module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.CategoryTheory.LocallyCartesianClosed.ExponentiableMorphism | {
"line": 165,
"column": 44
} | {
"line": 166,
"column": 58
} | {
"line": 168,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nI : C\ninst✝¹ : ChosenPullbacksAlong (𝟙 I)\ninst✝ : ExponentiableMorphism (𝟙 I)\n⊢ (pullbackPushforwardAdj (𝟙 I)).unit ≫ (pullback (𝟙 I)).whiskerLeft (pushforwardId I).hom =\n (pullbackPushforwardAdj (𝟙 I)).unit",
"ppTerm": "?m.72",
"assigned": tr... | [] | by
rw [pushforwardId, Adjunction.unit_rightAdjointUniq_hom] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Monad.Types | {
"line": 43,
"column": 26
} | {
"line": 43,
"column": 48
} | {
"line": 45,
"column": 0
} | [
{
"pp": "m : Type u → Type u\ninst✝¹ : _root_.Monad m\ninst✝ : LawfulMonad m\nx✝¹ : Type u\nx✝ : (ofTypeFunctor m).obj ((𝟭 (Type u)).obj x✝¹)\n⊢ (ConcreteCategory.hom\n ((ofTypeFunctor m).map ({ app := fun X ↦ ↾pure, naturality := ⋯ }.app x✝¹) ≫\n { app := fun X ↦ ↾joinM, naturality := ⋯ }.... | [] | exact joinM_map_pure _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Monoidal.Action.Basic | {
"line": 254,
"column": 29
} | {
"line": 254,
"column": 44
} | {
"line": 254,
"column": 45
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Category.{v_2, u_2} D\ninst✝¹ : MonoidalCategory C\ninst✝ : MonoidalLeftAction C D\nx : C\ny z : D\nf : y ≅ z\n⊢ x ⊴ₗ (f.hom ≫ f.inv) = 𝟙 (x ⊙ₗ y)",
"ppTerm": "?m.58",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"C : Type u_1\nD : Type u_2\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Category.{v_2, u_2} D\ninst✝¹ : MonoidalCategory C\ninst✝ : MonoidalLeftAction C D\nx : C\ny z : D\nf : y ≅ z\n⊢ x ⊴ₗ 𝟙 y = 𝟙 (x ⊙ₗ y)"
] | Iso.hom_inv_id, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Monoidal.Action.Basic | {
"line": 259,
"column": 28
} | {
"line": 259,
"column": 43
} | {
"line": 259,
"column": 44
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Category.{v_2, u_2} D\ninst✝¹ : MonoidalCategory C\ninst✝ : MonoidalLeftAction C D\nx y : C\nf : x ≅ y\nz : D\n⊢ (f.hom ≫ f.inv) ⊵ₗ z = 𝟙 (x ⊙ₗ z)",
"ppTerm": "?m.58",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"C : Type u_1\nD : Type u_2\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Category.{v_2, u_2} D\ninst✝¹ : MonoidalCategory C\ninst✝ : MonoidalLeftAction C D\nx y : C\nf : x ≅ y\nz : D\n⊢ 𝟙 x ⊵ₗ z = 𝟙 (x ⊙ₗ z)"
] | Iso.hom_inv_id, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Monoidal.Action.Basic | {
"line": 564,
"column": 29
} | {
"line": 564,
"column": 44
} | {
"line": 564,
"column": 45
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Category.{v_2, u_2} D\ninst✝¹ : MonoidalCategory C\ninst✝ : MonoidalRightAction C D\nx : D\ny z : C\nf : y ≅ z\n⊢ x ⊴ᵣ (f.hom ≫ f.inv) = 𝟙 (x ⊙ᵣ y)",
"ppTerm": "?m.58",
"assigned": true,
"usedConstants": [
"Eq.mpr",... | [
"C : Type u_1\nD : Type u_2\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Category.{v_2, u_2} D\ninst✝¹ : MonoidalCategory C\ninst✝ : MonoidalRightAction C D\nx : D\ny z : C\nf : y ≅ z\n⊢ x ⊴ᵣ 𝟙 y = 𝟙 (x ⊙ᵣ y)"
] | Iso.hom_inv_id, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Monoidal.Action.Basic | {
"line": 569,
"column": 28
} | {
"line": 569,
"column": 43
} | {
"line": 569,
"column": 44
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Category.{v_2, u_2} D\ninst✝¹ : MonoidalCategory C\ninst✝ : MonoidalRightAction C D\nx y : D\nf : x ≅ y\nz : C\n⊢ (f.hom ≫ f.inv) ⊵ᵣ z = 𝟙 (x ⊙ᵣ z)",
"ppTerm": "?m.58",
"assigned": true,
"usedConstants": [
"Eq.mpr",... | [
"C : Type u_1\nD : Type u_2\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Category.{v_2, u_2} D\ninst✝¹ : MonoidalCategory C\ninst✝ : MonoidalRightAction C D\nx y : D\nf : x ≅ y\nz : C\n⊢ 𝟙 x ⊵ᵣ z = 𝟙 (x ⊙ᵣ z)"
] | Iso.hom_inv_id, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Monoidal.DayConvolution.Braided | {
"line": 117,
"column": 2
} | {
"line": 117,
"column": 64
} | {
"line": 118,
"column": 2
} | [
{
"pp": "C : Type u₁\ninst✝⁹ : Category.{v₁, u₁} C\nV : Type u₂\ninst✝⁸ : Category.{v₂, u₂} V\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\ninst✝⁵ : MonoidalCategory V\ninst✝⁴ : BraidedCategory V\nF G : C ⥤ V\nη : F ⟶ G\nH : C ⥤ V\ninst✝³ : DayConvolution F H\ninst✝² : DayConvolution H F\ninst✝¹ : D... | [
"C : Type u₁\ninst✝⁹ : Category.{v₁, u₁} C\nV : Type u₂\ninst✝⁸ : Category.{v₂, u₂} V\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\ninst✝⁵ : MonoidalCategory V\ninst✝⁴ : BraidedCategory V\nF G : C ⥤ V\nη : F ⟶ G\nH : C ⥤ V\ninst✝³ : DayConvolution F H\ninst✝² : DayConvolution H F\ninst✝¹ : DayConvolutio... | apply Functor.hom_ext_of_isLeftKanExtension (F ⊛ H) (unit F H) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.CategoryTheory.Monoidal.Internal.Module | {
"line": 68,
"column": 10
} | {
"line": 68,
"column": 34
} | {
"line": 68,
"column": 35
} | [
{
"pp": "R : Type u\ninst✝¹ : CommRing R\nA : ModuleCat R\ninst✝ : MonObj A\nx : ↑A\n⊢ (ConcreteCategory.hom μ) (0 ⊗ₜ[R] x) = 0",
"ppTerm": "?m.522",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ModuleCat",
"congrArg",
"CategoryTheory.ConcreteCategory.hom",
"AddCommG... | [
"R : Type u\ninst✝¹ : CommRing R\nA : ModuleCat R\ninst✝ : MonObj A\nx : ↑A\n⊢ (ConcreteCategory.hom μ) 0 = 0"
] | TensorProduct.zero_tmul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Monoidal.Bimod | {
"line": 595,
"column": 2
} | {
"line": 595,
"column": 40
} | {
"line": 596,
"column": 2
} | [
{
"pp": "C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\ninst✝² : HasCoequalizers C\ninst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\nR S T U : Mon C\nP : Bimod R S\nQ : B... | [
"C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\ninst✝² : HasCoequalizers C\ninst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\nR S T U : Mon C\nP : Bimod R S\nQ : Bimod S T\nL ... | slice_rhs 2 3 => rw [Category.comp_id] | Mathlib.Tactic.Slice._aux_Mathlib_Tactic_CategoryTheory_Slice___macroRules_Mathlib_Tactic_Slice_sliceRHS_1 | Mathlib.Tactic.Slice.sliceRHS |
Mathlib.CategoryTheory.Monoidal.Bimod | {
"line": 611,
"column": 2
} | {
"line": 611,
"column": 40
} | {
"line": 612,
"column": 2
} | [
{
"pp": "C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\ninst✝² : HasCoequalizers C\ninst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\nR S T U : Mon C\nP : Bimod R S\nQ : B... | [
"C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\ninst✝² : HasCoequalizers C\ninst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\nR S T U : Mon C\nP : Bimod R S\nQ : Bimod S T\nL ... | slice_rhs 2 3 => rw [Category.comp_id] | Mathlib.Tactic.Slice._aux_Mathlib_Tactic_CategoryTheory_Slice___macroRules_Mathlib_Tactic_Slice_sliceRHS_1 | Mathlib.Tactic.Slice.sliceRHS |
Mathlib.CategoryTheory.Monoidal.Internal.FunctorCategory | {
"line": 169,
"column": 44
} | {
"line": 169,
"column": 79
} | {
"line": 169,
"column": 80
} | [
{
"pp": "C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\ninst✝¹ : MonoidalCategory D\nA : C ⥤ D\ninst✝ : ComonObj A\nX✝ Y✝ : C\nf : X✝ ⟶ Y✝\n⊢ A.map f ≫ Δ.app Y✝ = Δ.app X✝ ≫ (A.map f ⊗ₘ A.map f)",
"ppTerm": "?m.88",
"assigned": true,
"usedConstants": [
"C... | [] | rw [Δ[A].naturality, tensorObj_map] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Monoidal.DayConvolution | {
"line": 982,
"column": 10
} | {
"line": 982,
"column": 62
} | {
"line": 982,
"column": 62
} | [
{
"pp": "C✝ : Type u₁\ninst✝¹⁶ : Category.{v₁, u₁} C✝\nV✝ : Type u₂\ninst✝¹⁵ : Category.{v₂, u₂} V✝\ninst✝¹⁴ : MonoidalCategory C✝\ninst✝¹³ : MonoidalCategory V✝\nC : Type u₁\ninst✝¹² : Category.{v₁, u₁} C\nV : Type u₂\ninst✝¹¹ : Category.{v₂, u₂} V\ninst✝¹⁰ : MonoidalCategory C\ninst✝⁹ : MonoidalCategory V\nD ... | [
"C✝ : Type u₁\ninst✝¹⁶ : Category.{v₁, u₁} C✝\nV✝ : Type u₂\ninst✝¹⁵ : Category.{v₂, u₂} V✝\ninst✝¹⁴ : MonoidalCategory C✝\ninst✝¹³ : MonoidalCategory V✝\nC : Type u₁\ninst✝¹² : Category.{v₁, u₁} C\nV : Type u₂\ninst✝¹¹ : Category.{v₂, u₂} V\ninst✝¹⁰ : MonoidalCategory C\ninst✝⁹ : MonoidalCategory V\nD : Type u₃\ni... | convolutionExtensionUnit_comp_ι_map_whiskerRight_app | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Monoidal.Bimod | {
"line": 831,
"column": 2
} | {
"line": 831,
"column": 49
} | {
"line": 832,
"column": 2
} | [
{
"pp": "C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\ninst✝² : HasCoequalizers C\ninst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\nW X Y Z : Mon C\nM : Bimod W X\nN : B... | [
