module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 375
values | kind stringclasses 379
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.CategoryTheory.Topos.Sheaf | {
"line": 201,
"column": 49
} | {
"line": 201,
"column": 60
} | {
"line": 201,
"column": 61
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nJ : GrothendieckTopology C\nF G : Sheaf J (Type (max u v))\nm : F ⟶ G\ninst✝ : Mono m\n⊢ (m.hom ≫ Subfunctor.lift (Presheaf.χ m.hom) ⋯) ≫ (closedSieves J).ι =\n (((isTerminalTerminal J Types.isTerminalPUnit).from F).hom ≫ Subfunctor.lift (Presheaf.truth C) ⋯) ... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\nJ : GrothendieckTopology C\nF G : Sheaf J (Type (max u v))\nm : F ⟶ G\ninst✝ : Mono m\n⊢ m.hom ≫ Presheaf.χ m.hom = (isTerminalConst Cᵒᵖ Types.isTerminalPUnit).from F.obj ≫ Presheaf.truth C"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Triangulated.Adjunction | {
"line": 68,
"column": 6
} | {
"line": 68,
"column": 34
} | {
"line": 68,
"column": 35
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝¹⁵ : Category.{v_1, u_1} C\ninst✝¹⁴ : Category.{v_2, u_2} D\ninst✝¹³ : HasZeroObject C\ninst✝¹² : HasZeroObject D\ninst✝¹¹ : Preadditive C\ninst✝¹⁰ : Preadditive D\ninst✝⁹ : HasShift C ℤ\ninst✝⁸ : HasShift D ℤ\ninst✝⁷ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝⁶ : ∀... | [
"C : Type u_1\nD : Type u_2\ninst✝¹⁵ : Category.{v_1, u_1} C\ninst✝¹⁴ : Category.{v_2, u_2} D\ninst✝¹³ : HasZeroObject C\ninst✝¹² : HasZeroObject D\ninst✝¹¹ : Preadditive C\ninst✝¹⁰ : Preadditive D\ninst✝⁹ : HasShift C ℤ\ninst✝⁸ : HasShift D ℤ\ninst✝⁷ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝⁶ : ∀ (n : ℤ), (s... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Topos.Sheaf | {
"line": 217,
"column": 4
} | {
"line": 217,
"column": 15
} | {
"line": 217,
"column": 16
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nJ : GrothendieckTopology C\nF G : Sheaf J (Type (max u v))\nm : F ⟶ G\ninst✝ : Mono m\nχ' : G ⟶ Ω J\nhχ' : IsPullback m ((isTerminalTerminal J Types.isTerminalPUnit).from F) χ' (truth J)\npb : IsPullback (𝟙 G.obj) χ'.hom (χ'.hom ≫ (closedSieves J).ι) (closedSiev... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\nJ : GrothendieckTopology C\nF G : Sheaf J (Type (max u v))\nm : F ⟶ G\ninst✝ : Mono m\nχ' : G ⟶ Ω J\nhχ' : IsPullback m ((isTerminalTerminal J Types.isTerminalPUnit).from F) χ' (truth J)\npb : IsPullback (𝟙 G.obj) χ'.hom (χ'.hom ≫ (closedSieves J).ι) (closedSieves J).ι\n⊢ I... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Topos.Sheaf | {
"line": 218,
"column": 2
} | {
"line": 218,
"column": 13
} | {
"line": 218,
"column": 14
} | [
{
"pp": "case refine_2\nC : Type u\ninst✝¹ : Category.{v, u} C\nJ : GrothendieckTopology C\nF G : Sheaf J (Type (max u v))\nm : F ⟶ G\ninst✝ : Mono m\nχ' : G ⟶ Ω J\nhχ' : IsPullback m ((isTerminalTerminal J Types.isTerminalPUnit).from F) χ' (truth J)\npb : IsPullback (𝟙 G.obj) χ'.hom (χ'.hom ≫ (closedSieves J)... | [
"case refine_2\nC : Type u\ninst✝¹ : Category.{v, u} C\nJ : GrothendieckTopology C\nF G : Sheaf J (Type (max u v))\nm : F ⟶ G\ninst✝ : Mono m\nχ' : G ⟶ Ω J\nhχ' : IsPullback m ((isTerminalTerminal J Types.isTerminalPUnit).from F) χ' (truth J)\npb : IsPullback (𝟙 G.obj) χ'.hom (χ'.hom ≫ (closedSieves J).ι) (closedS... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Triangulated.Generators | {
"line": 155,
"column": 4
} | {
"line": 155,
"column": 40
} | {
"line": 156,
"column": 4
} | [
{
"pp": "C : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : HasZeroObject C\ninst✝³ : HasShift C ℤ\ninst✝² : Preadditive C\ninst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝ : Pretriangulated C\nP : ObjectProperty C\nX Y : C\nr : Retract X Y\nhY : P.triangEnvelope Y\n⊢ P.triangEnvelope X",
"ppTerm"... | [
"C : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : HasZeroObject C\ninst✝³ : HasShift C ℤ\ninst✝² : Preadditive C\ninst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝ : Pretriangulated C\nP : ObjectProperty C\nX Y : C\nr : Retract X Y\nhY : ∃ n, P.triangEnvelopeIter n Y\n⊢ ∃ n, P.triangEnvelopeIter n X"
] | rw [prop_triangEnvelope_iff] at hY ⊢ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Triangulated.Adjunction | {
"line": 86,
"column": 42
} | {
"line": 86,
"column": 93
} | {
"line": 86,
"column": 94
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝¹⁵ : Category.{v_1, u_1} C\ninst✝¹⁴ : Category.{v_2, u_2} D\ninst✝¹³ : HasZeroObject C\ninst✝¹² : HasZeroObject D\ninst✝¹¹ : Preadditive C\ninst✝¹⁰ : Preadditive D\ninst✝⁹ : HasShift C ℤ\ninst✝⁸ : HasShift D ℤ\ninst✝⁷ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝⁶ : ∀... | [
"C : Type u_1\nD : Type u_2\ninst✝¹⁵ : Category.{v_1, u_1} C\ninst✝¹⁴ : Category.{v_2, u_2} D\ninst✝¹³ : HasZeroObject C\ninst✝¹² : HasZeroObject D\ninst✝¹¹ : Preadditive C\ninst✝¹⁰ : Preadditive D\ninst✝⁹ : HasShift C ℤ\ninst✝⁸ : HasShift D ℤ\ninst✝⁷ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝⁶ : ∀ (n : ℤ), (s... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Triangulated.Opposite.Functor | {
"line": 139,
"column": 42
} | {
"line": 144,
"column": 6
} | {
"line": 146,
"column": 0
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : Category.{v_2, u_2} D\ninst✝² : HasShift C ℤ\ninst✝¹ : HasShift D ℤ\nF : C ⥤ D\ninst✝ : F.CommShift ℤ\nX : Cᵒᵖ\nn : ℤ\n⊢ F.map ((opShiftFunctorEquivalence C n).unitIso.inv.app X).unop =\n ((opShiftFunctorEquivalence D n).unitIso.in... | [] | by
rw [← cancel_mono (F.map ((opShiftFunctorEquivalence C n).unitIso.hom.app X).unop),
← F.map_comp, ← unop_comp, Iso.hom_inv_id_app,
map_opShiftFunctorEquivalence_unitIso_hom_app_unop, assoc, assoc,
Iso.inv_hom_id_app_assoc, ← Functor.map_comp_assoc, ← unop_comp]
simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Triangulated.TStructure.AbelianSubcategory | {
"line": 121,
"column": 24
} | {
"line": 123,
"column": 48
} | {
"line": 124,
"column": 2
} | [
{
"pp": "C : Type u_1\nA : Type u_2\ninst✝⁹ : Category.{v_1, u_1} C\ninst✝⁸ : HasZeroObject C\ninst✝⁷ : Preadditive C\ninst✝⁶ : HasShift C ℤ\ninst✝⁵ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝⁴ : Pretriangulated C\ninst✝³ : Category.{v_2, u_2} A\nι : A ⥤ C\nhι : ∀ ⦃X Y : A⦄ ⦃n : ℤ⦄ (f : ι.obj X ⟶ (shiftFunc... | [] | by
rw [← cancel_epi ((shiftFunctorAdd' C (1 : ℤ) 1 2 (by lia)).hom.app _), comp_zero]
exact eq_zero_of_hom_shift_pos hι _ (by lia) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Triangulated.Adjunction | {
"line": 117,
"column": 44
} | {
"line": 117,
"column": 88
} | {
"line": 117,
"column": 89
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝¹⁵ : Category.{v_1, u_1} C\ninst✝¹⁴ : Category.{v_2, u_2} D\ninst✝¹³ : HasZeroObject C\ninst✝¹² : HasZeroObject D\ninst✝¹¹ : Preadditive C\ninst✝¹⁰ : Preadditive D\ninst✝⁹ : HasShift C ℤ\ninst✝⁸ : HasShift D ℤ\ninst✝⁷ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝⁶ : ∀... | [
"C : Type u_1\nD : Type u_2\ninst✝¹⁵ : Category.{v_1, u_1} C\ninst✝¹⁴ : Category.{v_2, u_2} D\ninst✝¹³ : HasZeroObject C\ninst✝¹² : HasZeroObject D\ninst✝¹¹ : Preadditive C\ninst✝¹⁰ : Preadditive D\ninst✝⁹ : HasShift C ℤ\ninst✝⁸ : HasShift D ℤ\ninst✝⁷ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝⁶ : ∀ (n : ℤ), (s... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Triangulated.Adjunction | {
"line": 119,
"column": 8
} | {
"line": 119,
"column": 40
} | {
"line": 119,
"column": 41
} | [
{
"pp": "case right\nC : Type u_1\nD : Type u_2\ninst✝¹⁵ : Category.{v_1, u_1} C\ninst✝¹⁴ : Category.{v_2, u_2} D\ninst✝¹³ : HasZeroObject C\ninst✝¹² : HasZeroObject D\ninst✝¹¹ : Preadditive C\ninst✝¹⁰ : Preadditive D\ninst✝⁹ : HasShift C ℤ\ninst✝⁸ : HasShift D ℤ\ninst✝⁷ : ∀ (n : ℤ), (shiftFunctor C n).Additive... | [
"case right\nC : Type u_1\nD : Type u_2\ninst✝¹⁵ : Category.{v_1, u_1} C\ninst✝¹⁴ : Category.{v_2, u_2} D\ninst✝¹³ : HasZeroObject C\ninst✝¹² : HasZeroObject D\ninst✝¹¹ : Preadditive C\ninst✝¹⁰ : Preadditive D\ninst✝⁹ : HasShift C ℤ\ninst✝⁸ : HasShift D ℤ\ninst✝⁷ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝⁶ : ∀... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Triangulated.TStructure.AbelianSubcategory | {
"line": 136,
"column": 24
} | {
"line": 138,
"column": 48
} | {
"line": 139,
"column": 2
} | [
{
"pp": "C : Type u_1\nA : Type u_2\ninst✝⁹ : Category.{v_1, u_1} C\ninst✝⁸ : HasZeroObject C\ninst✝⁷ : Preadditive C\ninst✝⁶ : HasShift C ℤ\ninst✝⁵ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝⁴ : Pretriangulated C\ninst✝³ : Category.{v_2, u_2} A\nι : A ⥤ C\nhι : ∀ ⦃X Y : A⦄ ⦃n : ℤ⦄ (f : ι.obj X ⟶ (shiftFunc... | [] | by
