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Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "def P1 (x₁ x₂ x₃ y₁ y₂ y₃ : PGame) :=\n ⟦x₁ * y₁⟧ + ⟦x₂ * y₂⟧ - ⟦x₁ * y₂⟧ < ⟦x₃ * y₁⟧ + ⟦x₂ * y₃⟧ - (⟦x₃ * y₃⟧ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (x₁ x₂ y : PGame) := x₁ ≈ x₂ → ⟦x₁ * y⟧ = (⟦x₂ * y⟧ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (x₁ x₂ y₁ y₂ : PGame) := ⟦x₁ * y₂⟧ + ⟦x₂ * y₁⟧ < ⟦x₁ * y₁⟧ + (⟦x₂ * y₂⟧ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (x₁ x₂ y : PGame) :=\n x₁ < x₂ → (∀ i, P3 x₁ x₂ (y.moveLeft i) y) ∧ ∀ j, P3 x₁ x₂ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (x₁ x₂ y : PGame) : Prop := P2 x₁ x₂ y ∧ P4 x₁ x₂ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 x₁ x₂ y₁ y₂ ↔ P3 y₁ y₂ x₁ x₂ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (h₁ : P3 x₁ x₂ y₁ y₂) (h₂ : P3 x₂ x₃ y₁ y₂) : P3 x₁ x₃ y₁ y₂ := by\n rw [P3] at h₁ h₂\n rw [P3, ← add_lt_add_iff_left (⟦x₂ * y₁⟧ + ⟦x₂ * y₂⟧)]\n convert add_lt_add h₁ h₂ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 x₁ x₂ y₁ y₂ ↔ P3 (-x₂) (-x₁) y₁ y₂ := by\n simp_rw [P3, quot_neg_mul]\n rw [← _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 x₁ x₂ y ↔ P2 (-x₂) (-x₁) y := by\n rw [P2, P2]\n constructor\n · rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (· ·)\n · rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (· ·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 x₁ x₂ y ↔ P2 x₁ x₂ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 x₁ x₂ y ↔ P4 (-x₂) (-x₁) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', ← P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 x₁ x₂ y ↔ P4 x₁ x₂ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 x₁ x₂ y ↔ P24 (-x₂) (-x₁) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 x₁ x₂ y ↔ P24 x₁ x₂ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (⟦mulOption x y i k⟧ : Game) < -⟦mulOption x (-y) j l⟧ ↔\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n ⟦mulOption x y i j⟧ < (⟦x * y⟧ : Game) ↔ P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : x₁ ≈ x₃) (h₁ : P2 x₁ x₃ y₁) (h₃ : P2 x₁ x₃ y₃) (h3 : P3 x₁ x₂ y₂ y₃) :\n P1 x₁ x₂ x₃ y₁ y₂ y₃ := by\n rw [P1, ← h₁ he, ← h₃ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 ⟦x₁ * y₁⟧ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (h₁ : P3 x₃ x₂ y₂ y₃) (h₂ : P3 x₁ x₃ y₂ y₁) : P1 x₁ x₂ x₃ y₁ y₂ y₃ := by\n rw [P1, sub_lt_sub_iff, ← add_lt_add_iff_left ⟦x₃ * y₂⟧]\n convert add_lt_add h₁ h₂ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (x₁ x₂ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args → Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 x₁ x₂ y) => {x₁, x₂, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := ∀ x ∈ a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric ↔ x.Numeric ∧ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {x₁ x₂ y} :\n (Args.P24 x₁ x₂ y).Numeric ↔ x₁.Numeric ∧ x₂.Numeric ∧ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args → Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 x₁ x₂ y) => P24 x₁ x₂ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a → a.Numeric → a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption ⟨wf_isOption.isIrrefl.1⟩ _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n ∀ ⦃x₁ x₂ y'⦄, IsOption x₁ x → IsOption x₂ x → (y' = y ∨ IsOption y' y) → P24 x₁ x₂ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y → IH1 (-x) y :=\n fun h x₁ x₂ y' h₁ h₂ hy ↦ by\n rw [isOption_neg] at h₁ h₂\n exact P24_neg_left.2 (h h₂ h₁ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y → IH1 x (-y) :=\n fun h x₁ x₂ y' ↦ by\n rw [← neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro x₁ x₂ y' h₁ h₂ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left h₁ h₂)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih ⊢\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [← moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (⟦mulOption x y i k⟧ : Game) < -⟦mulOption x (-y) j l⟧ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (⟦mulOption x y i k⟧ : Game) < -⟦mulOption x (-y) j l⟧ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n · exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n · have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n · rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (x₁ x₂ y : PGame) : Prop :=\n ∀ ⦃z⦄, (IsOption z x₁ → P24 z x₂ y) ∧ (IsOption z x₂ → P24 x₁ z y) ∧ (IsOption z y → P24 x₁ x₂ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (x₁ x₂ y : PGame) : Prop :=\n ∀ ⦃z w⦄, IsOption w y → (IsOption z x₁ → P2 z x₂ w) ∧ (IsOption z x₂ → P2 x₁ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ih₁₂ : IH24 x₁ x₂ y := by\n rw [IH24]\n refine fun z ↦ ⟨?_, ?_, ?_⟩ <;>\n refine fun h ↦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n · exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n · exact (cutExpand_add_left {x₁}).2 (cutExpand_pair_left h)\n · exact (cutExpand_add_left {x₁}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ih₁₂",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ih₂₁ : IH24 x₂ x₁ y := ih₁₂ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' ⊢\n suffices {x₁, y, x₂} = {x₂, y, x₁} by rwa [← this]\n dsimp only [Multiset.insert_eq_cons, ← Multiset.singleton_add] at ih' ⊢\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ih₂₁",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 x₁ x₂ y := by\n refine fun z w h ↦ ⟨?_, ?_⟩\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {x₁}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (x₁ * y).Numeric ∧ (x₂ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n · exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {x₁}).2 cutExpand_zero\n · exact (cutExpand_add_right {x₂, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 x₁ x₂ y → IH24 (-x₂) (-x₁) y ∧ IH24 x₁ x₂ (-y) := by\n simp_rw [IH24, ← P24_neg_right, isOption_neg]\n refine fun h ↦ ⟨fun z ↦ ⟨?_, ?_, ?_⟩,\n fun z ↦ ⟨(@h z).1, (@h z).2.1, P24_neg_right.2 ∘ (@h <| -z).2.2⟩⟩\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 x₁ x₂ y → IH4 (-x₂) (-x₁) y ∧ IH4 x₁ x₂ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h ↦ ⟨fun z w h' ↦ ?_, fun z w h' ↦ ?_⟩\n · convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n · convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : x₁.Numeric) (h : IH24 x₁ x₂ y) (he : x₁ ≈ x₂) (i j) :\n ⟦mulOption x₁ y i j⟧ < (⟦x₂ * y⟧ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n · rw [← ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n · rw [← lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (h₁ : x₁.Numeric) (h₂ : x₂.Numeric)\n (h₁₂ : IH24 x₁ x₂ y) (h₂₁ : IH24 x₂ x₁ y) (he : x₁ ≈ x₂) : x₁ * y ≤ x₂ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := ∀ ⦃i j⦄, ⟦mulOption x y i j⟧ < (⟦x * y⟧ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y ∧ MulOptionsLTMul (-x) (-y)) ∧\n (MulOptionsLTMul x (-y) ∧ MulOptionsLTMul (-x) y) := by\n constructor\n · have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [← quot_neg_mul_neg]\n rfl\n · have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forall₂_imp fun a b ↦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (x₁ x' x₂ y₁ y₂ : PGame) : Prop :=\n P2 x₁ x' y₁ ∧ P2 x₁ x' y₂ ∧ P3 x' x₂ y₁ y₂ ∧ (x₁ < x' → P3 x₁ x' y₁ y₂)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 x₁ x₂ y) (h4 : IH4 x₁ x₂ y) (hl : MulOptionsLTMul x₂ y) (i j) :\n IH3 x₁ (x₂.moveLeft i) x₂ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n ⟨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l ↦ (h24.2 l).1 _⟩",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {y₁ y₂} (i) (h : IH3 x₁ (x₂.moveLeft i) x₂ y₁ y₂) (hl : x₁ ≤ x₂.moveLeft i) :\n P3 x₁ x₂ y₁ y₂ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n · exact (h.2.2.2 hl).trans h.2.2.1\n · rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {y₁ y₂} (h : ∀ i, IH3 x₁ (x₂.moveLeft i) x₂ y₁ y₂)\n (hs : ∀ i, IH3 (-x₂) ((-x₁).moveLeft i) (-x₁) y₁ y₂) (hl : x₁ < x₂) :\n P3 x₁ x₂ y₁ y₂",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric → P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 x₁ x₂ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : x₁ ≈ x₂) : x₁ * y ≈ x₂ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : y₁ ≈ y₂) : x * y₁ ≈ x * y₂",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : x₁ ≈ x₂) (hy : y₁ ≈ y₂) : x₁ * y₁ ≈ x₂ * y₂",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : x₁ < x₂) (hy : y₁ < y₂) : P3 x₁ x₂ y₁ y₂",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hp₁ : 0 < x₁) (hp₂ : 0 < x₂) : 0 < x₁ * x₂",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
] | 47 | SetTheory.PGame.P3_of_lt_of_lt | [
[
475,
74
],
[
489,
72
]
] | 8 | 13 | exact ⟨(P24 hx₁ hi hy₁).1, (P24 hx₁ hi hy₂).1,
P3_comm.2 <| ((P24 hy₁ hy₂ hx₂).2 hy).1 _,
ih _ (snd <| IsOption.moveLeft i) hx₁ hi⟩ | case mk.refine_1
x y y₁ y₂ : PGame
hx✝ : x.Numeric
hy✝ : y.Numeric
hy₁ : y₁.Numeric
hy₂ : y₂.Numeric
hy : y₁ < y₂
x₁ x₂ : PGame
ih :
∀ (y : PGame × PGame),
Prod.GameAdd IsOption IsOption y (x₁, x₂) → y.1.Numeric → y.2.Numeric → y.1 < y.2 → P3 y.1 y.2 y₁ y₂
hx₁ : (x₁, x₂).1.Numeric
hx₂ : (x₁, x₂).2.Numeric
hx : (x₁, x₂).1 < (x₁, x₂).2
i : (x₁, x₂).2.LeftMoves
hi : ((x₁, x₂).2.moveLeft i).Numeric
⊢ IH3 (x₁, x₂).1 ((x₁, x₂).2.moveLeft i) (x₁, x₂).2 y₁ y₂ | no goals |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "def P1 (x₁ x₂ x₃ y₁ y₂ y₃ : PGame) :=\n ⟦x₁ * y₁⟧ + ⟦x₂ * y₂⟧ - ⟦x₁ * y₂⟧ < ⟦x₃ * y₁⟧ + ⟦x₂ * y₃⟧ - (⟦x₃ * y₃⟧ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (x₁ x₂ y : PGame) := x₁ ≈ x₂ → ⟦x₁ * y⟧ = (⟦x₂ * y⟧ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (x₁ x₂ y₁ y₂ : PGame) := ⟦x₁ * y₂⟧ + ⟦x₂ * y₁⟧ < ⟦x₁ * y₁⟧ + (⟦x₂ * y₂⟧ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (x₁ x₂ y : PGame) :=\n x₁ < x₂ → (∀ i, P3 x₁ x₂ (y.moveLeft i) y) ∧ ∀ j, P3 x₁ x₂ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (x₁ x₂ y : PGame) : Prop := P2 x₁ x₂ y ∧ P4 x₁ x₂ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 x₁ x₂ y₁ y₂ ↔ P3 y₁ y₂ x₁ x₂ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (h₁ : P3 x₁ x₂ y₁ y₂) (h₂ : P3 x₂ x₃ y₁ y₂) : P3 x₁ x₃ y₁ y₂ := by\n rw [P3] at h₁ h₂\n rw [P3, ← add_lt_add_iff_left (⟦x₂ * y₁⟧ + ⟦x₂ * y₂⟧)]\n convert add_lt_add h₁ h₂ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 x₁ x₂ y₁ y₂ ↔ P3 (-x₂) (-x₁) y₁ y₂ := by\n simp_rw [P3, quot_neg_mul]\n rw [← _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 x₁ x₂ y ↔ P2 (-x₂) (-x₁) y := by\n rw [P2, P2]\n constructor\n · rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (· ·)\n · rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (· ·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 x₁ x₂ y ↔ P2 x₁ x₂ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 x₁ x₂ y ↔ P4 (-x₂) (-x₁) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', ← P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 x₁ x₂ y ↔ P4 x₁ x₂ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 x₁ x₂ y ↔ P24 (-x₂) (-x₁) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 x₁ x₂ y ↔ P24 x₁ x₂ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (⟦mulOption x y i k⟧ : Game) < -⟦mulOption x (-y) j l⟧ ↔\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n ⟦mulOption x y i j⟧ < (⟦x * y⟧ : Game) ↔ P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : x₁ ≈ x₃) (h₁ : P2 x₁ x₃ y₁) (h₃ : P2 x₁ x₃ y₃) (h3 : P3 x₁ x₂ y₂ y₃) :\n P1 x₁ x₂ x₃ y₁ y₂ y₃ := by\n rw [P1, ← h₁ he, ← h₃ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 ⟦x₁ * y₁⟧ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (h₁ : P3 x₃ x₂ y₂ y₃) (h₂ : P3 x₁ x₃ y₂ y₁) : P1 x₁ x₂ x₃ y₁ y₂ y₃ := by\n rw [P1, sub_lt_sub_iff, ← add_lt_add_iff_left ⟦x₃ * y₂⟧]\n convert add_lt_add h₁ h₂ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (x₁ x₂ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args → Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 x₁ x₂ y) => {x₁, x₂, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := ∀ x ∈ a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric ↔ x.Numeric ∧ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {x₁ x₂ y} :\n (Args.P24 x₁ x₂ y).Numeric ↔ x₁.Numeric ∧ x₂.Numeric ∧ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args → Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 x₁ x₂ y) => P24 x₁ x₂ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a → a.Numeric → a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption ⟨wf_isOption.isIrrefl.1⟩ _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n ∀ ⦃x₁ x₂ y'⦄, IsOption x₁ x → IsOption x₂ x → (y' = y ∨ IsOption y' y) → P24 x₁ x₂ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y → IH1 (-x) y :=\n fun h x₁ x₂ y' h₁ h₂ hy ↦ by\n rw [isOption_neg] at h₁ h₂\n exact P24_neg_left.2 (h h₂ h₁ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y → IH1 x (-y) :=\n fun h x₁ x₂ y' ↦ by\n rw [← neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro x₁ x₂ y' h₁ h₂ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left h₁ h₂)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih ⊢\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [← moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (⟦mulOption x y i k⟧ : Game) < -⟦mulOption x (-y) j l⟧ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (⟦mulOption x y i k⟧ : Game) < -⟦mulOption x (-y) j l⟧ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n · exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n · have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n · rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (x₁ x₂ y : PGame) : Prop :=\n ∀ ⦃z⦄, (IsOption z x₁ → P24 z x₂ y) ∧ (IsOption z x₂ → P24 x₁ z y) ∧ (IsOption z y → P24 x₁ x₂ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (x₁ x₂ y : PGame) : Prop :=\n ∀ ⦃z w⦄, IsOption w y → (IsOption z x₁ → P2 z x₂ w) ∧ (IsOption z x₂ → P2 x₁ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ih₁₂ : IH24 x₁ x₂ y := by\n rw [IH24]\n refine fun z ↦ ⟨?_, ?_, ?_⟩ <;>\n refine fun h ↦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n · exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n · exact (cutExpand_add_left {x₁}).2 (cutExpand_pair_left h)\n · exact (cutExpand_add_left {x₁}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ih₁₂",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ih₂₁ : IH24 x₂ x₁ y := ih₁₂ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' ⊢\n suffices {x₁, y, x₂} = {x₂, y, x₁} by rwa [← this]\n dsimp only [Multiset.insert_eq_cons, ← Multiset.singleton_add] at ih' ⊢\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ih₂₁",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 x₁ x₂ y := by\n refine fun z w h ↦ ⟨?_, ?_⟩\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {x₁}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (x₁ * y).Numeric ∧ (x₂ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n · exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {x₁}).2 cutExpand_zero\n · exact (cutExpand_add_right {x₂, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 x₁ x₂ y → IH24 (-x₂) (-x₁) y ∧ IH24 x₁ x₂ (-y) := by\n simp_rw [IH24, ← P24_neg_right, isOption_neg]\n refine fun h ↦ ⟨fun z ↦ ⟨?_, ?_, ?_⟩,\n fun z ↦ ⟨(@h z).1, (@h z).2.1, P24_neg_right.2 ∘ (@h <| -z).2.2⟩⟩\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 x₁ x₂ y → IH4 (-x₂) (-x₁) y ∧ IH4 x₁ x₂ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h ↦ ⟨fun z w h' ↦ ?_, fun z w h' ↦ ?_⟩\n · convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n · convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : x₁.Numeric) (h : IH24 x₁ x₂ y) (he : x₁ ≈ x₂) (i j) :\n ⟦mulOption x₁ y i j⟧ < (⟦x₂ * y⟧ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n · rw [← ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n · rw [← lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (h₁ : x₁.Numeric) (h₂ : x₂.Numeric)\n (h₁₂ : IH24 x₁ x₂ y) (h₂₁ : IH24 x₂ x₁ y) (he : x₁ ≈ x₂) : x₁ * y ≤ x₂ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := ∀ ⦃i j⦄, ⟦mulOption x y i j⟧ < (⟦x * y⟧ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y ∧ MulOptionsLTMul (-x) (-y)) ∧\n (MulOptionsLTMul x (-y) ∧ MulOptionsLTMul (-x) y) := by\n constructor\n · have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [← quot_neg_mul_neg]\n rfl\n · have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forall₂_imp fun a b ↦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (x₁ x' x₂ y₁ y₂ : PGame) : Prop :=\n P2 x₁ x' y₁ ∧ P2 x₁ x' y₂ ∧ P3 x' x₂ y₁ y₂ ∧ (x₁ < x' → P3 x₁ x' y₁ y₂)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 x₁ x₂ y) (h4 : IH4 x₁ x₂ y) (hl : MulOptionsLTMul x₂ y) (i j) :\n IH3 x₁ (x₂.moveLeft i) x₂ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n ⟨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l ↦ (h24.2 l).1 _⟩",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {y₁ y₂} (i) (h : IH3 x₁ (x₂.moveLeft i) x₂ y₁ y₂) (hl : x₁ ≤ x₂.moveLeft i) :\n P3 x₁ x₂ y₁ y₂ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n · exact (h.2.2.2 hl).trans h.2.2.1\n · rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {y₁ y₂} (h : ∀ i, IH3 x₁ (x₂.moveLeft i) x₂ y₁ y₂)\n (hs : ∀ i, IH3 (-x₂) ((-x₁).moveLeft i) (-x₁) y₁ y₂) (hl : x₁ < x₂) :\n P3 x₁ x₂ y₁ y₂",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric → P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 x₁ x₂ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : x₁ ≈ x₂) : x₁ * y ≈ x₂ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : y₁ ≈ y₂) : x * y₁ ≈ x * y₂",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : x₁ ≈ x₂) (hy : y₁ ≈ y₂) : x₁ * y₁ ≈ x₂ * y₂",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : x₁ < x₂) (hy : y₁ < y₂) : P3 x₁ x₂ y₁ y₂",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hp₁ : 0 < x₁) (hp₂ : 0 < x₂) : 0 < x₁ * x₂",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
] | 47 | SetTheory.PGame.P3_of_lt_of_lt | [
[
475,
74
],
[
489,
72
]
] | 9 | 13 | have hi := hx₁.neg.moveLeft i | case mk.refine_2
x y y₁ y₂ : PGame
hx✝ : x.Numeric
hy✝ : y.Numeric
hy₁ : y₁.Numeric
hy₂ : y₂.Numeric
hy : y₁ < y₂
x₁ x₂ : PGame
ih :
∀ (y : PGame × PGame),
Prod.GameAdd IsOption IsOption y (x₁, x₂) → y.1.Numeric → y.2.Numeric → y.1 < y.2 → P3 y.1 y.2 y₁ y₂
hx₁ : (x₁, x₂).1.Numeric
hx₂ : (x₁, x₂).2.Numeric
hx : (x₁, x₂).1 < (x₁, x₂).2
i : (-(x₁, x₂).1).LeftMoves
⊢ IH3 (-(x₁, x₂).2) ((-(x₁, x₂).1).moveLeft i) (-(x₁, x₂).1) y₁ y₂ | case mk.refine_2
x y y₁ y₂ : PGame
hx✝ : x.Numeric
hy✝ : y.Numeric
hy₁ : y₁.Numeric
hy₂ : y₂.Numeric
hy : y₁ < y₂
x₁ x₂ : PGame
ih :
∀ (y : PGame × PGame),
Prod.GameAdd IsOption IsOption y (x₁, x₂) → y.1.Numeric → y.2.Numeric → y.1 < y.2 → P3 y.1 y.2 y₁ y₂
hx₁ : (x₁, x₂).1.Numeric
hx₂ : (x₁, x₂).2.Numeric
hx : (x₁, x₂).1 < (x₁, x₂).2
i : (-(x₁, x₂).1).LeftMoves
hi : ((-(x₁, x₂).1).moveLeft i).Numeric
⊢ IH3 (-(x₁, x₂).2) ((-(x₁, x₂).1).moveLeft i) (-(x₁, x₂).1) y₁ y₂ |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "def P1 (x₁ x₂ x₃ y₁ y₂ y₃ : PGame) :=\n ⟦x₁ * y₁⟧ + ⟦x₂ * y₂⟧ - ⟦x₁ * y₂⟧ < ⟦x₃ * y₁⟧ + ⟦x₂ * y₃⟧ - (⟦x₃ * y₃⟧ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (x₁ x₂ y : PGame) := x₁ ≈ x₂ → ⟦x₁ * y⟧ = (⟦x₂ * y⟧ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (x₁ x₂ y₁ y₂ : PGame) := ⟦x₁ * y₂⟧ + ⟦x₂ * y₁⟧ < ⟦x₁ * y₁⟧ + (⟦x₂ * y₂⟧ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (x₁ x₂ y : PGame) :=\n x₁ < x₂ → (∀ i, P3 x₁ x₂ (y.moveLeft i) y) ∧ ∀ j, P3 x₁ x₂ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (x₁ x₂ y : PGame) : Prop := P2 x₁ x₂ y ∧ P4 x₁ x₂ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 x₁ x₂ y₁ y₂ ↔ P3 y₁ y₂ x₁ x₂ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (h₁ : P3 x₁ x₂ y₁ y₂) (h₂ : P3 x₂ x₃ y₁ y₂) : P3 x₁ x₃ y₁ y₂ := by\n rw [P3] at h₁ h₂\n rw [P3, ← add_lt_add_iff_left (⟦x₂ * y₁⟧ + ⟦x₂ * y₂⟧)]\n convert add_lt_add h₁ h₂ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 x₁ x₂ y₁ y₂ ↔ P3 (-x₂) (-x₁) y₁ y₂ := by\n simp_rw [P3, quot_neg_mul]\n rw [← _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 x₁ x₂ y ↔ P2 (-x₂) (-x₁) y := by\n rw [P2, P2]\n constructor\n · rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (· ·)\n · rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (· ·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 x₁ x₂ y ↔ P2 x₁ x₂ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 x₁ x₂ y ↔ P4 (-x₂) (-x₁) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', ← P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 x₁ x₂ y ↔ P4 x₁ x₂ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 x₁ x₂ y ↔ P24 (-x₂) (-x₁) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 x₁ x₂ y ↔ P24 x₁ x₂ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (⟦mulOption x y i k⟧ : Game) < -⟦mulOption x (-y) j l⟧ ↔\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n ⟦mulOption x y i j⟧ < (⟦x * y⟧ : Game) ↔ P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : x₁ ≈ x₃) (h₁ : P2 x₁ x₃ y₁) (h₃ : P2 x₁ x₃ y₃) (h3 : P3 x₁ x₂ y₂ y₃) :\n P1 x₁ x₂ x₃ y₁ y₂ y₃ := by\n rw [P1, ← h₁ he, ← h₃ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 ⟦x₁ * y₁⟧ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (h₁ : P3 x₃ x₂ y₂ y₃) (h₂ : P3 x₁ x₃ y₂ y₁) : P1 x₁ x₂ x₃ y₁ y₂ y₃ := by\n rw [P1, sub_lt_sub_iff, ← add_lt_add_iff_left ⟦x₃ * y₂⟧]\n convert add_lt_add h₁ h₂ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (x₁ x₂ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args → Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 x₁ x₂ y) => {x₁, x₂, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := ∀ x ∈ a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric ↔ x.Numeric ∧ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {x₁ x₂ y} :\n (Args.P24 x₁ x₂ y).Numeric ↔ x₁.Numeric ∧ x₂.Numeric ∧ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args → Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 x₁ x₂ y) => P24 x₁ x₂ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a → a.Numeric → a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption ⟨wf_isOption.isIrrefl.1⟩ _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n ∀ ⦃x₁ x₂ y'⦄, IsOption x₁ x → IsOption x₂ x → (y' = y ∨ IsOption y' y) → P24 x₁ x₂ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y → IH1 (-x) y :=\n fun h x₁ x₂ y' h₁ h₂ hy ↦ by\n rw [isOption_neg] at h₁ h₂\n exact P24_neg_left.2 (h h₂ h₁ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y → IH1 x (-y) :=\n fun h x₁ x₂ y' ↦ by\n rw [← neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro x₁ x₂ y' h₁ h₂ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left h₁ h₂)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih ⊢\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [← moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (⟦mulOption x y i k⟧ : Game) < -⟦mulOption x (-y) j l⟧ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (⟦mulOption x y i k⟧ : Game) < -⟦mulOption x (-y) j l⟧ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n · exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n · have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n · rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (x₁ x₂ y : PGame) : Prop :=\n ∀ ⦃z⦄, (IsOption z x₁ → P24 z x₂ y) ∧ (IsOption z x₂ → P24 x₁ z y) ∧ (IsOption z y → P24 x₁ x₂ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (x₁ x₂ y : PGame) : Prop :=\n ∀ ⦃z w⦄, IsOption w y → (IsOption z x₁ → P2 z x₂ w) ∧ (IsOption z x₂ → P2 x₁ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ih₁₂ : IH24 x₁ x₂ y := by\n rw [IH24]\n refine fun z ↦ ⟨?_, ?_, ?_⟩ <;>\n refine fun h ↦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n · exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n · exact (cutExpand_add_left {x₁}).2 (cutExpand_pair_left h)\n · exact (cutExpand_add_left {x₁}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ih₁₂",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ih₂₁ : IH24 x₂ x₁ y := ih₁₂ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' ⊢\n suffices {x₁, y, x₂} = {x₂, y, x₁} by rwa [← this]\n dsimp only [Multiset.insert_eq_cons, ← Multiset.singleton_add] at ih' ⊢\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ih₂₁",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 x₁ x₂ y := by\n refine fun z w h ↦ ⟨?_, ?_⟩\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {x₁}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (x₁ * y).Numeric ∧ (x₂ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n · exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {x₁}).2 cutExpand_zero\n · exact (cutExpand_add_right {x₂, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 x₁ x₂ y → IH24 (-x₂) (-x₁) y ∧ IH24 x₁ x₂ (-y) := by\n simp_rw [IH24, ← P24_neg_right, isOption_neg]\n refine fun h ↦ ⟨fun z ↦ ⟨?_, ?_, ?_⟩,\n fun z ↦ ⟨(@h z).1, (@h z).2.1, P24_neg_right.2 ∘ (@h <| -z).2.2⟩⟩\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 x₁ x₂ y → IH4 (-x₂) (-x₁) y ∧ IH4 x₁ x₂ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h ↦ ⟨fun z w h' ↦ ?_, fun z w h' ↦ ?_⟩\n · convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n · convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : x₁.Numeric) (h : IH24 x₁ x₂ y) (he : x₁ ≈ x₂) (i j) :\n ⟦mulOption x₁ y i j⟧ < (⟦x₂ * y⟧ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n · rw [← ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n · rw [← lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (h₁ : x₁.Numeric) (h₂ : x₂.Numeric)\n (h₁₂ : IH24 x₁ x₂ y) (h₂₁ : IH24 x₂ x₁ y) (he : x₁ ≈ x₂) : x₁ * y ≤ x₂ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := ∀ ⦃i j⦄, ⟦mulOption x y i j⟧ < (⟦x * y⟧ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y ∧ MulOptionsLTMul (-x) (-y)) ∧\n (MulOptionsLTMul x (-y) ∧ MulOptionsLTMul (-x) y) := by\n constructor\n · have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [← quot_neg_mul_neg]\n rfl\n · have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forall₂_imp fun a b ↦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (x₁ x' x₂ y₁ y₂ : PGame) : Prop :=\n P2 x₁ x' y₁ ∧ P2 x₁ x' y₂ ∧ P3 x' x₂ y₁ y₂ ∧ (x₁ < x' → P3 x₁ x' y₁ y₂)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 x₁ x₂ y) (h4 : IH4 x₁ x₂ y) (hl : MulOptionsLTMul x₂ y) (i j) :\n IH3 x₁ (x₂.moveLeft i) x₂ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n ⟨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l ↦ (h24.2 l).1 _⟩",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {y₁ y₂} (i) (h : IH3 x₁ (x₂.moveLeft i) x₂ y₁ y₂) (hl : x₁ ≤ x₂.moveLeft i) :\n P3 x₁ x₂ y₁ y₂ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n · exact (h.2.2.2 hl).trans h.2.2.1\n · rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {y₁ y₂} (h : ∀ i, IH3 x₁ (x₂.moveLeft i) x₂ y₁ y₂)\n (hs : ∀ i, IH3 (-x₂) ((-x₁).moveLeft i) (-x₁) y₁ y₂) (hl : x₁ < x₂) :\n P3 x₁ x₂ y₁ y₂",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric → P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 x₁ x₂ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : x₁ ≈ x₂) : x₁ * y ≈ x₂ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : y₁ ≈ y₂) : x * y₁ ≈ x * y₂",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : x₁ ≈ x₂) (hy : y₁ ≈ y₂) : x₁ * y₁ ≈ x₂ * y₂",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : x₁ < x₂) (hy : y₁ < y₂) : P3 x₁ x₂ y₁ y₂",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hp₁ : 0 < x₁) (hp₂ : 0 < x₂) : 0 < x₁ * x₂",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
] | 47 | SetTheory.PGame.P3_of_lt_of_lt | [
[
475,
74
],
[
489,
72
]
] | 10 | 13 | exact ⟨(P24 hx₂.neg hi hy₁).1, (P24 hx₂.neg hi hy₂).1,
P3_comm.2 <| ((P24 hy₁ hy₂ hx₁).2 hy).2 _, by
rw [moveLeft_neg', ← P3_neg, neg_lt_neg_iff]
exact ih _ (fst <| IsOption.moveRight _) (hx₁.moveRight _) hx₂⟩ | case mk.refine_2
x y y₁ y₂ : PGame
hx✝ : x.Numeric
hy✝ : y.Numeric
hy₁ : y₁.Numeric
hy₂ : y₂.Numeric
hy : y₁ < y₂
x₁ x₂ : PGame
ih :
∀ (y : PGame × PGame),
Prod.GameAdd IsOption IsOption y (x₁, x₂) → y.1.Numeric → y.2.Numeric → y.1 < y.2 → P3 y.1 y.2 y₁ y₂
hx₁ : (x₁, x₂).1.Numeric
hx₂ : (x₁, x₂).2.Numeric
hx : (x₁, x₂).1 < (x₁, x₂).2
i : (-(x₁, x₂).1).LeftMoves
hi : ((-(x₁, x₂).1).moveLeft i).Numeric
⊢ IH3 (-(x₁, x₂).2) ((-(x₁, x₂).1).moveLeft i) (-(x₁, x₂).1) y₁ y₂ | no goals |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "def P1 (x₁ x₂ x₃ y₁ y₂ y₃ : PGame) :=\n ⟦x₁ * y₁⟧ + ⟦x₂ * y₂⟧ - ⟦x₁ * y₂⟧ < ⟦x₃ * y₁⟧ + ⟦x₂ * y₃⟧ - (⟦x₃ * y₃⟧ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (x₁ x₂ y : PGame) := x₁ ≈ x₂ → ⟦x₁ * y⟧ = (⟦x₂ * y⟧ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (x₁ x₂ y₁ y₂ : PGame) := ⟦x₁ * y₂⟧ + ⟦x₂ * y₁⟧ < ⟦x₁ * y₁⟧ + (⟦x₂ * y₂⟧ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (x₁ x₂ y : PGame) :=\n x₁ < x₂ → (∀ i, P3 x₁ x₂ (y.moveLeft i) y) ∧ ∀ j, P3 x₁ x₂ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (x₁ x₂ y : PGame) : Prop := P2 x₁ x₂ y ∧ P4 x₁ x₂ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 x₁ x₂ y₁ y₂ ↔ P3 y₁ y₂ x₁ x₂ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (h₁ : P3 x₁ x₂ y₁ y₂) (h₂ : P3 x₂ x₃ y₁ y₂) : P3 x₁ x₃ y₁ y₂ := by\n rw [P3] at h₁ h₂\n rw [P3, ← add_lt_add_iff_left (⟦x₂ * y₁⟧ + ⟦x₂ * y₂⟧)]\n convert add_lt_add h₁ h₂ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 x₁ x₂ y₁ y₂ ↔ P3 (-x₂) (-x₁) y₁ y₂ := by\n simp_rw [P3, quot_neg_mul]\n rw [← _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 x₁ x₂ y ↔ P2 (-x₂) (-x₁) y := by\n rw [P2, P2]\n constructor\n · rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (· ·)\n · rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (· ·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 x₁ x₂ y ↔ P2 x₁ x₂ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 x₁ x₂ y ↔ P4 (-x₂) (-x₁) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', ← P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 x₁ x₂ y ↔ P4 x₁ x₂ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 x₁ x₂ y ↔ P24 (-x₂) (-x₁) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 x₁ x₂ y ↔ P24 x₁ x₂ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (⟦mulOption x y i k⟧ : Game) < -⟦mulOption x (-y) j l⟧ ↔\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n ⟦mulOption x y i j⟧ < (⟦x * y⟧ : Game) ↔ P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : x₁ ≈ x₃) (h₁ : P2 x₁ x₃ y₁) (h₃ : P2 x₁ x₃ y₃) (h3 : P3 x₁ x₂ y₂ y₃) :\n P1 x₁ x₂ x₃ y₁ y₂ y₃ := by\n rw [P1, ← h₁ he, ← h₃ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 ⟦x₁ * y₁⟧ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (h₁ : P3 x₃ x₂ y₂ y₃) (h₂ : P3 x₁ x₃ y₂ y₁) : P1 x₁ x₂ x₃ y₁ y₂ y₃ := by\n rw [P1, sub_lt_sub_iff, ← add_lt_add_iff_left ⟦x₃ * y₂⟧]\n convert add_lt_add h₁ h₂ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (x₁ x₂ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args → Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 x₁ x₂ y) => {x₁, x₂, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := ∀ x ∈ a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric ↔ x.Numeric ∧ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {x₁ x₂ y} :\n (Args.P24 x₁ x₂ y).Numeric ↔ x₁.Numeric ∧ x₂.Numeric ∧ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args → Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 x₁ x₂ y) => P24 x₁ x₂ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a → a.Numeric → a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption ⟨wf_isOption.isIrrefl.1⟩ _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n ∀ ⦃x₁ x₂ y'⦄, IsOption x₁ x → IsOption x₂ x → (y' = y ∨ IsOption y' y) → P24 x₁ x₂ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y → IH1 (-x) y :=\n fun h x₁ x₂ y' h₁ h₂ hy ↦ by\n rw [isOption_neg] at h₁ h₂\n exact P24_neg_left.2 (h h₂ h₁ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y → IH1 x (-y) :=\n fun h x₁ x₂ y' ↦ by\n rw [← neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro x₁ x₂ y' h₁ h₂ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left h₁ h₂)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih ⊢\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [← moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (⟦mulOption x y i k⟧ : Game) < -⟦mulOption x (-y) j l⟧ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (⟦mulOption x y i k⟧ : Game) < -⟦mulOption x (-y) j l⟧ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n · exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n · have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n · rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (x₁ x₂ y : PGame) : Prop :=\n ∀ ⦃z⦄, (IsOption z x₁ → P24 z x₂ y) ∧ (IsOption z x₂ → P24 x₁ z y) ∧ (IsOption z y → P24 x₁ x₂ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (x₁ x₂ y : PGame) : Prop :=\n ∀ ⦃z w⦄, IsOption w y → (IsOption z x₁ → P2 z x₂ w) ∧ (IsOption z x₂ → P2 x₁ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ih₁₂ : IH24 x₁ x₂ y := by\n rw [IH24]\n refine fun z ↦ ⟨?_, ?_, ?_⟩ <;>\n refine fun h ↦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n · exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n · exact (cutExpand_add_left {x₁}).2 (cutExpand_pair_left h)\n · exact (cutExpand_add_left {x₁}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ih₁₂",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ih₂₁ : IH24 x₂ x₁ y := ih₁₂ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' ⊢\n suffices {x₁, y, x₂} = {x₂, y, x₁} by rwa [← this]\n dsimp only [Multiset.insert_eq_cons, ← Multiset.singleton_add] at ih' ⊢\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ih₂₁",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 x₁ x₂ y := by\n refine fun z w h ↦ ⟨?_, ?_⟩\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {x₁}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (x₁ * y).Numeric ∧ (x₂ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n · exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {x₁}).2 cutExpand_zero\n · exact (cutExpand_add_right {x₂, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 x₁ x₂ y → IH24 (-x₂) (-x₁) y ∧ IH24 x₁ x₂ (-y) := by\n simp_rw [IH24, ← P24_neg_right, isOption_neg]\n refine fun h ↦ ⟨fun z ↦ ⟨?_, ?_, ?_⟩,\n fun z ↦ ⟨(@h z).1, (@h z).2.1, P24_neg_right.2 ∘ (@h <| -z).2.2⟩⟩\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 x₁ x₂ y → IH4 (-x₂) (-x₁) y ∧ IH4 x₁ x₂ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h ↦ ⟨fun z w h' ↦ ?_, fun z w h' ↦ ?_⟩\n · convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n · convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : x₁.Numeric) (h : IH24 x₁ x₂ y) (he : x₁ ≈ x₂) (i j) :\n ⟦mulOption x₁ y i j⟧ < (⟦x₂ * y⟧ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n · rw [← ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n · rw [← lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (h₁ : x₁.Numeric) (h₂ : x₂.Numeric)\n (h₁₂ : IH24 x₁ x₂ y) (h₂₁ : IH24 x₂ x₁ y) (he : x₁ ≈ x₂) : x₁ * y ≤ x₂ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := ∀ ⦃i j⦄, ⟦mulOption x y i j⟧ < (⟦x * y⟧ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y ∧ MulOptionsLTMul (-x) (-y)) ∧\n (MulOptionsLTMul x (-y) ∧ MulOptionsLTMul (-x) y) := by\n constructor\n · have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [← quot_neg_mul_neg]\n rfl\n · have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forall₂_imp fun a b ↦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (x₁ x' x₂ y₁ y₂ : PGame) : Prop :=\n P2 x₁ x' y₁ ∧ P2 x₁ x' y₂ ∧ P3 x' x₂ y₁ y₂ ∧ (x₁ < x' → P3 x₁ x' y₁ y₂)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 x₁ x₂ y) (h4 : IH4 x₁ x₂ y) (hl : MulOptionsLTMul x₂ y) (i j) :\n IH3 x₁ (x₂.moveLeft i) x₂ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n ⟨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l ↦ (h24.2 l).1 _⟩",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {y₁ y₂} (i) (h : IH3 x₁ (x₂.moveLeft i) x₂ y₁ y₂) (hl : x₁ ≤ x₂.moveLeft i) :\n P3 x₁ x₂ y₁ y₂ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n · exact (h.2.2.2 hl).trans h.2.2.1\n · rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {y₁ y₂} (h : ∀ i, IH3 x₁ (x₂.moveLeft i) x₂ y₁ y₂)\n (hs : ∀ i, IH3 (-x₂) ((-x₁).moveLeft i) (-x₁) y₁ y₂) (hl : x₁ < x₂) :\n P3 x₁ x₂ y₁ y₂",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric → P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 x₁ x₂ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : x₁ ≈ x₂) : x₁ * y ≈ x₂ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : y₁ ≈ y₂) : x * y₁ ≈ x * y₂",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : x₁ ≈ x₂) (hy : y₁ ≈ y₂) : x₁ * y₁ ≈ x₂ * y₂",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : x₁ < x₂) (hy : y₁ < y₂) : P3 x₁ x₂ y₁ y₂",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hp₁ : 0 < x₁) (hp₂ : 0 < x₂) : 0 < x₁ * x₂",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
] | 47 | SetTheory.PGame.P3_of_lt_of_lt | [
[
475,
74
],
[
489,
72
]
] | 11 | 13 | rw [moveLeft_neg', ← P3_neg, neg_lt_neg_iff] | x y y₁ y₂ : PGame
hx✝ : x.Numeric
hy✝ : y.Numeric
hy₁ : y₁.Numeric
hy₂ : y₂.Numeric
hy : y₁ < y₂
x₁ x₂ : PGame
ih :
∀ (y : PGame × PGame),
Prod.GameAdd IsOption IsOption y (x₁, x₂) → y.1.Numeric → y.2.Numeric → y.1 < y.2 → P3 y.1 y.2 y₁ y₂
hx₁ : (x₁, x₂).1.Numeric
hx₂ : (x₁, x₂).2.Numeric
hx : (x₁, x₂).1 < (x₁, x₂).2
i : (-(x₁, x₂).1).LeftMoves
hi : ((-(x₁, x₂).1).moveLeft i).Numeric
⊢ -(x₁, x₂).2 < (-(x₁, x₂).1).moveLeft i → P3 (-(x₁, x₂).2) ((-(x₁, x₂).1).moveLeft i) y₁ y₂ | x y y₁ y₂ : PGame
hx✝ : x.Numeric
hy✝ : y.Numeric
hy₁ : y₁.Numeric
hy₂ : y₂.Numeric
hy : y₁ < y₂
x₁ x₂ : PGame
ih :
∀ (y : PGame × PGame),
Prod.GameAdd IsOption IsOption y (x₁, x₂) → y.1.Numeric → y.2.Numeric → y.1 < y.2 → P3 y.1 y.2 y₁ y₂
hx₁ : (x₁, x₂).1.Numeric
hx₂ : (x₁, x₂).2.Numeric
hx : (x₁, x₂).1 < (x₁, x₂).2
i : (-(x₁, x₂).1).LeftMoves
hi : ((-(x₁, x₂).1).moveLeft i).Numeric
⊢ (x₁, x₂).1.moveRight (toLeftMovesNeg.symm i) < (x₁, x₂).2 →
P3 ((x₁, x₂).1.moveRight (toLeftMovesNeg.symm i)) (x₁, x₂).2 y₁ y₂ |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "def P1 (x₁ x₂ x₃ y₁ y₂ y₃ : PGame) :=\n ⟦x₁ * y₁⟧ + ⟦x₂ * y₂⟧ - ⟦x₁ * y₂⟧ < ⟦x₃ * y₁⟧ + ⟦x₂ * y₃⟧ - (⟦x₃ * y₃⟧ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (x₁ x₂ y : PGame) := x₁ ≈ x₂ → ⟦x₁ * y⟧ = (⟦x₂ * y⟧ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (x₁ x₂ y₁ y₂ : PGame) := ⟦x₁ * y₂⟧ + ⟦x₂ * y₁⟧ < ⟦x₁ * y₁⟧ + (⟦x₂ * y₂⟧ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (x₁ x₂ y : PGame) :=\n x₁ < x₂ → (∀ i, P3 x₁ x₂ (y.moveLeft i) y) ∧ ∀ j, P3 x₁ x₂ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (x₁ x₂ y : PGame) : Prop := P2 x₁ x₂ y ∧ P4 x₁ x₂ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 x₁ x₂ y₁ y₂ ↔ P3 y₁ y₂ x₁ x₂ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (h₁ : P3 x₁ x₂ y₁ y₂) (h₂ : P3 x₂ x₃ y₁ y₂) : P3 x₁ x₃ y₁ y₂ := by\n rw [P3] at h₁ h₂\n rw [P3, ← add_lt_add_iff_left (⟦x₂ * y₁⟧ + ⟦x₂ * y₂⟧)]\n convert add_lt_add h₁ h₂ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 x₁ x₂ y₁ y₂ ↔ P3 (-x₂) (-x₁) y₁ y₂ := by\n simp_rw [P3, quot_neg_mul]\n rw [← _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 x₁ x₂ y ↔ P2 (-x₂) (-x₁) y := by\n rw [P2, P2]\n constructor\n · rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (· ·)\n · rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (· ·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 x₁ x₂ y ↔ P2 x₁ x₂ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 x₁ x₂ y ↔ P4 (-x₂) (-x₁) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', ← P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 x₁ x₂ y ↔ P4 x₁ x₂ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 x₁ x₂ y ↔ P24 (-x₂) (-x₁) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 x₁ x₂ y ↔ P24 x₁ x₂ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (⟦mulOption x y i k⟧ : Game) < -⟦mulOption x (-y) j l⟧ ↔\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n ⟦mulOption x y i j⟧ < (⟦x * y⟧ : Game) ↔ P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : x₁ ≈ x₃) (h₁ : P2 x₁ x₃ y₁) (h₃ : P2 x₁ x₃ y₃) (h3 : P3 x₁ x₂ y₂ y₃) :\n P1 x₁ x₂ x₃ y₁ y₂ y₃ := by\n rw [P1, ← h₁ he, ← h₃ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 ⟦x₁ * y₁⟧ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (h₁ : P3 x₃ x₂ y₂ y₃) (h₂ : P3 x₁ x₃ y₂ y₁) : P1 x₁ x₂ x₃ y₁ y₂ y₃ := by\n rw [P1, sub_lt_sub_iff, ← add_lt_add_iff_left ⟦x₃ * y₂⟧]\n convert add_lt_add h₁ h₂ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (x₁ x₂ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args → Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 x₁ x₂ y) => {x₁, x₂, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := ∀ x ∈ a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric ↔ x.Numeric ∧ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {x₁ x₂ y} :\n (Args.P24 x₁ x₂ y).Numeric ↔ x₁.Numeric ∧ x₂.Numeric ∧ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args → Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 x₁ x₂ y) => P24 x₁ x₂ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a → a.Numeric → a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption ⟨wf_isOption.isIrrefl.1⟩ _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n ∀ ⦃x₁ x₂ y'⦄, IsOption x₁ x → IsOption x₂ x → (y' = y ∨ IsOption y' y) → P24 x₁ x₂ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y → IH1 (-x) y :=\n fun h x₁ x₂ y' h₁ h₂ hy ↦ by\n rw [isOption_neg] at h₁ h₂\n exact P24_neg_left.2 (h h₂ h₁ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y → IH1 x (-y) :=\n fun h x₁ x₂ y' ↦ by\n rw [← neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro x₁ x₂ y' h₁ h₂ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left h₁ h₂)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih ⊢\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [← moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (⟦mulOption x y i k⟧ : Game) < -⟦mulOption x (-y) j l⟧ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (⟦mulOption x y i k⟧ : Game) < -⟦mulOption x (-y) j l⟧ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n · exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n · have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n · rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (x₁ x₂ y : PGame) : Prop :=\n ∀ ⦃z⦄, (IsOption z x₁ → P24 z x₂ y) ∧ (IsOption z x₂ → P24 x₁ z y) ∧ (IsOption z y → P24 x₁ x₂ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (x₁ x₂ y : PGame) : Prop :=\n ∀ ⦃z w⦄, IsOption w y → (IsOption z x₁ → P2 z x₂ w) ∧ (IsOption z x₂ → P2 x₁ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ih₁₂ : IH24 x₁ x₂ y := by\n rw [IH24]\n refine fun z ↦ ⟨?_, ?_, ?_⟩ <;>\n refine fun h ↦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n · exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n · exact (cutExpand_add_left {x₁}).2 (cutExpand_pair_left h)\n · exact (cutExpand_add_left {x₁}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ih₁₂",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ih₂₁ : IH24 x₂ x₁ y := ih₁₂ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' ⊢\n suffices {x₁, y, x₂} = {x₂, y, x₁} by rwa [← this]\n dsimp only [Multiset.insert_eq_cons, ← Multiset.singleton_add] at ih' ⊢\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ih₂₁",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 x₁ x₂ y := by\n refine fun z w h ↦ ⟨?_, ?_⟩\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {x₁}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (x₁ * y).Numeric ∧ (x₂ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n · exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {x₁}).2 cutExpand_zero\n · exact (cutExpand_add_right {x₂, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 x₁ x₂ y → IH24 (-x₂) (-x₁) y ∧ IH24 x₁ x₂ (-y) := by\n simp_rw [IH24, ← P24_neg_right, isOption_neg]\n refine fun h ↦ ⟨fun z ↦ ⟨?_, ?_, ?_⟩,\n fun z ↦ ⟨(@h z).1, (@h z).2.1, P24_neg_right.2 ∘ (@h <| -z).2.2⟩⟩\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 x₁ x₂ y → IH4 (-x₂) (-x₁) y ∧ IH4 x₁ x₂ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h ↦ ⟨fun z w h' ↦ ?_, fun z w h' ↦ ?_⟩\n · convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n · convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : x₁.Numeric) (h : IH24 x₁ x₂ y) (he : x₁ ≈ x₂) (i j) :\n ⟦mulOption x₁ y i j⟧ < (⟦x₂ * y⟧ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n · rw [← ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n · rw [← lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (h₁ : x₁.Numeric) (h₂ : x₂.Numeric)\n (h₁₂ : IH24 x₁ x₂ y) (h₂₁ : IH24 x₂ x₁ y) (he : x₁ ≈ x₂) : x₁ * y ≤ x₂ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := ∀ ⦃i j⦄, ⟦mulOption x y i j⟧ < (⟦x * y⟧ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y ∧ MulOptionsLTMul (-x) (-y)) ∧\n (MulOptionsLTMul x (-y) ∧ MulOptionsLTMul (-x) y) := by\n constructor\n · have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [← quot_neg_mul_neg]\n rfl\n · have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forall₂_imp fun a b ↦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (x₁ x' x₂ y₁ y₂ : PGame) : Prop :=\n P2 x₁ x' y₁ ∧ P2 x₁ x' y₂ ∧ P3 x' x₂ y₁ y₂ ∧ (x₁ < x' → P3 x₁ x' y₁ y₂)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 x₁ x₂ y) (h4 : IH4 x₁ x₂ y) (hl : MulOptionsLTMul x₂ y) (i j) :\n IH3 x₁ (x₂.moveLeft i) x₂ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n ⟨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l ↦ (h24.2 l).1 _⟩",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {y₁ y₂} (i) (h : IH3 x₁ (x₂.moveLeft i) x₂ y₁ y₂) (hl : x₁ ≤ x₂.moveLeft i) :\n P3 x₁ x₂ y₁ y₂ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n · exact (h.2.2.2 hl).trans h.2.2.1\n · rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {y₁ y₂} (h : ∀ i, IH3 x₁ (x₂.moveLeft i) x₂ y₁ y₂)\n (hs : ∀ i, IH3 (-x₂) ((-x₁).moveLeft i) (-x₁) y₁ y₂) (hl : x₁ < x₂) :\n P3 x₁ x₂ y₁ y₂",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric → P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 x₁ x₂ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : x₁ ≈ x₂) : x₁ * y ≈ x₂ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : y₁ ≈ y₂) : x * y₁ ≈ x * y₂",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : x₁ ≈ x₂) (hy : y₁ ≈ y₂) : x₁ * y₁ ≈ x₂ * y₂",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : x₁ < x₂) (hy : y₁ < y₂) : P3 x₁ x₂ y₁ y₂",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hp₁ : 0 < x₁) (hp₂ : 0 < x₂) : 0 < x₁ * x₂",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
] | 47 | SetTheory.PGame.P3_of_lt_of_lt | [
[
475,
74
],
[
489,
72
]
] | 12 | 13 | exact ih _ (fst <| IsOption.moveRight _) (hx₁.moveRight _) hx₂ | x y y₁ y₂ : PGame
hx✝ : x.Numeric
hy✝ : y.Numeric
hy₁ : y₁.Numeric
hy₂ : y₂.Numeric
hy : y₁ < y₂
x₁ x₂ : PGame
ih :
∀ (y : PGame × PGame),
Prod.GameAdd IsOption IsOption y (x₁, x₂) → y.1.Numeric → y.2.Numeric → y.1 < y.2 → P3 y.1 y.2 y₁ y₂
hx₁ : (x₁, x₂).1.Numeric
hx₂ : (x₁, x₂).2.Numeric
hx : (x₁, x₂).1 < (x₁, x₂).2
i : (-(x₁, x₂).1).LeftMoves
hi : ((-(x₁, x₂).1).moveLeft i).Numeric
⊢ (x₁, x₂).1.moveRight (toLeftMovesNeg.symm i) < (x₁, x₂).2 →
P3 ((x₁, x₂).1.moveRight (toLeftMovesNeg.symm i)) (x₁, x₂).2 y₁ y₂ | no goals |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "def P1 (x₁ x₂ x₃ y₁ y₂ y₃ : PGame) :=\n ⟦x₁ * y₁⟧ + ⟦x₂ * y₂⟧ - ⟦x₁ * y₂⟧ < ⟦x₃ * y₁⟧ + ⟦x₂ * y₃⟧ - (⟦x₃ * y₃⟧ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (x₁ x₂ y : PGame) := x₁ ≈ x₂ → ⟦x₁ * y⟧ = (⟦x₂ * y⟧ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (x₁ x₂ y₁ y₂ : PGame) := ⟦x₁ * y₂⟧ + ⟦x₂ * y₁⟧ < ⟦x₁ * y₁⟧ + (⟦x₂ * y₂⟧ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (x₁ x₂ y : PGame) :=\n x₁ < x₂ → (∀ i, P3 x₁ x₂ (y.moveLeft i) y) ∧ ∀ j, P3 x₁ x₂ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (x₁ x₂ y : PGame) : Prop := P2 x₁ x₂ y ∧ P4 x₁ x₂ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 x₁ x₂ y₁ y₂ ↔ P3 y₁ y₂ x₁ x₂ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (h₁ : P3 x₁ x₂ y₁ y₂) (h₂ : P3 x₂ x₃ y₁ y₂) : P3 x₁ x₃ y₁ y₂ := by\n rw [P3] at h₁ h₂\n rw [P3, ← add_lt_add_iff_left (⟦x₂ * y₁⟧ + ⟦x₂ * y₂⟧)]\n convert add_lt_add h₁ h₂ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 x₁ x₂ y₁ y₂ ↔ P3 (-x₂) (-x₁) y₁ y₂ := by\n simp_rw [P3, quot_neg_mul]\n rw [← _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 x₁ x₂ y ↔ P2 (-x₂) (-x₁) y := by\n rw [P2, P2]\n constructor\n · rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (· ·)\n · rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (· ·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 x₁ x₂ y ↔ P2 x₁ x₂ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 x₁ x₂ y ↔ P4 (-x₂) (-x₁) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', ← P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 x₁ x₂ y ↔ P4 x₁ x₂ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 x₁ x₂ y ↔ P24 (-x₂) (-x₁) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 x₁ x₂ y ↔ P24 x₁ x₂ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (⟦mulOption x y i k⟧ : Game) < -⟦mulOption x (-y) j l⟧ ↔\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n ⟦mulOption x y i j⟧ < (⟦x * y⟧ : Game) ↔ P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : x₁ ≈ x₃) (h₁ : P2 x₁ x₃ y₁) (h₃ : P2 x₁ x₃ y₃) (h3 : P3 x₁ x₂ y₂ y₃) :\n P1 x₁ x₂ x₃ y₁ y₂ y₃ := by\n rw [P1, ← h₁ he, ← h₃ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 ⟦x₁ * y₁⟧ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (h₁ : P3 x₃ x₂ y₂ y₃) (h₂ : P3 x₁ x₃ y₂ y₁) : P1 x₁ x₂ x₃ y₁ y₂ y₃ := by\n rw [P1, sub_lt_sub_iff, ← add_lt_add_iff_left ⟦x₃ * y₂⟧]\n convert add_lt_add h₁ h₂ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (x₁ x₂ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args → Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 x₁ x₂ y) => {x₁, x₂, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := ∀ x ∈ a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric ↔ x.Numeric ∧ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {x₁ x₂ y} :\n (Args.P24 x₁ x₂ y).Numeric ↔ x₁.Numeric ∧ x₂.Numeric ∧ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args → Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 x₁ x₂ y) => P24 x₁ x₂ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a → a.Numeric → a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption ⟨wf_isOption.isIrrefl.1⟩ _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n ∀ ⦃x₁ x₂ y'⦄, IsOption x₁ x → IsOption x₂ x → (y' = y ∨ IsOption y' y) → P24 x₁ x₂ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y → IH1 (-x) y :=\n fun h x₁ x₂ y' h₁ h₂ hy ↦ by\n rw [isOption_neg] at h₁ h₂\n exact P24_neg_left.2 (h h₂ h₁ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y → IH1 x (-y) :=\n fun h x₁ x₂ y' ↦ by\n rw [← neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro x₁ x₂ y' h₁ h₂ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left h₁ h₂)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih ⊢\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [← moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (⟦mulOption x y i k⟧ : Game) < -⟦mulOption x (-y) j l⟧ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (⟦mulOption x y i k⟧ : Game) < -⟦mulOption x (-y) j l⟧ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n · exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n · have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n · rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (x₁ x₂ y : PGame) : Prop :=\n ∀ ⦃z⦄, (IsOption z x₁ → P24 z x₂ y) ∧ (IsOption z x₂ → P24 x₁ z y) ∧ (IsOption z y → P24 x₁ x₂ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (x₁ x₂ y : PGame) : Prop :=\n ∀ ⦃z w⦄, IsOption w y → (IsOption z x₁ → P2 z x₂ w) ∧ (IsOption z x₂ → P2 x₁ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ih₁₂ : IH24 x₁ x₂ y := by\n rw [IH24]\n refine fun z ↦ ⟨?_, ?_, ?_⟩ <;>\n refine fun h ↦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n · exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n · exact (cutExpand_add_left {x₁}).2 (cutExpand_pair_left h)\n · exact (cutExpand_add_left {x₁}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ih₁₂",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ih₂₁ : IH24 x₂ x₁ y := ih₁₂ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' ⊢\n suffices {x₁, y, x₂} = {x₂, y, x₁} by rwa [← this]\n dsimp only [Multiset.insert_eq_cons, ← Multiset.singleton_add] at ih' ⊢\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ih₂₁",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 x₁ x₂ y := by\n refine fun z w h ↦ ⟨?_, ?_⟩\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {x₁}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (x₁ * y).Numeric ∧ (x₂ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n · exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {x₁}).2 cutExpand_zero\n · exact (cutExpand_add_right {x₂, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 x₁ x₂ y → IH24 (-x₂) (-x₁) y ∧ IH24 x₁ x₂ (-y) := by\n simp_rw [IH24, ← P24_neg_right, isOption_neg]\n refine fun h ↦ ⟨fun z ↦ ⟨?_, ?_, ?_⟩,\n fun z ↦ ⟨(@h z).1, (@h z).2.1, P24_neg_right.2 ∘ (@h <| -z).2.2⟩⟩\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 x₁ x₂ y → IH4 (-x₂) (-x₁) y ∧ IH4 x₁ x₂ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h ↦ ⟨fun z w h' ↦ ?_, fun z w h' ↦ ?_⟩\n · convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n · convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : x₁.Numeric) (h : IH24 x₁ x₂ y) (he : x₁ ≈ x₂) (i j) :\n ⟦mulOption x₁ y i j⟧ < (⟦x₂ * y⟧ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n · rw [← ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n · rw [← lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (h₁ : x₁.Numeric) (h₂ : x₂.Numeric)\n (h₁₂ : IH24 x₁ x₂ y) (h₂₁ : IH24 x₂ x₁ y) (he : x₁ ≈ x₂) : x₁ * y ≤ x₂ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := ∀ ⦃i j⦄, ⟦mulOption x y i j⟧ < (⟦x * y⟧ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y ∧ MulOptionsLTMul (-x) (-y)) ∧\n (MulOptionsLTMul x (-y) ∧ MulOptionsLTMul (-x) y) := by\n constructor\n · have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [← quot_neg_mul_neg]\n rfl\n · have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forall₂_imp fun a b ↦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (x₁ x' x₂ y₁ y₂ : PGame) : Prop :=\n P2 x₁ x' y₁ ∧ P2 x₁ x' y₂ ∧ P3 x' x₂ y₁ y₂ ∧ (x₁ < x' → P3 x₁ x' y₁ y₂)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 x₁ x₂ y) (h4 : IH4 x₁ x₂ y) (hl : MulOptionsLTMul x₂ y) (i j) :\n IH3 x₁ (x₂.moveLeft i) x₂ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n ⟨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l ↦ (h24.2 l).1 _⟩",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {y₁ y₂} (i) (h : IH3 x₁ (x₂.moveLeft i) x₂ y₁ y₂) (hl : x₁ ≤ x₂.moveLeft i) :\n P3 x₁ x₂ y₁ y₂ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n · exact (h.2.2.2 hl).trans h.2.2.1\n · rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {y₁ y₂} (h : ∀ i, IH3 x₁ (x₂.moveLeft i) x₂ y₁ y₂)\n (hs : ∀ i, IH3 (-x₂) ((-x₁).moveLeft i) (-x₁) y₁ y₂) (hl : x₁ < x₂) :\n P3 x₁ x₂ y₁ y₂",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric → P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 x₁ x₂ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : x₁ ≈ x₂) : x₁ * y ≈ x₂ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : y₁ ≈ y₂) : x * y₁ ≈ x * y₂",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : x₁ ≈ x₂) (hy : y₁ ≈ y₂) : x₁ * y₁ ≈ x₂ * y₂",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : x₁ < x₂) (hy : y₁ < y₂) : P3 x₁ x₂ y₁ y₂",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hp₁ : 0 < x₁) (hp₂ : 0 < x₂) : 0 < x₁ * x₂",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
