mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.CategoryTheory.ObjectProperty.LimitsClosure

Goal state before:
pp: case succ
C : Type u
inst✝⁵ : Category.{v, u} C
P : ObjectProperty C
α : Type t
J : α → Type u'
inst✝⁴ : (a : α) → Category.{v', u'} (J a)
β : Type w'
inst✝³ : LinearOrder β
inst✝² : OrderBot β
inst✝¹ : SuccOrder β
inst✝ : WellFoundedLT β
b : β
hb : ¬IsMax b
hb' : P.strictLimitsClosureIter J b ≤ P.limitsClosure J
⊢ transfiniteIterate (fun Q ↦ Q.strictLimitsClosureStep J) b P ≤ P.limitsClosure J ∧
    ⨆ a, (transfiniteIterate (fun Q ↦ Q.strictLimitsClosureStep J) b P).strictLimitsOfShape (J a) ≤ P.limitsClosure J

Tactic:
iSup_le_iff

Goal state after:
case succ
C : Type u
inst✝⁵ : Category.{v, u} C
P : ObjectProperty C
α : Type t
J : α → Type u'
inst✝⁴ : (a : α) → Category.{v', u'} (J a)
β : Type w'
inst✝³ : LinearOrder β
inst✝² : OrderBot β
inst✝¹ : SuccOrder β
inst✝ : WellFoundedLT β
b : β
hb : ¬IsMax b
hb' : P.strictLimitsClosureIter J b ≤ P.limitsClosure J
⊢ transfiniteIterate (fun Q ↦ Q.strictLimitsClosureStep J) b P ≤ P.limitsClosure J ∧
    ∀ (i : α),
      (transfiniteIterate (fun Q ↦ Q.strictLimitsClosureStep J) b P).strictLimitsOfShape (J i) ≤ P.limitsClosure J
Module: Mathlib.CategoryTheory.ObjectProperty.LimitsClosure

Goal state before:
pp: case refine_1
C : Type u
inst✝³ : Category.{v, u} C
P : ObjectProperty C
α : Type t
J : α → Type u'
inst✝² : (a : α) → Category.{v', u'} (J a)
κ : Cardinal.{w}
inst✝¹ : Fact κ.IsRegular
h : ∀ (a : α), HasCardinalLT (J a) κ
hκ : κ.IsRegular
this : ∀ (a : α), Small.{w, u'} (J a)
a : α
F : J a ⥤ C
inst✝ : HasLimit F
o : J a → Ordinal.{w}
ho : ∀ (j : J a), o j < κ.ord
ho' : ∀ (j : J a), transfiniteIterate (fun Q ↦ Q.strictLimitsClosureStep J) (o j) P (F.obj j)
⊢ HasCardinalLT (J a) κ

Tactic:
exact h a

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.HomologicalFunctor

Goal state before:
pp: C : Type u_1
D : Type u_2
A : Type u_3
inst✝¹⁴ : Category.{v_1, u_1} C
inst✝¹³ : HasShift C ℤ
inst✝¹² : Category.{v_2, u_2} D
inst✝¹¹ : HasZeroObject D
inst✝¹⁰ : HasShift D ℤ
inst✝⁹ : Preadditive D
inst✝⁸ : ∀ (n : ℤ), (shiftFunctor D n).Additive
inst✝⁷ : Pretriangulated D
inst✝⁶ : Category.{v_3, u_3} A
F : C ⥤ A
inst✝⁵ : HasZeroObject C
inst✝⁴ : Preadditive C
inst✝³ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝² : Pretriangulated C
inst✝¹ : Abelian A
inst✝ : F.IsHomological
X₁ X₂ : C
this : HasBinaryBiproduct (F.obj X₁) (F.obj X₂)
Z : A
⊢ ∀ (g : Z ⟶ F.obj (X₁ ⊞ X₂)), g ≫ F.biprodComparison X₁ X₂ = 0 → g = 0