"C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\ninst✝² : HasCoequalizers C\ninst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\nW X Y Z : Mon C\nM : Bimod W X\nN : Bimod X Y\nP ... | dsimp [tensorHom, tensorBimod, associatorBimod] | Lean.Elab.Tactic.evalDSimp | Lean.Parser.Tactic.dsimp |
Mathlib.CategoryTheory.MorphismProperty.LocalClosure | {
"line": 125,
"column": 6
} | {
"line": 125,
"column": 23
} | {
"line": 126,
"column": 4
} | [
{
"pp": "case refine_1.comp.refine_2\nC : Type u\ninst✝⁶ : Category.{v, u} C\nK : Precoverage C\nP : MorphismProperty C\ninst✝⁵ : P.RespectsIso\ninst✝⁴ : P.RespectsLeft K.morphismProperty\ninst✝³ : K.HasIsos\ninst✝² : K.IsStableUnderBaseChange\ninst✝¹ : K.IsStableUnderComposition\ninst✝ : K.HasPullbacks\nX Y : ... | [] | · exact h _ _ ⟨i⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Monoidal.DayConvolution | {
"line": 1198,
"column": 2
} | {
"line": 1198,
"column": 19
} | {
"line": 1199,
"column": 2
} | [
{
"pp": "C : Type u₁\ninst✝⁹ : Category.{v₁, u₁} C\nV : Type u₂\ninst✝⁸ : Category.{v₂, u₂} V\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : MonoidalCategory V\nD : Type u₃\ninst✝⁵ : Category.{v₃, u₃} D\ninst✝⁴ : InducedLawfulDayConvolutionMonoidalCategoryStructCore C V D\ninst✝³ : ∀ (v : V) (d : C), Limits.PreservesCo... | [
"C : Type u₁\ninst✝⁹ : Category.{v₁, u₁} C\nV : Type u₂\ninst✝⁸ : Category.{v₂, u₂} V\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : MonoidalCategory V\nD : Type u₃\ninst✝⁵ : Category.{v₃, u₃} D\ninst✝⁴ : InducedLawfulDayConvolutionMonoidalCategoryStructCore C V D\ninst✝³ : ∀ (v : V) (d : C), Limits.PreservesColimitsOfShap... | rw [tensorHom_eq] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Subobject.NoetherianObject | {
"line": 91,
"column": 2
} | {
"line": 91,
"column": 54
} | {
"line": 92,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nX : C\n⊢ IsNoetherianObject X ↔ ∀ (F : ℕ ⥤ MonoOver X), IsFiltered.IsEventuallyConstant F",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Over",
"CategoryTheory.Functor",
"congrArg",
"Pa... | [
"C : Type u\ninst✝ : Category.{v, u} C\nX : C\n⊢ (∀ (f : ℕ →o Subobject X), ∃ n, ∀ (m : ℕ), n ≤ m → f n = f m) ↔\n ∀ (F : ℕ ⥤ MonoOver X), IsFiltered.IsEventuallyConstant F"
] | rw [isNoetherianObject_iff_monotone_chain_condition] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Subobject.NoetherianObject | {
"line": 110,
"column": 2
} | {
"line": 110,
"column": 54
} | {
"line": 111,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nX : C\nhX : IsZero X\n⊢ IsNoetherianObject X",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"PartialOrder.toPreorder",
"Exists",
"id",
"LE.le",
"CategoryTheory.instPartialOrderSu... | [
"C : Type u\ninst✝ : Category.{v, u} C\nX : C\nhX : IsZero X\n⊢ ∀ (f : ℕ →o Subobject X), ∃ n, ∀ (m : ℕ), n ≤ m → f n = f m"
] | rw [isNoetherianObject_iff_monotone_chain_condition] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Subobject.NoetherianObject | {
"line": 122,
"column": 2
} | {
"line": 122,
"column": 54
} | {
"line": 123,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nX Y : C\ni : X ⟶ Y\ninst✝¹ : Mono i\ninst✝ : IsNoetherianObject Y\n⊢ IsNoetherianObject X",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"PartialOrder.toPreorder",
"Exists",
"id",
"L... | [
"C : Type u\ninst✝² : Category.{v, u} C\nX Y : C\ni : X ⟶ Y\ninst✝¹ : Mono i\ninst✝ : IsNoetherianObject Y\n⊢ ∀ (f : ℕ →o Subobject X), ∃ n, ∀ (m : ℕ), n ≤ m → f n = f m"
] | rw [isNoetherianObject_iff_monotone_chain_condition] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Monoidal.Bimod | {
"line": 887,
"column": 2
} | {
"line": 887,
"column": 49
} | {
"line": 888,
"column": 2
} | [
{
"pp": "C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\ninst✝² : HasCoequalizers C\ninst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\nW X Y Z : Mon C\nM M' : Bimod W X\nf ... | [
"C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\ninst✝² : HasCoequalizers C\ninst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\nW X Y Z : Mon C\nM M' : Bimod W X\nf : M ⟶ M'\nN ... | dsimp [tensorHom, tensorBimod, associatorBimod] | Lean.Elab.Tactic.evalDSimp | Lean.Parser.Tactic.dsimp |
Mathlib.CategoryTheory.Preadditive.HomOrthogonal | {
"line": 203,
"column": 39
} | {
"line": 218,
"column": 13
} | {
"line": 220,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝⁵ : Category.{v, u} C\nι : Type u_1\ns : ι → C\ninst✝⁴ : Preadditive C\ninst✝³ : HasFiniteBiproducts C\ninst✝² : ∀ (i : ι), InvariantBasisNumber (End (s i))\no : HomOrthogonal s\nα β : Type\ninst✝¹ : Finite α\ninst✝ : Finite β\nf : α → ι\ng : β → ι\ni : (⨁ fun a ↦ s (f a)) ≅ ⨁ fun b ↦ ... | [] | by
classical
refine ⟨Equiv.ofPreimageEquiv ?_, fun a => Equiv.ofPreimageEquiv_map _ _⟩
intro c
apply Nonempty.some
apply Cardinal.eq.1
cases nonempty_fintype α; cases nonempty_fintype β
simp only [Cardinal.mk_fintype, Nat.cast_inj]
exact
Matrix.square_of_invertible (o.matrixDecomposition i.inv c) (o... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Monoidal.Bimod | {
"line": 916,
"column": 2
} | {
"line": 916,
"column": 49
} | {
"line": 917,
"column": 2
} | [
{
"pp": "C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\ninst✝² : HasCoequalizers C\ninst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\nW X Y Z : Mon C\nM : Bimod W X\nN N' ... | [
"C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\ninst✝² : HasCoequalizers C\ninst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\nW X Y Z : Mon C\nM : Bimod W X\nN N' : Bimod X Y\... | dsimp [tensorHom, tensorBimod, associatorBimod] | Lean.Elab.Tactic.evalDSimp | Lean.Parser.Tactic.dsimp |
Mathlib.CategoryTheory.Preadditive.Schur | {
"line": 172,
"column": 2
} | {
"line": 182,
"column": 7
} | {
"line": 184,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝⁹ : Category.{v_1, u_1} C\ninst✝⁸ : Preadditive C\n𝕜 : Type u_2\ninst✝⁷ : Field 𝕜\ninst✝⁶ : IsAlgClosed 𝕜\ninst✝⁵ : Linear 𝕜 C\ninst✝⁴ : HasKernels C\nX Y : C\ninst✝³ : FiniteDimensional 𝕜 (X ⟶ X)\ninst✝² : FiniteDimensional 𝕜 (X ⟶ Y)\ninst✝¹ : Simple X\ninst✝ : Simple Y\n⊢ fin... | [] | fconstructor
· intro h
rw [finrank_eq_one_iff'] at h
obtain ⟨f, nz, -⟩ := h
rw [← isIso_iff_nonzero] at nz
exact ⟨asIso f⟩
· rintro ⟨f⟩
have le_one := finrank_hom_simple_simple_le_one 𝕜 X Y
have zero_lt : 0 < finrank 𝕜 (X ⟶ Y) :=
finrank_pos_iff_exists_ne_zero.mpr ⟨f.hom, (isIso_iff_... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Preadditive.Schur | {
"line": 172,
"column": 2
} | {
"line": 182,
"column": 7
} | {
"line": 184,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝⁹ : Category.{v_1, u_1} C\ninst✝⁸ : Preadditive C\n𝕜 : Type u_2\ninst✝⁷ : Field 𝕜\ninst✝⁶ : IsAlgClosed 𝕜\ninst✝⁵ : Linear 𝕜 C\ninst✝⁴ : HasKernels C\nX Y : C\ninst✝³ : FiniteDimensional 𝕜 (X ⟶ X)\ninst✝² : FiniteDimensional 𝕜 (X ⟶ Y)\ninst✝¹ : Simple X\ninst✝ : Simple Y\n⊢ fin... | [] | fconstructor
· intro h
rw [finrank_eq_one_iff'] at h
obtain ⟨f, nz, -⟩ := h
rw [← isIso_iff_nonzero] at nz
exact ⟨asIso f⟩
· rintro ⟨f⟩
have le_one := finrank_hom_simple_simple_le_one 𝕜 X Y
have zero_lt : 0 < finrank 𝕜 (X ⟶ Y) :=
finrank_pos_iff_exists_ne_zero.mpr ⟨f.hom, (isIso_iff_... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Category.PartOrdEmb | {
"line": 308,
"column": 8
} | {
"line": 308,
"column": 11
} | {
"line": 308,
"column": 12
} | [
{
"pp": "J : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsFiltered J\nF : J ⥤ PartOrdEmb\nc : Cocone (F ⋙ forget PartOrdEmb)\nhc : IsColimit c\ns : Cocone F\nj : J\nx' y' : (F ⋙ forget PartOrdEmb).obj j\nhx :\n (ConcreteCategory.hom (hc.desc ((forget PartOrdEmb).mapCocone s))) ((ConcreteCategory.hom (c.ι.app j)... | [
"J : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsFiltered J\nF : J ⥤ PartOrdEmb\nc : Cocone (F ⋙ forget PartOrdEmb)\nhc : IsColimit c\ns : Cocone F\nj : J\nx' y' : (F ⋙ forget PartOrdEmb).obj j\nhx :\n (ConcreteCategory.hom (hc.desc ((forget PartOrdEmb).mapCocone s))) ((ConcreteCategory.hom (c.ι.app j)) x') =\n ... | hx, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Presentable.Type | {
"line": 108,
"column": 2
} | {
"line": 108,
"column": 40
} | {
"line": 109,
"column": 2
} | [
{
"pp": "X : Type u\nκ : Cardinal.{u}\nhκ : Cardinal.aleph0 ≤ κ\n⊢ IsColimit (cocone X κ)",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"ChainCompletePartialOrder.instOfCompleteLattice",
"HasCardinalLT",
"PartialOrder.toPreorder",
"Set.Elem",
"Preorder.smallC... | [
"X : Type u\nκ : Cardinal.{u}\nhκ : Cardinal.aleph0 ≤ κ\nthis : IsFiltered (HasCardinalLT.Set X κ)\n⊢ IsColimit (cocone X κ)"
