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

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
K L : CochainComplex C ℤ
f g : K ⟶ L
h : Homotopy ((opEquivalence C).functor.map f.op) ((opEquivalence C).functor.map g.op)
p q : ℤ
hpq : ¬(ComplexShape.up ℤ).Rel q p
⊢ (HomologicalComplex.XIsoOfEq K ⋯).hom ≫ (h.hom (-q) (-p)).unop ≫ (HomologicalComplex.XIsoOfEq L ⋯).hom = 0

Tactic:
h.zero,

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
K L : CochainComplex C ℤ
f g : K ⟶ L
h : Homotopy ((opEquivalence C).functor.map f.op) ((opEquivalence C).functor.map g.op)
p q : ℤ
hpq : ¬(ComplexShape.up ℤ).Rel q p
⊢ (HomologicalComplex.XIsoOfEq K ⋯).hom ≫ Quiver.Hom.unop 0 ≫ (HomologicalComplex.XIsoOfEq L ⋯).hom = 0

case a
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
K L : CochainComplex C ℤ
f g : K ⟶ L
h : Homotopy ((opEquivalence C).functor.map f.op) ((opEquivalence C).functor.map g.op)
p q : ℤ
hpq : ¬(ComplexShape.up ℤ).Rel q p
⊢ ¬(ComplexShape.up ℤ).Rel (-p) (-q)
Module: Mathlib.Algebra.Homology.BifunctorAssociator

Goal state before:
pp: case neg
C₁ : Type u_1
C₂ : Type u_2
C₁₂ : Type u_3
C₃ : Type u_5
C₄ : Type u_6
inst✝²⁰ : Category.{v_1, u_1} C₁
inst✝¹⁹ : Category.{v_2, u_2} C₂
inst✝¹⁸ : Category.{v_3, u_5} C₃
inst✝¹⁷ : Category.{v_4, u_6} C₄
inst✝¹⁶ : Category.{v_5, u_3} C₁₂
inst✝¹⁵ : HasZeroMorphisms C₁
inst✝¹⁴ : HasZeroMorphisms C₂
inst✝¹³ : HasZeroMorphisms C₃
inst✝¹² : Preadditive C₁₂
inst✝¹¹ : Preadditive C₄
F₁₂ : C₁ ⥤ C₂ ⥤ C₁₂
G : C₁₂ ⥤ C₃ ⥤ C₄
inst✝¹⁰ : F₁₂.PreservesZeroMorphisms
inst✝⁹ : ∀ (X₁ : C₁), (F₁₂.obj X₁).PreservesZeroMorphisms
inst✝⁸ : G.Additive
inst✝⁷ : ∀ (X₁₂ : C₁₂), (G.obj X₁₂).PreservesZeroMorphisms
ι₁ : Type u_7
ι₂ : Type u_8
ι₃ : Type u_9
ι₁₂ : Type u_10
ι₄ : Type u_12
inst✝⁶ : DecidableEq ι₄
c₁ : ComplexShape ι₁
c₂ : ComplexShape ι₂
c₃ : ComplexShape ι₃
K₁ : HomologicalComplex C₁ c₁
K₂ : HomologicalComplex C₂ c₂
K₃ : HomologicalComplex C₃ c₃
c₁₂ : ComplexShape ι₁₂
c₄ : ComplexShape ι₄
inst✝⁵ : TotalComplexShape c₁ c₂ c₁₂
inst✝⁴ : TotalComplexShape c₁₂ c₃ c₄
inst✝³ : K₁.HasMapBifunctor K₂ F₁₂ c₁₂
inst✝² : DecidableEq ι₁₂
inst✝¹ : (K₁.mapBifunctor K₂ F₁₂ c₁₂).HasMapBifunctor K₃ G c₄
j j' : ι₄
inst✝ : HasGoodTrifunctor₁₂Obj F₁₂ G K₁ K₂ K₃ c₁₂ c₄
i₁ : ι₁
i₂ : ι₂
i₃ : ι₃
h : c₁.r c₂ c₃ c₁₂ c₄ (i₁, i₂, i₃) = j
i₁₂ : ι₁₂ := c₁.π c₂ c₁₂ (i₁, i₂)
h₁ : c₁₂.Rel i₁₂ (c₁₂.next i₁₂)
h₂ : c₁₂.π c₃ c₄ (c₁₂.next i₁₂, i₃) = j'
h₃ : ¬c₂.Rel i₂ (c₂.next i₂)
⊢ c₁₂.ε₁ c₃ c₄ (c₁.π c₂ c₁₂ (i₁, i₂), i₃) •
      (G.map (mapBifunctor.d₂ K₁ K₂ F₁₂ c₁₂ i₁ i₂ (c₁₂.next (c₁.π c₂ c₁₂ (i₁, i₂))))).app (K₃.X i₃) ≫
        (K₁.mapBifunctor K₂ F₁₂ c₁₂).ιMapBifunctor K₃ G c₄ (c₁₂.next (c₁.π c₂ c₁₂ (i₁, i₂))) i₃ j' h₂ =
    d₂ F₁₂ G K₁ K₂ K₃ c₁₂ c₄ i₁ i₂ i₃ j'