Tactic:
f

Goal state after:
C : Type u_1
D : Type u_2
A : Type u_3
inst✝¹⁴ : Category.{v_1, u_1} C
inst✝¹³ : HasShift C ℤ
inst✝¹² : Category.{v_2, u_2} D
inst✝¹¹ : HasZeroObject D
inst✝¹⁰ : HasShift D ℤ
inst✝⁹ : Preadditive D
inst✝⁸ : ∀ (n : ℤ), (shiftFunctor D n).Additive
inst✝⁷ : Pretriangulated D
inst✝⁶ : Category.{v_3, u_3} A
F : C ⥤ A
inst✝⁵ : HasZeroObject C
inst✝⁴ : Preadditive C
inst✝³ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝² : Pretriangulated C
inst✝¹ : Abelian A
inst✝ : F.IsHomological
X₁ X₂ : C
this : HasBinaryBiproduct (F.obj X₁) (F.obj X₂)
Z : A
f : Z ⟶ F.obj (X₁ ⊞ X₂)
⊢ f ≫ F.biprodComparison X₁ X₂ = 0 → f = 0
Module: Mathlib.CategoryTheory.Localization.CalculusOfFractions

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} D
L : C ⥤ D
W✝ : MorphismProperty C
inst✝ : L.IsLocalization W✝
W : MorphismProperty Cᵒᵖ
h : W.HasRightCalculusOfFractions
X Y : C
φ : W.unop.RightFraction X Y
⊢ ∃ ψ, φ.f ≫ ψ.s = φ.s ≫ ψ.f

Tactic:
obtain ⟨ψ, eq⟩ := h.exists_rightFraction φ.op

Goal state after:
C : Type u_1
D : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} D
L : C ⥤ D
W✝ : MorphismProperty C
inst✝ : L.IsLocalization W✝
W : MorphismProperty Cᵒᵖ
h : W.HasRightCalculusOfFractions
X Y : C
φ : W.unop.RightFraction X Y
ψ : W.RightFraction (Opposite.op Y) (Opposite.op X)
eq : ψ.s ≫ φ.op.f = ψ.f ≫ φ.op.s
⊢ ∃ ψ, φ.f ≫ ψ.s = φ.s ≫ ψ.f
Module: Mathlib.CategoryTheory.MorphismProperty.Retract

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
P : MorphismProperty C
X Y : C
⊢ ∀ (f : X ⟶ Y), P f → P.retracts f

Tactic:
f

Goal state after:
C : Type u
inst✝ : Category.{v, u} C
P : MorphismProperty C
X Y : C
f : X ⟶ Y
⊢ P f → P.retracts f
Module: Mathlib.CategoryTheory.MorphismProperty.Retract

Goal state before:
pp: case mp
C : Type u
inst✝ : Category.{v, u} C
P : MorphismProperty C
h₁ : ∀ {X Y Z W : C} {f : X ⟶ Y} {g : Z ⟶ W} (h : RetractArrow f g), P g → P f
X Y : C
⊢ ∀ (f : X ⟶ Y), P.retracts f → P f

Tactic:
f

Goal state after:
case mp
C : Type u
inst✝ : Category.{v, u} C
P : MorphismProperty C
h₁ : ∀ {X Y Z W : C} {f : X ⟶ Y} {g : Z ⟶ W} (h : RetractArrow f g), P g → P f
X Y : C
f : X ⟶ Y
⊢ P.retracts f → P f
Module: Mathlib.Algebra.Homology.QuasiIso

Goal state before:
pp: case mp
ι : Type u_1
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroMorphisms C
c : ComplexShape ι
K L M : HomologicalComplex C c
φ : K ⟶ L
φ' : L ⟶ M
i : ι
inst✝² : K.HasHomology i
inst✝¹ : L.HasHomology i
inst✝ : M.HasHomology i
hφ : QuasiIsoAt φ i
a✝ : QuasiIsoAt (φ ≫ φ') i
⊢ QuasiIsoAt φ' i

Tactic:
exact quasiIsoAt_of_comp_left φ φ' i

Goal state after:
Module: Mathlib.Algebra.Homology.QuasiIso

Goal state before:
pp: ι : Type u_1
C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : HasZeroMorphisms C
c : ComplexShape ι
K L K' L' : HomologicalComplex C c
φ : K ⟶ L
φ' : K' ⟶ L'
e : Arrow.mk φ ≅ Arrow.mk φ'
inst✝³ : ∀ (i : ι), K.HasHomology i
inst✝² : ∀ (i : ι), L.HasHomology i
inst✝¹ : ∀ (i : ι), K'.HasHomology i
inst✝ : ∀ (i : ι), L'.HasHomology i
hφ : QuasiIso φ
⊢ QuasiIso φ'

Tactic:
by
  simpa only [← quasiIso_iff_of_arrow_mk_iso φ φ' e]

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.Subcategory

Goal state before:
pp: C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : HasZeroObject C
inst✝³ : HasShift C ℤ
inst✝² : Preadditive C
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
P Q : ObjectProperty C
h : P ≤ Q
X Y : C
⊢ ∀ (f : X ⟶ Y), P.trW f → Q.trW f

