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.Localization.Predicate | {
"line": 480,
"column": 49
} | {
"line": 480,
"column": 62
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : Category.{v_2, u_2} D\nL : C ⥤ D\nW : MorphismProperty C\ninst✝² : L.IsLocalization W\nP : MorphismProperty D\ninst✝¹ : P.RespectsIso\ninst✝ : P.IsMultiplicative\nh₁ : ∀ ⦃X Y : C⦄ (f : X ⟶ Y), P (L.map f)\nh₂ : ∀ ⦃X Y : C⦄ (f : X ⟶ Y)... | isoOfHom_hom, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Localization.CalculusOfFractions | {
"line": 268,
"column": 4
} | {
"line": 268,
"column": 24
} | [
{
"pp": "case refine_1\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\nW : MorphismProperty C\nX Y : C\nz₁ z₂ z₃ : W.LeftFraction X Y\ninst✝ : W.HasLeftCalculusOfFractions\nZ₄ : C\nt₁ : z₁.Y' ⟶ Z₄\nt₂ : z₂.Y' ⟶ Z₄\nhst : z₁.s ≫ t₁ = z₂.s ≫ t₂\nhft : z₁.f ≫ t₁ = z₂.f ≫ t₂\nht : W (z₁.s ≫ t₁)\nZ₅ : C\nu₂ : z₂.Y' ⟶... | rw [reassoc_of% fac] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Localization.CalculusOfFractions | {
"line": 268,
"column": 4
} | {
"line": 268,
"column": 24
} | [
{
"pp": "case refine_1\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\nW : MorphismProperty C\nX Y : C\nz₁ z₂ z₃ : W.LeftFraction X Y\ninst✝ : W.HasLeftCalculusOfFractions\nZ₄ : C\nt₁ : z₁.Y' ⟶ Z₄\nt₂ : z₂.Y' ⟶ Z₄\nhst : z₁.s ≫ t₁ = z₂.s ≫ t₂\nhft : z₁.f ≫ t₁ = z₂.f ≫ t₂\nht : W (z₁.s ≫ t₁)\nZ₅ : C\nu₂ : z₂.Y' ⟶... | rw [reassoc_of% fac] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Localization.CalculusOfFractions | {
"line": 268,
"column": 4
} | {
"line": 268,
"column": 24
} | [
{
"pp": "case refine_1\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\nW : MorphismProperty C\nX Y : C\nz₁ z₂ z₃ : W.LeftFraction X Y\ninst✝ : W.HasLeftCalculusOfFractions\nZ₄ : C\nt₁ : z₁.Y' ⟶ Z₄\nt₂ : z₂.Y' ⟶ Z₄\nhst : z₁.s ≫ t₁ = z₂.s ≫ t₂\nhft : z₁.f ≫ t₁ = z₂.f ≫ t₂\nht : W (z₁.s ≫ t₁)\nZ₅ : C\nu₂ : z₂.Y' ⟶... | rw [reassoc_of% fac] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Localization.CalculusOfFractions.Fractions | {
"line": 191,
"column": 2
} | {
"line": 191,
"column": 33
} | [
{
"pp": "C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nW : MorphismProperty C\nX Y : C\nz₁ z₂ : W.LeftFraction₂ X Y\nZ : C\nt₁ : z₁.Y' ⟶ Z\nt₂ : z₂.Y' ⟶ Z\nhst : z₁.s ≫ t₁ = z₂.s ≫ t₂\nhft : z₁.f ≫ t₁ = z₂.f ≫ t₂\nw✝ : z₁.f' ≫ t₁ = z₂.f' ≫ t₂\nht : W (z₁.s ≫ t₁)\n⊢ LeftFractionRel z₁.fst z₂.fst",
"usedConsta... | exact ⟨Z, t₁, t₂, hst, hft, ht⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Localization.CalculusOfFractions.Preadditive | {
"line": 152,
"column": 66
} | {
"line": 152,
"column": 68
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : Category.{v_2, u_2} D\ninst✝² : Preadditive C\nL : C ⥤ D\nW : MorphismProperty C\ninst✝¹ : L.IsLocalization W\ninst✝ : W.HasLeftCalculusOfFractions\nX Y : C\nf : L.obj X ⟶ L.obj Y\nα : W.LeftFraction X Y\nhα : f = α.map L ⋯\n⊢ { Y' :=... | hα | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Localization.LocalizerMorphism | {
"line": 179,
"column": 2
} | {
"line": 179,
"column": 23
} | [
{
"pp": "C₁ : Type u₁\nC₂ : Type u₂\nD₁ : Type u₄\nD₂ : Type u₅\ninst✝⁶ : Category.{v₁, u₁} C₁\ninst✝⁵ : Category.{v₂, u₂} C₂\ninst✝⁴ : Category.{v₄, u₄} D₁\ninst✝³ : Category.{v₅, u₅} D₂\nW₁ : MorphismProperty C₁\nW₂ : MorphismProperty C₂\nΦ : LocalizerMorphism W₁ W₂\nL₁ : C₁ ⥤ D₁\ninst✝² : L₁.IsLocalization W... | exact h.isEquivalence | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.ObjectProperty.Shift | {
"line": 88,
"column": 4
} | {
"line": 88,
"column": 66
} | [
{
"pp": "C : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\nP Q : ObjectProperty C\nA : Type u_2\ninst✝⁴ : AddMonoid A\ninst✝³ : HasShift C A\nE : Type u_3\ninst✝² : Category.{v_2, u_3} E\ninst✝¹ : HasShift E A\na : A\ninst✝ : P.IsStableUnderShiftBy a\nX Y : C\nhY : P Y\ne : X ≅ Y\n⊢ P.isoClosure.shift a X",
"us... | exact ⟨Y⟦a⟧, P.le_shift a _ hY, ⟨(shiftFunctor C a).mapIso e⟩⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Triangulated.HomologicalFunctor | {
"line": 141,
"column": 4
} | {
"line": 141,
"column": 98
} | [
{
"pp": "C : Type u_1\nD : Type u_2\nA : Type u_3\ninst✝¹⁴ : Category.{v_1, u_1} C\ninst✝¹³ : HasShift C ℤ\ninst✝¹² : Category.{v_2, u_2} D\ninst✝¹¹ : HasZeroObject D\ninst✝¹⁰ : HasShift D ℤ\ninst✝⁹ : Preadditive D\ninst✝⁸ : ∀ (n : ℤ), (shiftFunctor D n).Additive\ninst✝⁷ : Pretriangulated D\ninst✝⁶ : Category.{... | have ex : S.Exact := F.map_distinguished_exact _ (binaryBiproductTriangle_distinguished X₁ X₂) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.CategoryTheory.Shift.ShiftedHom | {
"line": 139,
"column": 24
} | {
"line": 139,
"column": 44
} | [
{
"pp": "C : Type u_1\ninst✝³ : Category.{v_1, u_1} C\nM : Type u_4\ninst✝² : AddMonoid M\ninst✝¹ : HasShift C M\nX Y Z : C\ninst✝ : Preadditive C\na b c : M\nα₁ α₂ : ShiftedHom X Y a\nβ : ShiftedHom Y Z b\nh : b + a = c\n⊢ (α₁ + α₂) ≫ (shiftFunctor C a).map β ≫ (shiftFunctorAdd' C b a c h).inv.app Z =\n α₁ ... | Preadditive.add_comp | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.Localization | {
"line": 70,
"column": 59
} | {
"line": 80,
"column": 70
} | [
{
"pp": "C : Type u_1\ninst✝³ : Category.{v_1, u_1} C\nι : Type u_2\nc : ComplexShape ι\ninst✝² : HasZeroMorphisms C\ninst✝¹ : CategoryWithHomology C\ninst✝ : (HomologicalComplex.quasiIso C c).HasLocalization\nK L : HomologicalComplex C c\nf : K ⟶ L\n⊢ IsIso (Q.map f) ↔ HomologicalComplex.quasiIso C c f",
"... | by
constructor
· intro h
rw [HomologicalComplex.mem_quasiIso_iff, quasiIso_iff]
intro i
rw [quasiIsoAt_iff_isIso_homologyMap]
refine (NatIso.isIso_map_iff (homologyFunctorFactors C c i) f).1 ?_
dsimp
infer_instance
· intro h
exact Localization.inverts Q (HomologicalComplex.quasiIso C c... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Triangulated.Subcategory | {
"line": 133,
"column": 2
} | {
"line": 133,
"column": 82
} | [
{
"pp": "C : Type u_1\ninst✝⁷ : Category.{v_1, u_1} C\ninst✝⁶ : HasZeroObject C\ninst✝⁵ : HasShift C ℤ\ninst✝⁴ : Preadditive C\ninst✝³ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝² : Pretriangulated C\nP : ObjectProperty C\ninst✝¹ : P.IsTriangulatedClosed₃\ninst✝ : P.IsClosedUnderIsomorphisms\nT : Triangle C... | simpa only [isoClosure_eq_self] using P.ext_of_isTriangulatedClosed₃' T hT h₁ h₂ | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.CategoryTheory.Triangulated.Subcategory | {
"line": 133,
"column": 2
} | {
"line": 133,
"column": 82
} | [
{