] | have := isFiltered_of_aleph0_le X κ hκ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.CategoryTheory.Presentable.Directed | {
"line": 474,
"column": 4
} | {
"line": 474,
"column": 32
} | {
"line": 475,
"column": 4
} | [
{
"pp": "case inl.inl\nJ : Type w\ninst✝² : SmallCategory J\nκ : Cardinal.{w}\ninst✝¹ : Fact κ.IsRegular\ninst✝ : IsCardinalFiltered J κ\nhJ : ∀ (e : J), ∃ m x, IsEmpty (m ⟶ e)\nthis✝¹ : IsCardinalFiltered (DiagramWithUniqueTerminal J κ) κ\nthis✝ : IsFiltered J\nthis : IsFiltered (DiagramWithUniqueTerminal J κ)... | [
"case inl.inr\nJ : Type w\ninst✝² : SmallCategory J\nκ : Cardinal.{w}\ninst✝¹ : Fact κ.IsRegular\ninst✝ : IsCardinalFiltered J κ\nhJ : ∀ (e : J), ∃ m x, IsEmpty (m ⟶ e)\nthis✝¹ : IsCardinalFiltered (DiagramWithUniqueTerminal J κ) κ\nthis✝ : IsFiltered J\nthis : IsFiltered (DiagramWithUniqueTerminal J κ)\nj : J\nD :... | · exact (h₁ (D.src hf)).elim | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Sites.CartesianMonoidal | {
"line": 36,
"column": 2
} | {
"line": 39,
"column": 46
} | {
"line": 41,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nA : Type u₂\ninst✝¹ : Category.{v₂, u₂} A\nJ : GrothendieckTopology C\ninst✝ : CartesianMonoidalCategory A\nX Y : Sheaf J A\n⊢ Presheaf.IsSheaf J (X.obj ⊗ Y.obj)",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"CategoryTheory.Functo... | [] | apply isSheaf_of_isLimit (E := (Cone.postcompose (pairComp X Y (sheafToPresheaf J A)).inv).obj
(BinaryFan.mk (fst X.obj Y.obj) (snd _ _)))
exact (IsLimit.postcomposeInvEquiv _ _).invFun
(tensorProductIsBinaryProduct X.obj Y.obj) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Sites.CartesianMonoidal | {
"line": 36,
"column": 2
} | {
"line": 39,
"column": 46
} | {
"line": 41,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nA : Type u₂\ninst✝¹ : Category.{v₂, u₂} A\nJ : GrothendieckTopology C\ninst✝ : CartesianMonoidalCategory A\nX Y : Sheaf J A\n⊢ Presheaf.IsSheaf J (X.obj ⊗ Y.obj)",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"CategoryTheory.Functo... | [] | apply isSheaf_of_isLimit (E := (Cone.postcompose (pairComp X Y (sheafToPresheaf J A)).inv).obj
(BinaryFan.mk (fst X.obj Y.obj) (snd _ _)))
exact (IsLimit.postcomposeInvEquiv _ _).invFun
(tensorProductIsBinaryProduct X.obj Y.obj) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Presentable.Directed | {
"line": 483,
"column": 8
} | {
"line": 483,
"column": 39
} | {
"line": 484,
"column": 8
} | [
{
"pp": "case inl\nJ : Type w\ninst✝² : SmallCategory J\nκ : Cardinal.{w}\ninst✝¹ : Fact κ.IsRegular\ninst✝ : IsCardinalFiltered J κ\nhJ : ∀ (e : J), ∃ m x, IsEmpty (m ⟶ e)\nthis✝¹ : IsCardinalFiltered (DiagramWithUniqueTerminal J κ) κ\nthis✝ : IsFiltered J\nthis : IsFiltered (DiagramWithUniqueTerminal J κ)\nj✝... | [
"case inr\nJ : Type w\ninst✝² : SmallCategory J\nκ : Cardinal.{w}\ninst✝¹ : Fact κ.IsRegular\ninst✝ : IsCardinalFiltered J κ\nhJ : ∀ (e : J), ∃ m x, IsEmpty (m ⟶ e)\nthis✝¹ : IsCardinalFiltered (DiagramWithUniqueTerminal J κ) κ\nthis✝ : IsFiltered J\nthis : IsFiltered (DiagramWithUniqueTerminal J κ)\nj : J\nD : Dia... | · exact ⟨φ ⟨_, hj⟩, Or.inr ⟨_⟩⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Sites.Coherent.CoherentTopology | {
"line": 66,
"column": 4
} | {
"line": 66,
"column": 34
} | {
"line": 67,
"column": 4
} | [
{
"pp": "case a\nC : Type u_1\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Precoherent C\nX : C\nα : Type\ninst✝¹ : Finite α\nY : α → C\nπ : (a : α) → Y a ⟶ X\nh : EffectiveEpiFamily Y π\nβ : α → Type\ninst✝ : ∀ (a : α), Finite (β a)\nY_n : (a : α) → β a → C\nπ_n : (a : α) → (b : β a) → Y_n a b ⟶ Y a\nH : ∀ (a : α... | [
"case a\nC : Type u_1\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Precoherent C\nX : C\nα : Type\ninst✝¹ : Finite α\nY : α → C\nπ : (a : α) → Y a ⟶ X\nh✝ : EffectiveEpiFamily Y π\nβ : α → Type\ninst✝ : ∀ (a : α), Finite (β a)\nY_n : (a : α) → β a → C\nπ_n : (a : α) → (b : β a) → Y_n a b ⟶ Y a\nH : ∀ (a : α), Effectiv... | intro V f ⟨Y₁, h, g, ⟨hY, hf⟩⟩ | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.CategoryTheory.Sites.Coherent.RegularTopology | {
"line": 59,
"column": 4
} | {
"line": 59,
"column": 34
} | {
"line": 60,
"column": 4
} | [
{
"pp": "case a\nC : Type u_1\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Preregular C\nX Y Y' : C\nπ : Y ⟶ X\ninst✝¹ : EffectiveEpi π\nπ' : Y' ⟶ Y\ninst✝ : EffectiveEpi π'\n⊢ ∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ X⦄,\n (Sieve.generate (Presieve.ofArrows (fun x ↦ Y) fun x ↦ π)).arrows f →\n (regularCoverage C).Saturate ... | [
"case a\nC : Type u_1\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Preregular C\nX Y Y' : C\nπ : Y ⟶ X\ninst✝¹ : EffectiveEpi π\nπ' : Y' ⟶ Y\ninst✝ : EffectiveEpi π'\nV : C\nf : V ⟶ X\nY₁ : C\nh : V ⟶ Y₁\ng : Y₁ ⟶ X\nhY : Presieve.ofArrows (fun x ↦ Y) (fun x ↦ π) g\nhf : h ≫ g = f\n⊢ (regularCoverage C).Saturate V (Si... | intro V f ⟨Y₁, h, g, ⟨hY, hf⟩⟩ | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.CategoryTheory.Sites.Coherent.RegularSheaves | {
"line": 178,
"column": 31
} | {
"line": 178,
"column": 53
} | {
"line": 178,
"column": 53
} | [
{
"pp": "case mp\nC : Type u_1\ninst✝² : Category.{v_1, u_1} C\nP : Cᵒᵖ ⥤ Type u_4\nhP : EqualizerCondition P\nX B : C\nπ : X ⟶ B\ninst✝¹ : EffectiveEpi π\ninst✝ : HasPullback π π\nh : IsIso (mapToEqualizer P π (pullback.fst π π) (pullback.snd π π) ⋯)\n⊢ IsIso (equalizer.lift (P.map π.op) ⋯)",
"ppTerm": "?m... | [
"case mp\nC : Type u_1\ninst✝² : Category.{v_1, u_1} C\nP : Cᵒᵖ ⥤ Type u_4\nhP : EqualizerCondition P\nX B : C\nπ : X ⟶ B\ninst✝¹ : EffectiveEpi π\ninst✝ : HasPullback π π\nh :\n IsIso\n (equalizer.lift (P.map π.op) ⋯ ≫\n (Types.equalizerIso (P.map (pullback.fst π π).op) (P.map (pullback.snd π π).op)).hom)... | mapToEqualizer_eq_comp | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Sites.Coherent.SheafComparison | {
"line": 170,
"column": 2
} | {
"line": 185,
"column": 37
} | {
"line": 187,
"column": 0
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝⁷ : Category.{v_1, u_1} C\ninst✝⁶ : Category.{v_2, u_2} D\nF : C ⥤ D\ninst✝⁵ : F.PreservesEffectiveEpis\ninst✝⁴ : F.ReflectsEffectiveEpis\ninst✝³ : F.Full\ninst✝² : F.Faithful\ninst✝¹ : F.EffectivelyEnough\ninst✝ : Preregular D\nX : C\nS : Sieve X\n⊢ (∃ Y π, EffectiveEp... | [] | refine ⟨fun ⟨Y, π, ⟨H₁, H₂⟩⟩ ↦ ?_, fun hS ↦ ?_⟩
· rw [mem_inducedTopology_iff_of_isCoverDense]
apply (mem_sieves_iff_hasEffectiveEpi (Sieve.functorPushforward _ S)).mpr
refine ⟨F.obj Y, F.map π, ⟨?_, Sieve.image_mem_functorPushforward F S H₂⟩⟩
exact F.map_effectiveEpi _
· rw [mem_inducedTopology_iff_of_... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Sites.Coherent.SheafComparison | {
"line": 170,
"column": 2
} | {
"line": 185,
"column": 37
} | {
"line": 187,
"column": 0
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝⁷ : Category.{v_1, u_1} C\ninst✝⁶ : Category.{v_2, u_2} D\nF : C ⥤ D\ninst✝⁵ : F.PreservesEffectiveEpis\ninst✝⁴ : F.ReflectsEffectiveEpis\ninst✝³ : F.Full\ninst✝² : F.Faithful\ninst✝¹ : F.EffectivelyEnough\ninst✝ : Preregular D\nX : C\nS : Sieve X\n⊢ (∃ Y π, EffectiveEp... | [] | refine ⟨fun ⟨Y, π, ⟨H₁, H₂⟩⟩ ↦ ?_, fun hS ↦ ?_⟩
· rw [mem_inducedTopology_iff_of_isCoverDense]
apply (mem_sieves_iff_hasEffectiveEpi (Sieve.functorPushforward _ S)).mpr
refine ⟨F.obj Y, F.map π, ⟨?_, Sieve.image_mem_functorPushforward F S H₂⟩⟩
exact F.map_effectiveEpi _
· rw [mem_inducedTopology_iff_of_... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Sites.Descent.Precoverage | {
"line": 227,
"column": 2
} | {
"line": 227,
"column": 70
} | {
"line": 228,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nF : LocallyDiscrete Cᵒᵖ ⥤ᵖ Cat\nι : Type t\nS : C\nX : ι → C\nf : (i : ι) → X i ⟶ S\nι' : Type t'\nX' : ι' → C\nf' : (j : ι') → X' j ⟶ S\nα : ι' → ι\np' : (j : ι') → X' j ⟶ X (α j)\nw : ∀ (j : ι'), p' j ≫ f (α j) = f' j\nD₁ D₂ : F.DescentData f\nφ : (pullFunctor F... | [
"C : Type u\ninst✝ : Category.{v, u} C\nF : LocallyDiscrete Cᵒᵖ ⥤ᵖ Cat\nι : Type t\nS : C\nX : ι → C\nf : (i : ι) → X i ⟶ S\nι' : Type t'\nX' : ι' → C\nf' : (j : ι') → X' j ⟶ S\nα : ι' → ι\np' : (j : ι') → X' j ⟶ X (α j)\nw : ∀ (j : ι'), p' j ≫ f (α j) = f' j\nD₁ D₂ : F.DescentData f\nφ : (pullFunctor F ⋯).obj D₁ ⟶... | obtain rfl : f₂ = Over.homMk Y₂.hom := by ext; simpa using Over.w f₂ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.CategoryTheory.Sites.Descent.Precoverage | {
"line": 334,
"column": 4
} | {
"line": 334,
"column": 66
} | {
"line": 335,
"column": 4
