module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.CategoryTheory.Groupoid.FreeGroupoidOfCategory | {
"line": 135,
"column": 49
} | {
"line": 135,
"column": 97
} | {
"line": 135,
"column": 97
} | [
{
"pp": "G : Type u₁\ninst✝ : Groupoid G\n⊢ of G ⋙ lift (𝟭 G) ⋙ of G = of G",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"CategoryTheory.FreeGroupoid",
"Eq.mpr",
"CategoryTheory.Functor",
"congrArg",
"CategoryTheory.Functor.assoc",
"CategoryTheory.Fu... | [] | rw [← Functor.assoc, lift_spec, Functor.id_comp] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Groupoid.FreeGroupoidOfCategory | {
"line": 135,
"column": 49
} | {
"line": 135,
"column": 97
} | {
"line": 135,
"column": 97
} | [
{
"pp": "G : Type u₁\ninst✝ : Groupoid G\n⊢ of G ⋙ lift (𝟭 G) ⋙ of G = of G",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"CategoryTheory.FreeGroupoid",
"Eq.mpr",
"CategoryTheory.Functor",
"congrArg",
"CategoryTheory.Functor.assoc",
"CategoryTheory.Fu... | [] | rw [← Functor.assoc, lift_spec, Functor.id_comp] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Groupoid.FreeGroupoidOfCategory | {
"line": 135,
"column": 49
} | {
"line": 135,
"column": 97
} | {
"line": 135,
"column": 97
} | [
{
"pp": "G : Type u₁\ninst✝ : Groupoid G\n⊢ of G ⋙ lift (𝟭 G) ⋙ of G = of G",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"CategoryTheory.FreeGroupoid",
"Eq.mpr",
"CategoryTheory.Functor",
"congrArg",
"CategoryTheory.Functor.assoc",
"CategoryTheory.Fu... | [] | rw [← Functor.assoc, lift_spec, Functor.id_comp] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Groupoid.VertexGroup | {
"line": 65,
"column": 4
} | {
"line": 66,
"column": 23
} | {
"line": 67,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝ : Groupoid C\nc d : C\nf : c ⟶ d\nγ : c ⟶ c\n⊢ (fun δ ↦ f ≫ δ ≫ inv f) ((fun γ ↦ inv f ≫ γ ≫ f) γ) = γ",
"ppTerm": "?m.53",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Category.assoc",
"CategoryTheory.CategoryStruct.toQuiver",
"Qu... | [] | simp_rw [Category.assoc, comp_inv, Category.comp_id, ← Category.assoc, comp_inv,
Category.id_comp] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.CategoryTheory.Groupoid.VertexGroup | {
"line": 65,
"column": 4
} | {
"line": 66,
"column": 23
} | {
"line": 67,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝ : Groupoid C\nc d : C\nf : c ⟶ d\nγ : c ⟶ c\n⊢ (fun δ ↦ f ≫ δ ≫ inv f) ((fun γ ↦ inv f ≫ γ ≫ f) γ) = γ",
"ppTerm": "?m.53",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Category.assoc",
"CategoryTheory.CategoryStruct.toQuiver",
"Qu... | [] | simp_rw [Category.assoc, comp_inv, Category.comp_id, ← Category.assoc, comp_inv,
Category.id_comp] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Groupoid.VertexGroup | {
"line": 65,
"column": 4
} | {
"line": 66,
"column": 23
} | {
"line": 67,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝ : Groupoid C\nc d : C\nf : c ⟶ d\nγ : c ⟶ c\n⊢ (fun δ ↦ f ≫ δ ≫ inv f) ((fun γ ↦ inv f ≫ γ ≫ f) γ) = γ",
"ppTerm": "?m.53",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Category.assoc",
"CategoryTheory.CategoryStruct.toQuiver",
"Qu... | [] | simp_rw [Category.assoc, comp_inv, Category.comp_id, ← Category.assoc, comp_inv,
Category.id_comp] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Groupoid.VertexGroup | {
"line": 68,
"column": 67
} | {
"line": 68,
"column": 84
} | {
"line": 69,
"column": 6
} | [
{
"pp": "C : Type u\ninst✝ : Groupoid C\nc d : C\nf : c ⟶ d\nδ : d ⟶ d\n⊢ (𝟙 d ≫ δ) ≫ 𝟙 d = δ",
"ppTerm": "?m.65",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrArg",
"CategoryTheory.CategoryStruct.id",
... | [
"C : Type u\ninst✝ : Groupoid C\nc d : C\nf : c ⟶ d\nδ : d ⟶ d\n⊢ δ ≫ 𝟙 d = δ"
] | Category.id_comp, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.CategoryTheory.Galois.EssSurj | {
"line": 212,
"column": 4
} | {
"line": 212,
"column": 34
} | {
"line": 213,
"column": 4
} | [
{
"pp": "C : Type u₁\ninst✝⁶ : Category.{u₂, u₁} C\nF : C ⥤ FintypeCat\ninst✝⁵ : GaloisCategory C\ninst✝⁴ : FiberFunctor F\nG : Type u_1\ninst✝³ : Group G\ninst✝² : TopologicalSpace G\ninst✝¹ : IsTopologicalGroup G\ninst✝ : CompactSpace G\nV U : OpenSubgroup (Aut F)\nh : (↑U).Normal\nA : C\nu : (functorToAction... | [
"C : Type u₁\ninst✝⁶ : Category.{u₂, u₁} C\nF : C ⥤ FintypeCat\ninst✝⁵ : GaloisCategory C\ninst✝⁴ : FiberFunctor F\nG : Type u_1\ninst✝³ : Group G\ninst✝² : TopologicalSpace G\ninst✝¹ : IsTopologicalGroup G\ninst✝ : CompactSpace G\nV U : OpenSubgroup (Aut F)\nh : (↑U).Normal\nA : C\nu : (functorToAction F).obj A ≅ ... | ext (x : Aut F ⧸ U.toSubgroup) | _private.Lean.Elab.Tactic.Ext.0.Lean.Elab.Tactic.Ext.evalExt | Lean.Elab.Tactic.Ext.ext |
Mathlib.CategoryTheory.LiftingProperties.PushoutProduct | {
"line": 121,
"column": 2
} | {
"line": 121,
"column": 43
} | {
"line": 122,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : HasPushouts C\ninst✝² : CartesianMonoidalCategory C\ninst✝¹ : MonoidalClosed C\ninst✝ : BraidedCategory C\nA B K L X Y : C\nf : A ⟶ B\ni : IsInitial K\nt : IsTerminal Y\n⊢ HasLiftingProperty (Arrow.mk f □ Arrow.mk (i.to L)).hom (t.from X) ↔ HasLiftingPro... | [
"C : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : HasPushouts C\ninst✝² : CartesianMonoidalCategory C\ninst✝¹ : MonoidalClosed C\ninst✝ : BraidedCategory C\nA B K L X Y : C\nf : A ⟶ B\ni : IsInitial K\nt : IsTerminal Y\n⊢ HasLiftingProperty f ((ihom L).map (t.from X)) ↔ HasLiftingProperty f (t.from (L ⟹ X))"
] | rw [hasLiftingProperty_mk_isInitial_iff'] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Limits.FunctorCategory.Shapes.Images | {
"line": 48,
"column": 4
} | {
"line": 50,
"column": 9
} | {
"line": 52,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nF G : C ⥤ Type u\nf : F ⟶ G\nH : MonoFactorisation f\n⊢ {\n app := fun X ↦\n ↾fun x ↦\n match x with\n | ⟨x, hx⟩ => (ConcreteCategory.hom (H.e.app X)) (Exists.choose hx),\n naturality := ⋯ } ≫\n H.m =\n (m... | [] | ext
simp
grind | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.FunctorCategory.Shapes.Images | {
"line": 48,
"column": 4
} | {
"line": 50,
"column": 9
} | {
"line": 52,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nF G : C ⥤ Type u\nf : F ⟶ G\nH : MonoFactorisation f\n⊢ {\n app := fun X ↦\n ↾fun x ↦\n match x with\n | ⟨x, hx⟩ => (ConcreteCategory.hom (H.e.app X)) (Exists.choose hx),\n naturality := ⋯ } ≫\n H.m =\n (m... | [] | ext
simp
grind | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Shapes.Pullback.IsPullback.BicartesianSq | {
"line": 191,
"column": 26
} | {
"line": 196,
"column": 25
} | {
"line": 196,
"column": 26
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : HasZeroObject C\ninst✝ : HasZeroMorphisms C\nX : C\ns : Cocone (span 0 (𝟙 X))\n⊢ ∀ (j : WalkingSpan), (PushoutCocone.mk 0 0 ⋯).ι.app j ≫ 0 = s.ι.app j",
"ppTerm": "?m.61",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Catego... | [] | by
have c :=
@PushoutCocone.coequalizer_ext _ _ _ _ _ _ _ s _ 0 (𝟙 _)
(by simp [eq_iff_true_of_subsingleton]) (by simpa using PushoutCocone.condition s)
dsimp at c
simpa using c | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Localization.Pi | {
"line": 71,
"column": 6
} | {
"line": 71,
"column": 17
} | {
"line": 72,
"column": 6
} | [
{
"pp": "case h_option.refine_1.none\nJ : Type w\ninst✝⁴ : Fintype J\nhJ :\n ∀ {C : J → Type u₁} {D : J → Type u₂} [inst : (j : J) → Category.{v₁, u₁} (C j)]\n [inst_1 : (j : J) → Category.{v₂, u₂} (D j)] (L : (j : J) → C j ⥤ D j) (W : (j : J) → MorphismProperty (C j))\n [∀ (j : J), (W j).ContainsIdentit... | [