"pp": "C : Type u_1\ninst✝⁷ : Category.{v_1, u_1} C\ninst✝⁶ : HasZeroObject C\ninst✝⁵ : HasShift C ℤ\ninst✝⁴ : Preadditive C\ninst✝³ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝² : Pretriangulated C\nP : ObjectProperty C\ninst✝¹ : P.IsTriangulatedClosed₃\ninst✝ : P.IsClosedUnderIsomorphisms\nT : Triangle C... | simpa only [isoClosure_eq_self] using P.ext_of_isTriangulatedClosed₃' T hT h₁ h₂ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Triangulated.Subcategory | {
"line": 133,
"column": 2
} | {
"line": 133,
"column": 82
} | [
{
"pp": "C : Type u_1\ninst✝⁷ : Category.{v_1, u_1} C\ninst✝⁶ : HasZeroObject C\ninst✝⁵ : HasShift C ℤ\ninst✝⁴ : Preadditive C\ninst✝³ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝² : Pretriangulated C\nP : ObjectProperty C\ninst✝¹ : P.IsTriangulatedClosed₃\ninst✝ : P.IsClosedUnderIsomorphisms\nT : Triangle C... | simpa only [isoClosure_eq_self] using P.ext_of_isTriangulatedClosed₃' T hT h₁ h₂ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Shift.SingleFunctors | {
"line": 189,
"column": 27
} | {
"line": 189,
"column": 42
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Category.{v_2, u_2} D\nA : Type u_5\ninst✝¹ : AddMonoid A\ninst✝ : HasShift D A\nF G : SingleFunctors C D A\ne : F ≅ G\nn : A\nX : C\n⊢ (e.hom.hom n ≫ e.inv.hom n).app X = 𝟙 ((F.functor n).obj X)",
"usedConstants": [
"Eq.mp... | hom_inv_id_hom, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.Embedding.Extend | {
"line": 134,
"column": 10
} | {
"line": 135,
"column": 73
} | [
{
"pp": "case inr.inr.inr\nι : Type u_1\nι' : Type u_2\nc : ComplexShape ι\nc' : ComplexShape ι'\nC : Type u_3\ninst✝² : Category.{v_1, u_3} C\ninst✝¹ : HasZeroObject C\ninst✝ : HasZeroMorphisms C\nK L M : HomologicalComplex C c\nφ : K ⟶ L\nφ' : L ⟶ M\ne : c.Embedding c'\ni' j' k' : ι'\nx✝¹ : c'.Rel i' j'\nx✝ :... | rw [extend.d_eq K hi hj, extend.d_eq K hj hk, assoc, assoc,
Iso.inv_hom_id_assoc, K.d_comp_d_assoc, zero_comp, comp_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Homology.Embedding.Extend | {
"line": 134,
"column": 10
} | {
"line": 135,
"column": 73
} | [
{
"pp": "case inr.inr.inr\nι : Type u_1\nι' : Type u_2\nc : ComplexShape ι\nc' : ComplexShape ι'\nC : Type u_3\ninst✝² : Category.{v_1, u_3} C\ninst✝¹ : HasZeroObject C\ninst✝ : HasZeroMorphisms C\nK L M : HomologicalComplex C c\nφ : K ⟶ L\nφ' : L ⟶ M\ne : c.Embedding c'\ni' j' k' : ι'\nx✝¹ : c'.Rel i' j'\nx✝ :... | rw [extend.d_eq K hi hj, extend.d_eq K hj hk, assoc, assoc,
Iso.inv_hom_id_assoc, K.d_comp_d_assoc, zero_comp, comp_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.Embedding.Extend | {
"line": 134,
"column": 10
} | {
"line": 135,
"column": 73
} | [
{
"pp": "case inr.inr.inr\nι : Type u_1\nι' : Type u_2\nc : ComplexShape ι\nc' : ComplexShape ι'\nC : Type u_3\ninst✝² : Category.{v_1, u_3} C\ninst✝¹ : HasZeroObject C\ninst✝ : HasZeroMorphisms C\nK L M : HomologicalComplex C c\nφ : K ⟶ L\nφ' : L ⟶ M\ne : c.Embedding c'\ni' j' k' : ι'\nx✝¹ : c'.Rel i' j'\nx✝ :... | rw [extend.d_eq K hi hj, extend.d_eq K hj hk, assoc, assoc,
Iso.inv_hom_id_assoc, K.d_comp_d_assoc, zero_comp, comp_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.Embedding.Boundary | {
"line": 90,
"column": 4
} | {
"line": 90,
"column": 29
} | [
{
"pp": "case neg.hj\nι : Type u_1\nι' : Type u_2\nc : ComplexShape ι\nc' : ComplexShape ι'\ne : c.Embedding c'\ninst✝ : e.IsRelIff\nj : ι\nhij : c.prev j = j\nhij' : ¬c.Rel j j\nhj' : c'.Rel (c'.prev (e.f j)) (e.f j)\nhj : c'.Rel (c'.prev (e.f j)) (e.f j) → ∃ x, c'.Rel (e.f x) (e.f j)\ni : ι\nhi : c.Rel i j\n⊢... | rw [c.prev_eq' hi] at hij | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Homology.Embedding.TruncGEHomology | {
"line": 117,
"column": 2
} | {
"line": 117,
"column": 74
} | [
{
"pp": "case pos\nι : Type u_1\nι' : Type u_2\nc : ComplexShape ι\nc' : ComplexShape ι'\nC : Type u_3\ninst✝⁴ : Category.{v_1, u_3} C\ninst✝³ : HasZeroMorphisms C\nK L : HomologicalComplex C c'\nφ : K ⟶ L\ne : c.Embedding c'\ninst✝² : e.IsTruncGE\ninst✝¹ : ∀ (i' : ι'), K.HasHomology i'\ninst✝ : ∀ (i' : ι'), L.... | · exact ShortComplex.HasHomology.mk' (homologyData K e _ _ _ rfl rfl hi) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Homology.Embedding.TruncLEHomology | {
"line": 130,
"column": 10
} | {
"line": 130,
"column": 24
} | [
{
"pp": "ι : Type u_1\nι' : Type u_2\nc : ComplexShape ι\nc' : ComplexShape ι'\nC : Type u_3\ninst✝² : Category.{v_1, u_3} C\ninst✝¹ : Abelian C\nK : HomologicalComplex C c'\ne : c.Embedding c'\ninst✝ : e.IsTruncLE\ni' j' : ι'\nhij' : c'.Rel i' j'\nhj : ¬∃ j, e.f j = j'\n⊢ ∀ (i : ι), e.f i ≠ j'",
"usedConst... | simpa using hj | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Algebra.Homology.Embedding.TruncLEHomology | {
"line": 130,
"column": 10
} | {
"line": 130,
"column": 24
} | [
{
"pp": "ι : Type u_1\nι' : Type u_2\nc : ComplexShape ι\nc' : ComplexShape ι'\nC : Type u_3\ninst✝² : Category.{v_1, u_3} C\ninst✝¹ : Abelian C\nK : HomologicalComplex C c'\ne : c.Embedding c'\ninst✝ : e.IsTruncLE\ni' j' : ι'\nhij' : c'.Rel i' j'\nhj : ¬∃ j, e.f j = j'\n⊢ ∀ (i : ι), e.f i ≠ j'",
"usedConst... | simpa using hj | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.Embedding.TruncLEHomology | {
"line": 130,
"column": 10
} | {
"line": 130,
"column": 24
} | [
{
"pp": "ι : Type u_1\nι' : Type u_2\nc : ComplexShape ι\nc' : ComplexShape ι'\nC : Type u_3\ninst✝² : Category.{v_1, u_3} C\ninst✝¹ : Abelian C\nK : HomologicalComplex C c'\ne : c.Embedding c'\ninst✝ : e.IsTruncLE\ni' j' : ι'\nhij' : c'.Rel i' j'\nhj : ¬∃ j, e.f j = j'\n⊢ ∀ (i : ι), e.f i ≠ j'",
"usedConst... | simpa using hj | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.Embedding.TruncGE | {
"line": 388,
"column": 4
} | {
"line": 391,
"column": 18
} | [
{
"pp": "case pos\nι : Type u_1\nι' : Type u_2\nc : ComplexShape ι\nc' : ComplexShape ι'\nC : Type u_3\ninst✝⁷ : Category.{v_1, u_3} C\ninst✝⁶ : HasZeroMorphisms C\nK L M : HomologicalComplex C c'\nφ : K ⟶ L\nφ' : L ⟶ M\ne : c.Embedding c'\ninst✝⁵ : e.IsTruncGE\ninst✝⁴ : ∀ (i' : ι'), K.HasHomology i'\ninst✝³ : ... | obtain ⟨i, hi⟩ := hn
dsimp [πTruncGE]
rw [e.isIso_liftExtend_f_iff _ _ hi]
infer_instance | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.Embedding.TruncGE | {
"line": 388,
"column": 4
} | {
"line": 391,
"column": 18
} | [
{
"pp": "case pos\nι : Type u_1\nι' : Type u_2\nc : ComplexShape ι\nc' : ComplexShape ι'\nC : Type u_3\ninst✝⁷ : Category.{v_1, u_3} C\ninst✝⁶ : HasZeroMorphisms C\nK L M : HomologicalComplex C c'\nφ : K ⟶ L\nφ' : L ⟶ M\ne : c.Embedding c'\ninst✝⁵ : e.IsTruncGE\ninst✝⁴ : ∀ (i' : ι'), K.HasHomology i'\ninst✝³ : ... | obtain ⟨i, hi⟩ := hn
