Datasets:

---

fumiyau

/

mathlib4-state-change

like 0

[ [ "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 ] } ]

[ [ 318, 62 ], [ 321, 57 ] ]

[ [ "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 ] } ]

[ [ 324, 73 ], [ 331, 73 ] ]

[ [ "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 ] } ]

[ [ 324, 73 ], [ 331, 73 ] ]

[ [ "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 ] } ]

[ [ 324, 73 ], [ 331, 73 ] ]

[ [ "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 ] } ]

[ [ 324, 73 ], [ 331, 73 ] ]

[ [ "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 ] } ]

[ [ 324, 73 ], [ 331, 73 ] ]

[ [ "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 ] } ]

[ [ 324, 73 ], [ 331, 73 ] ]

[ [ "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 ] } ]

[ [ 324, 73 ], [ 331, 73 ] ]

[ [ "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 ] } ]

[ [ 324, 73 ], [ 331, 73 ] ]

[ [ "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 ] } ]

[ [ 324, 73 ], [ 331, 73 ] ]

[ [ "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 ] } ]

[ [ 334, 69 ], [ 338, 47 ] ]

[ [ "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 ] } ]

[ [ 334, 69 ], [ 338, 47 ] ]

[ [ "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 ] } ]

[ [ 334, 69 ], [ 338, 47 ] ]

[ [ "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 ] } ]

[ [ 334, 69 ], [ 338, 47 ] ]

[ [ "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 ] } ]

[ [ 334, 69 ], [ 338, 47 ] ]

[ [ "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 ] } ]

[ [ 334, 69 ], [ 338, 47 ] ]

[ [ "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 ] } ]

[ [ 341, 49 ], [ 346, 25 ] ]

[ [ "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 ] } ]

[ [ 341, 49 ], [ 346, 25 ] ]

[ [ "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 ] } ]

[ [ 341, 49 ], [ 346, 25 ] ]

[ [ "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 ] } ]

[ [ 341, 49 ], [ 346, 25 ] ]

[ [ "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 ] } ]

[ [ 341, 49 ], [ 346, 25 ] ]

[ [ "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 ] } ]

[ [ 341, 49 ], [ 346, 25 ] ]

[ [ "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 ] } ]

[ [ 341, 49 ], [ 346, 25 ] ]

[ [ "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 ] } ]

[ [ 350, 83 ], [ 362, 73 ] ]

[ [ "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 ] } ]

[ [ 350, 83 ], [ 362, 73 ] ]

[ [ "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 ] } ]

[ [ 350, 83 ], [ 362, 73 ] ]

[ [ "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 ] } ]

[ [ 350, 83 ], [ 362, 73 ] ]

[ [ "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 ] } ]

[ [ 350, 83 ], [ 362, 73 ] ]

[ [ "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 ] } ]

[ [ 350, 83 ], [ 362, 73 ] ]

[ [ "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 ] } ]

[ [ 350, 83 ], [ 362, 73 ] ]

[ [ "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 ] } ]

[ [ 350, 83 ], [ 362, 73 ] ]

[ [ "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 ] } ]

[ [ 350, 83 ], [ 362, 73 ] ]

[ [ "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 ] } ]

[ [ 350, 83 ], [ 362, 73 ] ]

[ [ "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 ] } ]

[ [ 350, 83 ], [ 362, 73 ] ]

[ [ "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 ] } ]

[ [ 350, 83 ], [ 362, 73 ] ]

[ [ "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 ] } ]

[ [ 350, 83 ], [ 362, 73 ] ]

[ [ "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 ] } ]

[ [ 350, 83 ], [ 362, 73 ] ]

[ [ "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 ] } ]

[ [ 350, 83 ], [ 362, 73 ] ]

[ [ "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 ] } ]

[ [ 350, 83 ], [ 362, 73 ] ]

[ [ "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 ] } ]

[ [ 373, 58 ], [ 386, 15 ] ]

[ [ "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 ] } ]

[ [ 373, 58 ], [ 386, 15 ] ]

[ [ "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 ] } ]

[ [ 373, 58 ], [ 386, 15 ] ]

[ [ "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 ] } ]

[ [ 373, 58 ], [ 386, 15 ] ]

[ [ "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 ] } ]

[ [ 373, 58 ], [ 386, 15 ] ]

[ [ "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 ] } ]

[ [ 373, 58 ], [ 386, 15 ] ]

[ [ "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 ] } ]

[ [ 373, 58 ], [ 386, 15 ] ]

[ [ "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 ] } ]

[ [ 373, 58 ], [ 386, 15 ] ]

[ [ "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 ] } ]

[ [ 373, 58 ], [ 386, 15 ] ]

[ [ "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 ] } ]

[ [ 373, 58 ], [ 386, 15 ] ]

[ [ "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 ] } ]

