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

Goal state before:
pp: case pos
C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Preadditive C
I₁ : Type u_2
I₂ : Type u_3
I₁₂ : Type u_4
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
K : HomologicalComplex₂ C c₁ c₂
c₁₂ : ComplexShape I₁₂
inst✝² : TotalComplexShape c₁ c₂ c₁₂
inst✝¹ : DecidableEq I₁₂
inst✝ : K.HasTotal c₁₂
i₁₂ i₁₂' i₁₂'' : I₁₂
h₁ : c₁₂.Rel i₁₂ i₁₂'
h₂ : c₁₂.Rel i₁₂' i₁₂''
i₁ : I₁
i₂ : I₂
h : c₁.π c₂ c₁₂ (i₁, i₂) = i₁₂
h₃ : c₂.Rel i₂ (c₂.next i₂)
h₄ : c₂.Rel (c₂.next i₂) (c₂.next (c₂.next i₂))
⊢ c₁₂.next (c₁₂.next i₁₂) = i₁₂''

Tactic:
c₁₂.next_eq' h₁,

Goal state after:
case pos
C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Preadditive C
I₁ : Type u_2
I₂ : Type u_3
I₁₂ : Type u_4
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
K : HomologicalComplex₂ C c₁ c₂
c₁₂ : ComplexShape I₁₂
inst✝² : TotalComplexShape c₁ c₂ c₁₂
inst✝¹ : DecidableEq I₁₂
inst✝ : K.HasTotal c₁₂
i₁₂ i₁₂' i₁₂'' : I₁₂
h₁ : c₁₂.Rel i₁₂ i₁₂'
h₂ : c₁₂.Rel i₁₂' i₁₂''
i₁ : I₁
i₂ : I₂
h : c₁.π c₂ c₁₂ (i₁, i₂) = i₁₂
h₃ : c₂.Rel i₂ (c₂.next i₂)
h₄ : c₂.Rel (c₂.next i₂) (c₂.next (c₂.next i₂))
⊢ c₁₂.next i₁₂' = i₁₂''
Module: Mathlib.Algebra.Homology.BifunctorHomotopy

Goal state before:
pp: case neg
C₁ : Type u_1
C₂ : Type u_2
D : Type u_3
I₁ : Type u_4
I₂ : Type u_5
J : Type u_6
inst✝¹¹ : Category.{v_1, u_1} C₁
inst✝¹⁰ : Category.{v_2, u_2} C₂
inst✝⁹ : Category.{v_3, u_3} D
inst✝⁸ : Preadditive C₁
inst✝⁷ : Preadditive C₂
inst✝⁶ : Preadditive D
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
K₁ L₁ : HomologicalComplex C₁ c₁
f₁ f₁' : K₁ ⟶ L₁
h₁ : Homotopy f₁ f₁'
K₂ L₂ : HomologicalComplex C₂ c₂
f₂ : K₂ ⟶ L₂
F : C₁ ⥤ C₂ ⥤ D
inst✝⁵ : F.Additive
inst✝⁴ : ∀ (X₁ : C₁), (F.obj X₁).Additive
c : ComplexShape J
inst✝³ : DecidableEq J
inst✝² : TotalComplexShape c₁ c₂ c
inst✝¹ : K₁.HasMapBifunctor K₂ F c
inst✝ : L₁.HasMapBifunctor L₂ F c
j : J
i₁ : I₁
i₂ : I₂
h : c₁.π c₂ c (i₁, i₂) = j
this : ∀ {X Y : D} (a b c d e f : X ⟶ Y), a = c → b = e → f = -d → a + b = c + d + (e + f)
h₃ : c₁.Rel (c₁.prev i₁) i₁
h₄ : ¬c₂.Rel i₂ (c₂.next i₂)
⊢ c₁.ε₁ c₂ c (c₁.prev i₁, i₂) •
      (F.map (h₁.hom i₁ (c₁.prev i₁))).app (K₂.X i₂) ≫
        (F.obj (L₁.X (c₁.prev i₁))).map (f₂.f i₂) ≫
          (((F.mapBifunctorHomologicalComplex c₁ c₂).obj L₁).obj L₂).d₂ c (c₁.prev i₁) i₂ j =
    -(((F.mapBifunctorHomologicalComplex c₁ c₂).obj K₁).obj K₂).d₂ c i₁ i₂ (c.next j) ≫ hom₁ h₁ f₂ F c (c.next j) j