Tactic:
f

Goal state after:
C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : HasZeroObject C
inst✝³ : HasShift C ℤ
inst✝² : Preadditive C
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
P Q : ObjectProperty C
h : P ≤ Q
X Y : C
f : X ⟶ Y
⊢ P.trW f → Q.trW f
Module: Mathlib.Algebra.Homology.HomotopyCategory.ShiftSequence

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : CategoryWithHomology C
K L : CochainComplex C ℤ
φ : K ⟶ L
n i j : ℤ
h : n + i = j
⊢ QuasiIsoAt ((CategoryTheory.shiftFunctor (HomologicalComplex C (up ℤ)) n).map φ) i ↔ QuasiIsoAt φ j

Tactic:
by
  simp only [quasiIsoAt_iff_isIso_homologyMap]
  exact (NatIso.isIso_map_iff
    ((homologyFunctor C (ComplexShape.up ℤ) 0).shiftIso n i j h) φ)

Goal state after:
Module: Mathlib.Algebra.Homology.Localization

Goal state before:
pp: case mpr
C : Type u_1
inst✝² : Category.{v_1, u_1} C
ι : Type u_2
c : ComplexShape ι
inst✝¹ : Preadditive C
inst✝ : CategoryWithHomology C
K L : HomologicalComplex C c
f : K ⟶ L
K' L' : HomologicalComplex C c
g : K' ⟶ L'
h : quasiIso C c ((quotient C c).map g)
e : Arrow.mk ((quotient C c).map g) ≅ Arrow.mk ((quotient C c).map f)
⊢ quasiIso C c ((quotient C c).map f)

Tactic:
exact ((quasiIso C c).arrow_mk_iso_iff e).1 h

Goal state after:
Module: Mathlib.Algebra.Homology.Localization

Goal state before:
pp: C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
ι : Type u_2
c : ComplexShape ι
inst✝³ : Preadditive C
inst✝² : CategoryWithHomology C
inst✝¹ : (HomologicalComplex.quasiIso C c).HasLocalization
inst✝ : c.QFactorsThroughHomotopy C
K L : HomologicalComplex C c
⊢ ∀ (f₁ f₂ : K ⟶ L), homotopic C c f₁ f₂ → Q.map f₁ = Q.map f₂

Tactic:
f

Goal state after:
C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
ι : Type u_2
c : ComplexShape ι
inst✝³ : Preadditive C
inst✝² : CategoryWithHomology C
inst✝¹ : (HomologicalComplex.quasiIso C c).HasLocalization
inst✝ : c.QFactorsThroughHomotopy C
K L : HomologicalComplex C c
f : K ⟶ L
⊢ ∀ (f₂ : K ⟶ L), homotopic C c f f₂ → Q.map f = Q.map f₂
Module: Mathlib.Algebra.Homology.Localization

Goal state before:
pp: ι : Type u_1
c : ComplexShape ι
hc : ∀ (j : ι), ∃ i, c.Rel i j
C : Type u_2
inst✝³ : Category.{v_1, u_2} C
inst✝² : Preadditive C
inst✝¹ : HasBinaryBiproducts C
E : Type u_3
inst✝ : Category.{v_2, u_3} E
F : HomologicalComplex C c ⥤ E
hF : (HomologicalComplex.homotopyEquivalences C c).IsInvertedBy F
K L : HomologicalComplex C c
⊢ ∀ (f₁ f₂ : K ⟶ L), homotopic C c f₁ f₂ → F.map f₁ = F.map f₂

Tactic:
f

Goal state after:
ι : Type u_1
c : ComplexShape ι
hc : ∀ (j : ι), ∃ i, c.Rel i j
C : Type u_2
inst✝³ : Category.{v_1, u_2} C
inst✝² : Preadditive C
inst✝¹ : HasBinaryBiproducts C
E : Type u_3
inst✝ : Category.{v_2, u_3} E
F : HomologicalComplex C c ⥤ E
hF : (HomologicalComplex.homotopyEquivalences C c).IsInvertedBy F
K L : HomologicalComplex C c
f : K ⟶ L
⊢ ∀ (f₂ : K ⟶ L), homotopic C c f f₂ → F.map f = F.map f₂
Module: Mathlib.Algebra.Homology.Localization

