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

Goal state before:
pp: ⊢ stdSimplex.δ 1 = SSet.const (obj₀Equiv.symm 0)

Tactic:
by
  decide

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

Goal state before:
pp: ⊢ stdSimplex.δ 0 = SSet.const (obj₀Equiv.symm 1)

Tactic:
by
  decide

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

Goal state before:
pp: V : Type u
inst✝² : Category.{v, u} V
inst✝¹ : HasZeroObject V
inst✝ : HasZeroMorphisms V
C : CochainComplex V ℕ
⊢ C.d 0 1 ≫ (truncate.obj C).d 0 1 = 0

Tactic:
by simp

Goal state after:
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
⊢ n = ↑(Fin.rev 0)

Tactic:
by simp

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

Goal state before:
pp: n : ℕ
S : Finset (Fin (n + 1))
m : ℕ
e : Fin (m + 1) ≃o ↥S
j : SimplexCategory
x✝ : (face S).toSSet.obj (op j)
x : Δ[n].obj (op j)
hx : x ∈ (face S).obj (op j)
i : Fin (j.len + 1)
⊢ (Hom.toOrderHom (objEquiv x)) i ∈ Finset.image ⇑(Hom.toOrderHom (objEquiv x)) ⊤

Tactic:
by simp

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

Goal state before:
pp: ⊢ (faceSingletonIso 0).hom ≫ (face {0}).ι = stdSimplex.δ 1

Tactic:
by
  decide

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

Goal state before:
pp: ⊢ (faceSingletonIso 1).hom ≫ (face {1}).ι = stdSimplex.δ 0

Tactic:
by
  decide

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

Goal state before:
pp: n d : ℕ
x : ↑(Δ[n].nonDegenerate d)
i : Fin (d + 1)
⊢ i ∈ univ

Tactic:
by simp

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

Goal state before:
pp: case mp.inr
n d : ℕ
x : ↑(Δ[n].nonDegenerate d)
this : Function.Injective ⇑(Hom.toOrderHom (objEquiv ↑x))
a b : Fin (d + 1)
h : ¬↑x b < ↑x a
h' : b < a
h'' : (fun j ↦ ↑x j) b = (fun j ↦ ↑x j) a
⊢ ↑x b < ↑x a

Tactic:
simp only [this h'', lt_self_iff_false] at h'

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

Goal state before:
pp: n : ℕ
⊢ Subcomplex.ofSimplex (objEquiv.symm (𝟙 ⦋n⦌)) ≤ ⊤

Tactic:
by simp

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

Goal state before:
pp: case refine_2
n : ℕ
⊢ objEquiv.symm (𝟙 ⦋n⦌) ∈ Δ[n].nonDegenerate n

Tactic:
apply objEquiv_symm_id_mem_nonDegenerate

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

Goal state before:
pp: I₁ : Type u_1
I₂ : Type u_2
I₁₂ : Type u_4
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
c₁₂ : ComplexShape I₁₂
inst✝ : TotalComplexShape c₁ c₂ c₁₂
i₁ i₁' : I₁
i₂ i₂' : I₂
h₁ : c₁.Rel i₁ i₁'
h₂ : c₂.Rel i₂ i₂'
⊢ c₁.ε₁ c₂ c₁₂ (i₁, i₂) * (-c₁.ε₁ c₂ c₁₂ (i₁, i₂) * c₁.ε₂ c₂ c₁₂ (i₁', i₂) * c₁.ε₁ c₂ c₁₂ (i₁, i₂')) =
    -c₁.ε₂ c₂ c₁₂ (i₁', i₂) * c₁.ε₁ c₂ c₁₂ (i₁, i₂')

