mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Algebra.Homology.Embedding.ExtendHomotopy

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✝³ : Category.{v_1, u_3} C
inst✝² : HasZeroObject C
inst✝¹ : Preadditive C
K L : HomologicalComplex C c
f g : K ⟶ L
h : Homotopy f g
e : c.Embedding c'
inst✝ : e.IsRelIff
i j : ι
hij' : ¬c'.Rel (e.f j) (e.f i)
⊢ extend.hom e h.hom (e.f i) (e.f j) = 0

Tactic:
rw [extend.hom_eq _ _ rfl rfl, h.zero _ _ (by rwa [← e.rel_iff]),
          zero_comp, comp_zero]

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

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
e : c.Embedding c'
inst✝³ : e.IsRelIff
C : Type u_3
inst✝² : Category.{v_1, u_3} C
inst✝¹ : HasZeroObject C
inst✝ : Preadditive C
K L : HomologicalComplex C c
f₁ f₂ : K ⟶ L
h : Homotopy f₁ f₂
⊢ (e.extendFunctor C ⋙ HomotopyCategory.quotient C c').map f₁ =
    (e.extendFunctor C ⋙ HomotopyCategory.quotient C c').map f₂

Tactic:
exact HomotopyCategory.eq_of_homotopy _ _ (h.extend e)

Goal state after:
Module: Mathlib.AlgebraicTopology.ModelCategory.Instances

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : CategoryWithWeakEquivalences C
inst✝² : CategoryWithCofibrations C
inst✝¹ : CategoryWithFibrations C
inst✝ : (weakEquivalences C).ContainsIdentities
X : C
⊢ WeakEquivalence (𝟙 X)

Tactic:
by
  rw [weakEquivalence_iff]
  apply id_mem

Goal state after:
Module: Mathlib.AlgebraicTopology.ModelCategory.Basic

Goal state before:
pp: C : Type u
inst✝⁸ : Category.{v, u} C
inst✝⁷ : CategoryWithFibrations C
inst✝⁶ : CategoryWithCofibrations C
inst✝⁵ : CategoryWithWeakEquivalences C
inst✝⁴ : HasFiniteLimits C
inst✝³ : HasFiniteColimits C
inst✝² : (weakEquivalences C).HasTwoOutOfThreeProperty
inst✝¹ : (cofibrations C).IsWeakFactorizationSystem (trivialFibrations C)
inst✝ : (trivialCofibrations C).IsWeakFactorizationSystem (fibrations C)
A B X Y : C
f : A ⟶ B
w : X ⟶ Y
h : RetractArrow f w
hw : WeakEquivalence w
hf : (trivialCofibrations C).MapFactorizationData (fibrations C) f
this✝³ : Cofibration hf.i
this✝² : WeakEquivalence hf.i
φ : pushout hf.i (Arrow.Hom.left h.i) ⟶ Y := pushout.desc (hf.p ≫ Arrow.Hom.right h.i) w ⋯
this✝¹ : Fibration hf.p
this✝ : WeakEquivalence (pushout.inr hf.i (Arrow.Hom.left h.i) ≫ φ)
this : WeakEquivalence φ
hp : RetractArrow hf.p φ
⊢ weakEquivalences C f

Tactic:
have := mk'.cm3a_aux hp

Goal state after:
C : Type u
inst✝⁸ : Category.{v, u} C
inst✝⁷ : CategoryWithFibrations C
inst✝⁶ : CategoryWithCofibrations C
inst✝⁵ : CategoryWithWeakEquivalences C
inst✝⁴ : HasFiniteLimits C
inst✝³ : HasFiniteColimits C
inst✝² : (weakEquivalences C).HasTwoOutOfThreeProperty
inst✝¹ : (cofibrations C).IsWeakFactorizationSystem (trivialFibrations C)
inst✝ : (trivialCofibrations C).IsWeakFactorizationSystem (fibrations C)
A B X Y : C
f : A ⟶ B
w : X ⟶ Y
h : RetractArrow f w
hw : WeakEquivalence w
hf : (trivialCofibrations C).MapFactorizationData (fibrations C) f
this✝⁴ : Cofibration hf.i
this✝³ : WeakEquivalence hf.i
φ : pushout hf.i (Arrow.Hom.left h.i) ⟶ Y := pushout.desc (hf.p ≫ Arrow.Hom.right h.i) w ⋯
this✝² : Fibration hf.p
this✝¹ : WeakEquivalence (pushout.inr hf.i (Arrow.Hom.left h.i) ≫ φ)
this✝ : WeakEquivalence φ
hp : RetractArrow hf.p φ
this : WeakEquivalence hf.p
⊢ weakEquivalences C f
Module: Mathlib.AlgebraicTopology.ModelCategory.LeftHomotopy