dsimp [πTruncGE]
rw [e.isIso_liftExtend_f_iff _ _ hi]
infer_instance | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.Embedding.CochainComplex | {
"line": 306,
"column": 2
} | {
"line": 308,
"column": 31
} | [
{
"pp": "case mp\nC : Type u_1\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : HasZeroMorphisms C\nK L : CochainComplex C ℤ\nφ : K ⟶ L\ninst✝² : HasZeroObject C\ninst✝¹ : ∀ (i : ℤ), HasHomology K i\ninst✝ : ∀ (i : ℤ), HasHomology L i\nn : ℤ\n⊢ (∀ (i : ℕ) (i' : ℤ), (embeddingUpIntLE n).f i = i' → QuasiIsoAt φ i') → ∀ ... | · intro h i hi
obtain ⟨k, rfl⟩ := Int.le.dest hi
exact h k _ (by dsimp; lia) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Localization.HomEquiv | {
"line": 165,
"column": 62
} | {
"line": 165,
"column": 75
} | [
{
"pp": "C : Type u_1\nD₁ : Type u_5\nD₂ : Type u_6\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : Category.{v_5, u_5} D₁\ninst✝² : Category.{v_6, u_6} D₂\nW : MorphismProperty C\nL₁ : C ⥤ D₁\ninst✝¹ : L₁.IsLocalization W\nL₂ : C ⥤ D₂\ninst✝ : L₂.IsLocalization W\nX Y : C\nf : Y ⟶ X\nhf : W f\n⊢ (homEquiv W L₁ L₂) (... | isoOfHom_hom, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Abelian.DiagramLemmas.Four | {
"line": 75,
"column": 2
} | {
"line": 75,
"column": 77
} | [
{
"pp": "case h\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Abelian C\nR₁ R₂ : ComposableArrows C 3\nφ : R₁ ⟶ R₂\nhR₁ : R₁.map' 0 2 mono_of_epi_of_mono_of_mono'._proof_2 mono_of_epi_of_mono_of_mono'._proof_4 = 0\nhR₁' :\n (mk₂ (R₁.map' 1 2 mono_of_epi_of_mono_of_mono'._proof_6 mono_of_epi_of_mono_of... | obtain ⟨A₁, π₁, _, f₁, hf₁⟩ := (hR₁'.exact 0).exact_up_to_refinements f₂ h₂ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.CategoryTheory.Shift.Opposite | {
"line": 99,
"column": 2
} | {
"line": 100,
"column": 16
} | [
{
"pp": "C : Type u_1\ninst✝⁴ : Category.{v_1, u_1} C\nA : Type u_2\ninst✝³ : AddMonoid A\ninst✝² : HasShift C A\ninst✝¹ : Preadditive C\nn : A\ninst✝ : (shiftFunctor C n).Additive\n⊢ (shiftFunctor (OppositeShift C A) n).Additive",
"usedConstants": [
"CategoryTheory.Functor.op",
"Opposite",
... | change (shiftFunctor C n).op.Additive
infer_instance | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Shift.Opposite | {
"line": 99,
"column": 2
} | {
"line": 100,
"column": 16
} | [
{
"pp": "C : Type u_1\ninst✝⁴ : Category.{v_1, u_1} C\nA : Type u_2\ninst✝³ : AddMonoid A\ninst✝² : HasShift C A\ninst✝¹ : Preadditive C\nn : A\ninst✝ : (shiftFunctor C n).Additive\n⊢ (shiftFunctor (OppositeShift C A) n).Additive",
"usedConstants": [
"CategoryTheory.Functor.op",
"Opposite",
... | change (shiftFunctor C n).op.Additive
infer_instance | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Shift.ShiftedHomOpposite | {
"line": 146,
"column": 2
} | {
"line": 146,
"column": 48
} | [
{
"pp": "C : Type u_1\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : HasShift C ℤ\nX Y : C\ninst✝¹ : Preadditive C\ninst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive\nn : ℤ\nx y : ShiftedHom (Opposite.op Y) (Opposite.op X) n\n⊢ ((opShiftFunctorEquivalence C n).unitIso.inv.app (Opposite.op X)).unop ≫\n (shiftFuncto... | rw [← Preadditive.comp_add, ← Functor.map_add] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Shift.ShiftedHomOpposite | {
"line": 154,
"column": 24
} | {
"line": 154,
"column": 44
} | [
{
"pp": "C : Type u_1\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : HasShift C ℤ\nX Y : C\ninst✝¹ : Preadditive C\ninst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive\nn a : ℤ\nx y : Opposite.op ((shiftFunctor C a).obj Y) ⟶ (shiftFunctor Cᵒᵖ n).obj (Opposite.op X)\na' : ℤ\nh : n + a = a'\n⊢ ((opEquiv n).symm x + (opEqui... | Preadditive.add_comp | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Module.Presentation.Basic | {
"line": 314,
"column": 2
} | {
"line": 315,
"column": 21
} | [
{
"pp": "A : Type u\ninst✝² : Ring A\nrelations : Relations A\nM : Type v\ninst✝¹ : AddCommGroup M\ninst✝ : Module A M\nsolution : relations.Solution M\nh : solution.IsPresentation\n⊢ Function.Exact ⇑relations.map ⇑solution.π",
"usedConstants": [
"Function.Exact",
"Eq.mpr",
"Submodule",
... | rw [LinearMap.exact_iff, range_map, ← solution.injective_fromQuotient_iff_ker_π_eq_span]
exact h.bijective.1 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Module.Presentation.Basic | {
"line": 314,
"column": 2
} | {
"line": 315,
"column": 21
} | [
{
"pp": "A : Type u\ninst✝² : Ring A\nrelations : Relations A\nM : Type v\ninst✝¹ : AddCommGroup M\ninst✝ : Module A M\nsolution : relations.Solution M\nh : solution.IsPresentation\n⊢ Function.Exact ⇑relations.map ⇑solution.π",
"usedConstants": [
"Function.Exact",
"Eq.mpr",
"Submodule",
... | rw [LinearMap.exact_iff, range_map, ← solution.injective_fromQuotient_iff_ker_π_eq_span]
exact h.bijective.1 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Module.Presentation.Basic | {
"line": 453,
"column": 4
} | {
"line": 453,
"column": 75
} | [
{
"pp": "case h\nA : Type u\ninst✝² : Ring A\nrelations : Relations A\nM : Type v\ninst✝¹ : AddCommGroup M\ninst✝ : Module A M\nsolution : relations.Solution M\nh : solution.IsPresentationCore\nN : Type w''\nx✝¹ : AddCommGroup N\nx✝ : Module A N\nf f' : M →ₗ[A] N\nh' : solution.postcomp f = solution.postcomp f'... | have := congr_postcomp h' ULift.moduleEquiv.{_, _, w'}.symm.toLinearMap | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.LinearAlgebra.Matrix.Basis | {
"line": 238,
"column": 2
} | {
"line": 238,
"column": 27
} | [
{
"pp": "ι : Type u_1\nι' : Type u_2\nR : Type u_5\nM : Type u_6\ninst✝⁴ : CommSemiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nb : Basis ι R M\nb' : Basis ι' R M\ninst✝¹ : Fintype ι\ninst✝ : Finite ι'\nm : M\n⊢ b'.toMatrix ⇑b *ᵥ ⇑(b.repr m) = ⇑(b'.repr m)",
"usedConstants": [
"nonempty_fint... | cases nonempty_fintype ι' | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | Lean.Parser.Tactic.cases |
Mathlib.LinearAlgebra.Matrix.ToLinearEquiv | {
"line": 232,
"column": 4
} | {
"line": 232,
"column": 67
} | [
{
"pp": "case inr\nn : Type u_1\ninst✝⁴ : Fintype n\ninst✝³ : DecidableEq n\nS : Type u_2\ninst✝² : CommRing S\ninst✝¹ : LinearOrder S\ninst✝ : IsStrictOrderedRing S\nA : Matrix n n S\nh1 : Pairwise fun i j ↦ A i j < 0\nh✝ : Nonempty n\nh2 : A.det = 0\nv : n → S\nh_vnz : v ≠ 0\nh_vA : v ᵥ* A = 0\n⊢ ∃ j, ∑ i, A ... | wlog h_sup : 0 < Finset.sup' Finset.univ Finset.univ_nonempty v | Mathlib.Tactic._aux_Mathlib_Tactic_WLOG___elabRules_Mathlib_Tactic_wlog_1 | Mathlib.Tactic.wlog |
Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup | {
"line": 262,
"column": 4
} | {
"line": 262,
"column": 42
} | [
{
"pp": "case inr.refine_2\nn : Type u\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\nR : Type v\ninst✝ : CommRing R\nA : SpecialLinearGroup n R\ni : n\nh : A ∈ center (SpecialLinearGroup n R)\n⊢ (scalar n) (↑A i i) = ↑A",
"usedConstants": [
"Matrix.SpecialLinearGroup.scalar_eq_self_of_mem_center"
]... | exact scalar_eq_self_of_mem_center h i | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup | {
"line": 262,
"column": 4
} | {
"line": 262,
"column": 42
} | [
{
"pp": "case inr.refine_2\nn : Type u\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\nR : Type v\ninst✝ : CommRing R\nA : SpecialLinearGroup n R\ni : n\nh : A ∈ center (SpecialLinearGroup n R)\n⊢ (scalar n) (↑A i i) = ↑A",
"usedConstants": [
"Matrix.SpecialLinearGroup.scalar_eq_self_of_mem_center"
]... | exact scalar_eq_self_of_mem_center h i | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup | {
"line": 262,
"column": 4
} | {
"line": 262,
"column": 42
} | [
{
"pp": "case inr.refine_2\nn : Type u\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\nR : Type v\ninst✝ : CommRing R\nA : SpecialLinearGroup n R\ni : n\nh : A ∈ center (SpecialLinearGroup n R)\n⊢ (scalar n) (↑A i i) = ↑A",
"usedConstants": [
"Matrix.SpecialLinearGroup.scalar_eq_self_of_mem_center"
]... | exact scalar_eq_self_of_mem_center h i | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.TrivSqZeroExt.Basic | {
"line": 644,
"column": 6
} | {
"line": 644,
"column": 21
} | [
{
"pp": "case cons\nR : Type u\nM : Type v\ninst✝⁴ : Monoid R\ninst✝³ : AddCommMonoid M\ninst✝² : DistribMulAction R M\ninst✝¹ : DistribMulAction Rᵐᵒᵖ M\ninst✝ : SMulCommClass R Rᵐᵒᵖ M\nx : tsze R M\nxs : List (tsze R M)\nih :\n xs.prod.snd =\n (List.map (fun x ↦ (List.take x.2 (List.map fst xs)).prod •> x.... | List.take_zero, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.TrivSqZeroExt.Basic | {
"line": 716,
"column": 2
} | {
"line": 717,
"column": 21
} | [
{
"pp": "case h2\nR : Type u\nM : Type v\ninst✝³ : AddCommGroup M\ninst✝² : Semiring R\ninst✝¹ : Module Rᵐᵒᵖ M\ninst✝ : Module R M\nr : R\nx : tsze R M\nh : r * x.fst = 1\n⊢ (r + 0) •> x.snd + (0 + -(r •> x.snd <• r)) <• x.fst = 0",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
... | · rw [add_zero, zero_add, smul_neg, op_smul_op_smul, h, op_one, one_smul,
add_neg_cancel] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.TrivSqZeroExt.Basic | {
"line": 1012,
"column": 2
} | {
"line": 1012,
"column": 97
} | [
{
"pp": "case refine_2\nS : Type u_1\nR : Type u\nM : Type v\ninst✝⁹ : CommSemiring S\ninst✝⁸ : Semiring R\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : Algebra S R\ninst✝⁵ : Module S M\ninst✝⁴ : Module R M\ninst✝³ : Module Rᵐᵒᵖ M\ninst✝² : SMulCommClass R Rᵐᵒᵖ M\ninst✝¹ : IsScalarTower S R M\ninst✝ : IsScalarTower S Rᵐᵒ... | · exact le_sup_right (α := Subalgebra S _) <| Algebra.subset_adjoin <| Set.mem_range_self x.snd | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.DirectSum.Algebra | {
"line": 69,
"column": 4
} | {
"line": 70,
"column": 58
} | [
{
"pp": "ι : Type uι\nR : Type uR\nA : ι → Type uA\nB : Type uB\ninst✝⁷ : CommSemiring R\ninst✝⁶ : (i : ι) → AddCommMonoid (A i)\ninst✝⁵ : (i : ι) → Module R (A i)\ninst✝⁴ : AddMonoid ι\ninst✝³ : GSemiring A\ninst✝² : Semiring B\ninst✝¹ : GAlgebra R A\ninst✝ : Algebra R B\ns : R\nx y : GradedMonoid A\n⊢ (s • x)... | dsimp
rw [GAlgebra.smul_def, GAlgebra.smul_def, ← mul_assoc] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.DirectSum.Algebra | {
"line": 69,
"column": 4
} | {
"line": 70,
"column": 58
} | [
{
"pp": "ι : Type uι\nR : Type uR\nA : ι → Type uA\nB : Type uB\ninst✝⁷ : CommSemiring R\ninst✝⁶ : (i : ι) → AddCommMonoid (A i)\ninst✝⁵ : (i : ι) → Module R (A i)\ninst✝⁴ : AddMonoid ι\ninst✝³ : GSemiring A\ninst✝² : Semiring B\ninst✝¹ : GAlgebra R A\ninst✝ : Algebra R B\ns : R\nx y : GradedMonoid A\n⊢ (s • x)... | dsimp
rw [GAlgebra.smul_def, GAlgebra.smul_def, ← mul_assoc] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.DirectSum.Decomposition | {
"line": 64,
"column": 4
} | {
"line": 64,
"column": 26
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nσ : Type u_4\ninst✝³ : DecidableEq ι\ninst✝² : AddCommMonoid M\ninst✝¹ : SetLike σ M\ninst✝ : AddSubmonoidClass σ M\nℳ : ι → σ\nx y : Decomposition ℳ\n⊢ x = y",
"usedConstants": []
}
] | obtain ⟨_, _, xr⟩ := x | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Algebra.DirectSum.Internal | {
"line": 79,
"column": 30
} | {
"line": 79,
"column": 46
} | [
{
"pp": "case ofNat\nι : Type u_1\nσ : Type u_2\nR : Type u_4\ninst✝⁴ : Zero ι\ninst✝³ : AddGroupWithOne R\ninst✝² : SetLike σ R\ninst✝¹ : AddSubgroupClass σ R\nA : ι → σ\ninst✝ : GradedOne A\na✝ : ℕ\n⊢ ↑↑a✝ ∈ A 0",
"usedConstants": [
"Int.cast",
"Eq.mpr",
"Int.cast_natCast",
"congrA... | Int.cast_natCast | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.DirectSum.Internal | {
"line": 234,
"column": 4
} | {
"line": 235,
"column": 63
} | [
{
"pp": "ι : Type u_1\nσ : Type u_2\nR : Type u_4\ninst✝⁷ : DecidableEq ι\ninst✝⁶ : Semiring R\ninst✝⁵ : SetLike σ R\ninst✝⁴ : AddSubmonoidClass σ R\nA : ι → σ\ninst✝³ : AddCommMonoid ι\ninst✝² : PartialOrder ι\ninst✝¹ : CanonicallyOrderedAdd ι\ninst✝ : SetLike.GradedMonoid A\ni : ι\nr : ↥(A i)\nr' : ⨁ (i : ι),... | · rw [DFinsupp.sum, Finset.sum_ite_of_false, Finset.sum_const_zero]
exact fun x _ H => h ((self_le_add_right i x).trans_eq H) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.LinearAlgebra.QuadraticForm.Basic | {
"line": 289,
"column": 89
} | {
"line": 290,
"column": 69
} | [
{
"pp": "R : Type u_3\nM : Type u_4\nN : Type u_5\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R M\ninst✝ : Module R N\nQ : QuadraticMap R M N\nx x' y : M\n⊢ polar (⇑Q) (x - x') y = polar (⇑Q) x y - polar (⇑Q) x' y",
"usedConstants": [
"Eq.mpr",
"NegZer... | by
rw [sub_eq_add_neg, sub_eq_add_neg, polar_add_left, polar_neg_left] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Matrix.SesquilinearForm | {
"line": 839,
"column": 2
} | {
"line": 839,
"column": 45
} | [
{
"pp": "R : Type u_1\nn : Type u_11\ninst✝³ : CommRing R\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\ninst✝ : IsDomain R\nM : Matrix n n R\n⊢ ((toLinearMap₂' R) M).SeparatingRight ↔ M.det ≠ 0",