"case h_option.refine_1.some\nJ : Type w\ninst✝⁴ : Fintype J\nhJ :\n ∀ {C : J → Type u₁} {D : J → Type u₂} [inst : (j : J) → Category.{v₁, u₁} (C j)]\n [inst_1 : (j : J) → Category.{v₂, u₂} (D j)] (L : (j : J) → C j ⥤ D j) (W : (j : J) → MorphismProperty (C j))\n [∀ (j : J), (W j).ContainsIdentities] [∀ (j :... | · exact hf₁ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Monoidal.Closed.Ideal | {
"line": 148,
"column": 10
} | {
"line": 155,
"column": 48
} | {
"line": 156,
"column": 10
} | [
{
"pp": "case e\nC : Type u₁\nD : Type u₂\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : Category.{v₁, u₂} D\ni : D ⥤ C\ninst✝¹ : CartesianMonoidalCategory C\ninst✝ : Reflective i\nX Y : D\n⊢ (reflector i ⋙ i).obj (i.obj X ⊗ i.obj Y) ≅ (𝟭 C).obj (i.obj X ⊗ i.obj Y)",
"ppTerm": "?e",
"assigned": true,
"use... | [
"case e\nC : Type u₁\nD : Type u₂\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : Category.{v₁, u₂} D\ni : D ⥤ C\ninst✝¹ : CartesianMonoidalCategory C\ninst✝ : Reflective i\nX Y : D\nthis : IsIso ((reflectorAdjunction i).unit.app (i.obj X ⊗ i.obj Y)) := ofReflective._proof_2 i X Y\n⊢ (reflector i ⋙ i).obj (i.obj X ⊗ i.obj ... | letI : IsIso ((reflectorAdjunction i).unit.app (i.obj X ⊗ i.obj Y)) := by
apply Functor.essImage.unit_isIso
haveI := reflective_products i
use Limits.prod X Y
constructor
apply Limits.PreservesLimitPair.iso i _ _ |>.trans
refine Limits.IsLimit.cone... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLetI___1 | Lean.Parser.Tactic.tacticLetI__ |
Mathlib.CategoryTheory.Monoidal.Closed.Ideal | {
"line": 319,
"column": 13
} | {
"line": 319,
"column": 33
} | {
"line": 319,
"column": 34
} | [
{
"pp": "C : Type u₁\nD : Type u₂\ninst✝⁷ : Category.{v₁, u₁} C\ninst✝⁶ : Category.{v₁, u₂} D\ni : D ⥤ C\ninst✝⁵ : CartesianMonoidalCategory C\ninst✝⁴ : Reflective i\ninst✝³ : MonoidalClosed C\ninst✝² : CartesianMonoidalCategory D\ninst✝¹ : ExponentialIdeal i\ninst✝ : BraidedCategory C\nA B : C\n⊢ (bijection i ... | [
"C : Type u₁\nD : Type u₂\ninst✝⁷ : Category.{v₁, u₁} C\ninst✝⁶ : Category.{v₁, u₂} D\ni : D ⥤ C\ninst✝⁵ : CartesianMonoidalCategory C\ninst✝⁴ : Reflective i\ninst✝³ : MonoidalClosed C\ninst✝² : CartesianMonoidalCategory D\ninst✝¹ : ExponentialIdeal i\ninst✝ : BraidedCategory C\nA B : C\n⊢ (bijection i A B ((reflec... | ← bijection_natural, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Monoidal.Closed.Ideal | {
"line": 319,
"column": 34
} | {
"line": 319,
"column": 51
} | {
"line": 319,
"column": 52
} | [
{
"pp": "C : Type u₁\nD : Type u₂\ninst✝⁷ : Category.{v₁, u₁} C\ninst✝⁶ : Category.{v₁, u₂} D\ni : D ⥤ C\ninst✝⁵ : CartesianMonoidalCategory C\ninst✝⁴ : Reflective i\ninst✝³ : MonoidalClosed C\ninst✝² : CartesianMonoidalCategory D\ninst✝¹ : ExponentialIdeal i\ninst✝ : BraidedCategory C\nA B : C\n⊢ (bijection i ... | [
"C : Type u₁\nD : Type u₂\ninst✝⁷ : Category.{v₁, u₁} C\ninst✝⁶ : Category.{v₁, u₂} D\ni : D ⥤ C\ninst✝⁵ : CartesianMonoidalCategory C\ninst✝⁴ : Reflective i\ninst✝³ : MonoidalClosed C\ninst✝² : CartesianMonoidalCategory D\ninst✝¹ : ExponentialIdeal i\ninst✝ : BraidedCategory C\nA B : C\n⊢ (bijection i A B ((reflec... | Category.id_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Monoidal.DayConvolution.Closed | {
"line": 226,
"column": 4
} | {
"line": 232,
"column": 55
} | {
"line": 233,
"column": 4
} | [
{
"pp": "C : Type u₁\ninst✝⁵ : Category.{v₁, u₁} C\nV : Type u₂\ninst✝⁴ : Category.{v₂, u₂} V\ninst✝³ : MonoidalCategory C\ninst✝² : MonoidalCategory V\ninst✝¹ : MonoidalClosed V\nF G✝ H G : C ⥤ V\ninst✝ : DayConvolution F G\nℌ : DayConvolutionInternalHom F (F ⊛ G) H\nc c' : C\nf : c ⟶ c'\nj : C\n⊢ G.map f ≫\n ... | [
"C : Type u₁\ninst✝⁵ : Category.{v₁, u₁} C\nV : Type u₂\ninst✝⁴ : Category.{v₂, u₂} V\ninst✝³ : MonoidalCategory C\ninst✝² : MonoidalCategory V\ninst✝¹ : MonoidalClosed V\nF G✝ H G : C ⥤ V\ninst✝ : DayConvolution F G\nℌ : DayConvolutionInternalHom F (F ⊛ G) H\nc c' : C\nf : c ⟶ c'\nj : C\n⊢ G.map f ≫ MonoidalClosed... | rw [← Wedge.mk_ι
(F := dayConvolutionInternalHomDiagramFunctor F |>.obj _ |>.obj c)
(H.obj c) (ℌ.π c) (ℌ.hπ c),
← Wedge.mk_ι
(F := dayConvolutionInternalHomDiagramFunctor F |>.obj _ |>.obj c')
(H.obj c') (ℌ.π c') (ℌ.hπ c'),
Wedge.IsLimit.lift_ι_assoc, Wedge.IsLimit.lift_ι] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Monoidal.Hopf_ | {
"line": 154,
"column": 47
} | {
"line": 161,
"column": 33
} | {
"line": 163,
"column": 0
} | [
{
"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 ≫\n 𝒮 ▷ A ≫\n Δ ▷ A ≫\n (α_ A A A).hom ≫\n A ◁ A ◁ Δ ≫\n A ◁ (α_ A A A).inv ≫ A ◁ (β_ A A).hom ▷ A ≫ A ◁ (α_ A A A).hom ≫ (α_ A... | [] | by
slice_lhs 3 5 =>
rw [← associator_naturality_right, ← Category.assoc, ← tensorHom_def]
slice_lhs 3 9 =>
rw [Bimon.compatibility]
slice_lhs 1 3 =>
rw [antipode_left]
simp [MonObj.tensorObj.one_def] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Monoidal.Hopf_ | {
"line": 336,
"column": 47
} | {
"line": 419,
"column": 10
} | {
"line": 421,
"column": 0
} | [
{
"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 ... | [] | by
slice_lhs 7 8 =>
rw [associator_naturality_left]
slice_lhs 8 9 =>
rw [← whisker_exchange]
slice_lhs 9 10 =>
rw [← whisker_exchange]
slice_lhs 11 12 =>
rw [MonObj.mul_assoc_flip]
slice_lhs 10 11 =>
rw [associator_inv_naturality_left]
slice_lhs 11 12 =>
simp only [← comp_whiskerRigh... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Monoidal.Bimod | {
"line": 733,
"column": 23
} | {
"line": 733,
"column": 34
} | {
"line": 733,
"column": 34
} | [
{
"pp": "case a\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\ninst✝² : HasCoequalizers C\nR S : Mon C\nP : Bimod R S\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)\n|... | [
"case a\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\ninst✝² : HasCoequalizers C\nR S : Mon C\nP : Bimod R S\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)\n| (α_ P.X S.X... | right_assoc | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.CategoryTheory.ObjectProperty.Ind | {
"line": 65,
"column": 2
} | {
"line": 66,
"column": 87
} | {
"line": 68,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nP : ObjectProperty C\nh : P ≤ isFinitelyPresentable C\ninst✝ : LocallySmall.{w, v, u} C\nX : C\nJ : Type w\nJc : SmallCategory J\nJf : IsFiltered J\npres : ColimitPresentation J X\nK : J → Type w\nKc : (i : J) → SmallCategory (K i)\nKf : ∀ (i : J), IsFiltered (K ... | [] | exact ⟨_, inferInstance, inferInstance,
(pres.bind pres').reindex (ShrinkHoms.equivalence _).inverse, fun k ↦ by simp [hp]⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.ObjectProperty.Ind | {
"line": 114,
"column": 2
} | {
"line": 114,
"column": 52
} | {
"line": 116,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nD : Type u_1\ninst✝¹ : Category.{u_2, u_1} D\nP : ObjectProperty D\nF : C ⥤ D\ninst✝ : PreservesFilteredColimitsOfSize.{w, w, v, u_2, u, u_1} F\nX : C\nJ : Type w\nw✝¹ : SmallCategory J\nw✝ : IsFiltered J\npres : ColimitPresentation J X\nh : ∀ (i : J), P.inverseI... | [] | use J, inferInstance, inferInstance, pres.map F, h | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.CategoryTheory.Monoidal.DayConvolution | {
"line": 997,
"column": 6
} | {
"line": 1001,
"column": 51
} | {
"line": 1001,
"column": 51
} | [
{
"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 ... | [] | apply Functor.Faithful.map_injective (F := ι C V D)
simp only [Functor.map_comp, ι_map_tensorHom_hom_eq_tensorHom, Functor.map_id]
rw [ι_map_rightUnitor_hom_eq_rightUnitor_hom, ι_map_rightUnitor_hom_eq_rightUnitor_hom]
exact DayConvolutionUnit.rightUnitor_naturality
(ι C V D |>.obj <| 𝟙_ D) (... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Monoidal.DayConvolution | {