] | 48 | SetTheory.PGame.Numeric.mul_pos | [
[
491,
72
],
[
495,
13
]
] | 0 | 4 | rw [lt_iff_game_lt] | x x₁ x₂ y y₁ y₂ : PGame
hx : x.Numeric
hx₁ : x₁.Numeric
hx₂ : x₂.Numeric
hy : y.Numeric
hy₁ : y₁.Numeric
hy₂ : y₂.Numeric
hp₁ : 0 < x₁
hp₂ : 0 < x₂
⊢ 0 < x₁ * x₂ | x x₁ x₂ y y₁ y₂ : PGame
hx : x.Numeric
hx₁ : x₁.Numeric
hx₂ : x₂.Numeric
hy : y.Numeric
hy₁ : y₁.Numeric
hy₂ : y₂.Numeric
hp₁ : 0 < x₁
hp₂ : 0 < x₂
⊢ ⟦0⟧ < ⟦x₁ * x₂⟧ |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "def P1 (x₁ x₂ x₃ y₁ y₂ y₃ : PGame) :=\n ⟦x₁ * y₁⟧ + ⟦x₂ * y₂⟧ - ⟦x₁ * y₂⟧ < ⟦x₃ * y₁⟧ + ⟦x₂ * y₃⟧ - (⟦x₃ * y₃⟧ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (x₁ x₂ y : PGame) := x₁ ≈ x₂ → ⟦x₁ * y⟧ = (⟦x₂ * y⟧ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (x₁ x₂ y₁ y₂ : PGame) := ⟦x₁ * y₂⟧ + ⟦x₂ * y₁⟧ < ⟦x₁ * y₁⟧ + (⟦x₂ * y₂⟧ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (x₁ x₂ y : PGame) :=\n x₁ < x₂ → (∀ i, P3 x₁ x₂ (y.moveLeft i) y) ∧ ∀ j, P3 x₁ x₂ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (x₁ x₂ y : PGame) : Prop := P2 x₁ x₂ y ∧ P4 x₁ x₂ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 x₁ x₂ y₁ y₂ ↔ P3 y₁ y₂ x₁ x₂ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (h₁ : P3 x₁ x₂ y₁ y₂) (h₂ : P3 x₂ x₃ y₁ y₂) : P3 x₁ x₃ y₁ y₂ := by\n rw [P3] at h₁ h₂\n rw [P3, ← add_lt_add_iff_left (⟦x₂ * y₁⟧ + ⟦x₂ * y₂⟧)]\n convert add_lt_add h₁ h₂ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 x₁ x₂ y₁ y₂ ↔ P3 (-x₂) (-x₁) y₁ y₂ := by\n simp_rw [P3, quot_neg_mul]\n rw [← _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 x₁ x₂ y ↔ P2 (-x₂) (-x₁) y := by\n rw [P2, P2]\n constructor\n · rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (· ·)\n · rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (· ·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 x₁ x₂ y ↔ P2 x₁ x₂ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 x₁ x₂ y ↔ P4 (-x₂) (-x₁) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', ← P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 x₁ x₂ y ↔ P4 x₁ x₂ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 x₁ x₂ y ↔ P24 (-x₂) (-x₁) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 x₁ x₂ y ↔ P24 x₁ x₂ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (⟦mulOption x y i k⟧ : Game) < -⟦mulOption x (-y) j l⟧ ↔\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n ⟦mulOption x y i j⟧ < (⟦x * y⟧ : Game) ↔ P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : x₁ ≈ x₃) (h₁ : P2 x₁ x₃ y₁) (h₃ : P2 x₁ x₃ y₃) (h3 : P3 x₁ x₂ y₂ y₃) :\n P1 x₁ x₂ x₃ y₁ y₂ y₃ := by\n rw [P1, ← h₁ he, ← h₃ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 ⟦x₁ * y₁⟧ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (h₁ : P3 x₃ x₂ y₂ y₃) (h₂ : P3 x₁ x₃ y₂ y₁) : P1 x₁ x₂ x₃ y₁ y₂ y₃ := by\n rw [P1, sub_lt_sub_iff, ← add_lt_add_iff_left ⟦x₃ * y₂⟧]\n convert add_lt_add h₁ h₂ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (x₁ x₂ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args → Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 x₁ x₂ y) => {x₁, x₂, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := ∀ x ∈ a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric ↔ x.Numeric ∧ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {x₁ x₂ y} :\n (Args.P24 x₁ x₂ y).Numeric ↔ x₁.Numeric ∧ x₂.Numeric ∧ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args → Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 x₁ x₂ y) => P24 x₁ x₂ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a → a.Numeric → a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption ⟨wf_isOption.isIrrefl.1⟩ _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n ∀ ⦃x₁ x₂ y'⦄, IsOption x₁ x → IsOption x₂ x → (y' = y ∨ IsOption y' y) → P24 x₁ x₂ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y → IH1 (-x) y :=\n fun h x₁ x₂ y' h₁ h₂ hy ↦ by\n rw [isOption_neg] at h₁ h₂\n exact P24_neg_left.2 (h h₂ h₁ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y → IH1 x (-y) :=\n fun h x₁ x₂ y' ↦ by\n rw [← neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro x₁ x₂ y' h₁ h₂ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left h₁ h₂)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih ⊢\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [← moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (⟦mulOption x y i k⟧ : Game) < -⟦mulOption x (-y) j l⟧ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (⟦mulOption x y i k⟧ : Game) < -⟦mulOption x (-y) j l⟧ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n · exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n · have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n · rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (x₁ x₂ y : PGame) : Prop :=\n ∀ ⦃z⦄, (IsOption z x₁ → P24 z x₂ y) ∧ (IsOption z x₂ → P24 x₁ z y) ∧ (IsOption z y → P24 x₁ x₂ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (x₁ x₂ y : PGame) : Prop :=\n ∀ ⦃z w⦄, IsOption w y → (IsOption z x₁ → P2 z x₂ w) ∧ (IsOption z x₂ → P2 x₁ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ih₁₂ : IH24 x₁ x₂ y := by\n rw [IH24]\n refine fun z ↦ ⟨?_, ?_, ?_⟩ <;>\n refine fun h ↦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n · exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n · exact (cutExpand_add_left {x₁}).2 (cutExpand_pair_left h)\n · exact (cutExpand_add_left {x₁}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ih₁₂",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ih₂₁ : IH24 x₂ x₁ y := ih₁₂ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' ⊢\n suffices {x₁, y, x₂} = {x₂, y, x₁} by rwa [← this]\n dsimp only [Multiset.insert_eq_cons, ← Multiset.singleton_add] at ih' ⊢\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ih₂₁",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 x₁ x₂ y := by\n refine fun z w h ↦ ⟨?_, ?_⟩\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {x₁}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (x₁ * y).Numeric ∧ (x₂ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n · exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {x₁}).2 cutExpand_zero\n · exact (cutExpand_add_right {x₂, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 x₁ x₂ y → IH24 (-x₂) (-x₁) y ∧ IH24 x₁ x₂ (-y) := by\n simp_rw [IH24, ← P24_neg_right, isOption_neg]\n refine fun h ↦ ⟨fun z ↦ ⟨?_, ?_, ?_⟩,\n fun z ↦ ⟨(@h z).1, (@h z).2.1, P24_neg_right.2 ∘ (@h <| -z).2.2⟩⟩\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 x₁ x₂ y → IH4 (-x₂) (-x₁) y ∧ IH4 x₁ x₂ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h ↦ ⟨fun z w h' ↦ ?_, fun z w h' ↦ ?_⟩\n · convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n · convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : x₁.Numeric) (h : IH24 x₁ x₂ y) (he : x₁ ≈ x₂) (i j) :\n ⟦mulOption x₁ y i j⟧ < (⟦x₂ * y⟧ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n · rw [← ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n · rw [← lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (h₁ : x₁.Numeric) (h₂ : x₂.Numeric)\n (h₁₂ : IH24 x₁ x₂ y) (h₂₁ : IH24 x₂ x₁ y) (he : x₁ ≈ x₂) : x₁ * y ≤ x₂ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := ∀ ⦃i j⦄, ⟦mulOption x y i j⟧ < (⟦x * y⟧ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y ∧ MulOptionsLTMul (-x) (-y)) ∧\n (MulOptionsLTMul x (-y) ∧ MulOptionsLTMul (-x) y) := by\n constructor\n · have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [← quot_neg_mul_neg]\n rfl\n · have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forall₂_imp fun a b ↦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (x₁ x' x₂ y₁ y₂ : PGame) : Prop :=\n P2 x₁ x' y₁ ∧ P2 x₁ x' y₂ ∧ P3 x' x₂ y₁ y₂ ∧ (x₁ < x' → P3 x₁ x' y₁ y₂)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 x₁ x₂ y) (h4 : IH4 x₁ x₂ y) (hl : MulOptionsLTMul x₂ y) (i j) :\n IH3 x₁ (x₂.moveLeft i) x₂ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n ⟨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l ↦ (h24.2 l).1 _⟩",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {y₁ y₂} (i) (h : IH3 x₁ (x₂.moveLeft i) x₂ y₁ y₂) (hl : x₁ ≤ x₂.moveLeft i) :\n P3 x₁ x₂ y₁ y₂ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n · exact (h.2.2.2 hl).trans h.2.2.1\n · rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {y₁ y₂} (h : ∀ i, IH3 x₁ (x₂.moveLeft i) x₂ y₁ y₂)\n (hs : ∀ i, IH3 (-x₂) ((-x₁).moveLeft i) (-x₁) y₁ y₂) (hl : x₁ < x₂) :\n P3 x₁ x₂ y₁ y₂",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric → P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 x₁ x₂ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : x₁ ≈ x₂) : x₁ * y ≈ x₂ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : y₁ ≈ y₂) : x * y₁ ≈ x * y₂",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : x₁ ≈ x₂) (hy : y₁ ≈ y₂) : x₁ * y₁ ≈ x₂ * y₂",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : x₁ < x₂) (hy : y₁ < y₂) : P3 x₁ x₂ y₁ y₂",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hp₁ : 0 < x₁) (hp₂ : 0 < x₂) : 0 < x₁ * x₂",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
] | 48 | SetTheory.PGame.Numeric.mul_pos | [
[
491,
72
],
[
495,
13
]
] | 1 | 4 | have := P3_of_lt_of_lt numeric_zero hx₁ numeric_zero hx₂ hp₁ hp₂ | x x₁ x₂ y y₁ y₂ : PGame
hx : x.Numeric
hx₁ : x₁.Numeric
hx₂ : x₂.Numeric
hy : y.Numeric
hy₁ : y₁.Numeric
hy₂ : y₂.Numeric
hp₁ : 0 < x₁
hp₂ : 0 < x₂
⊢ ⟦0⟧ < ⟦x₁ * x₂⟧ | x x₁ x₂ y y₁ y₂ : PGame
hx : x.Numeric
hx₁ : x₁.Numeric
hx₂ : x₂.Numeric
hy : y.Numeric
hy₁ : y₁.Numeric
hy₂ : y₂.Numeric
hp₁ : 0 < x₁
hp₂ : 0 < x₂
this : P3 0 x₁ 0 x₂
⊢ ⟦0⟧ < ⟦x₁ * x₂⟧ |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "def P1 (x₁ x₂ x₃ y₁ y₂ y₃ : PGame) :=\n ⟦x₁ * y₁⟧ + ⟦x₂ * y₂⟧ - ⟦x₁ * y₂⟧ < ⟦x₃ * y₁⟧ + ⟦x₂ * y₃⟧ - (⟦x₃ * y₃⟧ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (x₁ x₂ y : PGame) := x₁ ≈ x₂ → ⟦x₁ * y⟧ = (⟦x₂ * y⟧ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (x₁ x₂ y₁ y₂ : PGame) := ⟦x₁ * y₂⟧ + ⟦x₂ * y₁⟧ < ⟦x₁ * y₁⟧ + (⟦x₂ * y₂⟧ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (x₁ x₂ y : PGame) :=\n x₁ < x₂ → (∀ i, P3 x₁ x₂ (y.moveLeft i) y) ∧ ∀ j, P3 x₁ x₂ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (x₁ x₂ y : PGame) : Prop := P2 x₁ x₂ y ∧ P4 x₁ x₂ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 x₁ x₂ y₁ y₂ ↔ P3 y₁ y₂ x₁ x₂ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (h₁ : P3 x₁ x₂ y₁ y₂) (h₂ : P3 x₂ x₃ y₁ y₂) : P3 x₁ x₃ y₁ y₂ := by\n rw [P3] at h₁ h₂\n rw [P3, ← add_lt_add_iff_left (⟦x₂ * y₁⟧ + ⟦x₂ * y₂⟧)]\n convert add_lt_add h₁ h₂ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 x₁ x₂ y₁ y₂ ↔ P3 (-x₂) (-x₁) y₁ y₂ := by\n simp_rw [P3, quot_neg_mul]\n rw [← _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 x₁ x₂ y ↔ P2 (-x₂) (-x₁) y := by\n rw [P2, P2]\n constructor\n · rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (· ·)\n · rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (· ·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 x₁ x₂ y ↔ P2 x₁ x₂ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 x₁ x₂ y ↔ P4 (-x₂) (-x₁) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', ← P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 x₁ x₂ y ↔ P4 x₁ x₂ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 x₁ x₂ y ↔ P24 (-x₂) (-x₁) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 x₁ x₂ y ↔ P24 x₁ x₂ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (⟦mulOption x y i k⟧ : Game) < -⟦mulOption x (-y) j l⟧ ↔\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n ⟦mulOption x y i j⟧ < (⟦x * y⟧ : Game) ↔ P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : x₁ ≈ x₃) (h₁ : P2 x₁ x₃ y₁) (h₃ : P2 x₁ x₃ y₃) (h3 : P3 x₁ x₂ y₂ y₃) :\n P1 x₁ x₂ x₃ y₁ y₂ y₃ := by\n rw [P1, ← h₁ he, ← h₃ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 ⟦x₁ * y₁⟧ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (h₁ : P3 x₃ x₂ y₂ y₃) (h₂ : P3 x₁ x₃ y₂ y₁) : P1 x₁ x₂ x₃ y₁ y₂ y₃ := by\n rw [P1, sub_lt_sub_iff, ← add_lt_add_iff_left ⟦x₃ * y₂⟧]\n convert add_lt_add h₁ h₂ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (x₁ x₂ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args → Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 x₁ x₂ y) => {x₁, x₂, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := ∀ x ∈ a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric ↔ x.Numeric ∧ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {x₁ x₂ y} :\n (Args.P24 x₁ x₂ y).Numeric ↔ x₁.Numeric ∧ x₂.Numeric ∧ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args → Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 x₁ x₂ y) => P24 x₁ x₂ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a → a.Numeric → a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption ⟨wf_isOption.isIrrefl.1⟩ _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n ∀ ⦃x₁ x₂ y'⦄, IsOption x₁ x → IsOption x₂ x → (y' = y ∨ IsOption y' y) → P24 x₁ x₂ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y → IH1 (-x) y :=\n fun h x₁ x₂ y' h₁ h₂ hy ↦ by\n rw [isOption_neg] at h₁ h₂\n exact P24_neg_left.2 (h h₂ h₁ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y → IH1 x (-y) :=\n fun h x₁ x₂ y' ↦ by\n rw [← neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro x₁ x₂ y' h₁ h₂ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left h₁ h₂)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih ⊢\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [← moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (⟦mulOption x y i k⟧ : Game) < -⟦mulOption x (-y) j l⟧ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (⟦mulOption x y i k⟧ : Game) < -⟦mulOption x (-y) j l⟧ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n · exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n · have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n · rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (x₁ x₂ y : PGame) : Prop :=\n ∀ ⦃z⦄, (IsOption z x₁ → P24 z x₂ y) ∧ (IsOption z x₂ → P24 x₁ z y) ∧ (IsOption z y → P24 x₁ x₂ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (x₁ x₂ y : PGame) : Prop :=\n ∀ ⦃z w⦄, IsOption w y → (IsOption z x₁ → P2 z x₂ w) ∧ (IsOption z x₂ → P2 x₁ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ih₁₂ : IH24 x₁ x₂ y := by\n rw [IH24]\n refine fun z ↦ ⟨?_, ?_, ?_⟩ <;>\n refine fun h ↦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n · exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n · exact (cutExpand_add_left {x₁}).2 (cutExpand_pair_left h)\n · exact (cutExpand_add_left {x₁}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ih₁₂",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ih₂₁ : IH24 x₂ x₁ y := ih₁₂ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' ⊢\n suffices {x₁, y, x₂} = {x₂, y, x₁} by rwa [← this]\n dsimp only [Multiset.insert_eq_cons, ← Multiset.singleton_add] at ih' ⊢\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ih₂₁",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 x₁ x₂ y := by\n refine fun z w h ↦ ⟨?_, ?_⟩\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {x₁}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (x₁ * y).Numeric ∧ (x₂ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n · exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {x₁}).2 cutExpand_zero\n · exact (cutExpand_add_right {x₂, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 x₁ x₂ y → IH24 (-x₂) (-x₁) y ∧ IH24 x₁ x₂ (-y) := by\n simp_rw [IH24, ← P24_neg_right, isOption_neg]\n refine fun h ↦ ⟨fun z ↦ ⟨?_, ?_, ?_⟩,\n fun z ↦ ⟨(@h z).1, (@h z).2.1, P24_neg_right.2 ∘ (@h <| -z).2.2⟩⟩\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 x₁ x₂ y → IH4 (-x₂) (-x₁) y ∧ IH4 x₁ x₂ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h ↦ ⟨fun z w h' ↦ ?_, fun z w h' ↦ ?_⟩\n · convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n · convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : x₁.Numeric) (h : IH24 x₁ x₂ y) (he : x₁ ≈ x₂) (i j) :\n ⟦mulOption x₁ y i j⟧ < (⟦x₂ * y⟧ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n · rw [← ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n · rw [← lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (h₁ : x₁.Numeric) (h₂ : x₂.Numeric)\n (h₁₂ : IH24 x₁ x₂ y) (h₂₁ : IH24 x₂ x₁ y) (he : x₁ ≈ x₂) : x₁ * y ≤ x₂ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := ∀ ⦃i j⦄, ⟦mulOption x y i j⟧ < (⟦x * y⟧ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y ∧ MulOptionsLTMul (-x) (-y)) ∧\n (MulOptionsLTMul x (-y) ∧ MulOptionsLTMul (-x) y) := by\n constructor\n · have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [← quot_neg_mul_neg]\n rfl\n · have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forall₂_imp fun a b ↦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (x₁ x' x₂ y₁ y₂ : PGame) : Prop :=\n P2 x₁ x' y₁ ∧ P2 x₁ x' y₂ ∧ P3 x' x₂ y₁ y₂ ∧ (x₁ < x' → P3 x₁ x' y₁ y₂)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 x₁ x₂ y) (h4 : IH4 x₁ x₂ y) (hl : MulOptionsLTMul x₂ y) (i j) :\n IH3 x₁ (x₂.moveLeft i) x₂ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n ⟨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l ↦ (h24.2 l).1 _⟩",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {y₁ y₂} (i) (h : IH3 x₁ (x₂.moveLeft i) x₂ y₁ y₂) (hl : x₁ ≤ x₂.moveLeft i) :\n P3 x₁ x₂ y₁ y₂ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n · exact (h.2.2.2 hl).trans h.2.2.1\n · rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {y₁ y₂} (h : ∀ i, IH3 x₁ (x₂.moveLeft i) x₂ y₁ y₂)\n (hs : ∀ i, IH3 (-x₂) ((-x₁).moveLeft i) (-x₁) y₁ y₂) (hl : x₁ < x₂) :\n P3 x₁ x₂ y₁ y₂",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric → P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 x₁ x₂ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : x₁ ≈ x₂) : x₁ * y ≈ x₂ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : y₁ ≈ y₂) : x * y₁ ≈ x * y₂",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : x₁ ≈ x₂) (hy : y₁ ≈ y₂) : x₁ * y₁ ≈ x₂ * y₂",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : x₁ < x₂) (hy : y₁ < y₂) : P3 x₁ x₂ y₁ y₂",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hp₁ : 0 < x₁) (hp₂ : 0 < x₂) : 0 < x₁ * x₂",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
] | 48 | SetTheory.PGame.Numeric.mul_pos | [
[
491,
72
],
[
495,
13
]
] | 2 | 4 | simp_rw [P3, quot_zero_mul, quot_mul_zero, add_lt_add_iff_left] at this | x x₁ x₂ y y₁ y₂ : PGame
hx : x.Numeric
hx₁ : x₁.Numeric
hx₂ : x₂.Numeric
hy : y.Numeric
hy₁ : y₁.Numeric
hy₂ : y₂.Numeric
hp₁ : 0 < x₁
hp₂ : 0 < x₂
this : P3 0 x₁ 0 x₂
⊢ ⟦0⟧ < ⟦x₁ * x₂⟧ | x x₁ x₂ y y₁ y₂ : PGame
hx : x.Numeric
hx₁ : x₁.Numeric
hx₂ : x₂.Numeric
hy : y.Numeric
hy₁ : y₁.Numeric
hy₂ : y₂.Numeric
hp₁ : 0 < x₁
hp₂ : 0 < x₂
this : ⟦0⟧ < ⟦x₁ * x₂⟧
⊢ ⟦0⟧ < ⟦x₁ * x₂⟧ |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "def P1 (x₁ x₂ x₃ y₁ y₂ y₃ : PGame) :=\n ⟦x₁ * y₁⟧ + ⟦x₂ * y₂⟧ - ⟦x₁ * y₂⟧ < ⟦x₃ * y₁⟧ + ⟦x₂ * y₃⟧ - (⟦x₃ * y₃⟧ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (x₁ x₂ y : PGame) := x₁ ≈ x₂ → ⟦x₁ * y⟧ = (⟦x₂ * y⟧ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (x₁ x₂ y₁ y₂ : PGame) := ⟦x₁ * y₂⟧ + ⟦x₂ * y₁⟧ < ⟦x₁ * y₁⟧ + (⟦x₂ * y₂⟧ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (x₁ x₂ y : PGame) :=\n x₁ < x₂ → (∀ i, P3 x₁ x₂ (y.moveLeft i) y) ∧ ∀ j, P3 x₁ x₂ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (x₁ x₂ y : PGame) : Prop := P2 x₁ x₂ y ∧ P4 x₁ x₂ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 x₁ x₂ y₁ y₂ ↔ P3 y₁ y₂ x₁ x₂ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (h₁ : P3 x₁ x₂ y₁ y₂) (h₂ : P3 x₂ x₃ y₁ y₂) : P3 x₁ x₃ y₁ y₂ := by\n rw [P3] at h₁ h₂\n rw [P3, ← add_lt_add_iff_left (⟦x₂ * y₁⟧ + ⟦x₂ * y₂⟧)]\n convert add_lt_add h₁ h₂ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 x₁ x₂ y₁ y₂ ↔ P3 (-x₂) (-x₁) y₁ y₂ := by\n simp_rw [P3, quot_neg_mul]\n rw [← _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 x₁ x₂ y ↔ P2 (-x₂) (-x₁) y := by\n rw [P2, P2]\n constructor\n · rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (· ·)\n · rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (· ·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 x₁ x₂ y ↔ P2 x₁ x₂ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 x₁ x₂ y ↔ P4 (-x₂) (-x₁) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', ← P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 x₁ x₂ y ↔ P4 x₁ x₂ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 x₁ x₂ y ↔ P24 (-x₂) (-x₁) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 x₁ x₂ y ↔ P24 x₁ x₂ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (⟦mulOption x y i k⟧ : Game) < -⟦mulOption x (-y) j l⟧ ↔\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n ⟦mulOption x y i j⟧ < (⟦x * y⟧ : Game) ↔ P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : x₁ ≈ x₃) (h₁ : P2 x₁ x₃ y₁) (h₃ : P2 x₁ x₃ y₃) (h3 : P3 x₁ x₂ y₂ y₃) :\n P1 x₁ x₂ x₃ y₁ y₂ y₃ := by\n rw [P1, ← h₁ he, ← h₃ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 ⟦x₁ * y₁⟧ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (h₁ : P3 x₃ x₂ y₂ y₃) (h₂ : P3 x₁ x₃ y₂ y₁) : P1 x₁ x₂ x₃ y₁ y₂ y₃ := by\n rw [P1, sub_lt_sub_iff, ← add_lt_add_iff_left ⟦x₃ * y₂⟧]\n convert add_lt_add h₁ h₂ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (x₁ x₂ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args → Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 x₁ x₂ y) => {x₁, x₂, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := ∀ x ∈ a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric ↔ x.Numeric ∧ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {x₁ x₂ y} :\n (Args.P24 x₁ x₂ y).Numeric ↔ x₁.Numeric ∧ x₂.Numeric ∧ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args → Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 x₁ x₂ y) => P24 x₁ x₂ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a → a.Numeric → a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption ⟨wf_isOption.isIrrefl.1⟩ _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n ∀ ⦃x₁ x₂ y'⦄, IsOption x₁ x → IsOption x₂ x → (y' = y ∨ IsOption y' y) → P24 x₁ x₂ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y → IH1 (-x) y :=\n fun h x₁ x₂ y' h₁ h₂ hy ↦ by\n rw [isOption_neg] at h₁ h₂\n exact P24_neg_left.2 (h h₂ h₁ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y → IH1 x (-y) :=\n fun h x₁ x₂ y' ↦ by\n rw [← neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro x₁ x₂ y' h₁ h₂ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left h₁ h₂)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih ⊢\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [← moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (⟦mulOption x y i k⟧ : Game) < -⟦mulOption x (-y) j l⟧ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (⟦mulOption x y i k⟧ : Game) < -⟦mulOption x (-y) j l⟧ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n · exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n · have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n · rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (x₁ x₂ y : PGame) : Prop :=\n ∀ ⦃z⦄, (IsOption z x₁ → P24 z x₂ y) ∧ (IsOption z x₂ → P24 x₁ z y) ∧ (IsOption z y → P24 x₁ x₂ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (x₁ x₂ y : PGame) : Prop :=\n ∀ ⦃z w⦄, IsOption w y → (IsOption z x₁ → P2 z x₂ w) ∧ (IsOption z x₂ → P2 x₁ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ih₁₂ : IH24 x₁ x₂ y := by\n rw [IH24]\n refine fun z ↦ ⟨?_, ?_, ?_⟩ <;>\n refine fun h ↦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n · exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n · exact (cutExpand_add_left {x₁}).2 (cutExpand_pair_left h)\n · exact (cutExpand_add_left {x₁}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ih₁₂",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ih₂₁ : IH24 x₂ x₁ y := ih₁₂ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' ⊢\n suffices {x₁, y, x₂} = {x₂, y, x₁} by rwa [← this]\n dsimp only [Multiset.insert_eq_cons, ← Multiset.singleton_add] at ih' ⊢\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ih₂₁",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 x₁ x₂ y := by\n refine fun z w h ↦ ⟨?_, ?_⟩\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {x₁}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (x₁ * y).Numeric ∧ (x₂ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n · exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {x₁}).2 cutExpand_zero\n · exact (cutExpand_add_right {x₂, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 x₁ x₂ y → IH24 (-x₂) (-x₁) y ∧ IH24 x₁ x₂ (-y) := by\n simp_rw [IH24, ← P24_neg_right, isOption_neg]\n refine fun h ↦ ⟨fun z ↦ ⟨?_, ?_, ?_⟩,\n fun z ↦ ⟨(@h z).1, (@h z).2.1, P24_neg_right.2 ∘ (@h <| -z).2.2⟩⟩\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 x₁ x₂ y → IH4 (-x₂) (-x₁) y ∧ IH4 x₁ x₂ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h ↦ ⟨fun z w h' ↦ ?_, fun z w h' ↦ ?_⟩\n · convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n · convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : x₁.Numeric) (h : IH24 x₁ x₂ y) (he : x₁ ≈ x₂) (i j) :\n ⟦mulOption x₁ y i j⟧ < (⟦x₂ * y⟧ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n · rw [← ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n · rw [← lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (h₁ : x₁.Numeric) (h₂ : x₂.Numeric)\n (h₁₂ : IH24 x₁ x₂ y) (h₂₁ : IH24 x₂ x₁ y) (he : x₁ ≈ x₂) : x₁ * y ≤ x₂ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := ∀ ⦃i j⦄, ⟦mulOption x y i j⟧ < (⟦x * y⟧ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y ∧ MulOptionsLTMul (-x) (-y)) ∧\n (MulOptionsLTMul x (-y) ∧ MulOptionsLTMul (-x) y) := by\n constructor\n · have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [← quot_neg_mul_neg]\n rfl\n · have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forall₂_imp fun a b ↦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (x₁ x' x₂ y₁ y₂ : PGame) : Prop :=\n P2 x₁ x' y₁ ∧ P2 x₁ x' y₂ ∧ P3 x' x₂ y₁ y₂ ∧ (x₁ < x' → P3 x₁ x' y₁ y₂)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 x₁ x₂ y) (h4 : IH4 x₁ x₂ y) (hl : MulOptionsLTMul x₂ y) (i j) :\n IH3 x₁ (x₂.moveLeft i) x₂ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n ⟨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l ↦ (h24.2 l).1 _⟩",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {y₁ y₂} (i) (h : IH3 x₁ (x₂.moveLeft i) x₂ y₁ y₂) (hl : x₁ ≤ x₂.moveLeft i) :\n P3 x₁ x₂ y₁ y₂ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n · exact (h.2.2.2 hl).trans h.2.2.1\n · rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {y₁ y₂} (h : ∀ i, IH3 x₁ (x₂.moveLeft i) x₂ y₁ y₂)\n (hs : ∀ i, IH3 (-x₂) ((-x₁).moveLeft i) (-x₁) y₁ y₂) (hl : x₁ < x₂) :\n P3 x₁ x₂ y₁ y₂",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric → P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 x₁ x₂ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : x₁ ≈ x₂) : x₁ * y ≈ x₂ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : y₁ ≈ y₂) : x * y₁ ≈ x * y₂",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : x₁ ≈ x₂) (hy : y₁ ≈ y₂) : x₁ * y₁ ≈ x₂ * y₂",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : x₁ < x₂) (hy : y₁ < y₂) : P3 x₁ x₂ y₁ y₂",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hp₁ : 0 < x₁) (hp₂ : 0 < x₂) : 0 < x₁ * x₂",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
] | 48 | SetTheory.PGame.Numeric.mul_pos | [
[
491,
72
],
[
495,
13
]
] | 3 | 4 | exact this | x x₁ x₂ y y₁ y₂ : PGame
hx : x.Numeric
hx₁ : x₁.Numeric
hx₂ : x₂.Numeric
hy : y.Numeric
hy₁ : y₁.Numeric
hy₂ : y₂.Numeric
hp₁ : 0 < x₁
hp₂ : 0 < x₂
this : ⟦0⟧ < ⟦x₁ * x₂⟧
⊢ ⟦0⟧ < ⟦x₁ * x₂⟧ | no goals |
Mathlib/SetTheory/Ordinal/CantorNormalForm.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Mathlib.SetTheory.Ordinal.Exponential",
"Mathlib/SetTheory/Ordinal/Exponential.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "@[elab_as_elim]\nnoncomputable def CNFRec (b : Ordinal) {C : Ordinal → Sort*} (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : ∀ o, C o := fun o ↦ by\n by_cases h : o = 0\n · rw [h]; exact H0\n · exact H o h (CNFRec _ H0 H (o % b ^ log b o))\n termination_by o => o\n decreasing_by exact mod_opow_log_lt_self b h",
"end": [
50,
49
],
"full_name": "Ordinal.CNFRec",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem CNFRec_zero {C : Ordinal → Sort*} (b : Ordinal) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : @CNFRec b C H0 H 0 = H0",
"end": [
58,
6
],
"full_name": "Ordinal.CNFRec_zero",
"kind": "commanddeclaration",
"start": [
54,
1
]
},
{
"code": "theorem CNFRec_pos (b : Ordinal) {o : Ordinal} {C : Ordinal → Sort*} (ho : o ≠ 0) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) :\n @CNFRec b C H0 H o = H o ho (@CNFRec b C H0 H _)",
"end": [
64,
83
],
"full_name": "Ordinal.CNFRec_pos",
"kind": "commanddeclaration",
"start": [
62,
1
]
},
{
"code": "@[pp_nodot]\ndef CNF (b o : Ordinal) : List (Ordinal × Ordinal) :=\n CNFRec b [] (fun o _ho IH ↦ (log b o, o / b ^ log b o)::IH) o",
"end": [
76,
64
],
"full_name": "Ordinal.CNF",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "@[simp]\ntheorem CNF_zero (b : Ordinal) : CNF b 0 = []",
"end": [
82,
20
],
"full_name": "Ordinal.CNF_zero",
"kind": "commanddeclaration",
"start": [
80,
1
]
},
{
"code": "theorem CNF_ne_zero {b o : Ordinal} (ho : o ≠ 0) :\n CNF b o = (log b o, o / b ^ log b o)::CNF b (o % b ^ log b o)",
"end": [
89,
22
],
"full_name": "Ordinal.CNF_ne_zero",
"kind": "commanddeclaration",
"start": [
86,
1
]
},
{
"code": "theorem zero_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 0 o = [⟨0, o⟩]",
"end": [
93,
93
],
"full_name": "Ordinal.zero_CNF",
"kind": "commanddeclaration",
"start": [
93,
1
]
},
{
"code": "theorem one_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 1 o = [⟨0, o⟩]",
"end": [
97,
92
],
"full_name": "Ordinal.one_CNF",
"kind": "commanddeclaration",
"start": [
97,
1
]
},
{
"code": "theorem CNF_of_le_one {b o : Ordinal} (hb : b ≤ 1) (ho : o ≠ 0) : CNF b o = [⟨0, o⟩]",
"end": [
104,
21
],
"full_name": "Ordinal.CNF_of_le_one",
"kind": "commanddeclaration",
"start": [
101,
1
]
},
{
"code": "theorem CNF_of_lt {b o : Ordinal} (ho : o ≠ 0) (hb : o < b) : CNF b o = [⟨0, o⟩]",
"end": [
109,
84
],
"full_name": "Ordinal.CNF_of_lt",
"kind": "commanddeclaration",
"start": [
108,
1
]
},
{
"code": "theorem CNF_foldr (b o : Ordinal) : (CNF b o).foldr (fun p r ↦ b ^ p.1 * p.2 + r) 0 = o",
"end": [
116,
74
],
"full_name": "Ordinal.CNF_foldr",
"kind": "commanddeclaration",
"start": [
113,
1
]
},
{
"code": "theorem CNF_fst_le_log {b o : Ordinal.{u}} {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.1 ≤ log b o",
"end": [
129,
74
],
"full_name": "Ordinal.CNF_fst_le_log",
"kind": "commanddeclaration",
"start": [
120,
1
]
},
{
"code": "theorem CNF_fst_le {b o : Ordinal.{u}} {x : Ordinal × Ordinal} (h : x ∈ CNF b o) : x.1 ≤ o",
"end": [
135,
46
],
"full_name": "Ordinal.CNF_fst_le",
"kind": "commanddeclaration",
"start": [
133,
1
]
},
{
"code": "theorem CNF_lt_snd {b o : Ordinal.{u}} {x : Ordinal × Ordinal} : x ∈ CNF b o → 0 < x.2",
"end": [
145,
15
],
"full_name": "Ordinal.CNF_lt_snd",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem CNF_snd_lt {b o : Ordinal.{u}} (hb : 1 < b) {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.2 < b",
"end": [
158,
17
],
"full_name": "Ordinal.CNF_snd_lt",
"kind": "commanddeclaration",
"start": [
149,
1
]
},
{
"code": "theorem CNF_sorted (b o : Ordinal) : ((CNF b o).map Prod.fst).Sorted (· > ·)",
"end": [
174,
83
],
"full_name": "Ordinal.CNF_sorted",
"kind": "commanddeclaration",
"start": [
162,
1
]
}
] | 0 | Ordinal.CNFRec_zero | [
[
56,
79
],
[
58,
6
]
] | 0 | 2 | rw [CNFRec, dif_pos rfl] | C : Ordinal.{u_2} → Sort u_1
b : Ordinal.{u_2}
H0 : C 0
H : (o : Ordinal.{u_2}) → o ≠ 0 → C (o % b ^ log b o) → C o
⊢ b.CNFRec H0 H 0 = H0 | C : Ordinal.{u_2} → Sort u_1
b : Ordinal.{u_2}
H0 : C 0
H : (o : Ordinal.{u_2}) → o ≠ 0 → C (o % b ^ log b o) → C o
⊢ ⋯.mpr H0 = H0 |
Mathlib/SetTheory/Ordinal/CantorNormalForm.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Mathlib.SetTheory.Ordinal.Exponential",
"Mathlib/SetTheory/Ordinal/Exponential.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "@[elab_as_elim]\nnoncomputable def CNFRec (b : Ordinal) {C : Ordinal → Sort*} (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : ∀ o, C o := fun o ↦ by\n by_cases h : o = 0\n · rw [h]; exact H0\n · exact H o h (CNFRec _ H0 H (o % b ^ log b o))\n termination_by o => o\n decreasing_by exact mod_opow_log_lt_self b h",
"end": [
50,
49
],
"full_name": "Ordinal.CNFRec",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem CNFRec_zero {C : Ordinal → Sort*} (b : Ordinal) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : @CNFRec b C H0 H 0 = H0",
"end": [
58,
6
],
"full_name": "Ordinal.CNFRec_zero",
"kind": "commanddeclaration",
"start": [
54,
1
]
},
{
"code": "theorem CNFRec_pos (b : Ordinal) {o : Ordinal} {C : Ordinal → Sort*} (ho : o ≠ 0) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) :\n @CNFRec b C H0 H o = H o ho (@CNFRec b C H0 H _)",
"end": [
64,
83
],
"full_name": "Ordinal.CNFRec_pos",
"kind": "commanddeclaration",
"start": [
62,
1
]
},
{
"code": "@[pp_nodot]\ndef CNF (b o : Ordinal) : List (Ordinal × Ordinal) :=\n CNFRec b [] (fun o _ho IH ↦ (log b o, o / b ^ log b o)::IH) o",
"end": [
76,
64
],
"full_name": "Ordinal.CNF",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "@[simp]\ntheorem CNF_zero (b : Ordinal) : CNF b 0 = []",
"end": [
82,
20
],
"full_name": "Ordinal.CNF_zero",
"kind": "commanddeclaration",
"start": [
80,
1
]
},
{
"code": "theorem CNF_ne_zero {b o : Ordinal} (ho : o ≠ 0) :\n CNF b o = (log b o, o / b ^ log b o)::CNF b (o % b ^ log b o)",
"end": [
89,
22
],
"full_name": "Ordinal.CNF_ne_zero",
"kind": "commanddeclaration",
"start": [
86,
1
]
},
{
"code": "theorem zero_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 0 o = [⟨0, o⟩]",
"end": [
93,
93
],
"full_name": "Ordinal.zero_CNF",
"kind": "commanddeclaration",
"start": [
93,
1
]
},
{
"code": "theorem one_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 1 o = [⟨0, o⟩]",
"end": [
97,
92
],
"full_name": "Ordinal.one_CNF",
"kind": "commanddeclaration",
"start": [
97,
1
]
},
{
"code": "theorem CNF_of_le_one {b o : Ordinal} (hb : b ≤ 1) (ho : o ≠ 0) : CNF b o = [⟨0, o⟩]",
"end": [
104,
21
],
"full_name": "Ordinal.CNF_of_le_one",
"kind": "commanddeclaration",
"start": [
101,
1
]
},
{
"code": "theorem CNF_of_lt {b o : Ordinal} (ho : o ≠ 0) (hb : o < b) : CNF b o = [⟨0, o⟩]",
"end": [
109,
84
],
"full_name": "Ordinal.CNF_of_lt",
"kind": "commanddeclaration",
"start": [
108,
1
]
},
{
"code": "theorem CNF_foldr (b o : Ordinal) : (CNF b o).foldr (fun p r ↦ b ^ p.1 * p.2 + r) 0 = o",
"end": [
116,
74
],
"full_name": "Ordinal.CNF_foldr",
"kind": "commanddeclaration",
"start": [
113,
1
]
},
{
"code": "theorem CNF_fst_le_log {b o : Ordinal.{u}} {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.1 ≤ log b o",
"end": [
129,
74
],
"full_name": "Ordinal.CNF_fst_le_log",
"kind": "commanddeclaration",
"start": [
120,
1
]
},
{
"code": "theorem CNF_fst_le {b o : Ordinal.{u}} {x : Ordinal × Ordinal} (h : x ∈ CNF b o) : x.1 ≤ o",
"end": [
135,
46
],
"full_name": "Ordinal.CNF_fst_le",
"kind": "commanddeclaration",
"start": [
133,
1
]
},
{
"code": "theorem CNF_lt_snd {b o : Ordinal.{u}} {x : Ordinal × Ordinal} : x ∈ CNF b o → 0 < x.2",
"end": [
145,
15
],
"full_name": "Ordinal.CNF_lt_snd",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem CNF_snd_lt {b o : Ordinal.{u}} (hb : 1 < b) {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.2 < b",
"end": [
158,
17
],
"full_name": "Ordinal.CNF_snd_lt",
"kind": "commanddeclaration",
"start": [
149,
1
]
},
{
"code": "theorem CNF_sorted (b o : Ordinal) : ((CNF b o).map Prod.fst).Sorted (· > ·)",
"end": [
174,
83
],
"full_name": "Ordinal.CNF_sorted",
"kind": "commanddeclaration",
"start": [
162,
1
]
}
] | 0 | Ordinal.CNFRec_zero | [
[
56,
79
],
[
58,
6
]
] | 1 | 2 | rfl | C : Ordinal.{u_2} → Sort u_1
b : Ordinal.{u_2}
H0 : C 0
H : (o : Ordinal.{u_2}) → o ≠ 0 → C (o % b ^ log b o) → C o
⊢ ⋯.mpr H0 = H0 | no goals |
Mathlib/SetTheory/Ordinal/CantorNormalForm.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Mathlib.SetTheory.Ordinal.Exponential",
"Mathlib/SetTheory/Ordinal/Exponential.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "@[elab_as_elim]\nnoncomputable def CNFRec (b : Ordinal) {C : Ordinal → Sort*} (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : ∀ o, C o := fun o ↦ by\n by_cases h : o = 0\n · rw [h]; exact H0\n · exact H o h (CNFRec _ H0 H (o % b ^ log b o))\n termination_by o => o\n decreasing_by exact mod_opow_log_lt_self b h",
"end": [
50,
49
],
"full_name": "Ordinal.CNFRec",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem CNFRec_zero {C : Ordinal → Sort*} (b : Ordinal) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : @CNFRec b C H0 H 0 = H0",
"end": [
58,
6
],
"full_name": "Ordinal.CNFRec_zero",
"kind": "commanddeclaration",
"start": [
54,
1
]
},
{
"code": "theorem CNFRec_pos (b : Ordinal) {o : Ordinal} {C : Ordinal → Sort*} (ho : o ≠ 0) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) :\n @CNFRec b C H0 H o = H o ho (@CNFRec b C H0 H _)",
"end": [
64,
83
],
"full_name": "Ordinal.CNFRec_pos",
"kind": "commanddeclaration",
"start": [
62,
1
]
},
{
"code": "@[pp_nodot]\ndef CNF (b o : Ordinal) : List (Ordinal × Ordinal) :=\n CNFRec b [] (fun o _ho IH ↦ (log b o, o / b ^ log b o)::IH) o",
"end": [
76,
64
],
"full_name": "Ordinal.CNF",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "@[simp]\ntheorem CNF_zero (b : Ordinal) : CNF b 0 = []",
"end": [
82,
20
],
"full_name": "Ordinal.CNF_zero",
"kind": "commanddeclaration",
"start": [
80,
1
]
},
{
"code": "theorem CNF_ne_zero {b o : Ordinal} (ho : o ≠ 0) :\n CNF b o = (log b o, o / b ^ log b o)::CNF b (o % b ^ log b o)",
"end": [
89,
22
],
"full_name": "Ordinal.CNF_ne_zero",
"kind": "commanddeclaration",
"start": [
86,
1
]
},
{
"code": "theorem zero_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 0 o = [⟨0, o⟩]",
"end": [
93,
93
],
"full_name": "Ordinal.zero_CNF",
"kind": "commanddeclaration",
"start": [
93,
1
]
},
{
"code": "theorem one_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 1 o = [⟨0, o⟩]",
"end": [
97,
92
],
"full_name": "Ordinal.one_CNF",
"kind": "commanddeclaration",
"start": [
97,
1
]
},
{
"code": "theorem CNF_of_le_one {b o : Ordinal} (hb : b ≤ 1) (ho : o ≠ 0) : CNF b o = [⟨0, o⟩]",
"end": [
104,
21
],
"full_name": "Ordinal.CNF_of_le_one",
"kind": "commanddeclaration",
"start": [
101,
1
]
},
{
"code": "theorem CNF_of_lt {b o : Ordinal} (ho : o ≠ 0) (hb : o < b) : CNF b o = [⟨0, o⟩]",
"end": [
109,
84
],
"full_name": "Ordinal.CNF_of_lt",
"kind": "commanddeclaration",
"start": [
108,
1
]
},
{
"code": "theorem CNF_foldr (b o : Ordinal) : (CNF b o).foldr (fun p r ↦ b ^ p.1 * p.2 + r) 0 = o",
"end": [
116,
74
],
"full_name": "Ordinal.CNF_foldr",
"kind": "commanddeclaration",
"start": [
113,
1
]
},
{
"code": "theorem CNF_fst_le_log {b o : Ordinal.{u}} {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.1 ≤ log b o",
"end": [
129,
74
],
"full_name": "Ordinal.CNF_fst_le_log",
"kind": "commanddeclaration",
"start": [
120,
1
]
},
{
"code": "theorem CNF_fst_le {b o : Ordinal.{u}} {x : Ordinal × Ordinal} (h : x ∈ CNF b o) : x.1 ≤ o",
"end": [
135,
46
],
"full_name": "Ordinal.CNF_fst_le",
"kind": "commanddeclaration",
"start": [
133,
1
]
},
{
"code": "theorem CNF_lt_snd {b o : Ordinal.{u}} {x : Ordinal × Ordinal} : x ∈ CNF b o → 0 < x.2",
"end": [
145,
15
],
"full_name": "Ordinal.CNF_lt_snd",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem CNF_snd_lt {b o : Ordinal.{u}} (hb : 1 < b) {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.2 < b",
"end": [
158,
17
],
"full_name": "Ordinal.CNF_snd_lt",
"kind": "commanddeclaration",
"start": [
149,
1
]
},
{
"code": "theorem CNF_sorted (b o : Ordinal) : ((CNF b o).map Prod.fst).Sorted (· > ·)",
"end": [
174,
83
],
"full_name": "Ordinal.CNF_sorted",
"kind": "commanddeclaration",
"start": [
162,
1
]
}
] | 1 | Ordinal.CNFRec_pos | [
[
64,
57
],
[
64,
83
]
] | 0 | 1 | rw [CNFRec, dif_neg ho] | b o : Ordinal.{u_2}
C : Ordinal.{u_2} → Sort u_1
ho : o ≠ 0
H0 : C 0
H : (o : Ordinal.{u_2}) → o ≠ 0 → C (o % b ^ log b o) → C o
⊢ b.CNFRec H0 H o = H o ho (b.CNFRec H0 H (o % b ^ log b o)) | no goals |
Mathlib/SetTheory/Ordinal/CantorNormalForm.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Mathlib.SetTheory.Ordinal.Exponential",
"Mathlib/SetTheory/Ordinal/Exponential.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "@[elab_as_elim]\nnoncomputable def CNFRec (b : Ordinal) {C : Ordinal → Sort*} (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : ∀ o, C o := fun o ↦ by\n by_cases h : o = 0\n · rw [h]; exact H0\n · exact H o h (CNFRec _ H0 H (o % b ^ log b o))\n termination_by o => o\n decreasing_by exact mod_opow_log_lt_self b h",
"end": [
50,
49
],
"full_name": "Ordinal.CNFRec",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem CNFRec_zero {C : Ordinal → Sort*} (b : Ordinal) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : @CNFRec b C H0 H 0 = H0",
"end": [
58,
6
],
"full_name": "Ordinal.CNFRec_zero",
"kind": "commanddeclaration",
"start": [
54,
1
]
},
{
"code": "theorem CNFRec_pos (b : Ordinal) {o : Ordinal} {C : Ordinal → Sort*} (ho : o ≠ 0) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) :\n @CNFRec b C H0 H o = H o ho (@CNFRec b C H0 H _)",
"end": [
64,
83
],
"full_name": "Ordinal.CNFRec_pos",
"kind": "commanddeclaration",
"start": [
62,
1
]
},
{
"code": "@[pp_nodot]\ndef CNF (b o : Ordinal) : List (Ordinal × Ordinal) :=\n CNFRec b [] (fun o _ho IH ↦ (log b o, o / b ^ log b o)::IH) o",
"end": [
76,
64
],
"full_name": "Ordinal.CNF",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "@[simp]\ntheorem CNF_zero (b : Ordinal) : CNF b 0 = []",
"end": [
82,
20
],
"full_name": "Ordinal.CNF_zero",
"kind": "commanddeclaration",
"start": [
80,
1
]
},
{
"code": "theorem CNF_ne_zero {b o : Ordinal} (ho : o ≠ 0) :\n CNF b o = (log b o, o / b ^ log b o)::CNF b (o % b ^ log b o)",
"end": [
89,
22
],
"full_name": "Ordinal.CNF_ne_zero",
"kind": "commanddeclaration",
"start": [
86,
1
]
},
{
"code": "theorem zero_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 0 o = [⟨0, o⟩]",
"end": [
93,
93
],
"full_name": "Ordinal.zero_CNF",
"kind": "commanddeclaration",
"start": [
93,
1
]
},
{
"code": "theorem one_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 1 o = [⟨0, o⟩]",
"end": [
97,
92
],
"full_name": "Ordinal.one_CNF",
"kind": "commanddeclaration",
"start": [
97,
1
]
},
{
"code": "theorem CNF_of_le_one {b o : Ordinal} (hb : b ≤ 1) (ho : o ≠ 0) : CNF b o = [⟨0, o⟩]",
"end": [
104,
21
],
"full_name": "Ordinal.CNF_of_le_one",
"kind": "commanddeclaration",
"start": [
101,
1
]
},
{
"code": "theorem CNF_of_lt {b o : Ordinal} (ho : o ≠ 0) (hb : o < b) : CNF b o = [⟨0, o⟩]",
"end": [
109,
84
],
"full_name": "Ordinal.CNF_of_lt",
"kind": "commanddeclaration",
"start": [
108,
1
]
},
{
"code": "theorem CNF_foldr (b o : Ordinal) : (CNF b o).foldr (fun p r ↦ b ^ p.1 * p.2 + r) 0 = o",
"end": [
116,
74
],
"full_name": "Ordinal.CNF_foldr",
"kind": "commanddeclaration",
"start": [
113,
1
]
},
{
"code": "theorem CNF_fst_le_log {b o : Ordinal.{u}} {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.1 ≤ log b o",
"end": [
129,
74
],
"full_name": "Ordinal.CNF_fst_le_log",
"kind": "commanddeclaration",
"start": [
120,
1
]
},
{
"code": "theorem CNF_fst_le {b o : Ordinal.{u}} {x : Ordinal × Ordinal} (h : x ∈ CNF b o) : x.1 ≤ o",
"end": [
135,
46
],
"full_name": "Ordinal.CNF_fst_le",
"kind": "commanddeclaration",
"start": [
133,
1
]
},
{
"code": "theorem CNF_lt_snd {b o : Ordinal.{u}} {x : Ordinal × Ordinal} : x ∈ CNF b o → 0 < x.2",
"end": [
145,
15
],
"full_name": "Ordinal.CNF_lt_snd",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem CNF_snd_lt {b o : Ordinal.{u}} (hb : 1 < b) {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.2 < b",
"end": [
158,
17
],
"full_name": "Ordinal.CNF_snd_lt",
"kind": "commanddeclaration",
"start": [
149,
1
]
},
{
"code": "theorem CNF_sorted (b o : Ordinal) : ((CNF b o).map Prod.fst).Sorted (· > ·)",
"end": [
174,
83
],
"full_name": "Ordinal.CNF_sorted",
"kind": "commanddeclaration",
"start": [
162,
1
]
}
] | 4 | Ordinal.zero_CNF | [
[
93,
69
],
[
93,
93
]
] | 0 | 1 | simp [CNF_ne_zero ho] | o : Ordinal.{u_1}
ho : o ≠ 0
⊢ CNF 0 o = [(0, o)] | no goals |
Mathlib/SetTheory/Ordinal/CantorNormalForm.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Mathlib.SetTheory.Ordinal.Exponential",
"Mathlib/SetTheory/Ordinal/Exponential.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "@[elab_as_elim]\nnoncomputable def CNFRec (b : Ordinal) {C : Ordinal → Sort*} (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : ∀ o, C o := fun o ↦ by\n by_cases h : o = 0\n · rw [h]; exact H0\n · exact H o h (CNFRec _ H0 H (o % b ^ log b o))\n termination_by o => o\n decreasing_by exact mod_opow_log_lt_self b h",
"end": [
50,
49
],
"full_name": "Ordinal.CNFRec",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem CNFRec_zero {C : Ordinal → Sort*} (b : Ordinal) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : @CNFRec b C H0 H 0 = H0",
"end": [
58,
6
],
"full_name": "Ordinal.CNFRec_zero",
"kind": "commanddeclaration",
"start": [
54,
1
]
},
{
"code": "theorem CNFRec_pos (b : Ordinal) {o : Ordinal} {C : Ordinal → Sort*} (ho : o ≠ 0) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) :\n @CNFRec b C H0 H o = H o ho (@CNFRec b C H0 H _)",
"end": [
64,
83
],
"full_name": "Ordinal.CNFRec_pos",
"kind": "commanddeclaration",
"start": [
62,
1
]
},
{
"code": "@[pp_nodot]\ndef CNF (b o : Ordinal) : List (Ordinal × Ordinal) :=\n CNFRec b [] (fun o _ho IH ↦ (log b o, o / b ^ log b o)::IH) o",
"end": [
76,
64
],
"full_name": "Ordinal.CNF",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "@[simp]\ntheorem CNF_zero (b : Ordinal) : CNF b 0 = []",
"end": [
82,
20
],
"full_name": "Ordinal.CNF_zero",
"kind": "commanddeclaration",
"start": [
80,
1
]
},
{
"code": "theorem CNF_ne_zero {b o : Ordinal} (ho : o ≠ 0) :\n CNF b o = (log b o, o / b ^ log b o)::CNF b (o % b ^ log b o)",
"end": [
89,
22
],
"full_name": "Ordinal.CNF_ne_zero",
"kind": "commanddeclaration",
"start": [
86,
1
]
},
{
"code": "theorem zero_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 0 o = [⟨0, o⟩]",
"end": [
93,
93
],
"full_name": "Ordinal.zero_CNF",
"kind": "commanddeclaration",
"start": [
93,
1
]
},
{
"code": "theorem one_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 1 o = [⟨0, o⟩]",
"end": [
97,
92
],
"full_name": "Ordinal.one_CNF",
"kind": "commanddeclaration",
"start": [
97,
1
]
},
{
"code": "theorem CNF_of_le_one {b o : Ordinal} (hb : b ≤ 1) (ho : o ≠ 0) : CNF b o = [⟨0, o⟩]",
"end": [
104,
21
],
"full_name": "Ordinal.CNF_of_le_one",
"kind": "commanddeclaration",
"start": [
101,
1
]
},
{
"code": "theorem CNF_of_lt {b o : Ordinal} (ho : o ≠ 0) (hb : o < b) : CNF b o = [⟨0, o⟩]",
"end": [
109,
84
],
"full_name": "Ordinal.CNF_of_lt",
"kind": "commanddeclaration",
"start": [
108,
1
]
},
{
"code": "theorem CNF_foldr (b o : Ordinal) : (CNF b o).foldr (fun p r ↦ b ^ p.1 * p.2 + r) 0 = o",
"end": [
116,
74
],
"full_name": "Ordinal.CNF_foldr",
"kind": "commanddeclaration",
"start": [
113,
1
]
},
{
"code": "theorem CNF_fst_le_log {b o : Ordinal.{u}} {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.1 ≤ log b o",
"end": [
129,
74
],
"full_name": "Ordinal.CNF_fst_le_log",
"kind": "commanddeclaration",
"start": [
120,
1
]
},
{
"code": "theorem CNF_fst_le {b o : Ordinal.{u}} {x : Ordinal × Ordinal} (h : x ∈ CNF b o) : x.1 ≤ o",
"end": [
135,
46
],
"full_name": "Ordinal.CNF_fst_le",
"kind": "commanddeclaration",
"start": [
133,
1
]
},
{
"code": "theorem CNF_lt_snd {b o : Ordinal.{u}} {x : Ordinal × Ordinal} : x ∈ CNF b o → 0 < x.2",
"end": [
145,
15
],
"full_name": "Ordinal.CNF_lt_snd",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem CNF_snd_lt {b o : Ordinal.{u}} (hb : 1 < b) {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.2 < b",
"end": [
158,
17
],
"full_name": "Ordinal.CNF_snd_lt",
"kind": "commanddeclaration",
"start": [
149,
1
]
},
{
"code": "theorem CNF_sorted (b o : Ordinal) : ((CNF b o).map Prod.fst).Sorted (· > ·)",