rw [← cancel_epi ((shiftFunctorAdd' C (1 : ℤ) 1 2 (by lia)).hom.app _), comp_zero]
exact eq_zero_of_hom_shift_pos hι _ (by lia) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Triangulated.TStructure.ETrunc | {
"line": 51,
"column": 53
} | {
"line": 51,
"column": 64
} | {
"line": 51,
"column": 65
} | [
{
"pp": "C : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : Preadditive C\ninst✝³ : HasZeroObject C\ninst✝² : HasShift C ℤ\ninst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝ : Pretriangulated C\nt : TStructure C\na b : ℤ\nf : WithBotTop.coe a ⟶ WithBotTop.coe b\n⊢ a ≤ b",
"ppTerm": "?m.141",
"a... | [
"C : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : Preadditive C\ninst✝³ : HasZeroObject C\ninst✝² : HasShift C ℤ\ninst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝ : Pretriangulated C\nt : TStructure C\na b : ℤ\nf : WithBotTop.coe a ⟶ WithBotTop.coe b\n⊢ a ≤ b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.AP.Three.Defs | {
"line": 154,
"column": 2
} | {
"line": 155,
"column": 91
} | {
"line": 157,
"column": 0
} | [
{
"pp": "F : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝³ : CommMonoid α\ninst✝² : CommMonoid β\ns : Set α\ninst✝¹ : FunLike F α β\ninst✝ : MulHomClass F α β\nf : F\nhf : InjOn (⇑f) (s * s)\nh : ThreeGPFree s\n⊢ ThreeGPFree (⇑f '' s)",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"E... | [] | rintro _ ⟨a, ha, rfl⟩ _ ⟨b, hb, rfl⟩ _ ⟨c, hc, rfl⟩ habc
rw [h ha hb hc (hf (mul_mem_mul ha hc) (mul_mem_mul hb hb) <| by rwa [map_mul, map_mul])] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Additive.AP.Three.Defs | {
"line": 154,
"column": 2
} | {
"line": 155,
"column": 91
} | {
"line": 157,
"column": 0
} | [
{
"pp": "F : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝³ : CommMonoid α\ninst✝² : CommMonoid β\ns : Set α\ninst✝¹ : FunLike F α β\ninst✝ : MulHomClass F α β\nf : F\nhf : InjOn (⇑f) (s * s)\nh : ThreeGPFree s\n⊢ ThreeGPFree (⇑f '' s)",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"E... | [] | rintro _ ⟨a, ha, rfl⟩ _ ⟨b, hb, rfl⟩ _ ⟨c, hc, rfl⟩ habc
rw [h ha hb hc (hf (mul_mem_mul ha hc) (mul_mem_mul hb hb) <| by rwa [map_mul, map_mul])] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Additive.AP.Three.Defs | {
"line": 167,
"column": 2
} | {
"line": 167,
"column": 13
} | {
"line": 167,
"column": 14
} | [
{
"pp": "α : Type u_2\ninst✝¹ : CommMonoid α\ninst✝ : IsCancelMul α\ns : Set α\nhs : ThreeGPFree s\na : α\nha : a ∈ s\nc : α\nhc : c ∈ s\nhb : a ∈ s\nhabc : a * c = a * a\n⊢ a = c",
"ppTerm": "?m.53",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_2\ninst✝¹ : CommMonoid α\ninst✝ : IsCancelMul α\ns : Set α\nhs : ThreeGPFree s\na : α\nha : a ∈ s\nc : α\nhc : c ∈ s\nhb : a ∈ s\nhabc : a * c = a * a\n⊢ a = c"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Triangulated.TStructure.AbelianSubcategory | {
"line": 326,
"column": 6
} | {
"line": 326,
"column": 72
} | {
"line": 327,
"column": 4
} | [
{
"pp": "C : Type u_1\nA : Type u_2\ninst✝¹² : Category.{v_1, u_1} C\ninst✝¹¹ : HasZeroObject C\ninst✝¹⁰ : Preadditive C\ninst✝⁹ : HasShift C ℤ\ninst✝⁸ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝⁷ : Pretriangulated C\ninst✝⁶ : Category.{v_2, u_2} A\nι : A ⥤ C\nhι : ∀ ⦃X Y : A⦄ ⦃n : ℤ⦄ (f : ι.obj X ⟶ (shiftF... | [] | exact Triangle.isoMk _ _ (-(Iso.refl _)) (Iso.refl _) (Iso.refl _) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Triangulated.TStructure.AbelianSubcategory | {
"line": 339,
"column": 60
} | {
"line": 339,
"column": 77
} | {
"line": 339,
"column": 78
} | [
{
"pp": "C : Type u_1\nA : Type u_2\ninst✝¹² : Category.{v_1, u_1} C\ninst✝¹¹ : HasZeroObject C\ninst✝¹⁰ : Preadditive C\ninst✝⁹ : HasShift C ℤ\ninst✝⁸ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝⁷ : Pretriangulated C\ninst✝⁶ : Category.{v_2, u_2} A\nι : A ⥤ C\nhι : ∀ ⦃X Y : A⦄ ⦃n : ℤ⦄ (f : ι.obj X ⟶ (shiftF... | [
"C : Type u_1\nA : Type u_2\ninst✝¹² : Category.{v_1, u_1} C\ninst✝¹¹ : HasZeroObject C\ninst✝¹⁰ : Preadditive C\ninst✝⁹ : HasShift C ℤ\ninst✝⁸ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝⁷ : Pretriangulated C\ninst✝⁶ : Category.{v_2, u_2} A\nι : A ⥤ C\nhι : ∀ ⦃X Y : A⦄ ⦃n : ℤ⦄ (f : ι.obj X ⟶ (shiftFunctor C n).... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.AP.Three.Defs | {
"line": 192,
"column": 47
} | {
"line": 192,
"column": 83
} | {
"line": 192,
"column": 84
} | [
{
"pp": "α : Type u_2\ninst✝¹ : CommMonoid α\ninst✝ : IsCancelMul α\ns : Set α\na : α\nhs : ThreeGPFree s\nb : α\nhb : b ∈ s\nc : α\nhc : c ∈ s\nd : α\nhd : d ∈ s\nh : (fun x ↦ a • x) b * (fun x ↦ a • x) d = (fun x ↦ a • x) c * (fun x ↦ a • x) c\n⊢ b * d = c * c",
"ppTerm": "?m.99",
"assigned": false,
... | [
"α : Type u_2\ninst✝¹ : CommMonoid α\ninst✝ : IsCancelMul α\ns : Set α\na : α\nhs : ThreeGPFree s\nb : α\nhb : b ∈ s\nc : α\nhc : c ∈ s\nd : α\nhd : d ∈ s\nh : (fun x ↦ a • x) b * (fun x ↦ a • x) d = (fun x ↦ a • x) c * (fun x ↦ a • x) c\n⊢ b * d = c * c"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.AP.Three.Defs | {
"line": 222,
"column": 47
} | {
"line": 222,
"column": 87
} | {
"line": 222,
"column": 88
} | [
{
"pp": "α : Type u_2\ninst✝² : CommMonoidWithZero α\ninst✝¹ : IsCancelMulZero α\ninst✝ : NoZeroDivisors α\ns : Set α\na : α\nhs : ThreeGPFree s\nha : a ≠ 0\nb : α\nhb : b ∈ s\nc : α\nhc : c ∈ s\nd : α\nhd : d ∈ s\nh : (fun x ↦ a • x) b * (fun x ↦ a • x) d = (fun x ↦ a • x) c * (fun x ↦ a • x) c\n⊢ b * d = c * ... | [
"α : Type u_2\ninst✝² : CommMonoidWithZero α\ninst✝¹ : IsCancelMulZero α\ninst✝ : NoZeroDivisors α\ns : Set α\na : α\nhs : ThreeGPFree s\nha : a ≠ 0\nb : α\nhb : b ∈ s\nc : α\nhc : c ∈ s\nd : α\nhd : d ∈ s\nh : (fun x ↦ a • x) b * (fun x ↦ a • x) d = (fun x ↦ a • x) c * (fun x ↦ a • x) c\n⊢ b * d = c * c"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.FreimanHom | {
"line": 153,
"column": 4
} | {
"line": 153,
"column": 19
} | {
"line": 155,
"column": 0
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\ninst✝¹ : CommMonoid α\ninst✝ : CommMonoid β\nA : Set α\nB : Set β\nf : α → β\nn : ℕ\ng : β → α\nhg₁ : MapsTo g B A\nhg₂ : RightInvOn g f B\nhf : IsMulFreimanIso n A B f\ns t : Multiset β\nhsB : ∀ ⦃x : β⦄, x ∈ s → x ∈ B\nhtB : ∀ ⦃x : β⦄, x ∈ t → x ∈ B\nhs : s.card = n\nht : t... | [] | all_goals aesop | Lean.Elab.Tactic.evalAllGoals | Lean.Parser.Tactic.allGoals |
Mathlib.Combinatorics.Additive.AP.Three.Defs | {
"line": 348,
"column": 2
} | {
"line": 348,
"column": 13
} | {
"line": 348,
"column": 14
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\ninst✝³ : DecidableEq α\ninst✝² : CommMonoid α\ninst✝¹ : CommMonoid β\ninst✝ : DecidableEq β\nA : Finset α\nB : Finset β\nf : α → β\nhf : IsMulFreimanHom 2 (↑A) (↑B) f\nhf' : Set.BijOn f ↑A ↑B\ns : Finset β\nhsB : s ⊆ B\nhcard : #s = mulRothNumber B\nhs : ThreeGPFree ↑s\nhsA ... | [
"α : Type u_2\nβ : Type u_3\ninst✝³ : DecidableEq α\ninst✝² : CommMonoid α\ninst✝¹ : CommMonoid β\ninst✝ : DecidableEq β\nA : Finset α\nB : Finset β\nf : α → β\nhf : IsMulFreimanHom 2 (↑A) (↑B) f\nhf' : Set.BijOn f ↑A ↑B\ns : Finset β\nhsB : s ⊆ B\nhcard : #s = mulRothNumber B\nhs : ThreeGPFree ↑s\nhsA : invFunOn f... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Pigeonhole | {
"line": 316,
"column": 27
} | {
"line": 316,
"column": 66
} | {
"line": 317,
"column": 4
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝² : DecidableEq β\ns : Finset α\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf : α → Finset β\nh₁ : s.Nonempty\nh₂ : ∀ j ∈ s, 0 < #(f j)\nk : ℕ := s.inf' h₁ fun j ↦ #(f j)\nhk : k = s.inf' h₁ fun j ↦ #(f j)\nh₃ : ∀ a ∈ s, ∀ x ∈ f a, #{j | j ∈ s ∧ x ∈ f j} ≤ k\n⊢ ∑ j ∈ s, k ≤... | [] | by gcongr with i hi; exact inf'_le _ hi | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.Additive.FreimanHom | {
"line": 193,
"column": 2
} | {
"line": 193,
"column": 13
} | {
"line": 193,
"column": 14
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\ninst✝¹ : CommMonoid α\ninst✝ : CommMonoid β\nA : Set α\nB : Set β\nf : α → β\nn : ℕ\nhf : IsMulFreimanHom n A B f\ns t : Finset α\nhsA : ↑s ⊆ A\nhtA : ↑t ⊆ A\nhs : s.card = n\nht : t.card = n\n⊢ ∏ i ∈ s, i = ∏ i ∈ t, i → ∏ i ∈ s, f i = ∏ i ∈ t, f i",
"ppTerm": "?m.35",
... | [
"α : Type u_2\nβ : Type u_3\ninst✝¹ : CommMonoid α\ninst✝ : CommMonoid β\nA : Set α\nB : Set β\nf : α → β\nn : ℕ\nhf : IsMulFreimanHom n A B f\ns t : Finset α\nhsA : ↑s ⊆ A\nhtA : ↑t ⊆ A\nhs : s.card = n\nht : t.card = n\n⊢ ∏ i ∈ s, i = ∏ i ∈ t, i → ∏ i ∈ s, f i = ∏ i ∈ t, f i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.FreimanHom | {
