below (m k : Nat) : Nat := match k with | 0 => 0 | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j def phi (m : Nat) : Nat := num_rp_below m m def prod_seq {m : Nat} (j k : Nat) (f : Nat → ZMod m) : ZMod m := match j with | 0 => [1]_m | n + 1 => prod_seq n k f * f (k + n) def maps_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i < n, g i < n def one_one_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i1 < n, ∀ i2 < n, g i1 = g i2 → i1 = i2 def onto_below (n : Nat) (g : Nat → Nat) : Prop := ∀ k < n, ∃ i < n, g i = k def perm_below (n : Nat) (g : Nat → Nat) : Prop := maps_below n g ∧ one_one_below n g ∧ onto_below n g def inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m) def swap (u v i : Nat) : Nat := if i = u then v else if i = v then u else i namespace Euler --For definitions specific to Euler's theorem def F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m def G (m a i : Nat) : Nat := (a * i) % m def Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i end Euler /- Section 7.1 -/ theorem dvd_mod_of_dvd_a_b {a b d : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ (a % b) := by set q : Nat := a / b have h3 : b * q + a % b = a := Nat.div_add_mod a b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 define --Goal : ∃ (c : Nat), a % b = d * c apply Exists.intro (j - k * q) show a % b = d * (j - k * q) from calc a % b _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm _ = a - b * q := by rw [h3] _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc] _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm done theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry #eval gcd 672 161 --Answer: 7 lemma gcd_base (a : Nat) : gcd a 0 = a := by rfl lemma gcd_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd a b = gcd b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1)) rfl done lemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b ≠ 0) : a % b < b := by have h1 : b > 0 := Nat.pos_of_ne_zero h show a % b < b from Nat.mod_lt a h1 done lemma dvd_self (n : Nat) : n ∣ n := by apply Exists.intro 1 ring done theorem gcd_dvd : ∀ (b a : Nat), (gcd a b) ∣ a ∧ (gcd a b) ∣ b := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd a b_1) ∣ a ∧ (gcd a b_1) ∣ b_1 fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_base] --Goal: a ∣ a ∧ a ∣ 0 apply And.intro (dvd_self a) define apply Exists.intro 0 rfl done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_nonzero a h1] --Goal : gcd b (a % b) ∣ a ∧ gcd b (a % b) ∣ b have h2 : a % b < b := mod_nonzero_lt a h1 have h3 : (gcd b (a % b)) ∣ b ∧ (gcd b (a % b)) ∣ (a % b) := ih (a % b) h2 b apply And.intro _ h3.left show (gcd b (a % b)) ∣ a from dvd_a_of_dvd_b_mod h3.left h3.right done done theorem gcd_dvd_left (a b : Nat) : (gcd a b) ∣ a := (gcd_dvd b a).left theorem gcd_dvd_right (a b : Nat) : (gcd a b) ∣ b := (gcd_dvd b a).right lemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl lemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c1 a b = gcd_c2 b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done lemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl lemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * ↑(a / b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done theorem gcd_lin_comb : ∀ (b a : Nat), (gcd_c1 a b) * ↑a + (gcd_c2 a b) * ↑b = ↑(gcd a b) := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd_c1 a b_1) * ↑a + (gcd_c2 a b_1) * ↑b_1 = ↑(gcd a b_1) fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base] --Goal : 1 * ↑a + 0 * ↑0 = ↑a ring done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1] --Goal : gcd_c2 b (a % b) * ↑a + -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * ↑(a / b)) * ↑b = -- ↑(gcd b (a % b)) set r : Nat := a % b set q : Nat := a / b set s : Int := gcd_c1 b r set t : Int := gcd_c2 b r --Goal : t * ↑a + (s - t * ↑q) * ↑b = ↑(gcd b r) have h2 : r < b := mod_nonzero_lt a h1 have h3 : s * ↑b + t * ↑r = ↑(gcd b r) := ih r h2 b have h4 : b * q + r = a := Nat.div_add_mod a b rewrite [←h3, ←h4] rewrite [Nat.cast_add, Nat.cast_mul] --Goal : t * (↑b * ↑q + ↑r) + (s - t * ↑q) * ↑b = s * ↑b + t * ↑r ring done done #eval gcd_c1 672 161 --Answer: 6 #eval gcd_c2 672 161 --Answer: -25 --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161 theorem Theorem_7_1_6 {d a b : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ gcd a b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑d ∣ ↑(gcd a b) set s : Int := gcd_c1 a b set t : Int := gcd_c2 a b have h3 : s * ↑a + t * ↑b = ↑(gcd a b) := gcd_lin_comb b a rewrite [←h3] --Goal : ↑d ∣ s * ↑a + t * ↑b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul] --Goal : ↑d ∣ s * (↑d * ↑j) + t * (↑d * ↑k) define apply Exists.intro (s * ↑j + t * ↑k) ring done /- Section 7.2 -/ theorem dvd_trans {a b c : Nat} (h1 : a ∣ b) (h2 : b ∣ c) : a ∣ c := by define at h1; define at h2; define obtain (m : Nat) (h3 : b = a * m) from h1 obtain (n : Nat) (h4 : c = b * n) from h2 rewrite [h3, mul_assoc] at h4 apply Exists.intro (m * n) show c = a * (m * n) from h4 done lemma exists_prime_factor : ∀ (n : Nat), 2 ≤ n → ∃ (p : Nat), prime_factor p n := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, 2 ≤ n_1 → ∃ (p : Nat), prime_factor p n_1 assume h1 : 2 ≤ n by_cases h2 : prime n · -- Case 1. h2 : prime n apply Exists.intro n define --Goal : prime n ∧ n ∣ n show prime n ∧ n ∣ n from And.intro h2 (dvd_self n) done · -- Case 2. h2 : ¬prime n define at h2 --h2 : ¬(2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n) demorgan at h2 disj_syll h2 h1 obtain (a : Nat) (h3 : ∃ (b : Nat), a * b = n ∧ a < n ∧ b < n) from h2 obtain (b : Nat) (h4 : a * b = n ∧ a < n ∧ b < n) from h3 have h5 : 2 ≤ a := by by_contra h6 have h7 : a ≤ 1 := by linarith have h8 : n ≤ b := calc n _ = a * b := h4.left.symm _ ≤ 1 * b := by rel [h7] _ = b := by ring linarith --n ≤ b contradicts b < n done have h6 : ∃ (p : Nat), prime_factor p a := ih a h4.right.left h5 obtain (p : Nat) (h7 : prime_factor p a) from h6 apply Exists.intro p define --Goal : prime p ∧ p ∣ n define at h7 --h7 : prime p ∧ p ∣ a apply And.intro h7.left have h8 : a ∣ n := by apply Exists.intro b show n = a * b from (h4.left).symm done show p ∣ n from dvd_trans h7.right h8 done done lemma exists_least_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q := by set S : Set Nat := {p : Nat | prime_factor p n} have h2 : ∃ (p : Nat), p ∈ S := exists_prime_factor n h show ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q from well_ord_princ S h2 done lemma all_prime_nil : all_prime [] := by define --Goal : ∀ p ∈ [], prime p fix p : Nat contrapos --Goal : ¬prime p → p ∉ [] assume h1 : ¬prime p show p ∉ [] from List.not_mem_nil p done lemma all_prime_cons (n : Nat) (L : List Nat) : all_prime (n :: L) ↔ prime n ∧ all_prime L := by apply Iff.intro · -- (→) assume h1 : all_prime (n :: L) --Goal : prime n ∧ all_prime L define at h1 --h1 : ∀ p ∈ n :: L, prime p apply And.intro (h1 n (List.mem_cons_self n L)) define --Goal : ∀ p ∈ L, prime p fix p : Nat assume h2 : p ∈ L show prime p from h1 p (List.mem_cons_of_mem n h2) done · -- (←) assume h1 : prime n ∧ all_prime L --Goal : all_prime (n :: l) define : all_prime L at h1 define fix p : Nat assume h2 : p ∈ n :: L rewrite [List.mem_cons] at h2 --h2 : p = n ∨ p ∈ L by_cases on h2 · -- Case 1. h2 : p = n rewrite [h2] show prime n from h1.left done · -- Case 2. h2 : p ∈ L show prime p from h1.right p h2 done done done lemma nondec_nil : nondec [] := by define --Goal : True trivial --trivial proves some obviously true statements, such as True done lemma nondec_cons (n : Nat) (L : List Nat) : nondec (n :: L) ↔ (∀ m ∈ L, n ≤ m) ∧ nondec L := by rfl lemma prod_nil : prod [] = 1 := by rfl lemma prod_cons : prod (n :: L) = n * (prod L) := by rfl lemma exists_cons_of_length_eq_succ {A : Type} {l : List A} {n : Nat} (h : l.length = n + 1) : ∃ (a : A) (L : List A), l = a :: L ∧ L.length = n := by have h1 : ¬l.length = 0 := by linarith rewrite [List.length_eq_zero] at h1 obtain (a : A) (h2 : ∃ (L : List A), l = a :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List A) (h3 : l = a :: L) from h2 apply Exists.intro a apply Exists.intro L apply And.intro h3 have h4 : (a :: L).length = L.length + 1 := List.length_cons a L rewrite [←h3, h] at h4 show L.length = n from (Nat.add_right_cancel h4).symm done lemma list_elt_dvd_prod_by_length (a : Nat) : ∀ (n : Nat), ∀ (l : List Nat), l.length = n → a ∈ l → a ∣ prod l := by by_induc · --Base Case fix l : List Nat assume h1 : l.length = 0 rewrite [List.length_eq_zero] at h1 --h1 : l = [] rewrite [h1] --Goal : a ∈ [] → a ∣ prod [] contrapos assume h2 : ¬a ∣ prod [] show a ∉ [] from List.not_mem_nil a done · -- Induction Step fix n : Nat assume ih : ∀ (l : List Nat), List.length l = n → a ∈ l → a ∣ prod l fix l : List Nat assume h1 : l.length = n + 1 --Goal : a ∈ l → a ∣ prod l obtain (b : Nat) (h2 : ∃ (L : List Nat), l = b :: L ∧ L.length = n) from exists_cons_of_length_eq_succ h1 obtain (L : List Nat) (h3 : l = b :: L ∧ L.length = n) from h2 have h4 : a ∈ L → a ∣ prod L := ih L h3.right assume h5 : a ∈ l rewrite [h3.left, prod_cons] --Goal : a ∣ b * prod L rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b ∨ a ∈ L by_cases on h5 · -- Case 1. h5 : a = b apply Exists.intro (prod L) rewrite [h5] rfl done · -- Case 2. h5 : a ∈ L have h6 : a ∣ prod L := h4 h5 have h7 : prod L ∣ b * prod L := by apply Exists.intro b ring done show a ∣ b * prod L from dvd_trans h6 h7 done done done lemma list_elt_dvd_prod {a : Nat} {l : List Nat} (h : a ∈ l) : a ∣ prod l := by set n : Nat := l.length have h1 : l.length = n := by rfl show a ∣ prod l from list_elt_dvd_prod_by_length a n l h1 h done lemma exists_prime_factorization : ∀ (n : Nat), n ≥ 1 → ∃ (l : List Nat), prime_factorization n l := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, n_1 ≥ 1 → ∃ (l : List Nat), prime_factorization n_1 l assume h1 : n ≥ 1 by_cases h2 : n = 1 · -- Case 1. h2 : n = 1 apply Exists.intro [] define apply And.intro · -- Proof of nondec_prime_list [] define show all_prime [] ∧ nondec [] from And.intro all_prime_nil nondec_nil done · -- Proof of prod [] = n rewrite [prod_nil, h2] rfl done done · -- Case 2. h2 : n ≠ 1 have h3 : n ≥ 2 := lt_of_le_of_ne' h1 h2 obtain (p : Nat) (h4 : prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q) from exists_least_prime_factor h3 have p_prime_factor : prime_factor p n := h4.left define at p_prime_factor have p_prime : prime p := p_prime_factor.left have p_dvd_n : p ∣ n := p_prime_factor.right have p_least : ∀ (q : Nat), prime_factor q n → p ≤ q := h4.right obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n have h5 : m ≠ 0 := by contradict h1 with h6 have h7 : n = 0 := calc n _ = p * m := n_eq_pm _ = p * 0 := by rw [h6] _ = 0 := by ring rewrite [h7] decide done have m_pos : 0 < m := Nat.pos_of_ne_zero h5 have m_lt_n : m < n := by define at p_prime show m < n from calc m _ < m + m := by linarith _ = 2 * m := by ring _ ≤ p * m := by rel [p_prime.left] _ = n := n_eq_pm.symm done obtain (L : List Nat) (h6 : prime_factorization m L) from ih m m_lt_n m_pos define at h6 have ndpl_L : nondec_prime_list L := h6.left define at ndpl_L apply Exists.intro (p :: L) define apply And.intro · -- Proof of nondec_prime_list (p :: L) define apply And.intro · -- Proof of all_prime (p :: L) rewrite [all_prime_cons] show prime p ∧ all_prime L from And.intro p_prime ndpl_L.left done · -- Proof of nondec (p :: L) rewrite [nondec_cons] apply And.intro _ ndpl_L.right fix q : Nat assume q_in_L : q ∈ L have h7 : q ∣ prod L := list_elt_dvd_prod q_in_L rewrite [h6.right] at h7 --h7 : q ∣ m have h8 : m ∣ n := by apply Exists.intro p rewrite [n_eq_pm] ring done have q_dvd_n : q ∣ n := dvd_trans h7 h8 have ap_L : all_prime L := ndpl_L.left define at ap_L have q_prime_factor : prime_factor q n := And.intro (ap_L q q_in_L) q_dvd_n show p ≤ q from p_least q q_prime_factor done done · -- Proof of prod (p :: L) = n rewrite [prod_cons, h6.right, n_eq_pm] rfl done done done theorem Theorem_7_2_2 {a b c : Nat} (h1 : c ∣ a * b) (h2 : rel_prime a c) : c ∣ b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑c ∣ ↑b define at h1; define at h2; define obtain (j : Nat) (h3 : a * b = c * j) from h1 set s : Int := gcd_c1 a c set t : Int := gcd_c2 a c have h4 : s * ↑a + t * ↑c = ↑(gcd a c) := gcd_lin_comb c a rewrite [h2, Nat.cast_one] at h4 --h4 : s * ↑a + t * ↑c = (1 : Int) apply Exists.intro (s * ↑j + t * ↑b) show ↑b = ↑c * (s * ↑j + t * ↑b) from calc ↑b _ = (1 : Int) * ↑b := (one_mul _).symm _ = (s * ↑a + t * ↑c) * ↑b := by rw [h4] _ = s * (↑a * ↑b) + t * ↑c * ↑b := by ring _ = s * (↑c * ↑j) + t * ↑c * ↑b := by rw [←Nat.cast_mul a b, h3, Nat.cast_mul c j] _ = ↑c * (s * ↑j + t * ↑b) := by ring done lemma le_nonzero_prod_left {a b : Nat} (h : a * b ≠ 0) : a ≤ a * b := by have h1 : b ≠ 0 := by contradict h with h1 rewrite [h1] ring done have h2 : 1 ≤ b := Nat.pos_of_ne_zero h1 show a ≤ a * b from calc a = a * 1 := (mul_one a).symm _ ≤ a * b := by rel [h2] done lemma le_nonzero_prod_right {a b : Nat} (h : a * b ≠ 0) : b ≤ a * b := by rewrite [mul_comm] rewrite [mul_comm] at h show b ≤ b * a from le_nonzero_prod_left h done lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry lemma rel_prime_of_prime_not_dvd {a p : Nat} (h1 : prime p) (h2 : ¬p ∣ a) : rel_prime a p := by have h3 : gcd a p ∣ a := gcd_dvd_left a p have h4 : gcd a p ∣ p := gcd_dvd_right a p have h5 : gcd a p = 1 ∨ gcd a p = p := dvd_prime h1 h4 have h6 : gcd a p ≠ p := by contradict h2 with h6 rewrite [h6] at h3 show p ∣ a from h3 done disj_syll h5 h6 show rel_prime a p from h5 done theorem Theorem_7_2_3 {a b p : Nat} (h1 : prime p) (h2 : p ∣ a * b) : p ∣ a ∨ p ∣ b := by or_right with h3 have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3 show p ∣ b from Theorem_7_2_2 h2 h4 done lemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a ≥ 1 := by have h1 : a ≠ 0 := by by_contra h1 rewrite [h1] at h contradict h linarith done show a ≥ 1 from Nat.pos_of_ne_zero h1 done lemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by have h1 : a ≥ 1 := ge_one_of_prod_one h have h2 : a * b ≠ 0 := by linarith have h3 : a ≤ a * b := le_nonzero_prod_left h2 rewrite [h] at h3 show a = 1 from Nat.le_antisymm h3 h1 done lemma eq_one_of_dvd_one {n : Nat} (h : n ∣ 1) : n = 1 := by obtain (j : Nat) (h1 : 1 = n * j) from h show n = 1 from eq_one_of_prod_one h1.symm done lemma prime_not_one {p : Nat} (h : prime p) : p ≠ 1 := by define at h linarith done theorem Theorem_7_2_4 {p : Nat} (h1 : prime p) : ∀ (l : List Nat), p ∣ prod l → ∃ a ∈ l, p ∣ a := by apply List.rec · -- Base Case. Goal : p ∣ prod [] → ∃ a ∈ [], p ∣ a rewrite [prod_nil] assume h2 : p ∣ 1 show ∃ a ∈ [], p ∣ a from absurd (eq_one_of_dvd_one h2) (prime_not_one h1) done · -- Induction Step fix b : Nat fix L : List Nat assume ih : p ∣ prod L → ∃ a ∈ L, p ∣ a --Goal : p ∣ prod (b :: L) → ∃ a ∈ b :: L, p ∣ a assume h2 : p ∣ prod (b :: L) rewrite [prod_cons] at h2 have h3 : p ∣ b ∨ p ∣ prod L := Theorem_7_2_3 h1 h2 by_cases on h3 · -- Case 1. h3 : p ∣ b apply Exists.intro b show b ∈ b :: L ∧ p ∣ b from And.intro (List.mem_cons_self b L) h3 done · -- Case 2. h3 : p ∣ prod L obtain (a : Nat) (h4 : a ∈ L ∧ p ∣ a) from ih h3 apply Exists.intro a show a ∈ b :: L ∧ p ∣ a from And.intro (List.mem_cons_of_mem b h4.left) h4.right done done done lemma prime_in_list {p : Nat} {l : List Nat} (h1 : prime p) (h2 : all_prime l) (h3 : p ∣ prod l) : p ∈ l := by obtain (a : Nat) (h4 : a ∈ l ∧ p ∣ a) from Theorem_7_2_4 h1 l h3 define at h2 have h5 : prime a := h2 a h4.left have h6 : p = 1 ∨ p = a := dvd_prime h5 h4.right disj_syll h6 (prime_not_one h1) rewrite [h6] show a ∈ l from h4.left done lemma first_le_first {p q : Nat} {l m : List Nat} (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m)) (h3 : prod (p :: l) = prod (q :: m)) : p ≤ q := by define at h1; define at h2 have h4 : q ∣ prod (p :: l) := by define apply Exists.intro (prod m) rewrite [←prod_cons] show prod (p :: l) = prod (q :: m) from h3 done have h5 : all_prime (q :: m) := h2.left rewrite [all_prime_cons] at h5 have h6 : q ∈ p :: l := prime_in_list h5.left h1.left h4 have h7 : nondec (p :: l) := h1.right rewrite [nondec_cons] at h7 rewrite [List.mem_cons] at h6 by_cases on h6 · -- Case 1. h6 : q = p linarith done · -- Case 2. h6 : q ∈ l have h8 : ∀ m ∈ l, p ≤ m := h7.left show p ≤ q from h8 q h6 done done lemma nondec_prime_list_tail {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by define at h define rewrite [all_prime_cons, nondec_cons] at h show all_prime l ∧ nondec l from And.intro h.left.right h.right.right done lemma cons_prod_not_one {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : prod (p :: l) ≠ 1 := by define at h have h1 : all_prime (p :: l) := h.left rewrite [all_prime_cons] at h1 rewrite [prod_cons] by_contra h2 show False from (prime_not_one h1.left) (eq_one_of_prod_one h2) done lemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) : l = [] ↔ prod l = 1 := by apply Iff.intro · -- (→) assume h1 : l = [] rewrite [h1] show prod [] = 1 from prod_nil done · -- (←) contrapos assume h1 : ¬l = [] obtain (p : Nat) (h2 : ∃ (L : List Nat), l = p :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List Nat) (h3 : l = p :: L) from h2 rewrite [h3] at h rewrite [h3] show ¬prod (p :: L) = 1 from cons_prod_not_one h done done lemma prime_pos {p : Nat} (h : prime p) : p > 0 := by define at h linarith done theorem Theorem_7_2_5 : ∀ (l1 l2 : List Nat), nondec_prime_list l1 → nondec_prime_list l2 → prod l1 = prod l2 → l1 = l2 := by apply List.rec · -- Base Case. Goal : ∀ (l2 : List Nat), nondec_prime_list [] → -- nondec_prime_list l2 → prod [] = prod l2 → [] = l2 fix l2 : List Nat assume h1 : nondec_prime_list [] assume h2 : nondec_prime_list l2 assume h3 : prod [] = prod l2 rewrite [prod_nil, eq_comm, ←list_nil_iff_prod_one h2] at h3 show [] = l2 from h3.symm done · -- Induction Step fix p : Nat fix L1 : List Nat assume ih : ∀ (L2 : List Nat), nondec_prime_list L1 → nondec_prime_list L2 → prod L1 = prod L2 → L1 = L2 -- Goal : ∀ (l2 : List Nat), nondec_prime_list (p :: L1) → -- nondec_prime_list l2 → prod (p :: L1) = prod l2 → p :: L1 = l2 fix l2 : List Nat assume h1 : nondec_prime_list (p :: L1) assume h2 : nondec_prime_list l2 assume h3 : prod (p :: L1) = prod l2 have h4 : ¬prod (p :: L1) = 1 := cons_prod_not_one h1 rewrite [h3, ←list_nil_iff_prod_one h2] at h4 obtain (q : Nat) (h5 : ∃ (L : List Nat), l2 = q :: L) from List.exists_cons_of_ne_nil h4 obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5 rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2) rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2) have h7 : p ≤ q := first_le_first h1 h2 h3 have h8 : q ≤ p := first_le_first h2 h1 h3.symm have h9 : p = q := by linarith rewrite [h9, prod_cons, prod_cons] at h3 --h3 : q * prod L1 = q * prod L2 have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1 have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2 define at h2 have h12 : all_prime (q :: L2) := h2.left rewrite [all_prime_cons] at h12 have h13 : q > 0 := prime_pos h12.left have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3 have h15 : L1 = L2 := ih L2 h10 h11 h14 rewrite [h6, h9, h15] rfl done done theorem fund_thm_arith (n : Nat) (h : n ≥ 1) : ∃! (l : List Nat), prime_factorization n l := by exists_unique · -- Existence show ∃ (l : List Nat), prime_factorization n l from exists_prime_factorization n h done · -- Uniqueness fix l1 : List Nat; fix l2 : List Nat assume h1 : prime_factorization n l1 assume h2 : prime_factorization n l2 define at h1; define at h2 have h3 : prod l1 = n := h1.right rewrite [←h2.right] at h3 show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3 done done /- Section 7.3 -/ theorem congr_refl (m : Nat) : ∀ (a : Int), a ≡ a (MOD m) := by fix a : Int define --Goal : ∃ (c : Int), a - a = ↑m * c apply Exists.intro 0 ring done theorem congr_symm {m : Nat} : ∀ {a b : Int}, a ≡ b (MOD m) → b ≡ a (MOD m) := by fix a : Int; fix b : Int assume h1 : a ≡ b (MOD m) define at h1 --h1 : ∃ (c : Int), a - b = ↑m * c define --Goal : ∃ (c : Int), b - a = ↑m * c obtain (c : Int) (h2 : a - b = m * c) from h1 apply Exists.intro (-c) show b - a = m * (-c) from calc b - a _ = -(a - b) := by ring _ = -(m * c) := by rw [h2] _ = m * (-c) := by ring done theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry /- Fundamental properties of congruence classes -/ lemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) : X = Y ↔ X.val = Y.val := Fin.ext_iff lemma val_nat_eq_mod (n k : Nat) : ([k]_(n + 1)).val = k % (n + 1) := by rfl lemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl theorem cc_rep {m : Nat} (X : ZMod m) : ∃ (a : Int), X = [a]_m := match m with | 0 => by apply Exists.intro X rfl done | n + 1 => by apply Exists.intro ↑(X.val) have h1 : X.val < n + 1 := Fin.prop X rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1] rfl done theorem add_class (m : Nat) (a b : Int) : [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm theorem mul_class (m : Nat) (a b : Int) : [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm lemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ [a - b]_m = [0]_m := by apply Iff.intro · -- (→) assume h1 : [a]_m = [b]_m have h2 : a - b = a + (-b) := by ring have h3 : b + (-b) = 0 := by ring show [a - b]_m = [0]_m from calc [a - b]_m _ = [a + (-b)]_m := by rw [h2] _ = [a]_m + [-b]_m := by rw [add_class] _ = [b]_m + [-b]_m := by rw [h1] _ = [b + -b]_m := by rw [add_class] _ = [0]_m := by rw [h3] done · -- (←) assume h1 : [a - b]_m = [0]_m have h2 : b + (a - b) = a := by ring have h3 : b + 0 = b := by ring show [a]_m = [b]_m from calc [a]_m _ = [b + (a - b)]_m := by rw [h2] _ = [b]_m + [a - b]_m := by rw [add_class] _ = [b]_m + [0]_m := by rw [h1] _ = [b + 0]_m := by rw [add_class] _ = [b]_m := by rw [h3] done done lemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) : [a]_m = [0]_m → [-a]_m = [0]_m := by assume h1 : [a]_m = [0]_m have h2 : 0 + (-a) = -a := by ring have h3 : a + (-a) = 0 := by ring show [-a]_m = [0]_m from calc [-a]_m _ = [0 + (-a)]_m := by rw [h2] _ = [0]_m + [-a]_m := by rw [add_class] _ = [a]_m + [-a]_m := by rw [h1] _ = [a + (-a)]_m := by rw [add_class] _ = [0]_m := by rw [h3] done lemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) : [-a]_m = [0]_m ↔ [a]_m = [0]_m := by apply Iff.intro _ (cc_neg_zero_of_cc_zero m a) assume h1 : [-a]_m = [0]_m have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1 have h3 : -(-a) = a := by ring rewrite [h3] at h2 show [a]_m = [0]_m from h2 done lemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl lemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m ↔ m ∣ k := match m with | 0 => by have h : (0 : Int) = (↑(0 : Nat) : Int) := by rfl rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj] apply Iff.intro · -- (→) assume h1 : k = 0 rewrite [h1] show 0 ∣ 0 from dvd_self 0 done · -- (←) assume h1 : 0 ∣ k obtain (c : Nat) (h2 : k = 0 * c) from h1 rewrite [h2] ring done done | n + 1 => by rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero] show k % (n + 1) = 0 ↔ n + 1 ∣ k from (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm done lemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m ↔ ↑m ∣ a := by obtain (k : Nat) (h1 : a = ↑k ∨ a = -↑k) from Int.eq_nat_or_neg a by_cases on h1 · -- Case 1. h1: a = ↑k rewrite [h1, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done · -- Case 2. h1: a = -↑k rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done done theorem cc_eq_iff_congr (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ a ≡ b (MOD m) := calc [a]_m = [b]_m _ ↔ [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b _ ↔ ↑m ∣ (a - b) := cc_zero_iff_dvd m (a - b) _ ↔ a ≡ b (MOD m) := by rfl /- End of fundamental properties of congruence classes -/ lemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m := by have h1 : (↑m : Int) ≠ 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m) show 0 ≤ a % m from Int.emod_nonneg a h1 done lemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m) have h2 : (↑m : Int) > 0 := (Nat.cast_pos).rtl h1 show a % m < m from Int.emod_lt_of_pos a h2 done lemma congr_mod_mod (m : Nat) (a : Int) : a ≡ a % m (MOD m) := by define have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m apply Exists.intro (a / m) show a - a % m = m * (a / m) from calc a - (a % m) _ = m * (a / m) + a % m - a % m := by rw [h1] _ = m * (a / m) := by ring done lemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) := And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a)) theorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) : ∃! (r : Int), 0 ≤ r ∧ r < m ∧ a ≡ r (MOD m) := by exists_unique · -- Existence apply Exists.intro (a % m) show 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) from mod_cmpl_res m a done · -- Uniqueness fix r1 : Int; fix r2 : Int assume h1 : 0 ≤ r1 ∧ r1 < m ∧ a ≡ r1 (MOD m) assume h2 : 0 ≤ r2 ∧ r2 < m ∧ a ≡ r2 (MOD m) have h3 : r1 ≡ r2 (MOD m) := congr_trans (congr_symm h1.right.right) h2.right.right obtain (d : Int) (h4 : r1 - r2 = m * d) from h3 have h5 : r1 - r2 < m * 1 := by linarith have h6 : m * (-1) < r1 - r2 := by linarith rewrite [h4] at h5 --h5 : m * d < m * 1 rewrite [h4] at h6 --h6 : m * -1 < m * d have h7 : (↑m : Int) ≥ 0 := Nat.cast_nonneg m have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7 have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7 have h10 : d = 0 := by linarith show r1 = r2 from calc r1 _ = r1 - r2 + r2 := by ring _ = m * 0 + r2 := by rw [h4, h10] _ = r2 := by ring done done lemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m := (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a) theorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y rewrite [h1, h2] have h3 : a + b = b + a := by ring show [a]_m + [b]_m = [b]_m + [a]_m from calc [a]_m + [b]_m _ = [a + b]_m := add_class m a b _ = [b + a]_m := by rw [h3] _ = [b]_m + [a]_m := (add_class m b a).symm done theorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X rewrite [h1] have h2 : a * 1 = a := by ring show [a]_m * [1]_m = [a]_m from calc [a]_m * [1]_m _ = [a * 1]_m := mul_class m a 1 _ = [a]_m := by rw [h2] done theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry lemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) : [a]_m * [gcd_c2 m a]_m = [1]_m := by set s : Int := gcd_c1 m a have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m define at h1 rewrite [h1, Nat.cast_one] at h2 --h2 : s * ↑m + gcd_c2 m a * ↑a = 1 rewrite [mul_class, cc_eq_iff_congr] define --Goal : ∃ (c : Int), ↑a * gcd_c2 m a - 1 = ↑m * c apply Exists.intro (-s) show a * (gcd_c2 m a) - 1 = m * (-s) from calc a * (gcd_c2 m a) - 1 _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring _ = 1 + m * (-s) - 1 := by rw [h2] _ = m * (-s) := by ring done theorem Theorem_7_3_7 (m a : Nat) : invertible [a]_m ↔ rel_prime m a := by apply Iff.intro · -- (→) assume h1 : invertible [a]_m define at h1 obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1 obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y rewrite [h3, mul_class, cc_eq_iff_congr] at h2 define at h2 obtain (c : Int) (h4 : a * b - 1 = m * c) from h2 rewrite [Exercise_7_2_6] --Goal : ∃ (s t : Int), s * ↑m + t * ↑a = 1 apply Exists.intro (-c) apply Exists.intro b show (-c) * m + b * a = 1 from calc (-c) * m + b * a _ = (-c) * m + (a * b - 1) + 1 := by ring _ = (-c) * m + m * c + 1 := by rw [h4] _ = 1 := by ring done · -- (←) assume h1 : rel_prime m a define show ∃ (Y : ZMod m), [a]_m * Y = [1]_m from Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1) done done /- Section 7.4 -/ section Euler open Euler lemma num_rp_below_base {m : Nat} : num_rp_below m 0 = 0 := by rfl lemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) : num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by have h1 : num_rp_below m (j + 1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : gcd m j = 1 rewrite [if_pos h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1 show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1 done lemma num_rp_below_step_not_rp {m j : Nat} (h : ¬rel_prime m j) : num_rp_below m (j + 1) = num_rp_below m j := by have h1 : num_rp_below m (j +1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : ¬gcd m j = 1 rewrite [if_neg h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j show num_rp_below m (j + 1) = num_rp_below m j from h1 done lemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl #eval phi 10 --Answer: 4 lemma prod_inv_iff_inv {m : Nat} {X : ZMod m} (h1 : invertible X) (Y : ZMod m) : invertible (X * Y) ↔ invertible Y := by apply Iff.intro · -- (→) assume h2 : invertible (X * Y) obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2 apply Exists.intro (X * Z) rewrite [←h3] --Goal : Y * (X * Z) = X * Y * Z ring --Note that ring can do algebra in ZMod m done · -- (←) assume h2 : invertible Y obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1 obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2 apply Exists.intro (Xi * Yi) show (X * Y) * (Xi * Yi) = [1]_m from calc X * Y * (Xi * Yi) _ = (X * Xi) * (Y * Yi) := by ring _ = [1]_m * [1]_m := by rw [h3, h4] _ = [1]_m := Theorem_7_3_6_7 [1]_m done done lemma F_rp_def {m i : Nat} (h : rel_prime m i) : F m i = [i]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h --h : gcd m i = 1 rewrite [if_pos h] at h1 show F m i = [i]_m from h1 done lemma F_not_rp_def {m i : Nat} (h : ¬rel_prime m i) : F m i = [1]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h rewrite [h1, if_neg h] rfl done lemma prod_seq_base {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 0 k f = [1]_m := by rfl lemma prod_seq_step {m : Nat} (n k : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl lemma prod_seq_zero_step {m : Nat} (n : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by rewrite [prod_seq_step, zero_add] rfl done lemma prod_one {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 1 k f = f k := by rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7] rfl done lemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl lemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m := calc [G m a i]_m _ = [(a * i) % m]_m := by rfl _ = [a * i]_m := (cc_eq_mod m (a * i)).symm _ = [a]_m * [i]_m := (mul_class m a i).symm lemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) : rel_prime m (G m a i) ↔ rel_prime m i := by have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1 show rel_prime m (G m a i) ↔ rel_prime m i from calc rel_prime m (G m a i) _ ↔ invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm _ ↔ invertible ([a]_m * [i]_m) := by rw [cc_G] _ ↔ invertible [i]_m := prod_inv_iff_inv h2 ([i]_m) _ ↔ rel_prime m i := Theorem_7_3_7 m i done lemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) : F m (G m a i) = [a]_m * F m i := by have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2 show F m (G m a i) = [a]_m * F m i from calc F m (G m a i) _ = [G m a i]_m := F_rp_def h3 _ = [a]_m * [i]_m := cc_G m a i _ = [a]_m * F m i := by rw [F_rp_def h2] done lemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : ¬rel_prime m i) : F m (G m a i) = [1]_m := by rewrite [←G_rp_iff h1 i] at h2 show F m (G m a i) = [1]_m from F_not_rp_def h2 done lemma FG_prod {m a : Nat} (h1 : rel_prime m a) : ∀ (k : Nat), prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by by_induc · -- Base Case show prod_seq 0 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from calc prod_seq 0 0 ((F m) ∘ (G m a)) _ = [1]_m := prod_seq_base _ _ _ = [a]_m ^ 0 * [1]_m := by ring _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by rw [num_rp_below_base, prod_seq_base] done · -- Induction Step fix k : Nat assume ih : prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) by_cases h2 : rel_prime m k · -- Case 1. h2 : rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([a]_m * F m k) := by rw [FG_rp h1 h2] _ = [a]_m ^ ((num_rp_below m k) + 1) * ((prod_seq k 0 (F m)) * F m k) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_rp h2, prod_seq_zero_step] done · -- Case 2. h2 : ¬rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([1]_m) := by rw [FG_not_rp h1 h2] _ = [a]_m ^ (num_rp_below m k) * (prod_seq k 0 (F m) * ([1]_m)) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_not_rp h2, prod_seq_zero_step, F_not_rp_def h2] done done done lemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by define --Goal : ∀ i < m, G m a i < m fix i : Nat assume h1 : i < m rewrite [G_def] --Goal : a * i % m < m show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m) done lemma left_inv_one_one_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g' (g i) = i) : one_one_below n g := sorry lemma right_inv_onto_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g (g' i) = i) (h2 : maps_below n g') : onto_below n g := by define at h2; define fix k : Nat assume h3 : k < n apply Exists.intro (g' k) show g' k < n ∧ g (g' k) = k from And.intro (h2 k h3) (h1 k h3) done lemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m * [inv_mod m a]_m = [1]_m := by have h2 : 0 ≤ (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a) show [a]_m * [inv_mod m a]_m = [1]_m from calc [a]_m * [inv_mod m a]_m _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2] _ = [a]_m * [gcd_c2 m a]_m := by rw [←cc_eq_mod] _ = [1]_m := gcd_c2_inv h1 done lemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m := calc a * (b % m) % m = a % m * (b % m % m) % m := Nat.mul_mod _ _ _ _ = a % m * (b % m) % m := by rw [Nat.mod_mod] _ = a * b % m := (Nat.mul_mod _ _ _).symm lemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm] rfl done theorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] : ↑a ≡ ↑b (MOD m) ↔ a % m = b % m := sorry lemma mul_inv_mod_cancel {m a i : Nat} [NeZero m] (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1 rewrite [mul_class, cc_eq_iff_congr, ←Nat.cast_mul, ←Nat.cast_one, congr_iff_mod_eq_Nat] at h3 show a * inv_mod m a * i % m = i from calc a * (inv_mod m a) * i % m _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = 1 % m * i % m := by rw [h3] _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = i % m := by rw [one_mul] _ = i := Nat.mod_eq_of_lt h2 done lemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl lemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, G m a (Ginv m a i) = i := by fix i : Nat assume h2 : i < m show G m a (Ginv m a i) = i from calc G m a (Ginv m a i) _ = a * ((inv_mod m a * i) % m) % m := by rfl _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, Ginv m a (G m a i) = i := by fix i : Nat assume h2 : i < m show Ginv m a (G m a i) = i from calc Ginv m a (G m a i) _ = inv_mod m a * ((a * i) % m) % m := by rfl _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc, mul_comm (inv_mod m a)] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_maps_below (m a : Nat) [NeZero m] : maps_below m (Ginv m a) := G_maps_below m (inv_mod m a) lemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : one_one_below m (G m a) := left_inv_one_one_below (Ginv_left_inv h1) lemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : onto_below m (G m a) := right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a) lemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : perm_below m (G m a) := And.intro (G_maps_below m a) (And.intro (G_one_one_below h1) (G_onto_below h1)) --Permuting a product of congruence classes doesn't change product lemma swap_fst (u v : Nat) : swap u v u = v := by define : swap u v u --Goal : (if u = u then v else if u = v then u else u) = v have h : u = u := by rfl rewrite [if_pos h] rfl done lemma swap_snd (u v : Nat) : swap u v v = u := by define : swap u v v by_cases h1 : v = u · -- Case 1. h1 : v = u rewrite [if_pos h1] show v = u from h1 done · -- Case 2. h1 : v ≠ u rewrite [if_neg h1] have h2 : v = v := by rfl rewrite [if_pos h2] rfl done done lemma swap_other {u v i : Nat} (h1 : i ≠ u) (h2 : i ≠ v) : swap u v i = i := by define : swap u v i rewrite [if_neg h1, if_neg h2] rfl done lemma swap_values (u v i : Nat) : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := by by_cases h1 : i = u · -- Case 1. h1 : i = u apply Or.inl rewrite [h1] show swap u v u = v from swap_fst u v done · -- Case 2. h1 : i ≠ u apply Or.inr by_cases h2 : i = v · -- Case 2.1. h2 : i = v apply Or.inl rewrite [h2] show swap u v v = u from swap_snd u v done · -- Case 2.2. h2 : i ≠ v apply Or.inr show swap u v i = i from swap_other h1 h2 done done done lemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by define fix i : Nat assume h3 : i < n have h4 : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := swap_values u v i by_cases on h4 · -- Case 1. h4 : swap u v i = v rewrite [h4] show v < n from h2 done · -- Case 2. by_cases on h4 · -- Case 2.1. h4 : swap u v i = u rewrite [h4] show u < n from h1 done · -- Case 2.2. h4 : swap u v i = i rewrite [h4] show i < n from h3 done done done lemma swap_swap (u v n : Nat) : ∀ i < n, swap u v (swap u v i) = i := by fix i : Nat assume h : i < n by_cases h1 : i = u · -- Case 1. h1 : i = u rewrite [h1, swap_fst, swap_snd] rfl done · -- Case 2. h1 : i ≠ u by_cases h2 : i = v · -- Case 2.1. h2 : i = v rewrite [h2, swap_snd, swap_fst] rfl done · -- Case 2.2. h2 : i ≠ v rewrite [swap_other h1 h2, swap_other h1 h2] rfl done done done lemma swap_one_one_below (u v n) : one_one_below n (swap u v) := left_inv_one_one_below (swap_swap u v n) lemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) := right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2) lemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) := And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2)) lemma comp_perm_below {n : Nat} {f g : Nat → Nat} (h1 : perm_below n f) (h2 : perm_below n g) : perm_below n (f ∘ g) := sorry lemma trivial_swap (u : Nat) : swap u u = id := by apply funext fix x : Nat by_cases h1 : x = u · -- Case 1. h1 : x = u rewrite [h1, swap_fst] rfl done · -- Case 2. h1 : x ≠ u rewrite [swap_other h1 h1] rfl done done lemma prod_eq_fun {m : Nat} (f g : Nat → ZMod m) (k : Nat) : ∀ (n : Nat), (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g := by by_induc · -- Base Case assume h : (∀ i < 0, f (k + i) = g (k + i)) rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g assume h1 : ∀ i < n + 1, f (k + i) = g (k + i) have h2 : ∀ i < n, f (k + i) = g (k + i) := by fix i : Nat assume h2 : i < n have h3 : i < n + 1 := by linarith show f (k + i) = g (k + i) from h1 i h3 done have h3 : prod_seq n k f = prod_seq n k g := ih h2 have h4 : n < n + 1 := Nat.lt_succ_self n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4] rfl done done lemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := by have h2 : ∀ (i : Nat), i < u → (f ∘ swap u n) (0 + i) = f (0 + i) := by fix i : Nat assume h2 : i < u have h3 : 0 + i ≠ u := by linarith have h4 : 0 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl done show prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f from prod_eq_fun (f ∘ swap u n) f 0 u h2 done lemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := by have h2 : ∀ i < j, (f ∘ swap u n) (u + 1 + i) = f (u + 1 + i) := by fix i : Nat assume h2 : i < j have h3 : u + 1 + i ≠ u := by linarith have h4 : u + 1 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl show prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f from prod_eq_fun (f ∘ swap u n) f (u + 1) j h2 done lemma break_prod {m : Nat} (n : Nat) (f : Nat → ZMod m) : ∀ (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by by_induc · -- Base Case have h : n + 0 = n := by rfl rewrite [prod_seq_base, h, Theorem_7_3_6_7] rfl done · -- Induction Step fix j : Nat assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f rewrite [←add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc] rfl done done lemma break_prod_twice {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f := break_prod n f 1 rewrite [prod_one] at h2 have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f := break_prod (u + 1) f j rewrite [←h1] at h3 have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f := break_prod u f 1 rewrite [prod_one] at h4 rewrite [h3, h4] at h2 show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2 done lemma swap_prod_eq_prod {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f := by by_cases h2 : u = n · -- Case 1. h2 : u = n rewrite [h2, trivial_swap n] --Goal : prod_seq (n + 1) 0 (f ∘ id) = prod_seq (n + 1) 0 f rfl done · -- Case 2. h2 : ¬u = n have h3 : u + 1 ≤ n := Nat.lt_of_le_of_ne h1 h2 obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3 have break_f : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := break_prod_twice f h4 have break_fs : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_prod_twice (f ∘ swap u n) h4 have f_eq_fs_below : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := swap_prod_eq_prod_below f h1 have f_eq_fs_btwn : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4 show prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f from calc prod_seq (n + 1) 0 (f ∘ swap u n) _ = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_fs _ = prod_seq u 0 f * (f ∘ swap u n) u * prod_seq j (u + 1) f * (f ∘ swap u n) n := by rw [f_eq_fs_below, f_eq_fs_btwn] _ = prod_seq u 0 f * f (swap u n u) * prod_seq j (u + 1) f * f (swap u n n) := by rfl _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by rw [swap_fst, swap_snd] _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring _ = prod_seq (n + 1) 0 f := break_f.symm done done lemma perm_below_fixed {n : Nat} {g : Nat → Nat} (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry lemma perm_prod {m : Nat} (f : Nat → ZMod m) : ∀ (n : Nat), ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) := by by_induc · -- Base Case fix g : Nat → Nat assume h1 : perm_below 0 g rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) fix g : Nat → Nat assume g_pb : perm_below (n + 1) g define at g_pb have g_ob : onto_below (n + 1) g := g_pb.right.right define at g_ob have h1 : n < n + 1 := by linarith obtain (u : Nat) (h2 : u < n + 1 ∧ g u = n) from g_ob n h1 have s_pb : perm_below (n + 1) (swap u n) := swap_perm_below h2.left h1 have gs_pb_n1 : perm_below (n + 1) (g ∘ swap u n) := comp_perm_below g_pb s_pb have gs_fix_n : (g ∘ swap u n) n = n := calc (g ∘ swap u n) n _ = g (swap u n n) := by rfl _ = g u := by rw [swap_snd] _ = n := h2.right have gs_pb_n : perm_below n (g ∘ swap u n) := perm_below_fixed gs_pb_n1 gs_fix_n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f ∘ (g ∘ swap u n)) := ih (g ∘ swap u n) gs_pb_n have h3 : u ≤ n := by linarith show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f ∘ g) from calc prod_seq (n + 1) 0 f _ = prod_seq n 0 f * f n := prod_seq_zero_step n f _ = prod_seq n 0 (f ∘ (g ∘ swap u n)) * f ((g ∘ swap u n) n) := by rw [gs_prod, gs_fix_n] _ = prod_seq n 0 (f ∘ g ∘ swap u n) * (f ∘ g ∘ swap u n) n := by rfl _ = prod_seq (n + 1) 0 (f ∘ g ∘ swap u n) := (prod_seq_zero_step n (f ∘ g ∘ swap u n)).symm _ = prod_seq (n + 1) 0 ((f ∘ g) ∘ swap u n) := by rfl _ = prod_seq (n + 1) 0 (f ∘ g) := swap_prod_eq_prod (f ∘ g) h3 done done lemma F_invertible (m i : Nat) : invertible (F m i) := by by_cases h : rel_prime m i · -- Case 1. h : rel_prime m i rewrite [F_rp_def h] show invertible [i]_m from (Theorem_7_3_7 m i).rtl h done · -- Case 2. h : ¬rel_prime m i rewrite [F_not_rp_def h] apply Exists.intro [1]_m show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m done done lemma Fprod_invertible (m : Nat) : ∀ (k : Nat), invertible (prod_seq k 0 (F m)) := by by_induc · -- Base Case apply Exists.intro [1]_m show prod_seq 0 0 (F m) * [1]_m = [1]_m from calc prod_seq 0 0 (F m) * [1]_m _ = [1]_m * [1]_m := by rw [prod_seq_base] _ = [1]_m := Theorem_7_3_6_7 ([1]_m) done · -- Induction Step fix k : Nat assume ih : invertible (prod_seq k 0 (F m)) rewrite [prod_seq_zero_step] show invertible (prod_seq k 0 (F m) * (F m k)) from (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k) done done theorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m ^ (phi m) = [1]_m := by have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2 show [a]_m ^ (phi m) = [1]_m from calc [a]_m ^ (phi m) _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3] _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring _ = prod_seq m 0 (F m ∘ G m a) * Y := by rw [FG_prod h1 m, phi_def] _ = prod_seq m 0 (F m) * Y := by rw [perm_prod (F m) m (G m a) (G_perm_below h1)] _ = [1]_m := by rw [h3] done lemma Exercise_7_4_5_Int (m : Nat) (a : Int) : ∀ (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry lemma Exercise_7_4_5_Nat (m a n : Nat) : [a]_m ^ n = [a ^ n]_m := by rewrite [Exercise_7_4_5_Int] rfl done theorem Euler's_theorem {m a : Nat} [NeZero m] (h1 : rel_prime m a) : a ^ (phi m) ≡ 1 (MOD m) := by have h2 : [a]_m ^ (phi m) = [1]_m := Theorem_7_4_2 h1 rewrite [Exercise_7_4_5_Nat m a (phi m)] at h2 --h2 : [a ^ phi m]_m = [1]_m show a ^ (phi m) ≡ 1 (MOD m) from (cc_eq_iff_congr _ _ _).ltr h2 done #eval gcd 10 7 --Answer: 1. So 10 and 7 are relatively prime #eval 7 ^ phi 10 --Answer: 2401, which is congruent to 1 mod 10. end Euler /- Section 7.5 -/ lemma num_rp_prime {p : Nat} (h1 : prime p) : ∀ k < p, num_rp_below p (k + 1) = k := sorry lemma phi_prime {p : Nat} (h1 : prime p) : phi p = p - 1 := by have h2 : 1 ≤ p := prime_pos h1 have h3 : p - 1 + 1 = p := Nat.sub_add_cancel h2 have h4 : p - 1 < p := by linarith have h5 : num_rp_below p (p - 1 + 1) = p - 1 := num_rp_prime h1 (p - 1) h4 rewrite [h3] at h5 show phi p = p - 1 from h5 done theorem Theorem_7_2_2_Int {a c : Nat} {b : Int} (h1 : ↑c ∣ ↑a * b) (h2 : rel_prime a c) : ↑c ∣ b := by rewrite [Int.natCast_dvd, Int.natAbs_mul, Int.natAbs_ofNat] at h1 --h1 : c ∣ a * Int.natAbs b rewrite [Int.natCast_dvd] --Goal : c ∣ Int.natAbs b show c ∣ Int.natAbs b from Theorem_7_2_2 h1 h2 done lemma Lemma_7_4_5 {m n : Nat} (a b : Int) (h1 : rel_prime m n) : a ≡ b (MOD m * n) ↔ a ≡ b (MOD m) ∧ a ≡ b (MOD n) := by apply Iff.intro · -- (→) assume h2 : a ≡ b (MOD m * n) obtain (j : Int) (h3 : a - b = (m * n) * j) from h2 apply And.intro · -- Proof of a ≡ b (MOD m) apply Exists.intro (n * j) show a - b = m * (n * j) from calc a - b _ = m * n * j := h3 _ = m * (n * j) := by ring done · -- Proof of a ≡ b (MOD n) apply Exists.intro (m * j) show a - b = n * (m * j) from calc a - b _ = m * n * j := h3 _ = n * (m * j) := by ring done done · -- (←) assume h2 : a ≡ b (MOD m) ∧ a ≡ b (MOD n) obtain (j : Int) (h3 : a - b = m * j) from h2.left have h4 : (↑n : Int) ∣ a - b := h2.right rewrite [h3] at h4 --h4 : ↑n ∣ ↑m * j have h5 : ↑n ∣ j := Theorem_7_2_2_Int h4 h1 obtain (k : Int) (h6 : j = n * k) from h5 apply Exists.intro k --Goal : a - b = ↑(m * n) * k rewrite [Nat.cast_mul] --Goal : a - b = ↑m * ↑n * k show a - b = (m * n) * k from calc a - b _ = m * j := h3 _ = m * (n * k) := by rw [h6] _ = (m * n) * k := by ring done done --From exercises of Section 7.2 theorem rel_prime_symm {a b : Nat} (h : rel_prime a b) : rel_prime b a := sorry lemma prime_NeZero {p : Nat} (h : prime p) : NeZero p := by rewrite [neZero_iff] --Goal : p ≠ 0 define at h linarith done lemma Lemma_7_5_1 {p e d m c s : Nat} {t : Int} (h1 : prime p) (h2 : e * d = (p - 1) * s + 1) (h3 : m ^ e - c = p * t) : c ^ d ≡ m (MOD p) := by have h4 : m ^ e ≡ c (MOD p) := Exists.intro t h3 have h5 : [m ^ e]_p = [c]_p := (cc_eq_iff_congr _ _ _).rtl h4 rewrite [←Exercise_7_4_5_Nat] at h5 --h5 : [m]_p ^ e = [c]_p by_cases h6 : p ∣ m · -- Case 1. h6 : p ∣ m have h7 : m ≡ 0 (MOD p) := by obtain (j : Nat) (h8 : m = p * j) from h6 apply Exists.intro (↑j : Int) --Goal : ↑m - 0 = ↑p * ↑j rewrite [h8, Nat.cast_mul] ring done have h8 : [m]_p = [0]_p := (cc_eq_iff_congr _ _ _).rtl h7 have h9 : e * d ≠ 0 := by rewrite [h2] show (p - 1) * s + 1 ≠ 0 from Nat.add_one_ne_zero _ done have h10 : (0 : Int) ^ (e * d) = 0 := zero_pow h9 have h11 : [c ^ d]_p = [m]_p := calc [c ^ d]_p _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat] _ = ([m]_p ^ e) ^ d := by rw [h5] _ = [m]_p ^ (e * d) := by ring _ = [0]_p ^ (e * d) := by rw [h8] _ = [0 ^ (e * d)]_p := Exercise_7_4_5_Int _ _ _ _ = [0]_p := by rw [h10] _ = [m]_p := by rw [h8] show c ^ d ≡ m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11 done · -- Case 2. h6 : ¬p ∣ m have h7 : rel_prime m p := rel_prime_of_prime_not_dvd h1 h6 have h8 : rel_prime p m := rel_prime_symm h7 have h9 : NeZero p := prime_NeZero h1 have h10 : (1 : Int) ^ s = 1 := by ring have h11 : [c ^ d]_p = [m]_p := calc [c ^ d]_p _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat] _ = ([m]_p ^ e) ^ d := by rw [h5] _ = [m]_p ^ (e * d) := by ring _ = [m]_p ^ ((p - 1) * s + 1) := by rw [h2] _ = ([m]_p ^ (p - 1)) ^ s * [m]_p := by ring _ = ([m]_p ^ (phi p)) ^ s * [m]_p := by rw [phi_prime h1] _ = [1]_p ^ s * [m]_p := by rw [Theorem_7_4_2 h8] _ = [1 ^ s]_p * [m]_p := by rw [Exercise_7_4_5_Int] _ = [1]_p * [m]_p := by rw [h10] _ = [m]_p * [1]_p := by ring _ = [m]_p := Theorem_7_3_6_7 _ show c ^ d ≡ m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11 done done theorem Theorem_7_5_1 (p q n e d k m c : Nat) (p_prime : prime p) (q_prime : prime q) (p_ne_q : p ≠ q) (n_pq : n = p * q) (ed_congr_1 : e * d = k * (p - 1) * (q - 1) + 1) (h1 : [m]_n ^ e = [c]_n) : [c]_n ^ d = [m]_n := by rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] at h1 --h1 : m ^ e ≡ c (MOD n) rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] --Goal : c ^ d ≡ m (MOD n) obtain (j : Int) (h2 : m ^ e - c = n * j) from h1 rewrite [n_pq, Nat.cast_mul] at h2 --h2 : m ^ e - c = p * q * j have h3 : e * d = (p - 1) * (k * (q - 1)) + 1 := by rewrite [ed_congr_1] ring done have h4 : m ^ e - c = p * (q * j) := by rewrite [h2] ring done have congr_p : c ^ d ≡ m (MOD p) := Lemma_7_5_1 p_prime h3 h4 have h5 : e * d = (q - 1) * (k * (p - 1)) + 1 := by rewrite [ed_congr_1] ring done have h6 : m ^ e - c = q * (p * j) := by rewrite [h2] ring done have congr_q : c ^ d ≡ m (MOD q) := Lemma_7_5_1 q_prime h5 h6 have h7 : ¬q ∣ p := by by_contra h8 have h9 : q = 1 ∨ q = p := dvd_prime p_prime h8 disj_syll h9 (prime_not_one q_prime) show False from p_ne_q h9.symm done have h8 : rel_prime p q := rel_prime_of_prime_not_dvd q_prime h7 rewrite [n_pq, Lemma_7_4_5 _ _ h8] show c ^ d ≡ m (MOD p) ∧ c ^ d ≡ m (MOD q) from And.intro congr_p congr_q done /- BEGIN EXERCISES -/ namespace Exercises /- Section 7.1 -/ -- 1. theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry -- 2. lemma gcd_comm_lt {a b : Nat} (h : a < b) : gcd a b = gcd b a := sorry theorem gcd_comm (a b : Nat) : gcd a b = gcd b a := sorry -- 3. theorem Exercise_7_1_5 (a b : Nat) (n : Int) : (∃ (s t : Int), s * a + t * b = n) ↔ (↑(gcd a b) : Int) ∣ n := sorry -- 4.

theorem Exercise_7_1_6 (a b c : Nat) : gcd a b = gcd (a + b * c) b

HTPI.Exercises.Exercise_7_1_6

htpi/HTPILib/Chap7.lean

htpi

/- Copyright 2023 Daniel J. Velleman -/ import HTPILib.Chap6 namespace HTPI /- Definitions -/ lemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by have h : n + 1 > 0 := Nat.succ_pos n show a % (n + 1) < n + 1 from Nat.mod_lt a h done def gcd (a b : Nat) : Nat := match b with | 0 => a | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd (n + 1) (a % (n + 1)) termination_by b mutual def gcd_c1 (a b : Nat) : Int := match b with | 0 => 1 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c2 (n + 1) (a % (n + 1)) --Corresponds to s = t' termination_by b def gcd_c2 (a b : Nat) : Int := match b with | 0 => 0 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c1 (n + 1) (a % (n + 1)) - (gcd_c2 (n + 1) (a % (n + 1))) * ↑(a / (n + 1)) --Corresponds to t = s' - t'q termination_by b end def prime (n : Nat) : Prop := 2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n def prime_factor (p n : Nat) : Prop := prime p ∧ p ∣ n def all_prime (l : List Nat) : Prop := ∀ p ∈ l, prime p def nondec (l : List Nat) : Prop := match l with | [] => True --Of course, True is a proposition that is always true | n :: L => (∀ m ∈ L, n ≤ m) ∧ nondec L def nondec_prime_list (l : List Nat) : Prop := all_prime l ∧ nondec l def prod (l : List Nat) : Nat := match l with | [] => 1 | n :: L => n * (prod L) def prime_factorization (n : Nat) (l : List Nat) : Prop := nondec_prime_list l ∧ prod l = n def rel_prime (a b : Nat) : Prop := gcd a b = 1 def congr_mod (m : Nat) (a b : Int) : Prop := (↑m : Int) ∣ (a - b) def cc (m : Nat) (a : Int) : ZMod m := (↑a : ZMod m) notation:50 a " ≡ " b " (MOD " m ")" => congr_mod m a b notation:max "["a"]_"m:max => cc m a def invertible {m : Nat} (X : ZMod m) : Prop := ∃ (Y : ZMod m), X * Y = [1]_m def num_rp_below (m k : Nat) : Nat := match k with | 0 => 0 | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j def phi (m : Nat) : Nat := num_rp_below m m def prod_seq {m : Nat} (j k : Nat) (f : Nat → ZMod m) : ZMod m := match j with | 0 => [1]_m | n + 1 => prod_seq n k f * f (k + n) def maps_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i < n, g i < n def one_one_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i1 < n, ∀ i2 < n, g i1 = g i2 → i1 = i2 def onto_below (n : Nat) (g : Nat → Nat) : Prop := ∀ k < n, ∃ i < n, g i = k def perm_below (n : Nat) (g : Nat → Nat) : Prop := maps_below n g ∧ one_one_below n g ∧ onto_below n g def inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m) def swap (u v i : Nat) : Nat := if i = u then v else if i = v then u else i namespace Euler --For definitions specific to Euler's theorem def F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m def G (m a i : Nat) : Nat := (a * i) % m def Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i end Euler /- Section 7.1 -/ theorem dvd_mod_of_dvd_a_b {a b d : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ (a % b) := by set q : Nat := a / b have h3 : b * q + a % b = a := Nat.div_add_mod a b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 define --Goal : ∃ (c : Nat), a % b = d * c apply Exists.intro (j - k * q) show a % b = d * (j - k * q) from calc a % b _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm _ = a - b * q := by rw [h3] _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc] _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm done theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry #eval gcd 672 161 --Answer: 7 lemma gcd_base (a : Nat) : gcd a 0 = a := by rfl lemma gcd_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd a b = gcd b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1)) rfl done lemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b ≠ 0) : a % b < b := by have h1 : b > 0 := Nat.pos_of_ne_zero h show a % b < b from Nat.mod_lt a h1 done lemma dvd_self (n : Nat) : n ∣ n := by apply Exists.intro 1 ring done theorem gcd_dvd : ∀ (b a : Nat), (gcd a b) ∣ a ∧ (gcd a b) ∣ b := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd a b_1) ∣ a ∧ (gcd a b_1) ∣ b_1 fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_base] --Goal: a ∣ a ∧ a ∣ 0 apply And.intro (dvd_self a) define apply Exists.intro 0 rfl done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_nonzero a h1] --Goal : gcd b (a % b) ∣ a ∧ gcd b (a % b) ∣ b have h2 : a % b < b := mod_nonzero_lt a h1 have h3 : (gcd b (a % b)) ∣ b ∧ (gcd b (a % b)) ∣ (a % b) := ih (a % b) h2 b apply And.intro _ h3.left show (gcd b (a % b)) ∣ a from dvd_a_of_dvd_b_mod h3.left h3.right done done theorem gcd_dvd_left (a b : Nat) : (gcd a b) ∣ a := (gcd_dvd b a).left theorem gcd_dvd_right (a b : Nat) : (gcd a b) ∣ b := (gcd_dvd b a).right lemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl lemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c1 a b = gcd_c2 b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done lemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl lemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * ↑(a / b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done theorem gcd_lin_comb : ∀ (b a : Nat), (gcd_c1 a b) * ↑a + (gcd_c2 a b) * ↑b = ↑(gcd a b) := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd_c1 a b_1) * ↑a + (gcd_c2 a b_1) * ↑b_1 = ↑(gcd a b_1) fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base] --Goal : 1 * ↑a + 0 * ↑0 = ↑a ring done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1] --Goal : gcd_c2 b (a % b) * ↑a + -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * ↑(a / b)) * ↑b = -- ↑(gcd b (a % b)) set r : Nat := a % b set q : Nat := a / b set s : Int := gcd_c1 b r set t : Int := gcd_c2 b r --Goal : t * ↑a + (s - t * ↑q) * ↑b = ↑(gcd b r) have h2 : r < b := mod_nonzero_lt a h1 have h3 : s * ↑b + t * ↑r = ↑(gcd b r) := ih r h2 b have h4 : b * q + r = a := Nat.div_add_mod a b rewrite [←h3, ←h4] rewrite [Nat.cast_add, Nat.cast_mul] --Goal : t * (↑b * ↑q + ↑r) + (s - t * ↑q) * ↑b = s * ↑b + t * ↑r ring done done #eval gcd_c1 672 161 --Answer: 6 #eval gcd_c2 672 161 --Answer: -25 --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161 theorem Theorem_7_1_6 {d a b : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ gcd a b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑d ∣ ↑(gcd a b) set s : Int := gcd_c1 a b set t : Int := gcd_c2 a b have h3 : s * ↑a + t * ↑b = ↑(gcd a b) := gcd_lin_comb b a rewrite [←h3] --Goal : ↑d ∣ s * ↑a + t * ↑b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul] --Goal : ↑d ∣ s * (↑d * ↑j) + t * (↑d * ↑k) define apply Exists.intro (s * ↑j + t * ↑k) ring done /- Section 7.2 -/ theorem dvd_trans {a b c : Nat} (h1 : a ∣ b) (h2 : b ∣ c) : a ∣ c := by define at h1; define at h2; define obtain (m : Nat) (h3 : b = a * m) from h1 obtain (n : Nat) (h4 : c = b * n) from h2 rewrite [h3, mul_assoc] at h4 apply Exists.intro (m * n) show c = a * (m * n) from h4 done lemma exists_prime_factor : ∀ (n : Nat), 2 ≤ n → ∃ (p : Nat), prime_factor p n := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, 2 ≤ n_1 → ∃ (p : Nat), prime_factor p n_1 assume h1 : 2 ≤ n by_cases h2 : prime n · -- Case 1. h2 : prime n apply Exists.intro n define --Goal : prime n ∧ n ∣ n show prime n ∧ n ∣ n from And.intro h2 (dvd_self n) done · -- Case 2. h2 : ¬prime n define at h2 --h2 : ¬(2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n) demorgan at h2 disj_syll h2 h1 obtain (a : Nat) (h3 : ∃ (b : Nat), a * b = n ∧ a < n ∧ b < n) from h2 obtain (b : Nat) (h4 : a * b = n ∧ a < n ∧ b < n) from h3 have h5 : 2 ≤ a := by by_contra h6 have h7 : a ≤ 1 := by linarith have h8 : n ≤ b := calc n _ = a * b := h4.left.symm _ ≤ 1 * b := by rel [h7] _ = b := by ring linarith --n ≤ b contradicts b < n done have h6 : ∃ (p : Nat), prime_factor p a := ih a h4.right.left h5 obtain (p : Nat) (h7 : prime_factor p a) from h6 apply Exists.intro p define --Goal : prime p ∧ p ∣ n define at h7 --h7 : prime p ∧ p ∣ a apply And.intro h7.left have h8 : a ∣ n := by apply Exists.intro b show n = a * b from (h4.left).symm done show p ∣ n from dvd_trans h7.right h8 done done lemma exists_least_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q := by set S : Set Nat := {p : Nat | prime_factor p n} have h2 : ∃ (p : Nat), p ∈ S := exists_prime_factor n h show ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q from well_ord_princ S h2 done lemma all_prime_nil : all_prime [] := by define --Goal : ∀ p ∈ [], prime p fix p : Nat contrapos --Goal : ¬prime p → p ∉ [] assume h1 : ¬prime p show p ∉ [] from List.not_mem_nil p done lemma all_prime_cons (n : Nat) (L : List Nat) : all_prime (n :: L) ↔ prime n ∧ all_prime L := by apply Iff.intro · -- (→) assume h1 : all_prime (n :: L) --Goal : prime n ∧ all_prime L define at h1 --h1 : ∀ p ∈ n :: L, prime p apply And.intro (h1 n (List.mem_cons_self n L)) define --Goal : ∀ p ∈ L, prime p fix p : Nat assume h2 : p ∈ L show prime p from h1 p (List.mem_cons_of_mem n h2) done · -- (←) assume h1 : prime n ∧ all_prime L --Goal : all_prime (n :: l) define : all_prime L at h1 define fix p : Nat assume h2 : p ∈ n :: L rewrite [List.mem_cons] at h2 --h2 : p = n ∨ p ∈ L by_cases on h2 · -- Case 1. h2 : p = n rewrite [h2] show prime n from h1.left done · -- Case 2. h2 : p ∈ L show prime p from h1.right p h2 done done done lemma nondec_nil : nondec [] := by define --Goal : True trivial --trivial proves some obviously true statements, such as True done lemma nondec_cons (n : Nat) (L : List Nat) : nondec (n :: L) ↔ (∀ m ∈ L, n ≤ m) ∧ nondec L := by rfl lemma prod_nil : prod [] = 1 := by rfl lemma prod_cons : prod (n :: L) = n * (prod L) := by rfl lemma exists_cons_of_length_eq_succ {A : Type} {l : List A} {n : Nat} (h : l.length = n + 1) : ∃ (a : A) (L : List A), l = a :: L ∧ L.length = n := by have h1 : ¬l.length = 0 := by linarith rewrite [List.length_eq_zero] at h1 obtain (a : A) (h2 : ∃ (L : List A), l = a :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List A) (h3 : l = a :: L) from h2 apply Exists.intro a apply Exists.intro L apply And.intro h3 have h4 : (a :: L).length = L.length + 1 := List.length_cons a L rewrite [←h3, h] at h4 show L.length = n from (Nat.add_right_cancel h4).symm done lemma list_elt_dvd_prod_by_length (a : Nat) : ∀ (n : Nat), ∀ (l : List Nat), l.length = n → a ∈ l → a ∣ prod l := by by_induc · --Base Case fix l : List Nat assume h1 : l.length = 0 rewrite [List.length_eq_zero] at h1 --h1 : l = [] rewrite [h1] --Goal : a ∈ [] → a ∣ prod [] contrapos assume h2 : ¬a ∣ prod [] show a ∉ [] from List.not_mem_nil a done · -- Induction Step fix n : Nat assume ih : ∀ (l : List Nat), List.length l = n → a ∈ l → a ∣ prod l fix l : List Nat assume h1 : l.length = n + 1 --Goal : a ∈ l → a ∣ prod l obtain (b : Nat) (h2 : ∃ (L : List Nat), l = b :: L ∧ L.length = n) from exists_cons_of_length_eq_succ h1 obtain (L : List Nat) (h3 : l = b :: L ∧ L.length = n) from h2 have h4 : a ∈ L → a ∣ prod L := ih L h3.right assume h5 : a ∈ l rewrite [h3.left, prod_cons] --Goal : a ∣ b * prod L rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b ∨ a ∈ L by_cases on h5 · -- Case 1. h5 : a = b apply Exists.intro (prod L) rewrite [h5] rfl done · -- Case 2. h5 : a ∈ L have h6 : a ∣ prod L := h4 h5 have h7 : prod L ∣ b * prod L := by apply Exists.intro b ring done show a ∣ b * prod L from dvd_trans h6 h7 done done done lemma list_elt_dvd_prod {a : Nat} {l : List Nat} (h : a ∈ l) : a ∣ prod l := by set n : Nat := l.length have h1 : l.length = n := by rfl show a ∣ prod l from list_elt_dvd_prod_by_length a n l h1 h done lemma exists_prime_factorization : ∀ (n : Nat), n ≥ 1 → ∃ (l : List Nat), prime_factorization n l := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, n_1 ≥ 1 → ∃ (l : List Nat), prime_factorization n_1 l assume h1 : n ≥ 1 by_cases h2 : n = 1 · -- Case 1. h2 : n = 1 apply Exists.intro [] define apply And.intro · -- Proof of nondec_prime_list [] define show all_prime [] ∧ nondec [] from And.intro all_prime_nil nondec_nil done · -- Proof of prod [] = n rewrite [prod_nil, h2] rfl done done · -- Case 2. h2 : n ≠ 1 have h3 : n ≥ 2 := lt_of_le_of_ne' h1 h2 obtain (p : Nat) (h4 : prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q) from exists_least_prime_factor h3 have p_prime_factor : prime_factor p n := h4.left define at p_prime_factor have p_prime : prime p := p_prime_factor.left have p_dvd_n : p ∣ n := p_prime_factor.right have p_least : ∀ (q : Nat), prime_factor q n → p ≤ q := h4.right obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n have h5 : m ≠ 0 := by contradict h1 with h6 have h7 : n = 0 := calc n _ = p * m := n_eq_pm _ = p * 0 := by rw [h6] _ = 0 := by ring rewrite [h7] decide done have m_pos : 0 < m := Nat.pos_of_ne_zero h5 have m_lt_n : m < n := by define at p_prime show m < n from calc m _ < m + m := by linarith _ = 2 * m := by ring _ ≤ p * m := by rel [p_prime.left] _ = n := n_eq_pm.symm done obtain (L : List Nat) (h6 : prime_factorization m L) from ih m m_lt_n m_pos define at h6 have ndpl_L : nondec_prime_list L := h6.left define at ndpl_L apply Exists.intro (p :: L) define apply And.intro · -- Proof of nondec_prime_list (p :: L) define apply And.intro · -- Proof of all_prime (p :: L) rewrite [all_prime_cons] show prime p ∧ all_prime L from And.intro p_prime ndpl_L.left done · -- Proof of nondec (p :: L) rewrite [nondec_cons] apply And.intro _ ndpl_L.right fix q : Nat assume q_in_L : q ∈ L have h7 : q ∣ prod L := list_elt_dvd_prod q_in_L rewrite [h6.right] at h7 --h7 : q ∣ m have h8 : m ∣ n := by apply Exists.intro p rewrite [n_eq_pm] ring done have q_dvd_n : q ∣ n := dvd_trans h7 h8 have ap_L : all_prime L := ndpl_L.left define at ap_L have q_prime_factor : prime_factor q n := And.intro (ap_L q q_in_L) q_dvd_n show p ≤ q from p_least q q_prime_factor done done · -- Proof of prod (p :: L) = n rewrite [prod_cons, h6.right, n_eq_pm] rfl done done done theorem Theorem_7_2_2 {a b c : Nat} (h1 : c ∣ a * b) (h2 : rel_prime a c) : c ∣ b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑c ∣ ↑b define at h1; define at h2; define obtain (j : Nat) (h3 : a * b = c * j) from h1 set s : Int := gcd_c1 a c set t : Int := gcd_c2 a c have h4 : s * ↑a + t * ↑c = ↑(gcd a c) := gcd_lin_comb c a rewrite [h2, Nat.cast_one] at h4 --h4 : s * ↑a + t * ↑c = (1 : Int) apply Exists.intro (s * ↑j + t * ↑b) show ↑b = ↑c * (s * ↑j + t * ↑b) from calc ↑b _ = (1 : Int) * ↑b := (one_mul _).symm _ = (s * ↑a + t * ↑c) * ↑b := by rw [h4] _ = s * (↑a * ↑b) + t * ↑c * ↑b := by ring _ = s * (↑c * ↑j) + t * ↑c * ↑b := by rw [←Nat.cast_mul a b, h3, Nat.cast_mul c j] _ = ↑c * (s * ↑j + t * ↑b) := by ring done lemma le_nonzero_prod_left {a b : Nat} (h : a * b ≠ 0) : a ≤ a * b := by have h1 : b ≠ 0 := by contradict h with h1 rewrite [h1] ring done have h2 : 1 ≤ b := Nat.pos_of_ne_zero h1 show a ≤ a * b from calc a = a * 1 := (mul_one a).symm _ ≤ a * b := by rel [h2] done lemma le_nonzero_prod_right {a b : Nat} (h : a * b ≠ 0) : b ≤ a * b := by rewrite [mul_comm] rewrite [mul_comm] at h show b ≤ b * a from le_nonzero_prod_left h done lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry lemma rel_prime_of_prime_not_dvd {a p : Nat} (h1 : prime p) (h2 : ¬p ∣ a) : rel_prime a p := by have h3 : gcd a p ∣ a := gcd_dvd_left a p have h4 : gcd a p ∣ p := gcd_dvd_right a p have h5 : gcd a p = 1 ∨ gcd a p = p := dvd_prime h1 h4 have h6 : gcd a p ≠ p := by contradict h2 with h6 rewrite [h6] at h3 show p ∣ a from h3 done disj_syll h5 h6 show rel_prime a p from h5 done theorem Theorem_7_2_3 {a b p : Nat} (h1 : prime p) (h2 : p ∣ a * b) : p ∣ a ∨ p ∣ b := by or_right with h3 have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3 show p ∣ b from Theorem_7_2_2 h2 h4 done lemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a ≥ 1 := by have h1 : a ≠ 0 := by by_contra h1 rewrite [h1] at h contradict h linarith done show a ≥ 1 from Nat.pos_of_ne_zero h1 done lemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by have h1 : a ≥ 1 := ge_one_of_prod_one h have h2 : a * b ≠ 0 := by linarith have h3 : a ≤ a * b := le_nonzero_prod_left h2 rewrite [h] at h3 show a = 1 from Nat.le_antisymm h3 h1 done lemma eq_one_of_dvd_one {n : Nat} (h : n ∣ 1) : n = 1 := by obtain (j : Nat) (h1 : 1 = n * j) from h show n = 1 from eq_one_of_prod_one h1.symm done lemma prime_not_one {p : Nat} (h : prime p) : p ≠ 1 := by define at h linarith done theorem Theorem_7_2_4 {p : Nat} (h1 : prime p) : ∀ (l : List Nat), p ∣ prod l → ∃ a ∈ l, p ∣ a := by apply List.rec · -- Base Case. Goal : p ∣ prod [] → ∃ a ∈ [], p ∣ a rewrite [prod_nil] assume h2 : p ∣ 1 show ∃ a ∈ [], p ∣ a from absurd (eq_one_of_dvd_one h2) (prime_not_one h1) done · -- Induction Step fix b : Nat fix L : List Nat assume ih : p ∣ prod L → ∃ a ∈ L, p ∣ a --Goal : p ∣ prod (b :: L) → ∃ a ∈ b :: L, p ∣ a assume h2 : p ∣ prod (b :: L) rewrite [prod_cons] at h2 have h3 : p ∣ b ∨ p ∣ prod L := Theorem_7_2_3 h1 h2 by_cases on h3 · -- Case 1. h3 : p ∣ b apply Exists.intro b show b ∈ b :: L ∧ p ∣ b from And.intro (List.mem_cons_self b L) h3 done · -- Case 2. h3 : p ∣ prod L obtain (a : Nat) (h4 : a ∈ L ∧ p ∣ a) from ih h3 apply Exists.intro a show a ∈ b :: L ∧ p ∣ a from And.intro (List.mem_cons_of_mem b h4.left) h4.right done done done lemma prime_in_list {p : Nat} {l : List Nat} (h1 : prime p) (h2 : all_prime l) (h3 : p ∣ prod l) : p ∈ l := by obtain (a : Nat) (h4 : a ∈ l ∧ p ∣ a) from Theorem_7_2_4 h1 l h3 define at h2 have h5 : prime a := h2 a h4.left have h6 : p = 1 ∨ p = a := dvd_prime h5 h4.right disj_syll h6 (prime_not_one h1) rewrite [h6] show a ∈ l from h4.left done lemma first_le_first {p q : Nat} {l m : List Nat} (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m)) (h3 : prod (p :: l) = prod (q :: m)) : p ≤ q := by define at h1; define at h2 have h4 : q ∣ prod (p :: l) := by define apply Exists.intro (prod m) rewrite [←prod_cons] show prod (p :: l) = prod (q :: m) from h3 done have h5 : all_prime (q :: m) := h2.left rewrite [all_prime_cons] at h5 have h6 : q ∈ p :: l := prime_in_list h5.left h1.left h4 have h7 : nondec (p :: l) := h1.right rewrite [nondec_cons] at h7 rewrite [List.mem_cons] at h6 by_cases on h6 · -- Case 1. h6 : q = p linarith done · -- Case 2. h6 : q ∈ l have h8 : ∀ m ∈ l, p ≤ m := h7.left show p ≤ q from h8 q h6 done done lemma nondec_prime_list_tail {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by define at h define rewrite [all_prime_cons, nondec_cons] at h show all_prime l ∧ nondec l from And.intro h.left.right h.right.right done lemma cons_prod_not_one {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : prod (p :: l) ≠ 1 := by define at h have h1 : all_prime (p :: l) := h.left rewrite [all_prime_cons] at h1 rewrite [prod_cons] by_contra h2 show False from (prime_not_one h1.left) (eq_one_of_prod_one h2) done lemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) : l = [] ↔ prod l = 1 := by apply Iff.intro · -- (→) assume h1 : l = [] rewrite [h1] show prod [] = 1 from prod_nil done · -- (←) contrapos assume h1 : ¬l = [] obtain (p : Nat) (h2 : ∃ (L : List Nat), l = p :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List Nat) (h3 : l = p :: L) from h2 rewrite [h3] at h rewrite [h3] show ¬prod (p :: L) = 1 from cons_prod_not_one h done done lemma prime_pos {p : Nat} (h : prime p) : p > 0 := by define at h linarith done theorem Theorem_7_2_5 : ∀ (l1 l2 : List Nat), nondec_prime_list l1 → nondec_prime_list l2 → prod l1 = prod l2 → l1 = l2 := by apply List.rec · -- Base Case. Goal : ∀ (l2 : List Nat), nondec_prime_list [] → -- nondec_prime_list l2 → prod [] = prod l2 → [] = l2 fix l2 : List Nat assume h1 : nondec_prime_list [] assume h2 : nondec_prime_list l2 assume h3 : prod [] = prod l2 rewrite [prod_nil, eq_comm, ←list_nil_iff_prod_one h2] at h3 show [] = l2 from h3.symm done · -- Induction Step fix p : Nat fix L1 : List Nat assume ih : ∀ (L2 : List Nat), nondec_prime_list L1 → nondec_prime_list L2 → prod L1 = prod L2 → L1 = L2 -- Goal : ∀ (l2 : List Nat), nondec_prime_list (p :: L1) → -- nondec_prime_list l2 → prod (p :: L1) = prod l2 → p :: L1 = l2 fix l2 : List Nat assume h1 : nondec_prime_list (p :: L1) assume h2 : nondec_prime_list l2 assume h3 : prod (p :: L1) = prod l2 have h4 : ¬prod (p :: L1) = 1 := cons_prod_not_one h1 rewrite [h3, ←list_nil_iff_prod_one h2] at h4 obtain (q : Nat) (h5 : ∃ (L : List Nat), l2 = q :: L) from List.exists_cons_of_ne_nil h4 obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5 rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2) rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2) have h7 : p ≤ q := first_le_first h1 h2 h3 have h8 : q ≤ p := first_le_first h2 h1 h3.symm have h9 : p = q := by linarith rewrite [h9, prod_cons, prod_cons] at h3 --h3 : q * prod L1 = q * prod L2 have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1 have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2 define at h2 have h12 : all_prime (q :: L2) := h2.left rewrite [all_prime_cons] at h12 have h13 : q > 0 := prime_pos h12.left have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3 have h15 : L1 = L2 := ih L2 h10 h11 h14 rewrite [h6, h9, h15] rfl done done theorem fund_thm_arith (n : Nat) (h : n ≥ 1) : ∃! (l : List Nat), prime_factorization n l := by exists_unique · -- Existence show ∃ (l : List Nat), prime_factorization n l from exists_prime_factorization n h done · -- Uniqueness fix l1 : List Nat; fix l2 : List Nat assume h1 : prime_factorization n l1 assume h2 : prime_factorization n l2 define at h1; define at h2 have h3 : prod l1 = n := h1.right rewrite [←h2.right] at h3 show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3 done done /- Section 7.3 -/ theorem congr_refl (m : Nat) : ∀ (a : Int), a ≡ a (MOD m) := by fix a : Int define --Goal : ∃ (c : Int), a - a = ↑m * c apply Exists.intro 0 ring done theorem congr_symm {m : Nat} : ∀ {a b : Int}, a ≡ b (MOD m) → b ≡ a (MOD m) := by fix a : Int; fix b : Int assume h1 : a ≡ b (MOD m) define at h1 --h1 : ∃ (c : Int), a - b = ↑m * c define --Goal : ∃ (c : Int), b - a = ↑m * c obtain (c : Int) (h2 : a - b = m * c) from h1 apply Exists.intro (-c) show b - a = m * (-c) from calc b - a _ = -(a - b) := by ring _ = -(m * c) := by rw [h2] _ = m * (-c) := by ring done theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry /- Fundamental properties of congruence classes -/ lemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) : X = Y ↔ X.val = Y.val := Fin.ext_iff lemma val_nat_eq_mod (n k : Nat) : ([k]_(n + 1)).val = k % (n + 1) := by rfl lemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl theorem cc_rep {m : Nat} (X : ZMod m) : ∃ (a : Int), X = [a]_m := match m with | 0 => by apply Exists.intro X rfl done | n + 1 => by apply Exists.intro ↑(X.val) have h1 : X.val < n + 1 := Fin.prop X rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1] rfl done theorem add_class (m : Nat) (a b : Int) : [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm theorem mul_class (m : Nat) (a b : Int) : [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm lemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ [a - b]_m = [0]_m := by apply Iff.intro · -- (→) assume h1 : [a]_m = [b]_m have h2 : a - b = a + (-b) := by ring have h3 : b + (-b) = 0 := by ring show [a - b]_m = [0]_m from calc [a - b]_m _ = [a + (-b)]_m := by rw [h2] _ = [a]_m + [-b]_m := by rw [add_class] _ = [b]_m + [-b]_m := by rw [h1] _ = [b + -b]_m := by rw [add_class] _ = [0]_m := by rw [h3] done · -- (←) assume h1 : [a - b]_m = [0]_m have h2 : b + (a - b) = a := by ring have h3 : b + 0 = b := by ring show [a]_m = [b]_m from calc [a]_m _ = [b + (a - b)]_m := by rw [h2] _ = [b]_m + [a - b]_m := by rw [add_class] _ = [b]_m + [0]_m := by rw [h1] _ = [b + 0]_m := by rw [add_class] _ = [b]_m := by rw [h3] done done lemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) : [a]_m = [0]_m → [-a]_m = [0]_m := by assume h1 : [a]_m = [0]_m have h2 : 0 + (-a) = -a := by ring have h3 : a + (-a) = 0 := by ring show [-a]_m = [0]_m from calc [-a]_m _ = [0 + (-a)]_m := by rw [h2] _ = [0]_m + [-a]_m := by rw [add_class] _ = [a]_m + [-a]_m := by rw [h1] _ = [a + (-a)]_m := by rw [add_class] _ = [0]_m := by rw [h3] done lemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) : [-a]_m = [0]_m ↔ [a]_m = [0]_m := by apply Iff.intro _ (cc_neg_zero_of_cc_zero m a) assume h1 : [-a]_m = [0]_m have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1 have h3 : -(-a) = a := by ring rewrite [h3] at h2 show [a]_m = [0]_m from h2 done lemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl lemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m ↔ m ∣ k := match m with | 0 => by have h : (0 : Int) = (↑(0 : Nat) : Int) := by rfl rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj] apply Iff.intro · -- (→) assume h1 : k = 0 rewrite [h1] show 0 ∣ 0 from dvd_self 0 done · -- (←) assume h1 : 0 ∣ k obtain (c : Nat) (h2 : k = 0 * c) from h1 rewrite [h2] ring done done | n + 1 => by rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero] show k % (n + 1) = 0 ↔ n + 1 ∣ k from (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm done lemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m ↔ ↑m ∣ a := by obtain (k : Nat) (h1 : a = ↑k ∨ a = -↑k) from Int.eq_nat_or_neg a by_cases on h1 · -- Case 1. h1: a = ↑k rewrite [h1, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done · -- Case 2. h1: a = -↑k rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done done theorem cc_eq_iff_congr (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ a ≡ b (MOD m) := calc [a]_m = [b]_m _ ↔ [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b _ ↔ ↑m ∣ (a - b) := cc_zero_iff_dvd m (a - b) _ ↔ a ≡ b (MOD m) := by rfl /- End of fundamental properties of congruence classes -/ lemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m := by have h1 : (↑m : Int) ≠ 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m) show 0 ≤ a % m from Int.emod_nonneg a h1 done lemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m) have h2 : (↑m : Int) > 0 := (Nat.cast_pos).rtl h1 show a % m < m from Int.emod_lt_of_pos a h2 done lemma congr_mod_mod (m : Nat) (a : Int) : a ≡ a % m (MOD m) := by define have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m apply Exists.intro (a / m) show a - a % m = m * (a / m) from calc a - (a % m) _ = m * (a / m) + a % m - a % m := by rw [h1] _ = m * (a / m) := by ring done lemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) := And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a)) theorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) : ∃! (r : Int), 0 ≤ r ∧ r < m ∧ a ≡ r (MOD m) := by exists_unique · -- Existence apply Exists.intro (a % m) show 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) from mod_cmpl_res m a done · -- Uniqueness fix r1 : Int; fix r2 : Int assume h1 : 0 ≤ r1 ∧ r1 < m ∧ a ≡ r1 (MOD m) assume h2 : 0 ≤ r2 ∧ r2 < m ∧ a ≡ r2 (MOD m) have h3 : r1 ≡ r2 (MOD m) := congr_trans (congr_symm h1.right.right) h2.right.right obtain (d : Int) (h4 : r1 - r2 = m * d) from h3 have h5 : r1 - r2 < m * 1 := by linarith have h6 : m * (-1) < r1 - r2 := by linarith rewrite [h4] at h5 --h5 : m * d < m * 1 rewrite [h4] at h6 --h6 : m * -1 < m * d have h7 : (↑m : Int) ≥ 0 := Nat.cast_nonneg m have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7 have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7 have h10 : d = 0 := by linarith show r1 = r2 from calc r1 _ = r1 - r2 + r2 := by ring _ = m * 0 + r2 := by rw [h4, h10] _ = r2 := by ring done done lemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m := (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a) theorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y rewrite [h1, h2] have h3 : a + b = b + a := by ring show [a]_m + [b]_m = [b]_m + [a]_m from calc [a]_m + [b]_m _ = [a + b]_m := add_class m a b _ = [b + a]_m := by rw [h3] _ = [b]_m + [a]_m := (add_class m b a).symm done theorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X rewrite [h1] have h2 : a * 1 = a := by ring show [a]_m * [1]_m = [a]_m from calc [a]_m * [1]_m _ = [a * 1]_m := mul_class m a 1 _ = [a]_m := by rw [h2] done theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry lemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) : [a]_m * [gcd_c2 m a]_m = [1]_m := by set s : Int := gcd_c1 m a have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m define at h1 rewrite [h1, Nat.cast_one] at h2 --h2 : s * ↑m + gcd_c2 m a * ↑a = 1 rewrite [mul_class, cc_eq_iff_congr] define --Goal : ∃ (c : Int), ↑a * gcd_c2 m a - 1 = ↑m * c apply Exists.intro (-s) show a * (gcd_c2 m a) - 1 = m * (-s) from calc a * (gcd_c2 m a) - 1 _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring _ = 1 + m * (-s) - 1 := by rw [h2] _ = m * (-s) := by ring done theorem Theorem_7_3_7 (m a : Nat) : invertible [a]_m ↔ rel_prime m a := by apply Iff.intro · -- (→) assume h1 : invertible [a]_m define at h1 obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1 obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y rewrite [h3, mul_class, cc_eq_iff_congr] at h2 define at h2 obtain (c : Int) (h4 : a * b - 1 = m * c) from h2 rewrite [Exercise_7_2_6] --Goal : ∃ (s t : Int), s * ↑m + t * ↑a = 1 apply Exists.intro (-c) apply Exists.intro b show (-c) * m + b * a = 1 from calc (-c) * m + b * a _ = (-c) * m + (a * b - 1) + 1 := by ring _ = (-c) * m + m * c + 1 := by rw [h4] _ = 1 := by ring done · -- (←) assume h1 : rel_prime m a define show ∃ (Y : ZMod m), [a]_m * Y = [1]_m from Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1) done done /- Section 7.4 -/ section Euler open Euler lemma num_rp_below_base {m : Nat} : num_rp_below m 0 = 0 := by rfl lemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) : num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by have h1 : num_rp_below m (j + 1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : gcd m j = 1 rewrite [if_pos h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1 show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1 done lemma num_rp_below_step_not_rp {m j : Nat} (h : ¬rel_prime m j) : num_rp_below m (j + 1) = num_rp_below m j := by have h1 : num_rp_below m (j +1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : ¬gcd m j = 1 rewrite [if_neg h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j show num_rp_below m (j + 1) = num_rp_below m j from h1 done lemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl #eval phi 10 --Answer: 4 lemma prod_inv_iff_inv {m : Nat} {X : ZMod m} (h1 : invertible X) (Y : ZMod m) : invertible (X * Y) ↔ invertible Y := by apply Iff.intro · -- (→) assume h2 : invertible (X * Y) obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2 apply Exists.intro (X * Z) rewrite [←h3] --Goal : Y * (X * Z) = X * Y * Z ring --Note that ring can do algebra in ZMod m done · -- (←) assume h2 : invertible Y obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1 obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2 apply Exists.intro (Xi * Yi) show (X * Y) * (Xi * Yi) = [1]_m from calc X * Y * (Xi * Yi) _ = (X * Xi) * (Y * Yi) := by ring _ = [1]_m * [1]_m := by rw [h3, h4] _ = [1]_m := Theorem_7_3_6_7 [1]_m done done lemma F_rp_def {m i : Nat} (h : rel_prime m i) : F m i = [i]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h --h : gcd m i = 1 rewrite [if_pos h] at h1 show F m i = [i]_m from h1 done lemma F_not_rp_def {m i : Nat} (h : ¬rel_prime m i) : F m i = [1]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h rewrite [h1, if_neg h] rfl done lemma prod_seq_base {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 0 k f = [1]_m := by rfl lemma prod_seq_step {m : Nat} (n k : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl lemma prod_seq_zero_step {m : Nat} (n : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by rewrite [prod_seq_step, zero_add] rfl done lemma prod_one {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 1 k f = f k := by rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7] rfl done lemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl lemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m := calc [G m a i]_m _ = [(a * i) % m]_m := by rfl _ = [a * i]_m := (cc_eq_mod m (a * i)).symm _ = [a]_m * [i]_m := (mul_class m a i).symm lemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) : rel_prime m (G m a i) ↔ rel_prime m i := by have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1 show rel_prime m (G m a i) ↔ rel_prime m i from calc rel_prime m (G m a i) _ ↔ invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm _ ↔ invertible ([a]_m * [i]_m) := by rw [cc_G] _ ↔ invertible [i]_m := prod_inv_iff_inv h2 ([i]_m) _ ↔ rel_prime m i := Theorem_7_3_7 m i done lemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) : F m (G m a i) = [a]_m * F m i := by have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2 show F m (G m a i) = [a]_m * F m i from calc F m (G m a i) _ = [G m a i]_m := F_rp_def h3 _ = [a]_m * [i]_m := cc_G m a i _ = [a]_m * F m i := by rw [F_rp_def h2] done lemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : ¬rel_prime m i) : F m (G m a i) = [1]_m := by rewrite [←G_rp_iff h1 i] at h2 show F m (G m a i) = [1]_m from F_not_rp_def h2 done lemma FG_prod {m a : Nat} (h1 : rel_prime m a) : ∀ (k : Nat), prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by by_induc · -- Base Case show prod_seq 0 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from calc prod_seq 0 0 ((F m) ∘ (G m a)) _ = [1]_m := prod_seq_base _ _ _ = [a]_m ^ 0 * [1]_m := by ring _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by rw [num_rp_below_base, prod_seq_base] done · -- Induction Step fix k : Nat assume ih : prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) by_cases h2 : rel_prime m k · -- Case 1. h2 : rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([a]_m * F m k) := by rw [FG_rp h1 h2] _ = [a]_m ^ ((num_rp_below m k) + 1) * ((prod_seq k 0 (F m)) * F m k) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_rp h2, prod_seq_zero_step] done · -- Case 2. h2 : ¬rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([1]_m) := by rw [FG_not_rp h1 h2] _ = [a]_m ^ (num_rp_below m k) * (prod_seq k 0 (F m) * ([1]_m)) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_not_rp h2, prod_seq_zero_step, F_not_rp_def h2] done done done lemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by define --Goal : ∀ i < m, G m a i < m fix i : Nat assume h1 : i < m rewrite [G_def] --Goal : a * i % m < m show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m) done lemma left_inv_one_one_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g' (g i) = i) : one_one_below n g := sorry lemma right_inv_onto_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g (g' i) = i) (h2 : maps_below n g') : onto_below n g := by define at h2; define fix k : Nat assume h3 : k < n apply Exists.intro (g' k) show g' k < n ∧ g (g' k) = k from And.intro (h2 k h3) (h1 k h3) done lemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m * [inv_mod m a]_m = [1]_m := by have h2 : 0 ≤ (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a) show [a]_m * [inv_mod m a]_m = [1]_m from calc [a]_m * [inv_mod m a]_m _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2] _ = [a]_m * [gcd_c2 m a]_m := by rw [←cc_eq_mod] _ = [1]_m := gcd_c2_inv h1 done lemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m := calc a * (b % m) % m = a % m * (b % m % m) % m := Nat.mul_mod _ _ _ _ = a % m * (b % m) % m := by rw [Nat.mod_mod] _ = a * b % m := (Nat.mul_mod _ _ _).symm lemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm] rfl done theorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] : ↑a ≡ ↑b (MOD m) ↔ a % m = b % m := sorry lemma mul_inv_mod_cancel {m a i : Nat} [NeZero m] (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1 rewrite [mul_class, cc_eq_iff_congr, ←Nat.cast_mul, ←Nat.cast_one, congr_iff_mod_eq_Nat] at h3 show a * inv_mod m a * i % m = i from calc a * (inv_mod m a) * i % m _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = 1 % m * i % m := by rw [h3] _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = i % m := by rw [one_mul] _ = i := Nat.mod_eq_of_lt h2 done lemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl lemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, G m a (Ginv m a i) = i := by fix i : Nat assume h2 : i < m show G m a (Ginv m a i) = i from calc G m a (Ginv m a i) _ = a * ((inv_mod m a * i) % m) % m := by rfl _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, Ginv m a (G m a i) = i := by fix i : Nat assume h2 : i < m show Ginv m a (G m a i) = i from calc Ginv m a (G m a i) _ = inv_mod m a * ((a * i) % m) % m := by rfl _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc, mul_comm (inv_mod m a)] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_maps_below (m a : Nat) [NeZero m] : maps_below m (Ginv m a) := G_maps_below m (inv_mod m a) lemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : one_one_below m (G m a) := left_inv_one_one_below (Ginv_left_inv h1) lemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : onto_below m (G m a) := right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a) lemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : perm_below m (G m a) := And.intro (G_maps_below m a) (And.intro (G_one_one_below h1) (G_onto_below h1)) --Permuting a product of congruence classes doesn't change product lemma swap_fst (u v : Nat) : swap u v u = v := by define : swap u v u --Goal : (if u = u then v else if u = v then u else u) = v have h : u = u := by rfl rewrite [if_pos h] rfl done lemma swap_snd (u v : Nat) : swap u v v = u := by define : swap u v v by_cases h1 : v = u · -- Case 1. h1 : v = u rewrite [if_pos h1] show v = u from h1 done · -- Case 2. h1 : v ≠ u rewrite [if_neg h1] have h2 : v = v := by rfl rewrite [if_pos h2] rfl done done lemma swap_other {u v i : Nat} (h1 : i ≠ u) (h2 : i ≠ v) : swap u v i = i := by define : swap u v i rewrite [if_neg h1, if_neg h2] rfl done lemma swap_values (u v i : Nat) : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := by by_cases h1 : i = u · -- Case 1. h1 : i = u apply Or.inl rewrite [h1] show swap u v u = v from swap_fst u v done · -- Case 2. h1 : i ≠ u apply Or.inr by_cases h2 : i = v · -- Case 2.1. h2 : i = v apply Or.inl rewrite [h2] show swap u v v = u from swap_snd u v done · -- Case 2.2. h2 : i ≠ v apply Or.inr show swap u v i = i from swap_other h1 h2 done done done lemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by define fix i : Nat assume h3 : i < n have h4 : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := swap_values u v i by_cases on h4 · -- Case 1. h4 : swap u v i = v rewrite [h4] show v < n from h2 done · -- Case 2. by_cases on h4 · -- Case 2.1. h4 : swap u v i = u rewrite [h4] show u < n from h1 done · -- Case 2.2. h4 : swap u v i = i rewrite [h4] show i < n from h3 done done done lemma swap_swap (u v n : Nat) : ∀ i < n, swap u v (swap u v i) = i := by fix i : Nat assume h : i < n by_cases h1 : i = u · -- Case 1. h1 : i = u rewrite [h1, swap_fst, swap_snd] rfl done · -- Case 2. h1 : i ≠ u by_cases h2 : i = v · -- Case 2.1. h2 : i = v rewrite [h2, swap_snd, swap_fst] rfl done · -- Case 2.2. h2 : i ≠ v rewrite [swap_other h1 h2, swap_other h1 h2] rfl done done done lemma swap_one_one_below (u v n) : one_one_below n (swap u v) := left_inv_one_one_below (swap_swap u v n) lemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) := right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2) lemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) := And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2)) lemma comp_perm_below {n : Nat} {f g : Nat → Nat} (h1 : perm_below n f) (h2 : perm_below n g) : perm_below n (f ∘ g) := sorry lemma trivial_swap (u : Nat) : swap u u = id := by apply funext fix x : Nat by_cases h1 : x = u · -- Case 1. h1 : x = u rewrite [h1, swap_fst] rfl done · -- Case 2. h1 : x ≠ u rewrite [swap_other h1 h1] rfl done done lemma prod_eq_fun {m : Nat} (f g : Nat → ZMod m) (k : Nat) : ∀ (n : Nat), (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g := by by_induc · -- Base Case assume h : (∀ i < 0, f (k + i) = g (k + i)) rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g assume h1 : ∀ i < n + 1, f (k + i) = g (k + i) have h2 : ∀ i < n, f (k + i) = g (k + i) := by fix i : Nat assume h2 : i < n have h3 : i < n + 1 := by linarith show f (k + i) = g (k + i) from h1 i h3 done have h3 : prod_seq n k f = prod_seq n k g := ih h2 have h4 : n < n + 1 := Nat.lt_succ_self n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4] rfl done done lemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := by have h2 : ∀ (i : Nat), i < u → (f ∘ swap u n) (0 + i) = f (0 + i) := by fix i : Nat assume h2 : i < u have h3 : 0 + i ≠ u := by linarith have h4 : 0 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl done show prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f from prod_eq_fun (f ∘ swap u n) f 0 u h2 done lemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := by have h2 : ∀ i < j, (f ∘ swap u n) (u + 1 + i) = f (u + 1 + i) := by fix i : Nat assume h2 : i < j have h3 : u + 1 + i ≠ u := by linarith have h4 : u + 1 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl show prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f from prod_eq_fun (f ∘ swap u n) f (u + 1) j h2 done lemma break_prod {m : Nat} (n : Nat) (f : Nat → ZMod m) : ∀ (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by by_induc · -- Base Case have h : n + 0 = n := by rfl rewrite [prod_seq_base, h, Theorem_7_3_6_7] rfl done · -- Induction Step fix j : Nat assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f rewrite [←add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc] rfl done done lemma break_prod_twice {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f := break_prod n f 1 rewrite [prod_one] at h2 have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f := break_prod (u + 1) f j rewrite [←h1] at h3 have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f := break_prod u f 1 rewrite [prod_one] at h4 rewrite [h3, h4] at h2 show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2 done lemma swap_prod_eq_prod {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f := by by_cases h2 : u = n · -- Case 1. h2 : u = n rewrite [h2, trivial_swap n] --Goal : prod_seq (n + 1) 0 (f ∘ id) = prod_seq (n + 1) 0 f rfl done · -- Case 2. h2 : ¬u = n have h3 : u + 1 ≤ n := Nat.lt_of_le_of_ne h1 h2 obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3 have break_f : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := break_prod_twice f h4 have break_fs : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_prod_twice (f ∘ swap u n) h4 have f_eq_fs_below : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := swap_prod_eq_prod_below f h1 have f_eq_fs_btwn : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4 show prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f from calc prod_seq (n + 1) 0 (f ∘ swap u n) _ = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_fs _ = prod_seq u 0 f * (f ∘ swap u n) u * prod_seq j (u + 1) f * (f ∘ swap u n) n := by rw [f_eq_fs_below, f_eq_fs_btwn] _ = prod_seq u 0 f * f (swap u n u) * prod_seq j (u + 1) f * f (swap u n n) := by rfl _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by rw [swap_fst, swap_snd] _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring _ = prod_seq (n + 1) 0 f := break_f.symm done done lemma perm_below_fixed {n : Nat} {g : Nat → Nat} (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry lemma perm_prod {m : Nat} (f : Nat → ZMod m) : ∀ (n : Nat), ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) := by by_induc · -- Base Case fix g : Nat → Nat assume h1 : perm_below 0 g rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) fix g : Nat → Nat assume g_pb : perm_below (n + 1) g define at g_pb have g_ob : onto_below (n + 1) g := g_pb.right.right define at g_ob have h1 : n < n + 1 := by linarith obtain (u : Nat) (h2 : u < n + 1 ∧ g u = n) from g_ob n h1 have s_pb : perm_below (n + 1) (swap u n) := swap_perm_below h2.left h1 have gs_pb_n1 : perm_below (n + 1) (g ∘ swap u n) := comp_perm_below g_pb s_pb have gs_fix_n : (g ∘ swap u n) n = n := calc (g ∘ swap u n) n _ = g (swap u n n) := by rfl _ = g u := by rw [swap_snd] _ = n := h2.right have gs_pb_n : perm_below n (g ∘ swap u n) := perm_below_fixed gs_pb_n1 gs_fix_n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f ∘ (g ∘ swap u n)) := ih (g ∘ swap u n) gs_pb_n have h3 : u ≤ n := by linarith show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f ∘ g) from calc prod_seq (n + 1) 0 f _ = prod_seq n 0 f * f n := prod_seq_zero_step n f _ = prod_seq n 0 (f ∘ (g ∘ swap u n)) * f ((g ∘ swap u n) n) := by rw [gs_prod, gs_fix_n] _ = prod_seq n 0 (f ∘ g ∘ swap u n) * (f ∘ g ∘ swap u n) n := by rfl _ = prod_seq (n + 1) 0 (f ∘ g ∘ swap u n) := (prod_seq_zero_step n (f ∘ g ∘ swap u n)).symm _ = prod_seq (n + 1) 0 ((f ∘ g) ∘ swap u n) := by rfl _ = prod_seq (n + 1) 0 (f ∘ g) := swap_prod_eq_prod (f ∘ g) h3 done done lemma F_invertible (m i : Nat) : invertible (F m i) := by by_cases h : rel_prime m i · -- Case 1. h : rel_prime m i rewrite [F_rp_def h] show invertible [i]_m from (Theorem_7_3_7 m i).rtl h done · -- Case 2. h : ¬rel_prime m i rewrite [F_not_rp_def h] apply Exists.intro [1]_m show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m done done lemma Fprod_invertible (m : Nat) : ∀ (k : Nat), invertible (prod_seq k 0 (F m)) := by by_induc · -- Base Case apply Exists.intro [1]_m show prod_seq 0 0 (F m) * [1]_m = [1]_m from calc prod_seq 0 0 (F m) * [1]_m _ = [1]_m * [1]_m := by rw [prod_seq_base] _ = [1]_m := Theorem_7_3_6_7 ([1]_m) done · -- Induction Step fix k : Nat assume ih : invertible (prod_seq k 0 (F m)) rewrite [prod_seq_zero_step] show invertible (prod_seq k 0 (F m) * (F m k)) from (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k) done done theorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m ^ (phi m) = [1]_m := by have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2 show [a]_m ^ (phi m) = [1]_m from calc [a]_m ^ (phi m) _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3] _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring _ = prod_seq m 0 (F m ∘ G m a) * Y := by rw [FG_prod h1 m, phi_def] _ = prod_seq m 0 (F m) * Y := by rw [perm_prod (F m) m (G m a) (G_perm_below h1)] _ = [1]_m := by rw [h3] done lemma Exercise_7_4_5_Int (m : Nat) (a : Int) : ∀ (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry lemma Exercise_7_4_5_Nat (m a n : Nat) : [a]_m ^ n = [a ^ n]_m := by rewrite [Exercise_7_4_5_Int] rfl done theorem Euler's_theorem {m a : Nat} [NeZero m] (h1 : rel_prime m a) : a ^ (phi m) ≡ 1 (MOD m) := by have h2 : [a]_m ^ (phi m) = [1]_m := Theorem_7_4_2 h1 rewrite [Exercise_7_4_5_Nat m a (phi m)] at h2 --h2 : [a ^ phi m]_m = [1]_m show a ^ (phi m) ≡ 1 (MOD m) from (cc_eq_iff_congr _ _ _).ltr h2 done #eval gcd 10 7 --Answer: 1. So 10 and 7 are relatively prime #eval 7 ^ phi 10 --Answer: 2401, which is congruent to 1 mod 10. end Euler /- Section 7.5 -/ lemma num_rp_prime {p : Nat} (h1 : prime p) : ∀ k < p, num_rp_below p (k + 1) = k := sorry lemma phi_prime {p : Nat} (h1 : prime p) : phi p = p - 1 := by have h2 : 1 ≤ p := prime_pos h1 have h3 : p - 1 + 1 = p := Nat.sub_add_cancel h2 have h4 : p - 1 < p := by linarith have h5 : num_rp_below p (p - 1 + 1) = p - 1 := num_rp_prime h1 (p - 1) h4 rewrite [h3] at h5 show phi p = p - 1 from h5 done theorem Theorem_7_2_2_Int {a c : Nat} {b : Int} (h1 : ↑c ∣ ↑a * b) (h2 : rel_prime a c) : ↑c ∣ b := by rewrite [Int.natCast_dvd, Int.natAbs_mul, Int.natAbs_ofNat] at h1 --h1 : c ∣ a * Int.natAbs b rewrite [Int.natCast_dvd] --Goal : c ∣ Int.natAbs b show c ∣ Int.natAbs b from Theorem_7_2_2 h1 h2 done lemma Lemma_7_4_5 {m n : Nat} (a b : Int) (h1 : rel_prime m n) : a ≡ b (MOD m * n) ↔ a ≡ b (MOD m) ∧ a ≡ b (MOD n) := by apply Iff.intro · -- (→) assume h2 : a ≡ b (MOD m * n) obtain (j : Int) (h3 : a - b = (m * n) * j) from h2 apply And.intro · -- Proof of a ≡ b (MOD m) apply Exists.intro (n * j) show a - b = m * (n * j) from calc a - b _ = m * n * j := h3 _ = m * (n * j) := by ring done · -- Proof of a ≡ b (MOD n) apply Exists.intro (m * j) show a - b = n * (m * j) from calc a - b _ = m * n * j := h3 _ = n * (m * j) := by ring done done · -- (←) assume h2 : a ≡ b (MOD m) ∧ a ≡ b (MOD n) obtain (j : Int) (h3 : a - b = m * j) from h2.left have h4 : (↑n : Int) ∣ a - b := h2.right rewrite [h3] at h4 --h4 : ↑n ∣ ↑m * j have h5 : ↑n ∣ j := Theorem_7_2_2_Int h4 h1 obtain (k : Int) (h6 : j = n * k) from h5 apply Exists.intro k --Goal : a - b = ↑(m * n) * k rewrite [Nat.cast_mul] --Goal : a - b = ↑m * ↑n * k show a - b = (m * n) * k from calc a - b _ = m * j := h3 _ = m * (n * k) := by rw [h6] _ = (m * n) * k := by ring done done --From exercises of Section 7.2 theorem rel_prime_symm {a b : Nat} (h : rel_prime a b) : rel_prime b a := sorry lemma prime_NeZero {p : Nat} (h : prime p) : NeZero p := by rewrite [neZero_iff] --Goal : p ≠ 0 define at h linarith done lemma Lemma_7_5_1 {p e d m c s : Nat} {t : Int} (h1 : prime p) (h2 : e * d = (p - 1) * s + 1) (h3 : m ^ e - c = p * t) : c ^ d ≡ m (MOD p) := by have h4 : m ^ e ≡ c (MOD p) := Exists.intro t h3 have h5 : [m ^ e]_p = [c]_p := (cc_eq_iff_congr _ _ _).rtl h4 rewrite [←Exercise_7_4_5_Nat] at h5 --h5 : [m]_p ^ e = [c]_p by_cases h6 : p ∣ m · -- Case 1. h6 : p ∣ m have h7 : m ≡ 0 (MOD p) := by obtain (j : Nat) (h8 : m = p * j) from h6 apply Exists.intro (↑j : Int) --Goal : ↑m - 0 = ↑p * ↑j rewrite [h8, Nat.cast_mul] ring done have h8 : [m]_p = [0]_p := (cc_eq_iff_congr _ _ _).rtl h7 have h9 : e * d ≠ 0 := by rewrite [h2] show (p - 1) * s + 1 ≠ 0 from Nat.add_one_ne_zero _ done have h10 : (0 : Int) ^ (e * d) = 0 := zero_pow h9 have h11 : [c ^ d]_p = [m]_p := calc [c ^ d]_p _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat] _ = ([m]_p ^ e) ^ d := by rw [h5] _ = [m]_p ^ (e * d) := by ring _ = [0]_p ^ (e * d) := by rw [h8] _ = [0 ^ (e * d)]_p := Exercise_7_4_5_Int _ _ _ _ = [0]_p := by rw [h10] _ = [m]_p := by rw [h8] show c ^ d ≡ m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11 done · -- Case 2. h6 : ¬p ∣ m have h7 : rel_prime m p := rel_prime_of_prime_not_dvd h1 h6 have h8 : rel_prime p m := rel_prime_symm h7 have h9 : NeZero p := prime_NeZero h1 have h10 : (1 : Int) ^ s = 1 := by ring have h11 : [c ^ d]_p = [m]_p := calc [c ^ d]_p _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat] _ = ([m]_p ^ e) ^ d := by rw [h5] _ = [m]_p ^ (e * d) := by ring _ = [m]_p ^ ((p - 1) * s + 1) := by rw [h2] _ = ([m]_p ^ (p - 1)) ^ s * [m]_p := by ring _ = ([m]_p ^ (phi p)) ^ s * [m]_p := by rw [phi_prime h1] _ = [1]_p ^ s * [m]_p := by rw [Theorem_7_4_2 h8] _ = [1 ^ s]_p * [m]_p := by rw [Exercise_7_4_5_Int] _ = [1]_p * [m]_p := by rw [h10] _ = [m]_p * [1]_p := by ring _ = [m]_p := Theorem_7_3_6_7 _ show c ^ d ≡ m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11 done done theorem Theorem_7_5_1 (p q n e d k m c : Nat) (p_prime : prime p) (q_prime : prime q) (p_ne_q : p ≠ q) (n_pq : n = p * q) (ed_congr_1 : e * d = k * (p - 1) * (q - 1) + 1) (h1 : [m]_n ^ e = [c]_n) : [c]_n ^ d = [m]_n := by rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] at h1 --h1 : m ^ e ≡ c (MOD n) rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] --Goal : c ^ d ≡ m (MOD n) obtain (j : Int) (h2 : m ^ e - c = n * j) from h1 rewrite [n_pq, Nat.cast_mul] at h2 --h2 : m ^ e - c = p * q * j have h3 : e * d = (p - 1) * (k * (q - 1)) + 1 := by rewrite [ed_congr_1] ring done have h4 : m ^ e - c = p * (q * j) := by rewrite [h2] ring done have congr_p : c ^ d ≡ m (MOD p) := Lemma_7_5_1 p_prime h3 h4 have h5 : e * d = (q - 1) * (k * (p - 1)) + 1 := by rewrite [ed_congr_1] ring done have h6 : m ^ e - c = q * (p * j) := by rewrite [h2] ring done have congr_q : c ^ d ≡ m (MOD q) := Lemma_7_5_1 q_prime h5 h6 have h7 : ¬q ∣ p := by by_contra h8 have h9 : q = 1 ∨ q = p := dvd_prime p_prime h8 disj_syll h9 (prime_not_one q_prime) show False from p_ne_q h9.symm done have h8 : rel_prime p q := rel_prime_of_prime_not_dvd q_prime h7 rewrite [n_pq, Lemma_7_4_5 _ _ h8] show c ^ d ≡ m (MOD p) ∧ c ^ d ≡ m (MOD q) from And.intro congr_p congr_q done /- BEGIN EXERCISES -/ namespace Exercises /- Section 7.1 -/ -- 1. theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry -- 2. lemma gcd_comm_lt {a b : Nat} (h : a < b) : gcd a b = gcd b a := sorry theorem gcd_comm (a b : Nat) : gcd a b = gcd b a := sorry -- 3. theorem Exercise_7_1_5 (a b : Nat) (n : Int) : (∃ (s t : Int), s * a + t * b = n) ↔ (↑(gcd a b) : Int) ∣ n := sorry -- 4. theorem Exercise_7_1_6 (a b c : Nat) : gcd a b = gcd (a + b * c) b := sorry -- 5.

theorem gcd_is_nonzero {a b : Nat} (h : a ≠ 0 ∨ b ≠ 0) : gcd a b ≠ 0

HTPI.Exercises.gcd_is_nonzero

htpi/HTPILib/Chap7.lean

htpi

/- Copyright 2023 Daniel J. Velleman -/ import HTPILib.Chap6 namespace HTPI /- Definitions -/ lemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by have h : n + 1 > 0 := Nat.succ_pos n show a % (n + 1) < n + 1 from Nat.mod_lt a h done def gcd (a b : Nat) : Nat := match b with | 0 => a | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd (n + 1) (a % (n + 1)) termination_by b mutual def gcd_c1 (a b : Nat) : Int := match b with | 0 => 1 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c2 (n + 1) (a % (n + 1)) --Corresponds to s = t' termination_by b def gcd_c2 (a b : Nat) : Int := match b with | 0 => 0 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c1 (n + 1) (a % (n + 1)) - (gcd_c2 (n + 1) (a % (n + 1))) * ↑(a / (n + 1)) --Corresponds to t = s' - t'q termination_by b end def prime (n : Nat) : Prop := 2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n def prime_factor (p n : Nat) : Prop := prime p ∧ p ∣ n def all_prime (l : List Nat) : Prop := ∀ p ∈ l, prime p def nondec (l : List Nat) : Prop := match l with | [] => True --Of course, True is a proposition that is always true | n :: L => (∀ m ∈ L, n ≤ m) ∧ nondec L def nondec_prime_list (l : List Nat) : Prop := all_prime l ∧ nondec l def prod (l : List Nat) : Nat := match l with | [] => 1 | n :: L => n * (prod L) def prime_factorization (n : Nat) (l : List Nat) : Prop := nondec_prime_list l ∧ prod l = n def rel_prime (a b : Nat) : Prop := gcd a b = 1 def congr_mod (m : Nat) (a b : Int) : Prop := (↑m : Int) ∣ (a - b) def cc (m : Nat) (a : Int) : ZMod m := (↑a : ZMod m) notation:50 a " ≡ " b " (MOD " m ")" => congr_mod m a b notation:max "["a"]_"m:max => cc m a def invertible {m : Nat} (X : ZMod m) : Prop := ∃ (Y : ZMod m), X * Y = [1]_m def num_rp_below (m k : Nat) : Nat := match k with | 0 => 0 | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j def phi (m : Nat) : Nat := num_rp_below m m def prod_seq {m : Nat} (j k : Nat) (f : Nat → ZMod m) : ZMod m := match j with | 0 => [1]_m | n + 1 => prod_seq n k f * f (k + n) def maps_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i < n, g i < n def one_one_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i1 < n, ∀ i2 < n, g i1 = g i2 → i1 = i2 def onto_below (n : Nat) (g : Nat → Nat) : Prop := ∀ k < n, ∃ i < n, g i = k def perm_below (n : Nat) (g : Nat → Nat) : Prop := maps_below n g ∧ one_one_below n g ∧ onto_below n g def inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m) def swap (u v i : Nat) : Nat := if i = u then v else if i = v then u else i namespace Euler --For definitions specific to Euler's theorem def F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m def G (m a i : Nat) : Nat := (a * i) % m def Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i end Euler /- Section 7.1 -/ theorem dvd_mod_of_dvd_a_b {a b d : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ (a % b) := by set q : Nat := a / b have h3 : b * q + a % b = a := Nat.div_add_mod a b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 define --Goal : ∃ (c : Nat), a % b = d * c apply Exists.intro (j - k * q) show a % b = d * (j - k * q) from calc a % b _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm _ = a - b * q := by rw [h3] _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc] _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm done theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry #eval gcd 672 161 --Answer: 7 lemma gcd_base (a : Nat) : gcd a 0 = a := by rfl lemma gcd_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd a b = gcd b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1)) rfl done lemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b ≠ 0) : a % b < b := by have h1 : b > 0 := Nat.pos_of_ne_zero h show a % b < b from Nat.mod_lt a h1 done lemma dvd_self (n : Nat) : n ∣ n := by apply Exists.intro 1 ring done theorem gcd_dvd : ∀ (b a : Nat), (gcd a b) ∣ a ∧ (gcd a b) ∣ b := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd a b_1) ∣ a ∧ (gcd a b_1) ∣ b_1 fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_base] --Goal: a ∣ a ∧ a ∣ 0 apply And.intro (dvd_self a) define apply Exists.intro 0 rfl done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_nonzero a h1] --Goal : gcd b (a % b) ∣ a ∧ gcd b (a % b) ∣ b have h2 : a % b < b := mod_nonzero_lt a h1 have h3 : (gcd b (a % b)) ∣ b ∧ (gcd b (a % b)) ∣ (a % b) := ih (a % b) h2 b apply And.intro _ h3.left show (gcd b (a % b)) ∣ a from dvd_a_of_dvd_b_mod h3.left h3.right done done theorem gcd_dvd_left (a b : Nat) : (gcd a b) ∣ a := (gcd_dvd b a).left theorem gcd_dvd_right (a b : Nat) : (gcd a b) ∣ b := (gcd_dvd b a).right lemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl lemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c1 a b = gcd_c2 b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done lemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl lemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * ↑(a / b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done theorem gcd_lin_comb : ∀ (b a : Nat), (gcd_c1 a b) * ↑a + (gcd_c2 a b) * ↑b = ↑(gcd a b) := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd_c1 a b_1) * ↑a + (gcd_c2 a b_1) * ↑b_1 = ↑(gcd a b_1) fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base] --Goal : 1 * ↑a + 0 * ↑0 = ↑a ring done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1] --Goal : gcd_c2 b (a % b) * ↑a + -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * ↑(a / b)) * ↑b = -- ↑(gcd b (a % b)) set r : Nat := a % b set q : Nat := a / b set s : Int := gcd_c1 b r set t : Int := gcd_c2 b r --Goal : t * ↑a + (s - t * ↑q) * ↑b = ↑(gcd b r) have h2 : r < b := mod_nonzero_lt a h1 have h3 : s * ↑b + t * ↑r = ↑(gcd b r) := ih r h2 b have h4 : b * q + r = a := Nat.div_add_mod a b rewrite [←h3, ←h4] rewrite [Nat.cast_add, Nat.cast_mul] --Goal : t * (↑b * ↑q + ↑r) + (s - t * ↑q) * ↑b = s * ↑b + t * ↑r ring done done #eval gcd_c1 672 161 --Answer: 6 #eval gcd_c2 672 161 --Answer: -25 --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161 theorem Theorem_7_1_6 {d a b : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ gcd a b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑d ∣ ↑(gcd a b) set s : Int := gcd_c1 a b set t : Int := gcd_c2 a b have h3 : s * ↑a + t * ↑b = ↑(gcd a b) := gcd_lin_comb b a rewrite [←h3] --Goal : ↑d ∣ s * ↑a + t * ↑b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul] --Goal : ↑d ∣ s * (↑d * ↑j) + t * (↑d * ↑k) define apply Exists.intro (s * ↑j + t * ↑k) ring done /- Section 7.2 -/ theorem dvd_trans {a b c : Nat} (h1 : a ∣ b) (h2 : b ∣ c) : a ∣ c := by define at h1; define at h2; define obtain (m : Nat) (h3 : b = a * m) from h1 obtain (n : Nat) (h4 : c = b * n) from h2 rewrite [h3, mul_assoc] at h4 apply Exists.intro (m * n) show c = a * (m * n) from h4 done lemma exists_prime_factor : ∀ (n : Nat), 2 ≤ n → ∃ (p : Nat), prime_factor p n := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, 2 ≤ n_1 → ∃ (p : Nat), prime_factor p n_1 assume h1 : 2 ≤ n by_cases h2 : prime n · -- Case 1. h2 : prime n apply Exists.intro n define --Goal : prime n ∧ n ∣ n show prime n ∧ n ∣ n from And.intro h2 (dvd_self n) done · -- Case 2. h2 : ¬prime n define at h2 --h2 : ¬(2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n) demorgan at h2 disj_syll h2 h1 obtain (a : Nat) (h3 : ∃ (b : Nat), a * b = n ∧ a < n ∧ b < n) from h2 obtain (b : Nat) (h4 : a * b = n ∧ a < n ∧ b < n) from h3 have h5 : 2 ≤ a := by by_contra h6 have h7 : a ≤ 1 := by linarith have h8 : n ≤ b := calc n _ = a * b := h4.left.symm _ ≤ 1 * b := by rel [h7] _ = b := by ring linarith --n ≤ b contradicts b < n done have h6 : ∃ (p : Nat), prime_factor p a := ih a h4.right.left h5 obtain (p : Nat) (h7 : prime_factor p a) from h6 apply Exists.intro p define --Goal : prime p ∧ p ∣ n define at h7 --h7 : prime p ∧ p ∣ a apply And.intro h7.left have h8 : a ∣ n := by apply Exists.intro b show n = a * b from (h4.left).symm done show p ∣ n from dvd_trans h7.right h8 done done lemma exists_least_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q := by set S : Set Nat := {p : Nat | prime_factor p n} have h2 : ∃ (p : Nat), p ∈ S := exists_prime_factor n h show ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q from well_ord_princ S h2 done lemma all_prime_nil : all_prime [] := by define --Goal : ∀ p ∈ [], prime p fix p : Nat contrapos --Goal : ¬prime p → p ∉ [] assume h1 : ¬prime p show p ∉ [] from List.not_mem_nil p done lemma all_prime_cons (n : Nat) (L : List Nat) : all_prime (n :: L) ↔ prime n ∧ all_prime L := by apply Iff.intro · -- (→) assume h1 : all_prime (n :: L) --Goal : prime n ∧ all_prime L define at h1 --h1 : ∀ p ∈ n :: L, prime p apply And.intro (h1 n (List.mem_cons_self n L)) define --Goal : ∀ p ∈ L, prime p fix p : Nat assume h2 : p ∈ L show prime p from h1 p (List.mem_cons_of_mem n h2) done · -- (←) assume h1 : prime n ∧ all_prime L --Goal : all_prime (n :: l) define : all_prime L at h1 define fix p : Nat assume h2 : p ∈ n :: L rewrite [List.mem_cons] at h2 --h2 : p = n ∨ p ∈ L by_cases on h2 · -- Case 1. h2 : p = n rewrite [h2] show prime n from h1.left done · -- Case 2. h2 : p ∈ L show prime p from h1.right p h2 done done done lemma nondec_nil : nondec [] := by define --Goal : True trivial --trivial proves some obviously true statements, such as True done lemma nondec_cons (n : Nat) (L : List Nat) : nondec (n :: L) ↔ (∀ m ∈ L, n ≤ m) ∧ nondec L := by rfl lemma prod_nil : prod [] = 1 := by rfl lemma prod_cons : prod (n :: L) = n * (prod L) := by rfl lemma exists_cons_of_length_eq_succ {A : Type} {l : List A} {n : Nat} (h : l.length = n + 1) : ∃ (a : A) (L : List A), l = a :: L ∧ L.length = n := by have h1 : ¬l.length = 0 := by linarith rewrite [List.length_eq_zero] at h1 obtain (a : A) (h2 : ∃ (L : List A), l = a :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List A) (h3 : l = a :: L) from h2 apply Exists.intro a apply Exists.intro L apply And.intro h3 have h4 : (a :: L).length = L.length + 1 := List.length_cons a L rewrite [←h3, h] at h4 show L.length = n from (Nat.add_right_cancel h4).symm done lemma list_elt_dvd_prod_by_length (a : Nat) : ∀ (n : Nat), ∀ (l : List Nat), l.length = n → a ∈ l → a ∣ prod l := by by_induc · --Base Case fix l : List Nat assume h1 : l.length = 0 rewrite [List.length_eq_zero] at h1 --h1 : l = [] rewrite [h1] --Goal : a ∈ [] → a ∣ prod [] contrapos assume h2 : ¬a ∣ prod [] show a ∉ [] from List.not_mem_nil a done · -- Induction Step fix n : Nat assume ih : ∀ (l : List Nat), List.length l = n → a ∈ l → a ∣ prod l fix l : List Nat assume h1 : l.length = n + 1 --Goal : a ∈ l → a ∣ prod l obtain (b : Nat) (h2 : ∃ (L : List Nat), l = b :: L ∧ L.length = n) from exists_cons_of_length_eq_succ h1 obtain (L : List Nat) (h3 : l = b :: L ∧ L.length = n) from h2 have h4 : a ∈ L → a ∣ prod L := ih L h3.right assume h5 : a ∈ l rewrite [h3.left, prod_cons] --Goal : a ∣ b * prod L rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b ∨ a ∈ L by_cases on h5 · -- Case 1. h5 : a = b apply Exists.intro (prod L) rewrite [h5] rfl done · -- Case 2. h5 : a ∈ L have h6 : a ∣ prod L := h4 h5 have h7 : prod L ∣ b * prod L := by apply Exists.intro b ring done show a ∣ b * prod L from dvd_trans h6 h7 done done done lemma list_elt_dvd_prod {a : Nat} {l : List Nat} (h : a ∈ l) : a ∣ prod l := by set n : Nat := l.length have h1 : l.length = n := by rfl show a ∣ prod l from list_elt_dvd_prod_by_length a n l h1 h done lemma exists_prime_factorization : ∀ (n : Nat), n ≥ 1 → ∃ (l : List Nat), prime_factorization n l := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, n_1 ≥ 1 → ∃ (l : List Nat), prime_factorization n_1 l assume h1 : n ≥ 1 by_cases h2 : n = 1 · -- Case 1. h2 : n = 1 apply Exists.intro [] define apply And.intro · -- Proof of nondec_prime_list [] define show all_prime [] ∧ nondec [] from And.intro all_prime_nil nondec_nil done · -- Proof of prod [] = n rewrite [prod_nil, h2] rfl done done · -- Case 2. h2 : n ≠ 1 have h3 : n ≥ 2 := lt_of_le_of_ne' h1 h2 obtain (p : Nat) (h4 : prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q) from exists_least_prime_factor h3 have p_prime_factor : prime_factor p n := h4.left define at p_prime_factor have p_prime : prime p := p_prime_factor.left have p_dvd_n : p ∣ n := p_prime_factor.right have p_least : ∀ (q : Nat), prime_factor q n → p ≤ q := h4.right obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n have h5 : m ≠ 0 := by contradict h1 with h6 have h7 : n = 0 := calc n _ = p * m := n_eq_pm _ = p * 0 := by rw [h6] _ = 0 := by ring rewrite [h7] decide done have m_pos : 0 < m := Nat.pos_of_ne_zero h5 have m_lt_n : m < n := by define at p_prime show m < n from calc m _ < m + m := by linarith _ = 2 * m := by ring _ ≤ p * m := by rel [p_prime.left] _ = n := n_eq_pm.symm done obtain (L : List Nat) (h6 : prime_factorization m L) from ih m m_lt_n m_pos define at h6 have ndpl_L : nondec_prime_list L := h6.left define at ndpl_L apply Exists.intro (p :: L) define apply And.intro · -- Proof of nondec_prime_list (p :: L) define apply And.intro · -- Proof of all_prime (p :: L) rewrite [all_prime_cons] show prime p ∧ all_prime L from And.intro p_prime ndpl_L.left done · -- Proof of nondec (p :: L) rewrite [nondec_cons] apply And.intro _ ndpl_L.right fix q : Nat assume q_in_L : q ∈ L have h7 : q ∣ prod L := list_elt_dvd_prod q_in_L rewrite [h6.right] at h7 --h7 : q ∣ m have h8 : m ∣ n := by apply Exists.intro p rewrite [n_eq_pm] ring done have q_dvd_n : q ∣ n := dvd_trans h7 h8 have ap_L : all_prime L := ndpl_L.left define at ap_L have q_prime_factor : prime_factor q n := And.intro (ap_L q q_in_L) q_dvd_n show p ≤ q from p_least q q_prime_factor done done · -- Proof of prod (p :: L) = n rewrite [prod_cons, h6.right, n_eq_pm] rfl done done done theorem Theorem_7_2_2 {a b c : Nat} (h1 : c ∣ a * b) (h2 : rel_prime a c) : c ∣ b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑c ∣ ↑b define at h1; define at h2; define obtain (j : Nat) (h3 : a * b = c * j) from h1 set s : Int := gcd_c1 a c set t : Int := gcd_c2 a c have h4 : s * ↑a + t * ↑c = ↑(gcd a c) := gcd_lin_comb c a rewrite [h2, Nat.cast_one] at h4 --h4 : s * ↑a + t * ↑c = (1 : Int) apply Exists.intro (s * ↑j + t * ↑b) show ↑b = ↑c * (s * ↑j + t * ↑b) from calc ↑b _ = (1 : Int) * ↑b := (one_mul _).symm _ = (s * ↑a + t * ↑c) * ↑b := by rw [h4] _ = s * (↑a * ↑b) + t * ↑c * ↑b := by ring _ = s * (↑c * ↑j) + t * ↑c * ↑b := by rw [←Nat.cast_mul a b, h3, Nat.cast_mul c j] _ = ↑c * (s * ↑j + t * ↑b) := by ring done lemma le_nonzero_prod_left {a b : Nat} (h : a * b ≠ 0) : a ≤ a * b := by have h1 : b ≠ 0 := by contradict h with h1 rewrite [h1] ring done have h2 : 1 ≤ b := Nat.pos_of_ne_zero h1 show a ≤ a * b from calc a = a * 1 := (mul_one a).symm _ ≤ a * b := by rel [h2] done lemma le_nonzero_prod_right {a b : Nat} (h : a * b ≠ 0) : b ≤ a * b := by rewrite [mul_comm] rewrite [mul_comm] at h show b ≤ b * a from le_nonzero_prod_left h done lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry lemma rel_prime_of_prime_not_dvd {a p : Nat} (h1 : prime p) (h2 : ¬p ∣ a) : rel_prime a p := by have h3 : gcd a p ∣ a := gcd_dvd_left a p have h4 : gcd a p ∣ p := gcd_dvd_right a p have h5 : gcd a p = 1 ∨ gcd a p = p := dvd_prime h1 h4 have h6 : gcd a p ≠ p := by contradict h2 with h6 rewrite [h6] at h3 show p ∣ a from h3 done disj_syll h5 h6 show rel_prime a p from h5 done theorem Theorem_7_2_3 {a b p : Nat} (h1 : prime p) (h2 : p ∣ a * b) : p ∣ a ∨ p ∣ b := by or_right with h3 have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3 show p ∣ b from Theorem_7_2_2 h2 h4 done lemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a ≥ 1 := by have h1 : a ≠ 0 := by by_contra h1 rewrite [h1] at h contradict h linarith done show a ≥ 1 from Nat.pos_of_ne_zero h1 done lemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by have h1 : a ≥ 1 := ge_one_of_prod_one h have h2 : a * b ≠ 0 := by linarith have h3 : a ≤ a * b := le_nonzero_prod_left h2 rewrite [h] at h3 show a = 1 from Nat.le_antisymm h3 h1 done lemma eq_one_of_dvd_one {n : Nat} (h : n ∣ 1) : n = 1 := by obtain (j : Nat) (h1 : 1 = n * j) from h show n = 1 from eq_one_of_prod_one h1.symm done lemma prime_not_one {p : Nat} (h : prime p) : p ≠ 1 := by define at h linarith done theorem Theorem_7_2_4 {p : Nat} (h1 : prime p) : ∀ (l : List Nat), p ∣ prod l → ∃ a ∈ l, p ∣ a := by apply List.rec · -- Base Case. Goal : p ∣ prod [] → ∃ a ∈ [], p ∣ a rewrite [prod_nil] assume h2 : p ∣ 1 show ∃ a ∈ [], p ∣ a from absurd (eq_one_of_dvd_one h2) (prime_not_one h1) done · -- Induction Step fix b : Nat fix L : List Nat assume ih : p ∣ prod L → ∃ a ∈ L, p ∣ a --Goal : p ∣ prod (b :: L) → ∃ a ∈ b :: L, p ∣ a assume h2 : p ∣ prod (b :: L) rewrite [prod_cons] at h2 have h3 : p ∣ b ∨ p ∣ prod L := Theorem_7_2_3 h1 h2 by_cases on h3 · -- Case 1. h3 : p ∣ b apply Exists.intro b show b ∈ b :: L ∧ p ∣ b from And.intro (List.mem_cons_self b L) h3 done · -- Case 2. h3 : p ∣ prod L obtain (a : Nat) (h4 : a ∈ L ∧ p ∣ a) from ih h3 apply Exists.intro a show a ∈ b :: L ∧ p ∣ a from And.intro (List.mem_cons_of_mem b h4.left) h4.right done done done lemma prime_in_list {p : Nat} {l : List Nat} (h1 : prime p) (h2 : all_prime l) (h3 : p ∣ prod l) : p ∈ l := by obtain (a : Nat) (h4 : a ∈ l ∧ p ∣ a) from Theorem_7_2_4 h1 l h3 define at h2 have h5 : prime a := h2 a h4.left have h6 : p = 1 ∨ p = a := dvd_prime h5 h4.right disj_syll h6 (prime_not_one h1) rewrite [h6] show a ∈ l from h4.left done lemma first_le_first {p q : Nat} {l m : List Nat} (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m)) (h3 : prod (p :: l) = prod (q :: m)) : p ≤ q := by define at h1; define at h2 have h4 : q ∣ prod (p :: l) := by define apply Exists.intro (prod m) rewrite [←prod_cons] show prod (p :: l) = prod (q :: m) from h3 done have h5 : all_prime (q :: m) := h2.left rewrite [all_prime_cons] at h5 have h6 : q ∈ p :: l := prime_in_list h5.left h1.left h4 have h7 : nondec (p :: l) := h1.right rewrite [nondec_cons] at h7 rewrite [List.mem_cons] at h6 by_cases on h6 · -- Case 1. h6 : q = p linarith done · -- Case 2. h6 : q ∈ l have h8 : ∀ m ∈ l, p ≤ m := h7.left show p ≤ q from h8 q h6 done done lemma nondec_prime_list_tail {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by define at h define rewrite [all_prime_cons, nondec_cons] at h show all_prime l ∧ nondec l from And.intro h.left.right h.right.right done lemma cons_prod_not_one {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : prod (p :: l) ≠ 1 := by define at h have h1 : all_prime (p :: l) := h.left rewrite [all_prime_cons] at h1 rewrite [prod_cons] by_contra h2 show False from (prime_not_one h1.left) (eq_one_of_prod_one h2) done lemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) : l = [] ↔ prod l = 1 := by apply Iff.intro · -- (→) assume h1 : l = [] rewrite [h1] show prod [] = 1 from prod_nil done · -- (←) contrapos assume h1 : ¬l = [] obtain (p : Nat) (h2 : ∃ (L : List Nat), l = p :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List Nat) (h3 : l = p :: L) from h2 rewrite [h3] at h rewrite [h3] show ¬prod (p :: L) = 1 from cons_prod_not_one h done done lemma prime_pos {p : Nat} (h : prime p) : p > 0 := by define at h linarith done theorem Theorem_7_2_5 : ∀ (l1 l2 : List Nat), nondec_prime_list l1 → nondec_prime_list l2 → prod l1 = prod l2 → l1 = l2 := by apply List.rec · -- Base Case. Goal : ∀ (l2 : List Nat), nondec_prime_list [] → -- nondec_prime_list l2 → prod [] = prod l2 → [] = l2 fix l2 : List Nat assume h1 : nondec_prime_list [] assume h2 : nondec_prime_list l2 assume h3 : prod [] = prod l2 rewrite [prod_nil, eq_comm, ←list_nil_iff_prod_one h2] at h3 show [] = l2 from h3.symm done · -- Induction Step fix p : Nat fix L1 : List Nat assume ih : ∀ (L2 : List Nat), nondec_prime_list L1 → nondec_prime_list L2 → prod L1 = prod L2 → L1 = L2 -- Goal : ∀ (l2 : List Nat), nondec_prime_list (p :: L1) → -- nondec_prime_list l2 → prod (p :: L1) = prod l2 → p :: L1 = l2 fix l2 : List Nat assume h1 : nondec_prime_list (p :: L1) assume h2 : nondec_prime_list l2 assume h3 : prod (p :: L1) = prod l2 have h4 : ¬prod (p :: L1) = 1 := cons_prod_not_one h1 rewrite [h3, ←list_nil_iff_prod_one h2] at h4 obtain (q : Nat) (h5 : ∃ (L : List Nat), l2 = q :: L) from List.exists_cons_of_ne_nil h4 obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5 rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2) rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2) have h7 : p ≤ q := first_le_first h1 h2 h3 have h8 : q ≤ p := first_le_first h2 h1 h3.symm have h9 : p = q := by linarith rewrite [h9, prod_cons, prod_cons] at h3 --h3 : q * prod L1 = q * prod L2 have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1 have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2 define at h2 have h12 : all_prime (q :: L2) := h2.left rewrite [all_prime_cons] at h12 have h13 : q > 0 := prime_pos h12.left have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3 have h15 : L1 = L2 := ih L2 h10 h11 h14 rewrite [h6, h9, h15] rfl done done theorem fund_thm_arith (n : Nat) (h : n ≥ 1) : ∃! (l : List Nat), prime_factorization n l := by exists_unique · -- Existence show ∃ (l : List Nat), prime_factorization n l from exists_prime_factorization n h done · -- Uniqueness fix l1 : List Nat; fix l2 : List Nat assume h1 : prime_factorization n l1 assume h2 : prime_factorization n l2 define at h1; define at h2 have h3 : prod l1 = n := h1.right rewrite [←h2.right] at h3 show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3 done done /- Section 7.3 -/ theorem congr_refl (m : Nat) : ∀ (a : Int), a ≡ a (MOD m) := by fix a : Int define --Goal : ∃ (c : Int), a - a = ↑m * c apply Exists.intro 0 ring done theorem congr_symm {m : Nat} : ∀ {a b : Int}, a ≡ b (MOD m) → b ≡ a (MOD m) := by fix a : Int; fix b : Int assume h1 : a ≡ b (MOD m) define at h1 --h1 : ∃ (c : Int), a - b = ↑m * c define --Goal : ∃ (c : Int), b - a = ↑m * c obtain (c : Int) (h2 : a - b = m * c) from h1 apply Exists.intro (-c) show b - a = m * (-c) from calc b - a _ = -(a - b) := by ring _ = -(m * c) := by rw [h2] _ = m * (-c) := by ring done theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry /- Fundamental properties of congruence classes -/ lemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) : X = Y ↔ X.val = Y.val := Fin.ext_iff lemma val_nat_eq_mod (n k : Nat) : ([k]_(n + 1)).val = k % (n + 1) := by rfl lemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl theorem cc_rep {m : Nat} (X : ZMod m) : ∃ (a : Int), X = [a]_m := match m with | 0 => by apply Exists.intro X rfl done | n + 1 => by apply Exists.intro ↑(X.val) have h1 : X.val < n + 1 := Fin.prop X rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1] rfl done theorem add_class (m : Nat) (a b : Int) : [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm theorem mul_class (m : Nat) (a b : Int) : [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm lemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ [a - b]_m = [0]_m := by apply Iff.intro · -- (→) assume h1 : [a]_m = [b]_m have h2 : a - b = a + (-b) := by ring have h3 : b + (-b) = 0 := by ring show [a - b]_m = [0]_m from calc [a - b]_m _ = [a + (-b)]_m := by rw [h2] _ = [a]_m + [-b]_m := by rw [add_class] _ = [b]_m + [-b]_m := by rw [h1] _ = [b + -b]_m := by rw [add_class] _ = [0]_m := by rw [h3] done · -- (←) assume h1 : [a - b]_m = [0]_m have h2 : b + (a - b) = a := by ring have h3 : b + 0 = b := by ring show [a]_m = [b]_m from calc [a]_m _ = [b + (a - b)]_m := by rw [h2] _ = [b]_m + [a - b]_m := by rw [add_class] _ = [b]_m + [0]_m := by rw [h1] _ = [b + 0]_m := by rw [add_class] _ = [b]_m := by rw [h3] done done lemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) : [a]_m = [0]_m → [-a]_m = [0]_m := by assume h1 : [a]_m = [0]_m have h2 : 0 + (-a) = -a := by ring have h3 : a + (-a) = 0 := by ring show [-a]_m = [0]_m from calc [-a]_m _ = [0 + (-a)]_m := by rw [h2] _ = [0]_m + [-a]_m := by rw [add_class] _ = [a]_m + [-a]_m := by rw [h1] _ = [a + (-a)]_m := by rw [add_class] _ = [0]_m := by rw [h3] done lemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) : [-a]_m = [0]_m ↔ [a]_m = [0]_m := by apply Iff.intro _ (cc_neg_zero_of_cc_zero m a) assume h1 : [-a]_m = [0]_m have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1 have h3 : -(-a) = a := by ring rewrite [h3] at h2 show [a]_m = [0]_m from h2 done lemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl lemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m ↔ m ∣ k := match m with | 0 => by have h : (0 : Int) = (↑(0 : Nat) : Int) := by rfl rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj] apply Iff.intro · -- (→) assume h1 : k = 0 rewrite [h1] show 0 ∣ 0 from dvd_self 0 done · -- (←) assume h1 : 0 ∣ k obtain (c : Nat) (h2 : k = 0 * c) from h1 rewrite [h2] ring done done | n + 1 => by rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero] show k % (n + 1) = 0 ↔ n + 1 ∣ k from (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm done lemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m ↔ ↑m ∣ a := by obtain (k : Nat) (h1 : a = ↑k ∨ a = -↑k) from Int.eq_nat_or_neg a by_cases on h1 · -- Case 1. h1: a = ↑k rewrite [h1, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done · -- Case 2. h1: a = -↑k rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done done theorem cc_eq_iff_congr (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ a ≡ b (MOD m) := calc [a]_m = [b]_m _ ↔ [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b _ ↔ ↑m ∣ (a - b) := cc_zero_iff_dvd m (a - b) _ ↔ a ≡ b (MOD m) := by rfl /- End of fundamental properties of congruence classes -/ lemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m := by have h1 : (↑m : Int) ≠ 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m) show 0 ≤ a % m from Int.emod_nonneg a h1 done lemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m) have h2 : (↑m : Int) > 0 := (Nat.cast_pos).rtl h1 show a % m < m from Int.emod_lt_of_pos a h2 done lemma congr_mod_mod (m : Nat) (a : Int) : a ≡ a % m (MOD m) := by define have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m apply Exists.intro (a / m) show a - a % m = m * (a / m) from calc a - (a % m) _ = m * (a / m) + a % m - a % m := by rw [h1] _ = m * (a / m) := by ring done lemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) := And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a)) theorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) : ∃! (r : Int), 0 ≤ r ∧ r < m ∧ a ≡ r (MOD m) := by exists_unique · -- Existence apply Exists.intro (a % m) show 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) from mod_cmpl_res m a done · -- Uniqueness fix r1 : Int; fix r2 : Int assume h1 : 0 ≤ r1 ∧ r1 < m ∧ a ≡ r1 (MOD m) assume h2 : 0 ≤ r2 ∧ r2 < m ∧ a ≡ r2 (MOD m) have h3 : r1 ≡ r2 (MOD m) := congr_trans (congr_symm h1.right.right) h2.right.right obtain (d : Int) (h4 : r1 - r2 = m * d) from h3 have h5 : r1 - r2 < m * 1 := by linarith have h6 : m * (-1) < r1 - r2 := by linarith rewrite [h4] at h5 --h5 : m * d < m * 1 rewrite [h4] at h6 --h6 : m * -1 < m * d have h7 : (↑m : Int) ≥ 0 := Nat.cast_nonneg m have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7 have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7 have h10 : d = 0 := by linarith show r1 = r2 from calc r1 _ = r1 - r2 + r2 := by ring _ = m * 0 + r2 := by rw [h4, h10] _ = r2 := by ring done done lemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m := (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a) theorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y rewrite [h1, h2] have h3 : a + b = b + a := by ring show [a]_m + [b]_m = [b]_m + [a]_m from calc [a]_m + [b]_m _ = [a + b]_m := add_class m a b _ = [b + a]_m := by rw [h3] _ = [b]_m + [a]_m := (add_class m b a).symm done theorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X rewrite [h1] have h2 : a * 1 = a := by ring show [a]_m * [1]_m = [a]_m from calc [a]_m * [1]_m _ = [a * 1]_m := mul_class m a 1 _ = [a]_m := by rw [h2] done theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry lemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) : [a]_m * [gcd_c2 m a]_m = [1]_m := by set s : Int := gcd_c1 m a have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m define at h1 rewrite [h1, Nat.cast_one] at h2 --h2 : s * ↑m + gcd_c2 m a * ↑a = 1 rewrite [mul_class, cc_eq_iff_congr] define --Goal : ∃ (c : Int), ↑a * gcd_c2 m a - 1 = ↑m * c apply Exists.intro (-s) show a * (gcd_c2 m a) - 1 = m * (-s) from calc a * (gcd_c2 m a) - 1 _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring _ = 1 + m * (-s) - 1 := by rw [h2] _ = m * (-s) := by ring done theorem Theorem_7_3_7 (m a : Nat) : invertible [a]_m ↔ rel_prime m a := by apply Iff.intro · -- (→) assume h1 : invertible [a]_m define at h1 obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1 obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y rewrite [h3, mul_class, cc_eq_iff_congr] at h2 define at h2 obtain (c : Int) (h4 : a * b - 1 = m * c) from h2 rewrite [Exercise_7_2_6] --Goal : ∃ (s t : Int), s * ↑m + t * ↑a = 1 apply Exists.intro (-c) apply Exists.intro b show (-c) * m + b * a = 1 from calc (-c) * m + b * a _ = (-c) * m + (a * b - 1) + 1 := by ring _ = (-c) * m + m * c + 1 := by rw [h4] _ = 1 := by ring done · -- (←) assume h1 : rel_prime m a define show ∃ (Y : ZMod m), [a]_m * Y = [1]_m from Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1) done done /- Section 7.4 -/ section Euler open Euler lemma num_rp_below_base {m : Nat} : num_rp_below m 0 = 0 := by rfl lemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) : num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by have h1 : num_rp_below m (j + 1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : gcd m j = 1 rewrite [if_pos h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1 show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1 done lemma num_rp_below_step_not_rp {m j : Nat} (h : ¬rel_prime m j) : num_rp_below m (j + 1) = num_rp_below m j := by have h1 : num_rp_below m (j +1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : ¬gcd m j = 1 rewrite [if_neg h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j show num_rp_below m (j + 1) = num_rp_below m j from h1 done lemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl #eval phi 10 --Answer: 4 lemma prod_inv_iff_inv {m : Nat} {X : ZMod m} (h1 : invertible X) (Y : ZMod m) : invertible (X * Y) ↔ invertible Y := by apply Iff.intro · -- (→) assume h2 : invertible (X * Y) obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2 apply Exists.intro (X * Z) rewrite [←h3] --Goal : Y * (X * Z) = X * Y * Z ring --Note that ring can do algebra in ZMod m done · -- (←) assume h2 : invertible Y obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1 obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2 apply Exists.intro (Xi * Yi) show (X * Y) * (Xi * Yi) = [1]_m from calc X * Y * (Xi * Yi) _ = (X * Xi) * (Y * Yi) := by ring _ = [1]_m * [1]_m := by rw [h3, h4] _ = [1]_m := Theorem_7_3_6_7 [1]_m done done lemma F_rp_def {m i : Nat} (h : rel_prime m i) : F m i = [i]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h --h : gcd m i = 1 rewrite [if_pos h] at h1 show F m i = [i]_m from h1 done lemma F_not_rp_def {m i : Nat} (h : ¬rel_prime m i) : F m i = [1]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h rewrite [h1, if_neg h] rfl done lemma prod_seq_base {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 0 k f = [1]_m := by rfl lemma prod_seq_step {m : Nat} (n k : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl lemma prod_seq_zero_step {m : Nat} (n : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by rewrite [prod_seq_step, zero_add] rfl done lemma prod_one {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 1 k f = f k := by rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7] rfl done lemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl lemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m := calc [G m a i]_m _ = [(a * i) % m]_m := by rfl _ = [a * i]_m := (cc_eq_mod m (a * i)).symm _ = [a]_m * [i]_m := (mul_class m a i).symm lemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) : rel_prime m (G m a i) ↔ rel_prime m i := by have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1 show rel_prime m (G m a i) ↔ rel_prime m i from calc rel_prime m (G m a i) _ ↔ invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm _ ↔ invertible ([a]_m * [i]_m) := by rw [cc_G] _ ↔ invertible [i]_m := prod_inv_iff_inv h2 ([i]_m) _ ↔ rel_prime m i := Theorem_7_3_7 m i done lemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) : F m (G m a i) = [a]_m * F m i := by have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2 show F m (G m a i) = [a]_m * F m i from calc F m (G m a i) _ = [G m a i]_m := F_rp_def h3 _ = [a]_m * [i]_m := cc_G m a i _ = [a]_m * F m i := by rw [F_rp_def h2] done lemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : ¬rel_prime m i) : F m (G m a i) = [1]_m := by rewrite [←G_rp_iff h1 i] at h2 show F m (G m a i) = [1]_m from F_not_rp_def h2 done lemma FG_prod {m a : Nat} (h1 : rel_prime m a) : ∀ (k : Nat), prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by by_induc · -- Base Case show prod_seq 0 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from calc prod_seq 0 0 ((F m) ∘ (G m a)) _ = [1]_m := prod_seq_base _ _ _ = [a]_m ^ 0 * [1]_m := by ring _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by rw [num_rp_below_base, prod_seq_base] done · -- Induction Step fix k : Nat assume ih : prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) by_cases h2 : rel_prime m k · -- Case 1. h2 : rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([a]_m * F m k) := by rw [FG_rp h1 h2] _ = [a]_m ^ ((num_rp_below m k) + 1) * ((prod_seq k 0 (F m)) * F m k) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_rp h2, prod_seq_zero_step] done · -- Case 2. h2 : ¬rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([1]_m) := by rw [FG_not_rp h1 h2] _ = [a]_m ^ (num_rp_below m k) * (prod_seq k 0 (F m) * ([1]_m)) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_not_rp h2, prod_seq_zero_step, F_not_rp_def h2] done done done lemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by define --Goal : ∀ i < m, G m a i < m fix i : Nat assume h1 : i < m rewrite [G_def] --Goal : a * i % m < m show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m) done lemma left_inv_one_one_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g' (g i) = i) : one_one_below n g := sorry lemma right_inv_onto_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g (g' i) = i) (h2 : maps_below n g') : onto_below n g := by define at h2; define fix k : Nat assume h3 : k < n apply Exists.intro (g' k) show g' k < n ∧ g (g' k) = k from And.intro (h2 k h3) (h1 k h3) done lemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m * [inv_mod m a]_m = [1]_m := by have h2 : 0 ≤ (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a) show [a]_m * [inv_mod m a]_m = [1]_m from calc [a]_m * [inv_mod m a]_m _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2] _ = [a]_m * [gcd_c2 m a]_m := by rw [←cc_eq_mod] _ = [1]_m := gcd_c2_inv h1 done lemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m := calc a * (b % m) % m = a % m * (b % m % m) % m := Nat.mul_mod _ _ _ _ = a % m * (b % m) % m := by rw [Nat.mod_mod] _ = a * b % m := (Nat.mul_mod _ _ _).symm lemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm] rfl done theorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] : ↑a ≡ ↑b (MOD m) ↔ a % m = b % m := sorry lemma mul_inv_mod_cancel {m a i : Nat} [NeZero m] (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1 rewrite [mul_class, cc_eq_iff_congr, ←Nat.cast_mul, ←Nat.cast_one, congr_iff_mod_eq_Nat] at h3 show a * inv_mod m a * i % m = i from calc a * (inv_mod m a) * i % m _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = 1 % m * i % m := by rw [h3] _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = i % m := by rw [one_mul] _ = i := Nat.mod_eq_of_lt h2 done lemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl lemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, G m a (Ginv m a i) = i := by fix i : Nat assume h2 : i < m show G m a (Ginv m a i) = i from calc G m a (Ginv m a i) _ = a * ((inv_mod m a * i) % m) % m := by rfl _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, Ginv m a (G m a i) = i := by fix i : Nat assume h2 : i < m show Ginv m a (G m a i) = i from calc Ginv m a (G m a i) _ = inv_mod m a * ((a * i) % m) % m := by rfl _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc, mul_comm (inv_mod m a)] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_maps_below (m a : Nat) [NeZero m] : maps_below m (Ginv m a) := G_maps_below m (inv_mod m a) lemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : one_one_below m (G m a) := left_inv_one_one_below (Ginv_left_inv h1) lemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : onto_below m (G m a) := right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a) lemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : perm_below m (G m a) := And.intro (G_maps_below m a) (And.intro (G_one_one_below h1) (G_onto_below h1)) --Permuting a product of congruence classes doesn't change product lemma swap_fst (u v : Nat) : swap u v u = v := by define : swap u v u --Goal : (if u = u then v else if u = v then u else u) = v have h : u = u := by rfl rewrite [if_pos h] rfl done lemma swap_snd (u v : Nat) : swap u v v = u := by define : swap u v v by_cases h1 : v = u · -- Case 1. h1 : v = u rewrite [if_pos h1] show v = u from h1 done · -- Case 2. h1 : v ≠ u rewrite [if_neg h1] have h2 : v = v := by rfl rewrite [if_pos h2] rfl done done lemma swap_other {u v i : Nat} (h1 : i ≠ u) (h2 : i ≠ v) : swap u v i = i := by define : swap u v i rewrite [if_neg h1, if_neg h2] rfl done lemma swap_values (u v i : Nat) : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := by by_cases h1 : i = u · -- Case 1. h1 : i = u apply Or.inl rewrite [h1] show swap u v u = v from swap_fst u v done · -- Case 2. h1 : i ≠ u apply Or.inr by_cases h2 : i = v · -- Case 2.1. h2 : i = v apply Or.inl rewrite [h2] show swap u v v = u from swap_snd u v done · -- Case 2.2. h2 : i ≠ v apply Or.inr show swap u v i = i from swap_other h1 h2 done done done lemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by define fix i : Nat assume h3 : i < n have h4 : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := swap_values u v i by_cases on h4 · -- Case 1. h4 : swap u v i = v rewrite [h4] show v < n from h2 done · -- Case 2. by_cases on h4 · -- Case 2.1. h4 : swap u v i = u rewrite [h4] show u < n from h1 done · -- Case 2.2. h4 : swap u v i = i rewrite [h4] show i < n from h3 done done done lemma swap_swap (u v n : Nat) : ∀ i < n, swap u v (swap u v i) = i := by fix i : Nat assume h : i < n by_cases h1 : i = u · -- Case 1. h1 : i = u rewrite [h1, swap_fst, swap_snd] rfl done · -- Case 2. h1 : i ≠ u by_cases h2 : i = v · -- Case 2.1. h2 : i = v rewrite [h2, swap_snd, swap_fst] rfl done · -- Case 2.2. h2 : i ≠ v rewrite [swap_other h1 h2, swap_other h1 h2] rfl done done done lemma swap_one_one_below (u v n) : one_one_below n (swap u v) := left_inv_one_one_below (swap_swap u v n) lemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) := right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2) lemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) := And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2)) lemma comp_perm_below {n : Nat} {f g : Nat → Nat} (h1 : perm_below n f) (h2 : perm_below n g) : perm_below n (f ∘ g) := sorry lemma trivial_swap (u : Nat) : swap u u = id := by apply funext fix x : Nat by_cases h1 : x = u · -- Case 1. h1 : x = u rewrite [h1, swap_fst] rfl done · -- Case 2. h1 : x ≠ u rewrite [swap_other h1 h1] rfl done done lemma prod_eq_fun {m : Nat} (f g : Nat → ZMod m) (k : Nat) : ∀ (n : Nat), (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g := by by_induc · -- Base Case assume h : (∀ i < 0, f (k + i) = g (k + i)) rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g assume h1 : ∀ i < n + 1, f (k + i) = g (k + i) have h2 : ∀ i < n, f (k + i) = g (k + i) := by fix i : Nat assume h2 : i < n have h3 : i < n + 1 := by linarith show f (k + i) = g (k + i) from h1 i h3 done have h3 : prod_seq n k f = prod_seq n k g := ih h2 have h4 : n < n + 1 := Nat.lt_succ_self n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4] rfl done done lemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := by have h2 : ∀ (i : Nat), i < u → (f ∘ swap u n) (0 + i) = f (0 + i) := by fix i : Nat assume h2 : i < u have h3 : 0 + i ≠ u := by linarith have h4 : 0 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl done show prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f from prod_eq_fun (f ∘ swap u n) f 0 u h2 done lemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := by have h2 : ∀ i < j, (f ∘ swap u n) (u + 1 + i) = f (u + 1 + i) := by fix i : Nat assume h2 : i < j have h3 : u + 1 + i ≠ u := by linarith have h4 : u + 1 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl show prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f from prod_eq_fun (f ∘ swap u n) f (u + 1) j h2 done lemma break_prod {m : Nat} (n : Nat) (f : Nat → ZMod m) : ∀ (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by by_induc · -- Base Case have h : n + 0 = n := by rfl rewrite [prod_seq_base, h, Theorem_7_3_6_7] rfl done · -- Induction Step fix j : Nat assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f rewrite [←add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc] rfl done done lemma break_prod_twice {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f := break_prod n f 1 rewrite [prod_one] at h2 have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f := break_prod (u + 1) f j rewrite [←h1] at h3 have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f := break_prod u f 1 rewrite [prod_one] at h4 rewrite [h3, h4] at h2 show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2 done lemma swap_prod_eq_prod {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f := by by_cases h2 : u = n · -- Case 1. h2 : u = n rewrite [h2, trivial_swap n] --Goal : prod_seq (n + 1) 0 (f ∘ id) = prod_seq (n + 1) 0 f rfl done · -- Case 2. h2 : ¬u = n have h3 : u + 1 ≤ n := Nat.lt_of_le_of_ne h1 h2 obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3 have break_f : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := break_prod_twice f h4 have break_fs : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_prod_twice (f ∘ swap u n) h4 have f_eq_fs_below : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := swap_prod_eq_prod_below f h1 have f_eq_fs_btwn : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4 show prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f from calc prod_seq (n + 1) 0 (f ∘ swap u n) _ = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_fs _ = prod_seq u 0 f * (f ∘ swap u n) u * prod_seq j (u + 1) f * (f ∘ swap u n) n := by rw [f_eq_fs_below, f_eq_fs_btwn] _ = prod_seq u 0 f * f (swap u n u) * prod_seq j (u + 1) f * f (swap u n n) := by rfl _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by rw [swap_fst, swap_snd] _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring _ = prod_seq (n + 1) 0 f := break_f.symm done done lemma perm_below_fixed {n : Nat} {g : Nat → Nat} (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry lemma perm_prod {m : Nat} (f : Nat → ZMod m) : ∀ (n : Nat), ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) := by by_induc · -- Base Case fix g : Nat → Nat assume h1 : perm_below 0 g rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) fix g : Nat → Nat assume g_pb : perm_below (n + 1) g define at g_pb have g_ob : onto_below (n + 1) g := g_pb.right.right define at g_ob have h1 : n < n + 1 := by linarith obtain (u : Nat) (h2 : u < n + 1 ∧ g u = n) from g_ob n h1 have s_pb : perm_below (n + 1) (swap u n) := swap_perm_below h2.left h1 have gs_pb_n1 : perm_below (n + 1) (g ∘ swap u n) := comp_perm_below g_pb s_pb have gs_fix_n : (g ∘ swap u n) n = n := calc (g ∘ swap u n) n _ = g (swap u n n) := by rfl _ = g u := by rw [swap_snd] _ = n := h2.right have gs_pb_n : perm_below n (g ∘ swap u n) := perm_below_fixed gs_pb_n1 gs_fix_n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f ∘ (g ∘ swap u n)) := ih (g ∘ swap u n) gs_pb_n have h3 : u ≤ n := by linarith show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f ∘ g) from calc prod_seq (n + 1) 0 f _ = prod_seq n 0 f * f n := prod_seq_zero_step n f _ = prod_seq n 0 (f ∘ (g ∘ swap u n)) * f ((g ∘ swap u n) n) := by rw [gs_prod, gs_fix_n] _ = prod_seq n 0 (f ∘ g ∘ swap u n) * (f ∘ g ∘ swap u n) n := by rfl _ = prod_seq (n + 1) 0 (f ∘ g ∘ swap u n) := (prod_seq_zero_step n (f ∘ g ∘ swap u n)).symm _ = prod_seq (n + 1) 0 ((f ∘ g) ∘ swap u n) := by rfl _ = prod_seq (n + 1) 0 (f ∘ g) := swap_prod_eq_prod (f ∘ g) h3 done done lemma F_invertible (m i : Nat) : invertible (F m i) := by by_cases h : rel_prime m i · -- Case 1. h : rel_prime m i rewrite [F_rp_def h] show invertible [i]_m from (Theorem_7_3_7 m i).rtl h done · -- Case 2. h : ¬rel_prime m i rewrite [F_not_rp_def h] apply Exists.intro [1]_m show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m done done lemma Fprod_invertible (m : Nat) : ∀ (k : Nat), invertible (prod_seq k 0 (F m)) := by by_induc · -- Base Case apply Exists.intro [1]_m show prod_seq 0 0 (F m) * [1]_m = [1]_m from calc prod_seq 0 0 (F m) * [1]_m _ = [1]_m * [1]_m := by rw [prod_seq_base] _ = [1]_m := Theorem_7_3_6_7 ([1]_m) done · -- Induction Step fix k : Nat assume ih : invertible (prod_seq k 0 (F m)) rewrite [prod_seq_zero_step] show invertible (prod_seq k 0 (F m) * (F m k)) from (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k) done done theorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m ^ (phi m) = [1]_m := by have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2 show [a]_m ^ (phi m) = [1]_m from calc [a]_m ^ (phi m) _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3] _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring _ = prod_seq m 0 (F m ∘ G m a) * Y := by rw [FG_prod h1 m, phi_def] _ = prod_seq m 0 (F m) * Y := by rw [perm_prod (F m) m (G m a) (G_perm_below h1)] _ = [1]_m := by rw [h3] done lemma Exercise_7_4_5_Int (m : Nat) (a : Int) : ∀ (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry lemma Exercise_7_4_5_Nat (m a n : Nat) : [a]_m ^ n = [a ^ n]_m := by rewrite [Exercise_7_4_5_Int] rfl done theorem Euler's_theorem {m a : Nat} [NeZero m] (h1 : rel_prime m a) : a ^ (phi m) ≡ 1 (MOD m) := by have h2 : [a]_m ^ (phi m) = [1]_m := Theorem_7_4_2 h1 rewrite [Exercise_7_4_5_Nat m a (phi m)] at h2 --h2 : [a ^ phi m]_m = [1]_m show a ^ (phi m) ≡ 1 (MOD m) from (cc_eq_iff_congr _ _ _).ltr h2 done #eval gcd 10 7 --Answer: 1. So 10 and 7 are relatively prime #eval 7 ^ phi 10 --Answer: 2401, which is congruent to 1 mod 10. end Euler /- Section 7.5 -/ lemma num_rp_prime {p : Nat} (h1 : prime p) : ∀ k < p, num_rp_below p (k + 1) = k := sorry lemma phi_prime {p : Nat} (h1 : prime p) : phi p = p - 1 := by have h2 : 1 ≤ p := prime_pos h1 have h3 : p - 1 + 1 = p := Nat.sub_add_cancel h2 have h4 : p - 1 < p := by linarith have h5 : num_rp_below p (p - 1 + 1) = p - 1 := num_rp_prime h1 (p - 1) h4 rewrite [h3] at h5 show phi p = p - 1 from h5 done theorem Theorem_7_2_2_Int {a c : Nat} {b : Int} (h1 : ↑c ∣ ↑a * b) (h2 : rel_prime a c) : ↑c ∣ b := by rewrite [Int.natCast_dvd, Int.natAbs_mul, Int.natAbs_ofNat] at h1 --h1 : c ∣ a * Int.natAbs b rewrite [Int.natCast_dvd] --Goal : c ∣ Int.natAbs b show c ∣ Int.natAbs b from Theorem_7_2_2 h1 h2 done lemma Lemma_7_4_5 {m n : Nat} (a b : Int) (h1 : rel_prime m n) : a ≡ b (MOD m * n) ↔ a ≡ b (MOD m) ∧ a ≡ b (MOD n) := by apply Iff.intro · -- (→) assume h2 : a ≡ b (MOD m * n) obtain (j : Int) (h3 : a - b = (m * n) * j) from h2 apply And.intro · -- Proof of a ≡ b (MOD m) apply Exists.intro (n * j) show a - b = m * (n * j) from calc a - b _ = m * n * j := h3 _ = m * (n * j) := by ring done · -- Proof of a ≡ b (MOD n) apply Exists.intro (m * j) show a - b = n * (m * j) from calc a - b _ = m * n * j := h3 _ = n * (m * j) := by ring done done · -- (←) assume h2 : a ≡ b (MOD m) ∧ a ≡ b (MOD n) obtain (j : Int) (h3 : a - b = m * j) from h2.left have h4 : (↑n : Int) ∣ a - b := h2.right rewrite [h3] at h4 --h4 : ↑n ∣ ↑m * j have h5 : ↑n ∣ j := Theorem_7_2_2_Int h4 h1 obtain (k : Int) (h6 : j = n * k) from h5 apply Exists.intro k --Goal : a - b = ↑(m * n) * k rewrite [Nat.cast_mul] --Goal : a - b = ↑m * ↑n * k show a - b = (m * n) * k from calc a - b _ = m * j := h3 _ = m * (n * k) := by rw [h6] _ = (m * n) * k := by ring done done --From exercises of Section 7.2 theorem rel_prime_symm {a b : Nat} (h : rel_prime a b) : rel_prime b a := sorry lemma prime_NeZero {p : Nat} (h : prime p) : NeZero p := by rewrite [neZero_iff] --Goal : p ≠ 0 define at h linarith done lemma Lemma_7_5_1 {p e d m c s : Nat} {t : Int} (h1 : prime p) (h2 : e * d = (p - 1) * s + 1) (h3 : m ^ e - c = p * t) : c ^ d ≡ m (MOD p) := by have h4 : m ^ e ≡ c (MOD p) := Exists.intro t h3 have h5 : [m ^ e]_p = [c]_p := (cc_eq_iff_congr _ _ _).rtl h4 rewrite [←Exercise_7_4_5_Nat] at h5 --h5 : [m]_p ^ e = [c]_p by_cases h6 : p ∣ m · -- Case 1. h6 : p ∣ m have h7 : m ≡ 0 (MOD p) := by obtain (j : Nat) (h8 : m = p * j) from h6 apply Exists.intro (↑j : Int) --Goal : ↑m - 0 = ↑p * ↑j rewrite [h8, Nat.cast_mul] ring done have h8 : [m]_p = [0]_p := (cc_eq_iff_congr _ _ _).rtl h7 have h9 : e * d ≠ 0 := by rewrite [h2] show (p - 1) * s + 1 ≠ 0 from Nat.add_one_ne_zero _ done have h10 : (0 : Int) ^ (e * d) = 0 := zero_pow h9 have h11 : [c ^ d]_p = [m]_p := calc [c ^ d]_p _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat] _ = ([m]_p ^ e) ^ d := by rw [h5] _ = [m]_p ^ (e * d) := by ring _ = [0]_p ^ (e * d) := by rw [h8] _ = [0 ^ (e * d)]_p := Exercise_7_4_5_Int _ _ _ _ = [0]_p := by rw [h10] _ = [m]_p := by rw [h8] show c ^ d ≡ m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11 done · -- Case 2. h6 : ¬p ∣ m have h7 : rel_prime m p := rel_prime_of_prime_not_dvd h1 h6 have h8 : rel_prime p m := rel_prime_symm h7 have h9 : NeZero p := prime_NeZero h1 have h10 : (1 : Int) ^ s = 1 := by ring have h11 : [c ^ d]_p = [m]_p := calc [c ^ d]_p _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat] _ = ([m]_p ^ e) ^ d := by rw [h5] _ = [m]_p ^ (e * d) := by ring _ = [m]_p ^ ((p - 1) * s + 1) := by rw [h2] _ = ([m]_p ^ (p - 1)) ^ s * [m]_p := by ring _ = ([m]_p ^ (phi p)) ^ s * [m]_p := by rw [phi_prime h1] _ = [1]_p ^ s * [m]_p := by rw [Theorem_7_4_2 h8] _ = [1 ^ s]_p * [m]_p := by rw [Exercise_7_4_5_Int] _ = [1]_p * [m]_p := by rw [h10] _ = [m]_p * [1]_p := by ring _ = [m]_p := Theorem_7_3_6_7 _ show c ^ d ≡ m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11 done done theorem Theorem_7_5_1 (p q n e d k m c : Nat) (p_prime : prime p) (q_prime : prime q) (p_ne_q : p ≠ q) (n_pq : n = p * q) (ed_congr_1 : e * d = k * (p - 1) * (q - 1) + 1) (h1 : [m]_n ^ e = [c]_n) : [c]_n ^ d = [m]_n := by rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] at h1 --h1 : m ^ e ≡ c (MOD n) rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] --Goal : c ^ d ≡ m (MOD n) obtain (j : Int) (h2 : m ^ e - c = n * j) from h1 rewrite [n_pq, Nat.cast_mul] at h2 --h2 : m ^ e - c = p * q * j have h3 : e * d = (p - 1) * (k * (q - 1)) + 1 := by rewrite [ed_congr_1] ring done have h4 : m ^ e - c = p * (q * j) := by rewrite [h2] ring done have congr_p : c ^ d ≡ m (MOD p) := Lemma_7_5_1 p_prime h3 h4 have h5 : e * d = (q - 1) * (k * (p - 1)) + 1 := by rewrite [ed_congr_1] ring done have h6 : m ^ e - c = q * (p * j) := by rewrite [h2] ring done have congr_q : c ^ d ≡ m (MOD q) := Lemma_7_5_1 q_prime h5 h6 have h7 : ¬q ∣ p := by by_contra h8 have h9 : q = 1 ∨ q = p := dvd_prime p_prime h8 disj_syll h9 (prime_not_one q_prime) show False from p_ne_q h9.symm done have h8 : rel_prime p q := rel_prime_of_prime_not_dvd q_prime h7 rewrite [n_pq, Lemma_7_4_5 _ _ h8] show c ^ d ≡ m (MOD p) ∧ c ^ d ≡ m (MOD q) from And.intro congr_p congr_q done /- BEGIN EXERCISES -/ namespace Exercises /- Section 7.1 -/ -- 1. theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry -- 2. lemma gcd_comm_lt {a b : Nat} (h : a < b) : gcd a b = gcd b a := sorry theorem gcd_comm (a b : Nat) : gcd a b = gcd b a := sorry -- 3. theorem Exercise_7_1_5 (a b : Nat) (n : Int) : (∃ (s t : Int), s * a + t * b = n) ↔ (↑(gcd a b) : Int) ∣ n := sorry -- 4. theorem Exercise_7_1_6 (a b c : Nat) : gcd a b = gcd (a + b * c) b := sorry -- 5. theorem gcd_is_nonzero {a b : Nat} (h : a ≠ 0 ∨ b ≠ 0) : gcd a b ≠ 0 := sorry -- 6.

theorem gcd_greatest {a b d : Nat} (h1 : gcd a b ≠ 0) (h2 : d ∣ a) (h3 : d ∣ b) : d ≤ gcd a b

HTPI.Exercises.gcd_greatest

htpi/HTPILib/Chap7.lean

htpi

/- Copyright 2023 Daniel J. Velleman -/ import HTPILib.Chap6 namespace HTPI /- Definitions -/ lemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by have h : n + 1 > 0 := Nat.succ_pos n show a % (n + 1) < n + 1 from Nat.mod_lt a h done def gcd (a b : Nat) : Nat := match b with | 0 => a | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd (n + 1) (a % (n + 1)) termination_by b mutual def gcd_c1 (a b : Nat) : Int := match b with | 0 => 1 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c2 (n + 1) (a % (n + 1)) --Corresponds to s = t' termination_by b def gcd_c2 (a b : Nat) : Int := match b with | 0 => 0 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c1 (n + 1) (a % (n + 1)) - (gcd_c2 (n + 1) (a % (n + 1))) * ↑(a / (n + 1)) --Corresponds to t = s' - t'q termination_by b end def prime (n : Nat) : Prop := 2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n def prime_factor (p n : Nat) : Prop := prime p ∧ p ∣ n def all_prime (l : List Nat) : Prop := ∀ p ∈ l, prime p def nondec (l : List Nat) : Prop := match l with | [] => True --Of course, True is a proposition that is always true | n :: L => (∀ m ∈ L, n ≤ m) ∧ nondec L def nondec_prime_list (l : List Nat) : Prop := all_prime l ∧ nondec l def prod (l : List Nat) : Nat := match l with | [] => 1 | n :: L => n * (prod L) def prime_factorization (n : Nat) (l : List Nat) : Prop := nondec_prime_list l ∧ prod l = n def rel_prime (a b : Nat) : Prop := gcd a b = 1 def congr_mod (m : Nat) (a b : Int) : Prop := (↑m : Int) ∣ (a - b) def cc (m : Nat) (a : Int) : ZMod m := (↑a : ZMod m) notation:50 a " ≡ " b " (MOD " m ")" => congr_mod m a b notation:max "["a"]_"m:max => cc m a def invertible {m : Nat} (X : ZMod m) : Prop := ∃ (Y : ZMod m), X * Y = [1]_m def num_rp_below (m k : Nat) : Nat := match k with | 0 => 0 | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j def phi (m : Nat) : Nat := num_rp_below m m def prod_seq {m : Nat} (j k : Nat) (f : Nat → ZMod m) : ZMod m := match j with | 0 => [1]_m | n + 1 => prod_seq n k f * f (k + n) def maps_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i < n, g i < n def one_one_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i1 < n, ∀ i2 < n, g i1 = g i2 → i1 = i2 def onto_below (n : Nat) (g : Nat → Nat) : Prop := ∀ k < n, ∃ i < n, g i = k def perm_below (n : Nat) (g : Nat → Nat) : Prop := maps_below n g ∧ one_one_below n g ∧ onto_below n g def inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m) def swap (u v i : Nat) : Nat := if i = u then v else if i = v then u else i namespace Euler --For definitions specific to Euler's theorem def F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m def G (m a i : Nat) : Nat := (a * i) % m def Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i end Euler /- Section 7.1 -/ theorem dvd_mod_of_dvd_a_b {a b d : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ (a % b) := by set q : Nat := a / b have h3 : b * q + a % b = a := Nat.div_add_mod a b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 define --Goal : ∃ (c : Nat), a % b = d * c apply Exists.intro (j - k * q) show a % b = d * (j - k * q) from calc a % b _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm _ = a - b * q := by rw [h3] _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc] _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm done theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry #eval gcd 672 161 --Answer: 7 lemma gcd_base (a : Nat) : gcd a 0 = a := by rfl lemma gcd_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd a b = gcd b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1)) rfl done lemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b ≠ 0) : a % b < b := by have h1 : b > 0 := Nat.pos_of_ne_zero h show a % b < b from Nat.mod_lt a h1 done lemma dvd_self (n : Nat) : n ∣ n := by apply Exists.intro 1 ring done theorem gcd_dvd : ∀ (b a : Nat), (gcd a b) ∣ a ∧ (gcd a b) ∣ b := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd a b_1) ∣ a ∧ (gcd a b_1) ∣ b_1 fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_base] --Goal: a ∣ a ∧ a ∣ 0 apply And.intro (dvd_self a) define apply Exists.intro 0 rfl done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_nonzero a h1] --Goal : gcd b (a % b) ∣ a ∧ gcd b (a % b) ∣ b have h2 : a % b < b := mod_nonzero_lt a h1 have h3 : (gcd b (a % b)) ∣ b ∧ (gcd b (a % b)) ∣ (a % b) := ih (a % b) h2 b apply And.intro _ h3.left show (gcd b (a % b)) ∣ a from dvd_a_of_dvd_b_mod h3.left h3.right done done theorem gcd_dvd_left (a b : Nat) : (gcd a b) ∣ a := (gcd_dvd b a).left theorem gcd_dvd_right (a b : Nat) : (gcd a b) ∣ b := (gcd_dvd b a).right lemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl lemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c1 a b = gcd_c2 b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done lemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl lemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * ↑(a / b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done theorem gcd_lin_comb : ∀ (b a : Nat), (gcd_c1 a b) * ↑a + (gcd_c2 a b) * ↑b = ↑(gcd a b) := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd_c1 a b_1) * ↑a + (gcd_c2 a b_1) * ↑b_1 = ↑(gcd a b_1) fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base] --Goal : 1 * ↑a + 0 * ↑0 = ↑a ring done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1] --Goal : gcd_c2 b (a % b) * ↑a + -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * ↑(a / b)) * ↑b = -- ↑(gcd b (a % b)) set r : Nat := a % b set q : Nat := a / b set s : Int := gcd_c1 b r set t : Int := gcd_c2 b r --Goal : t * ↑a + (s - t * ↑q) * ↑b = ↑(gcd b r) have h2 : r < b := mod_nonzero_lt a h1 have h3 : s * ↑b + t * ↑r = ↑(gcd b r) := ih r h2 b have h4 : b * q + r = a := Nat.div_add_mod a b rewrite [←h3, ←h4] rewrite [Nat.cast_add, Nat.cast_mul] --Goal : t * (↑b * ↑q + ↑r) + (s - t * ↑q) * ↑b = s * ↑b + t * ↑r ring done done #eval gcd_c1 672 161 --Answer: 6 #eval gcd_c2 672 161 --Answer: -25 --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161 theorem Theorem_7_1_6 {d a b : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ gcd a b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑d ∣ ↑(gcd a b) set s : Int := gcd_c1 a b set t : Int := gcd_c2 a b have h3 : s * ↑a + t * ↑b = ↑(gcd a b) := gcd_lin_comb b a rewrite [←h3] --Goal : ↑d ∣ s * ↑a + t * ↑b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul] --Goal : ↑d ∣ s * (↑d * ↑j) + t * (↑d * ↑k) define apply Exists.intro (s * ↑j + t * ↑k) ring done /- Section 7.2 -/ theorem dvd_trans {a b c : Nat} (h1 : a ∣ b) (h2 : b ∣ c) : a ∣ c := by define at h1; define at h2; define obtain (m : Nat) (h3 : b = a * m) from h1 obtain (n : Nat) (h4 : c = b * n) from h2 rewrite [h3, mul_assoc] at h4 apply Exists.intro (m * n) show c = a * (m * n) from h4 done lemma exists_prime_factor : ∀ (n : Nat), 2 ≤ n → ∃ (p : Nat), prime_factor p n := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, 2 ≤ n_1 → ∃ (p : Nat), prime_factor p n_1 assume h1 : 2 ≤ n by_cases h2 : prime n · -- Case 1. h2 : prime n apply Exists.intro n define --Goal : prime n ∧ n ∣ n show prime n ∧ n ∣ n from And.intro h2 (dvd_self n) done · -- Case 2. h2 : ¬prime n define at h2 --h2 : ¬(2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n) demorgan at h2 disj_syll h2 h1 obtain (a : Nat) (h3 : ∃ (b : Nat), a * b = n ∧ a < n ∧ b < n) from h2 obtain (b : Nat) (h4 : a * b = n ∧ a < n ∧ b < n) from h3 have h5 : 2 ≤ a := by by_contra h6 have h7 : a ≤ 1 := by linarith have h8 : n ≤ b := calc n _ = a * b := h4.left.symm _ ≤ 1 * b := by rel [h7] _ = b := by ring linarith --n ≤ b contradicts b < n done have h6 : ∃ (p : Nat), prime_factor p a := ih a h4.right.left h5 obtain (p : Nat) (h7 : prime_factor p a) from h6 apply Exists.intro p define --Goal : prime p ∧ p ∣ n define at h7 --h7 : prime p ∧ p ∣ a apply And.intro h7.left have h8 : a ∣ n := by apply Exists.intro b show n = a * b from (h4.left).symm done show p ∣ n from dvd_trans h7.right h8 done done lemma exists_least_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q := by set S : Set Nat := {p : Nat | prime_factor p n} have h2 : ∃ (p : Nat), p ∈ S := exists_prime_factor n h show ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q from well_ord_princ S h2 done lemma all_prime_nil : all_prime [] := by define --Goal : ∀ p ∈ [], prime p fix p : Nat contrapos --Goal : ¬prime p → p ∉ [] assume h1 : ¬prime p show p ∉ [] from List.not_mem_nil p done lemma all_prime_cons (n : Nat) (L : List Nat) : all_prime (n :: L) ↔ prime n ∧ all_prime L := by apply Iff.intro · -- (→) assume h1 : all_prime (n :: L) --Goal : prime n ∧ all_prime L define at h1 --h1 : ∀ p ∈ n :: L, prime p apply And.intro (h1 n (List.mem_cons_self n L)) define --Goal : ∀ p ∈ L, prime p fix p : Nat assume h2 : p ∈ L show prime p from h1 p (List.mem_cons_of_mem n h2) done · -- (←) assume h1 : prime n ∧ all_prime L --Goal : all_prime (n :: l) define : all_prime L at h1 define fix p : Nat assume h2 : p ∈ n :: L rewrite [List.mem_cons] at h2 --h2 : p = n ∨ p ∈ L by_cases on h2 · -- Case 1. h2 : p = n rewrite [h2] show prime n from h1.left done · -- Case 2. h2 : p ∈ L show prime p from h1.right p h2 done done done lemma nondec_nil : nondec [] := by define --Goal : True trivial --trivial proves some obviously true statements, such as True done lemma nondec_cons (n : Nat) (L : List Nat) : nondec (n :: L) ↔ (∀ m ∈ L, n ≤ m) ∧ nondec L := by rfl lemma prod_nil : prod [] = 1 := by rfl lemma prod_cons : prod (n :: L) = n * (prod L) := by rfl lemma exists_cons_of_length_eq_succ {A : Type} {l : List A} {n : Nat} (h : l.length = n + 1) : ∃ (a : A) (L : List A), l = a :: L ∧ L.length = n := by have h1 : ¬l.length = 0 := by linarith rewrite [List.length_eq_zero] at h1 obtain (a : A) (h2 : ∃ (L : List A), l = a :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List A) (h3 : l = a :: L) from h2 apply Exists.intro a apply Exists.intro L apply And.intro h3 have h4 : (a :: L).length = L.length + 1 := List.length_cons a L rewrite [←h3, h] at h4 show L.length = n from (Nat.add_right_cancel h4).symm done lemma list_elt_dvd_prod_by_length (a : Nat) : ∀ (n : Nat), ∀ (l : List Nat), l.length = n → a ∈ l → a ∣ prod l := by by_induc · --Base Case fix l : List Nat assume h1 : l.length = 0 rewrite [List.length_eq_zero] at h1 --h1 : l = [] rewrite [h1] --Goal : a ∈ [] → a ∣ prod [] contrapos assume h2 : ¬a ∣ prod [] show a ∉ [] from List.not_mem_nil a done · -- Induction Step fix n : Nat assume ih : ∀ (l : List Nat), List.length l = n → a ∈ l → a ∣ prod l fix l : List Nat assume h1 : l.length = n + 1 --Goal : a ∈ l → a ∣ prod l obtain (b : Nat) (h2 : ∃ (L : List Nat), l = b :: L ∧ L.length = n) from exists_cons_of_length_eq_succ h1 obtain (L : List Nat) (h3 : l = b :: L ∧ L.length = n) from h2 have h4 : a ∈ L → a ∣ prod L := ih L h3.right assume h5 : a ∈ l rewrite [h3.left, prod_cons] --Goal : a ∣ b * prod L rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b ∨ a ∈ L by_cases on h5 · -- Case 1. h5 : a = b apply Exists.intro (prod L) rewrite [h5] rfl done · -- Case 2. h5 : a ∈ L have h6 : a ∣ prod L := h4 h5 have h7 : prod L ∣ b * prod L := by apply Exists.intro b ring done show a ∣ b * prod L from dvd_trans h6 h7 done done done lemma list_elt_dvd_prod {a : Nat} {l : List Nat} (h : a ∈ l) : a ∣ prod l := by set n : Nat := l.length have h1 : l.length = n := by rfl show a ∣ prod l from list_elt_dvd_prod_by_length a n l h1 h done lemma exists_prime_factorization : ∀ (n : Nat), n ≥ 1 → ∃ (l : List Nat), prime_factorization n l := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, n_1 ≥ 1 → ∃ (l : List Nat), prime_factorization n_1 l assume h1 : n ≥ 1 by_cases h2 : n = 1 · -- Case 1. h2 : n = 1 apply Exists.intro [] define apply And.intro · -- Proof of nondec_prime_list [] define show all_prime [] ∧ nondec [] from And.intro all_prime_nil nondec_nil done · -- Proof of prod [] = n rewrite [prod_nil, h2] rfl done done · -- Case 2. h2 : n ≠ 1 have h3 : n ≥ 2 := lt_of_le_of_ne' h1 h2 obtain (p : Nat) (h4 : prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q) from exists_least_prime_factor h3 have p_prime_factor : prime_factor p n := h4.left define at p_prime_factor have p_prime : prime p := p_prime_factor.left have p_dvd_n : p ∣ n := p_prime_factor.right have p_least : ∀ (q : Nat), prime_factor q n → p ≤ q := h4.right obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n have h5 : m ≠ 0 := by contradict h1 with h6 have h7 : n = 0 := calc n _ = p * m := n_eq_pm _ = p * 0 := by rw [h6] _ = 0 := by ring rewrite [h7] decide done have m_pos : 0 < m := Nat.pos_of_ne_zero h5 have m_lt_n : m < n := by define at p_prime show m < n from calc m _ < m + m := by linarith _ = 2 * m := by ring _ ≤ p * m := by rel [p_prime.left] _ = n := n_eq_pm.symm done obtain (L : List Nat) (h6 : prime_factorization m L) from ih m m_lt_n m_pos define at h6 have ndpl_L : nondec_prime_list L := h6.left define at ndpl_L apply Exists.intro (p :: L) define apply And.intro · -- Proof of nondec_prime_list (p :: L) define apply And.intro · -- Proof of all_prime (p :: L) rewrite [all_prime_cons] show prime p ∧ all_prime L from And.intro p_prime ndpl_L.left done · -- Proof of nondec (p :: L) rewrite [nondec_cons] apply And.intro _ ndpl_L.right fix q : Nat assume q_in_L : q ∈ L have h7 : q ∣ prod L := list_elt_dvd_prod q_in_L rewrite [h6.right] at h7 --h7 : q ∣ m have h8 : m ∣ n := by apply Exists.intro p rewrite [n_eq_pm] ring done have q_dvd_n : q ∣ n := dvd_trans h7 h8 have ap_L : all_prime L := ndpl_L.left define at ap_L have q_prime_factor : prime_factor q n := And.intro (ap_L q q_in_L) q_dvd_n show p ≤ q from p_least q q_prime_factor done done · -- Proof of prod (p :: L) = n rewrite [prod_cons, h6.right, n_eq_pm] rfl done done done theorem Theorem_7_2_2 {a b c : Nat} (h1 : c ∣ a * b) (h2 : rel_prime a c) : c ∣ b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑c ∣ ↑b define at h1; define at h2; define obtain (j : Nat) (h3 : a * b = c * j) from h1 set s : Int := gcd_c1 a c set t : Int := gcd_c2 a c have h4 : s * ↑a + t * ↑c = ↑(gcd a c) := gcd_lin_comb c a rewrite [h2, Nat.cast_one] at h4 --h4 : s * ↑a + t * ↑c = (1 : Int) apply Exists.intro (s * ↑j + t * ↑b) show ↑b = ↑c * (s * ↑j + t * ↑b) from calc ↑b _ = (1 : Int) * ↑b := (one_mul _).symm _ = (s * ↑a + t * ↑c) * ↑b := by rw [h4] _ = s * (↑a * ↑b) + t * ↑c * ↑b := by ring _ = s * (↑c * ↑j) + t * ↑c * ↑b := by rw [←Nat.cast_mul a b, h3, Nat.cast_mul c j] _ = ↑c * (s * ↑j + t * ↑b) := by ring done lemma le_nonzero_prod_left {a b : Nat} (h : a * b ≠ 0) : a ≤ a * b := by have h1 : b ≠ 0 := by contradict h with h1 rewrite [h1] ring done have h2 : 1 ≤ b := Nat.pos_of_ne_zero h1 show a ≤ a * b from calc a = a * 1 := (mul_one a).symm _ ≤ a * b := by rel [h2] done lemma le_nonzero_prod_right {a b : Nat} (h : a * b ≠ 0) : b ≤ a * b := by rewrite [mul_comm] rewrite [mul_comm] at h show b ≤ b * a from le_nonzero_prod_left h done lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry lemma rel_prime_of_prime_not_dvd {a p : Nat} (h1 : prime p) (h2 : ¬p ∣ a) : rel_prime a p := by have h3 : gcd a p ∣ a := gcd_dvd_left a p have h4 : gcd a p ∣ p := gcd_dvd_right a p have h5 : gcd a p = 1 ∨ gcd a p = p := dvd_prime h1 h4 have h6 : gcd a p ≠ p := by contradict h2 with h6 rewrite [h6] at h3 show p ∣ a from h3 done disj_syll h5 h6 show rel_prime a p from h5 done theorem Theorem_7_2_3 {a b p : Nat} (h1 : prime p) (h2 : p ∣ a * b) : p ∣ a ∨ p ∣ b := by or_right with h3 have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3 show p ∣ b from Theorem_7_2_2 h2 h4 done lemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a ≥ 1 := by have h1 : a ≠ 0 := by by_contra h1 rewrite [h1] at h contradict h linarith done show a ≥ 1 from Nat.pos_of_ne_zero h1 done lemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by have h1 : a ≥ 1 := ge_one_of_prod_one h have h2 : a * b ≠ 0 := by linarith have h3 : a ≤ a * b := le_nonzero_prod_left h2 rewrite [h] at h3 show a = 1 from Nat.le_antisymm h3 h1 done lemma eq_one_of_dvd_one {n : Nat} (h : n ∣ 1) : n = 1 := by obtain (j : Nat) (h1 : 1 = n * j) from h show n = 1 from eq_one_of_prod_one h1.symm done lemma prime_not_one {p : Nat} (h : prime p) : p ≠ 1 := by define at h linarith done theorem Theorem_7_2_4 {p : Nat} (h1 : prime p) : ∀ (l : List Nat), p ∣ prod l → ∃ a ∈ l, p ∣ a := by apply List.rec · -- Base Case. Goal : p ∣ prod [] → ∃ a ∈ [], p ∣ a rewrite [prod_nil] assume h2 : p ∣ 1 show ∃ a ∈ [], p ∣ a from absurd (eq_one_of_dvd_one h2) (prime_not_one h1) done · -- Induction Step fix b : Nat fix L : List Nat assume ih : p ∣ prod L → ∃ a ∈ L, p ∣ a --Goal : p ∣ prod (b :: L) → ∃ a ∈ b :: L, p ∣ a assume h2 : p ∣ prod (b :: L) rewrite [prod_cons] at h2 have h3 : p ∣ b ∨ p ∣ prod L := Theorem_7_2_3 h1 h2 by_cases on h3 · -- Case 1. h3 : p ∣ b apply Exists.intro b show b ∈ b :: L ∧ p ∣ b from And.intro (List.mem_cons_self b L) h3 done · -- Case 2. h3 : p ∣ prod L obtain (a : Nat) (h4 : a ∈ L ∧ p ∣ a) from ih h3 apply Exists.intro a show a ∈ b :: L ∧ p ∣ a from And.intro (List.mem_cons_of_mem b h4.left) h4.right done done done lemma prime_in_list {p : Nat} {l : List Nat} (h1 : prime p) (h2 : all_prime l) (h3 : p ∣ prod l) : p ∈ l := by obtain (a : Nat) (h4 : a ∈ l ∧ p ∣ a) from Theorem_7_2_4 h1 l h3 define at h2 have h5 : prime a := h2 a h4.left have h6 : p = 1 ∨ p = a := dvd_prime h5 h4.right disj_syll h6 (prime_not_one h1) rewrite [h6] show a ∈ l from h4.left done lemma first_le_first {p q : Nat} {l m : List Nat} (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m)) (h3 : prod (p :: l) = prod (q :: m)) : p ≤ q := by define at h1; define at h2 have h4 : q ∣ prod (p :: l) := by define apply Exists.intro (prod m) rewrite [←prod_cons] show prod (p :: l) = prod (q :: m) from h3 done have h5 : all_prime (q :: m) := h2.left rewrite [all_prime_cons] at h5 have h6 : q ∈ p :: l := prime_in_list h5.left h1.left h4 have h7 : nondec (p :: l) := h1.right rewrite [nondec_cons] at h7 rewrite [List.mem_cons] at h6 by_cases on h6 · -- Case 1. h6 : q = p linarith done · -- Case 2. h6 : q ∈ l have h8 : ∀ m ∈ l, p ≤ m := h7.left show p ≤ q from h8 q h6 done done lemma nondec_prime_list_tail {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by define at h define rewrite [all_prime_cons, nondec_cons] at h show all_prime l ∧ nondec l from And.intro h.left.right h.right.right done lemma cons_prod_not_one {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : prod (p :: l) ≠ 1 := by define at h have h1 : all_prime (p :: l) := h.left rewrite [all_prime_cons] at h1 rewrite [prod_cons] by_contra h2 show False from (prime_not_one h1.left) (eq_one_of_prod_one h2) done lemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) : l = [] ↔ prod l = 1 := by apply Iff.intro · -- (→) assume h1 : l = [] rewrite [h1] show prod [] = 1 from prod_nil done · -- (←) contrapos assume h1 : ¬l = [] obtain (p : Nat) (h2 : ∃ (L : List Nat), l = p :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List Nat) (h3 : l = p :: L) from h2 rewrite [h3] at h rewrite [h3] show ¬prod (p :: L) = 1 from cons_prod_not_one h done done lemma prime_pos {p : Nat} (h : prime p) : p > 0 := by define at h linarith done theorem Theorem_7_2_5 : ∀ (l1 l2 : List Nat), nondec_prime_list l1 → nondec_prime_list l2 → prod l1 = prod l2 → l1 = l2 := by apply List.rec · -- Base Case. Goal : ∀ (l2 : List Nat), nondec_prime_list [] → -- nondec_prime_list l2 → prod [] = prod l2 → [] = l2 fix l2 : List Nat assume h1 : nondec_prime_list [] assume h2 : nondec_prime_list l2 assume h3 : prod [] = prod l2 rewrite [prod_nil, eq_comm, ←list_nil_iff_prod_one h2] at h3 show [] = l2 from h3.symm done · -- Induction Step fix p : Nat fix L1 : List Nat assume ih : ∀ (L2 : List Nat), nondec_prime_list L1 → nondec_prime_list L2 → prod L1 = prod L2 → L1 = L2 -- Goal : ∀ (l2 : List Nat), nondec_prime_list (p :: L1) → -- nondec_prime_list l2 → prod (p :: L1) = prod l2 → p :: L1 = l2 fix l2 : List Nat assume h1 : nondec_prime_list (p :: L1) assume h2 : nondec_prime_list l2 assume h3 : prod (p :: L1) = prod l2 have h4 : ¬prod (p :: L1) = 1 := cons_prod_not_one h1 rewrite [h3, ←list_nil_iff_prod_one h2] at h4 obtain (q : Nat) (h5 : ∃ (L : List Nat), l2 = q :: L) from List.exists_cons_of_ne_nil h4 obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5 rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2) rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2) have h7 : p ≤ q := first_le_first h1 h2 h3 have h8 : q ≤ p := first_le_first h2 h1 h3.symm have h9 : p = q := by linarith rewrite [h9, prod_cons, prod_cons] at h3 --h3 : q * prod L1 = q * prod L2 have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1 have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2 define at h2 have h12 : all_prime (q :: L2) := h2.left rewrite [all_prime_cons] at h12 have h13 : q > 0 := prime_pos h12.left have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3 have h15 : L1 = L2 := ih L2 h10 h11 h14 rewrite [h6, h9, h15] rfl done done theorem fund_thm_arith (n : Nat) (h : n ≥ 1) : ∃! (l : List Nat), prime_factorization n l := by exists_unique · -- Existence show ∃ (l : List Nat), prime_factorization n l from exists_prime_factorization n h done · -- Uniqueness fix l1 : List Nat; fix l2 : List Nat assume h1 : prime_factorization n l1 assume h2 : prime_factorization n l2 define at h1; define at h2 have h3 : prod l1 = n := h1.right rewrite [←h2.right] at h3 show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3 done done /- Section 7.3 -/ theorem congr_refl (m : Nat) : ∀ (a : Int), a ≡ a (MOD m) := by fix a : Int define --Goal : ∃ (c : Int), a - a = ↑m * c apply Exists.intro 0 ring done theorem congr_symm {m : Nat} : ∀ {a b : Int}, a ≡ b (MOD m) → b ≡ a (MOD m) := by fix a : Int; fix b : Int assume h1 : a ≡ b (MOD m) define at h1 --h1 : ∃ (c : Int), a - b = ↑m * c define --Goal : ∃ (c : Int), b - a = ↑m * c obtain (c : Int) (h2 : a - b = m * c) from h1 apply Exists.intro (-c) show b - a = m * (-c) from calc b - a _ = -(a - b) := by ring _ = -(m * c) := by rw [h2] _ = m * (-c) := by ring done theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry /- Fundamental properties of congruence classes -/ lemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) : X = Y ↔ X.val = Y.val := Fin.ext_iff lemma val_nat_eq_mod (n k : Nat) : ([k]_(n + 1)).val = k % (n + 1) := by rfl lemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl theorem cc_rep {m : Nat} (X : ZMod m) : ∃ (a : Int), X = [a]_m := match m with | 0 => by apply Exists.intro X rfl done | n + 1 => by apply Exists.intro ↑(X.val) have h1 : X.val < n + 1 := Fin.prop X rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1] rfl done theorem add_class (m : Nat) (a b : Int) : [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm theorem mul_class (m : Nat) (a b : Int) : [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm lemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ [a - b]_m = [0]_m := by apply Iff.intro · -- (→) assume h1 : [a]_m = [b]_m have h2 : a - b = a + (-b) := by ring have h3 : b + (-b) = 0 := by ring show [a - b]_m = [0]_m from calc [a - b]_m _ = [a + (-b)]_m := by rw [h2] _ = [a]_m + [-b]_m := by rw [add_class] _ = [b]_m + [-b]_m := by rw [h1] _ = [b + -b]_m := by rw [add_class] _ = [0]_m := by rw [h3] done · -- (←) assume h1 : [a - b]_m = [0]_m have h2 : b + (a - b) = a := by ring have h3 : b + 0 = b := by ring show [a]_m = [b]_m from calc [a]_m _ = [b + (a - b)]_m := by rw [h2] _ = [b]_m + [a - b]_m := by rw [add_class] _ = [b]_m + [0]_m := by rw [h1] _ = [b + 0]_m := by rw [add_class] _ = [b]_m := by rw [h3] done done lemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) : [a]_m = [0]_m → [-a]_m = [0]_m := by assume h1 : [a]_m = [0]_m have h2 : 0 + (-a) = -a := by ring have h3 : a + (-a) = 0 := by ring show [-a]_m = [0]_m from calc [-a]_m _ = [0 + (-a)]_m := by rw [h2] _ = [0]_m + [-a]_m := by rw [add_class] _ = [a]_m + [-a]_m := by rw [h1] _ = [a + (-a)]_m := by rw [add_class] _ = [0]_m := by rw [h3] done lemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) : [-a]_m = [0]_m ↔ [a]_m = [0]_m := by apply Iff.intro _ (cc_neg_zero_of_cc_zero m a) assume h1 : [-a]_m = [0]_m have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1 have h3 : -(-a) = a := by ring rewrite [h3] at h2 show [a]_m = [0]_m from h2 done lemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl lemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m ↔ m ∣ k := match m with | 0 => by have h : (0 : Int) = (↑(0 : Nat) : Int) := by rfl rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj] apply Iff.intro · -- (→) assume h1 : k = 0 rewrite [h1] show 0 ∣ 0 from dvd_self 0 done · -- (←) assume h1 : 0 ∣ k obtain (c : Nat) (h2 : k = 0 * c) from h1 rewrite [h2] ring done done | n + 1 => by rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero] show k % (n + 1) = 0 ↔ n + 1 ∣ k from (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm done lemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m ↔ ↑m ∣ a := by obtain (k : Nat) (h1 : a = ↑k ∨ a = -↑k) from Int.eq_nat_or_neg a by_cases on h1 · -- Case 1. h1: a = ↑k rewrite [h1, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done · -- Case 2. h1: a = -↑k rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done done theorem cc_eq_iff_congr (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ a ≡ b (MOD m) := calc [a]_m = [b]_m _ ↔ [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b _ ↔ ↑m ∣ (a - b) := cc_zero_iff_dvd m (a - b) _ ↔ a ≡ b (MOD m) := by rfl /- End of fundamental properties of congruence classes -/ lemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m := by have h1 : (↑m : Int) ≠ 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m) show 0 ≤ a % m from Int.emod_nonneg a h1 done lemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m) have h2 : (↑m : Int) > 0 := (Nat.cast_pos).rtl h1 show a % m < m from Int.emod_lt_of_pos a h2 done lemma congr_mod_mod (m : Nat) (a : Int) : a ≡ a % m (MOD m) := by define have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m apply Exists.intro (a / m) show a - a % m = m * (a / m) from calc a - (a % m) _ = m * (a / m) + a % m - a % m := by rw [h1] _ = m * (a / m) := by ring done lemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) := And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a)) theorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) : ∃! (r : Int), 0 ≤ r ∧ r < m ∧ a ≡ r (MOD m) := by exists_unique · -- Existence apply Exists.intro (a % m) show 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) from mod_cmpl_res m a done · -- Uniqueness fix r1 : Int; fix r2 : Int assume h1 : 0 ≤ r1 ∧ r1 < m ∧ a ≡ r1 (MOD m) assume h2 : 0 ≤ r2 ∧ r2 < m ∧ a ≡ r2 (MOD m) have h3 : r1 ≡ r2 (MOD m) := congr_trans (congr_symm h1.right.right) h2.right.right obtain (d : Int) (h4 : r1 - r2 = m * d) from h3 have h5 : r1 - r2 < m * 1 := by linarith have h6 : m * (-1) < r1 - r2 := by linarith rewrite [h4] at h5 --h5 : m * d < m * 1 rewrite [h4] at h6 --h6 : m * -1 < m * d have h7 : (↑m : Int) ≥ 0 := Nat.cast_nonneg m have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7 have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7 have h10 : d = 0 := by linarith show r1 = r2 from calc r1 _ = r1 - r2 + r2 := by ring _ = m * 0 + r2 := by rw [h4, h10] _ = r2 := by ring done done lemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m := (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a) theorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y rewrite [h1, h2] have h3 : a + b = b + a := by ring show [a]_m + [b]_m = [b]_m + [a]_m from calc [a]_m + [b]_m _ = [a + b]_m := add_class m a b _ = [b + a]_m := by rw [h3] _ = [b]_m + [a]_m := (add_class m b a).symm done theorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X rewrite [h1] have h2 : a * 1 = a := by ring show [a]_m * [1]_m = [a]_m from calc [a]_m * [1]_m _ = [a * 1]_m := mul_class m a 1 _ = [a]_m := by rw [h2] done theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry lemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) : [a]_m * [gcd_c2 m a]_m = [1]_m := by set s : Int := gcd_c1 m a have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m define at h1 rewrite [h1, Nat.cast_one] at h2 --h2 : s * ↑m + gcd_c2 m a * ↑a = 1 rewrite [mul_class, cc_eq_iff_congr] define --Goal : ∃ (c : Int), ↑a * gcd_c2 m a - 1 = ↑m * c apply Exists.intro (-s) show a * (gcd_c2 m a) - 1 = m * (-s) from calc a * (gcd_c2 m a) - 1 _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring _ = 1 + m * (-s) - 1 := by rw [h2] _ = m * (-s) := by ring done theorem Theorem_7_3_7 (m a : Nat) : invertible [a]_m ↔ rel_prime m a := by apply Iff.intro · -- (→) assume h1 : invertible [a]_m define at h1 obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1 obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y rewrite [h3, mul_class, cc_eq_iff_congr] at h2 define at h2 obtain (c : Int) (h4 : a * b - 1 = m * c) from h2 rewrite [Exercise_7_2_6] --Goal : ∃ (s t : Int), s * ↑m + t * ↑a = 1 apply Exists.intro (-c) apply Exists.intro b show (-c) * m + b * a = 1 from calc (-c) * m + b * a _ = (-c) * m + (a * b - 1) + 1 := by ring _ = (-c) * m + m * c + 1 := by rw [h4] _ = 1 := by ring done · -- (←) assume h1 : rel_prime m a define show ∃ (Y : ZMod m), [a]_m * Y = [1]_m from Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1) done done /- Section 7.4 -/ section Euler open Euler lemma num_rp_below_base {m : Nat} : num_rp_below m 0 = 0 := by rfl lemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) : num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by have h1 : num_rp_below m (j + 1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : gcd m j = 1 rewrite [if_pos h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1 show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1 done lemma num_rp_below_step_not_rp {m j : Nat} (h : ¬rel_prime m j) : num_rp_below m (j + 1) = num_rp_below m j := by have h1 : num_rp_below m (j +1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : ¬gcd m j = 1 rewrite [if_neg h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j show num_rp_below m (j + 1) = num_rp_below m j from h1 done lemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl #eval phi 10 --Answer: 4 lemma prod_inv_iff_inv {m : Nat} {X : ZMod m} (h1 : invertible X) (Y : ZMod m) : invertible (X * Y) ↔ invertible Y := by apply Iff.intro · -- (→) assume h2 : invertible (X * Y) obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2 apply Exists.intro (X * Z) rewrite [←h3] --Goal : Y * (X * Z) = X * Y * Z ring --Note that ring can do algebra in ZMod m done · -- (←) assume h2 : invertible Y obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1 obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2 apply Exists.intro (Xi * Yi) show (X * Y) * (Xi * Yi) = [1]_m from calc X * Y * (Xi * Yi) _ = (X * Xi) * (Y * Yi) := by ring _ = [1]_m * [1]_m := by rw [h3, h4] _ = [1]_m := Theorem_7_3_6_7 [1]_m done done lemma F_rp_def {m i : Nat} (h : rel_prime m i) : F m i = [i]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h --h : gcd m i = 1 rewrite [if_pos h] at h1 show F m i = [i]_m from h1 done lemma F_not_rp_def {m i : Nat} (h : ¬rel_prime m i) : F m i = [1]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h rewrite [h1, if_neg h] rfl done lemma prod_seq_base {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 0 k f = [1]_m := by rfl lemma prod_seq_step {m : Nat} (n k : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl lemma prod_seq_zero_step {m : Nat} (n : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by rewrite [prod_seq_step, zero_add] rfl done lemma prod_one {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 1 k f = f k := by rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7] rfl done lemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl lemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m := calc [G m a i]_m _ = [(a * i) % m]_m := by rfl _ = [a * i]_m := (cc_eq_mod m (a * i)).symm _ = [a]_m * [i]_m := (mul_class m a i).symm lemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) : rel_prime m (G m a i) ↔ rel_prime m i := by have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1 show rel_prime m (G m a i) ↔ rel_prime m i from calc rel_prime m (G m a i) _ ↔ invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm _ ↔ invertible ([a]_m * [i]_m) := by rw [cc_G] _ ↔ invertible [i]_m := prod_inv_iff_inv h2 ([i]_m) _ ↔ rel_prime m i := Theorem_7_3_7 m i done lemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) : F m (G m a i) = [a]_m * F m i := by have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2 show F m (G m a i) = [a]_m * F m i from calc F m (G m a i) _ = [G m a i]_m := F_rp_def h3 _ = [a]_m * [i]_m := cc_G m a i _ = [a]_m * F m i := by rw [F_rp_def h2] done lemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : ¬rel_prime m i) : F m (G m a i) = [1]_m := by rewrite [←G_rp_iff h1 i] at h2 show F m (G m a i) = [1]_m from F_not_rp_def h2 done lemma FG_prod {m a : Nat} (h1 : rel_prime m a) : ∀ (k : Nat), prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by by_induc · -- Base Case show prod_seq 0 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from calc prod_seq 0 0 ((F m) ∘ (G m a)) _ = [1]_m := prod_seq_base _ _ _ = [a]_m ^ 0 * [1]_m := by ring _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by rw [num_rp_below_base, prod_seq_base] done · -- Induction Step fix k : Nat assume ih : prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) by_cases h2 : rel_prime m k · -- Case 1. h2 : rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([a]_m * F m k) := by rw [FG_rp h1 h2] _ = [a]_m ^ ((num_rp_below m k) + 1) * ((prod_seq k 0 (F m)) * F m k) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_rp h2, prod_seq_zero_step] done · -- Case 2. h2 : ¬rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([1]_m) := by rw [FG_not_rp h1 h2] _ = [a]_m ^ (num_rp_below m k) * (prod_seq k 0 (F m) * ([1]_m)) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_not_rp h2, prod_seq_zero_step, F_not_rp_def h2] done done done lemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by define --Goal : ∀ i < m, G m a i < m fix i : Nat assume h1 : i < m rewrite [G_def] --Goal : a * i % m < m show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m) done lemma left_inv_one_one_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g' (g i) = i) : one_one_below n g := sorry lemma right_inv_onto_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g (g' i) = i) (h2 : maps_below n g') : onto_below n g := by define at h2; define fix k : Nat assume h3 : k < n apply Exists.intro (g' k) show g' k < n ∧ g (g' k) = k from And.intro (h2 k h3) (h1 k h3) done lemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m * [inv_mod m a]_m = [1]_m := by have h2 : 0 ≤ (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a) show [a]_m * [inv_mod m a]_m = [1]_m from calc [a]_m * [inv_mod m a]_m _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2] _ = [a]_m * [gcd_c2 m a]_m := by rw [←cc_eq_mod] _ = [1]_m := gcd_c2_inv h1 done lemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m := calc a * (b % m) % m = a % m * (b % m % m) % m := Nat.mul_mod _ _ _ _ = a % m * (b % m) % m := by rw [Nat.mod_mod] _ = a * b % m := (Nat.mul_mod _ _ _).symm lemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm] rfl done theorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] : ↑a ≡ ↑b (MOD m) ↔ a % m = b % m := sorry lemma mul_inv_mod_cancel {m a i : Nat} [NeZero m] (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1 rewrite [mul_class, cc_eq_iff_congr, ←Nat.cast_mul, ←Nat.cast_one, congr_iff_mod_eq_Nat] at h3 show a * inv_mod m a * i % m = i from calc a * (inv_mod m a) * i % m _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = 1 % m * i % m := by rw [h3] _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = i % m := by rw [one_mul] _ = i := Nat.mod_eq_of_lt h2 done lemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl lemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, G m a (Ginv m a i) = i := by fix i : Nat assume h2 : i < m show G m a (Ginv m a i) = i from calc G m a (Ginv m a i) _ = a * ((inv_mod m a * i) % m) % m := by rfl _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, Ginv m a (G m a i) = i := by fix i : Nat assume h2 : i < m show Ginv m a (G m a i) = i from calc Ginv m a (G m a i) _ = inv_mod m a * ((a * i) % m) % m := by rfl _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc, mul_comm (inv_mod m a)] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_maps_below (m a : Nat) [NeZero m] : maps_below m (Ginv m a) := G_maps_below m (inv_mod m a) lemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : one_one_below m (G m a) := left_inv_one_one_below (Ginv_left_inv h1) lemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : onto_below m (G m a) := right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a) lemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : perm_below m (G m a) := And.intro (G_maps_below m a) (And.intro (G_one_one_below h1) (G_onto_below h1)) --Permuting a product of congruence classes doesn't change product lemma swap_fst (u v : Nat) : swap u v u = v := by define : swap u v u --Goal : (if u = u then v else if u = v then u else u) = v have h : u = u := by rfl rewrite [if_pos h] rfl done lemma swap_snd (u v : Nat) : swap u v v = u := by define : swap u v v by_cases h1 : v = u · -- Case 1. h1 : v = u rewrite [if_pos h1] show v = u from h1 done · -- Case 2. h1 : v ≠ u rewrite [if_neg h1] have h2 : v = v := by rfl rewrite [if_pos h2] rfl done done lemma swap_other {u v i : Nat} (h1 : i ≠ u) (h2 : i ≠ v) : swap u v i = i := by define : swap u v i rewrite [if_neg h1, if_neg h2] rfl done lemma swap_values (u v i : Nat) : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := by by_cases h1 : i = u · -- Case 1. h1 : i = u apply Or.inl rewrite [h1] show swap u v u = v from swap_fst u v done · -- Case 2. h1 : i ≠ u apply Or.inr by_cases h2 : i = v · -- Case 2.1. h2 : i = v apply Or.inl rewrite [h2] show swap u v v = u from swap_snd u v done · -- Case 2.2. h2 : i ≠ v apply Or.inr show swap u v i = i from swap_other h1 h2 done done done lemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by define fix i : Nat assume h3 : i < n have h4 : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := swap_values u v i by_cases on h4 · -- Case 1. h4 : swap u v i = v rewrite [h4] show v < n from h2 done · -- Case 2. by_cases on h4 · -- Case 2.1. h4 : swap u v i = u rewrite [h4] show u < n from h1 done · -- Case 2.2. h4 : swap u v i = i rewrite [h4] show i < n from h3 done done done lemma swap_swap (u v n : Nat) : ∀ i < n, swap u v (swap u v i) = i := by fix i : Nat assume h : i < n by_cases h1 : i = u · -- Case 1. h1 : i = u rewrite [h1, swap_fst, swap_snd] rfl done · -- Case 2. h1 : i ≠ u by_cases h2 : i = v · -- Case 2.1. h2 : i = v rewrite [h2, swap_snd, swap_fst] rfl done · -- Case 2.2. h2 : i ≠ v rewrite [swap_other h1 h2, swap_other h1 h2] rfl done done done lemma swap_one_one_below (u v n) : one_one_below n (swap u v) := left_inv_one_one_below (swap_swap u v n) lemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) := right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2) lemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) := And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2)) lemma comp_perm_below {n : Nat} {f g : Nat → Nat} (h1 : perm_below n f) (h2 : perm_below n g) : perm_below n (f ∘ g) := sorry lemma trivial_swap (u : Nat) : swap u u = id := by apply funext fix x : Nat by_cases h1 : x = u · -- Case 1. h1 : x = u rewrite [h1, swap_fst] rfl done · -- Case 2. h1 : x ≠ u rewrite [swap_other h1 h1] rfl done done lemma prod_eq_fun {m : Nat} (f g : Nat → ZMod m) (k : Nat) : ∀ (n : Nat), (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g := by by_induc · -- Base Case assume h : (∀ i < 0, f (k + i) = g (k + i)) rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g assume h1 : ∀ i < n + 1, f (k + i) = g (k + i) have h2 : ∀ i < n, f (k + i) = g (k + i) := by fix i : Nat assume h2 : i < n have h3 : i < n + 1 := by linarith show f (k + i) = g (k + i) from h1 i h3 done have h3 : prod_seq n k f = prod_seq n k g := ih h2 have h4 : n < n + 1 := Nat.lt_succ_self n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4] rfl done done lemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := by have h2 : ∀ (i : Nat), i < u → (f ∘ swap u n) (0 + i) = f (0 + i) := by fix i : Nat assume h2 : i < u have h3 : 0 + i ≠ u := by linarith have h4 : 0 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl done show prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f from prod_eq_fun (f ∘ swap u n) f 0 u h2 done lemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := by have h2 : ∀ i < j, (f ∘ swap u n) (u + 1 + i) = f (u + 1 + i) := by fix i : Nat assume h2 : i < j have h3 : u + 1 + i ≠ u := by linarith have h4 : u + 1 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl show prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f from prod_eq_fun (f ∘ swap u n) f (u + 1) j h2 done lemma break_prod {m : Nat} (n : Nat) (f : Nat → ZMod m) : ∀ (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by by_induc · -- Base Case have h : n + 0 = n := by rfl rewrite [prod_seq_base, h, Theorem_7_3_6_7] rfl done · -- Induction Step fix j : Nat assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f rewrite [←add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc] rfl done done lemma break_prod_twice {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f := break_prod n f 1 rewrite [prod_one] at h2 have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f := break_prod (u + 1) f j rewrite [←h1] at h3 have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f := break_prod u f 1 rewrite [prod_one] at h4 rewrite [h3, h4] at h2 show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2 done lemma swap_prod_eq_prod {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f := by by_cases h2 : u = n · -- Case 1. h2 : u = n rewrite [h2, trivial_swap n] --Goal : prod_seq (n + 1) 0 (f ∘ id) = prod_seq (n + 1) 0 f rfl done · -- Case 2. h2 : ¬u = n have h3 : u + 1 ≤ n := Nat.lt_of_le_of_ne h1 h2 obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3 have break_f : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := break_prod_twice f h4 have break_fs : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_prod_twice (f ∘ swap u n) h4 have f_eq_fs_below : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := swap_prod_eq_prod_below f h1 have f_eq_fs_btwn : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4 show prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f from calc prod_seq (n + 1) 0 (f ∘ swap u n) _ = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_fs _ = prod_seq u 0 f * (f ∘ swap u n) u * prod_seq j (u + 1) f * (f ∘ swap u n) n := by rw [f_eq_fs_below, f_eq_fs_btwn] _ = prod_seq u 0 f * f (swap u n u) * prod_seq j (u + 1) f * f (swap u n n) := by rfl _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by rw [swap_fst, swap_snd] _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring _ = prod_seq (n + 1) 0 f := break_f.symm done done lemma perm_below_fixed {n : Nat} {g : Nat → Nat} (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry lemma perm_prod {m : Nat} (f : Nat → ZMod m) : ∀ (n : Nat), ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) := by by_induc · -- Base Case fix g : Nat → Nat assume h1 : perm_below 0 g rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) fix g : Nat → Nat assume g_pb : perm_below (n + 1) g define at g_pb have g_ob : onto_below (n + 1) g := g_pb.right.right define at g_ob have h1 : n < n + 1 := by linarith obtain (u : Nat) (h2 : u < n + 1 ∧ g u = n) from g_ob n h1 have s_pb : perm_below (n + 1) (swap u n) := swap_perm_below h2.left h1 have gs_pb_n1 : perm_below (n + 1) (g ∘ swap u n) := comp_perm_below g_pb s_pb have gs_fix_n : (g ∘ swap u n) n = n := calc (g ∘ swap u n) n _ = g (swap u n n) := by rfl _ = g u := by rw [swap_snd] _ = n := h2.right have gs_pb_n : perm_below n (g ∘ swap u n) := perm_below_fixed gs_pb_n1 gs_fix_n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f ∘ (g ∘ swap u n)) := ih (g ∘ swap u n) gs_pb_n have h3 : u ≤ n := by linarith show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f ∘ g) from calc prod_seq (n + 1) 0 f _ = prod_seq n 0 f * f n := prod_seq_zero_step n f _ = prod_seq n 0 (f ∘ (g ∘ swap u n)) * f ((g ∘ swap u n) n) := by rw [gs_prod, gs_fix_n] _ = prod_seq n 0 (f ∘ g ∘ swap u n) * (f ∘ g ∘ swap u n) n := by rfl _ = prod_seq (n + 1) 0 (f ∘ g ∘ swap u n) := (prod_seq_zero_step n (f ∘ g ∘ swap u n)).symm _ = prod_seq (n + 1) 0 ((f ∘ g) ∘ swap u n) := by rfl _ = prod_seq (n + 1) 0 (f ∘ g) := swap_prod_eq_prod (f ∘ g) h3 done done lemma F_invertible (m i : Nat) : invertible (F m i) := by by_cases h : rel_prime m i · -- Case 1. h : rel_prime m i rewrite [F_rp_def h] show invertible [i]_m from (Theorem_7_3_7 m i).rtl h done · -- Case 2. h : ¬rel_prime m i rewrite [F_not_rp_def h] apply Exists.intro [1]_m show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m done done lemma Fprod_invertible (m : Nat) : ∀ (k : Nat), invertible (prod_seq k 0 (F m)) := by by_induc · -- Base Case apply Exists.intro [1]_m show prod_seq 0 0 (F m) * [1]_m = [1]_m from calc prod_seq 0 0 (F m) * [1]_m _ = [1]_m * [1]_m := by rw [prod_seq_base] _ = [1]_m := Theorem_7_3_6_7 ([1]_m) done · -- Induction Step fix k : Nat assume ih : invertible (prod_seq k 0 (F m)) rewrite [prod_seq_zero_step] show invertible (prod_seq k 0 (F m) * (F m k)) from (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k) done done theorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m ^ (phi m) = [1]_m := by have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2 show [a]_m ^ (phi m) = [1]_m from calc [a]_m ^ (phi m) _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3] _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring _ = prod_seq m 0 (F m ∘ G m a) * Y := by rw [FG_prod h1 m, phi_def] _ = prod_seq m 0 (F m) * Y := by rw [perm_prod (F m) m (G m a) (G_perm_below h1)] _ = [1]_m := by rw [h3] done lemma Exercise_7_4_5_Int (m : Nat) (a : Int) : ∀ (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry lemma Exercise_7_4_5_Nat (m a n : Nat) : [a]_m ^ n = [a ^ n]_m := by rewrite [Exercise_7_4_5_Int] rfl done theorem Euler's_theorem {m a : Nat} [NeZero m] (h1 : rel_prime m a) : a ^ (phi m) ≡ 1 (MOD m) := by have h2 : [a]_m ^ (phi m) = [1]_m := Theorem_7_4_2 h1 rewrite [Exercise_7_4_5_Nat m a (phi m)] at h2 --h2 : [a ^ phi m]_m = [1]_m show a ^ (phi m) ≡ 1 (MOD m) from (cc_eq_iff_congr _ _ _).ltr h2 done #eval gcd 10 7 --Answer: 1. So 10 and 7 are relatively prime #eval 7 ^ phi 10 --Answer: 2401, which is congruent to 1 mod 10. end Euler /- Section 7.5 -/ lemma num_rp_prime {p : Nat} (h1 : prime p) : ∀ k < p, num_rp_below p (k + 1) = k := sorry lemma phi_prime {p : Nat} (h1 : prime p) : phi p = p - 1 := by have h2 : 1 ≤ p := prime_pos h1 have h3 : p - 1 + 1 = p := Nat.sub_add_cancel h2 have h4 : p - 1 < p := by linarith have h5 : num_rp_below p (p - 1 + 1) = p - 1 := num_rp_prime h1 (p - 1) h4 rewrite [h3] at h5 show phi p = p - 1 from h5 done theorem Theorem_7_2_2_Int {a c : Nat} {b : Int} (h1 : ↑c ∣ ↑a * b) (h2 : rel_prime a c) : ↑c ∣ b := by rewrite [Int.natCast_dvd, Int.natAbs_mul, Int.natAbs_ofNat] at h1 --h1 : c ∣ a * Int.natAbs b rewrite [Int.natCast_dvd] --Goal : c ∣ Int.natAbs b show c ∣ Int.natAbs b from Theorem_7_2_2 h1 h2 done lemma Lemma_7_4_5 {m n : Nat} (a b : Int) (h1 : rel_prime m n) : a ≡ b (MOD m * n) ↔ a ≡ b (MOD m) ∧ a ≡ b (MOD n) := by apply Iff.intro · -- (→) assume h2 : a ≡ b (MOD m * n) obtain (j : Int) (h3 : a - b = (m * n) * j) from h2 apply And.intro · -- Proof of a ≡ b (MOD m) apply Exists.intro (n * j) show a - b = m * (n * j) from calc a - b _ = m * n * j := h3 _ = m * (n * j) := by ring done · -- Proof of a ≡ b (MOD n) apply Exists.intro (m * j) show a - b = n * (m * j) from calc a - b _ = m * n * j := h3 _ = n * (m * j) := by ring done done · -- (←) assume h2 : a ≡ b (MOD m) ∧ a ≡ b (MOD n) obtain (j : Int) (h3 : a - b = m * j) from h2.left have h4 : (↑n : Int) ∣ a - b := h2.right rewrite [h3] at h4 --h4 : ↑n ∣ ↑m * j have h5 : ↑n ∣ j := Theorem_7_2_2_Int h4 h1 obtain (k : Int) (h6 : j = n * k) from h5 apply Exists.intro k --Goal : a - b = ↑(m * n) * k rewrite [Nat.cast_mul] --Goal : a - b = ↑m * ↑n * k show a - b = (m * n) * k from calc a - b _ = m * j := h3 _ = m * (n * k) := by rw [h6] _ = (m * n) * k := by ring done done --From exercises of Section 7.2 theorem rel_prime_symm {a b : Nat} (h : rel_prime a b) : rel_prime b a := sorry lemma prime_NeZero {p : Nat} (h : prime p) : NeZero p := by rewrite [neZero_iff] --Goal : p ≠ 0 define at h linarith done lemma Lemma_7_5_1 {p e d m c s : Nat} {t : Int} (h1 : prime p) (h2 : e * d = (p - 1) * s + 1) (h3 : m ^ e - c = p * t) : c ^ d ≡ m (MOD p) := by have h4 : m ^ e ≡ c (MOD p) := Exists.intro t h3 have h5 : [m ^ e]_p = [c]_p := (cc_eq_iff_congr _ _ _).rtl h4 rewrite [←Exercise_7_4_5_Nat] at h5 --h5 : [m]_p ^ e = [c]_p by_cases h6 : p ∣ m · -- Case 1. h6 : p ∣ m have h7 : m ≡ 0 (MOD p) := by obtain (j : Nat) (h8 : m = p * j) from h6 apply Exists.intro (↑j : Int) --Goal : ↑m - 0 = ↑p * ↑j rewrite [h8, Nat.cast_mul] ring done have h8 : [m]_p = [0]_p := (cc_eq_iff_congr _ _ _).rtl h7 have h9 : e * d ≠ 0 := by rewrite [h2] show (p - 1) * s + 1 ≠ 0 from Nat.add_one_ne_zero _ done have h10 : (0 : Int) ^ (e * d) = 0 := zero_pow h9 have h11 : [c ^ d]_p = [m]_p := calc [c ^ d]_p _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat] _ = ([m]_p ^ e) ^ d := by rw [h5] _ = [m]_p ^ (e * d) := by ring _ = [0]_p ^ (e * d) := by rw [h8] _ = [0 ^ (e * d)]_p := Exercise_7_4_5_Int _ _ _ _ = [0]_p := by rw [h10] _ = [m]_p := by rw [h8] show c ^ d ≡ m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11 done · -- Case 2. h6 : ¬p ∣ m have h7 : rel_prime m p := rel_prime_of_prime_not_dvd h1 h6 have h8 : rel_prime p m := rel_prime_symm h7 have h9 : NeZero p := prime_NeZero h1 have h10 : (1 : Int) ^ s = 1 := by ring have h11 : [c ^ d]_p = [m]_p := calc [c ^ d]_p _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat] _ = ([m]_p ^ e) ^ d := by rw [h5] _ = [m]_p ^ (e * d) := by ring _ = [m]_p ^ ((p - 1) * s + 1) := by rw [h2] _ = ([m]_p ^ (p - 1)) ^ s * [m]_p := by ring _ = ([m]_p ^ (phi p)) ^ s * [m]_p := by rw [phi_prime h1] _ = [1]_p ^ s * [m]_p := by rw [Theorem_7_4_2 h8] _ = [1 ^ s]_p * [m]_p := by rw [Exercise_7_4_5_Int] _ = [1]_p * [m]_p := by rw [h10] _ = [m]_p * [1]_p := by ring _ = [m]_p := Theorem_7_3_6_7 _ show c ^ d ≡ m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11 done done theorem Theorem_7_5_1 (p q n e d k m c : Nat) (p_prime : prime p) (q_prime : prime q) (p_ne_q : p ≠ q) (n_pq : n = p * q) (ed_congr_1 : e * d = k * (p - 1) * (q - 1) + 1) (h1 : [m]_n ^ e = [c]_n) : [c]_n ^ d = [m]_n := by rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] at h1 --h1 : m ^ e ≡ c (MOD n) rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] --Goal : c ^ d ≡ m (MOD n) obtain (j : Int) (h2 : m ^ e - c = n * j) from h1 rewrite [n_pq, Nat.cast_mul] at h2 --h2 : m ^ e - c = p * q * j have h3 : e * d = (p - 1) * (k * (q - 1)) + 1 := by rewrite [ed_congr_1] ring done have h4 : m ^ e - c = p * (q * j) := by rewrite [h2] ring done have congr_p : c ^ d ≡ m (MOD p) := Lemma_7_5_1 p_prime h3 h4 have h5 : e * d = (q - 1) * (k * (p - 1)) + 1 := by rewrite [ed_congr_1] ring done have h6 : m ^ e - c = q * (p * j) := by rewrite [h2] ring done have congr_q : c ^ d ≡ m (MOD q) := Lemma_7_5_1 q_prime h5 h6 have h7 : ¬q ∣ p := by by_contra h8 have h9 : q = 1 ∨ q = p := dvd_prime p_prime h8 disj_syll h9 (prime_not_one q_prime) show False from p_ne_q h9.symm done have h8 : rel_prime p q := rel_prime_of_prime_not_dvd q_prime h7 rewrite [n_pq, Lemma_7_4_5 _ _ h8] show c ^ d ≡ m (MOD p) ∧ c ^ d ≡ m (MOD q) from And.intro congr_p congr_q done /- BEGIN EXERCISES -/ namespace Exercises /- Section 7.1 -/ -- 1. theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry -- 2. lemma gcd_comm_lt {a b : Nat} (h : a < b) : gcd a b = gcd b a := sorry theorem gcd_comm (a b : Nat) : gcd a b = gcd b a := sorry -- 3. theorem Exercise_7_1_5 (a b : Nat) (n : Int) : (∃ (s t : Int), s * a + t * b = n) ↔ (↑(gcd a b) : Int) ∣ n := sorry -- 4. theorem Exercise_7_1_6 (a b c : Nat) : gcd a b = gcd (a + b * c) b := sorry -- 5. theorem gcd_is_nonzero {a b : Nat} (h : a ≠ 0 ∨ b ≠ 0) : gcd a b ≠ 0 := sorry -- 6. theorem gcd_greatest {a b d : Nat} (h1 : gcd a b ≠ 0) (h2 : d ∣ a) (h3 : d ∣ b) : d ≤ gcd a b := sorry -- 7.

lemma Lemma_7_1_10a {a b : Nat} (n : Nat) (h : a ∣ b) : (n * a) ∣ (n * b)

HTPI.Exercises.Lemma_7_1_10a

htpi/HTPILib/Chap7.lean

htpi

/- Copyright 2023 Daniel J. Velleman -/ import HTPILib.Chap6 namespace HTPI /- Definitions -/ lemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by have h : n + 1 > 0 := Nat.succ_pos n show a % (n + 1) < n + 1 from Nat.mod_lt a h done def gcd (a b : Nat) : Nat := match b with | 0 => a | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd (n + 1) (a % (n + 1)) termination_by b mutual def gcd_c1 (a b : Nat) : Int := match b with | 0 => 1 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c2 (n + 1) (a % (n + 1)) --Corresponds to s = t' termination_by b def gcd_c2 (a b : Nat) : Int := match b with | 0 => 0 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c1 (n + 1) (a % (n + 1)) - (gcd_c2 (n + 1) (a % (n + 1))) * ↑(a / (n + 1)) --Corresponds to t = s' - t'q termination_by b end def prime (n : Nat) : Prop := 2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n def prime_factor (p n : Nat) : Prop := prime p ∧ p ∣ n def all_prime (l : List Nat) : Prop := ∀ p ∈ l, prime p def nondec (l : List Nat) : Prop := match l with | [] => True --Of course, True is a proposition that is always true | n :: L => (∀ m ∈ L, n ≤ m) ∧ nondec L def nondec_prime_list (l : List Nat) : Prop := all_prime l ∧ nondec l def prod (l : List Nat) : Nat := match l with | [] => 1 | n :: L => n * (prod L) def prime_factorization (n : Nat) (l : List Nat) : Prop := nondec_prime_list l ∧ prod l = n def rel_prime (a b : Nat) : Prop := gcd a b = 1 def congr_mod (m : Nat) (a b : Int) : Prop := (↑m : Int) ∣ (a - b) def cc (m : Nat) (a : Int) : ZMod m := (↑a : ZMod m) notation:50 a " ≡ " b " (MOD " m ")" => congr_mod m a b notation:max "["a"]_"m:max => cc m a def invertible {m : Nat} (X : ZMod m) : Prop := ∃ (Y : ZMod m), X * Y = [1]_m def num_rp_below (m k : Nat) : Nat := match k with | 0 => 0 | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j def phi (m : Nat) : Nat := num_rp_below m m def prod_seq {m : Nat} (j k : Nat) (f : Nat → ZMod m) : ZMod m := match j with | 0 => [1]_m | n + 1 => prod_seq n k f * f (k + n) def maps_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i < n, g i < n def one_one_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i1 < n, ∀ i2 < n, g i1 = g i2 → i1 = i2 def onto_below (n : Nat) (g : Nat → Nat) : Prop := ∀ k < n, ∃ i < n, g i = k def perm_below (n : Nat) (g : Nat → Nat) : Prop := maps_below n g ∧ one_one_below n g ∧ onto_below n g def inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m) def swap (u v i : Nat) : Nat := if i = u then v else if i = v then u else i namespace Euler --For definitions specific to Euler's theorem def F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m def G (m a i : Nat) : Nat := (a * i) % m def Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i end Euler /- Section 7.1 -/ theorem dvd_mod_of_dvd_a_b {a b d : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ (a % b) := by set q : Nat := a / b have h3 : b * q + a % b = a := Nat.div_add_mod a b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 define --Goal : ∃ (c : Nat), a % b = d * c apply Exists.intro (j - k * q) show a % b = d * (j - k * q) from calc a % b _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm _ = a - b * q := by rw [h3] _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc] _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm done theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry #eval gcd 672 161 --Answer: 7 lemma gcd_base (a : Nat) : gcd a 0 = a := by rfl lemma gcd_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd a b = gcd b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1)) rfl done lemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b ≠ 0) : a % b < b := by have h1 : b > 0 := Nat.pos_of_ne_zero h show a % b < b from Nat.mod_lt a h1 done lemma dvd_self (n : Nat) : n ∣ n := by apply Exists.intro 1 ring done theorem gcd_dvd : ∀ (b a : Nat), (gcd a b) ∣ a ∧ (gcd a b) ∣ b := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd a b_1) ∣ a ∧ (gcd a b_1) ∣ b_1 fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_base] --Goal: a ∣ a ∧ a ∣ 0 apply And.intro (dvd_self a) define apply Exists.intro 0 rfl done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_nonzero a h1] --Goal : gcd b (a % b) ∣ a ∧ gcd b (a % b) ∣ b have h2 : a % b < b := mod_nonzero_lt a h1 have h3 : (gcd b (a % b)) ∣ b ∧ (gcd b (a % b)) ∣ (a % b) := ih (a % b) h2 b apply And.intro _ h3.left show (gcd b (a % b)) ∣ a from dvd_a_of_dvd_b_mod h3.left h3.right done done theorem gcd_dvd_left (a b : Nat) : (gcd a b) ∣ a := (gcd_dvd b a).left theorem gcd_dvd_right (a b : Nat) : (gcd a b) ∣ b := (gcd_dvd b a).right lemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl lemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c1 a b = gcd_c2 b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done lemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl lemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * ↑(a / b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done theorem gcd_lin_comb : ∀ (b a : Nat), (gcd_c1 a b) * ↑a + (gcd_c2 a b) * ↑b = ↑(gcd a b) := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd_c1 a b_1) * ↑a + (gcd_c2 a b_1) * ↑b_1 = ↑(gcd a b_1) fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base] --Goal : 1 * ↑a + 0 * ↑0 = ↑a ring done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1] --Goal : gcd_c2 b (a % b) * ↑a + -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * ↑(a / b)) * ↑b = -- ↑(gcd b (a % b)) set r : Nat := a % b set q : Nat := a / b set s : Int := gcd_c1 b r set t : Int := gcd_c2 b r --Goal : t * ↑a + (s - t * ↑q) * ↑b = ↑(gcd b r) have h2 : r < b := mod_nonzero_lt a h1 have h3 : s * ↑b + t * ↑r = ↑(gcd b r) := ih r h2 b have h4 : b * q + r = a := Nat.div_add_mod a b rewrite [←h3, ←h4] rewrite [Nat.cast_add, Nat.cast_mul] --Goal : t * (↑b * ↑q + ↑r) + (s - t * ↑q) * ↑b = s * ↑b + t * ↑r ring done done #eval gcd_c1 672 161 --Answer: 6 #eval gcd_c2 672 161 --Answer: -25 --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161 theorem Theorem_7_1_6 {d a b : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ gcd a b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑d ∣ ↑(gcd a b) set s : Int := gcd_c1 a b set t : Int := gcd_c2 a b have h3 : s * ↑a + t * ↑b = ↑(gcd a b) := gcd_lin_comb b a rewrite [←h3] --Goal : ↑d ∣ s * ↑a + t * ↑b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul] --Goal : ↑d ∣ s * (↑d * ↑j) + t * (↑d * ↑k) define apply Exists.intro (s * ↑j + t * ↑k) ring done /- Section 7.2 -/ theorem dvd_trans {a b c : Nat} (h1 : a ∣ b) (h2 : b ∣ c) : a ∣ c := by define at h1; define at h2; define obtain (m : Nat) (h3 : b = a * m) from h1 obtain (n : Nat) (h4 : c = b * n) from h2 rewrite [h3, mul_assoc] at h4 apply Exists.intro (m * n) show c = a * (m * n) from h4 done lemma exists_prime_factor : ∀ (n : Nat), 2 ≤ n → ∃ (p : Nat), prime_factor p n := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, 2 ≤ n_1 → ∃ (p : Nat), prime_factor p n_1 assume h1 : 2 ≤ n by_cases h2 : prime n · -- Case 1. h2 : prime n apply Exists.intro n define --Goal : prime n ∧ n ∣ n show prime n ∧ n ∣ n from And.intro h2 (dvd_self n) done · -- Case 2. h2 : ¬prime n define at h2 --h2 : ¬(2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n) demorgan at h2 disj_syll h2 h1 obtain (a : Nat) (h3 : ∃ (b : Nat), a * b = n ∧ a < n ∧ b < n) from h2 obtain (b : Nat) (h4 : a * b = n ∧ a < n ∧ b < n) from h3 have h5 : 2 ≤ a := by by_contra h6 have h7 : a ≤ 1 := by linarith have h8 : n ≤ b := calc n _ = a * b := h4.left.symm _ ≤ 1 * b := by rel [h7] _ = b := by ring linarith --n ≤ b contradicts b < n done have h6 : ∃ (p : Nat), prime_factor p a := ih a h4.right.left h5 obtain (p : Nat) (h7 : prime_factor p a) from h6 apply Exists.intro p define --Goal : prime p ∧ p ∣ n define at h7 --h7 : prime p ∧ p ∣ a apply And.intro h7.left have h8 : a ∣ n := by apply Exists.intro b show n = a * b from (h4.left).symm done show p ∣ n from dvd_trans h7.right h8 done done lemma exists_least_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q := by set S : Set Nat := {p : Nat | prime_factor p n} have h2 : ∃ (p : Nat), p ∈ S := exists_prime_factor n h show ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q from well_ord_princ S h2 done lemma all_prime_nil : all_prime [] := by define --Goal : ∀ p ∈ [], prime p fix p : Nat contrapos --Goal : ¬prime p → p ∉ [] assume h1 : ¬prime p show p ∉ [] from List.not_mem_nil p done lemma all_prime_cons (n : Nat) (L : List Nat) : all_prime (n :: L) ↔ prime n ∧ all_prime L := by apply Iff.intro · -- (→) assume h1 : all_prime (n :: L) --Goal : prime n ∧ all_prime L define at h1 --h1 : ∀ p ∈ n :: L, prime p apply And.intro (h1 n (List.mem_cons_self n L)) define --Goal : ∀ p ∈ L, prime p fix p : Nat assume h2 : p ∈ L show prime p from h1 p (List.mem_cons_of_mem n h2) done · -- (←) assume h1 : prime n ∧ all_prime L --Goal : all_prime (n :: l) define : all_prime L at h1 define fix p : Nat assume h2 : p ∈ n :: L rewrite [List.mem_cons] at h2 --h2 : p = n ∨ p ∈ L by_cases on h2 · -- Case 1. h2 : p = n rewrite [h2] show prime n from h1.left done · -- Case 2. h2 : p ∈ L show prime p from h1.right p h2 done done done lemma nondec_nil : nondec [] := by define --Goal : True trivial --trivial proves some obviously true statements, such as True done lemma nondec_cons (n : Nat) (L : List Nat) : nondec (n :: L) ↔ (∀ m ∈ L, n ≤ m) ∧ nondec L := by rfl lemma prod_nil : prod [] = 1 := by rfl lemma prod_cons : prod (n :: L) = n * (prod L) := by rfl lemma exists_cons_of_length_eq_succ {A : Type} {l : List A} {n : Nat} (h : l.length = n + 1) : ∃ (a : A) (L : List A), l = a :: L ∧ L.length = n := by have h1 : ¬l.length = 0 := by linarith rewrite [List.length_eq_zero] at h1 obtain (a : A) (h2 : ∃ (L : List A), l = a :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List A) (h3 : l = a :: L) from h2 apply Exists.intro a apply Exists.intro L apply And.intro h3 have h4 : (a :: L).length = L.length + 1 := List.length_cons a L rewrite [←h3, h] at h4 show L.length = n from (Nat.add_right_cancel h4).symm done lemma list_elt_dvd_prod_by_length (a : Nat) : ∀ (n : Nat), ∀ (l : List Nat), l.length = n → a ∈ l → a ∣ prod l := by by_induc · --Base Case fix l : List Nat assume h1 : l.length = 0 rewrite [List.length_eq_zero] at h1 --h1 : l = [] rewrite [h1] --Goal : a ∈ [] → a ∣ prod [] contrapos assume h2 : ¬a ∣ prod [] show a ∉ [] from List.not_mem_nil a done · -- Induction Step fix n : Nat assume ih : ∀ (l : List Nat), List.length l = n → a ∈ l → a ∣ prod l fix l : List Nat assume h1 : l.length = n + 1 --Goal : a ∈ l → a ∣ prod l obtain (b : Nat) (h2 : ∃ (L : List Nat), l = b :: L ∧ L.length = n) from exists_cons_of_length_eq_succ h1 obtain (L : List Nat) (h3 : l = b :: L ∧ L.length = n) from h2 have h4 : a ∈ L → a ∣ prod L := ih L h3.right assume h5 : a ∈ l rewrite [h3.left, prod_cons] --Goal : a ∣ b * prod L rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b ∨ a ∈ L by_cases on h5 · -- Case 1. h5 : a = b apply Exists.intro (prod L) rewrite [h5] rfl done · -- Case 2. h5 : a ∈ L have h6 : a ∣ prod L := h4 h5 have h7 : prod L ∣ b * prod L := by apply Exists.intro b ring done show a ∣ b * prod L from dvd_trans h6 h7 done done done lemma list_elt_dvd_prod {a : Nat} {l : List Nat} (h : a ∈ l) : a ∣ prod l := by set n : Nat := l.length have h1 : l.length = n := by rfl show a ∣ prod l from list_elt_dvd_prod_by_length a n l h1 h done lemma exists_prime_factorization : ∀ (n : Nat), n ≥ 1 → ∃ (l : List Nat), prime_factorization n l := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, n_1 ≥ 1 → ∃ (l : List Nat), prime_factorization n_1 l assume h1 : n ≥ 1 by_cases h2 : n = 1 · -- Case 1. h2 : n = 1 apply Exists.intro [] define apply And.intro · -- Proof of nondec_prime_list [] define show all_prime [] ∧ nondec [] from And.intro all_prime_nil nondec_nil done · -- Proof of prod [] = n rewrite [prod_nil, h2] rfl done done · -- Case 2. h2 : n ≠ 1 have h3 : n ≥ 2 := lt_of_le_of_ne' h1 h2 obtain (p : Nat) (h4 : prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q) from exists_least_prime_factor h3 have p_prime_factor : prime_factor p n := h4.left define at p_prime_factor have p_prime : prime p := p_prime_factor.left have p_dvd_n : p ∣ n := p_prime_factor.right have p_least : ∀ (q : Nat), prime_factor q n → p ≤ q := h4.right obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n have h5 : m ≠ 0 := by contradict h1 with h6 have h7 : n = 0 := calc n _ = p * m := n_eq_pm _ = p * 0 := by rw [h6] _ = 0 := by ring rewrite [h7] decide done have m_pos : 0 < m := Nat.pos_of_ne_zero h5 have m_lt_n : m < n := by define at p_prime show m < n from calc m _ < m + m := by linarith _ = 2 * m := by ring _ ≤ p * m := by rel [p_prime.left] _ = n := n_eq_pm.symm done obtain (L : List Nat) (h6 : prime_factorization m L) from ih m m_lt_n m_pos define at h6 have ndpl_L : nondec_prime_list L := h6.left define at ndpl_L apply Exists.intro (p :: L) define apply And.intro · -- Proof of nondec_prime_list (p :: L) define apply And.intro · -- Proof of all_prime (p :: L) rewrite [all_prime_cons] show prime p ∧ all_prime L from And.intro p_prime ndpl_L.left done · -- Proof of nondec (p :: L) rewrite [nondec_cons] apply And.intro _ ndpl_L.right fix q : Nat assume q_in_L : q ∈ L have h7 : q ∣ prod L := list_elt_dvd_prod q_in_L rewrite [h6.right] at h7 --h7 : q ∣ m have h8 : m ∣ n := by apply Exists.intro p rewrite [n_eq_pm] ring done have q_dvd_n : q ∣ n := dvd_trans h7 h8 have ap_L : all_prime L := ndpl_L.left define at ap_L have q_prime_factor : prime_factor q n := And.intro (ap_L q q_in_L) q_dvd_n show p ≤ q from p_least q q_prime_factor done done · -- Proof of prod (p :: L) = n rewrite [prod_cons, h6.right, n_eq_pm] rfl done done done theorem Theorem_7_2_2 {a b c : Nat} (h1 : c ∣ a * b) (h2 : rel_prime a c) : c ∣ b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑c ∣ ↑b define at h1; define at h2; define obtain (j : Nat) (h3 : a * b = c * j) from h1 set s : Int := gcd_c1 a c set t : Int := gcd_c2 a c have h4 : s * ↑a + t * ↑c = ↑(gcd a c) := gcd_lin_comb c a rewrite [h2, Nat.cast_one] at h4 --h4 : s * ↑a + t * ↑c = (1 : Int) apply Exists.intro (s * ↑j + t * ↑b) show ↑b = ↑c * (s * ↑j + t * ↑b) from calc ↑b _ = (1 : Int) * ↑b := (one_mul _).symm _ = (s * ↑a + t * ↑c) * ↑b := by rw [h4] _ = s * (↑a * ↑b) + t * ↑c * ↑b := by ring _ = s * (↑c * ↑j) + t * ↑c * ↑b := by rw [←Nat.cast_mul a b, h3, Nat.cast_mul c j] _ = ↑c * (s * ↑j + t * ↑b) := by ring done lemma le_nonzero_prod_left {a b : Nat} (h : a * b ≠ 0) : a ≤ a * b := by have h1 : b ≠ 0 := by contradict h with h1 rewrite [h1] ring done have h2 : 1 ≤ b := Nat.pos_of_ne_zero h1 show a ≤ a * b from calc a = a * 1 := (mul_one a).symm _ ≤ a * b := by rel [h2] done lemma le_nonzero_prod_right {a b : Nat} (h : a * b ≠ 0) : b ≤ a * b := by rewrite [mul_comm] rewrite [mul_comm] at h show b ≤ b * a from le_nonzero_prod_left h done lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry lemma rel_prime_of_prime_not_dvd {a p : Nat} (h1 : prime p) (h2 : ¬p ∣ a) : rel_prime a p := by have h3 : gcd a p ∣ a := gcd_dvd_left a p have h4 : gcd a p ∣ p := gcd_dvd_right a p have h5 : gcd a p = 1 ∨ gcd a p = p := dvd_prime h1 h4 have h6 : gcd a p ≠ p := by contradict h2 with h6 rewrite [h6] at h3 show p ∣ a from h3 done disj_syll h5 h6 show rel_prime a p from h5 done theorem Theorem_7_2_3 {a b p : Nat} (h1 : prime p) (h2 : p ∣ a * b) : p ∣ a ∨ p ∣ b := by or_right with h3 have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3 show p ∣ b from Theorem_7_2_2 h2 h4 done lemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a ≥ 1 := by have h1 : a ≠ 0 := by by_contra h1 rewrite [h1] at h contradict h linarith done show a ≥ 1 from Nat.pos_of_ne_zero h1 done lemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by have h1 : a ≥ 1 := ge_one_of_prod_one h have h2 : a * b ≠ 0 := by linarith have h3 : a ≤ a * b := le_nonzero_prod_left h2 rewrite [h] at h3 show a = 1 from Nat.le_antisymm h3 h1 done lemma eq_one_of_dvd_one {n : Nat} (h : n ∣ 1) : n = 1 := by obtain (j : Nat) (h1 : 1 = n * j) from h show n = 1 from eq_one_of_prod_one h1.symm done lemma prime_not_one {p : Nat} (h : prime p) : p ≠ 1 := by define at h linarith done theorem Theorem_7_2_4 {p : Nat} (h1 : prime p) : ∀ (l : List Nat), p ∣ prod l → ∃ a ∈ l, p ∣ a := by apply List.rec · -- Base Case. Goal : p ∣ prod [] → ∃ a ∈ [], p ∣ a rewrite [prod_nil] assume h2 : p ∣ 1 show ∃ a ∈ [], p ∣ a from absurd (eq_one_of_dvd_one h2) (prime_not_one h1) done · -- Induction Step fix b : Nat fix L : List Nat assume ih : p ∣ prod L → ∃ a ∈ L, p ∣ a --Goal : p ∣ prod (b :: L) → ∃ a ∈ b :: L, p ∣ a assume h2 : p ∣ prod (b :: L) rewrite [prod_cons] at h2 have h3 : p ∣ b ∨ p ∣ prod L := Theorem_7_2_3 h1 h2 by_cases on h3 · -- Case 1. h3 : p ∣ b apply Exists.intro b show b ∈ b :: L ∧ p ∣ b from And.intro (List.mem_cons_self b L) h3 done · -- Case 2. h3 : p ∣ prod L obtain (a : Nat) (h4 : a ∈ L ∧ p ∣ a) from ih h3 apply Exists.intro a show a ∈ b :: L ∧ p ∣ a from And.intro (List.mem_cons_of_mem b h4.left) h4.right done done done lemma prime_in_list {p : Nat} {l : List Nat} (h1 : prime p) (h2 : all_prime l) (h3 : p ∣ prod l) : p ∈ l := by obtain (a : Nat) (h4 : a ∈ l ∧ p ∣ a) from Theorem_7_2_4 h1 l h3 define at h2 have h5 : prime a := h2 a h4.left have h6 : p = 1 ∨ p = a := dvd_prime h5 h4.right disj_syll h6 (prime_not_one h1) rewrite [h6] show a ∈ l from h4.left done lemma first_le_first {p q : Nat} {l m : List Nat} (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m)) (h3 : prod (p :: l) = prod (q :: m)) : p ≤ q := by define at h1; define at h2 have h4 : q ∣ prod (p :: l) := by define apply Exists.intro (prod m) rewrite [←prod_cons] show prod (p :: l) = prod (q :: m) from h3 done have h5 : all_prime (q :: m) := h2.left rewrite [all_prime_cons] at h5 have h6 : q ∈ p :: l := prime_in_list h5.left h1.left h4 have h7 : nondec (p :: l) := h1.right rewrite [nondec_cons] at h7 rewrite [List.mem_cons] at h6 by_cases on h6 · -- Case 1. h6 : q = p linarith done · -- Case 2. h6 : q ∈ l have h8 : ∀ m ∈ l, p ≤ m := h7.left show p ≤ q from h8 q h6 done done lemma nondec_prime_list_tail {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by define at h define rewrite [all_prime_cons, nondec_cons] at h show all_prime l ∧ nondec l from And.intro h.left.right h.right.right done lemma cons_prod_not_one {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : prod (p :: l) ≠ 1 := by define at h have h1 : all_prime (p :: l) := h.left rewrite [all_prime_cons] at h1 rewrite [prod_cons] by_contra h2 show False from (prime_not_one h1.left) (eq_one_of_prod_one h2) done lemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) : l = [] ↔ prod l = 1 := by apply Iff.intro · -- (→) assume h1 : l = [] rewrite [h1] show prod [] = 1 from prod_nil done · -- (←) contrapos assume h1 : ¬l = [] obtain (p : Nat) (h2 : ∃ (L : List Nat), l = p :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List Nat) (h3 : l = p :: L) from h2 rewrite [h3] at h rewrite [h3] show ¬prod (p :: L) = 1 from cons_prod_not_one h done done lemma prime_pos {p : Nat} (h : prime p) : p > 0 := by define at h linarith done theorem Theorem_7_2_5 : ∀ (l1 l2 : List Nat), nondec_prime_list l1 → nondec_prime_list l2 → prod l1 = prod l2 → l1 = l2 := by apply List.rec · -- Base Case. Goal : ∀ (l2 : List Nat), nondec_prime_list [] → -- nondec_prime_list l2 → prod [] = prod l2 → [] = l2 fix l2 : List Nat assume h1 : nondec_prime_list [] assume h2 : nondec_prime_list l2 assume h3 : prod [] = prod l2 rewrite [prod_nil, eq_comm, ←list_nil_iff_prod_one h2] at h3 show [] = l2 from h3.symm done · -- Induction Step fix p : Nat fix L1 : List Nat assume ih : ∀ (L2 : List Nat), nondec_prime_list L1 → nondec_prime_list L2 → prod L1 = prod L2 → L1 = L2 -- Goal : ∀ (l2 : List Nat), nondec_prime_list (p :: L1) → -- nondec_prime_list l2 → prod (p :: L1) = prod l2 → p :: L1 = l2 fix l2 : List Nat assume h1 : nondec_prime_list (p :: L1) assume h2 : nondec_prime_list l2 assume h3 : prod (p :: L1) = prod l2 have h4 : ¬prod (p :: L1) = 1 := cons_prod_not_one h1 rewrite [h3, ←list_nil_iff_prod_one h2] at h4 obtain (q : Nat) (h5 : ∃ (L : List Nat), l2 = q :: L) from List.exists_cons_of_ne_nil h4 obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5 rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2) rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2) have h7 : p ≤ q := first_le_first h1 h2 h3 have h8 : q ≤ p := first_le_first h2 h1 h3.symm have h9 : p = q := by linarith rewrite [h9, prod_cons, prod_cons] at h3 --h3 : q * prod L1 = q * prod L2 have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1 have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2 define at h2 have h12 : all_prime (q :: L2) := h2.left rewrite [all_prime_cons] at h12 have h13 : q > 0 := prime_pos h12.left have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3 have h15 : L1 = L2 := ih L2 h10 h11 h14 rewrite [h6, h9, h15] rfl done done theorem fund_thm_arith (n : Nat) (h : n ≥ 1) : ∃! (l : List Nat), prime_factorization n l := by exists_unique · -- Existence show ∃ (l : List Nat), prime_factorization n l from exists_prime_factorization n h done · -- Uniqueness fix l1 : List Nat; fix l2 : List Nat assume h1 : prime_factorization n l1 assume h2 : prime_factorization n l2 define at h1; define at h2 have h3 : prod l1 = n := h1.right rewrite [←h2.right] at h3 show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3 done done /- Section 7.3 -/ theorem congr_refl (m : Nat) : ∀ (a : Int), a ≡ a (MOD m) := by fix a : Int define --Goal : ∃ (c : Int), a - a = ↑m * c apply Exists.intro 0 ring done theorem congr_symm {m : Nat} : ∀ {a b : Int}, a ≡ b (MOD m) → b ≡ a (MOD m) := by fix a : Int; fix b : Int assume h1 : a ≡ b (MOD m) define at h1 --h1 : ∃ (c : Int), a - b = ↑m * c define --Goal : ∃ (c : Int), b - a = ↑m * c obtain (c : Int) (h2 : a - b = m * c) from h1 apply Exists.intro (-c) show b - a = m * (-c) from calc b - a _ = -(a - b) := by ring _ = -(m * c) := by rw [h2] _ = m * (-c) := by ring done theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry /- Fundamental properties of congruence classes -/ lemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) : X = Y ↔ X.val = Y.val := Fin.ext_iff lemma val_nat_eq_mod (n k : Nat) : ([k]_(n + 1)).val = k % (n + 1) := by rfl lemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl theorem cc_rep {m : Nat} (X : ZMod m) : ∃ (a : Int), X = [a]_m := match m with | 0 => by apply Exists.intro X rfl done | n + 1 => by apply Exists.intro ↑(X.val) have h1 : X.val < n + 1 := Fin.prop X rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1] rfl done theorem add_class (m : Nat) (a b : Int) : [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm theorem mul_class (m : Nat) (a b : Int) : [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm lemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ [a - b]_m = [0]_m := by apply Iff.intro · -- (→) assume h1 : [a]_m = [b]_m have h2 : a - b = a + (-b) := by ring have h3 : b + (-b) = 0 := by ring show [a - b]_m = [0]_m from calc [a - b]_m _ = [a + (-b)]_m := by rw [h2] _ = [a]_m + [-b]_m := by rw [add_class] _ = [b]_m + [-b]_m := by rw [h1] _ = [b + -b]_m := by rw [add_class] _ = [0]_m := by rw [h3] done · -- (←) assume h1 : [a - b]_m = [0]_m have h2 : b + (a - b) = a := by ring have h3 : b + 0 = b := by ring show [a]_m = [b]_m from calc [a]_m _ = [b + (a - b)]_m := by rw [h2] _ = [b]_m + [a - b]_m := by rw [add_class] _ = [b]_m + [0]_m := by rw [h1] _ = [b + 0]_m := by rw [add_class] _ = [b]_m := by rw [h3] done done lemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) : [a]_m = [0]_m → [-a]_m = [0]_m := by assume h1 : [a]_m = [0]_m have h2 : 0 + (-a) = -a := by ring have h3 : a + (-a) = 0 := by ring show [-a]_m = [0]_m from calc [-a]_m _ = [0 + (-a)]_m := by rw [h2] _ = [0]_m + [-a]_m := by rw [add_class] _ = [a]_m + [-a]_m := by rw [h1] _ = [a + (-a)]_m := by rw [add_class] _ = [0]_m := by rw [h3] done lemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) : [-a]_m = [0]_m ↔ [a]_m = [0]_m := by apply Iff.intro _ (cc_neg_zero_of_cc_zero m a) assume h1 : [-a]_m = [0]_m have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1 have h3 : -(-a) = a := by ring rewrite [h3] at h2 show [a]_m = [0]_m from h2 done lemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl lemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m ↔ m ∣ k := match m with | 0 => by have h : (0 : Int) = (↑(0 : Nat) : Int) := by rfl rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj] apply Iff.intro · -- (→) assume h1 : k = 0 rewrite [h1] show 0 ∣ 0 from dvd_self 0 done · -- (←) assume h1 : 0 ∣ k obtain (c : Nat) (h2 : k = 0 * c) from h1 rewrite [h2] ring done done | n + 1 => by rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero] show k % (n + 1) = 0 ↔ n + 1 ∣ k from (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm done lemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m ↔ ↑m ∣ a := by obtain (k : Nat) (h1 : a = ↑k ∨ a = -↑k) from Int.eq_nat_or_neg a by_cases on h1 · -- Case 1. h1: a = ↑k rewrite [h1, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done · -- Case 2. h1: a = -↑k rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done done theorem cc_eq_iff_congr (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ a ≡ b (MOD m) := calc [a]_m = [b]_m _ ↔ [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b _ ↔ ↑m ∣ (a - b) := cc_zero_iff_dvd m (a - b) _ ↔ a ≡ b (MOD m) := by rfl /- End of fundamental properties of congruence classes -/ lemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m := by have h1 : (↑m : Int) ≠ 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m) show 0 ≤ a % m from Int.emod_nonneg a h1 done lemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m) have h2 : (↑m : Int) > 0 := (Nat.cast_pos).rtl h1 show a % m < m from Int.emod_lt_of_pos a h2 done lemma congr_mod_mod (m : Nat) (a : Int) : a ≡ a % m (MOD m) := by define have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m apply Exists.intro (a / m) show a - a % m = m * (a / m) from calc a - (a % m) _ = m * (a / m) + a % m - a % m := by rw [h1] _ = m * (a / m) := by ring done lemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) := And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a)) theorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) : ∃! (r : Int), 0 ≤ r ∧ r < m ∧ a ≡ r (MOD m) := by exists_unique · -- Existence apply Exists.intro (a % m) show 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) from mod_cmpl_res m a done · -- Uniqueness fix r1 : Int; fix r2 : Int assume h1 : 0 ≤ r1 ∧ r1 < m ∧ a ≡ r1 (MOD m) assume h2 : 0 ≤ r2 ∧ r2 < m ∧ a ≡ r2 (MOD m) have h3 : r1 ≡ r2 (MOD m) := congr_trans (congr_symm h1.right.right) h2.right.right obtain (d : Int) (h4 : r1 - r2 = m * d) from h3 have h5 : r1 - r2 < m * 1 := by linarith have h6 : m * (-1) < r1 - r2 := by linarith rewrite [h4] at h5 --h5 : m * d < m * 1 rewrite [h4] at h6 --h6 : m * -1 < m * d have h7 : (↑m : Int) ≥ 0 := Nat.cast_nonneg m have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7 have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7 have h10 : d = 0 := by linarith show r1 = r2 from calc r1 _ = r1 - r2 + r2 := by ring _ = m * 0 + r2 := by rw [h4, h10] _ = r2 := by ring done done lemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m := (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a) theorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y rewrite [h1, h2] have h3 : a + b = b + a := by ring show [a]_m + [b]_m = [b]_m + [a]_m from calc [a]_m + [b]_m _ = [a + b]_m := add_class m a b _ = [b + a]_m := by rw [h3] _ = [b]_m + [a]_m := (add_class m b a).symm done theorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X rewrite [h1] have h2 : a * 1 = a := by ring show [a]_m * [1]_m = [a]_m from calc [a]_m * [1]_m _ = [a * 1]_m := mul_class m a 1 _ = [a]_m := by rw [h2] done theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry lemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) : [a]_m * [gcd_c2 m a]_m = [1]_m := by set s : Int := gcd_c1 m a have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m define at h1 rewrite [h1, Nat.cast_one] at h2 --h2 : s * ↑m + gcd_c2 m a * ↑a = 1 rewrite [mul_class, cc_eq_iff_congr] define --Goal : ∃ (c : Int), ↑a * gcd_c2 m a - 1 = ↑m * c apply Exists.intro (-s) show a * (gcd_c2 m a) - 1 = m * (-s) from calc a * (gcd_c2 m a) - 1 _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring _ = 1 + m * (-s) - 1 := by rw [h2] _ = m * (-s) := by ring done theorem Theorem_7_3_7 (m a : Nat) : invertible [a]_m ↔ rel_prime m a := by apply Iff.intro · -- (→) assume h1 : invertible [a]_m define at h1 obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1 obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y rewrite [h3, mul_class, cc_eq_iff_congr] at h2 define at h2 obtain (c : Int) (h4 : a * b - 1 = m * c) from h2 rewrite [Exercise_7_2_6] --Goal : ∃ (s t : Int), s * ↑m + t * ↑a = 1 apply Exists.intro (-c) apply Exists.intro b show (-c) * m + b * a = 1 from calc (-c) * m + b * a _ = (-c) * m + (a * b - 1) + 1 := by ring _ = (-c) * m + m * c + 1 := by rw [h4] _ = 1 := by ring done · -- (←) assume h1 : rel_prime m a define show ∃ (Y : ZMod m), [a]_m * Y = [1]_m from Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1) done done /- Section 7.4 -/ section Euler open Euler lemma num_rp_below_base {m : Nat} : num_rp_below m 0 = 0 := by rfl lemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) : num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by have h1 : num_rp_below m (j + 1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : gcd m j = 1 rewrite [if_pos h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1 show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1 done lemma num_rp_below_step_not_rp {m j : Nat} (h : ¬rel_prime m j) : num_rp_below m (j + 1) = num_rp_below m j := by have h1 : num_rp_below m (j +1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : ¬gcd m j = 1 rewrite [if_neg h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j show num_rp_below m (j + 1) = num_rp_below m j from h1 done lemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl #eval phi 10 --Answer: 4 lemma prod_inv_iff_inv {m : Nat} {X : ZMod m} (h1 : invertible X) (Y : ZMod m) : invertible (X * Y) ↔ invertible Y := by apply Iff.intro · -- (→) assume h2 : invertible (X * Y) obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2 apply Exists.intro (X * Z) rewrite [←h3] --Goal : Y * (X * Z) = X * Y * Z ring --Note that ring can do algebra in ZMod m done · -- (←) assume h2 : invertible Y obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1 obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2 apply Exists.intro (Xi * Yi) show (X * Y) * (Xi * Yi) = [1]_m from calc X * Y * (Xi * Yi) _ = (X * Xi) * (Y * Yi) := by ring _ = [1]_m * [1]_m := by rw [h3, h4] _ = [1]_m := Theorem_7_3_6_7 [1]_m done done lemma F_rp_def {m i : Nat} (h : rel_prime m i) : F m i = [i]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h --h : gcd m i = 1 rewrite [if_pos h] at h1 show F m i = [i]_m from h1 done lemma F_not_rp_def {m i : Nat} (h : ¬rel_prime m i) : F m i = [1]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h rewrite [h1, if_neg h] rfl done lemma prod_seq_base {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 0 k f = [1]_m := by rfl lemma prod_seq_step {m : Nat} (n k : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl lemma prod_seq_zero_step {m : Nat} (n : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by rewrite [prod_seq_step, zero_add] rfl done lemma prod_one {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 1 k f = f k := by rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7] rfl done lemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl lemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m := calc [G m a i]_m _ = [(a * i) % m]_m := by rfl _ = [a * i]_m := (cc_eq_mod m (a * i)).symm _ = [a]_m * [i]_m := (mul_class m a i).symm lemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) : rel_prime m (G m a i) ↔ rel_prime m i := by have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1 show rel_prime m (G m a i) ↔ rel_prime m i from calc rel_prime m (G m a i) _ ↔ invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm _ ↔ invertible ([a]_m * [i]_m) := by rw [cc_G] _ ↔ invertible [i]_m := prod_inv_iff_inv h2 ([i]_m) _ ↔ rel_prime m i := Theorem_7_3_7 m i done lemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) : F m (G m a i) = [a]_m * F m i := by have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2 show F m (G m a i) = [a]_m * F m i from calc F m (G m a i) _ = [G m a i]_m := F_rp_def h3 _ = [a]_m * [i]_m := cc_G m a i _ = [a]_m * F m i := by rw [F_rp_def h2] done lemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : ¬rel_prime m i) : F m (G m a i) = [1]_m := by rewrite [←G_rp_iff h1 i] at h2 show F m (G m a i) = [1]_m from F_not_rp_def h2 done lemma FG_prod {m a : Nat} (h1 : rel_prime m a) : ∀ (k : Nat), prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by by_induc · -- Base Case show prod_seq 0 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from calc prod_seq 0 0 ((F m) ∘ (G m a)) _ = [1]_m := prod_seq_base _ _ _ = [a]_m ^ 0 * [1]_m := by ring _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by rw [num_rp_below_base, prod_seq_base] done · -- Induction Step fix k : Nat assume ih : prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) by_cases h2 : rel_prime m k · -- Case 1. h2 : rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([a]_m * F m k) := by rw [FG_rp h1 h2] _ = [a]_m ^ ((num_rp_below m k) + 1) * ((prod_seq k 0 (F m)) * F m k) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_rp h2, prod_seq_zero_step] done · -- Case 2. h2 : ¬rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([1]_m) := by rw [FG_not_rp h1 h2] _ = [a]_m ^ (num_rp_below m k) * (prod_seq k 0 (F m) * ([1]_m)) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_not_rp h2, prod_seq_zero_step, F_not_rp_def h2] done done done lemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by define --Goal : ∀ i < m, G m a i < m fix i : Nat assume h1 : i < m rewrite [G_def] --Goal : a * i % m < m show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m) done lemma left_inv_one_one_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g' (g i) = i) : one_one_below n g := sorry lemma right_inv_onto_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g (g' i) = i) (h2 : maps_below n g') : onto_below n g := by define at h2; define fix k : Nat assume h3 : k < n apply Exists.intro (g' k) show g' k < n ∧ g (g' k) = k from And.intro (h2 k h3) (h1 k h3) done lemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m * [inv_mod m a]_m = [1]_m := by have h2 : 0 ≤ (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a) show [a]_m * [inv_mod m a]_m = [1]_m from calc [a]_m * [inv_mod m a]_m _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2] _ = [a]_m * [gcd_c2 m a]_m := by rw [←cc_eq_mod] _ = [1]_m := gcd_c2_inv h1 done lemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m := calc a * (b % m) % m = a % m * (b % m % m) % m := Nat.mul_mod _ _ _ _ = a % m * (b % m) % m := by rw [Nat.mod_mod] _ = a * b % m := (Nat.mul_mod _ _ _).symm lemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm] rfl done theorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] : ↑a ≡ ↑b (MOD m) ↔ a % m = b % m := sorry lemma mul_inv_mod_cancel {m a i : Nat} [NeZero m] (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1 rewrite [mul_class, cc_eq_iff_congr, ←Nat.cast_mul, ←Nat.cast_one, congr_iff_mod_eq_Nat] at h3 show a * inv_mod m a * i % m = i from calc a * (inv_mod m a) * i % m _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = 1 % m * i % m := by rw [h3] _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = i % m := by rw [one_mul] _ = i := Nat.mod_eq_of_lt h2 done lemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl lemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, G m a (Ginv m a i) = i := by fix i : Nat assume h2 : i < m show G m a (Ginv m a i) = i from calc G m a (Ginv m a i) _ = a * ((inv_mod m a * i) % m) % m := by rfl _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, Ginv m a (G m a i) = i := by fix i : Nat assume h2 : i < m show Ginv m a (G m a i) = i from calc Ginv m a (G m a i) _ = inv_mod m a * ((a * i) % m) % m := by rfl _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc, mul_comm (inv_mod m a)] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_maps_below (m a : Nat) [NeZero m] : maps_below m (Ginv m a) := G_maps_below m (inv_mod m a) lemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : one_one_below m (G m a) := left_inv_one_one_below (Ginv_left_inv h1) lemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : onto_below m (G m a) := right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a) lemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : perm_below m (G m a) := And.intro (G_maps_below m a) (And.intro (G_one_one_below h1) (G_onto_below h1)) --Permuting a product of congruence classes doesn't change product lemma swap_fst (u v : Nat) : swap u v u = v := by define : swap u v u --Goal : (if u = u then v else if u = v then u else u) = v have h : u = u := by rfl rewrite [if_pos h] rfl done lemma swap_snd (u v : Nat) : swap u v v = u := by define : swap u v v by_cases h1 : v = u · -- Case 1. h1 : v = u rewrite [if_pos h1] show v = u from h1 done · -- Case 2. h1 : v ≠ u rewrite [if_neg h1] have h2 : v = v := by rfl rewrite [if_pos h2] rfl done done lemma swap_other {u v i : Nat} (h1 : i ≠ u) (h2 : i ≠ v) : swap u v i = i := by define : swap u v i rewrite [if_neg h1, if_neg h2] rfl done lemma swap_values (u v i : Nat) : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := by by_cases h1 : i = u · -- Case 1. h1 : i = u apply Or.inl rewrite [h1] show swap u v u = v from swap_fst u v done · -- Case 2. h1 : i ≠ u apply Or.inr by_cases h2 : i = v · -- Case 2.1. h2 : i = v apply Or.inl rewrite [h2] show swap u v v = u from swap_snd u v done · -- Case 2.2. h2 : i ≠ v apply Or.inr show swap u v i = i from swap_other h1 h2 done done done lemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by define fix i : Nat assume h3 : i < n have h4 : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := swap_values u v i by_cases on h4 · -- Case 1. h4 : swap u v i = v rewrite [h4] show v < n from h2 done · -- Case 2. by_cases on h4 · -- Case 2.1. h4 : swap u v i = u rewrite [h4] show u < n from h1 done · -- Case 2.2. h4 : swap u v i = i rewrite [h4] show i < n from h3 done done done lemma swap_swap (u v n : Nat) : ∀ i < n, swap u v (swap u v i) = i := by fix i : Nat assume h : i < n by_cases h1 : i = u · -- Case 1. h1 : i = u rewrite [h1, swap_fst, swap_snd] rfl done · -- Case 2. h1 : i ≠ u by_cases h2 : i = v · -- Case 2.1. h2 : i = v rewrite [h2, swap_snd, swap_fst] rfl done · -- Case 2.2. h2 : i ≠ v rewrite [swap_other h1 h2, swap_other h1 h2] rfl done done done lemma swap_one_one_below (u v n) : one_one_below n (swap u v) := left_inv_one_one_below (swap_swap u v n) lemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) := right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2) lemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) := And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2)) lemma comp_perm_below {n : Nat} {f g : Nat → Nat} (h1 : perm_below n f) (h2 : perm_below n g) : perm_below n (f ∘ g) := sorry lemma trivial_swap (u : Nat) : swap u u = id := by apply funext fix x : Nat by_cases h1 : x = u · -- Case 1. h1 : x = u rewrite [h1, swap_fst] rfl done · -- Case 2. h1 : x ≠ u rewrite [swap_other h1 h1] rfl done done lemma prod_eq_fun {m : Nat} (f g : Nat → ZMod m) (k : Nat) : ∀ (n : Nat), (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g := by by_induc · -- Base Case assume h : (∀ i < 0, f (k + i) = g (k + i)) rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g assume h1 : ∀ i < n + 1, f (k + i) = g (k + i) have h2 : ∀ i < n, f (k + i) = g (k + i) := by fix i : Nat assume h2 : i < n have h3 : i < n + 1 := by linarith show f (k + i) = g (k + i) from h1 i h3 done have h3 : prod_seq n k f = prod_seq n k g := ih h2 have h4 : n < n + 1 := Nat.lt_succ_self n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4] rfl done done lemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := by have h2 : ∀ (i : Nat), i < u → (f ∘ swap u n) (0 + i) = f (0 + i) := by fix i : Nat assume h2 : i < u have h3 : 0 + i ≠ u := by linarith have h4 : 0 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl done show prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f from prod_eq_fun (f ∘ swap u n) f 0 u h2 done lemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := by have h2 : ∀ i < j, (f ∘ swap u n) (u + 1 + i) = f (u + 1 + i) := by fix i : Nat assume h2 : i < j have h3 : u + 1 + i ≠ u := by linarith have h4 : u + 1 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl show prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f from prod_eq_fun (f ∘ swap u n) f (u + 1) j h2 done lemma break_prod {m : Nat} (n : Nat) (f : Nat → ZMod m) : ∀ (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by by_induc · -- Base Case have h : n + 0 = n := by rfl rewrite [prod_seq_base, h, Theorem_7_3_6_7] rfl done · -- Induction Step fix j : Nat assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f rewrite [←add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc] rfl done done lemma break_prod_twice {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f := break_prod n f 1 rewrite [prod_one] at h2 have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f := break_prod (u + 1) f j rewrite [←h1] at h3 have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f := break_prod u f 1 rewrite [prod_one] at h4 rewrite [h3, h4] at h2 show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2 done lemma swap_prod_eq_prod {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f := by by_cases h2 : u = n · -- Case 1. h2 : u = n rewrite [h2, trivial_swap n] --Goal : prod_seq (n + 1) 0 (f ∘ id) = prod_seq (n + 1) 0 f rfl done · -- Case 2. h2 : ¬u = n have h3 : u + 1 ≤ n := Nat.lt_of_le_of_ne h1 h2 obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3 have break_f : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := break_prod_twice f h4 have break_fs : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_prod_twice (f ∘ swap u n) h4 have f_eq_fs_below : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := swap_prod_eq_prod_below f h1 have f_eq_fs_btwn : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4 show prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f from calc prod_seq (n + 1) 0 (f ∘ swap u n) _ = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_fs _ = prod_seq u 0 f * (f ∘ swap u n) u * prod_seq j (u + 1) f * (f ∘ swap u n) n := by rw [f_eq_fs_below, f_eq_fs_btwn] _ = prod_seq u 0 f * f (swap u n u) * prod_seq j (u + 1) f * f (swap u n n) := by rfl _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by rw [swap_fst, swap_snd] _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring _ = prod_seq (n + 1) 0 f := break_f.symm done done lemma perm_below_fixed {n : Nat} {g : Nat → Nat} (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry lemma perm_prod {m : Nat} (f : Nat → ZMod m) : ∀ (n : Nat), ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) := by by_induc · -- Base Case fix g : Nat → Nat assume h1 : perm_below 0 g rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) fix g : Nat → Nat assume g_pb : perm_below (n + 1) g define at g_pb have g_ob : onto_below (n + 1) g := g_pb.right.right define at g_ob have h1 : n < n + 1 := by linarith obtain (u : Nat) (h2 : u < n + 1 ∧ g u = n) from g_ob n h1 have s_pb : perm_below (n + 1) (swap u n) := swap_perm_below h2.left h1 have gs_pb_n1 : perm_below (n + 1) (g ∘ swap u n) := comp_perm_below g_pb s_pb have gs_fix_n : (g ∘ swap u n) n = n := calc (g ∘ swap u n) n _ = g (swap u n n) := by rfl _ = g u := by rw [swap_snd] _ = n := h2.right have gs_pb_n : perm_below n (g ∘ swap u n) := perm_below_fixed gs_pb_n1 gs_fix_n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f ∘ (g ∘ swap u n)) := ih (g ∘ swap u n) gs_pb_n have h3 : u ≤ n := by linarith show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f ∘ g) from calc prod_seq (n + 1) 0 f _ = prod_seq n 0 f * f n := prod_seq_zero_step n f _ = prod_seq n 0 (f ∘ (g ∘ swap u n)) * f ((g ∘ swap u n) n) := by rw [gs_prod, gs_fix_n] _ = prod_seq n 0 (f ∘ g ∘ swap u n) * (f ∘ g ∘ swap u n) n := by rfl _ = prod_seq (n + 1) 0 (f ∘ g ∘ swap u n) := (prod_seq_zero_step n (f ∘ g ∘ swap u n)).symm _ = prod_seq (n + 1) 0 ((f ∘ g) ∘ swap u n) := by rfl _ = prod_seq (n + 1) 0 (f ∘ g) := swap_prod_eq_prod (f ∘ g) h3 done done lemma F_invertible (m i : Nat) : invertible (F m i) := by by_cases h : rel_prime m i · -- Case 1. h : rel_prime m i rewrite [F_rp_def h] show invertible [i]_m from (Theorem_7_3_7 m i).rtl h done · -- Case 2. h : ¬rel_prime m i rewrite [F_not_rp_def h] apply Exists.intro [1]_m show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m done done lemma Fprod_invertible (m : Nat) : ∀ (k : Nat), invertible (prod_seq k 0 (F m)) := by by_induc · -- Base Case apply Exists.intro [1]_m show prod_seq 0 0 (F m) * [1]_m = [1]_m from calc prod_seq 0 0 (F m) * [1]_m _ = [1]_m * [1]_m := by rw [prod_seq_base] _ = [1]_m := Theorem_7_3_6_7 ([1]_m) done · -- Induction Step fix k : Nat assume ih : invertible (prod_seq k 0 (F m)) rewrite [prod_seq_zero_step] show invertible (prod_seq k 0 (F m) * (F m k)) from (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k) done done theorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m ^ (phi m) = [1]_m := by have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2 show [a]_m ^ (phi m) = [1]_m from calc [a]_m ^ (phi m) _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3] _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring _ = prod_seq m 0 (F m ∘ G m a) * Y := by rw [FG_prod h1 m, phi_def] _ = prod_seq m 0 (F m) * Y := by rw [perm_prod (F m) m (G m a) (G_perm_below h1)] _ = [1]_m := by rw [h3] done lemma Exercise_7_4_5_Int (m : Nat) (a : Int) : ∀ (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry lemma Exercise_7_4_5_Nat (m a n : Nat) : [a]_m ^ n = [a ^ n]_m := by rewrite [Exercise_7_4_5_Int] rfl done theorem Euler's_theorem {m a : Nat} [NeZero m] (h1 : rel_prime m a) : a ^ (phi m) ≡ 1 (MOD m) := by have h2 : [a]_m ^ (phi m) = [1]_m := Theorem_7_4_2 h1 rewrite [Exercise_7_4_5_Nat m a (phi m)] at h2 --h2 : [a ^ phi m]_m = [1]_m show a ^ (phi m) ≡ 1 (MOD m) from (cc_eq_iff_congr _ _ _).ltr h2 done #eval gcd 10 7 --Answer: 1. So 10 and 7 are relatively prime #eval 7 ^ phi 10 --Answer: 2401, which is congruent to 1 mod 10. end Euler /- Section 7.5 -/ lemma num_rp_prime {p : Nat} (h1 : prime p) : ∀ k < p, num_rp_below p (k + 1) = k := sorry lemma phi_prime {p : Nat} (h1 : prime p) : phi p = p - 1 := by have h2 : 1 ≤ p := prime_pos h1 have h3 : p - 1 + 1 = p := Nat.sub_add_cancel h2 have h4 : p - 1 < p := by linarith have h5 : num_rp_below p (p - 1 + 1) = p - 1 := num_rp_prime h1 (p - 1) h4 rewrite [h3] at h5 show phi p = p - 1 from h5 done theorem Theorem_7_2_2_Int {a c : Nat} {b : Int} (h1 : ↑c ∣ ↑a * b) (h2 : rel_prime a c) : ↑c ∣ b := by rewrite [Int.natCast_dvd, Int.natAbs_mul, Int.natAbs_ofNat] at h1 --h1 : c ∣ a * Int.natAbs b rewrite [Int.natCast_dvd] --Goal : c ∣ Int.natAbs b show c ∣ Int.natAbs b from Theorem_7_2_2 h1 h2 done lemma Lemma_7_4_5 {m n : Nat} (a b : Int) (h1 : rel_prime m n) : a ≡ b (MOD m * n) ↔ a ≡ b (MOD m) ∧ a ≡ b (MOD n) := by apply Iff.intro · -- (→) assume h2 : a ≡ b (MOD m * n) obtain (j : Int) (h3 : a - b = (m * n) * j) from h2 apply And.intro · -- Proof of a ≡ b (MOD m) apply Exists.intro (n * j) show a - b = m * (n * j) from calc a - b _ = m * n * j := h3 _ = m * (n * j) := by ring done · -- Proof of a ≡ b (MOD n) apply Exists.intro (m * j) show a - b = n * (m * j) from calc a - b _ = m * n * j := h3 _ = n * (m * j) := by ring done done · -- (←) assume h2 : a ≡ b (MOD m) ∧ a ≡ b (MOD n) obtain (j : Int) (h3 : a - b = m * j) from h2.left have h4 : (↑n : Int) ∣ a - b := h2.right rewrite [h3] at h4 --h4 : ↑n ∣ ↑m * j have h5 : ↑n ∣ j := Theorem_7_2_2_Int h4 h1 obtain (k : Int) (h6 : j = n * k) from h5 apply Exists.intro k --Goal : a - b = ↑(m * n) * k rewrite [Nat.cast_mul] --Goal : a - b = ↑m * ↑n * k show a - b = (m * n) * k from calc a - b _ = m * j := h3 _ = m * (n * k) := by rw [h6] _ = (m * n) * k := by ring done done --From exercises of Section 7.2 theorem rel_prime_symm {a b : Nat} (h : rel_prime a b) : rel_prime b a := sorry lemma prime_NeZero {p : Nat} (h : prime p) : NeZero p := by rewrite [neZero_iff] --Goal : p ≠ 0 define at h linarith done lemma Lemma_7_5_1 {p e d m c s : Nat} {t : Int} (h1 : prime p) (h2 : e * d = (p - 1) * s + 1) (h3 : m ^ e - c = p * t) : c ^ d ≡ m (MOD p) := by have h4 : m ^ e ≡ c (MOD p) := Exists.intro t h3 have h5 : [m ^ e]_p = [c]_p := (cc_eq_iff_congr _ _ _).rtl h4 rewrite [←Exercise_7_4_5_Nat] at h5 --h5 : [m]_p ^ e = [c]_p by_cases h6 : p ∣ m · -- Case 1. h6 : p ∣ m have h7 : m ≡ 0 (MOD p) := by obtain (j : Nat) (h8 : m = p * j) from h6 apply Exists.intro (↑j : Int) --Goal : ↑m - 0 = ↑p * ↑j rewrite [h8, Nat.cast_mul] ring done have h8 : [m]_p = [0]_p := (cc_eq_iff_congr _ _ _).rtl h7 have h9 : e * d ≠ 0 := by rewrite [h2] show (p - 1) * s + 1 ≠ 0 from Nat.add_one_ne_zero _ done have h10 : (0 : Int) ^ (e * d) = 0 := zero_pow h9 have h11 : [c ^ d]_p = [m]_p := calc [c ^ d]_p _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat] _ = ([m]_p ^ e) ^ d := by rw [h5] _ = [m]_p ^ (e * d) := by ring _ = [0]_p ^ (e * d) := by rw [h8] _ = [0 ^ (e * d)]_p := Exercise_7_4_5_Int _ _ _ _ = [0]_p := by rw [h10] _ = [m]_p := by rw [h8] show c ^ d ≡ m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11 done · -- Case 2. h6 : ¬p ∣ m have h7 : rel_prime m p := rel_prime_of_prime_not_dvd h1 h6 have h8 : rel_prime p m := rel_prime_symm h7 have h9 : NeZero p := prime_NeZero h1 have h10 : (1 : Int) ^ s = 1 := by ring have h11 : [c ^ d]_p = [m]_p := calc [c ^ d]_p _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat] _ = ([m]_p ^ e) ^ d := by rw [h5] _ = [m]_p ^ (e * d) := by ring _ = [m]_p ^ ((p - 1) * s + 1) := by rw [h2] _ = ([m]_p ^ (p - 1)) ^ s * [m]_p := by ring _ = ([m]_p ^ (phi p)) ^ s * [m]_p := by rw [phi_prime h1] _ = [1]_p ^ s * [m]_p := by rw [Theorem_7_4_2 h8] _ = [1 ^ s]_p * [m]_p := by rw [Exercise_7_4_5_Int] _ = [1]_p * [m]_p := by rw [h10] _ = [m]_p * [1]_p := by ring _ = [m]_p := Theorem_7_3_6_7 _ show c ^ d ≡ m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11 done done theorem Theorem_7_5_1 (p q n e d k m c : Nat) (p_prime : prime p) (q_prime : prime q) (p_ne_q : p ≠ q) (n_pq : n = p * q) (ed_congr_1 : e * d = k * (p - 1) * (q - 1) + 1) (h1 : [m]_n ^ e = [c]_n) : [c]_n ^ d = [m]_n := by rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] at h1 --h1 : m ^ e ≡ c (MOD n) rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] --Goal : c ^ d ≡ m (MOD n) obtain (j : Int) (h2 : m ^ e - c = n * j) from h1 rewrite [n_pq, Nat.cast_mul] at h2 --h2 : m ^ e - c = p * q * j have h3 : e * d = (p - 1) * (k * (q - 1)) + 1 := by rewrite [ed_congr_1] ring done have h4 : m ^ e - c = p * (q * j) := by rewrite [h2] ring done have congr_p : c ^ d ≡ m (MOD p) := Lemma_7_5_1 p_prime h3 h4 have h5 : e * d = (q - 1) * (k * (p - 1)) + 1 := by rewrite [ed_congr_1] ring done have h6 : m ^ e - c = q * (p * j) := by rewrite [h2] ring done have congr_q : c ^ d ≡ m (MOD q) := Lemma_7_5_1 q_prime h5 h6 have h7 : ¬q ∣ p := by by_contra h8 have h9 : q = 1 ∨ q = p := dvd_prime p_prime h8 disj_syll h9 (prime_not_one q_prime) show False from p_ne_q h9.symm done have h8 : rel_prime p q := rel_prime_of_prime_not_dvd q_prime h7 rewrite [n_pq, Lemma_7_4_5 _ _ h8] show c ^ d ≡ m (MOD p) ∧ c ^ d ≡ m (MOD q) from And.intro congr_p congr_q done /- BEGIN EXERCISES -/ namespace Exercises /- Section 7.1 -/ -- 1. theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry -- 2. lemma gcd_comm_lt {a b : Nat} (h : a < b) : gcd a b = gcd b a := sorry theorem gcd_comm (a b : Nat) : gcd a b = gcd b a := sorry -- 3. theorem Exercise_7_1_5 (a b : Nat) (n : Int) : (∃ (s t : Int), s * a + t * b = n) ↔ (↑(gcd a b) : Int) ∣ n := sorry -- 4. theorem Exercise_7_1_6 (a b c : Nat) : gcd a b = gcd (a + b * c) b := sorry -- 5. theorem gcd_is_nonzero {a b : Nat} (h : a ≠ 0 ∨ b ≠ 0) : gcd a b ≠ 0 := sorry -- 6. theorem gcd_greatest {a b d : Nat} (h1 : gcd a b ≠ 0) (h2 : d ∣ a) (h3 : d ∣ b) : d ≤ gcd a b := sorry -- 7. lemma Lemma_7_1_10a {a b : Nat} (n : Nat) (h : a ∣ b) : (n * a) ∣ (n * b) := sorry

lemma Lemma_7_1_10b {a b n : Nat} (h1 : n ≠ 0) (h2 : (n * a) ∣ (n * b)) : a ∣ b

HTPI.Exercises.Lemma_7_1_10b

htpi/HTPILib/Chap7.lean

htpi

/- Copyright 2023 Daniel J. Velleman -/ import HTPILib.Chap6 namespace HTPI /- Definitions -/ lemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by have h : n + 1 > 0 := Nat.succ_pos n show a % (n + 1) < n + 1 from Nat.mod_lt a h done def gcd (a b : Nat) : Nat := match b with | 0 => a | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd (n + 1) (a % (n + 1)) termination_by b mutual def gcd_c1 (a b : Nat) : Int := match b with | 0 => 1 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c2 (n + 1) (a % (n + 1)) --Corresponds to s = t' termination_by b def gcd_c2 (a b : Nat) : Int := match b with | 0 => 0 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c1 (n + 1) (a % (n + 1)) - (gcd_c2 (n + 1) (a % (n + 1))) * ↑(a / (n + 1)) --Corresponds to t = s' - t'q termination_by b end def prime (n : Nat) : Prop := 2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n def prime_factor (p n : Nat) : Prop := prime p ∧ p ∣ n def all_prime (l : List Nat) : Prop := ∀ p ∈ l, prime p def nondec (l : List Nat) : Prop := match l with | [] => True --Of course, True is a proposition that is always true | n :: L => (∀ m ∈ L, n ≤ m) ∧ nondec L def nondec_prime_list (l : List Nat) : Prop := all_prime l ∧ nondec l def prod (l : List Nat) : Nat := match l with | [] => 1 | n :: L => n * (prod L) def prime_factorization (n : Nat) (l : List Nat) : Prop := nondec_prime_list l ∧ prod l = n def rel_prime (a b : Nat) : Prop := gcd a b = 1 def congr_mod (m : Nat) (a b : Int) : Prop := (↑m : Int) ∣ (a - b) def cc (m : Nat) (a : Int) : ZMod m := (↑a : ZMod m) notation:50 a " ≡ " b " (MOD " m ")" => congr_mod m a b notation:max "["a"]_"m:max => cc m a def invertible {m : Nat} (X : ZMod m) : Prop := ∃ (Y : ZMod m), X * Y = [1]_m def num_rp_below (m k : Nat) : Nat := match k with | 0 => 0 | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j def phi (m : Nat) : Nat := num_rp_below m m def prod_seq {m : Nat} (j k : Nat) (f : Nat → ZMod m) : ZMod m := match j with | 0 => [1]_m | n + 1 => prod_seq n k f * f (k + n) def maps_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i < n, g i < n def one_one_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i1 < n, ∀ i2 < n, g i1 = g i2 → i1 = i2 def onto_below (n : Nat) (g : Nat → Nat) : Prop := ∀ k < n, ∃ i < n, g i = k def perm_below (n : Nat) (g : Nat → Nat) : Prop := maps_below n g ∧ one_one_below n g ∧ onto_below n g def inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m) def swap (u v i : Nat) : Nat := if i = u then v else if i = v then u else i namespace Euler --For definitions specific to Euler's theorem def F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m def G (m a i : Nat) : Nat := (a * i) % m def Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i end Euler /- Section 7.1 -/ theorem dvd_mod_of_dvd_a_b {a b d : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ (a % b) := by set q : Nat := a / b have h3 : b * q + a % b = a := Nat.div_add_mod a b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 define --Goal : ∃ (c : Nat), a % b = d * c apply Exists.intro (j - k * q) show a % b = d * (j - k * q) from calc a % b _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm _ = a - b * q := by rw [h3] _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc] _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm done theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry #eval gcd 672 161 --Answer: 7 lemma gcd_base (a : Nat) : gcd a 0 = a := by rfl lemma gcd_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd a b = gcd b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1)) rfl done lemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b ≠ 0) : a % b < b := by have h1 : b > 0 := Nat.pos_of_ne_zero h show a % b < b from Nat.mod_lt a h1 done lemma dvd_self (n : Nat) : n ∣ n := by apply Exists.intro 1 ring done theorem gcd_dvd : ∀ (b a : Nat), (gcd a b) ∣ a ∧ (gcd a b) ∣ b := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd a b_1) ∣ a ∧ (gcd a b_1) ∣ b_1 fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_base] --Goal: a ∣ a ∧ a ∣ 0 apply And.intro (dvd_self a) define apply Exists.intro 0 rfl done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_nonzero a h1] --Goal : gcd b (a % b) ∣ a ∧ gcd b (a % b) ∣ b have h2 : a % b < b := mod_nonzero_lt a h1 have h3 : (gcd b (a % b)) ∣ b ∧ (gcd b (a % b)) ∣ (a % b) := ih (a % b) h2 b apply And.intro _ h3.left show (gcd b (a % b)) ∣ a from dvd_a_of_dvd_b_mod h3.left h3.right done done theorem gcd_dvd_left (a b : Nat) : (gcd a b) ∣ a := (gcd_dvd b a).left theorem gcd_dvd_right (a b : Nat) : (gcd a b) ∣ b := (gcd_dvd b a).right lemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl lemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c1 a b = gcd_c2 b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done lemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl lemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * ↑(a / b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done theorem gcd_lin_comb : ∀ (b a : Nat), (gcd_c1 a b) * ↑a + (gcd_c2 a b) * ↑b = ↑(gcd a b) := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd_c1 a b_1) * ↑a + (gcd_c2 a b_1) * ↑b_1 = ↑(gcd a b_1) fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base] --Goal : 1 * ↑a + 0 * ↑0 = ↑a ring done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1] --Goal : gcd_c2 b (a % b) * ↑a + -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * ↑(a / b)) * ↑b = -- ↑(gcd b (a % b)) set r : Nat := a % b set q : Nat := a / b set s : Int := gcd_c1 b r set t : Int := gcd_c2 b r --Goal : t * ↑a + (s - t * ↑q) * ↑b = ↑(gcd b r) have h2 : r < b := mod_nonzero_lt a h1 have h3 : s * ↑b + t * ↑r = ↑(gcd b r) := ih r h2 b have h4 : b * q + r = a := Nat.div_add_mod a b rewrite [←h3, ←h4] rewrite [Nat.cast_add, Nat.cast_mul] --Goal : t * (↑b * ↑q + ↑r) + (s - t * ↑q) * ↑b = s * ↑b + t * ↑r ring done done #eval gcd_c1 672 161 --Answer: 6 #eval gcd_c2 672 161 --Answer: -25 --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161 theorem Theorem_7_1_6 {d a b : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ gcd a b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑d ∣ ↑(gcd a b) set s : Int := gcd_c1 a b set t : Int := gcd_c2 a b have h3 : s * ↑a + t * ↑b = ↑(gcd a b) := gcd_lin_comb b a rewrite [←h3] --Goal : ↑d ∣ s * ↑a + t * ↑b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul] --Goal : ↑d ∣ s * (↑d * ↑j) + t * (↑d * ↑k) define apply Exists.intro (s * ↑j + t * ↑k) ring done /- Section 7.2 -/ theorem dvd_trans {a b c : Nat} (h1 : a ∣ b) (h2 : b ∣ c) : a ∣ c := by define at h1; define at h2; define obtain (m : Nat) (h3 : b = a * m) from h1 obtain (n : Nat) (h4 : c = b * n) from h2 rewrite [h3, mul_assoc] at h4 apply Exists.intro (m * n) show c = a * (m * n) from h4 done lemma exists_prime_factor : ∀ (n : Nat), 2 ≤ n → ∃ (p : Nat), prime_factor p n := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, 2 ≤ n_1 → ∃ (p : Nat), prime_factor p n_1 assume h1 : 2 ≤ n by_cases h2 : prime n · -- Case 1. h2 : prime n apply Exists.intro n define --Goal : prime n ∧ n ∣ n show prime n ∧ n ∣ n from And.intro h2 (dvd_self n) done · -- Case 2. h2 : ¬prime n define at h2 --h2 : ¬(2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n) demorgan at h2 disj_syll h2 h1 obtain (a : Nat) (h3 : ∃ (b : Nat), a * b = n ∧ a < n ∧ b < n) from h2 obtain (b : Nat) (h4 : a * b = n ∧ a < n ∧ b < n) from h3 have h5 : 2 ≤ a := by by_contra h6 have h7 : a ≤ 1 := by linarith have h8 : n ≤ b := calc n _ = a * b := h4.left.symm _ ≤ 1 * b := by rel [h7] _ = b := by ring linarith --n ≤ b contradicts b < n done have h6 : ∃ (p : Nat), prime_factor p a := ih a h4.right.left h5 obtain (p : Nat) (h7 : prime_factor p a) from h6 apply Exists.intro p define --Goal : prime p ∧ p ∣ n define at h7 --h7 : prime p ∧ p ∣ a apply And.intro h7.left have h8 : a ∣ n := by apply Exists.intro b show n = a * b from (h4.left).symm done show p ∣ n from dvd_trans h7.right h8 done done lemma exists_least_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q := by set S : Set Nat := {p : Nat | prime_factor p n} have h2 : ∃ (p : Nat), p ∈ S := exists_prime_factor n h show ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q from well_ord_princ S h2 done lemma all_prime_nil : all_prime [] := by define --Goal : ∀ p ∈ [], prime p fix p : Nat contrapos --Goal : ¬prime p → p ∉ [] assume h1 : ¬prime p show p ∉ [] from List.not_mem_nil p done lemma all_prime_cons (n : Nat) (L : List Nat) : all_prime (n :: L) ↔ prime n ∧ all_prime L := by apply Iff.intro · -- (→) assume h1 : all_prime (n :: L) --Goal : prime n ∧ all_prime L define at h1 --h1 : ∀ p ∈ n :: L, prime p apply And.intro (h1 n (List.mem_cons_self n L)) define --Goal : ∀ p ∈ L, prime p fix p : Nat assume h2 : p ∈ L show prime p from h1 p (List.mem_cons_of_mem n h2) done · -- (←) assume h1 : prime n ∧ all_prime L --Goal : all_prime (n :: l) define : all_prime L at h1 define fix p : Nat assume h2 : p ∈ n :: L rewrite [List.mem_cons] at h2 --h2 : p = n ∨ p ∈ L by_cases on h2 · -- Case 1. h2 : p = n rewrite [h2] show prime n from h1.left done · -- Case 2. h2 : p ∈ L show prime p from h1.right p h2 done done done lemma nondec_nil : nondec [] := by define --Goal : True trivial --trivial proves some obviously true statements, such as True done lemma nondec_cons (n : Nat) (L : List Nat) : nondec (n :: L) ↔ (∀ m ∈ L, n ≤ m) ∧ nondec L := by rfl lemma prod_nil : prod [] = 1 := by rfl lemma prod_cons : prod (n :: L) = n * (prod L) := by rfl lemma exists_cons_of_length_eq_succ {A : Type} {l : List A} {n : Nat} (h : l.length = n + 1) : ∃ (a : A) (L : List A), l = a :: L ∧ L.length = n := by have h1 : ¬l.length = 0 := by linarith rewrite [List.length_eq_zero] at h1 obtain (a : A) (h2 : ∃ (L : List A), l = a :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List A) (h3 : l = a :: L) from h2 apply Exists.intro a apply Exists.intro L apply And.intro h3 have h4 : (a :: L).length = L.length + 1 := List.length_cons a L rewrite [←h3, h] at h4 show L.length = n from (Nat.add_right_cancel h4).symm done lemma list_elt_dvd_prod_by_length (a : Nat) : ∀ (n : Nat), ∀ (l : List Nat), l.length = n → a ∈ l → a ∣ prod l := by by_induc · --Base Case fix l : List Nat assume h1 : l.length = 0 rewrite [List.length_eq_zero] at h1 --h1 : l = [] rewrite [h1] --Goal : a ∈ [] → a ∣ prod [] contrapos assume h2 : ¬a ∣ prod [] show a ∉ [] from List.not_mem_nil a done · -- Induction Step fix n : Nat assume ih : ∀ (l : List Nat), List.length l = n → a ∈ l → a ∣ prod l fix l : List Nat assume h1 : l.length = n + 1 --Goal : a ∈ l → a ∣ prod l obtain (b : Nat) (h2 : ∃ (L : List Nat), l = b :: L ∧ L.length = n) from exists_cons_of_length_eq_succ h1 obtain (L : List Nat) (h3 : l = b :: L ∧ L.length = n) from h2 have h4 : a ∈ L → a ∣ prod L := ih L h3.right assume h5 : a ∈ l rewrite [h3.left, prod_cons] --Goal : a ∣ b * prod L rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b ∨ a ∈ L by_cases on h5 · -- Case 1. h5 : a = b apply Exists.intro (prod L) rewrite [h5] rfl done · -- Case 2. h5 : a ∈ L have h6 : a ∣ prod L := h4 h5 have h7 : prod L ∣ b * prod L := by apply Exists.intro b ring done show a ∣ b * prod L from dvd_trans h6 h7 done done done lemma list_elt_dvd_prod {a : Nat} {l : List Nat} (h : a ∈ l) : a ∣ prod l := by set n : Nat := l.length have h1 : l.length = n := by rfl show a ∣ prod l from list_elt_dvd_prod_by_length a n l h1 h done lemma exists_prime_factorization : ∀ (n : Nat), n ≥ 1 → ∃ (l : List Nat), prime_factorization n l := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, n_1 ≥ 1 → ∃ (l : List Nat), prime_factorization n_1 l assume h1 : n ≥ 1 by_cases h2 : n = 1 · -- Case 1. h2 : n = 1 apply Exists.intro [] define apply And.intro · -- Proof of nondec_prime_list [] define show all_prime [] ∧ nondec [] from And.intro all_prime_nil nondec_nil done · -- Proof of prod [] = n rewrite [prod_nil, h2] rfl done done · -- Case 2. h2 : n ≠ 1 have h3 : n ≥ 2 := lt_of_le_of_ne' h1 h2 obtain (p : Nat) (h4 : prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q) from exists_least_prime_factor h3 have p_prime_factor : prime_factor p n := h4.left define at p_prime_factor have p_prime : prime p := p_prime_factor.left have p_dvd_n : p ∣ n := p_prime_factor.right have p_least : ∀ (q : Nat), prime_factor q n → p ≤ q := h4.right obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n have h5 : m ≠ 0 := by contradict h1 with h6 have h7 : n = 0 := calc n _ = p * m := n_eq_pm _ = p * 0 := by rw [h6] _ = 0 := by ring rewrite [h7] decide done have m_pos : 0 < m := Nat.pos_of_ne_zero h5 have m_lt_n : m < n := by define at p_prime show m < n from calc m _ < m + m := by linarith _ = 2 * m := by ring _ ≤ p * m := by rel [p_prime.left] _ = n := n_eq_pm.symm done obtain (L : List Nat) (h6 : prime_factorization m L) from ih m m_lt_n m_pos define at h6 have ndpl_L : nondec_prime_list L := h6.left define at ndpl_L apply Exists.intro (p :: L) define apply And.intro · -- Proof of nondec_prime_list (p :: L) define apply And.intro · -- Proof of all_prime (p :: L) rewrite [all_prime_cons] show prime p ∧ all_prime L from And.intro p_prime ndpl_L.left done · -- Proof of nondec (p :: L) rewrite [nondec_cons] apply And.intro _ ndpl_L.right fix q : Nat assume q_in_L : q ∈ L have h7 : q ∣ prod L := list_elt_dvd_prod q_in_L rewrite [h6.right] at h7 --h7 : q ∣ m have h8 : m ∣ n := by apply Exists.intro p rewrite [n_eq_pm] ring done have q_dvd_n : q ∣ n := dvd_trans h7 h8 have ap_L : all_prime L := ndpl_L.left define at ap_L have q_prime_factor : prime_factor q n := And.intro (ap_L q q_in_L) q_dvd_n show p ≤ q from p_least q q_prime_factor done done · -- Proof of prod (p :: L) = n rewrite [prod_cons, h6.right, n_eq_pm] rfl done done done theorem Theorem_7_2_2 {a b c : Nat} (h1 : c ∣ a * b) (h2 : rel_prime a c) : c ∣ b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑c ∣ ↑b define at h1; define at h2; define obtain (j : Nat) (h3 : a * b = c * j) from h1 set s : Int := gcd_c1 a c set t : Int := gcd_c2 a c have h4 : s * ↑a + t * ↑c = ↑(gcd a c) := gcd_lin_comb c a rewrite [h2, Nat.cast_one] at h4 --h4 : s * ↑a + t * ↑c = (1 : Int) apply Exists.intro (s * ↑j + t * ↑b) show ↑b = ↑c * (s * ↑j + t * ↑b) from calc ↑b _ = (1 : Int) * ↑b := (one_mul _).symm _ = (s * ↑a + t * ↑c) * ↑b := by rw [h4] _ = s * (↑a * ↑b) + t * ↑c * ↑b := by ring _ = s * (↑c * ↑j) + t * ↑c * ↑b := by rw [←Nat.cast_mul a b, h3, Nat.cast_mul c j] _ = ↑c * (s * ↑j + t * ↑b) := by ring done lemma le_nonzero_prod_left {a b : Nat} (h : a * b ≠ 0) : a ≤ a * b := by have h1 : b ≠ 0 := by contradict h with h1 rewrite [h1] ring done have h2 : 1 ≤ b := Nat.pos_of_ne_zero h1 show a ≤ a * b from calc a = a * 1 := (mul_one a).symm _ ≤ a * b := by rel [h2] done lemma le_nonzero_prod_right {a b : Nat} (h : a * b ≠ 0) : b ≤ a * b := by rewrite [mul_comm] rewrite [mul_comm] at h show b ≤ b * a from le_nonzero_prod_left h done lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry lemma rel_prime_of_prime_not_dvd {a p : Nat} (h1 : prime p) (h2 : ¬p ∣ a) : rel_prime a p := by have h3 : gcd a p ∣ a := gcd_dvd_left a p have h4 : gcd a p ∣ p := gcd_dvd_right a p have h5 : gcd a p = 1 ∨ gcd a p = p := dvd_prime h1 h4 have h6 : gcd a p ≠ p := by contradict h2 with h6 rewrite [h6] at h3 show p ∣ a from h3 done disj_syll h5 h6 show rel_prime a p from h5 done theorem Theorem_7_2_3 {a b p : Nat} (h1 : prime p) (h2 : p ∣ a * b) : p ∣ a ∨ p ∣ b := by or_right with h3 have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3 show p ∣ b from Theorem_7_2_2 h2 h4 done lemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a ≥ 1 := by have h1 : a ≠ 0 := by by_contra h1 rewrite [h1] at h contradict h linarith done show a ≥ 1 from Nat.pos_of_ne_zero h1 done lemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by have h1 : a ≥ 1 := ge_one_of_prod_one h have h2 : a * b ≠ 0 := by linarith have h3 : a ≤ a * b := le_nonzero_prod_left h2 rewrite [h] at h3 show a = 1 from Nat.le_antisymm h3 h1 done lemma eq_one_of_dvd_one {n : Nat} (h : n ∣ 1) : n = 1 := by obtain (j : Nat) (h1 : 1 = n * j) from h show n = 1 from eq_one_of_prod_one h1.symm done lemma prime_not_one {p : Nat} (h : prime p) : p ≠ 1 := by define at h linarith done theorem Theorem_7_2_4 {p : Nat} (h1 : prime p) : ∀ (l : List Nat), p ∣ prod l → ∃ a ∈ l, p ∣ a := by apply List.rec · -- Base Case. Goal : p ∣ prod [] → ∃ a ∈ [], p ∣ a rewrite [prod_nil] assume h2 : p ∣ 1 show ∃ a ∈ [], p ∣ a from absurd (eq_one_of_dvd_one h2) (prime_not_one h1) done · -- Induction Step fix b : Nat fix L : List Nat assume ih : p ∣ prod L → ∃ a ∈ L, p ∣ a --Goal : p ∣ prod (b :: L) → ∃ a ∈ b :: L, p ∣ a assume h2 : p ∣ prod (b :: L) rewrite [prod_cons] at h2 have h3 : p ∣ b ∨ p ∣ prod L := Theorem_7_2_3 h1 h2 by_cases on h3 · -- Case 1. h3 : p ∣ b apply Exists.intro b show b ∈ b :: L ∧ p ∣ b from And.intro (List.mem_cons_self b L) h3 done · -- Case 2. h3 : p ∣ prod L obtain (a : Nat) (h4 : a ∈ L ∧ p ∣ a) from ih h3 apply Exists.intro a show a ∈ b :: L ∧ p ∣ a from And.intro (List.mem_cons_of_mem b h4.left) h4.right done done done lemma prime_in_list {p : Nat} {l : List Nat} (h1 : prime p) (h2 : all_prime l) (h3 : p ∣ prod l) : p ∈ l := by obtain (a : Nat) (h4 : a ∈ l ∧ p ∣ a) from Theorem_7_2_4 h1 l h3 define at h2 have h5 : prime a := h2 a h4.left have h6 : p = 1 ∨ p = a := dvd_prime h5 h4.right disj_syll h6 (prime_not_one h1) rewrite [h6] show a ∈ l from h4.left done lemma first_le_first {p q : Nat} {l m : List Nat} (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m)) (h3 : prod (p :: l) = prod (q :: m)) : p ≤ q := by define at h1; define at h2 have h4 : q ∣ prod (p :: l) := by define apply Exists.intro (prod m) rewrite [←prod_cons] show prod (p :: l) = prod (q :: m) from h3 done have h5 : all_prime (q :: m) := h2.left rewrite [all_prime_cons] at h5 have h6 : q ∈ p :: l := prime_in_list h5.left h1.left h4 have h7 : nondec (p :: l) := h1.right rewrite [nondec_cons] at h7 rewrite [List.mem_cons] at h6 by_cases on h6 · -- Case 1. h6 : q = p linarith done · -- Case 2. h6 : q ∈ l have h8 : ∀ m ∈ l, p ≤ m := h7.left show p ≤ q from h8 q h6 done done lemma nondec_prime_list_tail {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by define at h define rewrite [all_prime_cons, nondec_cons] at h show all_prime l ∧ nondec l from And.intro h.left.right h.right.right done lemma cons_prod_not_one {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : prod (p :: l) ≠ 1 := by define at h have h1 : all_prime (p :: l) := h.left rewrite [all_prime_cons] at h1 rewrite [prod_cons] by_contra h2 show False from (prime_not_one h1.left) (eq_one_of_prod_one h2) done lemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) : l = [] ↔ prod l = 1 := by apply Iff.intro · -- (→) assume h1 : l = [] rewrite [h1] show prod [] = 1 from prod_nil done · -- (←) contrapos assume h1 : ¬l = [] obtain (p : Nat) (h2 : ∃ (L : List Nat), l = p :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List Nat) (h3 : l = p :: L) from h2 rewrite [h3] at h rewrite [h3] show ¬prod (p :: L) = 1 from cons_prod_not_one h done done lemma prime_pos {p : Nat} (h : prime p) : p > 0 := by define at h linarith done theorem Theorem_7_2_5 : ∀ (l1 l2 : List Nat), nondec_prime_list l1 → nondec_prime_list l2 → prod l1 = prod l2 → l1 = l2 := by apply List.rec · -- Base Case. Goal : ∀ (l2 : List Nat), nondec_prime_list [] → -- nondec_prime_list l2 → prod [] = prod l2 → [] = l2 fix l2 : List Nat assume h1 : nondec_prime_list [] assume h2 : nondec_prime_list l2 assume h3 : prod [] = prod l2 rewrite [prod_nil, eq_comm, ←list_nil_iff_prod_one h2] at h3 show [] = l2 from h3.symm done · -- Induction Step fix p : Nat fix L1 : List Nat assume ih : ∀ (L2 : List Nat), nondec_prime_list L1 → nondec_prime_list L2 → prod L1 = prod L2 → L1 = L2 -- Goal : ∀ (l2 : List Nat), nondec_prime_list (p :: L1) → -- nondec_prime_list l2 → prod (p :: L1) = prod l2 → p :: L1 = l2 fix l2 : List Nat assume h1 : nondec_prime_list (p :: L1) assume h2 : nondec_prime_list l2 assume h3 : prod (p :: L1) = prod l2 have h4 : ¬prod (p :: L1) = 1 := cons_prod_not_one h1 rewrite [h3, ←list_nil_iff_prod_one h2] at h4 obtain (q : Nat) (h5 : ∃ (L : List Nat), l2 = q :: L) from List.exists_cons_of_ne_nil h4 obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5 rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2) rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2) have h7 : p ≤ q := first_le_first h1 h2 h3 have h8 : q ≤ p := first_le_first h2 h1 h3.symm have h9 : p = q := by linarith rewrite [h9, prod_cons, prod_cons] at h3 --h3 : q * prod L1 = q * prod L2 have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1 have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2 define at h2 have h12 : all_prime (q :: L2) := h2.left rewrite [all_prime_cons] at h12 have h13 : q > 0 := prime_pos h12.left have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3 have h15 : L1 = L2 := ih L2 h10 h11 h14 rewrite [h6, h9, h15] rfl done done theorem fund_thm_arith (n : Nat) (h : n ≥ 1) : ∃! (l : List Nat), prime_factorization n l := by exists_unique · -- Existence show ∃ (l : List Nat), prime_factorization n l from exists_prime_factorization n h done · -- Uniqueness fix l1 : List Nat; fix l2 : List Nat assume h1 : prime_factorization n l1 assume h2 : prime_factorization n l2 define at h1; define at h2 have h3 : prod l1 = n := h1.right rewrite [←h2.right] at h3 show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3 done done /- Section 7.3 -/ theorem congr_refl (m : Nat) : ∀ (a : Int), a ≡ a (MOD m) := by fix a : Int define --Goal : ∃ (c : Int), a - a = ↑m * c apply Exists.intro 0 ring done theorem congr_symm {m : Nat} : ∀ {a b : Int}, a ≡ b (MOD m) → b ≡ a (MOD m) := by fix a : Int; fix b : Int assume h1 : a ≡ b (MOD m) define at h1 --h1 : ∃ (c : Int), a - b = ↑m * c define --Goal : ∃ (c : Int), b - a = ↑m * c obtain (c : Int) (h2 : a - b = m * c) from h1 apply Exists.intro (-c) show b - a = m * (-c) from calc b - a _ = -(a - b) := by ring _ = -(m * c) := by rw [h2] _ = m * (-c) := by ring done theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry /- Fundamental properties of congruence classes -/ lemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) : X = Y ↔ X.val = Y.val := Fin.ext_iff lemma val_nat_eq_mod (n k : Nat) : ([k]_(n + 1)).val = k % (n + 1) := by rfl lemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl theorem cc_rep {m : Nat} (X : ZMod m) : ∃ (a : Int), X = [a]_m := match m with | 0 => by apply Exists.intro X rfl done | n + 1 => by apply Exists.intro ↑(X.val) have h1 : X.val < n + 1 := Fin.prop X rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1] rfl done theorem add_class (m : Nat) (a b : Int) : [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm theorem mul_class (m : Nat) (a b : Int) : [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm lemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ [a - b]_m = [0]_m := by apply Iff.intro · -- (→) assume h1 : [a]_m = [b]_m have h2 : a - b = a + (-b) := by ring have h3 : b + (-b) = 0 := by ring show [a - b]_m = [0]_m from calc [a - b]_m _ = [a + (-b)]_m := by rw [h2] _ = [a]_m + [-b]_m := by rw [add_class] _ = [b]_m + [-b]_m := by rw [h1] _ = [b + -b]_m := by rw [add_class] _ = [0]_m := by rw [h3] done · -- (←) assume h1 : [a - b]_m = [0]_m have h2 : b + (a - b) = a := by ring have h3 : b + 0 = b := by ring show [a]_m = [b]_m from calc [a]_m _ = [b + (a - b)]_m := by rw [h2] _ = [b]_m + [a - b]_m := by rw [add_class] _ = [b]_m + [0]_m := by rw [h1] _ = [b + 0]_m := by rw [add_class] _ = [b]_m := by rw [h3] done done lemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) : [a]_m = [0]_m → [-a]_m = [0]_m := by assume h1 : [a]_m = [0]_m have h2 : 0 + (-a) = -a := by ring have h3 : a + (-a) = 0 := by ring show [-a]_m = [0]_m from calc [-a]_m _ = [0 + (-a)]_m := by rw [h2] _ = [0]_m + [-a]_m := by rw [add_class] _ = [a]_m + [-a]_m := by rw [h1] _ = [a + (-a)]_m := by rw [add_class] _ = [0]_m := by rw [h3] done lemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) : [-a]_m = [0]_m ↔ [a]_m = [0]_m := by apply Iff.intro _ (cc_neg_zero_of_cc_zero m a) assume h1 : [-a]_m = [0]_m have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1 have h3 : -(-a) = a := by ring rewrite [h3] at h2 show [a]_m = [0]_m from h2 done lemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl lemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m ↔ m ∣ k := match m with | 0 => by have h : (0 : Int) = (↑(0 : Nat) : Int) := by rfl rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj] apply Iff.intro · -- (→) assume h1 : k = 0 rewrite [h1] show 0 ∣ 0 from dvd_self 0 done · -- (←) assume h1 : 0 ∣ k obtain (c : Nat) (h2 : k = 0 * c) from h1 rewrite [h2] ring done done | n + 1 => by rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero] show k % (n + 1) = 0 ↔ n + 1 ∣ k from (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm done lemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m ↔ ↑m ∣ a := by obtain (k : Nat) (h1 : a = ↑k ∨ a = -↑k) from Int.eq_nat_or_neg a by_cases on h1 · -- Case 1. h1: a = ↑k rewrite [h1, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done · -- Case 2. h1: a = -↑k rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done done theorem cc_eq_iff_congr (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ a ≡ b (MOD m) := calc [a]_m = [b]_m _ ↔ [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b _ ↔ ↑m ∣ (a - b) := cc_zero_iff_dvd m (a - b) _ ↔ a ≡ b (MOD m) := by rfl /- End of fundamental properties of congruence classes -/ lemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m := by have h1 : (↑m : Int) ≠ 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m) show 0 ≤ a % m from Int.emod_nonneg a h1 done lemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m) have h2 : (↑m : Int) > 0 := (Nat.cast_pos).rtl h1 show a % m < m from Int.emod_lt_of_pos a h2 done lemma congr_mod_mod (m : Nat) (a : Int) : a ≡ a % m (MOD m) := by define have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m apply Exists.intro (a / m) show a - a % m = m * (a / m) from calc a - (a % m) _ = m * (a / m) + a % m - a % m := by rw [h1] _ = m * (a / m) := by ring done lemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) := And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a)) theorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) : ∃! (r : Int), 0 ≤ r ∧ r < m ∧ a ≡ r (MOD m) := by exists_unique · -- Existence apply Exists.intro (a % m) show 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) from mod_cmpl_res m a done · -- Uniqueness fix r1 : Int; fix r2 : Int assume h1 : 0 ≤ r1 ∧ r1 < m ∧ a ≡ r1 (MOD m) assume h2 : 0 ≤ r2 ∧ r2 < m ∧ a ≡ r2 (MOD m) have h3 : r1 ≡ r2 (MOD m) := congr_trans (congr_symm h1.right.right) h2.right.right obtain (d : Int) (h4 : r1 - r2 = m * d) from h3 have h5 : r1 - r2 < m * 1 := by linarith have h6 : m * (-1) < r1 - r2 := by linarith rewrite [h4] at h5 --h5 : m * d < m * 1 rewrite [h4] at h6 --h6 : m * -1 < m * d have h7 : (↑m : Int) ≥ 0 := Nat.cast_nonneg m have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7 have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7 have h10 : d = 0 := by linarith show r1 = r2 from calc r1 _ = r1 - r2 + r2 := by ring _ = m * 0 + r2 := by rw [h4, h10] _ = r2 := by ring done done lemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m := (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a) theorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y rewrite [h1, h2] have h3 : a + b = b + a := by ring show [a]_m + [b]_m = [b]_m + [a]_m from calc [a]_m + [b]_m _ = [a + b]_m := add_class m a b _ = [b + a]_m := by rw [h3] _ = [b]_m + [a]_m := (add_class m b a).symm done theorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X rewrite [h1] have h2 : a * 1 = a := by ring show [a]_m * [1]_m = [a]_m from calc [a]_m * [1]_m _ = [a * 1]_m := mul_class m a 1 _ = [a]_m := by rw [h2] done theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry lemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) : [a]_m * [gcd_c2 m a]_m = [1]_m := by set s : Int := gcd_c1 m a have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m define at h1 rewrite [h1, Nat.cast_one] at h2 --h2 : s * ↑m + gcd_c2 m a * ↑a = 1 rewrite [mul_class, cc_eq_iff_congr] define --Goal : ∃ (c : Int), ↑a * gcd_c2 m a - 1 = ↑m * c apply Exists.intro (-s) show a * (gcd_c2 m a) - 1 = m * (-s) from calc a * (gcd_c2 m a) - 1 _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring _ = 1 + m * (-s) - 1 := by rw [h2] _ = m * (-s) := by ring done theorem Theorem_7_3_7 (m a : Nat) : invertible [a]_m ↔ rel_prime m a := by apply Iff.intro · -- (→) assume h1 : invertible [a]_m define at h1 obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1 obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y rewrite [h3, mul_class, cc_eq_iff_congr] at h2 define at h2 obtain (c : Int) (h4 : a * b - 1 = m * c) from h2 rewrite [Exercise_7_2_6] --Goal : ∃ (s t : Int), s * ↑m + t * ↑a = 1 apply Exists.intro (-c) apply Exists.intro b show (-c) * m + b * a = 1 from calc (-c) * m + b * a _ = (-c) * m + (a * b - 1) + 1 := by ring _ = (-c) * m + m * c + 1 := by rw [h4] _ = 1 := by ring done · -- (←) assume h1 : rel_prime m a define show ∃ (Y : ZMod m), [a]_m * Y = [1]_m from Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1) done done /- Section 7.4 -/ section Euler open Euler lemma num_rp_below_base {m : Nat} : num_rp_below m 0 = 0 := by rfl lemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) : num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by have h1 : num_rp_below m (j + 1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : gcd m j = 1 rewrite [if_pos h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1 show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1 done lemma num_rp_below_step_not_rp {m j : Nat} (h : ¬rel_prime m j) : num_rp_below m (j + 1) = num_rp_below m j := by have h1 : num_rp_below m (j +1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : ¬gcd m j = 1 rewrite [if_neg h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j show num_rp_below m (j + 1) = num_rp_below m j from h1 done lemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl #eval phi 10 --Answer: 4 lemma prod_inv_iff_inv {m : Nat} {X : ZMod m} (h1 : invertible X) (Y : ZMod m) : invertible (X * Y) ↔ invertible Y := by apply Iff.intro · -- (→) assume h2 : invertible (X * Y) obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2 apply Exists.intro (X * Z) rewrite [←h3] --Goal : Y * (X * Z) = X * Y * Z ring --Note that ring can do algebra in ZMod m done · -- (←) assume h2 : invertible Y obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1 obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2 apply Exists.intro (Xi * Yi) show (X * Y) * (Xi * Yi) = [1]_m from calc X * Y * (Xi * Yi) _ = (X * Xi) * (Y * Yi) := by ring _ = [1]_m * [1]_m := by rw [h3, h4] _ = [1]_m := Theorem_7_3_6_7 [1]_m done done lemma F_rp_def {m i : Nat} (h : rel_prime m i) : F m i = [i]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h --h : gcd m i = 1 rewrite [if_pos h] at h1 show F m i = [i]_m from h1 done lemma F_not_rp_def {m i : Nat} (h : ¬rel_prime m i) : F m i = [1]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h rewrite [h1, if_neg h] rfl done lemma prod_seq_base {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 0 k f = [1]_m := by rfl lemma prod_seq_step {m : Nat} (n k : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl lemma prod_seq_zero_step {m : Nat} (n : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by rewrite [prod_seq_step, zero_add] rfl done lemma prod_one {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 1 k f = f k := by rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7] rfl done lemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl lemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m := calc [G m a i]_m _ = [(a * i) % m]_m := by rfl _ = [a * i]_m := (cc_eq_mod m (a * i)).symm _ = [a]_m * [i]_m := (mul_class m a i).symm lemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) : rel_prime m (G m a i) ↔ rel_prime m i := by have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1 show rel_prime m (G m a i) ↔ rel_prime m i from calc rel_prime m (G m a i) _ ↔ invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm _ ↔ invertible ([a]_m * [i]_m) := by rw [cc_G] _ ↔ invertible [i]_m := prod_inv_iff_inv h2 ([i]_m) _ ↔ rel_prime m i := Theorem_7_3_7 m i done lemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) : F m (G m a i) = [a]_m * F m i := by have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2 show F m (G m a i) = [a]_m * F m i from calc F m (G m a i) _ = [G m a i]_m := F_rp_def h3 _ = [a]_m * [i]_m := cc_G m a i _ = [a]_m * F m i := by rw [F_rp_def h2] done lemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : ¬rel_prime m i) : F m (G m a i) = [1]_m := by rewrite [←G_rp_iff h1 i] at h2 show F m (G m a i) = [1]_m from F_not_rp_def h2 done lemma FG_prod {m a : Nat} (h1 : rel_prime m a) : ∀ (k : Nat), prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by by_induc · -- Base Case show prod_seq 0 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from calc prod_seq 0 0 ((F m) ∘ (G m a)) _ = [1]_m := prod_seq_base _ _ _ = [a]_m ^ 0 * [1]_m := by ring _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by rw [num_rp_below_base, prod_seq_base] done · -- Induction Step fix k : Nat assume ih : prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) by_cases h2 : rel_prime m k · -- Case 1. h2 : rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([a]_m * F m k) := by rw [FG_rp h1 h2] _ = [a]_m ^ ((num_rp_below m k) + 1) * ((prod_seq k 0 (F m)) * F m k) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_rp h2, prod_seq_zero_step] done · -- Case 2. h2 : ¬rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([1]_m) := by rw [FG_not_rp h1 h2] _ = [a]_m ^ (num_rp_below m k) * (prod_seq k 0 (F m) * ([1]_m)) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_not_rp h2, prod_seq_zero_step, F_not_rp_def h2] done done done lemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by define --Goal : ∀ i < m, G m a i < m fix i : Nat assume h1 : i < m rewrite [G_def] --Goal : a * i % m < m show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m) done lemma left_inv_one_one_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g' (g i) = i) : one_one_below n g := sorry lemma right_inv_onto_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g (g' i) = i) (h2 : maps_below n g') : onto_below n g := by define at h2; define fix k : Nat assume h3 : k < n apply Exists.intro (g' k) show g' k < n ∧ g (g' k) = k from And.intro (h2 k h3) (h1 k h3) done lemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m * [inv_mod m a]_m = [1]_m := by have h2 : 0 ≤ (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a) show [a]_m * [inv_mod m a]_m = [1]_m from calc [a]_m * [inv_mod m a]_m _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2] _ = [a]_m * [gcd_c2 m a]_m := by rw [←cc_eq_mod] _ = [1]_m := gcd_c2_inv h1 done lemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m := calc a * (b % m) % m = a % m * (b % m % m) % m := Nat.mul_mod _ _ _ _ = a % m * (b % m) % m := by rw [Nat.mod_mod] _ = a * b % m := (Nat.mul_mod _ _ _).symm lemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm] rfl done theorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] : ↑a ≡ ↑b (MOD m) ↔ a % m = b % m := sorry lemma mul_inv_mod_cancel {m a i : Nat} [NeZero m] (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1 rewrite [mul_class, cc_eq_iff_congr, ←Nat.cast_mul, ←Nat.cast_one, congr_iff_mod_eq_Nat] at h3 show a * inv_mod m a * i % m = i from calc a * (inv_mod m a) * i % m _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = 1 % m * i % m := by rw [h3] _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = i % m := by rw [one_mul] _ = i := Nat.mod_eq_of_lt h2 done lemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl lemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, G m a (Ginv m a i) = i := by fix i : Nat assume h2 : i < m show G m a (Ginv m a i) = i from calc G m a (Ginv m a i) _ = a * ((inv_mod m a * i) % m) % m := by rfl _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, Ginv m a (G m a i) = i := by fix i : Nat assume h2 : i < m show Ginv m a (G m a i) = i from calc Ginv m a (G m a i) _ = inv_mod m a * ((a * i) % m) % m := by rfl _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc, mul_comm (inv_mod m a)] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_maps_below (m a : Nat) [NeZero m] : maps_below m (Ginv m a) := G_maps_below m (inv_mod m a) lemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : one_one_below m (G m a) := left_inv_one_one_below (Ginv_left_inv h1) lemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : onto_below m (G m a) := right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a) lemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : perm_below m (G m a) := And.intro (G_maps_below m a) (And.intro (G_one_one_below h1) (G_onto_below h1)) --Permuting a product of congruence classes doesn't change product lemma swap_fst (u v : Nat) : swap u v u = v := by define : swap u v u --Goal : (if u = u then v else if u = v then u else u) = v have h : u = u := by rfl rewrite [if_pos h] rfl done lemma swap_snd (u v : Nat) : swap u v v = u := by define : swap u v v by_cases h1 : v = u · -- Case 1. h1 : v = u rewrite [if_pos h1] show v = u from h1 done · -- Case 2. h1 : v ≠ u rewrite [if_neg h1] have h2 : v = v := by rfl rewrite [if_pos h2] rfl done done lemma swap_other {u v i : Nat} (h1 : i ≠ u) (h2 : i ≠ v) : swap u v i = i := by define : swap u v i rewrite [if_neg h1, if_neg h2] rfl done lemma swap_values (u v i : Nat) : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := by by_cases h1 : i = u · -- Case 1. h1 : i = u apply Or.inl rewrite [h1] show swap u v u = v from swap_fst u v done · -- Case 2. h1 : i ≠ u apply Or.inr by_cases h2 : i = v · -- Case 2.1. h2 : i = v apply Or.inl rewrite [h2] show swap u v v = u from swap_snd u v done · -- Case 2.2. h2 : i ≠ v apply Or.inr show swap u v i = i from swap_other h1 h2 done done done lemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by define fix i : Nat assume h3 : i < n have h4 : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := swap_values u v i by_cases on h4 · -- Case 1. h4 : swap u v i = v rewrite [h4] show v < n from h2 done · -- Case 2. by_cases on h4 · -- Case 2.1. h4 : swap u v i = u rewrite [h4] show u < n from h1 done · -- Case 2.2. h4 : swap u v i = i rewrite [h4] show i < n from h3 done done done lemma swap_swap (u v n : Nat) : ∀ i < n, swap u v (swap u v i) = i := by fix i : Nat assume h : i < n by_cases h1 : i = u · -- Case 1. h1 : i = u rewrite [h1, swap_fst, swap_snd] rfl done · -- Case 2. h1 : i ≠ u by_cases h2 : i = v · -- Case 2.1. h2 : i = v rewrite [h2, swap_snd, swap_fst] rfl done · -- Case 2.2. h2 : i ≠ v rewrite [swap_other h1 h2, swap_other h1 h2] rfl done done done lemma swap_one_one_below (u v n) : one_one_below n (swap u v) := left_inv_one_one_below (swap_swap u v n) lemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) := right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2) lemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) := And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2)) lemma comp_perm_below {n : Nat} {f g : Nat → Nat} (h1 : perm_below n f) (h2 : perm_below n g) : perm_below n (f ∘ g) := sorry lemma trivial_swap (u : Nat) : swap u u = id := by apply funext fix x : Nat by_cases h1 : x = u · -- Case 1. h1 : x = u rewrite [h1, swap_fst] rfl done · -- Case 2. h1 : x ≠ u rewrite [swap_other h1 h1] rfl done done lemma prod_eq_fun {m : Nat} (f g : Nat → ZMod m) (k : Nat) : ∀ (n : Nat), (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g := by by_induc · -- Base Case assume h : (∀ i < 0, f (k + i) = g (k + i)) rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g assume h1 : ∀ i < n + 1, f (k + i) = g (k + i) have h2 : ∀ i < n, f (k + i) = g (k + i) := by fix i : Nat assume h2 : i < n have h3 : i < n + 1 := by linarith show f (k + i) = g (k + i) from h1 i h3 done have h3 : prod_seq n k f = prod_seq n k g := ih h2 have h4 : n < n + 1 := Nat.lt_succ_self n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4] rfl done done lemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := by have h2 : ∀ (i : Nat), i < u → (f ∘ swap u n) (0 + i) = f (0 + i) := by fix i : Nat assume h2 : i < u have h3 : 0 + i ≠ u := by linarith have h4 : 0 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl done show prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f from prod_eq_fun (f ∘ swap u n) f 0 u h2 done lemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := by have h2 : ∀ i < j, (f ∘ swap u n) (u + 1 + i) = f (u + 1 + i) := by fix i : Nat assume h2 : i < j have h3 : u + 1 + i ≠ u := by linarith have h4 : u + 1 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl show prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f from prod_eq_fun (f ∘ swap u n) f (u + 1) j h2 done lemma break_prod {m : Nat} (n : Nat) (f : Nat → ZMod m) : ∀ (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by by_induc · -- Base Case have h : n + 0 = n := by rfl rewrite [prod_seq_base, h, Theorem_7_3_6_7] rfl done · -- Induction Step fix j : Nat assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f rewrite [←add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc] rfl done done lemma break_prod_twice {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f := break_prod n f 1 rewrite [prod_one] at h2 have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f := break_prod (u + 1) f j rewrite [←h1] at h3 have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f := break_prod u f 1 rewrite [prod_one] at h4 rewrite [h3, h4] at h2 show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2 done lemma swap_prod_eq_prod {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f := by by_cases h2 : u = n · -- Case 1. h2 : u = n rewrite [h2, trivial_swap n] --Goal : prod_seq (n + 1) 0 (f ∘ id) = prod_seq (n + 1) 0 f rfl done · -- Case 2. h2 : ¬u = n have h3 : u + 1 ≤ n := Nat.lt_of_le_of_ne h1 h2 obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3 have break_f : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := break_prod_twice f h4 have break_fs : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_prod_twice (f ∘ swap u n) h4 have f_eq_fs_below : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := swap_prod_eq_prod_below f h1 have f_eq_fs_btwn : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4 show prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f from calc prod_seq (n + 1) 0 (f ∘ swap u n) _ = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_fs _ = prod_seq u 0 f * (f ∘ swap u n) u * prod_seq j (u + 1) f * (f ∘ swap u n) n := by rw [f_eq_fs_below, f_eq_fs_btwn] _ = prod_seq u 0 f * f (swap u n u) * prod_seq j (u + 1) f * f (swap u n n) := by rfl _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by rw [swap_fst, swap_snd] _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring _ = prod_seq (n + 1) 0 f := break_f.symm done done lemma perm_below_fixed {n : Nat} {g : Nat → Nat} (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry lemma perm_prod {m : Nat} (f : Nat → ZMod m) : ∀ (n : Nat), ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) := by by_induc · -- Base Case fix g : Nat → Nat assume h1 : perm_below 0 g rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) fix g : Nat → Nat assume g_pb : perm_below (n + 1) g define at g_pb have g_ob : onto_below (n + 1) g := g_pb.right.right define at g_ob have h1 : n < n + 1 := by linarith obtain (u : Nat) (h2 : u < n + 1 ∧ g u = n) from g_ob n h1 have s_pb : perm_below (n + 1) (swap u n) := swap_perm_below h2.left h1 have gs_pb_n1 : perm_below (n + 1) (g ∘ swap u n) := comp_perm_below g_pb s_pb have gs_fix_n : (g ∘ swap u n) n = n := calc (g ∘ swap u n) n _ = g (swap u n n) := by rfl _ = g u := by rw [swap_snd] _ = n := h2.right have gs_pb_n : perm_below n (g ∘ swap u n) := perm_below_fixed gs_pb_n1 gs_fix_n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f ∘ (g ∘ swap u n)) := ih (g ∘ swap u n) gs_pb_n have h3 : u ≤ n := by linarith show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f ∘ g) from calc prod_seq (n + 1) 0 f _ = prod_seq n 0 f * f n := prod_seq_zero_step n f _ = prod_seq n 0 (f ∘ (g ∘ swap u n)) * f ((g ∘ swap u n) n) := by rw [gs_prod, gs_fix_n] _ = prod_seq n 0 (f ∘ g ∘ swap u n) * (f ∘ g ∘ swap u n) n := by rfl _ = prod_seq (n + 1) 0 (f ∘ g ∘ swap u n) := (prod_seq_zero_step n (f ∘ g ∘ swap u n)).symm _ = prod_seq (n + 1) 0 ((f ∘ g) ∘ swap u n) := by rfl _ = prod_seq (n + 1) 0 (f ∘ g) := swap_prod_eq_prod (f ∘ g) h3 done done lemma F_invertible (m i : Nat) : invertible (F m i) := by by_cases h : rel_prime m i · -- Case 1. h : rel_prime m i rewrite [F_rp_def h] show invertible [i]_m from (Theorem_7_3_7 m i).rtl h done · -- Case 2. h : ¬rel_prime m i rewrite [F_not_rp_def h] apply Exists.intro [1]_m show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m done done lemma Fprod_invertible (m : Nat) : ∀ (k : Nat), invertible (prod_seq k 0 (F m)) := by by_induc · -- Base Case apply Exists.intro [1]_m show prod_seq 0 0 (F m) * [1]_m = [1]_m from calc prod_seq 0 0 (F m) * [1]_m _ = [1]_m * [1]_m := by rw [prod_seq_base] _ = [1]_m := Theorem_7_3_6_7 ([1]_m) done · -- Induction Step fix k : Nat assume ih : invertible (prod_seq k 0 (F m)) rewrite [prod_seq_zero_step] show invertible (prod_seq k 0 (F m) * (F m k)) from (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k) done done theorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m ^ (phi m) = [1]_m := by have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2 show [a]_m ^ (phi m) = [1]_m from calc [a]_m ^ (phi m) _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3] _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring _ = prod_seq m 0 (F m ∘ G m a) * Y := by rw [FG_prod h1 m, phi_def] _ = prod_seq m 0 (F m) * Y := by rw [perm_prod (F m) m (G m a) (G_perm_below h1)] _ = [1]_m := by rw [h3] done lemma Exercise_7_4_5_Int (m : Nat) (a : Int) : ∀ (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry lemma Exercise_7_4_5_Nat (m a n : Nat) : [a]_m ^ n = [a ^ n]_m := by rewrite [Exercise_7_4_5_Int] rfl done theorem Euler's_theorem {m a : Nat} [NeZero m] (h1 : rel_prime m a) : a ^ (phi m) ≡ 1 (MOD m) := by have h2 : [a]_m ^ (phi m) = [1]_m := Theorem_7_4_2 h1 rewrite [Exercise_7_4_5_Nat m a (phi m)] at h2 --h2 : [a ^ phi m]_m = [1]_m show a ^ (phi m) ≡ 1 (MOD m) from (cc_eq_iff_congr _ _ _).ltr h2 done #eval gcd 10 7 --Answer: 1. So 10 and 7 are relatively prime #eval 7 ^ phi 10 --Answer: 2401, which is congruent to 1 mod 10. end Euler /- Section 7.5 -/ lemma num_rp_prime {p : Nat} (h1 : prime p) : ∀ k < p, num_rp_below p (k + 1) = k := sorry lemma phi_prime {p : Nat} (h1 : prime p) : phi p = p - 1 := by have h2 : 1 ≤ p := prime_pos h1 have h3 : p - 1 + 1 = p := Nat.sub_add_cancel h2 have h4 : p - 1 < p := by linarith have h5 : num_rp_below p (p - 1 + 1) = p - 1 := num_rp_prime h1 (p - 1) h4 rewrite [h3] at h5 show phi p = p - 1 from h5 done theorem Theorem_7_2_2_Int {a c : Nat} {b : Int} (h1 : ↑c ∣ ↑a * b) (h2 : rel_prime a c) : ↑c ∣ b := by rewrite [Int.natCast_dvd, Int.natAbs_mul, Int.natAbs_ofNat] at h1 --h1 : c ∣ a * Int.natAbs b rewrite [Int.natCast_dvd] --Goal : c ∣ Int.natAbs b show c ∣ Int.natAbs b from Theorem_7_2_2 h1 h2 done lemma Lemma_7_4_5 {m n : Nat} (a b : Int) (h1 : rel_prime m n) : a ≡ b (MOD m * n) ↔ a ≡ b (MOD m) ∧ a ≡ b (MOD n) := by apply Iff.intro · -- (→) assume h2 : a ≡ b (MOD m * n) obtain (j : Int) (h3 : a - b = (m * n) * j) from h2 apply And.intro · -- Proof of a ≡ b (MOD m) apply Exists.intro (n * j) show a - b = m * (n * j) from calc a - b _ = m * n * j := h3 _ = m * (n * j) := by ring done · -- Proof of a ≡ b (MOD n) apply Exists.intro (m * j) show a - b = n * (m * j) from calc a - b _ = m * n * j := h3 _ = n * (m * j) := by ring done done · -- (←) assume h2 : a ≡ b (MOD m) ∧ a ≡ b (MOD n) obtain (j : Int) (h3 : a - b = m * j) from h2.left have h4 : (↑n : Int) ∣ a - b := h2.right rewrite [h3] at h4 --h4 : ↑n ∣ ↑m * j have h5 : ↑n ∣ j := Theorem_7_2_2_Int h4 h1 obtain (k : Int) (h6 : j = n * k) from h5 apply Exists.intro k --Goal : a - b = ↑(m * n) * k rewrite [Nat.cast_mul] --Goal : a - b = ↑m * ↑n * k show a - b = (m * n) * k from calc a - b _ = m * j := h3 _ = m * (n * k) := by rw [h6] _ = (m * n) * k := by ring done done --From exercises of Section 7.2 theorem rel_prime_symm {a b : Nat} (h : rel_prime a b) : rel_prime b a := sorry lemma prime_NeZero {p : Nat} (h : prime p) : NeZero p := by rewrite [neZero_iff] --Goal : p ≠ 0 define at h linarith done lemma Lemma_7_5_1 {p e d m c s : Nat} {t : Int} (h1 : prime p) (h2 : e * d = (p - 1) * s + 1) (h3 : m ^ e - c = p * t) : c ^ d ≡ m (MOD p) := by have h4 : m ^ e ≡ c (MOD p) := Exists.intro t h3 have h5 : [m ^ e]_p = [c]_p := (cc_eq_iff_congr _ _ _).rtl h4 rewrite [←Exercise_7_4_5_Nat] at h5 --h5 : [m]_p ^ e = [c]_p by_cases h6 : p ∣ m · -- Case 1. h6 : p ∣ m have h7 : m ≡ 0 (MOD p) := by obtain (j : Nat) (h8 : m = p * j) from h6 apply Exists.intro (↑j : Int) --Goal : ↑m - 0 = ↑p * ↑j rewrite [h8, Nat.cast_mul] ring done have h8 : [m]_p = [0]_p := (cc_eq_iff_congr _ _ _).rtl h7 have h9 : e * d ≠ 0 := by rewrite [h2] show (p - 1) * s + 1 ≠ 0 from Nat.add_one_ne_zero _ done have h10 : (0 : Int) ^ (e * d) = 0 := zero_pow h9 have h11 : [c ^ d]_p = [m]_p := calc [c ^ d]_p _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat] _ = ([m]_p ^ e) ^ d := by rw [h5] _ = [m]_p ^ (e * d) := by ring _ = [0]_p ^ (e * d) := by rw [h8] _ = [0 ^ (e * d)]_p := Exercise_7_4_5_Int _ _ _ _ = [0]_p := by rw [h10] _ = [m]_p := by rw [h8] show c ^ d ≡ m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11 done · -- Case 2. h6 : ¬p ∣ m have h7 : rel_prime m p := rel_prime_of_prime_not_dvd h1 h6 have h8 : rel_prime p m := rel_prime_symm h7 have h9 : NeZero p := prime_NeZero h1 have h10 : (1 : Int) ^ s = 1 := by ring have h11 : [c ^ d]_p = [m]_p := calc [c ^ d]_p _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat] _ = ([m]_p ^ e) ^ d := by rw [h5] _ = [m]_p ^ (e * d) := by ring _ = [m]_p ^ ((p - 1) * s + 1) := by rw [h2] _ = ([m]_p ^ (p - 1)) ^ s * [m]_p := by ring _ = ([m]_p ^ (phi p)) ^ s * [m]_p := by rw [phi_prime h1] _ = [1]_p ^ s * [m]_p := by rw [Theorem_7_4_2 h8] _ = [1 ^ s]_p * [m]_p := by rw [Exercise_7_4_5_Int] _ = [1]_p * [m]_p := by rw [h10] _ = [m]_p * [1]_p := by ring _ = [m]_p := Theorem_7_3_6_7 _ show c ^ d ≡ m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11 done done theorem Theorem_7_5_1 (p q n e d k m c : Nat) (p_prime : prime p) (q_prime : prime q) (p_ne_q : p ≠ q) (n_pq : n = p * q) (ed_congr_1 : e * d = k * (p - 1) * (q - 1) + 1) (h1 : [m]_n ^ e = [c]_n) : [c]_n ^ d = [m]_n := by rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] at h1 --h1 : m ^ e ≡ c (MOD n) rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] --Goal : c ^ d ≡ m (MOD n) obtain (j : Int) (h2 : m ^ e - c = n * j) from h1 rewrite [n_pq, Nat.cast_mul] at h2 --h2 : m ^ e - c = p * q * j have h3 : e * d = (p - 1) * (k * (q - 1)) + 1 := by rewrite [ed_congr_1] ring done have h4 : m ^ e - c = p * (q * j) := by rewrite [h2] ring done have congr_p : c ^ d ≡ m (MOD p) := Lemma_7_5_1 p_prime h3 h4 have h5 : e * d = (q - 1) * (k * (p - 1)) + 1 := by rewrite [ed_congr_1] ring done have h6 : m ^ e - c = q * (p * j) := by rewrite [h2] ring done have congr_q : c ^ d ≡ m (MOD q) := Lemma_7_5_1 q_prime h5 h6 have h7 : ¬q ∣ p := by by_contra h8 have h9 : q = 1 ∨ q = p := dvd_prime p_prime h8 disj_syll h9 (prime_not_one q_prime) show False from p_ne_q h9.symm done have h8 : rel_prime p q := rel_prime_of_prime_not_dvd q_prime h7 rewrite [n_pq, Lemma_7_4_5 _ _ h8] show c ^ d ≡ m (MOD p) ∧ c ^ d ≡ m (MOD q) from And.intro congr_p congr_q done /- BEGIN EXERCISES -/ namespace Exercises /- Section 7.1 -/ -- 1. theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry -- 2. lemma gcd_comm_lt {a b : Nat} (h : a < b) : gcd a b = gcd b a := sorry theorem gcd_comm (a b : Nat) : gcd a b = gcd b a := sorry -- 3. theorem Exercise_7_1_5 (a b : Nat) (n : Int) : (∃ (s t : Int), s * a + t * b = n) ↔ (↑(gcd a b) : Int) ∣ n := sorry -- 4. theorem Exercise_7_1_6 (a b c : Nat) : gcd a b = gcd (a + b * c) b := sorry -- 5. theorem gcd_is_nonzero {a b : Nat} (h : a ≠ 0 ∨ b ≠ 0) : gcd a b ≠ 0 := sorry -- 6. theorem gcd_greatest {a b d : Nat} (h1 : gcd a b ≠ 0) (h2 : d ∣ a) (h3 : d ∣ b) : d ≤ gcd a b := sorry -- 7. lemma Lemma_7_1_10a {a b : Nat} (n : Nat) (h : a ∣ b) : (n * a) ∣ (n * b) := sorry lemma Lemma_7_1_10b {a b n : Nat} (h1 : n ≠ 0) (h2 : (n * a) ∣ (n * b)) : a ∣ b := sorry