"end": [
174,
83
],
"full_name": "Ordinal.CNF_sorted",
"kind": "commanddeclaration",
"start": [
162,
1
]
}
] | 5 | Ordinal.one_CNF | [
[
97,
68
],
[
97,
92
]
] | 0 | 1 | simp [CNF_ne_zero ho] | o : Ordinal.{u_1}
ho : o ≠ 0
⊢ CNF 1 o = [(0, o)] | no goals |
Mathlib/SetTheory/Ordinal/CantorNormalForm.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Mathlib.SetTheory.Ordinal.Exponential",
"Mathlib/SetTheory/Ordinal/Exponential.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "@[elab_as_elim]\nnoncomputable def CNFRec (b : Ordinal) {C : Ordinal → Sort*} (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : ∀ o, C o := fun o ↦ by\n by_cases h : o = 0\n · rw [h]; exact H0\n · exact H o h (CNFRec _ H0 H (o % b ^ log b o))\n termination_by o => o\n decreasing_by exact mod_opow_log_lt_self b h",
"end": [
50,
49
],
"full_name": "Ordinal.CNFRec",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem CNFRec_zero {C : Ordinal → Sort*} (b : Ordinal) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : @CNFRec b C H0 H 0 = H0",
"end": [
58,
6
],
"full_name": "Ordinal.CNFRec_zero",
"kind": "commanddeclaration",
"start": [
54,
1
]
},
{
"code": "theorem CNFRec_pos (b : Ordinal) {o : Ordinal} {C : Ordinal → Sort*} (ho : o ≠ 0) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) :\n @CNFRec b C H0 H o = H o ho (@CNFRec b C H0 H _)",
"end": [
64,
83
],
"full_name": "Ordinal.CNFRec_pos",
"kind": "commanddeclaration",
"start": [
62,
1
]
},
{
"code": "@[pp_nodot]\ndef CNF (b o : Ordinal) : List (Ordinal × Ordinal) :=\n CNFRec b [] (fun o _ho IH ↦ (log b o, o / b ^ log b o)::IH) o",
"end": [
76,
64
],
"full_name": "Ordinal.CNF",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "@[simp]\ntheorem CNF_zero (b : Ordinal) : CNF b 0 = []",
"end": [
82,
20
],
"full_name": "Ordinal.CNF_zero",
"kind": "commanddeclaration",
"start": [
80,
1
]
},
{
"code": "theorem CNF_ne_zero {b o : Ordinal} (ho : o ≠ 0) :\n CNF b o = (log b o, o / b ^ log b o)::CNF b (o % b ^ log b o)",
"end": [
89,
22
],
"full_name": "Ordinal.CNF_ne_zero",
"kind": "commanddeclaration",
"start": [
86,
1
]
},
{
"code": "theorem zero_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 0 o = [⟨0, o⟩]",
"end": [
93,
93
],
"full_name": "Ordinal.zero_CNF",
"kind": "commanddeclaration",
"start": [
93,
1
]
},
{
"code": "theorem one_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 1 o = [⟨0, o⟩]",
"end": [
97,
92
],
"full_name": "Ordinal.one_CNF",
"kind": "commanddeclaration",
"start": [
97,
1
]
},
{
"code": "theorem CNF_of_le_one {b o : Ordinal} (hb : b ≤ 1) (ho : o ≠ 0) : CNF b o = [⟨0, o⟩]",
"end": [
104,
21
],
"full_name": "Ordinal.CNF_of_le_one",
"kind": "commanddeclaration",
"start": [
101,
1
]
},
{
"code": "theorem CNF_of_lt {b o : Ordinal} (ho : o ≠ 0) (hb : o < b) : CNF b o = [⟨0, o⟩]",
"end": [
109,
84
],
"full_name": "Ordinal.CNF_of_lt",
"kind": "commanddeclaration",
"start": [
108,
1
]
},
{
"code": "theorem CNF_foldr (b o : Ordinal) : (CNF b o).foldr (fun p r ↦ b ^ p.1 * p.2 + r) 0 = o",
"end": [
116,
74
],
"full_name": "Ordinal.CNF_foldr",
"kind": "commanddeclaration",
"start": [
113,
1
]
},
{
"code": "theorem CNF_fst_le_log {b o : Ordinal.{u}} {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.1 ≤ log b o",
"end": [
129,
74
],
"full_name": "Ordinal.CNF_fst_le_log",
"kind": "commanddeclaration",
"start": [
120,
1
]
},
{
"code": "theorem CNF_fst_le {b o : Ordinal.{u}} {x : Ordinal × Ordinal} (h : x ∈ CNF b o) : x.1 ≤ o",
"end": [
135,
46
],
"full_name": "Ordinal.CNF_fst_le",
"kind": "commanddeclaration",
"start": [
133,
1
]
},
{
"code": "theorem CNF_lt_snd {b o : Ordinal.{u}} {x : Ordinal × Ordinal} : x ∈ CNF b o → 0 < x.2",
"end": [
145,
15
],
"full_name": "Ordinal.CNF_lt_snd",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem CNF_snd_lt {b o : Ordinal.{u}} (hb : 1 < b) {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.2 < b",
"end": [
158,
17
],
"full_name": "Ordinal.CNF_snd_lt",
"kind": "commanddeclaration",
"start": [
149,
1
]
},
{
"code": "theorem CNF_sorted (b o : Ordinal) : ((CNF b o).map Prod.fst).Sorted (· > ·)",
"end": [
174,
83
],
"full_name": "Ordinal.CNF_sorted",
"kind": "commanddeclaration",
"start": [
162,
1
]
}
] | 6 | Ordinal.CNF_of_le_one | [
[
101,
89
],
[
104,
21
]
] | 0 | 5 | rcases le_one_iff.1 hb with (rfl | rfl) | b o : Ordinal.{u_1}
hb : b ≤ 1
ho : o ≠ 0
⊢ CNF b o = [(0, o)] | case inl
o : Ordinal.{u_1}
ho : o ≠ 0
hb : 0 ≤ 1
⊢ CNF 0 o = [(0, o)]
case inr
o : Ordinal.{u_1}
ho : o ≠ 0
hb : 1 ≤ 1
⊢ CNF 1 o = [(0, o)] |
Mathlib/SetTheory/Ordinal/CantorNormalForm.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Mathlib.SetTheory.Ordinal.Exponential",
"Mathlib/SetTheory/Ordinal/Exponential.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "@[elab_as_elim]\nnoncomputable def CNFRec (b : Ordinal) {C : Ordinal → Sort*} (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : ∀ o, C o := fun o ↦ by\n by_cases h : o = 0\n · rw [h]; exact H0\n · exact H o h (CNFRec _ H0 H (o % b ^ log b o))\n termination_by o => o\n decreasing_by exact mod_opow_log_lt_self b h",
"end": [
50,
49
],
"full_name": "Ordinal.CNFRec",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem CNFRec_zero {C : Ordinal → Sort*} (b : Ordinal) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : @CNFRec b C H0 H 0 = H0",
"end": [
58,
6
],
"full_name": "Ordinal.CNFRec_zero",
"kind": "commanddeclaration",
"start": [
54,
1
]
},
{
"code": "theorem CNFRec_pos (b : Ordinal) {o : Ordinal} {C : Ordinal → Sort*} (ho : o ≠ 0) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) :\n @CNFRec b C H0 H o = H o ho (@CNFRec b C H0 H _)",
"end": [
64,
83
],
"full_name": "Ordinal.CNFRec_pos",
"kind": "commanddeclaration",
"start": [
62,
1
]
},
{
"code": "@[pp_nodot]\ndef CNF (b o : Ordinal) : List (Ordinal × Ordinal) :=\n CNFRec b [] (fun o _ho IH ↦ (log b o, o / b ^ log b o)::IH) o",
"end": [
76,
64
],
"full_name": "Ordinal.CNF",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "@[simp]\ntheorem CNF_zero (b : Ordinal) : CNF b 0 = []",
"end": [
82,
20
],
"full_name": "Ordinal.CNF_zero",
"kind": "commanddeclaration",
"start": [
80,
1
]
},
{
"code": "theorem CNF_ne_zero {b o : Ordinal} (ho : o ≠ 0) :\n CNF b o = (log b o, o / b ^ log b o)::CNF b (o % b ^ log b o)",
"end": [
89,
22
],
"full_name": "Ordinal.CNF_ne_zero",
"kind": "commanddeclaration",
"start": [
86,
1
]
},
{
"code": "theorem zero_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 0 o = [⟨0, o⟩]",
"end": [
93,
93
],
"full_name": "Ordinal.zero_CNF",
"kind": "commanddeclaration",
"start": [
93,
1
]
},
{
"code": "theorem one_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 1 o = [⟨0, o⟩]",
"end": [
97,
92
],
"full_name": "Ordinal.one_CNF",
"kind": "commanddeclaration",
"start": [
97,
1
]
},
{
"code": "theorem CNF_of_le_one {b o : Ordinal} (hb : b ≤ 1) (ho : o ≠ 0) : CNF b o = [⟨0, o⟩]",
"end": [
104,
21
],
"full_name": "Ordinal.CNF_of_le_one",
"kind": "commanddeclaration",
"start": [
101,
1
]
},
{
"code": "theorem CNF_of_lt {b o : Ordinal} (ho : o ≠ 0) (hb : o < b) : CNF b o = [⟨0, o⟩]",
"end": [
109,
84
],
"full_name": "Ordinal.CNF_of_lt",
"kind": "commanddeclaration",
"start": [
108,
1
]
},
{
"code": "theorem CNF_foldr (b o : Ordinal) : (CNF b o).foldr (fun p r ↦ b ^ p.1 * p.2 + r) 0 = o",
"end": [
116,
74
],
"full_name": "Ordinal.CNF_foldr",
"kind": "commanddeclaration",
"start": [
113,
1
]
},
{
"code": "theorem CNF_fst_le_log {b o : Ordinal.{u}} {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.1 ≤ log b o",
"end": [
129,
74
],
"full_name": "Ordinal.CNF_fst_le_log",
"kind": "commanddeclaration",
"start": [
120,
1
]
},
{
"code": "theorem CNF_fst_le {b o : Ordinal.{u}} {x : Ordinal × Ordinal} (h : x ∈ CNF b o) : x.1 ≤ o",
"end": [
135,
46
],
"full_name": "Ordinal.CNF_fst_le",
"kind": "commanddeclaration",
"start": [
133,
1
]
},
{
"code": "theorem CNF_lt_snd {b o : Ordinal.{u}} {x : Ordinal × Ordinal} : x ∈ CNF b o → 0 < x.2",
"end": [
145,
15
],
"full_name": "Ordinal.CNF_lt_snd",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem CNF_snd_lt {b o : Ordinal.{u}} (hb : 1 < b) {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.2 < b",
"end": [
158,
17
],
"full_name": "Ordinal.CNF_snd_lt",
"kind": "commanddeclaration",
"start": [
149,
1
]
},
{
"code": "theorem CNF_sorted (b o : Ordinal) : ((CNF b o).map Prod.fst).Sorted (· > ·)",
"end": [
174,
83
],
"full_name": "Ordinal.CNF_sorted",
"kind": "commanddeclaration",
"start": [
162,
1
]
}
] | 6 | Ordinal.CNF_of_le_one | [
[
101,
89
],
[
104,
21
]
] | 1 | 5 | · exact zero_CNF ho | case inl
o : Ordinal.{u_1}
ho : o ≠ 0
hb : 0 ≤ 1
⊢ CNF 0 o = [(0, o)]
case inr
o : Ordinal.{u_1}
ho : o ≠ 0
hb : 1 ≤ 1
⊢ CNF 1 o = [(0, o)] | case inr
o : Ordinal.{u_1}
ho : o ≠ 0
hb : 1 ≤ 1
⊢ CNF 1 o = [(0, o)] |
Mathlib/SetTheory/Ordinal/CantorNormalForm.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Mathlib.SetTheory.Ordinal.Exponential",
"Mathlib/SetTheory/Ordinal/Exponential.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "@[elab_as_elim]\nnoncomputable def CNFRec (b : Ordinal) {C : Ordinal → Sort*} (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : ∀ o, C o := fun o ↦ by\n by_cases h : o = 0\n · rw [h]; exact H0\n · exact H o h (CNFRec _ H0 H (o % b ^ log b o))\n termination_by o => o\n decreasing_by exact mod_opow_log_lt_self b h",
"end": [
50,
49
],
"full_name": "Ordinal.CNFRec",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem CNFRec_zero {C : Ordinal → Sort*} (b : Ordinal) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : @CNFRec b C H0 H 0 = H0",
"end": [
58,
6
],
"full_name": "Ordinal.CNFRec_zero",
"kind": "commanddeclaration",
"start": [
54,
1
]
},
{
"code": "theorem CNFRec_pos (b : Ordinal) {o : Ordinal} {C : Ordinal → Sort*} (ho : o ≠ 0) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) :\n @CNFRec b C H0 H o = H o ho (@CNFRec b C H0 H _)",
"end": [
64,
83
],
"full_name": "Ordinal.CNFRec_pos",
"kind": "commanddeclaration",
"start": [
62,
1
]
},
{
"code": "@[pp_nodot]\ndef CNF (b o : Ordinal) : List (Ordinal × Ordinal) :=\n CNFRec b [] (fun o _ho IH ↦ (log b o, o / b ^ log b o)::IH) o",
"end": [
76,
64
],
"full_name": "Ordinal.CNF",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "@[simp]\ntheorem CNF_zero (b : Ordinal) : CNF b 0 = []",
"end": [
82,
20
],
"full_name": "Ordinal.CNF_zero",
"kind": "commanddeclaration",
"start": [
80,
1
]
},
{
"code": "theorem CNF_ne_zero {b o : Ordinal} (ho : o ≠ 0) :\n CNF b o = (log b o, o / b ^ log b o)::CNF b (o % b ^ log b o)",
"end": [
89,
22
],
"full_name": "Ordinal.CNF_ne_zero",
"kind": "commanddeclaration",
"start": [
86,
1
]
},
{
"code": "theorem zero_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 0 o = [⟨0, o⟩]",
"end": [
93,
93
],
"full_name": "Ordinal.zero_CNF",
"kind": "commanddeclaration",
"start": [
93,
1
]
},
{
"code": "theorem one_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 1 o = [⟨0, o⟩]",
"end": [
97,
92
],
"full_name": "Ordinal.one_CNF",
"kind": "commanddeclaration",
"start": [
97,
1
]
},
{
"code": "theorem CNF_of_le_one {b o : Ordinal} (hb : b ≤ 1) (ho : o ≠ 0) : CNF b o = [⟨0, o⟩]",
"end": [
104,
21
],
"full_name": "Ordinal.CNF_of_le_one",
"kind": "commanddeclaration",
"start": [
101,
1
]
},
{
"code": "theorem CNF_of_lt {b o : Ordinal} (ho : o ≠ 0) (hb : o < b) : CNF b o = [⟨0, o⟩]",
"end": [
109,
84
],
"full_name": "Ordinal.CNF_of_lt",
"kind": "commanddeclaration",
"start": [
108,
1
]
},
{
"code": "theorem CNF_foldr (b o : Ordinal) : (CNF b o).foldr (fun p r ↦ b ^ p.1 * p.2 + r) 0 = o",
"end": [
116,
74
],
"full_name": "Ordinal.CNF_foldr",
"kind": "commanddeclaration",
"start": [
113,
1
]
},
{
"code": "theorem CNF_fst_le_log {b o : Ordinal.{u}} {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.1 ≤ log b o",
"end": [
129,
74
],
"full_name": "Ordinal.CNF_fst_le_log",
"kind": "commanddeclaration",
"start": [
120,
1
]
},
{
"code": "theorem CNF_fst_le {b o : Ordinal.{u}} {x : Ordinal × Ordinal} (h : x ∈ CNF b o) : x.1 ≤ o",
"end": [
135,
46
],
"full_name": "Ordinal.CNF_fst_le",
"kind": "commanddeclaration",
"start": [
133,
1
]
},
{
"code": "theorem CNF_lt_snd {b o : Ordinal.{u}} {x : Ordinal × Ordinal} : x ∈ CNF b o → 0 < x.2",
"end": [
145,
15
],
"full_name": "Ordinal.CNF_lt_snd",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem CNF_snd_lt {b o : Ordinal.{u}} (hb : 1 < b) {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.2 < b",
"end": [
158,
17
],
"full_name": "Ordinal.CNF_snd_lt",
"kind": "commanddeclaration",
"start": [
149,
1
]
},
{
"code": "theorem CNF_sorted (b o : Ordinal) : ((CNF b o).map Prod.fst).Sorted (· > ·)",
"end": [
174,
83
],
"full_name": "Ordinal.CNF_sorted",
"kind": "commanddeclaration",
"start": [
162,
1
]
}
] | 6 | Ordinal.CNF_of_le_one | [
[
101,
89
],
[
104,
21
]
] | 2 | 5 | · exact one_CNF ho | case inr
o : Ordinal.{u_1}
ho : o ≠ 0
hb : 1 ≤ 1
⊢ CNF 1 o = [(0, o)] | no goals |
Mathlib/SetTheory/Ordinal/CantorNormalForm.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Mathlib.SetTheory.Ordinal.Exponential",
"Mathlib/SetTheory/Ordinal/Exponential.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "@[elab_as_elim]\nnoncomputable def CNFRec (b : Ordinal) {C : Ordinal → Sort*} (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : ∀ o, C o := fun o ↦ by\n by_cases h : o = 0\n · rw [h]; exact H0\n · exact H o h (CNFRec _ H0 H (o % b ^ log b o))\n termination_by o => o\n decreasing_by exact mod_opow_log_lt_self b h",
"end": [
50,
49
],
"full_name": "Ordinal.CNFRec",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem CNFRec_zero {C : Ordinal → Sort*} (b : Ordinal) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : @CNFRec b C H0 H 0 = H0",
"end": [
58,
6
],
"full_name": "Ordinal.CNFRec_zero",
"kind": "commanddeclaration",
"start": [
54,
1
]
},
{
"code": "theorem CNFRec_pos (b : Ordinal) {o : Ordinal} {C : Ordinal → Sort*} (ho : o ≠ 0) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) :\n @CNFRec b C H0 H o = H o ho (@CNFRec b C H0 H _)",
"end": [
64,
83
],
"full_name": "Ordinal.CNFRec_pos",
"kind": "commanddeclaration",
"start": [
62,
1
]
},
{
"code": "@[pp_nodot]\ndef CNF (b o : Ordinal) : List (Ordinal × Ordinal) :=\n CNFRec b [] (fun o _ho IH ↦ (log b o, o / b ^ log b o)::IH) o",
"end": [
76,
64
],
"full_name": "Ordinal.CNF",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "@[simp]\ntheorem CNF_zero (b : Ordinal) : CNF b 0 = []",
"end": [
82,
20
],
"full_name": "Ordinal.CNF_zero",
"kind": "commanddeclaration",
"start": [
80,
1
]
},
{
"code": "theorem CNF_ne_zero {b o : Ordinal} (ho : o ≠ 0) :\n CNF b o = (log b o, o / b ^ log b o)::CNF b (o % b ^ log b o)",
"end": [
89,
22
],
"full_name": "Ordinal.CNF_ne_zero",
"kind": "commanddeclaration",
"start": [
86,
1
]
},
{
"code": "theorem zero_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 0 o = [⟨0, o⟩]",
"end": [
93,
93
],
"full_name": "Ordinal.zero_CNF",
"kind": "commanddeclaration",
"start": [
93,
1
]
},
{
"code": "theorem one_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 1 o = [⟨0, o⟩]",
"end": [
97,
92
],
"full_name": "Ordinal.one_CNF",
"kind": "commanddeclaration",
"start": [
97,
1
]
},
{
"code": "theorem CNF_of_le_one {b o : Ordinal} (hb : b ≤ 1) (ho : o ≠ 0) : CNF b o = [⟨0, o⟩]",
"end": [
104,
21
],
"full_name": "Ordinal.CNF_of_le_one",
"kind": "commanddeclaration",
"start": [
101,
1
]
},
{
"code": "theorem CNF_of_lt {b o : Ordinal} (ho : o ≠ 0) (hb : o < b) : CNF b o = [⟨0, o⟩]",
"end": [
109,
84
],
"full_name": "Ordinal.CNF_of_lt",
"kind": "commanddeclaration",
"start": [
108,
1
]
},
{
"code": "theorem CNF_foldr (b o : Ordinal) : (CNF b o).foldr (fun p r ↦ b ^ p.1 * p.2 + r) 0 = o",
"end": [
116,
74
],
"full_name": "Ordinal.CNF_foldr",
"kind": "commanddeclaration",
"start": [
113,
1
]
},
{
"code": "theorem CNF_fst_le_log {b o : Ordinal.{u}} {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.1 ≤ log b o",
"end": [
129,
74
],
"full_name": "Ordinal.CNF_fst_le_log",
"kind": "commanddeclaration",
"start": [
120,
1
]
},
{
"code": "theorem CNF_fst_le {b o : Ordinal.{u}} {x : Ordinal × Ordinal} (h : x ∈ CNF b o) : x.1 ≤ o",
"end": [
135,
46
],
"full_name": "Ordinal.CNF_fst_le",
"kind": "commanddeclaration",
"start": [
133,
1
]
},
{
"code": "theorem CNF_lt_snd {b o : Ordinal.{u}} {x : Ordinal × Ordinal} : x ∈ CNF b o → 0 < x.2",
"end": [
145,
15
],
"full_name": "Ordinal.CNF_lt_snd",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem CNF_snd_lt {b o : Ordinal.{u}} (hb : 1 < b) {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.2 < b",
"end": [
158,
17
],
"full_name": "Ordinal.CNF_snd_lt",
"kind": "commanddeclaration",
"start": [
149,
1
]
},
{
"code": "theorem CNF_sorted (b o : Ordinal) : ((CNF b o).map Prod.fst).Sorted (· > ·)",
"end": [
174,
83
],
"full_name": "Ordinal.CNF_sorted",
"kind": "commanddeclaration",
"start": [
162,
1
]
}
] | 6 | Ordinal.CNF_of_le_one | [
[
101,
89
],
[
104,
21
]
] | 3 | 5 | exact zero_CNF ho | case inl
o : Ordinal.{u_1}
ho : o ≠ 0
hb : 0 ≤ 1
⊢ CNF 0 o = [(0, o)] | no goals |
Mathlib/SetTheory/Ordinal/CantorNormalForm.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Mathlib.SetTheory.Ordinal.Exponential",
"Mathlib/SetTheory/Ordinal/Exponential.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "@[elab_as_elim]\nnoncomputable def CNFRec (b : Ordinal) {C : Ordinal → Sort*} (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : ∀ o, C o := fun o ↦ by\n by_cases h : o = 0\n · rw [h]; exact H0\n · exact H o h (CNFRec _ H0 H (o % b ^ log b o))\n termination_by o => o\n decreasing_by exact mod_opow_log_lt_self b h",
"end": [
50,
49
],
"full_name": "Ordinal.CNFRec",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem CNFRec_zero {C : Ordinal → Sort*} (b : Ordinal) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : @CNFRec b C H0 H 0 = H0",
"end": [
58,
6
],
"full_name": "Ordinal.CNFRec_zero",
"kind": "commanddeclaration",
"start": [
54,
1
]
},
{
"code": "theorem CNFRec_pos (b : Ordinal) {o : Ordinal} {C : Ordinal → Sort*} (ho : o ≠ 0) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) :\n @CNFRec b C H0 H o = H o ho (@CNFRec b C H0 H _)",
"end": [
64,
83
],
"full_name": "Ordinal.CNFRec_pos",
"kind": "commanddeclaration",
"start": [
62,
1
]
},
{
"code": "@[pp_nodot]\ndef CNF (b o : Ordinal) : List (Ordinal × Ordinal) :=\n CNFRec b [] (fun o _ho IH ↦ (log b o, o / b ^ log b o)::IH) o",
"end": [
76,
64
],
"full_name": "Ordinal.CNF",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "@[simp]\ntheorem CNF_zero (b : Ordinal) : CNF b 0 = []",
"end": [
82,
20
],
"full_name": "Ordinal.CNF_zero",
"kind": "commanddeclaration",
"start": [
80,
1
]
},
{
"code": "theorem CNF_ne_zero {b o : Ordinal} (ho : o ≠ 0) :\n CNF b o = (log b o, o / b ^ log b o)::CNF b (o % b ^ log b o)",
"end": [
89,
22
],
"full_name": "Ordinal.CNF_ne_zero",
"kind": "commanddeclaration",
"start": [
86,
1
]
},
{
"code": "theorem zero_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 0 o = [⟨0, o⟩]",
"end": [
93,
93
],
"full_name": "Ordinal.zero_CNF",
"kind": "commanddeclaration",
"start": [
93,
1
]
},
{
"code": "theorem one_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 1 o = [⟨0, o⟩]",
"end": [
97,
92
],
"full_name": "Ordinal.one_CNF",
"kind": "commanddeclaration",
"start": [
97,
1
]
},
{
"code": "theorem CNF_of_le_one {b o : Ordinal} (hb : b ≤ 1) (ho : o ≠ 0) : CNF b o = [⟨0, o⟩]",
"end": [
104,
21
],
"full_name": "Ordinal.CNF_of_le_one",
"kind": "commanddeclaration",
"start": [
101,
1
]
},
{
"code": "theorem CNF_of_lt {b o : Ordinal} (ho : o ≠ 0) (hb : o < b) : CNF b o = [⟨0, o⟩]",
"end": [
109,
84
],
"full_name": "Ordinal.CNF_of_lt",
"kind": "commanddeclaration",
"start": [
108,
1
]
},
{
"code": "theorem CNF_foldr (b o : Ordinal) : (CNF b o).foldr (fun p r ↦ b ^ p.1 * p.2 + r) 0 = o",
"end": [
116,
74
],
"full_name": "Ordinal.CNF_foldr",
"kind": "commanddeclaration",
"start": [
113,
1
]
},
{
"code": "theorem CNF_fst_le_log {b o : Ordinal.{u}} {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.1 ≤ log b o",
"end": [
129,
74
],
"full_name": "Ordinal.CNF_fst_le_log",
"kind": "commanddeclaration",
"start": [
120,
1
]
},
{
"code": "theorem CNF_fst_le {b o : Ordinal.{u}} {x : Ordinal × Ordinal} (h : x ∈ CNF b o) : x.1 ≤ o",
"end": [
135,
46
],
"full_name": "Ordinal.CNF_fst_le",
"kind": "commanddeclaration",
"start": [
133,
1
]
},
{
"code": "theorem CNF_lt_snd {b o : Ordinal.{u}} {x : Ordinal × Ordinal} : x ∈ CNF b o → 0 < x.2",
"end": [
145,
15
],
"full_name": "Ordinal.CNF_lt_snd",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem CNF_snd_lt {b o : Ordinal.{u}} (hb : 1 < b) {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.2 < b",
"end": [
158,
17
],
"full_name": "Ordinal.CNF_snd_lt",
"kind": "commanddeclaration",
"start": [
149,
1
]
},
{
"code": "theorem CNF_sorted (b o : Ordinal) : ((CNF b o).map Prod.fst).Sorted (· > ·)",
"end": [
174,
83
],
"full_name": "Ordinal.CNF_sorted",
"kind": "commanddeclaration",
"start": [
162,
1
]
}
] | 6 | Ordinal.CNF_of_le_one | [
[
101,
89
],
[
104,
21
]
] | 4 | 5 | exact one_CNF ho | case inr
o : Ordinal.{u_1}
ho : o ≠ 0
hb : 1 ≤ 1
⊢ CNF 1 o = [(0, o)] | no goals |
Mathlib/SetTheory/Ordinal/CantorNormalForm.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Mathlib.SetTheory.Ordinal.Exponential",
"Mathlib/SetTheory/Ordinal/Exponential.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "@[elab_as_elim]\nnoncomputable def CNFRec (b : Ordinal) {C : Ordinal → Sort*} (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : ∀ o, C o := fun o ↦ by\n by_cases h : o = 0\n · rw [h]; exact H0\n · exact H o h (CNFRec _ H0 H (o % b ^ log b o))\n termination_by o => o\n decreasing_by exact mod_opow_log_lt_self b h",
"end": [
50,
49
],
"full_name": "Ordinal.CNFRec",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem CNFRec_zero {C : Ordinal → Sort*} (b : Ordinal) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : @CNFRec b C H0 H 0 = H0",
"end": [
58,
6
],
"full_name": "Ordinal.CNFRec_zero",
"kind": "commanddeclaration",
"start": [
54,
1
]
},
{
"code": "theorem CNFRec_pos (b : Ordinal) {o : Ordinal} {C : Ordinal → Sort*} (ho : o ≠ 0) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) :\n @CNFRec b C H0 H o = H o ho (@CNFRec b C H0 H _)",
"end": [
64,
83
],
"full_name": "Ordinal.CNFRec_pos",
"kind": "commanddeclaration",
"start": [
62,
1
]
},
{
"code": "@[pp_nodot]\ndef CNF (b o : Ordinal) : List (Ordinal × Ordinal) :=\n CNFRec b [] (fun o _ho IH ↦ (log b o, o / b ^ log b o)::IH) o",
"end": [
76,
64
],
"full_name": "Ordinal.CNF",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "@[simp]\ntheorem CNF_zero (b : Ordinal) : CNF b 0 = []",
"end": [
82,
20
],
"full_name": "Ordinal.CNF_zero",
"kind": "commanddeclaration",
"start": [
80,
1
]
},
{
"code": "theorem CNF_ne_zero {b o : Ordinal} (ho : o ≠ 0) :\n CNF b o = (log b o, o / b ^ log b o)::CNF b (o % b ^ log b o)",
"end": [
89,
22
],
"full_name": "Ordinal.CNF_ne_zero",
"kind": "commanddeclaration",
"start": [
86,
1
]
},
{
"code": "theorem zero_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 0 o = [⟨0, o⟩]",
"end": [
93,
93
],
"full_name": "Ordinal.zero_CNF",
"kind": "commanddeclaration",
"start": [
93,
1
]
},
{
"code": "theorem one_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 1 o = [⟨0, o⟩]",
"end": [
97,
92
],
"full_name": "Ordinal.one_CNF",
"kind": "commanddeclaration",
"start": [
97,
1
]
},
{
"code": "theorem CNF_of_le_one {b o : Ordinal} (hb : b ≤ 1) (ho : o ≠ 0) : CNF b o = [⟨0, o⟩]",
"end": [
104,
21
],
"full_name": "Ordinal.CNF_of_le_one",
"kind": "commanddeclaration",
"start": [
101,
1
]
},
{
"code": "theorem CNF_of_lt {b o : Ordinal} (ho : o ≠ 0) (hb : o < b) : CNF b o = [⟨0, o⟩]",
"end": [
109,
84
],
"full_name": "Ordinal.CNF_of_lt",
"kind": "commanddeclaration",
"start": [
108,
1
]
},
{
"code": "theorem CNF_foldr (b o : Ordinal) : (CNF b o).foldr (fun p r ↦ b ^ p.1 * p.2 + r) 0 = o",
"end": [
116,
74
],
"full_name": "Ordinal.CNF_foldr",
"kind": "commanddeclaration",
"start": [
113,
1
]
},
{
"code": "theorem CNF_fst_le_log {b o : Ordinal.{u}} {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.1 ≤ log b o",
"end": [
129,
74
],
"full_name": "Ordinal.CNF_fst_le_log",
"kind": "commanddeclaration",
"start": [
120,
1
]
},
{
"code": "theorem CNF_fst_le {b o : Ordinal.{u}} {x : Ordinal × Ordinal} (h : x ∈ CNF b o) : x.1 ≤ o",
"end": [
135,
46
],
"full_name": "Ordinal.CNF_fst_le",
"kind": "commanddeclaration",
"start": [
133,
1
]
},
{
"code": "theorem CNF_lt_snd {b o : Ordinal.{u}} {x : Ordinal × Ordinal} : x ∈ CNF b o → 0 < x.2",
"end": [
145,
15
],
"full_name": "Ordinal.CNF_lt_snd",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem CNF_snd_lt {b o : Ordinal.{u}} (hb : 1 < b) {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.2 < b",
"end": [
158,
17
],
"full_name": "Ordinal.CNF_snd_lt",
"kind": "commanddeclaration",
"start": [
149,
1
]
},
{
"code": "theorem CNF_sorted (b o : Ordinal) : ((CNF b o).map Prod.fst).Sorted (· > ·)",
"end": [
174,
83
],
"full_name": "Ordinal.CNF_sorted",
"kind": "commanddeclaration",
"start": [
162,
1
]
}
] | 7 | Ordinal.CNF_of_lt | [
[
108,
85
],
[
109,
84
]
] | 0 | 1 | simp only [CNF_ne_zero ho, log_eq_zero hb, opow_zero, div_one, mod_one, CNF_zero] | b o : Ordinal.{u_1}
ho : o ≠ 0
hb : o < b
⊢ CNF b o = [(0, o)] | no goals |
Mathlib/SetTheory/Ordinal/CantorNormalForm.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Mathlib.SetTheory.Ordinal.Exponential",
"Mathlib/SetTheory/Ordinal/Exponential.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "@[elab_as_elim]\nnoncomputable def CNFRec (b : Ordinal) {C : Ordinal → Sort*} (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : ∀ o, C o := fun o ↦ by\n by_cases h : o = 0\n · rw [h]; exact H0\n · exact H o h (CNFRec _ H0 H (o % b ^ log b o))\n termination_by o => o\n decreasing_by exact mod_opow_log_lt_self b h",
"end": [
50,
49
],
"full_name": "Ordinal.CNFRec",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem CNFRec_zero {C : Ordinal → Sort*} (b : Ordinal) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : @CNFRec b C H0 H 0 = H0",
"end": [
58,
6
],
"full_name": "Ordinal.CNFRec_zero",
"kind": "commanddeclaration",
"start": [
54,
1
]
},
{
"code": "theorem CNFRec_pos (b : Ordinal) {o : Ordinal} {C : Ordinal → Sort*} (ho : o ≠ 0) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) :\n @CNFRec b C H0 H o = H o ho (@CNFRec b C H0 H _)",
"end": [
64,
83
],
"full_name": "Ordinal.CNFRec_pos",
"kind": "commanddeclaration",
"start": [
62,
1
]
},
{
"code": "@[pp_nodot]\ndef CNF (b o : Ordinal) : List (Ordinal × Ordinal) :=\n CNFRec b [] (fun o _ho IH ↦ (log b o, o / b ^ log b o)::IH) o",
"end": [
76,
64
],
"full_name": "Ordinal.CNF",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "@[simp]\ntheorem CNF_zero (b : Ordinal) : CNF b 0 = []",
"end": [
82,
20
],
"full_name": "Ordinal.CNF_zero",
"kind": "commanddeclaration",
"start": [
80,
1
]
},
{
"code": "theorem CNF_ne_zero {b o : Ordinal} (ho : o ≠ 0) :\n CNF b o = (log b o, o / b ^ log b o)::CNF b (o % b ^ log b o)",
"end": [
89,
22
],
"full_name": "Ordinal.CNF_ne_zero",
"kind": "commanddeclaration",
"start": [
86,
1
]
},
{
"code": "theorem zero_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 0 o = [⟨0, o⟩]",
"end": [
93,
93
],
"full_name": "Ordinal.zero_CNF",
"kind": "commanddeclaration",
"start": [
93,
1
]
},
{
"code": "theorem one_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 1 o = [⟨0, o⟩]",
"end": [
97,
92
],
"full_name": "Ordinal.one_CNF",
"kind": "commanddeclaration",
"start": [
97,
1
]
},
{
"code": "theorem CNF_of_le_one {b o : Ordinal} (hb : b ≤ 1) (ho : o ≠ 0) : CNF b o = [⟨0, o⟩]",
"end": [
104,
21
],
"full_name": "Ordinal.CNF_of_le_one",
"kind": "commanddeclaration",
"start": [
101,
1
]
},
{
"code": "theorem CNF_of_lt {b o : Ordinal} (ho : o ≠ 0) (hb : o < b) : CNF b o = [⟨0, o⟩]",
"end": [
109,
84
],
"full_name": "Ordinal.CNF_of_lt",
"kind": "commanddeclaration",
"start": [
108,
1
]
},
{
"code": "theorem CNF_foldr (b o : Ordinal) : (CNF b o).foldr (fun p r ↦ b ^ p.1 * p.2 + r) 0 = o",
"end": [
116,
74
],
"full_name": "Ordinal.CNF_foldr",
"kind": "commanddeclaration",
"start": [
113,
1
]
},
{
"code": "theorem CNF_fst_le_log {b o : Ordinal.{u}} {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.1 ≤ log b o",
"end": [
129,
74
],
"full_name": "Ordinal.CNF_fst_le_log",
"kind": "commanddeclaration",
"start": [
120,
1
]
},
{
"code": "theorem CNF_fst_le {b o : Ordinal.{u}} {x : Ordinal × Ordinal} (h : x ∈ CNF b o) : x.1 ≤ o",
"end": [
135,
46
],
"full_name": "Ordinal.CNF_fst_le",
"kind": "commanddeclaration",
"start": [
133,
1
]
},
{
"code": "theorem CNF_lt_snd {b o : Ordinal.{u}} {x : Ordinal × Ordinal} : x ∈ CNF b o → 0 < x.2",
"end": [
145,
15
],
"full_name": "Ordinal.CNF_lt_snd",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem CNF_snd_lt {b o : Ordinal.{u}} (hb : 1 < b) {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.2 < b",
"end": [
158,
17
],
"full_name": "Ordinal.CNF_snd_lt",
"kind": "commanddeclaration",
"start": [
149,
1
]
},
{
"code": "theorem CNF_sorted (b o : Ordinal) : ((CNF b o).map Prod.fst).Sorted (· > ·)",
"end": [
174,
83
],
"full_name": "Ordinal.CNF_sorted",
"kind": "commanddeclaration",
"start": [
162,
1
]
}
] | 9 | Ordinal.CNF_fst_le_log | [
[
122,
36
],
[
129,
74
]
] | 0 | 12 | refine CNFRec b ?_ (fun o ho H ↦ ?_) o | b o : Ordinal.{u}
x : Ordinal.{u} × Ordinal.{u}
⊢ x ∈ CNF b o → x.1 ≤ log b o | case refine_1
b o : Ordinal.{u}
x : Ordinal.{u} × Ordinal.{u}
⊢ x ∈ CNF b 0 → x.1 ≤ log b 0
case refine_2
b o✝ : Ordinal.{u}
x : Ordinal.{u} × Ordinal.{u}
o : Ordinal.{u}
ho : o ≠ 0
H : x ∈ CNF b (o % b ^ log b o) → x.1 ≤ log b (o % b ^ log b o)
⊢ x ∈ CNF b o → x.1 ≤ log b o |
Mathlib/SetTheory/Ordinal/CantorNormalForm.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Mathlib.SetTheory.Ordinal.Exponential",
"Mathlib/SetTheory/Ordinal/Exponential.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "@[elab_as_elim]\nnoncomputable def CNFRec (b : Ordinal) {C : Ordinal → Sort*} (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : ∀ o, C o := fun o ↦ by\n by_cases h : o = 0\n · rw [h]; exact H0\n · exact H o h (CNFRec _ H0 H (o % b ^ log b o))\n termination_by o => o\n decreasing_by exact mod_opow_log_lt_self b h",
"end": [
50,
49
],
"full_name": "Ordinal.CNFRec",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem CNFRec_zero {C : Ordinal → Sort*} (b : Ordinal) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : @CNFRec b C H0 H 0 = H0",
"end": [
58,
6
],
"full_name": "Ordinal.CNFRec_zero",
"kind": "commanddeclaration",
"start": [
54,
1
]
},
{
"code": "theorem CNFRec_pos (b : Ordinal) {o : Ordinal} {C : Ordinal → Sort*} (ho : o ≠ 0) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) :\n @CNFRec b C H0 H o = H o ho (@CNFRec b C H0 H _)",
"end": [
64,
83
],
"full_name": "Ordinal.CNFRec_pos",
"kind": "commanddeclaration",
"start": [
62,
1
]
},
{
"code": "@[pp_nodot]\ndef CNF (b o : Ordinal) : List (Ordinal × Ordinal) :=\n CNFRec b [] (fun o _ho IH ↦ (log b o, o / b ^ log b o)::IH) o",
"end": [
76,
64
],
"full_name": "Ordinal.CNF",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "@[simp]\ntheorem CNF_zero (b : Ordinal) : CNF b 0 = []",
"end": [
82,
20
],
"full_name": "Ordinal.CNF_zero",
"kind": "commanddeclaration",
"start": [
80,
1
]
},
{
"code": "theorem CNF_ne_zero {b o : Ordinal} (ho : o ≠ 0) :\n CNF b o = (log b o, o / b ^ log b o)::CNF b (o % b ^ log b o)",
"end": [
89,
22
],
"full_name": "Ordinal.CNF_ne_zero",
"kind": "commanddeclaration",
"start": [
86,
1
]
},
{
"code": "theorem zero_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 0 o = [⟨0, o⟩]",
"end": [
93,
93
],
"full_name": "Ordinal.zero_CNF",
"kind": "commanddeclaration",
"start": [
93,
1
]
},
{
"code": "theorem one_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 1 o = [⟨0, o⟩]",
"end": [
97,
92
],
"full_name": "Ordinal.one_CNF",
"kind": "commanddeclaration",
"start": [
97,
1
]
},
{
"code": "theorem CNF_of_le_one {b o : Ordinal} (hb : b ≤ 1) (ho : o ≠ 0) : CNF b o = [⟨0, o⟩]",
"end": [
104,
21
],
"full_name": "Ordinal.CNF_of_le_one",
"kind": "commanddeclaration",
"start": [
101,
1
]
},
{
"code": "theorem CNF_of_lt {b o : Ordinal} (ho : o ≠ 0) (hb : o < b) : CNF b o = [⟨0, o⟩]",
"end": [
109,
84
],
"full_name": "Ordinal.CNF_of_lt",
"kind": "commanddeclaration",
"start": [
108,
1
]
},
{
"code": "theorem CNF_foldr (b o : Ordinal) : (CNF b o).foldr (fun p r ↦ b ^ p.1 * p.2 + r) 0 = o",
"end": [
116,
74
],
"full_name": "Ordinal.CNF_foldr",
"kind": "commanddeclaration",
"start": [
113,
1
]
},
{
"code": "theorem CNF_fst_le_log {b o : Ordinal.{u}} {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.1 ≤ log b o",
"end": [
129,
74
],
"full_name": "Ordinal.CNF_fst_le_log",
"kind": "commanddeclaration",
"start": [
120,
1
]
},
{
"code": "theorem CNF_fst_le {b o : Ordinal.{u}} {x : Ordinal × Ordinal} (h : x ∈ CNF b o) : x.1 ≤ o",
"end": [
135,
46
],
"full_name": "Ordinal.CNF_fst_le",
"kind": "commanddeclaration",
"start": [
133,
1
]
},
{
"code": "theorem CNF_lt_snd {b o : Ordinal.{u}} {x : Ordinal × Ordinal} : x ∈ CNF b o → 0 < x.2",
"end": [
145,
15
],
"full_name": "Ordinal.CNF_lt_snd",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem CNF_snd_lt {b o : Ordinal.{u}} (hb : 1 < b) {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.2 < b",
"end": [
158,
17
],
"full_name": "Ordinal.CNF_snd_lt",
"kind": "commanddeclaration",
"start": [
149,
1
]
},
{
"code": "theorem CNF_sorted (b o : Ordinal) : ((CNF b o).map Prod.fst).Sorted (· > ·)",
"end": [
174,
83
],
"full_name": "Ordinal.CNF_sorted",
"kind": "commanddeclaration",
"start": [
162,
1
]
}
] | 9 | Ordinal.CNF_fst_le_log | [
[
122,
36
],
[
129,
74
]
] | 1 | 12 | · rw [CNF_zero]
intro contra; contradiction | case refine_1
b o : Ordinal.{u}
x : Ordinal.{u} × Ordinal.{u}
⊢ x ∈ CNF b 0 → x.1 ≤ log b 0
case refine_2
b o✝ : Ordinal.{u}
x : Ordinal.{u} × Ordinal.{u}
o : Ordinal.{u}
ho : o ≠ 0
H : x ∈ CNF b (o % b ^ log b o) → x.1 ≤ log b (o % b ^ log b o)
⊢ x ∈ CNF b o → x.1 ≤ log b o | case refine_2
b o✝ : Ordinal.{u}
x : Ordinal.{u} × Ordinal.{u}
o : Ordinal.{u}
ho : o ≠ 0
H : x ∈ CNF b (o % b ^ log b o) → x.1 ≤ log b (o % b ^ log b o)
⊢ x ∈ CNF b o → x.1 ≤ log b o |
Mathlib/SetTheory/Ordinal/CantorNormalForm.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Mathlib.SetTheory.Ordinal.Exponential",
"Mathlib/SetTheory/Ordinal/Exponential.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "@[elab_as_elim]\nnoncomputable def CNFRec (b : Ordinal) {C : Ordinal → Sort*} (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : ∀ o, C o := fun o ↦ by\n by_cases h : o = 0\n · rw [h]; exact H0\n · exact H o h (CNFRec _ H0 H (o % b ^ log b o))\n termination_by o => o\n decreasing_by exact mod_opow_log_lt_self b h",
"end": [
50,
49
],
"full_name": "Ordinal.CNFRec",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem CNFRec_zero {C : Ordinal → Sort*} (b : Ordinal) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : @CNFRec b C H0 H 0 = H0",
"end": [
58,
6
],
"full_name": "Ordinal.CNFRec_zero",
"kind": "commanddeclaration",
"start": [
54,
1
]
},
{
"code": "theorem CNFRec_pos (b : Ordinal) {o : Ordinal} {C : Ordinal → Sort*} (ho : o ≠ 0) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) :\n @CNFRec b C H0 H o = H o ho (@CNFRec b C H0 H _)",
"end": [
64,
83
],
"full_name": "Ordinal.CNFRec_pos",
"kind": "commanddeclaration",
"start": [
62,
1
]
},
{
"code": "@[pp_nodot]\ndef CNF (b o : Ordinal) : List (Ordinal × Ordinal) :=\n CNFRec b [] (fun o _ho IH ↦ (log b o, o / b ^ log b o)::IH) o",
"end": [
76,
64
],
"full_name": "Ordinal.CNF",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "@[simp]\ntheorem CNF_zero (b : Ordinal) : CNF b 0 = []",
"end": [
82,
20
],
"full_name": "Ordinal.CNF_zero",
"kind": "commanddeclaration",
"start": [
80,
1
]
},
{
"code": "theorem CNF_ne_zero {b o : Ordinal} (ho : o ≠ 0) :\n CNF b o = (log b o, o / b ^ log b o)::CNF b (o % b ^ log b o)",
"end": [
89,
22
],
"full_name": "Ordinal.CNF_ne_zero",
"kind": "commanddeclaration",
"start": [
86,
1
]
},
{
"code": "theorem zero_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 0 o = [⟨0, o⟩]",
"end": [
93,
93
],
"full_name": "Ordinal.zero_CNF",
"kind": "commanddeclaration",
"start": [
93,
1
]
},
{
"code": "theorem one_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 1 o = [⟨0, o⟩]",
"end": [
97,
92
],
"full_name": "Ordinal.one_CNF",
"kind": "commanddeclaration",
"start": [
97,
1
]
},
{
"code": "theorem CNF_of_le_one {b o : Ordinal} (hb : b ≤ 1) (ho : o ≠ 0) : CNF b o = [⟨0, o⟩]",
"end": [
104,
21
],
"full_name": "Ordinal.CNF_of_le_one",
"kind": "commanddeclaration",
"start": [
101,
1
]
},
{
"code": "theorem CNF_of_lt {b o : Ordinal} (ho : o ≠ 0) (hb : o < b) : CNF b o = [⟨0, o⟩]",
"end": [
109,
84
],
"full_name": "Ordinal.CNF_of_lt",
"kind": "commanddeclaration",
"start": [
108,
1
]
},
{
"code": "theorem CNF_foldr (b o : Ordinal) : (CNF b o).foldr (fun p r ↦ b ^ p.1 * p.2 + r) 0 = o",
"end": [
116,
74
],
"full_name": "Ordinal.CNF_foldr",
"kind": "commanddeclaration",
"start": [
113,
1
]
},
{
"code": "theorem CNF_fst_le_log {b o : Ordinal.{u}} {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.1 ≤ log b o",
"end": [
129,
74
],
"full_name": "Ordinal.CNF_fst_le_log",
"kind": "commanddeclaration",
"start": [
120,
1
]
},
{
"code": "theorem CNF_fst_le {b o : Ordinal.{u}} {x : Ordinal × Ordinal} (h : x ∈ CNF b o) : x.1 ≤ o",
"end": [
135,
46
],
"full_name": "Ordinal.CNF_fst_le",
"kind": "commanddeclaration",
"start": [
133,
1
]
},
{
"code": "theorem CNF_lt_snd {b o : Ordinal.{u}} {x : Ordinal × Ordinal} : x ∈ CNF b o → 0 < x.2",
"end": [
145,
15
],
"full_name": "Ordinal.CNF_lt_snd",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem CNF_snd_lt {b o : Ordinal.{u}} (hb : 1 < b) {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.2 < b",
"end": [
158,
17
],
"full_name": "Ordinal.CNF_snd_lt",
"kind": "commanddeclaration",
"start": [
149,
1
]
},
{
"code": "theorem CNF_sorted (b o : Ordinal) : ((CNF b o).map Prod.fst).Sorted (· > ·)",
"end": [
174,
83
],
"full_name": "Ordinal.CNF_sorted",
"kind": "commanddeclaration",
"start": [
162,
1
]
}
] | 9 | Ordinal.CNF_fst_le_log | [
[
122,
36
],
[
129,
74
]
] | 2 | 12 | · rw [CNF_ne_zero ho, mem_cons]
rintro (rfl | h)
· exact le_rfl
· exact (H h).trans (log_mono_right _ (mod_opow_log_lt_self b ho).le) | case refine_2
b o✝ : Ordinal.{u}
x : Ordinal.{u} × Ordinal.{u}
o : Ordinal.{u}
ho : o ≠ 0
H : x ∈ CNF b (o % b ^ log b o) → x.1 ≤ log b (o % b ^ log b o)
⊢ x ∈ CNF b o → x.1 ≤ log b o | no goals |
Mathlib/SetTheory/Ordinal/CantorNormalForm.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Mathlib.SetTheory.Ordinal.Exponential",
"Mathlib/SetTheory/Ordinal/Exponential.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "@[elab_as_elim]\nnoncomputable def CNFRec (b : Ordinal) {C : Ordinal → Sort*} (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : ∀ o, C o := fun o ↦ by\n by_cases h : o = 0\n · rw [h]; exact H0\n · exact H o h (CNFRec _ H0 H (o % b ^ log b o))\n termination_by o => o\n decreasing_by exact mod_opow_log_lt_self b h",
"end": [
50,
49
],
"full_name": "Ordinal.CNFRec",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem CNFRec_zero {C : Ordinal → Sort*} (b : Ordinal) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : @CNFRec b C H0 H 0 = H0",
"end": [
58,
6
],
"full_name": "Ordinal.CNFRec_zero",
"kind": "commanddeclaration",
"start": [
54,
1
]
},
{
"code": "theorem CNFRec_pos (b : Ordinal) {o : Ordinal} {C : Ordinal → Sort*} (ho : o ≠ 0) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) :\n @CNFRec b C H0 H o = H o ho (@CNFRec b C H0 H _)",
"end": [
64,
83
],
"full_name": "Ordinal.CNFRec_pos",
"kind": "commanddeclaration",
"start": [
62,
1
]
},
{
"code": "@[pp_nodot]\ndef CNF (b o : Ordinal) : List (Ordinal × Ordinal) :=\n CNFRec b [] (fun o _ho IH ↦ (log b o, o / b ^ log b o)::IH) o",
"end": [
76,
64
],
"full_name": "Ordinal.CNF",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "@[simp]\ntheorem CNF_zero (b : Ordinal) : CNF b 0 = []",
"end": [
82,
20
],
"full_name": "Ordinal.CNF_zero",
"kind": "commanddeclaration",
"start": [
80,
1
]
},
{
"code": "theorem CNF_ne_zero {b o : Ordinal} (ho : o ≠ 0) :\n CNF b o = (log b o, o / b ^ log b o)::CNF b (o % b ^ log b o)",
"end": [
89,
22
],
"full_name": "Ordinal.CNF_ne_zero",
"kind": "commanddeclaration",
"start": [
86,
1
]
},
{
"code": "theorem zero_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 0 o = [⟨0, o⟩]",
"end": [
93,
93
],
"full_name": "Ordinal.zero_CNF",
"kind": "commanddeclaration",
"start": [
93,
1
]
},
{
"code": "theorem one_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 1 o = [⟨0, o⟩]",
"end": [
97,
92
],
"full_name": "Ordinal.one_CNF",
"kind": "commanddeclaration",
"start": [
97,
1
]
},
{
"code": "theorem CNF_of_le_one {b o : Ordinal} (hb : b ≤ 1) (ho : o ≠ 0) : CNF b o = [⟨0, o⟩]",
"end": [
104,
21
],
"full_name": "Ordinal.CNF_of_le_one",
"kind": "commanddeclaration",
"start": [
101,
1
]
},
{
"code": "theorem CNF_of_lt {b o : Ordinal} (ho : o ≠ 0) (hb : o < b) : CNF b o = [⟨0, o⟩]",
"end": [
109,
84
],
"full_name": "Ordinal.CNF_of_lt",
"kind": "commanddeclaration",
"start": [
108,
1
]
},
{
"code": "theorem CNF_foldr (b o : Ordinal) : (CNF b o).foldr (fun p r ↦ b ^ p.1 * p.2 + r) 0 = o",
"end": [
116,
74
],
"full_name": "Ordinal.CNF_foldr",
"kind": "commanddeclaration",
"start": [
113,
1
]
},
{
"code": "theorem CNF_fst_le_log {b o : Ordinal.{u}} {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.1 ≤ log b o",
"end": [
129,
74
],
"full_name": "Ordinal.CNF_fst_le_log",
"kind": "commanddeclaration",
"start": [
120,
1
]
},
{
"code": "theorem CNF_fst_le {b o : Ordinal.{u}} {x : Ordinal × Ordinal} (h : x ∈ CNF b o) : x.1 ≤ o",
"end": [
135,
46
],
"full_name": "Ordinal.CNF_fst_le",
"kind": "commanddeclaration",
"start": [
133,
1
]
},
{
"code": "theorem CNF_lt_snd {b o : Ordinal.{u}} {x : Ordinal × Ordinal} : x ∈ CNF b o → 0 < x.2",
"end": [
145,
15
],
"full_name": "Ordinal.CNF_lt_snd",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem CNF_snd_lt {b o : Ordinal.{u}} (hb : 1 < b) {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.2 < b",
"end": [
158,
17
],
"full_name": "Ordinal.CNF_snd_lt",
"kind": "commanddeclaration",
"start": [
149,
1
]
},
{
"code": "theorem CNF_sorted (b o : Ordinal) : ((CNF b o).map Prod.fst).Sorted (· > ·)",
"end": [
174,
83
],
"full_name": "Ordinal.CNF_sorted",
"kind": "commanddeclaration",
"start": [
162,
1
]
}
] | 9 | Ordinal.CNF_fst_le_log | [
[
122,
36
],
[
129,
74
]
] | 3 | 12 | rw [CNF_zero] | case refine_1
b o : Ordinal.{u}
x : Ordinal.{u} × Ordinal.{u}
⊢ x ∈ CNF b 0 → x.1 ≤ log b 0 | case refine_1
b o : Ordinal.{u}
x : Ordinal.{u} × Ordinal.{u}
⊢ x ∈ [] → x.1 ≤ log b 0 |
Mathlib/SetTheory/Ordinal/CantorNormalForm.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Mathlib.SetTheory.Ordinal.Exponential",
"Mathlib/SetTheory/Ordinal/Exponential.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "@[elab_as_elim]\nnoncomputable def CNFRec (b : Ordinal) {C : Ordinal → Sort*} (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : ∀ o, C o := fun o ↦ by\n by_cases h : o = 0\n · rw [h]; exact H0\n · exact H o h (CNFRec _ H0 H (o % b ^ log b o))\n termination_by o => o\n decreasing_by exact mod_opow_log_lt_self b h",
"end": [
50,
49
],
"full_name": "Ordinal.CNFRec",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem CNFRec_zero {C : Ordinal → Sort*} (b : Ordinal) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : @CNFRec b C H0 H 0 = H0",
"end": [
58,
6
],
"full_name": "Ordinal.CNFRec_zero",
"kind": "commanddeclaration",
"start": [
54,
1
]
},
{
"code": "theorem CNFRec_pos (b : Ordinal) {o : Ordinal} {C : Ordinal → Sort*} (ho : o ≠ 0) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) :\n @CNFRec b C H0 H o = H o ho (@CNFRec b C H0 H _)",
"end": [
64,
83
],
"full_name": "Ordinal.CNFRec_pos",
"kind": "commanddeclaration",
"start": [
62,
1
]
},
{
"code": "@[pp_nodot]\ndef CNF (b o : Ordinal) : List (Ordinal × Ordinal) :=\n CNFRec b [] (fun o _ho IH ↦ (log b o, o / b ^ log b o)::IH) o",
"end": [
76,
64
],
"full_name": "Ordinal.CNF",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "@[simp]\ntheorem CNF_zero (b : Ordinal) : CNF b 0 = []",
"end": [
82,
20
],
"full_name": "Ordinal.CNF_zero",
"kind": "commanddeclaration",
"start": [
80,
1
]
},
{
"code": "theorem CNF_ne_zero {b o : Ordinal} (ho : o ≠ 0) :\n CNF b o = (log b o, o / b ^ log b o)::CNF b (o % b ^ log b o)",
"end": [
89,
22
],
"full_name": "Ordinal.CNF_ne_zero",
"kind": "commanddeclaration",
"start": [
86,
1
]
},
{
"code": "theorem zero_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 0 o = [⟨0, o⟩]",
"end": [
93,
93
],
"full_name": "Ordinal.zero_CNF",
"kind": "commanddeclaration",
"start": [
93,
1
]
},
{
"code": "theorem one_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 1 o = [⟨0, o⟩]",
"end": [
97,
92
],
"full_name": "Ordinal.one_CNF",
"kind": "commanddeclaration",
"start": [
97,
1
]
},
{
"code": "theorem CNF_of_le_one {b o : Ordinal} (hb : b ≤ 1) (ho : o ≠ 0) : CNF b o = [⟨0, o⟩]",
"end": [
104,
21
],
"full_name": "Ordinal.CNF_of_le_one",
"kind": "commanddeclaration",
"start": [
101,
1
]
},
{
"code": "theorem CNF_of_lt {b o : Ordinal} (ho : o ≠ 0) (hb : o < b) : CNF b o = [⟨0, o⟩]",
"end": [
109,
84
],
"full_name": "Ordinal.CNF_of_lt",
"kind": "commanddeclaration",
"start": [
108,
1
]
},
{
"code": "theorem CNF_foldr (b o : Ordinal) : (CNF b o).foldr (fun p r ↦ b ^ p.1 * p.2 + r) 0 = o",
"end": [
116,
74
],
"full_name": "Ordinal.CNF_foldr",
"kind": "commanddeclaration",
"start": [
113,
1
]
},
{
"code": "theorem CNF_fst_le_log {b o : Ordinal.{u}} {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.1 ≤ log b o",
"end": [
129,
74
],
"full_name": "Ordinal.CNF_fst_le_log",
"kind": "commanddeclaration",
"start": [
120,
1
]
},
{
"code": "theorem CNF_fst_le {b o : Ordinal.{u}} {x : Ordinal × Ordinal} (h : x ∈ CNF b o) : x.1 ≤ o",
"end": [
135,
46
],
"full_name": "Ordinal.CNF_fst_le",
"kind": "commanddeclaration",
"start": [
133,
1
]
},
{
"code": "theorem CNF_lt_snd {b o : Ordinal.{u}} {x : Ordinal × Ordinal} : x ∈ CNF b o → 0 < x.2",
"end": [
145,
15
],
"full_name": "Ordinal.CNF_lt_snd",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem CNF_snd_lt {b o : Ordinal.{u}} (hb : 1 < b) {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.2 < b",
"end": [
158,
17
],
"full_name": "Ordinal.CNF_snd_lt",
"kind": "commanddeclaration",
"start": [
149,
1
]
},
{
"code": "theorem CNF_sorted (b o : Ordinal) : ((CNF b o).map Prod.fst).Sorted (· > ·)",
"end": [
174,
83
],
"full_name": "Ordinal.CNF_sorted",
"kind": "commanddeclaration",
"start": [
162,
1
]
}
] | 9 | Ordinal.CNF_fst_le_log | [
[
122,
36
],
[
129,
74
]
] | 4 | 12 | intro contra | case refine_1
b o : Ordinal.{u}
x : Ordinal.{u} × Ordinal.{u}
⊢ x ∈ [] → x.1 ≤ log b 0 | case refine_1
b o : Ordinal.{u}
x : Ordinal.{u} × Ordinal.{u}
contra : x ∈ []
⊢ x.1 ≤ log b 0 |
Mathlib/SetTheory/Ordinal/CantorNormalForm.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Mathlib.SetTheory.Ordinal.Exponential",
"Mathlib/SetTheory/Ordinal/Exponential.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "@[elab_as_elim]\nnoncomputable def CNFRec (b : Ordinal) {C : Ordinal → Sort*} (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : ∀ o, C o := fun o ↦ by\n by_cases h : o = 0\n · rw [h]; exact H0\n · exact H o h (CNFRec _ H0 H (o % b ^ log b o))\n termination_by o => o\n decreasing_by exact mod_opow_log_lt_self b h",
"end": [
50,
49
],
"full_name": "Ordinal.CNFRec",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem CNFRec_zero {C : Ordinal → Sort*} (b : Ordinal) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : @CNFRec b C H0 H 0 = H0",
"end": [
58,
6
],
"full_name": "Ordinal.CNFRec_zero",
"kind": "commanddeclaration",
"start": [
54,
1
]
},
{
"code": "theorem CNFRec_pos (b : Ordinal) {o : Ordinal} {C : Ordinal → Sort*} (ho : o ≠ 0) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) :\n @CNFRec b C H0 H o = H o ho (@CNFRec b C H0 H _)",
"end": [
64,
83
],
"full_name": "Ordinal.CNFRec_pos",
"kind": "commanddeclaration",
"start": [
62,
1
]
},
{
"code": "@[pp_nodot]\ndef CNF (b o : Ordinal) : List (Ordinal × Ordinal) :=\n CNFRec b [] (fun o _ho IH ↦ (log b o, o / b ^ log b o)::IH) o",
"end": [
76,
64
],
"full_name": "Ordinal.CNF",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "@[simp]\ntheorem CNF_zero (b : Ordinal) : CNF b 0 = []",
"end": [
82,
20
],
"full_name": "Ordinal.CNF_zero",
"kind": "commanddeclaration",
"start": [
80,
1
]
},
{
"code": "theorem CNF_ne_zero {b o : Ordinal} (ho : o ≠ 0) :\n CNF b o = (log b o, o / b ^ log b o)::CNF b (o % b ^ log b o)",
"end": [
89,
22
],
"full_name": "Ordinal.CNF_ne_zero",
"kind": "commanddeclaration",
"start": [
86,
1
]
},
{
"code": "theorem zero_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 0 o = [⟨0, o⟩]",
"end": [
93,
93
],
"full_name": "Ordinal.zero_CNF",
"kind": "commanddeclaration",
"start": [
93,
1
]
},
{
"code": "theorem one_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 1 o = [⟨0, o⟩]",
"end": [
97,
92
],
"full_name": "Ordinal.one_CNF",
"kind": "commanddeclaration",
"start": [
97,
1
]
},
{
"code": "theorem CNF_of_le_one {b o : Ordinal} (hb : b ≤ 1) (ho : o ≠ 0) : CNF b o = [⟨0, o⟩]",
"end": [
104,
21
],
"full_name": "Ordinal.CNF_of_le_one",
"kind": "commanddeclaration",
"start": [
101,
1
]
},
{
"code": "theorem CNF_of_lt {b o : Ordinal} (ho : o ≠ 0) (hb : o < b) : CNF b o = [⟨0, o⟩]",
"end": [
109,
84
],
"full_name": "Ordinal.CNF_of_lt",
"kind": "commanddeclaration",
"start": [
108,
1
]
},
{
"code": "theorem CNF_foldr (b o : Ordinal) : (CNF b o).foldr (fun p r ↦ b ^ p.1 * p.2 + r) 0 = o",
"end": [
116,
74
],
"full_name": "Ordinal.CNF_foldr",
"kind": "commanddeclaration",
"start": [
113,
1
]
},
{
"code": "theorem CNF_fst_le_log {b o : Ordinal.{u}} {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.1 ≤ log b o",
"end": [
129,
74
],
"full_name": "Ordinal.CNF_fst_le_log",
"kind": "commanddeclaration",
"start": [
120,
1
]
},
{
"code": "theorem CNF_fst_le {b o : Ordinal.{u}} {x : Ordinal × Ordinal} (h : x ∈ CNF b o) : x.1 ≤ o",
"end": [
135,
46
],
"full_name": "Ordinal.CNF_fst_le",
"kind": "commanddeclaration",
"start": [
133,
1
]
},
{
"code": "theorem CNF_lt_snd {b o : Ordinal.{u}} {x : Ordinal × Ordinal} : x ∈ CNF b o → 0 < x.2",
"end": [
145,
15
],
"full_name": "Ordinal.CNF_lt_snd",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem CNF_snd_lt {b o : Ordinal.{u}} (hb : 1 < b) {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.2 < b",
"end": [
158,
17
],
"full_name": "Ordinal.CNF_snd_lt",
"kind": "commanddeclaration",
"start": [
149,
1
]
},
{
"code": "theorem CNF_sorted (b o : Ordinal) : ((CNF b o).map Prod.fst).Sorted (· > ·)",
"end": [
174,
83
],
"full_name": "Ordinal.CNF_sorted",
"kind": "commanddeclaration",
"start": [
162,
1
]
}
] | 9 | Ordinal.CNF_fst_le_log | [
[
122,
36
],
[
129,
74
]
] | 5 | 12 | contradiction | case refine_1
b o : Ordinal.{u}
x : Ordinal.{u} × Ordinal.{u}
contra : x ∈ []
⊢ x.1 ≤ log b 0 | no goals |
Mathlib/SetTheory/Ordinal/CantorNormalForm.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Mathlib.SetTheory.Ordinal.Exponential",
"Mathlib/SetTheory/Ordinal/Exponential.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "@[elab_as_elim]\nnoncomputable def CNFRec (b : Ordinal) {C : Ordinal → Sort*} (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : ∀ o, C o := fun o ↦ by\n by_cases h : o = 0\n · rw [h]; exact H0\n · exact H o h (CNFRec _ H0 H (o % b ^ log b o))\n termination_by o => o\n decreasing_by exact mod_opow_log_lt_self b h",
"end": [
50,
49
],
"full_name": "Ordinal.CNFRec",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem CNFRec_zero {C : Ordinal → Sort*} (b : Ordinal) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : @CNFRec b C H0 H 0 = H0",