Tactic:
HomologicalComplex₂.d₂_eq_zero _ _ _ _ _ h₄,

Goal state after:
case neg
C₁ : Type u_1
C₂ : Type u_2
D : Type u_3
I₁ : Type u_4
I₂ : Type u_5
J : Type u_6
inst✝¹¹ : Category.{v_1, u_1} C₁
inst✝¹⁰ : Category.{v_2, u_2} C₂
inst✝⁹ : Category.{v_3, u_3} D
inst✝⁸ : Preadditive C₁
inst✝⁷ : Preadditive C₂
inst✝⁶ : Preadditive D
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
K₁ L₁ : HomologicalComplex C₁ c₁
f₁ f₁' : K₁ ⟶ L₁
h₁ : Homotopy f₁ f₁'
K₂ L₂ : HomologicalComplex C₂ c₂
f₂ : K₂ ⟶ L₂
F : C₁ ⥤ C₂ ⥤ D
inst✝⁵ : F.Additive
inst✝⁴ : ∀ (X₁ : C₁), (F.obj X₁).Additive
c : ComplexShape J
inst✝³ : DecidableEq J
inst✝² : TotalComplexShape c₁ c₂ c
inst✝¹ : K₁.HasMapBifunctor K₂ F c
inst✝ : L₁.HasMapBifunctor L₂ F c
j : J
i₁ : I₁
i₂ : I₂
h : c₁.π c₂ c (i₁, i₂) = j
this : ∀ {X Y : D} (a b c d e f : X ⟶ Y), a = c → b = e → f = -d → a + b = c + d + (e + f)
h₃ : c₁.Rel (c₁.prev i₁) i₁
h₄ : ¬c₂.Rel i₂ (c₂.next i₂)
⊢ c₁.ε₁ c₂ c (c₁.prev i₁, i₂) •
      (F.map (h₁.hom i₁ (c₁.prev i₁))).app (K₂.X i₂) ≫ (F.obj (L₁.X (c₁.prev i₁))).map (f₂.f i₂) ≫ 0 =
    -(((F.mapBifunctorHomologicalComplex c₁ c₂).obj K₁).obj K₂).d₂ c i₁ i₂ (c.next j) ≫ hom₁ h₁ f₂ F c (c.next j) j
Module: Mathlib.Algebra.Homology.BifunctorHomotopy

Goal state before:
pp: case neg
C₁ : Type u_1
C₂ : Type u_2
D : Type u_3
I₁ : Type u_4
I₂ : Type u_5
J : Type u_6
inst✝¹¹ : Category.{v_1, u_1} C₁
inst✝¹⁰ : Category.{v_2, u_2} C₂
inst✝⁹ : Category.{v_3, u_3} D
inst✝⁸ : Preadditive C₁
inst✝⁷ : Preadditive C₂
inst✝⁶ : Preadditive D
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
K₁ L₁ : HomologicalComplex C₁ c₁
f₁ f₁' : K₁ ⟶ L₁
h₁ : Homotopy f₁ f₁'
K₂ L₂ : HomologicalComplex C₂ c₂
f₂ : K₂ ⟶ L₂
F : C₁ ⥤ C₂ ⥤ D
inst✝⁵ : F.Additive
inst✝⁴ : ∀ (X₁ : C₁), (F.obj X₁).Additive
c : ComplexShape J
inst✝³ : DecidableEq J
inst✝² : TotalComplexShape c₁ c₂ c
inst✝¹ : K₁.HasMapBifunctor K₂ F c
inst✝ : L₁.HasMapBifunctor L₂ F c
j : J
i₁ : I₁
i₂ : I₂
h : c₁.π c₂ c (i₁, i₂) = j
this : ∀ {X Y : D} (a b c d e f : X ⟶ Y), a = c → b = e → f = -d → a + b = c + d + (e + f)
h₃ : c₁.Rel (c₁.prev i₁) i₁
h₄ : ¬c₂.Rel i₂ (c₂.next i₂)
⊢ 0 = -(((F.mapBifunctorHomologicalComplex c₁ c₂).obj K₁).obj K₂).d₂ c i₁ i₂ (c.next j) ≫ hom₁ h₁ f₂ F c (c.next j) j