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁸ : Category.{v_1, u_1} C
inst✝⁷ : Category.{v_2, u_2} D
F : C ⥤ D
ι : Type u_3
c : ComplexShape ι
inst✝⁶ : Preadditive C
inst✝⁵ : Preadditive D
inst✝⁴ : CategoryWithHomology C
inst✝³ : CategoryWithHomology D
inst✝² : (HomologicalComplex.quasiIso D c).HasLocalization
inst✝¹ : F.Additive
inst✝ : F.PreservesHomology
⊢ (HomologicalComplex.quasiIso C c).IsInvertedBy (F.mapHomologicalComplex c ⋙ HomologicalComplexUpToQuasiIso.Q)

Tactic:
by
  apply (F.mapHomologicalComplexUpToQuasiIsoLocalizerMorphism c).inverts

Goal state after:
Module: Mathlib.Algebra.Homology.HomotopyCategory.SingleFunctors

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preadditive C
inst✝ : HasZeroObject C
n : ℤ
⊢ (singleFunctor C n).Additive

Tactic:
dsimp only [singleFunctor, singleFunctors, SingleFunctors.postcomp]

Goal state after:
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preadditive C
inst✝ : HasZeroObject C
n : ℤ
⊢ ((CochainComplex.singleFunctors C).functor n ⋙ quotient C (ComplexShape.up ℤ)).Additive
Module: Mathlib.CategoryTheory.Triangulated.Subcategory

Goal state before:
pp: C : Type u_1
inst✝¹⁵ : Category.{v_1, u_1} C
inst✝¹⁴ : HasZeroObject C
inst✝¹³ : HasShift C ℤ
inst✝¹² : Preadditive C
inst✝¹¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹⁰ : Pretriangulated C
D : Type u_2
inst✝⁹ : Category.{v_2, u_2} D
inst✝⁸ : Preadditive D
inst✝⁷ : HasZeroObject D
inst✝⁶ : HasShift D ℤ
inst✝⁵ : ∀ (n : ℤ), (shiftFunctor D n).Additive
inst✝⁴ : Pretriangulated D
E : Type u_3
inst✝³ : Category.{v_3, u_3} E
inst✝² : HasShift E ℤ
P : ObjectProperty C
inst✝¹ : IsTriangulated C
inst✝ : P.IsTriangulated
Y Z✝ Z' : C
f₁ f₂ : Y ⟶ Z✝
s : Z✝ ⟶ Z'
hf₁ : f₁ ≫ s = f₂ ≫ s
Z : C
g : Z ⟶ Z✝
h : Z' ⟶ (shiftFunctor C 1).obj Z
H : Triangle.mk g s h ∈ distinguishedTriangles
mem : P Z
⊢ ∃ X' t, ∃ (_ : P.trW t), t ≫ f₁ = t ≫ f₂

Tactic:
have hf₂ : (f₁ - f₂) ≫ s = 0 := by rw [sub_comp, hf₁, sub_self]

Goal state after:
C : Type u_1
inst✝¹⁵ : Category.{v_1, u_1} C
inst✝¹⁴ : HasZeroObject C
inst✝¹³ : HasShift C ℤ
inst✝¹² : Preadditive C
inst✝¹¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹⁰ : Pretriangulated C
D : Type u_2
inst✝⁹ : Category.{v_2, u_2} D
inst✝⁸ : Preadditive D
inst✝⁷ : HasZeroObject D
inst✝⁶ : HasShift D ℤ
inst✝⁵ : ∀ (n : ℤ), (shiftFunctor D n).Additive
inst✝⁴ : Pretriangulated D
E : Type u_3
inst✝³ : Category.{v_3, u_3} E
inst✝² : HasShift E ℤ
P : ObjectProperty C
inst✝¹ : IsTriangulated C
inst✝ : P.IsTriangulated
Y Z✝ Z' : C
f₁ f₂ : Y ⟶ Z✝
s : Z✝ ⟶ Z'
hf₁ : f₁ ≫ s = f₂ ≫ s
Z : C
g : Z ⟶ Z✝
h : Z' ⟶ (shiftFunctor C 1).obj Z
H : Triangle.mk g s h ∈ distinguishedTriangles
mem : P Z
hf₂ : (f₁ - f₂) ≫ s = 0
⊢ ∃ X' t, ∃ (_ : P.trW t), t ≫ f₁ = t ≫ f₂
Module: Mathlib.Algebra.Homology.DerivedCategory.Basic

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasDerivedCategory C
⊢ Q.IsLocalization (HomologicalComplex.quasiIso C (ComplexShape.up ℤ))