"end": [
58,
6
],
"full_name": "Ordinal.CNFRec_zero",
"kind": "commanddeclaration",
"start": [
54,
1
]
},
{
"code": "theorem CNFRec_pos (b : Ordinal) {o : Ordinal} {C : Ordinal → Sort*} (ho : o ≠ 0) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) :\n @CNFRec b C H0 H o = H o ho (@CNFRec b C H0 H _)",
"end": [
64,
83
],
"full_name": "Ordinal.CNFRec_pos",
"kind": "commanddeclaration",
"start": [
62,
1
]
},
{
"code": "@[pp_nodot]\ndef CNF (b o : Ordinal) : List (Ordinal × Ordinal) :=\n CNFRec b [] (fun o _ho IH ↦ (log b o, o / b ^ log b o)::IH) o",
"end": [
76,
64
],
"full_name": "Ordinal.CNF",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "@[simp]\ntheorem CNF_zero (b : Ordinal) : CNF b 0 = []",
"end": [
82,
20
],
"full_name": "Ordinal.CNF_zero",
"kind": "commanddeclaration",
"start": [
80,
1
]
},
{
"code": "theorem CNF_ne_zero {b o : Ordinal} (ho : o ≠ 0) :\n CNF b o = (log b o, o / b ^ log b o)::CNF b (o % b ^ log b o)",
"end": [
89,
22
],
"full_name": "Ordinal.CNF_ne_zero",
"kind": "commanddeclaration",
"start": [
86,
1
]
},
{
"code": "theorem zero_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 0 o = [⟨0, o⟩]",
"end": [
93,
93
],
"full_name": "Ordinal.zero_CNF",
"kind": "commanddeclaration",
"start": [
93,
1
]
},
{
"code": "theorem one_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 1 o = [⟨0, o⟩]",
"end": [
97,
92
],
"full_name": "Ordinal.one_CNF",
"kind": "commanddeclaration",
"start": [
97,
1
]
},
{
"code": "theorem CNF_of_le_one {b o : Ordinal} (hb : b ≤ 1) (ho : o ≠ 0) : CNF b o = [⟨0, o⟩]",
"end": [
104,
21
],
"full_name": "Ordinal.CNF_of_le_one",
"kind": "commanddeclaration",
"start": [
101,
1
]
},
{
"code": "theorem CNF_of_lt {b o : Ordinal} (ho : o ≠ 0) (hb : o < b) : CNF b o = [⟨0, o⟩]",
"end": [
109,
84
],
"full_name": "Ordinal.CNF_of_lt",
"kind": "commanddeclaration",
"start": [
108,
1
]
},
{
"code": "theorem CNF_foldr (b o : Ordinal) : (CNF b o).foldr (fun p r ↦ b ^ p.1 * p.2 + r) 0 = o",
"end": [
116,
74
],
"full_name": "Ordinal.CNF_foldr",
"kind": "commanddeclaration",
"start": [
113,
1
]
},
{
"code": "theorem CNF_fst_le_log {b o : Ordinal.{u}} {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.1 ≤ log b o",
"end": [
129,
74
],
"full_name": "Ordinal.CNF_fst_le_log",
"kind": "commanddeclaration",
"start": [
120,
1
]
},
{
"code": "theorem CNF_fst_le {b o : Ordinal.{u}} {x : Ordinal × Ordinal} (h : x ∈ CNF b o) : x.1 ≤ o",
"end": [
135,
46
],
"full_name": "Ordinal.CNF_fst_le",
"kind": "commanddeclaration",
"start": [
133,
1
]
},
{
"code": "theorem CNF_lt_snd {b o : Ordinal.{u}} {x : Ordinal × Ordinal} : x ∈ CNF b o → 0 < x.2",
"end": [
145,
15
],
"full_name": "Ordinal.CNF_lt_snd",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem CNF_snd_lt {b o : Ordinal.{u}} (hb : 1 < b) {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.2 < b",
"end": [
158,
17
],
"full_name": "Ordinal.CNF_snd_lt",
"kind": "commanddeclaration",
"start": [
149,
1
]
},
{
"code": "theorem CNF_sorted (b o : Ordinal) : ((CNF b o).map Prod.fst).Sorted (· > ·)",
"end": [
174,
83
],
"full_name": "Ordinal.CNF_sorted",
"kind": "commanddeclaration",
"start": [
162,
1
]
}
] | 9 | Ordinal.CNF_fst_le_log | [
[
122,
36
],
[
129,
74
]
] | 6 | 12 | rw [CNF_ne_zero ho, mem_cons] | case refine_2
b o✝ : Ordinal.{u}
x : Ordinal.{u} × Ordinal.{u}
o : Ordinal.{u}
ho : o ≠ 0
H : x ∈ CNF b (o % b ^ log b o) → x.1 ≤ log b (o % b ^ log b o)
⊢ x ∈ CNF b o → x.1 ≤ log b o | case refine_2
b o✝ : Ordinal.{u}
x : Ordinal.{u} × Ordinal.{u}
o : Ordinal.{u}
ho : o ≠ 0
H : x ∈ CNF b (o % b ^ log b o) → x.1 ≤ log b (o % b ^ log b o)
⊢ x = (log b o, o / b ^ log b o) ∨ x ∈ CNF b (o % b ^ log b o) → x.1 ≤ log b o |
Mathlib/SetTheory/Ordinal/CantorNormalForm.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Mathlib.SetTheory.Ordinal.Exponential",
"Mathlib/SetTheory/Ordinal/Exponential.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "@[elab_as_elim]\nnoncomputable def CNFRec (b : Ordinal) {C : Ordinal → Sort*} (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : ∀ o, C o := fun o ↦ by\n by_cases h : o = 0\n · rw [h]; exact H0\n · exact H o h (CNFRec _ H0 H (o % b ^ log b o))\n termination_by o => o\n decreasing_by exact mod_opow_log_lt_self b h",
"end": [
50,
49
],
"full_name": "Ordinal.CNFRec",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem CNFRec_zero {C : Ordinal → Sort*} (b : Ordinal) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : @CNFRec b C H0 H 0 = H0",
"end": [
58,
6
],
"full_name": "Ordinal.CNFRec_zero",
"kind": "commanddeclaration",
"start": [
54,
1
]
},
{
"code": "theorem CNFRec_pos (b : Ordinal) {o : Ordinal} {C : Ordinal → Sort*} (ho : o ≠ 0) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) :\n @CNFRec b C H0 H o = H o ho (@CNFRec b C H0 H _)",
"end": [
64,
83
],
"full_name": "Ordinal.CNFRec_pos",
"kind": "commanddeclaration",
"start": [
62,
1
]
},
{
"code": "@[pp_nodot]\ndef CNF (b o : Ordinal) : List (Ordinal × Ordinal) :=\n CNFRec b [] (fun o _ho IH ↦ (log b o, o / b ^ log b o)::IH) o",
"end": [
76,
64
],
"full_name": "Ordinal.CNF",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "@[simp]\ntheorem CNF_zero (b : Ordinal) : CNF b 0 = []",
"end": [
82,
20
],
"full_name": "Ordinal.CNF_zero",
"kind": "commanddeclaration",
"start": [
80,
1
]
},
{
"code": "theorem CNF_ne_zero {b o : Ordinal} (ho : o ≠ 0) :\n CNF b o = (log b o, o / b ^ log b o)::CNF b (o % b ^ log b o)",
"end": [
89,
22
],
"full_name": "Ordinal.CNF_ne_zero",
"kind": "commanddeclaration",
"start": [
86,
1
]
},
{
"code": "theorem zero_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 0 o = [⟨0, o⟩]",
"end": [
93,
93
],
"full_name": "Ordinal.zero_CNF",
"kind": "commanddeclaration",
"start": [
93,
1
]
},
{
"code": "theorem one_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 1 o = [⟨0, o⟩]",
"end": [
97,
92
],
"full_name": "Ordinal.one_CNF",
"kind": "commanddeclaration",
"start": [
97,
1
]
},
{
"code": "theorem CNF_of_le_one {b o : Ordinal} (hb : b ≤ 1) (ho : o ≠ 0) : CNF b o = [⟨0, o⟩]",
"end": [
104,
21
],
"full_name": "Ordinal.CNF_of_le_one",
"kind": "commanddeclaration",
"start": [
101,
1
]
},
{
"code": "theorem CNF_of_lt {b o : Ordinal} (ho : o ≠ 0) (hb : o < b) : CNF b o = [⟨0, o⟩]",
"end": [
109,
84
],
"full_name": "Ordinal.CNF_of_lt",
"kind": "commanddeclaration",
"start": [
108,
1
]
},
{
"code": "theorem CNF_foldr (b o : Ordinal) : (CNF b o).foldr (fun p r ↦ b ^ p.1 * p.2 + r) 0 = o",
"end": [
116,
74
],
"full_name": "Ordinal.CNF_foldr",
"kind": "commanddeclaration",
"start": [
113,
1
]
},
{
"code": "theorem CNF_fst_le_log {b o : Ordinal.{u}} {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.1 ≤ log b o",
"end": [
129,
74
],
"full_name": "Ordinal.CNF_fst_le_log",
"kind": "commanddeclaration",
"start": [
120,
1
]
},
{
"code": "theorem CNF_fst_le {b o : Ordinal.{u}} {x : Ordinal × Ordinal} (h : x ∈ CNF b o) : x.1 ≤ o",
"end": [
135,
46
],
"full_name": "Ordinal.CNF_fst_le",
"kind": "commanddeclaration",
"start": [
133,
1
]
},
{
"code": "theorem CNF_lt_snd {b o : Ordinal.{u}} {x : Ordinal × Ordinal} : x ∈ CNF b o → 0 < x.2",
"end": [
145,
15
],
"full_name": "Ordinal.CNF_lt_snd",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem CNF_snd_lt {b o : Ordinal.{u}} (hb : 1 < b) {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.2 < b",
"end": [
158,
17
],
"full_name": "Ordinal.CNF_snd_lt",
"kind": "commanddeclaration",
"start": [
149,
1
]
},
{
"code": "theorem CNF_sorted (b o : Ordinal) : ((CNF b o).map Prod.fst).Sorted (· > ·)",
"end": [
174,
83
],
"full_name": "Ordinal.CNF_sorted",
"kind": "commanddeclaration",
"start": [
162,
1
]
}
] | 9 | Ordinal.CNF_fst_le_log | [
[
122,
36
],
[
129,
74
]
] | 7 | 12 | rintro (rfl | h) | case refine_2
b o✝ : Ordinal.{u}
x : Ordinal.{u} × Ordinal.{u}
o : Ordinal.{u}
ho : o ≠ 0
H : x ∈ CNF b (o % b ^ log b o) → x.1 ≤ log b (o % b ^ log b o)
⊢ x = (log b o, o / b ^ log b o) ∨ x ∈ CNF b (o % b ^ log b o) → x.1 ≤ log b o | case refine_2.inl
b o✝ o : Ordinal.{u}
ho : o ≠ 0
H : (log b o, o / b ^ log b o) ∈ CNF b (o % b ^ log b o) → (log b o, o / b ^ log b o).1 ≤ log b (o % b ^ log b o)
⊢ (log b o, o / b ^ log b o).1 ≤ log b o
case refine_2.inr
b o✝ : Ordinal.{u}
x : Ordinal.{u} × Ordinal.{u}
o : Ordinal.{u}
ho : o ≠ 0
H : x ∈ CNF b (o % b ^ log b o) → x.1 ≤ log b (o % b ^ log b o)
h : x ∈ CNF b (o % b ^ log b o)
⊢ x.1 ≤ log b o |
Mathlib/SetTheory/Ordinal/CantorNormalForm.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Mathlib.SetTheory.Ordinal.Exponential",
"Mathlib/SetTheory/Ordinal/Exponential.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "@[elab_as_elim]\nnoncomputable def CNFRec (b : Ordinal) {C : Ordinal → Sort*} (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : ∀ o, C o := fun o ↦ by\n by_cases h : o = 0\n · rw [h]; exact H0\n · exact H o h (CNFRec _ H0 H (o % b ^ log b o))\n termination_by o => o\n decreasing_by exact mod_opow_log_lt_self b h",
"end": [
50,
49
],
"full_name": "Ordinal.CNFRec",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem CNFRec_zero {C : Ordinal → Sort*} (b : Ordinal) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : @CNFRec b C H0 H 0 = H0",
"end": [
58,
6
],
"full_name": "Ordinal.CNFRec_zero",
"kind": "commanddeclaration",
"start": [
54,
1
]
},
{
"code": "theorem CNFRec_pos (b : Ordinal) {o : Ordinal} {C : Ordinal → Sort*} (ho : o ≠ 0) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) :\n @CNFRec b C H0 H o = H o ho (@CNFRec b C H0 H _)",
"end": [
64,
83
],
"full_name": "Ordinal.CNFRec_pos",
"kind": "commanddeclaration",
"start": [
62,
1
]
},
{
"code": "@[pp_nodot]\ndef CNF (b o : Ordinal) : List (Ordinal × Ordinal) :=\n CNFRec b [] (fun o _ho IH ↦ (log b o, o / b ^ log b o)::IH) o",
"end": [
76,
64
],
"full_name": "Ordinal.CNF",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "@[simp]\ntheorem CNF_zero (b : Ordinal) : CNF b 0 = []",
"end": [
82,
20
],
"full_name": "Ordinal.CNF_zero",
"kind": "commanddeclaration",
"start": [
80,
1
]
},
{
"code": "theorem CNF_ne_zero {b o : Ordinal} (ho : o ≠ 0) :\n CNF b o = (log b o, o / b ^ log b o)::CNF b (o % b ^ log b o)",
"end": [
89,
22
],
"full_name": "Ordinal.CNF_ne_zero",
"kind": "commanddeclaration",
"start": [
86,
1
]
},
{
"code": "theorem zero_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 0 o = [⟨0, o⟩]",
"end": [
93,
93
],
"full_name": "Ordinal.zero_CNF",
"kind": "commanddeclaration",
"start": [
93,
1
]
},
{
"code": "theorem one_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 1 o = [⟨0, o⟩]",
"end": [
97,
92
],
"full_name": "Ordinal.one_CNF",
"kind": "commanddeclaration",
"start": [
97,
1
]
},
{
"code": "theorem CNF_of_le_one {b o : Ordinal} (hb : b ≤ 1) (ho : o ≠ 0) : CNF b o = [⟨0, o⟩]",
"end": [
104,
21
],
"full_name": "Ordinal.CNF_of_le_one",
"kind": "commanddeclaration",
"start": [
101,
1
]
},
{
"code": "theorem CNF_of_lt {b o : Ordinal} (ho : o ≠ 0) (hb : o < b) : CNF b o = [⟨0, o⟩]",
"end": [
109,
84
],
"full_name": "Ordinal.CNF_of_lt",
"kind": "commanddeclaration",
"start": [
108,
1
]
},
{
"code": "theorem CNF_foldr (b o : Ordinal) : (CNF b o).foldr (fun p r ↦ b ^ p.1 * p.2 + r) 0 = o",
"end": [
116,
74
],
"full_name": "Ordinal.CNF_foldr",
"kind": "commanddeclaration",
"start": [
113,
1
]
},
{
"code": "theorem CNF_fst_le_log {b o : Ordinal.{u}} {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.1 ≤ log b o",
"end": [
129,
74
],
"full_name": "Ordinal.CNF_fst_le_log",
"kind": "commanddeclaration",
"start": [
120,
1
]
},
{
"code": "theorem CNF_fst_le {b o : Ordinal.{u}} {x : Ordinal × Ordinal} (h : x ∈ CNF b o) : x.1 ≤ o",
"end": [
135,
46
],
"full_name": "Ordinal.CNF_fst_le",
"kind": "commanddeclaration",
"start": [
133,
1
]
},
{
"code": "theorem CNF_lt_snd {b o : Ordinal.{u}} {x : Ordinal × Ordinal} : x ∈ CNF b o → 0 < x.2",
"end": [
145,
15
],
"full_name": "Ordinal.CNF_lt_snd",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem CNF_snd_lt {b o : Ordinal.{u}} (hb : 1 < b) {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.2 < b",
"end": [
158,
17
],
"full_name": "Ordinal.CNF_snd_lt",
"kind": "commanddeclaration",
"start": [
149,
1
]
},
{
"code": "theorem CNF_sorted (b o : Ordinal) : ((CNF b o).map Prod.fst).Sorted (· > ·)",
"end": [
174,
83
],
"full_name": "Ordinal.CNF_sorted",
"kind": "commanddeclaration",
"start": [
162,
1
]
}
] | 9 | Ordinal.CNF_fst_le_log | [
[
122,
36
],
[
129,
74
]
] | 8 | 12 | · exact le_rfl | case refine_2.inl
b o✝ o : Ordinal.{u}
ho : o ≠ 0
H : (log b o, o / b ^ log b o) ∈ CNF b (o % b ^ log b o) → (log b o, o / b ^ log b o).1 ≤ log b (o % b ^ log b o)
⊢ (log b o, o / b ^ log b o).1 ≤ log b o
case refine_2.inr
b o✝ : Ordinal.{u}
x : Ordinal.{u} × Ordinal.{u}
o : Ordinal.{u}
ho : o ≠ 0
H : x ∈ CNF b (o % b ^ log b o) → x.1 ≤ log b (o % b ^ log b o)
h : x ∈ CNF b (o % b ^ log b o)
⊢ x.1 ≤ log b o | case refine_2.inr
b o✝ : Ordinal.{u}
x : Ordinal.{u} × Ordinal.{u}
o : Ordinal.{u}
ho : o ≠ 0
H : x ∈ CNF b (o % b ^ log b o) → x.1 ≤ log b (o % b ^ log b o)
h : x ∈ CNF b (o % b ^ log b o)
⊢ x.1 ≤ log b o |
Mathlib/SetTheory/Ordinal/CantorNormalForm.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Mathlib.SetTheory.Ordinal.Exponential",
"Mathlib/SetTheory/Ordinal/Exponential.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "@[elab_as_elim]\nnoncomputable def CNFRec (b : Ordinal) {C : Ordinal → Sort*} (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : ∀ o, C o := fun o ↦ by\n by_cases h : o = 0\n · rw [h]; exact H0\n · exact H o h (CNFRec _ H0 H (o % b ^ log b o))\n termination_by o => o\n decreasing_by exact mod_opow_log_lt_self b h",
"end": [
50,
49
],
"full_name": "Ordinal.CNFRec",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem CNFRec_zero {C : Ordinal → Sort*} (b : Ordinal) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : @CNFRec b C H0 H 0 = H0",
"end": [
58,
6
],
"full_name": "Ordinal.CNFRec_zero",
"kind": "commanddeclaration",
"start": [
54,
1
]
},
{
"code": "theorem CNFRec_pos (b : Ordinal) {o : Ordinal} {C : Ordinal → Sort*} (ho : o ≠ 0) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) :\n @CNFRec b C H0 H o = H o ho (@CNFRec b C H0 H _)",
"end": [
64,
83
],
"full_name": "Ordinal.CNFRec_pos",
"kind": "commanddeclaration",
"start": [
62,
1
]
},
{
"code": "@[pp_nodot]\ndef CNF (b o : Ordinal) : List (Ordinal × Ordinal) :=\n CNFRec b [] (fun o _ho IH ↦ (log b o, o / b ^ log b o)::IH) o",
"end": [
76,
64
],
"full_name": "Ordinal.CNF",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "@[simp]\ntheorem CNF_zero (b : Ordinal) : CNF b 0 = []",
"end": [
82,
20
],
"full_name": "Ordinal.CNF_zero",
"kind": "commanddeclaration",
"start": [
80,
1
]
},
{
"code": "theorem CNF_ne_zero {b o : Ordinal} (ho : o ≠ 0) :\n CNF b o = (log b o, o / b ^ log b o)::CNF b (o % b ^ log b o)",
"end": [
89,
22
],
"full_name": "Ordinal.CNF_ne_zero",
"kind": "commanddeclaration",
"start": [
86,
1
]
},
{
"code": "theorem zero_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 0 o = [⟨0, o⟩]",
"end": [
93,
93
],
"full_name": "Ordinal.zero_CNF",
"kind": "commanddeclaration",
"start": [
93,
1
]
},
{
"code": "theorem one_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 1 o = [⟨0, o⟩]",
"end": [
97,
92
],
"full_name": "Ordinal.one_CNF",
"kind": "commanddeclaration",
"start": [
97,
1
]
},
{
"code": "theorem CNF_of_le_one {b o : Ordinal} (hb : b ≤ 1) (ho : o ≠ 0) : CNF b o = [⟨0, o⟩]",
"end": [
104,
21
],
"full_name": "Ordinal.CNF_of_le_one",
"kind": "commanddeclaration",
"start": [
101,
1
]
},
{
"code": "theorem CNF_of_lt {b o : Ordinal} (ho : o ≠ 0) (hb : o < b) : CNF b o = [⟨0, o⟩]",
"end": [
109,
84
],
"full_name": "Ordinal.CNF_of_lt",
"kind": "commanddeclaration",
"start": [
108,
1
]
},
{
"code": "theorem CNF_foldr (b o : Ordinal) : (CNF b o).foldr (fun p r ↦ b ^ p.1 * p.2 + r) 0 = o",
"end": [
116,
74
],
"full_name": "Ordinal.CNF_foldr",
"kind": "commanddeclaration",
"start": [
113,
1
]
},
{
"code": "theorem CNF_fst_le_log {b o : Ordinal.{u}} {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.1 ≤ log b o",
"end": [
129,
74
],
"full_name": "Ordinal.CNF_fst_le_log",
"kind": "commanddeclaration",
"start": [
120,
1
]
},
{
"code": "theorem CNF_fst_le {b o : Ordinal.{u}} {x : Ordinal × Ordinal} (h : x ∈ CNF b o) : x.1 ≤ o",
"end": [
135,
46
],
"full_name": "Ordinal.CNF_fst_le",
"kind": "commanddeclaration",
"start": [
133,
1
]
},
{
"code": "theorem CNF_lt_snd {b o : Ordinal.{u}} {x : Ordinal × Ordinal} : x ∈ CNF b o → 0 < x.2",
"end": [
145,
15
],
"full_name": "Ordinal.CNF_lt_snd",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem CNF_snd_lt {b o : Ordinal.{u}} (hb : 1 < b) {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.2 < b",
"end": [
158,
17
],
"full_name": "Ordinal.CNF_snd_lt",
"kind": "commanddeclaration",
"start": [
149,
1
]
},
{
"code": "theorem CNF_sorted (b o : Ordinal) : ((CNF b o).map Prod.fst).Sorted (· > ·)",
"end": [
174,
83
],
"full_name": "Ordinal.CNF_sorted",
"kind": "commanddeclaration",
"start": [
162,
1
]
}
] | 9 | Ordinal.CNF_fst_le_log | [
[
122,
36
],
[
129,
74
]
] | 9 | 12 | · exact (H h).trans (log_mono_right _ (mod_opow_log_lt_self b ho).le) | case refine_2.inr
b o✝ : Ordinal.{u}
x : Ordinal.{u} × Ordinal.{u}
o : Ordinal.{u}
ho : o ≠ 0
H : x ∈ CNF b (o % b ^ log b o) → x.1 ≤ log b (o % b ^ log b o)
h : x ∈ CNF b (o % b ^ log b o)
⊢ x.1 ≤ log b o | no goals |
Mathlib/SetTheory/Ordinal/CantorNormalForm.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Mathlib.SetTheory.Ordinal.Exponential",
"Mathlib/SetTheory/Ordinal/Exponential.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "@[elab_as_elim]\nnoncomputable def CNFRec (b : Ordinal) {C : Ordinal → Sort*} (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : ∀ o, C o := fun o ↦ by\n by_cases h : o = 0\n · rw [h]; exact H0\n · exact H o h (CNFRec _ H0 H (o % b ^ log b o))\n termination_by o => o\n decreasing_by exact mod_opow_log_lt_self b h",
"end": [
50,
49
],
"full_name": "Ordinal.CNFRec",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem CNFRec_zero {C : Ordinal → Sort*} (b : Ordinal) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : @CNFRec b C H0 H 0 = H0",
"end": [
58,
6
],
"full_name": "Ordinal.CNFRec_zero",
"kind": "commanddeclaration",
"start": [
54,
1
]
},
{
"code": "theorem CNFRec_pos (b : Ordinal) {o : Ordinal} {C : Ordinal → Sort*} (ho : o ≠ 0) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) :\n @CNFRec b C H0 H o = H o ho (@CNFRec b C H0 H _)",
"end": [
64,
83
],
"full_name": "Ordinal.CNFRec_pos",
"kind": "commanddeclaration",
"start": [
62,
1
]
},
{
"code": "@[pp_nodot]\ndef CNF (b o : Ordinal) : List (Ordinal × Ordinal) :=\n CNFRec b [] (fun o _ho IH ↦ (log b o, o / b ^ log b o)::IH) o",
"end": [
76,
64
],
"full_name": "Ordinal.CNF",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "@[simp]\ntheorem CNF_zero (b : Ordinal) : CNF b 0 = []",
"end": [
82,
20
],
"full_name": "Ordinal.CNF_zero",
"kind": "commanddeclaration",
"start": [
80,
1
]
},
{
"code": "theorem CNF_ne_zero {b o : Ordinal} (ho : o ≠ 0) :\n CNF b o = (log b o, o / b ^ log b o)::CNF b (o % b ^ log b o)",
"end": [
89,
22
],
"full_name": "Ordinal.CNF_ne_zero",
"kind": "commanddeclaration",
"start": [
86,
1
]
},
{
"code": "theorem zero_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 0 o = [⟨0, o⟩]",
"end": [
93,
93
],
"full_name": "Ordinal.zero_CNF",
"kind": "commanddeclaration",
"start": [
93,
1
]
},
{
"code": "theorem one_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 1 o = [⟨0, o⟩]",
"end": [
97,
92
],
"full_name": "Ordinal.one_CNF",
"kind": "commanddeclaration",
"start": [
97,
1
]
},
{
"code": "theorem CNF_of_le_one {b o : Ordinal} (hb : b ≤ 1) (ho : o ≠ 0) : CNF b o = [⟨0, o⟩]",
"end": [
104,
21
],
"full_name": "Ordinal.CNF_of_le_one",
"kind": "commanddeclaration",
"start": [
101,
1
]
},
{
"code": "theorem CNF_of_lt {b o : Ordinal} (ho : o ≠ 0) (hb : o < b) : CNF b o = [⟨0, o⟩]",
"end": [
109,
84
],
"full_name": "Ordinal.CNF_of_lt",
"kind": "commanddeclaration",
"start": [
108,
1
]
},
{
"code": "theorem CNF_foldr (b o : Ordinal) : (CNF b o).foldr (fun p r ↦ b ^ p.1 * p.2 + r) 0 = o",
"end": [
116,
74
],
"full_name": "Ordinal.CNF_foldr",
"kind": "commanddeclaration",
"start": [
113,
1
]
},
{
"code": "theorem CNF_fst_le_log {b o : Ordinal.{u}} {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.1 ≤ log b o",
"end": [
129,
74
],
"full_name": "Ordinal.CNF_fst_le_log",
"kind": "commanddeclaration",
"start": [
120,
1
]
},
{
"code": "theorem CNF_fst_le {b o : Ordinal.{u}} {x : Ordinal × Ordinal} (h : x ∈ CNF b o) : x.1 ≤ o",
"end": [
135,
46
],
"full_name": "Ordinal.CNF_fst_le",
"kind": "commanddeclaration",
"start": [
133,
1
]
},
{
"code": "theorem CNF_lt_snd {b o : Ordinal.{u}} {x : Ordinal × Ordinal} : x ∈ CNF b o → 0 < x.2",
"end": [
145,
15
],
"full_name": "Ordinal.CNF_lt_snd",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem CNF_snd_lt {b o : Ordinal.{u}} (hb : 1 < b) {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.2 < b",
"end": [
158,
17
],
"full_name": "Ordinal.CNF_snd_lt",
"kind": "commanddeclaration",
"start": [
149,
1
]
},
{
"code": "theorem CNF_sorted (b o : Ordinal) : ((CNF b o).map Prod.fst).Sorted (· > ·)",
"end": [
174,
83
],
"full_name": "Ordinal.CNF_sorted",
"kind": "commanddeclaration",
"start": [
162,
1
]
}
] | 9 | Ordinal.CNF_fst_le_log | [
[
122,
36
],
[
129,
74
]
] | 10 | 12 | exact le_rfl | case refine_2.inl
b o✝ o : Ordinal.{u}
ho : o ≠ 0
H : (log b o, o / b ^ log b o) ∈ CNF b (o % b ^ log b o) → (log b o, o / b ^ log b o).1 ≤ log b (o % b ^ log b o)
⊢ (log b o, o / b ^ log b o).1 ≤ log b o | no goals |
Mathlib/SetTheory/Ordinal/CantorNormalForm.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Mathlib.SetTheory.Ordinal.Exponential",
"Mathlib/SetTheory/Ordinal/Exponential.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "@[elab_as_elim]\nnoncomputable def CNFRec (b : Ordinal) {C : Ordinal → Sort*} (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : ∀ o, C o := fun o ↦ by\n by_cases h : o = 0\n · rw [h]; exact H0\n · exact H o h (CNFRec _ H0 H (o % b ^ log b o))\n termination_by o => o\n decreasing_by exact mod_opow_log_lt_self b h",
"end": [
50,
49
],
"full_name": "Ordinal.CNFRec",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem CNFRec_zero {C : Ordinal → Sort*} (b : Ordinal) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : @CNFRec b C H0 H 0 = H0",
"end": [
58,
6
],
"full_name": "Ordinal.CNFRec_zero",
"kind": "commanddeclaration",
"start": [
54,
1
]
},
{
"code": "theorem CNFRec_pos (b : Ordinal) {o : Ordinal} {C : Ordinal → Sort*} (ho : o ≠ 0) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) :\n @CNFRec b C H0 H o = H o ho (@CNFRec b C H0 H _)",
"end": [
64,
83
],
"full_name": "Ordinal.CNFRec_pos",
"kind": "commanddeclaration",
"start": [
62,
1
]
},
{
"code": "@[pp_nodot]\ndef CNF (b o : Ordinal) : List (Ordinal × Ordinal) :=\n CNFRec b [] (fun o _ho IH ↦ (log b o, o / b ^ log b o)::IH) o",
"end": [
76,
64
],
"full_name": "Ordinal.CNF",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "@[simp]\ntheorem CNF_zero (b : Ordinal) : CNF b 0 = []",
"end": [
82,
20
],
"full_name": "Ordinal.CNF_zero",
"kind": "commanddeclaration",
"start": [
80,
1
]
},
{
"code": "theorem CNF_ne_zero {b o : Ordinal} (ho : o ≠ 0) :\n CNF b o = (log b o, o / b ^ log b o)::CNF b (o % b ^ log b o)",
"end": [
89,
22
],
"full_name": "Ordinal.CNF_ne_zero",
"kind": "commanddeclaration",
"start": [
86,
1
]
},
{
"code": "theorem zero_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 0 o = [⟨0, o⟩]",
"end": [
93,
93
],
"full_name": "Ordinal.zero_CNF",
"kind": "commanddeclaration",
"start": [
93,
1
]
},
{
"code": "theorem one_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 1 o = [⟨0, o⟩]",
"end": [
97,
92
],
"full_name": "Ordinal.one_CNF",
"kind": "commanddeclaration",
"start": [
97,
1
]
},
{
"code": "theorem CNF_of_le_one {b o : Ordinal} (hb : b ≤ 1) (ho : o ≠ 0) : CNF b o = [⟨0, o⟩]",
"end": [
104,
21
],
"full_name": "Ordinal.CNF_of_le_one",
"kind": "commanddeclaration",
"start": [
101,
1
]
},
{
"code": "theorem CNF_of_lt {b o : Ordinal} (ho : o ≠ 0) (hb : o < b) : CNF b o = [⟨0, o⟩]",
"end": [
109,
84
],
"full_name": "Ordinal.CNF_of_lt",
"kind": "commanddeclaration",
"start": [
108,
1
]
},
{
"code": "theorem CNF_foldr (b o : Ordinal) : (CNF b o).foldr (fun p r ↦ b ^ p.1 * p.2 + r) 0 = o",
"end": [
116,
74
],
"full_name": "Ordinal.CNF_foldr",
"kind": "commanddeclaration",
"start": [
113,
1
]
},
{
"code": "theorem CNF_fst_le_log {b o : Ordinal.{u}} {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.1 ≤ log b o",
"end": [
129,
74
],
"full_name": "Ordinal.CNF_fst_le_log",
"kind": "commanddeclaration",
"start": [
120,
1
]
},
{
"code": "theorem CNF_fst_le {b o : Ordinal.{u}} {x : Ordinal × Ordinal} (h : x ∈ CNF b o) : x.1 ≤ o",
"end": [
135,
46
],
"full_name": "Ordinal.CNF_fst_le",
"kind": "commanddeclaration",
"start": [
133,
1
]
},
{
"code": "theorem CNF_lt_snd {b o : Ordinal.{u}} {x : Ordinal × Ordinal} : x ∈ CNF b o → 0 < x.2",
"end": [
145,
15
],
"full_name": "Ordinal.CNF_lt_snd",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem CNF_snd_lt {b o : Ordinal.{u}} (hb : 1 < b) {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.2 < b",
"end": [
158,
17
],
"full_name": "Ordinal.CNF_snd_lt",
"kind": "commanddeclaration",
"start": [
149,
1
]
},
{
"code": "theorem CNF_sorted (b o : Ordinal) : ((CNF b o).map Prod.fst).Sorted (· > ·)",
"end": [
174,
83
],
"full_name": "Ordinal.CNF_sorted",
"kind": "commanddeclaration",
"start": [
162,
1
]
}
] | 9 | Ordinal.CNF_fst_le_log | [
[
122,
36
],
[
129,
74
]
] | 11 | 12 | exact (H h).trans (log_mono_right _ (mod_opow_log_lt_self b ho).le) | case refine_2.inr
b o✝ : Ordinal.{u}
x : Ordinal.{u} × Ordinal.{u}
o : Ordinal.{u}
ho : o ≠ 0
H : x ∈ CNF b (o % b ^ log b o) → x.1 ≤ log b (o % b ^ log b o)
h : x ∈ CNF b (o % b ^ log b o)
⊢ x.1 ≤ log b o | no goals |
Mathlib/SetTheory/Ordinal/CantorNormalForm.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Mathlib.SetTheory.Ordinal.Exponential",
"Mathlib/SetTheory/Ordinal/Exponential.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "@[elab_as_elim]\nnoncomputable def CNFRec (b : Ordinal) {C : Ordinal → Sort*} (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : ∀ o, C o := fun o ↦ by\n by_cases h : o = 0\n · rw [h]; exact H0\n · exact H o h (CNFRec _ H0 H (o % b ^ log b o))\n termination_by o => o\n decreasing_by exact mod_opow_log_lt_self b h",
"end": [
50,
49
],
"full_name": "Ordinal.CNFRec",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem CNFRec_zero {C : Ordinal → Sort*} (b : Ordinal) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : @CNFRec b C H0 H 0 = H0",
"end": [
58,
6
],
"full_name": "Ordinal.CNFRec_zero",
"kind": "commanddeclaration",
"start": [
54,
1
]
},
{
"code": "theorem CNFRec_pos (b : Ordinal) {o : Ordinal} {C : Ordinal → Sort*} (ho : o ≠ 0) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) :\n @CNFRec b C H0 H o = H o ho (@CNFRec b C H0 H _)",
"end": [
64,
83
],
"full_name": "Ordinal.CNFRec_pos",
"kind": "commanddeclaration",
"start": [
62,
1
]
},
{
"code": "@[pp_nodot]\ndef CNF (b o : Ordinal) : List (Ordinal × Ordinal) :=\n CNFRec b [] (fun o _ho IH ↦ (log b o, o / b ^ log b o)::IH) o",
"end": [
76,
64
],
"full_name": "Ordinal.CNF",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "@[simp]\ntheorem CNF_zero (b : Ordinal) : CNF b 0 = []",
"end": [
82,
20
],
"full_name": "Ordinal.CNF_zero",
"kind": "commanddeclaration",
"start": [
80,
1
]
},
{
"code": "theorem CNF_ne_zero {b o : Ordinal} (ho : o ≠ 0) :\n CNF b o = (log b o, o / b ^ log b o)::CNF b (o % b ^ log b o)",
"end": [
89,
22
],
"full_name": "Ordinal.CNF_ne_zero",
"kind": "commanddeclaration",
"start": [
86,
1
]
},
{
"code": "theorem zero_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 0 o = [⟨0, o⟩]",
"end": [
93,
93
],
"full_name": "Ordinal.zero_CNF",
"kind": "commanddeclaration",
"start": [
93,
1
]
},
{
"code": "theorem one_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 1 o = [⟨0, o⟩]",
"end": [
97,
92
],
"full_name": "Ordinal.one_CNF",
"kind": "commanddeclaration",
"start": [
97,
1
]
},
{
"code": "theorem CNF_of_le_one {b o : Ordinal} (hb : b ≤ 1) (ho : o ≠ 0) : CNF b o = [⟨0, o⟩]",
"end": [
104,
21
],
"full_name": "Ordinal.CNF_of_le_one",
"kind": "commanddeclaration",
"start": [
101,
1
]
},
{
"code": "theorem CNF_of_lt {b o : Ordinal} (ho : o ≠ 0) (hb : o < b) : CNF b o = [⟨0, o⟩]",
"end": [
109,
84
],
"full_name": "Ordinal.CNF_of_lt",
"kind": "commanddeclaration",
"start": [
108,
1
]
},
{
"code": "theorem CNF_foldr (b o : Ordinal) : (CNF b o).foldr (fun p r ↦ b ^ p.1 * p.2 + r) 0 = o",
"end": [
116,
74
],
"full_name": "Ordinal.CNF_foldr",
"kind": "commanddeclaration",
"start": [
113,
1
]
},
{
"code": "theorem CNF_fst_le_log {b o : Ordinal.{u}} {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.1 ≤ log b o",
"end": [
129,
74
],
"full_name": "Ordinal.CNF_fst_le_log",
"kind": "commanddeclaration",
"start": [
120,
1
]
},
{
"code": "theorem CNF_fst_le {b o : Ordinal.{u}} {x : Ordinal × Ordinal} (h : x ∈ CNF b o) : x.1 ≤ o",
"end": [
135,
46
],
"full_name": "Ordinal.CNF_fst_le",
"kind": "commanddeclaration",
"start": [
133,
1
]
},
{
"code": "theorem CNF_lt_snd {b o : Ordinal.{u}} {x : Ordinal × Ordinal} : x ∈ CNF b o → 0 < x.2",
"end": [
145,
15
],
"full_name": "Ordinal.CNF_lt_snd",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem CNF_snd_lt {b o : Ordinal.{u}} (hb : 1 < b) {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.2 < b",
"end": [
158,
17
],
"full_name": "Ordinal.CNF_snd_lt",
"kind": "commanddeclaration",
"start": [
149,
1
]
},
{
"code": "theorem CNF_sorted (b o : Ordinal) : ((CNF b o).map Prod.fst).Sorted (· > ·)",
"end": [
174,
83
],
"full_name": "Ordinal.CNF_sorted",
"kind": "commanddeclaration",
"start": [
162,
1
]
}
] | 11 | Ordinal.CNF_lt_snd | [
[
140,
91
],
[
145,
15
]
] | 0 | 8 | refine CNFRec b (by simp) (fun o ho IH ↦ ?_) o | b o : Ordinal.{u}
x : Ordinal.{u} × Ordinal.{u}
⊢ x ∈ CNF b o → 0 < x.2 | b o✝ : Ordinal.{u}
x : Ordinal.{u} × Ordinal.{u}
o : Ordinal.{u}
ho : o ≠ 0
IH : x ∈ CNF b (o % b ^ log b o) → 0 < x.2
⊢ x ∈ CNF b o → 0 < x.2 |
Mathlib/SetTheory/Ordinal/CantorNormalForm.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Mathlib.SetTheory.Ordinal.Exponential",
"Mathlib/SetTheory/Ordinal/Exponential.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "@[elab_as_elim]\nnoncomputable def CNFRec (b : Ordinal) {C : Ordinal → Sort*} (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : ∀ o, C o := fun o ↦ by\n by_cases h : o = 0\n · rw [h]; exact H0\n · exact H o h (CNFRec _ H0 H (o % b ^ log b o))\n termination_by o => o\n decreasing_by exact mod_opow_log_lt_self b h",
"end": [
50,
49
],
"full_name": "Ordinal.CNFRec",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem CNFRec_zero {C : Ordinal → Sort*} (b : Ordinal) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : @CNFRec b C H0 H 0 = H0",
"end": [
58,
6
],
"full_name": "Ordinal.CNFRec_zero",
"kind": "commanddeclaration",
"start": [
54,
1
]
},
{
"code": "theorem CNFRec_pos (b : Ordinal) {o : Ordinal} {C : Ordinal → Sort*} (ho : o ≠ 0) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) :\n @CNFRec b C H0 H o = H o ho (@CNFRec b C H0 H _)",
"end": [
64,
83
],
"full_name": "Ordinal.CNFRec_pos",
"kind": "commanddeclaration",
"start": [
62,
1
]
},
{
"code": "@[pp_nodot]\ndef CNF (b o : Ordinal) : List (Ordinal × Ordinal) :=\n CNFRec b [] (fun o _ho IH ↦ (log b o, o / b ^ log b o)::IH) o",
"end": [
76,
64
],
"full_name": "Ordinal.CNF",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "@[simp]\ntheorem CNF_zero (b : Ordinal) : CNF b 0 = []",
"end": [
82,
20
],
"full_name": "Ordinal.CNF_zero",
"kind": "commanddeclaration",
"start": [
80,
1
]
},
{
"code": "theorem CNF_ne_zero {b o : Ordinal} (ho : o ≠ 0) :\n CNF b o = (log b o, o / b ^ log b o)::CNF b (o % b ^ log b o)",
"end": [
89,
22
],
"full_name": "Ordinal.CNF_ne_zero",
"kind": "commanddeclaration",
"start": [
86,
1
]
},
{
"code": "theorem zero_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 0 o = [⟨0, o⟩]",
"end": [
93,
93
],
"full_name": "Ordinal.zero_CNF",
"kind": "commanddeclaration",
"start": [
93,
1
]
},
{
"code": "theorem one_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 1 o = [⟨0, o⟩]",
"end": [
97,
92
],
"full_name": "Ordinal.one_CNF",
"kind": "commanddeclaration",
"start": [
97,
1
]
},
{
"code": "theorem CNF_of_le_one {b o : Ordinal} (hb : b ≤ 1) (ho : o ≠ 0) : CNF b o = [⟨0, o⟩]",
"end": [
104,
21
],
"full_name": "Ordinal.CNF_of_le_one",
"kind": "commanddeclaration",
"start": [
101,
1
]
},
{
"code": "theorem CNF_of_lt {b o : Ordinal} (ho : o ≠ 0) (hb : o < b) : CNF b o = [⟨0, o⟩]",
"end": [
109,
84
],
"full_name": "Ordinal.CNF_of_lt",
"kind": "commanddeclaration",
"start": [
108,
1
]
},
{
"code": "theorem CNF_foldr (b o : Ordinal) : (CNF b o).foldr (fun p r ↦ b ^ p.1 * p.2 + r) 0 = o",
"end": [
116,
74
],
"full_name": "Ordinal.CNF_foldr",
"kind": "commanddeclaration",
"start": [
113,
1
]
},
{
"code": "theorem CNF_fst_le_log {b o : Ordinal.{u}} {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.1 ≤ log b o",
"end": [
129,
74
],
"full_name": "Ordinal.CNF_fst_le_log",
"kind": "commanddeclaration",
"start": [
120,
1
]
},
{
"code": "theorem CNF_fst_le {b o : Ordinal.{u}} {x : Ordinal × Ordinal} (h : x ∈ CNF b o) : x.1 ≤ o",
"end": [
135,
46
],
"full_name": "Ordinal.CNF_fst_le",
"kind": "commanddeclaration",
"start": [
133,
1
]
},
{
"code": "theorem CNF_lt_snd {b o : Ordinal.{u}} {x : Ordinal × Ordinal} : x ∈ CNF b o → 0 < x.2",
"end": [
145,
15
],
"full_name": "Ordinal.CNF_lt_snd",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem CNF_snd_lt {b o : Ordinal.{u}} (hb : 1 < b) {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.2 < b",
"end": [
158,
17
],
"full_name": "Ordinal.CNF_snd_lt",
"kind": "commanddeclaration",
"start": [
149,
1
]
},
{
"code": "theorem CNF_sorted (b o : Ordinal) : ((CNF b o).map Prod.fst).Sorted (· > ·)",
"end": [
174,
83
],
"full_name": "Ordinal.CNF_sorted",
"kind": "commanddeclaration",
"start": [
162,
1
]
}
] | 11 | Ordinal.CNF_lt_snd | [
[
140,
91
],
[
145,
15
]
] | 1 | 8 | rw [CNF_ne_zero ho] | b o✝ : Ordinal.{u}
x : Ordinal.{u} × Ordinal.{u}
o : Ordinal.{u}
ho : o ≠ 0
IH : x ∈ CNF b (o % b ^ log b o) → 0 < x.2
⊢ x ∈ CNF b o → 0 < x.2 | b o✝ : Ordinal.{u}
x : Ordinal.{u} × Ordinal.{u}
o : Ordinal.{u}
ho : o ≠ 0
IH : x ∈ CNF b (o % b ^ log b o) → 0 < x.2
⊢ x ∈ (log b o, o / b ^ log b o) :: CNF b (o % b ^ log b o) → 0 < x.2 |
Mathlib/SetTheory/Ordinal/CantorNormalForm.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Mathlib.SetTheory.Ordinal.Exponential",
"Mathlib/SetTheory/Ordinal/Exponential.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "@[elab_as_elim]\nnoncomputable def CNFRec (b : Ordinal) {C : Ordinal → Sort*} (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : ∀ o, C o := fun o ↦ by\n by_cases h : o = 0\n · rw [h]; exact H0\n · exact H o h (CNFRec _ H0 H (o % b ^ log b o))\n termination_by o => o\n decreasing_by exact mod_opow_log_lt_self b h",
"end": [
50,
49
],
"full_name": "Ordinal.CNFRec",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem CNFRec_zero {C : Ordinal → Sort*} (b : Ordinal) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : @CNFRec b C H0 H 0 = H0",
"end": [
58,
6
],
"full_name": "Ordinal.CNFRec_zero",
"kind": "commanddeclaration",
"start": [
54,
1
]
},
{
"code": "theorem CNFRec_pos (b : Ordinal) {o : Ordinal} {C : Ordinal → Sort*} (ho : o ≠ 0) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) :\n @CNFRec b C H0 H o = H o ho (@CNFRec b C H0 H _)",
"end": [
64,
83
],
"full_name": "Ordinal.CNFRec_pos",
"kind": "commanddeclaration",
"start": [
62,
1
]
},
{
"code": "@[pp_nodot]\ndef CNF (b o : Ordinal) : List (Ordinal × Ordinal) :=\n CNFRec b [] (fun o _ho IH ↦ (log b o, o / b ^ log b o)::IH) o",
"end": [
76,
64
],
"full_name": "Ordinal.CNF",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "@[simp]\ntheorem CNF_zero (b : Ordinal) : CNF b 0 = []",
"end": [
82,
20
],
"full_name": "Ordinal.CNF_zero",
"kind": "commanddeclaration",
"start": [
80,
1
]
},
{
"code": "theorem CNF_ne_zero {b o : Ordinal} (ho : o ≠ 0) :\n CNF b o = (log b o, o / b ^ log b o)::CNF b (o % b ^ log b o)",
"end": [
89,
22
],
"full_name": "Ordinal.CNF_ne_zero",
"kind": "commanddeclaration",
"start": [
86,
1
]
},
{
"code": "theorem zero_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 0 o = [⟨0, o⟩]",
"end": [
93,
93
],
"full_name": "Ordinal.zero_CNF",
"kind": "commanddeclaration",
"start": [
93,
1
]
},
{
"code": "theorem one_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 1 o = [⟨0, o⟩]",
"end": [
97,
92
],
"full_name": "Ordinal.one_CNF",
"kind": "commanddeclaration",
"start": [
97,
1
]
},
{
"code": "theorem CNF_of_le_one {b o : Ordinal} (hb : b ≤ 1) (ho : o ≠ 0) : CNF b o = [⟨0, o⟩]",
"end": [
104,
21
],
"full_name": "Ordinal.CNF_of_le_one",
"kind": "commanddeclaration",
"start": [
101,
1
]
},
{
"code": "theorem CNF_of_lt {b o : Ordinal} (ho : o ≠ 0) (hb : o < b) : CNF b o = [⟨0, o⟩]",
"end": [
109,
84
],
"full_name": "Ordinal.CNF_of_lt",
"kind": "commanddeclaration",
"start": [
108,
1
]
},
{
"code": "theorem CNF_foldr (b o : Ordinal) : (CNF b o).foldr (fun p r ↦ b ^ p.1 * p.2 + r) 0 = o",
"end": [
116,
74
],
"full_name": "Ordinal.CNF_foldr",
"kind": "commanddeclaration",
"start": [
113,
1
]
},
{
"code": "theorem CNF_fst_le_log {b o : Ordinal.{u}} {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.1 ≤ log b o",
"end": [
129,
74
],
"full_name": "Ordinal.CNF_fst_le_log",
"kind": "commanddeclaration",
"start": [
120,
1
]
},
{
"code": "theorem CNF_fst_le {b o : Ordinal.{u}} {x : Ordinal × Ordinal} (h : x ∈ CNF b o) : x.1 ≤ o",
"end": [
135,
46
],
"full_name": "Ordinal.CNF_fst_le",
"kind": "commanddeclaration",
"start": [
133,
1
]
},
{
"code": "theorem CNF_lt_snd {b o : Ordinal.{u}} {x : Ordinal × Ordinal} : x ∈ CNF b o → 0 < x.2",
"end": [
145,
15
],
"full_name": "Ordinal.CNF_lt_snd",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem CNF_snd_lt {b o : Ordinal.{u}} (hb : 1 < b) {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.2 < b",
"end": [
158,
17
],
"full_name": "Ordinal.CNF_snd_lt",
"kind": "commanddeclaration",
"start": [
149,
1
]
},
{
"code": "theorem CNF_sorted (b o : Ordinal) : ((CNF b o).map Prod.fst).Sorted (· > ·)",
"end": [
174,
83
],
"full_name": "Ordinal.CNF_sorted",
"kind": "commanddeclaration",
"start": [
162,
1
]
}
] | 11 | Ordinal.CNF_lt_snd | [
[
140,
91
],
[
145,
15
]
] | 2 | 8 | rintro (h | ⟨_, h⟩) | b o✝ : Ordinal.{u}
x : Ordinal.{u} × Ordinal.{u}
o : Ordinal.{u}
ho : o ≠ 0
IH : x ∈ CNF b (o % b ^ log b o) → 0 < x.2
⊢ x ∈ (log b o, o / b ^ log b o) :: CNF b (o % b ^ log b o) → 0 < x.2 | case head
b o✝ o : Ordinal.{u}
ho : o ≠ 0
IH : (log b o, o / b ^ log b o) ∈ CNF b (o % b ^ log b o) → 0 < (log b o, o / b ^ log b o).2
⊢ 0 < (log b o, o / b ^ log b o).2
case tail
b o✝ : Ordinal.{u}
x : Ordinal.{u} × Ordinal.{u}
o : Ordinal.{u}
ho : o ≠ 0
IH : x ∈ CNF b (o % b ^ log b o) → 0 < x.2
h : Mem x (CNF b (o % b ^ log b o))
⊢ 0 < x.2 |
Mathlib/SetTheory/Ordinal/CantorNormalForm.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Mathlib.SetTheory.Ordinal.Exponential",
"Mathlib/SetTheory/Ordinal/Exponential.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "@[elab_as_elim]\nnoncomputable def CNFRec (b : Ordinal) {C : Ordinal → Sort*} (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : ∀ o, C o := fun o ↦ by\n by_cases h : o = 0\n · rw [h]; exact H0\n · exact H o h (CNFRec _ H0 H (o % b ^ log b o))\n termination_by o => o\n decreasing_by exact mod_opow_log_lt_self b h",
"end": [
50,
49
],
"full_name": "Ordinal.CNFRec",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem CNFRec_zero {C : Ordinal → Sort*} (b : Ordinal) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : @CNFRec b C H0 H 0 = H0",
"end": [
58,
6
],
"full_name": "Ordinal.CNFRec_zero",
"kind": "commanddeclaration",
"start": [
54,
1
]
},
{
"code": "theorem CNFRec_pos (b : Ordinal) {o : Ordinal} {C : Ordinal → Sort*} (ho : o ≠ 0) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) :\n @CNFRec b C H0 H o = H o ho (@CNFRec b C H0 H _)",
"end": [
64,
83
],
"full_name": "Ordinal.CNFRec_pos",
"kind": "commanddeclaration",
"start": [
62,
1
]
},
{
"code": "@[pp_nodot]\ndef CNF (b o : Ordinal) : List (Ordinal × Ordinal) :=\n CNFRec b [] (fun o _ho IH ↦ (log b o, o / b ^ log b o)::IH) o",
"end": [
76,
64
],
"full_name": "Ordinal.CNF",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "@[simp]\ntheorem CNF_zero (b : Ordinal) : CNF b 0 = []",
"end": [
82,
20
],
"full_name": "Ordinal.CNF_zero",
"kind": "commanddeclaration",
"start": [
80,
1
]
},
{
"code": "theorem CNF_ne_zero {b o : Ordinal} (ho : o ≠ 0) :\n CNF b o = (log b o, o / b ^ log b o)::CNF b (o % b ^ log b o)",
"end": [
89,
22
],
"full_name": "Ordinal.CNF_ne_zero",
"kind": "commanddeclaration",
"start": [
86,
1
]
},
{
"code": "theorem zero_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 0 o = [⟨0, o⟩]",
"end": [
93,
93
],
"full_name": "Ordinal.zero_CNF",
"kind": "commanddeclaration",
"start": [
93,
1
]
},
{
"code": "theorem one_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 1 o = [⟨0, o⟩]",
"end": [
97,
92
],
"full_name": "Ordinal.one_CNF",
"kind": "commanddeclaration",
"start": [
97,
1
]
},
{
"code": "theorem CNF_of_le_one {b o : Ordinal} (hb : b ≤ 1) (ho : o ≠ 0) : CNF b o = [⟨0, o⟩]",
"end": [
104,
21
],
"full_name": "Ordinal.CNF_of_le_one",
"kind": "commanddeclaration",
"start": [
101,
1
]
},
{
"code": "theorem CNF_of_lt {b o : Ordinal} (ho : o ≠ 0) (hb : o < b) : CNF b o = [⟨0, o⟩]",
"end": [
109,
84
],
"full_name": "Ordinal.CNF_of_lt",
"kind": "commanddeclaration",
"start": [
108,
1
]
},
{
"code": "theorem CNF_foldr (b o : Ordinal) : (CNF b o).foldr (fun p r ↦ b ^ p.1 * p.2 + r) 0 = o",
"end": [
116,
74
],
"full_name": "Ordinal.CNF_foldr",
"kind": "commanddeclaration",
"start": [
113,
1
]
},
{
"code": "theorem CNF_fst_le_log {b o : Ordinal.{u}} {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.1 ≤ log b o",
"end": [
129,
74
],
"full_name": "Ordinal.CNF_fst_le_log",
"kind": "commanddeclaration",
"start": [
120,
1
]
},
{
"code": "theorem CNF_fst_le {b o : Ordinal.{u}} {x : Ordinal × Ordinal} (h : x ∈ CNF b o) : x.1 ≤ o",
"end": [
135,
46
],
"full_name": "Ordinal.CNF_fst_le",
"kind": "commanddeclaration",
"start": [
133,
1
]
},
{
"code": "theorem CNF_lt_snd {b o : Ordinal.{u}} {x : Ordinal × Ordinal} : x ∈ CNF b o → 0 < x.2",
"end": [
145,
15
],
"full_name": "Ordinal.CNF_lt_snd",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem CNF_snd_lt {b o : Ordinal.{u}} (hb : 1 < b) {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.2 < b",
"end": [
158,
17
],
"full_name": "Ordinal.CNF_snd_lt",
"kind": "commanddeclaration",
"start": [
149,
1
]
},
{
"code": "theorem CNF_sorted (b o : Ordinal) : ((CNF b o).map Prod.fst).Sorted (· > ·)",
"end": [
174,
83
],
"full_name": "Ordinal.CNF_sorted",
"kind": "commanddeclaration",
"start": [
162,
1
]
}
] | 11 | Ordinal.CNF_lt_snd | [
[
140,
91
],
[
145,
15
]
] | 3 | 8 | · exact div_opow_log_pos b ho | case head
b o✝ o : Ordinal.{u}
ho : o ≠ 0
IH : (log b o, o / b ^ log b o) ∈ CNF b (o % b ^ log b o) → 0 < (log b o, o / b ^ log b o).2
⊢ 0 < (log b o, o / b ^ log b o).2
case tail
b o✝ : Ordinal.{u}
x : Ordinal.{u} × Ordinal.{u}
o : Ordinal.{u}
ho : o ≠ 0
IH : x ∈ CNF b (o % b ^ log b o) → 0 < x.2
h : Mem x (CNF b (o % b ^ log b o))
⊢ 0 < x.2 | case tail
b o✝ : Ordinal.{u}
x : Ordinal.{u} × Ordinal.{u}
o : Ordinal.{u}
ho : o ≠ 0
IH : x ∈ CNF b (o % b ^ log b o) → 0 < x.2
h : Mem x (CNF b (o % b ^ log b o))
⊢ 0 < x.2 |
Mathlib/SetTheory/Ordinal/CantorNormalForm.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Mathlib.SetTheory.Ordinal.Exponential",
"Mathlib/SetTheory/Ordinal/Exponential.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "@[elab_as_elim]\nnoncomputable def CNFRec (b : Ordinal) {C : Ordinal → Sort*} (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : ∀ o, C o := fun o ↦ by\n by_cases h : o = 0\n · rw [h]; exact H0\n · exact H o h (CNFRec _ H0 H (o % b ^ log b o))\n termination_by o => o\n decreasing_by exact mod_opow_log_lt_self b h",
"end": [
50,
49
],
"full_name": "Ordinal.CNFRec",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem CNFRec_zero {C : Ordinal → Sort*} (b : Ordinal) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : @CNFRec b C H0 H 0 = H0",
"end": [
58,
6
],
"full_name": "Ordinal.CNFRec_zero",
"kind": "commanddeclaration",
"start": [
54,
1
]
},
{
"code": "theorem CNFRec_pos (b : Ordinal) {o : Ordinal} {C : Ordinal → Sort*} (ho : o ≠ 0) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) :\n @CNFRec b C H0 H o = H o ho (@CNFRec b C H0 H _)",
"end": [
64,
83
],
"full_name": "Ordinal.CNFRec_pos",
"kind": "commanddeclaration",
"start": [
62,
1
]
},
{
"code": "@[pp_nodot]\ndef CNF (b o : Ordinal) : List (Ordinal × Ordinal) :=\n CNFRec b [] (fun o _ho IH ↦ (log b o, o / b ^ log b o)::IH) o",
"end": [
76,
64
],
"full_name": "Ordinal.CNF",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "@[simp]\ntheorem CNF_zero (b : Ordinal) : CNF b 0 = []",
"end": [
82,
20
],
"full_name": "Ordinal.CNF_zero",
"kind": "commanddeclaration",
"start": [
80,
1
]
},
{
"code": "theorem CNF_ne_zero {b o : Ordinal} (ho : o ≠ 0) :\n CNF b o = (log b o, o / b ^ log b o)::CNF b (o % b ^ log b o)",
"end": [
89,
22
],
"full_name": "Ordinal.CNF_ne_zero",
"kind": "commanddeclaration",
"start": [
86,
1
]
},
{
"code": "theorem zero_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 0 o = [⟨0, o⟩]",
"end": [
93,
93
],
"full_name": "Ordinal.zero_CNF",
"kind": "commanddeclaration",
"start": [
93,
1
]
},
{
"code": "theorem one_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 1 o = [⟨0, o⟩]",
"end": [
97,
92
],
"full_name": "Ordinal.one_CNF",
"kind": "commanddeclaration",
"start": [
97,
1
]
},
{
"code": "theorem CNF_of_le_one {b o : Ordinal} (hb : b ≤ 1) (ho : o ≠ 0) : CNF b o = [⟨0, o⟩]",
"end": [
104,
21
],
"full_name": "Ordinal.CNF_of_le_one",
"kind": "commanddeclaration",
"start": [
101,
1
]
},
{
"code": "theorem CNF_of_lt {b o : Ordinal} (ho : o ≠ 0) (hb : o < b) : CNF b o = [⟨0, o⟩]",
"end": [
109,
84
],
"full_name": "Ordinal.CNF_of_lt",
"kind": "commanddeclaration",
"start": [
108,
1
]
},
{
"code": "theorem CNF_foldr (b o : Ordinal) : (CNF b o).foldr (fun p r ↦ b ^ p.1 * p.2 + r) 0 = o",
"end": [
116,
74
],
"full_name": "Ordinal.CNF_foldr",
"kind": "commanddeclaration",
"start": [
113,
1
]
},
{
"code": "theorem CNF_fst_le_log {b o : Ordinal.{u}} {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.1 ≤ log b o",
"end": [
129,
74
],
"full_name": "Ordinal.CNF_fst_le_log",
"kind": "commanddeclaration",
"start": [
120,
1
]
},
{
"code": "theorem CNF_fst_le {b o : Ordinal.{u}} {x : Ordinal × Ordinal} (h : x ∈ CNF b o) : x.1 ≤ o",
"end": [
135,
46
],
"full_name": "Ordinal.CNF_fst_le",
"kind": "commanddeclaration",
"start": [
133,
1
]
},
{
"code": "theorem CNF_lt_snd {b o : Ordinal.{u}} {x : Ordinal × Ordinal} : x ∈ CNF b o → 0 < x.2",
"end": [
145,
15
],
"full_name": "Ordinal.CNF_lt_snd",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem CNF_snd_lt {b o : Ordinal.{u}} (hb : 1 < b) {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.2 < b",
"end": [
158,
17
],
"full_name": "Ordinal.CNF_snd_lt",
"kind": "commanddeclaration",
"start": [
149,
1
]
},
{
"code": "theorem CNF_sorted (b o : Ordinal) : ((CNF b o).map Prod.fst).Sorted (· > ·)",
"end": [
174,
83
],
"full_name": "Ordinal.CNF_sorted",
"kind": "commanddeclaration",
"start": [
162,
1
]
}
] | 11 | Ordinal.CNF_lt_snd | [
[
140,
91
],
[
145,
15
]
] | 4 | 8 | · exact IH h | case tail
b o✝ : Ordinal.{u}
x : Ordinal.{u} × Ordinal.{u}
o : Ordinal.{u}
ho : o ≠ 0
IH : x ∈ CNF b (o % b ^ log b o) → 0 < x.2
h : Mem x (CNF b (o % b ^ log b o))
⊢ 0 < x.2 | no goals |
Mathlib/SetTheory/Ordinal/CantorNormalForm.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Mathlib.SetTheory.Ordinal.Exponential",
"Mathlib/SetTheory/Ordinal/Exponential.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "@[elab_as_elim]\nnoncomputable def CNFRec (b : Ordinal) {C : Ordinal → Sort*} (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : ∀ o, C o := fun o ↦ by\n by_cases h : o = 0\n · rw [h]; exact H0\n · exact H o h (CNFRec _ H0 H (o % b ^ log b o))\n termination_by o => o\n decreasing_by exact mod_opow_log_lt_self b h",
"end": [
50,
49
],
"full_name": "Ordinal.CNFRec",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem CNFRec_zero {C : Ordinal → Sort*} (b : Ordinal) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : @CNFRec b C H0 H 0 = H0",
"end": [
58,
6
],
"full_name": "Ordinal.CNFRec_zero",
"kind": "commanddeclaration",
"start": [
54,
1
]
},
{
"code": "theorem CNFRec_pos (b : Ordinal) {o : Ordinal} {C : Ordinal → Sort*} (ho : o ≠ 0) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) :\n @CNFRec b C H0 H o = H o ho (@CNFRec b C H0 H _)",
"end": [
64,
83
],
"full_name": "Ordinal.CNFRec_pos",
"kind": "commanddeclaration",
"start": [
62,
1
]
},
{
"code": "@[pp_nodot]\ndef CNF (b o : Ordinal) : List (Ordinal × Ordinal) :=\n CNFRec b [] (fun o _ho IH ↦ (log b o, o / b ^ log b o)::IH) o",
"end": [
76,
64
],
"full_name": "Ordinal.CNF",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "@[simp]\ntheorem CNF_zero (b : Ordinal) : CNF b 0 = []",
"end": [
82,
20
],
"full_name": "Ordinal.CNF_zero",
"kind": "commanddeclaration",
"start": [
80,
1
]
},
{
"code": "theorem CNF_ne_zero {b o : Ordinal} (ho : o ≠ 0) :\n CNF b o = (log b o, o / b ^ log b o)::CNF b (o % b ^ log b o)",
"end": [
89,
22
],
"full_name": "Ordinal.CNF_ne_zero",
"kind": "commanddeclaration",
"start": [
86,
1
]
},
{
"code": "theorem zero_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 0 o = [⟨0, o⟩]",
"end": [
93,
93
],
"full_name": "Ordinal.zero_CNF",
"kind": "commanddeclaration",
"start": [
93,
1
]
},
{
"code": "theorem one_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 1 o = [⟨0, o⟩]",
"end": [
97,
92
],
"full_name": "Ordinal.one_CNF",
"kind": "commanddeclaration",
"start": [
97,
1
]
},
{
"code": "theorem CNF_of_le_one {b o : Ordinal} (hb : b ≤ 1) (ho : o ≠ 0) : CNF b o = [⟨0, o⟩]",
"end": [
104,
21
],
"full_name": "Ordinal.CNF_of_le_one",
"kind": "commanddeclaration",
"start": [
101,
1
]
},
{
"code": "theorem CNF_of_lt {b o : Ordinal} (ho : o ≠ 0) (hb : o < b) : CNF b o = [⟨0, o⟩]",
"end": [
109,
84
],
"full_name": "Ordinal.CNF_of_lt",
"kind": "commanddeclaration",
"start": [
108,
1
]
},
{
"code": "theorem CNF_foldr (b o : Ordinal) : (CNF b o).foldr (fun p r ↦ b ^ p.1 * p.2 + r) 0 = o",
"end": [
116,
74
],
"full_name": "Ordinal.CNF_foldr",
"kind": "commanddeclaration",
"start": [
113,
1
]
},
{
"code": "theorem CNF_fst_le_log {b o : Ordinal.{u}} {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.1 ≤ log b o",
"end": [
129,
74
],
"full_name": "Ordinal.CNF_fst_le_log",
"kind": "commanddeclaration",
"start": [
120,
1
]
},
{
"code": "theorem CNF_fst_le {b o : Ordinal.{u}} {x : Ordinal × Ordinal} (h : x ∈ CNF b o) : x.1 ≤ o",
"end": [
135,
46
],
"full_name": "Ordinal.CNF_fst_le",
"kind": "commanddeclaration",
"start": [
133,
1
]
},
{
"code": "theorem CNF_lt_snd {b o : Ordinal.{u}} {x : Ordinal × Ordinal} : x ∈ CNF b o → 0 < x.2",
"end": [
145,
15
],
"full_name": "Ordinal.CNF_lt_snd",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem CNF_snd_lt {b o : Ordinal.{u}} (hb : 1 < b) {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.2 < b",
"end": [
158,
17
],
"full_name": "Ordinal.CNF_snd_lt",
"kind": "commanddeclaration",
"start": [
149,
1
]
},
{
"code": "theorem CNF_sorted (b o : Ordinal) : ((CNF b o).map Prod.fst).Sorted (· > ·)",
"end": [
174,
83
],
"full_name": "Ordinal.CNF_sorted",
