module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.CategoryTheory.Monoidal.DayConvolution.DayFunctor | {
"line": 199,
"column": 2
} | {
"line": 199,
"column": 40
} | [
{
"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\nhasDayConvolution : ∀ (F G : C ⥤ V), (tensor C).HasPointwiseLeftKanExtension (F ⊠ G)\nhasDayConvolutionUnit : (Functor.fromPUnit (𝟙_ C)).HasPointwiseLeftKanEx... | simp [η, isoPointwiseLeftKanExtension] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Monoidal.Bimod | {
"line": 868,
"column": 2
} | {
"line": 868,
"column": 80
} | [
{
"pp": "case h.h\nC : 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' : Bim... | slice_rhs 2 3 => rw [← comp_whiskerRight, ι_colimMap, parallelPairHom_app_one] | Mathlib.Tactic.Slice._aux_Mathlib_Tactic_CategoryTheory_Slice___macroRules_Mathlib_Tactic_Slice_sliceRHS_1 | Mathlib.Tactic.Slice.sliceRHS |
Mathlib.CategoryTheory.Monoidal.Hopf_ | {
"line": 292,
"column": 2
} | {
"line": 293,
"column": 27
} | [
{
"pp": "C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : MonoidalCategory C\ninst✝¹ : BraidedCategory C\nA : C\ninst✝ : HopfObj A\n⊢ (Δ ⊗ₘ Δ) ≫\n (α_ A A (A ⊗ A)).hom ≫\n A ◁ (α_ A A A).inv ≫\n A ◁ (β_ A A).hom ▷ A ≫\n (α_ A (A ⊗ A) A).inv ≫ (α_ A A A).inv ▷ A ≫ μ ▷ A ▷ A ≫ (... | slice_lhs 9 10 =>
rw [← whisker_exchange] | Mathlib.Tactic.Slice._aux_Mathlib_Tactic_CategoryTheory_Slice___macroRules_Mathlib_Tactic_Slice_sliceLHS_1 | Mathlib.Tactic.Slice.sliceLHS |
Mathlib.CategoryTheory.Monoidal.Hopf_ | {
"line": 341,
"column": 2
} | {
"line": 342,
"column": 27
} | [
{
"pp": "C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : MonoidalCategory C\ninst✝¹ : BraidedCategory C\nA : C\ninst✝ : HopfObj A\n⊢ (Δ ⊗ₘ Δ) ≫\n (α_ A A (A ⊗ A)).hom ≫\n A ◁ (α_ A A A).inv ≫\n A ◁ (β_ A A).hom ▷ A ≫\n (α_ A (A ⊗ A) A).inv ≫\n (α_ A A A).inv ▷ A ... | slice_lhs 9 10 =>
rw [← whisker_exchange] | Mathlib.Tactic.Slice._aux_Mathlib_Tactic_CategoryTheory_Slice___macroRules_Mathlib_Tactic_Slice_sliceLHS_1 | Mathlib.Tactic.Slice.sliceLHS |
Mathlib.CategoryTheory.Monoidal.Internal.Module | {
"line": 153,
"column": 4
} | {
"line": 153,
"column": 52
} | [
{
"pp": "case hf\nR : Type u\ninst✝ : CommRing R\nA : AlgCat R\nx y z : ↑(of R ↑A)\n⊢ (Hom.hom (↟(mul' R ↑A)))\n ((Hom.hom (MonoidalCategoryStruct.whiskerRight (↟(mul' R ↑A)) (of R ↑A)))\n (((TensorProduct.mk R ↑(MonoidalCategoryStruct.tensorObj (of R ↑A) (of R ↑A)) ↑A) (x ⊗ₜ[R] y)) z)) =\n (Hom.... | erw [LinearMap.mul'_apply, LinearMap.mul'_apply] | Lean.Parser.Tactic._aux_Init_Meta___macroRules_Lean_Parser_Tactic_tacticErw____1 | Lean.Parser.Tactic.tacticErw___ |
Mathlib.CategoryTheory.Monoidal.Opposite.Mon | {
"line": 70,
"column": 4
} | {
"line": 70,
"column": 58
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : MonoidalCategory C\nM : Cᴹᵒᵖ\ninst✝ : MonObj M\n⊢ MonObj.one.unmop ▷ M.unmop ≫ MonObj.mul.unmop = (λ_ M.unmop).hom",
"usedConstants": [
"CategoryTheory.Functor.FullyFaithful.map_injective",
"CategoryTheory.MonoidalOpposite.unmop",
... | apply mopEquiv C |>.fullyFaithfulFunctor.map_injective | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.CategoryTheory.Monoidal.Opposite.Mon | {
"line": 67,
"column": 4
} | {
"line": 67,
"column": 58
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : MonoidalCategory C\nM : Cᴹᵒᵖ\ninst✝ : MonObj M\n⊢ M.unmop ◁ MonObj.one.unmop ≫ MonObj.mul.unmop = (ρ_ M.unmop).hom",
"usedConstants": [
"CategoryTheory.Functor.FullyFaithful.map_injective",
"CategoryTheory.MonoidalCategoryStruct.whi... | apply mopEquiv C |>.fullyFaithfulFunctor.map_injective | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.CategoryTheory.Monoidal.Opposite.Mon | {
"line": 73,
"column": 4
} | {
"line": 73,
"column": 58
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : MonoidalCategory C\nM : Cᴹᵒᵖ\ninst✝ : MonObj M\n⊢ MonObj.mul.unmop ▷ M.unmop ≫ MonObj.mul.unmop =\n (α_ M.unmop M.unmop M.unmop).hom ≫ M.unmop ◁ MonObj.mul.unmop ≫ MonObj.mul.unmop",
"usedConstants": [
"CategoryTheory.Functor.FullyFait... | apply mopEquiv C |>.fullyFaithfulFunctor.map_injective | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.CategoryTheory.Monoidal.Opposite.Mon | {
"line": 85,
"column": 4
} | {
"line": 85,
"column": 58
} | [
{
"pp": "C : Type u_1\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : MonoidalCategory C\nM : Cᴹᵒᵖ\ninst✝² : MonObj M\nN : Cᴹᵒᵖ\ninst✝¹ : MonObj N\nf : M ⟶ N\ninst✝ : IsMonHom f\n⊢ MonObj.one ≫ f.unmop = MonObj.one",
"usedConstants": [
"CategoryTheory.Functor.FullyFaithful.map_injective",
"CategoryThe... | apply mopEquiv C |>.fullyFaithfulFunctor.map_injective | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.CategoryTheory.Monoidal.Opposite.Mon | {
"line": 82,
"column": 4
} | {
"line": 82,
"column": 58
} | [
{
"pp": "C : Type u_1\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : MonoidalCategory C\nM : Cᴹᵒᵖ\ninst✝² : MonObj M\nN : Cᴹᵒᵖ\ninst✝¹ : MonObj N\nf : M ⟶ N\ninst✝ : IsMonHom f\n⊢ MonObj.mul ≫ f.unmop = (f.unmop ⊗ₘ f.unmop) ≫ MonObj.mul",
"usedConstants": [
"CategoryTheory.Functor.FullyFaithful.map_injecti... | apply mopEquiv C |>.fullyFaithfulFunctor.map_injective | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.CategoryTheory.ObjectProperty.Ind | {
"line": 84,
"column": 4
} | {
"line": 84,
"column": 42
} | [
{
"pp": "case refine_1\nC : Type u\ninst✝¹ : Category.{v, u} C\nP : ObjectProperty C\nH : P ≤ isFinitelyPresentable C\ninst✝ : IsFinitelyAccessibleCategory C\nX : C\nx✝ : P.ind X\nZ : C\ng : Z ⟶ X\nhZ : IsFinitelyPresentable Z\nJ : Type w\nw✝¹ : SmallCategory J\nw✝ : IsFiltered J\npres : ColimitPresentation J X... | exact ⟨_, u, pres.ι.app j, hcomp, h j⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.ObjectProperty.Ind | {
"line": 98,
"column": 4
} | {
"line": 100,
"column": 39
} | [
{
"pp": "case refine_2\nC : Type u\ninst✝¹ : Category.{v, u} C\nP : ObjectProperty C\nH✝ : P ≤ isFinitelyPresentable C\ninst✝ : IsFinitelyAccessibleCategory C\nX : C\nhfac : ∀ {Z : C} (g : Z ⟶ X) [IsFinitelyPresentable Z], ∃ W u v, u ≫ v = g ∧ P W\nincl : P.FullSubcategory ⥤ (isFinitelyPresentable C).FullSubcat... | obtain ⟨hc⟩ : P.ι.isDenseAt X :=
Functor.IsDenseAt.of_final (F := (isFinitelyPresentable.{w} C).ι) incl
(Functor.IsDense.isDenseAt _ _) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.CategoryTheory.Presentable.ColimitPresentation | {
"line": 144,
"column": 8
} | {
"line": 145,
"column": 30