Tactic:
neg_mul,

Goal state after:
I₁ : Type u_1
I₂ : Type u_2
I₁₂ : Type u_4
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
c₁₂ : ComplexShape I₁₂
inst✝ : TotalComplexShape c₁ c₂ c₁₂
i₁ i₁' : I₁
i₂ i₂' : I₂
h₁ : c₁.Rel i₁ i₁'
h₂ : c₂.Rel i₂ i₂'
⊢ c₁.ε₁ c₂ c₁₂ (i₁, i₂) * (-(c₁.ε₁ c₂ c₁₂ (i₁, i₂) * c₁.ε₂ c₂ c₁₂ (i₁', i₂)) * c₁.ε₁ c₂ c₁₂ (i₁, i₂')) =
    -c₁.ε₂ c₂ c₁₂ (i₁', i₂) * c₁.ε₁ c₂ c₁₂ (i₁, i₂')
Module: Mathlib.Algebra.Homology.ComplexShapeSigns

Goal state before:
pp: I₁ : Type u_1
I₂ : Type u_2
I₁₂ : Type u_4
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
c₁₂ : ComplexShape I₁₂
inst✝ : TotalComplexShape c₁ c₂ c₁₂
i₁ i₁' : I₁
i₂ i₂' : I₂
h₁ : c₁.Rel i₁ i₁'
h₂ : c₂.Rel i₂ i₂'
⊢ c₁.ε₁ c₂ c₁₂ (i₁, i₂) * (-(c₁.ε₁ c₂ c₁₂ (i₁, i₂) * c₁.ε₂ c₂ c₁₂ (i₁', i₂)) * c₁.ε₁ c₂ c₁₂ (i₁, i₂')) =
    -c₁.ε₂ c₂ c₁₂ (i₁', i₂) * c₁.ε₁ c₂ c₁₂ (i₁, i₂')

Tactic:
neg_mul,

Goal state after:
I₁ : Type u_1
I₂ : Type u_2
I₁₂ : Type u_4
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
c₁₂ : ComplexShape I₁₂
inst✝ : TotalComplexShape c₁ c₂ c₁₂
i₁ i₁' : I₁
i₂ i₂' : I₂
h₁ : c₁.Rel i₁ i₁'
h₂ : c₂.Rel i₂ i₂'
⊢ c₁.ε₁ c₂ c₁₂ (i₁, i₂) * -(c₁.ε₁ c₂ c₁₂ (i₁, i₂) * c₁.ε₂ c₂ c₁₂ (i₁', i₂) * c₁.ε₁ c₂ c₁₂ (i₁, i₂')) =
    -c₁.ε₂ c₂ c₁₂ (i₁', i₂) * c₁.ε₁ c₂ c₁₂ (i₁, i₂')
Module: Mathlib.Algebra.Homology.ComplexShapeSigns

Goal state before:
pp: I₁ : Type u_1
I₂ : Type u_2
I₁₂ : Type u_4
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
c₁₂ : ComplexShape I₁₂
inst✝ : TotalComplexShape c₁ c₂ c₁₂
i₁ i₁' : I₁
i₂ i₂' : I₂
h₁ : c₁.Rel i₁ i₁'
h₂ : c₂.Rel i₂ i₂'
⊢ c₁.ε₁ c₂ c₁₂ (i₁, i₂) * -(c₁.ε₁ c₂ c₁₂ (i₁, i₂) * c₁.ε₂ c₂ c₁₂ (i₁', i₂) * c₁.ε₁ c₂ c₁₂ (i₁, i₂')) =
    -c₁.ε₂ c₂ c₁₂ (i₁', i₂) * c₁.ε₁ c₂ c₁₂ (i₁, i₂')

