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

Goal state before:
pp: ⊢ Skeleton.incl.IsEquivalence

Tactic:
by infer_instance

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

Goal state before:
pp: case right
C : Type u_1
inst✝ : Category.{v_1, u_1} C
a✝ : IsIdempotentComplete C
X : C
p : X ⟶ X
hp : p ≫ p = p
Y : C
i : Y ⟶ X
e : X ⟶ Y
h₁ : i ≫ e = 𝟙 Y
h₂ : e ≫ i = p
s : Fork (𝟙 X) p
⊢ ∀ {m : s.pt ⟶ (Fork.ofι i ⋯).pt}, m ≫ (Fork.ofι i ⋯).ι = s.ι → m = s.ι ≫ e

Tactic:
intro m hm
                rw [Fork.ι_ofι] at hm
                rw [← hm]
                simp only [assoc, h₁]
                exact (comp_id m).symm

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

Goal state before:
pp: case right
C : Type u_1
inst✝ : Category.{v_1, u_1} C
a✝ : IsIdempotentComplete C
X : C
p : X ⟶ X
hp : p ≫ p = p
Y : C
i : Y ⟶ X
e : X ⟶ Y
h₁ : i ≫ e = 𝟙 Y
h₂ : e ≫ i = p
s : Fork (𝟙 X) p
⊢ ∀ {m : s.pt ⟶ (Fork.ofι i ⋯).pt}, m ≫ (Fork.ofι i ⋯).ι = s.ι → m = s.ι ≫ e

Tactic:
· intro m hm
                rw [Fork.ι_ofι] at hm
                rw [← hm]
                simp only [assoc, h₁]
                exact (comp_id m).symm

Goal state after:
Module: Mathlib.CategoryTheory.Idempotents.Karoubi

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
P Q : Karoubi C
⊢ AddCommGroup (P ⟶ Q)

Tactic:
by infer_instance

Goal state after:
Module: Mathlib.AlgebraicTopology.AlternatingFaceMapComplex

Goal state before:
pp: case i_inj
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
n : ℕ
P : Type := Fin (n + 2) × Fin (n + 3)
S : Finset P := {ij | ↑ij.2 ≤ ↑ij.1}
φ : (ij : P) → ij ∈ S → P := fun ij hij ↦ (ij.2.castLT ⋯, ij.1.succ)
i : Fin (n + 2)
j : Fin (n + 3)
hij : (i, j) ∈ S
i' : Fin (n + 2)
j' : Fin (n + 3)
hij' : (i', j') ∈ S
h : φ (i, j) hij = φ (i', j') hij'
⊢ (i, j) = (i', j')

Tactic:
rw [Prod.mk_inj]

Goal state after:
case i_inj
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
n : ℕ
P : Type := Fin (n + 2) × Fin (n + 3)
S : Finset P := {ij | ↑ij.2 ≤ ↑ij.1}
φ : (ij : P) → ij ∈ S → P := fun ij hij ↦ (ij.2.castLT ⋯, ij.1.succ)
i : Fin (n + 2)
j : Fin (n + 3)
hij : (i, j) ∈ S
i' : Fin (n + 2)
j' : Fin (n + 3)
hij' : (i', j') ∈ S
h : φ (i, j) hij = φ (i', j') hij'
⊢ i = i' ∧ j = j'
Module: Mathlib.AlgebraicTopology.AlternatingFaceMapComplex

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
n : ℕ
⊢ ((alternatingFaceMapComplex C).obj X).d (n + 1) n = AlternatingFaceMapComplex.objD X n

Tactic:
by
  simp [alternatingFaceMapComplex]

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

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
X : SimplicialObject C
n : ℕ
i j : Fin (n + 1)
H : i ≤ j
⊢ X.σ j ≫ X.σ i.castSucc = X.σ i ≫ X.σ j.succ

Tactic:
dsimp [δ, σ]
  simp only [← X.map_comp, ← op_comp, SimplexCategory.σ_comp_σ H]

Goal state after:
Module: Mathlib.AlgebraicTopology.AlternatingFaceMapComplex

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : Preadditive C
A : Type u_2
inst✝¹ : Category.{v_2, u_2} A
inst✝ : Abelian A
X Y : CosimplicialObject C
f : X ⟶ Y
n : ℕ
⊢ f.app ⦋n⦌ ≫ (obj Y).d n (n + 1) = (obj X).d n (n + 1) ≫ f.app ⦋n + 1⦌