Tactic:
· rw [d₂_eq_zero _ _ _ _ _ _ _ _ _ _ _ h₃,
            mapBifunctor.d₂_eq_zero _ _ _ _ _ _ _ h₃,
            Functor.map_zero, zero_app, zero_comp, smul_zero]

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

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
X₁ X₂ X₃ X₄ : C
inst✝ : HasBinaryBiproduct X₂ X₃
f : X₁ ⟶ X₂
g : X₁ ⟶ X₃
inl : X₂ ⟶ X₄
inr : X₃ ⟶ X₄
h : IsPushout f g inl inr
R✝ : C
b : ⋯.shortComplex.X₃ ⟶ R✝
hb : ⋯.shortComplex.g ≫ b = 0
⊢ Cofork.π ⋯.cokernelCofork ≫ b = Cofork.π ⋯.cokernelCofork ≫ 0

Tactic:
simpa using hb

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

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
X₁ X₂ X₃ X₄ : C
inst✝ : HasBinaryBiproduct X₂ X₃
fst : X₁ ⟶ X₂
snd : X₁ ⟶ X₃
f : X₂ ⟶ X₄
g : X₃ ⟶ X₄
h : IsPullback fst snd f g
P✝ : C
b : P✝ ⟶ ⋯.shortComplex'.X₁
hb : b ≫ ⋯.shortComplex'.f = 0
⊢ b ≫ Fork.ι ⋯.kernelFork = 0 ≫ Fork.ι ⋯.kernelFork

Tactic:
simpa using hb

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

Goal state before:
pp: case succ
C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
t : TStructure C
H : ℕ → Prop := fun a ↦ ∀ (n : ℤ), t.le n ≤ t.le (n + ↑a)
H_zero : H 0
H_one : H 1
H_add : ∀ (a b c : ℕ), a + b = c → H a → H b → H c
a : ℕ
ha : H a
⊢ H (a + 1)

Tactic:
exact H_add a 1 _ rfl ha H_one

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

Goal state before:
pp: case succ
C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
t : TStructure C
H : ℕ → Prop := fun a ↦ ∀ (n : ℤ), t.ge (n + ↑a) ≤ t.ge n
H_zero : H 0
H_one : H 1
H_add : ∀ (a b c : ℕ), a + b = c → H a → H b → H c
a : ℕ
ha : H a
⊢ H (a + 1)

Tactic:
exact H_add a 1 _ rfl ha H_one

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

Goal state before:
pp: C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
t : TStructure C
X : C
n a n' : ℤ
hn' : a + n' = n
⊢ t.IsGE ((shiftFunctor C a).obj X) n' ↔ t.IsGE X n

Tactic:
constructor
  · intro
    exact t.isGE_of_shift X n a n' hn'
  · intro
    exact t.isGE_shift X n a n' hn'

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

Goal state before:
pp: C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
t : TStructure C
n : ℤ
⊢ t.minus ≤ t.minus.shift n

Tactic:
by
        rintro X ⟨i, _⟩
        exact ⟨i - n, t.isLE_shift _ i _ _ (by omega)⟩

Goal state after:
Module: Mathlib.CategoryTheory.LiftingProperties.Limits

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
X Y Z W : C
f : X ⟶ Y
s : X ⟶ Z
g : Z ⟶ W
t : Y ⟶ W
h : IsPullback s f g t
X' Y' : C
f' : X' ⟶ Y'
inst✝ : HasLiftingProperty f' g
u : X' ⟶ X
v : Y' ⟶ Y
sq : CommSq u f' f v
⊢ (u ≫ s) ≫ g = f' ≫ v ≫ t