"line": 227,
"column": 43
} | {
"line": 227,
"column": 54
} | {
"line": 227,
"column": 55
} | [
{
"pp": "α : Type u_2\ninst✝ : CommMonoid α\nA₁ A₂ : Set α\nn : ℕ\nhA : A₁ ⊆ A₂\ns t : Multiset α\nx✝³ : ∀ ⦃x : α⦄, x ∈ s → x ∈ A₁\nx✝² : ∀ ⦃x : α⦄, x ∈ t → x ∈ A₁\nx✝¹ : s.card = n\nx✝ : t.card = n\nh : s.prod = t.prod\n⊢ (map id s).prod = (map id t).prod",
"ppTerm": "?m.23",
"assigned": true,
"use... | [
"α : Type u_2\ninst✝ : CommMonoid α\nA₁ A₂ : Set α\nn : ℕ\nhA : A₁ ⊆ A₂\ns t : Multiset α\nx✝³ : ∀ ⦃x : α⦄, x ∈ s → x ∈ A₁\nx✝² : ∀ ⦃x : α⦄, x ∈ t → x ∈ A₁\nx✝¹ : s.card = n\nx✝ : t.card = n\nh : s.prod = t.prod\n⊢ s.prod = t.prod"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.FreimanHom | {
"line": 240,
"column": 6
} | {
"line": 240,
"column": 17
} | {
"line": 240,
"column": 18
} | [
{
"pp": "case refine_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝² : CommMonoid α\ninst✝¹ : CommMonoid β\ninst✝ : CommMonoid γ\nA : Set α\nB : Set β\nC : Set γ\nf : α → β\ng : β → γ\nn : ℕ\nhg : IsMulFreimanHom n B C g\nhf : IsMulFreimanHom n A B f\ns t : Multiset α\nhsA : ∀ ⦃x : α⦄, x ∈ s → x ∈ A\nhtA : ... | [
"case refine_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝² : CommMonoid α\ninst✝¹ : CommMonoid β\ninst✝ : CommMonoid γ\nA : Set α\nB : Set β\nC : Set γ\nf : α → β\ng : β → γ\nn : ℕ\nhg : IsMulFreimanHom n B C g\nhf : IsMulFreimanHom n A B f\ns t : Multiset α\nhsA : ∀ ⦃x : α⦄, x ∈ s → x ∈ A\nhtA : ∀ ⦃x : α⦄, x... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.FreimanHom | {
"line": 241,
"column": 6
} | {
"line": 241,
"column": 17
} | {
"line": 241,
"column": 18
} | [
{
"pp": "case refine_2\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝² : CommMonoid α\ninst✝¹ : CommMonoid β\ninst✝ : CommMonoid γ\nA : Set α\nB : Set β\nC : Set γ\nf : α → β\ng : β → γ\nn : ℕ\nhg : IsMulFreimanHom n B C g\nhf : IsMulFreimanHom n A B f\ns t : Multiset α\nhsA : ∀ ⦃x : α⦄, x ∈ s → x ∈ A\nhtA : ... | [
"case refine_2\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝² : CommMonoid α\ninst✝¹ : CommMonoid β\ninst✝ : CommMonoid γ\nA : Set α\nB : Set β\nC : Set γ\nf : α → β\ng : β → γ\nn : ℕ\nhg : IsMulFreimanHom n B C g\nhf : IsMulFreimanHom n A B f\ns t : Multiset α\nhsA : ∀ ⦃x : α⦄, x ∈ s → x ∈ A\nhtA : ∀ ⦃x : α⦄, x... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.FreimanHom | {
"line": 250,
"column": 6
} | {
"line": 250,
"column": 17
} | {
"line": 250,
"column": 18
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝² : CommMonoid α\ninst✝¹ : CommMonoid β\ninst✝ : CommMonoid γ\nA : Set α\nB : Set β\nC : Set γ\nf : α → β\ng : β → γ\nn : ℕ\nhg : IsMulFreimanIso n B C g\nhf : IsMulFreimanIso n A B f\ns t : Multiset α\nhsA : ∀ ⦃x : α⦄, x ∈ s → x ∈ A\nhtA : ∀ ⦃x : α⦄, x ∈ ... | [
"α : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝² : CommMonoid α\ninst✝¹ : CommMonoid β\ninst✝ : CommMonoid γ\nA : Set α\nB : Set β\nC : Set γ\nf : α → β\ng : β → γ\nn : ℕ\nhg : IsMulFreimanIso n B C g\nhf : IsMulFreimanIso n A B f\ns t : Multiset α\nhsA : ∀ ⦃x : α⦄, x ∈ s → x ∈ A\nhtA : ∀ ⦃x : α⦄, x ∈ t → x ∈ A\nh... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.FreimanHom | {
"line": 250,
"column": 6
} | {
"line": 250,
"column": 51
} | {
"line": 251,
"column": 4
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝² : CommMonoid α\ninst✝¹ : CommMonoid β\ninst✝ : CommMonoid γ\nA : Set α\nB : Set β\nC : Set γ\nf : α → β\ng : β → γ\nn : ℕ\nhg : IsMulFreimanIso n B C g\nhf : IsMulFreimanIso n A B f\ns t : Multiset α\nhsA : ∀ ⦃x : α⦄, x ∈ s → x ∈ A\nhtA : ∀ ⦃x : α⦄, x ∈ ... | [] | simpa using fun a h ↦ hf.bijOn.mapsTo (hsA h) | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Combinatorics.Additive.FreimanHom | {
"line": 250,
"column": 6
} | {
"line": 250,
"column": 51
} | {
"line": 251,
"column": 4
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝² : CommMonoid α\ninst✝¹ : CommMonoid β\ninst✝ : CommMonoid γ\nA : Set α\nB : Set β\nC : Set γ\nf : α → β\ng : β → γ\nn : ℕ\nhg : IsMulFreimanIso n B C g\nhf : IsMulFreimanIso n A B f\ns t : Multiset α\nhsA : ∀ ⦃x : α⦄, x ∈ s → x ∈ A\nhtA : ∀ ⦃x : α⦄, x ∈ ... | [] | simpa using fun a h ↦ hf.bijOn.mapsTo (hsA h) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Additive.FreimanHom | {
"line": 250,
"column": 6
} | {
"line": 250,
"column": 51
} | {
"line": 251,
"column": 4
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝² : CommMonoid α\ninst✝¹ : CommMonoid β\ninst✝ : CommMonoid γ\nA : Set α\nB : Set β\nC : Set γ\nf : α → β\ng : β → γ\nn : ℕ\nhg : IsMulFreimanIso n B C g\nhf : IsMulFreimanIso n A B f\ns t : Multiset α\nhsA : ∀ ⦃x : α⦄, x ∈ s → x ∈ A\nhtA : ∀ ⦃x : α⦄, x ∈ ... | [] | simpa using fun a h ↦ hf.bijOn.mapsTo (hsA h) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Additive.FreimanHom | {
"line": 251,
"column": 6
} | {
"line": 251,
"column": 17
} | {
"line": 251,
"column": 18
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝² : CommMonoid α\ninst✝¹ : CommMonoid β\ninst✝ : CommMonoid γ\nA : Set α\nB : Set β\nC : Set γ\nf : α → β\ng : β → γ\nn : ℕ\nhg : IsMulFreimanIso n B C g\nhf : IsMulFreimanIso n A B f\ns t : Multiset α\nhsA : ∀ ⦃x : α⦄, x ∈ s → x ∈ A\nhtA : ∀ ⦃x : α⦄, x ∈ ... | [
"α : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝² : CommMonoid α\ninst✝¹ : CommMonoid β\ninst✝ : CommMonoid γ\nA : Set α\nB : Set β\nC : Set γ\nf : α → β\ng : β → γ\nn : ℕ\nhg : IsMulFreimanIso n B C g\nhf : IsMulFreimanIso n A B f\ns t : Multiset α\nhsA : ∀ ⦃x : α⦄, x ∈ s → x ∈ A\nhtA : ∀ ⦃x : α⦄, x ∈ t → x ∈ A\nh... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.CovBySMul | {
"line": 69,
"column": 29
} | {
"line": 69,
"column": 40
} | {
"line": 69,
"column": 41
} | [
{
"pp": "M : Type u_1\nX : Type u_3\ninst✝¹ : Monoid M\ninst✝ : MulAction M X\nK : ℝ\nA₁ A₂ B : Set X\nhA : A₁ ⊆ A₂\nhAB : CovBySMul M K A₂ B\n⊢ CovBySMul M K A₁ B",
"ppTerm": "?m.11",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"M : Type u_1\nX : Type u_3\ninst✝¹ : Monoid M\ninst✝ : MulAction M X\nK : ℝ\nA₁ A₂ B : Set X\nhA : A₁ ⊆ A₂\nhAB : CovBySMul M K A₂ B\n⊢ CovBySMul M K A₁ B"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.CovBySMul | {
"line": 73,
"column": 29
} | {
"line": 73,
"column": 40
} | {
"line": 73,
"column": 41
} | [
{
"pp": "M : Type u_1\nX : Type u_3\ninst✝¹ : Monoid M\ninst✝ : MulAction M X\nK : ℝ\nA B₁ B₂ : Set X\nhB : B₁ ⊆ B₂\nhAB : CovBySMul M K A B₁\n⊢ CovBySMul M K A B₂",
"ppTerm": "?m.11",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"M : Type u_1\nX : Type u_3\ninst✝¹ : Monoid M\ninst✝ : MulAction M X\nK : ℝ\nA B₁ B₂ : Set X\nhB : B₁ ⊆ B₂\nhAB : CovBySMul M K A B₁\n⊢ CovBySMul M K A B₂"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.FreimanHom | {
"line": 344,
"column": 8
} | {
"line": 344,
"column": 23
} | {
"line": 344,
"column": 24
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\ninst✝¹ : CommMonoid α\ninst✝ : CancelCommMonoid β\nA : Set α\nB : Set β\nf : α → β\nn✝ : ℕ\ns t : Multiset α\nhsA : ∀ ⦃x : α⦄, x ∈ s → x ∈ A\nhtA : ∀ ⦃x : α⦄, x ∈ t → x ∈ A\nh : s.prod = t.prod\nn : ℕ\nhf : IsMulFreimanHom (n + 1 + 1) A B f\nhs : s.card = n + 1\nx✝ : t.card ... | [
"α : Type u_2\nβ : Type u_3\ninst✝¹ : CommMonoid α\ninst✝ : CancelCommMonoid β\nA : Set α\nB : Set β\nf : α → β\nn✝ : ℕ\ns t : Multiset α\nhsA : ∀ ⦃x : α⦄, x ∈ s → x ∈ A\nhtA : ∀ ⦃x : α⦄, x ∈ t → x ∈ A\nh : s.prod = t.prod\nn : ℕ\nhf : IsMulFreimanHom (n + 1 + 1) A B f\nhs : s.card = n + 1\nx✝ : t.card = n + 1\na :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.FreimanHom | {
"line": 378,
"column": 20
} | {
"line": 378,
"column": 33
} | {
"line": 378,
"column": 34
} | [
{
"pp": "α : Type u_2\ninst✝¹ : CommMonoid α\nA : Set α\nn : ℕ\nβ : Type u_5\ninst✝ : DivisionCommMonoid β\nB₁ B₂ : Set β\nf₁ f₂ : α → β\nh₁ : IsMulFreimanHom n A B₁ f₁\nh₂ : IsMulFreimanHom n A B₂ f₂\ns t : Multiset α\nhsA : ∀ ⦃x : α⦄, x ∈ s → x ∈ A\nhtA : ∀ ⦃x : α⦄, x ∈ t → x ∈ A\nhs : s.card = n\nht : t.card... | [
"α : Type u_2\ninst✝¹ : CommMonoid α\nA : Set α\nn : ℕ\nβ : Type u_5\ninst✝ : DivisionCommMonoid β\nB₁ B₂ : Set β\nf₁ f₂ : α → β\nh₁ : IsMulFreimanHom n A B₁ f₁\nh₂ : IsMulFreimanHom n A B₂ f₂\ns t : Multiset α\nhsA : ∀ ⦃x : α⦄, x ∈ s → x ∈ A\nhtA : ∀ ⦃x : α⦄, x ∈ t → x ∈ A\nhs : s.card = n\nht : t.card = n\nh : s.... | prod_map_div, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Additive.FreimanHom | {
"line": 378,
"column": 34
} | {
"line": 378,
"column": 47
} | {
"line": 378,
"column": 48
} | [
{