lemma Lemma_7_1_10c {a b : Nat} (h1 : a ∣ b) (h2 : b ∣ a) : a = b

HTPI.Exercises.Lemma_7_1_10c

htpi/HTPILib/Chap7.lean

htpi

/- Copyright 2023 Daniel J. Velleman -/ import HTPILib.Chap6 namespace HTPI /- Definitions -/ lemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by have h : n + 1 > 0 := Nat.succ_pos n show a % (n + 1) < n + 1 from Nat.mod_lt a h done def gcd (a b : Nat) : Nat := match b with | 0 => a | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd (n + 1) (a % (n + 1)) termination_by b mutual def gcd_c1 (a b : Nat) : Int := match b with | 0 => 1 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c2 (n + 1) (a % (n + 1)) --Corresponds to s = t' termination_by b def gcd_c2 (a b : Nat) : Int := match b with | 0 => 0 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c1 (n + 1) (a % (n + 1)) - (gcd_c2 (n + 1) (a % (n + 1))) * ↑(a / (n + 1)) --Corresponds to t = s' - t'q termination_by b end def prime (n : Nat) : Prop := 2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n def prime_factor (p n : Nat) : Prop := prime p ∧ p ∣ n def all_prime (l : List Nat) : Prop := ∀ p ∈ l, prime p def nondec (l : List Nat) : Prop := match l with | [] => True --Of course, True is a proposition that is always true | n :: L => (∀ m ∈ L, n ≤ m) ∧ nondec L def nondec_prime_list (l : List Nat) : Prop := all_prime l ∧ nondec l def prod (l : List Nat) : Nat := match l with | [] => 1 | n :: L => n * (prod L) def prime_factorization (n : Nat) (l : List Nat) : Prop := nondec_prime_list l ∧ prod l = n def rel_prime (a b : Nat) : Prop := gcd a b = 1 def congr_mod (m : Nat) (a b : Int) : Prop := (↑m : Int) ∣ (a - b) def cc (m : Nat) (a : Int) : ZMod m := (↑a : ZMod m) notation:50 a " ≡ " b " (MOD " m ")" => congr_mod m a b notation:max "["a"]_"m:max => cc m a def invertible {m : Nat} (X : ZMod m) : Prop := ∃ (Y : ZMod m), X * Y = [1]_m def num_rp_below (m k : Nat) : Nat := match k with | 0 => 0 | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j def phi (m : Nat) : Nat := num_rp_below m m def prod_seq {m : Nat} (j k : Nat) (f : Nat → ZMod m) : ZMod m := match j with | 0 => [1]_m | n + 1 => prod_seq n k f * f (k + n) def maps_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i < n, g i < n def one_one_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i1 < n, ∀ i2 < n, g i1 = g i2 → i1 = i2 def onto_below (n : Nat) (g : Nat → Nat) : Prop := ∀ k < n, ∃ i < n, g i = k def perm_below (n : Nat) (g : Nat → Nat) : Prop := maps_below n g ∧ one_one_below n g ∧ onto_below n g def inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m) def swap (u v i : Nat) : Nat := if i = u then v else if i = v then u else i namespace Euler --For definitions specific to Euler's theorem def F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m def G (m a i : Nat) : Nat := (a * i) % m def Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i end Euler /- Section 7.1 -/ theorem dvd_mod_of_dvd_a_b {a b d : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ (a % b) := by set q : Nat := a / b have h3 : b * q + a % b = a := Nat.div_add_mod a b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 define --Goal : ∃ (c : Nat), a % b = d * c apply Exists.intro (j - k * q) show a % b = d * (j - k * q) from calc a % b _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm _ = a - b * q := by rw [h3] _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc] _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm done theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry #eval gcd 672 161 --Answer: 7 lemma gcd_base (a : Nat) : gcd a 0 = a := by rfl lemma gcd_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd a b = gcd b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1)) rfl done lemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b ≠ 0) : a % b < b := by have h1 : b > 0 := Nat.pos_of_ne_zero h show a % b < b from Nat.mod_lt a h1 done lemma dvd_self (n : Nat) : n ∣ n := by apply Exists.intro 1 ring done theorem gcd_dvd : ∀ (b a : Nat), (gcd a b) ∣ a ∧ (gcd a b) ∣ b := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd a b_1) ∣ a ∧ (gcd a b_1) ∣ b_1 fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_base] --Goal: a ∣ a ∧ a ∣ 0 apply And.intro (dvd_self a) define apply Exists.intro 0 rfl done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_nonzero a h1] --Goal : gcd b (a % b) ∣ a ∧ gcd b (a % b) ∣ b have h2 : a % b < b := mod_nonzero_lt a h1 have h3 : (gcd b (a % b)) ∣ b ∧ (gcd b (a % b)) ∣ (a % b) := ih (a % b) h2 b apply And.intro _ h3.left show (gcd b (a % b)) ∣ a from dvd_a_of_dvd_b_mod h3.left h3.right done done theorem gcd_dvd_left (a b : Nat) : (gcd a b) ∣ a := (gcd_dvd b a).left theorem gcd_dvd_right (a b : Nat) : (gcd a b) ∣ b := (gcd_dvd b a).right lemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl lemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c1 a b = gcd_c2 b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done lemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl lemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * ↑(a / b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done theorem gcd_lin_comb : ∀ (b a : Nat), (gcd_c1 a b) * ↑a + (gcd_c2 a b) * ↑b = ↑(gcd a b) := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd_c1 a b_1) * ↑a + (gcd_c2 a b_1) * ↑b_1 = ↑(gcd a b_1) fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base] --Goal : 1 * ↑a + 0 * ↑0 = ↑a ring done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1] --Goal : gcd_c2 b (a % b) * ↑a + -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * ↑(a / b)) * ↑b = -- ↑(gcd b (a % b)) set r : Nat := a % b set q : Nat := a / b set s : Int := gcd_c1 b r set t : Int := gcd_c2 b r --Goal : t * ↑a + (s - t * ↑q) * ↑b = ↑(gcd b r) have h2 : r < b := mod_nonzero_lt a h1 have h3 : s * ↑b + t * ↑r = ↑(gcd b r) := ih r h2 b have h4 : b * q + r = a := Nat.div_add_mod a b rewrite [←h3, ←h4] rewrite [Nat.cast_add, Nat.cast_mul] --Goal : t * (↑b * ↑q + ↑r) + (s - t * ↑q) * ↑b = s * ↑b + t * ↑r ring done done #eval gcd_c1 672 161 --Answer: 6 #eval gcd_c2 672 161 --Answer: -25 --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161 theorem Theorem_7_1_6 {d a b : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ gcd a b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑d ∣ ↑(gcd a b) set s : Int := gcd_c1 a b set t : Int := gcd_c2 a b have h3 : s * ↑a + t * ↑b = ↑(gcd a b) := gcd_lin_comb b a rewrite [←h3] --Goal : ↑d ∣ s * ↑a + t * ↑b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul] --Goal : ↑d ∣ s * (↑d * ↑j) + t * (↑d * ↑k) define apply Exists.intro (s * ↑j + t * ↑k) ring done /- Section 7.2 -/ theorem dvd_trans {a b c : Nat} (h1 : a ∣ b) (h2 : b ∣ c) : a ∣ c := by define at h1; define at h2; define obtain (m : Nat) (h3 : b = a * m) from h1 obtain (n : Nat) (h4 : c = b * n) from h2 rewrite [h3, mul_assoc] at h4 apply Exists.intro (m * n) show c = a * (m * n) from h4 done lemma exists_prime_factor : ∀ (n : Nat), 2 ≤ n → ∃ (p : Nat), prime_factor p n := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, 2 ≤ n_1 → ∃ (p : Nat), prime_factor p n_1 assume h1 : 2 ≤ n by_cases h2 : prime n · -- Case 1. h2 : prime n apply Exists.intro n define --Goal : prime n ∧ n ∣ n show prime n ∧ n ∣ n from And.intro h2 (dvd_self n) done · -- Case 2. h2 : ¬prime n define at h2 --h2 : ¬(2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n) demorgan at h2 disj_syll h2 h1 obtain (a : Nat) (h3 : ∃ (b : Nat), a * b = n ∧ a < n ∧ b < n) from h2 obtain (b : Nat) (h4 : a * b = n ∧ a < n ∧ b < n) from h3 have h5 : 2 ≤ a := by by_contra h6 have h7 : a ≤ 1 := by linarith have h8 : n ≤ b := calc n _ = a * b := h4.left.symm _ ≤ 1 * b := by rel [h7] _ = b := by ring linarith --n ≤ b contradicts b < n done have h6 : ∃ (p : Nat), prime_factor p a := ih a h4.right.left h5 obtain (p : Nat) (h7 : prime_factor p a) from h6 apply Exists.intro p define --Goal : prime p ∧ p ∣ n define at h7 --h7 : prime p ∧ p ∣ a apply And.intro h7.left have h8 : a ∣ n := by apply Exists.intro b show n = a * b from (h4.left).symm done show p ∣ n from dvd_trans h7.right h8 done done lemma exists_least_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q := by set S : Set Nat := {p : Nat | prime_factor p n} have h2 : ∃ (p : Nat), p ∈ S := exists_prime_factor n h show ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q from well_ord_princ S h2 done lemma all_prime_nil : all_prime [] := by define --Goal : ∀ p ∈ [], prime p fix p : Nat contrapos --Goal : ¬prime p → p ∉ [] assume h1 : ¬prime p show p ∉ [] from List.not_mem_nil p done lemma all_prime_cons (n : Nat) (L : List Nat) : all_prime (n :: L) ↔ prime n ∧ all_prime L := by apply Iff.intro · -- (→) assume h1 : all_prime (n :: L) --Goal : prime n ∧ all_prime L define at h1 --h1 : ∀ p ∈ n :: L, prime p apply And.intro (h1 n (List.mem_cons_self n L)) define --Goal : ∀ p ∈ L, prime p fix p : Nat assume h2 : p ∈ L show prime p from h1 p (List.mem_cons_of_mem n h2) done · -- (←) assume h1 : prime n ∧ all_prime L --Goal : all_prime (n :: l) define : all_prime L at h1 define fix p : Nat assume h2 : p ∈ n :: L rewrite [List.mem_cons] at h2 --h2 : p = n ∨ p ∈ L by_cases on h2 · -- Case 1. h2 : p = n rewrite [h2] show prime n from h1.left done · -- Case 2. h2 : p ∈ L show prime p from h1.right p h2 done done done lemma nondec_nil : nondec [] := by define --Goal : True trivial --trivial proves some obviously true statements, such as True done lemma nondec_cons (n : Nat) (L : List Nat) : nondec (n :: L) ↔ (∀ m ∈ L, n ≤ m) ∧ nondec L := by rfl lemma prod_nil : prod [] = 1 := by rfl lemma prod_cons : prod (n :: L) = n * (prod L) := by rfl lemma exists_cons_of_length_eq_succ {A : Type} {l : List A} {n : Nat} (h : l.length = n + 1) : ∃ (a : A) (L : List A), l = a :: L ∧ L.length = n := by have h1 : ¬l.length = 0 := by linarith rewrite [List.length_eq_zero] at h1 obtain (a : A) (h2 : ∃ (L : List A), l = a :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List A) (h3 : l = a :: L) from h2 apply Exists.intro a apply Exists.intro L apply And.intro h3 have h4 : (a :: L).length = L.length + 1 := List.length_cons a L rewrite [←h3, h] at h4 show L.length = n from (Nat.add_right_cancel h4).symm done lemma list_elt_dvd_prod_by_length (a : Nat) : ∀ (n : Nat), ∀ (l : List Nat), l.length = n → a ∈ l → a ∣ prod l := by by_induc · --Base Case fix l : List Nat assume h1 : l.length = 0 rewrite [List.length_eq_zero] at h1 --h1 : l = [] rewrite [h1] --Goal : a ∈ [] → a ∣ prod [] contrapos assume h2 : ¬a ∣ prod [] show a ∉ [] from List.not_mem_nil a done · -- Induction Step fix n : Nat assume ih : ∀ (l : List Nat), List.length l = n → a ∈ l → a ∣ prod l fix l : List Nat assume h1 : l.length = n + 1 --Goal : a ∈ l → a ∣ prod l obtain (b : Nat) (h2 : ∃ (L : List Nat), l = b :: L ∧ L.length = n) from exists_cons_of_length_eq_succ h1 obtain (L : List Nat) (h3 : l = b :: L ∧ L.length = n) from h2 have h4 : a ∈ L → a ∣ prod L := ih L h3.right assume h5 : a ∈ l rewrite [h3.left, prod_cons] --Goal : a ∣ b * prod L rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b ∨ a ∈ L by_cases on h5 · -- Case 1. h5 : a = b apply Exists.intro (prod L) rewrite [h5] rfl done · -- Case 2. h5 : a ∈ L have h6 : a ∣ prod L := h4 h5 have h7 : prod L ∣ b * prod L := by apply Exists.intro b ring done show a ∣ b * prod L from dvd_trans h6 h7 done done done lemma list_elt_dvd_prod {a : Nat} {l : List Nat} (h : a ∈ l) : a ∣ prod l := by set n : Nat := l.length have h1 : l.length = n := by rfl show a ∣ prod l from list_elt_dvd_prod_by_length a n l h1 h done lemma exists_prime_factorization : ∀ (n : Nat), n ≥ 1 → ∃ (l : List Nat), prime_factorization n l := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, n_1 ≥ 1 → ∃ (l : List Nat), prime_factorization n_1 l assume h1 : n ≥ 1 by_cases h2 : n = 1 · -- Case 1. h2 : n = 1 apply Exists.intro [] define apply And.intro · -- Proof of nondec_prime_list [] define show all_prime [] ∧ nondec [] from And.intro all_prime_nil nondec_nil done · -- Proof of prod [] = n rewrite [prod_nil, h2] rfl done done · -- Case 2. h2 : n ≠ 1 have h3 : n ≥ 2 := lt_of_le_of_ne' h1 h2 obtain (p : Nat) (h4 : prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q) from exists_least_prime_factor h3 have p_prime_factor : prime_factor p n := h4.left define at p_prime_factor have p_prime : prime p := p_prime_factor.left have p_dvd_n : p ∣ n := p_prime_factor.right have p_least : ∀ (q : Nat), prime_factor q n → p ≤ q := h4.right obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n have h5 : m ≠ 0 := by contradict h1 with h6 have h7 : n = 0 := calc n _ = p * m := n_eq_pm _ = p * 0 := by rw [h6] _ = 0 := by ring rewrite [h7] decide done have m_pos : 0 < m := Nat.pos_of_ne_zero h5 have m_lt_n : m < n := by define at p_prime show m < n from calc m _ < m + m := by linarith _ = 2 * m := by ring _ ≤ p * m := by rel [p_prime.left] _ = n := n_eq_pm.symm done obtain (L : List Nat) (h6 : prime_factorization m L) from ih m m_lt_n m_pos define at h6 have ndpl_L : nondec_prime_list L := h6.left define at ndpl_L apply Exists.intro (p :: L) define apply And.intro · -- Proof of nondec_prime_list (p :: L) define apply And.intro · -- Proof of all_prime (p :: L) rewrite [all_prime_cons] show prime p ∧ all_prime L from And.intro p_prime ndpl_L.left done · -- Proof of nondec (p :: L) rewrite [nondec_cons] apply And.intro _ ndpl_L.right fix q : Nat assume q_in_L : q ∈ L have h7 : q ∣ prod L := list_elt_dvd_prod q_in_L rewrite [h6.right] at h7 --h7 : q ∣ m have h8 : m ∣ n := by apply Exists.intro p rewrite [n_eq_pm] ring done have q_dvd_n : q ∣ n := dvd_trans h7 h8 have ap_L : all_prime L := ndpl_L.left define at ap_L have q_prime_factor : prime_factor q n := And.intro (ap_L q q_in_L) q_dvd_n show p ≤ q from p_least q q_prime_factor done done · -- Proof of prod (p :: L) = n rewrite [prod_cons, h6.right, n_eq_pm] rfl done done done theorem Theorem_7_2_2 {a b c : Nat} (h1 : c ∣ a * b) (h2 : rel_prime a c) : c ∣ b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑c ∣ ↑b define at h1; define at h2; define obtain (j : Nat) (h3 : a * b = c * j) from h1 set s : Int := gcd_c1 a c set t : Int := gcd_c2 a c have h4 : s * ↑a + t * ↑c = ↑(gcd a c) := gcd_lin_comb c a rewrite [h2, Nat.cast_one] at h4 --h4 : s * ↑a + t * ↑c = (1 : Int) apply Exists.intro (s * ↑j + t * ↑b) show ↑b = ↑c * (s * ↑j + t * ↑b) from calc ↑b _ = (1 : Int) * ↑b := (one_mul _).symm _ = (s * ↑a + t * ↑c) * ↑b := by rw [h4] _ = s * (↑a * ↑b) + t * ↑c * ↑b := by ring _ = s * (↑c * ↑j) + t * ↑c * ↑b := by rw [←Nat.cast_mul a b, h3, Nat.cast_mul c j] _ = ↑c * (s * ↑j + t * ↑b) := by ring done lemma le_nonzero_prod_left {a b : Nat} (h : a * b ≠ 0) : a ≤ a * b := by have h1 : b ≠ 0 := by contradict h with h1 rewrite [h1] ring done have h2 : 1 ≤ b := Nat.pos_of_ne_zero h1 show a ≤ a * b from calc a = a * 1 := (mul_one a).symm _ ≤ a * b := by rel [h2] done lemma le_nonzero_prod_right {a b : Nat} (h : a * b ≠ 0) : b ≤ a * b := by rewrite [mul_comm] rewrite [mul_comm] at h show b ≤ b * a from le_nonzero_prod_left h done lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry lemma rel_prime_of_prime_not_dvd {a p : Nat} (h1 : prime p) (h2 : ¬p ∣ a) : rel_prime a p := by have h3 : gcd a p ∣ a := gcd_dvd_left a p have h4 : gcd a p ∣ p := gcd_dvd_right a p have h5 : gcd a p = 1 ∨ gcd a p = p := dvd_prime h1 h4 have h6 : gcd a p ≠ p := by contradict h2 with h6 rewrite [h6] at h3 show p ∣ a from h3 done disj_syll h5 h6 show rel_prime a p from h5 done theorem Theorem_7_2_3 {a b p : Nat} (h1 : prime p) (h2 : p ∣ a * b) : p ∣ a ∨ p ∣ b := by or_right with h3 have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3 show p ∣ b from Theorem_7_2_2 h2 h4 done lemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a ≥ 1 := by have h1 : a ≠ 0 := by by_contra h1 rewrite [h1] at h contradict h linarith done show a ≥ 1 from Nat.pos_of_ne_zero h1 done lemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by have h1 : a ≥ 1 := ge_one_of_prod_one h have h2 : a * b ≠ 0 := by linarith have h3 : a ≤ a * b := le_nonzero_prod_left h2 rewrite [h] at h3 show a = 1 from Nat.le_antisymm h3 h1 done lemma eq_one_of_dvd_one {n : Nat} (h : n ∣ 1) : n = 1 := by obtain (j : Nat) (h1 : 1 = n * j) from h show n = 1 from eq_one_of_prod_one h1.symm done lemma prime_not_one {p : Nat} (h : prime p) : p ≠ 1 := by define at h linarith done theorem Theorem_7_2_4 {p : Nat} (h1 : prime p) : ∀ (l : List Nat), p ∣ prod l → ∃ a ∈ l, p ∣ a := by apply List.rec · -- Base Case. Goal : p ∣ prod [] → ∃ a ∈ [], p ∣ a rewrite [prod_nil] assume h2 : p ∣ 1 show ∃ a ∈ [], p ∣ a from absurd (eq_one_of_dvd_one h2) (prime_not_one h1) done · -- Induction Step fix b : Nat fix L : List Nat assume ih : p ∣ prod L → ∃ a ∈ L, p ∣ a --Goal : p ∣ prod (b :: L) → ∃ a ∈ b :: L, p ∣ a assume h2 : p ∣ prod (b :: L) rewrite [prod_cons] at h2 have h3 : p ∣ b ∨ p ∣ prod L := Theorem_7_2_3 h1 h2 by_cases on h3 · -- Case 1. h3 : p ∣ b apply Exists.intro b show b ∈ b :: L ∧ p ∣ b from And.intro (List.mem_cons_self b L) h3 done · -- Case 2. h3 : p ∣ prod L obtain (a : Nat) (h4 : a ∈ L ∧ p ∣ a) from ih h3 apply Exists.intro a show a ∈ b :: L ∧ p ∣ a from And.intro (List.mem_cons_of_mem b h4.left) h4.right done done done lemma prime_in_list {p : Nat} {l : List Nat} (h1 : prime p) (h2 : all_prime l) (h3 : p ∣ prod l) : p ∈ l := by obtain (a : Nat) (h4 : a ∈ l ∧ p ∣ a) from Theorem_7_2_4 h1 l h3 define at h2 have h5 : prime a := h2 a h4.left have h6 : p = 1 ∨ p = a := dvd_prime h5 h4.right disj_syll h6 (prime_not_one h1) rewrite [h6] show a ∈ l from h4.left done lemma first_le_first {p q : Nat} {l m : List Nat} (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m)) (h3 : prod (p :: l) = prod (q :: m)) : p ≤ q := by define at h1; define at h2 have h4 : q ∣ prod (p :: l) := by define apply Exists.intro (prod m) rewrite [←prod_cons] show prod (p :: l) = prod (q :: m) from h3 done have h5 : all_prime (q :: m) := h2.left rewrite [all_prime_cons] at h5 have h6 : q ∈ p :: l := prime_in_list h5.left h1.left h4 have h7 : nondec (p :: l) := h1.right rewrite [nondec_cons] at h7 rewrite [List.mem_cons] at h6 by_cases on h6 · -- Case 1. h6 : q = p linarith done · -- Case 2. h6 : q ∈ l have h8 : ∀ m ∈ l, p ≤ m := h7.left show p ≤ q from h8 q h6 done done lemma nondec_prime_list_tail {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by define at h define rewrite [all_prime_cons, nondec_cons] at h show all_prime l ∧ nondec l from And.intro h.left.right h.right.right done lemma cons_prod_not_one {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : prod (p :: l) ≠ 1 := by define at h have h1 : all_prime (p :: l) := h.left rewrite [all_prime_cons] at h1 rewrite [prod_cons] by_contra h2 show False from (prime_not_one h1.left) (eq_one_of_prod_one h2) done lemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) : l = [] ↔ prod l = 1 := by apply Iff.intro · -- (→) assume h1 : l = [] rewrite [h1] show prod [] = 1 from prod_nil done · -- (←) contrapos assume h1 : ¬l = [] obtain (p : Nat) (h2 : ∃ (L : List Nat), l = p :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List Nat) (h3 : l = p :: L) from h2 rewrite [h3] at h rewrite [h3] show ¬prod (p :: L) = 1 from cons_prod_not_one h done done lemma prime_pos {p : Nat} (h : prime p) : p > 0 := by define at h linarith done theorem Theorem_7_2_5 : ∀ (l1 l2 : List Nat), nondec_prime_list l1 → nondec_prime_list l2 → prod l1 = prod l2 → l1 = l2 := by apply List.rec · -- Base Case. Goal : ∀ (l2 : List Nat), nondec_prime_list [] → -- nondec_prime_list l2 → prod [] = prod l2 → [] = l2 fix l2 : List Nat assume h1 : nondec_prime_list [] assume h2 : nondec_prime_list l2 assume h3 : prod [] = prod l2 rewrite [prod_nil, eq_comm, ←list_nil_iff_prod_one h2] at h3 show [] = l2 from h3.symm done · -- Induction Step fix p : Nat fix L1 : List Nat assume ih : ∀ (L2 : List Nat), nondec_prime_list L1 → nondec_prime_list L2 → prod L1 = prod L2 → L1 = L2 -- Goal : ∀ (l2 : List Nat), nondec_prime_list (p :: L1) → -- nondec_prime_list l2 → prod (p :: L1) = prod l2 → p :: L1 = l2 fix l2 : List Nat assume h1 : nondec_prime_list (p :: L1) assume h2 : nondec_prime_list l2 assume h3 : prod (p :: L1) = prod l2 have h4 : ¬prod (p :: L1) = 1 := cons_prod_not_one h1 rewrite [h3, ←list_nil_iff_prod_one h2] at h4 obtain (q : Nat) (h5 : ∃ (L : List Nat), l2 = q :: L) from List.exists_cons_of_ne_nil h4 obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5 rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2) rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2) have h7 : p ≤ q := first_le_first h1 h2 h3 have h8 : q ≤ p := first_le_first h2 h1 h3.symm have h9 : p = q := by linarith rewrite [h9, prod_cons, prod_cons] at h3 --h3 : q * prod L1 = q * prod L2 have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1 have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2 define at h2 have h12 : all_prime (q :: L2) := h2.left rewrite [all_prime_cons] at h12 have h13 : q > 0 := prime_pos h12.left have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3 have h15 : L1 = L2 := ih L2 h10 h11 h14 rewrite [h6, h9, h15] rfl done done theorem fund_thm_arith (n : Nat) (h : n ≥ 1) : ∃! (l : List Nat), prime_factorization n l := by exists_unique · -- Existence show ∃ (l : List Nat), prime_factorization n l from exists_prime_factorization n h done · -- Uniqueness fix l1 : List Nat; fix l2 : List Nat assume h1 : prime_factorization n l1 assume h2 : prime_factorization n l2 define at h1; define at h2 have h3 : prod l1 = n := h1.right rewrite [←h2.right] at h3 show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3 done done /- Section 7.3 -/ theorem congr_refl (m : Nat) : ∀ (a : Int), a ≡ a (MOD m) := by fix a : Int define --Goal : ∃ (c : Int), a - a = ↑m * c apply Exists.intro 0 ring done theorem congr_symm {m : Nat} : ∀ {a b : Int}, a ≡ b (MOD m) → b ≡ a (MOD m) := by fix a : Int; fix b : Int assume h1 : a ≡ b (MOD m) define at h1 --h1 : ∃ (c : Int), a - b = ↑m * c define --Goal : ∃ (c : Int), b - a = ↑m * c obtain (c : Int) (h2 : a - b = m * c) from h1 apply Exists.intro (-c) show b - a = m * (-c) from calc b - a _ = -(a - b) := by ring _ = -(m * c) := by rw [h2] _ = m * (-c) := by ring done theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry /- Fundamental properties of congruence classes -/ lemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) : X = Y ↔ X.val = Y.val := Fin.ext_iff lemma val_nat_eq_mod (n k : Nat) : ([k]_(n + 1)).val = k % (n + 1) := by rfl lemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl theorem cc_rep {m : Nat} (X : ZMod m) : ∃ (a : Int), X = [a]_m := match m with | 0 => by apply Exists.intro X rfl done | n + 1 => by apply Exists.intro ↑(X.val) have h1 : X.val < n + 1 := Fin.prop X rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1] rfl done theorem add_class (m : Nat) (a b : Int) : [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm theorem mul_class (m : Nat) (a b : Int) : [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm lemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ [a - b]_m = [0]_m := by apply Iff.intro · -- (→) assume h1 : [a]_m = [b]_m have h2 : a - b = a + (-b) := by ring have h3 : b + (-b) = 0 := by ring show [a - b]_m = [0]_m from calc [a - b]_m _ = [a + (-b)]_m := by rw [h2] _ = [a]_m + [-b]_m := by rw [add_class] _ = [b]_m + [-b]_m := by rw [h1] _ = [b + -b]_m := by rw [add_class] _ = [0]_m := by rw [h3] done · -- (←) assume h1 : [a - b]_m = [0]_m have h2 : b + (a - b) = a := by ring have h3 : b + 0 = b := by ring show [a]_m = [b]_m from calc [a]_m _ = [b + (a - b)]_m := by rw [h2] _ = [b]_m + [a - b]_m := by rw [add_class] _ = [b]_m + [0]_m := by rw [h1] _ = [b + 0]_m := by rw [add_class] _ = [b]_m := by rw [h3] done done lemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) : [a]_m = [0]_m → [-a]_m = [0]_m := by assume h1 : [a]_m = [0]_m have h2 : 0 + (-a) = -a := by ring have h3 : a + (-a) = 0 := by ring show [-a]_m = [0]_m from calc [-a]_m _ = [0 + (-a)]_m := by rw [h2] _ = [0]_m + [-a]_m := by rw [add_class] _ = [a]_m + [-a]_m := by rw [h1] _ = [a + (-a)]_m := by rw [add_class] _ = [0]_m := by rw [h3] done lemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) : [-a]_m = [0]_m ↔ [a]_m = [0]_m := by apply Iff.intro _ (cc_neg_zero_of_cc_zero m a) assume h1 : [-a]_m = [0]_m have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1 have h3 : -(-a) = a := by ring rewrite [h3] at h2 show [a]_m = [0]_m from h2 done lemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl lemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m ↔ m ∣ k := match m with | 0 => by have h : (0 : Int) = (↑(0 : Nat) : Int) := by rfl rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj] apply Iff.intro · -- (→) assume h1 : k = 0 rewrite [h1] show 0 ∣ 0 from dvd_self 0 done · -- (←) assume h1 : 0 ∣ k obtain (c : Nat) (h2 : k = 0 * c) from h1 rewrite [h2] ring done done | n + 1 => by rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero] show k % (n + 1) = 0 ↔ n + 1 ∣ k from (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm done lemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m ↔ ↑m ∣ a := by obtain (k : Nat) (h1 : a = ↑k ∨ a = -↑k) from Int.eq_nat_or_neg a by_cases on h1 · -- Case 1. h1: a = ↑k rewrite [h1, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done · -- Case 2. h1: a = -↑k rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done done theorem cc_eq_iff_congr (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ a ≡ b (MOD m) := calc [a]_m = [b]_m _ ↔ [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b _ ↔ ↑m ∣ (a - b) := cc_zero_iff_dvd m (a - b) _ ↔ a ≡ b (MOD m) := by rfl /- End of fundamental properties of congruence classes -/ lemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m := by have h1 : (↑m : Int) ≠ 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m) show 0 ≤ a % m from Int.emod_nonneg a h1 done lemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m) have h2 : (↑m : Int) > 0 := (Nat.cast_pos).rtl h1 show a % m < m from Int.emod_lt_of_pos a h2 done lemma congr_mod_mod (m : Nat) (a : Int) : a ≡ a % m (MOD m) := by define have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m apply Exists.intro (a / m) show a - a % m = m * (a / m) from calc a - (a % m) _ = m * (a / m) + a % m - a % m := by rw [h1] _ = m * (a / m) := by ring done lemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) := And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a)) theorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) : ∃! (r : Int), 0 ≤ r ∧ r < m ∧ a ≡ r (MOD m) := by exists_unique · -- Existence apply Exists.intro (a % m) show 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) from mod_cmpl_res m a done · -- Uniqueness fix r1 : Int; fix r2 : Int assume h1 : 0 ≤ r1 ∧ r1 < m ∧ a ≡ r1 (MOD m) assume h2 : 0 ≤ r2 ∧ r2 < m ∧ a ≡ r2 (MOD m) have h3 : r1 ≡ r2 (MOD m) := congr_trans (congr_symm h1.right.right) h2.right.right obtain (d : Int) (h4 : r1 - r2 = m * d) from h3 have h5 : r1 - r2 < m * 1 := by linarith have h6 : m * (-1) < r1 - r2 := by linarith rewrite [h4] at h5 --h5 : m * d < m * 1 rewrite [h4] at h6 --h6 : m * -1 < m * d have h7 : (↑m : Int) ≥ 0 := Nat.cast_nonneg m have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7 have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7 have h10 : d = 0 := by linarith show r1 = r2 from calc r1 _ = r1 - r2 + r2 := by ring _ = m * 0 + r2 := by rw [h4, h10] _ = r2 := by ring done done lemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m := (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a) theorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y rewrite [h1, h2] have h3 : a + b = b + a := by ring show [a]_m + [b]_m = [b]_m + [a]_m from calc [a]_m + [b]_m _ = [a + b]_m := add_class m a b _ = [b + a]_m := by rw [h3] _ = [b]_m + [a]_m := (add_class m b a).symm done theorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X rewrite [h1] have h2 : a * 1 = a := by ring show [a]_m * [1]_m = [a]_m from calc [a]_m * [1]_m _ = [a * 1]_m := mul_class m a 1 _ = [a]_m := by rw [h2] done theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry lemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) : [a]_m * [gcd_c2 m a]_m = [1]_m := by set s : Int := gcd_c1 m a have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m define at h1 rewrite [h1, Nat.cast_one] at h2 --h2 : s * ↑m + gcd_c2 m a * ↑a = 1 rewrite [mul_class, cc_eq_iff_congr] define --Goal : ∃ (c : Int), ↑a * gcd_c2 m a - 1 = ↑m * c apply Exists.intro (-s) show a * (gcd_c2 m a) - 1 = m * (-s) from calc a * (gcd_c2 m a) - 1 _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring _ = 1 + m * (-s) - 1 := by rw [h2] _ = m * (-s) := by ring done theorem Theorem_7_3_7 (m a : Nat) : invertible [a]_m ↔ rel_prime m a := by apply Iff.intro · -- (→) assume h1 : invertible [a]_m define at h1 obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1 obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y rewrite [h3, mul_class, cc_eq_iff_congr] at h2 define at h2 obtain (c : Int) (h4 : a * b - 1 = m * c) from h2 rewrite [Exercise_7_2_6] --Goal : ∃ (s t : Int), s * ↑m + t * ↑a = 1 apply Exists.intro (-c) apply Exists.intro b show (-c) * m + b * a = 1 from calc (-c) * m + b * a _ = (-c) * m + (a * b - 1) + 1 := by ring _ = (-c) * m + m * c + 1 := by rw [h4] _ = 1 := by ring done · -- (←) assume h1 : rel_prime m a define show ∃ (Y : ZMod m), [a]_m * Y = [1]_m from Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1) done done /- Section 7.4 -/ section Euler open Euler lemma num_rp_below_base {m : Nat} : num_rp_below m 0 = 0 := by rfl lemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) : num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by have h1 : num_rp_below m (j + 1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : gcd m j = 1 rewrite [if_pos h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1 show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1 done lemma num_rp_below_step_not_rp {m j : Nat} (h : ¬rel_prime m j) : num_rp_below m (j + 1) = num_rp_below m j := by have h1 : num_rp_below m (j +1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : ¬gcd m j = 1 rewrite [if_neg h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j show num_rp_below m (j + 1) = num_rp_below m j from h1 done lemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl #eval phi 10 --Answer: 4 lemma prod_inv_iff_inv {m : Nat} {X : ZMod m} (h1 : invertible X) (Y : ZMod m) : invertible (X * Y) ↔ invertible Y := by apply Iff.intro · -- (→) assume h2 : invertible (X * Y) obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2 apply Exists.intro (X * Z) rewrite [←h3] --Goal : Y * (X * Z) = X * Y * Z ring --Note that ring can do algebra in ZMod m done · -- (←) assume h2 : invertible Y obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1 obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2 apply Exists.intro (Xi * Yi) show (X * Y) * (Xi * Yi) = [1]_m from calc X * Y * (Xi * Yi) _ = (X * Xi) * (Y * Yi) := by ring _ = [1]_m * [1]_m := by rw [h3, h4] _ = [1]_m := Theorem_7_3_6_7 [1]_m done done lemma F_rp_def {m i : Nat} (h : rel_prime m i) : F m i = [i]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h --h : gcd m i = 1 rewrite [if_pos h] at h1 show F m i = [i]_m from h1 done lemma F_not_rp_def {m i : Nat} (h : ¬rel_prime m i) : F m i = [1]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h rewrite [h1, if_neg h] rfl done lemma prod_seq_base {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 0 k f = [1]_m := by rfl lemma prod_seq_step {m : Nat} (n k : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl lemma prod_seq_zero_step {m : Nat} (n : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by rewrite [prod_seq_step, zero_add] rfl done lemma prod_one {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 1 k f = f k := by rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7] rfl done lemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl lemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m := calc [G m a i]_m _ = [(a * i) % m]_m := by rfl _ = [a * i]_m := (cc_eq_mod m (a * i)).symm _ = [a]_m * [i]_m := (mul_class m a i).symm lemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) : rel_prime m (G m a i) ↔ rel_prime m i := by have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1 show rel_prime m (G m a i) ↔ rel_prime m i from calc rel_prime m (G m a i) _ ↔ invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm _ ↔ invertible ([a]_m * [i]_m) := by rw [cc_G] _ ↔ invertible [i]_m := prod_inv_iff_inv h2 ([i]_m) _ ↔ rel_prime m i := Theorem_7_3_7 m i done lemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) : F m (G m a i) = [a]_m * F m i := by have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2 show F m (G m a i) = [a]_m * F m i from calc F m (G m a i) _ = [G m a i]_m := F_rp_def h3 _ = [a]_m * [i]_m := cc_G m a i _ = [a]_m * F m i := by rw [F_rp_def h2] done lemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : ¬rel_prime m i) : F m (G m a i) = [1]_m := by rewrite [←G_rp_iff h1 i] at h2 show F m (G m a i) = [1]_m from F_not_rp_def h2 done lemma FG_prod {m a : Nat} (h1 : rel_prime m a) : ∀ (k : Nat), prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by by_induc · -- Base Case show prod_seq 0 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from calc prod_seq 0 0 ((F m) ∘ (G m a)) _ = [1]_m := prod_seq_base _ _ _ = [a]_m ^ 0 * [1]_m := by ring _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by rw [num_rp_below_base, prod_seq_base] done · -- Induction Step fix k : Nat assume ih : prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) by_cases h2 : rel_prime m k · -- Case 1. h2 : rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([a]_m * F m k) := by rw [FG_rp h1 h2] _ = [a]_m ^ ((num_rp_below m k) + 1) * ((prod_seq k 0 (F m)) * F m k) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_rp h2, prod_seq_zero_step] done · -- Case 2. h2 : ¬rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([1]_m) := by rw [FG_not_rp h1 h2] _ = [a]_m ^ (num_rp_below m k) * (prod_seq k 0 (F m) * ([1]_m)) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_not_rp h2, prod_seq_zero_step, F_not_rp_def h2] done done done lemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by define --Goal : ∀ i < m, G m a i < m fix i : Nat assume h1 : i < m rewrite [G_def] --Goal : a * i % m < m show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m) done lemma left_inv_one_one_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g' (g i) = i) : one_one_below n g := sorry lemma right_inv_onto_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g (g' i) = i) (h2 : maps_below n g') : onto_below n g := by define at h2; define fix k : Nat assume h3 : k < n apply Exists.intro (g' k) show g' k < n ∧ g (g' k) = k from And.intro (h2 k h3) (h1 k h3) done lemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m * [inv_mod m a]_m = [1]_m := by have h2 : 0 ≤ (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a) show [a]_m * [inv_mod m a]_m = [1]_m from calc [a]_m * [inv_mod m a]_m _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2] _ = [a]_m * [gcd_c2 m a]_m := by rw [←cc_eq_mod] _ = [1]_m := gcd_c2_inv h1 done lemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m := calc a * (b % m) % m = a % m * (b % m % m) % m := Nat.mul_mod _ _ _ _ = a % m * (b % m) % m := by rw [Nat.mod_mod] _ = a * b % m := (Nat.mul_mod _ _ _).symm lemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm] rfl done theorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] : ↑a ≡ ↑b (MOD m) ↔ a % m = b % m := sorry lemma mul_inv_mod_cancel {m a i : Nat} [NeZero m] (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1 rewrite [mul_class, cc_eq_iff_congr, ←Nat.cast_mul, ←Nat.cast_one, congr_iff_mod_eq_Nat] at h3 show a * inv_mod m a * i % m = i from calc a * (inv_mod m a) * i % m _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = 1 % m * i % m := by rw [h3] _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = i % m := by rw [one_mul] _ = i := Nat.mod_eq_of_lt h2 done lemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl lemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, G m a (Ginv m a i) = i := by fix i : Nat assume h2 : i < m show G m a (Ginv m a i) = i from calc G m a (Ginv m a i) _ = a * ((inv_mod m a * i) % m) % m := by rfl _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, Ginv m a (G m a i) = i := by fix i : Nat assume h2 : i < m show Ginv m a (G m a i) = i from calc Ginv m a (G m a i) _ = inv_mod m a * ((a * i) % m) % m := by rfl _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc, mul_comm (inv_mod m a)] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_maps_below (m a : Nat) [NeZero m] : maps_below m (Ginv m a) := G_maps_below m (inv_mod m a) lemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : one_one_below m (G m a) := left_inv_one_one_below (Ginv_left_inv h1) lemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : onto_below m (G m a) := right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a) lemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : perm_below m (G m a) := And.intro (G_maps_below m a) (And.intro (G_one_one_below h1) (G_onto_below h1)) --Permuting a product of congruence classes doesn't change product lemma swap_fst (u v : Nat) : swap u v u = v := by define : swap u v u --Goal : (if u = u then v else if u = v then u else u) = v have h : u = u := by rfl rewrite [if_pos h] rfl done lemma swap_snd (u v : Nat) : swap u v v = u := by define : swap u v v by_cases h1 : v = u · -- Case 1. h1 : v = u rewrite [if_pos h1] show v = u from h1 done · -- Case 2. h1 : v ≠ u rewrite [if_neg h1] have h2 : v = v := by rfl rewrite [if_pos h2] rfl done done lemma swap_other {u v i : Nat} (h1 : i ≠ u) (h2 : i ≠ v) : swap u v i = i := by define : swap u v i rewrite [if_neg h1, if_neg h2] rfl done lemma swap_values (u v i : Nat) : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := by by_cases h1 : i = u · -- Case 1. h1 : i = u apply Or.inl rewrite [h1] show swap u v u = v from swap_fst u v done · -- Case 2. h1 : i ≠ u apply Or.inr by_cases h2 : i = v · -- Case 2.1. h2 : i = v apply Or.inl rewrite [h2] show swap u v v = u from swap_snd u v done · -- Case 2.2. h2 : i ≠ v apply Or.inr show swap u v i = i from swap_other h1 h2 done done done lemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by define fix i : Nat assume h3 : i < n have h4 : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := swap_values u v i by_cases on h4 · -- Case 1. h4 : swap u v i = v rewrite [h4] show v < n from h2 done · -- Case 2. by_cases on h4 · -- Case 2.1. h4 : swap u v i = u rewrite [h4] show u < n from h1 done · -- Case 2.2. h4 : swap u v i = i rewrite [h4] show i < n from h3 done done done lemma swap_swap (u v n : Nat) : ∀ i < n, swap u v (swap u v i) = i := by fix i : Nat assume h : i < n by_cases h1 : i = u · -- Case 1. h1 : i = u rewrite [h1, swap_fst, swap_snd] rfl done · -- Case 2. h1 : i ≠ u by_cases h2 : i = v · -- Case 2.1. h2 : i = v rewrite [h2, swap_snd, swap_fst] rfl done · -- Case 2.2. h2 : i ≠ v rewrite [swap_other h1 h2, swap_other h1 h2] rfl done done done lemma swap_one_one_below (u v n) : one_one_below n (swap u v) := left_inv_one_one_below (swap_swap u v n) lemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) := right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2) lemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) := And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2)) lemma comp_perm_below {n : Nat} {f g : Nat → Nat} (h1 : perm_below n f) (h2 : perm_below n g) : perm_below n (f ∘ g) := sorry lemma trivial_swap (u : Nat) : swap u u = id := by apply funext fix x : Nat by_cases h1 : x = u · -- Case 1. h1 : x = u rewrite [h1, swap_fst] rfl done · -- Case 2. h1 : x ≠ u rewrite [swap_other h1 h1] rfl done done lemma prod_eq_fun {m : Nat} (f g : Nat → ZMod m) (k : Nat) : ∀ (n : Nat), (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g := by by_induc · -- Base Case assume h : (∀ i < 0, f (k + i) = g (k + i)) rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g assume h1 : ∀ i < n + 1, f (k + i) = g (k + i) have h2 : ∀ i < n, f (k + i) = g (k + i) := by fix i : Nat assume h2 : i < n have h3 : i < n + 1 := by linarith show f (k + i) = g (k + i) from h1 i h3 done have h3 : prod_seq n k f = prod_seq n k g := ih h2 have h4 : n < n + 1 := Nat.lt_succ_self n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4] rfl done done lemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := by have h2 : ∀ (i : Nat), i < u → (f ∘ swap u n) (0 + i) = f (0 + i) := by fix i : Nat assume h2 : i < u have h3 : 0 + i ≠ u := by linarith have h4 : 0 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl done show prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f from prod_eq_fun (f ∘ swap u n) f 0 u h2 done lemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := by have h2 : ∀ i < j, (f ∘ swap u n) (u + 1 + i) = f (u + 1 + i) := by fix i : Nat assume h2 : i < j have h3 : u + 1 + i ≠ u := by linarith have h4 : u + 1 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl show prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f from prod_eq_fun (f ∘ swap u n) f (u + 1) j h2 done lemma break_prod {m : Nat} (n : Nat) (f : Nat → ZMod m) : ∀ (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by by_induc · -- Base Case have h : n + 0 = n := by rfl rewrite [prod_seq_base, h, Theorem_7_3_6_7] rfl done · -- Induction Step fix j : Nat assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f rewrite [←add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc] rfl done done lemma break_prod_twice {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f := break_prod n f 1 rewrite [prod_one] at h2 have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f := break_prod (u + 1) f j rewrite [←h1] at h3 have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f := break_prod u f 1 rewrite [prod_one] at h4 rewrite [h3, h4] at h2 show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2 done lemma swap_prod_eq_prod {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f := by by_cases h2 : u = n · -- Case 1. h2 : u = n rewrite [h2, trivial_swap n] --Goal : prod_seq (n + 1) 0 (f ∘ id) = prod_seq (n + 1) 0 f rfl done · -- Case 2. h2 : ¬u = n have h3 : u + 1 ≤ n := Nat.lt_of_le_of_ne h1 h2 obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3 have break_f : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := break_prod_twice f h4 have break_fs : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_prod_twice (f ∘ swap u n) h4 have f_eq_fs_below : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := swap_prod_eq_prod_below f h1 have f_eq_fs_btwn : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4 show prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f from calc prod_seq (n + 1) 0 (f ∘ swap u n) _ = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_fs _ = prod_seq u 0 f * (f ∘ swap u n) u * prod_seq j (u + 1) f * (f ∘ swap u n) n := by rw [f_eq_fs_below, f_eq_fs_btwn] _ = prod_seq u 0 f * f (swap u n u) * prod_seq j (u + 1) f * f (swap u n n) := by rfl _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by rw [swap_fst, swap_snd] _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring _ = prod_seq (n + 1) 0 f := break_f.symm done done lemma perm_below_fixed {n : Nat} {g : Nat → Nat} (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry lemma perm_prod {m : Nat} (f : Nat → ZMod m) : ∀ (n : Nat), ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) := by by_induc · -- Base Case fix g : Nat → Nat assume h1 : perm_below 0 g rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) fix g : Nat → Nat assume g_pb : perm_below (n + 1) g define at g_pb have g_ob : onto_below (n + 1) g := g_pb.right.right define at g_ob have h1 : n < n + 1 := by linarith obtain (u : Nat) (h2 : u < n + 1 ∧ g u = n) from g_ob n h1 have s_pb : perm_below (n + 1) (swap u n) := swap_perm_below h2.left h1 have gs_pb_n1 : perm_below (n + 1) (g ∘ swap u n) := comp_perm_below g_pb s_pb have gs_fix_n : (g ∘ swap u n) n = n := calc (g ∘ swap u n) n _ = g (swap u n n) := by rfl _ = g u := by rw [swap_snd] _ = n := h2.right have gs_pb_n : perm_below n (g ∘ swap u n) := perm_below_fixed gs_pb_n1 gs_fix_n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f ∘ (g ∘ swap u n)) := ih (g ∘ swap u n) gs_pb_n have h3 : u ≤ n := by linarith show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f ∘ g) from calc prod_seq (n + 1) 0 f _ = prod_seq n 0 f * f n := prod_seq_zero_step n f _ = prod_seq n 0 (f ∘ (g ∘ swap u n)) * f ((g ∘ swap u n) n) := by rw [gs_prod, gs_fix_n] _ = prod_seq n 0 (f ∘ g ∘ swap u n) * (f ∘ g ∘ swap u n) n := by rfl _ = prod_seq (n + 1) 0 (f ∘ g ∘ swap u n) := (prod_seq_zero_step n (f ∘ g ∘ swap u n)).symm _ = prod_seq (n + 1) 0 ((f ∘ g) ∘ swap u n) := by rfl _ = prod_seq (n + 1) 0 (f ∘ g) := swap_prod_eq_prod (f ∘ g) h3 done done lemma F_invertible (m i : Nat) : invertible (F m i) := by by_cases h : rel_prime m i · -- Case 1. h : rel_prime m i rewrite [F_rp_def h] show invertible [i]_m from (Theorem_7_3_7 m i).rtl h done · -- Case 2. h : ¬rel_prime m i rewrite [F_not_rp_def h] apply Exists.intro [1]_m show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m done done lemma Fprod_invertible (m : Nat) : ∀ (k : Nat), invertible (prod_seq k 0 (F m)) := by by_induc · -- Base Case apply Exists.intro [1]_m show prod_seq 0 0 (F m) * [1]_m = [1]_m from calc prod_seq 0 0 (F m) * [1]_m _ = [1]_m * [1]_m := by rw [prod_seq_base] _ = [1]_m := Theorem_7_3_6_7 ([1]_m) done · -- Induction Step fix k : Nat assume ih : invertible (prod_seq k 0 (F m)) rewrite [prod_seq_zero_step] show invertible (prod_seq k 0 (F m) * (F m k)) from (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k) done done theorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m ^ (phi m) = [1]_m := by have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2 show [a]_m ^ (phi m) = [1]_m from calc [a]_m ^ (phi m) _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3] _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring _ = prod_seq m 0 (F m ∘ G m a) * Y := by rw [FG_prod h1 m, phi_def] _ = prod_seq m 0 (F m) * Y := by rw [perm_prod (F m) m (G m a) (G_perm_below h1)] _ = [1]_m := by rw [h3] done lemma Exercise_7_4_5_Int (m : Nat) (a : Int) : ∀ (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry lemma Exercise_7_4_5_Nat (m a n : Nat) : [a]_m ^ n = [a ^ n]_m := by rewrite [Exercise_7_4_5_Int] rfl done theorem Euler's_theorem {m a : Nat} [NeZero m] (h1 : rel_prime m a) : a ^ (phi m) ≡ 1 (MOD m) := by have h2 : [a]_m ^ (phi m) = [1]_m := Theorem_7_4_2 h1 rewrite [Exercise_7_4_5_Nat m a (phi m)] at h2 --h2 : [a ^ phi m]_m = [1]_m show a ^ (phi m) ≡ 1 (MOD m) from (cc_eq_iff_congr _ _ _).ltr h2 done #eval gcd 10 7 --Answer: 1. So 10 and 7 are relatively prime #eval 7 ^ phi 10 --Answer: 2401, which is congruent to 1 mod 10. end Euler /- Section 7.5 -/ lemma num_rp_prime {p : Nat} (h1 : prime p) : ∀ k < p, num_rp_below p (k + 1) = k := sorry lemma phi_prime {p : Nat} (h1 : prime p) : phi p = p - 1 := by have h2 : 1 ≤ p := prime_pos h1 have h3 : p - 1 + 1 = p := Nat.sub_add_cancel h2 have h4 : p - 1 < p := by linarith have h5 : num_rp_below p (p - 1 + 1) = p - 1 := num_rp_prime h1 (p - 1) h4 rewrite [h3] at h5 show phi p = p - 1 from h5 done theorem Theorem_7_2_2_Int {a c : Nat} {b : Int} (h1 : ↑c ∣ ↑a * b) (h2 : rel_prime a c) : ↑c ∣ b := by rewrite [Int.natCast_dvd, Int.natAbs_mul, Int.natAbs_ofNat] at h1 --h1 : c ∣ a * Int.natAbs b rewrite [Int.natCast_dvd] --Goal : c ∣ Int.natAbs b show c ∣ Int.natAbs b from Theorem_7_2_2 h1 h2 done lemma Lemma_7_4_5 {m n : Nat} (a b : Int) (h1 : rel_prime m n) : a ≡ b (MOD m * n) ↔ a ≡ b (MOD m) ∧ a ≡ b (MOD n) := by apply Iff.intro · -- (→) assume h2 : a ≡ b (MOD m * n) obtain (j : Int) (h3 : a - b = (m * n) * j) from h2 apply And.intro · -- Proof of a ≡ b (MOD m) apply Exists.intro (n * j) show a - b = m * (n * j) from calc a - b _ = m * n * j := h3 _ = m * (n * j) := by ring done · -- Proof of a ≡ b (MOD n) apply Exists.intro (m * j) show a - b = n * (m * j) from calc a - b _ = m * n * j := h3 _ = n * (m * j) := by ring done done · -- (←) assume h2 : a ≡ b (MOD m) ∧ a ≡ b (MOD n) obtain (j : Int) (h3 : a - b = m * j) from h2.left have h4 : (↑n : Int) ∣ a - b := h2.right rewrite [h3] at h4 --h4 : ↑n ∣ ↑m * j have h5 : ↑n ∣ j := Theorem_7_2_2_Int h4 h1 obtain (k : Int) (h6 : j = n * k) from h5 apply Exists.intro k --Goal : a - b = ↑(m * n) * k rewrite [Nat.cast_mul] --Goal : a - b = ↑m * ↑n * k show a - b = (m * n) * k from calc a - b _ = m * j := h3 _ = m * (n * k) := by rw [h6] _ = (m * n) * k := by ring done done --From exercises of Section 7.2 theorem rel_prime_symm {a b : Nat} (h : rel_prime a b) : rel_prime b a := sorry lemma prime_NeZero {p : Nat} (h : prime p) : NeZero p := by rewrite [neZero_iff] --Goal : p ≠ 0 define at h linarith done lemma Lemma_7_5_1 {p e d m c s : Nat} {t : Int} (h1 : prime p) (h2 : e * d = (p - 1) * s + 1) (h3 : m ^ e - c = p * t) : c ^ d ≡ m (MOD p) := by have h4 : m ^ e ≡ c (MOD p) := Exists.intro t h3 have h5 : [m ^ e]_p = [c]_p := (cc_eq_iff_congr _ _ _).rtl h4 rewrite [←Exercise_7_4_5_Nat] at h5 --h5 : [m]_p ^ e = [c]_p by_cases h6 : p ∣ m · -- Case 1. h6 : p ∣ m have h7 : m ≡ 0 (MOD p) := by obtain (j : Nat) (h8 : m = p * j) from h6 apply Exists.intro (↑j : Int) --Goal : ↑m - 0 = ↑p * ↑j rewrite [h8, Nat.cast_mul] ring done have h8 : [m]_p = [0]_p := (cc_eq_iff_congr _ _ _).rtl h7 have h9 : e * d ≠ 0 := by rewrite [h2] show (p - 1) * s + 1 ≠ 0 from Nat.add_one_ne_zero _ done have h10 : (0 : Int) ^ (e * d) = 0 := zero_pow h9 have h11 : [c ^ d]_p = [m]_p := calc [c ^ d]_p _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat] _ = ([m]_p ^ e) ^ d := by rw [h5] _ = [m]_p ^ (e * d) := by ring _ = [0]_p ^ (e * d) := by rw [h8] _ = [0 ^ (e * d)]_p := Exercise_7_4_5_Int _ _ _ _ = [0]_p := by rw [h10] _ = [m]_p := by rw [h8] show c ^ d ≡ m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11 done · -- Case 2. h6 : ¬p ∣ m have h7 : rel_prime m p := rel_prime_of_prime_not_dvd h1 h6 have h8 : rel_prime p m := rel_prime_symm h7 have h9 : NeZero p := prime_NeZero h1 have h10 : (1 : Int) ^ s = 1 := by ring have h11 : [c ^ d]_p = [m]_p := calc [c ^ d]_p _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat] _ = ([m]_p ^ e) ^ d := by rw [h5] _ = [m]_p ^ (e * d) := by ring _ = [m]_p ^ ((p - 1) * s + 1) := by rw [h2] _ = ([m]_p ^ (p - 1)) ^ s * [m]_p := by ring _ = ([m]_p ^ (phi p)) ^ s * [m]_p := by rw [phi_prime h1] _ = [1]_p ^ s * [m]_p := by rw [Theorem_7_4_2 h8] _ = [1 ^ s]_p * [m]_p := by rw [Exercise_7_4_5_Int] _ = [1]_p * [m]_p := by rw [h10] _ = [m]_p * [1]_p := by ring _ = [m]_p := Theorem_7_3_6_7 _ show c ^ d ≡ m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11 done done theorem Theorem_7_5_1 (p q n e d k m c : Nat) (p_prime : prime p) (q_prime : prime q) (p_ne_q : p ≠ q) (n_pq : n = p * q) (ed_congr_1 : e * d = k * (p - 1) * (q - 1) + 1) (h1 : [m]_n ^ e = [c]_n) : [c]_n ^ d = [m]_n := by rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] at h1 --h1 : m ^ e ≡ c (MOD n) rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] --Goal : c ^ d ≡ m (MOD n) obtain (j : Int) (h2 : m ^ e - c = n * j) from h1 rewrite [n_pq, Nat.cast_mul] at h2 --h2 : m ^ e - c = p * q * j have h3 : e * d = (p - 1) * (k * (q - 1)) + 1 := by rewrite [ed_congr_1] ring done have h4 : m ^ e - c = p * (q * j) := by rewrite [h2] ring done have congr_p : c ^ d ≡ m (MOD p) := Lemma_7_5_1 p_prime h3 h4 have h5 : e * d = (q - 1) * (k * (p - 1)) + 1 := by rewrite [ed_congr_1] ring done have h6 : m ^ e - c = q * (p * j) := by rewrite [h2] ring done have congr_q : c ^ d ≡ m (MOD q) := Lemma_7_5_1 q_prime h5 h6 have h7 : ¬q ∣ p := by by_contra h8 have h9 : q = 1 ∨ q = p := dvd_prime p_prime h8 disj_syll h9 (prime_not_one q_prime) show False from p_ne_q h9.symm done have h8 : rel_prime p q := rel_prime_of_prime_not_dvd q_prime h7 rewrite [n_pq, Lemma_7_4_5 _ _ h8] show c ^ d ≡ m (MOD p) ∧ c ^ d ≡ m (MOD q) from And.intro congr_p congr_q done /- BEGIN EXERCISES -/ namespace Exercises /- Section 7.1 -/ -- 1. theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry -- 2. lemma gcd_comm_lt {a b : Nat} (h : a < b) : gcd a b = gcd b a := sorry theorem gcd_comm (a b : Nat) : gcd a b = gcd b a := sorry -- 3. theorem Exercise_7_1_5 (a b : Nat) (n : Int) : (∃ (s t : Int), s * a + t * b = n) ↔ (↑(gcd a b) : Int) ∣ n := sorry -- 4. theorem Exercise_7_1_6 (a b c : Nat) : gcd a b = gcd (a + b * c) b := sorry -- 5. theorem gcd_is_nonzero {a b : Nat} (h : a ≠ 0 ∨ b ≠ 0) : gcd a b ≠ 0 := sorry -- 6. theorem gcd_greatest {a b d : Nat} (h1 : gcd a b ≠ 0) (h2 : d ∣ a) (h3 : d ∣ b) : d ≤ gcd a b := sorry -- 7. lemma Lemma_7_1_10a {a b : Nat} (n : Nat) (h : a ∣ b) : (n * a) ∣ (n * b) := sorry lemma Lemma_7_1_10b {a b n : Nat} (h1 : n ≠ 0) (h2 : (n * a) ∣ (n * b)) : a ∣ b := sorry lemma Lemma_7_1_10c {a b : Nat} (h1 : a ∣ b) (h2 : b ∣ a) : a = b := sorry

theorem Exercise_7_1_10 (a b n : Nat) : gcd (n * a) (n * b) = n * gcd a b

HTPI.Exercises.Exercise_7_1_10

htpi/HTPILib/Chap7.lean

htpi

/- Copyright 2023 Daniel J. Velleman -/ import HTPILib.Chap6 namespace HTPI /- Definitions -/ lemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by have h : n + 1 > 0 := Nat.succ_pos n show a % (n + 1) < n + 1 from Nat.mod_lt a h done def gcd (a b : Nat) : Nat := match b with | 0 => a | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd (n + 1) (a % (n + 1)) termination_by b mutual def gcd_c1 (a b : Nat) : Int := match b with | 0 => 1 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c2 (n + 1) (a % (n + 1)) --Corresponds to s = t' termination_by b def gcd_c2 (a b : Nat) : Int := match b with | 0 => 0 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c1 (n + 1) (a % (n + 1)) - (gcd_c2 (n + 1) (a % (n + 1))) * ↑(a / (n + 1)) --Corresponds to t = s' - t'q termination_by b end def prime (n : Nat) : Prop := 2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n def prime_factor (p n : Nat) : Prop := prime p ∧ p ∣ n def all_prime (l : List Nat) : Prop := ∀ p ∈ l, prime p def nondec (l : List Nat) : Prop := match l with | [] => True --Of course, True is a proposition that is always true | n :: L => (∀ m ∈ L, n ≤ m) ∧ nondec L def nondec_prime_list (l : List Nat) : Prop := all_prime l ∧ nondec l def prod (l : List Nat) : Nat := match l with | [] => 1 | n :: L => n * (prod L) def prime_factorization (n : Nat) (l : List Nat) : Prop := nondec_prime_list l ∧ prod l = n def rel_prime (a b : Nat) : Prop := gcd a b = 1 def congr_mod (m : Nat) (a b : Int) : Prop := (↑m : Int) ∣ (a - b) def cc (m : Nat) (a : Int) : ZMod m := (↑a : ZMod m) notation:50 a " ≡ " b " (MOD " m ")" => congr_mod m a b notation:max "["a"]_"m:max => cc m a def invertible {m : Nat} (X : ZMod m) : Prop := ∃ (Y : ZMod m), X * Y = [1]_m def num_rp_below (m k : Nat) : Nat := match k with | 0 => 0 | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j def phi (m : Nat) : Nat := num_rp_below m m def prod_seq {m : Nat} (j k : Nat) (f : Nat → ZMod m) : ZMod m := match j with | 0 => [1]_m | n + 1 => prod_seq n k f * f (k + n) def maps_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i < n, g i < n def one_one_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i1 < n, ∀ i2 < n, g i1 = g i2 → i1 = i2 def onto_below (n : Nat) (g : Nat → Nat) : Prop := ∀ k < n, ∃ i < n, g i = k def perm_below (n : Nat) (g : Nat → Nat) : Prop := maps_below n g ∧ one_one_below n g ∧ onto_below n g def inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m) def swap (u v i : Nat) : Nat := if i = u then v else if i = v then u else i namespace Euler --For definitions specific to Euler's theorem def F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m def G (m a i : Nat) : Nat := (a * i) % m def Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i end Euler /- Section 7.1 -/ theorem dvd_mod_of_dvd_a_b {a b d : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ (a % b) := by set q : Nat := a / b have h3 : b * q + a % b = a := Nat.div_add_mod a b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 define --Goal : ∃ (c : Nat), a % b = d * c apply Exists.intro (j - k * q) show a % b = d * (j - k * q) from calc a % b _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm _ = a - b * q := by rw [h3] _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc] _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm done theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry #eval gcd 672 161 --Answer: 7 lemma gcd_base (a : Nat) : gcd a 0 = a := by rfl lemma gcd_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd a b = gcd b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1)) rfl done lemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b ≠ 0) : a % b < b := by have h1 : b > 0 := Nat.pos_of_ne_zero h show a % b < b from Nat.mod_lt a h1 done lemma dvd_self (n : Nat) : n ∣ n := by apply Exists.intro 1 ring done theorem gcd_dvd : ∀ (b a : Nat), (gcd a b) ∣ a ∧ (gcd a b) ∣ b := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd a b_1) ∣ a ∧ (gcd a b_1) ∣ b_1 fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_base] --Goal: a ∣ a ∧ a ∣ 0 apply And.intro (dvd_self a) define apply Exists.intro 0 rfl done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_nonzero a h1] --Goal : gcd b (a % b) ∣ a ∧ gcd b (a % b) ∣ b have h2 : a % b < b := mod_nonzero_lt a h1 have h3 : (gcd b (a % b)) ∣ b ∧ (gcd b (a % b)) ∣ (a % b) := ih (a % b) h2 b apply And.intro _ h3.left show (gcd b (a % b)) ∣ a from dvd_a_of_dvd_b_mod h3.left h3.right done done theorem gcd_dvd_left (a b : Nat) : (gcd a b) ∣ a := (gcd_dvd b a).left theorem gcd_dvd_right (a b : Nat) : (gcd a b) ∣ b := (gcd_dvd b a).right lemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl lemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c1 a b = gcd_c2 b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done lemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl lemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * ↑(a / b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done theorem gcd_lin_comb : ∀ (b a : Nat), (gcd_c1 a b) * ↑a + (gcd_c2 a b) * ↑b = ↑(gcd a b) := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd_c1 a b_1) * ↑a + (gcd_c2 a b_1) * ↑b_1 = ↑(gcd a b_1) fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base] --Goal : 1 * ↑a + 0 * ↑0 = ↑a ring done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1] --Goal : gcd_c2 b (a % b) * ↑a + -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * ↑(a / b)) * ↑b = -- ↑(gcd b (a % b)) set r : Nat := a % b set q : Nat := a / b set s : Int := gcd_c1 b r set t : Int := gcd_c2 b r --Goal : t * ↑a + (s - t * ↑q) * ↑b = ↑(gcd b r) have h2 : r < b := mod_nonzero_lt a h1 have h3 : s * ↑b + t * ↑r = ↑(gcd b r) := ih r h2 b have h4 : b * q + r = a := Nat.div_add_mod a b rewrite [←h3, ←h4] rewrite [Nat.cast_add, Nat.cast_mul] --Goal : t * (↑b * ↑q + ↑r) + (s - t * ↑q) * ↑b = s * ↑b + t * ↑r ring done done #eval gcd_c1 672 161 --Answer: 6 #eval gcd_c2 672 161 --Answer: -25 --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161 theorem Theorem_7_1_6 {d a b : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ gcd a b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑d ∣ ↑(gcd a b) set s : Int := gcd_c1 a b set t : Int := gcd_c2 a b have h3 : s * ↑a + t * ↑b = ↑(gcd a b) := gcd_lin_comb b a rewrite [←h3] --Goal : ↑d ∣ s * ↑a + t * ↑b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul] --Goal : ↑d ∣ s * (↑d * ↑j) + t * (↑d * ↑k) define apply Exists.intro (s * ↑j + t * ↑k) ring done /- Section 7.2 -/ theorem dvd_trans {a b c : Nat} (h1 : a ∣ b) (h2 : b ∣ c) : a ∣ c := by define at h1; define at h2; define obtain (m : Nat) (h3 : b = a * m) from h1 obtain (n : Nat) (h4 : c = b * n) from h2 rewrite [h3, mul_assoc] at h4 apply Exists.intro (m * n) show c = a * (m * n) from h4 done lemma exists_prime_factor : ∀ (n : Nat), 2 ≤ n → ∃ (p : Nat), prime_factor p n := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, 2 ≤ n_1 → ∃ (p : Nat), prime_factor p n_1 assume h1 : 2 ≤ n by_cases h2 : prime n · -- Case 1. h2 : prime n apply Exists.intro n define --Goal : prime n ∧ n ∣ n show prime n ∧ n ∣ n from And.intro h2 (dvd_self n) done · -- Case 2. h2 : ¬prime n define at h2 --h2 : ¬(2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n) demorgan at h2 disj_syll h2 h1 obtain (a : Nat) (h3 : ∃ (b : Nat), a * b = n ∧ a < n ∧ b < n) from h2 obtain (b : Nat) (h4 : a * b = n ∧ a < n ∧ b < n) from h3 have h5 : 2 ≤ a := by by_contra h6 have h7 : a ≤ 1 := by linarith have h8 : n ≤ b := calc n _ = a * b := h4.left.symm _ ≤ 1 * b := by rel [h7] _ = b := by ring linarith --n ≤ b contradicts b < n done have h6 : ∃ (p : Nat), prime_factor p a := ih a h4.right.left h5 obtain (p : Nat) (h7 : prime_factor p a) from h6 apply Exists.intro p define --Goal : prime p ∧ p ∣ n define at h7 --h7 : prime p ∧ p ∣ a apply And.intro h7.left have h8 : a ∣ n := by apply Exists.intro b show n = a * b from (h4.left).symm done show p ∣ n from dvd_trans h7.right h8 done done lemma exists_least_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q := by set S : Set Nat := {p : Nat | prime_factor p n} have h2 : ∃ (p : Nat), p ∈ S := exists_prime_factor n h show ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q from well_ord_princ S h2 done lemma all_prime_nil : all_prime [] := by define --Goal : ∀ p ∈ [], prime p fix p : Nat contrapos --Goal : ¬prime p → p ∉ [] assume h1 : ¬prime p show p ∉ [] from List.not_mem_nil p done lemma all_prime_cons (n : Nat) (L : List Nat) : all_prime (n :: L) ↔ prime n ∧ all_prime L := by apply Iff.intro · -- (→) assume h1 : all_prime (n :: L) --Goal : prime n ∧ all_prime L define at h1 --h1 : ∀ p ∈ n :: L, prime p apply And.intro (h1 n (List.mem_cons_self n L)) define --Goal : ∀ p ∈ L, prime p fix p : Nat assume h2 : p ∈ L show prime p from h1 p (List.mem_cons_of_mem n h2) done · -- (←) assume h1 : prime n ∧ all_prime L --Goal : all_prime (n :: l) define : all_prime L at h1 define fix p : Nat assume h2 : p ∈ n :: L rewrite [List.mem_cons] at h2 --h2 : p = n ∨ p ∈ L by_cases on h2 · -- Case 1. h2 : p = n rewrite [h2] show prime n from h1.left done · -- Case 2. h2 : p ∈ L show prime p from h1.right p h2 done done done lemma nondec_nil : nondec [] := by define --Goal : True trivial --trivial proves some obviously true statements, such as True done lemma nondec_cons (n : Nat) (L : List Nat) : nondec (n :: L) ↔ (∀ m ∈ L, n ≤ m) ∧ nondec L := by rfl lemma prod_nil : prod [] = 1 := by rfl lemma prod_cons : prod (n :: L) = n * (prod L) := by rfl lemma exists_cons_of_length_eq_succ {A : Type} {l : List A} {n : Nat} (h : l.length = n + 1) : ∃ (a : A) (L : List A), l = a :: L ∧ L.length = n := by have h1 : ¬l.length = 0 := by linarith rewrite [List.length_eq_zero] at h1 obtain (a : A) (h2 : ∃ (L : List A), l = a :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List A) (h3 : l = a :: L) from h2 apply Exists.intro a apply Exists.intro L apply And.intro h3 have h4 : (a :: L).length = L.length + 1 := List.length_cons a L rewrite [←h3, h] at h4 show L.length = n from (Nat.add_right_cancel h4).symm done lemma list_elt_dvd_prod_by_length (a : Nat) : ∀ (n : Nat), ∀ (l : List Nat), l.length = n → a ∈ l → a ∣ prod l := by by_induc · --Base Case fix l : List Nat assume h1 : l.length = 0 rewrite [List.length_eq_zero] at h1 --h1 : l = [] rewrite [h1] --Goal : a ∈ [] → a ∣ prod [] contrapos assume h2 : ¬a ∣ prod [] show a ∉ [] from List.not_mem_nil a done · -- Induction Step fix n : Nat assume ih : ∀ (l : List Nat), List.length l = n → a ∈ l → a ∣ prod l fix l : List Nat assume h1 : l.length = n + 1 --Goal : a ∈ l → a ∣ prod l obtain (b : Nat) (h2 : ∃ (L : List Nat), l = b :: L ∧ L.length = n) from exists_cons_of_length_eq_succ h1 obtain (L : List Nat) (h3 : l = b :: L ∧ L.length = n) from h2 have h4 : a ∈ L → a ∣ prod L := ih L h3.right assume h5 : a ∈ l rewrite [h3.left, prod_cons] --Goal : a ∣ b * prod L rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b ∨ a ∈ L by_cases on h5 · -- Case 1. h5 : a = b apply Exists.intro (prod L) rewrite [h5] rfl done · -- Case 2. h5 : a ∈ L have h6 : a ∣ prod L := h4 h5 have h7 : prod L ∣ b * prod L := by apply Exists.intro b ring done show a ∣ b * prod L from dvd_trans h6 h7 done done done lemma list_elt_dvd_prod {a : Nat} {l : List Nat} (h : a ∈ l) : a ∣ prod l := by set n : Nat := l.length have h1 : l.length = n := by rfl show a ∣ prod l from list_elt_dvd_prod_by_length a n l h1 h done lemma exists_prime_factorization : ∀ (n : Nat), n ≥ 1 → ∃ (l : List Nat), prime_factorization n l := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, n_1 ≥ 1 → ∃ (l : List Nat), prime_factorization n_1 l assume h1 : n ≥ 1 by_cases h2 : n = 1 · -- Case 1. h2 : n = 1 apply Exists.intro [] define apply And.intro · -- Proof of nondec_prime_list [] define show all_prime [] ∧ nondec [] from And.intro all_prime_nil nondec_nil done · -- Proof of prod [] = n rewrite [prod_nil, h2] rfl done done · -- Case 2. h2 : n ≠ 1 have h3 : n ≥ 2 := lt_of_le_of_ne' h1 h2 obtain (p : Nat) (h4 : prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q) from exists_least_prime_factor h3 have p_prime_factor : prime_factor p n := h4.left define at p_prime_factor have p_prime : prime p := p_prime_factor.left have p_dvd_n : p ∣ n := p_prime_factor.right have p_least : ∀ (q : Nat), prime_factor q n → p ≤ q := h4.right obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n have h5 : m ≠ 0 := by contradict h1 with h6 have h7 : n = 0 := calc n _ = p * m := n_eq_pm _ = p * 0 := by rw [h6] _ = 0 := by ring rewrite [h7] decide done have m_pos : 0 < m := Nat.pos_of_ne_zero h5 have m_lt_n : m < n := by define at p_prime show m < n from calc m _ < m + m := by linarith _ = 2 * m := by ring _ ≤ p * m := by rel [p_prime.left] _ = n := n_eq_pm.symm done obtain (L : List Nat) (h6 : prime_factorization m L) from ih m m_lt_n m_pos define at h6 have ndpl_L : nondec_prime_list L := h6.left define at ndpl_L apply Exists.intro (p :: L) define apply And.intro · -- Proof of nondec_prime_list (p :: L) define apply And.intro · -- Proof of all_prime (p :: L) rewrite [all_prime_cons] show prime p ∧ all_prime L from And.intro p_prime ndpl_L.left done · -- Proof of nondec (p :: L) rewrite [nondec_cons] apply And.intro _ ndpl_L.right fix q : Nat assume q_in_L : q ∈ L have h7 : q ∣ prod L := list_elt_dvd_prod q_in_L rewrite [h6.right] at h7 --h7 : q ∣ m have h8 : m ∣ n := by apply Exists.intro p rewrite [n_eq_pm] ring done have q_dvd_n : q ∣ n := dvd_trans h7 h8 have ap_L : all_prime L := ndpl_L.left define at ap_L have q_prime_factor : prime_factor q n := And.intro (ap_L q q_in_L) q_dvd_n show p ≤ q from p_least q q_prime_factor done done · -- Proof of prod (p :: L) = n rewrite [prod_cons, h6.right, n_eq_pm] rfl done done done theorem Theorem_7_2_2 {a b c : Nat} (h1 : c ∣ a * b) (h2 : rel_prime a c) : c ∣ b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑c ∣ ↑b define at h1; define at h2; define obtain (j : Nat) (h3 : a * b = c * j) from h1 set s : Int := gcd_c1 a c set t : Int := gcd_c2 a c have h4 : s * ↑a + t * ↑c = ↑(gcd a c) := gcd_lin_comb c a rewrite [h2, Nat.cast_one] at h4 --h4 : s * ↑a + t * ↑c = (1 : Int) apply Exists.intro (s * ↑j + t * ↑b) show ↑b = ↑c * (s * ↑j + t * ↑b) from calc ↑b _ = (1 : Int) * ↑b := (one_mul _).symm _ = (s * ↑a + t * ↑c) * ↑b := by rw [h4] _ = s * (↑a * ↑b) + t * ↑c * ↑b := by ring _ = s * (↑c * ↑j) + t * ↑c * ↑b := by rw [←Nat.cast_mul a b, h3, Nat.cast_mul c j] _ = ↑c * (s * ↑j + t * ↑b) := by ring done lemma le_nonzero_prod_left {a b : Nat} (h : a * b ≠ 0) : a ≤ a * b := by have h1 : b ≠ 0 := by contradict h with h1 rewrite [h1] ring done have h2 : 1 ≤ b := Nat.pos_of_ne_zero h1 show a ≤ a * b from calc a = a * 1 := (mul_one a).symm _ ≤ a * b := by rel [h2] done lemma le_nonzero_prod_right {a b : Nat} (h : a * b ≠ 0) : b ≤ a * b := by rewrite [mul_comm] rewrite [mul_comm] at h show b ≤ b * a from le_nonzero_prod_left h done lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry lemma rel_prime_of_prime_not_dvd {a p : Nat} (h1 : prime p) (h2 : ¬p ∣ a) : rel_prime a p := by have h3 : gcd a p ∣ a := gcd_dvd_left a p have h4 : gcd a p ∣ p := gcd_dvd_right a p have h5 : gcd a p = 1 ∨ gcd a p = p := dvd_prime h1 h4 have h6 : gcd a p ≠ p := by contradict h2 with h6 rewrite [h6] at h3 show p ∣ a from h3 done disj_syll h5 h6 show rel_prime a p from h5 done theorem Theorem_7_2_3 {a b p : Nat} (h1 : prime p) (h2 : p ∣ a * b) : p ∣ a ∨ p ∣ b := by or_right with h3 have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3 show p ∣ b from Theorem_7_2_2 h2 h4 done lemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a ≥ 1 := by have h1 : a ≠ 0 := by by_contra h1 rewrite [h1] at h contradict h linarith done show a ≥ 1 from Nat.pos_of_ne_zero h1 done lemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by have h1 : a ≥ 1 := ge_one_of_prod_one h have h2 : a * b ≠ 0 := by linarith have h3 : a ≤ a * b := le_nonzero_prod_left h2 rewrite [h] at h3 show a = 1 from Nat.le_antisymm h3 h1 done lemma eq_one_of_dvd_one {n : Nat} (h : n ∣ 1) : n = 1 := by obtain (j : Nat) (h1 : 1 = n * j) from h show n = 1 from eq_one_of_prod_one h1.symm done lemma prime_not_one {p : Nat} (h : prime p) : p ≠ 1 := by define at h linarith done theorem Theorem_7_2_4 {p : Nat} (h1 : prime p) : ∀ (l : List Nat), p ∣ prod l → ∃ a ∈ l, p ∣ a := by apply List.rec · -- Base Case. Goal : p ∣ prod [] → ∃ a ∈ [], p ∣ a rewrite [prod_nil] assume h2 : p ∣ 1 show ∃ a ∈ [], p ∣ a from absurd (eq_one_of_dvd_one h2) (prime_not_one h1) done · -- Induction Step fix b : Nat fix L : List Nat assume ih : p ∣ prod L → ∃ a ∈ L, p ∣ a --Goal : p ∣ prod (b :: L) → ∃ a ∈ b :: L, p ∣ a assume h2 : p ∣ prod (b :: L) rewrite [prod_cons] at h2 have h3 : p ∣ b ∨ p ∣ prod L := Theorem_7_2_3 h1 h2 by_cases on h3 · -- Case 1. h3 : p ∣ b apply Exists.intro b show b ∈ b :: L ∧ p ∣ b from And.intro (List.mem_cons_self b L) h3 done · -- Case 2. h3 : p ∣ prod L obtain (a : Nat) (h4 : a ∈ L ∧ p ∣ a) from ih h3 apply Exists.intro a show a ∈ b :: L ∧ p ∣ a from And.intro (List.mem_cons_of_mem b h4.left) h4.right done done done lemma prime_in_list {p : Nat} {l : List Nat} (h1 : prime p) (h2 : all_prime l) (h3 : p ∣ prod l) : p ∈ l := by obtain (a : Nat) (h4 : a ∈ l ∧ p ∣ a) from Theorem_7_2_4 h1 l h3 define at h2 have h5 : prime a := h2 a h4.left have h6 : p = 1 ∨ p = a := dvd_prime h5 h4.right disj_syll h6 (prime_not_one h1) rewrite [h6] show a ∈ l from h4.left done lemma first_le_first {p q : Nat} {l m : List Nat} (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m)) (h3 : prod (p :: l) = prod (q :: m)) : p ≤ q := by define at h1; define at h2 have h4 : q ∣ prod (p :: l) := by define apply Exists.intro (prod m) rewrite [←prod_cons] show prod (p :: l) = prod (q :: m) from h3 done have h5 : all_prime (q :: m) := h2.left rewrite [all_prime_cons] at h5 have h6 : q ∈ p :: l := prime_in_list h5.left h1.left h4 have h7 : nondec (p :: l) := h1.right rewrite [nondec_cons] at h7 rewrite [List.mem_cons] at h6 by_cases on h6 · -- Case 1. h6 : q = p linarith done · -- Case 2. h6 : q ∈ l have h8 : ∀ m ∈ l, p ≤ m := h7.left show p ≤ q from h8 q h6 done done lemma nondec_prime_list_tail {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by define at h define rewrite [all_prime_cons, nondec_cons] at h show all_prime l ∧ nondec l from And.intro h.left.right h.right.right done lemma cons_prod_not_one {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : prod (p :: l) ≠ 1 := by define at h have h1 : all_prime (p :: l) := h.left rewrite [all_prime_cons] at h1 rewrite [prod_cons] by_contra h2 show False from (prime_not_one h1.left) (eq_one_of_prod_one h2) done lemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) : l = [] ↔ prod l = 1 := by apply Iff.intro · -- (→) assume h1 : l = [] rewrite [h1] show prod [] = 1 from prod_nil done · -- (←) contrapos assume h1 : ¬l = [] obtain (p : Nat) (h2 : ∃ (L : List Nat), l = p :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List Nat) (h3 : l = p :: L) from h2 rewrite [h3] at h rewrite [h3] show ¬prod (p :: L) = 1 from cons_prod_not_one h done done lemma prime_pos {p : Nat} (h : prime p) : p > 0 := by define at h linarith done theorem Theorem_7_2_5 : ∀ (l1 l2 : List Nat), nondec_prime_list l1 → nondec_prime_list l2 → prod l1 = prod l2 → l1 = l2 := by apply List.rec · -- Base Case. Goal : ∀ (l2 : List Nat), nondec_prime_list [] → -- nondec_prime_list l2 → prod [] = prod l2 → [] = l2 fix l2 : List Nat assume h1 : nondec_prime_list [] assume h2 : nondec_prime_list l2 assume h3 : prod [] = prod l2 rewrite [prod_nil, eq_comm, ←list_nil_iff_prod_one h2] at h3 show [] = l2 from h3.symm done · -- Induction Step fix p : Nat fix L1 : List Nat assume ih : ∀ (L2 : List Nat), nondec_prime_list L1 → nondec_prime_list L2 → prod L1 = prod L2 → L1 = L2 -- Goal : ∀ (l2 : List Nat), nondec_prime_list (p :: L1) → -- nondec_prime_list l2 → prod (p :: L1) = prod l2 → p :: L1 = l2 fix l2 : List Nat assume h1 : nondec_prime_list (p :: L1) assume h2 : nondec_prime_list l2 assume h3 : prod (p :: L1) = prod l2 have h4 : ¬prod (p :: L1) = 1 := cons_prod_not_one h1 rewrite [h3, ←list_nil_iff_prod_one h2] at h4 obtain (q : Nat) (h5 : ∃ (L : List Nat), l2 = q :: L) from List.exists_cons_of_ne_nil h4 obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5 rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2) rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2) have h7 : p ≤ q := first_le_first h1 h2 h3 have h8 : q ≤ p := first_le_first h2 h1 h3.symm have h9 : p = q := by linarith rewrite [h9, prod_cons, prod_cons] at h3 --h3 : q * prod L1 = q * prod L2 have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1 have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2 define at h2 have h12 : all_prime (q :: L2) := h2.left rewrite [all_prime_cons] at h12 have h13 : q > 0 := prime_pos h12.left have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3 have h15 : L1 = L2 := ih L2 h10 h11 h14 rewrite [h6, h9, h15] rfl done done theorem fund_thm_arith (n : Nat) (h : n ≥ 1) : ∃! (l : List Nat), prime_factorization n l := by exists_unique · -- Existence show ∃ (l : List Nat), prime_factorization n l from exists_prime_factorization n h done · -- Uniqueness fix l1 : List Nat; fix l2 : List Nat assume h1 : prime_factorization n l1 assume h2 : prime_factorization n l2 define at h1; define at h2 have h3 : prod l1 = n := h1.right rewrite [←h2.right] at h3 show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3 done done /- Section 7.3 -/ theorem congr_refl (m : Nat) : ∀ (a : Int), a ≡ a (MOD m) := by fix a : Int define --Goal : ∃ (c : Int), a - a = ↑m * c apply Exists.intro 0 ring done theorem congr_symm {m : Nat} : ∀ {a b : Int}, a ≡ b (MOD m) → b ≡ a (MOD m) := by fix a : Int; fix b : Int assume h1 : a ≡ b (MOD m) define at h1 --h1 : ∃ (c : Int), a - b = ↑m * c define --Goal : ∃ (c : Int), b - a = ↑m * c obtain (c : Int) (h2 : a - b = m * c) from h1 apply Exists.intro (-c) show b - a = m * (-c) from calc b - a _ = -(a - b) := by ring _ = -(m * c) := by rw [h2] _ = m * (-c) := by ring done theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry /- Fundamental properties of congruence classes -/ lemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) : X = Y ↔ X.val = Y.val := Fin.ext_iff lemma val_nat_eq_mod (n k : Nat) : ([k]_(n + 1)).val = k % (n + 1) := by rfl lemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl theorem cc_rep {m : Nat} (X : ZMod m) : ∃ (a : Int), X = [a]_m := match m with | 0 => by apply Exists.intro X rfl done | n + 1 => by apply Exists.intro ↑(X.val) have h1 : X.val < n + 1 := Fin.prop X rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1] rfl done theorem add_class (m : Nat) (a b : Int) : [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm theorem mul_class (m : Nat) (a b : Int) : [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm lemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ [a - b]_m = [0]_m := by apply Iff.intro · -- (→) assume h1 : [a]_m = [b]_m have h2 : a - b = a + (-b) := by ring have h3 : b + (-b) = 0 := by ring show [a - b]_m = [0]_m from calc [a - b]_m _ = [a + (-b)]_m := by rw [h2] _ = [a]_m + [-b]_m := by rw [add_class] _ = [b]_m + [-b]_m := by rw [h1] _ = [b + -b]_m := by rw [add_class] _ = [0]_m := by rw [h3] done · -- (←) assume h1 : [a - b]_m = [0]_m have h2 : b + (a - b) = a := by ring have h3 : b + 0 = b := by ring show [a]_m = [b]_m from calc [a]_m _ = [b + (a - b)]_m := by rw [h2] _ = [b]_m + [a - b]_m := by rw [add_class] _ = [b]_m + [0]_m := by rw [h1] _ = [b + 0]_m := by rw [add_class] _ = [b]_m := by rw [h3] done done lemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) : [a]_m = [0]_m → [-a]_m = [0]_m := by assume h1 : [a]_m = [0]_m have h2 : 0 + (-a) = -a := by ring have h3 : a + (-a) = 0 := by ring show [-a]_m = [0]_m from calc [-a]_m _ = [0 + (-a)]_m := by rw [h2] _ = [0]_m + [-a]_m := by rw [add_class] _ = [a]_m + [-a]_m := by rw [h1] _ = [a + (-a)]_m := by rw [add_class] _ = [0]_m := by rw [h3] done lemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) : [-a]_m = [0]_m ↔ [a]_m = [0]_m := by apply Iff.intro _ (cc_neg_zero_of_cc_zero m a) assume h1 : [-a]_m = [0]_m have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1 have h3 : -(-a) = a := by ring rewrite [h3] at h2 show [a]_m = [0]_m from h2 done lemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl lemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m ↔ m ∣ k := match m with | 0 => by have h : (0 : Int) = (↑(0 : Nat) : Int) := by rfl rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj] apply Iff.intro · -- (→) assume h1 : k = 0 rewrite [h1] show 0 ∣ 0 from dvd_self 0 done · -- (←) assume h1 : 0 ∣ k obtain (c : Nat) (h2 : k = 0 * c) from h1 rewrite [h2] ring done done | n + 1 => by rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero] show k % (n + 1) = 0 ↔ n + 1 ∣ k from (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm done lemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m ↔ ↑m ∣ a := by obtain (k : Nat) (h1 : a = ↑k ∨ a = -↑k) from Int.eq_nat_or_neg a by_cases on h1 · -- Case 1. h1: a = ↑k rewrite [h1, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done · -- Case 2. h1: a = -↑k rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done done theorem cc_eq_iff_congr (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ a ≡ b (MOD m) := calc [a]_m = [b]_m _ ↔ [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b _ ↔ ↑m ∣ (a - b) := cc_zero_iff_dvd m (a - b) _ ↔ a ≡ b (MOD m) := by rfl /- End of fundamental properties of congruence classes -/ lemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m := by have h1 : (↑m : Int) ≠ 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m) show 0 ≤ a % m from Int.emod_nonneg a h1 done lemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m) have h2 : (↑m : Int) > 0 := (Nat.cast_pos).rtl h1 show a % m < m from Int.emod_lt_of_pos a h2 done lemma congr_mod_mod (m : Nat) (a : Int) : a ≡ a % m (MOD m) := by define have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m apply Exists.intro (a / m) show a - a % m = m * (a / m) from calc a - (a % m) _ = m * (a / m) + a % m - a % m := by rw [h1] _ = m * (a / m) := by ring done lemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) := And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a)) theorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) : ∃! (r : Int), 0 ≤ r ∧ r < m ∧ a ≡ r (MOD m) := by exists_unique · -- Existence apply Exists.intro (a % m) show 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) from mod_cmpl_res m a done · -- Uniqueness fix r1 : Int; fix r2 : Int assume h1 : 0 ≤ r1 ∧ r1 < m ∧ a ≡ r1 (MOD m) assume h2 : 0 ≤ r2 ∧ r2 < m ∧ a ≡ r2 (MOD m) have h3 : r1 ≡ r2 (MOD m) := congr_trans (congr_symm h1.right.right) h2.right.right obtain (d : Int) (h4 : r1 - r2 = m * d) from h3 have h5 : r1 - r2 < m * 1 := by linarith have h6 : m * (-1) < r1 - r2 := by linarith rewrite [h4] at h5 --h5 : m * d < m * 1 rewrite [h4] at h6 --h6 : m * -1 < m * d have h7 : (↑m : Int) ≥ 0 := Nat.cast_nonneg m have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7 have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7 have h10 : d = 0 := by linarith show r1 = r2 from calc r1 _ = r1 - r2 + r2 := by ring _ = m * 0 + r2 := by rw [h4, h10] _ = r2 := by ring done done lemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m := (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a) theorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y rewrite [h1, h2] have h3 : a + b = b + a := by ring show [a]_m + [b]_m = [b]_m + [a]_m from calc [a]_m + [b]_m _ = [a + b]_m := add_class m a b _ = [b + a]_m := by rw [h3] _ = [b]_m + [a]_m := (add_class m b a).symm done theorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X rewrite [h1] have h2 : a * 1 = a := by ring show [a]_m * [1]_m = [a]_m from calc [a]_m * [1]_m _ = [a * 1]_m := mul_class m a 1 _ = [a]_m := by rw [h2] done theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry lemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) : [a]_m * [gcd_c2 m a]_m = [1]_m := by set s : Int := gcd_c1 m a have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m define at h1 rewrite [h1, Nat.cast_one] at h2 --h2 : s * ↑m + gcd_c2 m a * ↑a = 1 rewrite [mul_class, cc_eq_iff_congr] define --Goal : ∃ (c : Int), ↑a * gcd_c2 m a - 1 = ↑m * c apply Exists.intro (-s) show a * (gcd_c2 m a) - 1 = m * (-s) from calc a * (gcd_c2 m a) - 1 _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring _ = 1 + m * (-s) - 1 := by rw [h2] _ = m * (-s) := by ring done theorem Theorem_7_3_7 (m a : Nat) : invertible [a]_m ↔ rel_prime m a := by apply Iff.intro · -- (→) assume h1 : invertible [a]_m define at h1 obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1 obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y rewrite [h3, mul_class, cc_eq_iff_congr] at h2 define at h2 obtain (c : Int) (h4 : a * b - 1 = m * c) from h2 rewrite [Exercise_7_2_6] --Goal : ∃ (s t : Int), s * ↑m + t * ↑a = 1 apply Exists.intro (-c) apply Exists.intro b show (-c) * m + b * a = 1 from calc (-c) * m + b * a _ = (-c) * m + (a * b - 1) + 1 := by ring _ = (-c) * m + m * c + 1 := by rw [h4] _ = 1 := by ring done · -- (←) assume h1 : rel_prime m a define show ∃ (Y : ZMod m), [a]_m * Y = [1]_m from Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1) done done /- Section 7.4 -/ section Euler open Euler lemma num_rp_below_base {m : Nat} : num_rp_below m 0 = 0 := by rfl lemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) : num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by have h1 : num_rp_below m (j + 1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : gcd m j = 1 rewrite [if_pos h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1 show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1 done lemma num_rp_below_step_not_rp {m j : Nat} (h : ¬rel_prime m j) : num_rp_below m (j + 1) = num_rp_below m j := by have h1 : num_rp_below m (j +1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : ¬gcd m j = 1 rewrite [if_neg h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j show num_rp_below m (j + 1) = num_rp_below m j from h1 done lemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl #eval phi 10 --Answer: 4 lemma prod_inv_iff_inv {m : Nat} {X : ZMod m} (h1 : invertible X) (Y : ZMod m) : invertible (X * Y) ↔ invertible Y := by apply Iff.intro · -- (→) assume h2 : invertible (X * Y) obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2 apply Exists.intro (X * Z) rewrite [←h3] --Goal : Y * (X * Z) = X * Y * Z ring --Note that ring can do algebra in ZMod m done · -- (←) assume h2 : invertible Y obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1 obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2 apply Exists.intro (Xi * Yi) show (X * Y) * (Xi * Yi) = [1]_m from calc X * Y * (Xi * Yi) _ = (X * Xi) * (Y * Yi) := by ring _ = [1]_m * [1]_m := by rw [h3, h4] _ = [1]_m := Theorem_7_3_6_7 [1]_m done done lemma F_rp_def {m i : Nat} (h : rel_prime m i) : F m i = [i]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h --h : gcd m i = 1 rewrite [if_pos h] at h1 show F m i = [i]_m from h1 done lemma F_not_rp_def {m i : Nat} (h : ¬rel_prime m i) : F m i = [1]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h rewrite [h1, if_neg h] rfl done lemma prod_seq_base {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 0 k f = [1]_m := by rfl lemma prod_seq_step {m : Nat} (n k : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl lemma prod_seq_zero_step {m : Nat} (n : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by rewrite [prod_seq_step, zero_add] rfl done lemma prod_one {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 1 k f = f k := by rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7] rfl done lemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl lemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m := calc [G m a i]_m _ = [(a * i) % m]_m := by rfl _ = [a * i]_m := (cc_eq_mod m (a * i)).symm _ = [a]_m * [i]_m := (mul_class m a i).symm lemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) : rel_prime m (G m a i) ↔ rel_prime m i := by have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1 show rel_prime m (G m a i) ↔ rel_prime m i from calc rel_prime m (G m a i) _ ↔ invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm _ ↔ invertible ([a]_m * [i]_m) := by rw [cc_G] _ ↔ invertible [i]_m := prod_inv_iff_inv h2 ([i]_m) _ ↔ rel_prime m i := Theorem_7_3_7 m i done lemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) : F m (G m a i) = [a]_m * F m i := by have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2 show F m (G m a i) = [a]_m * F m i from calc F m (G m a i) _ = [G m a i]_m := F_rp_def h3 _ = [a]_m * [i]_m := cc_G m a i _ = [a]_m * F m i := by rw [F_rp_def h2] done lemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : ¬rel_prime m i) : F m (G m a i) = [1]_m := by rewrite [←G_rp_iff h1 i] at h2 show F m (G m a i) = [1]_m from F_not_rp_def h2 done lemma FG_prod {m a : Nat} (h1 : rel_prime m a) : ∀ (k : Nat), prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by by_induc · -- Base Case show prod_seq 0 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from calc prod_seq 0 0 ((F m) ∘ (G m a)) _ = [1]_m := prod_seq_base _ _ _ = [a]_m ^ 0 * [1]_m := by ring _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by rw [num_rp_below_base, prod_seq_base] done · -- Induction Step fix k : Nat assume ih : prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) by_cases h2 : rel_prime m k · -- Case 1. h2 : rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([a]_m * F m k) := by rw [FG_rp h1 h2] _ = [a]_m ^ ((num_rp_below m k) + 1) * ((prod_seq k 0 (F m)) * F m k) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_rp h2, prod_seq_zero_step] done · -- Case 2. h2 : ¬rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([1]_m) := by rw [FG_not_rp h1 h2] _ = [a]_m ^ (num_rp_below m k) * (prod_seq k 0 (F m) * ([1]_m)) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_not_rp h2, prod_seq_zero_step, F_not_rp_def h2] done done done lemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by define --Goal : ∀ i < m, G m a i < m fix i : Nat assume h1 : i < m rewrite [G_def] --Goal : a * i % m < m show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m) done lemma left_inv_one_one_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g' (g i) = i) : one_one_below n g := sorry lemma right_inv_onto_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g (g' i) = i) (h2 : maps_below n g') : onto_below n g := by define at h2; define fix k : Nat assume h3 : k < n apply Exists.intro (g' k) show g' k < n ∧ g (g' k) = k from And.intro (h2 k h3) (h1 k h3) done lemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m * [inv_mod m a]_m = [1]_m := by have h2 : 0 ≤ (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a) show [a]_m * [inv_mod m a]_m = [1]_m from calc [a]_m * [inv_mod m a]_m _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2] _ = [a]_m * [gcd_c2 m a]_m := by rw [←cc_eq_mod] _ = [1]_m := gcd_c2_inv h1 done lemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m := calc a * (b % m) % m = a % m * (b % m % m) % m := Nat.mul_mod _ _ _ _ = a % m * (b % m) % m := by rw [Nat.mod_mod] _ = a * b % m := (Nat.mul_mod _ _ _).symm lemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm] rfl done theorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] : ↑a ≡ ↑b (MOD m) ↔ a % m = b % m := sorry lemma mul_inv_mod_cancel {m a i : Nat} [NeZero m] (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1 rewrite [mul_class, cc_eq_iff_congr, ←Nat.cast_mul, ←Nat.cast_one, congr_iff_mod_eq_Nat] at h3 show a * inv_mod m a * i % m = i from calc a * (inv_mod m a) * i % m _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = 1 % m * i % m := by rw [h3] _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = i % m := by rw [one_mul] _ = i := Nat.mod_eq_of_lt h2 done lemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl lemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, G m a (Ginv m a i) = i := by fix i : Nat assume h2 : i < m show G m a (Ginv m a i) = i from calc G m a (Ginv m a i) _ = a * ((inv_mod m a * i) % m) % m := by rfl _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, Ginv m a (G m a i) = i := by fix i : Nat assume h2 : i < m show Ginv m a (G m a i) = i from calc Ginv m a (G m a i) _ = inv_mod m a * ((a * i) % m) % m := by rfl _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc, mul_comm (inv_mod m a)] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_maps_below (m a : Nat) [NeZero m] : maps_below m (Ginv m a) := G_maps_below m (inv_mod m a) lemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : one_one_below m (G m a) := left_inv_one_one_below (Ginv_left_inv h1) lemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : onto_below m (G m a) := right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a) lemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : perm_below m (G m a) := And.intro (G_maps_below m a) (And.intro (G_one_one_below h1) (G_onto_below h1)) --Permuting a product of congruence classes doesn't change product lemma swap_fst (u v : Nat) : swap u v u = v := by define : swap u v u --Goal : (if u = u then v else if u = v then u else u) = v have h : u = u := by rfl rewrite [if_pos h] rfl done lemma swap_snd (u v : Nat) : swap u v v = u := by define : swap u v v by_cases h1 : v = u · -- Case 1. h1 : v = u rewrite [if_pos h1] show v = u from h1 done · -- Case 2. h1 : v ≠ u rewrite [if_neg h1] have h2 : v = v := by rfl rewrite [if_pos h2] rfl done done lemma swap_other {u v i : Nat} (h1 : i ≠ u) (h2 : i ≠ v) : swap u v i = i := by define : swap u v i rewrite [if_neg h1, if_neg h2] rfl done lemma swap_values (u v i : Nat) : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := by by_cases h1 : i = u · -- Case 1. h1 : i = u apply Or.inl rewrite [h1] show swap u v u = v from swap_fst u v done · -- Case 2. h1 : i ≠ u apply Or.inr by_cases h2 : i = v · -- Case 2.1. h2 : i = v apply Or.inl rewrite [h2] show swap u v v = u from swap_snd u v done · -- Case 2.2. h2 : i ≠ v apply Or.inr show swap u v i = i from swap_other h1 h2 done done done lemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by define fix i : Nat assume h3 : i < n have h4 : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := swap_values u v i by_cases on h4 · -- Case 1. h4 : swap u v i = v rewrite [h4] show v < n from h2 done · -- Case 2. by_cases on h4 · -- Case 2.1. h4 : swap u v i = u rewrite [h4] show u < n from h1 done · -- Case 2.2. h4 : swap u v i = i rewrite [h4] show i < n from h3 done done done lemma swap_swap (u v n : Nat) : ∀ i < n, swap u v (swap u v i) = i := by fix i : Nat assume h : i < n by_cases h1 : i = u · -- Case 1. h1 : i = u rewrite [h1, swap_fst, swap_snd] rfl done · -- Case 2. h1 : i ≠ u by_cases h2 : i = v · -- Case 2.1. h2 : i = v rewrite [h2, swap_snd, swap_fst] rfl done · -- Case 2.2. h2 : i ≠ v rewrite [swap_other h1 h2, swap_other h1 h2] rfl done done done lemma swap_one_one_below (u v n) : one_one_below n (swap u v) := left_inv_one_one_below (swap_swap u v n) lemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) := right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2) lemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) := And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2)) lemma comp_perm_below {n : Nat} {f g : Nat → Nat} (h1 : perm_below n f) (h2 : perm_below n g) : perm_below n (f ∘ g) := sorry lemma trivial_swap (u : Nat) : swap u u = id := by apply funext fix x : Nat by_cases h1 : x = u · -- Case 1. h1 : x = u rewrite [h1, swap_fst] rfl done · -- Case 2. h1 : x ≠ u rewrite [swap_other h1 h1] rfl done done lemma prod_eq_fun {m : Nat} (f g : Nat → ZMod m) (k : Nat) : ∀ (n : Nat), (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g := by by_induc · -- Base Case assume h : (∀ i < 0, f (k + i) = g (k + i)) rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g assume h1 : ∀ i < n + 1, f (k + i) = g (k + i) have h2 : ∀ i < n, f (k + i) = g (k + i) := by fix i : Nat assume h2 : i < n have h3 : i < n + 1 := by linarith show f (k + i) = g (k + i) from h1 i h3 done have h3 : prod_seq n k f = prod_seq n k g := ih h2 have h4 : n < n + 1 := Nat.lt_succ_self n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4] rfl done done lemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := by have h2 : ∀ (i : Nat), i < u → (f ∘ swap u n) (0 + i) = f (0 + i) := by fix i : Nat assume h2 : i < u have h3 : 0 + i ≠ u := by linarith have h4 : 0 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl done show prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f from prod_eq_fun (f ∘ swap u n) f 0 u h2 done lemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := by have h2 : ∀ i < j, (f ∘ swap u n) (u + 1 + i) = f (u + 1 + i) := by fix i : Nat assume h2 : i < j have h3 : u + 1 + i ≠ u := by linarith have h4 : u + 1 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl show prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f from prod_eq_fun (f ∘ swap u n) f (u + 1) j h2 done lemma break_prod {m : Nat} (n : Nat) (f : Nat → ZMod m) : ∀ (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by by_induc · -- Base Case have h : n + 0 = n := by rfl rewrite [prod_seq_base, h, Theorem_7_3_6_7] rfl done · -- Induction Step fix j : Nat assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f rewrite [←add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc] rfl done done lemma break_prod_twice {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f := break_prod n f 1 rewrite [prod_one] at h2 have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f := break_prod (u + 1) f j rewrite [←h1] at h3 have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f := break_prod u f 1 rewrite [prod_one] at h4 rewrite [h3, h4] at h2 show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2 done lemma swap_prod_eq_prod {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f := by by_cases h2 : u = n · -- Case 1. h2 : u = n rewrite [h2, trivial_swap n] --Goal : prod_seq (n + 1) 0 (f ∘ id) = prod_seq (n + 1) 0 f rfl done · -- Case 2. h2 : ¬u = n have h3 : u + 1 ≤ n := Nat.lt_of_le_of_ne h1 h2 obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3 have break_f : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := break_prod_twice f h4 have break_fs : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_prod_twice (f ∘ swap u n) h4 have f_eq_fs_below : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := swap_prod_eq_prod_below f h1 have f_eq_fs_btwn : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4 show prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f from calc prod_seq (n + 1) 0 (f ∘ swap u n) _ = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_fs _ = prod_seq u 0 f * (f ∘ swap u n) u * prod_seq j (u + 1) f * (f ∘ swap u n) n := by rw [f_eq_fs_below, f_eq_fs_btwn] _ = prod_seq u 0 f * f (swap u n u) * prod_seq j (u + 1) f * f (swap u n n) := by rfl _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by rw [swap_fst, swap_snd] _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring _ = prod_seq (n + 1) 0 f := break_f.symm done done lemma perm_below_fixed {n : Nat} {g : Nat → Nat} (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry lemma perm_prod {m : Nat} (f : Nat → ZMod m) : ∀ (n : Nat), ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) := by by_induc · -- Base Case fix g : Nat → Nat assume h1 : perm_below 0 g rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) fix g : Nat → Nat assume g_pb : perm_below (n + 1) g define at g_pb have g_ob : onto_below (n + 1) g := g_pb.right.right define at g_ob have h1 : n < n + 1 := by linarith obtain (u : Nat) (h2 : u < n + 1 ∧ g u = n) from g_ob n h1 have s_pb : perm_below (n + 1) (swap u n) := swap_perm_below h2.left h1 have gs_pb_n1 : perm_below (n + 1) (g ∘ swap u n) := comp_perm_below g_pb s_pb have gs_fix_n : (g ∘ swap u n) n = n := calc (g ∘ swap u n) n _ = g (swap u n n) := by rfl _ = g u := by rw [swap_snd] _ = n := h2.right have gs_pb_n : perm_below n (g ∘ swap u n) := perm_below_fixed gs_pb_n1 gs_fix_n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f ∘ (g ∘ swap u n)) := ih (g ∘ swap u n) gs_pb_n have h3 : u ≤ n := by linarith show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f ∘ g) from calc prod_seq (n + 1) 0 f _ = prod_seq n 0 f * f n := prod_seq_zero_step n f _ = prod_seq n 0 (f ∘ (g ∘ swap u n)) * f ((g ∘ swap u n) n) := by rw [gs_prod, gs_fix_n] _ = prod_seq n 0 (f ∘ g ∘ swap u n) * (f ∘ g ∘ swap u n) n := by rfl _ = prod_seq (n + 1) 0 (f ∘ g ∘ swap u n) := (prod_seq_zero_step n (f ∘ g ∘ swap u n)).symm _ = prod_seq (n + 1) 0 ((f ∘ g) ∘ swap u n) := by rfl _ = prod_seq (n + 1) 0 (f ∘ g) := swap_prod_eq_prod (f ∘ g) h3 done done lemma F_invertible (m i : Nat) : invertible (F m i) := by by_cases h : rel_prime m i · -- Case 1. h : rel_prime m i rewrite [F_rp_def h] show invertible [i]_m from (Theorem_7_3_7 m i).rtl h done · -- Case 2. h : ¬rel_prime m i rewrite [F_not_rp_def h] apply Exists.intro [1]_m show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m done done lemma Fprod_invertible (m : Nat) : ∀ (k : Nat), invertible (prod_seq k 0 (F m)) := by by_induc · -- Base Case apply Exists.intro [1]_m show prod_seq 0 0 (F m) * [1]_m = [1]_m from calc prod_seq 0 0 (F m) * [1]_m _ = [1]_m * [1]_m := by rw [prod_seq_base] _ = [1]_m := Theorem_7_3_6_7 ([1]_m) done · -- Induction Step fix k : Nat assume ih : invertible (prod_seq k 0 (F m)) rewrite [prod_seq_zero_step] show invertible (prod_seq k 0 (F m) * (F m k)) from (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k) done done theorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m ^ (phi m) = [1]_m := by have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2 show [a]_m ^ (phi m) = [1]_m from calc [a]_m ^ (phi m) _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3] _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring _ = prod_seq m 0 (F m ∘ G m a) * Y := by rw [FG_prod h1 m, phi_def] _ = prod_seq m 0 (F m) * Y := by rw [perm_prod (F m) m (G m a) (G_perm_below h1)] _ = [1]_m := by rw [h3] done lemma Exercise_7_4_5_Int (m : Nat) (a : Int) : ∀ (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry lemma Exercise_7_4_5_Nat (m a n : Nat) : [a]_m ^ n = [a ^ n]_m := by rewrite [Exercise_7_4_5_Int] rfl done theorem Euler's_theorem {m a : Nat} [NeZero m] (h1 : rel_prime m a) : a ^ (phi m) ≡ 1 (MOD m) := by have h2 : [a]_m ^ (phi m) = [1]_m := Theorem_7_4_2 h1 rewrite [Exercise_7_4_5_Nat m a (phi m)] at h2 --h2 : [a ^ phi m]_m = [1]_m show a ^ (phi m) ≡ 1 (MOD m) from (cc_eq_iff_congr _ _ _).ltr h2 done #eval gcd 10 7 --Answer: 1. So 10 and 7 are relatively prime #eval 7 ^ phi 10 --Answer: 2401, which is congruent to 1 mod 10. end Euler /- Section 7.5 -/ lemma num_rp_prime {p : Nat} (h1 : prime p) : ∀ k < p, num_rp_below p (k + 1) = k := sorry lemma phi_prime {p : Nat} (h1 : prime p) : phi p = p - 1 := by have h2 : 1 ≤ p := prime_pos h1 have h3 : p - 1 + 1 = p := Nat.sub_add_cancel h2 have h4 : p - 1 < p := by linarith have h5 : num_rp_below p (p - 1 + 1) = p - 1 := num_rp_prime h1 (p - 1) h4 rewrite [h3] at h5 show phi p = p - 1 from h5 done theorem Theorem_7_2_2_Int {a c : Nat} {b : Int} (h1 : ↑c ∣ ↑a * b) (h2 : rel_prime a c) : ↑c ∣ b := by rewrite [Int.natCast_dvd, Int.natAbs_mul, Int.natAbs_ofNat] at h1 --h1 : c ∣ a * Int.natAbs b rewrite [Int.natCast_dvd] --Goal : c ∣ Int.natAbs b show c ∣ Int.natAbs b from Theorem_7_2_2 h1 h2 done lemma Lemma_7_4_5 {m n : Nat} (a b : Int) (h1 : rel_prime m n) : a ≡ b (MOD m * n) ↔ a ≡ b (MOD m) ∧ a ≡ b (MOD n) := by apply Iff.intro · -- (→) assume h2 : a ≡ b (MOD m * n) obtain (j : Int) (h3 : a - b = (m * n) * j) from h2 apply And.intro · -- Proof of a ≡ b (MOD m) apply Exists.intro (n * j) show a - b = m * (n * j) from calc a - b _ = m * n * j := h3 _ = m * (n * j) := by ring done · -- Proof of a ≡ b (MOD n) apply Exists.intro (m * j) show a - b = n * (m * j) from calc a - b _ = m * n * j := h3 _ = n * (m * j) := by ring done done · -- (←) assume h2 : a ≡ b (MOD m) ∧ a ≡ b (MOD n) obtain (j : Int) (h3 : a - b = m * j) from h2.left have h4 : (↑n : Int) ∣ a - b := h2.right rewrite [h3] at h4 --h4 : ↑n ∣ ↑m * j have h5 : ↑n ∣ j := Theorem_7_2_2_Int h4 h1 obtain (k : Int) (h6 : j = n * k) from h5 apply Exists.intro k --Goal : a - b = ↑(m * n) * k rewrite [Nat.cast_mul] --Goal : a - b = ↑m * ↑n * k show a - b = (m * n) * k from calc a - b _ = m * j := h3 _ = m * (n * k) := by rw [h6] _ = (m * n) * k := by ring done done --From exercises of Section 7.2 theorem rel_prime_symm {a b : Nat} (h : rel_prime a b) : rel_prime b a := sorry lemma prime_NeZero {p : Nat} (h : prime p) : NeZero p := by rewrite [neZero_iff] --Goal : p ≠ 0 define at h linarith done lemma Lemma_7_5_1 {p e d m c s : Nat} {t : Int} (h1 : prime p) (h2 : e * d = (p - 1) * s + 1) (h3 : m ^ e - c = p * t) : c ^ d ≡ m (MOD p) := by have h4 : m ^ e ≡ c (MOD p) := Exists.intro t h3 have h5 : [m ^ e]_p = [c]_p := (cc_eq_iff_congr _ _ _).rtl h4 rewrite [←Exercise_7_4_5_Nat] at h5 --h5 : [m]_p ^ e = [c]_p by_cases h6 : p ∣ m · -- Case 1. h6 : p ∣ m have h7 : m ≡ 0 (MOD p) := by obtain (j : Nat) (h8 : m = p * j) from h6 apply Exists.intro (↑j : Int) --Goal : ↑m - 0 = ↑p * ↑j rewrite [h8, Nat.cast_mul] ring done have h8 : [m]_p = [0]_p := (cc_eq_iff_congr _ _ _).rtl h7 have h9 : e * d ≠ 0 := by rewrite [h2] show (p - 1) * s + 1 ≠ 0 from Nat.add_one_ne_zero _ done have h10 : (0 : Int) ^ (e * d) = 0 := zero_pow h9 have h11 : [c ^ d]_p = [m]_p := calc [c ^ d]_p _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat] _ = ([m]_p ^ e) ^ d := by rw [h5] _ = [m]_p ^ (e * d) := by ring _ = [0]_p ^ (e * d) := by rw [h8] _ = [0 ^ (e * d)]_p := Exercise_7_4_5_Int _ _ _ _ = [0]_p := by rw [h10] _ = [m]_p := by rw [h8] show c ^ d ≡ m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11 done · -- Case 2. h6 : ¬p ∣ m have h7 : rel_prime m p := rel_prime_of_prime_not_dvd h1 h6 have h8 : rel_prime p m := rel_prime_symm h7 have h9 : NeZero p := prime_NeZero h1 have h10 : (1 : Int) ^ s = 1 := by ring have h11 : [c ^ d]_p = [m]_p := calc [c ^ d]_p _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat] _ = ([m]_p ^ e) ^ d := by rw [h5] _ = [m]_p ^ (e * d) := by ring _ = [m]_p ^ ((p - 1) * s + 1) := by rw [h2] _ = ([m]_p ^ (p - 1)) ^ s * [m]_p := by ring _ = ([m]_p ^ (phi p)) ^ s * [m]_p := by rw [phi_prime h1] _ = [1]_p ^ s * [m]_p := by rw [Theorem_7_4_2 h8] _ = [1 ^ s]_p * [m]_p := by rw [Exercise_7_4_5_Int] _ = [1]_p * [m]_p := by rw [h10] _ = [m]_p * [1]_p := by ring _ = [m]_p := Theorem_7_3_6_7 _ show c ^ d ≡ m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11 done done theorem Theorem_7_5_1 (p q n e d k m c : Nat) (p_prime : prime p) (q_prime : prime q) (p_ne_q : p ≠ q) (n_pq : n = p * q) (ed_congr_1 : e * d = k * (p - 1) * (q - 1) + 1) (h1 : [m]_n ^ e = [c]_n) : [c]_n ^ d = [m]_n := by rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] at h1 --h1 : m ^ e ≡ c (MOD n) rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] --Goal : c ^ d ≡ m (MOD n) obtain (j : Int) (h2 : m ^ e - c = n * j) from h1 rewrite [n_pq, Nat.cast_mul] at h2 --h2 : m ^ e - c = p * q * j have h3 : e * d = (p - 1) * (k * (q - 1)) + 1 := by rewrite [ed_congr_1] ring done have h4 : m ^ e - c = p * (q * j) := by rewrite [h2] ring done have congr_p : c ^ d ≡ m (MOD p) := Lemma_7_5_1 p_prime h3 h4 have h5 : e * d = (q - 1) * (k * (p - 1)) + 1 := by rewrite [ed_congr_1] ring done have h6 : m ^ e - c = q * (p * j) := by rewrite [h2] ring done have congr_q : c ^ d ≡ m (MOD q) := Lemma_7_5_1 q_prime h5 h6 have h7 : ¬q ∣ p := by by_contra h8 have h9 : q = 1 ∨ q = p := dvd_prime p_prime h8 disj_syll h9 (prime_not_one q_prime) show False from p_ne_q h9.symm done have h8 : rel_prime p q := rel_prime_of_prime_not_dvd q_prime h7 rewrite [n_pq, Lemma_7_4_5 _ _ h8] show c ^ d ≡ m (MOD p) ∧ c ^ d ≡ m (MOD q) from And.intro congr_p congr_q done /- BEGIN EXERCISES -/ namespace Exercises /- Section 7.1 -/ -- 1. theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry -- 2. lemma gcd_comm_lt {a b : Nat} (h : a < b) : gcd a b = gcd b a := sorry theorem gcd_comm (a b : Nat) : gcd a b = gcd b a := sorry -- 3. theorem Exercise_7_1_5 (a b : Nat) (n : Int) : (∃ (s t : Int), s * a + t * b = n) ↔ (↑(gcd a b) : Int) ∣ n := sorry -- 4. theorem Exercise_7_1_6 (a b c : Nat) : gcd a b = gcd (a + b * c) b := sorry -- 5. theorem gcd_is_nonzero {a b : Nat} (h : a ≠ 0 ∨ b ≠ 0) : gcd a b ≠ 0 := sorry -- 6. theorem gcd_greatest {a b d : Nat} (h1 : gcd a b ≠ 0) (h2 : d ∣ a) (h3 : d ∣ b) : d ≤ gcd a b := sorry -- 7. lemma Lemma_7_1_10a {a b : Nat} (n : Nat) (h : a ∣ b) : (n * a) ∣ (n * b) := sorry lemma Lemma_7_1_10b {a b n : Nat} (h1 : n ≠ 0) (h2 : (n * a) ∣ (n * b)) : a ∣ b := sorry lemma Lemma_7_1_10c {a b : Nat} (h1 : a ∣ b) (h2 : b ∣ a) : a = b := sorry theorem Exercise_7_1_10 (a b n : Nat) : gcd (n * a) (n * b) = n * gcd a b := sorry /- Section 7.2 -/ -- 1.

lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p

HTPI.Exercises.dvd_prime

htpi/HTPILib/Chap7.lean

htpi

/- Copyright 2023 Daniel J. Velleman -/ import HTPILib.Chap6 namespace HTPI /- Definitions -/ lemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by have h : n + 1 > 0 := Nat.succ_pos n show a % (n + 1) < n + 1 from Nat.mod_lt a h done def gcd (a b : Nat) : Nat := match b with | 0 => a | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd (n + 1) (a % (n + 1)) termination_by b mutual def gcd_c1 (a b : Nat) : Int := match b with | 0 => 1 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c2 (n + 1) (a % (n + 1)) --Corresponds to s = t' termination_by b def gcd_c2 (a b : Nat) : Int := match b with | 0 => 0 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c1 (n + 1) (a % (n + 1)) - (gcd_c2 (n + 1) (a % (n + 1))) * ↑(a / (n + 1)) --Corresponds to t = s' - t'q termination_by b end def prime (n : Nat) : Prop := 2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n def prime_factor (p n : Nat) : Prop := prime p ∧ p ∣ n def all_prime (l : List Nat) : Prop := ∀ p ∈ l, prime p def nondec (l : List Nat) : Prop := match l with | [] => True --Of course, True is a proposition that is always true | n :: L => (∀ m ∈ L, n ≤ m) ∧ nondec L def nondec_prime_list (l : List Nat) : Prop := all_prime l ∧ nondec l def prod (l : List Nat) : Nat := match l with | [] => 1 | n :: L => n * (prod L) def prime_factorization (n : Nat) (l : List Nat) : Prop := nondec_prime_list l ∧ prod l = n def rel_prime (a b : Nat) : Prop := gcd a b = 1 def congr_mod (m : Nat) (a b : Int) : Prop := (↑m : Int) ∣ (a - b) def cc (m : Nat) (a : Int) : ZMod m := (↑a : ZMod m) notation:50 a " ≡ " b " (MOD " m ")" => congr_mod m a b notation:max "["a"]_"m:max => cc m a def invertible {m : Nat} (X : ZMod m) : Prop := ∃ (Y : ZMod m), X * Y = [1]_m def num_rp_below (m k : Nat) : Nat := match k with | 0 => 0 | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j def phi (m : Nat) : Nat := num_rp_below m m def prod_seq {m : Nat} (j k : Nat) (f : Nat → ZMod m) : ZMod m := match j with | 0 => [1]_m | n + 1 => prod_seq n k f * f (k + n) def maps_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i < n, g i < n def one_one_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i1 < n, ∀ i2 < n, g i1 = g i2 → i1 = i2 def onto_below (n : Nat) (g : Nat → Nat) : Prop := ∀ k < n, ∃ i < n, g i = k def perm_below (n : Nat) (g : Nat → Nat) : Prop := maps_below n g ∧ one_one_below n g ∧ onto_below n g def inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m) def swap (u v i : Nat) : Nat := if i = u then v else if i = v then u else i namespace Euler --For definitions specific to Euler's theorem def F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m def G (m a i : Nat) : Nat := (a * i) % m def Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i end Euler /- Section 7.1 -/ theorem dvd_mod_of_dvd_a_b {a b d : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ (a % b) := by set q : Nat := a / b have h3 : b * q + a % b = a := Nat.div_add_mod a b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 define --Goal : ∃ (c : Nat), a % b = d * c apply Exists.intro (j - k * q) show a % b = d * (j - k * q) from calc a % b _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm _ = a - b * q := by rw [h3] _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc] _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm done theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry #eval gcd 672 161 --Answer: 7 lemma gcd_base (a : Nat) : gcd a 0 = a := by rfl lemma gcd_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd a b = gcd b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1)) rfl done lemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b ≠ 0) : a % b < b := by have h1 : b > 0 := Nat.pos_of_ne_zero h show a % b < b from Nat.mod_lt a h1 done lemma dvd_self (n : Nat) : n ∣ n := by apply Exists.intro 1 ring done theorem gcd_dvd : ∀ (b a : Nat), (gcd a b) ∣ a ∧ (gcd a b) ∣ b := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd a b_1) ∣ a ∧ (gcd a b_1) ∣ b_1 fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_base] --Goal: a ∣ a ∧ a ∣ 0 apply And.intro (dvd_self a) define apply Exists.intro 0 rfl done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_nonzero a h1] --Goal : gcd b (a % b) ∣ a ∧ gcd b (a % b) ∣ b have h2 : a % b < b := mod_nonzero_lt a h1 have h3 : (gcd b (a % b)) ∣ b ∧ (gcd b (a % b)) ∣ (a % b) := ih (a % b) h2 b apply And.intro _ h3.left show (gcd b (a % b)) ∣ a from dvd_a_of_dvd_b_mod h3.left h3.right done done theorem gcd_dvd_left (a b : Nat) : (gcd a b) ∣ a := (gcd_dvd b a).left theorem gcd_dvd_right (a b : Nat) : (gcd a b) ∣ b := (gcd_dvd b a).right lemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl lemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c1 a b = gcd_c2 b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done lemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl lemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * ↑(a / b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done theorem gcd_lin_comb : ∀ (b a : Nat), (gcd_c1 a b) * ↑a + (gcd_c2 a b) * ↑b = ↑(gcd a b) := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd_c1 a b_1) * ↑a + (gcd_c2 a b_1) * ↑b_1 = ↑(gcd a b_1) fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base] --Goal : 1 * ↑a + 0 * ↑0 = ↑a ring done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1] --Goal : gcd_c2 b (a % b) * ↑a + -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * ↑(a / b)) * ↑b = -- ↑(gcd b (a % b)) set r : Nat := a % b set q : Nat := a / b set s : Int := gcd_c1 b r set t : Int := gcd_c2 b r --Goal : t * ↑a + (s - t * ↑q) * ↑b = ↑(gcd b r) have h2 : r < b := mod_nonzero_lt a h1 have h3 : s * ↑b + t * ↑r = ↑(gcd b r) := ih r h2 b have h4 : b * q + r = a := Nat.div_add_mod a b rewrite [←h3, ←h4] rewrite [Nat.cast_add, Nat.cast_mul] --Goal : t * (↑b * ↑q + ↑r) + (s - t * ↑q) * ↑b = s * ↑b + t * ↑r ring done done #eval gcd_c1 672 161 --Answer: 6 #eval gcd_c2 672 161 --Answer: -25 --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161 theorem Theorem_7_1_6 {d a b : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ gcd a b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑d ∣ ↑(gcd a b) set s : Int := gcd_c1 a b set t : Int := gcd_c2 a b have h3 : s * ↑a + t * ↑b = ↑(gcd a b) := gcd_lin_comb b a rewrite [←h3] --Goal : ↑d ∣ s * ↑a + t * ↑b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul] --Goal : ↑d ∣ s * (↑d * ↑j) + t * (↑d * ↑k) define apply Exists.intro (s * ↑j + t * ↑k) ring done /- Section 7.2 -/ theorem dvd_trans {a b c : Nat} (h1 : a ∣ b) (h2 : b ∣ c) : a ∣ c := by define at h1; define at h2; define obtain (m : Nat) (h3 : b = a * m) from h1 obtain (n : Nat) (h4 : c = b * n) from h2 rewrite [h3, mul_assoc] at h4 apply Exists.intro (m * n) show c = a * (m * n) from h4 done lemma exists_prime_factor : ∀ (n : Nat), 2 ≤ n → ∃ (p : Nat), prime_factor p n := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, 2 ≤ n_1 → ∃ (p : Nat), prime_factor p n_1 assume h1 : 2 ≤ n by_cases h2 : prime n · -- Case 1. h2 : prime n apply Exists.intro n define --Goal : prime n ∧ n ∣ n show prime n ∧ n ∣ n from And.intro h2 (dvd_self n) done · -- Case 2. h2 : ¬prime n define at h2 --h2 : ¬(2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n) demorgan at h2 disj_syll h2 h1 obtain (a : Nat) (h3 : ∃ (b : Nat), a * b = n ∧ a < n ∧ b < n) from h2 obtain (b : Nat) (h4 : a * b = n ∧ a < n ∧ b < n) from h3 have h5 : 2 ≤ a := by by_contra h6 have h7 : a ≤ 1 := by linarith have h8 : n ≤ b := calc n _ = a * b := h4.left.symm _ ≤ 1 * b := by rel [h7] _ = b := by ring linarith --n ≤ b contradicts b < n done have h6 : ∃ (p : Nat), prime_factor p a := ih a h4.right.left h5 obtain (p : Nat) (h7 : prime_factor p a) from h6 apply Exists.intro p define --Goal : prime p ∧ p ∣ n define at h7 --h7 : prime p ∧ p ∣ a apply And.intro h7.left have h8 : a ∣ n := by apply Exists.intro b show n = a * b from (h4.left).symm done show p ∣ n from dvd_trans h7.right h8 done done lemma exists_least_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q := by set S : Set Nat := {p : Nat | prime_factor p n} have h2 : ∃ (p : Nat), p ∈ S := exists_prime_factor n h show ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q from well_ord_princ S h2 done lemma all_prime_nil : all_prime [] := by define --Goal : ∀ p ∈ [], prime p fix p : Nat contrapos --Goal : ¬prime p → p ∉ [] assume h1 : ¬prime p show p ∉ [] from List.not_mem_nil p done lemma all_prime_cons (n : Nat) (L : List Nat) : all_prime (n :: L) ↔ prime n ∧ all_prime L := by apply Iff.intro · -- (→) assume h1 : all_prime (n :: L) --Goal : prime n ∧ all_prime L define at h1 --h1 : ∀ p ∈ n :: L, prime p apply And.intro (h1 n (List.mem_cons_self n L)) define --Goal : ∀ p ∈ L, prime p fix p : Nat assume h2 : p ∈ L show prime p from h1 p (List.mem_cons_of_mem n h2) done · -- (←) assume h1 : prime n ∧ all_prime L --Goal : all_prime (n :: l) define : all_prime L at h1 define fix p : Nat assume h2 : p ∈ n :: L rewrite [List.mem_cons] at h2 --h2 : p = n ∨ p ∈ L by_cases on h2 · -- Case 1. h2 : p = n rewrite [h2] show prime n from h1.left done · -- Case 2. h2 : p ∈ L show prime p from h1.right p h2 done done done lemma nondec_nil : nondec [] := by define --Goal : True trivial --trivial proves some obviously true statements, such as True done lemma nondec_cons (n : Nat) (L : List Nat) : nondec (n :: L) ↔ (∀ m ∈ L, n ≤ m) ∧ nondec L := by rfl lemma prod_nil : prod [] = 1 := by rfl lemma prod_cons : prod (n :: L) = n * (prod L) := by rfl lemma exists_cons_of_length_eq_succ {A : Type} {l : List A} {n : Nat} (h : l.length = n + 1) : ∃ (a : A) (L : List A), l = a :: L ∧ L.length = n := by have h1 : ¬l.length = 0 := by linarith rewrite [List.length_eq_zero] at h1 obtain (a : A) (h2 : ∃ (L : List A), l = a :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List A) (h3 : l = a :: L) from h2 apply Exists.intro a apply Exists.intro L apply And.intro h3 have h4 : (a :: L).length = L.length + 1 := List.length_cons a L rewrite [←h3, h] at h4 show L.length = n from (Nat.add_right_cancel h4).symm done lemma list_elt_dvd_prod_by_length (a : Nat) : ∀ (n : Nat), ∀ (l : List Nat), l.length = n → a ∈ l → a ∣ prod l := by by_induc · --Base Case fix l : List Nat assume h1 : l.length = 0 rewrite [List.length_eq_zero] at h1 --h1 : l = [] rewrite [h1] --Goal : a ∈ [] → a ∣ prod [] contrapos assume h2 : ¬a ∣ prod [] show a ∉ [] from List.not_mem_nil a done · -- Induction Step fix n : Nat assume ih : ∀ (l : List Nat), List.length l = n → a ∈ l → a ∣ prod l fix l : List Nat assume h1 : l.length = n + 1 --Goal : a ∈ l → a ∣ prod l obtain (b : Nat) (h2 : ∃ (L : List Nat), l = b :: L ∧ L.length = n) from exists_cons_of_length_eq_succ h1 obtain (L : List Nat) (h3 : l = b :: L ∧ L.length = n) from h2 have h4 : a ∈ L → a ∣ prod L := ih L h3.right assume h5 : a ∈ l rewrite [h3.left, prod_cons] --Goal : a ∣ b * prod L rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b ∨ a ∈ L by_cases on h5 · -- Case 1. h5 : a = b apply Exists.intro (prod L) rewrite [h5] rfl done · -- Case 2. h5 : a ∈ L have h6 : a ∣ prod L := h4 h5 have h7 : prod L ∣ b * prod L := by apply Exists.intro b ring done show a ∣ b * prod L from dvd_trans h6 h7 done done done lemma list_elt_dvd_prod {a : Nat} {l : List Nat} (h : a ∈ l) : a ∣ prod l := by set n : Nat := l.length have h1 : l.length = n := by rfl show a ∣ prod l from list_elt_dvd_prod_by_length a n l h1 h done lemma exists_prime_factorization : ∀ (n : Nat), n ≥ 1 → ∃ (l : List Nat), prime_factorization n l := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, n_1 ≥ 1 → ∃ (l : List Nat), prime_factorization n_1 l assume h1 : n ≥ 1 by_cases h2 : n = 1 · -- Case 1. h2 : n = 1 apply Exists.intro [] define apply And.intro · -- Proof of nondec_prime_list [] define show all_prime [] ∧ nondec [] from And.intro all_prime_nil nondec_nil done · -- Proof of prod [] = n rewrite [prod_nil, h2] rfl done done · -- Case 2. h2 : n ≠ 1 have h3 : n ≥ 2 := lt_of_le_of_ne' h1 h2 obtain (p : Nat) (h4 : prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q) from exists_least_prime_factor h3 have p_prime_factor : prime_factor p n := h4.left define at p_prime_factor have p_prime : prime p := p_prime_factor.left have p_dvd_n : p ∣ n := p_prime_factor.right have p_least : ∀ (q : Nat), prime_factor q n → p ≤ q := h4.right obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n have h5 : m ≠ 0 := by contradict h1 with h6 have h7 : n = 0 := calc n _ = p * m := n_eq_pm _ = p * 0 := by rw [h6] _ = 0 := by ring rewrite [h7] decide done have m_pos : 0 < m := Nat.pos_of_ne_zero h5 have m_lt_n : m < n := by define at p_prime show m < n from calc m _ < m + m := by linarith _ = 2 * m := by ring _ ≤ p * m := by rel [p_prime.left] _ = n := n_eq_pm.symm done obtain (L : List Nat) (h6 : prime_factorization m L) from ih m m_lt_n m_pos define at h6 have ndpl_L : nondec_prime_list L := h6.left define at ndpl_L apply Exists.intro (p :: L) define apply And.intro · -- Proof of nondec_prime_list (p :: L) define apply And.intro · -- Proof of all_prime (p :: L) rewrite [all_prime_cons] show prime p ∧ all_prime L from And.intro p_prime ndpl_L.left done · -- Proof of nondec (p :: L) rewrite [nondec_cons] apply And.intro _ ndpl_L.right fix q : Nat assume q_in_L : q ∈ L have h7 : q ∣ prod L := list_elt_dvd_prod q_in_L rewrite [h6.right] at h7 --h7 : q ∣ m have h8 : m ∣ n := by apply Exists.intro p rewrite [n_eq_pm] ring done have q_dvd_n : q ∣ n := dvd_trans h7 h8 have ap_L : all_prime L := ndpl_L.left define at ap_L have q_prime_factor : prime_factor q n := And.intro (ap_L q q_in_L) q_dvd_n show p ≤ q from p_least q q_prime_factor done done · -- Proof of prod (p :: L) = n rewrite [prod_cons, h6.right, n_eq_pm] rfl done done done theorem Theorem_7_2_2 {a b c : Nat} (h1 : c ∣ a * b) (h2 : rel_prime a c) : c ∣ b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑c ∣ ↑b define at h1; define at h2; define obtain (j : Nat) (h3 : a * b = c * j) from h1 set s : Int := gcd_c1 a c set t : Int := gcd_c2 a c have h4 : s * ↑a + t * ↑c = ↑(gcd a c) := gcd_lin_comb c a rewrite [h2, Nat.cast_one] at h4 --h4 : s * ↑a + t * ↑c = (1 : Int) apply Exists.intro (s * ↑j + t * ↑b) show ↑b = ↑c * (s * ↑j + t * ↑b) from calc ↑b _ = (1 : Int) * ↑b := (one_mul _).symm _ = (s * ↑a + t * ↑c) * ↑b := by rw [h4] _ = s * (↑a * ↑b) + t * ↑c * ↑b := by ring _ = s * (↑c * ↑j) + t * ↑c * ↑b := by rw [←Nat.cast_mul a b, h3, Nat.cast_mul c j] _ = ↑c * (s * ↑j + t * ↑b) := by ring done lemma le_nonzero_prod_left {a b : Nat} (h : a * b ≠ 0) : a ≤ a * b := by have h1 : b ≠ 0 := by contradict h with h1 rewrite [h1] ring done have h2 : 1 ≤ b := Nat.pos_of_ne_zero h1 show a ≤ a * b from calc a = a * 1 := (mul_one a).symm _ ≤ a * b := by rel [h2] done lemma le_nonzero_prod_right {a b : Nat} (h : a * b ≠ 0) : b ≤ a * b := by rewrite [mul_comm] rewrite [mul_comm] at h show b ≤ b * a from le_nonzero_prod_left h done lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry lemma rel_prime_of_prime_not_dvd {a p : Nat} (h1 : prime p) (h2 : ¬p ∣ a) : rel_prime a p := by have h3 : gcd a p ∣ a := gcd_dvd_left a p have h4 : gcd a p ∣ p := gcd_dvd_right a p have h5 : gcd a p = 1 ∨ gcd a p = p := dvd_prime h1 h4 have h6 : gcd a p ≠ p := by contradict h2 with h6 rewrite [h6] at h3 show p ∣ a from h3 done disj_syll h5 h6 show rel_prime a p from h5 done theorem Theorem_7_2_3 {a b p : Nat} (h1 : prime p) (h2 : p ∣ a * b) : p ∣ a ∨ p ∣ b := by or_right with h3 have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3 show p ∣ b from Theorem_7_2_2 h2 h4 done lemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a ≥ 1 := by have h1 : a ≠ 0 := by by_contra h1 rewrite [h1] at h contradict h linarith done show a ≥ 1 from Nat.pos_of_ne_zero h1 done lemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by have h1 : a ≥ 1 := ge_one_of_prod_one h have h2 : a * b ≠ 0 := by linarith have h3 : a ≤ a * b := le_nonzero_prod_left h2 rewrite [h] at h3 show a = 1 from Nat.le_antisymm h3 h1 done lemma eq_one_of_dvd_one {n : Nat} (h : n ∣ 1) : n = 1 := by obtain (j : Nat) (h1 : 1 = n * j) from h show n = 1 from eq_one_of_prod_one h1.symm done lemma prime_not_one {p : Nat} (h : prime p) : p ≠ 1 := by define at h linarith done theorem Theorem_7_2_4 {p : Nat} (h1 : prime p) : ∀ (l : List Nat), p ∣ prod l → ∃ a ∈ l, p ∣ a := by apply List.rec · -- Base Case. Goal : p ∣ prod [] → ∃ a ∈ [], p ∣ a rewrite [prod_nil] assume h2 : p ∣ 1 show ∃ a ∈ [], p ∣ a from absurd (eq_one_of_dvd_one h2) (prime_not_one h1) done · -- Induction Step fix b : Nat fix L : List Nat assume ih : p ∣ prod L → ∃ a ∈ L, p ∣ a --Goal : p ∣ prod (b :: L) → ∃ a ∈ b :: L, p ∣ a assume h2 : p ∣ prod (b :: L) rewrite [prod_cons] at h2 have h3 : p ∣ b ∨ p ∣ prod L := Theorem_7_2_3 h1 h2 by_cases on h3 · -- Case 1. h3 : p ∣ b apply Exists.intro b show b ∈ b :: L ∧ p ∣ b from And.intro (List.mem_cons_self b L) h3 done · -- Case 2. h3 : p ∣ prod L obtain (a : Nat) (h4 : a ∈ L ∧ p ∣ a) from ih h3 apply Exists.intro a show a ∈ b :: L ∧ p ∣ a from And.intro (List.mem_cons_of_mem b h4.left) h4.right done done done lemma prime_in_list {p : Nat} {l : List Nat} (h1 : prime p) (h2 : all_prime l) (h3 : p ∣ prod l) : p ∈ l := by obtain (a : Nat) (h4 : a ∈ l ∧ p ∣ a) from Theorem_7_2_4 h1 l h3 define at h2 have h5 : prime a := h2 a h4.left have h6 : p = 1 ∨ p = a := dvd_prime h5 h4.right disj_syll h6 (prime_not_one h1) rewrite [h6] show a ∈ l from h4.left done lemma first_le_first {p q : Nat} {l m : List Nat} (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m)) (h3 : prod (p :: l) = prod (q :: m)) : p ≤ q := by define at h1; define at h2 have h4 : q ∣ prod (p :: l) := by define apply Exists.intro (prod m) rewrite [←prod_cons] show prod (p :: l) = prod (q :: m) from h3 done have h5 : all_prime (q :: m) := h2.left rewrite [all_prime_cons] at h5 have h6 : q ∈ p :: l := prime_in_list h5.left h1.left h4 have h7 : nondec (p :: l) := h1.right rewrite [nondec_cons] at h7 rewrite [List.mem_cons] at h6 by_cases on h6 · -- Case 1. h6 : q = p linarith done · -- Case 2. h6 : q ∈ l have h8 : ∀ m ∈ l, p ≤ m := h7.left show p ≤ q from h8 q h6 done done lemma nondec_prime_list_tail {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by define at h define rewrite [all_prime_cons, nondec_cons] at h show all_prime l ∧ nondec l from And.intro h.left.right h.right.right done lemma cons_prod_not_one {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : prod (p :: l) ≠ 1 := by define at h have h1 : all_prime (p :: l) := h.left rewrite [all_prime_cons] at h1 rewrite [prod_cons] by_contra h2 show False from (prime_not_one h1.left) (eq_one_of_prod_one h2) done lemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) : l = [] ↔ prod l = 1 := by apply Iff.intro · -- (→) assume h1 : l = [] rewrite [h1] show prod [] = 1 from prod_nil done · -- (←) contrapos assume h1 : ¬l = [] obtain (p : Nat) (h2 : ∃ (L : List Nat), l = p :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List Nat) (h3 : l = p :: L) from h2 rewrite [h3] at h rewrite [h3] show ¬prod (p :: L) = 1 from cons_prod_not_one h done done lemma prime_pos {p : Nat} (h : prime p) : p > 0 := by define at h linarith done theorem Theorem_7_2_5 : ∀ (l1 l2 : List Nat), nondec_prime_list l1 → nondec_prime_list l2 → prod l1 = prod l2 → l1 = l2 := by apply List.rec · -- Base Case. Goal : ∀ (l2 : List Nat), nondec_prime_list [] → -- nondec_prime_list l2 → prod [] = prod l2 → [] = l2 fix l2 : List Nat assume h1 : nondec_prime_list [] assume h2 : nondec_prime_list l2 assume h3 : prod [] = prod l2 rewrite [prod_nil, eq_comm, ←list_nil_iff_prod_one h2] at h3 show [] = l2 from h3.symm done · -- Induction Step fix p : Nat fix L1 : List Nat assume ih : ∀ (L2 : List Nat), nondec_prime_list L1 → nondec_prime_list L2 → prod L1 = prod L2 → L1 = L2 -- Goal : ∀ (l2 : List Nat), nondec_prime_list (p :: L1) → -- nondec_prime_list l2 → prod (p :: L1) = prod l2 → p :: L1 = l2 fix l2 : List Nat assume h1 : nondec_prime_list (p :: L1) assume h2 : nondec_prime_list l2 assume h3 : prod (p :: L1) = prod l2 have h4 : ¬prod (p :: L1) = 1 := cons_prod_not_one h1 rewrite [h3, ←list_nil_iff_prod_one h2] at h4 obtain (q : Nat) (h5 : ∃ (L : List Nat), l2 = q :: L) from List.exists_cons_of_ne_nil h4 obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5 rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2) rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2) have h7 : p ≤ q := first_le_first h1 h2 h3 have h8 : q ≤ p := first_le_first h2 h1 h3.symm have h9 : p = q := by linarith rewrite [h9, prod_cons, prod_cons] at h3 --h3 : q * prod L1 = q * prod L2 have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1 have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2 define at h2 have h12 : all_prime (q :: L2) := h2.left rewrite [all_prime_cons] at h12 have h13 : q > 0 := prime_pos h12.left have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3 have h15 : L1 = L2 := ih L2 h10 h11 h14 rewrite [h6, h9, h15] rfl done done theorem fund_thm_arith (n : Nat) (h : n ≥ 1) : ∃! (l : List Nat), prime_factorization n l := by exists_unique · -- Existence show ∃ (l : List Nat), prime_factorization n l from exists_prime_factorization n h done · -- Uniqueness fix l1 : List Nat; fix l2 : List Nat assume h1 : prime_factorization n l1 assume h2 : prime_factorization n l2 define at h1; define at h2 have h3 : prod l1 = n := h1.right rewrite [←h2.right] at h3 show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3 done done /- Section 7.3 -/ theorem congr_refl (m : Nat) : ∀ (a : Int), a ≡ a (MOD m) := by fix a : Int define --Goal : ∃ (c : Int), a - a = ↑m * c apply Exists.intro 0 ring done theorem congr_symm {m : Nat} : ∀ {a b : Int}, a ≡ b (MOD m) → b ≡ a (MOD m) := by fix a : Int; fix b : Int assume h1 : a ≡ b (MOD m) define at h1 --h1 : ∃ (c : Int), a - b = ↑m * c define --Goal : ∃ (c : Int), b - a = ↑m * c obtain (c : Int) (h2 : a - b = m * c) from h1 apply Exists.intro (-c) show b - a = m * (-c) from calc b - a _ = -(a - b) := by ring _ = -(m * c) := by rw [h2] _ = m * (-c) := by ring done theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry /- Fundamental properties of congruence classes -/ lemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) : X = Y ↔ X.val = Y.val := Fin.ext_iff lemma val_nat_eq_mod (n k : Nat) : ([k]_(n + 1)).val = k % (n + 1) := by rfl lemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl theorem cc_rep {m : Nat} (X : ZMod m) : ∃ (a : Int), X = [a]_m := match m with | 0 => by apply Exists.intro X rfl done | n + 1 => by apply Exists.intro ↑(X.val) have h1 : X.val < n + 1 := Fin.prop X rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1] rfl done theorem add_class (m : Nat) (a b : Int) : [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm theorem mul_class (m : Nat) (a b : Int) : [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm lemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ [a - b]_m = [0]_m := by apply Iff.intro · -- (→) assume h1 : [a]_m = [b]_m have h2 : a - b = a + (-b) := by ring have h3 : b + (-b) = 0 := by ring show [a - b]_m = [0]_m from calc [a - b]_m _ = [a + (-b)]_m := by rw [h2] _ = [a]_m + [-b]_m := by rw [add_class] _ = [b]_m + [-b]_m := by rw [h1] _ = [b + -b]_m := by rw [add_class] _ = [0]_m := by rw [h3] done · -- (←) assume h1 : [a - b]_m = [0]_m have h2 : b + (a - b) = a := by ring have h3 : b + 0 = b := by ring show [a]_m = [b]_m from calc [a]_m _ = [b + (a - b)]_m := by rw [h2] _ = [b]_m + [a - b]_m := by rw [add_class] _ = [b]_m + [0]_m := by rw [h1] _ = [b + 0]_m := by rw [add_class] _ = [b]_m := by rw [h3] done done lemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) : [a]_m = [0]_m → [-a]_m = [0]_m := by assume h1 : [a]_m = [0]_m have h2 : 0 + (-a) = -a := by ring have h3 : a + (-a) = 0 := by ring show [-a]_m = [0]_m from calc [-a]_m _ = [0 + (-a)]_m := by rw [h2] _ = [0]_m + [-a]_m := by rw [add_class] _ = [a]_m + [-a]_m := by rw [h1] _ = [a + (-a)]_m := by rw [add_class] _ = [0]_m := by rw [h3] done lemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) : [-a]_m = [0]_m ↔ [a]_m = [0]_m := by apply Iff.intro _ (cc_neg_zero_of_cc_zero m a) assume h1 : [-a]_m = [0]_m have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1 have h3 : -(-a) = a := by ring rewrite [h3] at h2 show [a]_m = [0]_m from h2 done lemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl lemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m ↔ m ∣ k := match m with | 0 => by have h : (0 : Int) = (↑(0 : Nat) : Int) := by rfl rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj] apply Iff.intro · -- (→) assume h1 : k = 0 rewrite [h1] show 0 ∣ 0 from dvd_self 0 done · -- (←) assume h1 : 0 ∣ k obtain (c : Nat) (h2 : k = 0 * c) from h1 rewrite [h2] ring done done | n + 1 => by rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero] show k % (n + 1) = 0 ↔ n + 1 ∣ k from (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm done lemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m ↔ ↑m ∣ a := by obtain (k : Nat) (h1 : a = ↑k ∨ a = -↑k) from Int.eq_nat_or_neg a by_cases on h1 · -- Case 1. h1: a = ↑k rewrite [h1, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done · -- Case 2. h1: a = -↑k rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done done theorem cc_eq_iff_congr (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ a ≡ b (MOD m) := calc [a]_m = [b]_m _ ↔ [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b _ ↔ ↑m ∣ (a - b) := cc_zero_iff_dvd m (a - b) _ ↔ a ≡ b (MOD m) := by rfl /- End of fundamental properties of congruence classes -/ lemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m := by have h1 : (↑m : Int) ≠ 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m) show 0 ≤ a % m from Int.emod_nonneg a h1 done lemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m) have h2 : (↑m : Int) > 0 := (Nat.cast_pos).rtl h1 show a % m < m from Int.emod_lt_of_pos a h2 done lemma congr_mod_mod (m : Nat) (a : Int) : a ≡ a % m (MOD m) := by define have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m apply Exists.intro (a / m) show a - a % m = m * (a / m) from calc a - (a % m) _ = m * (a / m) + a % m - a % m := by rw [h1] _ = m * (a / m) := by ring done lemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) := And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a)) theorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) : ∃! (r : Int), 0 ≤ r ∧ r < m ∧ a ≡ r (MOD m) := by exists_unique · -- Existence apply Exists.intro (a % m) show 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) from mod_cmpl_res m a done · -- Uniqueness fix r1 : Int; fix r2 : Int assume h1 : 0 ≤ r1 ∧ r1 < m ∧ a ≡ r1 (MOD m) assume h2 : 0 ≤ r2 ∧ r2 < m ∧ a ≡ r2 (MOD m) have h3 : r1 ≡ r2 (MOD m) := congr_trans (congr_symm h1.right.right) h2.right.right obtain (d : Int) (h4 : r1 - r2 = m * d) from h3 have h5 : r1 - r2 < m * 1 := by linarith have h6 : m * (-1) < r1 - r2 := by linarith rewrite [h4] at h5 --h5 : m * d < m * 1 rewrite [h4] at h6 --h6 : m * -1 < m * d have h7 : (↑m : Int) ≥ 0 := Nat.cast_nonneg m have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7 have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7 have h10 : d = 0 := by linarith show r1 = r2 from calc r1 _ = r1 - r2 + r2 := by ring _ = m * 0 + r2 := by rw [h4, h10] _ = r2 := by ring done done lemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m := (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a) theorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y rewrite [h1, h2] have h3 : a + b = b + a := by ring show [a]_m + [b]_m = [b]_m + [a]_m from calc [a]_m + [b]_m _ = [a + b]_m := add_class m a b _ = [b + a]_m := by rw [h3] _ = [b]_m + [a]_m := (add_class m b a).symm done theorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X rewrite [h1] have h2 : a * 1 = a := by ring show [a]_m * [1]_m = [a]_m from calc [a]_m * [1]_m _ = [a * 1]_m := mul_class m a 1 _ = [a]_m := by rw [h2] done theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry lemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) : [a]_m * [gcd_c2 m a]_m = [1]_m := by set s : Int := gcd_c1 m a have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m define at h1 rewrite [h1, Nat.cast_one] at h2 --h2 : s * ↑m + gcd_c2 m a * ↑a = 1 rewrite [mul_class, cc_eq_iff_congr] define --Goal : ∃ (c : Int), ↑a * gcd_c2 m a - 1 = ↑m * c apply Exists.intro (-s) show a * (gcd_c2 m a) - 1 = m * (-s) from calc a * (gcd_c2 m a) - 1 _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring _ = 1 + m * (-s) - 1 := by rw [h2] _ = m * (-s) := by ring done theorem Theorem_7_3_7 (m a : Nat) : invertible [a]_m ↔ rel_prime m a := by apply Iff.intro · -- (→) assume h1 : invertible [a]_m define at h1 obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1 obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y rewrite [h3, mul_class, cc_eq_iff_congr] at h2 define at h2 obtain (c : Int) (h4 : a * b - 1 = m * c) from h2 rewrite [Exercise_7_2_6] --Goal : ∃ (s t : Int), s * ↑m + t * ↑a = 1 apply Exists.intro (-c) apply Exists.intro b show (-c) * m + b * a = 1 from calc (-c) * m + b * a _ = (-c) * m + (a * b - 1) + 1 := by ring _ = (-c) * m + m * c + 1 := by rw [h4] _ = 1 := by ring done · -- (←) assume h1 : rel_prime m a define show ∃ (Y : ZMod m), [a]_m * Y = [1]_m from Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1) done done /- Section 7.4 -/ section Euler open Euler lemma num_rp_below_base {m : Nat} : num_rp_below m 0 = 0 := by rfl lemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) : num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by have h1 : num_rp_below m (j + 1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : gcd m j = 1 rewrite [if_pos h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1 show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1 done lemma num_rp_below_step_not_rp {m j : Nat} (h : ¬rel_prime m j) : num_rp_below m (j + 1) = num_rp_below m j := by have h1 : num_rp_below m (j +1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : ¬gcd m j = 1 rewrite [if_neg h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j show num_rp_below m (j + 1) = num_rp_below m j from h1 done lemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl #eval phi 10 --Answer: 4 lemma prod_inv_iff_inv {m : Nat} {X : ZMod m} (h1 : invertible X) (Y : ZMod m) : invertible (X * Y) ↔ invertible Y := by apply Iff.intro · -- (→) assume h2 : invertible (X * Y) obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2 apply Exists.intro (X * Z) rewrite [←h3] --Goal : Y * (X * Z) = X * Y * Z ring --Note that ring can do algebra in ZMod m done · -- (←) assume h2 : invertible Y obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1 obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2 apply Exists.intro (Xi * Yi) show (X * Y) * (Xi * Yi) = [1]_m from calc X * Y * (Xi * Yi) _ = (X * Xi) * (Y * Yi) := by ring _ = [1]_m * [1]_m := by rw [h3, h4] _ = [1]_m := Theorem_7_3_6_7 [1]_m done done lemma F_rp_def {m i : Nat} (h : rel_prime m i) : F m i = [i]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h --h : gcd m i = 1 rewrite [if_pos h] at h1 show F m i = [i]_m from h1 done lemma F_not_rp_def {m i : Nat} (h : ¬rel_prime m i) : F m i = [1]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h rewrite [h1, if_neg h] rfl done lemma prod_seq_base {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 0 k f = [1]_m := by rfl lemma prod_seq_step {m : Nat} (n k : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl lemma prod_seq_zero_step {m : Nat} (n : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by rewrite [prod_seq_step, zero_add] rfl done lemma prod_one {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 1 k f = f k := by rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7] rfl done lemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl lemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m := calc [G m a i]_m _ = [(a * i) % m]_m := by rfl _ = [a * i]_m := (cc_eq_mod m (a * i)).symm _ = [a]_m * [i]_m := (mul_class m a i).symm lemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) : rel_prime m (G m a i) ↔ rel_prime m i := by have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1 show rel_prime m (G m a i) ↔ rel_prime m i from calc rel_prime m (G m a i) _ ↔ invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm _ ↔ invertible ([a]_m * [i]_m) := by rw [cc_G] _ ↔ invertible [i]_m := prod_inv_iff_inv h2 ([i]_m) _ ↔ rel_prime m i := Theorem_7_3_7 m i done lemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) : F m (G m a i) = [a]_m * F m i := by have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2 show F m (G m a i) = [a]_m * F m i from calc F m (G m a i) _ = [G m a i]_m := F_rp_def h3 _ = [a]_m * [i]_m := cc_G m a i _ = [a]_m * F m i := by rw [F_rp_def h2] done lemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : ¬rel_prime m i) : F m (G m a i) = [1]_m := by rewrite [←G_rp_iff h1 i] at h2 show F m (G m a i) = [1]_m from F_not_rp_def h2 done lemma FG_prod {m a : Nat} (h1 : rel_prime m a) : ∀ (k : Nat), prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by by_induc · -- Base Case show prod_seq 0 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from calc prod_seq 0 0 ((F m) ∘ (G m a)) _ = [1]_m := prod_seq_base _ _ _ = [a]_m ^ 0 * [1]_m := by ring _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by rw [num_rp_below_base, prod_seq_base] done · -- Induction Step fix k : Nat assume ih : prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) by_cases h2 : rel_prime m k · -- Case 1. h2 : rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([a]_m * F m k) := by rw [FG_rp h1 h2] _ = [a]_m ^ ((num_rp_below m k) + 1) * ((prod_seq k 0 (F m)) * F m k) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_rp h2, prod_seq_zero_step] done · -- Case 2. h2 : ¬rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([1]_m) := by rw [FG_not_rp h1 h2] _ = [a]_m ^ (num_rp_below m k) * (prod_seq k 0 (F m) * ([1]_m)) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_not_rp h2, prod_seq_zero_step, F_not_rp_def h2] done done done lemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by define --Goal : ∀ i < m, G m a i < m fix i : Nat assume h1 : i < m rewrite [G_def] --Goal : a * i % m < m show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m) done lemma left_inv_one_one_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g' (g i) = i) : one_one_below n g := sorry lemma right_inv_onto_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g (g' i) = i) (h2 : maps_below n g') : onto_below n g := by define at h2; define fix k : Nat assume h3 : k < n apply Exists.intro (g' k) show g' k < n ∧ g (g' k) = k from And.intro (h2 k h3) (h1 k h3) done lemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m * [inv_mod m a]_m = [1]_m := by have h2 : 0 ≤ (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a) show [a]_m * [inv_mod m a]_m = [1]_m from calc [a]_m * [inv_mod m a]_m _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2] _ = [a]_m * [gcd_c2 m a]_m := by rw [←cc_eq_mod] _ = [1]_m := gcd_c2_inv h1 done lemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m := calc a * (b % m) % m = a % m * (b % m % m) % m := Nat.mul_mod _ _ _ _ = a % m * (b % m) % m := by rw [Nat.mod_mod] _ = a * b % m := (Nat.mul_mod _ _ _).symm lemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm] rfl done theorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] : ↑a ≡ ↑b (MOD m) ↔ a % m = b % m := sorry lemma mul_inv_mod_cancel {m a i : Nat} [NeZero m] (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1 rewrite [mul_class, cc_eq_iff_congr, ←Nat.cast_mul, ←Nat.cast_one, congr_iff_mod_eq_Nat] at h3 show a * inv_mod m a * i % m = i from calc a * (inv_mod m a) * i % m _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = 1 % m * i % m := by rw [h3] _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = i % m := by rw [one_mul] _ = i := Nat.mod_eq_of_lt h2 done lemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl lemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, G m a (Ginv m a i) = i := by fix i : Nat assume h2 : i < m show G m a (Ginv m a i) = i from calc G m a (Ginv m a i) _ = a * ((inv_mod m a * i) % m) % m := by rfl _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, Ginv m a (G m a i) = i := by fix i : Nat assume h2 : i < m show Ginv m a (G m a i) = i from calc Ginv m a (G m a i) _ = inv_mod m a * ((a * i) % m) % m := by rfl _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc, mul_comm (inv_mod m a)] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_maps_below (m a : Nat) [NeZero m] : maps_below m (Ginv m a) := G_maps_below m (inv_mod m a) lemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : one_one_below m (G m a) := left_inv_one_one_below (Ginv_left_inv h1) lemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : onto_below m (G m a) := right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a) lemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : perm_below m (G m a) := And.intro (G_maps_below m a) (And.intro (G_one_one_below h1) (G_onto_below h1)) --Permuting a product of congruence classes doesn't change product lemma swap_fst (u v : Nat) : swap u v u = v := by define : swap u v u --Goal : (if u = u then v else if u = v then u else u) = v have h : u = u := by rfl rewrite [if_pos h] rfl done lemma swap_snd (u v : Nat) : swap u v v = u := by define : swap u v v by_cases h1 : v = u · -- Case 1. h1 : v = u rewrite [if_pos h1] show v = u from h1 done · -- Case 2. h1 : v ≠ u rewrite [if_neg h1] have h2 : v = v := by rfl rewrite [if_pos h2] rfl done done lemma swap_other {u v i : Nat} (h1 : i ≠ u) (h2 : i ≠ v) : swap u v i = i := by define : swap u v i rewrite [if_neg h1, if_neg h2] rfl done lemma swap_values (u v i : Nat) : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := by by_cases h1 : i = u · -- Case 1. h1 : i = u apply Or.inl rewrite [h1] show swap u v u = v from swap_fst u v done · -- Case 2. h1 : i ≠ u apply Or.inr by_cases h2 : i = v · -- Case 2.1. h2 : i = v apply Or.inl rewrite [h2] show swap u v v = u from swap_snd u v done · -- Case 2.2. h2 : i ≠ v apply Or.inr show swap u v i = i from swap_other h1 h2 done done done lemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by define fix i : Nat assume h3 : i < n have h4 : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := swap_values u v i by_cases on h4 · -- Case 1. h4 : swap u v i = v rewrite [h4] show v < n from h2 done · -- Case 2. by_cases on h4 · -- Case 2.1. h4 : swap u v i = u rewrite [h4] show u < n from h1 done · -- Case 2.2. h4 : swap u v i = i rewrite [h4] show i < n from h3 done done done lemma swap_swap (u v n : Nat) : ∀ i < n, swap u v (swap u v i) = i := by fix i : Nat assume h : i < n by_cases h1 : i = u · -- Case 1. h1 : i = u rewrite [h1, swap_fst, swap_snd] rfl done · -- Case 2. h1 : i ≠ u by_cases h2 : i = v · -- Case 2.1. h2 : i = v rewrite [h2, swap_snd, swap_fst] rfl done · -- Case 2.2. h2 : i ≠ v rewrite [swap_other h1 h2, swap_other h1 h2] rfl done done done lemma swap_one_one_below (u v n) : one_one_below n (swap u v) := left_inv_one_one_below (swap_swap u v n) lemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) := right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2) lemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) := And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2)) lemma comp_perm_below {n : Nat} {f g : Nat → Nat} (h1 : perm_below n f) (h2 : perm_below n g) : perm_below n (f ∘ g) := sorry lemma trivial_swap (u : Nat) : swap u u = id := by apply funext fix x : Nat by_cases h1 : x = u · -- Case 1. h1 : x = u rewrite [h1, swap_fst] rfl done · -- Case 2. h1 : x ≠ u rewrite [swap_other h1 h1] rfl done done lemma prod_eq_fun {m : Nat} (f g : Nat → ZMod m) (k : Nat) : ∀ (n : Nat), (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g := by by_induc · -- Base Case assume h : (∀ i < 0, f (k + i) = g (k + i)) rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g assume h1 : ∀ i < n + 1, f (k + i) = g (k + i) have h2 : ∀ i < n, f (k + i) = g (k + i) := by fix i : Nat assume h2 : i < n have h3 : i < n + 1 := by linarith show f (k + i) = g (k + i) from h1 i h3 done have h3 : prod_seq n k f = prod_seq n k g := ih h2 have h4 : n < n + 1 := Nat.lt_succ_self n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4] rfl done done lemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := by have h2 : ∀ (i : Nat), i < u → (f ∘ swap u n) (0 + i) = f (0 + i) := by fix i : Nat assume h2 : i < u have h3 : 0 + i ≠ u := by linarith have h4 : 0 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl done show prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f from prod_eq_fun (f ∘ swap u n) f 0 u h2 done lemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := by have h2 : ∀ i < j, (f ∘ swap u n) (u + 1 + i) = f (u + 1 + i) := by fix i : Nat assume h2 : i < j have h3 : u + 1 + i ≠ u := by linarith have h4 : u + 1 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl show prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f from prod_eq_fun (f ∘ swap u n) f (u + 1) j h2 done lemma break_prod {m : Nat} (n : Nat) (f : Nat → ZMod m) : ∀ (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by by_induc · -- Base Case have h : n + 0 = n := by rfl rewrite [prod_seq_base, h, Theorem_7_3_6_7] rfl done · -- Induction Step fix j : Nat assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f rewrite [←add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc] rfl done done lemma break_prod_twice {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f := break_prod n f 1 rewrite [prod_one] at h2 have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f := break_prod (u + 1) f j rewrite [←h1] at h3 have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f := break_prod u f 1 rewrite [prod_one] at h4 rewrite [h3, h4] at h2 show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2 done lemma swap_prod_eq_prod {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f := by by_cases h2 : u = n · -- Case 1. h2 : u = n rewrite [h2, trivial_swap n] --Goal : prod_seq (n + 1) 0 (f ∘ id) = prod_seq (n + 1) 0 f rfl done · -- Case 2. h2 : ¬u = n have h3 : u + 1 ≤ n := Nat.lt_of_le_of_ne h1 h2 obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3 have break_f : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := break_prod_twice f h4 have break_fs : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_prod_twice (f ∘ swap u n) h4 have f_eq_fs_below : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := swap_prod_eq_prod_below f h1 have f_eq_fs_btwn : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4 show prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f from calc prod_seq (n + 1) 0 (f ∘ swap u n) _ = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_fs _ = prod_seq u 0 f * (f ∘ swap u n) u * prod_seq j (u + 1) f * (f ∘ swap u n) n := by rw [f_eq_fs_below, f_eq_fs_btwn] _ = prod_seq u 0 f * f (swap u n u) * prod_seq j (u + 1) f * f (swap u n n) := by rfl _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by rw [swap_fst, swap_snd] _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring _ = prod_seq (n + 1) 0 f := break_f.symm done done lemma perm_below_fixed {n : Nat} {g : Nat → Nat} (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry lemma perm_prod {m : Nat} (f : Nat → ZMod m) : ∀ (n : Nat), ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) := by by_induc · -- Base Case fix g : Nat → Nat assume h1 : perm_below 0 g rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) fix g : Nat → Nat assume g_pb : perm_below (n + 1) g define at g_pb have g_ob : onto_below (n + 1) g := g_pb.right.right define at g_ob have h1 : n < n + 1 := by linarith obtain (u : Nat) (h2 : u < n + 1 ∧ g u = n) from g_ob n h1 have s_pb : perm_below (n + 1) (swap u n) := swap_perm_below h2.left h1 have gs_pb_n1 : perm_below (n + 1) (g ∘ swap u n) := comp_perm_below g_pb s_pb have gs_fix_n : (g ∘ swap u n) n = n := calc (g ∘ swap u n) n _ = g (swap u n n) := by rfl _ = g u := by rw [swap_snd] _ = n := h2.right have gs_pb_n : perm_below n (g ∘ swap u n) := perm_below_fixed gs_pb_n1 gs_fix_n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f ∘ (g ∘ swap u n)) := ih (g ∘ swap u n) gs_pb_n have h3 : u ≤ n := by linarith show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f ∘ g) from calc prod_seq (n + 1) 0 f _ = prod_seq n 0 f * f n := prod_seq_zero_step n f _ = prod_seq n 0 (f ∘ (g ∘ swap u n)) * f ((g ∘ swap u n) n) := by rw [gs_prod, gs_fix_n] _ = prod_seq n 0 (f ∘ g ∘ swap u n) * (f ∘ g ∘ swap u n) n := by rfl _ = prod_seq (n + 1) 0 (f ∘ g ∘ swap u n) := (prod_seq_zero_step n (f ∘ g ∘ swap u n)).symm _ = prod_seq (n + 1) 0 ((f ∘ g) ∘ swap u n) := by rfl _ = prod_seq (n + 1) 0 (f ∘ g) := swap_prod_eq_prod (f ∘ g) h3 done done lemma F_invertible (m i : Nat) : invertible (F m i) := by by_cases h : rel_prime m i · -- Case 1. h : rel_prime m i rewrite [F_rp_def h] show invertible [i]_m from (Theorem_7_3_7 m i).rtl h done · -- Case 2. h : ¬rel_prime m i rewrite [F_not_rp_def h] apply Exists.intro [1]_m show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m done done lemma Fprod_invertible (m : Nat) : ∀ (k : Nat), invertible (prod_seq k 0 (F m)) := by by_induc · -- Base Case apply Exists.intro [1]_m show prod_seq 0 0 (F m) * [1]_m = [1]_m from calc prod_seq 0 0 (F m) * [1]_m _ = [1]_m * [1]_m := by rw [prod_seq_base] _ = [1]_m := Theorem_7_3_6_7 ([1]_m) done · -- Induction Step fix k : Nat assume ih : invertible (prod_seq k 0 (F m)) rewrite [prod_seq_zero_step] show invertible (prod_seq k 0 (F m) * (F m k)) from (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k) done done theorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m ^ (phi m) = [1]_m := by have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2 show [a]_m ^ (phi m) = [1]_m from calc [a]_m ^ (phi m) _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3] _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring _ = prod_seq m 0 (F m ∘ G m a) * Y := by rw [FG_prod h1 m, phi_def] _ = prod_seq m 0 (F m) * Y := by rw [perm_prod (F m) m (G m a) (G_perm_below h1)] _ = [1]_m := by rw [h3] done lemma Exercise_7_4_5_Int (m : Nat) (a : Int) : ∀ (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry lemma Exercise_7_4_5_Nat (m a n : Nat) : [a]_m ^ n = [a ^ n]_m := by rewrite [Exercise_7_4_5_Int] rfl done theorem Euler's_theorem {m a : Nat} [NeZero m] (h1 : rel_prime m a) : a ^ (phi m) ≡ 1 (MOD m) := by have h2 : [a]_m ^ (phi m) = [1]_m := Theorem_7_4_2 h1 rewrite [Exercise_7_4_5_Nat m a (phi m)] at h2 --h2 : [a ^ phi m]_m = [1]_m show a ^ (phi m) ≡ 1 (MOD m) from (cc_eq_iff_congr _ _ _).ltr h2 done #eval gcd 10 7 --Answer: 1. So 10 and 7 are relatively prime #eval 7 ^ phi 10 --Answer: 2401, which is congruent to 1 mod 10. end Euler /- Section 7.5 -/ lemma num_rp_prime {p : Nat} (h1 : prime p) : ∀ k < p, num_rp_below p (k + 1) = k := sorry lemma phi_prime {p : Nat} (h1 : prime p) : phi p = p - 1 := by have h2 : 1 ≤ p := prime_pos h1 have h3 : p - 1 + 1 = p := Nat.sub_add_cancel h2 have h4 : p - 1 < p := by linarith have h5 : num_rp_below p (p - 1 + 1) = p - 1 := num_rp_prime h1 (p - 1) h4 rewrite [h3] at h5 show phi p = p - 1 from h5 done theorem Theorem_7_2_2_Int {a c : Nat} {b : Int} (h1 : ↑c ∣ ↑a * b) (h2 : rel_prime a c) : ↑c ∣ b := by rewrite [Int.natCast_dvd, Int.natAbs_mul, Int.natAbs_ofNat] at h1 --h1 : c ∣ a * Int.natAbs b rewrite [Int.natCast_dvd] --Goal : c ∣ Int.natAbs b show c ∣ Int.natAbs b from Theorem_7_2_2 h1 h2 done lemma Lemma_7_4_5 {m n : Nat} (a b : Int) (h1 : rel_prime m n) : a ≡ b (MOD m * n) ↔ a ≡ b (MOD m) ∧ a ≡ b (MOD n) := by apply Iff.intro · -- (→) assume h2 : a ≡ b (MOD m * n) obtain (j : Int) (h3 : a - b = (m * n) * j) from h2 apply And.intro · -- Proof of a ≡ b (MOD m) apply Exists.intro (n * j) show a - b = m * (n * j) from calc a - b _ = m * n * j := h3 _ = m * (n * j) := by ring done · -- Proof of a ≡ b (MOD n) apply Exists.intro (m * j) show a - b = n * (m * j) from calc a - b _ = m * n * j := h3 _ = n * (m * j) := by ring done done · -- (←) assume h2 : a ≡ b (MOD m) ∧ a ≡ b (MOD n) obtain (j : Int) (h3 : a - b = m * j) from h2.left have h4 : (↑n : Int) ∣ a - b := h2.right rewrite [h3] at h4 --h4 : ↑n ∣ ↑m * j have h5 : ↑n ∣ j := Theorem_7_2_2_Int h4 h1 obtain (k : Int) (h6 : j = n * k) from h5 apply Exists.intro k --Goal : a - b = ↑(m * n) * k rewrite [Nat.cast_mul] --Goal : a - b = ↑m * ↑n * k show a - b = (m * n) * k from calc a - b _ = m * j := h3 _ = m * (n * k) := by rw [h6] _ = (m * n) * k := by ring done done --From exercises of Section 7.2 theorem rel_prime_symm {a b : Nat} (h : rel_prime a b) : rel_prime b a := sorry lemma prime_NeZero {p : Nat} (h : prime p) : NeZero p := by rewrite [neZero_iff] --Goal : p ≠ 0 define at h linarith done lemma Lemma_7_5_1 {p e d m c s : Nat} {t : Int} (h1 : prime p) (h2 : e * d = (p - 1) * s + 1) (h3 : m ^ e - c = p * t) : c ^ d ≡ m (MOD p) := by have h4 : m ^ e ≡ c (MOD p) := Exists.intro t h3 have h5 : [m ^ e]_p = [c]_p := (cc_eq_iff_congr _ _ _).rtl h4 rewrite [←Exercise_7_4_5_Nat] at h5 --h5 : [m]_p ^ e = [c]_p by_cases h6 : p ∣ m · -- Case 1. h6 : p ∣ m have h7 : m ≡ 0 (MOD p) := by obtain (j : Nat) (h8 : m = p * j) from h6 apply Exists.intro (↑j : Int) --Goal : ↑m - 0 = ↑p * ↑j rewrite [h8, Nat.cast_mul] ring done have h8 : [m]_p = [0]_p := (cc_eq_iff_congr _ _ _).rtl h7 have h9 : e * d ≠ 0 := by rewrite [h2] show (p - 1) * s + 1 ≠ 0 from Nat.add_one_ne_zero _ done have h10 : (0 : Int) ^ (e * d) = 0 := zero_pow h9 have h11 : [c ^ d]_p = [m]_p := calc [c ^ d]_p _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat] _ = ([m]_p ^ e) ^ d := by rw [h5] _ = [m]_p ^ (e * d) := by ring _ = [0]_p ^ (e * d) := by rw [h8] _ = [0 ^ (e * d)]_p := Exercise_7_4_5_Int _ _ _ _ = [0]_p := by rw [h10] _ = [m]_p := by rw [h8] show c ^ d ≡ m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11 done · -- Case 2. h6 : ¬p ∣ m have h7 : rel_prime m p := rel_prime_of_prime_not_dvd h1 h6 have h8 : rel_prime p m := rel_prime_symm h7 have h9 : NeZero p := prime_NeZero h1 have h10 : (1 : Int) ^ s = 1 := by ring have h11 : [c ^ d]_p = [m]_p := calc [c ^ d]_p _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat] _ = ([m]_p ^ e) ^ d := by rw [h5] _ = [m]_p ^ (e * d) := by ring _ = [m]_p ^ ((p - 1) * s + 1) := by rw [h2] _ = ([m]_p ^ (p - 1)) ^ s * [m]_p := by ring _ = ([m]_p ^ (phi p)) ^ s * [m]_p := by rw [phi_prime h1] _ = [1]_p ^ s * [m]_p := by rw [Theorem_7_4_2 h8] _ = [1 ^ s]_p * [m]_p := by rw [Exercise_7_4_5_Int] _ = [1]_p * [m]_p := by rw [h10] _ = [m]_p * [1]_p := by ring _ = [m]_p := Theorem_7_3_6_7 _ show c ^ d ≡ m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11 done done theorem Theorem_7_5_1 (p q n e d k m c : Nat) (p_prime : prime p) (q_prime : prime q) (p_ne_q : p ≠ q) (n_pq : n = p * q) (ed_congr_1 : e * d = k * (p - 1) * (q - 1) + 1) (h1 : [m]_n ^ e = [c]_n) : [c]_n ^ d = [m]_n := by rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] at h1 --h1 : m ^ e ≡ c (MOD n) rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] --Goal : c ^ d ≡ m (MOD n) obtain (j : Int) (h2 : m ^ e - c = n * j) from h1 rewrite [n_pq, Nat.cast_mul] at h2 --h2 : m ^ e - c = p * q * j have h3 : e * d = (p - 1) * (k * (q - 1)) + 1 := by rewrite [ed_congr_1] ring done have h4 : m ^ e - c = p * (q * j) := by rewrite [h2] ring done have congr_p : c ^ d ≡ m (MOD p) := Lemma_7_5_1 p_prime h3 h4 have h5 : e * d = (q - 1) * (k * (p - 1)) + 1 := by rewrite [ed_congr_1] ring done have h6 : m ^ e - c = q * (p * j) := by rewrite [h2] ring done have congr_q : c ^ d ≡ m (MOD q) := Lemma_7_5_1 q_prime h5 h6 have h7 : ¬q ∣ p := by by_contra h8 have h9 : q = 1 ∨ q = p := dvd_prime p_prime h8 disj_syll h9 (prime_not_one q_prime) show False from p_ne_q h9.symm done have h8 : rel_prime p q := rel_prime_of_prime_not_dvd q_prime h7 rewrite [n_pq, Lemma_7_4_5 _ _ h8] show c ^ d ≡ m (MOD p) ∧ c ^ d ≡ m (MOD q) from And.intro congr_p congr_q done /- BEGIN EXERCISES -/ namespace Exercises /- Section 7.1 -/ -- 1. theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry -- 2. lemma gcd_comm_lt {a b : Nat} (h : a < b) : gcd a b = gcd b a := sorry theorem gcd_comm (a b : Nat) : gcd a b = gcd b a := sorry -- 3. theorem Exercise_7_1_5 (a b : Nat) (n : Int) : (∃ (s t : Int), s * a + t * b = n) ↔ (↑(gcd a b) : Int) ∣ n := sorry -- 4. theorem Exercise_7_1_6 (a b c : Nat) : gcd a b = gcd (a + b * c) b := sorry -- 5. theorem gcd_is_nonzero {a b : Nat} (h : a ≠ 0 ∨ b ≠ 0) : gcd a b ≠ 0 := sorry -- 6. theorem gcd_greatest {a b d : Nat} (h1 : gcd a b ≠ 0) (h2 : d ∣ a) (h3 : d ∣ b) : d ≤ gcd a b := sorry -- 7. lemma Lemma_7_1_10a {a b : Nat} (n : Nat) (h : a ∣ b) : (n * a) ∣ (n * b) := sorry lemma Lemma_7_1_10b {a b n : Nat} (h1 : n ≠ 0) (h2 : (n * a) ∣ (n * b)) : a ∣ b := sorry lemma Lemma_7_1_10c {a b : Nat} (h1 : a ∣ b) (h2 : b ∣ a) : a = b := sorry theorem Exercise_7_1_10 (a b n : Nat) : gcd (n * a) (n * b) = n * gcd a b := sorry /- Section 7.2 -/ -- 1. lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry -- 2. -- Hints: Start with apply List.rec. You may find mul_ne_zero useful

theorem prod_nonzero_nonzero : ∀ (l : List Nat), (∀ a ∈ l, a ≠ 0) → prod l ≠ 0

HTPI.Exercises.prod_nonzero_nonzero

htpi/HTPILib/Chap7.lean

htpi

/- Copyright 2023 Daniel J. Velleman -/ import HTPILib.Chap6 namespace HTPI /- Definitions -/ lemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by have h : n + 1 > 0 := Nat.succ_pos n show a % (n + 1) < n + 1 from Nat.mod_lt a h done def gcd (a b : Nat) : Nat := match b with | 0 => a | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd (n + 1) (a % (n + 1)) termination_by b mutual def gcd_c1 (a b : Nat) : Int := match b with | 0 => 1 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c2 (n + 1) (a % (n + 1)) --Corresponds to s = t' termination_by b def gcd_c2 (a b : Nat) : Int := match b with | 0 => 0 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c1 (n + 1) (a % (n + 1)) - (gcd_c2 (n + 1) (a % (n + 1))) * ↑(a / (n + 1)) --Corresponds to t = s' - t'q termination_by b end def prime (n : Nat) : Prop := 2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n def prime_factor (p n : Nat) : Prop := prime p ∧ p ∣ n def all_prime (l : List Nat) : Prop := ∀ p ∈ l, prime p def nondec (l : List Nat) : Prop := match l with | [] => True --Of course, True is a proposition that is always true | n :: L => (∀ m ∈ L, n ≤ m) ∧ nondec L def nondec_prime_list (l : List Nat) : Prop := all_prime l ∧ nondec l def prod (l : List Nat) : Nat := match l with | [] => 1 | n :: L => n * (prod L) def prime_factorization (n : Nat) (l : List Nat) : Prop := nondec_prime_list l ∧ prod l = n def rel_prime (a b : Nat) : Prop := gcd a b = 1 def congr_mod (m : Nat) (a b : Int) : Prop := (↑m : Int) ∣ (a - b) def cc (m : Nat) (a : Int) : ZMod m := (↑a : ZMod m) notation:50 a " ≡ " b " (MOD " m ")" => congr_mod m a b notation:max "["a"]_"m:max => cc m a def invertible {m : Nat} (X : ZMod m) : Prop := ∃ (Y : ZMod m), X * Y = [1]_m def num_rp_below (m k : Nat) : Nat := match k with | 0 => 0 | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j def phi (m : Nat) : Nat := num_rp_below m m def prod_seq {m : Nat} (j k : Nat) (f : Nat → ZMod m) : ZMod m := match j with | 0 => [1]_m | n + 1 => prod_seq n k f * f (k + n) def maps_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i < n, g i < n def one_one_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i1 < n, ∀ i2 < n, g i1 = g i2 → i1 = i2 def onto_below (n : Nat) (g : Nat → Nat) : Prop := ∀ k < n, ∃ i < n, g i = k def perm_below (n : Nat) (g : Nat → Nat) : Prop := maps_below n g ∧ one_one_below n g ∧ onto_below n g def inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m) def swap (u v i : Nat) : Nat := if i = u then v else if i = v then u else i namespace Euler --For definitions specific to Euler's theorem def F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m def G (m a i : Nat) : Nat := (a * i) % m def Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i end Euler /- Section 7.1 -/ theorem dvd_mod_of_dvd_a_b {a b d : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ (a % b) := by set q : Nat := a / b have h3 : b * q + a % b = a := Nat.div_add_mod a b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 define --Goal : ∃ (c : Nat), a % b = d * c apply Exists.intro (j - k * q) show a % b = d * (j - k * q) from calc a % b _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm _ = a - b * q := by rw [h3] _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc] _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm done theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry #eval gcd 672 161 --Answer: 7 lemma gcd_base (a : Nat) : gcd a 0 = a := by rfl lemma gcd_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd a b = gcd b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1)) rfl done lemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b ≠ 0) : a % b < b := by have h1 : b > 0 := Nat.pos_of_ne_zero h show a % b < b from Nat.mod_lt a h1 done lemma dvd_self (n : Nat) : n ∣ n := by apply Exists.intro 1 ring done theorem gcd_dvd : ∀ (b a : Nat), (gcd a b) ∣ a ∧ (gcd a b) ∣ b := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd a b_1) ∣ a ∧ (gcd a b_1) ∣ b_1 fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_base] --Goal: a ∣ a ∧ a ∣ 0 apply And.intro (dvd_self a) define apply Exists.intro 0 rfl done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_nonzero a h1] --Goal : gcd b (a % b) ∣ a ∧ gcd b (a % b) ∣ b have h2 : a % b < b := mod_nonzero_lt a h1 have h3 : (gcd b (a % b)) ∣ b ∧ (gcd b (a % b)) ∣ (a % b) := ih (a % b) h2 b apply And.intro _ h3.left show (gcd b (a % b)) ∣ a from dvd_a_of_dvd_b_mod h3.left h3.right done done theorem gcd_dvd_left (a b : Nat) : (gcd a b) ∣ a := (gcd_dvd b a).left theorem gcd_dvd_right (a b : Nat) : (gcd a b) ∣ b := (gcd_dvd b a).right lemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl lemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c1 a b = gcd_c2 b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done lemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl lemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * ↑(a / b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done theorem gcd_lin_comb : ∀ (b a : Nat), (gcd_c1 a b) * ↑a + (gcd_c2 a b) * ↑b = ↑(gcd a b) := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd_c1 a b_1) * ↑a + (gcd_c2 a b_1) * ↑b_1 = ↑(gcd a b_1) fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base] --Goal : 1 * ↑a + 0 * ↑0 = ↑a ring done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1] --Goal : gcd_c2 b (a % b) * ↑a + -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * ↑(a / b)) * ↑b = -- ↑(gcd b (a % b)) set r : Nat := a % b set q : Nat := a / b set s : Int := gcd_c1 b r set t : Int := gcd_c2 b r --Goal : t * ↑a + (s - t * ↑q) * ↑b = ↑(gcd b r) have h2 : r < b := mod_nonzero_lt a h1 have h3 : s * ↑b + t * ↑r = ↑(gcd b r) := ih r h2 b have h4 : b * q + r = a := Nat.div_add_mod a b rewrite [←h3, ←h4] rewrite [Nat.cast_add, Nat.cast_mul] --Goal : t * (↑b * ↑q + ↑r) + (s - t * ↑q) * ↑b = s * ↑b + t * ↑r ring done done #eval gcd_c1 672 161 --Answer: 6 #eval gcd_c2 672 161 --Answer: -25 --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161 theorem Theorem_7_1_6 {d a b : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ gcd a b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑d ∣ ↑(gcd a b) set s : Int := gcd_c1 a b set t : Int := gcd_c2 a b have h3 : s * ↑a + t * ↑b = ↑(gcd a b) := gcd_lin_comb b a rewrite [←h3] --Goal : ↑d ∣ s * ↑a + t * ↑b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul] --Goal : ↑d ∣ s * (↑d * ↑j) + t * (↑d * ↑k) define apply Exists.intro (s * ↑j + t * ↑k) ring done /- Section 7.2 -/ theorem dvd_trans {a b c : Nat} (h1 : a ∣ b) (h2 : b ∣ c) : a ∣ c := by define at h1; define at h2; define obtain (m : Nat) (h3 : b = a * m) from h1 obtain (n : Nat) (h4 : c = b * n) from h2 rewrite [h3, mul_assoc] at h4 apply Exists.intro (m * n) show c = a * (m * n) from h4 done lemma exists_prime_factor : ∀ (n : Nat), 2 ≤ n → ∃ (p : Nat), prime_factor p n := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, 2 ≤ n_1 → ∃ (p : Nat), prime_factor p n_1 assume h1 : 2 ≤ n by_cases h2 : prime n · -- Case 1. h2 : prime n apply Exists.intro n define --Goal : prime n ∧ n ∣ n show prime n ∧ n ∣ n from And.intro h2 (dvd_self n) done · -- Case 2. h2 : ¬prime n define at h2 --h2 : ¬(2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n) demorgan at h2 disj_syll h2 h1 obtain (a : Nat) (h3 : ∃ (b : Nat), a * b = n ∧ a < n ∧ b < n) from h2 obtain (b : Nat) (h4 : a * b = n ∧ a < n ∧ b < n) from h3 have h5 : 2 ≤ a := by by_contra h6 have h7 : a ≤ 1 := by linarith have h8 : n ≤ b := calc n _ = a * b := h4.left.symm _ ≤ 1 * b := by rel [h7] _ = b := by ring linarith --n ≤ b contradicts b < n done have h6 : ∃ (p : Nat), prime_factor p a := ih a h4.right.left h5 obtain (p : Nat) (h7 : prime_factor p a) from h6 apply Exists.intro p define --Goal : prime p ∧ p ∣ n define at h7 --h7 : prime p ∧ p ∣ a apply And.intro h7.left have h8 : a ∣ n := by apply Exists.intro b show n = a * b from (h4.left).symm done show p ∣ n from dvd_trans h7.right h8 done done lemma exists_least_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q := by set S : Set Nat := {p : Nat | prime_factor p n} have h2 : ∃ (p : Nat), p ∈ S := exists_prime_factor n h show ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q from well_ord_princ S h2 done lemma all_prime_nil : all_prime [] := by define --Goal : ∀ p ∈ [], prime p fix p : Nat contrapos --Goal : ¬prime p → p ∉ [] assume h1 : ¬prime p show p ∉ [] from List.not_mem_nil p done lemma all_prime_cons (n : Nat) (L : List Nat) : all_prime (n :: L) ↔ prime n ∧ all_prime L := by apply Iff.intro · -- (→) assume h1 : all_prime (n :: L) --Goal : prime n ∧ all_prime L define at h1 --h1 : ∀ p ∈ n :: L, prime p apply And.intro (h1 n (List.mem_cons_self n L)) define --Goal : ∀ p ∈ L, prime p fix p : Nat assume h2 : p ∈ L show prime p from h1 p (List.mem_cons_of_mem n h2) done · -- (←) assume h1 : prime n ∧ all_prime L --Goal : all_prime (n :: l) define : all_prime L at h1 define fix p : Nat assume h2 : p ∈ n :: L rewrite [List.mem_cons] at h2 --h2 : p = n ∨ p ∈ L by_cases on h2 · -- Case 1. h2 : p = n rewrite [h2] show prime n from h1.left done · -- Case 2. h2 : p ∈ L show prime p from h1.right p h2 done done done lemma nondec_nil : nondec [] := by define --Goal : True trivial --trivial proves some obviously true statements, such as True done lemma nondec_cons (n : Nat) (L : List Nat) : nondec (n :: L) ↔ (∀ m ∈ L, n ≤ m) ∧ nondec L := by rfl lemma prod_nil : prod [] = 1 := by rfl lemma prod_cons : prod (n :: L) = n * (prod L) := by rfl lemma exists_cons_of_length_eq_succ {A : Type} {l : List A} {n : Nat} (h : l.length = n + 1) : ∃ (a : A) (L : List A), l = a :: L ∧ L.length = n := by have h1 : ¬l.length = 0 := by linarith rewrite [List.length_eq_zero] at h1 obtain (a : A) (h2 : ∃ (L : List A), l = a :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List A) (h3 : l = a :: L) from h2 apply Exists.intro a apply Exists.intro L apply And.intro h3 have h4 : (a :: L).length = L.length + 1 := List.length_cons a L rewrite [←h3, h] at h4 show L.length = n from (Nat.add_right_cancel h4).symm done lemma list_elt_dvd_prod_by_length (a : Nat) : ∀ (n : Nat), ∀ (l : List Nat), l.length = n → a ∈ l → a ∣ prod l := by by_induc · --Base Case fix l : List Nat assume h1 : l.length = 0 rewrite [List.length_eq_zero] at h1 --h1 : l = [] rewrite [h1] --Goal : a ∈ [] → a ∣ prod [] contrapos assume h2 : ¬a ∣ prod [] show a ∉ [] from List.not_mem_nil a done · -- Induction Step fix n : Nat assume ih : ∀ (l : List Nat), List.length l = n → a ∈ l → a ∣ prod l fix l : List Nat assume h1 : l.length = n + 1 --Goal : a ∈ l → a ∣ prod l obtain (b : Nat) (h2 : ∃ (L : List Nat), l = b :: L ∧ L.length = n) from exists_cons_of_length_eq_succ h1 obtain (L : List Nat) (h3 : l = b :: L ∧ L.length = n) from h2 have h4 : a ∈ L → a ∣ prod L := ih L h3.right assume h5 : a ∈ l rewrite [h3.left, prod_cons] --Goal : a ∣ b * prod L rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b ∨ a ∈ L by_cases on h5 · -- Case 1. h5 : a = b apply Exists.intro (prod L) rewrite [h5] rfl done · -- Case 2. h5 : a ∈ L have h6 : a ∣ prod L := h4 h5 have h7 : prod L ∣ b * prod L := by apply Exists.intro b ring done show a ∣ b * prod L from dvd_trans h6 h7 done done done lemma list_elt_dvd_prod {a : Nat} {l : List Nat} (h : a ∈ l) : a ∣ prod l := by set n : Nat := l.length have h1 : l.length = n := by rfl show a ∣ prod l from list_elt_dvd_prod_by_length a n l h1 h done lemma exists_prime_factorization : ∀ (n : Nat), n ≥ 1 → ∃ (l : List Nat), prime_factorization n l := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, n_1 ≥ 1 → ∃ (l : List Nat), prime_factorization n_1 l assume h1 : n ≥ 1 by_cases h2 : n = 1 · -- Case 1. h2 : n = 1 apply Exists.intro [] define apply And.intro · -- Proof of nondec_prime_list [] define show all_prime [] ∧ nondec [] from And.intro all_prime_nil nondec_nil done · -- Proof of prod [] = n rewrite [prod_nil, h2] rfl done done · -- Case 2. h2 : n ≠ 1 have h3 : n ≥ 2 := lt_of_le_of_ne' h1 h2 obtain (p : Nat) (h4 : prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q) from exists_least_prime_factor h3 have p_prime_factor : prime_factor p n := h4.left define at p_prime_factor have p_prime : prime p := p_prime_factor.left have p_dvd_n : p ∣ n := p_prime_factor.right have p_least : ∀ (q : Nat), prime_factor q n → p ≤ q := h4.right obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n have h5 : m ≠ 0 := by contradict h1 with h6 have h7 : n = 0 := calc n _ = p * m := n_eq_pm _ = p * 0 := by rw [h6] _ = 0 := by ring rewrite [h7] decide done have m_pos : 0 < m := Nat.pos_of_ne_zero h5 have m_lt_n : m < n := by define at p_prime show m < n from calc m _ < m + m := by linarith _ = 2 * m := by ring _ ≤ p * m := by rel [p_prime.left] _ = n := n_eq_pm.symm done obtain (L : List Nat) (h6 : prime_factorization m L) from ih m m_lt_n m_pos define at h6 have ndpl_L : nondec_prime_list L := h6.left define at ndpl_L apply Exists.intro (p :: L) define apply And.intro · -- Proof of nondec_prime_list (p :: L) define apply And.intro · -- Proof of all_prime (p :: L) rewrite [all_prime_cons] show prime p ∧ all_prime L from And.intro p_prime ndpl_L.left done · -- Proof of nondec (p :: L) rewrite [nondec_cons] apply And.intro _ ndpl_L.right fix q : Nat assume q_in_L : q ∈ L have h7 : q ∣ prod L := list_elt_dvd_prod q_in_L rewrite [h6.right] at h7 --h7 : q ∣ m have h8 : m ∣ n := by apply Exists.intro p rewrite [n_eq_pm] ring done have q_dvd_n : q ∣ n := dvd_trans h7 h8 have ap_L : all_prime L := ndpl_L.left define at ap_L have q_prime_factor : prime_factor q n := And.intro (ap_L q q_in_L) q_dvd_n show p ≤ q from p_least q q_prime_factor done done · -- Proof of prod (p :: L) = n rewrite [prod_cons, h6.right, n_eq_pm] rfl done done done theorem Theorem_7_2_2 {a b c : Nat} (h1 : c ∣ a * b) (h2 : rel_prime a c) : c ∣ b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑c ∣ ↑b define at h1; define at h2; define obtain (j : Nat) (h3 : a * b = c * j) from h1 set s : Int := gcd_c1 a c set t : Int := gcd_c2 a c have h4 : s * ↑a + t * ↑c = ↑(gcd a c) := gcd_lin_comb c a rewrite [h2, Nat.cast_one] at h4 --h4 : s * ↑a + t * ↑c = (1 : Int) apply Exists.intro (s * ↑j + t * ↑b) show ↑b = ↑c * (s * ↑j + t * ↑b) from calc ↑b _ = (1 : Int) * ↑b := (one_mul _).symm _ = (s * ↑a + t * ↑c) * ↑b := by rw [h4] _ = s * (↑a * ↑b) + t * ↑c * ↑b := by ring _ = s * (↑c * ↑j) + t * ↑c * ↑b := by rw [←Nat.cast_mul a b, h3, Nat.cast_mul c j] _ = ↑c * (s * ↑j + t * ↑b) := by ring done lemma le_nonzero_prod_left {a b : Nat} (h : a * b ≠ 0) : a ≤ a * b := by have h1 : b ≠ 0 := by contradict h with h1 rewrite [h1] ring done have h2 : 1 ≤ b := Nat.pos_of_ne_zero h1 show a ≤ a * b from calc a = a * 1 := (mul_one a).symm _ ≤ a * b := by rel [h2] done lemma le_nonzero_prod_right {a b : Nat} (h : a * b ≠ 0) : b ≤ a * b := by rewrite [mul_comm] rewrite [mul_comm] at h show b ≤ b * a from le_nonzero_prod_left h done lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry lemma rel_prime_of_prime_not_dvd {a p : Nat} (h1 : prime p) (h2 : ¬p ∣ a) : rel_prime a p := by have h3 : gcd a p ∣ a := gcd_dvd_left a p have h4 : gcd a p ∣ p := gcd_dvd_right a p have h5 : gcd a p = 1 ∨ gcd a p = p := dvd_prime h1 h4 have h6 : gcd a p ≠ p := by contradict h2 with h6 rewrite [h6] at h3 show p ∣ a from h3 done disj_syll h5 h6 show rel_prime a p from h5 done theorem Theorem_7_2_3 {a b p : Nat} (h1 : prime p) (h2 : p ∣ a * b) : p ∣ a ∨ p ∣ b := by or_right with h3 have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3 show p ∣ b from Theorem_7_2_2 h2 h4 done lemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a ≥ 1 := by have h1 : a ≠ 0 := by by_contra h1 rewrite [h1] at h contradict h linarith done show a ≥ 1 from Nat.pos_of_ne_zero h1 done lemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by have h1 : a ≥ 1 := ge_one_of_prod_one h have h2 : a * b ≠ 0 := by linarith have h3 : a ≤ a * b := le_nonzero_prod_left h2 rewrite [h] at h3 show a = 1 from Nat.le_antisymm h3 h1 done lemma eq_one_of_dvd_one {n : Nat} (h : n ∣ 1) : n = 1 := by obtain (j : Nat) (h1 : 1 = n * j) from h show n = 1 from eq_one_of_prod_one h1.symm done lemma prime_not_one {p : Nat} (h : prime p) : p ≠ 1 := by define at h linarith done theorem Theorem_7_2_4 {p : Nat} (h1 : prime p) : ∀ (l : List Nat), p ∣ prod l → ∃ a ∈ l, p ∣ a := by apply List.rec · -- Base Case. Goal : p ∣ prod [] → ∃ a ∈ [], p ∣ a rewrite [prod_nil] assume h2 : p ∣ 1 show ∃ a ∈ [], p ∣ a from absurd (eq_one_of_dvd_one h2) (prime_not_one h1) done · -- Induction Step fix b : Nat fix L : List Nat assume ih : p ∣ prod L → ∃ a ∈ L, p ∣ a --Goal : p ∣ prod (b :: L) → ∃ a ∈ b :: L, p ∣ a assume h2 : p ∣ prod (b :: L) rewrite [prod_cons] at h2 have h3 : p ∣ b ∨ p ∣ prod L := Theorem_7_2_3 h1 h2 by_cases on h3 · -- Case 1. h3 : p ∣ b apply Exists.intro b show b ∈ b :: L ∧ p ∣ b from And.intro (List.mem_cons_self b L) h3 done · -- Case 2. h3 : p ∣ prod L obtain (a : Nat) (h4 : a ∈ L ∧ p ∣ a) from ih h3 apply Exists.intro a show a ∈ b :: L ∧ p ∣ a from And.intro (List.mem_cons_of_mem b h4.left) h4.right done done done lemma prime_in_list {p : Nat} {l : List Nat} (h1 : prime p) (h2 : all_prime l) (h3 : p ∣ prod l) : p ∈ l := by obtain (a : Nat) (h4 : a ∈ l ∧ p ∣ a) from Theorem_7_2_4 h1 l h3 define at h2 have h5 : prime a := h2 a h4.left have h6 : p = 1 ∨ p = a := dvd_prime h5 h4.right disj_syll h6 (prime_not_one h1) rewrite [h6] show a ∈ l from h4.left done lemma first_le_first {p q : Nat} {l m : List Nat} (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m)) (h3 : prod (p :: l) = prod (q :: m)) : p ≤ q := by define at h1; define at h2 have h4 : q ∣ prod (p :: l) := by define apply Exists.intro (prod m) rewrite [←prod_cons] show prod (p :: l) = prod (q :: m) from h3 done have h5 : all_prime (q :: m) := h2.left rewrite [all_prime_cons] at h5 have h6 : q ∈ p :: l := prime_in_list h5.left h1.left h4 have h7 : nondec (p :: l) := h1.right rewrite [nondec_cons] at h7 rewrite [List.mem_cons] at h6 by_cases on h6 · -- Case 1. h6 : q = p linarith done · -- Case 2. h6 : q ∈ l have h8 : ∀ m ∈ l, p ≤ m := h7.left show p ≤ q from h8 q h6 done done lemma nondec_prime_list_tail {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by define at h define rewrite [all_prime_cons, nondec_cons] at h show all_prime l ∧ nondec l from And.intro h.left.right h.right.right done lemma cons_prod_not_one {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : prod (p :: l) ≠ 1 := by define at h have h1 : all_prime (p :: l) := h.left rewrite [all_prime_cons] at h1 rewrite [prod_cons] by_contra h2 show False from (prime_not_one h1.left) (eq_one_of_prod_one h2) done lemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) : l = [] ↔ prod l = 1 := by apply Iff.intro · -- (→) assume h1 : l = [] rewrite [h1] show prod [] = 1 from prod_nil done · -- (←) contrapos assume h1 : ¬l = [] obtain (p : Nat) (h2 : ∃ (L : List Nat), l = p :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List Nat) (h3 : l = p :: L) from h2 rewrite [h3] at h rewrite [h3] show ¬prod (p :: L) = 1 from cons_prod_not_one h done done lemma prime_pos {p : Nat} (h : prime p) : p > 0 := by define at h linarith done theorem Theorem_7_2_5 : ∀ (l1 l2 : List Nat), nondec_prime_list l1 → nondec_prime_list l2 → prod l1 = prod l2 → l1 = l2 := by apply List.rec · -- Base Case. Goal : ∀ (l2 : List Nat), nondec_prime_list [] → -- nondec_prime_list l2 → prod [] = prod l2 → [] = l2 fix l2 : List Nat assume h1 : nondec_prime_list [] assume h2 : nondec_prime_list l2 assume h3 : prod [] = prod l2 rewrite [prod_nil, eq_comm, ←list_nil_iff_prod_one h2] at h3 show [] = l2 from h3.symm done · -- Induction Step fix p : Nat fix L1 : List Nat assume ih : ∀ (L2 : List Nat), nondec_prime_list L1 → nondec_prime_list L2 → prod L1 = prod L2 → L1 = L2 -- Goal : ∀ (l2 : List Nat), nondec_prime_list (p :: L1) → -- nondec_prime_list l2 → prod (p :: L1) = prod l2 → p :: L1 = l2 fix l2 : List Nat assume h1 : nondec_prime_list (p :: L1) assume h2 : nondec_prime_list l2 assume h3 : prod (p :: L1) = prod l2 have h4 : ¬prod (p :: L1) = 1 := cons_prod_not_one h1 rewrite [h3, ←list_nil_iff_prod_one h2] at h4 obtain (q : Nat) (h5 : ∃ (L : List Nat), l2 = q :: L) from List.exists_cons_of_ne_nil h4 obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5 rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2) rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2) have h7 : p ≤ q := first_le_first h1 h2 h3 have h8 : q ≤ p := first_le_first h2 h1 h3.symm have h9 : p = q := by linarith rewrite [h9, prod_cons, prod_cons] at h3 --h3 : q * prod L1 = q * prod L2 have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1 have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2 define at h2 have h12 : all_prime (q :: L2) := h2.left rewrite [all_prime_cons] at h12 have h13 : q > 0 := prime_pos h12.left have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3 have h15 : L1 = L2 := ih L2 h10 h11 h14 rewrite [h6, h9, h15] rfl done done theorem fund_thm_arith (n : Nat) (h : n ≥ 1) : ∃! (l : List Nat), prime_factorization n l := by exists_unique · -- Existence show ∃ (l : List Nat), prime_factorization n l from exists_prime_factorization n h done · -- Uniqueness fix l1 : List Nat; fix l2 : List Nat assume h1 : prime_factorization n l1 assume h2 : prime_factorization n l2 define at h1; define at h2 have h3 : prod l1 = n := h1.right rewrite [←h2.right] at h3 show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3 done done /- Section 7.3 -/ theorem congr_refl (m : Nat) : ∀ (a : Int), a ≡ a (MOD m) := by fix a : Int define --Goal : ∃ (c : Int), a - a = ↑m * c apply Exists.intro 0 ring done theorem congr_symm {m : Nat} : ∀ {a b : Int}, a ≡ b (MOD m) → b ≡ a (MOD m) := by fix a : Int; fix b : Int assume h1 : a ≡ b (MOD m) define at h1 --h1 : ∃ (c : Int), a - b = ↑m * c define --Goal : ∃ (c : Int), b - a = ↑m * c obtain (c : Int) (h2 : a - b = m * c) from h1 apply Exists.intro (-c) show b - a = m * (-c) from calc b - a _ = -(a - b) := by ring _ = -(m * c) := by rw [h2] _ = m * (-c) := by ring done theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry /- Fundamental properties of congruence classes -/ lemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) : X = Y ↔ X.val = Y.val := Fin.ext_iff lemma val_nat_eq_mod (n k : Nat) : ([k]_(n + 1)).val = k % (n + 1) := by rfl lemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl theorem cc_rep {m : Nat} (X : ZMod m) : ∃ (a : Int), X = [a]_m := match m with | 0 => by apply Exists.intro X rfl done | n + 1 => by apply Exists.intro ↑(X.val) have h1 : X.val < n + 1 := Fin.prop X rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1] rfl done theorem add_class (m : Nat) (a b : Int) : [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm theorem mul_class (m : Nat) (a b : Int) : [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm lemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ [a - b]_m = [0]_m := by apply Iff.intro · -- (→) assume h1 : [a]_m = [b]_m have h2 : a - b = a + (-b) := by ring have h3 : b + (-b) = 0 := by ring show [a - b]_m = [0]_m from calc [a - b]_m _ = [a + (-b)]_m := by rw [h2] _ = [a]_m + [-b]_m := by rw [add_class] _ = [b]_m + [-b]_m := by rw [h1] _ = [b + -b]_m := by rw [add_class] _ = [0]_m := by rw [h3] done · -- (←) assume h1 : [a - b]_m = [0]_m have h2 : b + (a - b) = a := by ring have h3 : b + 0 = b := by ring show [a]_m = [b]_m from calc [a]_m _ = [b + (a - b)]_m := by rw [h2] _ = [b]_m + [a - b]_m := by rw [add_class] _ = [b]_m + [0]_m := by rw [h1] _ = [b + 0]_m := by rw [add_class] _ = [b]_m := by rw [h3] done done lemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) : [a]_m = [0]_m → [-a]_m = [0]_m := by assume h1 : [a]_m = [0]_m have h2 : 0 + (-a) = -a := by ring have h3 : a + (-a) = 0 := by ring show [-a]_m = [0]_m from calc [-a]_m _ = [0 + (-a)]_m := by rw [h2] _ = [0]_m + [-a]_m := by rw [add_class] _ = [a]_m + [-a]_m := by rw [h1] _ = [a + (-a)]_m := by rw [add_class] _ = [0]_m := by rw [h3] done lemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) : [-a]_m = [0]_m ↔ [a]_m = [0]_m := by apply Iff.intro _ (cc_neg_zero_of_cc_zero m a) assume h1 : [-a]_m = [0]_m have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1 have h3 : -(-a) = a := by ring rewrite [h3] at h2 show [a]_m = [0]_m from h2 done lemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl lemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m ↔ m ∣ k := match m with | 0 => by have h : (0 : Int) = (↑(0 : Nat) : Int) := by rfl rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj] apply Iff.intro · -- (→) assume h1 : k = 0 rewrite [h1] show 0 ∣ 0 from dvd_self 0 done · -- (←) assume h1 : 0 ∣ k obtain (c : Nat) (h2 : k = 0 * c) from h1 rewrite [h2] ring done done | n + 1 => by rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero] show k % (n + 1) = 0 ↔ n + 1 ∣ k from (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm done lemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m ↔ ↑m ∣ a := by obtain (k : Nat) (h1 : a = ↑k ∨ a = -↑k) from Int.eq_nat_or_neg a by_cases on h1 · -- Case 1. h1: a = ↑k rewrite [h1, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done · -- Case 2. h1: a = -↑k rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done done theorem cc_eq_iff_congr (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ a ≡ b (MOD m) := calc [a]_m = [b]_m _ ↔ [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b _ ↔ ↑m ∣ (a - b) := cc_zero_iff_dvd m (a - b) _ ↔ a ≡ b (MOD m) := by rfl /- End of fundamental properties of congruence classes -/ lemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m := by have h1 : (↑m : Int) ≠ 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m) show 0 ≤ a % m from Int.emod_nonneg a h1 done lemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m) have h2 : (↑m : Int) > 0 := (Nat.cast_pos).rtl h1 show a % m < m from Int.emod_lt_of_pos a h2 done lemma congr_mod_mod (m : Nat) (a : Int) : a ≡ a % m (MOD m) := by define have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m apply Exists.intro (a / m) show a - a % m = m * (a / m) from calc a - (a % m) _ = m * (a / m) + a % m - a % m := by rw [h1] _ = m * (a / m) := by ring done lemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) := And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a)) theorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) : ∃! (r : Int), 0 ≤ r ∧ r < m ∧ a ≡ r (MOD m) := by exists_unique · -- Existence apply Exists.intro (a % m) show 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) from mod_cmpl_res m a done · -- Uniqueness fix r1 : Int; fix r2 : Int assume h1 : 0 ≤ r1 ∧ r1 < m ∧ a ≡ r1 (MOD m) assume h2 : 0 ≤ r2 ∧ r2 < m ∧ a ≡ r2 (MOD m) have h3 : r1 ≡ r2 (MOD m) := congr_trans (congr_symm h1.right.right) h2.right.right obtain (d : Int) (h4 : r1 - r2 = m * d) from h3 have h5 : r1 - r2 < m * 1 := by linarith have h6 : m * (-1) < r1 - r2 := by linarith rewrite [h4] at h5 --h5 : m * d < m * 1 rewrite [h4] at h6 --h6 : m * -1 < m * d have h7 : (↑m : Int) ≥ 0 := Nat.cast_nonneg m have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7 have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7 have h10 : d = 0 := by linarith show r1 = r2 from calc r1 _ = r1 - r2 + r2 := by ring _ = m * 0 + r2 := by rw [h4, h10] _ = r2 := by ring done done lemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m := (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a) theorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y rewrite [h1, h2] have h3 : a + b = b + a := by ring show [a]_m + [b]_m = [b]_m + [a]_m from calc [a]_m + [b]_m _ = [a + b]_m := add_class m a b _ = [b + a]_m := by rw [h3] _ = [b]_m + [a]_m := (add_class m b a).symm done theorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X rewrite [h1] have h2 : a * 1 = a := by ring show [a]_m * [1]_m = [a]_m from calc [a]_m * [1]_m _ = [a * 1]_m := mul_class m a 1 _ = [a]_m := by rw [h2] done theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry lemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) : [a]_m * [gcd_c2 m a]_m = [1]_m := by set s : Int := gcd_c1 m a have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m define at h1 rewrite [h1, Nat.cast_one] at h2 --h2 : s * ↑m + gcd_c2 m a * ↑a = 1 rewrite [mul_class, cc_eq_iff_congr] define --Goal : ∃ (c : Int), ↑a * gcd_c2 m a - 1 = ↑m * c apply Exists.intro (-s) show a * (gcd_c2 m a) - 1 = m * (-s) from calc a * (gcd_c2 m a) - 1 _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring _ = 1 + m * (-s) - 1 := by rw [h2] _ = m * (-s) := by ring done theorem Theorem_7_3_7 (m a : Nat) : invertible [a]_m ↔ rel_prime m a := by apply Iff.intro · -- (→) assume h1 : invertible [a]_m define at h1 obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1 obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y rewrite [h3, mul_class, cc_eq_iff_congr] at h2 define at h2 obtain (c : Int) (h4 : a * b - 1 = m * c) from h2 rewrite [Exercise_7_2_6] --Goal : ∃ (s t : Int), s * ↑m + t * ↑a = 1 apply Exists.intro (-c) apply Exists.intro b show (-c) * m + b * a = 1 from calc (-c) * m + b * a _ = (-c) * m + (a * b - 1) + 1 := by ring _ = (-c) * m + m * c + 1 := by rw [h4] _ = 1 := by ring done · -- (←) assume h1 : rel_prime m a define show ∃ (Y : ZMod m), [a]_m * Y = [1]_m from Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1) done done /- Section 7.4 -/ section Euler open Euler lemma num_rp_below_base {m : Nat} : num_rp_below m 0 = 0 := by rfl lemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) : num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by have h1 : num_rp_below m (j + 1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : gcd m j = 1 rewrite [if_pos h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1 show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1 done lemma num_rp_below_step_not_rp {m j : Nat} (h : ¬rel_prime m j) : num_rp_below m (j + 1) = num_rp_below m j := by have h1 : num_rp_below m (j +1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : ¬gcd m j = 1 rewrite [if_neg h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j show num_rp_below m (j + 1) = num_rp_below m j from h1 done lemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl #eval phi 10 --Answer: 4 lemma prod_inv_iff_inv {m : Nat} {X : ZMod m} (h1 : invertible X) (Y : ZMod m) : invertible (X * Y) ↔ invertible Y := by apply Iff.intro · -- (→) assume h2 : invertible (X * Y) obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2 apply Exists.intro (X * Z) rewrite [←h3] --Goal : Y * (X * Z) = X * Y * Z ring --Note that ring can do algebra in ZMod m done · -- (←) assume h2 : invertible Y obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1 obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2 apply Exists.intro (Xi * Yi) show (X * Y) * (Xi * Yi) = [1]_m from calc X * Y * (Xi * Yi) _ = (X * Xi) * (Y * Yi) := by ring _ = [1]_m * [1]_m := by rw [h3, h4] _ = [1]_m := Theorem_7_3_6_7 [1]_m done done lemma F_rp_def {m i : Nat} (h : rel_prime m i) : F m i = [i]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h --h : gcd m i = 1 rewrite [if_pos h] at h1 show F m i = [i]_m from h1 done lemma F_not_rp_def {m i : Nat} (h : ¬rel_prime m i) : F m i = [1]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h rewrite [h1, if_neg h] rfl done lemma prod_seq_base {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 0 k f = [1]_m := by rfl lemma prod_seq_step {m : Nat} (n k : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl lemma prod_seq_zero_step {m : Nat} (n : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by rewrite [prod_seq_step, zero_add] rfl done lemma prod_one {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 1 k f = f k := by rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7] rfl done lemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl lemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m := calc [G m a i]_m _ = [(a * i) % m]_m := by rfl _ = [a * i]_m := (cc_eq_mod m (a * i)).symm _ = [a]_m * [i]_m := (mul_class m a i).symm lemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) : rel_prime m (G m a i) ↔ rel_prime m i := by have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1 show rel_prime m (G m a i) ↔ rel_prime m i from calc rel_prime m (G m a i) _ ↔ invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm _ ↔ invertible ([a]_m * [i]_m) := by rw [cc_G] _ ↔ invertible [i]_m := prod_inv_iff_inv h2 ([i]_m) _ ↔ rel_prime m i := Theorem_7_3_7 m i done lemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) : F m (G m a i) = [a]_m * F m i := by have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2 show F m (G m a i) = [a]_m * F m i from calc F m (G m a i) _ = [G m a i]_m := F_rp_def h3 _ = [a]_m * [i]_m := cc_G m a i _ = [a]_m * F m i := by rw [F_rp_def h2] done lemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : ¬rel_prime m i) : F m (G m a i) = [1]_m := by rewrite [←G_rp_iff h1 i] at h2 show F m (G m a i) = [1]_m from F_not_rp_def h2 done lemma FG_prod {m a : Nat} (h1 : rel_prime m a) : ∀ (k : Nat), prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by by_induc · -- Base Case show prod_seq 0 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from calc prod_seq 0 0 ((F m) ∘ (G m a)) _ = [1]_m := prod_seq_base _ _ _ = [a]_m ^ 0 * [1]_m := by ring _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by rw [num_rp_below_base, prod_seq_base] done · -- Induction Step fix k : Nat assume ih : prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) by_cases h2 : rel_prime m k · -- Case 1. h2 : rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([a]_m * F m k) := by rw [FG_rp h1 h2] _ = [a]_m ^ ((num_rp_below m k) + 1) * ((prod_seq k 0 (F m)) * F m k) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_rp h2, prod_seq_zero_step] done · -- Case 2. h2 : ¬rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([1]_m) := by rw [FG_not_rp h1 h2] _ = [a]_m ^ (num_rp_below m k) * (prod_seq k 0 (F m) * ([1]_m)) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_not_rp h2, prod_seq_zero_step, F_not_rp_def h2] done done done lemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by define --Goal : ∀ i < m, G m a i < m fix i : Nat assume h1 : i < m rewrite [G_def] --Goal : a * i % m < m show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m) done lemma left_inv_one_one_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g' (g i) = i) : one_one_below n g := sorry lemma right_inv_onto_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g (g' i) = i) (h2 : maps_below n g') : onto_below n g := by define at h2; define fix k : Nat assume h3 : k < n apply Exists.intro (g' k) show g' k < n ∧ g (g' k) = k from And.intro (h2 k h3) (h1 k h3) done lemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m * [inv_mod m a]_m = [1]_m := by have h2 : 0 ≤ (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a) show [a]_m * [inv_mod m a]_m = [1]_m from calc [a]_m * [inv_mod m a]_m _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2] _ = [a]_m * [gcd_c2 m a]_m := by rw [←cc_eq_mod] _ = [1]_m := gcd_c2_inv h1 done lemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m := calc a * (b % m) % m = a % m * (b % m % m) % m := Nat.mul_mod _ _ _ _ = a % m * (b % m) % m := by rw [Nat.mod_mod] _ = a * b % m := (Nat.mul_mod _ _ _).symm lemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm] rfl done theorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] : ↑a ≡ ↑b (MOD m) ↔ a % m = b % m := sorry lemma mul_inv_mod_cancel {m a i : Nat} [NeZero m] (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1 rewrite [mul_class, cc_eq_iff_congr, ←Nat.cast_mul, ←Nat.cast_one, congr_iff_mod_eq_Nat] at h3 show a * inv_mod m a * i % m = i from calc a * (inv_mod m a) * i % m _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = 1 % m * i % m := by rw [h3] _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = i % m := by rw [one_mul] _ = i := Nat.mod_eq_of_lt h2 done lemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl lemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, G m a (Ginv m a i) = i := by fix i : Nat assume h2 : i < m show G m a (Ginv m a i) = i from calc G m a (Ginv m a i) _ = a * ((inv_mod m a * i) % m) % m := by rfl _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, Ginv m a (G m a i) = i := by fix i : Nat assume h2 : i < m show Ginv m a (G m a i) = i from calc Ginv m a (G m a i) _ = inv_mod m a * ((a * i) % m) % m := by rfl _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc, mul_comm (inv_mod m a)] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_maps_below (m a : Nat) [NeZero m] : maps_below m (Ginv m a) := G_maps_below m (inv_mod m a) lemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : one_one_below m (G m a) := left_inv_one_one_below (Ginv_left_inv h1) lemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : onto_below m (G m a) := right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a) lemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : perm_below m (G m a) := And.intro (G_maps_below m a) (And.intro (G_one_one_below h1) (G_onto_below h1)) --Permuting a product of congruence classes doesn't change product lemma swap_fst (u v : Nat) : swap u v u = v := by define : swap u v u --Goal : (if u = u then v else if u = v then u else u) = v have h : u = u := by rfl rewrite [if_pos h] rfl done lemma swap_snd (u v : Nat) : swap u v v = u := by define : swap u v v by_cases h1 : v = u · -- Case 1. h1 : v = u rewrite [if_pos h1] show v = u from h1 done · -- Case 2. h1 : v ≠ u rewrite [if_neg h1] have h2 : v = v := by rfl rewrite [if_pos h2] rfl done done lemma swap_other {u v i : Nat} (h1 : i ≠ u) (h2 : i ≠ v) : swap u v i = i := by define : swap u v i rewrite [if_neg h1, if_neg h2] rfl done lemma swap_values (u v i : Nat) : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := by by_cases h1 : i = u · -- Case 1. h1 : i = u apply Or.inl rewrite [h1] show swap u v u = v from swap_fst u v done · -- Case 2. h1 : i ≠ u apply Or.inr by_cases h2 : i = v · -- Case 2.1. h2 : i = v apply Or.inl rewrite [h2] show swap u v v = u from swap_snd u v done · -- Case 2.2. h2 : i ≠ v apply Or.inr show swap u v i = i from swap_other h1 h2 done done done lemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by define fix i : Nat assume h3 : i < n have h4 : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := swap_values u v i by_cases on h4 · -- Case 1. h4 : swap u v i = v rewrite [h4] show v < n from h2 done · -- Case 2. by_cases on h4 · -- Case 2.1. h4 : swap u v i = u rewrite [h4] show u < n from h1 done · -- Case 2.2. h4 : swap u v i = i rewrite [h4] show i < n from h3 done done done lemma swap_swap (u v n : Nat) : ∀ i < n, swap u v (swap u v i) = i := by fix i : Nat assume h : i < n by_cases h1 : i = u · -- Case 1. h1 : i = u rewrite [h1, swap_fst, swap_snd] rfl done · -- Case 2. h1 : i ≠ u by_cases h2 : i = v · -- Case 2.1. h2 : i = v rewrite [h2, swap_snd, swap_fst] rfl done · -- Case 2.2. h2 : i ≠ v rewrite [swap_other h1 h2, swap_other h1 h2] rfl done done done lemma swap_one_one_below (u v n) : one_one_below n (swap u v) := left_inv_one_one_below (swap_swap u v n) lemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) := right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2) lemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) := And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2)) lemma comp_perm_below {n : Nat} {f g : Nat → Nat} (h1 : perm_below n f) (h2 : perm_below n g) : perm_below n (f ∘ g) := sorry lemma trivial_swap (u : Nat) : swap u u = id := by apply funext fix x : Nat by_cases h1 : x = u · -- Case 1. h1 : x = u rewrite [h1, swap_fst] rfl done · -- Case 2. h1 : x ≠ u rewrite [swap_other h1 h1] rfl done done lemma prod_eq_fun {m : Nat} (f g : Nat → ZMod m) (k : Nat) : ∀ (n : Nat), (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g := by by_induc · -- Base Case assume h : (∀ i < 0, f (k + i) = g (k + i)) rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g assume h1 : ∀ i < n + 1, f (k + i) = g (k + i) have h2 : ∀ i < n, f (k + i) = g (k + i) := by fix i : Nat assume h2 : i < n have h3 : i < n + 1 := by linarith show f (k + i) = g (k + i) from h1 i h3 done have h3 : prod_seq n k f = prod_seq n k g := ih h2 have h4 : n < n + 1 := Nat.lt_succ_self n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4] rfl done done lemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := by have h2 : ∀ (i : Nat), i < u → (f ∘ swap u n) (0 + i) = f (0 + i) := by fix i : Nat assume h2 : i < u have h3 : 0 + i ≠ u := by linarith have h4 : 0 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl done show prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f from prod_eq_fun (f ∘ swap u n) f 0 u h2 done lemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := by have h2 : ∀ i < j, (f ∘ swap u n) (u + 1 + i) = f (u + 1 + i) := by fix i : Nat assume h2 : i < j have h3 : u + 1 + i ≠ u := by linarith have h4 : u + 1 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl show prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f from prod_eq_fun (f ∘ swap u n) f (u + 1) j h2 done lemma break_prod {m : Nat} (n : Nat) (f : Nat → ZMod m) : ∀ (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by by_induc · -- Base Case have h : n + 0 = n := by rfl rewrite [prod_seq_base, h, Theorem_7_3_6_7] rfl done · -- Induction Step fix j : Nat assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f rewrite [←add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc] rfl done done lemma break_prod_twice {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f := break_prod n f 1 rewrite [prod_one] at h2 have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f := break_prod (u + 1) f j rewrite [←h1] at h3 have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f := break_prod u f 1 rewrite [prod_one] at h4 rewrite [h3, h4] at h2 show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2 done lemma swap_prod_eq_prod {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f := by by_cases h2 : u = n · -- Case 1. h2 : u = n rewrite [h2, trivial_swap n] --Goal : prod_seq (n + 1) 0 (f ∘ id) = prod_seq (n + 1) 0 f rfl done · -- Case 2. h2 : ¬u = n have h3 : u + 1 ≤ n := Nat.lt_of_le_of_ne h1 h2 obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3 have break_f : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := break_prod_twice f h4 have break_fs : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_prod_twice (f ∘ swap u n) h4 have f_eq_fs_below : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := swap_prod_eq_prod_below f h1 have f_eq_fs_btwn : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4 show prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f from calc prod_seq (n + 1) 0 (f ∘ swap u n) _ = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_fs _ = prod_seq u 0 f * (f ∘ swap u n) u * prod_seq j (u + 1) f * (f ∘ swap u n) n := by rw [f_eq_fs_below, f_eq_fs_btwn] _ = prod_seq u 0 f * f (swap u n u) * prod_seq j (u + 1) f * f (swap u n n) := by rfl _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by rw [swap_fst, swap_snd] _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring _ = prod_seq (n + 1) 0 f := break_f.symm done done lemma perm_below_fixed {n : Nat} {g : Nat → Nat} (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry lemma perm_prod {m : Nat} (f : Nat → ZMod m) : ∀ (n : Nat), ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) := by by_induc · -- Base Case fix g : Nat → Nat assume h1 : perm_below 0 g rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) fix g : Nat → Nat assume g_pb : perm_below (n + 1) g define at g_pb have g_ob : onto_below (n + 1) g := g_pb.right.right define at g_ob have h1 : n < n + 1 := by linarith obtain (u : Nat) (h2 : u < n + 1 ∧ g u = n) from g_ob n h1 have s_pb : perm_below (n + 1) (swap u n) := swap_perm_below h2.left h1 have gs_pb_n1 : perm_below (n + 1) (g ∘ swap u n) := comp_perm_below g_pb s_pb have gs_fix_n : (g ∘ swap u n) n = n := calc (g ∘ swap u n) n _ = g (swap u n n) := by rfl _ = g u := by rw [swap_snd] _ = n := h2.right have gs_pb_n : perm_below n (g ∘ swap u n) := perm_below_fixed gs_pb_n1 gs_fix_n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f ∘ (g ∘ swap u n)) := ih (g ∘ swap u n) gs_pb_n have h3 : u ≤ n := by linarith show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f ∘ g) from calc prod_seq (n + 1) 0 f _ = prod_seq n 0 f * f n := prod_seq_zero_step n f _ = prod_seq n 0 (f ∘ (g ∘ swap u n)) * f ((g ∘ swap u n) n) := by rw [gs_prod, gs_fix_n] _ = prod_seq n 0 (f ∘ g ∘ swap u n) * (f ∘ g ∘ swap u n) n := by rfl _ = prod_seq (n + 1) 0 (f ∘ g ∘ swap u n) := (prod_seq_zero_step n (f ∘ g ∘ swap u n)).symm _ = prod_seq (n + 1) 0 ((f ∘ g) ∘ swap u n) := by rfl _ = prod_seq (n + 1) 0 (f ∘ g) := swap_prod_eq_prod (f ∘ g) h3 done done lemma F_invertible (m i : Nat) : invertible (F m i) := by by_cases h : rel_prime m i · -- Case 1. h : rel_prime m i rewrite [F_rp_def h] show invertible [i]_m from (Theorem_7_3_7 m i).rtl h done · -- Case 2. h : ¬rel_prime m i rewrite [F_not_rp_def h] apply Exists.intro [1]_m show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m done done lemma Fprod_invertible (m : Nat) : ∀ (k : Nat), invertible (prod_seq k 0 (F m)) := by by_induc · -- Base Case apply Exists.intro [1]_m show prod_seq 0 0 (F m) * [1]_m = [1]_m from calc prod_seq 0 0 (F m) * [1]_m _ = [1]_m * [1]_m := by rw [prod_seq_base] _ = [1]_m := Theorem_7_3_6_7 ([1]_m) done · -- Induction Step fix k : Nat assume ih : invertible (prod_seq k 0 (F m)) rewrite [prod_seq_zero_step] show invertible (prod_seq k 0 (F m) * (F m k)) from (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k) done done theorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m ^ (phi m) = [1]_m := by have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2 show [a]_m ^ (phi m) = [1]_m from calc [a]_m ^ (phi m) _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3] _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring _ = prod_seq m 0 (F m ∘ G m a) * Y := by rw [FG_prod h1 m, phi_def] _ = prod_seq m 0 (F m) * Y := by rw [perm_prod (F m) m (G m a) (G_perm_below h1)] _ = [1]_m := by rw [h3] done lemma Exercise_7_4_5_Int (m : Nat) (a : Int) : ∀ (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry lemma Exercise_7_4_5_Nat (m a n : Nat) : [a]_m ^ n = [a ^ n]_m := by rewrite [Exercise_7_4_5_Int] rfl done theorem Euler's_theorem {m a : Nat} [NeZero m] (h1 : rel_prime m a) : a ^ (phi m) ≡ 1 (MOD m) := by have h2 : [a]_m ^ (phi m) = [1]_m := Theorem_7_4_2 h1 rewrite [Exercise_7_4_5_Nat m a (phi m)] at h2 --h2 : [a ^ phi m]_m = [1]_m show a ^ (phi m) ≡ 1 (MOD m) from (cc_eq_iff_congr _ _ _).ltr h2 done #eval gcd 10 7 --Answer: 1. So 10 and 7 are relatively prime #eval 7 ^ phi 10 --Answer: 2401, which is congruent to 1 mod 10. end Euler /- Section 7.5 -/ lemma num_rp_prime {p : Nat} (h1 : prime p) : ∀ k < p, num_rp_below p (k + 1) = k := sorry lemma phi_prime {p : Nat} (h1 : prime p) : phi p = p - 1 := by have h2 : 1 ≤ p := prime_pos h1 have h3 : p - 1 + 1 = p := Nat.sub_add_cancel h2 have h4 : p - 1 < p := by linarith have h5 : num_rp_below p (p - 1 + 1) = p - 1 := num_rp_prime h1 (p - 1) h4 rewrite [h3] at h5 show phi p = p - 1 from h5 done theorem Theorem_7_2_2_Int {a c : Nat} {b : Int} (h1 : ↑c ∣ ↑a * b) (h2 : rel_prime a c) : ↑c ∣ b := by rewrite [Int.natCast_dvd, Int.natAbs_mul, Int.natAbs_ofNat] at h1 --h1 : c ∣ a * Int.natAbs b rewrite [Int.natCast_dvd] --Goal : c ∣ Int.natAbs b show c ∣ Int.natAbs b from Theorem_7_2_2 h1 h2 done lemma Lemma_7_4_5 {m n : Nat} (a b : Int) (h1 : rel_prime m n) : a ≡ b (MOD m * n) ↔ a ≡ b (MOD m) ∧ a ≡ b (MOD n) := by apply Iff.intro · -- (→) assume h2 : a ≡ b (MOD m * n) obtain (j : Int) (h3 : a - b = (m * n) * j) from h2 apply And.intro · -- Proof of a ≡ b (MOD m) apply Exists.intro (n * j) show a - b = m * (n * j) from calc a - b _ = m * n * j := h3 _ = m * (n * j) := by ring done · -- Proof of a ≡ b (MOD n) apply Exists.intro (m * j) show a - b = n * (m * j) from calc a - b _ = m * n * j := h3 _ = n * (m * j) := by ring done done · -- (←) assume h2 : a ≡ b (MOD m) ∧ a ≡ b (MOD n) obtain (j : Int) (h3 : a - b = m * j) from h2.left have h4 : (↑n : Int) ∣ a - b := h2.right rewrite [h3] at h4 --h4 : ↑n ∣ ↑m * j have h5 : ↑n ∣ j := Theorem_7_2_2_Int h4 h1 obtain (k : Int) (h6 : j = n * k) from h5 apply Exists.intro k --Goal : a - b = ↑(m * n) * k rewrite [Nat.cast_mul] --Goal : a - b = ↑m * ↑n * k show a - b = (m * n) * k from calc a - b _ = m * j := h3 _ = m * (n * k) := by rw [h6] _ = (m * n) * k := by ring done done --From exercises of Section 7.2 theorem rel_prime_symm {a b : Nat} (h : rel_prime a b) : rel_prime b a := sorry lemma prime_NeZero {p : Nat} (h : prime p) : NeZero p := by rewrite [neZero_iff] --Goal : p ≠ 0 define at h linarith done lemma Lemma_7_5_1 {p e d m c s : Nat} {t : Int} (h1 : prime p) (h2 : e * d = (p - 1) * s + 1) (h3 : m ^ e - c = p * t) : c ^ d ≡ m (MOD p) := by have h4 : m ^ e ≡ c (MOD p) := Exists.intro t h3 have h5 : [m ^ e]_p = [c]_p := (cc_eq_iff_congr _ _ _).rtl h4 rewrite [←Exercise_7_4_5_Nat] at h5 --h5 : [m]_p ^ e = [c]_p by_cases h6 : p ∣ m · -- Case 1. h6 : p ∣ m have h7 : m ≡ 0 (MOD p) := by obtain (j : Nat) (h8 : m = p * j) from h6 apply Exists.intro (↑j : Int) --Goal : ↑m - 0 = ↑p * ↑j rewrite [h8, Nat.cast_mul] ring done have h8 : [m]_p = [0]_p := (cc_eq_iff_congr _ _ _).rtl h7 have h9 : e * d ≠ 0 := by rewrite [h2] show (p - 1) * s + 1 ≠ 0 from Nat.add_one_ne_zero _ done have h10 : (0 : Int) ^ (e * d) = 0 := zero_pow h9 have h11 : [c ^ d]_p = [m]_p := calc [c ^ d]_p _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat] _ = ([m]_p ^ e) ^ d := by rw [h5] _ = [m]_p ^ (e * d) := by ring _ = [0]_p ^ (e * d) := by rw [h8] _ = [0 ^ (e * d)]_p := Exercise_7_4_5_Int _ _ _ _ = [0]_p := by rw [h10] _ = [m]_p := by rw [h8] show c ^ d ≡ m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11 done · -- Case 2. h6 : ¬p ∣ m have h7 : rel_prime m p := rel_prime_of_prime_not_dvd h1 h6 have h8 : rel_prime p m := rel_prime_symm h7 have h9 : NeZero p := prime_NeZero h1 have h10 : (1 : Int) ^ s = 1 := by ring have h11 : [c ^ d]_p = [m]_p := calc [c ^ d]_p _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat] _ = ([m]_p ^ e) ^ d := by rw [h5] _ = [m]_p ^ (e * d) := by ring _ = [m]_p ^ ((p - 1) * s + 1) := by rw [h2] _ = ([m]_p ^ (p - 1)) ^ s * [m]_p := by ring _ = ([m]_p ^ (phi p)) ^ s * [m]_p := by rw [phi_prime h1] _ = [1]_p ^ s * [m]_p := by rw [Theorem_7_4_2 h8] _ = [1 ^ s]_p * [m]_p := by rw [Exercise_7_4_5_Int] _ = [1]_p * [m]_p := by rw [h10] _ = [m]_p * [1]_p := by ring _ = [m]_p := Theorem_7_3_6_7 _ show c ^ d ≡ m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11 done done theorem Theorem_7_5_1 (p q n e d k m c : Nat) (p_prime : prime p) (q_prime : prime q) (p_ne_q : p ≠ q) (n_pq : n = p * q) (ed_congr_1 : e * d = k * (p - 1) * (q - 1) + 1) (h1 : [m]_n ^ e = [c]_n) : [c]_n ^ d = [m]_n := by rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] at h1 --h1 : m ^ e ≡ c (MOD n) rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] --Goal : c ^ d ≡ m (MOD n) obtain (j : Int) (h2 : m ^ e - c = n * j) from h1 rewrite [n_pq, Nat.cast_mul] at h2 --h2 : m ^ e - c = p * q * j have h3 : e * d = (p - 1) * (k * (q - 1)) + 1 := by rewrite [ed_congr_1] ring done have h4 : m ^ e - c = p * (q * j) := by rewrite [h2] ring done have congr_p : c ^ d ≡ m (MOD p) := Lemma_7_5_1 p_prime h3 h4 have h5 : e * d = (q - 1) * (k * (p - 1)) + 1 := by rewrite [ed_congr_1] ring done have h6 : m ^ e - c = q * (p * j) := by rewrite [h2] ring done have congr_q : c ^ d ≡ m (MOD q) := Lemma_7_5_1 q_prime h5 h6 have h7 : ¬q ∣ p := by by_contra h8 have h9 : q = 1 ∨ q = p := dvd_prime p_prime h8 disj_syll h9 (prime_not_one q_prime) show False from p_ne_q h9.symm done have h8 : rel_prime p q := rel_prime_of_prime_not_dvd q_prime h7 rewrite [n_pq, Lemma_7_4_5 _ _ h8] show c ^ d ≡ m (MOD p) ∧ c ^ d ≡ m (MOD q) from And.intro congr_p congr_q done /- BEGIN EXERCISES -/ namespace Exercises /- Section 7.1 -/ -- 1. theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry -- 2. lemma gcd_comm_lt {a b : Nat} (h : a < b) : gcd a b = gcd b a := sorry theorem gcd_comm (a b : Nat) : gcd a b = gcd b a := sorry -- 3. theorem Exercise_7_1_5 (a b : Nat) (n : Int) : (∃ (s t : Int), s * a + t * b = n) ↔ (↑(gcd a b) : Int) ∣ n := sorry -- 4. theorem Exercise_7_1_6 (a b c : Nat) : gcd a b = gcd (a + b * c) b := sorry -- 5. theorem gcd_is_nonzero {a b : Nat} (h : a ≠ 0 ∨ b ≠ 0) : gcd a b ≠ 0 := sorry -- 6. theorem gcd_greatest {a b d : Nat} (h1 : gcd a b ≠ 0) (h2 : d ∣ a) (h3 : d ∣ b) : d ≤ gcd a b := sorry -- 7. lemma Lemma_7_1_10a {a b : Nat} (n : Nat) (h : a ∣ b) : (n * a) ∣ (n * b) := sorry lemma Lemma_7_1_10b {a b n : Nat} (h1 : n ≠ 0) (h2 : (n * a) ∣ (n * b)) : a ∣ b := sorry lemma Lemma_7_1_10c {a b : Nat} (h1 : a ∣ b) (h2 : b ∣ a) : a = b := sorry theorem Exercise_7_1_10 (a b n : Nat) : gcd (n * a) (n * b) = n * gcd a b := sorry /- Section 7.2 -/ -- 1. lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry -- 2. -- Hints: Start with apply List.rec. You may find mul_ne_zero useful theorem prod_nonzero_nonzero : ∀ (l : List Nat), (∀ a ∈ l, a ≠ 0) → prod l ≠ 0 := sorry -- 3.

theorem rel_prime_iff_no_common_factor (a b : Nat) : rel_prime a b ↔ ¬∃ (p : Nat), prime p ∧ p ∣ a ∧ p ∣ b

HTPI.Exercises.rel_prime_iff_no_common_factor

htpi/HTPILib/Chap7.lean

htpi

/- Copyright 2023 Daniel J. Velleman -/ import HTPILib.Chap6 namespace HTPI /- Definitions -/ lemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by have h : n + 1 > 0 := Nat.succ_pos n show a % (n + 1) < n + 1 from Nat.mod_lt a h done def gcd (a b : Nat) : Nat := match b with | 0 => a | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd (n + 1) (a % (n + 1)) termination_by b mutual def gcd_c1 (a b : Nat) : Int := match b with | 0 => 1 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c2 (n + 1) (a % (n + 1)) --Corresponds to s = t' termination_by b def gcd_c2 (a b : Nat) : Int := match b with | 0 => 0 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c1 (n + 1) (a % (n + 1)) - (gcd_c2 (n + 1) (a % (n + 1))) * ↑(a / (n + 1)) --Corresponds to t = s' - t'q termination_by b end def prime (n : Nat) : Prop := 2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n def prime_factor (p n : Nat) : Prop := prime p ∧ p ∣ n def all_prime (l : List Nat) : Prop := ∀ p ∈ l, prime p def nondec (l : List Nat) : Prop := match l with | [] => True --Of course, True is a proposition that is always true | n :: L => (∀ m ∈ L, n ≤ m) ∧ nondec L def nondec_prime_list (l : List Nat) : Prop := all_prime l ∧ nondec l def prod (l : List Nat) : Nat := match l with | [] => 1 | n :: L => n * (prod L) def prime_factorization (n : Nat) (l : List Nat) : Prop := nondec_prime_list l ∧ prod l = n def rel_prime (a b : Nat) : Prop := gcd a b = 1 def congr_mod (m : Nat) (a b : Int) : Prop := (↑m : Int) ∣ (a - b) def cc (m : Nat) (a : Int) : ZMod m := (↑a : ZMod m) notation:50 a " ≡ " b " (MOD " m ")" => congr_mod m a b notation:max "["a"]_"m:max => cc m a def invertible {m : Nat} (X : ZMod m) : Prop := ∃ (Y : ZMod m), X * Y = [1]_m def num_rp_below (m k : Nat) : Nat := match k with | 0 => 0 | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j def phi (m : Nat) : Nat := num_rp_below m m def prod_seq {m : Nat} (j k : Nat) (f : Nat → ZMod m) : ZMod m := match j with | 0 => [1]_m | n + 1 => prod_seq n k f * f (k + n) def maps_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i < n, g i < n def one_one_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i1 < n, ∀ i2 < n, g i1 = g i2 → i1 = i2 def onto_below (n : Nat) (g : Nat → Nat) : Prop := ∀ k < n, ∃ i < n, g i = k def perm_below (n : Nat) (g : Nat → Nat) : Prop := maps_below n g ∧ one_one_below n g ∧ onto_below n g def inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m) def swap (u v i : Nat) : Nat := if i = u then v else if i = v then u else i namespace Euler --For definitions specific to Euler's theorem def F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m def G (m a i : Nat) : Nat := (a * i) % m def Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i end Euler /- Section 7.1 -/ theorem dvd_mod_of_dvd_a_b {a b d : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ (a % b) := by set q : Nat := a / b have h3 : b * q + a % b = a := Nat.div_add_mod a b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 define --Goal : ∃ (c : Nat), a % b = d * c apply Exists.intro (j - k * q) show a % b = d * (j - k * q) from calc a % b _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm _ = a - b * q := by rw [h3] _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc] _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm done theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry #eval gcd 672 161 --Answer: 7 lemma gcd_base (a : Nat) : gcd a 0 = a := by rfl lemma gcd_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd a b = gcd b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1)) rfl done lemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b ≠ 0) : a % b < b := by have h1 : b > 0 := Nat.pos_of_ne_zero h show a % b < b from Nat.mod_lt a h1 done lemma dvd_self (n : Nat) : n ∣ n := by apply Exists.intro 1 ring done theorem gcd_dvd : ∀ (b a : Nat), (gcd a b) ∣ a ∧ (gcd a b) ∣ b := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd a b_1) ∣ a ∧ (gcd a b_1) ∣ b_1 fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_base] --Goal: a ∣ a ∧ a ∣ 0 apply And.intro (dvd_self a) define apply Exists.intro 0 rfl done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_nonzero a h1] --Goal : gcd b (a % b) ∣ a ∧ gcd b (a % b) ∣ b have h2 : a % b < b := mod_nonzero_lt a h1 have h3 : (gcd b (a % b)) ∣ b ∧ (gcd b (a % b)) ∣ (a % b) := ih (a % b) h2 b apply And.intro _ h3.left show (gcd b (a % b)) ∣ a from dvd_a_of_dvd_b_mod h3.left h3.right done done theorem gcd_dvd_left (a b : Nat) : (gcd a b) ∣ a := (gcd_dvd b a).left theorem gcd_dvd_right (a b : Nat) : (gcd a b) ∣ b := (gcd_dvd b a).right lemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl lemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c1 a b = gcd_c2 b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done lemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl lemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * ↑(a / b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done theorem gcd_lin_comb : ∀ (b a : Nat), (gcd_c1 a b) * ↑a + (gcd_c2 a b) * ↑b = ↑(gcd a b) := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd_c1 a b_1) * ↑a + (gcd_c2 a b_1) * ↑b_1 = ↑(gcd a b_1) fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base] --Goal : 1 * ↑a + 0 * ↑0 = ↑a ring done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1] --Goal : gcd_c2 b (a % b) * ↑a + -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * ↑(a / b)) * ↑b = -- ↑(gcd b (a % b)) set r : Nat := a % b set q : Nat := a / b set s : Int := gcd_c1 b r set t : Int := gcd_c2 b r --Goal : t * ↑a + (s - t * ↑q) * ↑b = ↑(gcd b r) have h2 : r < b := mod_nonzero_lt a h1 have h3 : s * ↑b + t * ↑r = ↑(gcd b r) := ih r h2 b have h4 : b * q + r = a := Nat.div_add_mod a b rewrite [←h3, ←h4] rewrite [Nat.cast_add, Nat.cast_mul] --Goal : t * (↑b * ↑q + ↑r) + (s - t * ↑q) * ↑b = s * ↑b + t * ↑r ring done done #eval gcd_c1 672 161 --Answer: 6 #eval gcd_c2 672 161 --Answer: -25 --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161 theorem Theorem_7_1_6 {d a b : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ gcd a b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑d ∣ ↑(gcd a b) set s : Int := gcd_c1 a b set t : Int := gcd_c2 a b have h3 : s * ↑a + t * ↑b = ↑(gcd a b) := gcd_lin_comb b a rewrite [←h3] --Goal : ↑d ∣ s * ↑a + t * ↑b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul] --Goal : ↑d ∣ s * (↑d * ↑j) + t * (↑d * ↑k) define apply Exists.intro (s * ↑j + t * ↑k) ring done /- Section 7.2 -/ theorem dvd_trans {a b c : Nat} (h1 : a ∣ b) (h2 : b ∣ c) : a ∣ c := by define at h1; define at h2; define obtain (m : Nat) (h3 : b = a * m) from h1 obtain (n : Nat) (h4 : c = b * n) from h2 rewrite [h3, mul_assoc] at h4 apply Exists.intro (m * n) show c = a * (m * n) from h4 done lemma exists_prime_factor : ∀ (n : Nat), 2 ≤ n → ∃ (p : Nat), prime_factor p n := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, 2 ≤ n_1 → ∃ (p : Nat), prime_factor p n_1 assume h1 : 2 ≤ n by_cases h2 : prime n · -- Case 1. h2 : prime n apply Exists.intro n define --Goal : prime n ∧ n ∣ n show prime n ∧ n ∣ n from And.intro h2 (dvd_self n) done · -- Case 2. h2 : ¬prime n define at h2 --h2 : ¬(2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n) demorgan at h2 disj_syll h2 h1 obtain (a : Nat) (h3 : ∃ (b : Nat), a * b = n ∧ a < n ∧ b < n) from h2 obtain (b : Nat) (h4 : a * b = n ∧ a < n ∧ b < n) from h3 have h5 : 2 ≤ a := by by_contra h6 have h7 : a ≤ 1 := by linarith have h8 : n ≤ b := calc n _ = a * b := h4.left.symm _ ≤ 1 * b := by rel [h7] _ = b := by ring linarith --n ≤ b contradicts b < n done have h6 : ∃ (p : Nat), prime_factor p a := ih a h4.right.left h5 obtain (p : Nat) (h7 : prime_factor p a) from h6 apply Exists.intro p define --Goal : prime p ∧ p ∣ n define at h7 --h7 : prime p ∧ p ∣ a apply And.intro h7.left have h8 : a ∣ n := by apply Exists.intro b show n = a * b from (h4.left).symm done show p ∣ n from dvd_trans h7.right h8 done done lemma exists_least_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q := by set S : Set Nat := {p : Nat | prime_factor p n} have h2 : ∃ (p : Nat), p ∈ S := exists_prime_factor n h show ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q from well_ord_princ S h2 done lemma all_prime_nil : all_prime [] := by define --Goal : ∀ p ∈ [], prime p fix p : Nat contrapos --Goal : ¬prime p → p ∉ [] assume h1 : ¬prime p show p ∉ [] from List.not_mem_nil p done lemma all_prime_cons (n : Nat) (L : List Nat) : all_prime (n :: L) ↔ prime n ∧ all_prime L := by apply Iff.intro · -- (→) assume h1 : all_prime (n :: L) --Goal : prime n ∧ all_prime L define at h1 --h1 : ∀ p ∈ n :: L, prime p apply And.intro (h1 n (List.mem_cons_self n L)) define --Goal : ∀ p ∈ L, prime p fix p : Nat assume h2 : p ∈ L show prime p from h1 p (List.mem_cons_of_mem n h2) done · -- (←) assume h1 : prime n ∧ all_prime L --Goal : all_prime (n :: l) define : all_prime L at h1 define fix p : Nat assume h2 : p ∈ n :: L rewrite [List.mem_cons] at h2 --h2 : p = n ∨ p ∈ L by_cases on h2 · -- Case 1. h2 : p = n rewrite [h2] show prime n from h1.left done · -- Case 2. h2 : p ∈ L show prime p from h1.right p h2 done done done lemma nondec_nil : nondec [] := by define --Goal : True trivial --trivial proves some obviously true statements, such as True done lemma nondec_cons (n : Nat) (L : List Nat) : nondec (n :: L) ↔ (∀ m ∈ L, n ≤ m) ∧ nondec L := by rfl lemma prod_nil : prod [] = 1 := by rfl lemma prod_cons : prod (n :: L) = n * (prod L) := by rfl lemma exists_cons_of_length_eq_succ {A : Type} {l : List A} {n : Nat} (h : l.length = n + 1) : ∃ (a : A) (L : List A), l = a :: L ∧ L.length = n := by have h1 : ¬l.length = 0 := by linarith rewrite [List.length_eq_zero] at h1 obtain (a : A) (h2 : ∃ (L : List A), l = a :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List A) (h3 : l = a :: L) from h2 apply Exists.intro a apply Exists.intro L apply And.intro h3 have h4 : (a :: L).length = L.length + 1 := List.length_cons a L rewrite [←h3, h] at h4 show L.length = n from (Nat.add_right_cancel h4).symm done lemma list_elt_dvd_prod_by_length (a : Nat) : ∀ (n : Nat), ∀ (l : List Nat), l.length = n → a ∈ l → a ∣ prod l := by by_induc · --Base Case fix l : List Nat assume h1 : l.length = 0 rewrite [List.length_eq_zero] at h1 --h1 : l = [] rewrite [h1] --Goal : a ∈ [] → a ∣ prod [] contrapos assume h2 : ¬a ∣ prod [] show a ∉ [] from List.not_mem_nil a done · -- Induction Step fix n : Nat assume ih : ∀ (l : List Nat), List.length l = n → a ∈ l → a ∣ prod l fix l : List Nat assume h1 : l.length = n + 1 --Goal : a ∈ l → a ∣ prod l obtain (b : Nat) (h2 : ∃ (L : List Nat), l = b :: L ∧ L.length = n) from exists_cons_of_length_eq_succ h1 obtain (L : List Nat) (h3 : l = b :: L ∧ L.length = n) from h2 have h4 : a ∈ L → a ∣ prod L := ih L h3.right assume h5 : a ∈ l rewrite [h3.left, prod_cons] --Goal : a ∣ b * prod L rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b ∨ a ∈ L by_cases on h5 · -- Case 1. h5 : a = b apply Exists.intro (prod L) rewrite [h5] rfl done · -- Case 2. h5 : a ∈ L have h6 : a ∣ prod L := h4 h5 have h7 : prod L ∣ b * prod L := by apply Exists.intro b ring done show a ∣ b * prod L from dvd_trans h6 h7 done done done lemma list_elt_dvd_prod {a : Nat} {l : List Nat} (h : a ∈ l) : a ∣ prod l := by set n : Nat := l.length have h1 : l.length = n := by rfl show a ∣ prod l from list_elt_dvd_prod_by_length a n l h1 h done lemma exists_prime_factorization : ∀ (n : Nat), n ≥ 1 → ∃ (l : List Nat), prime_factorization n l := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, n_1 ≥ 1 → ∃ (l : List Nat), prime_factorization n_1 l assume h1 : n ≥ 1 by_cases h2 : n = 1 · -- Case 1. h2 : n = 1 apply Exists.intro [] define apply And.intro · -- Proof of nondec_prime_list [] define show all_prime [] ∧ nondec [] from And.intro all_prime_nil nondec_nil done · -- Proof of prod [] = n rewrite [prod_nil, h2] rfl done done · -- Case 2. h2 : n ≠ 1 have h3 : n ≥ 2 := lt_of_le_of_ne' h1 h2 obtain (p : Nat) (h4 : prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q) from exists_least_prime_factor h3 have p_prime_factor : prime_factor p n := h4.left define at p_prime_factor have p_prime : prime p := p_prime_factor.left have p_dvd_n : p ∣ n := p_prime_factor.right have p_least : ∀ (q : Nat), prime_factor q n → p ≤ q := h4.right obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n have h5 : m ≠ 0 := by contradict h1 with h6 have h7 : n = 0 := calc n _ = p * m := n_eq_pm _ = p * 0 := by rw [h6] _ = 0 := by ring rewrite [h7] decide done have m_pos : 0 < m := Nat.pos_of_ne_zero h5 have m_lt_n : m < n := by define at p_prime show m < n from calc m _ < m + m := by linarith _ = 2 * m := by ring _ ≤ p * m := by rel [p_prime.left] _ = n := n_eq_pm.symm done obtain (L : List Nat) (h6 : prime_factorization m L) from ih m m_lt_n m_pos define at h6 have ndpl_L : nondec_prime_list L := h6.left define at ndpl_L apply Exists.intro (p :: L) define apply And.intro · -- Proof of nondec_prime_list (p :: L) define apply And.intro · -- Proof of all_prime (p :: L) rewrite [all_prime_cons] show prime p ∧ all_prime L from And.intro p_prime ndpl_L.left done · -- Proof of nondec (p :: L) rewrite [nondec_cons] apply And.intro _ ndpl_L.right fix q : Nat assume q_in_L : q ∈ L have h7 : q ∣ prod L := list_elt_dvd_prod q_in_L rewrite [h6.right] at h7 --h7 : q ∣ m have h8 : m ∣ n := by apply Exists.intro p rewrite [n_eq_pm] ring done have q_dvd_n : q ∣ n := dvd_trans h7 h8 have ap_L : all_prime L := ndpl_L.left define at ap_L have q_prime_factor : prime_factor q n := And.intro (ap_L q q_in_L) q_dvd_n show p ≤ q from p_least q q_prime_factor done done · -- Proof of prod (p :: L) = n rewrite [prod_cons, h6.right, n_eq_pm] rfl done done done theorem Theorem_7_2_2 {a b c : Nat} (h1 : c ∣ a * b) (h2 : rel_prime a c) : c ∣ b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑c ∣ ↑b define at h1; define at h2; define obtain (j : Nat) (h3 : a * b = c * j) from h1 set s : Int := gcd_c1 a c set t : Int := gcd_c2 a c have h4 : s * ↑a + t * ↑c = ↑(gcd a c) := gcd_lin_comb c a rewrite [h2, Nat.cast_one] at h4 --h4 : s * ↑a + t * ↑c = (1 : Int) apply Exists.intro (s * ↑j + t * ↑b) show ↑b = ↑c * (s * ↑j + t * ↑b) from calc ↑b _ = (1 : Int) * ↑b := (one_mul _).symm _ = (s * ↑a + t * ↑c) * ↑b := by rw [h4] _ = s * (↑a * ↑b) + t * ↑c * ↑b := by ring _ = s * (↑c * ↑j) + t * ↑c * ↑b := by rw [←Nat.cast_mul a b, h3, Nat.cast_mul c j] _ = ↑c * (s * ↑j + t * ↑b) := by ring done lemma le_nonzero_prod_left {a b : Nat} (h : a * b ≠ 0) : a ≤ a * b := by have h1 : b ≠ 0 := by contradict h with h1 rewrite [h1] ring done have h2 : 1 ≤ b := Nat.pos_of_ne_zero h1 show a ≤ a * b from calc a = a * 1 := (mul_one a).symm _ ≤ a * b := by rel [h2] done lemma le_nonzero_prod_right {a b : Nat} (h : a * b ≠ 0) : b ≤ a * b := by rewrite [mul_comm] rewrite [mul_comm] at h show b ≤ b * a from le_nonzero_prod_left h done lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry lemma rel_prime_of_prime_not_dvd {a p : Nat} (h1 : prime p) (h2 : ¬p ∣ a) : rel_prime a p := by have h3 : gcd a p ∣ a := gcd_dvd_left a p have h4 : gcd a p ∣ p := gcd_dvd_right a p have h5 : gcd a p = 1 ∨ gcd a p = p := dvd_prime h1 h4 have h6 : gcd a p ≠ p := by contradict h2 with h6 rewrite [h6] at h3 show p ∣ a from h3 done disj_syll h5 h6 show rel_prime a p from h5 done theorem Theorem_7_2_3 {a b p : Nat} (h1 : prime p) (h2 : p ∣ a * b) : p ∣ a ∨ p ∣ b := by or_right with h3 have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3 show p ∣ b from Theorem_7_2_2 h2 h4 done lemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a ≥ 1 := by have h1 : a ≠ 0 := by by_contra h1 rewrite [h1] at h contradict h linarith done show a ≥ 1 from Nat.pos_of_ne_zero h1 done lemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by have h1 : a ≥ 1 := ge_one_of_prod_one h have h2 : a * b ≠ 0 := by linarith have h3 : a ≤ a * b := le_nonzero_prod_left h2 rewrite [h] at h3 show a = 1 from Nat.le_antisymm h3 h1 done lemma eq_one_of_dvd_one {n : Nat} (h : n ∣ 1) : n = 1 := by obtain (j : Nat) (h1 : 1 = n * j) from h show n = 1 from eq_one_of_prod_one h1.symm done lemma prime_not_one {p : Nat} (h : prime p) : p ≠ 1 := by define at h linarith done theorem Theorem_7_2_4 {p : Nat} (h1 : prime p) : ∀ (l : List Nat), p ∣ prod l → ∃ a ∈ l, p ∣ a := by apply List.rec · -- Base Case. Goal : p ∣ prod [] → ∃ a ∈ [], p ∣ a rewrite [prod_nil] assume h2 : p ∣ 1 show ∃ a ∈ [], p ∣ a from absurd (eq_one_of_dvd_one h2) (prime_not_one h1) done · -- Induction Step fix b : Nat fix L : List Nat assume ih : p ∣ prod L → ∃ a ∈ L, p ∣ a --Goal : p ∣ prod (b :: L) → ∃ a ∈ b :: L, p ∣ a assume h2 : p ∣ prod (b :: L) rewrite [prod_cons] at h2 have h3 : p ∣ b ∨ p ∣ prod L := Theorem_7_2_3 h1 h2 by_cases on h3 · -- Case 1. h3 : p ∣ b apply Exists.intro b show b ∈ b :: L ∧ p ∣ b from And.intro (List.mem_cons_self b L) h3 done · -- Case 2. h3 : p ∣ prod L obtain (a : Nat) (h4 : a ∈ L ∧ p ∣ a) from ih h3 apply Exists.intro a show a ∈ b :: L ∧ p ∣ a from And.intro (List.mem_cons_of_mem b h4.left) h4.right done done done lemma prime_in_list {p : Nat} {l : List Nat} (h1 : prime p) (h2 : all_prime l) (h3 : p ∣ prod l) : p ∈ l := by obtain (a : Nat) (h4 : a ∈ l ∧ p ∣ a) from Theorem_7_2_4 h1 l h3 define at h2 have h5 : prime a := h2 a h4.left have h6 : p = 1 ∨ p = a := dvd_prime h5 h4.right disj_syll h6 (prime_not_one h1) rewrite [h6] show a ∈ l from h4.left done lemma first_le_first {p q : Nat} {l m : List Nat} (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m)) (h3 : prod (p :: l) = prod (q :: m)) : p ≤ q := by define at h1; define at h2 have h4 : q ∣ prod (p :: l) := by define apply Exists.intro (prod m) rewrite [←prod_cons] show prod (p :: l) = prod (q :: m) from h3 done have h5 : all_prime (q :: m) := h2.left rewrite [all_prime_cons] at h5 have h6 : q ∈ p :: l := prime_in_list h5.left h1.left h4 have h7 : nondec (p :: l) := h1.right rewrite [nondec_cons] at h7 rewrite [List.mem_cons] at h6 by_cases on h6 · -- Case 1. h6 : q = p linarith done · -- Case 2. h6 : q ∈ l have h8 : ∀ m ∈ l, p ≤ m := h7.left show p ≤ q from h8 q h6 done done lemma nondec_prime_list_tail {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by define at h define rewrite [all_prime_cons, nondec_cons] at h show all_prime l ∧ nondec l from And.intro h.left.right h.right.right done lemma cons_prod_not_one {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : prod (p :: l) ≠ 1 := by define at h have h1 : all_prime (p :: l) := h.left rewrite [all_prime_cons] at h1 rewrite [prod_cons] by_contra h2 show False from (prime_not_one h1.left) (eq_one_of_prod_one h2) done lemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) : l = [] ↔ prod l = 1 := by apply Iff.intro · -- (→) assume h1 : l = [] rewrite [h1] show prod [] = 1 from prod_nil done · -- (←) contrapos assume h1 : ¬l = [] obtain (p : Nat) (h2 : ∃ (L : List Nat), l = p :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List Nat) (h3 : l = p :: L) from h2 rewrite [h3] at h rewrite [h3] show ¬prod (p :: L) = 1 from cons_prod_not_one h done done lemma prime_pos {p : Nat} (h : prime p) : p > 0 := by define at h linarith done theorem Theorem_7_2_5 : ∀ (l1 l2 : List Nat), nondec_prime_list l1 → nondec_prime_list l2 → prod l1 = prod l2 → l1 = l2 := by apply List.rec · -- Base Case. Goal : ∀ (l2 : List Nat), nondec_prime_list [] → -- nondec_prime_list l2 → prod [] = prod l2 → [] = l2 fix l2 : List Nat assume h1 : nondec_prime_list [] assume h2 : nondec_prime_list l2 assume h3 : prod [] = prod l2 rewrite [prod_nil, eq_comm, ←list_nil_iff_prod_one h2] at h3 show [] = l2 from h3.symm done · -- Induction Step fix p : Nat fix L1 : List Nat assume ih : ∀ (L2 : List Nat), nondec_prime_list L1 → nondec_prime_list L2 → prod L1 = prod L2 → L1 = L2 -- Goal : ∀ (l2 : List Nat), nondec_prime_list (p :: L1) → -- nondec_prime_list l2 → prod (p :: L1) = prod l2 → p :: L1 = l2 fix l2 : List Nat assume h1 : nondec_prime_list (p :: L1) assume h2 : nondec_prime_list l2 assume h3 : prod (p :: L1) = prod l2 have h4 : ¬prod (p :: L1) = 1 := cons_prod_not_one h1 rewrite [h3, ←list_nil_iff_prod_one h2] at h4 obtain (q : Nat) (h5 : ∃ (L : List Nat), l2 = q :: L) from List.exists_cons_of_ne_nil h4 obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5 rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2) rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2) have h7 : p ≤ q := first_le_first h1 h2 h3 have h8 : q ≤ p := first_le_first h2 h1 h3.symm have h9 : p = q := by linarith rewrite [h9, prod_cons, prod_cons] at h3 --h3 : q * prod L1 = q * prod L2 have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1 have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2 define at h2 have h12 : all_prime (q :: L2) := h2.left rewrite [all_prime_cons] at h12 have h13 : q > 0 := prime_pos h12.left have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3 have h15 : L1 = L2 := ih L2 h10 h11 h14 rewrite [h6, h9, h15] rfl done done theorem fund_thm_arith (n : Nat) (h : n ≥ 1) : ∃! (l : List Nat), prime_factorization n l := by exists_unique · -- Existence show ∃ (l : List Nat), prime_factorization n l from exists_prime_factorization n h done · -- Uniqueness fix l1 : List Nat; fix l2 : List Nat assume h1 : prime_factorization n l1 assume h2 : prime_factorization n l2 define at h1; define at h2 have h3 : prod l1 = n := h1.right rewrite [←h2.right] at h3 show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3 done done /- Section 7.3 -/ theorem congr_refl (m : Nat) : ∀ (a : Int), a ≡ a (MOD m) := by fix a : Int define --Goal : ∃ (c : Int), a - a = ↑m * c apply Exists.intro 0 ring done theorem congr_symm {m : Nat} : ∀ {a b : Int}, a ≡ b (MOD m) → b ≡ a (MOD m) := by fix a : Int; fix b : Int assume h1 : a ≡ b (MOD m) define at h1 --h1 : ∃ (c : Int), a - b = ↑m * c define --Goal : ∃ (c : Int), b - a = ↑m * c obtain (c : Int) (h2 : a - b = m * c) from h1 apply Exists.intro (-c) show b - a = m * (-c) from calc b - a _ = -(a - b) := by ring _ = -(m * c) := by rw [h2] _ = m * (-c) := by ring done theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry /- Fundamental properties of congruence classes -/ lemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) : X = Y ↔ X.val = Y.val := Fin.ext_iff lemma val_nat_eq_mod (n k : Nat) : ([k]_(n + 1)).val = k % (n + 1) := by rfl lemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl theorem cc_rep {m : Nat} (X : ZMod m) : ∃ (a : Int), X = [a]_m := match m with | 0 => by apply Exists.intro X rfl done | n + 1 => by apply Exists.intro ↑(X.val) have h1 : X.val < n + 1 := Fin.prop X rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1] rfl done theorem add_class (m : Nat) (a b : Int) : [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm theorem mul_class (m : Nat) (a b : Int) : [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm lemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ [a - b]_m = [0]_m := by apply Iff.intro · -- (→) assume h1 : [a]_m = [b]_m have h2 : a - b = a + (-b) := by ring have h3 : b + (-b) = 0 := by ring show [a - b]_m = [0]_m from calc [a - b]_m _ = [a + (-b)]_m := by rw [h2] _ = [a]_m + [-b]_m := by rw [add_class] _ = [b]_m + [-b]_m := by rw [h1] _ = [b + -b]_m := by rw [add_class] _ = [0]_m := by rw [h3] done · -- (←) assume h1 : [a - b]_m = [0]_m have h2 : b + (a - b) = a := by ring have h3 : b + 0 = b := by ring show [a]_m = [b]_m from calc [a]_m _ = [b + (a - b)]_m := by rw [h2] _ = [b]_m + [a - b]_m := by rw [add_class] _ = [b]_m + [0]_m := by rw [h1] _ = [b + 0]_m := by rw [add_class] _ = [b]_m := by rw [h3] done done lemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) : [a]_m = [0]_m → [-a]_m = [0]_m := by assume h1 : [a]_m = [0]_m have h2 : 0 + (-a) = -a := by ring have h3 : a + (-a) = 0 := by ring show [-a]_m = [0]_m from calc [-a]_m _ = [0 + (-a)]_m := by rw [h2] _ = [0]_m + [-a]_m := by rw [add_class] _ = [a]_m + [-a]_m := by rw [h1] _ = [a + (-a)]_m := by rw [add_class] _ = [0]_m := by rw [h3] done lemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) : [-a]_m = [0]_m ↔ [a]_m = [0]_m := by apply Iff.intro _ (cc_neg_zero_of_cc_zero m a) assume h1 : [-a]_m = [0]_m have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1 have h3 : -(-a) = a := by ring rewrite [h3] at h2 show [a]_m = [0]_m from h2 done lemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl lemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m ↔ m ∣ k := match m with | 0 => by have h : (0 : Int) = (↑(0 : Nat) : Int) := by rfl rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj] apply Iff.intro · -- (→) assume h1 : k = 0 rewrite [h1] show 0 ∣ 0 from dvd_self 0 done · -- (←) assume h1 : 0 ∣ k obtain (c : Nat) (h2 : k = 0 * c) from h1 rewrite [h2] ring done done | n + 1 => by rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero] show k % (n + 1) = 0 ↔ n + 1 ∣ k from (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm done lemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m ↔ ↑m ∣ a := by obtain (k : Nat) (h1 : a = ↑k ∨ a = -↑k) from Int.eq_nat_or_neg a by_cases on h1 · -- Case 1. h1: a = ↑k rewrite [h1, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done · -- Case 2. h1: a = -↑k rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done done theorem cc_eq_iff_congr (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ a ≡ b (MOD m) := calc [a]_m = [b]_m _ ↔ [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b _ ↔ ↑m ∣ (a - b) := cc_zero_iff_dvd m (a - b) _ ↔ a ≡ b (MOD m) := by rfl /- End of fundamental properties of congruence classes -/ lemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m := by have h1 : (↑m : Int) ≠ 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m) show 0 ≤ a % m from Int.emod_nonneg a h1 done lemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m) have h2 : (↑m : Int) > 0 := (Nat.cast_pos).rtl h1 show a % m < m from Int.emod_lt_of_pos a h2 done lemma congr_mod_mod (m : Nat) (a : Int) : a ≡ a % m (MOD m) := by define have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m apply Exists.intro (a / m) show a - a % m = m * (a / m) from calc a - (a % m) _ = m * (a / m) + a % m - a % m := by rw [h1] _ = m * (a / m) := by ring done lemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) := And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a)) theorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) : ∃! (r : Int), 0 ≤ r ∧ r < m ∧ a ≡ r (MOD m) := by exists_unique · -- Existence apply Exists.intro (a % m) show 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) from mod_cmpl_res m a done · -- Uniqueness fix r1 : Int; fix r2 : Int assume h1 : 0 ≤ r1 ∧ r1 < m ∧ a ≡ r1 (MOD m) assume h2 : 0 ≤ r2 ∧ r2 < m ∧ a ≡ r2 (MOD m) have h3 : r1 ≡ r2 (MOD m) := congr_trans (congr_symm h1.right.right) h2.right.right obtain (d : Int) (h4 : r1 - r2 = m * d) from h3 have h5 : r1 - r2 < m * 1 := by linarith have h6 : m * (-1) < r1 - r2 := by linarith rewrite [h4] at h5 --h5 : m * d < m * 1 rewrite [h4] at h6 --h6 : m * -1 < m * d have h7 : (↑m : Int) ≥ 0 := Nat.cast_nonneg m have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7 have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7 have h10 : d = 0 := by linarith show r1 = r2 from calc r1 _ = r1 - r2 + r2 := by ring _ = m * 0 + r2 := by rw [h4, h10] _ = r2 := by ring done done lemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m := (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a) theorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y rewrite [h1, h2] have h3 : a + b = b + a := by ring show [a]_m + [b]_m = [b]_m + [a]_m from calc [a]_m + [b]_m _ = [a + b]_m := add_class m a b _ = [b + a]_m := by rw [h3] _ = [b]_m + [a]_m := (add_class m b a).symm done theorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X rewrite [h1] have h2 : a * 1 = a := by ring show [a]_m * [1]_m = [a]_m from calc [a]_m * [1]_m _ = [a * 1]_m := mul_class m a 1 _ = [a]_m := by rw [h2] done theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry lemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) : [a]_m * [gcd_c2 m a]_m = [1]_m := by set s : Int := gcd_c1 m a have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m define at h1 rewrite [h1, Nat.cast_one] at h2 --h2 : s * ↑m + gcd_c2 m a * ↑a = 1 rewrite [mul_class, cc_eq_iff_congr] define --Goal : ∃ (c : Int), ↑a * gcd_c2 m a - 1 = ↑m * c apply Exists.intro (-s) show a * (gcd_c2 m a) - 1 = m * (-s) from calc a * (gcd_c2 m a) - 1 _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring _ = 1 + m * (-s) - 1 := by rw [h2] _ = m * (-s) := by ring done theorem Theorem_7_3_7 (m a : Nat) : invertible [a]_m ↔ rel_prime m a := by apply Iff.intro · -- (→) assume h1 : invertible [a]_m define at h1 obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1 obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y rewrite [h3, mul_class, cc_eq_iff_congr] at h2 define at h2 obtain (c : Int) (h4 : a * b - 1 = m * c) from h2 rewrite [Exercise_7_2_6] --Goal : ∃ (s t : Int), s * ↑m + t * ↑a = 1 apply Exists.intro (-c) apply Exists.intro b show (-c) * m + b * a = 1 from calc (-c) * m + b * a _ = (-c) * m + (a * b - 1) + 1 := by ring _ = (-c) * m + m * c + 1 := by rw [h4] _ = 1 := by ring done · -- (←) assume h1 : rel_prime m a define show ∃ (Y : ZMod m), [a]_m * Y = [1]_m from Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1) done done /- Section 7.4 -/ section Euler open Euler lemma num_rp_below_base {m : Nat} : num_rp_below m 0 = 0 := by rfl lemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) : num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by have h1 : num_rp_below m (j + 1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : gcd m j = 1 rewrite [if_pos h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1 show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1 done lemma num_rp_below_step_not_rp {m j : Nat} (h : ¬rel_prime m j) : num_rp_below m (j + 1) = num_rp_below m j := by have h1 : num_rp_below m (j +1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : ¬gcd m j = 1 rewrite [if_neg h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j show num_rp_below m (j + 1) = num_rp_below m j from h1 done lemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl #eval phi 10 --Answer: 4 lemma prod_inv_iff_inv {m : Nat} {X : ZMod m} (h1 : invertible X) (Y : ZMod m) : invertible (X * Y) ↔ invertible Y := by apply Iff.intro · -- (→) assume h2 : invertible (X * Y) obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2 apply Exists.intro (X * Z) rewrite [←h3] --Goal : Y * (X * Z) = X * Y * Z ring --Note that ring can do algebra in ZMod m done · -- (←) assume h2 : invertible Y obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1 obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2 apply Exists.intro (Xi * Yi) show (X * Y) * (Xi * Yi) = [1]_m from calc X * Y * (Xi * Yi) _ = (X * Xi) * (Y * Yi) := by ring _ = [1]_m * [1]_m := by rw [h3, h4] _ = [1]_m := Theorem_7_3_6_7 [1]_m done done lemma F_rp_def {m i : Nat} (h : rel_prime m i) : F m i = [i]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h --h : gcd m i = 1 rewrite [if_pos h] at h1 show F m i = [i]_m from h1 done lemma F_not_rp_def {m i : Nat} (h : ¬rel_prime m i) : F m i = [1]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h rewrite [h1, if_neg h] rfl done lemma prod_seq_base {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 0 k f = [1]_m := by rfl lemma prod_seq_step {m : Nat} (n k : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl lemma prod_seq_zero_step {m : Nat} (n : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by rewrite [prod_seq_step, zero_add] rfl done lemma prod_one {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 1 k f = f k := by rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7] rfl done lemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl lemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m := calc [G m a i]_m _ = [(a * i) % m]_m := by rfl _ = [a * i]_m := (cc_eq_mod m (a * i)).symm _ = [a]_m * [i]_m := (mul_class m a i).symm lemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) : rel_prime m (G m a i) ↔ rel_prime m i := by have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1 show rel_prime m (G m a i) ↔ rel_prime m i from calc rel_prime m (G m a i) _ ↔ invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm _ ↔ invertible ([a]_m * [i]_m) := by rw [cc_G] _ ↔ invertible [i]_m := prod_inv_iff_inv h2 ([i]_m) _ ↔ rel_prime m i := Theorem_7_3_7 m i done lemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) : F m (G m a i) = [a]_m * F m i := by have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2 show F m (G m a i) = [a]_m * F m i from calc F m (G m a i) _ = [G m a i]_m := F_rp_def h3 _ = [a]_m * [i]_m := cc_G m a i _ = [a]_m * F m i := by rw [F_rp_def h2] done lemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : ¬rel_prime m i) : F m (G m a i) = [1]_m := by rewrite [←G_rp_iff h1 i] at h2 show F m (G m a i) = [1]_m from F_not_rp_def h2 done lemma FG_prod {m a : Nat} (h1 : rel_prime m a) : ∀ (k : Nat), prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by by_induc · -- Base Case show prod_seq 0 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from calc prod_seq 0 0 ((F m) ∘ (G m a)) _ = [1]_m := prod_seq_base _ _ _ = [a]_m ^ 0 * [1]_m := by ring _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by rw [num_rp_below_base, prod_seq_base] done · -- Induction Step fix k : Nat assume ih : prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) by_cases h2 : rel_prime m k · -- Case 1. h2 : rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([a]_m * F m k) := by rw [FG_rp h1 h2] _ = [a]_m ^ ((num_rp_below m k) + 1) * ((prod_seq k 0 (F m)) * F m k) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_rp h2, prod_seq_zero_step] done · -- Case 2. h2 : ¬rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([1]_m) := by rw [FG_not_rp h1 h2] _ = [a]_m ^ (num_rp_below m k) * (prod_seq k 0 (F m) * ([1]_m)) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_not_rp h2, prod_seq_zero_step, F_not_rp_def h2] done done done lemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by define --Goal : ∀ i < m, G m a i < m fix i : Nat assume h1 : i < m rewrite [G_def] --Goal : a * i % m < m show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m) done lemma left_inv_one_one_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g' (g i) = i) : one_one_below n g := sorry lemma right_inv_onto_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g (g' i) = i) (h2 : maps_below n g') : onto_below n g := by define at h2; define fix k : Nat assume h3 : k < n apply Exists.intro (g' k) show g' k < n ∧ g (g' k) = k from And.intro (h2 k h3) (h1 k h3) done lemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m * [inv_mod m a]_m = [1]_m := by have h2 : 0 ≤ (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a) show [a]_m * [inv_mod m a]_m = [1]_m from calc [a]_m * [inv_mod m a]_m _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2] _ = [a]_m * [gcd_c2 m a]_m := by rw [←cc_eq_mod] _ = [1]_m := gcd_c2_inv h1 done lemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m := calc a * (b % m) % m = a % m * (b % m % m) % m := Nat.mul_mod _ _ _ _ = a % m * (b % m) % m := by rw [Nat.mod_mod] _ = a * b % m := (Nat.mul_mod _ _ _).symm lemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm] rfl done theorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] : ↑a ≡ ↑b (MOD m) ↔ a % m = b % m := sorry lemma mul_inv_mod_cancel {m a i : Nat} [NeZero m] (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1 rewrite [mul_class, cc_eq_iff_congr, ←Nat.cast_mul, ←Nat.cast_one, congr_iff_mod_eq_Nat] at h3 show a * inv_mod m a * i % m = i from calc a * (inv_mod m a) * i % m _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = 1 % m * i % m := by rw [h3] _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = i % m := by rw [one_mul] _ = i := Nat.mod_eq_of_lt h2 done lemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl lemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, G m a (Ginv m a i) = i := by fix i : Nat assume h2 : i < m show G m a (Ginv m a i) = i from calc G m a (Ginv m a i) _ = a * ((inv_mod m a * i) % m) % m := by rfl _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, Ginv m a (G m a i) = i := by fix i : Nat assume h2 : i < m show Ginv m a (G m a i) = i from calc Ginv m a (G m a i) _ = inv_mod m a * ((a * i) % m) % m := by rfl _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc, mul_comm (inv_mod m a)] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_maps_below (m a : Nat) [NeZero m] : maps_below m (Ginv m a) := G_maps_below m (inv_mod m a) lemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : one_one_below m (G m a) := left_inv_one_one_below (Ginv_left_inv h1) lemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : onto_below m (G m a) := right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a) lemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : perm_below m (G m a) := And.intro (G_maps_below m a) (And.intro (G_one_one_below h1) (G_onto_below h1)) --Permuting a product of congruence classes doesn't change product lemma swap_fst (u v : Nat) : swap u v u = v := by define : swap u v u --Goal : (if u = u then v else if u = v then u else u) = v have h : u = u := by rfl rewrite [if_pos h] rfl done lemma swap_snd (u v : Nat) : swap u v v = u := by define : swap u v v by_cases h1 : v = u · -- Case 1. h1 : v = u rewrite [if_pos h1] show v = u from h1 done · -- Case 2. h1 : v ≠ u rewrite [if_neg h1] have h2 : v = v := by rfl rewrite [if_pos h2] rfl done done lemma swap_other {u v i : Nat} (h1 : i ≠ u) (h2 : i ≠ v) : swap u v i = i := by define : swap u v i rewrite [if_neg h1, if_neg h2] rfl done lemma swap_values (u v i : Nat) : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := by by_cases h1 : i = u · -- Case 1. h1 : i = u apply Or.inl rewrite [h1] show swap u v u = v from swap_fst u v done · -- Case 2. h1 : i ≠ u apply Or.inr by_cases h2 : i = v · -- Case 2.1. h2 : i = v apply Or.inl rewrite [h2] show swap u v v = u from swap_snd u v done · -- Case 2.2. h2 : i ≠ v apply Or.inr show swap u v i = i from swap_other h1 h2 done done done lemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by define fix i : Nat assume h3 : i < n have h4 : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := swap_values u v i by_cases on h4 · -- Case 1. h4 : swap u v i = v rewrite [h4] show v < n from h2 done · -- Case 2. by_cases on h4 · -- Case 2.1. h4 : swap u v i = u rewrite [h4] show u < n from h1 done · -- Case 2.2. h4 : swap u v i = i rewrite [h4] show i < n from h3 done done done lemma swap_swap (u v n : Nat) : ∀ i < n, swap u v (swap u v i) = i := by fix i : Nat assume h : i < n by_cases h1 : i = u · -- Case 1. h1 : i = u rewrite [h1, swap_fst, swap_snd] rfl done · -- Case 2. h1 : i ≠ u by_cases h2 : i = v · -- Case 2.1. h2 : i = v rewrite [h2, swap_snd, swap_fst] rfl done · -- Case 2.2. h2 : i ≠ v rewrite [swap_other h1 h2, swap_other h1 h2] rfl done done done lemma swap_one_one_below (u v n) : one_one_below n (swap u v) := left_inv_one_one_below (swap_swap u v n) lemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) := right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2) lemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) := And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2)) lemma comp_perm_below {n : Nat} {f g : Nat → Nat} (h1 : perm_below n f) (h2 : perm_below n g) : perm_below n (f ∘ g) := sorry lemma trivial_swap (u : Nat) : swap u u = id := by apply funext fix x : Nat by_cases h1 : x = u · -- Case 1. h1 : x = u rewrite [h1, swap_fst] rfl done · -- Case 2. h1 : x ≠ u rewrite [swap_other h1 h1] rfl done done lemma prod_eq_fun {m : Nat} (f g : Nat → ZMod m) (k : Nat) : ∀ (n : Nat), (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g := by by_induc · -- Base Case assume h : (∀ i < 0, f (k + i) = g (k + i)) rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g assume h1 : ∀ i < n + 1, f (k + i) = g (k + i) have h2 : ∀ i < n, f (k + i) = g (k + i) := by fix i : Nat assume h2 : i < n have h3 : i < n + 1 := by linarith show f (k + i) = g (k + i) from h1 i h3 done have h3 : prod_seq n k f = prod_seq n k g := ih h2 have h4 : n < n + 1 := Nat.lt_succ_self n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4] rfl done done lemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := by have h2 : ∀ (i : Nat), i < u → (f ∘ swap u n) (0 + i) = f (0 + i) := by fix i : Nat assume h2 : i < u have h3 : 0 + i ≠ u := by linarith have h4 : 0 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl done show prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f from prod_eq_fun (f ∘ swap u n) f 0 u h2 done lemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := by have h2 : ∀ i < j, (f ∘ swap u n) (u + 1 + i) = f (u + 1 + i) := by fix i : Nat assume h2 : i < j have h3 : u + 1 + i ≠ u := by linarith have h4 : u + 1 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl show prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f from prod_eq_fun (f ∘ swap u n) f (u + 1) j h2 done lemma break_prod {m : Nat} (n : Nat) (f : Nat → ZMod m) : ∀ (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by by_induc · -- Base Case have h : n + 0 = n := by rfl rewrite [prod_seq_base, h, Theorem_7_3_6_7] rfl done · -- Induction Step fix j : Nat assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f rewrite [←add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc] rfl done done lemma break_prod_twice {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f := break_prod n f 1 rewrite [prod_one] at h2 have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f := break_prod (u + 1) f j rewrite [←h1] at h3 have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f := break_prod u f 1 rewrite [prod_one] at h4 rewrite [h3, h4] at h2 show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2 done lemma swap_prod_eq_prod {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f := by by_cases h2 : u = n · -- Case 1. h2 : u = n rewrite [h2, trivial_swap n] --Goal : prod_seq (n + 1) 0 (f ∘ id) = prod_seq (n + 1) 0 f rfl done · -- Case 2. h2 : ¬u = n have h3 : u + 1 ≤ n := Nat.lt_of_le_of_ne h1 h2 obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3 have break_f : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := break_prod_twice f h4 have break_fs : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_prod_twice (f ∘ swap u n) h4 have f_eq_fs_below : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := swap_prod_eq_prod_below f h1 have f_eq_fs_btwn : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4 show prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f from calc prod_seq (n + 1) 0 (f ∘ swap u n) _ = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_fs _ = prod_seq u 0 f * (f ∘ swap u n) u * prod_seq j (u + 1) f * (f ∘ swap u n) n := by rw [f_eq_fs_below, f_eq_fs_btwn] _ = prod_seq u 0 f * f (swap u n u) * prod_seq j (u + 1) f * f (swap u n n) := by rfl _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by rw [swap_fst, swap_snd] _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring _ = prod_seq (n + 1) 0 f := break_f.symm done done lemma perm_below_fixed {n : Nat} {g : Nat → Nat} (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry lemma perm_prod {m : Nat} (f : Nat → ZMod m) : ∀ (n : Nat), ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) := by by_induc · -- Base Case fix g : Nat → Nat assume h1 : perm_below 0 g rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) fix g : Nat → Nat assume g_pb : perm_below (n + 1) g define at g_pb have g_ob : onto_below (n + 1) g := g_pb.right.right define at g_ob have h1 : n < n + 1 := by linarith obtain (u : Nat) (h2 : u < n + 1 ∧ g u = n) from g_ob n h1 have s_pb : perm_below (n + 1) (swap u n) := swap_perm_below h2.left h1 have gs_pb_n1 : perm_below (n + 1) (g ∘ swap u n) := comp_perm_below g_pb s_pb have gs_fix_n : (g ∘ swap u n) n = n := calc (g ∘ swap u n) n _ = g (swap u n n) := by rfl _ = g u := by rw [swap_snd] _ = n := h2.right have gs_pb_n : perm_below n (g ∘ swap u n) := perm_below_fixed gs_pb_n1 gs_fix_n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f ∘ (g ∘ swap u n)) := ih (g ∘ swap u n) gs_pb_n have h3 : u ≤ n := by linarith show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f ∘ g) from calc prod_seq (n + 1) 0 f _ = prod_seq n 0 f * f n := prod_seq_zero_step n f _ = prod_seq n 0 (f ∘ (g ∘ swap u n)) * f ((g ∘ swap u n) n) := by rw [gs_prod, gs_fix_n] _ = prod_seq n 0 (f ∘ g ∘ swap u n) * (f ∘ g ∘ swap u n) n := by rfl _ = prod_seq (n + 1) 0 (f ∘ g ∘ swap u n) := (prod_seq_zero_step n (f ∘ g ∘ swap u n)).symm _ = prod_seq (n + 1) 0 ((f ∘ g) ∘ swap u n) := by rfl _ = prod_seq (n + 1) 0 (f ∘ g) := swap_prod_eq_prod (f ∘ g) h3 done done lemma F_invertible (m i : Nat) : invertible (F m i) := by by_cases h : rel_prime m i · -- Case 1. h : rel_prime m i rewrite [F_rp_def h] show invertible [i]_m from (Theorem_7_3_7 m i).rtl h done · -- Case 2. h : ¬rel_prime m i rewrite [F_not_rp_def h] apply Exists.intro [1]_m show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m done done lemma Fprod_invertible (m : Nat) : ∀ (k : Nat), invertible (prod_seq k 0 (F m)) := by by_induc · -- Base Case apply Exists.intro [1]_m show prod_seq 0 0 (F m) * [1]_m = [1]_m from calc prod_seq 0 0 (F m) * [1]_m _ = [1]_m * [1]_m := by rw [prod_seq_base] _ = [1]_m := Theorem_7_3_6_7 ([1]_m) done · -- Induction Step fix k : Nat assume ih : invertible (prod_seq k 0 (F m)) rewrite [prod_seq_zero_step] show invertible (prod_seq k 0 (F m) * (F m k)) from (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k) done done theorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m ^ (phi m) = [1]_m := by have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2 show [a]_m ^ (phi m) = [1]_m from calc [a]_m ^ (phi m) _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3] _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring _ = prod_seq m 0 (F m ∘ G m a) * Y := by rw [FG_prod h1 m, phi_def] _ = prod_seq m 0 (F m) * Y := by rw [perm_prod (F m) m (G m a) (G_perm_below h1)] _ = [1]_m := by rw [h3] done lemma Exercise_7_4_5_Int (m : Nat) (a : Int) : ∀ (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry lemma Exercise_7_4_5_Nat (m a n : Nat) : [a]_m ^ n = [a ^ n]_m := by rewrite [Exercise_7_4_5_Int] rfl done theorem Euler's_theorem {m a : Nat} [NeZero m] (h1 : rel_prime m a) : a ^ (phi m) ≡ 1 (MOD m) := by have h2 : [a]_m ^ (phi m) = [1]_m := Theorem_7_4_2 h1 rewrite [Exercise_7_4_5_Nat m a (phi m)] at h2 --h2 : [a ^ phi m]_m = [1]_m show a ^ (phi m) ≡ 1 (MOD m) from (cc_eq_iff_congr _ _ _).ltr h2 done #eval gcd 10 7 --Answer: 1. So 10 and 7 are relatively prime #eval 7 ^ phi 10 --Answer: 2401, which is congruent to 1 mod 10. end Euler /- Section 7.5 -/ lemma num_rp_prime {p : Nat} (h1 : prime p) : ∀ k < p, num_rp_below p (k + 1) = k := sorry lemma phi_prime {p : Nat} (h1 : prime p) : phi p = p - 1 := by have h2 : 1 ≤ p := prime_pos h1 have h3 : p - 1 + 1 = p := Nat.sub_add_cancel h2 have h4 : p - 1 < p := by linarith have h5 : num_rp_below p (p - 1 + 1) = p - 1 := num_rp_prime h1 (p - 1) h4 rewrite [h3] at h5 show phi p = p - 1 from h5 done theorem Theorem_7_2_2_Int {a c : Nat} {b : Int} (h1 : ↑c ∣ ↑a * b) (h2 : rel_prime a c) : ↑c ∣ b := by rewrite [Int.natCast_dvd, Int.natAbs_mul, Int.natAbs_ofNat] at h1 --h1 : c ∣ a * Int.natAbs b rewrite [Int.natCast_dvd] --Goal : c ∣ Int.natAbs b show c ∣ Int.natAbs b from Theorem_7_2_2 h1 h2 done lemma Lemma_7_4_5 {m n : Nat} (a b : Int) (h1 : rel_prime m n) : a ≡ b (MOD m * n) ↔ a ≡ b (MOD m) ∧ a ≡ b (MOD n) := by apply Iff.intro · -- (→) assume h2 : a ≡ b (MOD m * n) obtain (j : Int) (h3 : a - b = (m * n) * j) from h2 apply And.intro · -- Proof of a ≡ b (MOD m) apply Exists.intro (n * j) show a - b = m * (n * j) from calc a - b _ = m * n * j := h3 _ = m * (n * j) := by ring done · -- Proof of a ≡ b (MOD n) apply Exists.intro (m * j) show a - b = n * (m * j) from calc a - b _ = m * n * j := h3 _ = n * (m * j) := by ring done done · -- (←) assume h2 : a ≡ b (MOD m) ∧ a ≡ b (MOD n) obtain (j : Int) (h3 : a - b = m * j) from h2.left have h4 : (↑n : Int) ∣ a - b := h2.right rewrite [h3] at h4 --h4 : ↑n ∣ ↑m * j have h5 : ↑n ∣ j := Theorem_7_2_2_Int h4 h1 obtain (k : Int) (h6 : j = n * k) from h5 apply Exists.intro k --Goal : a - b = ↑(m * n) * k rewrite [Nat.cast_mul] --Goal : a - b = ↑m * ↑n * k show a - b = (m * n) * k from calc a - b _ = m * j := h3 _ = m * (n * k) := by rw [h6] _ = (m * n) * k := by ring done done --From exercises of Section 7.2 theorem rel_prime_symm {a b : Nat} (h : rel_prime a b) : rel_prime b a := sorry lemma prime_NeZero {p : Nat} (h : prime p) : NeZero p := by rewrite [neZero_iff] --Goal : p ≠ 0 define at h linarith done lemma Lemma_7_5_1 {p e d m c s : Nat} {t : Int} (h1 : prime p) (h2 : e * d = (p - 1) * s + 1) (h3 : m ^ e - c = p * t) : c ^ d ≡ m (MOD p) := by have h4 : m ^ e ≡ c (MOD p) := Exists.intro t h3 have h5 : [m ^ e]_p = [c]_p := (cc_eq_iff_congr _ _ _).rtl h4 rewrite [←Exercise_7_4_5_Nat] at h5 --h5 : [m]_p ^ e = [c]_p by_cases h6 : p ∣ m · -- Case 1. h6 : p ∣ m have h7 : m ≡ 0 (MOD p) := by obtain (j : Nat) (h8 : m = p * j) from h6 apply Exists.intro (↑j : Int) --Goal : ↑m - 0 = ↑p * ↑j rewrite [h8, Nat.cast_mul] ring done have h8 : [m]_p = [0]_p := (cc_eq_iff_congr _ _ _).rtl h7 have h9 : e * d ≠ 0 := by rewrite [h2] show (p - 1) * s + 1 ≠ 0 from Nat.add_one_ne_zero _ done have h10 : (0 : Int) ^ (e * d) = 0 := zero_pow h9 have h11 : [c ^ d]_p = [m]_p := calc [c ^ d]_p _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat] _ = ([m]_p ^ e) ^ d := by rw [h5] _ = [m]_p ^ (e * d) := by ring _ = [0]_p ^ (e * d) := by rw [h8] _ = [0 ^ (e * d)]_p := Exercise_7_4_5_Int _ _ _ _ = [0]_p := by rw [h10] _ = [m]_p := by rw [h8] show c ^ d ≡ m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11 done · -- Case 2. h6 : ¬p ∣ m have h7 : rel_prime m p := rel_prime_of_prime_not_dvd h1 h6 have h8 : rel_prime p m := rel_prime_symm h7 have h9 : NeZero p := prime_NeZero h1 have h10 : (1 : Int) ^ s = 1 := by ring have h11 : [c ^ d]_p = [m]_p := calc [c ^ d]_p _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat] _ = ([m]_p ^ e) ^ d := by rw [h5] _ = [m]_p ^ (e * d) := by ring _ = [m]_p ^ ((p - 1) * s + 1) := by rw [h2] _ = ([m]_p ^ (p - 1)) ^ s * [m]_p := by ring _ = ([m]_p ^ (phi p)) ^ s * [m]_p := by rw [phi_prime h1] _ = [1]_p ^ s * [m]_p := by rw [Theorem_7_4_2 h8] _ = [1 ^ s]_p * [m]_p := by rw [Exercise_7_4_5_Int] _ = [1]_p * [m]_p := by rw [h10] _ = [m]_p * [1]_p := by ring _ = [m]_p := Theorem_7_3_6_7 _ show c ^ d ≡ m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11 done done theorem Theorem_7_5_1 (p q n e d k m c : Nat) (p_prime : prime p) (q_prime : prime q) (p_ne_q : p ≠ q) (n_pq : n = p * q) (ed_congr_1 : e * d = k * (p - 1) * (q - 1) + 1) (h1 : [m]_n ^ e = [c]_n) : [c]_n ^ d = [m]_n := by rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] at h1 --h1 : m ^ e ≡ c (MOD n) rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] --Goal : c ^ d ≡ m (MOD n) obtain (j : Int) (h2 : m ^ e - c = n * j) from h1 rewrite [n_pq, Nat.cast_mul] at h2 --h2 : m ^ e - c = p * q * j have h3 : e * d = (p - 1) * (k * (q - 1)) + 1 := by rewrite [ed_congr_1] ring done have h4 : m ^ e - c = p * (q * j) := by rewrite [h2] ring done have congr_p : c ^ d ≡ m (MOD p) := Lemma_7_5_1 p_prime h3 h4 have h5 : e * d = (q - 1) * (k * (p - 1)) + 1 := by rewrite [ed_congr_1] ring done have h6 : m ^ e - c = q * (p * j) := by rewrite [h2] ring done have congr_q : c ^ d ≡ m (MOD q) := Lemma_7_5_1 q_prime h5 h6 have h7 : ¬q ∣ p := by by_contra h8 have h9 : q = 1 ∨ q = p := dvd_prime p_prime h8 disj_syll h9 (prime_not_one q_prime) show False from p_ne_q h9.symm done have h8 : rel_prime p q := rel_prime_of_prime_not_dvd q_prime h7 rewrite [n_pq, Lemma_7_4_5 _ _ h8] show c ^ d ≡ m (MOD p) ∧ c ^ d ≡ m (MOD q) from And.intro congr_p congr_q done /- BEGIN EXERCISES -/ namespace Exercises /- Section 7.1 -/ -- 1. theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry -- 2. lemma gcd_comm_lt {a b : Nat} (h : a < b) : gcd a b = gcd b a := sorry theorem gcd_comm (a b : Nat) : gcd a b = gcd b a := sorry -- 3. theorem Exercise_7_1_5 (a b : Nat) (n : Int) : (∃ (s t : Int), s * a + t * b = n) ↔ (↑(gcd a b) : Int) ∣ n := sorry -- 4. theorem Exercise_7_1_6 (a b c : Nat) : gcd a b = gcd (a + b * c) b := sorry -- 5. theorem gcd_is_nonzero {a b : Nat} (h : a ≠ 0 ∨ b ≠ 0) : gcd a b ≠ 0 := sorry -- 6. theorem gcd_greatest {a b d : Nat} (h1 : gcd a b ≠ 0) (h2 : d ∣ a) (h3 : d ∣ b) : d ≤ gcd a b := sorry -- 7. lemma Lemma_7_1_10a {a b : Nat} (n : Nat) (h : a ∣ b) : (n * a) ∣ (n * b) := sorry lemma Lemma_7_1_10b {a b n : Nat} (h1 : n ≠ 0) (h2 : (n * a) ∣ (n * b)) : a ∣ b := sorry lemma Lemma_7_1_10c {a b : Nat} (h1 : a ∣ b) (h2 : b ∣ a) : a = b := sorry theorem Exercise_7_1_10 (a b n : Nat) : gcd (n * a) (n * b) = n * gcd a b := sorry /- Section 7.2 -/ -- 1. lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry -- 2. -- Hints: Start with apply List.rec. You may find mul_ne_zero useful theorem prod_nonzero_nonzero : ∀ (l : List Nat), (∀ a ∈ l, a ≠ 0) → prod l ≠ 0 := sorry -- 3. theorem rel_prime_iff_no_common_factor (a b : Nat) : rel_prime a b ↔ ¬∃ (p : Nat), prime p ∧ p ∣ a ∧ p ∣ b := sorry -- 4.

theorem rel_prime_symm {a b : Nat} (h : rel_prime a b) : rel_prime b a

HTPI.Exercises.rel_prime_symm

htpi/HTPILib/Chap7.lean

htpi

/- Copyright 2023 Daniel J. Velleman -/ import HTPILib.Chap6 namespace HTPI /- Definitions -/ lemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by have h : n + 1 > 0 := Nat.succ_pos n show a % (n + 1) < n + 1 from Nat.mod_lt a h done def gcd (a b : Nat) : Nat := match b with | 0 => a | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd (n + 1) (a % (n + 1)) termination_by b mutual def gcd_c1 (a b : Nat) : Int := match b with | 0 => 1 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c2 (n + 1) (a % (n + 1)) --Corresponds to s = t' termination_by b def gcd_c2 (a b : Nat) : Int := match b with | 0 => 0 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c1 (n + 1) (a % (n + 1)) - (gcd_c2 (n + 1) (a % (n + 1))) * ↑(a / (n + 1)) --Corresponds to t = s' - t'q termination_by b end def prime (n : Nat) : Prop := 2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n def prime_factor (p n : Nat) : Prop := prime p ∧ p ∣ n def all_prime (l : List Nat) : Prop := ∀ p ∈ l, prime p def nondec (l : List Nat) : Prop := match l with | [] => True --Of course, True is a proposition that is always true | n :: L => (∀ m ∈ L, n ≤ m) ∧ nondec L def nondec_prime_list (l : List Nat) : Prop := all_prime l ∧ nondec l def prod (l : List Nat) : Nat := match l with | [] => 1 | n :: L => n * (prod L) def prime_factorization (n : Nat) (l : List Nat) : Prop := nondec_prime_list l ∧ prod l = n def rel_prime (a b : Nat) : Prop := gcd a b = 1 def congr_mod (m : Nat) (a b : Int) : Prop := (↑m : Int) ∣ (a - b) def cc (m : Nat) (a : Int) : ZMod m := (↑a : ZMod m) notation:50 a " ≡ " b " (MOD " m ")" => congr_mod m a b notation:max "["a"]_"m:max => cc m a def invertible {m : Nat} (X : ZMod m) : Prop := ∃ (Y : ZMod m), X * Y = [1]_m def num_rp_below (m k : Nat) : Nat := match k with | 0 => 0 | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j def phi (m : Nat) : Nat := num_rp_below m m def prod_seq {m : Nat} (j k : Nat) (f : Nat → ZMod m) : ZMod m := match j with | 0 => [1]_m | n + 1 => prod_seq n k f * f (k + n) def maps_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i < n, g i < n def one_one_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i1 < n, ∀ i2 < n, g i1 = g i2 → i1 = i2 def onto_below (n : Nat) (g : Nat → Nat) : Prop := ∀ k < n, ∃ i < n, g i = k def perm_below (n : Nat) (g : Nat → Nat) : Prop := maps_below n g ∧ one_one_below n g ∧ onto_below n g def inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m) def swap (u v i : Nat) : Nat := if i = u then v else if i = v then u else i namespace Euler --For definitions specific to Euler's theorem def F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m def G (m a i : Nat) : Nat := (a * i) % m def Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i end Euler /- Section 7.1 -/ theorem dvd_mod_of_dvd_a_b {a b d : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ (a % b) := by set q : Nat := a / b have h3 : b * q + a % b = a := Nat.div_add_mod a b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 define --Goal : ∃ (c : Nat), a % b = d * c apply Exists.intro (j - k * q) show a % b = d * (j - k * q) from calc a % b _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm _ = a - b * q := by rw [h3] _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc] _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm done theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry #eval gcd 672 161 --Answer: 7 lemma gcd_base (a : Nat) : gcd a 0 = a := by rfl lemma gcd_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd a b = gcd b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1)) rfl done lemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b ≠ 0) : a % b < b := by have h1 : b > 0 := Nat.pos_of_ne_zero h show a % b < b from Nat.mod_lt a h1 done lemma dvd_self (n : Nat) : n ∣ n := by apply Exists.intro 1 ring done theorem gcd_dvd : ∀ (b a : Nat), (gcd a b) ∣ a ∧ (gcd a b) ∣ b := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd a b_1) ∣ a ∧ (gcd a b_1) ∣ b_1 fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_base] --Goal: a ∣ a ∧ a ∣ 0 apply And.intro (dvd_self a) define apply Exists.intro 0 rfl done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_nonzero a h1] --Goal : gcd b (a % b) ∣ a ∧ gcd b (a % b) ∣ b have h2 : a % b < b := mod_nonzero_lt a h1 have h3 : (gcd b (a % b)) ∣ b ∧ (gcd b (a % b)) ∣ (a % b) := ih (a % b) h2 b apply And.intro _ h3.left show (gcd b (a % b)) ∣ a from dvd_a_of_dvd_b_mod h3.left h3.right done done theorem gcd_dvd_left (a b : Nat) : (gcd a b) ∣ a := (gcd_dvd b a).left theorem gcd_dvd_right (a b : Nat) : (gcd a b) ∣ b := (gcd_dvd b a).right lemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl lemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c1 a b = gcd_c2 b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done lemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl lemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * ↑(a / b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done theorem gcd_lin_comb : ∀ (b a : Nat), (gcd_c1 a b) * ↑a + (gcd_c2 a b) * ↑b = ↑(gcd a b) := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd_c1 a b_1) * ↑a + (gcd_c2 a b_1) * ↑b_1 = ↑(gcd a b_1) fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base] --Goal : 1 * ↑a + 0 * ↑0 = ↑a ring done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1] --Goal : gcd_c2 b (a % b) * ↑a + -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * ↑(a / b)) * ↑b = -- ↑(gcd b (a % b)) set r : Nat := a % b set q : Nat := a / b set s : Int := gcd_c1 b r set t : Int := gcd_c2 b r --Goal : t * ↑a + (s - t * ↑q) * ↑b = ↑(gcd b r) have h2 : r < b := mod_nonzero_lt a h1 have h3 : s * ↑b + t * ↑r = ↑(gcd b r) := ih r h2 b have h4 : b * q + r = a := Nat.div_add_mod a b rewrite [←h3, ←h4] rewrite [Nat.cast_add, Nat.cast_mul] --Goal : t * (↑b * ↑q + ↑r) + (s - t * ↑q) * ↑b = s * ↑b + t * ↑r ring done done #eval gcd_c1 672 161 --Answer: 6 #eval gcd_c2 672 161 --Answer: -25 --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161 theorem Theorem_7_1_6 {d a b : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ gcd a b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑d ∣ ↑(gcd a b) set s : Int := gcd_c1 a b set t : Int := gcd_c2 a b have h3 : s * ↑a + t * ↑b = ↑(gcd a b) := gcd_lin_comb b a rewrite [←h3] --Goal : ↑d ∣ s * ↑a + t * ↑b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul] --Goal : ↑d ∣ s * (↑d * ↑j) + t * (↑d * ↑k) define apply Exists.intro (s * ↑j + t * ↑k) ring done /- Section 7.2 -/ theorem dvd_trans {a b c : Nat} (h1 : a ∣ b) (h2 : b ∣ c) : a ∣ c := by define at h1; define at h2; define obtain (m : Nat) (h3 : b = a * m) from h1 obtain (n : Nat) (h4 : c = b * n) from h2 rewrite [h3, mul_assoc] at h4 apply Exists.intro (m * n) show c = a * (m * n) from h4 done lemma exists_prime_factor : ∀ (n : Nat), 2 ≤ n → ∃ (p : Nat), prime_factor p n := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, 2 ≤ n_1 → ∃ (p : Nat), prime_factor p n_1 assume h1 : 2 ≤ n by_cases h2 : prime n · -- Case 1. h2 : prime n apply Exists.intro n define --Goal : prime n ∧ n ∣ n show prime n ∧ n ∣ n from And.intro h2 (dvd_self n) done · -- Case 2. h2 : ¬prime n define at h2 --h2 : ¬(2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n) demorgan at h2 disj_syll h2 h1 obtain (a : Nat) (h3 : ∃ (b : Nat), a * b = n ∧ a < n ∧ b < n) from h2 obtain (b : Nat) (h4 : a * b = n ∧ a < n ∧ b < n) from h3 have h5 : 2 ≤ a := by by_contra h6 have h7 : a ≤ 1 := by linarith have h8 : n ≤ b := calc n _ = a * b := h4.left.symm _ ≤ 1 * b := by rel [h7] _ = b := by ring linarith --n ≤ b contradicts b < n done have h6 : ∃ (p : Nat), prime_factor p a := ih a h4.right.left h5 obtain (p : Nat) (h7 : prime_factor p a) from h6 apply Exists.intro p define --Goal : prime p ∧ p ∣ n define at h7 --h7 : prime p ∧ p ∣ a apply And.intro h7.left have h8 : a ∣ n := by apply Exists.intro b show n = a * b from (h4.left).symm done show p ∣ n from dvd_trans h7.right h8 done done lemma exists_least_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q := by set S : Set Nat := {p : Nat | prime_factor p n} have h2 : ∃ (p : Nat), p ∈ S := exists_prime_factor n h show ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q from well_ord_princ S h2 done lemma all_prime_nil : all_prime [] := by define --Goal : ∀ p ∈ [], prime p fix p : Nat contrapos --Goal : ¬prime p → p ∉ [] assume h1 : ¬prime p show p ∉ [] from List.not_mem_nil p done lemma all_prime_cons (n : Nat) (L : List Nat) : all_prime (n :: L) ↔ prime n ∧ all_prime L := by apply Iff.intro · -- (→) assume h1 : all_prime (n :: L) --Goal : prime n ∧ all_prime L define at h1 --h1 : ∀ p ∈ n :: L, prime p apply And.intro (h1 n (List.mem_cons_self n L)) define --Goal : ∀ p ∈ L, prime p fix p : Nat assume h2 : p ∈ L show prime p from h1 p (List.mem_cons_of_mem n h2) done · -- (←) assume h1 : prime n ∧ all_prime L --Goal : all_prime (n :: l) define : all_prime L at h1 define fix p : Nat assume h2 : p ∈ n :: L rewrite [List.mem_cons] at h2 --h2 : p = n ∨ p ∈ L by_cases on h2 · -- Case 1. h2 : p = n rewrite [h2] show prime n from h1.left done · -- Case 2. h2 : p ∈ L show prime p from h1.right p h2 done done done lemma nondec_nil : nondec [] := by define --Goal : True trivial --trivial proves some obviously true statements, such as True done lemma nondec_cons (n : Nat) (L : List Nat) : nondec (n :: L) ↔ (∀ m ∈ L, n ≤ m) ∧ nondec L := by rfl lemma prod_nil : prod [] = 1 := by rfl lemma prod_cons : prod (n :: L) = n * (prod L) := by rfl lemma exists_cons_of_length_eq_succ {A : Type} {l : List A} {n : Nat} (h : l.length = n + 1) : ∃ (a : A) (L : List A), l = a :: L ∧ L.length = n := by have h1 : ¬l.length = 0 := by linarith rewrite [List.length_eq_zero] at h1 obtain (a : A) (h2 : ∃ (L : List A), l = a :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List A) (h3 : l = a :: L) from h2 apply Exists.intro a apply Exists.intro L apply And.intro h3 have h4 : (a :: L).length = L.length + 1 := List.length_cons a L rewrite [←h3, h] at h4 show L.length = n from (Nat.add_right_cancel h4).symm done lemma list_elt_dvd_prod_by_length (a : Nat) : ∀ (n : Nat), ∀ (l : List Nat), l.length = n → a ∈ l → a ∣ prod l := by by_induc · --Base Case fix l : List Nat assume h1 : l.length = 0 rewrite [List.length_eq_zero] at h1 --h1 : l = [] rewrite [h1] --Goal : a ∈ [] → a ∣ prod [] contrapos assume h2 : ¬a ∣ prod [] show a ∉ [] from List.not_mem_nil a done · -- Induction Step fix n : Nat assume ih : ∀ (l : List Nat), List.length l = n → a ∈ l → a ∣ prod l fix l : List Nat assume h1 : l.length = n + 1 --Goal : a ∈ l → a ∣ prod l obtain (b : Nat) (h2 : ∃ (L : List Nat), l = b :: L ∧ L.length = n) from exists_cons_of_length_eq_succ h1 obtain (L : List Nat) (h3 : l = b :: L ∧ L.length = n) from h2 have h4 : a ∈ L → a ∣ prod L := ih L h3.right assume h5 : a ∈ l rewrite [h3.left, prod_cons] --Goal : a ∣ b * prod L rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b ∨ a ∈ L by_cases on h5 · -- Case 1. h5 : a = b apply Exists.intro (prod L) rewrite [h5] rfl done · -- Case 2. h5 : a ∈ L have h6 : a ∣ prod L := h4 h5 have h7 : prod L ∣ b * prod L := by apply Exists.intro b ring done show a ∣ b * prod L from dvd_trans h6 h7 done done done lemma list_elt_dvd_prod {a : Nat} {l : List Nat} (h : a ∈ l) : a ∣ prod l := by set n : Nat := l.length have h1 : l.length = n := by rfl show a ∣ prod l from list_elt_dvd_prod_by_length a n l h1 h done lemma exists_prime_factorization : ∀ (n : Nat), n ≥ 1 → ∃ (l : List Nat), prime_factorization n l := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, n_1 ≥ 1 → ∃ (l : List Nat), prime_factorization n_1 l assume h1 : n ≥ 1 by_cases h2 : n = 1 · -- Case 1. h2 : n = 1 apply Exists.intro [] define apply And.intro · -- Proof of nondec_prime_list [] define show all_prime [] ∧ nondec [] from And.intro all_prime_nil nondec_nil done · -- Proof of prod [] = n rewrite [prod_nil, h2] rfl done done · -- Case 2. h2 : n ≠ 1 have h3 : n ≥ 2 := lt_of_le_of_ne' h1 h2 obtain (p : Nat) (h4 : prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q) from exists_least_prime_factor h3 have p_prime_factor : prime_factor p n := h4.left define at p_prime_factor have p_prime : prime p := p_prime_factor.left have p_dvd_n : p ∣ n := p_prime_factor.right have p_least : ∀ (q : Nat), prime_factor q n → p ≤ q := h4.right obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n have h5 : m ≠ 0 := by contradict h1 with h6 have h7 : n = 0 := calc n _ = p * m := n_eq_pm _ = p * 0 := by rw [h6] _ = 0 := by ring rewrite [h7] decide done have m_pos : 0 < m := Nat.pos_of_ne_zero h5 have m_lt_n : m < n := by define at p_prime show m < n from calc m _ < m + m := by linarith _ = 2 * m := by ring _ ≤ p * m := by rel [p_prime.left] _ = n := n_eq_pm.symm done obtain (L : List Nat) (h6 : prime_factorization m L) from ih m m_lt_n m_pos define at h6 have ndpl_L : nondec_prime_list L := h6.left define at ndpl_L apply Exists.intro (p :: L) define apply And.intro · -- Proof of nondec_prime_list (p :: L) define apply And.intro · -- Proof of all_prime (p :: L) rewrite [all_prime_cons] show prime p ∧ all_prime L from And.intro p_prime ndpl_L.left done · -- Proof of nondec (p :: L) rewrite [nondec_cons] apply And.intro _ ndpl_L.right fix q : Nat assume q_in_L : q ∈ L have h7 : q ∣ prod L := list_elt_dvd_prod q_in_L rewrite [h6.right] at h7 --h7 : q ∣ m have h8 : m ∣ n := by apply Exists.intro p rewrite [n_eq_pm] ring done have q_dvd_n : q ∣ n := dvd_trans h7 h8 have ap_L : all_prime L := ndpl_L.left define at ap_L have q_prime_factor : prime_factor q n := And.intro (ap_L q q_in_L) q_dvd_n show p ≤ q from p_least q q_prime_factor done done · -- Proof of prod (p :: L) = n rewrite [prod_cons, h6.right, n_eq_pm] rfl done done done theorem Theorem_7_2_2 {a b c : Nat} (h1 : c ∣ a * b) (h2 : rel_prime a c) : c ∣ b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑c ∣ ↑b define at h1; define at h2; define obtain (j : Nat) (h3 : a * b = c * j) from h1 set s : Int := gcd_c1 a c set t : Int := gcd_c2 a c have h4 : s * ↑a + t * ↑c = ↑(gcd a c) := gcd_lin_comb c a rewrite [h2, Nat.cast_one] at h4 --h4 : s * ↑a + t * ↑c = (1 : Int) apply Exists.intro (s * ↑j + t * ↑b) show ↑b = ↑c * (s * ↑j + t * ↑b) from calc ↑b _ = (1 : Int) * ↑b := (one_mul _).symm _ = (s * ↑a + t * ↑c) * ↑b := by rw [h4] _ = s * (↑a * ↑b) + t * ↑c * ↑b := by ring _ = s * (↑c * ↑j) + t * ↑c * ↑b := by rw [←Nat.cast_mul a b, h3, Nat.cast_mul c j] _ = ↑c * (s * ↑j + t * ↑b) := by ring done lemma le_nonzero_prod_left {a b : Nat} (h : a * b ≠ 0) : a ≤ a * b := by have h1 : b ≠ 0 := by contradict h with h1 rewrite [h1] ring done have h2 : 1 ≤ b := Nat.pos_of_ne_zero h1 show a ≤ a * b from calc a = a * 1 := (mul_one a).symm _ ≤ a * b := by rel [h2] done lemma le_nonzero_prod_right {a b : Nat} (h : a * b ≠ 0) : b ≤ a * b := by rewrite [mul_comm] rewrite [mul_comm] at h show b ≤ b * a from le_nonzero_prod_left h done lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry lemma rel_prime_of_prime_not_dvd {a p : Nat} (h1 : prime p) (h2 : ¬p ∣ a) : rel_prime a p := by have h3 : gcd a p ∣ a := gcd_dvd_left a p have h4 : gcd a p ∣ p := gcd_dvd_right a p have h5 : gcd a p = 1 ∨ gcd a p = p := dvd_prime h1 h4 have h6 : gcd a p ≠ p := by contradict h2 with h6 rewrite [h6] at h3 show p ∣ a from h3 done disj_syll h5 h6 show rel_prime a p from h5 done theorem Theorem_7_2_3 {a b p : Nat} (h1 : prime p) (h2 : p ∣ a * b) : p ∣ a ∨ p ∣ b := by or_right with h3 have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3 show p ∣ b from Theorem_7_2_2 h2 h4 done lemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a ≥ 1 := by have h1 : a ≠ 0 := by by_contra h1 rewrite [h1] at h contradict h linarith done show a ≥ 1 from Nat.pos_of_ne_zero h1 done lemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by have h1 : a ≥ 1 := ge_one_of_prod_one h have h2 : a * b ≠ 0 := by linarith have h3 : a ≤ a * b := le_nonzero_prod_left h2 rewrite [h] at h3 show a = 1 from Nat.le_antisymm h3 h1 done lemma eq_one_of_dvd_one {n : Nat} (h : n ∣ 1) : n = 1 := by obtain (j : Nat) (h1 : 1 = n * j) from h show n = 1 from eq_one_of_prod_one h1.symm done lemma prime_not_one {p : Nat} (h : prime p) : p ≠ 1 := by define at h linarith done theorem Theorem_7_2_4 {p : Nat} (h1 : prime p) : ∀ (l : List Nat), p ∣ prod l → ∃ a ∈ l, p ∣ a := by apply List.rec · -- Base Case. Goal : p ∣ prod [] → ∃ a ∈ [], p ∣ a rewrite [prod_nil] assume h2 : p ∣ 1 show ∃ a ∈ [], p ∣ a from absurd (eq_one_of_dvd_one h2) (prime_not_one h1) done · -- Induction Step fix b : Nat fix L : List Nat assume ih : p ∣ prod L → ∃ a ∈ L, p ∣ a --Goal : p ∣ prod (b :: L) → ∃ a ∈ b :: L, p ∣ a assume h2 : p ∣ prod (b :: L) rewrite [prod_cons] at h2 have h3 : p ∣ b ∨ p ∣ prod L := Theorem_7_2_3 h1 h2 by_cases on h3 · -- Case 1. h3 : p ∣ b apply Exists.intro b show b ∈ b :: L ∧ p ∣ b from And.intro (List.mem_cons_self b L) h3 done · -- Case 2. h3 : p ∣ prod L obtain (a : Nat) (h4 : a ∈ L ∧ p ∣ a) from ih h3 apply Exists.intro a show a ∈ b :: L ∧ p ∣ a from And.intro (List.mem_cons_of_mem b h4.left) h4.right done done done lemma prime_in_list {p : Nat} {l : List Nat} (h1 : prime p) (h2 : all_prime l) (h3 : p ∣ prod l) : p ∈ l := by obtain (a : Nat) (h4 : a ∈ l ∧ p ∣ a) from Theorem_7_2_4 h1 l h3 define at h2 have h5 : prime a := h2 a h4.left have h6 : p = 1 ∨ p = a := dvd_prime h5 h4.right disj_syll h6 (prime_not_one h1) rewrite [h6] show a ∈ l from h4.left done lemma first_le_first {p q : Nat} {l m : List Nat} (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m)) (h3 : prod (p :: l) = prod (q :: m)) : p ≤ q := by define at h1; define at h2 have h4 : q ∣ prod (p :: l) := by define apply Exists.intro (prod m) rewrite [←prod_cons] show prod (p :: l) = prod (q :: m) from h3 done have h5 : all_prime (q :: m) := h2.left rewrite [all_prime_cons] at h5 have h6 : q ∈ p :: l := prime_in_list h5.left h1.left h4 have h7 : nondec (p :: l) := h1.right rewrite [nondec_cons] at h7 rewrite [List.mem_cons] at h6 by_cases on h6 · -- Case 1. h6 : q = p linarith done · -- Case 2. h6 : q ∈ l have h8 : ∀ m ∈ l, p ≤ m := h7.left show p ≤ q from h8 q h6 done done lemma nondec_prime_list_tail {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by define at h define rewrite [all_prime_cons, nondec_cons] at h show all_prime l ∧ nondec l from And.intro h.left.right h.right.right done lemma cons_prod_not_one {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : prod (p :: l) ≠ 1 := by define at h have h1 : all_prime (p :: l) := h.left rewrite [all_prime_cons] at h1 rewrite [prod_cons] by_contra h2 show False from (prime_not_one h1.left) (eq_one_of_prod_one h2) done lemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) : l = [] ↔ prod l = 1 := by apply Iff.intro · -- (→) assume h1 : l = [] rewrite [h1] show prod [] = 1 from prod_nil done · -- (←) contrapos assume h1 : ¬l = [] obtain (p : Nat) (h2 : ∃ (L : List Nat), l = p :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List Nat) (h3 : l = p :: L) from h2 rewrite [h3] at h rewrite [h3] show ¬prod (p :: L) = 1 from cons_prod_not_one h done done lemma prime_pos {p : Nat} (h : prime p) : p > 0 := by define at h linarith done theorem Theorem_7_2_5 : ∀ (l1 l2 : List Nat), nondec_prime_list l1 → nondec_prime_list l2 → prod l1 = prod l2 → l1 = l2 := by apply List.rec · -- Base Case. Goal : ∀ (l2 : List Nat), nondec_prime_list [] → -- nondec_prime_list l2 → prod [] = prod l2 → [] = l2 fix l2 : List Nat assume h1 : nondec_prime_list [] assume h2 : nondec_prime_list l2 assume h3 : prod [] = prod l2 rewrite [prod_nil, eq_comm, ←list_nil_iff_prod_one h2] at h3 show [] = l2 from h3.symm done · -- Induction Step fix p : Nat fix L1 : List Nat assume ih : ∀ (L2 : List Nat), nondec_prime_list L1 → nondec_prime_list L2 → prod L1 = prod L2 → L1 = L2 -- Goal : ∀ (l2 : List Nat), nondec_prime_list (p :: L1) → -- nondec_prime_list l2 → prod (p :: L1) = prod l2 → p :: L1 = l2 fix l2 : List Nat assume h1 : nondec_prime_list (p :: L1) assume h2 : nondec_prime_list l2 assume h3 : prod (p :: L1) = prod l2 have h4 : ¬prod (p :: L1) = 1 := cons_prod_not_one h1 rewrite [h3, ←list_nil_iff_prod_one h2] at h4 obtain (q : Nat) (h5 : ∃ (L : List Nat), l2 = q :: L) from List.exists_cons_of_ne_nil h4 obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5 rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2) rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2) have h7 : p ≤ q := first_le_first h1 h2 h3 have h8 : q ≤ p := first_le_first h2 h1 h3.symm have h9 : p = q := by linarith rewrite [h9, prod_cons, prod_cons] at h3 --h3 : q * prod L1 = q * prod L2 have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1 have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2 define at h2 have h12 : all_prime (q :: L2) := h2.left rewrite [all_prime_cons] at h12 have h13 : q > 0 := prime_pos h12.left have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3 have h15 : L1 = L2 := ih L2 h10 h11 h14 rewrite [h6, h9, h15] rfl done done theorem fund_thm_arith (n : Nat) (h : n ≥ 1) : ∃! (l : List Nat), prime_factorization n l := by exists_unique · -- Existence show ∃ (l : List Nat), prime_factorization n l from exists_prime_factorization n h done · -- Uniqueness fix l1 : List Nat; fix l2 : List Nat assume h1 : prime_factorization n l1 assume h2 : prime_factorization n l2 define at h1; define at h2 have h3 : prod l1 = n := h1.right rewrite [←h2.right] at h3 show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3 done done /- Section 7.3 -/ theorem congr_refl (m : Nat) : ∀ (a : Int), a ≡ a (MOD m) := by fix a : Int define --Goal : ∃ (c : Int), a - a = ↑m * c apply Exists.intro 0 ring done theorem congr_symm {m : Nat} : ∀ {a b : Int}, a ≡ b (MOD m) → b ≡ a (MOD m) := by fix a : Int; fix b : Int assume h1 : a ≡ b (MOD m) define at h1 --h1 : ∃ (c : Int), a - b = ↑m * c define --Goal : ∃ (c : Int), b - a = ↑m * c obtain (c : Int) (h2 : a - b = m * c) from h1 apply Exists.intro (-c) show b - a = m * (-c) from calc b - a _ = -(a - b) := by ring _ = -(m * c) := by rw [h2] _ = m * (-c) := by ring done theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry /- Fundamental properties of congruence classes -/ lemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) : X = Y ↔ X.val = Y.val := Fin.ext_iff lemma val_nat_eq_mod (n k : Nat) : ([k]_(n + 1)).val = k % (n + 1) := by rfl lemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl theorem cc_rep {m : Nat} (X : ZMod m) : ∃ (a : Int), X = [a]_m := match m with | 0 => by apply Exists.intro X rfl done | n + 1 => by apply Exists.intro ↑(X.val) have h1 : X.val < n + 1 := Fin.prop X rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1] rfl done theorem add_class (m : Nat) (a b : Int) : [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm theorem mul_class (m : Nat) (a b : Int) : [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm lemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ [a - b]_m = [0]_m := by apply Iff.intro · -- (→) assume h1 : [a]_m = [b]_m have h2 : a - b = a + (-b) := by ring have h3 : b + (-b) = 0 := by ring show [a - b]_m = [0]_m from calc [a - b]_m _ = [a + (-b)]_m := by rw [h2] _ = [a]_m + [-b]_m := by rw [add_class] _ = [b]_m + [-b]_m := by rw [h1] _ = [b + -b]_m := by rw [add_class] _ = [0]_m := by rw [h3] done · -- (←) assume h1 : [a - b]_m = [0]_m have h2 : b + (a - b) = a := by ring have h3 : b + 0 = b := by ring show [a]_m = [b]_m from calc [a]_m _ = [b + (a - b)]_m := by rw [h2] _ = [b]_m + [a - b]_m := by rw [add_class] _ = [b]_m + [0]_m := by rw [h1] _ = [b + 0]_m := by rw [add_class] _ = [b]_m := by rw [h3] done done lemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) : [a]_m = [0]_m → [-a]_m = [0]_m := by assume h1 : [a]_m = [0]_m have h2 : 0 + (-a) = -a := by ring have h3 : a + (-a) = 0 := by ring show [-a]_m = [0]_m from calc [-a]_m _ = [0 + (-a)]_m := by rw [h2] _ = [0]_m + [-a]_m := by rw [add_class] _ = [a]_m + [-a]_m := by rw [h1] _ = [a + (-a)]_m := by rw [add_class] _ = [0]_m := by rw [h3] done lemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) : [-a]_m = [0]_m ↔ [a]_m = [0]_m := by apply Iff.intro _ (cc_neg_zero_of_cc_zero m a) assume h1 : [-a]_m = [0]_m have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1 have h3 : -(-a) = a := by ring rewrite [h3] at h2 show [a]_m = [0]_m from h2 done lemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl lemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m ↔ m ∣ k := match m with | 0 => by have h : (0 : Int) = (↑(0 : Nat) : Int) := by rfl rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj] apply Iff.intro · -- (→) assume h1 : k = 0 rewrite [h1] show 0 ∣ 0 from dvd_self 0 done · -- (←) assume h1 : 0 ∣ k obtain (c : Nat) (h2 : k = 0 * c) from h1 rewrite [h2] ring done done | n + 1 => by rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero] show k % (n + 1) = 0 ↔ n + 1 ∣ k from (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm done lemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m ↔ ↑m ∣ a := by obtain (k : Nat) (h1 : a = ↑k ∨ a = -↑k) from Int.eq_nat_or_neg a by_cases on h1 · -- Case 1. h1: a = ↑k rewrite [h1, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done · -- Case 2. h1: a = -↑k rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done done theorem cc_eq_iff_congr (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ a ≡ b (MOD m) := calc [a]_m = [b]_m _ ↔ [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b _ ↔ ↑m ∣ (a - b) := cc_zero_iff_dvd m (a - b) _ ↔ a ≡ b (MOD m) := by rfl /- End of fundamental properties of congruence classes -/ lemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m := by have h1 : (↑m : Int) ≠ 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m) show 0 ≤ a % m from Int.emod_nonneg a h1 done lemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m) have h2 : (↑m : Int) > 0 := (Nat.cast_pos).rtl h1 show a % m < m from Int.emod_lt_of_pos a h2 done lemma congr_mod_mod (m : Nat) (a : Int) : a ≡ a % m (MOD m) := by define have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m apply Exists.intro (a / m) show a - a % m = m * (a / m) from calc a - (a % m) _ = m * (a / m) + a % m - a % m := by rw [h1] _ = m * (a / m) := by ring done lemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) := And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a)) theorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) : ∃! (r : Int), 0 ≤ r ∧ r < m ∧ a ≡ r (MOD m) := by exists_unique · -- Existence apply Exists.intro (a % m) show 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) from mod_cmpl_res m a done · -- Uniqueness fix r1 : Int; fix r2 : Int assume h1 : 0 ≤ r1 ∧ r1 < m ∧ a ≡ r1 (MOD m) assume h2 : 0 ≤ r2 ∧ r2 < m ∧ a ≡ r2 (MOD m) have h3 : r1 ≡ r2 (MOD m) := congr_trans (congr_symm h1.right.right) h2.right.right obtain (d : Int) (h4 : r1 - r2 = m * d) from h3 have h5 : r1 - r2 < m * 1 := by linarith have h6 : m * (-1) < r1 - r2 := by linarith rewrite [h4] at h5 --h5 : m * d < m * 1 rewrite [h4] at h6 --h6 : m * -1 < m * d have h7 : (↑m : Int) ≥ 0 := Nat.cast_nonneg m have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7 have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7 have h10 : d = 0 := by linarith show r1 = r2 from calc r1 _ = r1 - r2 + r2 := by ring _ = m * 0 + r2 := by rw [h4, h10] _ = r2 := by ring done done lemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m := (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a) theorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y rewrite [h1, h2] have h3 : a + b = b + a := by ring show [a]_m + [b]_m = [b]_m + [a]_m from calc [a]_m + [b]_m _ = [a + b]_m := add_class m a b _ = [b + a]_m := by rw [h3] _ = [b]_m + [a]_m := (add_class m b a).symm done theorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X rewrite [h1] have h2 : a * 1 = a := by ring show [a]_m * [1]_m = [a]_m from calc [a]_m * [1]_m _ = [a * 1]_m := mul_class m a 1 _ = [a]_m := by rw [h2] done theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry lemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) : [a]_m * [gcd_c2 m a]_m = [1]_m := by set s : Int := gcd_c1 m a have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m define at h1 rewrite [h1, Nat.cast_one] at h2 --h2 : s * ↑m + gcd_c2 m a * ↑a = 1 rewrite [mul_class, cc_eq_iff_congr] define --Goal : ∃ (c : Int), ↑a * gcd_c2 m a - 1 = ↑m * c apply Exists.intro (-s) show a * (gcd_c2 m a) - 1 = m * (-s) from calc a * (gcd_c2 m a) - 1 _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring _ = 1 + m * (-s) - 1 := by rw [h2] _ = m * (-s) := by ring done theorem Theorem_7_3_7 (m a : Nat) : invertible [a]_m ↔ rel_prime m a := by apply Iff.intro · -- (→) assume h1 : invertible [a]_m define at h1 obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1 obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y rewrite [h3, mul_class, cc_eq_iff_congr] at h2 define at h2 obtain (c : Int) (h4 : a * b - 1 = m * c) from h2 rewrite [Exercise_7_2_6] --Goal : ∃ (s t : Int), s * ↑m + t * ↑a = 1 apply Exists.intro (-c) apply Exists.intro b show (-c) * m + b * a = 1 from calc (-c) * m + b * a _ = (-c) * m + (a * b - 1) + 1 := by ring _ = (-c) * m + m * c + 1 := by rw [h4] _ = 1 := by ring done · -- (←) assume h1 : rel_prime m a define show ∃ (Y : ZMod m), [a]_m * Y = [1]_m from Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1) done done /- Section 7.4 -/ section Euler open Euler lemma num_rp_below_base {m : Nat} : num_rp_below m 0 = 0 := by rfl lemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) : num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by have h1 : num_rp_below m (j + 1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : gcd m j = 1 rewrite [if_pos h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1 show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1 done lemma num_rp_below_step_not_rp {m j : Nat} (h : ¬rel_prime m j) : num_rp_below m (j + 1) = num_rp_below m j := by have h1 : num_rp_below m (j +1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : ¬gcd m j = 1 rewrite [if_neg h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j show num_rp_below m (j + 1) = num_rp_below m j from h1 done lemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl #eval phi 10 --Answer: 4 lemma prod_inv_iff_inv {m : Nat} {X : ZMod m} (h1 : invertible X) (Y : ZMod m) : invertible (X * Y) ↔ invertible Y := by apply Iff.intro · -- (→) assume h2 : invertible (X * Y) obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2 apply Exists.intro (X * Z) rewrite [←h3] --Goal : Y * (X * Z) = X * Y * Z ring --Note that ring can do algebra in ZMod m done · -- (←) assume h2 : invertible Y obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1 obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2 apply Exists.intro (Xi * Yi) show (X * Y) * (Xi * Yi) = [1]_m from calc X * Y * (Xi * Yi) _ = (X * Xi) * (Y * Yi) := by ring _ = [1]_m * [1]_m := by rw [h3, h4] _ = [1]_m := Theorem_7_3_6_7 [1]_m done done lemma F_rp_def {m i : Nat} (h : rel_prime m i) : F m i = [i]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h --h : gcd m i = 1 rewrite [if_pos h] at h1 show F m i = [i]_m from h1 done lemma F_not_rp_def {m i : Nat} (h : ¬rel_prime m i) : F m i = [1]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h rewrite [h1, if_neg h] rfl done lemma prod_seq_base {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 0 k f = [1]_m := by rfl lemma prod_seq_step {m : Nat} (n k : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl lemma prod_seq_zero_step {m : Nat} (n : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by rewrite [prod_seq_step, zero_add] rfl done lemma prod_one {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 1 k f = f k := by rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7] rfl done lemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl lemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m := calc [G m a i]_m _ = [(a * i) % m]_m := by rfl _ = [a * i]_m := (cc_eq_mod m (a * i)).symm _ = [a]_m * [i]_m := (mul_class m a i).symm lemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) : rel_prime m (G m a i) ↔ rel_prime m i := by have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1 show rel_prime m (G m a i) ↔ rel_prime m i from calc rel_prime m (G m a i) _ ↔ invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm _ ↔ invertible ([a]_m * [i]_m) := by rw [cc_G] _ ↔ invertible [i]_m := prod_inv_iff_inv h2 ([i]_m) _ ↔ rel_prime m i := Theorem_7_3_7 m i done lemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) : F m (G m a i) = [a]_m * F m i := by have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2 show F m (G m a i) = [a]_m * F m i from calc F m (G m a i) _ = [G m a i]_m := F_rp_def h3 _ = [a]_m * [i]_m := cc_G m a i _ = [a]_m * F m i := by rw [F_rp_def h2] done lemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : ¬rel_prime m i) : F m (G m a i) = [1]_m := by rewrite [←G_rp_iff h1 i] at h2 show F m (G m a i) = [1]_m from F_not_rp_def h2 done lemma FG_prod {m a : Nat} (h1 : rel_prime m a) : ∀ (k : Nat), prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by by_induc · -- Base Case show prod_seq 0 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from calc prod_seq 0 0 ((F m) ∘ (G m a)) _ = [1]_m := prod_seq_base _ _ _ = [a]_m ^ 0 * [1]_m := by ring _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by rw [num_rp_below_base, prod_seq_base] done · -- Induction Step fix k : Nat assume ih : prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) by_cases h2 : rel_prime m k · -- Case 1. h2 : rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([a]_m * F m k) := by rw [FG_rp h1 h2] _ = [a]_m ^ ((num_rp_below m k) + 1) * ((prod_seq k 0 (F m)) * F m k) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_rp h2, prod_seq_zero_step] done · -- Case 2. h2 : ¬rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([1]_m) := by rw [FG_not_rp h1 h2] _ = [a]_m ^ (num_rp_below m k) * (prod_seq k 0 (F m) * ([1]_m)) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_not_rp h2, prod_seq_zero_step, F_not_rp_def h2] done done done lemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by define --Goal : ∀ i < m, G m a i < m fix i : Nat assume h1 : i < m rewrite [G_def] --Goal : a * i % m < m show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m) done lemma left_inv_one_one_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g' (g i) = i) : one_one_below n g := sorry lemma right_inv_onto_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g (g' i) = i) (h2 : maps_below n g') : onto_below n g := by define at h2; define fix k : Nat assume h3 : k < n apply Exists.intro (g' k) show g' k < n ∧ g (g' k) = k from And.intro (h2 k h3) (h1 k h3) done lemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m * [inv_mod m a]_m = [1]_m := by have h2 : 0 ≤ (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a) show [a]_m * [inv_mod m a]_m = [1]_m from calc [a]_m * [inv_mod m a]_m _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2] _ = [a]_m * [gcd_c2 m a]_m := by rw [←cc_eq_mod] _ = [1]_m := gcd_c2_inv h1 done lemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m := calc a * (b % m) % m = a % m * (b % m % m) % m := Nat.mul_mod _ _ _ _ = a % m * (b % m) % m := by rw [Nat.mod_mod] _ = a * b % m := (Nat.mul_mod _ _ _).symm lemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm] rfl done theorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] : ↑a ≡ ↑b (MOD m) ↔ a % m = b % m := sorry lemma mul_inv_mod_cancel {m a i : Nat} [NeZero m] (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1 rewrite [mul_class, cc_eq_iff_congr, ←Nat.cast_mul, ←Nat.cast_one, congr_iff_mod_eq_Nat] at h3 show a * inv_mod m a * i % m = i from calc a * (inv_mod m a) * i % m _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = 1 % m * i % m := by rw [h3] _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = i % m := by rw [one_mul] _ = i := Nat.mod_eq_of_lt h2 done lemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl lemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, G m a (Ginv m a i) = i := by fix i : Nat assume h2 : i < m show G m a (Ginv m a i) = i from calc G m a (Ginv m a i) _ = a * ((inv_mod m a * i) % m) % m := by rfl _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, Ginv m a (G m a i) = i := by fix i : Nat assume h2 : i < m show Ginv m a (G m a i) = i from calc Ginv m a (G m a i) _ = inv_mod m a * ((a * i) % m) % m := by rfl _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc, mul_comm (inv_mod m a)] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_maps_below (m a : Nat) [NeZero m] : maps_below m (Ginv m a) := G_maps_below m (inv_mod m a) lemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : one_one_below m (G m a) := left_inv_one_one_below (Ginv_left_inv h1) lemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : onto_below m (G m a) := right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a) lemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : perm_below m (G m a) := And.intro (G_maps_below m a) (And.intro (G_one_one_below h1) (G_onto_below h1)) --Permuting a product of congruence classes doesn't change product lemma swap_fst (u v : Nat) : swap u v u = v := by define : swap u v u --Goal : (if u = u then v else if u = v then u else u) = v have h : u = u := by rfl rewrite [if_pos h] rfl done lemma swap_snd (u v : Nat) : swap u v v = u := by define : swap u v v by_cases h1 : v = u · -- Case 1. h1 : v = u rewrite [if_pos h1] show v = u from h1 done · -- Case 2. h1 : v ≠ u rewrite [if_neg h1] have h2 : v = v := by rfl rewrite [if_pos h2] rfl done done lemma swap_other {u v i : Nat} (h1 : i ≠ u) (h2 : i ≠ v) : swap u v i = i := by define : swap u v i rewrite [if_neg h1, if_neg h2] rfl done lemma swap_values (u v i : Nat) : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := by by_cases h1 : i = u · -- Case 1. h1 : i = u apply Or.inl rewrite [h1] show swap u v u = v from swap_fst u v done · -- Case 2. h1 : i ≠ u apply Or.inr by_cases h2 : i = v · -- Case 2.1. h2 : i = v apply Or.inl rewrite [h2] show swap u v v = u from swap_snd u v done · -- Case 2.2. h2 : i ≠ v apply Or.inr show swap u v i = i from swap_other h1 h2 done done done lemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by define fix i : Nat assume h3 : i < n have h4 : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := swap_values u v i by_cases on h4 · -- Case 1. h4 : swap u v i = v rewrite [h4] show v < n from h2 done · -- Case 2. by_cases on h4 · -- Case 2.1. h4 : swap u v i = u rewrite [h4] show u < n from h1 done · -- Case 2.2. h4 : swap u v i = i rewrite [h4] show i < n from h3 done done done lemma swap_swap (u v n : Nat) : ∀ i < n, swap u v (swap u v i) = i := by fix i : Nat assume h : i < n by_cases h1 : i = u · -- Case 1. h1 : i = u rewrite [h1, swap_fst, swap_snd] rfl done · -- Case 2. h1 : i ≠ u by_cases h2 : i = v · -- Case 2.1. h2 : i = v rewrite [h2, swap_snd, swap_fst] rfl done · -- Case 2.2. h2 : i ≠ v rewrite [swap_other h1 h2, swap_other h1 h2] rfl done done done lemma swap_one_one_below (u v n) : one_one_below n (swap u v) := left_inv_one_one_below (swap_swap u v n) lemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) := right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2) lemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) := And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2)) lemma comp_perm_below {n : Nat} {f g : Nat → Nat} (h1 : perm_below n f) (h2 : perm_below n g) : perm_below n (f ∘ g) := sorry lemma trivial_swap (u : Nat) : swap u u = id := by apply funext fix x : Nat by_cases h1 : x = u · -- Case 1. h1 : x = u rewrite [h1, swap_fst] rfl done · -- Case 2. h1 : x ≠ u rewrite [swap_other h1 h1] rfl done done lemma prod_eq_fun {m : Nat} (f g : Nat → ZMod m) (k : Nat) : ∀ (n : Nat), (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g := by by_induc · -- Base Case assume h : (∀ i < 0, f (k + i) = g (k + i)) rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g assume h1 : ∀ i < n + 1, f (k + i) = g (k + i) have h2 : ∀ i < n, f (k + i) = g (k + i) := by fix i : Nat assume h2 : i < n have h3 : i < n + 1 := by linarith show f (k + i) = g (k + i) from h1 i h3 done have h3 : prod_seq n k f = prod_seq n k g := ih h2 have h4 : n < n + 1 := Nat.lt_succ_self n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4] rfl done done lemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := by have h2 : ∀ (i : Nat), i < u → (f ∘ swap u n) (0 + i) = f (0 + i) := by fix i : Nat assume h2 : i < u have h3 : 0 + i ≠ u := by linarith have h4 : 0 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl done show prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f from prod_eq_fun (f ∘ swap u n) f 0 u h2 done lemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := by have h2 : ∀ i < j, (f ∘ swap u n) (u + 1 + i) = f (u + 1 + i) := by fix i : Nat assume h2 : i < j have h3 : u + 1 + i ≠ u := by linarith have h4 : u + 1 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl show prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f from prod_eq_fun (f ∘ swap u n) f (u + 1) j h2 done lemma break_prod {m : Nat} (n : Nat) (f : Nat → ZMod m) : ∀ (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by by_induc · -- Base Case have h : n + 0 = n := by rfl rewrite [prod_seq_base, h, Theorem_7_3_6_7] rfl done · -- Induction Step fix j : Nat assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f rewrite [←add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc] rfl done done lemma break_prod_twice {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f := break_prod n f 1 rewrite [prod_one] at h2 have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f := break_prod (u + 1) f j rewrite [←h1] at h3 have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f := break_prod u f 1 rewrite [prod_one] at h4 rewrite [h3, h4] at h2 show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2 done lemma swap_prod_eq_prod {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f := by by_cases h2 : u = n · -- Case 1. h2 : u = n rewrite [h2, trivial_swap n] --Goal : prod_seq (n + 1) 0 (f ∘ id) = prod_seq (n + 1) 0 f rfl done · -- Case 2. h2 : ¬u = n have h3 : u + 1 ≤ n := Nat.lt_of_le_of_ne h1 h2 obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3 have break_f : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := break_prod_twice f h4 have break_fs : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_prod_twice (f ∘ swap u n) h4 have f_eq_fs_below : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := swap_prod_eq_prod_below f h1 have f_eq_fs_btwn : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4 show prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f from calc prod_seq (n + 1) 0 (f ∘ swap u n) _ = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_fs _ = prod_seq u 0 f * (f ∘ swap u n) u * prod_seq j (u + 1) f * (f ∘ swap u n) n := by rw [f_eq_fs_below, f_eq_fs_btwn] _ = prod_seq u 0 f * f (swap u n u) * prod_seq j (u + 1) f * f (swap u n n) := by rfl _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by rw [swap_fst, swap_snd] _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring _ = prod_seq (n + 1) 0 f := break_f.symm done done lemma perm_below_fixed {n : Nat} {g : Nat → Nat} (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry lemma perm_prod {m : Nat} (f : Nat → ZMod m) : ∀ (n : Nat), ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) := by by_induc · -- Base Case fix g : Nat → Nat assume h1 : perm_below 0 g rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) fix g : Nat → Nat assume g_pb : perm_below (n + 1) g define at g_pb have g_ob : onto_below (n + 1) g := g_pb.right.right define at g_ob have h1 : n < n + 1 := by linarith obtain (u : Nat) (h2 : u < n + 1 ∧ g u = n) from g_ob n h1 have s_pb : perm_below (n + 1) (swap u n) := swap_perm_below h2.left h1 have gs_pb_n1 : perm_below (n + 1) (g ∘ swap u n) := comp_perm_below g_pb s_pb have gs_fix_n : (g ∘ swap u n) n = n := calc (g ∘ swap u n) n _ = g (swap u n n) := by rfl _ = g u := by rw [swap_snd] _ = n := h2.right have gs_pb_n : perm_below n (g ∘ swap u n) := perm_below_fixed gs_pb_n1 gs_fix_n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f ∘ (g ∘ swap u n)) := ih (g ∘ swap u n) gs_pb_n have h3 : u ≤ n := by linarith show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f ∘ g) from calc prod_seq (n + 1) 0 f _ = prod_seq n 0 f * f n := prod_seq_zero_step n f _ = prod_seq n 0 (f ∘ (g ∘ swap u n)) * f ((g ∘ swap u n) n) := by rw [gs_prod, gs_fix_n] _ = prod_seq n 0 (f ∘ g ∘ swap u n) * (f ∘ g ∘ swap u n) n := by rfl _ = prod_seq (n + 1) 0 (f ∘ g ∘ swap u n) := (prod_seq_zero_step n (f ∘ g ∘ swap u n)).symm _ = prod_seq (n + 1) 0 ((f ∘ g) ∘ swap u n) := by rfl _ = prod_seq (n + 1) 0 (f ∘ g) := swap_prod_eq_prod (f ∘ g) h3 done done lemma F_invertible (m i : Nat) : invertible (F m i) := by by_cases h : rel_prime m i · -- Case 1. h : rel_prime m i rewrite [F_rp_def h] show invertible [i]_m from (Theorem_7_3_7 m i).rtl h done · -- Case 2. h : ¬rel_prime m i rewrite [F_not_rp_def h] apply Exists.intro [1]_m show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m done done lemma Fprod_invertible (m : Nat) : ∀ (k : Nat), invertible (prod_seq k 0 (F m)) := by by_induc · -- Base Case apply Exists.intro [1]_m show prod_seq 0 0 (F m) * [1]_m = [1]_m from calc prod_seq 0 0 (F m) * [1]_m _ = [1]_m * [1]_m := by rw [prod_seq_base] _ = [1]_m := Theorem_7_3_6_7 ([1]_m) done · -- Induction Step fix k : Nat assume ih : invertible (prod_seq k 0 (F m)) rewrite [prod_seq_zero_step] show invertible (prod_seq k 0 (F m) * (F m k)) from (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k) done done theorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m ^ (phi m) = [1]_m := by have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2 show [a]_m ^ (phi m) = [1]_m from calc [a]_m ^ (phi m) _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3] _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring _ = prod_seq m 0 (F m ∘ G m a) * Y := by rw [FG_prod h1 m, phi_def] _ = prod_seq m 0 (F m) * Y := by rw [perm_prod (F m) m (G m a) (G_perm_below h1)] _ = [1]_m := by rw [h3] done lemma Exercise_7_4_5_Int (m : Nat) (a : Int) : ∀ (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry lemma Exercise_7_4_5_Nat (m a n : Nat) : [a]_m ^ n = [a ^ n]_m := by rewrite [Exercise_7_4_5_Int] rfl done theorem Euler's_theorem {m a : Nat} [NeZero m] (h1 : rel_prime m a) : a ^ (phi m) ≡ 1 (MOD m) := by have h2 : [a]_m ^ (phi m) = [1]_m := Theorem_7_4_2 h1 rewrite [Exercise_7_4_5_Nat m a (phi m)] at h2 --h2 : [a ^ phi m]_m = [1]_m show a ^ (phi m) ≡ 1 (MOD m) from (cc_eq_iff_congr _ _ _).ltr h2 done #eval gcd 10 7 --Answer: 1. So 10 and 7 are relatively prime #eval 7 ^ phi 10 --Answer: 2401, which is congruent to 1 mod 10. end Euler /- Section 7.5 -/ lemma num_rp_prime {p : Nat} (h1 : prime p) : ∀ k < p, num_rp_below p (k + 1) = k := sorry lemma phi_prime {p : Nat} (h1 : prime p) : phi p = p - 1 := by have h2 : 1 ≤ p := prime_pos h1 have h3 : p - 1 + 1 = p := Nat.sub_add_cancel h2 have h4 : p - 1 < p := by linarith have h5 : num_rp_below p (p - 1 + 1) = p - 1 := num_rp_prime h1 (p - 1) h4 rewrite [h3] at h5 show phi p = p - 1 from h5 done theorem Theorem_7_2_2_Int {a c : Nat} {b : Int} (h1 : ↑c ∣ ↑a * b) (h2 : rel_prime a c) : ↑c ∣ b := by rewrite [Int.natCast_dvd, Int.natAbs_mul, Int.natAbs_ofNat] at h1 --h1 : c ∣ a * Int.natAbs b rewrite [Int.natCast_dvd] --Goal : c ∣ Int.natAbs b show c ∣ Int.natAbs b from Theorem_7_2_2 h1 h2 done lemma Lemma_7_4_5 {m n : Nat} (a b : Int) (h1 : rel_prime m n) : a ≡ b (MOD m * n) ↔ a ≡ b (MOD m) ∧ a ≡ b (MOD n) := by apply Iff.intro · -- (→) assume h2 : a ≡ b (MOD m * n) obtain (j : Int) (h3 : a - b = (m * n) * j) from h2 apply And.intro · -- Proof of a ≡ b (MOD m) apply Exists.intro (n * j) show a - b = m * (n * j) from calc a - b _ = m * n * j := h3 _ = m * (n * j) := by ring done · -- Proof of a ≡ b (MOD n) apply Exists.intro (m * j) show a - b = n * (m * j) from calc a - b _ = m * n * j := h3 _ = n * (m * j) := by ring done done · -- (←) assume h2 : a ≡ b (MOD m) ∧ a ≡ b (MOD n) obtain (j : Int) (h3 : a - b = m * j) from h2.left have h4 : (↑n : Int) ∣ a - b := h2.right rewrite [h3] at h4 --h4 : ↑n ∣ ↑m * j have h5 : ↑n ∣ j := Theorem_7_2_2_Int h4 h1 obtain (k : Int) (h6 : j = n * k) from h5 apply Exists.intro k --Goal : a - b = ↑(m * n) * k rewrite [Nat.cast_mul] --Goal : a - b = ↑m * ↑n * k show a - b = (m * n) * k from calc a - b _ = m * j := h3 _ = m * (n * k) := by rw [h6] _ = (m * n) * k := by ring done done --From exercises of Section 7.2 theorem rel_prime_symm {a b : Nat} (h : rel_prime a b) : rel_prime b a := sorry lemma prime_NeZero {p : Nat} (h : prime p) : NeZero p := by rewrite [neZero_iff] --Goal : p ≠ 0 define at h linarith done lemma Lemma_7_5_1 {p e d m c s : Nat} {t : Int} (h1 : prime p) (h2 : e * d = (p - 1) * s + 1) (h3 : m ^ e - c = p * t) : c ^ d ≡ m (MOD p) := by have h4 : m ^ e ≡ c (MOD p) := Exists.intro t h3 have h5 : [m ^ e]_p = [c]_p := (cc_eq_iff_congr _ _ _).rtl h4 rewrite [←Exercise_7_4_5_Nat] at h5 --h5 : [m]_p ^ e = [c]_p by_cases h6 : p ∣ m · -- Case 1. h6 : p ∣ m have h7 : m ≡ 0 (MOD p) := by obtain (j : Nat) (h8 : m = p * j) from h6 apply Exists.intro (↑j : Int) --Goal : ↑m - 0 = ↑p * ↑j rewrite [h8, Nat.cast_mul] ring done have h8 : [m]_p = [0]_p := (cc_eq_iff_congr _ _ _).rtl h7 have h9 : e * d ≠ 0 := by rewrite [h2] show (p - 1) * s + 1 ≠ 0 from Nat.add_one_ne_zero _ done have h10 : (0 : Int) ^ (e * d) = 0 := zero_pow h9 have h11 : [c ^ d]_p = [m]_p := calc [c ^ d]_p _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat] _ = ([m]_p ^ e) ^ d := by rw [h5] _ = [m]_p ^ (e * d) := by ring _ = [0]_p ^ (e * d) := by rw [h8] _ = [0 ^ (e * d)]_p := Exercise_7_4_5_Int _ _ _ _ = [0]_p := by rw [h10] _ = [m]_p := by rw [h8] show c ^ d ≡ m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11 done · -- Case 2. h6 : ¬p ∣ m have h7 : rel_prime m p := rel_prime_of_prime_not_dvd h1 h6 have h8 : rel_prime p m := rel_prime_symm h7 have h9 : NeZero p := prime_NeZero h1 have h10 : (1 : Int) ^ s = 1 := by ring have h11 : [c ^ d]_p = [m]_p := calc [c ^ d]_p _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat] _ = ([m]_p ^ e) ^ d := by rw [h5] _ = [m]_p ^ (e * d) := by ring _ = [m]_p ^ ((p - 1) * s + 1) := by rw [h2] _ = ([m]_p ^ (p - 1)) ^ s * [m]_p := by ring _ = ([m]_p ^ (phi p)) ^ s * [m]_p := by rw [phi_prime h1] _ = [1]_p ^ s * [m]_p := by rw [Theorem_7_4_2 h8] _ = [1 ^ s]_p * [m]_p := by rw [Exercise_7_4_5_Int] _ = [1]_p * [m]_p := by rw [h10] _ = [m]_p * [1]_p := by ring _ = [m]_p := Theorem_7_3_6_7 _ show c ^ d ≡ m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11 done done theorem Theorem_7_5_1 (p q n e d k m c : Nat) (p_prime : prime p) (q_prime : prime q) (p_ne_q : p ≠ q) (n_pq : n = p * q) (ed_congr_1 : e * d = k * (p - 1) * (q - 1) + 1) (h1 : [m]_n ^ e = [c]_n) : [c]_n ^ d = [m]_n := by rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] at h1 --h1 : m ^ e ≡ c (MOD n) rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] --Goal : c ^ d ≡ m (MOD n) obtain (j : Int) (h2 : m ^ e - c = n * j) from h1 rewrite [n_pq, Nat.cast_mul] at h2 --h2 : m ^ e - c = p * q * j have h3 : e * d = (p - 1) * (k * (q - 1)) + 1 := by rewrite [ed_congr_1] ring done have h4 : m ^ e - c = p * (q * j) := by rewrite [h2] ring done have congr_p : c ^ d ≡ m (MOD p) := Lemma_7_5_1 p_prime h3 h4 have h5 : e * d = (q - 1) * (k * (p - 1)) + 1 := by rewrite [ed_congr_1] ring done have h6 : m ^ e - c = q * (p * j) := by rewrite [h2] ring done have congr_q : c ^ d ≡ m (MOD q) := Lemma_7_5_1 q_prime h5 h6 have h7 : ¬q ∣ p := by by_contra h8 have h9 : q = 1 ∨ q = p := dvd_prime p_prime h8 disj_syll h9 (prime_not_one q_prime) show False from p_ne_q h9.symm done have h8 : rel_prime p q := rel_prime_of_prime_not_dvd q_prime h7 rewrite [n_pq, Lemma_7_4_5 _ _ h8] show c ^ d ≡ m (MOD p) ∧ c ^ d ≡ m (MOD q) from And.intro congr_p congr_q done /- BEGIN EXERCISES -/ namespace Exercises /- Section 7.1 -/ -- 1. theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry -- 2. lemma gcd_comm_lt {a b : Nat} (h : a < b) : gcd a b = gcd b a := sorry theorem gcd_comm (a b : Nat) : gcd a b = gcd b a := sorry -- 3. theorem Exercise_7_1_5 (a b : Nat) (n : Int) : (∃ (s t : Int), s * a + t * b = n) ↔ (↑(gcd a b) : Int) ∣ n := sorry -- 4. theorem Exercise_7_1_6 (a b c : Nat) : gcd a b = gcd (a + b * c) b := sorry -- 5. theorem gcd_is_nonzero {a b : Nat} (h : a ≠ 0 ∨ b ≠ 0) : gcd a b ≠ 0 := sorry -- 6. theorem gcd_greatest {a b d : Nat} (h1 : gcd a b ≠ 0) (h2 : d ∣ a) (h3 : d ∣ b) : d ≤ gcd a b := sorry -- 7. lemma Lemma_7_1_10a {a b : Nat} (n : Nat) (h : a ∣ b) : (n * a) ∣ (n * b) := sorry lemma Lemma_7_1_10b {a b n : Nat} (h1 : n ≠ 0) (h2 : (n * a) ∣ (n * b)) : a ∣ b := sorry lemma Lemma_7_1_10c {a b : Nat} (h1 : a ∣ b) (h2 : b ∣ a) : a = b := sorry theorem Exercise_7_1_10 (a b n : Nat) : gcd (n * a) (n * b) = n * gcd a b := sorry /- Section 7.2 -/ -- 1. lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry -- 2. -- Hints: Start with apply List.rec. You may find mul_ne_zero useful theorem prod_nonzero_nonzero : ∀ (l : List Nat), (∀ a ∈ l, a ≠ 0) → prod l ≠ 0 := sorry -- 3. theorem rel_prime_iff_no_common_factor (a b : Nat) : rel_prime a b ↔ ¬∃ (p : Nat), prime p ∧ p ∣ a ∧ p ∣ b := sorry -- 4. theorem rel_prime_symm {a b : Nat} (h : rel_prime a b) : rel_prime b a := sorry -- 5.