"line": 997,
"column": 6
} | {
"line": 1001,
"column": 51
} | {
"line": 1001,
"column": 51
} | [
{
"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 ... | [] | apply Functor.Faithful.map_injective (F := ι C V D)
simp only [Functor.map_comp, ι_map_tensorHom_hom_eq_tensorHom, Functor.map_id]
rw [ι_map_rightUnitor_hom_eq_rightUnitor_hom, ι_map_rightUnitor_hom_eq_rightUnitor_hom]
exact DayConvolutionUnit.rightUnitor_naturality
(ι C V D |>.obj <| 𝟙_ D) (... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.ObjectProperty.SiteLocal | {
"line": 56,
"column": 4
} | {
"line": 60,
"column": 14
} | {
"line": 61,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nP : ObjectProperty C\nK : Precoverage C\ninst✝ : P.IsClosedUnderIsomorphisms\nH : ∀ ⦃X : C⦄ (𝒰 : K.ZeroHypercover X), P X ↔ ∀ (i : 𝒰.I₀), P (𝒰.X i)\nX : C\nR : Presieve X\nhR : R ∈ K.coverings X\nY : C\nf : Y ⟶ X\nhf : R f\nhX : P X\n⊢ P Y",
"ppTerm": "?m.... | [] | rw [CategoryTheory.Precoverage.mem_iff_exists_zeroHypercover] at hR
obtain ⟨𝒰, rfl⟩ := hR
rw [H 𝒰] at hX
obtain ⟨i⟩ := hf
exact hX i | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.ObjectProperty.SiteLocal | {
"line": 56,
"column": 4
} | {
"line": 60,
"column": 14
} | {
"line": 61,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nP : ObjectProperty C\nK : Precoverage C\ninst✝ : P.IsClosedUnderIsomorphisms\nH : ∀ ⦃X : C⦄ (𝒰 : K.ZeroHypercover X), P X ↔ ∀ (i : 𝒰.I₀), P (𝒰.X i)\nX : C\nR : Presieve X\nhR : R ∈ K.coverings X\nY : C\nf : Y ⟶ X\nhf : R f\nhX : P X\n⊢ P Y",
"ppTerm": "?m.... | [] | rw [CategoryTheory.Precoverage.mem_iff_exists_zeroHypercover] at hR
obtain ⟨𝒰, rfl⟩ := hR
rw [H 𝒰] at hX
obtain ⟨i⟩ := hf
exact hX i | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Preadditive.EilenbergMoore | {
"line": 47,
"column": 18
} | {
"line": 47,
"column": 86
} | {
"line": 47,
"column": 87
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Preadditive C\nT : Monad C\ninst✝ : T.Additive\nF G : T.Algebra\nα : F ⟶ G\n⊢ T.map (-α.f) ≫ G.a = F.a ≫ (-α.f)",
"ppTerm": "?m.353",
"assigned": true,
"usedConstants": [
"NegZeroClass.toNeg",
"CategoryTheory.Monad.Algebra.Hom.... | [] | simp only [Functor.map_neg, neg_comp, Monad.Algebra.Hom.h, comp_neg] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Preadditive.EilenbergMoore | {
"line": 47,
"column": 18
} | {
"line": 47,
"column": 86
} | {
"line": 47,
"column": 87
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Preadditive C\nT : Monad C\ninst✝ : T.Additive\nF G : T.Algebra\nα : F ⟶ G\n⊢ T.map (-α.f) ≫ G.a = F.a ≫ (-α.f)",
"ppTerm": "?m.353",
"assigned": true,
"usedConstants": [
"NegZeroClass.toNeg",
"CategoryTheory.Monad.Algebra.Hom.... | [] | simp only [Functor.map_neg, neg_comp, Monad.Algebra.Hom.h, comp_neg] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Preadditive.EilenbergMoore | {
"line": 47,
"column": 18
} | {
"line": 47,
"column": 86
} | {
"line": 47,
"column": 87
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Preadditive C\nT : Monad C\ninst✝ : T.Additive\nF G : T.Algebra\nα : F ⟶ G\n⊢ T.map (-α.f) ≫ G.a = F.a ≫ (-α.f)",
"ppTerm": "?m.353",
"assigned": true,
"usedConstants": [
"NegZeroClass.toNeg",
"CategoryTheory.Monad.Algebra.Hom.... | [] | simp only [Functor.map_neg, neg_comp, Monad.Algebra.Hom.h, comp_neg] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Preadditive.FreydCategory.RightFreyd | {
"line": 101,
"column": 4
} | {
"line": 101,
"column": 54
} | {
"line": 102,
"column": 4
} | [
{
"pp": "V : Type u_1\ninst✝² : Category.{v_1, u_1} V\ninst✝¹ : Preadditive V\nu v : Arrow V\nf : u ⟶ v\ninst✝ : IsIso (Hom.right f)\nZ✝ : RightFreyd V\ng₁ g₂ : (quotient V).obj v ⟶ Z✝\neq : (quotient V).map f ≫ g₁ = (quotient V).map f ≫ g₂\n⊢ g₁ = g₂",
"ppTerm": "?m.36",
"assigned": true,
"usedCons... | [
"V : Type u_1\ninst✝² : Category.{v_1, u_1} V\ninst✝¹ : Preadditive V\nu v : Arrow V\nf : u ⟶ v\ninst✝ : IsIso (Hom.right f)\nZ✝ : RightFreyd V\ng₂ : (quotient V).obj v ⟶ Z✝\ng₁ : v ⟶ Z✝.as\neq : (quotient V).map f ≫ (quotient V).map g₁ = (quotient V).map f ≫ g₂\n⊢ (quotient V).map g₁ = g₂"
] | obtain ⟨g₁, rfl⟩ := (quotient V).map_surjective g₁ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.CategoryTheory.Preadditive.Comma | {
"line": 74,
"column": 58
} | {
"line": 74,
"column": 92
} | {
"line": 75,
"column": 2
} | [
{
"pp": "A : Type u₁\ninst✝⁷ : Category.{v₁, u₁} A\ninst✝⁶ : Preadditive A\nB : Type u₂\ninst✝⁵ : Category.{v₂, u₂} B\ninst✝⁴ : Preadditive B\nT : Type u₃\ninst✝³ : Category.{v₃, u₃} T\ninst✝² : Preadditive T\nL : A ⥤ T\ninst✝¹ : L.Additive\nR : B ⥤ T\ninst✝ : R.Additive\nu v : Comma L R\nx✝¹ : ℕ\nx✝ : u ⟶ v\n⊢... | [] | ext <;> dsimp <;> simp [add_zsmul] | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.CategoryTheory.Preadditive.Comma | {
"line": 64,
"column": 18
} | {
"line": 64,
"column": 44
} | {
"line": 65,
"column": 2
} | [
{
"pp": "A : Type u₁\ninst✝⁷ : Category.{v₁, u₁} A\ninst✝⁶ : Preadditive A\nB : Type u₂\ninst✝⁵ : Category.{v₂, u₂} B\ninst✝⁴ : Preadditive B\nT : Type u₃\ninst✝³ : Category.{v₃, u₃} T\ninst✝² : Preadditive T\nL : A ⥤ T\ninst✝¹ : L.Additive\nR : B ⥤ T\ninst✝ : R.Additive\nu v : Comma L R\nx✝¹ x✝ : u ⟶ v\n⊢ x✝¹ ... | [] | by ext <;> simp [add_comm] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Preadditive.HomOrthogonal | {
"line": 141,
"column": 2
} | {
"line": 143,
"column": 46
} | {
"line": 145,
"column": 0
} | [
{
"pp": "case neg\nC : Type u\ninst✝³ : Category.{v, u} C\nι : Type u_1\ns : ι → C\ninst✝² : Preadditive C\ninst✝¹ : HasFiniteBiproducts C\no : HomOrthogonal s\nα : Type\ninst✝ : Finite α\nf : α → ι\nb a : α\nj_property✝ : a ∈ f ⁻¹' {f b}\nj_property : f a = f b\nh : ¬⟨b, ⋯⟩ = ⟨a, j_property✝⟩\n⊢ eqToHom ⋯ ≫ bi... | [] | · simp only [Subtype.mk.injEq] at h
convert! comp_zero
simpa using biproduct.ι_π_ne _ (Ne.symm h) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Presentable.Adjunction | {
"line": 123,
"column": 2
} | {
"line": 123,
"column": 60
} | {
"line": 125,
"column": 0
} | [
{
"pp": "C : Type u\nD : Type u'\ninst✝² : Category.{v, u} C\ninst✝¹ : Category.{v', u'} D\ne : C ≌ D\ninst✝ : IsLocallyPresentable.{w, v, u} C\nκ : Cardinal.{w}\nw✝ : Fact κ.IsRegular\nh✝ : IsCardinalLocallyPresentable C κ\n⊢ IsLocallyPresentable.{w, v', u'} D",
"ppTerm": "?m.23",
"assigned": true,
... | [] | exact ⟨κ, inferInstance, e.isCardinalLocallyPresentable κ⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Monoidal.Bimod | {
"line": 930,
"column": 2
} | {
"line": 930,
"column": 29
} | {
"line": 931,
"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 [AssociatorBimod.inv] | Lean.Elab.Tactic.evalDSimp | Lean.Parser.Tactic.dsimp |
Mathlib.Order.Category.PartOrdEmb | {
"line": 274,
"column": 28
} | {
"line": 285,
"column": 36
} | {
"line": 285,
"column": 37
} | [
{
"pp": "J : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsFiltered J\nF : J ⥤ PartOrdEmb\nc : Cocone (F ⋙ forget PartOrdEmb)\nhc : IsColimit c\nj : J\nx y : ↑(F.1 j)\n⊢ { toFun := ⇑(ConcreteCategory.hom (c.ι.app j)), inj' := ⋯ } x ≤\n { toFun := ⇑(ConcreteCategory.hom (c.ι.app j)), inj' := ⋯ } y ↔\n x ≤ ... | [] | by
refine ⟨?_, fun h ↦ ⟨j, x, y, rfl, rfl, h⟩⟩
rintro ⟨k, x', y', hx, hy, h⟩
obtain ⟨l₁, a₁, b₁, hl₁⟩ := (Types.FilteredColimit.isColimit_eq_iff _ hc).1 hx
obtain ⟨l₂, a₂, b₂, hl₂⟩ := (Types.FilteredColimit.isColimit_eq_iff _ hc).1 hy
dsimp at hx hy hl₁ hl₂
obtain ⟨m, d, ... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Presentable.Directed | {
"line": 198,
"column": 51
} | {
"line": 203,
"column": 78
} | {
"line": 203,
"column": 78
} | [
{