Tactic:
rw [Category.assoc, h.toCommSq.w, ← Category.assoc, ← Category.assoc, sq.w]

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✝ : (trivialCofibrations C).IsWeakFactorizationSystem (fibrations C)
J : Type w
⊢ (trivialCofibrations C).rlp.IsStableUnderProductsOfShape J

Tactic:
apply MorphismProperty.rlp_isStableUnderProductsOfShape

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

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : CategoryWithFibrations C
inst✝¹ : HasTerminal C
inst✝ : (fibrations C).RespectsIso
X Y : C
p : X ⟶ Y
hY : IsTerminal Y
⊢ Arrow.mk p ≅ Arrow.mk (terminal.from X)

Tactic:
exact Arrow.isoMk (Iso.refl _) (IsTerminal.uniqueUpToIso hY terminalIsTerminal)

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

Goal state before:
pp: C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : CategoryWithFibrations C
inst✝² : HasTerminal C
inst✝¹ : (fibrations C).IsStableUnderComposition
X Y : C
p : X ⟶ Y
inst✝ : Fibration p
hY : IsFibrant Y
⊢ IsFibrant X

Tactic:
rw [isFibrant_iff] at hY ⊢
  rw [Subsingleton.elim (terminal.from X) (p ≫ terminal.from Y)]
  infer_instance

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✝ : (cofibrations C).IsWeakFactorizationSystem (trivialFibrations C)
J : Type w
⊢ (cofibrations C).IsStableUnderCoproductsOfShape J

Tactic:
rw [← trivialFibrations_llp]
  apply MorphismProperty.llp_isStableUnderCoproductsOfShape

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✝ : (cofibrations C).IsWeakFactorizationSystem (trivialFibrations C)
J : Type w
⊢ (cofibrations C).rlp.IsStableUnderProductsOfShape J

Tactic:
apply MorphismProperty.rlp_isStableUnderProductsOfShape

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

Goal state before:
pp: C : Type u_1
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : HasShift C ℤ
inst✝⁴ : HasZeroObject C
inst✝³ : Preadditive C
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
inst✝ : IsTriangulated C
X₁ X₂ : Cᵒᵖ
⊢ ∀ {X₃ Z₁₂ Z₂₃ Z₁₃ : Cᵒᵖ} {u₁₂ : X₁ ⟶ X₂} {u₂₃ : X₂ ⟶ X₃} {u₁₃ : X₁ ⟶ X₃} (comm : u₁₂ ≫ u₂₃ = u₁₃) {v₁₂ : X₂ ⟶ Z₁₂}
    {w₁₂ : Z₁₂ ⟶ (shiftFunctor Cᵒᵖ 1).obj X₁} (h₁₂ : Triangle.mk u₁₂ v₁₂ w₁₂ ∈ Pretriangulated.distinguishedTriangles)
    {v₂₃ : X₃ ⟶ Z₂₃} {w₂₃ : Z₂₃ ⟶ (shiftFunctor Cᵒᵖ 1).obj X₂}
    (h₂₃ : Triangle.mk u₂₃ v₂₃ w₂₃ ∈ Pretriangulated.distinguishedTriangles) {v₁₃ : X₃ ⟶ Z₁₃}
    {w₁₃ : Z₁₃ ⟶ (shiftFunctor Cᵒᵖ 1).obj X₁} (h₁₃ : Triangle.mk u₁₃ v₁₃ w₁₃ ∈ Pretriangulated.distinguishedTriangles),
    Nonempty (Triangulated.Octahedron comm h₁₂ h₂₃ h₁₃)