Tactic:
by
    simp only [obj, CochainComplex.of_d, objD, Int.reduceNeg]
    rw [comp_sum, sum_comp]
    refine Finset.sum_congr rfl fun x _ => ?_
    rw [comp_zsmul, zsmul_comp]
    congr 1
    symm
    apply f.naturality

Goal state after:
Module: Mathlib.AlgebraicTopology.SimplexCategory.DeltaZeroIter

Goal state before:
pp: i n m : ℕ
h : n + (i + 1) = m
k : Fin (⦋m⦌.len + 1)
hk : i < ↑k
⊢ ↑(Fin.castSucc 0) < ↑((ConcreteCategory.hom (σ₀Iter i ⋯)) k)

Tactic:
Fin.castSucc_zero,

Goal state after:
i n m : ℕ
h : n + (i + 1) = m
k : Fin (⦋m⦌.len + 1)
hk : i < ↑k
⊢ ↑0 < ↑((ConcreteCategory.hom (σ₀Iter i ⋯)) k)
Module: Mathlib.AlgebraicTopology.SimplexCategory.Basic

Goal state before:
pp: Δ' Δ : SimplexCategory
i : Δ' ⟶ Δ
inst✝ : Mono i
hi' : Δ ≠ Δ'
⊢ Δ'.len < Δ.len

Tactic:
grind [→ len_le_of_mono, SimplexCategory.ext]

Goal state after:
Module: Mathlib.AlgebraicTopology.SimplicialSet.NonDegenerateSimplices

Goal state before:
pp: case refine_1.mk
X : SSet
A B : X.Subcomplex
h : ∀ (n : ℕ) (x : ↑(X.nonDegenerate n)), ↑x ∈ A.obj (Opposite.op ⦋n⦌) → ↑x ∈ B.obj (Opposite.op ⦋n⦌)
n : ℕ
x : ↑(X.nonDegenerate n)
⊢ (mk ↑x ⋯).subcomplex ≤ A → (mk ↑x ⋯).subcomplex ≤ B

Tactic:
intro hx
      simp only [Subfunctor.ofSection_le_iff, mk_simplex] at hx ⊢
      exact h _ _ hx

Goal state after:
Module: Mathlib.AlgebraicTopology.SimplicialSet.Finite

Goal state before:
pp: X : SSet
inst✝ : X.Finite
n : ℕ
φ : (m : Fin (n + 1)) × (_ : ⦋n⦌ ⟶ ⦋↑m⦌) × ↑(X.nonDegenerate ↑m) → X _⦋n⦌ :=
  fun x ↦
    match x with
    | ⟨m, ⟨f, x⟩⟩ => (ConcreteCategory.hom (X.map f.op)) ↑x
x : X _⦋n⦌
⊢ ∃ a, φ a = x

Tactic:
by
    obtain ⟨m, f, hf, y, rfl⟩ := X.exists_nonDegenerate x
    have := SimplexCategory.le_of_epi f
    exact ⟨⟨⟨m, by lia⟩, f, y⟩, rfl⟩

Goal state after:
Module: Mathlib.AlgebraicTopology.SimplicialSet.CompStructTruncated

Goal state before:
pp: X Y : Truncated 2
x y : X.obj (Opposite.op { obj := ⦋0⦌, property := Edge._proof_1 })
e : Edge x y
⊢ (ConcreteCategory.hom (X.map (δ₂ 0 Edge._proof_2 _proof_2).op))
      ((ConcreteCategory.hom (X.map (σ₂ 0 _proof_2 Edge._proof_2).op)) e.edge) =
    e.edge

Tactic:
by
    rw [← Functor.map_comp_apply, ← op_comp, δ₂_zero_comp_σ₂_zero]
    simp

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

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
I₁ : Type u_2
I₂ : Type u_3
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
K : HomologicalComplex₂ C c₁ c₂
i₁ i₁' : I₁
h : ¬c₁.Rel i₁ i₁'
i₂ : I₂
⊢ (K.d i₁ i₁').f i₂ = 0

Tactic:
by
  rw [K.shape _ _ h, zero_f]

