module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Algebra.Homology.ShortComplex.Abelian | {
"line": 142,
"column": 58
} | {
"line": 142,
"column": 82
} | {
"line": 142,
"column": 83
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Abelian C\nS : ShortComplex C\nD : Type u'\ninst✝¹ : Category.{v', u'} D\ninst✝ : HasZeroMorphisms D\nZ'✝ : C\nk : cokernel S.f ⟶ Z'✝\nhk : (kernel.ι S.g ≫ cokernel.π S.f) ≫ k = 0\n⊢ kernel.ι S.g ≫ cokernel.π S.f ≫ k = 0",
"ppTerm": "?m.71",
"ass... | [
"C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Abelian C\nS : ShortComplex C\nD : Type u'\ninst✝¹ : Category.{v', u'} D\ninst✝ : HasZeroMorphisms D\nZ'✝ : C\nk : cokernel S.f ⟶ Z'✝\nhk : (kernel.ι S.g ≫ cokernel.π S.f) ≫ k = 0\n⊢ kernel.ι S.g ≫ cokernel.π S.f ≫ k = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.ShortComplex.Abelian | {
"line": 142,
"column": 58
} | {
"line": 142,
"column": 85
} | {
"line": 142,
"column": 85
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Abelian C\nS : ShortComplex C\nD : Type u'\ninst✝¹ : Category.{v', u'} D\ninst✝ : HasZeroMorphisms D\nZ'✝ : C\nk : cokernel S.f ⟶ Z'✝\nhk : (kernel.ι S.g ≫ cokernel.π S.f) ≫ k = 0\n⊢ kernel.ι S.g ≫ cokernel.π S.f ≫ k = 0",
"ppTerm": "?m.71",
"ass... | [] | simpa only [assoc] using hk | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Algebra.Homology.ShortComplex.Abelian | {
"line": 142,
"column": 58
} | {
"line": 142,
"column": 85
} | {
"line": 142,
"column": 85
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Abelian C\nS : ShortComplex C\nD : Type u'\ninst✝¹ : Category.{v', u'} D\ninst✝ : HasZeroMorphisms D\nZ'✝ : C\nk : cokernel S.f ⟶ Z'✝\nhk : (kernel.ι S.g ≫ cokernel.π S.f) ≫ k = 0\n⊢ kernel.ι S.g ≫ cokernel.π S.f ≫ k = 0",
"ppTerm": "?m.71",
"ass... | [] | simpa only [assoc] using hk | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.ShortComplex.Abelian | {
"line": 142,
"column": 58
} | {
"line": 142,
"column": 85
} | {
"line": 142,
"column": 85
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Abelian C\nS : ShortComplex C\nD : Type u'\ninst✝¹ : Category.{v', u'} D\ninst✝ : HasZeroMorphisms D\nZ'✝ : C\nk : cokernel S.f ⟶ Z'✝\nhk : (kernel.ι S.g ≫ cokernel.π S.f) ≫ k = 0\n⊢ kernel.ι S.g ≫ cokernel.π S.f ≫ k = 0",
"ppTerm": "?m.71",
"ass... | [] | simpa only [assoc] using hk | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.ShortComplex.Abelian | {
"line": 170,
"column": 8
} | {
"line": 171,
"column": 31
} | {
"line": 171,
"column": 32
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Abelian C\nS : ShortComplex C\nD : Type u'\ninst✝¹ : Category.{v', u'} D\ninst✝ : HasZeroMorphisms D\nγ : kernel S.g ⟶ cokernel S.f := kernel.ι S.g ≫ cokernel.π S.f\ng' : cokernel S.f ⟶ S.X₃ := cokernel.desc S.f S.g ⋯\nhg' : g' = cokernel.π γ ≫ cokernel.... | [
"C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Abelian C\nS : ShortComplex C\nD : Type u'\ninst✝¹ : Category.{v', u'} D\ninst✝ : HasZeroMorphisms D\nγ : kernel S.g ⟶ cokernel S.f := kernel.ι S.g ≫ cokernel.π S.f\ng' : cokernel S.f ⟶ S.X₃ := cokernel.desc S.f S.g ⋯\nhg' : g' = cokernel.π γ ≫ cokernel.desc γ g' ⋯\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.ShortComplex.PreservesHomology | {
"line": 831,
"column": 2
} | {
"line": 832,
"column": 18
} | {
"line": 834,
"column": 0
} | [
{
"pp": "case mpr\nC : Type u_1\nD : Type u_2\ninst✝¹¹ : Category.{v_1, u_1} C\ninst✝¹⁰ : Category.{v_2, u_2} D\ninst✝⁹ : HasZeroMorphisms C\ninst✝⁸ : HasZeroMorphisms D\nF : C ⥤ D\ninst✝⁷ : F.PreservesZeroMorphisms\nS₁ S₂ : ShortComplex C\nφ : S₁ ⟶ S₂\ninst✝⁶ : S₁.HasHomology\ninst✝⁵ : S₂.HasHomology\ninst✝⁴ :... | [] | · intro
infer_instance | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Preadditive.Projective.Basic | {
"line": 267,
"column": 4
} | {
"line": 268,
"column": 56
} | {
"line": 270,
"column": 0
} | [
{
"pp": "case mpr\nC : Type u\ninst✝¹ : Category.{v, u} C\nD : Type u'\ninst✝ : Category.{v', u'} D\nF : C ≌ D\nH : EnoughProjectives D\nX : C\n⊢ ProjectivePresentation X",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"CategoryTheory.ProjectivePresentation",
"CategoryTheory.Eno... | [] | exact F.projectivePresentationOfMapProjectivePresentation X
(Nonempty.some (H.presentation (F.functor.obj X))) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Preadditive.Projective.Basic | {
"line": 267,
"column": 4
} | {
"line": 268,
"column": 56
} | {
"line": 270,
"column": 0
} | [
{
"pp": "case mpr\nC : Type u\ninst✝¹ : Category.{v, u} C\nD : Type u'\ninst✝ : Category.{v', u'} D\nF : C ≌ D\nH : EnoughProjectives D\nX : C\n⊢ ProjectivePresentation X",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"CategoryTheory.ProjectivePresentation",
"CategoryTheory.Eno... | [] | exact F.projectivePresentationOfMapProjectivePresentation X
(Nonempty.some (H.presentation (F.functor.obj X))) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Preadditive.Projective.Basic | {
"line": 267,
"column": 4
} | {
"line": 268,
"column": 56
} | {
"line": 270,
"column": 0
} | [
{
"pp": "case mpr\nC : Type u\ninst✝¹ : Category.{v, u} C\nD : Type u'\ninst✝ : Category.{v', u'} D\nF : C ≌ D\nH : EnoughProjectives D\nX : C\n⊢ ProjectivePresentation X",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"CategoryTheory.ProjectivePresentation",
"CategoryTheory.Eno... | [] | exact F.projectivePresentationOfMapProjectivePresentation X
(Nonempty.some (H.presentation (F.functor.obj X))) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.ShortComplex.PreservesHomology | {
"line": 855,
"column": 2
} | {
"line": 856,
"column": 18
} | {
"line": 858,
"column": 0
} | [
{
"pp": "case mpr\nC : Type u_1\nD : Type u_2\ninst✝¹¹ : Category.{v_1, u_1} C\ninst✝¹⁰ : Category.{v_2, u_2} D\ninst✝⁹ : HasZeroMorphisms C\ninst✝⁸ : HasZeroMorphisms D\nF : C ⥤ D\ninst✝⁷ : F.PreservesZeroMorphisms\nS₁ S₂ : ShortComplex C\nφ : S₁ ⟶ S₂\ninst✝⁶ : S₁.HasHomology\ninst✝⁵ : S₂.HasHomology\ninst✝⁴ :... | [] | · intro
infer_instance | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Preadditive.Injective.Basic | {
"line": 211,
"column": 2
} | {
"line": 212,
"column": 41
} | {
"line": 213,
"column": 2
} | [
{
"pp": "case pos\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : EnoughInjectives C\nX : C\nh : IsZero X\n⊢ ∃ p, IsZero X → IsZero p.J",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"CategoryTheory.Injective",
"CategoryTheory.Mono",
"CategoryTheory.InjectivePresenta... | [
"case neg\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : EnoughInjectives C\nX : C\nh : ¬IsZero X\n⊢ ∃ p, IsZero X → IsZero p.J"
] | · have := h.injective
exact ⟨{ J := X, f := 𝟙 X}, by tauto⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Homology.ShortComplex.Preadditive | {
"line": 582,
"column": 35
} | {
"line": 582,
"column": 60
} | {
"line": 582,
"column": 61
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nS₁ S₂ S₃ : ShortComplex C\nφ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂\nH₁ : S₁.LeftHomologyData\nH₂ : S₂.LeftHomologyData\nh₀ : S₁.X₁ ⟶ S₂.X₁\nh₀_f : h₀ ≫ S₂.f = 0\nh₁ : S₁.X₂ ⟶ S₂.X₁\nh₂ : S₁.X₃ ⟶ S₂.X₂\nh₃ : S₁.X₃ ⟶ S₂.X₃\ng_h₃ : S₁.g ≫ h₃ = 0\n⊢ H₁.f' ... | [
"C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nS₁ S₂ S₃ : ShortComplex C\nφ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂\nH₁ : S₁.LeftHomologyData\nH₂ : S₂.LeftHomologyData\nh₀ : S₁.X₁ ⟶ S₂.X₁\nh₀_f : h₀ ≫ S₂.f = 0\nh₁ : S₁.X₂ ⟶ S₂.X₁\nh₂ : S₁.X₃ ⟶ S₂.X₂\nh₃ : S₁.X₃ ⟶ S₂.X₃\ng_h₃ : S₁.g ≫ h₃ = 0\n⊢ H₁.f' ≫ H₁.i ≫ h₁ ... | LeftHomologyData.liftK_i, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.ShortComplex.Ab | {
"line": 115,
"column": 18
} | {
"line": 115,
"column": 74
} | {
"line": 115,
"column": 75
} | [
{
"pp": "S : ShortComplex Ab\nh : Function.Surjective ⇑S.abToCycles\nx₂ : ↑S.X₂\nhx₂ : (ConcreteCategory.hom S.g) x₂ = 0\nx₁ : ↑S.X₁\nhx₁ : S.abToCycles x₁ = ⟨x₂, hx₂⟩\n⊢ (ConcreteCategory.hom S.f) x₁ = x₂",
"ppTerm": "?m.54",
"assigned": true,
"usedConstants": [
"AddCommGrpCat.instCategory",
... | [
"S : ShortComplex Ab\nh : Function.Surjective ⇑S.abToCycles\nx₂ : ↑S.X₂\nhx₂ : (ConcreteCategory.hom S.g) x₂ = 0\nx₁ : ↑S.X₁\nhx₁ : S.abToCycles x₁ = ⟨x₂, hx₂⟩\n⊢ (ConcreteCategory.hom S.f) x₁ = x₂"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.ShortComplex.Ab | {
"line": 137,
"column": 6
} | {
"line": 137,
"column": 31
} | {
"line": 137,
"column": 31
} | [
{
"pp": "S : ShortComplex Ab\n⊢ S.Exact ↔ (AddCommGrpCat.Hom.hom S.f).range = (AddCommGrpCat.Hom.hom S.g).ker",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AddCommGrpCat.instCategory",
"congrArg",
"AddMonoid.toAddZeroClass",
"PartialOrder.toPreord... | [
"S : ShortComplex Ab\n⊢ (AddCommGrpCat.Hom.hom S.g).ker ≤ (AddCommGrpCat.Hom.hom S.f).range ↔\n (AddCommGrpCat.Hom.hom S.f).range = (AddCommGrpCat.Hom.hom S.g).ker"
] | ab_exact_iff_ker_le_range | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Adjunction.Restrict | {
"line": 55,
"column": 8
} | {
"line": 55,
"column": 33
} | {