"pp": "J : Type w\ninst✝¹ : SmallCategory J\nκ : Cardinal.{w}\ninst✝ : Fact κ.IsRegular\nj : J\n⊢ Function.Surjective fun x ↦ ⟨Arrow.mk (𝟙 j), ⋯⟩",
"ppTerm": "?m.221",
"assigned": true,
"usedConstants": [
"CategoryTheory.MorphismProperty.ofHoms_iff",
"Exists.choose_spec",
"Categ... | [] | by
rintro ⟨f, hf⟩
refine ⟨⟨⟩, ?_⟩
ext
exact ((MorphismProperty.ofHoms_iff _ _).1
((MorphismProperty.arrow_mk_mem_toSet_iff _ _).1 hf)).choose_spec.symm | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Presentable.Directed | {
"line": 229,
"column": 6
} | {
"line": 230,
"column": 43
} | {
"line": 231,
"column": 2
} | [
{
"pp": "case refine_3\nJ : Type w\ninst✝¹ : SmallCategory J\nκ : Cardinal.{w}\ninst✝ : Fact κ.IsRegular\nj x✝² x✝¹ : J\nx✝ : x✝² ⟶ x✝¹\nh : __Diagram✝.W x✝\n⊢ ∃ li lj, __Diagram✝.W li ∧ __Diagram✝.W lj ∧ x✝ ≫ lj = li",
"ppTerm": "?refine_3",
"assigned": true,
"usedConstants": [
"CategoryTheor... | [] | obtain ⟨⟨⟩⟩ := h
exact ⟨𝟙 _, 𝟙 _, ⟨⟨⟩⟩, ⟨⟨⟩⟩, by simp⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Presentable.Directed | {
"line": 229,
"column": 6
} | {
"line": 230,
"column": 43
} | {
"line": 231,
"column": 2
} | [
{
"pp": "case refine_3\nJ : Type w\ninst✝¹ : SmallCategory J\nκ : Cardinal.{w}\ninst✝ : Fact κ.IsRegular\nj x✝² x✝¹ : J\nx✝ : x✝² ⟶ x✝¹\nh : __Diagram✝.W x✝\n⊢ ∃ li lj, __Diagram✝.W li ∧ __Diagram✝.W lj ∧ x✝ ≫ lj = li",
"ppTerm": "?refine_3",
"assigned": true,
"usedConstants": [
"CategoryTheor... | [] | obtain ⟨⟨⟩⟩ := h
exact ⟨𝟙 _, 𝟙 _, ⟨⟨⟩⟩, ⟨⟨⟩⟩, by simp⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Preadditive.Mat | {
"line": 327,
"column": 6
} | {
"line": 328,
"column": 15
} | {
"line": 329,
"column": 4
} | [
{
"pp": "case h₁\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : Preadditive C\nM : Mat_ C\ni j : M.ι\n⊢ i ∉ Finset.univ →\n ((fun j x ↦ if h : j = i then eqToHom ⋯ else 0) ≫ fun x k ↦ if h : i = k then eqToHom ⋯ else 0) i j = 0",
"ppTerm": "?h₁",
"assigned": true,
"usedConstants": [
"F... | [] | intro h
simp at h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Preadditive.Mat | {
"line": 327,
"column": 6
} | {
"line": 328,
"column": 15
} | {
"line": 329,
"column": 4
} | [
{
"pp": "case h₁\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : Preadditive C\nM : Mat_ C\ni j : M.ι\n⊢ i ∉ Finset.univ →\n ((fun j x ↦ if h : j = i then eqToHom ⋯ else 0) ≫ fun x k ↦ if h : i = k then eqToHom ⋯ else 0) i j = 0",
"ppTerm": "?h₁",
"assigned": true,
"usedConstants": [
"F... | [] | intro h
simp at h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Presentable.Directed | {
"line": 384,
"column": 8
} | {
"line": 386,
"column": 40
} | {
"line": 387,
"column": 8
} | [
{
"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)\nι : Type w\nD : ι → DiagramWithUniqueTerminal J κ\nhι : HasCardinalLT ι κ\nm : J\nu : (i : ι) → (D i).top ⟶ m\nhm₀ : ∀ (i : ι), IsEmpt... | [
"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)\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)... | · simp only [MorphismProperty.iSup_iff] at hf
obtain ⟨i, hf⟩ := hf
exact (hD ((D i).tgt hf)).elim | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Sites.Coherent.CoherentTopology | {
"line": 71,
"column": 4
} | {
"line": 71,
"column": 20
} | {
"line": 72,
"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 : ... | [
"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), Effe... | obtain ⟨i⟩ := hY | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.CategoryTheory.Sites.Coherent.SheafComparison | {
"line": 192,
"column": 2
} | {
"line": 192,
"column": 48
} | {
"line": 193,
"column": 2
} | [
{
"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\nthis : Preregular C\n... | [
"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\nthis : Preregular C\n⊢ S ∈ (regul... | rw [← exists_effectiveEpi_iff_mem_induced F X] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Sites.Descent.Precoverage | {
"line": 225,
"column": 25
} | {
"line": 225,
"column": 27
} | {
"line": 225,
"column": 28
} | [
{
"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 ⟶... | f₂ | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.CategoryTheory.Sites.Descent.Precoverage | {
"line": 313,
"column": 8
} | {
"line": 313,
"column": 25
} | {
"line": 313,
"column": 26
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nF : LocallyDiscrete Cᵒᵖ ⥤ᵖ Cat\nJ : GrothendieckTopology C\ninst✝ : 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\nα : ι' → ι\np' : (j : ι') → X' j ⟶ X (α j)\nw : ∀ (j : ι'), p' j ≫ f (α j... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\nF : LocallyDiscrete Cᵒᵖ ⥤ᵖ Cat\nJ : GrothendieckTopology C\ninst✝ : 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\nα : ι' → ι\np' : (j : ι') → X' j ⟶ X (α j)\nw : ∀ (j : ι'), p' j ≫ f (α j) = f' j\nhf... | Category.id_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Sites.Descent.DescentData | {
"line": 223,
"column": 8
} | {
"line": 223,
"column": 88
} | {
"line": 224,
"column": 6
} | [
{
"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... | [
"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.DescentData... | F.mapComp'₀₁₃_inv_comp_mapComp'₀₂₃_hom_app_assoc _ _ _ _ _ _ _ _ (by cat_disch), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Sites.Point.Monoidal | {
"line": 65,
"column": 4
} | {
"line": 65,
"column": 61
} | {
"line": 67,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\nJ : GrothendieckTopology C\nΦ : J.Point\nA : Type u'\ninst✝² : Category.{v', u'} A\ninst✝¹ : MonoidalCategory A\ninst✝ : HasColimitsOfSize.{w, w, v', u'} A\nx✝¹ : Cᵒᵖ ⥤ A\nX✝ : C\nx✝ : Φ.fiber.obj X✝\n⊢ Φ.toPresheafFiber X✝ x✝ x✝¹ ≫ (ρ_ (Φ.presheafFiber.obj x✝¹))... | [] | simp [tensorHom_def, ← MonoidalCategory.whiskerLeft_comp] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Sites.Descent.DescentData | {
"line": 309,
"column": 23
} | {
"line": 309,
"column": 25
} | {
"line": 309,
"column": 26
} | [
{
"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\nβ : ι... | [
"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\nβ : ι' → ι\np'' :... | f₂ | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.CategoryTheory.Sites.Descent.DescentData | {
"line": 332,
"column": 23
} | {
"line": 332,
"column": 25
} | {
"line": 332,
"column": 26
} | [
{
"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... | [
"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.DescentData... | f₂ | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.CategoryTheory.Sites.Descent.DescentData | {
"line": 351,
"column": 25
} | {
"line": 351,
"column": 27
} | {
"line": 351,
"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\nS'' :... | [
"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\nS'' : C\nq : S'' ... | f₂ | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.CategoryTheory.Triangulated.Generators | {
"line": 75,
"column": 4
} | {
"line": 75,
"column": 67
} | {
"line": 75,
"column": 67
} | [
{
"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.shiftClosure ℤ).binaryProductsClosure.retractClosure.extensionProduct\n ... | [
"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.shiftClosure ℤ).binaryProductsClosure.retractClosure.retractClosure.extensionProduct\... | ← retractClosure_extensionProduct_retractClosure_retractClosure | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Triangulated.Generators | {
"line": 83,
"column": 4
} | {
"line": 83,
"column": 67
} | {
"line": 83,
"column": 67
} | [
{
"pp": "C : Type u_1\ninst✝⁶ : Category.{v_1, u_1} C\ninst✝⁵ : HasZeroObject C\ninst✝⁴ : HasShift C ℤ\ninst✝³ : Preadditive C\ninst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝¹ : Pretriangulated C\nP : ObjectProperty C\ninst✝ : IsTriangulated C\nn : ℕ\n⊢ (((P.shiftClosure ℤ).binaryProductsClosure.retract... | [