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nF : LocallyDiscrete Cᵒᵖ ⥤ᵖ Cat\nJ : GrothendieckTopology C\ninst✝¹ : F.IsPrestack J\nι : Type t\nS : C\nX : ι → C\nf : (i : ι) → X i ⟶ S\nι' : Type t'\nX' : ι' → C\nf' : (j : ι') → X' j ⟶ S\nh₁ : Sieve.ofArrows X' f' ∈ J S\nh₂ : Sieve.ofArrows X' f' ≤ Sieve.ofArr... | [
"C : Type u\ninst✝² : Category.{v, u} C\nF : LocallyDiscrete Cᵒᵖ ⥤ᵖ Cat\nJ : GrothendieckTopology C\ninst✝¹ : F.IsPrestack J\nι : Type t\nS : C\nX : ι → C\nf : (i : ι) → X i ⟶ S\nι' : Type t'\nX' : ι' → C\nf' : (j : ι') → X' j ⟶ S\nh₁ : Sieve.ofArrows X' f' ∈ J S\nh₂ : Sieve.ofArrows X' f' ≤ Sieve.ofArrows X f\nins... | obtain ⟨_, p, _, ⟨i⟩, fac⟩ := h₂ _ (Sieve.ofArrows_mk _ f' i') | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.CategoryTheory.Sites.SheafHom | {
"line": 87,
"column": 8
} | {
"line": 94,
"column": 11
} | {
"line": 94,
"column": 12
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst✝ : Category.{v', u'} A\nF G : Cᵒᵖ ⥤ A\ns : ↑(presheafHom F G).sections\n⊢ ∀ ⦃X Y : Cᵒᵖ⦄ (f : X ⟶ Y),\n F.map f ≫ (↑s Y).app (op (Over.mk (𝟙 (unop Y)))) = (↑s X).app (op (Over.mk (𝟙 (unop X)))) ≫ G.map f",
"pp... | [] | rintro ⟨X₁⟩ ⟨X₂⟩ ⟨f : X₂ ⟶ X₁⟩
dsimp
refine Eq.trans ?_ ((s.1 ⟨X₁⟩).naturality
(Over.homMk f : Over.mk f ⟶ Over.mk (𝟙 X₁)).op)
rw [← s.2 f.op]
dsimp
rw [presheafHom_map_app_op_mk_id]
rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Sites.SheafHom | {
"line": 87,
"column": 8
} | {
"line": 94,
"column": 11
} | {
"line": 94,
"column": 12
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst✝ : Category.{v', u'} A\nF G : Cᵒᵖ ⥤ A\ns : ↑(presheafHom F G).sections\n⊢ ∀ ⦃X Y : Cᵒᵖ⦄ (f : X ⟶ Y),\n F.map f ≫ (↑s Y).app (op (Over.mk (𝟙 (unop Y)))) = (↑s X).app (op (Over.mk (𝟙 (unop X)))) ≫ G.map f",
"pp... | [] | rintro ⟨X₁⟩ ⟨X₂⟩ ⟨f : X₂ ⟶ X₁⟩
dsimp
refine Eq.trans ?_ ((s.1 ⟨X₁⟩).naturality
(Over.homMk f : Over.mk f ⟶ Over.mk (𝟙 X₁)).op)
rw [← s.2 f.op]
dsimp
rw [presheafHom_map_app_op_mk_id]
rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Sites.MayerVietorisSquare | {
"line": 145,
"column": 8
} | {
"line": 145,
"column": 17
} | {
"line": 146,
"column": 6
} | [
{
"pp": "case refine_3.left\nC : Type u\ninst✝³ : Category.{v, u} C\nJ : GrothendieckTopology C\ninst✝² : HasWeakSheafify J (Type v)\nsq : Square C\ninst✝¹ : Mono sq.f₂₄\ninst✝ : Mono sq.f₃₄\nh₁ : sq.IsPullback\nh₂ : Sieve.ofTwoArrows sq.f₂₄ sq.f₃₄ ∈ J sq.X₄\nthis : Mono sq.f₁₃\nF : Sheaf J (Type v)\ns : Pullba... | [] | exact hm₁ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Sites.SheafHom | {
"line": 154,
"column": 22
} | {
"line": 154,
"column": 27
} | {
"line": 154,
"column": 28
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nA : Type u'\ninst✝ : Category.{v', u'} A\nF G : Cᵒᵖ ⥤ A\nX : C\nS : Sieve X\nhG : ⦃Y : C⦄ → (f : Y ⟶ X) → IsLimit (G.mapCone (Sieve.pullback f S).arrows.cocone.op)\nx : Presieve.FamilyOfElements (presheafHom F G) S.arrows\nY : C\nhx : x.Compatible\ng : Y ⟶ X\nZ₁ ... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\nA : Type u'\ninst✝ : Category.{v', u'} A\nF G : Cᵒᵖ ⥤ A\nX : C\nS : Sieve X\nhG : ⦃Y : C⦄ → (f : Y ⟶ X) → IsLimit (G.mapCone (Sieve.pullback f S).arrows.cocone.op)\nx : Presieve.FamilyOfElements (presheafHom F G) S.arrows\nY : C\nhx : x.Compatible\ng : Y ⟶ X\nZ₁ : Over Y\nhZ... | ← H', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Sites.Descent.DescentData | {
"line": 253,
"column": 12
} | {
"line": 253,
"column": 83
} | {
"line": 254,
"column": 10
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nF : Pseudofunctor (LocallyDiscrete Cᵒᵖ) Cat\nι : Type t\nS : C\nX : ι → C\nf : (i : ι) → X i ⟶ S\nS' : C\np : S' ⟶ S\nι' : Type t'\nX' : ι' → C\nf' : (j : ι') → X' j ⟶ S'\nα : ι' → ι\np' : (j : ι') → X' j ⟶ X (α j)\nw : ∀ (j : ι'), p' j ≫ f (α j) = f' j ≫ p\nD₁ D₂... | [
"C : Type u\ninst✝ : Category.{v, u} C\nF : Pseudofunctor (LocallyDiscrete Cᵒᵖ) Cat\nι : Type t\nS : C\nX : ι → C\nf : (i : ι) → X i ⟶ S\nS' : C\np : S' ⟶ S\nι' : Type t'\nX' : ι' → C\nf' : (j : ι') → X' j ⟶ S'\nα : ι' → ι\np' : (j : ι') → X' j ⟶ X (α j)\nw : ∀ (j : ι'), p' j ≫ f (α j) = f' j ≫ p\nD₁ D₂ : F.Descent... | pullFunctorObjHom_eq_assoc _ _ _ _ _ (q ≫ p) (f₁ ≫ p' j₁) (f₂ ≫ p' j₂), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Subobject.Presheaf | {
"line": 52,
"column": 26
} | {
"line": 52,
"column": 44
} | {
"line": 53,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Limits.HasPullbacks C\nx✝¹ : Cᵒᵖ\nx✝ : Subobject (Opposite.unop x✝¹)\n⊢ (ConcreteCategory.hom (↾(pullback (𝟙 x✝¹).unop).obj)).toFun x✝ =\n (ConcreteCategory.hom (𝟙 (Subobject (Opposite.unop x✝¹)))).toFun x✝",
"ppTerm": "?m.58",
"assigned": tr... | [] | simp [pullback_id] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Triangulated.TStructure.ETrunc | {
"line": 62,
"column": 6
} | {
"line": 62,
"column": 38
} | {
"line": 62,
"column": 39
} | [
{
"pp": "case bot.bot\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\nz : EInt\ng : ⊥ ⟶ z\ng' : ⊥ ≤ z\nf : ⊥ ⟶ ⊥\nf' : ⊥ ≤ ⊥\n⊢ WithBotTop.rec (mo... | [
"case bot.bot.bot\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\nf : ⊥ ⟶ ⊥\nf' : ⊥ ≤ ⊥\ng : ⊥ ⟶ ⊥\ng' : ⊥ ≤ ⊥\n⊢ WithBotTop.rec (motive := fun {x} ↦\... | induction z using WithBotTop.rec | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Combinatorics.Additive.AP.Three.Defs | {
"line": 328,
"column": 2
} | {
"line": 328,
"column": 10
} | {
"line": 329,
"column": 2
} | [
{
"pp": "α : Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : Monoid α\ns : Finset α\nn : ℕ\nh : ∀ t ∈ powersetCard n s, ¬ThreeGPFree ↑t\nt : Finset α\nhts : t ⊆ s\nhcard : #t = mulRothNumber s\nht : ThreeGPFree ↑t\n⊢ ¬n ≤ #t",
"ppTerm": "?m.47",
"assigned": true,
"usedConstants": [
"PartialOrder.toP... | [
"α : Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : Monoid α\ns : Finset α\nn : ℕ\nh : ∀ t ∈ powersetCard n s, ¬ThreeGPFree ↑t\nt : Finset α\nhts : t ⊆ s\nhcard : #t = mulRothNumber s\nht : ThreeGPFree ↑t\nhn : n ≤ #t\n⊢ False"
] | intro hn | Lean.Elab.Tactic.evalIntro | null |
Mathlib.Combinatorics.Additive.AP.Three.Defs | {
"line": 328,
"column": 2
} | {
"line": 328,
"column": 10
} | {
"line": 329,
"column": 2
} | [
{
"pp": "α : Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : Monoid α\ns : Finset α\nn : ℕ\nh : ∀ t ∈ powersetCard n s, ¬ThreeGPFree ↑t\nt : Finset α\nhts : t ⊆ s\nhcard : #t = mulRothNumber s\nht : ThreeGPFree ↑t\n⊢ ¬n ≤ #t",
"ppTerm": "?m.47",
"assigned": true,
"usedConstants": [
"PartialOrder.toP... | [
"α : Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : Monoid α\ns : Finset α\nn : ℕ\nh : ∀ t ∈ powersetCard n s, ¬ThreeGPFree ↑t\nt : Finset α\nhts : t ⊆ s\nhcard : #t = mulRothNumber s\nht : ThreeGPFree ↑t\nhn : n ≤ #t\n⊢ False"
] | intro hn | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.CategoryTheory.Triangulated.TStructure.ETrunc | {
"line": 62,
"column": 6
} | {
"line": 62,
"column": 38
} | {
"line": 62,
"column": 39
} | [
{
"pp": "case bot.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\nz : EInt\na✝ : ℤ\ng : WithBotTop.coe a✝ ⟶ z\ng' : WithBotTop.coe a✝ ≤ z\nf :... | [
"case bot.coe.bot\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\na✝ : ℤ\nf : ⊥ ⟶ WithBotTop.coe a✝\nf' : ⊥ ≤ WithBotTop.coe a✝\ng : WithBotTop.coe a✝... | induction z using WithBotTop.rec | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.CategoryTheory.Triangulated.TStructure.ETrunc | {
"line": 62,
"column": 6
} | {
"line": 62,
"column": 38
} | {
"line": 62,
"column": 39
} | [
{
"pp": "case bot.top\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\nz : EInt\ng : ⊤ ⟶ z\ng' : ⊤ ≤ z\nf : ⊥ ⟶ ⊤\nf' : ⊥ ≤ ⊤\n⊢ WithBotTop.rec (mo... | [
"case bot.top.bot\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\nf : ⊥ ⟶ ⊤\nf' : ⊥ ≤ ⊤\ng : ⊤ ⟶ ⊥\ng' : ⊤ ≤ ⊥\n⊢ WithBotTop.rec (motive := fun {x} ↦\... | induction z using WithBotTop.rec | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.CategoryTheory.Triangulated.TStructure.ETrunc | {
"line": 62,
"column": 6
} | {
"line": 62,
"column": 38
} | {
"line": 62,
"column": 39
} | [
{
"pp": "case coe.bot\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\nz : EInt\na✝ : ℤ\ng : ⊥ ⟶ z\ng' : ⊥ ≤ z\nf : WithBotTop.coe a✝ ⟶ ⊥\nf' : Wit... | [
"case coe.bot.bot\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\na✝ : ℤ\nf : WithBotTop.coe a✝ ⟶ ⊥\nf' : WithBotTop.coe a✝ ≤ ⊥\ng : ⊥ ⟶ ⊥\ng' : ⊥ ≤ ⊥... | induction z using WithBotTop.rec | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.CategoryTheory.Triangulated.TStructure.ETrunc | {
"line": 62,
"column": 6
} | {
"line": 62,
"column": 38