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
X Y : C
inst✝¹ : ModelCategory C
f g : X ⟶ Y
inst✝ : IsFibrant Y
h✝¹ : LeftHomotopyRel f g
Z : C
i : Z ⟶ X
P : Cylinder X
left✝ : P.IsVeryGood
h : P.LeftHomotopy f g
Q : Cylinder Z
h✝ : Q.IsVeryGood
sq : CommSq (coprod.desc (i ≫ P.i₀) (i ≫ P.i₁)) Q.i P.π (Q.π ≫ i)
⊢ LeftHomotopyRel (i ≫ f) (i ≫ g)

Tactic:
exact ⟨Q,
   ⟨{ h := sq.lift ≫ h.h
      h₀ := by
        have := coprod.inl ≫= sq.fac_left
        simp only [Q.inl_i_assoc, coprod.inl_desc] at this
        simp [reassoc_of% this]
      h₁ := by
        have := coprod.inr ≫= sq.fac_left
        simp only [Q.inr_i_assoc, coprod.inr_desc] at this
        simp [reassoc_of% this] }⟩⟩

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.TStructure.TruncLEGT

Goal state before:
pp: C : Type u_1
inst✝⁷ : Category.{v_1, u_1} C
inst✝⁶ : Preadditive C
inst✝⁵ : HasZeroObject C
inst✝⁴ : HasShift C ℤ
inst✝³ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝² : Pretriangulated C
t : TStructure C
inst✝¹ : IsTriangulated C
X : C
a b : ℤ
inst✝ : t.IsGE X a
⊢ t.IsGE ((t.truncLE b).obj X) a

Tactic:
by
  dsimp [truncLE]; infer_instance

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.TStructure.Induced

Goal state before:
pp: C : Type u_1
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : Preadditive C
inst✝⁴ : HasZeroObject C
inst✝³ : HasShift C ℤ
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
P : ObjectProperty C
t : TStructure C
inst✝ : P.IsTriangulated
h : P.HasInducedTStructure t
X Y : P.FullSubcategory
f : X ⟶ Y
hX : t.le 0 X.obj
hY : t.ge 1 Y.obj
⊢ P.ι.map f = 0

Tactic:
exact t.zero' (P.ι.map f) hX hY

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

Goal state before:
pp: C : Type u_1
J : Type u_2
ι : Type u_3
inst✝⁵ : Category.{u_4, u_1} C
inst✝⁴ : Category.{u_5, u_2} J
c : ComplexShape ι
inst✝³ : IsCofiltered J
inst✝² : HasZeroMorphisms C
F : J ⥤ HomologicalComplex C c
inst✝¹ : ∀ (j : ι), HasLimit (F ⋙ eval C c j)
cF : Cone F
hcF : IsLimit cF
inst✝ : CategoryWithHomology C
q₀ q₁ q₂ : ι
h₀ : c.prev q₁ = q₀
h₂ : c.next q₁ = q₂
j : J
hq₀ : (F ⋙ eval C c q₀).IsEventuallyConstantTo j
hq₁ : (F ⋙ eval C c q₁).IsEventuallyConstantTo j
hq₂ : (F ⋙ eval C c q₂).IsEventuallyConstantTo j
⊢ QuasiIsoAt (cF.π.app j) q₁

Tactic:
by
  rw [quasiIsoAt_iff' _ q₀ q₁ q₂ h₀ h₂]
  let φ := (shortComplexFunctor' C c q₀ q₁ q₂).map (cF.π.app j)
  have : IsIso φ.τ₁ := isIso_π_f_of_isLimit_of_isEventuallyConstantTo F hcF _ _ hq₀
  have : IsIso φ.τ₂ := isIso_π_f_of_isLimit_of_isEventuallyConstantTo F hcF _ _ hq₁
  have : IsIso φ.τ₃ := isIso_π_f_of_isLimit_of_isEventuallyConstantTo F hcF _ _ hq₂
  apply ShortComplex.quasiIso_of_epi_of_isIso_of_mono

Goal state after:
Module: Mathlib.CategoryTheory.Localization.Resolution

Goal state before:
pp: C₁ : Type u_1
C₂ : Type u_2
D₂ : Type u_4
inst✝⁴ : Category.{v_1, u_1} C₁
inst✝³ : Category.{v_2, u_2} C₂
inst✝² : Category.{v_4, u_4} D₂
W₁ : MorphismProperty C₁
W₂ : MorphismProperty C₂
Φ : LocalizerMorphism W₁ W₂
L₂ : C₂ ⥤ D₂
inst✝¹ : L₂.IsLocalization W₂
inst✝ : Φ.HasLeftResolutions
X₂ : D₂
this : L₂.EssSurj
⊢ (Φ.functor ⋙ L₂).essImage X₂