Tactic:
HomologicalComplex₂.d₂_eq_zero _ _ _ _ _ h₄,

Goal state after:
case neg
C₁ : Type u_1
C₂ : Type u_2
D : Type u_3
I₁ : Type u_4
I₂ : Type u_5
J : Type u_6
inst✝¹¹ : Category.{v_1, u_1} C₁
inst✝¹⁰ : Category.{v_2, u_2} C₂
inst✝⁹ : Category.{v_3, u_3} D
inst✝⁸ : Preadditive C₁
inst✝⁷ : Preadditive C₂
inst✝⁶ : Preadditive D
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
K₁ L₁ : HomologicalComplex C₁ c₁
f₁ f₁' : K₁ ⟶ L₁
h₁ : Homotopy f₁ f₁'
K₂ L₂ : HomologicalComplex C₂ c₂
f₂ : K₂ ⟶ L₂
F : C₁ ⥤ C₂ ⥤ D
inst✝⁵ : F.Additive
inst✝⁴ : ∀ (X₁ : C₁), (F.obj X₁).Additive
c : ComplexShape J
inst✝³ : DecidableEq J
inst✝² : TotalComplexShape c₁ c₂ c
inst✝¹ : K₁.HasMapBifunctor K₂ F c
inst✝ : L₁.HasMapBifunctor L₂ F c
j : J
i₁ : I₁
i₂ : I₂
h : c₁.π c₂ c (i₁, i₂) = j
this : ∀ {X Y : D} (a b c d e f : X ⟶ Y), a = c → b = e → f = -d → a + b = c + d + (e + f)
h₃ : c₁.Rel (c₁.prev i₁) i₁
h₄ : ¬c₂.Rel i₂ (c₂.next i₂)
⊢ 0 = -0 ≫ hom₁ h₁ f₂ F c (c.next j) j
Module: Mathlib.Algebra.Homology.TotalComplexShift

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : Preadditive C
K L : HomologicalComplex₂ C (up ℤ) (up ℤ)
f : K ⟶ L
x : ℤ
inst✝¹ : K.HasTotal (up ℤ)
inst✝ : L.HasTotal (up ℤ)
n i₁ i₂ : ℤ
h : i₁ + i₂ = n
⊢ (f.f (i₁ + x)).f i₂ ≫ L.ιTotal (up ℤ) (i₁ + x) i₂ (n + x) ⋯ ≫ 𝟙 ((L.total (up ℤ)).X (n + x)) =
    K.ιTotal (up ℤ) (i₁ + x) i₂ (n + x) ⋯ ≫ (total.map f (up ℤ)).f (n + x)

Tactic:
comp_id,

Goal state after:
C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : Preadditive C
K L : HomologicalComplex₂ C (up ℤ) (up ℤ)
f : K ⟶ L
x : ℤ
inst✝¹ : K.HasTotal (up ℤ)
inst✝ : L.HasTotal (up ℤ)
n i₁ i₂ : ℤ
h : i₁ + i₂ = n
⊢ (f.f (i₁ + x)).f i₂ ≫ L.ιTotal (up ℤ) (i₁ + x) i₂ (n + x) ⋯ =
    K.ιTotal (up ℤ) (i₁ + x) i₂ (n + x) ⋯ ≫ (total.map f (up ℤ)).f (n + x)
Module: Mathlib.Algebra.Homology.BifunctorHomotopy

Goal state before:
pp: case neg
C₁ : Type u_1
C₂ : Type u_2
D : Type u_3
I₁ : Type u_4
I₂ : Type u_5
J : Type u_6
inst✝¹¹ : Category.{v_1, u_1} C₁
inst✝¹⁰ : Category.{v_2, u_2} C₂
inst✝⁹ : Category.{v_3, u_3} D
inst✝⁸ : Preadditive C₁
inst✝⁷ : Preadditive C₂
inst✝⁶ : Preadditive D
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
K₁ L₁ : HomologicalComplex C₁ c₁
f₁ f₁' : K₁ ⟶ L₁
h₁ : Homotopy f₁ f₁'
K₂ L₂ : HomologicalComplex C₂ c₂
f₂ : K₂ ⟶ L₂
F : C₁ ⥤ C₂ ⥤ D
inst✝⁵ : F.Additive
inst✝⁴ : ∀ (X₁ : C₁), (F.obj X₁).Additive
c : ComplexShape J
inst✝³ : DecidableEq J
inst✝² : TotalComplexShape c₁ c₂ c
inst✝¹ : K₁.HasMapBifunctor K₂ F c
inst✝ : L₁.HasMapBifunctor L₂ F c
j : J
i₁ : I₁
i₂ : I₂
h : c₁.π c₂ c (i₁, i₂) = j
this : ∀ {X Y : D} (a b c d e f : X ⟶ Y), a = c → b = e → f = -d → a + b = c + d + (e + f)
h₃ : ¬c₁.Rel (c₁.prev i₁) i₁
h₄ : ¬c₂.Rel i₂ (c₂.next i₂)
⊢ (((F.mapBifunctorHomologicalComplex c₁ c₂).obj K₁).obj K₂).d₂ c i₁ i₂ (c.next j) ≫ hom₁ h₁ f₂ F c (c.next j) j = 0