"line": 55,
"column": 34
} | [
{
"pp": "C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\nC' : Type u₃\ninst✝¹ : Category.{v₃, u₃} C'\nD' : Type u₄\ninst✝ : Category.{v₄, u₄} D'\niC : C ⥤ C'\niD : D ⥤ D'\nL' : C' ⥤ D'\nR' : D' ⥤ C'\nadj : L' ⊣ R'\nhiC : iC.FullyFaithful\nhiD : iD.FullyFaithful\nL : C ⥤ D\n... | [
"C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\nC' : Type u₃\ninst✝¹ : Category.{v₃, u₃} C'\nD' : Type u₄\ninst✝ : Category.{v₄, u₄} D'\niC : C ⥤ C'\niD : D ⥤ D'\nL' : C' ⥤ D'\nR' : D' ⥤ C'\nadj : L' ⊣ R'\nhiC : iC.FullyFaithful\nhiD : iD.FullyFaithful\nL : C ⥤ D\nR : D ⥤ C\nc... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.ShortComplex.Exact | {
"line": 315,
"column": 4
} | {
"line": 315,
"column": 36
} | {
"line": 315,
"column": 37
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nS : ShortComplex C\nhS : S.Exact\nh : S.LeftHomologyData\nthis : S.HasHomology\n⊢ IsZero h.H",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"id",
"CategoryTheory.ShortComplex.Exact",
... | [
"C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nS : ShortComplex C\nhS : S.Exact\nh : S.LeftHomologyData\nthis : S.HasHomology\n⊢ S.Exact"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.ShortComplex.Exact | {
"line": 322,
"column": 4
} | {
"line": 322,
"column": 36
} | {
"line": 322,
"column": 37
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nS : ShortComplex C\nhS : S.Exact\nh : S.RightHomologyData\nthis : S.HasHomology\n⊢ IsZero h.H",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.ShortComplex.RightHomologyData.e... | [
"C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nS : ShortComplex C\nhS : S.Exact\nh : S.RightHomologyData\nthis : S.HasHomology\n⊢ S.Exact"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.ShortComplex.Exact | {
"line": 363,
"column": 4
} | {
"line": 369,
"column": 41
} | {
"line": 369,
"column": 41
} | [
{
"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\ninst✝¹ : Preadditive D\nS : ShortComplex C\nhS : S.Exact\ninst✝ : HasZeroObject C\nkf : KernelFork S.g\nhkf : IsLimit kf\n⊢ Epi ((hkf.ofIsoLimit (Fork.ext (Iso.refl kf.pt) ⋯)).lift (Kerne... | [] | have := hS.hasHomology
refine ((MorphismProperty.epimorphisms C).arrow_mk_iso_iff ?_).1
hS.epi_toCycles
refine Arrow.isoMk (Iso.refl _)
(IsLimit.conePointUniqueUpToIso S.cyclesIsKernel hkf) ?_
apply Fork.IsLimit.hom_ext hkf
simp [IsLimit.conePointUniqueUpToIso] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.ShortComplex.Exact | {
"line": 363,
"column": 4
} | {
"line": 369,
"column": 41
} | {
"line": 369,
"column": 41
} | [
{
"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\ninst✝¹ : Preadditive D\nS : ShortComplex C\nhS : S.Exact\ninst✝ : HasZeroObject C\nkf : KernelFork S.g\nhkf : IsLimit kf\n⊢ Epi ((hkf.ofIsoLimit (Fork.ext (Iso.refl kf.pt) ⋯)).lift (Kerne... | [] | have := hS.hasHomology
refine ((MorphismProperty.epimorphisms C).arrow_mk_iso_iff ?_).1
hS.epi_toCycles
refine Arrow.isoMk (Iso.refl _)
(IsLimit.conePointUniqueUpToIso S.cyclesIsKernel hkf) ?_
apply Fork.IsLimit.hom_ext hkf
simp [IsLimit.conePointUniqueUpToIso] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Subobject.FactorThru | {
"line": 137,
"column": 4
} | {
"line": 137,
"column": 15
} | {
"line": 137,
"column": 16
} | [
{
"pp": "case mp\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : HasZeroMorphisms C\nX Y : C\nP : Subobject Y\nf : X ⟶ Y\nh : P.Factors f\nw : P.factorThru f h ≫ P.arrow = 0 ≫ P.arrow\n⊢ f = 0",
"ppTerm": "?mp",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}... | [
"case mp\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : HasZeroMorphisms C\nX Y : C\nP : Subobject Y\nf : X ⟶ Y\nh : P.Factors f\nw : P.factorThru f h ≫ P.arrow = 0 ≫ P.arrow\n⊢ f = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.ShortComplex.Exact | {
"line": 570,
"column": 2
} | {
"line": 570,
"column": 48
} | {
"line": 571,
"column": 2
} | [
{
"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\ninst✝¹ : Preadditive D\nS : ShortComplex C\ninst✝ : HasZeroObject C\ns : S.Splitting\nhi : IsLimit (KernelFork.ofι S.f ⋯)\nf' : (KernelFork.ofι S.f ⋯).pt ⟶ (KernelFork.ofι S.f ⋯).pt := hi... | [
"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\ninst✝¹ : Preadditive D\nS : ShortComplex C\ninst✝ : HasZeroObject C\ns : S.Splitting\nhi : IsLimit (KernelFork.ofι S.f ⋯)\nf' : (KernelFork.ofι S.f ⋯).pt ⟶ (KernelFork.ofι S.f ⋯).pt := hi.lift (Kerne... | have wπ : f' ≫ (0 : S.X₁ ⟶ 0) = 0 := comp_zero | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.CategoryTheory.Subobject.Basic | {
"line": 303,
"column": 2
} | {
"line": 303,
"column": 13
} | {
"line": 303,
"column": 14
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nX✝ Y✝ Z✝ : C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nB : C\nX Y : Subobject B\nh : X ≤ Y\nZ : C\nf g : Z ⟶ underlying.obj X\nw : (f ≫ X.ofLE Y h) ≫ Y.arrow = (g ≫ X.ofLE Y h) ≫ Y.arrow\n⊢ f ≫ X.arrow = g ≫ X.arrow",
"ppTerm": "?m.50",
"assigned": fa... | [
"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nX✝ Y✝ Z✝ : C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nB : C\nX Y : Subobject B\nh : X ≤ Y\nZ : C\nf g : Z ⟶ underlying.obj X\nw : (f ≫ X.ofLE Y h) ≫ Y.arrow = (g ≫ X.ofLE Y h) ≫ Y.arrow\n⊢ f ≫ X.arrow = g ≫ X.arrow"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Subobject.MonoOver | {
"line": 216,
"column": 51
} | {
"line": 220,
"column": 21
} | {
"line": 220,
"column": 21
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nX✝ Y Z : C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nJ : Type u₃\ninst✝ : Category.{v₃, u₃} J\nX : C\nF : Over X\nx✝ : (Over.isMono X).limitsOfShape J F\np : LimitPresentation J F\nhp : ∀ (j : J), Over.isMono X (p.diag.obj j)\nZ✝ : C\ng h : Z✝ ⟶ F.left\ne : ... | [] | by
refine (WithTerminal.isLimitEquiv.invFun p.isLimit).hom_ext (fun j ↦ ?_)
cases j with
| of j => have := hp j; rw [← cancel_mono ((p.diag.obj j).hom)]; simpa
| star => exact e | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.ShortComplex.Exact | {
"line": 949,
"column": 2
} | {
"line": 949,
"column": 25
} | {
"line": 950,
"column": 2
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Abelian C\nS : ShortComplex C\nhS : S.Exact\nJ : C\nf : S.X₂ ⟶ J\ninst✝ : Injective J\nhf : S.f ≫ f = 0\n⊢ S.g ≫ hS.descToInjective f hf = f",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"CategoryTheory.Abelian.toPread... | [
"C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Abelian C\nS : ShortComplex C\nhS : S.Exact\nJ : C\nf : S.X₂ ⟶ J\ninst✝ : Injective J\nhf : S.f ≫ f = 0\n⊢ S.g ≫ Injective.factorThru (S.descOpcycles f hf) S.fromOpcycles = f"
] | dsimp [descToInjective] | Lean.Elab.Tactic.evalDSimp | Lean.Parser.Tactic.dsimp |
Mathlib.CategoryTheory.Subobject.Limits | {
"line": 103,
"column": 42
} | {
"line": 112,
"column": 52
} | {
"line": 114,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nX Y : C\nf g : X ⟶ Y\ninst✝¹ : HasEqualizer f g\nW : C\nh : W ⟶ X\ninst✝ : HasPullbacks C\n⊢ (Subobject.pullback h).obj (equalizerSubobject f g) = equalizerSubobject (h ≫ f) (h ≫ g)",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"Eq.m... | [] | by
refine skeletal _ ⟨iso_of_both_ways (homOfFactors ?_) (homOfFactors ?_)⟩
· apply equalizerSubobject_factors
have := (Subobject.isPullback h (equalizerSubobject f g)).w
rw [← reassoc_of% (Subobject.isPullback h (equalizerSubobject f g)).w,
← reassoc_of% (Subobject.isPullback h (equalizerSubobject f ... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Subobject.Basic | {
"line": 682,
"column": 8
} | {
"line": 682,
"column": 59
} | {
"line": 682,
"column": 60
} | [
{
"pp": "case mp\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nX Y Z : C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\ne : X ≅ Y\nA B : Subobject X\nh : (map e.inv).obj ((map e.hom).obj A) ≤ (map e.inv).obj ((map e.hom).obj B)\n⊢ A ≤ B",
"ppTerm": "?mp",
"assigned": false,
"usedConstants": [],
"usedF... | [
"case mp\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nX Y Z : C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\ne : X ≅ Y\nA B : Subobject X\nh : (map e.inv).obj ((map e.hom).obj A) ≤ (map e.inv).obj ((map e.hom).obj B)\n⊢ A ≤ B"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.ImageToKernel | {
"line": 102,
"column": 48
} | {
"line": 102,
"column": 59
} | {
"line": 102,
"column": 60
} | [
{
"pp": "ι : Type u_1\nV : Type u\ninst✝⁴ : Category.{v, u} V\ninst✝³ : HasZeroMorphisms V\nA B C : V\nf : A ⟶ B\ng : B ⟶ C\ninst✝² : HasKernels V\ninst✝¹ : HasImages V\nD : V\nh : C ⟶ D\ninst✝ : Mono h\nw : f ≫ g ≫ h = 0\n⊢ (f ≫ g) ≫ h = 0 ≫ h",
"ppTerm": "?m.90",
"assigned": true,
"usedConstants":... | [
"ι : Type u_1\nV : Type u\ninst✝⁴ : Category.{v, u} V\ninst✝³ : HasZeroMorphisms V\nA B C : V\nf : A ⟶ B\ng : B ⟶ C\ninst✝² : HasKernels V\ninst✝¹ : HasImages V\nD : V\nh : C ⟶ D\ninst✝ : Mono h\nw : f ≫ g ≫ h = 0\n⊢ f ≫ g ≫ h = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.ImageToKernel | {