Tactic:
neg_mul,

Goal state after:
I₁ : Type u_1
I₂ : Type u_2
I₁₂ : Type u_4
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
c₁₂ : ComplexShape I₁₂
inst✝ : TotalComplexShape c₁ c₂ c₁₂
i₁ i₁' : I₁
i₂ i₂' : I₂
h₁ : c₁.Rel i₁ i₁'
h₂ : c₂.Rel i₂ i₂'
⊢ c₁.ε₁ c₂ c₁₂ (i₁, i₂) * -(c₁.ε₁ c₂ c₁₂ (i₁, i₂) * c₁.ε₂ c₂ c₁₂ (i₁', i₂) * c₁.ε₁ c₂ c₁₂ (i₁, i₂')) =
    -(c₁.ε₂ c₂ c₁₂ (i₁', i₂) * c₁.ε₁ c₂ c₁₂ (i₁, i₂'))
Module: Mathlib.Algebra.Homology.ComplexShapeSigns

Goal state before:
pp: I₁ : Type u_1
I₂ : Type u_2
I₃ : Type u_3
I₁₂ : Type u_4
I₂₃ : Type u_5
J : Type u_6
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
c₃ : ComplexShape I₃
c₁₂ : ComplexShape I₁₂
c₂₃ : ComplexShape I₂₃
c✝ : ComplexShape J
inst✝² : TotalComplexShape c₁ c₂ c₁₂
I : Type u_7
inst✝¹ : AddMonoid I
c : ComplexShape I
inst✝ : c.TensorSigns
i₁✝ i₁'✝ i₂✝ i₂'✝ : I
h : c.Rel i₁✝ i₁'✝
x✝ : c.Rel i₂✝ i₂'✝
⊢ c.ε i₁✝ * 1 = -1 * c.ε i₁'✝

Tactic:
neg_mul,

Goal state after:
I₁ : Type u_1
I₂ : Type u_2
I₃ : Type u_3
I₁₂ : Type u_4
I₂₃ : Type u_5
J : Type u_6
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
c₃ : ComplexShape I₃
c₁₂ : ComplexShape I₁₂
c₂₃ : ComplexShape I₂₃
c✝ : ComplexShape J
inst✝² : TotalComplexShape c₁ c₂ c₁₂
I : Type u_7
inst✝¹ : AddMonoid I
c : ComplexShape I
inst✝ : c.TensorSigns
i₁✝ i₁'✝ i₂✝ i₂'✝ : I
h : c.Rel i₁✝ i₁'✝
x✝ : c.Rel i₂✝ i₂'✝
⊢ c.ε i₁✝ * 1 = -(1 * c.ε i₁'✝)
Module: Mathlib.Algebra.Homology.ComplexShapeSigns

Goal state before:
pp: I₁ : Type u_1
I₂ : Type u_2
I₃ : Type u_3
I₁₂ : Type u_4
I₂₃ : Type u_5
J : Type u_6
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
c₃ : ComplexShape I₃
c₁₂ : ComplexShape I₁₂
c₂₃ : ComplexShape I₂₃
c✝ : ComplexShape J
inst✝² : TotalComplexShape c₁ c₂ c₁₂
I : Type u_7
inst✝¹ : AddMonoid I
c : ComplexShape I
inst✝ : c.TensorSigns
x✝² x✝¹ x✝ : I
⊢ 1 = 1 * 1

Tactic:
rw [one_mul]

Goal state after:
Module: Mathlib.AlgebraicTopology.ExtraDegeneracy

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
f : Arrow C
inst✝ : ∀ (n : ℕ), HasWidePullback f.right (fun x ↦ f.left) fun x ↦ f.hom
S : SplitEpi f.hom
⊢ ∀ (j : Fin ((unop (op ⦋0⦌)).len + 1)), 𝟙 f.left ≫ f.hom = f.hom