Tactic:
have L : Φ.LeftResolution (L₂.objPreimage X₂) := Classical.arbitrary _

Goal state after:
C₁ : Type u_1
C₂ : Type u_2
D₂ : Type u_4
inst✝⁴ : Category.{v_1, u_1} C₁
inst✝³ : Category.{v_2, u_2} C₂
inst✝² : Category.{v_4, u_4} D₂
W₁ : MorphismProperty C₁
W₂ : MorphismProperty C₂
Φ : LocalizerMorphism W₁ W₂
L₂ : C₂ ⥤ D₂
inst✝¹ : L₂.IsLocalization W₂
inst✝ : Φ.HasLeftResolutions
X₂ : D₂
this : L₂.EssSurj
L : Φ.LeftResolution (L₂.objPreimage X₂)
⊢ (Φ.functor ⋙ L₂).essImage X₂
Module: Mathlib.CategoryTheory.Localization.DerivabilityStructure.Constructor

Goal state before:
pp: C₁ : Type u_1
C₂ : Type u_2
inst✝⁶ : Category.{v_1, u_1} C₁
inst✝⁵ : Category.{v_2, u_2} C₂
W₁ : MorphismProperty C₁
W₂ : MorphismProperty C₂
Φ : LocalizerMorphism W₁ W₂
inst✝⁴ : ∀ (X₂ : C₂), IsConnected (Φ.RightResolution X₂)
inst✝³ : Φ.arrow.HasRightResolutions
inst✝² : W₂.ContainsIdentities
D : Type u_3
inst✝¹ : Category.{v_3, u_3} D
L : C₂ ⥤ D
inst✝ : L.IsLocalization W₂
X₂ : C₂
X₃ : D
y : L.obj X₂ ⟶ X₃
w : TwoSquare Φ.functor (Φ.functor ⋙ L) L (𝟭 D) :=
  TwoSquare.mk Φ.functor (Φ.functor ⋙ L) L (𝟭 D) (Φ.functor ⋙ L).rightUnitor.inv
this : Nonempty (w.CostructuredArrowDownwards y)
⊢ IsConnected
    ((TwoSquare.mk Φ.functor (Φ.functor ⋙ L) L (𝟭 D) (Φ.functor ⋙ L).rightUnitor.inv).CostructuredArrowDownwards y)

Tactic:
suffices ∀ (X : w.CostructuredArrowDownwards y),
      ∃ Y, Zigzag X ((fromRightResolution Φ L y).obj Y) by
    refine zigzag_isConnected (fun X X' => ?_)
    obtain ⟨Y, hX⟩ := this X
    obtain ⟨Y', hX'⟩ := this X'
    exact hX.trans ((zigzag_obj_of_zigzag _ (isPreconnected_zigzag Y Y')).trans hX'.symm)

Goal state after:
C₁ : Type u_1
C₂ : Type u_2
inst✝⁶ : Category.{v_1, u_1} C₁
inst✝⁵ : Category.{v_2, u_2} C₂
W₁ : MorphismProperty C₁
W₂ : MorphismProperty C₂
Φ : LocalizerMorphism W₁ W₂
inst✝⁴ : ∀ (X₂ : C₂), IsConnected (Φ.RightResolution X₂)
inst✝³ : Φ.arrow.HasRightResolutions
inst✝² : W₂.ContainsIdentities
D : Type u_3
inst✝¹ : Category.{v_3, u_3} D
L : C₂ ⥤ D
inst✝ : L.IsLocalization W₂
X₂ : C₂
X₃ : D
y : L.obj X₂ ⟶ X₃
w : TwoSquare Φ.functor (Φ.functor ⋙ L) L (𝟭 D) :=
  TwoSquare.mk Φ.functor (Φ.functor ⋙ L) L (𝟭 D) (Φ.functor ⋙ L).rightUnitor.inv
this : Nonempty (w.CostructuredArrowDownwards y)
⊢ ∀ (X : w.CostructuredArrowDownwards y), ∃ Y, Zigzag X ((fromRightResolution Φ L y).obj Y)
Module: Mathlib.CategoryTheory.Localization.DerivabilityStructure.Constructor