"line": 111,
"column": 49
} | {
"line": 111,
"column": 60
} | {
"line": 111,
"column": 61
} | [
{
"pp": "ι : Type u_1\nV : Type u\ninst✝⁴ : Category.{v, u} V\ninst✝³ : HasZeroMorphisms V\nA B C : V\nf : A ⟶ B\ng : B ⟶ C\ninst✝² : HasKernels V\ninst✝¹ : HasImages V\nZ : V\nh : Z ⟶ A\ninst✝ : Epi h\nw : (h ≫ f) ≫ g = 0\n⊢ h ≫ f ≫ g = h ≫ 0",
"ppTerm": "?m.85",
"assigned": true,
"usedConstants": ... | [
"ι : Type u_1\nV : Type u\ninst✝⁴ : Category.{v, u} V\ninst✝³ : HasZeroMorphisms V\nA B C : V\nf : A ⟶ B\ng : B ⟶ C\ninst✝² : HasKernels V\ninst✝¹ : HasImages V\nZ : V\nh : Z ⟶ A\ninst✝ : Epi h\nw : (h ≫ f) ≫ g = 0\n⊢ h ≫ f ≫ g = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.ImageToKernel | {
"line": 121,
"column": 30
} | {
"line": 121,
"column": 41
} | {
"line": 121,
"column": 42
} | [
{
"pp": "ι : Type u_1\nV : Type u\ninst✝³ : Category.{v, u} V\ninst✝² : HasZeroMorphisms V\nA B C : V\nf : A ⟶ B\ng : B ⟶ C\ninst✝¹ : HasEqualizers V\ninst✝ : HasImages V\nZ : V\ni : B ≅ Z\nw : (f ≫ i.hom) ≫ i.inv ≫ g = 0\n⊢ f ≫ g = 0",
"ppTerm": "?m.88",
"assigned": false,
"usedConstants": [],
... | [
"ι : Type u_1\nV : Type u\ninst✝³ : Category.{v, u} V\ninst✝² : HasZeroMorphisms V\nA B C : V\nf : A ⟶ B\ng : B ⟶ C\ninst✝¹ : HasEqualizers V\ninst✝ : HasImages V\nZ : V\ni : B ≅ Z\nw : (f ≫ i.hom) ≫ i.inv ≫ g = 0\n⊢ f ≫ g = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.ImageToKernel | {
"line": 165,
"column": 4
} | {
"line": 165,
"column": 15
} | {
"line": 165,
"column": 16
} | [
{
"pp": "ι : Type u_1\nV : Type u\ninst✝³ : Category.{v, u} V\ninst✝² : HasZeroMorphisms V\nA B C : V\nf : A ⟶ B\ng : B ⟶ C\nw✝ : f ≫ g = 0\ninst✝¹ : HasKernels V\ninst✝ : HasImages V\nw : f ≫ g = 0\n⊢ factorThruImage f ≫ image.ι f ≫ g = factorThruImage f ≫ 0",
"ppTerm": "?m.62",
"assigned": true,
"... | [
"ι : Type u_1\nV : Type u\ninst✝³ : Category.{v, u} V\ninst✝² : HasZeroMorphisms V\nA B C : V\nf : A ⟶ B\ng : B ⟶ C\nw✝ : f ≫ g = 0\ninst✝¹ : HasKernels V\ninst✝ : HasImages V\nw : f ≫ g = 0\n⊢ f ≫ g = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Abelian.Exact | {
"line": 82,
"column": 2
} | {
"line": 83,
"column": 18
} | {
"line": 85,
"column": 0
} | [
{
"pp": "case mpr\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : Abelian C\nS : ShortComplex C\n⊢ IsIso (imageToKernel S.f S.g ⋯) → Epi (imageToKernel S.f S.g ⋯)",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"CategoryTheory.Abelian.toPreadditive",
"CategoryTheory.Limits.H... | [] | · intro
infer_instance | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Homology.ShortComplex.ExactFunctor | {
"line": 76,
"column": 14
} | {
"line": 76,
"column": 76
} | {
"line": 76,
"column": 76
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝⁸ : Category.{v_1, u_1} C\ninst✝⁷ : Category.{v_2, u_2} D\ninst✝⁶ : Preadditive C\ninst✝⁵ : Preadditive D\nF : C ⥤ D\ninst✝⁴ : F.Additive\ninst✝³ : F.PreservesHomology\ninst✝² : HasZeroObject C\ninst✝¹ : HasFiniteProducts C\ninst✝ : HasKernels C\nthis✝¹ : ∀ {X Y : C} (f... | [] | by rw [IsZero.iff_id_eq_zero, ← F.map_id, id_zero, F.map_zero] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.ShortComplex.ExactFunctor | {
"line": 86,
"column": 14
} | {
"line": 86,
"column": 76
} | {
"line": 86,
"column": 76
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝⁸ : Category.{v_1, u_1} C\ninst✝⁷ : Category.{v_2, u_2} D\ninst✝⁶ : Preadditive C\ninst✝⁵ : Preadditive D\nF : C ⥤ D\ninst✝⁴ : F.Additive\ninst✝³ : F.PreservesHomology\ninst✝² : HasZeroObject C\ninst✝¹ : HasFiniteCoproducts C\ninst✝ : HasCokernels C\nthis✝¹ : ∀ {X Y : C... | [] | by rw [IsZero.iff_id_eq_zero, ← F.map_id, id_zero, F.map_zero] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Subobject.Lattice | {
"line": 326,
"column": 15
} | {
"line": 326,
"column": 52
} | {
"line": 326,
"column": 53
} | [
{
"pp": "C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nX Y : C\ninst✝² : HasZeroObject C\ninst✝¹ : HasZeroMorphisms C\nf : X ⟶ Y\ninst✝ : Mono f\nh : mk f = ⊥\n⊢ f = 0",
"ppTerm": "?m.23",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nX Y : C\ninst✝² : HasZeroObject C\ninst✝¹ : HasZeroMorphisms C\nf : X ⟶ Y\ninst✝ : Mono f\nh : mk f = ⊥\n⊢ f = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.ShortComplex.ExactFunctor | {
"line": 126,
"column": 4
} | {
"line": 126,
"column": 30
} | {
"line": 127,
"column": 4
} | [
{
"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 : Abelian C\ninst✝¹ : Abelian D\nF : C ⥤ D\ninst✝ : F.Additive\nhF : ∀ (S : ShortComplex C), S.ShortExact → (S.map F).Exact ∧ Mono (F.map S.f)\nS : ShortComplex C\nhS : S.Exact\nhf : Mono S.f\nthis : F.Pr... | [
"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 : Abelian C\ninst✝¹ : Abelian D\nF : C ⥤ D\ninst✝ : F.Additive\nhF : ∀ (S : ShortComplex C), S.ShortExact → (S.map F).Exact ∧ Mono (F.map S.f)\nS : ShortComplex C\nhS : S.Exact\nhf : Mono S.f\nthis : F.PreservesMonom... | refine ⟨?_, inferInstance⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.CategoryTheory.Subobject.Lattice | {
"line": 402,
"column": 4
} | {
"line": 402,
"column": 15
} | {
"line": 402,
"column": 16
} | [
{
"pp": "case refine_1\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : HasPullbacks C\nA : C\nf g : Subobject A\ns : PullbackCone f.arrow g.arrow\n⊢ (f ⊓ g).Factors (s.fst ≫ f.arrow)",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"CategoryTheory.Subobject.Factors",
"Eq... | [
"case refine_1\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : HasPullbacks C\nA : C\nf g : Subobject A\ns : PullbackCone f.arrow g.arrow\n⊢ f.Factors (s.fst ≫ f.arrow) ∧ g.Factors (s.fst ≫ f.arrow)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Subobject.Lattice | {
"line": 402,
"column": 46
} | {
"line": 402,
"column": 71
} | {
"line": 402,
"column": 72
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : HasPullbacks C\nA : C\nf g : Subobject A\ns : PullbackCone f.arrow g.arrow\n⊢ g.Factors (s.fst ≫ f.arrow)",
"ppTerm": "?m.122",
"assigned": true,
"usedConstants": [
"CategoryTheory.Subobject.Factors",
"Eq.mpr",
"CategoryTh... | [
"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : HasPullbacks C\nA : C\nf g : Subobject A\ns : PullbackCone f.arrow g.arrow\n⊢ g.Factors (s.snd ≫ g.arrow)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Subobject.Limits | {
"line": 468,
"column": 2
} | {
"line": 468,
"column": 13
} | {
"line": 468,
"column": 14
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\nW X Y Z : C\nf : W ⟶ X\ninst✝² : HasImage f\ng : Y ⟶ Z\ninst✝¹ : HasImage g\nsq : Arrow.mk f ⟶ Arrow.mk g\ninst✝ : HasImageMap sq\n⊢ (image.map sq ≫ (imageSubobjectIso (Arrow.mk g).hom).inv) ≫ (imageSubobject (Arrow.mk g).hom).arrow =\n ((imageSubobjectIso f).... | [
"C : Type u\ninst✝³ : Category.{v, u} C\nW X Y Z : C\nf : W ⟶ X\ninst✝² : HasImage f\ng : Y ⟶ Z\ninst✝¹ : HasImage g\nsq : Arrow.mk f ⟶ Arrow.mk g\ninst✝ : HasImageMap sq\n⊢ image.map sq ≫ image.ι g = image.ι f ≫ Arrow.Hom.right sq"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.ShortComplex.ExactFunctor | {
"line": 186,
"column": 4
} | {
"line": 186,
"column": 30
} | {
"line": 187,
"column": 4
} | [
{
"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 : Abelian C\ninst✝¹ : Abelian D\nF : C ⥤ D\ninst✝ : F.Additive\nhF : ∀ (S : ShortComplex C), S.ShortExact → (S.map F).Exact ∧ Epi (F.map S.g)\nS : ShortComplex C\nhS : S.Exact\nhf : Epi S.g\nthis : F.Pres... | [
"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 : Abelian C\ninst✝¹ : Abelian D\nF : C ⥤ D\ninst✝ : F.Additive\nhF : ∀ (S : ShortComplex C), S.ShortExact → (S.map F).Exact ∧ Epi (F.map S.g)\nS : ShortComplex C\nhS : S.Exact\nhf : Epi S.g\nthis : F.PreservesEpimorp... | refine ⟨?_, inferInstance⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.CategoryTheory.Subobject.Lattice | {
"line": 570,
"column": 14
} | {
"line": 570,
"column": 25
} | {
"line": 570,
"column": 26
} | [
{
"pp": "C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nX Y Z : C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\ninst✝¹ : LocallySmall.{w, v₁, u₁} C\ninst✝ : WellPowered.{w, v₁, u₁} C\nA : C\ns : Set (Subobject A)\nf : Subobject A\nk : ∀ g ∈ s, f ≤ g\ng : Subobject A\nm : g ∈ s\n⊢ (equivShrink (Subobject A)).symm ↑⟨(e... | [
"C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nX Y Z : C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\ninst✝¹ : LocallySmall.{w, v₁, u₁} C\ninst✝ : WellPowered.{w, v₁, u₁} C\nA : C\ns : Set (Subobject A)\nf : Subobject A\nk : ∀ g ∈ s, f ≤ g\ng : Subobject A\nm : g ∈ s\n⊢ g ∈ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Subobject.Lattice | {
"line": 654,
"column": 67
} | {
"line": 654,
"column": 94
} | {
"line": 654,
"column": 95
} | [
{
"pp": "C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : LocallySmall.{w, v₁, u₁} C\ninst✝² : WellPowered.{w, v₁, u₁} C\ninst✝¹ : HasCoproducts C\ninst✝ : HasImages C\nA : C\ns : Set (Subobject A)\nf : Subobject A\nhf : f ∈ s\n⊢ (equivShrink (Subobject A)) f ∈ ⇑(equivShrink (Subobject A)) '' s",