Tactic:
by simp

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

Goal state before:
pp: I₁ : Type u_1
I₂ : Type u_2
I₃ : Type u_3
I₁₂ : Type u_4
I₂₃ : Type u_5
J : Type u_6
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
c₃ : ComplexShape I₃
c₁₂ : ComplexShape I₁₂
c₂₃ : ComplexShape I₂₃
c : ComplexShape J
inst✝ : TotalComplexShape c₁ c₂ c₁₂
i₁✝ i₁'✝ : I₂
i₂✝ i₂'✝ : I₁
h₂ : c₂.Rel i₁✝ i₁'✝
h₁ : c₁.Rel i₂✝ i₂'✝
⊢ ((match (i₁✝, i₂✝) with
      | (i₂, i₁) => c₁.ε₁ c₂ c₁₂ (i₁, i₂)) *
      match (i₁✝, i₂'✝) with
      | (i₂, i₁) => c₁.ε₂ c₂ c₁₂ (i₁, i₂)) =
    (-match (i₁✝, i₂✝) with
        | (i₂, i₁) => c₁.ε₂ c₂ c₁₂ (i₁, i₂)) *
      match (i₁'✝, i₂✝) with
      | (i₂, i₁) => c₁.ε₁ c₂ c₁₂ (i₁, i₂)

Tactic:
neg_mul,

Goal state after:
I₁ : Type u_1
I₂ : Type u_2
I₃ : Type u_3
I₁₂ : Type u_4
I₂₃ : Type u_5
J : Type u_6
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
c₃ : ComplexShape I₃
c₁₂ : ComplexShape I₁₂
c₂₃ : ComplexShape I₂₃
c : ComplexShape J
inst✝ : TotalComplexShape c₁ c₂ c₁₂
i₁✝ i₁'✝ : I₂
i₂✝ i₂'✝ : I₁
h₂ : c₂.Rel i₁✝ i₁'✝
h₁ : c₁.Rel i₂✝ i₂'✝
⊢ ((match (i₁✝, i₂✝) with
      | (i₂, i₁) => c₁.ε₁ c₂ c₁₂ (i₁, i₂)) *
      match (i₁✝, i₂'✝) with
      | (i₂, i₁) => c₁.ε₂ c₂ c₁₂ (i₁, i₂)) =
    -((match (i₁✝, i₂✝) with
        | (i₂, i₁) => c₁.ε₂ c₂ c₁₂ (i₁, i₂)) *
        match (i₁'✝, i₂✝) with
        | (i₂, i₁) => c₁.ε₁ c₂ c₁₂ (i₁, i₂))
Module: Mathlib.Algebra.Homology.ComplexShapeSigns

Goal state before:
pp: I₁ : Type u_1
I₂ : Type u_2
I₃ : Type u_3
I₁₂ : Type u_4
I₂₃ : Type u_5
J : Type u_6
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
c₃ : ComplexShape I₃
c₁₂ : ComplexShape I₁₂
c₂₃ : ComplexShape I₂₃
c : ComplexShape J
inst✝ : TotalComplexShape c₁ c₂ c₁₂
i₁✝ i₁'✝ : I₂
i₂✝ i₂'✝ : I₁
h₂ : c₂.Rel i₁✝ i₁'✝
h₁ : c₁.Rel i₂✝ i₂'✝
⊢ ((match (i₁✝, i₂✝) with
      | (i₂, i₁) => c₁.ε₁ c₂ c₁₂ (i₁, i₂)) *
      match (i₁✝, i₂'✝) with
      | (i₂, i₁) => c₁.ε₂ c₂ c₁₂ (i₁, i₂)) =
    -(-c₁.ε₁ c₂ c₁₂ (i₂✝, i₁✝) * c₁.ε₂ c₂ c₁₂ (i₂'✝, i₁✝))