"C : Type u_1\ninst✝⁶ : Category.{v_1, u_1} C\ninst✝⁵ : HasZeroObject C\ninst✝⁴ : HasShift C ℤ\ninst✝³ : Preadditive C\ninst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝¹ : Pretriangulated C\nP : ObjectProperty C\ninst✝ : IsTriangulated C\nn : ℕ\n⊢ (((P.shiftClosure ℤ).binaryProductsClosure.retractClosure.exte... | ← retractClosure_extensionProduct_retractClosure_retractClosure | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Triangulated.TStructure.SpectralObject | {
"line": 114,
"column": 68
} | {
"line": 114,
"column": 85
} | {
"line": 115,
"column": 6
} | [
{
"pp": "case refine_1\nC : Type u_1\ninst✝⁶ : Category.{v_1, u_1} C\ninst✝⁵ : Preadditive C\ninst✝⁴ : HasZeroObject C\ninst✝³ : HasShift C ℤ\ninst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝¹ : Pretriangulated C\nt : TStructure C\ninst✝ : IsTriangulated C\na b c : EInt\nhab : a ≤ b\nhbc : b ≤ c\nX : C\n⊢... | [
"case refine_1\nC : Type u_1\ninst✝⁶ : Category.{v_1, u_1} C\ninst✝⁵ : Preadditive C\ninst✝⁴ : HasZeroObject C\ninst✝³ : HasShift C ℤ\ninst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝¹ : Pretriangulated C\nt : TStructure C\ninst✝ : IsTriangulated C\na b c : EInt\nhab : a ≤ b\nhbc : b ≤ c\nX : C\n⊢ (t.eTruncGE... | Category.id_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Additive.FreimanHom | {
"line": 165,
"column": 18
} | {
"line": 165,
"column": 48
} | {
"line": 165,
"column": 49
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\ninst✝¹ : CommMonoid α\ninst✝ : CommMonoid β\nA : Set α\nB : Set β\nf : α → β\nn : ℕ\ng : β → α\nh : InvOn g f A B\nhf : IsMulFreimanHom n A B f\nhg : IsMulFreimanHom n B A g\ns t : Multiset α\nhsA : ∀ ⦃x : α⦄, x ∈ s → x ∈ A\nhtA : ∀ ⦃x : α⦄, x ∈ t → x ∈ A\nhs : s.card = n\nh... | [
"α : Type u_2\nβ : Type u_3\ninst✝¹ : CommMonoid α\ninst✝ : CommMonoid β\nA : Set α\nB : Set β\nf : α → β\nn : ℕ\ng : β → α\nh : InvOn g f A B\nhf : IsMulFreimanHom n A B f\nhg : IsMulFreimanHom n B A g\ns t : Multiset α\nhsA : ∀ ⦃x : α⦄, x ∈ s → x ∈ A\nhtA : ∀ ⦃x : α⦄, x ∈ t → x ∈ A\nhs : s.card = n\nht : t.card =... | map_congr rfl fun x hx => ?g1, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Additive.AP.Three.Behrend | {
"line": 383,
"column": 63
} | {
"line": 388,
"column": 29
} | {
"line": 389,
"column": 2
} | [
{
"pp": "N : ℕ\nhN : 2 ≤ N\nthis : (2 * dValue N - 1) ^ nValue N ≤ (2 * dValue N) ^ nValue N\ni : 2 * ↑(dValue N) ≤ ↑N ^ (↑(nValue N))⁻¹\n⊢ (2 * ↑(dValue N)) ^ nValue N ≤ ↑N",
"ppTerm": "?m.93",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"Eq.mpr",
"GroupWithZero... | [] | by
rw [← rpow_natCast]
apply (rpow_le_rpow (mul_nonneg zero_le_two (cast_nonneg _)) i (cast_nonneg _)).trans
rw [← rpow_mul (cast_nonneg _), inv_mul_cancel₀, rpow_one]
rw [cast_ne_zero]
apply (nValue_pos hN).ne' | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.Additive.AP.Three.Behrend | {
"line": 477,
"column": 2
} | {
"line": 477,
"column": 32
} | {
"line": 478,
"column": 2
} | [
{
"pp": "N : ℕ\nhN : 1 ≤ N\nhN' : N ≤ 4096\n⊢ ↑N * rexp (-4 * √(log ↑N)) ≤ 1",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"LE.le.eq_or_lt",
"Real.instLE",
"Real",
"Preorder.toLT",
"HMul.hMul",
"PartialOrder.toPreorder",
"Nat.instAtLeastTwoHAddOf... | [
"case inl\nhN : 1 ≤ 1\nhN' : 1 ≤ 4096\n⊢ ↑1 * rexp (-4 * √(log ↑1)) ≤ 1",
"case inr\nN : ℕ\nhN✝ : 1 ≤ N\nhN' : N ≤ 4096\nhN : 1 < N\n⊢ ↑N * rexp (-4 * √(log ↑N)) ≤ 1"
] | obtain rfl | hN := hN.eq_or_lt | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Combinatorics.Additive.CauchyDavenport | {
"line": 121,
"column": 2
} | {
"line": 121,
"column": 32
} | {
"line": 122,
"column": 2
} | [
{
"pp": "α : Type u_2\ninst✝¹ : Group α\ninst✝ : DecidableEq α\ns t : Finset α\nhs : s.Nonempty\nht : t.Nonempty\nx : Finset α × Finset α\nhx : x = (s, t)\n⊢ min (minOrder α) ↑(#s + #t - 1) ≤ ↑(#(s * t))",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"Finset",
"Eq.mp",
"... | [
"α : Type u_2\ninst✝¹ : Group α\ninst✝ : DecidableEq α\ns t : Finset α\nhs : s.Nonempty\nht : t.Nonempty\nx : Finset α × Finset α\nhx : x.1 = s ∧ x.2 = t\n⊢ min (minOrder α) ↑(#s + #t - 1) ≤ ↑(#(s * t))"
] | simp only [Prod.ext_iff] at hx | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Combinatorics.Additive.CauchyDavenport | {
"line": 214,
"column": 4
} | {
"line": 217,
"column": 69
} | {
"line": 218,
"column": 2
} | [
{
"pp": "α : Type u_2\ninst✝⁴ : LinearOrder α\ninst✝³ : Mul α\ninst✝² : IsCancelMul α\ninst✝¹ : MulLeftMono α\ninst✝ : MulRightMono α\ns t : Finset α\nhs : s.Nonempty\nht : t.Nonempty\nthis : s * {t.min' ht} ∩ ({s.max' hs} * t) = {s.max' hs * t.min' ht}\n⊢ #s + #t - 1 ≤ #(s * t)",
"ppTerm": "?m.61",
"as... | [] | rw [← card_singleton_mul (s.max' hs) t, ← card_mul_singleton s (t.min' ht),
← card_union_add_card_inter, ← card_singleton _, ← this, Nat.add_sub_cancel]
exact card_mono (union_subset (mul_subset_mul_left <| singleton_subset_iff.2 <| min'_mem _ _) <|
mul_subset_mul_right <| singleton_subset_iff.2 <| max'... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Additive.CauchyDavenport | {
"line": 214,
"column": 4
} | {
"line": 217,
"column": 69
} | {
"line": 218,
"column": 2
} | [
{
"pp": "α : Type u_2\ninst✝⁴ : LinearOrder α\ninst✝³ : Mul α\ninst✝² : IsCancelMul α\ninst✝¹ : MulLeftMono α\ninst✝ : MulRightMono α\ns t : Finset α\nhs : s.Nonempty\nht : t.Nonempty\nthis : s * {t.min' ht} ∩ ({s.max' hs} * t) = {s.max' hs * t.min' ht}\n⊢ #s + #t - 1 ≤ #(s * t)",
"ppTerm": "?m.61",
"as... | [] | rw [← card_singleton_mul (s.max' hs) t, ← card_mul_singleton s (t.min' ht),
← card_union_add_card_inter, ← card_singleton _, ← this, Nat.add_sub_cancel]
exact card_mono (union_subset (mul_subset_mul_left <| singleton_subset_iff.2 <| min'_mem _ _) <|
mul_subset_mul_right <| singleton_subset_iff.2 <| max'... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Partition.Finpartition | {
"line": 254,
"column": 4
} | {
"line": 254,
"column": 43
} | {
"line": 255,
"column": 4
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Lattice α\ninst✝ : OrderBot α\nb : α\nP✝ : Finpartition b\nha : IsAtom b\nP : Finpartition b\nh : ∀ b_1 ∈ P.parts, b_1 = b\n⊢ b ∈ P.parts",
"ppTerm": "?m.87",
"assigned": true,
"usedConstants": [
"Preorder.toLT",
"Finset",
"OrderBot.toBot",
"Pa... | [
"α : Type u_1\ninst✝¹ : Lattice α\ninst✝ : OrderBot α\nb : α\nP✝ : Finpartition b\nha : IsAtom b\nP : Finpartition b\nh : ∀ b_1 ∈ P.parts, b_1 = b\nc : α\nhc : c ∈ P.parts\n⊢ b ∈ P.parts"
] | obtain ⟨c, hc⟩ := P.parts_nonempty ha.1 | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Order.Partition.Finpartition | {
"line": 316,
"column": 56
} | {
"line": 316,
"column": 69
} | {
"line": 317,
"column": 4
} | [
{
"pp": "case refine_1\nα : Type u_1\ninst✝² : Lattice α\ninst✝¹ : OrderBot α\na : α\nP✝ : Finpartition a\ninst✝ : DecidableEq α\ns : Finset α\nP : Finpartition s\np : Finset α\nhp : p ∈ P.parts\n⊢ p ⊆ s",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Finset",
"Finset.inst... | [] | exact P.le hp | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Combinatorics.SimpleGraph.Regularity.Equitabilise | {
"line": 61,
"column": 2
} | {
"line": 67,
"column": 22
} | {
"line": 68,
"column": 2
} | [
{
"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 ... | [
"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) ∧ #... | · -- Rewrite using `← bot_eq_empty` because we have theorems about `Finpartition ⊥`,
-- and nothing about `Finpartition ∅`, even though they are defeq in this case.