"kind": "commanddeclaration",
"start": [
162,
1
]
}
] | 11 | Ordinal.CNF_lt_snd | [
[
140,
91
],
[
145,
15
]
] | 5 | 8 | simp | b o : Ordinal.{u}
x : Ordinal.{u} × Ordinal.{u}
⊢ x ∈ CNF b 0 → 0 < x.2 | no goals |
Mathlib/SetTheory/Ordinal/CantorNormalForm.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Mathlib.SetTheory.Ordinal.Exponential",
"Mathlib/SetTheory/Ordinal/Exponential.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "@[elab_as_elim]\nnoncomputable def CNFRec (b : Ordinal) {C : Ordinal → Sort*} (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : ∀ o, C o := fun o ↦ by\n by_cases h : o = 0\n · rw [h]; exact H0\n · exact H o h (CNFRec _ H0 H (o % b ^ log b o))\n termination_by o => o\n decreasing_by exact mod_opow_log_lt_self b h",
"end": [
50,
49
],
"full_name": "Ordinal.CNFRec",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem CNFRec_zero {C : Ordinal → Sort*} (b : Ordinal) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : @CNFRec b C H0 H 0 = H0",
"end": [
58,
6
],
"full_name": "Ordinal.CNFRec_zero",
"kind": "commanddeclaration",
"start": [
54,
1
]
},
{
"code": "theorem CNFRec_pos (b : Ordinal) {o : Ordinal} {C : Ordinal → Sort*} (ho : o ≠ 0) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) :\n @CNFRec b C H0 H o = H o ho (@CNFRec b C H0 H _)",
"end": [
64,
83
],
"full_name": "Ordinal.CNFRec_pos",
"kind": "commanddeclaration",
"start": [
62,
1
]
},
{
"code": "@[pp_nodot]\ndef CNF (b o : Ordinal) : List (Ordinal × Ordinal) :=\n CNFRec b [] (fun o _ho IH ↦ (log b o, o / b ^ log b o)::IH) o",
"end": [
76,
64
],
"full_name": "Ordinal.CNF",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "@[simp]\ntheorem CNF_zero (b : Ordinal) : CNF b 0 = []",
"end": [
82,
20
],
"full_name": "Ordinal.CNF_zero",
"kind": "commanddeclaration",
"start": [
80,
1
]
},
{
"code": "theorem CNF_ne_zero {b o : Ordinal} (ho : o ≠ 0) :\n CNF b o = (log b o, o / b ^ log b o)::CNF b (o % b ^ log b o)",
"end": [
89,
22
],
"full_name": "Ordinal.CNF_ne_zero",
"kind": "commanddeclaration",
"start": [
86,
1
]
},
{
"code": "theorem zero_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 0 o = [⟨0, o⟩]",
"end": [
93,
93
],
"full_name": "Ordinal.zero_CNF",
"kind": "commanddeclaration",
"start": [
93,
1
]
},
{
"code": "theorem one_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 1 o = [⟨0, o⟩]",
"end": [
97,
92
],
"full_name": "Ordinal.one_CNF",
"kind": "commanddeclaration",
"start": [
97,
1
]
},
{
"code": "theorem CNF_of_le_one {b o : Ordinal} (hb : b ≤ 1) (ho : o ≠ 0) : CNF b o = [⟨0, o⟩]",
"end": [
104,
21
],
"full_name": "Ordinal.CNF_of_le_one",
"kind": "commanddeclaration",
"start": [
101,
1
]
},
{
"code": "theorem CNF_of_lt {b o : Ordinal} (ho : o ≠ 0) (hb : o < b) : CNF b o = [⟨0, o⟩]",
"end": [
109,
84
],
"full_name": "Ordinal.CNF_of_lt",
"kind": "commanddeclaration",
"start": [
108,
1
]
},
{
"code": "theorem CNF_foldr (b o : Ordinal) : (CNF b o).foldr (fun p r ↦ b ^ p.1 * p.2 + r) 0 = o",
"end": [
116,
74
],
"full_name": "Ordinal.CNF_foldr",
"kind": "commanddeclaration",
"start": [
113,
1
]
},
{
"code": "theorem CNF_fst_le_log {b o : Ordinal.{u}} {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.1 ≤ log b o",
"end": [
129,
74
],
"full_name": "Ordinal.CNF_fst_le_log",
"kind": "commanddeclaration",
"start": [
120,
1
]
},
{
"code": "theorem CNF_fst_le {b o : Ordinal.{u}} {x : Ordinal × Ordinal} (h : x ∈ CNF b o) : x.1 ≤ o",
"end": [
135,
46
],
"full_name": "Ordinal.CNF_fst_le",
"kind": "commanddeclaration",
"start": [
133,
1
]
},
{
"code": "theorem CNF_lt_snd {b o : Ordinal.{u}} {x : Ordinal × Ordinal} : x ∈ CNF b o → 0 < x.2",
"end": [
145,
15
],
"full_name": "Ordinal.CNF_lt_snd",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem CNF_snd_lt {b o : Ordinal.{u}} (hb : 1 < b) {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.2 < b",
"end": [
158,
17
],
"full_name": "Ordinal.CNF_snd_lt",
"kind": "commanddeclaration",
"start": [
149,
1
]
},
{
"code": "theorem CNF_sorted (b o : Ordinal) : ((CNF b o).map Prod.fst).Sorted (· > ·)",
"end": [
174,
83
],
"full_name": "Ordinal.CNF_sorted",
"kind": "commanddeclaration",
"start": [
162,
1
]
}
] | 11 | Ordinal.CNF_lt_snd | [
[
140,
91
],
[
145,
15
]
] | 6 | 8 | exact div_opow_log_pos b ho | case head
b o✝ o : Ordinal.{u}
ho : o ≠ 0
IH : (log b o, o / b ^ log b o) ∈ CNF b (o % b ^ log b o) → 0 < (log b o, o / b ^ log b o).2
⊢ 0 < (log b o, o / b ^ log b o).2 | no goals |
Mathlib/SetTheory/Ordinal/CantorNormalForm.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Mathlib.SetTheory.Ordinal.Exponential",
"Mathlib/SetTheory/Ordinal/Exponential.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "@[elab_as_elim]\nnoncomputable def CNFRec (b : Ordinal) {C : Ordinal → Sort*} (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : ∀ o, C o := fun o ↦ by\n by_cases h : o = 0\n · rw [h]; exact H0\n · exact H o h (CNFRec _ H0 H (o % b ^ log b o))\n termination_by o => o\n decreasing_by exact mod_opow_log_lt_self b h",
"end": [
50,
49
],
"full_name": "Ordinal.CNFRec",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem CNFRec_zero {C : Ordinal → Sort*} (b : Ordinal) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : @CNFRec b C H0 H 0 = H0",
"end": [
58,
6
],
"full_name": "Ordinal.CNFRec_zero",
"kind": "commanddeclaration",
"start": [
54,
1
]
},
{
"code": "theorem CNFRec_pos (b : Ordinal) {o : Ordinal} {C : Ordinal → Sort*} (ho : o ≠ 0) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) :\n @CNFRec b C H0 H o = H o ho (@CNFRec b C H0 H _)",
"end": [
64,
83
],
"full_name": "Ordinal.CNFRec_pos",
"kind": "commanddeclaration",
"start": [
62,
1
]
},
{
"code": "@[pp_nodot]\ndef CNF (b o : Ordinal) : List (Ordinal × Ordinal) :=\n CNFRec b [] (fun o _ho IH ↦ (log b o, o / b ^ log b o)::IH) o",
"end": [
76,
64
],
"full_name": "Ordinal.CNF",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "@[simp]\ntheorem CNF_zero (b : Ordinal) : CNF b 0 = []",
"end": [
82,
20
],
"full_name": "Ordinal.CNF_zero",
"kind": "commanddeclaration",
"start": [
80,
1
]
},
{
"code": "theorem CNF_ne_zero {b o : Ordinal} (ho : o ≠ 0) :\n CNF b o = (log b o, o / b ^ log b o)::CNF b (o % b ^ log b o)",
"end": [
89,
22
],
"full_name": "Ordinal.CNF_ne_zero",
"kind": "commanddeclaration",
"start": [
86,
1
]
},
{
"code": "theorem zero_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 0 o = [⟨0, o⟩]",
"end": [
93,
93
],
"full_name": "Ordinal.zero_CNF",
"kind": "commanddeclaration",
"start": [
93,
1
]
},
{
"code": "theorem one_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 1 o = [⟨0, o⟩]",
"end": [
97,
92
],
"full_name": "Ordinal.one_CNF",
"kind": "commanddeclaration",
"start": [
97,
1
]
},
{
"code": "theorem CNF_of_le_one {b o : Ordinal} (hb : b ≤ 1) (ho : o ≠ 0) : CNF b o = [⟨0, o⟩]",
"end": [
104,
21
],
"full_name": "Ordinal.CNF_of_le_one",
"kind": "commanddeclaration",
"start": [
101,
1
]
},
{
"code": "theorem CNF_of_lt {b o : Ordinal} (ho : o ≠ 0) (hb : o < b) : CNF b o = [⟨0, o⟩]",
"end": [
109,
84
],
"full_name": "Ordinal.CNF_of_lt",
"kind": "commanddeclaration",
"start": [
108,
1
]
},
{
"code": "theorem CNF_foldr (b o : Ordinal) : (CNF b o).foldr (fun p r ↦ b ^ p.1 * p.2 + r) 0 = o",
"end": [
116,
74
],
"full_name": "Ordinal.CNF_foldr",
"kind": "commanddeclaration",
"start": [
113,
1
]
},
{
"code": "theorem CNF_fst_le_log {b o : Ordinal.{u}} {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.1 ≤ log b o",
"end": [
129,
74
],
"full_name": "Ordinal.CNF_fst_le_log",
"kind": "commanddeclaration",
"start": [
120,
1
]
},
{
"code": "theorem CNF_fst_le {b o : Ordinal.{u}} {x : Ordinal × Ordinal} (h : x ∈ CNF b o) : x.1 ≤ o",
"end": [
135,
46
],
"full_name": "Ordinal.CNF_fst_le",
"kind": "commanddeclaration",
"start": [
133,
1
]
},
{
"code": "theorem CNF_lt_snd {b o : Ordinal.{u}} {x : Ordinal × Ordinal} : x ∈ CNF b o → 0 < x.2",
"end": [
145,
15
],
"full_name": "Ordinal.CNF_lt_snd",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem CNF_snd_lt {b o : Ordinal.{u}} (hb : 1 < b) {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.2 < b",
"end": [
158,
17
],
"full_name": "Ordinal.CNF_snd_lt",
"kind": "commanddeclaration",
"start": [
149,
1
]
},
{
"code": "theorem CNF_sorted (b o : Ordinal) : ((CNF b o).map Prod.fst).Sorted (· > ·)",
"end": [
174,
83
],
"full_name": "Ordinal.CNF_sorted",
"kind": "commanddeclaration",
"start": [
162,
1
]
}
] | 11 | Ordinal.CNF_lt_snd | [
[
140,
91
],
[
145,
15
]
] | 7 | 8 | exact IH h | case tail
b o✝ : Ordinal.{u}
x : Ordinal.{u} × Ordinal.{u}
o : Ordinal.{u}
ho : o ≠ 0
IH : x ∈ CNF b (o % b ^ log b o) → 0 < x.2
h : Mem x (CNF b (o % b ^ log b o))
⊢ 0 < x.2 | no goals |
Mathlib/SetTheory/Ordinal/CantorNormalForm.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Mathlib.SetTheory.Ordinal.Exponential",
"Mathlib/SetTheory/Ordinal/Exponential.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "@[elab_as_elim]\nnoncomputable def CNFRec (b : Ordinal) {C : Ordinal → Sort*} (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : ∀ o, C o := fun o ↦ by\n by_cases h : o = 0\n · rw [h]; exact H0\n · exact H o h (CNFRec _ H0 H (o % b ^ log b o))\n termination_by o => o\n decreasing_by exact mod_opow_log_lt_self b h",
"end": [
50,
49
],
"full_name": "Ordinal.CNFRec",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem CNFRec_zero {C : Ordinal → Sort*} (b : Ordinal) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : @CNFRec b C H0 H 0 = H0",
"end": [
58,
6
],
"full_name": "Ordinal.CNFRec_zero",
"kind": "commanddeclaration",
"start": [
54,
1
]
},
{
"code": "theorem CNFRec_pos (b : Ordinal) {o : Ordinal} {C : Ordinal → Sort*} (ho : o ≠ 0) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) :\n @CNFRec b C H0 H o = H o ho (@CNFRec b C H0 H _)",
"end": [
64,
83
],
"full_name": "Ordinal.CNFRec_pos",
"kind": "commanddeclaration",
"start": [
62,
1
]
},
{
"code": "@[pp_nodot]\ndef CNF (b o : Ordinal) : List (Ordinal × Ordinal) :=\n CNFRec b [] (fun o _ho IH ↦ (log b o, o / b ^ log b o)::IH) o",
"end": [
76,
64
],
"full_name": "Ordinal.CNF",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "@[simp]\ntheorem CNF_zero (b : Ordinal) : CNF b 0 = []",
"end": [
82,
20
],
"full_name": "Ordinal.CNF_zero",
"kind": "commanddeclaration",
"start": [
80,
1
]
},
{
"code": "theorem CNF_ne_zero {b o : Ordinal} (ho : o ≠ 0) :\n CNF b o = (log b o, o / b ^ log b o)::CNF b (o % b ^ log b o)",
"end": [
89,
22
],
"full_name": "Ordinal.CNF_ne_zero",
"kind": "commanddeclaration",
"start": [
86,
1
]
},
{
"code": "theorem zero_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 0 o = [⟨0, o⟩]",
"end": [
93,
93
],
"full_name": "Ordinal.zero_CNF",
"kind": "commanddeclaration",
"start": [
93,
1
]
},
{
"code": "theorem one_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 1 o = [⟨0, o⟩]",
"end": [
97,
92
],
"full_name": "Ordinal.one_CNF",
"kind": "commanddeclaration",
"start": [
97,
1
]
},
{
"code": "theorem CNF_of_le_one {b o : Ordinal} (hb : b ≤ 1) (ho : o ≠ 0) : CNF b o = [⟨0, o⟩]",
"end": [
104,
21
],
"full_name": "Ordinal.CNF_of_le_one",
"kind": "commanddeclaration",
"start": [
101,
1
]
},
{
"code": "theorem CNF_of_lt {b o : Ordinal} (ho : o ≠ 0) (hb : o < b) : CNF b o = [⟨0, o⟩]",
"end": [
109,
84
],
"full_name": "Ordinal.CNF_of_lt",
"kind": "commanddeclaration",
"start": [
108,
1
]
},
{
"code": "theorem CNF_foldr (b o : Ordinal) : (CNF b o).foldr (fun p r ↦ b ^ p.1 * p.2 + r) 0 = o",
"end": [
116,
74
],
"full_name": "Ordinal.CNF_foldr",
"kind": "commanddeclaration",
"start": [
113,
1
]
},
{
"code": "theorem CNF_fst_le_log {b o : Ordinal.{u}} {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.1 ≤ log b o",
"end": [
129,
74
],
"full_name": "Ordinal.CNF_fst_le_log",
"kind": "commanddeclaration",
"start": [
120,
1
]
},
{
"code": "theorem CNF_fst_le {b o : Ordinal.{u}} {x : Ordinal × Ordinal} (h : x ∈ CNF b o) : x.1 ≤ o",
"end": [
135,
46
],
"full_name": "Ordinal.CNF_fst_le",
"kind": "commanddeclaration",
"start": [
133,
1
]
},
{
"code": "theorem CNF_lt_snd {b o : Ordinal.{u}} {x : Ordinal × Ordinal} : x ∈ CNF b o → 0 < x.2",
"end": [
145,
15
],
"full_name": "Ordinal.CNF_lt_snd",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem CNF_snd_lt {b o : Ordinal.{u}} (hb : 1 < b) {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.2 < b",
"end": [
158,
17
],
"full_name": "Ordinal.CNF_snd_lt",
"kind": "commanddeclaration",
"start": [
149,
1
]
},
{
"code": "theorem CNF_sorted (b o : Ordinal) : ((CNF b o).map Prod.fst).Sorted (· > ·)",
"end": [
174,
83
],
"full_name": "Ordinal.CNF_sorted",
"kind": "commanddeclaration",
"start": [
162,
1
]
}
] | 12 | Ordinal.CNF_snd_lt | [
[
151,
30
],
[
158,
17
]
] | 0 | 12 | refine CNFRec b ?_ (fun o ho IH ↦ ?_) o | b o : Ordinal.{u}
hb : 1 < b
x : Ordinal.{u} × Ordinal.{u}
⊢ x ∈ CNF b o → x.2 < b | case refine_1
b o : Ordinal.{u}
hb : 1 < b
x : Ordinal.{u} × Ordinal.{u}
⊢ x ∈ CNF b 0 → x.2 < b
case refine_2
b o✝ : Ordinal.{u}
hb : 1 < b
x : Ordinal.{u} × Ordinal.{u}
o : Ordinal.{u}
ho : o ≠ 0
IH : x ∈ CNF b (o % b ^ log b o) → x.2 < b
⊢ x ∈ CNF b o → x.2 < b |
Mathlib/SetTheory/Ordinal/CantorNormalForm.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Mathlib.SetTheory.Ordinal.Exponential",
"Mathlib/SetTheory/Ordinal/Exponential.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "@[elab_as_elim]\nnoncomputable def CNFRec (b : Ordinal) {C : Ordinal → Sort*} (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : ∀ o, C o := fun o ↦ by\n by_cases h : o = 0\n · rw [h]; exact H0\n · exact H o h (CNFRec _ H0 H (o % b ^ log b o))\n termination_by o => o\n decreasing_by exact mod_opow_log_lt_self b h",
"end": [
50,
49
],
"full_name": "Ordinal.CNFRec",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem CNFRec_zero {C : Ordinal → Sort*} (b : Ordinal) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : @CNFRec b C H0 H 0 = H0",
"end": [
58,
6
],
"full_name": "Ordinal.CNFRec_zero",
"kind": "commanddeclaration",
"start": [
54,
1
]
},
{
"code": "theorem CNFRec_pos (b : Ordinal) {o : Ordinal} {C : Ordinal → Sort*} (ho : o ≠ 0) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) :\n @CNFRec b C H0 H o = H o ho (@CNFRec b C H0 H _)",
"end": [
64,
83
],
"full_name": "Ordinal.CNFRec_pos",
"kind": "commanddeclaration",
"start": [
62,
1
]
},
{
"code": "@[pp_nodot]\ndef CNF (b o : Ordinal) : List (Ordinal × Ordinal) :=\n CNFRec b [] (fun o _ho IH ↦ (log b o, o / b ^ log b o)::IH) o",
"end": [
76,
64
],
"full_name": "Ordinal.CNF",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "@[simp]\ntheorem CNF_zero (b : Ordinal) : CNF b 0 = []",
"end": [
82,
20
],
"full_name": "Ordinal.CNF_zero",
"kind": "commanddeclaration",
"start": [
80,
1
]
},
{
"code": "theorem CNF_ne_zero {b o : Ordinal} (ho : o ≠ 0) :\n CNF b o = (log b o, o / b ^ log b o)::CNF b (o % b ^ log b o)",
"end": [
89,
22
],
"full_name": "Ordinal.CNF_ne_zero",
"kind": "commanddeclaration",
"start": [
86,
1
]
},
{
"code": "theorem zero_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 0 o = [⟨0, o⟩]",
"end": [
93,
93
],
"full_name": "Ordinal.zero_CNF",
"kind": "commanddeclaration",
"start": [
93,
1
]
},
{
"code": "theorem one_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 1 o = [⟨0, o⟩]",
"end": [
97,
92
],
"full_name": "Ordinal.one_CNF",
"kind": "commanddeclaration",
"start": [
97,
1
]
},
{
"code": "theorem CNF_of_le_one {b o : Ordinal} (hb : b ≤ 1) (ho : o ≠ 0) : CNF b o = [⟨0, o⟩]",
"end": [
104,
21
],
"full_name": "Ordinal.CNF_of_le_one",
"kind": "commanddeclaration",
"start": [
101,
1
]
},
{
"code": "theorem CNF_of_lt {b o : Ordinal} (ho : o ≠ 0) (hb : o < b) : CNF b o = [⟨0, o⟩]",
"end": [
109,
84
],
"full_name": "Ordinal.CNF_of_lt",
"kind": "commanddeclaration",
"start": [
108,
1
]
},
{
"code": "theorem CNF_foldr (b o : Ordinal) : (CNF b o).foldr (fun p r ↦ b ^ p.1 * p.2 + r) 0 = o",
"end": [
116,
74
],
"full_name": "Ordinal.CNF_foldr",
"kind": "commanddeclaration",
"start": [
113,
1
]
},
{
"code": "theorem CNF_fst_le_log {b o : Ordinal.{u}} {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.1 ≤ log b o",
"end": [
129,
74
],
"full_name": "Ordinal.CNF_fst_le_log",
"kind": "commanddeclaration",
"start": [
120,
1
]
},
{
"code": "theorem CNF_fst_le {b o : Ordinal.{u}} {x : Ordinal × Ordinal} (h : x ∈ CNF b o) : x.1 ≤ o",
"end": [
135,
46
],
"full_name": "Ordinal.CNF_fst_le",
"kind": "commanddeclaration",
"start": [
133,
1
]
},
{
"code": "theorem CNF_lt_snd {b o : Ordinal.{u}} {x : Ordinal × Ordinal} : x ∈ CNF b o → 0 < x.2",
"end": [
145,
15
],
"full_name": "Ordinal.CNF_lt_snd",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem CNF_snd_lt {b o : Ordinal.{u}} (hb : 1 < b) {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.2 < b",
"end": [
158,
17
],
"full_name": "Ordinal.CNF_snd_lt",
"kind": "commanddeclaration",
"start": [
149,
1
]
},
{
"code": "theorem CNF_sorted (b o : Ordinal) : ((CNF b o).map Prod.fst).Sorted (· > ·)",
"end": [
174,
83
],
"full_name": "Ordinal.CNF_sorted",
"kind": "commanddeclaration",
"start": [
162,
1
]
}
] | 12 | Ordinal.CNF_snd_lt | [
[
151,
30
],
[
158,
17
]
] | 1 | 12 | · simp only [CNF_zero, not_mem_nil, IsEmpty.forall_iff] | case refine_1
b o : Ordinal.{u}
hb : 1 < b
x : Ordinal.{u} × Ordinal.{u}
⊢ x ∈ CNF b 0 → x.2 < b
case refine_2
b o✝ : Ordinal.{u}
hb : 1 < b
x : Ordinal.{u} × Ordinal.{u}
o : Ordinal.{u}
ho : o ≠ 0
IH : x ∈ CNF b (o % b ^ log b o) → x.2 < b
⊢ x ∈ CNF b o → x.2 < b | case refine_2
b o✝ : Ordinal.{u}
hb : 1 < b
x : Ordinal.{u} × Ordinal.{u}
o : Ordinal.{u}
ho : o ≠ 0
IH : x ∈ CNF b (o % b ^ log b o) → x.2 < b
⊢ x ∈ CNF b o → x.2 < b |
Mathlib/SetTheory/Ordinal/CantorNormalForm.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Mathlib.SetTheory.Ordinal.Exponential",
"Mathlib/SetTheory/Ordinal/Exponential.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "@[elab_as_elim]\nnoncomputable def CNFRec (b : Ordinal) {C : Ordinal → Sort*} (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : ∀ o, C o := fun o ↦ by\n by_cases h : o = 0\n · rw [h]; exact H0\n · exact H o h (CNFRec _ H0 H (o % b ^ log b o))\n termination_by o => o\n decreasing_by exact mod_opow_log_lt_self b h",
"end": [
50,
49
],
"full_name": "Ordinal.CNFRec",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem CNFRec_zero {C : Ordinal → Sort*} (b : Ordinal) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : @CNFRec b C H0 H 0 = H0",
"end": [
58,
6
],
"full_name": "Ordinal.CNFRec_zero",
"kind": "commanddeclaration",
"start": [
54,
1
]
},
{
"code": "theorem CNFRec_pos (b : Ordinal) {o : Ordinal} {C : Ordinal → Sort*} (ho : o ≠ 0) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) :\n @CNFRec b C H0 H o = H o ho (@CNFRec b C H0 H _)",
"end": [
64,
83
],
"full_name": "Ordinal.CNFRec_pos",
"kind": "commanddeclaration",
"start": [
62,
1
]
},
{
"code": "@[pp_nodot]\ndef CNF (b o : Ordinal) : List (Ordinal × Ordinal) :=\n CNFRec b [] (fun o _ho IH ↦ (log b o, o / b ^ log b o)::IH) o",
"end": [
76,
64
],
"full_name": "Ordinal.CNF",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "@[simp]\ntheorem CNF_zero (b : Ordinal) : CNF b 0 = []",
"end": [
82,
20
],
"full_name": "Ordinal.CNF_zero",
"kind": "commanddeclaration",
"start": [
80,
1
]
},
{
"code": "theorem CNF_ne_zero {b o : Ordinal} (ho : o ≠ 0) :\n CNF b o = (log b o, o / b ^ log b o)::CNF b (o % b ^ log b o)",
"end": [
89,
22
],
"full_name": "Ordinal.CNF_ne_zero",
"kind": "commanddeclaration",
"start": [
86,
1
]
},
{
"code": "theorem zero_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 0 o = [⟨0, o⟩]",
"end": [
93,
93
],
"full_name": "Ordinal.zero_CNF",
"kind": "commanddeclaration",
"start": [
93,
1
]
},
{
"code": "theorem one_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 1 o = [⟨0, o⟩]",
"end": [
97,
92
],
"full_name": "Ordinal.one_CNF",
"kind": "commanddeclaration",
"start": [
97,
1
]
},
{
"code": "theorem CNF_of_le_one {b o : Ordinal} (hb : b ≤ 1) (ho : o ≠ 0) : CNF b o = [⟨0, o⟩]",
"end": [
104,
21
],
"full_name": "Ordinal.CNF_of_le_one",
"kind": "commanddeclaration",
"start": [
101,
1
]
},
{
"code": "theorem CNF_of_lt {b o : Ordinal} (ho : o ≠ 0) (hb : o < b) : CNF b o = [⟨0, o⟩]",
"end": [
109,
84
],
"full_name": "Ordinal.CNF_of_lt",
"kind": "commanddeclaration",
"start": [
108,
1
]
},
{
"code": "theorem CNF_foldr (b o : Ordinal) : (CNF b o).foldr (fun p r ↦ b ^ p.1 * p.2 + r) 0 = o",
"end": [
116,
74
],
"full_name": "Ordinal.CNF_foldr",
"kind": "commanddeclaration",
"start": [
113,
1
]
},
{
"code": "theorem CNF_fst_le_log {b o : Ordinal.{u}} {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.1 ≤ log b o",
"end": [
129,
74
],
"full_name": "Ordinal.CNF_fst_le_log",
"kind": "commanddeclaration",
"start": [
120,
1
]
},
{
"code": "theorem CNF_fst_le {b o : Ordinal.{u}} {x : Ordinal × Ordinal} (h : x ∈ CNF b o) : x.1 ≤ o",
"end": [
135,
46
],
"full_name": "Ordinal.CNF_fst_le",
"kind": "commanddeclaration",
"start": [
133,
1
]
},
{
"code": "theorem CNF_lt_snd {b o : Ordinal.{u}} {x : Ordinal × Ordinal} : x ∈ CNF b o → 0 < x.2",
"end": [
145,
15
],
"full_name": "Ordinal.CNF_lt_snd",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem CNF_snd_lt {b o : Ordinal.{u}} (hb : 1 < b) {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.2 < b",
"end": [
158,
17
],
"full_name": "Ordinal.CNF_snd_lt",
"kind": "commanddeclaration",
"start": [
149,
1
]
},
{
"code": "theorem CNF_sorted (b o : Ordinal) : ((CNF b o).map Prod.fst).Sorted (· > ·)",
"end": [
174,
83
],
"full_name": "Ordinal.CNF_sorted",
"kind": "commanddeclaration",
"start": [
162,
1
]
}
] | 12 | Ordinal.CNF_snd_lt | [
[
151,
30
],
[
158,
17
]
] | 2 | 12 | · rw [CNF_ne_zero ho]
intro h
cases' (mem_cons.mp h) with h h
· rw [h]; simpa only using div_opow_log_lt o hb
· exact IH h | case refine_2
b o✝ : Ordinal.{u}
hb : 1 < b
x : Ordinal.{u} × Ordinal.{u}
o : Ordinal.{u}
ho : o ≠ 0
IH : x ∈ CNF b (o % b ^ log b o) → x.2 < b
⊢ x ∈ CNF b o → x.2 < b | no goals |
Mathlib/SetTheory/Ordinal/CantorNormalForm.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Mathlib.SetTheory.Ordinal.Exponential",
"Mathlib/SetTheory/Ordinal/Exponential.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "@[elab_as_elim]\nnoncomputable def CNFRec (b : Ordinal) {C : Ordinal → Sort*} (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : ∀ o, C o := fun o ↦ by\n by_cases h : o = 0\n · rw [h]; exact H0\n · exact H o h (CNFRec _ H0 H (o % b ^ log b o))\n termination_by o => o\n decreasing_by exact mod_opow_log_lt_self b h",
"end": [
50,
49
],
"full_name": "Ordinal.CNFRec",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem CNFRec_zero {C : Ordinal → Sort*} (b : Ordinal) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : @CNFRec b C H0 H 0 = H0",
"end": [
58,
6
],
"full_name": "Ordinal.CNFRec_zero",
"kind": "commanddeclaration",
"start": [
54,
1
]
},
{
"code": "theorem CNFRec_pos (b : Ordinal) {o : Ordinal} {C : Ordinal → Sort*} (ho : o ≠ 0) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) :\n @CNFRec b C H0 H o = H o ho (@CNFRec b C H0 H _)",
"end": [
64,
83
],
"full_name": "Ordinal.CNFRec_pos",
"kind": "commanddeclaration",
"start": [
62,
1
]
},
{
"code": "@[pp_nodot]\ndef CNF (b o : Ordinal) : List (Ordinal × Ordinal) :=\n CNFRec b [] (fun o _ho IH ↦ (log b o, o / b ^ log b o)::IH) o",
"end": [
76,
64
],
"full_name": "Ordinal.CNF",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "@[simp]\ntheorem CNF_zero (b : Ordinal) : CNF b 0 = []",
"end": [
82,
20
],
"full_name": "Ordinal.CNF_zero",
"kind": "commanddeclaration",
"start": [
80,
1
]
},
{
"code": "theorem CNF_ne_zero {b o : Ordinal} (ho : o ≠ 0) :\n CNF b o = (log b o, o / b ^ log b o)::CNF b (o % b ^ log b o)",
"end": [
89,
22
],
"full_name": "Ordinal.CNF_ne_zero",
"kind": "commanddeclaration",
"start": [
86,
1
]
},
{
"code": "theorem zero_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 0 o = [⟨0, o⟩]",
"end": [
93,
93
],
"full_name": "Ordinal.zero_CNF",
"kind": "commanddeclaration",
"start": [
93,
1
]
},
{
"code": "theorem one_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 1 o = [⟨0, o⟩]",
"end": [
97,
92
],
"full_name": "Ordinal.one_CNF",
"kind": "commanddeclaration",
"start": [
97,
1
]
},
{
"code": "theorem CNF_of_le_one {b o : Ordinal} (hb : b ≤ 1) (ho : o ≠ 0) : CNF b o = [⟨0, o⟩]",
"end": [
104,
21
],
"full_name": "Ordinal.CNF_of_le_one",
"kind": "commanddeclaration",
"start": [
101,
1
]
},
{
"code": "theorem CNF_of_lt {b o : Ordinal} (ho : o ≠ 0) (hb : o < b) : CNF b o = [⟨0, o⟩]",
"end": [
109,
84
],
"full_name": "Ordinal.CNF_of_lt",
"kind": "commanddeclaration",
"start": [
108,
1
]
},
{
"code": "theorem CNF_foldr (b o : Ordinal) : (CNF b o).foldr (fun p r ↦ b ^ p.1 * p.2 + r) 0 = o",
"end": [
116,
74
],
"full_name": "Ordinal.CNF_foldr",
"kind": "commanddeclaration",
"start": [
113,
1
]
},
{
"code": "theorem CNF_fst_le_log {b o : Ordinal.{u}} {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.1 ≤ log b o",
"end": [
129,
74
],
"full_name": "Ordinal.CNF_fst_le_log",
"kind": "commanddeclaration",
"start": [
120,
1
]
},
{
"code": "theorem CNF_fst_le {b o : Ordinal.{u}} {x : Ordinal × Ordinal} (h : x ∈ CNF b o) : x.1 ≤ o",
"end": [
135,
46
],
"full_name": "Ordinal.CNF_fst_le",
"kind": "commanddeclaration",
"start": [
133,
1
]
},
{
"code": "theorem CNF_lt_snd {b o : Ordinal.{u}} {x : Ordinal × Ordinal} : x ∈ CNF b o → 0 < x.2",
"end": [
145,
15
],
"full_name": "Ordinal.CNF_lt_snd",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem CNF_snd_lt {b o : Ordinal.{u}} (hb : 1 < b) {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.2 < b",
"end": [
158,
17
],
"full_name": "Ordinal.CNF_snd_lt",
"kind": "commanddeclaration",
"start": [
149,
1
]
},
{
"code": "theorem CNF_sorted (b o : Ordinal) : ((CNF b o).map Prod.fst).Sorted (· > ·)",
"end": [
174,
83
],
"full_name": "Ordinal.CNF_sorted",
"kind": "commanddeclaration",
"start": [
162,
1
]
}
] | 12 | Ordinal.CNF_snd_lt | [
[
151,
30
],
[
158,
17
]
] | 3 | 12 | simp only [CNF_zero, not_mem_nil, IsEmpty.forall_iff] | case refine_1
b o : Ordinal.{u}
hb : 1 < b
x : Ordinal.{u} × Ordinal.{u}
⊢ x ∈ CNF b 0 → x.2 < b | no goals |
Mathlib/SetTheory/Ordinal/CantorNormalForm.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Mathlib.SetTheory.Ordinal.Exponential",
"Mathlib/SetTheory/Ordinal/Exponential.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "@[elab_as_elim]\nnoncomputable def CNFRec (b : Ordinal) {C : Ordinal → Sort*} (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : ∀ o, C o := fun o ↦ by\n by_cases h : o = 0\n · rw [h]; exact H0\n · exact H o h (CNFRec _ H0 H (o % b ^ log b o))\n termination_by o => o\n decreasing_by exact mod_opow_log_lt_self b h",
"end": [
50,
49
],
"full_name": "Ordinal.CNFRec",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem CNFRec_zero {C : Ordinal → Sort*} (b : Ordinal) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : @CNFRec b C H0 H 0 = H0",
"end": [
58,
6
],
"full_name": "Ordinal.CNFRec_zero",
"kind": "commanddeclaration",
"start": [
54,
1
]
},
{
"code": "theorem CNFRec_pos (b : Ordinal) {o : Ordinal} {C : Ordinal → Sort*} (ho : o ≠ 0) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) :\n @CNFRec b C H0 H o = H o ho (@CNFRec b C H0 H _)",
"end": [
64,
83
],
"full_name": "Ordinal.CNFRec_pos",
"kind": "commanddeclaration",
"start": [
62,
1
]
},
{
"code": "@[pp_nodot]\ndef CNF (b o : Ordinal) : List (Ordinal × Ordinal) :=\n CNFRec b [] (fun o _ho IH ↦ (log b o, o / b ^ log b o)::IH) o",
"end": [
76,
64
],
"full_name": "Ordinal.CNF",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "@[simp]\ntheorem CNF_zero (b : Ordinal) : CNF b 0 = []",
"end": [
82,
20
],
"full_name": "Ordinal.CNF_zero",
"kind": "commanddeclaration",
"start": [
80,
1
]
},
{
"code": "theorem CNF_ne_zero {b o : Ordinal} (ho : o ≠ 0) :\n CNF b o = (log b o, o / b ^ log b o)::CNF b (o % b ^ log b o)",
"end": [
89,
22
],
"full_name": "Ordinal.CNF_ne_zero",
"kind": "commanddeclaration",
"start": [
86,
1
]
},
{
"code": "theorem zero_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 0 o = [⟨0, o⟩]",
"end": [
93,
93
],
"full_name": "Ordinal.zero_CNF",
"kind": "commanddeclaration",
"start": [
93,
1
]
},
{
"code": "theorem one_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 1 o = [⟨0, o⟩]",
"end": [
97,
92
],
"full_name": "Ordinal.one_CNF",
"kind": "commanddeclaration",
"start": [
97,
1
]
},
{
"code": "theorem CNF_of_le_one {b o : Ordinal} (hb : b ≤ 1) (ho : o ≠ 0) : CNF b o = [⟨0, o⟩]",
"end": [
104,
21
],
"full_name": "Ordinal.CNF_of_le_one",
"kind": "commanddeclaration",
"start": [
101,
1
]
},
{
"code": "theorem CNF_of_lt {b o : Ordinal} (ho : o ≠ 0) (hb : o < b) : CNF b o = [⟨0, o⟩]",
"end": [
109,
84
],
"full_name": "Ordinal.CNF_of_lt",
"kind": "commanddeclaration",
"start": [
108,
1
]
},
{
"code": "theorem CNF_foldr (b o : Ordinal) : (CNF b o).foldr (fun p r ↦ b ^ p.1 * p.2 + r) 0 = o",
"end": [
116,
74
],
"full_name": "Ordinal.CNF_foldr",
"kind": "commanddeclaration",
"start": [
113,
1
]
},
{
"code": "theorem CNF_fst_le_log {b o : Ordinal.{u}} {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.1 ≤ log b o",
"end": [
129,
74
],
"full_name": "Ordinal.CNF_fst_le_log",
"kind": "commanddeclaration",
"start": [
120,
1
]
},
{
"code": "theorem CNF_fst_le {b o : Ordinal.{u}} {x : Ordinal × Ordinal} (h : x ∈ CNF b o) : x.1 ≤ o",
"end": [
135,
46
],
"full_name": "Ordinal.CNF_fst_le",
"kind": "commanddeclaration",
"start": [
133,
1
]
},
{
"code": "theorem CNF_lt_snd {b o : Ordinal.{u}} {x : Ordinal × Ordinal} : x ∈ CNF b o → 0 < x.2",
"end": [
145,
15
],
"full_name": "Ordinal.CNF_lt_snd",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem CNF_snd_lt {b o : Ordinal.{u}} (hb : 1 < b) {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.2 < b",
"end": [
158,
17
],
"full_name": "Ordinal.CNF_snd_lt",
"kind": "commanddeclaration",
"start": [
149,
1
]
},
{
"code": "theorem CNF_sorted (b o : Ordinal) : ((CNF b o).map Prod.fst).Sorted (· > ·)",
"end": [
174,
83
],
"full_name": "Ordinal.CNF_sorted",
"kind": "commanddeclaration",
"start": [
162,
1
]
}
] | 12 | Ordinal.CNF_snd_lt | [
[
151,
30
],
[
158,
17
]
] | 4 | 12 | rw [CNF_ne_zero ho] | case refine_2
b o✝ : Ordinal.{u}
hb : 1 < b
x : Ordinal.{u} × Ordinal.{u}
o : Ordinal.{u}
ho : o ≠ 0
IH : x ∈ CNF b (o % b ^ log b o) → x.2 < b
⊢ x ∈ CNF b o → x.2 < b | case refine_2
b o✝ : Ordinal.{u}
hb : 1 < b
x : Ordinal.{u} × Ordinal.{u}
o : Ordinal.{u}
ho : o ≠ 0
IH : x ∈ CNF b (o % b ^ log b o) → x.2 < b
⊢ x ∈ (log b o, o / b ^ log b o) :: CNF b (o % b ^ log b o) → x.2 < b |
Mathlib/SetTheory/Ordinal/CantorNormalForm.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Mathlib.SetTheory.Ordinal.Exponential",
"Mathlib/SetTheory/Ordinal/Exponential.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "@[elab_as_elim]\nnoncomputable def CNFRec (b : Ordinal) {C : Ordinal → Sort*} (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : ∀ o, C o := fun o ↦ by\n by_cases h : o = 0\n · rw [h]; exact H0\n · exact H o h (CNFRec _ H0 H (o % b ^ log b o))\n termination_by o => o\n decreasing_by exact mod_opow_log_lt_self b h",
"end": [
50,
49
],
"full_name": "Ordinal.CNFRec",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem CNFRec_zero {C : Ordinal → Sort*} (b : Ordinal) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : @CNFRec b C H0 H 0 = H0",
"end": [
58,
6
],
"full_name": "Ordinal.CNFRec_zero",
"kind": "commanddeclaration",
"start": [
54,
1
]
},
{
"code": "theorem CNFRec_pos (b : Ordinal) {o : Ordinal} {C : Ordinal → Sort*} (ho : o ≠ 0) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) :\n @CNFRec b C H0 H o = H o ho (@CNFRec b C H0 H _)",
"end": [
64,
83
],
"full_name": "Ordinal.CNFRec_pos",
"kind": "commanddeclaration",
"start": [
62,
1
]
},
{
"code": "@[pp_nodot]\ndef CNF (b o : Ordinal) : List (Ordinal × Ordinal) :=\n CNFRec b [] (fun o _ho IH ↦ (log b o, o / b ^ log b o)::IH) o",
"end": [
76,
64
],
"full_name": "Ordinal.CNF",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "@[simp]\ntheorem CNF_zero (b : Ordinal) : CNF b 0 = []",
"end": [
82,
20
],
"full_name": "Ordinal.CNF_zero",
"kind": "commanddeclaration",
"start": [
80,
1
]
},
{
"code": "theorem CNF_ne_zero {b o : Ordinal} (ho : o ≠ 0) :\n CNF b o = (log b o, o / b ^ log b o)::CNF b (o % b ^ log b o)",
"end": [
89,
22
],
"full_name": "Ordinal.CNF_ne_zero",
"kind": "commanddeclaration",
"start": [
86,
1
]
},
{
"code": "theorem zero_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 0 o = [⟨0, o⟩]",
"end": [
93,
93
],
"full_name": "Ordinal.zero_CNF",
"kind": "commanddeclaration",
"start": [
93,
1
]
},
{
"code": "theorem one_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 1 o = [⟨0, o⟩]",
"end": [
97,
92
],
"full_name": "Ordinal.one_CNF",
"kind": "commanddeclaration",
"start": [
97,
1
]
},
{
"code": "theorem CNF_of_le_one {b o : Ordinal} (hb : b ≤ 1) (ho : o ≠ 0) : CNF b o = [⟨0, o⟩]",
"end": [
104,
21
],
"full_name": "Ordinal.CNF_of_le_one",
"kind": "commanddeclaration",
"start": [
101,
1
]
},
{
"code": "theorem CNF_of_lt {b o : Ordinal} (ho : o ≠ 0) (hb : o < b) : CNF b o = [⟨0, o⟩]",
"end": [
109,
84
],
"full_name": "Ordinal.CNF_of_lt",
"kind": "commanddeclaration",
"start": [
108,
1
]
},
{
"code": "theorem CNF_foldr (b o : Ordinal) : (CNF b o).foldr (fun p r ↦ b ^ p.1 * p.2 + r) 0 = o",
"end": [
116,
74
],
"full_name": "Ordinal.CNF_foldr",
"kind": "commanddeclaration",
"start": [
113,
1
]
},
{
"code": "theorem CNF_fst_le_log {b o : Ordinal.{u}} {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.1 ≤ log b o",
"end": [
129,
74
],
"full_name": "Ordinal.CNF_fst_le_log",
"kind": "commanddeclaration",
"start": [
120,
1
]
},
{
"code": "theorem CNF_fst_le {b o : Ordinal.{u}} {x : Ordinal × Ordinal} (h : x ∈ CNF b o) : x.1 ≤ o",
"end": [
135,
46
],
"full_name": "Ordinal.CNF_fst_le",
"kind": "commanddeclaration",
"start": [
133,
1
]
},
{
"code": "theorem CNF_lt_snd {b o : Ordinal.{u}} {x : Ordinal × Ordinal} : x ∈ CNF b o → 0 < x.2",
"end": [
145,
15
],
"full_name": "Ordinal.CNF_lt_snd",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem CNF_snd_lt {b o : Ordinal.{u}} (hb : 1 < b) {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.2 < b",
"end": [
158,
17
],
"full_name": "Ordinal.CNF_snd_lt",
"kind": "commanddeclaration",
"start": [
149,
1
]
},
{
"code": "theorem CNF_sorted (b o : Ordinal) : ((CNF b o).map Prod.fst).Sorted (· > ·)",
"end": [
174,
83
],
"full_name": "Ordinal.CNF_sorted",
"kind": "commanddeclaration",
"start": [
162,
1
]
}
] | 12 | Ordinal.CNF_snd_lt | [
[
151,
30
],
[
158,
17
]
] | 5 | 12 | intro h | case refine_2
b o✝ : Ordinal.{u}
hb : 1 < b
x : Ordinal.{u} × Ordinal.{u}
o : Ordinal.{u}
ho : o ≠ 0
IH : x ∈ CNF b (o % b ^ log b o) → x.2 < b
⊢ x ∈ (log b o, o / b ^ log b o) :: CNF b (o % b ^ log b o) → x.2 < b | case refine_2
b o✝ : Ordinal.{u}
hb : 1 < b
x : Ordinal.{u} × Ordinal.{u}
o : Ordinal.{u}
ho : o ≠ 0
IH : x ∈ CNF b (o % b ^ log b o) → x.2 < b
h : x ∈ (log b o, o / b ^ log b o) :: CNF b (o % b ^ log b o)
⊢ x.2 < b |
Mathlib/SetTheory/Ordinal/CantorNormalForm.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Mathlib.SetTheory.Ordinal.Exponential",
"Mathlib/SetTheory/Ordinal/Exponential.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "@[elab_as_elim]\nnoncomputable def CNFRec (b : Ordinal) {C : Ordinal → Sort*} (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : ∀ o, C o := fun o ↦ by\n by_cases h : o = 0\n · rw [h]; exact H0\n · exact H o h (CNFRec _ H0 H (o % b ^ log b o))\n termination_by o => o\n decreasing_by exact mod_opow_log_lt_self b h",
"end": [
50,
49
],
"full_name": "Ordinal.CNFRec",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem CNFRec_zero {C : Ordinal → Sort*} (b : Ordinal) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : @CNFRec b C H0 H 0 = H0",
"end": [
58,
6
],
"full_name": "Ordinal.CNFRec_zero",
"kind": "commanddeclaration",
"start": [
54,
1
]
},
{
"code": "theorem CNFRec_pos (b : Ordinal) {o : Ordinal} {C : Ordinal → Sort*} (ho : o ≠ 0) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) :\n @CNFRec b C H0 H o = H o ho (@CNFRec b C H0 H _)",
"end": [
64,
83
],
"full_name": "Ordinal.CNFRec_pos",
"kind": "commanddeclaration",
"start": [
62,
1
]
},
{
"code": "@[pp_nodot]\ndef CNF (b o : Ordinal) : List (Ordinal × Ordinal) :=\n CNFRec b [] (fun o _ho IH ↦ (log b o, o / b ^ log b o)::IH) o",
"end": [
76,
64
],
"full_name": "Ordinal.CNF",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "@[simp]\ntheorem CNF_zero (b : Ordinal) : CNF b 0 = []",
"end": [
82,
20
],
"full_name": "Ordinal.CNF_zero",
"kind": "commanddeclaration",
"start": [
80,
1
]
},
{
"code": "theorem CNF_ne_zero {b o : Ordinal} (ho : o ≠ 0) :\n CNF b o = (log b o, o / b ^ log b o)::CNF b (o % b ^ log b o)",
"end": [
89,
22
],
"full_name": "Ordinal.CNF_ne_zero",
"kind": "commanddeclaration",
"start": [
86,
1
]
},
{
"code": "theorem zero_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 0 o = [⟨0, o⟩]",
"end": [
93,
93
],
"full_name": "Ordinal.zero_CNF",
"kind": "commanddeclaration",
"start": [
93,
1
]
},
{
"code": "theorem one_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 1 o = [⟨0, o⟩]",
"end": [
97,
92
],
"full_name": "Ordinal.one_CNF",
"kind": "commanddeclaration",
"start": [
97,
1
]
},
{
"code": "theorem CNF_of_le_one {b o : Ordinal} (hb : b ≤ 1) (ho : o ≠ 0) : CNF b o = [⟨0, o⟩]",
"end": [
104,
21
],
"full_name": "Ordinal.CNF_of_le_one",
"kind": "commanddeclaration",
"start": [
101,
1
]
},
{
"code": "theorem CNF_of_lt {b o : Ordinal} (ho : o ≠ 0) (hb : o < b) : CNF b o = [⟨0, o⟩]",
"end": [
109,
84
],
"full_name": "Ordinal.CNF_of_lt",
"kind": "commanddeclaration",
"start": [
108,
1
]
},
{
"code": "theorem CNF_foldr (b o : Ordinal) : (CNF b o).foldr (fun p r ↦ b ^ p.1 * p.2 + r) 0 = o",
"end": [
116,
74
],
"full_name": "Ordinal.CNF_foldr",
"kind": "commanddeclaration",
"start": [
113,
1
]
},
{
"code": "theorem CNF_fst_le_log {b o : Ordinal.{u}} {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.1 ≤ log b o",
"end": [
129,
74
],
"full_name": "Ordinal.CNF_fst_le_log",
"kind": "commanddeclaration",
"start": [
120,
1
]
},
{
"code": "theorem CNF_fst_le {b o : Ordinal.{u}} {x : Ordinal × Ordinal} (h : x ∈ CNF b o) : x.1 ≤ o",
"end": [
135,
46
],
"full_name": "Ordinal.CNF_fst_le",
"kind": "commanddeclaration",
"start": [
133,
1
]
},
{
"code": "theorem CNF_lt_snd {b o : Ordinal.{u}} {x : Ordinal × Ordinal} : x ∈ CNF b o → 0 < x.2",
"end": [
145,
15
],
"full_name": "Ordinal.CNF_lt_snd",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem CNF_snd_lt {b o : Ordinal.{u}} (hb : 1 < b) {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.2 < b",
"end": [
158,
17
],
"full_name": "Ordinal.CNF_snd_lt",
"kind": "commanddeclaration",
"start": [
149,
1
]
},
{
"code": "theorem CNF_sorted (b o : Ordinal) : ((CNF b o).map Prod.fst).Sorted (· > ·)",
"end": [
174,
83
],
"full_name": "Ordinal.CNF_sorted",
"kind": "commanddeclaration",
"start": [
162,
1
]
}
] | 12 | Ordinal.CNF_snd_lt | [
[
151,
30
],
[
158,
17
]
] | 6 | 12 | cases' (mem_cons.mp h) with h h | case refine_2
b o✝ : Ordinal.{u}
hb : 1 < b
x : Ordinal.{u} × Ordinal.{u}
o : Ordinal.{u}
ho : o ≠ 0
IH : x ∈ CNF b (o % b ^ log b o) → x.2 < b
h : x ∈ (log b o, o / b ^ log b o) :: CNF b (o % b ^ log b o)
⊢ x.2 < b | case refine_2.inl
b o✝ : Ordinal.{u}
hb : 1 < b
x : Ordinal.{u} × Ordinal.{u}
o : Ordinal.{u}
ho : o ≠ 0
IH : x ∈ CNF b (o % b ^ log b o) → x.2 < b
h✝ : x ∈ (log b o, o / b ^ log b o) :: CNF b (o % b ^ log b o)
h : x = (log b o, o / b ^ log b o)
⊢ x.2 < b
case refine_2.inr
b o✝ : Ordinal.{u}
hb : 1 < b
x : Ordinal.{u} × Ordinal.{u}
o : Ordinal.{u}
ho : o ≠ 0
IH : x ∈ CNF b (o % b ^ log b o) → x.2 < b
h✝ : x ∈ (log b o, o / b ^ log b o) :: CNF b (o % b ^ log b o)
h : x ∈ CNF b (o % b ^ log b o)
⊢ x.2 < b |
Mathlib/SetTheory/Ordinal/CantorNormalForm.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Mathlib.SetTheory.Ordinal.Exponential",
"Mathlib/SetTheory/Ordinal/Exponential.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "@[elab_as_elim]\nnoncomputable def CNFRec (b : Ordinal) {C : Ordinal → Sort*} (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : ∀ o, C o := fun o ↦ by\n by_cases h : o = 0\n · rw [h]; exact H0\n · exact H o h (CNFRec _ H0 H (o % b ^ log b o))\n termination_by o => o\n decreasing_by exact mod_opow_log_lt_self b h",
"end": [
50,
49
],
"full_name": "Ordinal.CNFRec",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem CNFRec_zero {C : Ordinal → Sort*} (b : Ordinal) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : @CNFRec b C H0 H 0 = H0",
"end": [
58,
6
],
"full_name": "Ordinal.CNFRec_zero",
"kind": "commanddeclaration",
"start": [
54,
1
]
},
{
"code": "theorem CNFRec_pos (b : Ordinal) {o : Ordinal} {C : Ordinal → Sort*} (ho : o ≠ 0) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) :\n @CNFRec b C H0 H o = H o ho (@CNFRec b C H0 H _)",
"end": [
64,
83
],
"full_name": "Ordinal.CNFRec_pos",
"kind": "commanddeclaration",
"start": [
62,
1
]
},
{
"code": "@[pp_nodot]\ndef CNF (b o : Ordinal) : List (Ordinal × Ordinal) :=\n CNFRec b [] (fun o _ho IH ↦ (log b o, o / b ^ log b o)::IH) o",
"end": [
76,
64
],
"full_name": "Ordinal.CNF",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "@[simp]\ntheorem CNF_zero (b : Ordinal) : CNF b 0 = []",
"end": [
82,
20
],
"full_name": "Ordinal.CNF_zero",
"kind": "commanddeclaration",
"start": [
80,
1
]
},
{
"code": "theorem CNF_ne_zero {b o : Ordinal} (ho : o ≠ 0) :\n CNF b o = (log b o, o / b ^ log b o)::CNF b (o % b ^ log b o)",
"end": [
89,
22
],
"full_name": "Ordinal.CNF_ne_zero",
"kind": "commanddeclaration",
"start": [
86,
1
]
},
{
"code": "theorem zero_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 0 o = [⟨0, o⟩]",
"end": [
93,
93
],
"full_name": "Ordinal.zero_CNF",
"kind": "commanddeclaration",
"start": [
93,
1
]
},
{
"code": "theorem one_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 1 o = [⟨0, o⟩]",
"end": [
97,
92
],
"full_name": "Ordinal.one_CNF",
"kind": "commanddeclaration",
"start": [
97,
1
]
},
{
"code": "theorem CNF_of_le_one {b o : Ordinal} (hb : b ≤ 1) (ho : o ≠ 0) : CNF b o = [⟨0, o⟩]",
"end": [
104,
21
],
"full_name": "Ordinal.CNF_of_le_one",
"kind": "commanddeclaration",
"start": [
101,
1
]
},
{
"code": "theorem CNF_of_lt {b o : Ordinal} (ho : o ≠ 0) (hb : o < b) : CNF b o = [⟨0, o⟩]",
"end": [
109,
84
],
"full_name": "Ordinal.CNF_of_lt",
"kind": "commanddeclaration",
"start": [
108,
1
]
},
{
"code": "theorem CNF_foldr (b o : Ordinal) : (CNF b o).foldr (fun p r ↦ b ^ p.1 * p.2 + r) 0 = o",
"end": [
116,
74
],
"full_name": "Ordinal.CNF_foldr",
"kind": "commanddeclaration",
"start": [
113,
1
]
},
{
"code": "theorem CNF_fst_le_log {b o : Ordinal.{u}} {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.1 ≤ log b o",
"end": [
129,
74
],
"full_name": "Ordinal.CNF_fst_le_log",
"kind": "commanddeclaration",
"start": [
120,
1
]
},
{
"code": "theorem CNF_fst_le {b o : Ordinal.{u}} {x : Ordinal × Ordinal} (h : x ∈ CNF b o) : x.1 ≤ o",
"end": [
135,
46
],
"full_name": "Ordinal.CNF_fst_le",
"kind": "commanddeclaration",
"start": [
133,
1
]
},
{
"code": "theorem CNF_lt_snd {b o : Ordinal.{u}} {x : Ordinal × Ordinal} : x ∈ CNF b o → 0 < x.2",
"end": [
145,
15
],
"full_name": "Ordinal.CNF_lt_snd",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem CNF_snd_lt {b o : Ordinal.{u}} (hb : 1 < b) {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.2 < b",
"end": [
158,
17
],
"full_name": "Ordinal.CNF_snd_lt",
"kind": "commanddeclaration",
"start": [
149,
1
]
},
{
"code": "theorem CNF_sorted (b o : Ordinal) : ((CNF b o).map Prod.fst).Sorted (· > ·)",
"end": [
174,
83
],
"full_name": "Ordinal.CNF_sorted",
"kind": "commanddeclaration",
"start": [
162,
1
]
}
] | 12 | Ordinal.CNF_snd_lt | [
[
151,
30
],
[
158,
17
]
] | 7 | 12 | · rw [h]; simpa only using div_opow_log_lt o hb | case refine_2.inl
b o✝ : Ordinal.{u}
hb : 1 < b
x : Ordinal.{u} × Ordinal.{u}
o : Ordinal.{u}
ho : o ≠ 0
IH : x ∈ CNF b (o % b ^ log b o) → x.2 < b
h✝ : x ∈ (log b o, o / b ^ log b o) :: CNF b (o % b ^ log b o)
h : x = (log b o, o / b ^ log b o)
⊢ x.2 < b
case refine_2.inr
b o✝ : Ordinal.{u}
hb : 1 < b
x : Ordinal.{u} × Ordinal.{u}
o : Ordinal.{u}
ho : o ≠ 0
IH : x ∈ CNF b (o % b ^ log b o) → x.2 < b
h✝ : x ∈ (log b o, o / b ^ log b o) :: CNF b (o % b ^ log b o)
h : x ∈ CNF b (o % b ^ log b o)
⊢ x.2 < b | case refine_2.inr
b o✝ : Ordinal.{u}
hb : 1 < b
x : Ordinal.{u} × Ordinal.{u}
o : Ordinal.{u}
ho : o ≠ 0
IH : x ∈ CNF b (o % b ^ log b o) → x.2 < b
h✝ : x ∈ (log b o, o / b ^ log b o) :: CNF b (o % b ^ log b o)
h : x ∈ CNF b (o % b ^ log b o)
⊢ x.2 < b |
Mathlib/SetTheory/Ordinal/CantorNormalForm.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Mathlib.SetTheory.Ordinal.Exponential",
"Mathlib/SetTheory/Ordinal/Exponential.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "@[elab_as_elim]\nnoncomputable def CNFRec (b : Ordinal) {C : Ordinal → Sort*} (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : ∀ o, C o := fun o ↦ by\n by_cases h : o = 0\n · rw [h]; exact H0\n · exact H o h (CNFRec _ H0 H (o % b ^ log b o))\n termination_by o => o\n decreasing_by exact mod_opow_log_lt_self b h",
"end": [
50,
49
],
"full_name": "Ordinal.CNFRec",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem CNFRec_zero {C : Ordinal → Sort*} (b : Ordinal) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : @CNFRec b C H0 H 0 = H0",
"end": [
58,
6
],
"full_name": "Ordinal.CNFRec_zero",
"kind": "commanddeclaration",
"start": [
54,
1
]
},
{
"code": "theorem CNFRec_pos (b : Ordinal) {o : Ordinal} {C : Ordinal → Sort*} (ho : o ≠ 0) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) :\n @CNFRec b C H0 H o = H o ho (@CNFRec b C H0 H _)",
"end": [
64,
83
],
"full_name": "Ordinal.CNFRec_pos",
"kind": "commanddeclaration",
"start": [
62,
1
]
},
{
"code": "@[pp_nodot]\ndef CNF (b o : Ordinal) : List (Ordinal × Ordinal) :=\n CNFRec b [] (fun o _ho IH ↦ (log b o, o / b ^ log b o)::IH) o",
"end": [
76,
64
],
"full_name": "Ordinal.CNF",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "@[simp]\ntheorem CNF_zero (b : Ordinal) : CNF b 0 = []",
"end": [
82,
20
],
"full_name": "Ordinal.CNF_zero",
"kind": "commanddeclaration",
"start": [
80,
1
]
},
{
"code": "theorem CNF_ne_zero {b o : Ordinal} (ho : o ≠ 0) :\n CNF b o = (log b o, o / b ^ log b o)::CNF b (o % b ^ log b o)",
"end": [
89,
22
],
"full_name": "Ordinal.CNF_ne_zero",
"kind": "commanddeclaration",
"start": [
86,
1
]
},
{
"code": "theorem zero_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 0 o = [⟨0, o⟩]",
"end": [
93,
93
],
"full_name": "Ordinal.zero_CNF",
"kind": "commanddeclaration",
"start": [
93,
1
]
},
{
"code": "theorem one_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 1 o = [⟨0, o⟩]",
"end": [
97,
92
],
"full_name": "Ordinal.one_CNF",
"kind": "commanddeclaration",
"start": [
97,
1
]
},
{
"code": "theorem CNF_of_le_one {b o : Ordinal} (hb : b ≤ 1) (ho : o ≠ 0) : CNF b o = [⟨0, o⟩]",
"end": [
104,
21
],
"full_name": "Ordinal.CNF_of_le_one",
"kind": "commanddeclaration",
"start": [
101,
1
]
},
{
"code": "theorem CNF_of_lt {b o : Ordinal} (ho : o ≠ 0) (hb : o < b) : CNF b o = [⟨0, o⟩]",
"end": [
109,
84
],
"full_name": "Ordinal.CNF_of_lt",
"kind": "commanddeclaration",
"start": [
108,
1
]
},
{
"code": "theorem CNF_foldr (b o : Ordinal) : (CNF b o).foldr (fun p r ↦ b ^ p.1 * p.2 + r) 0 = o",
"end": [
116,
74
],
"full_name": "Ordinal.CNF_foldr",
"kind": "commanddeclaration",
"start": [
113,
1
]
},
{
"code": "theorem CNF_fst_le_log {b o : Ordinal.{u}} {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.1 ≤ log b o",
"end": [
129,
74
],
"full_name": "Ordinal.CNF_fst_le_log",
"kind": "commanddeclaration",
"start": [
120,
1
]
},
{
"code": "theorem CNF_fst_le {b o : Ordinal.{u}} {x : Ordinal × Ordinal} (h : x ∈ CNF b o) : x.1 ≤ o",
"end": [
135,
46
],
"full_name": "Ordinal.CNF_fst_le",
"kind": "commanddeclaration",
"start": [
133,
1
]
},
{
"code": "theorem CNF_lt_snd {b o : Ordinal.{u}} {x : Ordinal × Ordinal} : x ∈ CNF b o → 0 < x.2",
"end": [
145,
15
],
"full_name": "Ordinal.CNF_lt_snd",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem CNF_snd_lt {b o : Ordinal.{u}} (hb : 1 < b) {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.2 < b",
"end": [
158,
17
],
"full_name": "Ordinal.CNF_snd_lt",
"kind": "commanddeclaration",
"start": [
149,
1
]
},
{
"code": "theorem CNF_sorted (b o : Ordinal) : ((CNF b o).map Prod.fst).Sorted (· > ·)",
"end": [
174,
83
],
"full_name": "Ordinal.CNF_sorted",
"kind": "commanddeclaration",
"start": [
162,
1
]
}
] | 12 | Ordinal.CNF_snd_lt | [
[
151,
30
],
[
158,
17
]
] | 8 | 12 | · exact IH h | case refine_2.inr
b o✝ : Ordinal.{u}
hb : 1 < b
x : Ordinal.{u} × Ordinal.{u}
o : Ordinal.{u}
ho : o ≠ 0
IH : x ∈ CNF b (o % b ^ log b o) → x.2 < b
h✝ : x ∈ (log b o, o / b ^ log b o) :: CNF b (o % b ^ log b o)
h : x ∈ CNF b (o % b ^ log b o)
⊢ x.2 < b | no goals |
Mathlib/SetTheory/Ordinal/CantorNormalForm.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Mathlib.SetTheory.Ordinal.Exponential",
"Mathlib/SetTheory/Ordinal/Exponential.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "@[elab_as_elim]\nnoncomputable def CNFRec (b : Ordinal) {C : Ordinal → Sort*} (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : ∀ o, C o := fun o ↦ by\n by_cases h : o = 0\n · rw [h]; exact H0\n · exact H o h (CNFRec _ H0 H (o % b ^ log b o))\n termination_by o => o\n decreasing_by exact mod_opow_log_lt_self b h",
"end": [
50,
49
],
"full_name": "Ordinal.CNFRec",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem CNFRec_zero {C : Ordinal → Sort*} (b : Ordinal) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : @CNFRec b C H0 H 0 = H0",
"end": [
58,
6
],
"full_name": "Ordinal.CNFRec_zero",
"kind": "commanddeclaration",
"start": [
54,
1
]
},
{
"code": "theorem CNFRec_pos (b : Ordinal) {o : Ordinal} {C : Ordinal → Sort*} (ho : o ≠ 0) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) :\n @CNFRec b C H0 H o = H o ho (@CNFRec b C H0 H _)",
"end": [
64,
83
],
"full_name": "Ordinal.CNFRec_pos",
"kind": "commanddeclaration",
"start": [
62,
1
]
},
{
"code": "@[pp_nodot]\ndef CNF (b o : Ordinal) : List (Ordinal × Ordinal) :=\n CNFRec b [] (fun o _ho IH ↦ (log b o, o / b ^ log b o)::IH) o",
"end": [
76,
64
],
"full_name": "Ordinal.CNF",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "@[simp]\ntheorem CNF_zero (b : Ordinal) : CNF b 0 = []",
"end": [
82,
20
],
"full_name": "Ordinal.CNF_zero",
"kind": "commanddeclaration",
"start": [
80,
1
]
},
{
"code": "theorem CNF_ne_zero {b o : Ordinal} (ho : o ≠ 0) :\n CNF b o = (log b o, o / b ^ log b o)::CNF b (o % b ^ log b o)",
"end": [
89,
22
],
"full_name": "Ordinal.CNF_ne_zero",
"kind": "commanddeclaration",
"start": [
86,
1
]
},
{
"code": "theorem zero_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 0 o = [⟨0, o⟩]",
"end": [
93,
93
],
"full_name": "Ordinal.zero_CNF",
"kind": "commanddeclaration",
"start": [
93,
1
]
},
{
"code": "theorem one_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 1 o = [⟨0, o⟩]",
"end": [
97,
92
],
"full_name": "Ordinal.one_CNF",
"kind": "commanddeclaration",
"start": [
97,
1
]
},
{
"code": "theorem CNF_of_le_one {b o : Ordinal} (hb : b ≤ 1) (ho : o ≠ 0) : CNF b o = [⟨0, o⟩]",
"end": [
104,
21
],
"full_name": "Ordinal.CNF_of_le_one",
"kind": "commanddeclaration",
"start": [
101,
1
]
},
{
"code": "theorem CNF_of_lt {b o : Ordinal} (ho : o ≠ 0) (hb : o < b) : CNF b o = [⟨0, o⟩]",
"end": [
109,
84
],
"full_name": "Ordinal.CNF_of_lt",
"kind": "commanddeclaration",
"start": [
108,
1
]
},
{
"code": "theorem CNF_foldr (b o : Ordinal) : (CNF b o).foldr (fun p r ↦ b ^ p.1 * p.2 + r) 0 = o",
"end": [
116,
74
],
"full_name": "Ordinal.CNF_foldr",
"kind": "commanddeclaration",
"start": [
113,
1
]
},
{
"code": "theorem CNF_fst_le_log {b o : Ordinal.{u}} {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.1 ≤ log b o",
"end": [
129,
74
],
"full_name": "Ordinal.CNF_fst_le_log",