"pp": "α : Type u_2\ninst✝¹ : CommMonoid α\nA : Set α\nn : ℕ\nβ : Type u_5\ninst✝ : DivisionCommMonoid β\nB₁ B₂ : Set β\nf₁ f₂ : α → β\nh₁ : IsMulFreimanHom n A B₁ f₁\nh₂ : IsMulFreimanHom n A B₂ f₂\ns t : Multiset α\nhsA : ∀ ⦃x : α⦄, x ∈ s → x ∈ A\nhtA : ∀ ⦃x : α⦄, x ∈ t → x ∈ A\nhs : s.card = n\nht : t.card... | [
"α : Type u_2\ninst✝¹ : CommMonoid α\nA : Set α\nn : ℕ\nβ : Type u_5\ninst✝ : DivisionCommMonoid β\nB₁ B₂ : Set β\nf₁ f₂ : α → β\nh₁ : IsMulFreimanHom n A B₁ f₁\nh₂ : IsMulFreimanHom n A B₂ f₂\ns t : Multiset α\nhsA : ∀ ⦃x : α⦄, x ∈ s → x ∈ A\nhtA : ∀ ⦃x : α⦄, x ∈ t → x ∈ A\nhs : s.card = n\nht : t.card = n\nh : s.... | prod_map_div, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Additive.FreimanHom | {
"line": 456,
"column": 6
} | {
"line": 456,
"column": 81
} | {
"line": 456,
"column": 82
} | [
{
"pp": "k m n : ℕ\nhm : m ≠ 0\nhkmn : m * k ≤ n\ns t : Multiset (Fin (n + 1))\nhsA : ∀ ⦃x : Fin (n + 1)⦄, x ∈ s → x ∈ Iic ↑k\nhtA : ∀ ⦃x : Fin (n + 1)⦄, x ∈ t → x ∈ Iic ↑k\nhs : s.card = m\nht : t.card = m\nthis : ∀ (u : Multiset (Fin (n + 1))), (Nat.castRingHom (Fin (n + 1))) (map val u).sum = u.sum\nu : Mult... | [
"k m n : ℕ\nhm : m ≠ 0\nhkmn : m * k ≤ n\ns t : Multiset (Fin (n + 1))\nhsA : ∀ ⦃x : Fin (n + 1)⦄, x ∈ s → x ∈ Iic ↑k\nhtA : ∀ ⦃x : Fin (n + 1)⦄, x ∈ t → x ∈ Iic ↑k\nhs : s.card = m\nht : t.card = m\nthis : ∀ (u : Multiset (Fin (n + 1))), (Nat.castRingHom (Fin (n + 1))) (map val u).sum = u.sum\nu : Multiset (Fin (n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.RuzsaCovering | {
"line": 69,
"column": 64
} | {
"line": 69,
"column": 94
} | {
"line": 69,
"column": 95
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nK : ℝ\nA B : Finset G\nhB₀ : (↑B).Nonempty\nhK : ↑(Nat.card ↑(↑A * ↑B)) ≤ K * ↑(Nat.card ↑↑B)\n⊢ ↑(?m.82 * B).card ≤ ?m.80 * ↑B.card",
"ppTerm": "?m.84",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"G : Type u_1\ninst✝ : Group G\nK : ℝ\nA B : Finset G\nhB₀ : (↑B).Nonempty\nhK : ↑(Nat.card ↑(↑A * ↑B)) ≤ K * ↑(Nat.card ↑↑B)\n⊢ ↑(?m.82 * B).card ≤ ?m.80 * ↑B.card"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.PluenneckeRuzsa | {
"line": 71,
"column": 2
} | {
"line": 71,
"column": 30
} | {
"line": 71,
"column": 31
} | [
{
"pp": "G : Type u_1\ninst✝¹ : DecidableEq G\ninst✝ : Group G\nA B C : Finset G\n⊢ #(A * C⁻¹) * #B ≤ #(A * B⁻¹) * #(C * B⁻¹)",
"ppTerm": "?m.33",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"G : Type u_1\ninst✝¹ : DecidableEq G\ninst✝ : Group G\nA B C : Finset G\n⊢ #(A * C⁻¹) * #B ≤ #(A * B⁻¹) * #(C * B⁻¹)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.PluenneckeRuzsa | {
"line": 77,
"column": 2
} | {
"line": 77,
"column": 68
} | {
"line": 78,
"column": 4
} | [
{
"pp": "G : Type u_1\ninst✝¹ : DecidableEq G\ninst✝ : Group G\nA B C : Finset G\n⊢ #B * #(A⁻¹ * C) ≤ #(B⁻¹ * A) * #(B⁻¹ * C)",
"ppTerm": "?m.33",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"G : Type u_1\ninst✝¹ : DecidableEq G\ninst✝ : Group G\nA B C : Finset G\n⊢ #B * #(A⁻¹ * C) ≤ #(B⁻¹ * A) * #(B⁻¹ * C)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.PluenneckeRuzsa | {
"line": 86,
"column": 2
} | {
"line": 86,
"column": 13
} | {
"line": 86,
"column": 14
} | [
{
"pp": "G : Type u_1\ninst✝¹ : DecidableEq G\ninst✝ : Group G\nA B C : Finset G\n⊢ #(A / C) * #B ≤ #(A * B) * #(C * B)",
"ppTerm": "?m.27",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"G : Type u_1\ninst✝¹ : DecidableEq G\ninst✝ : Group G\nA B C : Finset G\n⊢ #(A / C) * #B ≤ #(A * B) * #(C * B)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.PluenneckeRuzsa | {
"line": 92,
"column": 2
} | {
"line": 92,
"column": 13
} | {
"line": 92,
"column": 14
} | [
{
"pp": "G : Type u_1\ninst✝¹ : DecidableEq G\ninst✝ : Group G\nA B C : Finset G\n⊢ #(A * C⁻¹) * #B ≤ #(A * B) * #(C * B)",
"ppTerm": "?m.29",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"G : Type u_1\ninst✝¹ : DecidableEq G\ninst✝ : Group G\nA B C : Finset G\n⊢ #(A * C⁻¹) * #B ≤ #(A * B) * #(C * B)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.PluenneckeRuzsa | {
"line": 98,
"column": 2
} | {
"line": 98,
"column": 13
} | {
"line": 98,
"column": 14
} | [
{
"pp": "G : Type u_1\ninst✝¹ : DecidableEq G\ninst✝ : Group G\nA B C : Finset G\n⊢ #B * #(A⁻¹ * C) ≤ #(B * A) * #(B * C)",
"ppTerm": "?m.29",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"G : Type u_1\ninst✝¹ : DecidableEq G\ninst✝ : Group G\nA B C : Finset G\n⊢ #B * #(A⁻¹ * C) ≤ #(B * A) * #(B * C)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.PluenneckeRuzsa | {
"line": 105,
"column": 2
} | {
"line": 105,
"column": 30
} | {
"line": 105,
"column": 31
} | [
{
"pp": "G : Type u_1\ninst✝¹ : DecidableEq G\ninst✝ : Group G\nA B C : Finset G\n⊢ #B * #(A * C) ≤ #(B / A) * #(B * C)",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.toInv",
"instHDiv",
"HMul.hMul",
"Finset.divisionMonoid",
"Mon... | [
"G : Type u_1\ninst✝¹ : DecidableEq G\ninst✝ : Group G\nA B C : Finset G\n⊢ #B * #(A * C) ≤ #(B * A⁻¹) * #(B * C)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.PluenneckeRuzsa | {
"line": 111,
"column": 2
} | {
"line": 111,
"column": 30
} | {
"line": 111,
"column": 31
} | [
{
"pp": "G : Type u_1\ninst✝¹ : DecidableEq G\ninst✝ : Group G\nA B C : Finset G\n⊢ #B * #(A * C) ≤ #(B * A⁻¹) * #(B * C)",
"ppTerm": "?m.29",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"G : Type u_1\ninst✝¹ : DecidableEq G\ninst✝ : Group G\nA B C : Finset G\n⊢ #B * #(A * C) ≤ #(B * A⁻¹) * #(B * C)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.PluenneckeRuzsa | {
"line": 117,
"column": 2
} | {
"line": 117,
"column": 13
} | {
"line": 117,
"column": 14
} | [
{
"pp": "G : Type u_1\ninst✝¹ : DecidableEq G\ninst✝ : Group G\nA B C : Finset G\n⊢ #(A * C) * #B ≤ #(A * B) * #(C⁻¹ * B)",
"ppTerm": "?m.29",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"G : Type u_1\ninst✝¹ : DecidableEq G\ninst✝ : Group G\nA B C : Finset G\n⊢ #(A * C) * #B ≤ #(A * B) * #(C⁻¹ * B)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.ApproximateSubgroup | {
"line": 88,
"column": 12
} | {
"line": 88,
"column": 23
} | {
"line": 88,
"column": 24
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\nK : ℝ\ninst✝ : DecidableEq G\nA : Finset G\nhA : IsApproximateSubgroup K ↑A\n⊢ ↑(#(A ^ 0)) ≤ K ^ (0 - 1) * ↑(#A)",
"ppTerm": "?m.55",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.instCanonicallyOrderedAdd",
"MulOne.toOne",
"Real... | [
"G : Type u_1\ninst✝¹ : Group G\nK : ℝ\ninst✝ : DecidableEq G\nA : Finset G\nhA : IsApproximateSubgroup K ↑A\n⊢ A.Nonempty"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.ApproximateSubgroup | {
"line": 101,
"column": 28
} | {
"line": 101,
"column": 44
} | {
"line": 101,
"column": 45
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\nK : ℝ\ninst✝ : DecidableEq G\nA : Finset G\nhA : IsApproximateSubgroup K ↑A\n⊢ ↑(#(A * A)) ≤ K * ↑(#A)",
"ppTerm": "?m.15",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"G : Type u_1\ninst✝¹ : Group G\nK : ℝ\ninst✝ : DecidableEq G\nA : Finset G\nhA : IsApproximateSubgroup K ↑A\n⊢ ↑(#(A * A)) ≤ K * ↑(#A)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.ApproximateSubgroup | {
"line": 137,
"column": 32
} | {
"line": 137,
"column": 89
} | {
"line": 137,
"column": 90
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\nK : ℝ\ninst✝ : DecidableEq G\nA : Finset G\nhA₁ : 1 ∈ A\nhAsymm : A⁻¹ = A\nhA : ↑(#(A ^ 4 * A)) ≤ K ^ 3 * ↑(#A)\nhA₀ : A.Nonempty\nF : Finset G\nhF : ↑(#F) ≤ K ^ 3\nhAF : A ^ 4 ⊆ F * (A / A)\n⊢ (A ^ 2) ^ 2 ⊆ F • A ^ 2",
"ppTerm": "?m.188",
"assigned": true,
"... | [
"G : Type u_1\ninst✝¹ : Group G\nK : ℝ\ninst✝ : DecidableEq G\nA : Finset G\nhA₁ : 1 ∈ A\nhAsymm : A⁻¹ = A\nhA : ↑(#(A ^ 4 * A)) ≤ K ^ 3 * ↑(#A)\nhA₀ : A.Nonempty\nF : Finset G\nhF : ↑(#F) ≤ K ^ 3\nhAF : A ^ 4 ⊆ F * (A / A)\n⊢ A * (A * (A * A)) ⊆ F * (A * A)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Triangulated.TStructure.ETrunc | {
"line": 97,
"column": 53
} | {
"line": 97,
"column": 64
} | {
"line": 97,
"column": 65
} | [
{
"pp": "C : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : Preadditive C\ninst✝³ : HasZeroObject C\ninst✝² : HasShift C ℤ\ninst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝ : Pretriangulated C\nt : TStructure C\na b : ℤ\nf : WithBotTop.coe a ⟶ WithBotTop.coe b\n⊢ a ≤ b",
"ppTerm": "?m.146",
"a... | [
"C : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : Preadditive C\ninst✝³ : HasZeroObject C\ninst✝² : HasShift C ℤ\ninst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝ : Pretriangulated C\nt : TStructure C\na b : ℤ\nf : WithBotTop.coe a ⟶ WithBotTop.coe b\n⊢ a ≤ b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.ApproximateSubgroup | {