Tactic:
X₃

Goal state after:
C : Type u_1
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : HasShift C ℤ
inst✝⁴ : HasZeroObject C
inst✝³ : Preadditive C
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
inst✝ : IsTriangulated C
X₁ X₂ X₃ : Cᵒᵖ
⊢ ∀ {Z₁₂ Z₂₃ Z₁₃ : Cᵒᵖ} {u₁₂ : X₁ ⟶ X₂} {u₂₃ : X₂ ⟶ X₃} {u₁₃ : X₁ ⟶ X₃} (comm : u₁₂ ≫ u₂₃ = u₁₃) {v₁₂ : X₂ ⟶ Z₁₂}
    {w₁₂ : Z₁₂ ⟶ (shiftFunctor Cᵒᵖ 1).obj X₁} (h₁₂ : Triangle.mk u₁₂ v₁₂ w₁₂ ∈ Pretriangulated.distinguishedTriangles)
    {v₂₃ : X₃ ⟶ Z₂₃} {w₂₃ : Z₂₃ ⟶ (shiftFunctor Cᵒᵖ 1).obj X₂}
    (h₂₃ : Triangle.mk u₂₃ v₂₃ w₂₃ ∈ Pretriangulated.distinguishedTriangles) {v₁₃ : X₃ ⟶ Z₁₃}
    {w₁₃ : Z₁₃ ⟶ (shiftFunctor Cᵒᵖ 1).obj X₁} (h₁₃ : Triangle.mk u₁₃ v₁₃ w₁₃ ∈ Pretriangulated.distinguishedTriangles),
    Nonempty (Triangulated.Octahedron comm h₁₂ h₂₃ h₁₃)
Module: Mathlib.Algebra.Homology.HomotopyCategory.Plus

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁷ : Category.{v_1, u_1} C
inst✝⁶ : Category.{v_2, u_2} D
inst✝⁵ : Preadditive C
inst✝⁴ : Preadditive D
A : Type u_3
inst✝³ : Category.{v_3, u_3} A
inst✝² : Abelian A
inst✝¹ : HasZeroObject C
inst✝ : HasBinaryBiproducts C
⊢ ∀ ⦃X Y : CochainComplex.Plus C⦄ (f₀ f₁ : X ⟶ Y),
    (quotient C).map f₀ = (quotient C).map f₁ →
      ∃ P x, (homotopyEquivalences C (ComplexShape.up ℤ)).inverseImage (CochainComplex.Plus.ι C) P.π

Tactic:
rintro K L f₀ f₁ hf

Goal state after:
C : Type u_1
D : Type u_2
inst✝⁷ : Category.{v_1, u_1} C
inst✝⁶ : Category.{v_2, u_2} D
inst✝⁵ : Preadditive C
inst✝⁴ : Preadditive D
A : Type u_3
inst✝³ : Category.{v_3, u_3} A
inst✝² : Abelian A
inst✝¹ : HasZeroObject C
inst✝ : HasBinaryBiproducts C
K L : CochainComplex.Plus C
f₀ f₁ : K ⟶ L
hf : (quotient C).map f₀ = (quotient C).map f₁
⊢ ∃ P x, (homotopyEquivalences C (ComplexShape.up ℤ)).inverseImage (CochainComplex.Plus.ι C) P.π
Module: Mathlib.CategoryTheory.Triangulated.LocalizingSubcategory

Goal state before:
pp: case refine_2
C : Type u_1
inst✝⁸ : Category.{v_1, u_1} C
A B : ObjectProperty C
inst✝⁷ : HasZeroObject C
inst✝⁶ : HasShift C ℤ
inst✝⁵ : Preadditive C
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝³ : Pretriangulated C
inst✝² : A.IsTriangulated
inst✝¹ : B.IsTriangulated
inst✝ : B.IsClosedUnderIsomorphisms
hA : ∀ ⦃X Y : C⦄ (s : X ⟶ Y), A X → B.trW s → ∃ Z s' b, A Z ∧ (A ⊓ B).trW s' ∧ s ≫ b = s'
X Y : C
f : X ⟶ Y
hX : B X
hY : A Y
Z : C
s : Y ⟶ Z
b : Z ⟶ (shiftFunctor C 1).obj X
hT : Triangle.mk f s b ∈ Pretriangulated.distinguishedTriangles
hs : B.trW s
W : C
s' : Y ⟶ W
g : Z ⟶ W
hW : A W
hs' : (A ⊓ B).trW s'
fac : s ≫ g = s'
U : C
hU : A U
a : U ⟶ Y
c : W ⟶ (shiftFunctor C 1).obj U
hT' : Triangle.mk a s' c ∈ Pretriangulated.distinguishedTriangles
t : (Triangle.mk f s b).obj₁ ⟶ (Triangle.mk a s' c).obj₁
ht : (Triangle.mk f s b).mor₁ ≫ 𝟙 Y = t ≫ (Triangle.mk a s' c).mor₁
ht' : (Triangle.mk f s b).mor₃ ≫ (shiftFunctor C 1).map t = g ≫ (Triangle.mk a s' c).mor₃
⊢ ∃ Z a b, A Z ∧ B Z ∧ a ≫ b = f