Goal state before:
pp: case refine_2
C₁ : Type u_1
C₂ : Type u_2
inst✝⁶ : Category.{v_1, u_1} C₁
inst✝⁵ : Category.{v_2, u_2} C₂
W₁ : MorphismProperty C₁
W₂ : MorphismProperty C₂
Φ : LocalizerMorphism W₁ W₂
inst✝⁴ : ∀ (X₂ : C₂), IsConnected (Φ.RightResolution X₂)
inst✝³ : Φ.arrow.HasRightResolutions
inst✝² : W₂.ContainsIdentities
D : Type u_3
inst✝¹ : Category.{v_3, u_3} D
L : C₂ ⥤ D
inst✝ : L.IsLocalization W₂
X₂ : C₂
X₃ : D
y : L.obj X₂ ⟶ X₃
w : TwoSquare Φ.functor (Φ.functor ⋙ L) L (𝟭 D) := ⋯
this✝ : Nonempty (w.CostructuredArrowDownwards y)
c : C₁
g : X₂ ⟶ Φ.functor.obj c
x : L.obj (Φ.functor.obj c) ⟶ X₃
fac✝ : L.map g ≫ 𝟙 (L.obj (Φ.functor.obj c)) ≫ x = y
fac : L.map g ≫ x = y
ρ : Φ.arrow.RightResolution (Arrow.mk g) := ⋯
this :
  Zigzag
    ((fromRightResolution Φ L x ⋙ w.costructuredArrowDownwardsPrecomp x y g fac).obj
      { X₁ := c, w := 𝟙 (Φ.functor.obj c), hw := ⋯ })
    ((fromRightResolution Φ L x ⋙ w.costructuredArrowDownwardsPrecomp x y g fac).obj
      { X₁ := ρ.X₁.right, w := Arrow.Hom.right ρ.w, hw := ⋯ })
⊢ StructuredArrow.Hom.right
      (((TwoSquare.mk Φ.functor (Φ.functor ⋙ L) L (𝟭 D) (Φ.functor ⋙ L).rightUnitor.inv).structuredArrowDownwards
              X₂).map
          (StructuredArrow.homMk ρ.X₁.hom ⋯) ≫
        CostructuredArrow.hom
          ((fromRightResolution Φ L x ⋙ w.costructuredArrowDownwardsPrecomp x y g fac).obj
            { X₁ := ρ.X₁.right, w := Arrow.Hom.right ρ.w, hw := ⋯ })) =
    StructuredArrow.Hom.right
      (CostructuredArrow.hom ((fromRightResolution Φ L y).obj { X₁ := ρ.X₁.left, w := Arrow.Hom.left ρ.w, hw := ⋯ }))

Tactic:
simp [← cancel_epi (isoOfHom L W₂ ρ.w.left ρ.hw.1).hom, ← L.map_comp_assoc, fac]

Goal state after:
Module: Mathlib.CategoryTheory.Localization.StructuredArrow

Goal state before:
pp: case cons.inl
C : Type u_1
inst✝ : Category.{v_1, u_1} C
W : MorphismProperty C
X : C
P : StructuredArrow (W.Q.obj X) W.Q → Prop
hP₀ : P (StructuredArrow.mk (𝟙 (W.Q.obj X)))
hP₁ :
  ∀ ⦃Y₁ Y₂ : C⦄ (f : Y₁ ⟶ Y₂) (φ : W.Q.obj X ⟶ W.Q.obj Y₁),
    P (StructuredArrow.mk φ) → P (StructuredArrow.mk (φ ≫ W.Q.map f))
hP₂ :
  ∀ ⦃Y₁ Y₂ : C⦄ (w : Y₁ ⟶ Y₂) (hw : W w) (φ : W.Q.obj X ⟶ W.Q.obj Y₂),
    P (StructuredArrow.mk φ) → P (StructuredArrow.mk (φ ≫ (isoOfHom W.Q W w hw).inv))
g✝ : StructuredArrow (W.Q.obj X) W.Q
X₀ : Paths (LocQuiver W) := { obj := X }
Y₀ b✝ c✝ : Paths (LocQuiver W)
f : Quiver.Path X₀ b✝
hf : P (StructuredArrow.mk ((Quotient.functor (relations W)).map f))
g : b✝.obj ⟶ c✝.obj
⊢ P (StructuredArrow.mk ((Quotient.functor (relations W)).map (f.cons (Sum.inl g))))

pp: case cons.inr
C : Type u_1
inst✝ : Category.{v_1, u_1} C
W : MorphismProperty C
X : C
P : StructuredArrow (W.Q.obj X) W.Q → Prop
hP₀ : P (StructuredArrow.mk (𝟙 (W.Q.obj X)))
hP₁ :
  ∀ ⦃Y₁ Y₂ : C⦄ (f : Y₁ ⟶ Y₂) (φ : W.Q.obj X ⟶ W.Q.obj Y₁),
    P (StructuredArrow.mk φ) → P (StructuredArrow.mk (φ ≫ W.Q.map f))
hP₂ :
  ∀ ⦃Y₁ Y₂ : C⦄ (w : Y₁ ⟶ Y₂) (hw : W w) (φ : W.Q.obj X ⟶ W.Q.obj Y₂),
    P (StructuredArrow.mk φ) → P (StructuredArrow.mk (φ ≫ (isoOfHom W.Q W w hw).inv))
g : StructuredArrow (W.Q.obj X) W.Q
X₀ : Paths (LocQuiver W) := { obj := X }
Y₀ b✝ c✝ : Paths (LocQuiver W)
f : Quiver.Path X₀ b✝
hf : P (StructuredArrow.mk ((Quotient.functor (relations W)).map f))
w : c✝.obj ⟶ b✝.obj
hw : W w
⊢ P (StructuredArrow.mk ((Quotient.functor (relations W)).map (f.cons (Sum.inr ⟨w, hw⟩))))