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

Goal state before:
pp: C : Type u_1
I₁ : Type u_2
I₂ : Type u_3
J : Type u_4
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : Preadditive C
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
K : HomologicalComplex₂ C c₁ c₂
c : ComplexShape J
inst✝⁴ : TotalComplexShape c₁ c₂ c
inst✝³ : TotalComplexShape c₂ c₁ c
inst✝² : TotalComplexShapeSymmetry c₁ c₂ c
inst✝¹ : K.HasTotal c
inst✝ : DecidableEq J
j j' : J
x✝ : c.Rel j j'
⊢ (K.totalFlipIsoX c j).hom ≫ (K.total c).d j j' = (K.flip.total c).d j j' ≫ (K.totalFlipIsoX c j').hom

Tactic:
simp only [total_d, Preadditive.comp_add, totalFlipIsoX_hom_D₁,
      totalFlipIsoX_hom_D₂, Preadditive.add_comp]

Goal state after:
C : Type u_1
I₁ : Type u_2
I₂ : Type u_3
J : Type u_4
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : Preadditive C
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
K : HomologicalComplex₂ C c₁ c₂
c : ComplexShape J
inst✝⁴ : TotalComplexShape c₁ c₂ c
inst✝³ : TotalComplexShape c₂ c₁ c
inst✝² : TotalComplexShapeSymmetry c₁ c₂ c
inst✝¹ : K.HasTotal c
inst✝ : DecidableEq J
j j' : J
x✝ : c.Rel j j'
⊢ K.flip.D₂ c j j' ≫ (K.totalFlipIsoX c j').hom + K.flip.D₁ c j j' ≫ (K.totalFlipIsoX c j').hom =
    K.flip.D₁ c j j' ≫ (K.totalFlipIsoX c j').hom + K.flip.D₂ c j j' ≫ (K.totalFlipIsoX c j').hom
Module: Mathlib.Algebra.Homology.BifunctorHomotopy

Goal state before:
pp: 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' : J
h : ¬c.Rel j' j
⊢ hom₁ h₁ f₂ F c j j' = 0

Tactic:
by
  ext i₁ i₂ h'
  dsimp [hom₁]
  rw [comp_zero, HomologicalComplex₂.ι_totalDesc]
  by_cases h₃ : c₁.Rel (c₁.prev i₁) i₁
  · rw [ιMapBifunctorOrZero_eq_zero, comp_zero, comp_zero, smul_zero]
    intro h₄
    apply h
    rw [← h', ← h₄]
    exact ComplexShape.rel_π₁ c₂ c h₃ i₂
  · rw [h₁.zero _ _ h₃, Functor.map_zero, zero_app, zero_comp, smul_zero]

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 L : HomologicalComplex₂ C (up ℤ) (up ℤ)
f : K ⟶ L
x y : ℤ
inst✝ : K.HasTotal (up ℤ)
n n' : ℤ
h : n + x = n'
⊢ ((((shiftFunctor₁ C x).obj K).totalDesc fun p q hpq ↦ K.ιTotal (up ℤ) (p + x) q n' ⋯) ≫
      K.totalDesc fun p q hpq ↦
        (XXIsoOfEq C (up ℤ) (up ℤ) K ⋯ ⋯).inv ≫ ((shiftFunctor₁ C x).obj K).ιTotal (up ℤ) (p - x) q n ⋯) =
    𝟙 ((((shiftFunctor₁ C x).obj K).total (up ℤ)).X n)

Tactic:
ext p q h
    dsimp
    simp only [ι_totalDesc_assoc, CochainComplex.shiftFunctor_obj_X', ι_totalDesc, comp_id]
    exact ((shiftFunctor₁ C x).obj K).XXIsoOfEq_inv_ιTotal _ (by lia) rfl _ _

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

Goal state before:
pp: case pos
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
K : HomologicalComplex₂ C (up ℤ) (up ℤ)
x : ℤ
inst✝ : K.HasTotal (up ℤ)
n₀ n₁ n₀' n₁' : ℤ
h₀ : n₀ + x = n₀'
h₁ : n₁ + x = n₁'
h : (up ℤ).Rel n₀ n₁
p : ℤ
⊢ ∀ (i₂ : ℤ) (hi : (up ℤ).π (up ℤ) (up ℤ) (p, i₂) = n₀),
    ((shiftFunctor₁ C x).obj K).ιTotal (up ℤ) p i₂ n₀ hi ≫
        ((shiftFunctor₁ C x).obj K).D₁ (up ℤ) n₀ n₁ ≫ (K.totalShift₁XIso x n₁ n₁' h₁).hom =
      ((shiftFunctor₁ C x).obj K).ιTotal (up ℤ) p i₂ n₀ hi ≫
        (x.negOnePow • (K.totalShift₁XIso x n₀ n₀' h₀).hom ≫ K.D₁ (up ℤ) n₀' n₁')