} | [
{
"pp": "C : Type u\ninst✝⁶ : Category.{v, u} C\nJ✝ : Type u_1\nI✝ : J✝ → Type u_2\ninst✝⁵ : Category.{v_1, u_1} J✝\ninst✝⁴ : (j : J✝) → Category.{?u.49630, u_2} (I✝ j)\nD✝ : J✝ ⥤ C\nP✝¹ : (j : J✝) → ColimitPresentation (I✝ j) (D✝.obj j)\nJ : Type w\nI : J → Type w\ninst✝³ : SmallCategory J\ninst✝² : (j : J) → ... | simp only [Functor.const_obj_obj, Functor.const_obj_map, Category.comp_id,
(Q j).isColimit.fac] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.MorphismProperty.Ind | {
"line": 192,
"column": 32
} | {
"line": 221,
"column": 62
} | [
{
"pp": "C : Type u\ninst✝⁵ : Category.{v, u} C\nP : MorphismProperty C\ninst✝⁴ : ∀ (X : C), IsFinitelyAccessibleCategory (Under X)\ninst✝³ : HasPushouts C\ninst✝² : P.IsStableUnderComposition\ninst✝¹ : P.IsStableUnderCobaseChange\ninst✝ : P.PreIndSpreads\nH : P ≤ isFinitelyPresentable C\nX Y Z : C\nf : X ⟶ Y\n... | by
rw [ind_iff_exists H]
intro T p u hp hpu
obtain ⟨J₁, _, _, D₁, s₁, t₁, ht₁, h₁⟩ := hf
obtain ⟨J₂, _, _, D₂, s₂, t₂, ht₂, h₂⟩ := hg
have : IsFinitelyPresentable (CategoryTheory.Under.mk p) := hp
obtain ⟨j₂, q, hcomp, hu⟩ := IsFinitelyPresentable.exists_hom_of_isColimit_under
ht₂ p ((Fu... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Monoidal.Internal.FunctorCategory | {
"line": 67,
"column": 35
} | {
"line": 67,
"column": 53
} | [
{
"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✝ : MonObj A\nX✝ Y✝ : C\nf : X✝ ⟶ Y✝\n⊢ μ.app X✝ ≫ A.map f = (A.map f ⊗ₘ A.map f) ≫ μ.app Y✝",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Functor",
"... | ← μ[A].naturality, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Preadditive.Schur | {
"line": 179,
"column": 4
} | {
"line": 180,
"column": 92
} | [
{
"pp": "case mpr\nC : 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✝ : Simpl... | have zero_lt : 0 < finrank 𝕜 (X ⟶ Y) :=
finrank_pos_iff_exists_ne_zero.mpr ⟨f.hom, (isIso_iff_nonzero f.hom).mp inferInstance⟩ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.CategoryTheory.Presentable.Directed | {
"line": 158,
"column": 4
} | {
"line": 158,
"column": 33
} | [
{
"pp": "case hP\nJ : Type w\ninst✝ : SmallCategory J\nκ : Cardinal.{w}\nx✝¹ x✝ : DiagramWithUniqueTerminal J κ\nh₁ : x✝¹.W ≤ x✝.W ∧ x✝¹.P ≤ x✝.P\nh₂ : x✝.W ≤ x✝¹.W ∧ x✝.P ≤ x✝¹.P\n⊢ x✝¹.P = x✝.P",
"usedConstants": [
"CategoryTheory.MorphismProperty",
"ChainCompletePartialOrder.instOfCompleteLat... | · exact le_antisymm h₁.2 h₂.2 | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Presentable.OrthogonalReflection | {
"line": 466,
"column": 2
} | {
"line": 466,
"column": 39
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\nW : MorphismProperty C\nκ : Cardinal.{w}\ninst✝³ : Fact κ.IsRegular\ninst✝² : IsSmall.{w, v, u} W\ninst✝¹ : LocallySmall.{w, v, u} C\nhW : ∀ ⦃X Y : C⦄ (f : X ⟶ Y), W f → IsCardinalPresentable X κ ∧ IsCardinalPresentable Y κ\ninst✝ : HasColimitsOfSize.{w, w, v, u}... | have := D₁.hasCoproductsOfShape.{w} W | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.CategoryTheory.Sites.CartesianMonoidal | {
"line": 38,
"column": 2
} | {
"line": 39,
"column": 46
} | [
{
"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⊢ IsLimit\n ((Cone.postcompose (pairComp X Y (sheafToPresheaf J A)).inv).obj (BinaryFan.mk (fst X.obj Y.obj) (snd X.obj Y.obj)))",
... | exact (IsLimit.postcomposeInvEquiv _ _).invFun
(tensorProductIsBinaryProduct X.obj Y.obj) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Presentable.Directed | {
"line": 414,
"column": 10
} | {
"line": 414,
"column": 25
} | [
{
"pp": "J : 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)\nι : Type w\nD : ι → DiagramWithUniqueTerminal J κ\nhι : HasCardinalLT ι κ\nm : J\nu : (i : ι) → (D i).top ⟶ m\nhm₀ : ∀ (i : ι), IsEmpty (m ⟶ (D i).t... | have := hj.prop | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.CategoryTheory.Sites.Coherent.ExtensiveSheaves | {
"line": 68,
"column": 22
} | {
"line": 73,
"column": 59
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : FinitaryPreExtensive C\nW : C\n⊢ IsSheaf (extensiveTopology C) (yoneda.obj W)",
"usedConstants": [
"CategoryTheory.Presieve.IsSheaf",
"Eq.mpr",
"CategoryTheory.Functor",
"CategoryTheory.Limits.instPreservesFiniteProductsO... | by
rw [extensiveTopology, isSheaf_coverage]
intro X R ⟨Y, α, Z, π, hR, hi⟩
have : IsIso (Sigma.desc (Cofan.inj (Cofan.mk X π))) := hi
have : R.Extensive := ⟨Y, α, Z, π, hR, ⟨Cofan.isColimitOfIsIsoSigmaDesc (Cofan.mk X π)⟩⟩
exact isSheafFor_extensive_of_preservesFiniteProducts _ _ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Sites.Coherent.Comparison | {
"line": 67,
"column": 2
} | {
"line": 101,
"column": 91
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Preregular C\ninst✝ : FinitaryPreExtensive C\n⊢ (extensiveCoverage C ⊔ regularCoverage C).toGrothendieck = coherentTopology C",
"usedConstants": [
"Set.ext",
"CategoryTheory.Coverage",
"Eq.mpr",
"CategoryTheory.Category.... | ext B S
refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩
· induction h with
| of Y T hT =>
apply Coverage.Saturate.of
simp only [Coverage.sup_covering, Set.mem_union] at hT
exact Or.elim hT
(fun ⟨α, x, X, π, ⟨h, _⟩⟩ ↦ ⟨α, x, X, π, ⟨h, inferInstance⟩⟩)
(fun ⟨Z, f, ⟨h, _⟩⟩ ↦ ⟨Unit, inferInstance,... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Sites.Coherent.Comparison | {
"line": 67,
"column": 2
} | {
"line": 101,
"column": 91
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Preregular C\ninst✝ : FinitaryPreExtensive C\n⊢ (extensiveCoverage C ⊔ regularCoverage C).toGrothendieck = coherentTopology C",
"usedConstants": [
"Set.ext",
"CategoryTheory.Coverage",
"Eq.mpr",
"CategoryTheory.Category.... | ext B S
refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩
· induction h with
| of Y T hT =>
apply Coverage.Saturate.of
simp only [Coverage.sup_covering, Set.mem_union] at hT
exact Or.elim hT
(fun ⟨α, x, X, π, ⟨h, _⟩⟩ ↦ ⟨α, x, X, π, ⟨h, inferInstance⟩⟩)
(fun ⟨Z, f, ⟨h, _⟩⟩ ↦ ⟨Unit, inferInstance,... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Sites.Coherent.LocallySurjective | {
"line": 79,
"column": 2
} | {
"line": 79,
"column": 11
} | [
{
"pp": "case a\nC : Type u_1\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : FinitaryPreExtensive C\nF G : Cᵒᵖ ⥤ Type w\nf : F ⟶ G\ninst✝¹ : PreservesFiniteProducts F\ninst✝ : PreservesFiniteProducts G\nX : C\nx : G.obj (op X)\nα : Type\nw✝ : Finite α\nY : α → C\nπ : (a : α) → Y a ⟶ X\nh : Nonempty (IsColimit (Cofan... | intro ⟨a⟩ | Lean.Elab.Tactic.evalIntro | null |