"ppTerm... | [
"C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : LocallySmall.{w, v₁, u₁} C\ninst✝² : WellPowered.{w, v₁, u₁} C\ninst✝¹ : HasCoproducts C\ninst✝ : HasImages C\nA : C\ns : Set (Subobject A)\nf : Subobject A\nhf : f ∈ s\n⊢ f ∈ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Subobject.Lattice | {
"line": 663,
"column": 4
} | {
"line": 663,
"column": 15
} | {
"line": 663,
"column": 16
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ne : α ≃ β\ns : Set α\na : α\nm : a ∈ s\n⊢ e.symm (e a) ∈ s",
"ppTerm": "?m.52",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Equiv.instEquivLike",
"congrArg",
"Equiv.symm_apply_apply",
"Membership.mem",
"id",
"Equiv",
... | [
"α : Type u_1\nβ : Type u_2\ne : α ≃ β\ns : Set α\na : α\nm : a ∈ s\n⊢ a ∈ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Subobject.Lattice | {
"line": 673,
"column": 6
} | {
"line": 673,
"column": 17
} | {
"line": 673,
"column": 18
} | [
{
"pp": "case f.refine_1\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : LocallySmall.{w, v₁, u₁} C\ninst✝² : WellPowered.{w, v₁, u₁} C\ninst✝¹ : HasCoproducts C\ninst✝ : HasImages C\nA : C\ns : Set (Subobject A)\nf : Subobject A\nk : ∀ g ∈ s, g ≤ f\ng : Shrink.{w, max u₁ v₁} (Subobject A)\nm : g ∈ ⇑(equiv... | [
"case f.refine_1\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : LocallySmall.{w, v₁, u₁} C\ninst✝² : WellPowered.{w, v₁, u₁} C\ninst✝¹ : HasCoproducts C\ninst✝ : HasImages C\nA : C\ns : Set (Subobject A)\nf : Subobject A\nk : ∀ g ∈ s, g ≤ f\ng : Shrink.{w, max u₁ v₁} (Subobject A)\nm : g ∈ ⇑(equivShrink (Subo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Shapes.Opposites.Products | {
"line": 109,
"column": 2
} | {
"line": 114,
"column": 21
} | {
"line": 116,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nJ : Type u₂\ninst✝¹ : Category.{v₂, u₂} J\nX : Type v₂\nα : Type u_1\nZ : α → C\ninst✝ : HasCoproduct Z\nc : Cofan Z\nhc : IsColimit c\n⊢ IsLimit c.op",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"CategoryTheory.Functor.op",
... | [] | let e : Discrete.functor (Opposite.op <| Z ·) ≅ (Discrete.opposite α).inverse ⋙
(Discrete.functor Z).op := Discrete.natIso (fun _ ↦ Iso.refl _)
refine IsLimit.ofIsoLimit ((IsLimit.postcomposeInvEquiv e _).2
(IsLimit.whiskerEquivalence hc.op (Discrete.opposite α).symm))
(Cone.ext (Iso.refl _) (fun ⟨a⟩ ↦ ?_... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Shapes.Opposites.Products | {
"line": 109,
"column": 2
} | {
"line": 114,
"column": 21
} | {
"line": 116,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nJ : Type u₂\ninst✝¹ : Category.{v₂, u₂} J\nX : Type v₂\nα : Type u_1\nZ : α → C\ninst✝ : HasCoproduct Z\nc : Cofan Z\nhc : IsColimit c\n⊢ IsLimit c.op",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"CategoryTheory.Functor.op",
... | [] | let e : Discrete.functor (Opposite.op <| Z ·) ≅ (Discrete.opposite α).inverse ⋙
(Discrete.functor Z).op := Discrete.natIso (fun _ ↦ Iso.refl _)
refine IsLimit.ofIsoLimit ((IsLimit.postcomposeInvEquiv e _).2
(IsLimit.whiskerEquivalence hc.op (Discrete.opposite α).symm))
(Cone.ext (Iso.refl _) (fun ⟨a⟩ ↦ ?_... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Shapes.Opposites.Products | {
"line": 160,
"column": 2
} | {
"line": 160,
"column": 25
} | {
"line": 161,
"column": 2
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nα : Type u_1\nZ : α → C\nc c' : Cofan Z\nf : Fan fun x ↦ op (Z x)\nhc : IsColimit c\nhc' : IsColimit c'\nhf : IsLimit f\nj : α\n⊢ c'.ι.app { as := j } ≫ ((opCoproductIsoProduct' hc hf).hom ≫ (opCoproductIsoProduct' hc' hf).inv).unop =\n c'.ι.app { as := j } ... | [
"C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nα : Type u_1\nZ : α → C\nc c' : Cofan Z\nf : Fan fun x ↦ op (Z x)\nhc : IsColimit c\nhc' : IsColimit c'\nhf : IsLimit f\nj : α\n⊢ c'.inj j ≫ ((opCoproductIsoProduct' hc hf).hom ≫ (opCoproductIsoProduct' hc' hf).inv).unop =\n c'.ι.app { as := j } ≫ (hc.coconePointUniqueU... | change c'.inj _ ≫ _ = _ | Lean.Elab.Tactic.evalChange | Lean.Parser.Tactic.change |
Mathlib.CategoryTheory.Functor.KanExtension.Pointwise | {
"line": 265,
"column": 2
} | {
"line": 265,
"column": 13
} | {
"line": 265,
"column": 14
} | [
{
"pp": "C : Type u_1\nD : Type u_2\nH : Type u_4\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : Category.{v_2, u_2} D\ninst✝² : Category.{v_4, u_4} H\nL : C ⥤ D\nF : C ⥤ H\nE : L.LeftExtension F\nX : C\nh : E.IsPointwiseLeftKanExtensionAt (L.obj X)\ninst✝¹ : L.Full\ninst✝ : L.Faithful\n⊢ IsIso ((StructuredArrow.hom... | [
"C : Type u_1\nD : Type u_2\nH : Type u_4\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : Category.{v_2, u_2} D\ninst✝² : Category.{v_4, u_4} H\nL : C ⥤ D\nF : C ⥤ H\nE : L.LeftExtension F\nX : C\nh : E.IsPointwiseLeftKanExtensionAt (L.obj X)\ninst✝¹ : L.Full\ninst✝ : L.Faithful\n⊢ IsIso ((StructuredArrow.hom E).app X)"
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Functor.KanExtension.Pointwise | {
"line": 294,
"column": 24
} | {
"line": 294,
"column": 35
} | {
"line": 294,
"column": 36
} | [
{
"pp": "C : Type u_1\nD : Type u_2\nH : Type u_4\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Category.{v_2, u_2} D\ninst✝ : Category.{v_4, u_4} H\nL : C ⥤ D\nF : C ⥤ H\nE : L.LeftExtension F\nY : D\nh : E.IsPointwiseLeftKanExtensionAt Y\nT : H\nf g : (StructuredArrow.right E).obj Y ⟶ T\nhfg :\n ∀ ⦃X : C⦄ (φ : L... | [
"C : Type u_1\nD : Type u_2\nH : Type u_4\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Category.{v_2, u_2} D\ninst✝ : Category.{v_4, u_4} H\nL : C ⥤ D\nF : C ⥤ H\nE : L.LeftExtension F\nY : D\nh : E.IsPointwiseLeftKanExtensionAt Y\nT : H\nf g : (StructuredArrow.right E).obj Y ⟶ T\nhfg :\n ∀ ⦃X : C⦄ (φ : L.obj X ⟶ Y),... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Functor.KanExtension.Pointwise | {
"line": 302,
"column": 2
} | {
"line": 302,
"column": 13
} | {
"line": 302,
"column": 14
} | [
{
"pp": "C : Type u_1\nD : Type u_2\nH : Type u_4\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Category.{v_2, u_2} D\ninst✝ : Category.{v_4, u_4} H\nL : C ⥤ D\nF : C ⥤ H\nE : L.LeftExtension F\nY : D\nh : E.IsPointwiseLeftKanExtensionAt Y\nZ : H\nφ : CostructuredArrow.proj L Y ⋙ F ⟶ (const (CostructuredArrow L Y))... | [
"C : Type u_1\nD : Type u_2\nH : Type u_4\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Category.{v_2, u_2} D\ninst✝ : Category.{v_4, u_4} H\nL : C ⥤ D\nF : C ⥤ H\nE : L.LeftExtension F\nY : D\nh : E.IsPointwiseLeftKanExtensionAt Y\nZ : H\nφ : CostructuredArrow.proj L Y ⋙ F ⟶ (const (CostructuredArrow L Y)).obj Z\ng : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Functor.KanExtension.Pointwise | {
"line": 370,
"column": 8
} | {
"line": 370,
"column": 19
} | {
"line": 370,
"column": 20
} | [
{
"pp": "C : Type u_1\nD : Type u_2\nD' : Type u_3\nH : Type u_4\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Category.{v_2, u_2} D\ninst✝¹ : Category.{v_3, u_3} D'\ninst✝ : Category.{v_4, u_4} H\nL : C ⥤ D\nL' : C ⥤ D'\nF : C ⥤ H\nE : L.LeftExtension F\nE' : ?m.13\nh : E.IsPointwiseLeftKanExtension\nG : L.LeftExt... | [
"C : Type u_1\nD : Type u_2\nD' : Type u_3\nH : Type u_4\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Category.{v_2, u_2} D\ninst✝¹ : Category.{v_3, u_3} D'\ninst✝ : Category.{v_4, u_4} H\nL : C ⥤ D\nL' : C ⥤ D'\nF : C ⥤ H\nE : L.LeftExtension F\nE' : ?m.13\nh : E.IsPointwiseLeftKanExtension\nG : L.LeftExtension F\nY₁... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Functor.KanExtension.Pointwise | {
"line": 373,
"column": 6
} | {
"line": 373,
"column": 17
} | {
"line": 373,
"column": 18
} | [
{
"pp": "C : Type u_1\nD : Type u_2\nD' : Type u_3\nH : Type u_4\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Category.{v_2, u_2} D\ninst✝¹ : Category.{v_3, u_3} D'\ninst✝ : Category.{v_4, u_4} H\nL : C ⥤ D\nL' : C ⥤ D'\nF : C ⥤ H\nE : L.LeftExtension F\nE' : ?m.13\nh : E.IsPointwiseLeftKanExtension\nG : L.LeftExt... | [
"C : Type u_1\nD : Type u_2\nD' : Type u_3\nH : Type u_4\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Category.{v_2, u_2} D\ninst✝¹ : Category.{v_3, u_3} D'\ninst✝ : Category.{v_4, u_4} H\nL : C ⥤ D\nL' : C ⥤ D'\nF : C ⥤ H\nE : L.LeftExtension F\nE' : ?m.13\nh : E.IsPointwiseLeftKanExtension\nG : L.LeftExtension F\nX ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Functor.KanExtension.Pointwise | {
"line": 454,
"column": 2
} | {
"line": 454,
"column": 13
} | {
"line": 454,
"column": 14
} | [
{
"pp": "C : Type u_1\nD : Type u_2\nH : Type u_4\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : Category.{v_2, u_2} D\ninst✝² : Category.{v_4, u_4} H\nL : C ⥤ D\nF : C ⥤ H\nE : L.RightExtension F\nX : C\nh : E.IsPointwiseRightKanExtensionAt (L.obj X)\ninst✝¹ : L.Full\ninst✝ : L.Faithful\n⊢ IsIso ((CostructuredArrow... | [