"kind": "commanddeclaration",
"start": [
120,
1
]
},
{
"code": "theorem CNF_fst_le {b o : Ordinal.{u}} {x : Ordinal × Ordinal} (h : x ∈ CNF b o) : x.1 ≤ o",
"end": [
135,
46
],
"full_name": "Ordinal.CNF_fst_le",
"kind": "commanddeclaration",
"start": [
133,
1
]
},
{
"code": "theorem CNF_lt_snd {b o : Ordinal.{u}} {x : Ordinal × Ordinal} : x ∈ CNF b o → 0 < x.2",
"end": [
145,
15
],
"full_name": "Ordinal.CNF_lt_snd",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem CNF_snd_lt {b o : Ordinal.{u}} (hb : 1 < b) {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.2 < b",
"end": [
158,
17
],
"full_name": "Ordinal.CNF_snd_lt",
"kind": "commanddeclaration",
"start": [
149,
1
]
},
{
"code": "theorem CNF_sorted (b o : Ordinal) : ((CNF b o).map Prod.fst).Sorted (· > ·)",
"end": [
174,
83
],
"full_name": "Ordinal.CNF_sorted",
"kind": "commanddeclaration",
"start": [
162,
1
]
}
] | 12 | Ordinal.CNF_snd_lt | [
[
151,
30
],
[
158,
17
]
] | 9 | 12 | rw [h] | case refine_2.inl
b o✝ : Ordinal.{u}
hb : 1 < b
x : Ordinal.{u} × Ordinal.{u}
o : Ordinal.{u}
ho : o ≠ 0
IH : x ∈ CNF b (o % b ^ log b o) → x.2 < b
h✝ : x ∈ (log b o, o / b ^ log b o) :: CNF b (o % b ^ log b o)
h : x = (log b o, o / b ^ log b o)
⊢ x.2 < b | case refine_2.inl
b o✝ : Ordinal.{u}
hb : 1 < b
x : Ordinal.{u} × Ordinal.{u}
o : Ordinal.{u}
ho : o ≠ 0
IH : x ∈ CNF b (o % b ^ log b o) → x.2 < b
h✝ : x ∈ (log b o, o / b ^ log b o) :: CNF b (o % b ^ log b o)
h : x = (log b o, o / b ^ log b o)
⊢ (log b o, o / b ^ log b o).2 < b |
Mathlib/SetTheory/Ordinal/CantorNormalForm.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Mathlib.SetTheory.Ordinal.Exponential",
"Mathlib/SetTheory/Ordinal/Exponential.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "@[elab_as_elim]\nnoncomputable def CNFRec (b : Ordinal) {C : Ordinal → Sort*} (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : ∀ o, C o := fun o ↦ by\n by_cases h : o = 0\n · rw [h]; exact H0\n · exact H o h (CNFRec _ H0 H (o % b ^ log b o))\n termination_by o => o\n decreasing_by exact mod_opow_log_lt_self b h",
"end": [
50,
49
],
"full_name": "Ordinal.CNFRec",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem CNFRec_zero {C : Ordinal → Sort*} (b : Ordinal) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : @CNFRec b C H0 H 0 = H0",
"end": [
58,
6
],
"full_name": "Ordinal.CNFRec_zero",
"kind": "commanddeclaration",
"start": [
54,
1
]
},
{
"code": "theorem CNFRec_pos (b : Ordinal) {o : Ordinal} {C : Ordinal → Sort*} (ho : o ≠ 0) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) :\n @CNFRec b C H0 H o = H o ho (@CNFRec b C H0 H _)",
"end": [
64,
83
],
"full_name": "Ordinal.CNFRec_pos",
"kind": "commanddeclaration",
"start": [
62,
1
]
},
{
"code": "@[pp_nodot]\ndef CNF (b o : Ordinal) : List (Ordinal × Ordinal) :=\n CNFRec b [] (fun o _ho IH ↦ (log b o, o / b ^ log b o)::IH) o",
"end": [
76,
64
],
"full_name": "Ordinal.CNF",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "@[simp]\ntheorem CNF_zero (b : Ordinal) : CNF b 0 = []",
"end": [
82,
20
],
"full_name": "Ordinal.CNF_zero",
"kind": "commanddeclaration",
"start": [
80,
1
]
},
{
"code": "theorem CNF_ne_zero {b o : Ordinal} (ho : o ≠ 0) :\n CNF b o = (log b o, o / b ^ log b o)::CNF b (o % b ^ log b o)",
"end": [
89,
22
],
"full_name": "Ordinal.CNF_ne_zero",
"kind": "commanddeclaration",
"start": [
86,
1
]
},
{
"code": "theorem zero_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 0 o = [⟨0, o⟩]",
"end": [
93,
93
],
"full_name": "Ordinal.zero_CNF",
"kind": "commanddeclaration",
"start": [
93,
1
]
},
{
"code": "theorem one_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 1 o = [⟨0, o⟩]",
"end": [
97,
92
],
"full_name": "Ordinal.one_CNF",
"kind": "commanddeclaration",
"start": [
97,
1
]
},
{
"code": "theorem CNF_of_le_one {b o : Ordinal} (hb : b ≤ 1) (ho : o ≠ 0) : CNF b o = [⟨0, o⟩]",
"end": [
104,
21
],
"full_name": "Ordinal.CNF_of_le_one",
"kind": "commanddeclaration",
"start": [
101,
1
]
},
{
"code": "theorem CNF_of_lt {b o : Ordinal} (ho : o ≠ 0) (hb : o < b) : CNF b o = [⟨0, o⟩]",
"end": [
109,
84
],
"full_name": "Ordinal.CNF_of_lt",
"kind": "commanddeclaration",
"start": [
108,
1
]
},
{
"code": "theorem CNF_foldr (b o : Ordinal) : (CNF b o).foldr (fun p r ↦ b ^ p.1 * p.2 + r) 0 = o",
"end": [
116,
74
],
"full_name": "Ordinal.CNF_foldr",
"kind": "commanddeclaration",
"start": [
113,
1
]
},
{
"code": "theorem CNF_fst_le_log {b o : Ordinal.{u}} {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.1 ≤ log b o",
"end": [
129,
74
],
"full_name": "Ordinal.CNF_fst_le_log",
"kind": "commanddeclaration",
"start": [
120,
1
]
},
{
"code": "theorem CNF_fst_le {b o : Ordinal.{u}} {x : Ordinal × Ordinal} (h : x ∈ CNF b o) : x.1 ≤ o",
"end": [
135,
46
],
"full_name": "Ordinal.CNF_fst_le",
"kind": "commanddeclaration",
"start": [
133,
1
]
},
{
"code": "theorem CNF_lt_snd {b o : Ordinal.{u}} {x : Ordinal × Ordinal} : x ∈ CNF b o → 0 < x.2",
"end": [
145,
15
],
"full_name": "Ordinal.CNF_lt_snd",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem CNF_snd_lt {b o : Ordinal.{u}} (hb : 1 < b) {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.2 < b",
"end": [
158,
17
],
"full_name": "Ordinal.CNF_snd_lt",
"kind": "commanddeclaration",
"start": [
149,
1
]
},
{
"code": "theorem CNF_sorted (b o : Ordinal) : ((CNF b o).map Prod.fst).Sorted (· > ·)",
"end": [
174,
83
],
"full_name": "Ordinal.CNF_sorted",
"kind": "commanddeclaration",
"start": [
162,
1
]
}
] | 12 | Ordinal.CNF_snd_lt | [
[
151,
30
],
[
158,
17
]
] | 10 | 12 | simpa only using div_opow_log_lt o hb | case refine_2.inl
b o✝ : Ordinal.{u}
hb : 1 < b
x : Ordinal.{u} × Ordinal.{u}
o : Ordinal.{u}
ho : o ≠ 0
IH : x ∈ CNF b (o % b ^ log b o) → x.2 < b
h✝ : x ∈ (log b o, o / b ^ log b o) :: CNF b (o % b ^ log b o)
h : x = (log b o, o / b ^ log b o)
⊢ (log b o, o / b ^ log b o).2 < b | no goals |
Mathlib/SetTheory/Ordinal/CantorNormalForm.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Mathlib.SetTheory.Ordinal.Exponential",
"Mathlib/SetTheory/Ordinal/Exponential.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "@[elab_as_elim]\nnoncomputable def CNFRec (b : Ordinal) {C : Ordinal → Sort*} (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : ∀ o, C o := fun o ↦ by\n by_cases h : o = 0\n · rw [h]; exact H0\n · exact H o h (CNFRec _ H0 H (o % b ^ log b o))\n termination_by o => o\n decreasing_by exact mod_opow_log_lt_self b h",
"end": [
50,
49
],
"full_name": "Ordinal.CNFRec",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem CNFRec_zero {C : Ordinal → Sort*} (b : Ordinal) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : @CNFRec b C H0 H 0 = H0",
"end": [
58,
6
],
"full_name": "Ordinal.CNFRec_zero",
"kind": "commanddeclaration",
"start": [
54,
1
]
},
{
"code": "theorem CNFRec_pos (b : Ordinal) {o : Ordinal} {C : Ordinal → Sort*} (ho : o ≠ 0) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) :\n @CNFRec b C H0 H o = H o ho (@CNFRec b C H0 H _)",
"end": [
64,
83
],
"full_name": "Ordinal.CNFRec_pos",
"kind": "commanddeclaration",
"start": [
62,
1
]
},
{
"code": "@[pp_nodot]\ndef CNF (b o : Ordinal) : List (Ordinal × Ordinal) :=\n CNFRec b [] (fun o _ho IH ↦ (log b o, o / b ^ log b o)::IH) o",
"end": [
76,
64
],
"full_name": "Ordinal.CNF",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "@[simp]\ntheorem CNF_zero (b : Ordinal) : CNF b 0 = []",
"end": [
82,
20
],
"full_name": "Ordinal.CNF_zero",
"kind": "commanddeclaration",
"start": [
80,
1
]
},
{
"code": "theorem CNF_ne_zero {b o : Ordinal} (ho : o ≠ 0) :\n CNF b o = (log b o, o / b ^ log b o)::CNF b (o % b ^ log b o)",
"end": [
89,
22
],
"full_name": "Ordinal.CNF_ne_zero",
"kind": "commanddeclaration",
"start": [
86,
1
]
},
{
"code": "theorem zero_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 0 o = [⟨0, o⟩]",
"end": [
93,
93
],
"full_name": "Ordinal.zero_CNF",
"kind": "commanddeclaration",
"start": [
93,
1
]
},
{
"code": "theorem one_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 1 o = [⟨0, o⟩]",
"end": [
97,
92
],
"full_name": "Ordinal.one_CNF",
"kind": "commanddeclaration",
"start": [
97,
1
]
},
{
"code": "theorem CNF_of_le_one {b o : Ordinal} (hb : b ≤ 1) (ho : o ≠ 0) : CNF b o = [⟨0, o⟩]",
"end": [
104,
21
],
"full_name": "Ordinal.CNF_of_le_one",
"kind": "commanddeclaration",
"start": [
101,
1
]
},
{
"code": "theorem CNF_of_lt {b o : Ordinal} (ho : o ≠ 0) (hb : o < b) : CNF b o = [⟨0, o⟩]",
"end": [
109,
84
],
"full_name": "Ordinal.CNF_of_lt",
"kind": "commanddeclaration",
"start": [
108,
1
]
},
{
"code": "theorem CNF_foldr (b o : Ordinal) : (CNF b o).foldr (fun p r ↦ b ^ p.1 * p.2 + r) 0 = o",
"end": [
116,
74
],
"full_name": "Ordinal.CNF_foldr",
"kind": "commanddeclaration",
"start": [
113,
1
]
},
{
"code": "theorem CNF_fst_le_log {b o : Ordinal.{u}} {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.1 ≤ log b o",
"end": [
129,
74
],
"full_name": "Ordinal.CNF_fst_le_log",
"kind": "commanddeclaration",
"start": [
120,
1
]
},
{
"code": "theorem CNF_fst_le {b o : Ordinal.{u}} {x : Ordinal × Ordinal} (h : x ∈ CNF b o) : x.1 ≤ o",
"end": [
135,
46
],
"full_name": "Ordinal.CNF_fst_le",
"kind": "commanddeclaration",
"start": [
133,
1
]
},
{
"code": "theorem CNF_lt_snd {b o : Ordinal.{u}} {x : Ordinal × Ordinal} : x ∈ CNF b o → 0 < x.2",
"end": [
145,
15
],
"full_name": "Ordinal.CNF_lt_snd",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem CNF_snd_lt {b o : Ordinal.{u}} (hb : 1 < b) {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.2 < b",
"end": [
158,
17
],
"full_name": "Ordinal.CNF_snd_lt",
"kind": "commanddeclaration",
"start": [
149,
1
]
},
{
"code": "theorem CNF_sorted (b o : Ordinal) : ((CNF b o).map Prod.fst).Sorted (· > ·)",
"end": [
174,
83
],
"full_name": "Ordinal.CNF_sorted",
"kind": "commanddeclaration",
"start": [
162,
1
]
}
] | 12 | Ordinal.CNF_snd_lt | [
[
151,
30
],
[
158,
17
]
] | 11 | 12 | exact IH h | case refine_2.inr
b o✝ : Ordinal.{u}
hb : 1 < b
x : Ordinal.{u} × Ordinal.{u}
o : Ordinal.{u}
ho : o ≠ 0
IH : x ∈ CNF b (o % b ^ log b o) → x.2 < b
h✝ : x ∈ (log b o, o / b ^ log b o) :: CNF b (o % b ^ log b o)
h : x ∈ CNF b (o % b ^ log b o)
⊢ x.2 < b | no goals |
Mathlib/SetTheory/Ordinal/CantorNormalForm.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Mathlib.SetTheory.Ordinal.Exponential",
"Mathlib/SetTheory/Ordinal/Exponential.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "@[elab_as_elim]\nnoncomputable def CNFRec (b : Ordinal) {C : Ordinal → Sort*} (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : ∀ o, C o := fun o ↦ by\n by_cases h : o = 0\n · rw [h]; exact H0\n · exact H o h (CNFRec _ H0 H (o % b ^ log b o))\n termination_by o => o\n decreasing_by exact mod_opow_log_lt_self b h",
"end": [
50,
49
],
"full_name": "Ordinal.CNFRec",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem CNFRec_zero {C : Ordinal → Sort*} (b : Ordinal) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : @CNFRec b C H0 H 0 = H0",
"end": [
58,
6
],
"full_name": "Ordinal.CNFRec_zero",
"kind": "commanddeclaration",
"start": [
54,
1
]
},
{
"code": "theorem CNFRec_pos (b : Ordinal) {o : Ordinal} {C : Ordinal → Sort*} (ho : o ≠ 0) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) :\n @CNFRec b C H0 H o = H o ho (@CNFRec b C H0 H _)",
"end": [
64,
83
],
"full_name": "Ordinal.CNFRec_pos",
"kind": "commanddeclaration",
"start": [
62,
1
]
},
{
"code": "@[pp_nodot]\ndef CNF (b o : Ordinal) : List (Ordinal × Ordinal) :=\n CNFRec b [] (fun o _ho IH ↦ (log b o, o / b ^ log b o)::IH) o",
"end": [
76,
64
],
"full_name": "Ordinal.CNF",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "@[simp]\ntheorem CNF_zero (b : Ordinal) : CNF b 0 = []",
"end": [
82,
20
],
"full_name": "Ordinal.CNF_zero",
"kind": "commanddeclaration",
"start": [
80,
1
]
},
{
"code": "theorem CNF_ne_zero {b o : Ordinal} (ho : o ≠ 0) :\n CNF b o = (log b o, o / b ^ log b o)::CNF b (o % b ^ log b o)",
"end": [
89,
22
],
"full_name": "Ordinal.CNF_ne_zero",
"kind": "commanddeclaration",
"start": [
86,
1
]
},
{
"code": "theorem zero_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 0 o = [⟨0, o⟩]",
"end": [
93,
93
],
"full_name": "Ordinal.zero_CNF",
"kind": "commanddeclaration",
"start": [
93,
1
]
},
{
"code": "theorem one_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 1 o = [⟨0, o⟩]",
"end": [
97,
92
],
"full_name": "Ordinal.one_CNF",
"kind": "commanddeclaration",
"start": [
97,
1
]
},
{
"code": "theorem CNF_of_le_one {b o : Ordinal} (hb : b ≤ 1) (ho : o ≠ 0) : CNF b o = [⟨0, o⟩]",
"end": [
104,
21
],
"full_name": "Ordinal.CNF_of_le_one",
"kind": "commanddeclaration",
"start": [
101,
1
]
},
{
"code": "theorem CNF_of_lt {b o : Ordinal} (ho : o ≠ 0) (hb : o < b) : CNF b o = [⟨0, o⟩]",
"end": [
109,
84
],
"full_name": "Ordinal.CNF_of_lt",
"kind": "commanddeclaration",
"start": [
108,
1
]
},
{
"code": "theorem CNF_foldr (b o : Ordinal) : (CNF b o).foldr (fun p r ↦ b ^ p.1 * p.2 + r) 0 = o",
"end": [
116,
74
],
"full_name": "Ordinal.CNF_foldr",
"kind": "commanddeclaration",
"start": [
113,
1
]
},
{
"code": "theorem CNF_fst_le_log {b o : Ordinal.{u}} {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.1 ≤ log b o",
"end": [
129,
74
],
"full_name": "Ordinal.CNF_fst_le_log",
"kind": "commanddeclaration",
"start": [
120,
1
]
},
{
"code": "theorem CNF_fst_le {b o : Ordinal.{u}} {x : Ordinal × Ordinal} (h : x ∈ CNF b o) : x.1 ≤ o",
"end": [
135,
46
],
"full_name": "Ordinal.CNF_fst_le",
"kind": "commanddeclaration",
"start": [
133,
1
]
},
{
"code": "theorem CNF_lt_snd {b o : Ordinal.{u}} {x : Ordinal × Ordinal} : x ∈ CNF b o → 0 < x.2",
"end": [
145,
15
],
"full_name": "Ordinal.CNF_lt_snd",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem CNF_snd_lt {b o : Ordinal.{u}} (hb : 1 < b) {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.2 < b",
"end": [
158,
17
],
"full_name": "Ordinal.CNF_snd_lt",
"kind": "commanddeclaration",
"start": [
149,
1
]
},
{
"code": "theorem CNF_sorted (b o : Ordinal) : ((CNF b o).map Prod.fst).Sorted (· > ·)",
"end": [
174,
83
],
"full_name": "Ordinal.CNF_sorted",
"kind": "commanddeclaration",
"start": [
162,
1
]
}
] | 13 | Ordinal.CNF_sorted | [
[
163,
81
],
[
174,
83
]
] | 0 | 17 | refine CNFRec b ?_ (fun o ho IH ↦ ?_) o | b o : Ordinal.{u_1}
⊢ Sorted (fun x x_1 => x > x_1) (map Prod.fst (CNF b o)) | case refine_1
b o : Ordinal.{u_1}
⊢ Sorted (fun x x_1 => x > x_1) (map Prod.fst (CNF b 0))
case refine_2
b o✝ o : Ordinal.{u_1}
ho : o ≠ 0
IH : Sorted (fun x x_1 => x > x_1) (map Prod.fst (CNF b (o % b ^ log b o)))
⊢ Sorted (fun x x_1 => x > x_1) (map Prod.fst (CNF b o)) |
Mathlib/SetTheory/Ordinal/CantorNormalForm.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Mathlib.SetTheory.Ordinal.Exponential",
"Mathlib/SetTheory/Ordinal/Exponential.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "@[elab_as_elim]\nnoncomputable def CNFRec (b : Ordinal) {C : Ordinal → Sort*} (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : ∀ o, C o := fun o ↦ by\n by_cases h : o = 0\n · rw [h]; exact H0\n · exact H o h (CNFRec _ H0 H (o % b ^ log b o))\n termination_by o => o\n decreasing_by exact mod_opow_log_lt_self b h",
"end": [
50,
49
],
"full_name": "Ordinal.CNFRec",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem CNFRec_zero {C : Ordinal → Sort*} (b : Ordinal) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : @CNFRec b C H0 H 0 = H0",
"end": [
58,
6
],
"full_name": "Ordinal.CNFRec_zero",
"kind": "commanddeclaration",
"start": [
54,
1
]
},
{
"code": "theorem CNFRec_pos (b : Ordinal) {o : Ordinal} {C : Ordinal → Sort*} (ho : o ≠ 0) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) :\n @CNFRec b C H0 H o = H o ho (@CNFRec b C H0 H _)",
"end": [
64,
83
],
"full_name": "Ordinal.CNFRec_pos",
"kind": "commanddeclaration",
"start": [
62,
1
]
},
{
"code": "@[pp_nodot]\ndef CNF (b o : Ordinal) : List (Ordinal × Ordinal) :=\n CNFRec b [] (fun o _ho IH ↦ (log b o, o / b ^ log b o)::IH) o",
"end": [
76,
64
],
"full_name": "Ordinal.CNF",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "@[simp]\ntheorem CNF_zero (b : Ordinal) : CNF b 0 = []",
"end": [
82,
20
],
"full_name": "Ordinal.CNF_zero",
"kind": "commanddeclaration",
"start": [
80,
1
]
},
{
"code": "theorem CNF_ne_zero {b o : Ordinal} (ho : o ≠ 0) :\n CNF b o = (log b o, o / b ^ log b o)::CNF b (o % b ^ log b o)",
"end": [
89,
22
],
"full_name": "Ordinal.CNF_ne_zero",
"kind": "commanddeclaration",
"start": [
86,
1
]
},
{
"code": "theorem zero_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 0 o = [⟨0, o⟩]",
"end": [
93,
93
],
"full_name": "Ordinal.zero_CNF",
"kind": "commanddeclaration",
"start": [
93,
1
]
},
{
"code": "theorem one_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 1 o = [⟨0, o⟩]",
"end": [
97,
92
],
"full_name": "Ordinal.one_CNF",
"kind": "commanddeclaration",
"start": [
97,
1
]
},
{
"code": "theorem CNF_of_le_one {b o : Ordinal} (hb : b ≤ 1) (ho : o ≠ 0) : CNF b o = [⟨0, o⟩]",
"end": [
104,
21
],
"full_name": "Ordinal.CNF_of_le_one",
"kind": "commanddeclaration",
"start": [
101,
1
]
},
{
"code": "theorem CNF_of_lt {b o : Ordinal} (ho : o ≠ 0) (hb : o < b) : CNF b o = [⟨0, o⟩]",
"end": [
109,
84
],
"full_name": "Ordinal.CNF_of_lt",
"kind": "commanddeclaration",
"start": [
108,
1
]
},
{
"code": "theorem CNF_foldr (b o : Ordinal) : (CNF b o).foldr (fun p r ↦ b ^ p.1 * p.2 + r) 0 = o",
"end": [
116,
74
],
"full_name": "Ordinal.CNF_foldr",
"kind": "commanddeclaration",
"start": [
113,
1
]
},
{
"code": "theorem CNF_fst_le_log {b o : Ordinal.{u}} {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.1 ≤ log b o",
"end": [
129,
74
],
"full_name": "Ordinal.CNF_fst_le_log",
"kind": "commanddeclaration",
"start": [
120,
1
]
},
{
"code": "theorem CNF_fst_le {b o : Ordinal.{u}} {x : Ordinal × Ordinal} (h : x ∈ CNF b o) : x.1 ≤ o",
"end": [
135,
46
],
"full_name": "Ordinal.CNF_fst_le",
"kind": "commanddeclaration",
"start": [
133,
1
]
},
{
"code": "theorem CNF_lt_snd {b o : Ordinal.{u}} {x : Ordinal × Ordinal} : x ∈ CNF b o → 0 < x.2",
"end": [
145,
15
],
"full_name": "Ordinal.CNF_lt_snd",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem CNF_snd_lt {b o : Ordinal.{u}} (hb : 1 < b) {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.2 < b",
"end": [
158,
17
],
"full_name": "Ordinal.CNF_snd_lt",
"kind": "commanddeclaration",
"start": [
149,
1
]
},
{
"code": "theorem CNF_sorted (b o : Ordinal) : ((CNF b o).map Prod.fst).Sorted (· > ·)",
"end": [
174,
83
],
"full_name": "Ordinal.CNF_sorted",
"kind": "commanddeclaration",
"start": [
162,
1
]
}
] | 13 | Ordinal.CNF_sorted | [
[
163,
81
],
[
174,
83
]
] | 1 | 17 | · simp only [gt_iff_lt, CNF_zero, map_nil, sorted_nil] | case refine_1
b o : Ordinal.{u_1}
⊢ Sorted (fun x x_1 => x > x_1) (map Prod.fst (CNF b 0))
case refine_2
b o✝ o : Ordinal.{u_1}
ho : o ≠ 0
IH : Sorted (fun x x_1 => x > x_1) (map Prod.fst (CNF b (o % b ^ log b o)))
⊢ Sorted (fun x x_1 => x > x_1) (map Prod.fst (CNF b o)) | case refine_2
b o✝ o : Ordinal.{u_1}
ho : o ≠ 0
IH : Sorted (fun x x_1 => x > x_1) (map Prod.fst (CNF b (o % b ^ log b o)))
⊢ Sorted (fun x x_1 => x > x_1) (map Prod.fst (CNF b o)) |
Mathlib/SetTheory/Ordinal/CantorNormalForm.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Mathlib.SetTheory.Ordinal.Exponential",
"Mathlib/SetTheory/Ordinal/Exponential.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "@[elab_as_elim]\nnoncomputable def CNFRec (b : Ordinal) {C : Ordinal → Sort*} (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : ∀ o, C o := fun o ↦ by\n by_cases h : o = 0\n · rw [h]; exact H0\n · exact H o h (CNFRec _ H0 H (o % b ^ log b o))\n termination_by o => o\n decreasing_by exact mod_opow_log_lt_self b h",
"end": [
50,
49
],
"full_name": "Ordinal.CNFRec",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem CNFRec_zero {C : Ordinal → Sort*} (b : Ordinal) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : @CNFRec b C H0 H 0 = H0",
"end": [
58,
6
],
"full_name": "Ordinal.CNFRec_zero",
"kind": "commanddeclaration",
"start": [
54,
1
]
},
{
"code": "theorem CNFRec_pos (b : Ordinal) {o : Ordinal} {C : Ordinal → Sort*} (ho : o ≠ 0) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) :\n @CNFRec b C H0 H o = H o ho (@CNFRec b C H0 H _)",
"end": [
64,
83
],
"full_name": "Ordinal.CNFRec_pos",
"kind": "commanddeclaration",
"start": [
62,
1
]
},
{
"code": "@[pp_nodot]\ndef CNF (b o : Ordinal) : List (Ordinal × Ordinal) :=\n CNFRec b [] (fun o _ho IH ↦ (log b o, o / b ^ log b o)::IH) o",
"end": [
76,
64
],
"full_name": "Ordinal.CNF",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "@[simp]\ntheorem CNF_zero (b : Ordinal) : CNF b 0 = []",
"end": [
82,
20
],
"full_name": "Ordinal.CNF_zero",
"kind": "commanddeclaration",
"start": [
80,
1
]
},
{
"code": "theorem CNF_ne_zero {b o : Ordinal} (ho : o ≠ 0) :\n CNF b o = (log b o, o / b ^ log b o)::CNF b (o % b ^ log b o)",
"end": [
89,
22
],
"full_name": "Ordinal.CNF_ne_zero",
"kind": "commanddeclaration",
"start": [
86,
1
]
},
{
"code": "theorem zero_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 0 o = [⟨0, o⟩]",
"end": [
93,
93
],
"full_name": "Ordinal.zero_CNF",
"kind": "commanddeclaration",
"start": [
93,
1
]
},
{
"code": "theorem one_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 1 o = [⟨0, o⟩]",
"end": [
97,
92
],
"full_name": "Ordinal.one_CNF",
"kind": "commanddeclaration",
"start": [
97,
1
]
},
{
"code": "theorem CNF_of_le_one {b o : Ordinal} (hb : b ≤ 1) (ho : o ≠ 0) : CNF b o = [⟨0, o⟩]",
"end": [
104,
21
],
"full_name": "Ordinal.CNF_of_le_one",
"kind": "commanddeclaration",
"start": [
101,
1
]
},
{
"code": "theorem CNF_of_lt {b o : Ordinal} (ho : o ≠ 0) (hb : o < b) : CNF b o = [⟨0, o⟩]",
"end": [
109,
84
],
"full_name": "Ordinal.CNF_of_lt",
"kind": "commanddeclaration",
"start": [
108,
1
]
},
{
"code": "theorem CNF_foldr (b o : Ordinal) : (CNF b o).foldr (fun p r ↦ b ^ p.1 * p.2 + r) 0 = o",
"end": [
116,
74
],
"full_name": "Ordinal.CNF_foldr",
"kind": "commanddeclaration",
"start": [
113,
1
]
},
{
"code": "theorem CNF_fst_le_log {b o : Ordinal.{u}} {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.1 ≤ log b o",
"end": [
129,
74
],
"full_name": "Ordinal.CNF_fst_le_log",
"kind": "commanddeclaration",
"start": [
120,
1
]
},
{
"code": "theorem CNF_fst_le {b o : Ordinal.{u}} {x : Ordinal × Ordinal} (h : x ∈ CNF b o) : x.1 ≤ o",
"end": [
135,
46
],
"full_name": "Ordinal.CNF_fst_le",
"kind": "commanddeclaration",
"start": [
133,
1
]
},
{
"code": "theorem CNF_lt_snd {b o : Ordinal.{u}} {x : Ordinal × Ordinal} : x ∈ CNF b o → 0 < x.2",
"end": [
145,
15
],
"full_name": "Ordinal.CNF_lt_snd",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem CNF_snd_lt {b o : Ordinal.{u}} (hb : 1 < b) {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.2 < b",
"end": [
158,
17
],
"full_name": "Ordinal.CNF_snd_lt",
"kind": "commanddeclaration",
"start": [
149,
1
]
},
{
"code": "theorem CNF_sorted (b o : Ordinal) : ((CNF b o).map Prod.fst).Sorted (· > ·)",
"end": [
174,
83
],
"full_name": "Ordinal.CNF_sorted",
"kind": "commanddeclaration",
"start": [
162,
1
]
}
] | 13 | Ordinal.CNF_sorted | [
[
163,
81
],
[
174,
83
]
] | 2 | 17 | · rcases le_or_lt b 1 with hb | hb
· simp only [CNF_of_le_one hb ho, gt_iff_lt, map_cons, map, sorted_singleton]
· cases' lt_or_le o b with hob hbo
· simp only [CNF_of_lt ho hob, gt_iff_lt, map_cons, map, sorted_singleton]
· rw [CNF_ne_zero ho, map_cons, sorted_cons]
refine ⟨fun a H ↦ ?_, IH⟩
rw [mem_map] at H
rcases H with ⟨⟨a, a'⟩, H, rfl⟩
exact (CNF_fst_le_log H).trans_lt (log_mod_opow_log_lt_log_self hb ho hbo) | case refine_2
b o✝ o : Ordinal.{u_1}
ho : o ≠ 0
IH : Sorted (fun x x_1 => x > x_1) (map Prod.fst (CNF b (o % b ^ log b o)))
⊢ Sorted (fun x x_1 => x > x_1) (map Prod.fst (CNF b o)) | no goals |
Mathlib/SetTheory/Ordinal/CantorNormalForm.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Mathlib.SetTheory.Ordinal.Exponential",
"Mathlib/SetTheory/Ordinal/Exponential.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "@[elab_as_elim]\nnoncomputable def CNFRec (b : Ordinal) {C : Ordinal → Sort*} (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : ∀ o, C o := fun o ↦ by\n by_cases h : o = 0\n · rw [h]; exact H0\n · exact H o h (CNFRec _ H0 H (o % b ^ log b o))\n termination_by o => o\n decreasing_by exact mod_opow_log_lt_self b h",
"end": [
50,
49
],
"full_name": "Ordinal.CNFRec",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem CNFRec_zero {C : Ordinal → Sort*} (b : Ordinal) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : @CNFRec b C H0 H 0 = H0",
"end": [
58,
6
],
"full_name": "Ordinal.CNFRec_zero",
"kind": "commanddeclaration",
"start": [
54,
1
]
},
{
"code": "theorem CNFRec_pos (b : Ordinal) {o : Ordinal} {C : Ordinal → Sort*} (ho : o ≠ 0) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) :\n @CNFRec b C H0 H o = H o ho (@CNFRec b C H0 H _)",
"end": [
64,
83
],
"full_name": "Ordinal.CNFRec_pos",
"kind": "commanddeclaration",
"start": [
62,
1
]
},
{
"code": "@[pp_nodot]\ndef CNF (b o : Ordinal) : List (Ordinal × Ordinal) :=\n CNFRec b [] (fun o _ho IH ↦ (log b o, o / b ^ log b o)::IH) o",
"end": [
76,
64
],
"full_name": "Ordinal.CNF",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "@[simp]\ntheorem CNF_zero (b : Ordinal) : CNF b 0 = []",
"end": [
82,
20
],
"full_name": "Ordinal.CNF_zero",
"kind": "commanddeclaration",
"start": [
80,
1
]
},
{
"code": "theorem CNF_ne_zero {b o : Ordinal} (ho : o ≠ 0) :\n CNF b o = (log b o, o / b ^ log b o)::CNF b (o % b ^ log b o)",
"end": [
89,
22
],
"full_name": "Ordinal.CNF_ne_zero",
"kind": "commanddeclaration",
"start": [
86,
1
]
},
{
"code": "theorem zero_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 0 o = [⟨0, o⟩]",
"end": [
93,
93
],
"full_name": "Ordinal.zero_CNF",
"kind": "commanddeclaration",
"start": [
93,
1
]
},
{
"code": "theorem one_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 1 o = [⟨0, o⟩]",
"end": [
97,
92
],
"full_name": "Ordinal.one_CNF",
"kind": "commanddeclaration",
"start": [
97,
1
]
},
{
"code": "theorem CNF_of_le_one {b o : Ordinal} (hb : b ≤ 1) (ho : o ≠ 0) : CNF b o = [⟨0, o⟩]",
"end": [
104,
21
],
"full_name": "Ordinal.CNF_of_le_one",
"kind": "commanddeclaration",
"start": [
101,
1
]
},
{
"code": "theorem CNF_of_lt {b o : Ordinal} (ho : o ≠ 0) (hb : o < b) : CNF b o = [⟨0, o⟩]",
"end": [
109,
84
],
"full_name": "Ordinal.CNF_of_lt",
"kind": "commanddeclaration",
"start": [
108,
1
]
},
{
"code": "theorem CNF_foldr (b o : Ordinal) : (CNF b o).foldr (fun p r ↦ b ^ p.1 * p.2 + r) 0 = o",
"end": [
116,
74
],
"full_name": "Ordinal.CNF_foldr",
"kind": "commanddeclaration",
"start": [
113,
1
]
},
{
"code": "theorem CNF_fst_le_log {b o : Ordinal.{u}} {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.1 ≤ log b o",
"end": [
129,
74
],
"full_name": "Ordinal.CNF_fst_le_log",
"kind": "commanddeclaration",
"start": [
120,
1
]
},
{
"code": "theorem CNF_fst_le {b o : Ordinal.{u}} {x : Ordinal × Ordinal} (h : x ∈ CNF b o) : x.1 ≤ o",
"end": [
135,
46
],
"full_name": "Ordinal.CNF_fst_le",
"kind": "commanddeclaration",
"start": [
133,
1
]
},
{
"code": "theorem CNF_lt_snd {b o : Ordinal.{u}} {x : Ordinal × Ordinal} : x ∈ CNF b o → 0 < x.2",
"end": [
145,
15
],
"full_name": "Ordinal.CNF_lt_snd",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem CNF_snd_lt {b o : Ordinal.{u}} (hb : 1 < b) {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.2 < b",
"end": [
158,
17
],
"full_name": "Ordinal.CNF_snd_lt",
"kind": "commanddeclaration",
"start": [
149,
1
]
},
{
"code": "theorem CNF_sorted (b o : Ordinal) : ((CNF b o).map Prod.fst).Sorted (· > ·)",
"end": [
174,
83
],
"full_name": "Ordinal.CNF_sorted",
"kind": "commanddeclaration",
"start": [
162,
1
]
}
] | 13 | Ordinal.CNF_sorted | [
[
163,
81
],
[
174,
83
]
] | 3 | 17 | simp only [gt_iff_lt, CNF_zero, map_nil, sorted_nil] | case refine_1
b o : Ordinal.{u_1}
⊢ Sorted (fun x x_1 => x > x_1) (map Prod.fst (CNF b 0)) | no goals |
Mathlib/SetTheory/Ordinal/CantorNormalForm.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Mathlib.SetTheory.Ordinal.Exponential",
"Mathlib/SetTheory/Ordinal/Exponential.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "@[elab_as_elim]\nnoncomputable def CNFRec (b : Ordinal) {C : Ordinal → Sort*} (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : ∀ o, C o := fun o ↦ by\n by_cases h : o = 0\n · rw [h]; exact H0\n · exact H o h (CNFRec _ H0 H (o % b ^ log b o))\n termination_by o => o\n decreasing_by exact mod_opow_log_lt_self b h",
"end": [
50,
49
],
"full_name": "Ordinal.CNFRec",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem CNFRec_zero {C : Ordinal → Sort*} (b : Ordinal) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : @CNFRec b C H0 H 0 = H0",
"end": [
58,
6
],
"full_name": "Ordinal.CNFRec_zero",
"kind": "commanddeclaration",
"start": [
54,
1
]
},
{
"code": "theorem CNFRec_pos (b : Ordinal) {o : Ordinal} {C : Ordinal → Sort*} (ho : o ≠ 0) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) :\n @CNFRec b C H0 H o = H o ho (@CNFRec b C H0 H _)",
"end": [
64,
83
],
"full_name": "Ordinal.CNFRec_pos",
"kind": "commanddeclaration",
"start": [
62,
1
]
},
{
"code": "@[pp_nodot]\ndef CNF (b o : Ordinal) : List (Ordinal × Ordinal) :=\n CNFRec b [] (fun o _ho IH ↦ (log b o, o / b ^ log b o)::IH) o",
"end": [
76,
64
],
"full_name": "Ordinal.CNF",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "@[simp]\ntheorem CNF_zero (b : Ordinal) : CNF b 0 = []",
"end": [
82,
20
],
"full_name": "Ordinal.CNF_zero",
"kind": "commanddeclaration",
"start": [
80,
1
]
},
{
"code": "theorem CNF_ne_zero {b o : Ordinal} (ho : o ≠ 0) :\n CNF b o = (log b o, o / b ^ log b o)::CNF b (o % b ^ log b o)",
"end": [
89,
22
],
"full_name": "Ordinal.CNF_ne_zero",
"kind": "commanddeclaration",
"start": [
86,
1
]
},
{
"code": "theorem zero_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 0 o = [⟨0, o⟩]",
"end": [
93,
93
],
"full_name": "Ordinal.zero_CNF",
"kind": "commanddeclaration",
"start": [
93,
1
]
},
{
"code": "theorem one_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 1 o = [⟨0, o⟩]",
"end": [
97,
92
],
"full_name": "Ordinal.one_CNF",
"kind": "commanddeclaration",
"start": [
97,
1
]
},
{
"code": "theorem CNF_of_le_one {b o : Ordinal} (hb : b ≤ 1) (ho : o ≠ 0) : CNF b o = [⟨0, o⟩]",
"end": [
104,
21
],
"full_name": "Ordinal.CNF_of_le_one",
"kind": "commanddeclaration",
"start": [
101,
1
]
},
{
"code": "theorem CNF_of_lt {b o : Ordinal} (ho : o ≠ 0) (hb : o < b) : CNF b o = [⟨0, o⟩]",
"end": [
109,
84
],
"full_name": "Ordinal.CNF_of_lt",
"kind": "commanddeclaration",
"start": [
108,
1
]
},
{
"code": "theorem CNF_foldr (b o : Ordinal) : (CNF b o).foldr (fun p r ↦ b ^ p.1 * p.2 + r) 0 = o",
"end": [
116,
74
],
"full_name": "Ordinal.CNF_foldr",
"kind": "commanddeclaration",
"start": [
113,
1
]
},
{
"code": "theorem CNF_fst_le_log {b o : Ordinal.{u}} {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.1 ≤ log b o",
"end": [
129,
74
],
"full_name": "Ordinal.CNF_fst_le_log",
"kind": "commanddeclaration",
"start": [
120,
1
]
},
{
"code": "theorem CNF_fst_le {b o : Ordinal.{u}} {x : Ordinal × Ordinal} (h : x ∈ CNF b o) : x.1 ≤ o",
"end": [
135,
46
],
"full_name": "Ordinal.CNF_fst_le",
"kind": "commanddeclaration",
"start": [
133,
1
]
},
{
"code": "theorem CNF_lt_snd {b o : Ordinal.{u}} {x : Ordinal × Ordinal} : x ∈ CNF b o → 0 < x.2",
"end": [
145,
15
],
"full_name": "Ordinal.CNF_lt_snd",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem CNF_snd_lt {b o : Ordinal.{u}} (hb : 1 < b) {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.2 < b",
"end": [
158,
17
],
"full_name": "Ordinal.CNF_snd_lt",
"kind": "commanddeclaration",
"start": [
149,
1
]
},
{
"code": "theorem CNF_sorted (b o : Ordinal) : ((CNF b o).map Prod.fst).Sorted (· > ·)",
"end": [
174,
83
],
"full_name": "Ordinal.CNF_sorted",
"kind": "commanddeclaration",
"start": [
162,
1
]
}
] | 13 | Ordinal.CNF_sorted | [
[
163,
81
],
[
174,
83
]
] | 4 | 17 | rcases le_or_lt b 1 with hb | hb | case refine_2
b o✝ o : Ordinal.{u_1}
ho : o ≠ 0
IH : Sorted (fun x x_1 => x > x_1) (map Prod.fst (CNF b (o % b ^ log b o)))
⊢ Sorted (fun x x_1 => x > x_1) (map Prod.fst (CNF b o)) | case refine_2.inl
b o✝ o : Ordinal.{u_1}
ho : o ≠ 0
IH : Sorted (fun x x_1 => x > x_1) (map Prod.fst (CNF b (o % b ^ log b o)))
hb : b ≤ 1
⊢ Sorted (fun x x_1 => x > x_1) (map Prod.fst (CNF b o))
case refine_2.inr
b o✝ o : Ordinal.{u_1}
ho : o ≠ 0
IH : Sorted (fun x x_1 => x > x_1) (map Prod.fst (CNF b (o % b ^ log b o)))
hb : 1 < b
⊢ Sorted (fun x x_1 => x > x_1) (map Prod.fst (CNF b o)) |
Mathlib/SetTheory/Ordinal/CantorNormalForm.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Mathlib.SetTheory.Ordinal.Exponential",
"Mathlib/SetTheory/Ordinal/Exponential.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "@[elab_as_elim]\nnoncomputable def CNFRec (b : Ordinal) {C : Ordinal → Sort*} (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : ∀ o, C o := fun o ↦ by\n by_cases h : o = 0\n · rw [h]; exact H0\n · exact H o h (CNFRec _ H0 H (o % b ^ log b o))\n termination_by o => o\n decreasing_by exact mod_opow_log_lt_self b h",
"end": [
50,
49
],
"full_name": "Ordinal.CNFRec",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem CNFRec_zero {C : Ordinal → Sort*} (b : Ordinal) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : @CNFRec b C H0 H 0 = H0",
"end": [
58,
6
],
"full_name": "Ordinal.CNFRec_zero",
"kind": "commanddeclaration",
"start": [
54,
1
]
},
{
"code": "theorem CNFRec_pos (b : Ordinal) {o : Ordinal} {C : Ordinal → Sort*} (ho : o ≠ 0) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) :\n @CNFRec b C H0 H o = H o ho (@CNFRec b C H0 H _)",
"end": [
64,
83
],
"full_name": "Ordinal.CNFRec_pos",
"kind": "commanddeclaration",
"start": [
62,
1
]
},
{
"code": "@[pp_nodot]\ndef CNF (b o : Ordinal) : List (Ordinal × Ordinal) :=\n CNFRec b [] (fun o _ho IH ↦ (log b o, o / b ^ log b o)::IH) o",
"end": [
76,
64
],
"full_name": "Ordinal.CNF",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "@[simp]\ntheorem CNF_zero (b : Ordinal) : CNF b 0 = []",
"end": [
82,
20
],
"full_name": "Ordinal.CNF_zero",
"kind": "commanddeclaration",
"start": [
80,
1
]
},
{
"code": "theorem CNF_ne_zero {b o : Ordinal} (ho : o ≠ 0) :\n CNF b o = (log b o, o / b ^ log b o)::CNF b (o % b ^ log b o)",
"end": [
89,
22
],
"full_name": "Ordinal.CNF_ne_zero",
"kind": "commanddeclaration",
"start": [
86,
1
]
},
{
"code": "theorem zero_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 0 o = [⟨0, o⟩]",
"end": [
93,
93
],
"full_name": "Ordinal.zero_CNF",
"kind": "commanddeclaration",
"start": [
93,
1
]
},
{
"code": "theorem one_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 1 o = [⟨0, o⟩]",
"end": [
97,
92
],
"full_name": "Ordinal.one_CNF",
"kind": "commanddeclaration",
"start": [
97,
1
]
},
{
"code": "theorem CNF_of_le_one {b o : Ordinal} (hb : b ≤ 1) (ho : o ≠ 0) : CNF b o = [⟨0, o⟩]",
"end": [
104,
21
],
"full_name": "Ordinal.CNF_of_le_one",
"kind": "commanddeclaration",
"start": [
101,
1
]
},
{
"code": "theorem CNF_of_lt {b o : Ordinal} (ho : o ≠ 0) (hb : o < b) : CNF b o = [⟨0, o⟩]",
"end": [
109,
84
],
"full_name": "Ordinal.CNF_of_lt",
"kind": "commanddeclaration",
"start": [
108,
1
]
},
{
"code": "theorem CNF_foldr (b o : Ordinal) : (CNF b o).foldr (fun p r ↦ b ^ p.1 * p.2 + r) 0 = o",
"end": [
116,
74
],
"full_name": "Ordinal.CNF_foldr",
"kind": "commanddeclaration",
"start": [
113,
1
]
},
{
"code": "theorem CNF_fst_le_log {b o : Ordinal.{u}} {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.1 ≤ log b o",
"end": [
129,
74
],
"full_name": "Ordinal.CNF_fst_le_log",
"kind": "commanddeclaration",
"start": [
120,
1
]
},
{
"code": "theorem CNF_fst_le {b o : Ordinal.{u}} {x : Ordinal × Ordinal} (h : x ∈ CNF b o) : x.1 ≤ o",
"end": [
135,
46
],
"full_name": "Ordinal.CNF_fst_le",
"kind": "commanddeclaration",
"start": [
133,
1
]
},
{
"code": "theorem CNF_lt_snd {b o : Ordinal.{u}} {x : Ordinal × Ordinal} : x ∈ CNF b o → 0 < x.2",
"end": [
145,
15
],
"full_name": "Ordinal.CNF_lt_snd",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem CNF_snd_lt {b o : Ordinal.{u}} (hb : 1 < b) {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.2 < b",
"end": [
158,
17
],
"full_name": "Ordinal.CNF_snd_lt",
"kind": "commanddeclaration",
"start": [
149,
1
]
},
{
"code": "theorem CNF_sorted (b o : Ordinal) : ((CNF b o).map Prod.fst).Sorted (· > ·)",
"end": [
174,
83
],
"full_name": "Ordinal.CNF_sorted",
"kind": "commanddeclaration",
"start": [
162,
1
]
}
] | 13 | Ordinal.CNF_sorted | [
[
163,
81
],
[
174,
83
]
] | 5 | 17 | · simp only [CNF_of_le_one hb ho, gt_iff_lt, map_cons, map, sorted_singleton] | case refine_2.inl
b o✝ o : Ordinal.{u_1}
ho : o ≠ 0
IH : Sorted (fun x x_1 => x > x_1) (map Prod.fst (CNF b (o % b ^ log b o)))
hb : b ≤ 1
⊢ Sorted (fun x x_1 => x > x_1) (map Prod.fst (CNF b o))
case refine_2.inr
b o✝ o : Ordinal.{u_1}
ho : o ≠ 0
IH : Sorted (fun x x_1 => x > x_1) (map Prod.fst (CNF b (o % b ^ log b o)))
hb : 1 < b
⊢ Sorted (fun x x_1 => x > x_1) (map Prod.fst (CNF b o)) | case refine_2.inr
b o✝ o : Ordinal.{u_1}
ho : o ≠ 0
IH : Sorted (fun x x_1 => x > x_1) (map Prod.fst (CNF b (o % b ^ log b o)))
hb : 1 < b
⊢ Sorted (fun x x_1 => x > x_1) (map Prod.fst (CNF b o)) |
Mathlib/SetTheory/Ordinal/CantorNormalForm.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Mathlib.SetTheory.Ordinal.Exponential",
"Mathlib/SetTheory/Ordinal/Exponential.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "@[elab_as_elim]\nnoncomputable def CNFRec (b : Ordinal) {C : Ordinal → Sort*} (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : ∀ o, C o := fun o ↦ by\n by_cases h : o = 0\n · rw [h]; exact H0\n · exact H o h (CNFRec _ H0 H (o % b ^ log b o))\n termination_by o => o\n decreasing_by exact mod_opow_log_lt_self b h",
"end": [
50,
49
],
"full_name": "Ordinal.CNFRec",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem CNFRec_zero {C : Ordinal → Sort*} (b : Ordinal) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : @CNFRec b C H0 H 0 = H0",
"end": [
58,
6
],
"full_name": "Ordinal.CNFRec_zero",
"kind": "commanddeclaration",
"start": [
54,
1
]
},
{
"code": "theorem CNFRec_pos (b : Ordinal) {o : Ordinal} {C : Ordinal → Sort*} (ho : o ≠ 0) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) :\n @CNFRec b C H0 H o = H o ho (@CNFRec b C H0 H _)",
"end": [
64,
83
],
"full_name": "Ordinal.CNFRec_pos",
"kind": "commanddeclaration",
"start": [
62,
1
]
},
{
"code": "@[pp_nodot]\ndef CNF (b o : Ordinal) : List (Ordinal × Ordinal) :=\n CNFRec b [] (fun o _ho IH ↦ (log b o, o / b ^ log b o)::IH) o",
"end": [
76,
64
],
"full_name": "Ordinal.CNF",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "@[simp]\ntheorem CNF_zero (b : Ordinal) : CNF b 0 = []",
"end": [
82,
20
],
"full_name": "Ordinal.CNF_zero",
"kind": "commanddeclaration",
"start": [
80,
1
]
},
{
"code": "theorem CNF_ne_zero {b o : Ordinal} (ho : o ≠ 0) :\n CNF b o = (log b o, o / b ^ log b o)::CNF b (o % b ^ log b o)",
"end": [
89,
22
],
"full_name": "Ordinal.CNF_ne_zero",
"kind": "commanddeclaration",
"start": [
86,
1
]
},
{
"code": "theorem zero_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 0 o = [⟨0, o⟩]",
"end": [
93,
93
],
"full_name": "Ordinal.zero_CNF",
"kind": "commanddeclaration",
"start": [
93,
1
]
},
{
"code": "theorem one_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 1 o = [⟨0, o⟩]",
"end": [
97,
92
],
"full_name": "Ordinal.one_CNF",
"kind": "commanddeclaration",
"start": [
97,
1
]
},
{
"code": "theorem CNF_of_le_one {b o : Ordinal} (hb : b ≤ 1) (ho : o ≠ 0) : CNF b o = [⟨0, o⟩]",
"end": [
104,
21
],
"full_name": "Ordinal.CNF_of_le_one",
"kind": "commanddeclaration",
"start": [
101,
1
]
},
{
"code": "theorem CNF_of_lt {b o : Ordinal} (ho : o ≠ 0) (hb : o < b) : CNF b o = [⟨0, o⟩]",
"end": [
109,
84
],
"full_name": "Ordinal.CNF_of_lt",
"kind": "commanddeclaration",
"start": [
108,
1
]
},
{
"code": "theorem CNF_foldr (b o : Ordinal) : (CNF b o).foldr (fun p r ↦ b ^ p.1 * p.2 + r) 0 = o",
"end": [
116,
74
],
"full_name": "Ordinal.CNF_foldr",
"kind": "commanddeclaration",
"start": [
113,
1
]
},
{
"code": "theorem CNF_fst_le_log {b o : Ordinal.{u}} {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.1 ≤ log b o",
"end": [
129,
74
],
"full_name": "Ordinal.CNF_fst_le_log",
"kind": "commanddeclaration",
"start": [
120,
1
]
},
{
"code": "theorem CNF_fst_le {b o : Ordinal.{u}} {x : Ordinal × Ordinal} (h : x ∈ CNF b o) : x.1 ≤ o",
"end": [
135,
46
],
"full_name": "Ordinal.CNF_fst_le",
"kind": "commanddeclaration",
"start": [
133,
1
]
},
{
"code": "theorem CNF_lt_snd {b o : Ordinal.{u}} {x : Ordinal × Ordinal} : x ∈ CNF b o → 0 < x.2",
"end": [
145,
15
],
"full_name": "Ordinal.CNF_lt_snd",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem CNF_snd_lt {b o : Ordinal.{u}} (hb : 1 < b) {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.2 < b",
"end": [
158,
17
],
"full_name": "Ordinal.CNF_snd_lt",
"kind": "commanddeclaration",
"start": [
149,
1
]
},
{
"code": "theorem CNF_sorted (b o : Ordinal) : ((CNF b o).map Prod.fst).Sorted (· > ·)",
"end": [
174,
83
],
"full_name": "Ordinal.CNF_sorted",
"kind": "commanddeclaration",
"start": [
162,
1
]
}
] | 13 | Ordinal.CNF_sorted | [
[
163,
81
],
[
174,
83
]
] | 6 | 17 | · cases' lt_or_le o b with hob hbo
· simp only [CNF_of_lt ho hob, gt_iff_lt, map_cons, map, sorted_singleton]
· rw [CNF_ne_zero ho, map_cons, sorted_cons]
refine ⟨fun a H ↦ ?_, IH⟩
rw [mem_map] at H
rcases H with ⟨⟨a, a'⟩, H, rfl⟩
exact (CNF_fst_le_log H).trans_lt (log_mod_opow_log_lt_log_self hb ho hbo) | case refine_2.inr
b o✝ o : Ordinal.{u_1}
ho : o ≠ 0
IH : Sorted (fun x x_1 => x > x_1) (map Prod.fst (CNF b (o % b ^ log b o)))
hb : 1 < b
⊢ Sorted (fun x x_1 => x > x_1) (map Prod.fst (CNF b o)) | no goals |
Mathlib/SetTheory/Ordinal/CantorNormalForm.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Mathlib.SetTheory.Ordinal.Exponential",
"Mathlib/SetTheory/Ordinal/Exponential.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "@[elab_as_elim]\nnoncomputable def CNFRec (b : Ordinal) {C : Ordinal → Sort*} (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : ∀ o, C o := fun o ↦ by\n by_cases h : o = 0\n · rw [h]; exact H0\n · exact H o h (CNFRec _ H0 H (o % b ^ log b o))\n termination_by o => o\n decreasing_by exact mod_opow_log_lt_self b h",
"end": [
50,
49
],
"full_name": "Ordinal.CNFRec",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem CNFRec_zero {C : Ordinal → Sort*} (b : Ordinal) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : @CNFRec b C H0 H 0 = H0",
"end": [
58,
6
],
"full_name": "Ordinal.CNFRec_zero",
"kind": "commanddeclaration",
"start": [
54,
1
]
},
{
"code": "theorem CNFRec_pos (b : Ordinal) {o : Ordinal} {C : Ordinal → Sort*} (ho : o ≠ 0) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) :\n @CNFRec b C H0 H o = H o ho (@CNFRec b C H0 H _)",
"end": [
64,
83
],
"full_name": "Ordinal.CNFRec_pos",
"kind": "commanddeclaration",
"start": [
62,
1
]
},
{
"code": "@[pp_nodot]\ndef CNF (b o : Ordinal) : List (Ordinal × Ordinal) :=\n CNFRec b [] (fun o _ho IH ↦ (log b o, o / b ^ log b o)::IH) o",
"end": [
76,
64
],
"full_name": "Ordinal.CNF",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "@[simp]\ntheorem CNF_zero (b : Ordinal) : CNF b 0 = []",
"end": [
82,
20
],
"full_name": "Ordinal.CNF_zero",
"kind": "commanddeclaration",
"start": [
80,
1
]
},
{
"code": "theorem CNF_ne_zero {b o : Ordinal} (ho : o ≠ 0) :\n CNF b o = (log b o, o / b ^ log b o)::CNF b (o % b ^ log b o)",
"end": [
89,
22
],
"full_name": "Ordinal.CNF_ne_zero",
"kind": "commanddeclaration",
"start": [
86,
1
]
},
{
"code": "theorem zero_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 0 o = [⟨0, o⟩]",
"end": [
93,
93
],
"full_name": "Ordinal.zero_CNF",
"kind": "commanddeclaration",
"start": [
93,
1
]
},
{
"code": "theorem one_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 1 o = [⟨0, o⟩]",
"end": [
97,
92
],
"full_name": "Ordinal.one_CNF",
"kind": "commanddeclaration",
"start": [
97,
1
]
},
{
"code": "theorem CNF_of_le_one {b o : Ordinal} (hb : b ≤ 1) (ho : o ≠ 0) : CNF b o = [⟨0, o⟩]",
"end": [
104,
21
],
"full_name": "Ordinal.CNF_of_le_one",
"kind": "commanddeclaration",
"start": [
101,
1
]
},
{
"code": "theorem CNF_of_lt {b o : Ordinal} (ho : o ≠ 0) (hb : o < b) : CNF b o = [⟨0, o⟩]",
"end": [
109,
84
],
"full_name": "Ordinal.CNF_of_lt",
"kind": "commanddeclaration",
"start": [
108,
1
]
},
{
"code": "theorem CNF_foldr (b o : Ordinal) : (CNF b o).foldr (fun p r ↦ b ^ p.1 * p.2 + r) 0 = o",
"end": [
116,
74
],
"full_name": "Ordinal.CNF_foldr",
"kind": "commanddeclaration",
"start": [
113,
1
]
},
{
"code": "theorem CNF_fst_le_log {b o : Ordinal.{u}} {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.1 ≤ log b o",
"end": [
129,
74
],
"full_name": "Ordinal.CNF_fst_le_log",
"kind": "commanddeclaration",
"start": [
120,
1
]
},
{
"code": "theorem CNF_fst_le {b o : Ordinal.{u}} {x : Ordinal × Ordinal} (h : x ∈ CNF b o) : x.1 ≤ o",
"end": [
135,
46
],
"full_name": "Ordinal.CNF_fst_le",
"kind": "commanddeclaration",
"start": [
133,
1
]
},
{
"code": "theorem CNF_lt_snd {b o : Ordinal.{u}} {x : Ordinal × Ordinal} : x ∈ CNF b o → 0 < x.2",
"end": [
145,
15
],
"full_name": "Ordinal.CNF_lt_snd",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem CNF_snd_lt {b o : Ordinal.{u}} (hb : 1 < b) {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.2 < b",
"end": [
158,
17
],
"full_name": "Ordinal.CNF_snd_lt",
"kind": "commanddeclaration",
"start": [
149,
1
]
},
{
"code": "theorem CNF_sorted (b o : Ordinal) : ((CNF b o).map Prod.fst).Sorted (· > ·)",
"end": [
174,
83
],
"full_name": "Ordinal.CNF_sorted",
"kind": "commanddeclaration",
"start": [
162,
1
]
}
] | 13 | Ordinal.CNF_sorted | [
[
163,
81
],
[
174,
83
]
] | 7 | 17 | simp only [CNF_of_le_one hb ho, gt_iff_lt, map_cons, map, sorted_singleton] | case refine_2.inl
b o✝ o : Ordinal.{u_1}
ho : o ≠ 0
IH : Sorted (fun x x_1 => x > x_1) (map Prod.fst (CNF b (o % b ^ log b o)))
hb : b ≤ 1
⊢ Sorted (fun x x_1 => x > x_1) (map Prod.fst (CNF b o)) | no goals |
Mathlib/SetTheory/Ordinal/CantorNormalForm.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Mathlib.SetTheory.Ordinal.Exponential",
"Mathlib/SetTheory/Ordinal/Exponential.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "@[elab_as_elim]\nnoncomputable def CNFRec (b : Ordinal) {C : Ordinal → Sort*} (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : ∀ o, C o := fun o ↦ by\n by_cases h : o = 0\n · rw [h]; exact H0\n · exact H o h (CNFRec _ H0 H (o % b ^ log b o))\n termination_by o => o\n decreasing_by exact mod_opow_log_lt_self b h",
"end": [
50,
49
],
"full_name": "Ordinal.CNFRec",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem CNFRec_zero {C : Ordinal → Sort*} (b : Ordinal) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : @CNFRec b C H0 H 0 = H0",
"end": [
58,
6
],
"full_name": "Ordinal.CNFRec_zero",
"kind": "commanddeclaration",
"start": [
54,
1
]
},
{
"code": "theorem CNFRec_pos (b : Ordinal) {o : Ordinal} {C : Ordinal → Sort*} (ho : o ≠ 0) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) :\n @CNFRec b C H0 H o = H o ho (@CNFRec b C H0 H _)",
"end": [
64,
83
],
"full_name": "Ordinal.CNFRec_pos",
"kind": "commanddeclaration",
"start": [
62,
1
]
},
{
"code": "@[pp_nodot]\ndef CNF (b o : Ordinal) : List (Ordinal × Ordinal) :=\n CNFRec b [] (fun o _ho IH ↦ (log b o, o / b ^ log b o)::IH) o",
"end": [
76,
64
],
"full_name": "Ordinal.CNF",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "@[simp]\ntheorem CNF_zero (b : Ordinal) : CNF b 0 = []",
"end": [
82,
20
],
"full_name": "Ordinal.CNF_zero",
"kind": "commanddeclaration",
"start": [
80,
1
]
},
{
"code": "theorem CNF_ne_zero {b o : Ordinal} (ho : o ≠ 0) :\n CNF b o = (log b o, o / b ^ log b o)::CNF b (o % b ^ log b o)",
"end": [
89,
22
],
"full_name": "Ordinal.CNF_ne_zero",
"kind": "commanddeclaration",
"start": [
86,
1
]
},
{
"code": "theorem zero_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 0 o = [⟨0, o⟩]",
"end": [
93,
93
],
"full_name": "Ordinal.zero_CNF",
"kind": "commanddeclaration",
"start": [
93,
1
]
},
{
"code": "theorem one_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 1 o = [⟨0, o⟩]",
"end": [
97,
92
],
"full_name": "Ordinal.one_CNF",
"kind": "commanddeclaration",
"start": [
97,
1
]
},
{
"code": "theorem CNF_of_le_one {b o : Ordinal} (hb : b ≤ 1) (ho : o ≠ 0) : CNF b o = [⟨0, o⟩]",
"end": [
104,
21
],
"full_name": "Ordinal.CNF_of_le_one",
"kind": "commanddeclaration",
"start": [
101,
1
]
},
{
"code": "theorem CNF_of_lt {b o : Ordinal} (ho : o ≠ 0) (hb : o < b) : CNF b o = [⟨0, o⟩]",
"end": [
109,
84
],
"full_name": "Ordinal.CNF_of_lt",
"kind": "commanddeclaration",
"start": [
108,
1
]
},
{
"code": "theorem CNF_foldr (b o : Ordinal) : (CNF b o).foldr (fun p r ↦ b ^ p.1 * p.2 + r) 0 = o",
"end": [
116,
74
],
"full_name": "Ordinal.CNF_foldr",
"kind": "commanddeclaration",
"start": [
113,
1
]
},
{
"code": "theorem CNF_fst_le_log {b o : Ordinal.{u}} {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.1 ≤ log b o",
"end": [
129,
74
],
"full_name": "Ordinal.CNF_fst_le_log",
"kind": "commanddeclaration",
"start": [
120,
1
]
},
{
"code": "theorem CNF_fst_le {b o : Ordinal.{u}} {x : Ordinal × Ordinal} (h : x ∈ CNF b o) : x.1 ≤ o",
"end": [
135,
46
],
"full_name": "Ordinal.CNF_fst_le",
"kind": "commanddeclaration",
"start": [
133,
1
]
},
{
"code": "theorem CNF_lt_snd {b o : Ordinal.{u}} {x : Ordinal × Ordinal} : x ∈ CNF b o → 0 < x.2",
"end": [
145,
15
],
"full_name": "Ordinal.CNF_lt_snd",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem CNF_snd_lt {b o : Ordinal.{u}} (hb : 1 < b) {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.2 < b",
"end": [
158,
17
],
"full_name": "Ordinal.CNF_snd_lt",
"kind": "commanddeclaration",
"start": [
149,
1
]
},
{
"code": "theorem CNF_sorted (b o : Ordinal) : ((CNF b o).map Prod.fst).Sorted (· > ·)",
"end": [
174,
83
],
"full_name": "Ordinal.CNF_sorted",
"kind": "commanddeclaration",
"start": [
162,
1
]
}
] | 13 | Ordinal.CNF_sorted | [
[
163,
81
],
[
174,
83
]
] | 8 | 17 | cases' lt_or_le o b with hob hbo | case refine_2.inr
b o✝ o : Ordinal.{u_1}
ho : o ≠ 0
IH : Sorted (fun x x_1 => x > x_1) (map Prod.fst (CNF b (o % b ^ log b o)))
hb : 1 < b
⊢ Sorted (fun x x_1 => x > x_1) (map Prod.fst (CNF b o)) | case refine_2.inr.inl
b o✝ o : Ordinal.{u_1}
ho : o ≠ 0
IH : Sorted (fun x x_1 => x > x_1) (map Prod.fst (CNF b (o % b ^ log b o)))
hb : 1 < b
hob : o < b
⊢ Sorted (fun x x_1 => x > x_1) (map Prod.fst (CNF b o))
case refine_2.inr.inr
b o✝ o : Ordinal.{u_1}
ho : o ≠ 0
IH : Sorted (fun x x_1 => x > x_1) (map Prod.fst (CNF b (o % b ^ log b o)))
hb : 1 < b
hbo : b ≤ o
⊢ Sorted (fun x x_1 => x > x_1) (map Prod.fst (CNF b o)) |
Mathlib/SetTheory/Ordinal/CantorNormalForm.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Mathlib.SetTheory.Ordinal.Exponential",
"Mathlib/SetTheory/Ordinal/Exponential.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "@[elab_as_elim]\nnoncomputable def CNFRec (b : Ordinal) {C : Ordinal → Sort*} (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : ∀ o, C o := fun o ↦ by\n by_cases h : o = 0\n · rw [h]; exact H0\n · exact H o h (CNFRec _ H0 H (o % b ^ log b o))\n termination_by o => o\n decreasing_by exact mod_opow_log_lt_self b h",
"end": [
50,
49
],
"full_name": "Ordinal.CNFRec",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem CNFRec_zero {C : Ordinal → Sort*} (b : Ordinal) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : @CNFRec b C H0 H 0 = H0",
"end": [
58,
6
],
"full_name": "Ordinal.CNFRec_zero",
"kind": "commanddeclaration",
"start": [
54,
1
]
},
{
"code": "theorem CNFRec_pos (b : Ordinal) {o : Ordinal} {C : Ordinal → Sort*} (ho : o ≠ 0) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) :\n @CNFRec b C H0 H o = H o ho (@CNFRec b C H0 H _)",
"end": [
64,
83
],
"full_name": "Ordinal.CNFRec_pos",
