mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.AlgebraicTopology.SimplicialSet.NerveNondegenerate

Goal state before:
pp: case neg
X : Type u_1
inst✝ : PartialOrder X
n : ℕ
s : nerve X _⦋n + 1⦌
i : Fin (n + 1)
h : s.obj i.castSucc = s.obj i.succ
j : Fin ((Opposite.unop (Opposite.op ⦋n + 1⦌)).len + 1)
h₁ : ¬i.castSucc < j
⊢ s.obj (i.castSucc.succAbove (i.predAbove j)) = s.obj j

Tactic:
· simp only [not_lt] at h₁
      grind [→ Fin.succAbove_of_castSucc_lt,
        → Fin.predAbove_of_le_castSucc, Fin.castSucc_castPred, Fin.castPred_castSucc,
        Fin.succAbove_castSucc_self, → LE.le.lt_or_eq]

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

Goal state before:
pp: X : Truncated 2
s : X.obj (Opposite.op { obj := ⦋2⦌, property := _proof_1 })
⊢ ∃ x₀ x₁ x₂ e₀₁ e₁₂ e₀₂ h, h.simplex = s

Tactic:
refine ⟨X.map (Hom.tr (SimplexCategory.const _ _ 0)).op s,
    X.map (Hom.tr (SimplexCategory.const _ _ 1)).op s,
    X.map (Hom.tr (SimplexCategory.const _ _ 2)).op s,
    .mk _ ?_ ?_, .mk _ ?_ ?_, .mk _ ?_ ?_, .mk s rfl rfl rfl, rfl⟩
  all_goals
  · rw [← Functor.map_comp_apply, ← op_comp]
    apply congr_fun; congr
    decide

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

Goal state before:
pp: case succ
X : Type u_1
inst✝ : PartialOrder X
n : ℕ
s : nerve X _⦋n + 1⦌
⊢ s ∈ (nerve X).nonDegenerate (n + 1) ↔ StrictMono s.obj

Tactic:
rw [← not_iff_not, ← SSet.mem_degenerate_iff_notMem_nonDegenerate,
      Fin.strictMono_iff_lt_succ, SSet.degenerate_eq_iUnion_range_σ, Set.mem_iUnion]
    simp only [mem_range_nerve_σ_iff, not_forall]
    apply exists_congr
    intro i
    have := s.monotone i.castSucc_le_succ
    grind [lt_self_iff_false, LE.le.lt_or_eq]

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 2 Edge._proof_2 _proof_2).op))
      ((ConcreteCategory.hom (X.map (σ₂ 1 _proof_2 Edge._proof_2).op)) e.edge) =
    e.edge

Tactic:
rw [← Functor.map_comp_apply, ← op_comp, δ₂_two_comp_σ₂_one]

Goal state after:
X Y : Truncated 2
x y : X.obj (Opposite.op { obj := ⦋0⦌, property := Edge._proof_1 })
e : Edge x y
⊢ (ConcreteCategory.hom (X.map (𝟙 { obj := ⦋1⦌, property := δ₂_zero_comp_σ₂_one._proof_1 }).op)) e.edge = e.edge
Module: Mathlib.CategoryTheory.GradedObject.Bifunctor

Goal state before:
pp: C₁ : Type u_1
C₂ : Type u_2
C₃ : Type u_3
inst✝³ : Category.{v_1, u_1} C₁
inst✝² : Category.{v_2, u_2} C₂
inst✝¹ : Category.{v_3, u_3} C₃
F : C₁ ⥤ C₂ ⥤ C₃
I : Type u_4
J : Type u_5
K : Type u_6
p : I × J → K
X : GradedObject I C₁
Y : GradedObject J C₂
A : C₃
k : K
inst✝ : (((mapBifunctor F I J).obj X).obj Y).HasMap p
f g : mapBifunctorMapObj F p X Y k ⟶ A
h :
  ∀ (i : I) (j : J) (hij : p (i, j) = k),
    ιMapBifunctorMapObj F p X Y i j k hij ≫ f = ιMapBifunctorMapObj F p X Y i j k hij ≫ g
i : I
j : J
hij : p (i, j) = k
⊢ (((mapBifunctor F I J).obj X).obj Y).ιMapObj p (i, j) k hij ≫ f =
    (((mapBifunctor F I J).obj X).obj Y).ιMapObj p (i, j) k hij ≫ g

Tactic:
exact h i j hij

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

Goal state before:
pp: n : ℕ
a b : Fin (n + 1)
hab : a ≤ b
⊢ Monotone ![a, b]