} | {
"line": 62,
"column": 39
} | [
{
"pp": "case coe.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\nz : EInt\na✝¹ a✝ : ℤ\ng : WithBotTop.coe a✝ ⟶ z\ng' : WithBotTop.coe a✝ ≤ z\... | [
"case coe.coe.bot\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\na✝¹ a✝ : ℤ\nf : WithBotTop.coe a✝¹ ⟶ WithBotTop.coe a✝\nf' : WithBotTop.coe a✝¹ ≤ Wi... | induction z using WithBotTop.rec | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.CategoryTheory.Triangulated.TStructure.ETrunc | {
"line": 62,
"column": 6
} | {
"line": 62,
"column": 38
} | {
"line": 62,
"column": 39
} | [
{
"pp": "case coe.top\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\nz : EInt\na✝ : ℤ\ng : ⊤ ⟶ z\ng' : ⊤ ≤ z\nf : WithBotTop.coe a✝ ⟶ ⊤\nf' : Wit... | [
"case coe.top.bot\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\na✝ : ℤ\nf : WithBotTop.coe a✝ ⟶ ⊤\nf' : WithBotTop.coe a✝ ≤ ⊤\ng : ⊤ ⟶ ⊥\ng' : ⊤ ≤ ⊥... | induction z using WithBotTop.rec | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.CategoryTheory.Triangulated.TStructure.ETrunc | {
"line": 62,
"column": 6
} | {
"line": 62,
"column": 38
} | {
"line": 62,
"column": 39
} | [
{
"pp": "case top.bot\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\nz : EInt\ng : ⊥ ⟶ z\ng' : ⊥ ≤ z\nf : ⊤ ⟶ ⊥\nf' : ⊤ ≤ ⊥\n⊢ WithBotTop.rec (mo... | [
"case top.bot.bot\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\nf : ⊤ ⟶ ⊥\nf' : ⊤ ≤ ⊥\ng : ⊥ ⟶ ⊥\ng' : ⊥ ≤ ⊥\n⊢ WithBotTop.rec (motive := fun {x} ↦\... | induction z using WithBotTop.rec | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.CategoryTheory.Triangulated.TStructure.ETrunc | {
"line": 62,
"column": 6
} | {
"line": 62,
"column": 38
} | {
"line": 62,
"column": 39
} | [
{
"pp": "case top.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\nz : EInt\na✝ : ℤ\ng : WithBotTop.coe a✝ ⟶ z\ng' : WithBotTop.coe a✝ ≤ z\nf :... | [
"case top.coe.bot\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\na✝ : ℤ\nf : ⊤ ⟶ WithBotTop.coe a✝\nf' : ⊤ ≤ WithBotTop.coe a✝\ng : WithBotTop.coe a✝... | induction z using WithBotTop.rec | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Combinatorics.Additive.AP.Three.Behrend | {
"line": 187,
"column": 68
} | {
"line": 196,
"column": 71
} | {
"line": 198,
"column": 0
} | [
{
"pp": "n d k : ℕ\n⊢ ThreeAPFree ↑(image (⇑(map (2 * d - 1))) (sphere n d k))",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Nat.instOrderedSub",
"Preorder.toLT",
"Nat.instIsOrderedAddMonoid",
"AddM... | [] | by
rw [coe_image]
apply ThreeAPFree.image' (α := Fin n → ℕ) (β := ℕ) (s := sphere n d k) (map (2 * d - 1))
(map_injOn.mono _) threeAPFree_sphere
rw [Set.add_subset_iff]
rintro a ha b hb i
have hai := mem_box.1 (sphere_subset_box ha) i
have hbi := mem_box.1 (sphere_subset_box hb) i
rw [lt_tsub_iff_righ... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Triangulated.TStructure.ETrunc | {
"line": 62,
"column": 6
} | {
"line": 62,
"column": 38
} | {
"line": 62,
"column": 39
} | [
{
"pp": "case top.top\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\nz : EInt\ng : ⊤ ⟶ z\ng' : ⊤ ≤ z\nf : ⊤ ⟶ ⊤\nf' : ⊤ ≤ ⊤\n⊢ WithBotTop.rec (mo... | [
"case top.top.bot\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\nf : ⊤ ⟶ ⊤\nf' : ⊤ ≤ ⊤\ng : ⊤ ⟶ ⊥\ng' : ⊤ ≤ ⊥\n⊢ WithBotTop.rec (motive := fun {x} ↦\... | induction z using WithBotTop.rec | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Combinatorics.Additive.AP.Three.Behrend | {
"line": 367,
"column": 6
} | {
"line": 367,
"column": 73
} | {
"line": 368,
"column": 4
} | [
{
"pp": "N : ℕ\nhN₃ : 8 ≤ N\nhN₀ : 0 < ↑N\n⊢ rexp 1 ≤ ↑N",
"ppTerm": "?m.132",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"NNRat.divNat",
"Real.instNNRatCast",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real.partialOrder",
"Real",
"NNRatCast.toOfScient... | [] | exact (exp_one_lt_d9.le.trans <| by norm_num).trans (cast_le.2 hN₃) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Combinatorics.Additive.AP.Three.Behrend | {
"line": 442,
"column": 82
} | {
"line": 442,
"column": 96
} | {
"line": 443,
"column": 6
} | [
{
"pp": "case h₁\nN : ℕ\nhN : 4096 ≤ N\nn : ℕ := nValue N\nhn : 0 < ↑n\nhd : 0 < dValue N\nhN₀ : 0 < ↑N\nhn₂ : 2 < n\nthis✝ : (2 * dValue N - 1) ^ n ≤ N\nthis : rexp (-4 * √(log ↑N)) = rexp (-2 * √(log ↑N)) * rexp (-2 * √(log ↑N))\n⊢ -2 ≤ -(2 / ↑n) * log ↑N / √(log ↑N)",
"ppTerm": "?h₁",
"assigned": tru... | [
"case h₁\nN : ℕ\nhN : 4096 ≤ N\nn : ℕ := nValue N\nhn : 0 < ↑n\nhd : 0 < dValue N\nhN₀ : 0 < ↑N\nhn₂ : 2 < n\nthis✝ : (2 * dValue N - 1) ^ n ≤ N\nthis : rexp (-4 * √(log ↑N)) = rexp (-2 * √(log ↑N)) * rexp (-2 * √(log ↑N))\n⊢ -2 ≤ -(2 / ↑n) * (log ↑N / √(log ↑N))",
"case h₁\nN : ℕ\nhN : 4096 ≤ N\nn : ℕ := nValue ... | mul_div_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Additive.ApproximateSubgroup | {
"line": 159,
"column": 17
} | {
"line": 159,
"column": 31
} | {
"line": 159,
"column": 32
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nA B : Set G\nK L : ℝ\nm n : ℕ\nhA : IsApproximateSubgroup K A\nhB : IsApproximateSubgroup L B\nhm : 2 ≤ m\nhn : 2 ≤ n\nF₁ : Finset G\nhF₁ : ↑(#F₁) ≤ K\nhAF₁ : A ^ 2 ⊆ ↑F₁ • A\nF₂ : Finset G\nhF₂ : ↑(#F₂) ≤ L\nhBF₂ : B ^ 2 ⊆ ↑F₂ • B\nthis : 1 ≤ K\nf : G → G → G\nhf : ∀ (a ... | [
"G : Type u_1\ninst✝ : Group G\nA B : Set G\nK L : ℝ\nm n : ℕ\nhA : IsApproximateSubgroup K A\nhB : IsApproximateSubgroup L B\nhm : 2 ≤ m\nhn : 2 ≤ n\nF₁ : Finset G\nhF₁ : ↑(#F₁) ≤ K\nhAF₁ : A ^ 2 ⊆ ↑F₁ • A\nF₂ : Finset G\nhF₂ : ↑(#F₂) ≤ L\nhBF₂ : B ^ 2 ⊆ ↑F₂ • B\nthis : 1 ≤ K\nf : G → G → G\nhf : ∀ (a b : G), a • ... | ← smul_eq_mul, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Combinatorics.Additive.ApproximateSubgroup | {
"line": 164,
"column": 31
} | {
"line": 164,
"column": 45
} | {
"line": 164,
"column": 46
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nA B : Set G\nK L : ℝ\nm n : ℕ\nhA : IsApproximateSubgroup K A\nhB : IsApproximateSubgroup L B\nhm : 2 ≤ m\nhn : 2 ≤ n\nF₁ : Finset G\nhF₁ : ↑(#F₁) ≤ K\nhAF₁ : A ^ 2 ⊆ ↑F₁ • A\nF₂ : Finset G\nhF₂ : ↑(#F₂) ≤ L\nhBF₂ : B ^ 2 ⊆ ↑F₂ • B\nthis : 1 ≤ K\nf : G → G → G\nhf : ∀ (a ... | [
"G : Type u_1\ninst✝ : Group G\nA B : Set G\nK L : ℝ\nm n : ℕ\nhA : IsApproximateSubgroup K A\nhB : IsApproximateSubgroup L B\nhm : 2 ≤ m\nhn : 2 ≤ n\nF₁ : Finset G\nhF₁ : ↑(#F₁) ≤ K\nhAF₁ : A ^ 2 ⊆ ↑F₁ • A\nF₂ : Finset G\nhF₂ : ↑(#F₂) ≤ L\nhBF₂ : B ^ 2 ⊆ ↑F₂ • B\nthis : 1 ≤ K\nf : G → G → G\nhf : ∀ (a b : G), a • ... | ← smul_eq_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Triangulated.TStructure.ETrunc | {
"line": 107,
"column": 6
} | {
"line": 107,
"column": 38
} | {
"line": 107,
"column": 39
} | [
{
"pp": "case bot.bot\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\nz : EInt\ng : ⊥ ⟶ z\ng' : ⊥ ≤ z\nf : ⊥ ⟶ ⊥\nf' : ⊥ ≤ ⊥\n⊢ WithBotTop.rec (mo... | [
"case bot.bot.bot\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\nf : ⊥ ⟶ ⊥\nf' : ⊥ ≤ ⊥\ng : ⊥ ⟶ ⊥\ng' : ⊥ ≤ ⊥\n⊢ WithBotTop.rec (motive := fun {x} ↦\... | induction z using WithBotTop.rec | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.CategoryTheory.Triangulated.TStructure.ETrunc | {
"line": 107,
"column": 6
} | {
"line": 107,
"column": 38
} | {
"line": 107,
"column": 39
} | [
{
"pp": "case bot.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\nz : EInt\na✝ : ℤ\ng : WithBotTop.coe a✝ ⟶ z\ng' : WithBotTop.coe a✝ ≤ z\nf :... | [
"case bot.coe.bot\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\na✝ : ℤ\nf : ⊥ ⟶ WithBotTop.coe a✝\nf' : ⊥ ≤ WithBotTop.coe a✝\ng : WithBotTop.coe a✝... | induction z using WithBotTop.rec | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.CategoryTheory.Triangulated.TStructure.ETrunc | {
"line": 107,
"column": 6
} | {
"line": 107,
"column": 38
} | {
"line": 107,
"column": 39
} | [
{
"pp": "case bot.top\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\nz : EInt\ng : ⊤ ⟶ z\ng' : ⊤ ≤ z\nf : ⊥ ⟶ ⊤\nf' : ⊥ ≤ ⊤\n⊢ WithBotTop.rec (mo... | [
"case bot.top.bot\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\nf : ⊥ ⟶ ⊤\nf' : ⊥ ≤ ⊤\ng : ⊤ ⟶ ⊥\ng' : ⊤ ≤ ⊥\n⊢ WithBotTop.rec (motive := fun {x} ↦\... | induction z using WithBotTop.rec | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.CategoryTheory.Triangulated.TStructure.ETrunc | {
"line": 107,
"column": 6
} | {
"line": 107,
"column": 38
} | {
"line": 107,
"column": 39
} | [
{
"pp": "case coe.bot\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\nz : EInt\na✝ : ℤ\ng : ⊥ ⟶ z\ng' : ⊥ ≤ z\nf : WithBotTop.coe a✝ ⟶ ⊥\nf' : Wit... | [