lemma in_prime_factorization_iff_prime_factor {a : Nat} {l : List Nat} (h1 : prime_factorization a l) (p : Nat) : p ∈ l ↔ prime_factor p a

HTPI.Exercises.in_prime_factorization_iff_prime_factor

htpi/HTPILib/Chap7.lean

htpi

/- Copyright 2023 Daniel J. Velleman -/ import HTPILib.Chap6 namespace HTPI /- Definitions -/ lemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by have h : n + 1 > 0 := Nat.succ_pos n show a % (n + 1) < n + 1 from Nat.mod_lt a h done def gcd (a b : Nat) : Nat := match b with | 0 => a | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd (n + 1) (a % (n + 1)) termination_by b mutual def gcd_c1 (a b : Nat) : Int := match b with | 0 => 1 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c2 (n + 1) (a % (n + 1)) --Corresponds to s = t' termination_by b def gcd_c2 (a b : Nat) : Int := match b with | 0 => 0 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c1 (n + 1) (a % (n + 1)) - (gcd_c2 (n + 1) (a % (n + 1))) * ↑(a / (n + 1)) --Corresponds to t = s' - t'q termination_by b end def prime (n : Nat) : Prop := 2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n def prime_factor (p n : Nat) : Prop := prime p ∧ p ∣ n def all_prime (l : List Nat) : Prop := ∀ p ∈ l, prime p def nondec (l : List Nat) : Prop := match l with | [] => True --Of course, True is a proposition that is always true | n :: L => (∀ m ∈ L, n ≤ m) ∧ nondec L def nondec_prime_list (l : List Nat) : Prop := all_prime l ∧ nondec l def prod (l : List Nat) : Nat := match l with | [] => 1 | n :: L => n * (prod L) def prime_factorization (n : Nat) (l : List Nat) : Prop := nondec_prime_list l ∧ prod l = n def rel_prime (a b : Nat) : Prop := gcd a b = 1 def congr_mod (m : Nat) (a b : Int) : Prop := (↑m : Int) ∣ (a - b) def cc (m : Nat) (a : Int) : ZMod m := (↑a : ZMod m) notation:50 a " ≡ " b " (MOD " m ")" => congr_mod m a b notation:max "["a"]_"m:max => cc m a def invertible {m : Nat} (X : ZMod m) : Prop := ∃ (Y : ZMod m), X * Y = [1]_m def num_rp_below (m k : Nat) : Nat := match k with | 0 => 0 | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j def phi (m : Nat) : Nat := num_rp_below m m def prod_seq {m : Nat} (j k : Nat) (f : Nat → ZMod m) : ZMod m := match j with | 0 => [1]_m | n + 1 => prod_seq n k f * f (k + n) def maps_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i < n, g i < n def one_one_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i1 < n, ∀ i2 < n, g i1 = g i2 → i1 = i2 def onto_below (n : Nat) (g : Nat → Nat) : Prop := ∀ k < n, ∃ i < n, g i = k def perm_below (n : Nat) (g : Nat → Nat) : Prop := maps_below n g ∧ one_one_below n g ∧ onto_below n g def inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m) def swap (u v i : Nat) : Nat := if i = u then v else if i = v then u else i namespace Euler --For definitions specific to Euler's theorem def F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m def G (m a i : Nat) : Nat := (a * i) % m def Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i end Euler /- Section 7.1 -/ theorem dvd_mod_of_dvd_a_b {a b d : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ (a % b) := by set q : Nat := a / b have h3 : b * q + a % b = a := Nat.div_add_mod a b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 define --Goal : ∃ (c : Nat), a % b = d * c apply Exists.intro (j - k * q) show a % b = d * (j - k * q) from calc a % b _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm _ = a - b * q := by rw [h3] _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc] _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm done theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry #eval gcd 672 161 --Answer: 7 lemma gcd_base (a : Nat) : gcd a 0 = a := by rfl lemma gcd_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd a b = gcd b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1)) rfl done lemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b ≠ 0) : a % b < b := by have h1 : b > 0 := Nat.pos_of_ne_zero h show a % b < b from Nat.mod_lt a h1 done lemma dvd_self (n : Nat) : n ∣ n := by apply Exists.intro 1 ring done theorem gcd_dvd : ∀ (b a : Nat), (gcd a b) ∣ a ∧ (gcd a b) ∣ b := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd a b_1) ∣ a ∧ (gcd a b_1) ∣ b_1 fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_base] --Goal: a ∣ a ∧ a ∣ 0 apply And.intro (dvd_self a) define apply Exists.intro 0 rfl done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_nonzero a h1] --Goal : gcd b (a % b) ∣ a ∧ gcd b (a % b) ∣ b have h2 : a % b < b := mod_nonzero_lt a h1 have h3 : (gcd b (a % b)) ∣ b ∧ (gcd b (a % b)) ∣ (a % b) := ih (a % b) h2 b apply And.intro _ h3.left show (gcd b (a % b)) ∣ a from dvd_a_of_dvd_b_mod h3.left h3.right done done theorem gcd_dvd_left (a b : Nat) : (gcd a b) ∣ a := (gcd_dvd b a).left theorem gcd_dvd_right (a b : Nat) : (gcd a b) ∣ b := (gcd_dvd b a).right lemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl lemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c1 a b = gcd_c2 b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done lemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl lemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * ↑(a / b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done theorem gcd_lin_comb : ∀ (b a : Nat), (gcd_c1 a b) * ↑a + (gcd_c2 a b) * ↑b = ↑(gcd a b) := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd_c1 a b_1) * ↑a + (gcd_c2 a b_1) * ↑b_1 = ↑(gcd a b_1) fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base] --Goal : 1 * ↑a + 0 * ↑0 = ↑a ring done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1] --Goal : gcd_c2 b (a % b) * ↑a + -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * ↑(a / b)) * ↑b = -- ↑(gcd b (a % b)) set r : Nat := a % b set q : Nat := a / b set s : Int := gcd_c1 b r set t : Int := gcd_c2 b r --Goal : t * ↑a + (s - t * ↑q) * ↑b = ↑(gcd b r) have h2 : r < b := mod_nonzero_lt a h1 have h3 : s * ↑b + t * ↑r = ↑(gcd b r) := ih r h2 b have h4 : b * q + r = a := Nat.div_add_mod a b rewrite [←h3, ←h4] rewrite [Nat.cast_add, Nat.cast_mul] --Goal : t * (↑b * ↑q + ↑r) + (s - t * ↑q) * ↑b = s * ↑b + t * ↑r ring done done #eval gcd_c1 672 161 --Answer: 6 #eval gcd_c2 672 161 --Answer: -25 --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161 theorem Theorem_7_1_6 {d a b : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ gcd a b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑d ∣ ↑(gcd a b) set s : Int := gcd_c1 a b set t : Int := gcd_c2 a b have h3 : s * ↑a + t * ↑b = ↑(gcd a b) := gcd_lin_comb b a rewrite [←h3] --Goal : ↑d ∣ s * ↑a + t * ↑b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul] --Goal : ↑d ∣ s * (↑d * ↑j) + t * (↑d * ↑k) define apply Exists.intro (s * ↑j + t * ↑k) ring done /- Section 7.2 -/ theorem dvd_trans {a b c : Nat} (h1 : a ∣ b) (h2 : b ∣ c) : a ∣ c := by define at h1; define at h2; define obtain (m : Nat) (h3 : b = a * m) from h1 obtain (n : Nat) (h4 : c = b * n) from h2 rewrite [h3, mul_assoc] at h4 apply Exists.intro (m * n) show c = a * (m * n) from h4 done lemma exists_prime_factor : ∀ (n : Nat), 2 ≤ n → ∃ (p : Nat), prime_factor p n := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, 2 ≤ n_1 → ∃ (p : Nat), prime_factor p n_1 assume h1 : 2 ≤ n by_cases h2 : prime n · -- Case 1. h2 : prime n apply Exists.intro n define --Goal : prime n ∧ n ∣ n show prime n ∧ n ∣ n from And.intro h2 (dvd_self n) done · -- Case 2. h2 : ¬prime n define at h2 --h2 : ¬(2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n) demorgan at h2 disj_syll h2 h1 obtain (a : Nat) (h3 : ∃ (b : Nat), a * b = n ∧ a < n ∧ b < n) from h2 obtain (b : Nat) (h4 : a * b = n ∧ a < n ∧ b < n) from h3 have h5 : 2 ≤ a := by by_contra h6 have h7 : a ≤ 1 := by linarith have h8 : n ≤ b := calc n _ = a * b := h4.left.symm _ ≤ 1 * b := by rel [h7] _ = b := by ring linarith --n ≤ b contradicts b < n done have h6 : ∃ (p : Nat), prime_factor p a := ih a h4.right.left h5 obtain (p : Nat) (h7 : prime_factor p a) from h6 apply Exists.intro p define --Goal : prime p ∧ p ∣ n define at h7 --h7 : prime p ∧ p ∣ a apply And.intro h7.left have h8 : a ∣ n := by apply Exists.intro b show n = a * b from (h4.left).symm done show p ∣ n from dvd_trans h7.right h8 done done lemma exists_least_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q := by set S : Set Nat := {p : Nat | prime_factor p n} have h2 : ∃ (p : Nat), p ∈ S := exists_prime_factor n h show ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q from well_ord_princ S h2 done lemma all_prime_nil : all_prime [] := by define --Goal : ∀ p ∈ [], prime p fix p : Nat contrapos --Goal : ¬prime p → p ∉ [] assume h1 : ¬prime p show p ∉ [] from List.not_mem_nil p done lemma all_prime_cons (n : Nat) (L : List Nat) : all_prime (n :: L) ↔ prime n ∧ all_prime L := by apply Iff.intro · -- (→) assume h1 : all_prime (n :: L) --Goal : prime n ∧ all_prime L define at h1 --h1 : ∀ p ∈ n :: L, prime p apply And.intro (h1 n (List.mem_cons_self n L)) define --Goal : ∀ p ∈ L, prime p fix p : Nat assume h2 : p ∈ L show prime p from h1 p (List.mem_cons_of_mem n h2) done · -- (←) assume h1 : prime n ∧ all_prime L --Goal : all_prime (n :: l) define : all_prime L at h1 define fix p : Nat assume h2 : p ∈ n :: L rewrite [List.mem_cons] at h2 --h2 : p = n ∨ p ∈ L by_cases on h2 · -- Case 1. h2 : p = n rewrite [h2] show prime n from h1.left done · -- Case 2. h2 : p ∈ L show prime p from h1.right p h2 done done done lemma nondec_nil : nondec [] := by define --Goal : True trivial --trivial proves some obviously true statements, such as True done lemma nondec_cons (n : Nat) (L : List Nat) : nondec (n :: L) ↔ (∀ m ∈ L, n ≤ m) ∧ nondec L := by rfl lemma prod_nil : prod [] = 1 := by rfl lemma prod_cons : prod (n :: L) = n * (prod L) := by rfl lemma exists_cons_of_length_eq_succ {A : Type} {l : List A} {n : Nat} (h : l.length = n + 1) : ∃ (a : A) (L : List A), l = a :: L ∧ L.length = n := by have h1 : ¬l.length = 0 := by linarith rewrite [List.length_eq_zero] at h1 obtain (a : A) (h2 : ∃ (L : List A), l = a :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List A) (h3 : l = a :: L) from h2 apply Exists.intro a apply Exists.intro L apply And.intro h3 have h4 : (a :: L).length = L.length + 1 := List.length_cons a L rewrite [←h3, h] at h4 show L.length = n from (Nat.add_right_cancel h4).symm done lemma list_elt_dvd_prod_by_length (a : Nat) : ∀ (n : Nat), ∀ (l : List Nat), l.length = n → a ∈ l → a ∣ prod l := by by_induc · --Base Case fix l : List Nat assume h1 : l.length = 0 rewrite [List.length_eq_zero] at h1 --h1 : l = [] rewrite [h1] --Goal : a ∈ [] → a ∣ prod [] contrapos assume h2 : ¬a ∣ prod [] show a ∉ [] from List.not_mem_nil a done · -- Induction Step fix n : Nat assume ih : ∀ (l : List Nat), List.length l = n → a ∈ l → a ∣ prod l fix l : List Nat assume h1 : l.length = n + 1 --Goal : a ∈ l → a ∣ prod l obtain (b : Nat) (h2 : ∃ (L : List Nat), l = b :: L ∧ L.length = n) from exists_cons_of_length_eq_succ h1 obtain (L : List Nat) (h3 : l = b :: L ∧ L.length = n) from h2 have h4 : a ∈ L → a ∣ prod L := ih L h3.right assume h5 : a ∈ l rewrite [h3.left, prod_cons] --Goal : a ∣ b * prod L rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b ∨ a ∈ L by_cases on h5 · -- Case 1. h5 : a = b apply Exists.intro (prod L) rewrite [h5] rfl done · -- Case 2. h5 : a ∈ L have h6 : a ∣ prod L := h4 h5 have h7 : prod L ∣ b * prod L := by apply Exists.intro b ring done show a ∣ b * prod L from dvd_trans h6 h7 done done done lemma list_elt_dvd_prod {a : Nat} {l : List Nat} (h : a ∈ l) : a ∣ prod l := by set n : Nat := l.length have h1 : l.length = n := by rfl show a ∣ prod l from list_elt_dvd_prod_by_length a n l h1 h done lemma exists_prime_factorization : ∀ (n : Nat), n ≥ 1 → ∃ (l : List Nat), prime_factorization n l := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, n_1 ≥ 1 → ∃ (l : List Nat), prime_factorization n_1 l assume h1 : n ≥ 1 by_cases h2 : n = 1 · -- Case 1. h2 : n = 1 apply Exists.intro [] define apply And.intro · -- Proof of nondec_prime_list [] define show all_prime [] ∧ nondec [] from And.intro all_prime_nil nondec_nil done · -- Proof of prod [] = n rewrite [prod_nil, h2] rfl done done · -- Case 2. h2 : n ≠ 1 have h3 : n ≥ 2 := lt_of_le_of_ne' h1 h2 obtain (p : Nat) (h4 : prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q) from exists_least_prime_factor h3 have p_prime_factor : prime_factor p n := h4.left define at p_prime_factor have p_prime : prime p := p_prime_factor.left have p_dvd_n : p ∣ n := p_prime_factor.right have p_least : ∀ (q : Nat), prime_factor q n → p ≤ q := h4.right obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n have h5 : m ≠ 0 := by contradict h1 with h6 have h7 : n = 0 := calc n _ = p * m := n_eq_pm _ = p * 0 := by rw [h6] _ = 0 := by ring rewrite [h7] decide done have m_pos : 0 < m := Nat.pos_of_ne_zero h5 have m_lt_n : m < n := by define at p_prime show m < n from calc m _ < m + m := by linarith _ = 2 * m := by ring _ ≤ p * m := by rel [p_prime.left] _ = n := n_eq_pm.symm done obtain (L : List Nat) (h6 : prime_factorization m L) from ih m m_lt_n m_pos define at h6 have ndpl_L : nondec_prime_list L := h6.left define at ndpl_L apply Exists.intro (p :: L) define apply And.intro · -- Proof of nondec_prime_list (p :: L) define apply And.intro · -- Proof of all_prime (p :: L) rewrite [all_prime_cons] show prime p ∧ all_prime L from And.intro p_prime ndpl_L.left done · -- Proof of nondec (p :: L) rewrite [nondec_cons] apply And.intro _ ndpl_L.right fix q : Nat assume q_in_L : q ∈ L have h7 : q ∣ prod L := list_elt_dvd_prod q_in_L rewrite [h6.right] at h7 --h7 : q ∣ m have h8 : m ∣ n := by apply Exists.intro p rewrite [n_eq_pm] ring done have q_dvd_n : q ∣ n := dvd_trans h7 h8 have ap_L : all_prime L := ndpl_L.left define at ap_L have q_prime_factor : prime_factor q n := And.intro (ap_L q q_in_L) q_dvd_n show p ≤ q from p_least q q_prime_factor done done · -- Proof of prod (p :: L) = n rewrite [prod_cons, h6.right, n_eq_pm] rfl done done done theorem Theorem_7_2_2 {a b c : Nat} (h1 : c ∣ a * b) (h2 : rel_prime a c) : c ∣ b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑c ∣ ↑b define at h1; define at h2; define obtain (j : Nat) (h3 : a * b = c * j) from h1 set s : Int := gcd_c1 a c set t : Int := gcd_c2 a c have h4 : s * ↑a + t * ↑c = ↑(gcd a c) := gcd_lin_comb c a rewrite [h2, Nat.cast_one] at h4 --h4 : s * ↑a + t * ↑c = (1 : Int) apply Exists.intro (s * ↑j + t * ↑b) show ↑b = ↑c * (s * ↑j + t * ↑b) from calc ↑b _ = (1 : Int) * ↑b := (one_mul _).symm _ = (s * ↑a + t * ↑c) * ↑b := by rw [h4] _ = s * (↑a * ↑b) + t * ↑c * ↑b := by ring _ = s * (↑c * ↑j) + t * ↑c * ↑b := by rw [←Nat.cast_mul a b, h3, Nat.cast_mul c j] _ = ↑c * (s * ↑j + t * ↑b) := by ring done lemma le_nonzero_prod_left {a b : Nat} (h : a * b ≠ 0) : a ≤ a * b := by have h1 : b ≠ 0 := by contradict h with h1 rewrite [h1] ring done have h2 : 1 ≤ b := Nat.pos_of_ne_zero h1 show a ≤ a * b from calc a = a * 1 := (mul_one a).symm _ ≤ a * b := by rel [h2] done lemma le_nonzero_prod_right {a b : Nat} (h : a * b ≠ 0) : b ≤ a * b := by rewrite [mul_comm] rewrite [mul_comm] at h show b ≤ b * a from le_nonzero_prod_left h done lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry lemma rel_prime_of_prime_not_dvd {a p : Nat} (h1 : prime p) (h2 : ¬p ∣ a) : rel_prime a p := by have h3 : gcd a p ∣ a := gcd_dvd_left a p have h4 : gcd a p ∣ p := gcd_dvd_right a p have h5 : gcd a p = 1 ∨ gcd a p = p := dvd_prime h1 h4 have h6 : gcd a p ≠ p := by contradict h2 with h6 rewrite [h6] at h3 show p ∣ a from h3 done disj_syll h5 h6 show rel_prime a p from h5 done theorem Theorem_7_2_3 {a b p : Nat} (h1 : prime p) (h2 : p ∣ a * b) : p ∣ a ∨ p ∣ b := by or_right with h3 have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3 show p ∣ b from Theorem_7_2_2 h2 h4 done lemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a ≥ 1 := by have h1 : a ≠ 0 := by by_contra h1 rewrite [h1] at h contradict h linarith done show a ≥ 1 from Nat.pos_of_ne_zero h1 done lemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by have h1 : a ≥ 1 := ge_one_of_prod_one h have h2 : a * b ≠ 0 := by linarith have h3 : a ≤ a * b := le_nonzero_prod_left h2 rewrite [h] at h3 show a = 1 from Nat.le_antisymm h3 h1 done lemma eq_one_of_dvd_one {n : Nat} (h : n ∣ 1) : n = 1 := by obtain (j : Nat) (h1 : 1 = n * j) from h show n = 1 from eq_one_of_prod_one h1.symm done lemma prime_not_one {p : Nat} (h : prime p) : p ≠ 1 := by define at h linarith done theorem Theorem_7_2_4 {p : Nat} (h1 : prime p) : ∀ (l : List Nat), p ∣ prod l → ∃ a ∈ l, p ∣ a := by apply List.rec · -- Base Case. Goal : p ∣ prod [] → ∃ a ∈ [], p ∣ a rewrite [prod_nil] assume h2 : p ∣ 1 show ∃ a ∈ [], p ∣ a from absurd (eq_one_of_dvd_one h2) (prime_not_one h1) done · -- Induction Step fix b : Nat fix L : List Nat assume ih : p ∣ prod L → ∃ a ∈ L, p ∣ a --Goal : p ∣ prod (b :: L) → ∃ a ∈ b :: L, p ∣ a assume h2 : p ∣ prod (b :: L) rewrite [prod_cons] at h2 have h3 : p ∣ b ∨ p ∣ prod L := Theorem_7_2_3 h1 h2 by_cases on h3 · -- Case 1. h3 : p ∣ b apply Exists.intro b show b ∈ b :: L ∧ p ∣ b from And.intro (List.mem_cons_self b L) h3 done · -- Case 2. h3 : p ∣ prod L obtain (a : Nat) (h4 : a ∈ L ∧ p ∣ a) from ih h3 apply Exists.intro a show a ∈ b :: L ∧ p ∣ a from And.intro (List.mem_cons_of_mem b h4.left) h4.right done done done lemma prime_in_list {p : Nat} {l : List Nat} (h1 : prime p) (h2 : all_prime l) (h3 : p ∣ prod l) : p ∈ l := by obtain (a : Nat) (h4 : a ∈ l ∧ p ∣ a) from Theorem_7_2_4 h1 l h3 define at h2 have h5 : prime a := h2 a h4.left have h6 : p = 1 ∨ p = a := dvd_prime h5 h4.right disj_syll h6 (prime_not_one h1) rewrite [h6] show a ∈ l from h4.left done lemma first_le_first {p q : Nat} {l m : List Nat} (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m)) (h3 : prod (p :: l) = prod (q :: m)) : p ≤ q := by define at h1; define at h2 have h4 : q ∣ prod (p :: l) := by define apply Exists.intro (prod m) rewrite [←prod_cons] show prod (p :: l) = prod (q :: m) from h3 done have h5 : all_prime (q :: m) := h2.left rewrite [all_prime_cons] at h5 have h6 : q ∈ p :: l := prime_in_list h5.left h1.left h4 have h7 : nondec (p :: l) := h1.right rewrite [nondec_cons] at h7 rewrite [List.mem_cons] at h6 by_cases on h6 · -- Case 1. h6 : q = p linarith done · -- Case 2. h6 : q ∈ l have h8 : ∀ m ∈ l, p ≤ m := h7.left show p ≤ q from h8 q h6 done done lemma nondec_prime_list_tail {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by define at h define rewrite [all_prime_cons, nondec_cons] at h show all_prime l ∧ nondec l from And.intro h.left.right h.right.right done lemma cons_prod_not_one {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : prod (p :: l) ≠ 1 := by define at h have h1 : all_prime (p :: l) := h.left rewrite [all_prime_cons] at h1 rewrite [prod_cons] by_contra h2 show False from (prime_not_one h1.left) (eq_one_of_prod_one h2) done lemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) : l = [] ↔ prod l = 1 := by apply Iff.intro · -- (→) assume h1 : l = [] rewrite [h1] show prod [] = 1 from prod_nil done · -- (←) contrapos assume h1 : ¬l = [] obtain (p : Nat) (h2 : ∃ (L : List Nat), l = p :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List Nat) (h3 : l = p :: L) from h2 rewrite [h3] at h rewrite [h3] show ¬prod (p :: L) = 1 from cons_prod_not_one h done done lemma prime_pos {p : Nat} (h : prime p) : p > 0 := by define at h linarith done theorem Theorem_7_2_5 : ∀ (l1 l2 : List Nat), nondec_prime_list l1 → nondec_prime_list l2 → prod l1 = prod l2 → l1 = l2 := by apply List.rec · -- Base Case. Goal : ∀ (l2 : List Nat), nondec_prime_list [] → -- nondec_prime_list l2 → prod [] = prod l2 → [] = l2 fix l2 : List Nat assume h1 : nondec_prime_list [] assume h2 : nondec_prime_list l2 assume h3 : prod [] = prod l2 rewrite [prod_nil, eq_comm, ←list_nil_iff_prod_one h2] at h3 show [] = l2 from h3.symm done · -- Induction Step fix p : Nat fix L1 : List Nat assume ih : ∀ (L2 : List Nat), nondec_prime_list L1 → nondec_prime_list L2 → prod L1 = prod L2 → L1 = L2 -- Goal : ∀ (l2 : List Nat), nondec_prime_list (p :: L1) → -- nondec_prime_list l2 → prod (p :: L1) = prod l2 → p :: L1 = l2 fix l2 : List Nat assume h1 : nondec_prime_list (p :: L1) assume h2 : nondec_prime_list l2 assume h3 : prod (p :: L1) = prod l2 have h4 : ¬prod (p :: L1) = 1 := cons_prod_not_one h1 rewrite [h3, ←list_nil_iff_prod_one h2] at h4 obtain (q : Nat) (h5 : ∃ (L : List Nat), l2 = q :: L) from List.exists_cons_of_ne_nil h4 obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5 rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2) rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2) have h7 : p ≤ q := first_le_first h1 h2 h3 have h8 : q ≤ p := first_le_first h2 h1 h3.symm have h9 : p = q := by linarith rewrite [h9, prod_cons, prod_cons] at h3 --h3 : q * prod L1 = q * prod L2 have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1 have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2 define at h2 have h12 : all_prime (q :: L2) := h2.left rewrite [all_prime_cons] at h12 have h13 : q > 0 := prime_pos h12.left have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3 have h15 : L1 = L2 := ih L2 h10 h11 h14 rewrite [h6, h9, h15] rfl done done theorem fund_thm_arith (n : Nat) (h : n ≥ 1) : ∃! (l : List Nat), prime_factorization n l := by exists_unique · -- Existence show ∃ (l : List Nat), prime_factorization n l from exists_prime_factorization n h done · -- Uniqueness fix l1 : List Nat; fix l2 : List Nat assume h1 : prime_factorization n l1 assume h2 : prime_factorization n l2 define at h1; define at h2 have h3 : prod l1 = n := h1.right rewrite [←h2.right] at h3 show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3 done done /- Section 7.3 -/ theorem congr_refl (m : Nat) : ∀ (a : Int), a ≡ a (MOD m) := by fix a : Int define --Goal : ∃ (c : Int), a - a = ↑m * c apply Exists.intro 0 ring done theorem congr_symm {m : Nat} : ∀ {a b : Int}, a ≡ b (MOD m) → b ≡ a (MOD m) := by fix a : Int; fix b : Int assume h1 : a ≡ b (MOD m) define at h1 --h1 : ∃ (c : Int), a - b = ↑m * c define --Goal : ∃ (c : Int), b - a = ↑m * c obtain (c : Int) (h2 : a - b = m * c) from h1 apply Exists.intro (-c) show b - a = m * (-c) from calc b - a _ = -(a - b) := by ring _ = -(m * c) := by rw [h2] _ = m * (-c) := by ring done theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry /- Fundamental properties of congruence classes -/ lemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) : X = Y ↔ X.val = Y.val := Fin.ext_iff lemma val_nat_eq_mod (n k : Nat) : ([k]_(n + 1)).val = k % (n + 1) := by rfl lemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl theorem cc_rep {m : Nat} (X : ZMod m) : ∃ (a : Int), X = [a]_m := match m with | 0 => by apply Exists.intro X rfl done | n + 1 => by apply Exists.intro ↑(X.val) have h1 : X.val < n + 1 := Fin.prop X rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1] rfl done theorem add_class (m : Nat) (a b : Int) : [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm theorem mul_class (m : Nat) (a b : Int) : [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm lemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ [a - b]_m = [0]_m := by apply Iff.intro · -- (→) assume h1 : [a]_m = [b]_m have h2 : a - b = a + (-b) := by ring have h3 : b + (-b) = 0 := by ring show [a - b]_m = [0]_m from calc [a - b]_m _ = [a + (-b)]_m := by rw [h2] _ = [a]_m + [-b]_m := by rw [add_class] _ = [b]_m + [-b]_m := by rw [h1] _ = [b + -b]_m := by rw [add_class] _ = [0]_m := by rw [h3] done · -- (←) assume h1 : [a - b]_m = [0]_m have h2 : b + (a - b) = a := by ring have h3 : b + 0 = b := by ring show [a]_m = [b]_m from calc [a]_m _ = [b + (a - b)]_m := by rw [h2] _ = [b]_m + [a - b]_m := by rw [add_class] _ = [b]_m + [0]_m := by rw [h1] _ = [b + 0]_m := by rw [add_class] _ = [b]_m := by rw [h3] done done lemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) : [a]_m = [0]_m → [-a]_m = [0]_m := by assume h1 : [a]_m = [0]_m have h2 : 0 + (-a) = -a := by ring have h3 : a + (-a) = 0 := by ring show [-a]_m = [0]_m from calc [-a]_m _ = [0 + (-a)]_m := by rw [h2] _ = [0]_m + [-a]_m := by rw [add_class] _ = [a]_m + [-a]_m := by rw [h1] _ = [a + (-a)]_m := by rw [add_class] _ = [0]_m := by rw [h3] done lemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) : [-a]_m = [0]_m ↔ [a]_m = [0]_m := by apply Iff.intro _ (cc_neg_zero_of_cc_zero m a) assume h1 : [-a]_m = [0]_m have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1 have h3 : -(-a) = a := by ring rewrite [h3] at h2 show [a]_m = [0]_m from h2 done lemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl lemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m ↔ m ∣ k := match m with | 0 => by have h : (0 : Int) = (↑(0 : Nat) : Int) := by rfl rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj] apply Iff.intro · -- (→) assume h1 : k = 0 rewrite [h1] show 0 ∣ 0 from dvd_self 0 done · -- (←) assume h1 : 0 ∣ k obtain (c : Nat) (h2 : k = 0 * c) from h1 rewrite [h2] ring done done | n + 1 => by rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero] show k % (n + 1) = 0 ↔ n + 1 ∣ k from (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm done lemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m ↔ ↑m ∣ a := by obtain (k : Nat) (h1 : a = ↑k ∨ a = -↑k) from Int.eq_nat_or_neg a by_cases on h1 · -- Case 1. h1: a = ↑k rewrite [h1, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done · -- Case 2. h1: a = -↑k rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done done theorem cc_eq_iff_congr (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ a ≡ b (MOD m) := calc [a]_m = [b]_m _ ↔ [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b _ ↔ ↑m ∣ (a - b) := cc_zero_iff_dvd m (a - b) _ ↔ a ≡ b (MOD m) := by rfl /- End of fundamental properties of congruence classes -/ lemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m := by have h1 : (↑m : Int) ≠ 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m) show 0 ≤ a % m from Int.emod_nonneg a h1 done lemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m) have h2 : (↑m : Int) > 0 := (Nat.cast_pos).rtl h1 show a % m < m from Int.emod_lt_of_pos a h2 done lemma congr_mod_mod (m : Nat) (a : Int) : a ≡ a % m (MOD m) := by define have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m apply Exists.intro (a / m) show a - a % m = m * (a / m) from calc a - (a % m) _ = m * (a / m) + a % m - a % m := by rw [h1] _ = m * (a / m) := by ring done lemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) := And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a)) theorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) : ∃! (r : Int), 0 ≤ r ∧ r < m ∧ a ≡ r (MOD m) := by exists_unique · -- Existence apply Exists.intro (a % m) show 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) from mod_cmpl_res m a done · -- Uniqueness fix r1 : Int; fix r2 : Int assume h1 : 0 ≤ r1 ∧ r1 < m ∧ a ≡ r1 (MOD m) assume h2 : 0 ≤ r2 ∧ r2 < m ∧ a ≡ r2 (MOD m) have h3 : r1 ≡ r2 (MOD m) := congr_trans (congr_symm h1.right.right) h2.right.right obtain (d : Int) (h4 : r1 - r2 = m * d) from h3 have h5 : r1 - r2 < m * 1 := by linarith have h6 : m * (-1) < r1 - r2 := by linarith rewrite [h4] at h5 --h5 : m * d < m * 1 rewrite [h4] at h6 --h6 : m * -1 < m * d have h7 : (↑m : Int) ≥ 0 := Nat.cast_nonneg m have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7 have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7 have h10 : d = 0 := by linarith show r1 = r2 from calc r1 _ = r1 - r2 + r2 := by ring _ = m * 0 + r2 := by rw [h4, h10] _ = r2 := by ring done done lemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m := (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a) theorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y rewrite [h1, h2] have h3 : a + b = b + a := by ring show [a]_m + [b]_m = [b]_m + [a]_m from calc [a]_m + [b]_m _ = [a + b]_m := add_class m a b _ = [b + a]_m := by rw [h3] _ = [b]_m + [a]_m := (add_class m b a).symm done theorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X rewrite [h1] have h2 : a * 1 = a := by ring show [a]_m * [1]_m = [a]_m from calc [a]_m * [1]_m _ = [a * 1]_m := mul_class m a 1 _ = [a]_m := by rw [h2] done theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry lemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) : [a]_m * [gcd_c2 m a]_m = [1]_m := by set s : Int := gcd_c1 m a have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m define at h1 rewrite [h1, Nat.cast_one] at h2 --h2 : s * ↑m + gcd_c2 m a * ↑a = 1 rewrite [mul_class, cc_eq_iff_congr] define --Goal : ∃ (c : Int), ↑a * gcd_c2 m a - 1 = ↑m * c apply Exists.intro (-s) show a * (gcd_c2 m a) - 1 = m * (-s) from calc a * (gcd_c2 m a) - 1 _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring _ = 1 + m * (-s) - 1 := by rw [h2] _ = m * (-s) := by ring done theorem Theorem_7_3_7 (m a : Nat) : invertible [a]_m ↔ rel_prime m a := by apply Iff.intro · -- (→) assume h1 : invertible [a]_m define at h1 obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1 obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y rewrite [h3, mul_class, cc_eq_iff_congr] at h2 define at h2 obtain (c : Int) (h4 : a * b - 1 = m * c) from h2 rewrite [Exercise_7_2_6] --Goal : ∃ (s t : Int), s * ↑m + t * ↑a = 1 apply Exists.intro (-c) apply Exists.intro b show (-c) * m + b * a = 1 from calc (-c) * m + b * a _ = (-c) * m + (a * b - 1) + 1 := by ring _ = (-c) * m + m * c + 1 := by rw [h4] _ = 1 := by ring done · -- (←) assume h1 : rel_prime m a define show ∃ (Y : ZMod m), [a]_m * Y = [1]_m from Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1) done done /- Section 7.4 -/ section Euler open Euler lemma num_rp_below_base {m : Nat} : num_rp_below m 0 = 0 := by rfl lemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) : num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by have h1 : num_rp_below m (j + 1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : gcd m j = 1 rewrite [if_pos h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1 show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1 done lemma num_rp_below_step_not_rp {m j : Nat} (h : ¬rel_prime m j) : num_rp_below m (j + 1) = num_rp_below m j := by have h1 : num_rp_below m (j +1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : ¬gcd m j = 1 rewrite [if_neg h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j show num_rp_below m (j + 1) = num_rp_below m j from h1 done lemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl #eval phi 10 --Answer: 4 lemma prod_inv_iff_inv {m : Nat} {X : ZMod m} (h1 : invertible X) (Y : ZMod m) : invertible (X * Y) ↔ invertible Y := by apply Iff.intro · -- (→) assume h2 : invertible (X * Y) obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2 apply Exists.intro (X * Z) rewrite [←h3] --Goal : Y * (X * Z) = X * Y * Z ring --Note that ring can do algebra in ZMod m done · -- (←) assume h2 : invertible Y obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1 obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2 apply Exists.intro (Xi * Yi) show (X * Y) * (Xi * Yi) = [1]_m from calc X * Y * (Xi * Yi) _ = (X * Xi) * (Y * Yi) := by ring _ = [1]_m * [1]_m := by rw [h3, h4] _ = [1]_m := Theorem_7_3_6_7 [1]_m done done lemma F_rp_def {m i : Nat} (h : rel_prime m i) : F m i = [i]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h --h : gcd m i = 1 rewrite [if_pos h] at h1 show F m i = [i]_m from h1 done lemma F_not_rp_def {m i : Nat} (h : ¬rel_prime m i) : F m i = [1]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h rewrite [h1, if_neg h] rfl done lemma prod_seq_base {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 0 k f = [1]_m := by rfl lemma prod_seq_step {m : Nat} (n k : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl lemma prod_seq_zero_step {m : Nat} (n : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by rewrite [prod_seq_step, zero_add] rfl done lemma prod_one {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 1 k f = f k := by rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7] rfl done lemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl lemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m := calc [G m a i]_m _ = [(a * i) % m]_m := by rfl _ = [a * i]_m := (cc_eq_mod m (a * i)).symm _ = [a]_m * [i]_m := (mul_class m a i).symm lemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) : rel_prime m (G m a i) ↔ rel_prime m i := by have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1 show rel_prime m (G m a i) ↔ rel_prime m i from calc rel_prime m (G m a i) _ ↔ invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm _ ↔ invertible ([a]_m * [i]_m) := by rw [cc_G] _ ↔ invertible [i]_m := prod_inv_iff_inv h2 ([i]_m) _ ↔ rel_prime m i := Theorem_7_3_7 m i done lemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) : F m (G m a i) = [a]_m * F m i := by have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2 show F m (G m a i) = [a]_m * F m i from calc F m (G m a i) _ = [G m a i]_m := F_rp_def h3 _ = [a]_m * [i]_m := cc_G m a i _ = [a]_m * F m i := by rw [F_rp_def h2] done lemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : ¬rel_prime m i) : F m (G m a i) = [1]_m := by rewrite [←G_rp_iff h1 i] at h2 show F m (G m a i) = [1]_m from F_not_rp_def h2 done lemma FG_prod {m a : Nat} (h1 : rel_prime m a) : ∀ (k : Nat), prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by by_induc · -- Base Case show prod_seq 0 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from calc prod_seq 0 0 ((F m) ∘ (G m a)) _ = [1]_m := prod_seq_base _ _ _ = [a]_m ^ 0 * [1]_m := by ring _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by rw [num_rp_below_base, prod_seq_base] done · -- Induction Step fix k : Nat assume ih : prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) by_cases h2 : rel_prime m k · -- Case 1. h2 : rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([a]_m * F m k) := by rw [FG_rp h1 h2] _ = [a]_m ^ ((num_rp_below m k) + 1) * ((prod_seq k 0 (F m)) * F m k) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_rp h2, prod_seq_zero_step] done · -- Case 2. h2 : ¬rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([1]_m) := by rw [FG_not_rp h1 h2] _ = [a]_m ^ (num_rp_below m k) * (prod_seq k 0 (F m) * ([1]_m)) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_not_rp h2, prod_seq_zero_step, F_not_rp_def h2] done done done lemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by define --Goal : ∀ i < m, G m a i < m fix i : Nat assume h1 : i < m rewrite [G_def] --Goal : a * i % m < m show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m) done lemma left_inv_one_one_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g' (g i) = i) : one_one_below n g := sorry lemma right_inv_onto_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g (g' i) = i) (h2 : maps_below n g') : onto_below n g := by define at h2; define fix k : Nat assume h3 : k < n apply Exists.intro (g' k) show g' k < n ∧ g (g' k) = k from And.intro (h2 k h3) (h1 k h3) done lemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m * [inv_mod m a]_m = [1]_m := by have h2 : 0 ≤ (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a) show [a]_m * [inv_mod m a]_m = [1]_m from calc [a]_m * [inv_mod m a]_m _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2] _ = [a]_m * [gcd_c2 m a]_m := by rw [←cc_eq_mod] _ = [1]_m := gcd_c2_inv h1 done lemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m := calc a * (b % m) % m = a % m * (b % m % m) % m := Nat.mul_mod _ _ _ _ = a % m * (b % m) % m := by rw [Nat.mod_mod] _ = a * b % m := (Nat.mul_mod _ _ _).symm lemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm] rfl done theorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] : ↑a ≡ ↑b (MOD m) ↔ a % m = b % m := sorry lemma mul_inv_mod_cancel {m a i : Nat} [NeZero m] (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1 rewrite [mul_class, cc_eq_iff_congr, ←Nat.cast_mul, ←Nat.cast_one, congr_iff_mod_eq_Nat] at h3 show a * inv_mod m a * i % m = i from calc a * (inv_mod m a) * i % m _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = 1 % m * i % m := by rw [h3] _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = i % m := by rw [one_mul] _ = i := Nat.mod_eq_of_lt h2 done lemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl lemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, G m a (Ginv m a i) = i := by fix i : Nat assume h2 : i < m show G m a (Ginv m a i) = i from calc G m a (Ginv m a i) _ = a * ((inv_mod m a * i) % m) % m := by rfl _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, Ginv m a (G m a i) = i := by fix i : Nat assume h2 : i < m show Ginv m a (G m a i) = i from calc Ginv m a (G m a i) _ = inv_mod m a * ((a * i) % m) % m := by rfl _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc, mul_comm (inv_mod m a)] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_maps_below (m a : Nat) [NeZero m] : maps_below m (Ginv m a) := G_maps_below m (inv_mod m a) lemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : one_one_below m (G m a) := left_inv_one_one_below (Ginv_left_inv h1) lemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : onto_below m (G m a) := right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a) lemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : perm_below m (G m a) := And.intro (G_maps_below m a) (And.intro (G_one_one_below h1) (G_onto_below h1)) --Permuting a product of congruence classes doesn't change product lemma swap_fst (u v : Nat) : swap u v u = v := by define : swap u v u --Goal : (if u = u then v else if u = v then u else u) = v have h : u = u := by rfl rewrite [if_pos h] rfl done lemma swap_snd (u v : Nat) : swap u v v = u := by define : swap u v v by_cases h1 : v = u · -- Case 1. h1 : v = u rewrite [if_pos h1] show v = u from h1 done · -- Case 2. h1 : v ≠ u rewrite [if_neg h1] have h2 : v = v := by rfl rewrite [if_pos h2] rfl done done lemma swap_other {u v i : Nat} (h1 : i ≠ u) (h2 : i ≠ v) : swap u v i = i := by define : swap u v i rewrite [if_neg h1, if_neg h2] rfl done lemma swap_values (u v i : Nat) : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := by by_cases h1 : i = u · -- Case 1. h1 : i = u apply Or.inl rewrite [h1] show swap u v u = v from swap_fst u v done · -- Case 2. h1 : i ≠ u apply Or.inr by_cases h2 : i = v · -- Case 2.1. h2 : i = v apply Or.inl rewrite [h2] show swap u v v = u from swap_snd u v done · -- Case 2.2. h2 : i ≠ v apply Or.inr show swap u v i = i from swap_other h1 h2 done done done lemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by define fix i : Nat assume h3 : i < n have h4 : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := swap_values u v i by_cases on h4 · -- Case 1. h4 : swap u v i = v rewrite [h4] show v < n from h2 done · -- Case 2. by_cases on h4 · -- Case 2.1. h4 : swap u v i = u rewrite [h4] show u < n from h1 done · -- Case 2.2. h4 : swap u v i = i rewrite [h4] show i < n from h3 done done done lemma swap_swap (u v n : Nat) : ∀ i < n, swap u v (swap u v i) = i := by fix i : Nat assume h : i < n by_cases h1 : i = u · -- Case 1. h1 : i = u rewrite [h1, swap_fst, swap_snd] rfl done · -- Case 2. h1 : i ≠ u by_cases h2 : i = v · -- Case 2.1. h2 : i = v rewrite [h2, swap_snd, swap_fst] rfl done · -- Case 2.2. h2 : i ≠ v rewrite [swap_other h1 h2, swap_other h1 h2] rfl done done done lemma swap_one_one_below (u v n) : one_one_below n (swap u v) := left_inv_one_one_below (swap_swap u v n) lemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) := right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2) lemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) := And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2)) lemma comp_perm_below {n : Nat} {f g : Nat → Nat} (h1 : perm_below n f) (h2 : perm_below n g) : perm_below n (f ∘ g) := sorry lemma trivial_swap (u : Nat) : swap u u = id := by apply funext fix x : Nat by_cases h1 : x = u · -- Case 1. h1 : x = u rewrite [h1, swap_fst] rfl done · -- Case 2. h1 : x ≠ u rewrite [swap_other h1 h1] rfl done done lemma prod_eq_fun {m : Nat} (f g : Nat → ZMod m) (k : Nat) : ∀ (n : Nat), (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g := by by_induc · -- Base Case assume h : (∀ i < 0, f (k + i) = g (k + i)) rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g assume h1 : ∀ i < n + 1, f (k + i) = g (k + i) have h2 : ∀ i < n, f (k + i) = g (k + i) := by fix i : Nat assume h2 : i < n have h3 : i < n + 1 := by linarith show f (k + i) = g (k + i) from h1 i h3 done have h3 : prod_seq n k f = prod_seq n k g := ih h2 have h4 : n < n + 1 := Nat.lt_succ_self n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4] rfl done done lemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := by have h2 : ∀ (i : Nat), i < u → (f ∘ swap u n) (0 + i) = f (0 + i) := by fix i : Nat assume h2 : i < u have h3 : 0 + i ≠ u := by linarith have h4 : 0 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl done show prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f from prod_eq_fun (f ∘ swap u n) f 0 u h2 done lemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := by have h2 : ∀ i < j, (f ∘ swap u n) (u + 1 + i) = f (u + 1 + i) := by fix i : Nat assume h2 : i < j have h3 : u + 1 + i ≠ u := by linarith have h4 : u + 1 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl show prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f from prod_eq_fun (f ∘ swap u n) f (u + 1) j h2 done lemma break_prod {m : Nat} (n : Nat) (f : Nat → ZMod m) : ∀ (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by by_induc · -- Base Case have h : n + 0 = n := by rfl rewrite [prod_seq_base, h, Theorem_7_3_6_7] rfl done · -- Induction Step fix j : Nat assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f rewrite [←add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc] rfl done done lemma break_prod_twice {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f := break_prod n f 1 rewrite [prod_one] at h2 have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f := break_prod (u + 1) f j rewrite [←h1] at h3 have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f := break_prod u f 1 rewrite [prod_one] at h4 rewrite [h3, h4] at h2 show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2 done lemma swap_prod_eq_prod {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f := by by_cases h2 : u = n · -- Case 1. h2 : u = n rewrite [h2, trivial_swap n] --Goal : prod_seq (n + 1) 0 (f ∘ id) = prod_seq (n + 1) 0 f rfl done · -- Case 2. h2 : ¬u = n have h3 : u + 1 ≤ n := Nat.lt_of_le_of_ne h1 h2 obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3 have break_f : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := break_prod_twice f h4 have break_fs : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_prod_twice (f ∘ swap u n) h4 have f_eq_fs_below : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := swap_prod_eq_prod_below f h1 have f_eq_fs_btwn : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4 show prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f from calc prod_seq (n + 1) 0 (f ∘ swap u n) _ = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_fs _ = prod_seq u 0 f * (f ∘ swap u n) u * prod_seq j (u + 1) f * (f ∘ swap u n) n := by rw [f_eq_fs_below, f_eq_fs_btwn] _ = prod_seq u 0 f * f (swap u n u) * prod_seq j (u + 1) f * f (swap u n n) := by rfl _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by rw [swap_fst, swap_snd] _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring _ = prod_seq (n + 1) 0 f := break_f.symm done done lemma perm_below_fixed {n : Nat} {g : Nat → Nat} (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry lemma perm_prod {m : Nat} (f : Nat → ZMod m) : ∀ (n : Nat), ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) := by by_induc · -- Base Case fix g : Nat → Nat assume h1 : perm_below 0 g rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) fix g : Nat → Nat assume g_pb : perm_below (n + 1) g define at g_pb have g_ob : onto_below (n + 1) g := g_pb.right.right define at g_ob have h1 : n < n + 1 := by linarith obtain (u : Nat) (h2 : u < n + 1 ∧ g u = n) from g_ob n h1 have s_pb : perm_below (n + 1) (swap u n) := swap_perm_below h2.left h1 have gs_pb_n1 : perm_below (n + 1) (g ∘ swap u n) := comp_perm_below g_pb s_pb have gs_fix_n : (g ∘ swap u n) n = n := calc (g ∘ swap u n) n _ = g (swap u n n) := by rfl _ = g u := by rw [swap_snd] _ = n := h2.right have gs_pb_n : perm_below n (g ∘ swap u n) := perm_below_fixed gs_pb_n1 gs_fix_n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f ∘ (g ∘ swap u n)) := ih (g ∘ swap u n) gs_pb_n have h3 : u ≤ n := by linarith show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f ∘ g) from calc prod_seq (n + 1) 0 f _ = prod_seq n 0 f * f n := prod_seq_zero_step n f _ = prod_seq n 0 (f ∘ (g ∘ swap u n)) * f ((g ∘ swap u n) n) := by rw [gs_prod, gs_fix_n] _ = prod_seq n 0 (f ∘ g ∘ swap u n) * (f ∘ g ∘ swap u n) n := by rfl _ = prod_seq (n + 1) 0 (f ∘ g ∘ swap u n) := (prod_seq_zero_step n (f ∘ g ∘ swap u n)).symm _ = prod_seq (n + 1) 0 ((f ∘ g) ∘ swap u n) := by rfl _ = prod_seq (n + 1) 0 (f ∘ g) := swap_prod_eq_prod (f ∘ g) h3 done done lemma F_invertible (m i : Nat) : invertible (F m i) := by by_cases h : rel_prime m i · -- Case 1. h : rel_prime m i rewrite [F_rp_def h] show invertible [i]_m from (Theorem_7_3_7 m i).rtl h done · -- Case 2. h : ¬rel_prime m i rewrite [F_not_rp_def h] apply Exists.intro [1]_m show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m done done lemma Fprod_invertible (m : Nat) : ∀ (k : Nat), invertible (prod_seq k 0 (F m)) := by by_induc · -- Base Case apply Exists.intro [1]_m show prod_seq 0 0 (F m) * [1]_m = [1]_m from calc prod_seq 0 0 (F m) * [1]_m _ = [1]_m * [1]_m := by rw [prod_seq_base] _ = [1]_m := Theorem_7_3_6_7 ([1]_m) done · -- Induction Step fix k : Nat assume ih : invertible (prod_seq k 0 (F m)) rewrite [prod_seq_zero_step] show invertible (prod_seq k 0 (F m) * (F m k)) from (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k) done done theorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m ^ (phi m) = [1]_m := by have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2 show [a]_m ^ (phi m) = [1]_m from calc [a]_m ^ (phi m) _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3] _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring _ = prod_seq m 0 (F m ∘ G m a) * Y := by rw [FG_prod h1 m, phi_def] _ = prod_seq m 0 (F m) * Y := by rw [perm_prod (F m) m (G m a) (G_perm_below h1)] _ = [1]_m := by rw [h3] done lemma Exercise_7_4_5_Int (m : Nat) (a : Int) : ∀ (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry lemma Exercise_7_4_5_Nat (m a n : Nat) : [a]_m ^ n = [a ^ n]_m := by rewrite [Exercise_7_4_5_Int] rfl done theorem Euler's_theorem {m a : Nat} [NeZero m] (h1 : rel_prime m a) : a ^ (phi m) ≡ 1 (MOD m) := by have h2 : [a]_m ^ (phi m) = [1]_m := Theorem_7_4_2 h1 rewrite [Exercise_7_4_5_Nat m a (phi m)] at h2 --h2 : [a ^ phi m]_m = [1]_m show a ^ (phi m) ≡ 1 (MOD m) from (cc_eq_iff_congr _ _ _).ltr h2 done #eval gcd 10 7 --Answer: 1. So 10 and 7 are relatively prime #eval 7 ^ phi 10 --Answer: 2401, which is congruent to 1 mod 10. end Euler /- Section 7.5 -/ lemma num_rp_prime {p : Nat} (h1 : prime p) : ∀ k < p, num_rp_below p (k + 1) = k := sorry lemma phi_prime {p : Nat} (h1 : prime p) : phi p = p - 1 := by have h2 : 1 ≤ p := prime_pos h1 have h3 : p - 1 + 1 = p := Nat.sub_add_cancel h2 have h4 : p - 1 < p := by linarith have h5 : num_rp_below p (p - 1 + 1) = p - 1 := num_rp_prime h1 (p - 1) h4 rewrite [h3] at h5 show phi p = p - 1 from h5 done theorem Theorem_7_2_2_Int {a c : Nat} {b : Int} (h1 : ↑c ∣ ↑a * b) (h2 : rel_prime a c) : ↑c ∣ b := by rewrite [Int.natCast_dvd, Int.natAbs_mul, Int.natAbs_ofNat] at h1 --h1 : c ∣ a * Int.natAbs b rewrite [Int.natCast_dvd] --Goal : c ∣ Int.natAbs b show c ∣ Int.natAbs b from Theorem_7_2_2 h1 h2 done lemma Lemma_7_4_5 {m n : Nat} (a b : Int) (h1 : rel_prime m n) : a ≡ b (MOD m * n) ↔ a ≡ b (MOD m) ∧ a ≡ b (MOD n) := by apply Iff.intro · -- (→) assume h2 : a ≡ b (MOD m * n) obtain (j : Int) (h3 : a - b = (m * n) * j) from h2 apply And.intro · -- Proof of a ≡ b (MOD m) apply Exists.intro (n * j) show a - b = m * (n * j) from calc a - b _ = m * n * j := h3 _ = m * (n * j) := by ring done · -- Proof of a ≡ b (MOD n) apply Exists.intro (m * j) show a - b = n * (m * j) from calc a - b _ = m * n * j := h3 _ = n * (m * j) := by ring done done · -- (←) assume h2 : a ≡ b (MOD m) ∧ a ≡ b (MOD n) obtain (j : Int) (h3 : a - b = m * j) from h2.left have h4 : (↑n : Int) ∣ a - b := h2.right rewrite [h3] at h4 --h4 : ↑n ∣ ↑m * j have h5 : ↑n ∣ j := Theorem_7_2_2_Int h4 h1 obtain (k : Int) (h6 : j = n * k) from h5 apply Exists.intro k --Goal : a - b = ↑(m * n) * k rewrite [Nat.cast_mul] --Goal : a - b = ↑m * ↑n * k show a - b = (m * n) * k from calc a - b _ = m * j := h3 _ = m * (n * k) := by rw [h6] _ = (m * n) * k := by ring done done --From exercises of Section 7.2 theorem rel_prime_symm {a b : Nat} (h : rel_prime a b) : rel_prime b a := sorry lemma prime_NeZero {p : Nat} (h : prime p) : NeZero p := by rewrite [neZero_iff] --Goal : p ≠ 0 define at h linarith done lemma Lemma_7_5_1 {p e d m c s : Nat} {t : Int} (h1 : prime p) (h2 : e * d = (p - 1) * s + 1) (h3 : m ^ e - c = p * t) : c ^ d ≡ m (MOD p) := by have h4 : m ^ e ≡ c (MOD p) := Exists.intro t h3 have h5 : [m ^ e]_p = [c]_p := (cc_eq_iff_congr _ _ _).rtl h4 rewrite [←Exercise_7_4_5_Nat] at h5 --h5 : [m]_p ^ e = [c]_p by_cases h6 : p ∣ m · -- Case 1. h6 : p ∣ m have h7 : m ≡ 0 (MOD p) := by obtain (j : Nat) (h8 : m = p * j) from h6 apply Exists.intro (↑j : Int) --Goal : ↑m - 0 = ↑p * ↑j rewrite [h8, Nat.cast_mul] ring done have h8 : [m]_p = [0]_p := (cc_eq_iff_congr _ _ _).rtl h7 have h9 : e * d ≠ 0 := by rewrite [h2] show (p - 1) * s + 1 ≠ 0 from Nat.add_one_ne_zero _ done have h10 : (0 : Int) ^ (e * d) = 0 := zero_pow h9 have h11 : [c ^ d]_p = [m]_p := calc [c ^ d]_p _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat] _ = ([m]_p ^ e) ^ d := by rw [h5] _ = [m]_p ^ (e * d) := by ring _ = [0]_p ^ (e * d) := by rw [h8] _ = [0 ^ (e * d)]_p := Exercise_7_4_5_Int _ _ _ _ = [0]_p := by rw [h10] _ = [m]_p := by rw [h8] show c ^ d ≡ m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11 done · -- Case 2. h6 : ¬p ∣ m have h7 : rel_prime m p := rel_prime_of_prime_not_dvd h1 h6 have h8 : rel_prime p m := rel_prime_symm h7 have h9 : NeZero p := prime_NeZero h1 have h10 : (1 : Int) ^ s = 1 := by ring have h11 : [c ^ d]_p = [m]_p := calc [c ^ d]_p _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat] _ = ([m]_p ^ e) ^ d := by rw [h5] _ = [m]_p ^ (e * d) := by ring _ = [m]_p ^ ((p - 1) * s + 1) := by rw [h2] _ = ([m]_p ^ (p - 1)) ^ s * [m]_p := by ring _ = ([m]_p ^ (phi p)) ^ s * [m]_p := by rw [phi_prime h1] _ = [1]_p ^ s * [m]_p := by rw [Theorem_7_4_2 h8] _ = [1 ^ s]_p * [m]_p := by rw [Exercise_7_4_5_Int] _ = [1]_p * [m]_p := by rw [h10] _ = [m]_p * [1]_p := by ring _ = [m]_p := Theorem_7_3_6_7 _ show c ^ d ≡ m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11 done done theorem Theorem_7_5_1 (p q n e d k m c : Nat) (p_prime : prime p) (q_prime : prime q) (p_ne_q : p ≠ q) (n_pq : n = p * q) (ed_congr_1 : e * d = k * (p - 1) * (q - 1) + 1) (h1 : [m]_n ^ e = [c]_n) : [c]_n ^ d = [m]_n := by rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] at h1 --h1 : m ^ e ≡ c (MOD n) rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] --Goal : c ^ d ≡ m (MOD n) obtain (j : Int) (h2 : m ^ e - c = n * j) from h1 rewrite [n_pq, Nat.cast_mul] at h2 --h2 : m ^ e - c = p * q * j have h3 : e * d = (p - 1) * (k * (q - 1)) + 1 := by rewrite [ed_congr_1] ring done have h4 : m ^ e - c = p * (q * j) := by rewrite [h2] ring done have congr_p : c ^ d ≡ m (MOD p) := Lemma_7_5_1 p_prime h3 h4 have h5 : e * d = (q - 1) * (k * (p - 1)) + 1 := by rewrite [ed_congr_1] ring done have h6 : m ^ e - c = q * (p * j) := by rewrite [h2] ring done have congr_q : c ^ d ≡ m (MOD q) := Lemma_7_5_1 q_prime h5 h6 have h7 : ¬q ∣ p := by by_contra h8 have h9 : q = 1 ∨ q = p := dvd_prime p_prime h8 disj_syll h9 (prime_not_one q_prime) show False from p_ne_q h9.symm done have h8 : rel_prime p q := rel_prime_of_prime_not_dvd q_prime h7 rewrite [n_pq, Lemma_7_4_5 _ _ h8] show c ^ d ≡ m (MOD p) ∧ c ^ d ≡ m (MOD q) from And.intro congr_p congr_q done /- BEGIN EXERCISES -/ namespace Exercises /- Section 7.1 -/ -- 1. theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry -- 2. lemma gcd_comm_lt {a b : Nat} (h : a < b) : gcd a b = gcd b a := sorry theorem gcd_comm (a b : Nat) : gcd a b = gcd b a := sorry -- 3. theorem Exercise_7_1_5 (a b : Nat) (n : Int) : (∃ (s t : Int), s * a + t * b = n) ↔ (↑(gcd a b) : Int) ∣ n := sorry -- 4. theorem Exercise_7_1_6 (a b c : Nat) : gcd a b = gcd (a + b * c) b := sorry -- 5. theorem gcd_is_nonzero {a b : Nat} (h : a ≠ 0 ∨ b ≠ 0) : gcd a b ≠ 0 := sorry -- 6. theorem gcd_greatest {a b d : Nat} (h1 : gcd a b ≠ 0) (h2 : d ∣ a) (h3 : d ∣ b) : d ≤ gcd a b := sorry -- 7. lemma Lemma_7_1_10a {a b : Nat} (n : Nat) (h : a ∣ b) : (n * a) ∣ (n * b) := sorry lemma Lemma_7_1_10b {a b n : Nat} (h1 : n ≠ 0) (h2 : (n * a) ∣ (n * b)) : a ∣ b := sorry lemma Lemma_7_1_10c {a b : Nat} (h1 : a ∣ b) (h2 : b ∣ a) : a = b := sorry theorem Exercise_7_1_10 (a b n : Nat) : gcd (n * a) (n * b) = n * gcd a b := sorry /- Section 7.2 -/ -- 1. lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry -- 2. -- Hints: Start with apply List.rec. You may find mul_ne_zero useful theorem prod_nonzero_nonzero : ∀ (l : List Nat), (∀ a ∈ l, a ≠ 0) → prod l ≠ 0 := sorry -- 3. theorem rel_prime_iff_no_common_factor (a b : Nat) : rel_prime a b ↔ ¬∃ (p : Nat), prime p ∧ p ∣ a ∧ p ∣ b := sorry -- 4. theorem rel_prime_symm {a b : Nat} (h : rel_prime a b) : rel_prime b a := sorry -- 5. lemma in_prime_factorization_iff_prime_factor {a : Nat} {l : List Nat} (h1 : prime_factorization a l) (p : Nat) : p ∈ l ↔ prime_factor p a := sorry -- 6.

theorem Exercise_7_2_5 {a b : Nat} {l m : List Nat} (h1 : prime_factorization a l) (h2 : prime_factorization b m) : rel_prime a b ↔ (¬∃ (p : Nat), p ∈ l ∧ p ∈ m)

HTPI.Exercises.Exercise_7_2_5

htpi/HTPILib/Chap7.lean

htpi

/- Copyright 2023 Daniel J. Velleman -/ import HTPILib.Chap6 namespace HTPI /- Definitions -/ lemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by have h : n + 1 > 0 := Nat.succ_pos n show a % (n + 1) < n + 1 from Nat.mod_lt a h done def gcd (a b : Nat) : Nat := match b with | 0 => a | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd (n + 1) (a % (n + 1)) termination_by b mutual def gcd_c1 (a b : Nat) : Int := match b with | 0 => 1 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c2 (n + 1) (a % (n + 1)) --Corresponds to s = t' termination_by b def gcd_c2 (a b : Nat) : Int := match b with | 0 => 0 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c1 (n + 1) (a % (n + 1)) - (gcd_c2 (n + 1) (a % (n + 1))) * ↑(a / (n + 1)) --Corresponds to t = s' - t'q termination_by b end def prime (n : Nat) : Prop := 2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n def prime_factor (p n : Nat) : Prop := prime p ∧ p ∣ n def all_prime (l : List Nat) : Prop := ∀ p ∈ l, prime p def nondec (l : List Nat) : Prop := match l with | [] => True --Of course, True is a proposition that is always true | n :: L => (∀ m ∈ L, n ≤ m) ∧ nondec L def nondec_prime_list (l : List Nat) : Prop := all_prime l ∧ nondec l def prod (l : List Nat) : Nat := match l with | [] => 1 | n :: L => n * (prod L) def prime_factorization (n : Nat) (l : List Nat) : Prop := nondec_prime_list l ∧ prod l = n def rel_prime (a b : Nat) : Prop := gcd a b = 1 def congr_mod (m : Nat) (a b : Int) : Prop := (↑m : Int) ∣ (a - b) def cc (m : Nat) (a : Int) : ZMod m := (↑a : ZMod m) notation:50 a " ≡ " b " (MOD " m ")" => congr_mod m a b notation:max "["a"]_"m:max => cc m a def invertible {m : Nat} (X : ZMod m) : Prop := ∃ (Y : ZMod m), X * Y = [1]_m def num_rp_below (m k : Nat) : Nat := match k with | 0 => 0 | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j def phi (m : Nat) : Nat := num_rp_below m m def prod_seq {m : Nat} (j k : Nat) (f : Nat → ZMod m) : ZMod m := match j with | 0 => [1]_m | n + 1 => prod_seq n k f * f (k + n) def maps_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i < n, g i < n def one_one_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i1 < n, ∀ i2 < n, g i1 = g i2 → i1 = i2 def onto_below (n : Nat) (g : Nat → Nat) : Prop := ∀ k < n, ∃ i < n, g i = k def perm_below (n : Nat) (g : Nat → Nat) : Prop := maps_below n g ∧ one_one_below n g ∧ onto_below n g def inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m) def swap (u v i : Nat) : Nat := if i = u then v else if i = v then u else i namespace Euler --For definitions specific to Euler's theorem def F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m def G (m a i : Nat) : Nat := (a * i) % m def Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i end Euler /- Section 7.1 -/ theorem dvd_mod_of_dvd_a_b {a b d : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ (a % b) := by set q : Nat := a / b have h3 : b * q + a % b = a := Nat.div_add_mod a b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 define --Goal : ∃ (c : Nat), a % b = d * c apply Exists.intro (j - k * q) show a % b = d * (j - k * q) from calc a % b _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm _ = a - b * q := by rw [h3] _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc] _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm done theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry #eval gcd 672 161 --Answer: 7 lemma gcd_base (a : Nat) : gcd a 0 = a := by rfl lemma gcd_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd a b = gcd b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1)) rfl done lemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b ≠ 0) : a % b < b := by have h1 : b > 0 := Nat.pos_of_ne_zero h show a % b < b from Nat.mod_lt a h1 done lemma dvd_self (n : Nat) : n ∣ n := by apply Exists.intro 1 ring done theorem gcd_dvd : ∀ (b a : Nat), (gcd a b) ∣ a ∧ (gcd a b) ∣ b := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd a b_1) ∣ a ∧ (gcd a b_1) ∣ b_1 fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_base] --Goal: a ∣ a ∧ a ∣ 0 apply And.intro (dvd_self a) define apply Exists.intro 0 rfl done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_nonzero a h1] --Goal : gcd b (a % b) ∣ a ∧ gcd b (a % b) ∣ b have h2 : a % b < b := mod_nonzero_lt a h1 have h3 : (gcd b (a % b)) ∣ b ∧ (gcd b (a % b)) ∣ (a % b) := ih (a % b) h2 b apply And.intro _ h3.left show (gcd b (a % b)) ∣ a from dvd_a_of_dvd_b_mod h3.left h3.right done done theorem gcd_dvd_left (a b : Nat) : (gcd a b) ∣ a := (gcd_dvd b a).left theorem gcd_dvd_right (a b : Nat) : (gcd a b) ∣ b := (gcd_dvd b a).right lemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl lemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c1 a b = gcd_c2 b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done lemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl lemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * ↑(a / b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done theorem gcd_lin_comb : ∀ (b a : Nat), (gcd_c1 a b) * ↑a + (gcd_c2 a b) * ↑b = ↑(gcd a b) := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd_c1 a b_1) * ↑a + (gcd_c2 a b_1) * ↑b_1 = ↑(gcd a b_1) fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base] --Goal : 1 * ↑a + 0 * ↑0 = ↑a ring done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1] --Goal : gcd_c2 b (a % b) * ↑a + -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * ↑(a / b)) * ↑b = -- ↑(gcd b (a % b)) set r : Nat := a % b set q : Nat := a / b set s : Int := gcd_c1 b r set t : Int := gcd_c2 b r --Goal : t * ↑a + (s - t * ↑q) * ↑b = ↑(gcd b r) have h2 : r < b := mod_nonzero_lt a h1 have h3 : s * ↑b + t * ↑r = ↑(gcd b r) := ih r h2 b have h4 : b * q + r = a := Nat.div_add_mod a b rewrite [←h3, ←h4] rewrite [Nat.cast_add, Nat.cast_mul] --Goal : t * (↑b * ↑q + ↑r) + (s - t * ↑q) * ↑b = s * ↑b + t * ↑r ring done done #eval gcd_c1 672 161 --Answer: 6 #eval gcd_c2 672 161 --Answer: -25 --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161 theorem Theorem_7_1_6 {d a b : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ gcd a b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑d ∣ ↑(gcd a b) set s : Int := gcd_c1 a b set t : Int := gcd_c2 a b have h3 : s * ↑a + t * ↑b = ↑(gcd a b) := gcd_lin_comb b a rewrite [←h3] --Goal : ↑d ∣ s * ↑a + t * ↑b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul] --Goal : ↑d ∣ s * (↑d * ↑j) + t * (↑d * ↑k) define apply Exists.intro (s * ↑j + t * ↑k) ring done /- Section 7.2 -/ theorem dvd_trans {a b c : Nat} (h1 : a ∣ b) (h2 : b ∣ c) : a ∣ c := by define at h1; define at h2; define obtain (m : Nat) (h3 : b = a * m) from h1 obtain (n : Nat) (h4 : c = b * n) from h2 rewrite [h3, mul_assoc] at h4 apply Exists.intro (m * n) show c = a * (m * n) from h4 done lemma exists_prime_factor : ∀ (n : Nat), 2 ≤ n → ∃ (p : Nat), prime_factor p n := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, 2 ≤ n_1 → ∃ (p : Nat), prime_factor p n_1 assume h1 : 2 ≤ n by_cases h2 : prime n · -- Case 1. h2 : prime n apply Exists.intro n define --Goal : prime n ∧ n ∣ n show prime n ∧ n ∣ n from And.intro h2 (dvd_self n) done · -- Case 2. h2 : ¬prime n define at h2 --h2 : ¬(2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n) demorgan at h2 disj_syll h2 h1 obtain (a : Nat) (h3 : ∃ (b : Nat), a * b = n ∧ a < n ∧ b < n) from h2 obtain (b : Nat) (h4 : a * b = n ∧ a < n ∧ b < n) from h3 have h5 : 2 ≤ a := by by_contra h6 have h7 : a ≤ 1 := by linarith have h8 : n ≤ b := calc n _ = a * b := h4.left.symm _ ≤ 1 * b := by rel [h7] _ = b := by ring linarith --n ≤ b contradicts b < n done have h6 : ∃ (p : Nat), prime_factor p a := ih a h4.right.left h5 obtain (p : Nat) (h7 : prime_factor p a) from h6 apply Exists.intro p define --Goal : prime p ∧ p ∣ n define at h7 --h7 : prime p ∧ p ∣ a apply And.intro h7.left have h8 : a ∣ n := by apply Exists.intro b show n = a * b from (h4.left).symm done show p ∣ n from dvd_trans h7.right h8 done done lemma exists_least_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q := by set S : Set Nat := {p : Nat | prime_factor p n} have h2 : ∃ (p : Nat), p ∈ S := exists_prime_factor n h show ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q from well_ord_princ S h2 done lemma all_prime_nil : all_prime [] := by define --Goal : ∀ p ∈ [], prime p fix p : Nat contrapos --Goal : ¬prime p → p ∉ [] assume h1 : ¬prime p show p ∉ [] from List.not_mem_nil p done lemma all_prime_cons (n : Nat) (L : List Nat) : all_prime (n :: L) ↔ prime n ∧ all_prime L := by apply Iff.intro · -- (→) assume h1 : all_prime (n :: L) --Goal : prime n ∧ all_prime L define at h1 --h1 : ∀ p ∈ n :: L, prime p apply And.intro (h1 n (List.mem_cons_self n L)) define --Goal : ∀ p ∈ L, prime p fix p : Nat assume h2 : p ∈ L show prime p from h1 p (List.mem_cons_of_mem n h2) done · -- (←) assume h1 : prime n ∧ all_prime L --Goal : all_prime (n :: l) define : all_prime L at h1 define fix p : Nat assume h2 : p ∈ n :: L rewrite [List.mem_cons] at h2 --h2 : p = n ∨ p ∈ L by_cases on h2 · -- Case 1. h2 : p = n rewrite [h2] show prime n from h1.left done · -- Case 2. h2 : p ∈ L show prime p from h1.right p h2 done done done lemma nondec_nil : nondec [] := by define --Goal : True trivial --trivial proves some obviously true statements, such as True done lemma nondec_cons (n : Nat) (L : List Nat) : nondec (n :: L) ↔ (∀ m ∈ L, n ≤ m) ∧ nondec L := by rfl lemma prod_nil : prod [] = 1 := by rfl lemma prod_cons : prod (n :: L) = n * (prod L) := by rfl lemma exists_cons_of_length_eq_succ {A : Type} {l : List A} {n : Nat} (h : l.length = n + 1) : ∃ (a : A) (L : List A), l = a :: L ∧ L.length = n := by have h1 : ¬l.length = 0 := by linarith rewrite [List.length_eq_zero] at h1 obtain (a : A) (h2 : ∃ (L : List A), l = a :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List A) (h3 : l = a :: L) from h2 apply Exists.intro a apply Exists.intro L apply And.intro h3 have h4 : (a :: L).length = L.length + 1 := List.length_cons a L rewrite [←h3, h] at h4 show L.length = n from (Nat.add_right_cancel h4).symm done lemma list_elt_dvd_prod_by_length (a : Nat) : ∀ (n : Nat), ∀ (l : List Nat), l.length = n → a ∈ l → a ∣ prod l := by by_induc · --Base Case fix l : List Nat assume h1 : l.length = 0 rewrite [List.length_eq_zero] at h1 --h1 : l = [] rewrite [h1] --Goal : a ∈ [] → a ∣ prod [] contrapos assume h2 : ¬a ∣ prod [] show a ∉ [] from List.not_mem_nil a done · -- Induction Step fix n : Nat assume ih : ∀ (l : List Nat), List.length l = n → a ∈ l → a ∣ prod l fix l : List Nat assume h1 : l.length = n + 1 --Goal : a ∈ l → a ∣ prod l obtain (b : Nat) (h2 : ∃ (L : List Nat), l = b :: L ∧ L.length = n) from exists_cons_of_length_eq_succ h1 obtain (L : List Nat) (h3 : l = b :: L ∧ L.length = n) from h2 have h4 : a ∈ L → a ∣ prod L := ih L h3.right assume h5 : a ∈ l rewrite [h3.left, prod_cons] --Goal : a ∣ b * prod L rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b ∨ a ∈ L by_cases on h5 · -- Case 1. h5 : a = b apply Exists.intro (prod L) rewrite [h5] rfl done · -- Case 2. h5 : a ∈ L have h6 : a ∣ prod L := h4 h5 have h7 : prod L ∣ b * prod L := by apply Exists.intro b ring done show a ∣ b * prod L from dvd_trans h6 h7 done done done lemma list_elt_dvd_prod {a : Nat} {l : List Nat} (h : a ∈ l) : a ∣ prod l := by set n : Nat := l.length have h1 : l.length = n := by rfl show a ∣ prod l from list_elt_dvd_prod_by_length a n l h1 h done lemma exists_prime_factorization : ∀ (n : Nat), n ≥ 1 → ∃ (l : List Nat), prime_factorization n l := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, n_1 ≥ 1 → ∃ (l : List Nat), prime_factorization n_1 l assume h1 : n ≥ 1 by_cases h2 : n = 1 · -- Case 1. h2 : n = 1 apply Exists.intro [] define apply And.intro · -- Proof of nondec_prime_list [] define show all_prime [] ∧ nondec [] from And.intro all_prime_nil nondec_nil done · -- Proof of prod [] = n rewrite [prod_nil, h2] rfl done done · -- Case 2. h2 : n ≠ 1 have h3 : n ≥ 2 := lt_of_le_of_ne' h1 h2 obtain (p : Nat) (h4 : prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q) from exists_least_prime_factor h3 have p_prime_factor : prime_factor p n := h4.left define at p_prime_factor have p_prime : prime p := p_prime_factor.left have p_dvd_n : p ∣ n := p_prime_factor.right have p_least : ∀ (q : Nat), prime_factor q n → p ≤ q := h4.right obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n have h5 : m ≠ 0 := by contradict h1 with h6 have h7 : n = 0 := calc n _ = p * m := n_eq_pm _ = p * 0 := by rw [h6] _ = 0 := by ring rewrite [h7] decide done have m_pos : 0 < m := Nat.pos_of_ne_zero h5 have m_lt_n : m < n := by define at p_prime show m < n from calc m _ < m + m := by linarith _ = 2 * m := by ring _ ≤ p * m := by rel [p_prime.left] _ = n := n_eq_pm.symm done obtain (L : List Nat) (h6 : prime_factorization m L) from ih m m_lt_n m_pos define at h6 have ndpl_L : nondec_prime_list L := h6.left define at ndpl_L apply Exists.intro (p :: L) define apply And.intro · -- Proof of nondec_prime_list (p :: L) define apply And.intro · -- Proof of all_prime (p :: L) rewrite [all_prime_cons] show prime p ∧ all_prime L from And.intro p_prime ndpl_L.left done · -- Proof of nondec (p :: L) rewrite [nondec_cons] apply And.intro _ ndpl_L.right fix q : Nat assume q_in_L : q ∈ L have h7 : q ∣ prod L := list_elt_dvd_prod q_in_L rewrite [h6.right] at h7 --h7 : q ∣ m have h8 : m ∣ n := by apply Exists.intro p rewrite [n_eq_pm] ring done have q_dvd_n : q ∣ n := dvd_trans h7 h8 have ap_L : all_prime L := ndpl_L.left define at ap_L have q_prime_factor : prime_factor q n := And.intro (ap_L q q_in_L) q_dvd_n show p ≤ q from p_least q q_prime_factor done done · -- Proof of prod (p :: L) = n rewrite [prod_cons, h6.right, n_eq_pm] rfl done done done theorem Theorem_7_2_2 {a b c : Nat} (h1 : c ∣ a * b) (h2 : rel_prime a c) : c ∣ b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑c ∣ ↑b define at h1; define at h2; define obtain (j : Nat) (h3 : a * b = c * j) from h1 set s : Int := gcd_c1 a c set t : Int := gcd_c2 a c have h4 : s * ↑a + t * ↑c = ↑(gcd a c) := gcd_lin_comb c a rewrite [h2, Nat.cast_one] at h4 --h4 : s * ↑a + t * ↑c = (1 : Int) apply Exists.intro (s * ↑j + t * ↑b) show ↑b = ↑c * (s * ↑j + t * ↑b) from calc ↑b _ = (1 : Int) * ↑b := (one_mul _).symm _ = (s * ↑a + t * ↑c) * ↑b := by rw [h4] _ = s * (↑a * ↑b) + t * ↑c * ↑b := by ring _ = s * (↑c * ↑j) + t * ↑c * ↑b := by rw [←Nat.cast_mul a b, h3, Nat.cast_mul c j] _ = ↑c * (s * ↑j + t * ↑b) := by ring done lemma le_nonzero_prod_left {a b : Nat} (h : a * b ≠ 0) : a ≤ a * b := by have h1 : b ≠ 0 := by contradict h with h1 rewrite [h1] ring done have h2 : 1 ≤ b := Nat.pos_of_ne_zero h1 show a ≤ a * b from calc a = a * 1 := (mul_one a).symm _ ≤ a * b := by rel [h2] done lemma le_nonzero_prod_right {a b : Nat} (h : a * b ≠ 0) : b ≤ a * b := by rewrite [mul_comm] rewrite [mul_comm] at h show b ≤ b * a from le_nonzero_prod_left h done lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry lemma rel_prime_of_prime_not_dvd {a p : Nat} (h1 : prime p) (h2 : ¬p ∣ a) : rel_prime a p := by have h3 : gcd a p ∣ a := gcd_dvd_left a p have h4 : gcd a p ∣ p := gcd_dvd_right a p have h5 : gcd a p = 1 ∨ gcd a p = p := dvd_prime h1 h4 have h6 : gcd a p ≠ p := by contradict h2 with h6 rewrite [h6] at h3 show p ∣ a from h3 done disj_syll h5 h6 show rel_prime a p from h5 done theorem Theorem_7_2_3 {a b p : Nat} (h1 : prime p) (h2 : p ∣ a * b) : p ∣ a ∨ p ∣ b := by or_right with h3 have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3 show p ∣ b from Theorem_7_2_2 h2 h4 done lemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a ≥ 1 := by have h1 : a ≠ 0 := by by_contra h1 rewrite [h1] at h contradict h linarith done show a ≥ 1 from Nat.pos_of_ne_zero h1 done lemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by have h1 : a ≥ 1 := ge_one_of_prod_one h have h2 : a * b ≠ 0 := by linarith have h3 : a ≤ a * b := le_nonzero_prod_left h2 rewrite [h] at h3 show a = 1 from Nat.le_antisymm h3 h1 done lemma eq_one_of_dvd_one {n : Nat} (h : n ∣ 1) : n = 1 := by obtain (j : Nat) (h1 : 1 = n * j) from h show n = 1 from eq_one_of_prod_one h1.symm done lemma prime_not_one {p : Nat} (h : prime p) : p ≠ 1 := by define at h linarith done theorem Theorem_7_2_4 {p : Nat} (h1 : prime p) : ∀ (l : List Nat), p ∣ prod l → ∃ a ∈ l, p ∣ a := by apply List.rec · -- Base Case. Goal : p ∣ prod [] → ∃ a ∈ [], p ∣ a rewrite [prod_nil] assume h2 : p ∣ 1 show ∃ a ∈ [], p ∣ a from absurd (eq_one_of_dvd_one h2) (prime_not_one h1) done · -- Induction Step fix b : Nat fix L : List Nat assume ih : p ∣ prod L → ∃ a ∈ L, p ∣ a --Goal : p ∣ prod (b :: L) → ∃ a ∈ b :: L, p ∣ a assume h2 : p ∣ prod (b :: L) rewrite [prod_cons] at h2 have h3 : p ∣ b ∨ p ∣ prod L := Theorem_7_2_3 h1 h2 by_cases on h3 · -- Case 1. h3 : p ∣ b apply Exists.intro b show b ∈ b :: L ∧ p ∣ b from And.intro (List.mem_cons_self b L) h3 done · -- Case 2. h3 : p ∣ prod L obtain (a : Nat) (h4 : a ∈ L ∧ p ∣ a) from ih h3 apply Exists.intro a show a ∈ b :: L ∧ p ∣ a from And.intro (List.mem_cons_of_mem b h4.left) h4.right done done done lemma prime_in_list {p : Nat} {l : List Nat} (h1 : prime p) (h2 : all_prime l) (h3 : p ∣ prod l) : p ∈ l := by obtain (a : Nat) (h4 : a ∈ l ∧ p ∣ a) from Theorem_7_2_4 h1 l h3 define at h2 have h5 : prime a := h2 a h4.left have h6 : p = 1 ∨ p = a := dvd_prime h5 h4.right disj_syll h6 (prime_not_one h1) rewrite [h6] show a ∈ l from h4.left done lemma first_le_first {p q : Nat} {l m : List Nat} (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m)) (h3 : prod (p :: l) = prod (q :: m)) : p ≤ q := by define at h1; define at h2 have h4 : q ∣ prod (p :: l) := by define apply Exists.intro (prod m) rewrite [←prod_cons] show prod (p :: l) = prod (q :: m) from h3 done have h5 : all_prime (q :: m) := h2.left rewrite [all_prime_cons] at h5 have h6 : q ∈ p :: l := prime_in_list h5.left h1.left h4 have h7 : nondec (p :: l) := h1.right rewrite [nondec_cons] at h7 rewrite [List.mem_cons] at h6 by_cases on h6 · -- Case 1. h6 : q = p linarith done · -- Case 2. h6 : q ∈ l have h8 : ∀ m ∈ l, p ≤ m := h7.left show p ≤ q from h8 q h6 done done lemma nondec_prime_list_tail {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by define at h define rewrite [all_prime_cons, nondec_cons] at h show all_prime l ∧ nondec l from And.intro h.left.right h.right.right done lemma cons_prod_not_one {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : prod (p :: l) ≠ 1 := by define at h have h1 : all_prime (p :: l) := h.left rewrite [all_prime_cons] at h1 rewrite [prod_cons] by_contra h2 show False from (prime_not_one h1.left) (eq_one_of_prod_one h2) done lemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) : l = [] ↔ prod l = 1 := by apply Iff.intro · -- (→) assume h1 : l = [] rewrite [h1] show prod [] = 1 from prod_nil done · -- (←) contrapos assume h1 : ¬l = [] obtain (p : Nat) (h2 : ∃ (L : List Nat), l = p :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List Nat) (h3 : l = p :: L) from h2 rewrite [h3] at h rewrite [h3] show ¬prod (p :: L) = 1 from cons_prod_not_one h done done lemma prime_pos {p : Nat} (h : prime p) : p > 0 := by define at h linarith done theorem Theorem_7_2_5 : ∀ (l1 l2 : List Nat), nondec_prime_list l1 → nondec_prime_list l2 → prod l1 = prod l2 → l1 = l2 := by apply List.rec · -- Base Case. Goal : ∀ (l2 : List Nat), nondec_prime_list [] → -- nondec_prime_list l2 → prod [] = prod l2 → [] = l2 fix l2 : List Nat assume h1 : nondec_prime_list [] assume h2 : nondec_prime_list l2 assume h3 : prod [] = prod l2 rewrite [prod_nil, eq_comm, ←list_nil_iff_prod_one h2] at h3 show [] = l2 from h3.symm done · -- Induction Step fix p : Nat fix L1 : List Nat assume ih : ∀ (L2 : List Nat), nondec_prime_list L1 → nondec_prime_list L2 → prod L1 = prod L2 → L1 = L2 -- Goal : ∀ (l2 : List Nat), nondec_prime_list (p :: L1) → -- nondec_prime_list l2 → prod (p :: L1) = prod l2 → p :: L1 = l2 fix l2 : List Nat assume h1 : nondec_prime_list (p :: L1) assume h2 : nondec_prime_list l2 assume h3 : prod (p :: L1) = prod l2 have h4 : ¬prod (p :: L1) = 1 := cons_prod_not_one h1 rewrite [h3, ←list_nil_iff_prod_one h2] at h4 obtain (q : Nat) (h5 : ∃ (L : List Nat), l2 = q :: L) from List.exists_cons_of_ne_nil h4 obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5 rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2) rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2) have h7 : p ≤ q := first_le_first h1 h2 h3 have h8 : q ≤ p := first_le_first h2 h1 h3.symm have h9 : p = q := by linarith rewrite [h9, prod_cons, prod_cons] at h3 --h3 : q * prod L1 = q * prod L2 have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1 have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2 define at h2 have h12 : all_prime (q :: L2) := h2.left rewrite [all_prime_cons] at h12 have h13 : q > 0 := prime_pos h12.left have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3 have h15 : L1 = L2 := ih L2 h10 h11 h14 rewrite [h6, h9, h15] rfl done done theorem fund_thm_arith (n : Nat) (h : n ≥ 1) : ∃! (l : List Nat), prime_factorization n l := by exists_unique · -- Existence show ∃ (l : List Nat), prime_factorization n l from exists_prime_factorization n h done · -- Uniqueness fix l1 : List Nat; fix l2 : List Nat assume h1 : prime_factorization n l1 assume h2 : prime_factorization n l2 define at h1; define at h2 have h3 : prod l1 = n := h1.right rewrite [←h2.right] at h3 show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3 done done /- Section 7.3 -/ theorem congr_refl (m : Nat) : ∀ (a : Int), a ≡ a (MOD m) := by fix a : Int define --Goal : ∃ (c : Int), a - a = ↑m * c apply Exists.intro 0 ring done theorem congr_symm {m : Nat} : ∀ {a b : Int}, a ≡ b (MOD m) → b ≡ a (MOD m) := by fix a : Int; fix b : Int assume h1 : a ≡ b (MOD m) define at h1 --h1 : ∃ (c : Int), a - b = ↑m * c define --Goal : ∃ (c : Int), b - a = ↑m * c obtain (c : Int) (h2 : a - b = m * c) from h1 apply Exists.intro (-c) show b - a = m * (-c) from calc b - a _ = -(a - b) := by ring _ = -(m * c) := by rw [h2] _ = m * (-c) := by ring done theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry /- Fundamental properties of congruence classes -/ lemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) : X = Y ↔ X.val = Y.val := Fin.ext_iff lemma val_nat_eq_mod (n k : Nat) : ([k]_(n + 1)).val = k % (n + 1) := by rfl lemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl theorem cc_rep {m : Nat} (X : ZMod m) : ∃ (a : Int), X = [a]_m := match m with | 0 => by apply Exists.intro X rfl done | n + 1 => by apply Exists.intro ↑(X.val) have h1 : X.val < n + 1 := Fin.prop X rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1] rfl done theorem add_class (m : Nat) (a b : Int) : [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm theorem mul_class (m : Nat) (a b : Int) : [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm lemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ [a - b]_m = [0]_m := by apply Iff.intro · -- (→) assume h1 : [a]_m = [b]_m have h2 : a - b = a + (-b) := by ring have h3 : b + (-b) = 0 := by ring show [a - b]_m = [0]_m from calc [a - b]_m _ = [a + (-b)]_m := by rw [h2] _ = [a]_m + [-b]_m := by rw [add_class] _ = [b]_m + [-b]_m := by rw [h1] _ = [b + -b]_m := by rw [add_class] _ = [0]_m := by rw [h3] done · -- (←) assume h1 : [a - b]_m = [0]_m have h2 : b + (a - b) = a := by ring have h3 : b + 0 = b := by ring show [a]_m = [b]_m from calc [a]_m _ = [b + (a - b)]_m := by rw [h2] _ = [b]_m + [a - b]_m := by rw [add_class] _ = [b]_m + [0]_m := by rw [h1] _ = [b + 0]_m := by rw [add_class] _ = [b]_m := by rw [h3] done done lemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) : [a]_m = [0]_m → [-a]_m = [0]_m := by assume h1 : [a]_m = [0]_m have h2 : 0 + (-a) = -a := by ring have h3 : a + (-a) = 0 := by ring show [-a]_m = [0]_m from calc [-a]_m _ = [0 + (-a)]_m := by rw [h2] _ = [0]_m + [-a]_m := by rw [add_class] _ = [a]_m + [-a]_m := by rw [h1] _ = [a + (-a)]_m := by rw [add_class] _ = [0]_m := by rw [h3] done lemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) : [-a]_m = [0]_m ↔ [a]_m = [0]_m := by apply Iff.intro _ (cc_neg_zero_of_cc_zero m a) assume h1 : [-a]_m = [0]_m have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1 have h3 : -(-a) = a := by ring rewrite [h3] at h2 show [a]_m = [0]_m from h2 done lemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl lemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m ↔ m ∣ k := match m with | 0 => by have h : (0 : Int) = (↑(0 : Nat) : Int) := by rfl rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj] apply Iff.intro · -- (→) assume h1 : k = 0 rewrite [h1] show 0 ∣ 0 from dvd_self 0 done · -- (←) assume h1 : 0 ∣ k obtain (c : Nat) (h2 : k = 0 * c) from h1 rewrite [h2] ring done done | n + 1 => by rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero] show k % (n + 1) = 0 ↔ n + 1 ∣ k from (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm done lemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m ↔ ↑m ∣ a := by obtain (k : Nat) (h1 : a = ↑k ∨ a = -↑k) from Int.eq_nat_or_neg a by_cases on h1 · -- Case 1. h1: a = ↑k rewrite [h1, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done · -- Case 2. h1: a = -↑k rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done done theorem cc_eq_iff_congr (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ a ≡ b (MOD m) := calc [a]_m = [b]_m _ ↔ [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b _ ↔ ↑m ∣ (a - b) := cc_zero_iff_dvd m (a - b) _ ↔ a ≡ b (MOD m) := by rfl /- End of fundamental properties of congruence classes -/ lemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m := by have h1 : (↑m : Int) ≠ 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m) show 0 ≤ a % m from Int.emod_nonneg a h1 done lemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m) have h2 : (↑m : Int) > 0 := (Nat.cast_pos).rtl h1 show a % m < m from Int.emod_lt_of_pos a h2 done lemma congr_mod_mod (m : Nat) (a : Int) : a ≡ a % m (MOD m) := by define have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m apply Exists.intro (a / m) show a - a % m = m * (a / m) from calc a - (a % m) _ = m * (a / m) + a % m - a % m := by rw [h1] _ = m * (a / m) := by ring done lemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) := And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a)) theorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) : ∃! (r : Int), 0 ≤ r ∧ r < m ∧ a ≡ r (MOD m) := by exists_unique · -- Existence apply Exists.intro (a % m) show 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) from mod_cmpl_res m a done · -- Uniqueness fix r1 : Int; fix r2 : Int assume h1 : 0 ≤ r1 ∧ r1 < m ∧ a ≡ r1 (MOD m) assume h2 : 0 ≤ r2 ∧ r2 < m ∧ a ≡ r2 (MOD m) have h3 : r1 ≡ r2 (MOD m) := congr_trans (congr_symm h1.right.right) h2.right.right obtain (d : Int) (h4 : r1 - r2 = m * d) from h3 have h5 : r1 - r2 < m * 1 := by linarith have h6 : m * (-1) < r1 - r2 := by linarith rewrite [h4] at h5 --h5 : m * d < m * 1 rewrite [h4] at h6 --h6 : m * -1 < m * d have h7 : (↑m : Int) ≥ 0 := Nat.cast_nonneg m have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7 have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7 have h10 : d = 0 := by linarith show r1 = r2 from calc r1 _ = r1 - r2 + r2 := by ring _ = m * 0 + r2 := by rw [h4, h10] _ = r2 := by ring done done lemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m := (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a) theorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y rewrite [h1, h2] have h3 : a + b = b + a := by ring show [a]_m + [b]_m = [b]_m + [a]_m from calc [a]_m + [b]_m _ = [a + b]_m := add_class m a b _ = [b + a]_m := by rw [h3] _ = [b]_m + [a]_m := (add_class m b a).symm done theorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X rewrite [h1] have h2 : a * 1 = a := by ring show [a]_m * [1]_m = [a]_m from calc [a]_m * [1]_m _ = [a * 1]_m := mul_class m a 1 _ = [a]_m := by rw [h2] done theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry lemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) : [a]_m * [gcd_c2 m a]_m = [1]_m := by set s : Int := gcd_c1 m a have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m define at h1 rewrite [h1, Nat.cast_one] at h2 --h2 : s * ↑m + gcd_c2 m a * ↑a = 1 rewrite [mul_class, cc_eq_iff_congr] define --Goal : ∃ (c : Int), ↑a * gcd_c2 m a - 1 = ↑m * c apply Exists.intro (-s) show a * (gcd_c2 m a) - 1 = m * (-s) from calc a * (gcd_c2 m a) - 1 _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring _ = 1 + m * (-s) - 1 := by rw [h2] _ = m * (-s) := by ring done theorem Theorem_7_3_7 (m a : Nat) : invertible [a]_m ↔ rel_prime m a := by apply Iff.intro · -- (→) assume h1 : invertible [a]_m define at h1 obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1 obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y rewrite [h3, mul_class, cc_eq_iff_congr] at h2 define at h2 obtain (c : Int) (h4 : a * b - 1 = m * c) from h2 rewrite [Exercise_7_2_6] --Goal : ∃ (s t : Int), s * ↑m + t * ↑a = 1 apply Exists.intro (-c) apply Exists.intro b show (-c) * m + b * a = 1 from calc (-c) * m + b * a _ = (-c) * m + (a * b - 1) + 1 := by ring _ = (-c) * m + m * c + 1 := by rw [h4] _ = 1 := by ring done · -- (←) assume h1 : rel_prime m a define show ∃ (Y : ZMod m), [a]_m * Y = [1]_m from Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1) done done /- Section 7.4 -/ section Euler open Euler lemma num_rp_below_base {m : Nat} : num_rp_below m 0 = 0 := by rfl lemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) : num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by have h1 : num_rp_below m (j + 1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : gcd m j = 1 rewrite [if_pos h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1 show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1 done lemma num_rp_below_step_not_rp {m j : Nat} (h : ¬rel_prime m j) : num_rp_below m (j + 1) = num_rp_below m j := by have h1 : num_rp_below m (j +1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : ¬gcd m j = 1 rewrite [if_neg h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j show num_rp_below m (j + 1) = num_rp_below m j from h1 done lemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl #eval phi 10 --Answer: 4 lemma prod_inv_iff_inv {m : Nat} {X : ZMod m} (h1 : invertible X) (Y : ZMod m) : invertible (X * Y) ↔ invertible Y := by apply Iff.intro · -- (→) assume h2 : invertible (X * Y) obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2 apply Exists.intro (X * Z) rewrite [←h3] --Goal : Y * (X * Z) = X * Y * Z ring --Note that ring can do algebra in ZMod m done · -- (←) assume h2 : invertible Y obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1 obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2 apply Exists.intro (Xi * Yi) show (X * Y) * (Xi * Yi) = [1]_m from calc X * Y * (Xi * Yi) _ = (X * Xi) * (Y * Yi) := by ring _ = [1]_m * [1]_m := by rw [h3, h4] _ = [1]_m := Theorem_7_3_6_7 [1]_m done done lemma F_rp_def {m i : Nat} (h : rel_prime m i) : F m i = [i]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h --h : gcd m i = 1 rewrite [if_pos h] at h1 show F m i = [i]_m from h1 done lemma F_not_rp_def {m i : Nat} (h : ¬rel_prime m i) : F m i = [1]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h rewrite [h1, if_neg h] rfl done lemma prod_seq_base {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 0 k f = [1]_m := by rfl lemma prod_seq_step {m : Nat} (n k : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl lemma prod_seq_zero_step {m : Nat} (n : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by rewrite [prod_seq_step, zero_add] rfl done lemma prod_one {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 1 k f = f k := by rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7] rfl done lemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl lemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m := calc [G m a i]_m _ = [(a * i) % m]_m := by rfl _ = [a * i]_m := (cc_eq_mod m (a * i)).symm _ = [a]_m * [i]_m := (mul_class m a i).symm lemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) : rel_prime m (G m a i) ↔ rel_prime m i := by have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1 show rel_prime m (G m a i) ↔ rel_prime m i from calc rel_prime m (G m a i) _ ↔ invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm _ ↔ invertible ([a]_m * [i]_m) := by rw [cc_G] _ ↔ invertible [i]_m := prod_inv_iff_inv h2 ([i]_m) _ ↔ rel_prime m i := Theorem_7_3_7 m i done lemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) : F m (G m a i) = [a]_m * F m i := by have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2 show F m (G m a i) = [a]_m * F m i from calc F m (G m a i) _ = [G m a i]_m := F_rp_def h3 _ = [a]_m * [i]_m := cc_G m a i _ = [a]_m * F m i := by rw [F_rp_def h2] done lemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : ¬rel_prime m i) : F m (G m a i) = [1]_m := by rewrite [←G_rp_iff h1 i] at h2 show F m (G m a i) = [1]_m from F_not_rp_def h2 done lemma FG_prod {m a : Nat} (h1 : rel_prime m a) : ∀ (k : Nat), prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by by_induc · -- Base Case show prod_seq 0 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from calc prod_seq 0 0 ((F m) ∘ (G m a)) _ = [1]_m := prod_seq_base _ _ _ = [a]_m ^ 0 * [1]_m := by ring _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by rw [num_rp_below_base, prod_seq_base] done · -- Induction Step fix k : Nat assume ih : prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) by_cases h2 : rel_prime m k · -- Case 1. h2 : rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([a]_m * F m k) := by rw [FG_rp h1 h2] _ = [a]_m ^ ((num_rp_below m k) + 1) * ((prod_seq k 0 (F m)) * F m k) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_rp h2, prod_seq_zero_step] done · -- Case 2. h2 : ¬rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([1]_m) := by rw [FG_not_rp h1 h2] _ = [a]_m ^ (num_rp_below m k) * (prod_seq k 0 (F m) * ([1]_m)) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_not_rp h2, prod_seq_zero_step, F_not_rp_def h2] done done done lemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by define --Goal : ∀ i < m, G m a i < m fix i : Nat assume h1 : i < m rewrite [G_def] --Goal : a * i % m < m show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m) done lemma left_inv_one_one_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g' (g i) = i) : one_one_below n g := sorry lemma right_inv_onto_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g (g' i) = i) (h2 : maps_below n g') : onto_below n g := by define at h2; define fix k : Nat assume h3 : k < n apply Exists.intro (g' k) show g' k < n ∧ g (g' k) = k from And.intro (h2 k h3) (h1 k h3) done lemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m * [inv_mod m a]_m = [1]_m := by have h2 : 0 ≤ (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a) show [a]_m * [inv_mod m a]_m = [1]_m from calc [a]_m * [inv_mod m a]_m _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2] _ = [a]_m * [gcd_c2 m a]_m := by rw [←cc_eq_mod] _ = [1]_m := gcd_c2_inv h1 done lemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m := calc a * (b % m) % m = a % m * (b % m % m) % m := Nat.mul_mod _ _ _ _ = a % m * (b % m) % m := by rw [Nat.mod_mod] _ = a * b % m := (Nat.mul_mod _ _ _).symm lemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm] rfl done theorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] : ↑a ≡ ↑b (MOD m) ↔ a % m = b % m := sorry lemma mul_inv_mod_cancel {m a i : Nat} [NeZero m] (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1 rewrite [mul_class, cc_eq_iff_congr, ←Nat.cast_mul, ←Nat.cast_one, congr_iff_mod_eq_Nat] at h3 show a * inv_mod m a * i % m = i from calc a * (inv_mod m a) * i % m _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = 1 % m * i % m := by rw [h3] _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = i % m := by rw [one_mul] _ = i := Nat.mod_eq_of_lt h2 done lemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl lemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, G m a (Ginv m a i) = i := by fix i : Nat assume h2 : i < m show G m a (Ginv m a i) = i from calc G m a (Ginv m a i) _ = a * ((inv_mod m a * i) % m) % m := by rfl _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, Ginv m a (G m a i) = i := by fix i : Nat assume h2 : i < m show Ginv m a (G m a i) = i from calc Ginv m a (G m a i) _ = inv_mod m a * ((a * i) % m) % m := by rfl _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc, mul_comm (inv_mod m a)] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_maps_below (m a : Nat) [NeZero m] : maps_below m (Ginv m a) := G_maps_below m (inv_mod m a) lemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : one_one_below m (G m a) := left_inv_one_one_below (Ginv_left_inv h1) lemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : onto_below m (G m a) := right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a) lemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : perm_below m (G m a) := And.intro (G_maps_below m a) (And.intro (G_one_one_below h1) (G_onto_below h1)) --Permuting a product of congruence classes doesn't change product lemma swap_fst (u v : Nat) : swap u v u = v := by define : swap u v u --Goal : (if u = u then v else if u = v then u else u) = v have h : u = u := by rfl rewrite [if_pos h] rfl done lemma swap_snd (u v : Nat) : swap u v v = u := by define : swap u v v by_cases h1 : v = u · -- Case 1. h1 : v = u rewrite [if_pos h1] show v = u from h1 done · -- Case 2. h1 : v ≠ u rewrite [if_neg h1] have h2 : v = v := by rfl rewrite [if_pos h2] rfl done done lemma swap_other {u v i : Nat} (h1 : i ≠ u) (h2 : i ≠ v) : swap u v i = i := by define : swap u v i rewrite [if_neg h1, if_neg h2] rfl done lemma swap_values (u v i : Nat) : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := by by_cases h1 : i = u · -- Case 1. h1 : i = u apply Or.inl rewrite [h1] show swap u v u = v from swap_fst u v done · -- Case 2. h1 : i ≠ u apply Or.inr by_cases h2 : i = v · -- Case 2.1. h2 : i = v apply Or.inl rewrite [h2] show swap u v v = u from swap_snd u v done · -- Case 2.2. h2 : i ≠ v apply Or.inr show swap u v i = i from swap_other h1 h2 done done done lemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by define fix i : Nat assume h3 : i < n have h4 : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := swap_values u v i by_cases on h4 · -- Case 1. h4 : swap u v i = v rewrite [h4] show v < n from h2 done · -- Case 2. by_cases on h4 · -- Case 2.1. h4 : swap u v i = u rewrite [h4] show u < n from h1 done · -- Case 2.2. h4 : swap u v i = i rewrite [h4] show i < n from h3 done done done lemma swap_swap (u v n : Nat) : ∀ i < n, swap u v (swap u v i) = i := by fix i : Nat assume h : i < n by_cases h1 : i = u · -- Case 1. h1 : i = u rewrite [h1, swap_fst, swap_snd] rfl done · -- Case 2. h1 : i ≠ u by_cases h2 : i = v · -- Case 2.1. h2 : i = v rewrite [h2, swap_snd, swap_fst] rfl done · -- Case 2.2. h2 : i ≠ v rewrite [swap_other h1 h2, swap_other h1 h2] rfl done done done lemma swap_one_one_below (u v n) : one_one_below n (swap u v) := left_inv_one_one_below (swap_swap u v n) lemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) := right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2) lemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) := And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2)) lemma comp_perm_below {n : Nat} {f g : Nat → Nat} (h1 : perm_below n f) (h2 : perm_below n g) : perm_below n (f ∘ g) := sorry lemma trivial_swap (u : Nat) : swap u u = id := by apply funext fix x : Nat by_cases h1 : x = u · -- Case 1. h1 : x = u rewrite [h1, swap_fst] rfl done · -- Case 2. h1 : x ≠ u rewrite [swap_other h1 h1] rfl done done lemma prod_eq_fun {m : Nat} (f g : Nat → ZMod m) (k : Nat) : ∀ (n : Nat), (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g := by by_induc · -- Base Case assume h : (∀ i < 0, f (k + i) = g (k + i)) rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g assume h1 : ∀ i < n + 1, f (k + i) = g (k + i) have h2 : ∀ i < n, f (k + i) = g (k + i) := by fix i : Nat assume h2 : i < n have h3 : i < n + 1 := by linarith show f (k + i) = g (k + i) from h1 i h3 done have h3 : prod_seq n k f = prod_seq n k g := ih h2 have h4 : n < n + 1 := Nat.lt_succ_self n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4] rfl done done lemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := by have h2 : ∀ (i : Nat), i < u → (f ∘ swap u n) (0 + i) = f (0 + i) := by fix i : Nat assume h2 : i < u have h3 : 0 + i ≠ u := by linarith have h4 : 0 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl done show prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f from prod_eq_fun (f ∘ swap u n) f 0 u h2 done lemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := by have h2 : ∀ i < j, (f ∘ swap u n) (u + 1 + i) = f (u + 1 + i) := by fix i : Nat assume h2 : i < j have h3 : u + 1 + i ≠ u := by linarith have h4 : u + 1 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl show prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f from prod_eq_fun (f ∘ swap u n) f (u + 1) j h2 done lemma break_prod {m : Nat} (n : Nat) (f : Nat → ZMod m) : ∀ (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by by_induc · -- Base Case have h : n + 0 = n := by rfl rewrite [prod_seq_base, h, Theorem_7_3_6_7] rfl done · -- Induction Step fix j : Nat assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f rewrite [←add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc] rfl done done lemma break_prod_twice {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f := break_prod n f 1 rewrite [prod_one] at h2 have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f := break_prod (u + 1) f j rewrite [←h1] at h3 have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f := break_prod u f 1 rewrite [prod_one] at h4 rewrite [h3, h4] at h2 show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2 done lemma swap_prod_eq_prod {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f := by by_cases h2 : u = n · -- Case 1. h2 : u = n rewrite [h2, trivial_swap n] --Goal : prod_seq (n + 1) 0 (f ∘ id) = prod_seq (n + 1) 0 f rfl done · -- Case 2. h2 : ¬u = n have h3 : u + 1 ≤ n := Nat.lt_of_le_of_ne h1 h2 obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3 have break_f : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := break_prod_twice f h4 have break_fs : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_prod_twice (f ∘ swap u n) h4 have f_eq_fs_below : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := swap_prod_eq_prod_below f h1 have f_eq_fs_btwn : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4 show prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f from calc prod_seq (n + 1) 0 (f ∘ swap u n) _ = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_fs _ = prod_seq u 0 f * (f ∘ swap u n) u * prod_seq j (u + 1) f * (f ∘ swap u n) n := by rw [f_eq_fs_below, f_eq_fs_btwn] _ = prod_seq u 0 f * f (swap u n u) * prod_seq j (u + 1) f * f (swap u n n) := by rfl _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by rw [swap_fst, swap_snd] _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring _ = prod_seq (n + 1) 0 f := break_f.symm done done lemma perm_below_fixed {n : Nat} {g : Nat → Nat} (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry lemma perm_prod {m : Nat} (f : Nat → ZMod m) : ∀ (n : Nat), ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) := by by_induc · -- Base Case fix g : Nat → Nat assume h1 : perm_below 0 g rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) fix g : Nat → Nat assume g_pb : perm_below (n + 1) g define at g_pb have g_ob : onto_below (n + 1) g := g_pb.right.right define at g_ob have h1 : n < n + 1 := by linarith obtain (u : Nat) (h2 : u < n + 1 ∧ g u = n) from g_ob n h1 have s_pb : perm_below (n + 1) (swap u n) := swap_perm_below h2.left h1 have gs_pb_n1 : perm_below (n + 1) (g ∘ swap u n) := comp_perm_below g_pb s_pb have gs_fix_n : (g ∘ swap u n) n = n := calc (g ∘ swap u n) n _ = g (swap u n n) := by rfl _ = g u := by rw [swap_snd] _ = n := h2.right have gs_pb_n : perm_below n (g ∘ swap u n) := perm_below_fixed gs_pb_n1 gs_fix_n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f ∘ (g ∘ swap u n)) := ih (g ∘ swap u n) gs_pb_n have h3 : u ≤ n := by linarith show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f ∘ g) from calc prod_seq (n + 1) 0 f _ = prod_seq n 0 f * f n := prod_seq_zero_step n f _ = prod_seq n 0 (f ∘ (g ∘ swap u n)) * f ((g ∘ swap u n) n) := by rw [gs_prod, gs_fix_n] _ = prod_seq n 0 (f ∘ g ∘ swap u n) * (f ∘ g ∘ swap u n) n := by rfl _ = prod_seq (n + 1) 0 (f ∘ g ∘ swap u n) := (prod_seq_zero_step n (f ∘ g ∘ swap u n)).symm _ = prod_seq (n + 1) 0 ((f ∘ g) ∘ swap u n) := by rfl _ = prod_seq (n + 1) 0 (f ∘ g) := swap_prod_eq_prod (f ∘ g) h3 done done lemma F_invertible (m i : Nat) : invertible (F m i) := by by_cases h : rel_prime m i · -- Case 1. h : rel_prime m i rewrite [F_rp_def h] show invertible [i]_m from (Theorem_7_3_7 m i).rtl h done · -- Case 2. h : ¬rel_prime m i rewrite [F_not_rp_def h] apply Exists.intro [1]_m show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m done done lemma Fprod_invertible (m : Nat) : ∀ (k : Nat), invertible (prod_seq k 0 (F m)) := by by_induc · -- Base Case apply Exists.intro [1]_m show prod_seq 0 0 (F m) * [1]_m = [1]_m from calc prod_seq 0 0 (F m) * [1]_m _ = [1]_m * [1]_m := by rw [prod_seq_base] _ = [1]_m := Theorem_7_3_6_7 ([1]_m) done · -- Induction Step fix k : Nat assume ih : invertible (prod_seq k 0 (F m)) rewrite [prod_seq_zero_step] show invertible (prod_seq k 0 (F m) * (F m k)) from (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k) done done theorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m ^ (phi m) = [1]_m := by have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2 show [a]_m ^ (phi m) = [1]_m from calc [a]_m ^ (phi m) _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3] _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring _ = prod_seq m 0 (F m ∘ G m a) * Y := by rw [FG_prod h1 m, phi_def] _ = prod_seq m 0 (F m) * Y := by rw [perm_prod (F m) m (G m a) (G_perm_below h1)] _ = [1]_m := by rw [h3] done lemma Exercise_7_4_5_Int (m : Nat) (a : Int) : ∀ (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry lemma Exercise_7_4_5_Nat (m a n : Nat) : [a]_m ^ n = [a ^ n]_m := by rewrite [Exercise_7_4_5_Int] rfl done theorem Euler's_theorem {m a : Nat} [NeZero m] (h1 : rel_prime m a) : a ^ (phi m) ≡ 1 (MOD m) := by have h2 : [a]_m ^ (phi m) = [1]_m := Theorem_7_4_2 h1 rewrite [Exercise_7_4_5_Nat m a (phi m)] at h2 --h2 : [a ^ phi m]_m = [1]_m show a ^ (phi m) ≡ 1 (MOD m) from (cc_eq_iff_congr _ _ _).ltr h2 done #eval gcd 10 7 --Answer: 1. So 10 and 7 are relatively prime #eval 7 ^ phi 10 --Answer: 2401, which is congruent to 1 mod 10. end Euler /- Section 7.5 -/ lemma num_rp_prime {p : Nat} (h1 : prime p) : ∀ k < p, num_rp_below p (k + 1) = k := sorry lemma phi_prime {p : Nat} (h1 : prime p) : phi p = p - 1 := by have h2 : 1 ≤ p := prime_pos h1 have h3 : p - 1 + 1 = p := Nat.sub_add_cancel h2 have h4 : p - 1 < p := by linarith have h5 : num_rp_below p (p - 1 + 1) = p - 1 := num_rp_prime h1 (p - 1) h4 rewrite [h3] at h5 show phi p = p - 1 from h5 done theorem Theorem_7_2_2_Int {a c : Nat} {b : Int} (h1 : ↑c ∣ ↑a * b) (h2 : rel_prime a c) : ↑c ∣ b := by rewrite [Int.natCast_dvd, Int.natAbs_mul, Int.natAbs_ofNat] at h1 --h1 : c ∣ a * Int.natAbs b rewrite [Int.natCast_dvd] --Goal : c ∣ Int.natAbs b show c ∣ Int.natAbs b from Theorem_7_2_2 h1 h2 done lemma Lemma_7_4_5 {m n : Nat} (a b : Int) (h1 : rel_prime m n) : a ≡ b (MOD m * n) ↔ a ≡ b (MOD m) ∧ a ≡ b (MOD n) := by apply Iff.intro · -- (→) assume h2 : a ≡ b (MOD m * n) obtain (j : Int) (h3 : a - b = (m * n) * j) from h2 apply And.intro · -- Proof of a ≡ b (MOD m) apply Exists.intro (n * j) show a - b = m * (n * j) from calc a - b _ = m * n * j := h3 _ = m * (n * j) := by ring done · -- Proof of a ≡ b (MOD n) apply Exists.intro (m * j) show a - b = n * (m * j) from calc a - b _ = m * n * j := h3 _ = n * (m * j) := by ring done done · -- (←) assume h2 : a ≡ b (MOD m) ∧ a ≡ b (MOD n) obtain (j : Int) (h3 : a - b = m * j) from h2.left have h4 : (↑n : Int) ∣ a - b := h2.right rewrite [h3] at h4 --h4 : ↑n ∣ ↑m * j have h5 : ↑n ∣ j := Theorem_7_2_2_Int h4 h1 obtain (k : Int) (h6 : j = n * k) from h5 apply Exists.intro k --Goal : a - b = ↑(m * n) * k rewrite [Nat.cast_mul] --Goal : a - b = ↑m * ↑n * k show a - b = (m * n) * k from calc a - b _ = m * j := h3 _ = m * (n * k) := by rw [h6] _ = (m * n) * k := by ring done done --From exercises of Section 7.2 theorem rel_prime_symm {a b : Nat} (h : rel_prime a b) : rel_prime b a := sorry lemma prime_NeZero {p : Nat} (h : prime p) : NeZero p := by rewrite [neZero_iff] --Goal : p ≠ 0 define at h linarith done lemma Lemma_7_5_1 {p e d m c s : Nat} {t : Int} (h1 : prime p) (h2 : e * d = (p - 1) * s + 1) (h3 : m ^ e - c = p * t) : c ^ d ≡ m (MOD p) := by have h4 : m ^ e ≡ c (MOD p) := Exists.intro t h3 have h5 : [m ^ e]_p = [c]_p := (cc_eq_iff_congr _ _ _).rtl h4 rewrite [←Exercise_7_4_5_Nat] at h5 --h5 : [m]_p ^ e = [c]_p by_cases h6 : p ∣ m · -- Case 1. h6 : p ∣ m have h7 : m ≡ 0 (MOD p) := by obtain (j : Nat) (h8 : m = p * j) from h6 apply Exists.intro (↑j : Int) --Goal : ↑m - 0 = ↑p * ↑j rewrite [h8, Nat.cast_mul] ring done have h8 : [m]_p = [0]_p := (cc_eq_iff_congr _ _ _).rtl h7 have h9 : e * d ≠ 0 := by rewrite [h2] show (p - 1) * s + 1 ≠ 0 from Nat.add_one_ne_zero _ done have h10 : (0 : Int) ^ (e * d) = 0 := zero_pow h9 have h11 : [c ^ d]_p = [m]_p := calc [c ^ d]_p _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat] _ = ([m]_p ^ e) ^ d := by rw [h5] _ = [m]_p ^ (e * d) := by ring _ = [0]_p ^ (e * d) := by rw [h8] _ = [0 ^ (e * d)]_p := Exercise_7_4_5_Int _ _ _ _ = [0]_p := by rw [h10] _ = [m]_p := by rw [h8] show c ^ d ≡ m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11 done · -- Case 2. h6 : ¬p ∣ m have h7 : rel_prime m p := rel_prime_of_prime_not_dvd h1 h6 have h8 : rel_prime p m := rel_prime_symm h7 have h9 : NeZero p := prime_NeZero h1 have h10 : (1 : Int) ^ s = 1 := by ring have h11 : [c ^ d]_p = [m]_p := calc [c ^ d]_p _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat] _ = ([m]_p ^ e) ^ d := by rw [h5] _ = [m]_p ^ (e * d) := by ring _ = [m]_p ^ ((p - 1) * s + 1) := by rw [h2] _ = ([m]_p ^ (p - 1)) ^ s * [m]_p := by ring _ = ([m]_p ^ (phi p)) ^ s * [m]_p := by rw [phi_prime h1] _ = [1]_p ^ s * [m]_p := by rw [Theorem_7_4_2 h8] _ = [1 ^ s]_p * [m]_p := by rw [Exercise_7_4_5_Int] _ = [1]_p * [m]_p := by rw [h10] _ = [m]_p * [1]_p := by ring _ = [m]_p := Theorem_7_3_6_7 _ show c ^ d ≡ m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11 done done theorem Theorem_7_5_1 (p q n e d k m c : Nat) (p_prime : prime p) (q_prime : prime q) (p_ne_q : p ≠ q) (n_pq : n = p * q) (ed_congr_1 : e * d = k * (p - 1) * (q - 1) + 1) (h1 : [m]_n ^ e = [c]_n) : [c]_n ^ d = [m]_n := by rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] at h1 --h1 : m ^ e ≡ c (MOD n) rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] --Goal : c ^ d ≡ m (MOD n) obtain (j : Int) (h2 : m ^ e - c = n * j) from h1 rewrite [n_pq, Nat.cast_mul] at h2 --h2 : m ^ e - c = p * q * j have h3 : e * d = (p - 1) * (k * (q - 1)) + 1 := by rewrite [ed_congr_1] ring done have h4 : m ^ e - c = p * (q * j) := by rewrite [h2] ring done have congr_p : c ^ d ≡ m (MOD p) := Lemma_7_5_1 p_prime h3 h4 have h5 : e * d = (q - 1) * (k * (p - 1)) + 1 := by rewrite [ed_congr_1] ring done have h6 : m ^ e - c = q * (p * j) := by rewrite [h2] ring done have congr_q : c ^ d ≡ m (MOD q) := Lemma_7_5_1 q_prime h5 h6 have h7 : ¬q ∣ p := by by_contra h8 have h9 : q = 1 ∨ q = p := dvd_prime p_prime h8 disj_syll h9 (prime_not_one q_prime) show False from p_ne_q h9.symm done have h8 : rel_prime p q := rel_prime_of_prime_not_dvd q_prime h7 rewrite [n_pq, Lemma_7_4_5 _ _ h8] show c ^ d ≡ m (MOD p) ∧ c ^ d ≡ m (MOD q) from And.intro congr_p congr_q done /- BEGIN EXERCISES -/ namespace Exercises /- Section 7.1 -/ -- 1. theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry -- 2. lemma gcd_comm_lt {a b : Nat} (h : a < b) : gcd a b = gcd b a := sorry theorem gcd_comm (a b : Nat) : gcd a b = gcd b a := sorry -- 3. theorem Exercise_7_1_5 (a b : Nat) (n : Int) : (∃ (s t : Int), s * a + t * b = n) ↔ (↑(gcd a b) : Int) ∣ n := sorry -- 4. theorem Exercise_7_1_6 (a b c : Nat) : gcd a b = gcd (a + b * c) b := sorry -- 5. theorem gcd_is_nonzero {a b : Nat} (h : a ≠ 0 ∨ b ≠ 0) : gcd a b ≠ 0 := sorry -- 6. theorem gcd_greatest {a b d : Nat} (h1 : gcd a b ≠ 0) (h2 : d ∣ a) (h3 : d ∣ b) : d ≤ gcd a b := sorry -- 7. lemma Lemma_7_1_10a {a b : Nat} (n : Nat) (h : a ∣ b) : (n * a) ∣ (n * b) := sorry lemma Lemma_7_1_10b {a b n : Nat} (h1 : n ≠ 0) (h2 : (n * a) ∣ (n * b)) : a ∣ b := sorry lemma Lemma_7_1_10c {a b : Nat} (h1 : a ∣ b) (h2 : b ∣ a) : a = b := sorry theorem Exercise_7_1_10 (a b n : Nat) : gcd (n * a) (n * b) = n * gcd a b := sorry /- Section 7.2 -/ -- 1. lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry -- 2. -- Hints: Start with apply List.rec. You may find mul_ne_zero useful theorem prod_nonzero_nonzero : ∀ (l : List Nat), (∀ a ∈ l, a ≠ 0) → prod l ≠ 0 := sorry -- 3. theorem rel_prime_iff_no_common_factor (a b : Nat) : rel_prime a b ↔ ¬∃ (p : Nat), prime p ∧ p ∣ a ∧ p ∣ b := sorry -- 4. theorem rel_prime_symm {a b : Nat} (h : rel_prime a b) : rel_prime b a := sorry -- 5. lemma in_prime_factorization_iff_prime_factor {a : Nat} {l : List Nat} (h1 : prime_factorization a l) (p : Nat) : p ∈ l ↔ prime_factor p a := sorry -- 6. theorem Exercise_7_2_5 {a b : Nat} {l m : List Nat} (h1 : prime_factorization a l) (h2 : prime_factorization b m) : rel_prime a b ↔ (¬∃ (p : Nat), p ∈ l ∧ p ∈ m) := sorry -- 7.

theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1

HTPI.Exercises.Exercise_7_2_6

htpi/HTPILib/Chap7.lean

htpi

/- Copyright 2023 Daniel J. Velleman -/ import HTPILib.Chap6 namespace HTPI /- Definitions -/ lemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by have h : n + 1 > 0 := Nat.succ_pos n show a % (n + 1) < n + 1 from Nat.mod_lt a h done def gcd (a b : Nat) : Nat := match b with | 0 => a | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd (n + 1) (a % (n + 1)) termination_by b mutual def gcd_c1 (a b : Nat) : Int := match b with | 0 => 1 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c2 (n + 1) (a % (n + 1)) --Corresponds to s = t' termination_by b def gcd_c2 (a b : Nat) : Int := match b with | 0 => 0 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c1 (n + 1) (a % (n + 1)) - (gcd_c2 (n + 1) (a % (n + 1))) * ↑(a / (n + 1)) --Corresponds to t = s' - t'q termination_by b end def prime (n : Nat) : Prop := 2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n def prime_factor (p n : Nat) : Prop := prime p ∧ p ∣ n def all_prime (l : List Nat) : Prop := ∀ p ∈ l, prime p def nondec (l : List Nat) : Prop := match l with | [] => True --Of course, True is a proposition that is always true | n :: L => (∀ m ∈ L, n ≤ m) ∧ nondec L def nondec_prime_list (l : List Nat) : Prop := all_prime l ∧ nondec l def prod (l : List Nat) : Nat := match l with | [] => 1 | n :: L => n * (prod L) def prime_factorization (n : Nat) (l : List Nat) : Prop := nondec_prime_list l ∧ prod l = n def rel_prime (a b : Nat) : Prop := gcd a b = 1 def congr_mod (m : Nat) (a b : Int) : Prop := (↑m : Int) ∣ (a - b) def cc (m : Nat) (a : Int) : ZMod m := (↑a : ZMod m) notation:50 a " ≡ " b " (MOD " m ")" => congr_mod m a b notation:max "["a"]_"m:max => cc m a def invertible {m : Nat} (X : ZMod m) : Prop := ∃ (Y : ZMod m), X * Y = [1]_m def num_rp_below (m k : Nat) : Nat := match k with | 0 => 0 | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j def phi (m : Nat) : Nat := num_rp_below m m def prod_seq {m : Nat} (j k : Nat) (f : Nat → ZMod m) : ZMod m := match j with | 0 => [1]_m | n + 1 => prod_seq n k f * f (k + n) def maps_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i < n, g i < n def one_one_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i1 < n, ∀ i2 < n, g i1 = g i2 → i1 = i2 def onto_below (n : Nat) (g : Nat → Nat) : Prop := ∀ k < n, ∃ i < n, g i = k def perm_below (n : Nat) (g : Nat → Nat) : Prop := maps_below n g ∧ one_one_below n g ∧ onto_below n g def inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m) def swap (u v i : Nat) : Nat := if i = u then v else if i = v then u else i namespace Euler --For definitions specific to Euler's theorem def F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m def G (m a i : Nat) : Nat := (a * i) % m def Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i end Euler /- Section 7.1 -/ theorem dvd_mod_of_dvd_a_b {a b d : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ (a % b) := by set q : Nat := a / b have h3 : b * q + a % b = a := Nat.div_add_mod a b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 define --Goal : ∃ (c : Nat), a % b = d * c apply Exists.intro (j - k * q) show a % b = d * (j - k * q) from calc a % b _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm _ = a - b * q := by rw [h3] _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc] _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm done theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry #eval gcd 672 161 --Answer: 7 lemma gcd_base (a : Nat) : gcd a 0 = a := by rfl lemma gcd_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd a b = gcd b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1)) rfl done lemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b ≠ 0) : a % b < b := by have h1 : b > 0 := Nat.pos_of_ne_zero h show a % b < b from Nat.mod_lt a h1 done lemma dvd_self (n : Nat) : n ∣ n := by apply Exists.intro 1 ring done theorem gcd_dvd : ∀ (b a : Nat), (gcd a b) ∣ a ∧ (gcd a b) ∣ b := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd a b_1) ∣ a ∧ (gcd a b_1) ∣ b_1 fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_base] --Goal: a ∣ a ∧ a ∣ 0 apply And.intro (dvd_self a) define apply Exists.intro 0 rfl done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_nonzero a h1] --Goal : gcd b (a % b) ∣ a ∧ gcd b (a % b) ∣ b have h2 : a % b < b := mod_nonzero_lt a h1 have h3 : (gcd b (a % b)) ∣ b ∧ (gcd b (a % b)) ∣ (a % b) := ih (a % b) h2 b apply And.intro _ h3.left show (gcd b (a % b)) ∣ a from dvd_a_of_dvd_b_mod h3.left h3.right done done theorem gcd_dvd_left (a b : Nat) : (gcd a b) ∣ a := (gcd_dvd b a).left theorem gcd_dvd_right (a b : Nat) : (gcd a b) ∣ b := (gcd_dvd b a).right lemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl lemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c1 a b = gcd_c2 b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done lemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl lemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * ↑(a / b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done theorem gcd_lin_comb : ∀ (b a : Nat), (gcd_c1 a b) * ↑a + (gcd_c2 a b) * ↑b = ↑(gcd a b) := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd_c1 a b_1) * ↑a + (gcd_c2 a b_1) * ↑b_1 = ↑(gcd a b_1) fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base] --Goal : 1 * ↑a + 0 * ↑0 = ↑a ring done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1] --Goal : gcd_c2 b (a % b) * ↑a + -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * ↑(a / b)) * ↑b = -- ↑(gcd b (a % b)) set r : Nat := a % b set q : Nat := a / b set s : Int := gcd_c1 b r set t : Int := gcd_c2 b r --Goal : t * ↑a + (s - t * ↑q) * ↑b = ↑(gcd b r) have h2 : r < b := mod_nonzero_lt a h1 have h3 : s * ↑b + t * ↑r = ↑(gcd b r) := ih r h2 b have h4 : b * q + r = a := Nat.div_add_mod a b rewrite [←h3, ←h4] rewrite [Nat.cast_add, Nat.cast_mul] --Goal : t * (↑b * ↑q + ↑r) + (s - t * ↑q) * ↑b = s * ↑b + t * ↑r ring done done #eval gcd_c1 672 161 --Answer: 6 #eval gcd_c2 672 161 --Answer: -25 --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161 theorem Theorem_7_1_6 {d a b : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ gcd a b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑d ∣ ↑(gcd a b) set s : Int := gcd_c1 a b set t : Int := gcd_c2 a b have h3 : s * ↑a + t * ↑b = ↑(gcd a b) := gcd_lin_comb b a rewrite [←h3] --Goal : ↑d ∣ s * ↑a + t * ↑b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul] --Goal : ↑d ∣ s * (↑d * ↑j) + t * (↑d * ↑k) define apply Exists.intro (s * ↑j + t * ↑k) ring done /- Section 7.2 -/ theorem dvd_trans {a b c : Nat} (h1 : a ∣ b) (h2 : b ∣ c) : a ∣ c := by define at h1; define at h2; define obtain (m : Nat) (h3 : b = a * m) from h1 obtain (n : Nat) (h4 : c = b * n) from h2 rewrite [h3, mul_assoc] at h4 apply Exists.intro (m * n) show c = a * (m * n) from h4 done lemma exists_prime_factor : ∀ (n : Nat), 2 ≤ n → ∃ (p : Nat), prime_factor p n := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, 2 ≤ n_1 → ∃ (p : Nat), prime_factor p n_1 assume h1 : 2 ≤ n by_cases h2 : prime n · -- Case 1. h2 : prime n apply Exists.intro n define --Goal : prime n ∧ n ∣ n show prime n ∧ n ∣ n from And.intro h2 (dvd_self n) done · -- Case 2. h2 : ¬prime n define at h2 --h2 : ¬(2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n) demorgan at h2 disj_syll h2 h1 obtain (a : Nat) (h3 : ∃ (b : Nat), a * b = n ∧ a < n ∧ b < n) from h2 obtain (b : Nat) (h4 : a * b = n ∧ a < n ∧ b < n) from h3 have h5 : 2 ≤ a := by by_contra h6 have h7 : a ≤ 1 := by linarith have h8 : n ≤ b := calc n _ = a * b := h4.left.symm _ ≤ 1 * b := by rel [h7] _ = b := by ring linarith --n ≤ b contradicts b < n done have h6 : ∃ (p : Nat), prime_factor p a := ih a h4.right.left h5 obtain (p : Nat) (h7 : prime_factor p a) from h6 apply Exists.intro p define --Goal : prime p ∧ p ∣ n define at h7 --h7 : prime p ∧ p ∣ a apply And.intro h7.left have h8 : a ∣ n := by apply Exists.intro b show n = a * b from (h4.left).symm done show p ∣ n from dvd_trans h7.right h8 done done lemma exists_least_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q := by set S : Set Nat := {p : Nat | prime_factor p n} have h2 : ∃ (p : Nat), p ∈ S := exists_prime_factor n h show ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q from well_ord_princ S h2 done lemma all_prime_nil : all_prime [] := by define --Goal : ∀ p ∈ [], prime p fix p : Nat contrapos --Goal : ¬prime p → p ∉ [] assume h1 : ¬prime p show p ∉ [] from List.not_mem_nil p done lemma all_prime_cons (n : Nat) (L : List Nat) : all_prime (n :: L) ↔ prime n ∧ all_prime L := by apply Iff.intro · -- (→) assume h1 : all_prime (n :: L) --Goal : prime n ∧ all_prime L define at h1 --h1 : ∀ p ∈ n :: L, prime p apply And.intro (h1 n (List.mem_cons_self n L)) define --Goal : ∀ p ∈ L, prime p fix p : Nat assume h2 : p ∈ L show prime p from h1 p (List.mem_cons_of_mem n h2) done · -- (←) assume h1 : prime n ∧ all_prime L --Goal : all_prime (n :: l) define : all_prime L at h1 define fix p : Nat assume h2 : p ∈ n :: L rewrite [List.mem_cons] at h2 --h2 : p = n ∨ p ∈ L by_cases on h2 · -- Case 1. h2 : p = n rewrite [h2] show prime n from h1.left done · -- Case 2. h2 : p ∈ L show prime p from h1.right p h2 done done done lemma nondec_nil : nondec [] := by define --Goal : True trivial --trivial proves some obviously true statements, such as True done lemma nondec_cons (n : Nat) (L : List Nat) : nondec (n :: L) ↔ (∀ m ∈ L, n ≤ m) ∧ nondec L := by rfl lemma prod_nil : prod [] = 1 := by rfl lemma prod_cons : prod (n :: L) = n * (prod L) := by rfl lemma exists_cons_of_length_eq_succ {A : Type} {l : List A} {n : Nat} (h : l.length = n + 1) : ∃ (a : A) (L : List A), l = a :: L ∧ L.length = n := by have h1 : ¬l.length = 0 := by linarith rewrite [List.length_eq_zero] at h1 obtain (a : A) (h2 : ∃ (L : List A), l = a :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List A) (h3 : l = a :: L) from h2 apply Exists.intro a apply Exists.intro L apply And.intro h3 have h4 : (a :: L).length = L.length + 1 := List.length_cons a L rewrite [←h3, h] at h4 show L.length = n from (Nat.add_right_cancel h4).symm done lemma list_elt_dvd_prod_by_length (a : Nat) : ∀ (n : Nat), ∀ (l : List Nat), l.length = n → a ∈ l → a ∣ prod l := by by_induc · --Base Case fix l : List Nat assume h1 : l.length = 0 rewrite [List.length_eq_zero] at h1 --h1 : l = [] rewrite [h1] --Goal : a ∈ [] → a ∣ prod [] contrapos assume h2 : ¬a ∣ prod [] show a ∉ [] from List.not_mem_nil a done · -- Induction Step fix n : Nat assume ih : ∀ (l : List Nat), List.length l = n → a ∈ l → a ∣ prod l fix l : List Nat assume h1 : l.length = n + 1 --Goal : a ∈ l → a ∣ prod l obtain (b : Nat) (h2 : ∃ (L : List Nat), l = b :: L ∧ L.length = n) from exists_cons_of_length_eq_succ h1 obtain (L : List Nat) (h3 : l = b :: L ∧ L.length = n) from h2 have h4 : a ∈ L → a ∣ prod L := ih L h3.right assume h5 : a ∈ l rewrite [h3.left, prod_cons] --Goal : a ∣ b * prod L rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b ∨ a ∈ L by_cases on h5 · -- Case 1. h5 : a = b apply Exists.intro (prod L) rewrite [h5] rfl done · -- Case 2. h5 : a ∈ L have h6 : a ∣ prod L := h4 h5 have h7 : prod L ∣ b * prod L := by apply Exists.intro b ring done show a ∣ b * prod L from dvd_trans h6 h7 done done done lemma list_elt_dvd_prod {a : Nat} {l : List Nat} (h : a ∈ l) : a ∣ prod l := by set n : Nat := l.length have h1 : l.length = n := by rfl show a ∣ prod l from list_elt_dvd_prod_by_length a n l h1 h done lemma exists_prime_factorization : ∀ (n : Nat), n ≥ 1 → ∃ (l : List Nat), prime_factorization n l := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, n_1 ≥ 1 → ∃ (l : List Nat), prime_factorization n_1 l assume h1 : n ≥ 1 by_cases h2 : n = 1 · -- Case 1. h2 : n = 1 apply Exists.intro [] define apply And.intro · -- Proof of nondec_prime_list [] define show all_prime [] ∧ nondec [] from And.intro all_prime_nil nondec_nil done · -- Proof of prod [] = n rewrite [prod_nil, h2] rfl done done · -- Case 2. h2 : n ≠ 1 have h3 : n ≥ 2 := lt_of_le_of_ne' h1 h2 obtain (p : Nat) (h4 : prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q) from exists_least_prime_factor h3 have p_prime_factor : prime_factor p n := h4.left define at p_prime_factor have p_prime : prime p := p_prime_factor.left have p_dvd_n : p ∣ n := p_prime_factor.right have p_least : ∀ (q : Nat), prime_factor q n → p ≤ q := h4.right obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n have h5 : m ≠ 0 := by contradict h1 with h6 have h7 : n = 0 := calc n _ = p * m := n_eq_pm _ = p * 0 := by rw [h6] _ = 0 := by ring rewrite [h7] decide done have m_pos : 0 < m := Nat.pos_of_ne_zero h5 have m_lt_n : m < n := by define at p_prime show m < n from calc m _ < m + m := by linarith _ = 2 * m := by ring _ ≤ p * m := by rel [p_prime.left] _ = n := n_eq_pm.symm done obtain (L : List Nat) (h6 : prime_factorization m L) from ih m m_lt_n m_pos define at h6 have ndpl_L : nondec_prime_list L := h6.left define at ndpl_L apply Exists.intro (p :: L) define apply And.intro · -- Proof of nondec_prime_list (p :: L) define apply And.intro · -- Proof of all_prime (p :: L) rewrite [all_prime_cons] show prime p ∧ all_prime L from And.intro p_prime ndpl_L.left done · -- Proof of nondec (p :: L) rewrite [nondec_cons] apply And.intro _ ndpl_L.right fix q : Nat assume q_in_L : q ∈ L have h7 : q ∣ prod L := list_elt_dvd_prod q_in_L rewrite [h6.right] at h7 --h7 : q ∣ m have h8 : m ∣ n := by apply Exists.intro p rewrite [n_eq_pm] ring done have q_dvd_n : q ∣ n := dvd_trans h7 h8 have ap_L : all_prime L := ndpl_L.left define at ap_L have q_prime_factor : prime_factor q n := And.intro (ap_L q q_in_L) q_dvd_n show p ≤ q from p_least q q_prime_factor done done · -- Proof of prod (p :: L) = n rewrite [prod_cons, h6.right, n_eq_pm] rfl done done done theorem Theorem_7_2_2 {a b c : Nat} (h1 : c ∣ a * b) (h2 : rel_prime a c) : c ∣ b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑c ∣ ↑b define at h1; define at h2; define obtain (j : Nat) (h3 : a * b = c * j) from h1 set s : Int := gcd_c1 a c set t : Int := gcd_c2 a c have h4 : s * ↑a + t * ↑c = ↑(gcd a c) := gcd_lin_comb c a rewrite [h2, Nat.cast_one] at h4 --h4 : s * ↑a + t * ↑c = (1 : Int) apply Exists.intro (s * ↑j + t * ↑b) show ↑b = ↑c * (s * ↑j + t * ↑b) from calc ↑b _ = (1 : Int) * ↑b := (one_mul _).symm _ = (s * ↑a + t * ↑c) * ↑b := by rw [h4] _ = s * (↑a * ↑b) + t * ↑c * ↑b := by ring _ = s * (↑c * ↑j) + t * ↑c * ↑b := by rw [←Nat.cast_mul a b, h3, Nat.cast_mul c j] _ = ↑c * (s * ↑j + t * ↑b) := by ring done lemma le_nonzero_prod_left {a b : Nat} (h : a * b ≠ 0) : a ≤ a * b := by have h1 : b ≠ 0 := by contradict h with h1 rewrite [h1] ring done have h2 : 1 ≤ b := Nat.pos_of_ne_zero h1 show a ≤ a * b from calc a = a * 1 := (mul_one a).symm _ ≤ a * b := by rel [h2] done lemma le_nonzero_prod_right {a b : Nat} (h : a * b ≠ 0) : b ≤ a * b := by rewrite [mul_comm] rewrite [mul_comm] at h show b ≤ b * a from le_nonzero_prod_left h done lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry lemma rel_prime_of_prime_not_dvd {a p : Nat} (h1 : prime p) (h2 : ¬p ∣ a) : rel_prime a p := by have h3 : gcd a p ∣ a := gcd_dvd_left a p have h4 : gcd a p ∣ p := gcd_dvd_right a p have h5 : gcd a p = 1 ∨ gcd a p = p := dvd_prime h1 h4 have h6 : gcd a p ≠ p := by contradict h2 with h6 rewrite [h6] at h3 show p ∣ a from h3 done disj_syll h5 h6 show rel_prime a p from h5 done theorem Theorem_7_2_3 {a b p : Nat} (h1 : prime p) (h2 : p ∣ a * b) : p ∣ a ∨ p ∣ b := by or_right with h3 have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3 show p ∣ b from Theorem_7_2_2 h2 h4 done lemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a ≥ 1 := by have h1 : a ≠ 0 := by by_contra h1 rewrite [h1] at h contradict h linarith done show a ≥ 1 from Nat.pos_of_ne_zero h1 done lemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by have h1 : a ≥ 1 := ge_one_of_prod_one h have h2 : a * b ≠ 0 := by linarith have h3 : a ≤ a * b := le_nonzero_prod_left h2 rewrite [h] at h3 show a = 1 from Nat.le_antisymm h3 h1 done lemma eq_one_of_dvd_one {n : Nat} (h : n ∣ 1) : n = 1 := by obtain (j : Nat) (h1 : 1 = n * j) from h show n = 1 from eq_one_of_prod_one h1.symm done lemma prime_not_one {p : Nat} (h : prime p) : p ≠ 1 := by define at h linarith done theorem Theorem_7_2_4 {p : Nat} (h1 : prime p) : ∀ (l : List Nat), p ∣ prod l → ∃ a ∈ l, p ∣ a := by apply List.rec · -- Base Case. Goal : p ∣ prod [] → ∃ a ∈ [], p ∣ a rewrite [prod_nil] assume h2 : p ∣ 1 show ∃ a ∈ [], p ∣ a from absurd (eq_one_of_dvd_one h2) (prime_not_one h1) done · -- Induction Step fix b : Nat fix L : List Nat assume ih : p ∣ prod L → ∃ a ∈ L, p ∣ a --Goal : p ∣ prod (b :: L) → ∃ a ∈ b :: L, p ∣ a assume h2 : p ∣ prod (b :: L) rewrite [prod_cons] at h2 have h3 : p ∣ b ∨ p ∣ prod L := Theorem_7_2_3 h1 h2 by_cases on h3 · -- Case 1. h3 : p ∣ b apply Exists.intro b show b ∈ b :: L ∧ p ∣ b from And.intro (List.mem_cons_self b L) h3 done · -- Case 2. h3 : p ∣ prod L obtain (a : Nat) (h4 : a ∈ L ∧ p ∣ a) from ih h3 apply Exists.intro a show a ∈ b :: L ∧ p ∣ a from And.intro (List.mem_cons_of_mem b h4.left) h4.right done done done lemma prime_in_list {p : Nat} {l : List Nat} (h1 : prime p) (h2 : all_prime l) (h3 : p ∣ prod l) : p ∈ l := by obtain (a : Nat) (h4 : a ∈ l ∧ p ∣ a) from Theorem_7_2_4 h1 l h3 define at h2 have h5 : prime a := h2 a h4.left have h6 : p = 1 ∨ p = a := dvd_prime h5 h4.right disj_syll h6 (prime_not_one h1) rewrite [h6] show a ∈ l from h4.left done lemma first_le_first {p q : Nat} {l m : List Nat} (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m)) (h3 : prod (p :: l) = prod (q :: m)) : p ≤ q := by define at h1; define at h2 have h4 : q ∣ prod (p :: l) := by define apply Exists.intro (prod m) rewrite [←prod_cons] show prod (p :: l) = prod (q :: m) from h3 done have h5 : all_prime (q :: m) := h2.left rewrite [all_prime_cons] at h5 have h6 : q ∈ p :: l := prime_in_list h5.left h1.left h4 have h7 : nondec (p :: l) := h1.right rewrite [nondec_cons] at h7 rewrite [List.mem_cons] at h6 by_cases on h6 · -- Case 1. h6 : q = p linarith done · -- Case 2. h6 : q ∈ l have h8 : ∀ m ∈ l, p ≤ m := h7.left show p ≤ q from h8 q h6 done done lemma nondec_prime_list_tail {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by define at h define rewrite [all_prime_cons, nondec_cons] at h show all_prime l ∧ nondec l from And.intro h.left.right h.right.right done lemma cons_prod_not_one {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : prod (p :: l) ≠ 1 := by define at h have h1 : all_prime (p :: l) := h.left rewrite [all_prime_cons] at h1 rewrite [prod_cons] by_contra h2 show False from (prime_not_one h1.left) (eq_one_of_prod_one h2) done lemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) : l = [] ↔ prod l = 1 := by apply Iff.intro · -- (→) assume h1 : l = [] rewrite [h1] show prod [] = 1 from prod_nil done · -- (←) contrapos assume h1 : ¬l = [] obtain (p : Nat) (h2 : ∃ (L : List Nat), l = p :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List Nat) (h3 : l = p :: L) from h2 rewrite [h3] at h rewrite [h3] show ¬prod (p :: L) = 1 from cons_prod_not_one h done done lemma prime_pos {p : Nat} (h : prime p) : p > 0 := by define at h linarith done theorem Theorem_7_2_5 : ∀ (l1 l2 : List Nat), nondec_prime_list l1 → nondec_prime_list l2 → prod l1 = prod l2 → l1 = l2 := by apply List.rec · -- Base Case. Goal : ∀ (l2 : List Nat), nondec_prime_list [] → -- nondec_prime_list l2 → prod [] = prod l2 → [] = l2 fix l2 : List Nat assume h1 : nondec_prime_list [] assume h2 : nondec_prime_list l2 assume h3 : prod [] = prod l2 rewrite [prod_nil, eq_comm, ←list_nil_iff_prod_one h2] at h3 show [] = l2 from h3.symm done · -- Induction Step fix p : Nat fix L1 : List Nat assume ih : ∀ (L2 : List Nat), nondec_prime_list L1 → nondec_prime_list L2 → prod L1 = prod L2 → L1 = L2 -- Goal : ∀ (l2 : List Nat), nondec_prime_list (p :: L1) → -- nondec_prime_list l2 → prod (p :: L1) = prod l2 → p :: L1 = l2 fix l2 : List Nat assume h1 : nondec_prime_list (p :: L1) assume h2 : nondec_prime_list l2 assume h3 : prod (p :: L1) = prod l2 have h4 : ¬prod (p :: L1) = 1 := cons_prod_not_one h1 rewrite [h3, ←list_nil_iff_prod_one h2] at h4 obtain (q : Nat) (h5 : ∃ (L : List Nat), l2 = q :: L) from List.exists_cons_of_ne_nil h4 obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5 rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2) rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2) have h7 : p ≤ q := first_le_first h1 h2 h3 have h8 : q ≤ p := first_le_first h2 h1 h3.symm have h9 : p = q := by linarith rewrite [h9, prod_cons, prod_cons] at h3 --h3 : q * prod L1 = q * prod L2 have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1 have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2 define at h2 have h12 : all_prime (q :: L2) := h2.left rewrite [all_prime_cons] at h12 have h13 : q > 0 := prime_pos h12.left have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3 have h15 : L1 = L2 := ih L2 h10 h11 h14 rewrite [h6, h9, h15] rfl done done theorem fund_thm_arith (n : Nat) (h : n ≥ 1) : ∃! (l : List Nat), prime_factorization n l := by exists_unique · -- Existence show ∃ (l : List Nat), prime_factorization n l from exists_prime_factorization n h done · -- Uniqueness fix l1 : List Nat; fix l2 : List Nat assume h1 : prime_factorization n l1 assume h2 : prime_factorization n l2 define at h1; define at h2 have h3 : prod l1 = n := h1.right rewrite [←h2.right] at h3 show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3 done done /- Section 7.3 -/ theorem congr_refl (m : Nat) : ∀ (a : Int), a ≡ a (MOD m) := by fix a : Int define --Goal : ∃ (c : Int), a - a = ↑m * c apply Exists.intro 0 ring done theorem congr_symm {m : Nat} : ∀ {a b : Int}, a ≡ b (MOD m) → b ≡ a (MOD m) := by fix a : Int; fix b : Int assume h1 : a ≡ b (MOD m) define at h1 --h1 : ∃ (c : Int), a - b = ↑m * c define --Goal : ∃ (c : Int), b - a = ↑m * c obtain (c : Int) (h2 : a - b = m * c) from h1 apply Exists.intro (-c) show b - a = m * (-c) from calc b - a _ = -(a - b) := by ring _ = -(m * c) := by rw [h2] _ = m * (-c) := by ring done theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry /- Fundamental properties of congruence classes -/ lemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) : X = Y ↔ X.val = Y.val := Fin.ext_iff lemma val_nat_eq_mod (n k : Nat) : ([k]_(n + 1)).val = k % (n + 1) := by rfl lemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl theorem cc_rep {m : Nat} (X : ZMod m) : ∃ (a : Int), X = [a]_m := match m with | 0 => by apply Exists.intro X rfl done | n + 1 => by apply Exists.intro ↑(X.val) have h1 : X.val < n + 1 := Fin.prop X rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1] rfl done theorem add_class (m : Nat) (a b : Int) : [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm theorem mul_class (m : Nat) (a b : Int) : [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm lemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ [a - b]_m = [0]_m := by apply Iff.intro · -- (→) assume h1 : [a]_m = [b]_m have h2 : a - b = a + (-b) := by ring have h3 : b + (-b) = 0 := by ring show [a - b]_m = [0]_m from calc [a - b]_m _ = [a + (-b)]_m := by rw [h2] _ = [a]_m + [-b]_m := by rw [add_class] _ = [b]_m + [-b]_m := by rw [h1] _ = [b + -b]_m := by rw [add_class] _ = [0]_m := by rw [h3] done · -- (←) assume h1 : [a - b]_m = [0]_m have h2 : b + (a - b) = a := by ring have h3 : b + 0 = b := by ring show [a]_m = [b]_m from calc [a]_m _ = [b + (a - b)]_m := by rw [h2] _ = [b]_m + [a - b]_m := by rw [add_class] _ = [b]_m + [0]_m := by rw [h1] _ = [b + 0]_m := by rw [add_class] _ = [b]_m := by rw [h3] done done lemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) : [a]_m = [0]_m → [-a]_m = [0]_m := by assume h1 : [a]_m = [0]_m have h2 : 0 + (-a) = -a := by ring have h3 : a + (-a) = 0 := by ring show [-a]_m = [0]_m from calc [-a]_m _ = [0 + (-a)]_m := by rw [h2] _ = [0]_m + [-a]_m := by rw [add_class] _ = [a]_m + [-a]_m := by rw [h1] _ = [a + (-a)]_m := by rw [add_class] _ = [0]_m := by rw [h3] done lemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) : [-a]_m = [0]_m ↔ [a]_m = [0]_m := by apply Iff.intro _ (cc_neg_zero_of_cc_zero m a) assume h1 : [-a]_m = [0]_m have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1 have h3 : -(-a) = a := by ring rewrite [h3] at h2 show [a]_m = [0]_m from h2 done lemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl lemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m ↔ m ∣ k := match m with | 0 => by have h : (0 : Int) = (↑(0 : Nat) : Int) := by rfl rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj] apply Iff.intro · -- (→) assume h1 : k = 0 rewrite [h1] show 0 ∣ 0 from dvd_self 0 done · -- (←) assume h1 : 0 ∣ k obtain (c : Nat) (h2 : k = 0 * c) from h1 rewrite [h2] ring done done | n + 1 => by rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero] show k % (n + 1) = 0 ↔ n + 1 ∣ k from (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm done lemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m ↔ ↑m ∣ a := by obtain (k : Nat) (h1 : a = ↑k ∨ a = -↑k) from Int.eq_nat_or_neg a by_cases on h1 · -- Case 1. h1: a = ↑k rewrite [h1, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done · -- Case 2. h1: a = -↑k rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done done theorem cc_eq_iff_congr (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ a ≡ b (MOD m) := calc [a]_m = [b]_m _ ↔ [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b _ ↔ ↑m ∣ (a - b) := cc_zero_iff_dvd m (a - b) _ ↔ a ≡ b (MOD m) := by rfl /- End of fundamental properties of congruence classes -/ lemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m := by have h1 : (↑m : Int) ≠ 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m) show 0 ≤ a % m from Int.emod_nonneg a h1 done lemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m) have h2 : (↑m : Int) > 0 := (Nat.cast_pos).rtl h1 show a % m < m from Int.emod_lt_of_pos a h2 done lemma congr_mod_mod (m : Nat) (a : Int) : a ≡ a % m (MOD m) := by define have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m apply Exists.intro (a / m) show a - a % m = m * (a / m) from calc a - (a % m) _ = m * (a / m) + a % m - a % m := by rw [h1] _ = m * (a / m) := by ring done lemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) := And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a)) theorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) : ∃! (r : Int), 0 ≤ r ∧ r < m ∧ a ≡ r (MOD m) := by exists_unique · -- Existence apply Exists.intro (a % m) show 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) from mod_cmpl_res m a done · -- Uniqueness fix r1 : Int; fix r2 : Int assume h1 : 0 ≤ r1 ∧ r1 < m ∧ a ≡ r1 (MOD m) assume h2 : 0 ≤ r2 ∧ r2 < m ∧ a ≡ r2 (MOD m) have h3 : r1 ≡ r2 (MOD m) := congr_trans (congr_symm h1.right.right) h2.right.right obtain (d : Int) (h4 : r1 - r2 = m * d) from h3 have h5 : r1 - r2 < m * 1 := by linarith have h6 : m * (-1) < r1 - r2 := by linarith rewrite [h4] at h5 --h5 : m * d < m * 1 rewrite [h4] at h6 --h6 : m * -1 < m * d have h7 : (↑m : Int) ≥ 0 := Nat.cast_nonneg m have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7 have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7 have h10 : d = 0 := by linarith show r1 = r2 from calc r1 _ = r1 - r2 + r2 := by ring _ = m * 0 + r2 := by rw [h4, h10] _ = r2 := by ring done done lemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m := (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a) theorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y rewrite [h1, h2] have h3 : a + b = b + a := by ring show [a]_m + [b]_m = [b]_m + [a]_m from calc [a]_m + [b]_m _ = [a + b]_m := add_class m a b _ = [b + a]_m := by rw [h3] _ = [b]_m + [a]_m := (add_class m b a).symm done theorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X rewrite [h1] have h2 : a * 1 = a := by ring show [a]_m * [1]_m = [a]_m from calc [a]_m * [1]_m _ = [a * 1]_m := mul_class m a 1 _ = [a]_m := by rw [h2] done theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry lemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) : [a]_m * [gcd_c2 m a]_m = [1]_m := by set s : Int := gcd_c1 m a have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m define at h1 rewrite [h1, Nat.cast_one] at h2 --h2 : s * ↑m + gcd_c2 m a * ↑a = 1 rewrite [mul_class, cc_eq_iff_congr] define --Goal : ∃ (c : Int), ↑a * gcd_c2 m a - 1 = ↑m * c apply Exists.intro (-s) show a * (gcd_c2 m a) - 1 = m * (-s) from calc a * (gcd_c2 m a) - 1 _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring _ = 1 + m * (-s) - 1 := by rw [h2] _ = m * (-s) := by ring done theorem Theorem_7_3_7 (m a : Nat) : invertible [a]_m ↔ rel_prime m a := by apply Iff.intro · -- (→) assume h1 : invertible [a]_m define at h1 obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1 obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y rewrite [h3, mul_class, cc_eq_iff_congr] at h2 define at h2 obtain (c : Int) (h4 : a * b - 1 = m * c) from h2 rewrite [Exercise_7_2_6] --Goal : ∃ (s t : Int), s * ↑m + t * ↑a = 1 apply Exists.intro (-c) apply Exists.intro b show (-c) * m + b * a = 1 from calc (-c) * m + b * a _ = (-c) * m + (a * b - 1) + 1 := by ring _ = (-c) * m + m * c + 1 := by rw [h4] _ = 1 := by ring done · -- (←) assume h1 : rel_prime m a define show ∃ (Y : ZMod m), [a]_m * Y = [1]_m from Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1) done done /- Section 7.4 -/ section Euler open Euler lemma num_rp_below_base {m : Nat} : num_rp_below m 0 = 0 := by rfl lemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) : num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by have h1 : num_rp_below m (j + 1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : gcd m j = 1 rewrite [if_pos h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1 show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1 done lemma num_rp_below_step_not_rp {m j : Nat} (h : ¬rel_prime m j) : num_rp_below m (j + 1) = num_rp_below m j := by have h1 : num_rp_below m (j +1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : ¬gcd m j = 1 rewrite [if_neg h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j show num_rp_below m (j + 1) = num_rp_below m j from h1 done lemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl #eval phi 10 --Answer: 4 lemma prod_inv_iff_inv {m : Nat} {X : ZMod m} (h1 : invertible X) (Y : ZMod m) : invertible (X * Y) ↔ invertible Y := by apply Iff.intro · -- (→) assume h2 : invertible (X * Y) obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2 apply Exists.intro (X * Z) rewrite [←h3] --Goal : Y * (X * Z) = X * Y * Z ring --Note that ring can do algebra in ZMod m done · -- (←) assume h2 : invertible Y obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1 obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2 apply Exists.intro (Xi * Yi) show (X * Y) * (Xi * Yi) = [1]_m from calc X * Y * (Xi * Yi) _ = (X * Xi) * (Y * Yi) := by ring _ = [1]_m * [1]_m := by rw [h3, h4] _ = [1]_m := Theorem_7_3_6_7 [1]_m done done lemma F_rp_def {m i : Nat} (h : rel_prime m i) : F m i = [i]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h --h : gcd m i = 1 rewrite [if_pos h] at h1 show F m i = [i]_m from h1 done lemma F_not_rp_def {m i : Nat} (h : ¬rel_prime m i) : F m i = [1]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h rewrite [h1, if_neg h] rfl done lemma prod_seq_base {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 0 k f = [1]_m := by rfl lemma prod_seq_step {m : Nat} (n k : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl lemma prod_seq_zero_step {m : Nat} (n : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by rewrite [prod_seq_step, zero_add] rfl done lemma prod_one {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 1 k f = f k := by rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7] rfl done lemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl lemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m := calc [G m a i]_m _ = [(a * i) % m]_m := by rfl _ = [a * i]_m := (cc_eq_mod m (a * i)).symm _ = [a]_m * [i]_m := (mul_class m a i).symm lemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) : rel_prime m (G m a i) ↔ rel_prime m i := by have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1 show rel_prime m (G m a i) ↔ rel_prime m i from calc rel_prime m (G m a i) _ ↔ invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm _ ↔ invertible ([a]_m * [i]_m) := by rw [cc_G] _ ↔ invertible [i]_m := prod_inv_iff_inv h2 ([i]_m) _ ↔ rel_prime m i := Theorem_7_3_7 m i done lemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) : F m (G m a i) = [a]_m * F m i := by have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2 show F m (G m a i) = [a]_m * F m i from calc F m (G m a i) _ = [G m a i]_m := F_rp_def h3 _ = [a]_m * [i]_m := cc_G m a i _ = [a]_m * F m i := by rw [F_rp_def h2] done lemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : ¬rel_prime m i) : F m (G m a i) = [1]_m := by rewrite [←G_rp_iff h1 i] at h2 show F m (G m a i) = [1]_m from F_not_rp_def h2 done lemma FG_prod {m a : Nat} (h1 : rel_prime m a) : ∀ (k : Nat), prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by by_induc · -- Base Case show prod_seq 0 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from calc prod_seq 0 0 ((F m) ∘ (G m a)) _ = [1]_m := prod_seq_base _ _ _ = [a]_m ^ 0 * [1]_m := by ring _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by rw [num_rp_below_base, prod_seq_base] done · -- Induction Step fix k : Nat assume ih : prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) by_cases h2 : rel_prime m k · -- Case 1. h2 : rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([a]_m * F m k) := by rw [FG_rp h1 h2] _ = [a]_m ^ ((num_rp_below m k) + 1) * ((prod_seq k 0 (F m)) * F m k) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_rp h2, prod_seq_zero_step] done · -- Case 2. h2 : ¬rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([1]_m) := by rw [FG_not_rp h1 h2] _ = [a]_m ^ (num_rp_below m k) * (prod_seq k 0 (F m) * ([1]_m)) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_not_rp h2, prod_seq_zero_step, F_not_rp_def h2] done done done lemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by define --Goal : ∀ i < m, G m a i < m fix i : Nat assume h1 : i < m rewrite [G_def] --Goal : a * i % m < m show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m) done lemma left_inv_one_one_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g' (g i) = i) : one_one_below n g := sorry lemma right_inv_onto_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g (g' i) = i) (h2 : maps_below n g') : onto_below n g := by define at h2; define fix k : Nat assume h3 : k < n apply Exists.intro (g' k) show g' k < n ∧ g (g' k) = k from And.intro (h2 k h3) (h1 k h3) done lemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m * [inv_mod m a]_m = [1]_m := by have h2 : 0 ≤ (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a) show [a]_m * [inv_mod m a]_m = [1]_m from calc [a]_m * [inv_mod m a]_m _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2] _ = [a]_m * [gcd_c2 m a]_m := by rw [←cc_eq_mod] _ = [1]_m := gcd_c2_inv h1 done lemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m := calc a * (b % m) % m = a % m * (b % m % m) % m := Nat.mul_mod _ _ _ _ = a % m * (b % m) % m := by rw [Nat.mod_mod] _ = a * b % m := (Nat.mul_mod _ _ _).symm lemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm] rfl done theorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] : ↑a ≡ ↑b (MOD m) ↔ a % m = b % m := sorry lemma mul_inv_mod_cancel {m a i : Nat} [NeZero m] (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1 rewrite [mul_class, cc_eq_iff_congr, ←Nat.cast_mul, ←Nat.cast_one, congr_iff_mod_eq_Nat] at h3 show a * inv_mod m a * i % m = i from calc a * (inv_mod m a) * i % m _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = 1 % m * i % m := by rw [h3] _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = i % m := by rw [one_mul] _ = i := Nat.mod_eq_of_lt h2 done lemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl lemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, G m a (Ginv m a i) = i := by fix i : Nat assume h2 : i < m show G m a (Ginv m a i) = i from calc G m a (Ginv m a i) _ = a * ((inv_mod m a * i) % m) % m := by rfl _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, Ginv m a (G m a i) = i := by fix i : Nat assume h2 : i < m show Ginv m a (G m a i) = i from calc Ginv m a (G m a i) _ = inv_mod m a * ((a * i) % m) % m := by rfl _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc, mul_comm (inv_mod m a)] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_maps_below (m a : Nat) [NeZero m] : maps_below m (Ginv m a) := G_maps_below m (inv_mod m a) lemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : one_one_below m (G m a) := left_inv_one_one_below (Ginv_left_inv h1) lemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : onto_below m (G m a) := right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a) lemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : perm_below m (G m a) := And.intro (G_maps_below m a) (And.intro (G_one_one_below h1) (G_onto_below h1)) --Permuting a product of congruence classes doesn't change product lemma swap_fst (u v : Nat) : swap u v u = v := by define : swap u v u --Goal : (if u = u then v else if u = v then u else u) = v have h : u = u := by rfl rewrite [if_pos h] rfl done lemma swap_snd (u v : Nat) : swap u v v = u := by define : swap u v v by_cases h1 : v = u · -- Case 1. h1 : v = u rewrite [if_pos h1] show v = u from h1 done · -- Case 2. h1 : v ≠ u rewrite [if_neg h1] have h2 : v = v := by rfl rewrite [if_pos h2] rfl done done lemma swap_other {u v i : Nat} (h1 : i ≠ u) (h2 : i ≠ v) : swap u v i = i := by define : swap u v i rewrite [if_neg h1, if_neg h2] rfl done lemma swap_values (u v i : Nat) : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := by by_cases h1 : i = u · -- Case 1. h1 : i = u apply Or.inl rewrite [h1] show swap u v u = v from swap_fst u v done · -- Case 2. h1 : i ≠ u apply Or.inr by_cases h2 : i = v · -- Case 2.1. h2 : i = v apply Or.inl rewrite [h2] show swap u v v = u from swap_snd u v done · -- Case 2.2. h2 : i ≠ v apply Or.inr show swap u v i = i from swap_other h1 h2 done done done lemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by define fix i : Nat assume h3 : i < n have h4 : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := swap_values u v i by_cases on h4 · -- Case 1. h4 : swap u v i = v rewrite [h4] show v < n from h2 done · -- Case 2. by_cases on h4 · -- Case 2.1. h4 : swap u v i = u rewrite [h4] show u < n from h1 done · -- Case 2.2. h4 : swap u v i = i rewrite [h4] show i < n from h3 done done done lemma swap_swap (u v n : Nat) : ∀ i < n, swap u v (swap u v i) = i := by fix i : Nat assume h : i < n by_cases h1 : i = u · -- Case 1. h1 : i = u rewrite [h1, swap_fst, swap_snd] rfl done · -- Case 2. h1 : i ≠ u by_cases h2 : i = v · -- Case 2.1. h2 : i = v rewrite [h2, swap_snd, swap_fst] rfl done · -- Case 2.2. h2 : i ≠ v rewrite [swap_other h1 h2, swap_other h1 h2] rfl done done done lemma swap_one_one_below (u v n) : one_one_below n (swap u v) := left_inv_one_one_below (swap_swap u v n) lemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) := right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2) lemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) := And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2)) lemma comp_perm_below {n : Nat} {f g : Nat → Nat} (h1 : perm_below n f) (h2 : perm_below n g) : perm_below n (f ∘ g) := sorry lemma trivial_swap (u : Nat) : swap u u = id := by apply funext fix x : Nat by_cases h1 : x = u · -- Case 1. h1 : x = u rewrite [h1, swap_fst] rfl done · -- Case 2. h1 : x ≠ u rewrite [swap_other h1 h1] rfl done done lemma prod_eq_fun {m : Nat} (f g : Nat → ZMod m) (k : Nat) : ∀ (n : Nat), (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g := by by_induc · -- Base Case assume h : (∀ i < 0, f (k + i) = g (k + i)) rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g assume h1 : ∀ i < n + 1, f (k + i) = g (k + i) have h2 : ∀ i < n, f (k + i) = g (k + i) := by fix i : Nat assume h2 : i < n have h3 : i < n + 1 := by linarith show f (k + i) = g (k + i) from h1 i h3 done have h3 : prod_seq n k f = prod_seq n k g := ih h2 have h4 : n < n + 1 := Nat.lt_succ_self n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4] rfl done done lemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := by have h2 : ∀ (i : Nat), i < u → (f ∘ swap u n) (0 + i) = f (0 + i) := by fix i : Nat assume h2 : i < u have h3 : 0 + i ≠ u := by linarith have h4 : 0 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl done show prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f from prod_eq_fun (f ∘ swap u n) f 0 u h2 done lemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := by have h2 : ∀ i < j, (f ∘ swap u n) (u + 1 + i) = f (u + 1 + i) := by fix i : Nat assume h2 : i < j have h3 : u + 1 + i ≠ u := by linarith have h4 : u + 1 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl show prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f from prod_eq_fun (f ∘ swap u n) f (u + 1) j h2 done lemma break_prod {m : Nat} (n : Nat) (f : Nat → ZMod m) : ∀ (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by by_induc · -- Base Case have h : n + 0 = n := by rfl rewrite [prod_seq_base, h, Theorem_7_3_6_7] rfl done · -- Induction Step fix j : Nat assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f rewrite [←add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc] rfl done done lemma break_prod_twice {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f := break_prod n f 1 rewrite [prod_one] at h2 have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f := break_prod (u + 1) f j rewrite [←h1] at h3 have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f := break_prod u f 1 rewrite [prod_one] at h4 rewrite [h3, h4] at h2 show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2 done lemma swap_prod_eq_prod {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f := by by_cases h2 : u = n · -- Case 1. h2 : u = n rewrite [h2, trivial_swap n] --Goal : prod_seq (n + 1) 0 (f ∘ id) = prod_seq (n + 1) 0 f rfl done · -- Case 2. h2 : ¬u = n have h3 : u + 1 ≤ n := Nat.lt_of_le_of_ne h1 h2 obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3 have break_f : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := break_prod_twice f h4 have break_fs : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_prod_twice (f ∘ swap u n) h4 have f_eq_fs_below : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := swap_prod_eq_prod_below f h1 have f_eq_fs_btwn : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4 show prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f from calc prod_seq (n + 1) 0 (f ∘ swap u n) _ = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_fs _ = prod_seq u 0 f * (f ∘ swap u n) u * prod_seq j (u + 1) f * (f ∘ swap u n) n := by rw [f_eq_fs_below, f_eq_fs_btwn] _ = prod_seq u 0 f * f (swap u n u) * prod_seq j (u + 1) f * f (swap u n n) := by rfl _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by rw [swap_fst, swap_snd] _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring _ = prod_seq (n + 1) 0 f := break_f.symm done done lemma perm_below_fixed {n : Nat} {g : Nat → Nat} (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry lemma perm_prod {m : Nat} (f : Nat → ZMod m) : ∀ (n : Nat), ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) := by by_induc · -- Base Case fix g : Nat → Nat assume h1 : perm_below 0 g rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) fix g : Nat → Nat assume g_pb : perm_below (n + 1) g define at g_pb have g_ob : onto_below (n + 1) g := g_pb.right.right define at g_ob have h1 : n < n + 1 := by linarith obtain (u : Nat) (h2 : u < n + 1 ∧ g u = n) from g_ob n h1 have s_pb : perm_below (n + 1) (swap u n) := swap_perm_below h2.left h1 have gs_pb_n1 : perm_below (n + 1) (g ∘ swap u n) := comp_perm_below g_pb s_pb have gs_fix_n : (g ∘ swap u n) n = n := calc (g ∘ swap u n) n _ = g (swap u n n) := by rfl _ = g u := by rw [swap_snd] _ = n := h2.right have gs_pb_n : perm_below n (g ∘ swap u n) := perm_below_fixed gs_pb_n1 gs_fix_n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f ∘ (g ∘ swap u n)) := ih (g ∘ swap u n) gs_pb_n have h3 : u ≤ n := by linarith show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f ∘ g) from calc prod_seq (n + 1) 0 f _ = prod_seq n 0 f * f n := prod_seq_zero_step n f _ = prod_seq n 0 (f ∘ (g ∘ swap u n)) * f ((g ∘ swap u n) n) := by rw [gs_prod, gs_fix_n] _ = prod_seq n 0 (f ∘ g ∘ swap u n) * (f ∘ g ∘ swap u n) n := by rfl _ = prod_seq (n + 1) 0 (f ∘ g ∘ swap u n) := (prod_seq_zero_step n (f ∘ g ∘ swap u n)).symm _ = prod_seq (n + 1) 0 ((f ∘ g) ∘ swap u n) := by rfl _ = prod_seq (n + 1) 0 (f ∘ g) := swap_prod_eq_prod (f ∘ g) h3 done done lemma F_invertible (m i : Nat) : invertible (F m i) := by by_cases h : rel_prime m i · -- Case 1. h : rel_prime m i rewrite [F_rp_def h] show invertible [i]_m from (Theorem_7_3_7 m i).rtl h done · -- Case 2. h : ¬rel_prime m i rewrite [F_not_rp_def h] apply Exists.intro [1]_m show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m done done lemma Fprod_invertible (m : Nat) : ∀ (k : Nat), invertible (prod_seq k 0 (F m)) := by by_induc · -- Base Case apply Exists.intro [1]_m show prod_seq 0 0 (F m) * [1]_m = [1]_m from calc prod_seq 0 0 (F m) * [1]_m _ = [1]_m * [1]_m := by rw [prod_seq_base] _ = [1]_m := Theorem_7_3_6_7 ([1]_m) done · -- Induction Step fix k : Nat assume ih : invertible (prod_seq k 0 (F m)) rewrite [prod_seq_zero_step] show invertible (prod_seq k 0 (F m) * (F m k)) from (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k) done done theorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m ^ (phi m) = [1]_m := by have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2 show [a]_m ^ (phi m) = [1]_m from calc [a]_m ^ (phi m) _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3] _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring _ = prod_seq m 0 (F m ∘ G m a) * Y := by rw [FG_prod h1 m, phi_def] _ = prod_seq m 0 (F m) * Y := by rw [perm_prod (F m) m (G m a) (G_perm_below h1)] _ = [1]_m := by rw [h3] done lemma Exercise_7_4_5_Int (m : Nat) (a : Int) : ∀ (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry lemma Exercise_7_4_5_Nat (m a n : Nat) : [a]_m ^ n = [a ^ n]_m := by rewrite [Exercise_7_4_5_Int] rfl done theorem Euler's_theorem {m a : Nat} [NeZero m] (h1 : rel_prime m a) : a ^ (phi m) ≡ 1 (MOD m) := by have h2 : [a]_m ^ (phi m) = [1]_m := Theorem_7_4_2 h1 rewrite [Exercise_7_4_5_Nat m a (phi m)] at h2 --h2 : [a ^ phi m]_m = [1]_m show a ^ (phi m) ≡ 1 (MOD m) from (cc_eq_iff_congr _ _ _).ltr h2 done #eval gcd 10 7 --Answer: 1. So 10 and 7 are relatively prime #eval 7 ^ phi 10 --Answer: 2401, which is congruent to 1 mod 10. end Euler /- Section 7.5 -/ lemma num_rp_prime {p : Nat} (h1 : prime p) : ∀ k < p, num_rp_below p (k + 1) = k := sorry lemma phi_prime {p : Nat} (h1 : prime p) : phi p = p - 1 := by have h2 : 1 ≤ p := prime_pos h1 have h3 : p - 1 + 1 = p := Nat.sub_add_cancel h2 have h4 : p - 1 < p := by linarith have h5 : num_rp_below p (p - 1 + 1) = p - 1 := num_rp_prime h1 (p - 1) h4 rewrite [h3] at h5 show phi p = p - 1 from h5 done theorem Theorem_7_2_2_Int {a c : Nat} {b : Int} (h1 : ↑c ∣ ↑a * b) (h2 : rel_prime a c) : ↑c ∣ b := by rewrite [Int.natCast_dvd, Int.natAbs_mul, Int.natAbs_ofNat] at h1 --h1 : c ∣ a * Int.natAbs b rewrite [Int.natCast_dvd] --Goal : c ∣ Int.natAbs b show c ∣ Int.natAbs b from Theorem_7_2_2 h1 h2 done lemma Lemma_7_4_5 {m n : Nat} (a b : Int) (h1 : rel_prime m n) : a ≡ b (MOD m * n) ↔ a ≡ b (MOD m) ∧ a ≡ b (MOD n) := by apply Iff.intro · -- (→) assume h2 : a ≡ b (MOD m * n) obtain (j : Int) (h3 : a - b = (m * n) * j) from h2 apply And.intro · -- Proof of a ≡ b (MOD m) apply Exists.intro (n * j) show a - b = m * (n * j) from calc a - b _ = m * n * j := h3 _ = m * (n * j) := by ring done · -- Proof of a ≡ b (MOD n) apply Exists.intro (m * j) show a - b = n * (m * j) from calc a - b _ = m * n * j := h3 _ = n * (m * j) := by ring done done · -- (←) assume h2 : a ≡ b (MOD m) ∧ a ≡ b (MOD n) obtain (j : Int) (h3 : a - b = m * j) from h2.left have h4 : (↑n : Int) ∣ a - b := h2.right rewrite [h3] at h4 --h4 : ↑n ∣ ↑m * j have h5 : ↑n ∣ j := Theorem_7_2_2_Int h4 h1 obtain (k : Int) (h6 : j = n * k) from h5 apply Exists.intro k --Goal : a - b = ↑(m * n) * k rewrite [Nat.cast_mul] --Goal : a - b = ↑m * ↑n * k show a - b = (m * n) * k from calc a - b _ = m * j := h3 _ = m * (n * k) := by rw [h6] _ = (m * n) * k := by ring done done --From exercises of Section 7.2 theorem rel_prime_symm {a b : Nat} (h : rel_prime a b) : rel_prime b a := sorry lemma prime_NeZero {p : Nat} (h : prime p) : NeZero p := by rewrite [neZero_iff] --Goal : p ≠ 0 define at h linarith done lemma Lemma_7_5_1 {p e d m c s : Nat} {t : Int} (h1 : prime p) (h2 : e * d = (p - 1) * s + 1) (h3 : m ^ e - c = p * t) : c ^ d ≡ m (MOD p) := by have h4 : m ^ e ≡ c (MOD p) := Exists.intro t h3 have h5 : [m ^ e]_p = [c]_p := (cc_eq_iff_congr _ _ _).rtl h4 rewrite [←Exercise_7_4_5_Nat] at h5 --h5 : [m]_p ^ e = [c]_p by_cases h6 : p ∣ m · -- Case 1. h6 : p ∣ m have h7 : m ≡ 0 (MOD p) := by obtain (j : Nat) (h8 : m = p * j) from h6 apply Exists.intro (↑j : Int) --Goal : ↑m - 0 = ↑p * ↑j rewrite [h8, Nat.cast_mul] ring done have h8 : [m]_p = [0]_p := (cc_eq_iff_congr _ _ _).rtl h7 have h9 : e * d ≠ 0 := by rewrite [h2] show (p - 1) * s + 1 ≠ 0 from Nat.add_one_ne_zero _ done have h10 : (0 : Int) ^ (e * d) = 0 := zero_pow h9 have h11 : [c ^ d]_p = [m]_p := calc [c ^ d]_p _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat] _ = ([m]_p ^ e) ^ d := by rw [h5] _ = [m]_p ^ (e * d) := by ring _ = [0]_p ^ (e * d) := by rw [h8] _ = [0 ^ (e * d)]_p := Exercise_7_4_5_Int _ _ _ _ = [0]_p := by rw [h10] _ = [m]_p := by rw [h8] show c ^ d ≡ m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11 done · -- Case 2. h6 : ¬p ∣ m have h7 : rel_prime m p := rel_prime_of_prime_not_dvd h1 h6 have h8 : rel_prime p m := rel_prime_symm h7 have h9 : NeZero p := prime_NeZero h1 have h10 : (1 : Int) ^ s = 1 := by ring have h11 : [c ^ d]_p = [m]_p := calc [c ^ d]_p _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat] _ = ([m]_p ^ e) ^ d := by rw [h5] _ = [m]_p ^ (e * d) := by ring _ = [m]_p ^ ((p - 1) * s + 1) := by rw [h2] _ = ([m]_p ^ (p - 1)) ^ s * [m]_p := by ring _ = ([m]_p ^ (phi p)) ^ s * [m]_p := by rw [phi_prime h1] _ = [1]_p ^ s * [m]_p := by rw [Theorem_7_4_2 h8] _ = [1 ^ s]_p * [m]_p := by rw [Exercise_7_4_5_Int] _ = [1]_p * [m]_p := by rw [h10] _ = [m]_p * [1]_p := by ring _ = [m]_p := Theorem_7_3_6_7 _ show c ^ d ≡ m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11 done done theorem Theorem_7_5_1 (p q n e d k m c : Nat) (p_prime : prime p) (q_prime : prime q) (p_ne_q : p ≠ q) (n_pq : n = p * q) (ed_congr_1 : e * d = k * (p - 1) * (q - 1) + 1) (h1 : [m]_n ^ e = [c]_n) : [c]_n ^ d = [m]_n := by rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] at h1 --h1 : m ^ e ≡ c (MOD n) rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] --Goal : c ^ d ≡ m (MOD n) obtain (j : Int) (h2 : m ^ e - c = n * j) from h1 rewrite [n_pq, Nat.cast_mul] at h2 --h2 : m ^ e - c = p * q * j have h3 : e * d = (p - 1) * (k * (q - 1)) + 1 := by rewrite [ed_congr_1] ring done have h4 : m ^ e - c = p * (q * j) := by rewrite [h2] ring done have congr_p : c ^ d ≡ m (MOD p) := Lemma_7_5_1 p_prime h3 h4 have h5 : e * d = (q - 1) * (k * (p - 1)) + 1 := by rewrite [ed_congr_1] ring done have h6 : m ^ e - c = q * (p * j) := by rewrite [h2] ring done have congr_q : c ^ d ≡ m (MOD q) := Lemma_7_5_1 q_prime h5 h6 have h7 : ¬q ∣ p := by by_contra h8 have h9 : q = 1 ∨ q = p := dvd_prime p_prime h8 disj_syll h9 (prime_not_one q_prime) show False from p_ne_q h9.symm done have h8 : rel_prime p q := rel_prime_of_prime_not_dvd q_prime h7 rewrite [n_pq, Lemma_7_4_5 _ _ h8] show c ^ d ≡ m (MOD p) ∧ c ^ d ≡ m (MOD q) from And.intro congr_p congr_q done /- BEGIN EXERCISES -/ namespace Exercises /- Section 7.1 -/ -- 1. theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry -- 2. lemma gcd_comm_lt {a b : Nat} (h : a < b) : gcd a b = gcd b a := sorry theorem gcd_comm (a b : Nat) : gcd a b = gcd b a := sorry -- 3. theorem Exercise_7_1_5 (a b : Nat) (n : Int) : (∃ (s t : Int), s * a + t * b = n) ↔ (↑(gcd a b) : Int) ∣ n := sorry -- 4. theorem Exercise_7_1_6 (a b c : Nat) : gcd a b = gcd (a + b * c) b := sorry -- 5. theorem gcd_is_nonzero {a b : Nat} (h : a ≠ 0 ∨ b ≠ 0) : gcd a b ≠ 0 := sorry -- 6. theorem gcd_greatest {a b d : Nat} (h1 : gcd a b ≠ 0) (h2 : d ∣ a) (h3 : d ∣ b) : d ≤ gcd a b := sorry -- 7. lemma Lemma_7_1_10a {a b : Nat} (n : Nat) (h : a ∣ b) : (n * a) ∣ (n * b) := sorry lemma Lemma_7_1_10b {a b n : Nat} (h1 : n ≠ 0) (h2 : (n * a) ∣ (n * b)) : a ∣ b := sorry lemma Lemma_7_1_10c {a b : Nat} (h1 : a ∣ b) (h2 : b ∣ a) : a = b := sorry theorem Exercise_7_1_10 (a b n : Nat) : gcd (n * a) (n * b) = n * gcd a b := sorry /- Section 7.2 -/ -- 1. lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry -- 2. -- Hints: Start with apply List.rec. You may find mul_ne_zero useful theorem prod_nonzero_nonzero : ∀ (l : List Nat), (∀ a ∈ l, a ≠ 0) → prod l ≠ 0 := sorry -- 3. theorem rel_prime_iff_no_common_factor (a b : Nat) : rel_prime a b ↔ ¬∃ (p : Nat), prime p ∧ p ∣ a ∧ p ∣ b := sorry -- 4. theorem rel_prime_symm {a b : Nat} (h : rel_prime a b) : rel_prime b a := sorry -- 5. lemma in_prime_factorization_iff_prime_factor {a : Nat} {l : List Nat} (h1 : prime_factorization a l) (p : Nat) : p ∈ l ↔ prime_factor p a := sorry -- 6. theorem Exercise_7_2_5 {a b : Nat} {l m : List Nat} (h1 : prime_factorization a l) (h2 : prime_factorization b m) : rel_prime a b ↔ (¬∃ (p : Nat), p ∈ l ∧ p ∈ m) := sorry -- 7. theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry -- 8.

theorem Exercise_7_2_7 {a b a' b' : Nat} (h1 : rel_prime a b) (h2 : a' ∣ a) (h3 : b' ∣ b) : rel_prime a' b'

HTPI.Exercises.Exercise_7_2_7

htpi/HTPILib/Chap7.lean

htpi

/- Copyright 2023 Daniel J. Velleman -/ import HTPILib.Chap6 namespace HTPI /- Definitions -/ lemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by have h : n + 1 > 0 := Nat.succ_pos n show a % (n + 1) < n + 1 from Nat.mod_lt a h done def gcd (a b : Nat) : Nat := match b with | 0 => a | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd (n + 1) (a % (n + 1)) termination_by b mutual def gcd_c1 (a b : Nat) : Int := match b with | 0 => 1 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c2 (n + 1) (a % (n + 1)) --Corresponds to s = t' termination_by b def gcd_c2 (a b : Nat) : Int := match b with | 0 => 0 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c1 (n + 1) (a % (n + 1)) - (gcd_c2 (n + 1) (a % (n + 1))) * ↑(a / (n + 1)) --Corresponds to t = s' - t'q termination_by b end def prime (n : Nat) : Prop := 2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n def prime_factor (p n : Nat) : Prop := prime p ∧ p ∣ n def all_prime (l : List Nat) : Prop := ∀ p ∈ l, prime p def nondec (l : List Nat) : Prop := match l with | [] => True --Of course, True is a proposition that is always true | n :: L => (∀ m ∈ L, n ≤ m) ∧ nondec L def nondec_prime_list (l : List Nat) : Prop := all_prime l ∧ nondec l def prod (l : List Nat) : Nat := match l with | [] => 1 | n :: L => n * (prod L) def prime_factorization (n : Nat) (l : List Nat) : Prop := nondec_prime_list l ∧ prod l = n def rel_prime (a b : Nat) : Prop := gcd a b = 1 def congr_mod (m : Nat) (a b : Int) : Prop := (↑m : Int) ∣ (a - b) def cc (m : Nat) (a : Int) : ZMod m := (↑a : ZMod m) notation:50 a " ≡ " b " (MOD " m ")" => congr_mod m a b notation:max "["a"]_"m:max => cc m a def invertible {m : Nat} (X : ZMod m) : Prop := ∃ (Y : ZMod m), X * Y = [1]_m def num_rp_below (m k : Nat) : Nat := match k with | 0 => 0 | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j def phi (m : Nat) : Nat := num_rp_below m m def prod_seq {m : Nat} (j k : Nat) (f : Nat → ZMod m) : ZMod m := match j with | 0 => [1]_m | n + 1 => prod_seq n k f * f (k + n) def maps_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i < n, g i < n def one_one_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i1 < n, ∀ i2 < n, g i1 = g i2 → i1 = i2 def onto_below (n : Nat) (g : Nat → Nat) : Prop := ∀ k < n, ∃ i < n, g i = k def perm_below (n : Nat) (g : Nat → Nat) : Prop := maps_below n g ∧ one_one_below n g ∧ onto_below n g def inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m) def swap (u v i : Nat) : Nat := if i = u then v else if i = v then u else i namespace Euler --For definitions specific to Euler's theorem def F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m def G (m a i : Nat) : Nat := (a * i) % m def Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i end Euler /- Section 7.1 -/ theorem dvd_mod_of_dvd_a_b {a b d : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ (a % b) := by set q : Nat := a / b have h3 : b * q + a % b = a := Nat.div_add_mod a b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 define --Goal : ∃ (c : Nat), a % b = d * c apply Exists.intro (j - k * q) show a % b = d * (j - k * q) from calc a % b _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm _ = a - b * q := by rw [h3] _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc] _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm done theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry #eval gcd 672 161 --Answer: 7 lemma gcd_base (a : Nat) : gcd a 0 = a := by rfl lemma gcd_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd a b = gcd b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1)) rfl done lemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b ≠ 0) : a % b < b := by have h1 : b > 0 := Nat.pos_of_ne_zero h show a % b < b from Nat.mod_lt a h1 done lemma dvd_self (n : Nat) : n ∣ n := by apply Exists.intro 1 ring done theorem gcd_dvd : ∀ (b a : Nat), (gcd a b) ∣ a ∧ (gcd a b) ∣ b := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd a b_1) ∣ a ∧ (gcd a b_1) ∣ b_1 fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_base] --Goal: a ∣ a ∧ a ∣ 0 apply And.intro (dvd_self a) define apply Exists.intro 0 rfl done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_nonzero a h1] --Goal : gcd b (a % b) ∣ a ∧ gcd b (a % b) ∣ b have h2 : a % b < b := mod_nonzero_lt a h1 have h3 : (gcd b (a % b)) ∣ b ∧ (gcd b (a % b)) ∣ (a % b) := ih (a % b) h2 b apply And.intro _ h3.left show (gcd b (a % b)) ∣ a from dvd_a_of_dvd_b_mod h3.left h3.right done done theorem gcd_dvd_left (a b : Nat) : (gcd a b) ∣ a := (gcd_dvd b a).left theorem gcd_dvd_right (a b : Nat) : (gcd a b) ∣ b := (gcd_dvd b a).right lemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl lemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c1 a b = gcd_c2 b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done lemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl lemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * ↑(a / b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done theorem gcd_lin_comb : ∀ (b a : Nat), (gcd_c1 a b) * ↑a + (gcd_c2 a b) * ↑b = ↑(gcd a b) := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd_c1 a b_1) * ↑a + (gcd_c2 a b_1) * ↑b_1 = ↑(gcd a b_1) fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base] --Goal : 1 * ↑a + 0 * ↑0 = ↑a ring done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1] --Goal : gcd_c2 b (a % b) * ↑a + -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * ↑(a / b)) * ↑b = -- ↑(gcd b (a % b)) set r : Nat := a % b set q : Nat := a / b set s : Int := gcd_c1 b r set t : Int := gcd_c2 b r --Goal : t * ↑a + (s - t * ↑q) * ↑b = ↑(gcd b r) have h2 : r < b := mod_nonzero_lt a h1 have h3 : s * ↑b + t * ↑r = ↑(gcd b r) := ih r h2 b have h4 : b * q + r = a := Nat.div_add_mod a b rewrite [←h3, ←h4] rewrite [Nat.cast_add, Nat.cast_mul] --Goal : t * (↑b * ↑q + ↑r) + (s - t * ↑q) * ↑b = s * ↑b + t * ↑r ring done done #eval gcd_c1 672 161 --Answer: 6 #eval gcd_c2 672 161 --Answer: -25 --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161 theorem Theorem_7_1_6 {d a b : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ gcd a b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑d ∣ ↑(gcd a b) set s : Int := gcd_c1 a b set t : Int := gcd_c2 a b have h3 : s * ↑a + t * ↑b = ↑(gcd a b) := gcd_lin_comb b a rewrite [←h3] --Goal : ↑d ∣ s * ↑a + t * ↑b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul] --Goal : ↑d ∣ s * (↑d * ↑j) + t * (↑d * ↑k) define apply Exists.intro (s * ↑j + t * ↑k) ring done /- Section 7.2 -/ theorem dvd_trans {a b c : Nat} (h1 : a ∣ b) (h2 : b ∣ c) : a ∣ c := by define at h1; define at h2; define obtain (m : Nat) (h3 : b = a * m) from h1 obtain (n : Nat) (h4 : c = b * n) from h2 rewrite [h3, mul_assoc] at h4 apply Exists.intro (m * n) show c = a * (m * n) from h4 done lemma exists_prime_factor : ∀ (n : Nat), 2 ≤ n → ∃ (p : Nat), prime_factor p n := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, 2 ≤ n_1 → ∃ (p : Nat), prime_factor p n_1 assume h1 : 2 ≤ n by_cases h2 : prime n · -- Case 1. h2 : prime n apply Exists.intro n define --Goal : prime n ∧ n ∣ n show prime n ∧ n ∣ n from And.intro h2 (dvd_self n) done · -- Case 2. h2 : ¬prime n define at h2 --h2 : ¬(2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n) demorgan at h2 disj_syll h2 h1 obtain (a : Nat) (h3 : ∃ (b : Nat), a * b = n ∧ a < n ∧ b < n) from h2 obtain (b : Nat) (h4 : a * b = n ∧ a < n ∧ b < n) from h3 have h5 : 2 ≤ a := by by_contra h6 have h7 : a ≤ 1 := by linarith have h8 : n ≤ b := calc n _ = a * b := h4.left.symm _ ≤ 1 * b := by rel [h7] _ = b := by ring linarith --n ≤ b contradicts b < n done have h6 : ∃ (p : Nat), prime_factor p a := ih a h4.right.left h5 obtain (p : Nat) (h7 : prime_factor p a) from h6 apply Exists.intro p define --Goal : prime p ∧ p ∣ n define at h7 --h7 : prime p ∧ p ∣ a apply And.intro h7.left have h8 : a ∣ n := by apply Exists.intro b show n = a * b from (h4.left).symm done show p ∣ n from dvd_trans h7.right h8 done done lemma exists_least_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q := by set S : Set Nat := {p : Nat | prime_factor p n} have h2 : ∃ (p : Nat), p ∈ S := exists_prime_factor n h show ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q from well_ord_princ S h2 done lemma all_prime_nil : all_prime [] := by define --Goal : ∀ p ∈ [], prime p fix p : Nat contrapos --Goal : ¬prime p → p ∉ [] assume h1 : ¬prime p show p ∉ [] from List.not_mem_nil p done lemma all_prime_cons (n : Nat) (L : List Nat) : all_prime (n :: L) ↔ prime n ∧ all_prime L := by apply Iff.intro · -- (→) assume h1 : all_prime (n :: L) --Goal : prime n ∧ all_prime L define at h1 --h1 : ∀ p ∈ n :: L, prime p apply And.intro (h1 n (List.mem_cons_self n L)) define --Goal : ∀ p ∈ L, prime p fix p : Nat assume h2 : p ∈ L show prime p from h1 p (List.mem_cons_of_mem n h2) done · -- (←) assume h1 : prime n ∧ all_prime L --Goal : all_prime (n :: l) define : all_prime L at h1 define fix p : Nat assume h2 : p ∈ n :: L rewrite [List.mem_cons] at h2 --h2 : p = n ∨ p ∈ L by_cases on h2 · -- Case 1. h2 : p = n rewrite [h2] show prime n from h1.left done · -- Case 2. h2 : p ∈ L show prime p from h1.right p h2 done done done lemma nondec_nil : nondec [] := by define --Goal : True trivial --trivial proves some obviously true statements, such as True done lemma nondec_cons (n : Nat) (L : List Nat) : nondec (n :: L) ↔ (∀ m ∈ L, n ≤ m) ∧ nondec L := by rfl lemma prod_nil : prod [] = 1 := by rfl lemma prod_cons : prod (n :: L) = n * (prod L) := by rfl lemma exists_cons_of_length_eq_succ {A : Type} {l : List A} {n : Nat} (h : l.length = n + 1) : ∃ (a : A) (L : List A), l = a :: L ∧ L.length = n := by have h1 : ¬l.length = 0 := by linarith rewrite [List.length_eq_zero] at h1 obtain (a : A) (h2 : ∃ (L : List A), l = a :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List A) (h3 : l = a :: L) from h2 apply Exists.intro a apply Exists.intro L apply And.intro h3 have h4 : (a :: L).length = L.length + 1 := List.length_cons a L rewrite [←h3, h] at h4 show L.length = n from (Nat.add_right_cancel h4).symm done lemma list_elt_dvd_prod_by_length (a : Nat) : ∀ (n : Nat), ∀ (l : List Nat), l.length = n → a ∈ l → a ∣ prod l := by by_induc · --Base Case fix l : List Nat assume h1 : l.length = 0 rewrite [List.length_eq_zero] at h1 --h1 : l = [] rewrite [h1] --Goal : a ∈ [] → a ∣ prod [] contrapos assume h2 : ¬a ∣ prod [] show a ∉ [] from List.not_mem_nil a done · -- Induction Step fix n : Nat assume ih : ∀ (l : List Nat), List.length l = n → a ∈ l → a ∣ prod l fix l : List Nat assume h1 : l.length = n + 1 --Goal : a ∈ l → a ∣ prod l obtain (b : Nat) (h2 : ∃ (L : List Nat), l = b :: L ∧ L.length = n) from exists_cons_of_length_eq_succ h1 obtain (L : List Nat) (h3 : l = b :: L ∧ L.length = n) from h2 have h4 : a ∈ L → a ∣ prod L := ih L h3.right assume h5 : a ∈ l rewrite [h3.left, prod_cons] --Goal : a ∣ b * prod L rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b ∨ a ∈ L by_cases on h5 · -- Case 1. h5 : a = b apply Exists.intro (prod L) rewrite [h5] rfl done · -- Case 2. h5 : a ∈ L have h6 : a ∣ prod L := h4 h5 have h7 : prod L ∣ b * prod L := by apply Exists.intro b ring done show a ∣ b * prod L from dvd_trans h6 h7 done done done lemma list_elt_dvd_prod {a : Nat} {l : List Nat} (h : a ∈ l) : a ∣ prod l := by set n : Nat := l.length have h1 : l.length = n := by rfl show a ∣ prod l from list_elt_dvd_prod_by_length a n l h1 h done lemma exists_prime_factorization : ∀ (n : Nat), n ≥ 1 → ∃ (l : List Nat), prime_factorization n l := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, n_1 ≥ 1 → ∃ (l : List Nat), prime_factorization n_1 l assume h1 : n ≥ 1 by_cases h2 : n = 1 · -- Case 1. h2 : n = 1 apply Exists.intro [] define apply And.intro · -- Proof of nondec_prime_list [] define show all_prime [] ∧ nondec [] from And.intro all_prime_nil nondec_nil done · -- Proof of prod [] = n rewrite [prod_nil, h2] rfl done done · -- Case 2. h2 : n ≠ 1 have h3 : n ≥ 2 := lt_of_le_of_ne' h1 h2 obtain (p : Nat) (h4 : prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q) from exists_least_prime_factor h3 have p_prime_factor : prime_factor p n := h4.left define at p_prime_factor have p_prime : prime p := p_prime_factor.left have p_dvd_n : p ∣ n := p_prime_factor.right have p_least : ∀ (q : Nat), prime_factor q n → p ≤ q := h4.right obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n have h5 : m ≠ 0 := by contradict h1 with h6 have h7 : n = 0 := calc n _ = p * m := n_eq_pm _ = p * 0 := by rw [h6] _ = 0 := by ring rewrite [h7] decide done have m_pos : 0 < m := Nat.pos_of_ne_zero h5 have m_lt_n : m < n := by define at p_prime show m < n from calc m _ < m + m := by linarith _ = 2 * m := by ring _ ≤ p * m := by rel [p_prime.left] _ = n := n_eq_pm.symm done obtain (L : List Nat) (h6 : prime_factorization m L) from ih m m_lt_n m_pos define at h6 have ndpl_L : nondec_prime_list L := h6.left define at ndpl_L apply Exists.intro (p :: L) define apply And.intro · -- Proof of nondec_prime_list (p :: L) define apply And.intro · -- Proof of all_prime (p :: L) rewrite [all_prime_cons] show prime p ∧ all_prime L from And.intro p_prime ndpl_L.left done · -- Proof of nondec (p :: L) rewrite [nondec_cons] apply And.intro _ ndpl_L.right fix q : Nat assume q_in_L : q ∈ L have h7 : q ∣ prod L := list_elt_dvd_prod q_in_L rewrite [h6.right] at h7 --h7 : q ∣ m have h8 : m ∣ n := by apply Exists.intro p rewrite [n_eq_pm] ring done have q_dvd_n : q ∣ n := dvd_trans h7 h8 have ap_L : all_prime L := ndpl_L.left define at ap_L have q_prime_factor : prime_factor q n := And.intro (ap_L q q_in_L) q_dvd_n show p ≤ q from p_least q q_prime_factor done done · -- Proof of prod (p :: L) = n rewrite [prod_cons, h6.right, n_eq_pm] rfl done done done theorem Theorem_7_2_2 {a b c : Nat} (h1 : c ∣ a * b) (h2 : rel_prime a c) : c ∣ b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑c ∣ ↑b define at h1; define at h2; define obtain (j : Nat) (h3 : a * b = c * j) from h1 set s : Int := gcd_c1 a c set t : Int := gcd_c2 a c have h4 : s * ↑a + t * ↑c = ↑(gcd a c) := gcd_lin_comb c a rewrite [h2, Nat.cast_one] at h4 --h4 : s * ↑a + t * ↑c = (1 : Int) apply Exists.intro (s * ↑j + t * ↑b) show ↑b = ↑c * (s * ↑j + t * ↑b) from calc ↑b _ = (1 : Int) * ↑b := (one_mul _).symm _ = (s * ↑a + t * ↑c) * ↑b := by rw [h4] _ = s * (↑a * ↑b) + t * ↑c * ↑b := by ring _ = s * (↑c * ↑j) + t * ↑c * ↑b := by rw [←Nat.cast_mul a b, h3, Nat.cast_mul c j] _ = ↑c * (s * ↑j + t * ↑b) := by ring done lemma le_nonzero_prod_left {a b : Nat} (h : a * b ≠ 0) : a ≤ a * b := by have h1 : b ≠ 0 := by contradict h with h1 rewrite [h1] ring done have h2 : 1 ≤ b := Nat.pos_of_ne_zero h1 show a ≤ a * b from calc a = a * 1 := (mul_one a).symm _ ≤ a * b := by rel [h2] done lemma le_nonzero_prod_right {a b : Nat} (h : a * b ≠ 0) : b ≤ a * b := by rewrite [mul_comm] rewrite [mul_comm] at h show b ≤ b * a from le_nonzero_prod_left h done lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry lemma rel_prime_of_prime_not_dvd {a p : Nat} (h1 : prime p) (h2 : ¬p ∣ a) : rel_prime a p := by have h3 : gcd a p ∣ a := gcd_dvd_left a p have h4 : gcd a p ∣ p := gcd_dvd_right a p have h5 : gcd a p = 1 ∨ gcd a p = p := dvd_prime h1 h4 have h6 : gcd a p ≠ p := by contradict h2 with h6 rewrite [h6] at h3 show p ∣ a from h3 done disj_syll h5 h6 show rel_prime a p from h5 done theorem Theorem_7_2_3 {a b p : Nat} (h1 : prime p) (h2 : p ∣ a * b) : p ∣ a ∨ p ∣ b := by or_right with h3 have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3 show p ∣ b from Theorem_7_2_2 h2 h4 done lemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a ≥ 1 := by have h1 : a ≠ 0 := by by_contra h1 rewrite [h1] at h contradict h linarith done show a ≥ 1 from Nat.pos_of_ne_zero h1 done lemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by have h1 : a ≥ 1 := ge_one_of_prod_one h have h2 : a * b ≠ 0 := by linarith have h3 : a ≤ a * b := le_nonzero_prod_left h2 rewrite [h] at h3 show a = 1 from Nat.le_antisymm h3 h1 done lemma eq_one_of_dvd_one {n : Nat} (h : n ∣ 1) : n = 1 := by obtain (j : Nat) (h1 : 1 = n * j) from h show n = 1 from eq_one_of_prod_one h1.symm done lemma prime_not_one {p : Nat} (h : prime p) : p ≠ 1 := by define at h linarith done theorem Theorem_7_2_4 {p : Nat} (h1 : prime p) : ∀ (l : List Nat), p ∣ prod l → ∃ a ∈ l, p ∣ a := by apply List.rec · -- Base Case. Goal : p ∣ prod [] → ∃ a ∈ [], p ∣ a rewrite [prod_nil] assume h2 : p ∣ 1 show ∃ a ∈ [], p ∣ a from absurd (eq_one_of_dvd_one h2) (prime_not_one h1) done · -- Induction Step fix b : Nat fix L : List Nat assume ih : p ∣ prod L → ∃ a ∈ L, p ∣ a --Goal : p ∣ prod (b :: L) → ∃ a ∈ b :: L, p ∣ a assume h2 : p ∣ prod (b :: L) rewrite [prod_cons] at h2 have h3 : p ∣ b ∨ p ∣ prod L := Theorem_7_2_3 h1 h2 by_cases on h3 · -- Case 1. h3 : p ∣ b apply Exists.intro b show b ∈ b :: L ∧ p ∣ b from And.intro (List.mem_cons_self b L) h3 done · -- Case 2. h3 : p ∣ prod L obtain (a : Nat) (h4 : a ∈ L ∧ p ∣ a) from ih h3 apply Exists.intro a show a ∈ b :: L ∧ p ∣ a from And.intro (List.mem_cons_of_mem b h4.left) h4.right done done done lemma prime_in_list {p : Nat} {l : List Nat} (h1 : prime p) (h2 : all_prime l) (h3 : p ∣ prod l) : p ∈ l := by obtain (a : Nat) (h4 : a ∈ l ∧ p ∣ a) from Theorem_7_2_4 h1 l h3 define at h2 have h5 : prime a := h2 a h4.left have h6 : p = 1 ∨ p = a := dvd_prime h5 h4.right disj_syll h6 (prime_not_one h1) rewrite [h6] show a ∈ l from h4.left done lemma first_le_first {p q : Nat} {l m : List Nat} (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m)) (h3 : prod (p :: l) = prod (q :: m)) : p ≤ q := by define at h1; define at h2 have h4 : q ∣ prod (p :: l) := by define apply Exists.intro (prod m) rewrite [←prod_cons] show prod (p :: l) = prod (q :: m) from h3 done have h5 : all_prime (q :: m) := h2.left rewrite [all_prime_cons] at h5 have h6 : q ∈ p :: l := prime_in_list h5.left h1.left h4 have h7 : nondec (p :: l) := h1.right rewrite [nondec_cons] at h7 rewrite [List.mem_cons] at h6 by_cases on h6 · -- Case 1. h6 : q = p linarith done · -- Case 2. h6 : q ∈ l have h8 : ∀ m ∈ l, p ≤ m := h7.left show p ≤ q from h8 q h6 done done lemma nondec_prime_list_tail {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by define at h define rewrite [all_prime_cons, nondec_cons] at h show all_prime l ∧ nondec l from And.intro h.left.right h.right.right done lemma cons_prod_not_one {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : prod (p :: l) ≠ 1 := by define at h have h1 : all_prime (p :: l) := h.left rewrite [all_prime_cons] at h1 rewrite [prod_cons] by_contra h2 show False from (prime_not_one h1.left) (eq_one_of_prod_one h2) done lemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) : l = [] ↔ prod l = 1 := by apply Iff.intro · -- (→) assume h1 : l = [] rewrite [h1] show prod [] = 1 from prod_nil done · -- (←) contrapos assume h1 : ¬l = [] obtain (p : Nat) (h2 : ∃ (L : List Nat), l = p :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List Nat) (h3 : l = p :: L) from h2 rewrite [h3] at h rewrite [h3] show ¬prod (p :: L) = 1 from cons_prod_not_one h done done lemma prime_pos {p : Nat} (h : prime p) : p > 0 := by define at h linarith done theorem Theorem_7_2_5 : ∀ (l1 l2 : List Nat), nondec_prime_list l1 → nondec_prime_list l2 → prod l1 = prod l2 → l1 = l2 := by apply List.rec · -- Base Case. Goal : ∀ (l2 : List Nat), nondec_prime_list [] → -- nondec_prime_list l2 → prod [] = prod l2 → [] = l2 fix l2 : List Nat assume h1 : nondec_prime_list [] assume h2 : nondec_prime_list l2 assume h3 : prod [] = prod l2 rewrite [prod_nil, eq_comm, ←list_nil_iff_prod_one h2] at h3 show [] = l2 from h3.symm done · -- Induction Step fix p : Nat fix L1 : List Nat assume ih : ∀ (L2 : List Nat), nondec_prime_list L1 → nondec_prime_list L2 → prod L1 = prod L2 → L1 = L2 -- Goal : ∀ (l2 : List Nat), nondec_prime_list (p :: L1) → -- nondec_prime_list l2 → prod (p :: L1) = prod l2 → p :: L1 = l2 fix l2 : List Nat assume h1 : nondec_prime_list (p :: L1) assume h2 : nondec_prime_list l2 assume h3 : prod (p :: L1) = prod l2 have h4 : ¬prod (p :: L1) = 1 := cons_prod_not_one h1 rewrite [h3, ←list_nil_iff_prod_one h2] at h4 obtain (q : Nat) (h5 : ∃ (L : List Nat), l2 = q :: L) from List.exists_cons_of_ne_nil h4 obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5 rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2) rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2) have h7 : p ≤ q := first_le_first h1 h2 h3 have h8 : q ≤ p := first_le_first h2 h1 h3.symm have h9 : p = q := by linarith rewrite [h9, prod_cons, prod_cons] at h3 --h3 : q * prod L1 = q * prod L2 have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1 have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2 define at h2 have h12 : all_prime (q :: L2) := h2.left rewrite [all_prime_cons] at h12 have h13 : q > 0 := prime_pos h12.left have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3 have h15 : L1 = L2 := ih L2 h10 h11 h14 rewrite [h6, h9, h15] rfl done done theorem fund_thm_arith (n : Nat) (h : n ≥ 1) : ∃! (l : List Nat), prime_factorization n l := by exists_unique · -- Existence show ∃ (l : List Nat), prime_factorization n l from exists_prime_factorization n h done · -- Uniqueness fix l1 : List Nat; fix l2 : List Nat assume h1 : prime_factorization n l1 assume h2 : prime_factorization n l2 define at h1; define at h2 have h3 : prod l1 = n := h1.right rewrite [←h2.right] at h3 show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3 done done /- Section 7.3 -/ theorem congr_refl (m : Nat) : ∀ (a : Int), a ≡ a (MOD m) := by fix a : Int define --Goal : ∃ (c : Int), a - a = ↑m * c apply Exists.intro 0 ring done theorem congr_symm {m : Nat} : ∀ {a b : Int}, a ≡ b (MOD m) → b ≡ a (MOD m) := by fix a : Int; fix b : Int assume h1 : a ≡ b (MOD m) define at h1 --h1 : ∃ (c : Int), a - b = ↑m * c define --Goal : ∃ (c : Int), b - a = ↑m * c obtain (c : Int) (h2 : a - b = m * c) from h1 apply Exists.intro (-c) show b - a = m * (-c) from calc b - a _ = -(a - b) := by ring _ = -(m * c) := by rw [h2] _ = m * (-c) := by ring done theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry /- Fundamental properties of congruence classes -/ lemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) : X = Y ↔ X.val = Y.val := Fin.ext_iff lemma val_nat_eq_mod (n k : Nat) : ([k]_(n + 1)).val = k % (n + 1) := by rfl lemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl theorem cc_rep {m : Nat} (X : ZMod m) : ∃ (a : Int), X = [a]_m := match m with | 0 => by apply Exists.intro X rfl done | n + 1 => by apply Exists.intro ↑(X.val) have h1 : X.val < n + 1 := Fin.prop X rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1] rfl done theorem add_class (m : Nat) (a b : Int) : [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm theorem mul_class (m : Nat) (a b : Int) : [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm lemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ [a - b]_m = [0]_m := by apply Iff.intro · -- (→) assume h1 : [a]_m = [b]_m have h2 : a - b = a + (-b) := by ring have h3 : b + (-b) = 0 := by ring show [a - b]_m = [0]_m from calc [a - b]_m _ = [a + (-b)]_m := by rw [h2] _ = [a]_m + [-b]_m := by rw [add_class] _ = [b]_m + [-b]_m := by rw [h1] _ = [b + -b]_m := by rw [add_class] _ = [0]_m := by rw [h3] done · -- (←) assume h1 : [a - b]_m = [0]_m have h2 : b + (a - b) = a := by ring have h3 : b + 0 = b := by ring show [a]_m = [b]_m from calc [a]_m _ = [b + (a - b)]_m := by rw [h2] _ = [b]_m + [a - b]_m := by rw [add_class] _ = [b]_m + [0]_m := by rw [h1] _ = [b + 0]_m := by rw [add_class] _ = [b]_m := by rw [h3] done done lemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) : [a]_m = [0]_m → [-a]_m = [0]_m := by assume h1 : [a]_m = [0]_m have h2 : 0 + (-a) = -a := by ring have h3 : a + (-a) = 0 := by ring show [-a]_m = [0]_m from calc [-a]_m _ = [0 + (-a)]_m := by rw [h2] _ = [0]_m + [-a]_m := by rw [add_class] _ = [a]_m + [-a]_m := by rw [h1] _ = [a + (-a)]_m := by rw [add_class] _ = [0]_m := by rw [h3] done lemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) : [-a]_m = [0]_m ↔ [a]_m = [0]_m := by apply Iff.intro _ (cc_neg_zero_of_cc_zero m a) assume h1 : [-a]_m = [0]_m have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1 have h3 : -(-a) = a := by ring rewrite [h3] at h2 show [a]_m = [0]_m from h2 done lemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl lemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m ↔ m ∣ k := match m with | 0 => by have h : (0 : Int) = (↑(0 : Nat) : Int) := by rfl rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj] apply Iff.intro · -- (→) assume h1 : k = 0 rewrite [h1] show 0 ∣ 0 from dvd_self 0 done · -- (←) assume h1 : 0 ∣ k obtain (c : Nat) (h2 : k = 0 * c) from h1 rewrite [h2] ring done done | n + 1 => by rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero] show k % (n + 1) = 0 ↔ n + 1 ∣ k from (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm done lemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m ↔ ↑m ∣ a := by obtain (k : Nat) (h1 : a = ↑k ∨ a = -↑k) from Int.eq_nat_or_neg a by_cases on h1 · -- Case 1. h1: a = ↑k rewrite [h1, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done · -- Case 2. h1: a = -↑k rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done done theorem cc_eq_iff_congr (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ a ≡ b (MOD m) := calc [a]_m = [b]_m _ ↔ [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b _ ↔ ↑m ∣ (a - b) := cc_zero_iff_dvd m (a - b) _ ↔ a ≡ b (MOD m) := by rfl /- End of fundamental properties of congruence classes -/ lemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m := by have h1 : (↑m : Int) ≠ 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m) show 0 ≤ a % m from Int.emod_nonneg a h1 done lemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m) have h2 : (↑m : Int) > 0 := (Nat.cast_pos).rtl h1 show a % m < m from Int.emod_lt_of_pos a h2 done lemma congr_mod_mod (m : Nat) (a : Int) : a ≡ a % m (MOD m) := by define have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m apply Exists.intro (a / m) show a - a % m = m * (a / m) from calc a - (a % m) _ = m * (a / m) + a % m - a % m := by rw [h1] _ = m * (a / m) := by ring done lemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) := And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a)) theorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) : ∃! (r : Int), 0 ≤ r ∧ r < m ∧ a ≡ r (MOD m) := by exists_unique · -- Existence apply Exists.intro (a % m) show 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) from mod_cmpl_res m a done · -- Uniqueness fix r1 : Int; fix r2 : Int assume h1 : 0 ≤ r1 ∧ r1 < m ∧ a ≡ r1 (MOD m) assume h2 : 0 ≤ r2 ∧ r2 < m ∧ a ≡ r2 (MOD m) have h3 : r1 ≡ r2 (MOD m) := congr_trans (congr_symm h1.right.right) h2.right.right obtain (d : Int) (h4 : r1 - r2 = m * d) from h3 have h5 : r1 - r2 < m * 1 := by linarith have h6 : m * (-1) < r1 - r2 := by linarith rewrite [h4] at h5 --h5 : m * d < m * 1 rewrite [h4] at h6 --h6 : m * -1 < m * d have h7 : (↑m : Int) ≥ 0 := Nat.cast_nonneg m have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7 have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7 have h10 : d = 0 := by linarith show r1 = r2 from calc r1 _ = r1 - r2 + r2 := by ring _ = m * 0 + r2 := by rw [h4, h10] _ = r2 := by ring done done lemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m := (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a) theorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y rewrite [h1, h2] have h3 : a + b = b + a := by ring show [a]_m + [b]_m = [b]_m + [a]_m from calc [a]_m + [b]_m _ = [a + b]_m := add_class m a b _ = [b + a]_m := by rw [h3] _ = [b]_m + [a]_m := (add_class m b a).symm done theorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X rewrite [h1] have h2 : a * 1 = a := by ring show [a]_m * [1]_m = [a]_m from calc [a]_m * [1]_m _ = [a * 1]_m := mul_class m a 1 _ = [a]_m := by rw [h2] done theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry lemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) : [a]_m * [gcd_c2 m a]_m = [1]_m := by set s : Int := gcd_c1 m a have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m define at h1 rewrite [h1, Nat.cast_one] at h2 --h2 : s * ↑m + gcd_c2 m a * ↑a = 1 rewrite [mul_class, cc_eq_iff_congr] define --Goal : ∃ (c : Int), ↑a * gcd_c2 m a - 1 = ↑m * c apply Exists.intro (-s) show a * (gcd_c2 m a) - 1 = m * (-s) from calc a * (gcd_c2 m a) - 1 _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring _ = 1 + m * (-s) - 1 := by rw [h2] _ = m * (-s) := by ring done theorem Theorem_7_3_7 (m a : Nat) : invertible [a]_m ↔ rel_prime m a := by apply Iff.intro · -- (→) assume h1 : invertible [a]_m define at h1 obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1 obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y rewrite [h3, mul_class, cc_eq_iff_congr] at h2 define at h2 obtain (c : Int) (h4 : a * b - 1 = m * c) from h2 rewrite [Exercise_7_2_6] --Goal : ∃ (s t : Int), s * ↑m + t * ↑a = 1 apply Exists.intro (-c) apply Exists.intro b show (-c) * m + b * a = 1 from calc (-c) * m + b * a _ = (-c) * m + (a * b - 1) + 1 := by ring _ = (-c) * m + m * c + 1 := by rw [h4] _ = 1 := by ring done · -- (←) assume h1 : rel_prime m a define show ∃ (Y : ZMod m), [a]_m * Y = [1]_m from Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1) done done /- Section 7.4 -/ section Euler open Euler lemma num_rp_below_base {m : Nat} : num_rp_below m 0 = 0 := by rfl lemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) : num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by have h1 : num_rp_below m (j + 1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : gcd m j = 1 rewrite [if_pos h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1 show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1 done lemma num_rp_below_step_not_rp {m j : Nat} (h : ¬rel_prime m j) : num_rp_below m (j + 1) = num_rp_below m j := by have h1 : num_rp_below m (j +1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : ¬gcd m j = 1 rewrite [if_neg h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j show num_rp_below m (j + 1) = num_rp_below m j from h1 done lemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl #eval phi 10 --Answer: 4 lemma prod_inv_iff_inv {m : Nat} {X : ZMod m} (h1 : invertible X) (Y : ZMod m) : invertible (X * Y) ↔ invertible Y := by apply Iff.intro · -- (→) assume h2 : invertible (X * Y) obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2 apply Exists.intro (X * Z) rewrite [←h3] --Goal : Y * (X * Z) = X * Y * Z ring --Note that ring can do algebra in ZMod m done · -- (←) assume h2 : invertible Y obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1 obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2 apply Exists.intro (Xi * Yi) show (X * Y) * (Xi * Yi) = [1]_m from calc X * Y * (Xi * Yi) _ = (X * Xi) * (Y * Yi) := by ring _ = [1]_m * [1]_m := by rw [h3, h4] _ = [1]_m := Theorem_7_3_6_7 [1]_m done done lemma F_rp_def {m i : Nat} (h : rel_prime m i) : F m i = [i]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h --h : gcd m i = 1 rewrite [if_pos h] at h1 show F m i = [i]_m from h1 done lemma F_not_rp_def {m i : Nat} (h : ¬rel_prime m i) : F m i = [1]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h rewrite [h1, if_neg h] rfl done lemma prod_seq_base {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 0 k f = [1]_m := by rfl lemma prod_seq_step {m : Nat} (n k : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl lemma prod_seq_zero_step {m : Nat} (n : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by rewrite [prod_seq_step, zero_add] rfl done lemma prod_one {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 1 k f = f k := by rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7] rfl done lemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl lemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m := calc [G m a i]_m _ = [(a * i) % m]_m := by rfl _ = [a * i]_m := (cc_eq_mod m (a * i)).symm _ = [a]_m * [i]_m := (mul_class m a i).symm lemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) : rel_prime m (G m a i) ↔ rel_prime m i := by have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1 show rel_prime m (G m a i) ↔ rel_prime m i from calc rel_prime m (G m a i) _ ↔ invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm _ ↔ invertible ([a]_m * [i]_m) := by rw [cc_G] _ ↔ invertible [i]_m := prod_inv_iff_inv h2 ([i]_m) _ ↔ rel_prime m i := Theorem_7_3_7 m i done lemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) : F m (G m a i) = [a]_m * F m i := by have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2 show F m (G m a i) = [a]_m * F m i from calc F m (G m a i) _ = [G m a i]_m := F_rp_def h3 _ = [a]_m * [i]_m := cc_G m a i _ = [a]_m * F m i := by rw [F_rp_def h2] done lemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : ¬rel_prime m i) : F m (G m a i) = [1]_m := by rewrite [←G_rp_iff h1 i] at h2 show F m (G m a i) = [1]_m from F_not_rp_def h2 done lemma FG_prod {m a : Nat} (h1 : rel_prime m a) : ∀ (k : Nat), prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by by_induc · -- Base Case show prod_seq 0 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from calc prod_seq 0 0 ((F m) ∘ (G m a)) _ = [1]_m := prod_seq_base _ _ _ = [a]_m ^ 0 * [1]_m := by ring _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by rw [num_rp_below_base, prod_seq_base] done · -- Induction Step fix k : Nat assume ih : prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) by_cases h2 : rel_prime m k · -- Case 1. h2 : rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([a]_m * F m k) := by rw [FG_rp h1 h2] _ = [a]_m ^ ((num_rp_below m k) + 1) * ((prod_seq k 0 (F m)) * F m k) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_rp h2, prod_seq_zero_step] done · -- Case 2. h2 : ¬rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([1]_m) := by rw [FG_not_rp h1 h2] _ = [a]_m ^ (num_rp_below m k) * (prod_seq k 0 (F m) * ([1]_m)) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_not_rp h2, prod_seq_zero_step, F_not_rp_def h2] done done done lemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by define --Goal : ∀ i < m, G m a i < m fix i : Nat assume h1 : i < m rewrite [G_def] --Goal : a * i % m < m show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m) done lemma left_inv_one_one_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g' (g i) = i) : one_one_below n g := sorry lemma right_inv_onto_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g (g' i) = i) (h2 : maps_below n g') : onto_below n g := by define at h2; define fix k : Nat assume h3 : k < n apply Exists.intro (g' k) show g' k < n ∧ g (g' k) = k from And.intro (h2 k h3) (h1 k h3) done lemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m * [inv_mod m a]_m = [1]_m := by have h2 : 0 ≤ (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a) show [a]_m * [inv_mod m a]_m = [1]_m from calc [a]_m * [inv_mod m a]_m _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2] _ = [a]_m * [gcd_c2 m a]_m := by rw [←cc_eq_mod] _ = [1]_m := gcd_c2_inv h1 done lemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m := calc a * (b % m) % m = a % m * (b % m % m) % m := Nat.mul_mod _ _ _ _ = a % m * (b % m) % m := by rw [Nat.mod_mod] _ = a * b % m := (Nat.mul_mod _ _ _).symm lemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm] rfl done theorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] : ↑a ≡ ↑b (MOD m) ↔ a % m = b % m := sorry lemma mul_inv_mod_cancel {m a i : Nat} [NeZero m] (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1 rewrite [mul_class, cc_eq_iff_congr, ←Nat.cast_mul, ←Nat.cast_one, congr_iff_mod_eq_Nat] at h3 show a * inv_mod m a * i % m = i from calc a * (inv_mod m a) * i % m _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = 1 % m * i % m := by rw [h3] _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = i % m := by rw [one_mul] _ = i := Nat.mod_eq_of_lt h2 done lemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl lemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, G m a (Ginv m a i) = i := by fix i : Nat assume h2 : i < m show G m a (Ginv m a i) = i from calc G m a (Ginv m a i) _ = a * ((inv_mod m a * i) % m) % m := by rfl _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, Ginv m a (G m a i) = i := by fix i : Nat assume h2 : i < m show Ginv m a (G m a i) = i from calc Ginv m a (G m a i) _ = inv_mod m a * ((a * i) % m) % m := by rfl _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc, mul_comm (inv_mod m a)] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_maps_below (m a : Nat) [NeZero m] : maps_below m (Ginv m a) := G_maps_below m (inv_mod m a) lemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : one_one_below m (G m a) := left_inv_one_one_below (Ginv_left_inv h1) lemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : onto_below m (G m a) := right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a) lemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : perm_below m (G m a) := And.intro (G_maps_below m a) (And.intro (G_one_one_below h1) (G_onto_below h1)) --Permuting a product of congruence classes doesn't change product lemma swap_fst (u v : Nat) : swap u v u = v := by define : swap u v u --Goal : (if u = u then v else if u = v then u else u) = v have h : u = u := by rfl rewrite [if_pos h] rfl done lemma swap_snd (u v : Nat) : swap u v v = u := by define : swap u v v by_cases h1 : v = u · -- Case 1. h1 : v = u rewrite [if_pos h1] show v = u from h1 done · -- Case 2. h1 : v ≠ u rewrite [if_neg h1] have h2 : v = v := by rfl rewrite [if_pos h2] rfl done done lemma swap_other {u v i : Nat} (h1 : i ≠ u) (h2 : i ≠ v) : swap u v i = i := by define : swap u v i rewrite [if_neg h1, if_neg h2] rfl done lemma swap_values (u v i : Nat) : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := by by_cases h1 : i = u · -- Case 1. h1 : i = u apply Or.inl rewrite [h1] show swap u v u = v from swap_fst u v done · -- Case 2. h1 : i ≠ u apply Or.inr by_cases h2 : i = v · -- Case 2.1. h2 : i = v apply Or.inl rewrite [h2] show swap u v v = u from swap_snd u v done · -- Case 2.2. h2 : i ≠ v apply Or.inr show swap u v i = i from swap_other h1 h2 done done done lemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by define fix i : Nat assume h3 : i < n have h4 : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := swap_values u v i by_cases on h4 · -- Case 1. h4 : swap u v i = v rewrite [h4] show v < n from h2 done · -- Case 2. by_cases on h4 · -- Case 2.1. h4 : swap u v i = u rewrite [h4] show u < n from h1 done · -- Case 2.2. h4 : swap u v i = i rewrite [h4] show i < n from h3 done done done lemma swap_swap (u v n : Nat) : ∀ i < n, swap u v (swap u v i) = i := by fix i : Nat assume h : i < n by_cases h1 : i = u · -- Case 1. h1 : i = u rewrite [h1, swap_fst, swap_snd] rfl done · -- Case 2. h1 : i ≠ u by_cases h2 : i = v · -- Case 2.1. h2 : i = v rewrite [h2, swap_snd, swap_fst] rfl done · -- Case 2.2. h2 : i ≠ v rewrite [swap_other h1 h2, swap_other h1 h2] rfl done done done lemma swap_one_one_below (u v n) : one_one_below n (swap u v) := left_inv_one_one_below (swap_swap u v n) lemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) := right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2) lemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) := And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2)) lemma comp_perm_below {n : Nat} {f g : Nat → Nat} (h1 : perm_below n f) (h2 : perm_below n g) : perm_below n (f ∘ g) := sorry lemma trivial_swap (u : Nat) : swap u u = id := by apply funext fix x : Nat by_cases h1 : x = u · -- Case 1. h1 : x = u rewrite [h1, swap_fst] rfl done · -- Case 2. h1 : x ≠ u rewrite [swap_other h1 h1] rfl done done lemma prod_eq_fun {m : Nat} (f g : Nat → ZMod m) (k : Nat) : ∀ (n : Nat), (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g := by by_induc · -- Base Case assume h : (∀ i < 0, f (k + i) = g (k + i)) rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g assume h1 : ∀ i < n + 1, f (k + i) = g (k + i) have h2 : ∀ i < n, f (k + i) = g (k + i) := by fix i : Nat assume h2 : i < n have h3 : i < n + 1 := by linarith show f (k + i) = g (k + i) from h1 i h3 done have h3 : prod_seq n k f = prod_seq n k g := ih h2 have h4 : n < n + 1 := Nat.lt_succ_self n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4] rfl done done lemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := by have h2 : ∀ (i : Nat), i < u → (f ∘ swap u n) (0 + i) = f (0 + i) := by fix i : Nat assume h2 : i < u have h3 : 0 + i ≠ u := by linarith have h4 : 0 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl done show prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f from prod_eq_fun (f ∘ swap u n) f 0 u h2 done lemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := by have h2 : ∀ i < j, (f ∘ swap u n) (u + 1 + i) = f (u + 1 + i) := by fix i : Nat assume h2 : i < j have h3 : u + 1 + i ≠ u := by linarith have h4 : u + 1 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl show prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f from prod_eq_fun (f ∘ swap u n) f (u + 1) j h2 done lemma break_prod {m : Nat} (n : Nat) (f : Nat → ZMod m) : ∀ (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by by_induc · -- Base Case have h : n + 0 = n := by rfl rewrite [prod_seq_base, h, Theorem_7_3_6_7] rfl done · -- Induction Step fix j : Nat assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f rewrite [←add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc] rfl done done lemma break_prod_twice {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f := break_prod n f 1 rewrite [prod_one] at h2 have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f := break_prod (u + 1) f j rewrite [←h1] at h3 have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f := break_prod u f 1 rewrite [prod_one] at h4 rewrite [h3, h4] at h2 show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2 done lemma swap_prod_eq_prod {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f := by by_cases h2 : u = n · -- Case 1. h2 : u = n rewrite [h2, trivial_swap n] --Goal : prod_seq (n + 1) 0 (f ∘ id) = prod_seq (n + 1) 0 f rfl done · -- Case 2. h2 : ¬u = n have h3 : u + 1 ≤ n := Nat.lt_of_le_of_ne h1 h2 obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3 have break_f : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := break_prod_twice f h4 have break_fs : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_prod_twice (f ∘ swap u n) h4 have f_eq_fs_below : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := swap_prod_eq_prod_below f h1 have f_eq_fs_btwn : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4 show prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f from calc prod_seq (n + 1) 0 (f ∘ swap u n) _ = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_fs _ = prod_seq u 0 f * (f ∘ swap u n) u * prod_seq j (u + 1) f * (f ∘ swap u n) n := by rw [f_eq_fs_below, f_eq_fs_btwn] _ = prod_seq u 0 f * f (swap u n u) * prod_seq j (u + 1) f * f (swap u n n) := by rfl _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by rw [swap_fst, swap_snd] _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring _ = prod_seq (n + 1) 0 f := break_f.symm done done lemma perm_below_fixed {n : Nat} {g : Nat → Nat} (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry lemma perm_prod {m : Nat} (f : Nat → ZMod m) : ∀ (n : Nat), ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) := by by_induc · -- Base Case fix g : Nat → Nat assume h1 : perm_below 0 g rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) fix g : Nat → Nat assume g_pb : perm_below (n + 1) g define at g_pb have g_ob : onto_below (n + 1) g := g_pb.right.right define at g_ob have h1 : n < n + 1 := by linarith obtain (u : Nat) (h2 : u < n + 1 ∧ g u = n) from g_ob n h1 have s_pb : perm_below (n + 1) (swap u n) := swap_perm_below h2.left h1 have gs_pb_n1 : perm_below (n + 1) (g ∘ swap u n) := comp_perm_below g_pb s_pb have gs_fix_n : (g ∘ swap u n) n = n := calc (g ∘ swap u n) n _ = g (swap u n n) := by rfl _ = g u := by rw [swap_snd] _ = n := h2.right have gs_pb_n : perm_below n (g ∘ swap u n) := perm_below_fixed gs_pb_n1 gs_fix_n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f ∘ (g ∘ swap u n)) := ih (g ∘ swap u n) gs_pb_n have h3 : u ≤ n := by linarith show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f ∘ g) from calc prod_seq (n + 1) 0 f _ = prod_seq n 0 f * f n := prod_seq_zero_step n f _ = prod_seq n 0 (f ∘ (g ∘ swap u n)) * f ((g ∘ swap u n) n) := by rw [gs_prod, gs_fix_n] _ = prod_seq n 0 (f ∘ g ∘ swap u n) * (f ∘ g ∘ swap u n) n := by rfl _ = prod_seq (n + 1) 0 (f ∘ g ∘ swap u n) := (prod_seq_zero_step n (f ∘ g ∘ swap u n)).symm _ = prod_seq (n + 1) 0 ((f ∘ g) ∘ swap u n) := by rfl _ = prod_seq (n + 1) 0 (f ∘ g) := swap_prod_eq_prod (f ∘ g) h3 done done lemma F_invertible (m i : Nat) : invertible (F m i) := by by_cases h : rel_prime m i · -- Case 1. h : rel_prime m i rewrite [F_rp_def h] show invertible [i]_m from (Theorem_7_3_7 m i).rtl h done · -- Case 2. h : ¬rel_prime m i rewrite [F_not_rp_def h] apply Exists.intro [1]_m show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m done done lemma Fprod_invertible (m : Nat) : ∀ (k : Nat), invertible (prod_seq k 0 (F m)) := by by_induc · -- Base Case apply Exists.intro [1]_m show prod_seq 0 0 (F m) * [1]_m = [1]_m from calc prod_seq 0 0 (F m) * [1]_m _ = [1]_m * [1]_m := by rw [prod_seq_base] _ = [1]_m := Theorem_7_3_6_7 ([1]_m) done · -- Induction Step fix k : Nat assume ih : invertible (prod_seq k 0 (F m)) rewrite [prod_seq_zero_step] show invertible (prod_seq k 0 (F m) * (F m k)) from (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k) done done theorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m ^ (phi m) = [1]_m := by have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2 show [a]_m ^ (phi m) = [1]_m from calc [a]_m ^ (phi m) _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3] _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring _ = prod_seq m 0 (F m ∘ G m a) * Y := by rw [FG_prod h1 m, phi_def] _ = prod_seq m 0 (F m) * Y := by rw [perm_prod (F m) m (G m a) (G_perm_below h1)] _ = [1]_m := by rw [h3] done lemma Exercise_7_4_5_Int (m : Nat) (a : Int) : ∀ (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry lemma Exercise_7_4_5_Nat (m a n : Nat) : [a]_m ^ n = [a ^ n]_m := by rewrite [Exercise_7_4_5_Int] rfl done theorem Euler's_theorem {m a : Nat} [NeZero m] (h1 : rel_prime m a) : a ^ (phi m) ≡ 1 (MOD m) := by have h2 : [a]_m ^ (phi m) = [1]_m := Theorem_7_4_2 h1 rewrite [Exercise_7_4_5_Nat m a (phi m)] at h2 --h2 : [a ^ phi m]_m = [1]_m show a ^ (phi m) ≡ 1 (MOD m) from (cc_eq_iff_congr _ _ _).ltr h2 done #eval gcd 10 7 --Answer: 1. So 10 and 7 are relatively prime #eval 7 ^ phi 10 --Answer: 2401, which is congruent to 1 mod 10. end Euler /- Section 7.5 -/ lemma num_rp_prime {p : Nat} (h1 : prime p) : ∀ k < p, num_rp_below p (k + 1) = k := sorry lemma phi_prime {p : Nat} (h1 : prime p) : phi p = p - 1 := by have h2 : 1 ≤ p := prime_pos h1 have h3 : p - 1 + 1 = p := Nat.sub_add_cancel h2 have h4 : p - 1 < p := by linarith have h5 : num_rp_below p (p - 1 + 1) = p - 1 := num_rp_prime h1 (p - 1) h4 rewrite [h3] at h5 show phi p = p - 1 from h5 done theorem Theorem_7_2_2_Int {a c : Nat} {b : Int} (h1 : ↑c ∣ ↑a * b) (h2 : rel_prime a c) : ↑c ∣ b := by rewrite [Int.natCast_dvd, Int.natAbs_mul, Int.natAbs_ofNat] at h1 --h1 : c ∣ a * Int.natAbs b rewrite [Int.natCast_dvd] --Goal : c ∣ Int.natAbs b show c ∣ Int.natAbs b from Theorem_7_2_2 h1 h2 done lemma Lemma_7_4_5 {m n : Nat} (a b : Int) (h1 : rel_prime m n) : a ≡ b (MOD m * n) ↔ a ≡ b (MOD m) ∧ a ≡ b (MOD n) := by apply Iff.intro · -- (→) assume h2 : a ≡ b (MOD m * n) obtain (j : Int) (h3 : a - b = (m * n) * j) from h2 apply And.intro · -- Proof of a ≡ b (MOD m) apply Exists.intro (n * j) show a - b = m * (n * j) from calc a - b _ = m * n * j := h3 _ = m * (n * j) := by ring done · -- Proof of a ≡ b (MOD n) apply Exists.intro (m * j) show a - b = n * (m * j) from calc a - b _ = m * n * j := h3 _ = n * (m * j) := by ring done done · -- (←) assume h2 : a ≡ b (MOD m) ∧ a ≡ b (MOD n) obtain (j : Int) (h3 : a - b = m * j) from h2.left have h4 : (↑n : Int) ∣ a - b := h2.right rewrite [h3] at h4 --h4 : ↑n ∣ ↑m * j have h5 : ↑n ∣ j := Theorem_7_2_2_Int h4 h1 obtain (k : Int) (h6 : j = n * k) from h5 apply Exists.intro k --Goal : a - b = ↑(m * n) * k rewrite [Nat.cast_mul] --Goal : a - b = ↑m * ↑n * k show a - b = (m * n) * k from calc a - b _ = m * j := h3 _ = m * (n * k) := by rw [h6] _ = (m * n) * k := by ring done done --From exercises of Section 7.2 theorem rel_prime_symm {a b : Nat} (h : rel_prime a b) : rel_prime b a := sorry lemma prime_NeZero {p : Nat} (h : prime p) : NeZero p := by rewrite [neZero_iff] --Goal : p ≠ 0 define at h linarith done lemma Lemma_7_5_1 {p e d m c s : Nat} {t : Int} (h1 : prime p) (h2 : e * d = (p - 1) * s + 1) (h3 : m ^ e - c = p * t) : c ^ d ≡ m (MOD p) := by have h4 : m ^ e ≡ c (MOD p) := Exists.intro t h3 have h5 : [m ^ e]_p = [c]_p := (cc_eq_iff_congr _ _ _).rtl h4 rewrite [←Exercise_7_4_5_Nat] at h5 --h5 : [m]_p ^ e = [c]_p by_cases h6 : p ∣ m · -- Case 1. h6 : p ∣ m have h7 : m ≡ 0 (MOD p) := by obtain (j : Nat) (h8 : m = p * j) from h6 apply Exists.intro (↑j : Int) --Goal : ↑m - 0 = ↑p * ↑j rewrite [h8, Nat.cast_mul] ring done have h8 : [m]_p = [0]_p := (cc_eq_iff_congr _ _ _).rtl h7 have h9 : e * d ≠ 0 := by rewrite [h2] show (p - 1) * s + 1 ≠ 0 from Nat.add_one_ne_zero _ done have h10 : (0 : Int) ^ (e * d) = 0 := zero_pow h9 have h11 : [c ^ d]_p = [m]_p := calc [c ^ d]_p _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat] _ = ([m]_p ^ e) ^ d := by rw [h5] _ = [m]_p ^ (e * d) := by ring _ = [0]_p ^ (e * d) := by rw [h8] _ = [0 ^ (e * d)]_p := Exercise_7_4_5_Int _ _ _ _ = [0]_p := by rw [h10] _ = [m]_p := by rw [h8] show c ^ d ≡ m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11 done · -- Case 2. h6 : ¬p ∣ m have h7 : rel_prime m p := rel_prime_of_prime_not_dvd h1 h6 have h8 : rel_prime p m := rel_prime_symm h7 have h9 : NeZero p := prime_NeZero h1 have h10 : (1 : Int) ^ s = 1 := by ring have h11 : [c ^ d]_p = [m]_p := calc [c ^ d]_p _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat] _ = ([m]_p ^ e) ^ d := by rw [h5] _ = [m]_p ^ (e * d) := by ring _ = [m]_p ^ ((p - 1) * s + 1) := by rw [h2] _ = ([m]_p ^ (p - 1)) ^ s * [m]_p := by ring _ = ([m]_p ^ (phi p)) ^ s * [m]_p := by rw [phi_prime h1] _ = [1]_p ^ s * [m]_p := by rw [Theorem_7_4_2 h8] _ = [1 ^ s]_p * [m]_p := by rw [Exercise_7_4_5_Int] _ = [1]_p * [m]_p := by rw [h10] _ = [m]_p * [1]_p := by ring _ = [m]_p := Theorem_7_3_6_7 _ show c ^ d ≡ m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11 done done theorem Theorem_7_5_1 (p q n e d k m c : Nat) (p_prime : prime p) (q_prime : prime q) (p_ne_q : p ≠ q) (n_pq : n = p * q) (ed_congr_1 : e * d = k * (p - 1) * (q - 1) + 1) (h1 : [m]_n ^ e = [c]_n) : [c]_n ^ d = [m]_n := by rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] at h1 --h1 : m ^ e ≡ c (MOD n) rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] --Goal : c ^ d ≡ m (MOD n) obtain (j : Int) (h2 : m ^ e - c = n * j) from h1 rewrite [n_pq, Nat.cast_mul] at h2 --h2 : m ^ e - c = p * q * j have h3 : e * d = (p - 1) * (k * (q - 1)) + 1 := by rewrite [ed_congr_1] ring done have h4 : m ^ e - c = p * (q * j) := by rewrite [h2] ring done have congr_p : c ^ d ≡ m (MOD p) := Lemma_7_5_1 p_prime h3 h4 have h5 : e * d = (q - 1) * (k * (p - 1)) + 1 := by rewrite [ed_congr_1] ring done have h6 : m ^ e - c = q * (p * j) := by rewrite [h2] ring done have congr_q : c ^ d ≡ m (MOD q) := Lemma_7_5_1 q_prime h5 h6 have h7 : ¬q ∣ p := by by_contra h8 have h9 : q = 1 ∨ q = p := dvd_prime p_prime h8 disj_syll h9 (prime_not_one q_prime) show False from p_ne_q h9.symm done have h8 : rel_prime p q := rel_prime_of_prime_not_dvd q_prime h7 rewrite [n_pq, Lemma_7_4_5 _ _ h8] show c ^ d ≡ m (MOD p) ∧ c ^ d ≡ m (MOD q) from And.intro congr_p congr_q done /- BEGIN EXERCISES -/ namespace Exercises /- Section 7.1 -/ -- 1. theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry -- 2. lemma gcd_comm_lt {a b : Nat} (h : a < b) : gcd a b = gcd b a := sorry theorem gcd_comm (a b : Nat) : gcd a b = gcd b a := sorry -- 3. theorem Exercise_7_1_5 (a b : Nat) (n : Int) : (∃ (s t : Int), s * a + t * b = n) ↔ (↑(gcd a b) : Int) ∣ n := sorry -- 4. theorem Exercise_7_1_6 (a b c : Nat) : gcd a b = gcd (a + b * c) b := sorry -- 5. theorem gcd_is_nonzero {a b : Nat} (h : a ≠ 0 ∨ b ≠ 0) : gcd a b ≠ 0 := sorry -- 6. theorem gcd_greatest {a b d : Nat} (h1 : gcd a b ≠ 0) (h2 : d ∣ a) (h3 : d ∣ b) : d ≤ gcd a b := sorry -- 7. lemma Lemma_7_1_10a {a b : Nat} (n : Nat) (h : a ∣ b) : (n * a) ∣ (n * b) := sorry lemma Lemma_7_1_10b {a b n : Nat} (h1 : n ≠ 0) (h2 : (n * a) ∣ (n * b)) : a ∣ b := sorry lemma Lemma_7_1_10c {a b : Nat} (h1 : a ∣ b) (h2 : b ∣ a) : a = b := sorry theorem Exercise_7_1_10 (a b n : Nat) : gcd (n * a) (n * b) = n * gcd a b := sorry /- Section 7.2 -/ -- 1. lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry -- 2. -- Hints: Start with apply List.rec. You may find mul_ne_zero useful theorem prod_nonzero_nonzero : ∀ (l : List Nat), (∀ a ∈ l, a ≠ 0) → prod l ≠ 0 := sorry -- 3. theorem rel_prime_iff_no_common_factor (a b : Nat) : rel_prime a b ↔ ¬∃ (p : Nat), prime p ∧ p ∣ a ∧ p ∣ b := sorry -- 4. theorem rel_prime_symm {a b : Nat} (h : rel_prime a b) : rel_prime b a := sorry -- 5. lemma in_prime_factorization_iff_prime_factor {a : Nat} {l : List Nat} (h1 : prime_factorization a l) (p : Nat) : p ∈ l ↔ prime_factor p a := sorry -- 6. theorem Exercise_7_2_5 {a b : Nat} {l m : List Nat} (h1 : prime_factorization a l) (h2 : prime_factorization b m) : rel_prime a b ↔ (¬∃ (p : Nat), p ∈ l ∧ p ∈ m) := sorry -- 7. theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry -- 8. theorem Exercise_7_2_7 {a b a' b' : Nat} (h1 : rel_prime a b) (h2 : a' ∣ a) (h3 : b' ∣ b) : rel_prime a' b' := sorry -- 9.

theorem Exercise_7_2_9 {a b j k : Nat} (h1 : gcd a b ≠ 0) (h2 : a = j * gcd a b) (h3 : b = k * gcd a b) : rel_prime j k

HTPI.Exercises.Exercise_7_2_9

htpi/HTPILib/Chap7.lean

htpi

/- Copyright 2023 Daniel J. Velleman -/ import HTPILib.Chap6 namespace HTPI /- Definitions -/ lemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by have h : n + 1 > 0 := Nat.succ_pos n show a % (n + 1) < n + 1 from Nat.mod_lt a h done def gcd (a b : Nat) : Nat := match b with | 0 => a | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd (n + 1) (a % (n + 1)) termination_by b mutual def gcd_c1 (a b : Nat) : Int := match b with | 0 => 1 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c2 (n + 1) (a % (n + 1)) --Corresponds to s = t' termination_by b def gcd_c2 (a b : Nat) : Int := match b with | 0 => 0 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c1 (n + 1) (a % (n + 1)) - (gcd_c2 (n + 1) (a % (n + 1))) * ↑(a / (n + 1)) --Corresponds to t = s' - t'q termination_by b end def prime (n : Nat) : Prop := 2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n def prime_factor (p n : Nat) : Prop := prime p ∧ p ∣ n def all_prime (l : List Nat) : Prop := ∀ p ∈ l, prime p def nondec (l : List Nat) : Prop := match l with | [] => True --Of course, True is a proposition that is always true | n :: L => (∀ m ∈ L, n ≤ m) ∧ nondec L def nondec_prime_list (l : List Nat) : Prop := all_prime l ∧ nondec l def prod (l : List Nat) : Nat := match l with | [] => 1 | n :: L => n * (prod L) def prime_factorization (n : Nat) (l : List Nat) : Prop := nondec_prime_list l ∧ prod l = n def rel_prime (a b : Nat) : Prop := gcd a b = 1 def congr_mod (m : Nat) (a b : Int) : Prop := (↑m : Int) ∣ (a - b) def cc (m : Nat) (a : Int) : ZMod m := (↑a : ZMod m) notation:50 a " ≡ " b " (MOD " m ")" => congr_mod m a b notation:max "["a"]_"m:max => cc m a def invertible {m : Nat} (X : ZMod m) : Prop := ∃ (Y : ZMod m), X * Y = [1]_m def num_rp_below (m k : Nat) : Nat := match k with | 0 => 0 | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j def phi (m : Nat) : Nat := num_rp_below m m def prod_seq {m : Nat} (j k : Nat) (f : Nat → ZMod m) : ZMod m := match j with | 0 => [1]_m | n + 1 => prod_seq n k f * f (k + n) def maps_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i < n, g i < n def one_one_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i1 < n, ∀ i2 < n, g i1 = g i2 → i1 = i2 def onto_below (n : Nat) (g : Nat → Nat) : Prop := ∀ k < n, ∃ i < n, g i = k def perm_below (n : Nat) (g : Nat → Nat) : Prop := maps_below n g ∧ one_one_below n g ∧ onto_below n g def inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m) def swap (u v i : Nat) : Nat := if i = u then v else if i = v then u else i namespace Euler --For definitions specific to Euler's theorem def F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m def G (m a i : Nat) : Nat := (a * i) % m def Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i end Euler /- Section 7.1 -/ theorem dvd_mod_of_dvd_a_b {a b d : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ (a % b) := by set q : Nat := a / b have h3 : b * q + a % b = a := Nat.div_add_mod a b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 define --Goal : ∃ (c : Nat), a % b = d * c apply Exists.intro (j - k * q) show a % b = d * (j - k * q) from calc a % b _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm _ = a - b * q := by rw [h3] _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc] _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm done theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry #eval gcd 672 161 --Answer: 7 lemma gcd_base (a : Nat) : gcd a 0 = a := by rfl lemma gcd_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd a b = gcd b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1)) rfl done lemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b ≠ 0) : a % b < b := by have h1 : b > 0 := Nat.pos_of_ne_zero h show a % b < b from Nat.mod_lt a h1 done lemma dvd_self (n : Nat) : n ∣ n := by apply Exists.intro 1 ring done theorem gcd_dvd : ∀ (b a : Nat), (gcd a b) ∣ a ∧ (gcd a b) ∣ b := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd a b_1) ∣ a ∧ (gcd a b_1) ∣ b_1 fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_base] --Goal: a ∣ a ∧ a ∣ 0 apply And.intro (dvd_self a) define apply Exists.intro 0 rfl done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_nonzero a h1] --Goal : gcd b (a % b) ∣ a ∧ gcd b (a % b) ∣ b have h2 : a % b < b := mod_nonzero_lt a h1 have h3 : (gcd b (a % b)) ∣ b ∧ (gcd b (a % b)) ∣ (a % b) := ih (a % b) h2 b apply And.intro _ h3.left show (gcd b (a % b)) ∣ a from dvd_a_of_dvd_b_mod h3.left h3.right done done theorem gcd_dvd_left (a b : Nat) : (gcd a b) ∣ a := (gcd_dvd b a).left theorem gcd_dvd_right (a b : Nat) : (gcd a b) ∣ b := (gcd_dvd b a).right lemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl lemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c1 a b = gcd_c2 b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done lemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl lemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * ↑(a / b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done theorem gcd_lin_comb : ∀ (b a : Nat), (gcd_c1 a b) * ↑a + (gcd_c2 a b) * ↑b = ↑(gcd a b) := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd_c1 a b_1) * ↑a + (gcd_c2 a b_1) * ↑b_1 = ↑(gcd a b_1) fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base] --Goal : 1 * ↑a + 0 * ↑0 = ↑a ring done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1] --Goal : gcd_c2 b (a % b) * ↑a + -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * ↑(a / b)) * ↑b = -- ↑(gcd b (a % b)) set r : Nat := a % b set q : Nat := a / b set s : Int := gcd_c1 b r set t : Int := gcd_c2 b r --Goal : t * ↑a + (s - t * ↑q) * ↑b = ↑(gcd b r) have h2 : r < b := mod_nonzero_lt a h1 have h3 : s * ↑b + t * ↑r = ↑(gcd b r) := ih r h2 b have h4 : b * q + r = a := Nat.div_add_mod a b rewrite [←h3, ←h4] rewrite [Nat.cast_add, Nat.cast_mul] --Goal : t * (↑b * ↑q + ↑r) + (s - t * ↑q) * ↑b = s * ↑b + t * ↑r ring done done #eval gcd_c1 672 161 --Answer: 6 #eval gcd_c2 672 161 --Answer: -25 --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161 theorem Theorem_7_1_6 {d a b : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ gcd a b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑d ∣ ↑(gcd a b) set s : Int := gcd_c1 a b set t : Int := gcd_c2 a b have h3 : s * ↑a + t * ↑b = ↑(gcd a b) := gcd_lin_comb b a rewrite [←h3] --Goal : ↑d ∣ s * ↑a + t * ↑b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul] --Goal : ↑d ∣ s * (↑d * ↑j) + t * (↑d * ↑k) define apply Exists.intro (s * ↑j + t * ↑k) ring done /- Section 7.2 -/ theorem dvd_trans {a b c : Nat} (h1 : a ∣ b) (h2 : b ∣ c) : a ∣ c := by define at h1; define at h2; define obtain (m : Nat) (h3 : b = a * m) from h1 obtain (n : Nat) (h4 : c = b * n) from h2 rewrite [h3, mul_assoc] at h4 apply Exists.intro (m * n) show c = a * (m * n) from h4 done lemma exists_prime_factor : ∀ (n : Nat), 2 ≤ n → ∃ (p : Nat), prime_factor p n := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, 2 ≤ n_1 → ∃ (p : Nat), prime_factor p n_1 assume h1 : 2 ≤ n by_cases h2 : prime n · -- Case 1. h2 : prime n apply Exists.intro n define --Goal : prime n ∧ n ∣ n show prime n ∧ n ∣ n from And.intro h2 (dvd_self n) done · -- Case 2. h2 : ¬prime n define at h2 --h2 : ¬(2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n) demorgan at h2 disj_syll h2 h1 obtain (a : Nat) (h3 : ∃ (b : Nat), a * b = n ∧ a < n ∧ b < n) from h2 obtain (b : Nat) (h4 : a * b = n ∧ a < n ∧ b < n) from h3 have h5 : 2 ≤ a := by by_contra h6 have h7 : a ≤ 1 := by linarith have h8 : n ≤ b := calc n _ = a * b := h4.left.symm _ ≤ 1 * b := by rel [h7] _ = b := by ring linarith --n ≤ b contradicts b < n done have h6 : ∃ (p : Nat), prime_factor p a := ih a h4.right.left h5 obtain (p : Nat) (h7 : prime_factor p a) from h6 apply Exists.intro p define --Goal : prime p ∧ p ∣ n define at h7 --h7 : prime p ∧ p ∣ a apply And.intro h7.left have h8 : a ∣ n := by apply Exists.intro b show n = a * b from (h4.left).symm done show p ∣ n from dvd_trans h7.right h8 done done lemma exists_least_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q := by set S : Set Nat := {p : Nat | prime_factor p n} have h2 : ∃ (p : Nat), p ∈ S := exists_prime_factor n h show ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q from well_ord_princ S h2 done lemma all_prime_nil : all_prime [] := by define --Goal : ∀ p ∈ [], prime p fix p : Nat contrapos --Goal : ¬prime p → p ∉ [] assume h1 : ¬prime p show p ∉ [] from List.not_mem_nil p done lemma all_prime_cons (n : Nat) (L : List Nat) : all_prime (n :: L) ↔ prime n ∧ all_prime L := by apply Iff.intro · -- (→) assume h1 : all_prime (n :: L) --Goal : prime n ∧ all_prime L define at h1 --h1 : ∀ p ∈ n :: L, prime p apply And.intro (h1 n (List.mem_cons_self n L)) define --Goal : ∀ p ∈ L, prime p fix p : Nat assume h2 : p ∈ L show prime p from h1 p (List.mem_cons_of_mem n h2) done · -- (←) assume h1 : prime n ∧ all_prime L --Goal : all_prime (n :: l) define : all_prime L at h1 define fix p : Nat assume h2 : p ∈ n :: L rewrite [List.mem_cons] at h2 --h2 : p = n ∨ p ∈ L by_cases on h2 · -- Case 1. h2 : p = n rewrite [h2] show prime n from h1.left done · -- Case 2. h2 : p ∈ L show prime p from h1.right p h2 done done done lemma nondec_nil : nondec [] := by define --Goal : True trivial --trivial proves some obviously true statements, such as True done lemma nondec_cons (n : Nat) (L : List Nat) : nondec (n :: L) ↔ (∀ m ∈ L, n ≤ m) ∧ nondec L := by rfl lemma prod_nil : prod [] = 1 := by rfl lemma prod_cons : prod (n :: L) = n * (prod L) := by rfl lemma exists_cons_of_length_eq_succ {A : Type} {l : List A} {n : Nat} (h : l.length = n + 1) : ∃ (a : A) (L : List A), l = a :: L ∧ L.length = n := by have h1 : ¬l.length = 0 := by linarith rewrite [List.length_eq_zero] at h1 obtain (a : A) (h2 : ∃ (L : List A), l = a :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List A) (h3 : l = a :: L) from h2 apply Exists.intro a apply Exists.intro L apply And.intro h3 have h4 : (a :: L).length = L.length + 1 := List.length_cons a L rewrite [←h3, h] at h4 show L.length = n from (Nat.add_right_cancel h4).symm done lemma list_elt_dvd_prod_by_length (a : Nat) : ∀ (n : Nat), ∀ (l : List Nat), l.length = n → a ∈ l → a ∣ prod l := by by_induc · --Base Case fix l : List Nat assume h1 : l.length = 0 rewrite [List.length_eq_zero] at h1 --h1 : l = [] rewrite [h1] --Goal : a ∈ [] → a ∣ prod [] contrapos assume h2 : ¬a ∣ prod [] show a ∉ [] from List.not_mem_nil a done · -- Induction Step fix n : Nat assume ih : ∀ (l : List Nat), List.length l = n → a ∈ l → a ∣ prod l fix l : List Nat assume h1 : l.length = n + 1 --Goal : a ∈ l → a ∣ prod l obtain (b : Nat) (h2 : ∃ (L : List Nat), l = b :: L ∧ L.length = n) from exists_cons_of_length_eq_succ h1 obtain (L : List Nat) (h3 : l = b :: L ∧ L.length = n) from h2 have h4 : a ∈ L → a ∣ prod L := ih L h3.right assume h5 : a ∈ l rewrite [h3.left, prod_cons] --Goal : a ∣ b * prod L rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b ∨ a ∈ L by_cases on h5 · -- Case 1. h5 : a = b apply Exists.intro (prod L) rewrite [h5] rfl done · -- Case 2. h5 : a ∈ L have h6 : a ∣ prod L := h4 h5 have h7 : prod L ∣ b * prod L := by apply Exists.intro b ring done show a ∣ b * prod L from dvd_trans h6 h7 done done done lemma list_elt_dvd_prod {a : Nat} {l : List Nat} (h : a ∈ l) : a ∣ prod l := by set n : Nat := l.length have h1 : l.length = n := by rfl show a ∣ prod l from list_elt_dvd_prod_by_length a n l h1 h done lemma exists_prime_factorization : ∀ (n : Nat), n ≥ 1 → ∃ (l : List Nat), prime_factorization n l := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, n_1 ≥ 1 → ∃ (l : List Nat), prime_factorization n_1 l assume h1 : n ≥ 1 by_cases h2 : n = 1 · -- Case 1. h2 : n = 1 apply Exists.intro [] define apply And.intro · -- Proof of nondec_prime_list [] define show all_prime [] ∧ nondec [] from And.intro all_prime_nil nondec_nil done · -- Proof of prod [] = n rewrite [prod_nil, h2] rfl done done · -- Case 2. h2 : n ≠ 1 have h3 : n ≥ 2 := lt_of_le_of_ne' h1 h2 obtain (p : Nat) (h4 : prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q) from exists_least_prime_factor h3 have p_prime_factor : prime_factor p n := h4.left define at p_prime_factor have p_prime : prime p := p_prime_factor.left have p_dvd_n : p ∣ n := p_prime_factor.right have p_least : ∀ (q : Nat), prime_factor q n → p ≤ q := h4.right obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n have h5 : m ≠ 0 := by contradict h1 with h6 have h7 : n = 0 := calc n _ = p * m := n_eq_pm _ = p * 0 := by rw [h6] _ = 0 := by ring rewrite [h7] decide done have m_pos : 0 < m := Nat.pos_of_ne_zero h5 have m_lt_n : m < n := by define at p_prime show m < n from calc m _ < m + m := by linarith _ = 2 * m := by ring _ ≤ p * m := by rel [p_prime.left] _ = n := n_eq_pm.symm done obtain (L : List Nat) (h6 : prime_factorization m L) from ih m m_lt_n m_pos define at h6 have ndpl_L : nondec_prime_list L := h6.left define at ndpl_L apply Exists.intro (p :: L) define apply And.intro · -- Proof of nondec_prime_list (p :: L) define apply And.intro · -- Proof of all_prime (p :: L) rewrite [all_prime_cons] show prime p ∧ all_prime L from And.intro p_prime ndpl_L.left done · -- Proof of nondec (p :: L) rewrite [nondec_cons] apply And.intro _ ndpl_L.right fix q : Nat assume q_in_L : q ∈ L have h7 : q ∣ prod L := list_elt_dvd_prod q_in_L rewrite [h6.right] at h7 --h7 : q ∣ m have h8 : m ∣ n := by apply Exists.intro p rewrite [n_eq_pm] ring done have q_dvd_n : q ∣ n := dvd_trans h7 h8 have ap_L : all_prime L := ndpl_L.left define at ap_L have q_prime_factor : prime_factor q n := And.intro (ap_L q q_in_L) q_dvd_n show p ≤ q from p_least q q_prime_factor done done · -- Proof of prod (p :: L) = n rewrite [prod_cons, h6.right, n_eq_pm] rfl done done done theorem Theorem_7_2_2 {a b c : Nat} (h1 : c ∣ a * b) (h2 : rel_prime a c) : c ∣ b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑c ∣ ↑b define at h1; define at h2; define obtain (j : Nat) (h3 : a * b = c * j) from h1 set s : Int := gcd_c1 a c set t : Int := gcd_c2 a c have h4 : s * ↑a + t * ↑c = ↑(gcd a c) := gcd_lin_comb c a rewrite [h2, Nat.cast_one] at h4 --h4 : s * ↑a + t * ↑c = (1 : Int) apply Exists.intro (s * ↑j + t * ↑b) show ↑b = ↑c * (s * ↑j + t * ↑b) from calc ↑b _ = (1 : Int) * ↑b := (one_mul _).symm _ = (s * ↑a + t * ↑c) * ↑b := by rw [h4] _ = s * (↑a * ↑b) + t * ↑c * ↑b := by ring _ = s * (↑c * ↑j) + t * ↑c * ↑b := by rw [←Nat.cast_mul a b, h3, Nat.cast_mul c j] _ = ↑c * (s * ↑j + t * ↑b) := by ring done lemma le_nonzero_prod_left {a b : Nat} (h : a * b ≠ 0) : a ≤ a * b := by have h1 : b ≠ 0 := by contradict h with h1 rewrite [h1] ring done have h2 : 1 ≤ b := Nat.pos_of_ne_zero h1 show a ≤ a * b from calc a = a * 1 := (mul_one a).symm _ ≤ a * b := by rel [h2] done lemma le_nonzero_prod_right {a b : Nat} (h : a * b ≠ 0) : b ≤ a * b := by rewrite [mul_comm] rewrite [mul_comm] at h show b ≤ b * a from le_nonzero_prod_left h done lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry lemma rel_prime_of_prime_not_dvd {a p : Nat} (h1 : prime p) (h2 : ¬p ∣ a) : rel_prime a p := by have h3 : gcd a p ∣ a := gcd_dvd_left a p have h4 : gcd a p ∣ p := gcd_dvd_right a p have h5 : gcd a p = 1 ∨ gcd a p = p := dvd_prime h1 h4 have h6 : gcd a p ≠ p := by contradict h2 with h6 rewrite [h6] at h3 show p ∣ a from h3 done disj_syll h5 h6 show rel_prime a p from h5 done theorem Theorem_7_2_3 {a b p : Nat} (h1 : prime p) (h2 : p ∣ a * b) : p ∣ a ∨ p ∣ b := by or_right with h3 have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3 show p ∣ b from Theorem_7_2_2 h2 h4 done lemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a ≥ 1 := by have h1 : a ≠ 0 := by by_contra h1 rewrite [h1] at h contradict h linarith done show a ≥ 1 from Nat.pos_of_ne_zero h1 done lemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by have h1 : a ≥ 1 := ge_one_of_prod_one h have h2 : a * b ≠ 0 := by linarith have h3 : a ≤ a * b := le_nonzero_prod_left h2 rewrite [h] at h3 show a = 1 from Nat.le_antisymm h3 h1 done lemma eq_one_of_dvd_one {n : Nat} (h : n ∣ 1) : n = 1 := by obtain (j : Nat) (h1 : 1 = n * j) from h show n = 1 from eq_one_of_prod_one h1.symm done lemma prime_not_one {p : Nat} (h : prime p) : p ≠ 1 := by define at h linarith done theorem Theorem_7_2_4 {p : Nat} (h1 : prime p) : ∀ (l : List Nat), p ∣ prod l → ∃ a ∈ l, p ∣ a := by apply List.rec · -- Base Case. Goal : p ∣ prod [] → ∃ a ∈ [], p ∣ a rewrite [prod_nil] assume h2 : p ∣ 1 show ∃ a ∈ [], p ∣ a from absurd (eq_one_of_dvd_one h2) (prime_not_one h1) done · -- Induction Step fix b : Nat fix L : List Nat assume ih : p ∣ prod L → ∃ a ∈ L, p ∣ a --Goal : p ∣ prod (b :: L) → ∃ a ∈ b :: L, p ∣ a assume h2 : p ∣ prod (b :: L) rewrite [prod_cons] at h2 have h3 : p ∣ b ∨ p ∣ prod L := Theorem_7_2_3 h1 h2 by_cases on h3 · -- Case 1. h3 : p ∣ b apply Exists.intro b show b ∈ b :: L ∧ p ∣ b from And.intro (List.mem_cons_self b L) h3 done · -- Case 2. h3 : p ∣ prod L obtain (a : Nat) (h4 : a ∈ L ∧ p ∣ a) from ih h3 apply Exists.intro a show a ∈ b :: L ∧ p ∣ a from And.intro (List.mem_cons_of_mem b h4.left) h4.right done done done lemma prime_in_list {p : Nat} {l : List Nat} (h1 : prime p) (h2 : all_prime l) (h3 : p ∣ prod l) : p ∈ l := by obtain (a : Nat) (h4 : a ∈ l ∧ p ∣ a) from Theorem_7_2_4 h1 l h3 define at h2 have h5 : prime a := h2 a h4.left have h6 : p = 1 ∨ p = a := dvd_prime h5 h4.right disj_syll h6 (prime_not_one h1) rewrite [h6] show a ∈ l from h4.left done lemma first_le_first {p q : Nat} {l m : List Nat} (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m)) (h3 : prod (p :: l) = prod (q :: m)) : p ≤ q := by define at h1; define at h2 have h4 : q ∣ prod (p :: l) := by define apply Exists.intro (prod m) rewrite [←prod_cons] show prod (p :: l) = prod (q :: m) from h3 done have h5 : all_prime (q :: m) := h2.left rewrite [all_prime_cons] at h5 have h6 : q ∈ p :: l := prime_in_list h5.left h1.left h4 have h7 : nondec (p :: l) := h1.right rewrite [nondec_cons] at h7 rewrite [List.mem_cons] at h6 by_cases on h6 · -- Case 1. h6 : q = p linarith done · -- Case 2. h6 : q ∈ l have h8 : ∀ m ∈ l, p ≤ m := h7.left show p ≤ q from h8 q h6 done done lemma nondec_prime_list_tail {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by define at h define rewrite [all_prime_cons, nondec_cons] at h show all_prime l ∧ nondec l from And.intro h.left.right h.right.right done lemma cons_prod_not_one {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : prod (p :: l) ≠ 1 := by define at h have h1 : all_prime (p :: l) := h.left rewrite [all_prime_cons] at h1 rewrite [prod_cons] by_contra h2 show False from (prime_not_one h1.left) (eq_one_of_prod_one h2) done lemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) : l = [] ↔ prod l = 1 := by apply Iff.intro · -- (→) assume h1 : l = [] rewrite [h1] show prod [] = 1 from prod_nil done · -- (←) contrapos assume h1 : ¬l = [] obtain (p : Nat) (h2 : ∃ (L : List Nat), l = p :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List Nat) (h3 : l = p :: L) from h2 rewrite [h3] at h rewrite [h3] show ¬prod (p :: L) = 1 from cons_prod_not_one h done done lemma prime_pos {p : Nat} (h : prime p) : p > 0 := by define at h linarith done theorem Theorem_7_2_5 : ∀ (l1 l2 : List Nat), nondec_prime_list l1 → nondec_prime_list l2 → prod l1 = prod l2 → l1 = l2 := by apply List.rec · -- Base Case. Goal : ∀ (l2 : List Nat), nondec_prime_list [] → -- nondec_prime_list l2 → prod [] = prod l2 → [] = l2 fix l2 : List Nat assume h1 : nondec_prime_list [] assume h2 : nondec_prime_list l2 assume h3 : prod [] = prod l2 rewrite [prod_nil, eq_comm, ←list_nil_iff_prod_one h2] at h3 show [] = l2 from h3.symm done · -- Induction Step fix p : Nat fix L1 : List Nat assume ih : ∀ (L2 : List Nat), nondec_prime_list L1 → nondec_prime_list L2 → prod L1 = prod L2 → L1 = L2 -- Goal : ∀ (l2 : List Nat), nondec_prime_list (p :: L1) → -- nondec_prime_list l2 → prod (p :: L1) = prod l2 → p :: L1 = l2 fix l2 : List Nat assume h1 : nondec_prime_list (p :: L1) assume h2 : nondec_prime_list l2 assume h3 : prod (p :: L1) = prod l2 have h4 : ¬prod (p :: L1) = 1 := cons_prod_not_one h1 rewrite [h3, ←list_nil_iff_prod_one h2] at h4 obtain (q : Nat) (h5 : ∃ (L : List Nat), l2 = q :: L) from List.exists_cons_of_ne_nil h4 obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5 rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2) rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2) have h7 : p ≤ q := first_le_first h1 h2 h3 have h8 : q ≤ p := first_le_first h2 h1 h3.symm have h9 : p = q := by linarith rewrite [h9, prod_cons, prod_cons] at h3 --h3 : q * prod L1 = q * prod L2 have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1 have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2 define at h2 have h12 : all_prime (q :: L2) := h2.left rewrite [all_prime_cons] at h12 have h13 : q > 0 := prime_pos h12.left have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3 have h15 : L1 = L2 := ih L2 h10 h11 h14 rewrite [h6, h9, h15] rfl done done theorem fund_thm_arith (n : Nat) (h : n ≥ 1) : ∃! (l : List Nat), prime_factorization n l := by exists_unique · -- Existence show ∃ (l : List Nat), prime_factorization n l from exists_prime_factorization n h done · -- Uniqueness fix l1 : List Nat; fix l2 : List Nat assume h1 : prime_factorization n l1 assume h2 : prime_factorization n l2 define at h1; define at h2 have h3 : prod l1 = n := h1.right rewrite [←h2.right] at h3 show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3 done done /- Section 7.3 -/ theorem congr_refl (m : Nat) : ∀ (a : Int), a ≡ a (MOD m) := by fix a : Int define --Goal : ∃ (c : Int), a - a = ↑m * c apply Exists.intro 0 ring done theorem congr_symm {m : Nat} : ∀ {a b : Int}, a ≡ b (MOD m) → b ≡ a (MOD m) := by fix a : Int; fix b : Int assume h1 : a ≡ b (MOD m) define at h1 --h1 : ∃ (c : Int), a - b = ↑m * c define --Goal : ∃ (c : Int), b - a = ↑m * c obtain (c : Int) (h2 : a - b = m * c) from h1 apply Exists.intro (-c) show b - a = m * (-c) from calc b - a _ = -(a - b) := by ring _ = -(m * c) := by rw [h2] _ = m * (-c) := by ring done theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry /- Fundamental properties of congruence classes -/ lemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) : X = Y ↔ X.val = Y.val := Fin.ext_iff lemma val_nat_eq_mod (n k : Nat) : ([k]_(n + 1)).val = k % (n + 1) := by rfl lemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl theorem cc_rep {m : Nat} (X : ZMod m) : ∃ (a : Int), X = [a]_m := match m with | 0 => by apply Exists.intro X rfl done | n + 1 => by apply Exists.intro ↑(X.val) have h1 : X.val < n + 1 := Fin.prop X rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1] rfl done theorem add_class (m : Nat) (a b : Int) : [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm theorem mul_class (m : Nat) (a b : Int) : [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm lemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ [a - b]_m = [0]_m := by apply Iff.intro · -- (→) assume h1 : [a]_m = [b]_m have h2 : a - b = a + (-b) := by ring have h3 : b + (-b) = 0 := by ring show [a - b]_m = [0]_m from calc [a - b]_m _ = [a + (-b)]_m := by rw [h2] _ = [a]_m + [-b]_m := by rw [add_class] _ = [b]_m + [-b]_m := by rw [h1] _ = [b + -b]_m := by rw [add_class] _ = [0]_m := by rw [h3] done · -- (←) assume h1 : [a - b]_m = [0]_m have h2 : b + (a - b) = a := by ring have h3 : b + 0 = b := by ring show [a]_m = [b]_m from calc [a]_m _ = [b + (a - b)]_m := by rw [h2] _ = [b]_m + [a - b]_m := by rw [add_class] _ = [b]_m + [0]_m := by rw [h1] _ = [b + 0]_m := by rw [add_class] _ = [b]_m := by rw [h3] done done lemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) : [a]_m = [0]_m → [-a]_m = [0]_m := by assume h1 : [a]_m = [0]_m have h2 : 0 + (-a) = -a := by ring have h3 : a + (-a) = 0 := by ring show [-a]_m = [0]_m from calc [-a]_m _ = [0 + (-a)]_m := by rw [h2] _ = [0]_m + [-a]_m := by rw [add_class] _ = [a]_m + [-a]_m := by rw [h1] _ = [a + (-a)]_m := by rw [add_class] _ = [0]_m := by rw [h3] done lemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) : [-a]_m = [0]_m ↔ [a]_m = [0]_m := by apply Iff.intro _ (cc_neg_zero_of_cc_zero m a) assume h1 : [-a]_m = [0]_m have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1 have h3 : -(-a) = a := by ring rewrite [h3] at h2 show [a]_m = [0]_m from h2 done lemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl lemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m ↔ m ∣ k := match m with | 0 => by have h : (0 : Int) = (↑(0 : Nat) : Int) := by rfl rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj] apply Iff.intro · -- (→) assume h1 : k = 0 rewrite [h1] show 0 ∣ 0 from dvd_self 0 done · -- (←) assume h1 : 0 ∣ k obtain (c : Nat) (h2 : k = 0 * c) from h1 rewrite [h2] ring done done | n + 1 => by rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero] show k % (n + 1) = 0 ↔ n + 1 ∣ k from (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm done lemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m ↔ ↑m ∣ a := by obtain (k : Nat) (h1 : a = ↑k ∨ a = -↑k) from Int.eq_nat_or_neg a by_cases on h1 · -- Case 1. h1: a = ↑k rewrite [h1, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done · -- Case 2. h1: a = -↑k rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done done theorem cc_eq_iff_congr (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ a ≡ b (MOD m) := calc [a]_m = [b]_m _ ↔ [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b _ ↔ ↑m ∣ (a - b) := cc_zero_iff_dvd m (a - b) _ ↔ a ≡ b (MOD m) := by rfl /- End of fundamental properties of congruence classes -/ lemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m := by have h1 : (↑m : Int) ≠ 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m) show 0 ≤ a % m from Int.emod_nonneg a h1 done lemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m) have h2 : (↑m : Int) > 0 := (Nat.cast_pos).rtl h1 show a % m < m from Int.emod_lt_of_pos a h2 done lemma congr_mod_mod (m : Nat) (a : Int) : a ≡ a % m (MOD m) := by define have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m apply Exists.intro (a / m) show a - a % m = m * (a / m) from calc a - (a % m) _ = m * (a / m) + a % m - a % m := by rw [h1] _ = m * (a / m) := by ring done lemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) := And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a)) theorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) : ∃! (r : Int), 0 ≤ r ∧ r < m ∧ a ≡ r (MOD m) := by exists_unique · -- Existence apply Exists.intro (a % m) show 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) from mod_cmpl_res m a done · -- Uniqueness fix r1 : Int; fix r2 : Int assume h1 : 0 ≤ r1 ∧ r1 < m ∧ a ≡ r1 (MOD m) assume h2 : 0 ≤ r2 ∧ r2 < m ∧ a ≡ r2 (MOD m) have h3 : r1 ≡ r2 (MOD m) := congr_trans (congr_symm h1.right.right) h2.right.right obtain (d : Int) (h4 : r1 - r2 = m * d) from h3 have h5 : r1 - r2 < m * 1 := by linarith have h6 : m * (-1) < r1 - r2 := by linarith rewrite [h4] at h5 --h5 : m * d < m * 1 rewrite [h4] at h6 --h6 : m * -1 < m * d have h7 : (↑m : Int) ≥ 0 := Nat.cast_nonneg m have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7 have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7 have h10 : d = 0 := by linarith show r1 = r2 from calc r1 _ = r1 - r2 + r2 := by ring _ = m * 0 + r2 := by rw [h4, h10] _ = r2 := by ring done done lemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m := (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a) theorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y rewrite [h1, h2] have h3 : a + b = b + a := by ring show [a]_m + [b]_m = [b]_m + [a]_m from calc [a]_m + [b]_m _ = [a + b]_m := add_class m a b _ = [b + a]_m := by rw [h3] _ = [b]_m + [a]_m := (add_class m b a).symm done theorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X rewrite [h1] have h2 : a * 1 = a := by ring show [a]_m * [1]_m = [a]_m from calc [a]_m * [1]_m _ = [a * 1]_m := mul_class m a 1 _ = [a]_m := by rw [h2] done theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry lemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) : [a]_m * [gcd_c2 m a]_m = [1]_m := by set s : Int := gcd_c1 m a have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m define at h1 rewrite [h1, Nat.cast_one] at h2 --h2 : s * ↑m + gcd_c2 m a * ↑a = 1 rewrite [mul_class, cc_eq_iff_congr] define --Goal : ∃ (c : Int), ↑a * gcd_c2 m a - 1 = ↑m * c apply Exists.intro (-s) show a * (gcd_c2 m a) - 1 = m * (-s) from calc a * (gcd_c2 m a) - 1 _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring _ = 1 + m * (-s) - 1 := by rw [h2] _ = m * (-s) := by ring done theorem Theorem_7_3_7 (m a : Nat) : invertible [a]_m ↔ rel_prime m a := by apply Iff.intro · -- (→) assume h1 : invertible [a]_m define at h1 obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1 obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y rewrite [h3, mul_class, cc_eq_iff_congr] at h2 define at h2 obtain (c : Int) (h4 : a * b - 1 = m * c) from h2 rewrite [Exercise_7_2_6] --Goal : ∃ (s t : Int), s * ↑m + t * ↑a = 1 apply Exists.intro (-c) apply Exists.intro b show (-c) * m + b * a = 1 from calc (-c) * m + b * a _ = (-c) * m + (a * b - 1) + 1 := by ring _ = (-c) * m + m * c + 1 := by rw [h4] _ = 1 := by ring done · -- (←) assume h1 : rel_prime m a define show ∃ (Y : ZMod m), [a]_m * Y = [1]_m from Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1) done done /- Section 7.4 -/ section Euler open Euler lemma num_rp_below_base {m : Nat} : num_rp_below m 0 = 0 := by rfl lemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) : num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by have h1 : num_rp_below m (j + 1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : gcd m j = 1 rewrite [if_pos h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1 show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1 done lemma num_rp_below_step_not_rp {m j : Nat} (h : ¬rel_prime m j) : num_rp_below m (j + 1) = num_rp_below m j := by have h1 : num_rp_below m (j +1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : ¬gcd m j = 1 rewrite [if_neg h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j show num_rp_below m (j + 1) = num_rp_below m j from h1 done lemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl #eval phi 10 --Answer: 4 lemma prod_inv_iff_inv {m : Nat} {X : ZMod m} (h1 : invertible X) (Y : ZMod m) : invertible (X * Y) ↔ invertible Y := by apply Iff.intro · -- (→) assume h2 : invertible (X * Y) obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2 apply Exists.intro (X * Z) rewrite [←h3] --Goal : Y * (X * Z) = X * Y * Z ring --Note that ring can do algebra in ZMod m done · -- (←) assume h2 : invertible Y obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1 obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2 apply Exists.intro (Xi * Yi) show (X * Y) * (Xi * Yi) = [1]_m from calc X * Y * (Xi * Yi) _ = (X * Xi) * (Y * Yi) := by ring _ = [1]_m * [1]_m := by rw [h3, h4] _ = [1]_m := Theorem_7_3_6_7 [1]_m done done lemma F_rp_def {m i : Nat} (h : rel_prime m i) : F m i = [i]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h --h : gcd m i = 1 rewrite [if_pos h] at h1 show F m i = [i]_m from h1 done lemma F_not_rp_def {m i : Nat} (h : ¬rel_prime m i) : F m i = [1]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h rewrite [h1, if_neg h] rfl done lemma prod_seq_base {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 0 k f = [1]_m := by rfl lemma prod_seq_step {m : Nat} (n k : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl lemma prod_seq_zero_step {m : Nat} (n : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by rewrite [prod_seq_step, zero_add] rfl done lemma prod_one {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 1 k f = f k := by rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7] rfl done lemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl lemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m := calc [G m a i]_m _ = [(a * i) % m]_m := by rfl _ = [a * i]_m := (cc_eq_mod m (a * i)).symm _ = [a]_m * [i]_m := (mul_class m a i).symm lemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) : rel_prime m (G m a i) ↔ rel_prime m i := by have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1 show rel_prime m (G m a i) ↔ rel_prime m i from calc rel_prime m (G m a i) _ ↔ invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm _ ↔ invertible ([a]_m * [i]_m) := by rw [cc_G] _ ↔ invertible [i]_m := prod_inv_iff_inv h2 ([i]_m) _ ↔ rel_prime m i := Theorem_7_3_7 m i done lemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) : F m (G m a i) = [a]_m * F m i := by have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2 show F m (G m a i) = [a]_m * F m i from calc F m (G m a i) _ = [G m a i]_m := F_rp_def h3 _ = [a]_m * [i]_m := cc_G m a i _ = [a]_m * F m i := by rw [F_rp_def h2] done lemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : ¬rel_prime m i) : F m (G m a i) = [1]_m := by rewrite [←G_rp_iff h1 i] at h2 show F m (G m a i) = [1]_m from F_not_rp_def h2 done lemma FG_prod {m a : Nat} (h1 : rel_prime m a) : ∀ (k : Nat), prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by by_induc · -- Base Case show prod_seq 0 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from calc prod_seq 0 0 ((F m) ∘ (G m a)) _ = [1]_m := prod_seq_base _ _ _ = [a]_m ^ 0 * [1]_m := by ring _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by rw [num_rp_below_base, prod_seq_base] done · -- Induction Step fix k : Nat assume ih : prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) by_cases h2 : rel_prime m k · -- Case 1. h2 : rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([a]_m * F m k) := by rw [FG_rp h1 h2] _ = [a]_m ^ ((num_rp_below m k) + 1) * ((prod_seq k 0 (F m)) * F m k) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_rp h2, prod_seq_zero_step] done · -- Case 2. h2 : ¬rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([1]_m) := by rw [FG_not_rp h1 h2] _ = [a]_m ^ (num_rp_below m k) * (prod_seq k 0 (F m) * ([1]_m)) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_not_rp h2, prod_seq_zero_step, F_not_rp_def h2] done done done lemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by define --Goal : ∀ i < m, G m a i < m fix i : Nat assume h1 : i < m rewrite [G_def] --Goal : a * i % m < m show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m) done lemma left_inv_one_one_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g' (g i) = i) : one_one_below n g := sorry lemma right_inv_onto_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g (g' i) = i) (h2 : maps_below n g') : onto_below n g := by define at h2; define fix k : Nat assume h3 : k < n apply Exists.intro (g' k) show g' k < n ∧ g (g' k) = k from And.intro (h2 k h3) (h1 k h3) done lemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m * [inv_mod m a]_m = [1]_m := by have h2 : 0 ≤ (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a) show [a]_m * [inv_mod m a]_m = [1]_m from calc [a]_m * [inv_mod m a]_m _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2] _ = [a]_m * [gcd_c2 m a]_m := by rw [←cc_eq_mod] _ = [1]_m := gcd_c2_inv h1 done lemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m := calc a * (b % m) % m = a % m * (b % m % m) % m := Nat.mul_mod _ _ _ _ = a % m * (b % m) % m := by rw [Nat.mod_mod] _ = a * b % m := (Nat.mul_mod _ _ _).symm lemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm] rfl done theorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] : ↑a ≡ ↑b (MOD m) ↔ a % m = b % m := sorry lemma mul_inv_mod_cancel {m a i : Nat} [NeZero m] (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1 rewrite [mul_class, cc_eq_iff_congr, ←Nat.cast_mul, ←Nat.cast_one, congr_iff_mod_eq_Nat] at h3 show a * inv_mod m a * i % m = i from calc a * (inv_mod m a) * i % m _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = 1 % m * i % m := by rw [h3] _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = i % m := by rw [one_mul] _ = i := Nat.mod_eq_of_lt h2 done lemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl lemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, G m a (Ginv m a i) = i := by fix i : Nat assume h2 : i < m show G m a (Ginv m a i) = i from calc G m a (Ginv m a i) _ = a * ((inv_mod m a * i) % m) % m := by rfl _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, Ginv m a (G m a i) = i := by fix i : Nat assume h2 : i < m show Ginv m a (G m a i) = i from calc Ginv m a (G m a i) _ = inv_mod m a * ((a * i) % m) % m := by rfl _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc, mul_comm (inv_mod m a)] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_maps_below (m a : Nat) [NeZero m] : maps_below m (Ginv m a) := G_maps_below m (inv_mod m a) lemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : one_one_below m (G m a) := left_inv_one_one_below (Ginv_left_inv h1) lemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : onto_below m (G m a) := right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a) lemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : perm_below m (G m a) := And.intro (G_maps_below m a) (And.intro (G_one_one_below h1) (G_onto_below h1)) --Permuting a product of congruence classes doesn't change product lemma swap_fst (u v : Nat) : swap u v u = v := by define : swap u v u --Goal : (if u = u then v else if u = v then u else u) = v have h : u = u := by rfl rewrite [if_pos h] rfl done lemma swap_snd (u v : Nat) : swap u v v = u := by define : swap u v v by_cases h1 : v = u · -- Case 1. h1 : v = u rewrite [if_pos h1] show v = u from h1 done · -- Case 2. h1 : v ≠ u rewrite [if_neg h1] have h2 : v = v := by rfl rewrite [if_pos h2] rfl done done lemma swap_other {u v i : Nat} (h1 : i ≠ u) (h2 : i ≠ v) : swap u v i = i := by define : swap u v i rewrite [if_neg h1, if_neg h2] rfl done lemma swap_values (u v i : Nat) : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := by by_cases h1 : i = u · -- Case 1. h1 : i = u apply Or.inl rewrite [h1] show swap u v u = v from swap_fst u v done · -- Case 2. h1 : i ≠ u apply Or.inr by_cases h2 : i = v · -- Case 2.1. h2 : i = v apply Or.inl rewrite [h2] show swap u v v = u from swap_snd u v done · -- Case 2.2. h2 : i ≠ v apply Or.inr show swap u v i = i from swap_other h1 h2 done done done lemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by define fix i : Nat assume h3 : i < n have h4 : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := swap_values u v i by_cases on h4 · -- Case 1. h4 : swap u v i = v rewrite [h4] show v < n from h2 done · -- Case 2. by_cases on h4 · -- Case 2.1. h4 : swap u v i = u rewrite [h4] show u < n from h1 done · -- Case 2.2. h4 : swap u v i = i rewrite [h4] show i < n from h3 done done done lemma swap_swap (u v n : Nat) : ∀ i < n, swap u v (swap u v i) = i := by fix i : Nat assume h : i < n by_cases h1 : i = u · -- Case 1. h1 : i = u rewrite [h1, swap_fst, swap_snd] rfl done · -- Case 2. h1 : i ≠ u by_cases h2 : i = v · -- Case 2.1. h2 : i = v rewrite [h2, swap_snd, swap_fst] rfl done · -- Case 2.2. h2 : i ≠ v rewrite [swap_other h1 h2, swap_other h1 h2] rfl done done done lemma swap_one_one_below (u v n) : one_one_below n (swap u v) := left_inv_one_one_below (swap_swap u v n) lemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) := right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2) lemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) := And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2)) lemma comp_perm_below {n : Nat} {f g : Nat → Nat} (h1 : perm_below n f) (h2 : perm_below n g) : perm_below n (f ∘ g) := sorry lemma trivial_swap (u : Nat) : swap u u = id := by apply funext fix x : Nat by_cases h1 : x = u · -- Case 1. h1 : x = u rewrite [h1, swap_fst] rfl done · -- Case 2. h1 : x ≠ u rewrite [swap_other h1 h1] rfl done done lemma prod_eq_fun {m : Nat} (f g : Nat → ZMod m) (k : Nat) : ∀ (n : Nat), (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g := by by_induc · -- Base Case assume h : (∀ i < 0, f (k + i) = g (k + i)) rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g assume h1 : ∀ i < n + 1, f (k + i) = g (k + i) have h2 : ∀ i < n, f (k + i) = g (k + i) := by fix i : Nat assume h2 : i < n have h3 : i < n + 1 := by linarith show f (k + i) = g (k + i) from h1 i h3 done have h3 : prod_seq n k f = prod_seq n k g := ih h2 have h4 : n < n + 1 := Nat.lt_succ_self n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4] rfl done done lemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := by have h2 : ∀ (i : Nat), i < u → (f ∘ swap u n) (0 + i) = f (0 + i) := by fix i : Nat assume h2 : i < u have h3 : 0 + i ≠ u := by linarith have h4 : 0 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl done show prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f from prod_eq_fun (f ∘ swap u n) f 0 u h2 done lemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := by have h2 : ∀ i < j, (f ∘ swap u n) (u + 1 + i) = f (u + 1 + i) := by fix i : Nat assume h2 : i < j have h3 : u + 1 + i ≠ u := by linarith have h4 : u + 1 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl show prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f from prod_eq_fun (f ∘ swap u n) f (u + 1) j h2 done lemma break_prod {m : Nat} (n : Nat) (f : Nat → ZMod m) : ∀ (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by by_induc · -- Base Case have h : n + 0 = n := by rfl rewrite [prod_seq_base, h, Theorem_7_3_6_7] rfl done · -- Induction Step fix j : Nat assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f rewrite [←add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc] rfl done done lemma break_prod_twice {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f := break_prod n f 1 rewrite [prod_one] at h2 have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f := break_prod (u + 1) f j rewrite [←h1] at h3 have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f := break_prod u f 1 rewrite [prod_one] at h4 rewrite [h3, h4] at h2 show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2 done lemma swap_prod_eq_prod {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f := by by_cases h2 : u = n · -- Case 1. h2 : u = n rewrite [h2, trivial_swap n] --Goal : prod_seq (n + 1) 0 (f ∘ id) = prod_seq (n + 1) 0 f rfl done · -- Case 2. h2 : ¬u = n have h3 : u + 1 ≤ n := Nat.lt_of_le_of_ne h1 h2 obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3 have break_f : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := break_prod_twice f h4 have break_fs : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_prod_twice (f ∘ swap u n) h4 have f_eq_fs_below : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := swap_prod_eq_prod_below f h1 have f_eq_fs_btwn : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4 show prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f from calc prod_seq (n + 1) 0 (f ∘ swap u n) _ = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_fs _ = prod_seq u 0 f * (f ∘ swap u n) u * prod_seq j (u + 1) f * (f ∘ swap u n) n := by rw [f_eq_fs_below, f_eq_fs_btwn] _ = prod_seq u 0 f * f (swap u n u) * prod_seq j (u + 1) f * f (swap u n n) := by rfl _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by rw [swap_fst, swap_snd] _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring _ = prod_seq (n + 1) 0 f := break_f.symm done done lemma perm_below_fixed {n : Nat} {g : Nat → Nat} (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry lemma perm_prod {m : Nat} (f : Nat → ZMod m) : ∀ (n : Nat), ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) := by by_induc · -- Base Case fix g : Nat → Nat assume h1 : perm_below 0 g rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) fix g : Nat → Nat assume g_pb : perm_below (n + 1) g define at g_pb have g_ob : onto_below (n + 1) g := g_pb.right.right define at g_ob have h1 : n < n + 1 := by linarith obtain (u : Nat) (h2 : u < n + 1 ∧ g u = n) from g_ob n h1 have s_pb : perm_below (n + 1) (swap u n) := swap_perm_below h2.left h1 have gs_pb_n1 : perm_below (n + 1) (g ∘ swap u n) := comp_perm_below g_pb s_pb have gs_fix_n : (g ∘ swap u n) n = n := calc (g ∘ swap u n) n _ = g (swap u n n) := by rfl _ = g u := by rw [swap_snd] _ = n := h2.right have gs_pb_n : perm_below n (g ∘ swap u n) := perm_below_fixed gs_pb_n1 gs_fix_n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f ∘ (g ∘ swap u n)) := ih (g ∘ swap u n) gs_pb_n have h3 : u ≤ n := by linarith show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f ∘ g) from calc prod_seq (n + 1) 0 f _ = prod_seq n 0 f * f n := prod_seq_zero_step n f _ = prod_seq n 0 (f ∘ (g ∘ swap u n)) * f ((g ∘ swap u n) n) := by rw [gs_prod, gs_fix_n] _ = prod_seq n 0 (f ∘ g ∘ swap u n) * (f ∘ g ∘ swap u n) n := by rfl _ = prod_seq (n + 1) 0 (f ∘ g ∘ swap u n) := (prod_seq_zero_step n (f ∘ g ∘ swap u n)).symm _ = prod_seq (n + 1) 0 ((f ∘ g) ∘ swap u n) := by rfl _ = prod_seq (n + 1) 0 (f ∘ g) := swap_prod_eq_prod (f ∘ g) h3 done done lemma F_invertible (m i : Nat) : invertible (F m i) := by by_cases h : rel_prime m i · -- Case 1. h : rel_prime m i rewrite [F_rp_def h] show invertible [i]_m from (Theorem_7_3_7 m i).rtl h done · -- Case 2. h : ¬rel_prime m i rewrite [F_not_rp_def h] apply Exists.intro [1]_m show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m done done lemma Fprod_invertible (m : Nat) : ∀ (k : Nat), invertible (prod_seq k 0 (F m)) := by by_induc · -- Base Case apply Exists.intro [1]_m show prod_seq 0 0 (F m) * [1]_m = [1]_m from calc prod_seq 0 0 (F m) * [1]_m _ = [1]_m * [1]_m := by rw [prod_seq_base] _ = [1]_m := Theorem_7_3_6_7 ([1]_m) done · -- Induction Step fix k : Nat assume ih : invertible (prod_seq k 0 (F m)) rewrite [prod_seq_zero_step] show invertible (prod_seq k 0 (F m) * (F m k)) from (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k) done done theorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m ^ (phi m) = [1]_m := by have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2 show [a]_m ^ (phi m) = [1]_m from calc [a]_m ^ (phi m) _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3] _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring _ = prod_seq m 0 (F m ∘ G m a) * Y := by rw [FG_prod h1 m, phi_def] _ = prod_seq m 0 (F m) * Y := by rw [perm_prod (F m) m (G m a) (G_perm_below h1)] _ = [1]_m := by rw [h3] done lemma Exercise_7_4_5_Int (m : Nat) (a : Int) : ∀ (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry lemma Exercise_7_4_5_Nat (m a n : Nat) : [a]_m ^ n = [a ^ n]_m := by rewrite [Exercise_7_4_5_Int] rfl done theorem Euler's_theorem {m a : Nat} [NeZero m] (h1 : rel_prime m a) : a ^ (phi m) ≡ 1 (MOD m) := by have h2 : [a]_m ^ (phi m) = [1]_m := Theorem_7_4_2 h1 rewrite [Exercise_7_4_5_Nat m a (phi m)] at h2 --h2 : [a ^ phi m]_m = [1]_m show a ^ (phi m) ≡ 1 (MOD m) from (cc_eq_iff_congr _ _ _).ltr h2 done #eval gcd 10 7 --Answer: 1. So 10 and 7 are relatively prime #eval 7 ^ phi 10 --Answer: 2401, which is congruent to 1 mod 10. end Euler /- Section 7.5 -/ lemma num_rp_prime {p : Nat} (h1 : prime p) : ∀ k < p, num_rp_below p (k + 1) = k := sorry lemma phi_prime {p : Nat} (h1 : prime p) : phi p = p - 1 := by have h2 : 1 ≤ p := prime_pos h1 have h3 : p - 1 + 1 = p := Nat.sub_add_cancel h2 have h4 : p - 1 < p := by linarith have h5 : num_rp_below p (p - 1 + 1) = p - 1 := num_rp_prime h1 (p - 1) h4 rewrite [h3] at h5 show phi p = p - 1 from h5 done theorem Theorem_7_2_2_Int {a c : Nat} {b : Int} (h1 : ↑c ∣ ↑a * b) (h2 : rel_prime a c) : ↑c ∣ b := by rewrite [Int.natCast_dvd, Int.natAbs_mul, Int.natAbs_ofNat] at h1 --h1 : c ∣ a * Int.natAbs b rewrite [Int.natCast_dvd] --Goal : c ∣ Int.natAbs b show c ∣ Int.natAbs b from Theorem_7_2_2 h1 h2 done lemma Lemma_7_4_5 {m n : Nat} (a b : Int) (h1 : rel_prime m n) : a ≡ b (MOD m * n) ↔ a ≡ b (MOD m) ∧ a ≡ b (MOD n) := by apply Iff.intro · -- (→) assume h2 : a ≡ b (MOD m * n) obtain (j : Int) (h3 : a - b = (m * n) * j) from h2 apply And.intro · -- Proof of a ≡ b (MOD m) apply Exists.intro (n * j) show a - b = m * (n * j) from calc a - b _ = m * n * j := h3 _ = m * (n * j) := by ring done · -- Proof of a ≡ b (MOD n) apply Exists.intro (m * j) show a - b = n * (m * j) from calc a - b _ = m * n * j := h3 _ = n * (m * j) := by ring done done · -- (←) assume h2 : a ≡ b (MOD m) ∧ a ≡ b (MOD n) obtain (j : Int) (h3 : a - b = m * j) from h2.left have h4 : (↑n : Int) ∣ a - b := h2.right rewrite [h3] at h4 --h4 : ↑n ∣ ↑m * j have h5 : ↑n ∣ j := Theorem_7_2_2_Int h4 h1 obtain (k : Int) (h6 : j = n * k) from h5 apply Exists.intro k --Goal : a - b = ↑(m * n) * k rewrite [Nat.cast_mul] --Goal : a - b = ↑m * ↑n * k show a - b = (m * n) * k from calc a - b _ = m * j := h3 _ = m * (n * k) := by rw [h6] _ = (m * n) * k := by ring done done --From exercises of Section 7.2 theorem rel_prime_symm {a b : Nat} (h : rel_prime a b) : rel_prime b a := sorry lemma prime_NeZero {p : Nat} (h : prime p) : NeZero p := by rewrite [neZero_iff] --Goal : p ≠ 0 define at h linarith done lemma Lemma_7_5_1 {p e d m c s : Nat} {t : Int} (h1 : prime p) (h2 : e * d = (p - 1) * s + 1) (h3 : m ^ e - c = p * t) : c ^ d ≡ m (MOD p) := by have h4 : m ^ e ≡ c (MOD p) := Exists.intro t h3 have h5 : [m ^ e]_p = [c]_p := (cc_eq_iff_congr _ _ _).rtl h4 rewrite [←Exercise_7_4_5_Nat] at h5 --h5 : [m]_p ^ e = [c]_p by_cases h6 : p ∣ m · -- Case 1. h6 : p ∣ m have h7 : m ≡ 0 (MOD p) := by obtain (j : Nat) (h8 : m = p * j) from h6 apply Exists.intro (↑j : Int) --Goal : ↑m - 0 = ↑p * ↑j rewrite [h8, Nat.cast_mul] ring done have h8 : [m]_p = [0]_p := (cc_eq_iff_congr _ _ _).rtl h7 have h9 : e * d ≠ 0 := by rewrite [h2] show (p - 1) * s + 1 ≠ 0 from Nat.add_one_ne_zero _ done have h10 : (0 : Int) ^ (e * d) = 0 := zero_pow h9 have h11 : [c ^ d]_p = [m]_p := calc [c ^ d]_p _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat] _ = ([m]_p ^ e) ^ d := by rw [h5] _ = [m]_p ^ (e * d) := by ring _ = [0]_p ^ (e * d) := by rw [h8] _ = [0 ^ (e * d)]_p := Exercise_7_4_5_Int _ _ _ _ = [0]_p := by rw [h10] _ = [m]_p := by rw [h8] show c ^ d ≡ m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11 done · -- Case 2. h6 : ¬p ∣ m have h7 : rel_prime m p := rel_prime_of_prime_not_dvd h1 h6 have h8 : rel_prime p m := rel_prime_symm h7 have h9 : NeZero p := prime_NeZero h1 have h10 : (1 : Int) ^ s = 1 := by ring have h11 : [c ^ d]_p = [m]_p := calc [c ^ d]_p _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat] _ = ([m]_p ^ e) ^ d := by rw [h5] _ = [m]_p ^ (e * d) := by ring _ = [m]_p ^ ((p - 1) * s + 1) := by rw [h2] _ = ([m]_p ^ (p - 1)) ^ s * [m]_p := by ring _ = ([m]_p ^ (phi p)) ^ s * [m]_p := by rw [phi_prime h1] _ = [1]_p ^ s * [m]_p := by rw [Theorem_7_4_2 h8] _ = [1 ^ s]_p * [m]_p := by rw [Exercise_7_4_5_Int] _ = [1]_p * [m]_p := by rw [h10] _ = [m]_p * [1]_p := by ring _ = [m]_p := Theorem_7_3_6_7 _ show c ^ d ≡ m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11 done done theorem Theorem_7_5_1 (p q n e d k m c : Nat) (p_prime : prime p) (q_prime : prime q) (p_ne_q : p ≠ q) (n_pq : n = p * q) (ed_congr_1 : e * d = k * (p - 1) * (q - 1) + 1) (h1 : [m]_n ^ e = [c]_n) : [c]_n ^ d = [m]_n := by rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] at h1 --h1 : m ^ e ≡ c (MOD n) rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] --Goal : c ^ d ≡ m (MOD n) obtain (j : Int) (h2 : m ^ e - c = n * j) from h1 rewrite [n_pq, Nat.cast_mul] at h2 --h2 : m ^ e - c = p * q * j have h3 : e * d = (p - 1) * (k * (q - 1)) + 1 := by rewrite [ed_congr_1] ring done have h4 : m ^ e - c = p * (q * j) := by rewrite [h2] ring done have congr_p : c ^ d ≡ m (MOD p) := Lemma_7_5_1 p_prime h3 h4 have h5 : e * d = (q - 1) * (k * (p - 1)) + 1 := by rewrite [ed_congr_1] ring done have h6 : m ^ e - c = q * (p * j) := by rewrite [h2] ring done have congr_q : c ^ d ≡ m (MOD q) := Lemma_7_5_1 q_prime h5 h6 have h7 : ¬q ∣ p := by by_contra h8 have h9 : q = 1 ∨ q = p := dvd_prime p_prime h8 disj_syll h9 (prime_not_one q_prime) show False from p_ne_q h9.symm done have h8 : rel_prime p q := rel_prime_of_prime_not_dvd q_prime h7 rewrite [n_pq, Lemma_7_4_5 _ _ h8] show c ^ d ≡ m (MOD p) ∧ c ^ d ≡ m (MOD q) from And.intro congr_p congr_q done /- BEGIN EXERCISES -/ namespace Exercises /- Section 7.1 -/ -- 1. theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry -- 2. lemma gcd_comm_lt {a b : Nat} (h : a < b) : gcd a b = gcd b a := sorry theorem gcd_comm (a b : Nat) : gcd a b = gcd b a := sorry -- 3. theorem Exercise_7_1_5 (a b : Nat) (n : Int) : (∃ (s t : Int), s * a + t * b = n) ↔ (↑(gcd a b) : Int) ∣ n := sorry -- 4. theorem Exercise_7_1_6 (a b c : Nat) : gcd a b = gcd (a + b * c) b := sorry -- 5. theorem gcd_is_nonzero {a b : Nat} (h : a ≠ 0 ∨ b ≠ 0) : gcd a b ≠ 0 := sorry -- 6. theorem gcd_greatest {a b d : Nat} (h1 : gcd a b ≠ 0) (h2 : d ∣ a) (h3 : d ∣ b) : d ≤ gcd a b := sorry -- 7. lemma Lemma_7_1_10a {a b : Nat} (n : Nat) (h : a ∣ b) : (n * a) ∣ (n * b) := sorry lemma Lemma_7_1_10b {a b n : Nat} (h1 : n ≠ 0) (h2 : (n * a) ∣ (n * b)) : a ∣ b := sorry lemma Lemma_7_1_10c {a b : Nat} (h1 : a ∣ b) (h2 : b ∣ a) : a = b := sorry theorem Exercise_7_1_10 (a b n : Nat) : gcd (n * a) (n * b) = n * gcd a b := sorry /- Section 7.2 -/ -- 1. lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry -- 2. -- Hints: Start with apply List.rec. You may find mul_ne_zero useful theorem prod_nonzero_nonzero : ∀ (l : List Nat), (∀ a ∈ l, a ≠ 0) → prod l ≠ 0 := sorry -- 3. theorem rel_prime_iff_no_common_factor (a b : Nat) : rel_prime a b ↔ ¬∃ (p : Nat), prime p ∧ p ∣ a ∧ p ∣ b := sorry -- 4. theorem rel_prime_symm {a b : Nat} (h : rel_prime a b) : rel_prime b a := sorry -- 5. lemma in_prime_factorization_iff_prime_factor {a : Nat} {l : List Nat} (h1 : prime_factorization a l) (p : Nat) : p ∈ l ↔ prime_factor p a := sorry -- 6. theorem Exercise_7_2_5 {a b : Nat} {l m : List Nat} (h1 : prime_factorization a l) (h2 : prime_factorization b m) : rel_prime a b ↔ (¬∃ (p : Nat), p ∈ l ∧ p ∈ m) := sorry -- 7. theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry -- 8. theorem Exercise_7_2_7 {a b a' b' : Nat} (h1 : rel_prime a b) (h2 : a' ∣ a) (h3 : b' ∣ b) : rel_prime a' b' := sorry -- 9. theorem Exercise_7_2_9 {a b j k : Nat} (h1 : gcd a b ≠ 0) (h2 : a = j * gcd a b) (h3 : b = k * gcd a b) : rel_prime j k := sorry -- 10.

theorem Exercise_7_2_17a (a b c : Nat) : gcd a (b * c) ∣ gcd a b * gcd a c

HTPI.Exercises.Exercise_7_2_17a

htpi/HTPILib/Chap7.lean

htpi

/- Copyright 2023 Daniel J. Velleman -/ import HTPILib.Chap6 namespace HTPI /- Definitions -/ lemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by have h : n + 1 > 0 := Nat.succ_pos n show a % (n + 1) < n + 1 from Nat.mod_lt a h done def gcd (a b : Nat) : Nat := match b with | 0 => a | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd (n + 1) (a % (n + 1)) termination_by b mutual def gcd_c1 (a b : Nat) : Int := match b with | 0 => 1 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c2 (n + 1) (a % (n + 1)) --Corresponds to s = t' termination_by b def gcd_c2 (a b : Nat) : Int := match b with | 0 => 0 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c1 (n + 1) (a % (n + 1)) - (gcd_c2 (n + 1) (a % (n + 1))) * ↑(a / (n + 1)) --Corresponds to t = s' - t'q termination_by b end def prime (n : Nat) : Prop := 2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n def prime_factor (p n : Nat) : Prop := prime p ∧ p ∣ n def all_prime (l : List Nat) : Prop := ∀ p ∈ l, prime p def nondec (l : List Nat) : Prop := match l with | [] => True --Of course, True is a proposition that is always true | n :: L => (∀ m ∈ L, n ≤ m) ∧ nondec L def nondec_prime_list (l : List Nat) : Prop := all_prime l ∧ nondec l def prod (l : List Nat) : Nat := match l with | [] => 1 | n :: L => n * (prod L) def prime_factorization (n : Nat) (l : List Nat) : Prop := nondec_prime_list l ∧ prod l = n def rel_prime (a b : Nat) : Prop := gcd a b = 1 def congr_mod (m : Nat) (a b : Int) : Prop := (↑m : Int) ∣ (a - b) def cc (m : Nat) (a : Int) : ZMod m := (↑a : ZMod m) notation:50 a " ≡ " b " (MOD " m ")" => congr_mod m a b notation:max "["a"]_"m:max => cc m a def invertible {m : Nat} (X : ZMod m) : Prop := ∃ (Y : ZMod m), X * Y = [1]_m def num_rp_below (m k : Nat) : Nat := match k with | 0 => 0 | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j def phi (m : Nat) : Nat := num_rp_below m m def prod_seq {m : Nat} (j k : Nat) (f : Nat → ZMod m) : ZMod m := match j with | 0 => [1]_m | n + 1 => prod_seq n k f * f (k + n) def maps_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i < n, g i < n def one_one_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i1 < n, ∀ i2 < n, g i1 = g i2 → i1 = i2 def onto_below (n : Nat) (g : Nat → Nat) : Prop := ∀ k < n, ∃ i < n, g i = k def perm_below (n : Nat) (g : Nat → Nat) : Prop := maps_below n g ∧ one_one_below n g ∧ onto_below n g def inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m) def swap (u v i : Nat) : Nat := if i = u then v else if i = v then u else i namespace Euler --For definitions specific to Euler's theorem def F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m def G (m a i : Nat) : Nat := (a * i) % m def Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i end Euler /- Section 7.1 -/ theorem dvd_mod_of_dvd_a_b {a b d : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ (a % b) := by set q : Nat := a / b have h3 : b * q + a % b = a := Nat.div_add_mod a b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 define --Goal : ∃ (c : Nat), a % b = d * c apply Exists.intro (j - k * q) show a % b = d * (j - k * q) from calc a % b _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm _ = a - b * q := by rw [h3] _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc] _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm done theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry #eval gcd 672 161 --Answer: 7 lemma gcd_base (a : Nat) : gcd a 0 = a := by rfl lemma gcd_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd a b = gcd b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1)) rfl done lemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b ≠ 0) : a % b < b := by have h1 : b > 0 := Nat.pos_of_ne_zero h show a % b < b from Nat.mod_lt a h1 done lemma dvd_self (n : Nat) : n ∣ n := by apply Exists.intro 1 ring done theorem gcd_dvd : ∀ (b a : Nat), (gcd a b) ∣ a ∧ (gcd a b) ∣ b := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd a b_1) ∣ a ∧ (gcd a b_1) ∣ b_1 fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_base] --Goal: a ∣ a ∧ a ∣ 0 apply And.intro (dvd_self a) define apply Exists.intro 0 rfl done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_nonzero a h1] --Goal : gcd b (a % b) ∣ a ∧ gcd b (a % b) ∣ b have h2 : a % b < b := mod_nonzero_lt a h1 have h3 : (gcd b (a % b)) ∣ b ∧ (gcd b (a % b)) ∣ (a % b) := ih (a % b) h2 b apply And.intro _ h3.left show (gcd b (a % b)) ∣ a from dvd_a_of_dvd_b_mod h3.left h3.right done done theorem gcd_dvd_left (a b : Nat) : (gcd a b) ∣ a := (gcd_dvd b a).left theorem gcd_dvd_right (a b : Nat) : (gcd a b) ∣ b := (gcd_dvd b a).right lemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl lemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c1 a b = gcd_c2 b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done lemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl lemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * ↑(a / b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done theorem gcd_lin_comb : ∀ (b a : Nat), (gcd_c1 a b) * ↑a + (gcd_c2 a b) * ↑b = ↑(gcd a b) := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd_c1 a b_1) * ↑a + (gcd_c2 a b_1) * ↑b_1 = ↑(gcd a b_1) fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base] --Goal : 1 * ↑a + 0 * ↑0 = ↑a ring done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1] --Goal : gcd_c2 b (a % b) * ↑a + -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * ↑(a / b)) * ↑b = -- ↑(gcd b (a % b)) set r : Nat := a % b set q : Nat := a / b set s : Int := gcd_c1 b r set t : Int := gcd_c2 b r --Goal : t * ↑a + (s - t * ↑q) * ↑b = ↑(gcd b r) have h2 : r < b := mod_nonzero_lt a h1 have h3 : s * ↑b + t * ↑r = ↑(gcd b r) := ih r h2 b have h4 : b * q + r = a := Nat.div_add_mod a b rewrite [←h3, ←h4] rewrite [Nat.cast_add, Nat.cast_mul] --Goal : t * (↑b * ↑q + ↑r) + (s - t * ↑q) * ↑b = s * ↑b + t * ↑r ring done done #eval gcd_c1 672 161 --Answer: 6 #eval gcd_c2 672 161 --Answer: -25 --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161 theorem Theorem_7_1_6 {d a b : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ gcd a b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑d ∣ ↑(gcd a b) set s : Int := gcd_c1 a b set t : Int := gcd_c2 a b have h3 : s * ↑a + t * ↑b = ↑(gcd a b) := gcd_lin_comb b a rewrite [←h3] --Goal : ↑d ∣ s * ↑a + t * ↑b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul] --Goal : ↑d ∣ s * (↑d * ↑j) + t * (↑d * ↑k) define apply Exists.intro (s * ↑j + t * ↑k) ring done /- Section 7.2 -/ theorem dvd_trans {a b c : Nat} (h1 : a ∣ b) (h2 : b ∣ c) : a ∣ c := by define at h1; define at h2; define obtain (m : Nat) (h3 : b = a * m) from h1 obtain (n : Nat) (h4 : c = b * n) from h2 rewrite [h3, mul_assoc] at h4 apply Exists.intro (m * n) show c = a * (m * n) from h4 done lemma exists_prime_factor : ∀ (n : Nat), 2 ≤ n → ∃ (p : Nat), prime_factor p n := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, 2 ≤ n_1 → ∃ (p : Nat), prime_factor p n_1 assume h1 : 2 ≤ n by_cases h2 : prime n · -- Case 1. h2 : prime n apply Exists.intro n define --Goal : prime n ∧ n ∣ n show prime n ∧ n ∣ n from And.intro h2 (dvd_self n) done · -- Case 2. h2 : ¬prime n define at h2 --h2 : ¬(2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n) demorgan at h2 disj_syll h2 h1 obtain (a : Nat) (h3 : ∃ (b : Nat), a * b = n ∧ a < n ∧ b < n) from h2 obtain (b : Nat) (h4 : a * b = n ∧ a < n ∧ b < n) from h3 have h5 : 2 ≤ a := by by_contra h6 have h7 : a ≤ 1 := by linarith have h8 : n ≤ b := calc n _ = a * b := h4.left.symm _ ≤ 1 * b := by rel [h7] _ = b := by ring linarith --n ≤ b contradicts b < n done have h6 : ∃ (p : Nat), prime_factor p a := ih a h4.right.left h5 obtain (p : Nat) (h7 : prime_factor p a) from h6 apply Exists.intro p define --Goal : prime p ∧ p ∣ n define at h7 --h7 : prime p ∧ p ∣ a apply And.intro h7.left have h8 : a ∣ n := by apply Exists.intro b show n = a * b from (h4.left).symm done show p ∣ n from dvd_trans h7.right h8 done done lemma exists_least_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q := by set S : Set Nat := {p : Nat | prime_factor p n} have h2 : ∃ (p : Nat), p ∈ S := exists_prime_factor n h show ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q from well_ord_princ S h2 done lemma all_prime_nil : all_prime [] := by define --Goal : ∀ p ∈ [], prime p fix p : Nat contrapos --Goal : ¬prime p → p ∉ [] assume h1 : ¬prime p show p ∉ [] from List.not_mem_nil p done lemma all_prime_cons (n : Nat) (L : List Nat) : all_prime (n :: L) ↔ prime n ∧ all_prime L := by apply Iff.intro · -- (→) assume h1 : all_prime (n :: L) --Goal : prime n ∧ all_prime L define at h1 --h1 : ∀ p ∈ n :: L, prime p apply And.intro (h1 n (List.mem_cons_self n L)) define --Goal : ∀ p ∈ L, prime p fix p : Nat assume h2 : p ∈ L show prime p from h1 p (List.mem_cons_of_mem n h2) done · -- (←) assume h1 : prime n ∧ all_prime L --Goal : all_prime (n :: l) define : all_prime L at h1 define fix p : Nat assume h2 : p ∈ n :: L rewrite [List.mem_cons] at h2 --h2 : p = n ∨ p ∈ L by_cases on h2 · -- Case 1. h2 : p = n rewrite [h2] show prime n from h1.left done · -- Case 2. h2 : p ∈ L show prime p from h1.right p h2 done done done lemma nondec_nil : nondec [] := by define --Goal : True trivial --trivial proves some obviously true statements, such as True done lemma nondec_cons (n : Nat) (L : List Nat) : nondec (n :: L) ↔ (∀ m ∈ L, n ≤ m) ∧ nondec L := by rfl lemma prod_nil : prod [] = 1 := by rfl lemma prod_cons : prod (n :: L) = n * (prod L) := by rfl lemma exists_cons_of_length_eq_succ {A : Type} {l : List A} {n : Nat} (h : l.length = n + 1) : ∃ (a : A) (L : List A), l = a :: L ∧ L.length = n := by have h1 : ¬l.length = 0 := by linarith rewrite [List.length_eq_zero] at h1 obtain (a : A) (h2 : ∃ (L : List A), l = a :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List A) (h3 : l = a :: L) from h2 apply Exists.intro a apply Exists.intro L apply And.intro h3 have h4 : (a :: L).length = L.length + 1 := List.length_cons a L rewrite [←h3, h] at h4 show L.length = n from (Nat.add_right_cancel h4).symm done lemma list_elt_dvd_prod_by_length (a : Nat) : ∀ (n : Nat), ∀ (l : List Nat), l.length = n → a ∈ l → a ∣ prod l := by by_induc · --Base Case fix l : List Nat assume h1 : l.length = 0 rewrite [List.length_eq_zero] at h1 --h1 : l = [] rewrite [h1] --Goal : a ∈ [] → a ∣ prod [] contrapos assume h2 : ¬a ∣ prod [] show a ∉ [] from List.not_mem_nil a done · -- Induction Step fix n : Nat assume ih : ∀ (l : List Nat), List.length l = n → a ∈ l → a ∣ prod l fix l : List Nat assume h1 : l.length = n + 1 --Goal : a ∈ l → a ∣ prod l obtain (b : Nat) (h2 : ∃ (L : List Nat), l = b :: L ∧ L.length = n) from exists_cons_of_length_eq_succ h1 obtain (L : List Nat) (h3 : l = b :: L ∧ L.length = n) from h2 have h4 : a ∈ L → a ∣ prod L := ih L h3.right assume h5 : a ∈ l rewrite [h3.left, prod_cons] --Goal : a ∣ b * prod L rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b ∨ a ∈ L by_cases on h5 · -- Case 1. h5 : a = b apply Exists.intro (prod L) rewrite [h5] rfl done · -- Case 2. h5 : a ∈ L have h6 : a ∣ prod L := h4 h5 have h7 : prod L ∣ b * prod L := by apply Exists.intro b ring done show a ∣ b * prod L from dvd_trans h6 h7 done done done lemma list_elt_dvd_prod {a : Nat} {l : List Nat} (h : a ∈ l) : a ∣ prod l := by set n : Nat := l.length have h1 : l.length = n := by rfl show a ∣ prod l from list_elt_dvd_prod_by_length a n l h1 h done lemma exists_prime_factorization : ∀ (n : Nat), n ≥ 1 → ∃ (l : List Nat), prime_factorization n l := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, n_1 ≥ 1 → ∃ (l : List Nat), prime_factorization n_1 l assume h1 : n ≥ 1 by_cases h2 : n = 1 · -- Case 1. h2 : n = 1 apply Exists.intro [] define apply And.intro · -- Proof of nondec_prime_list [] define show all_prime [] ∧ nondec [] from And.intro all_prime_nil nondec_nil done · -- Proof of prod [] = n rewrite [prod_nil, h2] rfl done done · -- Case 2. h2 : n ≠ 1 have h3 : n ≥ 2 := lt_of_le_of_ne' h1 h2 obtain (p : Nat) (h4 : prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q) from exists_least_prime_factor h3 have p_prime_factor : prime_factor p n := h4.left define at p_prime_factor have p_prime : prime p := p_prime_factor.left have p_dvd_n : p ∣ n := p_prime_factor.right have p_least : ∀ (q : Nat), prime_factor q n → p ≤ q := h4.right obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n have h5 : m ≠ 0 := by contradict h1 with h6 have h7 : n = 0 := calc n _ = p * m := n_eq_pm _ = p * 0 := by rw [h6] _ = 0 := by ring rewrite [h7] decide done have m_pos : 0 < m := Nat.pos_of_ne_zero h5 have m_lt_n : m < n := by define at p_prime show m < n from calc m _ < m + m := by linarith _ = 2 * m := by ring _ ≤ p * m := by rel [p_prime.left] _ = n := n_eq_pm.symm done obtain (L : List Nat) (h6 : prime_factorization m L) from ih m m_lt_n m_pos define at h6 have ndpl_L : nondec_prime_list L := h6.left define at ndpl_L apply Exists.intro (p :: L) define apply And.intro · -- Proof of nondec_prime_list (p :: L) define apply And.intro · -- Proof of all_prime (p :: L) rewrite [all_prime_cons] show prime p ∧ all_prime L from And.intro p_prime ndpl_L.left done · -- Proof of nondec (p :: L) rewrite [nondec_cons] apply And.intro _ ndpl_L.right fix q : Nat assume q_in_L : q ∈ L have h7 : q ∣ prod L := list_elt_dvd_prod q_in_L rewrite [h6.right] at h7 --h7 : q ∣ m have h8 : m ∣ n := by apply Exists.intro p rewrite [n_eq_pm] ring done have q_dvd_n : q ∣ n := dvd_trans h7 h8 have ap_L : all_prime L := ndpl_L.left define at ap_L have q_prime_factor : prime_factor q n := And.intro (ap_L q q_in_L) q_dvd_n show p ≤ q from p_least q q_prime_factor done done · -- Proof of prod (p :: L) = n rewrite [prod_cons, h6.right, n_eq_pm] rfl done done done theorem Theorem_7_2_2 {a b c : Nat} (h1 : c ∣ a * b) (h2 : rel_prime a c) : c ∣ b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑c ∣ ↑b define at h1; define at h2; define obtain (j : Nat) (h3 : a * b = c * j) from h1 set s : Int := gcd_c1 a c set t : Int := gcd_c2 a c have h4 : s * ↑a + t * ↑c = ↑(gcd a c) := gcd_lin_comb c a rewrite [h2, Nat.cast_one] at h4 --h4 : s * ↑a + t * ↑c = (1 : Int) apply Exists.intro (s * ↑j + t * ↑b) show ↑b = ↑c * (s * ↑j + t * ↑b) from calc ↑b _ = (1 : Int) * ↑b := (one_mul _).symm _ = (s * ↑a + t * ↑c) * ↑b := by rw [h4] _ = s * (↑a * ↑b) + t * ↑c * ↑b := by ring _ = s * (↑c * ↑j) + t * ↑c * ↑b := by rw [←Nat.cast_mul a b, h3, Nat.cast_mul c j] _ = ↑c * (s * ↑j + t * ↑b) := by ring done lemma le_nonzero_prod_left {a b : Nat} (h : a * b ≠ 0) : a ≤ a * b := by have h1 : b ≠ 0 := by contradict h with h1 rewrite [h1] ring done have h2 : 1 ≤ b := Nat.pos_of_ne_zero h1 show a ≤ a * b from calc a = a * 1 := (mul_one a).symm _ ≤ a * b := by rel [h2] done lemma le_nonzero_prod_right {a b : Nat} (h : a * b ≠ 0) : b ≤ a * b := by rewrite [mul_comm] rewrite [mul_comm] at h show b ≤ b * a from le_nonzero_prod_left h done lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry lemma rel_prime_of_prime_not_dvd {a p : Nat} (h1 : prime p) (h2 : ¬p ∣ a) : rel_prime a p := by have h3 : gcd a p ∣ a := gcd_dvd_left a p have h4 : gcd a p ∣ p := gcd_dvd_right a p have h5 : gcd a p = 1 ∨ gcd a p = p := dvd_prime h1 h4 have h6 : gcd a p ≠ p := by contradict h2 with h6 rewrite [h6] at h3 show p ∣ a from h3 done disj_syll h5 h6 show rel_prime a p from h5 done theorem Theorem_7_2_3 {a b p : Nat} (h1 : prime p) (h2 : p ∣ a * b) : p ∣ a ∨ p ∣ b := by or_right with h3 have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3 show p ∣ b from Theorem_7_2_2 h2 h4 done lemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a ≥ 1 := by have h1 : a ≠ 0 := by by_contra h1 rewrite [h1] at h contradict h linarith done show a ≥ 1 from Nat.pos_of_ne_zero h1 done lemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by have h1 : a ≥ 1 := ge_one_of_prod_one h have h2 : a * b ≠ 0 := by linarith have h3 : a ≤ a * b := le_nonzero_prod_left h2 rewrite [h] at h3 show a = 1 from Nat.le_antisymm h3 h1 done lemma eq_one_of_dvd_one {n : Nat} (h : n ∣ 1) : n = 1 := by obtain (j : Nat) (h1 : 1 = n * j) from h show n = 1 from eq_one_of_prod_one h1.symm done lemma prime_not_one {p : Nat} (h : prime p) : p ≠ 1 := by define at h linarith done theorem Theorem_7_2_4 {p : Nat} (h1 : prime p) : ∀ (l : List Nat), p ∣ prod l → ∃ a ∈ l, p ∣ a := by apply List.rec · -- Base Case. Goal : p ∣ prod [] → ∃ a ∈ [], p ∣ a rewrite [prod_nil] assume h2 : p ∣ 1 show ∃ a ∈ [], p ∣ a from absurd (eq_one_of_dvd_one h2) (prime_not_one h1) done · -- Induction Step fix b : Nat fix L : List Nat assume ih : p ∣ prod L → ∃ a ∈ L, p ∣ a --Goal : p ∣ prod (b :: L) → ∃ a ∈ b :: L, p ∣ a assume h2 : p ∣ prod (b :: L) rewrite [prod_cons] at h2 have h3 : p ∣ b ∨ p ∣ prod L := Theorem_7_2_3 h1 h2 by_cases on h3 · -- Case 1. h3 : p ∣ b apply Exists.intro b show b ∈ b :: L ∧ p ∣ b from And.intro (List.mem_cons_self b L) h3 done · -- Case 2. h3 : p ∣ prod L obtain (a : Nat) (h4 : a ∈ L ∧ p ∣ a) from ih h3 apply Exists.intro a show a ∈ b :: L ∧ p ∣ a from And.intro (List.mem_cons_of_mem b h4.left) h4.right done done done lemma prime_in_list {p : Nat} {l : List Nat} (h1 : prime p) (h2 : all_prime l) (h3 : p ∣ prod l) : p ∈ l := by obtain (a : Nat) (h4 : a ∈ l ∧ p ∣ a) from Theorem_7_2_4 h1 l h3 define at h2 have h5 : prime a := h2 a h4.left have h6 : p = 1 ∨ p = a := dvd_prime h5 h4.right disj_syll h6 (prime_not_one h1) rewrite [h6] show a ∈ l from h4.left done lemma first_le_first {p q : Nat} {l m : List Nat} (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m)) (h3 : prod (p :: l) = prod (q :: m)) : p ≤ q := by define at h1; define at h2 have h4 : q ∣ prod (p :: l) := by define apply Exists.intro (prod m) rewrite [←prod_cons] show prod (p :: l) = prod (q :: m) from h3 done have h5 : all_prime (q :: m) := h2.left rewrite [all_prime_cons] at h5 have h6 : q ∈ p :: l := prime_in_list h5.left h1.left h4 have h7 : nondec (p :: l) := h1.right rewrite [nondec_cons] at h7 rewrite [List.mem_cons] at h6 by_cases on h6 · -- Case 1. h6 : q = p linarith done · -- Case 2. h6 : q ∈ l have h8 : ∀ m ∈ l, p ≤ m := h7.left show p ≤ q from h8 q h6 done done lemma nondec_prime_list_tail {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by define at h define rewrite [all_prime_cons, nondec_cons] at h show all_prime l ∧ nondec l from And.intro h.left.right h.right.right done lemma cons_prod_not_one {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : prod (p :: l) ≠ 1 := by define at h have h1 : all_prime (p :: l) := h.left rewrite [all_prime_cons] at h1 rewrite [prod_cons] by_contra h2 show False from (prime_not_one h1.left) (eq_one_of_prod_one h2) done lemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) : l = [] ↔ prod l = 1 := by apply Iff.intro · -- (→) assume h1 : l = [] rewrite [h1] show prod [] = 1 from prod_nil done · -- (←) contrapos assume h1 : ¬l = [] obtain (p : Nat) (h2 : ∃ (L : List Nat), l = p :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List Nat) (h3 : l = p :: L) from h2 rewrite [h3] at h rewrite [h3] show ¬prod (p :: L) = 1 from cons_prod_not_one h done done lemma prime_pos {p : Nat} (h : prime p) : p > 0 := by define at h linarith done theorem Theorem_7_2_5 : ∀ (l1 l2 : List Nat), nondec_prime_list l1 → nondec_prime_list l2 → prod l1 = prod l2 → l1 = l2 := by apply List.rec · -- Base Case. Goal : ∀ (l2 : List Nat), nondec_prime_list [] → -- nondec_prime_list l2 → prod [] = prod l2 → [] = l2 fix l2 : List Nat assume h1 : nondec_prime_list [] assume h2 : nondec_prime_list l2 assume h3 : prod [] = prod l2 rewrite [prod_nil, eq_comm, ←list_nil_iff_prod_one h2] at h3 show [] = l2 from h3.symm done · -- Induction Step fix p : Nat fix L1 : List Nat assume ih : ∀ (L2 : List Nat), nondec_prime_list L1 → nondec_prime_list L2 → prod L1 = prod L2 → L1 = L2 -- Goal : ∀ (l2 : List Nat), nondec_prime_list (p :: L1) → -- nondec_prime_list l2 → prod (p :: L1) = prod l2 → p :: L1 = l2 fix l2 : List Nat assume h1 : nondec_prime_list (p :: L1) assume h2 : nondec_prime_list l2 assume h3 : prod (p :: L1) = prod l2 have h4 : ¬prod (p :: L1) = 1 := cons_prod_not_one h1 rewrite [h3, ←list_nil_iff_prod_one h2] at h4 obtain (q : Nat) (h5 : ∃ (L : List Nat), l2 = q :: L) from List.exists_cons_of_ne_nil h4 obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5 rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2) rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2) have h7 : p ≤ q := first_le_first h1 h2 h3 have h8 : q ≤ p := first_le_first h2 h1 h3.symm have h9 : p = q := by linarith rewrite [h9, prod_cons, prod_cons] at h3 --h3 : q * prod L1 = q * prod L2 have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1 have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2 define at h2 have h12 : all_prime (q :: L2) := h2.left rewrite [all_prime_cons] at h12 have h13 : q > 0 := prime_pos h12.left have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3 have h15 : L1 = L2 := ih L2 h10 h11 h14 rewrite [h6, h9, h15] rfl done done theorem fund_thm_arith (n : Nat) (h : n ≥ 1) : ∃! (l : List Nat), prime_factorization n l := by exists_unique · -- Existence show ∃ (l : List Nat), prime_factorization n l from exists_prime_factorization n h done · -- Uniqueness fix l1 : List Nat; fix l2 : List Nat assume h1 : prime_factorization n l1 assume h2 : prime_factorization n l2 define at h1; define at h2 have h3 : prod l1 = n := h1.right rewrite [←h2.right] at h3 show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3 done done /- Section 7.3 -/ theorem congr_refl (m : Nat) : ∀ (a : Int), a ≡ a (MOD m) := by fix a : Int define --Goal : ∃ (c : Int), a - a = ↑m * c apply Exists.intro 0 ring done theorem congr_symm {m : Nat} : ∀ {a b : Int}, a ≡ b (MOD m) → b ≡ a (MOD m) := by fix a : Int; fix b : Int assume h1 : a ≡ b (MOD m) define at h1 --h1 : ∃ (c : Int), a - b = ↑m * c define --Goal : ∃ (c : Int), b - a = ↑m * c obtain (c : Int) (h2 : a - b = m * c) from h1 apply Exists.intro (-c) show b - a = m * (-c) from calc b - a _ = -(a - b) := by ring _ = -(m * c) := by rw [h2] _ = m * (-c) := by ring done theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry /- Fundamental properties of congruence classes -/ lemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) : X = Y ↔ X.val = Y.val := Fin.ext_iff lemma val_nat_eq_mod (n k : Nat) : ([k]_(n + 1)).val = k % (n + 1) := by rfl lemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl theorem cc_rep {m : Nat} (X : ZMod m) : ∃ (a : Int), X = [a]_m := match m with | 0 => by apply Exists.intro X rfl done | n + 1 => by apply Exists.intro ↑(X.val) have h1 : X.val < n + 1 := Fin.prop X rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1] rfl done theorem add_class (m : Nat) (a b : Int) : [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm theorem mul_class (m : Nat) (a b : Int) : [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm lemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ [a - b]_m = [0]_m := by apply Iff.intro · -- (→) assume h1 : [a]_m = [b]_m have h2 : a - b = a + (-b) := by ring have h3 : b + (-b) = 0 := by ring show [a - b]_m = [0]_m from calc [a - b]_m _ = [a + (-b)]_m := by rw [h2] _ = [a]_m + [-b]_m := by rw [add_class] _ = [b]_m + [-b]_m := by rw [h1] _ = [b + -b]_m := by rw [add_class] _ = [0]_m := by rw [h3] done · -- (←) assume h1 : [a - b]_m = [0]_m have h2 : b + (a - b) = a := by ring have h3 : b + 0 = b := by ring show [a]_m = [b]_m from calc [a]_m _ = [b + (a - b)]_m := by rw [h2] _ = [b]_m + [a - b]_m := by rw [add_class] _ = [b]_m + [0]_m := by rw [h1] _ = [b + 0]_m := by rw [add_class] _ = [b]_m := by rw [h3] done done lemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) : [a]_m = [0]_m → [-a]_m = [0]_m := by assume h1 : [a]_m = [0]_m have h2 : 0 + (-a) = -a := by ring have h3 : a + (-a) = 0 := by ring show [-a]_m = [0]_m from calc [-a]_m _ = [0 + (-a)]_m := by rw [h2] _ = [0]_m + [-a]_m := by rw [add_class] _ = [a]_m + [-a]_m := by rw [h1] _ = [a + (-a)]_m := by rw [add_class] _ = [0]_m := by rw [h3] done lemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) : [-a]_m = [0]_m ↔ [a]_m = [0]_m := by apply Iff.intro _ (cc_neg_zero_of_cc_zero m a) assume h1 : [-a]_m = [0]_m have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1 have h3 : -(-a) = a := by ring rewrite [h3] at h2 show [a]_m = [0]_m from h2 done lemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl lemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m ↔ m ∣ k := match m with | 0 => by have h : (0 : Int) = (↑(0 : Nat) : Int) := by rfl rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj] apply Iff.intro · -- (→) assume h1 : k = 0 rewrite [h1] show 0 ∣ 0 from dvd_self 0 done · -- (←) assume h1 : 0 ∣ k obtain (c : Nat) (h2 : k = 0 * c) from h1 rewrite [h2] ring done done | n + 1 => by rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero] show k % (n + 1) = 0 ↔ n + 1 ∣ k from (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm done lemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m ↔ ↑m ∣ a := by obtain (k : Nat) (h1 : a = ↑k ∨ a = -↑k) from Int.eq_nat_or_neg a by_cases on h1 · -- Case 1. h1: a = ↑k rewrite [h1, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done · -- Case 2. h1: a = -↑k rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done done theorem cc_eq_iff_congr (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ a ≡ b (MOD m) := calc [a]_m = [b]_m _ ↔ [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b _ ↔ ↑m ∣ (a - b) := cc_zero_iff_dvd m (a - b) _ ↔ a ≡ b (MOD m) := by rfl /- End of fundamental properties of congruence classes -/ lemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m := by have h1 : (↑m : Int) ≠ 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m) show 0 ≤ a % m from Int.emod_nonneg a h1 done lemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m) have h2 : (↑m : Int) > 0 := (Nat.cast_pos).rtl h1 show a % m < m from Int.emod_lt_of_pos a h2 done lemma congr_mod_mod (m : Nat) (a : Int) : a ≡ a % m (MOD m) := by define have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m apply Exists.intro (a / m) show a - a % m = m * (a / m) from calc a - (a % m) _ = m * (a / m) + a % m - a % m := by rw [h1] _ = m * (a / m) := by ring done lemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) := And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a)) theorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) : ∃! (r : Int), 0 ≤ r ∧ r < m ∧ a ≡ r (MOD m) := by exists_unique · -- Existence apply Exists.intro (a % m) show 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) from mod_cmpl_res m a done · -- Uniqueness fix r1 : Int; fix r2 : Int assume h1 : 0 ≤ r1 ∧ r1 < m ∧ a ≡ r1 (MOD m) assume h2 : 0 ≤ r2 ∧ r2 < m ∧ a ≡ r2 (MOD m) have h3 : r1 ≡ r2 (MOD m) := congr_trans (congr_symm h1.right.right) h2.right.right obtain (d : Int) (h4 : r1 - r2 = m * d) from h3 have h5 : r1 - r2 < m * 1 := by linarith have h6 : m * (-1) < r1 - r2 := by linarith rewrite [h4] at h5 --h5 : m * d < m * 1 rewrite [h4] at h6 --h6 : m * -1 < m * d have h7 : (↑m : Int) ≥ 0 := Nat.cast_nonneg m have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7 have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7 have h10 : d = 0 := by linarith show r1 = r2 from calc r1 _ = r1 - r2 + r2 := by ring _ = m * 0 + r2 := by rw [h4, h10] _ = r2 := by ring done done lemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m := (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a) theorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y rewrite [h1, h2] have h3 : a + b = b + a := by ring show [a]_m + [b]_m = [b]_m + [a]_m from calc [a]_m + [b]_m _ = [a + b]_m := add_class m a b _ = [b + a]_m := by rw [h3] _ = [b]_m + [a]_m := (add_class m b a).symm done theorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X rewrite [h1] have h2 : a * 1 = a := by ring show [a]_m * [1]_m = [a]_m from calc [a]_m * [1]_m _ = [a * 1]_m := mul_class m a 1 _ = [a]_m := by rw [h2] done theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry lemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) : [a]_m * [gcd_c2 m a]_m = [1]_m := by set s : Int := gcd_c1 m a have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m define at h1 rewrite [h1, Nat.cast_one] at h2 --h2 : s * ↑m + gcd_c2 m a * ↑a = 1 rewrite [mul_class, cc_eq_iff_congr] define --Goal : ∃ (c : Int), ↑a * gcd_c2 m a - 1 = ↑m * c apply Exists.intro (-s) show a * (gcd_c2 m a) - 1 = m * (-s) from calc a * (gcd_c2 m a) - 1 _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring _ = 1 + m * (-s) - 1 := by rw [h2] _ = m * (-s) := by ring done theorem Theorem_7_3_7 (m a : Nat) : invertible [a]_m ↔ rel_prime m a := by apply Iff.intro · -- (→) assume h1 : invertible [a]_m define at h1 obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1 obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y rewrite [h3, mul_class, cc_eq_iff_congr] at h2 define at h2 obtain (c : Int) (h4 : a * b - 1 = m * c) from h2 rewrite [Exercise_7_2_6] --Goal : ∃ (s t : Int), s * ↑m + t * ↑a = 1 apply Exists.intro (-c) apply Exists.intro b show (-c) * m + b * a = 1 from calc (-c) * m + b * a _ = (-c) * m + (a * b - 1) + 1 := by ring _ = (-c) * m + m * c + 1 := by rw [h4] _ = 1 := by ring done · -- (←) assume h1 : rel_prime m a define show ∃ (Y : ZMod m), [a]_m * Y = [1]_m from Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1) done done /- Section 7.4 -/ section Euler open Euler lemma num_rp_below_base {m : Nat} : num_rp_below m 0 = 0 := by rfl lemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) : num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by have h1 : num_rp_below m (j + 1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : gcd m j = 1 rewrite [if_pos h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1 show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1 done lemma num_rp_below_step_not_rp {m j : Nat} (h : ¬rel_prime m j) : num_rp_below m (j + 1) = num_rp_below m j := by have h1 : num_rp_below m (j +1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : ¬gcd m j = 1 rewrite [if_neg h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j show num_rp_below m (j + 1) = num_rp_below m j from h1 done lemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl #eval phi 10 --Answer: 4 lemma prod_inv_iff_inv {m : Nat} {X : ZMod m} (h1 : invertible X) (Y : ZMod m) : invertible (X * Y) ↔ invertible Y := by apply Iff.intro · -- (→) assume h2 : invertible (X * Y) obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2 apply Exists.intro (X * Z) rewrite [←h3] --Goal : Y * (X * Z) = X * Y * Z ring --Note that ring can do algebra in ZMod m done · -- (←) assume h2 : invertible Y obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1 obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2 apply Exists.intro (Xi * Yi) show (X * Y) * (Xi * Yi) = [1]_m from calc X * Y * (Xi * Yi) _ = (X * Xi) * (Y * Yi) := by ring _ = [1]_m * [1]_m := by rw [h3, h4] _ = [1]_m := Theorem_7_3_6_7 [1]_m done done lemma F_rp_def {m i : Nat} (h : rel_prime m i) : F m i = [i]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h --h : gcd m i = 1 rewrite [if_pos h] at h1 show F m i = [i]_m from h1 done lemma F_not_rp_def {m i : Nat} (h : ¬rel_prime m i) : F m i = [1]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h rewrite [h1, if_neg h] rfl done lemma prod_seq_base {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 0 k f = [1]_m := by rfl lemma prod_seq_step {m : Nat} (n k : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl lemma prod_seq_zero_step {m : Nat} (n : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by rewrite [prod_seq_step, zero_add] rfl done lemma prod_one {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 1 k f = f k := by rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7] rfl done lemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl lemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m := calc [G m a i]_m _ = [(a * i) % m]_m := by rfl _ = [a * i]_m := (cc_eq_mod m (a * i)).symm _ = [a]_m * [i]_m := (mul_class m a i).symm lemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) : rel_prime m (G m a i) ↔ rel_prime m i := by have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1 show rel_prime m (G m a i) ↔ rel_prime m i from calc rel_prime m (G m a i) _ ↔ invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm _ ↔ invertible ([a]_m * [i]_m) := by rw [cc_G] _ ↔ invertible [i]_m := prod_inv_iff_inv h2 ([i]_m) _ ↔ rel_prime m i := Theorem_7_3_7 m i done lemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) : F m (G m a i) = [a]_m * F m i := by have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2 show F m (G m a i) = [a]_m * F m i from calc F m (G m a i) _ = [G m a i]_m := F_rp_def h3 _ = [a]_m * [i]_m := cc_G m a i _ = [a]_m * F m i := by rw [F_rp_def h2] done lemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : ¬rel_prime m i) : F m (G m a i) = [1]_m := by rewrite [←G_rp_iff h1 i] at h2 show F m (G m a i) = [1]_m from F_not_rp_def h2 done lemma FG_prod {m a : Nat} (h1 : rel_prime m a) : ∀ (k : Nat), prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by by_induc · -- Base Case show prod_seq 0 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from calc prod_seq 0 0 ((F m) ∘ (G m a)) _ = [1]_m := prod_seq_base _ _ _ = [a]_m ^ 0 * [1]_m := by ring _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by rw [num_rp_below_base, prod_seq_base] done · -- Induction Step fix k : Nat assume ih : prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) by_cases h2 : rel_prime m k · -- Case 1. h2 : rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([a]_m * F m k) := by rw [FG_rp h1 h2] _ = [a]_m ^ ((num_rp_below m k) + 1) * ((prod_seq k 0 (F m)) * F m k) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_rp h2, prod_seq_zero_step] done · -- Case 2. h2 : ¬rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([1]_m) := by rw [FG_not_rp h1 h2] _ = [a]_m ^ (num_rp_below m k) * (prod_seq k 0 (F m) * ([1]_m)) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_not_rp h2, prod_seq_zero_step, F_not_rp_def h2] done done done lemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by define --Goal : ∀ i < m, G m a i < m fix i : Nat assume h1 : i < m rewrite [G_def] --Goal : a * i % m < m show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m) done lemma left_inv_one_one_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g' (g i) = i) : one_one_below n g := sorry lemma right_inv_onto_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g (g' i) = i) (h2 : maps_below n g') : onto_below n g := by define at h2; define fix k : Nat assume h3 : k < n apply Exists.intro (g' k) show g' k < n ∧ g (g' k) = k from And.intro (h2 k h3) (h1 k h3) done lemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m * [inv_mod m a]_m = [1]_m := by have h2 : 0 ≤ (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a) show [a]_m * [inv_mod m a]_m = [1]_m from calc [a]_m * [inv_mod m a]_m _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2] _ = [a]_m * [gcd_c2 m a]_m := by rw [←cc_eq_mod] _ = [1]_m := gcd_c2_inv h1 done lemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m := calc a * (b % m) % m = a % m * (b % m % m) % m := Nat.mul_mod _ _ _ _ = a % m * (b % m) % m := by rw [Nat.mod_mod] _ = a * b % m := (Nat.mul_mod _ _ _).symm lemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm] rfl done theorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] : ↑a ≡ ↑b (MOD m) ↔ a % m = b % m := sorry lemma mul_inv_mod_cancel {m a i : Nat} [NeZero m] (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1 rewrite [mul_class, cc_eq_iff_congr, ←Nat.cast_mul, ←Nat.cast_one, congr_iff_mod_eq_Nat] at h3 show a * inv_mod m a * i % m = i from calc a * (inv_mod m a) * i % m _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = 1 % m * i % m := by rw [h3] _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = i % m := by rw [one_mul] _ = i := Nat.mod_eq_of_lt h2 done lemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl lemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, G m a (Ginv m a i) = i := by fix i : Nat assume h2 : i < m show G m a (Ginv m a i) = i from calc G m a (Ginv m a i) _ = a * ((inv_mod m a * i) % m) % m := by rfl _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, Ginv m a (G m a i) = i := by fix i : Nat assume h2 : i < m show Ginv m a (G m a i) = i from calc Ginv m a (G m a i) _ = inv_mod m a * ((a * i) % m) % m := by rfl _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc, mul_comm (inv_mod m a)] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_maps_below (m a : Nat) [NeZero m] : maps_below m (Ginv m a) := G_maps_below m (inv_mod m a) lemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : one_one_below m (G m a) := left_inv_one_one_below (Ginv_left_inv h1) lemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : onto_below m (G m a) := right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a) lemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : perm_below m (G m a) := And.intro (G_maps_below m a) (And.intro (G_one_one_below h1) (G_onto_below h1)) --Permuting a product of congruence classes doesn't change product lemma swap_fst (u v : Nat) : swap u v u = v := by define : swap u v u --Goal : (if u = u then v else if u = v then u else u) = v have h : u = u := by rfl rewrite [if_pos h] rfl done lemma swap_snd (u v : Nat) : swap u v v = u := by define : swap u v v by_cases h1 : v = u · -- Case 1. h1 : v = u rewrite [if_pos h1] show v = u from h1 done · -- Case 2. h1 : v ≠ u rewrite [if_neg h1] have h2 : v = v := by rfl rewrite [if_pos h2] rfl done done lemma swap_other {u v i : Nat} (h1 : i ≠ u) (h2 : i ≠ v) : swap u v i = i := by define : swap u v i rewrite [if_neg h1, if_neg h2] rfl done lemma swap_values (u v i : Nat) : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := by by_cases h1 : i = u · -- Case 1. h1 : i = u apply Or.inl rewrite [h1] show swap u v u = v from swap_fst u v done · -- Case 2. h1 : i ≠ u apply Or.inr by_cases h2 : i = v · -- Case 2.1. h2 : i = v apply Or.inl rewrite [h2] show swap u v v = u from swap_snd u v done · -- Case 2.2. h2 : i ≠ v apply Or.inr show swap u v i = i from swap_other h1 h2 done done done lemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by define fix i : Nat assume h3 : i < n have h4 : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := swap_values u v i by_cases on h4 · -- Case 1. h4 : swap u v i = v rewrite [h4] show v < n from h2 done · -- Case 2. by_cases on h4 · -- Case 2.1. h4 : swap u v i = u rewrite [h4] show u < n from h1 done · -- Case 2.2. h4 : swap u v i = i rewrite [h4] show i < n from h3 done done done lemma swap_swap (u v n : Nat) : ∀ i < n, swap u v (swap u v i) = i := by fix i : Nat assume h : i < n by_cases h1 : i = u · -- Case 1. h1 : i = u rewrite [h1, swap_fst, swap_snd] rfl done · -- Case 2. h1 : i ≠ u by_cases h2 : i = v · -- Case 2.1. h2 : i = v rewrite [h2, swap_snd, swap_fst] rfl done · -- Case 2.2. h2 : i ≠ v rewrite [swap_other h1 h2, swap_other h1 h2] rfl done done done lemma swap_one_one_below (u v n) : one_one_below n (swap u v) := left_inv_one_one_below (swap_swap u v n) lemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) := right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2) lemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) := And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2)) lemma comp_perm_below {n : Nat} {f g : Nat → Nat} (h1 : perm_below n f) (h2 : perm_below n g) : perm_below n (f ∘ g) := sorry lemma trivial_swap (u : Nat) : swap u u = id := by apply funext fix x : Nat by_cases h1 : x = u · -- Case 1. h1 : x = u rewrite [h1, swap_fst] rfl done · -- Case 2. h1 : x ≠ u rewrite [swap_other h1 h1] rfl done done lemma prod_eq_fun {m : Nat} (f g : Nat → ZMod m) (k : Nat) : ∀ (n : Nat), (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g := by by_induc · -- Base Case assume h : (∀ i < 0, f (k + i) = g (k + i)) rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g assume h1 : ∀ i < n + 1, f (k + i) = g (k + i) have h2 : ∀ i < n, f (k + i) = g (k + i) := by fix i : Nat assume h2 : i < n have h3 : i < n + 1 := by linarith show f (k + i) = g (k + i) from h1 i h3 done have h3 : prod_seq n k f = prod_seq n k g := ih h2 have h4 : n < n + 1 := Nat.lt_succ_self n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4] rfl done done lemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := by have h2 : ∀ (i : Nat), i < u → (f ∘ swap u n) (0 + i) = f (0 + i) := by fix i : Nat assume h2 : i < u have h3 : 0 + i ≠ u := by linarith have h4 : 0 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl done show prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f from prod_eq_fun (f ∘ swap u n) f 0 u h2 done lemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := by have h2 : ∀ i < j, (f ∘ swap u n) (u + 1 + i) = f (u + 1 + i) := by fix i : Nat assume h2 : i < j have h3 : u + 1 + i ≠ u := by linarith have h4 : u + 1 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl show prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f from prod_eq_fun (f ∘ swap u n) f (u + 1) j h2 done lemma break_prod {m : Nat} (n : Nat) (f : Nat → ZMod m) : ∀ (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by by_induc · -- Base Case have h : n + 0 = n := by rfl rewrite [prod_seq_base, h, Theorem_7_3_6_7] rfl done · -- Induction Step fix j : Nat assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f rewrite [←add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc] rfl done done lemma break_prod_twice {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f := break_prod n f 1 rewrite [prod_one] at h2 have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f := break_prod (u + 1) f j rewrite [←h1] at h3 have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f := break_prod u f 1 rewrite [prod_one] at h4 rewrite [h3, h4] at h2 show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2 done lemma swap_prod_eq_prod {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f := by by_cases h2 : u = n · -- Case 1. h2 : u = n rewrite [h2, trivial_swap n] --Goal : prod_seq (n + 1) 0 (f ∘ id) = prod_seq (n + 1) 0 f rfl done · -- Case 2. h2 : ¬u = n have h3 : u + 1 ≤ n := Nat.lt_of_le_of_ne h1 h2 obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3 have break_f : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := break_prod_twice f h4 have break_fs : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_prod_twice (f ∘ swap u n) h4 have f_eq_fs_below : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := swap_prod_eq_prod_below f h1 have f_eq_fs_btwn : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4 show prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f from calc prod_seq (n + 1) 0 (f ∘ swap u n) _ = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_fs _ = prod_seq u 0 f * (f ∘ swap u n) u * prod_seq j (u + 1) f * (f ∘ swap u n) n := by rw [f_eq_fs_below, f_eq_fs_btwn] _ = prod_seq u 0 f * f (swap u n u) * prod_seq j (u + 1) f * f (swap u n n) := by rfl _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by rw [swap_fst, swap_snd] _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring _ = prod_seq (n + 1) 0 f := break_f.symm done done lemma perm_below_fixed {n : Nat} {g : Nat → Nat} (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry lemma perm_prod {m : Nat} (f : Nat → ZMod m) : ∀ (n : Nat), ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) := by by_induc · -- Base Case fix g : Nat → Nat assume h1 : perm_below 0 g rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) fix g : Nat → Nat assume g_pb : perm_below (n + 1) g define at g_pb have g_ob : onto_below (n + 1) g := g_pb.right.right define at g_ob have h1 : n < n + 1 := by linarith obtain (u : Nat) (h2 : u < n + 1 ∧ g u = n) from g_ob n h1 have s_pb : perm_below (n + 1) (swap u n) := swap_perm_below h2.left h1 have gs_pb_n1 : perm_below (n + 1) (g ∘ swap u n) := comp_perm_below g_pb s_pb have gs_fix_n : (g ∘ swap u n) n = n := calc (g ∘ swap u n) n _ = g (swap u n n) := by rfl _ = g u := by rw [swap_snd] _ = n := h2.right have gs_pb_n : perm_below n (g ∘ swap u n) := perm_below_fixed gs_pb_n1 gs_fix_n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f ∘ (g ∘ swap u n)) := ih (g ∘ swap u n) gs_pb_n have h3 : u ≤ n := by linarith show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f ∘ g) from calc prod_seq (n + 1) 0 f _ = prod_seq n 0 f * f n := prod_seq_zero_step n f _ = prod_seq n 0 (f ∘ (g ∘ swap u n)) * f ((g ∘ swap u n) n) := by rw [gs_prod, gs_fix_n] _ = prod_seq n 0 (f ∘ g ∘ swap u n) * (f ∘ g ∘ swap u n) n := by rfl _ = prod_seq (n + 1) 0 (f ∘ g ∘ swap u n) := (prod_seq_zero_step n (f ∘ g ∘ swap u n)).symm _ = prod_seq (n + 1) 0 ((f ∘ g) ∘ swap u n) := by rfl _ = prod_seq (n + 1) 0 (f ∘ g) := swap_prod_eq_prod (f ∘ g) h3 done done lemma F_invertible (m i : Nat) : invertible (F m i) := by by_cases h : rel_prime m i · -- Case 1. h : rel_prime m i rewrite [F_rp_def h] show invertible [i]_m from (Theorem_7_3_7 m i).rtl h done · -- Case 2. h : ¬rel_prime m i rewrite [F_not_rp_def h] apply Exists.intro [1]_m show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m done done lemma Fprod_invertible (m : Nat) : ∀ (k : Nat), invertible (prod_seq k 0 (F m)) := by by_induc · -- Base Case apply Exists.intro [1]_m show prod_seq 0 0 (F m) * [1]_m = [1]_m from calc prod_seq 0 0 (F m) * [1]_m _ = [1]_m * [1]_m := by rw [prod_seq_base] _ = [1]_m := Theorem_7_3_6_7 ([1]_m) done · -- Induction Step fix k : Nat assume ih : invertible (prod_seq k 0 (F m)) rewrite [prod_seq_zero_step] show invertible (prod_seq k 0 (F m) * (F m k)) from (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k) done done theorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m ^ (phi m) = [1]_m := by have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2 show [a]_m ^ (phi m) = [1]_m from calc [a]_m ^ (phi m) _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3] _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring _ = prod_seq m 0 (F m ∘ G m a) * Y := by rw [FG_prod h1 m, phi_def] _ = prod_seq m 0 (F m) * Y := by rw [perm_prod (F m) m (G m a) (G_perm_below h1)] _ = [1]_m := by rw [h3] done lemma Exercise_7_4_5_Int (m : Nat) (a : Int) : ∀ (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry lemma Exercise_7_4_5_Nat (m a n : Nat) : [a]_m ^ n = [a ^ n]_m := by rewrite [Exercise_7_4_5_Int] rfl done theorem Euler's_theorem {m a : Nat} [NeZero m] (h1 : rel_prime m a) : a ^ (phi m) ≡ 1 (MOD m) := by have h2 : [a]_m ^ (phi m) = [1]_m := Theorem_7_4_2 h1 rewrite [Exercise_7_4_5_Nat m a (phi m)] at h2 --h2 : [a ^ phi m]_m = [1]_m show a ^ (phi m) ≡ 1 (MOD m) from (cc_eq_iff_congr _ _ _).ltr h2 done #eval gcd 10 7 --Answer: 1. So 10 and 7 are relatively prime #eval 7 ^ phi 10 --Answer: 2401, which is congruent to 1 mod 10. end Euler /- Section 7.5 -/ lemma num_rp_prime {p : Nat} (h1 : prime p) : ∀ k < p, num_rp_below p (k + 1) = k := sorry lemma phi_prime {p : Nat} (h1 : prime p) : phi p = p - 1 := by have h2 : 1 ≤ p := prime_pos h1 have h3 : p - 1 + 1 = p := Nat.sub_add_cancel h2 have h4 : p - 1 < p := by linarith have h5 : num_rp_below p (p - 1 + 1) = p - 1 := num_rp_prime h1 (p - 1) h4 rewrite [h3] at h5 show phi p = p - 1 from h5 done theorem Theorem_7_2_2_Int {a c : Nat} {b : Int} (h1 : ↑c ∣ ↑a * b) (h2 : rel_prime a c) : ↑c ∣ b := by rewrite [Int.natCast_dvd, Int.natAbs_mul, Int.natAbs_ofNat] at h1 --h1 : c ∣ a * Int.natAbs b rewrite [Int.natCast_dvd] --Goal : c ∣ Int.natAbs b show c ∣ Int.natAbs b from Theorem_7_2_2 h1 h2 done lemma Lemma_7_4_5 {m n : Nat} (a b : Int) (h1 : rel_prime m n) : a ≡ b (MOD m * n) ↔ a ≡ b (MOD m) ∧ a ≡ b (MOD n) := by apply Iff.intro · -- (→) assume h2 : a ≡ b (MOD m * n) obtain (j : Int) (h3 : a - b = (m * n) * j) from h2 apply And.intro · -- Proof of a ≡ b (MOD m) apply Exists.intro (n * j) show a - b = m * (n * j) from calc a - b _ = m * n * j := h3 _ = m * (n * j) := by ring done · -- Proof of a ≡ b (MOD n) apply Exists.intro (m * j) show a - b = n * (m * j) from calc a - b _ = m * n * j := h3 _ = n * (m * j) := by ring done done · -- (←) assume h2 : a ≡ b (MOD m) ∧ a ≡ b (MOD n) obtain (j : Int) (h3 : a - b = m * j) from h2.left have h4 : (↑n : Int) ∣ a - b := h2.right rewrite [h3] at h4 --h4 : ↑n ∣ ↑m * j have h5 : ↑n ∣ j := Theorem_7_2_2_Int h4 h1 obtain (k : Int) (h6 : j = n * k) from h5 apply Exists.intro k --Goal : a - b = ↑(m * n) * k rewrite [Nat.cast_mul] --Goal : a - b = ↑m * ↑n * k show a - b = (m * n) * k from calc a - b _ = m * j := h3 _ = m * (n * k) := by rw [h6] _ = (m * n) * k := by ring done done --From exercises of Section 7.2 theorem rel_prime_symm {a b : Nat} (h : rel_prime a b) : rel_prime b a := sorry lemma prime_NeZero {p : Nat} (h : prime p) : NeZero p := by rewrite [neZero_iff] --Goal : p ≠ 0 define at h linarith done lemma Lemma_7_5_1 {p e d m c s : Nat} {t : Int} (h1 : prime p) (h2 : e * d = (p - 1) * s + 1) (h3 : m ^ e - c = p * t) : c ^ d ≡ m (MOD p) := by have h4 : m ^ e ≡ c (MOD p) := Exists.intro t h3 have h5 : [m ^ e]_p = [c]_p := (cc_eq_iff_congr _ _ _).rtl h4 rewrite [←Exercise_7_4_5_Nat] at h5 --h5 : [m]_p ^ e = [c]_p by_cases h6 : p ∣ m · -- Case 1. h6 : p ∣ m have h7 : m ≡ 0 (MOD p) := by obtain (j : Nat) (h8 : m = p * j) from h6 apply Exists.intro (↑j : Int) --Goal : ↑m - 0 = ↑p * ↑j rewrite [h8, Nat.cast_mul] ring done have h8 : [m]_p = [0]_p := (cc_eq_iff_congr _ _ _).rtl h7 have h9 : e * d ≠ 0 := by rewrite [h2] show (p - 1) * s + 1 ≠ 0 from Nat.add_one_ne_zero _ done have h10 : (0 : Int) ^ (e * d) = 0 := zero_pow h9 have h11 : [c ^ d]_p = [m]_p := calc [c ^ d]_p _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat] _ = ([m]_p ^ e) ^ d := by rw [h5] _ = [m]_p ^ (e * d) := by ring _ = [0]_p ^ (e * d) := by rw [h8] _ = [0 ^ (e * d)]_p := Exercise_7_4_5_Int _ _ _ _ = [0]_p := by rw [h10] _ = [m]_p := by rw [h8] show c ^ d ≡ m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11 done · -- Case 2. h6 : ¬p ∣ m have h7 : rel_prime m p := rel_prime_of_prime_not_dvd h1 h6 have h8 : rel_prime p m := rel_prime_symm h7 have h9 : NeZero p := prime_NeZero h1 have h10 : (1 : Int) ^ s = 1 := by ring have h11 : [c ^ d]_p = [m]_p := calc [c ^ d]_p _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat] _ = ([m]_p ^ e) ^ d := by rw [h5] _ = [m]_p ^ (e * d) := by ring _ = [m]_p ^ ((p - 1) * s + 1) := by rw [h2] _ = ([m]_p ^ (p - 1)) ^ s * [m]_p := by ring _ = ([m]_p ^ (phi p)) ^ s * [m]_p := by rw [phi_prime h1] _ = [1]_p ^ s * [m]_p := by rw [Theorem_7_4_2 h8] _ = [1 ^ s]_p * [m]_p := by rw [Exercise_7_4_5_Int] _ = [1]_p * [m]_p := by rw [h10] _ = [m]_p * [1]_p := by ring _ = [m]_p := Theorem_7_3_6_7 _ show c ^ d ≡ m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11 done done theorem Theorem_7_5_1 (p q n e d k m c : Nat) (p_prime : prime p) (q_prime : prime q) (p_ne_q : p ≠ q) (n_pq : n = p * q) (ed_congr_1 : e * d = k * (p - 1) * (q - 1) + 1) (h1 : [m]_n ^ e = [c]_n) : [c]_n ^ d = [m]_n := by rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] at h1 --h1 : m ^ e ≡ c (MOD n) rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] --Goal : c ^ d ≡ m (MOD n) obtain (j : Int) (h2 : m ^ e - c = n * j) from h1 rewrite [n_pq, Nat.cast_mul] at h2 --h2 : m ^ e - c = p * q * j have h3 : e * d = (p - 1) * (k * (q - 1)) + 1 := by rewrite [ed_congr_1] ring done have h4 : m ^ e - c = p * (q * j) := by rewrite [h2] ring done have congr_p : c ^ d ≡ m (MOD p) := Lemma_7_5_1 p_prime h3 h4 have h5 : e * d = (q - 1) * (k * (p - 1)) + 1 := by rewrite [ed_congr_1] ring done have h6 : m ^ e - c = q * (p * j) := by rewrite [h2] ring done have congr_q : c ^ d ≡ m (MOD q) := Lemma_7_5_1 q_prime h5 h6 have h7 : ¬q ∣ p := by by_contra h8 have h9 : q = 1 ∨ q = p := dvd_prime p_prime h8 disj_syll h9 (prime_not_one q_prime) show False from p_ne_q h9.symm done have h8 : rel_prime p q := rel_prime_of_prime_not_dvd q_prime h7 rewrite [n_pq, Lemma_7_4_5 _ _ h8] show c ^ d ≡ m (MOD p) ∧ c ^ d ≡ m (MOD q) from And.intro congr_p congr_q done /- BEGIN EXERCISES -/ namespace Exercises /- Section 7.1 -/ -- 1. theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry -- 2. lemma gcd_comm_lt {a b : Nat} (h : a < b) : gcd a b = gcd b a := sorry theorem gcd_comm (a b : Nat) : gcd a b = gcd b a := sorry -- 3. theorem Exercise_7_1_5 (a b : Nat) (n : Int) : (∃ (s t : Int), s * a + t * b = n) ↔ (↑(gcd a b) : Int) ∣ n := sorry -- 4. theorem Exercise_7_1_6 (a b c : Nat) : gcd a b = gcd (a + b * c) b := sorry -- 5. theorem gcd_is_nonzero {a b : Nat} (h : a ≠ 0 ∨ b ≠ 0) : gcd a b ≠ 0 := sorry -- 6. theorem gcd_greatest {a b d : Nat} (h1 : gcd a b ≠ 0) (h2 : d ∣ a) (h3 : d ∣ b) : d ≤ gcd a b := sorry -- 7. lemma Lemma_7_1_10a {a b : Nat} (n : Nat) (h : a ∣ b) : (n * a) ∣ (n * b) := sorry lemma Lemma_7_1_10b {a b n : Nat} (h1 : n ≠ 0) (h2 : (n * a) ∣ (n * b)) : a ∣ b := sorry lemma Lemma_7_1_10c {a b : Nat} (h1 : a ∣ b) (h2 : b ∣ a) : a = b := sorry theorem Exercise_7_1_10 (a b n : Nat) : gcd (n * a) (n * b) = n * gcd a b := sorry /- Section 7.2 -/ -- 1. lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry -- 2. -- Hints: Start with apply List.rec. You may find mul_ne_zero useful theorem prod_nonzero_nonzero : ∀ (l : List Nat), (∀ a ∈ l, a ≠ 0) → prod l ≠ 0 := sorry -- 3. theorem rel_prime_iff_no_common_factor (a b : Nat) : rel_prime a b ↔ ¬∃ (p : Nat), prime p ∧ p ∣ a ∧ p ∣ b := sorry -- 4. theorem rel_prime_symm {a b : Nat} (h : rel_prime a b) : rel_prime b a := sorry -- 5. lemma in_prime_factorization_iff_prime_factor {a : Nat} {l : List Nat} (h1 : prime_factorization a l) (p : Nat) : p ∈ l ↔ prime_factor p a := sorry -- 6. theorem Exercise_7_2_5 {a b : Nat} {l m : List Nat} (h1 : prime_factorization a l) (h2 : prime_factorization b m) : rel_prime a b ↔ (¬∃ (p : Nat), p ∈ l ∧ p ∈ m) := sorry -- 7. theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry -- 8. theorem Exercise_7_2_7 {a b a' b' : Nat} (h1 : rel_prime a b) (h2 : a' ∣ a) (h3 : b' ∣ b) : rel_prime a' b' := sorry -- 9. theorem Exercise_7_2_9 {a b j k : Nat} (h1 : gcd a b ≠ 0) (h2 : a = j * gcd a b) (h3 : b = k * gcd a b) : rel_prime j k := sorry -- 10. theorem Exercise_7_2_17a (a b c : Nat) : gcd a (b * c) ∣ gcd a b * gcd a c := sorry /- Section 7.3 -/ -- 1.

theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m)

HTPI.Exercises.congr_trans

htpi/HTPILib/Chap7.lean

htpi

/- Copyright 2023 Daniel J. Velleman -/ import HTPILib.Chap6 namespace HTPI /- Definitions -/ lemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by have h : n + 1 > 0 := Nat.succ_pos n show a % (n + 1) < n + 1 from Nat.mod_lt a h done def gcd (a b : Nat) : Nat := match b with | 0 => a | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd (n + 1) (a % (n + 1)) termination_by b mutual def gcd_c1 (a b : Nat) : Int := match b with | 0 => 1 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c2 (n + 1) (a % (n + 1)) --Corresponds to s = t' termination_by b def gcd_c2 (a b : Nat) : Int := match b with | 0 => 0 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c1 (n + 1) (a % (n + 1)) - (gcd_c2 (n + 1) (a % (n + 1))) * ↑(a / (n + 1)) --Corresponds to t = s' - t'q termination_by b end def prime (n : Nat) : Prop := 2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n def prime_factor (p n : Nat) : Prop := prime p ∧ p ∣ n def all_prime (l : List Nat) : Prop := ∀ p ∈ l, prime p def nondec (l : List Nat) : Prop := match l with | [] => True --Of course, True is a proposition that is always true | n :: L => (∀ m ∈ L, n ≤ m) ∧ nondec L def nondec_prime_list (l : List Nat) : Prop := all_prime l ∧ nondec l def prod (l : List Nat) : Nat := match l with | [] => 1 | n :: L => n * (prod L) def prime_factorization (n : Nat) (l : List Nat) : Prop := nondec_prime_list l ∧ prod l = n def rel_prime (a b : Nat) : Prop := gcd a b = 1 def congr_mod (m : Nat) (a b : Int) : Prop := (↑m : Int) ∣ (a - b) def cc (m : Nat) (a : Int) : ZMod m := (↑a : ZMod m) notation:50 a " ≡ " b " (MOD " m ")" => congr_mod m a b notation:max "["a"]_"m:max => cc m a def invertible {m : Nat} (X : ZMod m) : Prop := ∃ (Y : ZMod m), X * Y = [1]_m def num_rp_below (m k : Nat) : Nat := match k with | 0 => 0 | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j def phi (m : Nat) : Nat := num_rp_below m m def prod_seq {m : Nat} (j k : Nat) (f : Nat → ZMod m) : ZMod m := match j with | 0 => [1]_m | n + 1 => prod_seq n k f * f (k + n) def maps_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i < n, g i < n def one_one_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i1 < n, ∀ i2 < n, g i1 = g i2 → i1 = i2 def onto_below (n : Nat) (g : Nat → Nat) : Prop := ∀ k < n, ∃ i < n, g i = k def perm_below (n : Nat) (g : Nat → Nat) : Prop := maps_below n g ∧ one_one_below n g ∧ onto_below n g def inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m) def swap (u v i : Nat) : Nat := if i = u then v else if i = v then u else i namespace Euler --For definitions specific to Euler's theorem def F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m def G (m a i : Nat) : Nat := (a * i) % m def Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i end Euler /- Section 7.1 -/ theorem dvd_mod_of_dvd_a_b {a b d : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ (a % b) := by set q : Nat := a / b have h3 : b * q + a % b = a := Nat.div_add_mod a b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 define --Goal : ∃ (c : Nat), a % b = d * c apply Exists.intro (j - k * q) show a % b = d * (j - k * q) from calc a % b _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm _ = a - b * q := by rw [h3] _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc] _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm done theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry #eval gcd 672 161 --Answer: 7 lemma gcd_base (a : Nat) : gcd a 0 = a := by rfl lemma gcd_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd a b = gcd b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1)) rfl done lemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b ≠ 0) : a % b < b := by have h1 : b > 0 := Nat.pos_of_ne_zero h show a % b < b from Nat.mod_lt a h1 done lemma dvd_self (n : Nat) : n ∣ n := by apply Exists.intro 1 ring done theorem gcd_dvd : ∀ (b a : Nat), (gcd a b) ∣ a ∧ (gcd a b) ∣ b := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd a b_1) ∣ a ∧ (gcd a b_1) ∣ b_1 fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_base] --Goal: a ∣ a ∧ a ∣ 0 apply And.intro (dvd_self a) define apply Exists.intro 0 rfl done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_nonzero a h1] --Goal : gcd b (a % b) ∣ a ∧ gcd b (a % b) ∣ b have h2 : a % b < b := mod_nonzero_lt a h1 have h3 : (gcd b (a % b)) ∣ b ∧ (gcd b (a % b)) ∣ (a % b) := ih (a % b) h2 b apply And.intro _ h3.left show (gcd b (a % b)) ∣ a from dvd_a_of_dvd_b_mod h3.left h3.right done done theorem gcd_dvd_left (a b : Nat) : (gcd a b) ∣ a := (gcd_dvd b a).left theorem gcd_dvd_right (a b : Nat) : (gcd a b) ∣ b := (gcd_dvd b a).right lemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl lemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c1 a b = gcd_c2 b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done lemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl lemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * ↑(a / b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done theorem gcd_lin_comb : ∀ (b a : Nat), (gcd_c1 a b) * ↑a + (gcd_c2 a b) * ↑b = ↑(gcd a b) := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd_c1 a b_1) * ↑a + (gcd_c2 a b_1) * ↑b_1 = ↑(gcd a b_1) fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base] --Goal : 1 * ↑a + 0 * ↑0 = ↑a ring done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1] --Goal : gcd_c2 b (a % b) * ↑a + -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * ↑(a / b)) * ↑b = -- ↑(gcd b (a % b)) set r : Nat := a % b set q : Nat := a / b set s : Int := gcd_c1 b r set t : Int := gcd_c2 b r --Goal : t * ↑a + (s - t * ↑q) * ↑b = ↑(gcd b r) have h2 : r < b := mod_nonzero_lt a h1 have h3 : s * ↑b + t * ↑r = ↑(gcd b r) := ih r h2 b have h4 : b * q + r = a := Nat.div_add_mod a b rewrite [←h3, ←h4] rewrite [Nat.cast_add, Nat.cast_mul] --Goal : t * (↑b * ↑q + ↑r) + (s - t * ↑q) * ↑b = s * ↑b + t * ↑r ring done done #eval gcd_c1 672 161 --Answer: 6 #eval gcd_c2 672 161 --Answer: -25 --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161 theorem Theorem_7_1_6 {d a b : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ gcd a b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑d ∣ ↑(gcd a b) set s : Int := gcd_c1 a b set t : Int := gcd_c2 a b have h3 : s * ↑a + t * ↑b = ↑(gcd a b) := gcd_lin_comb b a rewrite [←h3] --Goal : ↑d ∣ s * ↑a + t * ↑b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul] --Goal : ↑d ∣ s * (↑d * ↑j) + t * (↑d * ↑k) define apply Exists.intro (s * ↑j + t * ↑k) ring done /- Section 7.2 -/ theorem dvd_trans {a b c : Nat} (h1 : a ∣ b) (h2 : b ∣ c) : a ∣ c := by define at h1; define at h2; define obtain (m : Nat) (h3 : b = a * m) from h1 obtain (n : Nat) (h4 : c = b * n) from h2 rewrite [h3, mul_assoc] at h4 apply Exists.intro (m * n) show c = a * (m * n) from h4 done lemma exists_prime_factor : ∀ (n : Nat), 2 ≤ n → ∃ (p : Nat), prime_factor p n := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, 2 ≤ n_1 → ∃ (p : Nat), prime_factor p n_1 assume h1 : 2 ≤ n by_cases h2 : prime n · -- Case 1. h2 : prime n apply Exists.intro n define --Goal : prime n ∧ n ∣ n show prime n ∧ n ∣ n from And.intro h2 (dvd_self n) done · -- Case 2. h2 : ¬prime n define at h2 --h2 : ¬(2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n) demorgan at h2 disj_syll h2 h1 obtain (a : Nat) (h3 : ∃ (b : Nat), a * b = n ∧ a < n ∧ b < n) from h2 obtain (b : Nat) (h4 : a * b = n ∧ a < n ∧ b < n) from h3 have h5 : 2 ≤ a := by by_contra h6 have h7 : a ≤ 1 := by linarith have h8 : n ≤ b := calc n _ = a * b := h4.left.symm _ ≤ 1 * b := by rel [h7] _ = b := by ring linarith --n ≤ b contradicts b < n done have h6 : ∃ (p : Nat), prime_factor p a := ih a h4.right.left h5 obtain (p : Nat) (h7 : prime_factor p a) from h6 apply Exists.intro p define --Goal : prime p ∧ p ∣ n define at h7 --h7 : prime p ∧ p ∣ a apply And.intro h7.left have h8 : a ∣ n := by apply Exists.intro b show n = a * b from (h4.left).symm done show p ∣ n from dvd_trans h7.right h8 done done lemma exists_least_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q := by set S : Set Nat := {p : Nat | prime_factor p n} have h2 : ∃ (p : Nat), p ∈ S := exists_prime_factor n h show ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q from well_ord_princ S h2 done lemma all_prime_nil : all_prime [] := by define --Goal : ∀ p ∈ [], prime p fix p : Nat contrapos --Goal : ¬prime p → p ∉ [] assume h1 : ¬prime p show p ∉ [] from List.not_mem_nil p done lemma all_prime_cons (n : Nat) (L : List Nat) : all_prime (n :: L) ↔ prime n ∧ all_prime L := by apply Iff.intro · -- (→) assume h1 : all_prime (n :: L) --Goal : prime n ∧ all_prime L define at h1 --h1 : ∀ p ∈ n :: L, prime p apply And.intro (h1 n (List.mem_cons_self n L)) define --Goal : ∀ p ∈ L, prime p fix p : Nat assume h2 : p ∈ L show prime p from h1 p (List.mem_cons_of_mem n h2) done · -- (←) assume h1 : prime n ∧ all_prime L --Goal : all_prime (n :: l) define : all_prime L at h1 define fix p : Nat assume h2 : p ∈ n :: L rewrite [List.mem_cons] at h2 --h2 : p = n ∨ p ∈ L by_cases on h2 · -- Case 1. h2 : p = n rewrite [h2] show prime n from h1.left done · -- Case 2. h2 : p ∈ L show prime p from h1.right p h2 done done done lemma nondec_nil : nondec [] := by define --Goal : True trivial --trivial proves some obviously true statements, such as True done lemma nondec_cons (n : Nat) (L : List Nat) : nondec (n :: L) ↔ (∀ m ∈ L, n ≤ m) ∧ nondec L := by rfl lemma prod_nil : prod [] = 1 := by rfl lemma prod_cons : prod (n :: L) = n * (prod L) := by rfl lemma exists_cons_of_length_eq_succ {A : Type} {l : List A} {n : Nat} (h : l.length = n + 1) : ∃ (a : A) (L : List A), l = a :: L ∧ L.length = n := by have h1 : ¬l.length = 0 := by linarith rewrite [List.length_eq_zero] at h1 obtain (a : A) (h2 : ∃ (L : List A), l = a :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List A) (h3 : l = a :: L) from h2 apply Exists.intro a apply Exists.intro L apply And.intro h3 have h4 : (a :: L).length = L.length + 1 := List.length_cons a L rewrite [←h3, h] at h4 show L.length = n from (Nat.add_right_cancel h4).symm done lemma list_elt_dvd_prod_by_length (a : Nat) : ∀ (n : Nat), ∀ (l : List Nat), l.length = n → a ∈ l → a ∣ prod l := by by_induc · --Base Case fix l : List Nat assume h1 : l.length = 0 rewrite [List.length_eq_zero] at h1 --h1 : l = [] rewrite [h1] --Goal : a ∈ [] → a ∣ prod [] contrapos assume h2 : ¬a ∣ prod [] show a ∉ [] from List.not_mem_nil a done · -- Induction Step fix n : Nat assume ih : ∀ (l : List Nat), List.length l = n → a ∈ l → a ∣ prod l fix l : List Nat assume h1 : l.length = n + 1 --Goal : a ∈ l → a ∣ prod l obtain (b : Nat) (h2 : ∃ (L : List Nat), l = b :: L ∧ L.length = n) from exists_cons_of_length_eq_succ h1 obtain (L : List Nat) (h3 : l = b :: L ∧ L.length = n) from h2 have h4 : a ∈ L → a ∣ prod L := ih L h3.right assume h5 : a ∈ l rewrite [h3.left, prod_cons] --Goal : a ∣ b * prod L rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b ∨ a ∈ L by_cases on h5 · -- Case 1. h5 : a = b apply Exists.intro (prod L) rewrite [h5] rfl done · -- Case 2. h5 : a ∈ L have h6 : a ∣ prod L := h4 h5 have h7 : prod L ∣ b * prod L := by apply Exists.intro b ring done show a ∣ b * prod L from dvd_trans h6 h7 done done done lemma list_elt_dvd_prod {a : Nat} {l : List Nat} (h : a ∈ l) : a ∣ prod l := by set n : Nat := l.length have h1 : l.length = n := by rfl show a ∣ prod l from list_elt_dvd_prod_by_length a n l h1 h done lemma exists_prime_factorization : ∀ (n : Nat), n ≥ 1 → ∃ (l : List Nat), prime_factorization n l := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, n_1 ≥ 1 → ∃ (l : List Nat), prime_factorization n_1 l assume h1 : n ≥ 1 by_cases h2 : n = 1 · -- Case 1. h2 : n = 1 apply Exists.intro [] define apply And.intro · -- Proof of nondec_prime_list [] define show all_prime [] ∧ nondec [] from And.intro all_prime_nil nondec_nil done · -- Proof of prod [] = n rewrite [prod_nil, h2] rfl done done · -- Case 2. h2 : n ≠ 1 have h3 : n ≥ 2 := lt_of_le_of_ne' h1 h2 obtain (p : Nat) (h4 : prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q) from exists_least_prime_factor h3 have p_prime_factor : prime_factor p n := h4.left define at p_prime_factor have p_prime : prime p := p_prime_factor.left have p_dvd_n : p ∣ n := p_prime_factor.right have p_least : ∀ (q : Nat), prime_factor q n → p ≤ q := h4.right obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n have h5 : m ≠ 0 := by contradict h1 with h6 have h7 : n = 0 := calc n _ = p * m := n_eq_pm _ = p * 0 := by rw [h6] _ = 0 := by ring rewrite [h7] decide done have m_pos : 0 < m := Nat.pos_of_ne_zero h5 have m_lt_n : m < n := by define at p_prime show m < n from calc m _ < m + m := by linarith _ = 2 * m := by ring _ ≤ p * m := by rel [p_prime.left] _ = n := n_eq_pm.symm done obtain (L : List Nat) (h6 : prime_factorization m L) from ih m m_lt_n m_pos define at h6 have ndpl_L : nondec_prime_list L := h6.left define at ndpl_L apply Exists.intro (p :: L) define apply And.intro · -- Proof of nondec_prime_list (p :: L) define apply And.intro · -- Proof of all_prime (p :: L) rewrite [all_prime_cons] show prime p ∧ all_prime L from And.intro p_prime ndpl_L.left done · -- Proof of nondec (p :: L) rewrite [nondec_cons] apply And.intro _ ndpl_L.right fix q : Nat assume q_in_L : q ∈ L have h7 : q ∣ prod L := list_elt_dvd_prod q_in_L rewrite [h6.right] at h7 --h7 : q ∣ m have h8 : m ∣ n := by apply Exists.intro p rewrite [n_eq_pm] ring done have q_dvd_n : q ∣ n := dvd_trans h7 h8 have ap_L : all_prime L := ndpl_L.left define at ap_L have q_prime_factor : prime_factor q n := And.intro (ap_L q q_in_L) q_dvd_n show p ≤ q from p_least q q_prime_factor done done · -- Proof of prod (p :: L) = n rewrite [prod_cons, h6.right, n_eq_pm] rfl done done done theorem Theorem_7_2_2 {a b c : Nat} (h1 : c ∣ a * b) (h2 : rel_prime a c) : c ∣ b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑c ∣ ↑b define at h1; define at h2; define obtain (j : Nat) (h3 : a * b = c * j) from h1 set s : Int := gcd_c1 a c set t : Int := gcd_c2 a c have h4 : s * ↑a + t * ↑c = ↑(gcd a c) := gcd_lin_comb c a rewrite [h2, Nat.cast_one] at h4 --h4 : s * ↑a + t * ↑c = (1 : Int) apply Exists.intro (s * ↑j + t * ↑b) show ↑b = ↑c * (s * ↑j + t * ↑b) from calc ↑b _ = (1 : Int) * ↑b := (one_mul _).symm _ = (s * ↑a + t * ↑c) * ↑b := by rw [h4] _ = s * (↑a * ↑b) + t * ↑c * ↑b := by ring _ = s * (↑c * ↑j) + t * ↑c * ↑b := by rw [←Nat.cast_mul a b, h3, Nat.cast_mul c j] _ = ↑c * (s * ↑j + t * ↑b) := by ring done lemma le_nonzero_prod_left {a b : Nat} (h : a * b ≠ 0) : a ≤ a * b := by have h1 : b ≠ 0 := by contradict h with h1 rewrite [h1] ring done have h2 : 1 ≤ b := Nat.pos_of_ne_zero h1 show a ≤ a * b from calc a = a * 1 := (mul_one a).symm _ ≤ a * b := by rel [h2] done lemma le_nonzero_prod_right {a b : Nat} (h : a * b ≠ 0) : b ≤ a * b := by rewrite [mul_comm] rewrite [mul_comm] at h show b ≤ b * a from le_nonzero_prod_left h done lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry lemma rel_prime_of_prime_not_dvd {a p : Nat} (h1 : prime p) (h2 : ¬p ∣ a) : rel_prime a p := by have h3 : gcd a p ∣ a := gcd_dvd_left a p have h4 : gcd a p ∣ p := gcd_dvd_right a p have h5 : gcd a p = 1 ∨ gcd a p = p := dvd_prime h1 h4 have h6 : gcd a p ≠ p := by contradict h2 with h6 rewrite [h6] at h3 show p ∣ a from h3 done disj_syll h5 h6 show rel_prime a p from h5 done theorem Theorem_7_2_3 {a b p : Nat} (h1 : prime p) (h2 : p ∣ a * b) : p ∣ a ∨ p ∣ b := by or_right with h3 have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3 show p ∣ b from Theorem_7_2_2 h2 h4 done lemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a ≥ 1 := by have h1 : a ≠ 0 := by by_contra h1 rewrite [h1] at h contradict h linarith done show a ≥ 1 from Nat.pos_of_ne_zero h1 done lemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by have h1 : a ≥ 1 := ge_one_of_prod_one h have h2 : a * b ≠ 0 := by linarith have h3 : a ≤ a * b := le_nonzero_prod_left h2 rewrite [h] at h3 show a = 1 from Nat.le_antisymm h3 h1 done lemma eq_one_of_dvd_one {n : Nat} (h : n ∣ 1) : n = 1 := by obtain (j : Nat) (h1 : 1 = n * j) from h show n = 1 from eq_one_of_prod_one h1.symm done lemma prime_not_one {p : Nat} (h : prime p) : p ≠ 1 := by define at h linarith done theorem Theorem_7_2_4 {p : Nat} (h1 : prime p) : ∀ (l : List Nat), p ∣ prod l → ∃ a ∈ l, p ∣ a := by apply List.rec · -- Base Case. Goal : p ∣ prod [] → ∃ a ∈ [], p ∣ a rewrite [prod_nil] assume h2 : p ∣ 1 show ∃ a ∈ [], p ∣ a from absurd (eq_one_of_dvd_one h2) (prime_not_one h1) done · -- Induction Step fix b : Nat fix L : List Nat assume ih : p ∣ prod L → ∃ a ∈ L, p ∣ a --Goal : p ∣ prod (b :: L) → ∃ a ∈ b :: L, p ∣ a assume h2 : p ∣ prod (b :: L) rewrite [prod_cons] at h2 have h3 : p ∣ b ∨ p ∣ prod L := Theorem_7_2_3 h1 h2 by_cases on h3 · -- Case 1. h3 : p ∣ b apply Exists.intro b show b ∈ b :: L ∧ p ∣ b from And.intro (List.mem_cons_self b L) h3 done · -- Case 2. h3 : p ∣ prod L obtain (a : Nat) (h4 : a ∈ L ∧ p ∣ a) from ih h3 apply Exists.intro a show a ∈ b :: L ∧ p ∣ a from And.intro (List.mem_cons_of_mem b h4.left) h4.right done done done lemma prime_in_list {p : Nat} {l : List Nat} (h1 : prime p) (h2 : all_prime l) (h3 : p ∣ prod l) : p ∈ l := by obtain (a : Nat) (h4 : a ∈ l ∧ p ∣ a) from Theorem_7_2_4 h1 l h3 define at h2 have h5 : prime a := h2 a h4.left have h6 : p = 1 ∨ p = a := dvd_prime h5 h4.right disj_syll h6 (prime_not_one h1) rewrite [h6] show a ∈ l from h4.left done lemma first_le_first {p q : Nat} {l m : List Nat} (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m)) (h3 : prod (p :: l) = prod (q :: m)) : p ≤ q := by define at h1; define at h2 have h4 : q ∣ prod (p :: l) := by define apply Exists.intro (prod m) rewrite [←prod_cons] show prod (p :: l) = prod (q :: m) from h3 done have h5 : all_prime (q :: m) := h2.left rewrite [all_prime_cons] at h5 have h6 : q ∈ p :: l := prime_in_list h5.left h1.left h4 have h7 : nondec (p :: l) := h1.right rewrite [nondec_cons] at h7 rewrite [List.mem_cons] at h6 by_cases on h6 · -- Case 1. h6 : q = p linarith done · -- Case 2. h6 : q ∈ l have h8 : ∀ m ∈ l, p ≤ m := h7.left show p ≤ q from h8 q h6 done done lemma nondec_prime_list_tail {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by define at h define rewrite [all_prime_cons, nondec_cons] at h show all_prime l ∧ nondec l from And.intro h.left.right h.right.right done lemma cons_prod_not_one {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : prod (p :: l) ≠ 1 := by define at h have h1 : all_prime (p :: l) := h.left rewrite [all_prime_cons] at h1 rewrite [prod_cons] by_contra h2 show False from (prime_not_one h1.left) (eq_one_of_prod_one h2) done lemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) : l = [] ↔ prod l = 1 := by apply Iff.intro · -- (→) assume h1 : l = [] rewrite [h1] show prod [] = 1 from prod_nil done · -- (←) contrapos assume h1 : ¬l = [] obtain (p : Nat) (h2 : ∃ (L : List Nat), l = p :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List Nat) (h3 : l = p :: L) from h2 rewrite [h3] at h rewrite [h3] show ¬prod (p :: L) = 1 from cons_prod_not_one h done done lemma prime_pos {p : Nat} (h : prime p) : p > 0 := by define at h linarith done theorem Theorem_7_2_5 : ∀ (l1 l2 : List Nat), nondec_prime_list l1 → nondec_prime_list l2 → prod l1 = prod l2 → l1 = l2 := by apply List.rec · -- Base Case. Goal : ∀ (l2 : List Nat), nondec_prime_list [] → -- nondec_prime_list l2 → prod [] = prod l2 → [] = l2 fix l2 : List Nat assume h1 : nondec_prime_list [] assume h2 : nondec_prime_list l2 assume h3 : prod [] = prod l2 rewrite [prod_nil, eq_comm, ←list_nil_iff_prod_one h2] at h3 show [] = l2 from h3.symm done · -- Induction Step fix p : Nat fix L1 : List Nat assume ih : ∀ (L2 : List Nat), nondec_prime_list L1 → nondec_prime_list L2 → prod L1 = prod L2 → L1 = L2 -- Goal : ∀ (l2 : List Nat), nondec_prime_list (p :: L1) → -- nondec_prime_list l2 → prod (p :: L1) = prod l2 → p :: L1 = l2 fix l2 : List Nat assume h1 : nondec_prime_list (p :: L1) assume h2 : nondec_prime_list l2 assume h3 : prod (p :: L1) = prod l2 have h4 : ¬prod (p :: L1) = 1 := cons_prod_not_one h1 rewrite [h3, ←list_nil_iff_prod_one h2] at h4 obtain (q : Nat) (h5 : ∃ (L : List Nat), l2 = q :: L) from List.exists_cons_of_ne_nil h4 obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5 rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2) rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2) have h7 : p ≤ q := first_le_first h1 h2 h3 have h8 : q ≤ p := first_le_first h2 h1 h3.symm have h9 : p = q := by linarith rewrite [h9, prod_cons, prod_cons] at h3 --h3 : q * prod L1 = q * prod L2 have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1 have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2 define at h2 have h12 : all_prime (q :: L2) := h2.left rewrite [all_prime_cons] at h12 have h13 : q > 0 := prime_pos h12.left have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3 have h15 : L1 = L2 := ih L2 h10 h11 h14 rewrite [h6, h9, h15] rfl done done theorem fund_thm_arith (n : Nat) (h : n ≥ 1) : ∃! (l : List Nat), prime_factorization n l := by exists_unique · -- Existence show ∃ (l : List Nat), prime_factorization n l from exists_prime_factorization n h done · -- Uniqueness fix l1 : List Nat; fix l2 : List Nat assume h1 : prime_factorization n l1 assume h2 : prime_factorization n l2 define at h1; define at h2 have h3 : prod l1 = n := h1.right rewrite [←h2.right] at h3 show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3 done done /- Section 7.3 -/ theorem congr_refl (m : Nat) : ∀ (a : Int), a ≡ a (MOD m) := by fix a : Int define --Goal : ∃ (c : Int), a - a = ↑m * c apply Exists.intro 0 ring done theorem congr_symm {m : Nat} : ∀ {a b : Int}, a ≡ b (MOD m) → b ≡ a (MOD m) := by fix a : Int; fix b : Int assume h1 : a ≡ b (MOD m) define at h1 --h1 : ∃ (c : Int), a - b = ↑m * c define --Goal : ∃ (c : Int), b - a = ↑m * c obtain (c : Int) (h2 : a - b = m * c) from h1 apply Exists.intro (-c) show b - a = m * (-c) from calc b - a _ = -(a - b) := by ring _ = -(m * c) := by rw [h2] _ = m * (-c) := by ring done theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry /- Fundamental properties of congruence classes -/ lemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) : X = Y ↔ X.val = Y.val := Fin.ext_iff lemma val_nat_eq_mod (n k : Nat) : ([k]_(n + 1)).val = k % (n + 1) := by rfl lemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl theorem cc_rep {m : Nat} (X : ZMod m) : ∃ (a : Int), X = [a]_m := match m with | 0 => by apply Exists.intro X rfl done | n + 1 => by apply Exists.intro ↑(X.val) have h1 : X.val < n + 1 := Fin.prop X rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1] rfl done theorem add_class (m : Nat) (a b : Int) : [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm theorem mul_class (m : Nat) (a b : Int) : [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm lemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ [a - b]_m = [0]_m := by apply Iff.intro · -- (→) assume h1 : [a]_m = [b]_m have h2 : a - b = a + (-b) := by ring have h3 : b + (-b) = 0 := by ring show [a - b]_m = [0]_m from calc [a - b]_m _ = [a + (-b)]_m := by rw [h2] _ = [a]_m + [-b]_m := by rw [add_class] _ = [b]_m + [-b]_m := by rw [h1] _ = [b + -b]_m := by rw [add_class] _ = [0]_m := by rw [h3] done · -- (←) assume h1 : [a - b]_m = [0]_m have h2 : b + (a - b) = a := by ring have h3 : b + 0 = b := by ring show [a]_m = [b]_m from calc [a]_m _ = [b + (a - b)]_m := by rw [h2] _ = [b]_m + [a - b]_m := by rw [add_class] _ = [b]_m + [0]_m := by rw [h1] _ = [b + 0]_m := by rw [add_class] _ = [b]_m := by rw [h3] done done lemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) : [a]_m = [0]_m → [-a]_m = [0]_m := by assume h1 : [a]_m = [0]_m have h2 : 0 + (-a) = -a := by ring have h3 : a + (-a) = 0 := by ring show [-a]_m = [0]_m from calc [-a]_m _ = [0 + (-a)]_m := by rw [h2] _ = [0]_m + [-a]_m := by rw [add_class] _ = [a]_m + [-a]_m := by rw [h1] _ = [a + (-a)]_m := by rw [add_class] _ = [0]_m := by rw [h3] done lemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) : [-a]_m = [0]_m ↔ [a]_m = [0]_m := by apply Iff.intro _ (cc_neg_zero_of_cc_zero m a) assume h1 : [-a]_m = [0]_m have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1 have h3 : -(-a) = a := by ring rewrite [h3] at h2 show [a]_m = [0]_m from h2 done lemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl lemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m ↔ m ∣ k := match m with | 0 => by have h : (0 : Int) = (↑(0 : Nat) : Int) := by rfl rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj] apply Iff.intro · -- (→) assume h1 : k = 0 rewrite [h1] show 0 ∣ 0 from dvd_self 0 done · -- (←) assume h1 : 0 ∣ k obtain (c : Nat) (h2 : k = 0 * c) from h1 rewrite [h2] ring done done | n + 1 => by rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero] show k % (n + 1) = 0 ↔ n + 1 ∣ k from (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm done lemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m ↔ ↑m ∣ a := by obtain (k : Nat) (h1 : a = ↑k ∨ a = -↑k) from Int.eq_nat_or_neg a by_cases on h1 · -- Case 1. h1: a = ↑k rewrite [h1, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done · -- Case 2. h1: a = -↑k rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done done theorem cc_eq_iff_congr (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ a ≡ b (MOD m) := calc [a]_m = [b]_m _ ↔ [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b _ ↔ ↑m ∣ (a - b) := cc_zero_iff_dvd m (a - b) _ ↔ a ≡ b (MOD m) := by rfl /- End of fundamental properties of congruence classes -/ lemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m := by have h1 : (↑m : Int) ≠ 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m) show 0 ≤ a % m from Int.emod_nonneg a h1 done lemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m) have h2 : (↑m : Int) > 0 := (Nat.cast_pos).rtl h1 show a % m < m from Int.emod_lt_of_pos a h2 done lemma congr_mod_mod (m : Nat) (a : Int) : a ≡ a % m (MOD m) := by define have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m apply Exists.intro (a / m) show a - a % m = m * (a / m) from calc a - (a % m) _ = m * (a / m) + a % m - a % m := by rw [h1] _ = m * (a / m) := by ring done lemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) := And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a)) theorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) : ∃! (r : Int), 0 ≤ r ∧ r < m ∧ a ≡ r (MOD m) := by exists_unique · -- Existence apply Exists.intro (a % m) show 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) from mod_cmpl_res m a done · -- Uniqueness fix r1 : Int; fix r2 : Int assume h1 : 0 ≤ r1 ∧ r1 < m ∧ a ≡ r1 (MOD m) assume h2 : 0 ≤ r2 ∧ r2 < m ∧ a ≡ r2 (MOD m) have h3 : r1 ≡ r2 (MOD m) := congr_trans (congr_symm h1.right.right) h2.right.right obtain (d : Int) (h4 : r1 - r2 = m * d) from h3 have h5 : r1 - r2 < m * 1 := by linarith have h6 : m * (-1) < r1 - r2 := by linarith rewrite [h4] at h5 --h5 : m * d < m * 1 rewrite [h4] at h6 --h6 : m * -1 < m * d have h7 : (↑m : Int) ≥ 0 := Nat.cast_nonneg m have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7 have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7 have h10 : d = 0 := by linarith show r1 = r2 from calc r1 _ = r1 - r2 + r2 := by ring _ = m * 0 + r2 := by rw [h4, h10] _ = r2 := by ring done done lemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m := (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a) theorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y rewrite [h1, h2] have h3 : a + b = b + a := by ring show [a]_m + [b]_m = [b]_m + [a]_m from calc [a]_m + [b]_m _ = [a + b]_m := add_class m a b _ = [b + a]_m := by rw [h3] _ = [b]_m + [a]_m := (add_class m b a).symm done theorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X rewrite [h1] have h2 : a * 1 = a := by ring show [a]_m * [1]_m = [a]_m from calc [a]_m * [1]_m _ = [a * 1]_m := mul_class m a 1 _ = [a]_m := by rw [h2] done theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry lemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) : [a]_m * [gcd_c2 m a]_m = [1]_m := by set s : Int := gcd_c1 m a have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m define at h1 rewrite [h1, Nat.cast_one] at h2 --h2 : s * ↑m + gcd_c2 m a * ↑a = 1 rewrite [mul_class, cc_eq_iff_congr] define --Goal : ∃ (c : Int), ↑a * gcd_c2 m a - 1 = ↑m * c apply Exists.intro (-s) show a * (gcd_c2 m a) - 1 = m * (-s) from calc a * (gcd_c2 m a) - 1 _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring _ = 1 + m * (-s) - 1 := by rw [h2] _ = m * (-s) := by ring done theorem Theorem_7_3_7 (m a : Nat) : invertible [a]_m ↔ rel_prime m a := by apply Iff.intro · -- (→) assume h1 : invertible [a]_m define at h1 obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1 obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y rewrite [h3, mul_class, cc_eq_iff_congr] at h2 define at h2 obtain (c : Int) (h4 : a * b - 1 = m * c) from h2 rewrite [Exercise_7_2_6] --Goal : ∃ (s t : Int), s * ↑m + t * ↑a = 1 apply Exists.intro (-c) apply Exists.intro b show (-c) * m + b * a = 1 from calc (-c) * m + b * a _ = (-c) * m + (a * b - 1) + 1 := by ring _ = (-c) * m + m * c + 1 := by rw [h4] _ = 1 := by ring done · -- (←) assume h1 : rel_prime m a define show ∃ (Y : ZMod m), [a]_m * Y = [1]_m from Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1) done done /- Section 7.4 -/ section Euler open Euler lemma num_rp_below_base {m : Nat} : num_rp_below m 0 = 0 := by rfl lemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) : num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by have h1 : num_rp_below m (j + 1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : gcd m j = 1 rewrite [if_pos h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1 show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1 done lemma num_rp_below_step_not_rp {m j : Nat} (h : ¬rel_prime m j) : num_rp_below m (j + 1) = num_rp_below m j := by have h1 : num_rp_below m (j +1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : ¬gcd m j = 1 rewrite [if_neg h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j show num_rp_below m (j + 1) = num_rp_below m j from h1 done lemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl #eval phi 10 --Answer: 4 lemma prod_inv_iff_inv {m : Nat} {X : ZMod m} (h1 : invertible X) (Y : ZMod m) : invertible (X * Y) ↔ invertible Y := by apply Iff.intro · -- (→) assume h2 : invertible (X * Y) obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2 apply Exists.intro (X * Z) rewrite [←h3] --Goal : Y * (X * Z) = X * Y * Z ring --Note that ring can do algebra in ZMod m done · -- (←) assume h2 : invertible Y obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1 obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2 apply Exists.intro (Xi * Yi) show (X * Y) * (Xi * Yi) = [1]_m from calc X * Y * (Xi * Yi) _ = (X * Xi) * (Y * Yi) := by ring _ = [1]_m * [1]_m := by rw [h3, h4] _ = [1]_m := Theorem_7_3_6_7 [1]_m done done lemma F_rp_def {m i : Nat} (h : rel_prime m i) : F m i = [i]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h --h : gcd m i = 1 rewrite [if_pos h] at h1 show F m i = [i]_m from h1 done lemma F_not_rp_def {m i : Nat} (h : ¬rel_prime m i) : F m i = [1]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h rewrite [h1, if_neg h] rfl done lemma prod_seq_base {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 0 k f = [1]_m := by rfl lemma prod_seq_step {m : Nat} (n k : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl lemma prod_seq_zero_step {m : Nat} (n : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by rewrite [prod_seq_step, zero_add] rfl done lemma prod_one {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 1 k f = f k := by rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7] rfl done lemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl lemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m := calc [G m a i]_m _ = [(a * i) % m]_m := by rfl _ = [a * i]_m := (cc_eq_mod m (a * i)).symm _ = [a]_m * [i]_m := (mul_class m a i).symm lemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) : rel_prime m (G m a i) ↔ rel_prime m i := by have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1 show rel_prime m (G m a i) ↔ rel_prime m i from calc rel_prime m (G m a i) _ ↔ invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm _ ↔ invertible ([a]_m * [i]_m) := by rw [cc_G] _ ↔ invertible [i]_m := prod_inv_iff_inv h2 ([i]_m) _ ↔ rel_prime m i := Theorem_7_3_7 m i done lemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) : F m (G m a i) = [a]_m * F m i := by have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2 show F m (G m a i) = [a]_m * F m i from calc F m (G m a i) _ = [G m a i]_m := F_rp_def h3 _ = [a]_m * [i]_m := cc_G m a i _ = [a]_m * F m i := by rw [F_rp_def h2] done lemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : ¬rel_prime m i) : F m (G m a i) = [1]_m := by rewrite [←G_rp_iff h1 i] at h2 show F m (G m a i) = [1]_m from F_not_rp_def h2 done lemma FG_prod {m a : Nat} (h1 : rel_prime m a) : ∀ (k : Nat), prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by by_induc · -- Base Case show prod_seq 0 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from calc prod_seq 0 0 ((F m) ∘ (G m a)) _ = [1]_m := prod_seq_base _ _ _ = [a]_m ^ 0 * [1]_m := by ring _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by rw [num_rp_below_base, prod_seq_base] done · -- Induction Step fix k : Nat assume ih : prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) by_cases h2 : rel_prime m k · -- Case 1. h2 : rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([a]_m * F m k) := by rw [FG_rp h1 h2] _ = [a]_m ^ ((num_rp_below m k) + 1) * ((prod_seq k 0 (F m)) * F m k) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_rp h2, prod_seq_zero_step] done · -- Case 2. h2 : ¬rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([1]_m) := by rw [FG_not_rp h1 h2] _ = [a]_m ^ (num_rp_below m k) * (prod_seq k 0 (F m) * ([1]_m)) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_not_rp h2, prod_seq_zero_step, F_not_rp_def h2] done done done lemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by define --Goal : ∀ i < m, G m a i < m fix i : Nat assume h1 : i < m rewrite [G_def] --Goal : a * i % m < m show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m) done lemma left_inv_one_one_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g' (g i) = i) : one_one_below n g := sorry lemma right_inv_onto_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g (g' i) = i) (h2 : maps_below n g') : onto_below n g := by define at h2; define fix k : Nat assume h3 : k < n apply Exists.intro (g' k) show g' k < n ∧ g (g' k) = k from And.intro (h2 k h3) (h1 k h3) done lemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m * [inv_mod m a]_m = [1]_m := by have h2 : 0 ≤ (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a) show [a]_m * [inv_mod m a]_m = [1]_m from calc [a]_m * [inv_mod m a]_m _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2] _ = [a]_m * [gcd_c2 m a]_m := by rw [←cc_eq_mod] _ = [1]_m := gcd_c2_inv h1 done lemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m := calc a * (b % m) % m = a % m * (b % m % m) % m := Nat.mul_mod _ _ _ _ = a % m * (b % m) % m := by rw [Nat.mod_mod] _ = a * b % m := (Nat.mul_mod _ _ _).symm lemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm] rfl done theorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] : ↑a ≡ ↑b (MOD m) ↔ a % m = b % m := sorry lemma mul_inv_mod_cancel {m a i : Nat} [NeZero m] (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1 rewrite [mul_class, cc_eq_iff_congr, ←Nat.cast_mul, ←Nat.cast_one, congr_iff_mod_eq_Nat] at h3 show a * inv_mod m a * i % m = i from calc a * (inv_mod m a) * i % m _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = 1 % m * i % m := by rw [h3] _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = i % m := by rw [one_mul] _ = i := Nat.mod_eq_of_lt h2 done lemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl lemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, G m a (Ginv m a i) = i := by fix i : Nat assume h2 : i < m show G m a (Ginv m a i) = i from calc G m a (Ginv m a i) _ = a * ((inv_mod m a * i) % m) % m := by rfl _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, Ginv m a (G m a i) = i := by fix i : Nat assume h2 : i < m show Ginv m a (G m a i) = i from calc Ginv m a (G m a i) _ = inv_mod m a * ((a * i) % m) % m := by rfl _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc, mul_comm (inv_mod m a)] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_maps_below (m a : Nat) [NeZero m] : maps_below m (Ginv m a) := G_maps_below m (inv_mod m a) lemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : one_one_below m (G m a) := left_inv_one_one_below (Ginv_left_inv h1) lemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : onto_below m (G m a) := right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a) lemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : perm_below m (G m a) := And.intro (G_maps_below m a) (And.intro (G_one_one_below h1) (G_onto_below h1)) --Permuting a product of congruence classes doesn't change product lemma swap_fst (u v : Nat) : swap u v u = v := by define : swap u v u --Goal : (if u = u then v else if u = v then u else u) = v have h : u = u := by rfl rewrite [if_pos h] rfl done lemma swap_snd (u v : Nat) : swap u v v = u := by define : swap u v v by_cases h1 : v = u · -- Case 1. h1 : v = u rewrite [if_pos h1] show v = u from h1 done · -- Case 2. h1 : v ≠ u rewrite [if_neg h1] have h2 : v = v := by rfl rewrite [if_pos h2] rfl done done lemma swap_other {u v i : Nat} (h1 : i ≠ u) (h2 : i ≠ v) : swap u v i = i := by define : swap u v i rewrite [if_neg h1, if_neg h2] rfl done lemma swap_values (u v i : Nat) : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := by by_cases h1 : i = u · -- Case 1. h1 : i = u apply Or.inl rewrite [h1] show swap u v u = v from swap_fst u v done · -- Case 2. h1 : i ≠ u apply Or.inr by_cases h2 : i = v · -- Case 2.1. h2 : i = v apply Or.inl rewrite [h2] show swap u v v = u from swap_snd u v done · -- Case 2.2. h2 : i ≠ v apply Or.inr show swap u v i = i from swap_other h1 h2 done done done lemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by define fix i : Nat assume h3 : i < n have h4 : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := swap_values u v i by_cases on h4 · -- Case 1. h4 : swap u v i = v rewrite [h4] show v < n from h2 done · -- Case 2. by_cases on h4 · -- Case 2.1. h4 : swap u v i = u rewrite [h4] show u < n from h1 done · -- Case 2.2. h4 : swap u v i = i rewrite [h4] show i < n from h3 done done done lemma swap_swap (u v n : Nat) : ∀ i < n, swap u v (swap u v i) = i := by fix i : Nat assume h : i < n by_cases h1 : i = u · -- Case 1. h1 : i = u rewrite [h1, swap_fst, swap_snd] rfl done · -- Case 2. h1 : i ≠ u by_cases h2 : i = v · -- Case 2.1. h2 : i = v rewrite [h2, swap_snd, swap_fst] rfl done · -- Case 2.2. h2 : i ≠ v rewrite [swap_other h1 h2, swap_other h1 h2] rfl done done done lemma swap_one_one_below (u v n) : one_one_below n (swap u v) := left_inv_one_one_below (swap_swap u v n) lemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) := right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2) lemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) := And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2)) lemma comp_perm_below {n : Nat} {f g : Nat → Nat} (h1 : perm_below n f) (h2 : perm_below n g) : perm_below n (f ∘ g) := sorry lemma trivial_swap (u : Nat) : swap u u = id := by apply funext fix x : Nat by_cases h1 : x = u · -- Case 1. h1 : x = u rewrite [h1, swap_fst] rfl done · -- Case 2. h1 : x ≠ u rewrite [swap_other h1 h1] rfl done done lemma prod_eq_fun {m : Nat} (f g : Nat → ZMod m) (k : Nat) : ∀ (n : Nat), (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g := by by_induc · -- Base Case assume h : (∀ i < 0, f (k + i) = g (k + i)) rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g assume h1 : ∀ i < n + 1, f (k + i) = g (k + i) have h2 : ∀ i < n, f (k + i) = g (k + i) := by fix i : Nat assume h2 : i < n have h3 : i < n + 1 := by linarith show f (k + i) = g (k + i) from h1 i h3 done have h3 : prod_seq n k f = prod_seq n k g := ih h2 have h4 : n < n + 1 := Nat.lt_succ_self n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4] rfl done done lemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := by have h2 : ∀ (i : Nat), i < u → (f ∘ swap u n) (0 + i) = f (0 + i) := by fix i : Nat assume h2 : i < u have h3 : 0 + i ≠ u := by linarith have h4 : 0 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl done show prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f from prod_eq_fun (f ∘ swap u n) f 0 u h2 done lemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := by have h2 : ∀ i < j, (f ∘ swap u n) (u + 1 + i) = f (u + 1 + i) := by fix i : Nat assume h2 : i < j have h3 : u + 1 + i ≠ u := by linarith have h4 : u + 1 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl show prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f from prod_eq_fun (f ∘ swap u n) f (u + 1) j h2 done lemma break_prod {m : Nat} (n : Nat) (f : Nat → ZMod m) : ∀ (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by by_induc · -- Base Case have h : n + 0 = n := by rfl rewrite [prod_seq_base, h, Theorem_7_3_6_7] rfl done · -- Induction Step fix j : Nat assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f rewrite [←add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc] rfl done done lemma break_prod_twice {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f := break_prod n f 1 rewrite [prod_one] at h2 have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f := break_prod (u + 1) f j rewrite [←h1] at h3 have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f := break_prod u f 1 rewrite [prod_one] at h4 rewrite [h3, h4] at h2 show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2 done lemma swap_prod_eq_prod {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f := by by_cases h2 : u = n · -- Case 1. h2 : u = n rewrite [h2, trivial_swap n] --Goal : prod_seq (n + 1) 0 (f ∘ id) = prod_seq (n + 1) 0 f rfl done · -- Case 2. h2 : ¬u = n have h3 : u + 1 ≤ n := Nat.lt_of_le_of_ne h1 h2 obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3 have break_f : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := break_prod_twice f h4 have break_fs : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_prod_twice (f ∘ swap u n) h4 have f_eq_fs_below : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := swap_prod_eq_prod_below f h1 have f_eq_fs_btwn : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4 show prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f from calc prod_seq (n + 1) 0 (f ∘ swap u n) _ = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_fs _ = prod_seq u 0 f * (f ∘ swap u n) u * prod_seq j (u + 1) f * (f ∘ swap u n) n := by rw [f_eq_fs_below, f_eq_fs_btwn] _ = prod_seq u 0 f * f (swap u n u) * prod_seq j (u + 1) f * f (swap u n n) := by rfl _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by rw [swap_fst, swap_snd] _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring _ = prod_seq (n + 1) 0 f := break_f.symm done done lemma perm_below_fixed {n : Nat} {g : Nat → Nat} (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry lemma perm_prod {m : Nat} (f : Nat → ZMod m) : ∀ (n : Nat), ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) := by by_induc · -- Base Case fix g : Nat → Nat assume h1 : perm_below 0 g rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) fix g : Nat → Nat assume g_pb : perm_below (n + 1) g define at g_pb have g_ob : onto_below (n + 1) g := g_pb.right.right define at g_ob have h1 : n < n + 1 := by linarith obtain (u : Nat) (h2 : u < n + 1 ∧ g u = n) from g_ob n h1 have s_pb : perm_below (n + 1) (swap u n) := swap_perm_below h2.left h1 have gs_pb_n1 : perm_below (n + 1) (g ∘ swap u n) := comp_perm_below g_pb s_pb have gs_fix_n : (g ∘ swap u n) n = n := calc (g ∘ swap u n) n _ = g (swap u n n) := by rfl _ = g u := by rw [swap_snd] _ = n := h2.right have gs_pb_n : perm_below n (g ∘ swap u n) := perm_below_fixed gs_pb_n1 gs_fix_n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f ∘ (g ∘ swap u n)) := ih (g ∘ swap u n) gs_pb_n have h3 : u ≤ n := by linarith show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f ∘ g) from calc prod_seq (n + 1) 0 f _ = prod_seq n 0 f * f n := prod_seq_zero_step n f _ = prod_seq n 0 (f ∘ (g ∘ swap u n)) * f ((g ∘ swap u n) n) := by rw [gs_prod, gs_fix_n] _ = prod_seq n 0 (f ∘ g ∘ swap u n) * (f ∘ g ∘ swap u n) n := by rfl _ = prod_seq (n + 1) 0 (f ∘ g ∘ swap u n) := (prod_seq_zero_step n (f ∘ g ∘ swap u n)).symm _ = prod_seq (n + 1) 0 ((f ∘ g) ∘ swap u n) := by rfl _ = prod_seq (n + 1) 0 (f ∘ g) := swap_prod_eq_prod (f ∘ g) h3 done done lemma F_invertible (m i : Nat) : invertible (F m i) := by by_cases h : rel_prime m i · -- Case 1. h : rel_prime m i rewrite [F_rp_def h] show invertible [i]_m from (Theorem_7_3_7 m i).rtl h done · -- Case 2. h : ¬rel_prime m i rewrite [F_not_rp_def h] apply Exists.intro [1]_m show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m done done lemma Fprod_invertible (m : Nat) : ∀ (k : Nat), invertible (prod_seq k 0 (F m)) := by by_induc · -- Base Case apply Exists.intro [1]_m show prod_seq 0 0 (F m) * [1]_m = [1]_m from calc prod_seq 0 0 (F m) * [1]_m _ = [1]_m * [1]_m := by rw [prod_seq_base] _ = [1]_m := Theorem_7_3_6_7 ([1]_m) done · -- Induction Step fix k : Nat assume ih : invertible (prod_seq k 0 (F m)) rewrite [prod_seq_zero_step] show invertible (prod_seq k 0 (F m) * (F m k)) from (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k) done done theorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m ^ (phi m) = [1]_m := by have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2 show [a]_m ^ (phi m) = [1]_m from calc [a]_m ^ (phi m) _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3] _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring _ = prod_seq m 0 (F m ∘ G m a) * Y := by rw [FG_prod h1 m, phi_def] _ = prod_seq m 0 (F m) * Y := by rw [perm_prod (F m) m (G m a) (G_perm_below h1)] _ = [1]_m := by rw [h3] done lemma Exercise_7_4_5_Int (m : Nat) (a : Int) : ∀ (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry lemma Exercise_7_4_5_Nat (m a n : Nat) : [a]_m ^ n = [a ^ n]_m := by rewrite [Exercise_7_4_5_Int] rfl done theorem Euler's_theorem {m a : Nat} [NeZero m] (h1 : rel_prime m a) : a ^ (phi m) ≡ 1 (MOD m) := by have h2 : [a]_m ^ (phi m) = [1]_m := Theorem_7_4_2 h1 rewrite [Exercise_7_4_5_Nat m a (phi m)] at h2 --h2 : [a ^ phi m]_m = [1]_m show a ^ (phi m) ≡ 1 (MOD m) from (cc_eq_iff_congr _ _ _).ltr h2 done #eval gcd 10 7 --Answer: 1. So 10 and 7 are relatively prime #eval 7 ^ phi 10 --Answer: 2401, which is congruent to 1 mod 10. end Euler /- Section 7.5 -/ lemma num_rp_prime {p : Nat} (h1 : prime p) : ∀ k < p, num_rp_below p (k + 1) = k := sorry lemma phi_prime {p : Nat} (h1 : prime p) : phi p = p - 1 := by have h2 : 1 ≤ p := prime_pos h1 have h3 : p - 1 + 1 = p := Nat.sub_add_cancel h2 have h4 : p - 1 < p := by linarith have h5 : num_rp_below p (p - 1 + 1) = p - 1 := num_rp_prime h1 (p - 1) h4 rewrite [h3] at h5 show phi p = p - 1 from h5 done theorem Theorem_7_2_2_Int {a c : Nat} {b : Int} (h1 : ↑c ∣ ↑a * b) (h2 : rel_prime a c) : ↑c ∣ b := by rewrite [Int.natCast_dvd, Int.natAbs_mul, Int.natAbs_ofNat] at h1 --h1 : c ∣ a * Int.natAbs b rewrite [Int.natCast_dvd] --Goal : c ∣ Int.natAbs b show c ∣ Int.natAbs b from Theorem_7_2_2 h1 h2 done lemma Lemma_7_4_5 {m n : Nat} (a b : Int) (h1 : rel_prime m n) : a ≡ b (MOD m * n) ↔ a ≡ b (MOD m) ∧ a ≡ b (MOD n) := by apply Iff.intro · -- (→) assume h2 : a ≡ b (MOD m * n) obtain (j : Int) (h3 : a - b = (m * n) * j) from h2 apply And.intro · -- Proof of a ≡ b (MOD m) apply Exists.intro (n * j) show a - b = m * (n * j) from calc a - b _ = m * n * j := h3 _ = m * (n * j) := by ring done · -- Proof of a ≡ b (MOD n) apply Exists.intro (m * j) show a - b = n * (m * j) from calc a - b _ = m * n * j := h3 _ = n * (m * j) := by ring done done · -- (←) assume h2 : a ≡ b (MOD m) ∧ a ≡ b (MOD n) obtain (j : Int) (h3 : a - b = m * j) from h2.left have h4 : (↑n : Int) ∣ a - b := h2.right rewrite [h3] at h4 --h4 : ↑n ∣ ↑m * j have h5 : ↑n ∣ j := Theorem_7_2_2_Int h4 h1 obtain (k : Int) (h6 : j = n * k) from h5 apply Exists.intro k --Goal : a - b = ↑(m * n) * k rewrite [Nat.cast_mul] --Goal : a - b = ↑m * ↑n * k show a - b = (m * n) * k from calc a - b _ = m * j := h3 _ = m * (n * k) := by rw [h6] _ = (m * n) * k := by ring done done --From exercises of Section 7.2 theorem rel_prime_symm {a b : Nat} (h : rel_prime a b) : rel_prime b a := sorry lemma prime_NeZero {p : Nat} (h : prime p) : NeZero p := by rewrite [neZero_iff] --Goal : p ≠ 0 define at h linarith done lemma Lemma_7_5_1 {p e d m c s : Nat} {t : Int} (h1 : prime p) (h2 : e * d = (p - 1) * s + 1) (h3 : m ^ e - c = p * t) : c ^ d ≡ m (MOD p) := by have h4 : m ^ e ≡ c (MOD p) := Exists.intro t h3 have h5 : [m ^ e]_p = [c]_p := (cc_eq_iff_congr _ _ _).rtl h4 rewrite [←Exercise_7_4_5_Nat] at h5 --h5 : [m]_p ^ e = [c]_p by_cases h6 : p ∣ m · -- Case 1. h6 : p ∣ m have h7 : m ≡ 0 (MOD p) := by obtain (j : Nat) (h8 : m = p * j) from h6 apply Exists.intro (↑j : Int) --Goal : ↑m - 0 = ↑p * ↑j rewrite [h8, Nat.cast_mul] ring done have h8 : [m]_p = [0]_p := (cc_eq_iff_congr _ _ _).rtl h7 have h9 : e * d ≠ 0 := by rewrite [h2] show (p - 1) * s + 1 ≠ 0 from Nat.add_one_ne_zero _ done have h10 : (0 : Int) ^ (e * d) = 0 := zero_pow h9 have h11 : [c ^ d]_p = [m]_p := calc [c ^ d]_p _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat] _ = ([m]_p ^ e) ^ d := by rw [h5] _ = [m]_p ^ (e * d) := by ring _ = [0]_p ^ (e * d) := by rw [h8] _ = [0 ^ (e * d)]_p := Exercise_7_4_5_Int _ _ _ _ = [0]_p := by rw [h10] _ = [m]_p := by rw [h8] show c ^ d ≡ m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11 done · -- Case 2. h6 : ¬p ∣ m have h7 : rel_prime m p := rel_prime_of_prime_not_dvd h1 h6 have h8 : rel_prime p m := rel_prime_symm h7 have h9 : NeZero p := prime_NeZero h1 have h10 : (1 : Int) ^ s = 1 := by ring have h11 : [c ^ d]_p = [m]_p := calc [c ^ d]_p _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat] _ = ([m]_p ^ e) ^ d := by rw [h5] _ = [m]_p ^ (e * d) := by ring _ = [m]_p ^ ((p - 1) * s + 1) := by rw [h2] _ = ([m]_p ^ (p - 1)) ^ s * [m]_p := by ring _ = ([m]_p ^ (phi p)) ^ s * [m]_p := by rw [phi_prime h1] _ = [1]_p ^ s * [m]_p := by rw [Theorem_7_4_2 h8] _ = [1 ^ s]_p * [m]_p := by rw [Exercise_7_4_5_Int] _ = [1]_p * [m]_p := by rw [h10] _ = [m]_p * [1]_p := by ring _ = [m]_p := Theorem_7_3_6_7 _ show c ^ d ≡ m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11 done done theorem Theorem_7_5_1 (p q n e d k m c : Nat) (p_prime : prime p) (q_prime : prime q) (p_ne_q : p ≠ q) (n_pq : n = p * q) (ed_congr_1 : e * d = k * (p - 1) * (q - 1) + 1) (h1 : [m]_n ^ e = [c]_n) : [c]_n ^ d = [m]_n := by rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] at h1 --h1 : m ^ e ≡ c (MOD n) rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] --Goal : c ^ d ≡ m (MOD n) obtain (j : Int) (h2 : m ^ e - c = n * j) from h1 rewrite [n_pq, Nat.cast_mul] at h2 --h2 : m ^ e - c = p * q * j have h3 : e * d = (p - 1) * (k * (q - 1)) + 1 := by rewrite [ed_congr_1] ring done have h4 : m ^ e - c = p * (q * j) := by rewrite [h2] ring done have congr_p : c ^ d ≡ m (MOD p) := Lemma_7_5_1 p_prime h3 h4 have h5 : e * d = (q - 1) * (k * (p - 1)) + 1 := by rewrite [ed_congr_1] ring done have h6 : m ^ e - c = q * (p * j) := by rewrite [h2] ring done have congr_q : c ^ d ≡ m (MOD q) := Lemma_7_5_1 q_prime h5 h6 have h7 : ¬q ∣ p := by by_contra h8 have h9 : q = 1 ∨ q = p := dvd_prime p_prime h8 disj_syll h9 (prime_not_one q_prime) show False from p_ne_q h9.symm done have h8 : rel_prime p q := rel_prime_of_prime_not_dvd q_prime h7 rewrite [n_pq, Lemma_7_4_5 _ _ h8] show c ^ d ≡ m (MOD p) ∧ c ^ d ≡ m (MOD q) from And.intro congr_p congr_q done /- BEGIN EXERCISES -/ namespace Exercises /- Section 7.1 -/ -- 1. theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry -- 2. lemma gcd_comm_lt {a b : Nat} (h : a < b) : gcd a b = gcd b a := sorry theorem gcd_comm (a b : Nat) : gcd a b = gcd b a := sorry -- 3. theorem Exercise_7_1_5 (a b : Nat) (n : Int) : (∃ (s t : Int), s * a + t * b = n) ↔ (↑(gcd a b) : Int) ∣ n := sorry -- 4. theorem Exercise_7_1_6 (a b c : Nat) : gcd a b = gcd (a + b * c) b := sorry -- 5. theorem gcd_is_nonzero {a b : Nat} (h : a ≠ 0 ∨ b ≠ 0) : gcd a b ≠ 0 := sorry -- 6. theorem gcd_greatest {a b d : Nat} (h1 : gcd a b ≠ 0) (h2 : d ∣ a) (h3 : d ∣ b) : d ≤ gcd a b := sorry -- 7. lemma Lemma_7_1_10a {a b : Nat} (n : Nat) (h : a ∣ b) : (n * a) ∣ (n * b) := sorry lemma Lemma_7_1_10b {a b n : Nat} (h1 : n ≠ 0) (h2 : (n * a) ∣ (n * b)) : a ∣ b := sorry lemma Lemma_7_1_10c {a b : Nat} (h1 : a ∣ b) (h2 : b ∣ a) : a = b := sorry theorem Exercise_7_1_10 (a b n : Nat) : gcd (n * a) (n * b) = n * gcd a b := sorry /- Section 7.2 -/ -- 1. lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry -- 2. -- Hints: Start with apply List.rec. You may find mul_ne_zero useful theorem prod_nonzero_nonzero : ∀ (l : List Nat), (∀ a ∈ l, a ≠ 0) → prod l ≠ 0 := sorry -- 3. theorem rel_prime_iff_no_common_factor (a b : Nat) : rel_prime a b ↔ ¬∃ (p : Nat), prime p ∧ p ∣ a ∧ p ∣ b := sorry -- 4. theorem rel_prime_symm {a b : Nat} (h : rel_prime a b) : rel_prime b a := sorry -- 5. lemma in_prime_factorization_iff_prime_factor {a : Nat} {l : List Nat} (h1 : prime_factorization a l) (p : Nat) : p ∈ l ↔ prime_factor p a := sorry -- 6. theorem Exercise_7_2_5 {a b : Nat} {l m : List Nat} (h1 : prime_factorization a l) (h2 : prime_factorization b m) : rel_prime a b ↔ (¬∃ (p : Nat), p ∈ l ∧ p ∈ m) := sorry -- 7. theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry -- 8. theorem Exercise_7_2_7 {a b a' b' : Nat} (h1 : rel_prime a b) (h2 : a' ∣ a) (h3 : b' ∣ b) : rel_prime a' b' := sorry -- 9. theorem Exercise_7_2_9 {a b j k : Nat} (h1 : gcd a b ≠ 0) (h2 : a = j * gcd a b) (h3 : b = k * gcd a b) : rel_prime j k := sorry -- 10. theorem Exercise_7_2_17a (a b c : Nat) : gcd a (b * c) ∣ gcd a b * gcd a c := sorry /- Section 7.3 -/ -- 1. theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry -- 2.

theorem Theorem_7_3_6_3 {m : Nat} (X : ZMod m) : X + [0]_m = X

HTPI.Exercises.Theorem_7_3_6_3

htpi/HTPILib/Chap7.lean

htpi

/- Copyright 2023 Daniel J. Velleman -/ import HTPILib.Chap6 namespace HTPI /- Definitions -/ lemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by have h : n + 1 > 0 := Nat.succ_pos n show a % (n + 1) < n + 1 from Nat.mod_lt a h done def gcd (a b : Nat) : Nat := match b with | 0 => a | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd (n + 1) (a % (n + 1)) termination_by b mutual def gcd_c1 (a b : Nat) : Int := match b with | 0 => 1 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c2 (n + 1) (a % (n + 1)) --Corresponds to s = t' termination_by b def gcd_c2 (a b : Nat) : Int := match b with | 0 => 0 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c1 (n + 1) (a % (n + 1)) - (gcd_c2 (n + 1) (a % (n + 1))) * ↑(a / (n + 1)) --Corresponds to t = s' - t'q termination_by b end def prime (n : Nat) : Prop := 2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n def prime_factor (p n : Nat) : Prop := prime p ∧ p ∣ n def all_prime (l : List Nat) : Prop := ∀ p ∈ l, prime p def nondec (l : List Nat) : Prop := match l with | [] => True --Of course, True is a proposition that is always true | n :: L => (∀ m ∈ L, n ≤ m) ∧ nondec L def nondec_prime_list (l : List Nat) : Prop := all_prime l ∧ nondec l def prod (l : List Nat) : Nat := match l with | [] => 1 | n :: L => n * (prod L) def prime_factorization (n : Nat) (l : List Nat) : Prop := nondec_prime_list l ∧ prod l = n def rel_prime (a b : Nat) : Prop := gcd a b = 1 def congr_mod (m : Nat) (a b : Int) : Prop := (↑m : Int) ∣ (a - b) def cc (m : Nat) (a : Int) : ZMod m := (↑a : ZMod m) notation:50 a " ≡ " b " (MOD " m ")" => congr_mod m a b notation:max "["a"]_"m:max => cc m a def invertible {m : Nat} (X : ZMod m) : Prop := ∃ (Y : ZMod m), X * Y = [1]_m def num_rp_below (m k : Nat) : Nat := match k with | 0 => 0 | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j def phi (m : Nat) : Nat := num_rp_below m m def prod_seq {m : Nat} (j k : Nat) (f : Nat → ZMod m) : ZMod m := match j with | 0 => [1]_m | n + 1 => prod_seq n k f * f (k + n) def maps_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i < n, g i < n def one_one_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i1 < n, ∀ i2 < n, g i1 = g i2 → i1 = i2 def onto_below (n : Nat) (g : Nat → Nat) : Prop := ∀ k < n, ∃ i < n, g i = k def perm_below (n : Nat) (g : Nat → Nat) : Prop := maps_below n g ∧ one_one_below n g ∧ onto_below n g def inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m) def swap (u v i : Nat) : Nat := if i = u then v else if i = v then u else i namespace Euler --For definitions specific to Euler's theorem def F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m def G (m a i : Nat) : Nat := (a * i) % m def Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i end Euler /- Section 7.1 -/ theorem dvd_mod_of_dvd_a_b {a b d : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ (a % b) := by set q : Nat := a / b have h3 : b * q + a % b = a := Nat.div_add_mod a b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 define --Goal : ∃ (c : Nat), a % b = d * c apply Exists.intro (j - k * q) show a % b = d * (j - k * q) from calc a % b _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm _ = a - b * q := by rw [h3] _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc] _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm done theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry #eval gcd 672 161 --Answer: 7 lemma gcd_base (a : Nat) : gcd a 0 = a := by rfl lemma gcd_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd a b = gcd b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1)) rfl done lemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b ≠ 0) : a % b < b := by have h1 : b > 0 := Nat.pos_of_ne_zero h show a % b < b from Nat.mod_lt a h1 done lemma dvd_self (n : Nat) : n ∣ n := by apply Exists.intro 1 ring done theorem gcd_dvd : ∀ (b a : Nat), (gcd a b) ∣ a ∧ (gcd a b) ∣ b := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd a b_1) ∣ a ∧ (gcd a b_1) ∣ b_1 fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_base] --Goal: a ∣ a ∧ a ∣ 0 apply And.intro (dvd_self a) define apply Exists.intro 0 rfl done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_nonzero a h1] --Goal : gcd b (a % b) ∣ a ∧ gcd b (a % b) ∣ b have h2 : a % b < b := mod_nonzero_lt a h1 have h3 : (gcd b (a % b)) ∣ b ∧ (gcd b (a % b)) ∣ (a % b) := ih (a % b) h2 b apply And.intro _ h3.left show (gcd b (a % b)) ∣ a from dvd_a_of_dvd_b_mod h3.left h3.right done done theorem gcd_dvd_left (a b : Nat) : (gcd a b) ∣ a := (gcd_dvd b a).left theorem gcd_dvd_right (a b : Nat) : (gcd a b) ∣ b := (gcd_dvd b a).right lemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl lemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c1 a b = gcd_c2 b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done lemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl lemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * ↑(a / b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done theorem gcd_lin_comb : ∀ (b a : Nat), (gcd_c1 a b) * ↑a + (gcd_c2 a b) * ↑b = ↑(gcd a b) := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd_c1 a b_1) * ↑a + (gcd_c2 a b_1) * ↑b_1 = ↑(gcd a b_1) fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base] --Goal : 1 * ↑a + 0 * ↑0 = ↑a ring done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1] --Goal : gcd_c2 b (a % b) * ↑a + -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * ↑(a / b)) * ↑b = -- ↑(gcd b (a % b)) set r : Nat := a % b set q : Nat := a / b set s : Int := gcd_c1 b r set t : Int := gcd_c2 b r --Goal : t * ↑a + (s - t * ↑q) * ↑b = ↑(gcd b r) have h2 : r < b := mod_nonzero_lt a h1 have h3 : s * ↑b + t * ↑r = ↑(gcd b r) := ih r h2 b have h4 : b * q + r = a := Nat.div_add_mod a b rewrite [←h3, ←h4] rewrite [Nat.cast_add, Nat.cast_mul] --Goal : t * (↑b * ↑q + ↑r) + (s - t * ↑q) * ↑b = s * ↑b + t * ↑r ring done done #eval gcd_c1 672 161 --Answer: 6 #eval gcd_c2 672 161 --Answer: -25 --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161 theorem Theorem_7_1_6 {d a b : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ gcd a b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑d ∣ ↑(gcd a b) set s : Int := gcd_c1 a b set t : Int := gcd_c2 a b have h3 : s * ↑a + t * ↑b = ↑(gcd a b) := gcd_lin_comb b a rewrite [←h3] --Goal : ↑d ∣ s * ↑a + t * ↑b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul] --Goal : ↑d ∣ s * (↑d * ↑j) + t * (↑d * ↑k) define apply Exists.intro (s * ↑j + t * ↑k) ring done /- Section 7.2 -/ theorem dvd_trans {a b c : Nat} (h1 : a ∣ b) (h2 : b ∣ c) : a ∣ c := by define at h1; define at h2; define obtain (m : Nat) (h3 : b = a * m) from h1 obtain (n : Nat) (h4 : c = b * n) from h2 rewrite [h3, mul_assoc] at h4 apply Exists.intro (m * n) show c = a * (m * n) from h4 done lemma exists_prime_factor : ∀ (n : Nat), 2 ≤ n → ∃ (p : Nat), prime_factor p n := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, 2 ≤ n_1 → ∃ (p : Nat), prime_factor p n_1 assume h1 : 2 ≤ n by_cases h2 : prime n · -- Case 1. h2 : prime n apply Exists.intro n define --Goal : prime n ∧ n ∣ n show prime n ∧ n ∣ n from And.intro h2 (dvd_self n) done · -- Case 2. h2 : ¬prime n define at h2 --h2 : ¬(2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n) demorgan at h2 disj_syll h2 h1 obtain (a : Nat) (h3 : ∃ (b : Nat), a * b = n ∧ a < n ∧ b < n) from h2 obtain (b : Nat) (h4 : a * b = n ∧ a < n ∧ b < n) from h3 have h5 : 2 ≤ a := by by_contra h6 have h7 : a ≤ 1 := by linarith have h8 : n ≤ b := calc n _ = a * b := h4.left.symm _ ≤ 1 * b := by rel [h7] _ = b := by ring linarith --n ≤ b contradicts b < n done have h6 : ∃ (p : Nat), prime_factor p a := ih a h4.right.left h5 obtain (p : Nat) (h7 : prime_factor p a) from h6 apply Exists.intro p define --Goal : prime p ∧ p ∣ n define at h7 --h7 : prime p ∧ p ∣ a apply And.intro h7.left have h8 : a ∣ n := by apply Exists.intro b show n = a * b from (h4.left).symm done show p ∣ n from dvd_trans h7.right h8 done done lemma exists_least_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q := by set S : Set Nat := {p : Nat | prime_factor p n} have h2 : ∃ (p : Nat), p ∈ S := exists_prime_factor n h show ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q from well_ord_princ S h2 done lemma all_prime_nil : all_prime [] := by define --Goal : ∀ p ∈ [], prime p fix p : Nat contrapos --Goal : ¬prime p → p ∉ [] assume h1 : ¬prime p show p ∉ [] from List.not_mem_nil p done lemma all_prime_cons (n : Nat) (L : List Nat) : all_prime (n :: L) ↔ prime n ∧ all_prime L := by apply Iff.intro · -- (→) assume h1 : all_prime (n :: L) --Goal : prime n ∧ all_prime L define at h1 --h1 : ∀ p ∈ n :: L, prime p apply And.intro (h1 n (List.mem_cons_self n L)) define --Goal : ∀ p ∈ L, prime p fix p : Nat assume h2 : p ∈ L show prime p from h1 p (List.mem_cons_of_mem n h2) done · -- (←) assume h1 : prime n ∧ all_prime L --Goal : all_prime (n :: l) define : all_prime L at h1 define fix p : Nat assume h2 : p ∈ n :: L rewrite [List.mem_cons] at h2 --h2 : p = n ∨ p ∈ L by_cases on h2 · -- Case 1. h2 : p = n rewrite [h2] show prime n from h1.left done · -- Case 2. h2 : p ∈ L show prime p from h1.right p h2 done done done lemma nondec_nil : nondec [] := by define --Goal : True trivial --trivial proves some obviously true statements, such as True done lemma nondec_cons (n : Nat) (L : List Nat) : nondec (n :: L) ↔ (∀ m ∈ L, n ≤ m) ∧ nondec L := by rfl lemma prod_nil : prod [] = 1 := by rfl lemma prod_cons : prod (n :: L) = n * (prod L) := by rfl lemma exists_cons_of_length_eq_succ {A : Type} {l : List A} {n : Nat} (h : l.length = n + 1) : ∃ (a : A) (L : List A), l = a :: L ∧ L.length = n := by have h1 : ¬l.length = 0 := by linarith rewrite [List.length_eq_zero] at h1 obtain (a : A) (h2 : ∃ (L : List A), l = a :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List A) (h3 : l = a :: L) from h2 apply Exists.intro a apply Exists.intro L apply And.intro h3 have h4 : (a :: L).length = L.length + 1 := List.length_cons a L rewrite [←h3, h] at h4 show L.length = n from (Nat.add_right_cancel h4).symm done lemma list_elt_dvd_prod_by_length (a : Nat) : ∀ (n : Nat), ∀ (l : List Nat), l.length = n → a ∈ l → a ∣ prod l := by by_induc · --Base Case fix l : List Nat assume h1 : l.length = 0 rewrite [List.length_eq_zero] at h1 --h1 : l = [] rewrite [h1] --Goal : a ∈ [] → a ∣ prod [] contrapos assume h2 : ¬a ∣ prod [] show a ∉ [] from List.not_mem_nil a done · -- Induction Step fix n : Nat assume ih : ∀ (l : List Nat), List.length l = n → a ∈ l → a ∣ prod l fix l : List Nat assume h1 : l.length = n + 1 --Goal : a ∈ l → a ∣ prod l obtain (b : Nat) (h2 : ∃ (L : List Nat), l = b :: L ∧ L.length = n) from exists_cons_of_length_eq_succ h1 obtain (L : List Nat) (h3 : l = b :: L ∧ L.length = n) from h2 have h4 : a ∈ L → a ∣ prod L := ih L h3.right assume h5 : a ∈ l rewrite [h3.left, prod_cons] --Goal : a ∣ b * prod L rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b ∨ a ∈ L by_cases on h5 · -- Case 1. h5 : a = b apply Exists.intro (prod L) rewrite [h5] rfl done · -- Case 2. h5 : a ∈ L have h6 : a ∣ prod L := h4 h5 have h7 : prod L ∣ b * prod L := by apply Exists.intro b ring done show a ∣ b * prod L from dvd_trans h6 h7 done done done lemma list_elt_dvd_prod {a : Nat} {l : List Nat} (h : a ∈ l) : a ∣ prod l := by set n : Nat := l.length have h1 : l.length = n := by rfl show a ∣ prod l from list_elt_dvd_prod_by_length a n l h1 h done lemma exists_prime_factorization : ∀ (n : Nat), n ≥ 1 → ∃ (l : List Nat), prime_factorization n l := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, n_1 ≥ 1 → ∃ (l : List Nat), prime_factorization n_1 l assume h1 : n ≥ 1 by_cases h2 : n = 1 · -- Case 1. h2 : n = 1 apply Exists.intro [] define apply And.intro · -- Proof of nondec_prime_list [] define show all_prime [] ∧ nondec [] from And.intro all_prime_nil nondec_nil done · -- Proof of prod [] = n rewrite [prod_nil, h2] rfl done done · -- Case 2. h2 : n ≠ 1 have h3 : n ≥ 2 := lt_of_le_of_ne' h1 h2 obtain (p : Nat) (h4 : prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q) from exists_least_prime_factor h3 have p_prime_factor : prime_factor p n := h4.left define at p_prime_factor have p_prime : prime p := p_prime_factor.left have p_dvd_n : p ∣ n := p_prime_factor.right have p_least : ∀ (q : Nat), prime_factor q n → p ≤ q := h4.right obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n have h5 : m ≠ 0 := by contradict h1 with h6 have h7 : n = 0 := calc n _ = p * m := n_eq_pm _ = p * 0 := by rw [h6] _ = 0 := by ring rewrite [h7] decide done have m_pos : 0 < m := Nat.pos_of_ne_zero h5 have m_lt_n : m < n := by define at p_prime show m < n from calc m _ < m + m := by linarith _ = 2 * m := by ring _ ≤ p * m := by rel [p_prime.left] _ = n := n_eq_pm.symm done obtain (L : List Nat) (h6 : prime_factorization m L) from ih m m_lt_n m_pos define at h6 have ndpl_L : nondec_prime_list L := h6.left define at ndpl_L apply Exists.intro (p :: L) define apply And.intro · -- Proof of nondec_prime_list (p :: L) define apply And.intro · -- Proof of all_prime (p :: L) rewrite [all_prime_cons] show prime p ∧ all_prime L from And.intro p_prime ndpl_L.left done · -- Proof of nondec (p :: L) rewrite [nondec_cons] apply And.intro _ ndpl_L.right fix q : Nat assume q_in_L : q ∈ L have h7 : q ∣ prod L := list_elt_dvd_prod q_in_L rewrite [h6.right] at h7 --h7 : q ∣ m have h8 : m ∣ n := by apply Exists.intro p rewrite [n_eq_pm] ring done have q_dvd_n : q ∣ n := dvd_trans h7 h8 have ap_L : all_prime L := ndpl_L.left define at ap_L have q_prime_factor : prime_factor q n := And.intro (ap_L q q_in_L) q_dvd_n show p ≤ q from p_least q q_prime_factor done done · -- Proof of prod (p :: L) = n rewrite [prod_cons, h6.right, n_eq_pm] rfl done done done theorem Theorem_7_2_2 {a b c : Nat} (h1 : c ∣ a * b) (h2 : rel_prime a c) : c ∣ b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑c ∣ ↑b define at h1; define at h2; define obtain (j : Nat) (h3 : a * b = c * j) from h1 set s : Int := gcd_c1 a c set t : Int := gcd_c2 a c have h4 : s * ↑a + t * ↑c = ↑(gcd a c) := gcd_lin_comb c a rewrite [h2, Nat.cast_one] at h4 --h4 : s * ↑a + t * ↑c = (1 : Int) apply Exists.intro (s * ↑j + t * ↑b) show ↑b = ↑c * (s * ↑j + t * ↑b) from calc ↑b _ = (1 : Int) * ↑b := (one_mul _).symm _ = (s * ↑a + t * ↑c) * ↑b := by rw [h4] _ = s * (↑a * ↑b) + t * ↑c * ↑b := by ring _ = s * (↑c * ↑j) + t * ↑c * ↑b := by rw [←Nat.cast_mul a b, h3, Nat.cast_mul c j] _ = ↑c * (s * ↑j + t * ↑b) := by ring done lemma le_nonzero_prod_left {a b : Nat} (h : a * b ≠ 0) : a ≤ a * b := by have h1 : b ≠ 0 := by contradict h with h1 rewrite [h1] ring done have h2 : 1 ≤ b := Nat.pos_of_ne_zero h1 show a ≤ a * b from calc a = a * 1 := (mul_one a).symm _ ≤ a * b := by rel [h2] done lemma le_nonzero_prod_right {a b : Nat} (h : a * b ≠ 0) : b ≤ a * b := by rewrite [mul_comm] rewrite [mul_comm] at h show b ≤ b * a from le_nonzero_prod_left h done lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry lemma rel_prime_of_prime_not_dvd {a p : Nat} (h1 : prime p) (h2 : ¬p ∣ a) : rel_prime a p := by have h3 : gcd a p ∣ a := gcd_dvd_left a p have h4 : gcd a p ∣ p := gcd_dvd_right a p have h5 : gcd a p = 1 ∨ gcd a p = p := dvd_prime h1 h4 have h6 : gcd a p ≠ p := by contradict h2 with h6 rewrite [h6] at h3 show p ∣ a from h3 done disj_syll h5 h6 show rel_prime a p from h5 done theorem Theorem_7_2_3 {a b p : Nat} (h1 : prime p) (h2 : p ∣ a * b) : p ∣ a ∨ p ∣ b := by or_right with h3 have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3 show p ∣ b from Theorem_7_2_2 h2 h4 done lemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a ≥ 1 := by have h1 : a ≠ 0 := by by_contra h1 rewrite [h1] at h contradict h linarith done show a ≥ 1 from Nat.pos_of_ne_zero h1 done lemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by have h1 : a ≥ 1 := ge_one_of_prod_one h have h2 : a * b ≠ 0 := by linarith have h3 : a ≤ a * b := le_nonzero_prod_left h2 rewrite [h] at h3 show a = 1 from Nat.le_antisymm h3 h1 done lemma eq_one_of_dvd_one {n : Nat} (h : n ∣ 1) : n = 1 := by obtain (j : Nat) (h1 : 1 = n * j) from h show n = 1 from eq_one_of_prod_one h1.symm done lemma prime_not_one {p : Nat} (h : prime p) : p ≠ 1 := by define at h linarith done theorem Theorem_7_2_4 {p : Nat} (h1 : prime p) : ∀ (l : List Nat), p ∣ prod l → ∃ a ∈ l, p ∣ a := by apply List.rec · -- Base Case. Goal : p ∣ prod [] → ∃ a ∈ [], p ∣ a rewrite [prod_nil] assume h2 : p ∣ 1 show ∃ a ∈ [], p ∣ a from absurd (eq_one_of_dvd_one h2) (prime_not_one h1) done · -- Induction Step fix b : Nat fix L : List Nat assume ih : p ∣ prod L → ∃ a ∈ L, p ∣ a --Goal : p ∣ prod (b :: L) → ∃ a ∈ b :: L, p ∣ a assume h2 : p ∣ prod (b :: L) rewrite [prod_cons] at h2 have h3 : p ∣ b ∨ p ∣ prod L := Theorem_7_2_3 h1 h2 by_cases on h3 · -- Case 1. h3 : p ∣ b apply Exists.intro b show b ∈ b :: L ∧ p ∣ b from And.intro (List.mem_cons_self b L) h3 done · -- Case 2. h3 : p ∣ prod L obtain (a : Nat) (h4 : a ∈ L ∧ p ∣ a) from ih h3 apply Exists.intro a show a ∈ b :: L ∧ p ∣ a from And.intro (List.mem_cons_of_mem b h4.left) h4.right done done done lemma prime_in_list {p : Nat} {l : List Nat} (h1 : prime p) (h2 : all_prime l) (h3 : p ∣ prod l) : p ∈ l := by obtain (a : Nat) (h4 : a ∈ l ∧ p ∣ a) from Theorem_7_2_4 h1 l h3 define at h2 have h5 : prime a := h2 a h4.left have h6 : p = 1 ∨ p = a := dvd_prime h5 h4.right disj_syll h6 (prime_not_one h1) rewrite [h6] show a ∈ l from h4.left done lemma first_le_first {p q : Nat} {l m : List Nat} (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m)) (h3 : prod (p :: l) = prod (q :: m)) : p ≤ q := by define at h1; define at h2 have h4 : q ∣ prod (p :: l) := by define apply Exists.intro (prod m) rewrite [←prod_cons] show prod (p :: l) = prod (q :: m) from h3 done have h5 : all_prime (q :: m) := h2.left rewrite [all_prime_cons] at h5 have h6 : q ∈ p :: l := prime_in_list h5.left h1.left h4 have h7 : nondec (p :: l) := h1.right rewrite [nondec_cons] at h7 rewrite [List.mem_cons] at h6 by_cases on h6 · -- Case 1. h6 : q = p linarith done · -- Case 2. h6 : q ∈ l have h8 : ∀ m ∈ l, p ≤ m := h7.left show p ≤ q from h8 q h6 done done lemma nondec_prime_list_tail {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by define at h define rewrite [all_prime_cons, nondec_cons] at h show all_prime l ∧ nondec l from And.intro h.left.right h.right.right done lemma cons_prod_not_one {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : prod (p :: l) ≠ 1 := by define at h have h1 : all_prime (p :: l) := h.left rewrite [all_prime_cons] at h1 rewrite [prod_cons] by_contra h2 show False from (prime_not_one h1.left) (eq_one_of_prod_one h2) done lemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) : l = [] ↔ prod l = 1 := by apply Iff.intro · -- (→) assume h1 : l = [] rewrite [h1] show prod [] = 1 from prod_nil done · -- (←) contrapos assume h1 : ¬l = [] obtain (p : Nat) (h2 : ∃ (L : List Nat), l = p :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List Nat) (h3 : l = p :: L) from h2 rewrite [h3] at h rewrite [h3] show ¬prod (p :: L) = 1 from cons_prod_not_one h done done lemma prime_pos {p : Nat} (h : prime p) : p > 0 := by define at h linarith done theorem Theorem_7_2_5 : ∀ (l1 l2 : List Nat), nondec_prime_list l1 → nondec_prime_list l2 → prod l1 = prod l2 → l1 = l2 := by apply List.rec · -- Base Case. Goal : ∀ (l2 : List Nat), nondec_prime_list [] → -- nondec_prime_list l2 → prod [] = prod l2 → [] = l2 fix l2 : List Nat assume h1 : nondec_prime_list [] assume h2 : nondec_prime_list l2 assume h3 : prod [] = prod l2 rewrite [prod_nil, eq_comm, ←list_nil_iff_prod_one h2] at h3 show [] = l2 from h3.symm done · -- Induction Step fix p : Nat fix L1 : List Nat assume ih : ∀ (L2 : List Nat), nondec_prime_list L1 → nondec_prime_list L2 → prod L1 = prod L2 → L1 = L2 -- Goal : ∀ (l2 : List Nat), nondec_prime_list (p :: L1) → -- nondec_prime_list l2 → prod (p :: L1) = prod l2 → p :: L1 = l2 fix l2 : List Nat assume h1 : nondec_prime_list (p :: L1) assume h2 : nondec_prime_list l2 assume h3 : prod (p :: L1) = prod l2 have h4 : ¬prod (p :: L1) = 1 := cons_prod_not_one h1 rewrite [h3, ←list_nil_iff_prod_one h2] at h4 obtain (q : Nat) (h5 : ∃ (L : List Nat), l2 = q :: L) from List.exists_cons_of_ne_nil h4 obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5 rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2) rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2) have h7 : p ≤ q := first_le_first h1 h2 h3 have h8 : q ≤ p := first_le_first h2 h1 h3.symm have h9 : p = q := by linarith rewrite [h9, prod_cons, prod_cons] at h3 --h3 : q * prod L1 = q * prod L2 have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1 have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2 define at h2 have h12 : all_prime (q :: L2) := h2.left rewrite [all_prime_cons] at h12 have h13 : q > 0 := prime_pos h12.left have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3 have h15 : L1 = L2 := ih L2 h10 h11 h14 rewrite [h6, h9, h15] rfl done done theorem fund_thm_arith (n : Nat) (h : n ≥ 1) : ∃! (l : List Nat), prime_factorization n l := by exists_unique · -- Existence show ∃ (l : List Nat), prime_factorization n l from exists_prime_factorization n h done · -- Uniqueness fix l1 : List Nat; fix l2 : List Nat assume h1 : prime_factorization n l1 assume h2 : prime_factorization n l2 define at h1; define at h2 have h3 : prod l1 = n := h1.right rewrite [←h2.right] at h3 show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3 done done /- Section 7.3 -/ theorem congr_refl (m : Nat) : ∀ (a : Int), a ≡ a (MOD m) := by fix a : Int define --Goal : ∃ (c : Int), a - a = ↑m * c apply Exists.intro 0 ring done theorem congr_symm {m : Nat} : ∀ {a b : Int}, a ≡ b (MOD m) → b ≡ a (MOD m) := by fix a : Int; fix b : Int assume h1 : a ≡ b (MOD m) define at h1 --h1 : ∃ (c : Int), a - b = ↑m * c define --Goal : ∃ (c : Int), b - a = ↑m * c obtain (c : Int) (h2 : a - b = m * c) from h1 apply Exists.intro (-c) show b - a = m * (-c) from calc b - a _ = -(a - b) := by ring _ = -(m * c) := by rw [h2] _ = m * (-c) := by ring done theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry /- Fundamental properties of congruence classes -/ lemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) : X = Y ↔ X.val = Y.val := Fin.ext_iff lemma val_nat_eq_mod (n k : Nat) : ([k]_(n + 1)).val = k % (n + 1) := by rfl lemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl theorem cc_rep {m : Nat} (X : ZMod m) : ∃ (a : Int), X = [a]_m := match m with | 0 => by apply Exists.intro X rfl done | n + 1 => by apply Exists.intro ↑(X.val) have h1 : X.val < n + 1 := Fin.prop X rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1] rfl done theorem add_class (m : Nat) (a b : Int) : [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm theorem mul_class (m : Nat) (a b : Int) : [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm lemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ [a - b]_m = [0]_m := by apply Iff.intro · -- (→) assume h1 : [a]_m = [b]_m have h2 : a - b = a + (-b) := by ring have h3 : b + (-b) = 0 := by ring show [a - b]_m = [0]_m from calc [a - b]_m _ = [a + (-b)]_m := by rw [h2] _ = [a]_m + [-b]_m := by rw [add_class] _ = [b]_m + [-b]_m := by rw [h1] _ = [b + -b]_m := by rw [add_class] _ = [0]_m := by rw [h3] done · -- (←) assume h1 : [a - b]_m = [0]_m have h2 : b + (a - b) = a := by ring have h3 : b + 0 = b := by ring show [a]_m = [b]_m from calc [a]_m _ = [b + (a - b)]_m := by rw [h2] _ = [b]_m + [a - b]_m := by rw [add_class] _ = [b]_m + [0]_m := by rw [h1] _ = [b + 0]_m := by rw [add_class] _ = [b]_m := by rw [h3] done done lemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) : [a]_m = [0]_m → [-a]_m = [0]_m := by assume h1 : [a]_m = [0]_m have h2 : 0 + (-a) = -a := by ring have h3 : a + (-a) = 0 := by ring show [-a]_m = [0]_m from calc [-a]_m _ = [0 + (-a)]_m := by rw [h2] _ = [0]_m + [-a]_m := by rw [add_class] _ = [a]_m + [-a]_m := by rw [h1] _ = [a + (-a)]_m := by rw [add_class] _ = [0]_m := by rw [h3] done lemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) : [-a]_m = [0]_m ↔ [a]_m = [0]_m := by apply Iff.intro _ (cc_neg_zero_of_cc_zero m a) assume h1 : [-a]_m = [0]_m have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1 have h3 : -(-a) = a := by ring rewrite [h3] at h2 show [a]_m = [0]_m from h2 done lemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl lemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m ↔ m ∣ k := match m with | 0 => by have h : (0 : Int) = (↑(0 : Nat) : Int) := by rfl rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj] apply Iff.intro · -- (→) assume h1 : k = 0 rewrite [h1] show 0 ∣ 0 from dvd_self 0 done · -- (←) assume h1 : 0 ∣ k obtain (c : Nat) (h2 : k = 0 * c) from h1 rewrite [h2] ring done done | n + 1 => by rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero] show k % (n + 1) = 0 ↔ n + 1 ∣ k from (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm done lemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m ↔ ↑m ∣ a := by obtain (k : Nat) (h1 : a = ↑k ∨ a = -↑k) from Int.eq_nat_or_neg a by_cases on h1 · -- Case 1. h1: a = ↑k rewrite [h1, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done · -- Case 2. h1: a = -↑k rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done done theorem cc_eq_iff_congr (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ a ≡ b (MOD m) := calc [a]_m = [b]_m _ ↔ [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b _ ↔ ↑m ∣ (a - b) := cc_zero_iff_dvd m (a - b) _ ↔ a ≡ b (MOD m) := by rfl /- End of fundamental properties of congruence classes -/ lemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m := by have h1 : (↑m : Int) ≠ 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m) show 0 ≤ a % m from Int.emod_nonneg a h1 done lemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m) have h2 : (↑m : Int) > 0 := (Nat.cast_pos).rtl h1 show a % m < m from Int.emod_lt_of_pos a h2 done lemma congr_mod_mod (m : Nat) (a : Int) : a ≡ a % m (MOD m) := by define have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m apply Exists.intro (a / m) show a - a % m = m * (a / m) from calc a - (a % m) _ = m * (a / m) + a % m - a % m := by rw [h1] _ = m * (a / m) := by ring done lemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) := And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a)) theorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) : ∃! (r : Int), 0 ≤ r ∧ r < m ∧ a ≡ r (MOD m) := by exists_unique · -- Existence apply Exists.intro (a % m) show 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) from mod_cmpl_res m a done · -- Uniqueness fix r1 : Int; fix r2 : Int assume h1 : 0 ≤ r1 ∧ r1 < m ∧ a ≡ r1 (MOD m) assume h2 : 0 ≤ r2 ∧ r2 < m ∧ a ≡ r2 (MOD m) have h3 : r1 ≡ r2 (MOD m) := congr_trans (congr_symm h1.right.right) h2.right.right obtain (d : Int) (h4 : r1 - r2 = m * d) from h3 have h5 : r1 - r2 < m * 1 := by linarith have h6 : m * (-1) < r1 - r2 := by linarith rewrite [h4] at h5 --h5 : m * d < m * 1 rewrite [h4] at h6 --h6 : m * -1 < m * d have h7 : (↑m : Int) ≥ 0 := Nat.cast_nonneg m have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7 have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7 have h10 : d = 0 := by linarith show r1 = r2 from calc r1 _ = r1 - r2 + r2 := by ring _ = m * 0 + r2 := by rw [h4, h10] _ = r2 := by ring done done lemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m := (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a) theorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y rewrite [h1, h2] have h3 : a + b = b + a := by ring show [a]_m + [b]_m = [b]_m + [a]_m from calc [a]_m + [b]_m _ = [a + b]_m := add_class m a b _ = [b + a]_m := by rw [h3] _ = [b]_m + [a]_m := (add_class m b a).symm done theorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X rewrite [h1] have h2 : a * 1 = a := by ring show [a]_m * [1]_m = [a]_m from calc [a]_m * [1]_m _ = [a * 1]_m := mul_class m a 1 _ = [a]_m := by rw [h2] done theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry lemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) : [a]_m * [gcd_c2 m a]_m = [1]_m := by set s : Int := gcd_c1 m a have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m define at h1 rewrite [h1, Nat.cast_one] at h2 --h2 : s * ↑m + gcd_c2 m a * ↑a = 1 rewrite [mul_class, cc_eq_iff_congr] define --Goal : ∃ (c : Int), ↑a * gcd_c2 m a - 1 = ↑m * c apply Exists.intro (-s) show a * (gcd_c2 m a) - 1 = m * (-s) from calc a * (gcd_c2 m a) - 1 _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring _ = 1 + m * (-s) - 1 := by rw [h2] _ = m * (-s) := by ring done theorem Theorem_7_3_7 (m a : Nat) : invertible [a]_m ↔ rel_prime m a := by apply Iff.intro · -- (→) assume h1 : invertible [a]_m define at h1 obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1 obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y rewrite [h3, mul_class, cc_eq_iff_congr] at h2 define at h2 obtain (c : Int) (h4 : a * b - 1 = m * c) from h2 rewrite [Exercise_7_2_6] --Goal : ∃ (s t : Int), s * ↑m + t * ↑a = 1 apply Exists.intro (-c) apply Exists.intro b show (-c) * m + b * a = 1 from calc (-c) * m + b * a _ = (-c) * m + (a * b - 1) + 1 := by ring _ = (-c) * m + m * c + 1 := by rw [h4] _ = 1 := by ring done · -- (←) assume h1 : rel_prime m a define show ∃ (Y : ZMod m), [a]_m * Y = [1]_m from Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1) done done /- Section 7.4 -/ section Euler open Euler lemma num_rp_below_base {m : Nat} : num_rp_below m 0 = 0 := by rfl lemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) : num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by have h1 : num_rp_below m (j + 1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : gcd m j = 1 rewrite [if_pos h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1 show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1 done lemma num_rp_below_step_not_rp {m j : Nat} (h : ¬rel_prime m j) : num_rp_below m (j + 1) = num_rp_below m j := by have h1 : num_rp_below m (j +1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : ¬gcd m j = 1 rewrite [if_neg h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j show num_rp_below m (j + 1) = num_rp_below m j from h1 done lemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl #eval phi 10 --Answer: 4 lemma prod_inv_iff_inv {m : Nat} {X : ZMod m} (h1 : invertible X) (Y : ZMod m) : invertible (X * Y) ↔ invertible Y := by apply Iff.intro · -- (→) assume h2 : invertible (X * Y) obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2 apply Exists.intro (X * Z) rewrite [←h3] --Goal : Y * (X * Z) = X * Y * Z ring --Note that ring can do algebra in ZMod m done · -- (←) assume h2 : invertible Y obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1 obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2 apply Exists.intro (Xi * Yi) show (X * Y) * (Xi * Yi) = [1]_m from calc X * Y * (Xi * Yi) _ = (X * Xi) * (Y * Yi) := by ring _ = [1]_m * [1]_m := by rw [h3, h4] _ = [1]_m := Theorem_7_3_6_7 [1]_m done done lemma F_rp_def {m i : Nat} (h : rel_prime m i) : F m i = [i]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h --h : gcd m i = 1 rewrite [if_pos h] at h1 show F m i = [i]_m from h1 done lemma F_not_rp_def {m i : Nat} (h : ¬rel_prime m i) : F m i = [1]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h rewrite [h1, if_neg h] rfl done lemma prod_seq_base {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 0 k f = [1]_m := by rfl lemma prod_seq_step {m : Nat} (n k : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl lemma prod_seq_zero_step {m : Nat} (n : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by rewrite [prod_seq_step, zero_add] rfl done lemma prod_one {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 1 k f = f k := by rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7] rfl done lemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl lemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m := calc [G m a i]_m _ = [(a * i) % m]_m := by rfl _ = [a * i]_m := (cc_eq_mod m (a * i)).symm _ = [a]_m * [i]_m := (mul_class m a i).symm lemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) : rel_prime m (G m a i) ↔ rel_prime m i := by have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1 show rel_prime m (G m a i) ↔ rel_prime m i from calc rel_prime m (G m a i) _ ↔ invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm _ ↔ invertible ([a]_m * [i]_m) := by rw [cc_G] _ ↔ invertible [i]_m := prod_inv_iff_inv h2 ([i]_m) _ ↔ rel_prime m i := Theorem_7_3_7 m i done lemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) : F m (G m a i) = [a]_m * F m i := by have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2 show F m (G m a i) = [a]_m * F m i from calc F m (G m a i) _ = [G m a i]_m := F_rp_def h3 _ = [a]_m * [i]_m := cc_G m a i _ = [a]_m * F m i := by rw [F_rp_def h2] done lemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : ¬rel_prime m i) : F m (G m a i) = [1]_m := by rewrite [←G_rp_iff h1 i] at h2 show F m (G m a i) = [1]_m from F_not_rp_def h2 done lemma FG_prod {m a : Nat} (h1 : rel_prime m a) : ∀ (k : Nat), prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by by_induc · -- Base Case show prod_seq 0 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from calc prod_seq 0 0 ((F m) ∘ (G m a)) _ = [1]_m := prod_seq_base _ _ _ = [a]_m ^ 0 * [1]_m := by ring _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by rw [num_rp_below_base, prod_seq_base] done · -- Induction Step fix k : Nat assume ih : prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) by_cases h2 : rel_prime m k · -- Case 1. h2 : rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([a]_m * F m k) := by rw [FG_rp h1 h2] _ = [a]_m ^ ((num_rp_below m k) + 1) * ((prod_seq k 0 (F m)) * F m k) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_rp h2, prod_seq_zero_step] done · -- Case 2. h2 : ¬rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([1]_m) := by rw [FG_not_rp h1 h2] _ = [a]_m ^ (num_rp_below m k) * (prod_seq k 0 (F m) * ([1]_m)) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_not_rp h2, prod_seq_zero_step, F_not_rp_def h2] done done done lemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by define --Goal : ∀ i < m, G m a i < m fix i : Nat assume h1 : i < m rewrite [G_def] --Goal : a * i % m < m show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m) done lemma left_inv_one_one_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g' (g i) = i) : one_one_below n g := sorry lemma right_inv_onto_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g (g' i) = i) (h2 : maps_below n g') : onto_below n g := by define at h2; define fix k : Nat assume h3 : k < n apply Exists.intro (g' k) show g' k < n ∧ g (g' k) = k from And.intro (h2 k h3) (h1 k h3) done lemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m * [inv_mod m a]_m = [1]_m := by have h2 : 0 ≤ (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a) show [a]_m * [inv_mod m a]_m = [1]_m from calc [a]_m * [inv_mod m a]_m _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2] _ = [a]_m * [gcd_c2 m a]_m := by rw [←cc_eq_mod] _ = [1]_m := gcd_c2_inv h1 done lemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m := calc a * (b % m) % m = a % m * (b % m % m) % m := Nat.mul_mod _ _ _ _ = a % m * (b % m) % m := by rw [Nat.mod_mod] _ = a * b % m := (Nat.mul_mod _ _ _).symm lemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm] rfl done theorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] : ↑a ≡ ↑b (MOD m) ↔ a % m = b % m := sorry lemma mul_inv_mod_cancel {m a i : Nat} [NeZero m] (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1 rewrite [mul_class, cc_eq_iff_congr, ←Nat.cast_mul, ←Nat.cast_one, congr_iff_mod_eq_Nat] at h3 show a * inv_mod m a * i % m = i from calc a * (inv_mod m a) * i % m _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = 1 % m * i % m := by rw [h3] _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = i % m := by rw [one_mul] _ = i := Nat.mod_eq_of_lt h2 done lemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl lemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, G m a (Ginv m a i) = i := by fix i : Nat assume h2 : i < m show G m a (Ginv m a i) = i from calc G m a (Ginv m a i) _ = a * ((inv_mod m a * i) % m) % m := by rfl _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, Ginv m a (G m a i) = i := by fix i : Nat assume h2 : i < m show Ginv m a (G m a i) = i from calc Ginv m a (G m a i) _ = inv_mod m a * ((a * i) % m) % m := by rfl _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc, mul_comm (inv_mod m a)] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_maps_below (m a : Nat) [NeZero m] : maps_below m (Ginv m a) := G_maps_below m (inv_mod m a) lemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : one_one_below m (G m a) := left_inv_one_one_below (Ginv_left_inv h1) lemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : onto_below m (G m a) := right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a) lemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : perm_below m (G m a) := And.intro (G_maps_below m a) (And.intro (G_one_one_below h1) (G_onto_below h1)) --Permuting a product of congruence classes doesn't change product lemma swap_fst (u v : Nat) : swap u v u = v := by define : swap u v u --Goal : (if u = u then v else if u = v then u else u) = v have h : u = u := by rfl rewrite [if_pos h] rfl done lemma swap_snd (u v : Nat) : swap u v v = u := by define : swap u v v by_cases h1 : v = u · -- Case 1. h1 : v = u rewrite [if_pos h1] show v = u from h1 done · -- Case 2. h1 : v ≠ u rewrite [if_neg h1] have h2 : v = v := by rfl rewrite [if_pos h2] rfl done done lemma swap_other {u v i : Nat} (h1 : i ≠ u) (h2 : i ≠ v) : swap u v i = i := by define : swap u v i rewrite [if_neg h1, if_neg h2] rfl done lemma swap_values (u v i : Nat) : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := by by_cases h1 : i = u · -- Case 1. h1 : i = u apply Or.inl rewrite [h1] show swap u v u = v from swap_fst u v done · -- Case 2. h1 : i ≠ u apply Or.inr by_cases h2 : i = v · -- Case 2.1. h2 : i = v apply Or.inl rewrite [h2] show swap u v v = u from swap_snd u v done · -- Case 2.2. h2 : i ≠ v apply Or.inr show swap u v i = i from swap_other h1 h2 done done done lemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by define fix i : Nat assume h3 : i < n have h4 : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := swap_values u v i by_cases on h4 · -- Case 1. h4 : swap u v i = v rewrite [h4] show v < n from h2 done · -- Case 2. by_cases on h4 · -- Case 2.1. h4 : swap u v i = u rewrite [h4] show u < n from h1 done · -- Case 2.2. h4 : swap u v i = i rewrite [h4] show i < n from h3 done done done lemma swap_swap (u v n : Nat) : ∀ i < n, swap u v (swap u v i) = i := by fix i : Nat assume h : i < n by_cases h1 : i = u · -- Case 1. h1 : i = u rewrite [h1, swap_fst, swap_snd] rfl done · -- Case 2. h1 : i ≠ u by_cases h2 : i = v · -- Case 2.1. h2 : i = v rewrite [h2, swap_snd, swap_fst] rfl done · -- Case 2.2. h2 : i ≠ v rewrite [swap_other h1 h2, swap_other h1 h2] rfl done done done lemma swap_one_one_below (u v n) : one_one_below n (swap u v) := left_inv_one_one_below (swap_swap u v n) lemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) := right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2) lemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) := And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2)) lemma comp_perm_below {n : Nat} {f g : Nat → Nat} (h1 : perm_below n f) (h2 : perm_below n g) : perm_below n (f ∘ g) := sorry lemma trivial_swap (u : Nat) : swap u u = id := by apply funext fix x : Nat by_cases h1 : x = u · -- Case 1. h1 : x = u rewrite [h1, swap_fst] rfl done · -- Case 2. h1 : x ≠ u rewrite [swap_other h1 h1] rfl done done lemma prod_eq_fun {m : Nat} (f g : Nat → ZMod m) (k : Nat) : ∀ (n : Nat), (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g := by by_induc · -- Base Case assume h : (∀ i < 0, f (k + i) = g (k + i)) rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g assume h1 : ∀ i < n + 1, f (k + i) = g (k + i) have h2 : ∀ i < n, f (k + i) = g (k + i) := by fix i : Nat assume h2 : i < n have h3 : i < n + 1 := by linarith show f (k + i) = g (k + i) from h1 i h3 done have h3 : prod_seq n k f = prod_seq n k g := ih h2 have h4 : n < n + 1 := Nat.lt_succ_self n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4] rfl done done lemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := by have h2 : ∀ (i : Nat), i < u → (f ∘ swap u n) (0 + i) = f (0 + i) := by fix i : Nat assume h2 : i < u have h3 : 0 + i ≠ u := by linarith have h4 : 0 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl done show prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f from prod_eq_fun (f ∘ swap u n) f 0 u h2 done lemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := by have h2 : ∀ i < j, (f ∘ swap u n) (u + 1 + i) = f (u + 1 + i) := by fix i : Nat assume h2 : i < j have h3 : u + 1 + i ≠ u := by linarith have h4 : u + 1 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl show prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f from prod_eq_fun (f ∘ swap u n) f (u + 1) j h2 done lemma break_prod {m : Nat} (n : Nat) (f : Nat → ZMod m) : ∀ (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by by_induc · -- Base Case have h : n + 0 = n := by rfl rewrite [prod_seq_base, h, Theorem_7_3_6_7] rfl done · -- Induction Step fix j : Nat assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f rewrite [←add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc] rfl done done lemma break_prod_twice {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f := break_prod n f 1 rewrite [prod_one] at h2 have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f := break_prod (u + 1) f j rewrite [←h1] at h3 have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f := break_prod u f 1 rewrite [prod_one] at h4 rewrite [h3, h4] at h2 show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2 done lemma swap_prod_eq_prod {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f := by by_cases h2 : u = n · -- Case 1. h2 : u = n rewrite [h2, trivial_swap n] --Goal : prod_seq (n + 1) 0 (f ∘ id) = prod_seq (n + 1) 0 f rfl done · -- Case 2. h2 : ¬u = n have h3 : u + 1 ≤ n := Nat.lt_of_le_of_ne h1 h2 obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3 have break_f : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := break_prod_twice f h4 have break_fs : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_prod_twice (f ∘ swap u n) h4 have f_eq_fs_below : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := swap_prod_eq_prod_below f h1 have f_eq_fs_btwn : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4 show prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f from calc prod_seq (n + 1) 0 (f ∘ swap u n) _ = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_fs _ = prod_seq u 0 f * (f ∘ swap u n) u * prod_seq j (u + 1) f * (f ∘ swap u n) n := by rw [f_eq_fs_below, f_eq_fs_btwn] _ = prod_seq u 0 f * f (swap u n u) * prod_seq j (u + 1) f * f (swap u n n) := by rfl _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by rw [swap_fst, swap_snd] _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring _ = prod_seq (n + 1) 0 f := break_f.symm done done lemma perm_below_fixed {n : Nat} {g : Nat → Nat} (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry lemma perm_prod {m : Nat} (f : Nat → ZMod m) : ∀ (n : Nat), ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) := by by_induc · -- Base Case fix g : Nat → Nat assume h1 : perm_below 0 g rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) fix g : Nat → Nat assume g_pb : perm_below (n + 1) g define at g_pb have g_ob : onto_below (n + 1) g := g_pb.right.right define at g_ob have h1 : n < n + 1 := by linarith obtain (u : Nat) (h2 : u < n + 1 ∧ g u = n) from g_ob n h1 have s_pb : perm_below (n + 1) (swap u n) := swap_perm_below h2.left h1 have gs_pb_n1 : perm_below (n + 1) (g ∘ swap u n) := comp_perm_below g_pb s_pb have gs_fix_n : (g ∘ swap u n) n = n := calc (g ∘ swap u n) n _ = g (swap u n n) := by rfl _ = g u := by rw [swap_snd] _ = n := h2.right have gs_pb_n : perm_below n (g ∘ swap u n) := perm_below_fixed gs_pb_n1 gs_fix_n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f ∘ (g ∘ swap u n)) := ih (g ∘ swap u n) gs_pb_n have h3 : u ≤ n := by linarith show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f ∘ g) from calc prod_seq (n + 1) 0 f _ = prod_seq n 0 f * f n := prod_seq_zero_step n f _ = prod_seq n 0 (f ∘ (g ∘ swap u n)) * f ((g ∘ swap u n) n) := by rw [gs_prod, gs_fix_n] _ = prod_seq n 0 (f ∘ g ∘ swap u n) * (f ∘ g ∘ swap u n) n := by rfl _ = prod_seq (n + 1) 0 (f ∘ g ∘ swap u n) := (prod_seq_zero_step n (f ∘ g ∘ swap u n)).symm _ = prod_seq (n + 1) 0 ((f ∘ g) ∘ swap u n) := by rfl _ = prod_seq (n + 1) 0 (f ∘ g) := swap_prod_eq_prod (f ∘ g) h3 done done lemma F_invertible (m i : Nat) : invertible (F m i) := by by_cases h : rel_prime m i · -- Case 1. h : rel_prime m i rewrite [F_rp_def h] show invertible [i]_m from (Theorem_7_3_7 m i).rtl h done · -- Case 2. h : ¬rel_prime m i rewrite [F_not_rp_def h] apply Exists.intro [1]_m show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m done done lemma Fprod_invertible (m : Nat) : ∀ (k : Nat), invertible (prod_seq k 0 (F m)) := by by_induc · -- Base Case apply Exists.intro [1]_m show prod_seq 0 0 (F m) * [1]_m = [1]_m from calc prod_seq 0 0 (F m) * [1]_m _ = [1]_m * [1]_m := by rw [prod_seq_base] _ = [1]_m := Theorem_7_3_6_7 ([1]_m) done · -- Induction Step fix k : Nat assume ih : invertible (prod_seq k 0 (F m)) rewrite [prod_seq_zero_step] show invertible (prod_seq k 0 (F m) * (F m k)) from (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k) done done theorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m ^ (phi m) = [1]_m := by have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2 show [a]_m ^ (phi m) = [1]_m from calc [a]_m ^ (phi m) _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3] _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring _ = prod_seq m 0 (F m ∘ G m a) * Y := by rw [FG_prod h1 m, phi_def] _ = prod_seq m 0 (F m) * Y := by rw [perm_prod (F m) m (G m a) (G_perm_below h1)] _ = [1]_m := by rw [h3] done lemma Exercise_7_4_5_Int (m : Nat) (a : Int) : ∀ (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry lemma Exercise_7_4_5_Nat (m a n : Nat) : [a]_m ^ n = [a ^ n]_m := by rewrite [Exercise_7_4_5_Int] rfl done theorem Euler's_theorem {m a : Nat} [NeZero m] (h1 : rel_prime m a) : a ^ (phi m) ≡ 1 (MOD m) := by have h2 : [a]_m ^ (phi m) = [1]_m := Theorem_7_4_2 h1 rewrite [Exercise_7_4_5_Nat m a (phi m)] at h2 --h2 : [a ^ phi m]_m = [1]_m show a ^ (phi m) ≡ 1 (MOD m) from (cc_eq_iff_congr _ _ _).ltr h2 done #eval gcd 10 7 --Answer: 1. So 10 and 7 are relatively prime #eval 7 ^ phi 10 --Answer: 2401, which is congruent to 1 mod 10. end Euler /- Section 7.5 -/ lemma num_rp_prime {p : Nat} (h1 : prime p) : ∀ k < p, num_rp_below p (k + 1) = k := sorry lemma phi_prime {p : Nat} (h1 : prime p) : phi p = p - 1 := by have h2 : 1 ≤ p := prime_pos h1 have h3 : p - 1 + 1 = p := Nat.sub_add_cancel h2 have h4 : p - 1 < p := by linarith have h5 : num_rp_below p (p - 1 + 1) = p - 1 := num_rp_prime h1 (p - 1) h4 rewrite [h3] at h5 show phi p = p - 1 from h5 done theorem Theorem_7_2_2_Int {a c : Nat} {b : Int} (h1 : ↑c ∣ ↑a * b) (h2 : rel_prime a c) : ↑c ∣ b := by rewrite [Int.natCast_dvd, Int.natAbs_mul, Int.natAbs_ofNat] at h1 --h1 : c ∣ a * Int.natAbs b rewrite [Int.natCast_dvd] --Goal : c ∣ Int.natAbs b show c ∣ Int.natAbs b from Theorem_7_2_2 h1 h2 done lemma Lemma_7_4_5 {m n : Nat} (a b : Int) (h1 : rel_prime m n) : a ≡ b (MOD m * n) ↔ a ≡ b (MOD m) ∧ a ≡ b (MOD n) := by apply Iff.intro · -- (→) assume h2 : a ≡ b (MOD m * n) obtain (j : Int) (h3 : a - b = (m * n) * j) from h2 apply And.intro · -- Proof of a ≡ b (MOD m) apply Exists.intro (n * j) show a - b = m * (n * j) from calc a - b _ = m * n * j := h3 _ = m * (n * j) := by ring done · -- Proof of a ≡ b (MOD n) apply Exists.intro (m * j) show a - b = n * (m * j) from calc a - b _ = m * n * j := h3 _ = n * (m * j) := by ring done done · -- (←) assume h2 : a ≡ b (MOD m) ∧ a ≡ b (MOD n) obtain (j : Int) (h3 : a - b = m * j) from h2.left have h4 : (↑n : Int) ∣ a - b := h2.right rewrite [h3] at h4 --h4 : ↑n ∣ ↑m * j have h5 : ↑n ∣ j := Theorem_7_2_2_Int h4 h1 obtain (k : Int) (h6 : j = n * k) from h5 apply Exists.intro k --Goal : a - b = ↑(m * n) * k rewrite [Nat.cast_mul] --Goal : a - b = ↑m * ↑n * k show a - b = (m * n) * k from calc a - b _ = m * j := h3 _ = m * (n * k) := by rw [h6] _ = (m * n) * k := by ring done done --From exercises of Section 7.2 theorem rel_prime_symm {a b : Nat} (h : rel_prime a b) : rel_prime b a := sorry lemma prime_NeZero {p : Nat} (h : prime p) : NeZero p := by rewrite [neZero_iff] --Goal : p ≠ 0 define at h linarith done lemma Lemma_7_5_1 {p e d m c s : Nat} {t : Int} (h1 : prime p) (h2 : e * d = (p - 1) * s + 1) (h3 : m ^ e - c = p * t) : c ^ d ≡ m (MOD p) := by have h4 : m ^ e ≡ c (MOD p) := Exists.intro t h3 have h5 : [m ^ e]_p = [c]_p := (cc_eq_iff_congr _ _ _).rtl h4 rewrite [←Exercise_7_4_5_Nat] at h5 --h5 : [m]_p ^ e = [c]_p by_cases h6 : p ∣ m · -- Case 1. h6 : p ∣ m have h7 : m ≡ 0 (MOD p) := by obtain (j : Nat) (h8 : m = p * j) from h6 apply Exists.intro (↑j : Int) --Goal : ↑m - 0 = ↑p * ↑j rewrite [h8, Nat.cast_mul] ring done have h8 : [m]_p = [0]_p := (cc_eq_iff_congr _ _ _).rtl h7 have h9 : e * d ≠ 0 := by rewrite [h2] show (p - 1) * s + 1 ≠ 0 from Nat.add_one_ne_zero _ done have h10 : (0 : Int) ^ (e * d) = 0 := zero_pow h9 have h11 : [c ^ d]_p = [m]_p := calc [c ^ d]_p _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat] _ = ([m]_p ^ e) ^ d := by rw [h5] _ = [m]_p ^ (e * d) := by ring _ = [0]_p ^ (e * d) := by rw [h8] _ = [0 ^ (e * d)]_p := Exercise_7_4_5_Int _ _ _ _ = [0]_p := by rw [h10] _ = [m]_p := by rw [h8] show c ^ d ≡ m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11 done · -- Case 2. h6 : ¬p ∣ m have h7 : rel_prime m p := rel_prime_of_prime_not_dvd h1 h6 have h8 : rel_prime p m := rel_prime_symm h7 have h9 : NeZero p := prime_NeZero h1 have h10 : (1 : Int) ^ s = 1 := by ring have h11 : [c ^ d]_p = [m]_p := calc [c ^ d]_p _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat] _ = ([m]_p ^ e) ^ d := by rw [h5] _ = [m]_p ^ (e * d) := by ring _ = [m]_p ^ ((p - 1) * s + 1) := by rw [h2] _ = ([m]_p ^ (p - 1)) ^ s * [m]_p := by ring _ = ([m]_p ^ (phi p)) ^ s * [m]_p := by rw [phi_prime h1] _ = [1]_p ^ s * [m]_p := by rw [Theorem_7_4_2 h8] _ = [1 ^ s]_p * [m]_p := by rw [Exercise_7_4_5_Int] _ = [1]_p * [m]_p := by rw [h10] _ = [m]_p * [1]_p := by ring _ = [m]_p := Theorem_7_3_6_7 _ show c ^ d ≡ m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11 done done theorem Theorem_7_5_1 (p q n e d k m c : Nat) (p_prime : prime p) (q_prime : prime q) (p_ne_q : p ≠ q) (n_pq : n = p * q) (ed_congr_1 : e * d = k * (p - 1) * (q - 1) + 1) (h1 : [m]_n ^ e = [c]_n) : [c]_n ^ d = [m]_n := by rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] at h1 --h1 : m ^ e ≡ c (MOD n) rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] --Goal : c ^ d ≡ m (MOD n) obtain (j : Int) (h2 : m ^ e - c = n * j) from h1 rewrite [n_pq, Nat.cast_mul] at h2 --h2 : m ^ e - c = p * q * j have h3 : e * d = (p - 1) * (k * (q - 1)) + 1 := by rewrite [ed_congr_1] ring done have h4 : m ^ e - c = p * (q * j) := by rewrite [h2] ring done have congr_p : c ^ d ≡ m (MOD p) := Lemma_7_5_1 p_prime h3 h4 have h5 : e * d = (q - 1) * (k * (p - 1)) + 1 := by rewrite [ed_congr_1] ring done have h6 : m ^ e - c = q * (p * j) := by rewrite [h2] ring done have congr_q : c ^ d ≡ m (MOD q) := Lemma_7_5_1 q_prime h5 h6 have h7 : ¬q ∣ p := by by_contra h8 have h9 : q = 1 ∨ q = p := dvd_prime p_prime h8 disj_syll h9 (prime_not_one q_prime) show False from p_ne_q h9.symm done have h8 : rel_prime p q := rel_prime_of_prime_not_dvd q_prime h7 rewrite [n_pq, Lemma_7_4_5 _ _ h8] show c ^ d ≡ m (MOD p) ∧ c ^ d ≡ m (MOD q) from And.intro congr_p congr_q done /- BEGIN EXERCISES -/ namespace Exercises /- Section 7.1 -/ -- 1. theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry -- 2. lemma gcd_comm_lt {a b : Nat} (h : a < b) : gcd a b = gcd b a := sorry theorem gcd_comm (a b : Nat) : gcd a b = gcd b a := sorry -- 3. theorem Exercise_7_1_5 (a b : Nat) (n : Int) : (∃ (s t : Int), s * a + t * b = n) ↔ (↑(gcd a b) : Int) ∣ n := sorry -- 4. theorem Exercise_7_1_6 (a b c : Nat) : gcd a b = gcd (a + b * c) b := sorry -- 5. theorem gcd_is_nonzero {a b : Nat} (h : a ≠ 0 ∨ b ≠ 0) : gcd a b ≠ 0 := sorry -- 6. theorem gcd_greatest {a b d : Nat} (h1 : gcd a b ≠ 0) (h2 : d ∣ a) (h3 : d ∣ b) : d ≤ gcd a b := sorry -- 7. lemma Lemma_7_1_10a {a b : Nat} (n : Nat) (h : a ∣ b) : (n * a) ∣ (n * b) := sorry lemma Lemma_7_1_10b {a b n : Nat} (h1 : n ≠ 0) (h2 : (n * a) ∣ (n * b)) : a ∣ b := sorry lemma Lemma_7_1_10c {a b : Nat} (h1 : a ∣ b) (h2 : b ∣ a) : a = b := sorry theorem Exercise_7_1_10 (a b n : Nat) : gcd (n * a) (n * b) = n * gcd a b := sorry /- Section 7.2 -/ -- 1. lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry -- 2. -- Hints: Start with apply List.rec. You may find mul_ne_zero useful theorem prod_nonzero_nonzero : ∀ (l : List Nat), (∀ a ∈ l, a ≠ 0) → prod l ≠ 0 := sorry -- 3. theorem rel_prime_iff_no_common_factor (a b : Nat) : rel_prime a b ↔ ¬∃ (p : Nat), prime p ∧ p ∣ a ∧ p ∣ b := sorry -- 4. theorem rel_prime_symm {a b : Nat} (h : rel_prime a b) : rel_prime b a := sorry -- 5. lemma in_prime_factorization_iff_prime_factor {a : Nat} {l : List Nat} (h1 : prime_factorization a l) (p : Nat) : p ∈ l ↔ prime_factor p a := sorry -- 6. theorem Exercise_7_2_5 {a b : Nat} {l m : List Nat} (h1 : prime_factorization a l) (h2 : prime_factorization b m) : rel_prime a b ↔ (¬∃ (p : Nat), p ∈ l ∧ p ∈ m) := sorry -- 7. theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry -- 8. theorem Exercise_7_2_7 {a b a' b' : Nat} (h1 : rel_prime a b) (h2 : a' ∣ a) (h3 : b' ∣ b) : rel_prime a' b' := sorry -- 9. theorem Exercise_7_2_9 {a b j k : Nat} (h1 : gcd a b ≠ 0) (h2 : a = j * gcd a b) (h3 : b = k * gcd a b) : rel_prime j k := sorry -- 10. theorem Exercise_7_2_17a (a b c : Nat) : gcd a (b * c) ∣ gcd a b * gcd a c := sorry /- Section 7.3 -/ -- 1. theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry -- 2. theorem Theorem_7_3_6_3 {m : Nat} (X : ZMod m) : X + [0]_m = X := sorry -- 3.

theorem Theorem_7_3_6_4 {m : Nat} (X : ZMod m) : ∃ (Y : ZMod m), X + Y = [0]_m

HTPI.Exercises.Theorem_7_3_6_4

htpi/HTPILib/Chap7.lean

htpi

/- Copyright 2023 Daniel J. Velleman -/ import HTPILib.Chap6 namespace HTPI /- Definitions -/ lemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by have h : n + 1 > 0 := Nat.succ_pos n show a % (n + 1) < n + 1 from Nat.mod_lt a h done def gcd (a b : Nat) : Nat := match b with | 0 => a | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd (n + 1) (a % (n + 1)) termination_by b mutual def gcd_c1 (a b : Nat) : Int := match b with | 0 => 1 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c2 (n + 1) (a % (n + 1)) --Corresponds to s = t' termination_by b def gcd_c2 (a b : Nat) : Int := match b with | 0 => 0 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c1 (n + 1) (a % (n + 1)) - (gcd_c2 (n + 1) (a % (n + 1))) * ↑(a / (n + 1)) --Corresponds to t = s' - t'q termination_by b end def prime (n : Nat) : Prop := 2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n def prime_factor (p n : Nat) : Prop := prime p ∧ p ∣ n def all_prime (l : List Nat) : Prop := ∀ p ∈ l, prime p def nondec (l : List Nat) : Prop := match l with | [] => True --Of course, True is a proposition that is always true | n :: L => (∀ m ∈ L, n ≤ m) ∧ nondec L def nondec_prime_list (l : List Nat) : Prop := all_prime l ∧ nondec l def prod (l : List Nat) : Nat := match l with | [] => 1 | n :: L => n * (prod L) def prime_factorization (n : Nat) (l : List Nat) : Prop := nondec_prime_list l ∧ prod l = n def rel_prime (a b : Nat) : Prop := gcd a b = 1 def congr_mod (m : Nat) (a b : Int) : Prop := (↑m : Int) ∣ (a - b) def cc (m : Nat) (a : Int) : ZMod m := (↑a : ZMod m) notation:50 a " ≡ " b " (MOD " m ")" => congr_mod m a b notation:max "["a"]_"m:max => cc m a def invertible {m : Nat} (X : ZMod m) : Prop := ∃ (Y : ZMod m), X * Y = [1]_m def num_rp_below (m k : Nat) : Nat := match k with | 0 => 0 | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j def phi (m : Nat) : Nat := num_rp_below m m def prod_seq {m : Nat} (j k : Nat) (f : Nat → ZMod m) : ZMod m := match j with | 0 => [1]_m | n + 1 => prod_seq n k f * f (k + n) def maps_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i < n, g i < n def one_one_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i1 < n, ∀ i2 < n, g i1 = g i2 → i1 = i2 def onto_below (n : Nat) (g : Nat → Nat) : Prop := ∀ k < n, ∃ i < n, g i = k def perm_below (n : Nat) (g : Nat → Nat) : Prop := maps_below n g ∧ one_one_below n g ∧ onto_below n g def inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m) def swap (u v i : Nat) : Nat := if i = u then v else if i = v then u else i namespace Euler --For definitions specific to Euler's theorem def F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m def G (m a i : Nat) : Nat := (a * i) % m def Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i end Euler /- Section 7.1 -/ theorem dvd_mod_of_dvd_a_b {a b d : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ (a % b) := by set q : Nat := a / b have h3 : b * q + a % b = a := Nat.div_add_mod a b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 define --Goal : ∃ (c : Nat), a % b = d * c apply Exists.intro (j - k * q) show a % b = d * (j - k * q) from calc a % b _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm _ = a - b * q := by rw [h3] _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc] _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm done theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry #eval gcd 672 161 --Answer: 7 lemma gcd_base (a : Nat) : gcd a 0 = a := by rfl lemma gcd_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd a b = gcd b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1)) rfl done lemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b ≠ 0) : a % b < b := by have h1 : b > 0 := Nat.pos_of_ne_zero h show a % b < b from Nat.mod_lt a h1 done lemma dvd_self (n : Nat) : n ∣ n := by apply Exists.intro 1 ring done theorem gcd_dvd : ∀ (b a : Nat), (gcd a b) ∣ a ∧ (gcd a b) ∣ b := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd a b_1) ∣ a ∧ (gcd a b_1) ∣ b_1 fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_base] --Goal: a ∣ a ∧ a ∣ 0 apply And.intro (dvd_self a) define apply Exists.intro 0 rfl done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_nonzero a h1] --Goal : gcd b (a % b) ∣ a ∧ gcd b (a % b) ∣ b have h2 : a % b < b := mod_nonzero_lt a h1 have h3 : (gcd b (a % b)) ∣ b ∧ (gcd b (a % b)) ∣ (a % b) := ih (a % b) h2 b apply And.intro _ h3.left show (gcd b (a % b)) ∣ a from dvd_a_of_dvd_b_mod h3.left h3.right done done theorem gcd_dvd_left (a b : Nat) : (gcd a b) ∣ a := (gcd_dvd b a).left theorem gcd_dvd_right (a b : Nat) : (gcd a b) ∣ b := (gcd_dvd b a).right lemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl lemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c1 a b = gcd_c2 b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done lemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl lemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * ↑(a / b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done theorem gcd_lin_comb : ∀ (b a : Nat), (gcd_c1 a b) * ↑a + (gcd_c2 a b) * ↑b = ↑(gcd a b) := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd_c1 a b_1) * ↑a + (gcd_c2 a b_1) * ↑b_1 = ↑(gcd a b_1) fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base] --Goal : 1 * ↑a + 0 * ↑0 = ↑a ring done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1] --Goal : gcd_c2 b (a % b) * ↑a + -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * ↑(a / b)) * ↑b = -- ↑(gcd b (a % b)) set r : Nat := a % b set q : Nat := a / b set s : Int := gcd_c1 b r set t : Int := gcd_c2 b r --Goal : t * ↑a + (s - t * ↑q) * ↑b = ↑(gcd b r) have h2 : r < b := mod_nonzero_lt a h1 have h3 : s * ↑b + t * ↑r = ↑(gcd b r) := ih r h2 b have h4 : b * q + r = a := Nat.div_add_mod a b rewrite [←h3, ←h4] rewrite [Nat.cast_add, Nat.cast_mul] --Goal : t * (↑b * ↑q + ↑r) + (s - t * ↑q) * ↑b = s * ↑b + t * ↑r ring done done #eval gcd_c1 672 161 --Answer: 6 #eval gcd_c2 672 161 --Answer: -25 --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161 theorem Theorem_7_1_6 {d a b : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ gcd a b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑d ∣ ↑(gcd a b) set s : Int := gcd_c1 a b set t : Int := gcd_c2 a b have h3 : s * ↑a + t * ↑b = ↑(gcd a b) := gcd_lin_comb b a rewrite [←h3] --Goal : ↑d ∣ s * ↑a + t * ↑b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul] --Goal : ↑d ∣ s * (↑d * ↑j) + t * (↑d * ↑k) define apply Exists.intro (s * ↑j + t * ↑k) ring done /- Section 7.2 -/ theorem dvd_trans {a b c : Nat} (h1 : a ∣ b) (h2 : b ∣ c) : a ∣ c := by define at h1; define at h2; define obtain (m : Nat) (h3 : b = a * m) from h1 obtain (n : Nat) (h4 : c = b * n) from h2 rewrite [h3, mul_assoc] at h4 apply Exists.intro (m * n) show c = a * (m * n) from h4 done lemma exists_prime_factor : ∀ (n : Nat), 2 ≤ n → ∃ (p : Nat), prime_factor p n := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, 2 ≤ n_1 → ∃ (p : Nat), prime_factor p n_1 assume h1 : 2 ≤ n by_cases h2 : prime n · -- Case 1. h2 : prime n apply Exists.intro n define --Goal : prime n ∧ n ∣ n show prime n ∧ n ∣ n from And.intro h2 (dvd_self n) done · -- Case 2. h2 : ¬prime n define at h2 --h2 : ¬(2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n) demorgan at h2 disj_syll h2 h1 obtain (a : Nat) (h3 : ∃ (b : Nat), a * b = n ∧ a < n ∧ b < n) from h2 obtain (b : Nat) (h4 : a * b = n ∧ a < n ∧ b < n) from h3 have h5 : 2 ≤ a := by by_contra h6 have h7 : a ≤ 1 := by linarith have h8 : n ≤ b := calc n _ = a * b := h4.left.symm _ ≤ 1 * b := by rel [h7] _ = b := by ring linarith --n ≤ b contradicts b < n done have h6 : ∃ (p : Nat), prime_factor p a := ih a h4.right.left h5 obtain (p : Nat) (h7 : prime_factor p a) from h6 apply Exists.intro p define --Goal : prime p ∧ p ∣ n define at h7 --h7 : prime p ∧ p ∣ a apply And.intro h7.left have h8 : a ∣ n := by apply Exists.intro b show n = a * b from (h4.left).symm done show p ∣ n from dvd_trans h7.right h8 done done lemma exists_least_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q := by set S : Set Nat := {p : Nat | prime_factor p n} have h2 : ∃ (p : Nat), p ∈ S := exists_prime_factor n h show ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q from well_ord_princ S h2 done lemma all_prime_nil : all_prime [] := by define --Goal : ∀ p ∈ [], prime p fix p : Nat contrapos --Goal : ¬prime p → p ∉ [] assume h1 : ¬prime p show p ∉ [] from List.not_mem_nil p done lemma all_prime_cons (n : Nat) (L : List Nat) : all_prime (n :: L) ↔ prime n ∧ all_prime L := by apply Iff.intro · -- (→) assume h1 : all_prime (n :: L) --Goal : prime n ∧ all_prime L define at h1 --h1 : ∀ p ∈ n :: L, prime p apply And.intro (h1 n (List.mem_cons_self n L)) define --Goal : ∀ p ∈ L, prime p fix p : Nat assume h2 : p ∈ L show prime p from h1 p (List.mem_cons_of_mem n h2) done · -- (←) assume h1 : prime n ∧ all_prime L --Goal : all_prime (n :: l) define : all_prime L at h1 define fix p : Nat assume h2 : p ∈ n :: L rewrite [List.mem_cons] at h2 --h2 : p = n ∨ p ∈ L by_cases on h2 · -- Case 1. h2 : p = n rewrite [h2] show prime n from h1.left done · -- Case 2. h2 : p ∈ L show prime p from h1.right p h2 done done done lemma nondec_nil : nondec [] := by define --Goal : True trivial --trivial proves some obviously true statements, such as True done lemma nondec_cons (n : Nat) (L : List Nat) : nondec (n :: L) ↔ (∀ m ∈ L, n ≤ m) ∧ nondec L := by rfl lemma prod_nil : prod [] = 1 := by rfl lemma prod_cons : prod (n :: L) = n * (prod L) := by rfl lemma exists_cons_of_length_eq_succ {A : Type} {l : List A} {n : Nat} (h : l.length = n + 1) : ∃ (a : A) (L : List A), l = a :: L ∧ L.length = n := by have h1 : ¬l.length = 0 := by linarith rewrite [List.length_eq_zero] at h1 obtain (a : A) (h2 : ∃ (L : List A), l = a :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List A) (h3 : l = a :: L) from h2 apply Exists.intro a apply Exists.intro L apply And.intro h3 have h4 : (a :: L).length = L.length + 1 := List.length_cons a L rewrite [←h3, h] at h4 show L.length = n from (Nat.add_right_cancel h4).symm done lemma list_elt_dvd_prod_by_length (a : Nat) : ∀ (n : Nat), ∀ (l : List Nat), l.length = n → a ∈ l → a ∣ prod l := by by_induc · --Base Case fix l : List Nat assume h1 : l.length = 0 rewrite [List.length_eq_zero] at h1 --h1 : l = [] rewrite [h1] --Goal : a ∈ [] → a ∣ prod [] contrapos assume h2 : ¬a ∣ prod [] show a ∉ [] from List.not_mem_nil a done · -- Induction Step fix n : Nat assume ih : ∀ (l : List Nat), List.length l = n → a ∈ l → a ∣ prod l fix l : List Nat assume h1 : l.length = n + 1 --Goal : a ∈ l → a ∣ prod l obtain (b : Nat) (h2 : ∃ (L : List Nat), l = b :: L ∧ L.length = n) from exists_cons_of_length_eq_succ h1 obtain (L : List Nat) (h3 : l = b :: L ∧ L.length = n) from h2 have h4 : a ∈ L → a ∣ prod L := ih L h3.right assume h5 : a ∈ l rewrite [h3.left, prod_cons] --Goal : a ∣ b * prod L rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b ∨ a ∈ L by_cases on h5 · -- Case 1. h5 : a = b apply Exists.intro (prod L) rewrite [h5] rfl done · -- Case 2. h5 : a ∈ L have h6 : a ∣ prod L := h4 h5 have h7 : prod L ∣ b * prod L := by apply Exists.intro b ring done show a ∣ b * prod L from dvd_trans h6 h7 done done done lemma list_elt_dvd_prod {a : Nat} {l : List Nat} (h : a ∈ l) : a ∣ prod l := by set n : Nat := l.length have h1 : l.length = n := by rfl show a ∣ prod l from list_elt_dvd_prod_by_length a n l h1 h done lemma exists_prime_factorization : ∀ (n : Nat), n ≥ 1 → ∃ (l : List Nat), prime_factorization n l := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, n_1 ≥ 1 → ∃ (l : List Nat), prime_factorization n_1 l assume h1 : n ≥ 1 by_cases h2 : n = 1 · -- Case 1. h2 : n = 1 apply Exists.intro [] define apply And.intro · -- Proof of nondec_prime_list [] define show all_prime [] ∧ nondec [] from And.intro all_prime_nil nondec_nil done · -- Proof of prod [] = n rewrite [prod_nil, h2] rfl done done · -- Case 2. h2 : n ≠ 1 have h3 : n ≥ 2 := lt_of_le_of_ne' h1 h2 obtain (p : Nat) (h4 : prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q) from exists_least_prime_factor h3 have p_prime_factor : prime_factor p n := h4.left define at p_prime_factor have p_prime : prime p := p_prime_factor.left have p_dvd_n : p ∣ n := p_prime_factor.right have p_least : ∀ (q : Nat), prime_factor q n → p ≤ q := h4.right obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n have h5 : m ≠ 0 := by contradict h1 with h6 have h7 : n = 0 := calc n _ = p * m := n_eq_pm _ = p * 0 := by rw [h6] _ = 0 := by ring rewrite [h7] decide done have m_pos : 0 < m := Nat.pos_of_ne_zero h5 have m_lt_n : m < n := by define at p_prime show m < n from calc m _ < m + m := by linarith _ = 2 * m := by ring _ ≤ p * m := by rel [p_prime.left] _ = n := n_eq_pm.symm done obtain (L : List Nat) (h6 : prime_factorization m L) from ih m m_lt_n m_pos define at h6 have ndpl_L : nondec_prime_list L := h6.left define at ndpl_L apply Exists.intro (p :: L) define apply And.intro · -- Proof of nondec_prime_list (p :: L) define apply And.intro · -- Proof of all_prime (p :: L) rewrite [all_prime_cons] show prime p ∧ all_prime L from And.intro p_prime ndpl_L.left done · -- Proof of nondec (p :: L) rewrite [nondec_cons] apply And.intro _ ndpl_L.right fix q : Nat assume q_in_L : q ∈ L have h7 : q ∣ prod L := list_elt_dvd_prod q_in_L rewrite [h6.right] at h7 --h7 : q ∣ m have h8 : m ∣ n := by apply Exists.intro p rewrite [n_eq_pm] ring done have q_dvd_n : q ∣ n := dvd_trans h7 h8 have ap_L : all_prime L := ndpl_L.left define at ap_L have q_prime_factor : prime_factor q n := And.intro (ap_L q q_in_L) q_dvd_n show p ≤ q from p_least q q_prime_factor done done · -- Proof of prod (p :: L) = n rewrite [prod_cons, h6.right, n_eq_pm] rfl done done done theorem Theorem_7_2_2 {a b c : Nat} (h1 : c ∣ a * b) (h2 : rel_prime a c) : c ∣ b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑c ∣ ↑b define at h1; define at h2; define obtain (j : Nat) (h3 : a * b = c * j) from h1 set s : Int := gcd_c1 a c set t : Int := gcd_c2 a c have h4 : s * ↑a + t * ↑c = ↑(gcd a c) := gcd_lin_comb c a rewrite [h2, Nat.cast_one] at h4 --h4 : s * ↑a + t * ↑c = (1 : Int) apply Exists.intro (s * ↑j + t * ↑b) show ↑b = ↑c * (s * ↑j + t * ↑b) from calc ↑b _ = (1 : Int) * ↑b := (one_mul _).symm _ = (s * ↑a + t * ↑c) * ↑b := by rw [h4] _ = s * (↑a * ↑b) + t * ↑c * ↑b := by ring _ = s * (↑c * ↑j) + t * ↑c * ↑b := by rw [←Nat.cast_mul a b, h3, Nat.cast_mul c j] _ = ↑c * (s * ↑j + t * ↑b) := by ring done lemma le_nonzero_prod_left {a b : Nat} (h : a * b ≠ 0) : a ≤ a * b := by have h1 : b ≠ 0 := by contradict h with h1 rewrite [h1] ring done have h2 : 1 ≤ b := Nat.pos_of_ne_zero h1 show a ≤ a * b from calc a = a * 1 := (mul_one a).symm _ ≤ a * b := by rel [h2] done lemma le_nonzero_prod_right {a b : Nat} (h : a * b ≠ 0) : b ≤ a * b := by rewrite [mul_comm] rewrite [mul_comm] at h show b ≤ b * a from le_nonzero_prod_left h done lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry lemma rel_prime_of_prime_not_dvd {a p : Nat} (h1 : prime p) (h2 : ¬p ∣ a) : rel_prime a p := by have h3 : gcd a p ∣ a := gcd_dvd_left a p have h4 : gcd a p ∣ p := gcd_dvd_right a p have h5 : gcd a p = 1 ∨ gcd a p = p := dvd_prime h1 h4 have h6 : gcd a p ≠ p := by contradict h2 with h6 rewrite [h6] at h3 show p ∣ a from h3 done disj_syll h5 h6 show rel_prime a p from h5 done theorem Theorem_7_2_3 {a b p : Nat} (h1 : prime p) (h2 : p ∣ a * b) : p ∣ a ∨ p ∣ b := by or_right with h3 have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3 show p ∣ b from Theorem_7_2_2 h2 h4 done lemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a ≥ 1 := by have h1 : a ≠ 0 := by by_contra h1 rewrite [h1] at h contradict h linarith done show a ≥ 1 from Nat.pos_of_ne_zero h1 done lemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by have h1 : a ≥ 1 := ge_one_of_prod_one h have h2 : a * b ≠ 0 := by linarith have h3 : a ≤ a * b := le_nonzero_prod_left h2 rewrite [h] at h3 show a = 1 from Nat.le_antisymm h3 h1 done lemma eq_one_of_dvd_one {n : Nat} (h : n ∣ 1) : n = 1 := by obtain (j : Nat) (h1 : 1 = n * j) from h show n = 1 from eq_one_of_prod_one h1.symm done lemma prime_not_one {p : Nat} (h : prime p) : p ≠ 1 := by define at h linarith done theorem Theorem_7_2_4 {p : Nat} (h1 : prime p) : ∀ (l : List Nat), p ∣ prod l → ∃ a ∈ l, p ∣ a := by apply List.rec · -- Base Case. Goal : p ∣ prod [] → ∃ a ∈ [], p ∣ a rewrite [prod_nil] assume h2 : p ∣ 1 show ∃ a ∈ [], p ∣ a from absurd (eq_one_of_dvd_one h2) (prime_not_one h1) done · -- Induction Step fix b : Nat fix L : List Nat assume ih : p ∣ prod L → ∃ a ∈ L, p ∣ a --Goal : p ∣ prod (b :: L) → ∃ a ∈ b :: L, p ∣ a assume h2 : p ∣ prod (b :: L) rewrite [prod_cons] at h2 have h3 : p ∣ b ∨ p ∣ prod L := Theorem_7_2_3 h1 h2 by_cases on h3 · -- Case 1. h3 : p ∣ b apply Exists.intro b show b ∈ b :: L ∧ p ∣ b from And.intro (List.mem_cons_self b L) h3 done · -- Case 2. h3 : p ∣ prod L obtain (a : Nat) (h4 : a ∈ L ∧ p ∣ a) from ih h3 apply Exists.intro a show a ∈ b :: L ∧ p ∣ a from And.intro (List.mem_cons_of_mem b h4.left) h4.right done done done lemma prime_in_list {p : Nat} {l : List Nat} (h1 : prime p) (h2 : all_prime l) (h3 : p ∣ prod l) : p ∈ l := by obtain (a : Nat) (h4 : a ∈ l ∧ p ∣ a) from Theorem_7_2_4 h1 l h3 define at h2 have h5 : prime a := h2 a h4.left have h6 : p = 1 ∨ p = a := dvd_prime h5 h4.right disj_syll h6 (prime_not_one h1) rewrite [h6] show a ∈ l from h4.left done lemma first_le_first {p q : Nat} {l m : List Nat} (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m)) (h3 : prod (p :: l) = prod (q :: m)) : p ≤ q := by define at h1; define at h2 have h4 : q ∣ prod (p :: l) := by define apply Exists.intro (prod m) rewrite [←prod_cons] show prod (p :: l) = prod (q :: m) from h3 done have h5 : all_prime (q :: m) := h2.left rewrite [all_prime_cons] at h5 have h6 : q ∈ p :: l := prime_in_list h5.left h1.left h4 have h7 : nondec (p :: l) := h1.right rewrite [nondec_cons] at h7 rewrite [List.mem_cons] at h6 by_cases on h6 · -- Case 1. h6 : q = p linarith done · -- Case 2. h6 : q ∈ l have h8 : ∀ m ∈ l, p ≤ m := h7.left show p ≤ q from h8 q h6 done done lemma nondec_prime_list_tail {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by define at h define rewrite [all_prime_cons, nondec_cons] at h show all_prime l ∧ nondec l from And.intro h.left.right h.right.right done lemma cons_prod_not_one {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : prod (p :: l) ≠ 1 := by define at h have h1 : all_prime (p :: l) := h.left rewrite [all_prime_cons] at h1 rewrite [prod_cons] by_contra h2 show False from (prime_not_one h1.left) (eq_one_of_prod_one h2) done lemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) : l = [] ↔ prod l = 1 := by apply Iff.intro · -- (→) assume h1 : l = [] rewrite [h1] show prod [] = 1 from prod_nil done · -- (←) contrapos assume h1 : ¬l = [] obtain (p : Nat) (h2 : ∃ (L : List Nat), l = p :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List Nat) (h3 : l = p :: L) from h2 rewrite [h3] at h rewrite [h3] show ¬prod (p :: L) = 1 from cons_prod_not_one h done done lemma prime_pos {p : Nat} (h : prime p) : p > 0 := by define at h linarith done theorem Theorem_7_2_5 : ∀ (l1 l2 : List Nat), nondec_prime_list l1 → nondec_prime_list l2 → prod l1 = prod l2 → l1 = l2 := by apply List.rec · -- Base Case. Goal : ∀ (l2 : List Nat), nondec_prime_list [] → -- nondec_prime_list l2 → prod [] = prod l2 → [] = l2 fix l2 : List Nat assume h1 : nondec_prime_list [] assume h2 : nondec_prime_list l2 assume h3 : prod [] = prod l2 rewrite [prod_nil, eq_comm, ←list_nil_iff_prod_one h2] at h3 show [] = l2 from h3.symm done · -- Induction Step fix p : Nat fix L1 : List Nat assume ih : ∀ (L2 : List Nat), nondec_prime_list L1 → nondec_prime_list L2 → prod L1 = prod L2 → L1 = L2 -- Goal : ∀ (l2 : List Nat), nondec_prime_list (p :: L1) → -- nondec_prime_list l2 → prod (p :: L1) = prod l2 → p :: L1 = l2 fix l2 : List Nat assume h1 : nondec_prime_list (p :: L1) assume h2 : nondec_prime_list l2 assume h3 : prod (p :: L1) = prod l2 have h4 : ¬prod (p :: L1) = 1 := cons_prod_not_one h1 rewrite [h3, ←list_nil_iff_prod_one h2] at h4 obtain (q : Nat) (h5 : ∃ (L : List Nat), l2 = q :: L) from List.exists_cons_of_ne_nil h4 obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5 rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2) rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2) have h7 : p ≤ q := first_le_first h1 h2 h3 have h8 : q ≤ p := first_le_first h2 h1 h3.symm have h9 : p = q := by linarith rewrite [h9, prod_cons, prod_cons] at h3 --h3 : q * prod L1 = q * prod L2 have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1 have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2 define at h2 have h12 : all_prime (q :: L2) := h2.left rewrite [all_prime_cons] at h12 have h13 : q > 0 := prime_pos h12.left have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3 have h15 : L1 = L2 := ih L2 h10 h11 h14 rewrite [h6, h9, h15] rfl done done theorem fund_thm_arith (n : Nat) (h : n ≥ 1) : ∃! (l : List Nat), prime_factorization n l := by exists_unique · -- Existence show ∃ (l : List Nat), prime_factorization n l from exists_prime_factorization n h done · -- Uniqueness fix l1 : List Nat; fix l2 : List Nat assume h1 : prime_factorization n l1 assume h2 : prime_factorization n l2 define at h1; define at h2 have h3 : prod l1 = n := h1.right rewrite [←h2.right] at h3 show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3 done done /- Section 7.3 -/ theorem congr_refl (m : Nat) : ∀ (a : Int), a ≡ a (MOD m) := by fix a : Int define --Goal : ∃ (c : Int), a - a = ↑m * c apply Exists.intro 0 ring done theorem congr_symm {m : Nat} : ∀ {a b : Int}, a ≡ b (MOD m) → b ≡ a (MOD m) := by fix a : Int; fix b : Int assume h1 : a ≡ b (MOD m) define at h1 --h1 : ∃ (c : Int), a - b = ↑m * c define --Goal : ∃ (c : Int), b - a = ↑m * c obtain (c : Int) (h2 : a - b = m * c) from h1 apply Exists.intro (-c) show b - a = m * (-c) from calc b - a _ = -(a - b) := by ring _ = -(m * c) := by rw [h2] _ = m * (-c) := by ring done theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry /- Fundamental properties of congruence classes -/ lemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) : X = Y ↔ X.val = Y.val := Fin.ext_iff lemma val_nat_eq_mod (n k : Nat) : ([k]_(n + 1)).val = k % (n + 1) := by rfl lemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl theorem cc_rep {m : Nat} (X : ZMod m) : ∃ (a : Int), X = [a]_m := match m with | 0 => by apply Exists.intro X rfl done | n + 1 => by apply Exists.intro ↑(X.val) have h1 : X.val < n + 1 := Fin.prop X rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1] rfl done theorem add_class (m : Nat) (a b : Int) : [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm theorem mul_class (m : Nat) (a b : Int) : [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm lemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ [a - b]_m = [0]_m := by apply Iff.intro · -- (→) assume h1 : [a]_m = [b]_m have h2 : a - b = a + (-b) := by ring have h3 : b + (-b) = 0 := by ring show [a - b]_m = [0]_m from calc [a - b]_m _ = [a + (-b)]_m := by rw [h2] _ = [a]_m + [-b]_m := by rw [add_class] _ = [b]_m + [-b]_m := by rw [h1] _ = [b + -b]_m := by rw [add_class] _ = [0]_m := by rw [h3] done · -- (←) assume h1 : [a - b]_m = [0]_m have h2 : b + (a - b) = a := by ring have h3 : b + 0 = b := by ring show [a]_m = [b]_m from calc [a]_m _ = [b + (a - b)]_m := by rw [h2] _ = [b]_m + [a - b]_m := by rw [add_class] _ = [b]_m + [0]_m := by rw [h1] _ = [b + 0]_m := by rw [add_class] _ = [b]_m := by rw [h3] done done lemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) : [a]_m = [0]_m → [-a]_m = [0]_m := by assume h1 : [a]_m = [0]_m have h2 : 0 + (-a) = -a := by ring have h3 : a + (-a) = 0 := by ring show [-a]_m = [0]_m from calc [-a]_m _ = [0 + (-a)]_m := by rw [h2] _ = [0]_m + [-a]_m := by rw [add_class] _ = [a]_m + [-a]_m := by rw [h1] _ = [a + (-a)]_m := by rw [add_class] _ = [0]_m := by rw [h3] done lemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) : [-a]_m = [0]_m ↔ [a]_m = [0]_m := by apply Iff.intro _ (cc_neg_zero_of_cc_zero m a) assume h1 : [-a]_m = [0]_m have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1 have h3 : -(-a) = a := by ring rewrite [h3] at h2 show [a]_m = [0]_m from h2 done lemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl lemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m ↔ m ∣ k := match m with | 0 => by have h : (0 : Int) = (↑(0 : Nat) : Int) := by rfl rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj] apply Iff.intro · -- (→) assume h1 : k = 0 rewrite [h1] show 0 ∣ 0 from dvd_self 0 done · -- (←) assume h1 : 0 ∣ k obtain (c : Nat) (h2 : k = 0 * c) from h1 rewrite [h2] ring done done | n + 1 => by rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero] show k % (n + 1) = 0 ↔ n + 1 ∣ k from (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm done lemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m ↔ ↑m ∣ a := by obtain (k : Nat) (h1 : a = ↑k ∨ a = -↑k) from Int.eq_nat_or_neg a by_cases on h1 · -- Case 1. h1: a = ↑k rewrite [h1, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done · -- Case 2. h1: a = -↑k rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done done theorem cc_eq_iff_congr (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ a ≡ b (MOD m) := calc [a]_m = [b]_m _ ↔ [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b _ ↔ ↑m ∣ (a - b) := cc_zero_iff_dvd m (a - b) _ ↔ a ≡ b (MOD m) := by rfl /- End of fundamental properties of congruence classes -/ lemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m := by have h1 : (↑m : Int) ≠ 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m) show 0 ≤ a % m from Int.emod_nonneg a h1 done lemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m) have h2 : (↑m : Int) > 0 := (Nat.cast_pos).rtl h1 show a % m < m from Int.emod_lt_of_pos a h2 done lemma congr_mod_mod (m : Nat) (a : Int) : a ≡ a % m (MOD m) := by define have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m apply Exists.intro (a / m) show a - a % m = m * (a / m) from calc a - (a % m) _ = m * (a / m) + a % m - a % m := by rw [h1] _ = m * (a / m) := by ring done lemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) := And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a)) theorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) : ∃! (r : Int), 0 ≤ r ∧ r < m ∧ a ≡ r (MOD m) := by exists_unique · -- Existence apply Exists.intro (a % m) show 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) from mod_cmpl_res m a done · -- Uniqueness fix r1 : Int; fix r2 : Int assume h1 : 0 ≤ r1 ∧ r1 < m ∧ a ≡ r1 (MOD m) assume h2 : 0 ≤ r2 ∧ r2 < m ∧ a ≡ r2 (MOD m) have h3 : r1 ≡ r2 (MOD m) := congr_trans (congr_symm h1.right.right) h2.right.right obtain (d : Int) (h4 : r1 - r2 = m * d) from h3 have h5 : r1 - r2 < m * 1 := by linarith have h6 : m * (-1) < r1 - r2 := by linarith rewrite [h4] at h5 --h5 : m * d < m * 1 rewrite [h4] at h6 --h6 : m * -1 < m * d have h7 : (↑m : Int) ≥ 0 := Nat.cast_nonneg m have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7 have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7 have h10 : d = 0 := by linarith show r1 = r2 from calc r1 _ = r1 - r2 + r2 := by ring _ = m * 0 + r2 := by rw [h4, h10] _ = r2 := by ring done done lemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m := (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a) theorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y rewrite [h1, h2] have h3 : a + b = b + a := by ring show [a]_m + [b]_m = [b]_m + [a]_m from calc [a]_m + [b]_m _ = [a + b]_m := add_class m a b _ = [b + a]_m := by rw [h3] _ = [b]_m + [a]_m := (add_class m b a).symm done theorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X rewrite [h1] have h2 : a * 1 = a := by ring show [a]_m * [1]_m = [a]_m from calc [a]_m * [1]_m _ = [a * 1]_m := mul_class m a 1 _ = [a]_m := by rw [h2] done theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry lemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) : [a]_m * [gcd_c2 m a]_m = [1]_m := by set s : Int := gcd_c1 m a have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m define at h1 rewrite [h1, Nat.cast_one] at h2 --h2 : s * ↑m + gcd_c2 m a * ↑a = 1 rewrite [mul_class, cc_eq_iff_congr] define --Goal : ∃ (c : Int), ↑a * gcd_c2 m a - 1 = ↑m * c apply Exists.intro (-s) show a * (gcd_c2 m a) - 1 = m * (-s) from calc a * (gcd_c2 m a) - 1 _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring _ = 1 + m * (-s) - 1 := by rw [h2] _ = m * (-s) := by ring done theorem Theorem_7_3_7 (m a : Nat) : invertible [a]_m ↔ rel_prime m a := by apply Iff.intro · -- (→) assume h1 : invertible [a]_m define at h1 obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1 obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y rewrite [h3, mul_class, cc_eq_iff_congr] at h2 define at h2 obtain (c : Int) (h4 : a * b - 1 = m * c) from h2 rewrite [Exercise_7_2_6] --Goal : ∃ (s t : Int), s * ↑m + t * ↑a = 1 apply Exists.intro (-c) apply Exists.intro b show (-c) * m + b * a = 1 from calc (-c) * m + b * a _ = (-c) * m + (a * b - 1) + 1 := by ring _ = (-c) * m + m * c + 1 := by rw [h4] _ = 1 := by ring done · -- (←) assume h1 : rel_prime m a define show ∃ (Y : ZMod m), [a]_m * Y = [1]_m from Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1) done done /- Section 7.4 -/ section Euler open Euler lemma num_rp_below_base {m : Nat} : num_rp_below m 0 = 0 := by rfl lemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) : num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by have h1 : num_rp_below m (j + 1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : gcd m j = 1 rewrite [if_pos h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1 show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1 done lemma num_rp_below_step_not_rp {m j : Nat} (h : ¬rel_prime m j) : num_rp_below m (j + 1) = num_rp_below m j := by have h1 : num_rp_below m (j +1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : ¬gcd m j = 1 rewrite [if_neg h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j show num_rp_below m (j + 1) = num_rp_below m j from h1 done lemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl #eval phi 10 --Answer: 4 lemma prod_inv_iff_inv {m : Nat} {X : ZMod m} (h1 : invertible X) (Y : ZMod m) : invertible (X * Y) ↔ invertible Y := by apply Iff.intro · -- (→) assume h2 : invertible (X * Y) obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2 apply Exists.intro (X * Z) rewrite [←h3] --Goal : Y * (X * Z) = X * Y * Z ring --Note that ring can do algebra in ZMod m done · -- (←) assume h2 : invertible Y obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1 obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2 apply Exists.intro (Xi * Yi) show (X * Y) * (Xi * Yi) = [1]_m from calc X * Y * (Xi * Yi) _ = (X * Xi) * (Y * Yi) := by ring _ = [1]_m * [1]_m := by rw [h3, h4] _ = [1]_m := Theorem_7_3_6_7 [1]_m done done lemma F_rp_def {m i : Nat} (h : rel_prime m i) : F m i = [i]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h --h : gcd m i = 1 rewrite [if_pos h] at h1 show F m i = [i]_m from h1 done lemma F_not_rp_def {m i : Nat} (h : ¬rel_prime m i) : F m i = [1]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h rewrite [h1, if_neg h] rfl done lemma prod_seq_base {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 0 k f = [1]_m := by rfl lemma prod_seq_step {m : Nat} (n k : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl lemma prod_seq_zero_step {m : Nat} (n : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by rewrite [prod_seq_step, zero_add] rfl done lemma prod_one {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 1 k f = f k := by rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7] rfl done lemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl lemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m := calc [G m a i]_m _ = [(a * i) % m]_m := by rfl _ = [a * i]_m := (cc_eq_mod m (a * i)).symm _ = [a]_m * [i]_m := (mul_class m a i).symm lemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) : rel_prime m (G m a i) ↔ rel_prime m i := by have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1 show rel_prime m (G m a i) ↔ rel_prime m i from calc rel_prime m (G m a i) _ ↔ invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm _ ↔ invertible ([a]_m * [i]_m) := by rw [cc_G] _ ↔ invertible [i]_m := prod_inv_iff_inv h2 ([i]_m) _ ↔ rel_prime m i := Theorem_7_3_7 m i done lemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) : F m (G m a i) = [a]_m * F m i := by have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2 show F m (G m a i) = [a]_m * F m i from calc F m (G m a i) _ = [G m a i]_m := F_rp_def h3 _ = [a]_m * [i]_m := cc_G m a i _ = [a]_m * F m i := by rw [F_rp_def h2] done lemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : ¬rel_prime m i) : F m (G m a i) = [1]_m := by rewrite [←G_rp_iff h1 i] at h2 show F m (G m a i) = [1]_m from F_not_rp_def h2 done lemma FG_prod {m a : Nat} (h1 : rel_prime m a) : ∀ (k : Nat), prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by by_induc · -- Base Case show prod_seq 0 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from calc prod_seq 0 0 ((F m) ∘ (G m a)) _ = [1]_m := prod_seq_base _ _ _ = [a]_m ^ 0 * [1]_m := by ring _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by rw [num_rp_below_base, prod_seq_base] done · -- Induction Step fix k : Nat assume ih : prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) by_cases h2 : rel_prime m k · -- Case 1. h2 : rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([a]_m * F m k) := by rw [FG_rp h1 h2] _ = [a]_m ^ ((num_rp_below m k) + 1) * ((prod_seq k 0 (F m)) * F m k) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_rp h2, prod_seq_zero_step] done · -- Case 2. h2 : ¬rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([1]_m) := by rw [FG_not_rp h1 h2] _ = [a]_m ^ (num_rp_below m k) * (prod_seq k 0 (F m) * ([1]_m)) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_not_rp h2, prod_seq_zero_step, F_not_rp_def h2] done done done lemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by define --Goal : ∀ i < m, G m a i < m fix i : Nat assume h1 : i < m rewrite [G_def] --Goal : a * i % m < m show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m) done lemma left_inv_one_one_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g' (g i) = i) : one_one_below n g := sorry lemma right_inv_onto_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g (g' i) = i) (h2 : maps_below n g') : onto_below n g := by define at h2; define fix k : Nat assume h3 : k < n apply Exists.intro (g' k) show g' k < n ∧ g (g' k) = k from And.intro (h2 k h3) (h1 k h3) done lemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m * [inv_mod m a]_m = [1]_m := by have h2 : 0 ≤ (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a) show [a]_m * [inv_mod m a]_m = [1]_m from calc [a]_m * [inv_mod m a]_m _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2] _ = [a]_m * [gcd_c2 m a]_m := by rw [←cc_eq_mod] _ = [1]_m := gcd_c2_inv h1 done lemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m := calc a * (b % m) % m = a % m * (b % m % m) % m := Nat.mul_mod _ _ _ _ = a % m * (b % m) % m := by rw [Nat.mod_mod] _ = a * b % m := (Nat.mul_mod _ _ _).symm lemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm] rfl done theorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] : ↑a ≡ ↑b (MOD m) ↔ a % m = b % m := sorry lemma mul_inv_mod_cancel {m a i : Nat} [NeZero m] (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1 rewrite [mul_class, cc_eq_iff_congr, ←Nat.cast_mul, ←Nat.cast_one, congr_iff_mod_eq_Nat] at h3 show a * inv_mod m a * i % m = i from calc a * (inv_mod m a) * i % m _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = 1 % m * i % m := by rw [h3] _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = i % m := by rw [one_mul] _ = i := Nat.mod_eq_of_lt h2 done lemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl lemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, G m a (Ginv m a i) = i := by fix i : Nat assume h2 : i < m show G m a (Ginv m a i) = i from calc G m a (Ginv m a i) _ = a * ((inv_mod m a * i) % m) % m := by rfl _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, Ginv m a (G m a i) = i := by fix i : Nat assume h2 : i < m show Ginv m a (G m a i) = i from calc Ginv m a (G m a i) _ = inv_mod m a * ((a * i) % m) % m := by rfl _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc, mul_comm (inv_mod m a)] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_maps_below (m a : Nat) [NeZero m] : maps_below m (Ginv m a) := G_maps_below m (inv_mod m a) lemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : one_one_below m (G m a) := left_inv_one_one_below (Ginv_left_inv h1) lemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : onto_below m (G m a) := right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a) lemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : perm_below m (G m a) := And.intro (G_maps_below m a) (And.intro (G_one_one_below h1) (G_onto_below h1)) --Permuting a product of congruence classes doesn't change product lemma swap_fst (u v : Nat) : swap u v u = v := by define : swap u v u --Goal : (if u = u then v else if u = v then u else u) = v have h : u = u := by rfl rewrite [if_pos h] rfl done lemma swap_snd (u v : Nat) : swap u v v = u := by define : swap u v v by_cases h1 : v = u · -- Case 1. h1 : v = u rewrite [if_pos h1] show v = u from h1 done · -- Case 2. h1 : v ≠ u rewrite [if_neg h1] have h2 : v = v := by rfl rewrite [if_pos h2] rfl done done lemma swap_other {u v i : Nat} (h1 : i ≠ u) (h2 : i ≠ v) : swap u v i = i := by define : swap u v i rewrite [if_neg h1, if_neg h2] rfl done lemma swap_values (u v i : Nat) : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := by by_cases h1 : i = u · -- Case 1. h1 : i = u apply Or.inl rewrite [h1] show swap u v u = v from swap_fst u v done · -- Case 2. h1 : i ≠ u apply Or.inr by_cases h2 : i = v · -- Case 2.1. h2 : i = v apply Or.inl rewrite [h2] show swap u v v = u from swap_snd u v done · -- Case 2.2. h2 : i ≠ v apply Or.inr show swap u v i = i from swap_other h1 h2 done done done lemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by define fix i : Nat assume h3 : i < n have h4 : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := swap_values u v i by_cases on h4 · -- Case 1. h4 : swap u v i = v rewrite [h4] show v < n from h2 done · -- Case 2. by_cases on h4 · -- Case 2.1. h4 : swap u v i = u rewrite [h4] show u < n from h1 done · -- Case 2.2. h4 : swap u v i = i rewrite [h4] show i < n from h3 done done done lemma swap_swap (u v n : Nat) : ∀ i < n, swap u v (swap u v i) = i := by fix i : Nat assume h : i < n by_cases h1 : i = u · -- Case 1. h1 : i = u rewrite [h1, swap_fst, swap_snd] rfl done · -- Case 2. h1 : i ≠ u by_cases h2 : i = v · -- Case 2.1. h2 : i = v rewrite [h2, swap_snd, swap_fst] rfl done · -- Case 2.2. h2 : i ≠ v rewrite [swap_other h1 h2, swap_other h1 h2] rfl done done done lemma swap_one_one_below (u v n) : one_one_below n (swap u v) := left_inv_one_one_below (swap_swap u v n) lemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) := right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2) lemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) := And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2)) lemma comp_perm_below {n : Nat} {f g : Nat → Nat} (h1 : perm_below n f) (h2 : perm_below n g) : perm_below n (f ∘ g) := sorry lemma trivial_swap (u : Nat) : swap u u = id := by apply funext fix x : Nat by_cases h1 : x = u · -- Case 1. h1 : x = u rewrite [h1, swap_fst] rfl done · -- Case 2. h1 : x ≠ u rewrite [swap_other h1 h1] rfl done done lemma prod_eq_fun {m : Nat} (f g : Nat → ZMod m) (k : Nat) : ∀ (n : Nat), (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g := by by_induc · -- Base Case assume h : (∀ i < 0, f (k + i) = g (k + i)) rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g assume h1 : ∀ i < n + 1, f (k + i) = g (k + i) have h2 : ∀ i < n, f (k + i) = g (k + i) := by fix i : Nat assume h2 : i < n have h3 : i < n + 1 := by linarith show f (k + i) = g (k + i) from h1 i h3 done have h3 : prod_seq n k f = prod_seq n k g := ih h2 have h4 : n < n + 1 := Nat.lt_succ_self n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4] rfl done done lemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := by have h2 : ∀ (i : Nat), i < u → (f ∘ swap u n) (0 + i) = f (0 + i) := by fix i : Nat assume h2 : i < u have h3 : 0 + i ≠ u := by linarith have h4 : 0 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl done show prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f from prod_eq_fun (f ∘ swap u n) f 0 u h2 done lemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := by have h2 : ∀ i < j, (f ∘ swap u n) (u + 1 + i) = f (u + 1 + i) := by fix i : Nat assume h2 : i < j have h3 : u + 1 + i ≠ u := by linarith have h4 : u + 1 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl show prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f from prod_eq_fun (f ∘ swap u n) f (u + 1) j h2 done lemma break_prod {m : Nat} (n : Nat) (f : Nat → ZMod m) : ∀ (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by by_induc · -- Base Case have h : n + 0 = n := by rfl rewrite [prod_seq_base, h, Theorem_7_3_6_7] rfl done · -- Induction Step fix j : Nat assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f rewrite [←add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc] rfl done done lemma break_prod_twice {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f := break_prod n f 1 rewrite [prod_one] at h2 have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f := break_prod (u + 1) f j rewrite [←h1] at h3 have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f := break_prod u f 1 rewrite [prod_one] at h4 rewrite [h3, h4] at h2 show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2 done lemma swap_prod_eq_prod {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f := by by_cases h2 : u = n · -- Case 1. h2 : u = n rewrite [h2, trivial_swap n] --Goal : prod_seq (n + 1) 0 (f ∘ id) = prod_seq (n + 1) 0 f rfl done · -- Case 2. h2 : ¬u = n have h3 : u + 1 ≤ n := Nat.lt_of_le_of_ne h1 h2 obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3 have break_f : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := break_prod_twice f h4 have break_fs : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_prod_twice (f ∘ swap u n) h4 have f_eq_fs_below : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := swap_prod_eq_prod_below f h1 have f_eq_fs_btwn : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4 show prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f from calc prod_seq (n + 1) 0 (f ∘ swap u n) _ = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_fs _ = prod_seq u 0 f * (f ∘ swap u n) u * prod_seq j (u + 1) f * (f ∘ swap u n) n := by rw [f_eq_fs_below, f_eq_fs_btwn] _ = prod_seq u 0 f * f (swap u n u) * prod_seq j (u + 1) f * f (swap u n n) := by rfl _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by rw [swap_fst, swap_snd] _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring _ = prod_seq (n + 1) 0 f := break_f.symm done done lemma perm_below_fixed {n : Nat} {g : Nat → Nat} (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry lemma perm_prod {m : Nat} (f : Nat → ZMod m) : ∀ (n : Nat), ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) := by by_induc · -- Base Case fix g : Nat → Nat assume h1 : perm_below 0 g rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) fix g : Nat → Nat assume g_pb : perm_below (n + 1) g define at g_pb have g_ob : onto_below (n + 1) g := g_pb.right.right define at g_ob have h1 : n < n + 1 := by linarith obtain (u : Nat) (h2 : u < n + 1 ∧ g u = n) from g_ob n h1 have s_pb : perm_below (n + 1) (swap u n) := swap_perm_below h2.left h1 have gs_pb_n1 : perm_below (n + 1) (g ∘ swap u n) := comp_perm_below g_pb s_pb have gs_fix_n : (g ∘ swap u n) n = n := calc (g ∘ swap u n) n _ = g (swap u n n) := by rfl _ = g u := by rw [swap_snd] _ = n := h2.right have gs_pb_n : perm_below n (g ∘ swap u n) := perm_below_fixed gs_pb_n1 gs_fix_n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f ∘ (g ∘ swap u n)) := ih (g ∘ swap u n) gs_pb_n have h3 : u ≤ n := by linarith show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f ∘ g) from calc prod_seq (n + 1) 0 f _ = prod_seq n 0 f * f n := prod_seq_zero_step n f _ = prod_seq n 0 (f ∘ (g ∘ swap u n)) * f ((g ∘ swap u n) n) := by rw [gs_prod, gs_fix_n] _ = prod_seq n 0 (f ∘ g ∘ swap u n) * (f ∘ g ∘ swap u n) n := by rfl _ = prod_seq (n + 1) 0 (f ∘ g ∘ swap u n) := (prod_seq_zero_step n (f ∘ g ∘ swap u n)).symm _ = prod_seq (n + 1) 0 ((f ∘ g) ∘ swap u n) := by rfl _ = prod_seq (n + 1) 0 (f ∘ g) := swap_prod_eq_prod (f ∘ g) h3 done done lemma F_invertible (m i : Nat) : invertible (F m i) := by by_cases h : rel_prime m i · -- Case 1. h : rel_prime m i rewrite [F_rp_def h] show invertible [i]_m from (Theorem_7_3_7 m i).rtl h done · -- Case 2. h : ¬rel_prime m i rewrite [F_not_rp_def h] apply Exists.intro [1]_m show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m done done lemma Fprod_invertible (m : Nat) : ∀ (k : Nat), invertible (prod_seq k 0 (F m)) := by by_induc · -- Base Case apply Exists.intro [1]_m show prod_seq 0 0 (F m) * [1]_m = [1]_m from calc prod_seq 0 0 (F m) * [1]_m _ = [1]_m * [1]_m := by rw [prod_seq_base] _ = [1]_m := Theorem_7_3_6_7 ([1]_m) done · -- Induction Step fix k : Nat assume ih : invertible (prod_seq k 0 (F m)) rewrite [prod_seq_zero_step] show invertible (prod_seq k 0 (F m) * (F m k)) from (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k) done done theorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m ^ (phi m) = [1]_m := by have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2 show [a]_m ^ (phi m) = [1]_m from calc [a]_m ^ (phi m) _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3] _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring _ = prod_seq m 0 (F m ∘ G m a) * Y := by rw [FG_prod h1 m, phi_def] _ = prod_seq m 0 (F m) * Y := by rw [perm_prod (F m) m (G m a) (G_perm_below h1)] _ = [1]_m := by rw [h3] done lemma Exercise_7_4_5_Int (m : Nat) (a : Int) : ∀ (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry lemma Exercise_7_4_5_Nat (m a n : Nat) : [a]_m ^ n = [a ^ n]_m := by rewrite [Exercise_7_4_5_Int] rfl done theorem Euler's_theorem {m a : Nat} [NeZero m] (h1 : rel_prime m a) : a ^ (phi m) ≡ 1 (MOD m) := by have h2 : [a]_m ^ (phi m) = [1]_m := Theorem_7_4_2 h1 rewrite [Exercise_7_4_5_Nat m a (phi m)] at h2 --h2 : [a ^ phi m]_m = [1]_m show a ^ (phi m) ≡ 1 (MOD m) from (cc_eq_iff_congr _ _ _).ltr h2 done #eval gcd 10 7 --Answer: 1. So 10 and 7 are relatively prime #eval 7 ^ phi 10 --Answer: 2401, which is congruent to 1 mod 10. end Euler /- Section 7.5 -/ lemma num_rp_prime {p : Nat} (h1 : prime p) : ∀ k < p, num_rp_below p (k + 1) = k := sorry lemma phi_prime {p : Nat} (h1 : prime p) : phi p = p - 1 := by have h2 : 1 ≤ p := prime_pos h1 have h3 : p - 1 + 1 = p := Nat.sub_add_cancel h2 have h4 : p - 1 < p := by linarith have h5 : num_rp_below p (p - 1 + 1) = p - 1 := num_rp_prime h1 (p - 1) h4 rewrite [h3] at h5 show phi p = p - 1 from h5 done theorem Theorem_7_2_2_Int {a c : Nat} {b : Int} (h1 : ↑c ∣ ↑a * b) (h2 : rel_prime a c) : ↑c ∣ b := by rewrite [Int.natCast_dvd, Int.natAbs_mul, Int.natAbs_ofNat] at h1 --h1 : c ∣ a * Int.natAbs b rewrite [Int.natCast_dvd] --Goal : c ∣ Int.natAbs b show c ∣ Int.natAbs b from Theorem_7_2_2 h1 h2 done lemma Lemma_7_4_5 {m n : Nat} (a b : Int) (h1 : rel_prime m n) : a ≡ b (MOD m * n) ↔ a ≡ b (MOD m) ∧ a ≡ b (MOD n) := by apply Iff.intro · -- (→) assume h2 : a ≡ b (MOD m * n) obtain (j : Int) (h3 : a - b = (m * n) * j) from h2 apply And.intro · -- Proof of a ≡ b (MOD m) apply Exists.intro (n * j) show a - b = m * (n * j) from calc a - b _ = m * n * j := h3 _ = m * (n * j) := by ring done · -- Proof of a ≡ b (MOD n) apply Exists.intro (m * j) show a - b = n * (m * j) from calc a - b _ = m * n * j := h3 _ = n * (m * j) := by ring done done · -- (←) assume h2 : a ≡ b (MOD m) ∧ a ≡ b (MOD n) obtain (j : Int) (h3 : a - b = m * j) from h2.left have h4 : (↑n : Int) ∣ a - b := h2.right rewrite [h3] at h4 --h4 : ↑n ∣ ↑m * j have h5 : ↑n ∣ j := Theorem_7_2_2_Int h4 h1 obtain (k : Int) (h6 : j = n * k) from h5 apply Exists.intro k --Goal : a - b = ↑(m * n) * k rewrite [Nat.cast_mul] --Goal : a - b = ↑m * ↑n * k show a - b = (m * n) * k from calc a - b _ = m * j := h3 _ = m * (n * k) := by rw [h6] _ = (m * n) * k := by ring done done --From exercises of Section 7.2 theorem rel_prime_symm {a b : Nat} (h : rel_prime a b) : rel_prime b a := sorry lemma prime_NeZero {p : Nat} (h : prime p) : NeZero p := by rewrite [neZero_iff] --Goal : p ≠ 0 define at h linarith done lemma Lemma_7_5_1 {p e d m c s : Nat} {t : Int} (h1 : prime p) (h2 : e * d = (p - 1) * s + 1) (h3 : m ^ e - c = p * t) : c ^ d ≡ m (MOD p) := by have h4 : m ^ e ≡ c (MOD p) := Exists.intro t h3 have h5 : [m ^ e]_p = [c]_p := (cc_eq_iff_congr _ _ _).rtl h4 rewrite [←Exercise_7_4_5_Nat] at h5 --h5 : [m]_p ^ e = [c]_p by_cases h6 : p ∣ m · -- Case 1. h6 : p ∣ m have h7 : m ≡ 0 (MOD p) := by obtain (j : Nat) (h8 : m = p * j) from h6 apply Exists.intro (↑j : Int) --Goal : ↑m - 0 = ↑p * ↑j rewrite [h8, Nat.cast_mul] ring done have h8 : [m]_p = [0]_p := (cc_eq_iff_congr _ _ _).rtl h7 have h9 : e * d ≠ 0 := by rewrite [h2] show (p - 1) * s + 1 ≠ 0 from Nat.add_one_ne_zero _ done have h10 : (0 : Int) ^ (e * d) = 0 := zero_pow h9 have h11 : [c ^ d]_p = [m]_p := calc [c ^ d]_p _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat] _ = ([m]_p ^ e) ^ d := by rw [h5] _ = [m]_p ^ (e * d) := by ring _ = [0]_p ^ (e * d) := by rw [h8] _ = [0 ^ (e * d)]_p := Exercise_7_4_5_Int _ _ _ _ = [0]_p := by rw [h10] _ = [m]_p := by rw [h8] show c ^ d ≡ m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11 done · -- Case 2. h6 : ¬p ∣ m have h7 : rel_prime m p := rel_prime_of_prime_not_dvd h1 h6 have h8 : rel_prime p m := rel_prime_symm h7 have h9 : NeZero p := prime_NeZero h1 have h10 : (1 : Int) ^ s = 1 := by ring have h11 : [c ^ d]_p = [m]_p := calc [c ^ d]_p _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat] _ = ([m]_p ^ e) ^ d := by rw [h5] _ = [m]_p ^ (e * d) := by ring _ = [m]_p ^ ((p - 1) * s + 1) := by rw [h2] _ = ([m]_p ^ (p - 1)) ^ s * [m]_p := by ring _ = ([m]_p ^ (phi p)) ^ s * [m]_p := by rw [phi_prime h1] _ = [1]_p ^ s * [m]_p := by rw [Theorem_7_4_2 h8] _ = [1 ^ s]_p * [m]_p := by rw [Exercise_7_4_5_Int] _ = [1]_p * [m]_p := by rw [h10] _ = [m]_p * [1]_p := by ring _ = [m]_p := Theorem_7_3_6_7 _ show c ^ d ≡ m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11 done done theorem Theorem_7_5_1 (p q n e d k m c : Nat) (p_prime : prime p) (q_prime : prime q) (p_ne_q : p ≠ q) (n_pq : n = p * q) (ed_congr_1 : e * d = k * (p - 1) * (q - 1) + 1) (h1 : [m]_n ^ e = [c]_n) : [c]_n ^ d = [m]_n := by rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] at h1 --h1 : m ^ e ≡ c (MOD n) rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] --Goal : c ^ d ≡ m (MOD n) obtain (j : Int) (h2 : m ^ e - c = n * j) from h1 rewrite [n_pq, Nat.cast_mul] at h2 --h2 : m ^ e - c = p * q * j have h3 : e * d = (p - 1) * (k * (q - 1)) + 1 := by rewrite [ed_congr_1] ring done have h4 : m ^ e - c = p * (q * j) := by rewrite [h2] ring done have congr_p : c ^ d ≡ m (MOD p) := Lemma_7_5_1 p_prime h3 h4 have h5 : e * d = (q - 1) * (k * (p - 1)) + 1 := by rewrite [ed_congr_1] ring done have h6 : m ^ e - c = q * (p * j) := by rewrite [h2] ring done have congr_q : c ^ d ≡ m (MOD q) := Lemma_7_5_1 q_prime h5 h6 have h7 : ¬q ∣ p := by by_contra h8 have h9 : q = 1 ∨ q = p := dvd_prime p_prime h8 disj_syll h9 (prime_not_one q_prime) show False from p_ne_q h9.symm done have h8 : rel_prime p q := rel_prime_of_prime_not_dvd q_prime h7 rewrite [n_pq, Lemma_7_4_5 _ _ h8] show c ^ d ≡ m (MOD p) ∧ c ^ d ≡ m (MOD q) from And.intro congr_p congr_q done /- BEGIN EXERCISES -/ namespace Exercises /- Section 7.1 -/ -- 1. theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry -- 2. lemma gcd_comm_lt {a b : Nat} (h : a < b) : gcd a b = gcd b a := sorry theorem gcd_comm (a b : Nat) : gcd a b = gcd b a := sorry -- 3. theorem Exercise_7_1_5 (a b : Nat) (n : Int) : (∃ (s t : Int), s * a + t * b = n) ↔ (↑(gcd a b) : Int) ∣ n := sorry -- 4. theorem Exercise_7_1_6 (a b c : Nat) : gcd a b = gcd (a + b * c) b := sorry -- 5. theorem gcd_is_nonzero {a b : Nat} (h : a ≠ 0 ∨ b ≠ 0) : gcd a b ≠ 0 := sorry -- 6. theorem gcd_greatest {a b d : Nat} (h1 : gcd a b ≠ 0) (h2 : d ∣ a) (h3 : d ∣ b) : d ≤ gcd a b := sorry -- 7. lemma Lemma_7_1_10a {a b : Nat} (n : Nat) (h : a ∣ b) : (n * a) ∣ (n * b) := sorry lemma Lemma_7_1_10b {a b n : Nat} (h1 : n ≠ 0) (h2 : (n * a) ∣ (n * b)) : a ∣ b := sorry lemma Lemma_7_1_10c {a b : Nat} (h1 : a ∣ b) (h2 : b ∣ a) : a = b := sorry theorem Exercise_7_1_10 (a b n : Nat) : gcd (n * a) (n * b) = n * gcd a b := sorry /- Section 7.2 -/ -- 1. lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry -- 2. -- Hints: Start with apply List.rec. You may find mul_ne_zero useful theorem prod_nonzero_nonzero : ∀ (l : List Nat), (∀ a ∈ l, a ≠ 0) → prod l ≠ 0 := sorry -- 3. theorem rel_prime_iff_no_common_factor (a b : Nat) : rel_prime a b ↔ ¬∃ (p : Nat), prime p ∧ p ∣ a ∧ p ∣ b := sorry -- 4. theorem rel_prime_symm {a b : Nat} (h : rel_prime a b) : rel_prime b a := sorry -- 5. lemma in_prime_factorization_iff_prime_factor {a : Nat} {l : List Nat} (h1 : prime_factorization a l) (p : Nat) : p ∈ l ↔ prime_factor p a := sorry -- 6. theorem Exercise_7_2_5 {a b : Nat} {l m : List Nat} (h1 : prime_factorization a l) (h2 : prime_factorization b m) : rel_prime a b ↔ (¬∃ (p : Nat), p ∈ l ∧ p ∈ m) := sorry -- 7. theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry -- 8. theorem Exercise_7_2_7 {a b a' b' : Nat} (h1 : rel_prime a b) (h2 : a' ∣ a) (h3 : b' ∣ b) : rel_prime a' b' := sorry -- 9. theorem Exercise_7_2_9 {a b j k : Nat} (h1 : gcd a b ≠ 0) (h2 : a = j * gcd a b) (h3 : b = k * gcd a b) : rel_prime j k := sorry -- 10. theorem Exercise_7_2_17a (a b c : Nat) : gcd a (b * c) ∣ gcd a b * gcd a c := sorry /- Section 7.3 -/ -- 1. theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry -- 2. theorem Theorem_7_3_6_3 {m : Nat} (X : ZMod m) : X + [0]_m = X := sorry -- 3. theorem Theorem_7_3_6_4 {m : Nat} (X : ZMod m) : ∃ (Y : ZMod m), X + Y = [0]_m := sorry -- 4.

theorem Exercise_7_3_4a {m : Nat} (Z1 Z2 : ZMod m) (h1 : ∀ (X : ZMod m), X + Z1 = X) (h2 : ∀ (X : ZMod m), X + Z2 = X) : Z1 = Z2

HTPI.Exercises.Exercise_7_3_4a

htpi/HTPILib/Chap7.lean

htpi

/- Copyright 2023 Daniel J. Velleman -/ import HTPILib.Chap6 namespace HTPI /- Definitions -/ lemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by have h : n + 1 > 0 := Nat.succ_pos n show a % (n + 1) < n + 1 from Nat.mod_lt a h done def gcd (a b : Nat) : Nat := match b with | 0 => a | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd (n + 1) (a % (n + 1)) termination_by b mutual def gcd_c1 (a b : Nat) : Int := match b with | 0 => 1 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c2 (n + 1) (a % (n + 1)) --Corresponds to s = t' termination_by b def gcd_c2 (a b : Nat) : Int := match b with | 0 => 0 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c1 (n + 1) (a % (n + 1)) - (gcd_c2 (n + 1) (a % (n + 1))) * ↑(a / (n + 1)) --Corresponds to t = s' - t'q termination_by b end def prime (n : Nat) : Prop := 2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n def prime_factor (p n : Nat) : Prop := prime p ∧ p ∣ n def all_prime (l : List Nat) : Prop := ∀ p ∈ l, prime p def nondec (l : List Nat) : Prop := match l with | [] => True --Of course, True is a proposition that is always true | n :: L => (∀ m ∈ L, n ≤ m) ∧ nondec L def nondec_prime_list (l : List Nat) : Prop := all_prime l ∧ nondec l def prod (l : List Nat) : Nat := match l with | [] => 1 | n :: L => n * (prod L) def prime_factorization (n : Nat) (l : List Nat) : Prop := nondec_prime_list l ∧ prod l = n def rel_prime (a b : Nat) : Prop := gcd a b = 1 def congr_mod (m : Nat) (a b : Int) : Prop := (↑m : Int) ∣ (a - b) def cc (m : Nat) (a : Int) : ZMod m := (↑a : ZMod m) notation:50 a " ≡ " b " (MOD " m ")" => congr_mod m a b notation:max "["a"]_"m:max => cc m a def invertible {m : Nat} (X : ZMod m) : Prop := ∃ (Y : ZMod m), X * Y = [1]_m def num_rp_below (m k : Nat) : Nat := match k with | 0 => 0 | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j def phi (m : Nat) : Nat := num_rp_below m m def prod_seq {m : Nat} (j k : Nat) (f : Nat → ZMod m) : ZMod m := match j with | 0 => [1]_m | n + 1 => prod_seq n k f * f (k + n) def maps_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i < n, g i < n def one_one_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i1 < n, ∀ i2 < n, g i1 = g i2 → i1 = i2 def onto_below (n : Nat) (g : Nat → Nat) : Prop := ∀ k < n, ∃ i < n, g i = k def perm_below (n : Nat) (g : Nat → Nat) : Prop := maps_below n g ∧ one_one_below n g ∧ onto_below n g def inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m) def swap (u v i : Nat) : Nat := if i = u then v else if i = v then u else i namespace Euler --For definitions specific to Euler's theorem def F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m def G (m a i : Nat) : Nat := (a * i) % m def Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i end Euler /- Section 7.1 -/ theorem dvd_mod_of_dvd_a_b {a b d : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ (a % b) := by set q : Nat := a / b have h3 : b * q + a % b = a := Nat.div_add_mod a b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 define --Goal : ∃ (c : Nat), a % b = d * c apply Exists.intro (j - k * q) show a % b = d * (j - k * q) from calc a % b _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm _ = a - b * q := by rw [h3] _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc] _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm done theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry #eval gcd 672 161 --Answer: 7 lemma gcd_base (a : Nat) : gcd a 0 = a := by rfl lemma gcd_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd a b = gcd b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1)) rfl done lemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b ≠ 0) : a % b < b := by have h1 : b > 0 := Nat.pos_of_ne_zero h show a % b < b from Nat.mod_lt a h1 done lemma dvd_self (n : Nat) : n ∣ n := by apply Exists.intro 1 ring done theorem gcd_dvd : ∀ (b a : Nat), (gcd a b) ∣ a ∧ (gcd a b) ∣ b := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd a b_1) ∣ a ∧ (gcd a b_1) ∣ b_1 fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_base] --Goal: a ∣ a ∧ a ∣ 0 apply And.intro (dvd_self a) define apply Exists.intro 0 rfl done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_nonzero a h1] --Goal : gcd b (a % b) ∣ a ∧ gcd b (a % b) ∣ b have h2 : a % b < b := mod_nonzero_lt a h1 have h3 : (gcd b (a % b)) ∣ b ∧ (gcd b (a % b)) ∣ (a % b) := ih (a % b) h2 b apply And.intro _ h3.left show (gcd b (a % b)) ∣ a from dvd_a_of_dvd_b_mod h3.left h3.right done done theorem gcd_dvd_left (a b : Nat) : (gcd a b) ∣ a := (gcd_dvd b a).left theorem gcd_dvd_right (a b : Nat) : (gcd a b) ∣ b := (gcd_dvd b a).right lemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl lemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c1 a b = gcd_c2 b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done lemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl lemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * ↑(a / b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done theorem gcd_lin_comb : ∀ (b a : Nat), (gcd_c1 a b) * ↑a + (gcd_c2 a b) * ↑b = ↑(gcd a b) := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd_c1 a b_1) * ↑a + (gcd_c2 a b_1) * ↑b_1 = ↑(gcd a b_1) fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base] --Goal : 1 * ↑a + 0 * ↑0 = ↑a ring done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1] --Goal : gcd_c2 b (a % b) * ↑a + -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * ↑(a / b)) * ↑b = -- ↑(gcd b (a % b)) set r : Nat := a % b set q : Nat := a / b set s : Int := gcd_c1 b r set t : Int := gcd_c2 b r --Goal : t * ↑a + (s - t * ↑q) * ↑b = ↑(gcd b r) have h2 : r < b := mod_nonzero_lt a h1 have h3 : s * ↑b + t * ↑r = ↑(gcd b r) := ih r h2 b have h4 : b * q + r = a := Nat.div_add_mod a b rewrite [←h3, ←h4] rewrite [Nat.cast_add, Nat.cast_mul] --Goal : t * (↑b * ↑q + ↑r) + (s - t * ↑q) * ↑b = s * ↑b + t * ↑r ring done done #eval gcd_c1 672 161 --Answer: 6 #eval gcd_c2 672 161 --Answer: -25 --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161 theorem Theorem_7_1_6 {d a b : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ gcd a b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑d ∣ ↑(gcd a b) set s : Int := gcd_c1 a b set t : Int := gcd_c2 a b have h3 : s * ↑a + t * ↑b = ↑(gcd a b) := gcd_lin_comb b a rewrite [←h3] --Goal : ↑d ∣ s * ↑a + t * ↑b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul] --Goal : ↑d ∣ s * (↑d * ↑j) + t * (↑d * ↑k) define apply Exists.intro (s * ↑j + t * ↑k) ring done /- Section 7.2 -/ theorem dvd_trans {a b c : Nat} (h1 : a ∣ b) (h2 : b ∣ c) : a ∣ c := by define at h1; define at h2; define obtain (m : Nat) (h3 : b = a * m) from h1 obtain (n : Nat) (h4 : c = b * n) from h2 rewrite [h3, mul_assoc] at h4 apply Exists.intro (m * n) show c = a * (m * n) from h4 done lemma exists_prime_factor : ∀ (n : Nat), 2 ≤ n → ∃ (p : Nat), prime_factor p n := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, 2 ≤ n_1 → ∃ (p : Nat), prime_factor p n_1 assume h1 : 2 ≤ n by_cases h2 : prime n · -- Case 1. h2 : prime n apply Exists.intro n define --Goal : prime n ∧ n ∣ n show prime n ∧ n ∣ n from And.intro h2 (dvd_self n) done · -- Case 2. h2 : ¬prime n define at h2 --h2 : ¬(2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n) demorgan at h2 disj_syll h2 h1 obtain (a : Nat) (h3 : ∃ (b : Nat), a * b = n ∧ a < n ∧ b < n) from h2 obtain (b : Nat) (h4 : a * b = n ∧ a < n ∧ b < n) from h3 have h5 : 2 ≤ a := by by_contra h6 have h7 : a ≤ 1 := by linarith have h8 : n ≤ b := calc n _ = a * b := h4.left.symm _ ≤ 1 * b := by rel [h7] _ = b := by ring linarith --n ≤ b contradicts b < n done have h6 : ∃ (p : Nat), prime_factor p a := ih a h4.right.left h5 obtain (p : Nat) (h7 : prime_factor p a) from h6 apply Exists.intro p define --Goal : prime p ∧ p ∣ n define at h7 --h7 : prime p ∧ p ∣ a apply And.intro h7.left have h8 : a ∣ n := by apply Exists.intro b show n = a * b from (h4.left).symm done show p ∣ n from dvd_trans h7.right h8 done done lemma exists_least_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q := by set S : Set Nat := {p : Nat | prime_factor p n} have h2 : ∃ (p : Nat), p ∈ S := exists_prime_factor n h show ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q from well_ord_princ S h2 done lemma all_prime_nil : all_prime [] := by define --Goal : ∀ p ∈ [], prime p fix p : Nat contrapos --Goal : ¬prime p → p ∉ [] assume h1 : ¬prime p show p ∉ [] from List.not_mem_nil p done lemma all_prime_cons (n : Nat) (L : List Nat) : all_prime (n :: L) ↔ prime n ∧ all_prime L := by apply Iff.intro · -- (→) assume h1 : all_prime (n :: L) --Goal : prime n ∧ all_prime L define at h1 --h1 : ∀ p ∈ n :: L, prime p apply And.intro (h1 n (List.mem_cons_self n L)) define --Goal : ∀ p ∈ L, prime p fix p : Nat assume h2 : p ∈ L show prime p from h1 p (List.mem_cons_of_mem n h2) done · -- (←) assume h1 : prime n ∧ all_prime L --Goal : all_prime (n :: l) define : all_prime L at h1 define fix p : Nat assume h2 : p ∈ n :: L rewrite [List.mem_cons] at h2 --h2 : p = n ∨ p ∈ L by_cases on h2 · -- Case 1. h2 : p = n rewrite [h2] show prime n from h1.left done · -- Case 2. h2 : p ∈ L show prime p from h1.right p h2 done done done lemma nondec_nil : nondec [] := by define --Goal : True trivial --trivial proves some obviously true statements, such as True done lemma nondec_cons (n : Nat) (L : List Nat) : nondec (n :: L) ↔ (∀ m ∈ L, n ≤ m) ∧ nondec L := by rfl lemma prod_nil : prod [] = 1 := by rfl lemma prod_cons : prod (n :: L) = n * (prod L) := by rfl lemma exists_cons_of_length_eq_succ {A : Type} {l : List A} {n : Nat} (h : l.length = n + 1) : ∃ (a : A) (L : List A), l = a :: L ∧ L.length = n := by have h1 : ¬l.length = 0 := by linarith rewrite [List.length_eq_zero] at h1 obtain (a : A) (h2 : ∃ (L : List A), l = a :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List A) (h3 : l = a :: L) from h2 apply Exists.intro a apply Exists.intro L apply And.intro h3 have h4 : (a :: L).length = L.length + 1 := List.length_cons a L rewrite [←h3, h] at h4 show L.length = n from (Nat.add_right_cancel h4).symm done lemma list_elt_dvd_prod_by_length (a : Nat) : ∀ (n : Nat), ∀ (l : List Nat), l.length = n → a ∈ l → a ∣ prod l := by by_induc · --Base Case fix l : List Nat assume h1 : l.length = 0 rewrite [List.length_eq_zero] at h1 --h1 : l = [] rewrite [h1] --Goal : a ∈ [] → a ∣ prod [] contrapos assume h2 : ¬a ∣ prod [] show a ∉ [] from List.not_mem_nil a done · -- Induction Step fix n : Nat assume ih : ∀ (l : List Nat), List.length l = n → a ∈ l → a ∣ prod l fix l : List Nat assume h1 : l.length = n + 1 --Goal : a ∈ l → a ∣ prod l obtain (b : Nat) (h2 : ∃ (L : List Nat), l = b :: L ∧ L.length = n) from exists_cons_of_length_eq_succ h1 obtain (L : List Nat) (h3 : l = b :: L ∧ L.length = n) from h2 have h4 : a ∈ L → a ∣ prod L := ih L h3.right assume h5 : a ∈ l rewrite [h3.left, prod_cons] --Goal : a ∣ b * prod L rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b ∨ a ∈ L by_cases on h5 · -- Case 1. h5 : a = b apply Exists.intro (prod L) rewrite [h5] rfl done · -- Case 2. h5 : a ∈ L have h6 : a ∣ prod L := h4 h5 have h7 : prod L ∣ b * prod L := by apply Exists.intro b ring done show a ∣ b * prod L from dvd_trans h6 h7 done done done lemma list_elt_dvd_prod {a : Nat} {l : List Nat} (h : a ∈ l) : a ∣ prod l := by set n : Nat := l.length have h1 : l.length = n := by rfl show a ∣ prod l from list_elt_dvd_prod_by_length a n l h1 h done lemma exists_prime_factorization : ∀ (n : Nat), n ≥ 1 → ∃ (l : List Nat), prime_factorization n l := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, n_1 ≥ 1 → ∃ (l : List Nat), prime_factorization n_1 l assume h1 : n ≥ 1 by_cases h2 : n = 1 · -- Case 1. h2 : n = 1 apply Exists.intro [] define apply And.intro · -- Proof of nondec_prime_list [] define show all_prime [] ∧ nondec [] from And.intro all_prime_nil nondec_nil done · -- Proof of prod [] = n rewrite [prod_nil, h2] rfl done done · -- Case 2. h2 : n ≠ 1 have h3 : n ≥ 2 := lt_of_le_of_ne' h1 h2 obtain (p : Nat) (h4 : prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q) from exists_least_prime_factor h3 have p_prime_factor : prime_factor p n := h4.left define at p_prime_factor have p_prime : prime p := p_prime_factor.left have p_dvd_n : p ∣ n := p_prime_factor.right have p_least : ∀ (q : Nat), prime_factor q n → p ≤ q := h4.right obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n have h5 : m ≠ 0 := by contradict h1 with h6 have h7 : n = 0 := calc n _ = p * m := n_eq_pm _ = p * 0 := by rw [h6] _ = 0 := by ring rewrite [h7] decide done have m_pos : 0 < m := Nat.pos_of_ne_zero h5 have m_lt_n : m < n := by define at p_prime show m < n from calc m _ < m + m := by linarith _ = 2 * m := by ring _ ≤ p * m := by rel [p_prime.left] _ = n := n_eq_pm.symm done obtain (L : List Nat) (h6 : prime_factorization m L) from ih m m_lt_n m_pos define at h6 have ndpl_L : nondec_prime_list L := h6.left define at ndpl_L apply Exists.intro (p :: L) define apply And.intro · -- Proof of nondec_prime_list (p :: L) define apply And.intro · -- Proof of all_prime (p :: L) rewrite [all_prime_cons] show prime p ∧ all_prime L from And.intro p_prime ndpl_L.left done · -- Proof of nondec (p :: L) rewrite [nondec_cons] apply And.intro _ ndpl_L.right fix q : Nat assume q_in_L : q ∈ L have h7 : q ∣ prod L := list_elt_dvd_prod q_in_L rewrite [h6.right] at h7 --h7 : q ∣ m have h8 : m ∣ n := by apply Exists.intro p rewrite [n_eq_pm] ring done have q_dvd_n : q ∣ n := dvd_trans h7 h8 have ap_L : all_prime L := ndpl_L.left define at ap_L have q_prime_factor : prime_factor q n := And.intro (ap_L q q_in_L) q_dvd_n show p ≤ q from p_least q q_prime_factor done done · -- Proof of prod (p :: L) = n rewrite [prod_cons, h6.right, n_eq_pm] rfl done done done theorem Theorem_7_2_2 {a b c : Nat} (h1 : c ∣ a * b) (h2 : rel_prime a c) : c ∣ b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑c ∣ ↑b define at h1; define at h2; define obtain (j : Nat) (h3 : a * b = c * j) from h1 set s : Int := gcd_c1 a c set t : Int := gcd_c2 a c have h4 : s * ↑a + t * ↑c = ↑(gcd a c) := gcd_lin_comb c a rewrite [h2, Nat.cast_one] at h4 --h4 : s * ↑a + t * ↑c = (1 : Int) apply Exists.intro (s * ↑j + t * ↑b) show ↑b = ↑c * (s * ↑j + t * ↑b) from calc ↑b _ = (1 : Int) * ↑b := (one_mul _).symm _ = (s * ↑a + t * ↑c) * ↑b := by rw [h4] _ = s * (↑a * ↑b) + t * ↑c * ↑b := by ring _ = s * (↑c * ↑j) + t * ↑c * ↑b := by rw [←Nat.cast_mul a b, h3, Nat.cast_mul c j] _ = ↑c * (s * ↑j + t * ↑b) := by ring done lemma le_nonzero_prod_left {a b : Nat} (h : a * b ≠ 0) : a ≤ a * b := by have h1 : b ≠ 0 := by contradict h with h1 rewrite [h1] ring done have h2 : 1 ≤ b := Nat.pos_of_ne_zero h1 show a ≤ a * b from calc a = a * 1 := (mul_one a).symm _ ≤ a * b := by rel [h2] done lemma le_nonzero_prod_right {a b : Nat} (h : a * b ≠ 0) : b ≤ a * b := by rewrite [mul_comm] rewrite [mul_comm] at h show b ≤ b * a from le_nonzero_prod_left h done lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry lemma rel_prime_of_prime_not_dvd {a p : Nat} (h1 : prime p) (h2 : ¬p ∣ a) : rel_prime a p := by have h3 : gcd a p ∣ a := gcd_dvd_left a p have h4 : gcd a p ∣ p := gcd_dvd_right a p have h5 : gcd a p = 1 ∨ gcd a p = p := dvd_prime h1 h4 have h6 : gcd a p ≠ p := by contradict h2 with h6 rewrite [h6] at h3 show p ∣ a from h3 done disj_syll h5 h6 show rel_prime a p from h5 done theorem Theorem_7_2_3 {a b p : Nat} (h1 : prime p) (h2 : p ∣ a * b) : p ∣ a ∨ p ∣ b := by or_right with h3 have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3 show p ∣ b from Theorem_7_2_2 h2 h4 done lemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a ≥ 1 := by have h1 : a ≠ 0 := by by_contra h1 rewrite [h1] at h contradict h linarith done show a ≥ 1 from Nat.pos_of_ne_zero h1 done lemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by have h1 : a ≥ 1 := ge_one_of_prod_one h have h2 : a * b ≠ 0 := by linarith have h3 : a ≤ a * b := le_nonzero_prod_left h2 rewrite [h] at h3 show a = 1 from Nat.le_antisymm h3 h1 done lemma eq_one_of_dvd_one {n : Nat} (h : n ∣ 1) : n = 1 := by obtain (j : Nat) (h1 : 1 = n * j) from h show n = 1 from eq_one_of_prod_one h1.symm done lemma prime_not_one {p : Nat} (h : prime p) : p ≠ 1 := by define at h linarith done theorem Theorem_7_2_4 {p : Nat} (h1 : prime p) : ∀ (l : List Nat), p ∣ prod l → ∃ a ∈ l, p ∣ a := by apply List.rec · -- Base Case. Goal : p ∣ prod [] → ∃ a ∈ [], p ∣ a rewrite [prod_nil] assume h2 : p ∣ 1 show ∃ a ∈ [], p ∣ a from absurd (eq_one_of_dvd_one h2) (prime_not_one h1) done · -- Induction Step fix b : Nat fix L : List Nat assume ih : p ∣ prod L → ∃ a ∈ L, p ∣ a --Goal : p ∣ prod (b :: L) → ∃ a ∈ b :: L, p ∣ a assume h2 : p ∣ prod (b :: L) rewrite [prod_cons] at h2 have h3 : p ∣ b ∨ p ∣ prod L := Theorem_7_2_3 h1 h2 by_cases on h3 · -- Case 1. h3 : p ∣ b apply Exists.intro b show b ∈ b :: L ∧ p ∣ b from And.intro (List.mem_cons_self b L) h3 done · -- Case 2. h3 : p ∣ prod L obtain (a : Nat) (h4 : a ∈ L ∧ p ∣ a) from ih h3 apply Exists.intro a show a ∈ b :: L ∧ p ∣ a from And.intro (List.mem_cons_of_mem b h4.left) h4.right done done done lemma prime_in_list {p : Nat} {l : List Nat} (h1 : prime p) (h2 : all_prime l) (h3 : p ∣ prod l) : p ∈ l := by obtain (a : Nat) (h4 : a ∈ l ∧ p ∣ a) from Theorem_7_2_4 h1 l h3 define at h2 have h5 : prime a := h2 a h4.left have h6 : p = 1 ∨ p = a := dvd_prime h5 h4.right disj_syll h6 (prime_not_one h1) rewrite [h6] show a ∈ l from h4.left done lemma first_le_first {p q : Nat} {l m : List Nat} (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m)) (h3 : prod (p :: l) = prod (q :: m)) : p ≤ q := by define at h1; define at h2 have h4 : q ∣ prod (p :: l) := by define apply Exists.intro (prod m) rewrite [←prod_cons] show prod (p :: l) = prod (q :: m) from h3 done have h5 : all_prime (q :: m) := h2.left rewrite [all_prime_cons] at h5 have h6 : q ∈ p :: l := prime_in_list h5.left h1.left h4 have h7 : nondec (p :: l) := h1.right rewrite [nondec_cons] at h7 rewrite [List.mem_cons] at h6 by_cases on h6 · -- Case 1. h6 : q = p linarith done · -- Case 2. h6 : q ∈ l have h8 : ∀ m ∈ l, p ≤ m := h7.left show p ≤ q from h8 q h6 done done lemma nondec_prime_list_tail {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by define at h define rewrite [all_prime_cons, nondec_cons] at h show all_prime l ∧ nondec l from And.intro h.left.right h.right.right done lemma cons_prod_not_one {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : prod (p :: l) ≠ 1 := by define at h have h1 : all_prime (p :: l) := h.left rewrite [all_prime_cons] at h1 rewrite [prod_cons] by_contra h2 show False from (prime_not_one h1.left) (eq_one_of_prod_one h2) done lemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) : l = [] ↔ prod l = 1 := by apply Iff.intro · -- (→) assume h1 : l = [] rewrite [h1] show prod [] = 1 from prod_nil done · -- (←) contrapos assume h1 : ¬l = [] obtain (p : Nat) (h2 : ∃ (L : List Nat), l = p :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List Nat) (h3 : l = p :: L) from h2 rewrite [h3] at h rewrite [h3] show ¬prod (p :: L) = 1 from cons_prod_not_one h done done lemma prime_pos {p : Nat} (h : prime p) : p > 0 := by define at h linarith done theorem Theorem_7_2_5 : ∀ (l1 l2 : List Nat), nondec_prime_list l1 → nondec_prime_list l2 → prod l1 = prod l2 → l1 = l2 := by apply List.rec · -- Base Case. Goal : ∀ (l2 : List Nat), nondec_prime_list [] → -- nondec_prime_list l2 → prod [] = prod l2 → [] = l2 fix l2 : List Nat assume h1 : nondec_prime_list [] assume h2 : nondec_prime_list l2 assume h3 : prod [] = prod l2 rewrite [prod_nil, eq_comm, ←list_nil_iff_prod_one h2] at h3 show [] = l2 from h3.symm done · -- Induction Step fix p : Nat fix L1 : List Nat assume ih : ∀ (L2 : List Nat), nondec_prime_list L1 → nondec_prime_list L2 → prod L1 = prod L2 → L1 = L2 -- Goal : ∀ (l2 : List Nat), nondec_prime_list (p :: L1) → -- nondec_prime_list l2 → prod (p :: L1) = prod l2 → p :: L1 = l2 fix l2 : List Nat assume h1 : nondec_prime_list (p :: L1) assume h2 : nondec_prime_list l2 assume h3 : prod (p :: L1) = prod l2 have h4 : ¬prod (p :: L1) = 1 := cons_prod_not_one h1 rewrite [h3, ←list_nil_iff_prod_one h2] at h4 obtain (q : Nat) (h5 : ∃ (L : List Nat), l2 = q :: L) from List.exists_cons_of_ne_nil h4 obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5 rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2) rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2) have h7 : p ≤ q := first_le_first h1 h2 h3 have h8 : q ≤ p := first_le_first h2 h1 h3.symm have h9 : p = q := by linarith rewrite [h9, prod_cons, prod_cons] at h3 --h3 : q * prod L1 = q * prod L2 have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1 have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2 define at h2 have h12 : all_prime (q :: L2) := h2.left rewrite [all_prime_cons] at h12 have h13 : q > 0 := prime_pos h12.left have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3 have h15 : L1 = L2 := ih L2 h10 h11 h14 rewrite [h6, h9, h15] rfl done done theorem fund_thm_arith (n : Nat) (h : n ≥ 1) : ∃! (l : List Nat), prime_factorization n l := by exists_unique · -- Existence show ∃ (l : List Nat), prime_factorization n l from exists_prime_factorization n h done · -- Uniqueness fix l1 : List Nat; fix l2 : List Nat assume h1 : prime_factorization n l1 assume h2 : prime_factorization n l2 define at h1; define at h2 have h3 : prod l1 = n := h1.right rewrite [←h2.right] at h3 show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3 done done /- Section 7.3 -/ theorem congr_refl (m : Nat) : ∀ (a : Int), a ≡ a (MOD m) := by fix a : Int define --Goal : ∃ (c : Int), a - a = ↑m * c apply Exists.intro 0 ring done theorem congr_symm {m : Nat} : ∀ {a b : Int}, a ≡ b (MOD m) → b ≡ a (MOD m) := by fix a : Int; fix b : Int assume h1 : a ≡ b (MOD m) define at h1 --h1 : ∃ (c : Int), a - b = ↑m * c define --Goal : ∃ (c : Int), b - a = ↑m * c obtain (c : Int) (h2 : a - b = m * c) from h1 apply Exists.intro (-c) show b - a = m * (-c) from calc b - a _ = -(a - b) := by ring _ = -(m * c) := by rw [h2] _ = m * (-c) := by ring done theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry /- Fundamental properties of congruence classes -/ lemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) : X = Y ↔ X.val = Y.val := Fin.ext_iff lemma val_nat_eq_mod (n k : Nat) : ([k]_(n + 1)).val = k % (n + 1) := by rfl lemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl theorem cc_rep {m : Nat} (X : ZMod m) : ∃ (a : Int), X = [a]_m := match m with | 0 => by apply Exists.intro X rfl done | n + 1 => by apply Exists.intro ↑(X.val) have h1 : X.val < n + 1 := Fin.prop X rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1] rfl done theorem add_class (m : Nat) (a b : Int) : [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm theorem mul_class (m : Nat) (a b : Int) : [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm lemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ [a - b]_m = [0]_m := by apply Iff.intro · -- (→) assume h1 : [a]_m = [b]_m have h2 : a - b = a + (-b) := by ring have h3 : b + (-b) = 0 := by ring show [a - b]_m = [0]_m from calc [a - b]_m _ = [a + (-b)]_m := by rw [h2] _ = [a]_m + [-b]_m := by rw [add_class] _ = [b]_m + [-b]_m := by rw [h1] _ = [b + -b]_m := by rw [add_class] _ = [0]_m := by rw [h3] done · -- (←) assume h1 : [a - b]_m = [0]_m have h2 : b + (a - b) = a := by ring have h3 : b + 0 = b := by ring show [a]_m = [b]_m from calc [a]_m _ = [b + (a - b)]_m := by rw [h2] _ = [b]_m + [a - b]_m := by rw [add_class] _ = [b]_m + [0]_m := by rw [h1] _ = [b + 0]_m := by rw [add_class] _ = [b]_m := by rw [h3] done done lemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) : [a]_m = [0]_m → [-a]_m = [0]_m := by assume h1 : [a]_m = [0]_m have h2 : 0 + (-a) = -a := by ring have h3 : a + (-a) = 0 := by ring show [-a]_m = [0]_m from calc [-a]_m _ = [0 + (-a)]_m := by rw [h2] _ = [0]_m + [-a]_m := by rw [add_class] _ = [a]_m + [-a]_m := by rw [h1] _ = [a + (-a)]_m := by rw [add_class] _ = [0]_m := by rw [h3] done lemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) : [-a]_m = [0]_m ↔ [a]_m = [0]_m := by apply Iff.intro _ (cc_neg_zero_of_cc_zero m a) assume h1 : [-a]_m = [0]_m have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1 have h3 : -(-a) = a := by ring rewrite [h3] at h2 show [a]_m = [0]_m from h2 done lemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl lemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m ↔ m ∣ k := match m with | 0 => by have h : (0 : Int) = (↑(0 : Nat) : Int) := by rfl rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj] apply Iff.intro · -- (→) assume h1 : k = 0 rewrite [h1] show 0 ∣ 0 from dvd_self 0 done · -- (←) assume h1 : 0 ∣ k obtain (c : Nat) (h2 : k = 0 * c) from h1 rewrite [h2] ring done done | n + 1 => by rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero] show k % (n + 1) = 0 ↔ n + 1 ∣ k from (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm done lemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m ↔ ↑m ∣ a := by obtain (k : Nat) (h1 : a = ↑k ∨ a = -↑k) from Int.eq_nat_or_neg a by_cases on h1 · -- Case 1. h1: a = ↑k rewrite [h1, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done · -- Case 2. h1: a = -↑k rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done done theorem cc_eq_iff_congr (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ a ≡ b (MOD m) := calc [a]_m = [b]_m _ ↔ [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b _ ↔ ↑m ∣ (a - b) := cc_zero_iff_dvd m (a - b) _ ↔ a ≡ b (MOD m) := by rfl /- End of fundamental properties of congruence classes -/ lemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m := by have h1 : (↑m : Int) ≠ 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m) show 0 ≤ a % m from Int.emod_nonneg a h1 done lemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m) have h2 : (↑m : Int) > 0 := (Nat.cast_pos).rtl h1 show a % m < m from Int.emod_lt_of_pos a h2 done lemma congr_mod_mod (m : Nat) (a : Int) : a ≡ a % m (MOD m) := by define have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m apply Exists.intro (a / m) show a - a % m = m * (a / m) from calc a - (a % m) _ = m * (a / m) + a % m - a % m := by rw [h1] _ = m * (a / m) := by ring done lemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) := And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a)) theorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) : ∃! (r : Int), 0 ≤ r ∧ r < m ∧ a ≡ r (MOD m) := by exists_unique · -- Existence apply Exists.intro (a % m) show 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) from mod_cmpl_res m a done · -- Uniqueness fix r1 : Int; fix r2 : Int assume h1 : 0 ≤ r1 ∧ r1 < m ∧ a ≡ r1 (MOD m) assume h2 : 0 ≤ r2 ∧ r2 < m ∧ a ≡ r2 (MOD m) have h3 : r1 ≡ r2 (MOD m) := congr_trans (congr_symm h1.right.right) h2.right.right obtain (d : Int) (h4 : r1 - r2 = m * d) from h3 have h5 : r1 - r2 < m * 1 := by linarith have h6 : m * (-1) < r1 - r2 := by linarith rewrite [h4] at h5 --h5 : m * d < m * 1 rewrite [h4] at h6 --h6 : m * -1 < m * d have h7 : (↑m : Int) ≥ 0 := Nat.cast_nonneg m have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7 have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7 have h10 : d = 0 := by linarith show r1 = r2 from calc r1 _ = r1 - r2 + r2 := by ring _ = m * 0 + r2 := by rw [h4, h10] _ = r2 := by ring done done lemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m := (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a) theorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y rewrite [h1, h2] have h3 : a + b = b + a := by ring show [a]_m + [b]_m = [b]_m + [a]_m from calc [a]_m + [b]_m _ = [a + b]_m := add_class m a b _ = [b + a]_m := by rw [h3] _ = [b]_m + [a]_m := (add_class m b a).symm done theorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X rewrite [h1] have h2 : a * 1 = a := by ring show [a]_m * [1]_m = [a]_m from calc [a]_m * [1]_m _ = [a * 1]_m := mul_class m a 1 _ = [a]_m := by rw [h2] done theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry lemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) : [a]_m * [gcd_c2 m a]_m = [1]_m := by set s : Int := gcd_c1 m a have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m define at h1 rewrite [h1, Nat.cast_one] at h2 --h2 : s * ↑m + gcd_c2 m a * ↑a = 1 rewrite [mul_class, cc_eq_iff_congr] define --Goal : ∃ (c : Int), ↑a * gcd_c2 m a - 1 = ↑m * c apply Exists.intro (-s) show a * (gcd_c2 m a) - 1 = m * (-s) from calc a * (gcd_c2 m a) - 1 _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring _ = 1 + m * (-s) - 1 := by rw [h2] _ = m * (-s) := by ring done theorem Theorem_7_3_7 (m a : Nat) : invertible [a]_m ↔ rel_prime m a := by apply Iff.intro · -- (→) assume h1 : invertible [a]_m define at h1 obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1 obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y rewrite [h3, mul_class, cc_eq_iff_congr] at h2 define at h2 obtain (c : Int) (h4 : a * b - 1 = m * c) from h2 rewrite [Exercise_7_2_6] --Goal : ∃ (s t : Int), s * ↑m + t * ↑a = 1 apply Exists.intro (-c) apply Exists.intro b show (-c) * m + b * a = 1 from calc (-c) * m + b * a _ = (-c) * m + (a * b - 1) + 1 := by ring _ = (-c) * m + m * c + 1 := by rw [h4] _ = 1 := by ring done · -- (←) assume h1 : rel_prime m a define show ∃ (Y : ZMod m), [a]_m * Y = [1]_m from Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1) done done /- Section 7.4 -/ section Euler open Euler lemma num_rp_below_base {m : Nat} : num_rp_below m 0 = 0 := by rfl lemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) : num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by have h1 : num_rp_below m (j + 1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : gcd m j = 1 rewrite [if_pos h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1 show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1 done lemma num_rp_below_step_not_rp {m j : Nat} (h : ¬rel_prime m j) : num_rp_below m (j + 1) = num_rp_below m j := by have h1 : num_rp_below m (j +1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : ¬gcd m j = 1 rewrite [if_neg h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j show num_rp_below m (j + 1) = num_rp_below m j from h1 done lemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl #eval phi 10 --Answer: 4 lemma prod_inv_iff_inv {m : Nat} {X : ZMod m} (h1 : invertible X) (Y : ZMod m) : invertible (X * Y) ↔ invertible Y := by apply Iff.intro · -- (→) assume h2 : invertible (X * Y) obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2 apply Exists.intro (X * Z) rewrite [←h3] --Goal : Y * (X * Z) = X * Y * Z ring --Note that ring can do algebra in ZMod m done · -- (←) assume h2 : invertible Y obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1 obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2 apply Exists.intro (Xi * Yi) show (X * Y) * (Xi * Yi) = [1]_m from calc X * Y * (Xi * Yi) _ = (X * Xi) * (Y * Yi) := by ring _ = [1]_m * [1]_m := by rw [h3, h4] _ = [1]_m := Theorem_7_3_6_7 [1]_m done done lemma F_rp_def {m i : Nat} (h : rel_prime m i) : F m i = [i]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h --h : gcd m i = 1 rewrite [if_pos h] at h1 show F m i = [i]_m from h1 done lemma F_not_rp_def {m i : Nat} (h : ¬rel_prime m i) : F m i = [1]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h rewrite [h1, if_neg h] rfl done lemma prod_seq_base {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 0 k f = [1]_m := by rfl lemma prod_seq_step {m : Nat} (n k : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl lemma prod_seq_zero_step {m : Nat} (n : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by rewrite [prod_seq_step, zero_add] rfl done lemma prod_one {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 1 k f = f k := by rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7] rfl done lemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl lemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m := calc [G m a i]_m _ = [(a * i) % m]_m := by rfl _ = [a * i]_m := (cc_eq_mod m (a * i)).symm _ = [a]_m * [i]_m := (mul_class m a i).symm lemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) : rel_prime m (G m a i) ↔ rel_prime m i := by have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1 show rel_prime m (G m a i) ↔ rel_prime m i from calc rel_prime m (G m a i) _ ↔ invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm _ ↔ invertible ([a]_m * [i]_m) := by rw [cc_G] _ ↔ invertible [i]_m := prod_inv_iff_inv h2 ([i]_m) _ ↔ rel_prime m i := Theorem_7_3_7 m i done lemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) : F m (G m a i) = [a]_m * F m i := by have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2 show F m (G m a i) = [a]_m * F m i from calc F m (G m a i) _ = [G m a i]_m := F_rp_def h3 _ = [a]_m * [i]_m := cc_G m a i _ = [a]_m * F m i := by rw [F_rp_def h2] done lemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : ¬rel_prime m i) : F m (G m a i) = [1]_m := by rewrite [←G_rp_iff h1 i] at h2 show F m (G m a i) = [1]_m from F_not_rp_def h2 done lemma FG_prod {m a : Nat} (h1 : rel_prime m a) : ∀ (k : Nat), prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by by_induc · -- Base Case show prod_seq 0 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from calc prod_seq 0 0 ((F m) ∘ (G m a)) _ = [1]_m := prod_seq_base _ _ _ = [a]_m ^ 0 * [1]_m := by ring _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by rw [num_rp_below_base, prod_seq_base] done · -- Induction Step fix k : Nat assume ih : prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) by_cases h2 : rel_prime m k · -- Case 1. h2 : rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([a]_m * F m k) := by rw [FG_rp h1 h2] _ = [a]_m ^ ((num_rp_below m k) + 1) * ((prod_seq k 0 (F m)) * F m k) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_rp h2, prod_seq_zero_step] done · -- Case 2. h2 : ¬rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([1]_m) := by rw [FG_not_rp h1 h2] _ = [a]_m ^ (num_rp_below m k) * (prod_seq k 0 (F m) * ([1]_m)) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_not_rp h2, prod_seq_zero_step, F_not_rp_def h2] done done done lemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by define --Goal : ∀ i < m, G m a i < m fix i : Nat assume h1 : i < m rewrite [G_def] --Goal : a * i % m < m show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m) done lemma left_inv_one_one_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g' (g i) = i) : one_one_below n g := sorry lemma right_inv_onto_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g (g' i) = i) (h2 : maps_below n g') : onto_below n g := by define at h2; define fix k : Nat assume h3 : k < n apply Exists.intro (g' k) show g' k < n ∧ g (g' k) = k from And.intro (h2 k h3) (h1 k h3) done lemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m * [inv_mod m a]_m = [1]_m := by have h2 : 0 ≤ (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a) show [a]_m * [inv_mod m a]_m = [1]_m from calc [a]_m * [inv_mod m a]_m _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2] _ = [a]_m * [gcd_c2 m a]_m := by rw [←cc_eq_mod] _ = [1]_m := gcd_c2_inv h1 done lemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m := calc a * (b % m) % m = a % m * (b % m % m) % m := Nat.mul_mod _ _ _ _ = a % m * (b % m) % m := by rw [Nat.mod_mod] _ = a * b % m := (Nat.mul_mod _ _ _).symm lemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm] rfl done theorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] : ↑a ≡ ↑b (MOD m) ↔ a % m = b % m := sorry lemma mul_inv_mod_cancel {m a i : Nat} [NeZero m] (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1 rewrite [mul_class, cc_eq_iff_congr, ←Nat.cast_mul, ←Nat.cast_one, congr_iff_mod_eq_Nat] at h3 show a * inv_mod m a * i % m = i from calc a * (inv_mod m a) * i % m _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = 1 % m * i % m := by rw [h3] _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = i % m := by rw [one_mul] _ = i := Nat.mod_eq_of_lt h2 done lemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl lemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, G m a (Ginv m a i) = i := by fix i : Nat assume h2 : i < m show G m a (Ginv m a i) = i from calc G m a (Ginv m a i) _ = a * ((inv_mod m a * i) % m) % m := by rfl _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, Ginv m a (G m a i) = i := by fix i : Nat assume h2 : i < m show Ginv m a (G m a i) = i from calc Ginv m a (G m a i) _ = inv_mod m a * ((a * i) % m) % m := by rfl _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc, mul_comm (inv_mod m a)] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_maps_below (m a : Nat) [NeZero m] : maps_below m (Ginv m a) := G_maps_below m (inv_mod m a) lemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : one_one_below m (G m a) := left_inv_one_one_below (Ginv_left_inv h1) lemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : onto_below m (G m a) := right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a) lemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : perm_below m (G m a) := And.intro (G_maps_below m a) (And.intro (G_one_one_below h1) (G_onto_below h1)) --Permuting a product of congruence classes doesn't change product lemma swap_fst (u v : Nat) : swap u v u = v := by define : swap u v u --Goal : (if u = u then v else if u = v then u else u) = v have h : u = u := by rfl rewrite [if_pos h] rfl done lemma swap_snd (u v : Nat) : swap u v v = u := by define : swap u v v by_cases h1 : v = u · -- Case 1. h1 : v = u rewrite [if_pos h1] show v = u from h1 done · -- Case 2. h1 : v ≠ u rewrite [if_neg h1] have h2 : v = v := by rfl rewrite [if_pos h2] rfl done done lemma swap_other {u v i : Nat} (h1 : i ≠ u) (h2 : i ≠ v) : swap u v i = i := by define : swap u v i rewrite [if_neg h1, if_neg h2] rfl done lemma swap_values (u v i : Nat) : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := by by_cases h1 : i = u · -- Case 1. h1 : i = u apply Or.inl rewrite [h1] show swap u v u = v from swap_fst u v done · -- Case 2. h1 : i ≠ u apply Or.inr by_cases h2 : i = v · -- Case 2.1. h2 : i = v apply Or.inl rewrite [h2] show swap u v v = u from swap_snd u v done · -- Case 2.2. h2 : i ≠ v apply Or.inr show swap u v i = i from swap_other h1 h2 done done done lemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by define fix i : Nat assume h3 : i < n have h4 : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := swap_values u v i by_cases on h4 · -- Case 1. h4 : swap u v i = v rewrite [h4] show v < n from h2 done · -- Case 2. by_cases on h4 · -- Case 2.1. h4 : swap u v i = u rewrite [h4] show u < n from h1 done · -- Case 2.2. h4 : swap u v i = i rewrite [h4] show i < n from h3 done done done lemma swap_swap (u v n : Nat) : ∀ i < n, swap u v (swap u v i) = i := by fix i : Nat assume h : i < n by_cases h1 : i = u · -- Case 1. h1 : i = u rewrite [h1, swap_fst, swap_snd] rfl done · -- Case 2. h1 : i ≠ u by_cases h2 : i = v · -- Case 2.1. h2 : i = v rewrite [h2, swap_snd, swap_fst] rfl done · -- Case 2.2. h2 : i ≠ v rewrite [swap_other h1 h2, swap_other h1 h2] rfl done done done lemma swap_one_one_below (u v n) : one_one_below n (swap u v) := left_inv_one_one_below (swap_swap u v n) lemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) := right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2) lemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) := And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2)) lemma comp_perm_below {n : Nat} {f g : Nat → Nat} (h1 : perm_below n f) (h2 : perm_below n g) : perm_below n (f ∘ g) := sorry lemma trivial_swap (u : Nat) : swap u u = id := by apply funext fix x : Nat by_cases h1 : x = u · -- Case 1. h1 : x = u rewrite [h1, swap_fst] rfl done · -- Case 2. h1 : x ≠ u rewrite [swap_other h1 h1] rfl done done lemma prod_eq_fun {m : Nat} (f g : Nat → ZMod m) (k : Nat) : ∀ (n : Nat), (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g := by by_induc · -- Base Case assume h : (∀ i < 0, f (k + i) = g (k + i)) rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g assume h1 : ∀ i < n + 1, f (k + i) = g (k + i) have h2 : ∀ i < n, f (k + i) = g (k + i) := by fix i : Nat assume h2 : i < n have h3 : i < n + 1 := by linarith show f (k + i) = g (k + i) from h1 i h3 done have h3 : prod_seq n k f = prod_seq n k g := ih h2 have h4 : n < n + 1 := Nat.lt_succ_self n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4] rfl done done lemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := by have h2 : ∀ (i : Nat), i < u → (f ∘ swap u n) (0 + i) = f (0 + i) := by fix i : Nat assume h2 : i < u have h3 : 0 + i ≠ u := by linarith have h4 : 0 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl done show prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f from prod_eq_fun (f ∘ swap u n) f 0 u h2 done lemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := by have h2 : ∀ i < j, (f ∘ swap u n) (u + 1 + i) = f (u + 1 + i) := by fix i : Nat assume h2 : i < j have h3 : u + 1 + i ≠ u := by linarith have h4 : u + 1 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl show prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f from prod_eq_fun (f ∘ swap u n) f (u + 1) j h2 done lemma break_prod {m : Nat} (n : Nat) (f : Nat → ZMod m) : ∀ (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by by_induc · -- Base Case have h : n + 0 = n := by rfl rewrite [prod_seq_base, h, Theorem_7_3_6_7] rfl done · -- Induction Step fix j : Nat assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f rewrite [←add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc] rfl done done lemma break_prod_twice {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f := break_prod n f 1 rewrite [prod_one] at h2 have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f := break_prod (u + 1) f j rewrite [←h1] at h3 have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f := break_prod u f 1 rewrite [prod_one] at h4 rewrite [h3, h4] at h2 show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2 done lemma swap_prod_eq_prod {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f := by by_cases h2 : u = n · -- Case 1. h2 : u = n rewrite [h2, trivial_swap n] --Goal : prod_seq (n + 1) 0 (f ∘ id) = prod_seq (n + 1) 0 f rfl done · -- Case 2. h2 : ¬u = n have h3 : u + 1 ≤ n := Nat.lt_of_le_of_ne h1 h2 obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3 have break_f : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := break_prod_twice f h4 have break_fs : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_prod_twice (f ∘ swap u n) h4 have f_eq_fs_below : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := swap_prod_eq_prod_below f h1 have f_eq_fs_btwn : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4 show prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f from calc prod_seq (n + 1) 0 (f ∘ swap u n) _ = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_fs _ = prod_seq u 0 f * (f ∘ swap u n) u * prod_seq j (u + 1) f * (f ∘ swap u n) n := by rw [f_eq_fs_below, f_eq_fs_btwn] _ = prod_seq u 0 f * f (swap u n u) * prod_seq j (u + 1) f * f (swap u n n) := by rfl _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by rw [swap_fst, swap_snd] _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring _ = prod_seq (n + 1) 0 f := break_f.symm done done lemma perm_below_fixed {n : Nat} {g : Nat → Nat} (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry lemma perm_prod {m : Nat} (f : Nat → ZMod m) : ∀ (n : Nat), ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) := by by_induc · -- Base Case fix g : Nat → Nat assume h1 : perm_below 0 g rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) fix g : Nat → Nat assume g_pb : perm_below (n + 1) g define at g_pb have g_ob : onto_below (n + 1) g := g_pb.right.right define at g_ob have h1 : n < n + 1 := by linarith obtain (u : Nat) (h2 : u < n + 1 ∧ g u = n) from g_ob n h1 have s_pb : perm_below (n + 1) (swap u n) := swap_perm_below h2.left h1 have gs_pb_n1 : perm_below (n + 1) (g ∘ swap u n) := comp_perm_below g_pb s_pb have gs_fix_n : (g ∘ swap u n) n = n := calc (g ∘ swap u n) n _ = g (swap u n n) := by rfl _ = g u := by rw [swap_snd] _ = n := h2.right have gs_pb_n : perm_below n (g ∘ swap u n) := perm_below_fixed gs_pb_n1 gs_fix_n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f ∘ (g ∘ swap u n)) := ih (g ∘ swap u n) gs_pb_n have h3 : u ≤ n := by linarith show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f ∘ g) from calc prod_seq (n + 1) 0 f _ = prod_seq n 0 f * f n := prod_seq_zero_step n f _ = prod_seq n 0 (f ∘ (g ∘ swap u n)) * f ((g ∘ swap u n) n) := by rw [gs_prod, gs_fix_n] _ = prod_seq n 0 (f ∘ g ∘ swap u n) * (f ∘ g ∘ swap u n) n := by rfl _ = prod_seq (n + 1) 0 (f ∘ g ∘ swap u n) := (prod_seq_zero_step n (f ∘ g ∘ swap u n)).symm _ = prod_seq (n + 1) 0 ((f ∘ g) ∘ swap u n) := by rfl _ = prod_seq (n + 1) 0 (f ∘ g) := swap_prod_eq_prod (f ∘ g) h3 done done lemma F_invertible (m i : Nat) : invertible (F m i) := by by_cases h : rel_prime m i · -- Case 1. h : rel_prime m i rewrite [F_rp_def h] show invertible [i]_m from (Theorem_7_3_7 m i).rtl h done · -- Case 2. h : ¬rel_prime m i rewrite [F_not_rp_def h] apply Exists.intro [1]_m show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m done done lemma Fprod_invertible (m : Nat) : ∀ (k : Nat), invertible (prod_seq k 0 (F m)) := by by_induc · -- Base Case apply Exists.intro [1]_m show prod_seq 0 0 (F m) * [1]_m = [1]_m from calc prod_seq 0 0 (F m) * [1]_m _ = [1]_m * [1]_m := by rw [prod_seq_base] _ = [1]_m := Theorem_7_3_6_7 ([1]_m) done · -- Induction Step fix k : Nat assume ih : invertible (prod_seq k 0 (F m)) rewrite [prod_seq_zero_step] show invertible (prod_seq k 0 (F m) * (F m k)) from (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k) done done theorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m ^ (phi m) = [1]_m := by have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2 show [a]_m ^ (phi m) = [1]_m from calc [a]_m ^ (phi m) _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3] _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring _ = prod_seq m 0 (F m ∘ G m a) * Y := by rw [FG_prod h1 m, phi_def] _ = prod_seq m 0 (F m) * Y := by rw [perm_prod (F m) m (G m a) (G_perm_below h1)] _ = [1]_m := by rw [h3] done lemma Exercise_7_4_5_Int (m : Nat) (a : Int) : ∀ (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry lemma Exercise_7_4_5_Nat (m a n : Nat) : [a]_m ^ n = [a ^ n]_m := by rewrite [Exercise_7_4_5_Int] rfl done theorem Euler's_theorem {m a : Nat} [NeZero m] (h1 : rel_prime m a) : a ^ (phi m) ≡ 1 (MOD m) := by have h2 : [a]_m ^ (phi m) = [1]_m := Theorem_7_4_2 h1 rewrite [Exercise_7_4_5_Nat m a (phi m)] at h2 --h2 : [a ^ phi m]_m = [1]_m show a ^ (phi m) ≡ 1 (MOD m) from (cc_eq_iff_congr _ _ _).ltr h2 done #eval gcd 10 7 --Answer: 1. So 10 and 7 are relatively prime #eval 7 ^ phi 10 --Answer: 2401, which is congruent to 1 mod 10. end Euler /- Section 7.5 -/ lemma num_rp_prime {p : Nat} (h1 : prime p) : ∀ k < p, num_rp_below p (k + 1) = k := sorry lemma phi_prime {p : Nat} (h1 : prime p) : phi p = p - 1 := by have h2 : 1 ≤ p := prime_pos h1 have h3 : p - 1 + 1 = p := Nat.sub_add_cancel h2 have h4 : p - 1 < p := by linarith have h5 : num_rp_below p (p - 1 + 1) = p - 1 := num_rp_prime h1 (p - 1) h4 rewrite [h3] at h5 show phi p = p - 1 from h5 done theorem Theorem_7_2_2_Int {a c : Nat} {b : Int} (h1 : ↑c ∣ ↑a * b) (h2 : rel_prime a c) : ↑c ∣ b := by rewrite [Int.natCast_dvd, Int.natAbs_mul, Int.natAbs_ofNat] at h1 --h1 : c ∣ a * Int.natAbs b rewrite [Int.natCast_dvd] --Goal : c ∣ Int.natAbs b show c ∣ Int.natAbs b from Theorem_7_2_2 h1 h2 done lemma Lemma_7_4_5 {m n : Nat} (a b : Int) (h1 : rel_prime m n) : a ≡ b (MOD m * n) ↔ a ≡ b (MOD m) ∧ a ≡ b (MOD n) := by apply Iff.intro · -- (→) assume h2 : a ≡ b (MOD m * n) obtain (j : Int) (h3 : a - b = (m * n) * j) from h2 apply And.intro · -- Proof of a ≡ b (MOD m) apply Exists.intro (n * j) show a - b = m * (n * j) from calc a - b _ = m * n * j := h3 _ = m * (n * j) := by ring done · -- Proof of a ≡ b (MOD n) apply Exists.intro (m * j) show a - b = n * (m * j) from calc a - b _ = m * n * j := h3 _ = n * (m * j) := by ring done done · -- (←) assume h2 : a ≡ b (MOD m) ∧ a ≡ b (MOD n) obtain (j : Int) (h3 : a - b = m * j) from h2.left have h4 : (↑n : Int) ∣ a - b := h2.right rewrite [h3] at h4 --h4 : ↑n ∣ ↑m * j have h5 : ↑n ∣ j := Theorem_7_2_2_Int h4 h1 obtain (k : Int) (h6 : j = n * k) from h5 apply Exists.intro k --Goal : a - b = ↑(m * n) * k rewrite [Nat.cast_mul] --Goal : a - b = ↑m * ↑n * k show a - b = (m * n) * k from calc a - b _ = m * j := h3 _ = m * (n * k) := by rw [h6] _ = (m * n) * k := by ring done done --From exercises of Section 7.2 theorem rel_prime_symm {a b : Nat} (h : rel_prime a b) : rel_prime b a := sorry lemma prime_NeZero {p : Nat} (h : prime p) : NeZero p := by rewrite [neZero_iff] --Goal : p ≠ 0 define at h linarith done lemma Lemma_7_5_1 {p e d m c s : Nat} {t : Int} (h1 : prime p) (h2 : e * d = (p - 1) * s + 1) (h3 : m ^ e - c = p * t) : c ^ d ≡ m (MOD p) := by have h4 : m ^ e ≡ c (MOD p) := Exists.intro t h3 have h5 : [m ^ e]_p = [c]_p := (cc_eq_iff_congr _ _ _).rtl h4 rewrite [←Exercise_7_4_5_Nat] at h5 --h5 : [m]_p ^ e = [c]_p by_cases h6 : p ∣ m · -- Case 1. h6 : p ∣ m have h7 : m ≡ 0 (MOD p) := by obtain (j : Nat) (h8 : m = p * j) from h6 apply Exists.intro (↑j : Int) --Goal : ↑m - 0 = ↑p * ↑j rewrite [h8, Nat.cast_mul] ring done have h8 : [m]_p = [0]_p := (cc_eq_iff_congr _ _ _).rtl h7 have h9 : e * d ≠ 0 := by rewrite [h2] show (p - 1) * s + 1 ≠ 0 from Nat.add_one_ne_zero _ done have h10 : (0 : Int) ^ (e * d) = 0 := zero_pow h9 have h11 : [c ^ d]_p = [m]_p := calc [c ^ d]_p _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat] _ = ([m]_p ^ e) ^ d := by rw [h5] _ = [m]_p ^ (e * d) := by ring _ = [0]_p ^ (e * d) := by rw [h8] _ = [0 ^ (e * d)]_p := Exercise_7_4_5_Int _ _ _ _ = [0]_p := by rw [h10] _ = [m]_p := by rw [h8] show c ^ d ≡ m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11 done · -- Case 2. h6 : ¬p ∣ m have h7 : rel_prime m p := rel_prime_of_prime_not_dvd h1 h6 have h8 : rel_prime p m := rel_prime_symm h7 have h9 : NeZero p := prime_NeZero h1 have h10 : (1 : Int) ^ s = 1 := by ring have h11 : [c ^ d]_p = [m]_p := calc [c ^ d]_p _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat] _ = ([m]_p ^ e) ^ d := by rw [h5] _ = [m]_p ^ (e * d) := by ring _ = [m]_p ^ ((p - 1) * s + 1) := by rw [h2] _ = ([m]_p ^ (p - 1)) ^ s * [m]_p := by ring _ = ([m]_p ^ (phi p)) ^ s * [m]_p := by rw [phi_prime h1] _ = [1]_p ^ s * [m]_p := by rw [Theorem_7_4_2 h8] _ = [1 ^ s]_p * [m]_p := by rw [Exercise_7_4_5_Int] _ = [1]_p * [m]_p := by rw [h10] _ = [m]_p * [1]_p := by ring _ = [m]_p := Theorem_7_3_6_7 _ show c ^ d ≡ m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11 done done theorem Theorem_7_5_1 (p q n e d k m c : Nat) (p_prime : prime p) (q_prime : prime q) (p_ne_q : p ≠ q) (n_pq : n = p * q) (ed_congr_1 : e * d = k * (p - 1) * (q - 1) + 1) (h1 : [m]_n ^ e = [c]_n) : [c]_n ^ d = [m]_n := by rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] at h1 --h1 : m ^ e ≡ c (MOD n) rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] --Goal : c ^ d ≡ m (MOD n) obtain (j : Int) (h2 : m ^ e - c = n * j) from h1 rewrite [n_pq, Nat.cast_mul] at h2 --h2 : m ^ e - c = p * q * j have h3 : e * d = (p - 1) * (k * (q - 1)) + 1 := by rewrite [ed_congr_1] ring done have h4 : m ^ e - c = p * (q * j) := by rewrite [h2] ring done have congr_p : c ^ d ≡ m (MOD p) := Lemma_7_5_1 p_prime h3 h4 have h5 : e * d = (q - 1) * (k * (p - 1)) + 1 := by rewrite [ed_congr_1] ring done have h6 : m ^ e - c = q * (p * j) := by rewrite [h2] ring done have congr_q : c ^ d ≡ m (MOD q) := Lemma_7_5_1 q_prime h5 h6 have h7 : ¬q ∣ p := by by_contra h8 have h9 : q = 1 ∨ q = p := dvd_prime p_prime h8 disj_syll h9 (prime_not_one q_prime) show False from p_ne_q h9.symm done have h8 : rel_prime p q := rel_prime_of_prime_not_dvd q_prime h7 rewrite [n_pq, Lemma_7_4_5 _ _ h8] show c ^ d ≡ m (MOD p) ∧ c ^ d ≡ m (MOD q) from And.intro congr_p congr_q done /- BEGIN EXERCISES -/ namespace Exercises /- Section 7.1 -/ -- 1. theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry -- 2. lemma gcd_comm_lt {a b : Nat} (h : a < b) : gcd a b = gcd b a := sorry theorem gcd_comm (a b : Nat) : gcd a b = gcd b a := sorry -- 3. theorem Exercise_7_1_5 (a b : Nat) (n : Int) : (∃ (s t : Int), s * a + t * b = n) ↔ (↑(gcd a b) : Int) ∣ n := sorry -- 4. theorem Exercise_7_1_6 (a b c : Nat) : gcd a b = gcd (a + b * c) b := sorry -- 5. theorem gcd_is_nonzero {a b : Nat} (h : a ≠ 0 ∨ b ≠ 0) : gcd a b ≠ 0 := sorry -- 6. theorem gcd_greatest {a b d : Nat} (h1 : gcd a b ≠ 0) (h2 : d ∣ a) (h3 : d ∣ b) : d ≤ gcd a b := sorry -- 7. lemma Lemma_7_1_10a {a b : Nat} (n : Nat) (h : a ∣ b) : (n * a) ∣ (n * b) := sorry lemma Lemma_7_1_10b {a b n : Nat} (h1 : n ≠ 0) (h2 : (n * a) ∣ (n * b)) : a ∣ b := sorry lemma Lemma_7_1_10c {a b : Nat} (h1 : a ∣ b) (h2 : b ∣ a) : a = b := sorry theorem Exercise_7_1_10 (a b n : Nat) : gcd (n * a) (n * b) = n * gcd a b := sorry /- Section 7.2 -/ -- 1. lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry -- 2. -- Hints: Start with apply List.rec. You may find mul_ne_zero useful theorem prod_nonzero_nonzero : ∀ (l : List Nat), (∀ a ∈ l, a ≠ 0) → prod l ≠ 0 := sorry -- 3. theorem rel_prime_iff_no_common_factor (a b : Nat) : rel_prime a b ↔ ¬∃ (p : Nat), prime p ∧ p ∣ a ∧ p ∣ b := sorry -- 4. theorem rel_prime_symm {a b : Nat} (h : rel_prime a b) : rel_prime b a := sorry -- 5. lemma in_prime_factorization_iff_prime_factor {a : Nat} {l : List Nat} (h1 : prime_factorization a l) (p : Nat) : p ∈ l ↔ prime_factor p a := sorry -- 6. theorem Exercise_7_2_5 {a b : Nat} {l m : List Nat} (h1 : prime_factorization a l) (h2 : prime_factorization b m) : rel_prime a b ↔ (¬∃ (p : Nat), p ∈ l ∧ p ∈ m) := sorry -- 7. theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry -- 8. theorem Exercise_7_2_7 {a b a' b' : Nat} (h1 : rel_prime a b) (h2 : a' ∣ a) (h3 : b' ∣ b) : rel_prime a' b' := sorry -- 9. theorem Exercise_7_2_9 {a b j k : Nat} (h1 : gcd a b ≠ 0) (h2 : a = j * gcd a b) (h3 : b = k * gcd a b) : rel_prime j k := sorry -- 10. theorem Exercise_7_2_17a (a b c : Nat) : gcd a (b * c) ∣ gcd a b * gcd a c := sorry /- Section 7.3 -/ -- 1. theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry -- 2. theorem Theorem_7_3_6_3 {m : Nat} (X : ZMod m) : X + [0]_m = X := sorry -- 3. theorem Theorem_7_3_6_4 {m : Nat} (X : ZMod m) : ∃ (Y : ZMod m), X + Y = [0]_m := sorry -- 4. theorem Exercise_7_3_4a {m : Nat} (Z1 Z2 : ZMod m) (h1 : ∀ (X : ZMod m), X + Z1 = X) (h2 : ∀ (X : ZMod m), X + Z2 = X) : Z1 = Z2 := sorry -- 5.

theorem Exercise_7_3_4b {m : Nat} (X Y1 Y2 : ZMod m) (h1 : X + Y1 = [0]_m) (h2 : X + Y2 = [0]_m) : Y1 = Y2

HTPI.Exercises.Exercise_7_3_4b

htpi/HTPILib/Chap7.lean

htpi

/- Copyright 2023 Daniel J. Velleman -/ import HTPILib.Chap6 namespace HTPI /- Definitions -/ lemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by have h : n + 1 > 0 := Nat.succ_pos n show a % (n + 1) < n + 1 from Nat.mod_lt a h done def gcd (a b : Nat) : Nat := match b with | 0 => a | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd (n + 1) (a % (n + 1)) termination_by b mutual def gcd_c1 (a b : Nat) : Int := match b with | 0 => 1 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c2 (n + 1) (a % (n + 1)) --Corresponds to s = t' termination_by b def gcd_c2 (a b : Nat) : Int := match b with | 0 => 0 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c1 (n + 1) (a % (n + 1)) - (gcd_c2 (n + 1) (a % (n + 1))) * ↑(a / (n + 1)) --Corresponds to t = s' - t'q termination_by b end def prime (n : Nat) : Prop := 2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n def prime_factor (p n : Nat) : Prop := prime p ∧ p ∣ n def all_prime (l : List Nat) : Prop := ∀ p ∈ l, prime p def nondec (l : List Nat) : Prop := match l with | [] => True --Of course, True is a proposition that is always true | n :: L => (∀ m ∈ L, n ≤ m) ∧ nondec L def nondec_prime_list (l : List Nat) : Prop := all_prime l ∧ nondec l def prod (l : List Nat) : Nat := match l with | [] => 1 | n :: L => n * (prod L) def prime_factorization (n : Nat) (l : List Nat) : Prop := nondec_prime_list l ∧ prod l = n def rel_prime (a b : Nat) : Prop := gcd a b = 1 def congr_mod (m : Nat) (a b : Int) : Prop := (↑m : Int) ∣ (a - b) def cc (m : Nat) (a : Int) : ZMod m := (↑a : ZMod m) notation:50 a " ≡ " b " (MOD " m ")" => congr_mod m a b notation:max "["a"]_"m:max => cc m a def invertible {m : Nat} (X : ZMod m) : Prop := ∃ (Y : ZMod m), X * Y = [1]_m def num_rp_below (m k : Nat) : Nat := match k with | 0 => 0 | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j def phi (m : Nat) : Nat := num_rp_below m m def prod_seq {m : Nat} (j k : Nat) (f : Nat → ZMod m) : ZMod m := match j with | 0 => [1]_m | n + 1 => prod_seq n k f * f (k + n) def maps_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i < n, g i < n def one_one_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i1 < n, ∀ i2 < n, g i1 = g i2 → i1 = i2 def onto_below (n : Nat) (g : Nat → Nat) : Prop := ∀ k < n, ∃ i < n, g i = k def perm_below (n : Nat) (g : Nat → Nat) : Prop := maps_below n g ∧ one_one_below n g ∧ onto_below n g def inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m) def swap (u v i : Nat) : Nat := if i = u then v else if i = v then u else i namespace Euler --For definitions specific to Euler's theorem def F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m def G (m a i : Nat) : Nat := (a * i) % m def Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i end Euler /- Section 7.1 -/ theorem dvd_mod_of_dvd_a_b {a b d : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ (a % b) := by set q : Nat := a / b have h3 : b * q + a % b = a := Nat.div_add_mod a b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 define --Goal : ∃ (c : Nat), a % b = d * c apply Exists.intro (j - k * q) show a % b = d * (j - k * q) from calc a % b _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm _ = a - b * q := by rw [h3] _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc] _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm done theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry #eval gcd 672 161 --Answer: 7 lemma gcd_base (a : Nat) : gcd a 0 = a := by rfl lemma gcd_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd a b = gcd b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1)) rfl done lemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b ≠ 0) : a % b < b := by have h1 : b > 0 := Nat.pos_of_ne_zero h show a % b < b from Nat.mod_lt a h1 done lemma dvd_self (n : Nat) : n ∣ n := by apply Exists.intro 1 ring done theorem gcd_dvd : ∀ (b a : Nat), (gcd a b) ∣ a ∧ (gcd a b) ∣ b := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd a b_1) ∣ a ∧ (gcd a b_1) ∣ b_1 fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_base] --Goal: a ∣ a ∧ a ∣ 0 apply And.intro (dvd_self a) define apply Exists.intro 0 rfl done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_nonzero a h1] --Goal : gcd b (a % b) ∣ a ∧ gcd b (a % b) ∣ b have h2 : a % b < b := mod_nonzero_lt a h1 have h3 : (gcd b (a % b)) ∣ b ∧ (gcd b (a % b)) ∣ (a % b) := ih (a % b) h2 b apply And.intro _ h3.left show (gcd b (a % b)) ∣ a from dvd_a_of_dvd_b_mod h3.left h3.right done done theorem gcd_dvd_left (a b : Nat) : (gcd a b) ∣ a := (gcd_dvd b a).left theorem gcd_dvd_right (a b : Nat) : (gcd a b) ∣ b := (gcd_dvd b a).right lemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl lemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c1 a b = gcd_c2 b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done lemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl lemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * ↑(a / b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done theorem gcd_lin_comb : ∀ (b a : Nat), (gcd_c1 a b) * ↑a + (gcd_c2 a b) * ↑b = ↑(gcd a b) := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd_c1 a b_1) * ↑a + (gcd_c2 a b_1) * ↑b_1 = ↑(gcd a b_1) fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base] --Goal : 1 * ↑a + 0 * ↑0 = ↑a ring done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1] --Goal : gcd_c2 b (a % b) * ↑a + -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * ↑(a / b)) * ↑b = -- ↑(gcd b (a % b)) set r : Nat := a % b set q : Nat := a / b set s : Int := gcd_c1 b r set t : Int := gcd_c2 b r --Goal : t * ↑a + (s - t * ↑q) * ↑b = ↑(gcd b r) have h2 : r < b := mod_nonzero_lt a h1 have h3 : s * ↑b + t * ↑r = ↑(gcd b r) := ih r h2 b have h4 : b * q + r = a := Nat.div_add_mod a b rewrite [←h3, ←h4] rewrite [Nat.cast_add, Nat.cast_mul] --Goal : t * (↑b * ↑q + ↑r) + (s - t * ↑q) * ↑b = s * ↑b + t * ↑r ring done done #eval gcd_c1 672 161 --Answer: 6 #eval gcd_c2 672 161 --Answer: -25 --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161 theorem Theorem_7_1_6 {d a b : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ gcd a b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑d ∣ ↑(gcd a b) set s : Int := gcd_c1 a b set t : Int := gcd_c2 a b have h3 : s * ↑a + t * ↑b = ↑(gcd a b) := gcd_lin_comb b a rewrite [←h3] --Goal : ↑d ∣ s * ↑a + t * ↑b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul] --Goal : ↑d ∣ s * (↑d * ↑j) + t * (↑d * ↑k) define apply Exists.intro (s * ↑j + t * ↑k) ring done /- Section 7.2 -/ theorem dvd_trans {a b c : Nat} (h1 : a ∣ b) (h2 : b ∣ c) : a ∣ c := by define at h1; define at h2; define obtain (m : Nat) (h3 : b = a * m) from h1 obtain (n : Nat) (h4 : c = b * n) from h2 rewrite [h3, mul_assoc] at h4 apply Exists.intro (m * n) show c = a * (m * n) from h4 done lemma exists_prime_factor : ∀ (n : Nat), 2 ≤ n → ∃ (p : Nat), prime_factor p n := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, 2 ≤ n_1 → ∃ (p : Nat), prime_factor p n_1 assume h1 : 2 ≤ n by_cases h2 : prime n · -- Case 1. h2 : prime n apply Exists.intro n define --Goal : prime n ∧ n ∣ n show prime n ∧ n ∣ n from And.intro h2 (dvd_self n) done · -- Case 2. h2 : ¬prime n define at h2 --h2 : ¬(2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n) demorgan at h2 disj_syll h2 h1 obtain (a : Nat) (h3 : ∃ (b : Nat), a * b = n ∧ a < n ∧ b < n) from h2 obtain (b : Nat) (h4 : a * b = n ∧ a < n ∧ b < n) from h3 have h5 : 2 ≤ a := by by_contra h6 have h7 : a ≤ 1 := by linarith have h8 : n ≤ b := calc n _ = a * b := h4.left.symm _ ≤ 1 * b := by rel [h7] _ = b := by ring linarith --n ≤ b contradicts b < n done have h6 : ∃ (p : Nat), prime_factor p a := ih a h4.right.left h5 obtain (p : Nat) (h7 : prime_factor p a) from h6 apply Exists.intro p define --Goal : prime p ∧ p ∣ n define at h7 --h7 : prime p ∧ p ∣ a apply And.intro h7.left have h8 : a ∣ n := by apply Exists.intro b show n = a * b from (h4.left).symm done show p ∣ n from dvd_trans h7.right h8 done done lemma exists_least_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q := by set S : Set Nat := {p : Nat | prime_factor p n} have h2 : ∃ (p : Nat), p ∈ S := exists_prime_factor n h show ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q from well_ord_princ S h2 done lemma all_prime_nil : all_prime [] := by define --Goal : ∀ p ∈ [], prime p fix p : Nat contrapos --Goal : ¬prime p → p ∉ [] assume h1 : ¬prime p show p ∉ [] from List.not_mem_nil p done lemma all_prime_cons (n : Nat) (L : List Nat) : all_prime (n :: L) ↔ prime n ∧ all_prime L := by apply Iff.intro · -- (→) assume h1 : all_prime (n :: L) --Goal : prime n ∧ all_prime L define at h1 --h1 : ∀ p ∈ n :: L, prime p apply And.intro (h1 n (List.mem_cons_self n L)) define --Goal : ∀ p ∈ L, prime p fix p : Nat assume h2 : p ∈ L show prime p from h1 p (List.mem_cons_of_mem n h2) done · -- (←) assume h1 : prime n ∧ all_prime L --Goal : all_prime (n :: l) define : all_prime L at h1 define fix p : Nat assume h2 : p ∈ n :: L rewrite [List.mem_cons] at h2 --h2 : p = n ∨ p ∈ L by_cases on h2 · -- Case 1. h2 : p = n rewrite [h2] show prime n from h1.left done · -- Case 2. h2 : p ∈ L show prime p from h1.right p h2 done done done lemma nondec_nil : nondec [] := by define --Goal : True trivial --trivial proves some obviously true statements, such as True done lemma nondec_cons (n : Nat) (L : List Nat) : nondec (n :: L) ↔ (∀ m ∈ L, n ≤ m) ∧ nondec L := by rfl lemma prod_nil : prod [] = 1 := by rfl lemma prod_cons : prod (n :: L) = n * (prod L) := by rfl lemma exists_cons_of_length_eq_succ {A : Type} {l : List A} {n : Nat} (h : l.length = n + 1) : ∃ (a : A) (L : List A), l = a :: L ∧ L.length = n := by have h1 : ¬l.length = 0 := by linarith rewrite [List.length_eq_zero] at h1 obtain (a : A) (h2 : ∃ (L : List A), l = a :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List A) (h3 : l = a :: L) from h2 apply Exists.intro a apply Exists.intro L apply And.intro h3 have h4 : (a :: L).length = L.length + 1 := List.length_cons a L rewrite [←h3, h] at h4 show L.length = n from (Nat.add_right_cancel h4).symm done lemma list_elt_dvd_prod_by_length (a : Nat) : ∀ (n : Nat), ∀ (l : List Nat), l.length = n → a ∈ l → a ∣ prod l := by by_induc · --Base Case fix l : List Nat assume h1 : l.length = 0 rewrite [List.length_eq_zero] at h1 --h1 : l = [] rewrite [h1] --Goal : a ∈ [] → a ∣ prod [] contrapos assume h2 : ¬a ∣ prod [] show a ∉ [] from List.not_mem_nil a done · -- Induction Step fix n : Nat assume ih : ∀ (l : List Nat), List.length l = n → a ∈ l → a ∣ prod l fix l : List Nat assume h1 : l.length = n + 1 --Goal : a ∈ l → a ∣ prod l obtain (b : Nat) (h2 : ∃ (L : List Nat), l = b :: L ∧ L.length = n) from exists_cons_of_length_eq_succ h1 obtain (L : List Nat) (h3 : l = b :: L ∧ L.length = n) from h2 have h4 : a ∈ L → a ∣ prod L := ih L h3.right assume h5 : a ∈ l rewrite [h3.left, prod_cons] --Goal : a ∣ b * prod L rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b ∨ a ∈ L by_cases on h5 · -- Case 1. h5 : a = b apply Exists.intro (prod L) rewrite [h5] rfl done · -- Case 2. h5 : a ∈ L have h6 : a ∣ prod L := h4 h5 have h7 : prod L ∣ b * prod L := by apply Exists.intro b ring done show a ∣ b * prod L from dvd_trans h6 h7 done done done lemma list_elt_dvd_prod {a : Nat} {l : List Nat} (h : a ∈ l) : a ∣ prod l := by set n : Nat := l.length have h1 : l.length = n := by rfl show a ∣ prod l from list_elt_dvd_prod_by_length a n l h1 h done lemma exists_prime_factorization : ∀ (n : Nat), n ≥ 1 → ∃ (l : List Nat), prime_factorization n l := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, n_1 ≥ 1 → ∃ (l : List Nat), prime_factorization n_1 l assume h1 : n ≥ 1 by_cases h2 : n = 1 · -- Case 1. h2 : n = 1 apply Exists.intro [] define apply And.intro · -- Proof of nondec_prime_list [] define show all_prime [] ∧ nondec [] from And.intro all_prime_nil nondec_nil done · -- Proof of prod [] = n rewrite [prod_nil, h2] rfl done done · -- Case 2. h2 : n ≠ 1 have h3 : n ≥ 2 := lt_of_le_of_ne' h1 h2 obtain (p : Nat) (h4 : prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q) from exists_least_prime_factor h3 have p_prime_factor : prime_factor p n := h4.left define at p_prime_factor have p_prime : prime p := p_prime_factor.left have p_dvd_n : p ∣ n := p_prime_factor.right have p_least : ∀ (q : Nat), prime_factor q n → p ≤ q := h4.right obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n have h5 : m ≠ 0 := by contradict h1 with h6 have h7 : n = 0 := calc n _ = p * m := n_eq_pm _ = p * 0 := by rw [h6] _ = 0 := by ring rewrite [h7] decide done have m_pos : 0 < m := Nat.pos_of_ne_zero h5 have m_lt_n : m < n := by define at p_prime show m < n from calc m _ < m + m := by linarith _ = 2 * m := by ring _ ≤ p * m := by rel [p_prime.left] _ = n := n_eq_pm.symm done obtain (L : List Nat) (h6 : prime_factorization m L) from ih m m_lt_n m_pos define at h6 have ndpl_L : nondec_prime_list L := h6.left define at ndpl_L apply Exists.intro (p :: L) define apply And.intro · -- Proof of nondec_prime_list (p :: L) define apply And.intro · -- Proof of all_prime (p :: L) rewrite [all_prime_cons] show prime p ∧ all_prime L from And.intro p_prime ndpl_L.left done · -- Proof of nondec (p :: L) rewrite [nondec_cons] apply And.intro _ ndpl_L.right fix q : Nat assume q_in_L : q ∈ L have h7 : q ∣ prod L := list_elt_dvd_prod q_in_L rewrite [h6.right] at h7 --h7 : q ∣ m have h8 : m ∣ n := by apply Exists.intro p rewrite [n_eq_pm] ring done have q_dvd_n : q ∣ n := dvd_trans h7 h8 have ap_L : all_prime L := ndpl_L.left define at ap_L have q_prime_factor : prime_factor q n := And.intro (ap_L q q_in_L) q_dvd_n show p ≤ q from p_least q q_prime_factor done done · -- Proof of prod (p :: L) = n rewrite [prod_cons, h6.right, n_eq_pm] rfl done done done theorem Theorem_7_2_2 {a b c : Nat} (h1 : c ∣ a * b) (h2 : rel_prime a c) : c ∣ b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑c ∣ ↑b define at h1; define at h2; define obtain (j : Nat) (h3 : a * b = c * j) from h1 set s : Int := gcd_c1 a c set t : Int := gcd_c2 a c have h4 : s * ↑a + t * ↑c = ↑(gcd a c) := gcd_lin_comb c a rewrite [h2, Nat.cast_one] at h4 --h4 : s * ↑a + t * ↑c = (1 : Int) apply Exists.intro (s * ↑j + t * ↑b) show ↑b = ↑c * (s * ↑j + t * ↑b) from calc ↑b _ = (1 : Int) * ↑b := (one_mul _).symm _ = (s * ↑a + t * ↑c) * ↑b := by rw [h4] _ = s * (↑a * ↑b) + t * ↑c * ↑b := by ring _ = s * (↑c * ↑j) + t * ↑c * ↑b := by rw [←Nat.cast_mul a b, h3, Nat.cast_mul c j] _ = ↑c * (s * ↑j + t * ↑b) := by ring done lemma le_nonzero_prod_left {a b : Nat} (h : a * b ≠ 0) : a ≤ a * b := by have h1 : b ≠ 0 := by contradict h with h1 rewrite [h1] ring done have h2 : 1 ≤ b := Nat.pos_of_ne_zero h1 show a ≤ a * b from calc a = a * 1 := (mul_one a).symm _ ≤ a * b := by rel [h2] done lemma le_nonzero_prod_right {a b : Nat} (h : a * b ≠ 0) : b ≤ a * b := by rewrite [mul_comm] rewrite [mul_comm] at h show b ≤ b * a from le_nonzero_prod_left h done lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry lemma rel_prime_of_prime_not_dvd {a p : Nat} (h1 : prime p) (h2 : ¬p ∣ a) : rel_prime a p := by have h3 : gcd a p ∣ a := gcd_dvd_left a p have h4 : gcd a p ∣ p := gcd_dvd_right a p have h5 : gcd a p = 1 ∨ gcd a p = p := dvd_prime h1 h4 have h6 : gcd a p ≠ p := by contradict h2 with h6 rewrite [h6] at h3 show p ∣ a from h3 done disj_syll h5 h6 show rel_prime a p from h5 done theorem Theorem_7_2_3 {a b p : Nat} (h1 : prime p) (h2 : p ∣ a * b) : p ∣ a ∨ p ∣ b := by or_right with h3 have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3 show p ∣ b from Theorem_7_2_2 h2 h4 done lemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a ≥ 1 := by have h1 : a ≠ 0 := by by_contra h1 rewrite [h1] at h contradict h linarith done show a ≥ 1 from Nat.pos_of_ne_zero h1 done lemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by have h1 : a ≥ 1 := ge_one_of_prod_one h have h2 : a * b ≠ 0 := by linarith have h3 : a ≤ a * b := le_nonzero_prod_left h2 rewrite [h] at h3 show a = 1 from Nat.le_antisymm h3 h1 done lemma eq_one_of_dvd_one {n : Nat} (h : n ∣ 1) : n = 1 := by obtain (j : Nat) (h1 : 1 = n * j) from h show n = 1 from eq_one_of_prod_one h1.symm done lemma prime_not_one {p : Nat} (h : prime p) : p ≠ 1 := by define at h linarith done theorem Theorem_7_2_4 {p : Nat} (h1 : prime p) : ∀ (l : List Nat), p ∣ prod l → ∃ a ∈ l, p ∣ a := by apply List.rec · -- Base Case. Goal : p ∣ prod [] → ∃ a ∈ [], p ∣ a rewrite [prod_nil] assume h2 : p ∣ 1 show ∃ a ∈ [], p ∣ a from absurd (eq_one_of_dvd_one h2) (prime_not_one h1) done · -- Induction Step fix b : Nat fix L : List Nat assume ih : p ∣ prod L → ∃ a ∈ L, p ∣ a --Goal : p ∣ prod (b :: L) → ∃ a ∈ b :: L, p ∣ a assume h2 : p ∣ prod (b :: L) rewrite [prod_cons] at h2 have h3 : p ∣ b ∨ p ∣ prod L := Theorem_7_2_3 h1 h2 by_cases on h3 · -- Case 1. h3 : p ∣ b apply Exists.intro b show b ∈ b :: L ∧ p ∣ b from And.intro (List.mem_cons_self b L) h3 done · -- Case 2. h3 : p ∣ prod L obtain (a : Nat) (h4 : a ∈ L ∧ p ∣ a) from ih h3 apply Exists.intro a show a ∈ b :: L ∧ p ∣ a from And.intro (List.mem_cons_of_mem b h4.left) h4.right done done done lemma prime_in_list {p : Nat} {l : List Nat} (h1 : prime p) (h2 : all_prime l) (h3 : p ∣ prod l) : p ∈ l := by obtain (a : Nat) (h4 : a ∈ l ∧ p ∣ a) from Theorem_7_2_4 h1 l h3 define at h2 have h5 : prime a := h2 a h4.left have h6 : p = 1 ∨ p = a := dvd_prime h5 h4.right disj_syll h6 (prime_not_one h1) rewrite [h6] show a ∈ l from h4.left done lemma first_le_first {p q : Nat} {l m : List Nat} (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m)) (h3 : prod (p :: l) = prod (q :: m)) : p ≤ q := by define at h1; define at h2 have h4 : q ∣ prod (p :: l) := by define apply Exists.intro (prod m) rewrite [←prod_cons] show prod (p :: l) = prod (q :: m) from h3 done have h5 : all_prime (q :: m) := h2.left rewrite [all_prime_cons] at h5 have h6 : q ∈ p :: l := prime_in_list h5.left h1.left h4 have h7 : nondec (p :: l) := h1.right rewrite [nondec_cons] at h7 rewrite [List.mem_cons] at h6 by_cases on h6 · -- Case 1. h6 : q = p linarith done · -- Case 2. h6 : q ∈ l have h8 : ∀ m ∈ l, p ≤ m := h7.left show p ≤ q from h8 q h6 done done lemma nondec_prime_list_tail {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by define at h define rewrite [all_prime_cons, nondec_cons] at h show all_prime l ∧ nondec l from And.intro h.left.right h.right.right done lemma cons_prod_not_one {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : prod (p :: l) ≠ 1 := by define at h have h1 : all_prime (p :: l) := h.left rewrite [all_prime_cons] at h1 rewrite [prod_cons] by_contra h2 show False from (prime_not_one h1.left) (eq_one_of_prod_one h2) done lemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) : l = [] ↔ prod l = 1 := by apply Iff.intro · -- (→) assume h1 : l = [] rewrite [h1] show prod [] = 1 from prod_nil done · -- (←) contrapos assume h1 : ¬l = [] obtain (p : Nat) (h2 : ∃ (L : List Nat), l = p :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List Nat) (h3 : l = p :: L) from h2 rewrite [h3] at h rewrite [h3] show ¬prod (p :: L) = 1 from cons_prod_not_one h done done lemma prime_pos {p : Nat} (h : prime p) : p > 0 := by define at h linarith done theorem Theorem_7_2_5 : ∀ (l1 l2 : List Nat), nondec_prime_list l1 → nondec_prime_list l2 → prod l1 = prod l2 → l1 = l2 := by apply List.rec · -- Base Case. Goal : ∀ (l2 : List Nat), nondec_prime_list [] → -- nondec_prime_list l2 → prod [] = prod l2 → [] = l2 fix l2 : List Nat assume h1 : nondec_prime_list [] assume h2 : nondec_prime_list l2 assume h3 : prod [] = prod l2 rewrite [prod_nil, eq_comm, ←list_nil_iff_prod_one h2] at h3 show [] = l2 from h3.symm done · -- Induction Step fix p : Nat fix L1 : List Nat assume ih : ∀ (L2 : List Nat), nondec_prime_list L1 → nondec_prime_list L2 → prod L1 = prod L2 → L1 = L2 -- Goal : ∀ (l2 : List Nat), nondec_prime_list (p :: L1) → -- nondec_prime_list l2 → prod (p :: L1) = prod l2 → p :: L1 = l2 fix l2 : List Nat assume h1 : nondec_prime_list (p :: L1) assume h2 : nondec_prime_list l2 assume h3 : prod (p :: L1) = prod l2 have h4 : ¬prod (p :: L1) = 1 := cons_prod_not_one h1 rewrite [h3, ←list_nil_iff_prod_one h2] at h4 obtain (q : Nat) (h5 : ∃ (L : List Nat), l2 = q :: L) from List.exists_cons_of_ne_nil h4 obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5 rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2) rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2) have h7 : p ≤ q := first_le_first h1 h2 h3 have h8 : q ≤ p := first_le_first h2 h1 h3.symm have h9 : p = q := by linarith rewrite [h9, prod_cons, prod_cons] at h3 --h3 : q * prod L1 = q * prod L2 have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1 have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2 define at h2 have h12 : all_prime (q :: L2) := h2.left rewrite [all_prime_cons] at h12 have h13 : q > 0 := prime_pos h12.left have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3 have h15 : L1 = L2 := ih L2 h10 h11 h14 rewrite [h6, h9, h15] rfl done done theorem fund_thm_arith (n : Nat) (h : n ≥ 1) : ∃! (l : List Nat), prime_factorization n l := by exists_unique · -- Existence show ∃ (l : List Nat), prime_factorization n l from exists_prime_factorization n h done · -- Uniqueness fix l1 : List Nat; fix l2 : List Nat assume h1 : prime_factorization n l1 assume h2 : prime_factorization n l2 define at h1; define at h2 have h3 : prod l1 = n := h1.right rewrite [←h2.right] at h3 show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3 done done /- Section 7.3 -/ theorem congr_refl (m : Nat) : ∀ (a : Int), a ≡ a (MOD m) := by fix a : Int define --Goal : ∃ (c : Int), a - a = ↑m * c apply Exists.intro 0 ring done theorem congr_symm {m : Nat} : ∀ {a b : Int}, a ≡ b (MOD m) → b ≡ a (MOD m) := by fix a : Int; fix b : Int assume h1 : a ≡ b (MOD m) define at h1 --h1 : ∃ (c : Int), a - b = ↑m * c define --Goal : ∃ (c : Int), b - a = ↑m * c obtain (c : Int) (h2 : a - b = m * c) from h1 apply Exists.intro (-c) show b - a = m * (-c) from calc b - a _ = -(a - b) := by ring _ = -(m * c) := by rw [h2] _ = m * (-c) := by ring done theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry /- Fundamental properties of congruence classes -/ lemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) : X = Y ↔ X.val = Y.val := Fin.ext_iff lemma val_nat_eq_mod (n k : Nat) : ([k]_(n + 1)).val = k % (n + 1) := by rfl lemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl theorem cc_rep {m : Nat} (X : ZMod m) : ∃ (a : Int), X = [a]_m := match m with | 0 => by apply Exists.intro X rfl done | n + 1 => by apply Exists.intro ↑(X.val) have h1 : X.val < n + 1 := Fin.prop X rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1] rfl done theorem add_class (m : Nat) (a b : Int) : [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm theorem mul_class (m : Nat) (a b : Int) : [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm lemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ [a - b]_m = [0]_m := by apply Iff.intro · -- (→) assume h1 : [a]_m = [b]_m have h2 : a - b = a + (-b) := by ring have h3 : b + (-b) = 0 := by ring show [a - b]_m = [0]_m from calc [a - b]_m _ = [a + (-b)]_m := by rw [h2] _ = [a]_m + [-b]_m := by rw [add_class] _ = [b]_m + [-b]_m := by rw [h1] _ = [b + -b]_m := by rw [add_class] _ = [0]_m := by rw [h3] done · -- (←) assume h1 : [a - b]_m = [0]_m have h2 : b + (a - b) = a := by ring have h3 : b + 0 = b := by ring show [a]_m = [b]_m from calc [a]_m _ = [b + (a - b)]_m := by rw [h2] _ = [b]_m + [a - b]_m := by rw [add_class] _ = [b]_m + [0]_m := by rw [h1] _ = [b + 0]_m := by rw [add_class] _ = [b]_m := by rw [h3] done done lemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) : [a]_m = [0]_m → [-a]_m = [0]_m := by assume h1 : [a]_m = [0]_m have h2 : 0 + (-a) = -a := by ring have h3 : a + (-a) = 0 := by ring show [-a]_m = [0]_m from calc [-a]_m _ = [0 + (-a)]_m := by rw [h2] _ = [0]_m + [-a]_m := by rw [add_class] _ = [a]_m + [-a]_m := by rw [h1] _ = [a + (-a)]_m := by rw [add_class] _ = [0]_m := by rw [h3] done lemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) : [-a]_m = [0]_m ↔ [a]_m = [0]_m := by apply Iff.intro _ (cc_neg_zero_of_cc_zero m a) assume h1 : [-a]_m = [0]_m have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1 have h3 : -(-a) = a := by ring rewrite [h3] at h2 show [a]_m = [0]_m from h2 done lemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl lemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m ↔ m ∣ k := match m with | 0 => by have h : (0 : Int) = (↑(0 : Nat) : Int) := by rfl rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj] apply Iff.intro · -- (→) assume h1 : k = 0 rewrite [h1] show 0 ∣ 0 from dvd_self 0 done · -- (←) assume h1 : 0 ∣ k obtain (c : Nat) (h2 : k = 0 * c) from h1 rewrite [h2] ring done done | n + 1 => by rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero] show k % (n + 1) = 0 ↔ n + 1 ∣ k from (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm done lemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m ↔ ↑m ∣ a := by obtain (k : Nat) (h1 : a = ↑k ∨ a = -↑k) from Int.eq_nat_or_neg a by_cases on h1 · -- Case 1. h1: a = ↑k rewrite [h1, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done · -- Case 2. h1: a = -↑k rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done done theorem cc_eq_iff_congr (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ a ≡ b (MOD m) := calc [a]_m = [b]_m _ ↔ [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b _ ↔ ↑m ∣ (a - b) := cc_zero_iff_dvd m (a - b) _ ↔ a ≡ b (MOD m) := by rfl /- End of fundamental properties of congruence classes -/ lemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m := by have h1 : (↑m : Int) ≠ 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m) show 0 ≤ a % m from Int.emod_nonneg a h1 done lemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m) have h2 : (↑m : Int) > 0 := (Nat.cast_pos).rtl h1 show a % m < m from Int.emod_lt_of_pos a h2 done lemma congr_mod_mod (m : Nat) (a : Int) : a ≡ a % m (MOD m) := by define have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m apply Exists.intro (a / m) show a - a % m = m * (a / m) from calc a - (a % m) _ = m * (a / m) + a % m - a % m := by rw [h1] _ = m * (a / m) := by ring done lemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) := And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a)) theorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) : ∃! (r : Int), 0 ≤ r ∧ r < m ∧ a ≡ r (MOD m) := by exists_unique · -- Existence apply Exists.intro (a % m) show 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) from mod_cmpl_res m a done · -- Uniqueness fix r1 : Int; fix r2 : Int assume h1 : 0 ≤ r1 ∧ r1 < m ∧ a ≡ r1 (MOD m) assume h2 : 0 ≤ r2 ∧ r2 < m ∧ a ≡ r2 (MOD m) have h3 : r1 ≡ r2 (MOD m) := congr_trans (congr_symm h1.right.right) h2.right.right obtain (d : Int) (h4 : r1 - r2 = m * d) from h3 have h5 : r1 - r2 < m * 1 := by linarith have h6 : m * (-1) < r1 - r2 := by linarith rewrite [h4] at h5 --h5 : m * d < m * 1 rewrite [h4] at h6 --h6 : m * -1 < m * d have h7 : (↑m : Int) ≥ 0 := Nat.cast_nonneg m have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7 have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7 have h10 : d = 0 := by linarith show r1 = r2 from calc r1 _ = r1 - r2 + r2 := by ring _ = m * 0 + r2 := by rw [h4, h10] _ = r2 := by ring done done lemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m := (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a) theorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y rewrite [h1, h2] have h3 : a + b = b + a := by ring show [a]_m + [b]_m = [b]_m + [a]_m from calc [a]_m + [b]_m _ = [a + b]_m := add_class m a b _ = [b + a]_m := by rw [h3] _ = [b]_m + [a]_m := (add_class m b a).symm done theorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X rewrite [h1] have h2 : a * 1 = a := by ring show [a]_m * [1]_m = [a]_m from calc [a]_m * [1]_m _ = [a * 1]_m := mul_class m a 1 _ = [a]_m := by rw [h2] done theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry lemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) : [a]_m * [gcd_c2 m a]_m = [1]_m := by set s : Int := gcd_c1 m a have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m define at h1 rewrite [h1, Nat.cast_one] at h2 --h2 : s * ↑m + gcd_c2 m a * ↑a = 1 rewrite [mul_class, cc_eq_iff_congr] define --Goal : ∃ (c : Int), ↑a * gcd_c2 m a - 1 = ↑m * c apply Exists.intro (-s) show a * (gcd_c2 m a) - 1 = m * (-s) from calc a * (gcd_c2 m a) - 1 _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring _ = 1 + m * (-s) - 1 := by rw [h2] _ = m * (-s) := by ring done theorem Theorem_7_3_7 (m a : Nat) : invertible [a]_m ↔ rel_prime m a := by apply Iff.intro · -- (→) assume h1 : invertible [a]_m define at h1 obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1 obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y rewrite [h3, mul_class, cc_eq_iff_congr] at h2 define at h2 obtain (c : Int) (h4 : a * b - 1 = m * c) from h2 rewrite [Exercise_7_2_6] --Goal : ∃ (s t : Int), s * ↑m + t * ↑a = 1 apply Exists.intro (-c) apply Exists.intro b show (-c) * m + b * a = 1 from calc (-c) * m + b * a _ = (-c) * m + (a * b - 1) + 1 := by ring _ = (-c) * m + m * c + 1 := by rw [h4] _ = 1 := by ring done · -- (←) assume h1 : rel_prime m a define show ∃ (Y : ZMod m), [a]_m * Y = [1]_m from Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1) done done /- Section 7.4 -/ section Euler open Euler lemma num_rp_below_base {m : Nat} : num_rp_below m 0 = 0 := by rfl lemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) : num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by have h1 : num_rp_below m (j + 1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : gcd m j = 1 rewrite [if_pos h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1 show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1 done lemma num_rp_below_step_not_rp {m j : Nat} (h : ¬rel_prime m j) : num_rp_below m (j + 1) = num_rp_below m j := by have h1 : num_rp_below m (j +1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : ¬gcd m j = 1 rewrite [if_neg h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j show num_rp_below m (j + 1) = num_rp_below m j from h1 done lemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl #eval phi 10 --Answer: 4 lemma prod_inv_iff_inv {m : Nat} {X : ZMod m} (h1 : invertible X) (Y : ZMod m) : invertible (X * Y) ↔ invertible Y := by apply Iff.intro · -- (→) assume h2 : invertible (X * Y) obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2 apply Exists.intro (X * Z) rewrite [←h3] --Goal : Y * (X * Z) = X * Y * Z ring --Note that ring can do algebra in ZMod m done · -- (←) assume h2 : invertible Y obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1 obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2 apply Exists.intro (Xi * Yi) show (X * Y) * (Xi * Yi) = [1]_m from calc X * Y * (Xi * Yi) _ = (X * Xi) * (Y * Yi) := by ring _ = [1]_m * [1]_m := by rw [h3, h4] _ = [1]_m := Theorem_7_3_6_7 [1]_m done done lemma F_rp_def {m i : Nat} (h : rel_prime m i) : F m i = [i]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h --h : gcd m i = 1 rewrite [if_pos h] at h1 show F m i = [i]_m from h1 done lemma F_not_rp_def {m i : Nat} (h : ¬rel_prime m i) : F m i = [1]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h rewrite [h1, if_neg h] rfl done lemma prod_seq_base {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 0 k f = [1]_m := by rfl lemma prod_seq_step {m : Nat} (n k : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl lemma prod_seq_zero_step {m : Nat} (n : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by rewrite [prod_seq_step, zero_add] rfl done lemma prod_one {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 1 k f = f k := by rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7] rfl done lemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl lemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m := calc [G m a i]_m _ = [(a * i) % m]_m := by rfl _ = [a * i]_m := (cc_eq_mod m (a * i)).symm _ = [a]_m * [i]_m := (mul_class m a i).symm lemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) : rel_prime m (G m a i) ↔ rel_prime m i := by have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1 show rel_prime m (G m a i) ↔ rel_prime m i from calc rel_prime m (G m a i) _ ↔ invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm _ ↔ invertible ([a]_m * [i]_m) := by rw [cc_G] _ ↔ invertible [i]_m := prod_inv_iff_inv h2 ([i]_m) _ ↔ rel_prime m i := Theorem_7_3_7 m i done lemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) : F m (G m a i) = [a]_m * F m i := by have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2 show F m (G m a i) = [a]_m * F m i from calc F m (G m a i) _ = [G m a i]_m := F_rp_def h3 _ = [a]_m * [i]_m := cc_G m a i _ = [a]_m * F m i := by rw [F_rp_def h2] done lemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : ¬rel_prime m i) : F m (G m a i) = [1]_m := by rewrite [←G_rp_iff h1 i] at h2 show F m (G m a i) = [1]_m from F_not_rp_def h2 done lemma FG_prod {m a : Nat} (h1 : rel_prime m a) : ∀ (k : Nat), prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by by_induc · -- Base Case show prod_seq 0 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from calc prod_seq 0 0 ((F m) ∘ (G m a)) _ = [1]_m := prod_seq_base _ _ _ = [a]_m ^ 0 * [1]_m := by ring _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by rw [num_rp_below_base, prod_seq_base] done · -- Induction Step fix k : Nat assume ih : prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) by_cases h2 : rel_prime m k · -- Case 1. h2 : rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([a]_m * F m k) := by rw [FG_rp h1 h2] _ = [a]_m ^ ((num_rp_below m k) + 1) * ((prod_seq k 0 (F m)) * F m k) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_rp h2, prod_seq_zero_step] done · -- Case 2. h2 : ¬rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([1]_m) := by rw [FG_not_rp h1 h2] _ = [a]_m ^ (num_rp_below m k) * (prod_seq k 0 (F m) * ([1]_m)) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_not_rp h2, prod_seq_zero_step, F_not_rp_def h2] done done done lemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by define --Goal : ∀ i < m, G m a i < m fix i : Nat assume h1 : i < m rewrite [G_def] --Goal : a * i % m < m show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m) done lemma left_inv_one_one_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g' (g i) = i) : one_one_below n g := sorry lemma right_inv_onto_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g (g' i) = i) (h2 : maps_below n g') : onto_below n g := by define at h2; define fix k : Nat assume h3 : k < n apply Exists.intro (g' k) show g' k < n ∧ g (g' k) = k from And.intro (h2 k h3) (h1 k h3) done lemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m * [inv_mod m a]_m = [1]_m := by have h2 : 0 ≤ (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a) show [a]_m * [inv_mod m a]_m = [1]_m from calc [a]_m * [inv_mod m a]_m _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2] _ = [a]_m * [gcd_c2 m a]_m := by rw [←cc_eq_mod] _ = [1]_m := gcd_c2_inv h1 done lemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m := calc a * (b % m) % m = a % m * (b % m % m) % m := Nat.mul_mod _ _ _ _ = a % m * (b % m) % m := by rw [Nat.mod_mod] _ = a * b % m := (Nat.mul_mod _ _ _).symm lemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm] rfl done theorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] : ↑a ≡ ↑b (MOD m) ↔ a % m = b % m := sorry lemma mul_inv_mod_cancel {m a i : Nat} [NeZero m] (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1 rewrite [mul_class, cc_eq_iff_congr, ←Nat.cast_mul, ←Nat.cast_one, congr_iff_mod_eq_Nat] at h3 show a * inv_mod m a * i % m = i from calc a * (inv_mod m a) * i % m _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = 1 % m * i % m := by rw [h3] _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = i % m := by rw [one_mul] _ = i := Nat.mod_eq_of_lt h2 done lemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl lemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, G m a (Ginv m a i) = i := by fix i : Nat assume h2 : i < m show G m a (Ginv m a i) = i from calc G m a (Ginv m a i) _ = a * ((inv_mod m a * i) % m) % m := by rfl _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, Ginv m a (G m a i) = i := by fix i : Nat assume h2 : i < m show Ginv m a (G m a i) = i from calc Ginv m a (G m a i) _ = inv_mod m a * ((a * i) % m) % m := by rfl _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc, mul_comm (inv_mod m a)] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_maps_below (m a : Nat) [NeZero m] : maps_below m (Ginv m a) := G_maps_below m (inv_mod m a) lemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : one_one_below m (G m a) := left_inv_one_one_below (Ginv_left_inv h1) lemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : onto_below m (G m a) := right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a) lemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : perm_below m (G m a) := And.intro (G_maps_below m a) (And.intro (G_one_one_below h1) (G_onto_below h1)) --Permuting a product of congruence classes doesn't change product lemma swap_fst (u v : Nat) : swap u v u = v := by define : swap u v u --Goal : (if u = u then v else if u = v then u else u) = v have h : u = u := by rfl rewrite [if_pos h] rfl done lemma swap_snd (u v : Nat) : swap u v v = u := by define : swap u v v by_cases h1 : v = u · -- Case 1. h1 : v = u rewrite [if_pos h1] show v = u from h1 done · -- Case 2. h1 : v ≠ u rewrite [if_neg h1] have h2 : v = v := by rfl rewrite [if_pos h2] rfl done done lemma swap_other {u v i : Nat} (h1 : i ≠ u) (h2 : i ≠ v) : swap u v i = i := by define : swap u v i rewrite [if_neg h1, if_neg h2] rfl done lemma swap_values (u v i : Nat) : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := by by_cases h1 : i = u · -- Case 1. h1 : i = u apply Or.inl rewrite [h1] show swap u v u = v from swap_fst u v done · -- Case 2. h1 : i ≠ u apply Or.inr by_cases h2 : i = v · -- Case 2.1. h2 : i = v apply Or.inl rewrite [h2] show swap u v v = u from swap_snd u v done · -- Case 2.2. h2 : i ≠ v apply Or.inr show swap u v i = i from swap_other h1 h2 done done done lemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by define fix i : Nat assume h3 : i < n have h4 : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := swap_values u v i by_cases on h4 · -- Case 1. h4 : swap u v i = v rewrite [h4] show v < n from h2 done · -- Case 2. by_cases on h4 · -- Case 2.1. h4 : swap u v i = u rewrite [h4] show u < n from h1 done · -- Case 2.2. h4 : swap u v i = i rewrite [h4] show i < n from h3 done done done lemma swap_swap (u v n : Nat) : ∀ i < n, swap u v (swap u v i) = i := by fix i : Nat assume h : i < n by_cases h1 : i = u · -- Case 1. h1 : i = u rewrite [h1, swap_fst, swap_snd] rfl done · -- Case 2. h1 : i ≠ u by_cases h2 : i = v · -- Case 2.1. h2 : i = v rewrite [h2, swap_snd, swap_fst] rfl done · -- Case 2.2. h2 : i ≠ v rewrite [swap_other h1 h2, swap_other h1 h2] rfl done done done lemma swap_one_one_below (u v n) : one_one_below n (swap u v) := left_inv_one_one_below (swap_swap u v n) lemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) := right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2) lemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) := And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2)) lemma comp_perm_below {n : Nat} {f g : Nat → Nat} (h1 : perm_below n f) (h2 : perm_below n g) : perm_below n (f ∘ g) := sorry lemma trivial_swap (u : Nat) : swap u u = id := by apply funext fix x : Nat by_cases h1 : x = u · -- Case 1. h1 : x = u rewrite [h1, swap_fst] rfl done · -- Case 2. h1 : x ≠ u rewrite [swap_other h1 h1] rfl done done lemma prod_eq_fun {m : Nat} (f g : Nat → ZMod m) (k : Nat) : ∀ (n : Nat), (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g := by by_induc · -- Base Case assume h : (∀ i < 0, f (k + i) = g (k + i)) rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g assume h1 : ∀ i < n + 1, f (k + i) = g (k + i) have h2 : ∀ i < n, f (k + i) = g (k + i) := by fix i : Nat assume h2 : i < n have h3 : i < n + 1 := by linarith show f (k + i) = g (k + i) from h1 i h3 done have h3 : prod_seq n k f = prod_seq n k g := ih h2 have h4 : n < n + 1 := Nat.lt_succ_self n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4] rfl done done lemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := by have h2 : ∀ (i : Nat), i < u → (f ∘ swap u n) (0 + i) = f (0 + i) := by fix i : Nat assume h2 : i < u have h3 : 0 + i ≠ u := by linarith have h4 : 0 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl done show prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f from prod_eq_fun (f ∘ swap u n) f 0 u h2 done lemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := by have h2 : ∀ i < j, (f ∘ swap u n) (u + 1 + i) = f (u + 1 + i) := by fix i : Nat assume h2 : i < j have h3 : u + 1 + i ≠ u := by linarith have h4 : u + 1 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl show prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f from prod_eq_fun (f ∘ swap u n) f (u + 1) j h2 done lemma break_prod {m : Nat} (n : Nat) (f : Nat → ZMod m) : ∀ (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by by_induc · -- Base Case have h : n + 0 = n := by rfl rewrite [prod_seq_base, h, Theorem_7_3_6_7] rfl done · -- Induction Step fix j : Nat assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f rewrite [←add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc] rfl done done lemma break_prod_twice {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f := break_prod n f 1 rewrite [prod_one] at h2 have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f := break_prod (u + 1) f j rewrite [←h1] at h3 have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f := break_prod u f 1 rewrite [prod_one] at h4 rewrite [h3, h4] at h2 show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2 done lemma swap_prod_eq_prod {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f := by by_cases h2 : u = n · -- Case 1. h2 : u = n rewrite [h2, trivial_swap n] --Goal : prod_seq (n + 1) 0 (f ∘ id) = prod_seq (n + 1) 0 f rfl done · -- Case 2. h2 : ¬u = n have h3 : u + 1 ≤ n := Nat.lt_of_le_of_ne h1 h2 obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3 have break_f : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := break_prod_twice f h4 have break_fs : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_prod_twice (f ∘ swap u n) h4 have f_eq_fs_below : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := swap_prod_eq_prod_below f h1 have f_eq_fs_btwn : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4 show prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f from calc prod_seq (n + 1) 0 (f ∘ swap u n) _ = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_fs _ = prod_seq u 0 f * (f ∘ swap u n) u * prod_seq j (u + 1) f * (f ∘ swap u n) n := by rw [f_eq_fs_below, f_eq_fs_btwn] _ = prod_seq u 0 f * f (swap u n u) * prod_seq j (u + 1) f * f (swap u n n) := by rfl _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by rw [swap_fst, swap_snd] _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring _ = prod_seq (n + 1) 0 f := break_f.symm done done lemma perm_below_fixed {n : Nat} {g : Nat → Nat} (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry lemma perm_prod {m : Nat} (f : Nat → ZMod m) : ∀ (n : Nat), ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) := by by_induc · -- Base Case fix g : Nat → Nat assume h1 : perm_below 0 g rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) fix g : Nat → Nat assume g_pb : perm_below (n + 1) g define at g_pb have g_ob : onto_below (n + 1) g := g_pb.right.right define at g_ob have h1 : n < n + 1 := by linarith obtain (u : Nat) (h2 : u < n + 1 ∧ g u = n) from g_ob n h1 have s_pb : perm_below (n + 1) (swap u n) := swap_perm_below h2.left h1 have gs_pb_n1 : perm_below (n + 1) (g ∘ swap u n) := comp_perm_below g_pb s_pb have gs_fix_n : (g ∘ swap u n) n = n := calc (g ∘ swap u n) n _ = g (swap u n n) := by rfl _ = g u := by rw [swap_snd] _ = n := h2.right have gs_pb_n : perm_below n (g ∘ swap u n) := perm_below_fixed gs_pb_n1 gs_fix_n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f ∘ (g ∘ swap u n)) := ih (g ∘ swap u n) gs_pb_n have h3 : u ≤ n := by linarith show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f ∘ g) from calc prod_seq (n + 1) 0 f _ = prod_seq n 0 f * f n := prod_seq_zero_step n f _ = prod_seq n 0 (f ∘ (g ∘ swap u n)) * f ((g ∘ swap u n) n) := by rw [gs_prod, gs_fix_n] _ = prod_seq n 0 (f ∘ g ∘ swap u n) * (f ∘ g ∘ swap u n) n := by rfl _ = prod_seq (n + 1) 0 (f ∘ g ∘ swap u n) := (prod_seq_zero_step n (f ∘ g ∘ swap u n)).symm _ = prod_seq (n + 1) 0 ((f ∘ g) ∘ swap u n) := by rfl _ = prod_seq (n + 1) 0 (f ∘ g) := swap_prod_eq_prod (f ∘ g) h3 done done lemma F_invertible (m i : Nat) : invertible (F m i) := by by_cases h : rel_prime m i · -- Case 1. h : rel_prime m i rewrite [F_rp_def h] show invertible [i]_m from (Theorem_7_3_7 m i).rtl h done · -- Case 2. h : ¬rel_prime m i rewrite [F_not_rp_def h] apply Exists.intro [1]_m show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m done done lemma Fprod_invertible (m : Nat) : ∀ (k : Nat), invertible (prod_seq k 0 (F m)) := by by_induc · -- Base Case apply Exists.intro [1]_m show prod_seq 0 0 (F m) * [1]_m = [1]_m from calc prod_seq 0 0 (F m) * [1]_m _ = [1]_m * [1]_m := by rw [prod_seq_base] _ = [1]_m := Theorem_7_3_6_7 ([1]_m) done · -- Induction Step fix k : Nat assume ih : invertible (prod_seq k 0 (F m)) rewrite [prod_seq_zero_step] show invertible (prod_seq k 0 (F m) * (F m k)) from (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k) done done theorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m ^ (phi m) = [1]_m := by have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2 show [a]_m ^ (phi m) = [1]_m from calc [a]_m ^ (phi m) _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3] _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring _ = prod_seq m 0 (F m ∘ G m a) * Y := by rw [FG_prod h1 m, phi_def] _ = prod_seq m 0 (F m) * Y := by rw [perm_prod (F m) m (G m a) (G_perm_below h1)] _ = [1]_m := by rw [h3] done lemma Exercise_7_4_5_Int (m : Nat) (a : Int) : ∀ (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry lemma Exercise_7_4_5_Nat (m a n : Nat) : [a]_m ^ n = [a ^ n]_m := by rewrite [Exercise_7_4_5_Int] rfl done theorem Euler's_theorem {m a : Nat} [NeZero m] (h1 : rel_prime m a) : a ^ (phi m) ≡ 1 (MOD m) := by have h2 : [a]_m ^ (phi m) = [1]_m := Theorem_7_4_2 h1 rewrite [Exercise_7_4_5_Nat m a (phi m)] at h2 --h2 : [a ^ phi m]_m = [1]_m show a ^ (phi m) ≡ 1 (MOD m) from (cc_eq_iff_congr _ _ _).ltr h2 done #eval gcd 10 7 --Answer: 1. So 10 and 7 are relatively prime #eval 7 ^ phi 10 --Answer: 2401, which is congruent to 1 mod 10. end Euler /- Section 7.5 -/ lemma num_rp_prime {p : Nat} (h1 : prime p) : ∀ k < p, num_rp_below p (k + 1) = k := sorry lemma phi_prime {p : Nat} (h1 : prime p) : phi p = p - 1 := by have h2 : 1 ≤ p := prime_pos h1 have h3 : p - 1 + 1 = p := Nat.sub_add_cancel h2 have h4 : p - 1 < p := by linarith have h5 : num_rp_below p (p - 1 + 1) = p - 1 := num_rp_prime h1 (p - 1) h4 rewrite [h3] at h5 show phi p = p - 1 from h5 done theorem Theorem_7_2_2_Int {a c : Nat} {b : Int} (h1 : ↑c ∣ ↑a * b) (h2 : rel_prime a c) : ↑c ∣ b := by rewrite [Int.natCast_dvd, Int.natAbs_mul, Int.natAbs_ofNat] at h1 --h1 : c ∣ a * Int.natAbs b rewrite [Int.natCast_dvd] --Goal : c ∣ Int.natAbs b show c ∣ Int.natAbs b from Theorem_7_2_2 h1 h2 done lemma Lemma_7_4_5 {m n : Nat} (a b : Int) (h1 : rel_prime m n) : a ≡ b (MOD m * n) ↔ a ≡ b (MOD m) ∧ a ≡ b (MOD n) := by apply Iff.intro · -- (→) assume h2 : a ≡ b (MOD m * n) obtain (j : Int) (h3 : a - b = (m * n) * j) from h2 apply And.intro · -- Proof of a ≡ b (MOD m) apply Exists.intro (n * j) show a - b = m * (n * j) from calc a - b _ = m * n * j := h3 _ = m * (n * j) := by ring done · -- Proof of a ≡ b (MOD n) apply Exists.intro (m * j) show a - b = n * (m * j) from calc a - b _ = m * n * j := h3 _ = n * (m * j) := by ring done done · -- (←) assume h2 : a ≡ b (MOD m) ∧ a ≡ b (MOD n) obtain (j : Int) (h3 : a - b = m * j) from h2.left have h4 : (↑n : Int) ∣ a - b := h2.right rewrite [h3] at h4 --h4 : ↑n ∣ ↑m * j have h5 : ↑n ∣ j := Theorem_7_2_2_Int h4 h1 obtain (k : Int) (h6 : j = n * k) from h5 apply Exists.intro k --Goal : a - b = ↑(m * n) * k rewrite [Nat.cast_mul] --Goal : a - b = ↑m * ↑n * k show a - b = (m * n) * k from calc a - b _ = m * j := h3 _ = m * (n * k) := by rw [h6] _ = (m * n) * k := by ring done done --From exercises of Section 7.2 theorem rel_prime_symm {a b : Nat} (h : rel_prime a b) : rel_prime b a := sorry lemma prime_NeZero {p : Nat} (h : prime p) : NeZero p := by rewrite [neZero_iff] --Goal : p ≠ 0 define at h linarith done lemma Lemma_7_5_1 {p e d m c s : Nat} {t : Int} (h1 : prime p) (h2 : e * d = (p - 1) * s + 1) (h3 : m ^ e - c = p * t) : c ^ d ≡ m (MOD p) := by have h4 : m ^ e ≡ c (MOD p) := Exists.intro t h3 have h5 : [m ^ e]_p = [c]_p := (cc_eq_iff_congr _ _ _).rtl h4 rewrite [←Exercise_7_4_5_Nat] at h5 --h5 : [m]_p ^ e = [c]_p by_cases h6 : p ∣ m · -- Case 1. h6 : p ∣ m have h7 : m ≡ 0 (MOD p) := by obtain (j : Nat) (h8 : m = p * j) from h6 apply Exists.intro (↑j : Int) --Goal : ↑m - 0 = ↑p * ↑j rewrite [h8, Nat.cast_mul] ring done have h8 : [m]_p = [0]_p := (cc_eq_iff_congr _ _ _).rtl h7 have h9 : e * d ≠ 0 := by rewrite [h2] show (p - 1) * s + 1 ≠ 0 from Nat.add_one_ne_zero _ done have h10 : (0 : Int) ^ (e * d) = 0 := zero_pow h9 have h11 : [c ^ d]_p = [m]_p := calc [c ^ d]_p _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat] _ = ([m]_p ^ e) ^ d := by rw [h5] _ = [m]_p ^ (e * d) := by ring _ = [0]_p ^ (e * d) := by rw [h8] _ = [0 ^ (e * d)]_p := Exercise_7_4_5_Int _ _ _ _ = [0]_p := by rw [h10] _ = [m]_p := by rw [h8] show c ^ d ≡ m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11 done · -- Case 2. h6 : ¬p ∣ m have h7 : rel_prime m p := rel_prime_of_prime_not_dvd h1 h6 have h8 : rel_prime p m := rel_prime_symm h7 have h9 : NeZero p := prime_NeZero h1 have h10 : (1 : Int) ^ s = 1 := by ring have h11 : [c ^ d]_p = [m]_p := calc [c ^ d]_p _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat] _ = ([m]_p ^ e) ^ d := by rw [h5] _ = [m]_p ^ (e * d) := by ring _ = [m]_p ^ ((p - 1) * s + 1) := by rw [h2] _ = ([m]_p ^ (p - 1)) ^ s * [m]_p := by ring _ = ([m]_p ^ (phi p)) ^ s * [m]_p := by rw [phi_prime h1] _ = [1]_p ^ s * [m]_p := by rw [Theorem_7_4_2 h8] _ = [1 ^ s]_p * [m]_p := by rw [Exercise_7_4_5_Int] _ = [1]_p * [m]_p := by rw [h10] _ = [m]_p * [1]_p := by ring _ = [m]_p := Theorem_7_3_6_7 _ show c ^ d ≡ m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11 done done theorem Theorem_7_5_1 (p q n e d k m c : Nat) (p_prime : prime p) (q_prime : prime q) (p_ne_q : p ≠ q) (n_pq : n = p * q) (ed_congr_1 : e * d = k * (p - 1) * (q - 1) + 1) (h1 : [m]_n ^ e = [c]_n) : [c]_n ^ d = [m]_n := by rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] at h1 --h1 : m ^ e ≡ c (MOD n) rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] --Goal : c ^ d ≡ m (MOD n) obtain (j : Int) (h2 : m ^ e - c = n * j) from h1 rewrite [n_pq, Nat.cast_mul] at h2 --h2 : m ^ e - c = p * q * j have h3 : e * d = (p - 1) * (k * (q - 1)) + 1 := by rewrite [ed_congr_1] ring done have h4 : m ^ e - c = p * (q * j) := by rewrite [h2] ring done have congr_p : c ^ d ≡ m (MOD p) := Lemma_7_5_1 p_prime h3 h4 have h5 : e * d = (q - 1) * (k * (p - 1)) + 1 := by rewrite [ed_congr_1] ring done have h6 : m ^ e - c = q * (p * j) := by rewrite [h2] ring done have congr_q : c ^ d ≡ m (MOD q) := Lemma_7_5_1 q_prime h5 h6 have h7 : ¬q ∣ p := by by_contra h8 have h9 : q = 1 ∨ q = p := dvd_prime p_prime h8 disj_syll h9 (prime_not_one q_prime) show False from p_ne_q h9.symm done have h8 : rel_prime p q := rel_prime_of_prime_not_dvd q_prime h7 rewrite [n_pq, Lemma_7_4_5 _ _ h8] show c ^ d ≡ m (MOD p) ∧ c ^ d ≡ m (MOD q) from And.intro congr_p congr_q done /- BEGIN EXERCISES -/ namespace Exercises /- Section 7.1 -/ -- 1. theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry -- 2. lemma gcd_comm_lt {a b : Nat} (h : a < b) : gcd a b = gcd b a := sorry theorem gcd_comm (a b : Nat) : gcd a b = gcd b a := sorry -- 3. theorem Exercise_7_1_5 (a b : Nat) (n : Int) : (∃ (s t : Int), s * a + t * b = n) ↔ (↑(gcd a b) : Int) ∣ n := sorry -- 4. theorem Exercise_7_1_6 (a b c : Nat) : gcd a b = gcd (a + b * c) b := sorry -- 5. theorem gcd_is_nonzero {a b : Nat} (h : a ≠ 0 ∨ b ≠ 0) : gcd a b ≠ 0 := sorry -- 6. theorem gcd_greatest {a b d : Nat} (h1 : gcd a b ≠ 0) (h2 : d ∣ a) (h3 : d ∣ b) : d ≤ gcd a b := sorry -- 7. lemma Lemma_7_1_10a {a b : Nat} (n : Nat) (h : a ∣ b) : (n * a) ∣ (n * b) := sorry lemma Lemma_7_1_10b {a b n : Nat} (h1 : n ≠ 0) (h2 : (n * a) ∣ (n * b)) : a ∣ b := sorry lemma Lemma_7_1_10c {a b : Nat} (h1 : a ∣ b) (h2 : b ∣ a) : a = b := sorry theorem Exercise_7_1_10 (a b n : Nat) : gcd (n * a) (n * b) = n * gcd a b := sorry /- Section 7.2 -/ -- 1. lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry -- 2. -- Hints: Start with apply List.rec. You may find mul_ne_zero useful theorem prod_nonzero_nonzero : ∀ (l : List Nat), (∀ a ∈ l, a ≠ 0) → prod l ≠ 0 := sorry -- 3. theorem rel_prime_iff_no_common_factor (a b : Nat) : rel_prime a b ↔ ¬∃ (p : Nat), prime p ∧ p ∣ a ∧ p ∣ b := sorry -- 4. theorem rel_prime_symm {a b : Nat} (h : rel_prime a b) : rel_prime b a := sorry -- 5. lemma in_prime_factorization_iff_prime_factor {a : Nat} {l : List Nat} (h1 : prime_factorization a l) (p : Nat) : p ∈ l ↔ prime_factor p a := sorry -- 6. theorem Exercise_7_2_5 {a b : Nat} {l m : List Nat} (h1 : prime_factorization a l) (h2 : prime_factorization b m) : rel_prime a b ↔ (¬∃ (p : Nat), p ∈ l ∧ p ∈ m) := sorry -- 7. theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry -- 8. theorem Exercise_7_2_7 {a b a' b' : Nat} (h1 : rel_prime a b) (h2 : a' ∣ a) (h3 : b' ∣ b) : rel_prime a' b' := sorry -- 9. theorem Exercise_7_2_9 {a b j k : Nat} (h1 : gcd a b ≠ 0) (h2 : a = j * gcd a b) (h3 : b = k * gcd a b) : rel_prime j k := sorry -- 10. theorem Exercise_7_2_17a (a b c : Nat) : gcd a (b * c) ∣ gcd a b * gcd a c := sorry /- Section 7.3 -/ -- 1. theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry -- 2. theorem Theorem_7_3_6_3 {m : Nat} (X : ZMod m) : X + [0]_m = X := sorry -- 3. theorem Theorem_7_3_6_4 {m : Nat} (X : ZMod m) : ∃ (Y : ZMod m), X + Y = [0]_m := sorry -- 4. theorem Exercise_7_3_4a {m : Nat} (Z1 Z2 : ZMod m) (h1 : ∀ (X : ZMod m), X + Z1 = X) (h2 : ∀ (X : ZMod m), X + Z2 = X) : Z1 = Z2 := sorry -- 5. theorem Exercise_7_3_4b {m : Nat} (X Y1 Y2 : ZMod m) (h1 : X + Y1 = [0]_m) (h2 : X + Y2 = [0]_m) : Y1 = Y2 := sorry -- 6.

theorem Theorem_7_3_10 (m a : Nat) (b : Int) : ¬(↑(gcd m a) : Int) ∣ b → ¬∃ (x : Int), a * x ≡ b (MOD m)

HTPI.Exercises.Theorem_7_3_10

htpi/HTPILib/Chap7.lean

htpi

/- Copyright 2023 Daniel J. Velleman -/ import HTPILib.Chap6 namespace HTPI /- Definitions -/ lemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by have h : n + 1 > 0 := Nat.succ_pos n show a % (n + 1) < n + 1 from Nat.mod_lt a h done def gcd (a b : Nat) : Nat := match b with | 0 => a | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd (n + 1) (a % (n + 1)) termination_by b mutual def gcd_c1 (a b : Nat) : Int := match b with | 0 => 1 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c2 (n + 1) (a % (n + 1)) --Corresponds to s = t' termination_by b def gcd_c2 (a b : Nat) : Int := match b with | 0 => 0 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c1 (n + 1) (a % (n + 1)) - (gcd_c2 (n + 1) (a % (n + 1))) * ↑(a / (n + 1)) --Corresponds to t = s' - t'q termination_by b end def prime (n : Nat) : Prop := 2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n def prime_factor (p n : Nat) : Prop := prime p ∧ p ∣ n def all_prime (l : List Nat) : Prop := ∀ p ∈ l, prime p def nondec (l : List Nat) : Prop := match l with | [] => True --Of course, True is a proposition that is always true | n :: L => (∀ m ∈ L, n ≤ m) ∧ nondec L def nondec_prime_list (l : List Nat) : Prop := all_prime l ∧ nondec l def prod (l : List Nat) : Nat := match l with | [] => 1 | n :: L => n * (prod L) def prime_factorization (n : Nat) (l : List Nat) : Prop := nondec_prime_list l ∧ prod l = n def rel_prime (a b : Nat) : Prop := gcd a b = 1 def congr_mod (m : Nat) (a b : Int) : Prop := (↑m : Int) ∣ (a - b) def cc (m : Nat) (a : Int) : ZMod m := (↑a : ZMod m) notation:50 a " ≡ " b " (MOD " m ")" => congr_mod m a b notation:max "["a"]_"m:max => cc m a def invertible {m : Nat} (X : ZMod m) : Prop := ∃ (Y : ZMod m), X * Y = [1]_m def num_rp_below (m k : Nat) : Nat := match k with | 0 => 0 | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j def phi (m : Nat) : Nat := num_rp_below m m def prod_seq {m : Nat} (j k : Nat) (f : Nat → ZMod m) : ZMod m := match j with | 0 => [1]_m | n + 1 => prod_seq n k f * f (k + n) def maps_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i < n, g i < n def one_one_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i1 < n, ∀ i2 < n, g i1 = g i2 → i1 = i2 def onto_below (n : Nat) (g : Nat → Nat) : Prop := ∀ k < n, ∃ i < n, g i = k def perm_below (n : Nat) (g : Nat → Nat) : Prop := maps_below n g ∧ one_one_below n g ∧ onto_below n g def inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m) def swap (u v i : Nat) : Nat := if i = u then v else if i = v then u else i namespace Euler --For definitions specific to Euler's theorem def F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m def G (m a i : Nat) : Nat := (a * i) % m def Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i end Euler /- Section 7.1 -/ theorem dvd_mod_of_dvd_a_b {a b d : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ (a % b) := by set q : Nat := a / b have h3 : b * q + a % b = a := Nat.div_add_mod a b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 define --Goal : ∃ (c : Nat), a % b = d * c apply Exists.intro (j - k * q) show a % b = d * (j - k * q) from calc a % b _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm _ = a - b * q := by rw [h3] _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc] _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm done theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry #eval gcd 672 161 --Answer: 7 lemma gcd_base (a : Nat) : gcd a 0 = a := by rfl lemma gcd_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd a b = gcd b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1)) rfl done lemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b ≠ 0) : a % b < b := by have h1 : b > 0 := Nat.pos_of_ne_zero h show a % b < b from Nat.mod_lt a h1 done lemma dvd_self (n : Nat) : n ∣ n := by apply Exists.intro 1 ring done theorem gcd_dvd : ∀ (b a : Nat), (gcd a b) ∣ a ∧ (gcd a b) ∣ b := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd a b_1) ∣ a ∧ (gcd a b_1) ∣ b_1 fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_base] --Goal: a ∣ a ∧ a ∣ 0 apply And.intro (dvd_self a) define apply Exists.intro 0 rfl done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_nonzero a h1] --Goal : gcd b (a % b) ∣ a ∧ gcd b (a % b) ∣ b have h2 : a % b < b := mod_nonzero_lt a h1 have h3 : (gcd b (a % b)) ∣ b ∧ (gcd b (a % b)) ∣ (a % b) := ih (a % b) h2 b apply And.intro _ h3.left show (gcd b (a % b)) ∣ a from dvd_a_of_dvd_b_mod h3.left h3.right done done theorem gcd_dvd_left (a b : Nat) : (gcd a b) ∣ a := (gcd_dvd b a).left theorem gcd_dvd_right (a b : Nat) : (gcd a b) ∣ b := (gcd_dvd b a).right lemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl lemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c1 a b = gcd_c2 b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done lemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl lemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * ↑(a / b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done theorem gcd_lin_comb : ∀ (b a : Nat), (gcd_c1 a b) * ↑a + (gcd_c2 a b) * ↑b = ↑(gcd a b) := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd_c1 a b_1) * ↑a + (gcd_c2 a b_1) * ↑b_1 = ↑(gcd a b_1) fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base] --Goal : 1 * ↑a + 0 * ↑0 = ↑a ring done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1] --Goal : gcd_c2 b (a % b) * ↑a + -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * ↑(a / b)) * ↑b = -- ↑(gcd b (a % b)) set r : Nat := a % b set q : Nat := a / b set s : Int := gcd_c1 b r set t : Int := gcd_c2 b r --Goal : t * ↑a + (s - t * ↑q) * ↑b = ↑(gcd b r) have h2 : r < b := mod_nonzero_lt a h1 have h3 : s * ↑b + t * ↑r = ↑(gcd b r) := ih r h2 b have h4 : b * q + r = a := Nat.div_add_mod a b rewrite [←h3, ←h4] rewrite [Nat.cast_add, Nat.cast_mul] --Goal : t * (↑b * ↑q + ↑r) + (s - t * ↑q) * ↑b = s * ↑b + t * ↑r ring done done #eval gcd_c1 672 161 --Answer: 6 #eval gcd_c2 672 161 --Answer: -25 --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161 theorem Theorem_7_1_6 {d a b : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ gcd a b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑d ∣ ↑(gcd a b) set s : Int := gcd_c1 a b set t : Int := gcd_c2 a b have h3 : s * ↑a + t * ↑b = ↑(gcd a b) := gcd_lin_comb b a rewrite [←h3] --Goal : ↑d ∣ s * ↑a + t * ↑b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul] --Goal : ↑d ∣ s * (↑d * ↑j) + t * (↑d * ↑k) define apply Exists.intro (s * ↑j + t * ↑k) ring done /- Section 7.2 -/ theorem dvd_trans {a b c : Nat} (h1 : a ∣ b) (h2 : b ∣ c) : a ∣ c := by define at h1; define at h2; define obtain (m : Nat) (h3 : b = a * m) from h1 obtain (n : Nat) (h4 : c = b * n) from h2 rewrite [h3, mul_assoc] at h4 apply Exists.intro (m * n) show c = a * (m * n) from h4 done lemma exists_prime_factor : ∀ (n : Nat), 2 ≤ n → ∃ (p : Nat), prime_factor p n := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, 2 ≤ n_1 → ∃ (p : Nat), prime_factor p n_1 assume h1 : 2 ≤ n by_cases h2 : prime n · -- Case 1. h2 : prime n apply Exists.intro n define --Goal : prime n ∧ n ∣ n show prime n ∧ n ∣ n from And.intro h2 (dvd_self n) done · -- Case 2. h2 : ¬prime n define at h2 --h2 : ¬(2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n) demorgan at h2 disj_syll h2 h1 obtain (a : Nat) (h3 : ∃ (b : Nat), a * b = n ∧ a < n ∧ b < n) from h2 obtain (b : Nat) (h4 : a * b = n ∧ a < n ∧ b < n) from h3 have h5 : 2 ≤ a := by by_contra h6 have h7 : a ≤ 1 := by linarith have h8 : n ≤ b := calc n _ = a * b := h4.left.symm _ ≤ 1 * b := by rel [h7] _ = b := by ring linarith --n ≤ b contradicts b < n done have h6 : ∃ (p : Nat), prime_factor p a := ih a h4.right.left h5 obtain (p : Nat) (h7 : prime_factor p a) from h6 apply Exists.intro p define --Goal : prime p ∧ p ∣ n define at h7 --h7 : prime p ∧ p ∣ a apply And.intro h7.left have h8 : a ∣ n := by apply Exists.intro b show n = a * b from (h4.left).symm done show p ∣ n from dvd_trans h7.right h8 done done lemma exists_least_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q := by set S : Set Nat := {p : Nat | prime_factor p n} have h2 : ∃ (p : Nat), p ∈ S := exists_prime_factor n h show ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q from well_ord_princ S h2 done lemma all_prime_nil : all_prime [] := by define --Goal : ∀ p ∈ [], prime p fix p : Nat contrapos --Goal : ¬prime p → p ∉ [] assume h1 : ¬prime p show p ∉ [] from List.not_mem_nil p done lemma all_prime_cons (n : Nat) (L : List Nat) : all_prime (n :: L) ↔ prime n ∧ all_prime L := by apply Iff.intro · -- (→) assume h1 : all_prime (n :: L) --Goal : prime n ∧ all_prime L define at h1 --h1 : ∀ p ∈ n :: L, prime p apply And.intro (h1 n (List.mem_cons_self n L)) define --Goal : ∀ p ∈ L, prime p fix p : Nat assume h2 : p ∈ L show prime p from h1 p (List.mem_cons_of_mem n h2) done · -- (←) assume h1 : prime n ∧ all_prime L --Goal : all_prime (n :: l) define : all_prime L at h1 define fix p : Nat assume h2 : p ∈ n :: L rewrite [List.mem_cons] at h2 --h2 : p = n ∨ p ∈ L by_cases on h2 · -- Case 1. h2 : p = n rewrite [h2] show prime n from h1.left done · -- Case 2. h2 : p ∈ L show prime p from h1.right p h2 done done done lemma nondec_nil : nondec [] := by define --Goal : True trivial --trivial proves some obviously true statements, such as True done lemma nondec_cons (n : Nat) (L : List Nat) : nondec (n :: L) ↔ (∀ m ∈ L, n ≤ m) ∧ nondec L := by rfl lemma prod_nil : prod [] = 1 := by rfl lemma prod_cons : prod (n :: L) = n * (prod L) := by rfl lemma exists_cons_of_length_eq_succ {A : Type} {l : List A} {n : Nat} (h : l.length = n + 1) : ∃ (a : A) (L : List A), l = a :: L ∧ L.length = n := by have h1 : ¬l.length = 0 := by linarith rewrite [List.length_eq_zero] at h1 obtain (a : A) (h2 : ∃ (L : List A), l = a :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List A) (h3 : l = a :: L) from h2 apply Exists.intro a apply Exists.intro L apply And.intro h3 have h4 : (a :: L).length = L.length + 1 := List.length_cons a L rewrite [←h3, h] at h4 show L.length = n from (Nat.add_right_cancel h4).symm done lemma list_elt_dvd_prod_by_length (a : Nat) : ∀ (n : Nat), ∀ (l : List Nat), l.length = n → a ∈ l → a ∣ prod l := by by_induc · --Base Case fix l : List Nat assume h1 : l.length = 0 rewrite [List.length_eq_zero] at h1 --h1 : l = [] rewrite [h1] --Goal : a ∈ [] → a ∣ prod [] contrapos assume h2 : ¬a ∣ prod [] show a ∉ [] from List.not_mem_nil a done · -- Induction Step fix n : Nat assume ih : ∀ (l : List Nat), List.length l = n → a ∈ l → a ∣ prod l fix l : List Nat assume h1 : l.length = n + 1 --Goal : a ∈ l → a ∣ prod l obtain (b : Nat) (h2 : ∃ (L : List Nat), l = b :: L ∧ L.length = n) from exists_cons_of_length_eq_succ h1 obtain (L : List Nat) (h3 : l = b :: L ∧ L.length = n) from h2 have h4 : a ∈ L → a ∣ prod L := ih L h3.right assume h5 : a ∈ l rewrite [h3.left, prod_cons] --Goal : a ∣ b * prod L rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b ∨ a ∈ L by_cases on h5 · -- Case 1. h5 : a = b apply Exists.intro (prod L) rewrite [h5] rfl done · -- Case 2. h5 : a ∈ L have h6 : a ∣ prod L := h4 h5 have h7 : prod L ∣ b * prod L := by apply Exists.intro b ring done show a ∣ b * prod L from dvd_trans h6 h7 done done done lemma list_elt_dvd_prod {a : Nat} {l : List Nat} (h : a ∈ l) : a ∣ prod l := by set n : Nat := l.length have h1 : l.length = n := by rfl show a ∣ prod l from list_elt_dvd_prod_by_length a n l h1 h done lemma exists_prime_factorization : ∀ (n : Nat), n ≥ 1 → ∃ (l : List Nat), prime_factorization n l := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, n_1 ≥ 1 → ∃ (l : List Nat), prime_factorization n_1 l assume h1 : n ≥ 1 by_cases h2 : n = 1 · -- Case 1. h2 : n = 1 apply Exists.intro [] define apply And.intro · -- Proof of nondec_prime_list [] define show all_prime [] ∧ nondec [] from And.intro all_prime_nil nondec_nil done · -- Proof of prod [] = n rewrite [prod_nil, h2] rfl done done · -- Case 2. h2 : n ≠ 1 have h3 : n ≥ 2 := lt_of_le_of_ne' h1 h2 obtain (p : Nat) (h4 : prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q) from exists_least_prime_factor h3 have p_prime_factor : prime_factor p n := h4.left define at p_prime_factor have p_prime : prime p := p_prime_factor.left have p_dvd_n : p ∣ n := p_prime_factor.right have p_least : ∀ (q : Nat), prime_factor q n → p ≤ q := h4.right obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n have h5 : m ≠ 0 := by contradict h1 with h6 have h7 : n = 0 := calc n _ = p * m := n_eq_pm _ = p * 0 := by rw [h6] _ = 0 := by ring rewrite [h7] decide done have m_pos : 0 < m := Nat.pos_of_ne_zero h5 have m_lt_n : m < n := by define at p_prime show m < n from calc m _ < m + m := by linarith _ = 2 * m := by ring _ ≤ p * m := by rel [p_prime.left] _ = n := n_eq_pm.symm done obtain (L : List Nat) (h6 : prime_factorization m L) from ih m m_lt_n m_pos define at h6 have ndpl_L : nondec_prime_list L := h6.left define at ndpl_L apply Exists.intro (p :: L) define apply And.intro · -- Proof of nondec_prime_list (p :: L) define apply And.intro · -- Proof of all_prime (p :: L) rewrite [all_prime_cons] show prime p ∧ all_prime L from And.intro p_prime ndpl_L.left done · -- Proof of nondec (p :: L) rewrite [nondec_cons] apply And.intro _ ndpl_L.right fix q : Nat assume q_in_L : q ∈ L have h7 : q ∣ prod L := list_elt_dvd_prod q_in_L rewrite [h6.right] at h7 --h7 : q ∣ m have h8 : m ∣ n := by apply Exists.intro p rewrite [n_eq_pm] ring done have q_dvd_n : q ∣ n := dvd_trans h7 h8 have ap_L : all_prime L := ndpl_L.left define at ap_L have q_prime_factor : prime_factor q n := And.intro (ap_L q q_in_L) q_dvd_n show p ≤ q from p_least q q_prime_factor done done · -- Proof of prod (p :: L) = n rewrite [prod_cons, h6.right, n_eq_pm] rfl done done done theorem Theorem_7_2_2 {a b c : Nat} (h1 : c ∣ a * b) (h2 : rel_prime a c) : c ∣ b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑c ∣ ↑b define at h1; define at h2; define obtain (j : Nat) (h3 : a * b = c * j) from h1 set s : Int := gcd_c1 a c set t : Int := gcd_c2 a c have h4 : s * ↑a + t * ↑c = ↑(gcd a c) := gcd_lin_comb c a rewrite [h2, Nat.cast_one] at h4 --h4 : s * ↑a + t * ↑c = (1 : Int) apply Exists.intro (s * ↑j + t * ↑b) show ↑b = ↑c * (s * ↑j + t * ↑b) from calc ↑b _ = (1 : Int) * ↑b := (one_mul _).symm _ = (s * ↑a + t * ↑c) * ↑b := by rw [h4] _ = s * (↑a * ↑b) + t * ↑c * ↑b := by ring _ = s * (↑c * ↑j) + t * ↑c * ↑b := by rw [←Nat.cast_mul a b, h3, Nat.cast_mul c j] _ = ↑c * (s * ↑j + t * ↑b) := by ring done lemma le_nonzero_prod_left {a b : Nat} (h : a * b ≠ 0) : a ≤ a * b := by have h1 : b ≠ 0 := by contradict h with h1 rewrite [h1] ring done have h2 : 1 ≤ b := Nat.pos_of_ne_zero h1 show a ≤ a * b from calc a = a * 1 := (mul_one a).symm _ ≤ a * b := by rel [h2] done lemma le_nonzero_prod_right {a b : Nat} (h : a * b ≠ 0) : b ≤ a * b := by rewrite [mul_comm] rewrite [mul_comm] at h show b ≤ b * a from le_nonzero_prod_left h done lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry lemma rel_prime_of_prime_not_dvd {a p : Nat} (h1 : prime p) (h2 : ¬p ∣ a) : rel_prime a p := by have h3 : gcd a p ∣ a := gcd_dvd_left a p have h4 : gcd a p ∣ p := gcd_dvd_right a p have h5 : gcd a p = 1 ∨ gcd a p = p := dvd_prime h1 h4 have h6 : gcd a p ≠ p := by contradict h2 with h6 rewrite [h6] at h3 show p ∣ a from h3 done disj_syll h5 h6 show rel_prime a p from h5 done theorem Theorem_7_2_3 {a b p : Nat} (h1 : prime p) (h2 : p ∣ a * b) : p ∣ a ∨ p ∣ b := by or_right with h3 have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3 show p ∣ b from Theorem_7_2_2 h2 h4 done lemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a ≥ 1 := by have h1 : a ≠ 0 := by by_contra h1 rewrite [h1] at h contradict h linarith done show a ≥ 1 from Nat.pos_of_ne_zero h1 done lemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by have h1 : a ≥ 1 := ge_one_of_prod_one h have h2 : a * b ≠ 0 := by linarith have h3 : a ≤ a * b := le_nonzero_prod_left h2 rewrite [h] at h3 show a = 1 from Nat.le_antisymm h3 h1 done lemma eq_one_of_dvd_one {n : Nat} (h : n ∣ 1) : n = 1 := by obtain (j : Nat) (h1 : 1 = n * j) from h show n = 1 from eq_one_of_prod_one h1.symm done lemma prime_not_one {p : Nat} (h : prime p) : p ≠ 1 := by define at h linarith done theorem Theorem_7_2_4 {p : Nat} (h1 : prime p) : ∀ (l : List Nat), p ∣ prod l → ∃ a ∈ l, p ∣ a := by apply List.rec · -- Base Case. Goal : p ∣ prod [] → ∃ a ∈ [], p ∣ a rewrite [prod_nil] assume h2 : p ∣ 1 show ∃ a ∈ [], p ∣ a from absurd (eq_one_of_dvd_one h2) (prime_not_one h1) done · -- Induction Step fix b : Nat fix L : List Nat assume ih : p ∣ prod L → ∃ a ∈ L, p ∣ a --Goal : p ∣ prod (b :: L) → ∃ a ∈ b :: L, p ∣ a assume h2 : p ∣ prod (b :: L) rewrite [prod_cons] at h2 have h3 : p ∣ b ∨ p ∣ prod L := Theorem_7_2_3 h1 h2 by_cases on h3 · -- Case 1. h3 : p ∣ b apply Exists.intro b show b ∈ b :: L ∧ p ∣ b from And.intro (List.mem_cons_self b L) h3 done · -- Case 2. h3 : p ∣ prod L obtain (a : Nat) (h4 : a ∈ L ∧ p ∣ a) from ih h3 apply Exists.intro a show a ∈ b :: L ∧ p ∣ a from And.intro (List.mem_cons_of_mem b h4.left) h4.right done done done lemma prime_in_list {p : Nat} {l : List Nat} (h1 : prime p) (h2 : all_prime l) (h3 : p ∣ prod l) : p ∈ l := by obtain (a : Nat) (h4 : a ∈ l ∧ p ∣ a) from Theorem_7_2_4 h1 l h3 define at h2 have h5 : prime a := h2 a h4.left have h6 : p = 1 ∨ p = a := dvd_prime h5 h4.right disj_syll h6 (prime_not_one h1) rewrite [h6] show a ∈ l from h4.left done lemma first_le_first {p q : Nat} {l m : List Nat} (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m)) (h3 : prod (p :: l) = prod (q :: m)) : p ≤ q := by define at h1; define at h2 have h4 : q ∣ prod (p :: l) := by define apply Exists.intro (prod m) rewrite [←prod_cons] show prod (p :: l) = prod (q :: m) from h3 done have h5 : all_prime (q :: m) := h2.left rewrite [all_prime_cons] at h5 have h6 : q ∈ p :: l := prime_in_list h5.left h1.left h4 have h7 : nondec (p :: l) := h1.right rewrite [nondec_cons] at h7 rewrite [List.mem_cons] at h6 by_cases on h6 · -- Case 1. h6 : q = p linarith done · -- Case 2. h6 : q ∈ l have h8 : ∀ m ∈ l, p ≤ m := h7.left show p ≤ q from h8 q h6 done done lemma nondec_prime_list_tail {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by define at h define rewrite [all_prime_cons, nondec_cons] at h show all_prime l ∧ nondec l from And.intro h.left.right h.right.right done lemma cons_prod_not_one {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : prod (p :: l) ≠ 1 := by define at h have h1 : all_prime (p :: l) := h.left rewrite [all_prime_cons] at h1 rewrite [prod_cons] by_contra h2 show False from (prime_not_one h1.left) (eq_one_of_prod_one h2) done lemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) : l = [] ↔ prod l = 1 := by apply Iff.intro · -- (→) assume h1 : l = [] rewrite [h1] show prod [] = 1 from prod_nil done · -- (←) contrapos assume h1 : ¬l = [] obtain (p : Nat) (h2 : ∃ (L : List Nat), l = p :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List Nat) (h3 : l = p :: L) from h2 rewrite [h3] at h rewrite [h3] show ¬prod (p :: L) = 1 from cons_prod_not_one h done done lemma prime_pos {p : Nat} (h : prime p) : p > 0 := by define at h linarith done theorem Theorem_7_2_5 : ∀ (l1 l2 : List Nat), nondec_prime_list l1 → nondec_prime_list l2 → prod l1 = prod l2 → l1 = l2 := by apply List.rec · -- Base Case. Goal : ∀ (l2 : List Nat), nondec_prime_list [] → -- nondec_prime_list l2 → prod [] = prod l2 → [] = l2 fix l2 : List Nat assume h1 : nondec_prime_list [] assume h2 : nondec_prime_list l2 assume h3 : prod [] = prod l2 rewrite [prod_nil, eq_comm, ←list_nil_iff_prod_one h2] at h3 show [] = l2 from h3.symm done · -- Induction Step fix p : Nat fix L1 : List Nat assume ih : ∀ (L2 : List Nat), nondec_prime_list L1 → nondec_prime_list L2 → prod L1 = prod L2 → L1 = L2 -- Goal : ∀ (l2 : List Nat), nondec_prime_list (p :: L1) → -- nondec_prime_list l2 → prod (p :: L1) = prod l2 → p :: L1 = l2 fix l2 : List Nat assume h1 : nondec_prime_list (p :: L1) assume h2 : nondec_prime_list l2 assume h3 : prod (p :: L1) = prod l2 have h4 : ¬prod (p :: L1) = 1 := cons_prod_not_one h1 rewrite [h3, ←list_nil_iff_prod_one h2] at h4 obtain (q : Nat) (h5 : ∃ (L : List Nat), l2 = q :: L) from List.exists_cons_of_ne_nil h4 obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5 rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2) rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2) have h7 : p ≤ q := first_le_first h1 h2 h3 have h8 : q ≤ p := first_le_first h2 h1 h3.symm have h9 : p = q := by linarith rewrite [h9, prod_cons, prod_cons] at h3 --h3 : q * prod L1 = q * prod L2 have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1 have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2 define at h2 have h12 : all_prime (q :: L2) := h2.left rewrite [all_prime_cons] at h12 have h13 : q > 0 := prime_pos h12.left have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3 have h15 : L1 = L2 := ih L2 h10 h11 h14 rewrite [h6, h9, h15] rfl done done theorem fund_thm_arith (n : Nat) (h : n ≥ 1) : ∃! (l : List Nat), prime_factorization n l := by exists_unique · -- Existence show ∃ (l : List Nat), prime_factorization n l from exists_prime_factorization n h done · -- Uniqueness fix l1 : List Nat; fix l2 : List Nat assume h1 : prime_factorization n l1 assume h2 : prime_factorization n l2 define at h1; define at h2 have h3 : prod l1 = n := h1.right rewrite [←h2.right] at h3 show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3 done done /- Section 7.3 -/ theorem congr_refl (m : Nat) : ∀ (a : Int), a ≡ a (MOD m) := by fix a : Int define --Goal : ∃ (c : Int), a - a = ↑m * c apply Exists.intro 0 ring done theorem congr_symm {m : Nat} : ∀ {a b : Int}, a ≡ b (MOD m) → b ≡ a (MOD m) := by fix a : Int; fix b : Int assume h1 : a ≡ b (MOD m) define at h1 --h1 : ∃ (c : Int), a - b = ↑m * c define --Goal : ∃ (c : Int), b - a = ↑m * c obtain (c : Int) (h2 : a - b = m * c) from h1 apply Exists.intro (-c) show b - a = m * (-c) from calc b - a _ = -(a - b) := by ring _ = -(m * c) := by rw [h2] _ = m * (-c) := by ring done theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry /- Fundamental properties of congruence classes -/ lemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) : X = Y ↔ X.val = Y.val := Fin.ext_iff lemma val_nat_eq_mod (n k : Nat) : ([k]_(n + 1)).val = k % (n + 1) := by rfl lemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl theorem cc_rep {m : Nat} (X : ZMod m) : ∃ (a : Int), X = [a]_m := match m with | 0 => by apply Exists.intro X rfl done | n + 1 => by apply Exists.intro ↑(X.val) have h1 : X.val < n + 1 := Fin.prop X rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1] rfl done theorem add_class (m : Nat) (a b : Int) : [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm theorem mul_class (m : Nat) (a b : Int) : [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm lemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ [a - b]_m = [0]_m := by apply Iff.intro · -- (→) assume h1 : [a]_m = [b]_m have h2 : a - b = a + (-b) := by ring have h3 : b + (-b) = 0 := by ring show [a - b]_m = [0]_m from calc [a - b]_m _ = [a + (-b)]_m := by rw [h2] _ = [a]_m + [-b]_m := by rw [add_class] _ = [b]_m + [-b]_m := by rw [h1] _ = [b + -b]_m := by rw [add_class] _ = [0]_m := by rw [h3] done · -- (←) assume h1 : [a - b]_m = [0]_m have h2 : b + (a - b) = a := by ring have h3 : b + 0 = b := by ring show [a]_m = [b]_m from calc [a]_m _ = [b + (a - b)]_m := by rw [h2] _ = [b]_m + [a - b]_m := by rw [add_class] _ = [b]_m + [0]_m := by rw [h1] _ = [b + 0]_m := by rw [add_class] _ = [b]_m := by rw [h3] done done lemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) : [a]_m = [0]_m → [-a]_m = [0]_m := by assume h1 : [a]_m = [0]_m have h2 : 0 + (-a) = -a := by ring have h3 : a + (-a) = 0 := by ring show [-a]_m = [0]_m from calc [-a]_m _ = [0 + (-a)]_m := by rw [h2] _ = [0]_m + [-a]_m := by rw [add_class] _ = [a]_m + [-a]_m := by rw [h1] _ = [a + (-a)]_m := by rw [add_class] _ = [0]_m := by rw [h3] done lemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) : [-a]_m = [0]_m ↔ [a]_m = [0]_m := by apply Iff.intro _ (cc_neg_zero_of_cc_zero m a) assume h1 : [-a]_m = [0]_m have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1 have h3 : -(-a) = a := by ring rewrite [h3] at h2 show [a]_m = [0]_m from h2 done lemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl lemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m ↔ m ∣ k := match m with | 0 => by have h : (0 : Int) = (↑(0 : Nat) : Int) := by rfl rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj] apply Iff.intro · -- (→) assume h1 : k = 0 rewrite [h1] show 0 ∣ 0 from dvd_self 0 done · -- (←) assume h1 : 0 ∣ k obtain (c : Nat) (h2 : k = 0 * c) from h1 rewrite [h2] ring done done | n + 1 => by rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero] show k % (n + 1) = 0 ↔ n + 1 ∣ k from (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm done lemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m ↔ ↑m ∣ a := by obtain (k : Nat) (h1 : a = ↑k ∨ a = -↑k) from Int.eq_nat_or_neg a by_cases on h1 · -- Case 1. h1: a = ↑k rewrite [h1, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done · -- Case 2. h1: a = -↑k rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done done theorem cc_eq_iff_congr (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ a ≡ b (MOD m) := calc [a]_m = [b]_m _ ↔ [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b _ ↔ ↑m ∣ (a - b) := cc_zero_iff_dvd m (a - b) _ ↔ a ≡ b (MOD m) := by rfl /- End of fundamental properties of congruence classes -/ lemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m := by have h1 : (↑m : Int) ≠ 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m) show 0 ≤ a % m from Int.emod_nonneg a h1 done lemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m) have h2 : (↑m : Int) > 0 := (Nat.cast_pos).rtl h1 show a % m < m from Int.emod_lt_of_pos a h2 done lemma congr_mod_mod (m : Nat) (a : Int) : a ≡ a % m (MOD m) := by define have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m apply Exists.intro (a / m) show a - a % m = m * (a / m) from calc a - (a % m) _ = m * (a / m) + a % m - a % m := by rw [h1] _ = m * (a / m) := by ring done lemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) := And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a)) theorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) : ∃! (r : Int), 0 ≤ r ∧ r < m ∧ a ≡ r (MOD m) := by exists_unique · -- Existence apply Exists.intro (a % m) show 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) from mod_cmpl_res m a done · -- Uniqueness fix r1 : Int; fix r2 : Int assume h1 : 0 ≤ r1 ∧ r1 < m ∧ a ≡ r1 (MOD m) assume h2 : 0 ≤ r2 ∧ r2 < m ∧ a ≡ r2 (MOD m) have h3 : r1 ≡ r2 (MOD m) := congr_trans (congr_symm h1.right.right) h2.right.right obtain (d : Int) (h4 : r1 - r2 = m * d) from h3 have h5 : r1 - r2 < m * 1 := by linarith have h6 : m * (-1) < r1 - r2 := by linarith rewrite [h4] at h5 --h5 : m * d < m * 1 rewrite [h4] at h6 --h6 : m * -1 < m * d have h7 : (↑m : Int) ≥ 0 := Nat.cast_nonneg m have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7 have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7 have h10 : d = 0 := by linarith show r1 = r2 from calc r1 _ = r1 - r2 + r2 := by ring _ = m * 0 + r2 := by rw [h4, h10] _ = r2 := by ring done done lemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m := (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a) theorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y rewrite [h1, h2] have h3 : a + b = b + a := by ring show [a]_m + [b]_m = [b]_m + [a]_m from calc [a]_m + [b]_m _ = [a + b]_m := add_class m a b _ = [b + a]_m := by rw [h3] _ = [b]_m + [a]_m := (add_class m b a).symm done theorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X rewrite [h1] have h2 : a * 1 = a := by ring show [a]_m * [1]_m = [a]_m from calc [a]_m * [1]_m _ = [a * 1]_m := mul_class m a 1 _ = [a]_m := by rw [h2] done theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry lemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) : [a]_m * [gcd_c2 m a]_m = [1]_m := by set s : Int := gcd_c1 m a have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m define at h1 rewrite [h1, Nat.cast_one] at h2 --h2 : s * ↑m + gcd_c2 m a * ↑a = 1 rewrite [mul_class, cc_eq_iff_congr] define --Goal : ∃ (c : Int), ↑a * gcd_c2 m a - 1 = ↑m * c apply Exists.intro (-s) show a * (gcd_c2 m a) - 1 = m * (-s) from calc a * (gcd_c2 m a) - 1 _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring _ = 1 + m * (-s) - 1 := by rw [h2] _ = m * (-s) := by ring done theorem Theorem_7_3_7 (m a : Nat) : invertible [a]_m ↔ rel_prime m a := by apply Iff.intro · -- (→) assume h1 : invertible [a]_m define at h1 obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1 obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y rewrite [h3, mul_class, cc_eq_iff_congr] at h2 define at h2 obtain (c : Int) (h4 : a * b - 1 = m * c) from h2 rewrite [Exercise_7_2_6] --Goal : ∃ (s t : Int), s * ↑m + t * ↑a = 1 apply Exists.intro (-c) apply Exists.intro b show (-c) * m + b * a = 1 from calc (-c) * m + b * a _ = (-c) * m + (a * b - 1) + 1 := by ring _ = (-c) * m + m * c + 1 := by rw [h4] _ = 1 := by ring done · -- (←) assume h1 : rel_prime m a define show ∃ (Y : ZMod m), [a]_m * Y = [1]_m from Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1) done done /- Section 7.4 -/ section Euler open Euler lemma num_rp_below_base {m : Nat} : num_rp_below m 0 = 0 := by rfl lemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) : num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by have h1 : num_rp_below m (j + 1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : gcd m j = 1 rewrite [if_pos h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1 show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1 done lemma num_rp_below_step_not_rp {m j : Nat} (h : ¬rel_prime m j) : num_rp_below m (j + 1) = num_rp_below m j := by have h1 : num_rp_below m (j +1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : ¬gcd m j = 1 rewrite [if_neg h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j show num_rp_below m (j + 1) = num_rp_below m j from h1 done lemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl #eval phi 10 --Answer: 4 lemma prod_inv_iff_inv {m : Nat} {X : ZMod m} (h1 : invertible X) (Y : ZMod m) : invertible (X * Y) ↔ invertible Y := by apply Iff.intro · -- (→) assume h2 : invertible (X * Y) obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2 apply Exists.intro (X * Z) rewrite [←h3] --Goal : Y * (X * Z) = X * Y * Z ring --Note that ring can do algebra in ZMod m done · -- (←) assume h2 : invertible Y obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1 obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2 apply Exists.intro (Xi * Yi) show (X * Y) * (Xi * Yi) = [1]_m from calc X * Y * (Xi * Yi) _ = (X * Xi) * (Y * Yi) := by ring _ = [1]_m * [1]_m := by rw [h3, h4] _ = [1]_m := Theorem_7_3_6_7 [1]_m done done lemma F_rp_def {m i : Nat} (h : rel_prime m i) : F m i = [i]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h --h : gcd m i = 1 rewrite [if_pos h] at h1 show F m i = [i]_m from h1 done lemma F_not_rp_def {m i : Nat} (h : ¬rel_prime m i) : F m i = [1]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h rewrite [h1, if_neg h] rfl done lemma prod_seq_base {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 0 k f = [1]_m := by rfl lemma prod_seq_step {m : Nat} (n k : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl lemma prod_seq_zero_step {m : Nat} (n : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by rewrite [prod_seq_step, zero_add] rfl done lemma prod_one {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 1 k f = f k := by rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7] rfl done lemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl lemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m := calc [G m a i]_m _ = [(a * i) % m]_m := by rfl _ = [a * i]_m := (cc_eq_mod m (a * i)).symm _ = [a]_m * [i]_m := (mul_class m a i).symm lemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) : rel_prime m (G m a i) ↔ rel_prime m i := by have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1 show rel_prime m (G m a i) ↔ rel_prime m i from calc rel_prime m (G m a i) _ ↔ invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm _ ↔ invertible ([a]_m * [i]_m) := by rw [cc_G] _ ↔ invertible [i]_m := prod_inv_iff_inv h2 ([i]_m) _ ↔ rel_prime m i := Theorem_7_3_7 m i done lemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) : F m (G m a i) = [a]_m * F m i := by have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2 show F m (G m a i) = [a]_m * F m i from calc F m (G m a i) _ = [G m a i]_m := F_rp_def h3 _ = [a]_m * [i]_m := cc_G m a i _ = [a]_m * F m i := by rw [F_rp_def h2] done lemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : ¬rel_prime m i) : F m (G m a i) = [1]_m := by rewrite [←G_rp_iff h1 i] at h2 show F m (G m a i) = [1]_m from F_not_rp_def h2 done lemma FG_prod {m a : Nat} (h1 : rel_prime m a) : ∀ (k : Nat), prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by by_induc · -- Base Case show prod_seq 0 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from calc prod_seq 0 0 ((F m) ∘ (G m a)) _ = [1]_m := prod_seq_base _ _ _ = [a]_m ^ 0 * [1]_m := by ring _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by rw [num_rp_below_base, prod_seq_base] done · -- Induction Step fix k : Nat assume ih : prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) by_cases h2 : rel_prime m k · -- Case 1. h2 : rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([a]_m * F m k) := by rw [FG_rp h1 h2] _ = [a]_m ^ ((num_rp_below m k) + 1) * ((prod_seq k 0 (F m)) * F m k) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_rp h2, prod_seq_zero_step] done · -- Case 2. h2 : ¬rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([1]_m) := by rw [FG_not_rp h1 h2] _ = [a]_m ^ (num_rp_below m k) * (prod_seq k 0 (F m) * ([1]_m)) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_not_rp h2, prod_seq_zero_step, F_not_rp_def h2] done done done lemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by define --Goal : ∀ i < m, G m a i < m fix i : Nat assume h1 : i < m rewrite [G_def] --Goal : a * i % m < m show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m) done lemma left_inv_one_one_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g' (g i) = i) : one_one_below n g := sorry lemma right_inv_onto_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g (g' i) = i) (h2 : maps_below n g') : onto_below n g := by define at h2; define fix k : Nat assume h3 : k < n apply Exists.intro (g' k) show g' k < n ∧ g (g' k) = k from And.intro (h2 k h3) (h1 k h3) done lemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m * [inv_mod m a]_m = [1]_m := by have h2 : 0 ≤ (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a) show [a]_m * [inv_mod m a]_m = [1]_m from calc [a]_m * [inv_mod m a]_m _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2] _ = [a]_m * [gcd_c2 m a]_m := by rw [←cc_eq_mod] _ = [1]_m := gcd_c2_inv h1 done lemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m := calc a * (b % m) % m = a % m * (b % m % m) % m := Nat.mul_mod _ _ _ _ = a % m * (b % m) % m := by rw [Nat.mod_mod] _ = a * b % m := (Nat.mul_mod _ _ _).symm lemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm] rfl done theorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] : ↑a ≡ ↑b (MOD m) ↔ a % m = b % m := sorry lemma mul_inv_mod_cancel {m a i : Nat} [NeZero m] (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1 rewrite [mul_class, cc_eq_iff_congr, ←Nat.cast_mul, ←Nat.cast_one, congr_iff_mod_eq_Nat] at h3 show a * inv_mod m a * i % m = i from calc a * (inv_mod m a) * i % m _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = 1 % m * i % m := by rw [h3] _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = i % m := by rw [one_mul] _ = i := Nat.mod_eq_of_lt h2 done lemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl lemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, G m a (Ginv m a i) = i := by fix i : Nat assume h2 : i < m show G m a (Ginv m a i) = i from calc G m a (Ginv m a i) _ = a * ((inv_mod m a * i) % m) % m := by rfl _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, Ginv m a (G m a i) = i := by fix i : Nat assume h2 : i < m show Ginv m a (G m a i) = i from calc Ginv m a (G m a i) _ = inv_mod m a * ((a * i) % m) % m := by rfl _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc, mul_comm (inv_mod m a)] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_maps_below (m a : Nat) [NeZero m] : maps_below m (Ginv m a) := G_maps_below m (inv_mod m a) lemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : one_one_below m (G m a) := left_inv_one_one_below (Ginv_left_inv h1) lemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : onto_below m (G m a) := right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a) lemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : perm_below m (G m a) := And.intro (G_maps_below m a) (And.intro (G_one_one_below h1) (G_onto_below h1)) --Permuting a product of congruence classes doesn't change product lemma swap_fst (u v : Nat) : swap u v u = v := by define : swap u v u --Goal : (if u = u then v else if u = v then u else u) = v have h : u = u := by rfl rewrite [if_pos h] rfl done lemma swap_snd (u v : Nat) : swap u v v = u := by define : swap u v v by_cases h1 : v = u · -- Case 1. h1 : v = u rewrite [if_pos h1] show v = u from h1 done · -- Case 2. h1 : v ≠ u rewrite [if_neg h1] have h2 : v = v := by rfl rewrite [if_pos h2] rfl done done lemma swap_other {u v i : Nat} (h1 : i ≠ u) (h2 : i ≠ v) : swap u v i = i := by define : swap u v i rewrite [if_neg h1, if_neg h2] rfl done lemma swap_values (u v i : Nat) : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := by by_cases h1 : i = u · -- Case 1. h1 : i = u apply Or.inl rewrite [h1] show swap u v u = v from swap_fst u v done · -- Case 2. h1 : i ≠ u apply Or.inr by_cases h2 : i = v · -- Case 2.1. h2 : i = v apply Or.inl rewrite [h2] show swap u v v = u from swap_snd u v done · -- Case 2.2. h2 : i ≠ v apply Or.inr show swap u v i = i from swap_other h1 h2 done done done lemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by define fix i : Nat assume h3 : i < n have h4 : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := swap_values u v i by_cases on h4 · -- Case 1. h4 : swap u v i = v rewrite [h4] show v < n from h2 done · -- Case 2. by_cases on h4 · -- Case 2.1. h4 : swap u v i = u rewrite [h4] show u < n from h1 done · -- Case 2.2. h4 : swap u v i = i rewrite [h4] show i < n from h3 done done done lemma swap_swap (u v n : Nat) : ∀ i < n, swap u v (swap u v i) = i := by fix i : Nat assume h : i < n by_cases h1 : i = u · -- Case 1. h1 : i = u rewrite [h1, swap_fst, swap_snd] rfl done · -- Case 2. h1 : i ≠ u by_cases h2 : i = v · -- Case 2.1. h2 : i = v rewrite [h2, swap_snd, swap_fst] rfl done · -- Case 2.2. h2 : i ≠ v rewrite [swap_other h1 h2, swap_other h1 h2] rfl done done done lemma swap_one_one_below (u v n) : one_one_below n (swap u v) := left_inv_one_one_below (swap_swap u v n) lemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) := right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2) lemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) := And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2)) lemma comp_perm_below {n : Nat} {f g : Nat → Nat} (h1 : perm_below n f) (h2 : perm_below n g) : perm_below n (f ∘ g) := sorry lemma trivial_swap (u : Nat) : swap u u = id := by apply funext fix x : Nat by_cases h1 : x = u · -- Case 1. h1 : x = u rewrite [h1, swap_fst] rfl done · -- Case 2. h1 : x ≠ u rewrite [swap_other h1 h1] rfl done done lemma prod_eq_fun {m : Nat} (f g : Nat → ZMod m) (k : Nat) : ∀ (n : Nat), (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g := by by_induc · -- Base Case assume h : (∀ i < 0, f (k + i) = g (k + i)) rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g assume h1 : ∀ i < n + 1, f (k + i) = g (k + i) have h2 : ∀ i < n, f (k + i) = g (k + i) := by fix i : Nat assume h2 : i < n have h3 : i < n + 1 := by linarith show f (k + i) = g (k + i) from h1 i h3 done have h3 : prod_seq n k f = prod_seq n k g := ih h2 have h4 : n < n + 1 := Nat.lt_succ_self n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4] rfl done done lemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := by have h2 : ∀ (i : Nat), i < u → (f ∘ swap u n) (0 + i) = f (0 + i) := by fix i : Nat assume h2 : i < u have h3 : 0 + i ≠ u := by linarith have h4 : 0 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl done show prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f from prod_eq_fun (f ∘ swap u n) f 0 u h2 done lemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := by have h2 : ∀ i < j, (f ∘ swap u n) (u + 1 + i) = f (u + 1 + i) := by fix i : Nat assume h2 : i < j have h3 : u + 1 + i ≠ u := by linarith have h4 : u + 1 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl show prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f from prod_eq_fun (f ∘ swap u n) f (u + 1) j h2 done lemma break_prod {m : Nat} (n : Nat) (f : Nat → ZMod m) : ∀ (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by by_induc · -- Base Case have h : n + 0 = n := by rfl rewrite [prod_seq_base, h, Theorem_7_3_6_7] rfl done · -- Induction Step fix j : Nat assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f rewrite [←add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc] rfl done done lemma break_prod_twice {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f := break_prod n f 1 rewrite [prod_one] at h2 have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f := break_prod (u + 1) f j rewrite [←h1] at h3 have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f := break_prod u f 1 rewrite [prod_one] at h4 rewrite [h3, h4] at h2 show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2 done lemma swap_prod_eq_prod {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f := by by_cases h2 : u = n · -- Case 1. h2 : u = n rewrite [h2, trivial_swap n] --Goal : prod_seq (n + 1) 0 (f ∘ id) = prod_seq (n + 1) 0 f rfl done · -- Case 2. h2 : ¬u = n have h3 : u + 1 ≤ n := Nat.lt_of_le_of_ne h1 h2 obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3 have break_f : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := break_prod_twice f h4 have break_fs : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_prod_twice (f ∘ swap u n) h4 have f_eq_fs_below : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := swap_prod_eq_prod_below f h1 have f_eq_fs_btwn : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4 show prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f from calc prod_seq (n + 1) 0 (f ∘ swap u n) _ = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_fs _ = prod_seq u 0 f * (f ∘ swap u n) u * prod_seq j (u + 1) f * (f ∘ swap u n) n := by rw [f_eq_fs_below, f_eq_fs_btwn] _ = prod_seq u 0 f * f (swap u n u) * prod_seq j (u + 1) f * f (swap u n n) := by rfl _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by rw [swap_fst, swap_snd] _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring _ = prod_seq (n + 1) 0 f := break_f.symm done done lemma perm_below_fixed {n : Nat} {g : Nat → Nat} (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry lemma perm_prod {m : Nat} (f : Nat → ZMod m) : ∀ (n : Nat), ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) := by by_induc · -- Base Case fix g : Nat → Nat assume h1 : perm_below 0 g rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) fix g : Nat → Nat assume g_pb : perm_below (n + 1) g define at g_pb have g_ob : onto_below (n + 1) g := g_pb.right.right define at g_ob have h1 : n < n + 1 := by linarith obtain (u : Nat) (h2 : u < n + 1 ∧ g u = n) from g_ob n h1 have s_pb : perm_below (n + 1) (swap u n) := swap_perm_below h2.left h1 have gs_pb_n1 : perm_below (n + 1) (g ∘ swap u n) := comp_perm_below g_pb s_pb have gs_fix_n : (g ∘ swap u n) n = n := calc (g ∘ swap u n) n _ = g (swap u n n) := by rfl _ = g u := by rw [swap_snd] _ = n := h2.right have gs_pb_n : perm_below n (g ∘ swap u n) := perm_below_fixed gs_pb_n1 gs_fix_n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f ∘ (g ∘ swap u n)) := ih (g ∘ swap u n) gs_pb_n have h3 : u ≤ n := by linarith show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f ∘ g) from calc prod_seq (n + 1) 0 f _ = prod_seq n 0 f * f n := prod_seq_zero_step n f _ = prod_seq n 0 (f ∘ (g ∘ swap u n)) * f ((g ∘ swap u n) n) := by rw [gs_prod, gs_fix_n] _ = prod_seq n 0 (f ∘ g ∘ swap u n) * (f ∘ g ∘ swap u n) n := by rfl _ = prod_seq (n + 1) 0 (f ∘ g ∘ swap u n) := (prod_seq_zero_step n (f ∘ g ∘ swap u n)).symm _ = prod_seq (n + 1) 0 ((f ∘ g) ∘ swap u n) := by rfl _ = prod_seq (n + 1) 0 (f ∘ g) := swap_prod_eq_prod (f ∘ g) h3 done done lemma F_invertible (m i : Nat) : invertible (F m i) := by by_cases h : rel_prime m i · -- Case 1. h : rel_prime m i rewrite [F_rp_def h] show invertible [i]_m from (Theorem_7_3_7 m i).rtl h done · -- Case 2. h : ¬rel_prime m i rewrite [F_not_rp_def h] apply Exists.intro [1]_m show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m done done lemma Fprod_invertible (m : Nat) : ∀ (k : Nat), invertible (prod_seq k 0 (F m)) := by by_induc · -- Base Case apply Exists.intro [1]_m show prod_seq 0 0 (F m) * [1]_m = [1]_m from calc prod_seq 0 0 (F m) * [1]_m _ = [1]_m * [1]_m := by rw [prod_seq_base] _ = [1]_m := Theorem_7_3_6_7 ([1]_m) done · -- Induction Step fix k : Nat assume ih : invertible (prod_seq k 0 (F m)) rewrite [prod_seq_zero_step] show invertible (prod_seq k 0 (F m) * (F m k)) from (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k) done done theorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m ^ (phi m) = [1]_m := by have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2 show [a]_m ^ (phi m) = [1]_m from calc [a]_m ^ (phi m) _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3] _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring _ = prod_seq m 0 (F m ∘ G m a) * Y := by rw [FG_prod h1 m, phi_def] _ = prod_seq m 0 (F m) * Y := by rw [perm_prod (F m) m (G m a) (G_perm_below h1)] _ = [1]_m := by rw [h3] done lemma Exercise_7_4_5_Int (m : Nat) (a : Int) : ∀ (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry lemma Exercise_7_4_5_Nat (m a n : Nat) : [a]_m ^ n = [a ^ n]_m := by rewrite [Exercise_7_4_5_Int] rfl done theorem Euler's_theorem {m a : Nat} [NeZero m] (h1 : rel_prime m a) : a ^ (phi m) ≡ 1 (MOD m) := by have h2 : [a]_m ^ (phi m) = [1]_m := Theorem_7_4_2 h1 rewrite [Exercise_7_4_5_Nat m a (phi m)] at h2 --h2 : [a ^ phi m]_m = [1]_m show a ^ (phi m) ≡ 1 (MOD m) from (cc_eq_iff_congr _ _ _).ltr h2 done #eval gcd 10 7 --Answer: 1. So 10 and 7 are relatively prime #eval 7 ^ phi 10 --Answer: 2401, which is congruent to 1 mod 10. end Euler /- Section 7.5 -/ lemma num_rp_prime {p : Nat} (h1 : prime p) : ∀ k < p, num_rp_below p (k + 1) = k := sorry lemma phi_prime {p : Nat} (h1 : prime p) : phi p = p - 1 := by have h2 : 1 ≤ p := prime_pos h1 have h3 : p - 1 + 1 = p := Nat.sub_add_cancel h2 have h4 : p - 1 < p := by linarith have h5 : num_rp_below p (p - 1 + 1) = p - 1 := num_rp_prime h1 (p - 1) h4 rewrite [h3] at h5 show phi p = p - 1 from h5 done theorem Theorem_7_2_2_Int {a c : Nat} {b : Int} (h1 : ↑c ∣ ↑a * b) (h2 : rel_prime a c) : ↑c ∣ b := by rewrite [Int.natCast_dvd, Int.natAbs_mul, Int.natAbs_ofNat] at h1 --h1 : c ∣ a * Int.natAbs b rewrite [Int.natCast_dvd] --Goal : c ∣ Int.natAbs b show c ∣ Int.natAbs b from Theorem_7_2_2 h1 h2 done lemma Lemma_7_4_5 {m n : Nat} (a b : Int) (h1 : rel_prime m n) : a ≡ b (MOD m * n) ↔ a ≡ b (MOD m) ∧ a ≡ b (MOD n) := by apply Iff.intro · -- (→) assume h2 : a ≡ b (MOD m * n) obtain (j : Int) (h3 : a - b = (m * n) * j) from h2 apply And.intro · -- Proof of a ≡ b (MOD m) apply Exists.intro (n * j) show a - b = m * (n * j) from calc a - b _ = m * n * j := h3 _ = m * (n * j) := by ring done · -- Proof of a ≡ b (MOD n) apply Exists.intro (m * j) show a - b = n * (m * j) from calc a - b _ = m * n * j := h3 _ = n * (m * j) := by ring done done · -- (←) assume h2 : a ≡ b (MOD m) ∧ a ≡ b (MOD n) obtain (j : Int) (h3 : a - b = m * j) from h2.left have h4 : (↑n : Int) ∣ a - b := h2.right rewrite [h3] at h4 --h4 : ↑n ∣ ↑m * j have h5 : ↑n ∣ j := Theorem_7_2_2_Int h4 h1 obtain (k : Int) (h6 : j = n * k) from h5 apply Exists.intro k --Goal : a - b = ↑(m * n) * k rewrite [Nat.cast_mul] --Goal : a - b = ↑m * ↑n * k show a - b = (m * n) * k from calc a - b _ = m * j := h3 _ = m * (n * k) := by rw [h6] _ = (m * n) * k := by ring done done --From exercises of Section 7.2 theorem rel_prime_symm {a b : Nat} (h : rel_prime a b) : rel_prime b a := sorry lemma prime_NeZero {p : Nat} (h : prime p) : NeZero p := by rewrite [neZero_iff] --Goal : p ≠ 0 define at h linarith done lemma Lemma_7_5_1 {p e d m c s : Nat} {t : Int} (h1 : prime p) (h2 : e * d = (p - 1) * s + 1) (h3 : m ^ e - c = p * t) : c ^ d ≡ m (MOD p) := by have h4 : m ^ e ≡ c (MOD p) := Exists.intro t h3 have h5 : [m ^ e]_p = [c]_p := (cc_eq_iff_congr _ _ _).rtl h4 rewrite [←Exercise_7_4_5_Nat] at h5 --h5 : [m]_p ^ e = [c]_p by_cases h6 : p ∣ m · -- Case 1. h6 : p ∣ m have h7 : m ≡ 0 (MOD p) := by obtain (j : Nat) (h8 : m = p * j) from h6 apply Exists.intro (↑j : Int) --Goal : ↑m - 0 = ↑p * ↑j rewrite [h8, Nat.cast_mul] ring done have h8 : [m]_p = [0]_p := (cc_eq_iff_congr _ _ _).rtl h7 have h9 : e * d ≠ 0 := by rewrite [h2] show (p - 1) * s + 1 ≠ 0 from Nat.add_one_ne_zero _ done have h10 : (0 : Int) ^ (e * d) = 0 := zero_pow h9 have h11 : [c ^ d]_p = [m]_p := calc [c ^ d]_p _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat] _ = ([m]_p ^ e) ^ d := by rw [h5] _ = [m]_p ^ (e * d) := by ring _ = [0]_p ^ (e * d) := by rw [h8] _ = [0 ^ (e * d)]_p := Exercise_7_4_5_Int _ _ _ _ = [0]_p := by rw [h10] _ = [m]_p := by rw [h8] show c ^ d ≡ m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11 done · -- Case 2. h6 : ¬p ∣ m have h7 : rel_prime m p := rel_prime_of_prime_not_dvd h1 h6 have h8 : rel_prime p m := rel_prime_symm h7 have h9 : NeZero p := prime_NeZero h1 have h10 : (1 : Int) ^ s = 1 := by ring have h11 : [c ^ d]_p = [m]_p := calc [c ^ d]_p _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat] _ = ([m]_p ^ e) ^ d := by rw [h5] _ = [m]_p ^ (e * d) := by ring _ = [m]_p ^ ((p - 1) * s + 1) := by rw [h2] _ = ([m]_p ^ (p - 1)) ^ s * [m]_p := by ring _ = ([m]_p ^ (phi p)) ^ s * [m]_p := by rw [phi_prime h1] _ = [1]_p ^ s * [m]_p := by rw [Theorem_7_4_2 h8] _ = [1 ^ s]_p * [m]_p := by rw [Exercise_7_4_5_Int] _ = [1]_p * [m]_p := by rw [h10] _ = [m]_p * [1]_p := by ring _ = [m]_p := Theorem_7_3_6_7 _ show c ^ d ≡ m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11 done done theorem Theorem_7_5_1 (p q n e d k m c : Nat) (p_prime : prime p) (q_prime : prime q) (p_ne_q : p ≠ q) (n_pq : n = p * q) (ed_congr_1 : e * d = k * (p - 1) * (q - 1) + 1) (h1 : [m]_n ^ e = [c]_n) : [c]_n ^ d = [m]_n := by rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] at h1 --h1 : m ^ e ≡ c (MOD n) rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] --Goal : c ^ d ≡ m (MOD n) obtain (j : Int) (h2 : m ^ e - c = n * j) from h1 rewrite [n_pq, Nat.cast_mul] at h2 --h2 : m ^ e - c = p * q * j have h3 : e * d = (p - 1) * (k * (q - 1)) + 1 := by rewrite [ed_congr_1] ring done have h4 : m ^ e - c = p * (q * j) := by rewrite [h2] ring done have congr_p : c ^ d ≡ m (MOD p) := Lemma_7_5_1 p_prime h3 h4 have h5 : e * d = (q - 1) * (k * (p - 1)) + 1 := by rewrite [ed_congr_1] ring done have h6 : m ^ e - c = q * (p * j) := by rewrite [h2] ring done have congr_q : c ^ d ≡ m (MOD q) := Lemma_7_5_1 q_prime h5 h6 have h7 : ¬q ∣ p := by by_contra h8 have h9 : q = 1 ∨ q = p := dvd_prime p_prime h8 disj_syll h9 (prime_not_one q_prime) show False from p_ne_q h9.symm done have h8 : rel_prime p q := rel_prime_of_prime_not_dvd q_prime h7 rewrite [n_pq, Lemma_7_4_5 _ _ h8] show c ^ d ≡ m (MOD p) ∧ c ^ d ≡ m (MOD q) from And.intro congr_p congr_q done /- BEGIN EXERCISES -/ namespace Exercises /- Section 7.1 -/ -- 1. theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry -- 2. lemma gcd_comm_lt {a b : Nat} (h : a < b) : gcd a b = gcd b a := sorry theorem gcd_comm (a b : Nat) : gcd a b = gcd b a := sorry -- 3. theorem Exercise_7_1_5 (a b : Nat) (n : Int) : (∃ (s t : Int), s * a + t * b = n) ↔ (↑(gcd a b) : Int) ∣ n := sorry -- 4. theorem Exercise_7_1_6 (a b c : Nat) : gcd a b = gcd (a + b * c) b := sorry -- 5. theorem gcd_is_nonzero {a b : Nat} (h : a ≠ 0 ∨ b ≠ 0) : gcd a b ≠ 0 := sorry -- 6. theorem gcd_greatest {a b d : Nat} (h1 : gcd a b ≠ 0) (h2 : d ∣ a) (h3 : d ∣ b) : d ≤ gcd a b := sorry -- 7. lemma Lemma_7_1_10a {a b : Nat} (n : Nat) (h : a ∣ b) : (n * a) ∣ (n * b) := sorry lemma Lemma_7_1_10b {a b n : Nat} (h1 : n ≠ 0) (h2 : (n * a) ∣ (n * b)) : a ∣ b := sorry lemma Lemma_7_1_10c {a b : Nat} (h1 : a ∣ b) (h2 : b ∣ a) : a = b := sorry theorem Exercise_7_1_10 (a b n : Nat) : gcd (n * a) (n * b) = n * gcd a b := sorry /- Section 7.2 -/ -- 1. lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry -- 2. -- Hints: Start with apply List.rec. You may find mul_ne_zero useful theorem prod_nonzero_nonzero : ∀ (l : List Nat), (∀ a ∈ l, a ≠ 0) → prod l ≠ 0 := sorry -- 3. theorem rel_prime_iff_no_common_factor (a b : Nat) : rel_prime a b ↔ ¬∃ (p : Nat), prime p ∧ p ∣ a ∧ p ∣ b := sorry -- 4. theorem rel_prime_symm {a b : Nat} (h : rel_prime a b) : rel_prime b a := sorry -- 5. lemma in_prime_factorization_iff_prime_factor {a : Nat} {l : List Nat} (h1 : prime_factorization a l) (p : Nat) : p ∈ l ↔ prime_factor p a := sorry -- 6. theorem Exercise_7_2_5 {a b : Nat} {l m : List Nat} (h1 : prime_factorization a l) (h2 : prime_factorization b m) : rel_prime a b ↔ (¬∃ (p : Nat), p ∈ l ∧ p ∈ m) := sorry -- 7. theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry -- 8. theorem Exercise_7_2_7 {a b a' b' : Nat} (h1 : rel_prime a b) (h2 : a' ∣ a) (h3 : b' ∣ b) : rel_prime a' b' := sorry -- 9. theorem Exercise_7_2_9 {a b j k : Nat} (h1 : gcd a b ≠ 0) (h2 : a = j * gcd a b) (h3 : b = k * gcd a b) : rel_prime j k := sorry -- 10. theorem Exercise_7_2_17a (a b c : Nat) : gcd a (b * c) ∣ gcd a b * gcd a c := sorry /- Section 7.3 -/ -- 1. theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry -- 2. theorem Theorem_7_3_6_3 {m : Nat} (X : ZMod m) : X + [0]_m = X := sorry -- 3. theorem Theorem_7_3_6_4 {m : Nat} (X : ZMod m) : ∃ (Y : ZMod m), X + Y = [0]_m := sorry -- 4. theorem Exercise_7_3_4a {m : Nat} (Z1 Z2 : ZMod m) (h1 : ∀ (X : ZMod m), X + Z1 = X) (h2 : ∀ (X : ZMod m), X + Z2 = X) : Z1 = Z2 := sorry -- 5. theorem Exercise_7_3_4b {m : Nat} (X Y1 Y2 : ZMod m) (h1 : X + Y1 = [0]_m) (h2 : X + Y2 = [0]_m) : Y1 = Y2 := sorry -- 6. theorem Theorem_7_3_10 (m a : Nat) (b : Int) : ¬(↑(gcd m a) : Int) ∣ b → ¬∃ (x : Int), a * x ≡ b (MOD m) := sorry -- 7.