"C : Type u_1\nD : Type u_2\nH : Type u_4\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : Category.{v_2, u_2} D\ninst✝² : Category.{v_4, u_4} H\nL : C ⥤ D\nF : C ⥤ H\nE : L.RightExtension F\nX : C\nh : E.IsPointwiseRightKanExtensionAt (L.obj X)\ninst✝¹ : L.Full\ninst✝ : L.Faithful\n⊢ IsIso ((CostructuredArrow.hom E).app ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Functor.KanExtension.Pointwise | {
"line": 549,
"column": 8
} | {
"line": 549,
"column": 19
} | {
"line": 549,
"column": 20
} | [
{
"pp": "C : Type u_1\nD : Type u_2\nD' : Type u_3\nH : Type u_4\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Category.{v_2, u_2} D\ninst✝¹ : Category.{v_3, u_3} D'\ninst✝ : Category.{v_4, u_4} H\nL : C ⥤ D\nL' : C ⥤ D'\nF : C ⥤ H\nE E' : L.RightExtension F\nh : E.IsPointwiseRightKanExtension\nG : L.RightExtension... | [
"C : Type u_1\nD : Type u_2\nD' : Type u_3\nH : Type u_4\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Category.{v_2, u_2} D\ninst✝¹ : Category.{v_3, u_3} D'\ninst✝ : Category.{v_4, u_4} H\nL : C ⥤ D\nL' : C ⥤ D'\nF : C ⥤ H\nE E' : L.RightExtension F\nh : E.IsPointwiseRightKanExtension\nG : L.RightExtension F\nY₁ Y₂ : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Functor.KanExtension.Adjunction | {
"line": 113,
"column": 2
} | {
"line": 113,
"column": 13
} | {
"line": 113,
"column": 14
} | [
{
"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\nL : C ⥤ D\nH : Type u_3\ninst✝¹ : Category.{v_3, u_3} H\nF : C ⥤ H\ninst✝ : L.HasPointwiseLeftKanExtension F\nX : C\n⊢ (L.leftKanExtensionUnit F).app X ≫ (L.leftKanExtensionObjIsoColimit F (L.obj X)).hom =\n ... | [
"C : Type u_1\nD : Type u_2\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Category.{v_2, u_2} D\nL : C ⥤ D\nH : Type u_3\ninst✝¹ : Category.{v_3, u_3} H\nF : C ⥤ H\ninst✝ : L.HasPointwiseLeftKanExtension F\nX : C\n⊢ (L.leftKanExtensionUnit F).app X ≫ (L.leftKanExtensionObjIsoColimit F (L.obj X)).hom =\n colimit.ι (p... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Functor.KanExtension.Pointwise | {
"line": 552,
"column": 6
} | {
"line": 552,
"column": 17
} | {
"line": 552,
"column": 18
} | [
{
"pp": "C : Type u_1\nD : Type u_2\nD' : Type u_3\nH : Type u_4\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Category.{v_2, u_2} D\ninst✝¹ : Category.{v_3, u_3} D'\ninst✝ : Category.{v_4, u_4} H\nL : C ⥤ D\nL' : C ⥤ D'\nF : C ⥤ H\nE E' : L.RightExtension F\nh : E.IsPointwiseRightKanExtension\nG : L.RightExtension... | [
"C : Type u_1\nD : Type u_2\nD' : Type u_3\nH : Type u_4\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Category.{v_2, u_2} D\ninst✝¹ : Category.{v_3, u_3} D'\ninst✝ : Category.{v_4, u_4} H\nL : C ⥤ D\nL' : C ⥤ D'\nF : C ⥤ H\nE E' : L.RightExtension F\nh : E.IsPointwiseRightKanExtension\nG : L.RightExtension F\nX : C\n⊢... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Functor.KanExtension.Pointwise | {
"line": 601,
"column": 12
} | {
"line": 601,
"column": 23
} | {
"line": 601,
"column": 24
} | [
{
"pp": "C : Type u_1\nD : Type u_2\nD' : Type u_3\nH : Type u_4\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : Category.{v_2, u_2} D\ninst✝² : Category.{v_3, u_3} D'\ninst✝¹ : Category.{v_4, u_4} H\nL : C ⥤ D\nL' : C ⥤ D'\nF : C ⥤ H\ninst✝ : L.HasPointwiseLeftKanExtension F\nY₁ Y₂ : D\nf : Y₁ ⟶ Y₂\ng₁ g₂ : Costruct... | [
"C : Type u_1\nD : Type u_2\nD' : Type u_3\nH : Type u_4\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : Category.{v_2, u_2} D\ninst✝² : Category.{v_3, u_3} D'\ninst✝¹ : Category.{v_4, u_4} H\nL : C ⥤ D\nL' : C ⥤ D'\nF : C ⥤ H\ninst✝ : L.HasPointwiseLeftKanExtension F\nY₁ Y₂ : D\nf : Y₁ ⟶ Y₂\ng₁ g₂ : CostructuredArrow L ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Comma.Presheaf.Basic | {
"line": 110,
"column": 2
} | {
"line": 110,
"column": 45
} | {
"line": 110,
"column": 46
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nA : Cᵒᵖ ⥤ Type v\nY : C\nη : yoneda.obj Y ⟶ A\nX : C\ns : yoneda.obj X ⟶ A\nf : X ⟶ Y\nhf : yoneda.map f ≫ η = s\n⊢ MakesOverArrow η s f",
"ppTerm": "?m.40",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"C : Type u\ninst✝ : Category.{v, u} C\nA : Cᵒᵖ ⥤ Type v\nY : C\nη : yoneda.obj Y ⟶ A\nX : C\ns : yoneda.obj X ⟶ A\nf : X ⟶ Y\nhf : yoneda.map f ≫ η = s\n⊢ MakesOverArrow η s f"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Functor.KanExtension.Pointwise | {
"line": 708,
"column": 12
} | {
"line": 708,
"column": 23
} | {
"line": 708,
"column": 24
} | [
{
"pp": "C : Type u_1\nD : Type u_2\nD' : Type u_3\nH : Type u_4\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : Category.{v_2, u_2} D\ninst✝² : Category.{v_3, u_3} D'\ninst✝¹ : Category.{v_4, u_4} H\nL : C ⥤ D\nL' : C ⥤ D'\nF : C ⥤ H\ninst✝ : L.HasPointwiseRightKanExtension F\nY₁ Y₂ : D\nf : Y₁ ⟶ Y₂\ng₁ g₂ : Structu... | [
"C : Type u_1\nD : Type u_2\nD' : Type u_3\nH : Type u_4\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : Category.{v_2, u_2} D\ninst✝² : Category.{v_3, u_3} D'\ninst✝¹ : Category.{v_4, u_4} H\nL : C ⥤ D\nL' : C ⥤ D'\nF : C ⥤ H\ninst✝ : L.HasPointwiseRightKanExtension F\nY₁ Y₂ : D\nf : Y₁ ⟶ Y₂\ng₁ g₂ : StructuredArrow Y₂ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Comma.Presheaf.Basic | {
"line": 469,
"column": 4
} | {
"line": 469,
"column": 43
} | {
"line": 469,
"column": 44
} | [
{
"pp": "case refine_1\nC : Type u\ninst✝ : Category.{v, u} C\nA F : Cᵒᵖ ⥤ Type v\nη : F ⟶ A\nX : C\np : YonedaCollection (restrictedYonedaObj η) X\n⊢ ((unitBackward η X ∘ unitForward η X) p).fst = (id p).fst",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"CategoryTheory.OverPre... | [
"case refine_1\nC : Type u\ninst✝ : Category.{v, u} C\nA F : Cᵒᵖ ⥤ Type v\nη : F ⟶ A\nX : C\np : YonedaCollection (restrictedYonedaObj η) X\n⊢ yonedaEquiv.symm ((ConcreteCategory.hom (η.app (op X))) (OverArrows.val p.snd)) = p.fst"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Comma.Presheaf.Basic | {
"line": 510,
"column": 2
} | {
"line": 510,
"column": 50
} | {
"line": 510,
"column": 51
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nA : Cᵒᵖ ⥤ Type v\nF : (CostructuredArrow yoneda A)ᵒᵖ ⥤ Type v\nX : C\ns : yoneda.obj X ⟶ A\np : OverArrows (yonedaCollectionPresheafToA F) s\n⊢ YonedaCollection.fst p.val = s",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"CategoryTheo... | [
"C : Type u\ninst✝ : Category.{v, u} C\nA : Cᵒᵖ ⥤ Type v\nF : (CostructuredArrow yoneda A)ᵒᵖ ⥤ Type v\nX : C\ns : yoneda.obj X ⟶ A\np : OverArrows (yonedaCollectionPresheafToA F) s\n⊢ YonedaCollection.fst p.val = s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Functor.KanExtension.Basic | {
"line": 877,
"column": 2
} | {
"line": 877,
"column": 13
} | {
"line": 877,
"column": 14
} | [
{
"pp": "C : Type u_1\nC' : Type u_2\nH : Type u_3\nD : Type u_4\nD' : Type u_5\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : Category.{v_2, u_2} C'\ninst✝² : Category.{v_3, u_3} H\ninst✝¹ : Category.{v_4, u_4} D\ninst✝ : Category.{v_5, u_5} D'\nL : C ≌ D\nF₀ : C ⥤ H\nF₁ : D ⥤ H\nα : F₀ ≅ L.functor ⋙ F₁\ny : Struct... | [
"C : Type u_1\nC' : Type u_2\nH : Type u_3\nD : Type u_4\nD' : Type u_5\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : Category.{v_2, u_2} C'\ninst✝² : Category.{v_3, u_3} H\ninst✝¹ : Category.{v_4, u_4} D\ninst✝ : Category.{v_5, u_5} D'\nL : C ≌ D\nF₀ : C ⥤ H\nF₁ : D ⥤ H\nα : F₀ ≅ L.functor ⋙ F₁\ny : StructuredArrow F₀... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Functor.KanExtension.Basic | {
"line": 896,
"column": 2
} | {
"line": 896,
"column": 13
} | {
"line": 896,
"column": 14
} | [
{
"pp": "C : Type u_1\nC' : Type u_2\nH : Type u_3\nD : Type u_4\nD' : Type u_5\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : Category.{v_2, u_2} C'\ninst✝² : Category.{v_3, u_3} H\ninst✝¹ : Category.{v_4, u_4} D\ninst✝ : Category.{v_5, u_5} D'\nL : C ≌ D\nF₀ : C ⥤ H\nF₁ : D ⥤ H\nα : L.functor ⋙ F₁ ≅ F₀\ny : Costru... | [
"C : Type u_1\nC' : Type u_2\nH : Type u_3\nD : Type u_4\nD' : Type u_5\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : Category.{v_2, u_2} C'\ninst✝² : Category.{v_3, u_3} H\ninst✝¹ : Category.{v_4, u_4} D\ninst✝ : Category.{v_5, u_5} D'\nL : C ≌ D\nF₀ : C ⥤ H\nF₁ : D ⥤ H\nα : L.functor ⋙ F₁ ≅ F₀\ny : CostructuredArrow ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Abelian.Transfer | {
"line": 62,
"column": 8
} | {
"line": 62,
"column": 19
} | {
"line": 62,
"column": 20
} | [
{
"pp": "C : Type u₁\ninst✝⁵ : Category.{v₁, u₁} C\ninst✝⁴ : Preadditive C\nD : Type u₂\ninst✝³ : Category.{v₂, u₂} D\ninst✝² : Abelian D\nF : C ⥤ D\nG : D ⥤ C\ninst✝¹ : G.PreservesZeroMorphisms\ninst✝ : PreservesFiniteLimits G\ni : F ⋙ G ≅ 𝟭 C\nX Y : C\nf : X ⟶ Y\n⊢ i.inv.app X ≫ G.map (F.map f) ≫ i.hom.app Y... | [
"C : Type u₁\ninst✝⁵ : Category.{v₁, u₁} C\ninst✝⁴ : Preadditive C\nD : Type u₂\ninst✝³ : Category.{v₂, u₂} D\ninst✝² : Abelian D\nF : C ⥤ D\nG : D ⥤ C\ninst✝¹ : G.PreservesZeroMorphisms\ninst✝ : PreservesFiniteLimits G\ni : F ⋙ G ≅ 𝟭 C\nX Y : C\nf : X ⟶ Y\n⊢ i.inv.app X ≫ G.map (F.map f) ≫ i.hom.app Y = f"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Abelian.Transfer | {
"line": 73,
"column": 8
} | {
"line": 73,
"column": 19
} | {
"line": 73,
"column": 20
} | [
{
"pp": "C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : Preadditive C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\ninst✝¹ : Abelian D\nF : C ⥤ D\nG : D ⥤ C\ninst✝ : G.PreservesZeroMorphisms\ni : F ⋙ G ≅ 𝟭 C\nadj : G ⊣ F\nX Y : C\nf : X ⟶ Y\nthis : PreservesColimits G\n⊢ i.inv.app X ≫ G.map (F.map f) ≫ i.ho... | [
"C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : Preadditive C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\ninst✝¹ : Abelian D\nF : C ⥤ D\nG : D ⥤ C\ninst✝ : G.PreservesZeroMorphisms\ni : F ⋙ G ≅ 𝟭 C\nadj : G ⊣ F\nX Y : C\nf : X ⟶ Y\nthis : PreservesColimits G\n⊢ i.inv.app X ≫ G.map (F.map f) ≫ i.hom.app Y = f"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Presheaf | {
"line": 434,
"column": 6
} | {
"line": 434,
"column": 49
} | {
"line": 435,
"column": 8
} | [
{
"pp": "C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nℰ : Type u₂\ninst✝² : Category.{v₂, u₂} ℰ\nA✝ A : C ⥤ ℰ\ninst✝¹ : uliftYoneda.{max w v₂, v₁, u₁}.HasPointwiseLeftKanExtension A\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF : C ⥤ D\nX : C\nY : F.op.LeftExtension (uliftYoneda.{max w v₂, v₁, u₁}.obj X)\nZ : Cᵒᵖ\... | [
"C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nℰ : Type u₂\ninst✝² : Category.{v₂, u₂} ℰ\nA✝ A : C ⥤ ℰ\ninst✝¹ : uliftYoneda.{max w v₂, v₁, u₁}.HasPointwiseLeftKanExtension A\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF : C ⥤ D\nX : C\nY : F.op.LeftExtension (uliftYoneda.{max w v₂, v₁, u₁}.obj X)\nZ : Cᵒᵖ\nf : (yoneda... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Generator.Basic | {
"line": 135,
"column": 4
} | {
"line": 135,
"column": 52
} | {
"line": 135,
"column": 53
} | [
{
"pp": "case refine_1\nC : Type u₁\ninst✝ : Category.{v₁, u₁} C\nP : ObjectProperty C\nhP : P.op.IsSeparating\nX Y : C\nf g : X ⟶ Y\nhfg : ∀ (G : C), P G → ∀ (h : Y ⟶ G), f ≫ h = g ≫ h\nG : Cᵒᵖ\nhG : P.op G\nh : G ⟶ op Y\n⊢ (h ≫ f.op).unop = (h ≫ g.op).unop",
"ppTerm": "?refine_1",
"assigned": true,
... | [
"case refine_1\nC : Type u₁\ninst✝ : Category.{v₁, u₁} C\nP : ObjectProperty C\nhP : P.op.IsSeparating\nX Y : C\nf g : X ⟶ Y\nhfg : ∀ (G : C), P G → ∀ (h : Y ⟶ G), f ≫ h = g ≫ h\nG : Cᵒᵖ\nhG : P.op G\nh : G ⟶ op Y\n⊢ f ≫ h.unop = g ≫ h.unop"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Generator.Basic | {
"line": 137,
"column": 4
} | {
"line": 137,
"column": 50
} | {
"line": 137,
"column": 51
} | [
{
"pp": "case refine_2\nC : Type u₁\ninst✝ : Category.{v₁, u₁} C\nP : ObjectProperty C\nhP : P.IsCoseparating\nX Y : Cᵒᵖ\nf g : X ⟶ Y\nhfg : ∀ (G : Cᵒᵖ), P.op G → ∀ (h : G ⟶ X), h ≫ f = h ≫ g\nG : C\nhG : P G\nh : unop X ⟶ G\n⊢ (f.unop ≫ h).op = (g.unop ≫ h).op",
"ppTerm": "?refine_2",
"assigned": true,... | [
"case refine_2\nC : Type u₁\ninst✝ : Category.{v₁, u₁} C\nP : ObjectProperty C\nhP : P.IsCoseparating\nX Y : Cᵒᵖ\nf g : X ⟶ Y\nhfg : ∀ (G : Cᵒᵖ), P.op G → ∀ (h : G ⟶ X), h ≫ f = h ≫ g\nG : C\nhG : P G\nh : unop X ⟶ G\n⊢ h.op ≫ f = h.op ≫ g"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Generator.Basic | {
"line": 142,
"column": 4
} | {
"line": 142,
"column": 52
} | {
"line": 142,
"column": 53
} | [
{
"pp": "case refine_1\nC : Type u₁\ninst✝ : Category.{v₁, u₁} C\nP : ObjectProperty C\nhP : P.op.IsCoseparating\nX Y : C\nf g : X ⟶ Y\nhfg : ∀ (G : C), P G → ∀ (h : G ⟶ X), h ≫ f = h ≫ g\nG : Cᵒᵖ\nhG : P.op G\nh : op X ⟶ G\n⊢ (f.op ≫ h).unop = (g.op ≫ h).unop",
"ppTerm": "?refine_1",
"assigned": true,
... | [
"case refine_1\nC : Type u₁\ninst✝ : Category.{v₁, u₁} C\nP : ObjectProperty C\nhP : P.op.IsCoseparating\nX Y : C\nf g : X ⟶ Y\nhfg : ∀ (G : C), P G → ∀ (h : G ⟶ X), h ≫ f = h ≫ g\nG : Cᵒᵖ\nhG : P.op G\nh : op X ⟶ G\n⊢ h.unop ≫ f = h.unop ≫ g"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Generator.Basic | {
"line": 144,
"column": 4
} | {
"line": 144,
"column": 50
} | {
"line": 144,
"column": 51
} | [
{
"pp": "case refine_2\nC : Type u₁\ninst✝ : Category.{v₁, u₁} C\nP : ObjectProperty C\nhP : P.IsSeparating\nX Y : Cᵒᵖ\nf g : X ⟶ Y\nhfg : ∀ (G : Cᵒᵖ), P.op G → ∀ (h : Y ⟶ G), f ≫ h = g ≫ h\nG : C\nhG : P G\nh : G ⟶ unop Y\n⊢ (h ≫ f.unop).op = (h ≫ g.unop).op",
"ppTerm": "?refine_2",
"assigned": true,
... | [
"case refine_2\nC : Type u₁\ninst✝ : Category.{v₁, u₁} C\nP : ObjectProperty C\nhP : P.IsSeparating\nX Y : Cᵒᵖ\nf g : X ⟶ Y\nhfg : ∀ (G : Cᵒᵖ), P.op G → ∀ (h : Y ⟶ G), f ≫ h = g ≫ h\nG : C\nhG : P G\nh : G ⟶ unop Y\n⊢ f ≫ h.op = g ≫ h.op"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Generator.Basic | {
"line": 160,
"column": 2
} | {
"line": 163,
"column": 55
} | {
"line": 165,
"column": 0
} | [
{
"pp": "case refine_2\nC : Type u₁\ninst✝ : Category.{v₁, u₁} C\nP : ObjectProperty C\nhP : P.IsCodetecting\nX Y : Cᵒᵖ\nf : X ⟶ Y\nhf : ∀ (G : Cᵒᵖ), P.op G → ∀ (h : G ⟶ Y), ∃! h', h' ≫ f = h\n⊢ IsIso f",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"CategoryTheory.isIso_unop_if... | [] | · refine (isIso_unop_iff _).1 (hP _ fun G hG h => ?_)
obtain ⟨t, ht, ht'⟩ := hf (op G) hG h.op
refine ⟨t.unop, Quiver.Hom.op_inj ht, fun y hy => Quiver.Hom.op_inj (ht' _ ?_)⟩
exact Quiver.Hom.unop_inj (by simpa only using! hy) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Generator.Basic | {
"line": 171,
"column": 2
} | {
"line": 174,
"column": 55
} | {
"line": 176,
"column": 0
} | [
{
"pp": "case refine_2\nC : Type u₁\ninst✝ : Category.{v₁, u₁} C\nP : ObjectProperty C\nhP : P.IsDetecting\nX Y : Cᵒᵖ\nf : X ⟶ Y\nhf : ∀ (G : Cᵒᵖ), P.op G → ∀ (h : X ⟶ G), ∃! h', f ≫ h' = h\n⊢ IsIso f",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"CategoryTheory.isIso_unop_iff"... | [] | · refine (isIso_unop_iff _).1 (hP _ fun G hG h => ?_)
obtain ⟨t, ht, ht'⟩ := hf (op G) hG h.op
refine ⟨t.unop, Quiver.Hom.op_inj ht, fun y hy => Quiver.Hom.op_inj (ht' _ ?_)⟩
exact Quiver.Hom.unop_inj (by simpa only using! hy) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Generator.Basic | {
"line": 193,
"column": 2
} | {
"line": 193,
"column": 64
} | {
"line": 193,
"column": 65
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nP : ObjectProperty C\ninst✝ : HasCoequalizers C\n⊢ P.IsCodetecting → P.IsCoseparating",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Opposite",
"CategoryTheory.ObjectProperty.IsCodetecting",
"id",
... | [
"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nP : ObjectProperty C\ninst✝ : HasCoequalizers C\n⊢ P.op.IsDetecting → P.op.IsSeparating"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Generator.Basic | {
"line": 197,
"column": 24
} | {
"line": 197,
"column": 49
} | {
"line": 197,
"column": 50
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nP : ObjectProperty C\nhP : P.IsSeparating\nX Y : C\nf : X ⟶ Y\nx✝⁴ : Mono f\nx✝³ : C\nx✝² : P x✝³\nx✝¹ x✝ : x✝³ ⟶ X\nh : x✝¹ ≫ f = x✝ ≫ f\n⊢ x✝¹ = x✝",
"ppTerm": "?m.34",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": ... | [
"C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nP : ObjectProperty C\nhP : P.IsSeparating\nX Y : C\nf : X ⟶ Y\nx✝⁴ : Mono f\nx✝³ : C\nx✝² : P x✝³\nx✝¹ x✝ : x✝³ ⟶ X\nh : x✝¹ ≫ f = x✝ ≫ f\n⊢ x✝¹ = x✝"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Generator.Basic | {
"line": 202,
"column": 24
} | {
"line": 202,
"column": 48
} | {
"line": 202,
"column": 49
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nP : ObjectProperty C\nhP : P.IsCoseparating\nX Y : C\nf : X ⟶ Y\nx✝⁴ : Epi f\nx✝³ : C\nx✝² : P x✝³\nx✝¹ x✝ : Y ⟶ x✝³\nh : f ≫ x✝¹ = f ≫ x✝\n⊢ x✝¹ = x✝",
"ppTerm": "?m.34",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals":... | [
"C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nP : ObjectProperty C\nhP : P.IsCoseparating\nX Y : C\nf : X ⟶ Y\nx✝⁴ : Epi f\nx✝³ : C\nx✝² : P x✝³\nx✝¹ x✝ : Y ⟶ x✝³\nh : f ≫ x✝¹ = f ≫ x✝\n⊢ x✝¹ = x✝"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Subobject.Comma | {
"line": 95,
"column": 59
} | {
"line": 95,
"column": 85
} | {
"line": 95,
"column": 86
} | [
{
"pp": "C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\nS : D\nT : C ⥤ D\ninst✝¹ : HasFiniteLimits C\ninst✝ : PreservesFiniteLimits T\nA P : StructuredArrow S T\nf : P ⟶ A\nhf : Mono f\nq : S ⟶ T.obj (Subobject.underlying.obj (projectSubobject (Subobject.mk f)))\nhq : q ≫ ... | [
"C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\nS : D\nT : C ⥤ D\ninst✝¹ : HasFiniteLimits C\ninst✝ : PreservesFiniteLimits T\nA P : StructuredArrow S T\nf : P ⟶ A\nhf : Mono f\nq : S ⟶ T.obj (Subobject.underlying.obj (projectSubobject (Subobject.mk f)))\nhq : q ≫ T.map (proje... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Presheaf | {
"line": 511,
"column": 10
} | {
"line": 511,
"column": 82
} | {
"line": 511,
"column": 83
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF : C ⥤ D\nG : (Cᵒᵖ ⥤ Type (max w v₁ v₂)) ⥤ Dᵒᵖ ⥤ Type (max w v₁ v₂)\nφ : F ⋙ uliftYoneda.{max w v₁, v₂, u₂} ⟶ uliftYoneda.{max w v₂, v₁, u₁} ⋙ G\nP : Cᵒᵖ ⥤ Type (max w v₁ v₂)\nx y : P.Elements\nf : x ⟶ y\n⊢ uliftYoned... | [