"line": 207,
"column": 6
} | {
"line": 207,
"column": 17
} | {
"line": 207,
"column": 18
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nA : Set G\nhA : IsApproximateSubgroup 1 A\nx : G\nhx : A * A ⊆ x • A\nhx' : x⁻¹ • (A * A) ⊆ A\n⊢ x⁻¹ ∈ A",
"ppTerm": "?m.195",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"G : Type u_1\ninst✝ : Group G\nA : Set G\nhA : IsApproximateSubgroup 1 A\nx : G\nhx : A * A ⊆ x • A\nhx' : x⁻¹ • (A * A) ⊆ A\n⊢ x⁻¹ ∈ A"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.ApproximateSubgroup | {
"line": 210,
"column": 6
} | {
"line": 210,
"column": 17
} | {
"line": 210,
"column": 18
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nA : Set G\nhA : IsApproximateSubgroup 1 A\nx : G\nhx : A * A ⊆ x • A\nhx' : x⁻¹ • (A * A) ⊆ A\nhx_inv : x⁻¹ ∈ A\n⊢ x * x ∈ A⁻¹",
"ppTerm": "?m.242",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.toInv",
"HMul.hMul",
"DivI... | [
"G : Type u_1\ninst✝ : Group G\nA : Set G\nhA : IsApproximateSubgroup 1 A\nx : G\nhx : A * A ⊆ x • A\nhx' : x⁻¹ • (A * A) ⊆ A\nhx_inv : x⁻¹ ∈ A\n⊢ x⁻¹ * x⁻¹ ∈ A"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.ApproximateSubgroup | {
"line": 211,
"column": 9
} | {
"line": 214,
"column": 18
} | {
"line": 215,
"column": 2
} | [] | [] | A * A ⊆ x • A := by assumption
_ = x⁻¹ • (x * x) • A := by simp [smul_smul]
_ ⊆ x⁻¹ • (A • A) := smul_set_mono (smul_set_subset_smul hx_sq)
_ ⊆ A := hx' | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcSteps |
Mathlib.GroupTheory.Order.Min | {
"line": 65,
"column": 4
} | {
"line": 65,
"column": 15
} | {
"line": 65,
"column": 16
} | [
{
"pp": "case refine_2\nG : Type u_1\ninst✝ : Group G\nn : ℕ∞\nh : ∀ ⦃s : Subgroup G⦄, s ≠ ⊥ → (↑s).Finite → n ≤ ↑(Nat.card ↥s)\na : G\nha : a ≠ 1\nha' : IsOfFinOrder a\n⊢ n ≤ ↑(orderOf a)",
"ppTerm": "?refine_2",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case refine_2\nG : Type u_1\ninst✝ : Group G\nn : ℕ∞\nh : ∀ ⦃s : Subgroup G⦄, s ≠ ⊥ → (↑s).Finite → n ≤ ↑(Nat.card ↥s)\na : G\nha : a ≠ 1\nha' : IsOfFinOrder a\n⊢ n ≤ ↑(orderOf a)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Order.Min | {
"line": 73,
"column": 2
} | {
"line": 73,
"column": 24
} | {
"line": 73,
"column": 25
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\ninst✝ : IsMulTorsionFree G\n⊢ minOrder G = ⊤",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"iInf_eq_top._simp_1",
"Eq.mpr",
"MulOne.toOne",
"False",
"iInf",
"instCompleteLinearOrderENat",
"ENat.instNatCast"... | [
"G : Type u_1\ninst✝¹ : Group G\ninst✝ : IsMulTorsionFree G\n⊢ ∀ (i : G), ¬i = 1 → ¬IsOfFinOrder i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.CauchyDavenport | {
"line": 131,
"column": 4
} | {
"line": 131,
"column": 56
} | {
"line": 132,
"column": 6
} | [
{
"pp": "case inl\nα : Type u_2\ninst✝¹ : Group α\ninst✝ : DecidableEq α\ns t : Finset α\nhs : s.Nonempty\nht : t.Nonempty\nih :\n ∀ (a b : Finset α),\n a.Nonempty → b.Nonempty → DevosMulRel (a, b) (s, t) → minOrder α ≤ ↑(#(a * b)) ∨ #a + #b ≤ #(a * b) + 1\nhts : #t < #s\n⊢ minOrder α ≤ ↑(#(s * t)) ∨ #s + #... | [
"case inl\nα : Type u_2\ninst✝¹ : Group α\ninst✝ : DecidableEq α\ns t : Finset α\nhs : s.Nonempty\nht : t.Nonempty\nih :\n ∀ (a b : Finset α),\n a.Nonempty → b.Nonempty → DevosMulRel (a, b) (s, t) → minOrder α ≤ ↑(#(a * b)) ∨ #a + #b ≤ #(a * b) + 1\nhts : #t < #s\n⊢ minOrder α ≤ ↑(#(s * t)) ∨ #s + #t ≤ #(s * t)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.SmallTripling | {
"line": 43,
"column": 12
} | {
"line": 43,
"column": 23
} | {
"line": 43,
"column": 24
} | [
{
"pp": "case base\nG : Type u_1\ninst✝¹ : DecidableEq G\ninst✝ : Group G\nA : Finset G\nk : ℝ\nm : ℕ\nh : ∀ (ε : Fin 3 → ℤ), (∀ (i : Fin 3), |ε i| = 1) → ↑(#(List.map (fun i ↦ A ^ ε i) (finRange 3)).prod) ≤ k * ↑(#A)\nε : Fin 3 → ℤ\nhε : ∀ (i : Fin 3), |ε i| = 1\n⊢ ↑(#(List.map (fun i ↦ A ^ ε i) (finRange 3)).... | [
"case base\nG : Type u_1\ninst✝¹ : DecidableEq G\ninst✝ : Group G\nA : Finset G\nk : ℝ\nm : ℕ\nh : ∀ (ε : Fin 3 → ℤ), (∀ (i : Fin 3), |ε i| = 1) → ↑(#(List.map (fun i ↦ A ^ ε i) (finRange 3)).prod) ≤ k * ↑(#A)\nε : Fin 3 → ℤ\nhε : ∀ (i : Fin 3), |ε i| = 1\n⊢ ↑(#(List.map (fun i ↦ A ^ ε i) (finRange 3)).prod) ≤ k * ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.SmallTripling | {
"line": 48,
"column": 39
} | {
"line": 48,
"column": 54
} | {
"line": 48,
"column": 55
} | [
{
"pp": "G : Type u_1\ninst✝¹ : DecidableEq G\ninst✝ : Group G\nA : Finset G\nk : ℝ\nm✝ : ℕ\nh✝ : ∀ (ε : Fin 3 → ℤ), (∀ (i : Fin 3), |ε i| = 1) → ↑(#(List.map (fun i ↦ A ^ ε i) (finRange 3)).prod) ≤ k * ↑(#A)\nm : ℕ\nhm : 3 ≤ m + 1\nih :\n ∀ (ε : Fin (m + 1) → ℤ),\n (∀ (i : Fin (m + 1)), |ε i| = 1) →\n ... | [
"G : Type u_1\ninst✝¹ : DecidableEq G\ninst✝ : Group G\nA : Finset G\nk : ℝ\nm✝ : ℕ\nh✝ : ∀ (ε : Fin 3 → ℤ), (∀ (i : Fin 3), |ε i| = 1) → ↑(#(List.map (fun i ↦ A ^ ε i) (finRange 3)).prod) ≤ k * ↑(#A)\nm : ℕ\nhm : 3 ≤ m + 1\nih :\n ∀ (ε : Fin (m + 1) → ℤ),\n (∀ (i : Fin (m + 1)), |ε i| = 1) →\n ↑(#(List.ma... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.CauchyDavenport | {
"line": 176,
"column": 4
} | {
"line": 177,
"column": 96
} | {
"line": 178,
"column": 2
} | [
{
"pp": "case inr.inr.inr.inr.inl\nα : Type u_2\ninst✝¹ : Group α\ninst✝ : DecidableEq α\ns t : Finset α\nhs : s.Nonempty\nht : t.Nonempty\nih :\n ∀ (a b : Finset α),\n a.Nonempty → b.Nonempty → DevosMulRel (a, b) (s, t) → minOrder α ≤ ↑(#(a * b)) ∨ #a + #b ≤ #(a * b) + 1\nhst : #s ≤ #t\na : α\nha : a ∈ ↑s\... | [] | exact (ih _ _ hgs (hgt.mono inter_subset_union) <| devosMulRel_of_le_of_le aux1 hstg hsg).imp
(WithTop.coe_le_coe.2 aux1).trans' fun h ↦ hstg.trans <| h.trans <| add_le_add_left aux1 _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Combinatorics.Additive.CauchyDavenport | {
"line": 176,
"column": 4
} | {
"line": 177,
"column": 96
} | {
"line": 178,
"column": 2
} | [
{
"pp": "case inr.inr.inr.inr.inl\nα : Type u_2\ninst✝¹ : Group α\ninst✝ : DecidableEq α\ns t : Finset α\nhs : s.Nonempty\nht : t.Nonempty\nih :\n ∀ (a b : Finset α),\n a.Nonempty → b.Nonempty → DevosMulRel (a, b) (s, t) → minOrder α ≤ ↑(#(a * b)) ∨ #a + #b ≤ #(a * b) + 1\nhst : #s ≤ #t\na : α\nha : a ∈ ↑s\... | [] | exact (ih _ _ hgs (hgt.mono inter_subset_union) <| devosMulRel_of_le_of_le aux1 hstg hsg).imp
(WithTop.coe_le_coe.2 aux1).trans' fun h ↦ hstg.trans <| h.trans <| add_le_add_left aux1 _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Additive.CauchyDavenport | {
"line": 176,
"column": 4
} | {
"line": 177,
"column": 96
} | {
"line": 178,
"column": 2
} | [
{
"pp": "case inr.inr.inr.inr.inl\nα : Type u_2\ninst✝¹ : Group α\ninst✝ : DecidableEq α\ns t : Finset α\nhs : s.Nonempty\nht : t.Nonempty\nih :\n ∀ (a b : Finset α),\n a.Nonempty → b.Nonempty → DevosMulRel (a, b) (s, t) → minOrder α ≤ ↑(#(a * b)) ∨ #a + #b ≤ #(a * b) + 1\nhst : #s ≤ #t\na : α\nha : a ∈ ↑s\... | [] | exact (ih _ _ hgs (hgt.mono inter_subset_union) <| devosMulRel_of_le_of_le aux1 hstg hsg).imp
(WithTop.coe_le_coe.2 aux1).trans' fun h ↦ hstg.trans <| h.trans <| add_le_add_left aux1 _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Additive.CauchyDavenport | {
"line": 191,
"column": 2
} | {
"line": 192,
"column": 9
} | {
"line": 192,
"column": 10
} | [
{
"pp": "G : Type u_1\ninst✝² : DecidableEq G\ninst✝¹ : Group G\ninst✝ : IsMulTorsionFree G\ns t : Finset G\nhs : s.Nonempty\nht : t.Nonempty\n⊢ #s + #t - 1 ≤ #(s * t)",
"ppTerm": "?m.23",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"G : Type u_1\ninst✝² : DecidableEq G\ninst✝¹ : Group G\ninst✝ : IsMulTorsionFree G\ns t : Finset G\nhs : s.Nonempty\nht : t.Nonempty\n⊢ #s + #t - 1 ≤ #(s * t)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.CauchyDavenport | {
"line": 200,
"column": 2
} | {
"line": 201,
"column": 9
} | {
"line": 201,
"column": 10
} | [
{
"pp": "p : ℕ\nhp : Nat.Prime p\ns t : Finset (ZMod p)\nhs : s.Nonempty\nht : t.Nonempty\n⊢ min p (#s + #t - 1) ≤ #(s + t)",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeSup",
"ZMod.commRing",
"CommSemiring.toSemiring",
"Finse... | [
"p : ℕ\nhp : Nat.Prime p\ns t : Finset (ZMod p)\nhs : s.Nonempty\nht : t.Nonempty\n⊢ p ≤ #(s + t) ∨ #s + #t - 1 ≤ #(s + t)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.Convolution | {
"line": 47,
"column": 2
} | {
"line": 47,
"column": 13
} | {
"line": 47,
"column": 14
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\ninst✝ : DecidableEq G\nA B : Finset G\nx : G\n⊢ #(A ∩ x •> B) = A.convolution B⁻¹ x",