"kind": "commanddeclaration",
"start": [
62,
1
]
},
{
"code": "@[pp_nodot]\ndef CNF (b o : Ordinal) : List (Ordinal × Ordinal) :=\n CNFRec b [] (fun o _ho IH ↦ (log b o, o / b ^ log b o)::IH) o",
"end": [
76,
64
],
"full_name": "Ordinal.CNF",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "@[simp]\ntheorem CNF_zero (b : Ordinal) : CNF b 0 = []",
"end": [
82,
20
],
"full_name": "Ordinal.CNF_zero",
"kind": "commanddeclaration",
"start": [
80,
1
]
},
{
"code": "theorem CNF_ne_zero {b o : Ordinal} (ho : o ≠ 0) :\n CNF b o = (log b o, o / b ^ log b o)::CNF b (o % b ^ log b o)",
"end": [
89,
22
],
"full_name": "Ordinal.CNF_ne_zero",
"kind": "commanddeclaration",
"start": [
86,
1
]
},
{
"code": "theorem zero_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 0 o = [⟨0, o⟩]",
"end": [
93,
93
],
"full_name": "Ordinal.zero_CNF",
"kind": "commanddeclaration",
"start": [
93,
1
]
},
{
"code": "theorem one_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 1 o = [⟨0, o⟩]",
"end": [
97,
92
],
"full_name": "Ordinal.one_CNF",
"kind": "commanddeclaration",
"start": [
97,
1
]
},
{
"code": "theorem CNF_of_le_one {b o : Ordinal} (hb : b ≤ 1) (ho : o ≠ 0) : CNF b o = [⟨0, o⟩]",
"end": [
104,
21
],
"full_name": "Ordinal.CNF_of_le_one",
"kind": "commanddeclaration",
"start": [
101,
1
]
},
{
"code": "theorem CNF_of_lt {b o : Ordinal} (ho : o ≠ 0) (hb : o < b) : CNF b o = [⟨0, o⟩]",
"end": [
109,
84
],
"full_name": "Ordinal.CNF_of_lt",
"kind": "commanddeclaration",
"start": [
108,
1
]
},
{
"code": "theorem CNF_foldr (b o : Ordinal) : (CNF b o).foldr (fun p r ↦ b ^ p.1 * p.2 + r) 0 = o",
"end": [
116,
74
],
"full_name": "Ordinal.CNF_foldr",
"kind": "commanddeclaration",
"start": [
113,
1
]
},
{
"code": "theorem CNF_fst_le_log {b o : Ordinal.{u}} {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.1 ≤ log b o",
"end": [
129,
74
],
"full_name": "Ordinal.CNF_fst_le_log",
"kind": "commanddeclaration",
"start": [
120,
1
]
},
{
"code": "theorem CNF_fst_le {b o : Ordinal.{u}} {x : Ordinal × Ordinal} (h : x ∈ CNF b o) : x.1 ≤ o",
"end": [
135,
46
],
"full_name": "Ordinal.CNF_fst_le",
"kind": "commanddeclaration",
"start": [
133,
1
]
},
{
"code": "theorem CNF_lt_snd {b o : Ordinal.{u}} {x : Ordinal × Ordinal} : x ∈ CNF b o → 0 < x.2",
"end": [
145,
15
],
"full_name": "Ordinal.CNF_lt_snd",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem CNF_snd_lt {b o : Ordinal.{u}} (hb : 1 < b) {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.2 < b",
"end": [
158,
17
],
"full_name": "Ordinal.CNF_snd_lt",
"kind": "commanddeclaration",
"start": [
149,
1
]
},
{
"code": "theorem CNF_sorted (b o : Ordinal) : ((CNF b o).map Prod.fst).Sorted (· > ·)",
"end": [
174,
83
],
"full_name": "Ordinal.CNF_sorted",
"kind": "commanddeclaration",
"start": [
162,
1
]
}
] | 13 | Ordinal.CNF_sorted | [
[
163,
81
],
[
174,
83
]
] | 9 | 17 | · simp only [CNF_of_lt ho hob, gt_iff_lt, map_cons, map, sorted_singleton] | case refine_2.inr.inl
b o✝ o : Ordinal.{u_1}
ho : o ≠ 0
IH : Sorted (fun x x_1 => x > x_1) (map Prod.fst (CNF b (o % b ^ log b o)))
hb : 1 < b
hob : o < b
⊢ Sorted (fun x x_1 => x > x_1) (map Prod.fst (CNF b o))
case refine_2.inr.inr
b o✝ o : Ordinal.{u_1}
ho : o ≠ 0
IH : Sorted (fun x x_1 => x > x_1) (map Prod.fst (CNF b (o % b ^ log b o)))
hb : 1 < b
hbo : b ≤ o
⊢ Sorted (fun x x_1 => x > x_1) (map Prod.fst (CNF b o)) | case refine_2.inr.inr
b o✝ o : Ordinal.{u_1}
ho : o ≠ 0
IH : Sorted (fun x x_1 => x > x_1) (map Prod.fst (CNF b (o % b ^ log b o)))
hb : 1 < b
hbo : b ≤ o
⊢ Sorted (fun x x_1 => x > x_1) (map Prod.fst (CNF b o)) |
Mathlib/SetTheory/Ordinal/CantorNormalForm.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Mathlib.SetTheory.Ordinal.Exponential",
"Mathlib/SetTheory/Ordinal/Exponential.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "@[elab_as_elim]\nnoncomputable def CNFRec (b : Ordinal) {C : Ordinal → Sort*} (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : ∀ o, C o := fun o ↦ by\n by_cases h : o = 0\n · rw [h]; exact H0\n · exact H o h (CNFRec _ H0 H (o % b ^ log b o))\n termination_by o => o\n decreasing_by exact mod_opow_log_lt_self b h",
"end": [
50,
49
],
"full_name": "Ordinal.CNFRec",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem CNFRec_zero {C : Ordinal → Sort*} (b : Ordinal) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : @CNFRec b C H0 H 0 = H0",
"end": [
58,
6
],
"full_name": "Ordinal.CNFRec_zero",
"kind": "commanddeclaration",
"start": [
54,
1
]
},
{
"code": "theorem CNFRec_pos (b : Ordinal) {o : Ordinal} {C : Ordinal → Sort*} (ho : o ≠ 0) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) :\n @CNFRec b C H0 H o = H o ho (@CNFRec b C H0 H _)",
"end": [
64,
83
],
"full_name": "Ordinal.CNFRec_pos",
"kind": "commanddeclaration",
"start": [
62,
1
]
},
{
"code": "@[pp_nodot]\ndef CNF (b o : Ordinal) : List (Ordinal × Ordinal) :=\n CNFRec b [] (fun o _ho IH ↦ (log b o, o / b ^ log b o)::IH) o",
"end": [
76,
64
],
"full_name": "Ordinal.CNF",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "@[simp]\ntheorem CNF_zero (b : Ordinal) : CNF b 0 = []",
"end": [
82,
20
],
"full_name": "Ordinal.CNF_zero",
"kind": "commanddeclaration",
"start": [
80,
1
]
},
{
"code": "theorem CNF_ne_zero {b o : Ordinal} (ho : o ≠ 0) :\n CNF b o = (log b o, o / b ^ log b o)::CNF b (o % b ^ log b o)",
"end": [
89,
22
],
"full_name": "Ordinal.CNF_ne_zero",
"kind": "commanddeclaration",
"start": [
86,
1
]
},
{
"code": "theorem zero_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 0 o = [⟨0, o⟩]",
"end": [
93,
93
],
"full_name": "Ordinal.zero_CNF",
"kind": "commanddeclaration",
"start": [
93,
1
]
},
{
"code": "theorem one_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 1 o = [⟨0, o⟩]",
"end": [
97,
92
],
"full_name": "Ordinal.one_CNF",
"kind": "commanddeclaration",
"start": [
97,
1
]
},
{
"code": "theorem CNF_of_le_one {b o : Ordinal} (hb : b ≤ 1) (ho : o ≠ 0) : CNF b o = [⟨0, o⟩]",
"end": [
104,
21
],
"full_name": "Ordinal.CNF_of_le_one",
"kind": "commanddeclaration",
"start": [
101,
1
]
},
{
"code": "theorem CNF_of_lt {b o : Ordinal} (ho : o ≠ 0) (hb : o < b) : CNF b o = [⟨0, o⟩]",
"end": [
109,
84
],
"full_name": "Ordinal.CNF_of_lt",
"kind": "commanddeclaration",
"start": [
108,
1
]
},
{
"code": "theorem CNF_foldr (b o : Ordinal) : (CNF b o).foldr (fun p r ↦ b ^ p.1 * p.2 + r) 0 = o",
"end": [
116,
74
],
"full_name": "Ordinal.CNF_foldr",
"kind": "commanddeclaration",
"start": [
113,
1
]
},
{
"code": "theorem CNF_fst_le_log {b o : Ordinal.{u}} {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.1 ≤ log b o",
"end": [
129,
74
],
"full_name": "Ordinal.CNF_fst_le_log",
"kind": "commanddeclaration",
"start": [
120,
1
]
},
{
"code": "theorem CNF_fst_le {b o : Ordinal.{u}} {x : Ordinal × Ordinal} (h : x ∈ CNF b o) : x.1 ≤ o",
"end": [
135,
46
],
"full_name": "Ordinal.CNF_fst_le",
"kind": "commanddeclaration",
"start": [
133,
1
]
},
{
"code": "theorem CNF_lt_snd {b o : Ordinal.{u}} {x : Ordinal × Ordinal} : x ∈ CNF b o → 0 < x.2",
"end": [
145,
15
],
"full_name": "Ordinal.CNF_lt_snd",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem CNF_snd_lt {b o : Ordinal.{u}} (hb : 1 < b) {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.2 < b",
"end": [
158,
17
],
"full_name": "Ordinal.CNF_snd_lt",
"kind": "commanddeclaration",
"start": [
149,
1
]
},
{
"code": "theorem CNF_sorted (b o : Ordinal) : ((CNF b o).map Prod.fst).Sorted (· > ·)",
"end": [
174,
83
],
"full_name": "Ordinal.CNF_sorted",
"kind": "commanddeclaration",
"start": [
162,
1
]
}
] | 13 | Ordinal.CNF_sorted | [
[
163,
81
],
[
174,
83
]
] | 10 | 17 | · rw [CNF_ne_zero ho, map_cons, sorted_cons]
refine ⟨fun a H ↦ ?_, IH⟩
rw [mem_map] at H
rcases H with ⟨⟨a, a'⟩, H, rfl⟩
exact (CNF_fst_le_log H).trans_lt (log_mod_opow_log_lt_log_self hb ho hbo) | case refine_2.inr.inr
b o✝ o : Ordinal.{u_1}
ho : o ≠ 0
IH : Sorted (fun x x_1 => x > x_1) (map Prod.fst (CNF b (o % b ^ log b o)))
hb : 1 < b
hbo : b ≤ o
⊢ Sorted (fun x x_1 => x > x_1) (map Prod.fst (CNF b o)) | no goals |
Mathlib/SetTheory/Ordinal/CantorNormalForm.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Mathlib.SetTheory.Ordinal.Exponential",
"Mathlib/SetTheory/Ordinal/Exponential.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "@[elab_as_elim]\nnoncomputable def CNFRec (b : Ordinal) {C : Ordinal → Sort*} (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : ∀ o, C o := fun o ↦ by\n by_cases h : o = 0\n · rw [h]; exact H0\n · exact H o h (CNFRec _ H0 H (o % b ^ log b o))\n termination_by o => o\n decreasing_by exact mod_opow_log_lt_self b h",
"end": [
50,
49
],
"full_name": "Ordinal.CNFRec",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem CNFRec_zero {C : Ordinal → Sort*} (b : Ordinal) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : @CNFRec b C H0 H 0 = H0",
"end": [
58,
6
],
"full_name": "Ordinal.CNFRec_zero",
"kind": "commanddeclaration",
"start": [
54,
1
]
},
{
"code": "theorem CNFRec_pos (b : Ordinal) {o : Ordinal} {C : Ordinal → Sort*} (ho : o ≠ 0) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) :\n @CNFRec b C H0 H o = H o ho (@CNFRec b C H0 H _)",
"end": [
64,
83
],
"full_name": "Ordinal.CNFRec_pos",
"kind": "commanddeclaration",
"start": [
62,
1
]
},
{
"code": "@[pp_nodot]\ndef CNF (b o : Ordinal) : List (Ordinal × Ordinal) :=\n CNFRec b [] (fun o _ho IH ↦ (log b o, o / b ^ log b o)::IH) o",
"end": [
76,
64
],
"full_name": "Ordinal.CNF",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "@[simp]\ntheorem CNF_zero (b : Ordinal) : CNF b 0 = []",
"end": [
82,
20
],
"full_name": "Ordinal.CNF_zero",
"kind": "commanddeclaration",
"start": [
80,
1
]
},
{
"code": "theorem CNF_ne_zero {b o : Ordinal} (ho : o ≠ 0) :\n CNF b o = (log b o, o / b ^ log b o)::CNF b (o % b ^ log b o)",
"end": [
89,
22
],
"full_name": "Ordinal.CNF_ne_zero",
"kind": "commanddeclaration",
"start": [
86,
1
]
},
{
"code": "theorem zero_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 0 o = [⟨0, o⟩]",
"end": [
93,
93
],
"full_name": "Ordinal.zero_CNF",
"kind": "commanddeclaration",
"start": [
93,
1
]
},
{
"code": "theorem one_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 1 o = [⟨0, o⟩]",
"end": [
97,
92
],
"full_name": "Ordinal.one_CNF",
"kind": "commanddeclaration",
"start": [
97,
1
]
},
{
"code": "theorem CNF_of_le_one {b o : Ordinal} (hb : b ≤ 1) (ho : o ≠ 0) : CNF b o = [⟨0, o⟩]",
"end": [
104,
21
],
"full_name": "Ordinal.CNF_of_le_one",
"kind": "commanddeclaration",
"start": [
101,
1
]
},
{
"code": "theorem CNF_of_lt {b o : Ordinal} (ho : o ≠ 0) (hb : o < b) : CNF b o = [⟨0, o⟩]",
"end": [
109,
84
],
"full_name": "Ordinal.CNF_of_lt",
"kind": "commanddeclaration",
"start": [
108,
1
]
},
{
"code": "theorem CNF_foldr (b o : Ordinal) : (CNF b o).foldr (fun p r ↦ b ^ p.1 * p.2 + r) 0 = o",
"end": [
116,
74
],
"full_name": "Ordinal.CNF_foldr",
"kind": "commanddeclaration",
"start": [
113,
1
]
},
{
"code": "theorem CNF_fst_le_log {b o : Ordinal.{u}} {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.1 ≤ log b o",
"end": [
129,
74
],
"full_name": "Ordinal.CNF_fst_le_log",
"kind": "commanddeclaration",
"start": [
120,
1
]
},
{
"code": "theorem CNF_fst_le {b o : Ordinal.{u}} {x : Ordinal × Ordinal} (h : x ∈ CNF b o) : x.1 ≤ o",
"end": [
135,
46
],
"full_name": "Ordinal.CNF_fst_le",
"kind": "commanddeclaration",
"start": [
133,
1
]
},
{
"code": "theorem CNF_lt_snd {b o : Ordinal.{u}} {x : Ordinal × Ordinal} : x ∈ CNF b o → 0 < x.2",
"end": [
145,
15
],
"full_name": "Ordinal.CNF_lt_snd",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem CNF_snd_lt {b o : Ordinal.{u}} (hb : 1 < b) {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.2 < b",
"end": [
158,
17
],
"full_name": "Ordinal.CNF_snd_lt",
"kind": "commanddeclaration",
"start": [
149,
1
]
},
{
"code": "theorem CNF_sorted (b o : Ordinal) : ((CNF b o).map Prod.fst).Sorted (· > ·)",
"end": [
174,
83
],
"full_name": "Ordinal.CNF_sorted",
"kind": "commanddeclaration",
"start": [
162,
1
]
}
] | 13 | Ordinal.CNF_sorted | [
[
163,
81
],
[
174,
83
]
] | 11 | 17 | simp only [CNF_of_lt ho hob, gt_iff_lt, map_cons, map, sorted_singleton] | case refine_2.inr.inl
b o✝ o : Ordinal.{u_1}
ho : o ≠ 0
IH : Sorted (fun x x_1 => x > x_1) (map Prod.fst (CNF b (o % b ^ log b o)))
hb : 1 < b
hob : o < b
⊢ Sorted (fun x x_1 => x > x_1) (map Prod.fst (CNF b o)) | no goals |
Mathlib/SetTheory/Ordinal/CantorNormalForm.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Mathlib.SetTheory.Ordinal.Exponential",
"Mathlib/SetTheory/Ordinal/Exponential.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "@[elab_as_elim]\nnoncomputable def CNFRec (b : Ordinal) {C : Ordinal → Sort*} (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : ∀ o, C o := fun o ↦ by\n by_cases h : o = 0\n · rw [h]; exact H0\n · exact H o h (CNFRec _ H0 H (o % b ^ log b o))\n termination_by o => o\n decreasing_by exact mod_opow_log_lt_self b h",
"end": [
50,
49
],
"full_name": "Ordinal.CNFRec",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem CNFRec_zero {C : Ordinal → Sort*} (b : Ordinal) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : @CNFRec b C H0 H 0 = H0",
"end": [
58,
6
],
"full_name": "Ordinal.CNFRec_zero",
"kind": "commanddeclaration",
"start": [
54,
1
]
},
{
"code": "theorem CNFRec_pos (b : Ordinal) {o : Ordinal} {C : Ordinal → Sort*} (ho : o ≠ 0) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) :\n @CNFRec b C H0 H o = H o ho (@CNFRec b C H0 H _)",
"end": [
64,
83
],
"full_name": "Ordinal.CNFRec_pos",
"kind": "commanddeclaration",
"start": [
62,
1
]
},
{
"code": "@[pp_nodot]\ndef CNF (b o : Ordinal) : List (Ordinal × Ordinal) :=\n CNFRec b [] (fun o _ho IH ↦ (log b o, o / b ^ log b o)::IH) o",
"end": [
76,
64
],
"full_name": "Ordinal.CNF",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "@[simp]\ntheorem CNF_zero (b : Ordinal) : CNF b 0 = []",
"end": [
82,
20
],
"full_name": "Ordinal.CNF_zero",
"kind": "commanddeclaration",
"start": [
80,
1
]
},
{
"code": "theorem CNF_ne_zero {b o : Ordinal} (ho : o ≠ 0) :\n CNF b o = (log b o, o / b ^ log b o)::CNF b (o % b ^ log b o)",
"end": [
89,
22
],
"full_name": "Ordinal.CNF_ne_zero",
"kind": "commanddeclaration",
"start": [
86,
1
]
},
{
"code": "theorem zero_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 0 o = [⟨0, o⟩]",
"end": [
93,
93
],
"full_name": "Ordinal.zero_CNF",
"kind": "commanddeclaration",
"start": [
93,
1
]
},
{
"code": "theorem one_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 1 o = [⟨0, o⟩]",
"end": [
97,
92
],
"full_name": "Ordinal.one_CNF",
"kind": "commanddeclaration",
"start": [
97,
1
]
},
{
"code": "theorem CNF_of_le_one {b o : Ordinal} (hb : b ≤ 1) (ho : o ≠ 0) : CNF b o = [⟨0, o⟩]",
"end": [
104,
21
],
"full_name": "Ordinal.CNF_of_le_one",
"kind": "commanddeclaration",
"start": [
101,
1
]
},
{
"code": "theorem CNF_of_lt {b o : Ordinal} (ho : o ≠ 0) (hb : o < b) : CNF b o = [⟨0, o⟩]",
"end": [
109,
84
],
"full_name": "Ordinal.CNF_of_lt",
"kind": "commanddeclaration",
"start": [
108,
1
]
},
{
"code": "theorem CNF_foldr (b o : Ordinal) : (CNF b o).foldr (fun p r ↦ b ^ p.1 * p.2 + r) 0 = o",
"end": [
116,
74
],
"full_name": "Ordinal.CNF_foldr",
"kind": "commanddeclaration",
"start": [
113,
1
]
},
{
"code": "theorem CNF_fst_le_log {b o : Ordinal.{u}} {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.1 ≤ log b o",
"end": [
129,
74
],
"full_name": "Ordinal.CNF_fst_le_log",
"kind": "commanddeclaration",
"start": [
120,
1
]
},
{
"code": "theorem CNF_fst_le {b o : Ordinal.{u}} {x : Ordinal × Ordinal} (h : x ∈ CNF b o) : x.1 ≤ o",
"end": [
135,
46
],
"full_name": "Ordinal.CNF_fst_le",
"kind": "commanddeclaration",
"start": [
133,
1
]
},
{
"code": "theorem CNF_lt_snd {b o : Ordinal.{u}} {x : Ordinal × Ordinal} : x ∈ CNF b o → 0 < x.2",
"end": [
145,
15
],
"full_name": "Ordinal.CNF_lt_snd",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem CNF_snd_lt {b o : Ordinal.{u}} (hb : 1 < b) {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.2 < b",
"end": [
158,
17
],
"full_name": "Ordinal.CNF_snd_lt",
"kind": "commanddeclaration",
"start": [
149,
1
]
},
{
"code": "theorem CNF_sorted (b o : Ordinal) : ((CNF b o).map Prod.fst).Sorted (· > ·)",
"end": [
174,
83
],
"full_name": "Ordinal.CNF_sorted",
"kind": "commanddeclaration",
"start": [
162,
1
]
}
] | 13 | Ordinal.CNF_sorted | [
[
163,
81
],
[
174,
83
]
] | 12 | 17 | rw [CNF_ne_zero ho, map_cons, sorted_cons] | case refine_2.inr.inr
b o✝ o : Ordinal.{u_1}
ho : o ≠ 0
IH : Sorted (fun x x_1 => x > x_1) (map Prod.fst (CNF b (o % b ^ log b o)))
hb : 1 < b
hbo : b ≤ o
⊢ Sorted (fun x x_1 => x > x_1) (map Prod.fst (CNF b o)) | case refine_2.inr.inr
b o✝ o : Ordinal.{u_1}
ho : o ≠ 0
IH : Sorted (fun x x_1 => x > x_1) (map Prod.fst (CNF b (o % b ^ log b o)))
hb : 1 < b
hbo : b ≤ o
⊢ (∀ b_1 ∈ map Prod.fst (CNF b (o % b ^ log b o)), (log b o, o / b ^ log b o).1 > b_1) ∧
Sorted (fun x x_1 => x > x_1) (map Prod.fst (CNF b (o % b ^ log b o))) |
Mathlib/SetTheory/Ordinal/CantorNormalForm.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Mathlib.SetTheory.Ordinal.Exponential",
"Mathlib/SetTheory/Ordinal/Exponential.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "@[elab_as_elim]\nnoncomputable def CNFRec (b : Ordinal) {C : Ordinal → Sort*} (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : ∀ o, C o := fun o ↦ by\n by_cases h : o = 0\n · rw [h]; exact H0\n · exact H o h (CNFRec _ H0 H (o % b ^ log b o))\n termination_by o => o\n decreasing_by exact mod_opow_log_lt_self b h",
"end": [
50,
49
],
"full_name": "Ordinal.CNFRec",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem CNFRec_zero {C : Ordinal → Sort*} (b : Ordinal) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : @CNFRec b C H0 H 0 = H0",
"end": [
58,
6
],
"full_name": "Ordinal.CNFRec_zero",
"kind": "commanddeclaration",
"start": [
54,
1
]
},
{
"code": "theorem CNFRec_pos (b : Ordinal) {o : Ordinal} {C : Ordinal → Sort*} (ho : o ≠ 0) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) :\n @CNFRec b C H0 H o = H o ho (@CNFRec b C H0 H _)",
"end": [
64,
83
],
"full_name": "Ordinal.CNFRec_pos",
"kind": "commanddeclaration",
"start": [
62,
1
]
},
{
"code": "@[pp_nodot]\ndef CNF (b o : Ordinal) : List (Ordinal × Ordinal) :=\n CNFRec b [] (fun o _ho IH ↦ (log b o, o / b ^ log b o)::IH) o",
"end": [
76,
64
],
"full_name": "Ordinal.CNF",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "@[simp]\ntheorem CNF_zero (b : Ordinal) : CNF b 0 = []",
"end": [
82,
20
],
"full_name": "Ordinal.CNF_zero",
"kind": "commanddeclaration",
"start": [
80,
1
]
},
{
"code": "theorem CNF_ne_zero {b o : Ordinal} (ho : o ≠ 0) :\n CNF b o = (log b o, o / b ^ log b o)::CNF b (o % b ^ log b o)",
"end": [
89,
22
],
"full_name": "Ordinal.CNF_ne_zero",
"kind": "commanddeclaration",
"start": [
86,
1
]
},
{
"code": "theorem zero_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 0 o = [⟨0, o⟩]",
"end": [
93,
93
],
"full_name": "Ordinal.zero_CNF",
"kind": "commanddeclaration",
"start": [
93,
1
]
},
{
"code": "theorem one_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 1 o = [⟨0, o⟩]",
"end": [
97,
92
],
"full_name": "Ordinal.one_CNF",
"kind": "commanddeclaration",
"start": [
97,
1
]
},
{
"code": "theorem CNF_of_le_one {b o : Ordinal} (hb : b ≤ 1) (ho : o ≠ 0) : CNF b o = [⟨0, o⟩]",
"end": [
104,
21
],
"full_name": "Ordinal.CNF_of_le_one",
"kind": "commanddeclaration",
"start": [
101,
1
]
},
{
"code": "theorem CNF_of_lt {b o : Ordinal} (ho : o ≠ 0) (hb : o < b) : CNF b o = [⟨0, o⟩]",
"end": [
109,
84
],
"full_name": "Ordinal.CNF_of_lt",
"kind": "commanddeclaration",
"start": [
108,
1
]
},
{
"code": "theorem CNF_foldr (b o : Ordinal) : (CNF b o).foldr (fun p r ↦ b ^ p.1 * p.2 + r) 0 = o",
"end": [
116,
74
],
"full_name": "Ordinal.CNF_foldr",
"kind": "commanddeclaration",
"start": [
113,
1
]
},
{
"code": "theorem CNF_fst_le_log {b o : Ordinal.{u}} {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.1 ≤ log b o",
"end": [
129,
74
],
"full_name": "Ordinal.CNF_fst_le_log",
"kind": "commanddeclaration",
"start": [
120,
1
]
},
{
"code": "theorem CNF_fst_le {b o : Ordinal.{u}} {x : Ordinal × Ordinal} (h : x ∈ CNF b o) : x.1 ≤ o",
"end": [
135,
46
],
"full_name": "Ordinal.CNF_fst_le",
"kind": "commanddeclaration",
"start": [
133,
1
]
},
{
"code": "theorem CNF_lt_snd {b o : Ordinal.{u}} {x : Ordinal × Ordinal} : x ∈ CNF b o → 0 < x.2",
"end": [
145,
15
],
"full_name": "Ordinal.CNF_lt_snd",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem CNF_snd_lt {b o : Ordinal.{u}} (hb : 1 < b) {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.2 < b",
"end": [
158,
17
],
"full_name": "Ordinal.CNF_snd_lt",
"kind": "commanddeclaration",
"start": [
149,
1
]
},
{
"code": "theorem CNF_sorted (b o : Ordinal) : ((CNF b o).map Prod.fst).Sorted (· > ·)",
"end": [
174,
83
],
"full_name": "Ordinal.CNF_sorted",
"kind": "commanddeclaration",
"start": [
162,
1
]
}
] | 13 | Ordinal.CNF_sorted | [
[
163,
81
],
[
174,
83
]
] | 13 | 17 | refine ⟨fun a H ↦ ?_, IH⟩ | case refine_2.inr.inr
b o✝ o : Ordinal.{u_1}
ho : o ≠ 0
IH : Sorted (fun x x_1 => x > x_1) (map Prod.fst (CNF b (o % b ^ log b o)))
hb : 1 < b
hbo : b ≤ o
⊢ (∀ b_1 ∈ map Prod.fst (CNF b (o % b ^ log b o)), (log b o, o / b ^ log b o).1 > b_1) ∧
Sorted (fun x x_1 => x > x_1) (map Prod.fst (CNF b (o % b ^ log b o))) | case refine_2.inr.inr
b o✝ o : Ordinal.{u_1}
ho : o ≠ 0
IH : Sorted (fun x x_1 => x > x_1) (map Prod.fst (CNF b (o % b ^ log b o)))
hb : 1 < b
hbo : b ≤ o
a : Ordinal.{u_1}
H : a ∈ map Prod.fst (CNF b (o % b ^ log b o))
⊢ (log b o, o / b ^ log b o).1 > a |
Mathlib/SetTheory/Ordinal/CantorNormalForm.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Mathlib.SetTheory.Ordinal.Exponential",
"Mathlib/SetTheory/Ordinal/Exponential.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "@[elab_as_elim]\nnoncomputable def CNFRec (b : Ordinal) {C : Ordinal → Sort*} (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : ∀ o, C o := fun o ↦ by\n by_cases h : o = 0\n · rw [h]; exact H0\n · exact H o h (CNFRec _ H0 H (o % b ^ log b o))\n termination_by o => o\n decreasing_by exact mod_opow_log_lt_self b h",
"end": [
50,
49
],
"full_name": "Ordinal.CNFRec",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem CNFRec_zero {C : Ordinal → Sort*} (b : Ordinal) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : @CNFRec b C H0 H 0 = H0",
"end": [
58,
6
],
"full_name": "Ordinal.CNFRec_zero",
"kind": "commanddeclaration",
"start": [
54,
1
]
},
{
"code": "theorem CNFRec_pos (b : Ordinal) {o : Ordinal} {C : Ordinal → Sort*} (ho : o ≠ 0) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) :\n @CNFRec b C H0 H o = H o ho (@CNFRec b C H0 H _)",
"end": [
64,
83
],
"full_name": "Ordinal.CNFRec_pos",
"kind": "commanddeclaration",
"start": [
62,
1
]
},
{
"code": "@[pp_nodot]\ndef CNF (b o : Ordinal) : List (Ordinal × Ordinal) :=\n CNFRec b [] (fun o _ho IH ↦ (log b o, o / b ^ log b o)::IH) o",
"end": [
76,
64
],
"full_name": "Ordinal.CNF",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "@[simp]\ntheorem CNF_zero (b : Ordinal) : CNF b 0 = []",
"end": [
82,
20
],
"full_name": "Ordinal.CNF_zero",
"kind": "commanddeclaration",
"start": [
80,
1
]
},
{
"code": "theorem CNF_ne_zero {b o : Ordinal} (ho : o ≠ 0) :\n CNF b o = (log b o, o / b ^ log b o)::CNF b (o % b ^ log b o)",
"end": [
89,
22
],
"full_name": "Ordinal.CNF_ne_zero",
"kind": "commanddeclaration",
"start": [
86,
1
]
},
{
"code": "theorem zero_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 0 o = [⟨0, o⟩]",
"end": [
93,
93
],
"full_name": "Ordinal.zero_CNF",
"kind": "commanddeclaration",
"start": [
93,
1
]
},
{
"code": "theorem one_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 1 o = [⟨0, o⟩]",
"end": [
97,
92
],
"full_name": "Ordinal.one_CNF",
"kind": "commanddeclaration",
"start": [
97,
1
]
},
{
"code": "theorem CNF_of_le_one {b o : Ordinal} (hb : b ≤ 1) (ho : o ≠ 0) : CNF b o = [⟨0, o⟩]",
"end": [
104,
21
],
"full_name": "Ordinal.CNF_of_le_one",
"kind": "commanddeclaration",
"start": [
101,
1
]
},
{
"code": "theorem CNF_of_lt {b o : Ordinal} (ho : o ≠ 0) (hb : o < b) : CNF b o = [⟨0, o⟩]",
"end": [
109,
84
],
"full_name": "Ordinal.CNF_of_lt",
"kind": "commanddeclaration",
"start": [
108,
1
]
},
{
"code": "theorem CNF_foldr (b o : Ordinal) : (CNF b o).foldr (fun p r ↦ b ^ p.1 * p.2 + r) 0 = o",
"end": [
116,
74
],
"full_name": "Ordinal.CNF_foldr",
"kind": "commanddeclaration",
"start": [
113,
1
]
},
{
"code": "theorem CNF_fst_le_log {b o : Ordinal.{u}} {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.1 ≤ log b o",
"end": [
129,
74
],
"full_name": "Ordinal.CNF_fst_le_log",
"kind": "commanddeclaration",
"start": [
120,
1
]
},
{
"code": "theorem CNF_fst_le {b o : Ordinal.{u}} {x : Ordinal × Ordinal} (h : x ∈ CNF b o) : x.1 ≤ o",
"end": [
135,
46
],
"full_name": "Ordinal.CNF_fst_le",
"kind": "commanddeclaration",
"start": [
133,
1
]
},
{
"code": "theorem CNF_lt_snd {b o : Ordinal.{u}} {x : Ordinal × Ordinal} : x ∈ CNF b o → 0 < x.2",
"end": [
145,
15
],
"full_name": "Ordinal.CNF_lt_snd",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem CNF_snd_lt {b o : Ordinal.{u}} (hb : 1 < b) {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.2 < b",
"end": [
158,
17
],
"full_name": "Ordinal.CNF_snd_lt",
"kind": "commanddeclaration",
"start": [
149,
1
]
},
{
"code": "theorem CNF_sorted (b o : Ordinal) : ((CNF b o).map Prod.fst).Sorted (· > ·)",
"end": [
174,
83
],
"full_name": "Ordinal.CNF_sorted",
"kind": "commanddeclaration",
"start": [
162,
1
]
}
] | 13 | Ordinal.CNF_sorted | [
[
163,
81
],
[
174,
83
]
] | 14 | 17 | rw [mem_map] at H | case refine_2.inr.inr
b o✝ o : Ordinal.{u_1}
ho : o ≠ 0
IH : Sorted (fun x x_1 => x > x_1) (map Prod.fst (CNF b (o % b ^ log b o)))
hb : 1 < b
hbo : b ≤ o
a : Ordinal.{u_1}
H : a ∈ map Prod.fst (CNF b (o % b ^ log b o))
⊢ (log b o, o / b ^ log b o).1 > a | case refine_2.inr.inr
b o✝ o : Ordinal.{u_1}
ho : o ≠ 0
IH : Sorted (fun x x_1 => x > x_1) (map Prod.fst (CNF b (o % b ^ log b o)))
hb : 1 < b
hbo : b ≤ o
a : Ordinal.{u_1}
H : ∃ a_1 ∈ CNF b (o % b ^ log b o), a_1.1 = a
⊢ (log b o, o / b ^ log b o).1 > a |
Mathlib/SetTheory/Ordinal/CantorNormalForm.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Mathlib.SetTheory.Ordinal.Exponential",
"Mathlib/SetTheory/Ordinal/Exponential.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "@[elab_as_elim]\nnoncomputable def CNFRec (b : Ordinal) {C : Ordinal → Sort*} (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : ∀ o, C o := fun o ↦ by\n by_cases h : o = 0\n · rw [h]; exact H0\n · exact H o h (CNFRec _ H0 H (o % b ^ log b o))\n termination_by o => o\n decreasing_by exact mod_opow_log_lt_self b h",
"end": [
50,
49
],
"full_name": "Ordinal.CNFRec",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem CNFRec_zero {C : Ordinal → Sort*} (b : Ordinal) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : @CNFRec b C H0 H 0 = H0",
"end": [
58,
6
],
"full_name": "Ordinal.CNFRec_zero",
"kind": "commanddeclaration",
"start": [
54,
1
]
},
{
"code": "theorem CNFRec_pos (b : Ordinal) {o : Ordinal} {C : Ordinal → Sort*} (ho : o ≠ 0) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) :\n @CNFRec b C H0 H o = H o ho (@CNFRec b C H0 H _)",
"end": [
64,
83
],
"full_name": "Ordinal.CNFRec_pos",
"kind": "commanddeclaration",
"start": [
62,
1
]
},
{
"code": "@[pp_nodot]\ndef CNF (b o : Ordinal) : List (Ordinal × Ordinal) :=\n CNFRec b [] (fun o _ho IH ↦ (log b o, o / b ^ log b o)::IH) o",
"end": [
76,
64
],
"full_name": "Ordinal.CNF",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "@[simp]\ntheorem CNF_zero (b : Ordinal) : CNF b 0 = []",
"end": [
82,
20
],
"full_name": "Ordinal.CNF_zero",
"kind": "commanddeclaration",
"start": [
80,
1
]
},
{
"code": "theorem CNF_ne_zero {b o : Ordinal} (ho : o ≠ 0) :\n CNF b o = (log b o, o / b ^ log b o)::CNF b (o % b ^ log b o)",
"end": [
89,
22
],
"full_name": "Ordinal.CNF_ne_zero",
"kind": "commanddeclaration",
"start": [
86,
1
]
},
{
"code": "theorem zero_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 0 o = [⟨0, o⟩]",
"end": [
93,
93
],
"full_name": "Ordinal.zero_CNF",
"kind": "commanddeclaration",
"start": [
93,
1
]
},
{
"code": "theorem one_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 1 o = [⟨0, o⟩]",
"end": [
97,
92
],
"full_name": "Ordinal.one_CNF",
"kind": "commanddeclaration",
"start": [
97,
1
]
},
{
"code": "theorem CNF_of_le_one {b o : Ordinal} (hb : b ≤ 1) (ho : o ≠ 0) : CNF b o = [⟨0, o⟩]",
"end": [
104,
21
],
"full_name": "Ordinal.CNF_of_le_one",
"kind": "commanddeclaration",
"start": [
101,
1
]
},
{
"code": "theorem CNF_of_lt {b o : Ordinal} (ho : o ≠ 0) (hb : o < b) : CNF b o = [⟨0, o⟩]",
"end": [
109,
84
],
"full_name": "Ordinal.CNF_of_lt",
"kind": "commanddeclaration",
"start": [
108,
1
]
},
{
"code": "theorem CNF_foldr (b o : Ordinal) : (CNF b o).foldr (fun p r ↦ b ^ p.1 * p.2 + r) 0 = o",
"end": [
116,
74
],
"full_name": "Ordinal.CNF_foldr",
"kind": "commanddeclaration",
"start": [
113,
1
]
},
{
"code": "theorem CNF_fst_le_log {b o : Ordinal.{u}} {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.1 ≤ log b o",
"end": [
129,
74
],
"full_name": "Ordinal.CNF_fst_le_log",
"kind": "commanddeclaration",
"start": [
120,
1
]
},
{
"code": "theorem CNF_fst_le {b o : Ordinal.{u}} {x : Ordinal × Ordinal} (h : x ∈ CNF b o) : x.1 ≤ o",
"end": [
135,
46
],
"full_name": "Ordinal.CNF_fst_le",
"kind": "commanddeclaration",
"start": [
133,
1
]
},
{
"code": "theorem CNF_lt_snd {b o : Ordinal.{u}} {x : Ordinal × Ordinal} : x ∈ CNF b o → 0 < x.2",
"end": [
145,
15
],
"full_name": "Ordinal.CNF_lt_snd",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem CNF_snd_lt {b o : Ordinal.{u}} (hb : 1 < b) {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.2 < b",
"end": [
158,
17
],
"full_name": "Ordinal.CNF_snd_lt",
"kind": "commanddeclaration",
"start": [
149,
1
]
},
{
"code": "theorem CNF_sorted (b o : Ordinal) : ((CNF b o).map Prod.fst).Sorted (· > ·)",
"end": [
174,
83
],
"full_name": "Ordinal.CNF_sorted",
"kind": "commanddeclaration",
"start": [
162,
1
]
}
] | 13 | Ordinal.CNF_sorted | [
[
163,
81
],
[
174,
83
]
] | 15 | 17 | rcases H with ⟨⟨a, a'⟩, H, rfl⟩ | case refine_2.inr.inr
b o✝ o : Ordinal.{u_1}
ho : o ≠ 0
IH : Sorted (fun x x_1 => x > x_1) (map Prod.fst (CNF b (o % b ^ log b o)))
hb : 1 < b
hbo : b ≤ o
a : Ordinal.{u_1}
H : ∃ a_1 ∈ CNF b (o % b ^ log b o), a_1.1 = a
⊢ (log b o, o / b ^ log b o).1 > a | case refine_2.inr.inr.intro.mk.intro
b o✝ o : Ordinal.{u_1}
ho : o ≠ 0
IH : Sorted (fun x x_1 => x > x_1) (map Prod.fst (CNF b (o % b ^ log b o)))
hb : 1 < b
hbo : b ≤ o
a a' : Ordinal.{u_1}
H : (a, a') ∈ CNF b (o % b ^ log b o)
⊢ (log b o, o / b ^ log b o).1 > (a, a').1 |
Mathlib/SetTheory/Ordinal/CantorNormalForm.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Mathlib.SetTheory.Ordinal.Exponential",
"Mathlib/SetTheory/Ordinal/Exponential.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "@[elab_as_elim]\nnoncomputable def CNFRec (b : Ordinal) {C : Ordinal → Sort*} (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : ∀ o, C o := fun o ↦ by\n by_cases h : o = 0\n · rw [h]; exact H0\n · exact H o h (CNFRec _ H0 H (o % b ^ log b o))\n termination_by o => o\n decreasing_by exact mod_opow_log_lt_self b h",
"end": [
50,
49
],
"full_name": "Ordinal.CNFRec",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem CNFRec_zero {C : Ordinal → Sort*} (b : Ordinal) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : @CNFRec b C H0 H 0 = H0",
"end": [
58,
6
],
"full_name": "Ordinal.CNFRec_zero",
"kind": "commanddeclaration",
"start": [
54,
1
]
},
{
"code": "theorem CNFRec_pos (b : Ordinal) {o : Ordinal} {C : Ordinal → Sort*} (ho : o ≠ 0) (H0 : C 0)\n (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) :\n @CNFRec b C H0 H o = H o ho (@CNFRec b C H0 H _)",
"end": [
64,
83
],
"full_name": "Ordinal.CNFRec_pos",
"kind": "commanddeclaration",
"start": [
62,
1
]
},
{
"code": "@[pp_nodot]\ndef CNF (b o : Ordinal) : List (Ordinal × Ordinal) :=\n CNFRec b [] (fun o _ho IH ↦ (log b o, o / b ^ log b o)::IH) o",
"end": [
76,
64
],
"full_name": "Ordinal.CNF",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "@[simp]\ntheorem CNF_zero (b : Ordinal) : CNF b 0 = []",
"end": [
82,
20
],
"full_name": "Ordinal.CNF_zero",
"kind": "commanddeclaration",
"start": [
80,
1
]
},
{
"code": "theorem CNF_ne_zero {b o : Ordinal} (ho : o ≠ 0) :\n CNF b o = (log b o, o / b ^ log b o)::CNF b (o % b ^ log b o)",
"end": [
89,
22
],
"full_name": "Ordinal.CNF_ne_zero",
"kind": "commanddeclaration",
"start": [
86,
1
]
},
{
"code": "theorem zero_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 0 o = [⟨0, o⟩]",
"end": [
93,
93
],
"full_name": "Ordinal.zero_CNF",
"kind": "commanddeclaration",
"start": [
93,
1
]
},
{
"code": "theorem one_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 1 o = [⟨0, o⟩]",
"end": [
97,
92
],
"full_name": "Ordinal.one_CNF",
"kind": "commanddeclaration",
"start": [
97,
1
]
},
{
"code": "theorem CNF_of_le_one {b o : Ordinal} (hb : b ≤ 1) (ho : o ≠ 0) : CNF b o = [⟨0, o⟩]",
"end": [
104,
21
],
"full_name": "Ordinal.CNF_of_le_one",
"kind": "commanddeclaration",
"start": [
101,
1
]
},
{
"code": "theorem CNF_of_lt {b o : Ordinal} (ho : o ≠ 0) (hb : o < b) : CNF b o = [⟨0, o⟩]",
"end": [
109,
84
],
"full_name": "Ordinal.CNF_of_lt",
"kind": "commanddeclaration",
"start": [
108,
1
]
},
{
"code": "theorem CNF_foldr (b o : Ordinal) : (CNF b o).foldr (fun p r ↦ b ^ p.1 * p.2 + r) 0 = o",
"end": [
116,
74
],
"full_name": "Ordinal.CNF_foldr",
"kind": "commanddeclaration",
"start": [
113,
1
]
},
{
"code": "theorem CNF_fst_le_log {b o : Ordinal.{u}} {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.1 ≤ log b o",
"end": [
129,
74
],
"full_name": "Ordinal.CNF_fst_le_log",
"kind": "commanddeclaration",
"start": [
120,
1
]
},
{
"code": "theorem CNF_fst_le {b o : Ordinal.{u}} {x : Ordinal × Ordinal} (h : x ∈ CNF b o) : x.1 ≤ o",
"end": [
135,
46
],
"full_name": "Ordinal.CNF_fst_le",
"kind": "commanddeclaration",
"start": [
133,
1
]
},
{
"code": "theorem CNF_lt_snd {b o : Ordinal.{u}} {x : Ordinal × Ordinal} : x ∈ CNF b o → 0 < x.2",
"end": [
145,
15
],
"full_name": "Ordinal.CNF_lt_snd",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem CNF_snd_lt {b o : Ordinal.{u}} (hb : 1 < b) {x : Ordinal × Ordinal} :\n x ∈ CNF b o → x.2 < b",
"end": [
158,
17
],
"full_name": "Ordinal.CNF_snd_lt",
"kind": "commanddeclaration",
"start": [
149,
1
]
},
{
"code": "theorem CNF_sorted (b o : Ordinal) : ((CNF b o).map Prod.fst).Sorted (· > ·)",
"end": [
174,
83
],
"full_name": "Ordinal.CNF_sorted",
"kind": "commanddeclaration",
"start": [
162,
1
]
}
] | 13 | Ordinal.CNF_sorted | [
[
163,
81
],
[
174,
83
]
] | 16 | 17 | exact (CNF_fst_le_log H).trans_lt (log_mod_opow_log_lt_log_self hb ho hbo) | case refine_2.inr.inr.intro.mk.intro
b o✝ o : Ordinal.{u_1}
ho : o ≠ 0
IH : Sorted (fun x x_1 => x > x_1) (map Prod.fst (CNF b (o % b ^ log b o)))
hb : 1 < b
hbo : b ≤ o
a a' : Ordinal.{u_1}
H : (a, a') ∈ CNF b (o % b ^ log b o)
⊢ (log b o, o / b ^ log b o).1 > (a, a').1 | no goals |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a ≠ 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a ≠ 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a ≠ 0) : a ^ b ≠ 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ ·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c ↔ b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b ≤ a ^ c ↔ b ≤ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c ↔ b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b → IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b ≠ 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a ≤ b) : a ^ c ≤ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a ≤ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b ≤ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b ≤ c) : a ^ b ∣ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b ∣ a ^ c ↔ b ≤ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : ¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b ≤ 1) : ∀ x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : ∀ b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : ∀ b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x ≠ 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x ≠ 0) : b ^ log b x ≤ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x ≠ 0) : b ^ c ≤ x ↔ c ≤ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x ≠ 0) : x < b ^ c ↔ log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o ≠ 0) (hbo : b ≤ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x ≤ y) : log b x ≤ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x ≤ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o ≠ 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o ≠ 0) (hbo : b ≤ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b ≠ 0) (u : Ordinal) (hv : v ≠ 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v ≠ 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o ≠ 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 1 | Ordinal.zero_opow' | [
[
42,
53
],
[
42,
85
]
] | 0 | 1 | simp only [opow_def, if_true] | a : Ordinal.{u_1}
⊢ 0 ^ a = 1 - a | no goals |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a ≠ 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a ≠ 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a ≠ 0) : a ^ b ≠ 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ ·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c ↔ b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b ≤ a ^ c ↔ b ≤ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c ↔ b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b → IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b ≠ 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a ≤ b) : a ^ c ≤ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a ≤ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b ≤ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b ≤ c) : a ^ b ∣ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b ∣ a ^ c ↔ b ≤ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : ¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b ≤ 1) : ∀ x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : ∀ b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : ∀ b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x ≠ 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x ≠ 0) : b ^ log b x ≤ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x ≠ 0) : b ^ c ≤ x ↔ c ≤ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x ≠ 0) : x < b ^ c ↔ log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o ≠ 0) (hbo : b ≤ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x ≤ y) : log b x ≤ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x ≤ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o ≠ 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o ≠ 0) (hbo : b ≤ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b ≠ 0) (u : Ordinal) (hv : v ≠ 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v ≠ 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o ≠ 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 2 | Ordinal.zero_opow | [
[
46,
73
],
[
47,
67
]
] | 0 | 1 | rwa [zero_opow', Ordinal.sub_eq_zero_iff_le, one_le_iff_ne_zero] | a : Ordinal.{u_1}
a0 : a ≠ 0
⊢ 0 ^ a = 0 | no goals |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a ≠ 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a ≠ 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a ≠ 0) : a ^ b ≠ 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ ·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c ↔ b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b ≤ a ^ c ↔ b ≤ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c ↔ b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b → IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b ≠ 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a ≤ b) : a ^ c ≤ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a ≤ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b ≤ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b ≤ c) : a ^ b ∣ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b ∣ a ^ c ↔ b ≤ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : ¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b ≤ 1) : ∀ x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : ∀ b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : ∀ b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x ≠ 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x ≠ 0) : b ^ log b x ≤ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x ≠ 0) : b ^ c ≤ x ↔ c ≤ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x ≠ 0) : x < b ^ c ↔ log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o ≠ 0) (hbo : b ≤ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x ≤ y) : log b x ≤ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x ≤ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o ≠ 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o ≠ 0) (hbo : b ≤ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b ≠ 0) (u : Ordinal) (hv : v ≠ 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v ≠ 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o ≠ 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 3 | Ordinal.opow_zero | [
[
51,
60
],
[
54,
52
]
] | 0 | 5 | by_cases h : a = 0 | a : Ordinal.{u_1}
⊢ a ^ 0 = 1 | case pos
a : Ordinal.{u_1}
h : a = 0
⊢ a ^ 0 = 1
case neg
a : Ordinal.{u_1}
h : ¬a = 0
⊢ a ^ 0 = 1 |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a ≠ 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a ≠ 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a ≠ 0) : a ^ b ≠ 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ ·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c ↔ b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b ≤ a ^ c ↔ b ≤ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c ↔ b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b → IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b ≠ 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a ≤ b) : a ^ c ≤ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a ≤ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b ≤ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b ≤ c) : a ^ b ∣ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b ∣ a ^ c ↔ b ≤ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : ¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b ≤ 1) : ∀ x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : ∀ b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : ∀ b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x ≠ 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x ≠ 0) : b ^ log b x ≤ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x ≠ 0) : b ^ c ≤ x ↔ c ≤ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x ≠ 0) : x < b ^ c ↔ log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o ≠ 0) (hbo : b ≤ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x ≤ y) : log b x ≤ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x ≤ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o ≠ 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o ≠ 0) (hbo : b ≤ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b ≠ 0) (u : Ordinal) (hv : v ≠ 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v ≠ 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o ≠ 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 3 | Ordinal.opow_zero | [
[
51,
60
],
[
54,
52
]
] | 1 | 5 | · simp only [opow_def, if_pos h, sub_zero] | case pos
a : Ordinal.{u_1}
h : a = 0
⊢ a ^ 0 = 1
case neg
a : Ordinal.{u_1}
h : ¬a = 0
⊢ a ^ 0 = 1 | case neg
a : Ordinal.{u_1}
h : ¬a = 0
⊢ a ^ 0 = 1 |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a ≠ 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a ≠ 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a ≠ 0) : a ^ b ≠ 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ ·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c ↔ b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b ≤ a ^ c ↔ b ≤ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c ↔ b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b → IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b ≠ 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a ≤ b) : a ^ c ≤ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a ≤ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b ≤ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b ≤ c) : a ^ b ∣ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b ∣ a ^ c ↔ b ≤ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : ¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b ≤ 1) : ∀ x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : ∀ b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : ∀ b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x ≠ 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x ≠ 0) : b ^ log b x ≤ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x ≠ 0) : b ^ c ≤ x ↔ c ≤ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x ≠ 0) : x < b ^ c ↔ log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o ≠ 0) (hbo : b ≤ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x ≤ y) : log b x ≤ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x ≤ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o ≠ 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o ≠ 0) (hbo : b ≤ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b ≠ 0) (u : Ordinal) (hv : v ≠ 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v ≠ 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o ≠ 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 3 | Ordinal.opow_zero | [
[
51,
60
],
[
54,
52
]
] | 2 | 5 | · simp only [opow_def, if_neg h, limitRecOn_zero] | case neg
a : Ordinal.{u_1}
h : ¬a = 0
⊢ a ^ 0 = 1 | no goals |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a ≠ 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a ≠ 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a ≠ 0) : a ^ b ≠ 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ ·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c ↔ b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b ≤ a ^ c ↔ b ≤ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c ↔ b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b → IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b ≠ 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a ≤ b) : a ^ c ≤ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a ≤ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b ≤ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b ≤ c) : a ^ b ∣ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b ∣ a ^ c ↔ b ≤ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : ¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b ≤ 1) : ∀ x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : ∀ b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : ∀ b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x ≠ 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x ≠ 0) : b ^ log b x ≤ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x ≠ 0) : b ^ c ≤ x ↔ c ≤ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x ≠ 0) : x < b ^ c ↔ log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o ≠ 0) (hbo : b ≤ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x ≤ y) : log b x ≤ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x ≤ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o ≠ 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o ≠ 0) (hbo : b ≤ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b ≠ 0) (u : Ordinal) (hv : v ≠ 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v ≠ 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o ≠ 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 3 | Ordinal.opow_zero | [
[
51,
60
],
[
54,
52
]
] | 3 | 5 | simp only [opow_def, if_pos h, sub_zero] | case pos
a : Ordinal.{u_1}
h : a = 0
⊢ a ^ 0 = 1 | no goals |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a ≠ 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a ≠ 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a ≠ 0) : a ^ b ≠ 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ ·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c ↔ b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b ≤ a ^ c ↔ b ≤ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c ↔ b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b → IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b ≠ 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a ≤ b) : a ^ c ≤ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a ≤ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b ≤ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b ≤ c) : a ^ b ∣ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b ∣ a ^ c ↔ b ≤ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : ¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b ≤ 1) : ∀ x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : ∀ b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : ∀ b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x ≠ 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x ≠ 0) : b ^ log b x ≤ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x ≠ 0) : b ^ c ≤ x ↔ c ≤ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x ≠ 0) : x < b ^ c ↔ log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o ≠ 0) (hbo : b ≤ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x ≤ y) : log b x ≤ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x ≤ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o ≠ 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o ≠ 0) (hbo : b ≤ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b ≠ 0) (u : Ordinal) (hv : v ≠ 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v ≠ 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o ≠ 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 3 | Ordinal.opow_zero | [
[
51,
60
],
[
54,
52
]
] | 4 | 5 | simp only [opow_def, if_neg h, limitRecOn_zero] | case neg
a : Ordinal.{u_1}
h : ¬a = 0
⊢ a ^ 0 = 1 | no goals |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a ≠ 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a ≠ 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a ≠ 0) : a ^ b ≠ 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ ·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c ↔ b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b ≤ a ^ c ↔ b ≤ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c ↔ b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b → IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b ≠ 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a ≤ b) : a ^ c ≤ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a ≤ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b ≤ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b ≤ c) : a ^ b ∣ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b ∣ a ^ c ↔ b ≤ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : ¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b ≤ 1) : ∀ x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : ∀ b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : ∀ b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x ≠ 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x ≠ 0) : b ^ log b x ≤ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x ≠ 0) : b ^ c ≤ x ↔ c ≤ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x ≠ 0) : x < b ^ c ↔ log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o ≠ 0) (hbo : b ≤ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x ≤ y) : log b x ≤ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x ≤ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o ≠ 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o ≠ 0) (hbo : b ≤ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b ≠ 0) (u : Ordinal) (hv : v ≠ 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v ≠ 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o ≠ 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 5 | Ordinal.opow_limit | [