Tactic:
rw [Fin.monotone_iff_le_succ]

Goal state after:
n : ℕ
a b : Fin (n + 1)
hab : a ≤ b
⊢ ∀ (i : Fin (unop (op ⦋1⦌)).len), ![a, b] i.castSucc ≤ ![a, b] i.succ
Module: Mathlib.AlgebraicTopology.SimplicialSet.StdSimplex

Goal state before:
pp: n : ℕ
a b c : Fin (n + 1)
hab : a ≤ b
hbc : b ≤ c
⊢ Monotone ![a, b, c]

Tactic:
rw [Fin.monotone_iff_le_succ]

Goal state after:
n : ℕ
a b c : Fin (n + 1)
hab : a ≤ b
hbc : b ≤ c
⊢ ∀ (i : Fin (unop (op ⦋2⦌)).len), ![a, b, c] i.castSucc ≤ ![a, b, c] i.succ
Module: Mathlib.AlgebraicTopology.ExtraDegeneracy

Goal state before:
pp: C : Type u_1
inst✝ : Category.{v_1, u_1} C
X : Augmented C
ed : X.ExtraDegeneracy
n : ℕ
i : Fin (n + 2)
j : Fin (n + 1)
hij : i ≤ j.castSucc
k : Fin (n + 1 + 1)
hk : j.succ.rev = k
⊢ X.left.δ₀Iter (↑j + 1) ⋯ ≫ ed.s ↑k ≫ X.left.σ₀Iter ↑j ⋯ =
    X.left.δ i ≫ X.left.δ₀Iter ↑j ⋯ ≫ ed.s ↑k ≫ X.left.σ₀Iter ↑j ⋯

Tactic:
X.left.δ_δ₀Iter_assoc _ _ (by grind)

Goal state after:
C : Type u_1
inst✝ : Category.{v_1, u_1} C
X : Augmented C
ed : X.ExtraDegeneracy
n : ℕ
i : Fin (n + 2)
j : Fin (n + 1)
hij : i ≤ j.castSucc
k : Fin (n + 1 + 1)
hk : j.succ.rev = k
⊢ X.left.δ₀Iter (↑j + 1) ⋯ ≫ ed.s ↑k ≫ X.left.σ₀Iter ↑j ⋯ = X.left.δ₀Iter (↑j + 1) ⋯ ≫ ed.s ↑k ≫ X.left.σ₀Iter ↑j ⋯
Module: Mathlib.Algebra.Homology.TotalComplex

Goal state before:
pp: 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
h₁₂ : ¬c₁₂.Rel i₁₂ i₁₂'
i₁ : I₁
i₂ : I₂
h : c₁.π c₂ c₁₂ (i₁, i₂) = i₁₂
h₁ : c₁.Rel i₁ (c₁.next i₁)
h₂ : c₁.π c₂ c₁₂ (c₁.next i₁, i₂) = i₁₂'
⊢ c₁₂.Rel i₁₂ i₁₂'

Tactic:
simpa only [← h, ← h₂] using ComplexShape.rel_π₁ c₂ c₁₂ h₁ i₂

Goal state after:
Module: Mathlib.CategoryTheory.GradedObject.Trifunctor

Goal state before:
pp: C₁ : Type u_1
C₂ : Type u_2
C₃ : Type u_3
C₄ : Type u_4
C₁₂ : Type u_5
C₂₃ : Type u_6
inst✝⁶ : Category.{v_1, u_1} C₁
inst✝⁵ : Category.{v_2, u_2} C₂
inst✝⁴ : Category.{v_3, u_3} C₃
inst✝³ : Category.{v_4, u_4} C₄
inst✝² : Category.{v_5, u_5} C₁₂
inst✝¹ : Category.{v_6, u_6} C₂₃
F : C₁ ⥤ C₂ ⥤ C₃ ⥤ C₄
I₁ : Type u_7
I₂ : Type u_8
I₃ : Type u_9
J : Type u_10
p : I₁ × I₂ × I₃ → J
inst✝ :
  ∀ (X₁ : GradedObject I₁ C₁) (X₂ : GradedObject I₂ C₂) (X₃ : GradedObject I₃ C₃),
    ((((mapTrifunctor F I₁ I₂ I₃).obj X₁).obj X₂).obj X₃).HasMap p
X₁ Y₁ : GradedObject I₁ C₁
φ : X₁ ⟶ Y₁
X₂ Y₂ : GradedObject I₂ C₂
ψ : X₂ ⟶ Y₂
⊢ (mapTrifunctorMapFunctorObj F p X₁).map ψ ≫
      { app := fun X₃ ↦ mapTrifunctorMapMap F p φ (𝟙 Y₂) (𝟙 X₃), naturality := ⋯ } =
    { app := fun X₃ ↦ mapTrifunctorMapMap F p φ (𝟙 X₂) (𝟙 X₃), naturality := ⋯ } ≫
      (mapTrifunctorMapFunctorObj F p Y₁).map ψ