"ppTerm": "?m.14",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"G : Type u_1\ninst✝¹ : Group G\ninst✝ : DecidableEq G\nA B : Finset G\nx : G\n⊢ #(A ∩ x •> B) = A.convolution B⁻¹ x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.Convolution | {
"line": 51,
"column": 2
} | {
"line": 51,
"column": 13
} | {
"line": 51,
"column": 14
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\ninst✝ : DecidableEq G\nA B : Finset G\nx : G\n⊢ #(x •> A ∩ B) = A.convolution B⁻¹ x⁻¹",
"ppTerm": "?m.16",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"G : Type u_1\ninst✝¹ : Group G\ninst✝ : DecidableEq G\nA B : Finset G\nx : G\n⊢ #(x •> A ∩ B) = A.convolution B⁻¹ x⁻¹"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.Corner.Defs | {
"line": 69,
"column": 32
} | {
"line": 69,
"column": 43
} | {
"line": 69,
"column": 44
} | [
{
"pp": "G : Type u_1\ninst✝ : AddCommMonoid G\nA : Set (G × G)\nhA : A.Subsingleton\n_x₁ _y₁ _x₂ _y₂ : G\nhxyd : IsCorner A _x₁ _y₁ _x₂ _y₂\n⊢ _x₁ = _x₂",
"ppTerm": "?m.9",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"G : Type u_1\ninst✝ : AddCommMonoid G\nA : Set (G × G)\nhA : A.Subsingleton\n_x₁ _y₁ _x₂ _y₂ : G\nhxyd : IsCorner A _x₁ _y₁ _x₂ _y₂\n⊢ _x₁ = _x₂"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.SmallTripling | {
"line": 91,
"column": 2
} | {
"line": 91,
"column": 25
} | {
"line": 91,
"column": 26
} | [
{
"pp": "G : Type u_1\ninst✝¹ : DecidableEq G\ninst✝ : Group G\nA : Finset G\nK : ℝ\nhA : ↑(#(A ^ 3)) ≤ K * ↑(#A)\n⊢ ↑(#(A⁻¹ * A⁻¹ * A)⁻¹) ≤ K ^ 2 * ↑(#A)",
"ppTerm": "?m.56",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Semigroup.toMul",
"Real",
"DivI... | [
"G : Type u_1\ninst✝¹ : DecidableEq G\ninst✝ : Group G\nA : Finset G\nK : ℝ\nhA : ↑(#(A ^ 3)) ≤ K * ↑(#A)\n⊢ ↑(#(A⁻¹ * (A * A))) ≤ K ^ 2 * ↑(#A)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.SmallTripling | {
"line": 95,
"column": 2
} | {
"line": 95,
"column": 13
} | {
"line": 95,
"column": 14
} | [
{
"pp": "G : Type u_1\ninst✝¹ : DecidableEq G\ninst✝ : Group G\nA : Finset G\nK : ℝ\nhA : ↑(#(A ^ 3)) ≤ K * ↑(#A)\n⊢ ↑(#(A * A⁻¹ * A⁻¹)) ≤ K ^ 2 * ↑(#A)",
"ppTerm": "?m.51",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"G : Type u_1\ninst✝¹ : DecidableEq G\ninst✝ : Group G\nA : Finset G\nK : ℝ\nhA : ↑(#(A ^ 3)) ≤ K * ↑(#A)\n⊢ ↑(#(A * A⁻¹ * A⁻¹)) ≤ K ^ 2 * ↑(#A)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.SmallTripling | {
"line": 100,
"column": 2
} | {
"line": 100,
"column": 25
} | {
"line": 100,
"column": 26
} | [
{
"pp": "G : Type u_1\ninst✝¹ : DecidableEq G\ninst✝ : Group G\nA : Finset G\nK : ℝ\nhA : ↑(#(A ^ 3)) ≤ K * ↑(#A)\n⊢ ↑(#(A * A * A⁻¹)⁻¹) ≤ K ^ 2 * ↑(#A)",
"ppTerm": "?m.54",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Semigroup.toMul",
"Real",
"DivInv... | [
"G : Type u_1\ninst✝¹ : DecidableEq G\ninst✝ : Group G\nA : Finset G\nK : ℝ\nhA : ↑(#(A ^ 3)) ≤ K * ↑(#A)\n⊢ ↑(#(A * (A⁻¹ * A⁻¹))) ≤ K ^ 2 * ↑(#A)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.SmallTripling | {
"line": 109,
"column": 17
} | {
"line": 109,
"column": 28
} | {
"line": 109,
"column": 29
} | [
{
"pp": "G : Type u_1\ninst✝¹ : DecidableEq G\ninst✝ : Group G\nA : Finset G\nK : ℝ\nhA : ↑(#(A ^ 3)) ≤ K * ↑(#A)\nhA₀ : A.Nonempty\n⊢ #A * #(A * A⁻¹ * A) ≤ #(A * (A * A⁻¹)) * #(A * A)",
"ppTerm": "?m.218",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"G : Type u_1\ninst✝¹ : DecidableEq G\ninst✝ : Group G\nA : Finset G\nK : ℝ\nhA : ↑(#(A ^ 3)) ≤ K * ↑(#A)\nhA₀ : A.Nonempty\n⊢ #A * #(A * A⁻¹ * A) ≤ #(A * (A * A⁻¹)) * #(A * A)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.SmallTripling | {
"line": 122,
"column": 2
} | {
"line": 122,
"column": 25
} | {
"line": 122,
"column": 26
} | [
{
"pp": "G : Type u_1\ninst✝¹ : DecidableEq G\ninst✝ : Group G\nA : Finset G\nK : ℝ\nhA : ↑(#(A ^ 3)) ≤ K * ↑(#A)\n⊢ ↑(#(A⁻¹ * A * A⁻¹)⁻¹) ≤ K ^ 3 * ↑(#A)",
"ppTerm": "?m.56",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Semigroup.toMul",
"Real",
"DivI... | [
"G : Type u_1\ninst✝¹ : DecidableEq G\ninst✝ : Group G\nA : Finset G\nK : ℝ\nhA : ↑(#(A ^ 3)) ≤ K * ↑(#A)\n⊢ ↑(#(A * (A⁻¹ * A))) ≤ K ^ 3 * ↑(#A)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.SmallTripling | {
"line": 141,
"column": 37
} | {
"line": 141,
"column": 52
} | {
"line": 141,
"column": 53
} | [
{
"pp": "G : Type u_1\ninst✝¹ : DecidableEq G\ninst✝ : Group G\nA : Finset G\nK : ℝ\nm : ℕ\nhm : 3 ≤ m\nhA : ↑(#(A ^ 3)) ≤ K * ↑(#A)\nε : Fin m → ℤ\nhε : ∀ (i : Fin m), |ε i| = 1\nhm₀ : m ≠ 0\ni : Fin m\nh : ε i = 0\n⊢ False",
"ppTerm": "?m.77",
"assigned": false,
"usedConstants": [],
"usedFVars... | [
"G : Type u_1\ninst✝¹ : DecidableEq G\ninst✝ : Group G\nA : Finset G\nK : ℝ\nm : ℕ\nhm : 3 ≤ m\nhA : ↑(#(A ^ 3)) ≤ K * ↑(#A)\nε : Fin m → ℤ\nhε : ∀ (i : Fin m), |ε i| = 1\nhm₀ : m ≠ 0\ni : Fin m\nh : ε i = 0\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.SmallTripling | {
"line": 184,
"column": 44
} | {
"line": 184,
"column": 82
} | {
"line": 184,
"column": 83
} | [
{
"pp": "G : Type u_1\ninst✝¹ : DecidableEq G\ninst✝ : Group G\nA : Finset G\nK : ℝ\nm : ℕ\nhm : 3 ≤ m\nhA : ↑(#(A ^ 3)) ≤ K * ↑(#A)\nhAsymm : A⁻¹ = A\nthis : ∀ (ε : ℤ), |ε| = 1 → A ^ ε = A\nδ : Fin 3 → ℤ\nhδ : ∀ (i : Fin 3), |δ i| = 1\n⊢ ↑(#(List.map (fun i ↦ A ^ δ i) (finRange 3)).prod) ≤ K * ↑(#A)",
"ppT... | [
"G : Type u_1\ninst✝¹ : DecidableEq G\ninst✝ : Group G\nA : Finset G\nK : ℝ\nm : ℕ\nhm : 3 ≤ m\nhA : ↑(#(A ^ 3)) ≤ K * ↑(#A)\nhAsymm : A⁻¹ = A\nthis : ∀ (ε : ℤ), |ε| = 1 → A ^ ε = A\nδ : Fin 3 → ℤ\nhδ : ∀ (i : Fin 3), |δ i| = 1\n⊢ ↑(#(A * (A * A))) ≤ K * ↑(#A)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.SmallTripling | {
"line": 178,
"column": 35
} | {
"line": 184,
"column": 98
} | {
"line": 186,
"column": 0
} | [
{
"pp": "G : Type u_1\ninst✝¹ : DecidableEq G\ninst✝ : Group G\nA : Finset G\nK : ℝ\nm : ℕ\nhm : 3 ≤ m\nhA : ↑(#(A ^ 3)) ≤ K * ↑(#A)\nhAsymm : A⁻¹ = A\n⊢ ↑(#(A ^ m)) ≤ K ^ (m - 2) * ↑(#A)",
"ppTerm": "?m.49",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
... | [] | by
have (ε : ℤ) (hε : |ε| = 1) : A ^ ε = A := by
obtain rfl | rfl := eq_or_eq_neg_of_abs_eq hε <;> simp [hAsymm]
calc
(#(A ^ m) : ℝ) = #((finRange m).map fun i ↦ A ^ 1).prod := by simp
_ ≤ K ^ (m - 2) * #A :=
inductive_claim_mul hm (fun δ hδ ↦ by simpa [this _ (hδ _), pow_succ'] using hA) _ (by si... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.SimpleGraph.DegreeSum | {
"line": 68,
"column": 2
} | {
"line": 68,
"column": 81
} | {
"line": 69,
"column": 4
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\ninst✝² : Fintype V\ninst✝¹ : DecidableRel G.Adj\ninst✝ : DecidableEq V\nv : V\n⊢ #{d | d.toProd.1 = v} = G.degree v",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"SimpleGraph.dartOfNeighborSet",
"Eq.mpr",
"Finset.univ",
"con... | [
"V : Type u\nG : SimpleGraph V\ninst✝² : Fintype V\ninst✝¹ : DecidableRel G.Adj\ninst✝ : DecidableEq V\nv : V\n⊢ #(image (G.dartOfNeighborSet v) univ) = G.degree v"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.DegreeSum | {
"line": 79,
"column": 26
} | {
"line": 79,
"column": 37
} | {
"line": 79,
"column": 38
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\ninst✝² : Fintype V\ninst✝¹ : DecidableRel G.Adj\ninst✝ : DecidableEq V\nd d' : G.Dart\n⊢ d' ∈ {d' | d'.edge = d.edge} ↔ d' ∈ {d, d.symm}",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Finset.mem_filter._simp_1",
"Finset.... | [
"V : Type u\nG : SimpleGraph V\ninst✝² : Fintype V\ninst✝¹ : DecidableRel G.Adj\ninst✝ : DecidableEq V\nd d' : G.Dart\n⊢ d'.edge = d.edge ↔ d' = d ∨ d' = d.symm"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Maps | {
"line": 568,
"column": 6
} | {
"line": 568,
"column": 36
} | {
"line": 568,
"column": 37
} | [
{
"pp": "case inr\nV : Type u_1\nW : Type u_2\nX : Type u_3\nY : Type u_4\nG : SimpleGraph V\nG' : SimpleGraph W\nu v✝ : V\nH : SimpleGraph W\nf✝ : G ↪g G'\nG'' : SimpleGraph X\nG''' : SimpleGraph Y\nf : Gᶜ ↪g Hᶜ\nv w : V\nhvw : v ≠ w\n⊢ H.Adj (f.toEmbedding v) (f.toEmbedding w) ↔ G.Adj v w",
"ppTerm": "?in... | [
"case inr\nV : Type u_1\nW : Type u_2\nX : Type u_3\nY : Type u_4\nG : SimpleGraph V\nG' : SimpleGraph W\nu v✝ : V\nH : SimpleGraph W\nf✝ : G ↪g G'\nG'' : SimpleGraph X\nG''' : SimpleGraph Y\nf : Gᶜ ↪g Hᶜ\nv w : V\nhvw : v ≠ w\n⊢ H.Adj (f v) (f w) ↔ G.Adj v w"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Maps | {
"line": 686,
"column": 6
} | {
"line": 686,
"column": 43
} | {
"line": 686,
"column": 44
} | [
{