"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF : C ⥤ D\nG : (Cᵒᵖ ⥤ Type (max w v₁ v₂)) ⥤ Dᵒᵖ ⥤ Type (max w v₁ v₂)\nφ : F ⋙ uliftYoneda.{max w v₁, v₂, u₂} ⟶ uliftYoneda.{max w v₂, v₁, u₁} ⋙ G\nP : Cᵒᵖ ⥤ Type (max w v₁ v₂)\nx y : P.Elements\nf : x ⟶ y\n⊢ (hom (P.map ↑f)) x.snd... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Generator.Basic | {
"line": 250,
"column": 2
} | {
"line": 250,
"column": 64
} | {
"line": 250,
"column": 65
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nP : ObjectProperty C\ninst✝ : Balanced C\n⊢ P.IsCoseparating → P.IsCodetecting",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Opposite",
"CategoryTheory.ObjectProperty.IsCodetecting",
"id",
"Catego... | [
"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nP : ObjectProperty C\ninst✝ : Balanced C\n⊢ P.op.IsSeparating → P.op.IsDetecting"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Generator.Basic | {
"line": 318,
"column": 16
} | {
"line": 318,
"column": 27
} | {
"line": 318,
"column": 28
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nP : ObjectProperty C\nhP : ∀ (X : C), ∃ ι s, ∃ (_ : ∀ (i : ι), P (s i)), ∃ c x p, Epi p\nX Y : C\nf g : X ⟶ Y\nh : ∀ (G : C), P G → ∀ (h : G ⟶ X), h ≫ f = h ≫ g\nι : Type w\ns : ι → C\nhs : ∀ (i : ι), P (s i)\nc : Cofan s\nhc : IsColimit c\np : c.pt ⟶ X\nh✝ : E... | [
"C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nP : ObjectProperty C\nhP : ∀ (X : C), ∃ ι s, ∃ (_ : ∀ (i : ι), P (s i)), ∃ c x p, Epi p\nX Y : C\nf g : X ⟶ Y\nh : ∀ (G : C), P G → ∀ (h : G ⟶ X), h ≫ f = h ≫ g\nι : Type w\ns : ι → C\nhs : ∀ (i : ι), P (s i)\nc : Cofan s\nhc : IsColimit c\np : c.pt ⟶ X\nh✝ : Epi p\ni : ι\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Generator.Basic | {
"line": 328,
"column": 16
} | {
"line": 328,
"column": 27
} | {
"line": 328,
"column": 28
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nP : ObjectProperty C\nhP : ∀ (X : C), ∃ ι s, ∃ (_ : ∀ (i : ι), P (s i)), ∃ c x j, Mono j\nX Y : C\nf g : X ⟶ Y\nh : ∀ (G : C), P G → ∀ (h : Y ⟶ G), f ≫ h = g ≫ h\nι : Type w\ns : ι → C\nhs : ∀ (i : ι), P (s i)\nc : Fan s\nhc : IsLimit c\nj : Y ⟶ c.pt\nh✝ : Mono... | [
"C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nP : ObjectProperty C\nhP : ∀ (X : C), ∃ ι s, ∃ (_ : ∀ (i : ι), P (s i)), ∃ c x j, Mono j\nX Y : C\nf g : X ⟶ Y\nh : ∀ (G : C), P G → ∀ (h : Y ⟶ G), f ≫ h = g ≫ h\nι : Type w\ns : ι → C\nhs : ∀ (i : ι), P (s i)\nc : Fan s\nhc : IsLimit c\nj : Y ⟶ c.pt\nh✝ : Mono j\ni : ι\n⊢... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Subobject.Comma | {
"line": 194,
"column": 10
} | {
"line": 194,
"column": 54
} | {
"line": 194,
"column": 55
} | [
{
"pp": "case i.refine_2\nC : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\nS : C ⥤ D\nT : D\ninst✝¹ : HasFiniteColimits C\ninst✝ : PreservesFiniteColimits S\nA : CostructuredArrow S T\nP : (CostructuredArrow S T)ᵒᵖ\nf : P ⟶ op A\nhf : Mono f\nq : S.obj (unop (Subobject.under... | [
"case i.refine_2\nC : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\nS : C ⥤ D\nT : D\ninst✝¹ : HasFiniteColimits C\ninst✝ : PreservesFiniteColimits S\nA : CostructuredArrow S T\nP : (CostructuredArrow S T)ᵒᵖ\nf : P ⟶ op A\nhf : Mono f\nq : S.obj (unop (Subobject.underlying.obj (p... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Generator.Basic | {
"line": 376,
"column": 28
} | {
"line": 376,
"column": 39
} | {
"line": 376,
"column": 40
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nP : ObjectProperty C\nhP : P.IsSeparating\nX : C\ninst✝ : HasCoproduct (P.coproductFromFamily X)\nZ✝ : C\nu v : X ⟶ Z✝\nhuv : P.coproductFrom X ≫ u = P.coproductFrom X ≫ v\nG : C\nhG : P G\nh : G ⟶ X\n⊢ h ≫ u = h ≫ v",
"ppTerm": "?m.31",
"assigned": fa... | [
"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nP : ObjectProperty C\nhP : P.IsSeparating\nX : C\ninst✝ : HasCoproduct (P.coproductFromFamily X)\nZ✝ : C\nu v : X ⟶ Z✝\nhuv : P.coproductFrom X ≫ u = P.coproductFrom X ≫ v\nG : C\nhG : P G\nh : G ⟶ X\n⊢ h ≫ u = h ≫ v"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Generator.Basic | {
"line": 415,
"column": 28
} | {
"line": 415,
"column": 39
} | {
"line": 415,
"column": 40
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nP : ObjectProperty C\nhP : P.IsCoseparating\nX : C\ninst✝ : HasProduct (P.productToFamily X)\nZ✝ : C\nu v : Z✝ ⟶ X\nhuv : u ≫ P.productTo X = v ≫ P.productTo X\nG : C\nhG : P G\nh : X ⟶ G\n⊢ u ≫ h = v ≫ h",
"ppTerm": "?m.31",
"assigned": false,
"us... | [
"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nP : ObjectProperty C\nhP : P.IsCoseparating\nX : C\ninst✝ : HasProduct (P.productToFamily X)\nZ✝ : C\nu v : Z✝ ⟶ X\nhuv : u ≫ P.productTo X = v ≫ P.productTo X\nG : C\nhG : P G\nh : X ⟶ G\n⊢ u ≫ h = v ≫ h"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Subobject.Comma | {
"line": 202,
"column": 2
} | {
"line": 204,
"column": 41
} | {
"line": 205,
"column": 2
} | [
{
"pp": "C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\nS : C ⥤ D\nT : D\nA : CostructuredArrow S T\nP Q : (CostructuredArrow S T)ᵒᵖ\nf : P ⟶ op A\ng : Q ⟶ op A\ninst✝¹ : Mono f.unop.left.op\ninst✝ : Mono g.unop.left.op\nh : Subobject.mk f.unop.left.op ≤ Subobject.mk g.uno... | [
"C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\nS : C ⥤ D\nT : D\nA : CostructuredArrow S T\nP Q : (CostructuredArrow S T)ᵒᵖ\nf : P ⟶ op A\ng : Q ⟶ op A\ninst✝¹ : Mono f.unop.left.op\ninst✝ : Mono g.unop.left.op\nh : Subobject.mk f.unop.left.op ≤ Subobject.mk g.unop.left.op\n⊢... | conv_lhs =>
congr
rw [← Quiver.Hom.unop_op g.unop.left] | Mathlib.Tactic.Conv._aux_Mathlib_Tactic_Conv___macroRules_Mathlib_Tactic_Conv_convLHS_1 | Mathlib.Tactic.Conv.convLHS |
Mathlib.CategoryTheory.Subobject.Comma | {
"line": 233,
"column": 6
} | {
"line": 235,
"column": 15
} | {
"line": 235,
"column": 16
} | [
{
"pp": "case refine_3\nC : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\nS : C ⥤ D\nT : D\ninst✝¹ : HasFiniteColimits C\ninst✝ : PreservesFiniteColimits S\nA : CostructuredArrow S T\nP Q : (CostructuredArrow S T)ᵒᵖ\nf : P ⟶ op A\ng : Q ⟶ op A\nhf : Mono f\nhg : Mono g\nh : S... | [
"case refine_3\nC : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\nS : C ⥤ D\nT : D\ninst✝¹ : HasFiniteColimits C\ninst✝ : PreservesFiniteColimits S\nA : CostructuredArrow S T\nP Q : (CostructuredArrow S T)ᵒᵖ\nf : P ⟶ op A\ng : Q ⟶ op A\nhf : Mono f\nhg : Mono g\nh : Subobject.mk ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Generator.Basic | {
"line": 503,
"column": 12
} | {
"line": 503,
"column": 51
} | {
"line": 503,
"column": 52
} | [
{
"pp": "C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : HasPullbacks C\n𝒢 : ObjectProperty C\ninst✝¹ : ObjectProperty.Small.{w, v₁, u₁} 𝒢\ninst✝ : LocallySmall.{w, v₁, u₁} C\nh𝒢 : 𝒢.IsDetecting\nX : C\nP Q : Subobject X\nh : (fun P ↦ {f | P.Factors f.snd}) P = (fun P ↦ {f | P.Factors f.snd}) Q\n⊢ ∀ (G ... | [
"C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : HasPullbacks C\n𝒢 : ObjectProperty C\ninst✝¹ : ObjectProperty.Small.{w, v₁, u₁} 𝒢\ninst✝ : LocallySmall.{w, v₁, u₁} C\nh𝒢 : 𝒢.IsDetecting\nX : C\nP Q : Subobject X\nh : (fun P ↦ {f | P.Factors f.snd}) P = (fun P ↦ {f | P.Factors f.snd}) Q\n⊢ ∀ (G : C), 𝒢 G →... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Shapes.WideEqualizers | {
"line": 475,
"column": 6
} | {
"line": 475,
"column": 17
} | {
"line": 475,
"column": 18
} | [
{
"pp": "case one\nJ : Type w\nC : Type u\ninst✝¹ : Category.{v, u} C\nX Y : C\nf : J → (X ⟶ Y)\ninst✝ : Nonempty J\ns t : Trident f\nk : s.pt ⟶ t.pt\nw : k ≫ t.ι = s.ι\n⊢ k ≫ t.π.app one = s.π.app one",
"ppTerm": "?one",
"assigned": true,
"usedConstants": [
"CategoryTheory.Limits.WalkingParal... | [
"case one\nJ : Type w\nC : Type u\ninst✝¹ : Category.{v, u} C\nX Y : C\nf : J → (X ⟶ Y)\ninst✝ : Nonempty J\ns t : Trident f\nk : s.pt ⟶ t.pt\nw : k ≫ t.ι = s.ι\n⊢ k ≫ t.π.app one = s.π.app one"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Shapes.WideEqualizers | {
"line": 496,
"column": 6
} | {
"line": 496,
"column": 17
} | {
"line": 496,
"column": 18
} | [
{
"pp": "case zero\nJ : Type w\nC : Type u\ninst✝¹ : Category.{v, u} C\nX Y : C\nf : J → (X ⟶ Y)\ninst✝ : Nonempty J\ns t : Cotrident f\nk : s.pt ⟶ t.pt\nw : s.π ≫ k = t.π\n⊢ s.ι.app zero ≫ k = t.ι.app zero",
"ppTerm": "?zero",
"assigned": true,
"usedConstants": [
"CategoryTheory.Functor",
... | [
"case zero\nJ : Type w\nC : Type u\ninst✝¹ : Category.{v, u} C\nX Y : C\nf : J → (X ⟶ Y)\ninst✝ : Nonempty J\ns t : Cotrident f\nk : s.pt ⟶ t.pt\nw : s.π ≫ k = t.π\n⊢ s.ι.app zero ≫ k = t.ι.app zero"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Shapes.WideEqualizers | {
"line": 496,
"column": 4
} | {
"line": 496,
"column": 48
} | {
"line": 497,
"column": 4
} | [
{
"pp": "case zero\nJ : Type w\nC : Type u\ninst✝¹ : Category.{v, u} C\nX Y : C\nf : J → (X ⟶ Y)\ninst✝ : Nonempty J\ns t : Cotrident f\nk : s.pt ⟶ t.pt\nw : s.π ≫ k = t.π\n⊢ s.ι.app zero ≫ k = t.ι.app zero",