Tactic:
ext X₃ j

Goal state after:
C₁ : Type u_1
C₂ : Type u_2
C₃ : Type u_3
C₄ : Type u_4
C₁₂ : Type u_5
C₂₃ : Type u_6
inst✝⁶ : Category.{v_1, u_1} C₁
inst✝⁵ : Category.{v_2, u_2} C₂
inst✝⁴ : Category.{v_3, u_3} C₃
inst✝³ : Category.{v_4, u_4} C₄
inst✝² : Category.{v_5, u_5} C₁₂
inst✝¹ : Category.{v_6, u_6} C₂₃
F : C₁ ⥤ C₂ ⥤ C₃ ⥤ C₄
I₁ : Type u_7
I₂ : Type u_8
I₃ : Type u_9
J : Type u_10
p : I₁ × I₂ × I₃ → J
inst✝ :
  ∀ (X₁ : GradedObject I₁ C₁) (X₂ : GradedObject I₂ C₂) (X₃ : GradedObject I₃ C₃),
    ((((mapTrifunctor F I₁ I₂ I₃).obj X₁).obj X₂).obj X₃).HasMap p
X₁ Y₁ : GradedObject I₁ C₁
φ : X₁ ⟶ Y₁
X₂ Y₂ : GradedObject I₂ C₂
ψ : X₂ ⟶ Y₂
X₃ : GradedObject I₃ C₃
j : J
⊢ ((mapTrifunctorMapFunctorObj F p X₁).map ψ ≫
          { app := fun X₃ ↦ mapTrifunctorMapMap F p φ (𝟙 Y₂) (𝟙 X₃), naturality := ⋯ }).app
      X₃ j =
    ({ app := fun X₃ ↦ mapTrifunctorMapMap F p φ (𝟙 X₂) (𝟙 X₃), naturality := ⋯ } ≫
          (mapTrifunctorMapFunctorObj F p Y₁).map ψ).app
      X₃ j
Module: Mathlib.Algebra.Homology.Bifunctor

Goal state before:
pp: C₁ : Type u_1
C₂ : Type u_2
D : Type u_3
inst✝⁷ : Category.{v_1, u_1} C₁
inst✝⁶ : Category.{v_2, u_2} C₂
inst✝⁵ : Category.{v_3, u_3} D
inst✝⁴ : HasZeroMorphisms C₁
inst✝³ : HasZeroMorphisms C₂
inst✝² : HasZeroMorphisms D
F : C₁ ⥤ C₂ ⥤ D
I₁ : Type u_4
I₂ : Type u_5
J : Type u_6
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
inst✝¹ : F.PreservesZeroMorphisms
inst✝ : ∀ (X₁ : C₁), (F.obj X₁).PreservesZeroMorphisms
K₁ K₁' : HomologicalComplex C₁ c₁
f : K₁ ⟶ K₁'
K₂ : HomologicalComplex C₂ c₂
⊢ ∀ (i₁ i₁' : I₁) (i₂ : I₂),
    c₁.Rel i₁ i₁' →
      ((GradedObject.mapBifunctor F I₁ I₂).map f.f).app K₂.X (i₁, i₂) ≫
          (((F.mapBifunctorHomologicalComplexObj c₂ K₁').obj K₂).d i₁ i₁').f i₂ =
        (((F.mapBifunctorHomologicalComplexObj c₂ K₁).obj K₂).d i₁ i₁').f i₂ ≫
          ((GradedObject.mapBifunctor F I₁ I₂).map f.f).app K₂.X (i₁', i₂)

Tactic:
by
          intros
          dsimp
          simp only [← NatTrans.comp_app, ← F.map_comp, f.comm]