Tactic:
q

Goal state after:
case pos
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
K : HomologicalComplex₂ C (up ℤ) (up ℤ)
x : ℤ
inst✝ : K.HasTotal (up ℤ)
n₀ n₁ n₀' n₁' : ℤ
h₀ : n₀ + x = n₀'
h₁ : n₁ + x = n₁'
h : (up ℤ).Rel n₀ n₁
p q : ℤ
⊢ ∀ (hi : (up ℤ).π (up ℤ) (up ℤ) (p, q) = n₀),
    ((shiftFunctor₁ C x).obj K).ιTotal (up ℤ) p q n₀ hi ≫
        ((shiftFunctor₁ C x).obj K).D₁ (up ℤ) n₀ n₁ ≫ (K.totalShift₁XIso x n₁ n₁' h₁).hom =
      ((shiftFunctor₁ C x).obj K).ιTotal (up ℤ) p q n₀ hi ≫
        (x.negOnePow • (K.totalShift₁XIso x n₀ n₀' h₀).hom ≫ K.D₁ (up ℤ) n₀' n₁')
Module: Mathlib.Algebra.Homology.TotalComplexShift

Goal state before:
pp: case pos
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
K : HomologicalComplex₂ C (up ℤ) (up ℤ)
x : ℤ
inst✝ : K.HasTotal (up ℤ)
n₀ n₁ n₀' n₁' : ℤ
h₀ : n₀ + x = n₀'
h₁ : n₁ + x = n₁'
h : (up ℤ).Rel n₀ n₁
p : ℤ
⊢ ∀ (i₂ : ℤ) (hi : (up ℤ).π (up ℤ) (up ℤ) (p, i₂) = n₀),
    ((shiftFunctor₁ C x).obj K).ιTotal (up ℤ) p i₂ n₀ hi ≫
        ((shiftFunctor₁ C x).obj K).D₂ (up ℤ) n₀ n₁ ≫ (K.totalShift₁XIso x n₁ n₁' h₁).hom =
      ((shiftFunctor₁ C x).obj K).ιTotal (up ℤ) p i₂ n₀ hi ≫
        (x.negOnePow • (K.totalShift₁XIso x n₀ n₀' h₀).hom ≫ K.D₂ (up ℤ) n₀' n₁')

Tactic:
q

Goal state after:
case pos
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
K : HomologicalComplex₂ C (up ℤ) (up ℤ)
x : ℤ
inst✝ : K.HasTotal (up ℤ)
n₀ n₁ n₀' n₁' : ℤ
h₀ : n₀ + x = n₀'
h₁ : n₁ + x = n₁'
h : (up ℤ).Rel n₀ n₁
p q : ℤ
⊢ ∀ (hi : (up ℤ).π (up ℤ) (up ℤ) (p, q) = n₀),
    ((shiftFunctor₁ C x).obj K).ιTotal (up ℤ) p q n₀ hi ≫
        ((shiftFunctor₁ C x).obj K).D₂ (up ℤ) n₀ n₁ ≫ (K.totalShift₁XIso x n₁ n₁' h₁).hom =
      ((shiftFunctor₁ C x).obj K).ιTotal (up ℤ) p q n₀ hi ≫
        (x.negOnePow • (K.totalShift₁XIso x n₀ n₀' h₀).hom ≫ K.D₂ (up ℤ) n₀' n₁')
Module: Mathlib.Algebra.Homology.BifunctorHomotopy