[ [ 373, 58 ], [ 386, 15 ] ]

[ [ "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 ] } ]

[ [ 373, 58 ], [ 386, 15 ] ]

[ [ "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 ] } ]

[ [ 373, 58 ], [ 386, 15 ] ]

[ [ "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 ] } ]

[ [ 373, 58 ], [ 386, 15 ] ]

[ [ "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 ] } ]

[ [ 373, 58 ], [ 386, 15 ] ]

[ [ "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 ] } ]

[ [ 373, 58 ], [ 386, 15 ] ]

[ [ "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 ] } ]

[ [ 373, 58 ], [ 386, 15 ] ]

[ [ "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 ] } ]

[ [ 404, 23 ], [ 408, 18 ] ]

[ [ "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 ] } ]

[ [ 404, 23 ], [ 408, 18 ] ]

[ [ "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 ] } ]

[ [ 404, 23 ], [ 408, 18 ] ]

[ [ "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 ] } ]

[ [ 404, 23 ], [ 408, 18 ] ]

[ [ "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 ] } ]

[ [ 404, 23 ], [ 408, 18 ] ]

[ [ "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 ] } ]

[ [ 404, 23 ], [ 408, 18 ] ]

[ [ "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 ] } ]

[ [ 414, 23 ], [ 419, 45 ] ]

[ [ "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 ] } ]

[ [ 414, 23 ], [ 419, 45 ] ]

[ [ "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 ] } ]

[ [ 414, 23 ], [ 419, 45 ] ]

[ [ "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 ] } ]

[ [ 414, 23 ], [ 419, 45 ] ]

[ [ "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 ] } ]

[ [ 414, 23 ], [ 419, 45 ] ]

[ [ "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 ] } ]

[ [ 414, 23 ], [ 419, 45 ] ]

[ [ "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 ] } ]

[ [ 414, 23 ], [ 419, 45 ] ]

[ [ "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 ] } ]

[ [ 422, 49 ], [ 448, 50 ] ]

[ [ "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 ] } ]

[ [ 422, 49 ], [ 448, 50 ] ]

[ [ "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 ] } ]

[ [ 422, 49 ], [ 448, 50 ] ]

[ [ "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 ] } ]

[ [ 422, 49 ], [ 448, 50 ] ]

[ [ "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 ] } ]

[ [ 422, 49 ], [ 448, 50 ] ]

[ [ "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 ] } ]

[ [ 422, 49 ], [ 448, 50 ] ]

[ [ "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 ] } ]

[ [ 422, 49 ], [ 448, 50 ] ]

[ [ "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 ] } ]

[ [ 422, 49 ], [ 448, 50 ] ]

[ [ "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 ] } ]

[ [ 422, 49 ], [ 448, 50 ] ]

[ [ "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 ] } ]

[ [ 422, 49 ], [ 448, 50 ] ]

[ [ "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 ] } ]

[ [ 422, 49 ], [ 448, 50 ] ]

[ [ "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 ] } ]

[ [ 422, 49 ], [ 448, 50 ] ]

[ [ "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 ] } ]

[ [ 422, 49 ], [ 448, 50 ] ]

[ [ "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 ] } ]

[ [ 422, 49 ], [ 448, 50 ] ]

[ [ "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 ] } ]

[ [ 422, 49 ], [ 448, 50 ] ]

[ [ "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 ] } ]

[ [ 422, 49 ], [ 448, 50 ] ]

[ [ "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 ] } ]

[ [ 422, 49 ], [ 448, 50 ] ]

[ [ "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 ] } ]

[ [ 422, 49 ], [ 448, 50 ] ]

[ [ "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 ] } ]

[ [ 422, 49 ], [ 448, 50 ] ]

[ [ "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 ] } ]

[ [ 422, 49 ], [ 448, 50 ] ]

[ [ "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 ] } ]

[ [ 422, 49 ], [ 448, 50 ] ]

[ [ "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 ] } ]

[ [ 422, 49 ], [ 448, 50 ] ]

[ [ "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 ] } ]

[ [ 422, 49 ], [ 448, 50 ] ]

[ [ "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 ] } ]

[ [ 475, 74 ], [ 489, 72 ] ]

[ [ "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 ] } ]

[ [ 475, 74 ], [ 489, 72 ] ]

[ [ "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 ] } ]

[ [ 475, 74 ], [ 489, 72 ] ]

[ [ "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 ] } ]

[ [ 475, 74 ], [ 489, 72 ] ]

[ [ "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 ] } ]

[ [ 475, 74 ], [ 489, 72 ] ]

[ [ "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 ] } ]

[ [ 475, 74 ], [ 489, 72 ] ]

[ [ "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 ] } ]

[ [ 475, 74 ], [ 489, 72 ] ]

[ [ "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 ] } ]

[ [ 475, 74 ], [ 489, 72 ] ]