Goal state after:
Module: Mathlib.CategoryTheory.GradedObject.Trifunctor

Goal state before:
pp: C₁ : Type u_1
C₂ : Type u_2
C₃ : Type u_3
C₄ : Type u_4
C₁₂ : Type u_5
C₂₃ : Type u_6
inst✝⁷ : Category.{v_1, u_1} C₁
inst✝⁶ : Category.{v_2, u_2} C₂
inst✝⁵ : Category.{v_3, u_3} C₃
inst✝⁴ : Category.{v_4, u_4} C₄
inst✝³ : Category.{v_5, u_5} C₁₂
inst✝² : Category.{v_6, u_6} C₂₃
F₁₂ : C₁ ⥤ C₂ ⥤ C₁₂
G : C₁₂ ⥤ C₃ ⥤ C₄
I₁ : Type u_7
I₂ : Type u_8
I₃ : Type u_9
J : Type u_10
r : I₁ × I₂ × I₃ → J
ρ₁₂ : BifunctorComp₁₂IndexData r
X₁ : GradedObject I₁ C₁
X₂ : GradedObject I₂ C₂
X₃ : GradedObject I₃ C₃
inst✝¹ : (((mapBifunctor F₁₂ I₁ I₂).obj X₁).obj X₂).HasMap ρ₁₂.p
inst✝ : (((mapBifunctor G ρ₁₂.I₁₂ I₃).obj (mapBifunctorMapObj F₁₂ ρ₁₂.p X₁ X₂)).obj X₃).HasMap ρ₁₂.q
H : HasGoodTrifunctor₁₂Obj F₁₂ G ρ₁₂ X₁ X₂ X₃
j : J
c₁₂ : (i₁₂ : ρ₁₂.I₁₂) → (((mapBifunctor F₁₂ I₁ I₂).obj X₁).obj X₂).CofanMapObjFun ρ₁₂.p i₁₂ :=
  fun i₁₂ ↦ (((mapBifunctor F₁₂ I₁ I₂).obj X₁).obj X₂).cofanMapObj ρ₁₂.p i₁₂
⊢ IsColimit (cofan₃MapBifunctor₁₂BifunctorMapObj F₁₂ G ρ₁₂ X₁ X₂ X₃ j)

Tactic:
have h₁₂ : ∀ i₁₂, IsColimit (c₁₂ i₁₂) := fun i₁₂ =>
    (((mapBifunctor F₁₂ I₁ I₂).obj X₁).obj X₂).isColimitCofanMapObj ρ₁₂.p i₁₂

Goal state after:
C₁ : Type u_1
C₂ : Type u_2
C₃ : Type u_3
C₄ : Type u_4
C₁₂ : Type u_5
C₂₃ : Type u_6
inst✝⁷ : Category.{v_1, u_1} C₁
inst✝⁶ : Category.{v_2, u_2} C₂
inst✝⁵ : Category.{v_3, u_3} C₃
inst✝⁴ : Category.{v_4, u_4} C₄
inst✝³ : Category.{v_5, u_5} C₁₂
inst✝² : Category.{v_6, u_6} C₂₃
F₁₂ : C₁ ⥤ C₂ ⥤ C₁₂
G : C₁₂ ⥤ C₃ ⥤ C₄
I₁ : Type u_7
I₂ : Type u_8
I₃ : Type u_9
J : Type u_10
r : I₁ × I₂ × I₃ → J
ρ₁₂ : BifunctorComp₁₂IndexData r
X₁ : GradedObject I₁ C₁
X₂ : GradedObject I₂ C₂
X₃ : GradedObject I₃ C₃
inst✝¹ : (((mapBifunctor F₁₂ I₁ I₂).obj X₁).obj X₂).HasMap ρ₁₂.p
inst✝ : (((mapBifunctor G ρ₁₂.I₁₂ I₃).obj (mapBifunctorMapObj F₁₂ ρ₁₂.p X₁ X₂)).obj X₃).HasMap ρ₁₂.q
H : HasGoodTrifunctor₁₂Obj F₁₂ G ρ₁₂ X₁ X₂ X₃
j : J
c₁₂ : (i₁₂ : ρ₁₂.I₁₂) → (((mapBifunctor F₁₂ I₁ I₂).obj X₁).obj X₂).CofanMapObjFun ρ₁₂.p i₁₂ :=
  fun i₁₂ ↦ (((mapBifunctor F₁₂ I₁ I₂).obj X₁).obj X₂).cofanMapObj ρ₁₂.p i₁₂
h₁₂ : (i₁₂ : ρ₁₂.I₁₂) → IsColimit (c₁₂ i₁₂)
⊢ IsColimit (cofan₃MapBifunctor₁₂BifunctorMapObj F₁₂ G ρ₁₂ X₁ X₂ X₃ j)
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
i₁ : I₁
i₂ : I₂
h : c₁.π c₂ c (i₁, i₂) = j
⊢ c₂.π c₁ c (i₂, i₁) = j