Tactic:
HomologicalComplex₂.d₂_eq_zero _ _ _ _ _ h₄,

Goal state after:
case neg
C₁ : Type u_1
C₂ : Type u_2
D : Type u_3
I₁ : Type u_4
I₂ : Type u_5
J : Type u_6
inst✝¹¹ : Category.{v_1, u_1} C₁
inst✝¹⁰ : Category.{v_2, u_2} C₂
inst✝⁹ : Category.{v_3, u_3} D
inst✝⁸ : Preadditive C₁
inst✝⁷ : Preadditive C₂
inst✝⁶ : Preadditive D
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
K₁ L₁ : HomologicalComplex C₁ c₁
f₁ f₁' : K₁ ⟶ L₁
h₁ : Homotopy f₁ f₁'
K₂ L₂ : HomologicalComplex C₂ c₂
f₂ : K₂ ⟶ L₂
F : C₁ ⥤ C₂ ⥤ D
inst✝⁵ : F.Additive
inst✝⁴ : ∀ (X₁ : C₁), (F.obj X₁).Additive
c : ComplexShape J
inst✝³ : DecidableEq J
inst✝² : TotalComplexShape c₁ c₂ c
inst✝¹ : K₁.HasMapBifunctor K₂ F c
inst✝ : L₁.HasMapBifunctor L₂ F c
j : J
i₁ : I₁
i₂ : I₂
h : c₁.π c₂ c (i₁, i₂) = j
this : ∀ {X Y : D} (a b c d e f : X ⟶ Y), a = c → b = e → f = -d → a + b = c + d + (e + f)
h₃ : ¬c₁.Rel (c₁.prev i₁) i₁
h₄ : ¬c₂.Rel i₂ (c₂.next i₂)
⊢ 0 ≫ hom₁ h₁ f₂ F c (c.next j) j = 0
Module: Mathlib.Algebra.Homology.TotalComplexShift

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : Preadditive C
K L : HomologicalComplex₂ C (up ℤ) (up ℤ)
f : K ⟶ L
y : ℤ
inst✝¹ : K.HasTotal (up ℤ)
inst✝ : L.HasTotal (up ℤ)
n i₁ i₂ : ℤ
h : i₁ + i₂ = n
⊢ (f.f i₁).f (i₂ + y) ≫ ((i₁ * y).negOnePow • L.ιTotal (up ℤ) i₁ (i₂ + y) (n + y) ⋯ ≫ 𝟙 ((L.total (up ℤ)).X (n + y))) =
    ((i₁ * y).negOnePow • K.ιTotal (up ℤ) i₁ (i₂ + y) (n + y) ⋯ ≫ 𝟙 ((K.total (up ℤ)).X (n + y))) ≫
      (total.map f (up ℤ)).f (n + y)

Tactic:
comp_id,

Goal state after:
C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : Preadditive C
K L : HomologicalComplex₂ C (up ℤ) (up ℤ)
f : K ⟶ L
y : ℤ
inst✝¹ : K.HasTotal (up ℤ)
inst✝ : L.HasTotal (up ℤ)
n i₁ i₂ : ℤ
h : i₁ + i₂ = n
⊢ (f.f i₁).f (i₂ + y) ≫ ((i₁ * y).negOnePow • L.ιTotal (up ℤ) i₁ (i₂ + y) (n + y) ⋯) =
    ((i₁ * y).negOnePow • K.ιTotal (up ℤ) i₁ (i₂ + y) (n + y) ⋯ ≫ 𝟙 ((K.total (up ℤ)).X (n + y))) ≫
      (total.map f (up ℤ)).f (n + y)
Module: Mathlib.Algebra.Homology.TotalComplexShift