"usedConstants": [
"Eq.mpr",
"Pi.Function.module",
"Algebra.to_smulCommClass",
"NonUnital... | simpa using separatingRight_iff_det_ne_zero | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.LinearAlgebra.QuadraticForm.Basic | {
"line": 1156,
"column": 10
} | {
"line": 1156,
"column": 13
} | [
{
"pp": "M : Type u_4\nN : Type u_5\nR₂ : Type u\ninst✝⁵ : CommSemiring R₂\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R₂ M\ninst✝² : PartialOrder N\ninst✝¹ : AddCommMonoid N\ninst✝ : Module R₂ N\nQ : QuadraticMap R₂ M N\nhQ : Q.PosDef\nx : M\nhQx : Q x = 0\nhx : ¬x = 0\nthis : 0 < Q x\n⊢ 0 < 0",
"usedConsta... | hQx | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Matrix.SesquilinearForm | {
"line": 839,
"column": 2
} | {
"line": 839,
"column": 45
} | [
{
"pp": "R : Type u_1\nn : Type u_11\ninst✝³ : CommRing R\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\ninst✝ : IsDomain R\nM : Matrix n n R\n⊢ ((toLinearMap₂' R) M).SeparatingRight ↔ M.det ≠ 0",
"usedConstants": [
"Eq.mpr",
"Pi.Function.module",
"Algebra.to_smulCommClass",
"NonUnital... | simpa using separatingRight_iff_det_ne_zero | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.CliffordAlgebra.Grading | {
"line": 170,
"column": 6
} | {
"line": 181,
"column": 33
} | [
{
"pp": "case mem_mul_mem.mem_mul\nR : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\nn : ZMod 2\nmotive : (x : CliffordAlgebra Q) → x ∈ evenOdd Q n → Prop\nrange_ι_pow : ∀ (v : CliffordAlgebra Q) (h : v ∈ (ι Q).range ^ n.val), motive v ⋯\nadd :\... | revert hx
simp_rw [pow_two]
intro hx2
induction hx2 using Submodule.mul_induction_on' with
| mem_mul_mem m hm n hn =>
simp_rw [LinearMap.mem_range] at hm hn
obtain ⟨m₁, rfl⟩ := hm; obtain ⟨m₂, rfl⟩ := hn
simp_rw [mul_assoc _ y b]
exact ι_mul_ι_mul _ _ _ _ ihy
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Matrix.SesquilinearForm | {
"line": 839,
"column": 2
} | {
"line": 839,
"column": 45
} | [
{
"pp": "R : Type u_1\nn : Type u_11\ninst✝³ : CommRing R\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\ninst✝ : IsDomain R\nM : Matrix n n R\n⊢ ((toLinearMap₂' R) M).SeparatingRight ↔ M.det ≠ 0",
"usedConstants": [
"Eq.mpr",
"Pi.Function.module",
"Algebra.to_smulCommClass",
"NonUnital... | simpa using separatingRight_iff_det_ne_zero | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.CliffordAlgebra.Grading | {
"line": 170,
"column": 6
} | {
"line": 181,
"column": 33
} | [
{
"pp": "case mem_mul_mem.mem_mul\nR : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\nn : ZMod 2\nmotive : (x : CliffordAlgebra Q) → x ∈ evenOdd Q n → Prop\nrange_ι_pow : ∀ (v : CliffordAlgebra Q) (h : v ∈ (ι Q).range ^ n.val), motive v ⋯\nadd :\... | revert hx
simp_rw [pow_two]
intro hx2
induction hx2 using Submodule.mul_induction_on' with
| mem_mul_mem m hm n hn =>
simp_rw [LinearMap.mem_range] at hm hn
obtain ⟨m₁, rfl⟩ := hm; obtain ⟨m₂, rfl⟩ := hn
simp_rw [mul_assoc _ y b]
exact ι_mul_ι_mul _ _ _ _ ihy
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.QuadraticForm.Basic | {
"line": 1395,
"column": 2
} | {
"line": 1395,
"column": 74
} | [
{
"pp": "case succ.inr\nK : Type v\ninst✝³ : Field K\nhK : Invertible 2\nd : ℕ\nih :\n ∀ {V : Type u} [inst : AddCommGroup V] [inst_1 : Module K V] [FiniteDimensional K V] {B : BilinForm K V},\n IsSymm B → finrank K V = d → ∃ v, IsOrthoᵢ B ⇑v\nV : Type u\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ ... | obtain ⟨v', hv₁⟩ := ih (hB₂.domRestrict _ : B'.IsSymm) (Nat.succ.inj hd) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.LinearAlgebra.Multilinear.Curry | {
"line": 204,
"column": 26
} | {
"line": 204,
"column": 50
} | [
{
"pp": "case H\nR : Type uR\nS : Type uS\nι : Type uι\nι' : Type uι'\nn : ℕ\nM : Fin n.succ → Type v\nM₁ : ι → Type v₁\nM₂ : Type v₂\nM₃ : Type v₃\nM' : Type v'\ninst✝⁶ : CommSemiring R\ninst✝⁵ : (i : Fin n.succ) → AddCommMonoid (M i)\ninst✝⁴ : AddCommMonoid M'\ninst✝³ : AddCommMonoid M₂\ninst✝² : (i : Fin n.s... | simp [map_insertNth_add] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Category.ModuleCat.FilteredColimits | {
"line": 137,
"column": 4
} | {
"line": 138,
"column": 46
} | [
{
"pp": "R : Type u\ninst✝² : Ring R\nJ : Type v\ninst✝¹ : SmallCategory J\ninst✝ : IsFiltered J\nF : J ⥤ ModuleCat R\nr s : R\nx : ↑(M F)\n⊢ (r + s) • x = r • x + s • x",
"usedConstants": [
"instHSMul",
"ModuleCat",
"congrArg",
"DistribMulAction.toDistribSMul",
"AddCommGroup.t... | obtain ⟨i, x, rfl⟩ := M.mk_surjective F x
simp [_root_.add_smul, colimit_add_mk_eq'] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Category.ModuleCat.FilteredColimits | {
"line": 137,
"column": 4
} | {
"line": 138,
"column": 46
} | [
{
"pp": "R : Type u\ninst✝² : Ring R\nJ : Type v\ninst✝¹ : SmallCategory J\ninst✝ : IsFiltered J\nF : J ⥤ ModuleCat R\nr s : R\nx : ↑(M F)\n⊢ (r + s) • x = r • x + s • x",
"usedConstants": [
"instHSMul",
"ModuleCat",
"congrArg",
"DistribMulAction.toDistribSMul",
"AddCommGroup.t... | obtain ⟨i, x, rfl⟩ := M.mk_surjective F x
simp [_root_.add_smul, colimit_add_mk_eq'] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.ExteriorAlgebra.Basic | {
"line": 230,
"column": 2
} | {
"line": 230,
"column": 78
} | [
{
"pp": "R : Type u1\ninst✝³ : CommRing R\nM : Type u2\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : Nontrivial R\nx : M\n⊢ (ι R) x ≠ 1",
"usedConstants": [
"ExteriorAlgebra.ι_eq_algebraMap_iff",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Semiring.toModule",
... | rw [← (algebraMap R (ExteriorAlgebra R M)).map_one, Ne, ι_eq_algebraMap_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.ExteriorAlgebra.Basic | {
"line": 352,
"column": 6
} | {
"line": 352,
"column": 11
} | [
{
"pp": "case e_xs.e_f.h\nR : Type u1\ninst✝² : CommRing R\nM : Type u2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nn : ℕ\nf : Fin n → ↑↑(ι R).range\nhu : (List.ofFn fun i ↦ ↑(f i)).prod ∈ ↑(ι R).range ^ n\ni : Fin n\nv : M\nhv : (ι R) v = ↑(f i)\n⊢ (ι R) (ιInv ↑(f i)) = ↑(f i)",
"usedConstants": [
... | ← hv, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.ExteriorPower.Basic | {
"line": 342,
"column": 4
} | {
"line": 342,
"column": 26
} | [
{
"pp": "R : Type u\ninst✝³ : CommRing R\nn : ℕ\nM : Type u_1\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nI : Type u_4\ninst✝ : LinearOrder I\nv : I → M\nα : Fin n → I\nα_inj : Injective α\nthis : ∃ σ, (ExteriorAlgebra.ιMulti R n) ((v ∘ α) ∘ ⇑σ) ∈ span R (range (ExteriorAlgebra.ιMulti_family R n v))\n⊢ (Exte... | obtain ⟨σ, hσ⟩ := this | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Algebra.Category.ModuleCat.Presheaf.Colimits | {