Tactic:
· exact hP₁ g _ hf

Goal state after:
case cons.inr
C : Type u_1
inst✝ : Category.{v_1, u_1} C
W : MorphismProperty C
X : C
P : StructuredArrow (W.Q.obj X) W.Q → Prop
hP₀ : P (StructuredArrow.mk (𝟙 (W.Q.obj X)))
hP₁ :
  ∀ ⦃Y₁ Y₂ : C⦄ (f : Y₁ ⟶ Y₂) (φ : W.Q.obj X ⟶ W.Q.obj Y₁),
    P (StructuredArrow.mk φ) → P (StructuredArrow.mk (φ ≫ W.Q.map f))
hP₂ :
  ∀ ⦃Y₁ Y₂ : C⦄ (w : Y₁ ⟶ Y₂) (hw : W w) (φ : W.Q.obj X ⟶ W.Q.obj Y₂),
    P (StructuredArrow.mk φ) → P (StructuredArrow.mk (φ ≫ (isoOfHom W.Q W w hw).inv))
g : StructuredArrow (W.Q.obj X) W.Q
X₀ : Paths (LocQuiver W) := { obj := X }
Y₀ b✝ c✝ : Paths (LocQuiver W)
f : Quiver.Path X₀ b✝
hf : P (StructuredArrow.mk ((Quotient.functor (relations W)).map f))
w : c✝.obj ⟶ b✝.obj
hw : W w
⊢ P (StructuredArrow.mk ((Quotient.functor (relations W)).map (f.cons (Sum.inr ⟨w, hw⟩))))
Module: Mathlib.CategoryTheory.Localization.StructuredArrow

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
Y : C
P : CostructuredArrow L (L.obj Y) → Prop
hP₀ : P (CostructuredArrow.mk (𝟙 (L.obj Y)))
hP₁ :
  ∀ ⦃X₁ X₂ : C⦄ (f : X₁ ⟶ X₂) (φ : L.obj X₂ ⟶ L.obj Y),
    P (CostructuredArrow.mk φ) → P (CostructuredArrow.mk (L.map f ≫ φ))
hP₂ :
  ∀ ⦃X₁ X₂ : C⦄ (w : X₁ ⟶ X₂) (hw : W w) (φ : L.obj X₁ ⟶ L.obj Y),
    P (CostructuredArrow.mk φ) → P (CostructuredArrow.mk ((isoOfHom L W w hw).inv ≫ φ))
g : CostructuredArrow L (L.obj Y)
g' : StructuredArrow (op (L.obj Y)) L.op := StructuredArrow.mk g.hom.op
x✝ : StructuredArrow (op (L.obj Y)) L.op
⊢ P (CostructuredArrow.mk x✝.hom.unop)

Tactic:
induction g' using induction_structuredArrow L.op W.op with

Goal state after:
Module: Mathlib.CategoryTheory.Localization.DerivabilityStructure.PointwiseRightDerived

Goal state before:
pp: case mp
C₁ : Type u₁
C₂ : Type u₂
H : Type u₃
inst✝³ : Category.{v₁, u₁} C₁
inst✝² : Category.{v₂, u₂} C₂
inst✝¹ : Category.{v₃, u₃} H
W₁ : MorphismProperty C₁
W₂ : MorphismProperty C₂
Φ : LocalizerMorphism W₁ W₂
F : C₂ ⥤ H
inst✝ : Φ.IsRightDerivabilityStructure
⊢ F.HasPointwiseRightDerivedFunctor W₂ → (Φ.functor ⋙ F).HasPointwiseRightDerivedFunctor W₁

Tactic:
intro hF X₁
    rw [hasPointwiseRightDerivedFunctorAt_iff_of_isRightDerivabilityStructure]
    apply hF