Goal state before:
pp: 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 f₂' : K₂ ⟶ L₂
h₂ : Homotopy f₂ f₂'
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
this✝² : TotalComplexShape c₂ c₁ c := TotalComplexShape.symm c₁ c₂ c
this✝¹ : TotalComplexShapeSymmetry c₁ c₂ c := TotalComplexShape.symmSymmetry c₁ c₂ c
this✝ : F.flip.Additive
this : ∀ (X₁ : C₂), (F.flip.obj X₁).Additive
H : Homotopy (mapBifunctorMap f₁ f₂ F c) (mapBifunctorMap f₁ f₂' F c) :=
  (Homotopy.ofEq ⋯).trans
    ((((mapBifunctorMapHomotopy₁ h₂ f₁ F.flip c).compRight (L₁.mapBifunctorFlipIso L₂ F c).hom).compLeft
          (K₁.mapBifunctorFlipIso K₂ F c).inv).trans
      (Homotopy.ofEq ⋯))
j j' : J
i₁ : I₁
i₂ : I₂
hj : c₁.π c₂ c (i₁, i₂) = j
⊢ K₁.ιMapBifunctor K₂ F c i₁ i₂ j hj ≫ hom₂ f₁ h₂ F c j j' =
    K₁.ιMapBifunctor K₂ F c i₁ i₂ j hj ≫
      (0 +
        (K₁.mapBifunctorFlipIso K₂ F c).inv.f j ≫ hom₁ h₂ f₁ F.flip c j j' ≫ (L₁.mapBifunctorFlipIso L₂ F c).hom.f j')

Tactic:
zero_add,

Goal state after:
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 f₂' : K₂ ⟶ L₂
h₂ : Homotopy f₂ f₂'
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
this✝² : TotalComplexShape c₂ c₁ c := TotalComplexShape.symm c₁ c₂ c
this✝¹ : TotalComplexShapeSymmetry c₁ c₂ c := TotalComplexShape.symmSymmetry c₁ c₂ c
this✝ : F.flip.Additive
this : ∀ (X₁ : C₂), (F.flip.obj X₁).Additive
H : Homotopy (mapBifunctorMap f₁ f₂ F c) (mapBifunctorMap f₁ f₂' F c) :=
  (Homotopy.ofEq ⋯).trans
    ((((mapBifunctorMapHomotopy₁ h₂ f₁ F.flip c).compRight (L₁.mapBifunctorFlipIso L₂ F c).hom).compLeft
          (K₁.mapBifunctorFlipIso K₂ F c).inv).trans
      (Homotopy.ofEq ⋯))
j j' : J
i₁ : I₁
i₂ : I₂
hj : c₁.π c₂ c (i₁, i₂) = j
⊢ K₁.ιMapBifunctor K₂ F c i₁ i₂ j hj ≫ hom₂ f₁ h₂ F c j j' =
    K₁.ιMapBifunctor K₂ F c i₁ i₂ j hj ≫
      (K₁.mapBifunctorFlipIso K₂ F c).inv.f j ≫ hom₁ h₂ f₁ F.flip c j j' ≫ (L₁.mapBifunctorFlipIso L₂ F c).hom.f j'
Module: Mathlib.Algebra.Homology.TotalComplexShift

Goal state before:
pp: case pos
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
K : HomologicalComplex₂ C (up ℤ) (up ℤ)
y : ℤ
inst✝ : K.HasTotal (up ℤ)
n₀ n₁ n₀' n₁' : ℤ
h₀ : n₀ + y = n₀'
h₁ : n₁ + y = n₁'
h : (up ℤ).Rel n₀ n₁
p : ℤ
⊢ ∀ (i₂ : ℤ) (hi : (up ℤ).π (up ℤ) (up ℤ) (p, i₂) = n₀),
    ((shiftFunctor₂ C y).obj K).ιTotal (up ℤ) p i₂ n₀ hi ≫
        ((shiftFunctor₂ C y).obj K).D₁ (up ℤ) n₀ n₁ ≫ (K.totalShift₂XIso y n₁ n₁' h₁).hom =
      ((shiftFunctor₂ C y).obj K).ιTotal (up ℤ) p i₂ n₀ hi ≫
        (y.negOnePow • (K.totalShift₂XIso y n₀ n₀' h₀).hom ≫ K.D₁ (up ℤ) n₀' n₁')