Tactic:
exact ⟨U, t, a, hU, (B.trW_iff_of_distinguished' _ hT').1 (trW_monotone (by simp) _ hs'),
      by cat_disch⟩

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.Injective.LiftingProperties

Goal state before:
pp: case mpr
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasZeroMorphisms C
I Z : C
hZ : IsZero Z
⊢ (MorphismProperty.monomorphisms C).rlp 0 → Injective I

Tactic:
intro h

Goal state after:
case mpr
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasZeroMorphisms C
I Z : C
hZ : IsZero Z
h : (MorphismProperty.monomorphisms C).rlp 0
⊢ Injective I
Module: Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
t : TStructure C
a : ℤ
⊢ triangleFunctorNatTransOfLE t a a ⋯ = 𝟙 (triangleFunctor t a)

Tactic:
apply NatTrans.ext
  ext1 X
  exact triangle_map_ext t (triangleFunctor_obj_distinguished _ _ _)
    (triangleFunctor_obj_distinguished _ _ _) (a - 1) a inferInstance inferInstance (by simp)

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

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
t : TStructure C
n₀ : ℤ
X : C
x✝ : t.IsLE X n₀
e : contractibleTriangle X ≅ (t.triangleLTGE (n₀ + 1)).obj X
he : e.hom.hom₂ = (Iso.refl X).hom
⊢ IsIso ((t.truncLTι (n₀ + 1)).app X)

Tactic:
have he' : e.inv.hom₂ = 𝟙 X := by
    rw [← cancel_mono e.hom.hom₂, ← comp_hom₂, e.inv_hom_id, he]
    simp

Goal state after:
C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
t : TStructure C
n₀ : ℤ
X : C
x✝ : t.IsLE X n₀
e : contractibleTriangle X ≅ (t.triangleLTGE (n₀ + 1)).obj X
he : e.hom.hom₂ = (Iso.refl X).hom
he' : e.inv.hom₂ = 𝟙 X
⊢ IsIso ((t.truncLTι (n₀ + 1)).app X)
Module: Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE

Goal state before:
pp: case mp
C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
t : TStructure C
n : ℤ
X : C
⊢ t.IsGE X n → IsIso ((t.truncGEπ n).app X)

Tactic:
intro h

Goal state after:
case mp
C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
t : TStructure C
n : ℤ
X : C
h : t.IsGE X n
⊢ IsIso ((t.truncGEπ n).app X)
Module: Mathlib.Algebra.Homology.Factorizations.CM5b

Goal state before:
pp: case refine_1
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
inst✝ : EnoughInjectives C
K L : CochainComplex C ℤ
n : ℤ
⊢ biprod.inl.f n ≫ (p K L).f n = 0

pp: case refine_2
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
inst✝ : EnoughInjectives C
K L : CochainComplex C ℤ
n : ℤ
⊢ { X₁ := (mappingCone (𝟙 (I K))).X n, X₂ := (mappingCone (𝟙 (I K)) ⊞ L).X n, X₃ := L.X n, f := biprod.inl.f n,
          g := (p K L).f n, zero := ?refine_1 }.f ≫
      biprod.fst.f n =
    𝟙
      { X₁ := (mappingCone (𝟙 (I K))).X n, X₂ := (mappingCone (𝟙 (I K)) ⊞ L).X n, X₃ := L.X n, f := biprod.inl.f n,
          g := (p K L).f n, zero := ?refine_1 }.X₁

pp: case refine_3
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
inst✝ : EnoughInjectives C
K L : CochainComplex C ℤ
n : ℤ
⊢ biprod.inr.f n ≫
      { X₁ := (mappingCone (𝟙 (I K))).X n, X₂ := (mappingCone (𝟙 (I K)) ⊞ L).X n, X₃ := L.X n, f := biprod.inl.f n,
          g := (p K L).f n, zero := ?refine_1 }.g =
    𝟙
      { X₁ := (mappingCone (𝟙 (I K))).X n, X₂ := (mappingCone (𝟙 (I K)) ⊞ L).X n, X₃ := L.X n, f := biprod.inl.f n,
          g := (p K L).f n, zero := ?refine_1 }.X₃