"ppTerm": "?zero",
"assigned": true,
"usedConstants": [
"CategoryTheory.Limits.Walkin... | [
"case one\nJ : Type w\nC : Type u\ninst✝¹ : Category.{v, u} C\nX Y : C\nf : J → (X ⟶ Y)\ninst✝ : Nonempty J\ns t : Cotrident f\nk : s.pt ⟶ t.pt\nw : s.π ≫ k = t.π\n⊢ s.ι.app one ≫ k = t.ι.app one"
] | · simpa using f (Classical.arbitrary J) ≫= w | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Generator.Basic | {
"line": 663,
"column": 6
} | {
"line": 663,
"column": 21
} | {
"line": 663,
"column": 21
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nG : C\ninst✝ : ∀ (A : C), HasCoproduct fun x ↦ G\n⊢ IsSeparator G ↔ ∀ (A : C), Epi (Sigma.desc fun f ↦ f)",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Epi",
"CategoryTheory.CategoryStruct.toQ... | [
"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nG : C\ninst✝ : ∀ (A : C), HasCoproduct fun x ↦ G\n⊢ (∀ ⦃X Y : C⦄ (f g : X ⟶ Y), (∀ (h : G ⟶ X), h ≫ f = h ≫ g) → f = g) ↔ ∀ (A : C), Epi (Sigma.desc fun f ↦ f)"
] | isSeparator_def | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Generator.Basic | {
"line": 665,
"column": 4
} | {
"line": 665,
"column": 15
} | {
"line": 665,
"column": 16
} | [
{
"pp": "case refine_1\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nG : C\ninst✝ : ∀ (A : C), HasCoproduct fun x ↦ G\nh : ∀ ⦃X Y : C⦄ (f g : X ⟶ Y), (∀ (h : G ⟶ X), h ≫ f = h ≫ g) → f = g\nA Z✝ : C\nu v : A ⟶ Z✝\nhuv : (Sigma.desc fun f ↦ f) ≫ u = (Sigma.desc fun f ↦ f) ≫ v\ni : G ⟶ A\n⊢ i ≫ u = i ≫ v",
"ppT... | [
"case refine_1\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nG : C\ninst✝ : ∀ (A : C), HasCoproduct fun x ↦ G\nh : ∀ ⦃X Y : C⦄ (f g : X ⟶ Y), (∀ (h : G ⟶ X), h ≫ f = h ≫ g) → f = g\nA Z✝ : C\nu v : A ⟶ Z✝\nhuv : (Sigma.desc fun f ↦ f) ≫ u = (Sigma.desc fun f ↦ f) ≫ v\ni : G ⟶ A\n⊢ i ≫ u = i ≫ v"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Generator.Basic | {
"line": 668,
"column": 4
} | {
"line": 668,
"column": 15
} | {
"line": 668,
"column": 16
} | [
{
"pp": "case refine_2\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nG : C\ninst✝ : ∀ (A : C), HasCoproduct fun x ↦ G\nh : ∀ (A : C), Epi (Sigma.desc fun f ↦ f)\nX Y : C\nf g : X ⟶ Y\nhh : ∀ (h : G ⟶ X), h ≫ f = h ≫ g\nthis : Epi (Sigma.desc fun f ↦ f)\nj : Discrete (G ⟶ X)\n⊢ colimit.ι (Discrete.functor fun f ↦ ... | [
"case refine_2\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nG : C\ninst✝ : ∀ (A : C), HasCoproduct fun x ↦ G\nh : ∀ (A : C), Epi (Sigma.desc fun f ↦ f)\nX Y : C\nf g : X ⟶ Y\nhh : ∀ (h : G ⟶ X), h ≫ f = h ≫ g\nthis : Epi (Sigma.desc fun f ↦ f)\nj : Discrete (G ⟶ X)\n⊢ j.as ≫ f = j.as ≫ g"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Generator.Basic | {
"line": 675,
"column": 4
} | {
"line": 675,
"column": 15
} | {
"line": 675,
"column": 16
} | [
{
"pp": "case refine_1\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nG : C\ninst✝ : ∀ (A : C), HasProduct fun x ↦ G\nh : ∀ ⦃X Y : C⦄ (f g : X ⟶ Y), (∀ (h : Y ⟶ G), f ≫ h = g ≫ h) → f = g\nA Z✝ : C\nu v : Z✝ ⟶ A\nhuv : (u ≫ Pi.lift fun f ↦ f) = v ≫ Pi.lift fun f ↦ f\ni : A ⟶ G\n⊢ u ≫ i = v ≫ i",
"ppTerm": "?re... | [
"case refine_1\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nG : C\ninst✝ : ∀ (A : C), HasProduct fun x ↦ G\nh : ∀ ⦃X Y : C⦄ (f g : X ⟶ Y), (∀ (h : Y ⟶ G), f ≫ h = g ≫ h) → f = g\nA Z✝ : C\nu v : Z✝ ⟶ A\nhuv : (u ≫ Pi.lift fun f ↦ f) = v ≫ Pi.lift fun f ↦ f\ni : A ⟶ G\n⊢ u ≫ i = v ≫ i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Generator.Basic | {
"line": 678,
"column": 4
} | {
"line": 678,
"column": 15
} | {
"line": 678,
"column": 16
} | [
{
"pp": "case refine_2\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nG : C\ninst✝ : ∀ (A : C), HasProduct fun x ↦ G\nh : ∀ (A : C), Mono (Pi.lift fun f ↦ f)\nX Y : C\nf g : X ⟶ Y\nhh : ∀ (h : Y ⟶ G), f ≫ h = g ≫ h\nthis : Mono (Pi.lift fun f ↦ f)\nj : Discrete (Y ⟶ G)\n⊢ (f ≫ Pi.lift fun f ↦ f) ≫ limit.π (Discret... | [
"case refine_2\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nG : C\ninst✝ : ∀ (A : C), HasProduct fun x ↦ G\nh : ∀ (A : C), Mono (Pi.lift fun f ↦ f)\nX Y : C\nf g : X ⟶ Y\nhh : ∀ (h : Y ⟶ G), f ≫ h = g ≫ h\nthis : Mono (Pi.lift fun f ↦ f)\nj : Discrete (Y ⟶ G)\n⊢ f ≫ j.as = g ≫ j.as"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Generator.Basic | {
"line": 688,
"column": 6
} | {
"line": 688,
"column": 21
} | {
"line": 688,
"column": 21
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : HasZeroMorphisms C\nβ : Type w\nf : β → C\nhf : (ObjectProperty.ofObj f).IsSeparating\nc : Cofan f\nhc : IsColimit c\n⊢ IsSeparator c.pt",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Categor... | [
"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : HasZeroMorphisms C\nβ : Type w\nf : β → C\nhf : (ObjectProperty.ofObj f).IsSeparating\nc : Cofan f\nhc : IsColimit c\n⊢ ∀ ⦃X Y : C⦄ (f_1 g : X ⟶ Y), (∀ (h : c.pt ⟶ X), h ≫ f_1 = h ≫ g) → f_1 = g"
] | isSeparator_def | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Generator.Basic | {
"line": 691,
"column": 12
} | {
"line": 691,
"column": 23
} | {
"line": 691,
"column": 24
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : HasZeroMorphisms C\nβ : Type w\nf : β → C\nhf : (ObjectProperty.ofObj f).IsSeparating\nc : Cofan f\nhc : IsColimit c\nx✝³ x✝² : C\nx✝¹ x✝ : x✝³ ⟶ x✝²\nhuv : ∀ (h : c.pt ⟶ x✝³), h ≫ x✝¹ = h ≫ x✝\nb : β\ng : f b ⟶ x✝³\n⊢ g ≫ x✝¹ = g ≫ x✝",
"ppTerm": ... | [
"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : HasZeroMorphisms C\nβ : Type w\nf : β → C\nhf : (ObjectProperty.ofObj f).IsSeparating\nc : Cofan f\nhc : IsColimit c\nx✝³ x✝² : C\nx✝¹ x✝ : x✝³ ⟶ x✝²\nhuv : ∀ (h : c.pt ⟶ x✝³), h ≫ x✝¹ = h ≫ x✝\nb : β\ng : f b ⟶ x✝³\n⊢ g ≫ x✝¹ = g ≫ x✝"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Generator.Basic | {
"line": 738,
"column": 12
} | {
"line": 738,
"column": 23
} | {
"line": 738,
"column": 24
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : HasZeroMorphisms C\nβ : Type w\nf : β → C\nhf : (ObjectProperty.ofObj f).IsCoseparating\nc : Fan f\nhc : IsLimit c\nx✝³ x✝² : C\nx✝¹ x✝ : x✝³ ⟶ x✝²\nhuv : ∀ (h : x✝² ⟶ c.pt), x✝¹ ≫ h = x✝ ≫ h\nb : β\ng : x✝² ⟶ f b\n⊢ x✝¹ ≫ g = x✝ ≫ g",
"ppTerm": "?... | [
"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : HasZeroMorphisms C\nβ : Type w\nf : β → C\nhf : (ObjectProperty.ofObj f).IsCoseparating\nc : Fan f\nhc : IsLimit c\nx✝³ x✝² : C\nx✝¹ x✝ : x✝³ ⟶ x✝²\nhuv : ∀ (h : x✝² ⟶ c.pt), x✝¹ ≫ h = x✝ ≫ h\nb : β\ng : x✝² ⟶ f b\n⊢ x✝¹ ≫ g = x✝ ≫ g"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Category.Grp.AB | {
"line": 90,
"column": 72
} | {
"line": 90,
"column": 83
} | {
"line": 90,
"column": 84
} | [
{
"pp": "J : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsFiltered J\nX Y : Discrete J ⥤ AddCommGrpCat\nf : X ⟶ Y\nhf : Epi f\niX : limit X ≅ AddCommGrpCat.of ((i : J) → ↑(X.obj { as := i })) :=\n (Pi.isoLimit X).symm ≪≫\n (limit.isLimit (Discrete.functor fun j ↦ X.obj { as := j })).conePointUniqueUpToIso\n ... | [
"J : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsFiltered J\nX Y : Discrete J ⥤ AddCommGrpCat\nf : X ⟶ Y\nhf : Epi f\niX : limit X ≅ AddCommGrpCat.of ((i : J) → ↑(X.obj { as := i })) :=\n (Pi.isoLimit X).symm ≪≫\n (limit.isLimit (Discrete.functor fun j ↦ X.obj { as := j })).conePointUniqueUpToIso\n (AddCom... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Types.Coproducts | {
"line": 52,
"column": 29
} | {
"line": 52,
"column": 40
} | {
"line": 52,
"column": 41
} | [
{
"pp": "C : Type u\nF : C → Type v\ni j : C\nf : { as := i } ⟶ { as := j }\n⊢ i = j",
"ppTerm": "?m.48",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"C : Type u\nF : C → Type v\ni j : C\nf : { as := i } ⟶ { as := j }\n⊢ i = j"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Types.Coproducts | {
"line": 80,
"column": 47
} | {
"line": 80,
"column": 58
} | {
"line": 80,
"column": 59
} | [
{
"pp": "C : Type u\nF : C → Type v\nx✝² : C\nx✝¹ x✝ : F x✝²\nh : ((sigma F).inj x✝² x✝¹).fst = ((sigma F).inj x✝² x✝).fst ∧ ((sigma F).inj x✝² x✝¹).snd ≍ ((sigma F).inj x✝² x✝).snd\n⊢ x✝¹ = x✝",
"ppTerm": "?m.33",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
... | [
"C : Type u\nF : C → Type v\nx✝² : C\nx✝¹ x✝ : F x✝²\nh : ((sigma F).inj x✝² x✝¹).fst = ((sigma F).inj x✝² x✝).fst ∧ ((sigma F).inj x✝² x✝¹).snd ≍ ((sigma F).inj x✝² x✝).snd\n⊢ x✝¹ = x✝"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Types.Coproducts | {
"line": 129,
"column": 2
} | {
"line": 129,
"column": 13
} | {
"line": 129,
"column": 14
} | [
{
"pp": "C : Type u\nF : C → Type v\nc : CofanTypes F\nhc : Functor.CoconeTypes.IsColimit c\ni : C\ny₁ y₂ : F i\nh : c.inj i y₁ = c.inj i y₂\n⊢ y₁ = y₂",
"ppTerm": "?m.14",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"C : Type u\nF : C → Type v\nc : CofanTypes F\nhc : Functor.CoconeTypes.IsColimit c\ni : C\ny₁ y₂ : F i\nh : c.inj i y₁ = c.inj i y₂\n⊢ y₁ = y₂"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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