Tactic:
q

Goal state after:
case pos
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
K : HomologicalComplex₂ C (up ℤ) (up ℤ)
y : ℤ
inst✝ : K.HasTotal (up ℤ)
n₀ n₁ n₀' n₁' : ℤ
h₀ : n₀ + y = n₀'
h₁ : n₁ + y = n₁'
h : (up ℤ).Rel n₀ n₁
p q : ℤ
⊢ ∀ (hi : (up ℤ).π (up ℤ) (up ℤ) (p, q) = n₀),
    ((shiftFunctor₂ C y).obj K).ιTotal (up ℤ) p q n₀ hi ≫
        ((shiftFunctor₂ C y).obj K).D₁ (up ℤ) n₀ n₁ ≫ (K.totalShift₂XIso y n₁ n₁' h₁).hom =
      ((shiftFunctor₂ C y).obj K).ιTotal (up ℤ) p q n₀ hi ≫
        (y.negOnePow • (K.totalShift₂XIso y n₀ n₀' h₀).hom ≫ K.D₁ (up ℤ) n₀' n₁')
Module: Mathlib.Algebra.Homology.TotalComplexShift

Goal state before:
pp: case pos
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
K : HomologicalComplex₂ C (up ℤ) (up ℤ)
y : ℤ
inst✝ : K.HasTotal (up ℤ)
n₀ n₁ n₀' n₁' : ℤ
h₀ : n₀ + y = n₀'
h₁ : n₁ + y = n₁'
h : n₀ + 1 = n₁
p q : ℤ
hpq : p + q = n₀
⊢ ((shiftFunctor₂ C y).obj K).ιTotal (up ℤ) p q n₀ hpq ≫
      ((shiftFunctor₂ C y).obj K).D₁ (up ℤ) n₀ n₁ ≫ (K.totalShift₂XIso y n₁ n₁' h₁).hom =
    ((shiftFunctor₂ C y).obj K).ιTotal (up ℤ) p q n₀ hpq ≫
      (y.negOnePow • (K.totalShift₂XIso y n₀ n₀' h₀).hom ≫ K.D₁ (up ℤ) n₀' n₁')

Tactic:
dsimp [totalShift₂XIso]

Goal state after:
case pos
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
K : HomologicalComplex₂ C (up ℤ) (up ℤ)
y : ℤ
inst✝ : K.HasTotal (up ℤ)
n₀ n₁ n₀' n₁' : ℤ
h₀ : n₀ + y = n₀'
h₁ : n₁ + y = n₁'
h : n₀ + 1 = n₁
p q : ℤ
hpq : p + q = n₀
⊢ (((shiftFunctor₂ C y).obj K).ιTotal (up ℤ) p q n₀ hpq ≫
      ((shiftFunctor₂ C y).obj K).D₁ (up ℤ) n₀ n₁ ≫
        ((shiftFunctor₂ C y).obj K).totalDesc fun p q hpq ↦ (p * y).negOnePow • K.ιTotal (up ℤ) p (q + y) n₁' ⋯) =
    ((shiftFunctor₂ C y).obj K).ιTotal (up ℤ) p q n₀ hpq ≫
      (y.negOnePow •
        (((shiftFunctor₂ C y).obj K).totalDesc fun p q hpq ↦ (p * y).negOnePow • K.ιTotal (up ℤ) p (q + y) n₀' ⋯) ≫
          K.D₁ (up ℤ) n₀' n₁')
Module: Mathlib.Algebra.Homology.TotalComplexShift

Goal state before:
pp: case pos
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
K : HomologicalComplex₂ C (up ℤ) (up ℤ)
y : ℤ
inst✝ : K.HasTotal (up ℤ)
n₀ n₁ n₀' n₁' : ℤ
h₀ : n₀ + y = n₀'
h₁ : n₁ + y = n₁'
h : (up ℤ).Rel n₀ n₁
p : ℤ
⊢ ∀ (i₂ : ℤ) (hi : (up ℤ).π (up ℤ) (up ℤ) (p, i₂) = n₀),
    ((shiftFunctor₂ C y).obj K).ιTotal (up ℤ) p i₂ n₀ hi ≫
        ((shiftFunctor₂ C y).obj K).D₂ (up ℤ) n₀ n₁ ≫ (K.totalShift₂XIso y n₁ n₁' h₁).hom =
      ((shiftFunctor₂ C y).obj K).ιTotal (up ℤ) p i₂ n₀ hi ≫
        (y.negOnePow • (K.totalShift₂XIso y n₀ n₀' h₀).hom ≫ K.D₂ (up ℤ) n₀' n₁')