Tactic:
by
    rw [ComplexShape.π_symm c₁ c₂ c i₁ i₂, h]

Goal state after:
Module: Mathlib.CategoryTheory.GradedObject.Trifunctor

Goal state before:
pp: C₁ : Type u_1
C₂ : Type u_2
C₃ : Type u_3
C₄ : Type u_4
C₁₂ : Type u_5
C₂₃ : Type u_6
inst✝⁷ : Category.{v_1, u_1} C₁
inst✝⁶ : Category.{v_2, u_2} C₂
inst✝⁵ : Category.{v_3, u_3} C₃
inst✝⁴ : Category.{v_4, u_4} C₄
inst✝³ : Category.{v_5, u_5} C₁₂
inst✝² : Category.{v_6, u_6} C₂₃
F : C₁ ⥤ C₂₃ ⥤ C₄
G₂₃ : C₂ ⥤ C₃ ⥤ C₂₃
I₁ : Type u_7
I₂ : Type u_8
I₃ : Type u_9
J : Type u_10
r : I₁ × I₂ × I₃ → J
ρ₂₃ : BifunctorComp₂₃IndexData r
X₁ : GradedObject I₁ C₁
X₂ : GradedObject I₂ C₂
X₃ : GradedObject I₃ C₃
inst✝¹ : (((mapBifunctor G₂₃ I₂ I₃).obj X₂).obj X₃).HasMap ρ₂₃.p
inst✝ : (((mapBifunctor F I₁ ρ₂₃.I₂₃).obj X₁).obj (mapBifunctorMapObj G₂₃ ρ₂₃.p X₂ X₃)).HasMap ρ₂₃.q
H : HasGoodTrifunctor₂₃Obj F G₂₃ ρ₂₃ X₁ X₂ X₃
j : J
⊢ IsColimit (cofan₃MapBifunctorBifunctor₂₃MapObj F G₂₃ ρ₂₃ X₁ X₂ X₃ j)

Tactic:
let c₂₃ := fun i₂₃ => (((mapBifunctor G₂₃ I₂ I₃).obj X₂).obj X₃).cofanMapObj ρ₂₃.p i₂₃

Goal state after:
C₁ : Type u_1
C₂ : Type u_2
C₃ : Type u_3
C₄ : Type u_4
C₁₂ : Type u_5
C₂₃ : Type u_6
inst✝⁷ : Category.{v_1, u_1} C₁
inst✝⁶ : Category.{v_2, u_2} C₂
inst✝⁵ : Category.{v_3, u_3} C₃
inst✝⁴ : Category.{v_4, u_4} C₄
inst✝³ : Category.{v_5, u_5} C₁₂
inst✝² : Category.{v_6, u_6} C₂₃
F : C₁ ⥤ C₂₃ ⥤ C₄
G₂₃ : C₂ ⥤ C₃ ⥤ C₂₃
I₁ : Type u_7
I₂ : Type u_8
I₃ : Type u_9
J : Type u_10
r : I₁ × I₂ × I₃ → J
ρ₂₃ : BifunctorComp₂₃IndexData r
X₁ : GradedObject I₁ C₁
X₂ : GradedObject I₂ C₂
X₃ : GradedObject I₃ C₃
inst✝¹ : (((mapBifunctor G₂₃ I₂ I₃).obj X₂).obj X₃).HasMap ρ₂₃.p
inst✝ : (((mapBifunctor F I₁ ρ₂₃.I₂₃).obj X₁).obj (mapBifunctorMapObj G₂₃ ρ₂₃.p X₂ X₃)).HasMap ρ₂₃.q
H : HasGoodTrifunctor₂₃Obj F G₂₃ ρ₂₃ X₁ X₂ X₃
j : J
c₂₃ : (i₂₃ : ρ₂₃.I₂₃) → (((mapBifunctor G₂₃ I₂ I₃).obj X₂).obj X₃).CofanMapObjFun ρ₂₃.p i₂₃ :=
  fun i₂₃ ↦ (((mapBifunctor G₂₃ I₂ I₃).obj X₂).obj X₃).cofanMapObj ρ₂₃.p i₂₃
⊢ IsColimit (cofan₃MapBifunctorBifunctor₂₃MapObj F G₂₃ ρ₂₃ X₁ X₂ X₃ j)
Module: Mathlib.Algebra.Homology.TotalComplex