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : Preadditive C
K L : HomologicalComplex₂ C (up ℤ) (up ℤ)
f : K ⟶ L
y : ℤ
inst✝¹ : K.HasTotal (up ℤ)
inst✝ : L.HasTotal (up ℤ)
n i₁ i₂ : ℤ
h : i₁ + i₂ = n
⊢ (f.f i₁).f (i₂ + y) ≫ ((i₁ * y).negOnePow • L.ιTotal (up ℤ) i₁ (i₂ + y) (n + y) ⋯) =
    ((i₁ * y).negOnePow • K.ιTotal (up ℤ) i₁ (i₂ + y) (n + y) ⋯ ≫ 𝟙 ((K.total (up ℤ)).X (n + y))) ≫
      (total.map f (up ℤ)).f (n + y)

Tactic:
comp_id,

Goal state after:
C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : Preadditive C
K L : HomologicalComplex₂ C (up ℤ) (up ℤ)
f : K ⟶ L
y : ℤ
inst✝¹ : K.HasTotal (up ℤ)
inst✝ : L.HasTotal (up ℤ)
n i₁ i₂ : ℤ
h : i₁ + i₂ = n
⊢ (f.f i₁).f (i₂ + y) ≫ ((i₁ * y).negOnePow • L.ιTotal (up ℤ) i₁ (i₂ + y) (n + y) ⋯) =
    ((i₁ * y).negOnePow • K.ιTotal (up ℤ) i₁ (i₂ + y) (n + y) ⋯) ≫ (total.map f (up ℤ)).f (n + y)
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 f g
n : ℤ
⊢ ((opEquivalence C).functor.map f.op).f n =
    (((dNext n) fun p q ↦ (h.hom (-q) (-p)).op) + (prevD n) fun p q ↦ (h.hom (-q) (-p)).op) +
      ((opEquivalence C).functor.map g.op).f n

Tactic:
by
    dsimp
    simp only [h.comm, op_add, add_left_inj]
    rw [add_comm]
    congr 1
    · rw [prevD_eq _ (j' := - (n + 1)) (by simp)]
      symm
      exact dNext_eq _ (i' := n + 1) (by simp)
    · rw [dNext_eq _ (i' := - (n - 1)) (by dsimp; lia)]
      symm
      exact prevD_eq _ (j' := n - 1) (by simp)

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

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
K : HomologicalComplex₂ C (up ℤ) (up ℤ)
x y : ℤ
inst✝ : K.HasTotal (up ℤ)
n n₁ n₂ : ℤ
h : n₁ + n₂ = n
⊢ ((shiftFunctor₂ C y).obj K).ιTotal (up ℤ) (n₁ + x) n₂ (n + x) ⋯ ≫ (K.totalShift₂Iso y).hom.f (n + x) =
    (x * y).negOnePow •
      (n₁ * y).negOnePow •
        (((shiftFunctor₁ C x).obj K).ιTotal (up ℤ) n₁ (n₂ + y) (n + y) ⋯ ≫
            𝟙 ((((shiftFunctor₁ C x).obj K).total (up ℤ)).X (n + y))) ≫
          (K.totalShift₁Iso x).hom.f (n + y) ≫
            ((shiftFunctorComm (CochainComplex C ℤ) x y).hom.app (K.total (up ℤ))).f n

Tactic:
comp_id,

Goal state after:
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
K : HomologicalComplex₂ C (up ℤ) (up ℤ)
x y : ℤ
inst✝ : K.HasTotal (up ℤ)
n n₁ n₂ : ℤ
h : n₁ + n₂ = n
⊢ ((shiftFunctor₂ C y).obj K).ιTotal (up ℤ) (n₁ + x) n₂ (n + x) ⋯ ≫ (K.totalShift₂Iso y).hom.f (n + x) =
    (x * y).negOnePow •
      (n₁ * y).negOnePow •
        ((shiftFunctor₁ C x).obj K).ιTotal (up ℤ) n₁ (n₂ + y) (n + y) ⋯ ≫
          (K.totalShift₁Iso x).hom.f (n + y) ≫
            ((shiftFunctorComm (CochainComplex C ℤ) x y).hom.app (K.total (up ℤ))).f n
Module: Mathlib.Algebra.Homology.TotalComplexShift