"line": 104,
"column": 46
} | {
"line": 104,
"column": 73
} | [
{
"pp": "C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nR : Cᵒᵖ ⥤ RingCat\nJ : Type u₂\ninst✝² : Category.{v₂, u₂} J\nF : J ⥤ PresheafOfModules R\ninst✝¹ :\n ∀ {X Y : Cᵒᵖ} (f : X ⟶ Y),\n PreservesColimit (F ⋙ evaluation R Y) (ModuleCat.restrictScalars (RingCat.Hom.hom (R.map f)))\ninst✝ : ∀ (X : Cᵒᵖ), HasColimi... | ← ι_preservesColimitIso_inv | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Limits.Sifted | {
"line": 109,
"column": 4
} | {
"line": 109,
"column": 47
} | [
{
"pp": "case out.h\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasBinaryCoproducts C\nc₁ c₂ : C\nthis : Nonempty (StructuredArrow (c₁, c₂) (Functor.diag C))\nleft✝¹ : Discrete PUnit.{1}\nc : C\nf : (fromPUnit (c₁, c₂)).obj left✝¹ ⟶ (Functor.diag C).obj c\nleft✝ : Discrete PUnit.{1}\nc' : C\ng : (fromPUnit... | dsimp only [const_obj_obj, diag_obj] at f g | Lean.Elab.Tactic.evalDSimp | Lean.Parser.Tactic.dsimp |
Mathlib.CategoryTheory.Filtered.Final | {
"line": 293,
"column": 2
} | {
"line": 293,
"column": 61
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nF : C ⥤ D\ninst✝ : IsFilteredOrEmpty C\n⊢ F.Final ↔ ∀ (d : D), IsFiltered (StructuredArrow d F)",
"usedConstants": [
"CategoryTheory.Functor.final_of_isFiltered_structuredArrow",
"Iff.intro",
"Ca... | refine ⟨?_, fun h => final_of_isFiltered_structuredArrow F⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.CategoryTheory.Filtered.Final | {
"line": 387,
"column": 2
} | {
"line": 387,
"column": 42
} | [
{
"pp": "C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\nF : C ⥤ D\ninst✝¹ : IsFiltered C\nS S' : D\nf : S ⟶ S'\nT : C ⥤ D\ninst✝ : T.Final\n⊢ (map f).Final",
"usedConstants": [
"CategoryTheory.Functor",
"CategoryTheory.NatIso.isIso_app_of_isIso",
"Cat... | haveI := NatIso.isIso_of_isIso_app (𝟙 T) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHaveI___1 | Lean.Parser.Tactic.tacticHaveI__ |
Mathlib.CategoryTheory.Limits.Fubini | {
"line": 156,
"column": 12
} | {
"line": 158,
"column": 29
} | [
{
"pp": "J : Type u_1\nK : Type u_2\ninst✝² : Category.{v_1, u_1} J\ninst✝¹ : Category.{v_2, u_2} K\nC : Type u_3\ninst✝ : Category.{v_3, u_3} C\nF : J ⥤ K ⥤ C\nG : J × K ⥤ C\nD : DiagramOfCocones F\nQ : (j : J) → IsColimit (D.obj j)\nc : Cocone (uncurry.obj F)\nj j' : J\nf : j ⟶ j'\nk : K\n⊢ (D.obj j).ι.app k ... | simp only [Limits.CoconeMorphism.w_assoc, Limits.Cocone.precompose_obj_ι,
Limits.IsColimit.fac, NatTrans.comp_app, Category.comp_id,
Category.assoc] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Limits.Fubini | {
"line": 480,
"column": 6
} | {
"line": 480,
"column": 27
} | [
{
"pp": "J : Type u_1\nK : Type u_2\ninst✝⁴ : Category.{v_1, u_1} J\ninst✝³ : Category.{v_2, u_2} K\nC : Type u_3\ninst✝² : Category.{v_3, u_3} C\nF : J ⥤ K ⥤ C\nG : J × K ⥤ C\ninst✝¹ : HasColimitsOfShape K C\ninst✝ : HasColimit (curry.obj G ⋙ colim)\nQ : DiagramOfCocones (curry.obj G) := DiagramOfCocones.mkOfH... | simp [← h ⟨j, k⟩, Q'] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Limits.Fubini | {
"line": 599,
"column": 2
} | {
"line": 599,
"column": 89
} | [
{
"pp": "J : Type u_1\nK : Type u_2\ninst✝⁴ : Category.{v_1, u_1} J\ninst✝³ : Category.{v_2, u_2} K\nC : Type u_3\ninst✝² : Category.{v_3, u_3} C\nF : J ⥤ K ⥤ C\nG : J × K ⥤ C\ninst✝¹ : HasLimitsOfShape K C\ninst✝ : HasLimit (curry.obj G ⋙ lim)\ni : G ≅ uncurry.obj (curry.obj G)\n⊢ limit G ≅ limit (curry.obj G ... | haveI : Limits.HasLimit (uncurry.obj ((@curry J _ K _ C _).obj G)) := hasLimit_of_iso i | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHaveI___1 | Lean.Parser.Tactic.tacticHaveI__ |
Mathlib.CategoryTheory.Subfunctor.Basic | {
"line": 169,
"column": 41
} | {
"line": 171,
"column": 5
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nF : C ⥤ Type w\nG G' : Subfunctor F\nh : G ≤ G'\n⊢ homOfLe h ≫ G'.ι = G.ι",
"usedConstants": [
"CategoryTheory.Functor",
"CategoryTheory.CategoryStruct.toQuiver",
"CategoryTheory.NatTrans.ext'",
"Quiver.Hom",
"CategoryTheory.Concr... | by
ext
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Sites.Grothendieck | {
"line": 226,
"column": 2
} | {
"line": 226,
"column": 28
} | [
{
"pp": "case h\nC : Type u\ninst✝ : Category.{v, u} C\nX Y : C\nJ : GrothendieckTopology C\nf : Y ⟶ X\nS R : Sieve X\nh : J.Covers S f\nk : ∀ {Z : C} (g : Z ⟶ X), S.arrows g → J.Covers R g\nZ : C\ng : Z ⟶ Y\nhg : (Sieve.pullback f S).arrows g\n⊢ Sieve.pullback g (Sieve.pullback f R) ∈ J Z",
"usedConstants"... | rw [← Sieve.pullback_comp] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Sites.Grothendieck | {
"line": 301,
"column": 2
} | {
"line": 301,
"column": 49
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\ns : Set (GrothendieckTopology C)\n⊢ IsGLB s (sInf s)",
"usedConstants": [
"IsGLB.of_image",
"Pi.preorder",
"CategoryTheory.GrothendieckTopology.sieves",
"PartialOrder.toPreorder",
"CompleteLattice.toConditionallyCompleteLattice",
... | refine @IsGLB.of_image _ _ _ _ sieves ?_ _ _ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.CategoryTheory.Sites.Sieves | {
"line": 970,
"column": 74
} | {
"line": 972,
"column": 5
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nX : C\nR : Sieve X\n⊢ functorPullback (𝟭 C) R = R",
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"Iff.rfl",
"CategoryTheory.Functor.id",
"CategoryTheory.Sieve.ext",
"CategoryTheory.Sieve.arr... | by
ext
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Sites.Sieves | {
"line": 975,
"column": 75
} | {
"line": 977,
"column": 5
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nF : C ⥤ D\nX : C\nE : Type u₃\ninst✝ : Category.{v₃, u₃} E\nG : D ⥤ E\nR : Sieve ((F ⋙ G).obj X)\n⊢ functorPullback (F ⋙ G) R = functorPullback F (functorPullback G R)",
"usedConstants": [
"CategoryTheory.Ca... | by
ext
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Sites.Sieves | {
"line": 1041,
"column": 4
} | {
"line": 1041,
"column": 30
} | [
{
"pp": "case mp.a\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF : C ⥤ D\nX✝ : C\nR : Sieve X✝\nS : Sieve (F.obj X✝)\nhle : functorPushforward F R ≤ S\nX : C\nf : X ⟶ X✝\nhf : R.arrows f\n⊢ (functorPushforward F R).arrows (F.map f)",
"usedConstants": [
"Catego... | refine ⟨X, f, 𝟙 _, hf, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.CategoryTheory.Sites.EqualizerSheafCondition | {