Goal state before:
pp: C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Preadditive C
I₁ : Type u_2
I₂ : Type u_3
I₁₂ : Type u_4
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
K L : HomologicalComplex₂ C c₁ c₂
φ : K ⟶ L
c₁₂ : ComplexShape I₁₂
inst✝³ : TotalComplexShape c₁ c₂ c₁₂
inst✝² : DecidableEq I₁₂
inst✝¹ : K.HasTotal c₁₂
inst✝ : L.HasTotal c₁₂
i₁ : I₁
i₂ : I₂
i₁₂ : I₁₂
⊢ K.d₂ c₁₂ i₁ i₂ i₁₂ ≫ GradedObject.mapMap (toGradedObjectMap φ) (c₁.π c₂ c₁₂) i₁₂ = (φ.f i₁).f i₂ ≫ L.d₂ c₁₂ i₁ i₂ i₁₂

Tactic:
by
  by_cases h : c₂.Rel i₂ (c₂.next i₂)
  · simp [totalAux.d₂_eq' _ c₁₂ i₁ h]
  · simp [d₂_eq_zero _ c₁₂ i₁ i₂ i₁₂ h]

Goal state after:
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
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)
h₅ : c.Rel j (c.next j)
⊢ c₁.π c₂ c (i₁, c₂.next i₂) = c.next j

Tactic:
by rw [← ComplexShape.next_π₂ c₁ c i₁ h₄, h]

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)
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
⊢ ((shiftFunctor₂ C y).obj K).D₁ (up ℤ) n₀ n₁ ≫ (K.totalShift₂XIso y n₁ n₁' h₁).hom =
    y.negOnePow • (K.totalShift₂XIso y n₀ n₀' h₀).hom ≫ K.D₁ (up ℤ) n₀' n₁'

pp: case neg
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
⊢ ((shiftFunctor₂ C y).obj K).D₁ (up ℤ) n₀ n₁ ≫ (K.totalShift₂XIso y n₁ n₁' h₁).hom =
    y.negOnePow • (K.totalShift₂XIso y n₀ n₀' h₀).hom ≫ K.D₁ (up ℤ) n₀' n₁'

Tactic:
· apply total.hom_ext
    intro p q hpq
    dsimp at h hpq
    dsimp [totalShift₂XIso]
    rw [ι_D₁_assoc, Linear.comp_units_smul, ι_totalDesc_assoc, Linear.units_smul_comp,
      ι_D₁, smul_smul, ((shiftFunctor₂ C y).obj K).d₁_eq _ rfl _ _ (by dsimp; lia),
      K.d₁_eq _ rfl _ _ (by dsimp; lia)]
    dsimp
    rw [one_smul, one_smul, Category.assoc, ι_totalDesc, Linear.comp_units_smul,
      ← Int.negOnePow_add]
    congr 2
    linarith

Goal state after:
case neg
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
⊢ ((shiftFunctor₂ C y).obj K).D₁ (up ℤ) n₀ n₁ ≫ (K.totalShift₂XIso y n₁ n₁' h₁).hom =
    y.negOnePow • (K.totalShift₂XIso y n₀ n₀' h₀).hom ≫ K.D₁ (up ℤ) n₀' n₁'
Module: Mathlib.CategoryTheory.Triangulated.Orthogonal

Goal state before:
pp: case refine_2.refine_1
C : Type u
inst✝⁶ : Category.{v, u} C
P : ObjectProperty C
inst✝⁵ : HasZeroObject C
inst✝⁴ : HasShift C ℤ
inst✝³ : Preadditive C
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
inst✝ : P.IsTriangulated
X : C
hX : P.leftOrthogonal X
Y₂ Y₃ : C
h : Y₂ ⟶ Y₃
Y₁ : C
f : Y₁ ⟶ Y₂
g : Y₃ ⟶ (shiftFunctor C 1).obj Y₁
hT : Triangle.mk f h g ∈ distinguishedTriangles
hY₁ : P Y₁
β : X ⟶ (Triangle.mk f h g).obj₁
hα : (β ≫ (Triangle.mk f h g).mor₁) ≫ h = 0
⊢ β ≫ (Triangle.mk f h g).mor₁ = 0

Tactic:
simp [hX β hY₁]