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
K : HomologicalComplex₂ C (up ℤ) (up ℤ)
x y : ℤ
inst✝ : K.HasTotal (up ℤ)
n n₁ n₂ : ℤ
h : n₁ + n₂ = n
⊢ (n₁ * y + x * y).negOnePow • K.ιTotal (up ℤ) (n₁ + x) (n₂ + y) (n + x + y) ⋯ ≫ 𝟙 ((K.total (up ℤ)).X (n + x + y)) =
    (n₁ * y + x * y).negOnePow •
      ((shiftFunctor₁ C x).obj K).ιTotal (up ℤ) n₁ (n₂ + y) (n + y) ⋯ ≫
        (K.totalShift₁Iso x).hom.f (n + y) ≫ ((shiftFunctorComm (CochainComplex C ℤ) x y).hom.app (K.total (up ℤ))).f n

Tactic:
comp_id,

Goal state after:
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
K : HomologicalComplex₂ C (up ℤ) (up ℤ)
x y : ℤ
inst✝ : K.HasTotal (up ℤ)
n n₁ n₂ : ℤ
h : n₁ + n₂ = n
⊢ (n₁ * y + x * y).negOnePow • K.ιTotal (up ℤ) (n₁ + x) (n₂ + y) (n + x + y) ⋯ =
    (n₁ * y + x * y).negOnePow •
      ((shiftFunctor₁ C x).obj K).ιTotal (up ℤ) n₁ (n₂ + y) (n + y) ⋯ ≫
        (K.totalShift₁Iso x).hom.f (n + y) ≫ ((shiftFunctorComm (CochainComplex C ℤ) x y).hom.app (K.total (up ℤ))).f n
Module: Mathlib.Algebra.Homology.TotalComplexShift

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
K : HomologicalComplex₂ C (up ℤ) (up ℤ)
x y : ℤ
inst✝ : K.HasTotal (up ℤ)
n n₁ n₂ : ℤ
h : n₁ + n₂ = n
⊢ (n₁ * y + x * y).negOnePow • K.ιTotal (up ℤ) (n₁ + x) (n₂ + y) (n + x + y) ⋯ =
    (n₁ * y + x * y).negOnePow •
      𝟙 ((K.X (n₁ + x)).X (n₂ + y)) ≫
        K.ιTotal (up ℤ) (n₁ + x) (n₂ + y) (n + x + y) ⋯ ≫ 𝟙 ((K.total (up ℤ)).X (n + x + y))

Tactic:
comp_id,

Goal state after:
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
K : HomologicalComplex₂ C (up ℤ) (up ℤ)
x y : ℤ
inst✝ : K.HasTotal (up ℤ)
n n₁ n₂ : ℤ
h : n₁ + n₂ = n
⊢ (n₁ * y + x * y).negOnePow • K.ιTotal (up ℤ) (n₁ + x) (n₂ + y) (n + x + y) ⋯ =
    (n₁ * y + x * y).negOnePow • 𝟙 ((K.X (n₁ + x)).X (n₂ + y)) ≫ K.ιTotal (up ℤ) (n₁ + x) (n₂ + y) (n + x + y) ⋯
Module: Mathlib.Algebra.Homology.BifunctorAssociator