"pp": "V : Type u_1\nW : Type u_2\nX : Type u_3\nY : Type u_4\nG : SimpleGraph V\nG' : SimpleGraph W\nu v✝ : V\nf : G ≃g G'\nv : V\nw : ↑(G'.neighborSet (f v))\n⊢ f.symm ↑w ∈ G.neighborSet v",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"SimpleGraph.Iso",
"S... | [
"V : Type u_1\nW : Type u_2\nX : Type u_3\nY : Type u_4\nG : SimpleGraph V\nG' : SimpleGraph W\nu v✝ : V\nf : G ≃g G'\nv : V\nw : ↑(G'.neighborSet (f v))\n⊢ G.Adj v (f.symm ↑w)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Finite | {
"line": 499,
"column": 2
} | {
"line": 500,
"column": 18
} | {
"line": 502,
"column": 0
} | [
{
"pp": "case inr\nV : Type u_1\nG : SimpleGraph V\ninst✝¹ : Fintype V\ninst✝ : DecidableRel G.Adj\nk : ℕ\nh : ∀ (v : V), G.degree v ≤ k\nh✝ : Nonempty V\n⊢ G.maxDegree ≤ k",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"SimpleGraph.maxDegree",
"Membership.mem",
"SimpleGr... | [] | · obtain ⟨_, hv⟩ := G.exists_maximal_degree_vertex
exact hv ▸ h _ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Combinatorics.SimpleGraph.Finite | {
"line": 627,
"column": 2
} | {
"line": 627,
"column": 13
} | {
"line": 627,
"column": 14
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\nW : Type u_2\nG' : SimpleGraph W\nf : G ≃g G'\ninst✝¹ : Fintype ↑G.edgeSet\ninst✝ : Fintype ↑G'.edgeSet\n⊢ ↥G.edgeFinset ≃ ↥G'.edgeFinset",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Finset",
"Membe... | [
"V : Type u_1\nG : SimpleGraph V\nW : Type u_2\nG' : SimpleGraph W\nf : G ≃g G'\ninst✝¹ : Fintype ↑G.edgeSet\ninst✝ : Fintype ↑G'.edgeSet\n⊢ { x // x ∈ G.edgeSet } ≃ { x // x ∈ G'.edgeSet }"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Finite | {
"line": 669,
"column": 2
} | {
"line": 669,
"column": 37
} | {
"line": 669,
"column": 38
} | [
{
"pp": "V : Type u_1\ns : Set V\ninst✝² : DecidablePred fun x ↦ x ∈ s\ninst✝¹ : Fintype V\nG : SimpleGraph V\ninst✝ : DecidableRel G.Adj\nh : G.support ⊆ s\n⊢ map (Embedding.subtype fun x ↦ x ∈ s).sym2Map (induce s G).edgeFinset = G.edgeFinset",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants":... | [
"V : Type u_1\ns : Set V\ninst✝² : DecidablePred fun x ↦ x ∈ s\ninst✝¹ : Fintype V\nG : SimpleGraph V\ninst✝ : DecidableRel G.Adj\nh : G.support ⊆ s\n⊢ G.edgeFinset ⊆ s.toFinset.sym2"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Partition.Finpartition | {
"line": 296,
"column": 6
} | {
"line": 296,
"column": 34
} | {
"line": 296,
"column": 35
} | [
{
"pp": "case pos\nα : Type u_1\ninst✝² : Lattice α\ninst✝¹ : OrderBot α\na : α\nP✝ : Finpartition a\ninst✝ : Decidable (a = ⊥)\nP : Finpartition a\nh : a = ⊥\nx : α\nhx : x ∈ P.parts\n⊢ ∃ c ∈ ((Finpartition.empty α).copy ⋯).parts, x ≤ c",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
... | [
"case pos\nα : Type u_1\ninst✝² : Lattice α\ninst✝¹ : OrderBot α\na : α\nP✝ : Finpartition a\ninst✝ : Decidable (a = ⊥)\nP : Finpartition a\nh : a = ⊥\nx : α\nhx : x ∈ P.parts\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Partition.Finpartition | {
"line": 362,
"column": 10
} | {
"line": 362,
"column": 56
} | {
"line": 362,
"column": 57
} | [
{
"pp": "case a\nα : Type u_1\ninst✝¹ : Lattice α\ninst✝ : OrderBot α\na : α\nP✝ : Finpartition a\ns : α\nP : Finpartition s\nPr : α → Prop\nPrsup : ∀ ⦃s t : α⦄, Pr s → Pr t → Pr (s ⊔ t)\nPrinf : ∀ ⦃s t : α⦄, Pr s → Pr t → Pr (s ⊓ t)\nPrbot : Pr ⊥\nhs : Pr s\nhP : ∀ p ∈ P.parts, Pr p\nthis✝¹ : Lattice (Subtype ... | [
"case a\nα : Type u_1\ninst✝¹ : Lattice α\ninst✝ : OrderBot α\na : α\nP✝ : Finpartition a\ns : α\nP : Finpartition s\nPr : α → Prop\nPrsup : ∀ ⦃s t : α⦄, Pr s → Pr t → Pr (s ⊔ t)\nPrinf : ∀ ⦃s t : α⦄, Pr s → Pr t → Pr (s ⊓ t)\nPrbot : Pr ⊥\nhs : Pr s\nhP : ∀ p ∈ P.parts, Pr p\nthis✝¹ : Lattice (Subtype Pr) := Subty... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Partition.Finpartition | {
"line": 363,
"column": 10
} | {
"line": 363,
"column": 21
} | {
"line": 363,
"column": 22
} | [
{
"pp": "case a\nα : Type u_1\ninst✝¹ : Lattice α\ninst✝ : OrderBot α\na : α\nP✝ : Finpartition a\ns : α\nP : Finpartition s\nPr : α → Prop\nPrsup : ∀ ⦃s t : α⦄, Pr s → Pr t → Pr (s ⊔ t)\nPrinf : ∀ ⦃s t : α⦄, Pr s → Pr t → Pr (s ⊓ t)\nPrbot : Pr ⊥\nhs : Pr s\nhP : ∀ p ∈ P.parts, Pr p\nthis✝¹ : Lattice (Subtype ... | [
"case a\nα : Type u_1\ninst✝¹ : Lattice α\ninst✝ : OrderBot α\na : α\nP✝ : Finpartition a\ns : α\nP : Finpartition s\nPr : α → Prop\nPrsup : ∀ ⦃s t : α⦄, Pr s → Pr t → Pr (s ⊔ t)\nPrinf : ∀ ⦃s t : α⦄, Pr s → Pr t → Pr (s ⊓ t)\nPrbot : Pr ⊥\nhs : Pr s\nhP : ∀ p ∈ P.parts, Pr p\nthis✝¹ : Lattice (Subtype Pr) := Subty... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Partition.Finpartition | {
"line": 364,
"column": 10
} | {
"line": 364,
"column": 34
} | {
"line": 364,
"column": 35
} | [
{
"pp": "case a\nα : Type u_1\ninst✝¹ : Lattice α\ninst✝ : OrderBot α\na : α\nP✝ : Finpartition a\ns : α\nP : Finpartition s\nPr : α → Prop\nPrsup : ∀ ⦃s t : α⦄, Pr s → Pr t → Pr (s ⊔ t)\nPrinf : ∀ ⦃s t : α⦄, Pr s → Pr t → Pr (s ⊓ t)\nPrbot : Pr ⊥\nhs : Pr s\nhP : ∀ p ∈ P.parts, Pr p\nthis✝¹ : Lattice (Subtype ... | [
"case a\nα : Type u_1\ninst✝¹ : Lattice α\ninst✝ : OrderBot α\na : α\nP✝ : Finpartition a\ns : α\nP : Finpartition s\nPr : α → Prop\nPrsup : ∀ ⦃s t : α⦄, Pr s → Pr t → Pr (s ⊔ t)\nPrinf : ∀ ⦃s t : α⦄, Pr s → Pr t → Pr (s ⊓ t)\nPrbot : Pr ⊥\nhs : Pr s\nhP : ∀ p ∈ P.parts, Pr p\nthis✝¹ : Lattice (Subtype Pr) := Subty... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Partition.Finpartition | {
"line": 367,
"column": 6
} | {
"line": 367,
"column": 61
} | {
"line": 367,
"column": 62
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Lattice α\ninst✝ : OrderBot α\na : α\nP✝ : Finpartition a\ns : α\nP : Finpartition s\nPr : α → Prop\nPrsup : ∀ ⦃s t : α⦄, Pr s → Pr t → Pr (s ⊔ t)\nPrinf : ∀ ⦃s t : α⦄, Pr s → Pr t → Pr (s ⊓ t)\nPrbot : Pr ⊥\nhs : Pr s\nhP : ∀ p ∈ P.parts, Pr p\nthis✝ : Lattice (Subtype Pr) := Su... | [
"α : Type u_1\ninst✝¹ : Lattice α\ninst✝ : OrderBot α\na : α\nP✝ : Finpartition a\ns : α\nP : Finpartition s\nPr : α → Prop\nPrsup : ∀ ⦃s t : α⦄, Pr s → Pr t → Pr (s ⊔ t)\nPrinf : ∀ ⦃s t : α⦄, Pr s → Pr t → Pr (s ⊓ t)\nPrbot : Pr ⊥\nhs : Pr s\nhP : ∀ p ∈ P.parts, Pr p\nthis✝ : Lattice (Subtype Pr) := Subtype.lattic... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Partition.Finpartition | {
"line": 368,
"column": 21
} | {
"line": 368,
"column": 70
} | {
"line": 368,
"column": 71
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Lattice α\ninst✝ : OrderBot α\na : α\nP✝ : Finpartition a\ns : α\nP : Finpartition s\nPr : α → Prop\nPrsup : ∀ ⦃s t : α⦄, Pr s → Pr t → Pr (s ⊔ t)\nPrinf : ∀ ⦃s t : α⦄, Pr s → Pr t → Pr (s ⊓ t)\nPrbot : Pr ⊥\nhs : Pr s\nhP : ∀ p ∈ P.parts, Pr p\nthis✝ : Lattice (Subtype Pr) := Su... | [
"α : Type u_1\ninst✝¹ : Lattice α\ninst✝ : OrderBot α\na : α\nP✝ : Finpartition a\ns : α\nP : Finpartition s\nPr : α → Prop\nPrsup : ∀ ⦃s t : α⦄, Pr s → Pr t → Pr (s ⊔ t)\nPrinf : ∀ ⦃s t : α⦄, Pr s → Pr t → Pr (s ⊓ t)\nPrbot : Pr ⊥\nhs : Pr s\nhP : ∀ p ∈ P.parts, Pr p\nthis✝ : Lattice (Subtype Pr) := Subtype.lattic... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Partition.Finpartition | {
"line": 442,
"column": 4
} | {
"line": 443,
"column": 11
} | {
"line": 443,
"column": 12
} | [
{
"pp": "α : Type u_1\ninst✝² : DistribLattice α\ninst✝¹ : OrderBot α\ninst✝ : DecidableEq α\na b c : α\nP : Finpartition a\nhb : b ≤ a\npx : α\nhpx : px ∈ ↑P.parts\nright✝¹ : ¬px ⊓ b = ⊥\npy : α\nhpy : py ∈ ↑P.parts\nhxy : px ⊓ b ≠ py ⊓ b\nright✝ : ¬py ⊓ b = ⊥\n⊢ (Disjoint on id) (px ⊓ b) (py ⊓ b)",
"ppTer... | [
"α : Type u_1\ninst✝² : DistribLattice α\ninst✝¹ : OrderBot α\ninst✝ : DecidableEq α\na b c : α\nP : Finpartition a\nhb : b ≤ a\npx : α\nhpx : px ∈ ↑P.parts\nright✝¹ : ¬px ⊓ b = ⊥\npy : α\nhpy : py ∈ ↑P.parts\nhxy : px ⊓ b ≠ py ⊓ b\nright✝ : ¬py ⊓ b = ⊥\n⊢ Disjoint (px ⊓ b) (py ⊓ b)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Triangulated.TStructure.ETrunc | {
"line": 234,
"column": 2
} | {
"line": 234,
"column": 12
} | {
"line": 234,
"column": 13
} | [
{
"pp": "case coe\nC : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : Preadditive C\ninst✝³ : HasZeroObject C\ninst✝² : HasShift C ℤ\ninst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝ : Pretriangulated C\nt : TStructure C\nX : C\nn : ℤ\n⊢ (t.eTriangleLTGE.obj (WithBotTop.coe n)).obj X ∈ distinguishedTr... | [] | | coe n => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.Order.Partition.Finpartition | {
"line": 615,
"column": 4
} | {
"line": 615,
"column": 69
} | {
"line": 615,
"column": 70
} | [
{