theorem Theorem_7_3_11 (m n : Nat) (a b : Int) (h1 : n ≠ 0) : n * a ≡ n * b (MOD n * m) ↔ a ≡ b (MOD m)

HTPI.Exercises.Theorem_7_3_11

htpi/HTPILib/Chap7.lean

htpi

/- Copyright 2023 Daniel J. Velleman -/ import HTPILib.Chap6 namespace HTPI /- Definitions -/ lemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by have h : n + 1 > 0 := Nat.succ_pos n show a % (n + 1) < n + 1 from Nat.mod_lt a h done def gcd (a b : Nat) : Nat := match b with | 0 => a | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd (n + 1) (a % (n + 1)) termination_by b mutual def gcd_c1 (a b : Nat) : Int := match b with | 0 => 1 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c2 (n + 1) (a % (n + 1)) --Corresponds to s = t' termination_by b def gcd_c2 (a b : Nat) : Int := match b with | 0 => 0 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c1 (n + 1) (a % (n + 1)) - (gcd_c2 (n + 1) (a % (n + 1))) * ↑(a / (n + 1)) --Corresponds to t = s' - t'q termination_by b end def prime (n : Nat) : Prop := 2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n def prime_factor (p n : Nat) : Prop := prime p ∧ p ∣ n def all_prime (l : List Nat) : Prop := ∀ p ∈ l, prime p def nondec (l : List Nat) : Prop := match l with | [] => True --Of course, True is a proposition that is always true | n :: L => (∀ m ∈ L, n ≤ m) ∧ nondec L def nondec_prime_list (l : List Nat) : Prop := all_prime l ∧ nondec l def prod (l : List Nat) : Nat := match l with | [] => 1 | n :: L => n * (prod L) def prime_factorization (n : Nat) (l : List Nat) : Prop := nondec_prime_list l ∧ prod l = n def rel_prime (a b : Nat) : Prop := gcd a b = 1 def congr_mod (m : Nat) (a b : Int) : Prop := (↑m : Int) ∣ (a - b) def cc (m : Nat) (a : Int) : ZMod m := (↑a : ZMod m) notation:50 a " ≡ " b " (MOD " m ")" => congr_mod m a b notation:max "["a"]_"m:max => cc m a def invertible {m : Nat} (X : ZMod m) : Prop := ∃ (Y : ZMod m), X * Y = [1]_m def num_rp_below (m k : Nat) : Nat := match k with | 0 => 0 | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j def phi (m : Nat) : Nat := num_rp_below m m def prod_seq {m : Nat} (j k : Nat) (f : Nat → ZMod m) : ZMod m := match j with | 0 => [1]_m | n + 1 => prod_seq n k f * f (k + n) def maps_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i < n, g i < n def one_one_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i1 < n, ∀ i2 < n, g i1 = g i2 → i1 = i2 def onto_below (n : Nat) (g : Nat → Nat) : Prop := ∀ k < n, ∃ i < n, g i = k def perm_below (n : Nat) (g : Nat → Nat) : Prop := maps_below n g ∧ one_one_below n g ∧ onto_below n g def inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m) def swap (u v i : Nat) : Nat := if i = u then v else if i = v then u else i namespace Euler --For definitions specific to Euler's theorem def F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m def G (m a i : Nat) : Nat := (a * i) % m def Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i end Euler /- Section 7.1 -/ theorem dvd_mod_of_dvd_a_b {a b d : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ (a % b) := by set q : Nat := a / b have h3 : b * q + a % b = a := Nat.div_add_mod a b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 define --Goal : ∃ (c : Nat), a % b = d * c apply Exists.intro (j - k * q) show a % b = d * (j - k * q) from calc a % b _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm _ = a - b * q := by rw [h3] _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc] _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm done theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry #eval gcd 672 161 --Answer: 7 lemma gcd_base (a : Nat) : gcd a 0 = a := by rfl lemma gcd_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd a b = gcd b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1)) rfl done lemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b ≠ 0) : a % b < b := by have h1 : b > 0 := Nat.pos_of_ne_zero h show a % b < b from Nat.mod_lt a h1 done lemma dvd_self (n : Nat) : n ∣ n := by apply Exists.intro 1 ring done theorem gcd_dvd : ∀ (b a : Nat), (gcd a b) ∣ a ∧ (gcd a b) ∣ b := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd a b_1) ∣ a ∧ (gcd a b_1) ∣ b_1 fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_base] --Goal: a ∣ a ∧ a ∣ 0 apply And.intro (dvd_self a) define apply Exists.intro 0 rfl done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_nonzero a h1] --Goal : gcd b (a % b) ∣ a ∧ gcd b (a % b) ∣ b have h2 : a % b < b := mod_nonzero_lt a h1 have h3 : (gcd b (a % b)) ∣ b ∧ (gcd b (a % b)) ∣ (a % b) := ih (a % b) h2 b apply And.intro _ h3.left show (gcd b (a % b)) ∣ a from dvd_a_of_dvd_b_mod h3.left h3.right done done theorem gcd_dvd_left (a b : Nat) : (gcd a b) ∣ a := (gcd_dvd b a).left theorem gcd_dvd_right (a b : Nat) : (gcd a b) ∣ b := (gcd_dvd b a).right lemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl lemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c1 a b = gcd_c2 b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done lemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl lemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * ↑(a / b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done theorem gcd_lin_comb : ∀ (b a : Nat), (gcd_c1 a b) * ↑a + (gcd_c2 a b) * ↑b = ↑(gcd a b) := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd_c1 a b_1) * ↑a + (gcd_c2 a b_1) * ↑b_1 = ↑(gcd a b_1) fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base] --Goal : 1 * ↑a + 0 * ↑0 = ↑a ring done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1] --Goal : gcd_c2 b (a % b) * ↑a + -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * ↑(a / b)) * ↑b = -- ↑(gcd b (a % b)) set r : Nat := a % b set q : Nat := a / b set s : Int := gcd_c1 b r set t : Int := gcd_c2 b r --Goal : t * ↑a + (s - t * ↑q) * ↑b = ↑(gcd b r) have h2 : r < b := mod_nonzero_lt a h1 have h3 : s * ↑b + t * ↑r = ↑(gcd b r) := ih r h2 b have h4 : b * q + r = a := Nat.div_add_mod a b rewrite [←h3, ←h4] rewrite [Nat.cast_add, Nat.cast_mul] --Goal : t * (↑b * ↑q + ↑r) + (s - t * ↑q) * ↑b = s * ↑b + t * ↑r ring done done #eval gcd_c1 672 161 --Answer: 6 #eval gcd_c2 672 161 --Answer: -25 --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161 theorem Theorem_7_1_6 {d a b : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ gcd a b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑d ∣ ↑(gcd a b) set s : Int := gcd_c1 a b set t : Int := gcd_c2 a b have h3 : s * ↑a + t * ↑b = ↑(gcd a b) := gcd_lin_comb b a rewrite [←h3] --Goal : ↑d ∣ s * ↑a + t * ↑b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul] --Goal : ↑d ∣ s * (↑d * ↑j) + t * (↑d * ↑k) define apply Exists.intro (s * ↑j + t * ↑k) ring done /- Section 7.2 -/ theorem dvd_trans {a b c : Nat} (h1 : a ∣ b) (h2 : b ∣ c) : a ∣ c := by define at h1; define at h2; define obtain (m : Nat) (h3 : b = a * m) from h1 obtain (n : Nat) (h4 : c = b * n) from h2 rewrite [h3, mul_assoc] at h4 apply Exists.intro (m * n) show c = a * (m * n) from h4 done lemma exists_prime_factor : ∀ (n : Nat), 2 ≤ n → ∃ (p : Nat), prime_factor p n := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, 2 ≤ n_1 → ∃ (p : Nat), prime_factor p n_1 assume h1 : 2 ≤ n by_cases h2 : prime n · -- Case 1. h2 : prime n apply Exists.intro n define --Goal : prime n ∧ n ∣ n show prime n ∧ n ∣ n from And.intro h2 (dvd_self n) done · -- Case 2. h2 : ¬prime n define at h2 --h2 : ¬(2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n) demorgan at h2 disj_syll h2 h1 obtain (a : Nat) (h3 : ∃ (b : Nat), a * b = n ∧ a < n ∧ b < n) from h2 obtain (b : Nat) (h4 : a * b = n ∧ a < n ∧ b < n) from h3 have h5 : 2 ≤ a := by by_contra h6 have h7 : a ≤ 1 := by linarith have h8 : n ≤ b := calc n _ = a * b := h4.left.symm _ ≤ 1 * b := by rel [h7] _ = b := by ring linarith --n ≤ b contradicts b < n done have h6 : ∃ (p : Nat), prime_factor p a := ih a h4.right.left h5 obtain (p : Nat) (h7 : prime_factor p a) from h6 apply Exists.intro p define --Goal : prime p ∧ p ∣ n define at h7 --h7 : prime p ∧ p ∣ a apply And.intro h7.left have h8 : a ∣ n := by apply Exists.intro b show n = a * b from (h4.left).symm done show p ∣ n from dvd_trans h7.right h8 done done lemma exists_least_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q := by set S : Set Nat := {p : Nat | prime_factor p n} have h2 : ∃ (p : Nat), p ∈ S := exists_prime_factor n h show ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q from well_ord_princ S h2 done lemma all_prime_nil : all_prime [] := by define --Goal : ∀ p ∈ [], prime p fix p : Nat contrapos --Goal : ¬prime p → p ∉ [] assume h1 : ¬prime p show p ∉ [] from List.not_mem_nil p done lemma all_prime_cons (n : Nat) (L : List Nat) : all_prime (n :: L) ↔ prime n ∧ all_prime L := by apply Iff.intro · -- (→) assume h1 : all_prime (n :: L) --Goal : prime n ∧ all_prime L define at h1 --h1 : ∀ p ∈ n :: L, prime p apply And.intro (h1 n (List.mem_cons_self n L)) define --Goal : ∀ p ∈ L, prime p fix p : Nat assume h2 : p ∈ L show prime p from h1 p (List.mem_cons_of_mem n h2) done · -- (←) assume h1 : prime n ∧ all_prime L --Goal : all_prime (n :: l) define : all_prime L at h1 define fix p : Nat assume h2 : p ∈ n :: L rewrite [List.mem_cons] at h2 --h2 : p = n ∨ p ∈ L by_cases on h2 · -- Case 1. h2 : p = n rewrite [h2] show prime n from h1.left done · -- Case 2. h2 : p ∈ L show prime p from h1.right p h2 done done done lemma nondec_nil : nondec [] := by define --Goal : True trivial --trivial proves some obviously true statements, such as True done lemma nondec_cons (n : Nat) (L : List Nat) : nondec (n :: L) ↔ (∀ m ∈ L, n ≤ m) ∧ nondec L := by rfl lemma prod_nil : prod [] = 1 := by rfl lemma prod_cons : prod (n :: L) = n * (prod L) := by rfl lemma exists_cons_of_length_eq_succ {A : Type} {l : List A} {n : Nat} (h : l.length = n + 1) : ∃ (a : A) (L : List A), l = a :: L ∧ L.length = n := by have h1 : ¬l.length = 0 := by linarith rewrite [List.length_eq_zero] at h1 obtain (a : A) (h2 : ∃ (L : List A), l = a :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List A) (h3 : l = a :: L) from h2 apply Exists.intro a apply Exists.intro L apply And.intro h3 have h4 : (a :: L).length = L.length + 1 := List.length_cons a L rewrite [←h3, h] at h4 show L.length = n from (Nat.add_right_cancel h4).symm done lemma list_elt_dvd_prod_by_length (a : Nat) : ∀ (n : Nat), ∀ (l : List Nat), l.length = n → a ∈ l → a ∣ prod l := by by_induc · --Base Case fix l : List Nat assume h1 : l.length = 0 rewrite [List.length_eq_zero] at h1 --h1 : l = [] rewrite [h1] --Goal : a ∈ [] → a ∣ prod [] contrapos assume h2 : ¬a ∣ prod [] show a ∉ [] from List.not_mem_nil a done · -- Induction Step fix n : Nat assume ih : ∀ (l : List Nat), List.length l = n → a ∈ l → a ∣ prod l fix l : List Nat assume h1 : l.length = n + 1 --Goal : a ∈ l → a ∣ prod l obtain (b : Nat) (h2 : ∃ (L : List Nat), l = b :: L ∧ L.length = n) from exists_cons_of_length_eq_succ h1 obtain (L : List Nat) (h3 : l = b :: L ∧ L.length = n) from h2 have h4 : a ∈ L → a ∣ prod L := ih L h3.right assume h5 : a ∈ l rewrite [h3.left, prod_cons] --Goal : a ∣ b * prod L rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b ∨ a ∈ L by_cases on h5 · -- Case 1. h5 : a = b apply Exists.intro (prod L) rewrite [h5] rfl done · -- Case 2. h5 : a ∈ L have h6 : a ∣ prod L := h4 h5 have h7 : prod L ∣ b * prod L := by apply Exists.intro b ring done show a ∣ b * prod L from dvd_trans h6 h7 done done done lemma list_elt_dvd_prod {a : Nat} {l : List Nat} (h : a ∈ l) : a ∣ prod l := by set n : Nat := l.length have h1 : l.length = n := by rfl show a ∣ prod l from list_elt_dvd_prod_by_length a n l h1 h done lemma exists_prime_factorization : ∀ (n : Nat), n ≥ 1 → ∃ (l : List Nat), prime_factorization n l := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, n_1 ≥ 1 → ∃ (l : List Nat), prime_factorization n_1 l assume h1 : n ≥ 1 by_cases h2 : n = 1 · -- Case 1. h2 : n = 1 apply Exists.intro [] define apply And.intro · -- Proof of nondec_prime_list [] define show all_prime [] ∧ nondec [] from And.intro all_prime_nil nondec_nil done · -- Proof of prod [] = n rewrite [prod_nil, h2] rfl done done · -- Case 2. h2 : n ≠ 1 have h3 : n ≥ 2 := lt_of_le_of_ne' h1 h2 obtain (p : Nat) (h4 : prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q) from exists_least_prime_factor h3 have p_prime_factor : prime_factor p n := h4.left define at p_prime_factor have p_prime : prime p := p_prime_factor.left have p_dvd_n : p ∣ n := p_prime_factor.right have p_least : ∀ (q : Nat), prime_factor q n → p ≤ q := h4.right obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n have h5 : m ≠ 0 := by contradict h1 with h6 have h7 : n = 0 := calc n _ = p * m := n_eq_pm _ = p * 0 := by rw [h6] _ = 0 := by ring rewrite [h7] decide done have m_pos : 0 < m := Nat.pos_of_ne_zero h5 have m_lt_n : m < n := by define at p_prime show m < n from calc m _ < m + m := by linarith _ = 2 * m := by ring _ ≤ p * m := by rel [p_prime.left] _ = n := n_eq_pm.symm done obtain (L : List Nat) (h6 : prime_factorization m L) from ih m m_lt_n m_pos define at h6 have ndpl_L : nondec_prime_list L := h6.left define at ndpl_L apply Exists.intro (p :: L) define apply And.intro · -- Proof of nondec_prime_list (p :: L) define apply And.intro · -- Proof of all_prime (p :: L) rewrite [all_prime_cons] show prime p ∧ all_prime L from And.intro p_prime ndpl_L.left done · -- Proof of nondec (p :: L) rewrite [nondec_cons] apply And.intro _ ndpl_L.right fix q : Nat assume q_in_L : q ∈ L have h7 : q ∣ prod L := list_elt_dvd_prod q_in_L rewrite [h6.right] at h7 --h7 : q ∣ m have h8 : m ∣ n := by apply Exists.intro p rewrite [n_eq_pm] ring done have q_dvd_n : q ∣ n := dvd_trans h7 h8 have ap_L : all_prime L := ndpl_L.left define at ap_L have q_prime_factor : prime_factor q n := And.intro (ap_L q q_in_L) q_dvd_n show p ≤ q from p_least q q_prime_factor done done · -- Proof of prod (p :: L) = n rewrite [prod_cons, h6.right, n_eq_pm] rfl done done done theorem Theorem_7_2_2 {a b c : Nat} (h1 : c ∣ a * b) (h2 : rel_prime a c) : c ∣ b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑c ∣ ↑b define at h1; define at h2; define obtain (j : Nat) (h3 : a * b = c * j) from h1 set s : Int := gcd_c1 a c set t : Int := gcd_c2 a c have h4 : s * ↑a + t * ↑c = ↑(gcd a c) := gcd_lin_comb c a rewrite [h2, Nat.cast_one] at h4 --h4 : s * ↑a + t * ↑c = (1 : Int) apply Exists.intro (s * ↑j + t * ↑b) show ↑b = ↑c * (s * ↑j + t * ↑b) from calc ↑b _ = (1 : Int) * ↑b := (one_mul _).symm _ = (s * ↑a + t * ↑c) * ↑b := by rw [h4] _ = s * (↑a * ↑b) + t * ↑c * ↑b := by ring _ = s * (↑c * ↑j) + t * ↑c * ↑b := by rw [←Nat.cast_mul a b, h3, Nat.cast_mul c j] _ = ↑c * (s * ↑j + t * ↑b) := by ring done lemma le_nonzero_prod_left {a b : Nat} (h : a * b ≠ 0) : a ≤ a * b := by have h1 : b ≠ 0 := by contradict h with h1 rewrite [h1] ring done have h2 : 1 ≤ b := Nat.pos_of_ne_zero h1 show a ≤ a * b from calc a = a * 1 := (mul_one a).symm _ ≤ a * b := by rel [h2] done lemma le_nonzero_prod_right {a b : Nat} (h : a * b ≠ 0) : b ≤ a * b := by rewrite [mul_comm] rewrite [mul_comm] at h show b ≤ b * a from le_nonzero_prod_left h done lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry lemma rel_prime_of_prime_not_dvd {a p : Nat} (h1 : prime p) (h2 : ¬p ∣ a) : rel_prime a p := by have h3 : gcd a p ∣ a := gcd_dvd_left a p have h4 : gcd a p ∣ p := gcd_dvd_right a p have h5 : gcd a p = 1 ∨ gcd a p = p := dvd_prime h1 h4 have h6 : gcd a p ≠ p := by contradict h2 with h6 rewrite [h6] at h3 show p ∣ a from h3 done disj_syll h5 h6 show rel_prime a p from h5 done theorem Theorem_7_2_3 {a b p : Nat} (h1 : prime p) (h2 : p ∣ a * b) : p ∣ a ∨ p ∣ b := by or_right with h3 have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3 show p ∣ b from Theorem_7_2_2 h2 h4 done lemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a ≥ 1 := by have h1 : a ≠ 0 := by by_contra h1 rewrite [h1] at h contradict h linarith done show a ≥ 1 from Nat.pos_of_ne_zero h1 done lemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by have h1 : a ≥ 1 := ge_one_of_prod_one h have h2 : a * b ≠ 0 := by linarith have h3 : a ≤ a * b := le_nonzero_prod_left h2 rewrite [h] at h3 show a = 1 from Nat.le_antisymm h3 h1 done lemma eq_one_of_dvd_one {n : Nat} (h : n ∣ 1) : n = 1 := by obtain (j : Nat) (h1 : 1 = n * j) from h show n = 1 from eq_one_of_prod_one h1.symm done lemma prime_not_one {p : Nat} (h : prime p) : p ≠ 1 := by define at h linarith done theorem Theorem_7_2_4 {p : Nat} (h1 : prime p) : ∀ (l : List Nat), p ∣ prod l → ∃ a ∈ l, p ∣ a := by apply List.rec · -- Base Case. Goal : p ∣ prod [] → ∃ a ∈ [], p ∣ a rewrite [prod_nil] assume h2 : p ∣ 1 show ∃ a ∈ [], p ∣ a from absurd (eq_one_of_dvd_one h2) (prime_not_one h1) done · -- Induction Step fix b : Nat fix L : List Nat assume ih : p ∣ prod L → ∃ a ∈ L, p ∣ a --Goal : p ∣ prod (b :: L) → ∃ a ∈ b :: L, p ∣ a assume h2 : p ∣ prod (b :: L) rewrite [prod_cons] at h2 have h3 : p ∣ b ∨ p ∣ prod L := Theorem_7_2_3 h1 h2 by_cases on h3 · -- Case 1. h3 : p ∣ b apply Exists.intro b show b ∈ b :: L ∧ p ∣ b from And.intro (List.mem_cons_self b L) h3 done · -- Case 2. h3 : p ∣ prod L obtain (a : Nat) (h4 : a ∈ L ∧ p ∣ a) from ih h3 apply Exists.intro a show a ∈ b :: L ∧ p ∣ a from And.intro (List.mem_cons_of_mem b h4.left) h4.right done done done lemma prime_in_list {p : Nat} {l : List Nat} (h1 : prime p) (h2 : all_prime l) (h3 : p ∣ prod l) : p ∈ l := by obtain (a : Nat) (h4 : a ∈ l ∧ p ∣ a) from Theorem_7_2_4 h1 l h3 define at h2 have h5 : prime a := h2 a h4.left have h6 : p = 1 ∨ p = a := dvd_prime h5 h4.right disj_syll h6 (prime_not_one h1) rewrite [h6] show a ∈ l from h4.left done lemma first_le_first {p q : Nat} {l m : List Nat} (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m)) (h3 : prod (p :: l) = prod (q :: m)) : p ≤ q := by define at h1; define at h2 have h4 : q ∣ prod (p :: l) := by define apply Exists.intro (prod m) rewrite [←prod_cons] show prod (p :: l) = prod (q :: m) from h3 done have h5 : all_prime (q :: m) := h2.left rewrite [all_prime_cons] at h5 have h6 : q ∈ p :: l := prime_in_list h5.left h1.left h4 have h7 : nondec (p :: l) := h1.right rewrite [nondec_cons] at h7 rewrite [List.mem_cons] at h6 by_cases on h6 · -- Case 1. h6 : q = p linarith done · -- Case 2. h6 : q ∈ l have h8 : ∀ m ∈ l, p ≤ m := h7.left show p ≤ q from h8 q h6 done done lemma nondec_prime_list_tail {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by define at h define rewrite [all_prime_cons, nondec_cons] at h show all_prime l ∧ nondec l from And.intro h.left.right h.right.right done lemma cons_prod_not_one {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : prod (p :: l) ≠ 1 := by define at h have h1 : all_prime (p :: l) := h.left rewrite [all_prime_cons] at h1 rewrite [prod_cons] by_contra h2 show False from (prime_not_one h1.left) (eq_one_of_prod_one h2) done lemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) : l = [] ↔ prod l = 1 := by apply Iff.intro · -- (→) assume h1 : l = [] rewrite [h1] show prod [] = 1 from prod_nil done · -- (←) contrapos assume h1 : ¬l = [] obtain (p : Nat) (h2 : ∃ (L : List Nat), l = p :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List Nat) (h3 : l = p :: L) from h2 rewrite [h3] at h rewrite [h3] show ¬prod (p :: L) = 1 from cons_prod_not_one h done done lemma prime_pos {p : Nat} (h : prime p) : p > 0 := by define at h linarith done theorem Theorem_7_2_5 : ∀ (l1 l2 : List Nat), nondec_prime_list l1 → nondec_prime_list l2 → prod l1 = prod l2 → l1 = l2 := by apply List.rec · -- Base Case. Goal : ∀ (l2 : List Nat), nondec_prime_list [] → -- nondec_prime_list l2 → prod [] = prod l2 → [] = l2 fix l2 : List Nat assume h1 : nondec_prime_list [] assume h2 : nondec_prime_list l2 assume h3 : prod [] = prod l2 rewrite [prod_nil, eq_comm, ←list_nil_iff_prod_one h2] at h3 show [] = l2 from h3.symm done · -- Induction Step fix p : Nat fix L1 : List Nat assume ih : ∀ (L2 : List Nat), nondec_prime_list L1 → nondec_prime_list L2 → prod L1 = prod L2 → L1 = L2 -- Goal : ∀ (l2 : List Nat), nondec_prime_list (p :: L1) → -- nondec_prime_list l2 → prod (p :: L1) = prod l2 → p :: L1 = l2 fix l2 : List Nat assume h1 : nondec_prime_list (p :: L1) assume h2 : nondec_prime_list l2 assume h3 : prod (p :: L1) = prod l2 have h4 : ¬prod (p :: L1) = 1 := cons_prod_not_one h1 rewrite [h3, ←list_nil_iff_prod_one h2] at h4 obtain (q : Nat) (h5 : ∃ (L : List Nat), l2 = q :: L) from List.exists_cons_of_ne_nil h4 obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5 rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2) rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2) have h7 : p ≤ q := first_le_first h1 h2 h3 have h8 : q ≤ p := first_le_first h2 h1 h3.symm have h9 : p = q := by linarith rewrite [h9, prod_cons, prod_cons] at h3 --h3 : q * prod L1 = q * prod L2 have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1 have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2 define at h2 have h12 : all_prime (q :: L2) := h2.left rewrite [all_prime_cons] at h12 have h13 : q > 0 := prime_pos h12.left have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3 have h15 : L1 = L2 := ih L2 h10 h11 h14 rewrite [h6, h9, h15] rfl done done theorem fund_thm_arith (n : Nat) (h : n ≥ 1) : ∃! (l : List Nat), prime_factorization n l := by exists_unique · -- Existence show ∃ (l : List Nat), prime_factorization n l from exists_prime_factorization n h done · -- Uniqueness fix l1 : List Nat; fix l2 : List Nat assume h1 : prime_factorization n l1 assume h2 : prime_factorization n l2 define at h1; define at h2 have h3 : prod l1 = n := h1.right rewrite [←h2.right] at h3 show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3 done done /- Section 7.3 -/ theorem congr_refl (m : Nat) : ∀ (a : Int), a ≡ a (MOD m) := by fix a : Int define --Goal : ∃ (c : Int), a - a = ↑m * c apply Exists.intro 0 ring done theorem congr_symm {m : Nat} : ∀ {a b : Int}, a ≡ b (MOD m) → b ≡ a (MOD m) := by fix a : Int; fix b : Int assume h1 : a ≡ b (MOD m) define at h1 --h1 : ∃ (c : Int), a - b = ↑m * c define --Goal : ∃ (c : Int), b - a = ↑m * c obtain (c : Int) (h2 : a - b = m * c) from h1 apply Exists.intro (-c) show b - a = m * (-c) from calc b - a _ = -(a - b) := by ring _ = -(m * c) := by rw [h2] _ = m * (-c) := by ring done theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry /- Fundamental properties of congruence classes -/ lemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) : X = Y ↔ X.val = Y.val := Fin.ext_iff lemma val_nat_eq_mod (n k : Nat) : ([k]_(n + 1)).val = k % (n + 1) := by rfl lemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl theorem cc_rep {m : Nat} (X : ZMod m) : ∃ (a : Int), X = [a]_m := match m with | 0 => by apply Exists.intro X rfl done | n + 1 => by apply Exists.intro ↑(X.val) have h1 : X.val < n + 1 := Fin.prop X rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1] rfl done theorem add_class (m : Nat) (a b : Int) : [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm theorem mul_class (m : Nat) (a b : Int) : [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm lemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ [a - b]_m = [0]_m := by apply Iff.intro · -- (→) assume h1 : [a]_m = [b]_m have h2 : a - b = a + (-b) := by ring have h3 : b + (-b) = 0 := by ring show [a - b]_m = [0]_m from calc [a - b]_m _ = [a + (-b)]_m := by rw [h2] _ = [a]_m + [-b]_m := by rw [add_class] _ = [b]_m + [-b]_m := by rw [h1] _ = [b + -b]_m := by rw [add_class] _ = [0]_m := by rw [h3] done · -- (←) assume h1 : [a - b]_m = [0]_m have h2 : b + (a - b) = a := by ring have h3 : b + 0 = b := by ring show [a]_m = [b]_m from calc [a]_m _ = [b + (a - b)]_m := by rw [h2] _ = [b]_m + [a - b]_m := by rw [add_class] _ = [b]_m + [0]_m := by rw [h1] _ = [b + 0]_m := by rw [add_class] _ = [b]_m := by rw [h3] done done lemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) : [a]_m = [0]_m → [-a]_m = [0]_m := by assume h1 : [a]_m = [0]_m have h2 : 0 + (-a) = -a := by ring have h3 : a + (-a) = 0 := by ring show [-a]_m = [0]_m from calc [-a]_m _ = [0 + (-a)]_m := by rw [h2] _ = [0]_m + [-a]_m := by rw [add_class] _ = [a]_m + [-a]_m := by rw [h1] _ = [a + (-a)]_m := by rw [add_class] _ = [0]_m := by rw [h3] done lemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) : [-a]_m = [0]_m ↔ [a]_m = [0]_m := by apply Iff.intro _ (cc_neg_zero_of_cc_zero m a) assume h1 : [-a]_m = [0]_m have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1 have h3 : -(-a) = a := by ring rewrite [h3] at h2 show [a]_m = [0]_m from h2 done lemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl lemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m ↔ m ∣ k := match m with | 0 => by have h : (0 : Int) = (↑(0 : Nat) : Int) := by rfl rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj] apply Iff.intro · -- (→) assume h1 : k = 0 rewrite [h1] show 0 ∣ 0 from dvd_self 0 done · -- (←) assume h1 : 0 ∣ k obtain (c : Nat) (h2 : k = 0 * c) from h1 rewrite [h2] ring done done | n + 1 => by rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero] show k % (n + 1) = 0 ↔ n + 1 ∣ k from (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm done lemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m ↔ ↑m ∣ a := by obtain (k : Nat) (h1 : a = ↑k ∨ a = -↑k) from Int.eq_nat_or_neg a by_cases on h1 · -- Case 1. h1: a = ↑k rewrite [h1, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done · -- Case 2. h1: a = -↑k rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done done theorem cc_eq_iff_congr (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ a ≡ b (MOD m) := calc [a]_m = [b]_m _ ↔ [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b _ ↔ ↑m ∣ (a - b) := cc_zero_iff_dvd m (a - b) _ ↔ a ≡ b (MOD m) := by rfl /- End of fundamental properties of congruence classes -/ lemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m := by have h1 : (↑m : Int) ≠ 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m) show 0 ≤ a % m from Int.emod_nonneg a h1 done lemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m) have h2 : (↑m : Int) > 0 := (Nat.cast_pos).rtl h1 show a % m < m from Int.emod_lt_of_pos a h2 done lemma congr_mod_mod (m : Nat) (a : Int) : a ≡ a % m (MOD m) := by define have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m apply Exists.intro (a / m) show a - a % m = m * (a / m) from calc a - (a % m) _ = m * (a / m) + a % m - a % m := by rw [h1] _ = m * (a / m) := by ring done lemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) := And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a)) theorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) : ∃! (r : Int), 0 ≤ r ∧ r < m ∧ a ≡ r (MOD m) := by exists_unique · -- Existence apply Exists.intro (a % m) show 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) from mod_cmpl_res m a done · -- Uniqueness fix r1 : Int; fix r2 : Int assume h1 : 0 ≤ r1 ∧ r1 < m ∧ a ≡ r1 (MOD m) assume h2 : 0 ≤ r2 ∧ r2 < m ∧ a ≡ r2 (MOD m) have h3 : r1 ≡ r2 (MOD m) := congr_trans (congr_symm h1.right.right) h2.right.right obtain (d : Int) (h4 : r1 - r2 = m * d) from h3 have h5 : r1 - r2 < m * 1 := by linarith have h6 : m * (-1) < r1 - r2 := by linarith rewrite [h4] at h5 --h5 : m * d < m * 1 rewrite [h4] at h6 --h6 : m * -1 < m * d have h7 : (↑m : Int) ≥ 0 := Nat.cast_nonneg m have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7 have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7 have h10 : d = 0 := by linarith show r1 = r2 from calc r1 _ = r1 - r2 + r2 := by ring _ = m * 0 + r2 := by rw [h4, h10] _ = r2 := by ring done done lemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m := (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a) theorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y rewrite [h1, h2] have h3 : a + b = b + a := by ring show [a]_m + [b]_m = [b]_m + [a]_m from calc [a]_m + [b]_m _ = [a + b]_m := add_class m a b _ = [b + a]_m := by rw [h3] _ = [b]_m + [a]_m := (add_class m b a).symm done theorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X rewrite [h1] have h2 : a * 1 = a := by ring show [a]_m * [1]_m = [a]_m from calc [a]_m * [1]_m _ = [a * 1]_m := mul_class m a 1 _ = [a]_m := by rw [h2] done theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry lemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) : [a]_m * [gcd_c2 m a]_m = [1]_m := by set s : Int := gcd_c1 m a have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m define at h1 rewrite [h1, Nat.cast_one] at h2 --h2 : s * ↑m + gcd_c2 m a * ↑a = 1 rewrite [mul_class, cc_eq_iff_congr] define --Goal : ∃ (c : Int), ↑a * gcd_c2 m a - 1 = ↑m * c apply Exists.intro (-s) show a * (gcd_c2 m a) - 1 = m * (-s) from calc a * (gcd_c2 m a) - 1 _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring _ = 1 + m * (-s) - 1 := by rw [h2] _ = m * (-s) := by ring done theorem Theorem_7_3_7 (m a : Nat) : invertible [a]_m ↔ rel_prime m a := by apply Iff.intro · -- (→) assume h1 : invertible [a]_m define at h1 obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1 obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y rewrite [h3, mul_class, cc_eq_iff_congr] at h2 define at h2 obtain (c : Int) (h4 : a * b - 1 = m * c) from h2 rewrite [Exercise_7_2_6] --Goal : ∃ (s t : Int), s * ↑m + t * ↑a = 1 apply Exists.intro (-c) apply Exists.intro b show (-c) * m + b * a = 1 from calc (-c) * m + b * a _ = (-c) * m + (a * b - 1) + 1 := by ring _ = (-c) * m + m * c + 1 := by rw [h4] _ = 1 := by ring done · -- (←) assume h1 : rel_prime m a define show ∃ (Y : ZMod m), [a]_m * Y = [1]_m from Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1) done done /- Section 7.4 -/ section Euler open Euler lemma num_rp_below_base {m : Nat} : num_rp_below m 0 = 0 := by rfl lemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) : num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by have h1 : num_rp_below m (j + 1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : gcd m j = 1 rewrite [if_pos h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1 show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1 done lemma num_rp_below_step_not_rp {m j : Nat} (h : ¬rel_prime m j) : num_rp_below m (j + 1) = num_rp_below m j := by have h1 : num_rp_below m (j +1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : ¬gcd m j = 1 rewrite [if_neg h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j show num_rp_below m (j + 1) = num_rp_below m j from h1 done lemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl #eval phi 10 --Answer: 4 lemma prod_inv_iff_inv {m : Nat} {X : ZMod m} (h1 : invertible X) (Y : ZMod m) : invertible (X * Y) ↔ invertible Y := by apply Iff.intro · -- (→) assume h2 : invertible (X * Y) obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2 apply Exists.intro (X * Z) rewrite [←h3] --Goal : Y * (X * Z) = X * Y * Z ring --Note that ring can do algebra in ZMod m done · -- (←) assume h2 : invertible Y obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1 obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2 apply Exists.intro (Xi * Yi) show (X * Y) * (Xi * Yi) = [1]_m from calc X * Y * (Xi * Yi) _ = (X * Xi) * (Y * Yi) := by ring _ = [1]_m * [1]_m := by rw [h3, h4] _ = [1]_m := Theorem_7_3_6_7 [1]_m done done lemma F_rp_def {m i : Nat} (h : rel_prime m i) : F m i = [i]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h --h : gcd m i = 1 rewrite [if_pos h] at h1 show F m i = [i]_m from h1 done lemma F_not_rp_def {m i : Nat} (h : ¬rel_prime m i) : F m i = [1]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h rewrite [h1, if_neg h] rfl done lemma prod_seq_base {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 0 k f = [1]_m := by rfl lemma prod_seq_step {m : Nat} (n k : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl lemma prod_seq_zero_step {m : Nat} (n : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by rewrite [prod_seq_step, zero_add] rfl done lemma prod_one {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 1 k f = f k := by rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7] rfl done lemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl lemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m := calc [G m a i]_m _ = [(a * i) % m]_m := by rfl _ = [a * i]_m := (cc_eq_mod m (a * i)).symm _ = [a]_m * [i]_m := (mul_class m a i).symm lemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) : rel_prime m (G m a i) ↔ rel_prime m i := by have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1 show rel_prime m (G m a i) ↔ rel_prime m i from calc rel_prime m (G m a i) _ ↔ invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm _ ↔ invertible ([a]_m * [i]_m) := by rw [cc_G] _ ↔ invertible [i]_m := prod_inv_iff_inv h2 ([i]_m) _ ↔ rel_prime m i := Theorem_7_3_7 m i done lemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) : F m (G m a i) = [a]_m * F m i := by have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2 show F m (G m a i) = [a]_m * F m i from calc F m (G m a i) _ = [G m a i]_m := F_rp_def h3 _ = [a]_m * [i]_m := cc_G m a i _ = [a]_m * F m i := by rw [F_rp_def h2] done lemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : ¬rel_prime m i) : F m (G m a i) = [1]_m := by rewrite [←G_rp_iff h1 i] at h2 show F m (G m a i) = [1]_m from F_not_rp_def h2 done lemma FG_prod {m a : Nat} (h1 : rel_prime m a) : ∀ (k : Nat), prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by by_induc · -- Base Case show prod_seq 0 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from calc prod_seq 0 0 ((F m) ∘ (G m a)) _ = [1]_m := prod_seq_base _ _ _ = [a]_m ^ 0 * [1]_m := by ring _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by rw [num_rp_below_base, prod_seq_base] done · -- Induction Step fix k : Nat assume ih : prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) by_cases h2 : rel_prime m k · -- Case 1. h2 : rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([a]_m * F m k) := by rw [FG_rp h1 h2] _ = [a]_m ^ ((num_rp_below m k) + 1) * ((prod_seq k 0 (F m)) * F m k) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_rp h2, prod_seq_zero_step] done · -- Case 2. h2 : ¬rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([1]_m) := by rw [FG_not_rp h1 h2] _ = [a]_m ^ (num_rp_below m k) * (prod_seq k 0 (F m) * ([1]_m)) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_not_rp h2, prod_seq_zero_step, F_not_rp_def h2] done done done lemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by define --Goal : ∀ i < m, G m a i < m fix i : Nat assume h1 : i < m rewrite [G_def] --Goal : a * i % m < m show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m) done lemma left_inv_one_one_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g' (g i) = i) : one_one_below n g := sorry lemma right_inv_onto_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g (g' i) = i) (h2 : maps_below n g') : onto_below n g := by define at h2; define fix k : Nat assume h3 : k < n apply Exists.intro (g' k) show g' k < n ∧ g (g' k) = k from And.intro (h2 k h3) (h1 k h3) done lemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m * [inv_mod m a]_m = [1]_m := by have h2 : 0 ≤ (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a) show [a]_m * [inv_mod m a]_m = [1]_m from calc [a]_m * [inv_mod m a]_m _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2] _ = [a]_m * [gcd_c2 m a]_m := by rw [←cc_eq_mod] _ = [1]_m := gcd_c2_inv h1 done lemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m := calc a * (b % m) % m = a % m * (b % m % m) % m := Nat.mul_mod _ _ _ _ = a % m * (b % m) % m := by rw [Nat.mod_mod] _ = a * b % m := (Nat.mul_mod _ _ _).symm lemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm] rfl done theorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] : ↑a ≡ ↑b (MOD m) ↔ a % m = b % m := sorry lemma mul_inv_mod_cancel {m a i : Nat} [NeZero m] (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1 rewrite [mul_class, cc_eq_iff_congr, ←Nat.cast_mul, ←Nat.cast_one, congr_iff_mod_eq_Nat] at h3 show a * inv_mod m a * i % m = i from calc a * (inv_mod m a) * i % m _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = 1 % m * i % m := by rw [h3] _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = i % m := by rw [one_mul] _ = i := Nat.mod_eq_of_lt h2 done lemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl lemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, G m a (Ginv m a i) = i := by fix i : Nat assume h2 : i < m show G m a (Ginv m a i) = i from calc G m a (Ginv m a i) _ = a * ((inv_mod m a * i) % m) % m := by rfl _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, Ginv m a (G m a i) = i := by fix i : Nat assume h2 : i < m show Ginv m a (G m a i) = i from calc Ginv m a (G m a i) _ = inv_mod m a * ((a * i) % m) % m := by rfl _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc, mul_comm (inv_mod m a)] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_maps_below (m a : Nat) [NeZero m] : maps_below m (Ginv m a) := G_maps_below m (inv_mod m a) lemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : one_one_below m (G m a) := left_inv_one_one_below (Ginv_left_inv h1) lemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : onto_below m (G m a) := right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a) lemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : perm_below m (G m a) := And.intro (G_maps_below m a) (And.intro (G_one_one_below h1) (G_onto_below h1)) --Permuting a product of congruence classes doesn't change product lemma swap_fst (u v : Nat) : swap u v u = v := by define : swap u v u --Goal : (if u = u then v else if u = v then u else u) = v have h : u = u := by rfl rewrite [if_pos h] rfl done lemma swap_snd (u v : Nat) : swap u v v = u := by define : swap u v v by_cases h1 : v = u · -- Case 1. h1 : v = u rewrite [if_pos h1] show v = u from h1 done · -- Case 2. h1 : v ≠ u rewrite [if_neg h1] have h2 : v = v := by rfl rewrite [if_pos h2] rfl done done lemma swap_other {u v i : Nat} (h1 : i ≠ u) (h2 : i ≠ v) : swap u v i = i := by define : swap u v i rewrite [if_neg h1, if_neg h2] rfl done lemma swap_values (u v i : Nat) : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := by by_cases h1 : i = u · -- Case 1. h1 : i = u apply Or.inl rewrite [h1] show swap u v u = v from swap_fst u v done · -- Case 2. h1 : i ≠ u apply Or.inr by_cases h2 : i = v · -- Case 2.1. h2 : i = v apply Or.inl rewrite [h2] show swap u v v = u from swap_snd u v done · -- Case 2.2. h2 : i ≠ v apply Or.inr show swap u v i = i from swap_other h1 h2 done done done lemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by define fix i : Nat assume h3 : i < n have h4 : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := swap_values u v i by_cases on h4 · -- Case 1. h4 : swap u v i = v rewrite [h4] show v < n from h2 done · -- Case 2. by_cases on h4 · -- Case 2.1. h4 : swap u v i = u rewrite [h4] show u < n from h1 done · -- Case 2.2. h4 : swap u v i = i rewrite [h4] show i < n from h3 done done done lemma swap_swap (u v n : Nat) : ∀ i < n, swap u v (swap u v i) = i := by fix i : Nat assume h : i < n by_cases h1 : i = u · -- Case 1. h1 : i = u rewrite [h1, swap_fst, swap_snd] rfl done · -- Case 2. h1 : i ≠ u by_cases h2 : i = v · -- Case 2.1. h2 : i = v rewrite [h2, swap_snd, swap_fst] rfl done · -- Case 2.2. h2 : i ≠ v rewrite [swap_other h1 h2, swap_other h1 h2] rfl done done done lemma swap_one_one_below (u v n) : one_one_below n (swap u v) := left_inv_one_one_below (swap_swap u v n) lemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) := right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2) lemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) := And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2)) lemma comp_perm_below {n : Nat} {f g : Nat → Nat} (h1 : perm_below n f) (h2 : perm_below n g) : perm_below n (f ∘ g) := sorry lemma trivial_swap (u : Nat) : swap u u = id := by apply funext fix x : Nat by_cases h1 : x = u · -- Case 1. h1 : x = u rewrite [h1, swap_fst] rfl done · -- Case 2. h1 : x ≠ u rewrite [swap_other h1 h1] rfl done done lemma prod_eq_fun {m : Nat} (f g : Nat → ZMod m) (k : Nat) : ∀ (n : Nat), (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g := by by_induc · -- Base Case assume h : (∀ i < 0, f (k + i) = g (k + i)) rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g assume h1 : ∀ i < n + 1, f (k + i) = g (k + i) have h2 : ∀ i < n, f (k + i) = g (k + i) := by fix i : Nat assume h2 : i < n have h3 : i < n + 1 := by linarith show f (k + i) = g (k + i) from h1 i h3 done have h3 : prod_seq n k f = prod_seq n k g := ih h2 have h4 : n < n + 1 := Nat.lt_succ_self n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4] rfl done done lemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := by have h2 : ∀ (i : Nat), i < u → (f ∘ swap u n) (0 + i) = f (0 + i) := by fix i : Nat assume h2 : i < u have h3 : 0 + i ≠ u := by linarith have h4 : 0 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl done show prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f from prod_eq_fun (f ∘ swap u n) f 0 u h2 done lemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := by have h2 : ∀ i < j, (f ∘ swap u n) (u + 1 + i) = f (u + 1 + i) := by fix i : Nat assume h2 : i < j have h3 : u + 1 + i ≠ u := by linarith have h4 : u + 1 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl show prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f from prod_eq_fun (f ∘ swap u n) f (u + 1) j h2 done lemma break_prod {m : Nat} (n : Nat) (f : Nat → ZMod m) : ∀ (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by by_induc · -- Base Case have h : n + 0 = n := by rfl rewrite [prod_seq_base, h, Theorem_7_3_6_7] rfl done · -- Induction Step fix j : Nat assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f rewrite [←add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc] rfl done done lemma break_prod_twice {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f := break_prod n f 1 rewrite [prod_one] at h2 have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f := break_prod (u + 1) f j rewrite [←h1] at h3 have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f := break_prod u f 1 rewrite [prod_one] at h4 rewrite [h3, h4] at h2 show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2 done lemma swap_prod_eq_prod {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f := by by_cases h2 : u = n · -- Case 1. h2 : u = n rewrite [h2, trivial_swap n] --Goal : prod_seq (n + 1) 0 (f ∘ id) = prod_seq (n + 1) 0 f rfl done · -- Case 2. h2 : ¬u = n have h3 : u + 1 ≤ n := Nat.lt_of_le_of_ne h1 h2 obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3 have break_f : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := break_prod_twice f h4 have break_fs : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_prod_twice (f ∘ swap u n) h4 have f_eq_fs_below : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := swap_prod_eq_prod_below f h1 have f_eq_fs_btwn : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4 show prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f from calc prod_seq (n + 1) 0 (f ∘ swap u n) _ = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_fs _ = prod_seq u 0 f * (f ∘ swap u n) u * prod_seq j (u + 1) f * (f ∘ swap u n) n := by rw [f_eq_fs_below, f_eq_fs_btwn] _ = prod_seq u 0 f * f (swap u n u) * prod_seq j (u + 1) f * f (swap u n n) := by rfl _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by rw [swap_fst, swap_snd] _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring _ = prod_seq (n + 1) 0 f := break_f.symm done done lemma perm_below_fixed {n : Nat} {g : Nat → Nat} (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry lemma perm_prod {m : Nat} (f : Nat → ZMod m) : ∀ (n : Nat), ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) := by by_induc · -- Base Case fix g : Nat → Nat assume h1 : perm_below 0 g rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) fix g : Nat → Nat assume g_pb : perm_below (n + 1) g define at g_pb have g_ob : onto_below (n + 1) g := g_pb.right.right define at g_ob have h1 : n < n + 1 := by linarith obtain (u : Nat) (h2 : u < n + 1 ∧ g u = n) from g_ob n h1 have s_pb : perm_below (n + 1) (swap u n) := swap_perm_below h2.left h1 have gs_pb_n1 : perm_below (n + 1) (g ∘ swap u n) := comp_perm_below g_pb s_pb have gs_fix_n : (g ∘ swap u n) n = n := calc (g ∘ swap u n) n _ = g (swap u n n) := by rfl _ = g u := by rw [swap_snd] _ = n := h2.right have gs_pb_n : perm_below n (g ∘ swap u n) := perm_below_fixed gs_pb_n1 gs_fix_n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f ∘ (g ∘ swap u n)) := ih (g ∘ swap u n) gs_pb_n have h3 : u ≤ n := by linarith show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f ∘ g) from calc prod_seq (n + 1) 0 f _ = prod_seq n 0 f * f n := prod_seq_zero_step n f _ = prod_seq n 0 (f ∘ (g ∘ swap u n)) * f ((g ∘ swap u n) n) := by rw [gs_prod, gs_fix_n] _ = prod_seq n 0 (f ∘ g ∘ swap u n) * (f ∘ g ∘ swap u n) n := by rfl _ = prod_seq (n + 1) 0 (f ∘ g ∘ swap u n) := (prod_seq_zero_step n (f ∘ g ∘ swap u n)).symm _ = prod_seq (n + 1) 0 ((f ∘ g) ∘ swap u n) := by rfl _ = prod_seq (n + 1) 0 (f ∘ g) := swap_prod_eq_prod (f ∘ g) h3 done done lemma F_invertible (m i : Nat) : invertible (F m i) := by by_cases h : rel_prime m i · -- Case 1. h : rel_prime m i rewrite [F_rp_def h] show invertible [i]_m from (Theorem_7_3_7 m i).rtl h done · -- Case 2. h : ¬rel_prime m i rewrite [F_not_rp_def h] apply Exists.intro [1]_m show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m done done lemma Fprod_invertible (m : Nat) : ∀ (k : Nat), invertible (prod_seq k 0 (F m)) := by by_induc · -- Base Case apply Exists.intro [1]_m show prod_seq 0 0 (F m) * [1]_m = [1]_m from calc prod_seq 0 0 (F m) * [1]_m _ = [1]_m * [1]_m := by rw [prod_seq_base] _ = [1]_m := Theorem_7_3_6_7 ([1]_m) done · -- Induction Step fix k : Nat assume ih : invertible (prod_seq k 0 (F m)) rewrite [prod_seq_zero_step] show invertible (prod_seq k 0 (F m) * (F m k)) from (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k) done done theorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m ^ (phi m) = [1]_m := by have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2 show [a]_m ^ (phi m) = [1]_m from calc [a]_m ^ (phi m) _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3] _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring _ = prod_seq m 0 (F m ∘ G m a) * Y := by rw [FG_prod h1 m, phi_def] _ = prod_seq m 0 (F m) * Y := by rw [perm_prod (F m) m (G m a) (G_perm_below h1)] _ = [1]_m := by rw [h3] done lemma Exercise_7_4_5_Int (m : Nat) (a : Int) : ∀ (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry lemma Exercise_7_4_5_Nat (m a n : Nat) : [a]_m ^ n = [a ^ n]_m := by rewrite [Exercise_7_4_5_Int] rfl done theorem Euler's_theorem {m a : Nat} [NeZero m] (h1 : rel_prime m a) : a ^ (phi m) ≡ 1 (MOD m) := by have h2 : [a]_m ^ (phi m) = [1]_m := Theorem_7_4_2 h1 rewrite [Exercise_7_4_5_Nat m a (phi m)] at h2 --h2 : [a ^ phi m]_m = [1]_m show a ^ (phi m) ≡ 1 (MOD m) from (cc_eq_iff_congr _ _ _).ltr h2 done #eval gcd 10 7 --Answer: 1. So 10 and 7 are relatively prime #eval 7 ^ phi 10 --Answer: 2401, which is congruent to 1 mod 10. end Euler /- Section 7.5 -/ lemma num_rp_prime {p : Nat} (h1 : prime p) : ∀ k < p, num_rp_below p (k + 1) = k := sorry lemma phi_prime {p : Nat} (h1 : prime p) : phi p = p - 1 := by have h2 : 1 ≤ p := prime_pos h1 have h3 : p - 1 + 1 = p := Nat.sub_add_cancel h2 have h4 : p - 1 < p := by linarith have h5 : num_rp_below p (p - 1 + 1) = p - 1 := num_rp_prime h1 (p - 1) h4 rewrite [h3] at h5 show phi p = p - 1 from h5 done theorem Theorem_7_2_2_Int {a c : Nat} {b : Int} (h1 : ↑c ∣ ↑a * b) (h2 : rel_prime a c) : ↑c ∣ b := by rewrite [Int.natCast_dvd, Int.natAbs_mul, Int.natAbs_ofNat] at h1 --h1 : c ∣ a * Int.natAbs b rewrite [Int.natCast_dvd] --Goal : c ∣ Int.natAbs b show c ∣ Int.natAbs b from Theorem_7_2_2 h1 h2 done lemma Lemma_7_4_5 {m n : Nat} (a b : Int) (h1 : rel_prime m n) : a ≡ b (MOD m * n) ↔ a ≡ b (MOD m) ∧ a ≡ b (MOD n) := by apply Iff.intro · -- (→) assume h2 : a ≡ b (MOD m * n) obtain (j : Int) (h3 : a - b = (m * n) * j) from h2 apply And.intro · -- Proof of a ≡ b (MOD m) apply Exists.intro (n * j) show a - b = m * (n * j) from calc a - b _ = m * n * j := h3 _ = m * (n * j) := by ring done · -- Proof of a ≡ b (MOD n) apply Exists.intro (m * j) show a - b = n * (m * j) from calc a - b _ = m * n * j := h3 _ = n * (m * j) := by ring done done · -- (←) assume h2 : a ≡ b (MOD m) ∧ a ≡ b (MOD n) obtain (j : Int) (h3 : a - b = m * j) from h2.left have h4 : (↑n : Int) ∣ a - b := h2.right rewrite [h3] at h4 --h4 : ↑n ∣ ↑m * j have h5 : ↑n ∣ j := Theorem_7_2_2_Int h4 h1 obtain (k : Int) (h6 : j = n * k) from h5 apply Exists.intro k --Goal : a - b = ↑(m * n) * k rewrite [Nat.cast_mul] --Goal : a - b = ↑m * ↑n * k show a - b = (m * n) * k from calc a - b _ = m * j := h3 _ = m * (n * k) := by rw [h6] _ = (m * n) * k := by ring done done --From exercises of Section 7.2 theorem rel_prime_symm {a b : Nat} (h : rel_prime a b) : rel_prime b a := sorry lemma prime_NeZero {p : Nat} (h : prime p) : NeZero p := by rewrite [neZero_iff] --Goal : p ≠ 0 define at h linarith done lemma Lemma_7_5_1 {p e d m c s : Nat} {t : Int} (h1 : prime p) (h2 : e * d = (p - 1) * s + 1) (h3 : m ^ e - c = p * t) : c ^ d ≡ m (MOD p) := by have h4 : m ^ e ≡ c (MOD p) := Exists.intro t h3 have h5 : [m ^ e]_p = [c]_p := (cc_eq_iff_congr _ _ _).rtl h4 rewrite [←Exercise_7_4_5_Nat] at h5 --h5 : [m]_p ^ e = [c]_p by_cases h6 : p ∣ m · -- Case 1. h6 : p ∣ m have h7 : m ≡ 0 (MOD p) := by obtain (j : Nat) (h8 : m = p * j) from h6 apply Exists.intro (↑j : Int) --Goal : ↑m - 0 = ↑p * ↑j rewrite [h8, Nat.cast_mul] ring done have h8 : [m]_p = [0]_p := (cc_eq_iff_congr _ _ _).rtl h7 have h9 : e * d ≠ 0 := by rewrite [h2] show (p - 1) * s + 1 ≠ 0 from Nat.add_one_ne_zero _ done have h10 : (0 : Int) ^ (e * d) = 0 := zero_pow h9 have h11 : [c ^ d]_p = [m]_p := calc [c ^ d]_p _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat] _ = ([m]_p ^ e) ^ d := by rw [h5] _ = [m]_p ^ (e * d) := by ring _ = [0]_p ^ (e * d) := by rw [h8] _ = [0 ^ (e * d)]_p := Exercise_7_4_5_Int _ _ _ _ = [0]_p := by rw [h10] _ = [m]_p := by rw [h8] show c ^ d ≡ m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11 done · -- Case 2. h6 : ¬p ∣ m have h7 : rel_prime m p := rel_prime_of_prime_not_dvd h1 h6 have h8 : rel_prime p m := rel_prime_symm h7 have h9 : NeZero p := prime_NeZero h1 have h10 : (1 : Int) ^ s = 1 := by ring have h11 : [c ^ d]_p = [m]_p := calc [c ^ d]_p _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat] _ = ([m]_p ^ e) ^ d := by rw [h5] _ = [m]_p ^ (e * d) := by ring _ = [m]_p ^ ((p - 1) * s + 1) := by rw [h2] _ = ([m]_p ^ (p - 1)) ^ s * [m]_p := by ring _ = ([m]_p ^ (phi p)) ^ s * [m]_p := by rw [phi_prime h1] _ = [1]_p ^ s * [m]_p := by rw [Theorem_7_4_2 h8] _ = [1 ^ s]_p * [m]_p := by rw [Exercise_7_4_5_Int] _ = [1]_p * [m]_p := by rw [h10] _ = [m]_p * [1]_p := by ring _ = [m]_p := Theorem_7_3_6_7 _ show c ^ d ≡ m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11 done done theorem Theorem_7_5_1 (p q n e d k m c : Nat) (p_prime : prime p) (q_prime : prime q) (p_ne_q : p ≠ q) (n_pq : n = p * q) (ed_congr_1 : e * d = k * (p - 1) * (q - 1) + 1) (h1 : [m]_n ^ e = [c]_n) : [c]_n ^ d = [m]_n := by rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] at h1 --h1 : m ^ e ≡ c (MOD n) rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] --Goal : c ^ d ≡ m (MOD n) obtain (j : Int) (h2 : m ^ e - c = n * j) from h1 rewrite [n_pq, Nat.cast_mul] at h2 --h2 : m ^ e - c = p * q * j have h3 : e * d = (p - 1) * (k * (q - 1)) + 1 := by rewrite [ed_congr_1] ring done have h4 : m ^ e - c = p * (q * j) := by rewrite [h2] ring done have congr_p : c ^ d ≡ m (MOD p) := Lemma_7_5_1 p_prime h3 h4 have h5 : e * d = (q - 1) * (k * (p - 1)) + 1 := by rewrite [ed_congr_1] ring done have h6 : m ^ e - c = q * (p * j) := by rewrite [h2] ring done have congr_q : c ^ d ≡ m (MOD q) := Lemma_7_5_1 q_prime h5 h6 have h7 : ¬q ∣ p := by by_contra h8 have h9 : q = 1 ∨ q = p := dvd_prime p_prime h8 disj_syll h9 (prime_not_one q_prime) show False from p_ne_q h9.symm done have h8 : rel_prime p q := rel_prime_of_prime_not_dvd q_prime h7 rewrite [n_pq, Lemma_7_4_5 _ _ h8] show c ^ d ≡ m (MOD p) ∧ c ^ d ≡ m (MOD q) from And.intro congr_p congr_q done /- BEGIN EXERCISES -/ namespace Exercises /- Section 7.1 -/ -- 1. theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry -- 2. lemma gcd_comm_lt {a b : Nat} (h : a < b) : gcd a b = gcd b a := sorry theorem gcd_comm (a b : Nat) : gcd a b = gcd b a := sorry -- 3. theorem Exercise_7_1_5 (a b : Nat) (n : Int) : (∃ (s t : Int), s * a + t * b = n) ↔ (↑(gcd a b) : Int) ∣ n := sorry -- 4. theorem Exercise_7_1_6 (a b c : Nat) : gcd a b = gcd (a + b * c) b := sorry -- 5. theorem gcd_is_nonzero {a b : Nat} (h : a ≠ 0 ∨ b ≠ 0) : gcd a b ≠ 0 := sorry -- 6. theorem gcd_greatest {a b d : Nat} (h1 : gcd a b ≠ 0) (h2 : d ∣ a) (h3 : d ∣ b) : d ≤ gcd a b := sorry -- 7. lemma Lemma_7_1_10a {a b : Nat} (n : Nat) (h : a ∣ b) : (n * a) ∣ (n * b) := sorry lemma Lemma_7_1_10b {a b n : Nat} (h1 : n ≠ 0) (h2 : (n * a) ∣ (n * b)) : a ∣ b := sorry lemma Lemma_7_1_10c {a b : Nat} (h1 : a ∣ b) (h2 : b ∣ a) : a = b := sorry theorem Exercise_7_1_10 (a b n : Nat) : gcd (n * a) (n * b) = n * gcd a b := sorry /- Section 7.2 -/ -- 1. lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry -- 2. -- Hints: Start with apply List.rec. You may find mul_ne_zero useful theorem prod_nonzero_nonzero : ∀ (l : List Nat), (∀ a ∈ l, a ≠ 0) → prod l ≠ 0 := sorry -- 3. theorem rel_prime_iff_no_common_factor (a b : Nat) : rel_prime a b ↔ ¬∃ (p : Nat), prime p ∧ p ∣ a ∧ p ∣ b := sorry -- 4. theorem rel_prime_symm {a b : Nat} (h : rel_prime a b) : rel_prime b a := sorry -- 5. lemma in_prime_factorization_iff_prime_factor {a : Nat} {l : List Nat} (h1 : prime_factorization a l) (p : Nat) : p ∈ l ↔ prime_factor p a := sorry -- 6. theorem Exercise_7_2_5 {a b : Nat} {l m : List Nat} (h1 : prime_factorization a l) (h2 : prime_factorization b m) : rel_prime a b ↔ (¬∃ (p : Nat), p ∈ l ∧ p ∈ m) := sorry -- 7. theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry -- 8. theorem Exercise_7_2_7 {a b a' b' : Nat} (h1 : rel_prime a b) (h2 : a' ∣ a) (h3 : b' ∣ b) : rel_prime a' b' := sorry -- 9. theorem Exercise_7_2_9 {a b j k : Nat} (h1 : gcd a b ≠ 0) (h2 : a = j * gcd a b) (h3 : b = k * gcd a b) : rel_prime j k := sorry -- 10. theorem Exercise_7_2_17a (a b c : Nat) : gcd a (b * c) ∣ gcd a b * gcd a c := sorry /- Section 7.3 -/ -- 1. theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry -- 2. theorem Theorem_7_3_6_3 {m : Nat} (X : ZMod m) : X + [0]_m = X := sorry -- 3. theorem Theorem_7_3_6_4 {m : Nat} (X : ZMod m) : ∃ (Y : ZMod m), X + Y = [0]_m := sorry -- 4. theorem Exercise_7_3_4a {m : Nat} (Z1 Z2 : ZMod m) (h1 : ∀ (X : ZMod m), X + Z1 = X) (h2 : ∀ (X : ZMod m), X + Z2 = X) : Z1 = Z2 := sorry -- 5. theorem Exercise_7_3_4b {m : Nat} (X Y1 Y2 : ZMod m) (h1 : X + Y1 = [0]_m) (h2 : X + Y2 = [0]_m) : Y1 = Y2 := sorry -- 6. theorem Theorem_7_3_10 (m a : Nat) (b : Int) : ¬(↑(gcd m a) : Int) ∣ b → ¬∃ (x : Int), a * x ≡ b (MOD m) := sorry -- 7. theorem Theorem_7_3_11 (m n : Nat) (a b : Int) (h1 : n ≠ 0) : n * a ≡ n * b (MOD n * m) ↔ a ≡ b (MOD m) := sorry -- 8.

theorem Exercise_7_3_16 {m : Nat} {a b : Int} (h : a ≡ b (MOD m)) : ∀ (n : Nat), a ^ n ≡ b ^ n (MOD m)

HTPI.Exercises.Exercise_7_3_16

htpi/HTPILib/Chap7.lean

htpi

/- Copyright 2023 Daniel J. Velleman -/ import HTPILib.Chap6 namespace HTPI /- Definitions -/ lemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by have h : n + 1 > 0 := Nat.succ_pos n show a % (n + 1) < n + 1 from Nat.mod_lt a h done def gcd (a b : Nat) : Nat := match b with | 0 => a | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd (n + 1) (a % (n + 1)) termination_by b mutual def gcd_c1 (a b : Nat) : Int := match b with | 0 => 1 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c2 (n + 1) (a % (n + 1)) --Corresponds to s = t' termination_by b def gcd_c2 (a b : Nat) : Int := match b with | 0 => 0 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c1 (n + 1) (a % (n + 1)) - (gcd_c2 (n + 1) (a % (n + 1))) * ↑(a / (n + 1)) --Corresponds to t = s' - t'q termination_by b end def prime (n : Nat) : Prop := 2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n def prime_factor (p n : Nat) : Prop := prime p ∧ p ∣ n def all_prime (l : List Nat) : Prop := ∀ p ∈ l, prime p def nondec (l : List Nat) : Prop := match l with | [] => True --Of course, True is a proposition that is always true | n :: L => (∀ m ∈ L, n ≤ m) ∧ nondec L def nondec_prime_list (l : List Nat) : Prop := all_prime l ∧ nondec l def prod (l : List Nat) : Nat := match l with | [] => 1 | n :: L => n * (prod L) def prime_factorization (n : Nat) (l : List Nat) : Prop := nondec_prime_list l ∧ prod l = n def rel_prime (a b : Nat) : Prop := gcd a b = 1 def congr_mod (m : Nat) (a b : Int) : Prop := (↑m : Int) ∣ (a - b) def cc (m : Nat) (a : Int) : ZMod m := (↑a : ZMod m) notation:50 a " ≡ " b " (MOD " m ")" => congr_mod m a b notation:max "["a"]_"m:max => cc m a def invertible {m : Nat} (X : ZMod m) : Prop := ∃ (Y : ZMod m), X * Y = [1]_m def num_rp_below (m k : Nat) : Nat := match k with | 0 => 0 | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j def phi (m : Nat) : Nat := num_rp_below m m def prod_seq {m : Nat} (j k : Nat) (f : Nat → ZMod m) : ZMod m := match j with | 0 => [1]_m | n + 1 => prod_seq n k f * f (k + n) def maps_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i < n, g i < n def one_one_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i1 < n, ∀ i2 < n, g i1 = g i2 → i1 = i2 def onto_below (n : Nat) (g : Nat → Nat) : Prop := ∀ k < n, ∃ i < n, g i = k def perm_below (n : Nat) (g : Nat → Nat) : Prop := maps_below n g ∧ one_one_below n g ∧ onto_below n g def inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m) def swap (u v i : Nat) : Nat := if i = u then v else if i = v then u else i namespace Euler --For definitions specific to Euler's theorem def F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m def G (m a i : Nat) : Nat := (a * i) % m def Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i end Euler /- Section 7.1 -/ theorem dvd_mod_of_dvd_a_b {a b d : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ (a % b) := by set q : Nat := a / b have h3 : b * q + a % b = a := Nat.div_add_mod a b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 define --Goal : ∃ (c : Nat), a % b = d * c apply Exists.intro (j - k * q) show a % b = d * (j - k * q) from calc a % b _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm _ = a - b * q := by rw [h3] _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc] _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm done theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry #eval gcd 672 161 --Answer: 7 lemma gcd_base (a : Nat) : gcd a 0 = a := by rfl lemma gcd_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd a b = gcd b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1)) rfl done lemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b ≠ 0) : a % b < b := by have h1 : b > 0 := Nat.pos_of_ne_zero h show a % b < b from Nat.mod_lt a h1 done lemma dvd_self (n : Nat) : n ∣ n := by apply Exists.intro 1 ring done theorem gcd_dvd : ∀ (b a : Nat), (gcd a b) ∣ a ∧ (gcd a b) ∣ b := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd a b_1) ∣ a ∧ (gcd a b_1) ∣ b_1 fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_base] --Goal: a ∣ a ∧ a ∣ 0 apply And.intro (dvd_self a) define apply Exists.intro 0 rfl done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_nonzero a h1] --Goal : gcd b (a % b) ∣ a ∧ gcd b (a % b) ∣ b have h2 : a % b < b := mod_nonzero_lt a h1 have h3 : (gcd b (a % b)) ∣ b ∧ (gcd b (a % b)) ∣ (a % b) := ih (a % b) h2 b apply And.intro _ h3.left show (gcd b (a % b)) ∣ a from dvd_a_of_dvd_b_mod h3.left h3.right done done theorem gcd_dvd_left (a b : Nat) : (gcd a b) ∣ a := (gcd_dvd b a).left theorem gcd_dvd_right (a b : Nat) : (gcd a b) ∣ b := (gcd_dvd b a).right lemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl lemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c1 a b = gcd_c2 b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done lemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl lemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * ↑(a / b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done theorem gcd_lin_comb : ∀ (b a : Nat), (gcd_c1 a b) * ↑a + (gcd_c2 a b) * ↑b = ↑(gcd a b) := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd_c1 a b_1) * ↑a + (gcd_c2 a b_1) * ↑b_1 = ↑(gcd a b_1) fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base] --Goal : 1 * ↑a + 0 * ↑0 = ↑a ring done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1] --Goal : gcd_c2 b (a % b) * ↑a + -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * ↑(a / b)) * ↑b = -- ↑(gcd b (a % b)) set r : Nat := a % b set q : Nat := a / b set s : Int := gcd_c1 b r set t : Int := gcd_c2 b r --Goal : t * ↑a + (s - t * ↑q) * ↑b = ↑(gcd b r) have h2 : r < b := mod_nonzero_lt a h1 have h3 : s * ↑b + t * ↑r = ↑(gcd b r) := ih r h2 b have h4 : b * q + r = a := Nat.div_add_mod a b rewrite [←h3, ←h4] rewrite [Nat.cast_add, Nat.cast_mul] --Goal : t * (↑b * ↑q + ↑r) + (s - t * ↑q) * ↑b = s * ↑b + t * ↑r ring done done #eval gcd_c1 672 161 --Answer: 6 #eval gcd_c2 672 161 --Answer: -25 --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161 theorem Theorem_7_1_6 {d a b : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ gcd a b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑d ∣ ↑(gcd a b) set s : Int := gcd_c1 a b set t : Int := gcd_c2 a b have h3 : s * ↑a + t * ↑b = ↑(gcd a b) := gcd_lin_comb b a rewrite [←h3] --Goal : ↑d ∣ s * ↑a + t * ↑b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul] --Goal : ↑d ∣ s * (↑d * ↑j) + t * (↑d * ↑k) define apply Exists.intro (s * ↑j + t * ↑k) ring done /- Section 7.2 -/ theorem dvd_trans {a b c : Nat} (h1 : a ∣ b) (h2 : b ∣ c) : a ∣ c := by define at h1; define at h2; define obtain (m : Nat) (h3 : b = a * m) from h1 obtain (n : Nat) (h4 : c = b * n) from h2 rewrite [h3, mul_assoc] at h4 apply Exists.intro (m * n) show c = a * (m * n) from h4 done lemma exists_prime_factor : ∀ (n : Nat), 2 ≤ n → ∃ (p : Nat), prime_factor p n := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, 2 ≤ n_1 → ∃ (p : Nat), prime_factor p n_1 assume h1 : 2 ≤ n by_cases h2 : prime n · -- Case 1. h2 : prime n apply Exists.intro n define --Goal : prime n ∧ n ∣ n show prime n ∧ n ∣ n from And.intro h2 (dvd_self n) done · -- Case 2. h2 : ¬prime n define at h2 --h2 : ¬(2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n) demorgan at h2 disj_syll h2 h1 obtain (a : Nat) (h3 : ∃ (b : Nat), a * b = n ∧ a < n ∧ b < n) from h2 obtain (b : Nat) (h4 : a * b = n ∧ a < n ∧ b < n) from h3 have h5 : 2 ≤ a := by by_contra h6 have h7 : a ≤ 1 := by linarith have h8 : n ≤ b := calc n _ = a * b := h4.left.symm _ ≤ 1 * b := by rel [h7] _ = b := by ring linarith --n ≤ b contradicts b < n done have h6 : ∃ (p : Nat), prime_factor p a := ih a h4.right.left h5 obtain (p : Nat) (h7 : prime_factor p a) from h6 apply Exists.intro p define --Goal : prime p ∧ p ∣ n define at h7 --h7 : prime p ∧ p ∣ a apply And.intro h7.left have h8 : a ∣ n := by apply Exists.intro b show n = a * b from (h4.left).symm done show p ∣ n from dvd_trans h7.right h8 done done lemma exists_least_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q := by set S : Set Nat := {p : Nat | prime_factor p n} have h2 : ∃ (p : Nat), p ∈ S := exists_prime_factor n h show ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q from well_ord_princ S h2 done lemma all_prime_nil : all_prime [] := by define --Goal : ∀ p ∈ [], prime p fix p : Nat contrapos --Goal : ¬prime p → p ∉ [] assume h1 : ¬prime p show p ∉ [] from List.not_mem_nil p done lemma all_prime_cons (n : Nat) (L : List Nat) : all_prime (n :: L) ↔ prime n ∧ all_prime L := by apply Iff.intro · -- (→) assume h1 : all_prime (n :: L) --Goal : prime n ∧ all_prime L define at h1 --h1 : ∀ p ∈ n :: L, prime p apply And.intro (h1 n (List.mem_cons_self n L)) define --Goal : ∀ p ∈ L, prime p fix p : Nat assume h2 : p ∈ L show prime p from h1 p (List.mem_cons_of_mem n h2) done · -- (←) assume h1 : prime n ∧ all_prime L --Goal : all_prime (n :: l) define : all_prime L at h1 define fix p : Nat assume h2 : p ∈ n :: L rewrite [List.mem_cons] at h2 --h2 : p = n ∨ p ∈ L by_cases on h2 · -- Case 1. h2 : p = n rewrite [h2] show prime n from h1.left done · -- Case 2. h2 : p ∈ L show prime p from h1.right p h2 done done done lemma nondec_nil : nondec [] := by define --Goal : True trivial --trivial proves some obviously true statements, such as True done lemma nondec_cons (n : Nat) (L : List Nat) : nondec (n :: L) ↔ (∀ m ∈ L, n ≤ m) ∧ nondec L := by rfl lemma prod_nil : prod [] = 1 := by rfl lemma prod_cons : prod (n :: L) = n * (prod L) := by rfl lemma exists_cons_of_length_eq_succ {A : Type} {l : List A} {n : Nat} (h : l.length = n + 1) : ∃ (a : A) (L : List A), l = a :: L ∧ L.length = n := by have h1 : ¬l.length = 0 := by linarith rewrite [List.length_eq_zero] at h1 obtain (a : A) (h2 : ∃ (L : List A), l = a :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List A) (h3 : l = a :: L) from h2 apply Exists.intro a apply Exists.intro L apply And.intro h3 have h4 : (a :: L).length = L.length + 1 := List.length_cons a L rewrite [←h3, h] at h4 show L.length = n from (Nat.add_right_cancel h4).symm done lemma list_elt_dvd_prod_by_length (a : Nat) : ∀ (n : Nat), ∀ (l : List Nat), l.length = n → a ∈ l → a ∣ prod l := by by_induc · --Base Case fix l : List Nat assume h1 : l.length = 0 rewrite [List.length_eq_zero] at h1 --h1 : l = [] rewrite [h1] --Goal : a ∈ [] → a ∣ prod [] contrapos assume h2 : ¬a ∣ prod [] show a ∉ [] from List.not_mem_nil a done · -- Induction Step fix n : Nat assume ih : ∀ (l : List Nat), List.length l = n → a ∈ l → a ∣ prod l fix l : List Nat assume h1 : l.length = n + 1 --Goal : a ∈ l → a ∣ prod l obtain (b : Nat) (h2 : ∃ (L : List Nat), l = b :: L ∧ L.length = n) from exists_cons_of_length_eq_succ h1 obtain (L : List Nat) (h3 : l = b :: L ∧ L.length = n) from h2 have h4 : a ∈ L → a ∣ prod L := ih L h3.right assume h5 : a ∈ l rewrite [h3.left, prod_cons] --Goal : a ∣ b * prod L rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b ∨ a ∈ L by_cases on h5 · -- Case 1. h5 : a = b apply Exists.intro (prod L) rewrite [h5] rfl done · -- Case 2. h5 : a ∈ L have h6 : a ∣ prod L := h4 h5 have h7 : prod L ∣ b * prod L := by apply Exists.intro b ring done show a ∣ b * prod L from dvd_trans h6 h7 done done done lemma list_elt_dvd_prod {a : Nat} {l : List Nat} (h : a ∈ l) : a ∣ prod l := by set n : Nat := l.length have h1 : l.length = n := by rfl show a ∣ prod l from list_elt_dvd_prod_by_length a n l h1 h done lemma exists_prime_factorization : ∀ (n : Nat), n ≥ 1 → ∃ (l : List Nat), prime_factorization n l := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, n_1 ≥ 1 → ∃ (l : List Nat), prime_factorization n_1 l assume h1 : n ≥ 1 by_cases h2 : n = 1 · -- Case 1. h2 : n = 1 apply Exists.intro [] define apply And.intro · -- Proof of nondec_prime_list [] define show all_prime [] ∧ nondec [] from And.intro all_prime_nil nondec_nil done · -- Proof of prod [] = n rewrite [prod_nil, h2] rfl done done · -- Case 2. h2 : n ≠ 1 have h3 : n ≥ 2 := lt_of_le_of_ne' h1 h2 obtain (p : Nat) (h4 : prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q) from exists_least_prime_factor h3 have p_prime_factor : prime_factor p n := h4.left define at p_prime_factor have p_prime : prime p := p_prime_factor.left have p_dvd_n : p ∣ n := p_prime_factor.right have p_least : ∀ (q : Nat), prime_factor q n → p ≤ q := h4.right obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n have h5 : m ≠ 0 := by contradict h1 with h6 have h7 : n = 0 := calc n _ = p * m := n_eq_pm _ = p * 0 := by rw [h6] _ = 0 := by ring rewrite [h7] decide done have m_pos : 0 < m := Nat.pos_of_ne_zero h5 have m_lt_n : m < n := by define at p_prime show m < n from calc m _ < m + m := by linarith _ = 2 * m := by ring _ ≤ p * m := by rel [p_prime.left] _ = n := n_eq_pm.symm done obtain (L : List Nat) (h6 : prime_factorization m L) from ih m m_lt_n m_pos define at h6 have ndpl_L : nondec_prime_list L := h6.left define at ndpl_L apply Exists.intro (p :: L) define apply And.intro · -- Proof of nondec_prime_list (p :: L) define apply And.intro · -- Proof of all_prime (p :: L) rewrite [all_prime_cons] show prime p ∧ all_prime L from And.intro p_prime ndpl_L.left done · -- Proof of nondec (p :: L) rewrite [nondec_cons] apply And.intro _ ndpl_L.right fix q : Nat assume q_in_L : q ∈ L have h7 : q ∣ prod L := list_elt_dvd_prod q_in_L rewrite [h6.right] at h7 --h7 : q ∣ m have h8 : m ∣ n := by apply Exists.intro p rewrite [n_eq_pm] ring done have q_dvd_n : q ∣ n := dvd_trans h7 h8 have ap_L : all_prime L := ndpl_L.left define at ap_L have q_prime_factor : prime_factor q n := And.intro (ap_L q q_in_L) q_dvd_n show p ≤ q from p_least q q_prime_factor done done · -- Proof of prod (p :: L) = n rewrite [prod_cons, h6.right, n_eq_pm] rfl done done done theorem Theorem_7_2_2 {a b c : Nat} (h1 : c ∣ a * b) (h2 : rel_prime a c) : c ∣ b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑c ∣ ↑b define at h1; define at h2; define obtain (j : Nat) (h3 : a * b = c * j) from h1 set s : Int := gcd_c1 a c set t : Int := gcd_c2 a c have h4 : s * ↑a + t * ↑c = ↑(gcd a c) := gcd_lin_comb c a rewrite [h2, Nat.cast_one] at h4 --h4 : s * ↑a + t * ↑c = (1 : Int) apply Exists.intro (s * ↑j + t * ↑b) show ↑b = ↑c * (s * ↑j + t * ↑b) from calc ↑b _ = (1 : Int) * ↑b := (one_mul _).symm _ = (s * ↑a + t * ↑c) * ↑b := by rw [h4] _ = s * (↑a * ↑b) + t * ↑c * ↑b := by ring _ = s * (↑c * ↑j) + t * ↑c * ↑b := by rw [←Nat.cast_mul a b, h3, Nat.cast_mul c j] _ = ↑c * (s * ↑j + t * ↑b) := by ring done lemma le_nonzero_prod_left {a b : Nat} (h : a * b ≠ 0) : a ≤ a * b := by have h1 : b ≠ 0 := by contradict h with h1 rewrite [h1] ring done have h2 : 1 ≤ b := Nat.pos_of_ne_zero h1 show a ≤ a * b from calc a = a * 1 := (mul_one a).symm _ ≤ a * b := by rel [h2] done lemma le_nonzero_prod_right {a b : Nat} (h : a * b ≠ 0) : b ≤ a * b := by rewrite [mul_comm] rewrite [mul_comm] at h show b ≤ b * a from le_nonzero_prod_left h done lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry lemma rel_prime_of_prime_not_dvd {a p : Nat} (h1 : prime p) (h2 : ¬p ∣ a) : rel_prime a p := by have h3 : gcd a p ∣ a := gcd_dvd_left a p have h4 : gcd a p ∣ p := gcd_dvd_right a p have h5 : gcd a p = 1 ∨ gcd a p = p := dvd_prime h1 h4 have h6 : gcd a p ≠ p := by contradict h2 with h6 rewrite [h6] at h3 show p ∣ a from h3 done disj_syll h5 h6 show rel_prime a p from h5 done theorem Theorem_7_2_3 {a b p : Nat} (h1 : prime p) (h2 : p ∣ a * b) : p ∣ a ∨ p ∣ b := by or_right with h3 have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3 show p ∣ b from Theorem_7_2_2 h2 h4 done lemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a ≥ 1 := by have h1 : a ≠ 0 := by by_contra h1 rewrite [h1] at h contradict h linarith done show a ≥ 1 from Nat.pos_of_ne_zero h1 done lemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by have h1 : a ≥ 1 := ge_one_of_prod_one h have h2 : a * b ≠ 0 := by linarith have h3 : a ≤ a * b := le_nonzero_prod_left h2 rewrite [h] at h3 show a = 1 from Nat.le_antisymm h3 h1 done lemma eq_one_of_dvd_one {n : Nat} (h : n ∣ 1) : n = 1 := by obtain (j : Nat) (h1 : 1 = n * j) from h show n = 1 from eq_one_of_prod_one h1.symm done lemma prime_not_one {p : Nat} (h : prime p) : p ≠ 1 := by define at h linarith done theorem Theorem_7_2_4 {p : Nat} (h1 : prime p) : ∀ (l : List Nat), p ∣ prod l → ∃ a ∈ l, p ∣ a := by apply List.rec · -- Base Case. Goal : p ∣ prod [] → ∃ a ∈ [], p ∣ a rewrite [prod_nil] assume h2 : p ∣ 1 show ∃ a ∈ [], p ∣ a from absurd (eq_one_of_dvd_one h2) (prime_not_one h1) done · -- Induction Step fix b : Nat fix L : List Nat assume ih : p ∣ prod L → ∃ a ∈ L, p ∣ a --Goal : p ∣ prod (b :: L) → ∃ a ∈ b :: L, p ∣ a assume h2 : p ∣ prod (b :: L) rewrite [prod_cons] at h2 have h3 : p ∣ b ∨ p ∣ prod L := Theorem_7_2_3 h1 h2 by_cases on h3 · -- Case 1. h3 : p ∣ b apply Exists.intro b show b ∈ b :: L ∧ p ∣ b from And.intro (List.mem_cons_self b L) h3 done · -- Case 2. h3 : p ∣ prod L obtain (a : Nat) (h4 : a ∈ L ∧ p ∣ a) from ih h3 apply Exists.intro a show a ∈ b :: L ∧ p ∣ a from And.intro (List.mem_cons_of_mem b h4.left) h4.right done done done lemma prime_in_list {p : Nat} {l : List Nat} (h1 : prime p) (h2 : all_prime l) (h3 : p ∣ prod l) : p ∈ l := by obtain (a : Nat) (h4 : a ∈ l ∧ p ∣ a) from Theorem_7_2_4 h1 l h3 define at h2 have h5 : prime a := h2 a h4.left have h6 : p = 1 ∨ p = a := dvd_prime h5 h4.right disj_syll h6 (prime_not_one h1) rewrite [h6] show a ∈ l from h4.left done lemma first_le_first {p q : Nat} {l m : List Nat} (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m)) (h3 : prod (p :: l) = prod (q :: m)) : p ≤ q := by define at h1; define at h2 have h4 : q ∣ prod (p :: l) := by define apply Exists.intro (prod m) rewrite [←prod_cons] show prod (p :: l) = prod (q :: m) from h3 done have h5 : all_prime (q :: m) := h2.left rewrite [all_prime_cons] at h5 have h6 : q ∈ p :: l := prime_in_list h5.left h1.left h4 have h7 : nondec (p :: l) := h1.right rewrite [nondec_cons] at h7 rewrite [List.mem_cons] at h6 by_cases on h6 · -- Case 1. h6 : q = p linarith done · -- Case 2. h6 : q ∈ l have h8 : ∀ m ∈ l, p ≤ m := h7.left show p ≤ q from h8 q h6 done done lemma nondec_prime_list_tail {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by define at h define rewrite [all_prime_cons, nondec_cons] at h show all_prime l ∧ nondec l from And.intro h.left.right h.right.right done lemma cons_prod_not_one {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : prod (p :: l) ≠ 1 := by define at h have h1 : all_prime (p :: l) := h.left rewrite [all_prime_cons] at h1 rewrite [prod_cons] by_contra h2 show False from (prime_not_one h1.left) (eq_one_of_prod_one h2) done lemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) : l = [] ↔ prod l = 1 := by apply Iff.intro · -- (→) assume h1 : l = [] rewrite [h1] show prod [] = 1 from prod_nil done · -- (←) contrapos assume h1 : ¬l = [] obtain (p : Nat) (h2 : ∃ (L : List Nat), l = p :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List Nat) (h3 : l = p :: L) from h2 rewrite [h3] at h rewrite [h3] show ¬prod (p :: L) = 1 from cons_prod_not_one h done done lemma prime_pos {p : Nat} (h : prime p) : p > 0 := by define at h linarith done theorem Theorem_7_2_5 : ∀ (l1 l2 : List Nat), nondec_prime_list l1 → nondec_prime_list l2 → prod l1 = prod l2 → l1 = l2 := by apply List.rec · -- Base Case. Goal : ∀ (l2 : List Nat), nondec_prime_list [] → -- nondec_prime_list l2 → prod [] = prod l2 → [] = l2 fix l2 : List Nat assume h1 : nondec_prime_list [] assume h2 : nondec_prime_list l2 assume h3 : prod [] = prod l2 rewrite [prod_nil, eq_comm, ←list_nil_iff_prod_one h2] at h3 show [] = l2 from h3.symm done · -- Induction Step fix p : Nat fix L1 : List Nat assume ih : ∀ (L2 : List Nat), nondec_prime_list L1 → nondec_prime_list L2 → prod L1 = prod L2 → L1 = L2 -- Goal : ∀ (l2 : List Nat), nondec_prime_list (p :: L1) → -- nondec_prime_list l2 → prod (p :: L1) = prod l2 → p :: L1 = l2 fix l2 : List Nat assume h1 : nondec_prime_list (p :: L1) assume h2 : nondec_prime_list l2 assume h3 : prod (p :: L1) = prod l2 have h4 : ¬prod (p :: L1) = 1 := cons_prod_not_one h1 rewrite [h3, ←list_nil_iff_prod_one h2] at h4 obtain (q : Nat) (h5 : ∃ (L : List Nat), l2 = q :: L) from List.exists_cons_of_ne_nil h4 obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5 rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2) rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2) have h7 : p ≤ q := first_le_first h1 h2 h3 have h8 : q ≤ p := first_le_first h2 h1 h3.symm have h9 : p = q := by linarith rewrite [h9, prod_cons, prod_cons] at h3 --h3 : q * prod L1 = q * prod L2 have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1 have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2 define at h2 have h12 : all_prime (q :: L2) := h2.left rewrite [all_prime_cons] at h12 have h13 : q > 0 := prime_pos h12.left have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3 have h15 : L1 = L2 := ih L2 h10 h11 h14 rewrite [h6, h9, h15] rfl done done theorem fund_thm_arith (n : Nat) (h : n ≥ 1) : ∃! (l : List Nat), prime_factorization n l := by exists_unique · -- Existence show ∃ (l : List Nat), prime_factorization n l from exists_prime_factorization n h done · -- Uniqueness fix l1 : List Nat; fix l2 : List Nat assume h1 : prime_factorization n l1 assume h2 : prime_factorization n l2 define at h1; define at h2 have h3 : prod l1 = n := h1.right rewrite [←h2.right] at h3 show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3 done done /- Section 7.3 -/ theorem congr_refl (m : Nat) : ∀ (a : Int), a ≡ a (MOD m) := by fix a : Int define --Goal : ∃ (c : Int), a - a = ↑m * c apply Exists.intro 0 ring done theorem congr_symm {m : Nat} : ∀ {a b : Int}, a ≡ b (MOD m) → b ≡ a (MOD m) := by fix a : Int; fix b : Int assume h1 : a ≡ b (MOD m) define at h1 --h1 : ∃ (c : Int), a - b = ↑m * c define --Goal : ∃ (c : Int), b - a = ↑m * c obtain (c : Int) (h2 : a - b = m * c) from h1 apply Exists.intro (-c) show b - a = m * (-c) from calc b - a _ = -(a - b) := by ring _ = -(m * c) := by rw [h2] _ = m * (-c) := by ring done theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry /- Fundamental properties of congruence classes -/ lemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) : X = Y ↔ X.val = Y.val := Fin.ext_iff lemma val_nat_eq_mod (n k : Nat) : ([k]_(n + 1)).val = k % (n + 1) := by rfl lemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl theorem cc_rep {m : Nat} (X : ZMod m) : ∃ (a : Int), X = [a]_m := match m with | 0 => by apply Exists.intro X rfl done | n + 1 => by apply Exists.intro ↑(X.val) have h1 : X.val < n + 1 := Fin.prop X rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1] rfl done theorem add_class (m : Nat) (a b : Int) : [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm theorem mul_class (m : Nat) (a b : Int) : [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm lemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ [a - b]_m = [0]_m := by apply Iff.intro · -- (→) assume h1 : [a]_m = [b]_m have h2 : a - b = a + (-b) := by ring have h3 : b + (-b) = 0 := by ring show [a - b]_m = [0]_m from calc [a - b]_m _ = [a + (-b)]_m := by rw [h2] _ = [a]_m + [-b]_m := by rw [add_class] _ = [b]_m + [-b]_m := by rw [h1] _ = [b + -b]_m := by rw [add_class] _ = [0]_m := by rw [h3] done · -- (←) assume h1 : [a - b]_m = [0]_m have h2 : b + (a - b) = a := by ring have h3 : b + 0 = b := by ring show [a]_m = [b]_m from calc [a]_m _ = [b + (a - b)]_m := by rw [h2] _ = [b]_m + [a - b]_m := by rw [add_class] _ = [b]_m + [0]_m := by rw [h1] _ = [b + 0]_m := by rw [add_class] _ = [b]_m := by rw [h3] done done lemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) : [a]_m = [0]_m → [-a]_m = [0]_m := by assume h1 : [a]_m = [0]_m have h2 : 0 + (-a) = -a := by ring have h3 : a + (-a) = 0 := by ring show [-a]_m = [0]_m from calc [-a]_m _ = [0 + (-a)]_m := by rw [h2] _ = [0]_m + [-a]_m := by rw [add_class] _ = [a]_m + [-a]_m := by rw [h1] _ = [a + (-a)]_m := by rw [add_class] _ = [0]_m := by rw [h3] done lemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) : [-a]_m = [0]_m ↔ [a]_m = [0]_m := by apply Iff.intro _ (cc_neg_zero_of_cc_zero m a) assume h1 : [-a]_m = [0]_m have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1 have h3 : -(-a) = a := by ring rewrite [h3] at h2 show [a]_m = [0]_m from h2 done lemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl lemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m ↔ m ∣ k := match m with | 0 => by have h : (0 : Int) = (↑(0 : Nat) : Int) := by rfl rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj] apply Iff.intro · -- (→) assume h1 : k = 0 rewrite [h1] show 0 ∣ 0 from dvd_self 0 done · -- (←) assume h1 : 0 ∣ k obtain (c : Nat) (h2 : k = 0 * c) from h1 rewrite [h2] ring done done | n + 1 => by rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero] show k % (n + 1) = 0 ↔ n + 1 ∣ k from (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm done lemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m ↔ ↑m ∣ a := by obtain (k : Nat) (h1 : a = ↑k ∨ a = -↑k) from Int.eq_nat_or_neg a by_cases on h1 · -- Case 1. h1: a = ↑k rewrite [h1, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done · -- Case 2. h1: a = -↑k rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done done theorem cc_eq_iff_congr (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ a ≡ b (MOD m) := calc [a]_m = [b]_m _ ↔ [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b _ ↔ ↑m ∣ (a - b) := cc_zero_iff_dvd m (a - b) _ ↔ a ≡ b (MOD m) := by rfl /- End of fundamental properties of congruence classes -/ lemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m := by have h1 : (↑m : Int) ≠ 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m) show 0 ≤ a % m from Int.emod_nonneg a h1 done lemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m) have h2 : (↑m : Int) > 0 := (Nat.cast_pos).rtl h1 show a % m < m from Int.emod_lt_of_pos a h2 done lemma congr_mod_mod (m : Nat) (a : Int) : a ≡ a % m (MOD m) := by define have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m apply Exists.intro (a / m) show a - a % m = m * (a / m) from calc a - (a % m) _ = m * (a / m) + a % m - a % m := by rw [h1] _ = m * (a / m) := by ring done lemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) := And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a)) theorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) : ∃! (r : Int), 0 ≤ r ∧ r < m ∧ a ≡ r (MOD m) := by exists_unique · -- Existence apply Exists.intro (a % m) show 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) from mod_cmpl_res m a done · -- Uniqueness fix r1 : Int; fix r2 : Int assume h1 : 0 ≤ r1 ∧ r1 < m ∧ a ≡ r1 (MOD m) assume h2 : 0 ≤ r2 ∧ r2 < m ∧ a ≡ r2 (MOD m) have h3 : r1 ≡ r2 (MOD m) := congr_trans (congr_symm h1.right.right) h2.right.right obtain (d : Int) (h4 : r1 - r2 = m * d) from h3 have h5 : r1 - r2 < m * 1 := by linarith have h6 : m * (-1) < r1 - r2 := by linarith rewrite [h4] at h5 --h5 : m * d < m * 1 rewrite [h4] at h6 --h6 : m * -1 < m * d have h7 : (↑m : Int) ≥ 0 := Nat.cast_nonneg m have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7 have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7 have h10 : d = 0 := by linarith show r1 = r2 from calc r1 _ = r1 - r2 + r2 := by ring _ = m * 0 + r2 := by rw [h4, h10] _ = r2 := by ring done done lemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m := (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a) theorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y rewrite [h1, h2] have h3 : a + b = b + a := by ring show [a]_m + [b]_m = [b]_m + [a]_m from calc [a]_m + [b]_m _ = [a + b]_m := add_class m a b _ = [b + a]_m := by rw [h3] _ = [b]_m + [a]_m := (add_class m b a).symm done theorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X rewrite [h1] have h2 : a * 1 = a := by ring show [a]_m * [1]_m = [a]_m from calc [a]_m * [1]_m _ = [a * 1]_m := mul_class m a 1 _ = [a]_m := by rw [h2] done theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry lemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) : [a]_m * [gcd_c2 m a]_m = [1]_m := by set s : Int := gcd_c1 m a have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m define at h1 rewrite [h1, Nat.cast_one] at h2 --h2 : s * ↑m + gcd_c2 m a * ↑a = 1 rewrite [mul_class, cc_eq_iff_congr] define --Goal : ∃ (c : Int), ↑a * gcd_c2 m a - 1 = ↑m * c apply Exists.intro (-s) show a * (gcd_c2 m a) - 1 = m * (-s) from calc a * (gcd_c2 m a) - 1 _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring _ = 1 + m * (-s) - 1 := by rw [h2] _ = m * (-s) := by ring done theorem Theorem_7_3_7 (m a : Nat) : invertible [a]_m ↔ rel_prime m a := by apply Iff.intro · -- (→) assume h1 : invertible [a]_m define at h1 obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1 obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y rewrite [h3, mul_class, cc_eq_iff_congr] at h2 define at h2 obtain (c : Int) (h4 : a * b - 1 = m * c) from h2 rewrite [Exercise_7_2_6] --Goal : ∃ (s t : Int), s * ↑m + t * ↑a = 1 apply Exists.intro (-c) apply Exists.intro b show (-c) * m + b * a = 1 from calc (-c) * m + b * a _ = (-c) * m + (a * b - 1) + 1 := by ring _ = (-c) * m + m * c + 1 := by rw [h4] _ = 1 := by ring done · -- (←) assume h1 : rel_prime m a define show ∃ (Y : ZMod m), [a]_m * Y = [1]_m from Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1) done done /- Section 7.4 -/ section Euler open Euler lemma num_rp_below_base {m : Nat} : num_rp_below m 0 = 0 := by rfl lemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) : num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by have h1 : num_rp_below m (j + 1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : gcd m j = 1 rewrite [if_pos h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1 show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1 done lemma num_rp_below_step_not_rp {m j : Nat} (h : ¬rel_prime m j) : num_rp_below m (j + 1) = num_rp_below m j := by have h1 : num_rp_below m (j +1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : ¬gcd m j = 1 rewrite [if_neg h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j show num_rp_below m (j + 1) = num_rp_below m j from h1 done lemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl #eval phi 10 --Answer: 4 lemma prod_inv_iff_inv {m : Nat} {X : ZMod m} (h1 : invertible X) (Y : ZMod m) : invertible (X * Y) ↔ invertible Y := by apply Iff.intro · -- (→) assume h2 : invertible (X * Y) obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2 apply Exists.intro (X * Z) rewrite [←h3] --Goal : Y * (X * Z) = X * Y * Z ring --Note that ring can do algebra in ZMod m done · -- (←) assume h2 : invertible Y obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1 obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2 apply Exists.intro (Xi * Yi) show (X * Y) * (Xi * Yi) = [1]_m from calc X * Y * (Xi * Yi) _ = (X * Xi) * (Y * Yi) := by ring _ = [1]_m * [1]_m := by rw [h3, h4] _ = [1]_m := Theorem_7_3_6_7 [1]_m done done lemma F_rp_def {m i : Nat} (h : rel_prime m i) : F m i = [i]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h --h : gcd m i = 1 rewrite [if_pos h] at h1 show F m i = [i]_m from h1 done lemma F_not_rp_def {m i : Nat} (h : ¬rel_prime m i) : F m i = [1]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h rewrite [h1, if_neg h] rfl done lemma prod_seq_base {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 0 k f = [1]_m := by rfl lemma prod_seq_step {m : Nat} (n k : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl lemma prod_seq_zero_step {m : Nat} (n : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by rewrite [prod_seq_step, zero_add] rfl done lemma prod_one {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 1 k f = f k := by rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7] rfl done lemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl lemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m := calc [G m a i]_m _ = [(a * i) % m]_m := by rfl _ = [a * i]_m := (cc_eq_mod m (a * i)).symm _ = [a]_m * [i]_m := (mul_class m a i).symm lemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) : rel_prime m (G m a i) ↔ rel_prime m i := by have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1 show rel_prime m (G m a i) ↔ rel_prime m i from calc rel_prime m (G m a i) _ ↔ invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm _ ↔ invertible ([a]_m * [i]_m) := by rw [cc_G] _ ↔ invertible [i]_m := prod_inv_iff_inv h2 ([i]_m) _ ↔ rel_prime m i := Theorem_7_3_7 m i done lemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) : F m (G m a i) = [a]_m * F m i := by have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2 show F m (G m a i) = [a]_m * F m i from calc F m (G m a i) _ = [G m a i]_m := F_rp_def h3 _ = [a]_m * [i]_m := cc_G m a i _ = [a]_m * F m i := by rw [F_rp_def h2] done lemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : ¬rel_prime m i) : F m (G m a i) = [1]_m := by rewrite [←G_rp_iff h1 i] at h2 show F m (G m a i) = [1]_m from F_not_rp_def h2 done lemma FG_prod {m a : Nat} (h1 : rel_prime m a) : ∀ (k : Nat), prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by by_induc · -- Base Case show prod_seq 0 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from calc prod_seq 0 0 ((F m) ∘ (G m a)) _ = [1]_m := prod_seq_base _ _ _ = [a]_m ^ 0 * [1]_m := by ring _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by rw [num_rp_below_base, prod_seq_base] done · -- Induction Step fix k : Nat assume ih : prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) by_cases h2 : rel_prime m k · -- Case 1. h2 : rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([a]_m * F m k) := by rw [FG_rp h1 h2] _ = [a]_m ^ ((num_rp_below m k) + 1) * ((prod_seq k 0 (F m)) * F m k) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_rp h2, prod_seq_zero_step] done · -- Case 2. h2 : ¬rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([1]_m) := by rw [FG_not_rp h1 h2] _ = [a]_m ^ (num_rp_below m k) * (prod_seq k 0 (F m) * ([1]_m)) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_not_rp h2, prod_seq_zero_step, F_not_rp_def h2] done done done lemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by define --Goal : ∀ i < m, G m a i < m fix i : Nat assume h1 : i < m rewrite [G_def] --Goal : a * i % m < m show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m) done lemma left_inv_one_one_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g' (g i) = i) : one_one_below n g := sorry lemma right_inv_onto_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g (g' i) = i) (h2 : maps_below n g') : onto_below n g := by define at h2; define fix k : Nat assume h3 : k < n apply Exists.intro (g' k) show g' k < n ∧ g (g' k) = k from And.intro (h2 k h3) (h1 k h3) done lemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m * [inv_mod m a]_m = [1]_m := by have h2 : 0 ≤ (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a) show [a]_m * [inv_mod m a]_m = [1]_m from calc [a]_m * [inv_mod m a]_m _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2] _ = [a]_m * [gcd_c2 m a]_m := by rw [←cc_eq_mod] _ = [1]_m := gcd_c2_inv h1 done lemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m := calc a * (b % m) % m = a % m * (b % m % m) % m := Nat.mul_mod _ _ _ _ = a % m * (b % m) % m := by rw [Nat.mod_mod] _ = a * b % m := (Nat.mul_mod _ _ _).symm lemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm] rfl done theorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] : ↑a ≡ ↑b (MOD m) ↔ a % m = b % m := sorry lemma mul_inv_mod_cancel {m a i : Nat} [NeZero m] (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1 rewrite [mul_class, cc_eq_iff_congr, ←Nat.cast_mul, ←Nat.cast_one, congr_iff_mod_eq_Nat] at h3 show a * inv_mod m a * i % m = i from calc a * (inv_mod m a) * i % m _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = 1 % m * i % m := by rw [h3] _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = i % m := by rw [one_mul] _ = i := Nat.mod_eq_of_lt h2 done lemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl lemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, G m a (Ginv m a i) = i := by fix i : Nat assume h2 : i < m show G m a (Ginv m a i) = i from calc G m a (Ginv m a i) _ = a * ((inv_mod m a * i) % m) % m := by rfl _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, Ginv m a (G m a i) = i := by fix i : Nat assume h2 : i < m show Ginv m a (G m a i) = i from calc Ginv m a (G m a i) _ = inv_mod m a * ((a * i) % m) % m := by rfl _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc, mul_comm (inv_mod m a)] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_maps_below (m a : Nat) [NeZero m] : maps_below m (Ginv m a) := G_maps_below m (inv_mod m a) lemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : one_one_below m (G m a) := left_inv_one_one_below (Ginv_left_inv h1) lemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : onto_below m (G m a) := right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a) lemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : perm_below m (G m a) := And.intro (G_maps_below m a) (And.intro (G_one_one_below h1) (G_onto_below h1)) --Permuting a product of congruence classes doesn't change product lemma swap_fst (u v : Nat) : swap u v u = v := by define : swap u v u --Goal : (if u = u then v else if u = v then u else u) = v have h : u = u := by rfl rewrite [if_pos h] rfl done lemma swap_snd (u v : Nat) : swap u v v = u := by define : swap u v v by_cases h1 : v = u · -- Case 1. h1 : v = u rewrite [if_pos h1] show v = u from h1 done · -- Case 2. h1 : v ≠ u rewrite [if_neg h1] have h2 : v = v := by rfl rewrite [if_pos h2] rfl done done lemma swap_other {u v i : Nat} (h1 : i ≠ u) (h2 : i ≠ v) : swap u v i = i := by define : swap u v i rewrite [if_neg h1, if_neg h2] rfl done lemma swap_values (u v i : Nat) : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := by by_cases h1 : i = u · -- Case 1. h1 : i = u apply Or.inl rewrite [h1] show swap u v u = v from swap_fst u v done · -- Case 2. h1 : i ≠ u apply Or.inr by_cases h2 : i = v · -- Case 2.1. h2 : i = v apply Or.inl rewrite [h2] show swap u v v = u from swap_snd u v done · -- Case 2.2. h2 : i ≠ v apply Or.inr show swap u v i = i from swap_other h1 h2 done done done lemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by define fix i : Nat assume h3 : i < n have h4 : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := swap_values u v i by_cases on h4 · -- Case 1. h4 : swap u v i = v rewrite [h4] show v < n from h2 done · -- Case 2. by_cases on h4 · -- Case 2.1. h4 : swap u v i = u rewrite [h4] show u < n from h1 done · -- Case 2.2. h4 : swap u v i = i rewrite [h4] show i < n from h3 done done done lemma swap_swap (u v n : Nat) : ∀ i < n, swap u v (swap u v i) = i := by fix i : Nat assume h : i < n by_cases h1 : i = u · -- Case 1. h1 : i = u rewrite [h1, swap_fst, swap_snd] rfl done · -- Case 2. h1 : i ≠ u by_cases h2 : i = v · -- Case 2.1. h2 : i = v rewrite [h2, swap_snd, swap_fst] rfl done · -- Case 2.2. h2 : i ≠ v rewrite [swap_other h1 h2, swap_other h1 h2] rfl done done done lemma swap_one_one_below (u v n) : one_one_below n (swap u v) := left_inv_one_one_below (swap_swap u v n) lemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) := right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2) lemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) := And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2)) lemma comp_perm_below {n : Nat} {f g : Nat → Nat} (h1 : perm_below n f) (h2 : perm_below n g) : perm_below n (f ∘ g) := sorry lemma trivial_swap (u : Nat) : swap u u = id := by apply funext fix x : Nat by_cases h1 : x = u · -- Case 1. h1 : x = u rewrite [h1, swap_fst] rfl done · -- Case 2. h1 : x ≠ u rewrite [swap_other h1 h1] rfl done done lemma prod_eq_fun {m : Nat} (f g : Nat → ZMod m) (k : Nat) : ∀ (n : Nat), (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g := by by_induc · -- Base Case assume h : (∀ i < 0, f (k + i) = g (k + i)) rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g assume h1 : ∀ i < n + 1, f (k + i) = g (k + i) have h2 : ∀ i < n, f (k + i) = g (k + i) := by fix i : Nat assume h2 : i < n have h3 : i < n + 1 := by linarith show f (k + i) = g (k + i) from h1 i h3 done have h3 : prod_seq n k f = prod_seq n k g := ih h2 have h4 : n < n + 1 := Nat.lt_succ_self n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4] rfl done done lemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := by have h2 : ∀ (i : Nat), i < u → (f ∘ swap u n) (0 + i) = f (0 + i) := by fix i : Nat assume h2 : i < u have h3 : 0 + i ≠ u := by linarith have h4 : 0 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl done show prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f from prod_eq_fun (f ∘ swap u n) f 0 u h2 done lemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := by have h2 : ∀ i < j, (f ∘ swap u n) (u + 1 + i) = f (u + 1 + i) := by fix i : Nat assume h2 : i < j have h3 : u + 1 + i ≠ u := by linarith have h4 : u + 1 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl show prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f from prod_eq_fun (f ∘ swap u n) f (u + 1) j h2 done lemma break_prod {m : Nat} (n : Nat) (f : Nat → ZMod m) : ∀ (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by by_induc · -- Base Case have h : n + 0 = n := by rfl rewrite [prod_seq_base, h, Theorem_7_3_6_7] rfl done · -- Induction Step fix j : Nat assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f rewrite [←add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc] rfl done done lemma break_prod_twice {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f := break_prod n f 1 rewrite [prod_one] at h2 have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f := break_prod (u + 1) f j rewrite [←h1] at h3 have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f := break_prod u f 1 rewrite [prod_one] at h4 rewrite [h3, h4] at h2 show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2 done lemma swap_prod_eq_prod {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f := by by_cases h2 : u = n · -- Case 1. h2 : u = n rewrite [h2, trivial_swap n] --Goal : prod_seq (n + 1) 0 (f ∘ id) = prod_seq (n + 1) 0 f rfl done · -- Case 2. h2 : ¬u = n have h3 : u + 1 ≤ n := Nat.lt_of_le_of_ne h1 h2 obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3 have break_f : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := break_prod_twice f h4 have break_fs : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_prod_twice (f ∘ swap u n) h4 have f_eq_fs_below : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := swap_prod_eq_prod_below f h1 have f_eq_fs_btwn : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4 show prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f from calc prod_seq (n + 1) 0 (f ∘ swap u n) _ = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_fs _ = prod_seq u 0 f * (f ∘ swap u n) u * prod_seq j (u + 1) f * (f ∘ swap u n) n := by rw [f_eq_fs_below, f_eq_fs_btwn] _ = prod_seq u 0 f * f (swap u n u) * prod_seq j (u + 1) f * f (swap u n n) := by rfl _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by rw [swap_fst, swap_snd] _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring _ = prod_seq (n + 1) 0 f := break_f.symm done done lemma perm_below_fixed {n : Nat} {g : Nat → Nat} (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry lemma perm_prod {m : Nat} (f : Nat → ZMod m) : ∀ (n : Nat), ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) := by by_induc · -- Base Case fix g : Nat → Nat assume h1 : perm_below 0 g rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) fix g : Nat → Nat assume g_pb : perm_below (n + 1) g define at g_pb have g_ob : onto_below (n + 1) g := g_pb.right.right define at g_ob have h1 : n < n + 1 := by linarith obtain (u : Nat) (h2 : u < n + 1 ∧ g u = n) from g_ob n h1 have s_pb : perm_below (n + 1) (swap u n) := swap_perm_below h2.left h1 have gs_pb_n1 : perm_below (n + 1) (g ∘ swap u n) := comp_perm_below g_pb s_pb have gs_fix_n : (g ∘ swap u n) n = n := calc (g ∘ swap u n) n _ = g (swap u n n) := by rfl _ = g u := by rw [swap_snd] _ = n := h2.right have gs_pb_n : perm_below n (g ∘ swap u n) := perm_below_fixed gs_pb_n1 gs_fix_n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f ∘ (g ∘ swap u n)) := ih (g ∘ swap u n) gs_pb_n have h3 : u ≤ n := by linarith show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f ∘ g) from calc prod_seq (n + 1) 0 f _ = prod_seq n 0 f * f n := prod_seq_zero_step n f _ = prod_seq n 0 (f ∘ (g ∘ swap u n)) * f ((g ∘ swap u n) n) := by rw [gs_prod, gs_fix_n] _ = prod_seq n 0 (f ∘ g ∘ swap u n) * (f ∘ g ∘ swap u n) n := by rfl _ = prod_seq (n + 1) 0 (f ∘ g ∘ swap u n) := (prod_seq_zero_step n (f ∘ g ∘ swap u n)).symm _ = prod_seq (n + 1) 0 ((f ∘ g) ∘ swap u n) := by rfl _ = prod_seq (n + 1) 0 (f ∘ g) := swap_prod_eq_prod (f ∘ g) h3 done done lemma F_invertible (m i : Nat) : invertible (F m i) := by by_cases h : rel_prime m i · -- Case 1. h : rel_prime m i rewrite [F_rp_def h] show invertible [i]_m from (Theorem_7_3_7 m i).rtl h done · -- Case 2. h : ¬rel_prime m i rewrite [F_not_rp_def h] apply Exists.intro [1]_m show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m done done lemma Fprod_invertible (m : Nat) : ∀ (k : Nat), invertible (prod_seq k 0 (F m)) := by by_induc · -- Base Case apply Exists.intro [1]_m show prod_seq 0 0 (F m) * [1]_m = [1]_m from calc prod_seq 0 0 (F m) * [1]_m _ = [1]_m * [1]_m := by rw [prod_seq_base] _ = [1]_m := Theorem_7_3_6_7 ([1]_m) done · -- Induction Step fix k : Nat assume ih : invertible (prod_seq k 0 (F m)) rewrite [prod_seq_zero_step] show invertible (prod_seq k 0 (F m) * (F m k)) from (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k) done done theorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m ^ (phi m) = [1]_m := by have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2 show [a]_m ^ (phi m) = [1]_m from calc [a]_m ^ (phi m) _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3] _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring _ = prod_seq m 0 (F m ∘ G m a) * Y := by rw [FG_prod h1 m, phi_def] _ = prod_seq m 0 (F m) * Y := by rw [perm_prod (F m) m (G m a) (G_perm_below h1)] _ = [1]_m := by rw [h3] done lemma Exercise_7_4_5_Int (m : Nat) (a : Int) : ∀ (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry lemma Exercise_7_4_5_Nat (m a n : Nat) : [a]_m ^ n = [a ^ n]_m := by rewrite [Exercise_7_4_5_Int] rfl done theorem Euler's_theorem {m a : Nat} [NeZero m] (h1 : rel_prime m a) : a ^ (phi m) ≡ 1 (MOD m) := by have h2 : [a]_m ^ (phi m) = [1]_m := Theorem_7_4_2 h1 rewrite [Exercise_7_4_5_Nat m a (phi m)] at h2 --h2 : [a ^ phi m]_m = [1]_m show a ^ (phi m) ≡ 1 (MOD m) from (cc_eq_iff_congr _ _ _).ltr h2 done #eval gcd 10 7 --Answer: 1. So 10 and 7 are relatively prime #eval 7 ^ phi 10 --Answer: 2401, which is congruent to 1 mod 10. end Euler /- Section 7.5 -/ lemma num_rp_prime {p : Nat} (h1 : prime p) : ∀ k < p, num_rp_below p (k + 1) = k := sorry lemma phi_prime {p : Nat} (h1 : prime p) : phi p = p - 1 := by have h2 : 1 ≤ p := prime_pos h1 have h3 : p - 1 + 1 = p := Nat.sub_add_cancel h2 have h4 : p - 1 < p := by linarith have h5 : num_rp_below p (p - 1 + 1) = p - 1 := num_rp_prime h1 (p - 1) h4 rewrite [h3] at h5 show phi p = p - 1 from h5 done theorem Theorem_7_2_2_Int {a c : Nat} {b : Int} (h1 : ↑c ∣ ↑a * b) (h2 : rel_prime a c) : ↑c ∣ b := by rewrite [Int.natCast_dvd, Int.natAbs_mul, Int.natAbs_ofNat] at h1 --h1 : c ∣ a * Int.natAbs b rewrite [Int.natCast_dvd] --Goal : c ∣ Int.natAbs b show c ∣ Int.natAbs b from Theorem_7_2_2 h1 h2 done lemma Lemma_7_4_5 {m n : Nat} (a b : Int) (h1 : rel_prime m n) : a ≡ b (MOD m * n) ↔ a ≡ b (MOD m) ∧ a ≡ b (MOD n) := by apply Iff.intro · -- (→) assume h2 : a ≡ b (MOD m * n) obtain (j : Int) (h3 : a - b = (m * n) * j) from h2 apply And.intro · -- Proof of a ≡ b (MOD m) apply Exists.intro (n * j) show a - b = m * (n * j) from calc a - b _ = m * n * j := h3 _ = m * (n * j) := by ring done · -- Proof of a ≡ b (MOD n) apply Exists.intro (m * j) show a - b = n * (m * j) from calc a - b _ = m * n * j := h3 _ = n * (m * j) := by ring done done · -- (←) assume h2 : a ≡ b (MOD m) ∧ a ≡ b (MOD n) obtain (j : Int) (h3 : a - b = m * j) from h2.left have h4 : (↑n : Int) ∣ a - b := h2.right rewrite [h3] at h4 --h4 : ↑n ∣ ↑m * j have h5 : ↑n ∣ j := Theorem_7_2_2_Int h4 h1 obtain (k : Int) (h6 : j = n * k) from h5 apply Exists.intro k --Goal : a - b = ↑(m * n) * k rewrite [Nat.cast_mul] --Goal : a - b = ↑m * ↑n * k show a - b = (m * n) * k from calc a - b _ = m * j := h3 _ = m * (n * k) := by rw [h6] _ = (m * n) * k := by ring done done --From exercises of Section 7.2 theorem rel_prime_symm {a b : Nat} (h : rel_prime a b) : rel_prime b a := sorry lemma prime_NeZero {p : Nat} (h : prime p) : NeZero p := by rewrite [neZero_iff] --Goal : p ≠ 0 define at h linarith done lemma Lemma_7_5_1 {p e d m c s : Nat} {t : Int} (h1 : prime p) (h2 : e * d = (p - 1) * s + 1) (h3 : m ^ e - c = p * t) : c ^ d ≡ m (MOD p) := by have h4 : m ^ e ≡ c (MOD p) := Exists.intro t h3 have h5 : [m ^ e]_p = [c]_p := (cc_eq_iff_congr _ _ _).rtl h4 rewrite [←Exercise_7_4_5_Nat] at h5 --h5 : [m]_p ^ e = [c]_p by_cases h6 : p ∣ m · -- Case 1. h6 : p ∣ m have h7 : m ≡ 0 (MOD p) := by obtain (j : Nat) (h8 : m = p * j) from h6 apply Exists.intro (↑j : Int) --Goal : ↑m - 0 = ↑p * ↑j rewrite [h8, Nat.cast_mul] ring done have h8 : [m]_p = [0]_p := (cc_eq_iff_congr _ _ _).rtl h7 have h9 : e * d ≠ 0 := by rewrite [h2] show (p - 1) * s + 1 ≠ 0 from Nat.add_one_ne_zero _ done have h10 : (0 : Int) ^ (e * d) = 0 := zero_pow h9 have h11 : [c ^ d]_p = [m]_p := calc [c ^ d]_p _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat] _ = ([m]_p ^ e) ^ d := by rw [h5] _ = [m]_p ^ (e * d) := by ring _ = [0]_p ^ (e * d) := by rw [h8] _ = [0 ^ (e * d)]_p := Exercise_7_4_5_Int _ _ _ _ = [0]_p := by rw [h10] _ = [m]_p := by rw [h8] show c ^ d ≡ m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11 done · -- Case 2. h6 : ¬p ∣ m have h7 : rel_prime m p := rel_prime_of_prime_not_dvd h1 h6 have h8 : rel_prime p m := rel_prime_symm h7 have h9 : NeZero p := prime_NeZero h1 have h10 : (1 : Int) ^ s = 1 := by ring have h11 : [c ^ d]_p = [m]_p := calc [c ^ d]_p _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat] _ = ([m]_p ^ e) ^ d := by rw [h5] _ = [m]_p ^ (e * d) := by ring _ = [m]_p ^ ((p - 1) * s + 1) := by rw [h2] _ = ([m]_p ^ (p - 1)) ^ s * [m]_p := by ring _ = ([m]_p ^ (phi p)) ^ s * [m]_p := by rw [phi_prime h1] _ = [1]_p ^ s * [m]_p := by rw [Theorem_7_4_2 h8] _ = [1 ^ s]_p * [m]_p := by rw [Exercise_7_4_5_Int] _ = [1]_p * [m]_p := by rw [h10] _ = [m]_p * [1]_p := by ring _ = [m]_p := Theorem_7_3_6_7 _ show c ^ d ≡ m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11 done done theorem Theorem_7_5_1 (p q n e d k m c : Nat) (p_prime : prime p) (q_prime : prime q) (p_ne_q : p ≠ q) (n_pq : n = p * q) (ed_congr_1 : e * d = k * (p - 1) * (q - 1) + 1) (h1 : [m]_n ^ e = [c]_n) : [c]_n ^ d = [m]_n := by rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] at h1 --h1 : m ^ e ≡ c (MOD n) rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] --Goal : c ^ d ≡ m (MOD n) obtain (j : Int) (h2 : m ^ e - c = n * j) from h1 rewrite [n_pq, Nat.cast_mul] at h2 --h2 : m ^ e - c = p * q * j have h3 : e * d = (p - 1) * (k * (q - 1)) + 1 := by rewrite [ed_congr_1] ring done have h4 : m ^ e - c = p * (q * j) := by rewrite [h2] ring done have congr_p : c ^ d ≡ m (MOD p) := Lemma_7_5_1 p_prime h3 h4 have h5 : e * d = (q - 1) * (k * (p - 1)) + 1 := by rewrite [ed_congr_1] ring done have h6 : m ^ e - c = q * (p * j) := by rewrite [h2] ring done have congr_q : c ^ d ≡ m (MOD q) := Lemma_7_5_1 q_prime h5 h6 have h7 : ¬q ∣ p := by by_contra h8 have h9 : q = 1 ∨ q = p := dvd_prime p_prime h8 disj_syll h9 (prime_not_one q_prime) show False from p_ne_q h9.symm done have h8 : rel_prime p q := rel_prime_of_prime_not_dvd q_prime h7 rewrite [n_pq, Lemma_7_4_5 _ _ h8] show c ^ d ≡ m (MOD p) ∧ c ^ d ≡ m (MOD q) from And.intro congr_p congr_q done /- BEGIN EXERCISES -/ namespace Exercises /- Section 7.1 -/ -- 1. theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry -- 2. lemma gcd_comm_lt {a b : Nat} (h : a < b) : gcd a b = gcd b a := sorry theorem gcd_comm (a b : Nat) : gcd a b = gcd b a := sorry -- 3. theorem Exercise_7_1_5 (a b : Nat) (n : Int) : (∃ (s t : Int), s * a + t * b = n) ↔ (↑(gcd a b) : Int) ∣ n := sorry -- 4. theorem Exercise_7_1_6 (a b c : Nat) : gcd a b = gcd (a + b * c) b := sorry -- 5. theorem gcd_is_nonzero {a b : Nat} (h : a ≠ 0 ∨ b ≠ 0) : gcd a b ≠ 0 := sorry -- 6. theorem gcd_greatest {a b d : Nat} (h1 : gcd a b ≠ 0) (h2 : d ∣ a) (h3 : d ∣ b) : d ≤ gcd a b := sorry -- 7. lemma Lemma_7_1_10a {a b : Nat} (n : Nat) (h : a ∣ b) : (n * a) ∣ (n * b) := sorry lemma Lemma_7_1_10b {a b n : Nat} (h1 : n ≠ 0) (h2 : (n * a) ∣ (n * b)) : a ∣ b := sorry lemma Lemma_7_1_10c {a b : Nat} (h1 : a ∣ b) (h2 : b ∣ a) : a = b := sorry theorem Exercise_7_1_10 (a b n : Nat) : gcd (n * a) (n * b) = n * gcd a b := sorry /- Section 7.2 -/ -- 1. lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry -- 2. -- Hints: Start with apply List.rec. You may find mul_ne_zero useful theorem prod_nonzero_nonzero : ∀ (l : List Nat), (∀ a ∈ l, a ≠ 0) → prod l ≠ 0 := sorry -- 3. theorem rel_prime_iff_no_common_factor (a b : Nat) : rel_prime a b ↔ ¬∃ (p : Nat), prime p ∧ p ∣ a ∧ p ∣ b := sorry -- 4. theorem rel_prime_symm {a b : Nat} (h : rel_prime a b) : rel_prime b a := sorry -- 5. lemma in_prime_factorization_iff_prime_factor {a : Nat} {l : List Nat} (h1 : prime_factorization a l) (p : Nat) : p ∈ l ↔ prime_factor p a := sorry -- 6. theorem Exercise_7_2_5 {a b : Nat} {l m : List Nat} (h1 : prime_factorization a l) (h2 : prime_factorization b m) : rel_prime a b ↔ (¬∃ (p : Nat), p ∈ l ∧ p ∈ m) := sorry -- 7. theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry -- 8. theorem Exercise_7_2_7 {a b a' b' : Nat} (h1 : rel_prime a b) (h2 : a' ∣ a) (h3 : b' ∣ b) : rel_prime a' b' := sorry -- 9. theorem Exercise_7_2_9 {a b j k : Nat} (h1 : gcd a b ≠ 0) (h2 : a = j * gcd a b) (h3 : b = k * gcd a b) : rel_prime j k := sorry -- 10. theorem Exercise_7_2_17a (a b c : Nat) : gcd a (b * c) ∣ gcd a b * gcd a c := sorry /- Section 7.3 -/ -- 1. theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry -- 2. theorem Theorem_7_3_6_3 {m : Nat} (X : ZMod m) : X + [0]_m = X := sorry -- 3. theorem Theorem_7_3_6_4 {m : Nat} (X : ZMod m) : ∃ (Y : ZMod m), X + Y = [0]_m := sorry -- 4. theorem Exercise_7_3_4a {m : Nat} (Z1 Z2 : ZMod m) (h1 : ∀ (X : ZMod m), X + Z1 = X) (h2 : ∀ (X : ZMod m), X + Z2 = X) : Z1 = Z2 := sorry -- 5. theorem Exercise_7_3_4b {m : Nat} (X Y1 Y2 : ZMod m) (h1 : X + Y1 = [0]_m) (h2 : X + Y2 = [0]_m) : Y1 = Y2 := sorry -- 6. theorem Theorem_7_3_10 (m a : Nat) (b : Int) : ¬(↑(gcd m a) : Int) ∣ b → ¬∃ (x : Int), a * x ≡ b (MOD m) := sorry -- 7. theorem Theorem_7_3_11 (m n : Nat) (a b : Int) (h1 : n ≠ 0) : n * a ≡ n * b (MOD n * m) ↔ a ≡ b (MOD m) := sorry -- 8. theorem Exercise_7_3_16 {m : Nat} {a b : Int} (h : a ≡ b (MOD m)) : ∀ (n : Nat), a ^ n ≡ b ^ n (MOD m) := sorry -- 9. example {m : Nat} [NeZero m] (X : ZMod m) : ∃! (a : Int), 0 ≤ a ∧ a < m ∧ X = [a]_m := sorry -- 10.

theorem congr_rel_prime {m a b : Nat} (h1 : a ≡ b (MOD m)) : rel_prime m a ↔ rel_prime m b

HTPI.Exercises.congr_rel_prime

htpi/HTPILib/Chap7.lean

htpi

/- Copyright 2023 Daniel J. Velleman -/ import HTPILib.Chap6 namespace HTPI /- Definitions -/ lemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by have h : n + 1 > 0 := Nat.succ_pos n show a % (n + 1) < n + 1 from Nat.mod_lt a h done def gcd (a b : Nat) : Nat := match b with | 0 => a | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd (n + 1) (a % (n + 1)) termination_by b mutual def gcd_c1 (a b : Nat) : Int := match b with | 0 => 1 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c2 (n + 1) (a % (n + 1)) --Corresponds to s = t' termination_by b def gcd_c2 (a b : Nat) : Int := match b with | 0 => 0 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c1 (n + 1) (a % (n + 1)) - (gcd_c2 (n + 1) (a % (n + 1))) * ↑(a / (n + 1)) --Corresponds to t = s' - t'q termination_by b end def prime (n : Nat) : Prop := 2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n def prime_factor (p n : Nat) : Prop := prime p ∧ p ∣ n def all_prime (l : List Nat) : Prop := ∀ p ∈ l, prime p def nondec (l : List Nat) : Prop := match l with | [] => True --Of course, True is a proposition that is always true | n :: L => (∀ m ∈ L, n ≤ m) ∧ nondec L def nondec_prime_list (l : List Nat) : Prop := all_prime l ∧ nondec l def prod (l : List Nat) : Nat := match l with | [] => 1 | n :: L => n * (prod L) def prime_factorization (n : Nat) (l : List Nat) : Prop := nondec_prime_list l ∧ prod l = n def rel_prime (a b : Nat) : Prop := gcd a b = 1 def congr_mod (m : Nat) (a b : Int) : Prop := (↑m : Int) ∣ (a - b) def cc (m : Nat) (a : Int) : ZMod m := (↑a : ZMod m) notation:50 a " ≡ " b " (MOD " m ")" => congr_mod m a b notation:max "["a"]_"m:max => cc m a def invertible {m : Nat} (X : ZMod m) : Prop := ∃ (Y : ZMod m), X * Y = [1]_m def num_rp_below (m k : Nat) : Nat := match k with | 0 => 0 | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j def phi (m : Nat) : Nat := num_rp_below m m def prod_seq {m : Nat} (j k : Nat) (f : Nat → ZMod m) : ZMod m := match j with | 0 => [1]_m | n + 1 => prod_seq n k f * f (k + n) def maps_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i < n, g i < n def one_one_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i1 < n, ∀ i2 < n, g i1 = g i2 → i1 = i2 def onto_below (n : Nat) (g : Nat → Nat) : Prop := ∀ k < n, ∃ i < n, g i = k def perm_below (n : Nat) (g : Nat → Nat) : Prop := maps_below n g ∧ one_one_below n g ∧ onto_below n g def inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m) def swap (u v i : Nat) : Nat := if i = u then v else if i = v then u else i namespace Euler --For definitions specific to Euler's theorem def F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m def G (m a i : Nat) : Nat := (a * i) % m def Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i end Euler /- Section 7.1 -/ theorem dvd_mod_of_dvd_a_b {a b d : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ (a % b) := by set q : Nat := a / b have h3 : b * q + a % b = a := Nat.div_add_mod a b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 define --Goal : ∃ (c : Nat), a % b = d * c apply Exists.intro (j - k * q) show a % b = d * (j - k * q) from calc a % b _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm _ = a - b * q := by rw [h3] _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc] _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm done theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry #eval gcd 672 161 --Answer: 7 lemma gcd_base (a : Nat) : gcd a 0 = a := by rfl lemma gcd_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd a b = gcd b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1)) rfl done lemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b ≠ 0) : a % b < b := by have h1 : b > 0 := Nat.pos_of_ne_zero h show a % b < b from Nat.mod_lt a h1 done lemma dvd_self (n : Nat) : n ∣ n := by apply Exists.intro 1 ring done theorem gcd_dvd : ∀ (b a : Nat), (gcd a b) ∣ a ∧ (gcd a b) ∣ b := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd a b_1) ∣ a ∧ (gcd a b_1) ∣ b_1 fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_base] --Goal: a ∣ a ∧ a ∣ 0 apply And.intro (dvd_self a) define apply Exists.intro 0 rfl done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_nonzero a h1] --Goal : gcd b (a % b) ∣ a ∧ gcd b (a % b) ∣ b have h2 : a % b < b := mod_nonzero_lt a h1 have h3 : (gcd b (a % b)) ∣ b ∧ (gcd b (a % b)) ∣ (a % b) := ih (a % b) h2 b apply And.intro _ h3.left show (gcd b (a % b)) ∣ a from dvd_a_of_dvd_b_mod h3.left h3.right done done theorem gcd_dvd_left (a b : Nat) : (gcd a b) ∣ a := (gcd_dvd b a).left theorem gcd_dvd_right (a b : Nat) : (gcd a b) ∣ b := (gcd_dvd b a).right lemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl lemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c1 a b = gcd_c2 b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done lemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl lemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * ↑(a / b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done theorem gcd_lin_comb : ∀ (b a : Nat), (gcd_c1 a b) * ↑a + (gcd_c2 a b) * ↑b = ↑(gcd a b) := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd_c1 a b_1) * ↑a + (gcd_c2 a b_1) * ↑b_1 = ↑(gcd a b_1) fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base] --Goal : 1 * ↑a + 0 * ↑0 = ↑a ring done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1] --Goal : gcd_c2 b (a % b) * ↑a + -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * ↑(a / b)) * ↑b = -- ↑(gcd b (a % b)) set r : Nat := a % b set q : Nat := a / b set s : Int := gcd_c1 b r set t : Int := gcd_c2 b r --Goal : t * ↑a + (s - t * ↑q) * ↑b = ↑(gcd b r) have h2 : r < b := mod_nonzero_lt a h1 have h3 : s * ↑b + t * ↑r = ↑(gcd b r) := ih r h2 b have h4 : b * q + r = a := Nat.div_add_mod a b rewrite [←h3, ←h4] rewrite [Nat.cast_add, Nat.cast_mul] --Goal : t * (↑b * ↑q + ↑r) + (s - t * ↑q) * ↑b = s * ↑b + t * ↑r ring done done #eval gcd_c1 672 161 --Answer: 6 #eval gcd_c2 672 161 --Answer: -25 --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161 theorem Theorem_7_1_6 {d a b : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ gcd a b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑d ∣ ↑(gcd a b) set s : Int := gcd_c1 a b set t : Int := gcd_c2 a b have h3 : s * ↑a + t * ↑b = ↑(gcd a b) := gcd_lin_comb b a rewrite [←h3] --Goal : ↑d ∣ s * ↑a + t * ↑b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul] --Goal : ↑d ∣ s * (↑d * ↑j) + t * (↑d * ↑k) define apply Exists.intro (s * ↑j + t * ↑k) ring done /- Section 7.2 -/ theorem dvd_trans {a b c : Nat} (h1 : a ∣ b) (h2 : b ∣ c) : a ∣ c := by define at h1; define at h2; define obtain (m : Nat) (h3 : b = a * m) from h1 obtain (n : Nat) (h4 : c = b * n) from h2 rewrite [h3, mul_assoc] at h4 apply Exists.intro (m * n) show c = a * (m * n) from h4 done lemma exists_prime_factor : ∀ (n : Nat), 2 ≤ n → ∃ (p : Nat), prime_factor p n := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, 2 ≤ n_1 → ∃ (p : Nat), prime_factor p n_1 assume h1 : 2 ≤ n by_cases h2 : prime n · -- Case 1. h2 : prime n apply Exists.intro n define --Goal : prime n ∧ n ∣ n show prime n ∧ n ∣ n from And.intro h2 (dvd_self n) done · -- Case 2. h2 : ¬prime n define at h2 --h2 : ¬(2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n) demorgan at h2 disj_syll h2 h1 obtain (a : Nat) (h3 : ∃ (b : Nat), a * b = n ∧ a < n ∧ b < n) from h2 obtain (b : Nat) (h4 : a * b = n ∧ a < n ∧ b < n) from h3 have h5 : 2 ≤ a := by by_contra h6 have h7 : a ≤ 1 := by linarith have h8 : n ≤ b := calc n _ = a * b := h4.left.symm _ ≤ 1 * b := by rel [h7] _ = b := by ring linarith --n ≤ b contradicts b < n done have h6 : ∃ (p : Nat), prime_factor p a := ih a h4.right.left h5 obtain (p : Nat) (h7 : prime_factor p a) from h6 apply Exists.intro p define --Goal : prime p ∧ p ∣ n define at h7 --h7 : prime p ∧ p ∣ a apply And.intro h7.left have h8 : a ∣ n := by apply Exists.intro b show n = a * b from (h4.left).symm done show p ∣ n from dvd_trans h7.right h8 done done lemma exists_least_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q := by set S : Set Nat := {p : Nat | prime_factor p n} have h2 : ∃ (p : Nat), p ∈ S := exists_prime_factor n h show ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q from well_ord_princ S h2 done lemma all_prime_nil : all_prime [] := by define --Goal : ∀ p ∈ [], prime p fix p : Nat contrapos --Goal : ¬prime p → p ∉ [] assume h1 : ¬prime p show p ∉ [] from List.not_mem_nil p done lemma all_prime_cons (n : Nat) (L : List Nat) : all_prime (n :: L) ↔ prime n ∧ all_prime L := by apply Iff.intro · -- (→) assume h1 : all_prime (n :: L) --Goal : prime n ∧ all_prime L define at h1 --h1 : ∀ p ∈ n :: L, prime p apply And.intro (h1 n (List.mem_cons_self n L)) define --Goal : ∀ p ∈ L, prime p fix p : Nat assume h2 : p ∈ L show prime p from h1 p (List.mem_cons_of_mem n h2) done · -- (←) assume h1 : prime n ∧ all_prime L --Goal : all_prime (n :: l) define : all_prime L at h1 define fix p : Nat assume h2 : p ∈ n :: L rewrite [List.mem_cons] at h2 --h2 : p = n ∨ p ∈ L by_cases on h2 · -- Case 1. h2 : p = n rewrite [h2] show prime n from h1.left done · -- Case 2. h2 : p ∈ L show prime p from h1.right p h2 done done done lemma nondec_nil : nondec [] := by define --Goal : True trivial --trivial proves some obviously true statements, such as True done lemma nondec_cons (n : Nat) (L : List Nat) : nondec (n :: L) ↔ (∀ m ∈ L, n ≤ m) ∧ nondec L := by rfl lemma prod_nil : prod [] = 1 := by rfl lemma prod_cons : prod (n :: L) = n * (prod L) := by rfl lemma exists_cons_of_length_eq_succ {A : Type} {l : List A} {n : Nat} (h : l.length = n + 1) : ∃ (a : A) (L : List A), l = a :: L ∧ L.length = n := by have h1 : ¬l.length = 0 := by linarith rewrite [List.length_eq_zero] at h1 obtain (a : A) (h2 : ∃ (L : List A), l = a :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List A) (h3 : l = a :: L) from h2 apply Exists.intro a apply Exists.intro L apply And.intro h3 have h4 : (a :: L).length = L.length + 1 := List.length_cons a L rewrite [←h3, h] at h4 show L.length = n from (Nat.add_right_cancel h4).symm done lemma list_elt_dvd_prod_by_length (a : Nat) : ∀ (n : Nat), ∀ (l : List Nat), l.length = n → a ∈ l → a ∣ prod l := by by_induc · --Base Case fix l : List Nat assume h1 : l.length = 0 rewrite [List.length_eq_zero] at h1 --h1 : l = [] rewrite [h1] --Goal : a ∈ [] → a ∣ prod [] contrapos assume h2 : ¬a ∣ prod [] show a ∉ [] from List.not_mem_nil a done · -- Induction Step fix n : Nat assume ih : ∀ (l : List Nat), List.length l = n → a ∈ l → a ∣ prod l fix l : List Nat assume h1 : l.length = n + 1 --Goal : a ∈ l → a ∣ prod l obtain (b : Nat) (h2 : ∃ (L : List Nat), l = b :: L ∧ L.length = n) from exists_cons_of_length_eq_succ h1 obtain (L : List Nat) (h3 : l = b :: L ∧ L.length = n) from h2 have h4 : a ∈ L → a ∣ prod L := ih L h3.right assume h5 : a ∈ l rewrite [h3.left, prod_cons] --Goal : a ∣ b * prod L rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b ∨ a ∈ L by_cases on h5 · -- Case 1. h5 : a = b apply Exists.intro (prod L) rewrite [h5] rfl done · -- Case 2. h5 : a ∈ L have h6 : a ∣ prod L := h4 h5 have h7 : prod L ∣ b * prod L := by apply Exists.intro b ring done show a ∣ b * prod L from dvd_trans h6 h7 done done done lemma list_elt_dvd_prod {a : Nat} {l : List Nat} (h : a ∈ l) : a ∣ prod l := by set n : Nat := l.length have h1 : l.length = n := by rfl show a ∣ prod l from list_elt_dvd_prod_by_length a n l h1 h done lemma exists_prime_factorization : ∀ (n : Nat), n ≥ 1 → ∃ (l : List Nat), prime_factorization n l := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, n_1 ≥ 1 → ∃ (l : List Nat), prime_factorization n_1 l assume h1 : n ≥ 1 by_cases h2 : n = 1 · -- Case 1. h2 : n = 1 apply Exists.intro [] define apply And.intro · -- Proof of nondec_prime_list [] define show all_prime [] ∧ nondec [] from And.intro all_prime_nil nondec_nil done · -- Proof of prod [] = n rewrite [prod_nil, h2] rfl done done · -- Case 2. h2 : n ≠ 1 have h3 : n ≥ 2 := lt_of_le_of_ne' h1 h2 obtain (p : Nat) (h4 : prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q) from exists_least_prime_factor h3 have p_prime_factor : prime_factor p n := h4.left define at p_prime_factor have p_prime : prime p := p_prime_factor.left have p_dvd_n : p ∣ n := p_prime_factor.right have p_least : ∀ (q : Nat), prime_factor q n → p ≤ q := h4.right obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n have h5 : m ≠ 0 := by contradict h1 with h6 have h7 : n = 0 := calc n _ = p * m := n_eq_pm _ = p * 0 := by rw [h6] _ = 0 := by ring rewrite [h7] decide done have m_pos : 0 < m := Nat.pos_of_ne_zero h5 have m_lt_n : m < n := by define at p_prime show m < n from calc m _ < m + m := by linarith _ = 2 * m := by ring _ ≤ p * m := by rel [p_prime.left] _ = n := n_eq_pm.symm done obtain (L : List Nat) (h6 : prime_factorization m L) from ih m m_lt_n m_pos define at h6 have ndpl_L : nondec_prime_list L := h6.left define at ndpl_L apply Exists.intro (p :: L) define apply And.intro · -- Proof of nondec_prime_list (p :: L) define apply And.intro · -- Proof of all_prime (p :: L) rewrite [all_prime_cons] show prime p ∧ all_prime L from And.intro p_prime ndpl_L.left done · -- Proof of nondec (p :: L) rewrite [nondec_cons] apply And.intro _ ndpl_L.right fix q : Nat assume q_in_L : q ∈ L have h7 : q ∣ prod L := list_elt_dvd_prod q_in_L rewrite [h6.right] at h7 --h7 : q ∣ m have h8 : m ∣ n := by apply Exists.intro p rewrite [n_eq_pm] ring done have q_dvd_n : q ∣ n := dvd_trans h7 h8 have ap_L : all_prime L := ndpl_L.left define at ap_L have q_prime_factor : prime_factor q n := And.intro (ap_L q q_in_L) q_dvd_n show p ≤ q from p_least q q_prime_factor done done · -- Proof of prod (p :: L) = n rewrite [prod_cons, h6.right, n_eq_pm] rfl done done done theorem Theorem_7_2_2 {a b c : Nat} (h1 : c ∣ a * b) (h2 : rel_prime a c) : c ∣ b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑c ∣ ↑b define at h1; define at h2; define obtain (j : Nat) (h3 : a * b = c * j) from h1 set s : Int := gcd_c1 a c set t : Int := gcd_c2 a c have h4 : s * ↑a + t * ↑c = ↑(gcd a c) := gcd_lin_comb c a rewrite [h2, Nat.cast_one] at h4 --h4 : s * ↑a + t * ↑c = (1 : Int) apply Exists.intro (s * ↑j + t * ↑b) show ↑b = ↑c * (s * ↑j + t * ↑b) from calc ↑b _ = (1 : Int) * ↑b := (one_mul _).symm _ = (s * ↑a + t * ↑c) * ↑b := by rw [h4] _ = s * (↑a * ↑b) + t * ↑c * ↑b := by ring _ = s * (↑c * ↑j) + t * ↑c * ↑b := by rw [←Nat.cast_mul a b, h3, Nat.cast_mul c j] _ = ↑c * (s * ↑j + t * ↑b) := by ring done lemma le_nonzero_prod_left {a b : Nat} (h : a * b ≠ 0) : a ≤ a * b := by have h1 : b ≠ 0 := by contradict h with h1 rewrite [h1] ring done have h2 : 1 ≤ b := Nat.pos_of_ne_zero h1 show a ≤ a * b from calc a = a * 1 := (mul_one a).symm _ ≤ a * b := by rel [h2] done lemma le_nonzero_prod_right {a b : Nat} (h : a * b ≠ 0) : b ≤ a * b := by rewrite [mul_comm] rewrite [mul_comm] at h show b ≤ b * a from le_nonzero_prod_left h done lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry lemma rel_prime_of_prime_not_dvd {a p : Nat} (h1 : prime p) (h2 : ¬p ∣ a) : rel_prime a p := by have h3 : gcd a p ∣ a := gcd_dvd_left a p have h4 : gcd a p ∣ p := gcd_dvd_right a p have h5 : gcd a p = 1 ∨ gcd a p = p := dvd_prime h1 h4 have h6 : gcd a p ≠ p := by contradict h2 with h6 rewrite [h6] at h3 show p ∣ a from h3 done disj_syll h5 h6 show rel_prime a p from h5 done theorem Theorem_7_2_3 {a b p : Nat} (h1 : prime p) (h2 : p ∣ a * b) : p ∣ a ∨ p ∣ b := by or_right with h3 have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3 show p ∣ b from Theorem_7_2_2 h2 h4 done lemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a ≥ 1 := by have h1 : a ≠ 0 := by by_contra h1 rewrite [h1] at h contradict h linarith done show a ≥ 1 from Nat.pos_of_ne_zero h1 done lemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by have h1 : a ≥ 1 := ge_one_of_prod_one h have h2 : a * b ≠ 0 := by linarith have h3 : a ≤ a * b := le_nonzero_prod_left h2 rewrite [h] at h3 show a = 1 from Nat.le_antisymm h3 h1 done lemma eq_one_of_dvd_one {n : Nat} (h : n ∣ 1) : n = 1 := by obtain (j : Nat) (h1 : 1 = n * j) from h show n = 1 from eq_one_of_prod_one h1.symm done lemma prime_not_one {p : Nat} (h : prime p) : p ≠ 1 := by define at h linarith done theorem Theorem_7_2_4 {p : Nat} (h1 : prime p) : ∀ (l : List Nat), p ∣ prod l → ∃ a ∈ l, p ∣ a := by apply List.rec · -- Base Case. Goal : p ∣ prod [] → ∃ a ∈ [], p ∣ a rewrite [prod_nil] assume h2 : p ∣ 1 show ∃ a ∈ [], p ∣ a from absurd (eq_one_of_dvd_one h2) (prime_not_one h1) done · -- Induction Step fix b : Nat fix L : List Nat assume ih : p ∣ prod L → ∃ a ∈ L, p ∣ a --Goal : p ∣ prod (b :: L) → ∃ a ∈ b :: L, p ∣ a assume h2 : p ∣ prod (b :: L) rewrite [prod_cons] at h2 have h3 : p ∣ b ∨ p ∣ prod L := Theorem_7_2_3 h1 h2 by_cases on h3 · -- Case 1. h3 : p ∣ b apply Exists.intro b show b ∈ b :: L ∧ p ∣ b from And.intro (List.mem_cons_self b L) h3 done · -- Case 2. h3 : p ∣ prod L obtain (a : Nat) (h4 : a ∈ L ∧ p ∣ a) from ih h3 apply Exists.intro a show a ∈ b :: L ∧ p ∣ a from And.intro (List.mem_cons_of_mem b h4.left) h4.right done done done lemma prime_in_list {p : Nat} {l : List Nat} (h1 : prime p) (h2 : all_prime l) (h3 : p ∣ prod l) : p ∈ l := by obtain (a : Nat) (h4 : a ∈ l ∧ p ∣ a) from Theorem_7_2_4 h1 l h3 define at h2 have h5 : prime a := h2 a h4.left have h6 : p = 1 ∨ p = a := dvd_prime h5 h4.right disj_syll h6 (prime_not_one h1) rewrite [h6] show a ∈ l from h4.left done lemma first_le_first {p q : Nat} {l m : List Nat} (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m)) (h3 : prod (p :: l) = prod (q :: m)) : p ≤ q := by define at h1; define at h2 have h4 : q ∣ prod (p :: l) := by define apply Exists.intro (prod m) rewrite [←prod_cons] show prod (p :: l) = prod (q :: m) from h3 done have h5 : all_prime (q :: m) := h2.left rewrite [all_prime_cons] at h5 have h6 : q ∈ p :: l := prime_in_list h5.left h1.left h4 have h7 : nondec (p :: l) := h1.right rewrite [nondec_cons] at h7 rewrite [List.mem_cons] at h6 by_cases on h6 · -- Case 1. h6 : q = p linarith done · -- Case 2. h6 : q ∈ l have h8 : ∀ m ∈ l, p ≤ m := h7.left show p ≤ q from h8 q h6 done done lemma nondec_prime_list_tail {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by define at h define rewrite [all_prime_cons, nondec_cons] at h show all_prime l ∧ nondec l from And.intro h.left.right h.right.right done lemma cons_prod_not_one {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : prod (p :: l) ≠ 1 := by define at h have h1 : all_prime (p :: l) := h.left rewrite [all_prime_cons] at h1 rewrite [prod_cons] by_contra h2 show False from (prime_not_one h1.left) (eq_one_of_prod_one h2) done lemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) : l = [] ↔ prod l = 1 := by apply Iff.intro · -- (→) assume h1 : l = [] rewrite [h1] show prod [] = 1 from prod_nil done · -- (←) contrapos assume h1 : ¬l = [] obtain (p : Nat) (h2 : ∃ (L : List Nat), l = p :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List Nat) (h3 : l = p :: L) from h2 rewrite [h3] at h rewrite [h3] show ¬prod (p :: L) = 1 from cons_prod_not_one h done done lemma prime_pos {p : Nat} (h : prime p) : p > 0 := by define at h linarith done theorem Theorem_7_2_5 : ∀ (l1 l2 : List Nat), nondec_prime_list l1 → nondec_prime_list l2 → prod l1 = prod l2 → l1 = l2 := by apply List.rec · -- Base Case. Goal : ∀ (l2 : List Nat), nondec_prime_list [] → -- nondec_prime_list l2 → prod [] = prod l2 → [] = l2 fix l2 : List Nat assume h1 : nondec_prime_list [] assume h2 : nondec_prime_list l2 assume h3 : prod [] = prod l2 rewrite [prod_nil, eq_comm, ←list_nil_iff_prod_one h2] at h3 show [] = l2 from h3.symm done · -- Induction Step fix p : Nat fix L1 : List Nat assume ih : ∀ (L2 : List Nat), nondec_prime_list L1 → nondec_prime_list L2 → prod L1 = prod L2 → L1 = L2 -- Goal : ∀ (l2 : List Nat), nondec_prime_list (p :: L1) → -- nondec_prime_list l2 → prod (p :: L1) = prod l2 → p :: L1 = l2 fix l2 : List Nat assume h1 : nondec_prime_list (p :: L1) assume h2 : nondec_prime_list l2 assume h3 : prod (p :: L1) = prod l2 have h4 : ¬prod (p :: L1) = 1 := cons_prod_not_one h1 rewrite [h3, ←list_nil_iff_prod_one h2] at h4 obtain (q : Nat) (h5 : ∃ (L : List Nat), l2 = q :: L) from List.exists_cons_of_ne_nil h4 obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5 rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2) rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2) have h7 : p ≤ q := first_le_first h1 h2 h3 have h8 : q ≤ p := first_le_first h2 h1 h3.symm have h9 : p = q := by linarith rewrite [h9, prod_cons, prod_cons] at h3 --h3 : q * prod L1 = q * prod L2 have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1 have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2 define at h2 have h12 : all_prime (q :: L2) := h2.left rewrite [all_prime_cons] at h12 have h13 : q > 0 := prime_pos h12.left have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3 have h15 : L1 = L2 := ih L2 h10 h11 h14 rewrite [h6, h9, h15] rfl done done theorem fund_thm_arith (n : Nat) (h : n ≥ 1) : ∃! (l : List Nat), prime_factorization n l := by exists_unique · -- Existence show ∃ (l : List Nat), prime_factorization n l from exists_prime_factorization n h done · -- Uniqueness fix l1 : List Nat; fix l2 : List Nat assume h1 : prime_factorization n l1 assume h2 : prime_factorization n l2 define at h1; define at h2 have h3 : prod l1 = n := h1.right rewrite [←h2.right] at h3 show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3 done done /- Section 7.3 -/ theorem congr_refl (m : Nat) : ∀ (a : Int), a ≡ a (MOD m) := by fix a : Int define --Goal : ∃ (c : Int), a - a = ↑m * c apply Exists.intro 0 ring done theorem congr_symm {m : Nat} : ∀ {a b : Int}, a ≡ b (MOD m) → b ≡ a (MOD m) := by fix a : Int; fix b : Int assume h1 : a ≡ b (MOD m) define at h1 --h1 : ∃ (c : Int), a - b = ↑m * c define --Goal : ∃ (c : Int), b - a = ↑m * c obtain (c : Int) (h2 : a - b = m * c) from h1 apply Exists.intro (-c) show b - a = m * (-c) from calc b - a _ = -(a - b) := by ring _ = -(m * c) := by rw [h2] _ = m * (-c) := by ring done theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry /- Fundamental properties of congruence classes -/ lemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) : X = Y ↔ X.val = Y.val := Fin.ext_iff lemma val_nat_eq_mod (n k : Nat) : ([k]_(n + 1)).val = k % (n + 1) := by rfl lemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl theorem cc_rep {m : Nat} (X : ZMod m) : ∃ (a : Int), X = [a]_m := match m with | 0 => by apply Exists.intro X rfl done | n + 1 => by apply Exists.intro ↑(X.val) have h1 : X.val < n + 1 := Fin.prop X rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1] rfl done theorem add_class (m : Nat) (a b : Int) : [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm theorem mul_class (m : Nat) (a b : Int) : [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm lemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ [a - b]_m = [0]_m := by apply Iff.intro · -- (→) assume h1 : [a]_m = [b]_m have h2 : a - b = a + (-b) := by ring have h3 : b + (-b) = 0 := by ring show [a - b]_m = [0]_m from calc [a - b]_m _ = [a + (-b)]_m := by rw [h2] _ = [a]_m + [-b]_m := by rw [add_class] _ = [b]_m + [-b]_m := by rw [h1] _ = [b + -b]_m := by rw [add_class] _ = [0]_m := by rw [h3] done · -- (←) assume h1 : [a - b]_m = [0]_m have h2 : b + (a - b) = a := by ring have h3 : b + 0 = b := by ring show [a]_m = [b]_m from calc [a]_m _ = [b + (a - b)]_m := by rw [h2] _ = [b]_m + [a - b]_m := by rw [add_class] _ = [b]_m + [0]_m := by rw [h1] _ = [b + 0]_m := by rw [add_class] _ = [b]_m := by rw [h3] done done lemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) : [a]_m = [0]_m → [-a]_m = [0]_m := by assume h1 : [a]_m = [0]_m have h2 : 0 + (-a) = -a := by ring have h3 : a + (-a) = 0 := by ring show [-a]_m = [0]_m from calc [-a]_m _ = [0 + (-a)]_m := by rw [h2] _ = [0]_m + [-a]_m := by rw [add_class] _ = [a]_m + [-a]_m := by rw [h1] _ = [a + (-a)]_m := by rw [add_class] _ = [0]_m := by rw [h3] done lemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) : [-a]_m = [0]_m ↔ [a]_m = [0]_m := by apply Iff.intro _ (cc_neg_zero_of_cc_zero m a) assume h1 : [-a]_m = [0]_m have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1 have h3 : -(-a) = a := by ring rewrite [h3] at h2 show [a]_m = [0]_m from h2 done lemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl lemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m ↔ m ∣ k := match m with | 0 => by have h : (0 : Int) = (↑(0 : Nat) : Int) := by rfl rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj] apply Iff.intro · -- (→) assume h1 : k = 0 rewrite [h1] show 0 ∣ 0 from dvd_self 0 done · -- (←) assume h1 : 0 ∣ k obtain (c : Nat) (h2 : k = 0 * c) from h1 rewrite [h2] ring done done | n + 1 => by rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero] show k % (n + 1) = 0 ↔ n + 1 ∣ k from (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm done lemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m ↔ ↑m ∣ a := by obtain (k : Nat) (h1 : a = ↑k ∨ a = -↑k) from Int.eq_nat_or_neg a by_cases on h1 · -- Case 1. h1: a = ↑k rewrite [h1, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done · -- Case 2. h1: a = -↑k rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done done theorem cc_eq_iff_congr (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ a ≡ b (MOD m) := calc [a]_m = [b]_m _ ↔ [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b _ ↔ ↑m ∣ (a - b) := cc_zero_iff_dvd m (a - b) _ ↔ a ≡ b (MOD m) := by rfl /- End of fundamental properties of congruence classes -/ lemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m := by have h1 : (↑m : Int) ≠ 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m) show 0 ≤ a % m from Int.emod_nonneg a h1 done lemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m) have h2 : (↑m : Int) > 0 := (Nat.cast_pos).rtl h1 show a % m < m from Int.emod_lt_of_pos a h2 done lemma congr_mod_mod (m : Nat) (a : Int) : a ≡ a % m (MOD m) := by define have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m apply Exists.intro (a / m) show a - a % m = m * (a / m) from calc a - (a % m) _ = m * (a / m) + a % m - a % m := by rw [h1] _ = m * (a / m) := by ring done lemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) := And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a)) theorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) : ∃! (r : Int), 0 ≤ r ∧ r < m ∧ a ≡ r (MOD m) := by exists_unique · -- Existence apply Exists.intro (a % m) show 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) from mod_cmpl_res m a done · -- Uniqueness fix r1 : Int; fix r2 : Int assume h1 : 0 ≤ r1 ∧ r1 < m ∧ a ≡ r1 (MOD m) assume h2 : 0 ≤ r2 ∧ r2 < m ∧ a ≡ r2 (MOD m) have h3 : r1 ≡ r2 (MOD m) := congr_trans (congr_symm h1.right.right) h2.right.right obtain (d : Int) (h4 : r1 - r2 = m * d) from h3 have h5 : r1 - r2 < m * 1 := by linarith have h6 : m * (-1) < r1 - r2 := by linarith rewrite [h4] at h5 --h5 : m * d < m * 1 rewrite [h4] at h6 --h6 : m * -1 < m * d have h7 : (↑m : Int) ≥ 0 := Nat.cast_nonneg m have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7 have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7 have h10 : d = 0 := by linarith show r1 = r2 from calc r1 _ = r1 - r2 + r2 := by ring _ = m * 0 + r2 := by rw [h4, h10] _ = r2 := by ring done done lemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m := (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a) theorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y rewrite [h1, h2] have h3 : a + b = b + a := by ring show [a]_m + [b]_m = [b]_m + [a]_m from calc [a]_m + [b]_m _ = [a + b]_m := add_class m a b _ = [b + a]_m := by rw [h3] _ = [b]_m + [a]_m := (add_class m b a).symm done theorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X rewrite [h1] have h2 : a * 1 = a := by ring show [a]_m * [1]_m = [a]_m from calc [a]_m * [1]_m _ = [a * 1]_m := mul_class m a 1 _ = [a]_m := by rw [h2] done theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry lemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) : [a]_m * [gcd_c2 m a]_m = [1]_m := by set s : Int := gcd_c1 m a have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m define at h1 rewrite [h1, Nat.cast_one] at h2 --h2 : s * ↑m + gcd_c2 m a * ↑a = 1 rewrite [mul_class, cc_eq_iff_congr] define --Goal : ∃ (c : Int), ↑a * gcd_c2 m a - 1 = ↑m * c apply Exists.intro (-s) show a * (gcd_c2 m a) - 1 = m * (-s) from calc a * (gcd_c2 m a) - 1 _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring _ = 1 + m * (-s) - 1 := by rw [h2] _ = m * (-s) := by ring done theorem Theorem_7_3_7 (m a : Nat) : invertible [a]_m ↔ rel_prime m a := by apply Iff.intro · -- (→) assume h1 : invertible [a]_m define at h1 obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1 obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y rewrite [h3, mul_class, cc_eq_iff_congr] at h2 define at h2 obtain (c : Int) (h4 : a * b - 1 = m * c) from h2 rewrite [Exercise_7_2_6] --Goal : ∃ (s t : Int), s * ↑m + t * ↑a = 1 apply Exists.intro (-c) apply Exists.intro b show (-c) * m + b * a = 1 from calc (-c) * m + b * a _ = (-c) * m + (a * b - 1) + 1 := by ring _ = (-c) * m + m * c + 1 := by rw [h4] _ = 1 := by ring done · -- (←) assume h1 : rel_prime m a define show ∃ (Y : ZMod m), [a]_m * Y = [1]_m from Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1) done done /- Section 7.4 -/ section Euler open Euler lemma num_rp_below_base {m : Nat} : num_rp_below m 0 = 0 := by rfl lemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) : num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by have h1 : num_rp_below m (j + 1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : gcd m j = 1 rewrite [if_pos h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1 show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1 done lemma num_rp_below_step_not_rp {m j : Nat} (h : ¬rel_prime m j) : num_rp_below m (j + 1) = num_rp_below m j := by have h1 : num_rp_below m (j +1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : ¬gcd m j = 1 rewrite [if_neg h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j show num_rp_below m (j + 1) = num_rp_below m j from h1 done lemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl #eval phi 10 --Answer: 4 lemma prod_inv_iff_inv {m : Nat} {X : ZMod m} (h1 : invertible X) (Y : ZMod m) : invertible (X * Y) ↔ invertible Y := by apply Iff.intro · -- (→) assume h2 : invertible (X * Y) obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2 apply Exists.intro (X * Z) rewrite [←h3] --Goal : Y * (X * Z) = X * Y * Z ring --Note that ring can do algebra in ZMod m done · -- (←) assume h2 : invertible Y obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1 obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2 apply Exists.intro (Xi * Yi) show (X * Y) * (Xi * Yi) = [1]_m from calc X * Y * (Xi * Yi) _ = (X * Xi) * (Y * Yi) := by ring _ = [1]_m * [1]_m := by rw [h3, h4] _ = [1]_m := Theorem_7_3_6_7 [1]_m done done lemma F_rp_def {m i : Nat} (h : rel_prime m i) : F m i = [i]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h --h : gcd m i = 1 rewrite [if_pos h] at h1 show F m i = [i]_m from h1 done lemma F_not_rp_def {m i : Nat} (h : ¬rel_prime m i) : F m i = [1]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h rewrite [h1, if_neg h] rfl done lemma prod_seq_base {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 0 k f = [1]_m := by rfl lemma prod_seq_step {m : Nat} (n k : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl lemma prod_seq_zero_step {m : Nat} (n : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by rewrite [prod_seq_step, zero_add] rfl done lemma prod_one {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 1 k f = f k := by rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7] rfl done lemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl lemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m := calc [G m a i]_m _ = [(a * i) % m]_m := by rfl _ = [a * i]_m := (cc_eq_mod m (a * i)).symm _ = [a]_m * [i]_m := (mul_class m a i).symm lemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) : rel_prime m (G m a i) ↔ rel_prime m i := by have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1 show rel_prime m (G m a i) ↔ rel_prime m i from calc rel_prime m (G m a i) _ ↔ invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm _ ↔ invertible ([a]_m * [i]_m) := by rw [cc_G] _ ↔ invertible [i]_m := prod_inv_iff_inv h2 ([i]_m) _ ↔ rel_prime m i := Theorem_7_3_7 m i done lemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) : F m (G m a i) = [a]_m * F m i := by have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2 show F m (G m a i) = [a]_m * F m i from calc F m (G m a i) _ = [G m a i]_m := F_rp_def h3 _ = [a]_m * [i]_m := cc_G m a i _ = [a]_m * F m i := by rw [F_rp_def h2] done lemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : ¬rel_prime m i) : F m (G m a i) = [1]_m := by rewrite [←G_rp_iff h1 i] at h2 show F m (G m a i) = [1]_m from F_not_rp_def h2 done lemma FG_prod {m a : Nat} (h1 : rel_prime m a) : ∀ (k : Nat), prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by by_induc · -- Base Case show prod_seq 0 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from calc prod_seq 0 0 ((F m) ∘ (G m a)) _ = [1]_m := prod_seq_base _ _ _ = [a]_m ^ 0 * [1]_m := by ring _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by rw [num_rp_below_base, prod_seq_base] done · -- Induction Step fix k : Nat assume ih : prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) by_cases h2 : rel_prime m k · -- Case 1. h2 : rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([a]_m * F m k) := by rw [FG_rp h1 h2] _ = [a]_m ^ ((num_rp_below m k) + 1) * ((prod_seq k 0 (F m)) * F m k) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_rp h2, prod_seq_zero_step] done · -- Case 2. h2 : ¬rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([1]_m) := by rw [FG_not_rp h1 h2] _ = [a]_m ^ (num_rp_below m k) * (prod_seq k 0 (F m) * ([1]_m)) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_not_rp h2, prod_seq_zero_step, F_not_rp_def h2] done done done lemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by define --Goal : ∀ i < m, G m a i < m fix i : Nat assume h1 : i < m rewrite [G_def] --Goal : a * i % m < m show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m) done lemma left_inv_one_one_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g' (g i) = i) : one_one_below n g := sorry lemma right_inv_onto_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g (g' i) = i) (h2 : maps_below n g') : onto_below n g := by define at h2; define fix k : Nat assume h3 : k < n apply Exists.intro (g' k) show g' k < n ∧ g (g' k) = k from And.intro (h2 k h3) (h1 k h3) done lemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m * [inv_mod m a]_m = [1]_m := by have h2 : 0 ≤ (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a) show [a]_m * [inv_mod m a]_m = [1]_m from calc [a]_m * [inv_mod m a]_m _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2] _ = [a]_m * [gcd_c2 m a]_m := by rw [←cc_eq_mod] _ = [1]_m := gcd_c2_inv h1 done lemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m := calc a * (b % m) % m = a % m * (b % m % m) % m := Nat.mul_mod _ _ _ _ = a % m * (b % m) % m := by rw [Nat.mod_mod] _ = a * b % m := (Nat.mul_mod _ _ _).symm lemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm] rfl done theorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] : ↑a ≡ ↑b (MOD m) ↔ a % m = b % m := sorry lemma mul_inv_mod_cancel {m a i : Nat} [NeZero m] (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1 rewrite [mul_class, cc_eq_iff_congr, ←Nat.cast_mul, ←Nat.cast_one, congr_iff_mod_eq_Nat] at h3 show a * inv_mod m a * i % m = i from calc a * (inv_mod m a) * i % m _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = 1 % m * i % m := by rw [h3] _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = i % m := by rw [one_mul] _ = i := Nat.mod_eq_of_lt h2 done lemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl lemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, G m a (Ginv m a i) = i := by fix i : Nat assume h2 : i < m show G m a (Ginv m a i) = i from calc G m a (Ginv m a i) _ = a * ((inv_mod m a * i) % m) % m := by rfl _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, Ginv m a (G m a i) = i := by fix i : Nat assume h2 : i < m show Ginv m a (G m a i) = i from calc Ginv m a (G m a i) _ = inv_mod m a * ((a * i) % m) % m := by rfl _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc, mul_comm (inv_mod m a)] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_maps_below (m a : Nat) [NeZero m] : maps_below m (Ginv m a) := G_maps_below m (inv_mod m a) lemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : one_one_below m (G m a) := left_inv_one_one_below (Ginv_left_inv h1) lemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : onto_below m (G m a) := right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a) lemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : perm_below m (G m a) := And.intro (G_maps_below m a) (And.intro (G_one_one_below h1) (G_onto_below h1)) --Permuting a product of congruence classes doesn't change product lemma swap_fst (u v : Nat) : swap u v u = v := by define : swap u v u --Goal : (if u = u then v else if u = v then u else u) = v have h : u = u := by rfl rewrite [if_pos h] rfl done lemma swap_snd (u v : Nat) : swap u v v = u := by define : swap u v v by_cases h1 : v = u · -- Case 1. h1 : v = u rewrite [if_pos h1] show v = u from h1 done · -- Case 2. h1 : v ≠ u rewrite [if_neg h1] have h2 : v = v := by rfl rewrite [if_pos h2] rfl done done lemma swap_other {u v i : Nat} (h1 : i ≠ u) (h2 : i ≠ v) : swap u v i = i := by define : swap u v i rewrite [if_neg h1, if_neg h2] rfl done lemma swap_values (u v i : Nat) : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := by by_cases h1 : i = u · -- Case 1. h1 : i = u apply Or.inl rewrite [h1] show swap u v u = v from swap_fst u v done · -- Case 2. h1 : i ≠ u apply Or.inr by_cases h2 : i = v · -- Case 2.1. h2 : i = v apply Or.inl rewrite [h2] show swap u v v = u from swap_snd u v done · -- Case 2.2. h2 : i ≠ v apply Or.inr show swap u v i = i from swap_other h1 h2 done done done lemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by define fix i : Nat assume h3 : i < n have h4 : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := swap_values u v i by_cases on h4 · -- Case 1. h4 : swap u v i = v rewrite [h4] show v < n from h2 done · -- Case 2. by_cases on h4 · -- Case 2.1. h4 : swap u v i = u rewrite [h4] show u < n from h1 done · -- Case 2.2. h4 : swap u v i = i rewrite [h4] show i < n from h3 done done done lemma swap_swap (u v n : Nat) : ∀ i < n, swap u v (swap u v i) = i := by fix i : Nat assume h : i < n by_cases h1 : i = u · -- Case 1. h1 : i = u rewrite [h1, swap_fst, swap_snd] rfl done · -- Case 2. h1 : i ≠ u by_cases h2 : i = v · -- Case 2.1. h2 : i = v rewrite [h2, swap_snd, swap_fst] rfl done · -- Case 2.2. h2 : i ≠ v rewrite [swap_other h1 h2, swap_other h1 h2] rfl done done done lemma swap_one_one_below (u v n) : one_one_below n (swap u v) := left_inv_one_one_below (swap_swap u v n) lemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) := right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2) lemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) := And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2)) lemma comp_perm_below {n : Nat} {f g : Nat → Nat} (h1 : perm_below n f) (h2 : perm_below n g) : perm_below n (f ∘ g) := sorry lemma trivial_swap (u : Nat) : swap u u = id := by apply funext fix x : Nat by_cases h1 : x = u · -- Case 1. h1 : x = u rewrite [h1, swap_fst] rfl done · -- Case 2. h1 : x ≠ u rewrite [swap_other h1 h1] rfl done done lemma prod_eq_fun {m : Nat} (f g : Nat → ZMod m) (k : Nat) : ∀ (n : Nat), (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g := by by_induc · -- Base Case assume h : (∀ i < 0, f (k + i) = g (k + i)) rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g assume h1 : ∀ i < n + 1, f (k + i) = g (k + i) have h2 : ∀ i < n, f (k + i) = g (k + i) := by fix i : Nat assume h2 : i < n have h3 : i < n + 1 := by linarith show f (k + i) = g (k + i) from h1 i h3 done have h3 : prod_seq n k f = prod_seq n k g := ih h2 have h4 : n < n + 1 := Nat.lt_succ_self n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4] rfl done done lemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := by have h2 : ∀ (i : Nat), i < u → (f ∘ swap u n) (0 + i) = f (0 + i) := by fix i : Nat assume h2 : i < u have h3 : 0 + i ≠ u := by linarith have h4 : 0 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl done show prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f from prod_eq_fun (f ∘ swap u n) f 0 u h2 done lemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := by have h2 : ∀ i < j, (f ∘ swap u n) (u + 1 + i) = f (u + 1 + i) := by fix i : Nat assume h2 : i < j have h3 : u + 1 + i ≠ u := by linarith have h4 : u + 1 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl show prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f from prod_eq_fun (f ∘ swap u n) f (u + 1) j h2 done lemma break_prod {m : Nat} (n : Nat) (f : Nat → ZMod m) : ∀ (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by by_induc · -- Base Case have h : n + 0 = n := by rfl rewrite [prod_seq_base, h, Theorem_7_3_6_7] rfl done · -- Induction Step fix j : Nat assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f rewrite [←add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc] rfl done done lemma break_prod_twice {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f := break_prod n f 1 rewrite [prod_one] at h2 have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f := break_prod (u + 1) f j rewrite [←h1] at h3 have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f := break_prod u f 1 rewrite [prod_one] at h4 rewrite [h3, h4] at h2 show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2 done lemma swap_prod_eq_prod {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f := by by_cases h2 : u = n · -- Case 1. h2 : u = n rewrite [h2, trivial_swap n] --Goal : prod_seq (n + 1) 0 (f ∘ id) = prod_seq (n + 1) 0 f rfl done · -- Case 2. h2 : ¬u = n have h3 : u + 1 ≤ n := Nat.lt_of_le_of_ne h1 h2 obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3 have break_f : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := break_prod_twice f h4 have break_fs : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_prod_twice (f ∘ swap u n) h4 have f_eq_fs_below : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := swap_prod_eq_prod_below f h1 have f_eq_fs_btwn : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4 show prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f from calc prod_seq (n + 1) 0 (f ∘ swap u n) _ = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_fs _ = prod_seq u 0 f * (f ∘ swap u n) u * prod_seq j (u + 1) f * (f ∘ swap u n) n := by rw [f_eq_fs_below, f_eq_fs_btwn] _ = prod_seq u 0 f * f (swap u n u) * prod_seq j (u + 1) f * f (swap u n n) := by rfl _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by rw [swap_fst, swap_snd] _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring _ = prod_seq (n + 1) 0 f := break_f.symm done done lemma perm_below_fixed {n : Nat} {g : Nat → Nat} (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry lemma perm_prod {m : Nat} (f : Nat → ZMod m) : ∀ (n : Nat), ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) := by by_induc · -- Base Case fix g : Nat → Nat assume h1 : perm_below 0 g rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) fix g : Nat → Nat assume g_pb : perm_below (n + 1) g define at g_pb have g_ob : onto_below (n + 1) g := g_pb.right.right define at g_ob have h1 : n < n + 1 := by linarith obtain (u : Nat) (h2 : u < n + 1 ∧ g u = n) from g_ob n h1 have s_pb : perm_below (n + 1) (swap u n) := swap_perm_below h2.left h1 have gs_pb_n1 : perm_below (n + 1) (g ∘ swap u n) := comp_perm_below g_pb s_pb have gs_fix_n : (g ∘ swap u n) n = n := calc (g ∘ swap u n) n _ = g (swap u n n) := by rfl _ = g u := by rw [swap_snd] _ = n := h2.right have gs_pb_n : perm_below n (g ∘ swap u n) := perm_below_fixed gs_pb_n1 gs_fix_n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f ∘ (g ∘ swap u n)) := ih (g ∘ swap u n) gs_pb_n have h3 : u ≤ n := by linarith show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f ∘ g) from calc prod_seq (n + 1) 0 f _ = prod_seq n 0 f * f n := prod_seq_zero_step n f _ = prod_seq n 0 (f ∘ (g ∘ swap u n)) * f ((g ∘ swap u n) n) := by rw [gs_prod, gs_fix_n] _ = prod_seq n 0 (f ∘ g ∘ swap u n) * (f ∘ g ∘ swap u n) n := by rfl _ = prod_seq (n + 1) 0 (f ∘ g ∘ swap u n) := (prod_seq_zero_step n (f ∘ g ∘ swap u n)).symm _ = prod_seq (n + 1) 0 ((f ∘ g) ∘ swap u n) := by rfl _ = prod_seq (n + 1) 0 (f ∘ g) := swap_prod_eq_prod (f ∘ g) h3 done done lemma F_invertible (m i : Nat) : invertible (F m i) := by by_cases h : rel_prime m i · -- Case 1. h : rel_prime m i rewrite [F_rp_def h] show invertible [i]_m from (Theorem_7_3_7 m i).rtl h done · -- Case 2. h : ¬rel_prime m i rewrite [F_not_rp_def h] apply Exists.intro [1]_m show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m done done lemma Fprod_invertible (m : Nat) : ∀ (k : Nat), invertible (prod_seq k 0 (F m)) := by by_induc · -- Base Case apply Exists.intro [1]_m show prod_seq 0 0 (F m) * [1]_m = [1]_m from calc prod_seq 0 0 (F m) * [1]_m _ = [1]_m * [1]_m := by rw [prod_seq_base] _ = [1]_m := Theorem_7_3_6_7 ([1]_m) done · -- Induction Step fix k : Nat assume ih : invertible (prod_seq k 0 (F m)) rewrite [prod_seq_zero_step] show invertible (prod_seq k 0 (F m) * (F m k)) from (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k) done done theorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m ^ (phi m) = [1]_m := by have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2 show [a]_m ^ (phi m) = [1]_m from calc [a]_m ^ (phi m) _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3] _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring _ = prod_seq m 0 (F m ∘ G m a) * Y := by rw [FG_prod h1 m, phi_def] _ = prod_seq m 0 (F m) * Y := by rw [perm_prod (F m) m (G m a) (G_perm_below h1)] _ = [1]_m := by rw [h3] done lemma Exercise_7_4_5_Int (m : Nat) (a : Int) : ∀ (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry lemma Exercise_7_4_5_Nat (m a n : Nat) : [a]_m ^ n = [a ^ n]_m := by rewrite [Exercise_7_4_5_Int] rfl done theorem Euler's_theorem {m a : Nat} [NeZero m] (h1 : rel_prime m a) : a ^ (phi m) ≡ 1 (MOD m) := by have h2 : [a]_m ^ (phi m) = [1]_m := Theorem_7_4_2 h1 rewrite [Exercise_7_4_5_Nat m a (phi m)] at h2 --h2 : [a ^ phi m]_m = [1]_m show a ^ (phi m) ≡ 1 (MOD m) from (cc_eq_iff_congr _ _ _).ltr h2 done #eval gcd 10 7 --Answer: 1. So 10 and 7 are relatively prime #eval 7 ^ phi 10 --Answer: 2401, which is congruent to 1 mod 10. end Euler /- Section 7.5 -/ lemma num_rp_prime {p : Nat} (h1 : prime p) : ∀ k < p, num_rp_below p (k + 1) = k := sorry lemma phi_prime {p : Nat} (h1 : prime p) : phi p = p - 1 := by have h2 : 1 ≤ p := prime_pos h1 have h3 : p - 1 + 1 = p := Nat.sub_add_cancel h2 have h4 : p - 1 < p := by linarith have h5 : num_rp_below p (p - 1 + 1) = p - 1 := num_rp_prime h1 (p - 1) h4 rewrite [h3] at h5 show phi p = p - 1 from h5 done theorem Theorem_7_2_2_Int {a c : Nat} {b : Int} (h1 : ↑c ∣ ↑a * b) (h2 : rel_prime a c) : ↑c ∣ b := by rewrite [Int.natCast_dvd, Int.natAbs_mul, Int.natAbs_ofNat] at h1 --h1 : c ∣ a * Int.natAbs b rewrite [Int.natCast_dvd] --Goal : c ∣ Int.natAbs b show c ∣ Int.natAbs b from Theorem_7_2_2 h1 h2 done lemma Lemma_7_4_5 {m n : Nat} (a b : Int) (h1 : rel_prime m n) : a ≡ b (MOD m * n) ↔ a ≡ b (MOD m) ∧ a ≡ b (MOD n) := by apply Iff.intro · -- (→) assume h2 : a ≡ b (MOD m * n) obtain (j : Int) (h3 : a - b = (m * n) * j) from h2 apply And.intro · -- Proof of a ≡ b (MOD m) apply Exists.intro (n * j) show a - b = m * (n * j) from calc a - b _ = m * n * j := h3 _ = m * (n * j) := by ring done · -- Proof of a ≡ b (MOD n) apply Exists.intro (m * j) show a - b = n * (m * j) from calc a - b _ = m * n * j := h3 _ = n * (m * j) := by ring done done · -- (←) assume h2 : a ≡ b (MOD m) ∧ a ≡ b (MOD n) obtain (j : Int) (h3 : a - b = m * j) from h2.left have h4 : (↑n : Int) ∣ a - b := h2.right rewrite [h3] at h4 --h4 : ↑n ∣ ↑m * j have h5 : ↑n ∣ j := Theorem_7_2_2_Int h4 h1 obtain (k : Int) (h6 : j = n * k) from h5 apply Exists.intro k --Goal : a - b = ↑(m * n) * k rewrite [Nat.cast_mul] --Goal : a - b = ↑m * ↑n * k show a - b = (m * n) * k from calc a - b _ = m * j := h3 _ = m * (n * k) := by rw [h6] _ = (m * n) * k := by ring done done --From exercises of Section 7.2 theorem rel_prime_symm {a b : Nat} (h : rel_prime a b) : rel_prime b a := sorry lemma prime_NeZero {p : Nat} (h : prime p) : NeZero p := by rewrite [neZero_iff] --Goal : p ≠ 0 define at h linarith done lemma Lemma_7_5_1 {p e d m c s : Nat} {t : Int} (h1 : prime p) (h2 : e * d = (p - 1) * s + 1) (h3 : m ^ e - c = p * t) : c ^ d ≡ m (MOD p) := by have h4 : m ^ e ≡ c (MOD p) := Exists.intro t h3 have h5 : [m ^ e]_p = [c]_p := (cc_eq_iff_congr _ _ _).rtl h4 rewrite [←Exercise_7_4_5_Nat] at h5 --h5 : [m]_p ^ e = [c]_p by_cases h6 : p ∣ m · -- Case 1. h6 : p ∣ m have h7 : m ≡ 0 (MOD p) := by obtain (j : Nat) (h8 : m = p * j) from h6 apply Exists.intro (↑j : Int) --Goal : ↑m - 0 = ↑p * ↑j rewrite [h8, Nat.cast_mul] ring done have h8 : [m]_p = [0]_p := (cc_eq_iff_congr _ _ _).rtl h7 have h9 : e * d ≠ 0 := by rewrite [h2] show (p - 1) * s + 1 ≠ 0 from Nat.add_one_ne_zero _ done have h10 : (0 : Int) ^ (e * d) = 0 := zero_pow h9 have h11 : [c ^ d]_p = [m]_p := calc [c ^ d]_p _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat] _ = ([m]_p ^ e) ^ d := by rw [h5] _ = [m]_p ^ (e * d) := by ring _ = [0]_p ^ (e * d) := by rw [h8] _ = [0 ^ (e * d)]_p := Exercise_7_4_5_Int _ _ _ _ = [0]_p := by rw [h10] _ = [m]_p := by rw [h8] show c ^ d ≡ m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11 done · -- Case 2. h6 : ¬p ∣ m have h7 : rel_prime m p := rel_prime_of_prime_not_dvd h1 h6 have h8 : rel_prime p m := rel_prime_symm h7 have h9 : NeZero p := prime_NeZero h1 have h10 : (1 : Int) ^ s = 1 := by ring have h11 : [c ^ d]_p = [m]_p := calc [c ^ d]_p _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat] _ = ([m]_p ^ e) ^ d := by rw [h5] _ = [m]_p ^ (e * d) := by ring _ = [m]_p ^ ((p - 1) * s + 1) := by rw [h2] _ = ([m]_p ^ (p - 1)) ^ s * [m]_p := by ring _ = ([m]_p ^ (phi p)) ^ s * [m]_p := by rw [phi_prime h1] _ = [1]_p ^ s * [m]_p := by rw [Theorem_7_4_2 h8] _ = [1 ^ s]_p * [m]_p := by rw [Exercise_7_4_5_Int] _ = [1]_p * [m]_p := by rw [h10] _ = [m]_p * [1]_p := by ring _ = [m]_p := Theorem_7_3_6_7 _ show c ^ d ≡ m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11 done done theorem Theorem_7_5_1 (p q n e d k m c : Nat) (p_prime : prime p) (q_prime : prime q) (p_ne_q : p ≠ q) (n_pq : n = p * q) (ed_congr_1 : e * d = k * (p - 1) * (q - 1) + 1) (h1 : [m]_n ^ e = [c]_n) : [c]_n ^ d = [m]_n := by rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] at h1 --h1 : m ^ e ≡ c (MOD n) rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] --Goal : c ^ d ≡ m (MOD n) obtain (j : Int) (h2 : m ^ e - c = n * j) from h1 rewrite [n_pq, Nat.cast_mul] at h2 --h2 : m ^ e - c = p * q * j have h3 : e * d = (p - 1) * (k * (q - 1)) + 1 := by rewrite [ed_congr_1] ring done have h4 : m ^ e - c = p * (q * j) := by rewrite [h2] ring done have congr_p : c ^ d ≡ m (MOD p) := Lemma_7_5_1 p_prime h3 h4 have h5 : e * d = (q - 1) * (k * (p - 1)) + 1 := by rewrite [ed_congr_1] ring done have h6 : m ^ e - c = q * (p * j) := by rewrite [h2] ring done have congr_q : c ^ d ≡ m (MOD q) := Lemma_7_5_1 q_prime h5 h6 have h7 : ¬q ∣ p := by by_contra h8 have h9 : q = 1 ∨ q = p := dvd_prime p_prime h8 disj_syll h9 (prime_not_one q_prime) show False from p_ne_q h9.symm done have h8 : rel_prime p q := rel_prime_of_prime_not_dvd q_prime h7 rewrite [n_pq, Lemma_7_4_5 _ _ h8] show c ^ d ≡ m (MOD p) ∧ c ^ d ≡ m (MOD q) from And.intro congr_p congr_q done /- BEGIN EXERCISES -/ namespace Exercises /- Section 7.1 -/ -- 1. theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry -- 2. lemma gcd_comm_lt {a b : Nat} (h : a < b) : gcd a b = gcd b a := sorry theorem gcd_comm (a b : Nat) : gcd a b = gcd b a := sorry -- 3. theorem Exercise_7_1_5 (a b : Nat) (n : Int) : (∃ (s t : Int), s * a + t * b = n) ↔ (↑(gcd a b) : Int) ∣ n := sorry -- 4. theorem Exercise_7_1_6 (a b c : Nat) : gcd a b = gcd (a + b * c) b := sorry -- 5. theorem gcd_is_nonzero {a b : Nat} (h : a ≠ 0 ∨ b ≠ 0) : gcd a b ≠ 0 := sorry -- 6. theorem gcd_greatest {a b d : Nat} (h1 : gcd a b ≠ 0) (h2 : d ∣ a) (h3 : d ∣ b) : d ≤ gcd a b := sorry -- 7. lemma Lemma_7_1_10a {a b : Nat} (n : Nat) (h : a ∣ b) : (n * a) ∣ (n * b) := sorry lemma Lemma_7_1_10b {a b n : Nat} (h1 : n ≠ 0) (h2 : (n * a) ∣ (n * b)) : a ∣ b := sorry lemma Lemma_7_1_10c {a b : Nat} (h1 : a ∣ b) (h2 : b ∣ a) : a = b := sorry theorem Exercise_7_1_10 (a b n : Nat) : gcd (n * a) (n * b) = n * gcd a b := sorry /- Section 7.2 -/ -- 1. lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry -- 2. -- Hints: Start with apply List.rec. You may find mul_ne_zero useful theorem prod_nonzero_nonzero : ∀ (l : List Nat), (∀ a ∈ l, a ≠ 0) → prod l ≠ 0 := sorry -- 3. theorem rel_prime_iff_no_common_factor (a b : Nat) : rel_prime a b ↔ ¬∃ (p : Nat), prime p ∧ p ∣ a ∧ p ∣ b := sorry -- 4. theorem rel_prime_symm {a b : Nat} (h : rel_prime a b) : rel_prime b a := sorry -- 5. lemma in_prime_factorization_iff_prime_factor {a : Nat} {l : List Nat} (h1 : prime_factorization a l) (p : Nat) : p ∈ l ↔ prime_factor p a := sorry -- 6. theorem Exercise_7_2_5 {a b : Nat} {l m : List Nat} (h1 : prime_factorization a l) (h2 : prime_factorization b m) : rel_prime a b ↔ (¬∃ (p : Nat), p ∈ l ∧ p ∈ m) := sorry -- 7. theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry -- 8. theorem Exercise_7_2_7 {a b a' b' : Nat} (h1 : rel_prime a b) (h2 : a' ∣ a) (h3 : b' ∣ b) : rel_prime a' b' := sorry -- 9. theorem Exercise_7_2_9 {a b j k : Nat} (h1 : gcd a b ≠ 0) (h2 : a = j * gcd a b) (h3 : b = k * gcd a b) : rel_prime j k := sorry -- 10. theorem Exercise_7_2_17a (a b c : Nat) : gcd a (b * c) ∣ gcd a b * gcd a c := sorry /- Section 7.3 -/ -- 1. theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry -- 2. theorem Theorem_7_3_6_3 {m : Nat} (X : ZMod m) : X + [0]_m = X := sorry -- 3. theorem Theorem_7_3_6_4 {m : Nat} (X : ZMod m) : ∃ (Y : ZMod m), X + Y = [0]_m := sorry -- 4. theorem Exercise_7_3_4a {m : Nat} (Z1 Z2 : ZMod m) (h1 : ∀ (X : ZMod m), X + Z1 = X) (h2 : ∀ (X : ZMod m), X + Z2 = X) : Z1 = Z2 := sorry -- 5. theorem Exercise_7_3_4b {m : Nat} (X Y1 Y2 : ZMod m) (h1 : X + Y1 = [0]_m) (h2 : X + Y2 = [0]_m) : Y1 = Y2 := sorry -- 6. theorem Theorem_7_3_10 (m a : Nat) (b : Int) : ¬(↑(gcd m a) : Int) ∣ b → ¬∃ (x : Int), a * x ≡ b (MOD m) := sorry -- 7. theorem Theorem_7_3_11 (m n : Nat) (a b : Int) (h1 : n ≠ 0) : n * a ≡ n * b (MOD n * m) ↔ a ≡ b (MOD m) := sorry -- 8. theorem Exercise_7_3_16 {m : Nat} {a b : Int} (h : a ≡ b (MOD m)) : ∀ (n : Nat), a ^ n ≡ b ^ n (MOD m) := sorry -- 9. example {m : Nat} [NeZero m] (X : ZMod m) : ∃! (a : Int), 0 ≤ a ∧ a < m ∧ X = [a]_m := sorry -- 10. theorem congr_rel_prime {m a b : Nat} (h1 : a ≡ b (MOD m)) : rel_prime m a ↔ rel_prime m b := sorry -- 11. --Hint: You may find the theorem Int.ofNat_mod_ofNat useful.

theorem rel_prime_mod (m a : Nat) : rel_prime m (a % m) ↔ rel_prime m a

HTPI.Exercises.rel_prime_mod

htpi/HTPILib/Chap7.lean

htpi

/- Copyright 2023 Daniel J. Velleman -/ import HTPILib.Chap6 namespace HTPI /- Definitions -/ lemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by have h : n + 1 > 0 := Nat.succ_pos n show a % (n + 1) < n + 1 from Nat.mod_lt a h done def gcd (a b : Nat) : Nat := match b with | 0 => a | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd (n + 1) (a % (n + 1)) termination_by b mutual def gcd_c1 (a b : Nat) : Int := match b with | 0 => 1 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c2 (n + 1) (a % (n + 1)) --Corresponds to s = t' termination_by b def gcd_c2 (a b : Nat) : Int := match b with | 0 => 0 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c1 (n + 1) (a % (n + 1)) - (gcd_c2 (n + 1) (a % (n + 1))) * ↑(a / (n + 1)) --Corresponds to t = s' - t'q termination_by b end def prime (n : Nat) : Prop := 2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n def prime_factor (p n : Nat) : Prop := prime p ∧ p ∣ n def all_prime (l : List Nat) : Prop := ∀ p ∈ l, prime p def nondec (l : List Nat) : Prop := match l with | [] => True --Of course, True is a proposition that is always true | n :: L => (∀ m ∈ L, n ≤ m) ∧ nondec L def nondec_prime_list (l : List Nat) : Prop := all_prime l ∧ nondec l def prod (l : List Nat) : Nat := match l with | [] => 1 | n :: L => n * (prod L) def prime_factorization (n : Nat) (l : List Nat) : Prop := nondec_prime_list l ∧ prod l = n def rel_prime (a b : Nat) : Prop := gcd a b = 1 def congr_mod (m : Nat) (a b : Int) : Prop := (↑m : Int) ∣ (a - b) def cc (m : Nat) (a : Int) : ZMod m := (↑a : ZMod m) notation:50 a " ≡ " b " (MOD " m ")" => congr_mod m a b notation:max "["a"]_"m:max => cc m a def invertible {m : Nat} (X : ZMod m) : Prop := ∃ (Y : ZMod m), X * Y = [1]_m def num_rp_below (m k : Nat) : Nat := match k with | 0 => 0 | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j def phi (m : Nat) : Nat := num_rp_below m m def prod_seq {m : Nat} (j k : Nat) (f : Nat → ZMod m) : ZMod m := match j with | 0 => [1]_m | n + 1 => prod_seq n k f * f (k + n) def maps_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i < n, g i < n def one_one_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i1 < n, ∀ i2 < n, g i1 = g i2 → i1 = i2 def onto_below (n : Nat) (g : Nat → Nat) : Prop := ∀ k < n, ∃ i < n, g i = k def perm_below (n : Nat) (g : Nat → Nat) : Prop := maps_below n g ∧ one_one_below n g ∧ onto_below n g def inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m) def swap (u v i : Nat) : Nat := if i = u then v else if i = v then u else i namespace Euler --For definitions specific to Euler's theorem def F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m def G (m a i : Nat) : Nat := (a * i) % m def Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i end Euler /- Section 7.1 -/ theorem dvd_mod_of_dvd_a_b {a b d : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ (a % b) := by set q : Nat := a / b have h3 : b * q + a % b = a := Nat.div_add_mod a b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 define --Goal : ∃ (c : Nat), a % b = d * c apply Exists.intro (j - k * q) show a % b = d * (j - k * q) from calc a % b _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm _ = a - b * q := by rw [h3] _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc] _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm done theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry #eval gcd 672 161 --Answer: 7 lemma gcd_base (a : Nat) : gcd a 0 = a := by rfl lemma gcd_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd a b = gcd b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1)) rfl done lemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b ≠ 0) : a % b < b := by have h1 : b > 0 := Nat.pos_of_ne_zero h show a % b < b from Nat.mod_lt a h1 done lemma dvd_self (n : Nat) : n ∣ n := by apply Exists.intro 1 ring done theorem gcd_dvd : ∀ (b a : Nat), (gcd a b) ∣ a ∧ (gcd a b) ∣ b := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd a b_1) ∣ a ∧ (gcd a b_1) ∣ b_1 fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_base] --Goal: a ∣ a ∧ a ∣ 0 apply And.intro (dvd_self a) define apply Exists.intro 0 rfl done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_nonzero a h1] --Goal : gcd b (a % b) ∣ a ∧ gcd b (a % b) ∣ b have h2 : a % b < b := mod_nonzero_lt a h1 have h3 : (gcd b (a % b)) ∣ b ∧ (gcd b (a % b)) ∣ (a % b) := ih (a % b) h2 b apply And.intro _ h3.left show (gcd b (a % b)) ∣ a from dvd_a_of_dvd_b_mod h3.left h3.right done done theorem gcd_dvd_left (a b : Nat) : (gcd a b) ∣ a := (gcd_dvd b a).left theorem gcd_dvd_right (a b : Nat) : (gcd a b) ∣ b := (gcd_dvd b a).right lemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl lemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c1 a b = gcd_c2 b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done lemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl lemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * ↑(a / b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done theorem gcd_lin_comb : ∀ (b a : Nat), (gcd_c1 a b) * ↑a + (gcd_c2 a b) * ↑b = ↑(gcd a b) := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd_c1 a b_1) * ↑a + (gcd_c2 a b_1) * ↑b_1 = ↑(gcd a b_1) fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base] --Goal : 1 * ↑a + 0 * ↑0 = ↑a ring done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1] --Goal : gcd_c2 b (a % b) * ↑a + -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * ↑(a / b)) * ↑b = -- ↑(gcd b (a % b)) set r : Nat := a % b set q : Nat := a / b set s : Int := gcd_c1 b r set t : Int := gcd_c2 b r --Goal : t * ↑a + (s - t * ↑q) * ↑b = ↑(gcd b r) have h2 : r < b := mod_nonzero_lt a h1 have h3 : s * ↑b + t * ↑r = ↑(gcd b r) := ih r h2 b have h4 : b * q + r = a := Nat.div_add_mod a b rewrite [←h3, ←h4] rewrite [Nat.cast_add, Nat.cast_mul] --Goal : t * (↑b * ↑q + ↑r) + (s - t * ↑q) * ↑b = s * ↑b + t * ↑r ring done done #eval gcd_c1 672 161 --Answer: 6 #eval gcd_c2 672 161 --Answer: -25 --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161 theorem Theorem_7_1_6 {d a b : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ gcd a b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑d ∣ ↑(gcd a b) set s : Int := gcd_c1 a b set t : Int := gcd_c2 a b have h3 : s * ↑a + t * ↑b = ↑(gcd a b) := gcd_lin_comb b a rewrite [←h3] --Goal : ↑d ∣ s * ↑a + t * ↑b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul] --Goal : ↑d ∣ s * (↑d * ↑j) + t * (↑d * ↑k) define apply Exists.intro (s * ↑j + t * ↑k) ring done /- Section 7.2 -/ theorem dvd_trans {a b c : Nat} (h1 : a ∣ b) (h2 : b ∣ c) : a ∣ c := by define at h1; define at h2; define obtain (m : Nat) (h3 : b = a * m) from h1 obtain (n : Nat) (h4 : c = b * n) from h2 rewrite [h3, mul_assoc] at h4 apply Exists.intro (m * n) show c = a * (m * n) from h4 done lemma exists_prime_factor : ∀ (n : Nat), 2 ≤ n → ∃ (p : Nat), prime_factor p n := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, 2 ≤ n_1 → ∃ (p : Nat), prime_factor p n_1 assume h1 : 2 ≤ n by_cases h2 : prime n · -- Case 1. h2 : prime n apply Exists.intro n define --Goal : prime n ∧ n ∣ n show prime n ∧ n ∣ n from And.intro h2 (dvd_self n) done · -- Case 2. h2 : ¬prime n define at h2 --h2 : ¬(2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n) demorgan at h2 disj_syll h2 h1 obtain (a : Nat) (h3 : ∃ (b : Nat), a * b = n ∧ a < n ∧ b < n) from h2 obtain (b : Nat) (h4 : a * b = n ∧ a < n ∧ b < n) from h3 have h5 : 2 ≤ a := by by_contra h6 have h7 : a ≤ 1 := by linarith have h8 : n ≤ b := calc n _ = a * b := h4.left.symm _ ≤ 1 * b := by rel [h7] _ = b := by ring linarith --n ≤ b contradicts b < n done have h6 : ∃ (p : Nat), prime_factor p a := ih a h4.right.left h5 obtain (p : Nat) (h7 : prime_factor p a) from h6 apply Exists.intro p define --Goal : prime p ∧ p ∣ n define at h7 --h7 : prime p ∧ p ∣ a apply And.intro h7.left have h8 : a ∣ n := by apply Exists.intro b show n = a * b from (h4.left).symm done show p ∣ n from dvd_trans h7.right h8 done done lemma exists_least_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q := by set S : Set Nat := {p : Nat | prime_factor p n} have h2 : ∃ (p : Nat), p ∈ S := exists_prime_factor n h show ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q from well_ord_princ S h2 done lemma all_prime_nil : all_prime [] := by define --Goal : ∀ p ∈ [], prime p fix p : Nat contrapos --Goal : ¬prime p → p ∉ [] assume h1 : ¬prime p show p ∉ [] from List.not_mem_nil p done lemma all_prime_cons (n : Nat) (L : List Nat) : all_prime (n :: L) ↔ prime n ∧ all_prime L := by apply Iff.intro · -- (→) assume h1 : all_prime (n :: L) --Goal : prime n ∧ all_prime L define at h1 --h1 : ∀ p ∈ n :: L, prime p apply And.intro (h1 n (List.mem_cons_self n L)) define --Goal : ∀ p ∈ L, prime p fix p : Nat assume h2 : p ∈ L show prime p from h1 p (List.mem_cons_of_mem n h2) done · -- (←) assume h1 : prime n ∧ all_prime L --Goal : all_prime (n :: l) define : all_prime L at h1 define fix p : Nat assume h2 : p ∈ n :: L rewrite [List.mem_cons] at h2 --h2 : p = n ∨ p ∈ L by_cases on h2 · -- Case 1. h2 : p = n rewrite [h2] show prime n from h1.left done · -- Case 2. h2 : p ∈ L show prime p from h1.right p h2 done done done lemma nondec_nil : nondec [] := by define --Goal : True trivial --trivial proves some obviously true statements, such as True done lemma nondec_cons (n : Nat) (L : List Nat) : nondec (n :: L) ↔ (∀ m ∈ L, n ≤ m) ∧ nondec L := by rfl lemma prod_nil : prod [] = 1 := by rfl lemma prod_cons : prod (n :: L) = n * (prod L) := by rfl lemma exists_cons_of_length_eq_succ {A : Type} {l : List A} {n : Nat} (h : l.length = n + 1) : ∃ (a : A) (L : List A), l = a :: L ∧ L.length = n := by have h1 : ¬l.length = 0 := by linarith rewrite [List.length_eq_zero] at h1 obtain (a : A) (h2 : ∃ (L : List A), l = a :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List A) (h3 : l = a :: L) from h2 apply Exists.intro a apply Exists.intro L apply And.intro h3 have h4 : (a :: L).length = L.length + 1 := List.length_cons a L rewrite [←h3, h] at h4 show L.length = n from (Nat.add_right_cancel h4).symm done lemma list_elt_dvd_prod_by_length (a : Nat) : ∀ (n : Nat), ∀ (l : List Nat), l.length = n → a ∈ l → a ∣ prod l := by by_induc · --Base Case fix l : List Nat assume h1 : l.length = 0 rewrite [List.length_eq_zero] at h1 --h1 : l = [] rewrite [h1] --Goal : a ∈ [] → a ∣ prod [] contrapos assume h2 : ¬a ∣ prod [] show a ∉ [] from List.not_mem_nil a done · -- Induction Step fix n : Nat assume ih : ∀ (l : List Nat), List.length l = n → a ∈ l → a ∣ prod l fix l : List Nat assume h1 : l.length = n + 1 --Goal : a ∈ l → a ∣ prod l obtain (b : Nat) (h2 : ∃ (L : List Nat), l = b :: L ∧ L.length = n) from exists_cons_of_length_eq_succ h1 obtain (L : List Nat) (h3 : l = b :: L ∧ L.length = n) from h2 have h4 : a ∈ L → a ∣ prod L := ih L h3.right assume h5 : a ∈ l rewrite [h3.left, prod_cons] --Goal : a ∣ b * prod L rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b ∨ a ∈ L by_cases on h5 · -- Case 1. h5 : a = b apply Exists.intro (prod L) rewrite [h5] rfl done · -- Case 2. h5 : a ∈ L have h6 : a ∣ prod L := h4 h5 have h7 : prod L ∣ b * prod L := by apply Exists.intro b ring done show a ∣ b * prod L from dvd_trans h6 h7 done done done lemma list_elt_dvd_prod {a : Nat} {l : List Nat} (h : a ∈ l) : a ∣ prod l := by set n : Nat := l.length have h1 : l.length = n := by rfl show a ∣ prod l from list_elt_dvd_prod_by_length a n l h1 h done lemma exists_prime_factorization : ∀ (n : Nat), n ≥ 1 → ∃ (l : List Nat), prime_factorization n l := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, n_1 ≥ 1 → ∃ (l : List Nat), prime_factorization n_1 l assume h1 : n ≥ 1 by_cases h2 : n = 1 · -- Case 1. h2 : n = 1 apply Exists.intro [] define apply And.intro · -- Proof of nondec_prime_list [] define show all_prime [] ∧ nondec [] from And.intro all_prime_nil nondec_nil done · -- Proof of prod [] = n rewrite [prod_nil, h2] rfl done done · -- Case 2. h2 : n ≠ 1 have h3 : n ≥ 2 := lt_of_le_of_ne' h1 h2 obtain (p : Nat) (h4 : prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q) from exists_least_prime_factor h3 have p_prime_factor : prime_factor p n := h4.left define at p_prime_factor have p_prime : prime p := p_prime_factor.left have p_dvd_n : p ∣ n := p_prime_factor.right have p_least : ∀ (q : Nat), prime_factor q n → p ≤ q := h4.right obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n have h5 : m ≠ 0 := by contradict h1 with h6 have h7 : n = 0 := calc n _ = p * m := n_eq_pm _ = p * 0 := by rw [h6] _ = 0 := by ring rewrite [h7] decide done have m_pos : 0 < m := Nat.pos_of_ne_zero h5 have m_lt_n : m < n := by define at p_prime show m < n from calc m _ < m + m := by linarith _ = 2 * m := by ring _ ≤ p * m := by rel [p_prime.left] _ = n := n_eq_pm.symm done obtain (L : List Nat) (h6 : prime_factorization m L) from ih m m_lt_n m_pos define at h6 have ndpl_L : nondec_prime_list L := h6.left define at ndpl_L apply Exists.intro (p :: L) define apply And.intro · -- Proof of nondec_prime_list (p :: L) define apply And.intro · -- Proof of all_prime (p :: L) rewrite [all_prime_cons] show prime p ∧ all_prime L from And.intro p_prime ndpl_L.left done · -- Proof of nondec (p :: L) rewrite [nondec_cons] apply And.intro _ ndpl_L.right fix q : Nat assume q_in_L : q ∈ L have h7 : q ∣ prod L := list_elt_dvd_prod q_in_L rewrite [h6.right] at h7 --h7 : q ∣ m have h8 : m ∣ n := by apply Exists.intro p rewrite [n_eq_pm] ring done have q_dvd_n : q ∣ n := dvd_trans h7 h8 have ap_L : all_prime L := ndpl_L.left define at ap_L have q_prime_factor : prime_factor q n := And.intro (ap_L q q_in_L) q_dvd_n show p ≤ q from p_least q q_prime_factor done done · -- Proof of prod (p :: L) = n rewrite [prod_cons, h6.right, n_eq_pm] rfl done done done theorem Theorem_7_2_2 {a b c : Nat} (h1 : c ∣ a * b) (h2 : rel_prime a c) : c ∣ b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑c ∣ ↑b define at h1; define at h2; define obtain (j : Nat) (h3 : a * b = c * j) from h1 set s : Int := gcd_c1 a c set t : Int := gcd_c2 a c have h4 : s * ↑a + t * ↑c = ↑(gcd a c) := gcd_lin_comb c a rewrite [h2, Nat.cast_one] at h4 --h4 : s * ↑a + t * ↑c = (1 : Int) apply Exists.intro (s * ↑j + t * ↑b) show ↑b = ↑c * (s * ↑j + t * ↑b) from calc ↑b _ = (1 : Int) * ↑b := (one_mul _).symm _ = (s * ↑a + t * ↑c) * ↑b := by rw [h4] _ = s * (↑a * ↑b) + t * ↑c * ↑b := by ring _ = s * (↑c * ↑j) + t * ↑c * ↑b := by rw [←Nat.cast_mul a b, h3, Nat.cast_mul c j] _ = ↑c * (s * ↑j + t * ↑b) := by ring done lemma le_nonzero_prod_left {a b : Nat} (h : a * b ≠ 0) : a ≤ a * b := by have h1 : b ≠ 0 := by contradict h with h1 rewrite [h1] ring done have h2 : 1 ≤ b := Nat.pos_of_ne_zero h1 show a ≤ a * b from calc a = a * 1 := (mul_one a).symm _ ≤ a * b := by rel [h2] done lemma le_nonzero_prod_right {a b : Nat} (h : a * b ≠ 0) : b ≤ a * b := by rewrite [mul_comm] rewrite [mul_comm] at h show b ≤ b * a from le_nonzero_prod_left h done lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry lemma rel_prime_of_prime_not_dvd {a p : Nat} (h1 : prime p) (h2 : ¬p ∣ a) : rel_prime a p := by have h3 : gcd a p ∣ a := gcd_dvd_left a p have h4 : gcd a p ∣ p := gcd_dvd_right a p have h5 : gcd a p = 1 ∨ gcd a p = p := dvd_prime h1 h4 have h6 : gcd a p ≠ p := by contradict h2 with h6 rewrite [h6] at h3 show p ∣ a from h3 done disj_syll h5 h6 show rel_prime a p from h5 done theorem Theorem_7_2_3 {a b p : Nat} (h1 : prime p) (h2 : p ∣ a * b) : p ∣ a ∨ p ∣ b := by or_right with h3 have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3 show p ∣ b from Theorem_7_2_2 h2 h4 done lemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a ≥ 1 := by have h1 : a ≠ 0 := by by_contra h1 rewrite [h1] at h contradict h linarith done show a ≥ 1 from Nat.pos_of_ne_zero h1 done lemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by have h1 : a ≥ 1 := ge_one_of_prod_one h have h2 : a * b ≠ 0 := by linarith have h3 : a ≤ a * b := le_nonzero_prod_left h2 rewrite [h] at h3 show a = 1 from Nat.le_antisymm h3 h1 done lemma eq_one_of_dvd_one {n : Nat} (h : n ∣ 1) : n = 1 := by obtain (j : Nat) (h1 : 1 = n * j) from h show n = 1 from eq_one_of_prod_one h1.symm done lemma prime_not_one {p : Nat} (h : prime p) : p ≠ 1 := by define at h linarith done theorem Theorem_7_2_4 {p : Nat} (h1 : prime p) : ∀ (l : List Nat), p ∣ prod l → ∃ a ∈ l, p ∣ a := by apply List.rec · -- Base Case. Goal : p ∣ prod [] → ∃ a ∈ [], p ∣ a rewrite [prod_nil] assume h2 : p ∣ 1 show ∃ a ∈ [], p ∣ a from absurd (eq_one_of_dvd_one h2) (prime_not_one h1) done · -- Induction Step fix b : Nat fix L : List Nat assume ih : p ∣ prod L → ∃ a ∈ L, p ∣ a --Goal : p ∣ prod (b :: L) → ∃ a ∈ b :: L, p ∣ a assume h2 : p ∣ prod (b :: L) rewrite [prod_cons] at h2 have h3 : p ∣ b ∨ p ∣ prod L := Theorem_7_2_3 h1 h2 by_cases on h3 · -- Case 1. h3 : p ∣ b apply Exists.intro b show b ∈ b :: L ∧ p ∣ b from And.intro (List.mem_cons_self b L) h3 done · -- Case 2. h3 : p ∣ prod L obtain (a : Nat) (h4 : a ∈ L ∧ p ∣ a) from ih h3 apply Exists.intro a show a ∈ b :: L ∧ p ∣ a from And.intro (List.mem_cons_of_mem b h4.left) h4.right done done done lemma prime_in_list {p : Nat} {l : List Nat} (h1 : prime p) (h2 : all_prime l) (h3 : p ∣ prod l) : p ∈ l := by obtain (a : Nat) (h4 : a ∈ l ∧ p ∣ a) from Theorem_7_2_4 h1 l h3 define at h2 have h5 : prime a := h2 a h4.left have h6 : p = 1 ∨ p = a := dvd_prime h5 h4.right disj_syll h6 (prime_not_one h1) rewrite [h6] show a ∈ l from h4.left done lemma first_le_first {p q : Nat} {l m : List Nat} (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m)) (h3 : prod (p :: l) = prod (q :: m)) : p ≤ q := by define at h1; define at h2 have h4 : q ∣ prod (p :: l) := by define apply Exists.intro (prod m) rewrite [←prod_cons] show prod (p :: l) = prod (q :: m) from h3 done have h5 : all_prime (q :: m) := h2.left rewrite [all_prime_cons] at h5 have h6 : q ∈ p :: l := prime_in_list h5.left h1.left h4 have h7 : nondec (p :: l) := h1.right rewrite [nondec_cons] at h7 rewrite [List.mem_cons] at h6 by_cases on h6 · -- Case 1. h6 : q = p linarith done · -- Case 2. h6 : q ∈ l have h8 : ∀ m ∈ l, p ≤ m := h7.left show p ≤ q from h8 q h6 done done lemma nondec_prime_list_tail {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by define at h define rewrite [all_prime_cons, nondec_cons] at h show all_prime l ∧ nondec l from And.intro h.left.right h.right.right done lemma cons_prod_not_one {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : prod (p :: l) ≠ 1 := by define at h have h1 : all_prime (p :: l) := h.left rewrite [all_prime_cons] at h1 rewrite [prod_cons] by_contra h2 show False from (prime_not_one h1.left) (eq_one_of_prod_one h2) done lemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) : l = [] ↔ prod l = 1 := by apply Iff.intro · -- (→) assume h1 : l = [] rewrite [h1] show prod [] = 1 from prod_nil done · -- (←) contrapos assume h1 : ¬l = [] obtain (p : Nat) (h2 : ∃ (L : List Nat), l = p :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List Nat) (h3 : l = p :: L) from h2 rewrite [h3] at h rewrite [h3] show ¬prod (p :: L) = 1 from cons_prod_not_one h done done lemma prime_pos {p : Nat} (h : prime p) : p > 0 := by define at h linarith done theorem Theorem_7_2_5 : ∀ (l1 l2 : List Nat), nondec_prime_list l1 → nondec_prime_list l2 → prod l1 = prod l2 → l1 = l2 := by apply List.rec · -- Base Case. Goal : ∀ (l2 : List Nat), nondec_prime_list [] → -- nondec_prime_list l2 → prod [] = prod l2 → [] = l2 fix l2 : List Nat assume h1 : nondec_prime_list [] assume h2 : nondec_prime_list l2 assume h3 : prod [] = prod l2 rewrite [prod_nil, eq_comm, ←list_nil_iff_prod_one h2] at h3 show [] = l2 from h3.symm done · -- Induction Step fix p : Nat fix L1 : List Nat assume ih : ∀ (L2 : List Nat), nondec_prime_list L1 → nondec_prime_list L2 → prod L1 = prod L2 → L1 = L2 -- Goal : ∀ (l2 : List Nat), nondec_prime_list (p :: L1) → -- nondec_prime_list l2 → prod (p :: L1) = prod l2 → p :: L1 = l2 fix l2 : List Nat assume h1 : nondec_prime_list (p :: L1) assume h2 : nondec_prime_list l2 assume h3 : prod (p :: L1) = prod l2 have h4 : ¬prod (p :: L1) = 1 := cons_prod_not_one h1 rewrite [h3, ←list_nil_iff_prod_one h2] at h4 obtain (q : Nat) (h5 : ∃ (L : List Nat), l2 = q :: L) from List.exists_cons_of_ne_nil h4 obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5 rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2) rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2) have h7 : p ≤ q := first_le_first h1 h2 h3 have h8 : q ≤ p := first_le_first h2 h1 h3.symm have h9 : p = q := by linarith rewrite [h9, prod_cons, prod_cons] at h3 --h3 : q * prod L1 = q * prod L2 have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1 have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2 define at h2 have h12 : all_prime (q :: L2) := h2.left rewrite [all_prime_cons] at h12 have h13 : q > 0 := prime_pos h12.left have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3 have h15 : L1 = L2 := ih L2 h10 h11 h14 rewrite [h6, h9, h15] rfl done done theorem fund_thm_arith (n : Nat) (h : n ≥ 1) : ∃! (l : List Nat), prime_factorization n l := by exists_unique · -- Existence show ∃ (l : List Nat), prime_factorization n l from exists_prime_factorization n h done · -- Uniqueness fix l1 : List Nat; fix l2 : List Nat assume h1 : prime_factorization n l1 assume h2 : prime_factorization n l2 define at h1; define at h2 have h3 : prod l1 = n := h1.right rewrite [←h2.right] at h3 show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3 done done /- Section 7.3 -/ theorem congr_refl (m : Nat) : ∀ (a : Int), a ≡ a (MOD m) := by fix a : Int define --Goal : ∃ (c : Int), a - a = ↑m * c apply Exists.intro 0 ring done theorem congr_symm {m : Nat} : ∀ {a b : Int}, a ≡ b (MOD m) → b ≡ a (MOD m) := by fix a : Int; fix b : Int assume h1 : a ≡ b (MOD m) define at h1 --h1 : ∃ (c : Int), a - b = ↑m * c define --Goal : ∃ (c : Int), b - a = ↑m * c obtain (c : Int) (h2 : a - b = m * c) from h1 apply Exists.intro (-c) show b - a = m * (-c) from calc b - a _ = -(a - b) := by ring _ = -(m * c) := by rw [h2] _ = m * (-c) := by ring done theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry /- Fundamental properties of congruence classes -/ lemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) : X = Y ↔ X.val = Y.val := Fin.ext_iff lemma val_nat_eq_mod (n k : Nat) : ([k]_(n + 1)).val = k % (n + 1) := by rfl lemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl theorem cc_rep {m : Nat} (X : ZMod m) : ∃ (a : Int), X = [a]_m := match m with | 0 => by apply Exists.intro X rfl done | n + 1 => by apply Exists.intro ↑(X.val) have h1 : X.val < n + 1 := Fin.prop X rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1] rfl done theorem add_class (m : Nat) (a b : Int) : [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm theorem mul_class (m : Nat) (a b : Int) : [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm lemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ [a - b]_m = [0]_m := by apply Iff.intro · -- (→) assume h1 : [a]_m = [b]_m have h2 : a - b = a + (-b) := by ring have h3 : b + (-b) = 0 := by ring show [a - b]_m = [0]_m from calc [a - b]_m _ = [a + (-b)]_m := by rw [h2] _ = [a]_m + [-b]_m := by rw [add_class] _ = [b]_m + [-b]_m := by rw [h1] _ = [b + -b]_m := by rw [add_class] _ = [0]_m := by rw [h3] done · -- (←) assume h1 : [a - b]_m = [0]_m have h2 : b + (a - b) = a := by ring have h3 : b + 0 = b := by ring show [a]_m = [b]_m from calc [a]_m _ = [b + (a - b)]_m := by rw [h2] _ = [b]_m + [a - b]_m := by rw [add_class] _ = [b]_m + [0]_m := by rw [h1] _ = [b + 0]_m := by rw [add_class] _ = [b]_m := by rw [h3] done done lemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) : [a]_m = [0]_m → [-a]_m = [0]_m := by assume h1 : [a]_m = [0]_m have h2 : 0 + (-a) = -a := by ring have h3 : a + (-a) = 0 := by ring show [-a]_m = [0]_m from calc [-a]_m _ = [0 + (-a)]_m := by rw [h2] _ = [0]_m + [-a]_m := by rw [add_class] _ = [a]_m + [-a]_m := by rw [h1] _ = [a + (-a)]_m := by rw [add_class] _ = [0]_m := by rw [h3] done lemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) : [-a]_m = [0]_m ↔ [a]_m = [0]_m := by apply Iff.intro _ (cc_neg_zero_of_cc_zero m a) assume h1 : [-a]_m = [0]_m have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1 have h3 : -(-a) = a := by ring rewrite [h3] at h2 show [a]_m = [0]_m from h2 done lemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl lemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m ↔ m ∣ k := match m with | 0 => by have h : (0 : Int) = (↑(0 : Nat) : Int) := by rfl rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj] apply Iff.intro · -- (→) assume h1 : k = 0 rewrite [h1] show 0 ∣ 0 from dvd_self 0 done · -- (←) assume h1 : 0 ∣ k obtain (c : Nat) (h2 : k = 0 * c) from h1 rewrite [h2] ring done done | n + 1 => by rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero] show k % (n + 1) = 0 ↔ n + 1 ∣ k from (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm done lemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m ↔ ↑m ∣ a := by obtain (k : Nat) (h1 : a = ↑k ∨ a = -↑k) from Int.eq_nat_or_neg a by_cases on h1 · -- Case 1. h1: a = ↑k rewrite [h1, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done · -- Case 2. h1: a = -↑k rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done done theorem cc_eq_iff_congr (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ a ≡ b (MOD m) := calc [a]_m = [b]_m _ ↔ [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b _ ↔ ↑m ∣ (a - b) := cc_zero_iff_dvd m (a - b) _ ↔ a ≡ b (MOD m) := by rfl /- End of fundamental properties of congruence classes -/ lemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m := by have h1 : (↑m : Int) ≠ 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m) show 0 ≤ a % m from Int.emod_nonneg a h1 done lemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m) have h2 : (↑m : Int) > 0 := (Nat.cast_pos).rtl h1 show a % m < m from Int.emod_lt_of_pos a h2 done lemma congr_mod_mod (m : Nat) (a : Int) : a ≡ a % m (MOD m) := by define have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m apply Exists.intro (a / m) show a - a % m = m * (a / m) from calc a - (a % m) _ = m * (a / m) + a % m - a % m := by rw [h1] _ = m * (a / m) := by ring done lemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) := And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a)) theorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) : ∃! (r : Int), 0 ≤ r ∧ r < m ∧ a ≡ r (MOD m) := by exists_unique · -- Existence apply Exists.intro (a % m) show 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) from mod_cmpl_res m a done · -- Uniqueness fix r1 : Int; fix r2 : Int assume h1 : 0 ≤ r1 ∧ r1 < m ∧ a ≡ r1 (MOD m) assume h2 : 0 ≤ r2 ∧ r2 < m ∧ a ≡ r2 (MOD m) have h3 : r1 ≡ r2 (MOD m) := congr_trans (congr_symm h1.right.right) h2.right.right obtain (d : Int) (h4 : r1 - r2 = m * d) from h3 have h5 : r1 - r2 < m * 1 := by linarith have h6 : m * (-1) < r1 - r2 := by linarith rewrite [h4] at h5 --h5 : m * d < m * 1 rewrite [h4] at h6 --h6 : m * -1 < m * d have h7 : (↑m : Int) ≥ 0 := Nat.cast_nonneg m have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7 have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7 have h10 : d = 0 := by linarith show r1 = r2 from calc r1 _ = r1 - r2 + r2 := by ring _ = m * 0 + r2 := by rw [h4, h10] _ = r2 := by ring done done lemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m := (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a) theorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y rewrite [h1, h2] have h3 : a + b = b + a := by ring show [a]_m + [b]_m = [b]_m + [a]_m from calc [a]_m + [b]_m _ = [a + b]_m := add_class m a b _ = [b + a]_m := by rw [h3] _ = [b]_m + [a]_m := (add_class m b a).symm done theorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X rewrite [h1] have h2 : a * 1 = a := by ring show [a]_m * [1]_m = [a]_m from calc [a]_m * [1]_m _ = [a * 1]_m := mul_class m a 1 _ = [a]_m := by rw [h2] done theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry lemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) : [a]_m * [gcd_c2 m a]_m = [1]_m := by set s : Int := gcd_c1 m a have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m define at h1 rewrite [h1, Nat.cast_one] at h2 --h2 : s * ↑m + gcd_c2 m a * ↑a = 1 rewrite [mul_class, cc_eq_iff_congr] define --Goal : ∃ (c : Int), ↑a * gcd_c2 m a - 1 = ↑m * c apply Exists.intro (-s) show a * (gcd_c2 m a) - 1 = m * (-s) from calc a * (gcd_c2 m a) - 1 _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring _ = 1 + m * (-s) - 1 := by rw [h2] _ = m * (-s) := by ring done theorem Theorem_7_3_7 (m a : Nat) : invertible [a]_m ↔ rel_prime m a := by apply Iff.intro · -- (→) assume h1 : invertible [a]_m define at h1 obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1 obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y rewrite [h3, mul_class, cc_eq_iff_congr] at h2 define at h2 obtain (c : Int) (h4 : a * b - 1 = m * c) from h2 rewrite [Exercise_7_2_6] --Goal : ∃ (s t : Int), s * ↑m + t * ↑a = 1 apply Exists.intro (-c) apply Exists.intro b show (-c) * m + b * a = 1 from calc (-c) * m + b * a _ = (-c) * m + (a * b - 1) + 1 := by ring _ = (-c) * m + m * c + 1 := by rw [h4] _ = 1 := by ring done · -- (←) assume h1 : rel_prime m a define show ∃ (Y : ZMod m), [a]_m * Y = [1]_m from Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1) done done /- Section 7.4 -/ section Euler open Euler lemma num_rp_below_base {m : Nat} : num_rp_below m 0 = 0 := by rfl lemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) : num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by have h1 : num_rp_below m (j + 1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : gcd m j = 1 rewrite [if_pos h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1 show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1 done lemma num_rp_below_step_not_rp {m j : Nat} (h : ¬rel_prime m j) : num_rp_below m (j + 1) = num_rp_below m j := by have h1 : num_rp_below m (j +1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : ¬gcd m j = 1 rewrite [if_neg h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j show num_rp_below m (j + 1) = num_rp_below m j from h1 done lemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl #eval phi 10 --Answer: 4 lemma prod_inv_iff_inv {m : Nat} {X : ZMod m} (h1 : invertible X) (Y : ZMod m) : invertible (X * Y) ↔ invertible Y := by apply Iff.intro · -- (→) assume h2 : invertible (X * Y) obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2 apply Exists.intro (X * Z) rewrite [←h3] --Goal : Y * (X * Z) = X * Y * Z ring --Note that ring can do algebra in ZMod m done · -- (←) assume h2 : invertible Y obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1 obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2 apply Exists.intro (Xi * Yi) show (X * Y) * (Xi * Yi) = [1]_m from calc X * Y * (Xi * Yi) _ = (X * Xi) * (Y * Yi) := by ring _ = [1]_m * [1]_m := by rw [h3, h4] _ = [1]_m := Theorem_7_3_6_7 [1]_m done done lemma F_rp_def {m i : Nat} (h : rel_prime m i) : F m i = [i]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h --h : gcd m i = 1 rewrite [if_pos h] at h1 show F m i = [i]_m from h1 done lemma F_not_rp_def {m i : Nat} (h : ¬rel_prime m i) : F m i = [1]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h rewrite [h1, if_neg h] rfl done lemma prod_seq_base {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 0 k f = [1]_m := by rfl lemma prod_seq_step {m : Nat} (n k : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl lemma prod_seq_zero_step {m : Nat} (n : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by rewrite [prod_seq_step, zero_add] rfl done lemma prod_one {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 1 k f = f k := by rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7] rfl done lemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl lemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m := calc [G m a i]_m _ = [(a * i) % m]_m := by rfl _ = [a * i]_m := (cc_eq_mod m (a * i)).symm _ = [a]_m * [i]_m := (mul_class m a i).symm lemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) : rel_prime m (G m a i) ↔ rel_prime m i := by have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1 show rel_prime m (G m a i) ↔ rel_prime m i from calc rel_prime m (G m a i) _ ↔ invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm _ ↔ invertible ([a]_m * [i]_m) := by rw [cc_G] _ ↔ invertible [i]_m := prod_inv_iff_inv h2 ([i]_m) _ ↔ rel_prime m i := Theorem_7_3_7 m i done lemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) : F m (G m a i) = [a]_m * F m i := by have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2 show F m (G m a i) = [a]_m * F m i from calc F m (G m a i) _ = [G m a i]_m := F_rp_def h3 _ = [a]_m * [i]_m := cc_G m a i _ = [a]_m * F m i := by rw [F_rp_def h2] done lemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : ¬rel_prime m i) : F m (G m a i) = [1]_m := by rewrite [←G_rp_iff h1 i] at h2 show F m (G m a i) = [1]_m from F_not_rp_def h2 done lemma FG_prod {m a : Nat} (h1 : rel_prime m a) : ∀ (k : Nat), prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by by_induc · -- Base Case show prod_seq 0 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from calc prod_seq 0 0 ((F m) ∘ (G m a)) _ = [1]_m := prod_seq_base _ _ _ = [a]_m ^ 0 * [1]_m := by ring _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by rw [num_rp_below_base, prod_seq_base] done · -- Induction Step fix k : Nat assume ih : prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) by_cases h2 : rel_prime m k · -- Case 1. h2 : rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([a]_m * F m k) := by rw [FG_rp h1 h2] _ = [a]_m ^ ((num_rp_below m k) + 1) * ((prod_seq k 0 (F m)) * F m k) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_rp h2, prod_seq_zero_step] done · -- Case 2. h2 : ¬rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([1]_m) := by rw [FG_not_rp h1 h2] _ = [a]_m ^ (num_rp_below m k) * (prod_seq k 0 (F m) * ([1]_m)) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_not_rp h2, prod_seq_zero_step, F_not_rp_def h2] done done done lemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by define --Goal : ∀ i < m, G m a i < m fix i : Nat assume h1 : i < m rewrite [G_def] --Goal : a * i % m < m show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m) done lemma left_inv_one_one_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g' (g i) = i) : one_one_below n g := sorry lemma right_inv_onto_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g (g' i) = i) (h2 : maps_below n g') : onto_below n g := by define at h2; define fix k : Nat assume h3 : k < n apply Exists.intro (g' k) show g' k < n ∧ g (g' k) = k from And.intro (h2 k h3) (h1 k h3) done lemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m * [inv_mod m a]_m = [1]_m := by have h2 : 0 ≤ (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a) show [a]_m * [inv_mod m a]_m = [1]_m from calc [a]_m * [inv_mod m a]_m _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2] _ = [a]_m * [gcd_c2 m a]_m := by rw [←cc_eq_mod] _ = [1]_m := gcd_c2_inv h1 done lemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m := calc a * (b % m) % m = a % m * (b % m % m) % m := Nat.mul_mod _ _ _ _ = a % m * (b % m) % m := by rw [Nat.mod_mod] _ = a * b % m := (Nat.mul_mod _ _ _).symm lemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm] rfl done theorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] : ↑a ≡ ↑b (MOD m) ↔ a % m = b % m := sorry lemma mul_inv_mod_cancel {m a i : Nat} [NeZero m] (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1 rewrite [mul_class, cc_eq_iff_congr, ←Nat.cast_mul, ←Nat.cast_one, congr_iff_mod_eq_Nat] at h3 show a * inv_mod m a * i % m = i from calc a * (inv_mod m a) * i % m _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = 1 % m * i % m := by rw [h3] _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = i % m := by rw [one_mul] _ = i := Nat.mod_eq_of_lt h2 done lemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl lemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, G m a (Ginv m a i) = i := by fix i : Nat assume h2 : i < m show G m a (Ginv m a i) = i from calc G m a (Ginv m a i) _ = a * ((inv_mod m a * i) % m) % m := by rfl _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, Ginv m a (G m a i) = i := by fix i : Nat assume h2 : i < m show Ginv m a (G m a i) = i from calc Ginv m a (G m a i) _ = inv_mod m a * ((a * i) % m) % m := by rfl _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc, mul_comm (inv_mod m a)] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_maps_below (m a : Nat) [NeZero m] : maps_below m (Ginv m a) := G_maps_below m (inv_mod m a) lemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : one_one_below m (G m a) := left_inv_one_one_below (Ginv_left_inv h1) lemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : onto_below m (G m a) := right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a) lemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : perm_below m (G m a) := And.intro (G_maps_below m a) (And.intro (G_one_one_below h1) (G_onto_below h1)) --Permuting a product of congruence classes doesn't change product lemma swap_fst (u v : Nat) : swap u v u = v := by define : swap u v u --Goal : (if u = u then v else if u = v then u else u) = v have h : u = u := by rfl rewrite [if_pos h] rfl done lemma swap_snd (u v : Nat) : swap u v v = u := by define : swap u v v by_cases h1 : v = u · -- Case 1. h1 : v = u rewrite [if_pos h1] show v = u from h1 done · -- Case 2. h1 : v ≠ u rewrite [if_neg h1] have h2 : v = v := by rfl rewrite [if_pos h2] rfl done done lemma swap_other {u v i : Nat} (h1 : i ≠ u) (h2 : i ≠ v) : swap u v i = i := by define : swap u v i rewrite [if_neg h1, if_neg h2] rfl done lemma swap_values (u v i : Nat) : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := by by_cases h1 : i = u · -- Case 1. h1 : i = u apply Or.inl rewrite [h1] show swap u v u = v from swap_fst u v done · -- Case 2. h1 : i ≠ u apply Or.inr by_cases h2 : i = v · -- Case 2.1. h2 : i = v apply Or.inl rewrite [h2] show swap u v v = u from swap_snd u v done · -- Case 2.2. h2 : i ≠ v apply Or.inr show swap u v i = i from swap_other h1 h2 done done done lemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by define fix i : Nat assume h3 : i < n have h4 : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := swap_values u v i by_cases on h4 · -- Case 1. h4 : swap u v i = v rewrite [h4] show v < n from h2 done · -- Case 2. by_cases on h4 · -- Case 2.1. h4 : swap u v i = u rewrite [h4] show u < n from h1 done · -- Case 2.2. h4 : swap u v i = i rewrite [h4] show i < n from h3 done done done lemma swap_swap (u v n : Nat) : ∀ i < n, swap u v (swap u v i) = i := by fix i : Nat assume h : i < n by_cases h1 : i = u · -- Case 1. h1 : i = u rewrite [h1, swap_fst, swap_snd] rfl done · -- Case 2. h1 : i ≠ u by_cases h2 : i = v · -- Case 2.1. h2 : i = v rewrite [h2, swap_snd, swap_fst] rfl done · -- Case 2.2. h2 : i ≠ v rewrite [swap_other h1 h2, swap_other h1 h2] rfl done done done lemma swap_one_one_below (u v n) : one_one_below n (swap u v) := left_inv_one_one_below (swap_swap u v n) lemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) := right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2) lemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) := And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2)) lemma comp_perm_below {n : Nat} {f g : Nat → Nat} (h1 : perm_below n f) (h2 : perm_below n g) : perm_below n (f ∘ g) := sorry lemma trivial_swap (u : Nat) : swap u u = id := by apply funext fix x : Nat by_cases h1 : x = u · -- Case 1. h1 : x = u rewrite [h1, swap_fst] rfl done · -- Case 2. h1 : x ≠ u rewrite [swap_other h1 h1] rfl done done lemma prod_eq_fun {m : Nat} (f g : Nat → ZMod m) (k : Nat) : ∀ (n : Nat), (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g := by by_induc · -- Base Case assume h : (∀ i < 0, f (k + i) = g (k + i)) rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g assume h1 : ∀ i < n + 1, f (k + i) = g (k + i) have h2 : ∀ i < n, f (k + i) = g (k + i) := by fix i : Nat assume h2 : i < n have h3 : i < n + 1 := by linarith show f (k + i) = g (k + i) from h1 i h3 done have h3 : prod_seq n k f = prod_seq n k g := ih h2 have h4 : n < n + 1 := Nat.lt_succ_self n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4] rfl done done lemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := by have h2 : ∀ (i : Nat), i < u → (f ∘ swap u n) (0 + i) = f (0 + i) := by fix i : Nat assume h2 : i < u have h3 : 0 + i ≠ u := by linarith have h4 : 0 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl done show prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f from prod_eq_fun (f ∘ swap u n) f 0 u h2 done lemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := by have h2 : ∀ i < j, (f ∘ swap u n) (u + 1 + i) = f (u + 1 + i) := by fix i : Nat assume h2 : i < j have h3 : u + 1 + i ≠ u := by linarith have h4 : u + 1 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl show prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f from prod_eq_fun (f ∘ swap u n) f (u + 1) j h2 done lemma break_prod {m : Nat} (n : Nat) (f : Nat → ZMod m) : ∀ (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by by_induc · -- Base Case have h : n + 0 = n := by rfl rewrite [prod_seq_base, h, Theorem_7_3_6_7] rfl done · -- Induction Step fix j : Nat assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f rewrite [←add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc] rfl done done lemma break_prod_twice {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f := break_prod n f 1 rewrite [prod_one] at h2 have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f := break_prod (u + 1) f j rewrite [←h1] at h3 have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f := break_prod u f 1 rewrite [prod_one] at h4 rewrite [h3, h4] at h2 show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2 done lemma swap_prod_eq_prod {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f := by by_cases h2 : u = n · -- Case 1. h2 : u = n rewrite [h2, trivial_swap n] --Goal : prod_seq (n + 1) 0 (f ∘ id) = prod_seq (n + 1) 0 f rfl done · -- Case 2. h2 : ¬u = n have h3 : u + 1 ≤ n := Nat.lt_of_le_of_ne h1 h2 obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3 have break_f : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := break_prod_twice f h4 have break_fs : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_prod_twice (f ∘ swap u n) h4 have f_eq_fs_below : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := swap_prod_eq_prod_below f h1 have f_eq_fs_btwn : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4 show prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f from calc prod_seq (n + 1) 0 (f ∘ swap u n) _ = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_fs _ = prod_seq u 0 f * (f ∘ swap u n) u * prod_seq j (u + 1) f * (f ∘ swap u n) n := by rw [f_eq_fs_below, f_eq_fs_btwn] _ = prod_seq u 0 f * f (swap u n u) * prod_seq j (u + 1) f * f (swap u n n) := by rfl _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by rw [swap_fst, swap_snd] _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring _ = prod_seq (n + 1) 0 f := break_f.symm done done lemma perm_below_fixed {n : Nat} {g : Nat → Nat} (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry lemma perm_prod {m : Nat} (f : Nat → ZMod m) : ∀ (n : Nat), ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) := by by_induc · -- Base Case fix g : Nat → Nat assume h1 : perm_below 0 g rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) fix g : Nat → Nat assume g_pb : perm_below (n + 1) g define at g_pb have g_ob : onto_below (n + 1) g := g_pb.right.right define at g_ob have h1 : n < n + 1 := by linarith obtain (u : Nat) (h2 : u < n + 1 ∧ g u = n) from g_ob n h1 have s_pb : perm_below (n + 1) (swap u n) := swap_perm_below h2.left h1 have gs_pb_n1 : perm_below (n + 1) (g ∘ swap u n) := comp_perm_below g_pb s_pb have gs_fix_n : (g ∘ swap u n) n = n := calc (g ∘ swap u n) n _ = g (swap u n n) := by rfl _ = g u := by rw [swap_snd] _ = n := h2.right have gs_pb_n : perm_below n (g ∘ swap u n) := perm_below_fixed gs_pb_n1 gs_fix_n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f ∘ (g ∘ swap u n)) := ih (g ∘ swap u n) gs_pb_n have h3 : u ≤ n := by linarith show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f ∘ g) from calc prod_seq (n + 1) 0 f _ = prod_seq n 0 f * f n := prod_seq_zero_step n f _ = prod_seq n 0 (f ∘ (g ∘ swap u n)) * f ((g ∘ swap u n) n) := by rw [gs_prod, gs_fix_n] _ = prod_seq n 0 (f ∘ g ∘ swap u n) * (f ∘ g ∘ swap u n) n := by rfl _ = prod_seq (n + 1) 0 (f ∘ g ∘ swap u n) := (prod_seq_zero_step n (f ∘ g ∘ swap u n)).symm _ = prod_seq (n + 1) 0 ((f ∘ g) ∘ swap u n) := by rfl _ = prod_seq (n + 1) 0 (f ∘ g) := swap_prod_eq_prod (f ∘ g) h3 done done lemma F_invertible (m i : Nat) : invertible (F m i) := by by_cases h : rel_prime m i · -- Case 1. h : rel_prime m i rewrite [F_rp_def h] show invertible [i]_m from (Theorem_7_3_7 m i).rtl h done · -- Case 2. h : ¬rel_prime m i rewrite [F_not_rp_def h] apply Exists.intro [1]_m show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m done done lemma Fprod_invertible (m : Nat) : ∀ (k : Nat), invertible (prod_seq k 0 (F m)) := by by_induc · -- Base Case apply Exists.intro [1]_m show prod_seq 0 0 (F m) * [1]_m = [1]_m from calc prod_seq 0 0 (F m) * [1]_m _ = [1]_m * [1]_m := by rw [prod_seq_base] _ = [1]_m := Theorem_7_3_6_7 ([1]_m) done · -- Induction Step fix k : Nat assume ih : invertible (prod_seq k 0 (F m)) rewrite [prod_seq_zero_step] show invertible (prod_seq k 0 (F m) * (F m k)) from (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k) done done theorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m ^ (phi m) = [1]_m := by have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2 show [a]_m ^ (phi m) = [1]_m from calc [a]_m ^ (phi m) _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3] _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring _ = prod_seq m 0 (F m ∘ G m a) * Y := by rw [FG_prod h1 m, phi_def] _ = prod_seq m 0 (F m) * Y := by rw [perm_prod (F m) m (G m a) (G_perm_below h1)] _ = [1]_m := by rw [h3] done lemma Exercise_7_4_5_Int (m : Nat) (a : Int) : ∀ (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry lemma Exercise_7_4_5_Nat (m a n : Nat) : [a]_m ^ n = [a ^ n]_m := by rewrite [Exercise_7_4_5_Int] rfl done theorem Euler's_theorem {m a : Nat} [NeZero m] (h1 : rel_prime m a) : a ^ (phi m) ≡ 1 (MOD m) := by have h2 : [a]_m ^ (phi m) = [1]_m := Theorem_7_4_2 h1 rewrite [Exercise_7_4_5_Nat m a (phi m)] at h2 --h2 : [a ^ phi m]_m = [1]_m show a ^ (phi m) ≡ 1 (MOD m) from (cc_eq_iff_congr _ _ _).ltr h2 done #eval gcd 10 7 --Answer: 1. So 10 and 7 are relatively prime #eval 7 ^ phi 10 --Answer: 2401, which is congruent to 1 mod 10. end Euler /- Section 7.5 -/ lemma num_rp_prime {p : Nat} (h1 : prime p) : ∀ k < p, num_rp_below p (k + 1) = k := sorry lemma phi_prime {p : Nat} (h1 : prime p) : phi p = p - 1 := by have h2 : 1 ≤ p := prime_pos h1 have h3 : p - 1 + 1 = p := Nat.sub_add_cancel h2 have h4 : p - 1 < p := by linarith have h5 : num_rp_below p (p - 1 + 1) = p - 1 := num_rp_prime h1 (p - 1) h4 rewrite [h3] at h5 show phi p = p - 1 from h5 done theorem Theorem_7_2_2_Int {a c : Nat} {b : Int} (h1 : ↑c ∣ ↑a * b) (h2 : rel_prime a c) : ↑c ∣ b := by rewrite [Int.natCast_dvd, Int.natAbs_mul, Int.natAbs_ofNat] at h1 --h1 : c ∣ a * Int.natAbs b rewrite [Int.natCast_dvd] --Goal : c ∣ Int.natAbs b show c ∣ Int.natAbs b from Theorem_7_2_2 h1 h2 done lemma Lemma_7_4_5 {m n : Nat} (a b : Int) (h1 : rel_prime m n) : a ≡ b (MOD m * n) ↔ a ≡ b (MOD m) ∧ a ≡ b (MOD n) := by apply Iff.intro · -- (→) assume h2 : a ≡ b (MOD m * n) obtain (j : Int) (h3 : a - b = (m * n) * j) from h2 apply And.intro · -- Proof of a ≡ b (MOD m) apply Exists.intro (n * j) show a - b = m * (n * j) from calc a - b _ = m * n * j := h3 _ = m * (n * j) := by ring done · -- Proof of a ≡ b (MOD n) apply Exists.intro (m * j) show a - b = n * (m * j) from calc a - b _ = m * n * j := h3 _ = n * (m * j) := by ring done done · -- (←) assume h2 : a ≡ b (MOD m) ∧ a ≡ b (MOD n) obtain (j : Int) (h3 : a - b = m * j) from h2.left have h4 : (↑n : Int) ∣ a - b := h2.right rewrite [h3] at h4 --h4 : ↑n ∣ ↑m * j have h5 : ↑n ∣ j := Theorem_7_2_2_Int h4 h1 obtain (k : Int) (h6 : j = n * k) from h5 apply Exists.intro k --Goal : a - b = ↑(m * n) * k rewrite [Nat.cast_mul] --Goal : a - b = ↑m * ↑n * k show a - b = (m * n) * k from calc a - b _ = m * j := h3 _ = m * (n * k) := by rw [h6] _ = (m * n) * k := by ring done done --From exercises of Section 7.2 theorem rel_prime_symm {a b : Nat} (h : rel_prime a b) : rel_prime b a := sorry lemma prime_NeZero {p : Nat} (h : prime p) : NeZero p := by rewrite [neZero_iff] --Goal : p ≠ 0 define at h linarith done lemma Lemma_7_5_1 {p e d m c s : Nat} {t : Int} (h1 : prime p) (h2 : e * d = (p - 1) * s + 1) (h3 : m ^ e - c = p * t) : c ^ d ≡ m (MOD p) := by have h4 : m ^ e ≡ c (MOD p) := Exists.intro t h3 have h5 : [m ^ e]_p = [c]_p := (cc_eq_iff_congr _ _ _).rtl h4 rewrite [←Exercise_7_4_5_Nat] at h5 --h5 : [m]_p ^ e = [c]_p by_cases h6 : p ∣ m · -- Case 1. h6 : p ∣ m have h7 : m ≡ 0 (MOD p) := by obtain (j : Nat) (h8 : m = p * j) from h6 apply Exists.intro (↑j : Int) --Goal : ↑m - 0 = ↑p * ↑j rewrite [h8, Nat.cast_mul] ring done have h8 : [m]_p = [0]_p := (cc_eq_iff_congr _ _ _).rtl h7 have h9 : e * d ≠ 0 := by rewrite [h2] show (p - 1) * s + 1 ≠ 0 from Nat.add_one_ne_zero _ done have h10 : (0 : Int) ^ (e * d) = 0 := zero_pow h9 have h11 : [c ^ d]_p = [m]_p := calc [c ^ d]_p _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat] _ = ([m]_p ^ e) ^ d := by rw [h5] _ = [m]_p ^ (e * d) := by ring _ = [0]_p ^ (e * d) := by rw [h8] _ = [0 ^ (e * d)]_p := Exercise_7_4_5_Int _ _ _ _ = [0]_p := by rw [h10] _ = [m]_p := by rw [h8] show c ^ d ≡ m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11 done · -- Case 2. h6 : ¬p ∣ m have h7 : rel_prime m p := rel_prime_of_prime_not_dvd h1 h6 have h8 : rel_prime p m := rel_prime_symm h7 have h9 : NeZero p := prime_NeZero h1 have h10 : (1 : Int) ^ s = 1 := by ring have h11 : [c ^ d]_p = [m]_p := calc [c ^ d]_p _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat] _ = ([m]_p ^ e) ^ d := by rw [h5] _ = [m]_p ^ (e * d) := by ring _ = [m]_p ^ ((p - 1) * s + 1) := by rw [h2] _ = ([m]_p ^ (p - 1)) ^ s * [m]_p := by ring _ = ([m]_p ^ (phi p)) ^ s * [m]_p := by rw [phi_prime h1] _ = [1]_p ^ s * [m]_p := by rw [Theorem_7_4_2 h8] _ = [1 ^ s]_p * [m]_p := by rw [Exercise_7_4_5_Int] _ = [1]_p * [m]_p := by rw [h10] _ = [m]_p * [1]_p := by ring _ = [m]_p := Theorem_7_3_6_7 _ show c ^ d ≡ m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11 done done theorem Theorem_7_5_1 (p q n e d k m c : Nat) (p_prime : prime p) (q_prime : prime q) (p_ne_q : p ≠ q) (n_pq : n = p * q) (ed_congr_1 : e * d = k * (p - 1) * (q - 1) + 1) (h1 : [m]_n ^ e = [c]_n) : [c]_n ^ d = [m]_n := by rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] at h1 --h1 : m ^ e ≡ c (MOD n) rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] --Goal : c ^ d ≡ m (MOD n) obtain (j : Int) (h2 : m ^ e - c = n * j) from h1 rewrite [n_pq, Nat.cast_mul] at h2 --h2 : m ^ e - c = p * q * j have h3 : e * d = (p - 1) * (k * (q - 1)) + 1 := by rewrite [ed_congr_1] ring done have h4 : m ^ e - c = p * (q * j) := by rewrite [h2] ring done have congr_p : c ^ d ≡ m (MOD p) := Lemma_7_5_1 p_prime h3 h4 have h5 : e * d = (q - 1) * (k * (p - 1)) + 1 := by rewrite [ed_congr_1] ring done have h6 : m ^ e - c = q * (p * j) := by rewrite [h2] ring done have congr_q : c ^ d ≡ m (MOD q) := Lemma_7_5_1 q_prime h5 h6 have h7 : ¬q ∣ p := by by_contra h8 have h9 : q = 1 ∨ q = p := dvd_prime p_prime h8 disj_syll h9 (prime_not_one q_prime) show False from p_ne_q h9.symm done have h8 : rel_prime p q := rel_prime_of_prime_not_dvd q_prime h7 rewrite [n_pq, Lemma_7_4_5 _ _ h8] show c ^ d ≡ m (MOD p) ∧ c ^ d ≡ m (MOD q) from And.intro congr_p congr_q done /- BEGIN EXERCISES -/ namespace Exercises /- Section 7.1 -/ -- 1. theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry -- 2. lemma gcd_comm_lt {a b : Nat} (h : a < b) : gcd a b = gcd b a := sorry theorem gcd_comm (a b : Nat) : gcd a b = gcd b a := sorry -- 3. theorem Exercise_7_1_5 (a b : Nat) (n : Int) : (∃ (s t : Int), s * a + t * b = n) ↔ (↑(gcd a b) : Int) ∣ n := sorry -- 4. theorem Exercise_7_1_6 (a b c : Nat) : gcd a b = gcd (a + b * c) b := sorry -- 5. theorem gcd_is_nonzero {a b : Nat} (h : a ≠ 0 ∨ b ≠ 0) : gcd a b ≠ 0 := sorry -- 6. theorem gcd_greatest {a b d : Nat} (h1 : gcd a b ≠ 0) (h2 : d ∣ a) (h3 : d ∣ b) : d ≤ gcd a b := sorry -- 7. lemma Lemma_7_1_10a {a b : Nat} (n : Nat) (h : a ∣ b) : (n * a) ∣ (n * b) := sorry lemma Lemma_7_1_10b {a b n : Nat} (h1 : n ≠ 0) (h2 : (n * a) ∣ (n * b)) : a ∣ b := sorry lemma Lemma_7_1_10c {a b : Nat} (h1 : a ∣ b) (h2 : b ∣ a) : a = b := sorry theorem Exercise_7_1_10 (a b n : Nat) : gcd (n * a) (n * b) = n * gcd a b := sorry /- Section 7.2 -/ -- 1. lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry -- 2. -- Hints: Start with apply List.rec. You may find mul_ne_zero useful theorem prod_nonzero_nonzero : ∀ (l : List Nat), (∀ a ∈ l, a ≠ 0) → prod l ≠ 0 := sorry -- 3. theorem rel_prime_iff_no_common_factor (a b : Nat) : rel_prime a b ↔ ¬∃ (p : Nat), prime p ∧ p ∣ a ∧ p ∣ b := sorry -- 4. theorem rel_prime_symm {a b : Nat} (h : rel_prime a b) : rel_prime b a := sorry -- 5. lemma in_prime_factorization_iff_prime_factor {a : Nat} {l : List Nat} (h1 : prime_factorization a l) (p : Nat) : p ∈ l ↔ prime_factor p a := sorry -- 6. theorem Exercise_7_2_5 {a b : Nat} {l m : List Nat} (h1 : prime_factorization a l) (h2 : prime_factorization b m) : rel_prime a b ↔ (¬∃ (p : Nat), p ∈ l ∧ p ∈ m) := sorry -- 7. theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry -- 8. theorem Exercise_7_2_7 {a b a' b' : Nat} (h1 : rel_prime a b) (h2 : a' ∣ a) (h3 : b' ∣ b) : rel_prime a' b' := sorry -- 9. theorem Exercise_7_2_9 {a b j k : Nat} (h1 : gcd a b ≠ 0) (h2 : a = j * gcd a b) (h3 : b = k * gcd a b) : rel_prime j k := sorry -- 10. theorem Exercise_7_2_17a (a b c : Nat) : gcd a (b * c) ∣ gcd a b * gcd a c := sorry /- Section 7.3 -/ -- 1. theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry -- 2. theorem Theorem_7_3_6_3 {m : Nat} (X : ZMod m) : X + [0]_m = X := sorry -- 3. theorem Theorem_7_3_6_4 {m : Nat} (X : ZMod m) : ∃ (Y : ZMod m), X + Y = [0]_m := sorry -- 4. theorem Exercise_7_3_4a {m : Nat} (Z1 Z2 : ZMod m) (h1 : ∀ (X : ZMod m), X + Z1 = X) (h2 : ∀ (X : ZMod m), X + Z2 = X) : Z1 = Z2 := sorry -- 5. theorem Exercise_7_3_4b {m : Nat} (X Y1 Y2 : ZMod m) (h1 : X + Y1 = [0]_m) (h2 : X + Y2 = [0]_m) : Y1 = Y2 := sorry -- 6. theorem Theorem_7_3_10 (m a : Nat) (b : Int) : ¬(↑(gcd m a) : Int) ∣ b → ¬∃ (x : Int), a * x ≡ b (MOD m) := sorry -- 7. theorem Theorem_7_3_11 (m n : Nat) (a b : Int) (h1 : n ≠ 0) : n * a ≡ n * b (MOD n * m) ↔ a ≡ b (MOD m) := sorry -- 8. theorem Exercise_7_3_16 {m : Nat} {a b : Int} (h : a ≡ b (MOD m)) : ∀ (n : Nat), a ^ n ≡ b ^ n (MOD m) := sorry -- 9. example {m : Nat} [NeZero m] (X : ZMod m) : ∃! (a : Int), 0 ≤ a ∧ a < m ∧ X = [a]_m := sorry -- 10. theorem congr_rel_prime {m a b : Nat} (h1 : a ≡ b (MOD m)) : rel_prime m a ↔ rel_prime m b := sorry -- 11. --Hint: You may find the theorem Int.ofNat_mod_ofNat useful. theorem rel_prime_mod (m a : Nat) : rel_prime m (a % m) ↔ rel_prime m a := sorry -- 12.

lemma congr_iff_mod_eq_Int (m : Nat) (a b : Int) [NeZero m] : a ≡ b (MOD m) ↔ a % ↑m = b % ↑m

HTPI.Exercises.congr_iff_mod_eq_Int

htpi/HTPILib/Chap7.lean

htpi

/- Copyright 2023 Daniel J. Velleman -/ import HTPILib.Chap6 namespace HTPI /- Definitions -/ lemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by have h : n + 1 > 0 := Nat.succ_pos n show a % (n + 1) < n + 1 from Nat.mod_lt a h done def gcd (a b : Nat) : Nat := match b with | 0 => a | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd (n + 1) (a % (n + 1)) termination_by b mutual def gcd_c1 (a b : Nat) : Int := match b with | 0 => 1 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c2 (n + 1) (a % (n + 1)) --Corresponds to s = t' termination_by b def gcd_c2 (a b : Nat) : Int := match b with | 0 => 0 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c1 (n + 1) (a % (n + 1)) - (gcd_c2 (n + 1) (a % (n + 1))) * ↑(a / (n + 1)) --Corresponds to t = s' - t'q termination_by b end def prime (n : Nat) : Prop := 2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n def prime_factor (p n : Nat) : Prop := prime p ∧ p ∣ n def all_prime (l : List Nat) : Prop := ∀ p ∈ l, prime p def nondec (l : List Nat) : Prop := match l with | [] => True --Of course, True is a proposition that is always true | n :: L => (∀ m ∈ L, n ≤ m) ∧ nondec L def nondec_prime_list (l : List Nat) : Prop := all_prime l ∧ nondec l def prod (l : List Nat) : Nat := match l with | [] => 1 | n :: L => n * (prod L) def prime_factorization (n : Nat) (l : List Nat) : Prop := nondec_prime_list l ∧ prod l = n def rel_prime (a b : Nat) : Prop := gcd a b = 1 def congr_mod (m : Nat) (a b : Int) : Prop := (↑m : Int) ∣ (a - b) def cc (m : Nat) (a : Int) : ZMod m := (↑a : ZMod m) notation:50 a " ≡ " b " (MOD " m ")" => congr_mod m a b notation:max "["a"]_"m:max => cc m a def invertible {m : Nat} (X : ZMod m) : Prop := ∃ (Y : ZMod m), X * Y = [1]_m def num_rp_below (m k : Nat) : Nat := match k with | 0 => 0 | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j def phi (m : Nat) : Nat := num_rp_below m m def prod_seq {m : Nat} (j k : Nat) (f : Nat → ZMod m) : ZMod m := match j with | 0 => [1]_m | n + 1 => prod_seq n k f * f (k + n) def maps_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i < n, g i < n def one_one_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i1 < n, ∀ i2 < n, g i1 = g i2 → i1 = i2 def onto_below (n : Nat) (g : Nat → Nat) : Prop := ∀ k < n, ∃ i < n, g i = k def perm_below (n : Nat) (g : Nat → Nat) : Prop := maps_below n g ∧ one_one_below n g ∧ onto_below n g def inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m) def swap (u v i : Nat) : Nat := if i = u then v else if i = v then u else i namespace Euler --For definitions specific to Euler's theorem def F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m def G (m a i : Nat) : Nat := (a * i) % m def Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i end Euler /- Section 7.1 -/ theorem dvd_mod_of_dvd_a_b {a b d : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ (a % b) := by set q : Nat := a / b have h3 : b * q + a % b = a := Nat.div_add_mod a b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 define --Goal : ∃ (c : Nat), a % b = d * c apply Exists.intro (j - k * q) show a % b = d * (j - k * q) from calc a % b _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm _ = a - b * q := by rw [h3] _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc] _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm done theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry #eval gcd 672 161 --Answer: 7 lemma gcd_base (a : Nat) : gcd a 0 = a := by rfl lemma gcd_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd a b = gcd b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1)) rfl done lemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b ≠ 0) : a % b < b := by have h1 : b > 0 := Nat.pos_of_ne_zero h show a % b < b from Nat.mod_lt a h1 done lemma dvd_self (n : Nat) : n ∣ n := by apply Exists.intro 1 ring done theorem gcd_dvd : ∀ (b a : Nat), (gcd a b) ∣ a ∧ (gcd a b) ∣ b := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd a b_1) ∣ a ∧ (gcd a b_1) ∣ b_1 fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_base] --Goal: a ∣ a ∧ a ∣ 0 apply And.intro (dvd_self a) define apply Exists.intro 0 rfl done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_nonzero a h1] --Goal : gcd b (a % b) ∣ a ∧ gcd b (a % b) ∣ b have h2 : a % b < b := mod_nonzero_lt a h1 have h3 : (gcd b (a % b)) ∣ b ∧ (gcd b (a % b)) ∣ (a % b) := ih (a % b) h2 b apply And.intro _ h3.left show (gcd b (a % b)) ∣ a from dvd_a_of_dvd_b_mod h3.left h3.right done done theorem gcd_dvd_left (a b : Nat) : (gcd a b) ∣ a := (gcd_dvd b a).left theorem gcd_dvd_right (a b : Nat) : (gcd a b) ∣ b := (gcd_dvd b a).right lemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl lemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c1 a b = gcd_c2 b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done lemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl lemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * ↑(a / b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done theorem gcd_lin_comb : ∀ (b a : Nat), (gcd_c1 a b) * ↑a + (gcd_c2 a b) * ↑b = ↑(gcd a b) := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd_c1 a b_1) * ↑a + (gcd_c2 a b_1) * ↑b_1 = ↑(gcd a b_1) fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base] --Goal : 1 * ↑a + 0 * ↑0 = ↑a ring done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1] --Goal : gcd_c2 b (a % b) * ↑a + -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * ↑(a / b)) * ↑b = -- ↑(gcd b (a % b)) set r : Nat := a % b set q : Nat := a / b set s : Int := gcd_c1 b r set t : Int := gcd_c2 b r --Goal : t * ↑a + (s - t * ↑q) * ↑b = ↑(gcd b r) have h2 : r < b := mod_nonzero_lt a h1 have h3 : s * ↑b + t * ↑r = ↑(gcd b r) := ih r h2 b have h4 : b * q + r = a := Nat.div_add_mod a b rewrite [←h3, ←h4] rewrite [Nat.cast_add, Nat.cast_mul] --Goal : t * (↑b * ↑q + ↑r) + (s - t * ↑q) * ↑b = s * ↑b + t * ↑r ring done done #eval gcd_c1 672 161 --Answer: 6 #eval gcd_c2 672 161 --Answer: -25 --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161 theorem Theorem_7_1_6 {d a b : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ gcd a b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑d ∣ ↑(gcd a b) set s : Int := gcd_c1 a b set t : Int := gcd_c2 a b have h3 : s * ↑a + t * ↑b = ↑(gcd a b) := gcd_lin_comb b a rewrite [←h3] --Goal : ↑d ∣ s * ↑a + t * ↑b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul] --Goal : ↑d ∣ s * (↑d * ↑j) + t * (↑d * ↑k) define apply Exists.intro (s * ↑j + t * ↑k) ring done /- Section 7.2 -/ theorem dvd_trans {a b c : Nat} (h1 : a ∣ b) (h2 : b ∣ c) : a ∣ c := by define at h1; define at h2; define obtain (m : Nat) (h3 : b = a * m) from h1 obtain (n : Nat) (h4 : c = b * n) from h2 rewrite [h3, mul_assoc] at h4 apply Exists.intro (m * n) show c = a * (m * n) from h4 done lemma exists_prime_factor : ∀ (n : Nat), 2 ≤ n → ∃ (p : Nat), prime_factor p n := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, 2 ≤ n_1 → ∃ (p : Nat), prime_factor p n_1 assume h1 : 2 ≤ n by_cases h2 : prime n · -- Case 1. h2 : prime n apply Exists.intro n define --Goal : prime n ∧ n ∣ n show prime n ∧ n ∣ n from And.intro h2 (dvd_self n) done · -- Case 2. h2 : ¬prime n define at h2 --h2 : ¬(2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n) demorgan at h2 disj_syll h2 h1 obtain (a : Nat) (h3 : ∃ (b : Nat), a * b = n ∧ a < n ∧ b < n) from h2 obtain (b : Nat) (h4 : a * b = n ∧ a < n ∧ b < n) from h3 have h5 : 2 ≤ a := by by_contra h6 have h7 : a ≤ 1 := by linarith have h8 : n ≤ b := calc n _ = a * b := h4.left.symm _ ≤ 1 * b := by rel [h7] _ = b := by ring linarith --n ≤ b contradicts b < n done have h6 : ∃ (p : Nat), prime_factor p a := ih a h4.right.left h5 obtain (p : Nat) (h7 : prime_factor p a) from h6 apply Exists.intro p define --Goal : prime p ∧ p ∣ n define at h7 --h7 : prime p ∧ p ∣ a apply And.intro h7.left have h8 : a ∣ n := by apply Exists.intro b show n = a * b from (h4.left).symm done show p ∣ n from dvd_trans h7.right h8 done done lemma exists_least_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q := by set S : Set Nat := {p : Nat | prime_factor p n} have h2 : ∃ (p : Nat), p ∈ S := exists_prime_factor n h show ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q from well_ord_princ S h2 done lemma all_prime_nil : all_prime [] := by define --Goal : ∀ p ∈ [], prime p fix p : Nat contrapos --Goal : ¬prime p → p ∉ [] assume h1 : ¬prime p show p ∉ [] from List.not_mem_nil p done lemma all_prime_cons (n : Nat) (L : List Nat) : all_prime (n :: L) ↔ prime n ∧ all_prime L := by apply Iff.intro · -- (→) assume h1 : all_prime (n :: L) --Goal : prime n ∧ all_prime L define at h1 --h1 : ∀ p ∈ n :: L, prime p apply And.intro (h1 n (List.mem_cons_self n L)) define --Goal : ∀ p ∈ L, prime p fix p : Nat assume h2 : p ∈ L show prime p from h1 p (List.mem_cons_of_mem n h2) done · -- (←) assume h1 : prime n ∧ all_prime L --Goal : all_prime (n :: l) define : all_prime L at h1 define fix p : Nat assume h2 : p ∈ n :: L rewrite [List.mem_cons] at h2 --h2 : p = n ∨ p ∈ L by_cases on h2 · -- Case 1. h2 : p = n rewrite [h2] show prime n from h1.left done · -- Case 2. h2 : p ∈ L show prime p from h1.right p h2 done done done lemma nondec_nil : nondec [] := by define --Goal : True trivial --trivial proves some obviously true statements, such as True done lemma nondec_cons (n : Nat) (L : List Nat) : nondec (n :: L) ↔ (∀ m ∈ L, n ≤ m) ∧ nondec L := by rfl lemma prod_nil : prod [] = 1 := by rfl lemma prod_cons : prod (n :: L) = n * (prod L) := by rfl lemma exists_cons_of_length_eq_succ {A : Type} {l : List A} {n : Nat} (h : l.length = n + 1) : ∃ (a : A) (L : List A), l = a :: L ∧ L.length = n := by have h1 : ¬l.length = 0 := by linarith rewrite [List.length_eq_zero] at h1 obtain (a : A) (h2 : ∃ (L : List A), l = a :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List A) (h3 : l = a :: L) from h2 apply Exists.intro a apply Exists.intro L apply And.intro h3 have h4 : (a :: L).length = L.length + 1 := List.length_cons a L rewrite [←h3, h] at h4 show L.length = n from (Nat.add_right_cancel h4).symm done lemma list_elt_dvd_prod_by_length (a : Nat) : ∀ (n : Nat), ∀ (l : List Nat), l.length = n → a ∈ l → a ∣ prod l := by by_induc · --Base Case fix l : List Nat assume h1 : l.length = 0 rewrite [List.length_eq_zero] at h1 --h1 : l = [] rewrite [h1] --Goal : a ∈ [] → a ∣ prod [] contrapos assume h2 : ¬a ∣ prod [] show a ∉ [] from List.not_mem_nil a done · -- Induction Step fix n : Nat assume ih : ∀ (l : List Nat), List.length l = n → a ∈ l → a ∣ prod l fix l : List Nat assume h1 : l.length = n + 1 --Goal : a ∈ l → a ∣ prod l obtain (b : Nat) (h2 : ∃ (L : List Nat), l = b :: L ∧ L.length = n) from exists_cons_of_length_eq_succ h1 obtain (L : List Nat) (h3 : l = b :: L ∧ L.length = n) from h2 have h4 : a ∈ L → a ∣ prod L := ih L h3.right assume h5 : a ∈ l rewrite [h3.left, prod_cons] --Goal : a ∣ b * prod L rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b ∨ a ∈ L by_cases on h5 · -- Case 1. h5 : a = b apply Exists.intro (prod L) rewrite [h5] rfl done · -- Case 2. h5 : a ∈ L have h6 : a ∣ prod L := h4 h5 have h7 : prod L ∣ b * prod L := by apply Exists.intro b ring done show a ∣ b * prod L from dvd_trans h6 h7 done done done lemma list_elt_dvd_prod {a : Nat} {l : List Nat} (h : a ∈ l) : a ∣ prod l := by set n : Nat := l.length have h1 : l.length = n := by rfl show a ∣ prod l from list_elt_dvd_prod_by_length a n l h1 h done lemma exists_prime_factorization : ∀ (n : Nat), n ≥ 1 → ∃ (l : List Nat), prime_factorization n l := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, n_1 ≥ 1 → ∃ (l : List Nat), prime_factorization n_1 l assume h1 : n ≥ 1 by_cases h2 : n = 1 · -- Case 1. h2 : n = 1 apply Exists.intro [] define apply And.intro · -- Proof of nondec_prime_list [] define show all_prime [] ∧ nondec [] from And.intro all_prime_nil nondec_nil done · -- Proof of prod [] = n rewrite [prod_nil, h2] rfl done done · -- Case 2. h2 : n ≠ 1 have h3 : n ≥ 2 := lt_of_le_of_ne' h1 h2 obtain (p : Nat) (h4 : prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q) from exists_least_prime_factor h3 have p_prime_factor : prime_factor p n := h4.left define at p_prime_factor have p_prime : prime p := p_prime_factor.left have p_dvd_n : p ∣ n := p_prime_factor.right have p_least : ∀ (q : Nat), prime_factor q n → p ≤ q := h4.right obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n have h5 : m ≠ 0 := by contradict h1 with h6 have h7 : n = 0 := calc n _ = p * m := n_eq_pm _ = p * 0 := by rw [h6] _ = 0 := by ring rewrite [h7] decide done have m_pos : 0 < m := Nat.pos_of_ne_zero h5 have m_lt_n : m < n := by define at p_prime show m < n from calc m _ < m + m := by linarith _ = 2 * m := by ring _ ≤ p * m := by rel [p_prime.left] _ = n := n_eq_pm.symm done obtain (L : List Nat) (h6 : prime_factorization m L) from ih m m_lt_n m_pos define at h6 have ndpl_L : nondec_prime_list L := h6.left define at ndpl_L apply Exists.intro (p :: L) define apply And.intro · -- Proof of nondec_prime_list (p :: L) define apply And.intro · -- Proof of all_prime (p :: L) rewrite [all_prime_cons] show prime p ∧ all_prime L from And.intro p_prime ndpl_L.left done · -- Proof of nondec (p :: L) rewrite [nondec_cons] apply And.intro _ ndpl_L.right fix q : Nat assume q_in_L : q ∈ L have h7 : q ∣ prod L := list_elt_dvd_prod q_in_L rewrite [h6.right] at h7 --h7 : q ∣ m have h8 : m ∣ n := by apply Exists.intro p rewrite [n_eq_pm] ring done have q_dvd_n : q ∣ n := dvd_trans h7 h8 have ap_L : all_prime L := ndpl_L.left define at ap_L have q_prime_factor : prime_factor q n := And.intro (ap_L q q_in_L) q_dvd_n show p ≤ q from p_least q q_prime_factor done done · -- Proof of prod (p :: L) = n rewrite [prod_cons, h6.right, n_eq_pm] rfl done done done theorem Theorem_7_2_2 {a b c : Nat} (h1 : c ∣ a * b) (h2 : rel_prime a c) : c ∣ b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑c ∣ ↑b define at h1; define at h2; define obtain (j : Nat) (h3 : a * b = c * j) from h1 set s : Int := gcd_c1 a c set t : Int := gcd_c2 a c have h4 : s * ↑a + t * ↑c = ↑(gcd a c) := gcd_lin_comb c a rewrite [h2, Nat.cast_one] at h4 --h4 : s * ↑a + t * ↑c = (1 : Int) apply Exists.intro (s * ↑j + t * ↑b) show ↑b = ↑c * (s * ↑j + t * ↑b) from calc ↑b _ = (1 : Int) * ↑b := (one_mul _).symm _ = (s * ↑a + t * ↑c) * ↑b := by rw [h4] _ = s * (↑a * ↑b) + t * ↑c * ↑b := by ring _ = s * (↑c * ↑j) + t * ↑c * ↑b := by rw [←Nat.cast_mul a b, h3, Nat.cast_mul c j] _ = ↑c * (s * ↑j + t * ↑b) := by ring done lemma le_nonzero_prod_left {a b : Nat} (h : a * b ≠ 0) : a ≤ a * b := by have h1 : b ≠ 0 := by contradict h with h1 rewrite [h1] ring done have h2 : 1 ≤ b := Nat.pos_of_ne_zero h1 show a ≤ a * b from calc a = a * 1 := (mul_one a).symm _ ≤ a * b := by rel [h2] done lemma le_nonzero_prod_right {a b : Nat} (h : a * b ≠ 0) : b ≤ a * b := by rewrite [mul_comm] rewrite [mul_comm] at h show b ≤ b * a from le_nonzero_prod_left h done lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry lemma rel_prime_of_prime_not_dvd {a p : Nat} (h1 : prime p) (h2 : ¬p ∣ a) : rel_prime a p := by have h3 : gcd a p ∣ a := gcd_dvd_left a p have h4 : gcd a p ∣ p := gcd_dvd_right a p have h5 : gcd a p = 1 ∨ gcd a p = p := dvd_prime h1 h4 have h6 : gcd a p ≠ p := by contradict h2 with h6 rewrite [h6] at h3 show p ∣ a from h3 done disj_syll h5 h6 show rel_prime a p from h5 done theorem Theorem_7_2_3 {a b p : Nat} (h1 : prime p) (h2 : p ∣ a * b) : p ∣ a ∨ p ∣ b := by or_right with h3 have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3 show p ∣ b from Theorem_7_2_2 h2 h4 done lemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a ≥ 1 := by have h1 : a ≠ 0 := by by_contra h1 rewrite [h1] at h contradict h linarith done show a ≥ 1 from Nat.pos_of_ne_zero h1 done lemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by have h1 : a ≥ 1 := ge_one_of_prod_one h have h2 : a * b ≠ 0 := by linarith have h3 : a ≤ a * b := le_nonzero_prod_left h2 rewrite [h] at h3 show a = 1 from Nat.le_antisymm h3 h1 done lemma eq_one_of_dvd_one {n : Nat} (h : n ∣ 1) : n = 1 := by obtain (j : Nat) (h1 : 1 = n * j) from h show n = 1 from eq_one_of_prod_one h1.symm done lemma prime_not_one {p : Nat} (h : prime p) : p ≠ 1 := by define at h linarith done theorem Theorem_7_2_4 {p : Nat} (h1 : prime p) : ∀ (l : List Nat), p ∣ prod l → ∃ a ∈ l, p ∣ a := by apply List.rec · -- Base Case. Goal : p ∣ prod [] → ∃ a ∈ [], p ∣ a rewrite [prod_nil] assume h2 : p ∣ 1 show ∃ a ∈ [], p ∣ a from absurd (eq_one_of_dvd_one h2) (prime_not_one h1) done · -- Induction Step fix b : Nat fix L : List Nat assume ih : p ∣ prod L → ∃ a ∈ L, p ∣ a --Goal : p ∣ prod (b :: L) → ∃ a ∈ b :: L, p ∣ a assume h2 : p ∣ prod (b :: L) rewrite [prod_cons] at h2 have h3 : p ∣ b ∨ p ∣ prod L := Theorem_7_2_3 h1 h2 by_cases on h3 · -- Case 1. h3 : p ∣ b apply Exists.intro b show b ∈ b :: L ∧ p ∣ b from And.intro (List.mem_cons_self b L) h3 done · -- Case 2. h3 : p ∣ prod L obtain (a : Nat) (h4 : a ∈ L ∧ p ∣ a) from ih h3 apply Exists.intro a show a ∈ b :: L ∧ p ∣ a from And.intro (List.mem_cons_of_mem b h4.left) h4.right done done done lemma prime_in_list {p : Nat} {l : List Nat} (h1 : prime p) (h2 : all_prime l) (h3 : p ∣ prod l) : p ∈ l := by obtain (a : Nat) (h4 : a ∈ l ∧ p ∣ a) from Theorem_7_2_4 h1 l h3 define at h2 have h5 : prime a := h2 a h4.left have h6 : p = 1 ∨ p = a := dvd_prime h5 h4.right disj_syll h6 (prime_not_one h1) rewrite [h6] show a ∈ l from h4.left done lemma first_le_first {p q : Nat} {l m : List Nat} (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m)) (h3 : prod (p :: l) = prod (q :: m)) : p ≤ q := by define at h1; define at h2 have h4 : q ∣ prod (p :: l) := by define apply Exists.intro (prod m) rewrite [←prod_cons] show prod (p :: l) = prod (q :: m) from h3 done have h5 : all_prime (q :: m) := h2.left rewrite [all_prime_cons] at h5 have h6 : q ∈ p :: l := prime_in_list h5.left h1.left h4 have h7 : nondec (p :: l) := h1.right rewrite [nondec_cons] at h7 rewrite [List.mem_cons] at h6 by_cases on h6 · -- Case 1. h6 : q = p linarith done · -- Case 2. h6 : q ∈ l have h8 : ∀ m ∈ l, p ≤ m := h7.left show p ≤ q from h8 q h6 done done lemma nondec_prime_list_tail {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by define at h define rewrite [all_prime_cons, nondec_cons] at h show all_prime l ∧ nondec l from And.intro h.left.right h.right.right done lemma cons_prod_not_one {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : prod (p :: l) ≠ 1 := by define at h have h1 : all_prime (p :: l) := h.left rewrite [all_prime_cons] at h1 rewrite [prod_cons] by_contra h2 show False from (prime_not_one h1.left) (eq_one_of_prod_one h2) done lemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) : l = [] ↔ prod l = 1 := by apply Iff.intro · -- (→) assume h1 : l = [] rewrite [h1] show prod [] = 1 from prod_nil done · -- (←) contrapos assume h1 : ¬l = [] obtain (p : Nat) (h2 : ∃ (L : List Nat), l = p :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List Nat) (h3 : l = p :: L) from h2 rewrite [h3] at h rewrite [h3] show ¬prod (p :: L) = 1 from cons_prod_not_one h done done lemma prime_pos {p : Nat} (h : prime p) : p > 0 := by define at h linarith done theorem Theorem_7_2_5 : ∀ (l1 l2 : List Nat), nondec_prime_list l1 → nondec_prime_list l2 → prod l1 = prod l2 → l1 = l2 := by apply List.rec · -- Base Case. Goal : ∀ (l2 : List Nat), nondec_prime_list [] → -- nondec_prime_list l2 → prod [] = prod l2 → [] = l2 fix l2 : List Nat assume h1 : nondec_prime_list [] assume h2 : nondec_prime_list l2 assume h3 : prod [] = prod l2 rewrite [prod_nil, eq_comm, ←list_nil_iff_prod_one h2] at h3 show [] = l2 from h3.symm done · -- Induction Step fix p : Nat fix L1 : List Nat assume ih : ∀ (L2 : List Nat), nondec_prime_list L1 → nondec_prime_list L2 → prod L1 = prod L2 → L1 = L2 -- Goal : ∀ (l2 : List Nat), nondec_prime_list (p :: L1) → -- nondec_prime_list l2 → prod (p :: L1) = prod l2 → p :: L1 = l2 fix l2 : List Nat assume h1 : nondec_prime_list (p :: L1) assume h2 : nondec_prime_list l2 assume h3 : prod (p :: L1) = prod l2 have h4 : ¬prod (p :: L1) = 1 := cons_prod_not_one h1 rewrite [h3, ←list_nil_iff_prod_one h2] at h4 obtain (q : Nat) (h5 : ∃ (L : List Nat), l2 = q :: L) from List.exists_cons_of_ne_nil h4 obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5 rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2) rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2) have h7 : p ≤ q := first_le_first h1 h2 h3 have h8 : q ≤ p := first_le_first h2 h1 h3.symm have h9 : p = q := by linarith rewrite [h9, prod_cons, prod_cons] at h3 --h3 : q * prod L1 = q * prod L2 have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1 have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2 define at h2 have h12 : all_prime (q :: L2) := h2.left rewrite [all_prime_cons] at h12 have h13 : q > 0 := prime_pos h12.left have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3 have h15 : L1 = L2 := ih L2 h10 h11 h14 rewrite [h6, h9, h15] rfl done done theorem fund_thm_arith (n : Nat) (h : n ≥ 1) : ∃! (l : List Nat), prime_factorization n l := by exists_unique · -- Existence show ∃ (l : List Nat), prime_factorization n l from exists_prime_factorization n h done · -- Uniqueness fix l1 : List Nat; fix l2 : List Nat assume h1 : prime_factorization n l1 assume h2 : prime_factorization n l2 define at h1; define at h2 have h3 : prod l1 = n := h1.right rewrite [←h2.right] at h3 show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3 done done /- Section 7.3 -/ theorem congr_refl (m : Nat) : ∀ (a : Int), a ≡ a (MOD m) := by fix a : Int define --Goal : ∃ (c : Int), a - a = ↑m * c apply Exists.intro 0 ring done theorem congr_symm {m : Nat} : ∀ {a b : Int}, a ≡ b (MOD m) → b ≡ a (MOD m) := by fix a : Int; fix b : Int assume h1 : a ≡ b (MOD m) define at h1 --h1 : ∃ (c : Int), a - b = ↑m * c define --Goal : ∃ (c : Int), b - a = ↑m * c obtain (c : Int) (h2 : a - b = m * c) from h1 apply Exists.intro (-c) show b - a = m * (-c) from calc b - a _ = -(a - b) := by ring _ = -(m * c) := by rw [h2] _ = m * (-c) := by ring done theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry /- Fundamental properties of congruence classes -/ lemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) : X = Y ↔ X.val = Y.val := Fin.ext_iff lemma val_nat_eq_mod (n k : Nat) : ([k]_(n + 1)).val = k % (n + 1) := by rfl lemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl theorem cc_rep {m : Nat} (X : ZMod m) : ∃ (a : Int), X = [a]_m := match m with | 0 => by apply Exists.intro X rfl done | n + 1 => by apply Exists.intro ↑(X.val) have h1 : X.val < n + 1 := Fin.prop X rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1] rfl done theorem add_class (m : Nat) (a b : Int) : [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm theorem mul_class (m : Nat) (a b : Int) : [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm lemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ [a - b]_m = [0]_m := by apply Iff.intro · -- (→) assume h1 : [a]_m = [b]_m have h2 : a - b = a + (-b) := by ring have h3 : b + (-b) = 0 := by ring show [a - b]_m = [0]_m from calc [a - b]_m _ = [a + (-b)]_m := by rw [h2] _ = [a]_m + [-b]_m := by rw [add_class] _ = [b]_m + [-b]_m := by rw [h1] _ = [b + -b]_m := by rw [add_class] _ = [0]_m := by rw [h3] done · -- (←) assume h1 : [a - b]_m = [0]_m have h2 : b + (a - b) = a := by ring have h3 : b + 0 = b := by ring show [a]_m = [b]_m from calc [a]_m _ = [b + (a - b)]_m := by rw [h2] _ = [b]_m + [a - b]_m := by rw [add_class] _ = [b]_m + [0]_m := by rw [h1] _ = [b + 0]_m := by rw [add_class] _ = [b]_m := by rw [h3] done done lemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) : [a]_m = [0]_m → [-a]_m = [0]_m := by assume h1 : [a]_m = [0]_m have h2 : 0 + (-a) = -a := by ring have h3 : a + (-a) = 0 := by ring show [-a]_m = [0]_m from calc [-a]_m _ = [0 + (-a)]_m := by rw [h2] _ = [0]_m + [-a]_m := by rw [add_class] _ = [a]_m + [-a]_m := by rw [h1] _ = [a + (-a)]_m := by rw [add_class] _ = [0]_m := by rw [h3] done lemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) : [-a]_m = [0]_m ↔ [a]_m = [0]_m := by apply Iff.intro _ (cc_neg_zero_of_cc_zero m a) assume h1 : [-a]_m = [0]_m have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1 have h3 : -(-a) = a := by ring rewrite [h3] at h2 show [a]_m = [0]_m from h2 done lemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl lemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m ↔ m ∣ k := match m with | 0 => by have h : (0 : Int) = (↑(0 : Nat) : Int) := by rfl rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj] apply Iff.intro · -- (→) assume h1 : k = 0 rewrite [h1] show 0 ∣ 0 from dvd_self 0 done · -- (←) assume h1 : 0 ∣ k obtain (c : Nat) (h2 : k = 0 * c) from h1 rewrite [h2] ring done done | n + 1 => by rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero] show k % (n + 1) = 0 ↔ n + 1 ∣ k from (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm done lemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m ↔ ↑m ∣ a := by obtain (k : Nat) (h1 : a = ↑k ∨ a = -↑k) from Int.eq_nat_or_neg a by_cases on h1 · -- Case 1. h1: a = ↑k rewrite [h1, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done · -- Case 2. h1: a = -↑k rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done done theorem cc_eq_iff_congr (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ a ≡ b (MOD m) := calc [a]_m = [b]_m _ ↔ [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b _ ↔ ↑m ∣ (a - b) := cc_zero_iff_dvd m (a - b) _ ↔ a ≡ b (MOD m) := by rfl /- End of fundamental properties of congruence classes -/ lemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m := by have h1 : (↑m : Int) ≠ 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m) show 0 ≤ a % m from Int.emod_nonneg a h1 done lemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m) have h2 : (↑m : Int) > 0 := (Nat.cast_pos).rtl h1 show a % m < m from Int.emod_lt_of_pos a h2 done lemma congr_mod_mod (m : Nat) (a : Int) : a ≡ a % m (MOD m) := by define have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m apply Exists.intro (a / m) show a - a % m = m * (a / m) from calc a - (a % m) _ = m * (a / m) + a % m - a % m := by rw [h1] _ = m * (a / m) := by ring done lemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) := And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a)) theorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) : ∃! (r : Int), 0 ≤ r ∧ r < m ∧ a ≡ r (MOD m) := by exists_unique · -- Existence apply Exists.intro (a % m) show 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) from mod_cmpl_res m a done · -- Uniqueness fix r1 : Int; fix r2 : Int assume h1 : 0 ≤ r1 ∧ r1 < m ∧ a ≡ r1 (MOD m) assume h2 : 0 ≤ r2 ∧ r2 < m ∧ a ≡ r2 (MOD m) have h3 : r1 ≡ r2 (MOD m) := congr_trans (congr_symm h1.right.right) h2.right.right obtain (d : Int) (h4 : r1 - r2 = m * d) from h3 have h5 : r1 - r2 < m * 1 := by linarith have h6 : m * (-1) < r1 - r2 := by linarith rewrite [h4] at h5 --h5 : m * d < m * 1 rewrite [h4] at h6 --h6 : m * -1 < m * d have h7 : (↑m : Int) ≥ 0 := Nat.cast_nonneg m have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7 have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7 have h10 : d = 0 := by linarith show r1 = r2 from calc r1 _ = r1 - r2 + r2 := by ring _ = m * 0 + r2 := by rw [h4, h10] _ = r2 := by ring done done lemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m := (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a) theorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y rewrite [h1, h2] have h3 : a + b = b + a := by ring show [a]_m + [b]_m = [b]_m + [a]_m from calc [a]_m + [b]_m _ = [a + b]_m := add_class m a b _ = [b + a]_m := by rw [h3] _ = [b]_m + [a]_m := (add_class m b a).symm done theorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X rewrite [h1] have h2 : a * 1 = a := by ring show [a]_m * [1]_m = [a]_m from calc [a]_m * [1]_m _ = [a * 1]_m := mul_class m a 1 _ = [a]_m := by rw [h2] done theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry lemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) : [a]_m * [gcd_c2 m a]_m = [1]_m := by set s : Int := gcd_c1 m a have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m define at h1 rewrite [h1, Nat.cast_one] at h2 --h2 : s * ↑m + gcd_c2 m a * ↑a = 1 rewrite [mul_class, cc_eq_iff_congr] define --Goal : ∃ (c : Int), ↑a * gcd_c2 m a - 1 = ↑m * c apply Exists.intro (-s) show a * (gcd_c2 m a) - 1 = m * (-s) from calc a * (gcd_c2 m a) - 1 _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring _ = 1 + m * (-s) - 1 := by rw [h2] _ = m * (-s) := by ring done theorem Theorem_7_3_7 (m a : Nat) : invertible [a]_m ↔ rel_prime m a := by apply Iff.intro · -- (→) assume h1 : invertible [a]_m define at h1 obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1 obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y rewrite [h3, mul_class, cc_eq_iff_congr] at h2 define at h2 obtain (c : Int) (h4 : a * b - 1 = m * c) from h2 rewrite [Exercise_7_2_6] --Goal : ∃ (s t : Int), s * ↑m + t * ↑a = 1 apply Exists.intro (-c) apply Exists.intro b show (-c) * m + b * a = 1 from calc (-c) * m + b * a _ = (-c) * m + (a * b - 1) + 1 := by ring _ = (-c) * m + m * c + 1 := by rw [h4] _ = 1 := by ring done · -- (←) assume h1 : rel_prime m a define show ∃ (Y : ZMod m), [a]_m * Y = [1]_m from Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1) done done /- Section 7.4 -/ section Euler open Euler lemma num_rp_below_base {m : Nat} : num_rp_below m 0 = 0 := by rfl lemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) : num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by have h1 : num_rp_below m (j + 1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : gcd m j = 1 rewrite [if_pos h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1 show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1 done lemma num_rp_below_step_not_rp {m j : Nat} (h : ¬rel_prime m j) : num_rp_below m (j + 1) = num_rp_below m j := by have h1 : num_rp_below m (j +1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : ¬gcd m j = 1 rewrite [if_neg h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j show num_rp_below m (j + 1) = num_rp_below m j from h1 done lemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl #eval phi 10 --Answer: 4 lemma prod_inv_iff_inv {m : Nat} {X : ZMod m} (h1 : invertible X) (Y : ZMod m) : invertible (X * Y) ↔ invertible Y := by apply Iff.intro · -- (→) assume h2 : invertible (X * Y) obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2 apply Exists.intro (X * Z) rewrite [←h3] --Goal : Y * (X * Z) = X * Y * Z ring --Note that ring can do algebra in ZMod m done · -- (←) assume h2 : invertible Y obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1 obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2 apply Exists.intro (Xi * Yi) show (X * Y) * (Xi * Yi) = [1]_m from calc X * Y * (Xi * Yi) _ = (X * Xi) * (Y * Yi) := by ring _ = [1]_m * [1]_m := by rw [h3, h4] _ = [1]_m := Theorem_7_3_6_7 [1]_m done done lemma F_rp_def {m i : Nat} (h : rel_prime m i) : F m i = [i]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h --h : gcd m i = 1 rewrite [if_pos h] at h1 show F m i = [i]_m from h1 done lemma F_not_rp_def {m i : Nat} (h : ¬rel_prime m i) : F m i = [1]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h rewrite [h1, if_neg h] rfl done lemma prod_seq_base {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 0 k f = [1]_m := by rfl lemma prod_seq_step {m : Nat} (n k : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl lemma prod_seq_zero_step {m : Nat} (n : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by rewrite [prod_seq_step, zero_add] rfl done lemma prod_one {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 1 k f = f k := by rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7] rfl done lemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl lemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m := calc [G m a i]_m _ = [(a * i) % m]_m := by rfl _ = [a * i]_m := (cc_eq_mod m (a * i)).symm _ = [a]_m * [i]_m := (mul_class m a i).symm lemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) : rel_prime m (G m a i) ↔ rel_prime m i := by have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1 show rel_prime m (G m a i) ↔ rel_prime m i from calc rel_prime m (G m a i) _ ↔ invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm _ ↔ invertible ([a]_m * [i]_m) := by rw [cc_G] _ ↔ invertible [i]_m := prod_inv_iff_inv h2 ([i]_m) _ ↔ rel_prime m i := Theorem_7_3_7 m i done lemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) : F m (G m a i) = [a]_m * F m i := by have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2 show F m (G m a i) = [a]_m * F m i from calc F m (G m a i) _ = [G m a i]_m := F_rp_def h3 _ = [a]_m * [i]_m := cc_G m a i _ = [a]_m * F m i := by rw [F_rp_def h2] done lemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : ¬rel_prime m i) : F m (G m a i) = [1]_m := by rewrite [←G_rp_iff h1 i] at h2 show F m (G m a i) = [1]_m from F_not_rp_def h2 done lemma FG_prod {m a : Nat} (h1 : rel_prime m a) : ∀ (k : Nat), prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by by_induc · -- Base Case show prod_seq 0 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from calc prod_seq 0 0 ((F m) ∘ (G m a)) _ = [1]_m := prod_seq_base _ _ _ = [a]_m ^ 0 * [1]_m := by ring _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by rw [num_rp_below_base, prod_seq_base] done · -- Induction Step fix k : Nat assume ih : prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) by_cases h2 : rel_prime m k · -- Case 1. h2 : rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([a]_m * F m k) := by rw [FG_rp h1 h2] _ = [a]_m ^ ((num_rp_below m k) + 1) * ((prod_seq k 0 (F m)) * F m k) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_rp h2, prod_seq_zero_step] done · -- Case 2. h2 : ¬rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([1]_m) := by rw [FG_not_rp h1 h2] _ = [a]_m ^ (num_rp_below m k) * (prod_seq k 0 (F m) * ([1]_m)) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_not_rp h2, prod_seq_zero_step, F_not_rp_def h2] done done done lemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by define --Goal : ∀ i < m, G m a i < m fix i : Nat assume h1 : i < m rewrite [G_def] --Goal : a * i % m < m show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m) done lemma left_inv_one_one_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g' (g i) = i) : one_one_below n g := sorry lemma right_inv_onto_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g (g' i) = i) (h2 : maps_below n g') : onto_below n g := by define at h2; define fix k : Nat assume h3 : k < n apply Exists.intro (g' k) show g' k < n ∧ g (g' k) = k from And.intro (h2 k h3) (h1 k h3) done lemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m * [inv_mod m a]_m = [1]_m := by have h2 : 0 ≤ (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a) show [a]_m * [inv_mod m a]_m = [1]_m from calc [a]_m * [inv_mod m a]_m _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2] _ = [a]_m * [gcd_c2 m a]_m := by rw [←cc_eq_mod] _ = [1]_m := gcd_c2_inv h1 done lemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m := calc a * (b % m) % m = a % m * (b % m % m) % m := Nat.mul_mod _ _ _ _ = a % m * (b % m) % m := by rw [Nat.mod_mod] _ = a * b % m := (Nat.mul_mod _ _ _).symm lemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm] rfl done theorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] : ↑a ≡ ↑b (MOD m) ↔ a % m = b % m := sorry lemma mul_inv_mod_cancel {m a i : Nat} [NeZero m] (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1 rewrite [mul_class, cc_eq_iff_congr, ←Nat.cast_mul, ←Nat.cast_one, congr_iff_mod_eq_Nat] at h3 show a * inv_mod m a * i % m = i from calc a * (inv_mod m a) * i % m _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = 1 % m * i % m := by rw [h3] _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = i % m := by rw [one_mul] _ = i := Nat.mod_eq_of_lt h2 done lemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl lemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, G m a (Ginv m a i) = i := by fix i : Nat assume h2 : i < m show G m a (Ginv m a i) = i from calc G m a (Ginv m a i) _ = a * ((inv_mod m a * i) % m) % m := by rfl _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, Ginv m a (G m a i) = i := by fix i : Nat assume h2 : i < m show Ginv m a (G m a i) = i from calc Ginv m a (G m a i) _ = inv_mod m a * ((a * i) % m) % m := by rfl _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc, mul_comm (inv_mod m a)] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_maps_below (m a : Nat) [NeZero m] : maps_below m (Ginv m a) := G_maps_below m (inv_mod m a) lemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : one_one_below m (G m a) := left_inv_one_one_below (Ginv_left_inv h1) lemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : onto_below m (G m a) := right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a) lemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : perm_below m (G m a) := And.intro (G_maps_below m a) (And.intro (G_one_one_below h1) (G_onto_below h1)) --Permuting a product of congruence classes doesn't change product lemma swap_fst (u v : Nat) : swap u v u = v := by define : swap u v u --Goal : (if u = u then v else if u = v then u else u) = v have h : u = u := by rfl rewrite [if_pos h] rfl done lemma swap_snd (u v : Nat) : swap u v v = u := by define : swap u v v by_cases h1 : v = u · -- Case 1. h1 : v = u rewrite [if_pos h1] show v = u from h1 done · -- Case 2. h1 : v ≠ u rewrite [if_neg h1] have h2 : v = v := by rfl rewrite [if_pos h2] rfl done done lemma swap_other {u v i : Nat} (h1 : i ≠ u) (h2 : i ≠ v) : swap u v i = i := by define : swap u v i rewrite [if_neg h1, if_neg h2] rfl done lemma swap_values (u v i : Nat) : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := by by_cases h1 : i = u · -- Case 1. h1 : i = u apply Or.inl rewrite [h1] show swap u v u = v from swap_fst u v done · -- Case 2. h1 : i ≠ u apply Or.inr by_cases h2 : i = v · -- Case 2.1. h2 : i = v apply Or.inl rewrite [h2] show swap u v v = u from swap_snd u v done · -- Case 2.2. h2 : i ≠ v apply Or.inr show swap u v i = i from swap_other h1 h2 done done done lemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by define fix i : Nat assume h3 : i < n have h4 : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := swap_values u v i by_cases on h4 · -- Case 1. h4 : swap u v i = v rewrite [h4] show v < n from h2 done · -- Case 2. by_cases on h4 · -- Case 2.1. h4 : swap u v i = u rewrite [h4] show u < n from h1 done · -- Case 2.2. h4 : swap u v i = i rewrite [h4] show i < n from h3 done done done lemma swap_swap (u v n : Nat) : ∀ i < n, swap u v (swap u v i) = i := by fix i : Nat assume h : i < n by_cases h1 : i = u · -- Case 1. h1 : i = u rewrite [h1, swap_fst, swap_snd] rfl done · -- Case 2. h1 : i ≠ u by_cases h2 : i = v · -- Case 2.1. h2 : i = v rewrite [h2, swap_snd, swap_fst] rfl done · -- Case 2.2. h2 : i ≠ v rewrite [swap_other h1 h2, swap_other h1 h2] rfl done done done lemma swap_one_one_below (u v n) : one_one_below n (swap u v) := left_inv_one_one_below (swap_swap u v n) lemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) := right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2) lemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) := And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2)) lemma comp_perm_below {n : Nat} {f g : Nat → Nat} (h1 : perm_below n f) (h2 : perm_below n g) : perm_below n (f ∘ g) := sorry lemma trivial_swap (u : Nat) : swap u u = id := by apply funext fix x : Nat by_cases h1 : x = u · -- Case 1. h1 : x = u rewrite [h1, swap_fst] rfl done · -- Case 2. h1 : x ≠ u rewrite [swap_other h1 h1] rfl done done lemma prod_eq_fun {m : Nat} (f g : Nat → ZMod m) (k : Nat) : ∀ (n : Nat), (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g := by by_induc · -- Base Case assume h : (∀ i < 0, f (k + i) = g (k + i)) rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g assume h1 : ∀ i < n + 1, f (k + i) = g (k + i) have h2 : ∀ i < n, f (k + i) = g (k + i) := by fix i : Nat assume h2 : i < n have h3 : i < n + 1 := by linarith show f (k + i) = g (k + i) from h1 i h3 done have h3 : prod_seq n k f = prod_seq n k g := ih h2 have h4 : n < n + 1 := Nat.lt_succ_self n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4] rfl done done lemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := by have h2 : ∀ (i : Nat), i < u → (f ∘ swap u n) (0 + i) = f (0 + i) := by fix i : Nat assume h2 : i < u have h3 : 0 + i ≠ u := by linarith have h4 : 0 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl done show prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f from prod_eq_fun (f ∘ swap u n) f 0 u h2 done lemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := by have h2 : ∀ i < j, (f ∘ swap u n) (u + 1 + i) = f (u + 1 + i) := by fix i : Nat assume h2 : i < j have h3 : u + 1 + i ≠ u := by linarith have h4 : u + 1 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl show prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f from prod_eq_fun (f ∘ swap u n) f (u + 1) j h2 done lemma break_prod {m : Nat} (n : Nat) (f : Nat → ZMod m) : ∀ (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by by_induc · -- Base Case have h : n + 0 = n := by rfl rewrite [prod_seq_base, h, Theorem_7_3_6_7] rfl done · -- Induction Step fix j : Nat assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f rewrite [←add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc] rfl done done lemma break_prod_twice {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f := break_prod n f 1 rewrite [prod_one] at h2 have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f := break_prod (u + 1) f j rewrite [←h1] at h3 have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f := break_prod u f 1 rewrite [prod_one] at h4 rewrite [h3, h4] at h2 show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2 done lemma swap_prod_eq_prod {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f := by by_cases h2 : u = n · -- Case 1. h2 : u = n rewrite [h2, trivial_swap n] --Goal : prod_seq (n + 1) 0 (f ∘ id) = prod_seq (n + 1) 0 f rfl done · -- Case 2. h2 : ¬u = n have h3 : u + 1 ≤ n := Nat.lt_of_le_of_ne h1 h2 obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3 have break_f : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := break_prod_twice f h4 have break_fs : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_prod_twice (f ∘ swap u n) h4 have f_eq_fs_below : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := swap_prod_eq_prod_below f h1 have f_eq_fs_btwn : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4 show prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f from calc prod_seq (n + 1) 0 (f ∘ swap u n) _ = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_fs _ = prod_seq u 0 f * (f ∘ swap u n) u * prod_seq j (u + 1) f * (f ∘ swap u n) n := by rw [f_eq_fs_below, f_eq_fs_btwn] _ = prod_seq u 0 f * f (swap u n u) * prod_seq j (u + 1) f * f (swap u n n) := by rfl _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by rw [swap_fst, swap_snd] _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring _ = prod_seq (n + 1) 0 f := break_f.symm done done lemma perm_below_fixed {n : Nat} {g : Nat → Nat} (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry lemma perm_prod {m : Nat} (f : Nat → ZMod m) : ∀ (n : Nat), ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) := by by_induc · -- Base Case fix g : Nat → Nat assume h1 : perm_below 0 g rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) fix g : Nat → Nat assume g_pb : perm_below (n + 1) g define at g_pb have g_ob : onto_below (n + 1) g := g_pb.right.right define at g_ob have h1 : n < n + 1 := by linarith obtain (u : Nat) (h2 : u < n + 1 ∧ g u = n) from g_ob n h1 have s_pb : perm_below (n + 1) (swap u n) := swap_perm_below h2.left h1 have gs_pb_n1 : perm_below (n + 1) (g ∘ swap u n) := comp_perm_below g_pb s_pb have gs_fix_n : (g ∘ swap u n) n = n := calc (g ∘ swap u n) n _ = g (swap u n n) := by rfl _ = g u := by rw [swap_snd] _ = n := h2.right have gs_pb_n : perm_below n (g ∘ swap u n) := perm_below_fixed gs_pb_n1 gs_fix_n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f ∘ (g ∘ swap u n)) := ih (g ∘ swap u n) gs_pb_n have h3 : u ≤ n := by linarith show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f ∘ g) from calc prod_seq (n + 1) 0 f _ = prod_seq n 0 f * f n := prod_seq_zero_step n f _ = prod_seq n 0 (f ∘ (g ∘ swap u n)) * f ((g ∘ swap u n) n) := by rw [gs_prod, gs_fix_n] _ = prod_seq n 0 (f ∘ g ∘ swap u n) * (f ∘ g ∘ swap u n) n := by rfl _ = prod_seq (n + 1) 0 (f ∘ g ∘ swap u n) := (prod_seq_zero_step n (f ∘ g ∘ swap u n)).symm _ = prod_seq (n + 1) 0 ((f ∘ g) ∘ swap u n) := by rfl _ = prod_seq (n + 1) 0 (f ∘ g) := swap_prod_eq_prod (f ∘ g) h3 done done lemma F_invertible (m i : Nat) : invertible (F m i) := by by_cases h : rel_prime m i · -- Case 1. h : rel_prime m i rewrite [F_rp_def h] show invertible [i]_m from (Theorem_7_3_7 m i).rtl h done · -- Case 2. h : ¬rel_prime m i rewrite [F_not_rp_def h] apply Exists.intro [1]_m show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m done done lemma Fprod_invertible (m : Nat) : ∀ (k : Nat), invertible (prod_seq k 0 (F m)) := by by_induc · -- Base Case apply Exists.intro [1]_m show prod_seq 0 0 (F m) * [1]_m = [1]_m from calc prod_seq 0 0 (F m) * [1]_m _ = [1]_m * [1]_m := by rw [prod_seq_base] _ = [1]_m := Theorem_7_3_6_7 ([1]_m) done · -- Induction Step fix k : Nat assume ih : invertible (prod_seq k 0 (F m)) rewrite [prod_seq_zero_step] show invertible (prod_seq k 0 (F m) * (F m k)) from (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k) done done theorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m ^ (phi m) = [1]_m := by have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2 show [a]_m ^ (phi m) = [1]_m from calc [a]_m ^ (phi m) _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3] _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring _ = prod_seq m 0 (F m ∘ G m a) * Y := by rw [FG_prod h1 m, phi_def] _ = prod_seq m 0 (F m) * Y := by rw [perm_prod (F m) m (G m a) (G_perm_below h1)] _ = [1]_m := by rw [h3] done lemma Exercise_7_4_5_Int (m : Nat) (a : Int) : ∀ (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry lemma Exercise_7_4_5_Nat (m a n : Nat) : [a]_m ^ n = [a ^ n]_m := by rewrite [Exercise_7_4_5_Int] rfl done theorem Euler's_theorem {m a : Nat} [NeZero m] (h1 : rel_prime m a) : a ^ (phi m) ≡ 1 (MOD m) := by have h2 : [a]_m ^ (phi m) = [1]_m := Theorem_7_4_2 h1 rewrite [Exercise_7_4_5_Nat m a (phi m)] at h2 --h2 : [a ^ phi m]_m = [1]_m show a ^ (phi m) ≡ 1 (MOD m) from (cc_eq_iff_congr _ _ _).ltr h2 done #eval gcd 10 7 --Answer: 1. So 10 and 7 are relatively prime #eval 7 ^ phi 10 --Answer: 2401, which is congruent to 1 mod 10. end Euler /- Section 7.5 -/ lemma num_rp_prime {p : Nat} (h1 : prime p) : ∀ k < p, num_rp_below p (k + 1) = k := sorry lemma phi_prime {p : Nat} (h1 : prime p) : phi p = p - 1 := by have h2 : 1 ≤ p := prime_pos h1 have h3 : p - 1 + 1 = p := Nat.sub_add_cancel h2 have h4 : p - 1 < p := by linarith have h5 : num_rp_below p (p - 1 + 1) = p - 1 := num_rp_prime h1 (p - 1) h4 rewrite [h3] at h5 show phi p = p - 1 from h5 done theorem Theorem_7_2_2_Int {a c : Nat} {b : Int} (h1 : ↑c ∣ ↑a * b) (h2 : rel_prime a c) : ↑c ∣ b := by rewrite [Int.natCast_dvd, Int.natAbs_mul, Int.natAbs_ofNat] at h1 --h1 : c ∣ a * Int.natAbs b rewrite [Int.natCast_dvd] --Goal : c ∣ Int.natAbs b show c ∣ Int.natAbs b from Theorem_7_2_2 h1 h2 done lemma Lemma_7_4_5 {m n : Nat} (a b : Int) (h1 : rel_prime m n) : a ≡ b (MOD m * n) ↔ a ≡ b (MOD m) ∧ a ≡ b (MOD n) := by apply Iff.intro · -- (→) assume h2 : a ≡ b (MOD m * n) obtain (j : Int) (h3 : a - b = (m * n) * j) from h2 apply And.intro · -- Proof of a ≡ b (MOD m) apply Exists.intro (n * j) show a - b = m * (n * j) from calc a - b _ = m * n * j := h3 _ = m * (n * j) := by ring done · -- Proof of a ≡ b (MOD n) apply Exists.intro (m * j) show a - b = n * (m * j) from calc a - b _ = m * n * j := h3 _ = n * (m * j) := by ring done done · -- (←) assume h2 : a ≡ b (MOD m) ∧ a ≡ b (MOD n) obtain (j : Int) (h3 : a - b = m * j) from h2.left have h4 : (↑n : Int) ∣ a - b := h2.right rewrite [h3] at h4 --h4 : ↑n ∣ ↑m * j have h5 : ↑n ∣ j := Theorem_7_2_2_Int h4 h1 obtain (k : Int) (h6 : j = n * k) from h5 apply Exists.intro k --Goal : a - b = ↑(m * n) * k rewrite [Nat.cast_mul] --Goal : a - b = ↑m * ↑n * k show a - b = (m * n) * k from calc a - b _ = m * j := h3 _ = m * (n * k) := by rw [h6] _ = (m * n) * k := by ring done done --From exercises of Section 7.2 theorem rel_prime_symm {a b : Nat} (h : rel_prime a b) : rel_prime b a := sorry lemma prime_NeZero {p : Nat} (h : prime p) : NeZero p := by rewrite [neZero_iff] --Goal : p ≠ 0 define at h linarith done lemma Lemma_7_5_1 {p e d m c s : Nat} {t : Int} (h1 : prime p) (h2 : e * d = (p - 1) * s + 1) (h3 : m ^ e - c = p * t) : c ^ d ≡ m (MOD p) := by have h4 : m ^ e ≡ c (MOD p) := Exists.intro t h3 have h5 : [m ^ e]_p = [c]_p := (cc_eq_iff_congr _ _ _).rtl h4 rewrite [←Exercise_7_4_5_Nat] at h5 --h5 : [m]_p ^ e = [c]_p by_cases h6 : p ∣ m · -- Case 1. h6 : p ∣ m have h7 : m ≡ 0 (MOD p) := by obtain (j : Nat) (h8 : m = p * j) from h6 apply Exists.intro (↑j : Int) --Goal : ↑m - 0 = ↑p * ↑j rewrite [h8, Nat.cast_mul] ring done have h8 : [m]_p = [0]_p := (cc_eq_iff_congr _ _ _).rtl h7 have h9 : e * d ≠ 0 := by rewrite [h2] show (p - 1) * s + 1 ≠ 0 from Nat.add_one_ne_zero _ done have h10 : (0 : Int) ^ (e * d) = 0 := zero_pow h9 have h11 : [c ^ d]_p = [m]_p := calc [c ^ d]_p _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat] _ = ([m]_p ^ e) ^ d := by rw [h5] _ = [m]_p ^ (e * d) := by ring _ = [0]_p ^ (e * d) := by rw [h8] _ = [0 ^ (e * d)]_p := Exercise_7_4_5_Int _ _ _ _ = [0]_p := by rw [h10] _ = [m]_p := by rw [h8] show c ^ d ≡ m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11 done · -- Case 2. h6 : ¬p ∣ m have h7 : rel_prime m p := rel_prime_of_prime_not_dvd h1 h6 have h8 : rel_prime p m := rel_prime_symm h7 have h9 : NeZero p := prime_NeZero h1 have h10 : (1 : Int) ^ s = 1 := by ring have h11 : [c ^ d]_p = [m]_p := calc [c ^ d]_p _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat] _ = ([m]_p ^ e) ^ d := by rw [h5] _ = [m]_p ^ (e * d) := by ring _ = [m]_p ^ ((p - 1) * s + 1) := by rw [h2] _ = ([m]_p ^ (p - 1)) ^ s * [m]_p := by ring _ = ([m]_p ^ (phi p)) ^ s * [m]_p := by rw [phi_prime h1] _ = [1]_p ^ s * [m]_p := by rw [Theorem_7_4_2 h8] _ = [1 ^ s]_p * [m]_p := by rw [Exercise_7_4_5_Int] _ = [1]_p * [m]_p := by rw [h10] _ = [m]_p * [1]_p := by ring _ = [m]_p := Theorem_7_3_6_7 _ show c ^ d ≡ m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11 done done theorem Theorem_7_5_1 (p q n e d k m c : Nat) (p_prime : prime p) (q_prime : prime q) (p_ne_q : p ≠ q) (n_pq : n = p * q) (ed_congr_1 : e * d = k * (p - 1) * (q - 1) + 1) (h1 : [m]_n ^ e = [c]_n) : [c]_n ^ d = [m]_n := by rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] at h1 --h1 : m ^ e ≡ c (MOD n) rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] --Goal : c ^ d ≡ m (MOD n) obtain (j : Int) (h2 : m ^ e - c = n * j) from h1 rewrite [n_pq, Nat.cast_mul] at h2 --h2 : m ^ e - c = p * q * j have h3 : e * d = (p - 1) * (k * (q - 1)) + 1 := by rewrite [ed_congr_1] ring done have h4 : m ^ e - c = p * (q * j) := by rewrite [h2] ring done have congr_p : c ^ d ≡ m (MOD p) := Lemma_7_5_1 p_prime h3 h4 have h5 : e * d = (q - 1) * (k * (p - 1)) + 1 := by rewrite [ed_congr_1] ring done have h6 : m ^ e - c = q * (p * j) := by rewrite [h2] ring done have congr_q : c ^ d ≡ m (MOD q) := Lemma_7_5_1 q_prime h5 h6 have h7 : ¬q ∣ p := by by_contra h8 have h9 : q = 1 ∨ q = p := dvd_prime p_prime h8 disj_syll h9 (prime_not_one q_prime) show False from p_ne_q h9.symm done have h8 : rel_prime p q := rel_prime_of_prime_not_dvd q_prime h7 rewrite [n_pq, Lemma_7_4_5 _ _ h8] show c ^ d ≡ m (MOD p) ∧ c ^ d ≡ m (MOD q) from And.intro congr_p congr_q done /- BEGIN EXERCISES -/ namespace Exercises /- Section 7.1 -/ -- 1. theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry -- 2. lemma gcd_comm_lt {a b : Nat} (h : a < b) : gcd a b = gcd b a := sorry theorem gcd_comm (a b : Nat) : gcd a b = gcd b a := sorry -- 3. theorem Exercise_7_1_5 (a b : Nat) (n : Int) : (∃ (s t : Int), s * a + t * b = n) ↔ (↑(gcd a b) : Int) ∣ n := sorry -- 4. theorem Exercise_7_1_6 (a b c : Nat) : gcd a b = gcd (a + b * c) b := sorry -- 5. theorem gcd_is_nonzero {a b : Nat} (h : a ≠ 0 ∨ b ≠ 0) : gcd a b ≠ 0 := sorry -- 6. theorem gcd_greatest {a b d : Nat} (h1 : gcd a b ≠ 0) (h2 : d ∣ a) (h3 : d ∣ b) : d ≤ gcd a b := sorry -- 7. lemma Lemma_7_1_10a {a b : Nat} (n : Nat) (h : a ∣ b) : (n * a) ∣ (n * b) := sorry lemma Lemma_7_1_10b {a b n : Nat} (h1 : n ≠ 0) (h2 : (n * a) ∣ (n * b)) : a ∣ b := sorry lemma Lemma_7_1_10c {a b : Nat} (h1 : a ∣ b) (h2 : b ∣ a) : a = b := sorry theorem Exercise_7_1_10 (a b n : Nat) : gcd (n * a) (n * b) = n * gcd a b := sorry /- Section 7.2 -/ -- 1. lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry -- 2. -- Hints: Start with apply List.rec. You may find mul_ne_zero useful theorem prod_nonzero_nonzero : ∀ (l : List Nat), (∀ a ∈ l, a ≠ 0) → prod l ≠ 0 := sorry -- 3. theorem rel_prime_iff_no_common_factor (a b : Nat) : rel_prime a b ↔ ¬∃ (p : Nat), prime p ∧ p ∣ a ∧ p ∣ b := sorry -- 4. theorem rel_prime_symm {a b : Nat} (h : rel_prime a b) : rel_prime b a := sorry -- 5. lemma in_prime_factorization_iff_prime_factor {a : Nat} {l : List Nat} (h1 : prime_factorization a l) (p : Nat) : p ∈ l ↔ prime_factor p a := sorry -- 6. theorem Exercise_7_2_5 {a b : Nat} {l m : List Nat} (h1 : prime_factorization a l) (h2 : prime_factorization b m) : rel_prime a b ↔ (¬∃ (p : Nat), p ∈ l ∧ p ∈ m) := sorry -- 7. theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry -- 8. theorem Exercise_7_2_7 {a b a' b' : Nat} (h1 : rel_prime a b) (h2 : a' ∣ a) (h3 : b' ∣ b) : rel_prime a' b' := sorry -- 9. theorem Exercise_7_2_9 {a b j k : Nat} (h1 : gcd a b ≠ 0) (h2 : a = j * gcd a b) (h3 : b = k * gcd a b) : rel_prime j k := sorry -- 10. theorem Exercise_7_2_17a (a b c : Nat) : gcd a (b * c) ∣ gcd a b * gcd a c := sorry /- Section 7.3 -/ -- 1. theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry -- 2. theorem Theorem_7_3_6_3 {m : Nat} (X : ZMod m) : X + [0]_m = X := sorry -- 3. theorem Theorem_7_3_6_4 {m : Nat} (X : ZMod m) : ∃ (Y : ZMod m), X + Y = [0]_m := sorry -- 4. theorem Exercise_7_3_4a {m : Nat} (Z1 Z2 : ZMod m) (h1 : ∀ (X : ZMod m), X + Z1 = X) (h2 : ∀ (X : ZMod m), X + Z2 = X) : Z1 = Z2 := sorry -- 5. theorem Exercise_7_3_4b {m : Nat} (X Y1 Y2 : ZMod m) (h1 : X + Y1 = [0]_m) (h2 : X + Y2 = [0]_m) : Y1 = Y2 := sorry -- 6. theorem Theorem_7_3_10 (m a : Nat) (b : Int) : ¬(↑(gcd m a) : Int) ∣ b → ¬∃ (x : Int), a * x ≡ b (MOD m) := sorry -- 7. theorem Theorem_7_3_11 (m n : Nat) (a b : Int) (h1 : n ≠ 0) : n * a ≡ n * b (MOD n * m) ↔ a ≡ b (MOD m) := sorry -- 8. theorem Exercise_7_3_16 {m : Nat} {a b : Int} (h : a ≡ b (MOD m)) : ∀ (n : Nat), a ^ n ≡ b ^ n (MOD m) := sorry -- 9. example {m : Nat} [NeZero m] (X : ZMod m) : ∃! (a : Int), 0 ≤ a ∧ a < m ∧ X = [a]_m := sorry -- 10. theorem congr_rel_prime {m a b : Nat} (h1 : a ≡ b (MOD m)) : rel_prime m a ↔ rel_prime m b := sorry -- 11. --Hint: You may find the theorem Int.ofNat_mod_ofNat useful. theorem rel_prime_mod (m a : Nat) : rel_prime m (a % m) ↔ rel_prime m a := sorry -- 12. lemma congr_iff_mod_eq_Int (m : Nat) (a b : Int) [NeZero m] : a ≡ b (MOD m) ↔ a % ↑m = b % ↑m := sorry --Hint for next theorem: Use the lemma above, --together with the theorems Int.ofNat_mod_ofNat and Nat.cast_inj.

theorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] : ↑a ≡ ↑b (MOD m) ↔ a % m = b % m