Mathlib.CategoryTheory.Sites.Coherent.LocallySurjective | {
"line": 79,
"column": 2
} | {
"line": 79,
"column": 11
} | [
{
"pp": "case a\nC : Type u_1\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : FinitaryPreExtensive C\nF G : Cᵒᵖ ⥤ Type w\nf : F ⟶ G\ninst✝¹ : PreservesFiniteProducts F\ninst✝ : PreservesFiniteProducts G\nX : C\nx : G.obj (op X)\nα : Type\nw✝ : Finite α\nY : α → C\nπ : (a : α) → Y a ⟶ X\nh : Nonempty (IsColimit (Cofan... | intro ⟨a⟩ | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.CategoryTheory.Sites.NonabelianCohomology.H1 | {
"line": 117,
"column": 15
} | {
"line": 117,
"column": 36
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nG : Cᵒᵖ ⥤ GrpCat\nI : Type w'\nU : I → C\nx✝ : OneCochain G U\n⊢ x✝ * 1 = x✝",
"usedConstants": [
"GrpCat",
"Opposite",
"HMul.hMul",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"GrpCat.str",
"Monoid.toMulOn... | by ext; apply mul_one | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Sites.Point.Over | {
"line": 80,
"column": 4
} | {
"line": 80,
"column": 31
} | [
{
"pp": "C : Type u\ninst✝⁵ : Category.{v, u} C\nJ : GrothendieckTopology C\ninst✝⁴ : LocallySmall.{w, v, u} C\nP : ObjectProperty J.Point\ninst✝³ : ObjectProperty.Small.{w, max u w, max (max u v) (w + 1)} P\ninst✝² : J.WEqualsLocallyBijective (Type w)\ninst✝¹ : HasSheafify J (Type w)\nhP : P.IsConservativeFami... | rw [Subtype.ext_iff] at hz₂ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Sites.Descent.DescentData | {
"line": 198,
"column": 21
} | {
"line": 198,
"column": 28
} | [
{
"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\nD : F... | ← hf₁', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Sites.Types | {
"line": 59,
"column": 6
} | {
"line": 60,
"column": 96
} | [
{
"pp": "α β : Type u\nx✝ : Sieve β\nhs : x✝ ∈ typesGrothendieckTopology β\nx : FamilyOfElements (yoneda.obj α) x✝.arrows\nhx : x.Compatible\nγ : Type u\nf : γ ⟶ β\nh : x✝.arrows f\nz : Opposite.unop (Opposite.op γ)\n⊢ (ConcreteCategory.hom\n ((ConcreteCategory.hom ((yoneda.obj α).map f.op)) (↾fun y ↦ (T... | convert!
ConcreteCategory.congr_hom (hx (𝟙 _) (↾fun _ => z) (hs <| f z) h rfl) PUnit.unit using 1 | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.CategoryTheory.Sites.Types | {
"line": 59,
"column": 6
} | {
"line": 60,
"column": 96
} | [
{
"pp": "α β : Type u\nx✝ : Sieve β\nhs : x✝ ∈ typesGrothendieckTopology β\nx : FamilyOfElements (yoneda.obj α) x✝.arrows\nhx : x.Compatible\nγ : Type u\nf : γ ⟶ β\nh : x✝.arrows f\nz : Opposite.unop (Opposite.op γ)\n⊢ (ConcreteCategory.hom\n ((ConcreteCategory.hom ((yoneda.obj α).map f.op)) (↾fun y ↦ (T... | convert!
ConcreteCategory.congr_hom (hx (𝟙 _) (↾fun _ => z) (hs <| f z) h rfl) PUnit.unit using 1 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Sites.Types | {
"line": 59,
"column": 6
} | {
"line": 60,
"column": 96
} | [
{
"pp": "α β : Type u\nx✝ : Sieve β\nhs : x✝ ∈ typesGrothendieckTopology β\nx : FamilyOfElements (yoneda.obj α) x✝.arrows\nhx : x.Compatible\nγ : Type u\nf : γ ⟶ β\nh : x✝.arrows f\nz : Opposite.unop (Opposite.op γ)\n⊢ (ConcreteCategory.hom\n ((ConcreteCategory.hom ((yoneda.obj α).map f.op)) (↾fun y ↦ (T... | convert!
ConcreteCategory.congr_hom (hx (𝟙 _) (↾fun _ => z) (hs <| f z) h rfl) PUnit.unit using 1 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Subfunctor.Finite | {
"line": 153,
"column": 76
} | {
"line": 154,
"column": 53
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nF : Cᵒᵖ ⥤ Type w\nι : Type w'\nX : ι → Cᵒᵖ\nx : (i : ι) → F.obj (X i)\nh : PresheafIsGeneratedBy F x\nF' : Cᵒᵖ ⥤ Type w\nf : F ⟶ F'\n⊢ (Subfunctor.range f).IsGeneratedBy fun i ↦ (ConcreteCategory.hom (f.app (X i))) (x i)",
"usedConstants": [
"Eq.mpr",
... | by
simpa only [← Subfunctor.image_top] using h.image f | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Sites.RegularEpi | {
"line": 97,
"column": 8
} | {
"line": 97,
"column": 81
} | [
{
"pp": "case e_a\nC : Type u_1\nD : Type u_2\ninst✝⁶ : Category.{u_3, u_1} C\ninst✝⁵ : Category.{u_4, u_2} D\nJ : GrothendieckTopology C\ninst✝⁴ : HasPullbacks D\ninst✝³ : HasPushouts D\ninst✝² : IsRegularEpiCategory D\nh : ∀ {F G : Sheaf J D} (f : F ⟶ G) [Epi f], ∃ I p i, Epi p ∧ Mono i ∧ p ≫ i = f.hom\ninst✝... | simp [hpi, ← ObjectProperty.FullSubcategory.comp_hom, pullback.condition] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Subobject.Classifier.Defs | {
"line": 601,
"column": 2
} | {
"line": 603,
"column": 47
} | [
{
"pp": "case mpr\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasTerminal C\ninst✝ : HasPullbacks C\nh : (Subobject.presheaf C).IsRepresentable\n⊢ HasSubobjectClassifier C",
"usedConstants": [
"Exists",
"Subobject.presheaf",
"CategoryTheory.Functor.RepresentableBy",
"Nonempty.i... | · obtain ⟨Ω, ⟨h⟩⟩ := h
constructor; constructor
exact SubobjectRepresentableBy.classifier h | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Topos.Sheaf | {
"line": 109,
"column": 2
} | {
"line": 109,
"column": 66
} | [
{
"pp": "case w.h.h.toFun.h\nC : Type u\ninst✝ : Category.{v, u} C\nF G : Cᵒᵖ ⥤ Type (max u v)\nm : F ⟶ G\nχ' : G ⟶ Functor.sieves C\nX : Cᵒᵖ\nx : G.obj X\nh₁ : ∀ (x : Cᵒᵖ), m.app x ≫ χ'.app x = Types.isTerminalPUnit.from (F.obj x) ≫ ↾fun x_1 ↦ ⊤\nh₂ : ∀ (x : Cᵒᵖ) (x₁ y₁ : F.obj x), (ConcreteCategory.hom (m.app... | rw [← dsimp% this, ← dsimp% NatTrans.naturality_apply χ' f.op x] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Sites.SheafCohomology.MayerVietoris | {
"line": 81,
"column": 2
} | {
"line": 81,
"column": 20
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\nJ : GrothendieckTopology C\ninst✝² : HasWeakSheafify J (Type v)\ninst✝¹ : HasSheafify J AddCommGrpCat\ninst✝ : HasExt (Sheaf J AddCommGrpCat)\nS : J.MayerVietorisSquare\nF : Sheaf J AddCommGrpCat\nn : ℕ\ny₁ : ↑(F.H' n S.X₂)\ny₂ : ↑(F.H' n S.X₃)\n⊢ (ConcreteCatego... | dsimp [fromBiprod] | Lean.Elab.Tactic.evalDSimp | Lean.Parser.Tactic.dsimp |
Mathlib.CategoryTheory.Triangulated.Generators | {
"line": 74,
"column": 2
} | {
"line": 75,
"column": 68
} | [
{
"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\nn : ℕ\n⊢ P.triangEnvelopeIter (n + 1) =\n ((P.shiftClosure ℤ).binaryProductsClosure... | rw [triangEnvelopeIter, extensionProductIter_succ,
← retractClosure_extensionProduct_retractClosure_retractClosure] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Triangulated.LocalizingSubcategory | {