Goal state after:
Module: Mathlib.CategoryTheory.GuitartExact.KanExtension

Goal state before:
pp: C₁ : Type u₁
C₂ : Type u₂
C₃ : Type u₃
C₄ : Type u₄
D : Type u₅
inst✝⁸ : Category.{v₁, u₁} C₁
inst✝⁷ : Category.{v₂, u₂} C₂
inst✝⁶ : Category.{v₃, u₃} C₃
inst✝⁵ : Category.{v₄, u₄} C₄
inst✝⁴ : Category.{v₅, u₅} D
T : C₁ ⥤ C₂
L : C₁ ⥤ C₃
R : C₂ ⥤ C₄
B : C₃ ⥤ C₄
w : TwoSquare T L R B
inst✝³ : ∀ (F : C₁ ⥤ D), L.HasLeftKanExtension F
inst✝² : ∀ (F : C₂ ⥤ D), R.HasLeftKanExtension F
F : C₂ ⥤ D
inst✝¹ : R.HasPointwiseLeftKanExtension F
inst✝ : w.GuitartExact
⊢ (StructuredArrow.right ((LeftExtension.mk (R.lan.obj F) (R.lanUnit.app F)).compTwoSquare w)).IsLeftKanExtension
    (StructuredArrow.hom ((LeftExtension.mk (R.lan.obj F) (R.lanUnit.app F)).compTwoSquare w))

Tactic:
let hF := isPointwiseLeftKanExtensionOfIsLeftKanExtension (F := F) _ (R.lanUnit.app F)

Goal state after:
C₁ : Type u₁
C₂ : Type u₂
C₃ : Type u₃
C₄ : Type u₄
D : Type u₅
inst✝⁸ : Category.{v₁, u₁} C₁
inst✝⁷ : Category.{v₂, u₂} C₂
inst✝⁶ : Category.{v₃, u₃} C₃
inst✝⁵ : Category.{v₄, u₄} C₄
inst✝⁴ : Category.{v₅, u₅} D
T : C₁ ⥤ C₂
L : C₁ ⥤ C₃
R : C₂ ⥤ C₄
B : C₃ ⥤ C₄
w : TwoSquare T L R B
inst✝³ : ∀ (F : C₁ ⥤ D), L.HasLeftKanExtension F
inst✝² : ∀ (F : C₂ ⥤ D), R.HasLeftKanExtension F
F : C₂ ⥤ D
inst✝¹ : R.HasPointwiseLeftKanExtension F
inst✝ : w.GuitartExact
hF : (LeftExtension.mk (R.lan.obj F) (R.lanUnit.app F)).IsPointwiseLeftKanExtension :=
  isPointwiseLeftKanExtensionOfIsLeftKanExtension (R.lan.obj F) (R.lanUnit.app F)
⊢ (StructuredArrow.right ((LeftExtension.mk (R.lan.obj F) (R.lanUnit.app F)).compTwoSquare w)).IsLeftKanExtension
    (StructuredArrow.hom ((LeftExtension.mk (R.lan.obj F) (R.lanUnit.app F)).compTwoSquare w))
Module: Mathlib.Algebra.Homology.Embedding.StupidTrunc

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✝³ : Category.{v_1, u_3} C
inst✝² : HasZeroMorphisms C
inst✝¹ : HasZeroObject C
K : HomologicalComplex C c'
e : c.Embedding c'
inst✝ : e.IsRelIff
⊢ IsZero (K.stupidTrunc e) → K.IsStrictlySupportedOutside e

Tactic:
exact fun h ↦ ⟨fun i ↦
      ((eval _ _ (e.f i)).map_isZero h).of_iso (K.stupidTruncXIso e rfl).symm⟩

Goal state after:
Module: Mathlib.CategoryTheory.DifferentialObject

Goal state before:
pp: S : Type u_1
inst✝⁶ : AddMonoidWithOne S
C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : HasZeroMorphisms C
inst✝³ : HasShift C S
D : Type u'
inst✝² : Category.{v', u'} D
inst✝¹ : HasZeroMorphisms D
inst✝ : HasShift D S
F : C ⥤ D
η : shiftFunctor C 1 ⋙ F ⟶ F ⋙ shiftFunctor D 1
hF : ∀ (c c' : C), F.map 0 = 0
X✝ Y✝ : DifferentialObject S C
f : X✝ ⟶ Y✝
⊢ F.map X✝.d ≫ (shiftFunctor C 1 ⋙ F).map f.f ≫ η.app Y✝.obj = F.map f.f ≫ F.map Y✝.d ≫ η.app Y✝.obj

Tactic:
slice_lhs 1 2 => rw [Functor.comp_map, ← F.map_comp, f.comm, F.map_comp]