HTPI.Exercises.congr_iff_mod_eq_Nat

htpi/HTPILib/Chap7.lean

htpi

/- Copyright 2023 Daniel J. Velleman -/ import HTPILib.Chap6 namespace HTPI /- Definitions -/ lemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by have h : n + 1 > 0 := Nat.succ_pos n show a % (n + 1) < n + 1 from Nat.mod_lt a h done def gcd (a b : Nat) : Nat := match b with | 0 => a | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd (n + 1) (a % (n + 1)) termination_by b mutual def gcd_c1 (a b : Nat) : Int := match b with | 0 => 1 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c2 (n + 1) (a % (n + 1)) --Corresponds to s = t' termination_by b def gcd_c2 (a b : Nat) : Int := match b with | 0 => 0 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c1 (n + 1) (a % (n + 1)) - (gcd_c2 (n + 1) (a % (n + 1))) * ↑(a / (n + 1)) --Corresponds to t = s' - t'q termination_by b end def prime (n : Nat) : Prop := 2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n def prime_factor (p n : Nat) : Prop := prime p ∧ p ∣ n def all_prime (l : List Nat) : Prop := ∀ p ∈ l, prime p def nondec (l : List Nat) : Prop := match l with | [] => True --Of course, True is a proposition that is always true | n :: L => (∀ m ∈ L, n ≤ m) ∧ nondec L def nondec_prime_list (l : List Nat) : Prop := all_prime l ∧ nondec l def prod (l : List Nat) : Nat := match l with | [] => 1 | n :: L => n * (prod L) def prime_factorization (n : Nat) (l : List Nat) : Prop := nondec_prime_list l ∧ prod l = n def rel_prime (a b : Nat) : Prop := gcd a b = 1 def congr_mod (m : Nat) (a b : Int) : Prop := (↑m : Int) ∣ (a - b) def cc (m : Nat) (a : Int) : ZMod m := (↑a : ZMod m) notation:50 a " ≡ " b " (MOD " m ")" => congr_mod m a b notation:max "["a"]_"m:max => cc m a def invertible {m : Nat} (X : ZMod m) : Prop := ∃ (Y : ZMod m), X * Y = [1]_m def num_rp_below (m k : Nat) : Nat := match k with | 0 => 0 | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j def phi (m : Nat) : Nat := num_rp_below m m def prod_seq {m : Nat} (j k : Nat) (f : Nat → ZMod m) : ZMod m := match j with | 0 => [1]_m | n + 1 => prod_seq n k f * f (k + n) def maps_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i < n, g i < n def one_one_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i1 < n, ∀ i2 < n, g i1 = g i2 → i1 = i2 def onto_below (n : Nat) (g : Nat → Nat) : Prop := ∀ k < n, ∃ i < n, g i = k def perm_below (n : Nat) (g : Nat → Nat) : Prop := maps_below n g ∧ one_one_below n g ∧ onto_below n g def inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m) def swap (u v i : Nat) : Nat := if i = u then v else if i = v then u else i namespace Euler --For definitions specific to Euler's theorem def F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m def G (m a i : Nat) : Nat := (a * i) % m def Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i end Euler /- Section 7.1 -/ theorem dvd_mod_of_dvd_a_b {a b d : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ (a % b) := by set q : Nat := a / b have h3 : b * q + a % b = a := Nat.div_add_mod a b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 define --Goal : ∃ (c : Nat), a % b = d * c apply Exists.intro (j - k * q) show a % b = d * (j - k * q) from calc a % b _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm _ = a - b * q := by rw [h3] _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc] _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm done theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry #eval gcd 672 161 --Answer: 7 lemma gcd_base (a : Nat) : gcd a 0 = a := by rfl lemma gcd_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd a b = gcd b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1)) rfl done lemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b ≠ 0) : a % b < b := by have h1 : b > 0 := Nat.pos_of_ne_zero h show a % b < b from Nat.mod_lt a h1 done lemma dvd_self (n : Nat) : n ∣ n := by apply Exists.intro 1 ring done theorem gcd_dvd : ∀ (b a : Nat), (gcd a b) ∣ a ∧ (gcd a b) ∣ b := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd a b_1) ∣ a ∧ (gcd a b_1) ∣ b_1 fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_base] --Goal: a ∣ a ∧ a ∣ 0 apply And.intro (dvd_self a) define apply Exists.intro 0 rfl done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_nonzero a h1] --Goal : gcd b (a % b) ∣ a ∧ gcd b (a % b) ∣ b have h2 : a % b < b := mod_nonzero_lt a h1 have h3 : (gcd b (a % b)) ∣ b ∧ (gcd b (a % b)) ∣ (a % b) := ih (a % b) h2 b apply And.intro _ h3.left show (gcd b (a % b)) ∣ a from dvd_a_of_dvd_b_mod h3.left h3.right done done theorem gcd_dvd_left (a b : Nat) : (gcd a b) ∣ a := (gcd_dvd b a).left theorem gcd_dvd_right (a b : Nat) : (gcd a b) ∣ b := (gcd_dvd b a).right lemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl lemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c1 a b = gcd_c2 b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done lemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl lemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * ↑(a / b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done theorem gcd_lin_comb : ∀ (b a : Nat), (gcd_c1 a b) * ↑a + (gcd_c2 a b) * ↑b = ↑(gcd a b) := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd_c1 a b_1) * ↑a + (gcd_c2 a b_1) * ↑b_1 = ↑(gcd a b_1) fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base] --Goal : 1 * ↑a + 0 * ↑0 = ↑a ring done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1] --Goal : gcd_c2 b (a % b) * ↑a + -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * ↑(a / b)) * ↑b = -- ↑(gcd b (a % b)) set r : Nat := a % b set q : Nat := a / b set s : Int := gcd_c1 b r set t : Int := gcd_c2 b r --Goal : t * ↑a + (s - t * ↑q) * ↑b = ↑(gcd b r) have h2 : r < b := mod_nonzero_lt a h1 have h3 : s * ↑b + t * ↑r = ↑(gcd b r) := ih r h2 b have h4 : b * q + r = a := Nat.div_add_mod a b rewrite [←h3, ←h4] rewrite [Nat.cast_add, Nat.cast_mul] --Goal : t * (↑b * ↑q + ↑r) + (s - t * ↑q) * ↑b = s * ↑b + t * ↑r ring done done #eval gcd_c1 672 161 --Answer: 6 #eval gcd_c2 672 161 --Answer: -25 --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161 theorem Theorem_7_1_6 {d a b : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ gcd a b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑d ∣ ↑(gcd a b) set s : Int := gcd_c1 a b set t : Int := gcd_c2 a b have h3 : s * ↑a + t * ↑b = ↑(gcd a b) := gcd_lin_comb b a rewrite [←h3] --Goal : ↑d ∣ s * ↑a + t * ↑b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul] --Goal : ↑d ∣ s * (↑d * ↑j) + t * (↑d * ↑k) define apply Exists.intro (s * ↑j + t * ↑k) ring done /- Section 7.2 -/ theorem dvd_trans {a b c : Nat} (h1 : a ∣ b) (h2 : b ∣ c) : a ∣ c := by define at h1; define at h2; define obtain (m : Nat) (h3 : b = a * m) from h1 obtain (n : Nat) (h4 : c = b * n) from h2 rewrite [h3, mul_assoc] at h4 apply Exists.intro (m * n) show c = a * (m * n) from h4 done lemma exists_prime_factor : ∀ (n : Nat), 2 ≤ n → ∃ (p : Nat), prime_factor p n := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, 2 ≤ n_1 → ∃ (p : Nat), prime_factor p n_1 assume h1 : 2 ≤ n by_cases h2 : prime n · -- Case 1. h2 : prime n apply Exists.intro n define --Goal : prime n ∧ n ∣ n show prime n ∧ n ∣ n from And.intro h2 (dvd_self n) done · -- Case 2. h2 : ¬prime n define at h2 --h2 : ¬(2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n) demorgan at h2 disj_syll h2 h1 obtain (a : Nat) (h3 : ∃ (b : Nat), a * b = n ∧ a < n ∧ b < n) from h2 obtain (b : Nat) (h4 : a * b = n ∧ a < n ∧ b < n) from h3 have h5 : 2 ≤ a := by by_contra h6 have h7 : a ≤ 1 := by linarith have h8 : n ≤ b := calc n _ = a * b := h4.left.symm _ ≤ 1 * b := by rel [h7] _ = b := by ring linarith --n ≤ b contradicts b < n done have h6 : ∃ (p : Nat), prime_factor p a := ih a h4.right.left h5 obtain (p : Nat) (h7 : prime_factor p a) from h6 apply Exists.intro p define --Goal : prime p ∧ p ∣ n define at h7 --h7 : prime p ∧ p ∣ a apply And.intro h7.left have h8 : a ∣ n := by apply Exists.intro b show n = a * b from (h4.left).symm done show p ∣ n from dvd_trans h7.right h8 done done lemma exists_least_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q := by set S : Set Nat := {p : Nat | prime_factor p n} have h2 : ∃ (p : Nat), p ∈ S := exists_prime_factor n h show ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q from well_ord_princ S h2 done lemma all_prime_nil : all_prime [] := by define --Goal : ∀ p ∈ [], prime p fix p : Nat contrapos --Goal : ¬prime p → p ∉ [] assume h1 : ¬prime p show p ∉ [] from List.not_mem_nil p done lemma all_prime_cons (n : Nat) (L : List Nat) : all_prime (n :: L) ↔ prime n ∧ all_prime L := by apply Iff.intro · -- (→) assume h1 : all_prime (n :: L) --Goal : prime n ∧ all_prime L define at h1 --h1 : ∀ p ∈ n :: L, prime p apply And.intro (h1 n (List.mem_cons_self n L)) define --Goal : ∀ p ∈ L, prime p fix p : Nat assume h2 : p ∈ L show prime p from h1 p (List.mem_cons_of_mem n h2) done · -- (←) assume h1 : prime n ∧ all_prime L --Goal : all_prime (n :: l) define : all_prime L at h1 define fix p : Nat assume h2 : p ∈ n :: L rewrite [List.mem_cons] at h2 --h2 : p = n ∨ p ∈ L by_cases on h2 · -- Case 1. h2 : p = n rewrite [h2] show prime n from h1.left done · -- Case 2. h2 : p ∈ L show prime p from h1.right p h2 done done done lemma nondec_nil : nondec [] := by define --Goal : True trivial --trivial proves some obviously true statements, such as True done lemma nondec_cons (n : Nat) (L : List Nat) : nondec (n :: L) ↔ (∀ m ∈ L, n ≤ m) ∧ nondec L := by rfl lemma prod_nil : prod [] = 1 := by rfl lemma prod_cons : prod (n :: L) = n * (prod L) := by rfl lemma exists_cons_of_length_eq_succ {A : Type} {l : List A} {n : Nat} (h : l.length = n + 1) : ∃ (a : A) (L : List A), l = a :: L ∧ L.length = n := by have h1 : ¬l.length = 0 := by linarith rewrite [List.length_eq_zero] at h1 obtain (a : A) (h2 : ∃ (L : List A), l = a :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List A) (h3 : l = a :: L) from h2 apply Exists.intro a apply Exists.intro L apply And.intro h3 have h4 : (a :: L).length = L.length + 1 := List.length_cons a L rewrite [←h3, h] at h4 show L.length = n from (Nat.add_right_cancel h4).symm done lemma list_elt_dvd_prod_by_length (a : Nat) : ∀ (n : Nat), ∀ (l : List Nat), l.length = n → a ∈ l → a ∣ prod l := by by_induc · --Base Case fix l : List Nat assume h1 : l.length = 0 rewrite [List.length_eq_zero] at h1 --h1 : l = [] rewrite [h1] --Goal : a ∈ [] → a ∣ prod [] contrapos assume h2 : ¬a ∣ prod [] show a ∉ [] from List.not_mem_nil a done · -- Induction Step fix n : Nat assume ih : ∀ (l : List Nat), List.length l = n → a ∈ l → a ∣ prod l fix l : List Nat assume h1 : l.length = n + 1 --Goal : a ∈ l → a ∣ prod l obtain (b : Nat) (h2 : ∃ (L : List Nat), l = b :: L ∧ L.length = n) from exists_cons_of_length_eq_succ h1 obtain (L : List Nat) (h3 : l = b :: L ∧ L.length = n) from h2 have h4 : a ∈ L → a ∣ prod L := ih L h3.right assume h5 : a ∈ l rewrite [h3.left, prod_cons] --Goal : a ∣ b * prod L rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b ∨ a ∈ L by_cases on h5 · -- Case 1. h5 : a = b apply Exists.intro (prod L) rewrite [h5] rfl done · -- Case 2. h5 : a ∈ L have h6 : a ∣ prod L := h4 h5 have h7 : prod L ∣ b * prod L := by apply Exists.intro b ring done show a ∣ b * prod L from dvd_trans h6 h7 done done done lemma list_elt_dvd_prod {a : Nat} {l : List Nat} (h : a ∈ l) : a ∣ prod l := by set n : Nat := l.length have h1 : l.length = n := by rfl show a ∣ prod l from list_elt_dvd_prod_by_length a n l h1 h done lemma exists_prime_factorization : ∀ (n : Nat), n ≥ 1 → ∃ (l : List Nat), prime_factorization n l := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, n_1 ≥ 1 → ∃ (l : List Nat), prime_factorization n_1 l assume h1 : n ≥ 1 by_cases h2 : n = 1 · -- Case 1. h2 : n = 1 apply Exists.intro [] define apply And.intro · -- Proof of nondec_prime_list [] define show all_prime [] ∧ nondec [] from And.intro all_prime_nil nondec_nil done · -- Proof of prod [] = n rewrite [prod_nil, h2] rfl done done · -- Case 2. h2 : n ≠ 1 have h3 : n ≥ 2 := lt_of_le_of_ne' h1 h2 obtain (p : Nat) (h4 : prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q) from exists_least_prime_factor h3 have p_prime_factor : prime_factor p n := h4.left define at p_prime_factor have p_prime : prime p := p_prime_factor.left have p_dvd_n : p ∣ n := p_prime_factor.right have p_least : ∀ (q : Nat), prime_factor q n → p ≤ q := h4.right obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n have h5 : m ≠ 0 := by contradict h1 with h6 have h7 : n = 0 := calc n _ = p * m := n_eq_pm _ = p * 0 := by rw [h6] _ = 0 := by ring rewrite [h7] decide done have m_pos : 0 < m := Nat.pos_of_ne_zero h5 have m_lt_n : m < n := by define at p_prime show m < n from calc m _ < m + m := by linarith _ = 2 * m := by ring _ ≤ p * m := by rel [p_prime.left] _ = n := n_eq_pm.symm done obtain (L : List Nat) (h6 : prime_factorization m L) from ih m m_lt_n m_pos define at h6 have ndpl_L : nondec_prime_list L := h6.left define at ndpl_L apply Exists.intro (p :: L) define apply And.intro · -- Proof of nondec_prime_list (p :: L) define apply And.intro · -- Proof of all_prime (p :: L) rewrite [all_prime_cons] show prime p ∧ all_prime L from And.intro p_prime ndpl_L.left done · -- Proof of nondec (p :: L) rewrite [nondec_cons] apply And.intro _ ndpl_L.right fix q : Nat assume q_in_L : q ∈ L have h7 : q ∣ prod L := list_elt_dvd_prod q_in_L rewrite [h6.right] at h7 --h7 : q ∣ m have h8 : m ∣ n := by apply Exists.intro p rewrite [n_eq_pm] ring done have q_dvd_n : q ∣ n := dvd_trans h7 h8 have ap_L : all_prime L := ndpl_L.left define at ap_L have q_prime_factor : prime_factor q n := And.intro (ap_L q q_in_L) q_dvd_n show p ≤ q from p_least q q_prime_factor done done · -- Proof of prod (p :: L) = n rewrite [prod_cons, h6.right, n_eq_pm] rfl done done done theorem Theorem_7_2_2 {a b c : Nat} (h1 : c ∣ a * b) (h2 : rel_prime a c) : c ∣ b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑c ∣ ↑b define at h1; define at h2; define obtain (j : Nat) (h3 : a * b = c * j) from h1 set s : Int := gcd_c1 a c set t : Int := gcd_c2 a c have h4 : s * ↑a + t * ↑c = ↑(gcd a c) := gcd_lin_comb c a rewrite [h2, Nat.cast_one] at h4 --h4 : s * ↑a + t * ↑c = (1 : Int) apply Exists.intro (s * ↑j + t * ↑b) show ↑b = ↑c * (s * ↑j + t * ↑b) from calc ↑b _ = (1 : Int) * ↑b := (one_mul _).symm _ = (s * ↑a + t * ↑c) * ↑b := by rw [h4] _ = s * (↑a * ↑b) + t * ↑c * ↑b := by ring _ = s * (↑c * ↑j) + t * ↑c * ↑b := by rw [←Nat.cast_mul a b, h3, Nat.cast_mul c j] _ = ↑c * (s * ↑j + t * ↑b) := by ring done lemma le_nonzero_prod_left {a b : Nat} (h : a * b ≠ 0) : a ≤ a * b := by have h1 : b ≠ 0 := by contradict h with h1 rewrite [h1] ring done have h2 : 1 ≤ b := Nat.pos_of_ne_zero h1 show a ≤ a * b from calc a = a * 1 := (mul_one a).symm _ ≤ a * b := by rel [h2] done lemma le_nonzero_prod_right {a b : Nat} (h : a * b ≠ 0) : b ≤ a * b := by rewrite [mul_comm] rewrite [mul_comm] at h show b ≤ b * a from le_nonzero_prod_left h done lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry lemma rel_prime_of_prime_not_dvd {a p : Nat} (h1 : prime p) (h2 : ¬p ∣ a) : rel_prime a p := by have h3 : gcd a p ∣ a := gcd_dvd_left a p have h4 : gcd a p ∣ p := gcd_dvd_right a p have h5 : gcd a p = 1 ∨ gcd a p = p := dvd_prime h1 h4 have h6 : gcd a p ≠ p := by contradict h2 with h6 rewrite [h6] at h3 show p ∣ a from h3 done disj_syll h5 h6 show rel_prime a p from h5 done theorem Theorem_7_2_3 {a b p : Nat} (h1 : prime p) (h2 : p ∣ a * b) : p ∣ a ∨ p ∣ b := by or_right with h3 have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3 show p ∣ b from Theorem_7_2_2 h2 h4 done lemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a ≥ 1 := by have h1 : a ≠ 0 := by by_contra h1 rewrite [h1] at h contradict h linarith done show a ≥ 1 from Nat.pos_of_ne_zero h1 done lemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by have h1 : a ≥ 1 := ge_one_of_prod_one h have h2 : a * b ≠ 0 := by linarith have h3 : a ≤ a * b := le_nonzero_prod_left h2 rewrite [h] at h3 show a = 1 from Nat.le_antisymm h3 h1 done lemma eq_one_of_dvd_one {n : Nat} (h : n ∣ 1) : n = 1 := by obtain (j : Nat) (h1 : 1 = n * j) from h show n = 1 from eq_one_of_prod_one h1.symm done lemma prime_not_one {p : Nat} (h : prime p) : p ≠ 1 := by define at h linarith done theorem Theorem_7_2_4 {p : Nat} (h1 : prime p) : ∀ (l : List Nat), p ∣ prod l → ∃ a ∈ l, p ∣ a := by apply List.rec · -- Base Case. Goal : p ∣ prod [] → ∃ a ∈ [], p ∣ a rewrite [prod_nil] assume h2 : p ∣ 1 show ∃ a ∈ [], p ∣ a from absurd (eq_one_of_dvd_one h2) (prime_not_one h1) done · -- Induction Step fix b : Nat fix L : List Nat assume ih : p ∣ prod L → ∃ a ∈ L, p ∣ a --Goal : p ∣ prod (b :: L) → ∃ a ∈ b :: L, p ∣ a assume h2 : p ∣ prod (b :: L) rewrite [prod_cons] at h2 have h3 : p ∣ b ∨ p ∣ prod L := Theorem_7_2_3 h1 h2 by_cases on h3 · -- Case 1. h3 : p ∣ b apply Exists.intro b show b ∈ b :: L ∧ p ∣ b from And.intro (List.mem_cons_self b L) h3 done · -- Case 2. h3 : p ∣ prod L obtain (a : Nat) (h4 : a ∈ L ∧ p ∣ a) from ih h3 apply Exists.intro a show a ∈ b :: L ∧ p ∣ a from And.intro (List.mem_cons_of_mem b h4.left) h4.right done done done lemma prime_in_list {p : Nat} {l : List Nat} (h1 : prime p) (h2 : all_prime l) (h3 : p ∣ prod l) : p ∈ l := by obtain (a : Nat) (h4 : a ∈ l ∧ p ∣ a) from Theorem_7_2_4 h1 l h3 define at h2 have h5 : prime a := h2 a h4.left have h6 : p = 1 ∨ p = a := dvd_prime h5 h4.right disj_syll h6 (prime_not_one h1) rewrite [h6] show a ∈ l from h4.left done lemma first_le_first {p q : Nat} {l m : List Nat} (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m)) (h3 : prod (p :: l) = prod (q :: m)) : p ≤ q := by define at h1; define at h2 have h4 : q ∣ prod (p :: l) := by define apply Exists.intro (prod m) rewrite [←prod_cons] show prod (p :: l) = prod (q :: m) from h3 done have h5 : all_prime (q :: m) := h2.left rewrite [all_prime_cons] at h5 have h6 : q ∈ p :: l := prime_in_list h5.left h1.left h4 have h7 : nondec (p :: l) := h1.right rewrite [nondec_cons] at h7 rewrite [List.mem_cons] at h6 by_cases on h6 · -- Case 1. h6 : q = p linarith done · -- Case 2. h6 : q ∈ l have h8 : ∀ m ∈ l, p ≤ m := h7.left show p ≤ q from h8 q h6 done done lemma nondec_prime_list_tail {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by define at h define rewrite [all_prime_cons, nondec_cons] at h show all_prime l ∧ nondec l from And.intro h.left.right h.right.right done lemma cons_prod_not_one {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : prod (p :: l) ≠ 1 := by define at h have h1 : all_prime (p :: l) := h.left rewrite [all_prime_cons] at h1 rewrite [prod_cons] by_contra h2 show False from (prime_not_one h1.left) (eq_one_of_prod_one h2) done lemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) : l = [] ↔ prod l = 1 := by apply Iff.intro · -- (→) assume h1 : l = [] rewrite [h1] show prod [] = 1 from prod_nil done · -- (←) contrapos assume h1 : ¬l = [] obtain (p : Nat) (h2 : ∃ (L : List Nat), l = p :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List Nat) (h3 : l = p :: L) from h2 rewrite [h3] at h rewrite [h3] show ¬prod (p :: L) = 1 from cons_prod_not_one h done done lemma prime_pos {p : Nat} (h : prime p) : p > 0 := by define at h linarith done theorem Theorem_7_2_5 : ∀ (l1 l2 : List Nat), nondec_prime_list l1 → nondec_prime_list l2 → prod l1 = prod l2 → l1 = l2 := by apply List.rec · -- Base Case. Goal : ∀ (l2 : List Nat), nondec_prime_list [] → -- nondec_prime_list l2 → prod [] = prod l2 → [] = l2 fix l2 : List Nat assume h1 : nondec_prime_list [] assume h2 : nondec_prime_list l2 assume h3 : prod [] = prod l2 rewrite [prod_nil, eq_comm, ←list_nil_iff_prod_one h2] at h3 show [] = l2 from h3.symm done · -- Induction Step fix p : Nat fix L1 : List Nat assume ih : ∀ (L2 : List Nat), nondec_prime_list L1 → nondec_prime_list L2 → prod L1 = prod L2 → L1 = L2 -- Goal : ∀ (l2 : List Nat), nondec_prime_list (p :: L1) → -- nondec_prime_list l2 → prod (p :: L1) = prod l2 → p :: L1 = l2 fix l2 : List Nat assume h1 : nondec_prime_list (p :: L1) assume h2 : nondec_prime_list l2 assume h3 : prod (p :: L1) = prod l2 have h4 : ¬prod (p :: L1) = 1 := cons_prod_not_one h1 rewrite [h3, ←list_nil_iff_prod_one h2] at h4 obtain (q : Nat) (h5 : ∃ (L : List Nat), l2 = q :: L) from List.exists_cons_of_ne_nil h4 obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5 rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2) rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2) have h7 : p ≤ q := first_le_first h1 h2 h3 have h8 : q ≤ p := first_le_first h2 h1 h3.symm have h9 : p = q := by linarith rewrite [h9, prod_cons, prod_cons] at h3 --h3 : q * prod L1 = q * prod L2 have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1 have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2 define at h2 have h12 : all_prime (q :: L2) := h2.left rewrite [all_prime_cons] at h12 have h13 : q > 0 := prime_pos h12.left have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3 have h15 : L1 = L2 := ih L2 h10 h11 h14 rewrite [h6, h9, h15] rfl done done theorem fund_thm_arith (n : Nat) (h : n ≥ 1) : ∃! (l : List Nat), prime_factorization n l := by exists_unique · -- Existence show ∃ (l : List Nat), prime_factorization n l from exists_prime_factorization n h done · -- Uniqueness fix l1 : List Nat; fix l2 : List Nat assume h1 : prime_factorization n l1 assume h2 : prime_factorization n l2 define at h1; define at h2 have h3 : prod l1 = n := h1.right rewrite [←h2.right] at h3 show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3 done done /- Section 7.3 -/ theorem congr_refl (m : Nat) : ∀ (a : Int), a ≡ a (MOD m) := by fix a : Int define --Goal : ∃ (c : Int), a - a = ↑m * c apply Exists.intro 0 ring done theorem congr_symm {m : Nat} : ∀ {a b : Int}, a ≡ b (MOD m) → b ≡ a (MOD m) := by fix a : Int; fix b : Int assume h1 : a ≡ b (MOD m) define at h1 --h1 : ∃ (c : Int), a - b = ↑m * c define --Goal : ∃ (c : Int), b - a = ↑m * c obtain (c : Int) (h2 : a - b = m * c) from h1 apply Exists.intro (-c) show b - a = m * (-c) from calc b - a _ = -(a - b) := by ring _ = -(m * c) := by rw [h2] _ = m * (-c) := by ring done theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry /- Fundamental properties of congruence classes -/ lemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) : X = Y ↔ X.val = Y.val := Fin.ext_iff lemma val_nat_eq_mod (n k : Nat) : ([k]_(n + 1)).val = k % (n + 1) := by rfl lemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl theorem cc_rep {m : Nat} (X : ZMod m) : ∃ (a : Int), X = [a]_m := match m with | 0 => by apply Exists.intro X rfl done | n + 1 => by apply Exists.intro ↑(X.val) have h1 : X.val < n + 1 := Fin.prop X rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1] rfl done theorem add_class (m : Nat) (a b : Int) : [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm theorem mul_class (m : Nat) (a b : Int) : [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm lemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ [a - b]_m = [0]_m := by apply Iff.intro · -- (→) assume h1 : [a]_m = [b]_m have h2 : a - b = a + (-b) := by ring have h3 : b + (-b) = 0 := by ring show [a - b]_m = [0]_m from calc [a - b]_m _ = [a + (-b)]_m := by rw [h2] _ = [a]_m + [-b]_m := by rw [add_class] _ = [b]_m + [-b]_m := by rw [h1] _ = [b + -b]_m := by rw [add_class] _ = [0]_m := by rw [h3] done · -- (←) assume h1 : [a - b]_m = [0]_m have h2 : b + (a - b) = a := by ring have h3 : b + 0 = b := by ring show [a]_m = [b]_m from calc [a]_m _ = [b + (a - b)]_m := by rw [h2] _ = [b]_m + [a - b]_m := by rw [add_class] _ = [b]_m + [0]_m := by rw [h1] _ = [b + 0]_m := by rw [add_class] _ = [b]_m := by rw [h3] done done lemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) : [a]_m = [0]_m → [-a]_m = [0]_m := by assume h1 : [a]_m = [0]_m have h2 : 0 + (-a) = -a := by ring have h3 : a + (-a) = 0 := by ring show [-a]_m = [0]_m from calc [-a]_m _ = [0 + (-a)]_m := by rw [h2] _ = [0]_m + [-a]_m := by rw [add_class] _ = [a]_m + [-a]_m := by rw [h1] _ = [a + (-a)]_m := by rw [add_class] _ = [0]_m := by rw [h3] done lemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) : [-a]_m = [0]_m ↔ [a]_m = [0]_m := by apply Iff.intro _ (cc_neg_zero_of_cc_zero m a) assume h1 : [-a]_m = [0]_m have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1 have h3 : -(-a) = a := by ring rewrite [h3] at h2 show [a]_m = [0]_m from h2 done lemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl lemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m ↔ m ∣ k := match m with | 0 => by have h : (0 : Int) = (↑(0 : Nat) : Int) := by rfl rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj] apply Iff.intro · -- (→) assume h1 : k = 0 rewrite [h1] show 0 ∣ 0 from dvd_self 0 done · -- (←) assume h1 : 0 ∣ k obtain (c : Nat) (h2 : k = 0 * c) from h1 rewrite [h2] ring done done | n + 1 => by rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero] show k % (n + 1) = 0 ↔ n + 1 ∣ k from (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm done lemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m ↔ ↑m ∣ a := by obtain (k : Nat) (h1 : a = ↑k ∨ a = -↑k) from Int.eq_nat_or_neg a by_cases on h1 · -- Case 1. h1: a = ↑k rewrite [h1, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done · -- Case 2. h1: a = -↑k rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done done theorem cc_eq_iff_congr (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ a ≡ b (MOD m) := calc [a]_m = [b]_m _ ↔ [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b _ ↔ ↑m ∣ (a - b) := cc_zero_iff_dvd m (a - b) _ ↔ a ≡ b (MOD m) := by rfl /- End of fundamental properties of congruence classes -/ lemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m := by have h1 : (↑m : Int) ≠ 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m) show 0 ≤ a % m from Int.emod_nonneg a h1 done lemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m) have h2 : (↑m : Int) > 0 := (Nat.cast_pos).rtl h1 show a % m < m from Int.emod_lt_of_pos a h2 done lemma congr_mod_mod (m : Nat) (a : Int) : a ≡ a % m (MOD m) := by define have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m apply Exists.intro (a / m) show a - a % m = m * (a / m) from calc a - (a % m) _ = m * (a / m) + a % m - a % m := by rw [h1] _ = m * (a / m) := by ring done lemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) := And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a)) theorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) : ∃! (r : Int), 0 ≤ r ∧ r < m ∧ a ≡ r (MOD m) := by exists_unique · -- Existence apply Exists.intro (a % m) show 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) from mod_cmpl_res m a done · -- Uniqueness fix r1 : Int; fix r2 : Int assume h1 : 0 ≤ r1 ∧ r1 < m ∧ a ≡ r1 (MOD m) assume h2 : 0 ≤ r2 ∧ r2 < m ∧ a ≡ r2 (MOD m) have h3 : r1 ≡ r2 (MOD m) := congr_trans (congr_symm h1.right.right) h2.right.right obtain (d : Int) (h4 : r1 - r2 = m * d) from h3 have h5 : r1 - r2 < m * 1 := by linarith have h6 : m * (-1) < r1 - r2 := by linarith rewrite [h4] at h5 --h5 : m * d < m * 1 rewrite [h4] at h6 --h6 : m * -1 < m * d have h7 : (↑m : Int) ≥ 0 := Nat.cast_nonneg m have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7 have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7 have h10 : d = 0 := by linarith show r1 = r2 from calc r1 _ = r1 - r2 + r2 := by ring _ = m * 0 + r2 := by rw [h4, h10] _ = r2 := by ring done done lemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m := (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a) theorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y rewrite [h1, h2] have h3 : a + b = b + a := by ring show [a]_m + [b]_m = [b]_m + [a]_m from calc [a]_m + [b]_m _ = [a + b]_m := add_class m a b _ = [b + a]_m := by rw [h3] _ = [b]_m + [a]_m := (add_class m b a).symm done theorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X rewrite [h1] have h2 : a * 1 = a := by ring show [a]_m * [1]_m = [a]_m from calc [a]_m * [1]_m _ = [a * 1]_m := mul_class m a 1 _ = [a]_m := by rw [h2] done theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry lemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) : [a]_m * [gcd_c2 m a]_m = [1]_m := by set s : Int := gcd_c1 m a have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m define at h1 rewrite [h1, Nat.cast_one] at h2 --h2 : s * ↑m + gcd_c2 m a * ↑a = 1 rewrite [mul_class, cc_eq_iff_congr] define --Goal : ∃ (c : Int), ↑a * gcd_c2 m a - 1 = ↑m * c apply Exists.intro (-s) show a * (gcd_c2 m a) - 1 = m * (-s) from calc a * (gcd_c2 m a) - 1 _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring _ = 1 + m * (-s) - 1 := by rw [h2] _ = m * (-s) := by ring done theorem Theorem_7_3_7 (m a : Nat) : invertible [a]_m ↔ rel_prime m a := by apply Iff.intro · -- (→) assume h1 : invertible [a]_m define at h1 obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1 obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y rewrite [h3, mul_class, cc_eq_iff_congr] at h2 define at h2 obtain (c : Int) (h4 : a * b - 1 = m * c) from h2 rewrite [Exercise_7_2_6] --Goal : ∃ (s t : Int), s * ↑m + t * ↑a = 1 apply Exists.intro (-c) apply Exists.intro b show (-c) * m + b * a = 1 from calc (-c) * m + b * a _ = (-c) * m + (a * b - 1) + 1 := by ring _ = (-c) * m + m * c + 1 := by rw [h4] _ = 1 := by ring done · -- (←) assume h1 : rel_prime m a define show ∃ (Y : ZMod m), [a]_m * Y = [1]_m from Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1) done done /- Section 7.4 -/ section Euler open Euler lemma num_rp_below_base {m : Nat} : num_rp_below m 0 = 0 := by rfl lemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) : num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by have h1 : num_rp_below m (j + 1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : gcd m j = 1 rewrite [if_pos h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1 show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1 done lemma num_rp_below_step_not_rp {m j : Nat} (h : ¬rel_prime m j) : num_rp_below m (j + 1) = num_rp_below m j := by have h1 : num_rp_below m (j +1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : ¬gcd m j = 1 rewrite [if_neg h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j show num_rp_below m (j + 1) = num_rp_below m j from h1 done lemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl #eval phi 10 --Answer: 4 lemma prod_inv_iff_inv {m : Nat} {X : ZMod m} (h1 : invertible X) (Y : ZMod m) : invertible (X * Y) ↔ invertible Y := by apply Iff.intro · -- (→) assume h2 : invertible (X * Y) obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2 apply Exists.intro (X * Z) rewrite [←h3] --Goal : Y * (X * Z) = X * Y * Z ring --Note that ring can do algebra in ZMod m done · -- (←) assume h2 : invertible Y obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1 obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2 apply Exists.intro (Xi * Yi) show (X * Y) * (Xi * Yi) = [1]_m from calc X * Y * (Xi * Yi) _ = (X * Xi) * (Y * Yi) := by ring _ = [1]_m * [1]_m := by rw [h3, h4] _ = [1]_m := Theorem_7_3_6_7 [1]_m done done lemma F_rp_def {m i : Nat} (h : rel_prime m i) : F m i = [i]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h --h : gcd m i = 1 rewrite [if_pos h] at h1 show F m i = [i]_m from h1 done lemma F_not_rp_def {m i : Nat} (h : ¬rel_prime m i) : F m i = [1]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h rewrite [h1, if_neg h] rfl done lemma prod_seq_base {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 0 k f = [1]_m := by rfl lemma prod_seq_step {m : Nat} (n k : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl lemma prod_seq_zero_step {m : Nat} (n : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by rewrite [prod_seq_step, zero_add] rfl done lemma prod_one {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 1 k f = f k := by rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7] rfl done lemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl lemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m := calc [G m a i]_m _ = [(a * i) % m]_m := by rfl _ = [a * i]_m := (cc_eq_mod m (a * i)).symm _ = [a]_m * [i]_m := (mul_class m a i).symm lemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) : rel_prime m (G m a i) ↔ rel_prime m i := by have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1 show rel_prime m (G m a i) ↔ rel_prime m i from calc rel_prime m (G m a i) _ ↔ invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm _ ↔ invertible ([a]_m * [i]_m) := by rw [cc_G] _ ↔ invertible [i]_m := prod_inv_iff_inv h2 ([i]_m) _ ↔ rel_prime m i := Theorem_7_3_7 m i done lemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) : F m (G m a i) = [a]_m * F m i := by have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2 show F m (G m a i) = [a]_m * F m i from calc F m (G m a i) _ = [G m a i]_m := F_rp_def h3 _ = [a]_m * [i]_m := cc_G m a i _ = [a]_m * F m i := by rw [F_rp_def h2] done lemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : ¬rel_prime m i) : F m (G m a i) = [1]_m := by rewrite [←G_rp_iff h1 i] at h2 show F m (G m a i) = [1]_m from F_not_rp_def h2 done lemma FG_prod {m a : Nat} (h1 : rel_prime m a) : ∀ (k : Nat), prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by by_induc · -- Base Case show prod_seq 0 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from calc prod_seq 0 0 ((F m) ∘ (G m a)) _ = [1]_m := prod_seq_base _ _ _ = [a]_m ^ 0 * [1]_m := by ring _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by rw [num_rp_below_base, prod_seq_base] done · -- Induction Step fix k : Nat assume ih : prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) by_cases h2 : rel_prime m k · -- Case 1. h2 : rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([a]_m * F m k) := by rw [FG_rp h1 h2] _ = [a]_m ^ ((num_rp_below m k) + 1) * ((prod_seq k 0 (F m)) * F m k) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_rp h2, prod_seq_zero_step] done · -- Case 2. h2 : ¬rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([1]_m) := by rw [FG_not_rp h1 h2] _ = [a]_m ^ (num_rp_below m k) * (prod_seq k 0 (F m) * ([1]_m)) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_not_rp h2, prod_seq_zero_step, F_not_rp_def h2] done done done lemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by define --Goal : ∀ i < m, G m a i < m fix i : Nat assume h1 : i < m rewrite [G_def] --Goal : a * i % m < m show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m) done lemma left_inv_one_one_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g' (g i) = i) : one_one_below n g := sorry lemma right_inv_onto_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g (g' i) = i) (h2 : maps_below n g') : onto_below n g := by define at h2; define fix k : Nat assume h3 : k < n apply Exists.intro (g' k) show g' k < n ∧ g (g' k) = k from And.intro (h2 k h3) (h1 k h3) done lemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m * [inv_mod m a]_m = [1]_m := by have h2 : 0 ≤ (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a) show [a]_m * [inv_mod m a]_m = [1]_m from calc [a]_m * [inv_mod m a]_m _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2] _ = [a]_m * [gcd_c2 m a]_m := by rw [←cc_eq_mod] _ = [1]_m := gcd_c2_inv h1 done lemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m := calc a * (b % m) % m = a % m * (b % m % m) % m := Nat.mul_mod _ _ _ _ = a % m * (b % m) % m := by rw [Nat.mod_mod] _ = a * b % m := (Nat.mul_mod _ _ _).symm lemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm] rfl done theorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] : ↑a ≡ ↑b (MOD m) ↔ a % m = b % m := sorry lemma mul_inv_mod_cancel {m a i : Nat} [NeZero m] (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1 rewrite [mul_class, cc_eq_iff_congr, ←Nat.cast_mul, ←Nat.cast_one, congr_iff_mod_eq_Nat] at h3 show a * inv_mod m a * i % m = i from calc a * (inv_mod m a) * i % m _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = 1 % m * i % m := by rw [h3] _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = i % m := by rw [one_mul] _ = i := Nat.mod_eq_of_lt h2 done lemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl lemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, G m a (Ginv m a i) = i := by fix i : Nat assume h2 : i < m show G m a (Ginv m a i) = i from calc G m a (Ginv m a i) _ = a * ((inv_mod m a * i) % m) % m := by rfl _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, Ginv m a (G m a i) = i := by fix i : Nat assume h2 : i < m show Ginv m a (G m a i) = i from calc Ginv m a (G m a i) _ = inv_mod m a * ((a * i) % m) % m := by rfl _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc, mul_comm (inv_mod m a)] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_maps_below (m a : Nat) [NeZero m] : maps_below m (Ginv m a) := G_maps_below m (inv_mod m a) lemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : one_one_below m (G m a) := left_inv_one_one_below (Ginv_left_inv h1) lemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : onto_below m (G m a) := right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a) lemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : perm_below m (G m a) := And.intro (G_maps_below m a) (And.intro (G_one_one_below h1) (G_onto_below h1)) --Permuting a product of congruence classes doesn't change product lemma swap_fst (u v : Nat) : swap u v u = v := by define : swap u v u --Goal : (if u = u then v else if u = v then u else u) = v have h : u = u := by rfl rewrite [if_pos h] rfl done lemma swap_snd (u v : Nat) : swap u v v = u := by define : swap u v v by_cases h1 : v = u · -- Case 1. h1 : v = u rewrite [if_pos h1] show v = u from h1 done · -- Case 2. h1 : v ≠ u rewrite [if_neg h1] have h2 : v = v := by rfl rewrite [if_pos h2] rfl done done lemma swap_other {u v i : Nat} (h1 : i ≠ u) (h2 : i ≠ v) : swap u v i = i := by define : swap u v i rewrite [if_neg h1, if_neg h2] rfl done lemma swap_values (u v i : Nat) : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := by by_cases h1 : i = u · -- Case 1. h1 : i = u apply Or.inl rewrite [h1] show swap u v u = v from swap_fst u v done · -- Case 2. h1 : i ≠ u apply Or.inr by_cases h2 : i = v · -- Case 2.1. h2 : i = v apply Or.inl rewrite [h2] show swap u v v = u from swap_snd u v done · -- Case 2.2. h2 : i ≠ v apply Or.inr show swap u v i = i from swap_other h1 h2 done done done lemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by define fix i : Nat assume h3 : i < n have h4 : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := swap_values u v i by_cases on h4 · -- Case 1. h4 : swap u v i = v rewrite [h4] show v < n from h2 done · -- Case 2. by_cases on h4 · -- Case 2.1. h4 : swap u v i = u rewrite [h4] show u < n from h1 done · -- Case 2.2. h4 : swap u v i = i rewrite [h4] show i < n from h3 done done done lemma swap_swap (u v n : Nat) : ∀ i < n, swap u v (swap u v i) = i := by fix i : Nat assume h : i < n by_cases h1 : i = u · -- Case 1. h1 : i = u rewrite [h1, swap_fst, swap_snd] rfl done · -- Case 2. h1 : i ≠ u by_cases h2 : i = v · -- Case 2.1. h2 : i = v rewrite [h2, swap_snd, swap_fst] rfl done · -- Case 2.2. h2 : i ≠ v rewrite [swap_other h1 h2, swap_other h1 h2] rfl done done done lemma swap_one_one_below (u v n) : one_one_below n (swap u v) := left_inv_one_one_below (swap_swap u v n) lemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) := right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2) lemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) := And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2)) lemma comp_perm_below {n : Nat} {f g : Nat → Nat} (h1 : perm_below n f) (h2 : perm_below n g) : perm_below n (f ∘ g) := sorry lemma trivial_swap (u : Nat) : swap u u = id := by apply funext fix x : Nat by_cases h1 : x = u · -- Case 1. h1 : x = u rewrite [h1, swap_fst] rfl done · -- Case 2. h1 : x ≠ u rewrite [swap_other h1 h1] rfl done done lemma prod_eq_fun {m : Nat} (f g : Nat → ZMod m) (k : Nat) : ∀ (n : Nat), (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g := by by_induc · -- Base Case assume h : (∀ i < 0, f (k + i) = g (k + i)) rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g assume h1 : ∀ i < n + 1, f (k + i) = g (k + i) have h2 : ∀ i < n, f (k + i) = g (k + i) := by fix i : Nat assume h2 : i < n have h3 : i < n + 1 := by linarith show f (k + i) = g (k + i) from h1 i h3 done have h3 : prod_seq n k f = prod_seq n k g := ih h2 have h4 : n < n + 1 := Nat.lt_succ_self n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4] rfl done done lemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := by have h2 : ∀ (i : Nat), i < u → (f ∘ swap u n) (0 + i) = f (0 + i) := by fix i : Nat assume h2 : i < u have h3 : 0 + i ≠ u := by linarith have h4 : 0 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl done show prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f from prod_eq_fun (f ∘ swap u n) f 0 u h2 done lemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := by have h2 : ∀ i < j, (f ∘ swap u n) (u + 1 + i) = f (u + 1 + i) := by fix i : Nat assume h2 : i < j have h3 : u + 1 + i ≠ u := by linarith have h4 : u + 1 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl show prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f from prod_eq_fun (f ∘ swap u n) f (u + 1) j h2 done lemma break_prod {m : Nat} (n : Nat) (f : Nat → ZMod m) : ∀ (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by by_induc · -- Base Case have h : n + 0 = n := by rfl rewrite [prod_seq_base, h, Theorem_7_3_6_7] rfl done · -- Induction Step fix j : Nat assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f rewrite [←add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc] rfl done done lemma break_prod_twice {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f := break_prod n f 1 rewrite [prod_one] at h2 have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f := break_prod (u + 1) f j rewrite [←h1] at h3 have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f := break_prod u f 1 rewrite [prod_one] at h4 rewrite [h3, h4] at h2 show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2 done lemma swap_prod_eq_prod {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f := by by_cases h2 : u = n · -- Case 1. h2 : u = n rewrite [h2, trivial_swap n] --Goal : prod_seq (n + 1) 0 (f ∘ id) = prod_seq (n + 1) 0 f rfl done · -- Case 2. h2 : ¬u = n have h3 : u + 1 ≤ n := Nat.lt_of_le_of_ne h1 h2 obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3 have break_f : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := break_prod_twice f h4 have break_fs : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_prod_twice (f ∘ swap u n) h4 have f_eq_fs_below : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := swap_prod_eq_prod_below f h1 have f_eq_fs_btwn : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4 show prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f from calc prod_seq (n + 1) 0 (f ∘ swap u n) _ = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_fs _ = prod_seq u 0 f * (f ∘ swap u n) u * prod_seq j (u + 1) f * (f ∘ swap u n) n := by rw [f_eq_fs_below, f_eq_fs_btwn] _ = prod_seq u 0 f * f (swap u n u) * prod_seq j (u + 1) f * f (swap u n n) := by rfl _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by rw [swap_fst, swap_snd] _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring _ = prod_seq (n + 1) 0 f := break_f.symm done done lemma perm_below_fixed {n : Nat} {g : Nat → Nat} (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry lemma perm_prod {m : Nat} (f : Nat → ZMod m) : ∀ (n : Nat), ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) := by by_induc · -- Base Case fix g : Nat → Nat assume h1 : perm_below 0 g rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) fix g : Nat → Nat assume g_pb : perm_below (n + 1) g define at g_pb have g_ob : onto_below (n + 1) g := g_pb.right.right define at g_ob have h1 : n < n + 1 := by linarith obtain (u : Nat) (h2 : u < n + 1 ∧ g u = n) from g_ob n h1 have s_pb : perm_below (n + 1) (swap u n) := swap_perm_below h2.left h1 have gs_pb_n1 : perm_below (n + 1) (g ∘ swap u n) := comp_perm_below g_pb s_pb have gs_fix_n : (g ∘ swap u n) n = n := calc (g ∘ swap u n) n _ = g (swap u n n) := by rfl _ = g u := by rw [swap_snd] _ = n := h2.right have gs_pb_n : perm_below n (g ∘ swap u n) := perm_below_fixed gs_pb_n1 gs_fix_n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f ∘ (g ∘ swap u n)) := ih (g ∘ swap u n) gs_pb_n have h3 : u ≤ n := by linarith show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f ∘ g) from calc prod_seq (n + 1) 0 f _ = prod_seq n 0 f * f n := prod_seq_zero_step n f _ = prod_seq n 0 (f ∘ (g ∘ swap u n)) * f ((g ∘ swap u n) n) := by rw [gs_prod, gs_fix_n] _ = prod_seq n 0 (f ∘ g ∘ swap u n) * (f ∘ g ∘ swap u n) n := by rfl _ = prod_seq (n + 1) 0 (f ∘ g ∘ swap u n) := (prod_seq_zero_step n (f ∘ g ∘ swap u n)).symm _ = prod_seq (n + 1) 0 ((f ∘ g) ∘ swap u n) := by rfl _ = prod_seq (n + 1) 0 (f ∘ g) := swap_prod_eq_prod (f ∘ g) h3 done done lemma F_invertible (m i : Nat) : invertible (F m i) := by by_cases h : rel_prime m i · -- Case 1. h : rel_prime m i rewrite [F_rp_def h] show invertible [i]_m from (Theorem_7_3_7 m i).rtl h done · -- Case 2. h : ¬rel_prime m i rewrite [F_not_rp_def h] apply Exists.intro [1]_m show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m done done lemma Fprod_invertible (m : Nat) : ∀ (k : Nat), invertible (prod_seq k 0 (F m)) := by by_induc · -- Base Case apply Exists.intro [1]_m show prod_seq 0 0 (F m) * [1]_m = [1]_m from calc prod_seq 0 0 (F m) * [1]_m _ = [1]_m * [1]_m := by rw [prod_seq_base] _ = [1]_m := Theorem_7_3_6_7 ([1]_m) done · -- Induction Step fix k : Nat assume ih : invertible (prod_seq k 0 (F m)) rewrite [prod_seq_zero_step] show invertible (prod_seq k 0 (F m) * (F m k)) from (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k) done done theorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m ^ (phi m) = [1]_m := by have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2 show [a]_m ^ (phi m) = [1]_m from calc [a]_m ^ (phi m) _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3] _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring _ = prod_seq m 0 (F m ∘ G m a) * Y := by rw [FG_prod h1 m, phi_def] _ = prod_seq m 0 (F m) * Y := by rw [perm_prod (F m) m (G m a) (G_perm_below h1)] _ = [1]_m := by rw [h3] done lemma Exercise_7_4_5_Int (m : Nat) (a : Int) : ∀ (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry lemma Exercise_7_4_5_Nat (m a n : Nat) : [a]_m ^ n = [a ^ n]_m := by rewrite [Exercise_7_4_5_Int] rfl done theorem Euler's_theorem {m a : Nat} [NeZero m] (h1 : rel_prime m a) : a ^ (phi m) ≡ 1 (MOD m) := by have h2 : [a]_m ^ (phi m) = [1]_m := Theorem_7_4_2 h1 rewrite [Exercise_7_4_5_Nat m a (phi m)] at h2 --h2 : [a ^ phi m]_m = [1]_m show a ^ (phi m) ≡ 1 (MOD m) from (cc_eq_iff_congr _ _ _).ltr h2 done #eval gcd 10 7 --Answer: 1. So 10 and 7 are relatively prime #eval 7 ^ phi 10 --Answer: 2401, which is congruent to 1 mod 10. end Euler /- Section 7.5 -/ lemma num_rp_prime {p : Nat} (h1 : prime p) : ∀ k < p, num_rp_below p (k + 1) = k := sorry lemma phi_prime {p : Nat} (h1 : prime p) : phi p = p - 1 := by have h2 : 1 ≤ p := prime_pos h1 have h3 : p - 1 + 1 = p := Nat.sub_add_cancel h2 have h4 : p - 1 < p := by linarith have h5 : num_rp_below p (p - 1 + 1) = p - 1 := num_rp_prime h1 (p - 1) h4 rewrite [h3] at h5 show phi p = p - 1 from h5 done theorem Theorem_7_2_2_Int {a c : Nat} {b : Int} (h1 : ↑c ∣ ↑a * b) (h2 : rel_prime a c) : ↑c ∣ b := by rewrite [Int.natCast_dvd, Int.natAbs_mul, Int.natAbs_ofNat] at h1 --h1 : c ∣ a * Int.natAbs b rewrite [Int.natCast_dvd] --Goal : c ∣ Int.natAbs b show c ∣ Int.natAbs b from Theorem_7_2_2 h1 h2 done lemma Lemma_7_4_5 {m n : Nat} (a b : Int) (h1 : rel_prime m n) : a ≡ b (MOD m * n) ↔ a ≡ b (MOD m) ∧ a ≡ b (MOD n) := by apply Iff.intro · -- (→) assume h2 : a ≡ b (MOD m * n) obtain (j : Int) (h3 : a - b = (m * n) * j) from h2 apply And.intro · -- Proof of a ≡ b (MOD m) apply Exists.intro (n * j) show a - b = m * (n * j) from calc a - b _ = m * n * j := h3 _ = m * (n * j) := by ring done · -- Proof of a ≡ b (MOD n) apply Exists.intro (m * j) show a - b = n * (m * j) from calc a - b _ = m * n * j := h3 _ = n * (m * j) := by ring done done · -- (←) assume h2 : a ≡ b (MOD m) ∧ a ≡ b (MOD n) obtain (j : Int) (h3 : a - b = m * j) from h2.left have h4 : (↑n : Int) ∣ a - b := h2.right rewrite [h3] at h4 --h4 : ↑n ∣ ↑m * j have h5 : ↑n ∣ j := Theorem_7_2_2_Int h4 h1 obtain (k : Int) (h6 : j = n * k) from h5 apply Exists.intro k --Goal : a - b = ↑(m * n) * k rewrite [Nat.cast_mul] --Goal : a - b = ↑m * ↑n * k show a - b = (m * n) * k from calc a - b _ = m * j := h3 _ = m * (n * k) := by rw [h6] _ = (m * n) * k := by ring done done --From exercises of Section 7.2 theorem rel_prime_symm {a b : Nat} (h : rel_prime a b) : rel_prime b a := sorry lemma prime_NeZero {p : Nat} (h : prime p) : NeZero p := by rewrite [neZero_iff] --Goal : p ≠ 0 define at h linarith done lemma Lemma_7_5_1 {p e d m c s : Nat} {t : Int} (h1 : prime p) (h2 : e * d = (p - 1) * s + 1) (h3 : m ^ e - c = p * t) : c ^ d ≡ m (MOD p) := by have h4 : m ^ e ≡ c (MOD p) := Exists.intro t h3 have h5 : [m ^ e]_p = [c]_p := (cc_eq_iff_congr _ _ _).rtl h4 rewrite [←Exercise_7_4_5_Nat] at h5 --h5 : [m]_p ^ e = [c]_p by_cases h6 : p ∣ m · -- Case 1. h6 : p ∣ m have h7 : m ≡ 0 (MOD p) := by obtain (j : Nat) (h8 : m = p * j) from h6 apply Exists.intro (↑j : Int) --Goal : ↑m - 0 = ↑p * ↑j rewrite [h8, Nat.cast_mul] ring done have h8 : [m]_p = [0]_p := (cc_eq_iff_congr _ _ _).rtl h7 have h9 : e * d ≠ 0 := by rewrite [h2] show (p - 1) * s + 1 ≠ 0 from Nat.add_one_ne_zero _ done have h10 : (0 : Int) ^ (e * d) = 0 := zero_pow h9 have h11 : [c ^ d]_p = [m]_p := calc [c ^ d]_p _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat] _ = ([m]_p ^ e) ^ d := by rw [h5] _ = [m]_p ^ (e * d) := by ring _ = [0]_p ^ (e * d) := by rw [h8] _ = [0 ^ (e * d)]_p := Exercise_7_4_5_Int _ _ _ _ = [0]_p := by rw [h10] _ = [m]_p := by rw [h8] show c ^ d ≡ m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11 done · -- Case 2. h6 : ¬p ∣ m have h7 : rel_prime m p := rel_prime_of_prime_not_dvd h1 h6 have h8 : rel_prime p m := rel_prime_symm h7 have h9 : NeZero p := prime_NeZero h1 have h10 : (1 : Int) ^ s = 1 := by ring have h11 : [c ^ d]_p = [m]_p := calc [c ^ d]_p _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat] _ = ([m]_p ^ e) ^ d := by rw [h5] _ = [m]_p ^ (e * d) := by ring _ = [m]_p ^ ((p - 1) * s + 1) := by rw [h2] _ = ([m]_p ^ (p - 1)) ^ s * [m]_p := by ring _ = ([m]_p ^ (phi p)) ^ s * [m]_p := by rw [phi_prime h1] _ = [1]_p ^ s * [m]_p := by rw [Theorem_7_4_2 h8] _ = [1 ^ s]_p * [m]_p := by rw [Exercise_7_4_5_Int] _ = [1]_p * [m]_p := by rw [h10] _ = [m]_p * [1]_p := by ring _ = [m]_p := Theorem_7_3_6_7 _ show c ^ d ≡ m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11 done done theorem Theorem_7_5_1 (p q n e d k m c : Nat) (p_prime : prime p) (q_prime : prime q) (p_ne_q : p ≠ q) (n_pq : n = p * q) (ed_congr_1 : e * d = k * (p - 1) * (q - 1) + 1) (h1 : [m]_n ^ e = [c]_n) : [c]_n ^ d = [m]_n := by rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] at h1 --h1 : m ^ e ≡ c (MOD n) rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] --Goal : c ^ d ≡ m (MOD n) obtain (j : Int) (h2 : m ^ e - c = n * j) from h1 rewrite [n_pq, Nat.cast_mul] at h2 --h2 : m ^ e - c = p * q * j have h3 : e * d = (p - 1) * (k * (q - 1)) + 1 := by rewrite [ed_congr_1] ring done have h4 : m ^ e - c = p * (q * j) := by rewrite [h2] ring done have congr_p : c ^ d ≡ m (MOD p) := Lemma_7_5_1 p_prime h3 h4 have h5 : e * d = (q - 1) * (k * (p - 1)) + 1 := by rewrite [ed_congr_1] ring done have h6 : m ^ e - c = q * (p * j) := by rewrite [h2] ring done have congr_q : c ^ d ≡ m (MOD q) := Lemma_7_5_1 q_prime h5 h6 have h7 : ¬q ∣ p := by by_contra h8 have h9 : q = 1 ∨ q = p := dvd_prime p_prime h8 disj_syll h9 (prime_not_one q_prime) show False from p_ne_q h9.symm done have h8 : rel_prime p q := rel_prime_of_prime_not_dvd q_prime h7 rewrite [n_pq, Lemma_7_4_5 _ _ h8] show c ^ d ≡ m (MOD p) ∧ c ^ d ≡ m (MOD q) from And.intro congr_p congr_q done /- BEGIN EXERCISES -/ namespace Exercises /- Section 7.1 -/ -- 1. theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry -- 2. lemma gcd_comm_lt {a b : Nat} (h : a < b) : gcd a b = gcd b a := sorry theorem gcd_comm (a b : Nat) : gcd a b = gcd b a := sorry -- 3. theorem Exercise_7_1_5 (a b : Nat) (n : Int) : (∃ (s t : Int), s * a + t * b = n) ↔ (↑(gcd a b) : Int) ∣ n := sorry -- 4. theorem Exercise_7_1_6 (a b c : Nat) : gcd a b = gcd (a + b * c) b := sorry -- 5. theorem gcd_is_nonzero {a b : Nat} (h : a ≠ 0 ∨ b ≠ 0) : gcd a b ≠ 0 := sorry -- 6. theorem gcd_greatest {a b d : Nat} (h1 : gcd a b ≠ 0) (h2 : d ∣ a) (h3 : d ∣ b) : d ≤ gcd a b := sorry -- 7. lemma Lemma_7_1_10a {a b : Nat} (n : Nat) (h : a ∣ b) : (n * a) ∣ (n * b) := sorry lemma Lemma_7_1_10b {a b n : Nat} (h1 : n ≠ 0) (h2 : (n * a) ∣ (n * b)) : a ∣ b := sorry lemma Lemma_7_1_10c {a b : Nat} (h1 : a ∣ b) (h2 : b ∣ a) : a = b := sorry theorem Exercise_7_1_10 (a b n : Nat) : gcd (n * a) (n * b) = n * gcd a b := sorry /- Section 7.2 -/ -- 1. lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry -- 2. -- Hints: Start with apply List.rec. You may find mul_ne_zero useful theorem prod_nonzero_nonzero : ∀ (l : List Nat), (∀ a ∈ l, a ≠ 0) → prod l ≠ 0 := sorry -- 3. theorem rel_prime_iff_no_common_factor (a b : Nat) : rel_prime a b ↔ ¬∃ (p : Nat), prime p ∧ p ∣ a ∧ p ∣ b := sorry -- 4. theorem rel_prime_symm {a b : Nat} (h : rel_prime a b) : rel_prime b a := sorry -- 5. lemma in_prime_factorization_iff_prime_factor {a : Nat} {l : List Nat} (h1 : prime_factorization a l) (p : Nat) : p ∈ l ↔ prime_factor p a := sorry -- 6. theorem Exercise_7_2_5 {a b : Nat} {l m : List Nat} (h1 : prime_factorization a l) (h2 : prime_factorization b m) : rel_prime a b ↔ (¬∃ (p : Nat), p ∈ l ∧ p ∈ m) := sorry -- 7. theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry -- 8. theorem Exercise_7_2_7 {a b a' b' : Nat} (h1 : rel_prime a b) (h2 : a' ∣ a) (h3 : b' ∣ b) : rel_prime a' b' := sorry -- 9. theorem Exercise_7_2_9 {a b j k : Nat} (h1 : gcd a b ≠ 0) (h2 : a = j * gcd a b) (h3 : b = k * gcd a b) : rel_prime j k := sorry -- 10. theorem Exercise_7_2_17a (a b c : Nat) : gcd a (b * c) ∣ gcd a b * gcd a c := sorry /- Section 7.3 -/ -- 1. theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry -- 2. theorem Theorem_7_3_6_3 {m : Nat} (X : ZMod m) : X + [0]_m = X := sorry -- 3. theorem Theorem_7_3_6_4 {m : Nat} (X : ZMod m) : ∃ (Y : ZMod m), X + Y = [0]_m := sorry -- 4. theorem Exercise_7_3_4a {m : Nat} (Z1 Z2 : ZMod m) (h1 : ∀ (X : ZMod m), X + Z1 = X) (h2 : ∀ (X : ZMod m), X + Z2 = X) : Z1 = Z2 := sorry -- 5. theorem Exercise_7_3_4b {m : Nat} (X Y1 Y2 : ZMod m) (h1 : X + Y1 = [0]_m) (h2 : X + Y2 = [0]_m) : Y1 = Y2 := sorry -- 6. theorem Theorem_7_3_10 (m a : Nat) (b : Int) : ¬(↑(gcd m a) : Int) ∣ b → ¬∃ (x : Int), a * x ≡ b (MOD m) := sorry -- 7. theorem Theorem_7_3_11 (m n : Nat) (a b : Int) (h1 : n ≠ 0) : n * a ≡ n * b (MOD n * m) ↔ a ≡ b (MOD m) := sorry -- 8. theorem Exercise_7_3_16 {m : Nat} {a b : Int} (h : a ≡ b (MOD m)) : ∀ (n : Nat), a ^ n ≡ b ^ n (MOD m) := sorry -- 9. example {m : Nat} [NeZero m] (X : ZMod m) : ∃! (a : Int), 0 ≤ a ∧ a < m ∧ X = [a]_m := sorry -- 10. theorem congr_rel_prime {m a b : Nat} (h1 : a ≡ b (MOD m)) : rel_prime m a ↔ rel_prime m b := sorry -- 11. --Hint: You may find the theorem Int.ofNat_mod_ofNat useful. theorem rel_prime_mod (m a : Nat) : rel_prime m (a % m) ↔ rel_prime m a := sorry -- 12. lemma congr_iff_mod_eq_Int (m : Nat) (a b : Int) [NeZero m] : a ≡ b (MOD m) ↔ a % ↑m = b % ↑m := sorry --Hint for next theorem: Use the lemma above, --together with the theorems Int.ofNat_mod_ofNat and Nat.cast_inj. theorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] : ↑a ≡ ↑b (MOD m) ↔ a % m = b % m := sorry /- Section 7.4 -/ -- 1. --Hint: Use induction. --For the base case, compute [a]_m ^ 0 * [1]_m in two ways: --by Theorem_7_3_6_7, [a] ^ 0 * [1]_m = [a]_m ^ 0 --by ring, [a]_m ^ 0 * [1]_m = [1]_m.

lemma Exercise_7_4_5_Int (m : Nat) (a : Int) : ∀ (n : Nat), [a]_m ^ n = [a ^ n]_m

HTPI.Exercises.Exercise_7_4_5_Int

htpi/HTPILib/Chap7.lean

htpi

/- Copyright 2023 Daniel J. Velleman -/ import HTPILib.Chap6 namespace HTPI /- Definitions -/ lemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by have h : n + 1 > 0 := Nat.succ_pos n show a % (n + 1) < n + 1 from Nat.mod_lt a h done def gcd (a b : Nat) : Nat := match b with | 0 => a | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd (n + 1) (a % (n + 1)) termination_by b mutual def gcd_c1 (a b : Nat) : Int := match b with | 0 => 1 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c2 (n + 1) (a % (n + 1)) --Corresponds to s = t' termination_by b def gcd_c2 (a b : Nat) : Int := match b with | 0 => 0 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c1 (n + 1) (a % (n + 1)) - (gcd_c2 (n + 1) (a % (n + 1))) * ↑(a / (n + 1)) --Corresponds to t = s' - t'q termination_by b end def prime (n : Nat) : Prop := 2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n def prime_factor (p n : Nat) : Prop := prime p ∧ p ∣ n def all_prime (l : List Nat) : Prop := ∀ p ∈ l, prime p def nondec (l : List Nat) : Prop := match l with | [] => True --Of course, True is a proposition that is always true | n :: L => (∀ m ∈ L, n ≤ m) ∧ nondec L def nondec_prime_list (l : List Nat) : Prop := all_prime l ∧ nondec l def prod (l : List Nat) : Nat := match l with | [] => 1 | n :: L => n * (prod L) def prime_factorization (n : Nat) (l : List Nat) : Prop := nondec_prime_list l ∧ prod l = n def rel_prime (a b : Nat) : Prop := gcd a b = 1 def congr_mod (m : Nat) (a b : Int) : Prop := (↑m : Int) ∣ (a - b) def cc (m : Nat) (a : Int) : ZMod m := (↑a : ZMod m) notation:50 a " ≡ " b " (MOD " m ")" => congr_mod m a b notation:max "["a"]_"m:max => cc m a def invertible {m : Nat} (X : ZMod m) : Prop := ∃ (Y : ZMod m), X * Y = [1]_m def num_rp_below (m k : Nat) : Nat := match k with | 0 => 0 | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j def phi (m : Nat) : Nat := num_rp_below m m def prod_seq {m : Nat} (j k : Nat) (f : Nat → ZMod m) : ZMod m := match j with | 0 => [1]_m | n + 1 => prod_seq n k f * f (k + n) def maps_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i < n, g i < n def one_one_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i1 < n, ∀ i2 < n, g i1 = g i2 → i1 = i2 def onto_below (n : Nat) (g : Nat → Nat) : Prop := ∀ k < n, ∃ i < n, g i = k def perm_below (n : Nat) (g : Nat → Nat) : Prop := maps_below n g ∧ one_one_below n g ∧ onto_below n g def inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m) def swap (u v i : Nat) : Nat := if i = u then v else if i = v then u else i namespace Euler --For definitions specific to Euler's theorem def F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m def G (m a i : Nat) : Nat := (a * i) % m def Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i end Euler /- Section 7.1 -/ theorem dvd_mod_of_dvd_a_b {a b d : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ (a % b) := by set q : Nat := a / b have h3 : b * q + a % b = a := Nat.div_add_mod a b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 define --Goal : ∃ (c : Nat), a % b = d * c apply Exists.intro (j - k * q) show a % b = d * (j - k * q) from calc a % b _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm _ = a - b * q := by rw [h3] _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc] _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm done theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry #eval gcd 672 161 --Answer: 7 lemma gcd_base (a : Nat) : gcd a 0 = a := by rfl lemma gcd_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd a b = gcd b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1)) rfl done lemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b ≠ 0) : a % b < b := by have h1 : b > 0 := Nat.pos_of_ne_zero h show a % b < b from Nat.mod_lt a h1 done lemma dvd_self (n : Nat) : n ∣ n := by apply Exists.intro 1 ring done theorem gcd_dvd : ∀ (b a : Nat), (gcd a b) ∣ a ∧ (gcd a b) ∣ b := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd a b_1) ∣ a ∧ (gcd a b_1) ∣ b_1 fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_base] --Goal: a ∣ a ∧ a ∣ 0 apply And.intro (dvd_self a) define apply Exists.intro 0 rfl done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_nonzero a h1] --Goal : gcd b (a % b) ∣ a ∧ gcd b (a % b) ∣ b have h2 : a % b < b := mod_nonzero_lt a h1 have h3 : (gcd b (a % b)) ∣ b ∧ (gcd b (a % b)) ∣ (a % b) := ih (a % b) h2 b apply And.intro _ h3.left show (gcd b (a % b)) ∣ a from dvd_a_of_dvd_b_mod h3.left h3.right done done theorem gcd_dvd_left (a b : Nat) : (gcd a b) ∣ a := (gcd_dvd b a).left theorem gcd_dvd_right (a b : Nat) : (gcd a b) ∣ b := (gcd_dvd b a).right lemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl lemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c1 a b = gcd_c2 b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done lemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl lemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * ↑(a / b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done theorem gcd_lin_comb : ∀ (b a : Nat), (gcd_c1 a b) * ↑a + (gcd_c2 a b) * ↑b = ↑(gcd a b) := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd_c1 a b_1) * ↑a + (gcd_c2 a b_1) * ↑b_1 = ↑(gcd a b_1) fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base] --Goal : 1 * ↑a + 0 * ↑0 = ↑a ring done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1] --Goal : gcd_c2 b (a % b) * ↑a + -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * ↑(a / b)) * ↑b = -- ↑(gcd b (a % b)) set r : Nat := a % b set q : Nat := a / b set s : Int := gcd_c1 b r set t : Int := gcd_c2 b r --Goal : t * ↑a + (s - t * ↑q) * ↑b = ↑(gcd b r) have h2 : r < b := mod_nonzero_lt a h1 have h3 : s * ↑b + t * ↑r = ↑(gcd b r) := ih r h2 b have h4 : b * q + r = a := Nat.div_add_mod a b rewrite [←h3, ←h4] rewrite [Nat.cast_add, Nat.cast_mul] --Goal : t * (↑b * ↑q + ↑r) + (s - t * ↑q) * ↑b = s * ↑b + t * ↑r ring done done #eval gcd_c1 672 161 --Answer: 6 #eval gcd_c2 672 161 --Answer: -25 --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161 theorem Theorem_7_1_6 {d a b : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ gcd a b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑d ∣ ↑(gcd a b) set s : Int := gcd_c1 a b set t : Int := gcd_c2 a b have h3 : s * ↑a + t * ↑b = ↑(gcd a b) := gcd_lin_comb b a rewrite [←h3] --Goal : ↑d ∣ s * ↑a + t * ↑b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul] --Goal : ↑d ∣ s * (↑d * ↑j) + t * (↑d * ↑k) define apply Exists.intro (s * ↑j + t * ↑k) ring done /- Section 7.2 -/ theorem dvd_trans {a b c : Nat} (h1 : a ∣ b) (h2 : b ∣ c) : a ∣ c := by define at h1; define at h2; define obtain (m : Nat) (h3 : b = a * m) from h1 obtain (n : Nat) (h4 : c = b * n) from h2 rewrite [h3, mul_assoc] at h4 apply Exists.intro (m * n) show c = a * (m * n) from h4 done lemma exists_prime_factor : ∀ (n : Nat), 2 ≤ n → ∃ (p : Nat), prime_factor p n := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, 2 ≤ n_1 → ∃ (p : Nat), prime_factor p n_1 assume h1 : 2 ≤ n by_cases h2 : prime n · -- Case 1. h2 : prime n apply Exists.intro n define --Goal : prime n ∧ n ∣ n show prime n ∧ n ∣ n from And.intro h2 (dvd_self n) done · -- Case 2. h2 : ¬prime n define at h2 --h2 : ¬(2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n) demorgan at h2 disj_syll h2 h1 obtain (a : Nat) (h3 : ∃ (b : Nat), a * b = n ∧ a < n ∧ b < n) from h2 obtain (b : Nat) (h4 : a * b = n ∧ a < n ∧ b < n) from h3 have h5 : 2 ≤ a := by by_contra h6 have h7 : a ≤ 1 := by linarith have h8 : n ≤ b := calc n _ = a * b := h4.left.symm _ ≤ 1 * b := by rel [h7] _ = b := by ring linarith --n ≤ b contradicts b < n done have h6 : ∃ (p : Nat), prime_factor p a := ih a h4.right.left h5 obtain (p : Nat) (h7 : prime_factor p a) from h6 apply Exists.intro p define --Goal : prime p ∧ p ∣ n define at h7 --h7 : prime p ∧ p ∣ a apply And.intro h7.left have h8 : a ∣ n := by apply Exists.intro b show n = a * b from (h4.left).symm done show p ∣ n from dvd_trans h7.right h8 done done lemma exists_least_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q := by set S : Set Nat := {p : Nat | prime_factor p n} have h2 : ∃ (p : Nat), p ∈ S := exists_prime_factor n h show ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q from well_ord_princ S h2 done lemma all_prime_nil : all_prime [] := by define --Goal : ∀ p ∈ [], prime p fix p : Nat contrapos --Goal : ¬prime p → p ∉ [] assume h1 : ¬prime p show p ∉ [] from List.not_mem_nil p done lemma all_prime_cons (n : Nat) (L : List Nat) : all_prime (n :: L) ↔ prime n ∧ all_prime L := by apply Iff.intro · -- (→) assume h1 : all_prime (n :: L) --Goal : prime n ∧ all_prime L define at h1 --h1 : ∀ p ∈ n :: L, prime p apply And.intro (h1 n (List.mem_cons_self n L)) define --Goal : ∀ p ∈ L, prime p fix p : Nat assume h2 : p ∈ L show prime p from h1 p (List.mem_cons_of_mem n h2) done · -- (←) assume h1 : prime n ∧ all_prime L --Goal : all_prime (n :: l) define : all_prime L at h1 define fix p : Nat assume h2 : p ∈ n :: L rewrite [List.mem_cons] at h2 --h2 : p = n ∨ p ∈ L by_cases on h2 · -- Case 1. h2 : p = n rewrite [h2] show prime n from h1.left done · -- Case 2. h2 : p ∈ L show prime p from h1.right p h2 done done done lemma nondec_nil : nondec [] := by define --Goal : True trivial --trivial proves some obviously true statements, such as True done lemma nondec_cons (n : Nat) (L : List Nat) : nondec (n :: L) ↔ (∀ m ∈ L, n ≤ m) ∧ nondec L := by rfl lemma prod_nil : prod [] = 1 := by rfl lemma prod_cons : prod (n :: L) = n * (prod L) := by rfl lemma exists_cons_of_length_eq_succ {A : Type} {l : List A} {n : Nat} (h : l.length = n + 1) : ∃ (a : A) (L : List A), l = a :: L ∧ L.length = n := by have h1 : ¬l.length = 0 := by linarith rewrite [List.length_eq_zero] at h1 obtain (a : A) (h2 : ∃ (L : List A), l = a :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List A) (h3 : l = a :: L) from h2 apply Exists.intro a apply Exists.intro L apply And.intro h3 have h4 : (a :: L).length = L.length + 1 := List.length_cons a L rewrite [←h3, h] at h4 show L.length = n from (Nat.add_right_cancel h4).symm done lemma list_elt_dvd_prod_by_length (a : Nat) : ∀ (n : Nat), ∀ (l : List Nat), l.length = n → a ∈ l → a ∣ prod l := by by_induc · --Base Case fix l : List Nat assume h1 : l.length = 0 rewrite [List.length_eq_zero] at h1 --h1 : l = [] rewrite [h1] --Goal : a ∈ [] → a ∣ prod [] contrapos assume h2 : ¬a ∣ prod [] show a ∉ [] from List.not_mem_nil a done · -- Induction Step fix n : Nat assume ih : ∀ (l : List Nat), List.length l = n → a ∈ l → a ∣ prod l fix l : List Nat assume h1 : l.length = n + 1 --Goal : a ∈ l → a ∣ prod l obtain (b : Nat) (h2 : ∃ (L : List Nat), l = b :: L ∧ L.length = n) from exists_cons_of_length_eq_succ h1 obtain (L : List Nat) (h3 : l = b :: L ∧ L.length = n) from h2 have h4 : a ∈ L → a ∣ prod L := ih L h3.right assume h5 : a ∈ l rewrite [h3.left, prod_cons] --Goal : a ∣ b * prod L rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b ∨ a ∈ L by_cases on h5 · -- Case 1. h5 : a = b apply Exists.intro (prod L) rewrite [h5] rfl done · -- Case 2. h5 : a ∈ L have h6 : a ∣ prod L := h4 h5 have h7 : prod L ∣ b * prod L := by apply Exists.intro b ring done show a ∣ b * prod L from dvd_trans h6 h7 done done done lemma list_elt_dvd_prod {a : Nat} {l : List Nat} (h : a ∈ l) : a ∣ prod l := by set n : Nat := l.length have h1 : l.length = n := by rfl show a ∣ prod l from list_elt_dvd_prod_by_length a n l h1 h done lemma exists_prime_factorization : ∀ (n : Nat), n ≥ 1 → ∃ (l : List Nat), prime_factorization n l := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, n_1 ≥ 1 → ∃ (l : List Nat), prime_factorization n_1 l assume h1 : n ≥ 1 by_cases h2 : n = 1 · -- Case 1. h2 : n = 1 apply Exists.intro [] define apply And.intro · -- Proof of nondec_prime_list [] define show all_prime [] ∧ nondec [] from And.intro all_prime_nil nondec_nil done · -- Proof of prod [] = n rewrite [prod_nil, h2] rfl done done · -- Case 2. h2 : n ≠ 1 have h3 : n ≥ 2 := lt_of_le_of_ne' h1 h2 obtain (p : Nat) (h4 : prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q) from exists_least_prime_factor h3 have p_prime_factor : prime_factor p n := h4.left define at p_prime_factor have p_prime : prime p := p_prime_factor.left have p_dvd_n : p ∣ n := p_prime_factor.right have p_least : ∀ (q : Nat), prime_factor q n → p ≤ q := h4.right obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n have h5 : m ≠ 0 := by contradict h1 with h6 have h7 : n = 0 := calc n _ = p * m := n_eq_pm _ = p * 0 := by rw [h6] _ = 0 := by ring rewrite [h7] decide done have m_pos : 0 < m := Nat.pos_of_ne_zero h5 have m_lt_n : m < n := by define at p_prime show m < n from calc m _ < m + m := by linarith _ = 2 * m := by ring _ ≤ p * m := by rel [p_prime.left] _ = n := n_eq_pm.symm done obtain (L : List Nat) (h6 : prime_factorization m L) from ih m m_lt_n m_pos define at h6 have ndpl_L : nondec_prime_list L := h6.left define at ndpl_L apply Exists.intro (p :: L) define apply And.intro · -- Proof of nondec_prime_list (p :: L) define apply And.intro · -- Proof of all_prime (p :: L) rewrite [all_prime_cons] show prime p ∧ all_prime L from And.intro p_prime ndpl_L.left done · -- Proof of nondec (p :: L) rewrite [nondec_cons] apply And.intro _ ndpl_L.right fix q : Nat assume q_in_L : q ∈ L have h7 : q ∣ prod L := list_elt_dvd_prod q_in_L rewrite [h6.right] at h7 --h7 : q ∣ m have h8 : m ∣ n := by apply Exists.intro p rewrite [n_eq_pm] ring done have q_dvd_n : q ∣ n := dvd_trans h7 h8 have ap_L : all_prime L := ndpl_L.left define at ap_L have q_prime_factor : prime_factor q n := And.intro (ap_L q q_in_L) q_dvd_n show p ≤ q from p_least q q_prime_factor done done · -- Proof of prod (p :: L) = n rewrite [prod_cons, h6.right, n_eq_pm] rfl done done done theorem Theorem_7_2_2 {a b c : Nat} (h1 : c ∣ a * b) (h2 : rel_prime a c) : c ∣ b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑c ∣ ↑b define at h1; define at h2; define obtain (j : Nat) (h3 : a * b = c * j) from h1 set s : Int := gcd_c1 a c set t : Int := gcd_c2 a c have h4 : s * ↑a + t * ↑c = ↑(gcd a c) := gcd_lin_comb c a rewrite [h2, Nat.cast_one] at h4 --h4 : s * ↑a + t * ↑c = (1 : Int) apply Exists.intro (s * ↑j + t * ↑b) show ↑b = ↑c * (s * ↑j + t * ↑b) from calc ↑b _ = (1 : Int) * ↑b := (one_mul _).symm _ = (s * ↑a + t * ↑c) * ↑b := by rw [h4] _ = s * (↑a * ↑b) + t * ↑c * ↑b := by ring _ = s * (↑c * ↑j) + t * ↑c * ↑b := by rw [←Nat.cast_mul a b, h3, Nat.cast_mul c j] _ = ↑c * (s * ↑j + t * ↑b) := by ring done lemma le_nonzero_prod_left {a b : Nat} (h : a * b ≠ 0) : a ≤ a * b := by have h1 : b ≠ 0 := by contradict h with h1 rewrite [h1] ring done have h2 : 1 ≤ b := Nat.pos_of_ne_zero h1 show a ≤ a * b from calc a = a * 1 := (mul_one a).symm _ ≤ a * b := by rel [h2] done lemma le_nonzero_prod_right {a b : Nat} (h : a * b ≠ 0) : b ≤ a * b := by rewrite [mul_comm] rewrite [mul_comm] at h show b ≤ b * a from le_nonzero_prod_left h done lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry lemma rel_prime_of_prime_not_dvd {a p : Nat} (h1 : prime p) (h2 : ¬p ∣ a) : rel_prime a p := by have h3 : gcd a p ∣ a := gcd_dvd_left a p have h4 : gcd a p ∣ p := gcd_dvd_right a p have h5 : gcd a p = 1 ∨ gcd a p = p := dvd_prime h1 h4 have h6 : gcd a p ≠ p := by contradict h2 with h6 rewrite [h6] at h3 show p ∣ a from h3 done disj_syll h5 h6 show rel_prime a p from h5 done theorem Theorem_7_2_3 {a b p : Nat} (h1 : prime p) (h2 : p ∣ a * b) : p ∣ a ∨ p ∣ b := by or_right with h3 have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3 show p ∣ b from Theorem_7_2_2 h2 h4 done lemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a ≥ 1 := by have h1 : a ≠ 0 := by by_contra h1 rewrite [h1] at h contradict h linarith done show a ≥ 1 from Nat.pos_of_ne_zero h1 done lemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by have h1 : a ≥ 1 := ge_one_of_prod_one h have h2 : a * b ≠ 0 := by linarith have h3 : a ≤ a * b := le_nonzero_prod_left h2 rewrite [h] at h3 show a = 1 from Nat.le_antisymm h3 h1 done lemma eq_one_of_dvd_one {n : Nat} (h : n ∣ 1) : n = 1 := by obtain (j : Nat) (h1 : 1 = n * j) from h show n = 1 from eq_one_of_prod_one h1.symm done lemma prime_not_one {p : Nat} (h : prime p) : p ≠ 1 := by define at h linarith done theorem Theorem_7_2_4 {p : Nat} (h1 : prime p) : ∀ (l : List Nat), p ∣ prod l → ∃ a ∈ l, p ∣ a := by apply List.rec · -- Base Case. Goal : p ∣ prod [] → ∃ a ∈ [], p ∣ a rewrite [prod_nil] assume h2 : p ∣ 1 show ∃ a ∈ [], p ∣ a from absurd (eq_one_of_dvd_one h2) (prime_not_one h1) done · -- Induction Step fix b : Nat fix L : List Nat assume ih : p ∣ prod L → ∃ a ∈ L, p ∣ a --Goal : p ∣ prod (b :: L) → ∃ a ∈ b :: L, p ∣ a assume h2 : p ∣ prod (b :: L) rewrite [prod_cons] at h2 have h3 : p ∣ b ∨ p ∣ prod L := Theorem_7_2_3 h1 h2 by_cases on h3 · -- Case 1. h3 : p ∣ b apply Exists.intro b show b ∈ b :: L ∧ p ∣ b from And.intro (List.mem_cons_self b L) h3 done · -- Case 2. h3 : p ∣ prod L obtain (a : Nat) (h4 : a ∈ L ∧ p ∣ a) from ih h3 apply Exists.intro a show a ∈ b :: L ∧ p ∣ a from And.intro (List.mem_cons_of_mem b h4.left) h4.right done done done lemma prime_in_list {p : Nat} {l : List Nat} (h1 : prime p) (h2 : all_prime l) (h3 : p ∣ prod l) : p ∈ l := by obtain (a : Nat) (h4 : a ∈ l ∧ p ∣ a) from Theorem_7_2_4 h1 l h3 define at h2 have h5 : prime a := h2 a h4.left have h6 : p = 1 ∨ p = a := dvd_prime h5 h4.right disj_syll h6 (prime_not_one h1) rewrite [h6] show a ∈ l from h4.left done lemma first_le_first {p q : Nat} {l m : List Nat} (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m)) (h3 : prod (p :: l) = prod (q :: m)) : p ≤ q := by define at h1; define at h2 have h4 : q ∣ prod (p :: l) := by define apply Exists.intro (prod m) rewrite [←prod_cons] show prod (p :: l) = prod (q :: m) from h3 done have h5 : all_prime (q :: m) := h2.left rewrite [all_prime_cons] at h5 have h6 : q ∈ p :: l := prime_in_list h5.left h1.left h4 have h7 : nondec (p :: l) := h1.right rewrite [nondec_cons] at h7 rewrite [List.mem_cons] at h6 by_cases on h6 · -- Case 1. h6 : q = p linarith done · -- Case 2. h6 : q ∈ l have h8 : ∀ m ∈ l, p ≤ m := h7.left show p ≤ q from h8 q h6 done done lemma nondec_prime_list_tail {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by define at h define rewrite [all_prime_cons, nondec_cons] at h show all_prime l ∧ nondec l from And.intro h.left.right h.right.right done lemma cons_prod_not_one {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : prod (p :: l) ≠ 1 := by define at h have h1 : all_prime (p :: l) := h.left rewrite [all_prime_cons] at h1 rewrite [prod_cons] by_contra h2 show False from (prime_not_one h1.left) (eq_one_of_prod_one h2) done lemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) : l = [] ↔ prod l = 1 := by apply Iff.intro · -- (→) assume h1 : l = [] rewrite [h1] show prod [] = 1 from prod_nil done · -- (←) contrapos assume h1 : ¬l = [] obtain (p : Nat) (h2 : ∃ (L : List Nat), l = p :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List Nat) (h3 : l = p :: L) from h2 rewrite [h3] at h rewrite [h3] show ¬prod (p :: L) = 1 from cons_prod_not_one h done done lemma prime_pos {p : Nat} (h : prime p) : p > 0 := by define at h linarith done theorem Theorem_7_2_5 : ∀ (l1 l2 : List Nat), nondec_prime_list l1 → nondec_prime_list l2 → prod l1 = prod l2 → l1 = l2 := by apply List.rec · -- Base Case. Goal : ∀ (l2 : List Nat), nondec_prime_list [] → -- nondec_prime_list l2 → prod [] = prod l2 → [] = l2 fix l2 : List Nat assume h1 : nondec_prime_list [] assume h2 : nondec_prime_list l2 assume h3 : prod [] = prod l2 rewrite [prod_nil, eq_comm, ←list_nil_iff_prod_one h2] at h3 show [] = l2 from h3.symm done · -- Induction Step fix p : Nat fix L1 : List Nat assume ih : ∀ (L2 : List Nat), nondec_prime_list L1 → nondec_prime_list L2 → prod L1 = prod L2 → L1 = L2 -- Goal : ∀ (l2 : List Nat), nondec_prime_list (p :: L1) → -- nondec_prime_list l2 → prod (p :: L1) = prod l2 → p :: L1 = l2 fix l2 : List Nat assume h1 : nondec_prime_list (p :: L1) assume h2 : nondec_prime_list l2 assume h3 : prod (p :: L1) = prod l2 have h4 : ¬prod (p :: L1) = 1 := cons_prod_not_one h1 rewrite [h3, ←list_nil_iff_prod_one h2] at h4 obtain (q : Nat) (h5 : ∃ (L : List Nat), l2 = q :: L) from List.exists_cons_of_ne_nil h4 obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5 rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2) rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2) have h7 : p ≤ q := first_le_first h1 h2 h3 have h8 : q ≤ p := first_le_first h2 h1 h3.symm have h9 : p = q := by linarith rewrite [h9, prod_cons, prod_cons] at h3 --h3 : q * prod L1 = q * prod L2 have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1 have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2 define at h2 have h12 : all_prime (q :: L2) := h2.left rewrite [all_prime_cons] at h12 have h13 : q > 0 := prime_pos h12.left have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3 have h15 : L1 = L2 := ih L2 h10 h11 h14 rewrite [h6, h9, h15] rfl done done theorem fund_thm_arith (n : Nat) (h : n ≥ 1) : ∃! (l : List Nat), prime_factorization n l := by exists_unique · -- Existence show ∃ (l : List Nat), prime_factorization n l from exists_prime_factorization n h done · -- Uniqueness fix l1 : List Nat; fix l2 : List Nat assume h1 : prime_factorization n l1 assume h2 : prime_factorization n l2 define at h1; define at h2 have h3 : prod l1 = n := h1.right rewrite [←h2.right] at h3 show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3 done done /- Section 7.3 -/ theorem congr_refl (m : Nat) : ∀ (a : Int), a ≡ a (MOD m) := by fix a : Int define --Goal : ∃ (c : Int), a - a = ↑m * c apply Exists.intro 0 ring done theorem congr_symm {m : Nat} : ∀ {a b : Int}, a ≡ b (MOD m) → b ≡ a (MOD m) := by fix a : Int; fix b : Int assume h1 : a ≡ b (MOD m) define at h1 --h1 : ∃ (c : Int), a - b = ↑m * c define --Goal : ∃ (c : Int), b - a = ↑m * c obtain (c : Int) (h2 : a - b = m * c) from h1 apply Exists.intro (-c) show b - a = m * (-c) from calc b - a _ = -(a - b) := by ring _ = -(m * c) := by rw [h2] _ = m * (-c) := by ring done theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry /- Fundamental properties of congruence classes -/ lemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) : X = Y ↔ X.val = Y.val := Fin.ext_iff lemma val_nat_eq_mod (n k : Nat) : ([k]_(n + 1)).val = k % (n + 1) := by rfl lemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl theorem cc_rep {m : Nat} (X : ZMod m) : ∃ (a : Int), X = [a]_m := match m with | 0 => by apply Exists.intro X rfl done | n + 1 => by apply Exists.intro ↑(X.val) have h1 : X.val < n + 1 := Fin.prop X rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1] rfl done theorem add_class (m : Nat) (a b : Int) : [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm theorem mul_class (m : Nat) (a b : Int) : [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm lemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ [a - b]_m = [0]_m := by apply Iff.intro · -- (→) assume h1 : [a]_m = [b]_m have h2 : a - b = a + (-b) := by ring have h3 : b + (-b) = 0 := by ring show [a - b]_m = [0]_m from calc [a - b]_m _ = [a + (-b)]_m := by rw [h2] _ = [a]_m + [-b]_m := by rw [add_class] _ = [b]_m + [-b]_m := by rw [h1] _ = [b + -b]_m := by rw [add_class] _ = [0]_m := by rw [h3] done · -- (←) assume h1 : [a - b]_m = [0]_m have h2 : b + (a - b) = a := by ring have h3 : b + 0 = b := by ring show [a]_m = [b]_m from calc [a]_m _ = [b + (a - b)]_m := by rw [h2] _ = [b]_m + [a - b]_m := by rw [add_class] _ = [b]_m + [0]_m := by rw [h1] _ = [b + 0]_m := by rw [add_class] _ = [b]_m := by rw [h3] done done lemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) : [a]_m = [0]_m → [-a]_m = [0]_m := by assume h1 : [a]_m = [0]_m have h2 : 0 + (-a) = -a := by ring have h3 : a + (-a) = 0 := by ring show [-a]_m = [0]_m from calc [-a]_m _ = [0 + (-a)]_m := by rw [h2] _ = [0]_m + [-a]_m := by rw [add_class] _ = [a]_m + [-a]_m := by rw [h1] _ = [a + (-a)]_m := by rw [add_class] _ = [0]_m := by rw [h3] done lemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) : [-a]_m = [0]_m ↔ [a]_m = [0]_m := by apply Iff.intro _ (cc_neg_zero_of_cc_zero m a) assume h1 : [-a]_m = [0]_m have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1 have h3 : -(-a) = a := by ring rewrite [h3] at h2 show [a]_m = [0]_m from h2 done lemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl lemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m ↔ m ∣ k := match m with | 0 => by have h : (0 : Int) = (↑(0 : Nat) : Int) := by rfl rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj] apply Iff.intro · -- (→) assume h1 : k = 0 rewrite [h1] show 0 ∣ 0 from dvd_self 0 done · -- (←) assume h1 : 0 ∣ k obtain (c : Nat) (h2 : k = 0 * c) from h1 rewrite [h2] ring done done | n + 1 => by rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero] show k % (n + 1) = 0 ↔ n + 1 ∣ k from (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm done lemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m ↔ ↑m ∣ a := by obtain (k : Nat) (h1 : a = ↑k ∨ a = -↑k) from Int.eq_nat_or_neg a by_cases on h1 · -- Case 1. h1: a = ↑k rewrite [h1, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done · -- Case 2. h1: a = -↑k rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done done theorem cc_eq_iff_congr (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ a ≡ b (MOD m) := calc [a]_m = [b]_m _ ↔ [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b _ ↔ ↑m ∣ (a - b) := cc_zero_iff_dvd m (a - b) _ ↔ a ≡ b (MOD m) := by rfl /- End of fundamental properties of congruence classes -/ lemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m := by have h1 : (↑m : Int) ≠ 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m) show 0 ≤ a % m from Int.emod_nonneg a h1 done lemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m) have h2 : (↑m : Int) > 0 := (Nat.cast_pos).rtl h1 show a % m < m from Int.emod_lt_of_pos a h2 done lemma congr_mod_mod (m : Nat) (a : Int) : a ≡ a % m (MOD m) := by define have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m apply Exists.intro (a / m) show a - a % m = m * (a / m) from calc a - (a % m) _ = m * (a / m) + a % m - a % m := by rw [h1] _ = m * (a / m) := by ring done lemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) := And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a)) theorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) : ∃! (r : Int), 0 ≤ r ∧ r < m ∧ a ≡ r (MOD m) := by exists_unique · -- Existence apply Exists.intro (a % m) show 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) from mod_cmpl_res m a done · -- Uniqueness fix r1 : Int; fix r2 : Int assume h1 : 0 ≤ r1 ∧ r1 < m ∧ a ≡ r1 (MOD m) assume h2 : 0 ≤ r2 ∧ r2 < m ∧ a ≡ r2 (MOD m) have h3 : r1 ≡ r2 (MOD m) := congr_trans (congr_symm h1.right.right) h2.right.right obtain (d : Int) (h4 : r1 - r2 = m * d) from h3 have h5 : r1 - r2 < m * 1 := by linarith have h6 : m * (-1) < r1 - r2 := by linarith rewrite [h4] at h5 --h5 : m * d < m * 1 rewrite [h4] at h6 --h6 : m * -1 < m * d have h7 : (↑m : Int) ≥ 0 := Nat.cast_nonneg m have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7 have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7 have h10 : d = 0 := by linarith show r1 = r2 from calc r1 _ = r1 - r2 + r2 := by ring _ = m * 0 + r2 := by rw [h4, h10] _ = r2 := by ring done done lemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m := (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a) theorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y rewrite [h1, h2] have h3 : a + b = b + a := by ring show [a]_m + [b]_m = [b]_m + [a]_m from calc [a]_m + [b]_m _ = [a + b]_m := add_class m a b _ = [b + a]_m := by rw [h3] _ = [b]_m + [a]_m := (add_class m b a).symm done theorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X rewrite [h1] have h2 : a * 1 = a := by ring show [a]_m * [1]_m = [a]_m from calc [a]_m * [1]_m _ = [a * 1]_m := mul_class m a 1 _ = [a]_m := by rw [h2] done theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry lemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) : [a]_m * [gcd_c2 m a]_m = [1]_m := by set s : Int := gcd_c1 m a have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m define at h1 rewrite [h1, Nat.cast_one] at h2 --h2 : s * ↑m + gcd_c2 m a * ↑a = 1 rewrite [mul_class, cc_eq_iff_congr] define --Goal : ∃ (c : Int), ↑a * gcd_c2 m a - 1 = ↑m * c apply Exists.intro (-s) show a * (gcd_c2 m a) - 1 = m * (-s) from calc a * (gcd_c2 m a) - 1 _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring _ = 1 + m * (-s) - 1 := by rw [h2] _ = m * (-s) := by ring done theorem Theorem_7_3_7 (m a : Nat) : invertible [a]_m ↔ rel_prime m a := by apply Iff.intro · -- (→) assume h1 : invertible [a]_m define at h1 obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1 obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y rewrite [h3, mul_class, cc_eq_iff_congr] at h2 define at h2 obtain (c : Int) (h4 : a * b - 1 = m * c) from h2 rewrite [Exercise_7_2_6] --Goal : ∃ (s t : Int), s * ↑m + t * ↑a = 1 apply Exists.intro (-c) apply Exists.intro b show (-c) * m + b * a = 1 from calc (-c) * m + b * a _ = (-c) * m + (a * b - 1) + 1 := by ring _ = (-c) * m + m * c + 1 := by rw [h4] _ = 1 := by ring done · -- (←) assume h1 : rel_prime m a define show ∃ (Y : ZMod m), [a]_m * Y = [1]_m from Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1) done done /- Section 7.4 -/ section Euler open Euler lemma num_rp_below_base {m : Nat} : num_rp_below m 0 = 0 := by rfl lemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) : num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by have h1 : num_rp_below m (j + 1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : gcd m j = 1 rewrite [if_pos h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1 show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1 done lemma num_rp_below_step_not_rp {m j : Nat} (h : ¬rel_prime m j) : num_rp_below m (j + 1) = num_rp_below m j := by have h1 : num_rp_below m (j +1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : ¬gcd m j = 1 rewrite [if_neg h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j show num_rp_below m (j + 1) = num_rp_below m j from h1 done lemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl #eval phi 10 --Answer: 4 lemma prod_inv_iff_inv {m : Nat} {X : ZMod m} (h1 : invertible X) (Y : ZMod m) : invertible (X * Y) ↔ invertible Y := by apply Iff.intro · -- (→) assume h2 : invertible (X * Y) obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2 apply Exists.intro (X * Z) rewrite [←h3] --Goal : Y * (X * Z) = X * Y * Z ring --Note that ring can do algebra in ZMod m done · -- (←) assume h2 : invertible Y obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1 obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2 apply Exists.intro (Xi * Yi) show (X * Y) * (Xi * Yi) = [1]_m from calc X * Y * (Xi * Yi) _ = (X * Xi) * (Y * Yi) := by ring _ = [1]_m * [1]_m := by rw [h3, h4] _ = [1]_m := Theorem_7_3_6_7 [1]_m done done lemma F_rp_def {m i : Nat} (h : rel_prime m i) : F m i = [i]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h --h : gcd m i = 1 rewrite [if_pos h] at h1 show F m i = [i]_m from h1 done lemma F_not_rp_def {m i : Nat} (h : ¬rel_prime m i) : F m i = [1]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h rewrite [h1, if_neg h] rfl done lemma prod_seq_base {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 0 k f = [1]_m := by rfl lemma prod_seq_step {m : Nat} (n k : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl lemma prod_seq_zero_step {m : Nat} (n : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by rewrite [prod_seq_step, zero_add] rfl done lemma prod_one {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 1 k f = f k := by rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7] rfl done lemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl lemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m := calc [G m a i]_m _ = [(a * i) % m]_m := by rfl _ = [a * i]_m := (cc_eq_mod m (a * i)).symm _ = [a]_m * [i]_m := (mul_class m a i).symm lemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) : rel_prime m (G m a i) ↔ rel_prime m i := by have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1 show rel_prime m (G m a i) ↔ rel_prime m i from calc rel_prime m (G m a i) _ ↔ invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm _ ↔ invertible ([a]_m * [i]_m) := by rw [cc_G] _ ↔ invertible [i]_m := prod_inv_iff_inv h2 ([i]_m) _ ↔ rel_prime m i := Theorem_7_3_7 m i done lemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) : F m (G m a i) = [a]_m * F m i := by have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2 show F m (G m a i) = [a]_m * F m i from calc F m (G m a i) _ = [G m a i]_m := F_rp_def h3 _ = [a]_m * [i]_m := cc_G m a i _ = [a]_m * F m i := by rw [F_rp_def h2] done lemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : ¬rel_prime m i) : F m (G m a i) = [1]_m := by rewrite [←G_rp_iff h1 i] at h2 show F m (G m a i) = [1]_m from F_not_rp_def h2 done lemma FG_prod {m a : Nat} (h1 : rel_prime m a) : ∀ (k : Nat), prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by by_induc · -- Base Case show prod_seq 0 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from calc prod_seq 0 0 ((F m) ∘ (G m a)) _ = [1]_m := prod_seq_base _ _ _ = [a]_m ^ 0 * [1]_m := by ring _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by rw [num_rp_below_base, prod_seq_base] done · -- Induction Step fix k : Nat assume ih : prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) by_cases h2 : rel_prime m k · -- Case 1. h2 : rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([a]_m * F m k) := by rw [FG_rp h1 h2] _ = [a]_m ^ ((num_rp_below m k) + 1) * ((prod_seq k 0 (F m)) * F m k) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_rp h2, prod_seq_zero_step] done · -- Case 2. h2 : ¬rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([1]_m) := by rw [FG_not_rp h1 h2] _ = [a]_m ^ (num_rp_below m k) * (prod_seq k 0 (F m) * ([1]_m)) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_not_rp h2, prod_seq_zero_step, F_not_rp_def h2] done done done lemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by define --Goal : ∀ i < m, G m a i < m fix i : Nat assume h1 : i < m rewrite [G_def] --Goal : a * i % m < m show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m) done lemma left_inv_one_one_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g' (g i) = i) : one_one_below n g := sorry lemma right_inv_onto_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g (g' i) = i) (h2 : maps_below n g') : onto_below n g := by define at h2; define fix k : Nat assume h3 : k < n apply Exists.intro (g' k) show g' k < n ∧ g (g' k) = k from And.intro (h2 k h3) (h1 k h3) done lemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m * [inv_mod m a]_m = [1]_m := by have h2 : 0 ≤ (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a) show [a]_m * [inv_mod m a]_m = [1]_m from calc [a]_m * [inv_mod m a]_m _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2] _ = [a]_m * [gcd_c2 m a]_m := by rw [←cc_eq_mod] _ = [1]_m := gcd_c2_inv h1 done lemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m := calc a * (b % m) % m = a % m * (b % m % m) % m := Nat.mul_mod _ _ _ _ = a % m * (b % m) % m := by rw [Nat.mod_mod] _ = a * b % m := (Nat.mul_mod _ _ _).symm lemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm] rfl done theorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] : ↑a ≡ ↑b (MOD m) ↔ a % m = b % m := sorry lemma mul_inv_mod_cancel {m a i : Nat} [NeZero m] (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1 rewrite [mul_class, cc_eq_iff_congr, ←Nat.cast_mul, ←Nat.cast_one, congr_iff_mod_eq_Nat] at h3 show a * inv_mod m a * i % m = i from calc a * (inv_mod m a) * i % m _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = 1 % m * i % m := by rw [h3] _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = i % m := by rw [one_mul] _ = i := Nat.mod_eq_of_lt h2 done lemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl lemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, G m a (Ginv m a i) = i := by fix i : Nat assume h2 : i < m show G m a (Ginv m a i) = i from calc G m a (Ginv m a i) _ = a * ((inv_mod m a * i) % m) % m := by rfl _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, Ginv m a (G m a i) = i := by fix i : Nat assume h2 : i < m show Ginv m a (G m a i) = i from calc Ginv m a (G m a i) _ = inv_mod m a * ((a * i) % m) % m := by rfl _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc, mul_comm (inv_mod m a)] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_maps_below (m a : Nat) [NeZero m] : maps_below m (Ginv m a) := G_maps_below m (inv_mod m a) lemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : one_one_below m (G m a) := left_inv_one_one_below (Ginv_left_inv h1) lemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : onto_below m (G m a) := right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a) lemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : perm_below m (G m a) := And.intro (G_maps_below m a) (And.intro (G_one_one_below h1) (G_onto_below h1)) --Permuting a product of congruence classes doesn't change product lemma swap_fst (u v : Nat) : swap u v u = v := by define : swap u v u --Goal : (if u = u then v else if u = v then u else u) = v have h : u = u := by rfl rewrite [if_pos h] rfl done lemma swap_snd (u v : Nat) : swap u v v = u := by define : swap u v v by_cases h1 : v = u · -- Case 1. h1 : v = u rewrite [if_pos h1] show v = u from h1 done · -- Case 2. h1 : v ≠ u rewrite [if_neg h1] have h2 : v = v := by rfl rewrite [if_pos h2] rfl done done lemma swap_other {u v i : Nat} (h1 : i ≠ u) (h2 : i ≠ v) : swap u v i = i := by define : swap u v i rewrite [if_neg h1, if_neg h2] rfl done lemma swap_values (u v i : Nat) : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := by by_cases h1 : i = u · -- Case 1. h1 : i = u apply Or.inl rewrite [h1] show swap u v u = v from swap_fst u v done · -- Case 2. h1 : i ≠ u apply Or.inr by_cases h2 : i = v · -- Case 2.1. h2 : i = v apply Or.inl rewrite [h2] show swap u v v = u from swap_snd u v done · -- Case 2.2. h2 : i ≠ v apply Or.inr show swap u v i = i from swap_other h1 h2 done done done lemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by define fix i : Nat assume h3 : i < n have h4 : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := swap_values u v i by_cases on h4 · -- Case 1. h4 : swap u v i = v rewrite [h4] show v < n from h2 done · -- Case 2. by_cases on h4 · -- Case 2.1. h4 : swap u v i = u rewrite [h4] show u < n from h1 done · -- Case 2.2. h4 : swap u v i = i rewrite [h4] show i < n from h3 done done done lemma swap_swap (u v n : Nat) : ∀ i < n, swap u v (swap u v i) = i := by fix i : Nat assume h : i < n by_cases h1 : i = u · -- Case 1. h1 : i = u rewrite [h1, swap_fst, swap_snd] rfl done · -- Case 2. h1 : i ≠ u by_cases h2 : i = v · -- Case 2.1. h2 : i = v rewrite [h2, swap_snd, swap_fst] rfl done · -- Case 2.2. h2 : i ≠ v rewrite [swap_other h1 h2, swap_other h1 h2] rfl done done done lemma swap_one_one_below (u v n) : one_one_below n (swap u v) := left_inv_one_one_below (swap_swap u v n) lemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) := right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2) lemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) := And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2)) lemma comp_perm_below {n : Nat} {f g : Nat → Nat} (h1 : perm_below n f) (h2 : perm_below n g) : perm_below n (f ∘ g) := sorry lemma trivial_swap (u : Nat) : swap u u = id := by apply funext fix x : Nat by_cases h1 : x = u · -- Case 1. h1 : x = u rewrite [h1, swap_fst] rfl done · -- Case 2. h1 : x ≠ u rewrite [swap_other h1 h1] rfl done done lemma prod_eq_fun {m : Nat} (f g : Nat → ZMod m) (k : Nat) : ∀ (n : Nat), (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g := by by_induc · -- Base Case assume h : (∀ i < 0, f (k + i) = g (k + i)) rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g assume h1 : ∀ i < n + 1, f (k + i) = g (k + i) have h2 : ∀ i < n, f (k + i) = g (k + i) := by fix i : Nat assume h2 : i < n have h3 : i < n + 1 := by linarith show f (k + i) = g (k + i) from h1 i h3 done have h3 : prod_seq n k f = prod_seq n k g := ih h2 have h4 : n < n + 1 := Nat.lt_succ_self n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4] rfl done done lemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := by have h2 : ∀ (i : Nat), i < u → (f ∘ swap u n) (0 + i) = f (0 + i) := by fix i : Nat assume h2 : i < u have h3 : 0 + i ≠ u := by linarith have h4 : 0 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl done show prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f from prod_eq_fun (f ∘ swap u n) f 0 u h2 done lemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := by have h2 : ∀ i < j, (f ∘ swap u n) (u + 1 + i) = f (u + 1 + i) := by fix i : Nat assume h2 : i < j have h3 : u + 1 + i ≠ u := by linarith have h4 : u + 1 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl show prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f from prod_eq_fun (f ∘ swap u n) f (u + 1) j h2 done lemma break_prod {m : Nat} (n : Nat) (f : Nat → ZMod m) : ∀ (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by by_induc · -- Base Case have h : n + 0 = n := by rfl rewrite [prod_seq_base, h, Theorem_7_3_6_7] rfl done · -- Induction Step fix j : Nat assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f rewrite [←add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc] rfl done done lemma break_prod_twice {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f := break_prod n f 1 rewrite [prod_one] at h2 have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f := break_prod (u + 1) f j rewrite [←h1] at h3 have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f := break_prod u f 1 rewrite [prod_one] at h4 rewrite [h3, h4] at h2 show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2 done lemma swap_prod_eq_prod {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f := by by_cases h2 : u = n · -- Case 1. h2 : u = n rewrite [h2, trivial_swap n] --Goal : prod_seq (n + 1) 0 (f ∘ id) = prod_seq (n + 1) 0 f rfl done · -- Case 2. h2 : ¬u = n have h3 : u + 1 ≤ n := Nat.lt_of_le_of_ne h1 h2 obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3 have break_f : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := break_prod_twice f h4 have break_fs : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_prod_twice (f ∘ swap u n) h4 have f_eq_fs_below : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := swap_prod_eq_prod_below f h1 have f_eq_fs_btwn : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4 show prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f from calc prod_seq (n + 1) 0 (f ∘ swap u n) _ = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_fs _ = prod_seq u 0 f * (f ∘ swap u n) u * prod_seq j (u + 1) f * (f ∘ swap u n) n := by rw [f_eq_fs_below, f_eq_fs_btwn] _ = prod_seq u 0 f * f (swap u n u) * prod_seq j (u + 1) f * f (swap u n n) := by rfl _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by rw [swap_fst, swap_snd] _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring _ = prod_seq (n + 1) 0 f := break_f.symm done done lemma perm_below_fixed {n : Nat} {g : Nat → Nat} (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry lemma perm_prod {m : Nat} (f : Nat → ZMod m) : ∀ (n : Nat), ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) := by by_induc · -- Base Case fix g : Nat → Nat assume h1 : perm_below 0 g rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) fix g : Nat → Nat assume g_pb : perm_below (n + 1) g define at g_pb have g_ob : onto_below (n + 1) g := g_pb.right.right define at g_ob have h1 : n < n + 1 := by linarith obtain (u : Nat) (h2 : u < n + 1 ∧ g u = n) from g_ob n h1 have s_pb : perm_below (n + 1) (swap u n) := swap_perm_below h2.left h1 have gs_pb_n1 : perm_below (n + 1) (g ∘ swap u n) := comp_perm_below g_pb s_pb have gs_fix_n : (g ∘ swap u n) n = n := calc (g ∘ swap u n) n _ = g (swap u n n) := by rfl _ = g u := by rw [swap_snd] _ = n := h2.right have gs_pb_n : perm_below n (g ∘ swap u n) := perm_below_fixed gs_pb_n1 gs_fix_n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f ∘ (g ∘ swap u n)) := ih (g ∘ swap u n) gs_pb_n have h3 : u ≤ n := by linarith show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f ∘ g) from calc prod_seq (n + 1) 0 f _ = prod_seq n 0 f * f n := prod_seq_zero_step n f _ = prod_seq n 0 (f ∘ (g ∘ swap u n)) * f ((g ∘ swap u n) n) := by rw [gs_prod, gs_fix_n] _ = prod_seq n 0 (f ∘ g ∘ swap u n) * (f ∘ g ∘ swap u n) n := by rfl _ = prod_seq (n + 1) 0 (f ∘ g ∘ swap u n) := (prod_seq_zero_step n (f ∘ g ∘ swap u n)).symm _ = prod_seq (n + 1) 0 ((f ∘ g) ∘ swap u n) := by rfl _ = prod_seq (n + 1) 0 (f ∘ g) := swap_prod_eq_prod (f ∘ g) h3 done done lemma F_invertible (m i : Nat) : invertible (F m i) := by by_cases h : rel_prime m i · -- Case 1. h : rel_prime m i rewrite [F_rp_def h] show invertible [i]_m from (Theorem_7_3_7 m i).rtl h done · -- Case 2. h : ¬rel_prime m i rewrite [F_not_rp_def h] apply Exists.intro [1]_m show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m done done lemma Fprod_invertible (m : Nat) : ∀ (k : Nat), invertible (prod_seq k 0 (F m)) := by by_induc · -- Base Case apply Exists.intro [1]_m show prod_seq 0 0 (F m) * [1]_m = [1]_m from calc prod_seq 0 0 (F m) * [1]_m _ = [1]_m * [1]_m := by rw [prod_seq_base] _ = [1]_m := Theorem_7_3_6_7 ([1]_m) done · -- Induction Step fix k : Nat assume ih : invertible (prod_seq k 0 (F m)) rewrite [prod_seq_zero_step] show invertible (prod_seq k 0 (F m) * (F m k)) from (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k) done done theorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m ^ (phi m) = [1]_m := by have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2 show [a]_m ^ (phi m) = [1]_m from calc [a]_m ^ (phi m) _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3] _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring _ = prod_seq m 0 (F m ∘ G m a) * Y := by rw [FG_prod h1 m, phi_def] _ = prod_seq m 0 (F m) * Y := by rw [perm_prod (F m) m (G m a) (G_perm_below h1)] _ = [1]_m := by rw [h3] done lemma Exercise_7_4_5_Int (m : Nat) (a : Int) : ∀ (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry lemma Exercise_7_4_5_Nat (m a n : Nat) : [a]_m ^ n = [a ^ n]_m := by rewrite [Exercise_7_4_5_Int] rfl done theorem Euler's_theorem {m a : Nat} [NeZero m] (h1 : rel_prime m a) : a ^ (phi m) ≡ 1 (MOD m) := by have h2 : [a]_m ^ (phi m) = [1]_m := Theorem_7_4_2 h1 rewrite [Exercise_7_4_5_Nat m a (phi m)] at h2 --h2 : [a ^ phi m]_m = [1]_m show a ^ (phi m) ≡ 1 (MOD m) from (cc_eq_iff_congr _ _ _).ltr h2 done #eval gcd 10 7 --Answer: 1. So 10 and 7 are relatively prime #eval 7 ^ phi 10 --Answer: 2401, which is congruent to 1 mod 10. end Euler /- Section 7.5 -/ lemma num_rp_prime {p : Nat} (h1 : prime p) : ∀ k < p, num_rp_below p (k + 1) = k := sorry lemma phi_prime {p : Nat} (h1 : prime p) : phi p = p - 1 := by have h2 : 1 ≤ p := prime_pos h1 have h3 : p - 1 + 1 = p := Nat.sub_add_cancel h2 have h4 : p - 1 < p := by linarith have h5 : num_rp_below p (p - 1 + 1) = p - 1 := num_rp_prime h1 (p - 1) h4 rewrite [h3] at h5 show phi p = p - 1 from h5 done theorem Theorem_7_2_2_Int {a c : Nat} {b : Int} (h1 : ↑c ∣ ↑a * b) (h2 : rel_prime a c) : ↑c ∣ b := by rewrite [Int.natCast_dvd, Int.natAbs_mul, Int.natAbs_ofNat] at h1 --h1 : c ∣ a * Int.natAbs b rewrite [Int.natCast_dvd] --Goal : c ∣ Int.natAbs b show c ∣ Int.natAbs b from Theorem_7_2_2 h1 h2 done lemma Lemma_7_4_5 {m n : Nat} (a b : Int) (h1 : rel_prime m n) : a ≡ b (MOD m * n) ↔ a ≡ b (MOD m) ∧ a ≡ b (MOD n) := by apply Iff.intro · -- (→) assume h2 : a ≡ b (MOD m * n) obtain (j : Int) (h3 : a - b = (m * n) * j) from h2 apply And.intro · -- Proof of a ≡ b (MOD m) apply Exists.intro (n * j) show a - b = m * (n * j) from calc a - b _ = m * n * j := h3 _ = m * (n * j) := by ring done · -- Proof of a ≡ b (MOD n) apply Exists.intro (m * j) show a - b = n * (m * j) from calc a - b _ = m * n * j := h3 _ = n * (m * j) := by ring done done · -- (←) assume h2 : a ≡ b (MOD m) ∧ a ≡ b (MOD n) obtain (j : Int) (h3 : a - b = m * j) from h2.left have h4 : (↑n : Int) ∣ a - b := h2.right rewrite [h3] at h4 --h4 : ↑n ∣ ↑m * j have h5 : ↑n ∣ j := Theorem_7_2_2_Int h4 h1 obtain (k : Int) (h6 : j = n * k) from h5 apply Exists.intro k --Goal : a - b = ↑(m * n) * k rewrite [Nat.cast_mul] --Goal : a - b = ↑m * ↑n * k show a - b = (m * n) * k from calc a - b _ = m * j := h3 _ = m * (n * k) := by rw [h6] _ = (m * n) * k := by ring done done --From exercises of Section 7.2 theorem rel_prime_symm {a b : Nat} (h : rel_prime a b) : rel_prime b a := sorry lemma prime_NeZero {p : Nat} (h : prime p) : NeZero p := by rewrite [neZero_iff] --Goal : p ≠ 0 define at h linarith done lemma Lemma_7_5_1 {p e d m c s : Nat} {t : Int} (h1 : prime p) (h2 : e * d = (p - 1) * s + 1) (h3 : m ^ e - c = p * t) : c ^ d ≡ m (MOD p) := by have h4 : m ^ e ≡ c (MOD p) := Exists.intro t h3 have h5 : [m ^ e]_p = [c]_p := (cc_eq_iff_congr _ _ _).rtl h4 rewrite [←Exercise_7_4_5_Nat] at h5 --h5 : [m]_p ^ e = [c]_p by_cases h6 : p ∣ m · -- Case 1. h6 : p ∣ m have h7 : m ≡ 0 (MOD p) := by obtain (j : Nat) (h8 : m = p * j) from h6 apply Exists.intro (↑j : Int) --Goal : ↑m - 0 = ↑p * ↑j rewrite [h8, Nat.cast_mul] ring done have h8 : [m]_p = [0]_p := (cc_eq_iff_congr _ _ _).rtl h7 have h9 : e * d ≠ 0 := by rewrite [h2] show (p - 1) * s + 1 ≠ 0 from Nat.add_one_ne_zero _ done have h10 : (0 : Int) ^ (e * d) = 0 := zero_pow h9 have h11 : [c ^ d]_p = [m]_p := calc [c ^ d]_p _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat] _ = ([m]_p ^ e) ^ d := by rw [h5] _ = [m]_p ^ (e * d) := by ring _ = [0]_p ^ (e * d) := by rw [h8] _ = [0 ^ (e * d)]_p := Exercise_7_4_5_Int _ _ _ _ = [0]_p := by rw [h10] _ = [m]_p := by rw [h8] show c ^ d ≡ m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11 done · -- Case 2. h6 : ¬p ∣ m have h7 : rel_prime m p := rel_prime_of_prime_not_dvd h1 h6 have h8 : rel_prime p m := rel_prime_symm h7 have h9 : NeZero p := prime_NeZero h1 have h10 : (1 : Int) ^ s = 1 := by ring have h11 : [c ^ d]_p = [m]_p := calc [c ^ d]_p _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat] _ = ([m]_p ^ e) ^ d := by rw [h5] _ = [m]_p ^ (e * d) := by ring _ = [m]_p ^ ((p - 1) * s + 1) := by rw [h2] _ = ([m]_p ^ (p - 1)) ^ s * [m]_p := by ring _ = ([m]_p ^ (phi p)) ^ s * [m]_p := by rw [phi_prime h1] _ = [1]_p ^ s * [m]_p := by rw [Theorem_7_4_2 h8] _ = [1 ^ s]_p * [m]_p := by rw [Exercise_7_4_5_Int] _ = [1]_p * [m]_p := by rw [h10] _ = [m]_p * [1]_p := by ring _ = [m]_p := Theorem_7_3_6_7 _ show c ^ d ≡ m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11 done done theorem Theorem_7_5_1 (p q n e d k m c : Nat) (p_prime : prime p) (q_prime : prime q) (p_ne_q : p ≠ q) (n_pq : n = p * q) (ed_congr_1 : e * d = k * (p - 1) * (q - 1) + 1) (h1 : [m]_n ^ e = [c]_n) : [c]_n ^ d = [m]_n := by rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] at h1 --h1 : m ^ e ≡ c (MOD n) rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] --Goal : c ^ d ≡ m (MOD n) obtain (j : Int) (h2 : m ^ e - c = n * j) from h1 rewrite [n_pq, Nat.cast_mul] at h2 --h2 : m ^ e - c = p * q * j have h3 : e * d = (p - 1) * (k * (q - 1)) + 1 := by rewrite [ed_congr_1] ring done have h4 : m ^ e - c = p * (q * j) := by rewrite [h2] ring done have congr_p : c ^ d ≡ m (MOD p) := Lemma_7_5_1 p_prime h3 h4 have h5 : e * d = (q - 1) * (k * (p - 1)) + 1 := by rewrite [ed_congr_1] ring done have h6 : m ^ e - c = q * (p * j) := by rewrite [h2] ring done have congr_q : c ^ d ≡ m (MOD q) := Lemma_7_5_1 q_prime h5 h6 have h7 : ¬q ∣ p := by by_contra h8 have h9 : q = 1 ∨ q = p := dvd_prime p_prime h8 disj_syll h9 (prime_not_one q_prime) show False from p_ne_q h9.symm done have h8 : rel_prime p q := rel_prime_of_prime_not_dvd q_prime h7 rewrite [n_pq, Lemma_7_4_5 _ _ h8] show c ^ d ≡ m (MOD p) ∧ c ^ d ≡ m (MOD q) from And.intro congr_p congr_q done /- BEGIN EXERCISES -/ namespace Exercises /- Section 7.1 -/ -- 1. theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry -- 2. lemma gcd_comm_lt {a b : Nat} (h : a < b) : gcd a b = gcd b a := sorry theorem gcd_comm (a b : Nat) : gcd a b = gcd b a := sorry -- 3. theorem Exercise_7_1_5 (a b : Nat) (n : Int) : (∃ (s t : Int), s * a + t * b = n) ↔ (↑(gcd a b) : Int) ∣ n := sorry -- 4. theorem Exercise_7_1_6 (a b c : Nat) : gcd a b = gcd (a + b * c) b := sorry -- 5. theorem gcd_is_nonzero {a b : Nat} (h : a ≠ 0 ∨ b ≠ 0) : gcd a b ≠ 0 := sorry -- 6. theorem gcd_greatest {a b d : Nat} (h1 : gcd a b ≠ 0) (h2 : d ∣ a) (h3 : d ∣ b) : d ≤ gcd a b := sorry -- 7. lemma Lemma_7_1_10a {a b : Nat} (n : Nat) (h : a ∣ b) : (n * a) ∣ (n * b) := sorry lemma Lemma_7_1_10b {a b n : Nat} (h1 : n ≠ 0) (h2 : (n * a) ∣ (n * b)) : a ∣ b := sorry lemma Lemma_7_1_10c {a b : Nat} (h1 : a ∣ b) (h2 : b ∣ a) : a = b := sorry theorem Exercise_7_1_10 (a b n : Nat) : gcd (n * a) (n * b) = n * gcd a b := sorry /- Section 7.2 -/ -- 1. lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry -- 2. -- Hints: Start with apply List.rec. You may find mul_ne_zero useful theorem prod_nonzero_nonzero : ∀ (l : List Nat), (∀ a ∈ l, a ≠ 0) → prod l ≠ 0 := sorry -- 3. theorem rel_prime_iff_no_common_factor (a b : Nat) : rel_prime a b ↔ ¬∃ (p : Nat), prime p ∧ p ∣ a ∧ p ∣ b := sorry -- 4. theorem rel_prime_symm {a b : Nat} (h : rel_prime a b) : rel_prime b a := sorry -- 5. lemma in_prime_factorization_iff_prime_factor {a : Nat} {l : List Nat} (h1 : prime_factorization a l) (p : Nat) : p ∈ l ↔ prime_factor p a := sorry -- 6. theorem Exercise_7_2_5 {a b : Nat} {l m : List Nat} (h1 : prime_factorization a l) (h2 : prime_factorization b m) : rel_prime a b ↔ (¬∃ (p : Nat), p ∈ l ∧ p ∈ m) := sorry -- 7. theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry -- 8. theorem Exercise_7_2_7 {a b a' b' : Nat} (h1 : rel_prime a b) (h2 : a' ∣ a) (h3 : b' ∣ b) : rel_prime a' b' := sorry -- 9. theorem Exercise_7_2_9 {a b j k : Nat} (h1 : gcd a b ≠ 0) (h2 : a = j * gcd a b) (h3 : b = k * gcd a b) : rel_prime j k := sorry -- 10. theorem Exercise_7_2_17a (a b c : Nat) : gcd a (b * c) ∣ gcd a b * gcd a c := sorry /- Section 7.3 -/ -- 1. theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry -- 2. theorem Theorem_7_3_6_3 {m : Nat} (X : ZMod m) : X + [0]_m = X := sorry -- 3. theorem Theorem_7_3_6_4 {m : Nat} (X : ZMod m) : ∃ (Y : ZMod m), X + Y = [0]_m := sorry -- 4. theorem Exercise_7_3_4a {m : Nat} (Z1 Z2 : ZMod m) (h1 : ∀ (X : ZMod m), X + Z1 = X) (h2 : ∀ (X : ZMod m), X + Z2 = X) : Z1 = Z2 := sorry -- 5. theorem Exercise_7_3_4b {m : Nat} (X Y1 Y2 : ZMod m) (h1 : X + Y1 = [0]_m) (h2 : X + Y2 = [0]_m) : Y1 = Y2 := sorry -- 6. theorem Theorem_7_3_10 (m a : Nat) (b : Int) : ¬(↑(gcd m a) : Int) ∣ b → ¬∃ (x : Int), a * x ≡ b (MOD m) := sorry -- 7. theorem Theorem_7_3_11 (m n : Nat) (a b : Int) (h1 : n ≠ 0) : n * a ≡ n * b (MOD n * m) ↔ a ≡ b (MOD m) := sorry -- 8. theorem Exercise_7_3_16 {m : Nat} {a b : Int} (h : a ≡ b (MOD m)) : ∀ (n : Nat), a ^ n ≡ b ^ n (MOD m) := sorry -- 9. example {m : Nat} [NeZero m] (X : ZMod m) : ∃! (a : Int), 0 ≤ a ∧ a < m ∧ X = [a]_m := sorry -- 10. theorem congr_rel_prime {m a b : Nat} (h1 : a ≡ b (MOD m)) : rel_prime m a ↔ rel_prime m b := sorry -- 11. --Hint: You may find the theorem Int.ofNat_mod_ofNat useful. theorem rel_prime_mod (m a : Nat) : rel_prime m (a % m) ↔ rel_prime m a := sorry -- 12. lemma congr_iff_mod_eq_Int (m : Nat) (a b : Int) [NeZero m] : a ≡ b (MOD m) ↔ a % ↑m = b % ↑m := sorry --Hint for next theorem: Use the lemma above, --together with the theorems Int.ofNat_mod_ofNat and Nat.cast_inj. theorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] : ↑a ≡ ↑b (MOD m) ↔ a % m = b % m := sorry /- Section 7.4 -/ -- 1. --Hint: Use induction. --For the base case, compute [a]_m ^ 0 * [1]_m in two ways: --by Theorem_7_3_6_7, [a] ^ 0 * [1]_m = [a]_m ^ 0 --by ring, [a]_m ^ 0 * [1]_m = [1]_m. lemma Exercise_7_4_5_Int (m : Nat) (a : Int) : ∀ (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry -- 2.

lemma left_inv_one_one_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g' (g i) = i) : one_one_below n g

HTPI.Exercises.left_inv_one_one_below

htpi/HTPILib/Chap7.lean

htpi

/- Copyright 2023 Daniel J. Velleman -/ import HTPILib.Chap6 namespace HTPI /- Definitions -/ lemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by have h : n + 1 > 0 := Nat.succ_pos n show a % (n + 1) < n + 1 from Nat.mod_lt a h done def gcd (a b : Nat) : Nat := match b with | 0 => a | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd (n + 1) (a % (n + 1)) termination_by b mutual def gcd_c1 (a b : Nat) : Int := match b with | 0 => 1 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c2 (n + 1) (a % (n + 1)) --Corresponds to s = t' termination_by b def gcd_c2 (a b : Nat) : Int := match b with | 0 => 0 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c1 (n + 1) (a % (n + 1)) - (gcd_c2 (n + 1) (a % (n + 1))) * ↑(a / (n + 1)) --Corresponds to t = s' - t'q termination_by b end def prime (n : Nat) : Prop := 2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n def prime_factor (p n : Nat) : Prop := prime p ∧ p ∣ n def all_prime (l : List Nat) : Prop := ∀ p ∈ l, prime p def nondec (l : List Nat) : Prop := match l with | [] => True --Of course, True is a proposition that is always true | n :: L => (∀ m ∈ L, n ≤ m) ∧ nondec L def nondec_prime_list (l : List Nat) : Prop := all_prime l ∧ nondec l def prod (l : List Nat) : Nat := match l with | [] => 1 | n :: L => n * (prod L) def prime_factorization (n : Nat) (l : List Nat) : Prop := nondec_prime_list l ∧ prod l = n def rel_prime (a b : Nat) : Prop := gcd a b = 1 def congr_mod (m : Nat) (a b : Int) : Prop := (↑m : Int) ∣ (a - b) def cc (m : Nat) (a : Int) : ZMod m := (↑a : ZMod m) notation:50 a " ≡ " b " (MOD " m ")" => congr_mod m a b notation:max "["a"]_"m:max => cc m a def invertible {m : Nat} (X : ZMod m) : Prop := ∃ (Y : ZMod m), X * Y = [1]_m def num_rp_below (m k : Nat) : Nat := match k with | 0 => 0 | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j def phi (m : Nat) : Nat := num_rp_below m m def prod_seq {m : Nat} (j k : Nat) (f : Nat → ZMod m) : ZMod m := match j with | 0 => [1]_m | n + 1 => prod_seq n k f * f (k + n) def maps_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i < n, g i < n def one_one_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i1 < n, ∀ i2 < n, g i1 = g i2 → i1 = i2 def onto_below (n : Nat) (g : Nat → Nat) : Prop := ∀ k < n, ∃ i < n, g i = k def perm_below (n : Nat) (g : Nat → Nat) : Prop := maps_below n g ∧ one_one_below n g ∧ onto_below n g def inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m) def swap (u v i : Nat) : Nat := if i = u then v else if i = v then u else i namespace Euler --For definitions specific to Euler's theorem def F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m def G (m a i : Nat) : Nat := (a * i) % m def Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i end Euler /- Section 7.1 -/ theorem dvd_mod_of_dvd_a_b {a b d : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ (a % b) := by set q : Nat := a / b have h3 : b * q + a % b = a := Nat.div_add_mod a b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 define --Goal : ∃ (c : Nat), a % b = d * c apply Exists.intro (j - k * q) show a % b = d * (j - k * q) from calc a % b _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm _ = a - b * q := by rw [h3] _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc] _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm done theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry #eval gcd 672 161 --Answer: 7 lemma gcd_base (a : Nat) : gcd a 0 = a := by rfl lemma gcd_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd a b = gcd b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1)) rfl done lemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b ≠ 0) : a % b < b := by have h1 : b > 0 := Nat.pos_of_ne_zero h show a % b < b from Nat.mod_lt a h1 done lemma dvd_self (n : Nat) : n ∣ n := by apply Exists.intro 1 ring done theorem gcd_dvd : ∀ (b a : Nat), (gcd a b) ∣ a ∧ (gcd a b) ∣ b := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd a b_1) ∣ a ∧ (gcd a b_1) ∣ b_1 fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_base] --Goal: a ∣ a ∧ a ∣ 0 apply And.intro (dvd_self a) define apply Exists.intro 0 rfl done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_nonzero a h1] --Goal : gcd b (a % b) ∣ a ∧ gcd b (a % b) ∣ b have h2 : a % b < b := mod_nonzero_lt a h1 have h3 : (gcd b (a % b)) ∣ b ∧ (gcd b (a % b)) ∣ (a % b) := ih (a % b) h2 b apply And.intro _ h3.left show (gcd b (a % b)) ∣ a from dvd_a_of_dvd_b_mod h3.left h3.right done done theorem gcd_dvd_left (a b : Nat) : (gcd a b) ∣ a := (gcd_dvd b a).left theorem gcd_dvd_right (a b : Nat) : (gcd a b) ∣ b := (gcd_dvd b a).right lemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl lemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c1 a b = gcd_c2 b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done lemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl lemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * ↑(a / b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done theorem gcd_lin_comb : ∀ (b a : Nat), (gcd_c1 a b) * ↑a + (gcd_c2 a b) * ↑b = ↑(gcd a b) := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd_c1 a b_1) * ↑a + (gcd_c2 a b_1) * ↑b_1 = ↑(gcd a b_1) fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base] --Goal : 1 * ↑a + 0 * ↑0 = ↑a ring done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1] --Goal : gcd_c2 b (a % b) * ↑a + -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * ↑(a / b)) * ↑b = -- ↑(gcd b (a % b)) set r : Nat := a % b set q : Nat := a / b set s : Int := gcd_c1 b r set t : Int := gcd_c2 b r --Goal : t * ↑a + (s - t * ↑q) * ↑b = ↑(gcd b r) have h2 : r < b := mod_nonzero_lt a h1 have h3 : s * ↑b + t * ↑r = ↑(gcd b r) := ih r h2 b have h4 : b * q + r = a := Nat.div_add_mod a b rewrite [←h3, ←h4] rewrite [Nat.cast_add, Nat.cast_mul] --Goal : t * (↑b * ↑q + ↑r) + (s - t * ↑q) * ↑b = s * ↑b + t * ↑r ring done done #eval gcd_c1 672 161 --Answer: 6 #eval gcd_c2 672 161 --Answer: -25 --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161 theorem Theorem_7_1_6 {d a b : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ gcd a b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑d ∣ ↑(gcd a b) set s : Int := gcd_c1 a b set t : Int := gcd_c2 a b have h3 : s * ↑a + t * ↑b = ↑(gcd a b) := gcd_lin_comb b a rewrite [←h3] --Goal : ↑d ∣ s * ↑a + t * ↑b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul] --Goal : ↑d ∣ s * (↑d * ↑j) + t * (↑d * ↑k) define apply Exists.intro (s * ↑j + t * ↑k) ring done /- Section 7.2 -/ theorem dvd_trans {a b c : Nat} (h1 : a ∣ b) (h2 : b ∣ c) : a ∣ c := by define at h1; define at h2; define obtain (m : Nat) (h3 : b = a * m) from h1 obtain (n : Nat) (h4 : c = b * n) from h2 rewrite [h3, mul_assoc] at h4 apply Exists.intro (m * n) show c = a * (m * n) from h4 done lemma exists_prime_factor : ∀ (n : Nat), 2 ≤ n → ∃ (p : Nat), prime_factor p n := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, 2 ≤ n_1 → ∃ (p : Nat), prime_factor p n_1 assume h1 : 2 ≤ n by_cases h2 : prime n · -- Case 1. h2 : prime n apply Exists.intro n define --Goal : prime n ∧ n ∣ n show prime n ∧ n ∣ n from And.intro h2 (dvd_self n) done · -- Case 2. h2 : ¬prime n define at h2 --h2 : ¬(2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n) demorgan at h2 disj_syll h2 h1 obtain (a : Nat) (h3 : ∃ (b : Nat), a * b = n ∧ a < n ∧ b < n) from h2 obtain (b : Nat) (h4 : a * b = n ∧ a < n ∧ b < n) from h3 have h5 : 2 ≤ a := by by_contra h6 have h7 : a ≤ 1 := by linarith have h8 : n ≤ b := calc n _ = a * b := h4.left.symm _ ≤ 1 * b := by rel [h7] _ = b := by ring linarith --n ≤ b contradicts b < n done have h6 : ∃ (p : Nat), prime_factor p a := ih a h4.right.left h5 obtain (p : Nat) (h7 : prime_factor p a) from h6 apply Exists.intro p define --Goal : prime p ∧ p ∣ n define at h7 --h7 : prime p ∧ p ∣ a apply And.intro h7.left have h8 : a ∣ n := by apply Exists.intro b show n = a * b from (h4.left).symm done show p ∣ n from dvd_trans h7.right h8 done done lemma exists_least_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q := by set S : Set Nat := {p : Nat | prime_factor p n} have h2 : ∃ (p : Nat), p ∈ S := exists_prime_factor n h show ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q from well_ord_princ S h2 done lemma all_prime_nil : all_prime [] := by define --Goal : ∀ p ∈ [], prime p fix p : Nat contrapos --Goal : ¬prime p → p ∉ [] assume h1 : ¬prime p show p ∉ [] from List.not_mem_nil p done lemma all_prime_cons (n : Nat) (L : List Nat) : all_prime (n :: L) ↔ prime n ∧ all_prime L := by apply Iff.intro · -- (→) assume h1 : all_prime (n :: L) --Goal : prime n ∧ all_prime L define at h1 --h1 : ∀ p ∈ n :: L, prime p apply And.intro (h1 n (List.mem_cons_self n L)) define --Goal : ∀ p ∈ L, prime p fix p : Nat assume h2 : p ∈ L show prime p from h1 p (List.mem_cons_of_mem n h2) done · -- (←) assume h1 : prime n ∧ all_prime L --Goal : all_prime (n :: l) define : all_prime L at h1 define fix p : Nat assume h2 : p ∈ n :: L rewrite [List.mem_cons] at h2 --h2 : p = n ∨ p ∈ L by_cases on h2 · -- Case 1. h2 : p = n rewrite [h2] show prime n from h1.left done · -- Case 2. h2 : p ∈ L show prime p from h1.right p h2 done done done lemma nondec_nil : nondec [] := by define --Goal : True trivial --trivial proves some obviously true statements, such as True done lemma nondec_cons (n : Nat) (L : List Nat) : nondec (n :: L) ↔ (∀ m ∈ L, n ≤ m) ∧ nondec L := by rfl lemma prod_nil : prod [] = 1 := by rfl lemma prod_cons : prod (n :: L) = n * (prod L) := by rfl lemma exists_cons_of_length_eq_succ {A : Type} {l : List A} {n : Nat} (h : l.length = n + 1) : ∃ (a : A) (L : List A), l = a :: L ∧ L.length = n := by have h1 : ¬l.length = 0 := by linarith rewrite [List.length_eq_zero] at h1 obtain (a : A) (h2 : ∃ (L : List A), l = a :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List A) (h3 : l = a :: L) from h2 apply Exists.intro a apply Exists.intro L apply And.intro h3 have h4 : (a :: L).length = L.length + 1 := List.length_cons a L rewrite [←h3, h] at h4 show L.length = n from (Nat.add_right_cancel h4).symm done lemma list_elt_dvd_prod_by_length (a : Nat) : ∀ (n : Nat), ∀ (l : List Nat), l.length = n → a ∈ l → a ∣ prod l := by by_induc · --Base Case fix l : List Nat assume h1 : l.length = 0 rewrite [List.length_eq_zero] at h1 --h1 : l = [] rewrite [h1] --Goal : a ∈ [] → a ∣ prod [] contrapos assume h2 : ¬a ∣ prod [] show a ∉ [] from List.not_mem_nil a done · -- Induction Step fix n : Nat assume ih : ∀ (l : List Nat), List.length l = n → a ∈ l → a ∣ prod l fix l : List Nat assume h1 : l.length = n + 1 --Goal : a ∈ l → a ∣ prod l obtain (b : Nat) (h2 : ∃ (L : List Nat), l = b :: L ∧ L.length = n) from exists_cons_of_length_eq_succ h1 obtain (L : List Nat) (h3 : l = b :: L ∧ L.length = n) from h2 have h4 : a ∈ L → a ∣ prod L := ih L h3.right assume h5 : a ∈ l rewrite [h3.left, prod_cons] --Goal : a ∣ b * prod L rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b ∨ a ∈ L by_cases on h5 · -- Case 1. h5 : a = b apply Exists.intro (prod L) rewrite [h5] rfl done · -- Case 2. h5 : a ∈ L have h6 : a ∣ prod L := h4 h5 have h7 : prod L ∣ b * prod L := by apply Exists.intro b ring done show a ∣ b * prod L from dvd_trans h6 h7 done done done lemma list_elt_dvd_prod {a : Nat} {l : List Nat} (h : a ∈ l) : a ∣ prod l := by set n : Nat := l.length have h1 : l.length = n := by rfl show a ∣ prod l from list_elt_dvd_prod_by_length a n l h1 h done lemma exists_prime_factorization : ∀ (n : Nat), n ≥ 1 → ∃ (l : List Nat), prime_factorization n l := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, n_1 ≥ 1 → ∃ (l : List Nat), prime_factorization n_1 l assume h1 : n ≥ 1 by_cases h2 : n = 1 · -- Case 1. h2 : n = 1 apply Exists.intro [] define apply And.intro · -- Proof of nondec_prime_list [] define show all_prime [] ∧ nondec [] from And.intro all_prime_nil nondec_nil done · -- Proof of prod [] = n rewrite [prod_nil, h2] rfl done done · -- Case 2. h2 : n ≠ 1 have h3 : n ≥ 2 := lt_of_le_of_ne' h1 h2 obtain (p : Nat) (h4 : prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q) from exists_least_prime_factor h3 have p_prime_factor : prime_factor p n := h4.left define at p_prime_factor have p_prime : prime p := p_prime_factor.left have p_dvd_n : p ∣ n := p_prime_factor.right have p_least : ∀ (q : Nat), prime_factor q n → p ≤ q := h4.right obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n have h5 : m ≠ 0 := by contradict h1 with h6 have h7 : n = 0 := calc n _ = p * m := n_eq_pm _ = p * 0 := by rw [h6] _ = 0 := by ring rewrite [h7] decide done have m_pos : 0 < m := Nat.pos_of_ne_zero h5 have m_lt_n : m < n := by define at p_prime show m < n from calc m _ < m + m := by linarith _ = 2 * m := by ring _ ≤ p * m := by rel [p_prime.left] _ = n := n_eq_pm.symm done obtain (L : List Nat) (h6 : prime_factorization m L) from ih m m_lt_n m_pos define at h6 have ndpl_L : nondec_prime_list L := h6.left define at ndpl_L apply Exists.intro (p :: L) define apply And.intro · -- Proof of nondec_prime_list (p :: L) define apply And.intro · -- Proof of all_prime (p :: L) rewrite [all_prime_cons] show prime p ∧ all_prime L from And.intro p_prime ndpl_L.left done · -- Proof of nondec (p :: L) rewrite [nondec_cons] apply And.intro _ ndpl_L.right fix q : Nat assume q_in_L : q ∈ L have h7 : q ∣ prod L := list_elt_dvd_prod q_in_L rewrite [h6.right] at h7 --h7 : q ∣ m have h8 : m ∣ n := by apply Exists.intro p rewrite [n_eq_pm] ring done have q_dvd_n : q ∣ n := dvd_trans h7 h8 have ap_L : all_prime L := ndpl_L.left define at ap_L have q_prime_factor : prime_factor q n := And.intro (ap_L q q_in_L) q_dvd_n show p ≤ q from p_least q q_prime_factor done done · -- Proof of prod (p :: L) = n rewrite [prod_cons, h6.right, n_eq_pm] rfl done done done theorem Theorem_7_2_2 {a b c : Nat} (h1 : c ∣ a * b) (h2 : rel_prime a c) : c ∣ b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑c ∣ ↑b define at h1; define at h2; define obtain (j : Nat) (h3 : a * b = c * j) from h1 set s : Int := gcd_c1 a c set t : Int := gcd_c2 a c have h4 : s * ↑a + t * ↑c = ↑(gcd a c) := gcd_lin_comb c a rewrite [h2, Nat.cast_one] at h4 --h4 : s * ↑a + t * ↑c = (1 : Int) apply Exists.intro (s * ↑j + t * ↑b) show ↑b = ↑c * (s * ↑j + t * ↑b) from calc ↑b _ = (1 : Int) * ↑b := (one_mul _).symm _ = (s * ↑a + t * ↑c) * ↑b := by rw [h4] _ = s * (↑a * ↑b) + t * ↑c * ↑b := by ring _ = s * (↑c * ↑j) + t * ↑c * ↑b := by rw [←Nat.cast_mul a b, h3, Nat.cast_mul c j] _ = ↑c * (s * ↑j + t * ↑b) := by ring done lemma le_nonzero_prod_left {a b : Nat} (h : a * b ≠ 0) : a ≤ a * b := by have h1 : b ≠ 0 := by contradict h with h1 rewrite [h1] ring done have h2 : 1 ≤ b := Nat.pos_of_ne_zero h1 show a ≤ a * b from calc a = a * 1 := (mul_one a).symm _ ≤ a * b := by rel [h2] done lemma le_nonzero_prod_right {a b : Nat} (h : a * b ≠ 0) : b ≤ a * b := by rewrite [mul_comm] rewrite [mul_comm] at h show b ≤ b * a from le_nonzero_prod_left h done lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry lemma rel_prime_of_prime_not_dvd {a p : Nat} (h1 : prime p) (h2 : ¬p ∣ a) : rel_prime a p := by have h3 : gcd a p ∣ a := gcd_dvd_left a p have h4 : gcd a p ∣ p := gcd_dvd_right a p have h5 : gcd a p = 1 ∨ gcd a p = p := dvd_prime h1 h4 have h6 : gcd a p ≠ p := by contradict h2 with h6 rewrite [h6] at h3 show p ∣ a from h3 done disj_syll h5 h6 show rel_prime a p from h5 done theorem Theorem_7_2_3 {a b p : Nat} (h1 : prime p) (h2 : p ∣ a * b) : p ∣ a ∨ p ∣ b := by or_right with h3 have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3 show p ∣ b from Theorem_7_2_2 h2 h4 done lemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a ≥ 1 := by have h1 : a ≠ 0 := by by_contra h1 rewrite [h1] at h contradict h linarith done show a ≥ 1 from Nat.pos_of_ne_zero h1 done lemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by have h1 : a ≥ 1 := ge_one_of_prod_one h have h2 : a * b ≠ 0 := by linarith have h3 : a ≤ a * b := le_nonzero_prod_left h2 rewrite [h] at h3 show a = 1 from Nat.le_antisymm h3 h1 done lemma eq_one_of_dvd_one {n : Nat} (h : n ∣ 1) : n = 1 := by obtain (j : Nat) (h1 : 1 = n * j) from h show n = 1 from eq_one_of_prod_one h1.symm done lemma prime_not_one {p : Nat} (h : prime p) : p ≠ 1 := by define at h linarith done theorem Theorem_7_2_4 {p : Nat} (h1 : prime p) : ∀ (l : List Nat), p ∣ prod l → ∃ a ∈ l, p ∣ a := by apply List.rec · -- Base Case. Goal : p ∣ prod [] → ∃ a ∈ [], p ∣ a rewrite [prod_nil] assume h2 : p ∣ 1 show ∃ a ∈ [], p ∣ a from absurd (eq_one_of_dvd_one h2) (prime_not_one h1) done · -- Induction Step fix b : Nat fix L : List Nat assume ih : p ∣ prod L → ∃ a ∈ L, p ∣ a --Goal : p ∣ prod (b :: L) → ∃ a ∈ b :: L, p ∣ a assume h2 : p ∣ prod (b :: L) rewrite [prod_cons] at h2 have h3 : p ∣ b ∨ p ∣ prod L := Theorem_7_2_3 h1 h2 by_cases on h3 · -- Case 1. h3 : p ∣ b apply Exists.intro b show b ∈ b :: L ∧ p ∣ b from And.intro (List.mem_cons_self b L) h3 done · -- Case 2. h3 : p ∣ prod L obtain (a : Nat) (h4 : a ∈ L ∧ p ∣ a) from ih h3 apply Exists.intro a show a ∈ b :: L ∧ p ∣ a from And.intro (List.mem_cons_of_mem b h4.left) h4.right done done done lemma prime_in_list {p : Nat} {l : List Nat} (h1 : prime p) (h2 : all_prime l) (h3 : p ∣ prod l) : p ∈ l := by obtain (a : Nat) (h4 : a ∈ l ∧ p ∣ a) from Theorem_7_2_4 h1 l h3 define at h2 have h5 : prime a := h2 a h4.left have h6 : p = 1 ∨ p = a := dvd_prime h5 h4.right disj_syll h6 (prime_not_one h1) rewrite [h6] show a ∈ l from h4.left done lemma first_le_first {p q : Nat} {l m : List Nat} (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m)) (h3 : prod (p :: l) = prod (q :: m)) : p ≤ q := by define at h1; define at h2 have h4 : q ∣ prod (p :: l) := by define apply Exists.intro (prod m) rewrite [←prod_cons] show prod (p :: l) = prod (q :: m) from h3 done have h5 : all_prime (q :: m) := h2.left rewrite [all_prime_cons] at h5 have h6 : q ∈ p :: l := prime_in_list h5.left h1.left h4 have h7 : nondec (p :: l) := h1.right rewrite [nondec_cons] at h7 rewrite [List.mem_cons] at h6 by_cases on h6 · -- Case 1. h6 : q = p linarith done · -- Case 2. h6 : q ∈ l have h8 : ∀ m ∈ l, p ≤ m := h7.left show p ≤ q from h8 q h6 done done lemma nondec_prime_list_tail {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by define at h define rewrite [all_prime_cons, nondec_cons] at h show all_prime l ∧ nondec l from And.intro h.left.right h.right.right done lemma cons_prod_not_one {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : prod (p :: l) ≠ 1 := by define at h have h1 : all_prime (p :: l) := h.left rewrite [all_prime_cons] at h1 rewrite [prod_cons] by_contra h2 show False from (prime_not_one h1.left) (eq_one_of_prod_one h2) done lemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) : l = [] ↔ prod l = 1 := by apply Iff.intro · -- (→) assume h1 : l = [] rewrite [h1] show prod [] = 1 from prod_nil done · -- (←) contrapos assume h1 : ¬l = [] obtain (p : Nat) (h2 : ∃ (L : List Nat), l = p :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List Nat) (h3 : l = p :: L) from h2 rewrite [h3] at h rewrite [h3] show ¬prod (p :: L) = 1 from cons_prod_not_one h done done lemma prime_pos {p : Nat} (h : prime p) : p > 0 := by define at h linarith done theorem Theorem_7_2_5 : ∀ (l1 l2 : List Nat), nondec_prime_list l1 → nondec_prime_list l2 → prod l1 = prod l2 → l1 = l2 := by apply List.rec · -- Base Case. Goal : ∀ (l2 : List Nat), nondec_prime_list [] → -- nondec_prime_list l2 → prod [] = prod l2 → [] = l2 fix l2 : List Nat assume h1 : nondec_prime_list [] assume h2 : nondec_prime_list l2 assume h3 : prod [] = prod l2 rewrite [prod_nil, eq_comm, ←list_nil_iff_prod_one h2] at h3 show [] = l2 from h3.symm done · -- Induction Step fix p : Nat fix L1 : List Nat assume ih : ∀ (L2 : List Nat), nondec_prime_list L1 → nondec_prime_list L2 → prod L1 = prod L2 → L1 = L2 -- Goal : ∀ (l2 : List Nat), nondec_prime_list (p :: L1) → -- nondec_prime_list l2 → prod (p :: L1) = prod l2 → p :: L1 = l2 fix l2 : List Nat assume h1 : nondec_prime_list (p :: L1) assume h2 : nondec_prime_list l2 assume h3 : prod (p :: L1) = prod l2 have h4 : ¬prod (p :: L1) = 1 := cons_prod_not_one h1 rewrite [h3, ←list_nil_iff_prod_one h2] at h4 obtain (q : Nat) (h5 : ∃ (L : List Nat), l2 = q :: L) from List.exists_cons_of_ne_nil h4 obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5 rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2) rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2) have h7 : p ≤ q := first_le_first h1 h2 h3 have h8 : q ≤ p := first_le_first h2 h1 h3.symm have h9 : p = q := by linarith rewrite [h9, prod_cons, prod_cons] at h3 --h3 : q * prod L1 = q * prod L2 have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1 have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2 define at h2 have h12 : all_prime (q :: L2) := h2.left rewrite [all_prime_cons] at h12 have h13 : q > 0 := prime_pos h12.left have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3 have h15 : L1 = L2 := ih L2 h10 h11 h14 rewrite [h6, h9, h15] rfl done done theorem fund_thm_arith (n : Nat) (h : n ≥ 1) : ∃! (l : List Nat), prime_factorization n l := by exists_unique · -- Existence show ∃ (l : List Nat), prime_factorization n l from exists_prime_factorization n h done · -- Uniqueness fix l1 : List Nat; fix l2 : List Nat assume h1 : prime_factorization n l1 assume h2 : prime_factorization n l2 define at h1; define at h2 have h3 : prod l1 = n := h1.right rewrite [←h2.right] at h3 show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3 done done /- Section 7.3 -/ theorem congr_refl (m : Nat) : ∀ (a : Int), a ≡ a (MOD m) := by fix a : Int define --Goal : ∃ (c : Int), a - a = ↑m * c apply Exists.intro 0 ring done theorem congr_symm {m : Nat} : ∀ {a b : Int}, a ≡ b (MOD m) → b ≡ a (MOD m) := by fix a : Int; fix b : Int assume h1 : a ≡ b (MOD m) define at h1 --h1 : ∃ (c : Int), a - b = ↑m * c define --Goal : ∃ (c : Int), b - a = ↑m * c obtain (c : Int) (h2 : a - b = m * c) from h1 apply Exists.intro (-c) show b - a = m * (-c) from calc b - a _ = -(a - b) := by ring _ = -(m * c) := by rw [h2] _ = m * (-c) := by ring done theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry /- Fundamental properties of congruence classes -/ lemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) : X = Y ↔ X.val = Y.val := Fin.ext_iff lemma val_nat_eq_mod (n k : Nat) : ([k]_(n + 1)).val = k % (n + 1) := by rfl lemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl theorem cc_rep {m : Nat} (X : ZMod m) : ∃ (a : Int), X = [a]_m := match m with | 0 => by apply Exists.intro X rfl done | n + 1 => by apply Exists.intro ↑(X.val) have h1 : X.val < n + 1 := Fin.prop X rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1] rfl done theorem add_class (m : Nat) (a b : Int) : [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm theorem mul_class (m : Nat) (a b : Int) : [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm lemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ [a - b]_m = [0]_m := by apply Iff.intro · -- (→) assume h1 : [a]_m = [b]_m have h2 : a - b = a + (-b) := by ring have h3 : b + (-b) = 0 := by ring show [a - b]_m = [0]_m from calc [a - b]_m _ = [a + (-b)]_m := by rw [h2] _ = [a]_m + [-b]_m := by rw [add_class] _ = [b]_m + [-b]_m := by rw [h1] _ = [b + -b]_m := by rw [add_class] _ = [0]_m := by rw [h3] done · -- (←) assume h1 : [a - b]_m = [0]_m have h2 : b + (a - b) = a := by ring have h3 : b + 0 = b := by ring show [a]_m = [b]_m from calc [a]_m _ = [b + (a - b)]_m := by rw [h2] _ = [b]_m + [a - b]_m := by rw [add_class] _ = [b]_m + [0]_m := by rw [h1] _ = [b + 0]_m := by rw [add_class] _ = [b]_m := by rw [h3] done done lemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) : [a]_m = [0]_m → [-a]_m = [0]_m := by assume h1 : [a]_m = [0]_m have h2 : 0 + (-a) = -a := by ring have h3 : a + (-a) = 0 := by ring show [-a]_m = [0]_m from calc [-a]_m _ = [0 + (-a)]_m := by rw [h2] _ = [0]_m + [-a]_m := by rw [add_class] _ = [a]_m + [-a]_m := by rw [h1] _ = [a + (-a)]_m := by rw [add_class] _ = [0]_m := by rw [h3] done lemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) : [-a]_m = [0]_m ↔ [a]_m = [0]_m := by apply Iff.intro _ (cc_neg_zero_of_cc_zero m a) assume h1 : [-a]_m = [0]_m have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1 have h3 : -(-a) = a := by ring rewrite [h3] at h2 show [a]_m = [0]_m from h2 done lemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl lemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m ↔ m ∣ k := match m with | 0 => by have h : (0 : Int) = (↑(0 : Nat) : Int) := by rfl rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj] apply Iff.intro · -- (→) assume h1 : k = 0 rewrite [h1] show 0 ∣ 0 from dvd_self 0 done · -- (←) assume h1 : 0 ∣ k obtain (c : Nat) (h2 : k = 0 * c) from h1 rewrite [h2] ring done done | n + 1 => by rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero] show k % (n + 1) = 0 ↔ n + 1 ∣ k from (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm done lemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m ↔ ↑m ∣ a := by obtain (k : Nat) (h1 : a = ↑k ∨ a = -↑k) from Int.eq_nat_or_neg a by_cases on h1 · -- Case 1. h1: a = ↑k rewrite [h1, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done · -- Case 2. h1: a = -↑k rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done done theorem cc_eq_iff_congr (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ a ≡ b (MOD m) := calc [a]_m = [b]_m _ ↔ [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b _ ↔ ↑m ∣ (a - b) := cc_zero_iff_dvd m (a - b) _ ↔ a ≡ b (MOD m) := by rfl /- End of fundamental properties of congruence classes -/ lemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m := by have h1 : (↑m : Int) ≠ 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m) show 0 ≤ a % m from Int.emod_nonneg a h1 done lemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m) have h2 : (↑m : Int) > 0 := (Nat.cast_pos).rtl h1 show a % m < m from Int.emod_lt_of_pos a h2 done lemma congr_mod_mod (m : Nat) (a : Int) : a ≡ a % m (MOD m) := by define have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m apply Exists.intro (a / m) show a - a % m = m * (a / m) from calc a - (a % m) _ = m * (a / m) + a % m - a % m := by rw [h1] _ = m * (a / m) := by ring done lemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) := And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a)) theorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) : ∃! (r : Int), 0 ≤ r ∧ r < m ∧ a ≡ r (MOD m) := by exists_unique · -- Existence apply Exists.intro (a % m) show 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) from mod_cmpl_res m a done · -- Uniqueness fix r1 : Int; fix r2 : Int assume h1 : 0 ≤ r1 ∧ r1 < m ∧ a ≡ r1 (MOD m) assume h2 : 0 ≤ r2 ∧ r2 < m ∧ a ≡ r2 (MOD m) have h3 : r1 ≡ r2 (MOD m) := congr_trans (congr_symm h1.right.right) h2.right.right obtain (d : Int) (h4 : r1 - r2 = m * d) from h3 have h5 : r1 - r2 < m * 1 := by linarith have h6 : m * (-1) < r1 - r2 := by linarith rewrite [h4] at h5 --h5 : m * d < m * 1 rewrite [h4] at h6 --h6 : m * -1 < m * d have h7 : (↑m : Int) ≥ 0 := Nat.cast_nonneg m have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7 have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7 have h10 : d = 0 := by linarith show r1 = r2 from calc r1 _ = r1 - r2 + r2 := by ring _ = m * 0 + r2 := by rw [h4, h10] _ = r2 := by ring done done lemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m := (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a) theorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y rewrite [h1, h2] have h3 : a + b = b + a := by ring show [a]_m + [b]_m = [b]_m + [a]_m from calc [a]_m + [b]_m _ = [a + b]_m := add_class m a b _ = [b + a]_m := by rw [h3] _ = [b]_m + [a]_m := (add_class m b a).symm done theorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X rewrite [h1] have h2 : a * 1 = a := by ring show [a]_m * [1]_m = [a]_m from calc [a]_m * [1]_m _ = [a * 1]_m := mul_class m a 1 _ = [a]_m := by rw [h2] done theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry lemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) : [a]_m * [gcd_c2 m a]_m = [1]_m := by set s : Int := gcd_c1 m a have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m define at h1 rewrite [h1, Nat.cast_one] at h2 --h2 : s * ↑m + gcd_c2 m a * ↑a = 1 rewrite [mul_class, cc_eq_iff_congr] define --Goal : ∃ (c : Int), ↑a * gcd_c2 m a - 1 = ↑m * c apply Exists.intro (-s) show a * (gcd_c2 m a) - 1 = m * (-s) from calc a * (gcd_c2 m a) - 1 _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring _ = 1 + m * (-s) - 1 := by rw [h2] _ = m * (-s) := by ring done theorem Theorem_7_3_7 (m a : Nat) : invertible [a]_m ↔ rel_prime m a := by apply Iff.intro · -- (→) assume h1 : invertible [a]_m define at h1 obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1 obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y rewrite [h3, mul_class, cc_eq_iff_congr] at h2 define at h2 obtain (c : Int) (h4 : a * b - 1 = m * c) from h2 rewrite [Exercise_7_2_6] --Goal : ∃ (s t : Int), s * ↑m + t * ↑a = 1 apply Exists.intro (-c) apply Exists.intro b show (-c) * m + b * a = 1 from calc (-c) * m + b * a _ = (-c) * m + (a * b - 1) + 1 := by ring _ = (-c) * m + m * c + 1 := by rw [h4] _ = 1 := by ring done · -- (←) assume h1 : rel_prime m a define show ∃ (Y : ZMod m), [a]_m * Y = [1]_m from Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1) done done /- Section 7.4 -/ section Euler open Euler lemma num_rp_below_base {m : Nat} : num_rp_below m 0 = 0 := by rfl lemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) : num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by have h1 : num_rp_below m (j + 1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : gcd m j = 1 rewrite [if_pos h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1 show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1 done lemma num_rp_below_step_not_rp {m j : Nat} (h : ¬rel_prime m j) : num_rp_below m (j + 1) = num_rp_below m j := by have h1 : num_rp_below m (j +1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : ¬gcd m j = 1 rewrite [if_neg h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j show num_rp_below m (j + 1) = num_rp_below m j from h1 done lemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl #eval phi 10 --Answer: 4 lemma prod_inv_iff_inv {m : Nat} {X : ZMod m} (h1 : invertible X) (Y : ZMod m) : invertible (X * Y) ↔ invertible Y := by apply Iff.intro · -- (→) assume h2 : invertible (X * Y) obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2 apply Exists.intro (X * Z) rewrite [←h3] --Goal : Y * (X * Z) = X * Y * Z ring --Note that ring can do algebra in ZMod m done · -- (←) assume h2 : invertible Y obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1 obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2 apply Exists.intro (Xi * Yi) show (X * Y) * (Xi * Yi) = [1]_m from calc X * Y * (Xi * Yi) _ = (X * Xi) * (Y * Yi) := by ring _ = [1]_m * [1]_m := by rw [h3, h4] _ = [1]_m := Theorem_7_3_6_7 [1]_m done done lemma F_rp_def {m i : Nat} (h : rel_prime m i) : F m i = [i]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h --h : gcd m i = 1 rewrite [if_pos h] at h1 show F m i = [i]_m from h1 done lemma F_not_rp_def {m i : Nat} (h : ¬rel_prime m i) : F m i = [1]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h rewrite [h1, if_neg h] rfl done lemma prod_seq_base {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 0 k f = [1]_m := by rfl lemma prod_seq_step {m : Nat} (n k : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl lemma prod_seq_zero_step {m : Nat} (n : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by rewrite [prod_seq_step, zero_add] rfl done lemma prod_one {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 1 k f = f k := by rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7] rfl done lemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl lemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m := calc [G m a i]_m _ = [(a * i) % m]_m := by rfl _ = [a * i]_m := (cc_eq_mod m (a * i)).symm _ = [a]_m * [i]_m := (mul_class m a i).symm lemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) : rel_prime m (G m a i) ↔ rel_prime m i := by have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1 show rel_prime m (G m a i) ↔ rel_prime m i from calc rel_prime m (G m a i) _ ↔ invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm _ ↔ invertible ([a]_m * [i]_m) := by rw [cc_G] _ ↔ invertible [i]_m := prod_inv_iff_inv h2 ([i]_m) _ ↔ rel_prime m i := Theorem_7_3_7 m i done lemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) : F m (G m a i) = [a]_m * F m i := by have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2 show F m (G m a i) = [a]_m * F m i from calc F m (G m a i) _ = [G m a i]_m := F_rp_def h3 _ = [a]_m * [i]_m := cc_G m a i _ = [a]_m * F m i := by rw [F_rp_def h2] done lemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : ¬rel_prime m i) : F m (G m a i) = [1]_m := by rewrite [←G_rp_iff h1 i] at h2 show F m (G m a i) = [1]_m from F_not_rp_def h2 done lemma FG_prod {m a : Nat} (h1 : rel_prime m a) : ∀ (k : Nat), prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by by_induc · -- Base Case show prod_seq 0 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from calc prod_seq 0 0 ((F m) ∘ (G m a)) _ = [1]_m := prod_seq_base _ _ _ = [a]_m ^ 0 * [1]_m := by ring _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by rw [num_rp_below_base, prod_seq_base] done · -- Induction Step fix k : Nat assume ih : prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) by_cases h2 : rel_prime m k · -- Case 1. h2 : rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([a]_m * F m k) := by rw [FG_rp h1 h2] _ = [a]_m ^ ((num_rp_below m k) + 1) * ((prod_seq k 0 (F m)) * F m k) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_rp h2, prod_seq_zero_step] done · -- Case 2. h2 : ¬rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([1]_m) := by rw [FG_not_rp h1 h2] _ = [a]_m ^ (num_rp_below m k) * (prod_seq k 0 (F m) * ([1]_m)) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_not_rp h2, prod_seq_zero_step, F_not_rp_def h2] done done done lemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by define --Goal : ∀ i < m, G m a i < m fix i : Nat assume h1 : i < m rewrite [G_def] --Goal : a * i % m < m show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m) done lemma left_inv_one_one_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g' (g i) = i) : one_one_below n g := sorry lemma right_inv_onto_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g (g' i) = i) (h2 : maps_below n g') : onto_below n g := by define at h2; define fix k : Nat assume h3 : k < n apply Exists.intro (g' k) show g' k < n ∧ g (g' k) = k from And.intro (h2 k h3) (h1 k h3) done lemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m * [inv_mod m a]_m = [1]_m := by have h2 : 0 ≤ (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a) show [a]_m * [inv_mod m a]_m = [1]_m from calc [a]_m * [inv_mod m a]_m _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2] _ = [a]_m * [gcd_c2 m a]_m := by rw [←cc_eq_mod] _ = [1]_m := gcd_c2_inv h1 done lemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m := calc a * (b % m) % m = a % m * (b % m % m) % m := Nat.mul_mod _ _ _ _ = a % m * (b % m) % m := by rw [Nat.mod_mod] _ = a * b % m := (Nat.mul_mod _ _ _).symm lemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm] rfl done theorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] : ↑a ≡ ↑b (MOD m) ↔ a % m = b % m := sorry lemma mul_inv_mod_cancel {m a i : Nat} [NeZero m] (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1 rewrite [mul_class, cc_eq_iff_congr, ←Nat.cast_mul, ←Nat.cast_one, congr_iff_mod_eq_Nat] at h3 show a * inv_mod m a * i % m = i from calc a * (inv_mod m a) * i % m _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = 1 % m * i % m := by rw [h3] _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = i % m := by rw [one_mul] _ = i := Nat.mod_eq_of_lt h2 done lemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl lemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, G m a (Ginv m a i) = i := by fix i : Nat assume h2 : i < m show G m a (Ginv m a i) = i from calc G m a (Ginv m a i) _ = a * ((inv_mod m a * i) % m) % m := by rfl _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, Ginv m a (G m a i) = i := by fix i : Nat assume h2 : i < m show Ginv m a (G m a i) = i from calc Ginv m a (G m a i) _ = inv_mod m a * ((a * i) % m) % m := by rfl _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc, mul_comm (inv_mod m a)] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_maps_below (m a : Nat) [NeZero m] : maps_below m (Ginv m a) := G_maps_below m (inv_mod m a) lemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : one_one_below m (G m a) := left_inv_one_one_below (Ginv_left_inv h1) lemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : onto_below m (G m a) := right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a) lemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : perm_below m (G m a) := And.intro (G_maps_below m a) (And.intro (G_one_one_below h1) (G_onto_below h1)) --Permuting a product of congruence classes doesn't change product lemma swap_fst (u v : Nat) : swap u v u = v := by define : swap u v u --Goal : (if u = u then v else if u = v then u else u) = v have h : u = u := by rfl rewrite [if_pos h] rfl done lemma swap_snd (u v : Nat) : swap u v v = u := by define : swap u v v by_cases h1 : v = u · -- Case 1. h1 : v = u rewrite [if_pos h1] show v = u from h1 done · -- Case 2. h1 : v ≠ u rewrite [if_neg h1] have h2 : v = v := by rfl rewrite [if_pos h2] rfl done done lemma swap_other {u v i : Nat} (h1 : i ≠ u) (h2 : i ≠ v) : swap u v i = i := by define : swap u v i rewrite [if_neg h1, if_neg h2] rfl done lemma swap_values (u v i : Nat) : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := by by_cases h1 : i = u · -- Case 1. h1 : i = u apply Or.inl rewrite [h1] show swap u v u = v from swap_fst u v done · -- Case 2. h1 : i ≠ u apply Or.inr by_cases h2 : i = v · -- Case 2.1. h2 : i = v apply Or.inl rewrite [h2] show swap u v v = u from swap_snd u v done · -- Case 2.2. h2 : i ≠ v apply Or.inr show swap u v i = i from swap_other h1 h2 done done done lemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by define fix i : Nat assume h3 : i < n have h4 : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := swap_values u v i by_cases on h4 · -- Case 1. h4 : swap u v i = v rewrite [h4] show v < n from h2 done · -- Case 2. by_cases on h4 · -- Case 2.1. h4 : swap u v i = u rewrite [h4] show u < n from h1 done · -- Case 2.2. h4 : swap u v i = i rewrite [h4] show i < n from h3 done done done lemma swap_swap (u v n : Nat) : ∀ i < n, swap u v (swap u v i) = i := by fix i : Nat assume h : i < n by_cases h1 : i = u · -- Case 1. h1 : i = u rewrite [h1, swap_fst, swap_snd] rfl done · -- Case 2. h1 : i ≠ u by_cases h2 : i = v · -- Case 2.1. h2 : i = v rewrite [h2, swap_snd, swap_fst] rfl done · -- Case 2.2. h2 : i ≠ v rewrite [swap_other h1 h2, swap_other h1 h2] rfl done done done lemma swap_one_one_below (u v n) : one_one_below n (swap u v) := left_inv_one_one_below (swap_swap u v n) lemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) := right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2) lemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) := And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2)) lemma comp_perm_below {n : Nat} {f g : Nat → Nat} (h1 : perm_below n f) (h2 : perm_below n g) : perm_below n (f ∘ g) := sorry lemma trivial_swap (u : Nat) : swap u u = id := by apply funext fix x : Nat by_cases h1 : x = u · -- Case 1. h1 : x = u rewrite [h1, swap_fst] rfl done · -- Case 2. h1 : x ≠ u rewrite [swap_other h1 h1] rfl done done lemma prod_eq_fun {m : Nat} (f g : Nat → ZMod m) (k : Nat) : ∀ (n : Nat), (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g := by by_induc · -- Base Case assume h : (∀ i < 0, f (k + i) = g (k + i)) rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g assume h1 : ∀ i < n + 1, f (k + i) = g (k + i) have h2 : ∀ i < n, f (k + i) = g (k + i) := by fix i : Nat assume h2 : i < n have h3 : i < n + 1 := by linarith show f (k + i) = g (k + i) from h1 i h3 done have h3 : prod_seq n k f = prod_seq n k g := ih h2 have h4 : n < n + 1 := Nat.lt_succ_self n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4] rfl done done lemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := by have h2 : ∀ (i : Nat), i < u → (f ∘ swap u n) (0 + i) = f (0 + i) := by fix i : Nat assume h2 : i < u have h3 : 0 + i ≠ u := by linarith have h4 : 0 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl done show prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f from prod_eq_fun (f ∘ swap u n) f 0 u h2 done lemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := by have h2 : ∀ i < j, (f ∘ swap u n) (u + 1 + i) = f (u + 1 + i) := by fix i : Nat assume h2 : i < j have h3 : u + 1 + i ≠ u := by linarith have h4 : u + 1 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl show prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f from prod_eq_fun (f ∘ swap u n) f (u + 1) j h2 done lemma break_prod {m : Nat} (n : Nat) (f : Nat → ZMod m) : ∀ (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by by_induc · -- Base Case have h : n + 0 = n := by rfl rewrite [prod_seq_base, h, Theorem_7_3_6_7] rfl done · -- Induction Step fix j : Nat assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f rewrite [←add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc] rfl done done lemma break_prod_twice {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f := break_prod n f 1 rewrite [prod_one] at h2 have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f := break_prod (u + 1) f j rewrite [←h1] at h3 have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f := break_prod u f 1 rewrite [prod_one] at h4 rewrite [h3, h4] at h2 show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2 done lemma swap_prod_eq_prod {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f := by by_cases h2 : u = n · -- Case 1. h2 : u = n rewrite [h2, trivial_swap n] --Goal : prod_seq (n + 1) 0 (f ∘ id) = prod_seq (n + 1) 0 f rfl done · -- Case 2. h2 : ¬u = n have h3 : u + 1 ≤ n := Nat.lt_of_le_of_ne h1 h2 obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3 have break_f : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := break_prod_twice f h4 have break_fs : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_prod_twice (f ∘ swap u n) h4 have f_eq_fs_below : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := swap_prod_eq_prod_below f h1 have f_eq_fs_btwn : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4 show prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f from calc prod_seq (n + 1) 0 (f ∘ swap u n) _ = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_fs _ = prod_seq u 0 f * (f ∘ swap u n) u * prod_seq j (u + 1) f * (f ∘ swap u n) n := by rw [f_eq_fs_below, f_eq_fs_btwn] _ = prod_seq u 0 f * f (swap u n u) * prod_seq j (u + 1) f * f (swap u n n) := by rfl _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by rw [swap_fst, swap_snd] _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring _ = prod_seq (n + 1) 0 f := break_f.symm done done lemma perm_below_fixed {n : Nat} {g : Nat → Nat} (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry lemma perm_prod {m : Nat} (f : Nat → ZMod m) : ∀ (n : Nat), ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) := by by_induc · -- Base Case fix g : Nat → Nat assume h1 : perm_below 0 g rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) fix g : Nat → Nat assume g_pb : perm_below (n + 1) g define at g_pb have g_ob : onto_below (n + 1) g := g_pb.right.right define at g_ob have h1 : n < n + 1 := by linarith obtain (u : Nat) (h2 : u < n + 1 ∧ g u = n) from g_ob n h1 have s_pb : perm_below (n + 1) (swap u n) := swap_perm_below h2.left h1 have gs_pb_n1 : perm_below (n + 1) (g ∘ swap u n) := comp_perm_below g_pb s_pb have gs_fix_n : (g ∘ swap u n) n = n := calc (g ∘ swap u n) n _ = g (swap u n n) := by rfl _ = g u := by rw [swap_snd] _ = n := h2.right have gs_pb_n : perm_below n (g ∘ swap u n) := perm_below_fixed gs_pb_n1 gs_fix_n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f ∘ (g ∘ swap u n)) := ih (g ∘ swap u n) gs_pb_n have h3 : u ≤ n := by linarith show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f ∘ g) from calc prod_seq (n + 1) 0 f _ = prod_seq n 0 f * f n := prod_seq_zero_step n f _ = prod_seq n 0 (f ∘ (g ∘ swap u n)) * f ((g ∘ swap u n) n) := by rw [gs_prod, gs_fix_n] _ = prod_seq n 0 (f ∘ g ∘ swap u n) * (f ∘ g ∘ swap u n) n := by rfl _ = prod_seq (n + 1) 0 (f ∘ g ∘ swap u n) := (prod_seq_zero_step n (f ∘ g ∘ swap u n)).symm _ = prod_seq (n + 1) 0 ((f ∘ g) ∘ swap u n) := by rfl _ = prod_seq (n + 1) 0 (f ∘ g) := swap_prod_eq_prod (f ∘ g) h3 done done lemma F_invertible (m i : Nat) : invertible (F m i) := by by_cases h : rel_prime m i · -- Case 1. h : rel_prime m i rewrite [F_rp_def h] show invertible [i]_m from (Theorem_7_3_7 m i).rtl h done · -- Case 2. h : ¬rel_prime m i rewrite [F_not_rp_def h] apply Exists.intro [1]_m show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m done done lemma Fprod_invertible (m : Nat) : ∀ (k : Nat), invertible (prod_seq k 0 (F m)) := by by_induc · -- Base Case apply Exists.intro [1]_m show prod_seq 0 0 (F m) * [1]_m = [1]_m from calc prod_seq 0 0 (F m) * [1]_m _ = [1]_m * [1]_m := by rw [prod_seq_base] _ = [1]_m := Theorem_7_3_6_7 ([1]_m) done · -- Induction Step fix k : Nat assume ih : invertible (prod_seq k 0 (F m)) rewrite [prod_seq_zero_step] show invertible (prod_seq k 0 (F m) * (F m k)) from (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k) done done theorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m ^ (phi m) = [1]_m := by have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2 show [a]_m ^ (phi m) = [1]_m from calc [a]_m ^ (phi m) _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3] _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring _ = prod_seq m 0 (F m ∘ G m a) * Y := by rw [FG_prod h1 m, phi_def] _ = prod_seq m 0 (F m) * Y := by rw [perm_prod (F m) m (G m a) (G_perm_below h1)] _ = [1]_m := by rw [h3] done lemma Exercise_7_4_5_Int (m : Nat) (a : Int) : ∀ (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry lemma Exercise_7_4_5_Nat (m a n : Nat) : [a]_m ^ n = [a ^ n]_m := by rewrite [Exercise_7_4_5_Int] rfl done theorem Euler's_theorem {m a : Nat} [NeZero m] (h1 : rel_prime m a) : a ^ (phi m) ≡ 1 (MOD m) := by have h2 : [a]_m ^ (phi m) = [1]_m := Theorem_7_4_2 h1 rewrite [Exercise_7_4_5_Nat m a (phi m)] at h2 --h2 : [a ^ phi m]_m = [1]_m show a ^ (phi m) ≡ 1 (MOD m) from (cc_eq_iff_congr _ _ _).ltr h2 done #eval gcd 10 7 --Answer: 1. So 10 and 7 are relatively prime #eval 7 ^ phi 10 --Answer: 2401, which is congruent to 1 mod 10. end Euler /- Section 7.5 -/ lemma num_rp_prime {p : Nat} (h1 : prime p) : ∀ k < p, num_rp_below p (k + 1) = k := sorry lemma phi_prime {p : Nat} (h1 : prime p) : phi p = p - 1 := by have h2 : 1 ≤ p := prime_pos h1 have h3 : p - 1 + 1 = p := Nat.sub_add_cancel h2 have h4 : p - 1 < p := by linarith have h5 : num_rp_below p (p - 1 + 1) = p - 1 := num_rp_prime h1 (p - 1) h4 rewrite [h3] at h5 show phi p = p - 1 from h5 done theorem Theorem_7_2_2_Int {a c : Nat} {b : Int} (h1 : ↑c ∣ ↑a * b) (h2 : rel_prime a c) : ↑c ∣ b := by rewrite [Int.natCast_dvd, Int.natAbs_mul, Int.natAbs_ofNat] at h1 --h1 : c ∣ a * Int.natAbs b rewrite [Int.natCast_dvd] --Goal : c ∣ Int.natAbs b show c ∣ Int.natAbs b from Theorem_7_2_2 h1 h2 done lemma Lemma_7_4_5 {m n : Nat} (a b : Int) (h1 : rel_prime m n) : a ≡ b (MOD m * n) ↔ a ≡ b (MOD m) ∧ a ≡ b (MOD n) := by apply Iff.intro · -- (→) assume h2 : a ≡ b (MOD m * n) obtain (j : Int) (h3 : a - b = (m * n) * j) from h2 apply And.intro · -- Proof of a ≡ b (MOD m) apply Exists.intro (n * j) show a - b = m * (n * j) from calc a - b _ = m * n * j := h3 _ = m * (n * j) := by ring done · -- Proof of a ≡ b (MOD n) apply Exists.intro (m * j) show a - b = n * (m * j) from calc a - b _ = m * n * j := h3 _ = n * (m * j) := by ring done done · -- (←) assume h2 : a ≡ b (MOD m) ∧ a ≡ b (MOD n) obtain (j : Int) (h3 : a - b = m * j) from h2.left have h4 : (↑n : Int) ∣ a - b := h2.right rewrite [h3] at h4 --h4 : ↑n ∣ ↑m * j have h5 : ↑n ∣ j := Theorem_7_2_2_Int h4 h1 obtain (k : Int) (h6 : j = n * k) from h5 apply Exists.intro k --Goal : a - b = ↑(m * n) * k rewrite [Nat.cast_mul] --Goal : a - b = ↑m * ↑n * k show a - b = (m * n) * k from calc a - b _ = m * j := h3 _ = m * (n * k) := by rw [h6] _ = (m * n) * k := by ring done done --From exercises of Section 7.2 theorem rel_prime_symm {a b : Nat} (h : rel_prime a b) : rel_prime b a := sorry lemma prime_NeZero {p : Nat} (h : prime p) : NeZero p := by rewrite [neZero_iff] --Goal : p ≠ 0 define at h linarith done lemma Lemma_7_5_1 {p e d m c s : Nat} {t : Int} (h1 : prime p) (h2 : e * d = (p - 1) * s + 1) (h3 : m ^ e - c = p * t) : c ^ d ≡ m (MOD p) := by have h4 : m ^ e ≡ c (MOD p) := Exists.intro t h3 have h5 : [m ^ e]_p = [c]_p := (cc_eq_iff_congr _ _ _).rtl h4 rewrite [←Exercise_7_4_5_Nat] at h5 --h5 : [m]_p ^ e = [c]_p by_cases h6 : p ∣ m · -- Case 1. h6 : p ∣ m have h7 : m ≡ 0 (MOD p) := by obtain (j : Nat) (h8 : m = p * j) from h6 apply Exists.intro (↑j : Int) --Goal : ↑m - 0 = ↑p * ↑j rewrite [h8, Nat.cast_mul] ring done have h8 : [m]_p = [0]_p := (cc_eq_iff_congr _ _ _).rtl h7 have h9 : e * d ≠ 0 := by rewrite [h2] show (p - 1) * s + 1 ≠ 0 from Nat.add_one_ne_zero _ done have h10 : (0 : Int) ^ (e * d) = 0 := zero_pow h9 have h11 : [c ^ d]_p = [m]_p := calc [c ^ d]_p _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat] _ = ([m]_p ^ e) ^ d := by rw [h5] _ = [m]_p ^ (e * d) := by ring _ = [0]_p ^ (e * d) := by rw [h8] _ = [0 ^ (e * d)]_p := Exercise_7_4_5_Int _ _ _ _ = [0]_p := by rw [h10] _ = [m]_p := by rw [h8] show c ^ d ≡ m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11 done · -- Case 2. h6 : ¬p ∣ m have h7 : rel_prime m p := rel_prime_of_prime_not_dvd h1 h6 have h8 : rel_prime p m := rel_prime_symm h7 have h9 : NeZero p := prime_NeZero h1 have h10 : (1 : Int) ^ s = 1 := by ring have h11 : [c ^ d]_p = [m]_p := calc [c ^ d]_p _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat] _ = ([m]_p ^ e) ^ d := by rw [h5] _ = [m]_p ^ (e * d) := by ring _ = [m]_p ^ ((p - 1) * s + 1) := by rw [h2] _ = ([m]_p ^ (p - 1)) ^ s * [m]_p := by ring _ = ([m]_p ^ (phi p)) ^ s * [m]_p := by rw [phi_prime h1] _ = [1]_p ^ s * [m]_p := by rw [Theorem_7_4_2 h8] _ = [1 ^ s]_p * [m]_p := by rw [Exercise_7_4_5_Int] _ = [1]_p * [m]_p := by rw [h10] _ = [m]_p * [1]_p := by ring _ = [m]_p := Theorem_7_3_6_7 _ show c ^ d ≡ m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11 done done theorem Theorem_7_5_1 (p q n e d k m c : Nat) (p_prime : prime p) (q_prime : prime q) (p_ne_q : p ≠ q) (n_pq : n = p * q) (ed_congr_1 : e * d = k * (p - 1) * (q - 1) + 1) (h1 : [m]_n ^ e = [c]_n) : [c]_n ^ d = [m]_n := by rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] at h1 --h1 : m ^ e ≡ c (MOD n) rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] --Goal : c ^ d ≡ m (MOD n) obtain (j : Int) (h2 : m ^ e - c = n * j) from h1 rewrite [n_pq, Nat.cast_mul] at h2 --h2 : m ^ e - c = p * q * j have h3 : e * d = (p - 1) * (k * (q - 1)) + 1 := by rewrite [ed_congr_1] ring done have h4 : m ^ e - c = p * (q * j) := by rewrite [h2] ring done have congr_p : c ^ d ≡ m (MOD p) := Lemma_7_5_1 p_prime h3 h4 have h5 : e * d = (q - 1) * (k * (p - 1)) + 1 := by rewrite [ed_congr_1] ring done have h6 : m ^ e - c = q * (p * j) := by rewrite [h2] ring done have congr_q : c ^ d ≡ m (MOD q) := Lemma_7_5_1 q_prime h5 h6 have h7 : ¬q ∣ p := by by_contra h8 have h9 : q = 1 ∨ q = p := dvd_prime p_prime h8 disj_syll h9 (prime_not_one q_prime) show False from p_ne_q h9.symm done have h8 : rel_prime p q := rel_prime_of_prime_not_dvd q_prime h7 rewrite [n_pq, Lemma_7_4_5 _ _ h8] show c ^ d ≡ m (MOD p) ∧ c ^ d ≡ m (MOD q) from And.intro congr_p congr_q done /- BEGIN EXERCISES -/ namespace Exercises /- Section 7.1 -/ -- 1. theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry -- 2. lemma gcd_comm_lt {a b : Nat} (h : a < b) : gcd a b = gcd b a := sorry theorem gcd_comm (a b : Nat) : gcd a b = gcd b a := sorry -- 3. theorem Exercise_7_1_5 (a b : Nat) (n : Int) : (∃ (s t : Int), s * a + t * b = n) ↔ (↑(gcd a b) : Int) ∣ n := sorry -- 4. theorem Exercise_7_1_6 (a b c : Nat) : gcd a b = gcd (a + b * c) b := sorry -- 5. theorem gcd_is_nonzero {a b : Nat} (h : a ≠ 0 ∨ b ≠ 0) : gcd a b ≠ 0 := sorry -- 6. theorem gcd_greatest {a b d : Nat} (h1 : gcd a b ≠ 0) (h2 : d ∣ a) (h3 : d ∣ b) : d ≤ gcd a b := sorry -- 7. lemma Lemma_7_1_10a {a b : Nat} (n : Nat) (h : a ∣ b) : (n * a) ∣ (n * b) := sorry lemma Lemma_7_1_10b {a b n : Nat} (h1 : n ≠ 0) (h2 : (n * a) ∣ (n * b)) : a ∣ b := sorry lemma Lemma_7_1_10c {a b : Nat} (h1 : a ∣ b) (h2 : b ∣ a) : a = b := sorry theorem Exercise_7_1_10 (a b n : Nat) : gcd (n * a) (n * b) = n * gcd a b := sorry /- Section 7.2 -/ -- 1. lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry -- 2. -- Hints: Start with apply List.rec. You may find mul_ne_zero useful theorem prod_nonzero_nonzero : ∀ (l : List Nat), (∀ a ∈ l, a ≠ 0) → prod l ≠ 0 := sorry -- 3. theorem rel_prime_iff_no_common_factor (a b : Nat) : rel_prime a b ↔ ¬∃ (p : Nat), prime p ∧ p ∣ a ∧ p ∣ b := sorry -- 4. theorem rel_prime_symm {a b : Nat} (h : rel_prime a b) : rel_prime b a := sorry -- 5. lemma in_prime_factorization_iff_prime_factor {a : Nat} {l : List Nat} (h1 : prime_factorization a l) (p : Nat) : p ∈ l ↔ prime_factor p a := sorry -- 6. theorem Exercise_7_2_5 {a b : Nat} {l m : List Nat} (h1 : prime_factorization a l) (h2 : prime_factorization b m) : rel_prime a b ↔ (¬∃ (p : Nat), p ∈ l ∧ p ∈ m) := sorry -- 7. theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry -- 8. theorem Exercise_7_2_7 {a b a' b' : Nat} (h1 : rel_prime a b) (h2 : a' ∣ a) (h3 : b' ∣ b) : rel_prime a' b' := sorry -- 9. theorem Exercise_7_2_9 {a b j k : Nat} (h1 : gcd a b ≠ 0) (h2 : a = j * gcd a b) (h3 : b = k * gcd a b) : rel_prime j k := sorry -- 10. theorem Exercise_7_2_17a (a b c : Nat) : gcd a (b * c) ∣ gcd a b * gcd a c := sorry /- Section 7.3 -/ -- 1. theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry -- 2. theorem Theorem_7_3_6_3 {m : Nat} (X : ZMod m) : X + [0]_m = X := sorry -- 3. theorem Theorem_7_3_6_4 {m : Nat} (X : ZMod m) : ∃ (Y : ZMod m), X + Y = [0]_m := sorry -- 4. theorem Exercise_7_3_4a {m : Nat} (Z1 Z2 : ZMod m) (h1 : ∀ (X : ZMod m), X + Z1 = X) (h2 : ∀ (X : ZMod m), X + Z2 = X) : Z1 = Z2 := sorry -- 5. theorem Exercise_7_3_4b {m : Nat} (X Y1 Y2 : ZMod m) (h1 : X + Y1 = [0]_m) (h2 : X + Y2 = [0]_m) : Y1 = Y2 := sorry -- 6. theorem Theorem_7_3_10 (m a : Nat) (b : Int) : ¬(↑(gcd m a) : Int) ∣ b → ¬∃ (x : Int), a * x ≡ b (MOD m) := sorry -- 7. theorem Theorem_7_3_11 (m n : Nat) (a b : Int) (h1 : n ≠ 0) : n * a ≡ n * b (MOD n * m) ↔ a ≡ b (MOD m) := sorry -- 8. theorem Exercise_7_3_16 {m : Nat} {a b : Int} (h : a ≡ b (MOD m)) : ∀ (n : Nat), a ^ n ≡ b ^ n (MOD m) := sorry -- 9. example {m : Nat} [NeZero m] (X : ZMod m) : ∃! (a : Int), 0 ≤ a ∧ a < m ∧ X = [a]_m := sorry -- 10. theorem congr_rel_prime {m a b : Nat} (h1 : a ≡ b (MOD m)) : rel_prime m a ↔ rel_prime m b := sorry -- 11. --Hint: You may find the theorem Int.ofNat_mod_ofNat useful. theorem rel_prime_mod (m a : Nat) : rel_prime m (a % m) ↔ rel_prime m a := sorry -- 12. lemma congr_iff_mod_eq_Int (m : Nat) (a b : Int) [NeZero m] : a ≡ b (MOD m) ↔ a % ↑m = b % ↑m := sorry --Hint for next theorem: Use the lemma above, --together with the theorems Int.ofNat_mod_ofNat and Nat.cast_inj. theorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] : ↑a ≡ ↑b (MOD m) ↔ a % m = b % m := sorry /- Section 7.4 -/ -- 1. --Hint: Use induction. --For the base case, compute [a]_m ^ 0 * [1]_m in two ways: --by Theorem_7_3_6_7, [a] ^ 0 * [1]_m = [a]_m ^ 0 --by ring, [a]_m ^ 0 * [1]_m = [1]_m. lemma Exercise_7_4_5_Int (m : Nat) (a : Int) : ∀ (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry -- 2. lemma left_inv_one_one_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g' (g i) = i) : one_one_below n g := sorry -- 3.

lemma comp_perm_below {n : Nat} {f g : Nat → Nat} (h1 : perm_below n f) (h2 : perm_below n g) : perm_below n (f ∘ g)

HTPI.Exercises.comp_perm_below

htpi/HTPILib/Chap7.lean

htpi

/- Copyright 2023 Daniel J. Velleman -/ import HTPILib.Chap6 namespace HTPI /- Definitions -/ lemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by have h : n + 1 > 0 := Nat.succ_pos n show a % (n + 1) < n + 1 from Nat.mod_lt a h done def gcd (a b : Nat) : Nat := match b with | 0 => a | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd (n + 1) (a % (n + 1)) termination_by b mutual def gcd_c1 (a b : Nat) : Int := match b with | 0 => 1 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c2 (n + 1) (a % (n + 1)) --Corresponds to s = t' termination_by b def gcd_c2 (a b : Nat) : Int := match b with | 0 => 0 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c1 (n + 1) (a % (n + 1)) - (gcd_c2 (n + 1) (a % (n + 1))) * ↑(a / (n + 1)) --Corresponds to t = s' - t'q termination_by b end def prime (n : Nat) : Prop := 2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n def prime_factor (p n : Nat) : Prop := prime p ∧ p ∣ n def all_prime (l : List Nat) : Prop := ∀ p ∈ l, prime p def nondec (l : List Nat) : Prop := match l with | [] => True --Of course, True is a proposition that is always true | n :: L => (∀ m ∈ L, n ≤ m) ∧ nondec L def nondec_prime_list (l : List Nat) : Prop := all_prime l ∧ nondec l def prod (l : List Nat) : Nat := match l with | [] => 1 | n :: L => n * (prod L) def prime_factorization (n : Nat) (l : List Nat) : Prop := nondec_prime_list l ∧ prod l = n def rel_prime (a b : Nat) : Prop := gcd a b = 1 def congr_mod (m : Nat) (a b : Int) : Prop := (↑m : Int) ∣ (a - b) def cc (m : Nat) (a : Int) : ZMod m := (↑a : ZMod m) notation:50 a " ≡ " b " (MOD " m ")" => congr_mod m a b notation:max "["a"]_"m:max => cc m a def invertible {m : Nat} (X : ZMod m) : Prop := ∃ (Y : ZMod m), X * Y = [1]_m def num_rp_below (m k : Nat) : Nat := match k with | 0 => 0 | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j def phi (m : Nat) : Nat := num_rp_below m m def prod_seq {m : Nat} (j k : Nat) (f : Nat → ZMod m) : ZMod m := match j with | 0 => [1]_m | n + 1 => prod_seq n k f * f (k + n) def maps_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i < n, g i < n def one_one_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i1 < n, ∀ i2 < n, g i1 = g i2 → i1 = i2 def onto_below (n : Nat) (g : Nat → Nat) : Prop := ∀ k < n, ∃ i < n, g i = k def perm_below (n : Nat) (g : Nat → Nat) : Prop := maps_below n g ∧ one_one_below n g ∧ onto_below n g def inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m) def swap (u v i : Nat) : Nat := if i = u then v else if i = v then u else i namespace Euler --For definitions specific to Euler's theorem def F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m def G (m a i : Nat) : Nat := (a * i) % m def Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i end Euler /- Section 7.1 -/ theorem dvd_mod_of_dvd_a_b {a b d : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ (a % b) := by set q : Nat := a / b have h3 : b * q + a % b = a := Nat.div_add_mod a b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 define --Goal : ∃ (c : Nat), a % b = d * c apply Exists.intro (j - k * q) show a % b = d * (j - k * q) from calc a % b _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm _ = a - b * q := by rw [h3] _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc] _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm done theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry #eval gcd 672 161 --Answer: 7 lemma gcd_base (a : Nat) : gcd a 0 = a := by rfl lemma gcd_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd a b = gcd b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1)) rfl done lemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b ≠ 0) : a % b < b := by have h1 : b > 0 := Nat.pos_of_ne_zero h show a % b < b from Nat.mod_lt a h1 done lemma dvd_self (n : Nat) : n ∣ n := by apply Exists.intro 1 ring done theorem gcd_dvd : ∀ (b a : Nat), (gcd a b) ∣ a ∧ (gcd a b) ∣ b := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd a b_1) ∣ a ∧ (gcd a b_1) ∣ b_1 fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_base] --Goal: a ∣ a ∧ a ∣ 0 apply And.intro (dvd_self a) define apply Exists.intro 0 rfl done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_nonzero a h1] --Goal : gcd b (a % b) ∣ a ∧ gcd b (a % b) ∣ b have h2 : a % b < b := mod_nonzero_lt a h1 have h3 : (gcd b (a % b)) ∣ b ∧ (gcd b (a % b)) ∣ (a % b) := ih (a % b) h2 b apply And.intro _ h3.left show (gcd b (a % b)) ∣ a from dvd_a_of_dvd_b_mod h3.left h3.right done done theorem gcd_dvd_left (a b : Nat) : (gcd a b) ∣ a := (gcd_dvd b a).left theorem gcd_dvd_right (a b : Nat) : (gcd a b) ∣ b := (gcd_dvd b a).right lemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl lemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c1 a b = gcd_c2 b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done lemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl lemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * ↑(a / b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done theorem gcd_lin_comb : ∀ (b a : Nat), (gcd_c1 a b) * ↑a + (gcd_c2 a b) * ↑b = ↑(gcd a b) := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd_c1 a b_1) * ↑a + (gcd_c2 a b_1) * ↑b_1 = ↑(gcd a b_1) fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base] --Goal : 1 * ↑a + 0 * ↑0 = ↑a ring done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1] --Goal : gcd_c2 b (a % b) * ↑a + -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * ↑(a / b)) * ↑b = -- ↑(gcd b (a % b)) set r : Nat := a % b set q : Nat := a / b set s : Int := gcd_c1 b r set t : Int := gcd_c2 b r --Goal : t * ↑a + (s - t * ↑q) * ↑b = ↑(gcd b r) have h2 : r < b := mod_nonzero_lt a h1 have h3 : s * ↑b + t * ↑r = ↑(gcd b r) := ih r h2 b have h4 : b * q + r = a := Nat.div_add_mod a b rewrite [←h3, ←h4] rewrite [Nat.cast_add, Nat.cast_mul] --Goal : t * (↑b * ↑q + ↑r) + (s - t * ↑q) * ↑b = s * ↑b + t * ↑r ring done done #eval gcd_c1 672 161 --Answer: 6 #eval gcd_c2 672 161 --Answer: -25 --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161 theorem Theorem_7_1_6 {d a b : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ gcd a b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑d ∣ ↑(gcd a b) set s : Int := gcd_c1 a b set t : Int := gcd_c2 a b have h3 : s * ↑a + t * ↑b = ↑(gcd a b) := gcd_lin_comb b a rewrite [←h3] --Goal : ↑d ∣ s * ↑a + t * ↑b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul] --Goal : ↑d ∣ s * (↑d * ↑j) + t * (↑d * ↑k) define apply Exists.intro (s * ↑j + t * ↑k) ring done /- Section 7.2 -/ theorem dvd_trans {a b c : Nat} (h1 : a ∣ b) (h2 : b ∣ c) : a ∣ c := by define at h1; define at h2; define obtain (m : Nat) (h3 : b = a * m) from h1 obtain (n : Nat) (h4 : c = b * n) from h2 rewrite [h3, mul_assoc] at h4 apply Exists.intro (m * n) show c = a * (m * n) from h4 done lemma exists_prime_factor : ∀ (n : Nat), 2 ≤ n → ∃ (p : Nat), prime_factor p n := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, 2 ≤ n_1 → ∃ (p : Nat), prime_factor p n_1 assume h1 : 2 ≤ n by_cases h2 : prime n · -- Case 1. h2 : prime n apply Exists.intro n define --Goal : prime n ∧ n ∣ n show prime n ∧ n ∣ n from And.intro h2 (dvd_self n) done · -- Case 2. h2 : ¬prime n define at h2 --h2 : ¬(2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n) demorgan at h2 disj_syll h2 h1 obtain (a : Nat) (h3 : ∃ (b : Nat), a * b = n ∧ a < n ∧ b < n) from h2 obtain (b : Nat) (h4 : a * b = n ∧ a < n ∧ b < n) from h3 have h5 : 2 ≤ a := by by_contra h6 have h7 : a ≤ 1 := by linarith have h8 : n ≤ b := calc n _ = a * b := h4.left.symm _ ≤ 1 * b := by rel [h7] _ = b := by ring linarith --n ≤ b contradicts b < n done have h6 : ∃ (p : Nat), prime_factor p a := ih a h4.right.left h5 obtain (p : Nat) (h7 : prime_factor p a) from h6 apply Exists.intro p define --Goal : prime p ∧ p ∣ n define at h7 --h7 : prime p ∧ p ∣ a apply And.intro h7.left have h8 : a ∣ n := by apply Exists.intro b show n = a * b from (h4.left).symm done show p ∣ n from dvd_trans h7.right h8 done done lemma exists_least_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q := by set S : Set Nat := {p : Nat | prime_factor p n} have h2 : ∃ (p : Nat), p ∈ S := exists_prime_factor n h show ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q from well_ord_princ S h2 done lemma all_prime_nil : all_prime [] := by define --Goal : ∀ p ∈ [], prime p fix p : Nat contrapos --Goal : ¬prime p → p ∉ [] assume h1 : ¬prime p show p ∉ [] from List.not_mem_nil p done lemma all_prime_cons (n : Nat) (L : List Nat) : all_prime (n :: L) ↔ prime n ∧ all_prime L := by apply Iff.intro · -- (→) assume h1 : all_prime (n :: L) --Goal : prime n ∧ all_prime L define at h1 --h1 : ∀ p ∈ n :: L, prime p apply And.intro (h1 n (List.mem_cons_self n L)) define --Goal : ∀ p ∈ L, prime p fix p : Nat assume h2 : p ∈ L show prime p from h1 p (List.mem_cons_of_mem n h2) done · -- (←) assume h1 : prime n ∧ all_prime L --Goal : all_prime (n :: l) define : all_prime L at h1 define fix p : Nat assume h2 : p ∈ n :: L rewrite [List.mem_cons] at h2 --h2 : p = n ∨ p ∈ L by_cases on h2 · -- Case 1. h2 : p = n rewrite [h2] show prime n from h1.left done · -- Case 2. h2 : p ∈ L show prime p from h1.right p h2 done done done lemma nondec_nil : nondec [] := by define --Goal : True trivial --trivial proves some obviously true statements, such as True done lemma nondec_cons (n : Nat) (L : List Nat) : nondec (n :: L) ↔ (∀ m ∈ L, n ≤ m) ∧ nondec L := by rfl lemma prod_nil : prod [] = 1 := by rfl lemma prod_cons : prod (n :: L) = n * (prod L) := by rfl lemma exists_cons_of_length_eq_succ {A : Type} {l : List A} {n : Nat} (h : l.length = n + 1) : ∃ (a : A) (L : List A), l = a :: L ∧ L.length = n := by have h1 : ¬l.length = 0 := by linarith rewrite [List.length_eq_zero] at h1 obtain (a : A) (h2 : ∃ (L : List A), l = a :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List A) (h3 : l = a :: L) from h2 apply Exists.intro a apply Exists.intro L apply And.intro h3 have h4 : (a :: L).length = L.length + 1 := List.length_cons a L rewrite [←h3, h] at h4 show L.length = n from (Nat.add_right_cancel h4).symm done lemma list_elt_dvd_prod_by_length (a : Nat) : ∀ (n : Nat), ∀ (l : List Nat), l.length = n → a ∈ l → a ∣ prod l := by by_induc · --Base Case fix l : List Nat assume h1 : l.length = 0 rewrite [List.length_eq_zero] at h1 --h1 : l = [] rewrite [h1] --Goal : a ∈ [] → a ∣ prod [] contrapos assume h2 : ¬a ∣ prod [] show a ∉ [] from List.not_mem_nil a done · -- Induction Step fix n : Nat assume ih : ∀ (l : List Nat), List.length l = n → a ∈ l → a ∣ prod l fix l : List Nat assume h1 : l.length = n + 1 --Goal : a ∈ l → a ∣ prod l obtain (b : Nat) (h2 : ∃ (L : List Nat), l = b :: L ∧ L.length = n) from exists_cons_of_length_eq_succ h1 obtain (L : List Nat) (h3 : l = b :: L ∧ L.length = n) from h2 have h4 : a ∈ L → a ∣ prod L := ih L h3.right assume h5 : a ∈ l rewrite [h3.left, prod_cons] --Goal : a ∣ b * prod L rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b ∨ a ∈ L by_cases on h5 · -- Case 1. h5 : a = b apply Exists.intro (prod L) rewrite [h5] rfl done · -- Case 2. h5 : a ∈ L have h6 : a ∣ prod L := h4 h5 have h7 : prod L ∣ b * prod L := by apply Exists.intro b ring done show a ∣ b * prod L from dvd_trans h6 h7 done done done lemma list_elt_dvd_prod {a : Nat} {l : List Nat} (h : a ∈ l) : a ∣ prod l := by set n : Nat := l.length have h1 : l.length = n := by rfl show a ∣ prod l from list_elt_dvd_prod_by_length a n l h1 h done lemma exists_prime_factorization : ∀ (n : Nat), n ≥ 1 → ∃ (l : List Nat), prime_factorization n l := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, n_1 ≥ 1 → ∃ (l : List Nat), prime_factorization n_1 l assume h1 : n ≥ 1 by_cases h2 : n = 1 · -- Case 1. h2 : n = 1 apply Exists.intro [] define apply And.intro · -- Proof of nondec_prime_list [] define show all_prime [] ∧ nondec [] from And.intro all_prime_nil nondec_nil done · -- Proof of prod [] = n rewrite [prod_nil, h2] rfl done done · -- Case 2. h2 : n ≠ 1 have h3 : n ≥ 2 := lt_of_le_of_ne' h1 h2 obtain (p : Nat) (h4 : prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q) from exists_least_prime_factor h3 have p_prime_factor : prime_factor p n := h4.left define at p_prime_factor have p_prime : prime p := p_prime_factor.left have p_dvd_n : p ∣ n := p_prime_factor.right have p_least : ∀ (q : Nat), prime_factor q n → p ≤ q := h4.right obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n have h5 : m ≠ 0 := by contradict h1 with h6 have h7 : n = 0 := calc n _ = p * m := n_eq_pm _ = p * 0 := by rw [h6] _ = 0 := by ring rewrite [h7] decide done have m_pos : 0 < m := Nat.pos_of_ne_zero h5 have m_lt_n : m < n := by define at p_prime show m < n from calc m _ < m + m := by linarith _ = 2 * m := by ring _ ≤ p * m := by rel [p_prime.left] _ = n := n_eq_pm.symm done obtain (L : List Nat) (h6 : prime_factorization m L) from ih m m_lt_n m_pos define at h6 have ndpl_L : nondec_prime_list L := h6.left define at ndpl_L apply Exists.intro (p :: L) define apply And.intro · -- Proof of nondec_prime_list (p :: L) define apply And.intro · -- Proof of all_prime (p :: L) rewrite [all_prime_cons] show prime p ∧ all_prime L from And.intro p_prime ndpl_L.left done · -- Proof of nondec (p :: L) rewrite [nondec_cons] apply And.intro _ ndpl_L.right fix q : Nat assume q_in_L : q ∈ L have h7 : q ∣ prod L := list_elt_dvd_prod q_in_L rewrite [h6.right] at h7 --h7 : q ∣ m have h8 : m ∣ n := by apply Exists.intro p rewrite [n_eq_pm] ring done have q_dvd_n : q ∣ n := dvd_trans h7 h8 have ap_L : all_prime L := ndpl_L.left define at ap_L have q_prime_factor : prime_factor q n := And.intro (ap_L q q_in_L) q_dvd_n show p ≤ q from p_least q q_prime_factor done done · -- Proof of prod (p :: L) = n rewrite [prod_cons, h6.right, n_eq_pm] rfl done done done theorem Theorem_7_2_2 {a b c : Nat} (h1 : c ∣ a * b) (h2 : rel_prime a c) : c ∣ b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑c ∣ ↑b define at h1; define at h2; define obtain (j : Nat) (h3 : a * b = c * j) from h1 set s : Int := gcd_c1 a c set t : Int := gcd_c2 a c have h4 : s * ↑a + t * ↑c = ↑(gcd a c) := gcd_lin_comb c a rewrite [h2, Nat.cast_one] at h4 --h4 : s * ↑a + t * ↑c = (1 : Int) apply Exists.intro (s * ↑j + t * ↑b) show ↑b = ↑c * (s * ↑j + t * ↑b) from calc ↑b _ = (1 : Int) * ↑b := (one_mul _).symm _ = (s * ↑a + t * ↑c) * ↑b := by rw [h4] _ = s * (↑a * ↑b) + t * ↑c * ↑b := by ring _ = s * (↑c * ↑j) + t * ↑c * ↑b := by rw [←Nat.cast_mul a b, h3, Nat.cast_mul c j] _ = ↑c * (s * ↑j + t * ↑b) := by ring done lemma le_nonzero_prod_left {a b : Nat} (h : a * b ≠ 0) : a ≤ a * b := by have h1 : b ≠ 0 := by contradict h with h1 rewrite [h1] ring done have h2 : 1 ≤ b := Nat.pos_of_ne_zero h1 show a ≤ a * b from calc a = a * 1 := (mul_one a).symm _ ≤ a * b := by rel [h2] done lemma le_nonzero_prod_right {a b : Nat} (h : a * b ≠ 0) : b ≤ a * b := by rewrite [mul_comm] rewrite [mul_comm] at h show b ≤ b * a from le_nonzero_prod_left h done lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry lemma rel_prime_of_prime_not_dvd {a p : Nat} (h1 : prime p) (h2 : ¬p ∣ a) : rel_prime a p := by have h3 : gcd a p ∣ a := gcd_dvd_left a p have h4 : gcd a p ∣ p := gcd_dvd_right a p have h5 : gcd a p = 1 ∨ gcd a p = p := dvd_prime h1 h4 have h6 : gcd a p ≠ p := by contradict h2 with h6 rewrite [h6] at h3 show p ∣ a from h3 done disj_syll h5 h6 show rel_prime a p from h5 done theorem Theorem_7_2_3 {a b p : Nat} (h1 : prime p) (h2 : p ∣ a * b) : p ∣ a ∨ p ∣ b := by or_right with h3 have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3 show p ∣ b from Theorem_7_2_2 h2 h4 done lemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a ≥ 1 := by have h1 : a ≠ 0 := by by_contra h1 rewrite [h1] at h contradict h linarith done show a ≥ 1 from Nat.pos_of_ne_zero h1 done lemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by have h1 : a ≥ 1 := ge_one_of_prod_one h have h2 : a * b ≠ 0 := by linarith have h3 : a ≤ a * b := le_nonzero_prod_left h2 rewrite [h] at h3 show a = 1 from Nat.le_antisymm h3 h1 done lemma eq_one_of_dvd_one {n : Nat} (h : n ∣ 1) : n = 1 := by obtain (j : Nat) (h1 : 1 = n * j) from h show n = 1 from eq_one_of_prod_one h1.symm done lemma prime_not_one {p : Nat} (h : prime p) : p ≠ 1 := by define at h linarith done theorem Theorem_7_2_4 {p : Nat} (h1 : prime p) : ∀ (l : List Nat), p ∣ prod l → ∃ a ∈ l, p ∣ a := by apply List.rec · -- Base Case. Goal : p ∣ prod [] → ∃ a ∈ [], p ∣ a rewrite [prod_nil] assume h2 : p ∣ 1 show ∃ a ∈ [], p ∣ a from absurd (eq_one_of_dvd_one h2) (prime_not_one h1) done · -- Induction Step fix b : Nat fix L : List Nat assume ih : p ∣ prod L → ∃ a ∈ L, p ∣ a --Goal : p ∣ prod (b :: L) → ∃ a ∈ b :: L, p ∣ a assume h2 : p ∣ prod (b :: L) rewrite [prod_cons] at h2 have h3 : p ∣ b ∨ p ∣ prod L := Theorem_7_2_3 h1 h2 by_cases on h3 · -- Case 1. h3 : p ∣ b apply Exists.intro b show b ∈ b :: L ∧ p ∣ b from And.intro (List.mem_cons_self b L) h3 done · -- Case 2. h3 : p ∣ prod L obtain (a : Nat) (h4 : a ∈ L ∧ p ∣ a) from ih h3 apply Exists.intro a show a ∈ b :: L ∧ p ∣ a from And.intro (List.mem_cons_of_mem b h4.left) h4.right done done done lemma prime_in_list {p : Nat} {l : List Nat} (h1 : prime p) (h2 : all_prime l) (h3 : p ∣ prod l) : p ∈ l := by obtain (a : Nat) (h4 : a ∈ l ∧ p ∣ a) from Theorem_7_2_4 h1 l h3 define at h2 have h5 : prime a := h2 a h4.left have h6 : p = 1 ∨ p = a := dvd_prime h5 h4.right disj_syll h6 (prime_not_one h1) rewrite [h6] show a ∈ l from h4.left done lemma first_le_first {p q : Nat} {l m : List Nat} (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m)) (h3 : prod (p :: l) = prod (q :: m)) : p ≤ q := by define at h1; define at h2 have h4 : q ∣ prod (p :: l) := by define apply Exists.intro (prod m) rewrite [←prod_cons] show prod (p :: l) = prod (q :: m) from h3 done have h5 : all_prime (q :: m) := h2.left rewrite [all_prime_cons] at h5 have h6 : q ∈ p :: l := prime_in_list h5.left h1.left h4 have h7 : nondec (p :: l) := h1.right rewrite [nondec_cons] at h7 rewrite [List.mem_cons] at h6 by_cases on h6 · -- Case 1. h6 : q = p linarith done · -- Case 2. h6 : q ∈ l have h8 : ∀ m ∈ l, p ≤ m := h7.left show p ≤ q from h8 q h6 done done lemma nondec_prime_list_tail {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by define at h define rewrite [all_prime_cons, nondec_cons] at h show all_prime l ∧ nondec l from And.intro h.left.right h.right.right done lemma cons_prod_not_one {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : prod (p :: l) ≠ 1 := by define at h have h1 : all_prime (p :: l) := h.left rewrite [all_prime_cons] at h1 rewrite [prod_cons] by_contra h2 show False from (prime_not_one h1.left) (eq_one_of_prod_one h2) done lemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) : l = [] ↔ prod l = 1 := by apply Iff.intro · -- (→) assume h1 : l = [] rewrite [h1] show prod [] = 1 from prod_nil done · -- (←) contrapos assume h1 : ¬l = [] obtain (p : Nat) (h2 : ∃ (L : List Nat), l = p :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List Nat) (h3 : l = p :: L) from h2 rewrite [h3] at h rewrite [h3] show ¬prod (p :: L) = 1 from cons_prod_not_one h done done lemma prime_pos {p : Nat} (h : prime p) : p > 0 := by define at h linarith done theorem Theorem_7_2_5 : ∀ (l1 l2 : List Nat), nondec_prime_list l1 → nondec_prime_list l2 → prod l1 = prod l2 → l1 = l2 := by apply List.rec · -- Base Case. Goal : ∀ (l2 : List Nat), nondec_prime_list [] → -- nondec_prime_list l2 → prod [] = prod l2 → [] = l2 fix l2 : List Nat assume h1 : nondec_prime_list [] assume h2 : nondec_prime_list l2 assume h3 : prod [] = prod l2 rewrite [prod_nil, eq_comm, ←list_nil_iff_prod_one h2] at h3 show [] = l2 from h3.symm done · -- Induction Step fix p : Nat fix L1 : List Nat assume ih : ∀ (L2 : List Nat), nondec_prime_list L1 → nondec_prime_list L2 → prod L1 = prod L2 → L1 = L2 -- Goal : ∀ (l2 : List Nat), nondec_prime_list (p :: L1) → -- nondec_prime_list l2 → prod (p :: L1) = prod l2 → p :: L1 = l2 fix l2 : List Nat assume h1 : nondec_prime_list (p :: L1) assume h2 : nondec_prime_list l2 assume h3 : prod (p :: L1) = prod l2 have h4 : ¬prod (p :: L1) = 1 := cons_prod_not_one h1 rewrite [h3, ←list_nil_iff_prod_one h2] at h4 obtain (q : Nat) (h5 : ∃ (L : List Nat), l2 = q :: L) from List.exists_cons_of_ne_nil h4 obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5 rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2) rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2) have h7 : p ≤ q := first_le_first h1 h2 h3 have h8 : q ≤ p := first_le_first h2 h1 h3.symm have h9 : p = q := by linarith rewrite [h9, prod_cons, prod_cons] at h3 --h3 : q * prod L1 = q * prod L2 have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1 have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2 define at h2 have h12 : all_prime (q :: L2) := h2.left rewrite [all_prime_cons] at h12 have h13 : q > 0 := prime_pos h12.left have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3 have h15 : L1 = L2 := ih L2 h10 h11 h14 rewrite [h6, h9, h15] rfl done done theorem fund_thm_arith (n : Nat) (h : n ≥ 1) : ∃! (l : List Nat), prime_factorization n l := by exists_unique · -- Existence show ∃ (l : List Nat), prime_factorization n l from exists_prime_factorization n h done · -- Uniqueness fix l1 : List Nat; fix l2 : List Nat assume h1 : prime_factorization n l1 assume h2 : prime_factorization n l2 define at h1; define at h2 have h3 : prod l1 = n := h1.right rewrite [←h2.right] at h3 show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3 done done /- Section 7.3 -/ theorem congr_refl (m : Nat) : ∀ (a : Int), a ≡ a (MOD m) := by fix a : Int define --Goal : ∃ (c : Int), a - a = ↑m * c apply Exists.intro 0 ring done theorem congr_symm {m : Nat} : ∀ {a b : Int}, a ≡ b (MOD m) → b ≡ a (MOD m) := by fix a : Int; fix b : Int assume h1 : a ≡ b (MOD m) define at h1 --h1 : ∃ (c : Int), a - b = ↑m * c define --Goal : ∃ (c : Int), b - a = ↑m * c obtain (c : Int) (h2 : a - b = m * c) from h1 apply Exists.intro (-c) show b - a = m * (-c) from calc b - a _ = -(a - b) := by ring _ = -(m * c) := by rw [h2] _ = m * (-c) := by ring done theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry /- Fundamental properties of congruence classes -/ lemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) : X = Y ↔ X.val = Y.val := Fin.ext_iff lemma val_nat_eq_mod (n k : Nat) : ([k]_(n + 1)).val = k % (n + 1) := by rfl lemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl theorem cc_rep {m : Nat} (X : ZMod m) : ∃ (a : Int), X = [a]_m := match m with | 0 => by apply Exists.intro X rfl done | n + 1 => by apply Exists.intro ↑(X.val) have h1 : X.val < n + 1 := Fin.prop X rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1] rfl done theorem add_class (m : Nat) (a b : Int) : [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm theorem mul_class (m : Nat) (a b : Int) : [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm lemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ [a - b]_m = [0]_m := by apply Iff.intro · -- (→) assume h1 : [a]_m = [b]_m have h2 : a - b = a + (-b) := by ring have h3 : b + (-b) = 0 := by ring show [a - b]_m = [0]_m from calc [a - b]_m _ = [a + (-b)]_m := by rw [h2] _ = [a]_m + [-b]_m := by rw [add_class] _ = [b]_m + [-b]_m := by rw [h1] _ = [b + -b]_m := by rw [add_class] _ = [0]_m := by rw [h3] done · -- (←) assume h1 : [a - b]_m = [0]_m have h2 : b + (a - b) = a := by ring have h3 : b + 0 = b := by ring show [a]_m = [b]_m from calc [a]_m _ = [b + (a - b)]_m := by rw [h2] _ = [b]_m + [a - b]_m := by rw [add_class] _ = [b]_m + [0]_m := by rw [h1] _ = [b + 0]_m := by rw [add_class] _ = [b]_m := by rw [h3] done done lemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) : [a]_m = [0]_m → [-a]_m = [0]_m := by assume h1 : [a]_m = [0]_m have h2 : 0 + (-a) = -a := by ring have h3 : a + (-a) = 0 := by ring show [-a]_m = [0]_m from calc [-a]_m _ = [0 + (-a)]_m := by rw [h2] _ = [0]_m + [-a]_m := by rw [add_class] _ = [a]_m + [-a]_m := by rw [h1] _ = [a + (-a)]_m := by rw [add_class] _ = [0]_m := by rw [h3] done lemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) : [-a]_m = [0]_m ↔ [a]_m = [0]_m := by apply Iff.intro _ (cc_neg_zero_of_cc_zero m a) assume h1 : [-a]_m = [0]_m have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1 have h3 : -(-a) = a := by ring rewrite [h3] at h2 show [a]_m = [0]_m from h2 done lemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl lemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m ↔ m ∣ k := match m with | 0 => by have h : (0 : Int) = (↑(0 : Nat) : Int) := by rfl rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj] apply Iff.intro · -- (→) assume h1 : k = 0 rewrite [h1] show 0 ∣ 0 from dvd_self 0 done · -- (←) assume h1 : 0 ∣ k obtain (c : Nat) (h2 : k = 0 * c) from h1 rewrite [h2] ring done done | n + 1 => by rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero] show k % (n + 1) = 0 ↔ n + 1 ∣ k from (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm done lemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m ↔ ↑m ∣ a := by obtain (k : Nat) (h1 : a = ↑k ∨ a = -↑k) from Int.eq_nat_or_neg a by_cases on h1 · -- Case 1. h1: a = ↑k rewrite [h1, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done · -- Case 2. h1: a = -↑k rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done done theorem cc_eq_iff_congr (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ a ≡ b (MOD m) := calc [a]_m = [b]_m _ ↔ [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b _ ↔ ↑m ∣ (a - b) := cc_zero_iff_dvd m (a - b) _ ↔ a ≡ b (MOD m) := by rfl /- End of fundamental properties of congruence classes -/ lemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m := by have h1 : (↑m : Int) ≠ 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m) show 0 ≤ a % m from Int.emod_nonneg a h1 done lemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m) have h2 : (↑m : Int) > 0 := (Nat.cast_pos).rtl h1 show a % m < m from Int.emod_lt_of_pos a h2 done lemma congr_mod_mod (m : Nat) (a : Int) : a ≡ a % m (MOD m) := by define have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m apply Exists.intro (a / m) show a - a % m = m * (a / m) from calc a - (a % m) _ = m * (a / m) + a % m - a % m := by rw [h1] _ = m * (a / m) := by ring done lemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) := And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a)) theorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) : ∃! (r : Int), 0 ≤ r ∧ r < m ∧ a ≡ r (MOD m) := by exists_unique · -- Existence apply Exists.intro (a % m) show 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) from mod_cmpl_res m a done · -- Uniqueness fix r1 : Int; fix r2 : Int assume h1 : 0 ≤ r1 ∧ r1 < m ∧ a ≡ r1 (MOD m) assume h2 : 0 ≤ r2 ∧ r2 < m ∧ a ≡ r2 (MOD m) have h3 : r1 ≡ r2 (MOD m) := congr_trans (congr_symm h1.right.right) h2.right.right obtain (d : Int) (h4 : r1 - r2 = m * d) from h3 have h5 : r1 - r2 < m * 1 := by linarith have h6 : m * (-1) < r1 - r2 := by linarith rewrite [h4] at h5 --h5 : m * d < m * 1 rewrite [h4] at h6 --h6 : m * -1 < m * d have h7 : (↑m : Int) ≥ 0 := Nat.cast_nonneg m have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7 have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7 have h10 : d = 0 := by linarith show r1 = r2 from calc r1 _ = r1 - r2 + r2 := by ring _ = m * 0 + r2 := by rw [h4, h10] _ = r2 := by ring done done lemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m := (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a) theorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y rewrite [h1, h2] have h3 : a + b = b + a := by ring show [a]_m + [b]_m = [b]_m + [a]_m from calc [a]_m + [b]_m _ = [a + b]_m := add_class m a b _ = [b + a]_m := by rw [h3] _ = [b]_m + [a]_m := (add_class m b a).symm done theorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X rewrite [h1] have h2 : a * 1 = a := by ring show [a]_m * [1]_m = [a]_m from calc [a]_m * [1]_m _ = [a * 1]_m := mul_class m a 1 _ = [a]_m := by rw [h2] done theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry lemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) : [a]_m * [gcd_c2 m a]_m = [1]_m := by set s : Int := gcd_c1 m a have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m define at h1 rewrite [h1, Nat.cast_one] at h2 --h2 : s * ↑m + gcd_c2 m a * ↑a = 1 rewrite [mul_class, cc_eq_iff_congr] define --Goal : ∃ (c : Int), ↑a * gcd_c2 m a - 1 = ↑m * c apply Exists.intro (-s) show a * (gcd_c2 m a) - 1 = m * (-s) from calc a * (gcd_c2 m a) - 1 _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring _ = 1 + m * (-s) - 1 := by rw [h2] _ = m * (-s) := by ring done theorem Theorem_7_3_7 (m a : Nat) : invertible [a]_m ↔ rel_prime m a := by apply Iff.intro · -- (→) assume h1 : invertible [a]_m define at h1 obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1 obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y rewrite [h3, mul_class, cc_eq_iff_congr] at h2 define at h2 obtain (c : Int) (h4 : a * b - 1 = m * c) from h2 rewrite [Exercise_7_2_6] --Goal : ∃ (s t : Int), s * ↑m + t * ↑a = 1 apply Exists.intro (-c) apply Exists.intro b show (-c) * m + b * a = 1 from calc (-c) * m + b * a _ = (-c) * m + (a * b - 1) + 1 := by ring _ = (-c) * m + m * c + 1 := by rw [h4] _ = 1 := by ring done · -- (←) assume h1 : rel_prime m a define show ∃ (Y : ZMod m), [a]_m * Y = [1]_m from Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1) done done /- Section 7.4 -/ section Euler open Euler lemma num_rp_below_base {m : Nat} : num_rp_below m 0 = 0 := by rfl lemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) : num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by have h1 : num_rp_below m (j + 1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : gcd m j = 1 rewrite [if_pos h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1 show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1 done lemma num_rp_below_step_not_rp {m j : Nat} (h : ¬rel_prime m j) : num_rp_below m (j + 1) = num_rp_below m j := by have h1 : num_rp_below m (j +1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : ¬gcd m j = 1 rewrite [if_neg h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j show num_rp_below m (j + 1) = num_rp_below m j from h1 done lemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl #eval phi 10 --Answer: 4 lemma prod_inv_iff_inv {m : Nat} {X : ZMod m} (h1 : invertible X) (Y : ZMod m) : invertible (X * Y) ↔ invertible Y := by apply Iff.intro · -- (→) assume h2 : invertible (X * Y) obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2 apply Exists.intro (X * Z) rewrite [←h3] --Goal : Y * (X * Z) = X * Y * Z ring --Note that ring can do algebra in ZMod m done · -- (←) assume h2 : invertible Y obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1 obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2 apply Exists.intro (Xi * Yi) show (X * Y) * (Xi * Yi) = [1]_m from calc X * Y * (Xi * Yi) _ = (X * Xi) * (Y * Yi) := by ring _ = [1]_m * [1]_m := by rw [h3, h4] _ = [1]_m := Theorem_7_3_6_7 [1]_m done done lemma F_rp_def {m i : Nat} (h : rel_prime m i) : F m i = [i]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h --h : gcd m i = 1 rewrite [if_pos h] at h1 show F m i = [i]_m from h1 done lemma F_not_rp_def {m i : Nat} (h : ¬rel_prime m i) : F m i = [1]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h rewrite [h1, if_neg h] rfl done lemma prod_seq_base {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 0 k f = [1]_m := by rfl lemma prod_seq_step {m : Nat} (n k : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl lemma prod_seq_zero_step {m : Nat} (n : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by rewrite [prod_seq_step, zero_add] rfl done lemma prod_one {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 1 k f = f k := by rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7] rfl done lemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl lemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m := calc [G m a i]_m _ = [(a * i) % m]_m := by rfl _ = [a * i]_m := (cc_eq_mod m (a * i)).symm _ = [a]_m * [i]_m := (mul_class m a i).symm lemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) : rel_prime m (G m a i) ↔ rel_prime m i := by have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1 show rel_prime m (G m a i) ↔ rel_prime m i from calc rel_prime m (G m a i) _ ↔ invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm _ ↔ invertible ([a]_m * [i]_m) := by rw [cc_G] _ ↔ invertible [i]_m := prod_inv_iff_inv h2 ([i]_m) _ ↔ rel_prime m i := Theorem_7_3_7 m i done lemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) : F m (G m a i) = [a]_m * F m i := by have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2 show F m (G m a i) = [a]_m * F m i from calc F m (G m a i) _ = [G m a i]_m := F_rp_def h3 _ = [a]_m * [i]_m := cc_G m a i _ = [a]_m * F m i := by rw [F_rp_def h2] done lemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : ¬rel_prime m i) : F m (G m a i) = [1]_m := by rewrite [←G_rp_iff h1 i] at h2 show F m (G m a i) = [1]_m from F_not_rp_def h2 done lemma FG_prod {m a : Nat} (h1 : rel_prime m a) : ∀ (k : Nat), prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by by_induc · -- Base Case show prod_seq 0 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from calc prod_seq 0 0 ((F m) ∘ (G m a)) _ = [1]_m := prod_seq_base _ _ _ = [a]_m ^ 0 * [1]_m := by ring _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by rw [num_rp_below_base, prod_seq_base] done · -- Induction Step fix k : Nat assume ih : prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) by_cases h2 : rel_prime m k · -- Case 1. h2 : rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([a]_m * F m k) := by rw [FG_rp h1 h2] _ = [a]_m ^ ((num_rp_below m k) + 1) * ((prod_seq k 0 (F m)) * F m k) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_rp h2, prod_seq_zero_step] done · -- Case 2. h2 : ¬rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([1]_m) := by rw [FG_not_rp h1 h2] _ = [a]_m ^ (num_rp_below m k) * (prod_seq k 0 (F m) * ([1]_m)) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_not_rp h2, prod_seq_zero_step, F_not_rp_def h2] done done done lemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by define --Goal : ∀ i < m, G m a i < m fix i : Nat assume h1 : i < m rewrite [G_def] --Goal : a * i % m < m show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m) done lemma left_inv_one_one_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g' (g i) = i) : one_one_below n g := sorry lemma right_inv_onto_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g (g' i) = i) (h2 : maps_below n g') : onto_below n g := by define at h2; define fix k : Nat assume h3 : k < n apply Exists.intro (g' k) show g' k < n ∧ g (g' k) = k from And.intro (h2 k h3) (h1 k h3) done lemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m * [inv_mod m a]_m = [1]_m := by have h2 : 0 ≤ (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a) show [a]_m * [inv_mod m a]_m = [1]_m from calc [a]_m * [inv_mod m a]_m _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2] _ = [a]_m * [gcd_c2 m a]_m := by rw [←cc_eq_mod] _ = [1]_m := gcd_c2_inv h1 done lemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m := calc a * (b % m) % m = a % m * (b % m % m) % m := Nat.mul_mod _ _ _ _ = a % m * (b % m) % m := by rw [Nat.mod_mod] _ = a * b % m := (Nat.mul_mod _ _ _).symm lemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm] rfl done theorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] : ↑a ≡ ↑b (MOD m) ↔ a % m = b % m := sorry lemma mul_inv_mod_cancel {m a i : Nat} [NeZero m] (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1 rewrite [mul_class, cc_eq_iff_congr, ←Nat.cast_mul, ←Nat.cast_one, congr_iff_mod_eq_Nat] at h3 show a * inv_mod m a * i % m = i from calc a * (inv_mod m a) * i % m _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = 1 % m * i % m := by rw [h3] _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = i % m := by rw [one_mul] _ = i := Nat.mod_eq_of_lt h2 done lemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl lemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, G m a (Ginv m a i) = i := by fix i : Nat assume h2 : i < m show G m a (Ginv m a i) = i from calc G m a (Ginv m a i) _ = a * ((inv_mod m a * i) % m) % m := by rfl _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, Ginv m a (G m a i) = i := by fix i : Nat assume h2 : i < m show Ginv m a (G m a i) = i from calc Ginv m a (G m a i) _ = inv_mod m a * ((a * i) % m) % m := by rfl _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc, mul_comm (inv_mod m a)] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_maps_below (m a : Nat) [NeZero m] : maps_below m (Ginv m a) := G_maps_below m (inv_mod m a) lemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : one_one_below m (G m a) := left_inv_one_one_below (Ginv_left_inv h1) lemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : onto_below m (G m a) := right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a) lemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : perm_below m (G m a) := And.intro (G_maps_below m a) (And.intro (G_one_one_below h1) (G_onto_below h1)) --Permuting a product of congruence classes doesn't change product lemma swap_fst (u v : Nat) : swap u v u = v := by define : swap u v u --Goal : (if u = u then v else if u = v then u else u) = v have h : u = u := by rfl rewrite [if_pos h] rfl done lemma swap_snd (u v : Nat) : swap u v v = u := by define : swap u v v by_cases h1 : v = u · -- Case 1. h1 : v = u rewrite [if_pos h1] show v = u from h1 done · -- Case 2. h1 : v ≠ u rewrite [if_neg h1] have h2 : v = v := by rfl rewrite [if_pos h2] rfl done done lemma swap_other {u v i : Nat} (h1 : i ≠ u) (h2 : i ≠ v) : swap u v i = i := by define : swap u v i rewrite [if_neg h1, if_neg h2] rfl done lemma swap_values (u v i : Nat) : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := by by_cases h1 : i = u · -- Case 1. h1 : i = u apply Or.inl rewrite [h1] show swap u v u = v from swap_fst u v done · -- Case 2. h1 : i ≠ u apply Or.inr by_cases h2 : i = v · -- Case 2.1. h2 : i = v apply Or.inl rewrite [h2] show swap u v v = u from swap_snd u v done · -- Case 2.2. h2 : i ≠ v apply Or.inr show swap u v i = i from swap_other h1 h2 done done done lemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by define fix i : Nat assume h3 : i < n have h4 : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := swap_values u v i by_cases on h4 · -- Case 1. h4 : swap u v i = v rewrite [h4] show v < n from h2 done · -- Case 2. by_cases on h4 · -- Case 2.1. h4 : swap u v i = u rewrite [h4] show u < n from h1 done · -- Case 2.2. h4 : swap u v i = i rewrite [h4] show i < n from h3 done done done lemma swap_swap (u v n : Nat) : ∀ i < n, swap u v (swap u v i) = i := by fix i : Nat assume h : i < n by_cases h1 : i = u · -- Case 1. h1 : i = u rewrite [h1, swap_fst, swap_snd] rfl done · -- Case 2. h1 : i ≠ u by_cases h2 : i = v · -- Case 2.1. h2 : i = v rewrite [h2, swap_snd, swap_fst] rfl done · -- Case 2.2. h2 : i ≠ v rewrite [swap_other h1 h2, swap_other h1 h2] rfl done done done lemma swap_one_one_below (u v n) : one_one_below n (swap u v) := left_inv_one_one_below (swap_swap u v n) lemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) := right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2) lemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) := And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2)) lemma comp_perm_below {n : Nat} {f g : Nat → Nat} (h1 : perm_below n f) (h2 : perm_below n g) : perm_below n (f ∘ g) := sorry lemma trivial_swap (u : Nat) : swap u u = id := by apply funext fix x : Nat by_cases h1 : x = u · -- Case 1. h1 : x = u rewrite [h1, swap_fst] rfl done · -- Case 2. h1 : x ≠ u rewrite [swap_other h1 h1] rfl done done lemma prod_eq_fun {m : Nat} (f g : Nat → ZMod m) (k : Nat) : ∀ (n : Nat), (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g := by by_induc · -- Base Case assume h : (∀ i < 0, f (k + i) = g (k + i)) rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g assume h1 : ∀ i < n + 1, f (k + i) = g (k + i) have h2 : ∀ i < n, f (k + i) = g (k + i) := by fix i : Nat assume h2 : i < n have h3 : i < n + 1 := by linarith show f (k + i) = g (k + i) from h1 i h3 done have h3 : prod_seq n k f = prod_seq n k g := ih h2 have h4 : n < n + 1 := Nat.lt_succ_self n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4] rfl done done lemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := by have h2 : ∀ (i : Nat), i < u → (f ∘ swap u n) (0 + i) = f (0 + i) := by fix i : Nat assume h2 : i < u have h3 : 0 + i ≠ u := by linarith have h4 : 0 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl done show prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f from prod_eq_fun (f ∘ swap u n) f 0 u h2 done lemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := by have h2 : ∀ i < j, (f ∘ swap u n) (u + 1 + i) = f (u + 1 + i) := by fix i : Nat assume h2 : i < j have h3 : u + 1 + i ≠ u := by linarith have h4 : u + 1 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl show prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f from prod_eq_fun (f ∘ swap u n) f (u + 1) j h2 done lemma break_prod {m : Nat} (n : Nat) (f : Nat → ZMod m) : ∀ (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by by_induc · -- Base Case have h : n + 0 = n := by rfl rewrite [prod_seq_base, h, Theorem_7_3_6_7] rfl done · -- Induction Step fix j : Nat assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f rewrite [←add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc] rfl done done lemma break_prod_twice {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f := break_prod n f 1 rewrite [prod_one] at h2 have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f := break_prod (u + 1) f j rewrite [←h1] at h3 have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f := break_prod u f 1 rewrite [prod_one] at h4 rewrite [h3, h4] at h2 show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2 done lemma swap_prod_eq_prod {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f := by by_cases h2 : u = n · -- Case 1. h2 : u = n rewrite [h2, trivial_swap n] --Goal : prod_seq (n + 1) 0 (f ∘ id) = prod_seq (n + 1) 0 f rfl done · -- Case 2. h2 : ¬u = n have h3 : u + 1 ≤ n := Nat.lt_of_le_of_ne h1 h2 obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3 have break_f : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := break_prod_twice f h4 have break_fs : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_prod_twice (f ∘ swap u n) h4 have f_eq_fs_below : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := swap_prod_eq_prod_below f h1 have f_eq_fs_btwn : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4 show prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f from calc prod_seq (n + 1) 0 (f ∘ swap u n) _ = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_fs _ = prod_seq u 0 f * (f ∘ swap u n) u * prod_seq j (u + 1) f * (f ∘ swap u n) n := by rw [f_eq_fs_below, f_eq_fs_btwn] _ = prod_seq u 0 f * f (swap u n u) * prod_seq j (u + 1) f * f (swap u n n) := by rfl _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by rw [swap_fst, swap_snd] _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring _ = prod_seq (n + 1) 0 f := break_f.symm done done lemma perm_below_fixed {n : Nat} {g : Nat → Nat} (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry lemma perm_prod {m : Nat} (f : Nat → ZMod m) : ∀ (n : Nat), ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) := by by_induc · -- Base Case fix g : Nat → Nat assume h1 : perm_below 0 g rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) fix g : Nat → Nat assume g_pb : perm_below (n + 1) g define at g_pb have g_ob : onto_below (n + 1) g := g_pb.right.right define at g_ob have h1 : n < n + 1 := by linarith obtain (u : Nat) (h2 : u < n + 1 ∧ g u = n) from g_ob n h1 have s_pb : perm_below (n + 1) (swap u n) := swap_perm_below h2.left h1 have gs_pb_n1 : perm_below (n + 1) (g ∘ swap u n) := comp_perm_below g_pb s_pb have gs_fix_n : (g ∘ swap u n) n = n := calc (g ∘ swap u n) n _ = g (swap u n n) := by rfl _ = g u := by rw [swap_snd] _ = n := h2.right have gs_pb_n : perm_below n (g ∘ swap u n) := perm_below_fixed gs_pb_n1 gs_fix_n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f ∘ (g ∘ swap u n)) := ih (g ∘ swap u n) gs_pb_n have h3 : u ≤ n := by linarith show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f ∘ g) from calc prod_seq (n + 1) 0 f _ = prod_seq n 0 f * f n := prod_seq_zero_step n f _ = prod_seq n 0 (f ∘ (g ∘ swap u n)) * f ((g ∘ swap u n) n) := by rw [gs_prod, gs_fix_n] _ = prod_seq n 0 (f ∘ g ∘ swap u n) * (f ∘ g ∘ swap u n) n := by rfl _ = prod_seq (n + 1) 0 (f ∘ g ∘ swap u n) := (prod_seq_zero_step n (f ∘ g ∘ swap u n)).symm _ = prod_seq (n + 1) 0 ((f ∘ g) ∘ swap u n) := by rfl _ = prod_seq (n + 1) 0 (f ∘ g) := swap_prod_eq_prod (f ∘ g) h3 done done lemma F_invertible (m i : Nat) : invertible (F m i) := by by_cases h : rel_prime m i · -- Case 1. h : rel_prime m i rewrite [F_rp_def h] show invertible [i]_m from (Theorem_7_3_7 m i).rtl h done · -- Case 2. h : ¬rel_prime m i rewrite [F_not_rp_def h] apply Exists.intro [1]_m show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m done done lemma Fprod_invertible (m : Nat) : ∀ (k : Nat), invertible (prod_seq k 0 (F m)) := by by_induc · -- Base Case apply Exists.intro [1]_m show prod_seq 0 0 (F m) * [1]_m = [1]_m from calc prod_seq 0 0 (F m) * [1]_m _ = [1]_m * [1]_m := by rw [prod_seq_base] _ = [1]_m := Theorem_7_3_6_7 ([1]_m) done · -- Induction Step fix k : Nat assume ih : invertible (prod_seq k 0 (F m)) rewrite [prod_seq_zero_step] show invertible (prod_seq k 0 (F m) * (F m k)) from (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k) done done theorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m ^ (phi m) = [1]_m := by have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2 show [a]_m ^ (phi m) = [1]_m from calc [a]_m ^ (phi m) _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3] _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring _ = prod_seq m 0 (F m ∘ G m a) * Y := by rw [FG_prod h1 m, phi_def] _ = prod_seq m 0 (F m) * Y := by rw [perm_prod (F m) m (G m a) (G_perm_below h1)] _ = [1]_m := by rw [h3] done lemma Exercise_7_4_5_Int (m : Nat) (a : Int) : ∀ (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry lemma Exercise_7_4_5_Nat (m a n : Nat) : [a]_m ^ n = [a ^ n]_m := by rewrite [Exercise_7_4_5_Int] rfl done theorem Euler's_theorem {m a : Nat} [NeZero m] (h1 : rel_prime m a) : a ^ (phi m) ≡ 1 (MOD m) := by have h2 : [a]_m ^ (phi m) = [1]_m := Theorem_7_4_2 h1 rewrite [Exercise_7_4_5_Nat m a (phi m)] at h2 --h2 : [a ^ phi m]_m = [1]_m show a ^ (phi m) ≡ 1 (MOD m) from (cc_eq_iff_congr _ _ _).ltr h2 done #eval gcd 10 7 --Answer: 1. So 10 and 7 are relatively prime #eval 7 ^ phi 10 --Answer: 2401, which is congruent to 1 mod 10. end Euler /- Section 7.5 -/ lemma num_rp_prime {p : Nat} (h1 : prime p) : ∀ k < p, num_rp_below p (k + 1) = k := sorry lemma phi_prime {p : Nat} (h1 : prime p) : phi p = p - 1 := by have h2 : 1 ≤ p := prime_pos h1 have h3 : p - 1 + 1 = p := Nat.sub_add_cancel h2 have h4 : p - 1 < p := by linarith have h5 : num_rp_below p (p - 1 + 1) = p - 1 := num_rp_prime h1 (p - 1) h4 rewrite [h3] at h5 show phi p = p - 1 from h5 done theorem Theorem_7_2_2_Int {a c : Nat} {b : Int} (h1 : ↑c ∣ ↑a * b) (h2 : rel_prime a c) : ↑c ∣ b := by rewrite [Int.natCast_dvd, Int.natAbs_mul, Int.natAbs_ofNat] at h1 --h1 : c ∣ a * Int.natAbs b rewrite [Int.natCast_dvd] --Goal : c ∣ Int.natAbs b show c ∣ Int.natAbs b from Theorem_7_2_2 h1 h2 done lemma Lemma_7_4_5 {m n : Nat} (a b : Int) (h1 : rel_prime m n) : a ≡ b (MOD m * n) ↔ a ≡ b (MOD m) ∧ a ≡ b (MOD n) := by apply Iff.intro · -- (→) assume h2 : a ≡ b (MOD m * n) obtain (j : Int) (h3 : a - b = (m * n) * j) from h2 apply And.intro · -- Proof of a ≡ b (MOD m) apply Exists.intro (n * j) show a - b = m * (n * j) from calc a - b _ = m * n * j := h3 _ = m * (n * j) := by ring done · -- Proof of a ≡ b (MOD n) apply Exists.intro (m * j) show a - b = n * (m * j) from calc a - b _ = m * n * j := h3 _ = n * (m * j) := by ring done done · -- (←) assume h2 : a ≡ b (MOD m) ∧ a ≡ b (MOD n) obtain (j : Int) (h3 : a - b = m * j) from h2.left have h4 : (↑n : Int) ∣ a - b := h2.right rewrite [h3] at h4 --h4 : ↑n ∣ ↑m * j have h5 : ↑n ∣ j := Theorem_7_2_2_Int h4 h1 obtain (k : Int) (h6 : j = n * k) from h5 apply Exists.intro k --Goal : a - b = ↑(m * n) * k rewrite [Nat.cast_mul] --Goal : a - b = ↑m * ↑n * k show a - b = (m * n) * k from calc a - b _ = m * j := h3 _ = m * (n * k) := by rw [h6] _ = (m * n) * k := by ring done done --From exercises of Section 7.2 theorem rel_prime_symm {a b : Nat} (h : rel_prime a b) : rel_prime b a := sorry lemma prime_NeZero {p : Nat} (h : prime p) : NeZero p := by rewrite [neZero_iff] --Goal : p ≠ 0 define at h linarith done lemma Lemma_7_5_1 {p e d m c s : Nat} {t : Int} (h1 : prime p) (h2 : e * d = (p - 1) * s + 1) (h3 : m ^ e - c = p * t) : c ^ d ≡ m (MOD p) := by have h4 : m ^ e ≡ c (MOD p) := Exists.intro t h3 have h5 : [m ^ e]_p = [c]_p := (cc_eq_iff_congr _ _ _).rtl h4 rewrite [←Exercise_7_4_5_Nat] at h5 --h5 : [m]_p ^ e = [c]_p by_cases h6 : p ∣ m · -- Case 1. h6 : p ∣ m have h7 : m ≡ 0 (MOD p) := by obtain (j : Nat) (h8 : m = p * j) from h6 apply Exists.intro (↑j : Int) --Goal : ↑m - 0 = ↑p * ↑j rewrite [h8, Nat.cast_mul] ring done have h8 : [m]_p = [0]_p := (cc_eq_iff_congr _ _ _).rtl h7 have h9 : e * d ≠ 0 := by rewrite [h2] show (p - 1) * s + 1 ≠ 0 from Nat.add_one_ne_zero _ done have h10 : (0 : Int) ^ (e * d) = 0 := zero_pow h9 have h11 : [c ^ d]_p = [m]_p := calc [c ^ d]_p _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat] _ = ([m]_p ^ e) ^ d := by rw [h5] _ = [m]_p ^ (e * d) := by ring _ = [0]_p ^ (e * d) := by rw [h8] _ = [0 ^ (e * d)]_p := Exercise_7_4_5_Int _ _ _ _ = [0]_p := by rw [h10] _ = [m]_p := by rw [h8] show c ^ d ≡ m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11 done · -- Case 2. h6 : ¬p ∣ m have h7 : rel_prime m p := rel_prime_of_prime_not_dvd h1 h6 have h8 : rel_prime p m := rel_prime_symm h7 have h9 : NeZero p := prime_NeZero h1 have h10 : (1 : Int) ^ s = 1 := by ring have h11 : [c ^ d]_p = [m]_p := calc [c ^ d]_p _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat] _ = ([m]_p ^ e) ^ d := by rw [h5] _ = [m]_p ^ (e * d) := by ring _ = [m]_p ^ ((p - 1) * s + 1) := by rw [h2] _ = ([m]_p ^ (p - 1)) ^ s * [m]_p := by ring _ = ([m]_p ^ (phi p)) ^ s * [m]_p := by rw [phi_prime h1] _ = [1]_p ^ s * [m]_p := by rw [Theorem_7_4_2 h8] _ = [1 ^ s]_p * [m]_p := by rw [Exercise_7_4_5_Int] _ = [1]_p * [m]_p := by rw [h10] _ = [m]_p * [1]_p := by ring _ = [m]_p := Theorem_7_3_6_7 _ show c ^ d ≡ m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11 done done theorem Theorem_7_5_1 (p q n e d k m c : Nat) (p_prime : prime p) (q_prime : prime q) (p_ne_q : p ≠ q) (n_pq : n = p * q) (ed_congr_1 : e * d = k * (p - 1) * (q - 1) + 1) (h1 : [m]_n ^ e = [c]_n) : [c]_n ^ d = [m]_n := by rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] at h1 --h1 : m ^ e ≡ c (MOD n) rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] --Goal : c ^ d ≡ m (MOD n) obtain (j : Int) (h2 : m ^ e - c = n * j) from h1 rewrite [n_pq, Nat.cast_mul] at h2 --h2 : m ^ e - c = p * q * j have h3 : e * d = (p - 1) * (k * (q - 1)) + 1 := by rewrite [ed_congr_1] ring done have h4 : m ^ e - c = p * (q * j) := by rewrite [h2] ring done have congr_p : c ^ d ≡ m (MOD p) := Lemma_7_5_1 p_prime h3 h4 have h5 : e * d = (q - 1) * (k * (p - 1)) + 1 := by rewrite [ed_congr_1] ring done have h6 : m ^ e - c = q * (p * j) := by rewrite [h2] ring done have congr_q : c ^ d ≡ m (MOD q) := Lemma_7_5_1 q_prime h5 h6 have h7 : ¬q ∣ p := by by_contra h8 have h9 : q = 1 ∨ q = p := dvd_prime p_prime h8 disj_syll h9 (prime_not_one q_prime) show False from p_ne_q h9.symm done have h8 : rel_prime p q := rel_prime_of_prime_not_dvd q_prime h7 rewrite [n_pq, Lemma_7_4_5 _ _ h8] show c ^ d ≡ m (MOD p) ∧ c ^ d ≡ m (MOD q) from And.intro congr_p congr_q done /- BEGIN EXERCISES -/ namespace Exercises /- Section 7.1 -/ -- 1. theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry -- 2. lemma gcd_comm_lt {a b : Nat} (h : a < b) : gcd a b = gcd b a := sorry theorem gcd_comm (a b : Nat) : gcd a b = gcd b a := sorry -- 3. theorem Exercise_7_1_5 (a b : Nat) (n : Int) : (∃ (s t : Int), s * a + t * b = n) ↔ (↑(gcd a b) : Int) ∣ n := sorry -- 4. theorem Exercise_7_1_6 (a b c : Nat) : gcd a b = gcd (a + b * c) b := sorry -- 5. theorem gcd_is_nonzero {a b : Nat} (h : a ≠ 0 ∨ b ≠ 0) : gcd a b ≠ 0 := sorry -- 6. theorem gcd_greatest {a b d : Nat} (h1 : gcd a b ≠ 0) (h2 : d ∣ a) (h3 : d ∣ b) : d ≤ gcd a b := sorry -- 7. lemma Lemma_7_1_10a {a b : Nat} (n : Nat) (h : a ∣ b) : (n * a) ∣ (n * b) := sorry lemma Lemma_7_1_10b {a b n : Nat} (h1 : n ≠ 0) (h2 : (n * a) ∣ (n * b)) : a ∣ b := sorry lemma Lemma_7_1_10c {a b : Nat} (h1 : a ∣ b) (h2 : b ∣ a) : a = b := sorry theorem Exercise_7_1_10 (a b n : Nat) : gcd (n * a) (n * b) = n * gcd a b := sorry /- Section 7.2 -/ -- 1. lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry -- 2. -- Hints: Start with apply List.rec. You may find mul_ne_zero useful theorem prod_nonzero_nonzero : ∀ (l : List Nat), (∀ a ∈ l, a ≠ 0) → prod l ≠ 0 := sorry -- 3. theorem rel_prime_iff_no_common_factor (a b : Nat) : rel_prime a b ↔ ¬∃ (p : Nat), prime p ∧ p ∣ a ∧ p ∣ b := sorry -- 4. theorem rel_prime_symm {a b : Nat} (h : rel_prime a b) : rel_prime b a := sorry -- 5. lemma in_prime_factorization_iff_prime_factor {a : Nat} {l : List Nat} (h1 : prime_factorization a l) (p : Nat) : p ∈ l ↔ prime_factor p a := sorry -- 6. theorem Exercise_7_2_5 {a b : Nat} {l m : List Nat} (h1 : prime_factorization a l) (h2 : prime_factorization b m) : rel_prime a b ↔ (¬∃ (p : Nat), p ∈ l ∧ p ∈ m) := sorry -- 7. theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry -- 8. theorem Exercise_7_2_7 {a b a' b' : Nat} (h1 : rel_prime a b) (h2 : a' ∣ a) (h3 : b' ∣ b) : rel_prime a' b' := sorry -- 9. theorem Exercise_7_2_9 {a b j k : Nat} (h1 : gcd a b ≠ 0) (h2 : a = j * gcd a b) (h3 : b = k * gcd a b) : rel_prime j k := sorry -- 10. theorem Exercise_7_2_17a (a b c : Nat) : gcd a (b * c) ∣ gcd a b * gcd a c := sorry /- Section 7.3 -/ -- 1. theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry -- 2. theorem Theorem_7_3_6_3 {m : Nat} (X : ZMod m) : X + [0]_m = X := sorry -- 3. theorem Theorem_7_3_6_4 {m : Nat} (X : ZMod m) : ∃ (Y : ZMod m), X + Y = [0]_m := sorry -- 4. theorem Exercise_7_3_4a {m : Nat} (Z1 Z2 : ZMod m) (h1 : ∀ (X : ZMod m), X + Z1 = X) (h2 : ∀ (X : ZMod m), X + Z2 = X) : Z1 = Z2 := sorry -- 5. theorem Exercise_7_3_4b {m : Nat} (X Y1 Y2 : ZMod m) (h1 : X + Y1 = [0]_m) (h2 : X + Y2 = [0]_m) : Y1 = Y2 := sorry -- 6. theorem Theorem_7_3_10 (m a : Nat) (b : Int) : ¬(↑(gcd m a) : Int) ∣ b → ¬∃ (x : Int), a * x ≡ b (MOD m) := sorry -- 7. theorem Theorem_7_3_11 (m n : Nat) (a b : Int) (h1 : n ≠ 0) : n * a ≡ n * b (MOD n * m) ↔ a ≡ b (MOD m) := sorry -- 8. theorem Exercise_7_3_16 {m : Nat} {a b : Int} (h : a ≡ b (MOD m)) : ∀ (n : Nat), a ^ n ≡ b ^ n (MOD m) := sorry -- 9. example {m : Nat} [NeZero m] (X : ZMod m) : ∃! (a : Int), 0 ≤ a ∧ a < m ∧ X = [a]_m := sorry -- 10. theorem congr_rel_prime {m a b : Nat} (h1 : a ≡ b (MOD m)) : rel_prime m a ↔ rel_prime m b := sorry -- 11. --Hint: You may find the theorem Int.ofNat_mod_ofNat useful. theorem rel_prime_mod (m a : Nat) : rel_prime m (a % m) ↔ rel_prime m a := sorry -- 12. lemma congr_iff_mod_eq_Int (m : Nat) (a b : Int) [NeZero m] : a ≡ b (MOD m) ↔ a % ↑m = b % ↑m := sorry --Hint for next theorem: Use the lemma above, --together with the theorems Int.ofNat_mod_ofNat and Nat.cast_inj. theorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] : ↑a ≡ ↑b (MOD m) ↔ a % m = b % m := sorry /- Section 7.4 -/ -- 1. --Hint: Use induction. --For the base case, compute [a]_m ^ 0 * [1]_m in two ways: --by Theorem_7_3_6_7, [a] ^ 0 * [1]_m = [a]_m ^ 0 --by ring, [a]_m ^ 0 * [1]_m = [1]_m. lemma Exercise_7_4_5_Int (m : Nat) (a : Int) : ∀ (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry -- 2. lemma left_inv_one_one_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g' (g i) = i) : one_one_below n g := sorry -- 3. lemma comp_perm_below {n : Nat} {f g : Nat → Nat} (h1 : perm_below n f) (h2 : perm_below n g) : perm_below n (f ∘ g) := sorry -- 4.

lemma perm_below_fixed {n : Nat} {g : Nat → Nat} (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g

HTPI.Exercises.perm_below_fixed

htpi/HTPILib/Chap7.lean

htpi

/- Copyright 2023 Daniel J. Velleman -/ import HTPILib.Chap6 namespace HTPI /- Definitions -/ lemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by have h : n + 1 > 0 := Nat.succ_pos n show a % (n + 1) < n + 1 from Nat.mod_lt a h done def gcd (a b : Nat) : Nat := match b with | 0 => a | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd (n + 1) (a % (n + 1)) termination_by b mutual def gcd_c1 (a b : Nat) : Int := match b with | 0 => 1 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c2 (n + 1) (a % (n + 1)) --Corresponds to s = t' termination_by b def gcd_c2 (a b : Nat) : Int := match b with | 0 => 0 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c1 (n + 1) (a % (n + 1)) - (gcd_c2 (n + 1) (a % (n + 1))) * ↑(a / (n + 1)) --Corresponds to t = s' - t'q termination_by b end def prime (n : Nat) : Prop := 2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n def prime_factor (p n : Nat) : Prop := prime p ∧ p ∣ n def all_prime (l : List Nat) : Prop := ∀ p ∈ l, prime p def nondec (l : List Nat) : Prop := match l with | [] => True --Of course, True is a proposition that is always true | n :: L => (∀ m ∈ L, n ≤ m) ∧ nondec L def nondec_prime_list (l : List Nat) : Prop := all_prime l ∧ nondec l def prod (l : List Nat) : Nat := match l with | [] => 1 | n :: L => n * (prod L) def prime_factorization (n : Nat) (l : List Nat) : Prop := nondec_prime_list l ∧ prod l = n def rel_prime (a b : Nat) : Prop := gcd a b = 1 def congr_mod (m : Nat) (a b : Int) : Prop := (↑m : Int) ∣ (a - b) def cc (m : Nat) (a : Int) : ZMod m := (↑a : ZMod m) notation:50 a " ≡ " b " (MOD " m ")" => congr_mod m a b notation:max "["a"]_"m:max => cc m a def invertible {m : Nat} (X : ZMod m) : Prop := ∃ (Y : ZMod m), X * Y = [1]_m def num_rp_below (m k : Nat) : Nat := match k with | 0 => 0 | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j def phi (m : Nat) : Nat := num_rp_below m m def prod_seq {m : Nat} (j k : Nat) (f : Nat → ZMod m) : ZMod m := match j with | 0 => [1]_m | n + 1 => prod_seq n k f * f (k + n) def maps_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i < n, g i < n def one_one_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i1 < n, ∀ i2 < n, g i1 = g i2 → i1 = i2 def onto_below (n : Nat) (g : Nat → Nat) : Prop := ∀ k < n, ∃ i < n, g i = k def perm_below (n : Nat) (g : Nat → Nat) : Prop := maps_below n g ∧ one_one_below n g ∧ onto_below n g def inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m) def swap (u v i : Nat) : Nat := if i = u then v else if i = v then u else i namespace Euler --For definitions specific to Euler's theorem def F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m def G (m a i : Nat) : Nat := (a * i) % m def Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i end Euler /- Section 7.1 -/ theorem dvd_mod_of_dvd_a_b {a b d : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ (a % b) := by set q : Nat := a / b have h3 : b * q + a % b = a := Nat.div_add_mod a b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 define --Goal : ∃ (c : Nat), a % b = d * c apply Exists.intro (j - k * q) show a % b = d * (j - k * q) from calc a % b _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm _ = a - b * q := by rw [h3] _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc] _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm done theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry #eval gcd 672 161 --Answer: 7 lemma gcd_base (a : Nat) : gcd a 0 = a := by rfl lemma gcd_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd a b = gcd b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1)) rfl done lemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b ≠ 0) : a % b < b := by have h1 : b > 0 := Nat.pos_of_ne_zero h show a % b < b from Nat.mod_lt a h1 done lemma dvd_self (n : Nat) : n ∣ n := by apply Exists.intro 1 ring done theorem gcd_dvd : ∀ (b a : Nat), (gcd a b) ∣ a ∧ (gcd a b) ∣ b := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd a b_1) ∣ a ∧ (gcd a b_1) ∣ b_1 fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_base] --Goal: a ∣ a ∧ a ∣ 0 apply And.intro (dvd_self a) define apply Exists.intro 0 rfl done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_nonzero a h1] --Goal : gcd b (a % b) ∣ a ∧ gcd b (a % b) ∣ b have h2 : a % b < b := mod_nonzero_lt a h1 have h3 : (gcd b (a % b)) ∣ b ∧ (gcd b (a % b)) ∣ (a % b) := ih (a % b) h2 b apply And.intro _ h3.left show (gcd b (a % b)) ∣ a from dvd_a_of_dvd_b_mod h3.left h3.right done done theorem gcd_dvd_left (a b : Nat) : (gcd a b) ∣ a := (gcd_dvd b a).left theorem gcd_dvd_right (a b : Nat) : (gcd a b) ∣ b := (gcd_dvd b a).right lemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl lemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c1 a b = gcd_c2 b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done lemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl lemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * ↑(a / b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done theorem gcd_lin_comb : ∀ (b a : Nat), (gcd_c1 a b) * ↑a + (gcd_c2 a b) * ↑b = ↑(gcd a b) := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd_c1 a b_1) * ↑a + (gcd_c2 a b_1) * ↑b_1 = ↑(gcd a b_1) fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base] --Goal : 1 * ↑a + 0 * ↑0 = ↑a ring done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1] --Goal : gcd_c2 b (a % b) * ↑a + -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * ↑(a / b)) * ↑b = -- ↑(gcd b (a % b)) set r : Nat := a % b set q : Nat := a / b set s : Int := gcd_c1 b r set t : Int := gcd_c2 b r --Goal : t * ↑a + (s - t * ↑q) * ↑b = ↑(gcd b r) have h2 : r < b := mod_nonzero_lt a h1 have h3 : s * ↑b + t * ↑r = ↑(gcd b r) := ih r h2 b have h4 : b * q + r = a := Nat.div_add_mod a b rewrite [←h3, ←h4] rewrite [Nat.cast_add, Nat.cast_mul] --Goal : t * (↑b * ↑q + ↑r) + (s - t * ↑q) * ↑b = s * ↑b + t * ↑r ring done done #eval gcd_c1 672 161 --Answer: 6 #eval gcd_c2 672 161 --Answer: -25 --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161 theorem Theorem_7_1_6 {d a b : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ gcd a b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑d ∣ ↑(gcd a b) set s : Int := gcd_c1 a b set t : Int := gcd_c2 a b have h3 : s * ↑a + t * ↑b = ↑(gcd a b) := gcd_lin_comb b a rewrite [←h3] --Goal : ↑d ∣ s * ↑a + t * ↑b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul] --Goal : ↑d ∣ s * (↑d * ↑j) + t * (↑d * ↑k) define apply Exists.intro (s * ↑j + t * ↑k) ring done /- Section 7.2 -/ theorem dvd_trans {a b c : Nat} (h1 : a ∣ b) (h2 : b ∣ c) : a ∣ c := by define at h1; define at h2; define obtain (m : Nat) (h3 : b = a * m) from h1 obtain (n : Nat) (h4 : c = b * n) from h2 rewrite [h3, mul_assoc] at h4 apply Exists.intro (m * n) show c = a * (m * n) from h4 done lemma exists_prime_factor : ∀ (n : Nat), 2 ≤ n → ∃ (p : Nat), prime_factor p n := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, 2 ≤ n_1 → ∃ (p : Nat), prime_factor p n_1 assume h1 : 2 ≤ n by_cases h2 : prime n · -- Case 1. h2 : prime n apply Exists.intro n define --Goal : prime n ∧ n ∣ n show prime n ∧ n ∣ n from And.intro h2 (dvd_self n) done · -- Case 2. h2 : ¬prime n define at h2 --h2 : ¬(2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n) demorgan at h2 disj_syll h2 h1 obtain (a : Nat) (h3 : ∃ (b : Nat), a * b = n ∧ a < n ∧ b < n) from h2 obtain (b : Nat) (h4 : a * b = n ∧ a < n ∧ b < n) from h3 have h5 : 2 ≤ a := by by_contra h6 have h7 : a ≤ 1 := by linarith have h8 : n ≤ b := calc n _ = a * b := h4.left.symm _ ≤ 1 * b := by rel [h7] _ = b := by ring linarith --n ≤ b contradicts b < n done have h6 : ∃ (p : Nat), prime_factor p a := ih a h4.right.left h5 obtain (p : Nat) (h7 : prime_factor p a) from h6 apply Exists.intro p define --Goal : prime p ∧ p ∣ n define at h7 --h7 : prime p ∧ p ∣ a apply And.intro h7.left have h8 : a ∣ n := by apply Exists.intro b show n = a * b from (h4.left).symm done show p ∣ n from dvd_trans h7.right h8 done done lemma exists_least_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q := by set S : Set Nat := {p : Nat | prime_factor p n} have h2 : ∃ (p : Nat), p ∈ S := exists_prime_factor n h show ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q from well_ord_princ S h2 done lemma all_prime_nil : all_prime [] := by define --Goal : ∀ p ∈ [], prime p fix p : Nat contrapos --Goal : ¬prime p → p ∉ [] assume h1 : ¬prime p show p ∉ [] from List.not_mem_nil p done lemma all_prime_cons (n : Nat) (L : List Nat) : all_prime (n :: L) ↔ prime n ∧ all_prime L := by apply Iff.intro · -- (→) assume h1 : all_prime (n :: L) --Goal : prime n ∧ all_prime L define at h1 --h1 : ∀ p ∈ n :: L, prime p apply And.intro (h1 n (List.mem_cons_self n L)) define --Goal : ∀ p ∈ L, prime p fix p : Nat assume h2 : p ∈ L show prime p from h1 p (List.mem_cons_of_mem n h2) done · -- (←) assume h1 : prime n ∧ all_prime L --Goal : all_prime (n :: l) define : all_prime L at h1 define fix p : Nat assume h2 : p ∈ n :: L rewrite [List.mem_cons] at h2 --h2 : p = n ∨ p ∈ L by_cases on h2 · -- Case 1. h2 : p = n rewrite [h2] show prime n from h1.left done · -- Case 2. h2 : p ∈ L show prime p from h1.right p h2 done done done lemma nondec_nil : nondec [] := by define --Goal : True trivial --trivial proves some obviously true statements, such as True done lemma nondec_cons (n : Nat) (L : List Nat) : nondec (n :: L) ↔ (∀ m ∈ L, n ≤ m) ∧ nondec L := by rfl lemma prod_nil : prod [] = 1 := by rfl lemma prod_cons : prod (n :: L) = n * (prod L) := by rfl lemma exists_cons_of_length_eq_succ {A : Type} {l : List A} {n : Nat} (h : l.length = n + 1) : ∃ (a : A) (L : List A), l = a :: L ∧ L.length = n := by have h1 : ¬l.length = 0 := by linarith rewrite [List.length_eq_zero] at h1 obtain (a : A) (h2 : ∃ (L : List A), l = a :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List A) (h3 : l = a :: L) from h2 apply Exists.intro a apply Exists.intro L apply And.intro h3 have h4 : (a :: L).length = L.length + 1 := List.length_cons a L rewrite [←h3, h] at h4 show L.length = n from (Nat.add_right_cancel h4).symm done lemma list_elt_dvd_prod_by_length (a : Nat) : ∀ (n : Nat), ∀ (l : List Nat), l.length = n → a ∈ l → a ∣ prod l := by by_induc · --Base Case fix l : List Nat assume h1 : l.length = 0 rewrite [List.length_eq_zero] at h1 --h1 : l = [] rewrite [h1] --Goal : a ∈ [] → a ∣ prod [] contrapos assume h2 : ¬a ∣ prod [] show a ∉ [] from List.not_mem_nil a done · -- Induction Step fix n : Nat assume ih : ∀ (l : List Nat), List.length l = n → a ∈ l → a ∣ prod l fix l : List Nat assume h1 : l.length = n + 1 --Goal : a ∈ l → a ∣ prod l obtain (b : Nat) (h2 : ∃ (L : List Nat), l = b :: L ∧ L.length = n) from exists_cons_of_length_eq_succ h1 obtain (L : List Nat) (h3 : l = b :: L ∧ L.length = n) from h2 have h4 : a ∈ L → a ∣ prod L := ih L h3.right assume h5 : a ∈ l rewrite [h3.left, prod_cons] --Goal : a ∣ b * prod L rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b ∨ a ∈ L by_cases on h5 · -- Case 1. h5 : a = b apply Exists.intro (prod L) rewrite [h5] rfl done · -- Case 2. h5 : a ∈ L have h6 : a ∣ prod L := h4 h5 have h7 : prod L ∣ b * prod L := by apply Exists.intro b ring done show a ∣ b * prod L from dvd_trans h6 h7 done done done lemma list_elt_dvd_prod {a : Nat} {l : List Nat} (h : a ∈ l) : a ∣ prod l := by set n : Nat := l.length have h1 : l.length = n := by rfl show a ∣ prod l from list_elt_dvd_prod_by_length a n l h1 h done lemma exists_prime_factorization : ∀ (n : Nat), n ≥ 1 → ∃ (l : List Nat), prime_factorization n l := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, n_1 ≥ 1 → ∃ (l : List Nat), prime_factorization n_1 l assume h1 : n ≥ 1 by_cases h2 : n = 1 · -- Case 1. h2 : n = 1 apply Exists.intro [] define apply And.intro · -- Proof of nondec_prime_list [] define show all_prime [] ∧ nondec [] from And.intro all_prime_nil nondec_nil done · -- Proof of prod [] = n rewrite [prod_nil, h2] rfl done done · -- Case 2. h2 : n ≠ 1 have h3 : n ≥ 2 := lt_of_le_of_ne' h1 h2 obtain (p : Nat) (h4 : prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q) from exists_least_prime_factor h3 have p_prime_factor : prime_factor p n := h4.left define at p_prime_factor have p_prime : prime p := p_prime_factor.left have p_dvd_n : p ∣ n := p_prime_factor.right have p_least : ∀ (q : Nat), prime_factor q n → p ≤ q := h4.right obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n have h5 : m ≠ 0 := by contradict h1 with h6 have h7 : n = 0 := calc n _ = p * m := n_eq_pm _ = p * 0 := by rw [h6] _ = 0 := by ring rewrite [h7] decide done have m_pos : 0 < m := Nat.pos_of_ne_zero h5 have m_lt_n : m < n := by define at p_prime show m < n from calc m _ < m + m := by linarith _ = 2 * m := by ring _ ≤ p * m := by rel [p_prime.left] _ = n := n_eq_pm.symm done obtain (L : List Nat) (h6 : prime_factorization m L) from ih m m_lt_n m_pos define at h6 have ndpl_L : nondec_prime_list L := h6.left define at ndpl_L apply Exists.intro (p :: L) define apply And.intro · -- Proof of nondec_prime_list (p :: L) define apply And.intro · -- Proof of all_prime (p :: L) rewrite [all_prime_cons] show prime p ∧ all_prime L from And.intro p_prime ndpl_L.left done · -- Proof of nondec (p :: L) rewrite [nondec_cons] apply And.intro _ ndpl_L.right fix q : Nat assume q_in_L : q ∈ L have h7 : q ∣ prod L := list_elt_dvd_prod q_in_L rewrite [h6.right] at h7 --h7 : q ∣ m have h8 : m ∣ n := by apply Exists.intro p rewrite [n_eq_pm] ring done have q_dvd_n : q ∣ n := dvd_trans h7 h8 have ap_L : all_prime L := ndpl_L.left define at ap_L have q_prime_factor : prime_factor q n := And.intro (ap_L q q_in_L) q_dvd_n show p ≤ q from p_least q q_prime_factor done done · -- Proof of prod (p :: L) = n rewrite [prod_cons, h6.right, n_eq_pm] rfl done done done theorem Theorem_7_2_2 {a b c : Nat} (h1 : c ∣ a * b) (h2 : rel_prime a c) : c ∣ b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑c ∣ ↑b define at h1; define at h2; define obtain (j : Nat) (h3 : a * b = c * j) from h1 set s : Int := gcd_c1 a c set t : Int := gcd_c2 a c have h4 : s * ↑a + t * ↑c = ↑(gcd a c) := gcd_lin_comb c a rewrite [h2, Nat.cast_one] at h4 --h4 : s * ↑a + t * ↑c = (1 : Int) apply Exists.intro (s * ↑j + t * ↑b) show ↑b = ↑c * (s * ↑j + t * ↑b) from calc ↑b _ = (1 : Int) * ↑b := (one_mul _).symm _ = (s * ↑a + t * ↑c) * ↑b := by rw [h4] _ = s * (↑a * ↑b) + t * ↑c * ↑b := by ring _ = s * (↑c * ↑j) + t * ↑c * ↑b := by rw [←Nat.cast_mul a b, h3, Nat.cast_mul c j] _ = ↑c * (s * ↑j + t * ↑b) := by ring done lemma le_nonzero_prod_left {a b : Nat} (h : a * b ≠ 0) : a ≤ a * b := by have h1 : b ≠ 0 := by contradict h with h1 rewrite [h1] ring done have h2 : 1 ≤ b := Nat.pos_of_ne_zero h1 show a ≤ a * b from calc a = a * 1 := (mul_one a).symm _ ≤ a * b := by rel [h2] done lemma le_nonzero_prod_right {a b : Nat} (h : a * b ≠ 0) : b ≤ a * b := by rewrite [mul_comm] rewrite [mul_comm] at h show b ≤ b * a from le_nonzero_prod_left h done lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry lemma rel_prime_of_prime_not_dvd {a p : Nat} (h1 : prime p) (h2 : ¬p ∣ a) : rel_prime a p := by have h3 : gcd a p ∣ a := gcd_dvd_left a p have h4 : gcd a p ∣ p := gcd_dvd_right a p have h5 : gcd a p = 1 ∨ gcd a p = p := dvd_prime h1 h4 have h6 : gcd a p ≠ p := by contradict h2 with h6 rewrite [h6] at h3 show p ∣ a from h3 done disj_syll h5 h6 show rel_prime a p from h5 done theorem Theorem_7_2_3 {a b p : Nat} (h1 : prime p) (h2 : p ∣ a * b) : p ∣ a ∨ p ∣ b := by or_right with h3 have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3 show p ∣ b from Theorem_7_2_2 h2 h4 done lemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a ≥ 1 := by have h1 : a ≠ 0 := by by_contra h1 rewrite [h1] at h contradict h linarith done show a ≥ 1 from Nat.pos_of_ne_zero h1 done lemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by have h1 : a ≥ 1 := ge_one_of_prod_one h have h2 : a * b ≠ 0 := by linarith have h3 : a ≤ a * b := le_nonzero_prod_left h2 rewrite [h] at h3 show a = 1 from Nat.le_antisymm h3 h1 done lemma eq_one_of_dvd_one {n : Nat} (h : n ∣ 1) : n = 1 := by obtain (j : Nat) (h1 : 1 = n * j) from h show n = 1 from eq_one_of_prod_one h1.symm done lemma prime_not_one {p : Nat} (h : prime p) : p ≠ 1 := by define at h linarith done theorem Theorem_7_2_4 {p : Nat} (h1 : prime p) : ∀ (l : List Nat), p ∣ prod l → ∃ a ∈ l, p ∣ a := by apply List.rec · -- Base Case. Goal : p ∣ prod [] → ∃ a ∈ [], p ∣ a rewrite [prod_nil] assume h2 : p ∣ 1 show ∃ a ∈ [], p ∣ a from absurd (eq_one_of_dvd_one h2) (prime_not_one h1) done · -- Induction Step fix b : Nat fix L : List Nat assume ih : p ∣ prod L → ∃ a ∈ L, p ∣ a --Goal : p ∣ prod (b :: L) → ∃ a ∈ b :: L, p ∣ a assume h2 : p ∣ prod (b :: L) rewrite [prod_cons] at h2 have h3 : p ∣ b ∨ p ∣ prod L := Theorem_7_2_3 h1 h2 by_cases on h3 · -- Case 1. h3 : p ∣ b apply Exists.intro b show b ∈ b :: L ∧ p ∣ b from And.intro (List.mem_cons_self b L) h3 done · -- Case 2. h3 : p ∣ prod L obtain (a : Nat) (h4 : a ∈ L ∧ p ∣ a) from ih h3 apply Exists.intro a show a ∈ b :: L ∧ p ∣ a from And.intro (List.mem_cons_of_mem b h4.left) h4.right done done done lemma prime_in_list {p : Nat} {l : List Nat} (h1 : prime p) (h2 : all_prime l) (h3 : p ∣ prod l) : p ∈ l := by obtain (a : Nat) (h4 : a ∈ l ∧ p ∣ a) from Theorem_7_2_4 h1 l h3 define at h2 have h5 : prime a := h2 a h4.left have h6 : p = 1 ∨ p = a := dvd_prime h5 h4.right disj_syll h6 (prime_not_one h1) rewrite [h6] show a ∈ l from h4.left done lemma first_le_first {p q : Nat} {l m : List Nat} (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m)) (h3 : prod (p :: l) = prod (q :: m)) : p ≤ q := by define at h1; define at h2 have h4 : q ∣ prod (p :: l) := by define apply Exists.intro (prod m) rewrite [←prod_cons] show prod (p :: l) = prod (q :: m) from h3 done have h5 : all_prime (q :: m) := h2.left rewrite [all_prime_cons] at h5 have h6 : q ∈ p :: l := prime_in_list h5.left h1.left h4 have h7 : nondec (p :: l) := h1.right rewrite [nondec_cons] at h7 rewrite [List.mem_cons] at h6 by_cases on h6 · -- Case 1. h6 : q = p linarith done · -- Case 2. h6 : q ∈ l have h8 : ∀ m ∈ l, p ≤ m := h7.left show p ≤ q from h8 q h6 done done lemma nondec_prime_list_tail {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by define at h define rewrite [all_prime_cons, nondec_cons] at h show all_prime l ∧ nondec l from And.intro h.left.right h.right.right done lemma cons_prod_not_one {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : prod (p :: l) ≠ 1 := by define at h have h1 : all_prime (p :: l) := h.left rewrite [all_prime_cons] at h1 rewrite [prod_cons] by_contra h2 show False from (prime_not_one h1.left) (eq_one_of_prod_one h2) done lemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) : l = [] ↔ prod l = 1 := by apply Iff.intro · -- (→) assume h1 : l = [] rewrite [h1] show prod [] = 1 from prod_nil done · -- (←) contrapos assume h1 : ¬l = [] obtain (p : Nat) (h2 : ∃ (L : List Nat), l = p :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List Nat) (h3 : l = p :: L) from h2 rewrite [h3] at h rewrite [h3] show ¬prod (p :: L) = 1 from cons_prod_not_one h done done lemma prime_pos {p : Nat} (h : prime p) : p > 0 := by define at h linarith done theorem Theorem_7_2_5 : ∀ (l1 l2 : List Nat), nondec_prime_list l1 → nondec_prime_list l2 → prod l1 = prod l2 → l1 = l2 := by apply List.rec · -- Base Case. Goal : ∀ (l2 : List Nat), nondec_prime_list [] → -- nondec_prime_list l2 → prod [] = prod l2 → [] = l2 fix l2 : List Nat assume h1 : nondec_prime_list [] assume h2 : nondec_prime_list l2 assume h3 : prod [] = prod l2 rewrite [prod_nil, eq_comm, ←list_nil_iff_prod_one h2] at h3 show [] = l2 from h3.symm done · -- Induction Step fix p : Nat fix L1 : List Nat assume ih : ∀ (L2 : List Nat), nondec_prime_list L1 → nondec_prime_list L2 → prod L1 = prod L2 → L1 = L2 -- Goal : ∀ (l2 : List Nat), nondec_prime_list (p :: L1) → -- nondec_prime_list l2 → prod (p :: L1) = prod l2 → p :: L1 = l2 fix l2 : List Nat assume h1 : nondec_prime_list (p :: L1) assume h2 : nondec_prime_list l2 assume h3 : prod (p :: L1) = prod l2 have h4 : ¬prod (p :: L1) = 1 := cons_prod_not_one h1 rewrite [h3, ←list_nil_iff_prod_one h2] at h4 obtain (q : Nat) (h5 : ∃ (L : List Nat), l2 = q :: L) from List.exists_cons_of_ne_nil h4 obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5 rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2) rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2) have h7 : p ≤ q := first_le_first h1 h2 h3 have h8 : q ≤ p := first_le_first h2 h1 h3.symm have h9 : p = q := by linarith rewrite [h9, prod_cons, prod_cons] at h3 --h3 : q * prod L1 = q * prod L2 have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1 have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2 define at h2 have h12 : all_prime (q :: L2) := h2.left rewrite [all_prime_cons] at h12 have h13 : q > 0 := prime_pos h12.left have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3 have h15 : L1 = L2 := ih L2 h10 h11 h14 rewrite [h6, h9, h15] rfl done done theorem fund_thm_arith (n : Nat) (h : n ≥ 1) : ∃! (l : List Nat), prime_factorization n l := by exists_unique · -- Existence show ∃ (l : List Nat), prime_factorization n l from exists_prime_factorization n h done · -- Uniqueness fix l1 : List Nat; fix l2 : List Nat assume h1 : prime_factorization n l1 assume h2 : prime_factorization n l2 define at h1; define at h2 have h3 : prod l1 = n := h1.right rewrite [←h2.right] at h3 show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3 done done /- Section 7.3 -/ theorem congr_refl (m : Nat) : ∀ (a : Int), a ≡ a (MOD m) := by fix a : Int define --Goal : ∃ (c : Int), a - a = ↑m * c apply Exists.intro 0 ring done theorem congr_symm {m : Nat} : ∀ {a b : Int}, a ≡ b (MOD m) → b ≡ a (MOD m) := by fix a : Int; fix b : Int assume h1 : a ≡ b (MOD m) define at h1 --h1 : ∃ (c : Int), a - b = ↑m * c define --Goal : ∃ (c : Int), b - a = ↑m * c obtain (c : Int) (h2 : a - b = m * c) from h1 apply Exists.intro (-c) show b - a = m * (-c) from calc b - a _ = -(a - b) := by ring _ = -(m * c) := by rw [h2] _ = m * (-c) := by ring done theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry /- Fundamental properties of congruence classes -/ lemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) : X = Y ↔ X.val = Y.val := Fin.ext_iff lemma val_nat_eq_mod (n k : Nat) : ([k]_(n + 1)).val = k % (n + 1) := by rfl lemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl theorem cc_rep {m : Nat} (X : ZMod m) : ∃ (a : Int), X = [a]_m := match m with | 0 => by apply Exists.intro X rfl done | n + 1 => by apply Exists.intro ↑(X.val) have h1 : X.val < n + 1 := Fin.prop X rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1] rfl done theorem add_class (m : Nat) (a b : Int) : [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm theorem mul_class (m : Nat) (a b : Int) : [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm lemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ [a - b]_m = [0]_m := by apply Iff.intro · -- (→) assume h1 : [a]_m = [b]_m have h2 : a - b = a + (-b) := by ring have h3 : b + (-b) = 0 := by ring show [a - b]_m = [0]_m from calc [a - b]_m _ = [a + (-b)]_m := by rw [h2] _ = [a]_m + [-b]_m := by rw [add_class] _ = [b]_m + [-b]_m := by rw [h1] _ = [b + -b]_m := by rw [add_class] _ = [0]_m := by rw [h3] done · -- (←) assume h1 : [a - b]_m = [0]_m have h2 : b + (a - b) = a := by ring have h3 : b + 0 = b := by ring show [a]_m = [b]_m from calc [a]_m _ = [b + (a - b)]_m := by rw [h2] _ = [b]_m + [a - b]_m := by rw [add_class] _ = [b]_m + [0]_m := by rw [h1] _ = [b + 0]_m := by rw [add_class] _ = [b]_m := by rw [h3] done done lemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) : [a]_m = [0]_m → [-a]_m = [0]_m := by assume h1 : [a]_m = [0]_m have h2 : 0 + (-a) = -a := by ring have h3 : a + (-a) = 0 := by ring show [-a]_m = [0]_m from calc [-a]_m _ = [0 + (-a)]_m := by rw [h2] _ = [0]_m + [-a]_m := by rw [add_class] _ = [a]_m + [-a]_m := by rw [h1] _ = [a + (-a)]_m := by rw [add_class] _ = [0]_m := by rw [h3] done lemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) : [-a]_m = [0]_m ↔ [a]_m = [0]_m := by apply Iff.intro _ (cc_neg_zero_of_cc_zero m a) assume h1 : [-a]_m = [0]_m have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1 have h3 : -(-a) = a := by ring rewrite [h3] at h2 show [a]_m = [0]_m from h2 done lemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl lemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m ↔ m ∣ k := match m with | 0 => by have h : (0 : Int) = (↑(0 : Nat) : Int) := by rfl rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj] apply Iff.intro · -- (→) assume h1 : k = 0 rewrite [h1] show 0 ∣ 0 from dvd_self 0 done · -- (←) assume h1 : 0 ∣ k obtain (c : Nat) (h2 : k = 0 * c) from h1 rewrite [h2] ring done done | n + 1 => by rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero] show k % (n + 1) = 0 ↔ n + 1 ∣ k from (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm done lemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m ↔ ↑m ∣ a := by obtain (k : Nat) (h1 : a = ↑k ∨ a = -↑k) from Int.eq_nat_or_neg a by_cases on h1 · -- Case 1. h1: a = ↑k rewrite [h1, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done · -- Case 2. h1: a = -↑k rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done done theorem cc_eq_iff_congr (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ a ≡ b (MOD m) := calc [a]_m = [b]_m _ ↔ [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b _ ↔ ↑m ∣ (a - b) := cc_zero_iff_dvd m (a - b) _ ↔ a ≡ b (MOD m) := by rfl /- End of fundamental properties of congruence classes -/ lemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m := by have h1 : (↑m : Int) ≠ 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m) show 0 ≤ a % m from Int.emod_nonneg a h1 done lemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m) have h2 : (↑m : Int) > 0 := (Nat.cast_pos).rtl h1 show a % m < m from Int.emod_lt_of_pos a h2 done lemma congr_mod_mod (m : Nat) (a : Int) : a ≡ a % m (MOD m) := by define have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m apply Exists.intro (a / m) show a - a % m = m * (a / m) from calc a - (a % m) _ = m * (a / m) + a % m - a % m := by rw [h1] _ = m * (a / m) := by ring done lemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) := And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a)) theorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) : ∃! (r : Int), 0 ≤ r ∧ r < m ∧ a ≡ r (MOD m) := by exists_unique · -- Existence apply Exists.intro (a % m) show 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) from mod_cmpl_res m a done · -- Uniqueness fix r1 : Int; fix r2 : Int assume h1 : 0 ≤ r1 ∧ r1 < m ∧ a ≡ r1 (MOD m) assume h2 : 0 ≤ r2 ∧ r2 < m ∧ a ≡ r2 (MOD m) have h3 : r1 ≡ r2 (MOD m) := congr_trans (congr_symm h1.right.right) h2.right.right obtain (d : Int) (h4 : r1 - r2 = m * d) from h3 have h5 : r1 - r2 < m * 1 := by linarith have h6 : m * (-1) < r1 - r2 := by linarith rewrite [h4] at h5 --h5 : m * d < m * 1 rewrite [h4] at h6 --h6 : m * -1 < m * d have h7 : (↑m : Int) ≥ 0 := Nat.cast_nonneg m have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7 have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7 have h10 : d = 0 := by linarith show r1 = r2 from calc r1 _ = r1 - r2 + r2 := by ring _ = m * 0 + r2 := by rw [h4, h10] _ = r2 := by ring done done lemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m := (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a) theorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y rewrite [h1, h2] have h3 : a + b = b + a := by ring show [a]_m + [b]_m = [b]_m + [a]_m from calc [a]_m + [b]_m _ = [a + b]_m := add_class m a b _ = [b + a]_m := by rw [h3] _ = [b]_m + [a]_m := (add_class m b a).symm done theorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X rewrite [h1] have h2 : a * 1 = a := by ring show [a]_m * [1]_m = [a]_m from calc [a]_m * [1]_m _ = [a * 1]_m := mul_class m a 1 _ = [a]_m := by rw [h2] done theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry lemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) : [a]_m * [gcd_c2 m a]_m = [1]_m := by set s : Int := gcd_c1 m a have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m define at h1 rewrite [h1, Nat.cast_one] at h2 --h2 : s * ↑m + gcd_c2 m a * ↑a = 1 rewrite [mul_class, cc_eq_iff_congr] define --Goal : ∃ (c : Int), ↑a * gcd_c2 m a - 1 = ↑m * c apply Exists.intro (-s) show a * (gcd_c2 m a) - 1 = m * (-s) from calc a * (gcd_c2 m a) - 1 _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring _ = 1 + m * (-s) - 1 := by rw [h2] _ = m * (-s) := by ring done theorem Theorem_7_3_7 (m a : Nat) : invertible [a]_m ↔ rel_prime m a := by apply Iff.intro · -- (→) assume h1 : invertible [a]_m define at h1 obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1 obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y rewrite [h3, mul_class, cc_eq_iff_congr] at h2 define at h2 obtain (c : Int) (h4 : a * b - 1 = m * c) from h2 rewrite [Exercise_7_2_6] --Goal : ∃ (s t : Int), s * ↑m + t * ↑a = 1 apply Exists.intro (-c) apply Exists.intro b show (-c) * m + b * a = 1 from calc (-c) * m + b * a _ = (-c) * m + (a * b - 1) + 1 := by ring _ = (-c) * m + m * c + 1 := by rw [h4] _ = 1 := by ring done · -- (←) assume h1 : rel_prime m a define show ∃ (Y : ZMod m), [a]_m * Y = [1]_m from Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1) done done /- Section 7.4 -/ section Euler open Euler lemma num_rp_below_base {m : Nat} : num_rp_below m 0 = 0 := by rfl lemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) : num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by have h1 : num_rp_below m (j + 1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : gcd m j = 1 rewrite [if_pos h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1 show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1 done lemma num_rp_below_step_not_rp {m j : Nat} (h : ¬rel_prime m j) : num_rp_below m (j + 1) = num_rp_below m j := by have h1 : num_rp_below m (j +1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : ¬gcd m j = 1 rewrite [if_neg h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j show num_rp_below m (j + 1) = num_rp_below m j from h1 done lemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl #eval phi 10 --Answer: 4 lemma prod_inv_iff_inv {m : Nat} {X : ZMod m} (h1 : invertible X) (Y : ZMod m) : invertible (X * Y) ↔ invertible Y := by apply Iff.intro · -- (→) assume h2 : invertible (X * Y) obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2 apply Exists.intro (X * Z) rewrite [←h3] --Goal : Y * (X * Z) = X * Y * Z ring --Note that ring can do algebra in ZMod m done · -- (←) assume h2 : invertible Y obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1 obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2 apply Exists.intro (Xi * Yi) show (X * Y) * (Xi * Yi) = [1]_m from calc X * Y * (Xi * Yi) _ = (X * Xi) * (Y * Yi) := by ring _ = [1]_m * [1]_m := by rw [h3, h4] _ = [1]_m := Theorem_7_3_6_7 [1]_m done done lemma F_rp_def {m i : Nat} (h : rel_prime m i) : F m i = [i]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h --h : gcd m i = 1 rewrite [if_pos h] at h1 show F m i = [i]_m from h1 done lemma F_not_rp_def {m i : Nat} (h : ¬rel_prime m i) : F m i = [1]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h rewrite [h1, if_neg h] rfl done lemma prod_seq_base {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 0 k f = [1]_m := by rfl lemma prod_seq_step {m : Nat} (n k : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl lemma prod_seq_zero_step {m : Nat} (n : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by rewrite [prod_seq_step, zero_add] rfl done lemma prod_one {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 1 k f = f k := by rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7] rfl done lemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl lemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m := calc [G m a i]_m _ = [(a * i) % m]_m := by rfl _ = [a * i]_m := (cc_eq_mod m (a * i)).symm _ = [a]_m * [i]_m := (mul_class m a i).symm lemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) : rel_prime m (G m a i) ↔ rel_prime m i := by have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1 show rel_prime m (G m a i) ↔ rel_prime m i from calc rel_prime m (G m a i) _ ↔ invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm _ ↔ invertible ([a]_m * [i]_m) := by rw [cc_G] _ ↔ invertible [i]_m := prod_inv_iff_inv h2 ([i]_m) _ ↔ rel_prime m i := Theorem_7_3_7 m i done lemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) : F m (G m a i) = [a]_m * F m i := by have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2 show F m (G m a i) = [a]_m * F m i from calc F m (G m a i) _ = [G m a i]_m := F_rp_def h3 _ = [a]_m * [i]_m := cc_G m a i _ = [a]_m * F m i := by rw [F_rp_def h2] done lemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : ¬rel_prime m i) : F m (G m a i) = [1]_m := by rewrite [←G_rp_iff h1 i] at h2 show F m (G m a i) = [1]_m from F_not_rp_def h2 done lemma FG_prod {m a : Nat} (h1 : rel_prime m a) : ∀ (k : Nat), prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by by_induc · -- Base Case show prod_seq 0 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from calc prod_seq 0 0 ((F m) ∘ (G m a)) _ = [1]_m := prod_seq_base _ _ _ = [a]_m ^ 0 * [1]_m := by ring _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by rw [num_rp_below_base, prod_seq_base] done · -- Induction Step fix k : Nat assume ih : prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) by_cases h2 : rel_prime m k · -- Case 1. h2 : rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([a]_m * F m k) := by rw [FG_rp h1 h2] _ = [a]_m ^ ((num_rp_below m k) + 1) * ((prod_seq k 0 (F m)) * F m k) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_rp h2, prod_seq_zero_step] done · -- Case 2. h2 : ¬rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([1]_m) := by rw [FG_not_rp h1 h2] _ = [a]_m ^ (num_rp_below m k) * (prod_seq k 0 (F m) * ([1]_m)) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_not_rp h2, prod_seq_zero_step, F_not_rp_def h2] done done done lemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by define --Goal : ∀ i < m, G m a i < m fix i : Nat assume h1 : i < m rewrite [G_def] --Goal : a * i % m < m show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m) done lemma left_inv_one_one_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g' (g i) = i) : one_one_below n g := sorry lemma right_inv_onto_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g (g' i) = i) (h2 : maps_below n g') : onto_below n g := by define at h2; define fix k : Nat assume h3 : k < n apply Exists.intro (g' k) show g' k < n ∧ g (g' k) = k from And.intro (h2 k h3) (h1 k h3) done lemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m * [inv_mod m a]_m = [1]_m := by have h2 : 0 ≤ (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a) show [a]_m * [inv_mod m a]_m = [1]_m from calc [a]_m * [inv_mod m a]_m _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2] _ = [a]_m * [gcd_c2 m a]_m := by rw [←cc_eq_mod] _ = [1]_m := gcd_c2_inv h1 done lemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m := calc a * (b % m) % m = a % m * (b % m % m) % m := Nat.mul_mod _ _ _ _ = a % m * (b % m) % m := by rw [Nat.mod_mod] _ = a * b % m := (Nat.mul_mod _ _ _).symm lemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm] rfl done theorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] : ↑a ≡ ↑b (MOD m) ↔ a % m = b % m := sorry lemma mul_inv_mod_cancel {m a i : Nat} [NeZero m] (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1 rewrite [mul_class, cc_eq_iff_congr, ←Nat.cast_mul, ←Nat.cast_one, congr_iff_mod_eq_Nat] at h3 show a * inv_mod m a * i % m = i from calc a * (inv_mod m a) * i % m _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = 1 % m * i % m := by rw [h3] _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = i % m := by rw [one_mul] _ = i := Nat.mod_eq_of_lt h2 done lemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl lemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, G m a (Ginv m a i) = i := by fix i : Nat assume h2 : i < m show G m a (Ginv m a i) = i from calc G m a (Ginv m a i) _ = a * ((inv_mod m a * i) % m) % m := by rfl _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, Ginv m a (G m a i) = i := by fix i : Nat assume h2 : i < m show Ginv m a (G m a i) = i from calc Ginv m a (G m a i) _ = inv_mod m a * ((a * i) % m) % m := by rfl _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc, mul_comm (inv_mod m a)] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_maps_below (m a : Nat) [NeZero m] : maps_below m (Ginv m a) := G_maps_below m (inv_mod m a) lemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : one_one_below m (G m a) := left_inv_one_one_below (Ginv_left_inv h1) lemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : onto_below m (G m a) := right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a) lemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : perm_below m (G m a) := And.intro (G_maps_below m a) (And.intro (G_one_one_below h1) (G_onto_below h1)) --Permuting a product of congruence classes doesn't change product lemma swap_fst (u v : Nat) : swap u v u = v := by define : swap u v u --Goal : (if u = u then v else if u = v then u else u) = v have h : u = u := by rfl rewrite [if_pos h] rfl done lemma swap_snd (u v : Nat) : swap u v v = u := by define : swap u v v by_cases h1 : v = u · -- Case 1. h1 : v = u rewrite [if_pos h1] show v = u from h1 done · -- Case 2. h1 : v ≠ u rewrite [if_neg h1] have h2 : v = v := by rfl rewrite [if_pos h2] rfl done done lemma swap_other {u v i : Nat} (h1 : i ≠ u) (h2 : i ≠ v) : swap u v i = i := by define : swap u v i rewrite [if_neg h1, if_neg h2] rfl done lemma swap_values (u v i : Nat) : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := by by_cases h1 : i = u · -- Case 1. h1 : i = u apply Or.inl rewrite [h1] show swap u v u = v from swap_fst u v done · -- Case 2. h1 : i ≠ u apply Or.inr by_cases h2 : i = v · -- Case 2.1. h2 : i = v apply Or.inl rewrite [h2] show swap u v v = u from swap_snd u v done · -- Case 2.2. h2 : i ≠ v apply Or.inr show swap u v i = i from swap_other h1 h2 done done done lemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by define fix i : Nat assume h3 : i < n have h4 : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := swap_values u v i by_cases on h4 · -- Case 1. h4 : swap u v i = v rewrite [h4] show v < n from h2 done · -- Case 2. by_cases on h4 · -- Case 2.1. h4 : swap u v i = u rewrite [h4] show u < n from h1 done · -- Case 2.2. h4 : swap u v i = i rewrite [h4] show i < n from h3 done done done lemma swap_swap (u v n : Nat) : ∀ i < n, swap u v (swap u v i) = i := by fix i : Nat assume h : i < n by_cases h1 : i = u · -- Case 1. h1 : i = u rewrite [h1, swap_fst, swap_snd] rfl done · -- Case 2. h1 : i ≠ u by_cases h2 : i = v · -- Case 2.1. h2 : i = v rewrite [h2, swap_snd, swap_fst] rfl done · -- Case 2.2. h2 : i ≠ v rewrite [swap_other h1 h2, swap_other h1 h2] rfl done done done lemma swap_one_one_below (u v n) : one_one_below n (swap u v) := left_inv_one_one_below (swap_swap u v n) lemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) := right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2) lemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) := And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2)) lemma comp_perm_below {n : Nat} {f g : Nat → Nat} (h1 : perm_below n f) (h2 : perm_below n g) : perm_below n (f ∘ g) := sorry lemma trivial_swap (u : Nat) : swap u u = id := by apply funext fix x : Nat by_cases h1 : x = u · -- Case 1. h1 : x = u rewrite [h1, swap_fst] rfl done · -- Case 2. h1 : x ≠ u rewrite [swap_other h1 h1] rfl done done lemma prod_eq_fun {m : Nat} (f g : Nat → ZMod m) (k : Nat) : ∀ (n : Nat), (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g := by by_induc · -- Base Case assume h : (∀ i < 0, f (k + i) = g (k + i)) rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g assume h1 : ∀ i < n + 1, f (k + i) = g (k + i) have h2 : ∀ i < n, f (k + i) = g (k + i) := by fix i : Nat assume h2 : i < n have h3 : i < n + 1 := by linarith show f (k + i) = g (k + i) from h1 i h3 done have h3 : prod_seq n k f = prod_seq n k g := ih h2 have h4 : n < n + 1 := Nat.lt_succ_self n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4] rfl done done lemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := by have h2 : ∀ (i : Nat), i < u → (f ∘ swap u n) (0 + i) = f (0 + i) := by fix i : Nat assume h2 : i < u have h3 : 0 + i ≠ u := by linarith have h4 : 0 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl done show prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f from prod_eq_fun (f ∘ swap u n) f 0 u h2 done lemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := by have h2 : ∀ i < j, (f ∘ swap u n) (u + 1 + i) = f (u + 1 + i) := by fix i : Nat assume h2 : i < j have h3 : u + 1 + i ≠ u := by linarith have h4 : u + 1 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl show prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f from prod_eq_fun (f ∘ swap u n) f (u + 1) j h2 done lemma break_prod {m : Nat} (n : Nat) (f : Nat → ZMod m) : ∀ (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by by_induc · -- Base Case have h : n + 0 = n := by rfl rewrite [prod_seq_base, h, Theorem_7_3_6_7] rfl done · -- Induction Step fix j : Nat assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f rewrite [←add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc] rfl done done lemma break_prod_twice {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f := break_prod n f 1 rewrite [prod_one] at h2 have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f := break_prod (u + 1) f j rewrite [←h1] at h3 have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f := break_prod u f 1 rewrite [prod_one] at h4 rewrite [h3, h4] at h2 show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2 done lemma swap_prod_eq_prod {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f := by by_cases h2 : u = n · -- Case 1. h2 : u = n rewrite [h2, trivial_swap n] --Goal : prod_seq (n + 1) 0 (f ∘ id) = prod_seq (n + 1) 0 f rfl done · -- Case 2. h2 : ¬u = n have h3 : u + 1 ≤ n := Nat.lt_of_le_of_ne h1 h2 obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3 have break_f : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := break_prod_twice f h4 have break_fs : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_prod_twice (f ∘ swap u n) h4 have f_eq_fs_below : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := swap_prod_eq_prod_below f h1 have f_eq_fs_btwn : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4 show prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f from calc prod_seq (n + 1) 0 (f ∘ swap u n) _ = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_fs _ = prod_seq u 0 f * (f ∘ swap u n) u * prod_seq j (u + 1) f * (f ∘ swap u n) n := by rw [f_eq_fs_below, f_eq_fs_btwn] _ = prod_seq u 0 f * f (swap u n u) * prod_seq j (u + 1) f * f (swap u n n) := by rfl _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by rw [swap_fst, swap_snd] _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring _ = prod_seq (n + 1) 0 f := break_f.symm done done lemma perm_below_fixed {n : Nat} {g : Nat → Nat} (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry lemma perm_prod {m : Nat} (f : Nat → ZMod m) : ∀ (n : Nat), ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) := by by_induc · -- Base Case fix g : Nat → Nat assume h1 : perm_below 0 g rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) fix g : Nat → Nat assume g_pb : perm_below (n + 1) g define at g_pb have g_ob : onto_below (n + 1) g := g_pb.right.right define at g_ob have h1 : n < n + 1 := by linarith obtain (u : Nat) (h2 : u < n + 1 ∧ g u = n) from g_ob n h1 have s_pb : perm_below (n + 1) (swap u n) := swap_perm_below h2.left h1 have gs_pb_n1 : perm_below (n + 1) (g ∘ swap u n) := comp_perm_below g_pb s_pb have gs_fix_n : (g ∘ swap u n) n = n := calc (g ∘ swap u n) n _ = g (swap u n n) := by rfl _ = g u := by rw [swap_snd] _ = n := h2.right have gs_pb_n : perm_below n (g ∘ swap u n) := perm_below_fixed gs_pb_n1 gs_fix_n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f ∘ (g ∘ swap u n)) := ih (g ∘ swap u n) gs_pb_n have h3 : u ≤ n := by linarith show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f ∘ g) from calc prod_seq (n + 1) 0 f _ = prod_seq n 0 f * f n := prod_seq_zero_step n f _ = prod_seq n 0 (f ∘ (g ∘ swap u n)) * f ((g ∘ swap u n) n) := by rw [gs_prod, gs_fix_n] _ = prod_seq n 0 (f ∘ g ∘ swap u n) * (f ∘ g ∘ swap u n) n := by rfl _ = prod_seq (n + 1) 0 (f ∘ g ∘ swap u n) := (prod_seq_zero_step n (f ∘ g ∘ swap u n)).symm _ = prod_seq (n + 1) 0 ((f ∘ g) ∘ swap u n) := by rfl _ = prod_seq (n + 1) 0 (f ∘ g) := swap_prod_eq_prod (f ∘ g) h3 done done lemma F_invertible (m i : Nat) : invertible (F m i) := by by_cases h : rel_prime m i · -- Case 1. h : rel_prime m i rewrite [F_rp_def h] show invertible [i]_m from (Theorem_7_3_7 m i).rtl h done · -- Case 2. h : ¬rel_prime m i rewrite [F_not_rp_def h] apply Exists.intro [1]_m show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m done done lemma Fprod_invertible (m : Nat) : ∀ (k : Nat), invertible (prod_seq k 0 (F m)) := by by_induc · -- Base Case apply Exists.intro [1]_m show prod_seq 0 0 (F m) * [1]_m = [1]_m from calc prod_seq 0 0 (F m) * [1]_m _ = [1]_m * [1]_m := by rw [prod_seq_base] _ = [1]_m := Theorem_7_3_6_7 ([1]_m) done · -- Induction Step fix k : Nat assume ih : invertible (prod_seq k 0 (F m)) rewrite [prod_seq_zero_step] show invertible (prod_seq k 0 (F m) * (F m k)) from (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k) done done theorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m ^ (phi m) = [1]_m := by have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2 show [a]_m ^ (phi m) = [1]_m from calc [a]_m ^ (phi m) _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3] _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring _ = prod_seq m 0 (F m ∘ G m a) * Y := by rw [FG_prod h1 m, phi_def] _ = prod_seq m 0 (F m) * Y := by rw [perm_prod (F m) m (G m a) (G_perm_below h1)] _ = [1]_m := by rw [h3] done lemma Exercise_7_4_5_Int (m : Nat) (a : Int) : ∀ (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry lemma Exercise_7_4_5_Nat (m a n : Nat) : [a]_m ^ n = [a ^ n]_m := by rewrite [Exercise_7_4_5_Int] rfl done theorem Euler's_theorem {m a : Nat} [NeZero m] (h1 : rel_prime m a) : a ^ (phi m) ≡ 1 (MOD m) := by have h2 : [a]_m ^ (phi m) = [1]_m := Theorem_7_4_2 h1 rewrite [Exercise_7_4_5_Nat m a (phi m)] at h2 --h2 : [a ^ phi m]_m = [1]_m show a ^ (phi m) ≡ 1 (MOD m) from (cc_eq_iff_congr _ _ _).ltr h2 done #eval gcd 10 7 --Answer: 1. So 10 and 7 are relatively prime #eval 7 ^ phi 10 --Answer: 2401, which is congruent to 1 mod 10. end Euler /- Section 7.5 -/ lemma num_rp_prime {p : Nat} (h1 : prime p) : ∀ k < p, num_rp_below p (k + 1) = k := sorry lemma phi_prime {p : Nat} (h1 : prime p) : phi p = p - 1 := by have h2 : 1 ≤ p := prime_pos h1 have h3 : p - 1 + 1 = p := Nat.sub_add_cancel h2 have h4 : p - 1 < p := by linarith have h5 : num_rp_below p (p - 1 + 1) = p - 1 := num_rp_prime h1 (p - 1) h4 rewrite [h3] at h5 show phi p = p - 1 from h5 done theorem Theorem_7_2_2_Int {a c : Nat} {b : Int} (h1 : ↑c ∣ ↑a * b) (h2 : rel_prime a c) : ↑c ∣ b := by rewrite [Int.natCast_dvd, Int.natAbs_mul, Int.natAbs_ofNat] at h1 --h1 : c ∣ a * Int.natAbs b rewrite [Int.natCast_dvd] --Goal : c ∣ Int.natAbs b show c ∣ Int.natAbs b from Theorem_7_2_2 h1 h2 done lemma Lemma_7_4_5 {m n : Nat} (a b : Int) (h1 : rel_prime m n) : a ≡ b (MOD m * n) ↔ a ≡ b (MOD m) ∧ a ≡ b (MOD n) := by apply Iff.intro · -- (→) assume h2 : a ≡ b (MOD m * n) obtain (j : Int) (h3 : a - b = (m * n) * j) from h2 apply And.intro · -- Proof of a ≡ b (MOD m) apply Exists.intro (n * j) show a - b = m * (n * j) from calc a - b _ = m * n * j := h3 _ = m * (n * j) := by ring done · -- Proof of a ≡ b (MOD n) apply Exists.intro (m * j) show a - b = n * (m * j) from calc a - b _ = m * n * j := h3 _ = n * (m * j) := by ring done done · -- (←) assume h2 : a ≡ b (MOD m) ∧ a ≡ b (MOD n) obtain (j : Int) (h3 : a - b = m * j) from h2.left have h4 : (↑n : Int) ∣ a - b := h2.right rewrite [h3] at h4 --h4 : ↑n ∣ ↑m * j have h5 : ↑n ∣ j := Theorem_7_2_2_Int h4 h1 obtain (k : Int) (h6 : j = n * k) from h5 apply Exists.intro k --Goal : a - b = ↑(m * n) * k rewrite [Nat.cast_mul] --Goal : a - b = ↑m * ↑n * k show a - b = (m * n) * k from calc a - b _ = m * j := h3 _ = m * (n * k) := by rw [h6] _ = (m * n) * k := by ring done done --From exercises of Section 7.2 theorem rel_prime_symm {a b : Nat} (h : rel_prime a b) : rel_prime b a := sorry lemma prime_NeZero {p : Nat} (h : prime p) : NeZero p := by rewrite [neZero_iff] --Goal : p ≠ 0 define at h linarith done lemma Lemma_7_5_1 {p e d m c s : Nat} {t : Int} (h1 : prime p) (h2 : e * d = (p - 1) * s + 1) (h3 : m ^ e - c = p * t) : c ^ d ≡ m (MOD p) := by have h4 : m ^ e ≡ c (MOD p) := Exists.intro t h3 have h5 : [m ^ e]_p = [c]_p := (cc_eq_iff_congr _ _ _).rtl h4 rewrite [←Exercise_7_4_5_Nat] at h5 --h5 : [m]_p ^ e = [c]_p by_cases h6 : p ∣ m · -- Case 1. h6 : p ∣ m have h7 : m ≡ 0 (MOD p) := by obtain (j : Nat) (h8 : m = p * j) from h6 apply Exists.intro (↑j : Int) --Goal : ↑m - 0 = ↑p * ↑j rewrite [h8, Nat.cast_mul] ring done have h8 : [m]_p = [0]_p := (cc_eq_iff_congr _ _ _).rtl h7 have h9 : e * d ≠ 0 := by rewrite [h2] show (p - 1) * s + 1 ≠ 0 from Nat.add_one_ne_zero _ done have h10 : (0 : Int) ^ (e * d) = 0 := zero_pow h9 have h11 : [c ^ d]_p = [m]_p := calc [c ^ d]_p _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat] _ = ([m]_p ^ e) ^ d := by rw [h5] _ = [m]_p ^ (e * d) := by ring _ = [0]_p ^ (e * d) := by rw [h8] _ = [0 ^ (e * d)]_p := Exercise_7_4_5_Int _ _ _ _ = [0]_p := by rw [h10] _ = [m]_p := by rw [h8] show c ^ d ≡ m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11 done · -- Case 2. h6 : ¬p ∣ m have h7 : rel_prime m p := rel_prime_of_prime_not_dvd h1 h6 have h8 : rel_prime p m := rel_prime_symm h7 have h9 : NeZero p := prime_NeZero h1 have h10 : (1 : Int) ^ s = 1 := by ring have h11 : [c ^ d]_p = [m]_p := calc [c ^ d]_p _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat] _ = ([m]_p ^ e) ^ d := by rw [h5] _ = [m]_p ^ (e * d) := by ring _ = [m]_p ^ ((p - 1) * s + 1) := by rw [h2] _ = ([m]_p ^ (p - 1)) ^ s * [m]_p := by ring _ = ([m]_p ^ (phi p)) ^ s * [m]_p := by rw [phi_prime h1] _ = [1]_p ^ s * [m]_p := by rw [Theorem_7_4_2 h8] _ = [1 ^ s]_p * [m]_p := by rw [Exercise_7_4_5_Int] _ = [1]_p * [m]_p := by rw [h10] _ = [m]_p * [1]_p := by ring _ = [m]_p := Theorem_7_3_6_7 _ show c ^ d ≡ m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11 done done theorem Theorem_7_5_1 (p q n e d k m c : Nat) (p_prime : prime p) (q_prime : prime q) (p_ne_q : p ≠ q) (n_pq : n = p * q) (ed_congr_1 : e * d = k * (p - 1) * (q - 1) + 1) (h1 : [m]_n ^ e = [c]_n) : [c]_n ^ d = [m]_n := by rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] at h1 --h1 : m ^ e ≡ c (MOD n) rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] --Goal : c ^ d ≡ m (MOD n) obtain (j : Int) (h2 : m ^ e - c = n * j) from h1 rewrite [n_pq, Nat.cast_mul] at h2 --h2 : m ^ e - c = p * q * j have h3 : e * d = (p - 1) * (k * (q - 1)) + 1 := by rewrite [ed_congr_1] ring done have h4 : m ^ e - c = p * (q * j) := by rewrite [h2] ring done have congr_p : c ^ d ≡ m (MOD p) := Lemma_7_5_1 p_prime h3 h4 have h5 : e * d = (q - 1) * (k * (p - 1)) + 1 := by rewrite [ed_congr_1] ring done have h6 : m ^ e - c = q * (p * j) := by rewrite [h2] ring done have congr_q : c ^ d ≡ m (MOD q) := Lemma_7_5_1 q_prime h5 h6 have h7 : ¬q ∣ p := by by_contra h8 have h9 : q = 1 ∨ q = p := dvd_prime p_prime h8 disj_syll h9 (prime_not_one q_prime) show False from p_ne_q h9.symm done have h8 : rel_prime p q := rel_prime_of_prime_not_dvd q_prime h7 rewrite [n_pq, Lemma_7_4_5 _ _ h8] show c ^ d ≡ m (MOD p) ∧ c ^ d ≡ m (MOD q) from And.intro congr_p congr_q done /- BEGIN EXERCISES -/ namespace Exercises /- Section 7.1 -/ -- 1. theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry -- 2. lemma gcd_comm_lt {a b : Nat} (h : a < b) : gcd a b = gcd b a := sorry theorem gcd_comm (a b : Nat) : gcd a b = gcd b a := sorry -- 3. theorem Exercise_7_1_5 (a b : Nat) (n : Int) : (∃ (s t : Int), s * a + t * b = n) ↔ (↑(gcd a b) : Int) ∣ n := sorry -- 4. theorem Exercise_7_1_6 (a b c : Nat) : gcd a b = gcd (a + b * c) b := sorry -- 5. theorem gcd_is_nonzero {a b : Nat} (h : a ≠ 0 ∨ b ≠ 0) : gcd a b ≠ 0 := sorry -- 6. theorem gcd_greatest {a b d : Nat} (h1 : gcd a b ≠ 0) (h2 : d ∣ a) (h3 : d ∣ b) : d ≤ gcd a b := sorry -- 7. lemma Lemma_7_1_10a {a b : Nat} (n : Nat) (h : a ∣ b) : (n * a) ∣ (n * b) := sorry lemma Lemma_7_1_10b {a b n : Nat} (h1 : n ≠ 0) (h2 : (n * a) ∣ (n * b)) : a ∣ b := sorry lemma Lemma_7_1_10c {a b : Nat} (h1 : a ∣ b) (h2 : b ∣ a) : a = b := sorry theorem Exercise_7_1_10 (a b n : Nat) : gcd (n * a) (n * b) = n * gcd a b := sorry /- Section 7.2 -/ -- 1. lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry -- 2. -- Hints: Start with apply List.rec. You may find mul_ne_zero useful theorem prod_nonzero_nonzero : ∀ (l : List Nat), (∀ a ∈ l, a ≠ 0) → prod l ≠ 0 := sorry -- 3. theorem rel_prime_iff_no_common_factor (a b : Nat) : rel_prime a b ↔ ¬∃ (p : Nat), prime p ∧ p ∣ a ∧ p ∣ b := sorry -- 4. theorem rel_prime_symm {a b : Nat} (h : rel_prime a b) : rel_prime b a := sorry -- 5. lemma in_prime_factorization_iff_prime_factor {a : Nat} {l : List Nat} (h1 : prime_factorization a l) (p : Nat) : p ∈ l ↔ prime_factor p a := sorry -- 6. theorem Exercise_7_2_5 {a b : Nat} {l m : List Nat} (h1 : prime_factorization a l) (h2 : prime_factorization b m) : rel_prime a b ↔ (¬∃ (p : Nat), p ∈ l ∧ p ∈ m) := sorry -- 7. theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry -- 8. theorem Exercise_7_2_7 {a b a' b' : Nat} (h1 : rel_prime a b) (h2 : a' ∣ a) (h3 : b' ∣ b) : rel_prime a' b' := sorry -- 9. theorem Exercise_7_2_9 {a b j k : Nat} (h1 : gcd a b ≠ 0) (h2 : a = j * gcd a b) (h3 : b = k * gcd a b) : rel_prime j k := sorry -- 10. theorem Exercise_7_2_17a (a b c : Nat) : gcd a (b * c) ∣ gcd a b * gcd a c := sorry /- Section 7.3 -/ -- 1. theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry -- 2. theorem Theorem_7_3_6_3 {m : Nat} (X : ZMod m) : X + [0]_m = X := sorry -- 3. theorem Theorem_7_3_6_4 {m : Nat} (X : ZMod m) : ∃ (Y : ZMod m), X + Y = [0]_m := sorry -- 4. theorem Exercise_7_3_4a {m : Nat} (Z1 Z2 : ZMod m) (h1 : ∀ (X : ZMod m), X + Z1 = X) (h2 : ∀ (X : ZMod m), X + Z2 = X) : Z1 = Z2 := sorry -- 5. theorem Exercise_7_3_4b {m : Nat} (X Y1 Y2 : ZMod m) (h1 : X + Y1 = [0]_m) (h2 : X + Y2 = [0]_m) : Y1 = Y2 := sorry -- 6. theorem Theorem_7_3_10 (m a : Nat) (b : Int) : ¬(↑(gcd m a) : Int) ∣ b → ¬∃ (x : Int), a * x ≡ b (MOD m) := sorry -- 7. theorem Theorem_7_3_11 (m n : Nat) (a b : Int) (h1 : n ≠ 0) : n * a ≡ n * b (MOD n * m) ↔ a ≡ b (MOD m) := sorry -- 8. theorem Exercise_7_3_16 {m : Nat} {a b : Int} (h : a ≡ b (MOD m)) : ∀ (n : Nat), a ^ n ≡ b ^ n (MOD m) := sorry -- 9. example {m : Nat} [NeZero m] (X : ZMod m) : ∃! (a : Int), 0 ≤ a ∧ a < m ∧ X = [a]_m := sorry -- 10. theorem congr_rel_prime {m a b : Nat} (h1 : a ≡ b (MOD m)) : rel_prime m a ↔ rel_prime m b := sorry -- 11. --Hint: You may find the theorem Int.ofNat_mod_ofNat useful. theorem rel_prime_mod (m a : Nat) : rel_prime m (a % m) ↔ rel_prime m a := sorry -- 12. lemma congr_iff_mod_eq_Int (m : Nat) (a b : Int) [NeZero m] : a ≡ b (MOD m) ↔ a % ↑m = b % ↑m := sorry --Hint for next theorem: Use the lemma above, --together with the theorems Int.ofNat_mod_ofNat and Nat.cast_inj. theorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] : ↑a ≡ ↑b (MOD m) ↔ a % m = b % m := sorry /- Section 7.4 -/ -- 1. --Hint: Use induction. --For the base case, compute [a]_m ^ 0 * [1]_m in two ways: --by Theorem_7_3_6_7, [a] ^ 0 * [1]_m = [a]_m ^ 0 --by ring, [a]_m ^ 0 * [1]_m = [1]_m. lemma Exercise_7_4_5_Int (m : Nat) (a : Int) : ∀ (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry -- 2. lemma left_inv_one_one_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g' (g i) = i) : one_one_below n g := sorry -- 3. lemma comp_perm_below {n : Nat} {f g : Nat → Nat} (h1 : perm_below n f) (h2 : perm_below n g) : perm_below n (f ∘ g) := sorry -- 4. lemma perm_below_fixed {n : Nat} {g : Nat → Nat} (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry -- 5.

lemma Lemma_7_4_6 {a b c : Nat} : rel_prime (a * b) c ↔ rel_prime a c ∧ rel_prime b c

HTPI.Exercises.Lemma_7_4_6

htpi/HTPILib/Chap7.lean

htpi

/- Copyright 2023 Daniel J. Velleman -/ import HTPILib.Chap6 namespace HTPI /- Definitions -/ lemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by have h : n + 1 > 0 := Nat.succ_pos n show a % (n + 1) < n + 1 from Nat.mod_lt a h done def gcd (a b : Nat) : Nat := match b with | 0 => a | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd (n + 1) (a % (n + 1)) termination_by b mutual def gcd_c1 (a b : Nat) : Int := match b with | 0 => 1 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c2 (n + 1) (a % (n + 1)) --Corresponds to s = t' termination_by b def gcd_c2 (a b : Nat) : Int := match b with | 0 => 0 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c1 (n + 1) (a % (n + 1)) - (gcd_c2 (n + 1) (a % (n + 1))) * ↑(a / (n + 1)) --Corresponds to t = s' - t'q termination_by b end def prime (n : Nat) : Prop := 2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n def prime_factor (p n : Nat) : Prop := prime p ∧ p ∣ n def all_prime (l : List Nat) : Prop := ∀ p ∈ l, prime p def nondec (l : List Nat) : Prop := match l with | [] => True --Of course, True is a proposition that is always true | n :: L => (∀ m ∈ L, n ≤ m) ∧ nondec L def nondec_prime_list (l : List Nat) : Prop := all_prime l ∧ nondec l def prod (l : List Nat) : Nat := match l with | [] => 1 | n :: L => n * (prod L) def prime_factorization (n : Nat) (l : List Nat) : Prop := nondec_prime_list l ∧ prod l = n def rel_prime (a b : Nat) : Prop := gcd a b = 1 def congr_mod (m : Nat) (a b : Int) : Prop := (↑m : Int) ∣ (a - b) def cc (m : Nat) (a : Int) : ZMod m := (↑a : ZMod m) notation:50 a " ≡ " b " (MOD " m ")" => congr_mod m a b notation:max "["a"]_"m:max => cc m a def invertible {m : Nat} (X : ZMod m) : Prop := ∃ (Y : ZMod m), X * Y = [1]_m def num_rp_below (m k : Nat) : Nat := match k with | 0 => 0 | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j def phi (m : Nat) : Nat := num_rp_below m m def prod_seq {m : Nat} (j k : Nat) (f : Nat → ZMod m) : ZMod m := match j with | 0 => [1]_m | n + 1 => prod_seq n k f * f (k + n) def maps_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i < n, g i < n def one_one_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i1 < n, ∀ i2 < n, g i1 = g i2 → i1 = i2 def onto_below (n : Nat) (g : Nat → Nat) : Prop := ∀ k < n, ∃ i < n, g i = k def perm_below (n : Nat) (g : Nat → Nat) : Prop := maps_below n g ∧ one_one_below n g ∧ onto_below n g def inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m) def swap (u v i : Nat) : Nat := if i = u then v else if i = v then u else i namespace Euler --For definitions specific to Euler's theorem def F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m def G (m a i : Nat) : Nat := (a * i) % m def Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i end Euler /- Section 7.1 -/ theorem dvd_mod_of_dvd_a_b {a b d : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ (a % b) := by set q : Nat := a / b have h3 : b * q + a % b = a := Nat.div_add_mod a b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 define --Goal : ∃ (c : Nat), a % b = d * c apply Exists.intro (j - k * q) show a % b = d * (j - k * q) from calc a % b _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm _ = a - b * q := by rw [h3] _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc] _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm done theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry #eval gcd 672 161 --Answer: 7 lemma gcd_base (a : Nat) : gcd a 0 = a := by rfl lemma gcd_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd a b = gcd b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1)) rfl done lemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b ≠ 0) : a % b < b := by have h1 : b > 0 := Nat.pos_of_ne_zero h show a % b < b from Nat.mod_lt a h1 done lemma dvd_self (n : Nat) : n ∣ n := by apply Exists.intro 1 ring done theorem gcd_dvd : ∀ (b a : Nat), (gcd a b) ∣ a ∧ (gcd a b) ∣ b := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd a b_1) ∣ a ∧ (gcd a b_1) ∣ b_1 fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_base] --Goal: a ∣ a ∧ a ∣ 0 apply And.intro (dvd_self a) define apply Exists.intro 0 rfl done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_nonzero a h1] --Goal : gcd b (a % b) ∣ a ∧ gcd b (a % b) ∣ b have h2 : a % b < b := mod_nonzero_lt a h1 have h3 : (gcd b (a % b)) ∣ b ∧ (gcd b (a % b)) ∣ (a % b) := ih (a % b) h2 b apply And.intro _ h3.left show (gcd b (a % b)) ∣ a from dvd_a_of_dvd_b_mod h3.left h3.right done done theorem gcd_dvd_left (a b : Nat) : (gcd a b) ∣ a := (gcd_dvd b a).left theorem gcd_dvd_right (a b : Nat) : (gcd a b) ∣ b := (gcd_dvd b a).right lemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl lemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c1 a b = gcd_c2 b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done lemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl lemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * ↑(a / b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done theorem gcd_lin_comb : ∀ (b a : Nat), (gcd_c1 a b) * ↑a + (gcd_c2 a b) * ↑b = ↑(gcd a b) := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd_c1 a b_1) * ↑a + (gcd_c2 a b_1) * ↑b_1 = ↑(gcd a b_1) fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base] --Goal : 1 * ↑a + 0 * ↑0 = ↑a ring done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1] --Goal : gcd_c2 b (a % b) * ↑a + -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * ↑(a / b)) * ↑b = -- ↑(gcd b (a % b)) set r : Nat := a % b set q : Nat := a / b set s : Int := gcd_c1 b r set t : Int := gcd_c2 b r --Goal : t * ↑a + (s - t * ↑q) * ↑b = ↑(gcd b r) have h2 : r < b := mod_nonzero_lt a h1 have h3 : s * ↑b + t * ↑r = ↑(gcd b r) := ih r h2 b have h4 : b * q + r = a := Nat.div_add_mod a b rewrite [←h3, ←h4] rewrite [Nat.cast_add, Nat.cast_mul] --Goal : t * (↑b * ↑q + ↑r) + (s - t * ↑q) * ↑b = s * ↑b + t * ↑r ring done done #eval gcd_c1 672 161 --Answer: 6 #eval gcd_c2 672 161 --Answer: -25 --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161 theorem Theorem_7_1_6 {d a b : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ gcd a b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑d ∣ ↑(gcd a b) set s : Int := gcd_c1 a b set t : Int := gcd_c2 a b have h3 : s * ↑a + t * ↑b = ↑(gcd a b) := gcd_lin_comb b a rewrite [←h3] --Goal : ↑d ∣ s * ↑a + t * ↑b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul] --Goal : ↑d ∣ s * (↑d * ↑j) + t * (↑d * ↑k) define apply Exists.intro (s * ↑j + t * ↑k) ring done /- Section 7.2 -/ theorem dvd_trans {a b c : Nat} (h1 : a ∣ b) (h2 : b ∣ c) : a ∣ c := by define at h1; define at h2; define obtain (m : Nat) (h3 : b = a * m) from h1 obtain (n : Nat) (h4 : c = b * n) from h2 rewrite [h3, mul_assoc] at h4 apply Exists.intro (m * n) show c = a * (m * n) from h4 done lemma exists_prime_factor : ∀ (n : Nat), 2 ≤ n → ∃ (p : Nat), prime_factor p n := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, 2 ≤ n_1 → ∃ (p : Nat), prime_factor p n_1 assume h1 : 2 ≤ n by_cases h2 : prime n · -- Case 1. h2 : prime n apply Exists.intro n define --Goal : prime n ∧ n ∣ n show prime n ∧ n ∣ n from And.intro h2 (dvd_self n) done · -- Case 2. h2 : ¬prime n define at h2 --h2 : ¬(2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n) demorgan at h2 disj_syll h2 h1 obtain (a : Nat) (h3 : ∃ (b : Nat), a * b = n ∧ a < n ∧ b < n) from h2 obtain (b : Nat) (h4 : a * b = n ∧ a < n ∧ b < n) from h3 have h5 : 2 ≤ a := by by_contra h6 have h7 : a ≤ 1 := by linarith have h8 : n ≤ b := calc n _ = a * b := h4.left.symm _ ≤ 1 * b := by rel [h7] _ = b := by ring linarith --n ≤ b contradicts b < n done have h6 : ∃ (p : Nat), prime_factor p a := ih a h4.right.left h5 obtain (p : Nat) (h7 : prime_factor p a) from h6 apply Exists.intro p define --Goal : prime p ∧ p ∣ n define at h7 --h7 : prime p ∧ p ∣ a apply And.intro h7.left have h8 : a ∣ n := by apply Exists.intro b show n = a * b from (h4.left).symm done show p ∣ n from dvd_trans h7.right h8 done done lemma exists_least_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q := by set S : Set Nat := {p : Nat | prime_factor p n} have h2 : ∃ (p : Nat), p ∈ S := exists_prime_factor n h show ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q from well_ord_princ S h2 done lemma all_prime_nil : all_prime [] := by define --Goal : ∀ p ∈ [], prime p fix p : Nat contrapos --Goal : ¬prime p → p ∉ [] assume h1 : ¬prime p show p ∉ [] from List.not_mem_nil p done lemma all_prime_cons (n : Nat) (L : List Nat) : all_prime (n :: L) ↔ prime n ∧ all_prime L := by apply Iff.intro · -- (→) assume h1 : all_prime (n :: L) --Goal : prime n ∧ all_prime L define at h1 --h1 : ∀ p ∈ n :: L, prime p apply And.intro (h1 n (List.mem_cons_self n L)) define --Goal : ∀ p ∈ L, prime p fix p : Nat assume h2 : p ∈ L show prime p from h1 p (List.mem_cons_of_mem n h2) done · -- (←) assume h1 : prime n ∧ all_prime L --Goal : all_prime (n :: l) define : all_prime L at h1 define fix p : Nat assume h2 : p ∈ n :: L rewrite [List.mem_cons] at h2 --h2 : p = n ∨ p ∈ L by_cases on h2 · -- Case 1. h2 : p = n rewrite [h2] show prime n from h1.left done · -- Case 2. h2 : p ∈ L show prime p from h1.right p h2 done done done lemma nondec_nil : nondec [] := by define --Goal : True trivial --trivial proves some obviously true statements, such as True done lemma nondec_cons (n : Nat) (L : List Nat) : nondec (n :: L) ↔ (∀ m ∈ L, n ≤ m) ∧ nondec L := by rfl lemma prod_nil : prod [] = 1 := by rfl lemma prod_cons : prod (n :: L) = n * (prod L) := by rfl lemma exists_cons_of_length_eq_succ {A : Type} {l : List A} {n : Nat} (h : l.length = n + 1) : ∃ (a : A) (L : List A), l = a :: L ∧ L.length = n := by have h1 : ¬l.length = 0 := by linarith rewrite [List.length_eq_zero] at h1 obtain (a : A) (h2 : ∃ (L : List A), l = a :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List A) (h3 : l = a :: L) from h2 apply Exists.intro a apply Exists.intro L apply And.intro h3 have h4 : (a :: L).length = L.length + 1 := List.length_cons a L rewrite [←h3, h] at h4 show L.length = n from (Nat.add_right_cancel h4).symm done lemma list_elt_dvd_prod_by_length (a : Nat) : ∀ (n : Nat), ∀ (l : List Nat), l.length = n → a ∈ l → a ∣ prod l := by by_induc · --Base Case fix l : List Nat assume h1 : l.length = 0 rewrite [List.length_eq_zero] at h1 --h1 : l = [] rewrite [h1] --Goal : a ∈ [] → a ∣ prod [] contrapos assume h2 : ¬a ∣ prod [] show a ∉ [] from List.not_mem_nil a done · -- Induction Step fix n : Nat assume ih : ∀ (l : List Nat), List.length l = n → a ∈ l → a ∣ prod l fix l : List Nat assume h1 : l.length = n + 1 --Goal : a ∈ l → a ∣ prod l obtain (b : Nat) (h2 : ∃ (L : List Nat), l = b :: L ∧ L.length = n) from exists_cons_of_length_eq_succ h1 obtain (L : List Nat) (h3 : l = b :: L ∧ L.length = n) from h2 have h4 : a ∈ L → a ∣ prod L := ih L h3.right assume h5 : a ∈ l rewrite [h3.left, prod_cons] --Goal : a ∣ b * prod L rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b ∨ a ∈ L by_cases on h5 · -- Case 1. h5 : a = b apply Exists.intro (prod L) rewrite [h5] rfl done · -- Case 2. h5 : a ∈ L have h6 : a ∣ prod L := h4 h5 have h7 : prod L ∣ b * prod L := by apply Exists.intro b ring done show a ∣ b * prod L from dvd_trans h6 h7 done done done lemma list_elt_dvd_prod {a : Nat} {l : List Nat} (h : a ∈ l) : a ∣ prod l := by set n : Nat := l.length have h1 : l.length = n := by rfl show a ∣ prod l from list_elt_dvd_prod_by_length a n l h1 h done lemma exists_prime_factorization : ∀ (n : Nat), n ≥ 1 → ∃ (l : List Nat), prime_factorization n l := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, n_1 ≥ 1 → ∃ (l : List Nat), prime_factorization n_1 l assume h1 : n ≥ 1 by_cases h2 : n = 1 · -- Case 1. h2 : n = 1 apply Exists.intro [] define apply And.intro · -- Proof of nondec_prime_list [] define show all_prime [] ∧ nondec [] from And.intro all_prime_nil nondec_nil done · -- Proof of prod [] = n rewrite [prod_nil, h2] rfl done done · -- Case 2. h2 : n ≠ 1 have h3 : n ≥ 2 := lt_of_le_of_ne' h1 h2 obtain (p : Nat) (h4 : prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q) from exists_least_prime_factor h3 have p_prime_factor : prime_factor p n := h4.left define at p_prime_factor have p_prime : prime p := p_prime_factor.left have p_dvd_n : p ∣ n := p_prime_factor.right have p_least : ∀ (q : Nat), prime_factor q n → p ≤ q := h4.right obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n have h5 : m ≠ 0 := by contradict h1 with h6 have h7 : n = 0 := calc n _ = p * m := n_eq_pm _ = p * 0 := by rw [h6] _ = 0 := by ring rewrite [h7] decide done have m_pos : 0 < m := Nat.pos_of_ne_zero h5 have m_lt_n : m < n := by define at p_prime show m < n from calc m _ < m + m := by linarith _ = 2 * m := by ring _ ≤ p * m := by rel [p_prime.left] _ = n := n_eq_pm.symm done obtain (L : List Nat) (h6 : prime_factorization m L) from ih m m_lt_n m_pos define at h6 have ndpl_L : nondec_prime_list L := h6.left define at ndpl_L apply Exists.intro (p :: L) define apply And.intro · -- Proof of nondec_prime_list (p :: L) define apply And.intro · -- Proof of all_prime (p :: L) rewrite [all_prime_cons] show prime p ∧ all_prime L from And.intro p_prime ndpl_L.left done · -- Proof of nondec (p :: L) rewrite [nondec_cons] apply And.intro _ ndpl_L.right fix q : Nat assume q_in_L : q ∈ L have h7 : q ∣ prod L := list_elt_dvd_prod q_in_L rewrite [h6.right] at h7 --h7 : q ∣ m have h8 : m ∣ n := by apply Exists.intro p rewrite [n_eq_pm] ring done have q_dvd_n : q ∣ n := dvd_trans h7 h8 have ap_L : all_prime L := ndpl_L.left define at ap_L have q_prime_factor : prime_factor q n := And.intro (ap_L q q_in_L) q_dvd_n show p ≤ q from p_least q q_prime_factor done done · -- Proof of prod (p :: L) = n rewrite [prod_cons, h6.right, n_eq_pm] rfl done done done theorem Theorem_7_2_2 {a b c : Nat} (h1 : c ∣ a * b) (h2 : rel_prime a c) : c ∣ b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑c ∣ ↑b define at h1; define at h2; define obtain (j : Nat) (h3 : a * b = c * j) from h1 set s : Int := gcd_c1 a c set t : Int := gcd_c2 a c have h4 : s * ↑a + t * ↑c = ↑(gcd a c) := gcd_lin_comb c a rewrite [h2, Nat.cast_one] at h4 --h4 : s * ↑a + t * ↑c = (1 : Int) apply Exists.intro (s * ↑j + t * ↑b) show ↑b = ↑c * (s * ↑j + t * ↑b) from calc ↑b _ = (1 : Int) * ↑b := (one_mul _).symm _ = (s * ↑a + t * ↑c) * ↑b := by rw [h4] _ = s * (↑a * ↑b) + t * ↑c * ↑b := by ring _ = s * (↑c * ↑j) + t * ↑c * ↑b := by rw [←Nat.cast_mul a b, h3, Nat.cast_mul c j] _ = ↑c * (s * ↑j + t * ↑b) := by ring done lemma le_nonzero_prod_left {a b : Nat} (h : a * b ≠ 0) : a ≤ a * b := by have h1 : b ≠ 0 := by contradict h with h1 rewrite [h1] ring done have h2 : 1 ≤ b := Nat.pos_of_ne_zero h1 show a ≤ a * b from calc a = a * 1 := (mul_one a).symm _ ≤ a * b := by rel [h2] done lemma le_nonzero_prod_right {a b : Nat} (h : a * b ≠ 0) : b ≤ a * b := by rewrite [mul_comm] rewrite [mul_comm] at h show b ≤ b * a from le_nonzero_prod_left h done lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry lemma rel_prime_of_prime_not_dvd {a p : Nat} (h1 : prime p) (h2 : ¬p ∣ a) : rel_prime a p := by have h3 : gcd a p ∣ a := gcd_dvd_left a p have h4 : gcd a p ∣ p := gcd_dvd_right a p have h5 : gcd a p = 1 ∨ gcd a p = p := dvd_prime h1 h4 have h6 : gcd a p ≠ p := by contradict h2 with h6 rewrite [h6] at h3 show p ∣ a from h3 done disj_syll h5 h6 show rel_prime a p from h5 done theorem Theorem_7_2_3 {a b p : Nat} (h1 : prime p) (h2 : p ∣ a * b) : p ∣ a ∨ p ∣ b := by or_right with h3 have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3 show p ∣ b from Theorem_7_2_2 h2 h4 done lemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a ≥ 1 := by have h1 : a ≠ 0 := by by_contra h1 rewrite [h1] at h contradict h linarith done show a ≥ 1 from Nat.pos_of_ne_zero h1 done lemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by have h1 : a ≥ 1 := ge_one_of_prod_one h have h2 : a * b ≠ 0 := by linarith have h3 : a ≤ a * b := le_nonzero_prod_left h2 rewrite [h] at h3 show a = 1 from Nat.le_antisymm h3 h1 done lemma eq_one_of_dvd_one {n : Nat} (h : n ∣ 1) : n = 1 := by obtain (j : Nat) (h1 : 1 = n * j) from h show n = 1 from eq_one_of_prod_one h1.symm done lemma prime_not_one {p : Nat} (h : prime p) : p ≠ 1 := by define at h linarith done theorem Theorem_7_2_4 {p : Nat} (h1 : prime p) : ∀ (l : List Nat), p ∣ prod l → ∃ a ∈ l, p ∣ a := by apply List.rec · -- Base Case. Goal : p ∣ prod [] → ∃ a ∈ [], p ∣ a rewrite [prod_nil] assume h2 : p ∣ 1 show ∃ a ∈ [], p ∣ a from absurd (eq_one_of_dvd_one h2) (prime_not_one h1) done · -- Induction Step fix b : Nat fix L : List Nat assume ih : p ∣ prod L → ∃ a ∈ L, p ∣ a --Goal : p ∣ prod (b :: L) → ∃ a ∈ b :: L, p ∣ a assume h2 : p ∣ prod (b :: L) rewrite [prod_cons] at h2 have h3 : p ∣ b ∨ p ∣ prod L := Theorem_7_2_3 h1 h2 by_cases on h3 · -- Case 1. h3 : p ∣ b apply Exists.intro b show b ∈ b :: L ∧ p ∣ b from And.intro (List.mem_cons_self b L) h3 done · -- Case 2. h3 : p ∣ prod L obtain (a : Nat) (h4 : a ∈ L ∧ p ∣ a) from ih h3 apply Exists.intro a show a ∈ b :: L ∧ p ∣ a from And.intro (List.mem_cons_of_mem b h4.left) h4.right done done done lemma prime_in_list {p : Nat} {l : List Nat} (h1 : prime p) (h2 : all_prime l) (h3 : p ∣ prod l) : p ∈ l := by obtain (a : Nat) (h4 : a ∈ l ∧ p ∣ a) from Theorem_7_2_4 h1 l h3 define at h2 have h5 : prime a := h2 a h4.left have h6 : p = 1 ∨ p = a := dvd_prime h5 h4.right disj_syll h6 (prime_not_one h1) rewrite [h6] show a ∈ l from h4.left done lemma first_le_first {p q : Nat} {l m : List Nat} (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m)) (h3 : prod (p :: l) = prod (q :: m)) : p ≤ q := by define at h1; define at h2 have h4 : q ∣ prod (p :: l) := by define apply Exists.intro (prod m) rewrite [←prod_cons] show prod (p :: l) = prod (q :: m) from h3 done have h5 : all_prime (q :: m) := h2.left rewrite [all_prime_cons] at h5 have h6 : q ∈ p :: l := prime_in_list h5.left h1.left h4 have h7 : nondec (p :: l) := h1.right rewrite [nondec_cons] at h7 rewrite [List.mem_cons] at h6 by_cases on h6 · -- Case 1. h6 : q = p linarith done · -- Case 2. h6 : q ∈ l have h8 : ∀ m ∈ l, p ≤ m := h7.left show p ≤ q from h8 q h6 done done lemma nondec_prime_list_tail {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by define at h define rewrite [all_prime_cons, nondec_cons] at h show all_prime l ∧ nondec l from And.intro h.left.right h.right.right done lemma cons_prod_not_one {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : prod (p :: l) ≠ 1 := by define at h have h1 : all_prime (p :: l) := h.left rewrite [all_prime_cons] at h1 rewrite [prod_cons] by_contra h2 show False from (prime_not_one h1.left) (eq_one_of_prod_one h2) done lemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) : l = [] ↔ prod l = 1 := by apply Iff.intro · -- (→) assume h1 : l = [] rewrite [h1] show prod [] = 1 from prod_nil done · -- (←) contrapos assume h1 : ¬l = [] obtain (p : Nat) (h2 : ∃ (L : List Nat), l = p :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List Nat) (h3 : l = p :: L) from h2 rewrite [h3] at h rewrite [h3] show ¬prod (p :: L) = 1 from cons_prod_not_one h done done lemma prime_pos {p : Nat} (h : prime p) : p > 0 := by define at h linarith done theorem Theorem_7_2_5 : ∀ (l1 l2 : List Nat), nondec_prime_list l1 → nondec_prime_list l2 → prod l1 = prod l2 → l1 = l2 := by apply List.rec · -- Base Case. Goal : ∀ (l2 : List Nat), nondec_prime_list [] → -- nondec_prime_list l2 → prod [] = prod l2 → [] = l2 fix l2 : List Nat assume h1 : nondec_prime_list [] assume h2 : nondec_prime_list l2 assume h3 : prod [] = prod l2 rewrite [prod_nil, eq_comm, ←list_nil_iff_prod_one h2] at h3 show [] = l2 from h3.symm done · -- Induction Step fix p : Nat fix L1 : List Nat assume ih : ∀ (L2 : List Nat), nondec_prime_list L1 → nondec_prime_list L2 → prod L1 = prod L2 → L1 = L2 -- Goal : ∀ (l2 : List Nat), nondec_prime_list (p :: L1) → -- nondec_prime_list l2 → prod (p :: L1) = prod l2 → p :: L1 = l2 fix l2 : List Nat assume h1 : nondec_prime_list (p :: L1) assume h2 : nondec_prime_list l2 assume h3 : prod (p :: L1) = prod l2 have h4 : ¬prod (p :: L1) = 1 := cons_prod_not_one h1 rewrite [h3, ←list_nil_iff_prod_one h2] at h4 obtain (q : Nat) (h5 : ∃ (L : List Nat), l2 = q :: L) from List.exists_cons_of_ne_nil h4 obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5 rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2) rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2) have h7 : p ≤ q := first_le_first h1 h2 h3 have h8 : q ≤ p := first_le_first h2 h1 h3.symm have h9 : p = q := by linarith rewrite [h9, prod_cons, prod_cons] at h3 --h3 : q * prod L1 = q * prod L2 have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1 have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2 define at h2 have h12 : all_prime (q :: L2) := h2.left rewrite [all_prime_cons] at h12 have h13 : q > 0 := prime_pos h12.left have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3 have h15 : L1 = L2 := ih L2 h10 h11 h14 rewrite [h6, h9, h15] rfl done done theorem fund_thm_arith (n : Nat) (h : n ≥ 1) : ∃! (l : List Nat), prime_factorization n l := by exists_unique · -- Existence show ∃ (l : List Nat), prime_factorization n l from exists_prime_factorization n h done · -- Uniqueness fix l1 : List Nat; fix l2 : List Nat assume h1 : prime_factorization n l1 assume h2 : prime_factorization n l2 define at h1; define at h2 have h3 : prod l1 = n := h1.right rewrite [←h2.right] at h3 show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3 done done /- Section 7.3 -/ theorem congr_refl (m : Nat) : ∀ (a : Int), a ≡ a (MOD m) := by fix a : Int define --Goal : ∃ (c : Int), a - a = ↑m * c apply Exists.intro 0 ring done theorem congr_symm {m : Nat} : ∀ {a b : Int}, a ≡ b (MOD m) → b ≡ a (MOD m) := by fix a : Int; fix b : Int assume h1 : a ≡ b (MOD m) define at h1 --h1 : ∃ (c : Int), a - b = ↑m * c define --Goal : ∃ (c : Int), b - a = ↑m * c obtain (c : Int) (h2 : a - b = m * c) from h1 apply Exists.intro (-c) show b - a = m * (-c) from calc b - a _ = -(a - b) := by ring _ = -(m * c) := by rw [h2] _ = m * (-c) := by ring done theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry /- Fundamental properties of congruence classes -/ lemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) : X = Y ↔ X.val = Y.val := Fin.ext_iff lemma val_nat_eq_mod (n k : Nat) : ([k]_(n + 1)).val = k % (n + 1) := by rfl lemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl theorem cc_rep {m : Nat} (X : ZMod m) : ∃ (a : Int), X = [a]_m := match m with | 0 => by apply Exists.intro X rfl done | n + 1 => by apply Exists.intro ↑(X.val) have h1 : X.val < n + 1 := Fin.prop X rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1] rfl done theorem add_class (m : Nat) (a b : Int) : [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm theorem mul_class (m : Nat) (a b : Int) : [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm lemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ [a - b]_m = [0]_m := by apply Iff.intro · -- (→) assume h1 : [a]_m = [b]_m have h2 : a - b = a + (-b) := by ring have h3 : b + (-b) = 0 := by ring show [a - b]_m = [0]_m from calc [a - b]_m _ = [a + (-b)]_m := by rw [h2] _ = [a]_m + [-b]_m := by rw [add_class] _ = [b]_m + [-b]_m := by rw [h1] _ = [b + -b]_m := by rw [add_class] _ = [0]_m := by rw [h3] done · -- (←) assume h1 : [a - b]_m = [0]_m have h2 : b + (a - b) = a := by ring have h3 : b + 0 = b := by ring show [a]_m = [b]_m from calc [a]_m _ = [b + (a - b)]_m := by rw [h2] _ = [b]_m + [a - b]_m := by rw [add_class] _ = [b]_m + [0]_m := by rw [h1] _ = [b + 0]_m := by rw [add_class] _ = [b]_m := by rw [h3] done done lemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) : [a]_m = [0]_m → [-a]_m = [0]_m := by assume h1 : [a]_m = [0]_m have h2 : 0 + (-a) = -a := by ring have h3 : a + (-a) = 0 := by ring show [-a]_m = [0]_m from calc [-a]_m _ = [0 + (-a)]_m := by rw [h2] _ = [0]_m + [-a]_m := by rw [add_class] _ = [a]_m + [-a]_m := by rw [h1] _ = [a + (-a)]_m := by rw [add_class] _ = [0]_m := by rw [h3] done lemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) : [-a]_m = [0]_m ↔ [a]_m = [0]_m := by apply Iff.intro _ (cc_neg_zero_of_cc_zero m a) assume h1 : [-a]_m = [0]_m have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1 have h3 : -(-a) = a := by ring rewrite [h3] at h2 show [a]_m = [0]_m from h2 done lemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl lemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m ↔ m ∣ k := match m with | 0 => by have h : (0 : Int) = (↑(0 : Nat) : Int) := by rfl rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj] apply Iff.intro · -- (→) assume h1 : k = 0 rewrite [h1] show 0 ∣ 0 from dvd_self 0 done · -- (←) assume h1 : 0 ∣ k obtain (c : Nat) (h2 : k = 0 * c) from h1 rewrite [h2] ring done done | n + 1 => by rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero] show k % (n + 1) = 0 ↔ n + 1 ∣ k from (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm done lemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m ↔ ↑m ∣ a := by obtain (k : Nat) (h1 : a = ↑k ∨ a = -↑k) from Int.eq_nat_or_neg a by_cases on h1 · -- Case 1. h1: a = ↑k rewrite [h1, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done · -- Case 2. h1: a = -↑k rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done done theorem cc_eq_iff_congr (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ a ≡ b (MOD m) := calc [a]_m = [b]_m _ ↔ [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b _ ↔ ↑m ∣ (a - b) := cc_zero_iff_dvd m (a - b) _ ↔ a ≡ b (MOD m) := by rfl /- End of fundamental properties of congruence classes -/ lemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m := by have h1 : (↑m : Int) ≠ 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m) show 0 ≤ a % m from Int.emod_nonneg a h1 done lemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m) have h2 : (↑m : Int) > 0 := (Nat.cast_pos).rtl h1 show a % m < m from Int.emod_lt_of_pos a h2 done lemma congr_mod_mod (m : Nat) (a : Int) : a ≡ a % m (MOD m) := by define have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m apply Exists.intro (a / m) show a - a % m = m * (a / m) from calc a - (a % m) _ = m * (a / m) + a % m - a % m := by rw [h1] _ = m * (a / m) := by ring done lemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) := And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a)) theorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) : ∃! (r : Int), 0 ≤ r ∧ r < m ∧ a ≡ r (MOD m) := by exists_unique · -- Existence apply Exists.intro (a % m) show 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) from mod_cmpl_res m a done · -- Uniqueness fix r1 : Int; fix r2 : Int assume h1 : 0 ≤ r1 ∧ r1 < m ∧ a ≡ r1 (MOD m) assume h2 : 0 ≤ r2 ∧ r2 < m ∧ a ≡ r2 (MOD m) have h3 : r1 ≡ r2 (MOD m) := congr_trans (congr_symm h1.right.right) h2.right.right obtain (d : Int) (h4 : r1 - r2 = m * d) from h3 have h5 : r1 - r2 < m * 1 := by linarith have h6 : m * (-1) < r1 - r2 := by linarith rewrite [h4] at h5 --h5 : m * d < m * 1 rewrite [h4] at h6 --h6 : m * -1 < m * d have h7 : (↑m : Int) ≥ 0 := Nat.cast_nonneg m have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7 have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7 have h10 : d = 0 := by linarith show r1 = r2 from calc r1 _ = r1 - r2 + r2 := by ring _ = m * 0 + r2 := by rw [h4, h10] _ = r2 := by ring done done lemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m := (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a) theorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y rewrite [h1, h2] have h3 : a + b = b + a := by ring show [a]_m + [b]_m = [b]_m + [a]_m from calc [a]_m + [b]_m _ = [a + b]_m := add_class m a b _ = [b + a]_m := by rw [h3] _ = [b]_m + [a]_m := (add_class m b a).symm done theorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X rewrite [h1] have h2 : a * 1 = a := by ring show [a]_m * [1]_m = [a]_m from calc [a]_m * [1]_m _ = [a * 1]_m := mul_class m a 1 _ = [a]_m := by rw [h2] done theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry lemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) : [a]_m * [gcd_c2 m a]_m = [1]_m := by set s : Int := gcd_c1 m a have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m define at h1 rewrite [h1, Nat.cast_one] at h2 --h2 : s * ↑m + gcd_c2 m a * ↑a = 1 rewrite [mul_class, cc_eq_iff_congr] define --Goal : ∃ (c : Int), ↑a * gcd_c2 m a - 1 = ↑m * c apply Exists.intro (-s) show a * (gcd_c2 m a) - 1 = m * (-s) from calc a * (gcd_c2 m a) - 1 _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring _ = 1 + m * (-s) - 1 := by rw [h2] _ = m * (-s) := by ring done theorem Theorem_7_3_7 (m a : Nat) : invertible [a]_m ↔ rel_prime m a := by apply Iff.intro · -- (→) assume h1 : invertible [a]_m define at h1 obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1 obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y rewrite [h3, mul_class, cc_eq_iff_congr] at h2 define at h2 obtain (c : Int) (h4 : a * b - 1 = m * c) from h2 rewrite [Exercise_7_2_6] --Goal : ∃ (s t : Int), s * ↑m + t * ↑a = 1 apply Exists.intro (-c) apply Exists.intro b show (-c) * m + b * a = 1 from calc (-c) * m + b * a _ = (-c) * m + (a * b - 1) + 1 := by ring _ = (-c) * m + m * c + 1 := by rw [h4] _ = 1 := by ring done · -- (←) assume h1 : rel_prime m a define show ∃ (Y : ZMod m), [a]_m * Y = [1]_m from Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1) done done /- Section 7.4 -/ section Euler open Euler lemma num_rp_below_base {m : Nat} : num_rp_below m 0 = 0 := by rfl lemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) : num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by have h1 : num_rp_below m (j + 1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : gcd m j = 1 rewrite [if_pos h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1 show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1 done lemma num_rp_below_step_not_rp {m j : Nat} (h : ¬rel_prime m j) : num_rp_below m (j + 1) = num_rp_below m j := by have h1 : num_rp_below m (j +1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : ¬gcd m j = 1 rewrite [if_neg h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j show num_rp_below m (j + 1) = num_rp_below m j from h1 done lemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl #eval phi 10 --Answer: 4 lemma prod_inv_iff_inv {m : Nat} {X : ZMod m} (h1 : invertible X) (Y : ZMod m) : invertible (X * Y) ↔ invertible Y := by apply Iff.intro · -- (→) assume h2 : invertible (X * Y) obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2 apply Exists.intro (X * Z) rewrite [←h3] --Goal : Y * (X * Z) = X * Y * Z ring --Note that ring can do algebra in ZMod m done · -- (←) assume h2 : invertible Y obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1 obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2 apply Exists.intro (Xi * Yi) show (X * Y) * (Xi * Yi) = [1]_m from calc X * Y * (Xi * Yi) _ = (X * Xi) * (Y * Yi) := by ring _ = [1]_m * [1]_m := by rw [h3, h4] _ = [1]_m := Theorem_7_3_6_7 [1]_m done done lemma F_rp_def {m i : Nat} (h : rel_prime m i) : F m i = [i]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h --h : gcd m i = 1 rewrite [if_pos h] at h1 show F m i = [i]_m from h1 done lemma F_not_rp_def {m i : Nat} (h : ¬rel_prime m i) : F m i = [1]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h rewrite [h1, if_neg h] rfl done lemma prod_seq_base {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 0 k f = [1]_m := by rfl lemma prod_seq_step {m : Nat} (n k : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl lemma prod_seq_zero_step {m : Nat} (n : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by rewrite [prod_seq_step, zero_add] rfl done lemma prod_one {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 1 k f = f k := by rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7] rfl done lemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl lemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m := calc [G m a i]_m _ = [(a * i) % m]_m := by rfl _ = [a * i]_m := (cc_eq_mod m (a * i)).symm _ = [a]_m * [i]_m := (mul_class m a i).symm lemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) : rel_prime m (G m a i) ↔ rel_prime m i := by have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1 show rel_prime m (G m a i) ↔ rel_prime m i from calc rel_prime m (G m a i) _ ↔ invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm _ ↔ invertible ([a]_m * [i]_m) := by rw [cc_G] _ ↔ invertible [i]_m := prod_inv_iff_inv h2 ([i]_m) _ ↔ rel_prime m i := Theorem_7_3_7 m i done lemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) : F m (G m a i) = [a]_m * F m i := by have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2 show F m (G m a i) = [a]_m * F m i from calc F m (G m a i) _ = [G m a i]_m := F_rp_def h3 _ = [a]_m * [i]_m := cc_G m a i _ = [a]_m * F m i := by rw [F_rp_def h2] done lemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : ¬rel_prime m i) : F m (G m a i) = [1]_m := by rewrite [←G_rp_iff h1 i] at h2 show F m (G m a i) = [1]_m from F_not_rp_def h2 done lemma FG_prod {m a : Nat} (h1 : rel_prime m a) : ∀ (k : Nat), prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by by_induc · -- Base Case show prod_seq 0 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from calc prod_seq 0 0 ((F m) ∘ (G m a)) _ = [1]_m := prod_seq_base _ _ _ = [a]_m ^ 0 * [1]_m := by ring _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by rw [num_rp_below_base, prod_seq_base] done · -- Induction Step fix k : Nat assume ih : prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) by_cases h2 : rel_prime m k · -- Case 1. h2 : rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([a]_m * F m k) := by rw [FG_rp h1 h2] _ = [a]_m ^ ((num_rp_below m k) + 1) * ((prod_seq k 0 (F m)) * F m k) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_rp h2, prod_seq_zero_step] done · -- Case 2. h2 : ¬rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([1]_m) := by rw [FG_not_rp h1 h2] _ = [a]_m ^ (num_rp_below m k) * (prod_seq k 0 (F m) * ([1]_m)) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_not_rp h2, prod_seq_zero_step, F_not_rp_def h2] done done done lemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by define --Goal : ∀ i < m, G m a i < m fix i : Nat assume h1 : i < m rewrite [G_def] --Goal : a * i % m < m show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m) done lemma left_inv_one_one_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g' (g i) = i) : one_one_below n g := sorry lemma right_inv_onto_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g (g' i) = i) (h2 : maps_below n g') : onto_below n g := by define at h2; define fix k : Nat assume h3 : k < n apply Exists.intro (g' k) show g' k < n ∧ g (g' k) = k from And.intro (h2 k h3) (h1 k h3) done lemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m * [inv_mod m a]_m = [1]_m := by have h2 : 0 ≤ (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a) show [a]_m * [inv_mod m a]_m = [1]_m from calc [a]_m * [inv_mod m a]_m _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2] _ = [a]_m * [gcd_c2 m a]_m := by rw [←cc_eq_mod] _ = [1]_m := gcd_c2_inv h1 done lemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m := calc a * (b % m) % m = a % m * (b % m % m) % m := Nat.mul_mod _ _ _ _ = a % m * (b % m) % m := by rw [Nat.mod_mod] _ = a * b % m := (Nat.mul_mod _ _ _).symm lemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm] rfl done theorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] : ↑a ≡ ↑b (MOD m) ↔ a % m = b % m := sorry lemma mul_inv_mod_cancel {m a i : Nat} [NeZero m] (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1 rewrite [mul_class, cc_eq_iff_congr, ←Nat.cast_mul, ←Nat.cast_one, congr_iff_mod_eq_Nat] at h3 show a * inv_mod m a * i % m = i from calc a * (inv_mod m a) * i % m _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = 1 % m * i % m := by rw [h3] _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = i % m := by rw [one_mul] _ = i := Nat.mod_eq_of_lt h2 done lemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl lemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, G m a (Ginv m a i) = i := by fix i : Nat assume h2 : i < m show G m a (Ginv m a i) = i from calc G m a (Ginv m a i) _ = a * ((inv_mod m a * i) % m) % m := by rfl _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, Ginv m a (G m a i) = i := by fix i : Nat assume h2 : i < m show Ginv m a (G m a i) = i from calc Ginv m a (G m a i) _ = inv_mod m a * ((a * i) % m) % m := by rfl _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc, mul_comm (inv_mod m a)] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_maps_below (m a : Nat) [NeZero m] : maps_below m (Ginv m a) := G_maps_below m (inv_mod m a) lemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : one_one_below m (G m a) := left_inv_one_one_below (Ginv_left_inv h1) lemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : onto_below m (G m a) := right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a) lemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : perm_below m (G m a) := And.intro (G_maps_below m a) (And.intro (G_one_one_below h1) (G_onto_below h1)) --Permuting a product of congruence classes doesn't change product lemma swap_fst (u v : Nat) : swap u v u = v := by define : swap u v u --Goal : (if u = u then v else if u = v then u else u) = v have h : u = u := by rfl rewrite [if_pos h] rfl done lemma swap_snd (u v : Nat) : swap u v v = u := by define : swap u v v by_cases h1 : v = u · -- Case 1. h1 : v = u rewrite [if_pos h1] show v = u from h1 done · -- Case 2. h1 : v ≠ u rewrite [if_neg h1] have h2 : v = v := by rfl rewrite [if_pos h2] rfl done done lemma swap_other {u v i : Nat} (h1 : i ≠ u) (h2 : i ≠ v) : swap u v i = i := by define : swap u v i rewrite [if_neg h1, if_neg h2] rfl done lemma swap_values (u v i : Nat) : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := by by_cases h1 : i = u · -- Case 1. h1 : i = u apply Or.inl rewrite [h1] show swap u v u = v from swap_fst u v done · -- Case 2. h1 : i ≠ u apply Or.inr by_cases h2 : i = v · -- Case 2.1. h2 : i = v apply Or.inl rewrite [h2] show swap u v v = u from swap_snd u v done · -- Case 2.2. h2 : i ≠ v apply Or.inr show swap u v i = i from swap_other h1 h2 done done done lemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by define fix i : Nat assume h3 : i < n have h4 : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := swap_values u v i by_cases on h4 · -- Case 1. h4 : swap u v i = v rewrite [h4] show v < n from h2 done · -- Case 2. by_cases on h4 · -- Case 2.1. h4 : swap u v i = u rewrite [h4] show u < n from h1 done · -- Case 2.2. h4 : swap u v i = i rewrite [h4] show i < n from h3 done done done lemma swap_swap (u v n : Nat) : ∀ i < n, swap u v (swap u v i) = i := by fix i : Nat assume h : i < n by_cases h1 : i = u · -- Case 1. h1 : i = u rewrite [h1, swap_fst, swap_snd] rfl done · -- Case 2. h1 : i ≠ u by_cases h2 : i = v · -- Case 2.1. h2 : i = v rewrite [h2, swap_snd, swap_fst] rfl done · -- Case 2.2. h2 : i ≠ v rewrite [swap_other h1 h2, swap_other h1 h2] rfl done done done lemma swap_one_one_below (u v n) : one_one_below n (swap u v) := left_inv_one_one_below (swap_swap u v n) lemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) := right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2) lemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) := And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2)) lemma comp_perm_below {n : Nat} {f g : Nat → Nat} (h1 : perm_below n f) (h2 : perm_below n g) : perm_below n (f ∘ g) := sorry lemma trivial_swap (u : Nat) : swap u u = id := by apply funext fix x : Nat by_cases h1 : x = u · -- Case 1. h1 : x = u rewrite [h1, swap_fst] rfl done · -- Case 2. h1 : x ≠ u rewrite [swap_other h1 h1] rfl done done lemma prod_eq_fun {m : Nat} (f g : Nat → ZMod m) (k : Nat) : ∀ (n : Nat), (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g := by by_induc · -- Base Case assume h : (∀ i < 0, f (k + i) = g (k + i)) rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g assume h1 : ∀ i < n + 1, f (k + i) = g (k + i) have h2 : ∀ i < n, f (k + i) = g (k + i) := by fix i : Nat assume h2 : i < n have h3 : i < n + 1 := by linarith show f (k + i) = g (k + i) from h1 i h3 done have h3 : prod_seq n k f = prod_seq n k g := ih h2 have h4 : n < n + 1 := Nat.lt_succ_self n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4] rfl done done lemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := by have h2 : ∀ (i : Nat), i < u → (f ∘ swap u n) (0 + i) = f (0 + i) := by fix i : Nat assume h2 : i < u have h3 : 0 + i ≠ u := by linarith have h4 : 0 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl done show prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f from prod_eq_fun (f ∘ swap u n) f 0 u h2 done lemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := by have h2 : ∀ i < j, (f ∘ swap u n) (u + 1 + i) = f (u + 1 + i) := by fix i : Nat assume h2 : i < j have h3 : u + 1 + i ≠ u := by linarith have h4 : u + 1 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl show prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f from prod_eq_fun (f ∘ swap u n) f (u + 1) j h2 done lemma break_prod {m : Nat} (n : Nat) (f : Nat → ZMod m) : ∀ (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by by_induc · -- Base Case have h : n + 0 = n := by rfl rewrite [prod_seq_base, h, Theorem_7_3_6_7] rfl done · -- Induction Step fix j : Nat assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f rewrite [←add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc] rfl done done lemma break_prod_twice {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f := break_prod n f 1 rewrite [prod_one] at h2 have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f := break_prod (u + 1) f j rewrite [←h1] at h3 have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f := break_prod u f 1 rewrite [prod_one] at h4 rewrite [h3, h4] at h2 show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2 done lemma swap_prod_eq_prod {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f := by by_cases h2 : u = n · -- Case 1. h2 : u = n rewrite [h2, trivial_swap n] --Goal : prod_seq (n + 1) 0 (f ∘ id) = prod_seq (n + 1) 0 f rfl done · -- Case 2. h2 : ¬u = n have h3 : u + 1 ≤ n := Nat.lt_of_le_of_ne h1 h2 obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3 have break_f : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := break_prod_twice f h4 have break_fs : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_prod_twice (f ∘ swap u n) h4 have f_eq_fs_below : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := swap_prod_eq_prod_below f h1 have f_eq_fs_btwn : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4 show prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f from calc prod_seq (n + 1) 0 (f ∘ swap u n) _ = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_fs _ = prod_seq u 0 f * (f ∘ swap u n) u * prod_seq j (u + 1) f * (f ∘ swap u n) n := by rw [f_eq_fs_below, f_eq_fs_btwn] _ = prod_seq u 0 f * f (swap u n u) * prod_seq j (u + 1) f * f (swap u n n) := by rfl _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by rw [swap_fst, swap_snd] _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring _ = prod_seq (n + 1) 0 f := break_f.symm done done lemma perm_below_fixed {n : Nat} {g : Nat → Nat} (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry lemma perm_prod {m : Nat} (f : Nat → ZMod m) : ∀ (n : Nat), ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) := by by_induc · -- Base Case fix g : Nat → Nat assume h1 : perm_below 0 g rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) fix g : Nat → Nat assume g_pb : perm_below (n + 1) g define at g_pb have g_ob : onto_below (n + 1) g := g_pb.right.right define at g_ob have h1 : n < n + 1 := by linarith obtain (u : Nat) (h2 : u < n + 1 ∧ g u = n) from g_ob n h1 have s_pb : perm_below (n + 1) (swap u n) := swap_perm_below h2.left h1 have gs_pb_n1 : perm_below (n + 1) (g ∘ swap u n) := comp_perm_below g_pb s_pb have gs_fix_n : (g ∘ swap u n) n = n := calc (g ∘ swap u n) n _ = g (swap u n n) := by rfl _ = g u := by rw [swap_snd] _ = n := h2.right have gs_pb_n : perm_below n (g ∘ swap u n) := perm_below_fixed gs_pb_n1 gs_fix_n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f ∘ (g ∘ swap u n)) := ih (g ∘ swap u n) gs_pb_n have h3 : u ≤ n := by linarith show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f ∘ g) from calc prod_seq (n + 1) 0 f _ = prod_seq n 0 f * f n := prod_seq_zero_step n f _ = prod_seq n 0 (f ∘ (g ∘ swap u n)) * f ((g ∘ swap u n) n) := by rw [gs_prod, gs_fix_n] _ = prod_seq n 0 (f ∘ g ∘ swap u n) * (f ∘ g ∘ swap u n) n := by rfl _ = prod_seq (n + 1) 0 (f ∘ g ∘ swap u n) := (prod_seq_zero_step n (f ∘ g ∘ swap u n)).symm _ = prod_seq (n + 1) 0 ((f ∘ g) ∘ swap u n) := by rfl _ = prod_seq (n + 1) 0 (f ∘ g) := swap_prod_eq_prod (f ∘ g) h3 done done lemma F_invertible (m i : Nat) : invertible (F m i) := by by_cases h : rel_prime m i · -- Case 1. h : rel_prime m i rewrite [F_rp_def h] show invertible [i]_m from (Theorem_7_3_7 m i).rtl h done · -- Case 2. h : ¬rel_prime m i rewrite [F_not_rp_def h] apply Exists.intro [1]_m show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m done done lemma Fprod_invertible (m : Nat) : ∀ (k : Nat), invertible (prod_seq k 0 (F m)) := by by_induc · -- Base Case apply Exists.intro [1]_m show prod_seq 0 0 (F m) * [1]_m = [1]_m from calc prod_seq 0 0 (F m) * [1]_m _ = [1]_m * [1]_m := by rw [prod_seq_base] _ = [1]_m := Theorem_7_3_6_7 ([1]_m) done · -- Induction Step fix k : Nat assume ih : invertible (prod_seq k 0 (F m)) rewrite [prod_seq_zero_step] show invertible (prod_seq k 0 (F m) * (F m k)) from (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k) done done theorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m ^ (phi m) = [1]_m := by have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2 show [a]_m ^ (phi m) = [1]_m from calc [a]_m ^ (phi m) _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3] _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring _ = prod_seq m 0 (F m ∘ G m a) * Y := by rw [FG_prod h1 m, phi_def] _ = prod_seq m 0 (F m) * Y := by rw [perm_prod (F m) m (G m a) (G_perm_below h1)] _ = [1]_m := by rw [h3] done lemma Exercise_7_4_5_Int (m : Nat) (a : Int) : ∀ (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry lemma Exercise_7_4_5_Nat (m a n : Nat) : [a]_m ^ n = [a ^ n]_m := by rewrite [Exercise_7_4_5_Int] rfl done theorem Euler's_theorem {m a : Nat} [NeZero m] (h1 : rel_prime m a) : a ^ (phi m) ≡ 1 (MOD m) := by have h2 : [a]_m ^ (phi m) = [1]_m := Theorem_7_4_2 h1 rewrite [Exercise_7_4_5_Nat m a (phi m)] at h2 --h2 : [a ^ phi m]_m = [1]_m show a ^ (phi m) ≡ 1 (MOD m) from (cc_eq_iff_congr _ _ _).ltr h2 done #eval gcd 10 7 --Answer: 1. So 10 and 7 are relatively prime #eval 7 ^ phi 10 --Answer: 2401, which is congruent to 1 mod 10. end Euler /- Section 7.5 -/ lemma num_rp_prime {p : Nat} (h1 : prime p) : ∀ k < p, num_rp_below p (k + 1) = k := sorry lemma phi_prime {p : Nat} (h1 : prime p) : phi p = p - 1 := by have h2 : 1 ≤ p := prime_pos h1 have h3 : p - 1 + 1 = p := Nat.sub_add_cancel h2 have h4 : p - 1 < p := by linarith have h5 : num_rp_below p (p - 1 + 1) = p - 1 := num_rp_prime h1 (p - 1) h4 rewrite [h3] at h5 show phi p = p - 1 from h5 done theorem Theorem_7_2_2_Int {a c : Nat} {b : Int} (h1 : ↑c ∣ ↑a * b) (h2 : rel_prime a c) : ↑c ∣ b := by rewrite [Int.natCast_dvd, Int.natAbs_mul, Int.natAbs_ofNat] at h1 --h1 : c ∣ a * Int.natAbs b rewrite [Int.natCast_dvd] --Goal : c ∣ Int.natAbs b show c ∣ Int.natAbs b from Theorem_7_2_2 h1 h2 done lemma Lemma_7_4_5 {m n : Nat} (a b : Int) (h1 : rel_prime m n) : a ≡ b (MOD m * n) ↔ a ≡ b (MOD m) ∧ a ≡ b (MOD n) := by apply Iff.intro · -- (→) assume h2 : a ≡ b (MOD m * n) obtain (j : Int) (h3 : a - b = (m * n) * j) from h2 apply And.intro · -- Proof of a ≡ b (MOD m) apply Exists.intro (n * j) show a - b = m * (n * j) from calc a - b _ = m * n * j := h3 _ = m * (n * j) := by ring done · -- Proof of a ≡ b (MOD n) apply Exists.intro (m * j) show a - b = n * (m * j) from calc a - b _ = m * n * j := h3 _ = n * (m * j) := by ring done done · -- (←) assume h2 : a ≡ b (MOD m) ∧ a ≡ b (MOD n) obtain (j : Int) (h3 : a - b = m * j) from h2.left have h4 : (↑n : Int) ∣ a - b := h2.right rewrite [h3] at h4 --h4 : ↑n ∣ ↑m * j have h5 : ↑n ∣ j := Theorem_7_2_2_Int h4 h1 obtain (k : Int) (h6 : j = n * k) from h5 apply Exists.intro k --Goal : a - b = ↑(m * n) * k rewrite [Nat.cast_mul] --Goal : a - b = ↑m * ↑n * k show a - b = (m * n) * k from calc a - b _ = m * j := h3 _ = m * (n * k) := by rw [h6] _ = (m * n) * k := by ring done done --From exercises of Section 7.2 theorem rel_prime_symm {a b : Nat} (h : rel_prime a b) : rel_prime b a := sorry lemma prime_NeZero {p : Nat} (h : prime p) : NeZero p := by rewrite [neZero_iff] --Goal : p ≠ 0 define at h linarith done lemma Lemma_7_5_1 {p e d m c s : Nat} {t : Int} (h1 : prime p) (h2 : e * d = (p - 1) * s + 1) (h3 : m ^ e - c = p * t) : c ^ d ≡ m (MOD p) := by have h4 : m ^ e ≡ c (MOD p) := Exists.intro t h3 have h5 : [m ^ e]_p = [c]_p := (cc_eq_iff_congr _ _ _).rtl h4 rewrite [←Exercise_7_4_5_Nat] at h5 --h5 : [m]_p ^ e = [c]_p by_cases h6 : p ∣ m · -- Case 1. h6 : p ∣ m have h7 : m ≡ 0 (MOD p) := by obtain (j : Nat) (h8 : m = p * j) from h6 apply Exists.intro (↑j : Int) --Goal : ↑m - 0 = ↑p * ↑j rewrite [h8, Nat.cast_mul] ring done have h8 : [m]_p = [0]_p := (cc_eq_iff_congr _ _ _).rtl h7 have h9 : e * d ≠ 0 := by rewrite [h2] show (p - 1) * s + 1 ≠ 0 from Nat.add_one_ne_zero _ done have h10 : (0 : Int) ^ (e * d) = 0 := zero_pow h9 have h11 : [c ^ d]_p = [m]_p := calc [c ^ d]_p _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat] _ = ([m]_p ^ e) ^ d := by rw [h5] _ = [m]_p ^ (e * d) := by ring _ = [0]_p ^ (e * d) := by rw [h8] _ = [0 ^ (e * d)]_p := Exercise_7_4_5_Int _ _ _ _ = [0]_p := by rw [h10] _ = [m]_p := by rw [h8] show c ^ d ≡ m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11 done · -- Case 2. h6 : ¬p ∣ m have h7 : rel_prime m p := rel_prime_of_prime_not_dvd h1 h6 have h8 : rel_prime p m := rel_prime_symm h7 have h9 : NeZero p := prime_NeZero h1 have h10 : (1 : Int) ^ s = 1 := by ring have h11 : [c ^ d]_p = [m]_p := calc [c ^ d]_p _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat] _ = ([m]_p ^ e) ^ d := by rw [h5] _ = [m]_p ^ (e * d) := by ring _ = [m]_p ^ ((p - 1) * s + 1) := by rw [h2] _ = ([m]_p ^ (p - 1)) ^ s * [m]_p := by ring _ = ([m]_p ^ (phi p)) ^ s * [m]_p := by rw [phi_prime h1] _ = [1]_p ^ s * [m]_p := by rw [Theorem_7_4_2 h8] _ = [1 ^ s]_p * [m]_p := by rw [Exercise_7_4_5_Int] _ = [1]_p * [m]_p := by rw [h10] _ = [m]_p * [1]_p := by ring _ = [m]_p := Theorem_7_3_6_7 _ show c ^ d ≡ m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11 done done theorem Theorem_7_5_1 (p q n e d k m c : Nat) (p_prime : prime p) (q_prime : prime q) (p_ne_q : p ≠ q) (n_pq : n = p * q) (ed_congr_1 : e * d = k * (p - 1) * (q - 1) + 1) (h1 : [m]_n ^ e = [c]_n) : [c]_n ^ d = [m]_n := by rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] at h1 --h1 : m ^ e ≡ c (MOD n) rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] --Goal : c ^ d ≡ m (MOD n) obtain (j : Int) (h2 : m ^ e - c = n * j) from h1 rewrite [n_pq, Nat.cast_mul] at h2 --h2 : m ^ e - c = p * q * j have h3 : e * d = (p - 1) * (k * (q - 1)) + 1 := by rewrite [ed_congr_1] ring done have h4 : m ^ e - c = p * (q * j) := by rewrite [h2] ring done have congr_p : c ^ d ≡ m (MOD p) := Lemma_7_5_1 p_prime h3 h4 have h5 : e * d = (q - 1) * (k * (p - 1)) + 1 := by rewrite [ed_congr_1] ring done have h6 : m ^ e - c = q * (p * j) := by rewrite [h2] ring done have congr_q : c ^ d ≡ m (MOD q) := Lemma_7_5_1 q_prime h5 h6 have h7 : ¬q ∣ p := by by_contra h8 have h9 : q = 1 ∨ q = p := dvd_prime p_prime h8 disj_syll h9 (prime_not_one q_prime) show False from p_ne_q h9.symm done have h8 : rel_prime p q := rel_prime_of_prime_not_dvd q_prime h7 rewrite [n_pq, Lemma_7_4_5 _ _ h8] show c ^ d ≡ m (MOD p) ∧ c ^ d ≡ m (MOD q) from And.intro congr_p congr_q done /- BEGIN EXERCISES -/ namespace Exercises /- Section 7.1 -/ -- 1. theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry -- 2. lemma gcd_comm_lt {a b : Nat} (h : a < b) : gcd a b = gcd b a := sorry theorem gcd_comm (a b : Nat) : gcd a b = gcd b a := sorry -- 3. theorem Exercise_7_1_5 (a b : Nat) (n : Int) : (∃ (s t : Int), s * a + t * b = n) ↔ (↑(gcd a b) : Int) ∣ n := sorry -- 4. theorem Exercise_7_1_6 (a b c : Nat) : gcd a b = gcd (a + b * c) b := sorry -- 5. theorem gcd_is_nonzero {a b : Nat} (h : a ≠ 0 ∨ b ≠ 0) : gcd a b ≠ 0 := sorry -- 6. theorem gcd_greatest {a b d : Nat} (h1 : gcd a b ≠ 0) (h2 : d ∣ a) (h3 : d ∣ b) : d ≤ gcd a b := sorry -- 7. lemma Lemma_7_1_10a {a b : Nat} (n : Nat) (h : a ∣ b) : (n * a) ∣ (n * b) := sorry lemma Lemma_7_1_10b {a b n : Nat} (h1 : n ≠ 0) (h2 : (n * a) ∣ (n * b)) : a ∣ b := sorry lemma Lemma_7_1_10c {a b : Nat} (h1 : a ∣ b) (h2 : b ∣ a) : a = b := sorry theorem Exercise_7_1_10 (a b n : Nat) : gcd (n * a) (n * b) = n * gcd a b := sorry /- Section 7.2 -/ -- 1. lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry -- 2. -- Hints: Start with apply List.rec. You may find mul_ne_zero useful theorem prod_nonzero_nonzero : ∀ (l : List Nat), (∀ a ∈ l, a ≠ 0) → prod l ≠ 0 := sorry -- 3. theorem rel_prime_iff_no_common_factor (a b : Nat) : rel_prime a b ↔ ¬∃ (p : Nat), prime p ∧ p ∣ a ∧ p ∣ b := sorry -- 4. theorem rel_prime_symm {a b : Nat} (h : rel_prime a b) : rel_prime b a := sorry -- 5. lemma in_prime_factorization_iff_prime_factor {a : Nat} {l : List Nat} (h1 : prime_factorization a l) (p : Nat) : p ∈ l ↔ prime_factor p a := sorry -- 6. theorem Exercise_7_2_5 {a b : Nat} {l m : List Nat} (h1 : prime_factorization a l) (h2 : prime_factorization b m) : rel_prime a b ↔ (¬∃ (p : Nat), p ∈ l ∧ p ∈ m) := sorry -- 7. theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry -- 8. theorem Exercise_7_2_7 {a b a' b' : Nat} (h1 : rel_prime a b) (h2 : a' ∣ a) (h3 : b' ∣ b) : rel_prime a' b' := sorry -- 9. theorem Exercise_7_2_9 {a b j k : Nat} (h1 : gcd a b ≠ 0) (h2 : a = j * gcd a b) (h3 : b = k * gcd a b) : rel_prime j k := sorry -- 10. theorem Exercise_7_2_17a (a b c : Nat) : gcd a (b * c) ∣ gcd a b * gcd a c := sorry /- Section 7.3 -/ -- 1. theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry -- 2. theorem Theorem_7_3_6_3 {m : Nat} (X : ZMod m) : X + [0]_m = X := sorry -- 3. theorem Theorem_7_3_6_4 {m : Nat} (X : ZMod m) : ∃ (Y : ZMod m), X + Y = [0]_m := sorry -- 4. theorem Exercise_7_3_4a {m : Nat} (Z1 Z2 : ZMod m) (h1 : ∀ (X : ZMod m), X + Z1 = X) (h2 : ∀ (X : ZMod m), X + Z2 = X) : Z1 = Z2 := sorry -- 5. theorem Exercise_7_3_4b {m : Nat} (X Y1 Y2 : ZMod m) (h1 : X + Y1 = [0]_m) (h2 : X + Y2 = [0]_m) : Y1 = Y2 := sorry -- 6. theorem Theorem_7_3_10 (m a : Nat) (b : Int) : ¬(↑(gcd m a) : Int) ∣ b → ¬∃ (x : Int), a * x ≡ b (MOD m) := sorry -- 7. theorem Theorem_7_3_11 (m n : Nat) (a b : Int) (h1 : n ≠ 0) : n * a ≡ n * b (MOD n * m) ↔ a ≡ b (MOD m) := sorry -- 8. theorem Exercise_7_3_16 {m : Nat} {a b : Int} (h : a ≡ b (MOD m)) : ∀ (n : Nat), a ^ n ≡ b ^ n (MOD m) := sorry -- 9. example {m : Nat} [NeZero m] (X : ZMod m) : ∃! (a : Int), 0 ≤ a ∧ a < m ∧ X = [a]_m := sorry -- 10. theorem congr_rel_prime {m a b : Nat} (h1 : a ≡ b (MOD m)) : rel_prime m a ↔ rel_prime m b := sorry -- 11. --Hint: You may find the theorem Int.ofNat_mod_ofNat useful. theorem rel_prime_mod (m a : Nat) : rel_prime m (a % m) ↔ rel_prime m a := sorry -- 12. lemma congr_iff_mod_eq_Int (m : Nat) (a b : Int) [NeZero m] : a ≡ b (MOD m) ↔ a % ↑m = b % ↑m := sorry --Hint for next theorem: Use the lemma above, --together with the theorems Int.ofNat_mod_ofNat and Nat.cast_inj. theorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] : ↑a ≡ ↑b (MOD m) ↔ a % m = b % m := sorry /- Section 7.4 -/ -- 1. --Hint: Use induction. --For the base case, compute [a]_m ^ 0 * [1]_m in two ways: --by Theorem_7_3_6_7, [a] ^ 0 * [1]_m = [a]_m ^ 0 --by ring, [a]_m ^ 0 * [1]_m = [1]_m. lemma Exercise_7_4_5_Int (m : Nat) (a : Int) : ∀ (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry -- 2. lemma left_inv_one_one_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g' (g i) = i) : one_one_below n g := sorry -- 3. lemma comp_perm_below {n : Nat} {f g : Nat → Nat} (h1 : perm_below n f) (h2 : perm_below n g) : perm_below n (f ∘ g) := sorry -- 4. lemma perm_below_fixed {n : Nat} {g : Nat → Nat} (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry -- 5. lemma Lemma_7_4_6 {a b c : Nat} : rel_prime (a * b) c ↔ rel_prime a c ∧ rel_prime b c := sorry -- 6. example {m a : Nat} [NeZero m] (h1 : rel_prime m a) : a ^ (phi m + 1) ≡ a (MOD m) := sorry -- 7.