-- TODO: specialize the `Finpartition ⊥` lemmas to `Finpartition ∅`?
simp only [hab.1, hab.2, add_zero, zero_mul, eq_comm, card_eq_zero, ← bot_... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Combinatorics.SimpleGraph.Regularity.Bound | {
"line": 152,
"column": 6
} | {
"line": 152,
"column": 28
} | {
"line": 152,
"column": 28
} | [
{
"pp": "α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nP : Finpartition univ\nu : Finset α\nhP : P.IsEquipartition\nhu : u ∈ P.parts\nhucard : #u ≠ m * 4 ^ #P.parts + a\nthis : m * 4 ^ #P.parts ≤ Fintype.card α / #P.parts\n⊢ (4 ^ #P.parts - (a + 1)) * m + (a + 1) * (m + 1) = #u",
"ppTerm": "?m.27... | [
"α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nP : Finpartition univ\nu : Finset α\nhP : P.IsEquipartition\nhu : u ∈ P.parts\nhucard : #u ≠ Fintype.card α / #P.parts\nthis : m * 4 ^ #P.parts ≤ Fintype.card α / #P.parts\n⊢ (4 ^ #P.parts - (a + 1)) * m + (a + 1) * (m + 1) = #u"
] | Nat.add_sub_of_le this | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Triangulated.TStructure.ETrunc | {
"line": 392,
"column": 2
} | {
"line": 395,
"column": 55
} | {
"line": 397,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝⁶ : Category.{v_1, u_1} C\ninst✝⁵ : Preadditive C\ninst✝⁴ : HasZeroObject C\ninst✝³ : HasShift C ℤ\ninst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝¹ : Pretriangulated C\nt : TStructure C\ninst✝ : IsTriangulated C\na b : EInt\nhab : b ≤ a\n⊢ IsIso (t.eTruncLTLTToLT a b)",
"... | [] | rw [NatTrans.isIso_iff_isIso_app]
intro
simp only [eTruncLTLTToLT_app]
exact t.isIso_eTruncLT_obj_map_truncLTπ_app _ _ hab _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Triangulated.TStructure.ETrunc | {
"line": 392,
"column": 2
} | {
"line": 395,
"column": 55
} | {
"line": 397,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝⁶ : Category.{v_1, u_1} C\ninst✝⁵ : Preadditive C\ninst✝⁴ : HasZeroObject C\ninst✝³ : HasShift C ℤ\ninst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝¹ : Pretriangulated C\nt : TStructure C\ninst✝ : IsTriangulated C\na b : EInt\nhab : b ≤ a\n⊢ IsIso (t.eTruncLTLTToLT a b)",
"... | [] | rw [NatTrans.isIso_iff_isIso_app]
intro
simp only [eTruncLTLTToLT_app]
exact t.isIso_eTruncLT_obj_map_truncLTπ_app _ _ hab _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Partition.Finpartition | {
"line": 586,
"column": 2
} | {
"line": 587,
"column": 27
} | {
"line": 588,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝¹ : GeneralizedBooleanAlgebra α\ninst✝ : DecidableEq α\na b c : α\nP : Finpartition a\n⊢ c ∈ (P.avoid b).parts ↔ ∃ d ∈ P.parts, ¬d ≤ b ∧ d \\ b = c",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Finset",
"OrderBot.... | [
"α : Type u_1\ninst✝¹ : GeneralizedBooleanAlgebra α\ninst✝ : DecidableEq α\na b c : α\nP : Finpartition a\n⊢ (∃ x ∈ P.parts, ¬c = ⊥ ∧ x \\ b = c) ↔ ∃ d ∈ P.parts, ¬d ≤ b ∧ d \\ b = c"
] | simp only [avoid, ofErase, mem_erase, Ne, mem_image, ← exists_and_left,
@and_left_comm (c ≠ ⊥)] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Combinatorics.SimpleGraph.Density | {
"line": 64,
"column": 2
} | {
"line": 64,
"column": 53
} | {
"line": 66,
"column": 0
} | [
{
"pp": "α : Type u_4\nβ : Type u_5\nr : α → β → Prop\ninst✝ : (a : α) → DecidablePred (r a)\nt : Finset β\n⊢ interedges r ∅ t = ∅",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Finset.empty_product",
"Rel.interedges",
"SProd.sprod",
"congrArg",
... | [] | rw [interedges, Finset.empty_product, filter_empty] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Combinatorics.SimpleGraph.Density | {
"line": 64,
"column": 2
} | {
"line": 64,
"column": 53
} | {
"line": 66,
"column": 0
} | [
{
"pp": "α : Type u_4\nβ : Type u_5\nr : α → β → Prop\ninst✝ : (a : α) → DecidablePred (r a)\nt : Finset β\n⊢ interedges r ∅ t = ∅",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Finset.empty_product",
"Rel.interedges",
"SProd.sprod",
"congrArg",
... | [] | rw [interedges, Finset.empty_product, filter_empty] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Density | {
"line": 64,
"column": 2
} | {
"line": 64,
"column": 53
} | {
"line": 66,
"column": 0
} | [
{
"pp": "α : Type u_4\nβ : Type u_5\nr : α → β → Prop\ninst✝ : (a : α) → DecidablePred (r a)\nt : Finset β\n⊢ interedges r ∅ t = ∅",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Finset.empty_product",
"Rel.interedges",
"SProd.sprod",
"congrArg",
... | [] | rw [interedges, Finset.empty_product, filter_empty] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Partition.Finpartition | {
"line": 705,
"column": 2
} | {
"line": 705,
"column": 32
} | {
"line": 707,
"column": 0
} | [
{
"pp": "case neg\nα : Type u_1\ninst✝ : DecidableEq α\ns : Finset α\nP : Finpartition s\na : α\nha : a ∉ s\n⊢ a ∈ P.part a ↔ a ∈ s",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"False",
"eq_false",
"congrArg",
"Finset",
"Finpartition.part",... | [] | · simp [P.part_eq_empty.2, ha] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Combinatorics.SimpleGraph.Density | {
"line": 129,
"column": 2
} | {
"line": 129,
"column": 25
} | {
"line": 130,
"column": 2
} | [
{
"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\n⊢ edgeDensity r s t ≤ 1",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"div_le_one_of_le₀",
"Rat.instMul",
"HMul.hMul",
"Rel.interedges",
... | [
"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)",
"case hb\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⊢ 0 ≤ ↑(#s) * ↑(#t)"
] | apply div_le_one_of_le₀ | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Combinatorics.SimpleGraph.Regularity.Increment | {
"line": 114,
"column": 2
} | {
"line": 114,
"column": 27
} | {
"line": 115,
"column": 2
} | [
{
"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\ns₁ s₂ : Finset α\nhs : (s₁, s₂) ∈ P.parts.offDiag\na✝¹ : ⟨(s₁, s₂), hs⟩ ∈ ↑P.parts.offDiag.attach\nt₁ t₂ : Finset α\nht : (t₁, t₂) ∈ P.parts.off... | [
"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\ns₁ s₂ : Finset α\nhs✝ : (s₁, s₂) ∈ P.parts.offDiag\nhs : (s₁, s₂).1 ∈ P.parts ∧ (s₁, s₂).2 ∈ P.parts ∧ (s₁, s₂).1 ≠ (s₁, s₂).2\na✝¹ : ⟨(s₁, s₂), hs✝⟩ ∈ ↑P.p... | rw [mem_offDiag] at hs ht | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Combinatorics.SimpleGraph.Regularity.Chunk | {
"line": 119,
"column": 46
} | {
"line": 119,
"column": 59
} | {
"line": 119,
"column": 59
} | [
{
"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\nh₂ : ¬G.IsUniform ε U V\nhX : G.nonuniformWitness ε U V ∈ P.nonunif... | [
"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\nh₂ : ¬G.IsUniform ε U V\nhX : G.nonuniformWitness ε U V ∈ P.nonuniformWitnesses... | filter_subset | Mathlib.Tactic.GRewrite.evalGRewriteSeq | null |
Mathlib.Combinatorics.SimpleGraph.Regularity.Increment | {
"line": 144,
"column": 4
} | {
"line": 157,
"column": 92
} | {
"line": 158,
"column": 2
} | [] | [
"α : Type u_1\ninst✝³ : Fintype α\ninst✝² : DecidableEq α\nP : Finpartition univ\nG : SimpleGraph α\ninst✝¹ : DecidableRel G.Adj\nε : ℝ\ninst✝ : Nonempty α\nhP : P.IsEquipartition\nhP₇ : 7 ≤ #P.parts\nhPε : 100 ≤ 4 ^ #P.parts * ε ^ 5\nhPα : #P.parts * 16 ^ #P.parts ≤ Fintype.card α\nhPG : ¬P.IsUniform G ε\nhε₀ : 0 ... | _ = (∑ x ∈ P.parts.offDiag, (G.edgeDensity x.1 x.2 : ℝ) ^ 2 +
#P.parts ^ 2 * (ε ^ 5 / 4) : ℝ) / #P.parts ^ 2 := by
rw [coe_energy, add_div, mul_div_cancel_left₀]; positivity
_ ≤ (∑ x ∈ P.parts.offDiag.attach, (∑ i ∈ distinctPairs hP G ε x,
G.edgeDensity i.1 i.2 ^ 2 : ℝ) / 16 ^ #P.parts) ... | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcSteps |
Mathlib.Combinatorics.SimpleGraph.DeleteEdges | {