[
64,
47
],
[
65,
67
]
] | 0 | 2 | simp only [opow_def, if_neg a0] | a b : Ordinal.{u}
a0 : a ≠ 0
h : b.IsLimit
⊢ a ^ b = b.bsup fun c x => a ^ c | a b : Ordinal.{u}
a0 : a ≠ 0
h : b.IsLimit
⊢ (b.limitRecOn 1 (fun x IH => IH * a) fun b x => b.bsup) =
b.bsup fun c x => c.limitRecOn 1 (fun x IH => IH * a) fun b x => b.bsup |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a ≠ 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a ≠ 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a ≠ 0) : a ^ b ≠ 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ ·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c ↔ b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b ≤ a ^ c ↔ b ≤ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c ↔ b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b → IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b ≠ 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a ≤ b) : a ^ c ≤ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a ≤ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b ≤ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b ≤ c) : a ^ b ∣ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b ∣ a ^ c ↔ b ≤ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : ¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b ≤ 1) : ∀ x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : ∀ b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : ∀ b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x ≠ 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x ≠ 0) : b ^ log b x ≤ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x ≠ 0) : b ^ c ≤ x ↔ c ≤ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x ≠ 0) : x < b ^ c ↔ log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o ≠ 0) (hbo : b ≤ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x ≤ y) : log b x ≤ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x ≤ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o ≠ 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o ≠ 0) (hbo : b ≤ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b ≠ 0) (u : Ordinal) (hv : v ≠ 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v ≠ 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o ≠ 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 5 | Ordinal.opow_limit | [
[
64,
47
],
[
65,
67
]
] | 1 | 2 | rw [limitRecOn_limit _ _ _ _ h] | a b : Ordinal.{u}
a0 : a ≠ 0
h : b.IsLimit
⊢ (b.limitRecOn 1 (fun x IH => IH * a) fun b x => b.bsup) =
b.bsup fun c x => c.limitRecOn 1 (fun x IH => IH * a) fun b x => b.bsup | no goals |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a ≠ 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a ≠ 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a ≠ 0) : a ^ b ≠ 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ ·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c ↔ b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b ≤ a ^ c ↔ b ≤ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c ↔ b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b → IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b ≠ 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a ≤ b) : a ^ c ≤ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a ≤ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b ≤ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b ≤ c) : a ^ b ∣ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b ∣ a ^ c ↔ b ≤ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : ¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b ≤ 1) : ∀ x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : ∀ b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : ∀ b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x ≠ 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x ≠ 0) : b ^ log b x ≤ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x ≠ 0) : b ^ c ≤ x ↔ c ≤ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x ≠ 0) : x < b ^ c ↔ log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o ≠ 0) (hbo : b ≤ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x ≤ y) : log b x ≤ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x ≤ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o ≠ 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o ≠ 0) (hbo : b ≤ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b ≠ 0) (u : Ordinal) (hv : v ≠ 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v ≠ 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o ≠ 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 6 | Ordinal.opow_le_of_limit | [
[
69,
41
],
[
69,
77
]
] | 0 | 1 | rw [opow_limit a0 h, bsup_le_iff] | a b c : Ordinal.{u_1}
a0 : a ≠ 0
h : b.IsLimit
⊢ a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ c | no goals |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a ≠ 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a ≠ 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a ≠ 0) : a ^ b ≠ 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ ·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c ↔ b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b ≤ a ^ c ↔ b ≤ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c ↔ b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b → IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b ≠ 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a ≤ b) : a ^ c ≤ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a ≤ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b ≤ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b ≤ c) : a ^ b ∣ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b ∣ a ^ c ↔ b ≤ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : ¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b ≤ 1) : ∀ x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : ∀ b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : ∀ b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x ≠ 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x ≠ 0) : b ^ log b x ≤ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x ≠ 0) : b ^ c ≤ x ↔ c ≤ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x ≠ 0) : x < b ^ c ↔ log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o ≠ 0) (hbo : b ≤ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x ≤ y) : log b x ≤ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x ≤ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o ≠ 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o ≠ 0) (hbo : b ≤ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b ≠ 0) (u : Ordinal) (hv : v ≠ 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v ≠ 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o ≠ 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 7 | Ordinal.lt_opow_of_limit | [
[
73,
41
],
[
74,
98
]
] | 0 | 2 | rw [← not_iff_not, not_exists] | a b c : Ordinal.{u_1}
b0 : b ≠ 0
h : c.IsLimit
⊢ a < b ^ c ↔ ∃ c' < c, a < b ^ c' | a b c : Ordinal.{u_1}
b0 : b ≠ 0
h : c.IsLimit
⊢ ¬a < b ^ c ↔ ∀ (x : Ordinal.{u_1}), ¬(x < c ∧ a < b ^ x) |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a ≠ 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a ≠ 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a ≠ 0) : a ^ b ≠ 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ ·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c ↔ b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b ≤ a ^ c ↔ b ≤ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c ↔ b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b → IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b ≠ 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a ≤ b) : a ^ c ≤ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a ≤ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b ≤ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b ≤ c) : a ^ b ∣ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b ∣ a ^ c ↔ b ≤ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : ¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b ≤ 1) : ∀ x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : ∀ b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : ∀ b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x ≠ 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x ≠ 0) : b ^ log b x ≤ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x ≠ 0) : b ^ c ≤ x ↔ c ≤ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x ≠ 0) : x < b ^ c ↔ log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o ≠ 0) (hbo : b ≤ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x ≤ y) : log b x ≤ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x ≤ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o ≠ 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o ≠ 0) (hbo : b ≤ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b ≠ 0) (u : Ordinal) (hv : v ≠ 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v ≠ 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o ≠ 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 7 | Ordinal.lt_opow_of_limit | [
[
73,
41
],
[
74,
98
]
] | 1 | 2 | simp only [not_lt, opow_le_of_limit b0 h, exists_prop, not_and] | a b c : Ordinal.{u_1}
b0 : b ≠ 0
h : c.IsLimit
⊢ ¬a < b ^ c ↔ ∀ (x : Ordinal.{u_1}), ¬(x < c ∧ a < b ^ x) | no goals |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a ≠ 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a ≠ 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a ≠ 0) : a ^ b ≠ 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ ·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c ↔ b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b ≤ a ^ c ↔ b ≤ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c ↔ b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b → IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b ≠ 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a ≤ b) : a ^ c ≤ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a ≤ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b ≤ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b ≤ c) : a ^ b ∣ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b ∣ a ^ c ↔ b ≤ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : ¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b ≤ 1) : ∀ x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : ∀ b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : ∀ b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x ≠ 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x ≠ 0) : b ^ log b x ≤ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x ≠ 0) : b ^ c ≤ x ↔ c ≤ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x ≠ 0) : x < b ^ c ↔ log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o ≠ 0) (hbo : b ≤ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x ≤ y) : log b x ≤ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x ≤ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o ≠ 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o ≠ 0) (hbo : b ≤ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b ≠ 0) (u : Ordinal) (hv : v ≠ 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v ≠ 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o ≠ 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 8 | Ordinal.opow_one | [
[
78,
59
],
[
79,
62
]
] | 0 | 2 | rw [← succ_zero, opow_succ] | a : Ordinal.{u_1}
⊢ a ^ 1 = a | a : Ordinal.{u_1}
⊢ a ^ 0 * a = a |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a ≠ 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a ≠ 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a ≠ 0) : a ^ b ≠ 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ ·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c ↔ b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b ≤ a ^ c ↔ b ≤ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c ↔ b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b → IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b ≠ 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a ≤ b) : a ^ c ≤ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a ≤ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b ≤ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b ≤ c) : a ^ b ∣ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b ∣ a ^ c ↔ b ≤ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : ¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b ≤ 1) : ∀ x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : ∀ b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : ∀ b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x ≠ 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x ≠ 0) : b ^ log b x ≤ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x ≠ 0) : b ^ c ≤ x ↔ c ≤ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x ≠ 0) : x < b ^ c ↔ log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o ≠ 0) (hbo : b ≤ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x ≤ y) : log b x ≤ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x ≤ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o ≠ 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o ≠ 0) (hbo : b ≤ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b ≠ 0) (u : Ordinal) (hv : v ≠ 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v ≠ 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o ≠ 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 8 | Ordinal.opow_one | [
[
78,
59
],
[
79,
62
]
] | 1 | 2 | simp only [opow_zero, one_mul] | a : Ordinal.{u_1}
⊢ a ^ 0 * a = a | no goals |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a ≠ 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a ≠ 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a ≠ 0) : a ^ b ≠ 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ ·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c ↔ b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b ≤ a ^ c ↔ b ≤ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c ↔ b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b → IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b ≠ 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a ≤ b) : a ^ c ≤ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a ≤ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b ≤ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b ≤ c) : a ^ b ∣ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b ∣ a ^ c ↔ b ≤ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : ¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b ≤ 1) : ∀ x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : ∀ b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : ∀ b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x ≠ 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x ≠ 0) : b ^ log b x ≤ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x ≠ 0) : b ^ c ≤ x ↔ c ≤ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x ≠ 0) : x < b ^ c ↔ log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o ≠ 0) (hbo : b ≤ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x ≤ y) : log b x ≤ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x ≤ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o ≠ 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o ≠ 0) (hbo : b ≤ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b ≠ 0) (u : Ordinal) (hv : v ≠ 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v ≠ 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o ≠ 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 9 | Ordinal.one_opow | [
[
83,
59
],
[
91,
94
]
] | 0 | 8 | induction a using limitRecOn with
| H₁ => simp only [opow_zero]
| H₂ _ ih =>
simp only [opow_succ, ih, mul_one]
| H₃ b l IH =>
refine eq_of_forall_ge_iff fun c => ?_
rw [opow_le_of_limit Ordinal.one_ne_zero l]
exact ⟨fun H => by simpa only [opow_zero] using H 0 l.pos, fun H b' h => by rwa [IH _ h]⟩ | a : Ordinal.{u_1}
⊢ 1 ^ a = 1 | no goals |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a ≠ 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a ≠ 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a ≠ 0) : a ^ b ≠ 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ ·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c ↔ b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b ≤ a ^ c ↔ b ≤ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c ↔ b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b → IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b ≠ 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a ≤ b) : a ^ c ≤ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a ≤ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b ≤ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b ≤ c) : a ^ b ∣ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b ∣ a ^ c ↔ b ≤ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : ¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b ≤ 1) : ∀ x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : ∀ b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : ∀ b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x ≠ 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x ≠ 0) : b ^ log b x ≤ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x ≠ 0) : b ^ c ≤ x ↔ c ≤ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x ≠ 0) : x < b ^ c ↔ log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o ≠ 0) (hbo : b ≤ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x ≤ y) : log b x ≤ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x ≤ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o ≠ 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o ≠ 0) (hbo : b ≤ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b ≠ 0) (u : Ordinal) (hv : v ≠ 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v ≠ 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o ≠ 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 9 | Ordinal.one_opow | [
[
83,
59
],
[
91,
94
]
] | 1 | 8 | simp only [opow_zero] | case H₁
⊢ 1 ^ 0 = 1 | no goals |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a ≠ 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a ≠ 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a ≠ 0) : a ^ b ≠ 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ ·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c ↔ b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b ≤ a ^ c ↔ b ≤ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c ↔ b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b → IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b ≠ 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a ≤ b) : a ^ c ≤ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a ≤ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b ≤ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b ≤ c) : a ^ b ∣ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b ∣ a ^ c ↔ b ≤ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : ¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b ≤ 1) : ∀ x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : ∀ b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : ∀ b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x ≠ 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x ≠ 0) : b ^ log b x ≤ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x ≠ 0) : b ^ c ≤ x ↔ c ≤ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x ≠ 0) : x < b ^ c ↔ log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o ≠ 0) (hbo : b ≤ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x ≤ y) : log b x ≤ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x ≤ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o ≠ 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o ≠ 0) (hbo : b ≤ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b ≠ 0) (u : Ordinal) (hv : v ≠ 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v ≠ 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o ≠ 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 9 | Ordinal.one_opow | [
[
83,
59
],
[
91,
94
]
] | 2 | 8 | simp only [opow_succ, ih, mul_one] | case H₂
o✝ : Ordinal.{u_1}
ih : 1 ^ o✝ = 1
⊢ 1 ^ succ o✝ = 1 | no goals |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a ≠ 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a ≠ 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a ≠ 0) : a ^ b ≠ 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ ·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c ↔ b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b ≤ a ^ c ↔ b ≤ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c ↔ b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b → IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b ≠ 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a ≤ b) : a ^ c ≤ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a ≤ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b ≤ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b ≤ c) : a ^ b ∣ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b ∣ a ^ c ↔ b ≤ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : ¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b ≤ 1) : ∀ x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : ∀ b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : ∀ b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x ≠ 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x ≠ 0) : b ^ log b x ≤ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x ≠ 0) : b ^ c ≤ x ↔ c ≤ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x ≠ 0) : x < b ^ c ↔ log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o ≠ 0) (hbo : b ≤ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x ≤ y) : log b x ≤ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x ≤ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o ≠ 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o ≠ 0) (hbo : b ≤ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b ≠ 0) (u : Ordinal) (hv : v ≠ 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v ≠ 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o ≠ 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 9 | Ordinal.one_opow | [
[
83,
59
],
[
91,
94
]
] | 3 | 8 | refine eq_of_forall_ge_iff fun c => ?_ | case H₃
b : Ordinal.{u_1}
l : b.IsLimit
IH : ∀ o' < b, 1 ^ o' = 1
⊢ 1 ^ b = 1 | case H₃
b : Ordinal.{u_1}
l : b.IsLimit
IH : ∀ o' < b, 1 ^ o' = 1
c : Ordinal.{u_1}
⊢ 1 ^ b ≤ c ↔ 1 ≤ c |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a ≠ 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a ≠ 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a ≠ 0) : a ^ b ≠ 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ ·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c ↔ b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b ≤ a ^ c ↔ b ≤ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c ↔ b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b → IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b ≠ 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a ≤ b) : a ^ c ≤ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a ≤ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b ≤ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b ≤ c) : a ^ b ∣ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b ∣ a ^ c ↔ b ≤ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : ¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b ≤ 1) : ∀ x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : ∀ b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : ∀ b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x ≠ 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x ≠ 0) : b ^ log b x ≤ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x ≠ 0) : b ^ c ≤ x ↔ c ≤ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x ≠ 0) : x < b ^ c ↔ log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o ≠ 0) (hbo : b ≤ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x ≤ y) : log b x ≤ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x ≤ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o ≠ 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o ≠ 0) (hbo : b ≤ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b ≠ 0) (u : Ordinal) (hv : v ≠ 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v ≠ 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o ≠ 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 9 | Ordinal.one_opow | [
[
83,
59
],
[
91,
94
]
] | 4 | 8 | rw [opow_le_of_limit Ordinal.one_ne_zero l] | case H₃
b : Ordinal.{u_1}
l : b.IsLimit
IH : ∀ o' < b, 1 ^ o' = 1
c : Ordinal.{u_1}
⊢ 1 ^ b ≤ c ↔ 1 ≤ c | case H₃
b : Ordinal.{u_1}
l : b.IsLimit
IH : ∀ o' < b, 1 ^ o' = 1
c : Ordinal.{u_1}
⊢ (∀ b' < b, 1 ^ b' ≤ c) ↔ 1 ≤ c |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a ≠ 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a ≠ 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a ≠ 0) : a ^ b ≠ 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ ·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c ↔ b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b ≤ a ^ c ↔ b ≤ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c ↔ b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b → IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b ≠ 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a ≤ b) : a ^ c ≤ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a ≤ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b ≤ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b ≤ c) : a ^ b ∣ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b ∣ a ^ c ↔ b ≤ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : ¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b ≤ 1) : ∀ x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : ∀ b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : ∀ b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x ≠ 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x ≠ 0) : b ^ log b x ≤ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x ≠ 0) : b ^ c ≤ x ↔ c ≤ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x ≠ 0) : x < b ^ c ↔ log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o ≠ 0) (hbo : b ≤ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x ≤ y) : log b x ≤ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x ≤ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o ≠ 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o ≠ 0) (hbo : b ≤ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b ≠ 0) (u : Ordinal) (hv : v ≠ 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v ≠ 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o ≠ 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 9 | Ordinal.one_opow | [
[
83,
59
],
[
91,
94
]
] | 5 | 8 | exact ⟨fun H => by simpa only [opow_zero] using H 0 l.pos, fun H b' h => by rwa [IH _ h]⟩ | case H₃
b : Ordinal.{u_1}
l : b.IsLimit
IH : ∀ o' < b, 1 ^ o' = 1
c : Ordinal.{u_1}
⊢ (∀ b' < b, 1 ^ b' ≤ c) ↔ 1 ≤ c | no goals |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a ≠ 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a ≠ 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a ≠ 0) : a ^ b ≠ 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ ·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c ↔ b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b ≤ a ^ c ↔ b ≤ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c ↔ b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b → IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b ≠ 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a ≤ b) : a ^ c ≤ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a ≤ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b ≤ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b ≤ c) : a ^ b ∣ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b ∣ a ^ c ↔ b ≤ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : ¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b ≤ 1) : ∀ x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : ∀ b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : ∀ b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x ≠ 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x ≠ 0) : b ^ log b x ≤ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x ≠ 0) : b ^ c ≤ x ↔ c ≤ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x ≠ 0) : x < b ^ c ↔ log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o ≠ 0) (hbo : b ≤ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x ≤ y) : log b x ≤ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x ≤ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o ≠ 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o ≠ 0) (hbo : b ≤ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b ≠ 0) (u : Ordinal) (hv : v ≠ 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v ≠ 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o ≠ 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 9 | Ordinal.one_opow | [
[
83,
59
],
[
91,
94
]
] | 6 | 8 | simpa only [opow_zero] using H 0 l.pos | b : Ordinal.{u_1}
l : b.IsLimit
IH : ∀ o' < b, 1 ^ o' = 1
c : Ordinal.{u_1}
H : ∀ b' < b, 1 ^ b' ≤ c
⊢ 1 ≤ c | no goals |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a ≠ 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a ≠ 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a ≠ 0) : a ^ b ≠ 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ ·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c ↔ b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b ≤ a ^ c ↔ b ≤ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c ↔ b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b → IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b ≠ 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a ≤ b) : a ^ c ≤ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a ≤ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b ≤ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b ≤ c) : a ^ b ∣ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b ∣ a ^ c ↔ b ≤ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : ¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b ≤ 1) : ∀ x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : ∀ b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : ∀ b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x ≠ 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x ≠ 0) : b ^ log b x ≤ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x ≠ 0) : b ^ c ≤ x ↔ c ≤ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x ≠ 0) : x < b ^ c ↔ log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o ≠ 0) (hbo : b ≤ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x ≤ y) : log b x ≤ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x ≤ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o ≠ 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o ≠ 0) (hbo : b ≤ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b ≠ 0) (u : Ordinal) (hv : v ≠ 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v ≠ 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o ≠ 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 9 | Ordinal.one_opow | [
[
83,
59
],
[
91,
94
]
] | 7 | 8 | rwa [IH _ h] | b : Ordinal.{u_1}
l : b.IsLimit
IH : ∀ o' < b, 1 ^ o' = 1
c : Ordinal.{u_1}
H : 1 ≤ c
b' : Ordinal.{u_1}
h : b' < b
⊢ 1 ^ b' ≤ c | no goals |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a ≠ 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a ≠ 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a ≠ 0) : a ^ b ≠ 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ ·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c ↔ b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b ≤ a ^ c ↔ b ≤ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c ↔ b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b → IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b ≠ 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a ≤ b) : a ^ c ≤ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a ≤ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b ≤ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b ≤ c) : a ^ b ∣ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b ∣ a ^ c ↔ b ≤ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : ¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b ≤ 1) : ∀ x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : ∀ b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : ∀ b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x ≠ 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x ≠ 0) : b ^ log b x ≤ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x ≠ 0) : b ^ c ≤ x ↔ c ≤ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x ≠ 0) : x < b ^ c ↔ log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o ≠ 0) (hbo : b ≤ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x ≤ y) : log b x ≤ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x ≤ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o ≠ 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o ≠ 0) (hbo : b ≤ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b ≠ 0) (u : Ordinal) (hv : v ≠ 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v ≠ 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o ≠ 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 10 | Ordinal.opow_pos | [
[
94,
74
],
[
102,
79
]
] | 0 | 7 | have h0 : 0 < a ^ (0 : Ordinal) := by simp only [opow_zero, zero_lt_one] | a b : Ordinal.{u_1}
a0 : 0 < a
⊢ 0 < a ^ b | a b : Ordinal.{u_1}
a0 : 0 < a
h0 : 0 < a ^ 0
⊢ 0 < a ^ b |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a ≠ 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a ≠ 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a ≠ 0) : a ^ b ≠ 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ ·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c ↔ b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b ≤ a ^ c ↔ b ≤ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c ↔ b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b → IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b ≠ 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a ≤ b) : a ^ c ≤ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a ≤ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b ≤ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b ≤ c) : a ^ b ∣ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b ∣ a ^ c ↔ b ≤ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : ¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b ≤ 1) : ∀ x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : ∀ b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : ∀ b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x ≠ 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x ≠ 0) : b ^ log b x ≤ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x ≠ 0) : b ^ c ≤ x ↔ c ≤ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x ≠ 0) : x < b ^ c ↔ log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o ≠ 0) (hbo : b ≤ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x ≤ y) : log b x ≤ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x ≤ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o ≠ 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o ≠ 0) (hbo : b ≤ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b ≠ 0) (u : Ordinal) (hv : v ≠ 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v ≠ 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o ≠ 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 10 | Ordinal.opow_pos | [
[
94,
74
],
[
102,
79
]
] | 1 | 7 | induction b using limitRecOn with
| H₁ => exact h0
| H₂ b IH =>
rw [opow_succ]
exact mul_pos IH a0
| H₃ b l _ =>
exact (lt_opow_of_limit (Ordinal.pos_iff_ne_zero.1 a0) l).2 ⟨0, l.pos, h0⟩ | a b : Ordinal.{u_1}
a0 : 0 < a
h0 : 0 < a ^ 0
⊢ 0 < a ^ b | no goals |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a ≠ 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a ≠ 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a ≠ 0) : a ^ b ≠ 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ ·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c ↔ b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b ≤ a ^ c ↔ b ≤ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c ↔ b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b → IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b ≠ 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a ≤ b) : a ^ c ≤ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a ≤ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b ≤ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b ≤ c) : a ^ b ∣ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b ∣ a ^ c ↔ b ≤ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : ¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b ≤ 1) : ∀ x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : ∀ b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : ∀ b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x ≠ 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x ≠ 0) : b ^ log b x ≤ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x ≠ 0) : b ^ c ≤ x ↔ c ≤ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x ≠ 0) : x < b ^ c ↔ log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o ≠ 0) (hbo : b ≤ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x ≤ y) : log b x ≤ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x ≤ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o ≠ 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o ≠ 0) (hbo : b ≤ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b ≠ 0) (u : Ordinal) (hv : v ≠ 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v ≠ 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o ≠ 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 10 | Ordinal.opow_pos | [
[
94,
74
],
[
102,
79
]
] | 2 | 7 | simp only [opow_zero, zero_lt_one] | a b : Ordinal.{u_1}
a0 : 0 < a
⊢ 0 < a ^ 0 | no goals |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a ≠ 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a ≠ 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a ≠ 0) : a ^ b ≠ 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ ·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c ↔ b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b ≤ a ^ c ↔ b ≤ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c ↔ b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b → IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b ≠ 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a ≤ b) : a ^ c ≤ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a ≤ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b ≤ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b ≤ c) : a ^ b ∣ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b ∣ a ^ c ↔ b ≤ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : ¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b ≤ 1) : ∀ x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : ∀ b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : ∀ b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x ≠ 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x ≠ 0) : b ^ log b x ≤ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x ≠ 0) : b ^ c ≤ x ↔ c ≤ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x ≠ 0) : x < b ^ c ↔ log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o ≠ 0) (hbo : b ≤ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x ≤ y) : log b x ≤ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x ≤ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o ≠ 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o ≠ 0) (hbo : b ≤ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b ≠ 0) (u : Ordinal) (hv : v ≠ 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v ≠ 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o ≠ 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 10 | Ordinal.opow_pos | [
[
94,
74
],
[
102,
79
]
] | 3 | 7 | exact h0 | case H₁
a : Ordinal.{u_1}
a0 : 0 < a
h0 : 0 < a ^ 0
⊢ 0 < a ^ 0 | no goals |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a ≠ 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a ≠ 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a ≠ 0) : a ^ b ≠ 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ ·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c ↔ b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b ≤ a ^ c ↔ b ≤ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c ↔ b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b → IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b ≠ 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a ≤ b) : a ^ c ≤ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a ≤ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b ≤ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b ≤ c) : a ^ b ∣ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b ∣ a ^ c ↔ b ≤ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : ¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b ≤ 1) : ∀ x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : ∀ b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : ∀ b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x ≠ 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x ≠ 0) : b ^ log b x ≤ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x ≠ 0) : b ^ c ≤ x ↔ c ≤ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x ≠ 0) : x < b ^ c ↔ log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o ≠ 0) (hbo : b ≤ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x ≤ y) : log b x ≤ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x ≤ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o ≠ 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o ≠ 0) (hbo : b ≤ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b ≠ 0) (u : Ordinal) (hv : v ≠ 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v ≠ 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o ≠ 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 10 | Ordinal.opow_pos | [
[
94,
74
],
[
102,
79
]
] | 4 | 7 | rw [opow_succ] | case H₂
a : Ordinal.{u_1}
a0 : 0 < a
h0 : 0 < a ^ 0
b : Ordinal.{u_1}
IH : 0 < a ^ b
⊢ 0 < a ^ succ b | case H₂
a : Ordinal.{u_1}
a0 : 0 < a
h0 : 0 < a ^ 0
b : Ordinal.{u_1}
IH : 0 < a ^ b
⊢ 0 < a ^ b * a |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a ≠ 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a ≠ 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a ≠ 0) : a ^ b ≠ 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ ·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c ↔ b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b ≤ a ^ c ↔ b ≤ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c ↔ b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b → IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b ≠ 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a ≤ b) : a ^ c ≤ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a ≤ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b ≤ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b ≤ c) : a ^ b ∣ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b ∣ a ^ c ↔ b ≤ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : ¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b ≤ 1) : ∀ x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : ∀ b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : ∀ b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x ≠ 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x ≠ 0) : b ^ log b x ≤ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x ≠ 0) : b ^ c ≤ x ↔ c ≤ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x ≠ 0) : x < b ^ c ↔ log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o ≠ 0) (hbo : b ≤ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x ≤ y) : log b x ≤ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x ≤ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o ≠ 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o ≠ 0) (hbo : b ≤ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b ≠ 0) (u : Ordinal) (hv : v ≠ 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v ≠ 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o ≠ 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 10 | Ordinal.opow_pos | [
[
94,
74
],
[
102,
79
]
] | 5 | 7 | exact mul_pos IH a0 | case H₂
a : Ordinal.{u_1}
a0 : 0 < a
h0 : 0 < a ^ 0
b : Ordinal.{u_1}
IH : 0 < a ^ b
⊢ 0 < a ^ b * a | no goals |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a ≠ 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a ≠ 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a ≠ 0) : a ^ b ≠ 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ ·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c ↔ b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b ≤ a ^ c ↔ b ≤ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c ↔ b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b → IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b ≠ 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a ≤ b) : a ^ c ≤ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a ≤ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b ≤ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b ≤ c) : a ^ b ∣ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b ∣ a ^ c ↔ b ≤ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : ¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b ≤ 1) : ∀ x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : ∀ b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : ∀ b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x ≠ 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x ≠ 0) : b ^ log b x ≤ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x ≠ 0) : b ^ c ≤ x ↔ c ≤ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x ≠ 0) : x < b ^ c ↔ log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o ≠ 0) (hbo : b ≤ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x ≤ y) : log b x ≤ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x ≤ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o ≠ 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o ≠ 0) (hbo : b ≤ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b ≠ 0) (u : Ordinal) (hv : v ≠ 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v ≠ 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o ≠ 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 10 | Ordinal.opow_pos | [
[
94,
74
],
[
102,
79
]
] | 6 | 7 | exact (lt_opow_of_limit (Ordinal.pos_iff_ne_zero.1 a0) l).2 ⟨0, l.pos, h0⟩ | case H₃
a : Ordinal.{u_1}
a0 : 0 < a
h0 : 0 < a ^ 0
b : Ordinal.{u_1}
l : b.IsLimit
a✝ : ∀ o' < b, 0 < a ^ o'
⊢ 0 < a ^ b | no goals |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a ≠ 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a ≠ 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a ≠ 0) : a ^ b ≠ 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ ·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c ↔ b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b ≤ a ^ c ↔ b ≤ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c ↔ b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b → IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b ≠ 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a ≤ b) : a ^ c ≤ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a ≤ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b ≤ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b ≤ c) : a ^ b ∣ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b ∣ a ^ c ↔ b ≤ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : ¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b ≤ 1) : ∀ x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : ∀ b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : ∀ b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x ≠ 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x ≠ 0) : b ^ log b x ≤ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x ≠ 0) : b ^ c ≤ x ↔ c ≤ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x ≠ 0) : x < b ^ c ↔ log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o ≠ 0) (hbo : b ≤ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x ≤ y) : log b x ≤ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x ≤ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o ≠ 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o ≠ 0) (hbo : b ≤ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b ≠ 0) (u : Ordinal) (hv : v ≠ 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v ≠ 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o ≠ 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 17 | Ordinal.opow_isLimit_left | [
[
131,
93
],
[
136,
35
]
] | 0 | 8 | rcases zero_or_succ_or_limit b with (e | ⟨b, rfl⟩ | l') | a b : Ordinal.{u_1}
l : a.IsLimit
hb : b ≠ 0
⊢ (a ^ b).IsLimit | case inl
a b : Ordinal.{u_1}
l : a.IsLimit
hb : b ≠ 0
e : b = 0
⊢ (a ^ b).IsLimit
case inr.inl.intro
a : Ordinal.{u_1}
l : a.IsLimit
b : Ordinal.{u_1}
hb : succ b ≠ 0
⊢ (a ^ succ b).IsLimit
case inr.inr
a b : Ordinal.{u_1}
l : a.IsLimit
hb : b ≠ 0
l' : b.IsLimit
⊢ (a ^ b).IsLimit |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a ≠ 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a ≠ 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a ≠ 0) : a ^ b ≠ 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ ·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c ↔ b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b ≤ a ^ c ↔ b ≤ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c ↔ b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b → IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b ≠ 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a ≤ b) : a ^ c ≤ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a ≤ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b ≤ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b ≤ c) : a ^ b ∣ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b ∣ a ^ c ↔ b ≤ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : ¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b ≤ 1) : ∀ x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : ∀ b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : ∀ b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x ≠ 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x ≠ 0) : b ^ log b x ≤ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x ≠ 0) : b ^ c ≤ x ↔ c ≤ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x ≠ 0) : x < b ^ c ↔ log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o ≠ 0) (hbo : b ≤ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x ≤ y) : log b x ≤ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x ≤ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o ≠ 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o ≠ 0) (hbo : b ≤ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b ≠ 0) (u : Ordinal) (hv : v ≠ 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v ≠ 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o ≠ 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 17 | Ordinal.opow_isLimit_left | [
[
131,
93
],
[
136,
35
]
] | 1 | 8 | · exact absurd e hb | case inl
a b : Ordinal.{u_1}
l : a.IsLimit
hb : b ≠ 0
e : b = 0
⊢ (a ^ b).IsLimit
case inr.inl.intro
a : Ordinal.{u_1}
l : a.IsLimit
b : Ordinal.{u_1}
hb : succ b ≠ 0
⊢ (a ^ succ b).IsLimit
case inr.inr
a b : Ordinal.{u_1}
l : a.IsLimit
hb : b ≠ 0
l' : b.IsLimit
⊢ (a ^ b).IsLimit | case inr.inl.intro
a : Ordinal.{u_1}
l : a.IsLimit
b : Ordinal.{u_1}
hb : succ b ≠ 0
⊢ (a ^ succ b).IsLimit
case inr.inr
a b : Ordinal.{u_1}
l : a.IsLimit
hb : b ≠ 0
l' : b.IsLimit
⊢ (a ^ b).IsLimit |
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