pp: case refine_4
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
inst✝ : EnoughInjectives C
K L : CochainComplex C ℤ
n : ℤ
⊢ biprod.fst.f n ≫
        { X₁ := (mappingCone (𝟙 (I K))).X n, X₂ := (mappingCone (𝟙 (I K)) ⊞ L).X n, X₃ := L.X n, f := biprod.inl.f n,
            g := (p K L).f n, zero := ?refine_1 }.f +
      { X₁ := (mappingCone (𝟙 (I K))).X n, X₂ := (mappingCone (𝟙 (I K)) ⊞ L).X n, X₃ := L.X n, f := biprod.inl.f n,
            g := (p K L).f n, zero := ?refine_1 }.g ≫
        biprod.inr.f n =
    𝟙
      { X₁ := (mappingCone (𝟙 (I K))).X n, X₂ := (mappingCone (𝟙 (I K)) ⊞ L).X n, X₃ := L.X n, f := biprod.inl.f n,
          g := (p K L).f n, zero := ?refine_1 }.X₂

Tactic:
all_goals simp [← HomologicalComplex.comp_f, ← HomologicalComplex.add_f_apply]

Goal state after:
Module: Mathlib.CategoryTheory.Functor.OfSequence

Goal state before:
pp: case zero.succ.succ.succ
C : Type u_1
inst✝ : Category.{v_1, u_1} C
j : ℕ
hj :
  ∀ {X : ℕ → C} (f : (n : ℕ) → X n ⟶ X (n + 1)) (k : ℕ) (hij : 0 ≤ j) (hjk : j ≤ k),
    map f 0 k ⋯ = map f 0 j hij ≫ map f j k hjk
X : ℕ → C
f : (n : ℕ) → X n ⟶ X (n + 1)
hij : 0 ≤ j + 1
k : ℕ
hjk : j + 1 ≤ k + 1 + 1
⊢ map f 0 (k + 1 + 1) ⋯ = map f 0 (j + 1) hij ≫ map f (j + 1) (k + 1 + 1) hjk

Tactic:
simp only [map, Nat.reduceAdd]
            rw [hj (fun n ↦ f (n + 1)) (k + 1) (by lia) (by lia)]
            obtain _ | j := j
            all_goals simp [map]

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

Goal state before:
pp: case left
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : ModelCategory C
X Y Z : C
inst✝² : IsCofibrant X
g : Y ⟶ Z
inst✝¹ : Fibration g
inst✝ : WeakEquivalence g
f₀ f₁ : LeftHomotopyClass X Y
h : (fun f ↦ f.postcomp g) f₀ = (fun f ↦ f.postcomp g) f₁
⊢ f₀ = f₁

Tactic:
obtain ⟨f₀, rfl⟩ := f₀.mk_surjective

Goal state after:
case left
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : ModelCategory C
X Y Z : C
inst✝² : IsCofibrant X
g : Y ⟶ Z
inst✝¹ : Fibration g
inst✝ : WeakEquivalence g
f₁ : LeftHomotopyClass X Y
f₀ : X ⟶ Y
h : (fun f ↦ f.postcomp g) (mk f₀) = (fun f ↦ f.postcomp g) f₁
⊢ mk f₀ = f₁
Module: Mathlib.AlgebraicTopology.ModelCategory.Homotopy

Goal state before:
pp: case left
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : ModelCategory C
X Y Z : C
inst✝² : IsFibrant Z
f : X ⟶ Y
inst✝¹ : Cofibration f
inst✝ : WeakEquivalence f
f₀ f₁ : RightHomotopyClass Y Z
h : (fun g ↦ g.precomp f) f₀ = (fun g ↦ g.precomp f) f₁
⊢ f₀ = f₁

Tactic:
obtain ⟨f₀, rfl⟩ := f₀.mk_surjective

Goal state after:
case left
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : ModelCategory C
X Y Z : C
inst✝² : IsFibrant Z
f : X ⟶ Y
inst✝¹ : Cofibration f
inst✝ : WeakEquivalence f
f₁ : RightHomotopyClass Y Z
f₀ : Y ⟶ Z
h : (fun g ↦ g.precomp f) (mk f₀) = (fun g ↦ g.precomp f) f₁
⊢ mk f₀ = f₁
Module: Mathlib.AlgebraicTopology.ModelCategory.FibrantObjectHomotopy