Goal state after:
S : Type u_1
inst✝⁶ : AddMonoidWithOne S
C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : HasZeroMorphisms C
inst✝³ : HasShift C S
D : Type u'
inst✝² : Category.{v', u'} D
inst✝¹ : HasZeroMorphisms D
inst✝ : HasShift D S
F : C ⥤ D
η : shiftFunctor C 1 ⋙ F ⟶ F ⋙ shiftFunctor D 1
hF : ∀ (c c' : C), F.map 0 = 0
X✝ Y✝ : DifferentialObject S C
f : X✝ ⟶ Y✝
⊢ (F.map f.f ≫ F.map Y✝.d) ≫ η.app Y✝.obj = F.map f.f ≫ F.map Y✝.d ≫ η.app Y✝.obj
Module: Mathlib.CategoryTheory.Idempotents.FunctorExtension

Goal state before:
pp: C : Type u_1
D : Type u_2
E : Type u_3
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} D
inst✝ : Category.{v_3, u_3} E
F G : Karoubi C ⥤ D
τ :
  ((whiskeringLeft C (Karoubi C) D).obj (toKaroubi C)).obj F ⟶
    ((whiskeringLeft C (Karoubi C) D).obj (toKaroubi C)).obj G
X : C
⊢ τ.app X ≫ G.map (𝟙 { X := ((toKaroubi C).obj X).X, p := 𝟙 ((toKaroubi C).obj X).X, idem := ⋯ }) = τ.app X

Tactic:
change _ ≫ G.map (𝟙 _) = _

Goal state after:
C : Type u_1
D : Type u_2
E : Type u_3
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} D
inst✝ : Category.{v_3, u_3} E
F G : Karoubi C ⥤ D
τ :
  ((whiskeringLeft C (Karoubi C) D).obj (toKaroubi C)).obj F ⟶
    ((whiskeringLeft C (Karoubi C) D).obj (toKaroubi C)).obj G
X : C
⊢ τ.app X ≫ G.map (𝟙 ((toKaroubi C).obj X)) = τ.app X
Module: Mathlib.CategoryTheory.Abelian.Projective.Resolution

Goal state before:
pp: case zero
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : EnoughProjectives C
Z : C
⊢ { X₁ := ((HomologicalComplex.shortComplexFunctor' C (ComplexShape.down ℕ) 1 0 0).obj (ofComplex Z)).X₁,
        X₂ := ((HomologicalComplex.shortComplexFunctor' C (ComplexShape.down ℕ) 1 0 0).obj (ofComplex Z)).X₂,
        X₃ :=
          ((HomologicalComplex.shortComplexFunctor' C (ComplexShape.down ℕ) 1 0 0).obj
              ((ChainComplex.single₀ C).obj Z)).X₂,
        f := ((HomologicalComplex.shortComplexFunctor' C (ComplexShape.down ℕ) 1 0 0).obj (ofComplex Z)).f,
        g :=
          ((HomologicalComplex.shortComplexFunctor' C (ComplexShape.down ℕ) 1 0 0).map
              (((ofComplex Z).toSingle₀Equiv Z).symm ⟨Projective.π Z, ⋯⟩)).τ₂,
        zero := ⋯ }.Exact ∧
    Epi
      ((HomologicalComplex.shortComplexFunctor' C (ComplexShape.down ℕ) 1 0 0).map
          (((ofComplex Z).toSingle₀Equiv Z).symm ⟨Projective.π Z, ⋯⟩)).τ₂

pp: case zero.hg₁
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : EnoughProjectives C
Z : C
⊢ ((HomologicalComplex.shortComplexFunctor' C (ComplexShape.down ℕ) 1 0 0).obj (ofComplex Z)).g = 0

pp: case zero.hf₂
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : EnoughProjectives C
Z : C
⊢ ((HomologicalComplex.shortComplexFunctor' C (ComplexShape.down ℕ) 1 0 0).obj ((ChainComplex.single₀ C).obj Z)).f = 0

pp: case zero.hg₂
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : EnoughProjectives C
Z : C
⊢ ((HomologicalComplex.shortComplexFunctor' C (ComplexShape.down ℕ) 1 0 0).obj ((ChainComplex.single₀ C).obj Z)).g = 0

Tactic:
· dsimp
        refine (ShortComplex.exact_and_epi_g_iff_of_iso ?_).2
          ⟨exact_d_f (Projective.π Z), by dsimp; infer_instance⟩
        exact ShortComplex.isoMk (Iso.refl _) (Iso.refl _) (Iso.refl _)
          (by simp [ofComplex]) (by simp)

Goal state after:
case zero.hg₁
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : EnoughProjectives C
Z : C
⊢ ((HomologicalComplex.shortComplexFunctor' C (ComplexShape.down ℕ) 1 0 0).obj (ofComplex Z)).g = 0

case zero.hf₂
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : EnoughProjectives C
Z : C
⊢ ((HomologicalComplex.shortComplexFunctor' C (ComplexShape.down ℕ) 1 0 0).obj ((ChainComplex.single₀ C).obj Z)).f = 0

case zero.hg₂
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : EnoughProjectives C
Z : C
⊢ ((HomologicalComplex.shortComplexFunctor' C (ComplexShape.down ℕ) 1 0 0).obj ((ChainComplex.single₀ C).obj Z)).g = 0
Module: Mathlib.Algebra.Homology.Monoidal