"line": 250,
"column": 6
} | {
"line": 250,
"column": 51
} | [
{
"pp": "C : Type u_1\nD : Type u_2\nD₁ : Type u_3\nD₂ : Type u_4\ninst✝¹⁵ : Category.{v_1, u_1} C\ninst✝¹⁴ : Category.{v_2, u_2} D\ninst✝¹³ : Category.{v_3, u_3} D₁\ninst✝¹² : Category.{v_4, u_4} D₂\nA B : ObjectProperty C\ninst✝¹¹ : HasZeroObject C\ninst✝¹⁰ : HasShift C ℤ\ninst✝⁹ : Preadditive C\ninst✝⁸ : ∀ (... | MorphismProperty.map_eq_iff_postcomp L₂ B.trW | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Triangulated.TStructure.AbelianSubcategory | {
"line": 276,
"column": 38
} | {
"line": 276,
"column": 47
} | [
{
"pp": "C : Type u_1\nA : Type u_2\ninst✝¹¹ : Category.{v_1, u_1} C\ninst✝¹⁰ : HasZeroObject C\ninst✝⁹ : Preadditive C\ninst✝⁸ : HasShift C ℤ\ninst✝⁷ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝⁶ : Pretriangulated C\ninst✝⁵ : Category.{v_2, u_2} A\nι : A ⥤ C\nhι : ∀ ⦃X Y : A⦄ ⦃n : ℤ⦄ (f : ι.obj X ⟶ (shiftFu... | simp [hA] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Triangulated.TStructure.AbelianSubcategory | {
"line": 305,
"column": 41
} | {
"line": 305,
"column": 50
} | [
{
"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... | simp [hA] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE | {
"line": 636,
"column": 4
} | {
"line": 637,
"column": 48
} | [
{
"pp": "C : Type u\ninst✝⁵ : Category.{v, u} 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\nT : Triangle C\nhT : T ∈ distinguishedTriangles\ni₁ : ℤ\nhi₁ : t.IsGE T.obj₁ i₁\ni₃ : ℤ\nhi₃ : t... | exact ⟨min i₁ i₃, t.isGE₂ T hT _ (t.isGE_of_ge _ _ _ (min_le_left i₁ i₃))
(t.isGE_of_ge _ _ _ (min_le_right i₁ i₃))⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE | {
"line": 687,
"column": 2
} | {
"line": 687,
"column": 31
} | [
{
"pp": "case pos\nC : Type u\ninst✝⁶ : Category.{v, u} 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\na b : ℤ\ninst✝ : t.IsGE X a\nh : a ≤ b\n⊢ t.IsGE ((t.truncGE b).obj X) a",
... | · exact t.isGE_truncGE_obj .. | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Combinatorics.Additive.AP.Three.Defs | {
"line": 237,
"column": 2
} | {
"line": 237,
"column": 80
} | [
{
"pp": "s : Set ℕ\n⊢ ThreeAPFree s ↔ ∀ ⦃a : ℕ⦄, a ∈ s → ∀ ⦃b : ℕ⦄, b ∈ s → ∀ ⦃c : ℕ⦄, c ∈ s → a + c = b + b → a = c",
"usedConstants": [
"AddMonoid.toAddZeroClass",
"forall₃_congr",
"Nat.instAddMonoid",
"Membership.mem",
"AddZeroClass.toAddZero",
"instHAdd",
"HAdd.... | refine forall₄_congr fun a _ha b hb => forall₃_congr fun c hc habc => ⟨?_, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Combinatorics.Additive.AP.Three.Defs | {
"line": 382,
"column": 83
} | {
"line": 382,
"column": 100
} | [
{
"pp": "α : Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : CancelCommMonoid α\ns : Finset α\na : α\nu : Finset α\nhus : u ⊆ s\nhcard : #u = mulRothNumber s\nhu : ThreeGPFree ↑u\n⊢ ThreeGPFree (⇑(mulLeftEmbedding a) '' ↑u)",
"usedConstants": [
"CancelCommMonoid.toCommMonoid",
"Finset",
"CancelC... | exact hu.smul_set | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE | {
"line": 809,
"column": 68
} | {
"line": 812,
"column": 61
} | [
{
"pp": "C : Type u\ninst✝⁶ : Category.{v, u} 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 b : ℤ\nX : C\n⊢ (t.truncLTι b).app ((t.truncGE a).obj ((t.truncLT b... | by
rw [← cancel_epi (inv ((t.truncLTι b).app ((t.truncGE a).obj ((t.truncLT b).obj X)))),
IsIso.inv_hom_id_assoc]
exact t.truncGELTToLTGE_app_pentagon_uniqueness _ (by simp) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.Additive.RuzsaCovering | {
"line": 33,
"column": 41
} | {
"line": 54,
"column": 80
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\nK : ℝ\ninst✝ : DecidableEq G\nA B : Finset G\nhB : B.Nonempty\nhK : ↑(#(A * B)) ≤ K * ↑(#B)\n⊢ ∃ F ⊆ A, ↑(#F) ≤ K ∧ A ⊆ F * (B / B)",
"usedConstants": [
"Mathlib.Tactic.Push.not_forall_eq",
"Iff.mpr",
"Set.instSProd",
"_private.Mathlib.Combina... | by
haveI : ∀ F, Decidable ((F : Set G).PairwiseDisjoint (· • B)) := fun F ↦ Classical.dec _
set C := {F ∈ A.powerset | (SetLike.coe F).PairwiseDisjoint (· • B)}
obtain ⟨F, hFmax⟩ := C.exists_maximal <| filter_nonempty_iff.2
⟨∅, empty_mem_powerset _, by simp [coe_empty]⟩
simp only [C, mem_filter, mem_powerse... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Triangulated.TStructure.ETrunc | {
"line": 274,
"column": 35
} | {
"line": 274,
"column": 49
} | [
{
"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 ≤ ... | simpa using hj | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.CategoryTheory.Triangulated.TStructure.ETrunc | {
"line": 274,
"column": 35
} | {
"line": 274,
"column": 49
} | [
{
"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 ≤ ... | simpa using hj | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Triangulated.TStructure.ETrunc | {
"line": 274,
"column": 35
} | {
"line": 274,
"column": 49
} | [
{
"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 ≤ ... | simpa using hj | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Additive.CauchyDavenport | {
"line": 136,
"column": 2
} | {
"line": 136,
"column": 79
} | [
{
"pp": "case inr\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\n⊢ minOrder α ≤ ↑(#(s * t)) ∨ #s + #... | obtain ⟨a, rfl⟩ | ⟨a, ha, b, hb, hab⟩ := hs.exists_eq_singleton_or_nontrivial | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Combinatorics.Additive.PluenneckeRuzsa | {
"line": 149,
"column": 8
} | {
"line": 149,
"column": 56
} | [
{
"pp": "G : Type u_1\ninst✝¹ : DecidableEq G\ninst✝ : Group G\nA B : Finset G\nhA : ∀ A' ⊆ A, #(A * B) * #A' ≤ #(A' * B) * #A\nx : G\nC : Finset G\na✝ : x ∉ C\nih : #(C * A * B) * #A ≤ #(A * B) * #(C * A)\nA' : Finset G := A ∩ ({x}⁻¹ * C * A)\nhA' : A' = A ∩ ({x}⁻¹ * C * A)\nC' : Finset G := insert x C\nhC' : ... | gcongr ?_ + _ - ?_; exact hA _ inter_subset_left | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Additive.PluenneckeRuzsa | {
"line": 149,
"column": 8
} | {
"line": 149,
"column": 56
} | [
{
"pp": "G : Type u_1\ninst✝¹ : DecidableEq G\ninst✝ : Group G\nA B : Finset G\nhA : ∀ A' ⊆ A, #(A * B) * #A' ≤ #(A' * B) * #A\nx : G\nC : Finset G\na✝ : x ∉ C\nih : #(C * A * B) * #A ≤ #(A * B) * #(C * A)\nA' : Finset G := A ∩ ({x}⁻¹ * C * A)\nhA' : A' = A ∩ ({x}⁻¹ * C * A)\nC' : Finset G := insert x C\nhC' : ... | gcongr ?_ + _ - ?_; exact hA _ inter_subset_left | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Dart | {
"line": 38,
"column": 2
} | {
"line": 38,
"column": 10