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : ModelCategory C
X Y : FibrantObject C
inst✝ : IsCofibrant X.obj
f g : X ⟶ Y
⊢ (Quotient.functor (homRel C)).map f = (Quotient.functor (homRel C)).map g ↔ homRel C f g

Tactic:
rw [← Functor.homRel_iff, Quotient.functor_homRel_eq_compClosure_eqvGen,
    HomRel.compClosure_eq_self, homRel_equivalence_of_isCofibrant_src.eqvGen_eq]

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

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : ModelCategory C
X Y : C
f : X ⟶ Y
⊢ ∃ g, iResolutionObj X ≫ g = f ≫ iResolutionObj Y

Tactic:
have sq : CommSq (f ≫ iResolutionObj Y) (iResolutionObj X)
    (terminal.from _) (terminal.from _) := ⟨by simp⟩

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : ModelCategory C
X Y : C
f : X ⟶ Y
sq :
  CommSq (f ≫ iResolutionObj Y) (iResolutionObj X) (terminal.from (resolutionObj Y)) (terminal.from (resolutionObj X))
⊢ ∃ g, iResolutionObj X ≫ g = f ≫ iResolutionObj Y
Module: Mathlib.Algebra.Homology.Factorizations.CM5a

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
K L : CochainComplex C ℤ
f : K ⟶ L
inst✝ : EnoughInjectives C
n : ℤ
this : IsIso ((S f n).homologyπ n)
⊢ Mono (homologyMap (p f n) n)

Tactic:
have : Mono ((truncGE (cokernel f) n).homologyπ n ≫ homologyMap (p f n) n) := by
    rw [homologyπ_naturality (p f n) n]
    infer_instance

Goal state after:
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
K L : CochainComplex C ℤ
f : K ⟶ L
inst✝ : EnoughInjectives C
n : ℤ
this✝ : IsIso ((S f n).homologyπ n)
this : Mono (homologyπ ((cokernel f).truncGE n) n ≫ homologyMap (p f n) n)
⊢ Mono (homologyMap (p f n) n)
Module: Mathlib.CategoryTheory.GuitartExact.VerticalComposition

Goal state before:
pp: case mp
C₁ : Type u_1
C₂ : Type u_2
D₁ : Type u_4
D₂ : Type u_5
inst✝³ : Category.{v_1, u_1} C₁
inst✝² : Category.{v_2, u_2} C₂
inst✝¹ : Category.{v_4, u_4} D₁
inst✝ : Category.{v_5, u_5} D₂
T : C₁ ⥤ D₁
L : C₁ ⥤ C₂
R : D₁ ⥤ D₂
B : C₂ ⥤ D₂
w : TwoSquare T L R B
L' : C₁ ⥤ C₂
R' : D₁ ⥤ D₂
α : L ≅ L'
β : R ≅ R'
⊢ (w.whiskerVertical α.hom β.inv).GuitartExact → w.GuitartExact

Tactic:
intro h

Goal state after:
case mp
C₁ : Type u_1
C₂ : Type u_2
D₁ : Type u_4
D₂ : Type u_5
inst✝³ : Category.{v_1, u_1} C₁
inst✝² : Category.{v_2, u_2} C₂
inst✝¹ : Category.{v_4, u_4} D₁
inst✝ : Category.{v_5, u_5} D₂
T : C₁ ⥤ D₁
L : C₁ ⥤ C₂
R : D₁ ⥤ D₂
B : C₂ ⥤ D₂
w : TwoSquare T L R B
L' : C₁ ⥤ C₂
R' : D₁ ⥤ D₂
α : L ≅ L'
β : R ≅ R'
h : (w.whiskerVertical α.hom β.inv).GuitartExact
⊢ w.GuitartExact
Module: Mathlib.CategoryTheory.GuitartExact.VerticalComposition