"case coe.bot.bot\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\na✝ : ℤ\nf : WithBotTop.coe a✝ ⟶ ⊥\nf' : WithBotTop.coe a✝ ≤ ⊥\ng : ⊥ ⟶ ⊥\ng' : ⊥ ≤ ⊥... | induction z using WithBotTop.rec | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.CategoryTheory.Triangulated.TStructure.ETrunc | {
"line": 107,
"column": 6
} | {
"line": 107,
"column": 38
} | {
"line": 107,
"column": 39
} | [
{
"pp": "case coe.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\nz : EInt\na✝¹ a✝ : ℤ\ng : WithBotTop.coe a✝ ⟶ z\ng' : WithBotTop.coe a✝ ≤ z\... | [
"case coe.coe.bot\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\na✝¹ a✝ : ℤ\nf : WithBotTop.coe a✝¹ ⟶ WithBotTop.coe a✝\nf' : WithBotTop.coe a✝¹ ≤ Wi... | induction z using WithBotTop.rec | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.CategoryTheory.Triangulated.TStructure.ETrunc | {
"line": 107,
"column": 6
} | {
"line": 107,
"column": 38
} | {
"line": 107,
"column": 39
} | [
{
"pp": "case coe.top\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\nz : EInt\na✝ : ℤ\ng : ⊤ ⟶ z\ng' : ⊤ ≤ z\nf : WithBotTop.coe a✝ ⟶ ⊤\nf' : Wit... | [
"case coe.top.bot\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\na✝ : ℤ\nf : WithBotTop.coe a✝ ⟶ ⊤\nf' : WithBotTop.coe a✝ ≤ ⊤\ng : ⊤ ⟶ ⊥\ng' : ⊤ ≤ ⊥... | induction z using WithBotTop.rec | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.CategoryTheory.Triangulated.TStructure.ETrunc | {
"line": 107,
"column": 6
} | {
"line": 107,
"column": 38
} | {
"line": 107,
"column": 39
} | [
{
"pp": "case top.bot\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\nz : EInt\ng : ⊥ ⟶ z\ng' : ⊥ ≤ z\nf : ⊤ ⟶ ⊥\nf' : ⊤ ≤ ⊥\n⊢ WithBotTop.rec (mo... | [
"case top.bot.bot\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\nf : ⊤ ⟶ ⊥\nf' : ⊤ ≤ ⊥\ng : ⊥ ⟶ ⊥\ng' : ⊥ ≤ ⊥\n⊢ WithBotTop.rec (motive := fun {x} ↦\... | induction z using WithBotTop.rec | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.CategoryTheory.Triangulated.TStructure.ETrunc | {
"line": 107,
"column": 6
} | {
"line": 107,
"column": 38
} | {
"line": 107,
"column": 39
} | [
{
"pp": "case top.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\nz : EInt\na✝ : ℤ\ng : WithBotTop.coe a✝ ⟶ z\ng' : WithBotTop.coe a✝ ≤ z\nf :... | [
"case top.coe.bot\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\na✝ : ℤ\nf : ⊤ ⟶ WithBotTop.coe a✝\nf' : ⊤ ≤ WithBotTop.coe a✝\ng : WithBotTop.coe a✝... | induction z using WithBotTop.rec | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.CategoryTheory.Triangulated.TStructure.ETrunc | {
"line": 107,
"column": 6
} | {
"line": 107,
"column": 38
} | {
"line": 107,
"column": 39
} | [
{
"pp": "case top.top\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\nz : EInt\ng : ⊤ ⟶ z\ng' : ⊤ ≤ z\nf : ⊤ ⟶ ⊤\nf' : ⊤ ≤ ⊤\n⊢ WithBotTop.rec (mo... | [
"case top.top.bot\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\nf : ⊤ ⟶ ⊤\nf' : ⊤ ≤ ⊤\ng : ⊤ ⟶ ⊥\ng' : ⊤ ≤ ⊥\n⊢ WithBotTop.rec (motive := fun {x} ↦\... | induction z using WithBotTop.rec | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Combinatorics.SimpleGraph.Finite | {
"line": 111,
"column": 34
} | {
"line": 111,
"column": 49
} | {
"line": 111,
"column": 50
} | [
{
"pp": "V : Type u_1\nG₁ G₂ : SimpleGraph V\ninst✝¹ : Fintype ↑G₁.edgeSet\ninst✝ : Fintype ↑G₂.edgeSet\n⊢ Disjoint ↑G₁.edgeFinset ↑G₂.edgeFinset ↔ Disjoint G₁ G₂",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CompleteBooleanAlgebra.toCompleteDistribLattice",
... | [
"V : Type u_1\nG₁ G₂ : SimpleGraph V\ninst✝¹ : Fintype ↑G₁.edgeSet\ninst✝ : Fintype ↑G₂.edgeSet\n⊢ Disjoint G₁.edgeSet G₂.edgeSet ↔ Disjoint G₁ G₂"
] | coe_edgeFinset, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Data.Set.Equitable | {
"line": 42,
"column": 42
} | {
"line": 42,
"column": 72
} | {
"line": 42,
"column": 72
} | [
{
"pp": "α : Type u_1\ns : Set α\nf : α → ℕ\nx✝ : ∃ b, ∀ a ∈ s, b ≤ f a ∧ f a ≤ b + 1\nx y : α\nhx : x ∈ s\nhy : y ∈ s\nb : ℕ\nhb : ∀ a ∈ s, b ≤ f a ∧ f a ≤ b + 1\n⊢ f x ≤ f y + 1",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"le_refl",
"Nat.instIsOrderedAddMonoid",
"Na... | [] | grw [(hb x hx).2, (hb y hy).1] | Mathlib.Tactic.GRewrite._aux_Mathlib_Tactic_GRewrite_Elab___macroRules_Mathlib_Tactic_GRewrite_grwSeq_1 | Mathlib.Tactic.GRewrite.grwSeq |
Mathlib.Data.Set.Equitable | {
"line": 42,
"column": 42
} | {
"line": 42,
"column": 72
} | {
"line": 42,
"column": 72
} | [
{
"pp": "α : Type u_1\ns : Set α\nf : α → ℕ\nx✝ : ∃ b, ∀ a ∈ s, b ≤ f a ∧ f a ≤ b + 1\nx y : α\nhx : x ∈ s\nhy : y ∈ s\nb : ℕ\nhb : ∀ a ∈ s, b ≤ f a ∧ f a ≤ b + 1\n⊢ f x ≤ f y + 1",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"le_refl",
"Nat.instIsOrderedAddMonoid",
"Na... | [] | grw [(hb x hx).2, (hb y hy).1] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Set.Equitable | {
"line": 42,
"column": 42
} | {
"line": 42,
"column": 72
} | {
"line": 42,
"column": 72
} | [
{
"pp": "α : Type u_1\ns : Set α\nf : α → ℕ\nx✝ : ∃ b, ∀ a ∈ s, b ≤ f a ∧ f a ≤ b + 1\nx y : α\nhx : x ∈ s\nhy : y ∈ s\nb : ℕ\nhb : ∀ a ∈ s, b ≤ f a ∧ f a ≤ b + 1\n⊢ f x ≤ f y + 1",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"le_refl",
"Nat.instIsOrderedAddMonoid",
"Na... | [] | grw [(hb x hx).2, (hb y hy).1] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.DegreeSum | {
"line": 91,
"column": 2
} | {
"line": 98,
"column": 24
} | {
"line": 100,
"column": 0
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\ninst✝¹ : Fintype V\ninst✝ : DecidableRel G.Adj\n⊢ Fintype.card G.Dart = 2 * #G.edgeFinset",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Finset.card_univ",
"Eq.mpr",
"HMul.hMul",
"SimpleGraph.dart_edge_fiber_card",
"Fi... | [] | classical
rw [← card_univ]
rw [@card_eq_sum_card_fiberwise _ _ _ Dart.edge _ G.edgeFinset fun d _h =>
by rw [mem_coe, mem_edgeFinset]; apply Dart.edge_mem]
rw [← mul_comm, sum_const_nat]
intro e h
apply G.dart_edge_fiber_card e
rwa [← mem_edgeFinset] | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.Combinatorics.SimpleGraph.DegreeSum | {
"line": 91,
"column": 2
} | {
"line": 98,
"column": 24
} | {
"line": 100,
"column": 0
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\ninst✝¹ : Fintype V\ninst✝ : DecidableRel G.Adj\n⊢ Fintype.card G.Dart = 2 * #G.edgeFinset",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Finset.card_univ",
"Eq.mpr",
"HMul.hMul",
"SimpleGraph.dart_edge_fiber_card",
"Fi... | [] | classical
rw [← card_univ]
rw [@card_eq_sum_card_fiberwise _ _ _ Dart.edge _ G.edgeFinset fun d _h =>
by rw [mem_coe, mem_edgeFinset]; apply Dart.edge_mem]
rw [← mul_comm, sum_const_nat]
intro e h
apply G.dart_edge_fiber_card e
rwa [← mem_edgeFinset] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.DegreeSum | {
"line": 91,
"column": 2
} | {
"line": 98,
"column": 24
} | {
"line": 100,
"column": 0
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\ninst✝¹ : Fintype V\ninst✝ : DecidableRel G.Adj\n⊢ Fintype.card G.Dart = 2 * #G.edgeFinset",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Finset.card_univ",
"Eq.mpr",
"HMul.hMul",
"SimpleGraph.dart_edge_fiber_card",
"Fi... | [] | classical
rw [← card_univ]
rw [@card_eq_sum_card_fiberwise _ _ _ Dart.edge _ G.edgeFinset fun d _h =>
by rw [mem_coe, mem_edgeFinset]; apply Dart.edge_mem]
rw [← mul_comm, sum_const_nat]
intro e h
apply G.dart_edge_fiber_card e
rwa [← mem_edgeFinset] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Finset.Pairwise | {
"line": 68,
"column": 2
} | {
"line": 68,
"column": 42
} | {
"line": 69,
"column": 2
} | [
{
"pp": "α : Type u_1\nι : Type u_2\nι' : Type u_3\ninst✝¹ : Lattice α\ninst✝ : OrderBot α\ns : Set ι'\ng : ι' → Finset ι\nf : ι → α\nhs : s.PairwiseDisjoint fun i' ↦ (g i').sup f\nhg : ∀ i ∈ s, (↑(g i)).PairwiseDisjoint f\na b : ι\nhab : a ≠ b\nc : ι'\nhc : c ∈ s\nha : a ∈ ↑(g c)\nd : ι'\nhd : d ∈ s\nhb : b ∈ ... | [
"case inl\nα : Type u_1\nι : Type u_2\nι' : Type u_3\ninst✝¹ : Lattice α\ninst✝ : OrderBot α\ns : Set ι'\ng : ι' → Finset ι\nf : ι → α\nhs : s.PairwiseDisjoint fun i' ↦ (g i').sup f\nhg : ∀ i ∈ s, (↑(g i)).PairwiseDisjoint f\na b : ι\nhab : a ≠ b\nc : ι'\nhc : c ∈ s\nha : a ∈ ↑(g c)\nd : ι'\nhd : d ∈ s\nhb : b ∈ ↑(... | obtain hcd | hcd := eq_or_ne (g c) (g d) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Combinatorics.SimpleGraph.Finite | {
"line": 566,
"column": 2
} | {
"line": 566,
"column": 84
} | {
"line": 568,
"column": 0
} | [
{
"pp": "case refine_2\nV : Type u_1\nG : SimpleGraph V\ninst✝¹ : Fintype V\ninst✝ : DecidableRel G.Adj\nv : V\nh : G.IsUniversal v\n⊢ G.degree v = Fintype.card V - 1",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"SimpleGraph.neighborFinset_eq_erase_univ",