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\nd₁ d₂ : G.Dart\n⊢ d₁ = d₂ ↔ d₁.toProd = d₂.toProd",
"usedConstants": [
"SimpleGraph.Dart",
"Eq.refl"
]
}
] | cases d₁ | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | Lean.Parser.Tactic.cases |
Mathlib.Combinatorics.Additive.PluenneckeRuzsa | {
"line": 227,
"column": 38
} | {
"line": 227,
"column": 77
} | [
{
"pp": "G : Type u_1\ninst✝¹ : DecidableEq G\ninst✝ : CommGroup G\nA B : Finset G\nhAB : ∀ A' ⊆ A, #(A * B) * #A' ≤ #(A' * B) * #A\nhA : A.Nonempty\nn : ℕ\nih : ↑(#(A * B ^ n)) ≤ (↑(#(A * B)) / ↑(#A)) ^ n * ↑(#A)\n⊢ ↑(#(A * B ^ (n + 1))) * ↑(#A) = ↑(#(B ^ n * A * B)) * ↑(#A)",
"usedConstants": [
"Eq.... | rw [pow_succ, mul_left_comm, mul_assoc] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Combinatorics.Additive.PluenneckeRuzsa | {
"line": 227,
"column": 38
} | {
"line": 227,
"column": 77
} | [
{
"pp": "G : Type u_1\ninst✝¹ : DecidableEq G\ninst✝ : CommGroup G\nA B : Finset G\nhAB : ∀ A' ⊆ A, #(A * B) * #A' ≤ #(A' * B) * #A\nhA : A.Nonempty\nn : ℕ\nih : ↑(#(A * B ^ n)) ≤ (↑(#(A * B)) / ↑(#A)) ^ n * ↑(#A)\n⊢ ↑(#(A * B ^ (n + 1))) * ↑(#A) = ↑(#(B ^ n * A * B)) * ↑(#A)",
"usedConstants": [
"Eq.... | rw [pow_succ, mul_left_comm, mul_assoc] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Additive.PluenneckeRuzsa | {
"line": 227,
"column": 38
} | {
"line": 227,
"column": 77
} | [
{
"pp": "G : Type u_1\ninst✝¹ : DecidableEq G\ninst✝ : CommGroup G\nA B : Finset G\nhAB : ∀ A' ⊆ A, #(A * B) * #A' ≤ #(A' * B) * #A\nhA : A.Nonempty\nn : ℕ\nih : ↑(#(A * B ^ n)) ≤ (↑(#(A * B)) / ↑(#A)) ^ n * ↑(#A)\n⊢ ↑(#(A * B ^ (n + 1))) * ↑(#A) = ↑(#(B ^ n * A * B)) * ↑(#A)",
"usedConstants": [
"Eq.... | rw [pow_succ, mul_left_comm, mul_assoc] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Additive.Convolution | {
"line": 98,
"column": 2
} | {
"line": 99,
"column": 55
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\ninst✝ : DecidableEq G\nA B : Finset G\nx : G\n⊢ A.convolution B x⁻¹ = B⁻¹.convolution A⁻¹ x",
"usedConstants": [
"Eq.mpr",
"instHSMul",
"instSMulOfMul",
"DivInvOneMonoid.toInvOneClass",
"Finset.divisionMonoid",
"Monoid.toMulOneClas... | nth_rw 1 [← inv_inv B]
rw [← card_smul_inter, ← card_inter_smul, inter_comm] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Additive.Convolution | {
"line": 98,
"column": 2
} | {
"line": 99,
"column": 55
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\ninst✝ : DecidableEq G\nA B : Finset G\nx : G\n⊢ A.convolution B x⁻¹ = B⁻¹.convolution A⁻¹ x",
"usedConstants": [
"Eq.mpr",
"instHSMul",
"instSMulOfMul",
"DivInvOneMonoid.toInvOneClass",
"Finset.divisionMonoid",
"Monoid.toMulOneClas... | nth_rw 1 [← inv_inv B]
rw [← card_smul_inter, ← card_inter_smul, inter_comm] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.DegreeSum | {
"line": 62,
"column": 24
} | {
"line": 62,
"column": 35
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\ninst✝² : Fintype V\ninst✝¹ : DecidableRel G.Adj\ninst✝ : DecidableEq V\nd : G.Dart\n⊢ G.dartOfNeighborSet d.toProd.1 ⟨d.toProd.2, ⋯⟩ = d",
"usedConstants": [
"SimpleGraph.dartOfNeighborSet",
"SimpleGraph.Dart.ext",
"Membership.mem",
"SimpleGrap... | ext <;> rfl | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Combinatorics.SimpleGraph.DegreeSum | {
"line": 62,
"column": 24
} | {
"line": 62,
"column": 35
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\ninst✝² : Fintype V\ninst✝¹ : DecidableRel G.Adj\ninst✝ : DecidableEq V\nd : G.Dart\n⊢ G.dartOfNeighborSet d.toProd.1 ⟨d.toProd.2, ⋯⟩ = d",
"usedConstants": [
"SimpleGraph.dartOfNeighborSet",
"SimpleGraph.Dart.ext",
"Membership.mem",
"SimpleGrap... | ext <;> rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.DegreeSum | {
"line": 62,
"column": 24
} | {
"line": 62,
"column": 35
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\ninst✝² : Fintype V\ninst✝¹ : DecidableRel G.Adj\ninst✝ : DecidableEq V\nd : G.Dart\n⊢ G.dartOfNeighborSet d.toProd.1 ⟨d.toProd.2, ⋯⟩ = d",
"usedConstants": [
"SimpleGraph.dartOfNeighborSet",
"SimpleGraph.Dart.ext",
"Membership.mem",
"SimpleGrap... | ext <;> rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.DegreeSum | {
"line": 98,
"column": 2
} | {
"line": 98,
"column": 24
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\ninst✝¹ : Fintype V\ninst✝ : DecidableRel G.Adj\ne : Sym2 V\nh : e ∈ G.edgeFinset\n⊢ e ∈ G.edgeSet",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Finset",
"SimpleGraph.fintypeEdgeSet",
"Membership.mem",
"id",
"SimpleGraph.edgeSet"... | rwa [← mem_edgeFinset] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.Combinatorics.SimpleGraph.Finite | {
"line": 298,
"column": 6
} | {
"line": 298,
"column": 38
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\nv : V\ninst✝¹ : Fintype ↑(G.neighborSet v)\ninst✝ : Fintype ↑G.edgeSet\n⊢ G.degree v ≤ #G.edgeFinset",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Classical.propDecidable",
"id",
"LE.le",
"instLENat",
"SimpleGraph.edgeFinset",... | ← card_incidenceFinset_eq_degree | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.Partition.Finpartition | {
"line": 343,
"column": 8
} | {
"line": 346,
"column": 38
} | [
{
"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✝¹ : Lattice (Subtype Pr) := S... | apply P.supIndep
· simpa [image_subset_iff_subset_preimage] using ht
· simpa using hi
· simpa [i.property] using hi' | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Partition.Finpartition | {
"line": 343,
"column": 8
} | {
"line": 346,
"column": 38
} | [
{
"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✝¹ : Lattice (Subtype Pr) := S... | apply P.supIndep
· simpa [image_subset_iff_subset_preimage] using ht
· simpa using hi
· simpa [i.property] using hi' | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Density | {
"line": 130,
"column": 2
} | {
"line": 130,
"column": 47
} | [
{
"pp": "case h\nα : Type u_4\nβ : Type u_5\nr : α → β → Prop\ninst✝ : (a : α) → DecidablePred (r a)\ns : Finset α\nt : Finset β\n⊢ ↑(#(interedges r s t)) ≤ ↑(#s) * ↑(#t)",
"usedConstants": [
"Nat.cast_mul._simp_1",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Rel.card_interedges_le_mul",
... | · exact mod_cast card_interedges_le_mul r s t | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Combinatorics.SimpleGraph.Density | {
"line": 157,
"column": 2
} | {
"line": 157,
"column": 59
} | [
{