Goal state before:
pp: case pos
C : Type u_1
inst✝⁹ : Category.{v_1, u_1} C
inst✝⁸ : MonoidalCategory C
inst✝⁷ : Preadditive C
inst✝⁶ : HasZeroObject C
inst✝⁵ : (curriedTensor C).Additive
inst✝⁴ : ∀ (X₁ : C), ((curriedTensor C).obj X₁).Additive
I : Type u_2
inst✝³ : AddMonoid I
c : ComplexShape I
inst✝² : c.TensorSigns
K : HomologicalComplex C c
inst✝¹ : DecidableEq I
inst✝ : ∀ (X₁ : C), PreservesColimit (Functor.empty C) ((curriedTensor C).obj X₁)
i₁ i₂ j : I
h₁ : c.Rel i₂ (c.next i₂)
h₂ : c.π c c (i₁, c.next i₂) = j
⊢ mapBifunctor.d₂ K (tensorUnit C c) (curriedTensor C) c i₁ i₂ j = 0

Tactic:
mapBifunctor.d₂_eq _ _ _ _ _ h₁ _ h₂,

Goal state after:
case pos
C : Type u_1
inst✝⁹ : Category.{v_1, u_1} C
inst✝⁸ : MonoidalCategory C
inst✝⁷ : Preadditive C
inst✝⁶ : HasZeroObject C
inst✝⁵ : (curriedTensor C).Additive
inst✝⁴ : ∀ (X₁ : C), ((curriedTensor C).obj X₁).Additive
I : Type u_2
inst✝³ : AddMonoid I
c : ComplexShape I
inst✝² : c.TensorSigns
K : HomologicalComplex C c
inst✝¹ : DecidableEq I
inst✝ : ∀ (X₁ : C), PreservesColimit (Functor.empty C) ((curriedTensor C).obj X₁)
i₁ i₂ j : I
h₁ : c.Rel i₂ (c.next i₂)
h₂ : c.π c c (i₁, c.next i₂) = j
⊢ c.ε₂ c c (i₁, i₂) •
      ((curriedTensor C).obj (K.X i₁)).map ((tensorUnit C c).d i₂ (c.next i₂)) ≫
        K.ιMapBifunctor (tensorUnit C c) (curriedTensor C) c i₁ (c.next i₂) j h₂ =
    0
Module: Mathlib.Algebra.Homology.ShortComplex.Linear

Goal state before:
pp: R : Type u_1
C : Type u_2
inst✝³ : Semiring R
inst✝² : Category.{v_1, u_2} C
inst✝¹ : Preadditive C
inst✝ : Linear R C
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
h₁ : S₁.LeftHomologyData
h₂ : S₂.LeftHomologyData
a : R
⊢ leftHomologyMap' (a • φ) h₁ h₂ = a • leftHomologyMap' φ h₁ h₂

Tactic:
have γ : LeftHomologyMapData φ h₁ h₂ := default

Goal state after:
R : Type u_1
C : Type u_2
inst✝³ : Semiring R
inst✝² : Category.{v_1, u_2} C
inst✝¹ : Preadditive C
inst✝ : Linear R C
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
h₁ : S₁.LeftHomologyData
h₂ : S₂.LeftHomologyData
a : R
γ : LeftHomologyMapData φ h₁ h₂
⊢ leftHomologyMap' (a • φ) h₁ h₂ = a • leftHomologyMap' φ h₁ h₂
Module: Mathlib.Algebra.Homology.ShortComplex.Linear

Goal state before:
pp: R : Type u_1
C : Type u_2
inst✝³ : Semiring R
inst✝² : Category.{v_1, u_2} C
inst✝¹ : Preadditive C
inst✝ : Linear R C
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
h₁ : S₁.LeftHomologyData
h₂ : S₂.LeftHomologyData
a : R
⊢ cyclesMap' (a • φ) h₁ h₂ = a • cyclesMap' φ h₁ h₂

Tactic:
have γ : LeftHomologyMapData φ h₁ h₂ := default

Goal state after:
R : Type u_1
C : Type u_2
inst✝³ : Semiring R
inst✝² : Category.{v_1, u_2} C
inst✝¹ : Preadditive C
inst✝ : Linear R C
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
h₁ : S₁.LeftHomologyData
h₂ : S₂.LeftHomologyData
a : R
γ : LeftHomologyMapData φ h₁ h₂
⊢ cyclesMap' (a • φ) h₁ h₂ = a • cyclesMap' φ h₁ h₂