theorem Like_Exercise_7_4_11 {m a p : Nat} [NeZero m] (h1 : rel_prime m a) (h2 : p + 1 = phi m) : [a]_m * [a ^ p]_m = [1]_m

HTPI.Exercises.Like_Exercise_7_4_11

htpi/HTPILib/Chap7.lean

htpi

/- Copyright 2023 Daniel J. Velleman -/ import HTPILib.Chap6 namespace HTPI /- Definitions -/ lemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by have h : n + 1 > 0 := Nat.succ_pos n show a % (n + 1) < n + 1 from Nat.mod_lt a h done def gcd (a b : Nat) : Nat := match b with | 0 => a | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd (n + 1) (a % (n + 1)) termination_by b mutual def gcd_c1 (a b : Nat) : Int := match b with | 0 => 1 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c2 (n + 1) (a % (n + 1)) --Corresponds to s = t' termination_by b def gcd_c2 (a b : Nat) : Int := match b with | 0 => 0 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c1 (n + 1) (a % (n + 1)) - (gcd_c2 (n + 1) (a % (n + 1))) * ↑(a / (n + 1)) --Corresponds to t = s' - t'q termination_by b end def prime (n : Nat) : Prop := 2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n def prime_factor (p n : Nat) : Prop := prime p ∧ p ∣ n def all_prime (l : List Nat) : Prop := ∀ p ∈ l, prime p def nondec (l : List Nat) : Prop := match l with | [] => True --Of course, True is a proposition that is always true | n :: L => (∀ m ∈ L, n ≤ m) ∧ nondec L def nondec_prime_list (l : List Nat) : Prop := all_prime l ∧ nondec l def prod (l : List Nat) : Nat := match l with | [] => 1 | n :: L => n * (prod L) def prime_factorization (n : Nat) (l : List Nat) : Prop := nondec_prime_list l ∧ prod l = n def rel_prime (a b : Nat) : Prop := gcd a b = 1 def congr_mod (m : Nat) (a b : Int) : Prop := (↑m : Int) ∣ (a - b) def cc (m : Nat) (a : Int) : ZMod m := (↑a : ZMod m) notation:50 a " ≡ " b " (MOD " m ")" => congr_mod m a b notation:max "["a"]_"m:max => cc m a def invertible {m : Nat} (X : ZMod m) : Prop := ∃ (Y : ZMod m), X * Y = [1]_m def num_rp_below (m k : Nat) : Nat := match k with | 0 => 0 | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j def phi (m : Nat) : Nat := num_rp_below m m def prod_seq {m : Nat} (j k : Nat) (f : Nat → ZMod m) : ZMod m := match j with | 0 => [1]_m | n + 1 => prod_seq n k f * f (k + n) def maps_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i < n, g i < n def one_one_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i1 < n, ∀ i2 < n, g i1 = g i2 → i1 = i2 def onto_below (n : Nat) (g : Nat → Nat) : Prop := ∀ k < n, ∃ i < n, g i = k def perm_below (n : Nat) (g : Nat → Nat) : Prop := maps_below n g ∧ one_one_below n g ∧ onto_below n g def inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m) def swap (u v i : Nat) : Nat := if i = u then v else if i = v then u else i namespace Euler --For definitions specific to Euler's theorem def F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m def G (m a i : Nat) : Nat := (a * i) % m def Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i end Euler /- Section 7.1 -/ theorem dvd_mod_of_dvd_a_b {a b d : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ (a % b) := by set q : Nat := a / b have h3 : b * q + a % b = a := Nat.div_add_mod a b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 define --Goal : ∃ (c : Nat), a % b = d * c apply Exists.intro (j - k * q) show a % b = d * (j - k * q) from calc a % b _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm _ = a - b * q := by rw [h3] _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc] _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm done theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry #eval gcd 672 161 --Answer: 7 lemma gcd_base (a : Nat) : gcd a 0 = a := by rfl lemma gcd_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd a b = gcd b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1)) rfl done lemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b ≠ 0) : a % b < b := by have h1 : b > 0 := Nat.pos_of_ne_zero h show a % b < b from Nat.mod_lt a h1 done lemma dvd_self (n : Nat) : n ∣ n := by apply Exists.intro 1 ring done theorem gcd_dvd : ∀ (b a : Nat), (gcd a b) ∣ a ∧ (gcd a b) ∣ b := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd a b_1) ∣ a ∧ (gcd a b_1) ∣ b_1 fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_base] --Goal: a ∣ a ∧ a ∣ 0 apply And.intro (dvd_self a) define apply Exists.intro 0 rfl done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_nonzero a h1] --Goal : gcd b (a % b) ∣ a ∧ gcd b (a % b) ∣ b have h2 : a % b < b := mod_nonzero_lt a h1 have h3 : (gcd b (a % b)) ∣ b ∧ (gcd b (a % b)) ∣ (a % b) := ih (a % b) h2 b apply And.intro _ h3.left show (gcd b (a % b)) ∣ a from dvd_a_of_dvd_b_mod h3.left h3.right done done theorem gcd_dvd_left (a b : Nat) : (gcd a b) ∣ a := (gcd_dvd b a).left theorem gcd_dvd_right (a b : Nat) : (gcd a b) ∣ b := (gcd_dvd b a).right lemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl lemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c1 a b = gcd_c2 b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done lemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl lemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * ↑(a / b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done theorem gcd_lin_comb : ∀ (b a : Nat), (gcd_c1 a b) * ↑a + (gcd_c2 a b) * ↑b = ↑(gcd a b) := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd_c1 a b_1) * ↑a + (gcd_c2 a b_1) * ↑b_1 = ↑(gcd a b_1) fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base] --Goal : 1 * ↑a + 0 * ↑0 = ↑a ring done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1] --Goal : gcd_c2 b (a % b) * ↑a + -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * ↑(a / b)) * ↑b = -- ↑(gcd b (a % b)) set r : Nat := a % b set q : Nat := a / b set s : Int := gcd_c1 b r set t : Int := gcd_c2 b r --Goal : t * ↑a + (s - t * ↑q) * ↑b = ↑(gcd b r) have h2 : r < b := mod_nonzero_lt a h1 have h3 : s * ↑b + t * ↑r = ↑(gcd b r) := ih r h2 b have h4 : b * q + r = a := Nat.div_add_mod a b rewrite [←h3, ←h4] rewrite [Nat.cast_add, Nat.cast_mul] --Goal : t * (↑b * ↑q + ↑r) + (s - t * ↑q) * ↑b = s * ↑b + t * ↑r ring done done #eval gcd_c1 672 161 --Answer: 6 #eval gcd_c2 672 161 --Answer: -25 --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161 theorem Theorem_7_1_6 {d a b : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ gcd a b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑d ∣ ↑(gcd a b) set s : Int := gcd_c1 a b set t : Int := gcd_c2 a b have h3 : s * ↑a + t * ↑b = ↑(gcd a b) := gcd_lin_comb b a rewrite [←h3] --Goal : ↑d ∣ s * ↑a + t * ↑b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul] --Goal : ↑d ∣ s * (↑d * ↑j) + t * (↑d * ↑k) define apply Exists.intro (s * ↑j + t * ↑k) ring done /- Section 7.2 -/ theorem dvd_trans {a b c : Nat} (h1 : a ∣ b) (h2 : b ∣ c) : a ∣ c := by define at h1; define at h2; define obtain (m : Nat) (h3 : b = a * m) from h1 obtain (n : Nat) (h4 : c = b * n) from h2 rewrite [h3, mul_assoc] at h4 apply Exists.intro (m * n) show c = a * (m * n) from h4 done lemma exists_prime_factor : ∀ (n : Nat), 2 ≤ n → ∃ (p : Nat), prime_factor p n := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, 2 ≤ n_1 → ∃ (p : Nat), prime_factor p n_1 assume h1 : 2 ≤ n by_cases h2 : prime n · -- Case 1. h2 : prime n apply Exists.intro n define --Goal : prime n ∧ n ∣ n show prime n ∧ n ∣ n from And.intro h2 (dvd_self n) done · -- Case 2. h2 : ¬prime n define at h2 --h2 : ¬(2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n) demorgan at h2 disj_syll h2 h1 obtain (a : Nat) (h3 : ∃ (b : Nat), a * b = n ∧ a < n ∧ b < n) from h2 obtain (b : Nat) (h4 : a * b = n ∧ a < n ∧ b < n) from h3 have h5 : 2 ≤ a := by by_contra h6 have h7 : a ≤ 1 := by linarith have h8 : n ≤ b := calc n _ = a * b := h4.left.symm _ ≤ 1 * b := by rel [h7] _ = b := by ring linarith --n ≤ b contradicts b < n done have h6 : ∃ (p : Nat), prime_factor p a := ih a h4.right.left h5 obtain (p : Nat) (h7 : prime_factor p a) from h6 apply Exists.intro p define --Goal : prime p ∧ p ∣ n define at h7 --h7 : prime p ∧ p ∣ a apply And.intro h7.left have h8 : a ∣ n := by apply Exists.intro b show n = a * b from (h4.left).symm done show p ∣ n from dvd_trans h7.right h8 done done lemma exists_least_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q := by set S : Set Nat := {p : Nat | prime_factor p n} have h2 : ∃ (p : Nat), p ∈ S := exists_prime_factor n h show ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q from well_ord_princ S h2 done lemma all_prime_nil : all_prime [] := by define --Goal : ∀ p ∈ [], prime p fix p : Nat contrapos --Goal : ¬prime p → p ∉ [] assume h1 : ¬prime p show p ∉ [] from List.not_mem_nil p done lemma all_prime_cons (n : Nat) (L : List Nat) : all_prime (n :: L) ↔ prime n ∧ all_prime L := by apply Iff.intro · -- (→) assume h1 : all_prime (n :: L) --Goal : prime n ∧ all_prime L define at h1 --h1 : ∀ p ∈ n :: L, prime p apply And.intro (h1 n (List.mem_cons_self n L)) define --Goal : ∀ p ∈ L, prime p fix p : Nat assume h2 : p ∈ L show prime p from h1 p (List.mem_cons_of_mem n h2) done · -- (←) assume h1 : prime n ∧ all_prime L --Goal : all_prime (n :: l) define : all_prime L at h1 define fix p : Nat assume h2 : p ∈ n :: L rewrite [List.mem_cons] at h2 --h2 : p = n ∨ p ∈ L by_cases on h2 · -- Case 1. h2 : p = n rewrite [h2] show prime n from h1.left done · -- Case 2. h2 : p ∈ L show prime p from h1.right p h2 done done done lemma nondec_nil : nondec [] := by define --Goal : True trivial --trivial proves some obviously true statements, such as True done lemma nondec_cons (n : Nat) (L : List Nat) : nondec (n :: L) ↔ (∀ m ∈ L, n ≤ m) ∧ nondec L := by rfl lemma prod_nil : prod [] = 1 := by rfl lemma prod_cons : prod (n :: L) = n * (prod L) := by rfl lemma exists_cons_of_length_eq_succ {A : Type} {l : List A} {n : Nat} (h : l.length = n + 1) : ∃ (a : A) (L : List A), l = a :: L ∧ L.length = n := by have h1 : ¬l.length = 0 := by linarith rewrite [List.length_eq_zero] at h1 obtain (a : A) (h2 : ∃ (L : List A), l = a :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List A) (h3 : l = a :: L) from h2 apply Exists.intro a apply Exists.intro L apply And.intro h3 have h4 : (a :: L).length = L.length + 1 := List.length_cons a L rewrite [←h3, h] at h4 show L.length = n from (Nat.add_right_cancel h4).symm done lemma list_elt_dvd_prod_by_length (a : Nat) : ∀ (n : Nat), ∀ (l : List Nat), l.length = n → a ∈ l → a ∣ prod l := by by_induc · --Base Case fix l : List Nat assume h1 : l.length = 0 rewrite [List.length_eq_zero] at h1 --h1 : l = [] rewrite [h1] --Goal : a ∈ [] → a ∣ prod [] contrapos assume h2 : ¬a ∣ prod [] show a ∉ [] from List.not_mem_nil a done · -- Induction Step fix n : Nat assume ih : ∀ (l : List Nat), List.length l = n → a ∈ l → a ∣ prod l fix l : List Nat assume h1 : l.length = n + 1 --Goal : a ∈ l → a ∣ prod l obtain (b : Nat) (h2 : ∃ (L : List Nat), l = b :: L ∧ L.length = n) from exists_cons_of_length_eq_succ h1 obtain (L : List Nat) (h3 : l = b :: L ∧ L.length = n) from h2 have h4 : a ∈ L → a ∣ prod L := ih L h3.right assume h5 : a ∈ l rewrite [h3.left, prod_cons] --Goal : a ∣ b * prod L rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b ∨ a ∈ L by_cases on h5 · -- Case 1. h5 : a = b apply Exists.intro (prod L) rewrite [h5] rfl done · -- Case 2. h5 : a ∈ L have h6 : a ∣ prod L := h4 h5 have h7 : prod L ∣ b * prod L := by apply Exists.intro b ring done show a ∣ b * prod L from dvd_trans h6 h7 done done done lemma list_elt_dvd_prod {a : Nat} {l : List Nat} (h : a ∈ l) : a ∣ prod l := by set n : Nat := l.length have h1 : l.length = n := by rfl show a ∣ prod l from list_elt_dvd_prod_by_length a n l h1 h done lemma exists_prime_factorization : ∀ (n : Nat), n ≥ 1 → ∃ (l : List Nat), prime_factorization n l := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, n_1 ≥ 1 → ∃ (l : List Nat), prime_factorization n_1 l assume h1 : n ≥ 1 by_cases h2 : n = 1 · -- Case 1. h2 : n = 1 apply Exists.intro [] define apply And.intro · -- Proof of nondec_prime_list [] define show all_prime [] ∧ nondec [] from And.intro all_prime_nil nondec_nil done · -- Proof of prod [] = n rewrite [prod_nil, h2] rfl done done · -- Case 2. h2 : n ≠ 1 have h3 : n ≥ 2 := lt_of_le_of_ne' h1 h2 obtain (p : Nat) (h4 : prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q) from exists_least_prime_factor h3 have p_prime_factor : prime_factor p n := h4.left define at p_prime_factor have p_prime : prime p := p_prime_factor.left have p_dvd_n : p ∣ n := p_prime_factor.right have p_least : ∀ (q : Nat), prime_factor q n → p ≤ q := h4.right obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n have h5 : m ≠ 0 := by contradict h1 with h6 have h7 : n = 0 := calc n _ = p * m := n_eq_pm _ = p * 0 := by rw [h6] _ = 0 := by ring rewrite [h7] decide done have m_pos : 0 < m := Nat.pos_of_ne_zero h5 have m_lt_n : m < n := by define at p_prime show m < n from calc m _ < m + m := by linarith _ = 2 * m := by ring _ ≤ p * m := by rel [p_prime.left] _ = n := n_eq_pm.symm done obtain (L : List Nat) (h6 : prime_factorization m L) from ih m m_lt_n m_pos define at h6 have ndpl_L : nondec_prime_list L := h6.left define at ndpl_L apply Exists.intro (p :: L) define apply And.intro · -- Proof of nondec_prime_list (p :: L) define apply And.intro · -- Proof of all_prime (p :: L) rewrite [all_prime_cons] show prime p ∧ all_prime L from And.intro p_prime ndpl_L.left done · -- Proof of nondec (p :: L) rewrite [nondec_cons] apply And.intro _ ndpl_L.right fix q : Nat assume q_in_L : q ∈ L have h7 : q ∣ prod L := list_elt_dvd_prod q_in_L rewrite [h6.right] at h7 --h7 : q ∣ m have h8 : m ∣ n := by apply Exists.intro p rewrite [n_eq_pm] ring done have q_dvd_n : q ∣ n := dvd_trans h7 h8 have ap_L : all_prime L := ndpl_L.left define at ap_L have q_prime_factor : prime_factor q n := And.intro (ap_L q q_in_L) q_dvd_n show p ≤ q from p_least q q_prime_factor done done · -- Proof of prod (p :: L) = n rewrite [prod_cons, h6.right, n_eq_pm] rfl done done done theorem Theorem_7_2_2 {a b c : Nat} (h1 : c ∣ a * b) (h2 : rel_prime a c) : c ∣ b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑c ∣ ↑b define at h1; define at h2; define obtain (j : Nat) (h3 : a * b = c * j) from h1 set s : Int := gcd_c1 a c set t : Int := gcd_c2 a c have h4 : s * ↑a + t * ↑c = ↑(gcd a c) := gcd_lin_comb c a rewrite [h2, Nat.cast_one] at h4 --h4 : s * ↑a + t * ↑c = (1 : Int) apply Exists.intro (s * ↑j + t * ↑b) show ↑b = ↑c * (s * ↑j + t * ↑b) from calc ↑b _ = (1 : Int) * ↑b := (one_mul _).symm _ = (s * ↑a + t * ↑c) * ↑b := by rw [h4] _ = s * (↑a * ↑b) + t * ↑c * ↑b := by ring _ = s * (↑c * ↑j) + t * ↑c * ↑b := by rw [←Nat.cast_mul a b, h3, Nat.cast_mul c j] _ = ↑c * (s * ↑j + t * ↑b) := by ring done lemma le_nonzero_prod_left {a b : Nat} (h : a * b ≠ 0) : a ≤ a * b := by have h1 : b ≠ 0 := by contradict h with h1 rewrite [h1] ring done have h2 : 1 ≤ b := Nat.pos_of_ne_zero h1 show a ≤ a * b from calc a = a * 1 := (mul_one a).symm _ ≤ a * b := by rel [h2] done lemma le_nonzero_prod_right {a b : Nat} (h : a * b ≠ 0) : b ≤ a * b := by rewrite [mul_comm] rewrite [mul_comm] at h show b ≤ b * a from le_nonzero_prod_left h done lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry lemma rel_prime_of_prime_not_dvd {a p : Nat} (h1 : prime p) (h2 : ¬p ∣ a) : rel_prime a p := by have h3 : gcd a p ∣ a := gcd_dvd_left a p have h4 : gcd a p ∣ p := gcd_dvd_right a p have h5 : gcd a p = 1 ∨ gcd a p = p := dvd_prime h1 h4 have h6 : gcd a p ≠ p := by contradict h2 with h6 rewrite [h6] at h3 show p ∣ a from h3 done disj_syll h5 h6 show rel_prime a p from h5 done theorem Theorem_7_2_3 {a b p : Nat} (h1 : prime p) (h2 : p ∣ a * b) : p ∣ a ∨ p ∣ b := by or_right with h3 have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3 show p ∣ b from Theorem_7_2_2 h2 h4 done lemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a ≥ 1 := by have h1 : a ≠ 0 := by by_contra h1 rewrite [h1] at h contradict h linarith done show a ≥ 1 from Nat.pos_of_ne_zero h1 done lemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by have h1 : a ≥ 1 := ge_one_of_prod_one h have h2 : a * b ≠ 0 := by linarith have h3 : a ≤ a * b := le_nonzero_prod_left h2 rewrite [h] at h3 show a = 1 from Nat.le_antisymm h3 h1 done lemma eq_one_of_dvd_one {n : Nat} (h : n ∣ 1) : n = 1 := by obtain (j : Nat) (h1 : 1 = n * j) from h show n = 1 from eq_one_of_prod_one h1.symm done lemma prime_not_one {p : Nat} (h : prime p) : p ≠ 1 := by define at h linarith done theorem Theorem_7_2_4 {p : Nat} (h1 : prime p) : ∀ (l : List Nat), p ∣ prod l → ∃ a ∈ l, p ∣ a := by apply List.rec · -- Base Case. Goal : p ∣ prod [] → ∃ a ∈ [], p ∣ a rewrite [prod_nil] assume h2 : p ∣ 1 show ∃ a ∈ [], p ∣ a from absurd (eq_one_of_dvd_one h2) (prime_not_one h1) done · -- Induction Step fix b : Nat fix L : List Nat assume ih : p ∣ prod L → ∃ a ∈ L, p ∣ a --Goal : p ∣ prod (b :: L) → ∃ a ∈ b :: L, p ∣ a assume h2 : p ∣ prod (b :: L) rewrite [prod_cons] at h2 have h3 : p ∣ b ∨ p ∣ prod L := Theorem_7_2_3 h1 h2 by_cases on h3 · -- Case 1. h3 : p ∣ b apply Exists.intro b show b ∈ b :: L ∧ p ∣ b from And.intro (List.mem_cons_self b L) h3 done · -- Case 2. h3 : p ∣ prod L obtain (a : Nat) (h4 : a ∈ L ∧ p ∣ a) from ih h3 apply Exists.intro a show a ∈ b :: L ∧ p ∣ a from And.intro (List.mem_cons_of_mem b h4.left) h4.right done done done lemma prime_in_list {p : Nat} {l : List Nat} (h1 : prime p) (h2 : all_prime l) (h3 : p ∣ prod l) : p ∈ l := by obtain (a : Nat) (h4 : a ∈ l ∧ p ∣ a) from Theorem_7_2_4 h1 l h3 define at h2 have h5 : prime a := h2 a h4.left have h6 : p = 1 ∨ p = a := dvd_prime h5 h4.right disj_syll h6 (prime_not_one h1) rewrite [h6] show a ∈ l from h4.left done lemma first_le_first {p q : Nat} {l m : List Nat} (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m)) (h3 : prod (p :: l) = prod (q :: m)) : p ≤ q := by define at h1; define at h2 have h4 : q ∣ prod (p :: l) := by define apply Exists.intro (prod m) rewrite [←prod_cons] show prod (p :: l) = prod (q :: m) from h3 done have h5 : all_prime (q :: m) := h2.left rewrite [all_prime_cons] at h5 have h6 : q ∈ p :: l := prime_in_list h5.left h1.left h4 have h7 : nondec (p :: l) := h1.right rewrite [nondec_cons] at h7 rewrite [List.mem_cons] at h6 by_cases on h6 · -- Case 1. h6 : q = p linarith done · -- Case 2. h6 : q ∈ l have h8 : ∀ m ∈ l, p ≤ m := h7.left show p ≤ q from h8 q h6 done done lemma nondec_prime_list_tail {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by define at h define rewrite [all_prime_cons, nondec_cons] at h show all_prime l ∧ nondec l from And.intro h.left.right h.right.right done lemma cons_prod_not_one {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : prod (p :: l) ≠ 1 := by define at h have h1 : all_prime (p :: l) := h.left rewrite [all_prime_cons] at h1 rewrite [prod_cons] by_contra h2 show False from (prime_not_one h1.left) (eq_one_of_prod_one h2) done lemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) : l = [] ↔ prod l = 1 := by apply Iff.intro · -- (→) assume h1 : l = [] rewrite [h1] show prod [] = 1 from prod_nil done · -- (←) contrapos assume h1 : ¬l = [] obtain (p : Nat) (h2 : ∃ (L : List Nat), l = p :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List Nat) (h3 : l = p :: L) from h2 rewrite [h3] at h rewrite [h3] show ¬prod (p :: L) = 1 from cons_prod_not_one h done done lemma prime_pos {p : Nat} (h : prime p) : p > 0 := by define at h linarith done theorem Theorem_7_2_5 : ∀ (l1 l2 : List Nat), nondec_prime_list l1 → nondec_prime_list l2 → prod l1 = prod l2 → l1 = l2 := by apply List.rec · -- Base Case. Goal : ∀ (l2 : List Nat), nondec_prime_list [] → -- nondec_prime_list l2 → prod [] = prod l2 → [] = l2 fix l2 : List Nat assume h1 : nondec_prime_list [] assume h2 : nondec_prime_list l2 assume h3 : prod [] = prod l2 rewrite [prod_nil, eq_comm, ←list_nil_iff_prod_one h2] at h3 show [] = l2 from h3.symm done · -- Induction Step fix p : Nat fix L1 : List Nat assume ih : ∀ (L2 : List Nat), nondec_prime_list L1 → nondec_prime_list L2 → prod L1 = prod L2 → L1 = L2 -- Goal : ∀ (l2 : List Nat), nondec_prime_list (p :: L1) → -- nondec_prime_list l2 → prod (p :: L1) = prod l2 → p :: L1 = l2 fix l2 : List Nat assume h1 : nondec_prime_list (p :: L1) assume h2 : nondec_prime_list l2 assume h3 : prod (p :: L1) = prod l2 have h4 : ¬prod (p :: L1) = 1 := cons_prod_not_one h1 rewrite [h3, ←list_nil_iff_prod_one h2] at h4 obtain (q : Nat) (h5 : ∃ (L : List Nat), l2 = q :: L) from List.exists_cons_of_ne_nil h4 obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5 rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2) rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2) have h7 : p ≤ q := first_le_first h1 h2 h3 have h8 : q ≤ p := first_le_first h2 h1 h3.symm have h9 : p = q := by linarith rewrite [h9, prod_cons, prod_cons] at h3 --h3 : q * prod L1 = q * prod L2 have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1 have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2 define at h2 have h12 : all_prime (q :: L2) := h2.left rewrite [all_prime_cons] at h12 have h13 : q > 0 := prime_pos h12.left have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3 have h15 : L1 = L2 := ih L2 h10 h11 h14 rewrite [h6, h9, h15] rfl done done theorem fund_thm_arith (n : Nat) (h : n ≥ 1) : ∃! (l : List Nat), prime_factorization n l := by exists_unique · -- Existence show ∃ (l : List Nat), prime_factorization n l from exists_prime_factorization n h done · -- Uniqueness fix l1 : List Nat; fix l2 : List Nat assume h1 : prime_factorization n l1 assume h2 : prime_factorization n l2 define at h1; define at h2 have h3 : prod l1 = n := h1.right rewrite [←h2.right] at h3 show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3 done done /- Section 7.3 -/ theorem congr_refl (m : Nat) : ∀ (a : Int), a ≡ a (MOD m) := by fix a : Int define --Goal : ∃ (c : Int), a - a = ↑m * c apply Exists.intro 0 ring done theorem congr_symm {m : Nat} : ∀ {a b : Int}, a ≡ b (MOD m) → b ≡ a (MOD m) := by fix a : Int; fix b : Int assume h1 : a ≡ b (MOD m) define at h1 --h1 : ∃ (c : Int), a - b = ↑m * c define --Goal : ∃ (c : Int), b - a = ↑m * c obtain (c : Int) (h2 : a - b = m * c) from h1 apply Exists.intro (-c) show b - a = m * (-c) from calc b - a _ = -(a - b) := by ring _ = -(m * c) := by rw [h2] _ = m * (-c) := by ring done theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry /- Fundamental properties of congruence classes -/ lemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) : X = Y ↔ X.val = Y.val := Fin.ext_iff lemma val_nat_eq_mod (n k : Nat) : ([k]_(n + 1)).val = k % (n + 1) := by rfl lemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl theorem cc_rep {m : Nat} (X : ZMod m) : ∃ (a : Int), X = [a]_m := match m with | 0 => by apply Exists.intro X rfl done | n + 1 => by apply Exists.intro ↑(X.val) have h1 : X.val < n + 1 := Fin.prop X rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1] rfl done theorem add_class (m : Nat) (a b : Int) : [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm theorem mul_class (m : Nat) (a b : Int) : [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm lemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ [a - b]_m = [0]_m := by apply Iff.intro · -- (→) assume h1 : [a]_m = [b]_m have h2 : a - b = a + (-b) := by ring have h3 : b + (-b) = 0 := by ring show [a - b]_m = [0]_m from calc [a - b]_m _ = [a + (-b)]_m := by rw [h2] _ = [a]_m + [-b]_m := by rw [add_class] _ = [b]_m + [-b]_m := by rw [h1] _ = [b + -b]_m := by rw [add_class] _ = [0]_m := by rw [h3] done · -- (←) assume h1 : [a - b]_m = [0]_m have h2 : b + (a - b) = a := by ring have h3 : b + 0 = b := by ring show [a]_m = [b]_m from calc [a]_m _ = [b + (a - b)]_m := by rw [h2] _ = [b]_m + [a - b]_m := by rw [add_class] _ = [b]_m + [0]_m := by rw [h1] _ = [b + 0]_m := by rw [add_class] _ = [b]_m := by rw [h3] done done lemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) : [a]_m = [0]_m → [-a]_m = [0]_m := by assume h1 : [a]_m = [0]_m have h2 : 0 + (-a) = -a := by ring have h3 : a + (-a) = 0 := by ring show [-a]_m = [0]_m from calc [-a]_m _ = [0 + (-a)]_m := by rw [h2] _ = [0]_m + [-a]_m := by rw [add_class] _ = [a]_m + [-a]_m := by rw [h1] _ = [a + (-a)]_m := by rw [add_class] _ = [0]_m := by rw [h3] done lemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) : [-a]_m = [0]_m ↔ [a]_m = [0]_m := by apply Iff.intro _ (cc_neg_zero_of_cc_zero m a) assume h1 : [-a]_m = [0]_m have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1 have h3 : -(-a) = a := by ring rewrite [h3] at h2 show [a]_m = [0]_m from h2 done lemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl lemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m ↔ m ∣ k := match m with | 0 => by have h : (0 : Int) = (↑(0 : Nat) : Int) := by rfl rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj] apply Iff.intro · -- (→) assume h1 : k = 0 rewrite [h1] show 0 ∣ 0 from dvd_self 0 done · -- (←) assume h1 : 0 ∣ k obtain (c : Nat) (h2 : k = 0 * c) from h1 rewrite [h2] ring done done | n + 1 => by rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero] show k % (n + 1) = 0 ↔ n + 1 ∣ k from (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm done lemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m ↔ ↑m ∣ a := by obtain (k : Nat) (h1 : a = ↑k ∨ a = -↑k) from Int.eq_nat_or_neg a by_cases on h1 · -- Case 1. h1: a = ↑k rewrite [h1, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done · -- Case 2. h1: a = -↑k rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done done theorem cc_eq_iff_congr (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ a ≡ b (MOD m) := calc [a]_m = [b]_m _ ↔ [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b _ ↔ ↑m ∣ (a - b) := cc_zero_iff_dvd m (a - b) _ ↔ a ≡ b (MOD m) := by rfl /- End of fundamental properties of congruence classes -/ lemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m := by have h1 : (↑m : Int) ≠ 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m) show 0 ≤ a % m from Int.emod_nonneg a h1 done lemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m) have h2 : (↑m : Int) > 0 := (Nat.cast_pos).rtl h1 show a % m < m from Int.emod_lt_of_pos a h2 done lemma congr_mod_mod (m : Nat) (a : Int) : a ≡ a % m (MOD m) := by define have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m apply Exists.intro (a / m) show a - a % m = m * (a / m) from calc a - (a % m) _ = m * (a / m) + a % m - a % m := by rw [h1] _ = m * (a / m) := by ring done lemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) := And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a)) theorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) : ∃! (r : Int), 0 ≤ r ∧ r < m ∧ a ≡ r (MOD m) := by exists_unique · -- Existence apply Exists.intro (a % m) show 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) from mod_cmpl_res m a done · -- Uniqueness fix r1 : Int; fix r2 : Int assume h1 : 0 ≤ r1 ∧ r1 < m ∧ a ≡ r1 (MOD m) assume h2 : 0 ≤ r2 ∧ r2 < m ∧ a ≡ r2 (MOD m) have h3 : r1 ≡ r2 (MOD m) := congr_trans (congr_symm h1.right.right) h2.right.right obtain (d : Int) (h4 : r1 - r2 = m * d) from h3 have h5 : r1 - r2 < m * 1 := by linarith have h6 : m * (-1) < r1 - r2 := by linarith rewrite [h4] at h5 --h5 : m * d < m * 1 rewrite [h4] at h6 --h6 : m * -1 < m * d have h7 : (↑m : Int) ≥ 0 := Nat.cast_nonneg m have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7 have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7 have h10 : d = 0 := by linarith show r1 = r2 from calc r1 _ = r1 - r2 + r2 := by ring _ = m * 0 + r2 := by rw [h4, h10] _ = r2 := by ring done done lemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m := (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a) theorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y rewrite [h1, h2] have h3 : a + b = b + a := by ring show [a]_m + [b]_m = [b]_m + [a]_m from calc [a]_m + [b]_m _ = [a + b]_m := add_class m a b _ = [b + a]_m := by rw [h3] _ = [b]_m + [a]_m := (add_class m b a).symm done theorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X rewrite [h1] have h2 : a * 1 = a := by ring show [a]_m * [1]_m = [a]_m from calc [a]_m * [1]_m _ = [a * 1]_m := mul_class m a 1 _ = [a]_m := by rw [h2] done theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry lemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) : [a]_m * [gcd_c2 m a]_m = [1]_m := by set s : Int := gcd_c1 m a have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m define at h1 rewrite [h1, Nat.cast_one] at h2 --h2 : s * ↑m + gcd_c2 m a * ↑a = 1 rewrite [mul_class, cc_eq_iff_congr] define --Goal : ∃ (c : Int), ↑a * gcd_c2 m a - 1 = ↑m * c apply Exists.intro (-s) show a * (gcd_c2 m a) - 1 = m * (-s) from calc a * (gcd_c2 m a) - 1 _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring _ = 1 + m * (-s) - 1 := by rw [h2] _ = m * (-s) := by ring done theorem Theorem_7_3_7 (m a : Nat) : invertible [a]_m ↔ rel_prime m a := by apply Iff.intro · -- (→) assume h1 : invertible [a]_m define at h1 obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1 obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y rewrite [h3, mul_class, cc_eq_iff_congr] at h2 define at h2 obtain (c : Int) (h4 : a * b - 1 = m * c) from h2 rewrite [Exercise_7_2_6] --Goal : ∃ (s t : Int), s * ↑m + t * ↑a = 1 apply Exists.intro (-c) apply Exists.intro b show (-c) * m + b * a = 1 from calc (-c) * m + b * a _ = (-c) * m + (a * b - 1) + 1 := by ring _ = (-c) * m + m * c + 1 := by rw [h4] _ = 1 := by ring done · -- (←) assume h1 : rel_prime m a define show ∃ (Y : ZMod m), [a]_m * Y = [1]_m from Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1) done done /- Section 7.4 -/ section Euler open Euler lemma num_rp_below_base {m : Nat} : num_rp_below m 0 = 0 := by rfl lemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) : num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by have h1 : num_rp_below m (j + 1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : gcd m j = 1 rewrite [if_pos h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1 show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1 done lemma num_rp_below_step_not_rp {m j : Nat} (h : ¬rel_prime m j) : num_rp_below m (j + 1) = num_rp_below m j := by have h1 : num_rp_below m (j +1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : ¬gcd m j = 1 rewrite [if_neg h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j show num_rp_below m (j + 1) = num_rp_below m j from h1 done lemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl #eval phi 10 --Answer: 4 lemma prod_inv_iff_inv {m : Nat} {X : ZMod m} (h1 : invertible X) (Y : ZMod m) : invertible (X * Y) ↔ invertible Y := by apply Iff.intro · -- (→) assume h2 : invertible (X * Y) obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2 apply Exists.intro (X * Z) rewrite [←h3] --Goal : Y * (X * Z) = X * Y * Z ring --Note that ring can do algebra in ZMod m done · -- (←) assume h2 : invertible Y obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1 obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2 apply Exists.intro (Xi * Yi) show (X * Y) * (Xi * Yi) = [1]_m from calc X * Y * (Xi * Yi) _ = (X * Xi) * (Y * Yi) := by ring _ = [1]_m * [1]_m := by rw [h3, h4] _ = [1]_m := Theorem_7_3_6_7 [1]_m done done lemma F_rp_def {m i : Nat} (h : rel_prime m i) : F m i = [i]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h --h : gcd m i = 1 rewrite [if_pos h] at h1 show F m i = [i]_m from h1 done lemma F_not_rp_def {m i : Nat} (h : ¬rel_prime m i) : F m i = [1]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h rewrite [h1, if_neg h] rfl done lemma prod_seq_base {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 0 k f = [1]_m := by rfl lemma prod_seq_step {m : Nat} (n k : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl lemma prod_seq_zero_step {m : Nat} (n : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by rewrite [prod_seq_step, zero_add] rfl done lemma prod_one {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 1 k f = f k := by rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7] rfl done lemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl lemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m := calc [G m a i]_m _ = [(a * i) % m]_m := by rfl _ = [a * i]_m := (cc_eq_mod m (a * i)).symm _ = [a]_m * [i]_m := (mul_class m a i).symm lemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) : rel_prime m (G m a i) ↔ rel_prime m i := by have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1 show rel_prime m (G m a i) ↔ rel_prime m i from calc rel_prime m (G m a i) _ ↔ invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm _ ↔ invertible ([a]_m * [i]_m) := by rw [cc_G] _ ↔ invertible [i]_m := prod_inv_iff_inv h2 ([i]_m) _ ↔ rel_prime m i := Theorem_7_3_7 m i done lemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) : F m (G m a i) = [a]_m * F m i := by have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2 show F m (G m a i) = [a]_m * F m i from calc F m (G m a i) _ = [G m a i]_m := F_rp_def h3 _ = [a]_m * [i]_m := cc_G m a i _ = [a]_m * F m i := by rw [F_rp_def h2] done lemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : ¬rel_prime m i) : F m (G m a i) = [1]_m := by rewrite [←G_rp_iff h1 i] at h2 show F m (G m a i) = [1]_m from F_not_rp_def h2 done lemma FG_prod {m a : Nat} (h1 : rel_prime m a) : ∀ (k : Nat), prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by by_induc · -- Base Case show prod_seq 0 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from calc prod_seq 0 0 ((F m) ∘ (G m a)) _ = [1]_m := prod_seq_base _ _ _ = [a]_m ^ 0 * [1]_m := by ring _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by rw [num_rp_below_base, prod_seq_base] done · -- Induction Step fix k : Nat assume ih : prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) by_cases h2 : rel_prime m k · -- Case 1. h2 : rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([a]_m * F m k) := by rw [FG_rp h1 h2] _ = [a]_m ^ ((num_rp_below m k) + 1) * ((prod_seq k 0 (F m)) * F m k) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_rp h2, prod_seq_zero_step] done · -- Case 2. h2 : ¬rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([1]_m) := by rw [FG_not_rp h1 h2] _ = [a]_m ^ (num_rp_below m k) * (prod_seq k 0 (F m) * ([1]_m)) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_not_rp h2, prod_seq_zero_step, F_not_rp_def h2] done done done lemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by define --Goal : ∀ i < m, G m a i < m fix i : Nat assume h1 : i < m rewrite [G_def] --Goal : a * i % m < m show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m) done lemma left_inv_one_one_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g' (g i) = i) : one_one_below n g := sorry lemma right_inv_onto_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g (g' i) = i) (h2 : maps_below n g') : onto_below n g := by define at h2; define fix k : Nat assume h3 : k < n apply Exists.intro (g' k) show g' k < n ∧ g (g' k) = k from And.intro (h2 k h3) (h1 k h3) done lemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m * [inv_mod m a]_m = [1]_m := by have h2 : 0 ≤ (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a) show [a]_m * [inv_mod m a]_m = [1]_m from calc [a]_m * [inv_mod m a]_m _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2] _ = [a]_m * [gcd_c2 m a]_m := by rw [←cc_eq_mod] _ = [1]_m := gcd_c2_inv h1 done lemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m := calc a * (b % m) % m = a % m * (b % m % m) % m := Nat.mul_mod _ _ _ _ = a % m * (b % m) % m := by rw [Nat.mod_mod] _ = a * b % m := (Nat.mul_mod _ _ _).symm lemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm] rfl done theorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] : ↑a ≡ ↑b (MOD m) ↔ a % m = b % m := sorry lemma mul_inv_mod_cancel {m a i : Nat} [NeZero m] (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1 rewrite [mul_class, cc_eq_iff_congr, ←Nat.cast_mul, ←Nat.cast_one, congr_iff_mod_eq_Nat] at h3 show a * inv_mod m a * i % m = i from calc a * (inv_mod m a) * i % m _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = 1 % m * i % m := by rw [h3] _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = i % m := by rw [one_mul] _ = i := Nat.mod_eq_of_lt h2 done lemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl lemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, G m a (Ginv m a i) = i := by fix i : Nat assume h2 : i < m show G m a (Ginv m a i) = i from calc G m a (Ginv m a i) _ = a * ((inv_mod m a * i) % m) % m := by rfl _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, Ginv m a (G m a i) = i := by fix i : Nat assume h2 : i < m show Ginv m a (G m a i) = i from calc Ginv m a (G m a i) _ = inv_mod m a * ((a * i) % m) % m := by rfl _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc, mul_comm (inv_mod m a)] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_maps_below (m a : Nat) [NeZero m] : maps_below m (Ginv m a) := G_maps_below m (inv_mod m a) lemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : one_one_below m (G m a) := left_inv_one_one_below (Ginv_left_inv h1) lemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : onto_below m (G m a) := right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a) lemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : perm_below m (G m a) := And.intro (G_maps_below m a) (And.intro (G_one_one_below h1) (G_onto_below h1)) --Permuting a product of congruence classes doesn't change product lemma swap_fst (u v : Nat) : swap u v u = v := by define : swap u v u --Goal : (if u = u then v else if u = v then u else u) = v have h : u = u := by rfl rewrite [if_pos h] rfl done lemma swap_snd (u v : Nat) : swap u v v = u := by define : swap u v v by_cases h1 : v = u · -- Case 1. h1 : v = u rewrite [if_pos h1] show v = u from h1 done · -- Case 2. h1 : v ≠ u rewrite [if_neg h1] have h2 : v = v := by rfl rewrite [if_pos h2] rfl done done lemma swap_other {u v i : Nat} (h1 : i ≠ u) (h2 : i ≠ v) : swap u v i = i := by define : swap u v i rewrite [if_neg h1, if_neg h2] rfl done lemma swap_values (u v i : Nat) : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := by by_cases h1 : i = u · -- Case 1. h1 : i = u apply Or.inl rewrite [h1] show swap u v u = v from swap_fst u v done · -- Case 2. h1 : i ≠ u apply Or.inr by_cases h2 : i = v · -- Case 2.1. h2 : i = v apply Or.inl rewrite [h2] show swap u v v = u from swap_snd u v done · -- Case 2.2. h2 : i ≠ v apply Or.inr show swap u v i = i from swap_other h1 h2 done done done lemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by define fix i : Nat assume h3 : i < n have h4 : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := swap_values u v i by_cases on h4 · -- Case 1. h4 : swap u v i = v rewrite [h4] show v < n from h2 done · -- Case 2. by_cases on h4 · -- Case 2.1. h4 : swap u v i = u rewrite [h4] show u < n from h1 done · -- Case 2.2. h4 : swap u v i = i rewrite [h4] show i < n from h3 done done done lemma swap_swap (u v n : Nat) : ∀ i < n, swap u v (swap u v i) = i := by fix i : Nat assume h : i < n by_cases h1 : i = u · -- Case 1. h1 : i = u rewrite [h1, swap_fst, swap_snd] rfl done · -- Case 2. h1 : i ≠ u by_cases h2 : i = v · -- Case 2.1. h2 : i = v rewrite [h2, swap_snd, swap_fst] rfl done · -- Case 2.2. h2 : i ≠ v rewrite [swap_other h1 h2, swap_other h1 h2] rfl done done done lemma swap_one_one_below (u v n) : one_one_below n (swap u v) := left_inv_one_one_below (swap_swap u v n) lemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) := right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2) lemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) := And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2)) lemma comp_perm_below {n : Nat} {f g : Nat → Nat} (h1 : perm_below n f) (h2 : perm_below n g) : perm_below n (f ∘ g) := sorry lemma trivial_swap (u : Nat) : swap u u = id := by apply funext fix x : Nat by_cases h1 : x = u · -- Case 1. h1 : x = u rewrite [h1, swap_fst] rfl done · -- Case 2. h1 : x ≠ u rewrite [swap_other h1 h1] rfl done done lemma prod_eq_fun {m : Nat} (f g : Nat → ZMod m) (k : Nat) : ∀ (n : Nat), (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g := by by_induc · -- Base Case assume h : (∀ i < 0, f (k + i) = g (k + i)) rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g assume h1 : ∀ i < n + 1, f (k + i) = g (k + i) have h2 : ∀ i < n, f (k + i) = g (k + i) := by fix i : Nat assume h2 : i < n have h3 : i < n + 1 := by linarith show f (k + i) = g (k + i) from h1 i h3 done have h3 : prod_seq n k f = prod_seq n k g := ih h2 have h4 : n < n + 1 := Nat.lt_succ_self n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4] rfl done done lemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := by have h2 : ∀ (i : Nat), i < u → (f ∘ swap u n) (0 + i) = f (0 + i) := by fix i : Nat assume h2 : i < u have h3 : 0 + i ≠ u := by linarith have h4 : 0 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl done show prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f from prod_eq_fun (f ∘ swap u n) f 0 u h2 done lemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := by have h2 : ∀ i < j, (f ∘ swap u n) (u + 1 + i) = f (u + 1 + i) := by fix i : Nat assume h2 : i < j have h3 : u + 1 + i ≠ u := by linarith have h4 : u + 1 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl show prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f from prod_eq_fun (f ∘ swap u n) f (u + 1) j h2 done lemma break_prod {m : Nat} (n : Nat) (f : Nat → ZMod m) : ∀ (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by by_induc · -- Base Case have h : n + 0 = n := by rfl rewrite [prod_seq_base, h, Theorem_7_3_6_7] rfl done · -- Induction Step fix j : Nat assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f rewrite [←add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc] rfl done done lemma break_prod_twice {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f := break_prod n f 1 rewrite [prod_one] at h2 have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f := break_prod (u + 1) f j rewrite [←h1] at h3 have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f := break_prod u f 1 rewrite [prod_one] at h4 rewrite [h3, h4] at h2 show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2 done lemma swap_prod_eq_prod {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f := by by_cases h2 : u = n · -- Case 1. h2 : u = n rewrite [h2, trivial_swap n] --Goal : prod_seq (n + 1) 0 (f ∘ id) = prod_seq (n + 1) 0 f rfl done · -- Case 2. h2 : ¬u = n have h3 : u + 1 ≤ n := Nat.lt_of_le_of_ne h1 h2 obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3 have break_f : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := break_prod_twice f h4 have break_fs : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_prod_twice (f ∘ swap u n) h4 have f_eq_fs_below : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := swap_prod_eq_prod_below f h1 have f_eq_fs_btwn : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4 show prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f from calc prod_seq (n + 1) 0 (f ∘ swap u n) _ = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_fs _ = prod_seq u 0 f * (f ∘ swap u n) u * prod_seq j (u + 1) f * (f ∘ swap u n) n := by rw [f_eq_fs_below, f_eq_fs_btwn] _ = prod_seq u 0 f * f (swap u n u) * prod_seq j (u + 1) f * f (swap u n n) := by rfl _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by rw [swap_fst, swap_snd] _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring _ = prod_seq (n + 1) 0 f := break_f.symm done done lemma perm_below_fixed {n : Nat} {g : Nat → Nat} (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry lemma perm_prod {m : Nat} (f : Nat → ZMod m) : ∀ (n : Nat), ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) := by by_induc · -- Base Case fix g : Nat → Nat assume h1 : perm_below 0 g rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) fix g : Nat → Nat assume g_pb : perm_below (n + 1) g define at g_pb have g_ob : onto_below (n + 1) g := g_pb.right.right define at g_ob have h1 : n < n + 1 := by linarith obtain (u : Nat) (h2 : u < n + 1 ∧ g u = n) from g_ob n h1 have s_pb : perm_below (n + 1) (swap u n) := swap_perm_below h2.left h1 have gs_pb_n1 : perm_below (n + 1) (g ∘ swap u n) := comp_perm_below g_pb s_pb have gs_fix_n : (g ∘ swap u n) n = n := calc (g ∘ swap u n) n _ = g (swap u n n) := by rfl _ = g u := by rw [swap_snd] _ = n := h2.right have gs_pb_n : perm_below n (g ∘ swap u n) := perm_below_fixed gs_pb_n1 gs_fix_n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f ∘ (g ∘ swap u n)) := ih (g ∘ swap u n) gs_pb_n have h3 : u ≤ n := by linarith show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f ∘ g) from calc prod_seq (n + 1) 0 f _ = prod_seq n 0 f * f n := prod_seq_zero_step n f _ = prod_seq n 0 (f ∘ (g ∘ swap u n)) * f ((g ∘ swap u n) n) := by rw [gs_prod, gs_fix_n] _ = prod_seq n 0 (f ∘ g ∘ swap u n) * (f ∘ g ∘ swap u n) n := by rfl _ = prod_seq (n + 1) 0 (f ∘ g ∘ swap u n) := (prod_seq_zero_step n (f ∘ g ∘ swap u n)).symm _ = prod_seq (n + 1) 0 ((f ∘ g) ∘ swap u n) := by rfl _ = prod_seq (n + 1) 0 (f ∘ g) := swap_prod_eq_prod (f ∘ g) h3 done done lemma F_invertible (m i : Nat) : invertible (F m i) := by by_cases h : rel_prime m i · -- Case 1. h : rel_prime m i rewrite [F_rp_def h] show invertible [i]_m from (Theorem_7_3_7 m i).rtl h done · -- Case 2. h : ¬rel_prime m i rewrite [F_not_rp_def h] apply Exists.intro [1]_m show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m done done lemma Fprod_invertible (m : Nat) : ∀ (k : Nat), invertible (prod_seq k 0 (F m)) := by by_induc · -- Base Case apply Exists.intro [1]_m show prod_seq 0 0 (F m) * [1]_m = [1]_m from calc prod_seq 0 0 (F m) * [1]_m _ = [1]_m * [1]_m := by rw [prod_seq_base] _ = [1]_m := Theorem_7_3_6_7 ([1]_m) done · -- Induction Step fix k : Nat assume ih : invertible (prod_seq k 0 (F m)) rewrite [prod_seq_zero_step] show invertible (prod_seq k 0 (F m) * (F m k)) from (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k) done done theorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m ^ (phi m) = [1]_m := by have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2 show [a]_m ^ (phi m) = [1]_m from calc [a]_m ^ (phi m) _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3] _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring _ = prod_seq m 0 (F m ∘ G m a) * Y := by rw [FG_prod h1 m, phi_def] _ = prod_seq m 0 (F m) * Y := by rw [perm_prod (F m) m (G m a) (G_perm_below h1)] _ = [1]_m := by rw [h3] done lemma Exercise_7_4_5_Int (m : Nat) (a : Int) : ∀ (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry lemma Exercise_7_4_5_Nat (m a n : Nat) : [a]_m ^ n = [a ^ n]_m := by rewrite [Exercise_7_4_5_Int] rfl done theorem Euler's_theorem {m a : Nat} [NeZero m] (h1 : rel_prime m a) : a ^ (phi m) ≡ 1 (MOD m) := by have h2 : [a]_m ^ (phi m) = [1]_m := Theorem_7_4_2 h1 rewrite [Exercise_7_4_5_Nat m a (phi m)] at h2 --h2 : [a ^ phi m]_m = [1]_m show a ^ (phi m) ≡ 1 (MOD m) from (cc_eq_iff_congr _ _ _).ltr h2 done #eval gcd 10 7 --Answer: 1. So 10 and 7 are relatively prime #eval 7 ^ phi 10 --Answer: 2401, which is congruent to 1 mod 10. end Euler /- Section 7.5 -/ lemma num_rp_prime {p : Nat} (h1 : prime p) : ∀ k < p, num_rp_below p (k + 1) = k := sorry lemma phi_prime {p : Nat} (h1 : prime p) : phi p = p - 1 := by have h2 : 1 ≤ p := prime_pos h1 have h3 : p - 1 + 1 = p := Nat.sub_add_cancel h2 have h4 : p - 1 < p := by linarith have h5 : num_rp_below p (p - 1 + 1) = p - 1 := num_rp_prime h1 (p - 1) h4 rewrite [h3] at h5 show phi p = p - 1 from h5 done theorem Theorem_7_2_2_Int {a c : Nat} {b : Int} (h1 : ↑c ∣ ↑a * b) (h2 : rel_prime a c) : ↑c ∣ b := by rewrite [Int.natCast_dvd, Int.natAbs_mul, Int.natAbs_ofNat] at h1 --h1 : c ∣ a * Int.natAbs b rewrite [Int.natCast_dvd] --Goal : c ∣ Int.natAbs b show c ∣ Int.natAbs b from Theorem_7_2_2 h1 h2 done lemma Lemma_7_4_5 {m n : Nat} (a b : Int) (h1 : rel_prime m n) : a ≡ b (MOD m * n) ↔ a ≡ b (MOD m) ∧ a ≡ b (MOD n) := by apply Iff.intro · -- (→) assume h2 : a ≡ b (MOD m * n) obtain (j : Int) (h3 : a - b = (m * n) * j) from h2 apply And.intro · -- Proof of a ≡ b (MOD m) apply Exists.intro (n * j) show a - b = m * (n * j) from calc a - b _ = m * n * j := h3 _ = m * (n * j) := by ring done · -- Proof of a ≡ b (MOD n) apply Exists.intro (m * j) show a - b = n * (m * j) from calc a - b _ = m * n * j := h3 _ = n * (m * j) := by ring done done · -- (←) assume h2 : a ≡ b (MOD m) ∧ a ≡ b (MOD n) obtain (j : Int) (h3 : a - b = m * j) from h2.left have h4 : (↑n : Int) ∣ a - b := h2.right rewrite [h3] at h4 --h4 : ↑n ∣ ↑m * j have h5 : ↑n ∣ j := Theorem_7_2_2_Int h4 h1 obtain (k : Int) (h6 : j = n * k) from h5 apply Exists.intro k --Goal : a - b = ↑(m * n) * k rewrite [Nat.cast_mul] --Goal : a - b = ↑m * ↑n * k show a - b = (m * n) * k from calc a - b _ = m * j := h3 _ = m * (n * k) := by rw [h6] _ = (m * n) * k := by ring done done --From exercises of Section 7.2 theorem rel_prime_symm {a b : Nat} (h : rel_prime a b) : rel_prime b a := sorry lemma prime_NeZero {p : Nat} (h : prime p) : NeZero p := by rewrite [neZero_iff] --Goal : p ≠ 0 define at h linarith done lemma Lemma_7_5_1 {p e d m c s : Nat} {t : Int} (h1 : prime p) (h2 : e * d = (p - 1) * s + 1) (h3 : m ^ e - c = p * t) : c ^ d ≡ m (MOD p) := by have h4 : m ^ e ≡ c (MOD p) := Exists.intro t h3 have h5 : [m ^ e]_p = [c]_p := (cc_eq_iff_congr _ _ _).rtl h4 rewrite [←Exercise_7_4_5_Nat] at h5 --h5 : [m]_p ^ e = [c]_p by_cases h6 : p ∣ m · -- Case 1. h6 : p ∣ m have h7 : m ≡ 0 (MOD p) := by obtain (j : Nat) (h8 : m = p * j) from h6 apply Exists.intro (↑j : Int) --Goal : ↑m - 0 = ↑p * ↑j rewrite [h8, Nat.cast_mul] ring done have h8 : [m]_p = [0]_p := (cc_eq_iff_congr _ _ _).rtl h7 have h9 : e * d ≠ 0 := by rewrite [h2] show (p - 1) * s + 1 ≠ 0 from Nat.add_one_ne_zero _ done have h10 : (0 : Int) ^ (e * d) = 0 := zero_pow h9 have h11 : [c ^ d]_p = [m]_p := calc [c ^ d]_p _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat] _ = ([m]_p ^ e) ^ d := by rw [h5] _ = [m]_p ^ (e * d) := by ring _ = [0]_p ^ (e * d) := by rw [h8] _ = [0 ^ (e * d)]_p := Exercise_7_4_5_Int _ _ _ _ = [0]_p := by rw [h10] _ = [m]_p := by rw [h8] show c ^ d ≡ m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11 done · -- Case 2. h6 : ¬p ∣ m have h7 : rel_prime m p := rel_prime_of_prime_not_dvd h1 h6 have h8 : rel_prime p m := rel_prime_symm h7 have h9 : NeZero p := prime_NeZero h1 have h10 : (1 : Int) ^ s = 1 := by ring have h11 : [c ^ d]_p = [m]_p := calc [c ^ d]_p _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat] _ = ([m]_p ^ e) ^ d := by rw [h5] _ = [m]_p ^ (e * d) := by ring _ = [m]_p ^ ((p - 1) * s + 1) := by rw [h2] _ = ([m]_p ^ (p - 1)) ^ s * [m]_p := by ring _ = ([m]_p ^ (phi p)) ^ s * [m]_p := by rw [phi_prime h1] _ = [1]_p ^ s * [m]_p := by rw [Theorem_7_4_2 h8] _ = [1 ^ s]_p * [m]_p := by rw [Exercise_7_4_5_Int] _ = [1]_p * [m]_p := by rw [h10] _ = [m]_p * [1]_p := by ring _ = [m]_p := Theorem_7_3_6_7 _ show c ^ d ≡ m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11 done done theorem Theorem_7_5_1 (p q n e d k m c : Nat) (p_prime : prime p) (q_prime : prime q) (p_ne_q : p ≠ q) (n_pq : n = p * q) (ed_congr_1 : e * d = k * (p - 1) * (q - 1) + 1) (h1 : [m]_n ^ e = [c]_n) : [c]_n ^ d = [m]_n := by rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] at h1 --h1 : m ^ e ≡ c (MOD n) rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] --Goal : c ^ d ≡ m (MOD n) obtain (j : Int) (h2 : m ^ e - c = n * j) from h1 rewrite [n_pq, Nat.cast_mul] at h2 --h2 : m ^ e - c = p * q * j have h3 : e * d = (p - 1) * (k * (q - 1)) + 1 := by rewrite [ed_congr_1] ring done have h4 : m ^ e - c = p * (q * j) := by rewrite [h2] ring done have congr_p : c ^ d ≡ m (MOD p) := Lemma_7_5_1 p_prime h3 h4 have h5 : e * d = (q - 1) * (k * (p - 1)) + 1 := by rewrite [ed_congr_1] ring done have h6 : m ^ e - c = q * (p * j) := by rewrite [h2] ring done have congr_q : c ^ d ≡ m (MOD q) := Lemma_7_5_1 q_prime h5 h6 have h7 : ¬q ∣ p := by by_contra h8 have h9 : q = 1 ∨ q = p := dvd_prime p_prime h8 disj_syll h9 (prime_not_one q_prime) show False from p_ne_q h9.symm done have h8 : rel_prime p q := rel_prime_of_prime_not_dvd q_prime h7 rewrite [n_pq, Lemma_7_4_5 _ _ h8] show c ^ d ≡ m (MOD p) ∧ c ^ d ≡ m (MOD q) from And.intro congr_p congr_q done /- BEGIN EXERCISES -/ namespace Exercises /- Section 7.1 -/ -- 1. theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry -- 2. lemma gcd_comm_lt {a b : Nat} (h : a < b) : gcd a b = gcd b a := sorry theorem gcd_comm (a b : Nat) : gcd a b = gcd b a := sorry -- 3. theorem Exercise_7_1_5 (a b : Nat) (n : Int) : (∃ (s t : Int), s * a + t * b = n) ↔ (↑(gcd a b) : Int) ∣ n := sorry -- 4. theorem Exercise_7_1_6 (a b c : Nat) : gcd a b = gcd (a + b * c) b := sorry -- 5. theorem gcd_is_nonzero {a b : Nat} (h : a ≠ 0 ∨ b ≠ 0) : gcd a b ≠ 0 := sorry -- 6. theorem gcd_greatest {a b d : Nat} (h1 : gcd a b ≠ 0) (h2 : d ∣ a) (h3 : d ∣ b) : d ≤ gcd a b := sorry -- 7. lemma Lemma_7_1_10a {a b : Nat} (n : Nat) (h : a ∣ b) : (n * a) ∣ (n * b) := sorry lemma Lemma_7_1_10b {a b n : Nat} (h1 : n ≠ 0) (h2 : (n * a) ∣ (n * b)) : a ∣ b := sorry lemma Lemma_7_1_10c {a b : Nat} (h1 : a ∣ b) (h2 : b ∣ a) : a = b := sorry theorem Exercise_7_1_10 (a b n : Nat) : gcd (n * a) (n * b) = n * gcd a b := sorry /- Section 7.2 -/ -- 1. lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry -- 2. -- Hints: Start with apply List.rec. You may find mul_ne_zero useful theorem prod_nonzero_nonzero : ∀ (l : List Nat), (∀ a ∈ l, a ≠ 0) → prod l ≠ 0 := sorry -- 3. theorem rel_prime_iff_no_common_factor (a b : Nat) : rel_prime a b ↔ ¬∃ (p : Nat), prime p ∧ p ∣ a ∧ p ∣ b := sorry -- 4. theorem rel_prime_symm {a b : Nat} (h : rel_prime a b) : rel_prime b a := sorry -- 5. lemma in_prime_factorization_iff_prime_factor {a : Nat} {l : List Nat} (h1 : prime_factorization a l) (p : Nat) : p ∈ l ↔ prime_factor p a := sorry -- 6. theorem Exercise_7_2_5 {a b : Nat} {l m : List Nat} (h1 : prime_factorization a l) (h2 : prime_factorization b m) : rel_prime a b ↔ (¬∃ (p : Nat), p ∈ l ∧ p ∈ m) := sorry -- 7. theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry -- 8. theorem Exercise_7_2_7 {a b a' b' : Nat} (h1 : rel_prime a b) (h2 : a' ∣ a) (h3 : b' ∣ b) : rel_prime a' b' := sorry -- 9. theorem Exercise_7_2_9 {a b j k : Nat} (h1 : gcd a b ≠ 0) (h2 : a = j * gcd a b) (h3 : b = k * gcd a b) : rel_prime j k := sorry -- 10. theorem Exercise_7_2_17a (a b c : Nat) : gcd a (b * c) ∣ gcd a b * gcd a c := sorry /- Section 7.3 -/ -- 1. theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry -- 2. theorem Theorem_7_3_6_3 {m : Nat} (X : ZMod m) : X + [0]_m = X := sorry -- 3. theorem Theorem_7_3_6_4 {m : Nat} (X : ZMod m) : ∃ (Y : ZMod m), X + Y = [0]_m := sorry -- 4. theorem Exercise_7_3_4a {m : Nat} (Z1 Z2 : ZMod m) (h1 : ∀ (X : ZMod m), X + Z1 = X) (h2 : ∀ (X : ZMod m), X + Z2 = X) : Z1 = Z2 := sorry -- 5. theorem Exercise_7_3_4b {m : Nat} (X Y1 Y2 : ZMod m) (h1 : X + Y1 = [0]_m) (h2 : X + Y2 = [0]_m) : Y1 = Y2 := sorry -- 6. theorem Theorem_7_3_10 (m a : Nat) (b : Int) : ¬(↑(gcd m a) : Int) ∣ b → ¬∃ (x : Int), a * x ≡ b (MOD m) := sorry -- 7. theorem Theorem_7_3_11 (m n : Nat) (a b : Int) (h1 : n ≠ 0) : n * a ≡ n * b (MOD n * m) ↔ a ≡ b (MOD m) := sorry -- 8. theorem Exercise_7_3_16 {m : Nat} {a b : Int} (h : a ≡ b (MOD m)) : ∀ (n : Nat), a ^ n ≡ b ^ n (MOD m) := sorry -- 9. example {m : Nat} [NeZero m] (X : ZMod m) : ∃! (a : Int), 0 ≤ a ∧ a < m ∧ X = [a]_m := sorry -- 10. theorem congr_rel_prime {m a b : Nat} (h1 : a ≡ b (MOD m)) : rel_prime m a ↔ rel_prime m b := sorry -- 11. --Hint: You may find the theorem Int.ofNat_mod_ofNat useful. theorem rel_prime_mod (m a : Nat) : rel_prime m (a % m) ↔ rel_prime m a := sorry -- 12. lemma congr_iff_mod_eq_Int (m : Nat) (a b : Int) [NeZero m] : a ≡ b (MOD m) ↔ a % ↑m = b % ↑m := sorry --Hint for next theorem: Use the lemma above, --together with the theorems Int.ofNat_mod_ofNat and Nat.cast_inj. theorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] : ↑a ≡ ↑b (MOD m) ↔ a % m = b % m := sorry /- Section 7.4 -/ -- 1. --Hint: Use induction. --For the base case, compute [a]_m ^ 0 * [1]_m in two ways: --by Theorem_7_3_6_7, [a] ^ 0 * [1]_m = [a]_m ^ 0 --by ring, [a]_m ^ 0 * [1]_m = [1]_m. lemma Exercise_7_4_5_Int (m : Nat) (a : Int) : ∀ (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry -- 2. lemma left_inv_one_one_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g' (g i) = i) : one_one_below n g := sorry -- 3. lemma comp_perm_below {n : Nat} {f g : Nat → Nat} (h1 : perm_below n f) (h2 : perm_below n g) : perm_below n (f ∘ g) := sorry -- 4. lemma perm_below_fixed {n : Nat} {g : Nat → Nat} (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry -- 5. lemma Lemma_7_4_6 {a b c : Nat} : rel_prime (a * b) c ↔ rel_prime a c ∧ rel_prime b c := sorry -- 6. example {m a : Nat} [NeZero m] (h1 : rel_prime m a) : a ^ (phi m + 1) ≡ a (MOD m) := sorry -- 7. theorem Like_Exercise_7_4_11 {m a p : Nat} [NeZero m] (h1 : rel_prime m a) (h2 : p + 1 = phi m) : [a]_m * [a ^ p]_m = [1]_m := sorry -- 8.

theorem Like_Exercise_7_4_12 {m a p q k : Nat} [NeZero m] (h1 : rel_prime m a) (h2 : p = q + (phi m) * k) : a ^ p ≡ a ^ q (MOD m)

HTPI.Exercises.Like_Exercise_7_4_12

htpi/HTPILib/Chap7.lean

htpi

/- Copyright 2023 Daniel J. Velleman -/ import HTPILib.Chap6 namespace HTPI /- Definitions -/ lemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by have h : n + 1 > 0 := Nat.succ_pos n show a % (n + 1) < n + 1 from Nat.mod_lt a h done def gcd (a b : Nat) : Nat := match b with | 0 => a | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd (n + 1) (a % (n + 1)) termination_by b mutual def gcd_c1 (a b : Nat) : Int := match b with | 0 => 1 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c2 (n + 1) (a % (n + 1)) --Corresponds to s = t' termination_by b def gcd_c2 (a b : Nat) : Int := match b with | 0 => 0 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c1 (n + 1) (a % (n + 1)) - (gcd_c2 (n + 1) (a % (n + 1))) * ↑(a / (n + 1)) --Corresponds to t = s' - t'q termination_by b end def prime (n : Nat) : Prop := 2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n def prime_factor (p n : Nat) : Prop := prime p ∧ p ∣ n def all_prime (l : List Nat) : Prop := ∀ p ∈ l, prime p def nondec (l : List Nat) : Prop := match l with | [] => True --Of course, True is a proposition that is always true | n :: L => (∀ m ∈ L, n ≤ m) ∧ nondec L def nondec_prime_list (l : List Nat) : Prop := all_prime l ∧ nondec l def prod (l : List Nat) : Nat := match l with | [] => 1 | n :: L => n * (prod L) def prime_factorization (n : Nat) (l : List Nat) : Prop := nondec_prime_list l ∧ prod l = n def rel_prime (a b : Nat) : Prop := gcd a b = 1 def congr_mod (m : Nat) (a b : Int) : Prop := (↑m : Int) ∣ (a - b) def cc (m : Nat) (a : Int) : ZMod m := (↑a : ZMod m) notation:50 a " ≡ " b " (MOD " m ")" => congr_mod m a b notation:max "["a"]_"m:max => cc m a def invertible {m : Nat} (X : ZMod m) : Prop := ∃ (Y : ZMod m), X * Y = [1]_m def num_rp_below (m k : Nat) : Nat := match k with | 0 => 0 | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j def phi (m : Nat) : Nat := num_rp_below m m def prod_seq {m : Nat} (j k : Nat) (f : Nat → ZMod m) : ZMod m := match j with | 0 => [1]_m | n + 1 => prod_seq n k f * f (k + n) def maps_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i < n, g i < n def one_one_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i1 < n, ∀ i2 < n, g i1 = g i2 → i1 = i2 def onto_below (n : Nat) (g : Nat → Nat) : Prop := ∀ k < n, ∃ i < n, g i = k def perm_below (n : Nat) (g : Nat → Nat) : Prop := maps_below n g ∧ one_one_below n g ∧ onto_below n g def inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m) def swap (u v i : Nat) : Nat := if i = u then v else if i = v then u else i namespace Euler --For definitions specific to Euler's theorem def F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m def G (m a i : Nat) : Nat := (a * i) % m def Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i end Euler /- Section 7.1 -/ theorem dvd_mod_of_dvd_a_b {a b d : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ (a % b) := by set q : Nat := a / b have h3 : b * q + a % b = a := Nat.div_add_mod a b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 define --Goal : ∃ (c : Nat), a % b = d * c apply Exists.intro (j - k * q) show a % b = d * (j - k * q) from calc a % b _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm _ = a - b * q := by rw [h3] _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc] _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm done theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry #eval gcd 672 161 --Answer: 7 lemma gcd_base (a : Nat) : gcd a 0 = a := by rfl lemma gcd_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd a b = gcd b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1)) rfl done lemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b ≠ 0) : a % b < b := by have h1 : b > 0 := Nat.pos_of_ne_zero h show a % b < b from Nat.mod_lt a h1 done lemma dvd_self (n : Nat) : n ∣ n := by apply Exists.intro 1 ring done theorem gcd_dvd : ∀ (b a : Nat), (gcd a b) ∣ a ∧ (gcd a b) ∣ b := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd a b_1) ∣ a ∧ (gcd a b_1) ∣ b_1 fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_base] --Goal: a ∣ a ∧ a ∣ 0 apply And.intro (dvd_self a) define apply Exists.intro 0 rfl done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_nonzero a h1] --Goal : gcd b (a % b) ∣ a ∧ gcd b (a % b) ∣ b have h2 : a % b < b := mod_nonzero_lt a h1 have h3 : (gcd b (a % b)) ∣ b ∧ (gcd b (a % b)) ∣ (a % b) := ih (a % b) h2 b apply And.intro _ h3.left show (gcd b (a % b)) ∣ a from dvd_a_of_dvd_b_mod h3.left h3.right done done theorem gcd_dvd_left (a b : Nat) : (gcd a b) ∣ a := (gcd_dvd b a).left theorem gcd_dvd_right (a b : Nat) : (gcd a b) ∣ b := (gcd_dvd b a).right lemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl lemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c1 a b = gcd_c2 b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done lemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl lemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * ↑(a / b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done theorem gcd_lin_comb : ∀ (b a : Nat), (gcd_c1 a b) * ↑a + (gcd_c2 a b) * ↑b = ↑(gcd a b) := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd_c1 a b_1) * ↑a + (gcd_c2 a b_1) * ↑b_1 = ↑(gcd a b_1) fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base] --Goal : 1 * ↑a + 0 * ↑0 = ↑a ring done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1] --Goal : gcd_c2 b (a % b) * ↑a + -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * ↑(a / b)) * ↑b = -- ↑(gcd b (a % b)) set r : Nat := a % b set q : Nat := a / b set s : Int := gcd_c1 b r set t : Int := gcd_c2 b r --Goal : t * ↑a + (s - t * ↑q) * ↑b = ↑(gcd b r) have h2 : r < b := mod_nonzero_lt a h1 have h3 : s * ↑b + t * ↑r = ↑(gcd b r) := ih r h2 b have h4 : b * q + r = a := Nat.div_add_mod a b rewrite [←h3, ←h4] rewrite [Nat.cast_add, Nat.cast_mul] --Goal : t * (↑b * ↑q + ↑r) + (s - t * ↑q) * ↑b = s * ↑b + t * ↑r ring done done #eval gcd_c1 672 161 --Answer: 6 #eval gcd_c2 672 161 --Answer: -25 --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161 theorem Theorem_7_1_6 {d a b : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ gcd a b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑d ∣ ↑(gcd a b) set s : Int := gcd_c1 a b set t : Int := gcd_c2 a b have h3 : s * ↑a + t * ↑b = ↑(gcd a b) := gcd_lin_comb b a rewrite [←h3] --Goal : ↑d ∣ s * ↑a + t * ↑b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul] --Goal : ↑d ∣ s * (↑d * ↑j) + t * (↑d * ↑k) define apply Exists.intro (s * ↑j + t * ↑k) ring done /- Section 7.2 -/ theorem dvd_trans {a b c : Nat} (h1 : a ∣ b) (h2 : b ∣ c) : a ∣ c := by define at h1; define at h2; define obtain (m : Nat) (h3 : b = a * m) from h1 obtain (n : Nat) (h4 : c = b * n) from h2 rewrite [h3, mul_assoc] at h4 apply Exists.intro (m * n) show c = a * (m * n) from h4 done lemma exists_prime_factor : ∀ (n : Nat), 2 ≤ n → ∃ (p : Nat), prime_factor p n := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, 2 ≤ n_1 → ∃ (p : Nat), prime_factor p n_1 assume h1 : 2 ≤ n by_cases h2 : prime n · -- Case 1. h2 : prime n apply Exists.intro n define --Goal : prime n ∧ n ∣ n show prime n ∧ n ∣ n from And.intro h2 (dvd_self n) done · -- Case 2. h2 : ¬prime n define at h2 --h2 : ¬(2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n) demorgan at h2 disj_syll h2 h1 obtain (a : Nat) (h3 : ∃ (b : Nat), a * b = n ∧ a < n ∧ b < n) from h2 obtain (b : Nat) (h4 : a * b = n ∧ a < n ∧ b < n) from h3 have h5 : 2 ≤ a := by by_contra h6 have h7 : a ≤ 1 := by linarith have h8 : n ≤ b := calc n _ = a * b := h4.left.symm _ ≤ 1 * b := by rel [h7] _ = b := by ring linarith --n ≤ b contradicts b < n done have h6 : ∃ (p : Nat), prime_factor p a := ih a h4.right.left h5 obtain (p : Nat) (h7 : prime_factor p a) from h6 apply Exists.intro p define --Goal : prime p ∧ p ∣ n define at h7 --h7 : prime p ∧ p ∣ a apply And.intro h7.left have h8 : a ∣ n := by apply Exists.intro b show n = a * b from (h4.left).symm done show p ∣ n from dvd_trans h7.right h8 done done lemma exists_least_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q := by set S : Set Nat := {p : Nat | prime_factor p n} have h2 : ∃ (p : Nat), p ∈ S := exists_prime_factor n h show ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q from well_ord_princ S h2 done lemma all_prime_nil : all_prime [] := by define --Goal : ∀ p ∈ [], prime p fix p : Nat contrapos --Goal : ¬prime p → p ∉ [] assume h1 : ¬prime p show p ∉ [] from List.not_mem_nil p done lemma all_prime_cons (n : Nat) (L : List Nat) : all_prime (n :: L) ↔ prime n ∧ all_prime L := by apply Iff.intro · -- (→) assume h1 : all_prime (n :: L) --Goal : prime n ∧ all_prime L define at h1 --h1 : ∀ p ∈ n :: L, prime p apply And.intro (h1 n (List.mem_cons_self n L)) define --Goal : ∀ p ∈ L, prime p fix p : Nat assume h2 : p ∈ L show prime p from h1 p (List.mem_cons_of_mem n h2) done · -- (←) assume h1 : prime n ∧ all_prime L --Goal : all_prime (n :: l) define : all_prime L at h1 define fix p : Nat assume h2 : p ∈ n :: L rewrite [List.mem_cons] at h2 --h2 : p = n ∨ p ∈ L by_cases on h2 · -- Case 1. h2 : p = n rewrite [h2] show prime n from h1.left done · -- Case 2. h2 : p ∈ L show prime p from h1.right p h2 done done done lemma nondec_nil : nondec [] := by define --Goal : True trivial --trivial proves some obviously true statements, such as True done lemma nondec_cons (n : Nat) (L : List Nat) : nondec (n :: L) ↔ (∀ m ∈ L, n ≤ m) ∧ nondec L := by rfl lemma prod_nil : prod [] = 1 := by rfl lemma prod_cons : prod (n :: L) = n * (prod L) := by rfl lemma exists_cons_of_length_eq_succ {A : Type} {l : List A} {n : Nat} (h : l.length = n + 1) : ∃ (a : A) (L : List A), l = a :: L ∧ L.length = n := by have h1 : ¬l.length = 0 := by linarith rewrite [List.length_eq_zero] at h1 obtain (a : A) (h2 : ∃ (L : List A), l = a :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List A) (h3 : l = a :: L) from h2 apply Exists.intro a apply Exists.intro L apply And.intro h3 have h4 : (a :: L).length = L.length + 1 := List.length_cons a L rewrite [←h3, h] at h4 show L.length = n from (Nat.add_right_cancel h4).symm done lemma list_elt_dvd_prod_by_length (a : Nat) : ∀ (n : Nat), ∀ (l : List Nat), l.length = n → a ∈ l → a ∣ prod l := by by_induc · --Base Case fix l : List Nat assume h1 : l.length = 0 rewrite [List.length_eq_zero] at h1 --h1 : l = [] rewrite [h1] --Goal : a ∈ [] → a ∣ prod [] contrapos assume h2 : ¬a ∣ prod [] show a ∉ [] from List.not_mem_nil a done · -- Induction Step fix n : Nat assume ih : ∀ (l : List Nat), List.length l = n → a ∈ l → a ∣ prod l fix l : List Nat assume h1 : l.length = n + 1 --Goal : a ∈ l → a ∣ prod l obtain (b : Nat) (h2 : ∃ (L : List Nat), l = b :: L ∧ L.length = n) from exists_cons_of_length_eq_succ h1 obtain (L : List Nat) (h3 : l = b :: L ∧ L.length = n) from h2 have h4 : a ∈ L → a ∣ prod L := ih L h3.right assume h5 : a ∈ l rewrite [h3.left, prod_cons] --Goal : a ∣ b * prod L rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b ∨ a ∈ L by_cases on h5 · -- Case 1. h5 : a = b apply Exists.intro (prod L) rewrite [h5] rfl done · -- Case 2. h5 : a ∈ L have h6 : a ∣ prod L := h4 h5 have h7 : prod L ∣ b * prod L := by apply Exists.intro b ring done show a ∣ b * prod L from dvd_trans h6 h7 done done done lemma list_elt_dvd_prod {a : Nat} {l : List Nat} (h : a ∈ l) : a ∣ prod l := by set n : Nat := l.length have h1 : l.length = n := by rfl show a ∣ prod l from list_elt_dvd_prod_by_length a n l h1 h done lemma exists_prime_factorization : ∀ (n : Nat), n ≥ 1 → ∃ (l : List Nat), prime_factorization n l := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, n_1 ≥ 1 → ∃ (l : List Nat), prime_factorization n_1 l assume h1 : n ≥ 1 by_cases h2 : n = 1 · -- Case 1. h2 : n = 1 apply Exists.intro [] define apply And.intro · -- Proof of nondec_prime_list [] define show all_prime [] ∧ nondec [] from And.intro all_prime_nil nondec_nil done · -- Proof of prod [] = n rewrite [prod_nil, h2] rfl done done · -- Case 2. h2 : n ≠ 1 have h3 : n ≥ 2 := lt_of_le_of_ne' h1 h2 obtain (p : Nat) (h4 : prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q) from exists_least_prime_factor h3 have p_prime_factor : prime_factor p n := h4.left define at p_prime_factor have p_prime : prime p := p_prime_factor.left have p_dvd_n : p ∣ n := p_prime_factor.right have p_least : ∀ (q : Nat), prime_factor q n → p ≤ q := h4.right obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n have h5 : m ≠ 0 := by contradict h1 with h6 have h7 : n = 0 := calc n _ = p * m := n_eq_pm _ = p * 0 := by rw [h6] _ = 0 := by ring rewrite [h7] decide done have m_pos : 0 < m := Nat.pos_of_ne_zero h5 have m_lt_n : m < n := by define at p_prime show m < n from calc m _ < m + m := by linarith _ = 2 * m := by ring _ ≤ p * m := by rel [p_prime.left] _ = n := n_eq_pm.symm done obtain (L : List Nat) (h6 : prime_factorization m L) from ih m m_lt_n m_pos define at h6 have ndpl_L : nondec_prime_list L := h6.left define at ndpl_L apply Exists.intro (p :: L) define apply And.intro · -- Proof of nondec_prime_list (p :: L) define apply And.intro · -- Proof of all_prime (p :: L) rewrite [all_prime_cons] show prime p ∧ all_prime L from And.intro p_prime ndpl_L.left done · -- Proof of nondec (p :: L) rewrite [nondec_cons] apply And.intro _ ndpl_L.right fix q : Nat assume q_in_L : q ∈ L have h7 : q ∣ prod L := list_elt_dvd_prod q_in_L rewrite [h6.right] at h7 --h7 : q ∣ m have h8 : m ∣ n := by apply Exists.intro p rewrite [n_eq_pm] ring done have q_dvd_n : q ∣ n := dvd_trans h7 h8 have ap_L : all_prime L := ndpl_L.left define at ap_L have q_prime_factor : prime_factor q n := And.intro (ap_L q q_in_L) q_dvd_n show p ≤ q from p_least q q_prime_factor done done · -- Proof of prod (p :: L) = n rewrite [prod_cons, h6.right, n_eq_pm] rfl done done done theorem Theorem_7_2_2 {a b c : Nat} (h1 : c ∣ a * b) (h2 : rel_prime a c) : c ∣ b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑c ∣ ↑b define at h1; define at h2; define obtain (j : Nat) (h3 : a * b = c * j) from h1 set s : Int := gcd_c1 a c set t : Int := gcd_c2 a c have h4 : s * ↑a + t * ↑c = ↑(gcd a c) := gcd_lin_comb c a rewrite [h2, Nat.cast_one] at h4 --h4 : s * ↑a + t * ↑c = (1 : Int) apply Exists.intro (s * ↑j + t * ↑b) show ↑b = ↑c * (s * ↑j + t * ↑b) from calc ↑b _ = (1 : Int) * ↑b := (one_mul _).symm _ = (s * ↑a + t * ↑c) * ↑b := by rw [h4] _ = s * (↑a * ↑b) + t * ↑c * ↑b := by ring _ = s * (↑c * ↑j) + t * ↑c * ↑b := by rw [←Nat.cast_mul a b, h3, Nat.cast_mul c j] _ = ↑c * (s * ↑j + t * ↑b) := by ring done lemma le_nonzero_prod_left {a b : Nat} (h : a * b ≠ 0) : a ≤ a * b := by have h1 : b ≠ 0 := by contradict h with h1 rewrite [h1] ring done have h2 : 1 ≤ b := Nat.pos_of_ne_zero h1 show a ≤ a * b from calc a = a * 1 := (mul_one a).symm _ ≤ a * b := by rel [h2] done lemma le_nonzero_prod_right {a b : Nat} (h : a * b ≠ 0) : b ≤ a * b := by rewrite [mul_comm] rewrite [mul_comm] at h show b ≤ b * a from le_nonzero_prod_left h done lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry lemma rel_prime_of_prime_not_dvd {a p : Nat} (h1 : prime p) (h2 : ¬p ∣ a) : rel_prime a p := by have h3 : gcd a p ∣ a := gcd_dvd_left a p have h4 : gcd a p ∣ p := gcd_dvd_right a p have h5 : gcd a p = 1 ∨ gcd a p = p := dvd_prime h1 h4 have h6 : gcd a p ≠ p := by contradict h2 with h6 rewrite [h6] at h3 show p ∣ a from h3 done disj_syll h5 h6 show rel_prime a p from h5 done theorem Theorem_7_2_3 {a b p : Nat} (h1 : prime p) (h2 : p ∣ a * b) : p ∣ a ∨ p ∣ b := by or_right with h3 have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3 show p ∣ b from Theorem_7_2_2 h2 h4 done lemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a ≥ 1 := by have h1 : a ≠ 0 := by by_contra h1 rewrite [h1] at h contradict h linarith done show a ≥ 1 from Nat.pos_of_ne_zero h1 done lemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by have h1 : a ≥ 1 := ge_one_of_prod_one h have h2 : a * b ≠ 0 := by linarith have h3 : a ≤ a * b := le_nonzero_prod_left h2 rewrite [h] at h3 show a = 1 from Nat.le_antisymm h3 h1 done lemma eq_one_of_dvd_one {n : Nat} (h : n ∣ 1) : n = 1 := by obtain (j : Nat) (h1 : 1 = n * j) from h show n = 1 from eq_one_of_prod_one h1.symm done lemma prime_not_one {p : Nat} (h : prime p) : p ≠ 1 := by define at h linarith done theorem Theorem_7_2_4 {p : Nat} (h1 : prime p) : ∀ (l : List Nat), p ∣ prod l → ∃ a ∈ l, p ∣ a := by apply List.rec · -- Base Case. Goal : p ∣ prod [] → ∃ a ∈ [], p ∣ a rewrite [prod_nil] assume h2 : p ∣ 1 show ∃ a ∈ [], p ∣ a from absurd (eq_one_of_dvd_one h2) (prime_not_one h1) done · -- Induction Step fix b : Nat fix L : List Nat assume ih : p ∣ prod L → ∃ a ∈ L, p ∣ a --Goal : p ∣ prod (b :: L) → ∃ a ∈ b :: L, p ∣ a assume h2 : p ∣ prod (b :: L) rewrite [prod_cons] at h2 have h3 : p ∣ b ∨ p ∣ prod L := Theorem_7_2_3 h1 h2 by_cases on h3 · -- Case 1. h3 : p ∣ b apply Exists.intro b show b ∈ b :: L ∧ p ∣ b from And.intro (List.mem_cons_self b L) h3 done · -- Case 2. h3 : p ∣ prod L obtain (a : Nat) (h4 : a ∈ L ∧ p ∣ a) from ih h3 apply Exists.intro a show a ∈ b :: L ∧ p ∣ a from And.intro (List.mem_cons_of_mem b h4.left) h4.right done done done lemma prime_in_list {p : Nat} {l : List Nat} (h1 : prime p) (h2 : all_prime l) (h3 : p ∣ prod l) : p ∈ l := by obtain (a : Nat) (h4 : a ∈ l ∧ p ∣ a) from Theorem_7_2_4 h1 l h3 define at h2 have h5 : prime a := h2 a h4.left have h6 : p = 1 ∨ p = a := dvd_prime h5 h4.right disj_syll h6 (prime_not_one h1) rewrite [h6] show a ∈ l from h4.left done lemma first_le_first {p q : Nat} {l m : List Nat} (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m)) (h3 : prod (p :: l) = prod (q :: m)) : p ≤ q := by define at h1; define at h2 have h4 : q ∣ prod (p :: l) := by define apply Exists.intro (prod m) rewrite [←prod_cons] show prod (p :: l) = prod (q :: m) from h3 done have h5 : all_prime (q :: m) := h2.left rewrite [all_prime_cons] at h5 have h6 : q ∈ p :: l := prime_in_list h5.left h1.left h4 have h7 : nondec (p :: l) := h1.right rewrite [nondec_cons] at h7 rewrite [List.mem_cons] at h6 by_cases on h6 · -- Case 1. h6 : q = p linarith done · -- Case 2. h6 : q ∈ l have h8 : ∀ m ∈ l, p ≤ m := h7.left show p ≤ q from h8 q h6 done done lemma nondec_prime_list_tail {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by define at h define rewrite [all_prime_cons, nondec_cons] at h show all_prime l ∧ nondec l from And.intro h.left.right h.right.right done lemma cons_prod_not_one {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : prod (p :: l) ≠ 1 := by define at h have h1 : all_prime (p :: l) := h.left rewrite [all_prime_cons] at h1 rewrite [prod_cons] by_contra h2 show False from (prime_not_one h1.left) (eq_one_of_prod_one h2) done lemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) : l = [] ↔ prod l = 1 := by apply Iff.intro · -- (→) assume h1 : l = [] rewrite [h1] show prod [] = 1 from prod_nil done · -- (←) contrapos assume h1 : ¬l = [] obtain (p : Nat) (h2 : ∃ (L : List Nat), l = p :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List Nat) (h3 : l = p :: L) from h2 rewrite [h3] at h rewrite [h3] show ¬prod (p :: L) = 1 from cons_prod_not_one h done done lemma prime_pos {p : Nat} (h : prime p) : p > 0 := by define at h linarith done theorem Theorem_7_2_5 : ∀ (l1 l2 : List Nat), nondec_prime_list l1 → nondec_prime_list l2 → prod l1 = prod l2 → l1 = l2 := by apply List.rec · -- Base Case. Goal : ∀ (l2 : List Nat), nondec_prime_list [] → -- nondec_prime_list l2 → prod [] = prod l2 → [] = l2 fix l2 : List Nat assume h1 : nondec_prime_list [] assume h2 : nondec_prime_list l2 assume h3 : prod [] = prod l2 rewrite [prod_nil, eq_comm, ←list_nil_iff_prod_one h2] at h3 show [] = l2 from h3.symm done · -- Induction Step fix p : Nat fix L1 : List Nat assume ih : ∀ (L2 : List Nat), nondec_prime_list L1 → nondec_prime_list L2 → prod L1 = prod L2 → L1 = L2 -- Goal : ∀ (l2 : List Nat), nondec_prime_list (p :: L1) → -- nondec_prime_list l2 → prod (p :: L1) = prod l2 → p :: L1 = l2 fix l2 : List Nat assume h1 : nondec_prime_list (p :: L1) assume h2 : nondec_prime_list l2 assume h3 : prod (p :: L1) = prod l2 have h4 : ¬prod (p :: L1) = 1 := cons_prod_not_one h1 rewrite [h3, ←list_nil_iff_prod_one h2] at h4 obtain (q : Nat) (h5 : ∃ (L : List Nat), l2 = q :: L) from List.exists_cons_of_ne_nil h4 obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5 rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2) rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2) have h7 : p ≤ q := first_le_first h1 h2 h3 have h8 : q ≤ p := first_le_first h2 h1 h3.symm have h9 : p = q := by linarith rewrite [h9, prod_cons, prod_cons] at h3 --h3 : q * prod L1 = q * prod L2 have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1 have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2 define at h2 have h12 : all_prime (q :: L2) := h2.left rewrite [all_prime_cons] at h12 have h13 : q > 0 := prime_pos h12.left have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3 have h15 : L1 = L2 := ih L2 h10 h11 h14 rewrite [h6, h9, h15] rfl done done theorem fund_thm_arith (n : Nat) (h : n ≥ 1) : ∃! (l : List Nat), prime_factorization n l := by exists_unique · -- Existence show ∃ (l : List Nat), prime_factorization n l from exists_prime_factorization n h done · -- Uniqueness fix l1 : List Nat; fix l2 : List Nat assume h1 : prime_factorization n l1 assume h2 : prime_factorization n l2 define at h1; define at h2 have h3 : prod l1 = n := h1.right rewrite [←h2.right] at h3 show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3 done done /- Section 7.3 -/ theorem congr_refl (m : Nat) : ∀ (a : Int), a ≡ a (MOD m) := by fix a : Int define --Goal : ∃ (c : Int), a - a = ↑m * c apply Exists.intro 0 ring done theorem congr_symm {m : Nat} : ∀ {a b : Int}, a ≡ b (MOD m) → b ≡ a (MOD m) := by fix a : Int; fix b : Int assume h1 : a ≡ b (MOD m) define at h1 --h1 : ∃ (c : Int), a - b = ↑m * c define --Goal : ∃ (c : Int), b - a = ↑m * c obtain (c : Int) (h2 : a - b = m * c) from h1 apply Exists.intro (-c) show b - a = m * (-c) from calc b - a _ = -(a - b) := by ring _ = -(m * c) := by rw [h2] _ = m * (-c) := by ring done theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry /- Fundamental properties of congruence classes -/ lemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) : X = Y ↔ X.val = Y.val := Fin.ext_iff lemma val_nat_eq_mod (n k : Nat) : ([k]_(n + 1)).val = k % (n + 1) := by rfl lemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl theorem cc_rep {m : Nat} (X : ZMod m) : ∃ (a : Int), X = [a]_m := match m with | 0 => by apply Exists.intro X rfl done | n + 1 => by apply Exists.intro ↑(X.val) have h1 : X.val < n + 1 := Fin.prop X rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1] rfl done theorem add_class (m : Nat) (a b : Int) : [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm theorem mul_class (m : Nat) (a b : Int) : [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm lemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ [a - b]_m = [0]_m := by apply Iff.intro · -- (→) assume h1 : [a]_m = [b]_m have h2 : a - b = a + (-b) := by ring have h3 : b + (-b) = 0 := by ring show [a - b]_m = [0]_m from calc [a - b]_m _ = [a + (-b)]_m := by rw [h2] _ = [a]_m + [-b]_m := by rw [add_class] _ = [b]_m + [-b]_m := by rw [h1] _ = [b + -b]_m := by rw [add_class] _ = [0]_m := by rw [h3] done · -- (←) assume h1 : [a - b]_m = [0]_m have h2 : b + (a - b) = a := by ring have h3 : b + 0 = b := by ring show [a]_m = [b]_m from calc [a]_m _ = [b + (a - b)]_m := by rw [h2] _ = [b]_m + [a - b]_m := by rw [add_class] _ = [b]_m + [0]_m := by rw [h1] _ = [b + 0]_m := by rw [add_class] _ = [b]_m := by rw [h3] done done lemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) : [a]_m = [0]_m → [-a]_m = [0]_m := by assume h1 : [a]_m = [0]_m have h2 : 0 + (-a) = -a := by ring have h3 : a + (-a) = 0 := by ring show [-a]_m = [0]_m from calc [-a]_m _ = [0 + (-a)]_m := by rw [h2] _ = [0]_m + [-a]_m := by rw [add_class] _ = [a]_m + [-a]_m := by rw [h1] _ = [a + (-a)]_m := by rw [add_class] _ = [0]_m := by rw [h3] done lemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) : [-a]_m = [0]_m ↔ [a]_m = [0]_m := by apply Iff.intro _ (cc_neg_zero_of_cc_zero m a) assume h1 : [-a]_m = [0]_m have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1 have h3 : -(-a) = a := by ring rewrite [h3] at h2 show [a]_m = [0]_m from h2 done lemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl lemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m ↔ m ∣ k := match m with | 0 => by have h : (0 : Int) = (↑(0 : Nat) : Int) := by rfl rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj] apply Iff.intro · -- (→) assume h1 : k = 0 rewrite [h1] show 0 ∣ 0 from dvd_self 0 done · -- (←) assume h1 : 0 ∣ k obtain (c : Nat) (h2 : k = 0 * c) from h1 rewrite [h2] ring done done | n + 1 => by rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero] show k % (n + 1) = 0 ↔ n + 1 ∣ k from (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm done lemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m ↔ ↑m ∣ a := by obtain (k : Nat) (h1 : a = ↑k ∨ a = -↑k) from Int.eq_nat_or_neg a by_cases on h1 · -- Case 1. h1: a = ↑k rewrite [h1, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done · -- Case 2. h1: a = -↑k rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done done theorem cc_eq_iff_congr (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ a ≡ b (MOD m) := calc [a]_m = [b]_m _ ↔ [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b _ ↔ ↑m ∣ (a - b) := cc_zero_iff_dvd m (a - b) _ ↔ a ≡ b (MOD m) := by rfl /- End of fundamental properties of congruence classes -/ lemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m := by have h1 : (↑m : Int) ≠ 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m) show 0 ≤ a % m from Int.emod_nonneg a h1 done lemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m) have h2 : (↑m : Int) > 0 := (Nat.cast_pos).rtl h1 show a % m < m from Int.emod_lt_of_pos a h2 done lemma congr_mod_mod (m : Nat) (a : Int) : a ≡ a % m (MOD m) := by define have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m apply Exists.intro (a / m) show a - a % m = m * (a / m) from calc a - (a % m) _ = m * (a / m) + a % m - a % m := by rw [h1] _ = m * (a / m) := by ring done lemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) := And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a)) theorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) : ∃! (r : Int), 0 ≤ r ∧ r < m ∧ a ≡ r (MOD m) := by exists_unique · -- Existence apply Exists.intro (a % m) show 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) from mod_cmpl_res m a done · -- Uniqueness fix r1 : Int; fix r2 : Int assume h1 : 0 ≤ r1 ∧ r1 < m ∧ a ≡ r1 (MOD m) assume h2 : 0 ≤ r2 ∧ r2 < m ∧ a ≡ r2 (MOD m) have h3 : r1 ≡ r2 (MOD m) := congr_trans (congr_symm h1.right.right) h2.right.right obtain (d : Int) (h4 : r1 - r2 = m * d) from h3 have h5 : r1 - r2 < m * 1 := by linarith have h6 : m * (-1) < r1 - r2 := by linarith rewrite [h4] at h5 --h5 : m * d < m * 1 rewrite [h4] at h6 --h6 : m * -1 < m * d have h7 : (↑m : Int) ≥ 0 := Nat.cast_nonneg m have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7 have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7 have h10 : d = 0 := by linarith show r1 = r2 from calc r1 _ = r1 - r2 + r2 := by ring _ = m * 0 + r2 := by rw [h4, h10] _ = r2 := by ring done done lemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m := (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a) theorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y rewrite [h1, h2] have h3 : a + b = b + a := by ring show [a]_m + [b]_m = [b]_m + [a]_m from calc [a]_m + [b]_m _ = [a + b]_m := add_class m a b _ = [b + a]_m := by rw [h3] _ = [b]_m + [a]_m := (add_class m b a).symm done theorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X rewrite [h1] have h2 : a * 1 = a := by ring show [a]_m * [1]_m = [a]_m from calc [a]_m * [1]_m _ = [a * 1]_m := mul_class m a 1 _ = [a]_m := by rw [h2] done theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry lemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) : [a]_m * [gcd_c2 m a]_m = [1]_m := by set s : Int := gcd_c1 m a have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m define at h1 rewrite [h1, Nat.cast_one] at h2 --h2 : s * ↑m + gcd_c2 m a * ↑a = 1 rewrite [mul_class, cc_eq_iff_congr] define --Goal : ∃ (c : Int), ↑a * gcd_c2 m a - 1 = ↑m * c apply Exists.intro (-s) show a * (gcd_c2 m a) - 1 = m * (-s) from calc a * (gcd_c2 m a) - 1 _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring _ = 1 + m * (-s) - 1 := by rw [h2] _ = m * (-s) := by ring done theorem Theorem_7_3_7 (m a : Nat) : invertible [a]_m ↔ rel_prime m a := by apply Iff.intro · -- (→) assume h1 : invertible [a]_m define at h1 obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1 obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y rewrite [h3, mul_class, cc_eq_iff_congr] at h2 define at h2 obtain (c : Int) (h4 : a * b - 1 = m * c) from h2 rewrite [Exercise_7_2_6] --Goal : ∃ (s t : Int), s * ↑m + t * ↑a = 1 apply Exists.intro (-c) apply Exists.intro b show (-c) * m + b * a = 1 from calc (-c) * m + b * a _ = (-c) * m + (a * b - 1) + 1 := by ring _ = (-c) * m + m * c + 1 := by rw [h4] _ = 1 := by ring done · -- (←) assume h1 : rel_prime m a define show ∃ (Y : ZMod m), [a]_m * Y = [1]_m from Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1) done done /- Section 7.4 -/ section Euler open Euler lemma num_rp_below_base {m : Nat} : num_rp_below m 0 = 0 := by rfl lemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) : num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by have h1 : num_rp_below m (j + 1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : gcd m j = 1 rewrite [if_pos h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1 show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1 done lemma num_rp_below_step_not_rp {m j : Nat} (h : ¬rel_prime m j) : num_rp_below m (j + 1) = num_rp_below m j := by have h1 : num_rp_below m (j +1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : ¬gcd m j = 1 rewrite [if_neg h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j show num_rp_below m (j + 1) = num_rp_below m j from h1 done lemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl #eval phi 10 --Answer: 4 lemma prod_inv_iff_inv {m : Nat} {X : ZMod m} (h1 : invertible X) (Y : ZMod m) : invertible (X * Y) ↔ invertible Y := by apply Iff.intro · -- (→) assume h2 : invertible (X * Y) obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2 apply Exists.intro (X * Z) rewrite [←h3] --Goal : Y * (X * Z) = X * Y * Z ring --Note that ring can do algebra in ZMod m done · -- (←) assume h2 : invertible Y obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1 obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2 apply Exists.intro (Xi * Yi) show (X * Y) * (Xi * Yi) = [1]_m from calc X * Y * (Xi * Yi) _ = (X * Xi) * (Y * Yi) := by ring _ = [1]_m * [1]_m := by rw [h3, h4] _ = [1]_m := Theorem_7_3_6_7 [1]_m done done lemma F_rp_def {m i : Nat} (h : rel_prime m i) : F m i = [i]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h --h : gcd m i = 1 rewrite [if_pos h] at h1 show F m i = [i]_m from h1 done lemma F_not_rp_def {m i : Nat} (h : ¬rel_prime m i) : F m i = [1]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h rewrite [h1, if_neg h] rfl done lemma prod_seq_base {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 0 k f = [1]_m := by rfl lemma prod_seq_step {m : Nat} (n k : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl lemma prod_seq_zero_step {m : Nat} (n : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by rewrite [prod_seq_step, zero_add] rfl done lemma prod_one {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 1 k f = f k := by rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7] rfl done lemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl lemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m := calc [G m a i]_m _ = [(a * i) % m]_m := by rfl _ = [a * i]_m := (cc_eq_mod m (a * i)).symm _ = [a]_m * [i]_m := (mul_class m a i).symm lemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) : rel_prime m (G m a i) ↔ rel_prime m i := by have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1 show rel_prime m (G m a i) ↔ rel_prime m i from calc rel_prime m (G m a i) _ ↔ invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm _ ↔ invertible ([a]_m * [i]_m) := by rw [cc_G] _ ↔ invertible [i]_m := prod_inv_iff_inv h2 ([i]_m) _ ↔ rel_prime m i := Theorem_7_3_7 m i done lemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) : F m (G m a i) = [a]_m * F m i := by have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2 show F m (G m a i) = [a]_m * F m i from calc F m (G m a i) _ = [G m a i]_m := F_rp_def h3 _ = [a]_m * [i]_m := cc_G m a i _ = [a]_m * F m i := by rw [F_rp_def h2] done lemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : ¬rel_prime m i) : F m (G m a i) = [1]_m := by rewrite [←G_rp_iff h1 i] at h2 show F m (G m a i) = [1]_m from F_not_rp_def h2 done lemma FG_prod {m a : Nat} (h1 : rel_prime m a) : ∀ (k : Nat), prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by by_induc · -- Base Case show prod_seq 0 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from calc prod_seq 0 0 ((F m) ∘ (G m a)) _ = [1]_m := prod_seq_base _ _ _ = [a]_m ^ 0 * [1]_m := by ring _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by rw [num_rp_below_base, prod_seq_base] done · -- Induction Step fix k : Nat assume ih : prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) by_cases h2 : rel_prime m k · -- Case 1. h2 : rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([a]_m * F m k) := by rw [FG_rp h1 h2] _ = [a]_m ^ ((num_rp_below m k) + 1) * ((prod_seq k 0 (F m)) * F m k) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_rp h2, prod_seq_zero_step] done · -- Case 2. h2 : ¬rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([1]_m) := by rw [FG_not_rp h1 h2] _ = [a]_m ^ (num_rp_below m k) * (prod_seq k 0 (F m) * ([1]_m)) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_not_rp h2, prod_seq_zero_step, F_not_rp_def h2] done done done lemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by define --Goal : ∀ i < m, G m a i < m fix i : Nat assume h1 : i < m rewrite [G_def] --Goal : a * i % m < m show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m) done lemma left_inv_one_one_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g' (g i) = i) : one_one_below n g := sorry lemma right_inv_onto_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g (g' i) = i) (h2 : maps_below n g') : onto_below n g := by define at h2; define fix k : Nat assume h3 : k < n apply Exists.intro (g' k) show g' k < n ∧ g (g' k) = k from And.intro (h2 k h3) (h1 k h3) done lemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m * [inv_mod m a]_m = [1]_m := by have h2 : 0 ≤ (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a) show [a]_m * [inv_mod m a]_m = [1]_m from calc [a]_m * [inv_mod m a]_m _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2] _ = [a]_m * [gcd_c2 m a]_m := by rw [←cc_eq_mod] _ = [1]_m := gcd_c2_inv h1 done lemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m := calc a * (b % m) % m = a % m * (b % m % m) % m := Nat.mul_mod _ _ _ _ = a % m * (b % m) % m := by rw [Nat.mod_mod] _ = a * b % m := (Nat.mul_mod _ _ _).symm lemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm] rfl done theorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] : ↑a ≡ ↑b (MOD m) ↔ a % m = b % m := sorry lemma mul_inv_mod_cancel {m a i : Nat} [NeZero m] (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1 rewrite [mul_class, cc_eq_iff_congr, ←Nat.cast_mul, ←Nat.cast_one, congr_iff_mod_eq_Nat] at h3 show a * inv_mod m a * i % m = i from calc a * (inv_mod m a) * i % m _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = 1 % m * i % m := by rw [h3] _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = i % m := by rw [one_mul] _ = i := Nat.mod_eq_of_lt h2 done lemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl lemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, G m a (Ginv m a i) = i := by fix i : Nat assume h2 : i < m show G m a (Ginv m a i) = i from calc G m a (Ginv m a i) _ = a * ((inv_mod m a * i) % m) % m := by rfl _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, Ginv m a (G m a i) = i := by fix i : Nat assume h2 : i < m show Ginv m a (G m a i) = i from calc Ginv m a (G m a i) _ = inv_mod m a * ((a * i) % m) % m := by rfl _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc, mul_comm (inv_mod m a)] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_maps_below (m a : Nat) [NeZero m] : maps_below m (Ginv m a) := G_maps_below m (inv_mod m a) lemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : one_one_below m (G m a) := left_inv_one_one_below (Ginv_left_inv h1) lemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : onto_below m (G m a) := right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a) lemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : perm_below m (G m a) := And.intro (G_maps_below m a) (And.intro (G_one_one_below h1) (G_onto_below h1)) --Permuting a product of congruence classes doesn't change product lemma swap_fst (u v : Nat) : swap u v u = v := by define : swap u v u --Goal : (if u = u then v else if u = v then u else u) = v have h : u = u := by rfl rewrite [if_pos h] rfl done lemma swap_snd (u v : Nat) : swap u v v = u := by define : swap u v v by_cases h1 : v = u · -- Case 1. h1 : v = u rewrite [if_pos h1] show v = u from h1 done · -- Case 2. h1 : v ≠ u rewrite [if_neg h1] have h2 : v = v := by rfl rewrite [if_pos h2] rfl done done lemma swap_other {u v i : Nat} (h1 : i ≠ u) (h2 : i ≠ v) : swap u v i = i := by define : swap u v i rewrite [if_neg h1, if_neg h2] rfl done lemma swap_values (u v i : Nat) : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := by by_cases h1 : i = u · -- Case 1. h1 : i = u apply Or.inl rewrite [h1] show swap u v u = v from swap_fst u v done · -- Case 2. h1 : i ≠ u apply Or.inr by_cases h2 : i = v · -- Case 2.1. h2 : i = v apply Or.inl rewrite [h2] show swap u v v = u from swap_snd u v done · -- Case 2.2. h2 : i ≠ v apply Or.inr show swap u v i = i from swap_other h1 h2 done done done lemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by define fix i : Nat assume h3 : i < n have h4 : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := swap_values u v i by_cases on h4 · -- Case 1. h4 : swap u v i = v rewrite [h4] show v < n from h2 done · -- Case 2. by_cases on h4 · -- Case 2.1. h4 : swap u v i = u rewrite [h4] show u < n from h1 done · -- Case 2.2. h4 : swap u v i = i rewrite [h4] show i < n from h3 done done done lemma swap_swap (u v n : Nat) : ∀ i < n, swap u v (swap u v i) = i := by fix i : Nat assume h : i < n by_cases h1 : i = u · -- Case 1. h1 : i = u rewrite [h1, swap_fst, swap_snd] rfl done · -- Case 2. h1 : i ≠ u by_cases h2 : i = v · -- Case 2.1. h2 : i = v rewrite [h2, swap_snd, swap_fst] rfl done · -- Case 2.2. h2 : i ≠ v rewrite [swap_other h1 h2, swap_other h1 h2] rfl done done done lemma swap_one_one_below (u v n) : one_one_below n (swap u v) := left_inv_one_one_below (swap_swap u v n) lemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) := right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2) lemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) := And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2)) lemma comp_perm_below {n : Nat} {f g : Nat → Nat} (h1 : perm_below n f) (h2 : perm_below n g) : perm_below n (f ∘ g) := sorry lemma trivial_swap (u : Nat) : swap u u = id := by apply funext fix x : Nat by_cases h1 : x = u · -- Case 1. h1 : x = u rewrite [h1, swap_fst] rfl done · -- Case 2. h1 : x ≠ u rewrite [swap_other h1 h1] rfl done done lemma prod_eq_fun {m : Nat} (f g : Nat → ZMod m) (k : Nat) : ∀ (n : Nat), (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g := by by_induc · -- Base Case assume h : (∀ i < 0, f (k + i) = g (k + i)) rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g assume h1 : ∀ i < n + 1, f (k + i) = g (k + i) have h2 : ∀ i < n, f (k + i) = g (k + i) := by fix i : Nat assume h2 : i < n have h3 : i < n + 1 := by linarith show f (k + i) = g (k + i) from h1 i h3 done have h3 : prod_seq n k f = prod_seq n k g := ih h2 have h4 : n < n + 1 := Nat.lt_succ_self n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4] rfl done done lemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := by have h2 : ∀ (i : Nat), i < u → (f ∘ swap u n) (0 + i) = f (0 + i) := by fix i : Nat assume h2 : i < u have h3 : 0 + i ≠ u := by linarith have h4 : 0 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl done show prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f from prod_eq_fun (f ∘ swap u n) f 0 u h2 done lemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := by have h2 : ∀ i < j, (f ∘ swap u n) (u + 1 + i) = f (u + 1 + i) := by fix i : Nat assume h2 : i < j have h3 : u + 1 + i ≠ u := by linarith have h4 : u + 1 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl show prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f from prod_eq_fun (f ∘ swap u n) f (u + 1) j h2 done lemma break_prod {m : Nat} (n : Nat) (f : Nat → ZMod m) : ∀ (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by by_induc · -- Base Case have h : n + 0 = n := by rfl rewrite [prod_seq_base, h, Theorem_7_3_6_7] rfl done · -- Induction Step fix j : Nat assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f rewrite [←add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc] rfl done done lemma break_prod_twice {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f := break_prod n f 1 rewrite [prod_one] at h2 have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f := break_prod (u + 1) f j rewrite [←h1] at h3 have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f := break_prod u f 1 rewrite [prod_one] at h4 rewrite [h3, h4] at h2 show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2 done lemma swap_prod_eq_prod {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f := by by_cases h2 : u = n · -- Case 1. h2 : u = n rewrite [h2, trivial_swap n] --Goal : prod_seq (n + 1) 0 (f ∘ id) = prod_seq (n + 1) 0 f rfl done · -- Case 2. h2 : ¬u = n have h3 : u + 1 ≤ n := Nat.lt_of_le_of_ne h1 h2 obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3 have break_f : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := break_prod_twice f h4 have break_fs : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_prod_twice (f ∘ swap u n) h4 have f_eq_fs_below : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := swap_prod_eq_prod_below f h1 have f_eq_fs_btwn : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4 show prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f from calc prod_seq (n + 1) 0 (f ∘ swap u n) _ = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_fs _ = prod_seq u 0 f * (f ∘ swap u n) u * prod_seq j (u + 1) f * (f ∘ swap u n) n := by rw [f_eq_fs_below, f_eq_fs_btwn] _ = prod_seq u 0 f * f (swap u n u) * prod_seq j (u + 1) f * f (swap u n n) := by rfl _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by rw [swap_fst, swap_snd] _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring _ = prod_seq (n + 1) 0 f := break_f.symm done done lemma perm_below_fixed {n : Nat} {g : Nat → Nat} (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry lemma perm_prod {m : Nat} (f : Nat → ZMod m) : ∀ (n : Nat), ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) := by by_induc · -- Base Case fix g : Nat → Nat assume h1 : perm_below 0 g rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) fix g : Nat → Nat assume g_pb : perm_below (n + 1) g define at g_pb have g_ob : onto_below (n + 1) g := g_pb.right.right define at g_ob have h1 : n < n + 1 := by linarith obtain (u : Nat) (h2 : u < n + 1 ∧ g u = n) from g_ob n h1 have s_pb : perm_below (n + 1) (swap u n) := swap_perm_below h2.left h1 have gs_pb_n1 : perm_below (n + 1) (g ∘ swap u n) := comp_perm_below g_pb s_pb have gs_fix_n : (g ∘ swap u n) n = n := calc (g ∘ swap u n) n _ = g (swap u n n) := by rfl _ = g u := by rw [swap_snd] _ = n := h2.right have gs_pb_n : perm_below n (g ∘ swap u n) := perm_below_fixed gs_pb_n1 gs_fix_n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f ∘ (g ∘ swap u n)) := ih (g ∘ swap u n) gs_pb_n have h3 : u ≤ n := by linarith show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f ∘ g) from calc prod_seq (n + 1) 0 f _ = prod_seq n 0 f * f n := prod_seq_zero_step n f _ = prod_seq n 0 (f ∘ (g ∘ swap u n)) * f ((g ∘ swap u n) n) := by rw [gs_prod, gs_fix_n] _ = prod_seq n 0 (f ∘ g ∘ swap u n) * (f ∘ g ∘ swap u n) n := by rfl _ = prod_seq (n + 1) 0 (f ∘ g ∘ swap u n) := (prod_seq_zero_step n (f ∘ g ∘ swap u n)).symm _ = prod_seq (n + 1) 0 ((f ∘ g) ∘ swap u n) := by rfl _ = prod_seq (n + 1) 0 (f ∘ g) := swap_prod_eq_prod (f ∘ g) h3 done done lemma F_invertible (m i : Nat) : invertible (F m i) := by by_cases h : rel_prime m i · -- Case 1. h : rel_prime m i rewrite [F_rp_def h] show invertible [i]_m from (Theorem_7_3_7 m i).rtl h done · -- Case 2. h : ¬rel_prime m i rewrite [F_not_rp_def h] apply Exists.intro [1]_m show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m done done lemma Fprod_invertible (m : Nat) : ∀ (k : Nat), invertible (prod_seq k 0 (F m)) := by by_induc · -- Base Case apply Exists.intro [1]_m show prod_seq 0 0 (F m) * [1]_m = [1]_m from calc prod_seq 0 0 (F m) * [1]_m _ = [1]_m * [1]_m := by rw [prod_seq_base] _ = [1]_m := Theorem_7_3_6_7 ([1]_m) done · -- Induction Step fix k : Nat assume ih : invertible (prod_seq k 0 (F m)) rewrite [prod_seq_zero_step] show invertible (prod_seq k 0 (F m) * (F m k)) from (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k) done done theorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m ^ (phi m) = [1]_m := by have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2 show [a]_m ^ (phi m) = [1]_m from calc [a]_m ^ (phi m) _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3] _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring _ = prod_seq m 0 (F m ∘ G m a) * Y := by rw [FG_prod h1 m, phi_def] _ = prod_seq m 0 (F m) * Y := by rw [perm_prod (F m) m (G m a) (G_perm_below h1)] _ = [1]_m := by rw [h3] done lemma Exercise_7_4_5_Int (m : Nat) (a : Int) : ∀ (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry lemma Exercise_7_4_5_Nat (m a n : Nat) : [a]_m ^ n = [a ^ n]_m := by rewrite [Exercise_7_4_5_Int] rfl done theorem Euler's_theorem {m a : Nat} [NeZero m] (h1 : rel_prime m a) : a ^ (phi m) ≡ 1 (MOD m) := by have h2 : [a]_m ^ (phi m) = [1]_m := Theorem_7_4_2 h1 rewrite [Exercise_7_4_5_Nat m a (phi m)] at h2 --h2 : [a ^ phi m]_m = [1]_m show a ^ (phi m) ≡ 1 (MOD m) from (cc_eq_iff_congr _ _ _).ltr h2 done #eval gcd 10 7 --Answer: 1. So 10 and 7 are relatively prime #eval 7 ^ phi 10 --Answer: 2401, which is congruent to 1 mod 10. end Euler /- Section 7.5 -/ lemma num_rp_prime {p : Nat} (h1 : prime p) : ∀ k < p, num_rp_below p (k + 1) = k := sorry lemma phi_prime {p : Nat} (h1 : prime p) : phi p = p - 1 := by have h2 : 1 ≤ p := prime_pos h1 have h3 : p - 1 + 1 = p := Nat.sub_add_cancel h2 have h4 : p - 1 < p := by linarith have h5 : num_rp_below p (p - 1 + 1) = p - 1 := num_rp_prime h1 (p - 1) h4 rewrite [h3] at h5 show phi p = p - 1 from h5 done theorem Theorem_7_2_2_Int {a c : Nat} {b : Int} (h1 : ↑c ∣ ↑a * b) (h2 : rel_prime a c) : ↑c ∣ b := by rewrite [Int.natCast_dvd, Int.natAbs_mul, Int.natAbs_ofNat] at h1 --h1 : c ∣ a * Int.natAbs b rewrite [Int.natCast_dvd] --Goal : c ∣ Int.natAbs b show c ∣ Int.natAbs b from Theorem_7_2_2 h1 h2 done lemma Lemma_7_4_5 {m n : Nat} (a b : Int) (h1 : rel_prime m n) : a ≡ b (MOD m * n) ↔ a ≡ b (MOD m) ∧ a ≡ b (MOD n) := by apply Iff.intro · -- (→) assume h2 : a ≡ b (MOD m * n) obtain (j : Int) (h3 : a - b = (m * n) * j) from h2 apply And.intro · -- Proof of a ≡ b (MOD m) apply Exists.intro (n * j) show a - b = m * (n * j) from calc a - b _ = m * n * j := h3 _ = m * (n * j) := by ring done · -- Proof of a ≡ b (MOD n) apply Exists.intro (m * j) show a - b = n * (m * j) from calc a - b _ = m * n * j := h3 _ = n * (m * j) := by ring done done · -- (←) assume h2 : a ≡ b (MOD m) ∧ a ≡ b (MOD n) obtain (j : Int) (h3 : a - b = m * j) from h2.left have h4 : (↑n : Int) ∣ a - b := h2.right rewrite [h3] at h4 --h4 : ↑n ∣ ↑m * j have h5 : ↑n ∣ j := Theorem_7_2_2_Int h4 h1 obtain (k : Int) (h6 : j = n * k) from h5 apply Exists.intro k --Goal : a - b = ↑(m * n) * k rewrite [Nat.cast_mul] --Goal : a - b = ↑m * ↑n * k show a - b = (m * n) * k from calc a - b _ = m * j := h3 _ = m * (n * k) := by rw [h6] _ = (m * n) * k := by ring done done --From exercises of Section 7.2 theorem rel_prime_symm {a b : Nat} (h : rel_prime a b) : rel_prime b a := sorry lemma prime_NeZero {p : Nat} (h : prime p) : NeZero p := by rewrite [neZero_iff] --Goal : p ≠ 0 define at h linarith done lemma Lemma_7_5_1 {p e d m c s : Nat} {t : Int} (h1 : prime p) (h2 : e * d = (p - 1) * s + 1) (h3 : m ^ e - c = p * t) : c ^ d ≡ m (MOD p) := by have h4 : m ^ e ≡ c (MOD p) := Exists.intro t h3 have h5 : [m ^ e]_p = [c]_p := (cc_eq_iff_congr _ _ _).rtl h4 rewrite [←Exercise_7_4_5_Nat] at h5 --h5 : [m]_p ^ e = [c]_p by_cases h6 : p ∣ m · -- Case 1. h6 : p ∣ m have h7 : m ≡ 0 (MOD p) := by obtain (j : Nat) (h8 : m = p * j) from h6 apply Exists.intro (↑j : Int) --Goal : ↑m - 0 = ↑p * ↑j rewrite [h8, Nat.cast_mul] ring done have h8 : [m]_p = [0]_p := (cc_eq_iff_congr _ _ _).rtl h7 have h9 : e * d ≠ 0 := by rewrite [h2] show (p - 1) * s + 1 ≠ 0 from Nat.add_one_ne_zero _ done have h10 : (0 : Int) ^ (e * d) = 0 := zero_pow h9 have h11 : [c ^ d]_p = [m]_p := calc [c ^ d]_p _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat] _ = ([m]_p ^ e) ^ d := by rw [h5] _ = [m]_p ^ (e * d) := by ring _ = [0]_p ^ (e * d) := by rw [h8] _ = [0 ^ (e * d)]_p := Exercise_7_4_5_Int _ _ _ _ = [0]_p := by rw [h10] _ = [m]_p := by rw [h8] show c ^ d ≡ m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11 done · -- Case 2. h6 : ¬p ∣ m have h7 : rel_prime m p := rel_prime_of_prime_not_dvd h1 h6 have h8 : rel_prime p m := rel_prime_symm h7 have h9 : NeZero p := prime_NeZero h1 have h10 : (1 : Int) ^ s = 1 := by ring have h11 : [c ^ d]_p = [m]_p := calc [c ^ d]_p _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat] _ = ([m]_p ^ e) ^ d := by rw [h5] _ = [m]_p ^ (e * d) := by ring _ = [m]_p ^ ((p - 1) * s + 1) := by rw [h2] _ = ([m]_p ^ (p - 1)) ^ s * [m]_p := by ring _ = ([m]_p ^ (phi p)) ^ s * [m]_p := by rw [phi_prime h1] _ = [1]_p ^ s * [m]_p := by rw [Theorem_7_4_2 h8] _ = [1 ^ s]_p * [m]_p := by rw [Exercise_7_4_5_Int] _ = [1]_p * [m]_p := by rw [h10] _ = [m]_p * [1]_p := by ring _ = [m]_p := Theorem_7_3_6_7 _ show c ^ d ≡ m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11 done done theorem Theorem_7_5_1 (p q n e d k m c : Nat) (p_prime : prime p) (q_prime : prime q) (p_ne_q : p ≠ q) (n_pq : n = p * q) (ed_congr_1 : e * d = k * (p - 1) * (q - 1) + 1) (h1 : [m]_n ^ e = [c]_n) : [c]_n ^ d = [m]_n := by rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] at h1 --h1 : m ^ e ≡ c (MOD n) rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] --Goal : c ^ d ≡ m (MOD n) obtain (j : Int) (h2 : m ^ e - c = n * j) from h1 rewrite [n_pq, Nat.cast_mul] at h2 --h2 : m ^ e - c = p * q * j have h3 : e * d = (p - 1) * (k * (q - 1)) + 1 := by rewrite [ed_congr_1] ring done have h4 : m ^ e - c = p * (q * j) := by rewrite [h2] ring done have congr_p : c ^ d ≡ m (MOD p) := Lemma_7_5_1 p_prime h3 h4 have h5 : e * d = (q - 1) * (k * (p - 1)) + 1 := by rewrite [ed_congr_1] ring done have h6 : m ^ e - c = q * (p * j) := by rewrite [h2] ring done have congr_q : c ^ d ≡ m (MOD q) := Lemma_7_5_1 q_prime h5 h6 have h7 : ¬q ∣ p := by by_contra h8 have h9 : q = 1 ∨ q = p := dvd_prime p_prime h8 disj_syll h9 (prime_not_one q_prime) show False from p_ne_q h9.symm done have h8 : rel_prime p q := rel_prime_of_prime_not_dvd q_prime h7 rewrite [n_pq, Lemma_7_4_5 _ _ h8] show c ^ d ≡ m (MOD p) ∧ c ^ d ≡ m (MOD q) from And.intro congr_p congr_q done /- BEGIN EXERCISES -/ namespace Exercises /- Section 7.1 -/ -- 1. theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry -- 2. lemma gcd_comm_lt {a b : Nat} (h : a < b) : gcd a b = gcd b a := sorry theorem gcd_comm (a b : Nat) : gcd a b = gcd b a := sorry -- 3. theorem Exercise_7_1_5 (a b : Nat) (n : Int) : (∃ (s t : Int), s * a + t * b = n) ↔ (↑(gcd a b) : Int) ∣ n := sorry -- 4. theorem Exercise_7_1_6 (a b c : Nat) : gcd a b = gcd (a + b * c) b := sorry -- 5. theorem gcd_is_nonzero {a b : Nat} (h : a ≠ 0 ∨ b ≠ 0) : gcd a b ≠ 0 := sorry -- 6. theorem gcd_greatest {a b d : Nat} (h1 : gcd a b ≠ 0) (h2 : d ∣ a) (h3 : d ∣ b) : d ≤ gcd a b := sorry -- 7. lemma Lemma_7_1_10a {a b : Nat} (n : Nat) (h : a ∣ b) : (n * a) ∣ (n * b) := sorry lemma Lemma_7_1_10b {a b n : Nat} (h1 : n ≠ 0) (h2 : (n * a) ∣ (n * b)) : a ∣ b := sorry lemma Lemma_7_1_10c {a b : Nat} (h1 : a ∣ b) (h2 : b ∣ a) : a = b := sorry theorem Exercise_7_1_10 (a b n : Nat) : gcd (n * a) (n * b) = n * gcd a b := sorry /- Section 7.2 -/ -- 1. lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry -- 2. -- Hints: Start with apply List.rec. You may find mul_ne_zero useful theorem prod_nonzero_nonzero : ∀ (l : List Nat), (∀ a ∈ l, a ≠ 0) → prod l ≠ 0 := sorry -- 3. theorem rel_prime_iff_no_common_factor (a b : Nat) : rel_prime a b ↔ ¬∃ (p : Nat), prime p ∧ p ∣ a ∧ p ∣ b := sorry -- 4. theorem rel_prime_symm {a b : Nat} (h : rel_prime a b) : rel_prime b a := sorry -- 5. lemma in_prime_factorization_iff_prime_factor {a : Nat} {l : List Nat} (h1 : prime_factorization a l) (p : Nat) : p ∈ l ↔ prime_factor p a := sorry -- 6. theorem Exercise_7_2_5 {a b : Nat} {l m : List Nat} (h1 : prime_factorization a l) (h2 : prime_factorization b m) : rel_prime a b ↔ (¬∃ (p : Nat), p ∈ l ∧ p ∈ m) := sorry -- 7. theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry -- 8. theorem Exercise_7_2_7 {a b a' b' : Nat} (h1 : rel_prime a b) (h2 : a' ∣ a) (h3 : b' ∣ b) : rel_prime a' b' := sorry -- 9. theorem Exercise_7_2_9 {a b j k : Nat} (h1 : gcd a b ≠ 0) (h2 : a = j * gcd a b) (h3 : b = k * gcd a b) : rel_prime j k := sorry -- 10. theorem Exercise_7_2_17a (a b c : Nat) : gcd a (b * c) ∣ gcd a b * gcd a c := sorry /- Section 7.3 -/ -- 1. theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry -- 2. theorem Theorem_7_3_6_3 {m : Nat} (X : ZMod m) : X + [0]_m = X := sorry -- 3. theorem Theorem_7_3_6_4 {m : Nat} (X : ZMod m) : ∃ (Y : ZMod m), X + Y = [0]_m := sorry -- 4. theorem Exercise_7_3_4a {m : Nat} (Z1 Z2 : ZMod m) (h1 : ∀ (X : ZMod m), X + Z1 = X) (h2 : ∀ (X : ZMod m), X + Z2 = X) : Z1 = Z2 := sorry -- 5. theorem Exercise_7_3_4b {m : Nat} (X Y1 Y2 : ZMod m) (h1 : X + Y1 = [0]_m) (h2 : X + Y2 = [0]_m) : Y1 = Y2 := sorry -- 6. theorem Theorem_7_3_10 (m a : Nat) (b : Int) : ¬(↑(gcd m a) : Int) ∣ b → ¬∃ (x : Int), a * x ≡ b (MOD m) := sorry -- 7. theorem Theorem_7_3_11 (m n : Nat) (a b : Int) (h1 : n ≠ 0) : n * a ≡ n * b (MOD n * m) ↔ a ≡ b (MOD m) := sorry -- 8. theorem Exercise_7_3_16 {m : Nat} {a b : Int} (h : a ≡ b (MOD m)) : ∀ (n : Nat), a ^ n ≡ b ^ n (MOD m) := sorry -- 9. example {m : Nat} [NeZero m] (X : ZMod m) : ∃! (a : Int), 0 ≤ a ∧ a < m ∧ X = [a]_m := sorry -- 10. theorem congr_rel_prime {m a b : Nat} (h1 : a ≡ b (MOD m)) : rel_prime m a ↔ rel_prime m b := sorry -- 11. --Hint: You may find the theorem Int.ofNat_mod_ofNat useful. theorem rel_prime_mod (m a : Nat) : rel_prime m (a % m) ↔ rel_prime m a := sorry -- 12. lemma congr_iff_mod_eq_Int (m : Nat) (a b : Int) [NeZero m] : a ≡ b (MOD m) ↔ a % ↑m = b % ↑m := sorry --Hint for next theorem: Use the lemma above, --together with the theorems Int.ofNat_mod_ofNat and Nat.cast_inj. theorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] : ↑a ≡ ↑b (MOD m) ↔ a % m = b % m := sorry /- Section 7.4 -/ -- 1. --Hint: Use induction. --For the base case, compute [a]_m ^ 0 * [1]_m in two ways: --by Theorem_7_3_6_7, [a] ^ 0 * [1]_m = [a]_m ^ 0 --by ring, [a]_m ^ 0 * [1]_m = [1]_m. lemma Exercise_7_4_5_Int (m : Nat) (a : Int) : ∀ (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry -- 2. lemma left_inv_one_one_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g' (g i) = i) : one_one_below n g := sorry -- 3. lemma comp_perm_below {n : Nat} {f g : Nat → Nat} (h1 : perm_below n f) (h2 : perm_below n g) : perm_below n (f ∘ g) := sorry -- 4. lemma perm_below_fixed {n : Nat} {g : Nat → Nat} (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry -- 5. lemma Lemma_7_4_6 {a b c : Nat} : rel_prime (a * b) c ↔ rel_prime a c ∧ rel_prime b c := sorry -- 6. example {m a : Nat} [NeZero m] (h1 : rel_prime m a) : a ^ (phi m + 1) ≡ a (MOD m) := sorry -- 7. theorem Like_Exercise_7_4_11 {m a p : Nat} [NeZero m] (h1 : rel_prime m a) (h2 : p + 1 = phi m) : [a]_m * [a ^ p]_m = [1]_m := sorry -- 8. theorem Like_Exercise_7_4_12 {m a p q k : Nat} [NeZero m] (h1 : rel_prime m a) (h2 : p = q + (phi m) * k) : a ^ p ≡ a ^ q (MOD m) := sorry /- Section 7.5 -/ -- 1. --Hint: Use induction.

lemma num_rp_prime {p : Nat} (h1 : prime p) : ∀ k < p, num_rp_below p (k + 1) = k

HTPI.Exercises.num_rp_prime

htpi/HTPILib/Chap7.lean

htpi

/- Copyright 2023 Daniel J. Velleman -/ import HTPILib.Chap6 namespace HTPI /- Definitions -/ lemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by have h : n + 1 > 0 := Nat.succ_pos n show a % (n + 1) < n + 1 from Nat.mod_lt a h done def gcd (a b : Nat) : Nat := match b with | 0 => a | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd (n + 1) (a % (n + 1)) termination_by b mutual def gcd_c1 (a b : Nat) : Int := match b with | 0 => 1 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c2 (n + 1) (a % (n + 1)) --Corresponds to s = t' termination_by b def gcd_c2 (a b : Nat) : Int := match b with | 0 => 0 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c1 (n + 1) (a % (n + 1)) - (gcd_c2 (n + 1) (a % (n + 1))) * ↑(a / (n + 1)) --Corresponds to t = s' - t'q termination_by b end def prime (n : Nat) : Prop := 2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n def prime_factor (p n : Nat) : Prop := prime p ∧ p ∣ n def all_prime (l : List Nat) : Prop := ∀ p ∈ l, prime p def nondec (l : List Nat) : Prop := match l with | [] => True --Of course, True is a proposition that is always true | n :: L => (∀ m ∈ L, n ≤ m) ∧ nondec L def nondec_prime_list (l : List Nat) : Prop := all_prime l ∧ nondec l def prod (l : List Nat) : Nat := match l with | [] => 1 | n :: L => n * (prod L) def prime_factorization (n : Nat) (l : List Nat) : Prop := nondec_prime_list l ∧ prod l = n def rel_prime (a b : Nat) : Prop := gcd a b = 1 def congr_mod (m : Nat) (a b : Int) : Prop := (↑m : Int) ∣ (a - b) def cc (m : Nat) (a : Int) : ZMod m := (↑a : ZMod m) notation:50 a " ≡ " b " (MOD " m ")" => congr_mod m a b notation:max "["a"]_"m:max => cc m a def invertible {m : Nat} (X : ZMod m) : Prop := ∃ (Y : ZMod m), X * Y = [1]_m def num_rp_below (m k : Nat) : Nat := match k with | 0 => 0 | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j def phi (m : Nat) : Nat := num_rp_below m m def prod_seq {m : Nat} (j k : Nat) (f : Nat → ZMod m) : ZMod m := match j with | 0 => [1]_m | n + 1 => prod_seq n k f * f (k + n) def maps_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i < n, g i < n def one_one_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i1 < n, ∀ i2 < n, g i1 = g i2 → i1 = i2 def onto_below (n : Nat) (g : Nat → Nat) : Prop := ∀ k < n, ∃ i < n, g i = k def perm_below (n : Nat) (g : Nat → Nat) : Prop := maps_below n g ∧ one_one_below n g ∧ onto_below n g def inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m) def swap (u v i : Nat) : Nat := if i = u then v else if i = v then u else i namespace Euler --For definitions specific to Euler's theorem def F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m def G (m a i : Nat) : Nat := (a * i) % m def Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i end Euler /- Section 7.1 -/ theorem dvd_mod_of_dvd_a_b {a b d : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ (a % b) := by set q : Nat := a / b have h3 : b * q + a % b = a := Nat.div_add_mod a b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 define --Goal : ∃ (c : Nat), a % b = d * c apply Exists.intro (j - k * q) show a % b = d * (j - k * q) from calc a % b _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm _ = a - b * q := by rw [h3] _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc] _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm done theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry #eval gcd 672 161 --Answer: 7 lemma gcd_base (a : Nat) : gcd a 0 = a := by rfl lemma gcd_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd a b = gcd b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1)) rfl done lemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b ≠ 0) : a % b < b := by have h1 : b > 0 := Nat.pos_of_ne_zero h show a % b < b from Nat.mod_lt a h1 done lemma dvd_self (n : Nat) : n ∣ n := by apply Exists.intro 1 ring done theorem gcd_dvd : ∀ (b a : Nat), (gcd a b) ∣ a ∧ (gcd a b) ∣ b := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd a b_1) ∣ a ∧ (gcd a b_1) ∣ b_1 fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_base] --Goal: a ∣ a ∧ a ∣ 0 apply And.intro (dvd_self a) define apply Exists.intro 0 rfl done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_nonzero a h1] --Goal : gcd b (a % b) ∣ a ∧ gcd b (a % b) ∣ b have h2 : a % b < b := mod_nonzero_lt a h1 have h3 : (gcd b (a % b)) ∣ b ∧ (gcd b (a % b)) ∣ (a % b) := ih (a % b) h2 b apply And.intro _ h3.left show (gcd b (a % b)) ∣ a from dvd_a_of_dvd_b_mod h3.left h3.right done done theorem gcd_dvd_left (a b : Nat) : (gcd a b) ∣ a := (gcd_dvd b a).left theorem gcd_dvd_right (a b : Nat) : (gcd a b) ∣ b := (gcd_dvd b a).right lemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl lemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c1 a b = gcd_c2 b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done lemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl lemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * ↑(a / b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done theorem gcd_lin_comb : ∀ (b a : Nat), (gcd_c1 a b) * ↑a + (gcd_c2 a b) * ↑b = ↑(gcd a b) := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd_c1 a b_1) * ↑a + (gcd_c2 a b_1) * ↑b_1 = ↑(gcd a b_1) fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base] --Goal : 1 * ↑a + 0 * ↑0 = ↑a ring done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1] --Goal : gcd_c2 b (a % b) * ↑a + -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * ↑(a / b)) * ↑b = -- ↑(gcd b (a % b)) set r : Nat := a % b set q : Nat := a / b set s : Int := gcd_c1 b r set t : Int := gcd_c2 b r --Goal : t * ↑a + (s - t * ↑q) * ↑b = ↑(gcd b r) have h2 : r < b := mod_nonzero_lt a h1 have h3 : s * ↑b + t * ↑r = ↑(gcd b r) := ih r h2 b have h4 : b * q + r = a := Nat.div_add_mod a b rewrite [←h3, ←h4] rewrite [Nat.cast_add, Nat.cast_mul] --Goal : t * (↑b * ↑q + ↑r) + (s - t * ↑q) * ↑b = s * ↑b + t * ↑r ring done done #eval gcd_c1 672 161 --Answer: 6 #eval gcd_c2 672 161 --Answer: -25 --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161 theorem Theorem_7_1_6 {d a b : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ gcd a b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑d ∣ ↑(gcd a b) set s : Int := gcd_c1 a b set t : Int := gcd_c2 a b have h3 : s * ↑a + t * ↑b = ↑(gcd a b) := gcd_lin_comb b a rewrite [←h3] --Goal : ↑d ∣ s * ↑a + t * ↑b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul] --Goal : ↑d ∣ s * (↑d * ↑j) + t * (↑d * ↑k) define apply Exists.intro (s * ↑j + t * ↑k) ring done /- Section 7.2 -/ theorem dvd_trans {a b c : Nat} (h1 : a ∣ b) (h2 : b ∣ c) : a ∣ c := by define at h1; define at h2; define obtain (m : Nat) (h3 : b = a * m) from h1 obtain (n : Nat) (h4 : c = b * n) from h2 rewrite [h3, mul_assoc] at h4 apply Exists.intro (m * n) show c = a * (m * n) from h4 done lemma exists_prime_factor : ∀ (n : Nat), 2 ≤ n → ∃ (p : Nat), prime_factor p n := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, 2 ≤ n_1 → ∃ (p : Nat), prime_factor p n_1 assume h1 : 2 ≤ n by_cases h2 : prime n · -- Case 1. h2 : prime n apply Exists.intro n define --Goal : prime n ∧ n ∣ n show prime n ∧ n ∣ n from And.intro h2 (dvd_self n) done · -- Case 2. h2 : ¬prime n define at h2 --h2 : ¬(2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n) demorgan at h2 disj_syll h2 h1 obtain (a : Nat) (h3 : ∃ (b : Nat), a * b = n ∧ a < n ∧ b < n) from h2 obtain (b : Nat) (h4 : a * b = n ∧ a < n ∧ b < n) from h3 have h5 : 2 ≤ a := by by_contra h6 have h7 : a ≤ 1 := by linarith have h8 : n ≤ b := calc n _ = a * b := h4.left.symm _ ≤ 1 * b := by rel [h7] _ = b := by ring linarith --n ≤ b contradicts b < n done have h6 : ∃ (p : Nat), prime_factor p a := ih a h4.right.left h5 obtain (p : Nat) (h7 : prime_factor p a) from h6 apply Exists.intro p define --Goal : prime p ∧ p ∣ n define at h7 --h7 : prime p ∧ p ∣ a apply And.intro h7.left have h8 : a ∣ n := by apply Exists.intro b show n = a * b from (h4.left).symm done show p ∣ n from dvd_trans h7.right h8 done done lemma exists_least_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q := by set S : Set Nat := {p : Nat | prime_factor p n} have h2 : ∃ (p : Nat), p ∈ S := exists_prime_factor n h show ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q from well_ord_princ S h2 done lemma all_prime_nil : all_prime [] := by define --Goal : ∀ p ∈ [], prime p fix p : Nat contrapos --Goal : ¬prime p → p ∉ [] assume h1 : ¬prime p show p ∉ [] from List.not_mem_nil p done lemma all_prime_cons (n : Nat) (L : List Nat) : all_prime (n :: L) ↔ prime n ∧ all_prime L := by apply Iff.intro · -- (→) assume h1 : all_prime (n :: L) --Goal : prime n ∧ all_prime L define at h1 --h1 : ∀ p ∈ n :: L, prime p apply And.intro (h1 n (List.mem_cons_self n L)) define --Goal : ∀ p ∈ L, prime p fix p : Nat assume h2 : p ∈ L show prime p from h1 p (List.mem_cons_of_mem n h2) done · -- (←) assume h1 : prime n ∧ all_prime L --Goal : all_prime (n :: l) define : all_prime L at h1 define fix p : Nat assume h2 : p ∈ n :: L rewrite [List.mem_cons] at h2 --h2 : p = n ∨ p ∈ L by_cases on h2 · -- Case 1. h2 : p = n rewrite [h2] show prime n from h1.left done · -- Case 2. h2 : p ∈ L show prime p from h1.right p h2 done done done lemma nondec_nil : nondec [] := by define --Goal : True trivial --trivial proves some obviously true statements, such as True done lemma nondec_cons (n : Nat) (L : List Nat) : nondec (n :: L) ↔ (∀ m ∈ L, n ≤ m) ∧ nondec L := by rfl lemma prod_nil : prod [] = 1 := by rfl lemma prod_cons : prod (n :: L) = n * (prod L) := by rfl lemma exists_cons_of_length_eq_succ {A : Type} {l : List A} {n : Nat} (h : l.length = n + 1) : ∃ (a : A) (L : List A), l = a :: L ∧ L.length = n := by have h1 : ¬l.length = 0 := by linarith rewrite [List.length_eq_zero] at h1 obtain (a : A) (h2 : ∃ (L : List A), l = a :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List A) (h3 : l = a :: L) from h2 apply Exists.intro a apply Exists.intro L apply And.intro h3 have h4 : (a :: L).length = L.length + 1 := List.length_cons a L rewrite [←h3, h] at h4 show L.length = n from (Nat.add_right_cancel h4).symm done lemma list_elt_dvd_prod_by_length (a : Nat) : ∀ (n : Nat), ∀ (l : List Nat), l.length = n → a ∈ l → a ∣ prod l := by by_induc · --Base Case fix l : List Nat assume h1 : l.length = 0 rewrite [List.length_eq_zero] at h1 --h1 : l = [] rewrite [h1] --Goal : a ∈ [] → a ∣ prod [] contrapos assume h2 : ¬a ∣ prod [] show a ∉ [] from List.not_mem_nil a done · -- Induction Step fix n : Nat assume ih : ∀ (l : List Nat), List.length l = n → a ∈ l → a ∣ prod l fix l : List Nat assume h1 : l.length = n + 1 --Goal : a ∈ l → a ∣ prod l obtain (b : Nat) (h2 : ∃ (L : List Nat), l = b :: L ∧ L.length = n) from exists_cons_of_length_eq_succ h1 obtain (L : List Nat) (h3 : l = b :: L ∧ L.length = n) from h2 have h4 : a ∈ L → a ∣ prod L := ih L h3.right assume h5 : a ∈ l rewrite [h3.left, prod_cons] --Goal : a ∣ b * prod L rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b ∨ a ∈ L by_cases on h5 · -- Case 1. h5 : a = b apply Exists.intro (prod L) rewrite [h5] rfl done · -- Case 2. h5 : a ∈ L have h6 : a ∣ prod L := h4 h5 have h7 : prod L ∣ b * prod L := by apply Exists.intro b ring done show a ∣ b * prod L from dvd_trans h6 h7 done done done lemma list_elt_dvd_prod {a : Nat} {l : List Nat} (h : a ∈ l) : a ∣ prod l := by set n : Nat := l.length have h1 : l.length = n := by rfl show a ∣ prod l from list_elt_dvd_prod_by_length a n l h1 h done lemma exists_prime_factorization : ∀ (n : Nat), n ≥ 1 → ∃ (l : List Nat), prime_factorization n l := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, n_1 ≥ 1 → ∃ (l : List Nat), prime_factorization n_1 l assume h1 : n ≥ 1 by_cases h2 : n = 1 · -- Case 1. h2 : n = 1 apply Exists.intro [] define apply And.intro · -- Proof of nondec_prime_list [] define show all_prime [] ∧ nondec [] from And.intro all_prime_nil nondec_nil done · -- Proof of prod [] = n rewrite [prod_nil, h2] rfl done done · -- Case 2. h2 : n ≠ 1 have h3 : n ≥ 2 := lt_of_le_of_ne' h1 h2 obtain (p : Nat) (h4 : prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q) from exists_least_prime_factor h3 have p_prime_factor : prime_factor p n := h4.left define at p_prime_factor have p_prime : prime p := p_prime_factor.left have p_dvd_n : p ∣ n := p_prime_factor.right have p_least : ∀ (q : Nat), prime_factor q n → p ≤ q := h4.right obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n have h5 : m ≠ 0 := by contradict h1 with h6 have h7 : n = 0 := calc n _ = p * m := n_eq_pm _ = p * 0 := by rw [h6] _ = 0 := by ring rewrite [h7] decide done have m_pos : 0 < m := Nat.pos_of_ne_zero h5 have m_lt_n : m < n := by define at p_prime show m < n from calc m _ < m + m := by linarith _ = 2 * m := by ring _ ≤ p * m := by rel [p_prime.left] _ = n := n_eq_pm.symm done obtain (L : List Nat) (h6 : prime_factorization m L) from ih m m_lt_n m_pos define at h6 have ndpl_L : nondec_prime_list L := h6.left define at ndpl_L apply Exists.intro (p :: L) define apply And.intro · -- Proof of nondec_prime_list (p :: L) define apply And.intro · -- Proof of all_prime (p :: L) rewrite [all_prime_cons] show prime p ∧ all_prime L from And.intro p_prime ndpl_L.left done · -- Proof of nondec (p :: L) rewrite [nondec_cons] apply And.intro _ ndpl_L.right fix q : Nat assume q_in_L : q ∈ L have h7 : q ∣ prod L := list_elt_dvd_prod q_in_L rewrite [h6.right] at h7 --h7 : q ∣ m have h8 : m ∣ n := by apply Exists.intro p rewrite [n_eq_pm] ring done have q_dvd_n : q ∣ n := dvd_trans h7 h8 have ap_L : all_prime L := ndpl_L.left define at ap_L have q_prime_factor : prime_factor q n := And.intro (ap_L q q_in_L) q_dvd_n show p ≤ q from p_least q q_prime_factor done done · -- Proof of prod (p :: L) = n rewrite [prod_cons, h6.right, n_eq_pm] rfl done done done theorem Theorem_7_2_2 {a b c : Nat} (h1 : c ∣ a * b) (h2 : rel_prime a c) : c ∣ b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑c ∣ ↑b define at h1; define at h2; define obtain (j : Nat) (h3 : a * b = c * j) from h1 set s : Int := gcd_c1 a c set t : Int := gcd_c2 a c have h4 : s * ↑a + t * ↑c = ↑(gcd a c) := gcd_lin_comb c a rewrite [h2, Nat.cast_one] at h4 --h4 : s * ↑a + t * ↑c = (1 : Int) apply Exists.intro (s * ↑j + t * ↑b) show ↑b = ↑c * (s * ↑j + t * ↑b) from calc ↑b _ = (1 : Int) * ↑b := (one_mul _).symm _ = (s * ↑a + t * ↑c) * ↑b := by rw [h4] _ = s * (↑a * ↑b) + t * ↑c * ↑b := by ring _ = s * (↑c * ↑j) + t * ↑c * ↑b := by rw [←Nat.cast_mul a b, h3, Nat.cast_mul c j] _ = ↑c * (s * ↑j + t * ↑b) := by ring done lemma le_nonzero_prod_left {a b : Nat} (h : a * b ≠ 0) : a ≤ a * b := by have h1 : b ≠ 0 := by contradict h with h1 rewrite [h1] ring done have h2 : 1 ≤ b := Nat.pos_of_ne_zero h1 show a ≤ a * b from calc a = a * 1 := (mul_one a).symm _ ≤ a * b := by rel [h2] done lemma le_nonzero_prod_right {a b : Nat} (h : a * b ≠ 0) : b ≤ a * b := by rewrite [mul_comm] rewrite [mul_comm] at h show b ≤ b * a from le_nonzero_prod_left h done lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry lemma rel_prime_of_prime_not_dvd {a p : Nat} (h1 : prime p) (h2 : ¬p ∣ a) : rel_prime a p := by have h3 : gcd a p ∣ a := gcd_dvd_left a p have h4 : gcd a p ∣ p := gcd_dvd_right a p have h5 : gcd a p = 1 ∨ gcd a p = p := dvd_prime h1 h4 have h6 : gcd a p ≠ p := by contradict h2 with h6 rewrite [h6] at h3 show p ∣ a from h3 done disj_syll h5 h6 show rel_prime a p from h5 done theorem Theorem_7_2_3 {a b p : Nat} (h1 : prime p) (h2 : p ∣ a * b) : p ∣ a ∨ p ∣ b := by or_right with h3 have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3 show p ∣ b from Theorem_7_2_2 h2 h4 done lemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a ≥ 1 := by have h1 : a ≠ 0 := by by_contra h1 rewrite [h1] at h contradict h linarith done show a ≥ 1 from Nat.pos_of_ne_zero h1 done lemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by have h1 : a ≥ 1 := ge_one_of_prod_one h have h2 : a * b ≠ 0 := by linarith have h3 : a ≤ a * b := le_nonzero_prod_left h2 rewrite [h] at h3 show a = 1 from Nat.le_antisymm h3 h1 done lemma eq_one_of_dvd_one {n : Nat} (h : n ∣ 1) : n = 1 := by obtain (j : Nat) (h1 : 1 = n * j) from h show n = 1 from eq_one_of_prod_one h1.symm done lemma prime_not_one {p : Nat} (h : prime p) : p ≠ 1 := by define at h linarith done theorem Theorem_7_2_4 {p : Nat} (h1 : prime p) : ∀ (l : List Nat), p ∣ prod l → ∃ a ∈ l, p ∣ a := by apply List.rec · -- Base Case. Goal : p ∣ prod [] → ∃ a ∈ [], p ∣ a rewrite [prod_nil] assume h2 : p ∣ 1 show ∃ a ∈ [], p ∣ a from absurd (eq_one_of_dvd_one h2) (prime_not_one h1) done · -- Induction Step fix b : Nat fix L : List Nat assume ih : p ∣ prod L → ∃ a ∈ L, p ∣ a --Goal : p ∣ prod (b :: L) → ∃ a ∈ b :: L, p ∣ a assume h2 : p ∣ prod (b :: L) rewrite [prod_cons] at h2 have h3 : p ∣ b ∨ p ∣ prod L := Theorem_7_2_3 h1 h2 by_cases on h3 · -- Case 1. h3 : p ∣ b apply Exists.intro b show b ∈ b :: L ∧ p ∣ b from And.intro (List.mem_cons_self b L) h3 done · -- Case 2. h3 : p ∣ prod L obtain (a : Nat) (h4 : a ∈ L ∧ p ∣ a) from ih h3 apply Exists.intro a show a ∈ b :: L ∧ p ∣ a from And.intro (List.mem_cons_of_mem b h4.left) h4.right done done done lemma prime_in_list {p : Nat} {l : List Nat} (h1 : prime p) (h2 : all_prime l) (h3 : p ∣ prod l) : p ∈ l := by obtain (a : Nat) (h4 : a ∈ l ∧ p ∣ a) from Theorem_7_2_4 h1 l h3 define at h2 have h5 : prime a := h2 a h4.left have h6 : p = 1 ∨ p = a := dvd_prime h5 h4.right disj_syll h6 (prime_not_one h1) rewrite [h6] show a ∈ l from h4.left done lemma first_le_first {p q : Nat} {l m : List Nat} (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m)) (h3 : prod (p :: l) = prod (q :: m)) : p ≤ q := by define at h1; define at h2 have h4 : q ∣ prod (p :: l) := by define apply Exists.intro (prod m) rewrite [←prod_cons] show prod (p :: l) = prod (q :: m) from h3 done have h5 : all_prime (q :: m) := h2.left rewrite [all_prime_cons] at h5 have h6 : q ∈ p :: l := prime_in_list h5.left h1.left h4 have h7 : nondec (p :: l) := h1.right rewrite [nondec_cons] at h7 rewrite [List.mem_cons] at h6 by_cases on h6 · -- Case 1. h6 : q = p linarith done · -- Case 2. h6 : q ∈ l have h8 : ∀ m ∈ l, p ≤ m := h7.left show p ≤ q from h8 q h6 done done lemma nondec_prime_list_tail {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by define at h define rewrite [all_prime_cons, nondec_cons] at h show all_prime l ∧ nondec l from And.intro h.left.right h.right.right done lemma cons_prod_not_one {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : prod (p :: l) ≠ 1 := by define at h have h1 : all_prime (p :: l) := h.left rewrite [all_prime_cons] at h1 rewrite [prod_cons] by_contra h2 show False from (prime_not_one h1.left) (eq_one_of_prod_one h2) done lemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) : l = [] ↔ prod l = 1 := by apply Iff.intro · -- (→) assume h1 : l = [] rewrite [h1] show prod [] = 1 from prod_nil done · -- (←) contrapos assume h1 : ¬l = [] obtain (p : Nat) (h2 : ∃ (L : List Nat), l = p :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List Nat) (h3 : l = p :: L) from h2 rewrite [h3] at h rewrite [h3] show ¬prod (p :: L) = 1 from cons_prod_not_one h done done lemma prime_pos {p : Nat} (h : prime p) : p > 0 := by define at h linarith done theorem Theorem_7_2_5 : ∀ (l1 l2 : List Nat), nondec_prime_list l1 → nondec_prime_list l2 → prod l1 = prod l2 → l1 = l2 := by apply List.rec · -- Base Case. Goal : ∀ (l2 : List Nat), nondec_prime_list [] → -- nondec_prime_list l2 → prod [] = prod l2 → [] = l2 fix l2 : List Nat assume h1 : nondec_prime_list [] assume h2 : nondec_prime_list l2 assume h3 : prod [] = prod l2 rewrite [prod_nil, eq_comm, ←list_nil_iff_prod_one h2] at h3 show [] = l2 from h3.symm done · -- Induction Step fix p : Nat fix L1 : List Nat assume ih : ∀ (L2 : List Nat), nondec_prime_list L1 → nondec_prime_list L2 → prod L1 = prod L2 → L1 = L2 -- Goal : ∀ (l2 : List Nat), nondec_prime_list (p :: L1) → -- nondec_prime_list l2 → prod (p :: L1) = prod l2 → p :: L1 = l2 fix l2 : List Nat assume h1 : nondec_prime_list (p :: L1) assume h2 : nondec_prime_list l2 assume h3 : prod (p :: L1) = prod l2 have h4 : ¬prod (p :: L1) = 1 := cons_prod_not_one h1 rewrite [h3, ←list_nil_iff_prod_one h2] at h4 obtain (q : Nat) (h5 : ∃ (L : List Nat), l2 = q :: L) from List.exists_cons_of_ne_nil h4 obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5 rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2) rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2) have h7 : p ≤ q := first_le_first h1 h2 h3 have h8 : q ≤ p := first_le_first h2 h1 h3.symm have h9 : p = q := by linarith rewrite [h9, prod_cons, prod_cons] at h3 --h3 : q * prod L1 = q * prod L2 have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1 have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2 define at h2 have h12 : all_prime (q :: L2) := h2.left rewrite [all_prime_cons] at h12 have h13 : q > 0 := prime_pos h12.left have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3 have h15 : L1 = L2 := ih L2 h10 h11 h14 rewrite [h6, h9, h15] rfl done done theorem fund_thm_arith (n : Nat) (h : n ≥ 1) : ∃! (l : List Nat), prime_factorization n l := by exists_unique · -- Existence show ∃ (l : List Nat), prime_factorization n l from exists_prime_factorization n h done · -- Uniqueness fix l1 : List Nat; fix l2 : List Nat assume h1 : prime_factorization n l1 assume h2 : prime_factorization n l2 define at h1; define at h2 have h3 : prod l1 = n := h1.right rewrite [←h2.right] at h3 show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3 done done /- Section 7.3 -/ theorem congr_refl (m : Nat) : ∀ (a : Int), a ≡ a (MOD m) := by fix a : Int define --Goal : ∃ (c : Int), a - a = ↑m * c apply Exists.intro 0 ring done theorem congr_symm {m : Nat} : ∀ {a b : Int}, a ≡ b (MOD m) → b ≡ a (MOD m) := by fix a : Int; fix b : Int assume h1 : a ≡ b (MOD m) define at h1 --h1 : ∃ (c : Int), a - b = ↑m * c define --Goal : ∃ (c : Int), b - a = ↑m * c obtain (c : Int) (h2 : a - b = m * c) from h1 apply Exists.intro (-c) show b - a = m * (-c) from calc b - a _ = -(a - b) := by ring _ = -(m * c) := by rw [h2] _ = m * (-c) := by ring done theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry /- Fundamental properties of congruence classes -/ lemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) : X = Y ↔ X.val = Y.val := Fin.ext_iff lemma val_nat_eq_mod (n k : Nat) : ([k]_(n + 1)).val = k % (n + 1) := by rfl lemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl theorem cc_rep {m : Nat} (X : ZMod m) : ∃ (a : Int), X = [a]_m := match m with | 0 => by apply Exists.intro X rfl done | n + 1 => by apply Exists.intro ↑(X.val) have h1 : X.val < n + 1 := Fin.prop X rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1] rfl done theorem add_class (m : Nat) (a b : Int) : [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm theorem mul_class (m : Nat) (a b : Int) : [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm lemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ [a - b]_m = [0]_m := by apply Iff.intro · -- (→) assume h1 : [a]_m = [b]_m have h2 : a - b = a + (-b) := by ring have h3 : b + (-b) = 0 := by ring show [a - b]_m = [0]_m from calc [a - b]_m _ = [a + (-b)]_m := by rw [h2] _ = [a]_m + [-b]_m := by rw [add_class] _ = [b]_m + [-b]_m := by rw [h1] _ = [b + -b]_m := by rw [add_class] _ = [0]_m := by rw [h3] done · -- (←) assume h1 : [a - b]_m = [0]_m have h2 : b + (a - b) = a := by ring have h3 : b + 0 = b := by ring show [a]_m = [b]_m from calc [a]_m _ = [b + (a - b)]_m := by rw [h2] _ = [b]_m + [a - b]_m := by rw [add_class] _ = [b]_m + [0]_m := by rw [h1] _ = [b + 0]_m := by rw [add_class] _ = [b]_m := by rw [h3] done done lemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) : [a]_m = [0]_m → [-a]_m = [0]_m := by assume h1 : [a]_m = [0]_m have h2 : 0 + (-a) = -a := by ring have h3 : a + (-a) = 0 := by ring show [-a]_m = [0]_m from calc [-a]_m _ = [0 + (-a)]_m := by rw [h2] _ = [0]_m + [-a]_m := by rw [add_class] _ = [a]_m + [-a]_m := by rw [h1] _ = [a + (-a)]_m := by rw [add_class] _ = [0]_m := by rw [h3] done lemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) : [-a]_m = [0]_m ↔ [a]_m = [0]_m := by apply Iff.intro _ (cc_neg_zero_of_cc_zero m a) assume h1 : [-a]_m = [0]_m have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1 have h3 : -(-a) = a := by ring rewrite [h3] at h2 show [a]_m = [0]_m from h2 done lemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl lemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m ↔ m ∣ k := match m with | 0 => by have h : (0 : Int) = (↑(0 : Nat) : Int) := by rfl rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj] apply Iff.intro · -- (→) assume h1 : k = 0 rewrite [h1] show 0 ∣ 0 from dvd_self 0 done · -- (←) assume h1 : 0 ∣ k obtain (c : Nat) (h2 : k = 0 * c) from h1 rewrite [h2] ring done done | n + 1 => by rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero] show k % (n + 1) = 0 ↔ n + 1 ∣ k from (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm done lemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m ↔ ↑m ∣ a := by obtain (k : Nat) (h1 : a = ↑k ∨ a = -↑k) from Int.eq_nat_or_neg a by_cases on h1 · -- Case 1. h1: a = ↑k rewrite [h1, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done · -- Case 2. h1: a = -↑k rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done done theorem cc_eq_iff_congr (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ a ≡ b (MOD m) := calc [a]_m = [b]_m _ ↔ [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b _ ↔ ↑m ∣ (a - b) := cc_zero_iff_dvd m (a - b) _ ↔ a ≡ b (MOD m) := by rfl /- End of fundamental properties of congruence classes -/ lemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m := by have h1 : (↑m : Int) ≠ 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m) show 0 ≤ a % m from Int.emod_nonneg a h1 done lemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m) have h2 : (↑m : Int) > 0 := (Nat.cast_pos).rtl h1 show a % m < m from Int.emod_lt_of_pos a h2 done lemma congr_mod_mod (m : Nat) (a : Int) : a ≡ a % m (MOD m) := by define have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m apply Exists.intro (a / m) show a - a % m = m * (a / m) from calc a - (a % m) _ = m * (a / m) + a % m - a % m := by rw [h1] _ = m * (a / m) := by ring done lemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) := And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a)) theorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) : ∃! (r : Int), 0 ≤ r ∧ r < m ∧ a ≡ r (MOD m) := by exists_unique · -- Existence apply Exists.intro (a % m) show 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) from mod_cmpl_res m a done · -- Uniqueness fix r1 : Int; fix r2 : Int assume h1 : 0 ≤ r1 ∧ r1 < m ∧ a ≡ r1 (MOD m) assume h2 : 0 ≤ r2 ∧ r2 < m ∧ a ≡ r2 (MOD m) have h3 : r1 ≡ r2 (MOD m) := congr_trans (congr_symm h1.right.right) h2.right.right obtain (d : Int) (h4 : r1 - r2 = m * d) from h3 have h5 : r1 - r2 < m * 1 := by linarith have h6 : m * (-1) < r1 - r2 := by linarith rewrite [h4] at h5 --h5 : m * d < m * 1 rewrite [h4] at h6 --h6 : m * -1 < m * d have h7 : (↑m : Int) ≥ 0 := Nat.cast_nonneg m have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7 have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7 have h10 : d = 0 := by linarith show r1 = r2 from calc r1 _ = r1 - r2 + r2 := by ring _ = m * 0 + r2 := by rw [h4, h10] _ = r2 := by ring done done lemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m := (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a) theorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y rewrite [h1, h2] have h3 : a + b = b + a := by ring show [a]_m + [b]_m = [b]_m + [a]_m from calc [a]_m + [b]_m _ = [a + b]_m := add_class m a b _ = [b + a]_m := by rw [h3] _ = [b]_m + [a]_m := (add_class m b a).symm done theorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X rewrite [h1] have h2 : a * 1 = a := by ring show [a]_m * [1]_m = [a]_m from calc [a]_m * [1]_m _ = [a * 1]_m := mul_class m a 1 _ = [a]_m := by rw [h2] done theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry lemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) : [a]_m * [gcd_c2 m a]_m = [1]_m := by set s : Int := gcd_c1 m a have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m define at h1 rewrite [h1, Nat.cast_one] at h2 --h2 : s * ↑m + gcd_c2 m a * ↑a = 1 rewrite [mul_class, cc_eq_iff_congr] define --Goal : ∃ (c : Int), ↑a * gcd_c2 m a - 1 = ↑m * c apply Exists.intro (-s) show a * (gcd_c2 m a) - 1 = m * (-s) from calc a * (gcd_c2 m a) - 1 _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring _ = 1 + m * (-s) - 1 := by rw [h2] _ = m * (-s) := by ring done theorem Theorem_7_3_7 (m a : Nat) : invertible [a]_m ↔ rel_prime m a := by apply Iff.intro · -- (→) assume h1 : invertible [a]_m define at h1 obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1 obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y rewrite [h3, mul_class, cc_eq_iff_congr] at h2 define at h2 obtain (c : Int) (h4 : a * b - 1 = m * c) from h2 rewrite [Exercise_7_2_6] --Goal : ∃ (s t : Int), s * ↑m + t * ↑a = 1 apply Exists.intro (-c) apply Exists.intro b show (-c) * m + b * a = 1 from calc (-c) * m + b * a _ = (-c) * m + (a * b - 1) + 1 := by ring _ = (-c) * m + m * c + 1 := by rw [h4] _ = 1 := by ring done · -- (←) assume h1 : rel_prime m a define show ∃ (Y : ZMod m), [a]_m * Y = [1]_m from Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1) done done /- Section 7.4 -/ section Euler open Euler lemma num_rp_below_base {m : Nat} : num_rp_below m 0 = 0 := by rfl lemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) : num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by have h1 : num_rp_below m (j + 1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : gcd m j = 1 rewrite [if_pos h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1 show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1 done lemma num_rp_below_step_not_rp {m j : Nat} (h : ¬rel_prime m j) : num_rp_below m (j + 1) = num_rp_below m j := by have h1 : num_rp_below m (j +1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : ¬gcd m j = 1 rewrite [if_neg h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j show num_rp_below m (j + 1) = num_rp_below m j from h1 done lemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl #eval phi 10 --Answer: 4 lemma prod_inv_iff_inv {m : Nat} {X : ZMod m} (h1 : invertible X) (Y : ZMod m) : invertible (X * Y) ↔ invertible Y := by apply Iff.intro · -- (→) assume h2 : invertible (X * Y) obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2 apply Exists.intro (X * Z) rewrite [←h3] --Goal : Y * (X * Z) = X * Y * Z ring --Note that ring can do algebra in ZMod m done · -- (←) assume h2 : invertible Y obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1 obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2 apply Exists.intro (Xi * Yi) show (X * Y) * (Xi * Yi) = [1]_m from calc X * Y * (Xi * Yi) _ = (X * Xi) * (Y * Yi) := by ring _ = [1]_m * [1]_m := by rw [h3, h4] _ = [1]_m := Theorem_7_3_6_7 [1]_m done done lemma F_rp_def {m i : Nat} (h : rel_prime m i) : F m i = [i]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h --h : gcd m i = 1 rewrite [if_pos h] at h1 show F m i = [i]_m from h1 done lemma F_not_rp_def {m i : Nat} (h : ¬rel_prime m i) : F m i = [1]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h rewrite [h1, if_neg h] rfl done lemma prod_seq_base {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 0 k f = [1]_m := by rfl lemma prod_seq_step {m : Nat} (n k : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl lemma prod_seq_zero_step {m : Nat} (n : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by rewrite [prod_seq_step, zero_add] rfl done lemma prod_one {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 1 k f = f k := by rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7] rfl done lemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl lemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m := calc [G m a i]_m _ = [(a * i) % m]_m := by rfl _ = [a * i]_m := (cc_eq_mod m (a * i)).symm _ = [a]_m * [i]_m := (mul_class m a i).symm lemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) : rel_prime m (G m a i) ↔ rel_prime m i := by have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1 show rel_prime m (G m a i) ↔ rel_prime m i from calc rel_prime m (G m a i) _ ↔ invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm _ ↔ invertible ([a]_m * [i]_m) := by rw [cc_G] _ ↔ invertible [i]_m := prod_inv_iff_inv h2 ([i]_m) _ ↔ rel_prime m i := Theorem_7_3_7 m i done lemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) : F m (G m a i) = [a]_m * F m i := by have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2 show F m (G m a i) = [a]_m * F m i from calc F m (G m a i) _ = [G m a i]_m := F_rp_def h3 _ = [a]_m * [i]_m := cc_G m a i _ = [a]_m * F m i := by rw [F_rp_def h2] done lemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : ¬rel_prime m i) : F m (G m a i) = [1]_m := by rewrite [←G_rp_iff h1 i] at h2 show F m (G m a i) = [1]_m from F_not_rp_def h2 done lemma FG_prod {m a : Nat} (h1 : rel_prime m a) : ∀ (k : Nat), prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by by_induc · -- Base Case show prod_seq 0 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from calc prod_seq 0 0 ((F m) ∘ (G m a)) _ = [1]_m := prod_seq_base _ _ _ = [a]_m ^ 0 * [1]_m := by ring _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by rw [num_rp_below_base, prod_seq_base] done · -- Induction Step fix k : Nat assume ih : prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) by_cases h2 : rel_prime m k · -- Case 1. h2 : rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([a]_m * F m k) := by rw [FG_rp h1 h2] _ = [a]_m ^ ((num_rp_below m k) + 1) * ((prod_seq k 0 (F m)) * F m k) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_rp h2, prod_seq_zero_step] done · -- Case 2. h2 : ¬rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([1]_m) := by rw [FG_not_rp h1 h2] _ = [a]_m ^ (num_rp_below m k) * (prod_seq k 0 (F m) * ([1]_m)) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_not_rp h2, prod_seq_zero_step, F_not_rp_def h2] done done done lemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by define --Goal : ∀ i < m, G m a i < m fix i : Nat assume h1 : i < m rewrite [G_def] --Goal : a * i % m < m show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m) done lemma left_inv_one_one_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g' (g i) = i) : one_one_below n g := sorry lemma right_inv_onto_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g (g' i) = i) (h2 : maps_below n g') : onto_below n g := by define at h2; define fix k : Nat assume h3 : k < n apply Exists.intro (g' k) show g' k < n ∧ g (g' k) = k from And.intro (h2 k h3) (h1 k h3) done lemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m * [inv_mod m a]_m = [1]_m := by have h2 : 0 ≤ (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a) show [a]_m * [inv_mod m a]_m = [1]_m from calc [a]_m * [inv_mod m a]_m _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2] _ = [a]_m * [gcd_c2 m a]_m := by rw [←cc_eq_mod] _ = [1]_m := gcd_c2_inv h1 done lemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m := calc a * (b % m) % m = a % m * (b % m % m) % m := Nat.mul_mod _ _ _ _ = a % m * (b % m) % m := by rw [Nat.mod_mod] _ = a * b % m := (Nat.mul_mod _ _ _).symm lemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm] rfl done theorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] : ↑a ≡ ↑b (MOD m) ↔ a % m = b % m := sorry lemma mul_inv_mod_cancel {m a i : Nat} [NeZero m] (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1 rewrite [mul_class, cc_eq_iff_congr, ←Nat.cast_mul, ←Nat.cast_one, congr_iff_mod_eq_Nat] at h3 show a * inv_mod m a * i % m = i from calc a * (inv_mod m a) * i % m _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = 1 % m * i % m := by rw [h3] _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = i % m := by rw [one_mul] _ = i := Nat.mod_eq_of_lt h2 done lemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl lemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, G m a (Ginv m a i) = i := by fix i : Nat assume h2 : i < m show G m a (Ginv m a i) = i from calc G m a (Ginv m a i) _ = a * ((inv_mod m a * i) % m) % m := by rfl _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, Ginv m a (G m a i) = i := by fix i : Nat assume h2 : i < m show Ginv m a (G m a i) = i from calc Ginv m a (G m a i) _ = inv_mod m a * ((a * i) % m) % m := by rfl _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc, mul_comm (inv_mod m a)] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_maps_below (m a : Nat) [NeZero m] : maps_below m (Ginv m a) := G_maps_below m (inv_mod m a) lemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : one_one_below m (G m a) := left_inv_one_one_below (Ginv_left_inv h1) lemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : onto_below m (G m a) := right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a) lemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : perm_below m (G m a) := And.intro (G_maps_below m a) (And.intro (G_one_one_below h1) (G_onto_below h1)) --Permuting a product of congruence classes doesn't change product lemma swap_fst (u v : Nat) : swap u v u = v := by define : swap u v u --Goal : (if u = u then v else if u = v then u else u) = v have h : u = u := by rfl rewrite [if_pos h] rfl done lemma swap_snd (u v : Nat) : swap u v v = u := by define : swap u v v by_cases h1 : v = u · -- Case 1. h1 : v = u rewrite [if_pos h1] show v = u from h1 done · -- Case 2. h1 : v ≠ u rewrite [if_neg h1] have h2 : v = v := by rfl rewrite [if_pos h2] rfl done done lemma swap_other {u v i : Nat} (h1 : i ≠ u) (h2 : i ≠ v) : swap u v i = i := by define : swap u v i rewrite [if_neg h1, if_neg h2] rfl done lemma swap_values (u v i : Nat) : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := by by_cases h1 : i = u · -- Case 1. h1 : i = u apply Or.inl rewrite [h1] show swap u v u = v from swap_fst u v done · -- Case 2. h1 : i ≠ u apply Or.inr by_cases h2 : i = v · -- Case 2.1. h2 : i = v apply Or.inl rewrite [h2] show swap u v v = u from swap_snd u v done · -- Case 2.2. h2 : i ≠ v apply Or.inr show swap u v i = i from swap_other h1 h2 done done done lemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by define fix i : Nat assume h3 : i < n have h4 : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := swap_values u v i by_cases on h4 · -- Case 1. h4 : swap u v i = v rewrite [h4] show v < n from h2 done · -- Case 2. by_cases on h4 · -- Case 2.1. h4 : swap u v i = u rewrite [h4] show u < n from h1 done · -- Case 2.2. h4 : swap u v i = i rewrite [h4] show i < n from h3 done done done lemma swap_swap (u v n : Nat) : ∀ i < n, swap u v (swap u v i) = i := by fix i : Nat assume h : i < n by_cases h1 : i = u · -- Case 1. h1 : i = u rewrite [h1, swap_fst, swap_snd] rfl done · -- Case 2. h1 : i ≠ u by_cases h2 : i = v · -- Case 2.1. h2 : i = v rewrite [h2, swap_snd, swap_fst] rfl done · -- Case 2.2. h2 : i ≠ v rewrite [swap_other h1 h2, swap_other h1 h2] rfl done done done lemma swap_one_one_below (u v n) : one_one_below n (swap u v) := left_inv_one_one_below (swap_swap u v n) lemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) := right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2) lemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) := And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2)) lemma comp_perm_below {n : Nat} {f g : Nat → Nat} (h1 : perm_below n f) (h2 : perm_below n g) : perm_below n (f ∘ g) := sorry lemma trivial_swap (u : Nat) : swap u u = id := by apply funext fix x : Nat by_cases h1 : x = u · -- Case 1. h1 : x = u rewrite [h1, swap_fst] rfl done · -- Case 2. h1 : x ≠ u rewrite [swap_other h1 h1] rfl done done lemma prod_eq_fun {m : Nat} (f g : Nat → ZMod m) (k : Nat) : ∀ (n : Nat), (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g := by by_induc · -- Base Case assume h : (∀ i < 0, f (k + i) = g (k + i)) rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g assume h1 : ∀ i < n + 1, f (k + i) = g (k + i) have h2 : ∀ i < n, f (k + i) = g (k + i) := by fix i : Nat assume h2 : i < n have h3 : i < n + 1 := by linarith show f (k + i) = g (k + i) from h1 i h3 done have h3 : prod_seq n k f = prod_seq n k g := ih h2 have h4 : n < n + 1 := Nat.lt_succ_self n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4] rfl done done lemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := by have h2 : ∀ (i : Nat), i < u → (f ∘ swap u n) (0 + i) = f (0 + i) := by fix i : Nat assume h2 : i < u have h3 : 0 + i ≠ u := by linarith have h4 : 0 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl done show prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f from prod_eq_fun (f ∘ swap u n) f 0 u h2 done lemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := by have h2 : ∀ i < j, (f ∘ swap u n) (u + 1 + i) = f (u + 1 + i) := by fix i : Nat assume h2 : i < j have h3 : u + 1 + i ≠ u := by linarith have h4 : u + 1 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl show prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f from prod_eq_fun (f ∘ swap u n) f (u + 1) j h2 done lemma break_prod {m : Nat} (n : Nat) (f : Nat → ZMod m) : ∀ (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by by_induc · -- Base Case have h : n + 0 = n := by rfl rewrite [prod_seq_base, h, Theorem_7_3_6_7] rfl done · -- Induction Step fix j : Nat assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f rewrite [←add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc] rfl done done lemma break_prod_twice {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f := break_prod n f 1 rewrite [prod_one] at h2 have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f := break_prod (u + 1) f j rewrite [←h1] at h3 have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f := break_prod u f 1 rewrite [prod_one] at h4 rewrite [h3, h4] at h2 show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2 done lemma swap_prod_eq_prod {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f := by by_cases h2 : u = n · -- Case 1. h2 : u = n rewrite [h2, trivial_swap n] --Goal : prod_seq (n + 1) 0 (f ∘ id) = prod_seq (n + 1) 0 f rfl done · -- Case 2. h2 : ¬u = n have h3 : u + 1 ≤ n := Nat.lt_of_le_of_ne h1 h2 obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3 have break_f : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := break_prod_twice f h4 have break_fs : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_prod_twice (f ∘ swap u n) h4 have f_eq_fs_below : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := swap_prod_eq_prod_below f h1 have f_eq_fs_btwn : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4 show prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f from calc prod_seq (n + 1) 0 (f ∘ swap u n) _ = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_fs _ = prod_seq u 0 f * (f ∘ swap u n) u * prod_seq j (u + 1) f * (f ∘ swap u n) n := by rw [f_eq_fs_below, f_eq_fs_btwn] _ = prod_seq u 0 f * f (swap u n u) * prod_seq j (u + 1) f * f (swap u n n) := by rfl _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by rw [swap_fst, swap_snd] _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring _ = prod_seq (n + 1) 0 f := break_f.symm done done lemma perm_below_fixed {n : Nat} {g : Nat → Nat} (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry lemma perm_prod {m : Nat} (f : Nat → ZMod m) : ∀ (n : Nat), ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) := by by_induc · -- Base Case fix g : Nat → Nat assume h1 : perm_below 0 g rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) fix g : Nat → Nat assume g_pb : perm_below (n + 1) g define at g_pb have g_ob : onto_below (n + 1) g := g_pb.right.right define at g_ob have h1 : n < n + 1 := by linarith obtain (u : Nat) (h2 : u < n + 1 ∧ g u = n) from g_ob n h1 have s_pb : perm_below (n + 1) (swap u n) := swap_perm_below h2.left h1 have gs_pb_n1 : perm_below (n + 1) (g ∘ swap u n) := comp_perm_below g_pb s_pb have gs_fix_n : (g ∘ swap u n) n = n := calc (g ∘ swap u n) n _ = g (swap u n n) := by rfl _ = g u := by rw [swap_snd] _ = n := h2.right have gs_pb_n : perm_below n (g ∘ swap u n) := perm_below_fixed gs_pb_n1 gs_fix_n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f ∘ (g ∘ swap u n)) := ih (g ∘ swap u n) gs_pb_n have h3 : u ≤ n := by linarith show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f ∘ g) from calc prod_seq (n + 1) 0 f _ = prod_seq n 0 f * f n := prod_seq_zero_step n f _ = prod_seq n 0 (f ∘ (g ∘ swap u n)) * f ((g ∘ swap u n) n) := by rw [gs_prod, gs_fix_n] _ = prod_seq n 0 (f ∘ g ∘ swap u n) * (f ∘ g ∘ swap u n) n := by rfl _ = prod_seq (n + 1) 0 (f ∘ g ∘ swap u n) := (prod_seq_zero_step n (f ∘ g ∘ swap u n)).symm _ = prod_seq (n + 1) 0 ((f ∘ g) ∘ swap u n) := by rfl _ = prod_seq (n + 1) 0 (f ∘ g) := swap_prod_eq_prod (f ∘ g) h3 done done lemma F_invertible (m i : Nat) : invertible (F m i) := by by_cases h : rel_prime m i · -- Case 1. h : rel_prime m i rewrite [F_rp_def h] show invertible [i]_m from (Theorem_7_3_7 m i).rtl h done · -- Case 2. h : ¬rel_prime m i rewrite [F_not_rp_def h] apply Exists.intro [1]_m show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m done done lemma Fprod_invertible (m : Nat) : ∀ (k : Nat), invertible (prod_seq k 0 (F m)) := by by_induc · -- Base Case apply Exists.intro [1]_m show prod_seq 0 0 (F m) * [1]_m = [1]_m from calc prod_seq 0 0 (F m) * [1]_m _ = [1]_m * [1]_m := by rw [prod_seq_base] _ = [1]_m := Theorem_7_3_6_7 ([1]_m) done · -- Induction Step fix k : Nat assume ih : invertible (prod_seq k 0 (F m)) rewrite [prod_seq_zero_step] show invertible (prod_seq k 0 (F m) * (F m k)) from (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k) done done theorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m ^ (phi m) = [1]_m := by have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2 show [a]_m ^ (phi m) = [1]_m from calc [a]_m ^ (phi m) _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3] _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring _ = prod_seq m 0 (F m ∘ G m a) * Y := by rw [FG_prod h1 m, phi_def] _ = prod_seq m 0 (F m) * Y := by rw [perm_prod (F m) m (G m a) (G_perm_below h1)] _ = [1]_m := by rw [h3] done lemma Exercise_7_4_5_Int (m : Nat) (a : Int) : ∀ (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry lemma Exercise_7_4_5_Nat (m a n : Nat) : [a]_m ^ n = [a ^ n]_m := by rewrite [Exercise_7_4_5_Int] rfl done theorem Euler's_theorem {m a : Nat} [NeZero m] (h1 : rel_prime m a) : a ^ (phi m) ≡ 1 (MOD m) := by have h2 : [a]_m ^ (phi m) = [1]_m := Theorem_7_4_2 h1 rewrite [Exercise_7_4_5_Nat m a (phi m)] at h2 --h2 : [a ^ phi m]_m = [1]_m show a ^ (phi m) ≡ 1 (MOD m) from (cc_eq_iff_congr _ _ _).ltr h2 done #eval gcd 10 7 --Answer: 1. So 10 and 7 are relatively prime #eval 7 ^ phi 10 --Answer: 2401, which is congruent to 1 mod 10. end Euler /- Section 7.5 -/ lemma num_rp_prime {p : Nat} (h1 : prime p) : ∀ k < p, num_rp_below p (k + 1) = k := sorry lemma phi_prime {p : Nat} (h1 : prime p) : phi p = p - 1 := by have h2 : 1 ≤ p := prime_pos h1 have h3 : p - 1 + 1 = p := Nat.sub_add_cancel h2 have h4 : p - 1 < p := by linarith have h5 : num_rp_below p (p - 1 + 1) = p - 1 := num_rp_prime h1 (p - 1) h4 rewrite [h3] at h5 show phi p = p - 1 from h5 done theorem Theorem_7_2_2_Int {a c : Nat} {b : Int} (h1 : ↑c ∣ ↑a * b) (h2 : rel_prime a c) : ↑c ∣ b := by rewrite [Int.natCast_dvd, Int.natAbs_mul, Int.natAbs_ofNat] at h1 --h1 : c ∣ a * Int.natAbs b rewrite [Int.natCast_dvd] --Goal : c ∣ Int.natAbs b show c ∣ Int.natAbs b from Theorem_7_2_2 h1 h2 done lemma Lemma_7_4_5 {m n : Nat} (a b : Int) (h1 : rel_prime m n) : a ≡ b (MOD m * n) ↔ a ≡ b (MOD m) ∧ a ≡ b (MOD n) := by apply Iff.intro · -- (→) assume h2 : a ≡ b (MOD m * n) obtain (j : Int) (h3 : a - b = (m * n) * j) from h2 apply And.intro · -- Proof of a ≡ b (MOD m) apply Exists.intro (n * j) show a - b = m * (n * j) from calc a - b _ = m * n * j := h3 _ = m * (n * j) := by ring done · -- Proof of a ≡ b (MOD n) apply Exists.intro (m * j) show a - b = n * (m * j) from calc a - b _ = m * n * j := h3 _ = n * (m * j) := by ring done done · -- (←) assume h2 : a ≡ b (MOD m) ∧ a ≡ b (MOD n) obtain (j : Int) (h3 : a - b = m * j) from h2.left have h4 : (↑n : Int) ∣ a - b := h2.right rewrite [h3] at h4 --h4 : ↑n ∣ ↑m * j have h5 : ↑n ∣ j := Theorem_7_2_2_Int h4 h1 obtain (k : Int) (h6 : j = n * k) from h5 apply Exists.intro k --Goal : a - b = ↑(m * n) * k rewrite [Nat.cast_mul] --Goal : a - b = ↑m * ↑n * k show a - b = (m * n) * k from calc a - b _ = m * j := h3 _ = m * (n * k) := by rw [h6] _ = (m * n) * k := by ring done done --From exercises of Section 7.2 theorem rel_prime_symm {a b : Nat} (h : rel_prime a b) : rel_prime b a := sorry lemma prime_NeZero {p : Nat} (h : prime p) : NeZero p := by rewrite [neZero_iff] --Goal : p ≠ 0 define at h linarith done lemma Lemma_7_5_1 {p e d m c s : Nat} {t : Int} (h1 : prime p) (h2 : e * d = (p - 1) * s + 1) (h3 : m ^ e - c = p * t) : c ^ d ≡ m (MOD p) := by have h4 : m ^ e ≡ c (MOD p) := Exists.intro t h3 have h5 : [m ^ e]_p = [c]_p := (cc_eq_iff_congr _ _ _).rtl h4 rewrite [←Exercise_7_4_5_Nat] at h5 --h5 : [m]_p ^ e = [c]_p by_cases h6 : p ∣ m · -- Case 1. h6 : p ∣ m have h7 : m ≡ 0 (MOD p) := by obtain (j : Nat) (h8 : m = p * j) from h6 apply Exists.intro (↑j : Int) --Goal : ↑m - 0 = ↑p * ↑j rewrite [h8, Nat.cast_mul] ring done have h8 : [m]_p = [0]_p := (cc_eq_iff_congr _ _ _).rtl h7 have h9 : e * d ≠ 0 := by rewrite [h2] show (p - 1) * s + 1 ≠ 0 from Nat.add_one_ne_zero _ done have h10 : (0 : Int) ^ (e * d) = 0 := zero_pow h9 have h11 : [c ^ d]_p = [m]_p := calc [c ^ d]_p _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat] _ = ([m]_p ^ e) ^ d := by rw [h5] _ = [m]_p ^ (e * d) := by ring _ = [0]_p ^ (e * d) := by rw [h8] _ = [0 ^ (e * d)]_p := Exercise_7_4_5_Int _ _ _ _ = [0]_p := by rw [h10] _ = [m]_p := by rw [h8] show c ^ d ≡ m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11 done · -- Case 2. h6 : ¬p ∣ m have h7 : rel_prime m p := rel_prime_of_prime_not_dvd h1 h6 have h8 : rel_prime p m := rel_prime_symm h7 have h9 : NeZero p := prime_NeZero h1 have h10 : (1 : Int) ^ s = 1 := by ring have h11 : [c ^ d]_p = [m]_p := calc [c ^ d]_p _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat] _ = ([m]_p ^ e) ^ d := by rw [h5] _ = [m]_p ^ (e * d) := by ring _ = [m]_p ^ ((p - 1) * s + 1) := by rw [h2] _ = ([m]_p ^ (p - 1)) ^ s * [m]_p := by ring _ = ([m]_p ^ (phi p)) ^ s * [m]_p := by rw [phi_prime h1] _ = [1]_p ^ s * [m]_p := by rw [Theorem_7_4_2 h8] _ = [1 ^ s]_p * [m]_p := by rw [Exercise_7_4_5_Int] _ = [1]_p * [m]_p := by rw [h10] _ = [m]_p * [1]_p := by ring _ = [m]_p := Theorem_7_3_6_7 _ show c ^ d ≡ m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11 done done theorem Theorem_7_5_1 (p q n e d k m c : Nat) (p_prime : prime p) (q_prime : prime q) (p_ne_q : p ≠ q) (n_pq : n = p * q) (ed_congr_1 : e * d = k * (p - 1) * (q - 1) + 1) (h1 : [m]_n ^ e = [c]_n) : [c]_n ^ d = [m]_n := by rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] at h1 --h1 : m ^ e ≡ c (MOD n) rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] --Goal : c ^ d ≡ m (MOD n) obtain (j : Int) (h2 : m ^ e - c = n * j) from h1 rewrite [n_pq, Nat.cast_mul] at h2 --h2 : m ^ e - c = p * q * j have h3 : e * d = (p - 1) * (k * (q - 1)) + 1 := by rewrite [ed_congr_1] ring done have h4 : m ^ e - c = p * (q * j) := by rewrite [h2] ring done have congr_p : c ^ d ≡ m (MOD p) := Lemma_7_5_1 p_prime h3 h4 have h5 : e * d = (q - 1) * (k * (p - 1)) + 1 := by rewrite [ed_congr_1] ring done have h6 : m ^ e - c = q * (p * j) := by rewrite [h2] ring done have congr_q : c ^ d ≡ m (MOD q) := Lemma_7_5_1 q_prime h5 h6 have h7 : ¬q ∣ p := by by_contra h8 have h9 : q = 1 ∨ q = p := dvd_prime p_prime h8 disj_syll h9 (prime_not_one q_prime) show False from p_ne_q h9.symm done have h8 : rel_prime p q := rel_prime_of_prime_not_dvd q_prime h7 rewrite [n_pq, Lemma_7_4_5 _ _ h8] show c ^ d ≡ m (MOD p) ∧ c ^ d ≡ m (MOD q) from And.intro congr_p congr_q done /- BEGIN EXERCISES -/ namespace Exercises /- Section 7.1 -/ -- 1. theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry -- 2. lemma gcd_comm_lt {a b : Nat} (h : a < b) : gcd a b = gcd b a := sorry theorem gcd_comm (a b : Nat) : gcd a b = gcd b a := sorry -- 3. theorem Exercise_7_1_5 (a b : Nat) (n : Int) : (∃ (s t : Int), s * a + t * b = n) ↔ (↑(gcd a b) : Int) ∣ n := sorry -- 4. theorem Exercise_7_1_6 (a b c : Nat) : gcd a b = gcd (a + b * c) b := sorry -- 5. theorem gcd_is_nonzero {a b : Nat} (h : a ≠ 0 ∨ b ≠ 0) : gcd a b ≠ 0 := sorry -- 6. theorem gcd_greatest {a b d : Nat} (h1 : gcd a b ≠ 0) (h2 : d ∣ a) (h3 : d ∣ b) : d ≤ gcd a b := sorry -- 7. lemma Lemma_7_1_10a {a b : Nat} (n : Nat) (h : a ∣ b) : (n * a) ∣ (n * b) := sorry lemma Lemma_7_1_10b {a b n : Nat} (h1 : n ≠ 0) (h2 : (n * a) ∣ (n * b)) : a ∣ b := sorry lemma Lemma_7_1_10c {a b : Nat} (h1 : a ∣ b) (h2 : b ∣ a) : a = b := sorry theorem Exercise_7_1_10 (a b n : Nat) : gcd (n * a) (n * b) = n * gcd a b := sorry /- Section 7.2 -/ -- 1. lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry -- 2. -- Hints: Start with apply List.rec. You may find mul_ne_zero useful theorem prod_nonzero_nonzero : ∀ (l : List Nat), (∀ a ∈ l, a ≠ 0) → prod l ≠ 0 := sorry -- 3. theorem rel_prime_iff_no_common_factor (a b : Nat) : rel_prime a b ↔ ¬∃ (p : Nat), prime p ∧ p ∣ a ∧ p ∣ b := sorry -- 4. theorem rel_prime_symm {a b : Nat} (h : rel_prime a b) : rel_prime b a := sorry -- 5. lemma in_prime_factorization_iff_prime_factor {a : Nat} {l : List Nat} (h1 : prime_factorization a l) (p : Nat) : p ∈ l ↔ prime_factor p a := sorry -- 6. theorem Exercise_7_2_5 {a b : Nat} {l m : List Nat} (h1 : prime_factorization a l) (h2 : prime_factorization b m) : rel_prime a b ↔ (¬∃ (p : Nat), p ∈ l ∧ p ∈ m) := sorry -- 7. theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry -- 8. theorem Exercise_7_2_7 {a b a' b' : Nat} (h1 : rel_prime a b) (h2 : a' ∣ a) (h3 : b' ∣ b) : rel_prime a' b' := sorry -- 9. theorem Exercise_7_2_9 {a b j k : Nat} (h1 : gcd a b ≠ 0) (h2 : a = j * gcd a b) (h3 : b = k * gcd a b) : rel_prime j k := sorry -- 10. theorem Exercise_7_2_17a (a b c : Nat) : gcd a (b * c) ∣ gcd a b * gcd a c := sorry /- Section 7.3 -/ -- 1. theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry -- 2. theorem Theorem_7_3_6_3 {m : Nat} (X : ZMod m) : X + [0]_m = X := sorry -- 3. theorem Theorem_7_3_6_4 {m : Nat} (X : ZMod m) : ∃ (Y : ZMod m), X + Y = [0]_m := sorry -- 4. theorem Exercise_7_3_4a {m : Nat} (Z1 Z2 : ZMod m) (h1 : ∀ (X : ZMod m), X + Z1 = X) (h2 : ∀ (X : ZMod m), X + Z2 = X) : Z1 = Z2 := sorry -- 5. theorem Exercise_7_3_4b {m : Nat} (X Y1 Y2 : ZMod m) (h1 : X + Y1 = [0]_m) (h2 : X + Y2 = [0]_m) : Y1 = Y2 := sorry -- 6. theorem Theorem_7_3_10 (m a : Nat) (b : Int) : ¬(↑(gcd m a) : Int) ∣ b → ¬∃ (x : Int), a * x ≡ b (MOD m) := sorry -- 7. theorem Theorem_7_3_11 (m n : Nat) (a b : Int) (h1 : n ≠ 0) : n * a ≡ n * b (MOD n * m) ↔ a ≡ b (MOD m) := sorry -- 8. theorem Exercise_7_3_16 {m : Nat} {a b : Int} (h : a ≡ b (MOD m)) : ∀ (n : Nat), a ^ n ≡ b ^ n (MOD m) := sorry -- 9. example {m : Nat} [NeZero m] (X : ZMod m) : ∃! (a : Int), 0 ≤ a ∧ a < m ∧ X = [a]_m := sorry -- 10. theorem congr_rel_prime {m a b : Nat} (h1 : a ≡ b (MOD m)) : rel_prime m a ↔ rel_prime m b := sorry -- 11. --Hint: You may find the theorem Int.ofNat_mod_ofNat useful. theorem rel_prime_mod (m a : Nat) : rel_prime m (a % m) ↔ rel_prime m a := sorry -- 12. lemma congr_iff_mod_eq_Int (m : Nat) (a b : Int) [NeZero m] : a ≡ b (MOD m) ↔ a % ↑m = b % ↑m := sorry --Hint for next theorem: Use the lemma above, --together with the theorems Int.ofNat_mod_ofNat and Nat.cast_inj. theorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] : ↑a ≡ ↑b (MOD m) ↔ a % m = b % m := sorry /- Section 7.4 -/ -- 1. --Hint: Use induction. --For the base case, compute [a]_m ^ 0 * [1]_m in two ways: --by Theorem_7_3_6_7, [a] ^ 0 * [1]_m = [a]_m ^ 0 --by ring, [a]_m ^ 0 * [1]_m = [1]_m. lemma Exercise_7_4_5_Int (m : Nat) (a : Int) : ∀ (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry -- 2. lemma left_inv_one_one_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g' (g i) = i) : one_one_below n g := sorry -- 3. lemma comp_perm_below {n : Nat} {f g : Nat → Nat} (h1 : perm_below n f) (h2 : perm_below n g) : perm_below n (f ∘ g) := sorry -- 4. lemma perm_below_fixed {n : Nat} {g : Nat → Nat} (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry -- 5. lemma Lemma_7_4_6 {a b c : Nat} : rel_prime (a * b) c ↔ rel_prime a c ∧ rel_prime b c := sorry -- 6. example {m a : Nat} [NeZero m] (h1 : rel_prime m a) : a ^ (phi m + 1) ≡ a (MOD m) := sorry -- 7. theorem Like_Exercise_7_4_11 {m a p : Nat} [NeZero m] (h1 : rel_prime m a) (h2 : p + 1 = phi m) : [a]_m * [a ^ p]_m = [1]_m := sorry -- 8. theorem Like_Exercise_7_4_12 {m a p q k : Nat} [NeZero m] (h1 : rel_prime m a) (h2 : p = q + (phi m) * k) : a ^ p ≡ a ^ q (MOD m) := sorry /- Section 7.5 -/ -- 1. --Hint: Use induction. lemma num_rp_prime {p : Nat} (h1 : prime p) : ∀ k < p, num_rp_below p (k + 1) = k := sorry -- 2.

lemma three_prime : prime 3

HTPI.Exercises.three_prime

htpi/HTPILib/Chap7.lean

htpi

/- Copyright 2023 Daniel J. Velleman -/ import HTPILib.Chap6 namespace HTPI /- Definitions -/ lemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by have h : n + 1 > 0 := Nat.succ_pos n show a % (n + 1) < n + 1 from Nat.mod_lt a h done def gcd (a b : Nat) : Nat := match b with | 0 => a | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd (n + 1) (a % (n + 1)) termination_by b mutual def gcd_c1 (a b : Nat) : Int := match b with | 0 => 1 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c2 (n + 1) (a % (n + 1)) --Corresponds to s = t' termination_by b def gcd_c2 (a b : Nat) : Int := match b with | 0 => 0 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c1 (n + 1) (a % (n + 1)) - (gcd_c2 (n + 1) (a % (n + 1))) * ↑(a / (n + 1)) --Corresponds to t = s' - t'q termination_by b end def prime (n : Nat) : Prop := 2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n def prime_factor (p n : Nat) : Prop := prime p ∧ p ∣ n def all_prime (l : List Nat) : Prop := ∀ p ∈ l, prime p def nondec (l : List Nat) : Prop := match l with | [] => True --Of course, True is a proposition that is always true | n :: L => (∀ m ∈ L, n ≤ m) ∧ nondec L def nondec_prime_list (l : List Nat) : Prop := all_prime l ∧ nondec l def prod (l : List Nat) : Nat := match l with | [] => 1 | n :: L => n * (prod L) def prime_factorization (n : Nat) (l : List Nat) : Prop := nondec_prime_list l ∧ prod l = n def rel_prime (a b : Nat) : Prop := gcd a b = 1 def congr_mod (m : Nat) (a b : Int) : Prop := (↑m : Int) ∣ (a - b) def cc (m : Nat) (a : Int) : ZMod m := (↑a : ZMod m) notation:50 a " ≡ " b " (MOD " m ")" => congr_mod m a b notation:max "["a"]_"m:max => cc m a def invertible {m : Nat} (X : ZMod m) : Prop := ∃ (Y : ZMod m), X * Y = [1]_m def num_rp_below (m k : Nat) : Nat := match k with | 0 => 0 | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j def phi (m : Nat) : Nat := num_rp_below m m def prod_seq {m : Nat} (j k : Nat) (f : Nat → ZMod m) : ZMod m := match j with | 0 => [1]_m | n + 1 => prod_seq n k f * f (k + n) def maps_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i < n, g i < n def one_one_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i1 < n, ∀ i2 < n, g i1 = g i2 → i1 = i2 def onto_below (n : Nat) (g : Nat → Nat) : Prop := ∀ k < n, ∃ i < n, g i = k def perm_below (n : Nat) (g : Nat → Nat) : Prop := maps_below n g ∧ one_one_below n g ∧ onto_below n g def inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m) def swap (u v i : Nat) : Nat := if i = u then v else if i = v then u else i namespace Euler --For definitions specific to Euler's theorem def F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m def G (m a i : Nat) : Nat := (a * i) % m def Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i end Euler /- Section 7.1 -/ theorem dvd_mod_of_dvd_a_b {a b d : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ (a % b) := by set q : Nat := a / b have h3 : b * q + a % b = a := Nat.div_add_mod a b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 define --Goal : ∃ (c : Nat), a % b = d * c apply Exists.intro (j - k * q) show a % b = d * (j - k * q) from calc a % b _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm _ = a - b * q := by rw [h3] _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc] _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm done theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry #eval gcd 672 161 --Answer: 7 lemma gcd_base (a : Nat) : gcd a 0 = a := by rfl lemma gcd_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd a b = gcd b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1)) rfl done lemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b ≠ 0) : a % b < b := by have h1 : b > 0 := Nat.pos_of_ne_zero h show a % b < b from Nat.mod_lt a h1 done lemma dvd_self (n : Nat) : n ∣ n := by apply Exists.intro 1 ring done theorem gcd_dvd : ∀ (b a : Nat), (gcd a b) ∣ a ∧ (gcd a b) ∣ b := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd a b_1) ∣ a ∧ (gcd a b_1) ∣ b_1 fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_base] --Goal: a ∣ a ∧ a ∣ 0 apply And.intro (dvd_self a) define apply Exists.intro 0 rfl done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_nonzero a h1] --Goal : gcd b (a % b) ∣ a ∧ gcd b (a % b) ∣ b have h2 : a % b < b := mod_nonzero_lt a h1 have h3 : (gcd b (a % b)) ∣ b ∧ (gcd b (a % b)) ∣ (a % b) := ih (a % b) h2 b apply And.intro _ h3.left show (gcd b (a % b)) ∣ a from dvd_a_of_dvd_b_mod h3.left h3.right done done theorem gcd_dvd_left (a b : Nat) : (gcd a b) ∣ a := (gcd_dvd b a).left theorem gcd_dvd_right (a b : Nat) : (gcd a b) ∣ b := (gcd_dvd b a).right lemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl lemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c1 a b = gcd_c2 b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done lemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl lemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * ↑(a / b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done theorem gcd_lin_comb : ∀ (b a : Nat), (gcd_c1 a b) * ↑a + (gcd_c2 a b) * ↑b = ↑(gcd a b) := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd_c1 a b_1) * ↑a + (gcd_c2 a b_1) * ↑b_1 = ↑(gcd a b_1) fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base] --Goal : 1 * ↑a + 0 * ↑0 = ↑a ring done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1] --Goal : gcd_c2 b (a % b) * ↑a + -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * ↑(a / b)) * ↑b = -- ↑(gcd b (a % b)) set r : Nat := a % b set q : Nat := a / b set s : Int := gcd_c1 b r set t : Int := gcd_c2 b r --Goal : t * ↑a + (s - t * ↑q) * ↑b = ↑(gcd b r) have h2 : r < b := mod_nonzero_lt a h1 have h3 : s * ↑b + t * ↑r = ↑(gcd b r) := ih r h2 b have h4 : b * q + r = a := Nat.div_add_mod a b rewrite [←h3, ←h4] rewrite [Nat.cast_add, Nat.cast_mul] --Goal : t * (↑b * ↑q + ↑r) + (s - t * ↑q) * ↑b = s * ↑b + t * ↑r ring done done #eval gcd_c1 672 161 --Answer: 6 #eval gcd_c2 672 161 --Answer: -25 --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161 theorem Theorem_7_1_6 {d a b : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ gcd a b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑d ∣ ↑(gcd a b) set s : Int := gcd_c1 a b set t : Int := gcd_c2 a b have h3 : s * ↑a + t * ↑b = ↑(gcd a b) := gcd_lin_comb b a rewrite [←h3] --Goal : ↑d ∣ s * ↑a + t * ↑b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul] --Goal : ↑d ∣ s * (↑d * ↑j) + t * (↑d * ↑k) define apply Exists.intro (s * ↑j + t * ↑k) ring done /- Section 7.2 -/ theorem dvd_trans {a b c : Nat} (h1 : a ∣ b) (h2 : b ∣ c) : a ∣ c := by define at h1; define at h2; define obtain (m : Nat) (h3 : b = a * m) from h1 obtain (n : Nat) (h4 : c = b * n) from h2 rewrite [h3, mul_assoc] at h4 apply Exists.intro (m * n) show c = a * (m * n) from h4 done lemma exists_prime_factor : ∀ (n : Nat), 2 ≤ n → ∃ (p : Nat), prime_factor p n := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, 2 ≤ n_1 → ∃ (p : Nat), prime_factor p n_1 assume h1 : 2 ≤ n by_cases h2 : prime n · -- Case 1. h2 : prime n apply Exists.intro n define --Goal : prime n ∧ n ∣ n show prime n ∧ n ∣ n from And.intro h2 (dvd_self n) done · -- Case 2. h2 : ¬prime n define at h2 --h2 : ¬(2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n) demorgan at h2 disj_syll h2 h1 obtain (a : Nat) (h3 : ∃ (b : Nat), a * b = n ∧ a < n ∧ b < n) from h2 obtain (b : Nat) (h4 : a * b = n ∧ a < n ∧ b < n) from h3 have h5 : 2 ≤ a := by by_contra h6 have h7 : a ≤ 1 := by linarith have h8 : n ≤ b := calc n _ = a * b := h4.left.symm _ ≤ 1 * b := by rel [h7] _ = b := by ring linarith --n ≤ b contradicts b < n done have h6 : ∃ (p : Nat), prime_factor p a := ih a h4.right.left h5 obtain (p : Nat) (h7 : prime_factor p a) from h6 apply Exists.intro p define --Goal : prime p ∧ p ∣ n define at h7 --h7 : prime p ∧ p ∣ a apply And.intro h7.left have h8 : a ∣ n := by apply Exists.intro b show n = a * b from (h4.left).symm done show p ∣ n from dvd_trans h7.right h8 done done lemma exists_least_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q := by set S : Set Nat := {p : Nat | prime_factor p n} have h2 : ∃ (p : Nat), p ∈ S := exists_prime_factor n h show ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q from well_ord_princ S h2 done lemma all_prime_nil : all_prime [] := by define --Goal : ∀ p ∈ [], prime p fix p : Nat contrapos --Goal : ¬prime p → p ∉ [] assume h1 : ¬prime p show p ∉ [] from List.not_mem_nil p done lemma all_prime_cons (n : Nat) (L : List Nat) : all_prime (n :: L) ↔ prime n ∧ all_prime L := by apply Iff.intro · -- (→) assume h1 : all_prime (n :: L) --Goal : prime n ∧ all_prime L define at h1 --h1 : ∀ p ∈ n :: L, prime p apply And.intro (h1 n (List.mem_cons_self n L)) define --Goal : ∀ p ∈ L, prime p fix p : Nat assume h2 : p ∈ L show prime p from h1 p (List.mem_cons_of_mem n h2) done · -- (←) assume h1 : prime n ∧ all_prime L --Goal : all_prime (n :: l) define : all_prime L at h1 define fix p : Nat assume h2 : p ∈ n :: L rewrite [List.mem_cons] at h2 --h2 : p = n ∨ p ∈ L by_cases on h2 · -- Case 1. h2 : p = n rewrite [h2] show prime n from h1.left done · -- Case 2. h2 : p ∈ L show prime p from h1.right p h2 done done done lemma nondec_nil : nondec [] := by define --Goal : True trivial --trivial proves some obviously true statements, such as True done lemma nondec_cons (n : Nat) (L : List Nat) : nondec (n :: L) ↔ (∀ m ∈ L, n ≤ m) ∧ nondec L := by rfl lemma prod_nil : prod [] = 1 := by rfl lemma prod_cons : prod (n :: L) = n * (prod L) := by rfl lemma exists_cons_of_length_eq_succ {A : Type} {l : List A} {n : Nat} (h : l.length = n + 1) : ∃ (a : A) (L : List A), l = a :: L ∧ L.length = n := by have h1 : ¬l.length = 0 := by linarith rewrite [List.length_eq_zero] at h1 obtain (a : A) (h2 : ∃ (L : List A), l = a :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List A) (h3 : l = a :: L) from h2 apply Exists.intro a apply Exists.intro L apply And.intro h3 have h4 : (a :: L).length = L.length + 1 := List.length_cons a L rewrite [←h3, h] at h4 show L.length = n from (Nat.add_right_cancel h4).symm done lemma list_elt_dvd_prod_by_length (a : Nat) : ∀ (n : Nat), ∀ (l : List Nat), l.length = n → a ∈ l → a ∣ prod l := by by_induc · --Base Case fix l : List Nat assume h1 : l.length = 0 rewrite [List.length_eq_zero] at h1 --h1 : l = [] rewrite [h1] --Goal : a ∈ [] → a ∣ prod [] contrapos assume h2 : ¬a ∣ prod [] show a ∉ [] from List.not_mem_nil a done · -- Induction Step fix n : Nat assume ih : ∀ (l : List Nat), List.length l = n → a ∈ l → a ∣ prod l fix l : List Nat assume h1 : l.length = n + 1 --Goal : a ∈ l → a ∣ prod l obtain (b : Nat) (h2 : ∃ (L : List Nat), l = b :: L ∧ L.length = n) from exists_cons_of_length_eq_succ h1 obtain (L : List Nat) (h3 : l = b :: L ∧ L.length = n) from h2 have h4 : a ∈ L → a ∣ prod L := ih L h3.right assume h5 : a ∈ l rewrite [h3.left, prod_cons] --Goal : a ∣ b * prod L rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b ∨ a ∈ L by_cases on h5 · -- Case 1. h5 : a = b apply Exists.intro (prod L) rewrite [h5] rfl done · -- Case 2. h5 : a ∈ L have h6 : a ∣ prod L := h4 h5 have h7 : prod L ∣ b * prod L := by apply Exists.intro b ring done show a ∣ b * prod L from dvd_trans h6 h7 done done done lemma list_elt_dvd_prod {a : Nat} {l : List Nat} (h : a ∈ l) : a ∣ prod l := by set n : Nat := l.length have h1 : l.length = n := by rfl show a ∣ prod l from list_elt_dvd_prod_by_length a n l h1 h done lemma exists_prime_factorization : ∀ (n : Nat), n ≥ 1 → ∃ (l : List Nat), prime_factorization n l := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, n_1 ≥ 1 → ∃ (l : List Nat), prime_factorization n_1 l assume h1 : n ≥ 1 by_cases h2 : n = 1 · -- Case 1. h2 : n = 1 apply Exists.intro [] define apply And.intro · -- Proof of nondec_prime_list [] define show all_prime [] ∧ nondec [] from And.intro all_prime_nil nondec_nil done · -- Proof of prod [] = n rewrite [prod_nil, h2] rfl done done · -- Case 2. h2 : n ≠ 1 have h3 : n ≥ 2 := lt_of_le_of_ne' h1 h2 obtain (p : Nat) (h4 : prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q) from exists_least_prime_factor h3 have p_prime_factor : prime_factor p n := h4.left define at p_prime_factor have p_prime : prime p := p_prime_factor.left have p_dvd_n : p ∣ n := p_prime_factor.right have p_least : ∀ (q : Nat), prime_factor q n → p ≤ q := h4.right obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n have h5 : m ≠ 0 := by contradict h1 with h6 have h7 : n = 0 := calc n _ = p * m := n_eq_pm _ = p * 0 := by rw [h6] _ = 0 := by ring rewrite [h7] decide done have m_pos : 0 < m := Nat.pos_of_ne_zero h5 have m_lt_n : m < n := by define at p_prime show m < n from calc m _ < m + m := by linarith _ = 2 * m := by ring _ ≤ p * m := by rel [p_prime.left] _ = n := n_eq_pm.symm done obtain (L : List Nat) (h6 : prime_factorization m L) from ih m m_lt_n m_pos define at h6 have ndpl_L : nondec_prime_list L := h6.left define at ndpl_L apply Exists.intro (p :: L) define apply And.intro · -- Proof of nondec_prime_list (p :: L) define apply And.intro · -- Proof of all_prime (p :: L) rewrite [all_prime_cons] show prime p ∧ all_prime L from And.intro p_prime ndpl_L.left done · -- Proof of nondec (p :: L) rewrite [nondec_cons] apply And.intro _ ndpl_L.right fix q : Nat assume q_in_L : q ∈ L have h7 : q ∣ prod L := list_elt_dvd_prod q_in_L rewrite [h6.right] at h7 --h7 : q ∣ m have h8 : m ∣ n := by apply Exists.intro p rewrite [n_eq_pm] ring done have q_dvd_n : q ∣ n := dvd_trans h7 h8 have ap_L : all_prime L := ndpl_L.left define at ap_L have q_prime_factor : prime_factor q n := And.intro (ap_L q q_in_L) q_dvd_n show p ≤ q from p_least q q_prime_factor done done · -- Proof of prod (p :: L) = n rewrite [prod_cons, h6.right, n_eq_pm] rfl done done done theorem Theorem_7_2_2 {a b c : Nat} (h1 : c ∣ a * b) (h2 : rel_prime a c) : c ∣ b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑c ∣ ↑b define at h1; define at h2; define obtain (j : Nat) (h3 : a * b = c * j) from h1 set s : Int := gcd_c1 a c set t : Int := gcd_c2 a c have h4 : s * ↑a + t * ↑c = ↑(gcd a c) := gcd_lin_comb c a rewrite [h2, Nat.cast_one] at h4 --h4 : s * ↑a + t * ↑c = (1 : Int) apply Exists.intro (s * ↑j + t * ↑b) show ↑b = ↑c * (s * ↑j + t * ↑b) from calc ↑b _ = (1 : Int) * ↑b := (one_mul _).symm _ = (s * ↑a + t * ↑c) * ↑b := by rw [h4] _ = s * (↑a * ↑b) + t * ↑c * ↑b := by ring _ = s * (↑c * ↑j) + t * ↑c * ↑b := by rw [←Nat.cast_mul a b, h3, Nat.cast_mul c j] _ = ↑c * (s * ↑j + t * ↑b) := by ring done lemma le_nonzero_prod_left {a b : Nat} (h : a * b ≠ 0) : a ≤ a * b := by have h1 : b ≠ 0 := by contradict h with h1 rewrite [h1] ring done have h2 : 1 ≤ b := Nat.pos_of_ne_zero h1 show a ≤ a * b from calc a = a * 1 := (mul_one a).symm _ ≤ a * b := by rel [h2] done lemma le_nonzero_prod_right {a b : Nat} (h : a * b ≠ 0) : b ≤ a * b := by rewrite [mul_comm] rewrite [mul_comm] at h show b ≤ b * a from le_nonzero_prod_left h done lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry lemma rel_prime_of_prime_not_dvd {a p : Nat} (h1 : prime p) (h2 : ¬p ∣ a) : rel_prime a p := by have h3 : gcd a p ∣ a := gcd_dvd_left a p have h4 : gcd a p ∣ p := gcd_dvd_right a p have h5 : gcd a p = 1 ∨ gcd a p = p := dvd_prime h1 h4 have h6 : gcd a p ≠ p := by contradict h2 with h6 rewrite [h6] at h3 show p ∣ a from h3 done disj_syll h5 h6 show rel_prime a p from h5 done theorem Theorem_7_2_3 {a b p : Nat} (h1 : prime p) (h2 : p ∣ a * b) : p ∣ a ∨ p ∣ b := by or_right with h3 have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3 show p ∣ b from Theorem_7_2_2 h2 h4 done lemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a ≥ 1 := by have h1 : a ≠ 0 := by by_contra h1 rewrite [h1] at h contradict h linarith done show a ≥ 1 from Nat.pos_of_ne_zero h1 done lemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by have h1 : a ≥ 1 := ge_one_of_prod_one h have h2 : a * b ≠ 0 := by linarith have h3 : a ≤ a * b := le_nonzero_prod_left h2 rewrite [h] at h3 show a = 1 from Nat.le_antisymm h3 h1 done lemma eq_one_of_dvd_one {n : Nat} (h : n ∣ 1) : n = 1 := by obtain (j : Nat) (h1 : 1 = n * j) from h show n = 1 from eq_one_of_prod_one h1.symm done lemma prime_not_one {p : Nat} (h : prime p) : p ≠ 1 := by define at h linarith done theorem Theorem_7_2_4 {p : Nat} (h1 : prime p) : ∀ (l : List Nat), p ∣ prod l → ∃ a ∈ l, p ∣ a := by apply List.rec · -- Base Case. Goal : p ∣ prod [] → ∃ a ∈ [], p ∣ a rewrite [prod_nil] assume h2 : p ∣ 1 show ∃ a ∈ [], p ∣ a from absurd (eq_one_of_dvd_one h2) (prime_not_one h1) done · -- Induction Step fix b : Nat fix L : List Nat assume ih : p ∣ prod L → ∃ a ∈ L, p ∣ a --Goal : p ∣ prod (b :: L) → ∃ a ∈ b :: L, p ∣ a assume h2 : p ∣ prod (b :: L) rewrite [prod_cons] at h2 have h3 : p ∣ b ∨ p ∣ prod L := Theorem_7_2_3 h1 h2 by_cases on h3 · -- Case 1. h3 : p ∣ b apply Exists.intro b show b ∈ b :: L ∧ p ∣ b from And.intro (List.mem_cons_self b L) h3 done · -- Case 2. h3 : p ∣ prod L obtain (a : Nat) (h4 : a ∈ L ∧ p ∣ a) from ih h3 apply Exists.intro a show a ∈ b :: L ∧ p ∣ a from And.intro (List.mem_cons_of_mem b h4.left) h4.right done done done lemma prime_in_list {p : Nat} {l : List Nat} (h1 : prime p) (h2 : all_prime l) (h3 : p ∣ prod l) : p ∈ l := by obtain (a : Nat) (h4 : a ∈ l ∧ p ∣ a) from Theorem_7_2_4 h1 l h3 define at h2 have h5 : prime a := h2 a h4.left have h6 : p = 1 ∨ p = a := dvd_prime h5 h4.right disj_syll h6 (prime_not_one h1) rewrite [h6] show a ∈ l from h4.left done lemma first_le_first {p q : Nat} {l m : List Nat} (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m)) (h3 : prod (p :: l) = prod (q :: m)) : p ≤ q := by define at h1; define at h2 have h4 : q ∣ prod (p :: l) := by define apply Exists.intro (prod m) rewrite [←prod_cons] show prod (p :: l) = prod (q :: m) from h3 done have h5 : all_prime (q :: m) := h2.left rewrite [all_prime_cons] at h5 have h6 : q ∈ p :: l := prime_in_list h5.left h1.left h4 have h7 : nondec (p :: l) := h1.right rewrite [nondec_cons] at h7 rewrite [List.mem_cons] at h6 by_cases on h6 · -- Case 1. h6 : q = p linarith done · -- Case 2. h6 : q ∈ l have h8 : ∀ m ∈ l, p ≤ m := h7.left show p ≤ q from h8 q h6 done done lemma nondec_prime_list_tail {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by define at h define rewrite [all_prime_cons, nondec_cons] at h show all_prime l ∧ nondec l from And.intro h.left.right h.right.right done lemma cons_prod_not_one {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : prod (p :: l) ≠ 1 := by define at h have h1 : all_prime (p :: l) := h.left rewrite [all_prime_cons] at h1 rewrite [prod_cons] by_contra h2 show False from (prime_not_one h1.left) (eq_one_of_prod_one h2) done lemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) : l = [] ↔ prod l = 1 := by apply Iff.intro · -- (→) assume h1 : l = [] rewrite [h1] show prod [] = 1 from prod_nil done · -- (←) contrapos assume h1 : ¬l = [] obtain (p : Nat) (h2 : ∃ (L : List Nat), l = p :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List Nat) (h3 : l = p :: L) from h2 rewrite [h3] at h rewrite [h3] show ¬prod (p :: L) = 1 from cons_prod_not_one h done done lemma prime_pos {p : Nat} (h : prime p) : p > 0 := by define at h linarith done theorem Theorem_7_2_5 : ∀ (l1 l2 : List Nat), nondec_prime_list l1 → nondec_prime_list l2 → prod l1 = prod l2 → l1 = l2 := by apply List.rec · -- Base Case. Goal : ∀ (l2 : List Nat), nondec_prime_list [] → -- nondec_prime_list l2 → prod [] = prod l2 → [] = l2 fix l2 : List Nat assume h1 : nondec_prime_list [] assume h2 : nondec_prime_list l2 assume h3 : prod [] = prod l2 rewrite [prod_nil, eq_comm, ←list_nil_iff_prod_one h2] at h3 show [] = l2 from h3.symm done · -- Induction Step fix p : Nat fix L1 : List Nat assume ih : ∀ (L2 : List Nat), nondec_prime_list L1 → nondec_prime_list L2 → prod L1 = prod L2 → L1 = L2 -- Goal : ∀ (l2 : List Nat), nondec_prime_list (p :: L1) → -- nondec_prime_list l2 → prod (p :: L1) = prod l2 → p :: L1 = l2 fix l2 : List Nat assume h1 : nondec_prime_list (p :: L1) assume h2 : nondec_prime_list l2 assume h3 : prod (p :: L1) = prod l2 have h4 : ¬prod (p :: L1) = 1 := cons_prod_not_one h1 rewrite [h3, ←list_nil_iff_prod_one h2] at h4 obtain (q : Nat) (h5 : ∃ (L : List Nat), l2 = q :: L) from List.exists_cons_of_ne_nil h4 obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5 rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2) rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2) have h7 : p ≤ q := first_le_first h1 h2 h3 have h8 : q ≤ p := first_le_first h2 h1 h3.symm have h9 : p = q := by linarith rewrite [h9, prod_cons, prod_cons] at h3 --h3 : q * prod L1 = q * prod L2 have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1 have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2 define at h2 have h12 : all_prime (q :: L2) := h2.left rewrite [all_prime_cons] at h12 have h13 : q > 0 := prime_pos h12.left have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3 have h15 : L1 = L2 := ih L2 h10 h11 h14 rewrite [h6, h9, h15] rfl done done theorem fund_thm_arith (n : Nat) (h : n ≥ 1) : ∃! (l : List Nat), prime_factorization n l := by exists_unique · -- Existence show ∃ (l : List Nat), prime_factorization n l from exists_prime_factorization n h done · -- Uniqueness fix l1 : List Nat; fix l2 : List Nat assume h1 : prime_factorization n l1 assume h2 : prime_factorization n l2 define at h1; define at h2 have h3 : prod l1 = n := h1.right rewrite [←h2.right] at h3 show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3 done done /- Section 7.3 -/ theorem congr_refl (m : Nat) : ∀ (a : Int), a ≡ a (MOD m) := by fix a : Int define --Goal : ∃ (c : Int), a - a = ↑m * c apply Exists.intro 0 ring done theorem congr_symm {m : Nat} : ∀ {a b : Int}, a ≡ b (MOD m) → b ≡ a (MOD m) := by fix a : Int; fix b : Int assume h1 : a ≡ b (MOD m) define at h1 --h1 : ∃ (c : Int), a - b = ↑m * c define --Goal : ∃ (c : Int), b - a = ↑m * c obtain (c : Int) (h2 : a - b = m * c) from h1 apply Exists.intro (-c) show b - a = m * (-c) from calc b - a _ = -(a - b) := by ring _ = -(m * c) := by rw [h2] _ = m * (-c) := by ring done theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry /- Fundamental properties of congruence classes -/ lemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) : X = Y ↔ X.val = Y.val := Fin.ext_iff lemma val_nat_eq_mod (n k : Nat) : ([k]_(n + 1)).val = k % (n + 1) := by rfl lemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl theorem cc_rep {m : Nat} (X : ZMod m) : ∃ (a : Int), X = [a]_m := match m with | 0 => by apply Exists.intro X rfl done | n + 1 => by apply Exists.intro ↑(X.val) have h1 : X.val < n + 1 := Fin.prop X rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1] rfl done theorem add_class (m : Nat) (a b : Int) : [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm theorem mul_class (m : Nat) (a b : Int) : [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm lemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ [a - b]_m = [0]_m := by apply Iff.intro · -- (→) assume h1 : [a]_m = [b]_m have h2 : a - b = a + (-b) := by ring have h3 : b + (-b) = 0 := by ring show [a - b]_m = [0]_m from calc [a - b]_m _ = [a + (-b)]_m := by rw [h2] _ = [a]_m + [-b]_m := by rw [add_class] _ = [b]_m + [-b]_m := by rw [h1] _ = [b + -b]_m := by rw [add_class] _ = [0]_m := by rw [h3] done · -- (←) assume h1 : [a - b]_m = [0]_m have h2 : b + (a - b) = a := by ring have h3 : b + 0 = b := by ring show [a]_m = [b]_m from calc [a]_m _ = [b + (a - b)]_m := by rw [h2] _ = [b]_m + [a - b]_m := by rw [add_class] _ = [b]_m + [0]_m := by rw [h1] _ = [b + 0]_m := by rw [add_class] _ = [b]_m := by rw [h3] done done lemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) : [a]_m = [0]_m → [-a]_m = [0]_m := by assume h1 : [a]_m = [0]_m have h2 : 0 + (-a) = -a := by ring have h3 : a + (-a) = 0 := by ring show [-a]_m = [0]_m from calc [-a]_m _ = [0 + (-a)]_m := by rw [h2] _ = [0]_m + [-a]_m := by rw [add_class] _ = [a]_m + [-a]_m := by rw [h1] _ = [a + (-a)]_m := by rw [add_class] _ = [0]_m := by rw [h3] done lemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) : [-a]_m = [0]_m ↔ [a]_m = [0]_m := by apply Iff.intro _ (cc_neg_zero_of_cc_zero m a) assume h1 : [-a]_m = [0]_m have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1 have h3 : -(-a) = a := by ring rewrite [h3] at h2 show [a]_m = [0]_m from h2 done lemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl lemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m ↔ m ∣ k := match m with | 0 => by have h : (0 : Int) = (↑(0 : Nat) : Int) := by rfl rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj] apply Iff.intro · -- (→) assume h1 : k = 0 rewrite [h1] show 0 ∣ 0 from dvd_self 0 done · -- (←) assume h1 : 0 ∣ k obtain (c : Nat) (h2 : k = 0 * c) from h1 rewrite [h2] ring done done | n + 1 => by rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero] show k % (n + 1) = 0 ↔ n + 1 ∣ k from (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm done lemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m ↔ ↑m ∣ a := by obtain (k : Nat) (h1 : a = ↑k ∨ a = -↑k) from Int.eq_nat_or_neg a by_cases on h1 · -- Case 1. h1: a = ↑k rewrite [h1, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done · -- Case 2. h1: a = -↑k rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done done theorem cc_eq_iff_congr (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ a ≡ b (MOD m) := calc [a]_m = [b]_m _ ↔ [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b _ ↔ ↑m ∣ (a - b) := cc_zero_iff_dvd m (a - b) _ ↔ a ≡ b (MOD m) := by rfl /- End of fundamental properties of congruence classes -/ lemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m := by have h1 : (↑m : Int) ≠ 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m) show 0 ≤ a % m from Int.emod_nonneg a h1 done lemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m) have h2 : (↑m : Int) > 0 := (Nat.cast_pos).rtl h1 show a % m < m from Int.emod_lt_of_pos a h2 done lemma congr_mod_mod (m : Nat) (a : Int) : a ≡ a % m (MOD m) := by define have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m apply Exists.intro (a / m) show a - a % m = m * (a / m) from calc a - (a % m) _ = m * (a / m) + a % m - a % m := by rw [h1] _ = m * (a / m) := by ring done lemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) := And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a)) theorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) : ∃! (r : Int), 0 ≤ r ∧ r < m ∧ a ≡ r (MOD m) := by exists_unique · -- Existence apply Exists.intro (a % m) show 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) from mod_cmpl_res m a done · -- Uniqueness fix r1 : Int; fix r2 : Int assume h1 : 0 ≤ r1 ∧ r1 < m ∧ a ≡ r1 (MOD m) assume h2 : 0 ≤ r2 ∧ r2 < m ∧ a ≡ r2 (MOD m) have h3 : r1 ≡ r2 (MOD m) := congr_trans (congr_symm h1.right.right) h2.right.right obtain (d : Int) (h4 : r1 - r2 = m * d) from h3 have h5 : r1 - r2 < m * 1 := by linarith have h6 : m * (-1) < r1 - r2 := by linarith rewrite [h4] at h5 --h5 : m * d < m * 1 rewrite [h4] at h6 --h6 : m * -1 < m * d have h7 : (↑m : Int) ≥ 0 := Nat.cast_nonneg m have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7 have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7 have h10 : d = 0 := by linarith show r1 = r2 from calc r1 _ = r1 - r2 + r2 := by ring _ = m * 0 + r2 := by rw [h4, h10] _ = r2 := by ring done done lemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m := (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a) theorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y rewrite [h1, h2] have h3 : a + b = b + a := by ring show [a]_m + [b]_m = [b]_m + [a]_m from calc [a]_m + [b]_m _ = [a + b]_m := add_class m a b _ = [b + a]_m := by rw [h3] _ = [b]_m + [a]_m := (add_class m b a).symm done theorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X rewrite [h1] have h2 : a * 1 = a := by ring show [a]_m * [1]_m = [a]_m from calc [a]_m * [1]_m _ = [a * 1]_m := mul_class m a 1 _ = [a]_m := by rw [h2] done theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry lemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) : [a]_m * [gcd_c2 m a]_m = [1]_m := by set s : Int := gcd_c1 m a have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m define at h1 rewrite [h1, Nat.cast_one] at h2 --h2 : s * ↑m + gcd_c2 m a * ↑a = 1 rewrite [mul_class, cc_eq_iff_congr] define --Goal : ∃ (c : Int), ↑a * gcd_c2 m a - 1 = ↑m * c apply Exists.intro (-s) show a * (gcd_c2 m a) - 1 = m * (-s) from calc a * (gcd_c2 m a) - 1 _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring _ = 1 + m * (-s) - 1 := by rw [h2] _ = m * (-s) := by ring done theorem Theorem_7_3_7 (m a : Nat) : invertible [a]_m ↔ rel_prime m a := by apply Iff.intro · -- (→) assume h1 : invertible [a]_m define at h1 obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1 obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y rewrite [h3, mul_class, cc_eq_iff_congr] at h2 define at h2 obtain (c : Int) (h4 : a * b - 1 = m * c) from h2 rewrite [Exercise_7_2_6] --Goal : ∃ (s t : Int), s * ↑m + t * ↑a = 1 apply Exists.intro (-c) apply Exists.intro b show (-c) * m + b * a = 1 from calc (-c) * m + b * a _ = (-c) * m + (a * b - 1) + 1 := by ring _ = (-c) * m + m * c + 1 := by rw [h4] _ = 1 := by ring done · -- (←) assume h1 : rel_prime m a define show ∃ (Y : ZMod m), [a]_m * Y = [1]_m from Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1) done done /- Section 7.4 -/ section Euler open Euler lemma num_rp_below_base {m : Nat} : num_rp_below m 0 = 0 := by rfl lemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) : num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by have h1 : num_rp_below m (j + 1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : gcd m j = 1 rewrite [if_pos h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1 show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1 done lemma num_rp_below_step_not_rp {m j : Nat} (h : ¬rel_prime m j) : num_rp_below m (j + 1) = num_rp_below m j := by have h1 : num_rp_below m (j +1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : ¬gcd m j = 1 rewrite [if_neg h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j show num_rp_below m (j + 1) = num_rp_below m j from h1 done lemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl #eval phi 10 --Answer: 4 lemma prod_inv_iff_inv {m : Nat} {X : ZMod m} (h1 : invertible X) (Y : ZMod m) : invertible (X * Y) ↔ invertible Y := by apply Iff.intro · -- (→) assume h2 : invertible (X * Y) obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2 apply Exists.intro (X * Z) rewrite [←h3] --Goal : Y * (X * Z) = X * Y * Z ring --Note that ring can do algebra in ZMod m done · -- (←) assume h2 : invertible Y obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1 obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2 apply Exists.intro (Xi * Yi) show (X * Y) * (Xi * Yi) = [1]_m from calc X * Y * (Xi * Yi) _ = (X * Xi) * (Y * Yi) := by ring _ = [1]_m * [1]_m := by rw [h3, h4] _ = [1]_m := Theorem_7_3_6_7 [1]_m done done lemma F_rp_def {m i : Nat} (h : rel_prime m i) : F m i = [i]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h --h : gcd m i = 1 rewrite [if_pos h] at h1 show F m i = [i]_m from h1 done lemma F_not_rp_def {m i : Nat} (h : ¬rel_prime m i) : F m i = [1]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h rewrite [h1, if_neg h] rfl done lemma prod_seq_base {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 0 k f = [1]_m := by rfl lemma prod_seq_step {m : Nat} (n k : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl lemma prod_seq_zero_step {m : Nat} (n : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by rewrite [prod_seq_step, zero_add] rfl done lemma prod_one {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 1 k f = f k := by rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7] rfl done lemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl lemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m := calc [G m a i]_m _ = [(a * i) % m]_m := by rfl _ = [a * i]_m := (cc_eq_mod m (a * i)).symm _ = [a]_m * [i]_m := (mul_class m a i).symm lemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) : rel_prime m (G m a i) ↔ rel_prime m i := by have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1 show rel_prime m (G m a i) ↔ rel_prime m i from calc rel_prime m (G m a i) _ ↔ invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm _ ↔ invertible ([a]_m * [i]_m) := by rw [cc_G] _ ↔ invertible [i]_m := prod_inv_iff_inv h2 ([i]_m) _ ↔ rel_prime m i := Theorem_7_3_7 m i done lemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) : F m (G m a i) = [a]_m * F m i := by have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2 show F m (G m a i) = [a]_m * F m i from calc F m (G m a i) _ = [G m a i]_m := F_rp_def h3 _ = [a]_m * [i]_m := cc_G m a i _ = [a]_m * F m i := by rw [F_rp_def h2] done lemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : ¬rel_prime m i) : F m (G m a i) = [1]_m := by rewrite [←G_rp_iff h1 i] at h2 show F m (G m a i) = [1]_m from F_not_rp_def h2 done lemma FG_prod {m a : Nat} (h1 : rel_prime m a) : ∀ (k : Nat), prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by by_induc · -- Base Case show prod_seq 0 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from calc prod_seq 0 0 ((F m) ∘ (G m a)) _ = [1]_m := prod_seq_base _ _ _ = [a]_m ^ 0 * [1]_m := by ring _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by rw [num_rp_below_base, prod_seq_base] done · -- Induction Step fix k : Nat assume ih : prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) by_cases h2 : rel_prime m k · -- Case 1. h2 : rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([a]_m * F m k) := by rw [FG_rp h1 h2] _ = [a]_m ^ ((num_rp_below m k) + 1) * ((prod_seq k 0 (F m)) * F m k) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_rp h2, prod_seq_zero_step] done · -- Case 2. h2 : ¬rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([1]_m) := by rw [FG_not_rp h1 h2] _ = [a]_m ^ (num_rp_below m k) * (prod_seq k 0 (F m) * ([1]_m)) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_not_rp h2, prod_seq_zero_step, F_not_rp_def h2] done done done lemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by define --Goal : ∀ i < m, G m a i < m fix i : Nat assume h1 : i < m rewrite [G_def] --Goal : a * i % m < m show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m) done lemma left_inv_one_one_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g' (g i) = i) : one_one_below n g := sorry lemma right_inv_onto_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g (g' i) = i) (h2 : maps_below n g') : onto_below n g := by define at h2; define fix k : Nat assume h3 : k < n apply Exists.intro (g' k) show g' k < n ∧ g (g' k) = k from And.intro (h2 k h3) (h1 k h3) done lemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m * [inv_mod m a]_m = [1]_m := by have h2 : 0 ≤ (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a) show [a]_m * [inv_mod m a]_m = [1]_m from calc [a]_m * [inv_mod m a]_m _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2] _ = [a]_m * [gcd_c2 m a]_m := by rw [←cc_eq_mod] _ = [1]_m := gcd_c2_inv h1 done lemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m := calc a * (b % m) % m = a % m * (b % m % m) % m := Nat.mul_mod _ _ _ _ = a % m * (b % m) % m := by rw [Nat.mod_mod] _ = a * b % m := (Nat.mul_mod _ _ _).symm lemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm] rfl done theorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] : ↑a ≡ ↑b (MOD m) ↔ a % m = b % m := sorry lemma mul_inv_mod_cancel {m a i : Nat} [NeZero m] (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1 rewrite [mul_class, cc_eq_iff_congr, ←Nat.cast_mul, ←Nat.cast_one, congr_iff_mod_eq_Nat] at h3 show a * inv_mod m a * i % m = i from calc a * (inv_mod m a) * i % m _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = 1 % m * i % m := by rw [h3] _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = i % m := by rw [one_mul] _ = i := Nat.mod_eq_of_lt h2 done lemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl lemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, G m a (Ginv m a i) = i := by fix i : Nat assume h2 : i < m show G m a (Ginv m a i) = i from calc G m a (Ginv m a i) _ = a * ((inv_mod m a * i) % m) % m := by rfl _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, Ginv m a (G m a i) = i := by fix i : Nat assume h2 : i < m show Ginv m a (G m a i) = i from calc Ginv m a (G m a i) _ = inv_mod m a * ((a * i) % m) % m := by rfl _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc, mul_comm (inv_mod m a)] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_maps_below (m a : Nat) [NeZero m] : maps_below m (Ginv m a) := G_maps_below m (inv_mod m a) lemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : one_one_below m (G m a) := left_inv_one_one_below (Ginv_left_inv h1) lemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : onto_below m (G m a) := right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a) lemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : perm_below m (G m a) := And.intro (G_maps_below m a) (And.intro (G_one_one_below h1) (G_onto_below h1)) --Permuting a product of congruence classes doesn't change product lemma swap_fst (u v : Nat) : swap u v u = v := by define : swap u v u --Goal : (if u = u then v else if u = v then u else u) = v have h : u = u := by rfl rewrite [if_pos h] rfl done lemma swap_snd (u v : Nat) : swap u v v = u := by define : swap u v v by_cases h1 : v = u · -- Case 1. h1 : v = u rewrite [if_pos h1] show v = u from h1 done · -- Case 2. h1 : v ≠ u rewrite [if_neg h1] have h2 : v = v := by rfl rewrite [if_pos h2] rfl done done lemma swap_other {u v i : Nat} (h1 : i ≠ u) (h2 : i ≠ v) : swap u v i = i := by define : swap u v i rewrite [if_neg h1, if_neg h2] rfl done lemma swap_values (u v i : Nat) : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := by by_cases h1 : i = u · -- Case 1. h1 : i = u apply Or.inl rewrite [h1] show swap u v u = v from swap_fst u v done · -- Case 2. h1 : i ≠ u apply Or.inr by_cases h2 : i = v · -- Case 2.1. h2 : i = v apply Or.inl rewrite [h2] show swap u v v = u from swap_snd u v done · -- Case 2.2. h2 : i ≠ v apply Or.inr show swap u v i = i from swap_other h1 h2 done done done lemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by define fix i : Nat assume h3 : i < n have h4 : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := swap_values u v i by_cases on h4 · -- Case 1. h4 : swap u v i = v rewrite [h4] show v < n from h2 done · -- Case 2. by_cases on h4 · -- Case 2.1. h4 : swap u v i = u rewrite [h4] show u < n from h1 done · -- Case 2.2. h4 : swap u v i = i rewrite [h4] show i < n from h3 done done done lemma swap_swap (u v n : Nat) : ∀ i < n, swap u v (swap u v i) = i := by fix i : Nat assume h : i < n by_cases h1 : i = u · -- Case 1. h1 : i = u rewrite [h1, swap_fst, swap_snd] rfl done · -- Case 2. h1 : i ≠ u by_cases h2 : i = v · -- Case 2.1. h2 : i = v rewrite [h2, swap_snd, swap_fst] rfl done · -- Case 2.2. h2 : i ≠ v rewrite [swap_other h1 h2, swap_other h1 h2] rfl done done done lemma swap_one_one_below (u v n) : one_one_below n (swap u v) := left_inv_one_one_below (swap_swap u v n) lemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) := right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2) lemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) := And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2)) lemma comp_perm_below {n : Nat} {f g : Nat → Nat} (h1 : perm_below n f) (h2 : perm_below n g) : perm_below n (f ∘ g) := sorry lemma trivial_swap (u : Nat) : swap u u = id := by apply funext fix x : Nat by_cases h1 : x = u · -- Case 1. h1 : x = u rewrite [h1, swap_fst] rfl done · -- Case 2. h1 : x ≠ u rewrite [swap_other h1 h1] rfl done done lemma prod_eq_fun {m : Nat} (f g : Nat → ZMod m) (k : Nat) : ∀ (n : Nat), (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g := by by_induc · -- Base Case assume h : (∀ i < 0, f (k + i) = g (k + i)) rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g assume h1 : ∀ i < n + 1, f (k + i) = g (k + i) have h2 : ∀ i < n, f (k + i) = g (k + i) := by fix i : Nat assume h2 : i < n have h3 : i < n + 1 := by linarith show f (k + i) = g (k + i) from h1 i h3 done have h3 : prod_seq n k f = prod_seq n k g := ih h2 have h4 : n < n + 1 := Nat.lt_succ_self n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4] rfl done done lemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := by have h2 : ∀ (i : Nat), i < u → (f ∘ swap u n) (0 + i) = f (0 + i) := by fix i : Nat assume h2 : i < u have h3 : 0 + i ≠ u := by linarith have h4 : 0 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl done show prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f from prod_eq_fun (f ∘ swap u n) f 0 u h2 done lemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := by have h2 : ∀ i < j, (f ∘ swap u n) (u + 1 + i) = f (u + 1 + i) := by fix i : Nat assume h2 : i < j have h3 : u + 1 + i ≠ u := by linarith have h4 : u + 1 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl show prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f from prod_eq_fun (f ∘ swap u n) f (u + 1) j h2 done lemma break_prod {m : Nat} (n : Nat) (f : Nat → ZMod m) : ∀ (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by by_induc · -- Base Case have h : n + 0 = n := by rfl rewrite [prod_seq_base, h, Theorem_7_3_6_7] rfl done · -- Induction Step fix j : Nat assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f rewrite [←add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc] rfl done done lemma break_prod_twice {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f := break_prod n f 1 rewrite [prod_one] at h2 have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f := break_prod (u + 1) f j rewrite [←h1] at h3 have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f := break_prod u f 1 rewrite [prod_one] at h4 rewrite [h3, h4] at h2 show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2 done lemma swap_prod_eq_prod {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f := by by_cases h2 : u = n · -- Case 1. h2 : u = n rewrite [h2, trivial_swap n] --Goal : prod_seq (n + 1) 0 (f ∘ id) = prod_seq (n + 1) 0 f rfl done · -- Case 2. h2 : ¬u = n have h3 : u + 1 ≤ n := Nat.lt_of_le_of_ne h1 h2 obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3 have break_f : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := break_prod_twice f h4 have break_fs : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_prod_twice (f ∘ swap u n) h4 have f_eq_fs_below : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := swap_prod_eq_prod_below f h1 have f_eq_fs_btwn : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4 show prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f from calc prod_seq (n + 1) 0 (f ∘ swap u n) _ = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_fs _ = prod_seq u 0 f * (f ∘ swap u n) u * prod_seq j (u + 1) f * (f ∘ swap u n) n := by rw [f_eq_fs_below, f_eq_fs_btwn] _ = prod_seq u 0 f * f (swap u n u) * prod_seq j (u + 1) f * f (swap u n n) := by rfl _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by rw [swap_fst, swap_snd] _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring _ = prod_seq (n + 1) 0 f := break_f.symm done done lemma perm_below_fixed {n : Nat} {g : Nat → Nat} (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry lemma perm_prod {m : Nat} (f : Nat → ZMod m) : ∀ (n : Nat), ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) := by by_induc · -- Base Case fix g : Nat → Nat assume h1 : perm_below 0 g rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) fix g : Nat → Nat assume g_pb : perm_below (n + 1) g define at g_pb have g_ob : onto_below (n + 1) g := g_pb.right.right define at g_ob have h1 : n < n + 1 := by linarith obtain (u : Nat) (h2 : u < n + 1 ∧ g u = n) from g_ob n h1 have s_pb : perm_below (n + 1) (swap u n) := swap_perm_below h2.left h1 have gs_pb_n1 : perm_below (n + 1) (g ∘ swap u n) := comp_perm_below g_pb s_pb have gs_fix_n : (g ∘ swap u n) n = n := calc (g ∘ swap u n) n _ = g (swap u n n) := by rfl _ = g u := by rw [swap_snd] _ = n := h2.right have gs_pb_n : perm_below n (g ∘ swap u n) := perm_below_fixed gs_pb_n1 gs_fix_n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f ∘ (g ∘ swap u n)) := ih (g ∘ swap u n) gs_pb_n have h3 : u ≤ n := by linarith show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f ∘ g) from calc prod_seq (n + 1) 0 f _ = prod_seq n 0 f * f n := prod_seq_zero_step n f _ = prod_seq n 0 (f ∘ (g ∘ swap u n)) * f ((g ∘ swap u n) n) := by rw [gs_prod, gs_fix_n] _ = prod_seq n 0 (f ∘ g ∘ swap u n) * (f ∘ g ∘ swap u n) n := by rfl _ = prod_seq (n + 1) 0 (f ∘ g ∘ swap u n) := (prod_seq_zero_step n (f ∘ g ∘ swap u n)).symm _ = prod_seq (n + 1) 0 ((f ∘ g) ∘ swap u n) := by rfl _ = prod_seq (n + 1) 0 (f ∘ g) := swap_prod_eq_prod (f ∘ g) h3 done done lemma F_invertible (m i : Nat) : invertible (F m i) := by by_cases h : rel_prime m i · -- Case 1. h : rel_prime m i rewrite [F_rp_def h] show invertible [i]_m from (Theorem_7_3_7 m i).rtl h done · -- Case 2. h : ¬rel_prime m i rewrite [F_not_rp_def h] apply Exists.intro [1]_m show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m done done lemma Fprod_invertible (m : Nat) : ∀ (k : Nat), invertible (prod_seq k 0 (F m)) := by by_induc · -- Base Case apply Exists.intro [1]_m show prod_seq 0 0 (F m) * [1]_m = [1]_m from calc prod_seq 0 0 (F m) * [1]_m _ = [1]_m * [1]_m := by rw [prod_seq_base] _ = [1]_m := Theorem_7_3_6_7 ([1]_m) done · -- Induction Step fix k : Nat assume ih : invertible (prod_seq k 0 (F m)) rewrite [prod_seq_zero_step] show invertible (prod_seq k 0 (F m) * (F m k)) from (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k) done done theorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m ^ (phi m) = [1]_m := by have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2 show [a]_m ^ (phi m) = [1]_m from calc [a]_m ^ (phi m) _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3] _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring _ = prod_seq m 0 (F m ∘ G m a) * Y := by rw [FG_prod h1 m, phi_def] _ = prod_seq m 0 (F m) * Y := by rw [perm_prod (F m) m (G m a) (G_perm_below h1)] _ = [1]_m := by rw [h3] done lemma Exercise_7_4_5_Int (m : Nat) (a : Int) : ∀ (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry lemma Exercise_7_4_5_Nat (m a n : Nat) : [a]_m ^ n = [a ^ n]_m := by rewrite [Exercise_7_4_5_Int] rfl done theorem Euler's_theorem {m a : Nat} [NeZero m] (h1 : rel_prime m a) : a ^ (phi m) ≡ 1 (MOD m) := by have h2 : [a]_m ^ (phi m) = [1]_m := Theorem_7_4_2 h1 rewrite [Exercise_7_4_5_Nat m a (phi m)] at h2 --h2 : [a ^ phi m]_m = [1]_m show a ^ (phi m) ≡ 1 (MOD m) from (cc_eq_iff_congr _ _ _).ltr h2 done #eval gcd 10 7 --Answer: 1. So 10 and 7 are relatively prime #eval 7 ^ phi 10 --Answer: 2401, which is congruent to 1 mod 10. end Euler /- Section 7.5 -/ lemma num_rp_prime {p : Nat} (h1 : prime p) : ∀ k < p, num_rp_below p (k + 1) = k := sorry lemma phi_prime {p : Nat} (h1 : prime p) : phi p = p - 1 := by have h2 : 1 ≤ p := prime_pos h1 have h3 : p - 1 + 1 = p := Nat.sub_add_cancel h2 have h4 : p - 1 < p := by linarith have h5 : num_rp_below p (p - 1 + 1) = p - 1 := num_rp_prime h1 (p - 1) h4 rewrite [h3] at h5 show phi p = p - 1 from h5 done theorem Theorem_7_2_2_Int {a c : Nat} {b : Int} (h1 : ↑c ∣ ↑a * b) (h2 : rel_prime a c) : ↑c ∣ b := by rewrite [Int.natCast_dvd, Int.natAbs_mul, Int.natAbs_ofNat] at h1 --h1 : c ∣ a * Int.natAbs b rewrite [Int.natCast_dvd] --Goal : c ∣ Int.natAbs b show c ∣ Int.natAbs b from Theorem_7_2_2 h1 h2 done lemma Lemma_7_4_5 {m n : Nat} (a b : Int) (h1 : rel_prime m n) : a ≡ b (MOD m * n) ↔ a ≡ b (MOD m) ∧ a ≡ b (MOD n) := by apply Iff.intro · -- (→) assume h2 : a ≡ b (MOD m * n) obtain (j : Int) (h3 : a - b = (m * n) * j) from h2 apply And.intro · -- Proof of a ≡ b (MOD m) apply Exists.intro (n * j) show a - b = m * (n * j) from calc a - b _ = m * n * j := h3 _ = m * (n * j) := by ring done · -- Proof of a ≡ b (MOD n) apply Exists.intro (m * j) show a - b = n * (m * j) from calc a - b _ = m * n * j := h3 _ = n * (m * j) := by ring done done · -- (←) assume h2 : a ≡ b (MOD m) ∧ a ≡ b (MOD n) obtain (j : Int) (h3 : a - b = m * j) from h2.left have h4 : (↑n : Int) ∣ a - b := h2.right rewrite [h3] at h4 --h4 : ↑n ∣ ↑m * j have h5 : ↑n ∣ j := Theorem_7_2_2_Int h4 h1 obtain (k : Int) (h6 : j = n * k) from h5 apply Exists.intro k --Goal : a - b = ↑(m * n) * k rewrite [Nat.cast_mul] --Goal : a - b = ↑m * ↑n * k show a - b = (m * n) * k from calc a - b _ = m * j := h3 _ = m * (n * k) := by rw [h6] _ = (m * n) * k := by ring done done --From exercises of Section 7.2 theorem rel_prime_symm {a b : Nat} (h : rel_prime a b) : rel_prime b a := sorry lemma prime_NeZero {p : Nat} (h : prime p) : NeZero p := by rewrite [neZero_iff] --Goal : p ≠ 0 define at h linarith done lemma Lemma_7_5_1 {p e d m c s : Nat} {t : Int} (h1 : prime p) (h2 : e * d = (p - 1) * s + 1) (h3 : m ^ e - c = p * t) : c ^ d ≡ m (MOD p) := by have h4 : m ^ e ≡ c (MOD p) := Exists.intro t h3 have h5 : [m ^ e]_p = [c]_p := (cc_eq_iff_congr _ _ _).rtl h4 rewrite [←Exercise_7_4_5_Nat] at h5 --h5 : [m]_p ^ e = [c]_p by_cases h6 : p ∣ m · -- Case 1. h6 : p ∣ m have h7 : m ≡ 0 (MOD p) := by obtain (j : Nat) (h8 : m = p * j) from h6 apply Exists.intro (↑j : Int) --Goal : ↑m - 0 = ↑p * ↑j rewrite [h8, Nat.cast_mul] ring done have h8 : [m]_p = [0]_p := (cc_eq_iff_congr _ _ _).rtl h7 have h9 : e * d ≠ 0 := by rewrite [h2] show (p - 1) * s + 1 ≠ 0 from Nat.add_one_ne_zero _ done have h10 : (0 : Int) ^ (e * d) = 0 := zero_pow h9 have h11 : [c ^ d]_p = [m]_p := calc [c ^ d]_p _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat] _ = ([m]_p ^ e) ^ d := by rw [h5] _ = [m]_p ^ (e * d) := by ring _ = [0]_p ^ (e * d) := by rw [h8] _ = [0 ^ (e * d)]_p := Exercise_7_4_5_Int _ _ _ _ = [0]_p := by rw [h10] _ = [m]_p := by rw [h8] show c ^ d ≡ m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11 done · -- Case 2. h6 : ¬p ∣ m have h7 : rel_prime m p := rel_prime_of_prime_not_dvd h1 h6 have h8 : rel_prime p m := rel_prime_symm h7 have h9 : NeZero p := prime_NeZero h1 have h10 : (1 : Int) ^ s = 1 := by ring have h11 : [c ^ d]_p = [m]_p := calc [c ^ d]_p _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat] _ = ([m]_p ^ e) ^ d := by rw [h5] _ = [m]_p ^ (e * d) := by ring _ = [m]_p ^ ((p - 1) * s + 1) := by rw [h2] _ = ([m]_p ^ (p - 1)) ^ s * [m]_p := by ring _ = ([m]_p ^ (phi p)) ^ s * [m]_p := by rw [phi_prime h1] _ = [1]_p ^ s * [m]_p := by rw [Theorem_7_4_2 h8] _ = [1 ^ s]_p * [m]_p := by rw [Exercise_7_4_5_Int] _ = [1]_p * [m]_p := by rw [h10] _ = [m]_p * [1]_p := by ring _ = [m]_p := Theorem_7_3_6_7 _ show c ^ d ≡ m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11 done done theorem Theorem_7_5_1 (p q n e d k m c : Nat) (p_prime : prime p) (q_prime : prime q) (p_ne_q : p ≠ q) (n_pq : n = p * q) (ed_congr_1 : e * d = k * (p - 1) * (q - 1) + 1) (h1 : [m]_n ^ e = [c]_n) : [c]_n ^ d = [m]_n := by rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] at h1 --h1 : m ^ e ≡ c (MOD n) rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] --Goal : c ^ d ≡ m (MOD n) obtain (j : Int) (h2 : m ^ e - c = n * j) from h1 rewrite [n_pq, Nat.cast_mul] at h2 --h2 : m ^ e - c = p * q * j have h3 : e * d = (p - 1) * (k * (q - 1)) + 1 := by rewrite [ed_congr_1] ring done have h4 : m ^ e - c = p * (q * j) := by rewrite [h2] ring done have congr_p : c ^ d ≡ m (MOD p) := Lemma_7_5_1 p_prime h3 h4 have h5 : e * d = (q - 1) * (k * (p - 1)) + 1 := by rewrite [ed_congr_1] ring done have h6 : m ^ e - c = q * (p * j) := by rewrite [h2] ring done have congr_q : c ^ d ≡ m (MOD q) := Lemma_7_5_1 q_prime h5 h6 have h7 : ¬q ∣ p := by by_contra h8 have h9 : q = 1 ∨ q = p := dvd_prime p_prime h8 disj_syll h9 (prime_not_one q_prime) show False from p_ne_q h9.symm done have h8 : rel_prime p q := rel_prime_of_prime_not_dvd q_prime h7 rewrite [n_pq, Lemma_7_4_5 _ _ h8] show c ^ d ≡ m (MOD p) ∧ c ^ d ≡ m (MOD q) from And.intro congr_p congr_q done /- BEGIN EXERCISES -/ namespace Exercises /- Section 7.1 -/ -- 1. theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry -- 2. lemma gcd_comm_lt {a b : Nat} (h : a < b) : gcd a b = gcd b a := sorry theorem gcd_comm (a b : Nat) : gcd a b = gcd b a := sorry -- 3. theorem Exercise_7_1_5 (a b : Nat) (n : Int) : (∃ (s t : Int), s * a + t * b = n) ↔ (↑(gcd a b) : Int) ∣ n := sorry -- 4. theorem Exercise_7_1_6 (a b c : Nat) : gcd a b = gcd (a + b * c) b := sorry -- 5. theorem gcd_is_nonzero {a b : Nat} (h : a ≠ 0 ∨ b ≠ 0) : gcd a b ≠ 0 := sorry -- 6. theorem gcd_greatest {a b d : Nat} (h1 : gcd a b ≠ 0) (h2 : d ∣ a) (h3 : d ∣ b) : d ≤ gcd a b := sorry -- 7. lemma Lemma_7_1_10a {a b : Nat} (n : Nat) (h : a ∣ b) : (n * a) ∣ (n * b) := sorry lemma Lemma_7_1_10b {a b n : Nat} (h1 : n ≠ 0) (h2 : (n * a) ∣ (n * b)) : a ∣ b := sorry lemma Lemma_7_1_10c {a b : Nat} (h1 : a ∣ b) (h2 : b ∣ a) : a = b := sorry theorem Exercise_7_1_10 (a b n : Nat) : gcd (n * a) (n * b) = n * gcd a b := sorry /- Section 7.2 -/ -- 1. lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry -- 2. -- Hints: Start with apply List.rec. You may find mul_ne_zero useful theorem prod_nonzero_nonzero : ∀ (l : List Nat), (∀ a ∈ l, a ≠ 0) → prod l ≠ 0 := sorry -- 3. theorem rel_prime_iff_no_common_factor (a b : Nat) : rel_prime a b ↔ ¬∃ (p : Nat), prime p ∧ p ∣ a ∧ p ∣ b := sorry -- 4. theorem rel_prime_symm {a b : Nat} (h : rel_prime a b) : rel_prime b a := sorry -- 5. lemma in_prime_factorization_iff_prime_factor {a : Nat} {l : List Nat} (h1 : prime_factorization a l) (p : Nat) : p ∈ l ↔ prime_factor p a := sorry -- 6. theorem Exercise_7_2_5 {a b : Nat} {l m : List Nat} (h1 : prime_factorization a l) (h2 : prime_factorization b m) : rel_prime a b ↔ (¬∃ (p : Nat), p ∈ l ∧ p ∈ m) := sorry -- 7. theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry -- 8. theorem Exercise_7_2_7 {a b a' b' : Nat} (h1 : rel_prime a b) (h2 : a' ∣ a) (h3 : b' ∣ b) : rel_prime a' b' := sorry -- 9. theorem Exercise_7_2_9 {a b j k : Nat} (h1 : gcd a b ≠ 0) (h2 : a = j * gcd a b) (h3 : b = k * gcd a b) : rel_prime j k := sorry -- 10. theorem Exercise_7_2_17a (a b c : Nat) : gcd a (b * c) ∣ gcd a b * gcd a c := sorry /- Section 7.3 -/ -- 1. theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry -- 2. theorem Theorem_7_3_6_3 {m : Nat} (X : ZMod m) : X + [0]_m = X := sorry -- 3. theorem Theorem_7_3_6_4 {m : Nat} (X : ZMod m) : ∃ (Y : ZMod m), X + Y = [0]_m := sorry -- 4. theorem Exercise_7_3_4a {m : Nat} (Z1 Z2 : ZMod m) (h1 : ∀ (X : ZMod m), X + Z1 = X) (h2 : ∀ (X : ZMod m), X + Z2 = X) : Z1 = Z2 := sorry -- 5. theorem Exercise_7_3_4b {m : Nat} (X Y1 Y2 : ZMod m) (h1 : X + Y1 = [0]_m) (h2 : X + Y2 = [0]_m) : Y1 = Y2 := sorry -- 6. theorem Theorem_7_3_10 (m a : Nat) (b : Int) : ¬(↑(gcd m a) : Int) ∣ b → ¬∃ (x : Int), a * x ≡ b (MOD m) := sorry -- 7. theorem Theorem_7_3_11 (m n : Nat) (a b : Int) (h1 : n ≠ 0) : n * a ≡ n * b (MOD n * m) ↔ a ≡ b (MOD m) := sorry -- 8. theorem Exercise_7_3_16 {m : Nat} {a b : Int} (h : a ≡ b (MOD m)) : ∀ (n : Nat), a ^ n ≡ b ^ n (MOD m) := sorry -- 9. example {m : Nat} [NeZero m] (X : ZMod m) : ∃! (a : Int), 0 ≤ a ∧ a < m ∧ X = [a]_m := sorry -- 10. theorem congr_rel_prime {m a b : Nat} (h1 : a ≡ b (MOD m)) : rel_prime m a ↔ rel_prime m b := sorry -- 11. --Hint: You may find the theorem Int.ofNat_mod_ofNat useful. theorem rel_prime_mod (m a : Nat) : rel_prime m (a % m) ↔ rel_prime m a := sorry -- 12. lemma congr_iff_mod_eq_Int (m : Nat) (a b : Int) [NeZero m] : a ≡ b (MOD m) ↔ a % ↑m = b % ↑m := sorry --Hint for next theorem: Use the lemma above, --together with the theorems Int.ofNat_mod_ofNat and Nat.cast_inj. theorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] : ↑a ≡ ↑b (MOD m) ↔ a % m = b % m := sorry /- Section 7.4 -/ -- 1. --Hint: Use induction. --For the base case, compute [a]_m ^ 0 * [1]_m in two ways: --by Theorem_7_3_6_7, [a] ^ 0 * [1]_m = [a]_m ^ 0 --by ring, [a]_m ^ 0 * [1]_m = [1]_m. lemma Exercise_7_4_5_Int (m : Nat) (a : Int) : ∀ (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry -- 2. lemma left_inv_one_one_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g' (g i) = i) : one_one_below n g := sorry -- 3. lemma comp_perm_below {n : Nat} {f g : Nat → Nat} (h1 : perm_below n f) (h2 : perm_below n g) : perm_below n (f ∘ g) := sorry -- 4. lemma perm_below_fixed {n : Nat} {g : Nat → Nat} (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry -- 5. lemma Lemma_7_4_6 {a b c : Nat} : rel_prime (a * b) c ↔ rel_prime a c ∧ rel_prime b c := sorry -- 6. example {m a : Nat} [NeZero m] (h1 : rel_prime m a) : a ^ (phi m + 1) ≡ a (MOD m) := sorry -- 7. theorem Like_Exercise_7_4_11 {m a p : Nat} [NeZero m] (h1 : rel_prime m a) (h2 : p + 1 = phi m) : [a]_m * [a ^ p]_m = [1]_m := sorry -- 8. theorem Like_Exercise_7_4_12 {m a p q k : Nat} [NeZero m] (h1 : rel_prime m a) (h2 : p = q + (phi m) * k) : a ^ p ≡ a ^ q (MOD m) := sorry /- Section 7.5 -/ -- 1. --Hint: Use induction. lemma num_rp_prime {p : Nat} (h1 : prime p) : ∀ k < p, num_rp_below p (k + 1) = k := sorry -- 2. lemma three_prime : prime 3 := sorry -- 3. --Hint: Use the previous exercise, Exercise_7_2_7, and Theorem_7_4_2.

theorem Exercise_7_5_13a (a : Nat) (h1 : rel_prime 561 a) : a ^ 560 ≡ 1 (MOD 3)

HTPI.Exercises.Exercise_7_5_13a

htpi/HTPILib/Chap7.lean

htpi

/- Copyright 2023 Daniel J. Velleman -/ import HTPILib.Chap6 namespace HTPI /- Definitions -/ lemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by have h : n + 1 > 0 := Nat.succ_pos n show a % (n + 1) < n + 1 from Nat.mod_lt a h done def gcd (a b : Nat) : Nat := match b with | 0 => a | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd (n + 1) (a % (n + 1)) termination_by b mutual def gcd_c1 (a b : Nat) : Int := match b with | 0 => 1 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c2 (n + 1) (a % (n + 1)) --Corresponds to s = t' termination_by b def gcd_c2 (a b : Nat) : Int := match b with | 0 => 0 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c1 (n + 1) (a % (n + 1)) - (gcd_c2 (n + 1) (a % (n + 1))) * ↑(a / (n + 1)) --Corresponds to t = s' - t'q termination_by b end def prime (n : Nat) : Prop := 2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n def prime_factor (p n : Nat) : Prop := prime p ∧ p ∣ n def all_prime (l : List Nat) : Prop := ∀ p ∈ l, prime p def nondec (l : List Nat) : Prop := match l with | [] => True --Of course, True is a proposition that is always true | n :: L => (∀ m ∈ L, n ≤ m) ∧ nondec L def nondec_prime_list (l : List Nat) : Prop := all_prime l ∧ nondec l def prod (l : List Nat) : Nat := match l with | [] => 1 | n :: L => n * (prod L) def prime_factorization (n : Nat) (l : List Nat) : Prop := nondec_prime_list l ∧ prod l = n def rel_prime (a b : Nat) : Prop := gcd a b = 1 def congr_mod (m : Nat) (a b : Int) : Prop := (↑m : Int) ∣ (a - b) def cc (m : Nat) (a : Int) : ZMod m := (↑a : ZMod m) notation:50 a " ≡ " b " (MOD " m ")" => congr_mod m a b notation:max "["a"]_"m:max => cc m a def invertible {m : Nat} (X : ZMod m) : Prop := ∃ (Y : ZMod m), X * Y = [1]_m def num_rp_below (m k : Nat) : Nat := match k with | 0 => 0 | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j def phi (m : Nat) : Nat := num_rp_below m m def prod_seq {m : Nat} (j k : Nat) (f : Nat → ZMod m) : ZMod m := match j with | 0 => [1]_m | n + 1 => prod_seq n k f * f (k + n) def maps_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i < n, g i < n def one_one_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i1 < n, ∀ i2 < n, g i1 = g i2 → i1 = i2 def onto_below (n : Nat) (g : Nat → Nat) : Prop := ∀ k < n, ∃ i < n, g i = k def perm_below (n : Nat) (g : Nat → Nat) : Prop := maps_below n g ∧ one_one_below n g ∧ onto_below n g def inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m) def swap (u v i : Nat) : Nat := if i = u then v else if i = v then u else i namespace Euler --For definitions specific to Euler's theorem def F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m def G (m a i : Nat) : Nat := (a * i) % m def Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i end Euler /- Section 7.1 -/ theorem dvd_mod_of_dvd_a_b {a b d : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ (a % b) := by set q : Nat := a / b have h3 : b * q + a % b = a := Nat.div_add_mod a b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 define --Goal : ∃ (c : Nat), a % b = d * c apply Exists.intro (j - k * q) show a % b = d * (j - k * q) from calc a % b _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm _ = a - b * q := by rw [h3] _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc] _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm done theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry #eval gcd 672 161 --Answer: 7 lemma gcd_base (a : Nat) : gcd a 0 = a := by rfl lemma gcd_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd a b = gcd b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1)) rfl done lemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b ≠ 0) : a % b < b := by have h1 : b > 0 := Nat.pos_of_ne_zero h show a % b < b from Nat.mod_lt a h1 done lemma dvd_self (n : Nat) : n ∣ n := by apply Exists.intro 1 ring done theorem gcd_dvd : ∀ (b a : Nat), (gcd a b) ∣ a ∧ (gcd a b) ∣ b := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd a b_1) ∣ a ∧ (gcd a b_1) ∣ b_1 fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_base] --Goal: a ∣ a ∧ a ∣ 0 apply And.intro (dvd_self a) define apply Exists.intro 0 rfl done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_nonzero a h1] --Goal : gcd b (a % b) ∣ a ∧ gcd b (a % b) ∣ b have h2 : a % b < b := mod_nonzero_lt a h1 have h3 : (gcd b (a % b)) ∣ b ∧ (gcd b (a % b)) ∣ (a % b) := ih (a % b) h2 b apply And.intro _ h3.left show (gcd b (a % b)) ∣ a from dvd_a_of_dvd_b_mod h3.left h3.right done done theorem gcd_dvd_left (a b : Nat) : (gcd a b) ∣ a := (gcd_dvd b a).left theorem gcd_dvd_right (a b : Nat) : (gcd a b) ∣ b := (gcd_dvd b a).right lemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl lemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c1 a b = gcd_c2 b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done lemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl lemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * ↑(a / b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done theorem gcd_lin_comb : ∀ (b a : Nat), (gcd_c1 a b) * ↑a + (gcd_c2 a b) * ↑b = ↑(gcd a b) := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd_c1 a b_1) * ↑a + (gcd_c2 a b_1) * ↑b_1 = ↑(gcd a b_1) fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base] --Goal : 1 * ↑a + 0 * ↑0 = ↑a ring done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1] --Goal : gcd_c2 b (a % b) * ↑a + -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * ↑(a / b)) * ↑b = -- ↑(gcd b (a % b)) set r : Nat := a % b set q : Nat := a / b set s : Int := gcd_c1 b r set t : Int := gcd_c2 b r --Goal : t * ↑a + (s - t * ↑q) * ↑b = ↑(gcd b r) have h2 : r < b := mod_nonzero_lt a h1 have h3 : s * ↑b + t * ↑r = ↑(gcd b r) := ih r h2 b have h4 : b * q + r = a := Nat.div_add_mod a b rewrite [←h3, ←h4] rewrite [Nat.cast_add, Nat.cast_mul] --Goal : t * (↑b * ↑q + ↑r) + (s - t * ↑q) * ↑b = s * ↑b + t * ↑r ring done done #eval gcd_c1 672 161 --Answer: 6 #eval gcd_c2 672 161 --Answer: -25 --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161 theorem Theorem_7_1_6 {d a b : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ gcd a b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑d ∣ ↑(gcd a b) set s : Int := gcd_c1 a b set t : Int := gcd_c2 a b have h3 : s * ↑a + t * ↑b = ↑(gcd a b) := gcd_lin_comb b a rewrite [←h3] --Goal : ↑d ∣ s * ↑a + t * ↑b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul] --Goal : ↑d ∣ s * (↑d * ↑j) + t * (↑d * ↑k) define apply Exists.intro (s * ↑j + t * ↑k) ring done /- Section 7.2 -/ theorem dvd_trans {a b c : Nat} (h1 : a ∣ b) (h2 : b ∣ c) : a ∣ c := by define at h1; define at h2; define obtain (m : Nat) (h3 : b = a * m) from h1 obtain (n : Nat) (h4 : c = b * n) from h2 rewrite [h3, mul_assoc] at h4 apply Exists.intro (m * n) show c = a * (m * n) from h4 done lemma exists_prime_factor : ∀ (n : Nat), 2 ≤ n → ∃ (p : Nat), prime_factor p n := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, 2 ≤ n_1 → ∃ (p : Nat), prime_factor p n_1 assume h1 : 2 ≤ n by_cases h2 : prime n · -- Case 1. h2 : prime n apply Exists.intro n define --Goal : prime n ∧ n ∣ n show prime n ∧ n ∣ n from And.intro h2 (dvd_self n) done · -- Case 2. h2 : ¬prime n define at h2 --h2 : ¬(2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n) demorgan at h2 disj_syll h2 h1 obtain (a : Nat) (h3 : ∃ (b : Nat), a * b = n ∧ a < n ∧ b < n) from h2 obtain (b : Nat) (h4 : a * b = n ∧ a < n ∧ b < n) from h3 have h5 : 2 ≤ a := by by_contra h6 have h7 : a ≤ 1 := by linarith have h8 : n ≤ b := calc n _ = a * b := h4.left.symm _ ≤ 1 * b := by rel [h7] _ = b := by ring linarith --n ≤ b contradicts b < n done have h6 : ∃ (p : Nat), prime_factor p a := ih a h4.right.left h5 obtain (p : Nat) (h7 : prime_factor p a) from h6 apply Exists.intro p define --Goal : prime p ∧ p ∣ n define at h7 --h7 : prime p ∧ p ∣ a apply And.intro h7.left have h8 : a ∣ n := by apply Exists.intro b show n = a * b from (h4.left).symm done show p ∣ n from dvd_trans h7.right h8 done done lemma exists_least_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q := by set S : Set Nat := {p : Nat | prime_factor p n} have h2 : ∃ (p : Nat), p ∈ S := exists_prime_factor n h show ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q from well_ord_princ S h2 done lemma all_prime_nil : all_prime [] := by define --Goal : ∀ p ∈ [], prime p fix p : Nat contrapos --Goal : ¬prime p → p ∉ [] assume h1 : ¬prime p show p ∉ [] from List.not_mem_nil p done lemma all_prime_cons (n : Nat) (L : List Nat) : all_prime (n :: L) ↔ prime n ∧ all_prime L := by apply Iff.intro · -- (→) assume h1 : all_prime (n :: L) --Goal : prime n ∧ all_prime L define at h1 --h1 : ∀ p ∈ n :: L, prime p apply And.intro (h1 n (List.mem_cons_self n L)) define --Goal : ∀ p ∈ L, prime p fix p : Nat assume h2 : p ∈ L show prime p from h1 p (List.mem_cons_of_mem n h2) done · -- (←) assume h1 : prime n ∧ all_prime L --Goal : all_prime (n :: l) define : all_prime L at h1 define fix p : Nat assume h2 : p ∈ n :: L rewrite [List.mem_cons] at h2 --h2 : p = n ∨ p ∈ L by_cases on h2 · -- Case 1. h2 : p = n rewrite [h2] show prime n from h1.left done · -- Case 2. h2 : p ∈ L show prime p from h1.right p h2 done done done lemma nondec_nil : nondec [] := by define --Goal : True trivial --trivial proves some obviously true statements, such as True done lemma nondec_cons (n : Nat) (L : List Nat) : nondec (n :: L) ↔ (∀ m ∈ L, n ≤ m) ∧ nondec L := by rfl lemma prod_nil : prod [] = 1 := by rfl lemma prod_cons : prod (n :: L) = n * (prod L) := by rfl lemma exists_cons_of_length_eq_succ {A : Type} {l : List A} {n : Nat} (h : l.length = n + 1) : ∃ (a : A) (L : List A), l = a :: L ∧ L.length = n := by have h1 : ¬l.length = 0 := by linarith rewrite [List.length_eq_zero] at h1 obtain (a : A) (h2 : ∃ (L : List A), l = a :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List A) (h3 : l = a :: L) from h2 apply Exists.intro a apply Exists.intro L apply And.intro h3 have h4 : (a :: L).length = L.length + 1 := List.length_cons a L rewrite [←h3, h] at h4 show L.length = n from (Nat.add_right_cancel h4).symm done lemma list_elt_dvd_prod_by_length (a : Nat) : ∀ (n : Nat), ∀ (l : List Nat), l.length = n → a ∈ l → a ∣ prod l := by by_induc · --Base Case fix l : List Nat assume h1 : l.length = 0 rewrite [List.length_eq_zero] at h1 --h1 : l = [] rewrite [h1] --Goal : a ∈ [] → a ∣ prod [] contrapos assume h2 : ¬a ∣ prod [] show a ∉ [] from List.not_mem_nil a done · -- Induction Step fix n : Nat assume ih : ∀ (l : List Nat), List.length l = n → a ∈ l → a ∣ prod l fix l : List Nat assume h1 : l.length = n + 1 --Goal : a ∈ l → a ∣ prod l obtain (b : Nat) (h2 : ∃ (L : List Nat), l = b :: L ∧ L.length = n) from exists_cons_of_length_eq_succ h1 obtain (L : List Nat) (h3 : l = b :: L ∧ L.length = n) from h2 have h4 : a ∈ L → a ∣ prod L := ih L h3.right assume h5 : a ∈ l rewrite [h3.left, prod_cons] --Goal : a ∣ b * prod L rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b ∨ a ∈ L by_cases on h5 · -- Case 1. h5 : a = b apply Exists.intro (prod L) rewrite [h5] rfl done · -- Case 2. h5 : a ∈ L have h6 : a ∣ prod L := h4 h5 have h7 : prod L ∣ b * prod L := by apply Exists.intro b ring done show a ∣ b * prod L from dvd_trans h6 h7 done done done lemma list_elt_dvd_prod {a : Nat} {l : List Nat} (h : a ∈ l) : a ∣ prod l := by set n : Nat := l.length have h1 : l.length = n := by rfl show a ∣ prod l from list_elt_dvd_prod_by_length a n l h1 h done lemma exists_prime_factorization : ∀ (n : Nat), n ≥ 1 → ∃ (l : List Nat), prime_factorization n l := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, n_1 ≥ 1 → ∃ (l : List Nat), prime_factorization n_1 l assume h1 : n ≥ 1 by_cases h2 : n = 1 · -- Case 1. h2 : n = 1 apply Exists.intro [] define apply And.intro · -- Proof of nondec_prime_list [] define show all_prime [] ∧ nondec [] from And.intro all_prime_nil nondec_nil done · -- Proof of prod [] = n rewrite [prod_nil, h2] rfl done done · -- Case 2. h2 : n ≠ 1 have h3 : n ≥ 2 := lt_of_le_of_ne' h1 h2 obtain (p : Nat) (h4 : prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q) from exists_least_prime_factor h3 have p_prime_factor : prime_factor p n := h4.left define at p_prime_factor have p_prime : prime p := p_prime_factor.left have p_dvd_n : p ∣ n := p_prime_factor.right have p_least : ∀ (q : Nat), prime_factor q n → p ≤ q := h4.right obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n have h5 : m ≠ 0 := by contradict h1 with h6 have h7 : n = 0 := calc n _ = p * m := n_eq_pm _ = p * 0 := by rw [h6] _ = 0 := by ring rewrite [h7] decide done have m_pos : 0 < m := Nat.pos_of_ne_zero h5 have m_lt_n : m < n := by define at p_prime show m < n from calc m _ < m + m := by linarith _ = 2 * m := by ring _ ≤ p * m := by rel [p_prime.left] _ = n := n_eq_pm.symm done obtain (L : List Nat) (h6 : prime_factorization m L) from ih m m_lt_n m_pos define at h6 have ndpl_L : nondec_prime_list L := h6.left define at ndpl_L apply Exists.intro (p :: L) define apply And.intro · -- Proof of nondec_prime_list (p :: L) define apply And.intro · -- Proof of all_prime (p :: L) rewrite [all_prime_cons] show prime p ∧ all_prime L from And.intro p_prime ndpl_L.left done · -- Proof of nondec (p :: L) rewrite [nondec_cons] apply And.intro _ ndpl_L.right fix q : Nat assume q_in_L : q ∈ L have h7 : q ∣ prod L := list_elt_dvd_prod q_in_L rewrite [h6.right] at h7 --h7 : q ∣ m have h8 : m ∣ n := by apply Exists.intro p rewrite [n_eq_pm] ring done have q_dvd_n : q ∣ n := dvd_trans h7 h8 have ap_L : all_prime L := ndpl_L.left define at ap_L have q_prime_factor : prime_factor q n := And.intro (ap_L q q_in_L) q_dvd_n show p ≤ q from p_least q q_prime_factor done done · -- Proof of prod (p :: L) = n rewrite [prod_cons, h6.right, n_eq_pm] rfl done done done theorem Theorem_7_2_2 {a b c : Nat} (h1 : c ∣ a * b) (h2 : rel_prime a c) : c ∣ b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑c ∣ ↑b define at h1; define at h2; define obtain (j : Nat) (h3 : a * b = c * j) from h1 set s : Int := gcd_c1 a c set t : Int := gcd_c2 a c have h4 : s * ↑a + t * ↑c = ↑(gcd a c) := gcd_lin_comb c a rewrite [h2, Nat.cast_one] at h4 --h4 : s * ↑a + t * ↑c = (1 : Int) apply Exists.intro (s * ↑j + t * ↑b) show ↑b = ↑c * (s * ↑j + t * ↑b) from calc ↑b _ = (1 : Int) * ↑b := (one_mul _).symm _ = (s * ↑a + t * ↑c) * ↑b := by rw [h4] _ = s * (↑a * ↑b) + t * ↑c * ↑b := by ring _ = s * (↑c * ↑j) + t * ↑c * ↑b := by rw [←Nat.cast_mul a b, h3, Nat.cast_mul c j] _ = ↑c * (s * ↑j + t * ↑b) := by ring done lemma le_nonzero_prod_left {a b : Nat} (h : a * b ≠ 0) : a ≤ a * b := by have h1 : b ≠ 0 := by contradict h with h1 rewrite [h1] ring done have h2 : 1 ≤ b := Nat.pos_of_ne_zero h1 show a ≤ a * b from calc a = a * 1 := (mul_one a).symm _ ≤ a * b := by rel [h2] done lemma le_nonzero_prod_right {a b : Nat} (h : a * b ≠ 0) : b ≤ a * b := by rewrite [mul_comm] rewrite [mul_comm] at h show b ≤ b * a from le_nonzero_prod_left h done lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry lemma rel_prime_of_prime_not_dvd {a p : Nat} (h1 : prime p) (h2 : ¬p ∣ a) : rel_prime a p := by have h3 : gcd a p ∣ a := gcd_dvd_left a p have h4 : gcd a p ∣ p := gcd_dvd_right a p have h5 : gcd a p = 1 ∨ gcd a p = p := dvd_prime h1 h4 have h6 : gcd a p ≠ p := by contradict h2 with h6 rewrite [h6] at h3 show p ∣ a from h3 done disj_syll h5 h6 show rel_prime a p from h5 done theorem Theorem_7_2_3 {a b p : Nat} (h1 : prime p) (h2 : p ∣ a * b) : p ∣ a ∨ p ∣ b := by or_right with h3 have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3 show p ∣ b from Theorem_7_2_2 h2 h4 done lemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a ≥ 1 := by have h1 : a ≠ 0 := by by_contra h1 rewrite [h1] at h contradict h linarith done show a ≥ 1 from Nat.pos_of_ne_zero h1 done lemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by have h1 : a ≥ 1 := ge_one_of_prod_one h have h2 : a * b ≠ 0 := by linarith have h3 : a ≤ a * b := le_nonzero_prod_left h2 rewrite [h] at h3 show a = 1 from Nat.le_antisymm h3 h1 done lemma eq_one_of_dvd_one {n : Nat} (h : n ∣ 1) : n = 1 := by obtain (j : Nat) (h1 : 1 = n * j) from h show n = 1 from eq_one_of_prod_one h1.symm done lemma prime_not_one {p : Nat} (h : prime p) : p ≠ 1 := by define at h linarith done theorem Theorem_7_2_4 {p : Nat} (h1 : prime p) : ∀ (l : List Nat), p ∣ prod l → ∃ a ∈ l, p ∣ a := by apply List.rec · -- Base Case. Goal : p ∣ prod [] → ∃ a ∈ [], p ∣ a rewrite [prod_nil] assume h2 : p ∣ 1 show ∃ a ∈ [], p ∣ a from absurd (eq_one_of_dvd_one h2) (prime_not_one h1) done · -- Induction Step fix b : Nat fix L : List Nat assume ih : p ∣ prod L → ∃ a ∈ L, p ∣ a --Goal : p ∣ prod (b :: L) → ∃ a ∈ b :: L, p ∣ a assume h2 : p ∣ prod (b :: L) rewrite [prod_cons] at h2 have h3 : p ∣ b ∨ p ∣ prod L := Theorem_7_2_3 h1 h2 by_cases on h3 · -- Case 1. h3 : p ∣ b apply Exists.intro b show b ∈ b :: L ∧ p ∣ b from And.intro (List.mem_cons_self b L) h3 done · -- Case 2. h3 : p ∣ prod L obtain (a : Nat) (h4 : a ∈ L ∧ p ∣ a) from ih h3 apply Exists.intro a show a ∈ b :: L ∧ p ∣ a from And.intro (List.mem_cons_of_mem b h4.left) h4.right done done done lemma prime_in_list {p : Nat} {l : List Nat} (h1 : prime p) (h2 : all_prime l) (h3 : p ∣ prod l) : p ∈ l := by obtain (a : Nat) (h4 : a ∈ l ∧ p ∣ a) from Theorem_7_2_4 h1 l h3 define at h2 have h5 : prime a := h2 a h4.left have h6 : p = 1 ∨ p = a := dvd_prime h5 h4.right disj_syll h6 (prime_not_one h1) rewrite [h6] show a ∈ l from h4.left done lemma first_le_first {p q : Nat} {l m : List Nat} (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m)) (h3 : prod (p :: l) = prod (q :: m)) : p ≤ q := by define at h1; define at h2 have h4 : q ∣ prod (p :: l) := by define apply Exists.intro (prod m) rewrite [←prod_cons] show prod (p :: l) = prod (q :: m) from h3 done have h5 : all_prime (q :: m) := h2.left rewrite [all_prime_cons] at h5 have h6 : q ∈ p :: l := prime_in_list h5.left h1.left h4 have h7 : nondec (p :: l) := h1.right rewrite [nondec_cons] at h7 rewrite [List.mem_cons] at h6 by_cases on h6 · -- Case 1. h6 : q = p linarith done · -- Case 2. h6 : q ∈ l have h8 : ∀ m ∈ l, p ≤ m := h7.left show p ≤ q from h8 q h6 done done lemma nondec_prime_list_tail {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by define at h define rewrite [all_prime_cons, nondec_cons] at h show all_prime l ∧ nondec l from And.intro h.left.right h.right.right done lemma cons_prod_not_one {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : prod (p :: l) ≠ 1 := by define at h have h1 : all_prime (p :: l) := h.left rewrite [all_prime_cons] at h1 rewrite [prod_cons] by_contra h2 show False from (prime_not_one h1.left) (eq_one_of_prod_one h2) done lemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) : l = [] ↔ prod l = 1 := by apply Iff.intro · -- (→) assume h1 : l = [] rewrite [h1] show prod [] = 1 from prod_nil done · -- (←) contrapos assume h1 : ¬l = [] obtain (p : Nat) (h2 : ∃ (L : List Nat), l = p :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List Nat) (h3 : l = p :: L) from h2 rewrite [h3] at h rewrite [h3] show ¬prod (p :: L) = 1 from cons_prod_not_one h done done lemma prime_pos {p : Nat} (h : prime p) : p > 0 := by define at h linarith done theorem Theorem_7_2_5 : ∀ (l1 l2 : List Nat), nondec_prime_list l1 → nondec_prime_list l2 → prod l1 = prod l2 → l1 = l2 := by apply List.rec · -- Base Case. Goal : ∀ (l2 : List Nat), nondec_prime_list [] → -- nondec_prime_list l2 → prod [] = prod l2 → [] = l2 fix l2 : List Nat assume h1 : nondec_prime_list [] assume h2 : nondec_prime_list l2 assume h3 : prod [] = prod l2 rewrite [prod_nil, eq_comm, ←list_nil_iff_prod_one h2] at h3 show [] = l2 from h3.symm done · -- Induction Step fix p : Nat fix L1 : List Nat assume ih : ∀ (L2 : List Nat), nondec_prime_list L1 → nondec_prime_list L2 → prod L1 = prod L2 → L1 = L2 -- Goal : ∀ (l2 : List Nat), nondec_prime_list (p :: L1) → -- nondec_prime_list l2 → prod (p :: L1) = prod l2 → p :: L1 = l2 fix l2 : List Nat assume h1 : nondec_prime_list (p :: L1) assume h2 : nondec_prime_list l2 assume h3 : prod (p :: L1) = prod l2 have h4 : ¬prod (p :: L1) = 1 := cons_prod_not_one h1 rewrite [h3, ←list_nil_iff_prod_one h2] at h4 obtain (q : Nat) (h5 : ∃ (L : List Nat), l2 = q :: L) from List.exists_cons_of_ne_nil h4 obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5 rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2) rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2) have h7 : p ≤ q := first_le_first h1 h2 h3 have h8 : q ≤ p := first_le_first h2 h1 h3.symm have h9 : p = q := by linarith rewrite [h9, prod_cons, prod_cons] at h3 --h3 : q * prod L1 = q * prod L2 have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1 have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2 define at h2 have h12 : all_prime (q :: L2) := h2.left rewrite [all_prime_cons] at h12 have h13 : q > 0 := prime_pos h12.left have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3 have h15 : L1 = L2 := ih L2 h10 h11 h14 rewrite [h6, h9, h15] rfl done done theorem fund_thm_arith (n : Nat) (h : n ≥ 1) : ∃! (l : List Nat), prime_factorization n l := by exists_unique · -- Existence show ∃ (l : List Nat), prime_factorization n l from exists_prime_factorization n h done · -- Uniqueness fix l1 : List Nat; fix l2 : List Nat assume h1 : prime_factorization n l1 assume h2 : prime_factorization n l2 define at h1; define at h2 have h3 : prod l1 = n := h1.right rewrite [←h2.right] at h3 show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3 done done /- Section 7.3 -/ theorem congr_refl (m : Nat) : ∀ (a : Int), a ≡ a (MOD m) := by fix a : Int define --Goal : ∃ (c : Int), a - a = ↑m * c apply Exists.intro 0 ring done theorem congr_symm {m : Nat} : ∀ {a b : Int}, a ≡ b (MOD m) → b ≡ a (MOD m) := by fix a : Int; fix b : Int assume h1 : a ≡ b (MOD m) define at h1 --h1 : ∃ (c : Int), a - b = ↑m * c define --Goal : ∃ (c : Int), b - a = ↑m * c obtain (c : Int) (h2 : a - b = m * c) from h1 apply Exists.intro (-c) show b - a = m * (-c) from calc b - a _ = -(a - b) := by ring _ = -(m * c) := by rw [h2] _ = m * (-c) := by ring done theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry /- Fundamental properties of congruence classes -/ lemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) : X = Y ↔ X.val = Y.val := Fin.ext_iff lemma val_nat_eq_mod (n k : Nat) : ([k]_(n + 1)).val = k % (n + 1) := by rfl lemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl theorem cc_rep {m : Nat} (X : ZMod m) : ∃ (a : Int), X = [a]_m := match m with | 0 => by apply Exists.intro X rfl done | n + 1 => by apply Exists.intro ↑(X.val) have h1 : X.val < n + 1 := Fin.prop X rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1] rfl done theorem add_class (m : Nat) (a b : Int) : [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm theorem mul_class (m : Nat) (a b : Int) : [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm lemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ [a - b]_m = [0]_m := by apply Iff.intro · -- (→) assume h1 : [a]_m = [b]_m have h2 : a - b = a + (-b) := by ring have h3 : b + (-b) = 0 := by ring show [a - b]_m = [0]_m from calc [a - b]_m _ = [a + (-b)]_m := by rw [h2] _ = [a]_m + [-b]_m := by rw [add_class] _ = [b]_m + [-b]_m := by rw [h1] _ = [b + -b]_m := by rw [add_class] _ = [0]_m := by rw [h3] done · -- (←) assume h1 : [a - b]_m = [0]_m have h2 : b + (a - b) = a := by ring have h3 : b + 0 = b := by ring show [a]_m = [b]_m from calc [a]_m _ = [b + (a - b)]_m := by rw [h2] _ = [b]_m + [a - b]_m := by rw [add_class] _ = [b]_m + [0]_m := by rw [h1] _ = [b + 0]_m := by rw [add_class] _ = [b]_m := by rw [h3] done done lemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) : [a]_m = [0]_m → [-a]_m = [0]_m := by assume h1 : [a]_m = [0]_m have h2 : 0 + (-a) = -a := by ring have h3 : a + (-a) = 0 := by ring show [-a]_m = [0]_m from calc [-a]_m _ = [0 + (-a)]_m := by rw [h2] _ = [0]_m + [-a]_m := by rw [add_class] _ = [a]_m + [-a]_m := by rw [h1] _ = [a + (-a)]_m := by rw [add_class] _ = [0]_m := by rw [h3] done lemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) : [-a]_m = [0]_m ↔ [a]_m = [0]_m := by apply Iff.intro _ (cc_neg_zero_of_cc_zero m a) assume h1 : [-a]_m = [0]_m have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1 have h3 : -(-a) = a := by ring rewrite [h3] at h2 show [a]_m = [0]_m from h2 done lemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl lemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m ↔ m ∣ k := match m with | 0 => by have h : (0 : Int) = (↑(0 : Nat) : Int) := by rfl rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj] apply Iff.intro · -- (→) assume h1 : k = 0 rewrite [h1] show 0 ∣ 0 from dvd_self 0 done · -- (←) assume h1 : 0 ∣ k obtain (c : Nat) (h2 : k = 0 * c) from h1 rewrite [h2] ring done done | n + 1 => by rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero] show k % (n + 1) = 0 ↔ n + 1 ∣ k from (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm done lemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m ↔ ↑m ∣ a := by obtain (k : Nat) (h1 : a = ↑k ∨ a = -↑k) from Int.eq_nat_or_neg a by_cases on h1 · -- Case 1. h1: a = ↑k rewrite [h1, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done · -- Case 2. h1: a = -↑k rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done done theorem cc_eq_iff_congr (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ a ≡ b (MOD m) := calc [a]_m = [b]_m _ ↔ [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b _ ↔ ↑m ∣ (a - b) := cc_zero_iff_dvd m (a - b) _ ↔ a ≡ b (MOD m) := by rfl /- End of fundamental properties of congruence classes -/ lemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m := by have h1 : (↑m : Int) ≠ 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m) show 0 ≤ a % m from Int.emod_nonneg a h1 done lemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m) have h2 : (↑m : Int) > 0 := (Nat.cast_pos).rtl h1 show a % m < m from Int.emod_lt_of_pos a h2 done lemma congr_mod_mod (m : Nat) (a : Int) : a ≡ a % m (MOD m) := by define have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m apply Exists.intro (a / m) show a - a % m = m * (a / m) from calc a - (a % m) _ = m * (a / m) + a % m - a % m := by rw [h1] _ = m * (a / m) := by ring done lemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) := And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a)) theorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) : ∃! (r : Int), 0 ≤ r ∧ r < m ∧ a ≡ r (MOD m) := by exists_unique · -- Existence apply Exists.intro (a % m) show 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) from mod_cmpl_res m a done · -- Uniqueness fix r1 : Int; fix r2 : Int assume h1 : 0 ≤ r1 ∧ r1 < m ∧ a ≡ r1 (MOD m) assume h2 : 0 ≤ r2 ∧ r2 < m ∧ a ≡ r2 (MOD m) have h3 : r1 ≡ r2 (MOD m) := congr_trans (congr_symm h1.right.right) h2.right.right obtain (d : Int) (h4 : r1 - r2 = m * d) from h3 have h5 : r1 - r2 < m * 1 := by linarith have h6 : m * (-1) < r1 - r2 := by linarith rewrite [h4] at h5 --h5 : m * d < m * 1 rewrite [h4] at h6 --h6 : m * -1 < m * d have h7 : (↑m : Int) ≥ 0 := Nat.cast_nonneg m have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7 have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7 have h10 : d = 0 := by linarith show r1 = r2 from calc r1 _ = r1 - r2 + r2 := by ring _ = m * 0 + r2 := by rw [h4, h10] _ = r2 := by ring done done lemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m := (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a) theorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y rewrite [h1, h2] have h3 : a + b = b + a := by ring show [a]_m + [b]_m = [b]_m + [a]_m from calc [a]_m + [b]_m _ = [a + b]_m := add_class m a b _ = [b + a]_m := by rw [h3] _ = [b]_m + [a]_m := (add_class m b a).symm done theorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X rewrite [h1] have h2 : a * 1 = a := by ring show [a]_m * [1]_m = [a]_m from calc [a]_m * [1]_m _ = [a * 1]_m := mul_class m a 1 _ = [a]_m := by rw [h2] done theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry lemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) : [a]_m * [gcd_c2 m a]_m = [1]_m := by set s : Int := gcd_c1 m a have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m define at h1 rewrite [h1, Nat.cast_one] at h2 --h2 : s * ↑m + gcd_c2 m a * ↑a = 1 rewrite [mul_class, cc_eq_iff_congr] define --Goal : ∃ (c : Int), ↑a * gcd_c2 m a - 1 = ↑m * c apply Exists.intro (-s) show a * (gcd_c2 m a) - 1 = m * (-s) from calc a * (gcd_c2 m a) - 1 _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring _ = 1 + m * (-s) - 1 := by rw [h2] _ = m * (-s) := by ring done theorem Theorem_7_3_7 (m a : Nat) : invertible [a]_m ↔ rel_prime m a := by apply Iff.intro · -- (→) assume h1 : invertible [a]_m define at h1 obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1 obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y rewrite [h3, mul_class, cc_eq_iff_congr] at h2 define at h2 obtain (c : Int) (h4 : a * b - 1 = m * c) from h2 rewrite [Exercise_7_2_6] --Goal : ∃ (s t : Int), s * ↑m + t * ↑a = 1 apply Exists.intro (-c) apply Exists.intro b show (-c) * m + b * a = 1 from calc (-c) * m + b * a _ = (-c) * m + (a * b - 1) + 1 := by ring _ = (-c) * m + m * c + 1 := by rw [h4] _ = 1 := by ring done · -- (←) assume h1 : rel_prime m a define show ∃ (Y : ZMod m), [a]_m * Y = [1]_m from Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1) done done /- Section 7.4 -/ section Euler open Euler lemma num_rp_below_base {m : Nat} : num_rp_below m 0 = 0 := by rfl lemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) : num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by have h1 : num_rp_below m (j + 1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : gcd m j = 1 rewrite [if_pos h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1 show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1 done lemma num_rp_below_step_not_rp {m j : Nat} (h : ¬rel_prime m j) : num_rp_below m (j + 1) = num_rp_below m j := by have h1 : num_rp_below m (j +1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : ¬gcd m j = 1 rewrite [if_neg h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j show num_rp_below m (j + 1) = num_rp_below m j from h1 done lemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl #eval phi 10 --Answer: 4 lemma prod_inv_iff_inv {m : Nat} {X : ZMod m} (h1 : invertible X) (Y : ZMod m) : invertible (X * Y) ↔ invertible Y := by apply Iff.intro · -- (→) assume h2 : invertible (X * Y) obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2 apply Exists.intro (X * Z) rewrite [←h3] --Goal : Y * (X * Z) = X * Y * Z ring --Note that ring can do algebra in ZMod m done · -- (←) assume h2 : invertible Y obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1 obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2 apply Exists.intro (Xi * Yi) show (X * Y) * (Xi * Yi) = [1]_m from calc X * Y * (Xi * Yi) _ = (X * Xi) * (Y * Yi) := by ring _ = [1]_m * [1]_m := by rw [h3, h4] _ = [1]_m := Theorem_7_3_6_7 [1]_m done done lemma F_rp_def {m i : Nat} (h : rel_prime m i) : F m i = [i]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h --h : gcd m i = 1 rewrite [if_pos h] at h1 show F m i = [i]_m from h1 done lemma F_not_rp_def {m i : Nat} (h : ¬rel_prime m i) : F m i = [1]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h rewrite [h1, if_neg h] rfl done lemma prod_seq_base {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 0 k f = [1]_m := by rfl lemma prod_seq_step {m : Nat} (n k : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl lemma prod_seq_zero_step {m : Nat} (n : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by rewrite [prod_seq_step, zero_add] rfl done lemma prod_one {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 1 k f = f k := by rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7] rfl done lemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl lemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m := calc [G m a i]_m _ = [(a * i) % m]_m := by rfl _ = [a * i]_m := (cc_eq_mod m (a * i)).symm _ = [a]_m * [i]_m := (mul_class m a i).symm lemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) : rel_prime m (G m a i) ↔ rel_prime m i := by have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1 show rel_prime m (G m a i) ↔ rel_prime m i from calc rel_prime m (G m a i) _ ↔ invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm _ ↔ invertible ([a]_m * [i]_m) := by rw [cc_G] _ ↔ invertible [i]_m := prod_inv_iff_inv h2 ([i]_m) _ ↔ rel_prime m i := Theorem_7_3_7 m i done lemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) : F m (G m a i) = [a]_m * F m i := by have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2 show F m (G m a i) = [a]_m * F m i from calc F m (G m a i) _ = [G m a i]_m := F_rp_def h3 _ = [a]_m * [i]_m := cc_G m a i _ = [a]_m * F m i := by rw [F_rp_def h2] done lemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : ¬rel_prime m i) : F m (G m a i) = [1]_m := by rewrite [←G_rp_iff h1 i] at h2 show F m (G m a i) = [1]_m from F_not_rp_def h2 done lemma FG_prod {m a : Nat} (h1 : rel_prime m a) : ∀ (k : Nat), prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by by_induc · -- Base Case show prod_seq 0 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from calc prod_seq 0 0 ((F m) ∘ (G m a)) _ = [1]_m := prod_seq_base _ _ _ = [a]_m ^ 0 * [1]_m := by ring _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by rw [num_rp_below_base, prod_seq_base] done · -- Induction Step fix k : Nat assume ih : prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) by_cases h2 : rel_prime m k · -- Case 1. h2 : rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([a]_m * F m k) := by rw [FG_rp h1 h2] _ = [a]_m ^ ((num_rp_below m k) + 1) * ((prod_seq k 0 (F m)) * F m k) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_rp h2, prod_seq_zero_step] done · -- Case 2. h2 : ¬rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([1]_m) := by rw [FG_not_rp h1 h2] _ = [a]_m ^ (num_rp_below m k) * (prod_seq k 0 (F m) * ([1]_m)) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_not_rp h2, prod_seq_zero_step, F_not_rp_def h2] done done done lemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by define --Goal : ∀ i < m, G m a i < m fix i : Nat assume h1 : i < m rewrite [G_def] --Goal : a * i % m < m show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m) done lemma left_inv_one_one_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g' (g i) = i) : one_one_below n g := sorry lemma right_inv_onto_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g (g' i) = i) (h2 : maps_below n g') : onto_below n g := by define at h2; define fix k : Nat assume h3 : k < n apply Exists.intro (g' k) show g' k < n ∧ g (g' k) = k from And.intro (h2 k h3) (h1 k h3) done lemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m * [inv_mod m a]_m = [1]_m := by have h2 : 0 ≤ (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a) show [a]_m * [inv_mod m a]_m = [1]_m from calc [a]_m * [inv_mod m a]_m _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2] _ = [a]_m * [gcd_c2 m a]_m := by rw [←cc_eq_mod] _ = [1]_m := gcd_c2_inv h1 done lemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m := calc a * (b % m) % m = a % m * (b % m % m) % m := Nat.mul_mod _ _ _ _ = a % m * (b % m) % m := by rw [Nat.mod_mod] _ = a * b % m := (Nat.mul_mod _ _ _).symm lemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm] rfl done theorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] : ↑a ≡ ↑b (MOD m) ↔ a % m = b % m := sorry lemma mul_inv_mod_cancel {m a i : Nat} [NeZero m] (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1 rewrite [mul_class, cc_eq_iff_congr, ←Nat.cast_mul, ←Nat.cast_one, congr_iff_mod_eq_Nat] at h3 show a * inv_mod m a * i % m = i from calc a * (inv_mod m a) * i % m _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = 1 % m * i % m := by rw [h3] _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = i % m := by rw [one_mul] _ = i := Nat.mod_eq_of_lt h2 done lemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl lemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, G m a (Ginv m a i) = i := by fix i : Nat assume h2 : i < m show G m a (Ginv m a i) = i from calc G m a (Ginv m a i) _ = a * ((inv_mod m a * i) % m) % m := by rfl _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, Ginv m a (G m a i) = i := by fix i : Nat assume h2 : i < m show Ginv m a (G m a i) = i from calc Ginv m a (G m a i) _ = inv_mod m a * ((a * i) % m) % m := by rfl _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc, mul_comm (inv_mod m a)] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_maps_below (m a : Nat) [NeZero m] : maps_below m (Ginv m a) := G_maps_below m (inv_mod m a) lemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : one_one_below m (G m a) := left_inv_one_one_below (Ginv_left_inv h1) lemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : onto_below m (G m a) := right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a) lemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : perm_below m (G m a) := And.intro (G_maps_below m a) (And.intro (G_one_one_below h1) (G_onto_below h1)) --Permuting a product of congruence classes doesn't change product lemma swap_fst (u v : Nat) : swap u v u = v := by define : swap u v u --Goal : (if u = u then v else if u = v then u else u) = v have h : u = u := by rfl rewrite [if_pos h] rfl done lemma swap_snd (u v : Nat) : swap u v v = u := by define : swap u v v by_cases h1 : v = u · -- Case 1. h1 : v = u rewrite [if_pos h1] show v = u from h1 done · -- Case 2. h1 : v ≠ u rewrite [if_neg h1] have h2 : v = v := by rfl rewrite [if_pos h2] rfl done done lemma swap_other {u v i : Nat} (h1 : i ≠ u) (h2 : i ≠ v) : swap u v i = i := by define : swap u v i rewrite [if_neg h1, if_neg h2] rfl done lemma swap_values (u v i : Nat) : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := by by_cases h1 : i = u · -- Case 1. h1 : i = u apply Or.inl rewrite [h1] show swap u v u = v from swap_fst u v done · -- Case 2. h1 : i ≠ u apply Or.inr by_cases h2 : i = v · -- Case 2.1. h2 : i = v apply Or.inl rewrite [h2] show swap u v v = u from swap_snd u v done · -- Case 2.2. h2 : i ≠ v apply Or.inr show swap u v i = i from swap_other h1 h2 done done done lemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by define fix i : Nat assume h3 : i < n have h4 : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := swap_values u v i by_cases on h4 · -- Case 1. h4 : swap u v i = v rewrite [h4] show v < n from h2 done · -- Case 2. by_cases on h4 · -- Case 2.1. h4 : swap u v i = u rewrite [h4] show u < n from h1 done · -- Case 2.2. h4 : swap u v i = i rewrite [h4] show i < n from h3 done done done lemma swap_swap (u v n : Nat) : ∀ i < n, swap u v (swap u v i) = i := by fix i : Nat assume h : i < n by_cases h1 : i = u · -- Case 1. h1 : i = u rewrite [h1, swap_fst, swap_snd] rfl done · -- Case 2. h1 : i ≠ u by_cases h2 : i = v · -- Case 2.1. h2 : i = v rewrite [h2, swap_snd, swap_fst] rfl done · -- Case 2.2. h2 : i ≠ v rewrite [swap_other h1 h2, swap_other h1 h2] rfl done done done lemma swap_one_one_below (u v n) : one_one_below n (swap u v) := left_inv_one_one_below (swap_swap u v n) lemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) := right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2) lemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) := And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2)) lemma comp_perm_below {n : Nat} {f g : Nat → Nat} (h1 : perm_below n f) (h2 : perm_below n g) : perm_below n (f ∘ g) := sorry lemma trivial_swap (u : Nat) : swap u u = id := by apply funext fix x : Nat by_cases h1 : x = u · -- Case 1. h1 : x = u rewrite [h1, swap_fst] rfl done · -- Case 2. h1 : x ≠ u rewrite [swap_other h1 h1] rfl done done lemma prod_eq_fun {m : Nat} (f g : Nat → ZMod m) (k : Nat) : ∀ (n : Nat), (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g := by by_induc · -- Base Case assume h : (∀ i < 0, f (k + i) = g (k + i)) rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g assume h1 : ∀ i < n + 1, f (k + i) = g (k + i) have h2 : ∀ i < n, f (k + i) = g (k + i) := by fix i : Nat assume h2 : i < n have h3 : i < n + 1 := by linarith show f (k + i) = g (k + i) from h1 i h3 done have h3 : prod_seq n k f = prod_seq n k g := ih h2 have h4 : n < n + 1 := Nat.lt_succ_self n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4] rfl done done lemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := by have h2 : ∀ (i : Nat), i < u → (f ∘ swap u n) (0 + i) = f (0 + i) := by fix i : Nat assume h2 : i < u have h3 : 0 + i ≠ u := by linarith have h4 : 0 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl done show prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f from prod_eq_fun (f ∘ swap u n) f 0 u h2 done lemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := by have h2 : ∀ i < j, (f ∘ swap u n) (u + 1 + i) = f (u + 1 + i) := by fix i : Nat assume h2 : i < j have h3 : u + 1 + i ≠ u := by linarith have h4 : u + 1 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl show prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f from prod_eq_fun (f ∘ swap u n) f (u + 1) j h2 done lemma break_prod {m : Nat} (n : Nat) (f : Nat → ZMod m) : ∀ (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by by_induc · -- Base Case have h : n + 0 = n := by rfl rewrite [prod_seq_base, h, Theorem_7_3_6_7] rfl done · -- Induction Step fix j : Nat assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f rewrite [←add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc] rfl done done lemma break_prod_twice {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f := break_prod n f 1 rewrite [prod_one] at h2 have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f := break_prod (u + 1) f j rewrite [←h1] at h3 have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f := break_prod u f 1 rewrite [prod_one] at h4 rewrite [h3, h4] at h2 show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2 done lemma swap_prod_eq_prod {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f := by by_cases h2 : u = n · -- Case 1. h2 : u = n rewrite [h2, trivial_swap n] --Goal : prod_seq (n + 1) 0 (f ∘ id) = prod_seq (n + 1) 0 f rfl done · -- Case 2. h2 : ¬u = n have h3 : u + 1 ≤ n := Nat.lt_of_le_of_ne h1 h2 obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3 have break_f : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := break_prod_twice f h4 have break_fs : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_prod_twice (f ∘ swap u n) h4 have f_eq_fs_below : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := swap_prod_eq_prod_below f h1 have f_eq_fs_btwn : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4 show prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f from calc prod_seq (n + 1) 0 (f ∘ swap u n) _ = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_fs _ = prod_seq u 0 f * (f ∘ swap u n) u * prod_seq j (u + 1) f * (f ∘ swap u n) n := by rw [f_eq_fs_below, f_eq_fs_btwn] _ = prod_seq u 0 f * f (swap u n u) * prod_seq j (u + 1) f * f (swap u n n) := by rfl _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by rw [swap_fst, swap_snd] _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring _ = prod_seq (n + 1) 0 f := break_f.symm done done lemma perm_below_fixed {n : Nat} {g : Nat → Nat} (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry lemma perm_prod {m : Nat} (f : Nat → ZMod m) : ∀ (n : Nat), ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) := by by_induc · -- Base Case fix g : Nat → Nat assume h1 : perm_below 0 g rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) fix g : Nat → Nat assume g_pb : perm_below (n + 1) g define at g_pb have g_ob : onto_below (n + 1) g := g_pb.right.right define at g_ob have h1 : n < n + 1 := by linarith obtain (u : Nat) (h2 : u < n + 1 ∧ g u = n) from g_ob n h1 have s_pb : perm_below (n + 1) (swap u n) := swap_perm_below h2.left h1 have gs_pb_n1 : perm_below (n + 1) (g ∘ swap u n) := comp_perm_below g_pb s_pb have gs_fix_n : (g ∘ swap u n) n = n := calc (g ∘ swap u n) n _ = g (swap u n n) := by rfl _ = g u := by rw [swap_snd] _ = n := h2.right have gs_pb_n : perm_below n (g ∘ swap u n) := perm_below_fixed gs_pb_n1 gs_fix_n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f ∘ (g ∘ swap u n)) := ih (g ∘ swap u n) gs_pb_n have h3 : u ≤ n := by linarith show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f ∘ g) from calc prod_seq (n + 1) 0 f _ = prod_seq n 0 f * f n := prod_seq_zero_step n f _ = prod_seq n 0 (f ∘ (g ∘ swap u n)) * f ((g ∘ swap u n) n) := by rw [gs_prod, gs_fix_n] _ = prod_seq n 0 (f ∘ g ∘ swap u n) * (f ∘ g ∘ swap u n) n := by rfl _ = prod_seq (n + 1) 0 (f ∘ g ∘ swap u n) := (prod_seq_zero_step n (f ∘ g ∘ swap u n)).symm _ = prod_seq (n + 1) 0 ((f ∘ g) ∘ swap u n) := by rfl _ = prod_seq (n + 1) 0 (f ∘ g) := swap_prod_eq_prod (f ∘ g) h3 done done lemma F_invertible (m i : Nat) : invertible (F m i) := by by_cases h : rel_prime m i · -- Case 1. h : rel_prime m i rewrite [F_rp_def h] show invertible [i]_m from (Theorem_7_3_7 m i).rtl h done · -- Case 2. h : ¬rel_prime m i rewrite [F_not_rp_def h] apply Exists.intro [1]_m show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m done done lemma Fprod_invertible (m : Nat) : ∀ (k : Nat), invertible (prod_seq k 0 (F m)) := by by_induc · -- Base Case apply Exists.intro [1]_m show prod_seq 0 0 (F m) * [1]_m = [1]_m from calc prod_seq 0 0 (F m) * [1]_m _ = [1]_m * [1]_m := by rw [prod_seq_base] _ = [1]_m := Theorem_7_3_6_7 ([1]_m) done · -- Induction Step fix k : Nat assume ih : invertible (prod_seq k 0 (F m)) rewrite [prod_seq_zero_step] show invertible (prod_seq k 0 (F m) * (F m k)) from (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k) done done theorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m ^ (phi m) = [1]_m := by have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2 show [a]_m ^ (phi m) = [1]_m from calc [a]_m ^ (phi m) _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3] _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring _ = prod_seq m 0 (F m ∘ G m a) * Y := by rw [FG_prod h1 m, phi_def] _ = prod_seq m 0 (F m) * Y := by rw [perm_prod (F m) m (G m a) (G_perm_below h1)] _ = [1]_m := by rw [h3] done lemma Exercise_7_4_5_Int (m : Nat) (a : Int) : ∀ (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry lemma Exercise_7_4_5_Nat (m a n : Nat) : [a]_m ^ n = [a ^ n]_m := by rewrite [Exercise_7_4_5_Int] rfl done theorem Euler's_theorem {m a : Nat} [NeZero m] (h1 : rel_prime m a) : a ^ (phi m) ≡ 1 (MOD m) := by have h2 : [a]_m ^ (phi m) = [1]_m := Theorem_7_4_2 h1 rewrite [Exercise_7_4_5_Nat m a (phi m)] at h2 --h2 : [a ^ phi m]_m = [1]_m show a ^ (phi m) ≡ 1 (MOD m) from (cc_eq_iff_congr _ _ _).ltr h2 done #eval gcd 10 7 --Answer: 1. So 10 and 7 are relatively prime #eval 7 ^ phi 10 --Answer: 2401, which is congruent to 1 mod 10. end Euler /- Section 7.5 -/ lemma num_rp_prime {p : Nat} (h1 : prime p) : ∀ k < p, num_rp_below p (k + 1) = k := sorry lemma phi_prime {p : Nat} (h1 : prime p) : phi p = p - 1 := by have h2 : 1 ≤ p := prime_pos h1 have h3 : p - 1 + 1 = p := Nat.sub_add_cancel h2 have h4 : p - 1 < p := by linarith have h5 : num_rp_below p (p - 1 + 1) = p - 1 := num_rp_prime h1 (p - 1) h4 rewrite [h3] at h5 show phi p = p - 1 from h5 done theorem Theorem_7_2_2_Int {a c : Nat} {b : Int} (h1 : ↑c ∣ ↑a * b) (h2 : rel_prime a c) : ↑c ∣ b := by rewrite [Int.natCast_dvd, Int.natAbs_mul, Int.natAbs_ofNat] at h1 --h1 : c ∣ a * Int.natAbs b rewrite [Int.natCast_dvd] --Goal : c ∣ Int.natAbs b show c ∣ Int.natAbs b from Theorem_7_2_2 h1 h2 done lemma Lemma_7_4_5 {m n : Nat} (a b : Int) (h1 : rel_prime m n) : a ≡ b (MOD m * n) ↔ a ≡ b (MOD m) ∧ a ≡ b (MOD n) := by apply Iff.intro · -- (→) assume h2 : a ≡ b (MOD m * n) obtain (j : Int) (h3 : a - b = (m * n) * j) from h2 apply And.intro · -- Proof of a ≡ b (MOD m) apply Exists.intro (n * j) show a - b = m * (n * j) from calc a - b _ = m * n * j := h3 _ = m * (n * j) := by ring done · -- Proof of a ≡ b (MOD n) apply Exists.intro (m * j) show a - b = n * (m * j) from calc a - b _ = m * n * j := h3 _ = n * (m * j) := by ring done done · -- (←) assume h2 : a ≡ b (MOD m) ∧ a ≡ b (MOD n) obtain (j : Int) (h3 : a - b = m * j) from h2.left have h4 : (↑n : Int) ∣ a - b := h2.right rewrite [h3] at h4 --h4 : ↑n ∣ ↑m * j have h5 : ↑n ∣ j := Theorem_7_2_2_Int h4 h1 obtain (k : Int) (h6 : j = n * k) from h5 apply Exists.intro k --Goal : a - b = ↑(m * n) * k rewrite [Nat.cast_mul] --Goal : a - b = ↑m * ↑n * k show a - b = (m * n) * k from calc a - b _ = m * j := h3 _ = m * (n * k) := by rw [h6] _ = (m * n) * k := by ring done done --From exercises of Section 7.2 theorem rel_prime_symm {a b : Nat} (h : rel_prime a b) : rel_prime b a := sorry lemma prime_NeZero {p : Nat} (h : prime p) : NeZero p := by rewrite [neZero_iff] --Goal : p ≠ 0 define at h linarith done lemma Lemma_7_5_1 {p e d m c s : Nat} {t : Int} (h1 : prime p) (h2 : e * d = (p - 1) * s + 1) (h3 : m ^ e - c = p * t) : c ^ d ≡ m (MOD p) := by have h4 : m ^ e ≡ c (MOD p) := Exists.intro t h3 have h5 : [m ^ e]_p = [c]_p := (cc_eq_iff_congr _ _ _).rtl h4 rewrite [←Exercise_7_4_5_Nat] at h5 --h5 : [m]_p ^ e = [c]_p by_cases h6 : p ∣ m · -- Case 1. h6 : p ∣ m have h7 : m ≡ 0 (MOD p) := by obtain (j : Nat) (h8 : m = p * j) from h6 apply Exists.intro (↑j : Int) --Goal : ↑m - 0 = ↑p * ↑j rewrite [h8, Nat.cast_mul] ring done have h8 : [m]_p = [0]_p := (cc_eq_iff_congr _ _ _).rtl h7 have h9 : e * d ≠ 0 := by rewrite [h2] show (p - 1) * s + 1 ≠ 0 from Nat.add_one_ne_zero _ done have h10 : (0 : Int) ^ (e * d) = 0 := zero_pow h9 have h11 : [c ^ d]_p = [m]_p := calc [c ^ d]_p _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat] _ = ([m]_p ^ e) ^ d := by rw [h5] _ = [m]_p ^ (e * d) := by ring _ = [0]_p ^ (e * d) := by rw [h8] _ = [0 ^ (e * d)]_p := Exercise_7_4_5_Int _ _ _ _ = [0]_p := by rw [h10] _ = [m]_p := by rw [h8] show c ^ d ≡ m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11 done · -- Case 2. h6 : ¬p ∣ m have h7 : rel_prime m p := rel_prime_of_prime_not_dvd h1 h6 have h8 : rel_prime p m := rel_prime_symm h7 have h9 : NeZero p := prime_NeZero h1 have h10 : (1 : Int) ^ s = 1 := by ring have h11 : [c ^ d]_p = [m]_p := calc [c ^ d]_p _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat] _ = ([m]_p ^ e) ^ d := by rw [h5] _ = [m]_p ^ (e * d) := by ring _ = [m]_p ^ ((p - 1) * s + 1) := by rw [h2] _ = ([m]_p ^ (p - 1)) ^ s * [m]_p := by ring _ = ([m]_p ^ (phi p)) ^ s * [m]_p := by rw [phi_prime h1] _ = [1]_p ^ s * [m]_p := by rw [Theorem_7_4_2 h8] _ = [1 ^ s]_p * [m]_p := by rw [Exercise_7_4_5_Int] _ = [1]_p * [m]_p := by rw [h10] _ = [m]_p * [1]_p := by ring _ = [m]_p := Theorem_7_3_6_7 _ show c ^ d ≡ m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11 done done theorem Theorem_7_5_1 (p q n e d k m c : Nat) (p_prime : prime p) (q_prime : prime q) (p_ne_q : p ≠ q) (n_pq : n = p * q) (ed_congr_1 : e * d = k * (p - 1) * (q - 1) + 1) (h1 : [m]_n ^ e = [c]_n) : [c]_n ^ d = [m]_n := by rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] at h1 --h1 : m ^ e ≡ c (MOD n) rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] --Goal : c ^ d ≡ m (MOD n) obtain (j : Int) (h2 : m ^ e - c = n * j) from h1 rewrite [n_pq, Nat.cast_mul] at h2 --h2 : m ^ e - c = p * q * j have h3 : e * d = (p - 1) * (k * (q - 1)) + 1 := by rewrite [ed_congr_1] ring done have h4 : m ^ e - c = p * (q * j) := by rewrite [h2] ring done have congr_p : c ^ d ≡ m (MOD p) := Lemma_7_5_1 p_prime h3 h4 have h5 : e * d = (q - 1) * (k * (p - 1)) + 1 := by rewrite [ed_congr_1] ring done have h6 : m ^ e - c = q * (p * j) := by rewrite [h2] ring done have congr_q : c ^ d ≡ m (MOD q) := Lemma_7_5_1 q_prime h5 h6 have h7 : ¬q ∣ p := by by_contra h8 have h9 : q = 1 ∨ q = p := dvd_prime p_prime h8 disj_syll h9 (prime_not_one q_prime) show False from p_ne_q h9.symm done have h8 : rel_prime p q := rel_prime_of_prime_not_dvd q_prime h7 rewrite [n_pq, Lemma_7_4_5 _ _ h8] show c ^ d ≡ m (MOD p) ∧ c ^ d ≡ m (MOD q) from And.intro congr_p congr_q done /- BEGIN EXERCISES -/ namespace Exercises /- Section 7.1 -/ -- 1. theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry -- 2. lemma gcd_comm_lt {a b : Nat} (h : a < b) : gcd a b = gcd b a := sorry theorem gcd_comm (a b : Nat) : gcd a b = gcd b a := sorry -- 3. theorem Exercise_7_1_5 (a b : Nat) (n : Int) : (∃ (s t : Int), s * a + t * b = n) ↔ (↑(gcd a b) : Int) ∣ n := sorry -- 4. theorem Exercise_7_1_6 (a b c : Nat) : gcd a b = gcd (a + b * c) b := sorry -- 5. theorem gcd_is_nonzero {a b : Nat} (h : a ≠ 0 ∨ b ≠ 0) : gcd a b ≠ 0 := sorry -- 6. theorem gcd_greatest {a b d : Nat} (h1 : gcd a b ≠ 0) (h2 : d ∣ a) (h3 : d ∣ b) : d ≤ gcd a b := sorry -- 7. lemma Lemma_7_1_10a {a b : Nat} (n : Nat) (h : a ∣ b) : (n * a) ∣ (n * b) := sorry lemma Lemma_7_1_10b {a b n : Nat} (h1 : n ≠ 0) (h2 : (n * a) ∣ (n * b)) : a ∣ b := sorry lemma Lemma_7_1_10c {a b : Nat} (h1 : a ∣ b) (h2 : b ∣ a) : a = b := sorry theorem Exercise_7_1_10 (a b n : Nat) : gcd (n * a) (n * b) = n * gcd a b := sorry /- Section 7.2 -/ -- 1. lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry -- 2. -- Hints: Start with apply List.rec. You may find mul_ne_zero useful theorem prod_nonzero_nonzero : ∀ (l : List Nat), (∀ a ∈ l, a ≠ 0) → prod l ≠ 0 := sorry -- 3. theorem rel_prime_iff_no_common_factor (a b : Nat) : rel_prime a b ↔ ¬∃ (p : Nat), prime p ∧ p ∣ a ∧ p ∣ b := sorry -- 4. theorem rel_prime_symm {a b : Nat} (h : rel_prime a b) : rel_prime b a := sorry -- 5. lemma in_prime_factorization_iff_prime_factor {a : Nat} {l : List Nat} (h1 : prime_factorization a l) (p : Nat) : p ∈ l ↔ prime_factor p a := sorry -- 6. theorem Exercise_7_2_5 {a b : Nat} {l m : List Nat} (h1 : prime_factorization a l) (h2 : prime_factorization b m) : rel_prime a b ↔ (¬∃ (p : Nat), p ∈ l ∧ p ∈ m) := sorry -- 7. theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry -- 8. theorem Exercise_7_2_7 {a b a' b' : Nat} (h1 : rel_prime a b) (h2 : a' ∣ a) (h3 : b' ∣ b) : rel_prime a' b' := sorry -- 9. theorem Exercise_7_2_9 {a b j k : Nat} (h1 : gcd a b ≠ 0) (h2 : a = j * gcd a b) (h3 : b = k * gcd a b) : rel_prime j k := sorry -- 10. theorem Exercise_7_2_17a (a b c : Nat) : gcd a (b * c) ∣ gcd a b * gcd a c := sorry /- Section 7.3 -/ -- 1. theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry -- 2. theorem Theorem_7_3_6_3 {m : Nat} (X : ZMod m) : X + [0]_m = X := sorry -- 3. theorem Theorem_7_3_6_4 {m : Nat} (X : ZMod m) : ∃ (Y : ZMod m), X + Y = [0]_m := sorry -- 4. theorem Exercise_7_3_4a {m : Nat} (Z1 Z2 : ZMod m) (h1 : ∀ (X : ZMod m), X + Z1 = X) (h2 : ∀ (X : ZMod m), X + Z2 = X) : Z1 = Z2 := sorry -- 5. theorem Exercise_7_3_4b {m : Nat} (X Y1 Y2 : ZMod m) (h1 : X + Y1 = [0]_m) (h2 : X + Y2 = [0]_m) : Y1 = Y2 := sorry -- 6. theorem Theorem_7_3_10 (m a : Nat) (b : Int) : ¬(↑(gcd m a) : Int) ∣ b → ¬∃ (x : Int), a * x ≡ b (MOD m) := sorry -- 7. theorem Theorem_7_3_11 (m n : Nat) (a b : Int) (h1 : n ≠ 0) : n * a ≡ n * b (MOD n * m) ↔ a ≡ b (MOD m) := sorry -- 8. theorem Exercise_7_3_16 {m : Nat} {a b : Int} (h : a ≡ b (MOD m)) : ∀ (n : Nat), a ^ n ≡ b ^ n (MOD m) := sorry -- 9. example {m : Nat} [NeZero m] (X : ZMod m) : ∃! (a : Int), 0 ≤ a ∧ a < m ∧ X = [a]_m := sorry -- 10. theorem congr_rel_prime {m a b : Nat} (h1 : a ≡ b (MOD m)) : rel_prime m a ↔ rel_prime m b := sorry -- 11. --Hint: You may find the theorem Int.ofNat_mod_ofNat useful. theorem rel_prime_mod (m a : Nat) : rel_prime m (a % m) ↔ rel_prime m a := sorry -- 12. lemma congr_iff_mod_eq_Int (m : Nat) (a b : Int) [NeZero m] : a ≡ b (MOD m) ↔ a % ↑m = b % ↑m := sorry --Hint for next theorem: Use the lemma above, --together with the theorems Int.ofNat_mod_ofNat and Nat.cast_inj. theorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] : ↑a ≡ ↑b (MOD m) ↔ a % m = b % m := sorry /- Section 7.4 -/ -- 1. --Hint: Use induction. --For the base case, compute [a]_m ^ 0 * [1]_m in two ways: --by Theorem_7_3_6_7, [a] ^ 0 * [1]_m = [a]_m ^ 0 --by ring, [a]_m ^ 0 * [1]_m = [1]_m. lemma Exercise_7_4_5_Int (m : Nat) (a : Int) : ∀ (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry -- 2. lemma left_inv_one_one_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g' (g i) = i) : one_one_below n g := sorry -- 3. lemma comp_perm_below {n : Nat} {f g : Nat → Nat} (h1 : perm_below n f) (h2 : perm_below n g) : perm_below n (f ∘ g) := sorry -- 4. lemma perm_below_fixed {n : Nat} {g : Nat → Nat} (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry -- 5. lemma Lemma_7_4_6 {a b c : Nat} : rel_prime (a * b) c ↔ rel_prime a c ∧ rel_prime b c := sorry -- 6. example {m a : Nat} [NeZero m] (h1 : rel_prime m a) : a ^ (phi m + 1) ≡ a (MOD m) := sorry -- 7. theorem Like_Exercise_7_4_11 {m a p : Nat} [NeZero m] (h1 : rel_prime m a) (h2 : p + 1 = phi m) : [a]_m * [a ^ p]_m = [1]_m := sorry -- 8. theorem Like_Exercise_7_4_12 {m a p q k : Nat} [NeZero m] (h1 : rel_prime m a) (h2 : p = q + (phi m) * k) : a ^ p ≡ a ^ q (MOD m) := sorry /- Section 7.5 -/ -- 1. --Hint: Use induction. lemma num_rp_prime {p : Nat} (h1 : prime p) : ∀ k < p, num_rp_below p (k + 1) = k := sorry -- 2. lemma three_prime : prime 3 := sorry -- 3. --Hint: Use the previous exercise, Exercise_7_2_7, and Theorem_7_4_2. theorem Exercise_7_5_13a (a : Nat) (h1 : rel_prime 561 a) : a ^ 560 ≡ 1 (MOD 3) := sorry -- 4. --Hint: Imitate the way Theorem_7_2_2_Int was proven from Theorem_7_2_2.

lemma Theorem_7_2_3_Int {p : Nat} {a b : Int} (h1 : prime p) (h2 : ↑p ∣ a * b) : ↑p ∣ a ∨ ↑p ∣ b