"line": 231,
"column": 2
} | {
"line": 231,
"column": 52
} | {
"line": 232,
"column": 2
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\n𝕜 : Type u_2\ninst✝³ : Ring 𝕜\ninst✝² : PartialOrder 𝕜\ninst✝¹ : Fintype ↑G.edgeSet\np : SimpleGraph V → Prop\nr : 𝕜\ninst✝ : Fintype (Sym2 V)\n⊢ G.DeleteFar p r ↔\n ∀ ⦃H : SimpleGraph V⦄ [inst : DecidableRel H.Adj], H ≤ G → p H → r ≤ ↑(#G.edgeFinset) - ↑(#H.edge... | [
"case refine_1\nV : Type u_1\nG : SimpleGraph V\n𝕜 : Type u_2\ninst✝³ : Ring 𝕜\ninst✝² : PartialOrder 𝕜\ninst✝¹ : Fintype ↑G.edgeSet\np : SimpleGraph V → Prop\nr : 𝕜\ninst✝ : Fintype (Sym2 V)\nh : G.DeleteFar p r\nH : SimpleGraph V\nx✝ : DecidableRel H.Adj\nhHG : H ≤ G\nhH : p H\n⊢ r ≤ ↑(#G.edgeFinset) - ↑(#H.e... | refine ⟨fun h H _ hHG hH ↦ ?_, fun h s hs hG ↦ ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Combinatorics.SimpleGraph.Subgraph | {
"line": 652,
"column": 2
} | {
"line": 658,
"column": 35
} | {
"line": 660,
"column": 0
} | [
{
"pp": "V : Type u\nW : Type v\nG : SimpleGraph V\nG' : SimpleGraph W\nf : G →g G'\nH₁ H₂ : G.Subgraph\nhH : H₁ ≤ H₂\n⊢ Subgraph.map f H₁ ≤ Subgraph.map f H₂",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"RelHom.instFunLike",
"congrArg",
"SimpleGraph.Ad... | [] | constructor
· intro
simp only [map_verts, Set.mem_image, forall_exists_index, and_imp]
rintro v hv rfl
exact ⟨_, hH.1 hv, rfl⟩
· rintro _ _ ⟨u, v, ha, rfl, rfl⟩
exact ⟨_, _, hH.2 ha, rfl, rfl⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Subgraph | {
"line": 652,
"column": 2
} | {
"line": 658,
"column": 35
} | {
"line": 660,
"column": 0
} | [
{
"pp": "V : Type u\nW : Type v\nG : SimpleGraph V\nG' : SimpleGraph W\nf : G →g G'\nH₁ H₂ : G.Subgraph\nhH : H₁ ≤ H₂\n⊢ Subgraph.map f H₁ ≤ Subgraph.map f H₂",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"RelHom.instFunLike",
"congrArg",
"SimpleGraph.Ad... | [] | constructor
· intro
simp only [map_verts, Set.mem_image, forall_exists_index, and_imp]
rintro v hv rfl
exact ⟨_, hH.1 hv, rfl⟩
· rintro _ _ ⟨u, v, ha, rfl, rfl⟩
exact ⟨_, _, hH.2 ha, rfl, rfl⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Subgraph | {
"line": 720,
"column": 19
} | {
"line": 720,
"column": 42
} | {
"line": 720,
"column": 42
} | [
{
"pp": "ι : 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\nH : G.Subgraph\na b : V\n⊢ (H.verts.toFinset, fun a b ↦ decide (H.Adj a b)).2 a b = true → a ∈ (H.verts.toFinset, fun a b ↦ decide (H.Adj a b))... | [] | simpa using H.edge_vert | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Combinatorics.SimpleGraph.Subgraph | {
"line": 720,
"column": 19
} | {
"line": 720,
"column": 42
} | {
"line": 720,
"column": 42
} | [
{
"pp": "ι : 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\nH : G.Subgraph\na b : V\n⊢ (H.verts.toFinset, fun a b ↦ decide (H.Adj a b)).2 a b = true → a ∈ (H.verts.toFinset, fun a b ↦ decide (H.Adj a b))... | [] | simpa using H.edge_vert | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Subgraph | {
"line": 720,
"column": 19
} | {
"line": 720,
"column": 42
} | {
"line": 720,
"column": 42
} | [
{
"pp": "ι : 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\nH : G.Subgraph\na b : V\n⊢ (H.verts.toFinset, fun a b ↦ decide (H.Adj a b)).2 a b = true → a ∈ (H.verts.toFinset, fun a b ↦ decide (H.Adj a b))... | [] | simpa using H.edge_vert | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Subgraph | {
"line": 1110,
"column": 2
} | {
"line": 1122,
"column": 21
} | {
"line": 1124,
"column": 0
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nG' : G.Subgraph\ns : Set (Sym2 ↑G'.verts)\n⊢ G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map Subtype.val '' s)).coe",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"SimpleGraph.deleteEdges",
"Eq.mpr",
"Sym2.Rel",
"Sym2.map",
... | [] | ext ⟨v, hv⟩ ⟨w, hw⟩
simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists,
not_and, and_congr_right_iff]
intro
constructor
· intro hs
refine Sym2.ind ?_
rintro ⟨v', hv'⟩ ⟨w', hw'⟩
simp only [Sym2.map_mk, Sym2.eq]
contrapose
rintro (_ | _) <;> simpa o... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Subgraph | {
"line": 1110,
"column": 2
} | {
"line": 1122,
"column": 21
} | {
"line": 1124,
"column": 0
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nG' : G.Subgraph\ns : Set (Sym2 ↑G'.verts)\n⊢ G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map Subtype.val '' s)).coe",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"SimpleGraph.deleteEdges",
"Eq.mpr",
"Sym2.Rel",
"Sym2.map",
... | [] | ext ⟨v, hv⟩ ⟨w, hw⟩
simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists,
not_and, and_congr_right_iff]
intro
constructor
· intro hs
refine Sym2.ind ?_
rintro ⟨v', hv'⟩ ⟨w', hw'⟩
simp only [Sym2.map_mk, Sym2.eq]
contrapose
rintro (_ | _) <;> simpa o... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Subgraph | {
"line": 1260,
"column": 6
} | {
"line": 1260,
"column": 66
} | {
"line": 1261,
"column": 4
} | [
{
"pp": "case Adj.mp\nV : Type u\nG : SimpleGraph V\nv w : V\nhvw : G.Adj v w\nx✝¹ x✝ : V\nh : v = x✝¹ ∧ w = x✝ ∨ v = x✝ ∧ w = x✝¹\n⊢ (⊤.induce {v, w}).Adj x✝¹ x✝",
"ppTerm": "?Adj.mp",
"assigned": true,
"usedConstants": [
"SimpleGraph.Adj.symm",
"SimpleGraph.Subgraph.induce_adj",
... | [] | obtain ⟨rfl, rfl⟩ | ⟨rfl, rfl⟩ := h <;> simp [hvw, hvw.symm] | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Combinatorics.SimpleGraph.Paths | {
"line": 234,
"column": 2
} | {
"line": 234,
"column": 23
} | {
"line": 235,
"column": 2
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nv w v' w' : V\np₁ : G.Walk v w\np₂ : G.Walk v' w'\nh : p₁.IsSubwalk p₂\nh₂ : p₂.IsTrail\n⊢ p₁.IsTrail",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"SimpleGraph.Walk",
"Exists",
"Exists.casesOn",
"SimpleGraph.Walk.IsSubwalk"... | [
"V : Type u\nG : SimpleGraph V\nv w v' w' : V\np₁ : G.Walk v w\np₂ : G.Walk v' w'\nh₂ : p₂.IsTrail\nw✝¹ : G.Walk v' v\nw✝ : G.Walk w w'\nh : p₂ = (w✝¹.append p₁).append w✝\n⊢ p₁.IsTrail"
] | obtain ⟨_, _, h⟩ := h | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Combinatorics.SimpleGraph.Paths | {
"line": 240,
"column": 2
} | {
"line": 240,
"column": 23
} | {
"line": 241,
"column": 2
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nv w v' w' : V\np₁ : G.Walk v w\np₂ : G.Walk v' w'\nh : p₁.IsSubwalk p₂\nh₂ : p₂.IsPath\n⊢ p₁.IsPath",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"SimpleGraph.Walk",
"Exists",
"Exists.casesOn",
"SimpleGraph.Walk.IsSubwalk",
... | [
"V : Type u\nG : SimpleGraph V\nv w v' w' : V\np₁ : G.Walk v w\np₂ : G.Walk v' w'\nh₂ : p₂.IsPath\nw✝¹ : G.Walk v' v\nw✝ : G.Walk w w'\nh : p₂ = (w✝¹.append p₁).append w✝\n⊢ p₁.IsPath"
] | obtain ⟨_, _, h⟩ := h | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Combinatorics.SimpleGraph.Paths | {
"line": 249,
"column": 65
} | {
"line": 249,
"column": 80
} | {
"line": 249,
"column": 81
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v w : V\np : G.Walk u v\nh : G.Adj v w\n⊢ (p.concat h).reverse.IsPath ↔ p.reverse.IsPath ∧ w ∉ p.support",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"SimpleGraph.Adj.symm",
"congrArg",
"SimpleGraph.Walk.support... | [
"V : Type u\nG : SimpleGraph V\nu v w : V\np : G.Walk u v\nh : G.Adj v w\n⊢ (cons ⋯ p.reverse).IsPath ↔ p.reverse.IsPath ∧ w ∉ p.support"
] | reverse_concat, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SimpleGraph.Paths | {
"line": 298,
"column": 2
} | {
"line": 298,
"column": 55
} | {
"line": 300,
"column": 0