"line": 141,
"column": 6
} | {
"line": 141,
"column": 45
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nP : Cᵒᵖ ⥤ Type (max v u)\nX : C\nS : Sieve X\nx : FirstObj P S.arrows\n⊢ Presieve.FamilyOfElements.Compatible ((ConcreteCategory.hom (firstObjEqFamily P S.arrows).hom) x) ↔\n (ConcreteCategory.hom (firstMap P S)) x = (ConcreteCategory.hom (secondMap P S)) x",
... | Presieve.compatible_iff_sieveCompatible | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Sites.IsSheafFor | {
"line": 424,
"column": 4
} | {
"line": 424,
"column": 46
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nP : Cᵒᵖ ⥤ Type w\nX : C\nR : Presieve X\nh : IsSeparatedFor P R\nx : FamilyOfElements P (generate R).arrows\nt₁ t₂ : P.obj (op X)\nht₁ : x.IsAmalgamation t₁\nht₂ : x.IsAmalgamation t₂\n⊢ (FamilyOfElements.restrict ⋯ x).IsAmalgamation t₂",
"usedConstants": [... | · exact isAmalgamation_restrict _ x t₂ ht₂ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Sites.IsSheafFor | {
"line": 471,
"column": 8
} | {
"line": 471,
"column": 47
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nP Q U : Cᵒᵖ ⥤ Type w\nX Y : C\nS : Sieve X\nR : Presieve X\ninst✝ : LocallySmall.{w, v₁, u₁} C\nF : Cᵒᵖ ⥤ Type w\nt : (shrinkFunctor.{w, v₁, u₁} S).toFunctor ⟶ F\n⊢ FamilyOfElements.Compatible fun Y f hf ↦ (ConcreteCategory.hom (t.app (op Y))) ⟨shrinkYonedaObj... | Presieve.compatible_iff_sieveCompatible | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Sites.IsSheafFor | {
"line": 954,
"column": 53
} | {
"line": 954,
"column": 71
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nP : Cᵒᵖ ⥤ Type w\nX Y : C\nf : X ⟶ Y\n⊢ (∀ (b : P.obj (op X)) (t₁ t₂ : P.obj (op Y)),\n (ConcreteCategory.hom (P.map f.op)) t₁ = b → (ConcreteCategory.hom (P.map f.op)) t₂ = b → t₁ = t₂) ↔\n Function.Injective ⇑(ConcreteCategory.hom (P.map f.op))",
... | Function.Injective | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.CategoryTheory.Sites.IsSheafFor | {
"line": 994,
"column": 8
} | {
"line": 994,
"column": 47
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nX : C\nP : Cᵒᵖ ⥤ Type u_1\nU : Sieve X\nB : ⦃Y : C⦄ → ⦃f : Y ⟶ X⦄ → U.arrows f → Sieve Y\nhU : IsSheafFor P U.arrows\nhB : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄ (hf : U.arrows f), IsSheafFor P (B hf).arrows\nhB' : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄ (h : U.arrows f) ⦃Z : C⦄ (g : Z ⟶ Y), IsS... | Presieve.compatible_iff_sieveCompatible | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Sites.IsSheafFor | {
"line": 981,
"column": 74
} | {
"line": 1019,
"column": 39
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nX : C\nP : Cᵒᵖ ⥤ Type u_1\nU : Sieve X\nB : ⦃Y : C⦄ → ⦃f : Y ⟶ X⦄ → U.arrows f → Sieve Y\nhU : IsSheafFor P U.arrows\nhB : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄ (hf : U.arrows f), IsSheafFor P (B hf).arrows\nhB' : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄ (h : U.arrows f) ⦃Z : C⦄ (g : Z ⟶ Y), IsS... | by
intro s hs
let y : ∀ ⦃Y⦄ ⦃f : Y ⟶ X⦄ (hf : U f), Presieve.FamilyOfElements P (B hf : Presieve Y) :=
fun Y f hf Z g hg => s _ (Presieve.bind_comp _ _ hg)
have hy : ∀ ⦃Y⦄ ⦃f : Y ⟶ X⦄ (hf : U f), (y hf).Compatible := by
intro Y f H Y₁ Y₂ Z g₁ g₂ f₁ f₂ hf₁ hf₂ comm
apply hs
apply reassoc_of% comm
... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Sites.Subsheaf | {
"line": 85,
"column": 4
} | {
"line": 86,
"column": 72
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nJ : GrothendieckTopology C\nF : Cᵒᵖ ⥤ Type w\nG : Subfunctor F\nh : Presieve.IsSheaf J F\nhG : Presieve.IsSheaf J G.toFunctor\nU : Cᵒᵖ\ns : F.obj U\nhs : s ∈ (sheafify J G).obj U\nthis : ↑(⋯.amalgamate (G.familyOfElementsOfSection s) ⋯) = s\n⊢ s ∈ G.obj U",
"u... | rw [← this]
exact ((hG _ hs).amalgamate _ (G.family_of_elements_compatible s)).2 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Sites.Subsheaf | {
"line": 85,
"column": 4
} | {
"line": 86,
"column": 72
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nJ : GrothendieckTopology C\nF : Cᵒᵖ ⥤ Type w\nG : Subfunctor F\nh : Presieve.IsSheaf J F\nhG : Presieve.IsSheaf J G.toFunctor\nU : Cᵒᵖ\ns : F.obj U\nhs : s ∈ (sheafify J G).obj U\nthis : ↑(⋯.amalgamate (G.familyOfElementsOfSection s) ⋯) = s\n⊢ s ∈ G.obj U",
"u... | rw [← this]
exact ((hG _ hs).amalgamate _ (G.family_of_elements_compatible s)).2 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Sites.Sheafification | {
"line": 205,
"column": 56
} | {
"line": 207,
"column": 24
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nD : Type u_1\ninst✝¹ : Category.{v_1, u_1} D\ninst✝ : HasWeakSheafify J D\nP : Cᵒᵖ ⥤ D\nhP : Presheaf.IsSheaf J P\n⊢ (isoSheafify J hP).inv = sheafifyLift J (𝟙 P) hP",
"usedConstants": [
"CategoryTheory.Functor",
"O... | by
apply sheafifyLift_unique
simp [Iso.comp_inv_eq] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Sites.ConcreteSheafification | {
"line": 592,
"column": 46
} | {
"line": 594,
"column": 27
} | [
{
"pp": "C : Type u\ninst✝⁷ : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst✝⁶ : Category.{w', w} D\nFD : D → D → Type u_1\nCD : D → Type t\ninst✝⁵ : (X Y : D) → FunLike (FD X Y) (CD X) (CD Y)\ninstCC : ConcreteCategory D FD\ninst✝⁴ : ∀ {X : C} (S : J.Cover X), PreservesLimitsOfShape (WalkingMu... | by
dsimp
rw [Category.assoc] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Category.TopCat.EpiMono | {
"line": 40,
"column": 14
} | {
"line": 40,
"column": 47
} | [
{
"pp": "X Y : TopCat\nf : X ⟶ Y\nthis : Mono f ↔ Mono ((forget TopCat).map f)\n⊢ Mono ((forget TopCat).map f) ↔ Function.Injective ⇑(ConcreteCategory.hom f)",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Mono",
"congrArg",
"CategoryTheory.ConcreteCategory.hom",
"TopCat.instCat... | CategoryTheory.mono_iff_injective | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Category.TopCat.Limits.Pullbacks | {
"line": 108,
"column": 6
} | {
"line": 108,
"column": 24
} | [
{
"pp": "X Y Z : TopCat\nf : X ⟶ Z\ng : Y ⟶ Z\n⊢ (pullbackIsoProdSubtype f g).hom ≫ pullbackFst f g = pullback.fst f g",
"usedConstants": [
"CategoryTheory.Limits.hasFiniteLimits_of_hasLimits",
"Eq.mpr",
"CategoryTheory.Limits.pullback",
"CategoryTheory.CategoryStruct.toQuiver",
... | ← Iso.eq_inv_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Category.TopCat.Limits.Pullbacks | {
"line": 112,
"column": 6
} | {
"line": 112,
"column": 24
} | [
{
"pp": "X Y Z : TopCat\nf : X ⟶ Z\ng : Y ⟶ Z\n⊢ (pullbackIsoProdSubtype f g).hom ≫ pullbackSnd f g = pullback.snd f g",
"usedConstants": [