"pp": "α : Type u_4\nβ : Type u_5\nr : α → β → Prop\ninst✝¹ : (a : α) → DecidablePred (r a)\nt : Finset β\ninst✝ : DecidableEq β\ns : Finset α\nP : Finpartition t\n⊢ #(interedges r s t) = ∑ b ∈ P.parts, #(interedges r s b)",
"usedConstants": [
"Eq.mpr",
"Rel.interedges",
"instDecidableEq... | simp_rw [← P.biUnion_parts, interedges_biUnion_right, id] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Order.Partition.Finpartition | {
"line": 646,
"column": 14
} | {
"line": 651,
"column": 60
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\ns t u : Finset α\nP : Finpartition s\na : α\nparts : Finset (Finset α)\nh : ∀ p ∈ parts, p ⊆ s\nh' : ∀ a ∈ s, ∃! t, t ∈ parts ∧ a ∈ t\nh'' : ∅ ∉ parts\n⊢ parts.SupIndep id",
"usedConstants": [
"Eq.mpr",
"Function.onFun",
"congrArg",
"Fins... | by
simp only [supIndep_iff_pairwiseDisjoint]
intro a ha b hb hab
rw [Function.onFun, Finset.disjoint_left]
intro x hx hx'
exact hab ((h' x (h _ ha hx)).unique ⟨ha, hx⟩ ⟨hb, hx'⟩) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.SimpleGraph.Regularity.Equitabilise | {
"line": 98,
"column": 6
} | {
"line": 98,
"column": 87
} | [
{
"pp": "case pos.refine_2\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}... | exact fun x hx => (card_le_card sdiff_subset).trans (Nat.lt_succ_iff.1 <| h _ hx) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Combinatorics.SimpleGraph.Regularity.Equitabilise | {
"line": 98,
"column": 6
} | {
"line": 98,
"column": 87
} | [
{
"pp": "case pos.refine_2\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}... | exact fun x hx => (card_le_card sdiff_subset).trans (Nat.lt_succ_iff.1 <| h _ hx) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Regularity.Equitabilise | {
"line": 98,
"column": 6
} | {
"line": 98,
"column": 87
} | [
{
"pp": "case pos.refine_2\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}... | exact fun x hx => (card_le_card sdiff_subset).trans (Nat.lt_succ_iff.1 <| h _ hx) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Regularity.Equitabilise | {
"line": 88,
"column": 4
} | {
"line": 104,
"column": 82
} | [
{
"pp": "case pos\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 ... | obtain ⟨t, hts, htn⟩ := exists_subset_card_eq (hn₂.trans_eq hs)
have ht : t.Nonempty := by rwa [← card_pos, htn]
have hcard : ite (0 < a) (a - 1) a * m + ite (0 < a) b (b - 1) * (m + 1) = #(s \ t) := by
rw [card_sdiff_of_subset ‹t ⊆ s›, htn, hn₃]
obtain ⟨R, hR₁, _, hR₃⟩ :=
@ih (s \ t) (sdiff_ssu... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Regularity.Equitabilise | {
"line": 88,
"column": 4
} | {
"line": 104,
"column": 82
} | [
{
"pp": "case pos\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 ... | obtain ⟨t, hts, htn⟩ := exists_subset_card_eq (hn₂.trans_eq hs)
have ht : t.Nonempty := by rwa [← card_pos, htn]
have hcard : ite (0 < a) (a - 1) a * m + ite (0 < a) b (b - 1) * (m + 1) = #(s \ t) := by
rw [card_sdiff_of_subset ‹t ⊆ s›, htn, hn₃]
obtain ⟨R, hR₁, _, hR₃⟩ :=
@ih (s \ t) (sdiff_ssu... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Regularity.Equitabilise | {
"line": 168,
"column": 2
} | {
"line": 168,
"column": 87
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\ns : Finset α\nm a b : ℕ\nP : Finpartition s\nh : a * m + b * (m + 1) = #s\nhm : m ≠ 0\n⊢ #({u ∈ (equitabilise h).parts | #u = m}) = a",
"usedConstants": [
"Nat.instMulZeroClass",
"Nat.instIsOrderedAddMonoid",
"HMul.hMul",
"AddLeftCancelSe... | refine (mul_eq_mul_right_iff.1 <| (add_left_inj (b * (m + 1))).1 ?_).resolve_right hm | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Combinatorics.SimpleGraph.DeleteEdges | {
"line": 171,
"column": 4
} | {
"line": 171,
"column": 63
} | [
{
"pp": "V : Type u_1\ninst✝² : Fintype V\ninst✝¹ : DecidableEq V\nG : SimpleGraph V\ninst✝ : DecidableRel G.Adj\nx : V\nh_notMem : x ∉ {x}ᶜ\n⊢ G.support \\ {x} ⊆ Set.univ \\ {x}",
"usedConstants": [
"Set.subset_univ",
"Set.univ",
"Set.instSingletonSet",
"Set.diff_subset_diff_left",
... | exact Set.diff_subset_diff_left (Set.subset_univ G.support) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Combinatorics.SimpleGraph.Regularity.Lemma | {
"line": 100,
"column": 2
} | {
"line": 115,
"column": 41
} | [
{
"pp": "case inr.inr\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\nhd... | suffices h : ∀ i, ∃ P : Finpartition (univ : Finset α), P.IsEquipartition ∧ t ≤ #P.parts ∧
#P.parts ≤ stepBound^[i] t ∧ (P.IsUniform G ε ∨ ε ^ 5 / 4 * i ≤ P.energy G) by
-- For `i > 4 / ε ^ 5` we know that the partition we get can't have energy `≥ ε ^ 5 / 4 * i > 1`,
-- so it must instead be `ε`-uniform and we ... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.Combinatorics.SimpleGraph.Subgraph | {
"line": 720,
"column": 4
} | {
"line": 720,
"column": 34
} | [
{
"pp": "case refine_1\nι : Sort u_1\nV : Type u\nW : Type v\nG : SimpleGraph V\nG₁ G₂ : G.Subgraph\na b : V\ninst✝² : DecidableEq V\ninst✝¹ : Fintype V\ninst✝ : DecidableRel G.Adj\n⊢ Function.Injective fun H ↦\n { verts := ↑(↑H).1, Adj := fun a b ↦ (↑H).2 a b = true, adj_sub := ⋯, edge_vert := ⋯, symm := ⋯ ... | rintro ⟨⟨_, _⟩, -⟩ ⟨⟨_, _⟩, -⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Combinatorics.SimpleGraph.Walk.Decomp | {
"line": 243,
"column": 14
} | {
"line": 243,
"column": 51
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nw : V\ninst✝ : DecidableEq V\nu v : V\nn : ℕ\np : G.Walk u v\nhw : w ∈ p.support\nhn : n ≤ (p.takeUntil w hw).length\n| p.getVert n",
"usedConstants": [
"congrArg",
"SimpleGraph.Walk.length",
"HSub.hSub",
"SimpleGraph.Walk.getVert_append",
... | rw [← take_spec p hw, getVert_append] | Lean.Parser.Tactic.Conv._aux_Init_Conv___macroRules_Lean_Parser_Tactic_Conv_convRw___1 | Lean.Parser.Tactic.Conv.convRw__ |
Mathlib.Combinatorics.SimpleGraph.Walk.Decomp | {
"line": 243,
"column": 14
} | {
"line": 243,
"column": 51
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nw : V\ninst✝ : DecidableEq V\nu v : V\nn : ℕ\np : G.Walk u v\nhw : w ∈ p.support\nhn : n ≤ (p.takeUntil w hw).length\n| p.getVert n",
"usedConstants": [
"congrArg",
"SimpleGraph.Walk.length",
"HSub.hSub",
"SimpleGraph.Walk.getVert_append",
... | rw [← take_spec p hw, getVert_append] | Lean.Elab.Tactic.Conv.evalConvSeq1Indented | Lean.Parser.Tactic.Conv.convSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Walk.Decomp | {
"line": 243,
"column": 14
} | {
"line": 243,
"column": 51
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nw : V\ninst✝ : DecidableEq V\nu v : V\nn : ℕ\np : G.Walk u v\nhw : w ∈ p.support\nhn : n ≤ (p.takeUntil w hw).length\n| p.getVert n",
"usedConstants": [
"congrArg",
"SimpleGraph.Walk.length",
"HSub.hSub",
"SimpleGraph.Walk.getVert_append",
... | rw [← take_spec p hw, getVert_append] | Lean.Elab.Tactic.Conv.evalConvSeq | Lean.Parser.Tactic.Conv.convSeq |
Mathlib.Combinatorics.SimpleGraph.Walk.Decomp | {
"line": 260,
"column": 2
} | {
"line": 266,
"column": 39
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nv w u : V\ninst✝ : DecidableEq V\nn : ℕ\np : G.Walk v w\nh : u ∈ p.support\nhn : n < (p.takeUntil u h).length\n⊢ p.getVert n ≠ u",
"usedConstants": [
"_private.Mathlib.Combinatorics.SimpleGraph.Walk.Decomp.0.SimpleGraph.Walk.getVert_lt_length_takeUntil_ne._proof... | rintro rfl
have h₁ : n < (p.takeUntil _ h).support.dropLast.length := by simpa
have : p.getVert n ∈ (p.takeUntil _ h).support.dropLast := by
simp_rw [p.getVert_takeUntil h hn.le ▸ getVert_eq_support_getElem _ hn.le,
← List.getElem_dropLast h₁, List.getElem_mem h₁]
have := dropLast_support_concat _ ▸ p.c... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Walk.Decomp | {
"line": 260,
"column": 2
} | {
"line": 266,
"column": 39
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nv w u : V\ninst✝ : DecidableEq V\nn : ℕ\np : G.Walk v w\nh : u ∈ p.support\nhn : n < (p.takeUntil u h).length\n⊢ p.getVert n ≠ u",
"usedConstants": [
"_private.Mathlib.Combinatorics.SimpleGraph.Walk.Decomp.0.SimpleGraph.Walk.getVert_lt_length_takeUntil_ne._proof... | rintro rfl
have h₁ : n < (p.takeUntil _ h).support.dropLast.length := by simpa
have : p.getVert n ∈ (p.takeUntil _ h).support.dropLast := by
simp_rw [p.getVert_takeUntil h hn.le ▸ getVert_eq_support_getElem _ hn.le,
← List.getElem_dropLast h₁, List.getElem_mem h₁]
have := dropLast_support_concat _ ▸ p.c... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Regularity.Chunk | {
"line": 249,
"column": 2
} | {
"line": 278,
"column": 93
} | [
{
"pp": "α : Type u_1\ninst✝³ : Fintype α\ninst✝² : DecidableEq α\nP : Finpartition univ\nhP : P.IsEquipartition\nG : SimpleGraph α\ninst✝¹ : DecidableRel G.Adj\nε : ℝ\nU V : Finset α\ninst✝ : Nonempty α\nhPα : #P.parts * 16 ^ #P.parts ≤ Fintype.card α\nhPε : 100 ≤ 4 ^ #P.parts * ε ^ 5\nhU : U ∈ P.parts\nhV : V... | have : ↑(G.edgeDensity (A.biUnion id) (B.biUnion id)) - ε ^ 5 / ↑50 ≤
(↑1 - ε ^ 5 / 50) * G.edgeDensity (A.biUnion id) (B.biUnion id) := by
rw [sub_mul, one_mul, sub_le_sub_iff_left]
refine mul_le_of_le_one_right (by sz_positivity) ?_
exact mod_cast G.edgeDensity_le_one _ _
refine this.trans ?_
co... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Regularity.Chunk | {
"line": 249,
"column": 2
} | {
"line": 278,
"column": 93
} | [
{
"pp": "α : Type u_1\ninst✝³ : Fintype α\ninst✝² : DecidableEq α\nP : Finpartition univ\nhP : P.IsEquipartition\nG : SimpleGraph α\ninst✝¹ : DecidableRel G.Adj\nε : ℝ\nU V : Finset α\ninst✝ : Nonempty α\nhPα : #P.parts * 16 ^ #P.parts ≤ Fintype.card α\nhPε : 100 ≤ 4 ^ #P.parts * ε ^ 5\nhU : U ∈ P.parts\nhV : V... | have : ↑(G.edgeDensity (A.biUnion id) (B.biUnion id)) - ε ^ 5 / ↑50 ≤
(↑1 - ε ^ 5 / 50) * G.edgeDensity (A.biUnion id) (B.biUnion id) := by
rw [sub_mul, one_mul, sub_le_sub_iff_left]
refine mul_le_of_le_one_right (by sz_positivity) ?_
exact mod_cast G.edgeDensity_le_one _ _
refine this.trans ?_
co... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Paths | {
"line": 595,
"column": 44
} | {
"line": 599,
"column": 7
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v : V\ninst✝ : DecidableEq V\nc : G.Walk v v\nhc : c.IsCycle\nhu : u ∈ c.support\nhv : u ≠ v\n⊢ List.count u c.support = 1",
"usedConstants": [
"List.head?",
"List.head",
"List.head?_eq_some_head",
"List.cons_head_tail",
"SimpleGraph.Wa... | by
have := List.eq_or_mem_of_mem_cons <| List.cons_head_tail c.support_ne_nil ▸ hu
have := List.count_eq_one_of_mem hc.support_nodup <| this.resolve_left <| head_support _ ▸ hv
have := c.head_support ▸ List.head?_eq_some_head c.support_ne_nil
grind | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.SimpleGraph.Triangle.Basic | {
"line": 274,
"column": 2
} | {
"line": 276,
"column": 90
} | [
{
"pp": "α : Type u_1\n𝕜 : Type u_3\ninst✝⁵ : Field 𝕜\ninst✝⁴ : LinearOrder 𝕜\ninst✝³ : IsStrictOrderedRing 𝕜\nG : SimpleGraph α\nε : 𝕜\ninst✝² : Fintype α\ninst✝¹ : DecidableRel G.Adj\ninst✝ : Nonempty α\nh₀ : G.FarFromTriangleFree ε\nh₁ : G.CliqueFree 3\n⊢ ε ≤ 0",
"usedConstants": [
"SimpleGrap... | have := h₀ (empty_subset _)
rw [coe_empty, Finset.card_empty, cast_zero, deleteEdges_empty] at this
exact nonpos_of_mul_nonpos_left (this h₁) (cast_pos.2 <| sq_pos_of_pos Fintype.card_pos) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Triangle.Basic | {
"line": 274,
"column": 2
} | {
"line": 276,
"column": 90
} | [
{
"pp": "α : Type u_1\n𝕜 : Type u_3\ninst✝⁵ : Field 𝕜\ninst✝⁴ : LinearOrder 𝕜\ninst✝³ : IsStrictOrderedRing 𝕜\nG : SimpleGraph α\nε : 𝕜\ninst✝² : Fintype α\ninst✝¹ : DecidableRel G.Adj\ninst✝ : Nonempty α\nh₀ : G.FarFromTriangleFree ε\nh₁ : G.CliqueFree 3\n⊢ ε ≤ 0",
"usedConstants": [
"SimpleGrap... | have := h₀ (empty_subset _)
rw [coe_empty, Finset.card_empty, cast_zero, deleteEdges_empty] at this
exact nonpos_of_mul_nonpos_left (this h₁) (cast_pos.2 <| sq_pos_of_pos Fintype.card_pos) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Paths | {
"line": 762,
"column": 6
} | {
"line": 763,
"column": 38
} | [
{
"pp": "case neg\nV : Type u\nG : SimpleGraph V\ninst✝ : DecidableEq V\nu v u✝ v✝ w✝ : V\nh✝ : G.Adj u✝ v✝\np : G.Walk v✝ w✝\nih : p.length ≤ p.bypass.length → p.bypass = p\nh : (cons h✝ p).length ≤ (cons h✝ p).bypass.length\nhb : u✝ ∉ p.bypass.support\n⊢ cons h✝ p.bypass = cons h✝ p",
"usedConstants": [
... | simp only [hb, Walk.bypass, Walk.length_cons, not_false_iff, dif_neg,
Nat.add_le_add_iff_right] at h | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Combinatorics.SimpleGraph.Triangle.Counting | {
"line": 118,
"column": 2
} | {
"line": 129,
"column": 10
} | [
{
"pp": "α : Type u_1\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : ℝ\ns t u : Finset α\ndst : 2 * ε ≤ ↑(G.edgeDensity s t)\nhst : G.IsUniform ε s t\ndsu : 2 * ε ≤ ↑(G.edgeDensity s u)\nusu : G.IsUniform ε s u\ndtu : 2 * ε ≤ ↑(G.edgeDensity t u)\nutu : G.IsUniform ε t u\nh₁ : ↑(#(G.badVertices ε s t)) ≤ ↑... | · apply le_trans _ (card_nsmul_le_sum X' _ _ <| G.good_vertices_triangle_card dst dsu dtu utu)
rw [nsmul_eq_mul]
have := hst.pos.le
suffices hX' : (1 - 2 * ε) * #s ≤ #X' by
exact Eq.trans_le (by ring) (mul_le_mul_of_nonneg_right hX' <| by positivity)
have i : badVertices G ε s t ∪ badVertices G ε ... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Combinatorics.SimpleGraph.Regularity.Chunk | {
"line": 357,
"column": 6
} | {
"line": 357,
"column": 54
} | [
{
"pp": "case hab\nα : Type u_1\ninst✝³ : Fintype α\ninst✝² : DecidableEq α\nP : Finpartition univ\nG : SimpleGraph α\ninst✝¹ : DecidableRel G.Adj\nε : ℝ\nU V : Finset α\ninst✝ : Nonempty α\nhP : P.IsEquipartition\nhPα : #P.parts * 16 ^ #P.parts ≤ Fintype.card α\nhPε : 100 ≤ 4 ^ #P.parts * ε ^ 5\nhU : U ∈ P.par... | have rflV := Subset.refl (chunk hP G ε hV).parts | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Combinatorics.SimpleGraph.Clique | {
"line": 530,
"column": 2
} | {
"line": 530,
"column": 62
} | [
{
"pp": "case right\nα : Type u_1\nG : SimpleGraph α\nn : ℕ\ninst✝ : DecidableEq α\nh : Maximal (fun H ↦ H.CliqueFree (n + 1)) G\nx y : α\nhne : x ≠ y\nhn : ¬G.Adj x y\nt : Finset α\nhc : (G ⊔ edge x y).IsNClique (n + 1) t\nh1 : x ∈ t\n⊢ G.IsNClique n (insert x ((t.erase x).erase y)) ∧ G.IsNClique n (insert y (... | have h2 := h.1.mem_of_sup_edge_isNClique (edge_comm .. ▸ hc) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.FieldTheory.ChevalleyWarning | {
"line": 76,
"column": 2
} | {
"line": 81,
"column": 61
} | [
{
"pp": "K : Type u_1\nσ : Type u_2\ninst✝³ : Fintype K\ninst✝² : Field K\ninst✝¹ : Fintype σ\ninst✝ : DecidableEq σ\nf : MvPolynomial σ K\nh : f.totalDegree < (q - 1) * Fintype.card σ\nthis : DecidableEq K\nd : σ →₀ ℕ\nhd : d ∈ f.support\ni : σ\nhi : d i < q - 1\nx₀ : { j // j ≠ i } → K\ne : K ≃ { x // x ∘ Sub... | calc
(∑ x : { x : σ → K // x ∘ (↑) = x₀ }, ∏ j, (x : σ → K) j ^ d j) =
∑ a : K, ∏ j : σ, (e a : σ → K) j ^ d j := (e.sum_comp _).symm
_ = ∑ a : K, (∏ j, x₀ j ^ d j) * a ^ d i := Fintype.sum_congr _ _ ?_
_ = (∏ j, x₀ j ^ d j) * ∑ a : K, a ^ d i := by rw [mul_sum]
_ = 0 := by rw [sum_pow_lt_card_s... | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcTactic |
Mathlib.FieldTheory.ChevalleyWarning | {
"line": 111,
"column": 77
} | {
"line": 111,
"column": 86
} | [
{
"pp": "K : Type u_1\nσ : Type u_2\nι : Type u_3\ninst✝⁵ : Fintype K\ninst✝⁴ : Field K\ninst✝³ : Fintype σ\ninst✝² : DecidableEq σ\ninst✝¹ : DecidableEq K\np : ℕ\ninst✝ : CharP K p\ns : Finset ι\nf : ι → MvPolynomial σ K\nh : ∑ i ∈ s, (f i).totalDegree < Fintype.card σ\n⊢ Nonempty Kˣ",
"usedConstants": [
... | exact ⟨1⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Combinatorics.Additive.Energy | {
"line": 180,
"column": 2
} | {
"line": 180,
"column": 76
} | [
{
"pp": "α : Type u_1\ninst✝² : DecidableEq α\ninst✝¹ : CommGroup α\ninst✝ : Fintype α\nt : Finset α\n⊢ Eₘ[univ, t] = Fintype.card α * #t ^ 2",
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"Finset.univ",
"Monoid.toMulOneClass",
"SProd.sprod",
"congrArg",
"Finset",
... | simp only [mulEnergy, univ_product_univ, Fintype.card, sq, ← card_product] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Combinatorics.Additive.ErdosGinzburgZiv | {
"line": 141,
"column": 8
} | {
"line": 141,
"column": 96
} | [
{
"pp": "case refine_1\nm : ℕ\nhm : 2 ≤ m\nihm : ∀ {ι : Type u_1} {s : Finset ι} (a : ι → ℤ), 2 * m - 1 ≤ #s → ∃ t ⊆ s, #t = m ∧ ↑m ∣ ∑ i ∈ t, a i\nn : ℕ\nhn : 2 ≤ n\nihn : ∀ {ι : Type u_1} {s : Finset ι} (a : ι → ℤ), 2 * n - 1 ≤ #s → ∃ t ⊆ s, #t = n ∧ ↑n ∣ ∑ i ∈ t, a i\nι : Type u_1\ns : Finset ι\na : ι → ℤ\nh... | rw [card_biUnion (h𝒜disj.mono hℬ𝒜), sum_const_nat fun t ht ↦ (h𝒜 <| hℬ𝒜 ht).2.1, hℬcard] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Combinatorics.Additive.ErdosGinzburgZiv | {
"line": 141,
"column": 8
} | {
"line": 141,
"column": 96
} | [
{
"pp": "case refine_1\nm : ℕ\nhm : 2 ≤ m\nihm : ∀ {ι : Type u_1} {s : Finset ι} (a : ι → ℤ), 2 * m - 1 ≤ #s → ∃ t ⊆ s, #t = m ∧ ↑m ∣ ∑ i ∈ t, a i\nn : ℕ\nhn : 2 ≤ n\nihn : ∀ {ι : Type u_1} {s : Finset ι} (a : ι → ℤ), 2 * n - 1 ≤ #s → ∃ t ⊆ s, #t = n ∧ ↑n ∣ ∑ i ∈ t, a i\nι : Type u_1\ns : Finset ι\na : ι → ℤ\nh... | rw [card_biUnion (h𝒜disj.mono hℬ𝒜), sum_const_nat fun t ht ↦ (h𝒜 <| hℬ𝒜 ht).2.1, hℬcard] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Additive.ErdosGinzburgZiv | {
"line": 141,
"column": 8
} | {
"line": 141,
"column": 96
} | [
{
"pp": "case refine_1\nm : ℕ\nhm : 2 ≤ m\nihm : ∀ {ι : Type u_1} {s : Finset ι} (a : ι → ℤ), 2 * m - 1 ≤ #s → ∃ t ⊆ s, #t = m ∧ ↑m ∣ ∑ i ∈ t, a i\nn : ℕ\nhn : 2 ≤ n\nihn : ∀ {ι : Type u_1} {s : Finset ι} (a : ι → ℤ), 2 * n - 1 ≤ #s → ∃ t ⊆ s, #t = n ∧ ↑n ∣ ∑ i ∈ t, a i\nι : Type u_1\ns : Finset ι\na : ι → ℤ\nh... | rw [card_biUnion (h𝒜disj.mono hℬ𝒜), sum_const_nat fun t ht ↦ (h𝒜 <| hℬ𝒜 ht).2.1, hℬcard] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Nat.BitIndices | {
"line": 108,
"column": 2
} | {
"line": 108,
"column": 49
} | [
{
"pp": "case cons\na : ℕ\nL : List ℕ\nhL : (a :: L).SortedLT\n⊢ (map (fun i ↦ 2 ^ i) (a :: L)).sum.bitIndices = a :: L",
"usedConstants": [
"Preorder.toLT",
"List.Pairwise",
"List.SortedLT.pairwise",
"Membership.mem",
"List.cons",
"List.pairwise_cons",
"List",
... | obtain ⟨haL, hL⟩ := pairwise_cons.1 hL.pairwise | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Combinatorics.Colex | {
"line": 302,
"column": 89
} | {
"line": 313,
"column": 75
} | [
{
"pp": "α : Type u_1\ninst✝ : LinearOrder α\ns t : Finset α\n⊢ toColex s ≤ toColex t ↔ ∀ (hst : s ≠ t), (s ∆ t).max' ⋯ ∈ t",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"False",
"Preorder.toLT",
"Lattice.toSemilatticeSup",
"lt_irrefl",
"Equiv.instEquivLike",
"Col... | by
refine ⟨fun h hst ↦ ?_, fun h a has hat ↦ ?_⟩
· set m := (s ∆ t).max' (symmDiff_nonempty.2 hst)
by_contra hmt
have hms : m ∈ s := by
simpa [m, mem_symmDiff, hmt] using max'_mem _ <| symmDiff_nonempty.2 hst
have ⟨b, hbt, hbs, hmb⟩ := h hms hmt
exact lt_irrefl _ <| (max'_lt_iff _ _).1 (hmb.lt... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.Configuration | {
"line": 497,
"column": 22
} | {
"line": 497,
"column": 38
} | [
{
"pp": "K : Type u_3\ninst✝ : Field K\na b c d : Fin 3 → K\nhac : a ⬝ᵥ c = 0\nhbc : b ⬝ᵥ c = 0\nhad : a ⬝ᵥ d = 0\nhbd : b ⬝ᵥ d = 0\nh : LinearIndependent K (of ![a, b]).row ∧ LinearIndependent K (of ![c, d]).row\nA : Matrix (Fin 2) (Fin 3) K := of ![a, b]\nB : Matrix (Fin 2) (Fin 3) K := of ![c, d]\nhAB : A.ra... | h.1.rank_matrix, | Lean.Elab.Tactic.evalRewriteSeq | null |
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