Goal state before:
pp: case pos.e_a
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'
⊢ c₁₂.ε₁ c₃ c₄ (c₁.π c₂ c₁₂ (i₁, i₂), i₃) •
      (G.map (K₁.ιMapBifunctor K₂ F₁₂ c₁₂ i₁ i₂ (c₁.π c₂ c₁₂ (i₁, i₂)) ⋯)).app (K₃.X i₃) ≫
        (G.map (mapBifunctor.D₂ K₁ K₂ F₁₂ c₁₂ (c₁.π c₂ 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 [← NatTrans.comp_app_assoc, ← Functor.map_comp,
          mapBifunctor.ι_D₂]
        by_cases h₃ : c₂.Rel i₂ (c₂.next i₂)
        · have h₄ := (ComplexShape.next_π₂ c₁ c₁₂ i₁ h₃).symm
          rw [mapBifunctor.d₂_eq _ _ _ _ _ h₃ _ h₄,
            d₂_eq _ _ _ _ _ _ _ _ h₃,
            ιOrZero_eq _ _ _ _ _ _ _ _ _ _ _ (by rw [← h₂, ← h₄]; rfl),
            ι_eq _ _ _ _ _ _ _ _ _ _ (c₁₂.next i₁₂) _ h₄ h₂,
            Functor.map_units_smul, Functor.map_comp, NatTrans.app_units_zsmul,
            NatTrans.comp_app, Linear.units_smul_comp, assoc, smul_smul]
        · 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: case h₀
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
sq : CommSq f g inl inr
h : IsColimit (PushoutCocone.mk inl inr ⋯)
s : Cofork (biprod.lift f (-g)) 0
m : sq.cokernelCofork.pt ⟶ s.pt
hm : Cofork.π sq.cokernelCofork ≫ m = s.π
⊢ (PushoutCocone.mk inl inr ⋯).inl ≫ m =
    (PushoutCocone.mk inl inr ⋯).inl ≫ PushoutCocone.IsColimit.desc h (biprod.inl ≫ s.π) (biprod.inr ≫ s.π) ⋯

Tactic:
replace hm := biprod.inl ≫= hm

Goal state after:
case h₀
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
sq : CommSq f g inl inr
h : IsColimit (PushoutCocone.mk inl inr ⋯)
s : Cofork (biprod.lift f (-g)) 0
m : sq.cokernelCofork.pt ⟶ s.pt
hm : biprod.inl ≫ Cofork.π sq.cokernelCofork ≫ m = biprod.inl ≫ s.π
⊢ (PushoutCocone.mk inl inr ⋯).inl ≫ m =
    (PushoutCocone.mk inl inr ⋯).inl ≫ PushoutCocone.IsColimit.desc h (biprod.inl ≫ s.π) (biprod.inr ≫ s.π) ⋯
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
H : ℕ → Prop := fun a ↦ ∀ (n : ℤ), t.le n ≤ t.le (n + ↑a)
this : ∀ (a : ℕ), H a
n₀ : ℤ
a : ℕ
h : n₀ ≤ n₀ + ↑a
ha : n₀ + ↑a - n₀ = ↑a
⊢ t.le n₀ ≤ t.le (n₀ + ↑a)

Tactic:
apply this

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
H : ℕ → Prop := fun a ↦ ∀ (n : ℤ), t.ge (n + ↑a) ≤ t.ge n
this : ∀ (a : ℕ), H a
n₀ : ℤ
a : ℕ
h : n₀ ≤ n₀ + ↑a
ha : n₀ + ↑a - n₀ = ↑a
⊢ t.ge (n₀ + ↑a) ≤ t.ge n₀

Tactic:
apply this

Goal state after:
Module: Mathlib.CategoryTheory.MorphismProperty.RetractArgument

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
W₁ W₂ : MorphismProperty C
inst✝¹ : W₁.HasFactorization W₂
inst✝ : W₂.IsStableUnderRetracts
h₂ : W₂ ≤ W₁.rlp
X Y : C
p : X ⟶ Y
hp : W₁.rlp p
h : W₁.MapFactorizationData W₂ p
this : HasLiftingProperty h.i p
⊢ W₂ p

Tactic:
simpa using of_retract (RetractArrow.ofRightLiftingProperty h.fac) h.hp

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

Goal state before:
pp: C : Type u
inst✝⁷ : Category.{v, u} C
inst✝⁶ : CategoryWithWeakEquivalences C
inst✝⁵ : CategoryWithCofibrations C
inst✝⁴ : CategoryWithFibrations C
X Y Z : C
f : X ⟶ Y
g : X ⟶ Z
inst✝³ : HasPushout f g
inst✝² : (trivialCofibrations C).IsStableUnderCobaseChange
inst✝¹ : Cofibration f
inst✝ : WeakEquivalence f
⊢ WeakEquivalence (pushout.inr f g)

Tactic:
rw [weakEquivalence_iff]
  exact (MorphismProperty.of_isPushout (IsPushout.of_hasPushout f g).flip
    (mem_trivialCofibrations f)).2