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

Goal state before:
pp: case refine_2
C : Type u
inst✝⁹ : Category.{v, u} C
D : Type u'
inst✝⁸ : Category.{v', u'} D
P : 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 : P.IsTriangulated
inst✝¹ : IsTriangulated C
X : C
hX : P.trW.isColocal X
L : C ⥤ D
inst✝ : L.IsLocalization P.trW
Y : C
g : L.obj X ⟶ L.obj Y
φ : P.trW.LeftFraction X Y
hφ : g = φ.map L ⋯
⊢ ∃ a, L.map a = g

Tactic:
obtain ⟨α, hα⟩ := (hX _ φ.hs).2 φ.f

Goal state after:
case refine_2
C : Type u
inst✝⁹ : Category.{v, u} C
D : Type u'
inst✝⁸ : Category.{v', u'} D
P : 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 : P.IsTriangulated
inst✝¹ : IsTriangulated C
X : C
hX : P.trW.isColocal X
L : C ⥤ D
inst✝ : L.IsLocalization P.trW
Y : C
g : L.obj X ⟶ L.obj Y
φ : P.trW.LeftFraction X Y
hφ : g = φ.map L ⋯
α : X ⟶ Y
hα : (fun f ↦ f ≫ φ.s) α = φ.f
⊢ ∃ a, L.map a = g
Module: Mathlib.Algebra.Homology.HomotopyCategory.KInjective

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
n : ℤ
z : Cocycle K L n
inst✝ : L.IsKInjective
hK : HomologicalComplex.Acyclic K
m : ℤ
hm : m + 1 = n
φ : K ⟶ (CategoryTheory.shiftFunctor (CochainComplex C ℤ) n).obj L
hφ : Cochain.ofHom φ = (↑z).rightShift n 0 ⋯
h : Homotopy φ 0
f : Cochain K ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) n).obj L) (-1)
hf : Cochain.ofHom φ = δ (-1) 0 f
⊢ (δ m n (n.negOnePow • f.rightUnshift m ⋯)).rightShift n 0 ⋯ = (↑z).rightShift n 0 ⋯

Tactic:
← hφ,

Goal state after:
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
K L : CochainComplex C ℤ
n : ℤ
z : Cocycle K L n
inst✝ : L.IsKInjective
hK : HomologicalComplex.Acyclic K
m : ℤ
hm : m + 1 = n
φ : K ⟶ (CategoryTheory.shiftFunctor (CochainComplex C ℤ) n).obj L
hφ : Cochain.ofHom φ = (↑z).rightShift n 0 ⋯
h : Homotopy φ 0
f : Cochain K ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) n).obj L) (-1)
hf : Cochain.ofHom φ = δ (-1) 0 f
⊢ (δ m n (n.negOnePow • f.rightUnshift m ⋯)).rightShift n 0 ⋯ = Cochain.ofHom φ
Module: Mathlib.AlgebraicTopology.ModelCategory.IsCofibrant

Goal state before:
pp: C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : CategoryWithCofibrations C
inst✝² : HasInitial C
X Y : C
inst✝¹ : (cofibrations C).IsStableUnderCobaseChange
inst✝ : HasBinaryCoproduct X Y
hX : Cofibration (initial.to X)
⊢ Cofibration coprod.inr

Tactic:
rw [cofibration_iff] at hX ⊢

Goal state after:
C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : CategoryWithCofibrations C
inst✝² : HasInitial C
X Y : C
inst✝¹ : (cofibrations C).IsStableUnderCobaseChange
inst✝ : HasBinaryCoproduct X Y
hX : cofibrations C (initial.to X)
⊢ cofibrations C coprod.inr
Module: Mathlib.AlgebraicTopology.ModelCategory.RightHomotopy

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
X Y : C
inst✝¹ : ModelCategory C
inst✝ : IsFibrant Y
f g : X ⟶ Y
⊢ mk f = mk g ↔ RightHomotopyRel f g

Tactic:
rw [← (RightHomotopyRel.equivalence X Y).eqvGen_iff]
  exact Quot.eq

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
T T' : Triangle C
f₁ f₂ : T ⟶ T'
hT : T ∈ distinguishedTriangles
hT' : T' ∈ distinguishedTriangles
a b : ℤ
h₀ : t.IsLE T.obj₁ a
h₁ : t.IsGE T'.obj₃ b
H : f₁.hom₂ = f₂.hom₂
hab : a ≤ b
this : ∀ (f : T ⟶ T'), f.hom₂ = 0 → f = 0
⊢ f₁ = f₂