"pp": "case neg\nα : Type u_1\ninst✝¹ : GeneralizedBooleanAlgebra α\ninst✝ : DecidableEq α\na b : α\nP : Finpartition a\nhab : a ≤ b\np : α\nhp : p ∈ (P.extendOfLE hab).parts\nh : ¬a < b\n⊢ p ∈ P.parts",
"ppTerm": "?neg✝",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals... | [
"case neg\nα : Type u_1\ninst✝¹ : GeneralizedBooleanAlgebra α\ninst✝ : DecidableEq α\na b : α\nP : Finpartition a\nhab : a ≤ b\np : α\nhp : p ∈ (P.extendOfLE hab).parts\nh : ¬a < b\n⊢ p ∈ P.parts"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Triangulated.TStructure.ETrunc | {
"line": 260,
"column": 33
} | {
"line": 260,
"column": 44
} | {
"line": 260,
"column": 45
} | [
{
"pp": "C : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : Preadditive C\ninst✝³ : HasZeroObject C\ninst✝² : HasShift C ℤ\ninst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝ : Pretriangulated C\nt : TStructure C\nn : ℤ\nX : C\ni : ℤ\nh : WithBotTop.coe n ≤ WithBotTop.coe i\n⊢ n ≤ ?m.80",
"ppTerm": ... | [
"C : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : Preadditive C\ninst✝³ : HasZeroObject C\ninst✝² : HasShift C ℤ\ninst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝ : Pretriangulated C\nt : TStructure C\nn : ℤ\nX : C\ni : ℤ\nh : WithBotTop.coe n ≤ WithBotTop.coe i\n⊢ n ≤ ?m.80"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Triangulated.TStructure.ETrunc | {
"line": 278,
"column": 35
} | {
"line": 278,
"column": 46
} | {
"line": 278,
"column": 47
} | [
{
"pp": "C : Type u_1\ninst✝⁶ : Category.{v_1, u_1} C\ninst✝⁵ : Preadditive C\ninst✝⁴ : HasZeroObject C\ninst✝³ : HasShift C ℤ\ninst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝¹ : Pretriangulated C\nt : TStructure C\nX : C\nn : ℤ\ninst✝ : t.IsGE X n\nj : ℤ\nhj : WithBotTop.coe j ≤ WithBotTop.coe n\n⊢ j ≤ ... | [
"C : Type u_1\ninst✝⁶ : Category.{v_1, u_1} C\ninst✝⁵ : Preadditive C\ninst✝⁴ : HasZeroObject C\ninst✝³ : HasShift C ℤ\ninst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝¹ : Pretriangulated C\nt : TStructure C\nX : C\nn : ℤ\ninst✝ : t.IsGE X n\nj : ℤ\nhj : WithBotTop.coe j ≤ WithBotTop.coe n\n⊢ j ≤ n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Triangulated.TStructure.ETrunc | {
"line": 300,
"column": 29
} | {
"line": 300,
"column": 40
} | {
"line": 300,
"column": 41
} | [
{
"pp": "C : Type u_1\ninst✝⁶ : Category.{v_1, u_1} C\ninst✝⁵ : Preadditive C\ninst✝⁴ : HasZeroObject C\ninst✝³ : HasShift C ℤ\ninst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝¹ : Pretriangulated C\nt : TStructure C\ninst✝ : IsTriangulated C\na : EInt\nX : C\nh : a ≤ ⊥\n⊢ a = ⊥",
"ppTerm": "?m.87",
... | [
"C : Type u_1\ninst✝⁶ : Category.{v_1, u_1} C\ninst✝⁵ : Preadditive C\ninst✝⁴ : HasZeroObject C\ninst✝³ : HasShift C ℤ\ninst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝¹ : Pretriangulated C\nt : TStructure C\ninst✝ : IsTriangulated C\na : EInt\nX : C\nh : a ≤ ⊥\n⊢ a = ⊥"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Triangulated.TStructure.ETrunc | {
"line": 305,
"column": 62
} | {
"line": 305,
"column": 73
} | {
"line": 305,
"column": 74
} | [
{
"pp": "C : Type u_1\ninst✝⁶ : Category.{v_1, u_1} C\ninst✝⁵ : Preadditive C\ninst✝⁴ : HasZeroObject C\ninst✝³ : HasShift C ℤ\ninst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝¹ : Pretriangulated C\nt : TStructure C\ninst✝ : IsTriangulated C\nX : C\nb a : ℤ\nh : WithBotTop.coe a ≤ WithBotTop.coe b\n⊢ a ≤ ... | [
"C : Type u_1\ninst✝⁶ : Category.{v_1, u_1} C\ninst✝⁵ : Preadditive C\ninst✝⁴ : HasZeroObject C\ninst✝³ : HasShift C ℤ\ninst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝¹ : Pretriangulated C\nt : TStructure C\ninst✝ : IsTriangulated C\nX : C\nb a : ℤ\nh : WithBotTop.coe a ≤ WithBotTop.coe b\n⊢ a ≤ b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Partition.Finpartition | {
"line": 811,
"column": 4
} | {
"line": 817,
"column": 96
} | {
"line": 818,
"column": 4
} | [
{
"pp": "α : Type u_1\ninst✝¹ : DecidableEq α\ns✝ t u : Finset α\nP : Finpartition s✝\na : α\ns : Setoid α\nx : Finset α\ninst✝ : DecidableRel ⇑s\n⊢ (image (fun a ↦ {b ∈ x | s a b}) x).SupIndep id",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"_private.Mathlib.Order... | [
"α : Type u_1\ninst✝¹ : DecidableEq α\ns✝ t u : Finset α\nP : Finpartition s✝\na : α\ns : Setoid α\nx : Finset α\ninst✝ : DecidableRel ⇑s\n⊢ ∀ (a b c d : α), s a d → s b d → (s a c ↔ s b c)"
] | suffices ∀ (a b c d : α), s a d → s b d → (s a c ↔ s b c) by
simp only [supIndep_iff_pairwiseDisjoint, Set.PairwiseDisjoint, Set.Pairwise, coe_image,
Set.mem_image, mem_coe, ne_eq, onFun, id_eq, disjoint_iff_ne, forall_mem_not_eq,
forall_exists_index, and_imp, forall_apply_eq_imp_iff₂, mem_filter,... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.Order.Partition.Finpartition | {
"line": 830,
"column": 2
} | {
"line": 831,
"column": 50
} | {
"line": 832,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝¹ : DecidableEq α\na : α\ns : Setoid α\nx : Finset α\ninst✝ : DecidableRel ⇑s\nb : α\n⊢ b ∈ (ofSetSetoid s x).part a ↔ a ∈ x ∧ b ∈ x ∧ s a b",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Finset.mem_filter._simp_1",
"congrArg",
... | [
"α : Type u_1\ninst✝¹ : DecidableEq α\na : α\ns : Setoid α\nx : Finset α\ninst✝ : DecidableRel ⇑s\nb : α\n⊢ (∃ a₁ ∈ x, (b ∈ x ∧ s a₁ b) ∧ a ∈ x ∧ s a₁ a) ↔ a ∈ x ∧ b ∈ x ∧ s a b"
] | suffices (∃ a₁ ∈ x, (b ∈ x ∧ s a₁ b) ∧ a ∈ x ∧ s a₁ a) ↔ a ∈ x ∧ b ∈ x ∧ s a b by
simpa [mem_part_iff_exists, ofSetSetoid_parts] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.CategoryTheory.Triangulated.TStructure.ETrunc | {
"line": 317,
"column": 53
} | {
"line": 317,
"column": 64
} | {
"line": 317,
"column": 65
} | [
{
"pp": "C : Type u_1\ninst✝⁶ : Category.{v_1, u_1} C\ninst✝⁵ : Preadditive C\ninst✝⁴ : HasZeroObject C\ninst✝³ : HasShift C ℤ\ninst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝¹ : Pretriangulated C\nt : TStructure C\ninst✝ : IsTriangulated C\nX : C\na b : ℤ\nh : WithBotTop.coe a ≤ WithBotTop.coe b\n⊢ a ≤ ... | [
"C : Type u_1\ninst✝⁶ : Category.{v_1, u_1} C\ninst✝⁵ : Preadditive C\ninst✝⁴ : HasZeroObject C\ninst✝³ : HasShift C ℤ\ninst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝¹ : Pretriangulated C\nt : TStructure C\ninst✝ : IsTriangulated C\nX : C\na b : ℤ\nh : WithBotTop.coe a ≤ WithBotTop.coe b\n⊢ a ≤ b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Regularity.Bound | {
"line": 217,
"column": 4
} | {
"line": 217,
"column": 26
} | {
"line": 217,
"column": 27
} | [
{
"pp": "case inl\nι : Type u_2\n𝕜 : Type u_3\ninst✝² : Field 𝕜\ninst✝¹ : LinearOrder 𝕜\ninst✝ : IsStrictOrderedRing 𝕜\ns t : Finset ι\nx : 𝕜\nhst : s ⊆ t\nf : ι → 𝕜\nd : 𝕜\nhx : 0 ≤ x\nhs : x ≤ |(∑ i ∈ s, f i) / ↑(#s) - (∑ i ∈ t, f i) / ↑(#t)|\nht : d ≤ ((∑ i ∈ t, f i) / ↑(#t)) ^ 2\nhscard : 0 = ↑(#s)\n... | [
"case inl\nι : Type u_2\n𝕜 : Type u_3\ninst✝² : Field 𝕜\ninst✝¹ : LinearOrder 𝕜\ninst✝ : IsStrictOrderedRing 𝕜\ns t : Finset ι\nx : 𝕜\nhst : s ⊆ t\nf : ι → 𝕜\nd : 𝕜\nhx : 0 ≤ x\nhs : x ≤ |(∑ i ∈ s, f i) / ↑(#s) - (∑ i ∈ t, f i) / ↑(#t)|\nht : d ≤ ((∑ i ∈ t, f i) / ↑(#t)) ^ 2\nhscard : 0 = ↑(#s)\n⊢ d ≤ (∑ i ∈... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Triangulated.TStructure.ETrunc | {
"line": 320,
"column": 29
} | {
"line": 320,
"column": 40
} | {
"line": 320,
"column": 41
} | [
{
"pp": "C : Type u_1\ninst✝⁶ : Category.{v_1, u_1} C\ninst✝⁵ : Preadditive C\ninst✝⁴ : HasZeroObject C\ninst✝³ : HasShift C ℤ\ninst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝¹ : Pretriangulated C\nt : TStructure C\ninst✝ : IsTriangulated C\nb : EInt\nX : C\nh : ⊤ ≤ b\n⊢ b = ⊤",
"ppTerm": "?m.163",
... | [
"C : Type u_1\ninst✝⁶ : Category.{v_1, u_1} C\ninst✝⁵ : Preadditive C\ninst✝⁴ : HasZeroObject C\ninst✝³ : HasShift C ℤ\ninst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝¹ : Pretriangulated C\nt : TStructure C\ninst✝ : IsTriangulated C\nb : EInt\nX : C\nh : ⊤ ≤ b\n⊢ b = ⊤"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Regularity.Equitabilise | {
"line": 52,
"column": 31
} | {
"line": 52,
"column": 70
} | {
"line": 52,
"column": 71
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\ns : Finset α\na b : ℕ\nP : Finpartition s\nhs : a * 0 + b * (0 + 1) = #s\n⊢ #({i ∈ ⊥.parts | #i = 0 + 1}) = b",
"ppTerm": "?m.127",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Iff.of_eq",
"congrArg",
"Finset",
"AddMonoid... | [
"α : Type u_1\ninst✝ : DecidableEq α\ns : Finset α\na b : ℕ\nP : Finpartition s\nhs : a * 0 + b * (0 + 1) = #s\n⊢ #s = b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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