HTPI.Exercises.Theorem_7_2_3_Int

htpi/HTPILib/Chap7.lean

htpi

/- Copyright 2023 Daniel J. Velleman -/ import HTPILib.Chap6 namespace HTPI /- Definitions -/ lemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by have h : n + 1 > 0 := Nat.succ_pos n show a % (n + 1) < n + 1 from Nat.mod_lt a h done def gcd (a b : Nat) : Nat := match b with | 0 => a | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd (n + 1) (a % (n + 1)) termination_by b mutual def gcd_c1 (a b : Nat) : Int := match b with | 0 => 1 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c2 (n + 1) (a % (n + 1)) --Corresponds to s = t' termination_by b def gcd_c2 (a b : Nat) : Int := match b with | 0 => 0 | n + 1 => have : a % (n + 1) < n + 1 := mod_succ_lt a n gcd_c1 (n + 1) (a % (n + 1)) - (gcd_c2 (n + 1) (a % (n + 1))) * ↑(a / (n + 1)) --Corresponds to t = s' - t'q termination_by b end def prime (n : Nat) : Prop := 2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n def prime_factor (p n : Nat) : Prop := prime p ∧ p ∣ n def all_prime (l : List Nat) : Prop := ∀ p ∈ l, prime p def nondec (l : List Nat) : Prop := match l with | [] => True --Of course, True is a proposition that is always true | n :: L => (∀ m ∈ L, n ≤ m) ∧ nondec L def nondec_prime_list (l : List Nat) : Prop := all_prime l ∧ nondec l def prod (l : List Nat) : Nat := match l with | [] => 1 | n :: L => n * (prod L) def prime_factorization (n : Nat) (l : List Nat) : Prop := nondec_prime_list l ∧ prod l = n def rel_prime (a b : Nat) : Prop := gcd a b = 1 def congr_mod (m : Nat) (a b : Int) : Prop := (↑m : Int) ∣ (a - b) def cc (m : Nat) (a : Int) : ZMod m := (↑a : ZMod m) notation:50 a " ≡ " b " (MOD " m ")" => congr_mod m a b notation:max "["a"]_"m:max => cc m a def invertible {m : Nat} (X : ZMod m) : Prop := ∃ (Y : ZMod m), X * Y = [1]_m def num_rp_below (m k : Nat) : Nat := match k with | 0 => 0 | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j def phi (m : Nat) : Nat := num_rp_below m m def prod_seq {m : Nat} (j k : Nat) (f : Nat → ZMod m) : ZMod m := match j with | 0 => [1]_m | n + 1 => prod_seq n k f * f (k + n) def maps_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i < n, g i < n def one_one_below (n : Nat) (g : Nat → Nat) : Prop := ∀ i1 < n, ∀ i2 < n, g i1 = g i2 → i1 = i2 def onto_below (n : Nat) (g : Nat → Nat) : Prop := ∀ k < n, ∃ i < n, g i = k def perm_below (n : Nat) (g : Nat → Nat) : Prop := maps_below n g ∧ one_one_below n g ∧ onto_below n g def inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m) def swap (u v i : Nat) : Nat := if i = u then v else if i = v then u else i namespace Euler --For definitions specific to Euler's theorem def F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m def G (m a i : Nat) : Nat := (a * i) % m def Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i end Euler /- Section 7.1 -/ theorem dvd_mod_of_dvd_a_b {a b d : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ (a % b) := by set q : Nat := a / b have h3 : b * q + a % b = a := Nat.div_add_mod a b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 define --Goal : ∃ (c : Nat), a % b = d * c apply Exists.intro (j - k * q) show a % b = d * (j - k * q) from calc a % b _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm _ = a - b * q := by rw [h3] _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc] _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm done theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry #eval gcd 672 161 --Answer: 7 lemma gcd_base (a : Nat) : gcd a 0 = a := by rfl lemma gcd_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd a b = gcd b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1)) rfl done lemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b ≠ 0) : a % b < b := by have h1 : b > 0 := Nat.pos_of_ne_zero h show a % b < b from Nat.mod_lt a h1 done lemma dvd_self (n : Nat) : n ∣ n := by apply Exists.intro 1 ring done theorem gcd_dvd : ∀ (b a : Nat), (gcd a b) ∣ a ∧ (gcd a b) ∣ b := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd a b_1) ∣ a ∧ (gcd a b_1) ∣ b_1 fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_base] --Goal: a ∣ a ∧ a ∣ 0 apply And.intro (dvd_self a) define apply Exists.intro 0 rfl done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_nonzero a h1] --Goal : gcd b (a % b) ∣ a ∧ gcd b (a % b) ∣ b have h2 : a % b < b := mod_nonzero_lt a h1 have h3 : (gcd b (a % b)) ∣ b ∧ (gcd b (a % b)) ∣ (a % b) := ih (a % b) h2 b apply And.intro _ h3.left show (gcd b (a % b)) ∣ a from dvd_a_of_dvd_b_mod h3.left h3.right done done theorem gcd_dvd_left (a b : Nat) : (gcd a b) ∣ a := (gcd_dvd b a).left theorem gcd_dvd_right (a b : Nat) : (gcd a b) ∣ b := (gcd_dvd b a).right lemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl lemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c1 a b = gcd_c2 b (a % b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done lemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl lemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b ≠ 0) : gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * ↑(a / b) := by obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h rewrite [h2] rfl done theorem gcd_lin_comb : ∀ (b a : Nat), (gcd_c1 a b) * ↑a + (gcd_c2 a b) * ↑b = ↑(gcd a b) := by by_strong_induc fix b : Nat assume ih : ∀ b_1 < b, ∀ (a : Nat), (gcd_c1 a b_1) * ↑a + (gcd_c2 a b_1) * ↑b_1 = ↑(gcd a b_1) fix a : Nat by_cases h1 : b = 0 · -- Case 1. h1 : b = 0 rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base] --Goal : 1 * ↑a + 0 * ↑0 = ↑a ring done · -- Case 2. h1 : b ≠ 0 rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1] --Goal : gcd_c2 b (a % b) * ↑a + -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * ↑(a / b)) * ↑b = -- ↑(gcd b (a % b)) set r : Nat := a % b set q : Nat := a / b set s : Int := gcd_c1 b r set t : Int := gcd_c2 b r --Goal : t * ↑a + (s - t * ↑q) * ↑b = ↑(gcd b r) have h2 : r < b := mod_nonzero_lt a h1 have h3 : s * ↑b + t * ↑r = ↑(gcd b r) := ih r h2 b have h4 : b * q + r = a := Nat.div_add_mod a b rewrite [←h3, ←h4] rewrite [Nat.cast_add, Nat.cast_mul] --Goal : t * (↑b * ↑q + ↑r) + (s - t * ↑q) * ↑b = s * ↑b + t * ↑r ring done done #eval gcd_c1 672 161 --Answer: 6 #eval gcd_c2 672 161 --Answer: -25 --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161 theorem Theorem_7_1_6 {d a b : Nat} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ gcd a b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑d ∣ ↑(gcd a b) set s : Int := gcd_c1 a b set t : Int := gcd_c2 a b have h3 : s * ↑a + t * ↑b = ↑(gcd a b) := gcd_lin_comb b a rewrite [←h3] --Goal : ↑d ∣ s * ↑a + t * ↑b obtain (j : Nat) (h4 : a = d * j) from h1 obtain (k : Nat) (h5 : b = d * k) from h2 rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul] --Goal : ↑d ∣ s * (↑d * ↑j) + t * (↑d * ↑k) define apply Exists.intro (s * ↑j + t * ↑k) ring done /- Section 7.2 -/ theorem dvd_trans {a b c : Nat} (h1 : a ∣ b) (h2 : b ∣ c) : a ∣ c := by define at h1; define at h2; define obtain (m : Nat) (h3 : b = a * m) from h1 obtain (n : Nat) (h4 : c = b * n) from h2 rewrite [h3, mul_assoc] at h4 apply Exists.intro (m * n) show c = a * (m * n) from h4 done lemma exists_prime_factor : ∀ (n : Nat), 2 ≤ n → ∃ (p : Nat), prime_factor p n := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, 2 ≤ n_1 → ∃ (p : Nat), prime_factor p n_1 assume h1 : 2 ≤ n by_cases h2 : prime n · -- Case 1. h2 : prime n apply Exists.intro n define --Goal : prime n ∧ n ∣ n show prime n ∧ n ∣ n from And.intro h2 (dvd_self n) done · -- Case 2. h2 : ¬prime n define at h2 --h2 : ¬(2 ≤ n ∧ ¬∃ (a b : Nat), a * b = n ∧ a < n ∧ b < n) demorgan at h2 disj_syll h2 h1 obtain (a : Nat) (h3 : ∃ (b : Nat), a * b = n ∧ a < n ∧ b < n) from h2 obtain (b : Nat) (h4 : a * b = n ∧ a < n ∧ b < n) from h3 have h5 : 2 ≤ a := by by_contra h6 have h7 : a ≤ 1 := by linarith have h8 : n ≤ b := calc n _ = a * b := h4.left.symm _ ≤ 1 * b := by rel [h7] _ = b := by ring linarith --n ≤ b contradicts b < n done have h6 : ∃ (p : Nat), prime_factor p a := ih a h4.right.left h5 obtain (p : Nat) (h7 : prime_factor p a) from h6 apply Exists.intro p define --Goal : prime p ∧ p ∣ n define at h7 --h7 : prime p ∧ p ∣ a apply And.intro h7.left have h8 : a ∣ n := by apply Exists.intro b show n = a * b from (h4.left).symm done show p ∣ n from dvd_trans h7.right h8 done done lemma exists_least_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q := by set S : Set Nat := {p : Nat | prime_factor p n} have h2 : ∃ (p : Nat), p ∈ S := exists_prime_factor n h show ∃ (p : Nat), prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q from well_ord_princ S h2 done lemma all_prime_nil : all_prime [] := by define --Goal : ∀ p ∈ [], prime p fix p : Nat contrapos --Goal : ¬prime p → p ∉ [] assume h1 : ¬prime p show p ∉ [] from List.not_mem_nil p done lemma all_prime_cons (n : Nat) (L : List Nat) : all_prime (n :: L) ↔ prime n ∧ all_prime L := by apply Iff.intro · -- (→) assume h1 : all_prime (n :: L) --Goal : prime n ∧ all_prime L define at h1 --h1 : ∀ p ∈ n :: L, prime p apply And.intro (h1 n (List.mem_cons_self n L)) define --Goal : ∀ p ∈ L, prime p fix p : Nat assume h2 : p ∈ L show prime p from h1 p (List.mem_cons_of_mem n h2) done · -- (←) assume h1 : prime n ∧ all_prime L --Goal : all_prime (n :: l) define : all_prime L at h1 define fix p : Nat assume h2 : p ∈ n :: L rewrite [List.mem_cons] at h2 --h2 : p = n ∨ p ∈ L by_cases on h2 · -- Case 1. h2 : p = n rewrite [h2] show prime n from h1.left done · -- Case 2. h2 : p ∈ L show prime p from h1.right p h2 done done done lemma nondec_nil : nondec [] := by define --Goal : True trivial --trivial proves some obviously true statements, such as True done lemma nondec_cons (n : Nat) (L : List Nat) : nondec (n :: L) ↔ (∀ m ∈ L, n ≤ m) ∧ nondec L := by rfl lemma prod_nil : prod [] = 1 := by rfl lemma prod_cons : prod (n :: L) = n * (prod L) := by rfl lemma exists_cons_of_length_eq_succ {A : Type} {l : List A} {n : Nat} (h : l.length = n + 1) : ∃ (a : A) (L : List A), l = a :: L ∧ L.length = n := by have h1 : ¬l.length = 0 := by linarith rewrite [List.length_eq_zero] at h1 obtain (a : A) (h2 : ∃ (L : List A), l = a :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List A) (h3 : l = a :: L) from h2 apply Exists.intro a apply Exists.intro L apply And.intro h3 have h4 : (a :: L).length = L.length + 1 := List.length_cons a L rewrite [←h3, h] at h4 show L.length = n from (Nat.add_right_cancel h4).symm done lemma list_elt_dvd_prod_by_length (a : Nat) : ∀ (n : Nat), ∀ (l : List Nat), l.length = n → a ∈ l → a ∣ prod l := by by_induc · --Base Case fix l : List Nat assume h1 : l.length = 0 rewrite [List.length_eq_zero] at h1 --h1 : l = [] rewrite [h1] --Goal : a ∈ [] → a ∣ prod [] contrapos assume h2 : ¬a ∣ prod [] show a ∉ [] from List.not_mem_nil a done · -- Induction Step fix n : Nat assume ih : ∀ (l : List Nat), List.length l = n → a ∈ l → a ∣ prod l fix l : List Nat assume h1 : l.length = n + 1 --Goal : a ∈ l → a ∣ prod l obtain (b : Nat) (h2 : ∃ (L : List Nat), l = b :: L ∧ L.length = n) from exists_cons_of_length_eq_succ h1 obtain (L : List Nat) (h3 : l = b :: L ∧ L.length = n) from h2 have h4 : a ∈ L → a ∣ prod L := ih L h3.right assume h5 : a ∈ l rewrite [h3.left, prod_cons] --Goal : a ∣ b * prod L rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b ∨ a ∈ L by_cases on h5 · -- Case 1. h5 : a = b apply Exists.intro (prod L) rewrite [h5] rfl done · -- Case 2. h5 : a ∈ L have h6 : a ∣ prod L := h4 h5 have h7 : prod L ∣ b * prod L := by apply Exists.intro b ring done show a ∣ b * prod L from dvd_trans h6 h7 done done done lemma list_elt_dvd_prod {a : Nat} {l : List Nat} (h : a ∈ l) : a ∣ prod l := by set n : Nat := l.length have h1 : l.length = n := by rfl show a ∣ prod l from list_elt_dvd_prod_by_length a n l h1 h done lemma exists_prime_factorization : ∀ (n : Nat), n ≥ 1 → ∃ (l : List Nat), prime_factorization n l := by by_strong_induc fix n : Nat assume ih : ∀ n_1 < n, n_1 ≥ 1 → ∃ (l : List Nat), prime_factorization n_1 l assume h1 : n ≥ 1 by_cases h2 : n = 1 · -- Case 1. h2 : n = 1 apply Exists.intro [] define apply And.intro · -- Proof of nondec_prime_list [] define show all_prime [] ∧ nondec [] from And.intro all_prime_nil nondec_nil done · -- Proof of prod [] = n rewrite [prod_nil, h2] rfl done done · -- Case 2. h2 : n ≠ 1 have h3 : n ≥ 2 := lt_of_le_of_ne' h1 h2 obtain (p : Nat) (h4 : prime_factor p n ∧ ∀ (q : Nat), prime_factor q n → p ≤ q) from exists_least_prime_factor h3 have p_prime_factor : prime_factor p n := h4.left define at p_prime_factor have p_prime : prime p := p_prime_factor.left have p_dvd_n : p ∣ n := p_prime_factor.right have p_least : ∀ (q : Nat), prime_factor q n → p ≤ q := h4.right obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n have h5 : m ≠ 0 := by contradict h1 with h6 have h7 : n = 0 := calc n _ = p * m := n_eq_pm _ = p * 0 := by rw [h6] _ = 0 := by ring rewrite [h7] decide done have m_pos : 0 < m := Nat.pos_of_ne_zero h5 have m_lt_n : m < n := by define at p_prime show m < n from calc m _ < m + m := by linarith _ = 2 * m := by ring _ ≤ p * m := by rel [p_prime.left] _ = n := n_eq_pm.symm done obtain (L : List Nat) (h6 : prime_factorization m L) from ih m m_lt_n m_pos define at h6 have ndpl_L : nondec_prime_list L := h6.left define at ndpl_L apply Exists.intro (p :: L) define apply And.intro · -- Proof of nondec_prime_list (p :: L) define apply And.intro · -- Proof of all_prime (p :: L) rewrite [all_prime_cons] show prime p ∧ all_prime L from And.intro p_prime ndpl_L.left done · -- Proof of nondec (p :: L) rewrite [nondec_cons] apply And.intro _ ndpl_L.right fix q : Nat assume q_in_L : q ∈ L have h7 : q ∣ prod L := list_elt_dvd_prod q_in_L rewrite [h6.right] at h7 --h7 : q ∣ m have h8 : m ∣ n := by apply Exists.intro p rewrite [n_eq_pm] ring done have q_dvd_n : q ∣ n := dvd_trans h7 h8 have ap_L : all_prime L := ndpl_L.left define at ap_L have q_prime_factor : prime_factor q n := And.intro (ap_L q q_in_L) q_dvd_n show p ≤ q from p_least q q_prime_factor done done · -- Proof of prod (p :: L) = n rewrite [prod_cons, h6.right, n_eq_pm] rfl done done done theorem Theorem_7_2_2 {a b c : Nat} (h1 : c ∣ a * b) (h2 : rel_prime a c) : c ∣ b := by rewrite [←Int.natCast_dvd_natCast] --Goal : ↑c ∣ ↑b define at h1; define at h2; define obtain (j : Nat) (h3 : a * b = c * j) from h1 set s : Int := gcd_c1 a c set t : Int := gcd_c2 a c have h4 : s * ↑a + t * ↑c = ↑(gcd a c) := gcd_lin_comb c a rewrite [h2, Nat.cast_one] at h4 --h4 : s * ↑a + t * ↑c = (1 : Int) apply Exists.intro (s * ↑j + t * ↑b) show ↑b = ↑c * (s * ↑j + t * ↑b) from calc ↑b _ = (1 : Int) * ↑b := (one_mul _).symm _ = (s * ↑a + t * ↑c) * ↑b := by rw [h4] _ = s * (↑a * ↑b) + t * ↑c * ↑b := by ring _ = s * (↑c * ↑j) + t * ↑c * ↑b := by rw [←Nat.cast_mul a b, h3, Nat.cast_mul c j] _ = ↑c * (s * ↑j + t * ↑b) := by ring done lemma le_nonzero_prod_left {a b : Nat} (h : a * b ≠ 0) : a ≤ a * b := by have h1 : b ≠ 0 := by contradict h with h1 rewrite [h1] ring done have h2 : 1 ≤ b := Nat.pos_of_ne_zero h1 show a ≤ a * b from calc a = a * 1 := (mul_one a).symm _ ≤ a * b := by rel [h2] done lemma le_nonzero_prod_right {a b : Nat} (h : a * b ≠ 0) : b ≤ a * b := by rewrite [mul_comm] rewrite [mul_comm] at h show b ≤ b * a from le_nonzero_prod_left h done lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry lemma rel_prime_of_prime_not_dvd {a p : Nat} (h1 : prime p) (h2 : ¬p ∣ a) : rel_prime a p := by have h3 : gcd a p ∣ a := gcd_dvd_left a p have h4 : gcd a p ∣ p := gcd_dvd_right a p have h5 : gcd a p = 1 ∨ gcd a p = p := dvd_prime h1 h4 have h6 : gcd a p ≠ p := by contradict h2 with h6 rewrite [h6] at h3 show p ∣ a from h3 done disj_syll h5 h6 show rel_prime a p from h5 done theorem Theorem_7_2_3 {a b p : Nat} (h1 : prime p) (h2 : p ∣ a * b) : p ∣ a ∨ p ∣ b := by or_right with h3 have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3 show p ∣ b from Theorem_7_2_2 h2 h4 done lemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a ≥ 1 := by have h1 : a ≠ 0 := by by_contra h1 rewrite [h1] at h contradict h linarith done show a ≥ 1 from Nat.pos_of_ne_zero h1 done lemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by have h1 : a ≥ 1 := ge_one_of_prod_one h have h2 : a * b ≠ 0 := by linarith have h3 : a ≤ a * b := le_nonzero_prod_left h2 rewrite [h] at h3 show a = 1 from Nat.le_antisymm h3 h1 done lemma eq_one_of_dvd_one {n : Nat} (h : n ∣ 1) : n = 1 := by obtain (j : Nat) (h1 : 1 = n * j) from h show n = 1 from eq_one_of_prod_one h1.symm done lemma prime_not_one {p : Nat} (h : prime p) : p ≠ 1 := by define at h linarith done theorem Theorem_7_2_4 {p : Nat} (h1 : prime p) : ∀ (l : List Nat), p ∣ prod l → ∃ a ∈ l, p ∣ a := by apply List.rec · -- Base Case. Goal : p ∣ prod [] → ∃ a ∈ [], p ∣ a rewrite [prod_nil] assume h2 : p ∣ 1 show ∃ a ∈ [], p ∣ a from absurd (eq_one_of_dvd_one h2) (prime_not_one h1) done · -- Induction Step fix b : Nat fix L : List Nat assume ih : p ∣ prod L → ∃ a ∈ L, p ∣ a --Goal : p ∣ prod (b :: L) → ∃ a ∈ b :: L, p ∣ a assume h2 : p ∣ prod (b :: L) rewrite [prod_cons] at h2 have h3 : p ∣ b ∨ p ∣ prod L := Theorem_7_2_3 h1 h2 by_cases on h3 · -- Case 1. h3 : p ∣ b apply Exists.intro b show b ∈ b :: L ∧ p ∣ b from And.intro (List.mem_cons_self b L) h3 done · -- Case 2. h3 : p ∣ prod L obtain (a : Nat) (h4 : a ∈ L ∧ p ∣ a) from ih h3 apply Exists.intro a show a ∈ b :: L ∧ p ∣ a from And.intro (List.mem_cons_of_mem b h4.left) h4.right done done done lemma prime_in_list {p : Nat} {l : List Nat} (h1 : prime p) (h2 : all_prime l) (h3 : p ∣ prod l) : p ∈ l := by obtain (a : Nat) (h4 : a ∈ l ∧ p ∣ a) from Theorem_7_2_4 h1 l h3 define at h2 have h5 : prime a := h2 a h4.left have h6 : p = 1 ∨ p = a := dvd_prime h5 h4.right disj_syll h6 (prime_not_one h1) rewrite [h6] show a ∈ l from h4.left done lemma first_le_first {p q : Nat} {l m : List Nat} (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m)) (h3 : prod (p :: l) = prod (q :: m)) : p ≤ q := by define at h1; define at h2 have h4 : q ∣ prod (p :: l) := by define apply Exists.intro (prod m) rewrite [←prod_cons] show prod (p :: l) = prod (q :: m) from h3 done have h5 : all_prime (q :: m) := h2.left rewrite [all_prime_cons] at h5 have h6 : q ∈ p :: l := prime_in_list h5.left h1.left h4 have h7 : nondec (p :: l) := h1.right rewrite [nondec_cons] at h7 rewrite [List.mem_cons] at h6 by_cases on h6 · -- Case 1. h6 : q = p linarith done · -- Case 2. h6 : q ∈ l have h8 : ∀ m ∈ l, p ≤ m := h7.left show p ≤ q from h8 q h6 done done lemma nondec_prime_list_tail {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by define at h define rewrite [all_prime_cons, nondec_cons] at h show all_prime l ∧ nondec l from And.intro h.left.right h.right.right done lemma cons_prod_not_one {p : Nat} {l : List Nat} (h : nondec_prime_list (p :: l)) : prod (p :: l) ≠ 1 := by define at h have h1 : all_prime (p :: l) := h.left rewrite [all_prime_cons] at h1 rewrite [prod_cons] by_contra h2 show False from (prime_not_one h1.left) (eq_one_of_prod_one h2) done lemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) : l = [] ↔ prod l = 1 := by apply Iff.intro · -- (→) assume h1 : l = [] rewrite [h1] show prod [] = 1 from prod_nil done · -- (←) contrapos assume h1 : ¬l = [] obtain (p : Nat) (h2 : ∃ (L : List Nat), l = p :: L) from List.exists_cons_of_ne_nil h1 obtain (L : List Nat) (h3 : l = p :: L) from h2 rewrite [h3] at h rewrite [h3] show ¬prod (p :: L) = 1 from cons_prod_not_one h done done lemma prime_pos {p : Nat} (h : prime p) : p > 0 := by define at h linarith done theorem Theorem_7_2_5 : ∀ (l1 l2 : List Nat), nondec_prime_list l1 → nondec_prime_list l2 → prod l1 = prod l2 → l1 = l2 := by apply List.rec · -- Base Case. Goal : ∀ (l2 : List Nat), nondec_prime_list [] → -- nondec_prime_list l2 → prod [] = prod l2 → [] = l2 fix l2 : List Nat assume h1 : nondec_prime_list [] assume h2 : nondec_prime_list l2 assume h3 : prod [] = prod l2 rewrite [prod_nil, eq_comm, ←list_nil_iff_prod_one h2] at h3 show [] = l2 from h3.symm done · -- Induction Step fix p : Nat fix L1 : List Nat assume ih : ∀ (L2 : List Nat), nondec_prime_list L1 → nondec_prime_list L2 → prod L1 = prod L2 → L1 = L2 -- Goal : ∀ (l2 : List Nat), nondec_prime_list (p :: L1) → -- nondec_prime_list l2 → prod (p :: L1) = prod l2 → p :: L1 = l2 fix l2 : List Nat assume h1 : nondec_prime_list (p :: L1) assume h2 : nondec_prime_list l2 assume h3 : prod (p :: L1) = prod l2 have h4 : ¬prod (p :: L1) = 1 := cons_prod_not_one h1 rewrite [h3, ←list_nil_iff_prod_one h2] at h4 obtain (q : Nat) (h5 : ∃ (L : List Nat), l2 = q :: L) from List.exists_cons_of_ne_nil h4 obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5 rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2) rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2) have h7 : p ≤ q := first_le_first h1 h2 h3 have h8 : q ≤ p := first_le_first h2 h1 h3.symm have h9 : p = q := by linarith rewrite [h9, prod_cons, prod_cons] at h3 --h3 : q * prod L1 = q * prod L2 have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1 have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2 define at h2 have h12 : all_prime (q :: L2) := h2.left rewrite [all_prime_cons] at h12 have h13 : q > 0 := prime_pos h12.left have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3 have h15 : L1 = L2 := ih L2 h10 h11 h14 rewrite [h6, h9, h15] rfl done done theorem fund_thm_arith (n : Nat) (h : n ≥ 1) : ∃! (l : List Nat), prime_factorization n l := by exists_unique · -- Existence show ∃ (l : List Nat), prime_factorization n l from exists_prime_factorization n h done · -- Uniqueness fix l1 : List Nat; fix l2 : List Nat assume h1 : prime_factorization n l1 assume h2 : prime_factorization n l2 define at h1; define at h2 have h3 : prod l1 = n := h1.right rewrite [←h2.right] at h3 show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3 done done /- Section 7.3 -/ theorem congr_refl (m : Nat) : ∀ (a : Int), a ≡ a (MOD m) := by fix a : Int define --Goal : ∃ (c : Int), a - a = ↑m * c apply Exists.intro 0 ring done theorem congr_symm {m : Nat} : ∀ {a b : Int}, a ≡ b (MOD m) → b ≡ a (MOD m) := by fix a : Int; fix b : Int assume h1 : a ≡ b (MOD m) define at h1 --h1 : ∃ (c : Int), a - b = ↑m * c define --Goal : ∃ (c : Int), b - a = ↑m * c obtain (c : Int) (h2 : a - b = m * c) from h1 apply Exists.intro (-c) show b - a = m * (-c) from calc b - a _ = -(a - b) := by ring _ = -(m * c) := by rw [h2] _ = m * (-c) := by ring done theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry /- Fundamental properties of congruence classes -/ lemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) : X = Y ↔ X.val = Y.val := Fin.ext_iff lemma val_nat_eq_mod (n k : Nat) : ([k]_(n + 1)).val = k % (n + 1) := by rfl lemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl theorem cc_rep {m : Nat} (X : ZMod m) : ∃ (a : Int), X = [a]_m := match m with | 0 => by apply Exists.intro X rfl done | n + 1 => by apply Exists.intro ↑(X.val) have h1 : X.val < n + 1 := Fin.prop X rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1] rfl done theorem add_class (m : Nat) (a b : Int) : [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm theorem mul_class (m : Nat) (a b : Int) : [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm lemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ [a - b]_m = [0]_m := by apply Iff.intro · -- (→) assume h1 : [a]_m = [b]_m have h2 : a - b = a + (-b) := by ring have h3 : b + (-b) = 0 := by ring show [a - b]_m = [0]_m from calc [a - b]_m _ = [a + (-b)]_m := by rw [h2] _ = [a]_m + [-b]_m := by rw [add_class] _ = [b]_m + [-b]_m := by rw [h1] _ = [b + -b]_m := by rw [add_class] _ = [0]_m := by rw [h3] done · -- (←) assume h1 : [a - b]_m = [0]_m have h2 : b + (a - b) = a := by ring have h3 : b + 0 = b := by ring show [a]_m = [b]_m from calc [a]_m _ = [b + (a - b)]_m := by rw [h2] _ = [b]_m + [a - b]_m := by rw [add_class] _ = [b]_m + [0]_m := by rw [h1] _ = [b + 0]_m := by rw [add_class] _ = [b]_m := by rw [h3] done done lemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) : [a]_m = [0]_m → [-a]_m = [0]_m := by assume h1 : [a]_m = [0]_m have h2 : 0 + (-a) = -a := by ring have h3 : a + (-a) = 0 := by ring show [-a]_m = [0]_m from calc [-a]_m _ = [0 + (-a)]_m := by rw [h2] _ = [0]_m + [-a]_m := by rw [add_class] _ = [a]_m + [-a]_m := by rw [h1] _ = [a + (-a)]_m := by rw [add_class] _ = [0]_m := by rw [h3] done lemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) : [-a]_m = [0]_m ↔ [a]_m = [0]_m := by apply Iff.intro _ (cc_neg_zero_of_cc_zero m a) assume h1 : [-a]_m = [0]_m have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1 have h3 : -(-a) = a := by ring rewrite [h3] at h2 show [a]_m = [0]_m from h2 done lemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl lemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m ↔ m ∣ k := match m with | 0 => by have h : (0 : Int) = (↑(0 : Nat) : Int) := by rfl rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj] apply Iff.intro · -- (→) assume h1 : k = 0 rewrite [h1] show 0 ∣ 0 from dvd_self 0 done · -- (←) assume h1 : 0 ∣ k obtain (c : Nat) (h2 : k = 0 * c) from h1 rewrite [h2] ring done done | n + 1 => by rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero] show k % (n + 1) = 0 ↔ n + 1 ∣ k from (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm done lemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m ↔ ↑m ∣ a := by obtain (k : Nat) (h1 : a = ↑k ∨ a = -↑k) from Int.eq_nat_or_neg a by_cases on h1 · -- Case 1. h1: a = ↑k rewrite [h1, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done · -- Case 2. h1: a = -↑k rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast] show [↑k]_m = [0]_m ↔ m ∣ k from cc_nat_zero_iff_dvd m k done done theorem cc_eq_iff_congr (m : Nat) (a b : Int) : [a]_m = [b]_m ↔ a ≡ b (MOD m) := calc [a]_m = [b]_m _ ↔ [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b _ ↔ ↑m ∣ (a - b) := cc_zero_iff_dvd m (a - b) _ ↔ a ≡ b (MOD m) := by rfl /- End of fundamental properties of congruence classes -/ lemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m := by have h1 : (↑m : Int) ≠ 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m) show 0 ≤ a % m from Int.emod_nonneg a h1 done lemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m) have h2 : (↑m : Int) > 0 := (Nat.cast_pos).rtl h1 show a % m < m from Int.emod_lt_of_pos a h2 done lemma congr_mod_mod (m : Nat) (a : Int) : a ≡ a % m (MOD m) := by define have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m apply Exists.intro (a / m) show a - a % m = m * (a / m) from calc a - (a % m) _ = m * (a / m) + a % m - a % m := by rw [h1] _ = m * (a / m) := by ring done lemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) : 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) := And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a)) theorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) : ∃! (r : Int), 0 ≤ r ∧ r < m ∧ a ≡ r (MOD m) := by exists_unique · -- Existence apply Exists.intro (a % m) show 0 ≤ a % m ∧ a % m < m ∧ a ≡ a % m (MOD m) from mod_cmpl_res m a done · -- Uniqueness fix r1 : Int; fix r2 : Int assume h1 : 0 ≤ r1 ∧ r1 < m ∧ a ≡ r1 (MOD m) assume h2 : 0 ≤ r2 ∧ r2 < m ∧ a ≡ r2 (MOD m) have h3 : r1 ≡ r2 (MOD m) := congr_trans (congr_symm h1.right.right) h2.right.right obtain (d : Int) (h4 : r1 - r2 = m * d) from h3 have h5 : r1 - r2 < m * 1 := by linarith have h6 : m * (-1) < r1 - r2 := by linarith rewrite [h4] at h5 --h5 : m * d < m * 1 rewrite [h4] at h6 --h6 : m * -1 < m * d have h7 : (↑m : Int) ≥ 0 := Nat.cast_nonneg m have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7 have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7 have h10 : d = 0 := by linarith show r1 = r2 from calc r1 _ = r1 - r2 + r2 := by ring _ = m * 0 + r2 := by rw [h4, h10] _ = r2 := by ring done done lemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m := (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a) theorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y rewrite [h1, h2] have h3 : a + b = b + a := by ring show [a]_m + [b]_m = [b]_m + [a]_m from calc [a]_m + [b]_m _ = [a + b]_m := add_class m a b _ = [b + a]_m := by rw [h3] _ = [b]_m + [a]_m := (add_class m b a).symm done theorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by obtain (a : Int) (h1 : X = [a]_m) from cc_rep X rewrite [h1] have h2 : a * 1 = a := by ring show [a]_m * [1]_m = [a]_m from calc [a]_m * [1]_m _ = [a * 1]_m := mul_class m a 1 _ = [a]_m := by rw [h2] done theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry lemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) : [a]_m * [gcd_c2 m a]_m = [1]_m := by set s : Int := gcd_c1 m a have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m define at h1 rewrite [h1, Nat.cast_one] at h2 --h2 : s * ↑m + gcd_c2 m a * ↑a = 1 rewrite [mul_class, cc_eq_iff_congr] define --Goal : ∃ (c : Int), ↑a * gcd_c2 m a - 1 = ↑m * c apply Exists.intro (-s) show a * (gcd_c2 m a) - 1 = m * (-s) from calc a * (gcd_c2 m a) - 1 _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring _ = 1 + m * (-s) - 1 := by rw [h2] _ = m * (-s) := by ring done theorem Theorem_7_3_7 (m a : Nat) : invertible [a]_m ↔ rel_prime m a := by apply Iff.intro · -- (→) assume h1 : invertible [a]_m define at h1 obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1 obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y rewrite [h3, mul_class, cc_eq_iff_congr] at h2 define at h2 obtain (c : Int) (h4 : a * b - 1 = m * c) from h2 rewrite [Exercise_7_2_6] --Goal : ∃ (s t : Int), s * ↑m + t * ↑a = 1 apply Exists.intro (-c) apply Exists.intro b show (-c) * m + b * a = 1 from calc (-c) * m + b * a _ = (-c) * m + (a * b - 1) + 1 := by ring _ = (-c) * m + m * c + 1 := by rw [h4] _ = 1 := by ring done · -- (←) assume h1 : rel_prime m a define show ∃ (Y : ZMod m), [a]_m * Y = [1]_m from Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1) done done /- Section 7.4 -/ section Euler open Euler lemma num_rp_below_base {m : Nat} : num_rp_below m 0 = 0 := by rfl lemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) : num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by have h1 : num_rp_below m (j + 1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : gcd m j = 1 rewrite [if_pos h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1 show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1 done lemma num_rp_below_step_not_rp {m j : Nat} (h : ¬rel_prime m j) : num_rp_below m (j + 1) = num_rp_below m j := by have h1 : num_rp_below m (j +1) = if gcd m j = 1 then (num_rp_below m j) + 1 else num_rp_below m j := by rfl define at h --h : ¬gcd m j = 1 rewrite [if_neg h] at h1 --h1 : num_rp_below m (j + 1) = num_rp_below m j show num_rp_below m (j + 1) = num_rp_below m j from h1 done lemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl #eval phi 10 --Answer: 4 lemma prod_inv_iff_inv {m : Nat} {X : ZMod m} (h1 : invertible X) (Y : ZMod m) : invertible (X * Y) ↔ invertible Y := by apply Iff.intro · -- (→) assume h2 : invertible (X * Y) obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2 apply Exists.intro (X * Z) rewrite [←h3] --Goal : Y * (X * Z) = X * Y * Z ring --Note that ring can do algebra in ZMod m done · -- (←) assume h2 : invertible Y obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1 obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2 apply Exists.intro (Xi * Yi) show (X * Y) * (Xi * Yi) = [1]_m from calc X * Y * (Xi * Yi) _ = (X * Xi) * (Y * Yi) := by ring _ = [1]_m * [1]_m := by rw [h3, h4] _ = [1]_m := Theorem_7_3_6_7 [1]_m done done lemma F_rp_def {m i : Nat} (h : rel_prime m i) : F m i = [i]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h --h : gcd m i = 1 rewrite [if_pos h] at h1 show F m i = [i]_m from h1 done lemma F_not_rp_def {m i : Nat} (h : ¬rel_prime m i) : F m i = [1]_m := by have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl define at h rewrite [h1, if_neg h] rfl done lemma prod_seq_base {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 0 k f = [1]_m := by rfl lemma prod_seq_step {m : Nat} (n k : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl lemma prod_seq_zero_step {m : Nat} (n : Nat) (f : Nat → ZMod m) : prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by rewrite [prod_seq_step, zero_add] rfl done lemma prod_one {m : Nat} (k : Nat) (f : Nat → ZMod m) : prod_seq 1 k f = f k := by rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7] rfl done lemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl lemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m := calc [G m a i]_m _ = [(a * i) % m]_m := by rfl _ = [a * i]_m := (cc_eq_mod m (a * i)).symm _ = [a]_m * [i]_m := (mul_class m a i).symm lemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) : rel_prime m (G m a i) ↔ rel_prime m i := by have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1 show rel_prime m (G m a i) ↔ rel_prime m i from calc rel_prime m (G m a i) _ ↔ invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm _ ↔ invertible ([a]_m * [i]_m) := by rw [cc_G] _ ↔ invertible [i]_m := prod_inv_iff_inv h2 ([i]_m) _ ↔ rel_prime m i := Theorem_7_3_7 m i done lemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) : F m (G m a i) = [a]_m * F m i := by have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2 show F m (G m a i) = [a]_m * F m i from calc F m (G m a i) _ = [G m a i]_m := F_rp_def h3 _ = [a]_m * [i]_m := cc_G m a i _ = [a]_m * F m i := by rw [F_rp_def h2] done lemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : ¬rel_prime m i) : F m (G m a i) = [1]_m := by rewrite [←G_rp_iff h1 i] at h2 show F m (G m a i) = [1]_m from F_not_rp_def h2 done lemma FG_prod {m a : Nat} (h1 : rel_prime m a) : ∀ (k : Nat), prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by by_induc · -- Base Case show prod_seq 0 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from calc prod_seq 0 0 ((F m) ∘ (G m a)) _ = [1]_m := prod_seq_base _ _ _ = [a]_m ^ 0 * [1]_m := by ring _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by rw [num_rp_below_base, prod_seq_base] done · -- Induction Step fix k : Nat assume ih : prod_seq k 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) by_cases h2 : rel_prime m k · -- Case 1. h2 : rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([a]_m * F m k) := by rw [FG_rp h1 h2] _ = [a]_m ^ ((num_rp_below m k) + 1) * ((prod_seq k 0 (F m)) * F m k) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_rp h2, prod_seq_zero_step] done · -- Case 2. h2 : ¬rel_prime m k show prod_seq (k + 1) 0 ((F m) ∘ (G m a)) = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) from calc prod_seq (k + 1) 0 ((F m) ∘ (G m a)) _ = prod_seq k 0 ((F m) ∘ (G m a)) * F m (G m a k) := prod_seq_zero_step _ _ _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * F m (G m a k) := by rw [ih] _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) * ([1]_m) := by rw [FG_not_rp h1 h2] _ = [a]_m ^ (num_rp_below m k) * (prod_seq k 0 (F m) * ([1]_m)) := by ring _ = [a]_m ^ (num_rp_below m (k + 1)) * prod_seq (k + 1) 0 (F m) := by rw [num_rp_below_step_not_rp h2, prod_seq_zero_step, F_not_rp_def h2] done done done lemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by define --Goal : ∀ i < m, G m a i < m fix i : Nat assume h1 : i < m rewrite [G_def] --Goal : a * i % m < m show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m) done lemma left_inv_one_one_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g' (g i) = i) : one_one_below n g := sorry lemma right_inv_onto_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g (g' i) = i) (h2 : maps_below n g') : onto_below n g := by define at h2; define fix k : Nat assume h3 : k < n apply Exists.intro (g' k) show g' k < n ∧ g (g' k) = k from And.intro (h2 k h3) (h1 k h3) done lemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m * [inv_mod m a]_m = [1]_m := by have h2 : 0 ≤ (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a) show [a]_m * [inv_mod m a]_m = [1]_m from calc [a]_m * [inv_mod m a]_m _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2] _ = [a]_m * [gcd_c2 m a]_m := by rw [←cc_eq_mod] _ = [1]_m := gcd_c2_inv h1 done lemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m := calc a * (b % m) % m = a % m * (b % m % m) % m := Nat.mul_mod _ _ _ _ = a % m * (b % m) % m := by rw [Nat.mod_mod] _ = a * b % m := (Nat.mul_mod _ _ _).symm lemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm] rfl done theorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] : ↑a ≡ ↑b (MOD m) ↔ a % m = b % m := sorry lemma mul_inv_mod_cancel {m a i : Nat} [NeZero m] (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1 rewrite [mul_class, cc_eq_iff_congr, ←Nat.cast_mul, ←Nat.cast_one, congr_iff_mod_eq_Nat] at h3 show a * inv_mod m a * i % m = i from calc a * (inv_mod m a) * i % m _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = 1 % m * i % m := by rw [h3] _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod] _ = i % m := by rw [one_mul] _ = i := Nat.mod_eq_of_lt h2 done lemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl lemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, G m a (Ginv m a i) = i := by fix i : Nat assume h2 : i < m show G m a (Ginv m a i) = i from calc G m a (Ginv m a i) _ = a * ((inv_mod m a * i) % m) % m := by rfl _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) : ∀ i < m, Ginv m a (G m a i) = i := by fix i : Nat assume h2 : i < m show Ginv m a (G m a i) = i from calc Ginv m a (G m a i) _ = inv_mod m a * ((a * i) % m) % m := by rfl _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod] _ = a * inv_mod m a * i % m := by rw [←mul_assoc, mul_comm (inv_mod m a)] _ = i := mul_inv_mod_cancel h1 h2 done lemma Ginv_maps_below (m a : Nat) [NeZero m] : maps_below m (Ginv m a) := G_maps_below m (inv_mod m a) lemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : one_one_below m (G m a) := left_inv_one_one_below (Ginv_left_inv h1) lemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : onto_below m (G m a) := right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a) lemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) : perm_below m (G m a) := And.intro (G_maps_below m a) (And.intro (G_one_one_below h1) (G_onto_below h1)) --Permuting a product of congruence classes doesn't change product lemma swap_fst (u v : Nat) : swap u v u = v := by define : swap u v u --Goal : (if u = u then v else if u = v then u else u) = v have h : u = u := by rfl rewrite [if_pos h] rfl done lemma swap_snd (u v : Nat) : swap u v v = u := by define : swap u v v by_cases h1 : v = u · -- Case 1. h1 : v = u rewrite [if_pos h1] show v = u from h1 done · -- Case 2. h1 : v ≠ u rewrite [if_neg h1] have h2 : v = v := by rfl rewrite [if_pos h2] rfl done done lemma swap_other {u v i : Nat} (h1 : i ≠ u) (h2 : i ≠ v) : swap u v i = i := by define : swap u v i rewrite [if_neg h1, if_neg h2] rfl done lemma swap_values (u v i : Nat) : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := by by_cases h1 : i = u · -- Case 1. h1 : i = u apply Or.inl rewrite [h1] show swap u v u = v from swap_fst u v done · -- Case 2. h1 : i ≠ u apply Or.inr by_cases h2 : i = v · -- Case 2.1. h2 : i = v apply Or.inl rewrite [h2] show swap u v v = u from swap_snd u v done · -- Case 2.2. h2 : i ≠ v apply Or.inr show swap u v i = i from swap_other h1 h2 done done done lemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by define fix i : Nat assume h3 : i < n have h4 : swap u v i = v ∨ swap u v i = u ∨ swap u v i = i := swap_values u v i by_cases on h4 · -- Case 1. h4 : swap u v i = v rewrite [h4] show v < n from h2 done · -- Case 2. by_cases on h4 · -- Case 2.1. h4 : swap u v i = u rewrite [h4] show u < n from h1 done · -- Case 2.2. h4 : swap u v i = i rewrite [h4] show i < n from h3 done done done lemma swap_swap (u v n : Nat) : ∀ i < n, swap u v (swap u v i) = i := by fix i : Nat assume h : i < n by_cases h1 : i = u · -- Case 1. h1 : i = u rewrite [h1, swap_fst, swap_snd] rfl done · -- Case 2. h1 : i ≠ u by_cases h2 : i = v · -- Case 2.1. h2 : i = v rewrite [h2, swap_snd, swap_fst] rfl done · -- Case 2.2. h2 : i ≠ v rewrite [swap_other h1 h2, swap_other h1 h2] rfl done done done lemma swap_one_one_below (u v n) : one_one_below n (swap u v) := left_inv_one_one_below (swap_swap u v n) lemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) := right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2) lemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) := And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2)) lemma comp_perm_below {n : Nat} {f g : Nat → Nat} (h1 : perm_below n f) (h2 : perm_below n g) : perm_below n (f ∘ g) := sorry lemma trivial_swap (u : Nat) : swap u u = id := by apply funext fix x : Nat by_cases h1 : x = u · -- Case 1. h1 : x = u rewrite [h1, swap_fst] rfl done · -- Case 2. h1 : x ≠ u rewrite [swap_other h1 h1] rfl done done lemma prod_eq_fun {m : Nat} (f g : Nat → ZMod m) (k : Nat) : ∀ (n : Nat), (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g := by by_induc · -- Base Case assume h : (∀ i < 0, f (k + i) = g (k + i)) rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : (∀ i < n, f (k + i) = g (k + i)) → prod_seq n k f = prod_seq n k g assume h1 : ∀ i < n + 1, f (k + i) = g (k + i) have h2 : ∀ i < n, f (k + i) = g (k + i) := by fix i : Nat assume h2 : i < n have h3 : i < n + 1 := by linarith show f (k + i) = g (k + i) from h1 i h3 done have h3 : prod_seq n k f = prod_seq n k g := ih h2 have h4 : n < n + 1 := Nat.lt_succ_self n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4] rfl done done lemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := by have h2 : ∀ (i : Nat), i < u → (f ∘ swap u n) (0 + i) = f (0 + i) := by fix i : Nat assume h2 : i < u have h3 : 0 + i ≠ u := by linarith have h4 : 0 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl done show prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f from prod_eq_fun (f ∘ swap u n) f 0 u h2 done lemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := by have h2 : ∀ i < j, (f ∘ swap u n) (u + 1 + i) = f (u + 1 + i) := by fix i : Nat assume h2 : i < j have h3 : u + 1 + i ≠ u := by linarith have h4 : u + 1 + i ≠ n := by linarith rewrite [comp_def, swap_other h3 h4] rfl show prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f from prod_eq_fun (f ∘ swap u n) f (u + 1) j h2 done lemma break_prod {m : Nat} (n : Nat) (f : Nat → ZMod m) : ∀ (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by by_induc · -- Base Case have h : n + 0 = n := by rfl rewrite [prod_seq_base, h, Theorem_7_3_6_7] rfl done · -- Induction Step fix j : Nat assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f rewrite [←add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc] rfl done done lemma break_prod_twice {m u j n : Nat} (f : Nat → ZMod m) (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f := break_prod n f 1 rewrite [prod_one] at h2 have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f := break_prod (u + 1) f j rewrite [←h1] at h3 have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f := break_prod u f 1 rewrite [prod_one] at h4 rewrite [h3, h4] at h2 show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2 done lemma swap_prod_eq_prod {m u n : Nat} (f : Nat → ZMod m) (h1 : u ≤ n) : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f := by by_cases h2 : u = n · -- Case 1. h2 : u = n rewrite [h2, trivial_swap n] --Goal : prod_seq (n + 1) 0 (f ∘ id) = prod_seq (n + 1) 0 f rfl done · -- Case 2. h2 : ¬u = n have h3 : u + 1 ≤ n := Nat.lt_of_le_of_ne h1 h2 obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3 have break_f : prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := break_prod_twice f h4 have break_fs : prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_prod_twice (f ∘ swap u n) h4 have f_eq_fs_below : prod_seq u 0 (f ∘ swap u n) = prod_seq u 0 f := swap_prod_eq_prod_below f h1 have f_eq_fs_btwn : prod_seq j (u + 1) (f ∘ swap u n) = prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4 show prod_seq (n + 1) 0 (f ∘ swap u n) = prod_seq (n + 1) 0 f from calc prod_seq (n + 1) 0 (f ∘ swap u n) _ = prod_seq u 0 (f ∘ swap u n) * (f ∘ swap u n) u * prod_seq j (u + 1) (f ∘ swap u n) * (f ∘ swap u n) n := break_fs _ = prod_seq u 0 f * (f ∘ swap u n) u * prod_seq j (u + 1) f * (f ∘ swap u n) n := by rw [f_eq_fs_below, f_eq_fs_btwn] _ = prod_seq u 0 f * f (swap u n u) * prod_seq j (u + 1) f * f (swap u n n) := by rfl _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by rw [swap_fst, swap_snd] _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring _ = prod_seq (n + 1) 0 f := break_f.symm done done lemma perm_below_fixed {n : Nat} {g : Nat → Nat} (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry lemma perm_prod {m : Nat} (f : Nat → ZMod m) : ∀ (n : Nat), ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) := by by_induc · -- Base Case fix g : Nat → Nat assume h1 : perm_below 0 g rewrite [prod_seq_base, prod_seq_base] rfl done · -- Induction Step fix n : Nat assume ih : ∀ (g : Nat → Nat), perm_below n g → prod_seq n 0 f = prod_seq n 0 (f ∘ g) fix g : Nat → Nat assume g_pb : perm_below (n + 1) g define at g_pb have g_ob : onto_below (n + 1) g := g_pb.right.right define at g_ob have h1 : n < n + 1 := by linarith obtain (u : Nat) (h2 : u < n + 1 ∧ g u = n) from g_ob n h1 have s_pb : perm_below (n + 1) (swap u n) := swap_perm_below h2.left h1 have gs_pb_n1 : perm_below (n + 1) (g ∘ swap u n) := comp_perm_below g_pb s_pb have gs_fix_n : (g ∘ swap u n) n = n := calc (g ∘ swap u n) n _ = g (swap u n n) := by rfl _ = g u := by rw [swap_snd] _ = n := h2.right have gs_pb_n : perm_below n (g ∘ swap u n) := perm_below_fixed gs_pb_n1 gs_fix_n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f ∘ (g ∘ swap u n)) := ih (g ∘ swap u n) gs_pb_n have h3 : u ≤ n := by linarith show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f ∘ g) from calc prod_seq (n + 1) 0 f _ = prod_seq n 0 f * f n := prod_seq_zero_step n f _ = prod_seq n 0 (f ∘ (g ∘ swap u n)) * f ((g ∘ swap u n) n) := by rw [gs_prod, gs_fix_n] _ = prod_seq n 0 (f ∘ g ∘ swap u n) * (f ∘ g ∘ swap u n) n := by rfl _ = prod_seq (n + 1) 0 (f ∘ g ∘ swap u n) := (prod_seq_zero_step n (f ∘ g ∘ swap u n)).symm _ = prod_seq (n + 1) 0 ((f ∘ g) ∘ swap u n) := by rfl _ = prod_seq (n + 1) 0 (f ∘ g) := swap_prod_eq_prod (f ∘ g) h3 done done lemma F_invertible (m i : Nat) : invertible (F m i) := by by_cases h : rel_prime m i · -- Case 1. h : rel_prime m i rewrite [F_rp_def h] show invertible [i]_m from (Theorem_7_3_7 m i).rtl h done · -- Case 2. h : ¬rel_prime m i rewrite [F_not_rp_def h] apply Exists.intro [1]_m show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m done done lemma Fprod_invertible (m : Nat) : ∀ (k : Nat), invertible (prod_seq k 0 (F m)) := by by_induc · -- Base Case apply Exists.intro [1]_m show prod_seq 0 0 (F m) * [1]_m = [1]_m from calc prod_seq 0 0 (F m) * [1]_m _ = [1]_m * [1]_m := by rw [prod_seq_base] _ = [1]_m := Theorem_7_3_6_7 ([1]_m) done · -- Induction Step fix k : Nat assume ih : invertible (prod_seq k 0 (F m)) rewrite [prod_seq_zero_step] show invertible (prod_seq k 0 (F m) * (F m k)) from (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k) done done theorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) : [a]_m ^ (phi m) = [1]_m := by have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2 show [a]_m ^ (phi m) = [1]_m from calc [a]_m ^ (phi m) _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3] _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring _ = prod_seq m 0 (F m ∘ G m a) * Y := by rw [FG_prod h1 m, phi_def] _ = prod_seq m 0 (F m) * Y := by rw [perm_prod (F m) m (G m a) (G_perm_below h1)] _ = [1]_m := by rw [h3] done lemma Exercise_7_4_5_Int (m : Nat) (a : Int) : ∀ (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry lemma Exercise_7_4_5_Nat (m a n : Nat) : [a]_m ^ n = [a ^ n]_m := by rewrite [Exercise_7_4_5_Int] rfl done theorem Euler's_theorem {m a : Nat} [NeZero m] (h1 : rel_prime m a) : a ^ (phi m) ≡ 1 (MOD m) := by have h2 : [a]_m ^ (phi m) = [1]_m := Theorem_7_4_2 h1 rewrite [Exercise_7_4_5_Nat m a (phi m)] at h2 --h2 : [a ^ phi m]_m = [1]_m show a ^ (phi m) ≡ 1 (MOD m) from (cc_eq_iff_congr _ _ _).ltr h2 done #eval gcd 10 7 --Answer: 1. So 10 and 7 are relatively prime #eval 7 ^ phi 10 --Answer: 2401, which is congruent to 1 mod 10. end Euler /- Section 7.5 -/ lemma num_rp_prime {p : Nat} (h1 : prime p) : ∀ k < p, num_rp_below p (k + 1) = k := sorry lemma phi_prime {p : Nat} (h1 : prime p) : phi p = p - 1 := by have h2 : 1 ≤ p := prime_pos h1 have h3 : p - 1 + 1 = p := Nat.sub_add_cancel h2 have h4 : p - 1 < p := by linarith have h5 : num_rp_below p (p - 1 + 1) = p - 1 := num_rp_prime h1 (p - 1) h4 rewrite [h3] at h5 show phi p = p - 1 from h5 done theorem Theorem_7_2_2_Int {a c : Nat} {b : Int} (h1 : ↑c ∣ ↑a * b) (h2 : rel_prime a c) : ↑c ∣ b := by rewrite [Int.natCast_dvd, Int.natAbs_mul, Int.natAbs_ofNat] at h1 --h1 : c ∣ a * Int.natAbs b rewrite [Int.natCast_dvd] --Goal : c ∣ Int.natAbs b show c ∣ Int.natAbs b from Theorem_7_2_2 h1 h2 done lemma Lemma_7_4_5 {m n : Nat} (a b : Int) (h1 : rel_prime m n) : a ≡ b (MOD m * n) ↔ a ≡ b (MOD m) ∧ a ≡ b (MOD n) := by apply Iff.intro · -- (→) assume h2 : a ≡ b (MOD m * n) obtain (j : Int) (h3 : a - b = (m * n) * j) from h2 apply And.intro · -- Proof of a ≡ b (MOD m) apply Exists.intro (n * j) show a - b = m * (n * j) from calc a - b _ = m * n * j := h3 _ = m * (n * j) := by ring done · -- Proof of a ≡ b (MOD n) apply Exists.intro (m * j) show a - b = n * (m * j) from calc a - b _ = m * n * j := h3 _ = n * (m * j) := by ring done done · -- (←) assume h2 : a ≡ b (MOD m) ∧ a ≡ b (MOD n) obtain (j : Int) (h3 : a - b = m * j) from h2.left have h4 : (↑n : Int) ∣ a - b := h2.right rewrite [h3] at h4 --h4 : ↑n ∣ ↑m * j have h5 : ↑n ∣ j := Theorem_7_2_2_Int h4 h1 obtain (k : Int) (h6 : j = n * k) from h5 apply Exists.intro k --Goal : a - b = ↑(m * n) * k rewrite [Nat.cast_mul] --Goal : a - b = ↑m * ↑n * k show a - b = (m * n) * k from calc a - b _ = m * j := h3 _ = m * (n * k) := by rw [h6] _ = (m * n) * k := by ring done done --From exercises of Section 7.2 theorem rel_prime_symm {a b : Nat} (h : rel_prime a b) : rel_prime b a := sorry lemma prime_NeZero {p : Nat} (h : prime p) : NeZero p := by rewrite [neZero_iff] --Goal : p ≠ 0 define at h linarith done lemma Lemma_7_5_1 {p e d m c s : Nat} {t : Int} (h1 : prime p) (h2 : e * d = (p - 1) * s + 1) (h3 : m ^ e - c = p * t) : c ^ d ≡ m (MOD p) := by have h4 : m ^ e ≡ c (MOD p) := Exists.intro t h3 have h5 : [m ^ e]_p = [c]_p := (cc_eq_iff_congr _ _ _).rtl h4 rewrite [←Exercise_7_4_5_Nat] at h5 --h5 : [m]_p ^ e = [c]_p by_cases h6 : p ∣ m · -- Case 1. h6 : p ∣ m have h7 : m ≡ 0 (MOD p) := by obtain (j : Nat) (h8 : m = p * j) from h6 apply Exists.intro (↑j : Int) --Goal : ↑m - 0 = ↑p * ↑j rewrite [h8, Nat.cast_mul] ring done have h8 : [m]_p = [0]_p := (cc_eq_iff_congr _ _ _).rtl h7 have h9 : e * d ≠ 0 := by rewrite [h2] show (p - 1) * s + 1 ≠ 0 from Nat.add_one_ne_zero _ done have h10 : (0 : Int) ^ (e * d) = 0 := zero_pow h9 have h11 : [c ^ d]_p = [m]_p := calc [c ^ d]_p _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat] _ = ([m]_p ^ e) ^ d := by rw [h5] _ = [m]_p ^ (e * d) := by ring _ = [0]_p ^ (e * d) := by rw [h8] _ = [0 ^ (e * d)]_p := Exercise_7_4_5_Int _ _ _ _ = [0]_p := by rw [h10] _ = [m]_p := by rw [h8] show c ^ d ≡ m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11 done · -- Case 2. h6 : ¬p ∣ m have h7 : rel_prime m p := rel_prime_of_prime_not_dvd h1 h6 have h8 : rel_prime p m := rel_prime_symm h7 have h9 : NeZero p := prime_NeZero h1 have h10 : (1 : Int) ^ s = 1 := by ring have h11 : [c ^ d]_p = [m]_p := calc [c ^ d]_p _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat] _ = ([m]_p ^ e) ^ d := by rw [h5] _ = [m]_p ^ (e * d) := by ring _ = [m]_p ^ ((p - 1) * s + 1) := by rw [h2] _ = ([m]_p ^ (p - 1)) ^ s * [m]_p := by ring _ = ([m]_p ^ (phi p)) ^ s * [m]_p := by rw [phi_prime h1] _ = [1]_p ^ s * [m]_p := by rw [Theorem_7_4_2 h8] _ = [1 ^ s]_p * [m]_p := by rw [Exercise_7_4_5_Int] _ = [1]_p * [m]_p := by rw [h10] _ = [m]_p * [1]_p := by ring _ = [m]_p := Theorem_7_3_6_7 _ show c ^ d ≡ m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11 done done theorem Theorem_7_5_1 (p q n e d k m c : Nat) (p_prime : prime p) (q_prime : prime q) (p_ne_q : p ≠ q) (n_pq : n = p * q) (ed_congr_1 : e * d = k * (p - 1) * (q - 1) + 1) (h1 : [m]_n ^ e = [c]_n) : [c]_n ^ d = [m]_n := by rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] at h1 --h1 : m ^ e ≡ c (MOD n) rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] --Goal : c ^ d ≡ m (MOD n) obtain (j : Int) (h2 : m ^ e - c = n * j) from h1 rewrite [n_pq, Nat.cast_mul] at h2 --h2 : m ^ e - c = p * q * j have h3 : e * d = (p - 1) * (k * (q - 1)) + 1 := by rewrite [ed_congr_1] ring done have h4 : m ^ e - c = p * (q * j) := by rewrite [h2] ring done have congr_p : c ^ d ≡ m (MOD p) := Lemma_7_5_1 p_prime h3 h4 have h5 : e * d = (q - 1) * (k * (p - 1)) + 1 := by rewrite [ed_congr_1] ring done have h6 : m ^ e - c = q * (p * j) := by rewrite [h2] ring done have congr_q : c ^ d ≡ m (MOD q) := Lemma_7_5_1 q_prime h5 h6 have h7 : ¬q ∣ p := by by_contra h8 have h9 : q = 1 ∨ q = p := dvd_prime p_prime h8 disj_syll h9 (prime_not_one q_prime) show False from p_ne_q h9.symm done have h8 : rel_prime p q := rel_prime_of_prime_not_dvd q_prime h7 rewrite [n_pq, Lemma_7_4_5 _ _ h8] show c ^ d ≡ m (MOD p) ∧ c ^ d ≡ m (MOD q) from And.intro congr_p congr_q done /- BEGIN EXERCISES -/ namespace Exercises /- Section 7.1 -/ -- 1. theorem dvd_a_of_dvd_b_mod {a b d : Nat} (h1 : d ∣ b) (h2 : d ∣ (a % b)) : d ∣ a := sorry -- 2. lemma gcd_comm_lt {a b : Nat} (h : a < b) : gcd a b = gcd b a := sorry theorem gcd_comm (a b : Nat) : gcd a b = gcd b a := sorry -- 3. theorem Exercise_7_1_5 (a b : Nat) (n : Int) : (∃ (s t : Int), s * a + t * b = n) ↔ (↑(gcd a b) : Int) ∣ n := sorry -- 4. theorem Exercise_7_1_6 (a b c : Nat) : gcd a b = gcd (a + b * c) b := sorry -- 5. theorem gcd_is_nonzero {a b : Nat} (h : a ≠ 0 ∨ b ≠ 0) : gcd a b ≠ 0 := sorry -- 6. theorem gcd_greatest {a b d : Nat} (h1 : gcd a b ≠ 0) (h2 : d ∣ a) (h3 : d ∣ b) : d ≤ gcd a b := sorry -- 7. lemma Lemma_7_1_10a {a b : Nat} (n : Nat) (h : a ∣ b) : (n * a) ∣ (n * b) := sorry lemma Lemma_7_1_10b {a b n : Nat} (h1 : n ≠ 0) (h2 : (n * a) ∣ (n * b)) : a ∣ b := sorry lemma Lemma_7_1_10c {a b : Nat} (h1 : a ∣ b) (h2 : b ∣ a) : a = b := sorry theorem Exercise_7_1_10 (a b n : Nat) : gcd (n * a) (n * b) = n * gcd a b := sorry /- Section 7.2 -/ -- 1. lemma dvd_prime {a p : Nat} (h1 : prime p) (h2 : a ∣ p) : a = 1 ∨ a = p := sorry -- 2. -- Hints: Start with apply List.rec. You may find mul_ne_zero useful theorem prod_nonzero_nonzero : ∀ (l : List Nat), (∀ a ∈ l, a ≠ 0) → prod l ≠ 0 := sorry -- 3. theorem rel_prime_iff_no_common_factor (a b : Nat) : rel_prime a b ↔ ¬∃ (p : Nat), prime p ∧ p ∣ a ∧ p ∣ b := sorry -- 4. theorem rel_prime_symm {a b : Nat} (h : rel_prime a b) : rel_prime b a := sorry -- 5. lemma in_prime_factorization_iff_prime_factor {a : Nat} {l : List Nat} (h1 : prime_factorization a l) (p : Nat) : p ∈ l ↔ prime_factor p a := sorry -- 6. theorem Exercise_7_2_5 {a b : Nat} {l m : List Nat} (h1 : prime_factorization a l) (h2 : prime_factorization b m) : rel_prime a b ↔ (¬∃ (p : Nat), p ∈ l ∧ p ∈ m) := sorry -- 7. theorem Exercise_7_2_6 (a b : Nat) : rel_prime a b ↔ ∃ (s t : Int), s * a + t * b = 1 := sorry -- 8. theorem Exercise_7_2_7 {a b a' b' : Nat} (h1 : rel_prime a b) (h2 : a' ∣ a) (h3 : b' ∣ b) : rel_prime a' b' := sorry -- 9. theorem Exercise_7_2_9 {a b j k : Nat} (h1 : gcd a b ≠ 0) (h2 : a = j * gcd a b) (h3 : b = k * gcd a b) : rel_prime j k := sorry -- 10. theorem Exercise_7_2_17a (a b c : Nat) : gcd a (b * c) ∣ gcd a b * gcd a c := sorry /- Section 7.3 -/ -- 1. theorem congr_trans {m : Nat} : ∀ {a b c : Int}, a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m) := sorry -- 2. theorem Theorem_7_3_6_3 {m : Nat} (X : ZMod m) : X + [0]_m = X := sorry -- 3. theorem Theorem_7_3_6_4 {m : Nat} (X : ZMod m) : ∃ (Y : ZMod m), X + Y = [0]_m := sorry -- 4. theorem Exercise_7_3_4a {m : Nat} (Z1 Z2 : ZMod m) (h1 : ∀ (X : ZMod m), X + Z1 = X) (h2 : ∀ (X : ZMod m), X + Z2 = X) : Z1 = Z2 := sorry -- 5. theorem Exercise_7_3_4b {m : Nat} (X Y1 Y2 : ZMod m) (h1 : X + Y1 = [0]_m) (h2 : X + Y2 = [0]_m) : Y1 = Y2 := sorry -- 6. theorem Theorem_7_3_10 (m a : Nat) (b : Int) : ¬(↑(gcd m a) : Int) ∣ b → ¬∃ (x : Int), a * x ≡ b (MOD m) := sorry -- 7. theorem Theorem_7_3_11 (m n : Nat) (a b : Int) (h1 : n ≠ 0) : n * a ≡ n * b (MOD n * m) ↔ a ≡ b (MOD m) := sorry -- 8. theorem Exercise_7_3_16 {m : Nat} {a b : Int} (h : a ≡ b (MOD m)) : ∀ (n : Nat), a ^ n ≡ b ^ n (MOD m) := sorry -- 9. example {m : Nat} [NeZero m] (X : ZMod m) : ∃! (a : Int), 0 ≤ a ∧ a < m ∧ X = [a]_m := sorry -- 10. theorem congr_rel_prime {m a b : Nat} (h1 : a ≡ b (MOD m)) : rel_prime m a ↔ rel_prime m b := sorry -- 11. --Hint: You may find the theorem Int.ofNat_mod_ofNat useful. theorem rel_prime_mod (m a : Nat) : rel_prime m (a % m) ↔ rel_prime m a := sorry -- 12. lemma congr_iff_mod_eq_Int (m : Nat) (a b : Int) [NeZero m] : a ≡ b (MOD m) ↔ a % ↑m = b % ↑m := sorry --Hint for next theorem: Use the lemma above, --together with the theorems Int.ofNat_mod_ofNat and Nat.cast_inj. theorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] : ↑a ≡ ↑b (MOD m) ↔ a % m = b % m := sorry /- Section 7.4 -/ -- 1. --Hint: Use induction. --For the base case, compute [a]_m ^ 0 * [1]_m in two ways: --by Theorem_7_3_6_7, [a] ^ 0 * [1]_m = [a]_m ^ 0 --by ring, [a]_m ^ 0 * [1]_m = [1]_m. lemma Exercise_7_4_5_Int (m : Nat) (a : Int) : ∀ (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry -- 2. lemma left_inv_one_one_below {n : Nat} {g g' : Nat → Nat} (h1 : ∀ i < n, g' (g i) = i) : one_one_below n g := sorry -- 3. lemma comp_perm_below {n : Nat} {f g : Nat → Nat} (h1 : perm_below n f) (h2 : perm_below n g) : perm_below n (f ∘ g) := sorry -- 4. lemma perm_below_fixed {n : Nat} {g : Nat → Nat} (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry -- 5. lemma Lemma_7_4_6 {a b c : Nat} : rel_prime (a * b) c ↔ rel_prime a c ∧ rel_prime b c := sorry -- 6. example {m a : Nat} [NeZero m] (h1 : rel_prime m a) : a ^ (phi m + 1) ≡ a (MOD m) := sorry -- 7. theorem Like_Exercise_7_4_11 {m a p : Nat} [NeZero m] (h1 : rel_prime m a) (h2 : p + 1 = phi m) : [a]_m * [a ^ p]_m = [1]_m := sorry -- 8. theorem Like_Exercise_7_4_12 {m a p q k : Nat} [NeZero m] (h1 : rel_prime m a) (h2 : p = q + (phi m) * k) : a ^ p ≡ a ^ q (MOD m) := sorry /- Section 7.5 -/ -- 1. --Hint: Use induction. lemma num_rp_prime {p : Nat} (h1 : prime p) : ∀ k < p, num_rp_below p (k + 1) = k := sorry -- 2. lemma three_prime : prime 3 := sorry -- 3. --Hint: Use the previous exercise, Exercise_7_2_7, and Theorem_7_4_2. theorem Exercise_7_5_13a (a : Nat) (h1 : rel_prime 561 a) : a ^ 560 ≡ 1 (MOD 3) := sorry -- 4. --Hint: Imitate the way Theorem_7_2_2_Int was proven from Theorem_7_2_2. lemma Theorem_7_2_3_Int {p : Nat} {a b : Int} (h1 : prime p) (h2 : ↑p ∣ a * b) : ↑p ∣ a ∨ ↑p ∣ b := sorry -- 5. --Hint: Use the previous exercise.

theorem Exercise_7_5_14b (n : Nat) (b : Int) (h1 : prime n) (h2 : b ^ 2 ≡ 1 (MOD n)) : b ≡ 1 (MOD n) ∨ b ≡ -1 (MOD n)

HTPI.Exercises.Exercise_7_5_14b

htpi/HTPILib/Chap7.lean

htpi

/- Copyright (c) 2019 Johan Commelin. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Johan Commelin, Kenny Lau -/ import Mathlib.Algebra.Polynomial.AlgebraMap import Mathlib.Algebra.Polynomial.Basic import Mathlib.RingTheory.MvPowerSeries.Basic #align_import ring_theory.power_series.basic from "leanprover-community/mathlib"@"2d5739b61641ee4e7e53eca5688a08f66f2e6a60" /-! # Formal power series (in one variable) This file defines (univariate) formal power series and develops the basic properties of these objects. A formal power series is to a polynomial like an infinite sum is to a finite sum. Formal power series in one variable are defined from multivariate power series as `PowerSeries R := MvPowerSeries Unit R`. The file sets up the (semi)ring structure on univariate power series. We provide the natural inclusion from polynomials to formal power series. Additional results can be found in: * `Mathlib.RingTheory.PowerSeries.Trunc`, truncation of power series; * `Mathlib.RingTheory.PowerSeries.Inverse`, about inverses of power series, and the fact that power series over a local ring form a local ring; * `Mathlib.RingTheory.PowerSeries.Order`, the order of a power series at 0, and application to the fact that power series over an integral domain form an integral domain. ## Implementation notes Because of its definition, `PowerSeries R := MvPowerSeries Unit R`. a lot of proofs and properties from the multivariate case can be ported to the single variable case. However, it means that formal power series are indexed by `Unit →₀ ℕ`, which is of course canonically isomorphic to `ℕ`. We then build some glue to treat formal power series as if they were indexed by `ℕ`. Occasionally this leads to proofs that are uglier than expected. -/ noncomputable section open BigOperators open Finset (antidiagonal mem_antidiagonal) /-- Formal power series over a coefficient type `R` -/ def PowerSeries (R : Type*) := MvPowerSeries Unit R #align power_series PowerSeries namespace PowerSeries open Finsupp (single) variable {R : Type*} section -- Porting note: not available in Lean 4 -- local reducible PowerSeries /-- `R⟦X⟧` is notation for `PowerSeries R`, the semiring of formal power series in one variable over a semiring `R`. -/ scoped notation:9000 R "⟦X⟧" => PowerSeries R instance [Inhabited R] : Inhabited R⟦X⟧ := by dsimp only [PowerSeries] infer_instance instance [Zero R] : Zero R⟦X⟧ := by dsimp only [PowerSeries] infer_instance instance [AddMonoid R] : AddMonoid R⟦X⟧ := by dsimp only [PowerSeries] infer_instance instance [AddGroup R] : AddGroup R⟦X⟧ := by dsimp only [PowerSeries] infer_instance instance [AddCommMonoid R] : AddCommMonoid R⟦X⟧ := by dsimp only [PowerSeries] infer_instance instance [AddCommGroup R] : AddCommGroup R⟦X⟧ := by dsimp only [PowerSeries] infer_instance instance [Semiring R] : Semiring R⟦X⟧ := by dsimp only [PowerSeries] infer_instance instance [CommSemiring R] : CommSemiring R⟦X⟧ := by dsimp only [PowerSeries] infer_instance instance [Ring R] : Ring R⟦X⟧ := by dsimp only [PowerSeries] infer_instance instance [CommRing R] : CommRing R⟦X⟧ := by dsimp only [PowerSeries] infer_instance instance [Nontrivial R] : Nontrivial R⟦X⟧ := by dsimp only [PowerSeries] infer_instance instance {A} [Semiring R] [AddCommMonoid A] [Module R A] : Module R A⟦X⟧ := by dsimp only [PowerSeries] infer_instance instance {A S} [Semiring R] [Semiring S] [AddCommMonoid A] [Module R A] [Module S A] [SMul R S] [IsScalarTower R S A] : IsScalarTower R S A⟦X⟧ := Pi.isScalarTower instance {A} [Semiring A] [CommSemiring R] [Algebra R A] : Algebra R A⟦X⟧ := by dsimp only [PowerSeries] infer_instance end section Semiring variable (R) [Semiring R] /-- The `n`th coefficient of a formal power series. -/ def coeff (n : ℕ) : R⟦X⟧ →ₗ[R] R := MvPowerSeries.coeff R (single () n) #align power_series.coeff PowerSeries.coeff /-- The `n`th monomial with coefficient `a` as formal power series. -/ def monomial (n : ℕ) : R →ₗ[R] R⟦X⟧ := MvPowerSeries.monomial R (single () n) #align power_series.monomial PowerSeries.monomial variable {R} theorem coeff_def {s : Unit →₀ ℕ} {n : ℕ} (h : s () = n) : coeff R n = MvPowerSeries.coeff R s := by erw [coeff, ← h, ← Finsupp.unique_single s] #align power_series.coeff_def PowerSeries.coeff_def /-- Two formal power series are equal if all their coefficients are equal. -/ @[ext] theorem ext {φ ψ : R⟦X⟧} (h : ∀ n, coeff R n φ = coeff R n ψ) : φ = ψ := MvPowerSeries.ext fun n => by rw [← coeff_def] · apply h rfl #align power_series.ext PowerSeries.ext /-- Two formal power series are equal if all their coefficients are equal. -/ theorem ext_iff {φ ψ : R⟦X⟧} : φ = ψ ↔ ∀ n, coeff R n φ = coeff R n ψ := ⟨fun h n => congr_arg (coeff R n) h, ext⟩ #align power_series.ext_iff PowerSeries.ext_iff instance [Subsingleton R] : Subsingleton R⟦X⟧ := by simp only [subsingleton_iff, ext_iff] exact fun _ _ _ ↦ (subsingleton_iff).mp (by infer_instance) _ _ /-- Constructor for formal power series. -/ def mk {R} (f : ℕ → R) : R⟦X⟧ := fun s => f (s ()) #align power_series.mk PowerSeries.mk @[simp] theorem coeff_mk (n : ℕ) (f : ℕ → R) : coeff R n (mk f) = f n := congr_arg f Finsupp.single_eq_same #align power_series.coeff_mk PowerSeries.coeff_mk theorem coeff_monomial (m n : ℕ) (a : R) : coeff R m (monomial R n a) = if m = n then a else 0 := calc coeff R m (monomial R n a) = _ := MvPowerSeries.coeff_monomial _ _ _ _ = if m = n then a else 0 := by simp only [Finsupp.unique_single_eq_iff] #align power_series.coeff_monomial PowerSeries.coeff_monomial theorem monomial_eq_mk (n : ℕ) (a : R) : monomial R n a = mk fun m => if m = n then a else 0 := ext fun m => by rw [coeff_monomial, coeff_mk] #align power_series.monomial_eq_mk PowerSeries.monomial_eq_mk @[simp] theorem coeff_monomial_same (n : ℕ) (a : R) : coeff R n (monomial R n a) = a := MvPowerSeries.coeff_monomial_same _ _ #align power_series.coeff_monomial_same PowerSeries.coeff_monomial_same @[simp] theorem coeff_comp_monomial (n : ℕ) : (coeff R n).comp (monomial R n) = LinearMap.id := LinearMap.ext <| coeff_monomial_same n #align power_series.coeff_comp_monomial PowerSeries.coeff_comp_monomial variable (R) /-- The constant coefficient of a formal power series. -/ def constantCoeff : R⟦X⟧ →+* R := MvPowerSeries.constantCoeff Unit R #align power_series.constant_coeff PowerSeries.constantCoeff /-- The constant formal power series. -/ def C : R →+* R⟦X⟧ := MvPowerSeries.C Unit R set_option linter.uppercaseLean3 false in #align power_series.C PowerSeries.C variable {R} /-- The variable of the formal power series ring. -/ def X : R⟦X⟧ := MvPowerSeries.X () set_option linter.uppercaseLean3 false in #align power_series.X PowerSeries.X theorem commute_X (φ : R⟦X⟧) : Commute φ X := MvPowerSeries.commute_X _ _ set_option linter.uppercaseLean3 false in #align power_series.commute_X PowerSeries.commute_X @[simp] theorem coeff_zero_eq_constantCoeff : ⇑(coeff R 0) = constantCoeff R := by rw [coeff, Finsupp.single_zero] rfl #align power_series.coeff_zero_eq_constant_coeff PowerSeries.coeff_zero_eq_constantCoeff theorem coeff_zero_eq_constantCoeff_apply (φ : R⟦X⟧) : coeff R 0 φ = constantCoeff R φ := by rw [coeff_zero_eq_constantCoeff] #align power_series.coeff_zero_eq_constant_coeff_apply PowerSeries.coeff_zero_eq_constantCoeff_apply @[simp] theorem monomial_zero_eq_C : ⇑(monomial R 0) = C R := by -- This used to be `rw`, but we need `erw` after leanprover/lean4#2644 erw [monomial, Finsupp.single_zero, MvPowerSeries.monomial_zero_eq_C] set_option linter.uppercaseLean3 false in #align power_series.monomial_zero_eq_C PowerSeries.monomial_zero_eq_C theorem monomial_zero_eq_C_apply (a : R) : monomial R 0 a = C R a := by simp set_option linter.uppercaseLean3 false in #align power_series.monomial_zero_eq_C_apply PowerSeries.monomial_zero_eq_C_apply theorem coeff_C (n : ℕ) (a : R) : coeff R n (C R a : R⟦X⟧) = if n = 0 then a else 0 := by rw [← monomial_zero_eq_C_apply, coeff_monomial] set_option linter.uppercaseLean3 false in #align power_series.coeff_C PowerSeries.coeff_C @[simp] theorem coeff_zero_C (a : R) : coeff R 0 (C R a) = a := by rw [coeff_C, if_pos rfl] set_option linter.uppercaseLean3 false in #align power_series.coeff_zero_C PowerSeries.coeff_zero_C theorem coeff_ne_zero_C {a : R} {n : ℕ} (h : n ≠ 0) : coeff R n (C R a) = 0 := by rw [coeff_C, if_neg h] @[simp] theorem coeff_succ_C {a : R} {n : ℕ} : coeff R (n + 1) (C R a) = 0 := coeff_ne_zero_C n.succ_ne_zero

theorem C_injective : Function.Injective (C R)

C_injective

2019

85a4719

mathlib4/Mathlib/RingTheory/PowerSeries/Basic.lean

mathlib

/- Copyright (c) 2019 Johan Commelin. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Johan Commelin, Kenny Lau -/ import Mathlib.Algebra.Polynomial.AlgebraMap import Mathlib.Algebra.Polynomial.Basic import Mathlib.RingTheory.MvPowerSeries.Basic #align_import ring_theory.power_series.basic from "leanprover-community/mathlib"@"2d5739b61641ee4e7e53eca5688a08f66f2e6a60" /-! # Formal power series (in one variable) This file defines (univariate) formal power series and develops the basic properties of these objects. A formal power series is to a polynomial like an infinite sum is to a finite sum. Formal power series in one variable are defined from multivariate power series as `PowerSeries R := MvPowerSeries Unit R`. The file sets up the (semi)ring structure on univariate power series. We provide the natural inclusion from polynomials to formal power series. Additional results can be found in: * `Mathlib.RingTheory.PowerSeries.Trunc`, truncation of power series; * `Mathlib.RingTheory.PowerSeries.Inverse`, about inverses of power series, and the fact that power series over a local ring form a local ring; * `Mathlib.RingTheory.PowerSeries.Order`, the order of a power series at 0, and application to the fact that power series over an integral domain form an integral domain. ## Implementation notes Because of its definition, `PowerSeries R := MvPowerSeries Unit R`. a lot of proofs and properties from the multivariate case can be ported to the single variable case. However, it means that formal power series are indexed by `Unit →₀ ℕ`, which is of course canonically isomorphic to `ℕ`. We then build some glue to treat formal power series as if they were indexed by `ℕ`. Occasionally this leads to proofs that are uglier than expected. -/ noncomputable section open BigOperators open Finset (antidiagonal mem_antidiagonal) /-- Formal power series over a coefficient type `R` -/ def PowerSeries (R : Type*) := MvPowerSeries Unit R #align power_series PowerSeries namespace PowerSeries open Finsupp (single) variable {R : Type*} section -- Porting note: not available in Lean 4 -- local reducible PowerSeries /-- `R⟦X⟧` is notation for `PowerSeries R`, the semiring of formal power series in one variable over a semiring `R`. -/ scoped notation:9000 R "⟦X⟧" => PowerSeries R instance [Inhabited R] : Inhabited R⟦X⟧ := by dsimp only [PowerSeries] infer_instance instance [Zero R] : Zero R⟦X⟧ := by dsimp only [PowerSeries] infer_instance instance [AddMonoid R] : AddMonoid R⟦X⟧ := by dsimp only [PowerSeries] infer_instance instance [AddGroup R] : AddGroup R⟦X⟧ := by dsimp only [PowerSeries] infer_instance instance [AddCommMonoid R] : AddCommMonoid R⟦X⟧ := by dsimp only [PowerSeries] infer_instance instance [AddCommGroup R] : AddCommGroup R⟦X⟧ := by dsimp only [PowerSeries] infer_instance instance [Semiring R] : Semiring R⟦X⟧ := by dsimp only [PowerSeries] infer_instance instance [CommSemiring R] : CommSemiring R⟦X⟧ := by dsimp only [PowerSeries] infer_instance instance [Ring R] : Ring R⟦X⟧ := by dsimp only [PowerSeries] infer_instance instance [CommRing R] : CommRing R⟦X⟧ := by dsimp only [PowerSeries] infer_instance instance [Nontrivial R] : Nontrivial R⟦X⟧ := by dsimp only [PowerSeries] infer_instance instance {A} [Semiring R] [AddCommMonoid A] [Module R A] : Module R A⟦X⟧ := by dsimp only [PowerSeries] infer_instance instance {A S} [Semiring R] [Semiring S] [AddCommMonoid A] [Module R A] [Module S A] [SMul R S] [IsScalarTower R S A] : IsScalarTower R S A⟦X⟧ := Pi.isScalarTower instance {A} [Semiring A] [CommSemiring R] [Algebra R A] : Algebra R A⟦X⟧ := by dsimp only [PowerSeries] infer_instance end section Semiring variable (R) [Semiring R] /-- The `n`th coefficient of a formal power series. -/ def coeff (n : ℕ) : R⟦X⟧ →ₗ[R] R := MvPowerSeries.coeff R (single () n) #align power_series.coeff PowerSeries.coeff /-- The `n`th monomial with coefficient `a` as formal power series. -/ def monomial (n : ℕ) : R →ₗ[R] R⟦X⟧ := MvPowerSeries.monomial R (single () n) #align power_series.monomial PowerSeries.monomial variable {R} theorem coeff_def {s : Unit →₀ ℕ} {n : ℕ} (h : s () = n) : coeff R n = MvPowerSeries.coeff R s := by erw [coeff, ← h, ← Finsupp.unique_single s] #align power_series.coeff_def PowerSeries.coeff_def /-- Two formal power series are equal if all their coefficients are equal. -/ @[ext] theorem ext {φ ψ : R⟦X⟧} (h : ∀ n, coeff R n φ = coeff R n ψ) : φ = ψ := MvPowerSeries.ext fun n => by rw [← coeff_def] · apply h rfl #align power_series.ext PowerSeries.ext /-- Two formal power series are equal if all their coefficients are equal. -/ theorem ext_iff {φ ψ : R⟦X⟧} : φ = ψ ↔ ∀ n, coeff R n φ = coeff R n ψ := ⟨fun h n => congr_arg (coeff R n) h, ext⟩ #align power_series.ext_iff PowerSeries.ext_iff instance [Subsingleton R] : Subsingleton R⟦X⟧ := by simp only [subsingleton_iff, ext_iff] exact fun _ _ _ ↦ (subsingleton_iff).mp (by infer_instance) _ _ /-- Constructor for formal power series. -/ def mk {R} (f : ℕ → R) : R⟦X⟧ := fun s => f (s ()) #align power_series.mk PowerSeries.mk @[simp] theorem coeff_mk (n : ℕ) (f : ℕ → R) : coeff R n (mk f) = f n := congr_arg f Finsupp.single_eq_same #align power_series.coeff_mk PowerSeries.coeff_mk theorem coeff_monomial (m n : ℕ) (a : R) : coeff R m (monomial R n a) = if m = n then a else 0 := calc coeff R m (monomial R n a) = _ := MvPowerSeries.coeff_monomial _ _ _ _ = if m = n then a else 0 := by simp only [Finsupp.unique_single_eq_iff] #align power_series.coeff_monomial PowerSeries.coeff_monomial theorem monomial_eq_mk (n : ℕ) (a : R) : monomial R n a = mk fun m => if m = n then a else 0 := ext fun m => by rw [coeff_monomial, coeff_mk] #align power_series.monomial_eq_mk PowerSeries.monomial_eq_mk @[simp] theorem coeff_monomial_same (n : ℕ) (a : R) : coeff R n (monomial R n a) = a := MvPowerSeries.coeff_monomial_same _ _ #align power_series.coeff_monomial_same PowerSeries.coeff_monomial_same @[simp] theorem coeff_comp_monomial (n : ℕ) : (coeff R n).comp (monomial R n) = LinearMap.id := LinearMap.ext <| coeff_monomial_same n #align power_series.coeff_comp_monomial PowerSeries.coeff_comp_monomial variable (R) /-- The constant coefficient of a formal power series. -/ def constantCoeff : R⟦X⟧ →+* R := MvPowerSeries.constantCoeff Unit R #align power_series.constant_coeff PowerSeries.constantCoeff /-- The constant formal power series. -/ def C : R →+* R⟦X⟧ := MvPowerSeries.C Unit R set_option linter.uppercaseLean3 false in #align power_series.C PowerSeries.C variable {R} /-- The variable of the formal power series ring. -/ def X : R⟦X⟧ := MvPowerSeries.X () set_option linter.uppercaseLean3 false in #align power_series.X PowerSeries.X theorem commute_X (φ : R⟦X⟧) : Commute φ X := MvPowerSeries.commute_X _ _ set_option linter.uppercaseLean3 false in #align power_series.commute_X PowerSeries.commute_X @[simp] theorem coeff_zero_eq_constantCoeff : ⇑(coeff R 0) = constantCoeff R := by rw [coeff, Finsupp.single_zero] rfl #align power_series.coeff_zero_eq_constant_coeff PowerSeries.coeff_zero_eq_constantCoeff theorem coeff_zero_eq_constantCoeff_apply (φ : R⟦X⟧) : coeff R 0 φ = constantCoeff R φ := by rw [coeff_zero_eq_constantCoeff] #align power_series.coeff_zero_eq_constant_coeff_apply PowerSeries.coeff_zero_eq_constantCoeff_apply @[simp] theorem monomial_zero_eq_C : ⇑(monomial R 0) = C R := by -- This used to be `rw`, but we need `erw` after leanprover/lean4#2644 erw [monomial, Finsupp.single_zero, MvPowerSeries.monomial_zero_eq_C] set_option linter.uppercaseLean3 false in #align power_series.monomial_zero_eq_C PowerSeries.monomial_zero_eq_C theorem monomial_zero_eq_C_apply (a : R) : monomial R 0 a = C R a := by simp set_option linter.uppercaseLean3 false in #align power_series.monomial_zero_eq_C_apply PowerSeries.monomial_zero_eq_C_apply theorem coeff_C (n : ℕ) (a : R) : coeff R n (C R a : R⟦X⟧) = if n = 0 then a else 0 := by rw [← monomial_zero_eq_C_apply, coeff_monomial] set_option linter.uppercaseLean3 false in #align power_series.coeff_C PowerSeries.coeff_C @[simp] theorem coeff_zero_C (a : R) : coeff R 0 (C R a) = a := by rw [coeff_C, if_pos rfl] set_option linter.uppercaseLean3 false in #align power_series.coeff_zero_C PowerSeries.coeff_zero_C theorem coeff_ne_zero_C {a : R} {n : ℕ} (h : n ≠ 0) : coeff R n (C R a) = 0 := by rw [coeff_C, if_neg h] @[simp] theorem coeff_succ_C {a : R} {n : ℕ} : coeff R (n + 1) (C R a) = 0 := coeff_ne_zero_C n.succ_ne_zero theorem C_injective : Function.Injective (C R) := by intro a b H have := (ext_iff (φ := C R a) (ψ := C R b)).mp H 0 rwa [coeff_zero_C, coeff_zero_C] at this

protected theorem subsingleton_iff : Subsingleton R⟦X⟧ ↔ Subsingleton R

subsingleton_iff

2019

85a4719

mathlib4/Mathlib/RingTheory/PowerSeries/Basic.lean

mathlib

/- Copyright (c) 2019 Johan Commelin. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Johan Commelin, Kenny Lau -/ import Mathlib.Algebra.Polynomial.AlgebraMap import Mathlib.Algebra.Polynomial.Basic import Mathlib.RingTheory.MvPowerSeries.Basic #align_import ring_theory.power_series.basic from "leanprover-community/mathlib"@"2d5739b61641ee4e7e53eca5688a08f66f2e6a60" /-! # Formal power series (in one variable) This file defines (univariate) formal power series and develops the basic properties of these objects. A formal power series is to a polynomial like an infinite sum is to a finite sum. Formal power series in one variable are defined from multivariate power series as `PowerSeries R := MvPowerSeries Unit R`. The file sets up the (semi)ring structure on univariate power series. We provide the natural inclusion from polynomials to formal power series. Additional results can be found in: * `Mathlib.RingTheory.PowerSeries.Trunc`, truncation of power series; * `Mathlib.RingTheory.PowerSeries.Inverse`, about inverses of power series, and the fact that power series over a local ring form a local ring; * `Mathlib.RingTheory.PowerSeries.Order`, the order of a power series at 0, and application to the fact that power series over an integral domain form an integral domain. ## Implementation notes Because of its definition, `PowerSeries R := MvPowerSeries Unit R`. a lot of proofs and properties from the multivariate case can be ported to the single variable case. However, it means that formal power series are indexed by `Unit →₀ ℕ`, which is of course canonically isomorphic to `ℕ`. We then build some glue to treat formal power series as if they were indexed by `ℕ`. Occasionally this leads to proofs that are uglier than expected. -/ noncomputable section open BigOperators open Finset (antidiagonal mem_antidiagonal) /-- Formal power series over a coefficient type `R` -/ def PowerSeries (R : Type*) := MvPowerSeries Unit R #align power_series PowerSeries namespace PowerSeries open Finsupp (single) variable {R : Type*} section -- Porting note: not available in Lean 4 -- local reducible PowerSeries /-- `R⟦X⟧` is notation for `PowerSeries R`, the semiring of formal power series in one variable over a semiring `R`. -/ scoped notation:9000 R "⟦X⟧" => PowerSeries R instance [Inhabited R] : Inhabited R⟦X⟧ := by dsimp only [PowerSeries] infer_instance instance [Zero R] : Zero R⟦X⟧ := by dsimp only [PowerSeries] infer_instance instance [AddMonoid R] : AddMonoid R⟦X⟧ := by dsimp only [PowerSeries] infer_instance instance [AddGroup R] : AddGroup R⟦X⟧ := by dsimp only [PowerSeries] infer_instance instance [AddCommMonoid R] : AddCommMonoid R⟦X⟧ := by dsimp only [PowerSeries] infer_instance instance [AddCommGroup R] : AddCommGroup R⟦X⟧ := by dsimp only [PowerSeries] infer_instance instance [Semiring R] : Semiring R⟦X⟧ := by dsimp only [PowerSeries] infer_instance instance [CommSemiring R] : CommSemiring R⟦X⟧ := by dsimp only [PowerSeries] infer_instance instance [Ring R] : Ring R⟦X⟧ := by dsimp only [PowerSeries] infer_instance instance [CommRing R] : CommRing R⟦X⟧ := by dsimp only [PowerSeries] infer_instance instance [Nontrivial R] : Nontrivial R⟦X⟧ := by dsimp only [PowerSeries] infer_instance instance {A} [Semiring R] [AddCommMonoid A] [Module R A] : Module R A⟦X⟧ := by dsimp only [PowerSeries] infer_instance instance {A S} [Semiring R] [Semiring S] [AddCommMonoid A] [Module R A] [Module S A] [SMul R S] [IsScalarTower R S A] : IsScalarTower R S A⟦X⟧ := Pi.isScalarTower instance {A} [Semiring A] [CommSemiring R] [Algebra R A] : Algebra R A⟦X⟧ := by dsimp only [PowerSeries] infer_instance end section Semiring variable (R) [Semiring R] /-- The `n`th coefficient of a formal power series. -/ def coeff (n : ℕ) : R⟦X⟧ →ₗ[R] R := MvPowerSeries.coeff R (single () n) #align power_series.coeff PowerSeries.coeff /-- The `n`th monomial with coefficient `a` as formal power series. -/ def monomial (n : ℕ) : R →ₗ[R] R⟦X⟧ := MvPowerSeries.monomial R (single () n) #align power_series.monomial PowerSeries.monomial variable {R} theorem coeff_def {s : Unit →₀ ℕ} {n : ℕ} (h : s () = n) : coeff R n = MvPowerSeries.coeff R s := by erw [coeff, ← h, ← Finsupp.unique_single s] #align power_series.coeff_def PowerSeries.coeff_def /-- Two formal power series are equal if all their coefficients are equal. -/ @[ext] theorem ext {φ ψ : R⟦X⟧} (h : ∀ n, coeff R n φ = coeff R n ψ) : φ = ψ := MvPowerSeries.ext fun n => by rw [← coeff_def] · apply h rfl #align power_series.ext PowerSeries.ext /-- Two formal power series are equal if all their coefficients are equal. -/ theorem ext_iff {φ ψ : R⟦X⟧} : φ = ψ ↔ ∀ n, coeff R n φ = coeff R n ψ := ⟨fun h n => congr_arg (coeff R n) h, ext⟩ #align power_series.ext_iff PowerSeries.ext_iff instance [Subsingleton R] : Subsingleton R⟦X⟧ := by simp only [subsingleton_iff, ext_iff] exact fun _ _ _ ↦ (subsingleton_iff).mp (by infer_instance) _ _ /-- Constructor for formal power series. -/ def mk {R} (f : ℕ → R) : R⟦X⟧ := fun s => f (s ()) #align power_series.mk PowerSeries.mk @[simp] theorem coeff_mk (n : ℕ) (f : ℕ → R) : coeff R n (mk f) = f n := congr_arg f Finsupp.single_eq_same #align power_series.coeff_mk PowerSeries.coeff_mk theorem coeff_monomial (m n : ℕ) (a : R) : coeff R m (monomial R n a) = if m = n then a else 0 := calc coeff R m (monomial R n a) = _ := MvPowerSeries.coeff_monomial _ _ _ _ = if m = n then a else 0 := by simp only [Finsupp.unique_single_eq_iff] #align power_series.coeff_monomial PowerSeries.coeff_monomial theorem monomial_eq_mk (n : ℕ) (a : R) : monomial R n a = mk fun m => if m = n then a else 0 := ext fun m => by rw [coeff_monomial, coeff_mk] #align power_series.monomial_eq_mk PowerSeries.monomial_eq_mk @[simp] theorem coeff_monomial_same (n : ℕ) (a : R) : coeff R n (monomial R n a) = a := MvPowerSeries.coeff_monomial_same _ _ #align power_series.coeff_monomial_same PowerSeries.coeff_monomial_same @[simp] theorem coeff_comp_monomial (n : ℕ) : (coeff R n).comp (monomial R n) = LinearMap.id := LinearMap.ext <| coeff_monomial_same n #align power_series.coeff_comp_monomial PowerSeries.coeff_comp_monomial variable (R) /-- The constant coefficient of a formal power series. -/ def constantCoeff : R⟦X⟧ →+* R := MvPowerSeries.constantCoeff Unit R #align power_series.constant_coeff PowerSeries.constantCoeff /-- The constant formal power series. -/ def C : R →+* R⟦X⟧ := MvPowerSeries.C Unit R set_option linter.uppercaseLean3 false in #align power_series.C PowerSeries.C variable {R} /-- The variable of the formal power series ring. -/ def X : R⟦X⟧ := MvPowerSeries.X () set_option linter.uppercaseLean3 false in #align power_series.X PowerSeries.X theorem commute_X (φ : R⟦X⟧) : Commute φ X := MvPowerSeries.commute_X _ _ set_option linter.uppercaseLean3 false in #align power_series.commute_X PowerSeries.commute_X @[simp] theorem coeff_zero_eq_constantCoeff : ⇑(coeff R 0) = constantCoeff R := by rw [coeff, Finsupp.single_zero] rfl #align power_series.coeff_zero_eq_constant_coeff PowerSeries.coeff_zero_eq_constantCoeff theorem coeff_zero_eq_constantCoeff_apply (φ : R⟦X⟧) : coeff R 0 φ = constantCoeff R φ := by rw [coeff_zero_eq_constantCoeff] #align power_series.coeff_zero_eq_constant_coeff_apply PowerSeries.coeff_zero_eq_constantCoeff_apply @[simp] theorem monomial_zero_eq_C : ⇑(monomial R 0) = C R := by -- This used to be `rw`, but we need `erw` after leanprover/lean4#2644 erw [monomial, Finsupp.single_zero, MvPowerSeries.monomial_zero_eq_C] set_option linter.uppercaseLean3 false in #align power_series.monomial_zero_eq_C PowerSeries.monomial_zero_eq_C theorem monomial_zero_eq_C_apply (a : R) : monomial R 0 a = C R a := by simp set_option linter.uppercaseLean3 false in #align power_series.monomial_zero_eq_C_apply PowerSeries.monomial_zero_eq_C_apply theorem coeff_C (n : ℕ) (a : R) : coeff R n (C R a : R⟦X⟧) = if n = 0 then a else 0 := by rw [← monomial_zero_eq_C_apply, coeff_monomial] set_option linter.uppercaseLean3 false in #align power_series.coeff_C PowerSeries.coeff_C @[simp] theorem coeff_zero_C (a : R) : coeff R 0 (C R a) = a := by rw [coeff_C, if_pos rfl] set_option linter.uppercaseLean3 false in #align power_series.coeff_zero_C PowerSeries.coeff_zero_C theorem coeff_ne_zero_C {a : R} {n : ℕ} (h : n ≠ 0) : coeff R n (C R a) = 0 := by rw [coeff_C, if_neg h] @[simp] theorem coeff_succ_C {a : R} {n : ℕ} : coeff R (n + 1) (C R a) = 0 := coeff_ne_zero_C n.succ_ne_zero theorem C_injective : Function.Injective (C R) := by intro a b H have := (ext_iff (φ := C R a) (ψ := C R b)).mp H 0 rwa [coeff_zero_C, coeff_zero_C] at this protected theorem subsingleton_iff : Subsingleton R⟦X⟧ ↔ Subsingleton R := by refine ⟨fun h ↦ ?_, fun _ ↦ inferInstance⟩ rw [subsingleton_iff] at h ⊢ exact fun a b ↦ C_injective (h (C R a) (C R b)) theorem X_eq : (X : R⟦X⟧) = monomial R 1 1 := rfl set_option linter.uppercaseLean3 false in #align power_series.X_eq PowerSeries.X_eq theorem coeff_X (n : ℕ) : coeff R n (X : R⟦X⟧) = if n = 1 then 1 else 0 := by rw [X_eq, coeff_monomial] set_option linter.uppercaseLean3 false in #align power_series.coeff_X PowerSeries.coeff_X @[simp] theorem coeff_zero_X : coeff R 0 (X : R⟦X⟧) = 0 := by -- This used to be `rw`, but we need `erw` after leanprover/lean4#2644 erw [coeff, Finsupp.single_zero, X, MvPowerSeries.coeff_zero_X] set_option linter.uppercaseLean3 false in #align power_series.coeff_zero_X PowerSeries.coeff_zero_X @[simp] theorem coeff_one_X : coeff R 1 (X : R⟦X⟧) = 1 := by rw [coeff_X, if_pos rfl] set_option linter.uppercaseLean3 false in #align power_series.coeff_one_X PowerSeries.coeff_one_X @[simp] theorem X_ne_zero [Nontrivial R] : (X : R⟦X⟧) ≠ 0 := fun H => by simpa only [coeff_one_X, one_ne_zero, map_zero] using congr_arg (coeff R 1) H set_option linter.uppercaseLean3 false in #align power_series.X_ne_zero PowerSeries.X_ne_zero theorem X_pow_eq (n : ℕ) : (X : R⟦X⟧) ^ n = monomial R n 1 := MvPowerSeries.X_pow_eq _ n set_option linter.uppercaseLean3 false in #align power_series.X_pow_eq PowerSeries.X_pow_eq theorem coeff_X_pow (m n : ℕ) : coeff R m ((X : R⟦X⟧) ^ n) = if m = n then 1 else 0 := by rw [X_pow_eq, coeff_monomial] set_option linter.uppercaseLean3 false in #align power_series.coeff_X_pow PowerSeries.coeff_X_pow @[simp] theorem coeff_X_pow_self (n : ℕ) : coeff R n ((X : R⟦X⟧) ^ n) = 1 := by rw [coeff_X_pow, if_pos rfl] set_option linter.uppercaseLean3 false in #align power_series.coeff_X_pow_self PowerSeries.coeff_X_pow_self @[simp] theorem coeff_one (n : ℕ) : coeff R n (1 : R⟦X⟧) = if n = 0 then 1 else 0 := coeff_C n 1 #align power_series.coeff_one PowerSeries.coeff_one theorem coeff_zero_one : coeff R 0 (1 : R⟦X⟧) = 1 := coeff_zero_C 1 #align power_series.coeff_zero_one PowerSeries.coeff_zero_one theorem coeff_mul (n : ℕ) (φ ψ : R⟦X⟧) : coeff R n (φ * ψ) = ∑ p in antidiagonal n, coeff R p.1 φ * coeff R p.2 ψ := by -- `rw` can't see that `PowerSeries = MvPowerSeries Unit`, so use `.trans` refine (MvPowerSeries.coeff_mul _ φ ψ).trans ?_ rw [Finsupp.antidiagonal_single, Finset.sum_map] rfl #align power_series.coeff_mul PowerSeries.coeff_mul @[simp] theorem coeff_mul_C (n : ℕ) (φ : R⟦X⟧) (a : R) : coeff R n (φ * C R a) = coeff R n φ * a := MvPowerSeries.coeff_mul_C _ φ a set_option linter.uppercaseLean3 false in #align power_series.coeff_mul_C PowerSeries.coeff_mul_C @[simp] theorem coeff_C_mul (n : ℕ) (φ : R⟦X⟧) (a : R) : coeff R n (C R a * φ) = a * coeff R n φ := MvPowerSeries.coeff_C_mul _ φ a set_option linter.uppercaseLean3 false in #align power_series.coeff_C_mul PowerSeries.coeff_C_mul @[simp] theorem coeff_smul {S : Type*} [Semiring S] [Module R S] (n : ℕ) (φ : PowerSeries S) (a : R) : coeff S n (a • φ) = a • coeff S n φ := rfl #align power_series.coeff_smul PowerSeries.coeff_smul theorem smul_eq_C_mul (f : R⟦X⟧) (a : R) : a • f = C R a * f := by ext simp set_option linter.uppercaseLean3 false in #align power_series.smul_eq_C_mul PowerSeries.smul_eq_C_mul @[simp] theorem coeff_succ_mul_X (n : ℕ) (φ : R⟦X⟧) : coeff R (n + 1) (φ * X) = coeff R n φ := by simp only [coeff, Finsupp.single_add] convert φ.coeff_add_mul_monomial (single () n) (single () 1) _ rw [mul_one]; rfl set_option linter.uppercaseLean3 false in #align power_series.coeff_succ_mul_X PowerSeries.coeff_succ_mul_X @[simp] theorem coeff_succ_X_mul (n : ℕ) (φ : R⟦X⟧) : coeff R (n + 1) (X * φ) = coeff R n φ := by simp only [coeff, Finsupp.single_add, add_comm n 1] convert φ.coeff_add_monomial_mul (single () 1) (single () n) _ rw [one_mul]; rfl set_option linter.uppercaseLean3 false in #align power_series.coeff_succ_X_mul PowerSeries.coeff_succ_X_mul @[simp] theorem constantCoeff_C (a : R) : constantCoeff R (C R a) = a := rfl set_option linter.uppercaseLean3 false in #align power_series.constant_coeff_C PowerSeries.constantCoeff_C @[simp] theorem constantCoeff_comp_C : (constantCoeff R).comp (C R) = RingHom.id R := rfl set_option linter.uppercaseLean3 false in #align power_series.constant_coeff_comp_C PowerSeries.constantCoeff_comp_C -- Porting note (#10618): simp can prove this. -- @[simp] theorem constantCoeff_zero : constantCoeff R 0 = 0 := rfl #align power_series.constant_coeff_zero PowerSeries.constantCoeff_zero -- Porting note (#10618): simp can prove this. -- @[simp] theorem constantCoeff_one : constantCoeff R 1 = 1 := rfl #align power_series.constant_coeff_one PowerSeries.constantCoeff_one @[simp] theorem constantCoeff_X : constantCoeff R X = 0 := MvPowerSeries.coeff_zero_X _ set_option linter.uppercaseLean3 false in #align power_series.constant_coeff_X PowerSeries.constantCoeff_X theorem coeff_zero_mul_X (φ : R⟦X⟧) : coeff R 0 (φ * X) = 0 := by simp set_option linter.uppercaseLean3 false in #align power_series.coeff_zero_mul_X PowerSeries.coeff_zero_mul_X theorem coeff_zero_X_mul (φ : R⟦X⟧) : coeff R 0 (X * φ) = 0 := by simp set_option linter.uppercaseLean3 false in #align power_series.coeff_zero_X_mul PowerSeries.coeff_zero_X_mul

theorem constantCoeff_surj : Function.Surjective (constantCoeff R)

constantCoeff_surj

2019

85a4719

mathlib4/Mathlib/RingTheory/PowerSeries/Basic.lean

mathlib

/- Copyright (c) 2019 Johan Commelin. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Johan Commelin, Kenny Lau -/ import Mathlib.Algebra.Polynomial.AlgebraMap import Mathlib.Algebra.Polynomial.Basic import Mathlib.RingTheory.MvPowerSeries.Basic #align_import ring_theory.power_series.basic from "leanprover-community/mathlib"@"2d5739b61641ee4e7e53eca5688a08f66f2e6a60" /-! # Formal power series (in one variable) This file defines (univariate) formal power series and develops the basic properties of these objects. A formal power series is to a polynomial like an infinite sum is to a finite sum. Formal power series in one variable are defined from multivariate power series as `PowerSeries R := MvPowerSeries Unit R`. The file sets up the (semi)ring structure on univariate power series. We provide the natural inclusion from polynomials to formal power series. Additional results can be found in: * `Mathlib.RingTheory.PowerSeries.Trunc`, truncation of power series; * `Mathlib.RingTheory.PowerSeries.Inverse`, about inverses of power series, and the fact that power series over a local ring form a local ring; * `Mathlib.RingTheory.PowerSeries.Order`, the order of a power series at 0, and application to the fact that power series over an integral domain form an integral domain. ## Implementation notes Because of its definition, `PowerSeries R := MvPowerSeries Unit R`. a lot of proofs and properties from the multivariate case can be ported to the single variable case. However, it means that formal power series are indexed by `Unit →₀ ℕ`, which is of course canonically isomorphic to `ℕ`. We then build some glue to treat formal power series as if they were indexed by `ℕ`. Occasionally this leads to proofs that are uglier than expected. -/ noncomputable section open BigOperators open Finset (antidiagonal mem_antidiagonal) /-- Formal power series over a coefficient type `R` -/ def PowerSeries (R : Type*) := MvPowerSeries Unit R #align power_series PowerSeries namespace PowerSeries open Finsupp (single) variable {R : Type*} section -- Porting note: not available in Lean 4 -- local reducible PowerSeries /-- `R⟦X⟧` is notation for `PowerSeries R`, the semiring of formal power series in one variable over a semiring `R`. -/ scoped notation:9000 R "⟦X⟧" => PowerSeries R instance [Inhabited R] : Inhabited R⟦X⟧ := by dsimp only [PowerSeries] infer_instance instance [Zero R] : Zero R⟦X⟧ := by dsimp only [PowerSeries] infer_instance instance [AddMonoid R] : AddMonoid R⟦X⟧ := by dsimp only [PowerSeries] infer_instance instance [AddGroup R] : AddGroup R⟦X⟧ := by dsimp only [PowerSeries] infer_instance instance [AddCommMonoid R] : AddCommMonoid R⟦X⟧ := by dsimp only [PowerSeries] infer_instance instance [AddCommGroup R] : AddCommGroup R⟦X⟧ := by dsimp only [PowerSeries] infer_instance instance [Semiring R] : Semiring R⟦X⟧ := by dsimp only [PowerSeries] infer_instance instance [CommSemiring R] : CommSemiring R⟦X⟧ := by dsimp only [PowerSeries] infer_instance instance [Ring R] : Ring R⟦X⟧ := by dsimp only [PowerSeries] infer_instance instance [CommRing R] : CommRing R⟦X⟧ := by dsimp only [PowerSeries] infer_instance instance [Nontrivial R] : Nontrivial R⟦X⟧ := by dsimp only [PowerSeries] infer_instance instance {A} [Semiring R] [AddCommMonoid A] [Module R A] : Module R A⟦X⟧ := by dsimp only [PowerSeries] infer_instance instance {A S} [Semiring R] [Semiring S] [AddCommMonoid A] [Module R A] [Module S A] [SMul R S] [IsScalarTower R S A] : IsScalarTower R S A⟦X⟧ := Pi.isScalarTower instance {A} [Semiring A] [CommSemiring R] [Algebra R A] : Algebra R A⟦X⟧ := by dsimp only [PowerSeries] infer_instance end section Semiring variable (R) [Semiring R] /-- The `n`th coefficient of a formal power series. -/ def coeff (n : ℕ) : R⟦X⟧ →ₗ[R] R := MvPowerSeries.coeff R (single () n) #align power_series.coeff PowerSeries.coeff /-- The `n`th monomial with coefficient `a` as formal power series. -/ def monomial (n : ℕ) : R →ₗ[R] R⟦X⟧ := MvPowerSeries.monomial R (single () n) #align power_series.monomial PowerSeries.monomial variable {R} theorem coeff_def {s : Unit →₀ ℕ} {n : ℕ} (h : s () = n) : coeff R n = MvPowerSeries.coeff R s := by erw [coeff, ← h, ← Finsupp.unique_single s] #align power_series.coeff_def PowerSeries.coeff_def /-- Two formal power series are equal if all their coefficients are equal. -/ @[ext] theorem ext {φ ψ : R⟦X⟧} (h : ∀ n, coeff R n φ = coeff R n ψ) : φ = ψ := MvPowerSeries.ext fun n => by rw [← coeff_def] · apply h rfl #align power_series.ext PowerSeries.ext /-- Two formal power series are equal if all their coefficients are equal. -/ theorem ext_iff {φ ψ : R⟦X⟧} : φ = ψ ↔ ∀ n, coeff R n φ = coeff R n ψ := ⟨fun h n => congr_arg (coeff R n) h, ext⟩ #align power_series.ext_iff PowerSeries.ext_iff instance [Subsingleton R] : Subsingleton R⟦X⟧ := by simp only [subsingleton_iff, ext_iff] exact fun _ _ _ ↦ (subsingleton_iff).mp (by infer_instance) _ _ /-- Constructor for formal power series. -/ def mk {R} (f : ℕ → R) : R⟦X⟧ := fun s => f (s ()) #align power_series.mk PowerSeries.mk @[simp] theorem coeff_mk (n : ℕ) (f : ℕ → R) : coeff R n (mk f) = f n := congr_arg f Finsupp.single_eq_same #align power_series.coeff_mk PowerSeries.coeff_mk theorem coeff_monomial (m n : ℕ) (a : R) : coeff R m (monomial R n a) = if m = n then a else 0 := calc coeff R m (monomial R n a) = _ := MvPowerSeries.coeff_monomial _ _ _ _ = if m = n then a else 0 := by simp only [Finsupp.unique_single_eq_iff] #align power_series.coeff_monomial PowerSeries.coeff_monomial theorem monomial_eq_mk (n : ℕ) (a : R) : monomial R n a = mk fun m => if m = n then a else 0 := ext fun m => by rw [coeff_monomial, coeff_mk] #align power_series.monomial_eq_mk PowerSeries.monomial_eq_mk @[simp] theorem coeff_monomial_same (n : ℕ) (a : R) : coeff R n (monomial R n a) = a := MvPowerSeries.coeff_monomial_same _ _ #align power_series.coeff_monomial_same PowerSeries.coeff_monomial_same @[simp] theorem coeff_comp_monomial (n : ℕ) : (coeff R n).comp (monomial R n) = LinearMap.id := LinearMap.ext <| coeff_monomial_same n #align power_series.coeff_comp_monomial PowerSeries.coeff_comp_monomial variable (R) /-- The constant coefficient of a formal power series. -/ def constantCoeff : R⟦X⟧ →+* R := MvPowerSeries.constantCoeff Unit R #align power_series.constant_coeff PowerSeries.constantCoeff /-- The constant formal power series. -/ def C : R →+* R⟦X⟧ := MvPowerSeries.C Unit R set_option linter.uppercaseLean3 false in #align power_series.C PowerSeries.C variable {R} /-- The variable of the formal power series ring. -/ def X : R⟦X⟧ := MvPowerSeries.X () set_option linter.uppercaseLean3 false in #align power_series.X PowerSeries.X theorem commute_X (φ : R⟦X⟧) : Commute φ X := MvPowerSeries.commute_X _ _ set_option linter.uppercaseLean3 false in #align power_series.commute_X PowerSeries.commute_X @[simp] theorem coeff_zero_eq_constantCoeff : ⇑(coeff R 0) = constantCoeff R := by rw [coeff, Finsupp.single_zero] rfl #align power_series.coeff_zero_eq_constant_coeff PowerSeries.coeff_zero_eq_constantCoeff theorem coeff_zero_eq_constantCoeff_apply (φ : R⟦X⟧) : coeff R 0 φ = constantCoeff R φ := by rw [coeff_zero_eq_constantCoeff] #align power_series.coeff_zero_eq_constant_coeff_apply PowerSeries.coeff_zero_eq_constantCoeff_apply @[simp] theorem monomial_zero_eq_C : ⇑(monomial R 0) = C R := by -- This used to be `rw`, but we need `erw` after leanprover/lean4#2644 erw [monomial, Finsupp.single_zero, MvPowerSeries.monomial_zero_eq_C] set_option linter.uppercaseLean3 false in #align power_series.monomial_zero_eq_C PowerSeries.monomial_zero_eq_C theorem monomial_zero_eq_C_apply (a : R) : monomial R 0 a = C R a := by simp set_option linter.uppercaseLean3 false in #align power_series.monomial_zero_eq_C_apply PowerSeries.monomial_zero_eq_C_apply theorem coeff_C (n : ℕ) (a : R) : coeff R n (C R a : R⟦X⟧) = if n = 0 then a else 0 := by rw [← monomial_zero_eq_C_apply, coeff_monomial] set_option linter.uppercaseLean3 false in #align power_series.coeff_C PowerSeries.coeff_C @[simp] theorem coeff_zero_C (a : R) : coeff R 0 (C R a) = a := by rw [coeff_C, if_pos rfl] set_option linter.uppercaseLean3 false in #align power_series.coeff_zero_C PowerSeries.coeff_zero_C theorem coeff_ne_zero_C {a : R} {n : ℕ} (h : n ≠ 0) : coeff R n (C R a) = 0 := by rw [coeff_C, if_neg h] @[simp] theorem coeff_succ_C {a : R} {n : ℕ} : coeff R (n + 1) (C R a) = 0 := coeff_ne_zero_C n.succ_ne_zero theorem C_injective : Function.Injective (C R) := by intro a b H have := (ext_iff (φ := C R a) (ψ := C R b)).mp H 0 rwa [coeff_zero_C, coeff_zero_C] at this protected theorem subsingleton_iff : Subsingleton R⟦X⟧ ↔ Subsingleton R := by refine ⟨fun h ↦ ?_, fun _ ↦ inferInstance⟩ rw [subsingleton_iff] at h ⊢ exact fun a b ↦ C_injective (h (C R a) (C R b)) theorem X_eq : (X : R⟦X⟧) = monomial R 1 1 := rfl set_option linter.uppercaseLean3 false in #align power_series.X_eq PowerSeries.X_eq theorem coeff_X (n : ℕ) : coeff R n (X : R⟦X⟧) = if n = 1 then 1 else 0 := by rw [X_eq, coeff_monomial] set_option linter.uppercaseLean3 false in #align power_series.coeff_X PowerSeries.coeff_X @[simp] theorem coeff_zero_X : coeff R 0 (X : R⟦X⟧) = 0 := by -- This used to be `rw`, but we need `erw` after leanprover/lean4#2644 erw [coeff, Finsupp.single_zero, X, MvPowerSeries.coeff_zero_X] set_option linter.uppercaseLean3 false in #align power_series.coeff_zero_X PowerSeries.coeff_zero_X @[simp] theorem coeff_one_X : coeff R 1 (X : R⟦X⟧) = 1 := by rw [coeff_X, if_pos rfl] set_option linter.uppercaseLean3 false in #align power_series.coeff_one_X PowerSeries.coeff_one_X @[simp] theorem X_ne_zero [Nontrivial R] : (X : R⟦X⟧) ≠ 0 := fun H => by simpa only [coeff_one_X, one_ne_zero, map_zero] using congr_arg (coeff R 1) H set_option linter.uppercaseLean3 false in #align power_series.X_ne_zero PowerSeries.X_ne_zero theorem X_pow_eq (n : ℕ) : (X : R⟦X⟧) ^ n = monomial R n 1 := MvPowerSeries.X_pow_eq _ n set_option linter.uppercaseLean3 false in #align power_series.X_pow_eq PowerSeries.X_pow_eq theorem coeff_X_pow (m n : ℕ) : coeff R m ((X : R⟦X⟧) ^ n) = if m = n then 1 else 0 := by rw [X_pow_eq, coeff_monomial] set_option linter.uppercaseLean3 false in #align power_series.coeff_X_pow PowerSeries.coeff_X_pow @[simp] theorem coeff_X_pow_self (n : ℕ) : coeff R n ((X : R⟦X⟧) ^ n) = 1 := by rw [coeff_X_pow, if_pos rfl] set_option linter.uppercaseLean3 false in #align power_series.coeff_X_pow_self PowerSeries.coeff_X_pow_self @[simp] theorem coeff_one (n : ℕ) : coeff R n (1 : R⟦X⟧) = if n = 0 then 1 else 0 := coeff_C n 1 #align power_series.coeff_one PowerSeries.coeff_one theorem coeff_zero_one : coeff R 0 (1 : R⟦X⟧) = 1 := coeff_zero_C 1 #align power_series.coeff_zero_one PowerSeries.coeff_zero_one theorem coeff_mul (n : ℕ) (φ ψ : R⟦X⟧) : coeff R n (φ * ψ) = ∑ p in antidiagonal n, coeff R p.1 φ * coeff R p.2 ψ := by -- `rw` can't see that `PowerSeries = MvPowerSeries Unit`, so use `.trans` refine (MvPowerSeries.coeff_mul _ φ ψ).trans ?_ rw [Finsupp.antidiagonal_single, Finset.sum_map] rfl #align power_series.coeff_mul PowerSeries.coeff_mul @[simp] theorem coeff_mul_C (n : ℕ) (φ : R⟦X⟧) (a : R) : coeff R n (φ * C R a) = coeff R n φ * a := MvPowerSeries.coeff_mul_C _ φ a set_option linter.uppercaseLean3 false in #align power_series.coeff_mul_C PowerSeries.coeff_mul_C @[simp] theorem coeff_C_mul (n : ℕ) (φ : R⟦X⟧) (a : R) : coeff R n (C R a * φ) = a * coeff R n φ := MvPowerSeries.coeff_C_mul _ φ a set_option linter.uppercaseLean3 false in #align power_series.coeff_C_mul PowerSeries.coeff_C_mul @[simp] theorem coeff_smul {S : Type*} [Semiring S] [Module R S] (n : ℕ) (φ : PowerSeries S) (a : R) : coeff S n (a • φ) = a • coeff S n φ := rfl #align power_series.coeff_smul PowerSeries.coeff_smul theorem smul_eq_C_mul (f : R⟦X⟧) (a : R) : a • f = C R a * f := by ext simp set_option linter.uppercaseLean3 false in #align power_series.smul_eq_C_mul PowerSeries.smul_eq_C_mul @[simp] theorem coeff_succ_mul_X (n : ℕ) (φ : R⟦X⟧) : coeff R (n + 1) (φ * X) = coeff R n φ := by simp only [coeff, Finsupp.single_add] convert φ.coeff_add_mul_monomial (single () n) (single () 1) _ rw [mul_one]; rfl set_option linter.uppercaseLean3 false in #align power_series.coeff_succ_mul_X PowerSeries.coeff_succ_mul_X @[simp] theorem coeff_succ_X_mul (n : ℕ) (φ : R⟦X⟧) : coeff R (n + 1) (X * φ) = coeff R n φ := by simp only [coeff, Finsupp.single_add, add_comm n 1] convert φ.coeff_add_monomial_mul (single () 1) (single () n) _ rw [one_mul]; rfl set_option linter.uppercaseLean3 false in #align power_series.coeff_succ_X_mul PowerSeries.coeff_succ_X_mul @[simp] theorem constantCoeff_C (a : R) : constantCoeff R (C R a) = a := rfl set_option linter.uppercaseLean3 false in #align power_series.constant_coeff_C PowerSeries.constantCoeff_C @[simp] theorem constantCoeff_comp_C : (constantCoeff R).comp (C R) = RingHom.id R := rfl set_option linter.uppercaseLean3 false in #align power_series.constant_coeff_comp_C PowerSeries.constantCoeff_comp_C -- Porting note (#10618): simp can prove this. -- @[simp] theorem constantCoeff_zero : constantCoeff R 0 = 0 := rfl #align power_series.constant_coeff_zero PowerSeries.constantCoeff_zero -- Porting note (#10618): simp can prove this. -- @[simp] theorem constantCoeff_one : constantCoeff R 1 = 1 := rfl #align power_series.constant_coeff_one PowerSeries.constantCoeff_one @[simp] theorem constantCoeff_X : constantCoeff R X = 0 := MvPowerSeries.coeff_zero_X _ set_option linter.uppercaseLean3 false in #align power_series.constant_coeff_X PowerSeries.constantCoeff_X theorem coeff_zero_mul_X (φ : R⟦X⟧) : coeff R 0 (φ * X) = 0 := by simp set_option linter.uppercaseLean3 false in #align power_series.coeff_zero_mul_X PowerSeries.coeff_zero_mul_X theorem coeff_zero_X_mul (φ : R⟦X⟧) : coeff R 0 (X * φ) = 0 := by simp set_option linter.uppercaseLean3 false in #align power_series.coeff_zero_X_mul PowerSeries.coeff_zero_X_mul theorem constantCoeff_surj : Function.Surjective (constantCoeff R) := fun r => ⟨(C R) r, constantCoeff_C r⟩ -- The following section duplicates the API of `Data.Polynomial.Coeff` and should attempt to keep -- up to date with that section theorem coeff_C_mul_X_pow (x : R) (k n : ℕ) : coeff R n (C R x * X ^ k : R⟦X⟧) = if n = k then x else 0 := by simp [X_pow_eq, coeff_monomial] set_option linter.uppercaseLean3 false in #align power_series.coeff_C_mul_X_pow PowerSeries.coeff_C_mul_X_pow @[simp] theorem coeff_mul_X_pow (p : R⟦X⟧) (n d : ℕ) : coeff R (d + n) (p * X ^ n) = coeff R d p := by rw [coeff_mul, Finset.sum_eq_single (d, n), coeff_X_pow, if_pos rfl, mul_one] · rintro ⟨i, j⟩ h1 h2 rw [coeff_X_pow, if_neg, mul_zero] rintro rfl apply h2 rw [mem_antidiagonal, add_right_cancel_iff] at h1 subst h1 rfl · exact fun h1 => (h1 (mem_antidiagonal.2 rfl)).elim set_option linter.uppercaseLean3 false in #align power_series.coeff_mul_X_pow PowerSeries.coeff_mul_X_pow @[simp] theorem coeff_X_pow_mul (p : R⟦X⟧) (n d : ℕ) : coeff R (d + n) (X ^ n * p) = coeff R d p := by rw [coeff_mul, Finset.sum_eq_single (n, d), coeff_X_pow, if_pos rfl, one_mul] · rintro ⟨i, j⟩ h1 h2 rw [coeff_X_pow, if_neg, zero_mul] rintro rfl apply h2 rw [mem_antidiagonal, add_comm, add_right_cancel_iff] at h1 subst h1 rfl · rw [add_comm] exact fun h1 => (h1 (mem_antidiagonal.2 rfl)).elim set_option linter.uppercaseLean3 false in #align power_series.coeff_X_pow_mul PowerSeries.coeff_X_pow_mul theorem coeff_mul_X_pow' (p : R⟦X⟧) (n d : ℕ) : coeff R d (p * X ^ n) = ite (n ≤ d) (coeff R (d - n) p) 0 := by split_ifs with h · rw [← tsub_add_cancel_of_le h, coeff_mul_X_pow, add_tsub_cancel_right] · refine' (coeff_mul _ _ _).trans (Finset.sum_eq_zero fun x hx => _) rw [coeff_X_pow, if_neg, mul_zero] exact ((le_of_add_le_right (mem_antidiagonal.mp hx).le).trans_lt <| not_le.mp h).ne set_option linter.uppercaseLean3 false in #align power_series.coeff_mul_X_pow' PowerSeries.coeff_mul_X_pow' theorem coeff_X_pow_mul' (p : R⟦X⟧) (n d : ℕ) : coeff R d (X ^ n * p) = ite (n ≤ d) (coeff R (d - n) p) 0 := by split_ifs with h · rw [← tsub_add_cancel_of_le h, coeff_X_pow_mul] simp · refine' (coeff_mul _ _ _).trans (Finset.sum_eq_zero fun x hx => _) rw [coeff_X_pow, if_neg, zero_mul] have := mem_antidiagonal.mp hx rw [add_comm] at this exact ((le_of_add_le_right this.le).trans_lt <| not_le.mp h).ne set_option linter.uppercaseLean3 false in #align power_series.coeff_X_pow_mul' PowerSeries.coeff_X_pow_mul' end /-- If a formal power series is invertible, then so is its constant coefficient. -/ theorem isUnit_constantCoeff (φ : R⟦X⟧) (h : IsUnit φ) : IsUnit (constantCoeff R φ) := MvPowerSeries.isUnit_constantCoeff φ h #align power_series.is_unit_constant_coeff PowerSeries.isUnit_constantCoeff /-- Split off the constant coefficient. -/ theorem eq_shift_mul_X_add_const (φ : R⟦X⟧) : φ = (mk fun p => coeff R (p + 1) φ) * X + C R (constantCoeff R φ) := by ext (_ | n) · simp only [Nat.zero_eq, coeff_zero_eq_constantCoeff, map_add, map_mul, constantCoeff_X, mul_zero, coeff_zero_C, zero_add] · simp only [coeff_succ_mul_X, coeff_mk, LinearMap.map_add, coeff_C, n.succ_ne_zero, sub_zero, if_false, add_zero] set_option linter.uppercaseLean3 false in #align power_series.eq_shift_mul_X_add_const PowerSeries.eq_shift_mul_X_add_const /-- Split off the constant coefficient. -/ theorem eq_X_mul_shift_add_const (φ : R⟦X⟧) : φ = (X * mk fun p => coeff R (p + 1) φ) + C R (constantCoeff R φ) := by ext (_ | n) · simp only [Nat.zero_eq, coeff_zero_eq_constantCoeff, map_add, map_mul, constantCoeff_X, zero_mul, coeff_zero_C, zero_add] · simp only [coeff_succ_X_mul, coeff_mk, LinearMap.map_add, coeff_C, n.succ_ne_zero, sub_zero, if_false, add_zero] set_option linter.uppercaseLean3 false in #align power_series.eq_X_mul_shift_add_const PowerSeries.eq_X_mul_shift_add_const section Map variable {S : Type*} {T : Type*} [Semiring S] [Semiring T] variable (f : R →+* S) (g : S →+* T) /-- The map between formal power series induced by a map on the coefficients. -/ def map : R⟦X⟧ →+* S⟦X⟧ := MvPowerSeries.map _ f #align power_series.map PowerSeries.map @[simp] theorem map_id : (map (RingHom.id R) : R⟦X⟧ → R⟦X⟧) = id := rfl #align power_series.map_id PowerSeries.map_id theorem map_comp : map (g.comp f) = (map g).comp (map f) := rfl #align power_series.map_comp PowerSeries.map_comp @[simp] theorem coeff_map (n : ℕ) (φ : R⟦X⟧) : coeff S n (map f φ) = f (coeff R n φ) := rfl #align power_series.coeff_map PowerSeries.coeff_map @[simp] theorem map_C (r : R) : map f (C _ r) = C _ (f r) := by ext simp [coeff_C, apply_ite f] set_option linter.uppercaseLean3 false in #align power_series.map_C PowerSeries.map_C @[simp] theorem map_X : map f X = X := by ext simp [coeff_X, apply_ite f] set_option linter.uppercaseLean3 false in #align power_series.map_X PowerSeries.map_X end Map theorem X_pow_dvd_iff {n : ℕ} {φ : R⟦X⟧} : (X : R⟦X⟧) ^ n ∣ φ ↔ ∀ m, m < n → coeff R m φ = 0 := by [email protected]_pow_dvd_iff Unit R _ () n φ constructor <;> intro h m hm · rw [Finsupp.unique_single m] convert h _ hm · apply h simpa only [Finsupp.single_eq_same] using hm set_option linter.uppercaseLean3 false in #align power_series.X_pow_dvd_iff PowerSeries.X_pow_dvd_iff theorem X_dvd_iff {φ : R⟦X⟧} : (X : R⟦X⟧) ∣ φ ↔ constantCoeff R φ = 0 := by rw [← pow_one (X : R⟦X⟧), X_pow_dvd_iff, ← coeff_zero_eq_constantCoeff_apply] constructor <;> intro h · exact h 0 zero_lt_one · intro m hm rwa [Nat.eq_zero_of_le_zero (Nat.le_of_succ_le_succ hm)] set_option linter.uppercaseLean3 false in #align power_series.X_dvd_iff PowerSeries.X_dvd_iff end Semiring section CommSemiring variable [CommSemiring R] open Finset Nat /-- The ring homomorphism taking a power series `f(X)` to `f(aX)`. -/ noncomputable def rescale (a : R) : R⟦X⟧ →+* R⟦X⟧ where toFun f := PowerSeries.mk fun n => a ^ n * PowerSeries.coeff R n f map_zero' := by ext simp only [LinearMap.map_zero, PowerSeries.coeff_mk, mul_zero] map_one' := by ext1 simp only [mul_boole, PowerSeries.coeff_mk, PowerSeries.coeff_one] split_ifs with h · rw [h, pow_zero a] rfl map_add' := by intros ext dsimp only exact mul_add _ _ _ map_mul' f g := by ext rw [PowerSeries.coeff_mul, PowerSeries.coeff_mk, PowerSeries.coeff_mul, Finset.mul_sum] apply sum_congr rfl simp only [coeff_mk, Prod.forall, mem_antidiagonal] intro b c H rw [← H, pow_add, mul_mul_mul_comm] #align power_series.rescale PowerSeries.rescale @[simp] theorem coeff_rescale (f : R⟦X⟧) (a : R) (n : ℕ) : coeff R n (rescale a f) = a ^ n * coeff R n f := coeff_mk n (fun n ↦ a ^ n * (coeff R n) f) #align power_series.coeff_rescale PowerSeries.coeff_rescale @[simp] theorem rescale_zero : rescale 0 = (C R).comp (constantCoeff R) := by ext x n simp only [Function.comp_apply, RingHom.coe_comp, rescale, RingHom.coe_mk, PowerSeries.coeff_mk _ _, coeff_C] split_ifs with h <;> simp [h] #align power_series.rescale_zero PowerSeries.rescale_zero theorem rescale_zero_apply : rescale 0 X = C R (constantCoeff R X) := by simp #align power_series.rescale_zero_apply PowerSeries.rescale_zero_apply @[simp] theorem rescale_one : rescale 1 = RingHom.id R⟦X⟧ := by ext simp only [coeff_rescale, one_pow, one_mul, RingHom.id_apply] #align power_series.rescale_one PowerSeries.rescale_one theorem rescale_mk (f : ℕ → R) (a : R) : rescale a (mk f) = mk fun n : ℕ => a ^ n * f n := by ext rw [coeff_rescale, coeff_mk, coeff_mk] #align power_series.rescale_mk PowerSeries.rescale_mk theorem rescale_rescale (f : R⟦X⟧) (a b : R) : rescale b (rescale a f) = rescale (a * b) f := by ext n simp_rw [coeff_rescale] rw [mul_pow, mul_comm _ (b ^ n), mul_assoc] #align power_series.rescale_rescale PowerSeries.rescale_rescale theorem rescale_mul (a b : R) : rescale (a * b) = (rescale b).comp (rescale a) := by ext simp [← rescale_rescale] #align power_series.rescale_mul PowerSeries.rescale_mul end CommSemiring section CommSemiring open Finset.HasAntidiagonal Finset variable {R : Type*} [CommSemiring R] {ι : Type*} [DecidableEq ι] /-- Coefficients of a product of power series -/ theorem coeff_prod (f : ι → PowerSeries R) (d : ℕ) (s : Finset ι) : coeff R d (∏ j in s, f j) = ∑ l in piAntidiagonal s d, ∏ i in s, coeff R (l i) (f i) := by simp only [coeff] convert MvPowerSeries.coeff_prod _ _ _ rw [← AddEquiv.finsuppUnique_symm d, ← mapRange_piAntidiagonal_eq, sum_map, sum_congr rfl] intro x _ apply prod_congr rfl intro i _ congr 2 simp only [AddEquiv.toEquiv_eq_coe, Finsupp.mapRange.addEquiv_toEquiv, AddEquiv.toEquiv_symm, Equiv.coe_toEmbedding, Finsupp.mapRange.equiv_apply, AddEquiv.coe_toEquiv_symm, Finsupp.mapRange_apply, AddEquiv.finsuppUnique_symm] end CommSemiring section CommRing variable {A : Type*} [CommRing A]

theorem not_isField : ¬IsField A⟦X⟧

not_isField

2019

85a4719

mathlib4/Mathlib/RingTheory/PowerSeries/Basic.lean

mathlib

/- Copyright (c) 2019 Johan Commelin. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Johan Commelin, Kenny Lau -/ import Mathlib.Algebra.Polynomial.AlgebraMap import Mathlib.Algebra.Polynomial.Basic import Mathlib.RingTheory.MvPowerSeries.Basic #align_import ring_theory.power_series.basic from "leanprover-community/mathlib"@"2d5739b61641ee4e7e53eca5688a08f66f2e6a60" /-! # Formal power series (in one variable) This file defines (univariate) formal power series and develops the basic properties of these objects. A formal power series is to a polynomial like an infinite sum is to a finite sum. Formal power series in one variable are defined from multivariate power series as `PowerSeries R := MvPowerSeries Unit R`. The file sets up the (semi)ring structure on univariate power series. We provide the natural inclusion from polynomials to formal power series. Additional results can be found in: * `Mathlib.RingTheory.PowerSeries.Trunc`, truncation of power series; * `Mathlib.RingTheory.PowerSeries.Inverse`, about inverses of power series, and the fact that power series over a local ring form a local ring; * `Mathlib.RingTheory.PowerSeries.Order`, the order of a power series at 0, and application to the fact that power series over an integral domain form an integral domain. ## Implementation notes Because of its definition, `PowerSeries R := MvPowerSeries Unit R`. a lot of proofs and properties from the multivariate case can be ported to the single variable case. However, it means that formal power series are indexed by `Unit →₀ ℕ`, which is of course canonically isomorphic to `ℕ`. We then build some glue to treat formal power series as if they were indexed by `ℕ`. Occasionally this leads to proofs that are uglier than expected. -/ noncomputable section open BigOperators open Finset (antidiagonal mem_antidiagonal) /-- Formal power series over a coefficient type `R` -/ def PowerSeries (R : Type*) := MvPowerSeries Unit R #align power_series PowerSeries namespace PowerSeries open Finsupp (single) variable {R : Type*} section -- Porting note: not available in Lean 4 -- local reducible PowerSeries /-- `R⟦X⟧` is notation for `PowerSeries R`, the semiring of formal power series in one variable over a semiring `R`. -/ scoped notation:9000 R "⟦X⟧" => PowerSeries R instance [Inhabited R] : Inhabited R⟦X⟧ := by dsimp only [PowerSeries] infer_instance instance [Zero R] : Zero R⟦X⟧ := by dsimp only [PowerSeries] infer_instance instance [AddMonoid R] : AddMonoid R⟦X⟧ := by dsimp only [PowerSeries] infer_instance instance [AddGroup R] : AddGroup R⟦X⟧ := by dsimp only [PowerSeries] infer_instance instance [AddCommMonoid R] : AddCommMonoid R⟦X⟧ := by dsimp only [PowerSeries] infer_instance instance [AddCommGroup R] : AddCommGroup R⟦X⟧ := by dsimp only [PowerSeries] infer_instance instance [Semiring R] : Semiring R⟦X⟧ := by dsimp only [PowerSeries] infer_instance instance [CommSemiring R] : CommSemiring R⟦X⟧ := by dsimp only [PowerSeries] infer_instance instance [Ring R] : Ring R⟦X⟧ := by dsimp only [PowerSeries] infer_instance instance [CommRing R] : CommRing R⟦X⟧ := by dsimp only [PowerSeries] infer_instance instance [Nontrivial R] : Nontrivial R⟦X⟧ := by dsimp only [PowerSeries] infer_instance instance {A} [Semiring R] [AddCommMonoid A] [Module R A] : Module R A⟦X⟧ := by dsimp only [PowerSeries] infer_instance instance {A S} [Semiring R] [Semiring S] [AddCommMonoid A] [Module R A] [Module S A] [SMul R S] [IsScalarTower R S A] : IsScalarTower R S A⟦X⟧ := Pi.isScalarTower instance {A} [Semiring A] [CommSemiring R] [Algebra R A] : Algebra R A⟦X⟧ := by dsimp only [PowerSeries] infer_instance end section Semiring variable (R) [Semiring R] /-- The `n`th coefficient of a formal power series. -/ def coeff (n : ℕ) : R⟦X⟧ →ₗ[R] R := MvPowerSeries.coeff R (single () n) #align power_series.coeff PowerSeries.coeff /-- The `n`th monomial with coefficient `a` as formal power series. -/ def monomial (n : ℕ) : R →ₗ[R] R⟦X⟧ := MvPowerSeries.monomial R (single () n) #align power_series.monomial PowerSeries.monomial variable {R} theorem coeff_def {s : Unit →₀ ℕ} {n : ℕ} (h : s () = n) : coeff R n = MvPowerSeries.coeff R s := by erw [coeff, ← h, ← Finsupp.unique_single s] #align power_series.coeff_def PowerSeries.coeff_def /-- Two formal power series are equal if all their coefficients are equal. -/ @[ext] theorem ext {φ ψ : R⟦X⟧} (h : ∀ n, coeff R n φ = coeff R n ψ) : φ = ψ := MvPowerSeries.ext fun n => by rw [← coeff_def] · apply h rfl #align power_series.ext PowerSeries.ext /-- Two formal power series are equal if all their coefficients are equal. -/ theorem ext_iff {φ ψ : R⟦X⟧} : φ = ψ ↔ ∀ n, coeff R n φ = coeff R n ψ := ⟨fun h n => congr_arg (coeff R n) h, ext⟩ #align power_series.ext_iff PowerSeries.ext_iff instance [Subsingleton R] : Subsingleton R⟦X⟧ := by simp only [subsingleton_iff, ext_iff] exact fun _ _ _ ↦ (subsingleton_iff).mp (by infer_instance) _ _ /-- Constructor for formal power series. -/ def mk {R} (f : ℕ → R) : R⟦X⟧ := fun s => f (s ()) #align power_series.mk PowerSeries.mk @[simp] theorem coeff_mk (n : ℕ) (f : ℕ → R) : coeff R n (mk f) = f n := congr_arg f Finsupp.single_eq_same #align power_series.coeff_mk PowerSeries.coeff_mk theorem coeff_monomial (m n : ℕ) (a : R) : coeff R m (monomial R n a) = if m = n then a else 0 := calc coeff R m (monomial R n a) = _ := MvPowerSeries.coeff_monomial _ _ _ _ = if m = n then a else 0 := by simp only [Finsupp.unique_single_eq_iff] #align power_series.coeff_monomial PowerSeries.coeff_monomial theorem monomial_eq_mk (n : ℕ) (a : R) : monomial R n a = mk fun m => if m = n then a else 0 := ext fun m => by rw [coeff_monomial, coeff_mk] #align power_series.monomial_eq_mk PowerSeries.monomial_eq_mk @[simp] theorem coeff_monomial_same (n : ℕ) (a : R) : coeff R n (monomial R n a) = a := MvPowerSeries.coeff_monomial_same _ _ #align power_series.coeff_monomial_same PowerSeries.coeff_monomial_same @[simp] theorem coeff_comp_monomial (n : ℕ) : (coeff R n).comp (monomial R n) = LinearMap.id := LinearMap.ext <| coeff_monomial_same n #align power_series.coeff_comp_monomial PowerSeries.coeff_comp_monomial variable (R) /-- The constant coefficient of a formal power series. -/ def constantCoeff : R⟦X⟧ →+* R := MvPowerSeries.constantCoeff Unit R #align power_series.constant_coeff PowerSeries.constantCoeff /-- The constant formal power series. -/ def C : R →+* R⟦X⟧ := MvPowerSeries.C Unit R set_option linter.uppercaseLean3 false in #align power_series.C PowerSeries.C variable {R} /-- The variable of the formal power series ring. -/ def X : R⟦X⟧ := MvPowerSeries.X () set_option linter.uppercaseLean3 false in #align power_series.X PowerSeries.X theorem commute_X (φ : R⟦X⟧) : Commute φ X := MvPowerSeries.commute_X _ _ set_option linter.uppercaseLean3 false in #align power_series.commute_X PowerSeries.commute_X @[simp] theorem coeff_zero_eq_constantCoeff : ⇑(coeff R 0) = constantCoeff R := by rw [coeff, Finsupp.single_zero] rfl #align power_series.coeff_zero_eq_constant_coeff PowerSeries.coeff_zero_eq_constantCoeff theorem coeff_zero_eq_constantCoeff_apply (φ : R⟦X⟧) : coeff R 0 φ = constantCoeff R φ := by rw [coeff_zero_eq_constantCoeff] #align power_series.coeff_zero_eq_constant_coeff_apply PowerSeries.coeff_zero_eq_constantCoeff_apply @[simp] theorem monomial_zero_eq_C : ⇑(monomial R 0) = C R := by -- This used to be `rw`, but we need `erw` after leanprover/lean4#2644 erw [monomial, Finsupp.single_zero, MvPowerSeries.monomial_zero_eq_C] set_option linter.uppercaseLean3 false in #align power_series.monomial_zero_eq_C PowerSeries.monomial_zero_eq_C theorem monomial_zero_eq_C_apply (a : R) : monomial R 0 a = C R a := by simp set_option linter.uppercaseLean3 false in #align power_series.monomial_zero_eq_C_apply PowerSeries.monomial_zero_eq_C_apply theorem coeff_C (n : ℕ) (a : R) : coeff R n (C R a : R⟦X⟧) = if n = 0 then a else 0 := by rw [← monomial_zero_eq_C_apply, coeff_monomial] set_option linter.uppercaseLean3 false in #align power_series.coeff_C PowerSeries.coeff_C @[simp] theorem coeff_zero_C (a : R) : coeff R 0 (C R a) = a := by rw [coeff_C, if_pos rfl] set_option linter.uppercaseLean3 false in #align power_series.coeff_zero_C PowerSeries.coeff_zero_C theorem coeff_ne_zero_C {a : R} {n : ℕ} (h : n ≠ 0) : coeff R n (C R a) = 0 := by rw [coeff_C, if_neg h] @[simp] theorem coeff_succ_C {a : R} {n : ℕ} : coeff R (n + 1) (C R a) = 0 := coeff_ne_zero_C n.succ_ne_zero theorem C_injective : Function.Injective (C R) := by intro a b H have := (ext_iff (φ := C R a) (ψ := C R b)).mp H 0 rwa [coeff_zero_C, coeff_zero_C] at this protected theorem subsingleton_iff : Subsingleton R⟦X⟧ ↔ Subsingleton R := by refine ⟨fun h ↦ ?_, fun _ ↦ inferInstance⟩ rw [subsingleton_iff] at h ⊢ exact fun a b ↦ C_injective (h (C R a) (C R b)) theorem X_eq : (X : R⟦X⟧) = monomial R 1 1 := rfl set_option linter.uppercaseLean3 false in #align power_series.X_eq PowerSeries.X_eq theorem coeff_X (n : ℕ) : coeff R n (X : R⟦X⟧) = if n = 1 then 1 else 0 := by rw [X_eq, coeff_monomial] set_option linter.uppercaseLean3 false in #align power_series.coeff_X PowerSeries.coeff_X @[simp] theorem coeff_zero_X : coeff R 0 (X : R⟦X⟧) = 0 := by -- This used to be `rw`, but we need `erw` after leanprover/lean4#2644 erw [coeff, Finsupp.single_zero, X, MvPowerSeries.coeff_zero_X] set_option linter.uppercaseLean3 false in #align power_series.coeff_zero_X PowerSeries.coeff_zero_X @[simp] theorem coeff_one_X : coeff R 1 (X : R⟦X⟧) = 1 := by rw [coeff_X, if_pos rfl] set_option linter.uppercaseLean3 false in #align power_series.coeff_one_X PowerSeries.coeff_one_X @[simp] theorem X_ne_zero [Nontrivial R] : (X : R⟦X⟧) ≠ 0 := fun H => by simpa only [coeff_one_X, one_ne_zero, map_zero] using congr_arg (coeff R 1) H set_option linter.uppercaseLean3 false in #align power_series.X_ne_zero PowerSeries.X_ne_zero theorem X_pow_eq (n : ℕ) : (X : R⟦X⟧) ^ n = monomial R n 1 := MvPowerSeries.X_pow_eq _ n set_option linter.uppercaseLean3 false in #align power_series.X_pow_eq PowerSeries.X_pow_eq theorem coeff_X_pow (m n : ℕ) : coeff R m ((X : R⟦X⟧) ^ n) = if m = n then 1 else 0 := by rw [X_pow_eq, coeff_monomial] set_option linter.uppercaseLean3 false in #align power_series.coeff_X_pow PowerSeries.coeff_X_pow @[simp] theorem coeff_X_pow_self (n : ℕ) : coeff R n ((X : R⟦X⟧) ^ n) = 1 := by rw [coeff_X_pow, if_pos rfl] set_option linter.uppercaseLean3 false in #align power_series.coeff_X_pow_self PowerSeries.coeff_X_pow_self @[simp] theorem coeff_one (n : ℕ) : coeff R n (1 : R⟦X⟧) = if n = 0 then 1 else 0 := coeff_C n 1 #align power_series.coeff_one PowerSeries.coeff_one theorem coeff_zero_one : coeff R 0 (1 : R⟦X⟧) = 1 := coeff_zero_C 1 #align power_series.coeff_zero_one PowerSeries.coeff_zero_one theorem coeff_mul (n : ℕ) (φ ψ : R⟦X⟧) : coeff R n (φ * ψ) = ∑ p in antidiagonal n, coeff R p.1 φ * coeff R p.2 ψ := by -- `rw` can't see that `PowerSeries = MvPowerSeries Unit`, so use `.trans` refine (MvPowerSeries.coeff_mul _ φ ψ).trans ?_ rw [Finsupp.antidiagonal_single, Finset.sum_map] rfl #align power_series.coeff_mul PowerSeries.coeff_mul @[simp] theorem coeff_mul_C (n : ℕ) (φ : R⟦X⟧) (a : R) : coeff R n (φ * C R a) = coeff R n φ * a := MvPowerSeries.coeff_mul_C _ φ a set_option linter.uppercaseLean3 false in #align power_series.coeff_mul_C PowerSeries.coeff_mul_C @[simp] theorem coeff_C_mul (n : ℕ) (φ : R⟦X⟧) (a : R) : coeff R n (C R a * φ) = a * coeff R n φ := MvPowerSeries.coeff_C_mul _ φ a set_option linter.uppercaseLean3 false in #align power_series.coeff_C_mul PowerSeries.coeff_C_mul @[simp] theorem coeff_smul {S : Type*} [Semiring S] [Module R S] (n : ℕ) (φ : PowerSeries S) (a : R) : coeff S n (a • φ) = a • coeff S n φ := rfl #align power_series.coeff_smul PowerSeries.coeff_smul theorem smul_eq_C_mul (f : R⟦X⟧) (a : R) : a • f = C R a * f := by ext simp set_option linter.uppercaseLean3 false in #align power_series.smul_eq_C_mul PowerSeries.smul_eq_C_mul @[simp] theorem coeff_succ_mul_X (n : ℕ) (φ : R⟦X⟧) : coeff R (n + 1) (φ * X) = coeff R n φ := by simp only [coeff, Finsupp.single_add] convert φ.coeff_add_mul_monomial (single () n) (single () 1) _ rw [mul_one]; rfl set_option linter.uppercaseLean3 false in #align power_series.coeff_succ_mul_X PowerSeries.coeff_succ_mul_X @[simp] theorem coeff_succ_X_mul (n : ℕ) (φ : R⟦X⟧) : coeff R (n + 1) (X * φ) = coeff R n φ := by simp only [coeff, Finsupp.single_add, add_comm n 1] convert φ.coeff_add_monomial_mul (single () 1) (single () n) _ rw [one_mul]; rfl set_option linter.uppercaseLean3 false in #align power_series.coeff_succ_X_mul PowerSeries.coeff_succ_X_mul @[simp] theorem constantCoeff_C (a : R) : constantCoeff R (C R a) = a := rfl set_option linter.uppercaseLean3 false in #align power_series.constant_coeff_C PowerSeries.constantCoeff_C @[simp] theorem constantCoeff_comp_C : (constantCoeff R).comp (C R) = RingHom.id R := rfl set_option linter.uppercaseLean3 false in #align power_series.constant_coeff_comp_C PowerSeries.constantCoeff_comp_C -- Porting note (#10618): simp can prove this. -- @[simp] theorem constantCoeff_zero : constantCoeff R 0 = 0 := rfl #align power_series.constant_coeff_zero PowerSeries.constantCoeff_zero -- Porting note (#10618): simp can prove this. -- @[simp] theorem constantCoeff_one : constantCoeff R 1 = 1 := rfl #align power_series.constant_coeff_one PowerSeries.constantCoeff_one @[simp] theorem constantCoeff_X : constantCoeff R X = 0 := MvPowerSeries.coeff_zero_X _ set_option linter.uppercaseLean3 false in #align power_series.constant_coeff_X PowerSeries.constantCoeff_X theorem coeff_zero_mul_X (φ : R⟦X⟧) : coeff R 0 (φ * X) = 0 := by simp set_option linter.uppercaseLean3 false in #align power_series.coeff_zero_mul_X PowerSeries.coeff_zero_mul_X theorem coeff_zero_X_mul (φ : R⟦X⟧) : coeff R 0 (X * φ) = 0 := by simp set_option linter.uppercaseLean3 false in #align power_series.coeff_zero_X_mul PowerSeries.coeff_zero_X_mul theorem constantCoeff_surj : Function.Surjective (constantCoeff R) := fun r => ⟨(C R) r, constantCoeff_C r⟩ -- The following section duplicates the API of `Data.Polynomial.Coeff` and should attempt to keep -- up to date with that section theorem coeff_C_mul_X_pow (x : R) (k n : ℕ) : coeff R n (C R x * X ^ k : R⟦X⟧) = if n = k then x else 0 := by simp [X_pow_eq, coeff_monomial] set_option linter.uppercaseLean3 false in #align power_series.coeff_C_mul_X_pow PowerSeries.coeff_C_mul_X_pow @[simp] theorem coeff_mul_X_pow (p : R⟦X⟧) (n d : ℕ) : coeff R (d + n) (p * X ^ n) = coeff R d p := by rw [coeff_mul, Finset.sum_eq_single (d, n), coeff_X_pow, if_pos rfl, mul_one] · rintro ⟨i, j⟩ h1 h2 rw [coeff_X_pow, if_neg, mul_zero] rintro rfl apply h2 rw [mem_antidiagonal, add_right_cancel_iff] at h1 subst h1 rfl · exact fun h1 => (h1 (mem_antidiagonal.2 rfl)).elim set_option linter.uppercaseLean3 false in #align power_series.coeff_mul_X_pow PowerSeries.coeff_mul_X_pow @[simp] theorem coeff_X_pow_mul (p : R⟦X⟧) (n d : ℕ) : coeff R (d + n) (X ^ n * p) = coeff R d p := by rw [coeff_mul, Finset.sum_eq_single (n, d), coeff_X_pow, if_pos rfl, one_mul] · rintro ⟨i, j⟩ h1 h2 rw [coeff_X_pow, if_neg, zero_mul] rintro rfl apply h2 rw [mem_antidiagonal, add_comm, add_right_cancel_iff] at h1 subst h1 rfl · rw [add_comm] exact fun h1 => (h1 (mem_antidiagonal.2 rfl)).elim set_option linter.uppercaseLean3 false in #align power_series.coeff_X_pow_mul PowerSeries.coeff_X_pow_mul theorem coeff_mul_X_pow' (p : R⟦X⟧) (n d : ℕ) : coeff R d (p * X ^ n) = ite (n ≤ d) (coeff R (d - n) p) 0 := by split_ifs with h · rw [← tsub_add_cancel_of_le h, coeff_mul_X_pow, add_tsub_cancel_right] · refine' (coeff_mul _ _ _).trans (Finset.sum_eq_zero fun x hx => _) rw [coeff_X_pow, if_neg, mul_zero] exact ((le_of_add_le_right (mem_antidiagonal.mp hx).le).trans_lt <| not_le.mp h).ne set_option linter.uppercaseLean3 false in #align power_series.coeff_mul_X_pow' PowerSeries.coeff_mul_X_pow' theorem coeff_X_pow_mul' (p : R⟦X⟧) (n d : ℕ) : coeff R d (X ^ n * p) = ite (n ≤ d) (coeff R (d - n) p) 0 := by split_ifs with h · rw [← tsub_add_cancel_of_le h, coeff_X_pow_mul] simp · refine' (coeff_mul _ _ _).trans (Finset.sum_eq_zero fun x hx => _) rw [coeff_X_pow, if_neg, zero_mul] have := mem_antidiagonal.mp hx rw [add_comm] at this exact ((le_of_add_le_right this.le).trans_lt <| not_le.mp h).ne set_option linter.uppercaseLean3 false in #align power_series.coeff_X_pow_mul' PowerSeries.coeff_X_pow_mul' end /-- If a formal power series is invertible, then so is its constant coefficient. -/ theorem isUnit_constantCoeff (φ : R⟦X⟧) (h : IsUnit φ) : IsUnit (constantCoeff R φ) := MvPowerSeries.isUnit_constantCoeff φ h #align power_series.is_unit_constant_coeff PowerSeries.isUnit_constantCoeff /-- Split off the constant coefficient. -/ theorem eq_shift_mul_X_add_const (φ : R⟦X⟧) : φ = (mk fun p => coeff R (p + 1) φ) * X + C R (constantCoeff R φ) := by ext (_ | n) · simp only [Nat.zero_eq, coeff_zero_eq_constantCoeff, map_add, map_mul, constantCoeff_X, mul_zero, coeff_zero_C, zero_add] · simp only [coeff_succ_mul_X, coeff_mk, LinearMap.map_add, coeff_C, n.succ_ne_zero, sub_zero, if_false, add_zero] set_option linter.uppercaseLean3 false in #align power_series.eq_shift_mul_X_add_const PowerSeries.eq_shift_mul_X_add_const /-- Split off the constant coefficient. -/ theorem eq_X_mul_shift_add_const (φ : R⟦X⟧) : φ = (X * mk fun p => coeff R (p + 1) φ) + C R (constantCoeff R φ) := by ext (_ | n) · simp only [Nat.zero_eq, coeff_zero_eq_constantCoeff, map_add, map_mul, constantCoeff_X, zero_mul, coeff_zero_C, zero_add] · simp only [coeff_succ_X_mul, coeff_mk, LinearMap.map_add, coeff_C, n.succ_ne_zero, sub_zero, if_false, add_zero] set_option linter.uppercaseLean3 false in #align power_series.eq_X_mul_shift_add_const PowerSeries.eq_X_mul_shift_add_const section Map variable {S : Type*} {T : Type*} [Semiring S] [Semiring T] variable (f : R →+* S) (g : S →+* T) /-- The map between formal power series induced by a map on the coefficients. -/ def map : R⟦X⟧ →+* S⟦X⟧ := MvPowerSeries.map _ f #align power_series.map PowerSeries.map @[simp] theorem map_id : (map (RingHom.id R) : R⟦X⟧ → R⟦X⟧) = id := rfl #align power_series.map_id PowerSeries.map_id theorem map_comp : map (g.comp f) = (map g).comp (map f) := rfl #align power_series.map_comp PowerSeries.map_comp @[simp] theorem coeff_map (n : ℕ) (φ : R⟦X⟧) : coeff S n (map f φ) = f (coeff R n φ) := rfl #align power_series.coeff_map PowerSeries.coeff_map @[simp] theorem map_C (r : R) : map f (C _ r) = C _ (f r) := by ext simp [coeff_C, apply_ite f] set_option linter.uppercaseLean3 false in #align power_series.map_C PowerSeries.map_C @[simp] theorem map_X : map f X = X := by ext simp [coeff_X, apply_ite f] set_option linter.uppercaseLean3 false in #align power_series.map_X PowerSeries.map_X end Map theorem X_pow_dvd_iff {n : ℕ} {φ : R⟦X⟧} : (X : R⟦X⟧) ^ n ∣ φ ↔ ∀ m, m < n → coeff R m φ = 0 := by [email protected]_pow_dvd_iff Unit R _ () n φ constructor <;> intro h m hm · rw [Finsupp.unique_single m] convert h _ hm · apply h simpa only [Finsupp.single_eq_same] using hm set_option linter.uppercaseLean3 false in #align power_series.X_pow_dvd_iff PowerSeries.X_pow_dvd_iff theorem X_dvd_iff {φ : R⟦X⟧} : (X : R⟦X⟧) ∣ φ ↔ constantCoeff R φ = 0 := by rw [← pow_one (X : R⟦X⟧), X_pow_dvd_iff, ← coeff_zero_eq_constantCoeff_apply] constructor <;> intro h · exact h 0 zero_lt_one · intro m hm rwa [Nat.eq_zero_of_le_zero (Nat.le_of_succ_le_succ hm)] set_option linter.uppercaseLean3 false in #align power_series.X_dvd_iff PowerSeries.X_dvd_iff end Semiring section CommSemiring variable [CommSemiring R] open Finset Nat /-- The ring homomorphism taking a power series `f(X)` to `f(aX)`. -/ noncomputable def rescale (a : R) : R⟦X⟧ →+* R⟦X⟧ where toFun f := PowerSeries.mk fun n => a ^ n * PowerSeries.coeff R n f map_zero' := by ext simp only [LinearMap.map_zero, PowerSeries.coeff_mk, mul_zero] map_one' := by ext1 simp only [mul_boole, PowerSeries.coeff_mk, PowerSeries.coeff_one] split_ifs with h · rw [h, pow_zero a] rfl map_add' := by intros ext dsimp only exact mul_add _ _ _ map_mul' f g := by ext rw [PowerSeries.coeff_mul, PowerSeries.coeff_mk, PowerSeries.coeff_mul, Finset.mul_sum] apply sum_congr rfl simp only [coeff_mk, Prod.forall, mem_antidiagonal] intro b c H rw [← H, pow_add, mul_mul_mul_comm] #align power_series.rescale PowerSeries.rescale @[simp] theorem coeff_rescale (f : R⟦X⟧) (a : R) (n : ℕ) : coeff R n (rescale a f) = a ^ n * coeff R n f := coeff_mk n (fun n ↦ a ^ n * (coeff R n) f) #align power_series.coeff_rescale PowerSeries.coeff_rescale @[simp] theorem rescale_zero : rescale 0 = (C R).comp (constantCoeff R) := by ext x n simp only [Function.comp_apply, RingHom.coe_comp, rescale, RingHom.coe_mk, PowerSeries.coeff_mk _ _, coeff_C] split_ifs with h <;> simp [h] #align power_series.rescale_zero PowerSeries.rescale_zero theorem rescale_zero_apply : rescale 0 X = C R (constantCoeff R X) := by simp #align power_series.rescale_zero_apply PowerSeries.rescale_zero_apply @[simp] theorem rescale_one : rescale 1 = RingHom.id R⟦X⟧ := by ext simp only [coeff_rescale, one_pow, one_mul, RingHom.id_apply] #align power_series.rescale_one PowerSeries.rescale_one theorem rescale_mk (f : ℕ → R) (a : R) : rescale a (mk f) = mk fun n : ℕ => a ^ n * f n := by ext rw [coeff_rescale, coeff_mk, coeff_mk] #align power_series.rescale_mk PowerSeries.rescale_mk theorem rescale_rescale (f : R⟦X⟧) (a b : R) : rescale b (rescale a f) = rescale (a * b) f := by ext n simp_rw [coeff_rescale] rw [mul_pow, mul_comm _ (b ^ n), mul_assoc] #align power_series.rescale_rescale PowerSeries.rescale_rescale theorem rescale_mul (a b : R) : rescale (a * b) = (rescale b).comp (rescale a) := by ext simp [← rescale_rescale] #align power_series.rescale_mul PowerSeries.rescale_mul end CommSemiring section CommSemiring open Finset.HasAntidiagonal Finset variable {R : Type*} [CommSemiring R] {ι : Type*} [DecidableEq ι] /-- Coefficients of a product of power series -/ theorem coeff_prod (f : ι → PowerSeries R) (d : ℕ) (s : Finset ι) : coeff R d (∏ j in s, f j) = ∑ l in piAntidiagonal s d, ∏ i in s, coeff R (l i) (f i) := by simp only [coeff] convert MvPowerSeries.coeff_prod _ _ _ rw [← AddEquiv.finsuppUnique_symm d, ← mapRange_piAntidiagonal_eq, sum_map, sum_congr rfl] intro x _ apply prod_congr rfl intro i _ congr 2 simp only [AddEquiv.toEquiv_eq_coe, Finsupp.mapRange.addEquiv_toEquiv, AddEquiv.toEquiv_symm, Equiv.coe_toEmbedding, Finsupp.mapRange.equiv_apply, AddEquiv.coe_toEquiv_symm, Finsupp.mapRange_apply, AddEquiv.finsuppUnique_symm] end CommSemiring section CommRing variable {A : Type*} [CommRing A] theorem not_isField : ¬IsField A⟦X⟧ := by by_cases hA : Subsingleton A · exact not_isField_of_subsingleton _ · nontriviality A rw [Ring.not_isField_iff_exists_ideal_bot_lt_and_lt_top] use Ideal.span {X} constructor · rw [bot_lt_iff_ne_bot, Ne.def, Ideal.span_singleton_eq_bot] exact X_ne_zero · rw [lt_top_iff_ne_top, Ne.def, Ideal.eq_top_iff_one, Ideal.mem_span_singleton, X_dvd_iff, constantCoeff_one] exact one_ne_zero @[simp] theorem rescale_X (a : A) : rescale a X = C A a * X := by ext simp only [coeff_rescale, coeff_C_mul, coeff_X] split_ifs with h <;> simp [h] set_option linter.uppercaseLean3 false in #align power_series.rescale_X PowerSeries.rescale_X theorem rescale_neg_one_X : rescale (-1 : A) X = -X := by rw [rescale_X, map_neg, map_one, neg_one_mul] set_option linter.uppercaseLean3 false in #align power_series.rescale_neg_one_X PowerSeries.rescale_neg_one_X /-- The ring homomorphism taking a power series `f(X)` to `f(-X)`. -/ noncomputable def evalNegHom : A⟦X⟧ →+* A⟦X⟧ := rescale (-1 : A) #align power_series.eval_neg_hom PowerSeries.evalNegHom @[simp] theorem evalNegHom_X : evalNegHom (X : A⟦X⟧) = -X := rescale_neg_one_X set_option linter.uppercaseLean3 false in #align power_series.eval_neg_hom_X PowerSeries.evalNegHom_X end CommRing section Domain variable [Ring R] theorem eq_zero_or_eq_zero_of_mul_eq_zero [NoZeroDivisors R] (φ ψ : R⟦X⟧) (h : φ * ψ = 0) : φ = 0 ∨ ψ = 0 := by classical rw [or_iff_not_imp_left] intro H have ex : ∃ m, coeff R m φ ≠ 0 := by contrapose! H exact ext H let m := Nat.find ex have hm₁ : coeff R m φ ≠ 0 := Nat.find_spec ex have hm₂ : ∀ k < m, ¬coeff R k φ ≠ 0 := fun k => Nat.find_min ex ext n rw [(coeff R n).map_zero] induction' n using Nat.strong_induction_on with n ih replace h := congr_arg (coeff R (m + n)) h rw [LinearMap.map_zero, coeff_mul, Finset.sum_eq_single (m, n)] at h · replace h := NoZeroDivisors.eq_zero_or_eq_zero_of_mul_eq_zero h rw [or_iff_not_imp_left] at h exact h hm₁ · rintro ⟨i, j⟩ hij hne by_cases hj : j < n · rw [ih j hj, mul_zero] by_cases hi : i < m · specialize hm₂ _ hi push_neg at hm₂ rw [hm₂, zero_mul] rw [mem_antidiagonal] at hij push_neg at hi hj suffices m < i by have : m + n < i + j := add_lt_add_of_lt_of_le this hj exfalso exact ne_of_lt this hij.symm contrapose! hne obtain rfl := le_antisymm hi hne simpa [Ne, Prod.mk.inj_iff] using (add_right_inj m).mp hij · contrapose! intro rw [mem_antidiagonal] #align power_series.eq_zero_or_eq_zero_of_mul_eq_zero PowerSeries.eq_zero_or_eq_zero_of_mul_eq_zero instance [NoZeroDivisors R] : NoZeroDivisors R⟦X⟧ where eq_zero_or_eq_zero_of_mul_eq_zero := eq_zero_or_eq_zero_of_mul_eq_zero _ _ instance [IsDomain R] : IsDomain R⟦X⟧ := NoZeroDivisors.to_isDomain _ end Domain section IsDomain variable [CommRing R] [IsDomain R] /-- The ideal spanned by the variable in the power series ring over an integral domain is a prime ideal. -/ theorem span_X_isPrime : (Ideal.span ({X} : Set R⟦X⟧)).IsPrime := by suffices Ideal.span ({X} : Set R⟦X⟧) = RingHom.ker (constantCoeff R) by rw [this] exact RingHom.ker_isPrime _ apply Ideal.ext intro φ rw [RingHom.mem_ker, Ideal.mem_span_singleton, X_dvd_iff] set_option linter.uppercaseLean3 false in #align power_series.span_X_is_prime PowerSeries.span_X_isPrime /-- The variable of the power series ring over an integral domain is prime. -/ theorem X_prime : Prime (X : R⟦X⟧) := by rw [← Ideal.span_singleton_prime] · exact span_X_isPrime · intro h simpa [map_zero (coeff R 1)] using congr_arg (coeff R 1) h set_option linter.uppercaseLean3 false in #align power_series.X_prime PowerSeries.X_prime /-- The variable of the power series ring over an integral domain is irreducible. -/

theorem X_irreducible : Irreducible (X : R⟦X⟧)

X_irreducible

2019

85a4719

mathlib4/Mathlib/RingTheory/PowerSeries/Basic.lean

mathlib

/- Copyright (c) 2019 Johan Commelin. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Johan Commelin, Kenny Lau -/ import Mathlib.RingTheory.PowerSeries.Basic import Mathlib.Algebra.CharP.Basic #align_import ring_theory.power_series.basic from "leanprover-community/mathlib"@"2d5739b61641ee4e7e53eca5688a08f66f2e6a60" /-! # Formal power series (in one variable) - Order The `PowerSeries.order` of a formal power series `φ` is the multiplicity of the variable `X` in `φ`. If the coefficients form an integral domain, then `PowerSeries.order` is an additive valuation (`PowerSeries.order_mul`, `PowerSeries.le_order_add`). We prove that if the commutative ring `R` of coefficients is an integral domain, then the ring `R⟦X⟧` of formal power series in one variable over `R` is an integral domain. Given a non-zero power series `f`, `divided_by_X_pow_order f` is the power series obtained by dividing out the largest power of X that divides `f`, that is its order. This is useful when proving that `R⟦X⟧` is a normalization monoid, which is done in `PowerSeries.Inverse`. -/ noncomputable section open BigOperators Polynomial open Finset (antidiagonal mem_antidiagonal) namespace PowerSeries open Finsupp (single) variable {R : Type*} section OrderBasic open multiplicity variable [Semiring R] {φ : R⟦X⟧} theorem exists_coeff_ne_zero_iff_ne_zero : (∃ n : ℕ, coeff R n φ ≠ 0) ↔ φ ≠ 0 := by refine' not_iff_not.mp _ push_neg -- FIXME: the `FunLike.coe` doesn't seem to be picked up in the expression after #8386? simp [PowerSeries.ext_iff, (coeff R _).map_zero] #align power_series.exists_coeff_ne_zero_iff_ne_zero PowerSeries.exists_coeff_ne_zero_iff_ne_zero /-- The order of a formal power series `φ` is the greatest `n : PartENat` such that `X^n` divides `φ`. The order is `⊤` if and only if `φ = 0`. -/ def order (φ : R⟦X⟧) : PartENat := letI := Classical.decEq R letI := Classical.decEq R⟦X⟧ if h : φ = 0 then ⊤ else Nat.find (exists_coeff_ne_zero_iff_ne_zero.mpr h) #align power_series.order PowerSeries.order /-- The order of the `0` power series is infinite. -/ @[simp] theorem order_zero : order (0 : R⟦X⟧) = ⊤ := dif_pos rfl #align power_series.order_zero PowerSeries.order_zero theorem order_finite_iff_ne_zero : (order φ).Dom ↔ φ ≠ 0 := by simp only [order] constructor · split_ifs with h <;> intro H · simp only [PartENat.top_eq_none, Part.not_none_dom] at H · exact h · intro h simp [h] #align power_series.order_finite_iff_ne_zero PowerSeries.order_finite_iff_ne_zero /-- If the order of a formal power series is finite, then the coefficient indexed by the order is nonzero. -/ theorem coeff_order (h : (order φ).Dom) : coeff R (φ.order.get h) φ ≠ 0 := by classical simp only [order, order_finite_iff_ne_zero.mp h, not_false_iff, dif_neg, PartENat.get_natCast'] generalize_proofs h exact Nat.find_spec h #align power_series.coeff_order PowerSeries.coeff_order /-- If the `n`th coefficient of a formal power series is nonzero, then the order of the power series is less than or equal to `n`. -/ theorem order_le (n : ℕ) (h : coeff R n φ ≠ 0) : order φ ≤ n := by classical rw [order, dif_neg] · simp only [PartENat.coe_le_coe] exact Nat.find_le h · exact exists_coeff_ne_zero_iff_ne_zero.mp ⟨n, h⟩ #align power_series.order_le PowerSeries.order_le /-- The `n`th coefficient of a formal power series is `0` if `n` is strictly smaller than the order of the power series. -/ theorem coeff_of_lt_order (n : ℕ) (h : ↑n < order φ) : coeff R n φ = 0 := by contrapose! h exact order_le _ h #align power_series.coeff_of_lt_order PowerSeries.coeff_of_lt_order /-- The `0` power series is the unique power series with infinite order. -/ @[simp] theorem order_eq_top {φ : R⟦X⟧} : φ.order = ⊤ ↔ φ = 0 := PartENat.not_dom_iff_eq_top.symm.trans order_finite_iff_ne_zero.not_left #align power_series.order_eq_top PowerSeries.order_eq_top /-- The order of a formal power series is at least `n` if the `i`th coefficient is `0` for all `i < n`. -/ theorem nat_le_order (φ : R⟦X⟧) (n : ℕ) (h : ∀ i < n, coeff R i φ = 0) : ↑n ≤ order φ := by by_contra H; rw [not_le] at H have : (order φ).Dom := PartENat.dom_of_le_natCast H.le rw [← PartENat.natCast_get this, PartENat.coe_lt_coe] at H exact coeff_order this (h _ H) #align power_series.nat_le_order PowerSeries.nat_le_order /-- The order of a formal power series is at least `n` if the `i`th coefficient is `0` for all `i < n`. -/ theorem le_order (φ : R⟦X⟧) (n : PartENat) (h : ∀ i : ℕ, ↑i < n → coeff R i φ = 0) : n ≤ order φ := by induction n using PartENat.casesOn · show _ ≤ _ rw [top_le_iff, order_eq_top] ext i exact h _ (PartENat.natCast_lt_top i) · apply nat_le_order simpa only [PartENat.coe_lt_coe] using h #align power_series.le_order PowerSeries.le_order /-- The order of a formal power series is exactly `n` if the `n`th coefficient is nonzero, and the `i`th coefficient is `0` for all `i < n`. -/ theorem order_eq_nat {φ : R⟦X⟧} {n : ℕ} : order φ = n ↔ coeff R n φ ≠ 0 ∧ ∀ i, i < n → coeff R i φ = 0 := by classical rcases eq_or_ne φ 0 with (rfl | hφ) · simpa [(coeff R _).map_zero] using (PartENat.natCast_ne_top _).symm simp [order, dif_neg hφ, Nat.find_eq_iff] #align power_series.order_eq_nat PowerSeries.order_eq_nat /-- The order of a formal power series is exactly `n` if the `n`th coefficient is nonzero, and the `i`th coefficient is `0` for all `i < n`. -/ theorem order_eq {φ : R⟦X⟧} {n : PartENat} : order φ = n ↔ (∀ i : ℕ, ↑i = n → coeff R i φ ≠ 0) ∧ ∀ i : ℕ, ↑i < n → coeff R i φ = 0 := by induction n using PartENat.casesOn · rw [order_eq_top] constructor · rintro rfl constructor <;> intros · exfalso exact PartENat.natCast_ne_top ‹_› ‹_› · exact (coeff _ _).map_zero · rintro ⟨_h₁, h₂⟩ ext i exact h₂ i (PartENat.natCast_lt_top i) · simpa [PartENat.natCast_inj] using order_eq_nat #align power_series.order_eq PowerSeries.order_eq /-- The order of the sum of two formal power series is at least the minimum of their orders. -/ theorem le_order_add (φ ψ : R⟦X⟧) : min (order φ) (order ψ) ≤ order (φ + ψ) := by refine' le_order _ _ _ simp (config := { contextual := true }) [coeff_of_lt_order] #align power_series.le_order_add PowerSeries.le_order_add private theorem order_add_of_order_eq.aux (φ ψ : R⟦X⟧) (_h : order φ ≠ order ψ) (H : order φ < order ψ) : order (φ + ψ) ≤ order φ ⊓ order ψ := by suffices order (φ + ψ) = order φ by rw [le_inf_iff, this] exact ⟨le_rfl, le_of_lt H⟩ · rw [order_eq] constructor · intro i hi rw [← hi] at H rw [(coeff _ _).map_add, coeff_of_lt_order i H, add_zero] exact (order_eq_nat.1 hi.symm).1 · intro i hi rw [(coeff _ _).map_add, coeff_of_lt_order i hi, coeff_of_lt_order i (lt_trans hi H), zero_add] -- #align power_series.order_add_of_order_eq.aux power_series.order_add_of_order_eq.aux /-- The order of the sum of two formal power series is the minimum of their orders if their orders differ. -/ theorem order_add_of_order_eq (φ ψ : R⟦X⟧) (h : order φ ≠ order ψ) : order (φ + ψ) = order φ ⊓ order ψ := by refine' le_antisymm _ (le_order_add _ _) by_cases H₁ : order φ < order ψ · apply order_add_of_order_eq.aux _ _ h H₁ by_cases H₂ : order ψ < order φ · simpa only [add_comm, inf_comm] using order_add_of_order_eq.aux _ _ h.symm H₂ exfalso; exact h (le_antisymm (not_lt.1 H₂) (not_lt.1 H₁)) #align power_series.order_add_of_order_eq PowerSeries.order_add_of_order_eq /-- The order of the product of two formal power series is at least the sum of their orders. -/ theorem order_mul_ge (φ ψ : R⟦X⟧) : order φ + order ψ ≤ order (φ * ψ) := by apply le_order intro n hn; rw [coeff_mul, Finset.sum_eq_zero] rintro ⟨i, j⟩ hij by_cases hi : ↑i < order φ · rw [coeff_of_lt_order i hi, zero_mul] by_cases hj : ↑j < order ψ · rw [coeff_of_lt_order j hj, mul_zero] rw [not_lt] at hi hj; rw [mem_antidiagonal] at hij exfalso apply ne_of_lt (lt_of_lt_of_le hn <| add_le_add hi hj) rw [← Nat.cast_add, hij] #align power_series.order_mul_ge PowerSeries.order_mul_ge /-- The order of the monomial `a*X^n` is infinite if `a = 0` and `n` otherwise. -/ theorem order_monomial (n : ℕ) (a : R) [Decidable (a = 0)] : order (monomial R n a) = if a = 0 then (⊤ : PartENat) else n := by split_ifs with h · rw [h, order_eq_top, LinearMap.map_zero] · rw [order_eq] constructor <;> intro i hi · rw [PartENat.natCast_inj] at hi rwa [hi, coeff_monomial_same] · rw [PartENat.coe_lt_coe] at hi rw [coeff_monomial, if_neg] exact ne_of_lt hi #align power_series.order_monomial PowerSeries.order_monomial /-- The order of the monomial `a*X^n` is `n` if `a ≠ 0`. -/ theorem order_monomial_of_ne_zero (n : ℕ) (a : R) (h : a ≠ 0) : order (monomial R n a) = n := by classical rw [order_monomial, if_neg h] #align power_series.order_monomial_of_ne_zero PowerSeries.order_monomial_of_ne_zero /-- If `n` is strictly smaller than the order of `ψ`, then the `n`th coefficient of its product with any other power series is `0`. -/ theorem coeff_mul_of_lt_order {φ ψ : R⟦X⟧} {n : ℕ} (h : ↑n < ψ.order) : coeff R n (φ * ψ) = 0 := by suffices coeff R n (φ * ψ) = ∑ p in antidiagonal n, 0 by rw [this, Finset.sum_const_zero] rw [coeff_mul] apply Finset.sum_congr rfl intro x hx refine' mul_eq_zero_of_right (coeff R x.fst φ) (coeff_of_lt_order x.snd (lt_of_le_of_lt _ h)) rw [mem_antidiagonal] at hx norm_cast omega #align power_series.coeff_mul_of_lt_order PowerSeries.coeff_mul_of_lt_order theorem coeff_mul_one_sub_of_lt_order {R : Type*} [CommRing R] {φ ψ : R⟦X⟧} (n : ℕ) (h : ↑n < ψ.order) : coeff R n (φ * (1 - ψ)) = coeff R n φ := by simp [coeff_mul_of_lt_order h, mul_sub] #align power_series.coeff_mul_one_sub_of_lt_order PowerSeries.coeff_mul_one_sub_of_lt_order theorem coeff_mul_prod_one_sub_of_lt_order {R ι : Type*} [CommRing R] (k : ℕ) (s : Finset ι) (φ : R⟦X⟧) (f : ι → R⟦X⟧) : (∀ i ∈ s, ↑k < (f i).order) → coeff R k (φ * ∏ i in s, (1 - f i)) = coeff R k φ := by classical induction' s using Finset.induction_on with a s ha ih t · simp · intro t simp only [Finset.mem_insert, forall_eq_or_imp] at t rw [Finset.prod_insert ha, ← mul_assoc, mul_right_comm, coeff_mul_one_sub_of_lt_order _ t.1] exact ih t.2 #align power_series.coeff_mul_prod_one_sub_of_lt_order PowerSeries.coeff_mul_prod_one_sub_of_lt_order -- TODO: link with `X_pow_dvd_iff` theorem X_pow_order_dvd (h : (order φ).Dom) : X ^ (order φ).get h ∣ φ := by refine' ⟨PowerSeries.mk fun n => coeff R (n + (order φ).get h) φ, _⟩ ext n simp only [coeff_mul, coeff_X_pow, coeff_mk, boole_mul, Finset.sum_ite, Finset.sum_const_zero, add_zero] rw [Finset.filter_fst_eq_antidiagonal n (Part.get (order φ) h)] split_ifs with hn · simp [tsub_add_cancel_of_le hn] · simp only [Finset.sum_empty] refine' coeff_of_lt_order _ _ simpa [PartENat.coe_lt_iff] using fun _ => hn set_option linter.uppercaseLean3 false in #align power_series.X_pow_order_dvd PowerSeries.X_pow_order_dvd theorem order_eq_multiplicity_X {R : Type*} [Semiring R] [@DecidableRel R⟦X⟧ (· ∣ ·)] (φ : R⟦X⟧) : order φ = multiplicity X φ := by classical rcases eq_or_ne φ 0 with (rfl | hφ) · simp induction' ho : order φ using PartENat.casesOn with n · simp [hφ] at ho have hn : φ.order.get (order_finite_iff_ne_zero.mpr hφ) = n := by simp [ho] rw [← hn] refine' le_antisymm (le_multiplicity_of_pow_dvd <| X_pow_order_dvd (order_finite_iff_ne_zero.mpr hφ)) (PartENat.find_le _ _ _) rintro ⟨ψ, H⟩ have := congr_arg (coeff R n) H rw [← (ψ.commute_X.pow_right _).eq, coeff_mul_of_lt_order, ← hn] at this · exact coeff_order _ this · rw [X_pow_eq, order_monomial] split_ifs · exact PartENat.natCast_lt_top _ · rw [← hn, PartENat.coe_lt_coe] exact Nat.lt_succ_self _ set_option linter.uppercaseLean3 false in #align power_series.order_eq_multiplicity_X PowerSeries.order_eq_multiplicity_X /-- Given a non-zero power series `f`, `divided_by_X_pow_order f` is the power series obtained by dividing out the largest power of X that divides `f`, that is its order-/ def divided_by_X_pow_order {f : PowerSeries R} (hf : f ≠ 0) : R⟦X⟧ := (exists_eq_mul_right_of_dvd (X_pow_order_dvd (order_finite_iff_ne_zero.2 hf))).choose

theorem self_eq_X_pow_order_mul_divided_by_X_pow_order {f : R⟦X⟧} (hf : f ≠ 0) : X ^ f.order.get (order_finite_iff_ne_zero.mpr hf) * divided_by_X_pow_order hf = f

self_eq_X_pow_order_mul_divided_by_X_pow_order

2019

85a4719

mathlib4/Mathlib/RingTheory/PowerSeries/Order.lean

mathlib

/- Copyright (c) 2019 Johan Commelin. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Johan Commelin, Kenny Lau -/ import Mathlib.RingTheory.PowerSeries.Basic import Mathlib.Algebra.CharP.Basic #align_import ring_theory.power_series.basic from "leanprover-community/mathlib"@"2d5739b61641ee4e7e53eca5688a08f66f2e6a60" /-! # Formal power series (in one variable) - Order The `PowerSeries.order` of a formal power series `φ` is the multiplicity of the variable `X` in `φ`. If the coefficients form an integral domain, then `PowerSeries.order` is an additive valuation (`PowerSeries.order_mul`, `PowerSeries.le_order_add`). We prove that if the commutative ring `R` of coefficients is an integral domain, then the ring `R⟦X⟧` of formal power series in one variable over `R` is an integral domain. Given a non-zero power series `f`, `divided_by_X_pow_order f` is the power series obtained by dividing out the largest power of X that divides `f`, that is its order. This is useful when proving that `R⟦X⟧` is a normalization monoid, which is done in `PowerSeries.Inverse`. -/ noncomputable section open BigOperators Polynomial open Finset (antidiagonal mem_antidiagonal) namespace PowerSeries open Finsupp (single) variable {R : Type*} section OrderBasic open multiplicity variable [Semiring R] {φ : R⟦X⟧} theorem exists_coeff_ne_zero_iff_ne_zero : (∃ n : ℕ, coeff R n φ ≠ 0) ↔ φ ≠ 0 := by refine' not_iff_not.mp _ push_neg -- FIXME: the `FunLike.coe` doesn't seem to be picked up in the expression after #8386? simp [PowerSeries.ext_iff, (coeff R _).map_zero] #align power_series.exists_coeff_ne_zero_iff_ne_zero PowerSeries.exists_coeff_ne_zero_iff_ne_zero /-- The order of a formal power series `φ` is the greatest `n : PartENat` such that `X^n` divides `φ`. The order is `⊤` if and only if `φ = 0`. -/ def order (φ : R⟦X⟧) : PartENat := letI := Classical.decEq R letI := Classical.decEq R⟦X⟧ if h : φ = 0 then ⊤ else Nat.find (exists_coeff_ne_zero_iff_ne_zero.mpr h) #align power_series.order PowerSeries.order /-- The order of the `0` power series is infinite. -/ @[simp] theorem order_zero : order (0 : R⟦X⟧) = ⊤ := dif_pos rfl #align power_series.order_zero PowerSeries.order_zero theorem order_finite_iff_ne_zero : (order φ).Dom ↔ φ ≠ 0 := by simp only [order] constructor · split_ifs with h <;> intro H · simp only [PartENat.top_eq_none, Part.not_none_dom] at H · exact h · intro h simp [h] #align power_series.order_finite_iff_ne_zero PowerSeries.order_finite_iff_ne_zero /-- If the order of a formal power series is finite, then the coefficient indexed by the order is nonzero. -/ theorem coeff_order (h : (order φ).Dom) : coeff R (φ.order.get h) φ ≠ 0 := by classical simp only [order, order_finite_iff_ne_zero.mp h, not_false_iff, dif_neg, PartENat.get_natCast'] generalize_proofs h exact Nat.find_spec h #align power_series.coeff_order PowerSeries.coeff_order /-- If the `n`th coefficient of a formal power series is nonzero, then the order of the power series is less than or equal to `n`. -/ theorem order_le (n : ℕ) (h : coeff R n φ ≠ 0) : order φ ≤ n := by classical rw [order, dif_neg] · simp only [PartENat.coe_le_coe] exact Nat.find_le h · exact exists_coeff_ne_zero_iff_ne_zero.mp ⟨n, h⟩ #align power_series.order_le PowerSeries.order_le /-- The `n`th coefficient of a formal power series is `0` if `n` is strictly smaller than the order of the power series. -/ theorem coeff_of_lt_order (n : ℕ) (h : ↑n < order φ) : coeff R n φ = 0 := by contrapose! h exact order_le _ h #align power_series.coeff_of_lt_order PowerSeries.coeff_of_lt_order /-- The `0` power series is the unique power series with infinite order. -/ @[simp] theorem order_eq_top {φ : R⟦X⟧} : φ.order = ⊤ ↔ φ = 0 := PartENat.not_dom_iff_eq_top.symm.trans order_finite_iff_ne_zero.not_left #align power_series.order_eq_top PowerSeries.order_eq_top /-- The order of a formal power series is at least `n` if the `i`th coefficient is `0` for all `i < n`. -/ theorem nat_le_order (φ : R⟦X⟧) (n : ℕ) (h : ∀ i < n, coeff R i φ = 0) : ↑n ≤ order φ := by by_contra H; rw [not_le] at H have : (order φ).Dom := PartENat.dom_of_le_natCast H.le rw [← PartENat.natCast_get this, PartENat.coe_lt_coe] at H exact coeff_order this (h _ H) #align power_series.nat_le_order PowerSeries.nat_le_order /-- The order of a formal power series is at least `n` if the `i`th coefficient is `0` for all `i < n`. -/ theorem le_order (φ : R⟦X⟧) (n : PartENat) (h : ∀ i : ℕ, ↑i < n → coeff R i φ = 0) : n ≤ order φ := by induction n using PartENat.casesOn · show _ ≤ _ rw [top_le_iff, order_eq_top] ext i exact h _ (PartENat.natCast_lt_top i) · apply nat_le_order simpa only [PartENat.coe_lt_coe] using h #align power_series.le_order PowerSeries.le_order /-- The order of a formal power series is exactly `n` if the `n`th coefficient is nonzero, and the `i`th coefficient is `0` for all `i < n`. -/ theorem order_eq_nat {φ : R⟦X⟧} {n : ℕ} : order φ = n ↔ coeff R n φ ≠ 0 ∧ ∀ i, i < n → coeff R i φ = 0 := by classical rcases eq_or_ne φ 0 with (rfl | hφ) · simpa [(coeff R _).map_zero] using (PartENat.natCast_ne_top _).symm simp [order, dif_neg hφ, Nat.find_eq_iff] #align power_series.order_eq_nat PowerSeries.order_eq_nat /-- The order of a formal power series is exactly `n` if the `n`th coefficient is nonzero, and the `i`th coefficient is `0` for all `i < n`. -/ theorem order_eq {φ : R⟦X⟧} {n : PartENat} : order φ = n ↔ (∀ i : ℕ, ↑i = n → coeff R i φ ≠ 0) ∧ ∀ i : ℕ, ↑i < n → coeff R i φ = 0 := by induction n using PartENat.casesOn · rw [order_eq_top] constructor · rintro rfl constructor <;> intros · exfalso exact PartENat.natCast_ne_top ‹_› ‹_› · exact (coeff _ _).map_zero · rintro ⟨_h₁, h₂⟩ ext i exact h₂ i (PartENat.natCast_lt_top i) · simpa [PartENat.natCast_inj] using order_eq_nat #align power_series.order_eq PowerSeries.order_eq /-- The order of the sum of two formal power series is at least the minimum of their orders. -/ theorem le_order_add (φ ψ : R⟦X⟧) : min (order φ) (order ψ) ≤ order (φ + ψ) := by refine' le_order _ _ _ simp (config := { contextual := true }) [coeff_of_lt_order] #align power_series.le_order_add PowerSeries.le_order_add private theorem order_add_of_order_eq.aux (φ ψ : R⟦X⟧) (_h : order φ ≠ order ψ) (H : order φ < order ψ) : order (φ + ψ) ≤ order φ ⊓ order ψ := by suffices order (φ + ψ) = order φ by rw [le_inf_iff, this] exact ⟨le_rfl, le_of_lt H⟩ · rw [order_eq] constructor · intro i hi rw [← hi] at H rw [(coeff _ _).map_add, coeff_of_lt_order i H, add_zero] exact (order_eq_nat.1 hi.symm).1 · intro i hi rw [(coeff _ _).map_add, coeff_of_lt_order i hi, coeff_of_lt_order i (lt_trans hi H), zero_add] -- #align power_series.order_add_of_order_eq.aux power_series.order_add_of_order_eq.aux /-- The order of the sum of two formal power series is the minimum of their orders if their orders differ. -/ theorem order_add_of_order_eq (φ ψ : R⟦X⟧) (h : order φ ≠ order ψ) : order (φ + ψ) = order φ ⊓ order ψ := by refine' le_antisymm _ (le_order_add _ _) by_cases H₁ : order φ < order ψ · apply order_add_of_order_eq.aux _ _ h H₁ by_cases H₂ : order ψ < order φ · simpa only [add_comm, inf_comm] using order_add_of_order_eq.aux _ _ h.symm H₂ exfalso; exact h (le_antisymm (not_lt.1 H₂) (not_lt.1 H₁)) #align power_series.order_add_of_order_eq PowerSeries.order_add_of_order_eq /-- The order of the product of two formal power series is at least the sum of their orders. -/ theorem order_mul_ge (φ ψ : R⟦X⟧) : order φ + order ψ ≤ order (φ * ψ) := by apply le_order intro n hn; rw [coeff_mul, Finset.sum_eq_zero] rintro ⟨i, j⟩ hij by_cases hi : ↑i < order φ · rw [coeff_of_lt_order i hi, zero_mul] by_cases hj : ↑j < order ψ · rw [coeff_of_lt_order j hj, mul_zero] rw [not_lt] at hi hj; rw [mem_antidiagonal] at hij exfalso apply ne_of_lt (lt_of_lt_of_le hn <| add_le_add hi hj) rw [← Nat.cast_add, hij] #align power_series.order_mul_ge PowerSeries.order_mul_ge /-- The order of the monomial `a*X^n` is infinite if `a = 0` and `n` otherwise. -/ theorem order_monomial (n : ℕ) (a : R) [Decidable (a = 0)] : order (monomial R n a) = if a = 0 then (⊤ : PartENat) else n := by split_ifs with h · rw [h, order_eq_top, LinearMap.map_zero] · rw [order_eq] constructor <;> intro i hi · rw [PartENat.natCast_inj] at hi rwa [hi, coeff_monomial_same] · rw [PartENat.coe_lt_coe] at hi rw [coeff_monomial, if_neg] exact ne_of_lt hi #align power_series.order_monomial PowerSeries.order_monomial /-- The order of the monomial `a*X^n` is `n` if `a ≠ 0`. -/ theorem order_monomial_of_ne_zero (n : ℕ) (a : R) (h : a ≠ 0) : order (monomial R n a) = n := by classical rw [order_monomial, if_neg h] #align power_series.order_monomial_of_ne_zero PowerSeries.order_monomial_of_ne_zero /-- If `n` is strictly smaller than the order of `ψ`, then the `n`th coefficient of its product with any other power series is `0`. -/ theorem coeff_mul_of_lt_order {φ ψ : R⟦X⟧} {n : ℕ} (h : ↑n < ψ.order) : coeff R n (φ * ψ) = 0 := by suffices coeff R n (φ * ψ) = ∑ p in antidiagonal n, 0 by rw [this, Finset.sum_const_zero] rw [coeff_mul] apply Finset.sum_congr rfl intro x hx refine' mul_eq_zero_of_right (coeff R x.fst φ) (coeff_of_lt_order x.snd (lt_of_le_of_lt _ h)) rw [mem_antidiagonal] at hx norm_cast omega #align power_series.coeff_mul_of_lt_order PowerSeries.coeff_mul_of_lt_order theorem coeff_mul_one_sub_of_lt_order {R : Type*} [CommRing R] {φ ψ : R⟦X⟧} (n : ℕ) (h : ↑n < ψ.order) : coeff R n (φ * (1 - ψ)) = coeff R n φ := by simp [coeff_mul_of_lt_order h, mul_sub] #align power_series.coeff_mul_one_sub_of_lt_order PowerSeries.coeff_mul_one_sub_of_lt_order theorem coeff_mul_prod_one_sub_of_lt_order {R ι : Type*} [CommRing R] (k : ℕ) (s : Finset ι) (φ : R⟦X⟧) (f : ι → R⟦X⟧) : (∀ i ∈ s, ↑k < (f i).order) → coeff R k (φ * ∏ i in s, (1 - f i)) = coeff R k φ := by classical induction' s using Finset.induction_on with a s ha ih t · simp · intro t simp only [Finset.mem_insert, forall_eq_or_imp] at t rw [Finset.prod_insert ha, ← mul_assoc, mul_right_comm, coeff_mul_one_sub_of_lt_order _ t.1] exact ih t.2 #align power_series.coeff_mul_prod_one_sub_of_lt_order PowerSeries.coeff_mul_prod_one_sub_of_lt_order -- TODO: link with `X_pow_dvd_iff` theorem X_pow_order_dvd (h : (order φ).Dom) : X ^ (order φ).get h ∣ φ := by refine' ⟨PowerSeries.mk fun n => coeff R (n + (order φ).get h) φ, _⟩ ext n simp only [coeff_mul, coeff_X_pow, coeff_mk, boole_mul, Finset.sum_ite, Finset.sum_const_zero, add_zero] rw [Finset.filter_fst_eq_antidiagonal n (Part.get (order φ) h)] split_ifs with hn · simp [tsub_add_cancel_of_le hn] · simp only [Finset.sum_empty] refine' coeff_of_lt_order _ _ simpa [PartENat.coe_lt_iff] using fun _ => hn set_option linter.uppercaseLean3 false in #align power_series.X_pow_order_dvd PowerSeries.X_pow_order_dvd theorem order_eq_multiplicity_X {R : Type*} [Semiring R] [@DecidableRel R⟦X⟧ (· ∣ ·)] (φ : R⟦X⟧) : order φ = multiplicity X φ := by classical rcases eq_or_ne φ 0 with (rfl | hφ) · simp induction' ho : order φ using PartENat.casesOn with n · simp [hφ] at ho have hn : φ.order.get (order_finite_iff_ne_zero.mpr hφ) = n := by simp [ho] rw [← hn] refine' le_antisymm (le_multiplicity_of_pow_dvd <| X_pow_order_dvd (order_finite_iff_ne_zero.mpr hφ)) (PartENat.find_le _ _ _) rintro ⟨ψ, H⟩ have := congr_arg (coeff R n) H rw [← (ψ.commute_X.pow_right _).eq, coeff_mul_of_lt_order, ← hn] at this · exact coeff_order _ this · rw [X_pow_eq, order_monomial] split_ifs · exact PartENat.natCast_lt_top _ · rw [← hn, PartENat.coe_lt_coe] exact Nat.lt_succ_self _ set_option linter.uppercaseLean3 false in #align power_series.order_eq_multiplicity_X PowerSeries.order_eq_multiplicity_X /-- Given a non-zero power series `f`, `divided_by_X_pow_order f` is the power series obtained by dividing out the largest power of X that divides `f`, that is its order-/ def divided_by_X_pow_order {f : PowerSeries R} (hf : f ≠ 0) : R⟦X⟧ := (exists_eq_mul_right_of_dvd (X_pow_order_dvd (order_finite_iff_ne_zero.2 hf))).choose theorem self_eq_X_pow_order_mul_divided_by_X_pow_order {f : R⟦X⟧} (hf : f ≠ 0) : X ^ f.order.get (order_finite_iff_ne_zero.mpr hf) * divided_by_X_pow_order hf = f := haveI dvd := X_pow_order_dvd (order_finite_iff_ne_zero.mpr hf) (exists_eq_mul_right_of_dvd dvd).choose_spec.symm end OrderBasic section OrderZeroNeOne variable [Semiring R] [Nontrivial R] /-- The order of the formal power series `1` is `0`. -/ @[simp] theorem order_one : order (1 : R⟦X⟧) = 0 := by simpa using order_monomial_of_ne_zero 0 (1 : R) one_ne_zero #align power_series.order_one PowerSeries.order_one /-- The order of an invertible power series is `0`. -/

theorem order_zero_of_unit {f : PowerSeries R} : IsUnit f → f.order = 0

order_zero_of_unit

2019

85a4719

mathlib4/Mathlib/RingTheory/PowerSeries/Order.lean

mathlib

/- Copyright (c) 2019 Johan Commelin. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Johan Commelin, Kenny Lau -/ import Mathlib.RingTheory.PowerSeries.Basic import Mathlib.Algebra.CharP.Basic #align_import ring_theory.power_series.basic from "leanprover-community/mathlib"@"2d5739b61641ee4e7e53eca5688a08f66f2e6a60" /-! # Formal power series (in one variable) - Order The `PowerSeries.order` of a formal power series `φ` is the multiplicity of the variable `X` in `φ`. If the coefficients form an integral domain, then `PowerSeries.order` is an additive valuation (`PowerSeries.order_mul`, `PowerSeries.le_order_add`). We prove that if the commutative ring `R` of coefficients is an integral domain, then the ring `R⟦X⟧` of formal power series in one variable over `R` is an integral domain. Given a non-zero power series `f`, `divided_by_X_pow_order f` is the power series obtained by dividing out the largest power of X that divides `f`, that is its order. This is useful when proving that `R⟦X⟧` is a normalization monoid, which is done in `PowerSeries.Inverse`. -/ noncomputable section open BigOperators Polynomial open Finset (antidiagonal mem_antidiagonal) namespace PowerSeries open Finsupp (single) variable {R : Type*} section OrderBasic open multiplicity variable [Semiring R] {φ : R⟦X⟧} theorem exists_coeff_ne_zero_iff_ne_zero : (∃ n : ℕ, coeff R n φ ≠ 0) ↔ φ ≠ 0 := by refine' not_iff_not.mp _ push_neg -- FIXME: the `FunLike.coe` doesn't seem to be picked up in the expression after #8386? simp [PowerSeries.ext_iff, (coeff R _).map_zero] #align power_series.exists_coeff_ne_zero_iff_ne_zero PowerSeries.exists_coeff_ne_zero_iff_ne_zero /-- The order of a formal power series `φ` is the greatest `n : PartENat` such that `X^n` divides `φ`. The order is `⊤` if and only if `φ = 0`. -/ def order (φ : R⟦X⟧) : PartENat := letI := Classical.decEq R letI := Classical.decEq R⟦X⟧ if h : φ = 0 then ⊤ else Nat.find (exists_coeff_ne_zero_iff_ne_zero.mpr h) #align power_series.order PowerSeries.order /-- The order of the `0` power series is infinite. -/ @[simp] theorem order_zero : order (0 : R⟦X⟧) = ⊤ := dif_pos rfl #align power_series.order_zero PowerSeries.order_zero theorem order_finite_iff_ne_zero : (order φ).Dom ↔ φ ≠ 0 := by simp only [order] constructor · split_ifs with h <;> intro H · simp only [PartENat.top_eq_none, Part.not_none_dom] at H · exact h · intro h simp [h] #align power_series.order_finite_iff_ne_zero PowerSeries.order_finite_iff_ne_zero /-- If the order of a formal power series is finite, then the coefficient indexed by the order is nonzero. -/ theorem coeff_order (h : (order φ).Dom) : coeff R (φ.order.get h) φ ≠ 0 := by classical simp only [order, order_finite_iff_ne_zero.mp h, not_false_iff, dif_neg, PartENat.get_natCast'] generalize_proofs h exact Nat.find_spec h #align power_series.coeff_order PowerSeries.coeff_order /-- If the `n`th coefficient of a formal power series is nonzero, then the order of the power series is less than or equal to `n`. -/ theorem order_le (n : ℕ) (h : coeff R n φ ≠ 0) : order φ ≤ n := by classical rw [order, dif_neg] · simp only [PartENat.coe_le_coe] exact Nat.find_le h · exact exists_coeff_ne_zero_iff_ne_zero.mp ⟨n, h⟩ #align power_series.order_le PowerSeries.order_le /-- The `n`th coefficient of a formal power series is `0` if `n` is strictly smaller than the order of the power series. -/ theorem coeff_of_lt_order (n : ℕ) (h : ↑n < order φ) : coeff R n φ = 0 := by contrapose! h exact order_le _ h #align power_series.coeff_of_lt_order PowerSeries.coeff_of_lt_order /-- The `0` power series is the unique power series with infinite order. -/ @[simp] theorem order_eq_top {φ : R⟦X⟧} : φ.order = ⊤ ↔ φ = 0 := PartENat.not_dom_iff_eq_top.symm.trans order_finite_iff_ne_zero.not_left #align power_series.order_eq_top PowerSeries.order_eq_top /-- The order of a formal power series is at least `n` if the `i`th coefficient is `0` for all `i < n`. -/ theorem nat_le_order (φ : R⟦X⟧) (n : ℕ) (h : ∀ i < n, coeff R i φ = 0) : ↑n ≤ order φ := by by_contra H; rw [not_le] at H have : (order φ).Dom := PartENat.dom_of_le_natCast H.le rw [← PartENat.natCast_get this, PartENat.coe_lt_coe] at H exact coeff_order this (h _ H) #align power_series.nat_le_order PowerSeries.nat_le_order /-- The order of a formal power series is at least `n` if the `i`th coefficient is `0` for all `i < n`. -/ theorem le_order (φ : R⟦X⟧) (n : PartENat) (h : ∀ i : ℕ, ↑i < n → coeff R i φ = 0) : n ≤ order φ := by induction n using PartENat.casesOn · show _ ≤ _ rw [top_le_iff, order_eq_top] ext i exact h _ (PartENat.natCast_lt_top i) · apply nat_le_order simpa only [PartENat.coe_lt_coe] using h #align power_series.le_order PowerSeries.le_order /-- The order of a formal power series is exactly `n` if the `n`th coefficient is nonzero, and the `i`th coefficient is `0` for all `i < n`. -/ theorem order_eq_nat {φ : R⟦X⟧} {n : ℕ} : order φ = n ↔ coeff R n φ ≠ 0 ∧ ∀ i, i < n → coeff R i φ = 0 := by classical rcases eq_or_ne φ 0 with (rfl | hφ) · simpa [(coeff R _).map_zero] using (PartENat.natCast_ne_top _).symm simp [order, dif_neg hφ, Nat.find_eq_iff] #align power_series.order_eq_nat PowerSeries.order_eq_nat /-- The order of a formal power series is exactly `n` if the `n`th coefficient is nonzero, and the `i`th coefficient is `0` for all `i < n`. -/ theorem order_eq {φ : R⟦X⟧} {n : PartENat} : order φ = n ↔ (∀ i : ℕ, ↑i = n → coeff R i φ ≠ 0) ∧ ∀ i : ℕ, ↑i < n → coeff R i φ = 0 := by induction n using PartENat.casesOn · rw [order_eq_top] constructor · rintro rfl constructor <;> intros · exfalso exact PartENat.natCast_ne_top ‹_› ‹_› · exact (coeff _ _).map_zero · rintro ⟨_h₁, h₂⟩ ext i exact h₂ i (PartENat.natCast_lt_top i) · simpa [PartENat.natCast_inj] using order_eq_nat #align power_series.order_eq PowerSeries.order_eq /-- The order of the sum of two formal power series is at least the minimum of their orders. -/ theorem le_order_add (φ ψ : R⟦X⟧) : min (order φ) (order ψ) ≤ order (φ + ψ) := by refine' le_order _ _ _ simp (config := { contextual := true }) [coeff_of_lt_order] #align power_series.le_order_add PowerSeries.le_order_add private theorem order_add_of_order_eq.aux (φ ψ : R⟦X⟧) (_h : order φ ≠ order ψ) (H : order φ < order ψ) : order (φ + ψ) ≤ order φ ⊓ order ψ := by suffices order (φ + ψ) = order φ by rw [le_inf_iff, this] exact ⟨le_rfl, le_of_lt H⟩ · rw [order_eq] constructor · intro i hi rw [← hi] at H rw [(coeff _ _).map_add, coeff_of_lt_order i H, add_zero] exact (order_eq_nat.1 hi.symm).1 · intro i hi rw [(coeff _ _).map_add, coeff_of_lt_order i hi, coeff_of_lt_order i (lt_trans hi H), zero_add] -- #align power_series.order_add_of_order_eq.aux power_series.order_add_of_order_eq.aux /-- The order of the sum of two formal power series is the minimum of their orders if their orders differ. -/ theorem order_add_of_order_eq (φ ψ : R⟦X⟧) (h : order φ ≠ order ψ) : order (φ + ψ) = order φ ⊓ order ψ := by refine' le_antisymm _ (le_order_add _ _) by_cases H₁ : order φ < order ψ · apply order_add_of_order_eq.aux _ _ h H₁ by_cases H₂ : order ψ < order φ · simpa only [add_comm, inf_comm] using order_add_of_order_eq.aux _ _ h.symm H₂ exfalso; exact h (le_antisymm (not_lt.1 H₂) (not_lt.1 H₁)) #align power_series.order_add_of_order_eq PowerSeries.order_add_of_order_eq /-- The order of the product of two formal power series is at least the sum of their orders. -/ theorem order_mul_ge (φ ψ : R⟦X⟧) : order φ + order ψ ≤ order (φ * ψ) := by apply le_order intro n hn; rw [coeff_mul, Finset.sum_eq_zero] rintro ⟨i, j⟩ hij by_cases hi : ↑i < order φ · rw [coeff_of_lt_order i hi, zero_mul] by_cases hj : ↑j < order ψ · rw [coeff_of_lt_order j hj, mul_zero] rw [not_lt] at hi hj; rw [mem_antidiagonal] at hij exfalso apply ne_of_lt (lt_of_lt_of_le hn <| add_le_add hi hj) rw [← Nat.cast_add, hij] #align power_series.order_mul_ge PowerSeries.order_mul_ge /-- The order of the monomial `a*X^n` is infinite if `a = 0` and `n` otherwise. -/ theorem order_monomial (n : ℕ) (a : R) [Decidable (a = 0)] : order (monomial R n a) = if a = 0 then (⊤ : PartENat) else n := by split_ifs with h · rw [h, order_eq_top, LinearMap.map_zero] · rw [order_eq] constructor <;> intro i hi · rw [PartENat.natCast_inj] at hi rwa [hi, coeff_monomial_same] · rw [PartENat.coe_lt_coe] at hi rw [coeff_monomial, if_neg] exact ne_of_lt hi #align power_series.order_monomial PowerSeries.order_monomial /-- The order of the monomial `a*X^n` is `n` if `a ≠ 0`. -/ theorem order_monomial_of_ne_zero (n : ℕ) (a : R) (h : a ≠ 0) : order (monomial R n a) = n := by classical rw [order_monomial, if_neg h] #align power_series.order_monomial_of_ne_zero PowerSeries.order_monomial_of_ne_zero /-- If `n` is strictly smaller than the order of `ψ`, then the `n`th coefficient of its product with any other power series is `0`. -/ theorem coeff_mul_of_lt_order {φ ψ : R⟦X⟧} {n : ℕ} (h : ↑n < ψ.order) : coeff R n (φ * ψ) = 0 := by suffices coeff R n (φ * ψ) = ∑ p in antidiagonal n, 0 by rw [this, Finset.sum_const_zero] rw [coeff_mul] apply Finset.sum_congr rfl intro x hx refine' mul_eq_zero_of_right (coeff R x.fst φ) (coeff_of_lt_order x.snd (lt_of_le_of_lt _ h)) rw [mem_antidiagonal] at hx norm_cast omega #align power_series.coeff_mul_of_lt_order PowerSeries.coeff_mul_of_lt_order theorem coeff_mul_one_sub_of_lt_order {R : Type*} [CommRing R] {φ ψ : R⟦X⟧} (n : ℕ) (h : ↑n < ψ.order) : coeff R n (φ * (1 - ψ)) = coeff R n φ := by simp [coeff_mul_of_lt_order h, mul_sub] #align power_series.coeff_mul_one_sub_of_lt_order PowerSeries.coeff_mul_one_sub_of_lt_order theorem coeff_mul_prod_one_sub_of_lt_order {R ι : Type*} [CommRing R] (k : ℕ) (s : Finset ι) (φ : R⟦X⟧) (f : ι → R⟦X⟧) : (∀ i ∈ s, ↑k < (f i).order) → coeff R k (φ * ∏ i in s, (1 - f i)) = coeff R k φ := by classical induction' s using Finset.induction_on with a s ha ih t · simp · intro t simp only [Finset.mem_insert, forall_eq_or_imp] at t rw [Finset.prod_insert ha, ← mul_assoc, mul_right_comm, coeff_mul_one_sub_of_lt_order _ t.1] exact ih t.2 #align power_series.coeff_mul_prod_one_sub_of_lt_order PowerSeries.coeff_mul_prod_one_sub_of_lt_order -- TODO: link with `X_pow_dvd_iff` theorem X_pow_order_dvd (h : (order φ).Dom) : X ^ (order φ).get h ∣ φ := by refine' ⟨PowerSeries.mk fun n => coeff R (n + (order φ).get h) φ, _⟩ ext n simp only [coeff_mul, coeff_X_pow, coeff_mk, boole_mul, Finset.sum_ite, Finset.sum_const_zero, add_zero] rw [Finset.filter_fst_eq_antidiagonal n (Part.get (order φ) h)] split_ifs with hn · simp [tsub_add_cancel_of_le hn] · simp only [Finset.sum_empty] refine' coeff_of_lt_order _ _ simpa [PartENat.coe_lt_iff] using fun _ => hn set_option linter.uppercaseLean3 false in #align power_series.X_pow_order_dvd PowerSeries.X_pow_order_dvd theorem order_eq_multiplicity_X {R : Type*} [Semiring R] [@DecidableRel R⟦X⟧ (· ∣ ·)] (φ : R⟦X⟧) : order φ = multiplicity X φ := by classical rcases eq_or_ne φ 0 with (rfl | hφ) · simp induction' ho : order φ using PartENat.casesOn with n · simp [hφ] at ho have hn : φ.order.get (order_finite_iff_ne_zero.mpr hφ) = n := by simp [ho] rw [← hn] refine' le_antisymm (le_multiplicity_of_pow_dvd <| X_pow_order_dvd (order_finite_iff_ne_zero.mpr hφ)) (PartENat.find_le _ _ _) rintro ⟨ψ, H⟩ have := congr_arg (coeff R n) H rw [← (ψ.commute_X.pow_right _).eq, coeff_mul_of_lt_order, ← hn] at this · exact coeff_order _ this · rw [X_pow_eq, order_monomial] split_ifs · exact PartENat.natCast_lt_top _ · rw [← hn, PartENat.coe_lt_coe] exact Nat.lt_succ_self _ set_option linter.uppercaseLean3 false in #align power_series.order_eq_multiplicity_X PowerSeries.order_eq_multiplicity_X /-- Given a non-zero power series `f`, `divided_by_X_pow_order f` is the power series obtained by dividing out the largest power of X that divides `f`, that is its order-/ def divided_by_X_pow_order {f : PowerSeries R} (hf : f ≠ 0) : R⟦X⟧ := (exists_eq_mul_right_of_dvd (X_pow_order_dvd (order_finite_iff_ne_zero.2 hf))).choose theorem self_eq_X_pow_order_mul_divided_by_X_pow_order {f : R⟦X⟧} (hf : f ≠ 0) : X ^ f.order.get (order_finite_iff_ne_zero.mpr hf) * divided_by_X_pow_order hf = f := haveI dvd := X_pow_order_dvd (order_finite_iff_ne_zero.mpr hf) (exists_eq_mul_right_of_dvd dvd).choose_spec.symm end OrderBasic section OrderZeroNeOne variable [Semiring R] [Nontrivial R] /-- The order of the formal power series `1` is `0`. -/ @[simp] theorem order_one : order (1 : R⟦X⟧) = 0 := by simpa using order_monomial_of_ne_zero 0 (1 : R) one_ne_zero #align power_series.order_one PowerSeries.order_one /-- The order of an invertible power series is `0`. -/ theorem order_zero_of_unit {f : PowerSeries R} : IsUnit f → f.order = 0 := by rintro ⟨⟨u, v, hu, hv⟩, hf⟩ apply And.left rw [← add_eq_zero_iff, ← hf, ← nonpos_iff_eq_zero, ← @order_one R _ _, ← hu] exact order_mul_ge _ _ /-- The order of the formal power series `X` is `1`. -/ @[simp] theorem order_X : order (X : R⟦X⟧) = 1 := by simpa only [Nat.cast_one] using order_monomial_of_ne_zero 1 (1 : R) one_ne_zero set_option linter.uppercaseLean3 false in #align power_series.order_X PowerSeries.order_X /-- The order of the formal power series `X^n` is `n`. -/ @[simp] theorem order_X_pow (n : ℕ) : order ((X : R⟦X⟧) ^ n) = n := by rw [X_pow_eq, order_monomial_of_ne_zero] exact one_ne_zero set_option linter.uppercaseLean3 false in #align power_series.order_X_pow PowerSeries.order_X_pow end OrderZeroNeOne section OrderIsDomain -- TODO: generalize to `[Semiring R] [NoZeroDivisors R]` variable [CommRing R] [IsDomain R] /-- The order of the product of two formal power series over an integral domain is the sum of their orders. -/ theorem order_mul (φ ψ : R⟦X⟧) : order (φ * ψ) = order φ + order ψ := by classical simp_rw [order_eq_multiplicity_X] exact multiplicity.mul X_prime #align power_series.order_mul PowerSeries.order_mul -- Dividing `X` by the maximal power of `X` dividing it leaves `1`.

@[simp] theorem divided_by_X_pow_order_of_X_eq_one : divided_by_X_pow_order X_ne_zero = (1 : R⟦X⟧)

divided_by_X_pow_order_of_X_eq_one

2019

85a4719

mathlib4/Mathlib/RingTheory/PowerSeries/Order.lean

mathlib

/- Copyright (c) 2019 Johan Commelin. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Johan Commelin, Kenny Lau -/ import Mathlib.RingTheory.PowerSeries.Basic import Mathlib.Algebra.CharP.Basic #align_import ring_theory.power_series.basic from "leanprover-community/mathlib"@"2d5739b61641ee4e7e53eca5688a08f66f2e6a60" /-! # Formal power series (in one variable) - Order The `PowerSeries.order` of a formal power series `φ` is the multiplicity of the variable `X` in `φ`. If the coefficients form an integral domain, then `PowerSeries.order` is an additive valuation (`PowerSeries.order_mul`, `PowerSeries.le_order_add`). We prove that if the commutative ring `R` of coefficients is an integral domain, then the ring `R⟦X⟧` of formal power series in one variable over `R` is an integral domain. Given a non-zero power series `f`, `divided_by_X_pow_order f` is the power series obtained by dividing out the largest power of X that divides `f`, that is its order. This is useful when proving that `R⟦X⟧` is a normalization monoid, which is done in `PowerSeries.Inverse`. -/ noncomputable section open BigOperators Polynomial open Finset (antidiagonal mem_antidiagonal) namespace PowerSeries open Finsupp (single) variable {R : Type*} section OrderBasic open multiplicity variable [Semiring R] {φ : R⟦X⟧} theorem exists_coeff_ne_zero_iff_ne_zero : (∃ n : ℕ, coeff R n φ ≠ 0) ↔ φ ≠ 0 := by refine' not_iff_not.mp _ push_neg -- FIXME: the `FunLike.coe` doesn't seem to be picked up in the expression after #8386? simp [PowerSeries.ext_iff, (coeff R _).map_zero] #align power_series.exists_coeff_ne_zero_iff_ne_zero PowerSeries.exists_coeff_ne_zero_iff_ne_zero /-- The order of a formal power series `φ` is the greatest `n : PartENat` such that `X^n` divides `φ`. The order is `⊤` if and only if `φ = 0`. -/ def order (φ : R⟦X⟧) : PartENat := letI := Classical.decEq R letI := Classical.decEq R⟦X⟧ if h : φ = 0 then ⊤ else Nat.find (exists_coeff_ne_zero_iff_ne_zero.mpr h) #align power_series.order PowerSeries.order /-- The order of the `0` power series is infinite. -/ @[simp] theorem order_zero : order (0 : R⟦X⟧) = ⊤ := dif_pos rfl #align power_series.order_zero PowerSeries.order_zero theorem order_finite_iff_ne_zero : (order φ).Dom ↔ φ ≠ 0 := by simp only [order] constructor · split_ifs with h <;> intro H · simp only [PartENat.top_eq_none, Part.not_none_dom] at H · exact h · intro h simp [h] #align power_series.order_finite_iff_ne_zero PowerSeries.order_finite_iff_ne_zero /-- If the order of a formal power series is finite, then the coefficient indexed by the order is nonzero. -/ theorem coeff_order (h : (order φ).Dom) : coeff R (φ.order.get h) φ ≠ 0 := by classical simp only [order, order_finite_iff_ne_zero.mp h, not_false_iff, dif_neg, PartENat.get_natCast'] generalize_proofs h exact Nat.find_spec h #align power_series.coeff_order PowerSeries.coeff_order /-- If the `n`th coefficient of a formal power series is nonzero, then the order of the power series is less than or equal to `n`. -/ theorem order_le (n : ℕ) (h : coeff R n φ ≠ 0) : order φ ≤ n := by classical rw [order, dif_neg] · simp only [PartENat.coe_le_coe] exact Nat.find_le h · exact exists_coeff_ne_zero_iff_ne_zero.mp ⟨n, h⟩ #align power_series.order_le PowerSeries.order_le /-- The `n`th coefficient of a formal power series is `0` if `n` is strictly smaller than the order of the power series. -/ theorem coeff_of_lt_order (n : ℕ) (h : ↑n < order φ) : coeff R n φ = 0 := by contrapose! h exact order_le _ h #align power_series.coeff_of_lt_order PowerSeries.coeff_of_lt_order /-- The `0` power series is the unique power series with infinite order. -/ @[simp] theorem order_eq_top {φ : R⟦X⟧} : φ.order = ⊤ ↔ φ = 0 := PartENat.not_dom_iff_eq_top.symm.trans order_finite_iff_ne_zero.not_left #align power_series.order_eq_top PowerSeries.order_eq_top /-- The order of a formal power series is at least `n` if the `i`th coefficient is `0` for all `i < n`. -/ theorem nat_le_order (φ : R⟦X⟧) (n : ℕ) (h : ∀ i < n, coeff R i φ = 0) : ↑n ≤ order φ := by by_contra H; rw [not_le] at H have : (order φ).Dom := PartENat.dom_of_le_natCast H.le rw [← PartENat.natCast_get this, PartENat.coe_lt_coe] at H exact coeff_order this (h _ H) #align power_series.nat_le_order PowerSeries.nat_le_order /-- The order of a formal power series is at least `n` if the `i`th coefficient is `0` for all `i < n`. -/ theorem le_order (φ : R⟦X⟧) (n : PartENat) (h : ∀ i : ℕ, ↑i < n → coeff R i φ = 0) : n ≤ order φ := by induction n using PartENat.casesOn · show _ ≤ _ rw [top_le_iff, order_eq_top] ext i exact h _ (PartENat.natCast_lt_top i) · apply nat_le_order simpa only [PartENat.coe_lt_coe] using h #align power_series.le_order PowerSeries.le_order /-- The order of a formal power series is exactly `n` if the `n`th coefficient is nonzero, and the `i`th coefficient is `0` for all `i < n`. -/ theorem order_eq_nat {φ : R⟦X⟧} {n : ℕ} : order φ = n ↔ coeff R n φ ≠ 0 ∧ ∀ i, i < n → coeff R i φ = 0 := by classical rcases eq_or_ne φ 0 with (rfl | hφ) · simpa [(coeff R _).map_zero] using (PartENat.natCast_ne_top _).symm simp [order, dif_neg hφ, Nat.find_eq_iff] #align power_series.order_eq_nat PowerSeries.order_eq_nat /-- The order of a formal power series is exactly `n` if the `n`th coefficient is nonzero, and the `i`th coefficient is `0` for all `i < n`. -/ theorem order_eq {φ : R⟦X⟧} {n : PartENat} : order φ = n ↔ (∀ i : ℕ, ↑i = n → coeff R i φ ≠ 0) ∧ ∀ i : ℕ, ↑i < n → coeff R i φ = 0 := by induction n using PartENat.casesOn · rw [order_eq_top] constructor · rintro rfl constructor <;> intros · exfalso exact PartENat.natCast_ne_top ‹_› ‹_› · exact (coeff _ _).map_zero · rintro ⟨_h₁, h₂⟩ ext i exact h₂ i (PartENat.natCast_lt_top i) · simpa [PartENat.natCast_inj] using order_eq_nat #align power_series.order_eq PowerSeries.order_eq /-- The order of the sum of two formal power series is at least the minimum of their orders. -/ theorem le_order_add (φ ψ : R⟦X⟧) : min (order φ) (order ψ) ≤ order (φ + ψ) := by refine' le_order _ _ _ simp (config := { contextual := true }) [coeff_of_lt_order] #align power_series.le_order_add PowerSeries.le_order_add private theorem order_add_of_order_eq.aux (φ ψ : R⟦X⟧) (_h : order φ ≠ order ψ) (H : order φ < order ψ) : order (φ + ψ) ≤ order φ ⊓ order ψ := by suffices order (φ + ψ) = order φ by rw [le_inf_iff, this] exact ⟨le_rfl, le_of_lt H⟩ · rw [order_eq] constructor · intro i hi rw [← hi] at H rw [(coeff _ _).map_add, coeff_of_lt_order i H, add_zero] exact (order_eq_nat.1 hi.symm).1 · intro i hi rw [(coeff _ _).map_add, coeff_of_lt_order i hi, coeff_of_lt_order i (lt_trans hi H), zero_add] -- #align power_series.order_add_of_order_eq.aux power_series.order_add_of_order_eq.aux /-- The order of the sum of two formal power series is the minimum of their orders if their orders differ. -/ theorem order_add_of_order_eq (φ ψ : R⟦X⟧) (h : order φ ≠ order ψ) : order (φ + ψ) = order φ ⊓ order ψ := by refine' le_antisymm _ (le_order_add _ _) by_cases H₁ : order φ < order ψ · apply order_add_of_order_eq.aux _ _ h H₁ by_cases H₂ : order ψ < order φ · simpa only [add_comm, inf_comm] using order_add_of_order_eq.aux _ _ h.symm H₂ exfalso; exact h (le_antisymm (not_lt.1 H₂) (not_lt.1 H₁)) #align power_series.order_add_of_order_eq PowerSeries.order_add_of_order_eq /-- The order of the product of two formal power series is at least the sum of their orders. -/ theorem order_mul_ge (φ ψ : R⟦X⟧) : order φ + order ψ ≤ order (φ * ψ) := by apply le_order intro n hn; rw [coeff_mul, Finset.sum_eq_zero] rintro ⟨i, j⟩ hij by_cases hi : ↑i < order φ · rw [coeff_of_lt_order i hi, zero_mul] by_cases hj : ↑j < order ψ · rw [coeff_of_lt_order j hj, mul_zero] rw [not_lt] at hi hj; rw [mem_antidiagonal] at hij exfalso apply ne_of_lt (lt_of_lt_of_le hn <| add_le_add hi hj) rw [← Nat.cast_add, hij] #align power_series.order_mul_ge PowerSeries.order_mul_ge /-- The order of the monomial `a*X^n` is infinite if `a = 0` and `n` otherwise. -/ theorem order_monomial (n : ℕ) (a : R) [Decidable (a = 0)] : order (monomial R n a) = if a = 0 then (⊤ : PartENat) else n := by split_ifs with h · rw [h, order_eq_top, LinearMap.map_zero] · rw [order_eq] constructor <;> intro i hi · rw [PartENat.natCast_inj] at hi rwa [hi, coeff_monomial_same] · rw [PartENat.coe_lt_coe] at hi rw [coeff_monomial, if_neg] exact ne_of_lt hi #align power_series.order_monomial PowerSeries.order_monomial /-- The order of the monomial `a*X^n` is `n` if `a ≠ 0`. -/ theorem order_monomial_of_ne_zero (n : ℕ) (a : R) (h : a ≠ 0) : order (monomial R n a) = n := by classical rw [order_monomial, if_neg h] #align power_series.order_monomial_of_ne_zero PowerSeries.order_monomial_of_ne_zero /-- If `n` is strictly smaller than the order of `ψ`, then the `n`th coefficient of its product with any other power series is `0`. -/ theorem coeff_mul_of_lt_order {φ ψ : R⟦X⟧} {n : ℕ} (h : ↑n < ψ.order) : coeff R n (φ * ψ) = 0 := by suffices coeff R n (φ * ψ) = ∑ p in antidiagonal n, 0 by rw [this, Finset.sum_const_zero] rw [coeff_mul] apply Finset.sum_congr rfl intro x hx refine' mul_eq_zero_of_right (coeff R x.fst φ) (coeff_of_lt_order x.snd (lt_of_le_of_lt _ h)) rw [mem_antidiagonal] at hx norm_cast omega #align power_series.coeff_mul_of_lt_order PowerSeries.coeff_mul_of_lt_order theorem coeff_mul_one_sub_of_lt_order {R : Type*} [CommRing R] {φ ψ : R⟦X⟧} (n : ℕ) (h : ↑n < ψ.order) : coeff R n (φ * (1 - ψ)) = coeff R n φ := by simp [coeff_mul_of_lt_order h, mul_sub] #align power_series.coeff_mul_one_sub_of_lt_order PowerSeries.coeff_mul_one_sub_of_lt_order theorem coeff_mul_prod_one_sub_of_lt_order {R ι : Type*} [CommRing R] (k : ℕ) (s : Finset ι) (φ : R⟦X⟧) (f : ι → R⟦X⟧) : (∀ i ∈ s, ↑k < (f i).order) → coeff R k (φ * ∏ i in s, (1 - f i)) = coeff R k φ := by classical induction' s using Finset.induction_on with a s ha ih t · simp · intro t simp only [Finset.mem_insert, forall_eq_or_imp] at t rw [Finset.prod_insert ha, ← mul_assoc, mul_right_comm, coeff_mul_one_sub_of_lt_order _ t.1] exact ih t.2 #align power_series.coeff_mul_prod_one_sub_of_lt_order PowerSeries.coeff_mul_prod_one_sub_of_lt_order -- TODO: link with `X_pow_dvd_iff` theorem X_pow_order_dvd (h : (order φ).Dom) : X ^ (order φ).get h ∣ φ := by refine' ⟨PowerSeries.mk fun n => coeff R (n + (order φ).get h) φ, _⟩ ext n simp only [coeff_mul, coeff_X_pow, coeff_mk, boole_mul, Finset.sum_ite, Finset.sum_const_zero, add_zero] rw [Finset.filter_fst_eq_antidiagonal n (Part.get (order φ) h)] split_ifs with hn · simp [tsub_add_cancel_of_le hn] · simp only [Finset.sum_empty] refine' coeff_of_lt_order _ _ simpa [PartENat.coe_lt_iff] using fun _ => hn set_option linter.uppercaseLean3 false in #align power_series.X_pow_order_dvd PowerSeries.X_pow_order_dvd theorem order_eq_multiplicity_X {R : Type*} [Semiring R] [@DecidableRel R⟦X⟧ (· ∣ ·)] (φ : R⟦X⟧) : order φ = multiplicity X φ := by classical rcases eq_or_ne φ 0 with (rfl | hφ) · simp induction' ho : order φ using PartENat.casesOn with n · simp [hφ] at ho have hn : φ.order.get (order_finite_iff_ne_zero.mpr hφ) = n := by simp [ho] rw [← hn] refine' le_antisymm (le_multiplicity_of_pow_dvd <| X_pow_order_dvd (order_finite_iff_ne_zero.mpr hφ)) (PartENat.find_le _ _ _) rintro ⟨ψ, H⟩ have := congr_arg (coeff R n) H rw [← (ψ.commute_X.pow_right _).eq, coeff_mul_of_lt_order, ← hn] at this · exact coeff_order _ this · rw [X_pow_eq, order_monomial] split_ifs · exact PartENat.natCast_lt_top _ · rw [← hn, PartENat.coe_lt_coe] exact Nat.lt_succ_self _ set_option linter.uppercaseLean3 false in #align power_series.order_eq_multiplicity_X PowerSeries.order_eq_multiplicity_X /-- Given a non-zero power series `f`, `divided_by_X_pow_order f` is the power series obtained by dividing out the largest power of X that divides `f`, that is its order-/ def divided_by_X_pow_order {f : PowerSeries R} (hf : f ≠ 0) : R⟦X⟧ := (exists_eq_mul_right_of_dvd (X_pow_order_dvd (order_finite_iff_ne_zero.2 hf))).choose theorem self_eq_X_pow_order_mul_divided_by_X_pow_order {f : R⟦X⟧} (hf : f ≠ 0) : X ^ f.order.get (order_finite_iff_ne_zero.mpr hf) * divided_by_X_pow_order hf = f := haveI dvd := X_pow_order_dvd (order_finite_iff_ne_zero.mpr hf) (exists_eq_mul_right_of_dvd dvd).choose_spec.symm end OrderBasic section OrderZeroNeOne variable [Semiring R] [Nontrivial R] /-- The order of the formal power series `1` is `0`. -/ @[simp] theorem order_one : order (1 : R⟦X⟧) = 0 := by simpa using order_monomial_of_ne_zero 0 (1 : R) one_ne_zero #align power_series.order_one PowerSeries.order_one /-- The order of an invertible power series is `0`. -/ theorem order_zero_of_unit {f : PowerSeries R} : IsUnit f → f.order = 0 := by rintro ⟨⟨u, v, hu, hv⟩, hf⟩ apply And.left rw [← add_eq_zero_iff, ← hf, ← nonpos_iff_eq_zero, ← @order_one R _ _, ← hu] exact order_mul_ge _ _ /-- The order of the formal power series `X` is `1`. -/ @[simp] theorem order_X : order (X : R⟦X⟧) = 1 := by simpa only [Nat.cast_one] using order_monomial_of_ne_zero 1 (1 : R) one_ne_zero set_option linter.uppercaseLean3 false in #align power_series.order_X PowerSeries.order_X /-- The order of the formal power series `X^n` is `n`. -/ @[simp] theorem order_X_pow (n : ℕ) : order ((X : R⟦X⟧) ^ n) = n := by rw [X_pow_eq, order_monomial_of_ne_zero] exact one_ne_zero set_option linter.uppercaseLean3 false in #align power_series.order_X_pow PowerSeries.order_X_pow end OrderZeroNeOne section OrderIsDomain -- TODO: generalize to `[Semiring R] [NoZeroDivisors R]` variable [CommRing R] [IsDomain R] /-- The order of the product of two formal power series over an integral domain is the sum of their orders. -/ theorem order_mul (φ ψ : R⟦X⟧) : order (φ * ψ) = order φ + order ψ := by classical simp_rw [order_eq_multiplicity_X] exact multiplicity.mul X_prime #align power_series.order_mul PowerSeries.order_mul -- Dividing `X` by the maximal power of `X` dividing it leaves `1`. @[simp] theorem divided_by_X_pow_order_of_X_eq_one : divided_by_X_pow_order X_ne_zero = (1 : R⟦X⟧) := by rw [← mul_eq_left₀ X_ne_zero] simpa only [order_X, X_ne_zero, PartENat.get_one, pow_one, Ne.def, not_false_iff] using self_eq_X_pow_order_mul_divided_by_X_pow_order (@X_ne_zero R _ _) -- Dividing a power series by the maximal power of `X` dividing it, respects multiplication.

theorem divided_by_X_pow_orderMul {f g : R⟦X⟧} (hf : f ≠ 0) (hg : g ≠ 0) : divided_by_X_pow_order hf * divided_by_X_pow_order hg = divided_by_X_pow_order (mul_ne_zero hf hg)

divided_by_X_pow_orderMul

2019

85a4719

mathlib4/Mathlib/RingTheory/PowerSeries/Order.lean

mathlib

/- Copyright (c) 2021 Kyle Miller. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kyle Miller -/ import Mathlib.Combinatorics.SimpleGraph.Subgraph import Mathlib.Data.List.Rotate #align_import combinatorics.simple_graph.connectivity from "leanprover-community/mathlib"@"b99e2d58a5e6861833fa8de11e51a81144258db4" /-! # Graph connectivity In a simple graph, * A *walk* is a finite sequence of adjacent vertices, and can be thought of equally well as a sequence of directed edges. * A *trail* is a walk whose edges each appear no more than once. * A *path* is a trail whose vertices appear no more than once. * A *cycle* is a nonempty trail whose first and last vertices are the same and whose vertices except for the first appear no more than once. **Warning:** graph theorists mean something different by "path" than do homotopy theorists. A "walk" in graph theory is a "path" in homotopy theory. Another warning: some graph theorists use "path" and "simple path" for "walk" and "path." Some definitions and theorems have inspiration from multigraph counterparts in [Chou1994]. ## Main definitions * `SimpleGraph.Walk` (with accompanying pattern definitions `SimpleGraph.Walk.nil'` and `SimpleGraph.Walk.cons'`) * `SimpleGraph.Walk.IsTrail`, `SimpleGraph.Walk.IsPath`, and `SimpleGraph.Walk.IsCycle`. * `SimpleGraph.Path` * `SimpleGraph.Walk.map` and `SimpleGraph.Path.map` for the induced map on walks, given an (injective) graph homomorphism. * `SimpleGraph.Reachable` for the relation of whether there exists a walk between a given pair of vertices * `SimpleGraph.Preconnected` and `SimpleGraph.Connected` are predicates on simple graphs for whether every vertex can be reached from every other, and in the latter case, whether the vertex type is nonempty. * `SimpleGraph.ConnectedComponent` is the type of connected components of a given graph. * `SimpleGraph.IsBridge` for whether an edge is a bridge edge ## Main statements * `SimpleGraph.isBridge_iff_mem_and_forall_cycle_not_mem` characterizes bridge edges in terms of there being no cycle containing them. ## Tags walks, trails, paths, circuits, cycles, bridge edges -/ open Function universe u v w namespace SimpleGraph variable {V : Type u} {V' : Type v} {V'' : Type w} variable (G : SimpleGraph V) (G' : SimpleGraph V') (G'' : SimpleGraph V'') /-- A walk is a sequence of adjacent vertices. For vertices `u v : V`, the type `walk u v` consists of all walks starting at `u` and ending at `v`. We say that a walk *visits* the vertices it contains. The set of vertices a walk visits is `SimpleGraph.Walk.support`. See `SimpleGraph.Walk.nil'` and `SimpleGraph.Walk.cons'` for patterns that can be useful in definitions since they make the vertices explicit. -/ inductive Walk : V → V → Type u | nil {u : V} : Walk u u | cons {u v w : V} (h : G.Adj u v) (p : Walk v w) : Walk u w deriving DecidableEq #align simple_graph.walk SimpleGraph.Walk attribute [refl] Walk.nil @[simps] instance Walk.instInhabited (v : V) : Inhabited (G.Walk v v) := ⟨Walk.nil⟩ #align simple_graph.walk.inhabited SimpleGraph.Walk.instInhabited /-- The one-edge walk associated to a pair of adjacent vertices. -/ @[match_pattern, reducible] def Adj.toWalk {G : SimpleGraph V} {u v : V} (h : G.Adj u v) : G.Walk u v := Walk.cons h Walk.nil #align simple_graph.adj.to_walk SimpleGraph.Adj.toWalk namespace Walk variable {G} /-- Pattern to get `Walk.nil` with the vertex as an explicit argument. -/ @[match_pattern] abbrev nil' (u : V) : G.Walk u u := Walk.nil #align simple_graph.walk.nil' SimpleGraph.Walk.nil' /-- Pattern to get `Walk.cons` with the vertices as explicit arguments. -/ @[match_pattern] abbrev cons' (u v w : V) (h : G.Adj u v) (p : G.Walk v w) : G.Walk u w := Walk.cons h p #align simple_graph.walk.cons' SimpleGraph.Walk.cons' /-- Change the endpoints of a walk using equalities. This is helpful for relaxing definitional equality constraints and to be able to state otherwise difficult-to-state lemmas. While this is a simple wrapper around `Eq.rec`, it gives a canonical way to write it. The simp-normal form is for the `copy` to be pushed outward. That way calculations can occur within the "copy context." -/ protected def copy {u v u' v'} (p : G.Walk u v) (hu : u = u') (hv : v = v') : G.Walk u' v' := hu ▸ hv ▸ p #align simple_graph.walk.copy SimpleGraph.Walk.copy @[simp] theorem copy_rfl_rfl {u v} (p : G.Walk u v) : p.copy rfl rfl = p := rfl #align simple_graph.walk.copy_rfl_rfl SimpleGraph.Walk.copy_rfl_rfl @[simp] theorem copy_copy {u v u' v' u'' v''} (p : G.Walk u v) (hu : u = u') (hv : v = v') (hu' : u' = u'') (hv' : v' = v'') : (p.copy hu hv).copy hu' hv' = p.copy (hu.trans hu') (hv.trans hv') := by subst_vars rfl #align simple_graph.walk.copy_copy SimpleGraph.Walk.copy_copy @[simp] theorem copy_nil {u u'} (hu : u = u') : (Walk.nil : G.Walk u u).copy hu hu = Walk.nil := by subst_vars rfl #align simple_graph.walk.copy_nil SimpleGraph.Walk.copy_nil theorem copy_cons {u v w u' w'} (h : G.Adj u v) (p : G.Walk v w) (hu : u = u') (hw : w = w') : (Walk.cons h p).copy hu hw = Walk.cons (hu ▸ h) (p.copy rfl hw) := by subst_vars rfl #align simple_graph.walk.copy_cons SimpleGraph.Walk.copy_cons @[simp] theorem cons_copy {u v w v' w'} (h : G.Adj u v) (p : G.Walk v' w') (hv : v' = v) (hw : w' = w) : Walk.cons h (p.copy hv hw) = (Walk.cons (hv ▸ h) p).copy rfl hw := by subst_vars rfl #align simple_graph.walk.cons_copy SimpleGraph.Walk.cons_copy theorem exists_eq_cons_of_ne {u v : V} (hne : u ≠ v) : ∀ (p : G.Walk u v), ∃ (w : V) (h : G.Adj u w) (p' : G.Walk w v), p = cons h p' | nil => (hne rfl).elim | cons h p' => ⟨_, h, p', rfl⟩ #align simple_graph.walk.exists_eq_cons_of_ne SimpleGraph.Walk.exists_eq_cons_of_ne /-- The length of a walk is the number of edges/darts along it. -/ def length {u v : V} : G.Walk u v → ℕ | nil => 0 | cons _ q => q.length.succ #align simple_graph.walk.length SimpleGraph.Walk.length /-- The concatenation of two compatible walks. -/ @[trans] def append {u v w : V} : G.Walk u v → G.Walk v w → G.Walk u w | nil, q => q | cons h p, q => cons h (p.append q) #align simple_graph.walk.append SimpleGraph.Walk.append /-- The reversed version of `SimpleGraph.Walk.cons`, concatenating an edge to the end of a walk. -/ def concat {u v w : V} (p : G.Walk u v) (h : G.Adj v w) : G.Walk u w := p.append (cons h nil) #align simple_graph.walk.concat SimpleGraph.Walk.concat theorem concat_eq_append {u v w : V} (p : G.Walk u v) (h : G.Adj v w) : p.concat h = p.append (cons h nil) := rfl #align simple_graph.walk.concat_eq_append SimpleGraph.Walk.concat_eq_append /-- The concatenation of the reverse of the first walk with the second walk. -/ protected def reverseAux {u v w : V} : G.Walk u v → G.Walk u w → G.Walk v w | nil, q => q | cons h p, q => Walk.reverseAux p (cons (G.symm h) q) #align simple_graph.walk.reverse_aux SimpleGraph.Walk.reverseAux /-- The walk in reverse. -/ @[symm] def reverse {u v : V} (w : G.Walk u v) : G.Walk v u := w.reverseAux nil #align simple_graph.walk.reverse SimpleGraph.Walk.reverse /-- Get the `n`th vertex from a walk, where `n` is generally expected to be between `0` and `p.length`, inclusive. If `n` is greater than or equal to `p.length`, the result is the path's endpoint. -/ def getVert {u v : V} : G.Walk u v → ℕ → V | nil, _ => u | cons _ _, 0 => u | cons _ q, n + 1 => q.getVert n #align simple_graph.walk.get_vert SimpleGraph.Walk.getVert @[simp] theorem getVert_zero {u v} (w : G.Walk u v) : w.getVert 0 = u := by cases w <;> rfl #align simple_graph.walk.get_vert_zero SimpleGraph.Walk.getVert_zero theorem getVert_of_length_le {u v} (w : G.Walk u v) {i : ℕ} (hi : w.length ≤ i) : w.getVert i = v := by induction w generalizing i with | nil => rfl | cons _ _ ih => cases i · cases hi · exact ih (Nat.succ_le_succ_iff.1 hi) #align simple_graph.walk.get_vert_of_length_le SimpleGraph.Walk.getVert_of_length_le @[simp] theorem getVert_length {u v} (w : G.Walk u v) : w.getVert w.length = v := w.getVert_of_length_le rfl.le #align simple_graph.walk.get_vert_length SimpleGraph.Walk.getVert_length theorem adj_getVert_succ {u v} (w : G.Walk u v) {i : ℕ} (hi : i < w.length) : G.Adj (w.getVert i) (w.getVert (i + 1)) := by induction w generalizing i with | nil => cases hi | cons hxy _ ih => cases i · simp [getVert, hxy] · exact ih (Nat.succ_lt_succ_iff.1 hi) #align simple_graph.walk.adj_get_vert_succ SimpleGraph.Walk.adj_getVert_succ @[simp] theorem cons_append {u v w x : V} (h : G.Adj u v) (p : G.Walk v w) (q : G.Walk w x) : (cons h p).append q = cons h (p.append q) := rfl #align simple_graph.walk.cons_append SimpleGraph.Walk.cons_append @[simp] theorem cons_nil_append {u v w : V} (h : G.Adj u v) (p : G.Walk v w) : (cons h nil).append p = cons h p := rfl #align simple_graph.walk.cons_nil_append SimpleGraph.Walk.cons_nil_append @[simp] theorem append_nil {u v : V} (p : G.Walk u v) : p.append nil = p := by induction p with | nil => rfl | cons _ _ ih => rw [cons_append, ih] #align simple_graph.walk.append_nil SimpleGraph.Walk.append_nil @[simp] theorem nil_append {u v : V} (p : G.Walk u v) : nil.append p = p := rfl #align simple_graph.walk.nil_append SimpleGraph.Walk.nil_append theorem append_assoc {u v w x : V} (p : G.Walk u v) (q : G.Walk v w) (r : G.Walk w x) : p.append (q.append r) = (p.append q).append r := by induction p with | nil => rfl | cons h p' ih => dsimp only [append] rw [ih] #align simple_graph.walk.append_assoc SimpleGraph.Walk.append_assoc @[simp] theorem append_copy_copy {u v w u' v' w'} (p : G.Walk u v) (q : G.Walk v w) (hu : u = u') (hv : v = v') (hw : w = w') : (p.copy hu hv).append (q.copy hv hw) = (p.append q).copy hu hw := by subst_vars rfl #align simple_graph.walk.append_copy_copy SimpleGraph.Walk.append_copy_copy theorem concat_nil {u v : V} (h : G.Adj u v) : nil.concat h = cons h nil := rfl #align simple_graph.walk.concat_nil SimpleGraph.Walk.concat_nil @[simp] theorem concat_cons {u v w x : V} (h : G.Adj u v) (p : G.Walk v w) (h' : G.Adj w x) : (cons h p).concat h' = cons h (p.concat h') := rfl #align simple_graph.walk.concat_cons SimpleGraph.Walk.concat_cons theorem append_concat {u v w x : V} (p : G.Walk u v) (q : G.Walk v w) (h : G.Adj w x) : p.append (q.concat h) = (p.append q).concat h := append_assoc _ _ _ #align simple_graph.walk.append_concat SimpleGraph.Walk.append_concat theorem concat_append {u v w x : V} (p : G.Walk u v) (h : G.Adj v w) (q : G.Walk w x) : (p.concat h).append q = p.append (cons h q) := by rw [concat_eq_append, ← append_assoc, cons_nil_append] #align simple_graph.walk.concat_append SimpleGraph.Walk.concat_append /-- A non-trivial `cons` walk is representable as a `concat` walk. -/ theorem exists_cons_eq_concat {u v w : V} (h : G.Adj u v) (p : G.Walk v w) : ∃ (x : V) (q : G.Walk u x) (h' : G.Adj x w), cons h p = q.concat h' := by induction p generalizing u with | nil => exact ⟨_, nil, h, rfl⟩ | cons h' p ih => obtain ⟨y, q, h'', hc⟩ := ih h' refine' ⟨y, cons h q, h'', _⟩ rw [concat_cons, hc] #align simple_graph.walk.exists_cons_eq_concat SimpleGraph.Walk.exists_cons_eq_concat /-- A non-trivial `concat` walk is representable as a `cons` walk. -/ theorem exists_concat_eq_cons {u v w : V} : ∀ (p : G.Walk u v) (h : G.Adj v w), ∃ (x : V) (h' : G.Adj u x) (q : G.Walk x w), p.concat h = cons h' q | nil, h => ⟨_, h, nil, rfl⟩ | cons h' p, h => ⟨_, h', Walk.concat p h, concat_cons _ _ _⟩ #align simple_graph.walk.exists_concat_eq_cons SimpleGraph.Walk.exists_concat_eq_cons @[simp] theorem reverse_nil {u : V} : (nil : G.Walk u u).reverse = nil := rfl #align simple_graph.walk.reverse_nil SimpleGraph.Walk.reverse_nil theorem reverse_singleton {u v : V} (h : G.Adj u v) : (cons h nil).reverse = cons (G.symm h) nil := rfl #align simple_graph.walk.reverse_singleton SimpleGraph.Walk.reverse_singleton @[simp] theorem cons_reverseAux {u v w x : V} (p : G.Walk u v) (q : G.Walk w x) (h : G.Adj w u) : (cons h p).reverseAux q = p.reverseAux (cons (G.symm h) q) := rfl #align simple_graph.walk.cons_reverse_aux SimpleGraph.Walk.cons_reverseAux @[simp] protected theorem append_reverseAux {u v w x : V} (p : G.Walk u v) (q : G.Walk v w) (r : G.Walk u x) : (p.append q).reverseAux r = q.reverseAux (p.reverseAux r) := by induction p with | nil => rfl | cons h _ ih => exact ih q (cons (G.symm h) r) #align simple_graph.walk.append_reverse_aux SimpleGraph.Walk.append_reverseAux @[simp] protected theorem reverseAux_append {u v w x : V} (p : G.Walk u v) (q : G.Walk u w) (r : G.Walk w x) : (p.reverseAux q).append r = p.reverseAux (q.append r) := by induction p with | nil => rfl | cons h _ ih => simp [ih (cons (G.symm h) q)] #align simple_graph.walk.reverse_aux_append SimpleGraph.Walk.reverseAux_append protected theorem reverseAux_eq_reverse_append {u v w : V} (p : G.Walk u v) (q : G.Walk u w) : p.reverseAux q = p.reverse.append q := by simp [reverse] #align simple_graph.walk.reverse_aux_eq_reverse_append SimpleGraph.Walk.reverseAux_eq_reverse_append @[simp] theorem reverse_cons {u v w : V} (h : G.Adj u v) (p : G.Walk v w) : (cons h p).reverse = p.reverse.append (cons (G.symm h) nil) := by simp [reverse] #align simple_graph.walk.reverse_cons SimpleGraph.Walk.reverse_cons @[simp] theorem reverse_copy {u v u' v'} (p : G.Walk u v) (hu : u = u') (hv : v = v') : (p.copy hu hv).reverse = p.reverse.copy hv hu := by subst_vars rfl #align simple_graph.walk.reverse_copy SimpleGraph.Walk.reverse_copy @[simp] theorem reverse_append {u v w : V} (p : G.Walk u v) (q : G.Walk v w) : (p.append q).reverse = q.reverse.append p.reverse := by simp [reverse] #align simple_graph.walk.reverse_append SimpleGraph.Walk.reverse_append @[simp] theorem reverse_concat {u v w : V} (p : G.Walk u v) (h : G.Adj v w) : (p.concat h).reverse = cons (G.symm h) p.reverse := by simp [concat_eq_append] #align simple_graph.walk.reverse_concat SimpleGraph.Walk.reverse_concat @[simp] theorem reverse_reverse {u v : V} (p : G.Walk u v) : p.reverse.reverse = p := by induction p with | nil => rfl | cons _ _ ih => simp [ih] #align simple_graph.walk.reverse_reverse SimpleGraph.Walk.reverse_reverse @[simp] theorem length_nil {u : V} : (nil : G.Walk u u).length = 0 := rfl #align simple_graph.walk.length_nil SimpleGraph.Walk.length_nil @[simp] theorem length_cons {u v w : V} (h : G.Adj u v) (p : G.Walk v w) : (cons h p).length = p.length + 1 := rfl #align simple_graph.walk.length_cons SimpleGraph.Walk.length_cons @[simp] theorem length_copy {u v u' v'} (p : G.Walk u v) (hu : u = u') (hv : v = v') : (p.copy hu hv).length = p.length := by subst_vars rfl #align simple_graph.walk.length_copy SimpleGraph.Walk.length_copy @[simp] theorem length_append {u v w : V} (p : G.Walk u v) (q : G.Walk v w) : (p.append q).length = p.length + q.length := by induction p with | nil => simp | cons _ _ ih => simp [ih, add_comm, add_left_comm, add_assoc] #align simple_graph.walk.length_append SimpleGraph.Walk.length_append @[simp] theorem length_concat {u v w : V} (p : G.Walk u v) (h : G.Adj v w) : (p.concat h).length = p.length + 1 := length_append _ _ #align simple_graph.walk.length_concat SimpleGraph.Walk.length_concat @[simp] protected theorem length_reverseAux {u v w : V} (p : G.Walk u v) (q : G.Walk u w) : (p.reverseAux q).length = p.length + q.length := by induction p with | nil => simp! | cons _ _ ih => simp [ih, Nat.add_succ, Nat.succ_add] #align simple_graph.walk.length_reverse_aux SimpleGraph.Walk.length_reverseAux @[simp] theorem length_reverse {u v : V} (p : G.Walk u v) : p.reverse.length = p.length := by simp [reverse] #align simple_graph.walk.length_reverse SimpleGraph.Walk.length_reverse theorem eq_of_length_eq_zero {u v : V} : ∀ {p : G.Walk u v}, p.length = 0 → u = v | nil, _ => rfl #align simple_graph.walk.eq_of_length_eq_zero SimpleGraph.Walk.eq_of_length_eq_zero @[simp] theorem exists_length_eq_zero_iff {u v : V} : (∃ p : G.Walk u v, p.length = 0) ↔ u = v := by constructor · rintro ⟨p, hp⟩ exact eq_of_length_eq_zero hp · rintro rfl exact ⟨nil, rfl⟩ #align simple_graph.walk.exists_length_eq_zero_iff SimpleGraph.Walk.exists_length_eq_zero_iff @[simp] theorem length_eq_zero_iff {u : V} {p : G.Walk u u} : p.length = 0 ↔ p = nil := by cases p <;> simp #align simple_graph.walk.length_eq_zero_iff SimpleGraph.Walk.length_eq_zero_iff section ConcatRec variable {motive : ∀ u v : V, G.Walk u v → Sort*} (Hnil : ∀ {u : V}, motive u u nil) (Hconcat : ∀ {u v w : V} (p : G.Walk u v) (h : G.Adj v w), motive u v p → motive u w (p.concat h)) /-- Auxiliary definition for `SimpleGraph.Walk.concatRec` -/ def concatRecAux {u v : V} : (p : G.Walk u v) → motive v u p.reverse | nil => Hnil | cons h p => reverse_cons h p ▸ Hconcat p.reverse h.symm (concatRecAux p) #align simple_graph.walk.concat_rec_aux SimpleGraph.Walk.concatRecAux /-- Recursor on walks by inducting on `SimpleGraph.Walk.concat`. This is inducting from the opposite end of the walk compared to `SimpleGraph.Walk.rec`, which inducts on `SimpleGraph.Walk.cons`. -/ @[elab_as_elim] def concatRec {u v : V} (p : G.Walk u v) : motive u v p := reverse_reverse p ▸ concatRecAux @Hnil @Hconcat p.reverse #align simple_graph.walk.concat_rec SimpleGraph.Walk.concatRec @[simp] theorem concatRec_nil (u : V) : @concatRec _ _ motive @Hnil @Hconcat _ _ (nil : G.Walk u u) = Hnil := rfl #align simple_graph.walk.concat_rec_nil SimpleGraph.Walk.concatRec_nil @[simp] theorem concatRec_concat {u v w : V} (p : G.Walk u v) (h : G.Adj v w) : @concatRec _ _ motive @Hnil @Hconcat _ _ (p.concat h) = Hconcat p h (concatRec @Hnil @Hconcat p) := by simp only [concatRec] apply eq_of_heq apply rec_heq_of_heq trans concatRecAux @Hnil @Hconcat (cons h.symm p.reverse) · congr simp · rw [concatRecAux, rec_heq_iff_heq] congr <;> simp [heq_rec_iff_heq] #align simple_graph.walk.concat_rec_concat SimpleGraph.Walk.concatRec_concat end ConcatRec theorem concat_ne_nil {u v : V} (p : G.Walk u v) (h : G.Adj v u) : p.concat h ≠ nil := by cases p <;> simp [concat] #align simple_graph.walk.concat_ne_nil SimpleGraph.Walk.concat_ne_nil theorem concat_inj {u v v' w : V} {p : G.Walk u v} {h : G.Adj v w} {p' : G.Walk u v'} {h' : G.Adj v' w} (he : p.concat h = p'.concat h') : ∃ hv : v = v', p.copy rfl hv = p' := by induction p with | nil => cases p' · exact ⟨rfl, rfl⟩ · exfalso simp only [concat_nil, concat_cons, cons.injEq] at he obtain ⟨rfl, he⟩ := he simp only [heq_iff_eq] at he exact concat_ne_nil _ _ he.symm | cons _ _ ih => rw [concat_cons] at he cases p' · exfalso simp only [concat_nil, cons.injEq] at he obtain ⟨rfl, he⟩ := he rw [heq_iff_eq] at he exact concat_ne_nil _ _ he · rw [concat_cons, cons.injEq] at he obtain ⟨rfl, he⟩ := he rw [heq_iff_eq] at he obtain ⟨rfl, rfl⟩ := ih he exact ⟨rfl, rfl⟩ #align simple_graph.walk.concat_inj SimpleGraph.Walk.concat_inj /-- The `support` of a walk is the list of vertices it visits in order. -/ def support {u v : V} : G.Walk u v → List V | nil => [u] | cons _ p => u :: p.support #align simple_graph.walk.support SimpleGraph.Walk.support /-- The `darts` of a walk is the list of darts it visits in order. -/ def darts {u v : V} : G.Walk u v → List G.Dart | nil => [] | cons h p => ⟨(u, _), h⟩ :: p.darts #align simple_graph.walk.darts SimpleGraph.Walk.darts /-- The `edges` of a walk is the list of edges it visits in order. This is defined to be the list of edges underlying `SimpleGraph.Walk.darts`. -/ def edges {u v : V} (p : G.Walk u v) : List (Sym2 V) := p.darts.map Dart.edge #align simple_graph.walk.edges SimpleGraph.Walk.edges @[simp] theorem support_nil {u : V} : (nil : G.Walk u u).support = [u] := rfl #align simple_graph.walk.support_nil SimpleGraph.Walk.support_nil @[simp] theorem support_cons {u v w : V} (h : G.Adj u v) (p : G.Walk v w) : (cons h p).support = u :: p.support := rfl #align simple_graph.walk.support_cons SimpleGraph.Walk.support_cons @[simp] theorem support_concat {u v w : V} (p : G.Walk u v) (h : G.Adj v w) : (p.concat h).support = p.support.concat w := by induction p <;> simp [*, concat_nil] #align simple_graph.walk.support_concat SimpleGraph.Walk.support_concat @[simp] theorem support_copy {u v u' v'} (p : G.Walk u v) (hu : u = u') (hv : v = v') : (p.copy hu hv).support = p.support := by subst_vars rfl #align simple_graph.walk.support_copy SimpleGraph.Walk.support_copy theorem support_append {u v w : V} (p : G.Walk u v) (p' : G.Walk v w) : (p.append p').support = p.support ++ p'.support.tail := by induction p <;> cases p' <;> simp [*] #align simple_graph.walk.support_append SimpleGraph.Walk.support_append @[simp] theorem support_reverse {u v : V} (p : G.Walk u v) : p.reverse.support = p.support.reverse := by induction p <;> simp [support_append, *] #align simple_graph.walk.support_reverse SimpleGraph.Walk.support_reverse @[simp] theorem support_ne_nil {u v : V} (p : G.Walk u v) : p.support ≠ [] := by cases p <;> simp #align simple_graph.walk.support_ne_nil SimpleGraph.Walk.support_ne_nil theorem tail_support_append {u v w : V} (p : G.Walk u v) (p' : G.Walk v w) : (p.append p').support.tail = p.support.tail ++ p'.support.tail := by rw [support_append, List.tail_append_of_ne_nil _ _ (support_ne_nil _)] #align simple_graph.walk.tail_support_append SimpleGraph.Walk.tail_support_append theorem support_eq_cons {u v : V} (p : G.Walk u v) : p.support = u :: p.support.tail := by cases p <;> simp #align simple_graph.walk.support_eq_cons SimpleGraph.Walk.support_eq_cons @[simp] theorem start_mem_support {u v : V} (p : G.Walk u v) : u ∈ p.support := by cases p <;> simp #align simple_graph.walk.start_mem_support SimpleGraph.Walk.start_mem_support @[simp] theorem end_mem_support {u v : V} (p : G.Walk u v) : v ∈ p.support := by induction p <;> simp [*] #align simple_graph.walk.end_mem_support SimpleGraph.Walk.end_mem_support @[simp] theorem support_nonempty {u v : V} (p : G.Walk u v) : { w | w ∈ p.support }.Nonempty := ⟨u, by simp⟩ #align simple_graph.walk.support_nonempty SimpleGraph.Walk.support_nonempty theorem mem_support_iff {u v w : V} (p : G.Walk u v) : w ∈ p.support ↔ w = u ∨ w ∈ p.support.tail := by cases p <;> simp #align simple_graph.walk.mem_support_iff SimpleGraph.Walk.mem_support_iff theorem mem_support_nil_iff {u v : V} : u ∈ (nil : G.Walk v v).support ↔ u = v := by simp #align simple_graph.walk.mem_support_nil_iff SimpleGraph.Walk.mem_support_nil_iff @[simp] theorem mem_tail_support_append_iff {t u v w : V} (p : G.Walk u v) (p' : G.Walk v w) : t ∈ (p.append p').support.tail ↔ t ∈ p.support.tail ∨ t ∈ p'.support.tail := by rw [tail_support_append, List.mem_append] #align simple_graph.walk.mem_tail_support_append_iff SimpleGraph.Walk.mem_tail_support_append_iff @[simp] theorem end_mem_tail_support_of_ne {u v : V} (h : u ≠ v) (p : G.Walk u v) : v ∈ p.support.tail := by obtain ⟨_, _, _, rfl⟩ := exists_eq_cons_of_ne h p simp #align simple_graph.walk.end_mem_tail_support_of_ne SimpleGraph.Walk.end_mem_tail_support_of_ne @[simp, nolint unusedHavesSuffices] theorem mem_support_append_iff {t u v w : V} (p : G.Walk u v) (p' : G.Walk v w) : t ∈ (p.append p').support ↔ t ∈ p.support ∨ t ∈ p'.support := by simp only [mem_support_iff, mem_tail_support_append_iff] obtain rfl | h := eq_or_ne t v <;> obtain rfl | h' := eq_or_ne t u <;> -- this `have` triggers the unusedHavesSuffices linter: (try have := h'.symm) <;> simp [*] #align simple_graph.walk.mem_support_append_iff SimpleGraph.Walk.mem_support_append_iff @[simp] theorem subset_support_append_left {V : Type u} {G : SimpleGraph V} {u v w : V} (p : G.Walk u v) (q : G.Walk v w) : p.support ⊆ (p.append q).support := by simp only [Walk.support_append, List.subset_append_left] #align simple_graph.walk.subset_support_append_left SimpleGraph.Walk.subset_support_append_left @[simp] theorem subset_support_append_right {V : Type u} {G : SimpleGraph V} {u v w : V} (p : G.Walk u v) (q : G.Walk v w) : q.support ⊆ (p.append q).support := by intro h simp (config := { contextual := true }) only [mem_support_append_iff, or_true_iff, imp_true_iff] #align simple_graph.walk.subset_support_append_right SimpleGraph.Walk.subset_support_append_right theorem coe_support {u v : V} (p : G.Walk u v) : (p.support : Multiset V) = {u} + p.support.tail := by cases p <;> rfl #align simple_graph.walk.coe_support SimpleGraph.Walk.coe_support theorem coe_support_append {u v w : V} (p : G.Walk u v) (p' : G.Walk v w) : ((p.append p').support : Multiset V) = {u} + p.support.tail + p'.support.tail := by rw [support_append, ← Multiset.coe_add, coe_support] #align simple_graph.walk.coe_support_append SimpleGraph.Walk.coe_support_append theorem coe_support_append' [DecidableEq V] {u v w : V} (p : G.Walk u v) (p' : G.Walk v w) : ((p.append p').support : Multiset V) = p.support + p'.support - {v} := by rw [support_append, ← Multiset.coe_add] simp only [coe_support] rw [add_comm ({v} : Multiset V)] simp only [← add_assoc, add_tsub_cancel_right] #align simple_graph.walk.coe_support_append' SimpleGraph.Walk.coe_support_append' theorem chain_adj_support {u v w : V} (h : G.Adj u v) : ∀ (p : G.Walk v w), List.Chain G.Adj u p.support | nil => List.Chain.cons h List.Chain.nil | cons h' p => List.Chain.cons h (chain_adj_support h' p) #align simple_graph.walk.chain_adj_support SimpleGraph.Walk.chain_adj_support theorem chain'_adj_support {u v : V} : ∀ (p : G.Walk u v), List.Chain' G.Adj p.support | nil => List.Chain.nil | cons h p => chain_adj_support h p #align simple_graph.walk.chain'_adj_support SimpleGraph.Walk.chain'_adj_support theorem chain_dartAdj_darts {d : G.Dart} {v w : V} (h : d.snd = v) (p : G.Walk v w) : List.Chain G.DartAdj d p.darts := by induction p generalizing d with | nil => exact List.Chain.nil -- Porting note: needed to defer `h` and `rfl` to help elaboration | cons h' p ih => exact List.Chain.cons (by exact h) (ih (by rfl)) #align simple_graph.walk.chain_dart_adj_darts SimpleGraph.Walk.chain_dartAdj_darts theorem chain'_dartAdj_darts {u v : V} : ∀ (p : G.Walk u v), List.Chain' G.DartAdj p.darts | nil => trivial -- Porting note: needed to defer `rfl` to help elaboration | cons h p => chain_dartAdj_darts (by rfl) p #align simple_graph.walk.chain'_dart_adj_darts SimpleGraph.Walk.chain'_dartAdj_darts /-- Every edge in a walk's edge list is an edge of the graph. It is written in this form (rather than using `⊆`) to avoid unsightly coercions. -/ theorem edges_subset_edgeSet {u v : V} : ∀ (p : G.Walk u v) ⦃e : Sym2 V⦄, e ∈ p.edges → e ∈ G.edgeSet | cons h' p', e, h => by cases h · exact h' next h' => exact edges_subset_edgeSet p' h' #align simple_graph.walk.edges_subset_edge_set SimpleGraph.Walk.edges_subset_edgeSet theorem adj_of_mem_edges {u v x y : V} (p : G.Walk u v) (h : s(x, y) ∈ p.edges) : G.Adj x y := edges_subset_edgeSet p h #align simple_graph.walk.adj_of_mem_edges SimpleGraph.Walk.adj_of_mem_edges @[simp] theorem darts_nil {u : V} : (nil : G.Walk u u).darts = [] := rfl #align simple_graph.walk.darts_nil SimpleGraph.Walk.darts_nil @[simp] theorem darts_cons {u v w : V} (h : G.Adj u v) (p : G.Walk v w) : (cons h p).darts = ⟨(u, v), h⟩ :: p.darts := rfl #align simple_graph.walk.darts_cons SimpleGraph.Walk.darts_cons @[simp] theorem darts_concat {u v w : V} (p : G.Walk u v) (h : G.Adj v w) : (p.concat h).darts = p.darts.concat ⟨(v, w), h⟩ := by induction p <;> simp [*, concat_nil] #align simple_graph.walk.darts_concat SimpleGraph.Walk.darts_concat @[simp] theorem darts_copy {u v u' v'} (p : G.Walk u v) (hu : u = u') (hv : v = v') : (p.copy hu hv).darts = p.darts := by subst_vars rfl #align simple_graph.walk.darts_copy SimpleGraph.Walk.darts_copy @[simp] theorem darts_append {u v w : V} (p : G.Walk u v) (p' : G.Walk v w) : (p.append p').darts = p.darts ++ p'.darts := by induction p <;> simp [*] #align simple_graph.walk.darts_append SimpleGraph.Walk.darts_append @[simp] theorem darts_reverse {u v : V} (p : G.Walk u v) : p.reverse.darts = (p.darts.map Dart.symm).reverse := by induction p <;> simp [*, Sym2.eq_swap] #align simple_graph.walk.darts_reverse SimpleGraph.Walk.darts_reverse theorem mem_darts_reverse {u v : V} {d : G.Dart} {p : G.Walk u v} : d ∈ p.reverse.darts ↔ d.symm ∈ p.darts := by simp #align simple_graph.walk.mem_darts_reverse SimpleGraph.Walk.mem_darts_reverse theorem cons_map_snd_darts {u v : V} (p : G.Walk u v) : (u :: p.darts.map (·.snd)) = p.support := by induction p <;> simp! [*] #align simple_graph.walk.cons_map_snd_darts SimpleGraph.Walk.cons_map_snd_darts theorem map_snd_darts {u v : V} (p : G.Walk u v) : p.darts.map (·.snd) = p.support.tail := by simpa using congr_arg List.tail (cons_map_snd_darts p) #align simple_graph.walk.map_snd_darts SimpleGraph.Walk.map_snd_darts theorem map_fst_darts_append {u v : V} (p : G.Walk u v) : p.darts.map (·.fst) ++ [v] = p.support := by induction p <;> simp! [*] #align simple_graph.walk.map_fst_darts_append SimpleGraph.Walk.map_fst_darts_append theorem map_fst_darts {u v : V} (p : G.Walk u v) : p.darts.map (·.fst) = p.support.dropLast := by simpa! using congr_arg List.dropLast (map_fst_darts_append p) #align simple_graph.walk.map_fst_darts SimpleGraph.Walk.map_fst_darts @[simp] theorem edges_nil {u : V} : (nil : G.Walk u u).edges = [] := rfl #align simple_graph.walk.edges_nil SimpleGraph.Walk.edges_nil @[simp] theorem edges_cons {u v w : V} (h : G.Adj u v) (p : G.Walk v w) : (cons h p).edges = s(u, v) :: p.edges := rfl #align simple_graph.walk.edges_cons SimpleGraph.Walk.edges_cons @[simp] theorem edges_concat {u v w : V} (p : G.Walk u v) (h : G.Adj v w) : (p.concat h).edges = p.edges.concat s(v, w) := by simp [edges] #align simple_graph.walk.edges_concat SimpleGraph.Walk.edges_concat @[simp] theorem edges_copy {u v u' v'} (p : G.Walk u v) (hu : u = u') (hv : v = v') : (p.copy hu hv).edges = p.edges := by subst_vars rfl #align simple_graph.walk.edges_copy SimpleGraph.Walk.edges_copy @[simp] theorem edges_append {u v w : V} (p : G.Walk u v) (p' : G.Walk v w) : (p.append p').edges = p.edges ++ p'.edges := by simp [edges] #align simple_graph.walk.edges_append SimpleGraph.Walk.edges_append @[simp] theorem edges_reverse {u v : V} (p : G.Walk u v) : p.reverse.edges = p.edges.reverse := by simp [edges, List.map_reverse] #align simple_graph.walk.edges_reverse SimpleGraph.Walk.edges_reverse @[simp] theorem length_support {u v : V} (p : G.Walk u v) : p.support.length = p.length + 1 := by induction p <;> simp [*] #align simple_graph.walk.length_support SimpleGraph.Walk.length_support @[simp] theorem length_darts {u v : V} (p : G.Walk u v) : p.darts.length = p.length := by induction p <;> simp [*] #align simple_graph.walk.length_darts SimpleGraph.Walk.length_darts @[simp] theorem length_edges {u v : V} (p : G.Walk u v) : p.edges.length = p.length := by simp [edges] #align simple_graph.walk.length_edges SimpleGraph.Walk.length_edges theorem dart_fst_mem_support_of_mem_darts {u v : V} : ∀ (p : G.Walk u v) {d : G.Dart}, d ∈ p.darts → d.fst ∈ p.support | cons h p', d, hd => by simp only [support_cons, darts_cons, List.mem_cons] at hd ⊢ rcases hd with (rfl | hd) · exact Or.inl rfl · exact Or.inr (dart_fst_mem_support_of_mem_darts _ hd) #align simple_graph.walk.dart_fst_mem_support_of_mem_darts SimpleGraph.Walk.dart_fst_mem_support_of_mem_darts theorem dart_snd_mem_support_of_mem_darts {u v : V} (p : G.Walk u v) {d : G.Dart} (h : d ∈ p.darts) : d.snd ∈ p.support := by simpa using p.reverse.dart_fst_mem_support_of_mem_darts (by simp [h] : d.symm ∈ p.reverse.darts) #align simple_graph.walk.dart_snd_mem_support_of_mem_darts SimpleGraph.Walk.dart_snd_mem_support_of_mem_darts theorem fst_mem_support_of_mem_edges {t u v w : V} (p : G.Walk v w) (he : s(t, u) ∈ p.edges) : t ∈ p.support := by obtain ⟨d, hd, he⟩ := List.mem_map.mp he rw [dart_edge_eq_mk'_iff'] at he rcases he with (⟨rfl, rfl⟩ | ⟨rfl, rfl⟩) · exact dart_fst_mem_support_of_mem_darts _ hd · exact dart_snd_mem_support_of_mem_darts _ hd #align simple_graph.walk.fst_mem_support_of_mem_edges SimpleGraph.Walk.fst_mem_support_of_mem_edges theorem snd_mem_support_of_mem_edges {t u v w : V} (p : G.Walk v w) (he : s(t, u) ∈ p.edges) : u ∈ p.support := by rw [Sym2.eq_swap] at he exact p.fst_mem_support_of_mem_edges he #align simple_graph.walk.snd_mem_support_of_mem_edges SimpleGraph.Walk.snd_mem_support_of_mem_edges theorem darts_nodup_of_support_nodup {u v : V} {p : G.Walk u v} (h : p.support.Nodup) : p.darts.Nodup := by induction p with | nil => simp | cons _ p' ih => simp only [darts_cons, support_cons, List.nodup_cons] at h ⊢ exact ⟨fun h' => h.1 (dart_fst_mem_support_of_mem_darts p' h'), ih h.2⟩ #align simple_graph.walk.darts_nodup_of_support_nodup SimpleGraph.Walk.darts_nodup_of_support_nodup theorem edges_nodup_of_support_nodup {u v : V} {p : G.Walk u v} (h : p.support.Nodup) : p.edges.Nodup := by induction p with | nil => simp | cons _ p' ih => simp only [edges_cons, support_cons, List.nodup_cons] at h ⊢ exact ⟨fun h' => h.1 (fst_mem_support_of_mem_edges p' h'), ih h.2⟩ #align simple_graph.walk.edges_nodup_of_support_nodup SimpleGraph.Walk.edges_nodup_of_support_nodup /-- Predicate for the empty walk. Solves the dependent type problem where `p = G.Walk.nil` typechecks only if `p` has defeq endpoints. -/ inductive Nil : {v w : V} → G.Walk v w → Prop | nil {u : V} : Nil (nil : G.Walk u u) variable {u v w : V} @[simp] lemma nil_nil : (nil : G.Walk u u).Nil := Nil.nil @[simp] lemma not_nil_cons {h : G.Adj u v} {p : G.Walk v w} : ¬ (cons h p).Nil := nofun instance (p : G.Walk v w) : Decidable p.Nil := match p with | nil => isTrue .nil | cons _ _ => isFalse nofun protected lemma Nil.eq {p : G.Walk v w} : p.Nil → v = w | .nil => rfl lemma not_nil_of_ne {p : G.Walk v w} : v ≠ w → ¬ p.Nil := mt Nil.eq lemma nil_iff_support_eq {p : G.Walk v w} : p.Nil ↔ p.support = [v] := by cases p <;> simp lemma nil_iff_length_eq {p : G.Walk v w} : p.Nil ↔ p.length = 0 := by cases p <;> simp lemma not_nil_iff {p : G.Walk v w} : ¬ p.Nil ↔ ∃ (u : V) (h : G.Adj v u) (q : G.Walk u w), p = cons h q := by cases p <;> simp [*] @[elab_as_elim] def notNilRec {motive : {u w : V} → (p : G.Walk u w) → (h : ¬ p.Nil) → Sort*} (cons : {u v w : V} → (h : G.Adj u v) → (q : G.Walk v w) → motive (cons h q) not_nil_cons) (p : G.Walk u w) : (hp : ¬ p.Nil) → motive p hp := match p with | nil => fun hp => absurd .nil hp | .cons h q => fun _ => cons h q /-- The second vertex along a non-nil walk. -/ def sndOfNotNil (p : G.Walk v w) (hp : ¬ p.Nil) : V := p.notNilRec (@fun _ u _ _ _ => u) hp @[simp] lemma adj_sndOfNotNil {p : G.Walk v w} (hp : ¬ p.Nil) : G.Adj v (p.sndOfNotNil hp) := p.notNilRec (fun h _ => h) hp /-- The walk obtained by removing the first dart of a non-nil walk. -/ def tail (p : G.Walk u v) (hp : ¬ p.Nil) : G.Walk (p.sndOfNotNil hp) v := p.notNilRec (fun _ q => q) hp /-- The first dart of a walk. -/ @[simps] def firstDart (p : G.Walk v w) (hp : ¬ p.Nil) : G.Dart where fst := v snd := p.sndOfNotNil hp is_adj := p.adj_sndOfNotNil hp lemma edge_firstDart (p : G.Walk v w) (hp : ¬ p.Nil) : (p.firstDart hp).edge = s(v, p.sndOfNotNil hp) := rfl variable {x y : V} -- TODO: rename to u, v, w instead? @[simp] lemma cons_tail_eq (p : G.Walk x y) (hp : ¬ p.Nil) : cons (p.adj_sndOfNotNil hp) (p.tail hp) = p := p.notNilRec (fun _ _ => rfl) hp @[simp] lemma cons_support_tail (p : G.Walk x y) (hp : ¬ p.Nil) : x :: (p.tail hp).support = p.support := by rw [← support_cons, cons_tail_eq] @[simp] lemma length_tail_add_one {p : G.Walk x y} (hp : ¬ p.Nil) : (p.tail hp).length + 1 = p.length := by rw [← length_cons, cons_tail_eq] @[simp] lemma nil_copy {x' y' : V} {p : G.Walk x y} (hx : x = x') (hy : y = y') : (p.copy hx hy).Nil = p.Nil := by subst_vars; rfl /-! ### Trails, paths, circuits, cycles -/ /-- A *trail* is a walk with no repeating edges. -/ @[mk_iff isTrail_def] structure IsTrail {u v : V} (p : G.Walk u v) : Prop where edges_nodup : p.edges.Nodup #align simple_graph.walk.is_trail SimpleGraph.Walk.IsTrail #align simple_graph.walk.is_trail_def SimpleGraph.Walk.isTrail_def /-- A *path* is a walk with no repeating vertices. Use `SimpleGraph.Walk.IsPath.mk'` for a simpler constructor. -/ structure IsPath {u v : V} (p : G.Walk u v) extends IsTrail p : Prop where support_nodup : p.support.Nodup #align simple_graph.walk.is_path SimpleGraph.Walk.IsPath -- Porting note: used to use `extends to_trail : is_trail p` in structure protected lemma IsPath.isTrail {p : Walk G u v}(h : IsPath p) : IsTrail p := h.toIsTrail #align simple_graph.walk.is_path.to_trail SimpleGraph.Walk.IsPath.isTrail /-- A *circuit* at `u : V` is a nonempty trail beginning and ending at `u`. -/ @[mk_iff isCircuit_def] structure IsCircuit {u : V} (p : G.Walk u u) extends IsTrail p : Prop where ne_nil : p ≠ nil #align simple_graph.walk.is_circuit SimpleGraph.Walk.IsCircuit #align simple_graph.walk.is_circuit_def SimpleGraph.Walk.isCircuit_def -- Porting note: used to use `extends to_trail : is_trail p` in structure protected lemma IsCircuit.isTrail {p : Walk G u u} (h : IsCircuit p) : IsTrail p := h.toIsTrail #align simple_graph.walk.is_circuit.to_trail SimpleGraph.Walk.IsCircuit.isTrail /-- A *cycle* at `u : V` is a circuit at `u` whose only repeating vertex is `u` (which appears exactly twice). -/ structure IsCycle {u : V} (p : G.Walk u u) extends IsCircuit p : Prop where support_nodup : p.support.tail.Nodup #align simple_graph.walk.is_cycle SimpleGraph.Walk.IsCycle -- Porting note: used to use `extends to_circuit : is_circuit p` in structure protected lemma IsCycle.isCircuit {p : Walk G u u} (h : IsCycle p) : IsCircuit p := h.toIsCircuit #align simple_graph.walk.is_cycle.to_circuit SimpleGraph.Walk.IsCycle.isCircuit @[simp] theorem isTrail_copy {u v u' v'} (p : G.Walk u v) (hu : u = u') (hv : v = v') : (p.copy hu hv).IsTrail ↔ p.IsTrail := by subst_vars rfl #align simple_graph.walk.is_trail_copy SimpleGraph.Walk.isTrail_copy theorem IsPath.mk' {u v : V} {p : G.Walk u v} (h : p.support.Nodup) : p.IsPath := ⟨⟨edges_nodup_of_support_nodup h⟩, h⟩ #align simple_graph.walk.is_path.mk' SimpleGraph.Walk.IsPath.mk' theorem isPath_def {u v : V} (p : G.Walk u v) : p.IsPath ↔ p.support.Nodup := ⟨IsPath.support_nodup, IsPath.mk'⟩ #align simple_graph.walk.is_path_def SimpleGraph.Walk.isPath_def @[simp] theorem isPath_copy {u v u' v'} (p : G.Walk u v) (hu : u = u') (hv : v = v') : (p.copy hu hv).IsPath ↔ p.IsPath := by subst_vars rfl #align simple_graph.walk.is_path_copy SimpleGraph.Walk.isPath_copy @[simp] theorem isCircuit_copy {u u'} (p : G.Walk u u) (hu : u = u') : (p.copy hu hu).IsCircuit ↔ p.IsCircuit := by subst_vars rfl #align simple_graph.walk.is_circuit_copy SimpleGraph.Walk.isCircuit_copy theorem isCycle_def {u : V} (p : G.Walk u u) : p.IsCycle ↔ p.IsTrail ∧ p ≠ nil ∧ p.support.tail.Nodup := Iff.intro (fun h => ⟨h.1.1, h.1.2, h.2⟩) fun h => ⟨⟨h.1, h.2.1⟩, h.2.2⟩ #align simple_graph.walk.is_cycle_def SimpleGraph.Walk.isCycle_def @[simp] theorem isCycle_copy {u u'} (p : G.Walk u u) (hu : u = u') : (p.copy hu hu).IsCycle ↔ p.IsCycle := by subst_vars rfl #align simple_graph.walk.is_cycle_copy SimpleGraph.Walk.isCycle_copy @[simp] theorem IsTrail.nil {u : V} : (nil : G.Walk u u).IsTrail := ⟨by simp [edges]⟩ #align simple_graph.walk.is_trail.nil SimpleGraph.Walk.IsTrail.nil theorem IsTrail.of_cons {u v w : V} {h : G.Adj u v} {p : G.Walk v w} : (cons h p).IsTrail → p.IsTrail := by simp [isTrail_def] #align simple_graph.walk.is_trail.of_cons SimpleGraph.Walk.IsTrail.of_cons @[simp] theorem cons_isTrail_iff {u v w : V} (h : G.Adj u v) (p : G.Walk v w) : (cons h p).IsTrail ↔ p.IsTrail ∧ s(u, v) ∉ p.edges := by simp [isTrail_def, and_comm] #align simple_graph.walk.cons_is_trail_iff SimpleGraph.Walk.cons_isTrail_iff theorem IsTrail.reverse {u v : V} (p : G.Walk u v) (h : p.IsTrail) : p.reverse.IsTrail := by simpa [isTrail_def] using h #align simple_graph.walk.is_trail.reverse SimpleGraph.Walk.IsTrail.reverse @[simp] theorem reverse_isTrail_iff {u v : V} (p : G.Walk u v) : p.reverse.IsTrail ↔ p.IsTrail := by constructor <;> · intro h convert h.reverse _ try rw [reverse_reverse] #align simple_graph.walk.reverse_is_trail_iff SimpleGraph.Walk.reverse_isTrail_iff theorem IsTrail.of_append_left {u v w : V} {p : G.Walk u v} {q : G.Walk v w} (h : (p.append q).IsTrail) : p.IsTrail := by rw [isTrail_def, edges_append, List.nodup_append] at h exact ⟨h.1⟩ #align simple_graph.walk.is_trail.of_append_left SimpleGraph.Walk.IsTrail.of_append_left theorem IsTrail.of_append_right {u v w : V} {p : G.Walk u v} {q : G.Walk v w} (h : (p.append q).IsTrail) : q.IsTrail := by rw [isTrail_def, edges_append, List.nodup_append] at h exact ⟨h.2.1⟩ #align simple_graph.walk.is_trail.of_append_right SimpleGraph.Walk.IsTrail.of_append_right theorem IsTrail.count_edges_le_one [DecidableEq V] {u v : V} {p : G.Walk u v} (h : p.IsTrail) (e : Sym2 V) : p.edges.count e ≤ 1 := List.nodup_iff_count_le_one.mp h.edges_nodup e #align simple_graph.walk.is_trail.count_edges_le_one SimpleGraph.Walk.IsTrail.count_edges_le_one theorem IsTrail.count_edges_eq_one [DecidableEq V] {u v : V} {p : G.Walk u v} (h : p.IsTrail) {e : Sym2 V} (he : e ∈ p.edges) : p.edges.count e = 1 := List.count_eq_one_of_mem h.edges_nodup he #align simple_graph.walk.is_trail.count_edges_eq_one SimpleGraph.Walk.IsTrail.count_edges_eq_one theorem IsPath.nil {u : V} : (nil : G.Walk u u).IsPath := by constructor <;> simp #align simple_graph.walk.is_path.nil SimpleGraph.Walk.IsPath.nil theorem IsPath.of_cons {u v w : V} {h : G.Adj u v} {p : G.Walk v w} : (cons h p).IsPath → p.IsPath := by simp [isPath_def] #align simple_graph.walk.is_path.of_cons SimpleGraph.Walk.IsPath.of_cons @[simp] theorem cons_isPath_iff {u v w : V} (h : G.Adj u v) (p : G.Walk v w) : (cons h p).IsPath ↔ p.IsPath ∧ u ∉ p.support := by constructor <;> simp (config := { contextual := true }) [isPath_def] #align simple_graph.walk.cons_is_path_iff SimpleGraph.Walk.cons_isPath_iff protected lemma IsPath.cons {p : Walk G v w} (hp : p.IsPath) (hu : u ∉ p.support) {h : G.Adj u v} : (cons h p).IsPath := (cons_isPath_iff _ _).2 ⟨hp, hu⟩ @[simp] theorem isPath_iff_eq_nil {u : V} (p : G.Walk u u) : p.IsPath ↔ p = nil := by cases p <;> simp [IsPath.nil] #align simple_graph.walk.is_path_iff_eq_nil SimpleGraph.Walk.isPath_iff_eq_nil theorem IsPath.reverse {u v : V} {p : G.Walk u v} (h : p.IsPath) : p.reverse.IsPath := by simpa [isPath_def] using h #align simple_graph.walk.is_path.reverse SimpleGraph.Walk.IsPath.reverse @[simp] theorem isPath_reverse_iff {u v : V} (p : G.Walk u v) : p.reverse.IsPath ↔ p.IsPath := by constructor <;> intro h <;> convert h.reverse; simp #align simple_graph.walk.is_path_reverse_iff SimpleGraph.Walk.isPath_reverse_iff theorem IsPath.of_append_left {u v w : V} {p : G.Walk u v} {q : G.Walk v w} : (p.append q).IsPath → p.IsPath := by simp only [isPath_def, support_append] exact List.Nodup.of_append_left #align simple_graph.walk.is_path.of_append_left SimpleGraph.Walk.IsPath.of_append_left theorem IsPath.of_append_right {u v w : V} {p : G.Walk u v} {q : G.Walk v w} (h : (p.append q).IsPath) : q.IsPath := by rw [← isPath_reverse_iff] at h ⊢ rw [reverse_append] at h apply h.of_append_left #align simple_graph.walk.is_path.of_append_right SimpleGraph.Walk.IsPath.of_append_right @[simp] theorem IsCycle.not_of_nil {u : V} : ¬(nil : G.Walk u u).IsCycle := fun h => h.ne_nil rfl #align simple_graph.walk.is_cycle.not_of_nil SimpleGraph.Walk.IsCycle.not_of_nil lemma IsCycle.ne_bot : ∀ {p : G.Walk u u}, p.IsCycle → G ≠ ⊥ | nil, hp => by cases hp.ne_nil rfl | cons h _, hp => by rintro rfl; exact h lemma IsCycle.three_le_length {v : V} {p : G.Walk v v} (hp : p.IsCycle) : 3 ≤ p.length := by have ⟨⟨hp, hp'⟩, _⟩ := hp match p with | .nil => simp at hp' | .cons h .nil => simp at h | .cons _ (.cons _ .nil) => simp at hp | .cons _ (.cons _ (.cons _ _)) => simp_rw [SimpleGraph.Walk.length_cons]; omega theorem cons_isCycle_iff {u v : V} (p : G.Walk v u) (h : G.Adj u v) : (Walk.cons h p).IsCycle ↔ p.IsPath ∧ ¬s(u, v) ∈ p.edges := by simp only [Walk.isCycle_def, Walk.isPath_def, Walk.isTrail_def, edges_cons, List.nodup_cons, support_cons, List.tail_cons] have : p.support.Nodup → p.edges.Nodup := edges_nodup_of_support_nodup tauto #align simple_graph.walk.cons_is_cycle_iff SimpleGraph.Walk.cons_isCycle_iff lemma IsPath.tail {p : G.Walk u v} (hp : p.IsPath) (hp' : ¬ p.Nil) : (p.tail hp').IsPath := by rw [Walk.isPath_def] at hp ⊢ rw [← cons_support_tail _ hp', List.nodup_cons] at hp exact hp.2 /-! ### About paths -/ instance [DecidableEq V] {u v : V} (p : G.Walk u v) : Decidable p.IsPath := by rw [isPath_def] infer_instance theorem IsPath.length_lt [Fintype V] {u v : V} {p : G.Walk u v} (hp : p.IsPath) : p.length < Fintype.card V := by rw [Nat.lt_iff_add_one_le, ← length_support] exact hp.support_nodup.length_le_card #align simple_graph.walk.is_path.length_lt SimpleGraph.Walk.IsPath.length_lt /-! ### Walk decompositions -/ section WalkDecomp variable [DecidableEq V] /-- Given a vertex in the support of a path, give the path up until (and including) that vertex. -/ def takeUntil {v w : V} : ∀ (p : G.Walk v w) (u : V), u ∈ p.support → G.Walk v u | nil, u, h => by rw [mem_support_nil_iff.mp h] | cons r p, u, h => if hx : v = u then by subst u; exact Walk.nil else cons r (takeUntil p u <| by cases h; exact (hx rfl).elim; assumption) #align simple_graph.walk.take_until SimpleGraph.Walk.takeUntil /-- Given a vertex in the support of a path, give the path from (and including) that vertex to the end. In other words, drop vertices from the front of a path until (and not including) that vertex. -/ def dropUntil {v w : V} : ∀ (p : G.Walk v w) (u : V), u ∈ p.support → G.Walk u w | nil, u, h => by rw [mem_support_nil_iff.mp h] | cons r p, u, h => if hx : v = u then by subst u exact cons r p else dropUntil p u <| by cases h; exact (hx rfl).elim; assumption #align simple_graph.walk.drop_until SimpleGraph.Walk.dropUntil /-- The `takeUntil` and `dropUntil` functions split a walk into two pieces. The lemma `SimpleGraph.Walk.count_support_takeUntil_eq_one` specifies where this split occurs. -/ @[simp] theorem take_spec {u v w : V} (p : G.Walk v w) (h : u ∈ p.support) : (p.takeUntil u h).append (p.dropUntil u h) = p := by induction p · rw [mem_support_nil_iff] at h subst u rfl · cases h · simp! · simp! only split_ifs with h' <;> subst_vars <;> simp [*] #align simple_graph.walk.take_spec SimpleGraph.Walk.take_spec theorem mem_support_iff_exists_append {V : Type u} {G : SimpleGraph V} {u v w : V} {p : G.Walk u v} : w ∈ p.support ↔ ∃ (q : G.Walk u w) (r : G.Walk w v), p = q.append r := by classical constructor · exact fun h => ⟨_, _, (p.take_spec h).symm⟩ · rintro ⟨q, r, rfl⟩ simp only [mem_support_append_iff, end_mem_support, start_mem_support, or_self_iff] #align simple_graph.walk.mem_support_iff_exists_append SimpleGraph.Walk.mem_support_iff_exists_append @[simp] theorem count_support_takeUntil_eq_one {u v w : V} (p : G.Walk v w) (h : u ∈ p.support) : (p.takeUntil u h).support.count u = 1 := by induction p · rw [mem_support_nil_iff] at h subst u simp! · cases h · simp! · simp! only split_ifs with h' <;> rw [eq_comm] at h' <;> subst_vars <;> simp! [*, List.count_cons] #align simple_graph.walk.count_support_take_until_eq_one SimpleGraph.Walk.count_support_takeUntil_eq_one theorem count_edges_takeUntil_le_one {u v w : V} (p : G.Walk v w) (h : u ∈ p.support) (x : V) : (p.takeUntil u h).edges.count s(u, x) ≤ 1 := by induction' p with u' u' v' w' ha p' ih · rw [mem_support_nil_iff] at h subst u simp! · cases h · simp! · simp! only split_ifs with h' · subst h' simp · rw [edges_cons, List.count_cons] split_ifs with h'' · rw [Sym2.eq_iff] at h'' obtain ⟨rfl, rfl⟩ | ⟨rfl, rfl⟩ := h'' · exact (h' rfl).elim · cases p' <;> simp! · apply ih #align simple_graph.walk.count_edges_take_until_le_one SimpleGraph.Walk.count_edges_takeUntil_le_one @[simp] theorem takeUntil_copy {u v w v' w'} (p : G.Walk v w) (hv : v = v') (hw : w = w') (h : u ∈ (p.copy hv hw).support) : (p.copy hv hw).takeUntil u h = (p.takeUntil u (by subst_vars; exact h)).copy hv rfl := by subst_vars rfl #align simple_graph.walk.take_until_copy SimpleGraph.Walk.takeUntil_copy @[simp] theorem dropUntil_copy {u v w v' w'} (p : G.Walk v w) (hv : v = v') (hw : w = w') (h : u ∈ (p.copy hv hw).support) : (p.copy hv hw).dropUntil u h = (p.dropUntil u (by subst_vars; exact h)).copy rfl hw := by subst_vars rfl #align simple_graph.walk.drop_until_copy SimpleGraph.Walk.dropUntil_copy theorem support_takeUntil_subset {u v w : V} (p : G.Walk v w) (h : u ∈ p.support) : (p.takeUntil u h).support ⊆ p.support := fun x hx => by rw [← take_spec p h, mem_support_append_iff] exact Or.inl hx #align simple_graph.walk.support_take_until_subset SimpleGraph.Walk.support_takeUntil_subset theorem support_dropUntil_subset {u v w : V} (p : G.Walk v w) (h : u ∈ p.support) : (p.dropUntil u h).support ⊆ p.support := fun x hx => by rw [← take_spec p h, mem_support_append_iff] exact Or.inr hx #align simple_graph.walk.support_drop_until_subset SimpleGraph.Walk.support_dropUntil_subset theorem darts_takeUntil_subset {u v w : V} (p : G.Walk v w) (h : u ∈ p.support) : (p.takeUntil u h).darts ⊆ p.darts := fun x hx => by rw [← take_spec p h, darts_append, List.mem_append] exact Or.inl hx #align simple_graph.walk.darts_take_until_subset SimpleGraph.Walk.darts_takeUntil_subset theorem darts_dropUntil_subset {u v w : V} (p : G.Walk v w) (h : u ∈ p.support) : (p.dropUntil u h).darts ⊆ p.darts := fun x hx => by rw [← take_spec p h, darts_append, List.mem_append] exact Or.inr hx #align simple_graph.walk.darts_drop_until_subset SimpleGraph.Walk.darts_dropUntil_subset theorem edges_takeUntil_subset {u v w : V} (p : G.Walk v w) (h : u ∈ p.support) : (p.takeUntil u h).edges ⊆ p.edges := List.map_subset _ (p.darts_takeUntil_subset h) #align simple_graph.walk.edges_take_until_subset SimpleGraph.Walk.edges_takeUntil_subset theorem edges_dropUntil_subset {u v w : V} (p : G.Walk v w) (h : u ∈ p.support) : (p.dropUntil u h).edges ⊆ p.edges := List.map_subset _ (p.darts_dropUntil_subset h) #align simple_graph.walk.edges_drop_until_subset SimpleGraph.Walk.edges_dropUntil_subset theorem length_takeUntil_le {u v w : V} (p : G.Walk v w) (h : u ∈ p.support) : (p.takeUntil u h).length ≤ p.length := by have := congr_arg Walk.length (p.take_spec h) rw [length_append] at this exact Nat.le.intro this #align simple_graph.walk.length_take_until_le SimpleGraph.Walk.length_takeUntil_le theorem length_dropUntil_le {u v w : V} (p : G.Walk v w) (h : u ∈ p.support) : (p.dropUntil u h).length ≤ p.length := by have := congr_arg Walk.length (p.take_spec h) rw [length_append, add_comm] at this exact Nat.le.intro this #align simple_graph.walk.length_drop_until_le SimpleGraph.Walk.length_dropUntil_le protected theorem IsTrail.takeUntil {u v w : V} {p : G.Walk v w} (hc : p.IsTrail) (h : u ∈ p.support) : (p.takeUntil u h).IsTrail := IsTrail.of_append_left (by rwa [← take_spec _ h] at hc) #align simple_graph.walk.is_trail.take_until SimpleGraph.Walk.IsTrail.takeUntil protected theorem IsTrail.dropUntil {u v w : V} {p : G.Walk v w} (hc : p.IsTrail) (h : u ∈ p.support) : (p.dropUntil u h).IsTrail := IsTrail.of_append_right (by rwa [← take_spec _ h] at hc) #align simple_graph.walk.is_trail.drop_until SimpleGraph.Walk.IsTrail.dropUntil protected theorem IsPath.takeUntil {u v w : V} {p : G.Walk v w} (hc : p.IsPath) (h : u ∈ p.support) : (p.takeUntil u h).IsPath := IsPath.of_append_left (by rwa [← take_spec _ h] at hc) #align simple_graph.walk.is_path.take_until SimpleGraph.Walk.IsPath.takeUntil -- Porting note: p was previously accidentally an explicit argument protected theorem IsPath.dropUntil {u v w : V} {p : G.Walk v w} (hc : p.IsPath) (h : u ∈ p.support) : (p.dropUntil u h).IsPath := IsPath.of_append_right (by rwa [← take_spec _ h] at hc) #align simple_graph.walk.is_path.drop_until SimpleGraph.Walk.IsPath.dropUntil /-- Rotate a loop walk such that it is centered at the given vertex. -/ def rotate {u v : V} (c : G.Walk v v) (h : u ∈ c.support) : G.Walk u u := (c.dropUntil u h).append (c.takeUntil u h) #align simple_graph.walk.rotate SimpleGraph.Walk.rotate @[simp] theorem support_rotate {u v : V} (c : G.Walk v v) (h : u ∈ c.support) : (c.rotate h).support.tail ~r c.support.tail := by simp only [rotate, tail_support_append] apply List.IsRotated.trans List.isRotated_append rw [← tail_support_append, take_spec] #align simple_graph.walk.support_rotate SimpleGraph.Walk.support_rotate theorem rotate_darts {u v : V} (c : G.Walk v v) (h : u ∈ c.support) : (c.rotate h).darts ~r c.darts := by simp only [rotate, darts_append] apply List.IsRotated.trans List.isRotated_append rw [← darts_append, take_spec] #align simple_graph.walk.rotate_darts SimpleGraph.Walk.rotate_darts theorem rotate_edges {u v : V} (c : G.Walk v v) (h : u ∈ c.support) : (c.rotate h).edges ~r c.edges := (rotate_darts c h).map _ #align simple_graph.walk.rotate_edges SimpleGraph.Walk.rotate_edges protected theorem IsTrail.rotate {u v : V} {c : G.Walk v v} (hc : c.IsTrail) (h : u ∈ c.support) : (c.rotate h).IsTrail := by rw [isTrail_def, (c.rotate_edges h).perm.nodup_iff] exact hc.edges_nodup #align simple_graph.walk.is_trail.rotate SimpleGraph.Walk.IsTrail.rotate protected theorem IsCircuit.rotate {u v : V} {c : G.Walk v v} (hc : c.IsCircuit) (h : u ∈ c.support) : (c.rotate h).IsCircuit := by refine ⟨hc.isTrail.rotate _, ?_⟩ cases c · exact (hc.ne_nil rfl).elim · intro hn have hn' := congr_arg length hn rw [rotate, length_append, add_comm, ← length_append, take_spec] at hn' simp at hn' #align simple_graph.walk.is_circuit.rotate SimpleGraph.Walk.IsCircuit.rotate protected theorem IsCycle.rotate {u v : V} {c : G.Walk v v} (hc : c.IsCycle) (h : u ∈ c.support) : (c.rotate h).IsCycle := by refine ⟨hc.isCircuit.rotate _, ?_⟩ rw [List.IsRotated.nodup_iff (support_rotate _ _)] exact hc.support_nodup #align simple_graph.walk.is_cycle.rotate SimpleGraph.Walk.IsCycle.rotate end WalkDecomp /-- Given a set `S` and a walk `w` from `u` to `v` such that `u ∈ S` but `v ∉ S`, there exists a dart in the walk whose start is in `S` but whose end is not. -/ theorem exists_boundary_dart {u v : V} (p : G.Walk u v) (S : Set V) (uS : u ∈ S) (vS : v ∉ S) : ∃ d : G.Dart, d ∈ p.darts ∧ d.fst ∈ S ∧ d.snd ∉ S := by induction' p with _ x y w a p' ih · cases vS uS · by_cases h : y ∈ S · obtain ⟨d, hd, hcd⟩ := ih h vS exact ⟨d, List.Mem.tail _ hd, hcd⟩ · exact ⟨⟨(x, y), a⟩, List.Mem.head _, uS, h⟩ #align simple_graph.walk.exists_boundary_dart SimpleGraph.Walk.exists_boundary_dart end Walk /-! ### Type of paths -/ /-- The type for paths between two vertices. -/ abbrev Path (u v : V) := { p : G.Walk u v // p.IsPath } #align simple_graph.path SimpleGraph.Path namespace Path variable {G G'} @[simp] protected theorem isPath {u v : V} (p : G.Path u v) : (p : G.Walk u v).IsPath := p.property #align simple_graph.path.is_path SimpleGraph.Path.isPath @[simp] protected theorem isTrail {u v : V} (p : G.Path u v) : (p : G.Walk u v).IsTrail := p.property.isTrail #align simple_graph.path.is_trail SimpleGraph.Path.isTrail /-- The length-0 path at a vertex. -/ @[refl, simps] protected def nil {u : V} : G.Path u u := ⟨Walk.nil, Walk.IsPath.nil⟩ #align simple_graph.path.nil SimpleGraph.Path.nil /-- The length-1 path between a pair of adjacent vertices. -/ @[simps] def singleton {u v : V} (h : G.Adj u v) : G.Path u v := ⟨Walk.cons h Walk.nil, by simp [h.ne]⟩ #align simple_graph.path.singleton SimpleGraph.Path.singleton theorem mk'_mem_edges_singleton {u v : V} (h : G.Adj u v) : s(u, v) ∈ (singleton h : G.Walk u v).edges := by simp [singleton] #align simple_graph.path.mk_mem_edges_singleton SimpleGraph.Path.mk'_mem_edges_singleton /-- The reverse of a path is another path. See also `SimpleGraph.Walk.reverse`. -/ @[symm, simps] def reverse {u v : V} (p : G.Path u v) : G.Path v u := ⟨Walk.reverse p, p.property.reverse⟩ #align simple_graph.path.reverse SimpleGraph.Path.reverse theorem count_support_eq_one [DecidableEq V] {u v w : V} {p : G.Path u v} (hw : w ∈ (p : G.Walk u v).support) : (p : G.Walk u v).support.count w = 1 := List.count_eq_one_of_mem p.property.support_nodup hw #align simple_graph.path.count_support_eq_one SimpleGraph.Path.count_support_eq_one theorem count_edges_eq_one [DecidableEq V] {u v : V} {p : G.Path u v} (e : Sym2 V) (hw : e ∈ (p : G.Walk u v).edges) : (p : G.Walk u v).edges.count e = 1 := List.count_eq_one_of_mem p.property.isTrail.edges_nodup hw #align simple_graph.path.count_edges_eq_one SimpleGraph.Path.count_edges_eq_one @[simp] theorem nodup_support {u v : V} (p : G.Path u v) : (p : G.Walk u v).support.Nodup := (Walk.isPath_def _).mp p.property #align simple_graph.path.nodup_support SimpleGraph.Path.nodup_support theorem loop_eq {v : V} (p : G.Path v v) : p = Path.nil := by obtain ⟨_ | _, h⟩ := p · rfl · simp at h #align simple_graph.path.loop_eq SimpleGraph.Path.loop_eq theorem not_mem_edges_of_loop {v : V} {e : Sym2 V} {p : G.Path v v} : ¬e ∈ (p : G.Walk v v).edges := by simp [p.loop_eq] #align simple_graph.path.not_mem_edges_of_loop SimpleGraph.Path.not_mem_edges_of_loop theorem cons_isCycle {u v : V} (p : G.Path v u) (h : G.Adj u v) (he : ¬s(u, v) ∈ (p : G.Walk v u).edges) : (Walk.cons h ↑p).IsCycle := by simp [Walk.isCycle_def, Walk.cons_isTrail_iff, he] #align simple_graph.path.cons_is_cycle SimpleGraph.Path.cons_isCycle end Path /-! ### Walks to paths -/ namespace Walk variable {G} [DecidableEq V] /-- Given a walk, produces a walk from it by bypassing subwalks between repeated vertices. The result is a path, as shown in `SimpleGraph.Walk.bypass_isPath`. This is packaged up in `SimpleGraph.Walk.toPath`. -/ def bypass {u v : V} : G.Walk u v → G.Walk u v | nil => nil | cons ha p => let p' := p.bypass if hs : u ∈ p'.support then p'.dropUntil u hs else cons ha p' #align simple_graph.walk.bypass SimpleGraph.Walk.bypass @[simp] theorem bypass_copy {u v u' v'} (p : G.Walk u v) (hu : u = u') (hv : v = v') : (p.copy hu hv).bypass = p.bypass.copy hu hv := by subst_vars rfl #align simple_graph.walk.bypass_copy SimpleGraph.Walk.bypass_copy theorem bypass_isPath {u v : V} (p : G.Walk u v) : p.bypass.IsPath := by induction p with | nil => simp! | cons _ p' ih => simp only [bypass] split_ifs with hs · exact ih.dropUntil hs · simp [*, cons_isPath_iff] #align simple_graph.walk.bypass_is_path SimpleGraph.Walk.bypass_isPath theorem length_bypass_le {u v : V} (p : G.Walk u v) : p.bypass.length ≤ p.length := by induction p with | nil => rfl | cons _ _ ih => simp only [bypass] split_ifs · trans apply length_dropUntil_le rw [length_cons] exact le_add_right ih · rw [length_cons, length_cons] exact add_le_add_right ih 1 #align simple_graph.walk.length_bypass_le SimpleGraph.Walk.length_bypass_le

lemma bypass_eq_self_of_length_le {u v : V} (p : G.Walk u v) (h : p.length ≤ p.bypass.length) : p.bypass = p

bypass_eq_self_of_length_le

2021

2e8aeee

mathlib4/Mathlib/Combinatorics/SimpleGraph/Connectivity.lean

mathlib

/- Copyright (c) 2022 Kyle Miller. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kyle Miller, Vincent Beffara -/ import Mathlib.Combinatorics.SimpleGraph.Connectivity import Mathlib.Data.Nat.Lattice #align_import combinatorics.simple_graph.metric from "leanprover-community/mathlib"@"352ecfe114946c903338006dd3287cb5a9955ff2" /-! # Graph metric This module defines the `SimpleGraph.dist` function, which takes pairs of vertices to the length of the shortest walk between them. ## Main definitions - `SimpleGraph.dist` is the graph metric. ## Todo - Provide an additional computable version of `SimpleGraph.dist` for when `G` is connected. - Evaluate `Nat` vs `ENat` for the codomain of `dist`, or potentially having an additional `edist` when the objects under consideration are disconnected graphs. - When directed graphs exist, a directed notion of distance, likely `ENat`-valued. ## Tags graph metric, distance -/ namespace SimpleGraph variable {V : Type*} (G : SimpleGraph V) /-! ## Metric -/ /-- The distance between two vertices is the length of the shortest walk between them. If no such walk exists, this uses the junk value of `0`. -/ noncomputable def dist (u v : V) : ℕ := sInf (Set.range (Walk.length : G.Walk u v → ℕ)) #align simple_graph.dist SimpleGraph.dist variable {G} protected theorem Reachable.exists_walk_of_dist {u v : V} (hr : G.Reachable u v) : ∃ p : G.Walk u v, p.length = G.dist u v := Nat.sInf_mem (Set.range_nonempty_iff_nonempty.mpr hr) #align simple_graph.reachable.exists_walk_of_dist SimpleGraph.Reachable.exists_walk_of_dist protected theorem Connected.exists_walk_of_dist (hconn : G.Connected) (u v : V) : ∃ p : G.Walk u v, p.length = G.dist u v := (hconn u v).exists_walk_of_dist #align simple_graph.connected.exists_walk_of_dist SimpleGraph.Connected.exists_walk_of_dist theorem dist_le {u v : V} (p : G.Walk u v) : G.dist u v ≤ p.length := Nat.sInf_le ⟨p, rfl⟩ #align simple_graph.dist_le SimpleGraph.dist_le @[simp] theorem dist_eq_zero_iff_eq_or_not_reachable {u v : V} : G.dist u v = 0 ↔ u = v ∨ ¬G.Reachable u v := by simp [dist, Nat.sInf_eq_zero, Reachable] #align simple_graph.dist_eq_zero_iff_eq_or_not_reachable SimpleGraph.dist_eq_zero_iff_eq_or_not_reachable theorem dist_self {v : V} : dist G v v = 0 := by simp #align simple_graph.dist_self SimpleGraph.dist_self protected theorem Reachable.dist_eq_zero_iff {u v : V} (hr : G.Reachable u v) : G.dist u v = 0 ↔ u = v := by simp [hr] #align simple_graph.reachable.dist_eq_zero_iff SimpleGraph.Reachable.dist_eq_zero_iff protected theorem Reachable.pos_dist_of_ne {u v : V} (h : G.Reachable u v) (hne : u ≠ v) : 0 < G.dist u v := Nat.pos_of_ne_zero (by simp [h, hne]) #align simple_graph.reachable.pos_dist_of_ne SimpleGraph.Reachable.pos_dist_of_ne protected theorem Connected.dist_eq_zero_iff (hconn : G.Connected) {u v : V} : G.dist u v = 0 ↔ u = v := by simp [hconn u v] #align simple_graph.connected.dist_eq_zero_iff SimpleGraph.Connected.dist_eq_zero_iff protected theorem Connected.pos_dist_of_ne {u v : V} (hconn : G.Connected) (hne : u ≠ v) : 0 < G.dist u v := Nat.pos_of_ne_zero (by intro h; exact False.elim (hne (hconn.dist_eq_zero_iff.mp h))) #align simple_graph.connected.pos_dist_of_ne SimpleGraph.Connected.pos_dist_of_ne theorem dist_eq_zero_of_not_reachable {u v : V} (h : ¬G.Reachable u v) : G.dist u v = 0 := by simp [h] #align simple_graph.dist_eq_zero_of_not_reachable SimpleGraph.dist_eq_zero_of_not_reachable theorem nonempty_of_pos_dist {u v : V} (h : 0 < G.dist u v) : (Set.univ : Set (G.Walk u v)).Nonempty := by simpa [Set.range_nonempty_iff_nonempty, Set.nonempty_iff_univ_nonempty] using Nat.nonempty_of_pos_sInf h #align simple_graph.nonempty_of_pos_dist SimpleGraph.nonempty_of_pos_dist protected theorem Connected.dist_triangle (hconn : G.Connected) {u v w : V} : G.dist u w ≤ G.dist u v + G.dist v w := by obtain ⟨p, hp⟩ := hconn.exists_walk_of_dist u v obtain ⟨q, hq⟩ := hconn.exists_walk_of_dist v w rw [← hp, ← hq, ← Walk.length_append] apply dist_le #align simple_graph.connected.dist_triangle SimpleGraph.Connected.dist_triangle private theorem dist_comm_aux {u v : V} (h : G.Reachable u v) : G.dist u v ≤ G.dist v u := by obtain ⟨p, hp⟩ := h.symm.exists_walk_of_dist rw [← hp, ← Walk.length_reverse] apply dist_le theorem dist_comm {u v : V} : G.dist u v = G.dist v u := by by_cases h : G.Reachable u v · apply le_antisymm (dist_comm_aux h) (dist_comm_aux h.symm) · have h' : ¬G.Reachable v u := fun h' => absurd h'.symm h simp [h, h', dist_eq_zero_of_not_reachable] #align simple_graph.dist_comm SimpleGraph.dist_comm

theorem Walk.isPath_of_length_eq_dist {u v : V} (p : G.Walk u v) (hp : p.length = G.dist u v) : p.IsPath

Walk.isPath_of_length_eq_dist

2022

2e8aeee

mathlib4/Mathlib/Combinatorics/SimpleGraph/Metric.lean

mathlib

/- Copyright (c) 2022 Kyle Miller. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kyle Miller, Vincent Beffara -/ import Mathlib.Combinatorics.SimpleGraph.Connectivity import Mathlib.Data.Nat.Lattice #align_import combinatorics.simple_graph.metric from "leanprover-community/mathlib"@"352ecfe114946c903338006dd3287cb5a9955ff2" /-! # Graph metric This module defines the `SimpleGraph.dist` function, which takes pairs of vertices to the length of the shortest walk between them. ## Main definitions - `SimpleGraph.dist` is the graph metric. ## Todo - Provide an additional computable version of `SimpleGraph.dist` for when `G` is connected. - Evaluate `Nat` vs `ENat` for the codomain of `dist`, or potentially having an additional `edist` when the objects under consideration are disconnected graphs. - When directed graphs exist, a directed notion of distance, likely `ENat`-valued. ## Tags graph metric, distance -/ namespace SimpleGraph variable {V : Type*} (G : SimpleGraph V) /-! ## Metric -/ /-- The distance between two vertices is the length of the shortest walk between them. If no such walk exists, this uses the junk value of `0`. -/ noncomputable def dist (u v : V) : ℕ := sInf (Set.range (Walk.length : G.Walk u v → ℕ)) #align simple_graph.dist SimpleGraph.dist variable {G} protected theorem Reachable.exists_walk_of_dist {u v : V} (hr : G.Reachable u v) : ∃ p : G.Walk u v, p.length = G.dist u v := Nat.sInf_mem (Set.range_nonempty_iff_nonempty.mpr hr) #align simple_graph.reachable.exists_walk_of_dist SimpleGraph.Reachable.exists_walk_of_dist protected theorem Connected.exists_walk_of_dist (hconn : G.Connected) (u v : V) : ∃ p : G.Walk u v, p.length = G.dist u v := (hconn u v).exists_walk_of_dist #align simple_graph.connected.exists_walk_of_dist SimpleGraph.Connected.exists_walk_of_dist theorem dist_le {u v : V} (p : G.Walk u v) : G.dist u v ≤ p.length := Nat.sInf_le ⟨p, rfl⟩ #align simple_graph.dist_le SimpleGraph.dist_le @[simp] theorem dist_eq_zero_iff_eq_or_not_reachable {u v : V} : G.dist u v = 0 ↔ u = v ∨ ¬G.Reachable u v := by simp [dist, Nat.sInf_eq_zero, Reachable] #align simple_graph.dist_eq_zero_iff_eq_or_not_reachable SimpleGraph.dist_eq_zero_iff_eq_or_not_reachable theorem dist_self {v : V} : dist G v v = 0 := by simp #align simple_graph.dist_self SimpleGraph.dist_self protected theorem Reachable.dist_eq_zero_iff {u v : V} (hr : G.Reachable u v) : G.dist u v = 0 ↔ u = v := by simp [hr] #align simple_graph.reachable.dist_eq_zero_iff SimpleGraph.Reachable.dist_eq_zero_iff protected theorem Reachable.pos_dist_of_ne {u v : V} (h : G.Reachable u v) (hne : u ≠ v) : 0 < G.dist u v := Nat.pos_of_ne_zero (by simp [h, hne]) #align simple_graph.reachable.pos_dist_of_ne SimpleGraph.Reachable.pos_dist_of_ne protected theorem Connected.dist_eq_zero_iff (hconn : G.Connected) {u v : V} : G.dist u v = 0 ↔ u = v := by simp [hconn u v] #align simple_graph.connected.dist_eq_zero_iff SimpleGraph.Connected.dist_eq_zero_iff protected theorem Connected.pos_dist_of_ne {u v : V} (hconn : G.Connected) (hne : u ≠ v) : 0 < G.dist u v := Nat.pos_of_ne_zero (by intro h; exact False.elim (hne (hconn.dist_eq_zero_iff.mp h))) #align simple_graph.connected.pos_dist_of_ne SimpleGraph.Connected.pos_dist_of_ne theorem dist_eq_zero_of_not_reachable {u v : V} (h : ¬G.Reachable u v) : G.dist u v = 0 := by simp [h] #align simple_graph.dist_eq_zero_of_not_reachable SimpleGraph.dist_eq_zero_of_not_reachable theorem nonempty_of_pos_dist {u v : V} (h : 0 < G.dist u v) : (Set.univ : Set (G.Walk u v)).Nonempty := by simpa [Set.range_nonempty_iff_nonempty, Set.nonempty_iff_univ_nonempty] using Nat.nonempty_of_pos_sInf h #align simple_graph.nonempty_of_pos_dist SimpleGraph.nonempty_of_pos_dist protected theorem Connected.dist_triangle (hconn : G.Connected) {u v w : V} : G.dist u w ≤ G.dist u v + G.dist v w := by obtain ⟨p, hp⟩ := hconn.exists_walk_of_dist u v obtain ⟨q, hq⟩ := hconn.exists_walk_of_dist v w rw [← hp, ← hq, ← Walk.length_append] apply dist_le #align simple_graph.connected.dist_triangle SimpleGraph.Connected.dist_triangle private theorem dist_comm_aux {u v : V} (h : G.Reachable u v) : G.dist u v ≤ G.dist v u := by obtain ⟨p, hp⟩ := h.symm.exists_walk_of_dist rw [← hp, ← Walk.length_reverse] apply dist_le theorem dist_comm {u v : V} : G.dist u v = G.dist v u := by by_cases h : G.Reachable u v · apply le_antisymm (dist_comm_aux h) (dist_comm_aux h.symm) · have h' : ¬G.Reachable v u := fun h' => absurd h'.symm h simp [h, h', dist_eq_zero_of_not_reachable] #align simple_graph.dist_comm SimpleGraph.dist_comm theorem Walk.isPath_of_length_eq_dist {u v : V} (p : G.Walk u v) (hp : p.length = G.dist u v) : p.IsPath := by classical have : p.bypass = p := by apply Walk.bypass_eq_self_of_length_le calc p.length _ = G.dist u v := hp _ ≤ p.bypass.length := dist_le p.bypass rw [← this] apply Walk.bypass_isPath

lemma Reachable.exists_path_of_dist {u v : V} (hr : G.Reachable u v) : ∃ (p : G.Walk u v), p.IsPath ∧ p.length = G.dist u v

Reachable.exists_path_of_dist

2022

2e8aeee

mathlib4/Mathlib/Combinatorics/SimpleGraph/Metric.lean

mathlib

/- Copyright (c) 2022 Kyle Miller. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kyle Miller, Vincent Beffara -/ import Mathlib.Combinatorics.SimpleGraph.Connectivity import Mathlib.Data.Nat.Lattice #align_import combinatorics.simple_graph.metric from "leanprover-community/mathlib"@"352ecfe114946c903338006dd3287cb5a9955ff2" /-! # Graph metric This module defines the `SimpleGraph.dist` function, which takes pairs of vertices to the length of the shortest walk between them. ## Main definitions - `SimpleGraph.dist` is the graph metric. ## Todo - Provide an additional computable version of `SimpleGraph.dist` for when `G` is connected. - Evaluate `Nat` vs `ENat` for the codomain of `dist`, or potentially having an additional `edist` when the objects under consideration are disconnected graphs. - When directed graphs exist, a directed notion of distance, likely `ENat`-valued. ## Tags graph metric, distance -/ namespace SimpleGraph variable {V : Type*} (G : SimpleGraph V) /-! ## Metric -/ /-- The distance between two vertices is the length of the shortest walk between them. If no such walk exists, this uses the junk value of `0`. -/ noncomputable def dist (u v : V) : ℕ := sInf (Set.range (Walk.length : G.Walk u v → ℕ)) #align simple_graph.dist SimpleGraph.dist variable {G} protected theorem Reachable.exists_walk_of_dist {u v : V} (hr : G.Reachable u v) : ∃ p : G.Walk u v, p.length = G.dist u v := Nat.sInf_mem (Set.range_nonempty_iff_nonempty.mpr hr) #align simple_graph.reachable.exists_walk_of_dist SimpleGraph.Reachable.exists_walk_of_dist protected theorem Connected.exists_walk_of_dist (hconn : G.Connected) (u v : V) : ∃ p : G.Walk u v, p.length = G.dist u v := (hconn u v).exists_walk_of_dist #align simple_graph.connected.exists_walk_of_dist SimpleGraph.Connected.exists_walk_of_dist theorem dist_le {u v : V} (p : G.Walk u v) : G.dist u v ≤ p.length := Nat.sInf_le ⟨p, rfl⟩ #align simple_graph.dist_le SimpleGraph.dist_le @[simp] theorem dist_eq_zero_iff_eq_or_not_reachable {u v : V} : G.dist u v = 0 ↔ u = v ∨ ¬G.Reachable u v := by simp [dist, Nat.sInf_eq_zero, Reachable] #align simple_graph.dist_eq_zero_iff_eq_or_not_reachable SimpleGraph.dist_eq_zero_iff_eq_or_not_reachable theorem dist_self {v : V} : dist G v v = 0 := by simp #align simple_graph.dist_self SimpleGraph.dist_self protected theorem Reachable.dist_eq_zero_iff {u v : V} (hr : G.Reachable u v) : G.dist u v = 0 ↔ u = v := by simp [hr] #align simple_graph.reachable.dist_eq_zero_iff SimpleGraph.Reachable.dist_eq_zero_iff protected theorem Reachable.pos_dist_of_ne {u v : V} (h : G.Reachable u v) (hne : u ≠ v) : 0 < G.dist u v := Nat.pos_of_ne_zero (by simp [h, hne]) #align simple_graph.reachable.pos_dist_of_ne SimpleGraph.Reachable.pos_dist_of_ne protected theorem Connected.dist_eq_zero_iff (hconn : G.Connected) {u v : V} : G.dist u v = 0 ↔ u = v := by simp [hconn u v] #align simple_graph.connected.dist_eq_zero_iff SimpleGraph.Connected.dist_eq_zero_iff protected theorem Connected.pos_dist_of_ne {u v : V} (hconn : G.Connected) (hne : u ≠ v) : 0 < G.dist u v := Nat.pos_of_ne_zero (by intro h; exact False.elim (hne (hconn.dist_eq_zero_iff.mp h))) #align simple_graph.connected.pos_dist_of_ne SimpleGraph.Connected.pos_dist_of_ne theorem dist_eq_zero_of_not_reachable {u v : V} (h : ¬G.Reachable u v) : G.dist u v = 0 := by simp [h] #align simple_graph.dist_eq_zero_of_not_reachable SimpleGraph.dist_eq_zero_of_not_reachable theorem nonempty_of_pos_dist {u v : V} (h : 0 < G.dist u v) : (Set.univ : Set (G.Walk u v)).Nonempty := by simpa [Set.range_nonempty_iff_nonempty, Set.nonempty_iff_univ_nonempty] using Nat.nonempty_of_pos_sInf h #align simple_graph.nonempty_of_pos_dist SimpleGraph.nonempty_of_pos_dist protected theorem Connected.dist_triangle (hconn : G.Connected) {u v w : V} : G.dist u w ≤ G.dist u v + G.dist v w := by obtain ⟨p, hp⟩ := hconn.exists_walk_of_dist u v obtain ⟨q, hq⟩ := hconn.exists_walk_of_dist v w rw [← hp, ← hq, ← Walk.length_append] apply dist_le #align simple_graph.connected.dist_triangle SimpleGraph.Connected.dist_triangle private theorem dist_comm_aux {u v : V} (h : G.Reachable u v) : G.dist u v ≤ G.dist v u := by obtain ⟨p, hp⟩ := h.symm.exists_walk_of_dist rw [← hp, ← Walk.length_reverse] apply dist_le theorem dist_comm {u v : V} : G.dist u v = G.dist v u := by by_cases h : G.Reachable u v · apply le_antisymm (dist_comm_aux h) (dist_comm_aux h.symm) · have h' : ¬G.Reachable v u := fun h' => absurd h'.symm h simp [h, h', dist_eq_zero_of_not_reachable] #align simple_graph.dist_comm SimpleGraph.dist_comm theorem Walk.isPath_of_length_eq_dist {u v : V} (p : G.Walk u v) (hp : p.length = G.dist u v) : p.IsPath := by classical have : p.bypass = p := by apply Walk.bypass_eq_self_of_length_le calc p.length _ = G.dist u v := hp _ ≤ p.bypass.length := dist_le p.bypass rw [← this] apply Walk.bypass_isPath lemma Reachable.exists_path_of_dist {u v : V} (hr : G.Reachable u v) : ∃ (p : G.Walk u v), p.IsPath ∧ p.length = G.dist u v := by obtain ⟨p, h⟩ := hr.exists_walk_of_dist exact ⟨p, p.isPath_of_length_eq_dist h, h⟩

lemma Connected.exists_path_of_dist (hconn : G.Connected) (u v : V) : ∃ (p : G.Walk u v), p.IsPath ∧ p.length = G.dist u v

Connected.exists_path_of_dist

2022

2e8aeee

mathlib4/Mathlib/Combinatorics/SimpleGraph/Metric.lean

mathlib

/- Copyright (c) 2020 Yury Kudryashov. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Alexander Bentkamp, Yury Kudryashov -/ import Mathlib.Analysis.Convex.Combination import Mathlib.Analysis.Convex.Strict import Mathlib.Topology.Connected.PathConnected import Mathlib.Topology.Algebra.Affine import Mathlib.Topology.Algebra.Module.Basic #align_import analysis.convex.topology from "leanprover-community/mathlib"@"0e3aacdc98d25e0afe035c452d876d28cbffaa7e" /-! # Topological properties of convex sets We prove the following facts: * `Convex.interior` : interior of a convex set is convex; * `Convex.closure` : closure of a convex set is convex; * `Set.Finite.isCompact_convexHull` : convex hull of a finite set is compact; * `Set.Finite.isClosed_convexHull` : convex hull of a finite set is closed. -/ assert_not_exists Norm open Metric Bornology Set Pointwise Convex variable {ι 𝕜 E : Type*} theorem Real.convex_iff_isPreconnected {s : Set ℝ} : Convex ℝ s ↔ IsPreconnected s := convex_iff_ordConnected.trans isPreconnected_iff_ordConnected.symm #align real.convex_iff_is_preconnected Real.convex_iff_isPreconnected alias ⟨_, IsPreconnected.convex⟩ := Real.convex_iff_isPreconnected #align is_preconnected.convex IsPreconnected.convex /-! ### Standard simplex -/ section stdSimplex variable [Fintype ι] /-- Every vector in `stdSimplex 𝕜 ι` has `max`-norm at most `1`. -/ theorem stdSimplex_subset_closedBall : stdSimplex ℝ ι ⊆ Metric.closedBall 0 1 := fun f hf ↦ by rw [Metric.mem_closedBall, dist_pi_le_iff zero_le_one] intro x rw [Pi.zero_apply, Real.dist_0_eq_abs, abs_of_nonneg <| hf.1 x] exact (mem_Icc_of_mem_stdSimplex hf x).2 #align std_simplex_subset_closed_ball stdSimplex_subset_closedBall variable (ι) /-- `stdSimplex ℝ ι` is bounded. -/ theorem bounded_stdSimplex : IsBounded (stdSimplex ℝ ι) := (Metric.isBounded_iff_subset_closedBall 0).2 ⟨1, stdSimplex_subset_closedBall⟩ #align bounded_std_simplex bounded_stdSimplex /-- `stdSimplex ℝ ι` is closed. -/ theorem isClosed_stdSimplex : IsClosed (stdSimplex ℝ ι) := (stdSimplex_eq_inter ℝ ι).symm ▸ IsClosed.inter (isClosed_iInter fun i => isClosed_le continuous_const (continuous_apply i)) (isClosed_eq (continuous_finset_sum _ fun x _ => continuous_apply x) continuous_const) #align is_closed_std_simplex isClosed_stdSimplex /-- `stdSimplex ℝ ι` is compact. -/ theorem isCompact_stdSimplex : IsCompact (stdSimplex ℝ ι) := Metric.isCompact_iff_isClosed_bounded.2 ⟨isClosed_stdSimplex ι, bounded_stdSimplex ι⟩ #align is_compact_std_simplex isCompact_stdSimplex instance stdSimplex.instCompactSpace_coe : CompactSpace ↥(stdSimplex ℝ ι) := isCompact_iff_compactSpace.mp <| isCompact_stdSimplex _ /-- The standard one-dimensional simplex in `ℝ² = Fin 2 → ℝ` is homeomorphic to the unit interval. -/ @[simps! (config := .asFn)] def stdSimplexHomeomorphUnitInterval : stdSimplex ℝ (Fin 2) ≃ₜ unitInterval where toEquiv := stdSimplexEquivIcc ℝ continuous_toFun := .subtype_mk ((continuous_apply 0).comp continuous_subtype_val) _ continuous_invFun := by apply Continuous.subtype_mk exact (continuous_pi <| Fin.forall_fin_two.2 ⟨continuous_subtype_val, continuous_const.sub continuous_subtype_val⟩) end stdSimplex /-! ### Topological vector spaces -/ section TopologicalSpace variable [LinearOrderedRing 𝕜] [DenselyOrdered 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] [AddCommGroup E] [TopologicalSpace E] [ContinuousAdd E] [Module 𝕜 E] [ContinuousSMul 𝕜 E] {x y : E} theorem segment_subset_closure_openSegment : [x -[𝕜] y] ⊆ closure (openSegment 𝕜 x y) := by rw [segment_eq_image, openSegment_eq_image, ← closure_Ioo (zero_ne_one' 𝕜)] exact image_closure_subset_closure_image (by continuity) #align segment_subset_closure_open_segment segment_subset_closure_openSegment end TopologicalSpace section PseudoMetricSpace variable [LinearOrderedRing 𝕜] [DenselyOrdered 𝕜] [PseudoMetricSpace 𝕜] [OrderTopology 𝕜] [ProperSpace 𝕜] [CompactIccSpace 𝕜] [AddCommGroup E] [TopologicalSpace E] [T2Space E] [ContinuousAdd E] [Module 𝕜 E] [ContinuousSMul 𝕜 E] @[simp] theorem closure_openSegment (x y : E) : closure (openSegment 𝕜 x y) = [x -[𝕜] y] := by rw [segment_eq_image, openSegment_eq_image, ← closure_Ioo (zero_ne_one' 𝕜)] exact (image_closure_of_isCompact (isBounded_Ioo _ _).isCompact_closure <| Continuous.continuousOn <| by continuity).symm #align closure_open_segment closure_openSegment end PseudoMetricSpace section ContinuousConstSMul variable [LinearOrderedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousConstSMul 𝕜 E] /-- If `s` is a convex set, then `a • interior s + b • closure s ⊆ interior s` for all `0 < a`, `0 ≤ b`, `a + b = 1`. See also `Convex.combo_interior_self_subset_interior` for a weaker version. -/ theorem Convex.combo_interior_closure_subset_interior {s : Set E} (hs : Convex 𝕜 s) {a b : 𝕜} (ha : 0 < a) (hb : 0 ≤ b) (hab : a + b = 1) : a • interior s + b • closure s ⊆ interior s := interior_smul₀ ha.ne' s ▸ calc interior (a • s) + b • closure s ⊆ interior (a • s) + closure (b • s) := add_subset_add Subset.rfl (smul_closure_subset b s) _ = interior (a • s) + b • s := by rw [isOpen_interior.add_closure (b • s)] _ ⊆ interior (a • s + b • s) := subset_interior_add_left _ ⊆ interior s := interior_mono <| hs.set_combo_subset ha.le hb hab #align convex.combo_interior_closure_subset_interior Convex.combo_interior_closure_subset_interior /-- If `s` is a convex set, then `a • interior s + b • s ⊆ interior s` for all `0 < a`, `0 ≤ b`, `a + b = 1`. See also `Convex.combo_interior_closure_subset_interior` for a stronger version. -/ theorem Convex.combo_interior_self_subset_interior {s : Set E} (hs : Convex 𝕜 s) {a b : 𝕜} (ha : 0 < a) (hb : 0 ≤ b) (hab : a + b = 1) : a • interior s + b • s ⊆ interior s := calc a • interior s + b • s ⊆ a • interior s + b • closure s := add_subset_add Subset.rfl <| image_subset _ subset_closure _ ⊆ interior s := hs.combo_interior_closure_subset_interior ha hb hab #align convex.combo_interior_self_subset_interior Convex.combo_interior_self_subset_interior /-- If `s` is a convex set, then `a • closure s + b • interior s ⊆ interior s` for all `0 ≤ a`, `0 < b`, `a + b = 1`. See also `Convex.combo_self_interior_subset_interior` for a weaker version. -/ theorem Convex.combo_closure_interior_subset_interior {s : Set E} (hs : Convex 𝕜 s) {a b : 𝕜} (ha : 0 ≤ a) (hb : 0 < b) (hab : a + b = 1) : a • closure s + b • interior s ⊆ interior s := by rw [add_comm] exact hs.combo_interior_closure_subset_interior hb ha (add_comm a b ▸ hab) #align convex.combo_closure_interior_subset_interior Convex.combo_closure_interior_subset_interior /-- If `s` is a convex set, then `a • s + b • interior s ⊆ interior s` for all `0 ≤ a`, `0 < b`, `a + b = 1`. See also `Convex.combo_closure_interior_subset_interior` for a stronger version. -/ theorem Convex.combo_self_interior_subset_interior {s : Set E} (hs : Convex 𝕜 s) {a b : 𝕜} (ha : 0 ≤ a) (hb : 0 < b) (hab : a + b = 1) : a • s + b • interior s ⊆ interior s := by rw [add_comm] exact hs.combo_interior_self_subset_interior hb ha (add_comm a b ▸ hab) #align convex.combo_self_interior_subset_interior Convex.combo_self_interior_subset_interior theorem Convex.combo_interior_closure_mem_interior {s : Set E} (hs : Convex 𝕜 s) {x y : E} (hx : x ∈ interior s) (hy : y ∈ closure s) {a b : 𝕜} (ha : 0 < a) (hb : 0 ≤ b) (hab : a + b = 1) : a • x + b • y ∈ interior s := hs.combo_interior_closure_subset_interior ha hb hab <| add_mem_add (smul_mem_smul_set hx) (smul_mem_smul_set hy) #align convex.combo_interior_closure_mem_interior Convex.combo_interior_closure_mem_interior theorem Convex.combo_interior_self_mem_interior {s : Set E} (hs : Convex 𝕜 s) {x y : E} (hx : x ∈ interior s) (hy : y ∈ s) {a b : 𝕜} (ha : 0 < a) (hb : 0 ≤ b) (hab : a + b = 1) : a • x + b • y ∈ interior s := hs.combo_interior_closure_mem_interior hx (subset_closure hy) ha hb hab #align convex.combo_interior_self_mem_interior Convex.combo_interior_self_mem_interior theorem Convex.combo_closure_interior_mem_interior {s : Set E} (hs : Convex 𝕜 s) {x y : E} (hx : x ∈ closure s) (hy : y ∈ interior s) {a b : 𝕜} (ha : 0 ≤ a) (hb : 0 < b) (hab : a + b = 1) : a • x + b • y ∈ interior s := hs.combo_closure_interior_subset_interior ha hb hab <| add_mem_add (smul_mem_smul_set hx) (smul_mem_smul_set hy) #align convex.combo_closure_interior_mem_interior Convex.combo_closure_interior_mem_interior theorem Convex.combo_self_interior_mem_interior {s : Set E} (hs : Convex 𝕜 s) {x y : E} (hx : x ∈ s) (hy : y ∈ interior s) {a b : 𝕜} (ha : 0 ≤ a) (hb : 0 < b) (hab : a + b = 1) : a • x + b • y ∈ interior s := hs.combo_closure_interior_mem_interior (subset_closure hx) hy ha hb hab #align convex.combo_self_interior_mem_interior Convex.combo_self_interior_mem_interior theorem Convex.openSegment_interior_closure_subset_interior {s : Set E} (hs : Convex 𝕜 s) {x y : E} (hx : x ∈ interior s) (hy : y ∈ closure s) : openSegment 𝕜 x y ⊆ interior s := by rintro _ ⟨a, b, ha, hb, hab, rfl⟩ exact hs.combo_interior_closure_mem_interior hx hy ha hb.le hab #align convex.open_segment_interior_closure_subset_interior Convex.openSegment_interior_closure_subset_interior theorem Convex.openSegment_interior_self_subset_interior {s : Set E} (hs : Convex 𝕜 s) {x y : E} (hx : x ∈ interior s) (hy : y ∈ s) : openSegment 𝕜 x y ⊆ interior s := hs.openSegment_interior_closure_subset_interior hx (subset_closure hy) #align convex.open_segment_interior_self_subset_interior Convex.openSegment_interior_self_subset_interior theorem Convex.openSegment_closure_interior_subset_interior {s : Set E} (hs : Convex 𝕜 s) {x y : E} (hx : x ∈ closure s) (hy : y ∈ interior s) : openSegment 𝕜 x y ⊆ interior s := by rintro _ ⟨a, b, ha, hb, hab, rfl⟩ exact hs.combo_closure_interior_mem_interior hx hy ha.le hb hab #align convex.open_segment_closure_interior_subset_interior Convex.openSegment_closure_interior_subset_interior theorem Convex.openSegment_self_interior_subset_interior {s : Set E} (hs : Convex 𝕜 s) {x y : E} (hx : x ∈ s) (hy : y ∈ interior s) : openSegment 𝕜 x y ⊆ interior s := hs.openSegment_closure_interior_subset_interior (subset_closure hx) hy #align convex.open_segment_self_interior_subset_interior Convex.openSegment_self_interior_subset_interior /-- If `x ∈ closure s` and `y ∈ interior s`, then the segment `(x, y]` is included in `interior s`. -/ theorem Convex.add_smul_sub_mem_interior' {s : Set E} (hs : Convex 𝕜 s) {x y : E} (hx : x ∈ closure s) (hy : y ∈ interior s) {t : 𝕜} (ht : t ∈ Ioc (0 : 𝕜) 1) : x + t • (y - x) ∈ interior s := by simpa only [sub_smul, smul_sub, one_smul, add_sub, add_comm] using hs.combo_interior_closure_mem_interior hy hx ht.1 (sub_nonneg.mpr ht.2) (add_sub_cancel _ _) #align convex.add_smul_sub_mem_interior' Convex.add_smul_sub_mem_interior' /-- If `x ∈ s` and `y ∈ interior s`, then the segment `(x, y]` is included in `interior s`. -/ theorem Convex.add_smul_sub_mem_interior {s : Set E} (hs : Convex 𝕜 s) {x y : E} (hx : x ∈ s) (hy : y ∈ interior s) {t : 𝕜} (ht : t ∈ Ioc (0 : 𝕜) 1) : x + t • (y - x) ∈ interior s := hs.add_smul_sub_mem_interior' (subset_closure hx) hy ht #align convex.add_smul_sub_mem_interior Convex.add_smul_sub_mem_interior /-- If `x ∈ closure s` and `x + y ∈ interior s`, then `x + t y ∈ interior s` for `t ∈ (0, 1]`. -/ theorem Convex.add_smul_mem_interior' {s : Set E} (hs : Convex 𝕜 s) {x y : E} (hx : x ∈ closure s) (hy : x + y ∈ interior s) {t : 𝕜} (ht : t ∈ Ioc (0 : 𝕜) 1) : x + t • y ∈ interior s := by simpa only [add_sub_cancel_left] using hs.add_smul_sub_mem_interior' hx hy ht #align convex.add_smul_mem_interior' Convex.add_smul_mem_interior' /-- If `x ∈ s` and `x + y ∈ interior s`, then `x + t y ∈ interior s` for `t ∈ (0, 1]`. -/ theorem Convex.add_smul_mem_interior {s : Set E} (hs : Convex 𝕜 s) {x y : E} (hx : x ∈ s) (hy : x + y ∈ interior s) {t : 𝕜} (ht : t ∈ Ioc (0 : 𝕜) 1) : x + t • y ∈ interior s := hs.add_smul_mem_interior' (subset_closure hx) hy ht #align convex.add_smul_mem_interior Convex.add_smul_mem_interior /-- In a topological vector space, the interior of a convex set is convex. -/ protected theorem Convex.interior {s : Set E} (hs : Convex 𝕜 s) : Convex 𝕜 (interior s) := convex_iff_openSegment_subset.mpr fun _ hx _ hy => hs.openSegment_closure_interior_subset_interior (interior_subset_closure hx) hy #align convex.interior Convex.interior /-- In a topological vector space, the closure of a convex set is convex. -/ protected theorem Convex.closure {s : Set E} (hs : Convex 𝕜 s) : Convex 𝕜 (closure s) := fun x hx y hy a b ha hb hab => let f : E → E → E := fun x' y' => a • x' + b • y' have hf : Continuous (Function.uncurry f) := (continuous_fst.const_smul _).add (continuous_snd.const_smul _) show f x y ∈ closure s from map_mem_closure₂ hf hx hy fun _ hx' _ hy' => hs hx' hy' ha hb hab #align convex.closure Convex.closure open AffineMap /-- A convex set `s` is strictly convex provided that for any two distinct points of `s \ interior s`, the line passing through these points has nonempty intersection with `interior s`. -/ protected theorem Convex.strictConvex' {s : Set E} (hs : Convex 𝕜 s) (h : (s \ interior s).Pairwise fun x y => ∃ c : 𝕜, lineMap x y c ∈ interior s) : StrictConvex 𝕜 s := by refine' strictConvex_iff_openSegment_subset.2 _ intro x hx y hy hne by_cases hx' : x ∈ interior s · exact hs.openSegment_interior_self_subset_interior hx' hy by_cases hy' : y ∈ interior s · exact hs.openSegment_self_interior_subset_interior hx hy' rcases h ⟨hx, hx'⟩ ⟨hy, hy'⟩ hne with ⟨c, hc⟩ refine' (openSegment_subset_union x y ⟨c, rfl⟩).trans (insert_subset_iff.2 ⟨hc, union_subset _ _⟩) exacts [hs.openSegment_self_interior_subset_interior hx hc, hs.openSegment_interior_self_subset_interior hc hy] #align convex.strict_convex' Convex.strictConvex' /-- A convex set `s` is strictly convex provided that for any two distinct points `x`, `y` of `s \ interior s`, the segment with endpoints `x`, `y` has nonempty intersection with `interior s`. -/ protected theorem Convex.strictConvex {s : Set E} (hs : Convex 𝕜 s) (h : (s \ interior s).Pairwise fun x y => ([x -[𝕜] y] \ frontier s).Nonempty) : StrictConvex 𝕜 s := by refine' hs.strictConvex' <| h.imp_on fun x hx y hy _ => _ simp only [segment_eq_image_lineMap, ← self_diff_frontier] rintro ⟨_, ⟨⟨c, hc, rfl⟩, hcs⟩⟩ refine' ⟨c, hs.segment_subset hx.1 hy.1 _, hcs⟩ exact (segment_eq_image_lineMap 𝕜 x y).symm ▸ mem_image_of_mem _ hc #align convex.strict_convex Convex.strictConvex end ContinuousConstSMul section ContinuousSMul variable [AddCommGroup E] [Module ℝ E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousSMul ℝ E] /-- Convex hull of a finite set is compact. -/ theorem Set.Finite.isCompact_convexHull {s : Set E} (hs : s.Finite) : IsCompact (convexHull ℝ s) := by rw [hs.convexHull_eq_image] apply (@isCompact_stdSimplex _ hs.fintype).image haveI := hs.fintype apply LinearMap.continuous_on_pi #align set.finite.compact_convex_hull Set.Finite.isCompact_convexHull /-- Convex hull of a finite set is closed. -/ theorem Set.Finite.isClosed_convexHull [T2Space E] {s : Set E} (hs : s.Finite) : IsClosed (convexHull ℝ s) := hs.isCompact_convexHull.isClosed #align set.finite.is_closed_convex_hull Set.Finite.isClosed_convexHull open AffineMap /-- If we dilate the interior of a convex set about a point in its interior by a scale `t > 1`, the result includes the closure of the original set. TODO Generalise this from convex sets to sets that are balanced / star-shaped about `x`. -/ theorem Convex.closure_subset_image_homothety_interior_of_one_lt {s : Set E} (hs : Convex ℝ s) {x : E} (hx : x ∈ interior s) (t : ℝ) (ht : 1 < t) : closure s ⊆ homothety x t '' interior s := by intro y hy have hne : t ≠ 0 := (one_pos.trans ht).ne' refine' ⟨homothety x t⁻¹ y, hs.openSegment_interior_closure_subset_interior hx hy _, (AffineEquiv.homothetyUnitsMulHom x (Units.mk0 t hne)).apply_symm_apply y⟩ rw [openSegment_eq_image_lineMap, ← inv_one, ← inv_Ioi (zero_lt_one' ℝ), ← image_inv, image_image, homothety_eq_lineMap] exact mem_image_of_mem _ ht #align convex.closure_subset_image_homothety_interior_of_one_lt Convex.closure_subset_image_homothety_interior_of_one_lt /-- If we dilate a convex set about a point in its interior by a scale `t > 1`, the interior of the result includes the closure of the original set. TODO Generalise this from convex sets to sets that are balanced / star-shaped about `x`. -/ theorem Convex.closure_subset_interior_image_homothety_of_one_lt {s : Set E} (hs : Convex ℝ s) {x : E} (hx : x ∈ interior s) (t : ℝ) (ht : 1 < t) : closure s ⊆ interior (homothety x t '' s) := (hs.closure_subset_image_homothety_interior_of_one_lt hx t ht).trans <| (homothety_isOpenMap x t (one_pos.trans ht).ne').image_interior_subset _ #align convex.closure_subset_interior_image_homothety_of_one_lt Convex.closure_subset_interior_image_homothety_of_one_lt /-- If we dilate a convex set about a point in its interior by a scale `t > 1`, the interior of the result includes the closure of the original set. TODO Generalise this from convex sets to sets that are balanced / star-shaped about `x`. -/ theorem Convex.subset_interior_image_homothety_of_one_lt {s : Set E} (hs : Convex ℝ s) {x : E} (hx : x ∈ interior s) (t : ℝ) (ht : 1 < t) : s ⊆ interior (homothety x t '' s) := subset_closure.trans <| hs.closure_subset_interior_image_homothety_of_one_lt hx t ht #align convex.subset_interior_image_homothety_of_one_lt Convex.subset_interior_image_homothety_of_one_lt theorem JoinedIn.of_segment_subset {E : Type*} [AddCommGroup E] [Module ℝ E] [TopologicalSpace E] [ContinuousAdd E] [ContinuousSMul ℝ E] {x y : E} {s : Set E} (h : [x -[ℝ] y] ⊆ s) : JoinedIn s x y := by have A : Continuous (fun t ↦ (1 - t) • x + t • y : ℝ → E) := by continuity apply JoinedIn.ofLine A.continuousOn (by simp) (by simp) convert h rw [segment_eq_image ℝ x y] /-- A nonempty convex set is path connected. -/ protected theorem Convex.isPathConnected {s : Set E} (hconv : Convex ℝ s) (hne : s.Nonempty) : IsPathConnected s := by refine' isPathConnected_iff.mpr ⟨hne, _⟩ intro x x_in y y_in exact JoinedIn.of_segment_subset ((segment_subset_iff ℝ).2 (hconv x_in y_in)) #align convex.is_path_connected Convex.isPathConnected /-- A nonempty convex set is connected. -/ protected theorem Convex.isConnected {s : Set E} (h : Convex ℝ s) (hne : s.Nonempty) : IsConnected s := (h.isPathConnected hne).isConnected #align convex.is_connected Convex.isConnected /-- A convex set is preconnected. -/ protected theorem Convex.isPreconnected {s : Set E} (h : Convex ℝ s) : IsPreconnected s := s.eq_empty_or_nonempty.elim (fun h => h.symm ▸ isPreconnected_empty) fun hne => (h.isConnected hne).isPreconnected #align convex.is_preconnected Convex.isPreconnected /-- Every topological vector space over ℝ is path connected. Not an instance, because it creates enormous TC subproblems (turn on `pp.all`). -/ protected theorem TopologicalAddGroup.pathConnectedSpace : PathConnectedSpace E := pathConnectedSpace_iff_univ.mpr <| convex_univ.isPathConnected ⟨(0 : E), trivial⟩ #align topological_add_group.path_connected TopologicalAddGroup.pathConnectedSpace end ContinuousSMul section ComplementsConnected variable {E : Type*} [AddCommGroup E] [Module ℝ E] [TopologicalSpace E] [TopologicalAddGroup E] local notation "π" => Submodule.linearProjOfIsCompl _ _ attribute [local instance 100] TopologicalAddGroup.pathConnectedSpace /-- Given two complementary subspaces `p` and `q` in `E`, if the complement of `{0}` is path connected in `p` then the complement of `q` is path connected in `E`. -/

theorem isPathConnected_compl_of_isPathConnected_compl_zero [ContinuousSMul ℝ E] {p q : Submodule ℝ E} (hpq : IsCompl p q) (hpc : IsPathConnected ({0}ᶜ : Set p)) : IsPathConnected (qᶜ : Set E)

isPathConnected_compl_of_isPathConnected_compl_zero

2020

6efcbba

mathlib4/Mathlib/Analysis/Convex/Topology.lean

mathlib

/- Copyright (c) 2023 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel -/ import Mathlib.Analysis.Convex.Topology import Mathlib.LinearAlgebra.Dimension.DivisionRing import Mathlib.Topology.Algebra.Module.Cardinality /-! # Connectedness of subsets of vector spaces We show several results related to the (path)-connectedness of subsets of real vector spaces: * `Set.Countable.isPathConnected_compl_of_one_lt_rank` asserts that the complement of a countable set is path-connected in a space of dimension `> 1`. * `isPathConnected_compl_singleton_of_one_lt_rank` is the special case of the complement of a singleton. * `isPathConnected_sphere` shows that any sphere is path-connected in dimension `> 1`. * `isPathConnected_compl_of_one_lt_codim` shows that the complement of a subspace of codimension `> 1` is path-connected. Statements with connectedness instead of path-connectedness are also given. -/ open Convex Set Metric section TopologicalVectorSpace variable {E : Type*} [AddCommGroup E] [Module ℝ E] [TopologicalSpace E] [ContinuousAdd E] [ContinuousSMul ℝ E] /-- In a real vector space of dimension `> 1`, the complement of any countable set is path connected. -/ theorem Set.Countable.isPathConnected_compl_of_one_lt_rank (h : 1 < Module.rank ℝ E) {s : Set E} (hs : s.Countable) : IsPathConnected sᶜ := by have : Nontrivial E := (rank_pos_iff_nontrivial (R := ℝ)).1 (zero_lt_one.trans h) -- the set `sᶜ` is dense, therefore nonempty. Pick `a ∈ sᶜ`. We have to show that any -- `b ∈ sᶜ` can be joined to `a`. obtain ⟨a, ha⟩ : sᶜ.Nonempty := (hs.dense_compl ℝ).nonempty refine ⟨a, ha, ?_⟩ intro b hb rcases eq_or_ne a b with rfl|hab · exact JoinedIn.refl ha /- Assume `b ≠ a`. Write `a = c - x` and `b = c + x` for some nonzero `x`. Choose `y` which is linearly independent from `x`. Then the segments joining `a = c - x` to `c + ty` are pairwise disjoint for varying `t` (except for the endpoint `a`) so only countably many of them can intersect `s`. In the same way, there are countably many `t`s for which the segment from `b = c + x` to `c + ty` intersects `s`. Choosing `t` outside of these countable exceptions, one gets a path in the complement of `s` from `a` to `z = c + ty` and then to `b`. -/ let c := (2 : ℝ)⁻¹ • (a + b) let x := (2 : ℝ)⁻¹ • (b - a) have Ia : c - x = a := by simp only [c, x, smul_add, smul_sub] abel_nf simp [zsmul_eq_smul_cast ℝ 2] have Ib : c + x = b := by simp only [c, x, smul_add, smul_sub] abel_nf simp [zsmul_eq_smul_cast ℝ 2] have x_ne_zero : x ≠ 0 := by simpa [x] using sub_ne_zero.2 hab.symm obtain ⟨y, hy⟩ : ∃ y, LinearIndependent ℝ ![x, y] := exists_linearIndependent_pair_of_one_lt_rank h x_ne_zero have A : Set.Countable {t : ℝ | ([c + x -[ℝ] c + t • y] ∩ s).Nonempty} := by apply countable_setOf_nonempty_of_disjoint _ (fun t ↦ inter_subset_right _ _) hs intro t t' htt' apply disjoint_iff_inter_eq_empty.2 have N : {c + x} ∩ s = ∅ := by simpa only [singleton_inter_eq_empty, mem_compl_iff, Ib] using hb rw [inter_assoc, inter_comm s, inter_assoc, inter_self, ← inter_assoc, ← subset_empty_iff, ← N] apply inter_subset_inter_left apply Eq.subset apply segment_inter_eq_endpoint_of_linearIndependent_of_ne hy htt'.symm have B : Set.Countable {t : ℝ | ([c - x -[ℝ] c + t • y] ∩ s).Nonempty} := by apply countable_setOf_nonempty_of_disjoint _ (fun t ↦ inter_subset_right _ _) hs intro t t' htt' apply disjoint_iff_inter_eq_empty.2 have N : {c - x} ∩ s = ∅ := by simpa only [singleton_inter_eq_empty, mem_compl_iff, Ia] using ha rw [inter_assoc, inter_comm s, inter_assoc, inter_self, ← inter_assoc, ← subset_empty_iff, ← N] apply inter_subset_inter_left rw [sub_eq_add_neg _ x] apply Eq.subset apply segment_inter_eq_endpoint_of_linearIndependent_of_ne _ htt'.symm convert hy.units_smul ![-1, 1] simp [← List.ofFn_inj] obtain ⟨t, ht⟩ : Set.Nonempty ({t : ℝ | ([c + x -[ℝ] c + t • y] ∩ s).Nonempty} ∪ {t : ℝ | ([c - x -[ℝ] c + t • y] ∩ s).Nonempty})ᶜ := ((A.union B).dense_compl ℝ).nonempty let z := c + t • y simp only [compl_union, mem_inter_iff, mem_compl_iff, mem_setOf_eq, not_nonempty_iff_eq_empty] at ht have JA : JoinedIn sᶜ a z := by apply JoinedIn.of_segment_subset rw [subset_compl_iff_disjoint_right, disjoint_iff_inter_eq_empty] convert ht.2 exact Ia.symm have JB : JoinedIn sᶜ b z := by apply JoinedIn.of_segment_subset rw [subset_compl_iff_disjoint_right, disjoint_iff_inter_eq_empty] convert ht.1 exact Ib.symm exact JA.trans JB.symm /-- In a real vector space of dimension `> 1`, the complement of any countable set is connected. -/ theorem Set.Countable.isConnected_compl_of_one_lt_rank (h : 1 < Module.rank ℝ E) {s : Set E} (hs : s.Countable) : IsConnected sᶜ := (hs.isPathConnected_compl_of_one_lt_rank h).isConnected /-- In a real vector space of dimension `> 1`, the complement of any singleton is path-connected. -/ theorem isPathConnected_compl_singleton_of_one_lt_rank (h : 1 < Module.rank ℝ E) (x : E) : IsPathConnected {x}ᶜ := Set.Countable.isPathConnected_compl_of_one_lt_rank h (countable_singleton x) /-- In a real vector space of dimension `> 1`, the complement of a singleton is connected. -/ theorem isConnected_compl_singleton_of_one_lt_rank (h : 1 < Module.rank ℝ E) (x : E) : IsConnected {x}ᶜ := (isPathConnected_compl_singleton_of_one_lt_rank h x).isConnected end TopologicalVectorSpace section NormedSpace variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] /-- In a real vector space of dimension `> 1`, any sphere of nonnegative radius is path connected. -/ theorem isPathConnected_sphere (h : 1 < Module.rank ℝ E) (x : E) {r : ℝ} (hr : 0 ≤ r) : IsPathConnected (sphere x r) := by /- when `r > 0`, we write the sphere as the image of `{0}ᶜ` under the map `y ↦ x + (r * ‖y‖⁻¹) • y`. Since the image under a continuous map of a path connected set is path connected, this concludes the proof. -/ rcases hr.eq_or_lt with rfl|rpos · simpa using isPathConnected_singleton x let f : E → E := fun y ↦ x + (r * ‖y‖⁻¹) • y have A : ContinuousOn f {0}ᶜ := by intro y hy apply (continuousAt_const.add _).continuousWithinAt apply (continuousAt_const.mul (ContinuousAt.inv₀ continuousAt_id.norm ?_)).smul continuousAt_id simpa using hy have B : IsPathConnected ({0}ᶜ : Set E) := isPathConnected_compl_singleton_of_one_lt_rank h 0 have C : IsPathConnected (f '' {0}ᶜ) := B.image' A have : f '' {0}ᶜ = sphere x r := by apply Subset.antisymm · rintro - ⟨y, hy, rfl⟩ have : ‖y‖ ≠ 0 := by simpa using hy simp [f, norm_smul, abs_of_nonneg hr, mul_assoc, inv_mul_cancel this] · intro y hy refine ⟨y - x, ?_, ?_⟩ · intro H simp only [mem_singleton_iff, sub_eq_zero] at H simp only [H, mem_sphere_iff_norm, sub_self, norm_zero] at hy exact rpos.ne hy · simp [f, mem_sphere_iff_norm.1 hy, mul_inv_cancel rpos.ne'] rwa [this] at C /-- In a real vector space of dimension `> 1`, any sphere of nonnegative radius is connected. -/ theorem isConnected_sphere (h : 1 < Module.rank ℝ E) (x : E) {r : ℝ} (hr : 0 ≤ r) : IsConnected (sphere x r) := (isPathConnected_sphere h x hr).isConnected /-- In a real vector space of dimension `> 1`, any sphere is preconnected. -/ theorem isPreconnected_sphere (h : 1 < Module.rank ℝ E) (x : E) (r : ℝ) : IsPreconnected (sphere x r) := by rcases le_or_lt 0 r with hr|hr · exact (isConnected_sphere h x hr).isPreconnected · simpa [hr] using isPreconnected_empty end NormedSpace section variable {F : Type*} [AddCommGroup F] [Module ℝ F] [TopologicalSpace F] [TopologicalAddGroup F] [ContinuousSMul ℝ F] /-- Let `E` be a linear subspace in a real vector space. If `E` has codimension at least two, its complement is path-connected. -/

theorem isPathConnected_compl_of_one_lt_codim {E : Submodule ℝ F} (hcodim : 1 < Module.rank ℝ (F ⧸ E)) : IsPathConnected (Eᶜ : Set F)

isPathConnected_compl_of_one_lt_codim

2023

6efcbba

mathlib4/Mathlib/Analysis/NormedSpace/Connected.lean

mathlib