Goal state before:
pp: case mpr
C₁ : Type u_1
C₂ : Type u_2
D₁ : Type u_4
D₂ : Type u_5
inst✝³ : Category.{v_1, u_1} C₁
inst✝² : Category.{v_2, u_2} C₂
inst✝¹ : Category.{v_4, u_4} D₁
inst✝ : Category.{v_5, u_5} D₂
T : C₁ ⥤ D₁
L : C₁ ⥤ C₂
R : D₁ ⥤ D₂
B : C₂ ⥤ D₂
w : TwoSquare T L R B
L' : C₁ ⥤ C₂
R' : D₁ ⥤ D₂
α : L ≅ L'
β : R ≅ R'
⊢ w.GuitartExact → (w.whiskerVertical α.hom β.inv).GuitartExact

Tactic:
intro h

Goal state after:
case mpr
C₁ : Type u_1
C₂ : Type u_2
D₁ : Type u_4
D₂ : Type u_5
inst✝³ : Category.{v_1, u_1} C₁
inst✝² : Category.{v_2, u_2} C₂
inst✝¹ : Category.{v_4, u_4} D₁
inst✝ : Category.{v_5, u_5} D₂
T : C₁ ⥤ D₁
L : C₁ ⥤ C₂
R : D₁ ⥤ D₂
B : C₂ ⥤ D₂
w : TwoSquare T L R B
L' : C₁ ⥤ C₂
R' : D₁ ⥤ D₂
α : L ≅ L'
β : R ≅ R'
h : w.GuitartExact
⊢ (w.whiskerVertical α.hom β.inv).GuitartExact
Module: Mathlib.CategoryTheory.GuitartExact.Basic

Goal state before:
pp: C₁ : Type u₁
C₂ : Type u₂
C₃ : Type u₃
C₄ : Type u₄
inst✝³ : Category.{v₁, u₁} C₁
inst✝² : Category.{v₂, u₂} C₂
inst✝¹ : Category.{v₃, u₃} C₃
inst✝ : Category.{v₄, u₄} C₄
T : C₁ ⥤ C₂
L : C₁ ⥤ C₃
R : C₂ ⥤ C₄
B : C₃ ⥤ C₄
w : TwoSquare T L R B
X₂ : C₂
X₃ : C₃
g : R.obj X₂ ⟶ B.obj X₃
X₁ : C₁
a : X₂ ⟶ T.obj X₁
b : ((w.structuredArrowDownwards X₂).obj (StructuredArrow.mk a)).right ⟶ (StructuredArrow.mk g).right
hb : ((w.structuredArrowDownwards X₂).obj (StructuredArrow.mk a)).hom ≫ B.map b = (StructuredArrow.mk g).hom
⊢ R.map a ≫ w.app X₁ ≫ B.map b = g

Tactic:
simpa using hb

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

Goal state before:
pp: C₁ : Type u₁
C₂ : Type u₂
C₃ : Type u₃
C₄ : Type u₄
inst✝³ : Category.{v₁, u₁} C₁
inst✝² : Category.{v₂, u₂} C₂
inst✝¹ : Category.{v₃, u₃} C₃
inst✝ : Category.{v₄, u₄} C₄
T : C₁ ⥤ C₂
L : C₁ ⥤ C₃
R : C₂ ⥤ C₄
B : C₃ ⥤ C₄
w : TwoSquare T L R B
F : C₁ ⥤ C₂
X₂ : C₁
⊢ ∀ {X₃ : C₂} (g : F.obj X₂ ⟶ (𝟭 C₂).obj X₃), IsConnected ((mk (𝟭 C₁) F F (𝟭 C₂) (𝟙 F)).StructuredArrowRightwards g)

Tactic:
X₃

Goal state after:
C₁ : Type u₁
C₂ : Type u₂
C₃ : Type u₃
C₄ : Type u₄
inst✝³ : Category.{v₁, u₁} C₁
inst✝² : Category.{v₂, u₂} C₂
inst✝¹ : Category.{v₃, u₃} C₃
inst✝ : Category.{v₄, u₄} C₄
T : C₁ ⥤ C₂
L : C₁ ⥤ C₃
R : C₂ ⥤ C₄
B : C₃ ⥤ C₄
w : TwoSquare T L R B
F : C₁ ⥤ C₂
X₂ : C₁
X₃ : C₂
⊢ ∀ (g : F.obj X₂ ⟶ (𝟭 C₂).obj X₃), IsConnected ((mk (𝟭 C₁) F F (𝟭 C₂) (𝟙 F)).StructuredArrowRightwards g)