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

Goal state before:
pp: C : Type u
inst✝⁷ : Category.{v, u} C
inst✝⁶ : CategoryWithWeakEquivalences C
inst✝⁵ : CategoryWithCofibrations C
inst✝⁴ : CategoryWithFibrations C
J✝ : Type w
J : Type u_1
X Y : J → C
f : (i : J) → X i ⟶ Y i
inst✝³ : HasProduct X
inst✝² : HasProduct Y
h : ∀ (i : J), Fibration (f i)
inst✝¹ : (cofibrations C).IsWeakFactorizationSystem (trivialFibrations C)
inst✝ : ∀ (i : J), WeakEquivalence (f i)
⊢ WeakEquivalence (Limits.Pi.map f)

Tactic:
by
  rw [weakEquivalence_iff]
  exact (MorphismProperty.limMap (W := (trivialFibrations C)) _
    (fun ⟨i⟩ ↦ mem_trivialFibrations (f i))).2

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

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
X✝ : C
P✝ : Precylinder X✝
Y✝ : C
P : ObjectProperty C
X Y : P.FullSubcategory
Q : Precylinder X
f g : X ⟶ Y
h : (Q.map P.ι).LeftHomotopy f.hom g.hom
⊢ Q.i₀ ≫ ObjectProperty.homMk h.h = f

Tactic:
by ext; exact h.h₀

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

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
Y✝ : C
P✝ : PrepathObject Y✝
X✝ : C
P : ObjectProperty C
X Y : P.FullSubcategory
Q : PrepathObject Y
f g : X ⟶ Y
h : (Q.map P.ι).RightHomotopy f.hom g.hom
⊢ ObjectProperty.homMk h.h ≫ Q.p₀ = f

Tactic:
by ext; exact h.h₀

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 n₁ : ℤ
h : n₀ + 1 = n₁
X : C
hX : ∀ (Y : C) (f : X ⟶ Y), t.IsGE Y n₁ → f = 0
⊢ t.IsLE X n₀

Tactic:
t.isLE_iff_isZero_truncGE_obj n₀ n₁ h,

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 n₁ : ℤ
h : n₀ + 1 = n₁
X : C
hX : ∀ (Y : C) (f : X ⟶ Y), t.IsGE Y n₁ → f = 0
⊢ IsZero ((t.truncGE n₁).obj X)
Module: Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} 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
t : TStructure C
inst✝ : IsTriangulated C
a b : ℤ
h : a ≤ b
X : C
this : IsIso ((t.truncLT a).map ((t.truncLTι b).app X))
⊢ (t.natTransTruncLTOfLE a b h).app X ≫ 𝟙 ((t.truncLT b).obj X) =
    inv ((t.truncLT a).map ((t.truncLTι b).app X)) ≫ (t.truncLTι a).app ((t.truncLT b).obj X)

Tactic:
simp only [Category.comp_id, IsIso.eq_inv_comp]

Goal state after:
C : Type u
inst✝⁶ : Category.{v, u} 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
t : TStructure C
inst✝ : IsTriangulated C
a b : ℤ
h : a ≤ b
X : C
this : IsIso ((t.truncLT a).map ((t.truncLTι b).app X))
⊢ (t.truncLT a).map ((t.truncLTι b).app X) ≫ (t.natTransTruncLTOfLE a b h).app X =
    (t.truncLTι a).app ((t.truncLT b).obj X)
Module: Mathlib.AlgebraicTopology.ModelCategory.Homotopy

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : ModelCategory C
X Y : C
inst✝⁴ : IsCofibrant X
inst✝³ : IsCofibrant Y
inst✝² : IsFibrant X
inst✝¹ : IsFibrant Y
f : X ⟶ Y
inst✝ : WeakEquivalence f
g : Y ⟶ X
hg : mk (f ≫ g) = mk (𝟙 X)
⊢ ∃ g, RightHomotopyRel (f ≫ g) (𝟙 X) ∧ RightHomotopyRel (g ≫ f) (𝟙 Y)

Tactic:
refine ⟨g, by rwa [← mk_eq_mk_iff], ?_⟩

Goal state after:
C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : ModelCategory C
X Y : C
inst✝⁴ : IsCofibrant X
inst✝³ : IsCofibrant Y
inst✝² : IsFibrant X
inst✝¹ : IsFibrant Y
f : X ⟶ Y
inst✝ : WeakEquivalence f
g : Y ⟶ X
hg : mk (f ≫ g) = mk (𝟙 X)
⊢ RightHomotopyRel (g ≫ f) (𝟙 Y)