Tactic:
dsimp only [Q, DerivedCategory]
  infer_instance

Goal state after:
Module: Mathlib.Algebra.Homology.DerivedCategory.Basic

Goal state before:
pp: case mp
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasDerivedCategory C
T : Triangle (DerivedCategory C)
⊢ T ∈ distinguishedTriangles → ∃ X Y f, Nonempty (T ≅ Q.mapTriangle.obj (CochainComplex.mappingCone.triangle f))

Tactic:
rintro ⟨T', e, ⟨X, Y, f, ⟨e'⟩⟩⟩

Goal state after:
case mp
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasDerivedCategory C
T : Triangle (DerivedCategory C)
T' : Triangle (HomotopyCategory C (ComplexShape.up ℤ))
e : T ≅ Qh.mapTriangle.obj T'
X Y : CochainComplex C ℤ
f : X ⟶ Y
e' : T' ≅ CochainComplex.mappingCone.triangleh f
⊢ ∃ X Y f, Nonempty (T ≅ Q.mapTriangle.obj (CochainComplex.mappingCone.triangle f))
Module: Mathlib.CategoryTheory.Triangulated.Subcategory

Goal state before:
pp: C : Type u_1
inst✝¹⁴ : Category.{v_1, u_1} C
inst✝¹³ : HasZeroObject C
inst✝¹² : HasShift C ℤ
inst✝¹¹ : Preadditive C
inst✝¹⁰ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁹ : Pretriangulated C
D : Type u_2
inst✝⁸ : Category.{v_2, u_2} D
inst✝⁷ : Preadditive D
inst✝⁶ : HasZeroObject D
inst✝⁵ : HasShift D ℤ
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor D n).Additive
inst✝³ : Pretriangulated D
E : Type u_3
inst✝² : Category.{v_3, u_3} E
inst✝¹ : HasShift E ℤ
P : ObjectProperty C
inst✝ : P.IsTriangulated
J : Type
x✝ : Finite J
X₁✝ X₂✝ : J → C
X₁ : HasProduct X₁✝
X₂ : HasProduct X₂✝
⊢ ∀ (f : (j : J) → X₁✝ j ⟶ X₂✝ j), (∀ (j : J), P.isoClosure.trW (f j)) → P.isoClosure.trW (Limits.Pi.map f)

Tactic:
f

Goal state after:
C : Type u_1
inst✝¹⁴ : Category.{v_1, u_1} C
inst✝¹³ : HasZeroObject C
inst✝¹² : HasShift C ℤ
inst✝¹¹ : Preadditive C
inst✝¹⁰ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁹ : Pretriangulated C
D : Type u_2
inst✝⁸ : Category.{v_2, u_2} D
inst✝⁷ : Preadditive D
inst✝⁶ : HasZeroObject D
inst✝⁵ : HasShift D ℤ
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor D n).Additive
inst✝³ : Pretriangulated D
E : Type u_3
inst✝² : Category.{v_3, u_3} E
inst✝¹ : HasShift E ℤ
P : ObjectProperty C
inst✝ : P.IsTriangulated
J : Type
x✝ : Finite J
X₁✝ X₂✝ : J → C
X₁ : HasProduct X₁✝
X₂ : HasProduct X₂✝
f : (j : J) → X₁✝ j ⟶ X₂✝ j
⊢ (∀ (j : J), P.isoClosure.trW (f j)) → P.isoClosure.trW (Limits.Pi.map f)
Module: Mathlib.Algebra.Homology.HomotopyCategory.MappingCocone

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
K L : CochainComplex C ℤ
φ : K ⟶ L
inst✝ : HasHomotopyCofiber φ
p : ℤ
hpq : p + -1 = p + -1
⊢ (fst φ).f p ≫ (inl φ).v p p ⋯ + (snd φ).v p (p + -1) hpq ≫ (↑(inr φ)).v (p + -1) p ⋯ = 𝟙 ((mappingCocone φ).X p)

Tactic:
simp [fst, inl, snd, inr, mappingCocone,
    Cochain.leftShift_v (n := 1) _ _ _ _ _ p _ (p + -1) (by lia),
    Cochain.rightShift_v _ _ _ _ _ _ _ _ hpq,
    Cochain.leftShift_v _ _ _ _ _ _ _ _ (add_zero (p + -1)),
    Cochain.rightShift_v _ _ _ _ _ _ _ _ (add_zero (p + -1)),
    Int.negOnePow_even 2 ⟨1, rfl⟩,
    mappingCone.id_X φ (p + -1) p (by lia)]