... | [] | · simp [← card_neighborFinset_eq_degree, (G.neighborFinset_eq_erase_univ v).mpr h] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Combinatorics.SimpleGraph.Regularity.Bound | {
"line": 135,
"column": 67
} | {
"line": 140,
"column": 51
} | {
"line": 142,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nP : Finpartition univ\n⊢ a + 1 ≤ 4 ^ #P.parts",
"ppTerm": "?m.64",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"SzemerediRegularity.stepBound",
"Nat.instOrderedSub",
"Preorder.toLT",
"instHDiv",
... | [] | by
have h : 1 ≤ 4 ^ #P.parts := one_le_pow₀ (by simp)
rw [stepBound, ← Nat.div_div_eq_div_mul]
conv_rhs => rw [← Nat.sub_add_cancel h]
rw [add_le_add_iff_right, tsub_le_iff_left, ← Nat.add_sub_assoc h]
exact Nat.le_sub_one_of_lt (Nat.lt_div_mul_add h) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.SimpleGraph.Regularity.Equitabilise | {
"line": 72,
"column": 2
} | {
"line": 80,
"column": 89
} | {
"line": 87,
"column": 2
} | [
{
"pp": "case neg\nα : Type u_1\ninst✝ : DecidableEq α\nm : ℕ\nm_pos : m > 0\ns : Finset α\nih :\n ∀ t ⊂ s,\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = #t →\n ∃ Q,\n (∀ x ∈ Q.parts, #x = m ∨ #x = m + 1) ∧\n (∀ x ∈ P.parts, #(x \\ {y ∈ Q.parts | y ⊆ x}.biUnion ... | [
"case neg\nα : Type u_1\ninst✝ : DecidableEq α\nm : ℕ\nm_pos : m > 0\ns : Finset α\nih :\n ∀ t ⊂ s,\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = #t →\n ∃ Q,\n (∀ x ∈ Q.parts, #x = m ∨ #x = m + 1) ∧\n (∀ x ∈ P.parts, #(x \\ {y ∈ Q.parts | y ⊆ x}.biUnion id) ≤ m) ∧ #... | obtain ⟨hn₀, hn₁, hn₂, hn₃⟩ : 0 < n ∧ n ≤ m + 1 ∧ n ≤ a * m + b * (m + 1) ∧
ite (0 < a) (a - 1) a * m + ite (0 < a) b (b - 1) * (m + 1) = #s - n := by
rw [hn, ← hs]
split_ifs with h <;> rw [tsub_mul, one_mul]
· refine ⟨m_pos, le_succ _, le_add_right (Nat.le_mul_of_pos_left _ ‹0 < a›), ?_⟩
rw [ts... | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Combinatorics.SimpleGraph.Density | {
"line": 57,
"column": 50
} | {
"line": 57,
"column": 59
} | {
"line": 57,
"column": 59
} | [
{
"pp": "α : Type u_4\nβ : Type u_5\nr : α → β → Prop\ninst✝ : (a : α) → DecidablePred (r a)\ns : Finset α\nt : Finset β\nx : α × β\n⊢ (x.1 ∈ s ∧ x.2 ∈ t) ∧ r x.1 x.2 ↔ x.1 ∈ s ∧ x.2 ∈ t ∧ r x.1 x.2",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Fi... | [
"α : Type u_4\nβ : Type u_5\nr : α → β → Prop\ninst✝ : (a : α) → DecidablePred (r a)\ns : Finset α\nt : Finset β\nx : α × β\n⊢ x.1 ∈ s ∧ x.2 ∈ t ∧ r x.1 x.2 ↔ x.1 ∈ s ∧ x.2 ∈ t ∧ r x.1 x.2"
] | and_assoc | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SimpleGraph.Density | {
"line": 178,
"column": 79
} | {
"line": 185,
"column": 86
} | {
"line": 187,
"column": 0
} | [
{
"pp": "α : Type u_4\nβ : Type u_5\nr : α → β → Prop\ninst✝ : (a : α) → DecidablePred (r a)\ns₁ s₂ : Finset α\nt₁ t₂ : Finset β\nhs : s₂ ⊆ s₁\nht : t₂ ⊆ t₁\nhs₂ : s₂.Nonempty\nht₂ : t₂.Nonempty\n⊢ edgeDensity r s₂ t₂ - edgeDensity r s₁ t₁ ≤ 1 - ↑(#s₂) / ↑(#s₁) * (↑(#t₂) / ↑(#t₁))",
"ppTerm": "?m.45",
"... | [] | by
refine (sub_le_sub_left (mul_edgeDensity_le_edgeDensity r hs ht hs₂ ht₂) _).trans ?_
refine le_trans ?_ (mul_le_of_le_one_right ?_ (edgeDensity_le_one r s₂ t₂))
· rw [sub_mul, one_mul]
refine sub_nonneg_of_le (mul_le_one₀ ?_ ?_ ?_)
· exact div_le_one_of_le₀ ((@Nat.cast_le ℚ).2 (card_le_card hs)) (Nat.cast_... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.SimpleGraph.Regularity.Uniform | {
"line": 208,
"column": 55
} | {
"line": 208,
"column": 64
} | {
"line": 208,
"column": 64
} | [
{
"pp": "α : Type u_1\n𝕜 : Type u_2\ninst✝³ : Field 𝕜\ninst✝² : LinearOrder 𝕜\ninst✝¹ : DecidableEq α\nA : Finset α\nP : Finpartition A\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nu v : Finset α\nε : 𝕜\n⊢ ((u, v).1 ∈ P.parts ∧\n ((u, v).2 ∈ P.parts ∧ (u, v).1 ≠ (u, v).2) ∧\n match (u, v) wit... | [
"α : Type u_1\n𝕜 : Type u_2\ninst✝³ : Field 𝕜\ninst✝² : LinearOrder 𝕜\ninst✝¹ : DecidableEq α\nA : Finset α\nP : Finpartition A\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nu v : Finset α\nε : 𝕜\n⊢ ((u, v).1 ∈ P.parts ∧\n (u, v).2 ∈ P.parts ∧\n (u, v).1 ≠ (u, v).2 ∧\n match (u, v) with\n... | and_assoc | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SimpleGraph.Regularity.Uniform | {
"line": 222,
"column": 55
} | {
"line": 222,
"column": 64
} | {
"line": 222,
"column": 64
} | [
{
"pp": "α : Type u_1\n𝕜 : Type u_2\ninst✝³ : Field 𝕜\ninst✝² : LinearOrder 𝕜\ninst✝¹ : DecidableEq α\nA : Finset α\nP : Finpartition A\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : 𝕜\nu v : Finset α\n⊢ ((u, v).1 ∈ P.parts ∧\n ((u, v).2 ∈ P.parts ∧ (u, v).1 ≠ (u, v).2) ∧\n match (u, v) wit... | [
"α : Type u_1\n𝕜 : Type u_2\ninst✝³ : Field 𝕜\ninst✝² : LinearOrder 𝕜\ninst✝¹ : DecidableEq α\nA : Finset α\nP : Finpartition A\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : 𝕜\nu v : Finset α\n⊢ ((u, v).1 ∈ P.parts ∧\n (u, v).2 ∈ P.parts ∧\n (u, v).1 ≠ (u, v).2 ∧\n match (u, v) with\n... | and_assoc | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SimpleGraph.Regularity.Equitabilise | {
"line": 115,
"column": 4
} | {
"line": 115,
"column": 85
} | {
"line": 116,
"column": 4
} | [
{
"pp": "case neg.refine_1\nα : Type u_1\ninst✝ : DecidableEq α\nm : ℕ\nm_pos : m > 0\ns : Finset α\nih :\n ∀ t ⊂ s,\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = #t →\n ∃ Q,\n (∀ x ∈ Q.parts, #x = m ∨ #x = m + 1) ∧\n (∀ x ∈ P.parts, #(x \\ {y ∈ Q.parts | y ⊆ x}... | [
"case neg.refine_1\nα : Type u_1\ninst✝ : DecidableEq α\nm : ℕ\nm_pos : m > 0\ns : Finset α\nih :\n ∀ t ⊂ s,\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = #t →\n ∃ Q,\n (∀ x ∈ Q.parts, #x = m ∨ #x = m + 1) ∧\n (∀ x ∈ P.parts, #(x \\ {y ∈ Q.parts | y ⊆ x}.biUnion id)... | simp only [mem_insert, forall_eq_or_imp, extend_parts, and_iff_left hR₁, htn, hn] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Combinatorics.SimpleGraph.Regularity.Uniform | {
"line": 441,
"column": 4
} | {
"line": 441,
"column": 48
} | {
"line": 443,
"column": 0
} | [
{
"pp": "case neg\nα : Type u_1\n𝕜 : Type u_2\ninst✝³ : Field 𝕜\ninst✝² : LinearOrder 𝕜\ninst✝¹ : DecidableEq α\nA : Finset α\nP : Finpartition A\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : 𝕜\nx y : α\nhx✝ : x ∈ A\nhy✝ : y ∈ A\nh : G.Adj x y\nh' :\n ∀ x_1 ∈ P.parts,\n ∀ x_2 ∈ P.parts, x ∈ x_1 → ... | [] | exact Or.inl ⟨U, V, hU, hV, hUV, h₂, hx, hy⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Combinatorics.SimpleGraph.Regularity.Uniform | {
"line": 441,
"column": 4
} | {
"line": 441,
"column": 48
} | {
"line": 443,
"column": 0
} | [
{
"pp": "case neg\nα : Type u_1\n𝕜 : Type u_2\ninst✝³ : Field 𝕜\ninst✝² : LinearOrder 𝕜\ninst✝¹ : DecidableEq α\nA : Finset α\nP : Finpartition A\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : 𝕜\nx y : α\nhx✝ : x ∈ A\nhy✝ : y ∈ A\nh : G.Adj x y\nh' :\n ∀ x_1 ∈ P.parts,\n ∀ x_2 ∈ P.parts, x ∈ x_1 → ... | [] | exact Or.inl ⟨U, V, hU, hV, hUV, h₂, hx, hy⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Regularity.Uniform | {
"line": 441,
"column": 4
} | {
"line": 441,
"column": 48
} | {
"line": 443,
"column": 0
} | [
{
"pp": "case neg\nα : Type u_1\n𝕜 : Type u_2\ninst✝³ : Field 𝕜\ninst✝² : LinearOrder 𝕜\ninst✝¹ : DecidableEq α\nA : Finset α\nP : Finpartition A\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : 𝕜\nx y : α\nhx✝ : x ∈ A\nhy✝ : y ∈ A\nh : G.Adj x y\nh' :\n ∀ x_1 ∈ P.parts,\n ∀ x_2 ∈ P.parts, x ∈ x_1 → ... | [] | exact Or.inl ⟨U, V, hU, hV, hUV, h₂, hx, hy⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Regularity.Lemma | {
"line": 107,
"column": 10
} | {
"line": 107,
"column": 30
} | {
"line": 107,
"column": 31
} | [
{
"pp": "case refine_1\nα : Type u_1\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : ℝ\nl : ℕ\nhε : 0 < ε\nhl : l ≤ Fintype.card α\nhα : bound ε l ≤ Fintype.card α\nt : ℕ := initialBound ε l\nhtα : t ≤ #univ\ndum : Finpartition univ\nhdum₁ : dum.IsEquipartition\nh... | [
"case refine_1\nα : Type u_1\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : ℝ\nl : ℕ\nhε : 0 < ε\nhl : l ≤ Fintype.card α\nhα : bound ε l ≤ Fintype.card α\nt : ℕ := initialBound ε l\nhtα : t ≤ #univ\ndum : Finpartition univ\nhdum₁ : dum.IsEquipartition\nhdum₂ : #dum.... | iterate_succ_apply', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SimpleGraph.Regularity.Lemma | {
"line": 122,
"column": 4
} | {
"line": 122,
"column": 32
} | {
"line": 123,
"column": 4
} | [
{
"pp": "case inr.inr.zero\nα : Type u_1\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : ℝ\nl : ℕ\nhε : 0 < ε\nhl : l ≤ Fintype.card α\nhα : bound ε l ≤ Fintype.card α\nt : ℕ := initialBound ε l\nhtα : t ≤ #univ\ndum : Finpartition univ\nhdum₁ : dum.IsEquipartitio... | [