Tactic:
rw [← sub_eq_zero]

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
T T' : Triangle C
f₁ f₂ : T ⟶ T'
hT : T ∈ distinguishedTriangles
hT' : T' ∈ distinguishedTriangles
a b : ℤ
h₀ : t.IsLE T.obj₁ a
h₁ : t.IsGE T'.obj₃ b
H : f₁.hom₂ = f₂.hom₂
hab : a ≤ b
this : ∀ (f : T ⟶ T'), f.hom₂ = 0 → f = 0
⊢ f₁ - f₂ = 0
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
n : ℤ
X Y : C
f₁ f₂ : (t.truncGE n).obj X ⟶ Y
h : (t.truncGEπ n).app X ≫ f₁ = (t.truncGEπ n).app X ≫ f₂
inst✝ : t.IsGE Y n
⊢ f₁ = f₂

Tactic:
suffices ∀ (f : (t.truncGE n).obj X ⟶ Y), (t.truncGEπ n).app X ≫ f = 0 → f = 0 by
    rw [← sub_eq_zero, this (f₁ - f₂) (by cat_disch)]

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
n : ℤ
X Y : C
f₁ f₂ : (t.truncGE n).obj X ⟶ Y
h : (t.truncGEπ n).app X ≫ f₁ = (t.truncGEπ n).app X ≫ f₂
inst✝ : t.IsGE Y n
⊢ ∀ (f : (t.truncGE n).obj X ⟶ Y), (t.truncGEπ n).app X ≫ f = 0 → f = 0
Module: Mathlib.CategoryTheory.Functor.OfSequence

Goal state before:
pp: C : Type u_1
inst✝ : Category.{v_1, u_1} C
X : ℕ → C
f : (n : ℕ) → X n ⟶ X (n + 1)
i : ℕ
⊢ map f i i ⋯ = 𝟙 (X i)

Tactic:
revert X f
  induction i with
  | zero => intros; rfl
  | succ _ hi =>
      intro X f
      apply hi

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

Goal state before:
pp: C : Type u_1
inst✝ : Category.{v_1, u_1} C
X : ℕ → C
f : (n : ℕ) → X n ⟶ X (n + 1)
i : ℕ
⊢ map f i (i + 1) ⋯ = f i

Tactic:
revert X f
  induction i with
  | zero => intros; rfl
  | succ _ hi =>
      intro X f
      apply hi

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

Goal state before:
pp: case zero.succ.succ.zero
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
hjk : j + 1 ≤ 0 + 1
⊢ map f 0 (0 + 1) ⋯ = map f 0 (j + 1) hij ≫ map f (j + 1) (0 + 1) hjk

Tactic:
obtain rfl : j = 0 := by lia

Goal state after:
case zero.succ.succ.zero
C : Type u_1
inst✝ : Category.{v_1, u_1} C
X : ℕ → C
f : (n : ℕ) → X n ⟶ X (n + 1)
hj :
  ∀ {X : ℕ → C} (f : (n : ℕ) → X n ⟶ X (n + 1)) (k : ℕ) (hij : 0 ≤ 0) (hjk : 0 ≤ k),
    map f 0 k ⋯ = map f 0 0 hij ≫ map f 0 k hjk
hij : 0 ≤ 0 + 1
hjk : 0 + 1 ≤ 0 + 1
⊢ map f 0 (0 + 1) ⋯ = map f 0 (0 + 1) hij ≫ map f (0 + 1) (0 + 1) hjk
Module: Mathlib.Algebra.Homology.Factorizations.CM5a

Goal state before:
pp: case inr
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
K L : CochainComplex C ℤ
inst✝ : EnoughInjectives C
n₁ i : ℤ
hn₁ : i + 1 = n₁
hi : i ≤ i
⊢ IsIso (homologyMap (π K L n₁) i)

Tactic:
have : homologyMap (biprod.inl : _ ⟶ mid K L n₁) i = 0 :=
      (ShortComplex.isZero_homology_of_isZero_X₂ _
        (isZero_single_obj_X (.up ℤ) _ _ _ (by lia))).eq_of_src _ _

Goal state after:
case inr
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
K L : CochainComplex C ℤ
inst✝ : EnoughInjectives C
n₁ i : ℤ
hn₁ : i + 1 = n₁
hi : i ≤ i
this : homologyMap biprod.inl i = 0
⊢ IsIso (homologyMap (π K L n₁) i)