Goal state after:
Module: Mathlib.Algebra.Homology.HomotopyCategory.MappingCocone

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
K L : CochainComplex C ℤ
φ : K ⟶ L
inst✝ : HasHomotopyCofiber φ
M : CochainComplex C ℤ
n m : ℤ
α : Cochain M K n
β : Cochain M L m
h : m + 1 = n
⊢ (liftCochain φ α β h).comp (snd φ) ⋯ = β

Tactic:
ext p q hpq
  simp [Cochain.comp_v (n₁ := n) (n₂ := -1) (n₁₂ := m) _ _ _ p _ _ (by lia)
    (Int.add_neg_cancel_right q 1)]

Goal state after:
Module: Mathlib.Algebra.Homology.Embedding.Basic

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
e : c.Embedding c'
inst✝ : e.IsRelIff
i₁ i₂ : ι
⊢ c'.Rel (e.f i₁) (e.f i₂) ↔ c.Rel i₁ i₂

Tactic:
constructor
  · apply IsRelIff.rel'
  · exact e.rel

Goal state after:
Module: Mathlib.Algebra.Homology.Opposite

Goal state before:
pp: ι : Type u_1
V : Type u_2
inst✝¹ : Category.{v_1, u_2} V
c : ComplexShape ι
inst✝ : HasZeroMorphisms V
X Y : (HomologicalComplex V c)ᵒᵖ
⊢ ∀ (f : X ⟶ Y),
    (𝟭 (HomologicalComplex V c)ᵒᵖ).map f ≫
        (Hom.isoOfComponents (fun x ↦ Iso.refl ((unop Y).op.unopSymm.X x)) ⋯).op.hom =
      (Hom.isoOfComponents (fun x ↦ Iso.refl ((unop X).op.unopSymm.X x)) ⋯).op.hom ≫
        (opFunctor V c ⋙ opInverse V c).map f

Tactic:
f

Goal state after:
ι : Type u_1
V : Type u_2
inst✝¹ : Category.{v_1, u_2} V
c : ComplexShape ι
inst✝ : HasZeroMorphisms V
X Y : (HomologicalComplex V c)ᵒᵖ
f : X ⟶ Y
⊢ (𝟭 (HomologicalComplex V c)ᵒᵖ).map f ≫ (Hom.isoOfComponents (fun x ↦ Iso.refl ((unop Y).op.unopSymm.X x)) ⋯).op.hom =
    (Hom.isoOfComponents (fun x ↦ Iso.refl ((unop X).op.unopSymm.X x)) ⋯).op.hom ≫ (opFunctor V c ⋙ opInverse V c).map f
Module: Mathlib.Algebra.Homology.Opposite

Goal state before:
pp: ι : Type u_1
V : Type u_2
inst✝¹ : Category.{v_1, u_2} V
c : ComplexShape ι
inst✝ : HasZeroMorphisms V
X Y : (HomologicalComplex Vᵒᵖ c)ᵒᵖ
⊢ ∀ (f : X ⟶ Y),
    (𝟭 (HomologicalComplex Vᵒᵖ c)ᵒᵖ).map f ≫
        (Hom.isoOfComponents (fun x ↦ Iso.refl ((unop Y).op.unopSymm.X x)) ⋯).op.hom =
      (Hom.isoOfComponents (fun x ↦ Iso.refl ((unop X).op.unopSymm.X x)) ⋯).op.hom ≫
        (unopFunctor V c ⋙ unopInverse V c).map f

Tactic:
f

Goal state after:
ι : Type u_1
V : Type u_2
inst✝¹ : Category.{v_1, u_2} V
c : ComplexShape ι
inst✝ : HasZeroMorphisms V
X Y : (HomologicalComplex Vᵒᵖ c)ᵒᵖ
f : X ⟶ Y
⊢ (𝟭 (HomologicalComplex Vᵒᵖ c)ᵒᵖ).map f ≫
      (Hom.isoOfComponents (fun x ↦ Iso.refl ((unop Y).op.unopSymm.X x)) ⋯).op.hom =
    (Hom.isoOfComponents (fun x ↦ Iso.refl ((unop X).op.unopSymm.X x)) ⋯).op.hom ≫
      (unopFunctor V c ⋙ unopInverse V c).map f
Module: Mathlib.Algebra.Homology.Embedding.Extend