"case inr.inr.zero\nα : Type u_1\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : ℝ\nl : ℕ\nhε : 0 < ε\nhl : l ≤ Fintype.card α\nhα : bound ε l ≤ Fintype.card α\nt : ℕ := initialBound ε l\nhtα : t ≤ #univ\ndum : Finpartition univ\nhdum₁ : dum.IsEquipartition\nhdum₂ : #... | rw [Nat.cast_zero, mul_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Combinatorics.SimpleGraph.Regularity.Chunk | {
"line": 147,
"column": 51
} | {
"line": 147,
"column": 65
} | {
"line": 147,
"column": 66
} | [
{
"pp": "α : Type u_1\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\nP : Finpartition univ\nhP : P.IsEquipartition\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : ℝ\nU : Finset α\nhU : U ∈ P.parts\nV : Finset α\nhV : V ∈ P.parts\nhUV : U ≠ V\nhunif : ¬G.IsUniform ε U V\nhPε : 100 ≤ 4 ^ #P.parts * ε ^ 5\nhε₁ :... | [
"α : Type u_1\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\nP : Finpartition univ\nhP : P.IsEquipartition\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : ℝ\nU : Finset α\nhU : U ∈ P.parts\nV : Finset α\nhV : V ∈ P.parts\nhUV : U ≠ V\nhunif : ¬G.IsUniform ε U V\nhPε : 100 ≤ 4 ^ #P.parts * ε ^ 5\nhε₁ : ε ≤ 1\nhP₁ ... | mul_div_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SimpleGraph.Operations | {
"line": 45,
"column": 22
} | {
"line": 45,
"column": 51
} | {
"line": 47,
"column": 0
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\ns t : V\ninst✝ : DecidableEq V\nv w : V\n⊢ (if v = t then if w = t then False else G.Adj s w else if w = t then G.Adj v s else G.Adj v w) →\n if w = t then if v = t then False else G.Adj s v else if v = t then G.Adj w s else G.Adj w v",
"ppTerm": "?m.26",
"as... | [] | split_ifs <;> simp [adj_comm] | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Combinatorics.SimpleGraph.Operations | {
"line": 45,
"column": 22
} | {
"line": 45,
"column": 51
} | {
"line": 47,
"column": 0
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\ns t : V\ninst✝ : DecidableEq V\nv w : V\n⊢ (if v = t then if w = t then False else G.Adj s w else if w = t then G.Adj v s else G.Adj v w) →\n if w = t then if v = t then False else G.Adj s v else if v = t then G.Adj w s else G.Adj w v",
"ppTerm": "?m.26",
"as... | [] | split_ifs <;> simp [adj_comm] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Operations | {
"line": 45,
"column": 22
} | {
"line": 45,
"column": 51
} | {
"line": 47,
"column": 0
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\ns t : V\ninst✝ : DecidableEq V\nv w : V\n⊢ (if v = t then if w = t then False else G.Adj s w else if w = t then G.Adj v s else G.Adj v w) →\n if w = t then if v = t then False else G.Adj s v else if v = t then G.Adj w s else G.Adj w v",
"ppTerm": "?m.26",
"as... | [] | split_ifs <;> simp [adj_comm] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Subgraph | {
"line": 618,
"column": 2
} | {
"line": 622,
"column": 65
} | {
"line": 623,
"column": 2
} | [
{
"pp": "case mp\nV : Type u\nG : SimpleGraph V\nH H' : G.Subgraph\n⊢ Disjoint H.verts H'.verts → Disjoint H H'",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"le_bot_iff",
"SimpleGraph.Subgraph.edge_vert",
"Set.ext",
"Eq.mpr",
"CompleteBooleanAlgebra.toComplet... | [
"case mpr\nV : Type u\nG : SimpleGraph V\nH H' : G.Subgraph\n⊢ Disjoint H H' → Disjoint H.verts H'.verts"
] | · rintro hdisj M' ⟨hsub₀, _⟩ ⟨hsub₁, _⟩
rw [le_bot_iff]
ext
· grind [verts_bot]
· exact ⟨(hdisj hsub₀ hsub₁ <| M'.edge_vert · :), False.elim⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Combinatorics.SimpleGraph.Regularity.Chunk | {
"line": 200,
"column": 4
} | {
"line": 200,
"column": 82
} | {
"line": 201,
"column": 4
} | [
{
"pp": "case h₁\nα : Type u_1\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\nP : Finpartition univ\nhP : P.IsEquipartition\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : ℝ\nU : Finset α\nhU : U ∈ P.parts\n𝒜 : Finset (Finset α)\ns : Finset α\nh𝒜 : 𝒜 ⊆ (chunk hP G ε hU).parts\nhs : s ∈ 𝒜\n⊢ ↑(#𝒜) * ↑m ≤ ... | [
"case h₁\nα : Type u_1\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\nP : Finpartition univ\nhP : P.IsEquipartition\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : ℝ\nU : Finset α\nhU : U ∈ P.parts\n𝒜 : Finset (Finset α)\ns : Finset α\nh𝒜 : 𝒜 ⊆ (chunk hP G ε hU).parts\nhs : s ∈ 𝒜\n⊢ #𝒜 * m ≤ ∑ i ∈ 𝒜, #i"
] | rw [← (ofSubset _ h𝒜 rfl).sum_card_parts, ofSubset_parts, ← cast_mul, cast_le] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Combinatorics.SimpleGraph.Walk.Basic | {
"line": 275,
"column": 2
} | {
"line": 281,
"column": 54
} | {
"line": 283,
"column": 0
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v u' v' : V\np : G.Walk u v\nh : G.Adj u' v'\n⊢ { fst := u', snd := v', adj := h } ∈ p.darts ↔ [u', v'] <:+: p.support",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"outParam",
"HEq.refl",
"SimpleGraph.Adj",
... | [] | refine .trans ⟨fun h ↦ ?_, fun ⟨i, hi, h⟩ ↦ ?_⟩ List.infix_iff_getElem?.symm
· have ⟨i, hi, h⟩ := List.getElem_of_mem h
exact ⟨i, by grind, fun j hj ↦ by grind [fst_darts_getElem, snd_darts_getElem]⟩
· have := h 0
have := h 1
convert! p.darts.getElem_mem (n := i) (by grind)
<;> grind [fst_darts_ge... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Walk.Basic | {
"line": 275,
"column": 2
} | {
"line": 281,
"column": 54
} | {
"line": 283,
"column": 0
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v u' v' : V\np : G.Walk u v\nh : G.Adj u' v'\n⊢ { fst := u', snd := v', adj := h } ∈ p.darts ↔ [u', v'] <:+: p.support",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"outParam",
"HEq.refl",
"SimpleGraph.Adj",
... | [] | refine .trans ⟨fun h ↦ ?_, fun ⟨i, hi, h⟩ ↦ ?_⟩ List.infix_iff_getElem?.symm
· have ⟨i, hi, h⟩ := List.getElem_of_mem h
exact ⟨i, by grind, fun j hj ↦ by grind [fst_darts_getElem, snd_darts_getElem]⟩
· have := h 0
have := h 1
convert! p.darts.getElem_mem (n := i) (by grind)
<;> grind [fst_darts_ge... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Walk.Subwalks | {
"line": 48,
"column": 2
} | {
"line": 50,
"column": 6
} | {
"line": 52,
"column": 0
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\nu v u' v' w : V\np : G.Walk u v\nq : G.Walk u' v'\nhpq : p.IsSubwalk q\nh : G.Adj w u'\n⊢ p.IsSubwalk (cons h q)",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"SimpleGraph.Walk",
"Exists",
"SimpleGraph.Walk.cons",
"Exists.... | [] | obtain ⟨r1, r2, rfl⟩ := hpq
use r1.cons h, r2
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Walk.Subwalks | {
"line": 48,
"column": 2
} | {
"line": 50,
"column": 6
} | {
"line": 52,
"column": 0
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\nu v u' v' w : V\np : G.Walk u v\nq : G.Walk u' v'\nhpq : p.IsSubwalk q\nh : G.Adj w u'\n⊢ p.IsSubwalk (cons h q)",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"SimpleGraph.Walk",
"Exists",
"SimpleGraph.Walk.cons",
"Exists.... | [] | obtain ⟨r1, r2, rfl⟩ := hpq
use r1.cons h, r2
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Walk.Operations | {
"line": 254,
"column": 4
} | {
"line": 261,
"column": 12
} | {
"line": 263,
"column": 0
} | [
{
"pp": "case cons\nV : Type u\nG : SimpleGraph V\nu v : V\ni : ℕ\nu✝ v✝ w✝ : V\nh : G.Adj u✝ v✝\np : G.Walk v✝ w✝\nih : p.reverse.getVert i = p.getVert (p.length - i)\n⊢ (cons h p).reverse.getVert i = (cons h p).getVert ((cons h p).length - i)",
"ppTerm": "?cons",
"assigned": true,
"usedConstants":... | [] | simp only [reverse_cons, getVert_append, length_reverse, ih, length_cons]
split_ifs
next hi => simp [Nat.succ_sub hi.le]
next hi =>
obtain rfl | hi' := eq_or_gt_of_not_lt hi
· simp
· rw [Nat.eq_add_of_sub_eq (Nat.sub_pos_of_lt hi') rfl, Nat.sub_eq_zero_of_le hi']
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Walk.Operations | {
"line": 254,
"column": 4
} | {
"line": 261,
"column": 12
} | {
"line": 263,
"column": 0
} | [
{
"pp": "case cons\nV : Type u\nG : SimpleGraph V\nu v : V\ni : ℕ\nu✝ v✝ w✝ : V\nh : G.Adj u✝ v✝\np : G.Walk v✝ w✝\nih : p.reverse.getVert i = p.getVert (p.length - i)\n⊢ (cons h p).reverse.getVert i = (cons h p).getVert ((cons h p).length - i)",
"ppTerm": "?cons",
"assigned": true,
"usedConstants":... | [] | simp only [reverse_cons, getVert_append, length_reverse, ih, length_cons]
split_ifs
next hi => simp [Nat.succ_sub hi.le]
next hi =>
obtain rfl | hi' := eq_or_gt_of_not_lt hi
· simp
· rw [Nat.eq_add_of_sub_eq (Nat.sub_pos_of_lt hi') rfl, Nat.sub_eq_zero_of_le hi']
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
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