Tactic:
q

Goal state after:
case pos
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
K : HomologicalComplex₂ C (up ℤ) (up ℤ)
y : ℤ
inst✝ : K.HasTotal (up ℤ)
n₀ n₁ n₀' n₁' : ℤ
h₀ : n₀ + y = n₀'
h₁ : n₁ + y = n₁'
h : (up ℤ).Rel n₀ n₁
p q : ℤ
⊢ ∀ (hi : (up ℤ).π (up ℤ) (up ℤ) (p, q) = n₀),
    ((shiftFunctor₂ C y).obj K).ιTotal (up ℤ) p q n₀ hi ≫
        ((shiftFunctor₂ C y).obj K).D₂ (up ℤ) n₀ n₁ ≫ (K.totalShift₂XIso y n₁ n₁' h₁).hom =
      ((shiftFunctor₂ C y).obj K).ιTotal (up ℤ) p q n₀ hi ≫
        (y.negOnePow • (K.totalShift₂XIso y n₀ n₀' h₀).hom ≫ K.D₂ (up ℤ) n₀' n₁')
Module: Mathlib.Algebra.Homology.TotalComplexShift

Goal state before:
pp: case pos
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
K : HomologicalComplex₂ C (up ℤ) (up ℤ)
y : ℤ
inst✝ : K.HasTotal (up ℤ)
n₀ n₁ n₀' n₁' : ℤ
h₀ : n₀ + y = n₀'
h₁ : n₁ + y = n₁'
h : n₀ + 1 = n₁
p q : ℤ
hpq : p + q = n₀
⊢ ((shiftFunctor₂ C y).obj K).ιTotal (up ℤ) p q n₀ hpq ≫
      ((shiftFunctor₂ C y).obj K).D₂ (up ℤ) n₀ n₁ ≫ (K.totalShift₂XIso y n₁ n₁' h₁).hom =
    ((shiftFunctor₂ C y).obj K).ιTotal (up ℤ) p q n₀ hpq ≫
      (y.negOnePow • (K.totalShift₂XIso y n₀ n₀' h₀).hom ≫ K.D₂ (up ℤ) n₀' n₁')

Tactic:
dsimp [totalShift₂XIso]

Goal state after:
case pos
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
K : HomologicalComplex₂ C (up ℤ) (up ℤ)
y : ℤ
inst✝ : K.HasTotal (up ℤ)
n₀ n₁ n₀' n₁' : ℤ
h₀ : n₀ + y = n₀'
h₁ : n₁ + y = n₁'
h : n₀ + 1 = n₁
p q : ℤ
hpq : p + q = n₀
⊢ (((shiftFunctor₂ C y).obj K).ιTotal (up ℤ) p q n₀ hpq ≫
      ((shiftFunctor₂ C y).obj K).D₂ (up ℤ) n₀ n₁ ≫
        ((shiftFunctor₂ C y).obj K).totalDesc fun p q hpq ↦ (p * y).negOnePow • K.ιTotal (up ℤ) p (q + y) n₁' ⋯) =
    ((shiftFunctor₂ C y).obj K).ιTotal (up ℤ) p q n₀ hpq ≫
      (y.negOnePow •
        (((shiftFunctor₂ C y).obj K).totalDesc fun p q hpq ↦ (p * y).negOnePow • K.ιTotal (up ℤ) p (q + y) n₀' ⋯) ≫
          K.D₂ (up ℤ) n₀' n₁')
Module: Mathlib.Algebra.Homology.BifunctorAssociator

Goal state before:
pp: 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 : c₁.Rel i₁ (c₁.next i₁)
h₄ : c₁.π c₂ c₁₂ (c₁.next i₁, i₂) = c₁₂.next (c₁.π c₂ c₁₂ (i₁, i₂))
⊢ c₁.r c₂ c₃ c₁₂ c₄ (c₁.next i₁, i₂, i₃) = j'

Tactic:
rw [← h₂, ← h₄]

Goal state after:
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 : c₁.Rel i₁ (c₁.next i₁)
h₄ : c₁.π c₂ c₁₂ (c₁.next i₁, i₂) = c₁₂.next (c₁.π c₂ c₁₂ (i₁, i₂))
⊢ c₁.r c₂ c₃ c₁₂ c₄ (c₁.next i₁, i₂, i₃) = c₁₂.π c₃ c₄ (c₁.π c₂ c₁₂ (c₁.next i₁, i₂), i₃)