Goal state before:
pp: case inl
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝² : Category.{v_1, u_3} C
inst✝¹ : HasZeroObject C
inst✝ : HasZeroMorphisms C
K L M : HomologicalComplex C c
φ : K ⟶ L
φ' : L ⟶ M
e : c.Embedding c'
i' j' : ι'
h : ¬c'.Rel i' j'
hi' : e.r i' = none
⊢ extend.d K (e.r i') (e.r j') = 0

Tactic:
extend.d_none_eq_zero K _ _ hi'

Goal state after:
case inl
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝² : Category.{v_1, u_3} C
inst✝¹ : HasZeroObject C
inst✝ : HasZeroMorphisms C
K L M : HomologicalComplex C c
φ : K ⟶ L
φ' : L ⟶ M
e : c.Embedding c'
i' j' : ι'
h : ¬c'.Rel i' j'
hi' : e.r i' = none
⊢ 0 = 0
Module: Mathlib.Algebra.Homology.Embedding.Extend

Goal state before:
pp: case pos
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝⁴ : Category.{v_1, u_3} C
inst✝³ : HasZeroObject C
inst✝² : HasZeroMorphisms C
inst✝¹ : DecidableEq ι
e : c.Embedding c'
X : C
inst✝ : DecidableEq ι'
i : ι
i' : ι'
h : e.f i = i'
j' : ι'
j : ι
hj : e.f j = j'
hij : j = i
⊢ (mkHomToSingle ((((single C c i).obj X).extendXIso e h).hom ≫ (singleObjXSelf c i X).hom) ⋯ ≫
          mkHomFromSingle ((singleObjXSelf c i X).inv ≫ (((single C c i).obj X).extendXIso e h).inv) ⋯).f
      j' =
    (𝟙 (((single C c i).obj X).extend e)).f j'

Tactic:
obtain rfl : i' = j' := by rw [← hj, hij, h]

Goal state after:
case pos
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝⁴ : Category.{v_1, u_3} C
inst✝³ : HasZeroObject C
inst✝² : HasZeroMorphisms C
inst✝¹ : DecidableEq ι
e : c.Embedding c'
X : C
inst✝ : DecidableEq ι'
i : ι
i' : ι'
h : e.f i = i'
j : ι
hij : j = i
hj : e.f j = i'
⊢ (mkHomToSingle ((((single C c i).obj X).extendXIso e h).hom ≫ (singleObjXSelf c i X).hom) ⋯ ≫
          mkHomFromSingle ((singleObjXSelf c i X).inv ≫ (((single C c i).obj X).extendXIso e h).inv) ⋯).f
      i' =
    (𝟙 (((single C c i).obj X).extend e)).f i'
Module: Mathlib.Algebra.Homology.Embedding.TruncGEHomology

Goal state before:
pp: case mp
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝⁴ : Category.{v_1, u_3} C
inst✝³ : HasZeroMorphisms C
K : HomologicalComplex C c'
e : c.Embedding c'
inst✝² : e.IsTruncGE
inst✝¹ : ∀ (i' : ι'), K.HasHomology i'
inst✝ : HasZeroObject C
⊢ QuasiIso (K.πTruncGE e) → K.IsSupported e

pp: case mpr
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝⁴ : Category.{v_1, u_3} C
inst✝³ : HasZeroMorphisms C
K : HomologicalComplex C c'
e : c.Embedding c'
inst✝² : e.IsTruncGE
inst✝¹ : ∀ (i' : ι'), K.HasHomology i'
inst✝ : HasZeroObject C
⊢ K.IsSupported e → QuasiIso (K.πTruncGE e)

Tactic:
· intro
    refine ⟨fun i' hi' => ?_⟩
    rw [exactAt_iff_of_quasiIsoAt (K.πTruncGE e) i']
    exact (K.truncGE e).exactAt_of_isSupported e i' hi'

Goal state after:
case mpr
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝⁴ : Category.{v_1, u_3} C
inst✝³ : HasZeroMorphisms C
K : HomologicalComplex C c'
e : c.Embedding c'
inst✝² : e.IsTruncGE
inst✝¹ : ∀ (i' : ι'), K.HasHomology i'
inst✝ : HasZeroObject C
⊢ K.IsSupported e → QuasiIso (K.πTruncGE e)
Module: Mathlib.Algebra.Homology.Embedding.AreComplementary