"CategoryTheory.Limits.hasFiniteLimits_of_hasLimits",
"Eq.mpr",
"CategoryTheory.Limits.pullback",
"CategoryTheory.CategoryStruct.toQuiver",
... | ← Iso.eq_inv_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Category.TopCat.Limits.Products | {
"line": 292,
"column": 10
} | {
"line": 292,
"column": 37
} | [
{
"pp": "case neg\nX Y : TopCat\nc : BinaryCofan X Y\nh₁ : IsOpenEmbedding ⇑(ConcreteCategory.hom c.inl)\nh₂ : IsOpenEmbedding ⇑(ConcreteCategory.hom c.inr)\nh₃ : IsCompl (range ⇑(ConcreteCategory.hom c.inl)) (range ⇑(ConcreteCategory.hom c.inr))\nthis :\n ∀ (x : ↑(((Functor.const (Discrete WalkingPair)).obj c... | simp only [← mem_compl_iff] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.Category.TopCat.Limits.Pullbacks | {
"line": 208,
"column": 2
} | {
"line": 215,
"column": 45
} | [
{
"pp": "X Y S : TopCat\nf : X ⟶ S\ng : Y ⟶ S\n⊢ Set.range ⇑(ConcreteCategory.hom (pullback.fst f g)) =\n {x | ∃ y, (ConcreteCategory.hom f) x = (ConcreteCategory.hom g) y}",
"usedConstants": [
"CategoryTheory.Limits.hasFiniteLimits_of_hasLimits",
"Set.ext",
"Eq.mpr",
"CategoryThe... | ext x
constructor
· rintro ⟨y, rfl⟩
use pullback.snd f g y
exact CategoryTheory.congr_fun pullback.condition y
· rintro ⟨y, eq⟩
use (TopCat.pullbackIsoProdSubtype f g).inv ⟨⟨x, y⟩, eq⟩
rw [pullbackIsoProdSubtype_inv_fst_apply] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Category.TopCat.Limits.Pullbacks | {
"line": 208,
"column": 2
} | {
"line": 215,
"column": 45
} | [
{
"pp": "X Y S : TopCat\nf : X ⟶ S\ng : Y ⟶ S\n⊢ Set.range ⇑(ConcreteCategory.hom (pullback.fst f g)) =\n {x | ∃ y, (ConcreteCategory.hom f) x = (ConcreteCategory.hom g) y}",
"usedConstants": [
"CategoryTheory.Limits.hasFiniteLimits_of_hasLimits",
"Set.ext",
"Eq.mpr",
"CategoryThe... | ext x
constructor
· rintro ⟨y, rfl⟩
use pullback.snd f g y
exact CategoryTheory.congr_fun pullback.condition y
· rintro ⟨y, eq⟩
use (TopCat.pullbackIsoProdSubtype f g).inv ⟨⟨x, y⟩, eq⟩
rw [pullbackIsoProdSubtype_inv_fst_apply] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Adhesive.Basic | {
"line": 94,
"column": 26
} | {
"line": 94,
"column": 28
} | [
{
"pp": "case mp\nC : Type u\ninst✝ : Category.{v, u} C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH✝ : IsPushout f g h i\nH : H✝.IsVanKampen\nF' : WalkingSpan ⥤ C\nc' : Cocone F'\nα : F' ⟶ span f g\nfα : c'.pt ⟶ (PushoutCocone.mk h i ⋯).pt\neα : α ≫ (PushoutCocone.mk h i ⋯).ι = c'.ι ≫ (Functor.c... | hα | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.CategoryTheory.Limits.VanKampen | {
"line": 81,
"column": 21
} | {
"line": 81,
"column": 23
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasStrictInitialObjects C\nX : C\nh : IsInitial X\nF' : Discrete PEmpty.{1} ⥤ C\nc' : Cocone F'\nα : F' ⟶ empty C\nf : c'.pt ⟶ (asEmptyCocone X).pt\nhf : α ≫ (asEmptyCocone X).ι = c'.ι ≫ (const (Discrete PEmpty.{1})).map f\n⊢ NatTrans.Equifibered α →\n ... | hα | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.CategoryTheory.Limits.VanKampen | {
"line": 92,
"column": 21
} | {
"line": 92,
"column": 23
} | [
{
"pp": "J : Type v'\ninst✝¹ : Category.{u', v'} J\nC : Type u\ninst✝ : Category.{v, u} C\nF : J ⥤ C\nc c' : Cocone F\nhc : IsUniversalColimit c\ne : c ≅ c'\nF' : J ⥤ C\nc'' : Cocone F'\nα : F' ⟶ F\nf : c''.pt ⟶ c'.pt\nh : α ≫ c'.ι = c''.ι ≫ (const J).map f\n⊢ NatTrans.Equifibered α → (∀ (j : J), IsPullback (c'... | hα | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.CategoryTheory.Extensive | {
"line": 155,
"column": 2
} | {
"line": 155,
"column": 95
} | [
{
"pp": "J : Type v'\ninst✝⁵ : Category.{u', v'} J\nC : Type u\ninst✝⁴ : Category.{v, u} C\nD✝ : Type u''\ninst✝³ : Category.{v'', u''} D✝\nX✝ Y✝ : C\nD : Type u_1\ninst✝² : Category.{v_1, u_1} D\ninst✝¹ : HasBinaryCoproducts C\nF : C ⥤ D\ninst✝ : PreservesPullbacksOfInclusions F\nX Y Z : C\nf : Z ⟶ X ⨿ Y\n⊢ Pr... | apply preservesLimit_of_iso_diagram (K₁ := cospan (f ≫ (coprod.braiding X Y).hom) coprod.inl) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.CategoryTheory.Limits.VanKampen | {
"line": 103,
"column": 21
} | {
"line": 103,
"column": 23
} | [
{
"pp": "J : Type v'\ninst✝¹ : Category.{u', v'} J\nC : Type u\ninst✝ : Category.{v, u} C\nF : J ⥤ C\nc c' : Cocone F\nH : IsVanKampenColimit c\ne : c ≅ c'\nF' : J ⥤ C\nc'' : Cocone F'\nα : F' ⟶ F\nf : c''.pt ⟶ c'.pt\nh : α ≫ c'.ι = c''.ι ≫ (const J).map f\n⊢ NatTrans.Equifibered α → (Nonempty (IsColimit c'') ↔... | hα | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.CategoryTheory.Limits.VanKampen | {
"line": 118,
"column": 21
} | {
"line": 118,
"column": 23
} | [
{
"pp": "J : Type v'\ninst✝² : Category.{u', v'} J\nC : Type u\ninst✝¹ : Category.{v, u} C\nF G : J ⥤ C\nα : F ⟶ G\ninst✝ : IsIso α\nc : Cocone G\nhc : IsVanKampenColimit c\nF' : J ⥤ C\nc' : Cocone F'\nα' : F' ⟶ F\nf : c'.pt ⟶ ((Cocone.precompose α).obj c).pt\ne : α' ≫ ((Cocone.precompose α).obj c).ι = c'.ι ≫ (... | hα | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.CategoryTheory.Limits.VanKampen | {
"line": 133,
"column": 21
} | {
"line": 133,
"column": 23
} | [
{
"pp": "J : Type v'\ninst✝² : Category.{u', v'} J\nC : Type u\ninst✝¹ : Category.{v, u} C\nF G : J ⥤ C\nα : F ⟶ G\ninst✝ : IsIso α\nc : Cocone G\nhc : IsUniversalColimit c\nF' : J ⥤ C\nc' : Cocone F'\nα' : F' ⟶ F\nf : c'.pt ⟶ ((Cocone.precompose α).obj c).pt\ne : α' ≫ ((Cocone.precompose α).obj c).ι = c'.ι ≫ (... | hα | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.CategoryTheory.Adhesive.Basic | {
"line": 131,
"column": 17
} | {
"line": 131,
"column": 68
} | [
{
"pp": "case mpr.refine_5.mp\nC : Type u\ninst✝ : Category.{v, u} C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh✝ : X ⟶ Z\ni : Y ⟶ Z\nH✝ : IsPushout f g h✝ i\nH : IsVanKampenColimit (PushoutCocone.mk h✝ i ⋯)\nW' X' Y' Z' : C\nf' : W' ⟶ X'\ng' : W' ⟶ Y'\nh' : X' ⟶ Z'\ni' : Y' ⟶ Z'\nαW : W' ⟶ W\nαX : X' ⟶ X\nαY : Y' ⟶ ... | exact ⟨h WalkingCospan.left, h WalkingCospan.right⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Limits.VanKampen | {
"line": 161,
"column": 20
} | {
"line": 161,
"column": 22
} | [
{
"pp": "J : Type v'\ninst✝⁷ : Category.{u', v'} J\nC : Type u\ninst✝⁶ : Category.{v, u} C\nD : Type u_2\ninst✝⁵ : Category.{v_2, u_2} D\nG : C ⥤ D\nF : J ⥤ C\nc : Cocone F\ninst✝⁴ : ∀ (i j : J) (X : C) (f : X ⟶ F.obj j) (g : i ⟶ j), PreservesLimit (cospan f (F.map g)) G\ninst✝³ : ∀ (i : J) (X : C) (f : X ⟶ c.p... | hα | Lean.Elab.Tactic.evalIntro | ident |
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