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v w : V\np : G.Walk u v\nq : G.Walk v w\nhpq : (p.append q).IsPath\nx : V\nhx : x ∈ p.support\nhyv : x ≠ v\nhy : x ∈ q.support\nhq : ¬q.Nil\nhx' : x ∈ q.tail.support\n⊢ False",
"ppTerm": "?m.63",
"assigned": true,
"usedConstants": [
"SimpleGraph.Walk.I... | [] | exact IsPath.disjoint_support_of_append hpq hq hx hx' | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Combinatorics.SimpleGraph.Paths | {
"line": 347,
"column": 39
} | {
"line": 347,
"column": 54
} | {
"line": 347,
"column": 54
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v : V\np : G.Walk u v\nq : G.Walk v u\nh : ¬p.Nil\nhcyc : (p.append q).IsCycle\nthis : p.support.tail.Nodup ∧ q.support.tail.Nodup ∧ p.support.tail.Disjoint q.support.tail\n⊢ (v :: q.support.tail).Nodup",
"ppTerm": "?m.59",
"assigned": true,
"usedConstants":... | [
"V : Type u\nG : SimpleGraph V\nu v : V\np : G.Walk u v\nq : G.Walk v u\nh : ¬p.Nil\nhcyc : (p.append q).IsCycle\nthis : p.support.tail.Nodup ∧ q.support.tail.Nodup ∧ p.support.tail.Disjoint q.support.tail\n⊢ v ∉ q.support.tail ∧ q.support.tail.Nodup"
] | List.nodup_cons | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SimpleGraph.Regularity.Chunk | {
"line": 356,
"column": 6
} | {
"line": 356,
"column": 54
} | {
"line": 357,
"column": 6
} | [
{
"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... | [
"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.parts\nhV : V ∈... | have rflU := Subset.refl (chunk hP G ε hU).parts | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Combinatorics.SimpleGraph.Triangle.Basic | {
"line": 174,
"column": 6
} | {
"line": 174,
"column": 25
} | {
"line": 174,
"column": 25
} | [
{
"pp": "α : Type u_1\nG : SimpleGraph α\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\ninst✝ : DecidableRel G.Adj\nhG : G.LocallyLinear\na b c : α\nhab : G.Adj a b\nhac : G.Adj a c\nhbc : G.Adj b c\n⊢ #{s(b, c)} + 1 = 2",
"ppTerm": "?m.144",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [] | rw [card_singleton] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Combinatorics.SimpleGraph.Triangle.Basic | {
"line": 174,
"column": 6
} | {
"line": 174,
"column": 25
} | {
"line": 174,
"column": 25
} | [
{
"pp": "α : Type u_1\nG : SimpleGraph α\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\ninst✝ : DecidableRel G.Adj\nhG : G.LocallyLinear\na b c : α\nhab : G.Adj a b\nhac : G.Adj a c\nhbc : G.Adj b c\n⊢ #{s(b, c)} + 1 = 2",
"ppTerm": "?m.144",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [] | rw [card_singleton] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Triangle.Basic | {
"line": 174,
"column": 6
} | {
"line": 174,
"column": 25
} | {
"line": 174,
"column": 25
} | [
{
"pp": "α : Type u_1\nG : SimpleGraph α\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\ninst✝ : DecidableRel G.Adj\nhG : G.LocallyLinear\na b c : α\nhab : G.Adj a b\nhac : G.Adj a c\nhbc : G.Adj b c\n⊢ #{s(b, c)} + 1 = 2",
"ppTerm": "?m.144",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [] | rw [card_singleton] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Clique | {
"line": 637,
"column": 32
} | {
"line": 637,
"column": 45
} | {
"line": 637,
"column": 45
} | [
{
"pp": "α : Type u_1\nG : SimpleGraph α\ns : Set α\nn : ℕ\nh : ∀ (t : Finset ↑s), ¬G.IsNClique n (map (Embedding.subtype fun x ↦ x ∈ s) t)\nt : Finset α\nht : ↑t ⊆ s\nthis : ¬G.IsNClique n (map (Embedding.subtype fun x ↦ x ∈ s) (Finset.subtype (fun x ↦ x ∈ s) t))\n⊢ ¬G.IsNClique n (filter (Membership.mem s) t)... | [
"α : Type u_1\nG : SimpleGraph α\ns : Set α\nn : ℕ\nh : ∀ (t : Finset ↑s), ¬G.IsNClique n (map (Embedding.subtype fun x ↦ x ∈ s) t)\nt : Finset α\nht : ↑t ⊆ s\nthis : ¬G.IsNClique n (map (Embedding.subtype fun x ↦ x ∈ s) (Finset.subtype (fun x ↦ x ∈ s) t))\n⊢ ¬G.IsNClique n (map (Embedding.subtype (Membership.mem s... | ← subtype_map | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SimpleGraph.Triangle.Basic | {
"line": 261,
"column": 2
} | {
"line": 261,
"column": 85
} | {
"line": 262,
"column": 2
} | [
{
"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\nhε : G.FarFromTriangleFree ε\n⊢ ε < 2⁻¹",
"ppTerm": "?m.21",
"assigned": true,
"usedConst... | [
"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\nhε : G.FarFromTriangleFree ε\n⊢ ε * ↑(Fintype.card α) ^ 2 < 2⁻¹ * ↑(Fintype.card α) ^ 2"
] | refine lt_of_mul_lt_mul_right (α := 𝕜) (a := Fintype.card α ^ 2) ?_ (by positivity) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Combinatorics.SimpleGraph.Triangle.Removal | {
"line": 91,
"column": 2
} | {
"line": 91,
"column": 34
} | {
"line": 92,
"column": 2
} | [
{
"pp": "case refl.refl.refl\nα : Type u_1\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nP : Finpartition univ\nε : ℝ\nhε : 0 < ε\nhε₁ : ε ≤ 1\nhP₁ : P.IsEquipartition\nhP₃ : #P.parts ≤ bound (ε / 8) ⌈4 / ε⌉₊\nx y z : α\ns : Finset α\nhX : s ∈ P.parts\nY : Finset α\... | [
"case refl.refl.refl\nα : Type u_1\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nP : Finpartition univ\nε : ℝ\nhε : 0 < ε\nhε₁ : ε ≤ 1\nhP₁ : P.IsEquipartition\nhP₃ : #P.parts ≤ bound (ε / 8) ⌈4 / ε⌉₊\nx y z : α\ns : Finset α\nhX : s ∈ P.parts\nY : Finset α\nhY : Y ∈ P.... | have dXY := P.disjoint hX hY nXY | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Combinatorics.SimpleGraph.Triangle.Removal | {
"line": 115,
"column": 2
} | {
"line": 132,
"column": 54
} | {
"line": 134,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝³ : DecidableEq α\ninst✝² : Fintype α\nG : SimpleGraph α\ninst✝¹ : DecidableRel G.Adj\nP : Finpartition univ\nε : ℝ\ninst✝ : Nonempty α\nhε : 0 < ε\nhP : P.IsEquipartition\nhPε : P.IsUniform G (ε / 8)\nhP' : 4 / ε ≤ ↑(#P.parts)\nA : Finset (α × α) :=\n (P.nonUniforms G (ε / 8)).biUn... | [] | calc
_ = (#((univ ×ˢ univ).filter fun (x, y) ↦
G.Adj x y ∧ ¬(G.regularityReduced P (ε / 8) (ε / 4)).Adj x y) : ℝ) := by
rw [univ_product_univ, mul_sub, filter_and_not, cast_card_sdiff]
· norm_cast
rw [two_mul_card_edgeFinset, two_mul_card_edgeFinset]
· gcongr with xy _
ex... | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcTactic |
Mathlib.Combinatorics.Additive.Energy | {
"line": 186,
"column": 4
} | {
"line": 186,
"column": 29
} | {
"line": 187,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝² : DecidableEq α\ninst✝¹ : CommGroup α\ninst✝ : Fintype α\nt : Finset α\nf : α × α × α → (α × α) × α × α := fun x ↦ ((x.1 * x.2.2, x.1 * x.2.1), x.2)\na₁ b₁ c₁ : α\na✝ : (a₁, b₁, c₁) ∈ ↑(univ ×ˢ t ×ˢ t)\na₂ : α\nh₂ : (a₂, b₁, c₁) ∈ ↑(univ ×ˢ t ×ˢ t)\nh : a₁ * c₁ = a₂ * c₁ ∧ a₁ * b₁ ... | [] | rw [mul_right_cancel h.1] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.Finset.Slice | {
"line": 67,
"column": 59
} | {
"line": 68,
"column": 30
} | {
"line": 70,
"column": 0
} | [
{
"pp": "α : Type u_1\nι : Sort u_2\nκ : ι → Sort u_3\nr : ℕ\nf : (i : ι) → κ i → Set (Finset α)\n⊢ Sized r (⋃ i, ⋃ j, f i j) ↔ ∀ (i : ι) (j : κ i), Sized r (f i j)",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Set.Sized",
"congrArg",
"Finset",
"iff_self",
... | [] | by
simp only [Set.sized_iUnion] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.Colex | {
"line": 250,
"column": 59
} | {
"line": 251,
"column": 69
} | {
"line": 253,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝¹ : PartialOrder α\ns : Finset α\na b : α\ninst✝ : DecidableEq α\nha : a ∉ s\nhb : b ∉ s\n⊢ toColex (insert a s) < toColex (insert b s) ↔ a < b",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"Equiv.instEquivLike",
... | [] | by
rw [← cons_eq_insert _ _ ha, ← cons_eq_insert _ _ hb, cons_lt_cons] | [anonymous] | Lean.Parser.Term.byTactic |
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