Goal state before:
pp: ι : Type u_1
ι₁ : Type u_2
ι₂ : Type u_3
c : ComplexShape ι
c₁ : ComplexShape ι₁
c₂ : ComplexShape ι₂
C : Type u_4
inst✝¹ : Category.{v_1, u_4} C
inst✝ : HasZeroMorphisms C
e₁ : c₁.Embedding c
e₂ : c₂.Embedding c
ac : e₁.AreComplementary e₂
X : ι → Type u_5
x₁ : (i₁ : ι₁) → X (e₁.f i₁)
x₂ : (i₂ : ι₂) → X (e₂.f i₂)
i j : ι
hij : i = j
⊢ X i ≃ X j

Tactic:
subst hij
  rfl

Goal state after:
Module: Mathlib.Algebra.Homology.Embedding.TruncLEHomology

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝⁵ : Category.{v_1, u_3} C
inst✝⁴ : HasZeroMorphisms C
K L : HomologicalComplex C c'
φ : K ⟶ L
e : c.Embedding c'
inst✝³ : e.IsTruncLE
inst✝² : ∀ (i' : ι'), K.HasHomology i'
inst✝¹ : ∀ (i' : ι'), L.HasHomology i'
inst✝ : HasZeroObject C
⊢ QuasiIso (truncLEMap φ e) ↔ ∀ (i : ι) (i' : ι'), e.f i = i' → QuasiIsoAt φ i'

Tactic:
rw [← quasiIso_opFunctor_map_iff]
  simp only [← quasiIsoAt_opFunctor_map_iff φ]
  apply quasiIso_truncGEMap_iff

Goal state after:
Module: Mathlib.Algebra.Homology.Embedding.CochainComplex

Goal state before:
pp: C✝ : Type u_1
inst✝⁵ : Category.{v_1, u_1} C✝
C : Type u_2
inst✝⁴ : Category.{?u.7, u_2} C
inst✝³ : HasZeroMorphisms C
inst✝² : HasZeroObject C
K L : CochainComplex C ℤ
φ : K ⟶ L
e : K ≅ L
inst✝¹ : ∀ (i : ℤ), HasHomology K i
inst✝ : ∀ (i : ℤ), HasHomology L i
n✝ n i : ℤ
hi : i < n
⊢ (embeddingUpIntLE n).f (n - i).natAbs = i

Tactic:
dsimp
      rw [Int.natAbs_of_nonneg (by lia), sub_sub_cancel]

Goal state after:
Module: Mathlib.Algebra.Homology.HomologySequenceLemmas

Goal state before:
pp: C : Type u_1
ι : Type u_2
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
c : ComplexShape ι
S₁ S₂ : ShortComplex (HomologicalComplex C c)
φ : S₁ ⟶ S₂
hS₁ : S₁.ShortExact
hS₂ : S₂.ShortExact
i : ι
h₁ : Epi (homologyMap φ.τ₁ i)
h₂ : IsIso (homologyMap φ.τ₂ i)
h₃ : ∀ (j : ι), c.Rel i j → IsIso (homologyMap φ.τ₁ j)
h₄ : ∀ (j : ι), c.Rel i j → Mono (homologyMap φ.τ₂ j)
⊢ IsIso (homologyMap φ.τ₃ i)

Tactic:
have := mono_homologyMap_τ₃ φ hS₁ hS₂ i h₁ (IsIso.mono_of_iso _) (fun j hij => by
    have := h₃ j hij
    infer_instance)
  have := epi_homologyMap_τ₃ φ hS₁ hS₂ i inferInstance (fun j hij => by
    have := h₃ j hij
    infer_instance) h₄
  apply isIso_of_mono_of_epi

Goal state after:
Module: Mathlib.Algebra.Homology.DerivedCategory.Ext.Basic

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasExt C
n : ℕ
X₁ X₂ : Cᵒᵖ
f : X₁ ⟶ X₂
Y₁ Y₂ : C
g : Y₁ ⟶ Y₂
⊢ (extFunctorObj (Opposite.unop X₁) n).map g ≫
      AddCommGrpCat.ofHom (AddMonoidHom.mk' (fun α ↦ (Ext.mk₀ f.unop).comp α ⋯) ⋯) =
    AddCommGrpCat.ofHom (AddMonoidHom.mk' (fun α ↦ (Ext.mk₀ f.unop).comp α ⋯) ⋯) ≫
      (extFunctorObj (Opposite.unop X₂) n).map g

Tactic:
by
        ext α
        dsimp
        symm
        apply Ext.comp_assoc
        all_goals lia

Goal state after: