case a
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝¹ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
a p : Associates α
hp : Irreducible p
inst✝ : (n : ℕ) → Decidable (a ∣ p ^ n)
n : ℕ
h : a ∣ p ^ n
⊢ Nat.find (_ : ∃ n, (fun n => a ∣ p ^ n) n) ≤ count p (factors a)

case a
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝¹ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
a p : Associates α
hp : Irreducible p
inst✝ : (n : ℕ) → Decidable (a ∣ p ^ n)
n : ℕ
h : a ∣ p ^ n
⊢ count p (factors a) ≤ Nat.find (_ : ∃ n, (fun n => a ∣ p ^ n) n)
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝¹ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
a p : Associates α
hp : Irreducible p
inst✝ : (n : ℕ) → Decidable (a ∣ p ^ n)
n : ℕ
h : a ∣ p ^ n
⊢ Nat.find (_ : ∃ n, (fun n => a ∣ p ^ n) n) = count p (factors a)
apply le_antisymm
case a
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝¹ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
a p : Associates α
hp : Irreducible p
inst✝ : (n : ℕ) → Decidable (a ∣ p ^ n)
n : ℕ
h : a ∣ p ^ n
⊢ p ^ count p (factors a) = a * 1
case a
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝¹ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
a p : Associates α
hp : Irreducible p
inst✝ : (n : ℕ) → Decidable (a ∣ p ^ n)
n : ℕ
h : a ∣ p ^ n
⊢ Nat.find (_ : ∃ n, (fun n => a ∣ p ^ n) n) ≤ count p (factors a)
refine' Nat.find_le ⟨1, _⟩
case a
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝¹ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
a p : Associates α
hp : Irreducible p
inst✝ : (n : ℕ) → Decidable (a ∣ p ^ n)
n : ℕ
h : a ∣ p ^ n
⊢ p ^ count p (factors a) = a
case a
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝¹ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
a p : Associates α
hp : Irreducible p
inst✝ : (n : ℕ) → Decidable (a ∣ p ^ n)
n : ℕ
h : a ∣ p ^ n
⊢ p ^ count p (factors a) = a * 1
rw [mul_one]
case a
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝¹ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
a p : Associates α
hp : Irreducible p
inst✝ : (n : ℕ) → Decidable (a ∣ p ^ n)
n : ℕ
h : a ∣ p ^ n
⊢ a = p ^ count p (factors a)
case a
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝¹ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
a p : Associates α
hp : Irreducible p
inst✝ : (n : ℕ) → Decidable (a ∣ p ^ n)
n : ℕ
h : a ∣ p ^ n
⊢ p ^ count p (factors a) = a
symm
no goals
case a
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝¹ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
a p : Associates α
hp : Irreducible p
inst✝ : (n : ℕ) → Decidable (a ∣ p ^ n)
n : ℕ
h : a ∣ p ^ n
⊢ a = p ^ count p (factors a)
exact eq_pow_count_factors_of_dvd_pow hp h
case a
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝¹ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
a p : Associates α
hp : Irreducible p
inst✝ : (n : ℕ) → Decidable (a ∣ p ^ n)
n : ℕ
h : a ∣ p ^ n
hph : p ^ Nat.find (_ : ∃ n, (fun n => a ∣ p ^ n) n) ≠ 0
⊢ count p (factors a) ≤ Nat.find (_ : ∃ n, (fun n => a ∣ p ^ n) n)
case a
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝¹ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
a p : Associates α
hp : Irreducible p
inst✝ : (n : ℕ) → Decidable (a ∣ p ^ n)
n : ℕ
h : a ∣ p ^ n
⊢ count p (factors a) ≤ Nat.find (_ : ∃ n, (fun n => a ∣ p ^ n) n)
have hph := pow_ne_zero (@Nat.find (fun n => a ∣ p ^ n) _ ⟨n, h⟩) hp.ne_zero
case a.inl
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝¹ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
a p : Associates α
hp : Irreducible p
inst✝ : (n : ℕ) → Decidable (a ∣ p ^ n)
n : ℕ
h : a ∣ p ^ n
hph : p ^ Nat.find (_ : ∃ n, (fun n => a ∣ p ^ n) n) ≠ 0
hα : Subsingleton α
⊢ count p (factors a) ≤ Nat.find (_ : ∃ n, (fun n => a ∣ p ^ n) n)

case a.inr
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝¹ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
a p : Associates α
hp : Irreducible p
inst✝ : (n : ℕ) → Decidable (a ∣ p ^ n)
n : ℕ
h : a ∣ p ^ n
hph : p ^ Nat.find (_ : ∃ n, (fun n => a ∣ p ^ n) n) ≠ 0
hα : Nontrivial α
⊢ count p (factors a) ≤ Nat.find (_ : ∃ n, (fun n => a ∣ p ^ n) n)
case a
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝¹ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
a p : Associates α
hp : Irreducible p
inst✝ : (n : ℕ) → Decidable (a ∣ p ^ n)
n : ℕ
h : a ∣ p ^ n
hph : p ^ Nat.find (_ : ∃ n, (fun n => a ∣ p ^ n) n) ≠ 0
⊢ count p (factors a) ≤ Nat.find (_ : ∃ n, (fun n => a ∣ p ^ n) n)
cases' subsingleton_or_nontrivial α with hα hα
case h.e'_4
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝¹ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
a p : Associates α
hp : Irreducible p
inst✝ : (n : ℕ) → Decidable (a ∣ p ^ n)
n : ℕ
h : a ∣ p ^ n
hph : p ^ Nat.find (_ : ∃ n, (fun n => a ∣ p ^ n) n) ≠ 0
hα : Nontrivial α
⊢ Nat.find (_ : ∃ n, (fun n => a ∣ p ^ n) n) = count p (factors (p ^ Nat.find (_ : ∃ n, (fun n => a ∣ p ^ n) n)))
case a.inr
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝¹ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
a p : Associates α
hp : Irreducible p
inst✝ : (n : ℕ) → Decidable (a ∣ p ^ n)
n : ℕ
h : a ∣ p ^ n
hph : p ^ Nat.find (_ : ∃ n, (fun n => a ∣ p ^ n) n) ≠ 0
hα : Nontrivial α
⊢ count p (factors a) ≤ Nat.find (_ : ∃ n, (fun n => a ∣ p ^ n) n)
convert count_le_count_of_le hph hp (@Nat.find_spec (fun n => a ∣ p ^ n) _ ⟨n, h⟩)
no goals
case h.e'_4
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝¹ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
a p : Associates α
hp : Irreducible p
inst✝ : (n : ℕ) → Decidable (a ∣ p ^ n)
n : ℕ
h : a ∣ p ^ n
hph : p ^ Nat.find (_ : ∃ n, (fun n => a ∣ p ^ n) n) ≠ 0
hα : Nontrivial α
⊢ Nat.find (_ : ∃ n, (fun n => a ∣ p ^ n) n) = count p (factors (p ^ Nat.find (_ : ∃ n, (fun n => a ∣ p ^ n) n)))
rw [count_pow hp.ne_zero hp, count_self hp, mul_one]
no goals
case a.inl
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝¹ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
a p : Associates α
hp : Irreducible p
inst✝ : (n : ℕ) → Decidable (a ∣ p ^ n)
n : ℕ
h : a ∣ p ^ n
hph : p ^ Nat.find (_ : ∃ n, (fun n => a ∣ p ^ n) n) ≠ 0
hα : Subsingleton α
⊢ count p (factors a) ≤ Nat.find (_ : ∃ n, (fun n => a ∣ p ^ n) n)
simp at hph