Tactic:
neg_mul,

Goal state after:
I₁ : Type u_1
I₂ : Type u_2
I₃ : Type u_3
I₁₂ : Type u_4
I₂₃ : Type u_5
J : Type u_6
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
c₃ : ComplexShape I₃
c₁₂ : ComplexShape I₁₂
c₂₃ : ComplexShape I₂₃
c : ComplexShape J
inst✝ : TotalComplexShape c₁ c₂ c₁₂
i₁✝ i₁'✝ : I₂
i₂✝ i₂'✝ : I₁
h₂ : c₂.Rel i₁✝ i₁'✝
h₁ : c₁.Rel i₂✝ i₂'✝
⊢ ((match (i₁✝, i₂✝) with
      | (i₂, i₁) => c₁.ε₁ c₂ c₁₂ (i₁, i₂)) *
      match (i₁✝, i₂'✝) with
      | (i₂, i₁) => c₁.ε₂ c₂ c₁₂ (i₁, i₂)) =
    - -(c₁.ε₁ c₂ c₁₂ (i₂✝, i₁✝) * c₁.ε₂ c₂ c₁₂ (i₂'✝, i₁✝))
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
⊢ ∀ (i j : I₂),
    c₂.Rel i j →
      (Iso.refl ((((𝟭 (HomologicalComplex₂ C c₁ c₂)).obj K).X i₁).X i)).hom ≫
          (((flipFunctor C c₁ c₂ ⋙ flipFunctor C c₂ c₁).obj K).X i₁).d i j =
        (((𝟭 (HomologicalComplex₂ C c₁ c₂)).obj K).X i₁).d i j ≫
          (Iso.refl ((((𝟭 (HomologicalComplex₂ C c₁ c₂)).obj K).X i₁).X j)).hom

Tactic:
by 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 c₂ c₁
i₂ : I₂
⊢ ∀ (i j : I₁),
    c₁.Rel i j →
      (Iso.refl ((((flipFunctor C c₂ c₁ ⋙ flipFunctor C c₁ c₂).obj K).X i₂).X i)).hom ≫
          (((𝟭 (HomologicalComplex₂ C c₂ c₁)).obj K).X i₂).d i j =
        (((flipFunctor C c₂ c₁ ⋙ flipFunctor C c₁ c₂).obj K).X i₂).d i j ≫
          (Iso.refl ((((flipFunctor C c₂ c₁ ⋙ flipFunctor C c₁ c₂).obj K).X i₂).X j)).hom

Tactic:
by simp

Goal state after:
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
i : I₁ × I₂
h : c₁.π c₂ c₁₂ i = i₁₂
⊢ K.toGradedObject.ιMapObj (c₁.π c₂ c₁₂) i i₁₂ h ≫ K.D₂ c₁₂ i₁₂ i₁₂' = K.d₂ c₁₂ i.1 i.2 i₁₂'

Tactic:
by
  simp [D₂]

Goal state after:
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) (i₂ i₂' : I₂),
    c₂.Rel i₂ i₂' →
      ((GradedObject.mapBifunctor F I₁ I₂).map f.f).app K₂.X (i₁, i₂) ≫
          (((F.mapBifunctorHomologicalComplexObj c₂ K₁').obj K₂).X i₁).d i₂ i₂' =
        (((F.mapBifunctorHomologicalComplexObj c₂ K₁).obj K₂).X i₁).d i₂ i₂' ≫
          ((GradedObject.mapBifunctor F I₁ I₂).map f.f).app K₂.X (i₁, i₂')

Tactic:
by simp

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
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_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
i₁ : I₁
i₂ : I₂
i₃ : I₃
j : J
h : r (i₁, i₂, i₃) = j
i₂₃ : ρ₂₃.I₂₃
h₂₃ : ρ₂₃.p (i₂, i₃) = i₂₃
⊢ ιMapBifunctorBifunctor₂₃MapObj F G₂₃ ρ₂₃ X₁ X₂ X₃ i₁ i₂ i₃ j h =
    (F.obj (X₁ i₁)).map (ιMapBifunctorMapObj G₂₃ ρ₂₃.p X₂ X₃ i₂ i₃ i₂₃ h₂₃) ≫
      ιMapBifunctorMapObj F ρ₂₃.q X₁ (mapBifunctorMapObj G₂₃ ρ₂₃.p X₂ X₃) i₁ i₂₃ j ⋯

Tactic:
subst h₂₃
  rfl

Goal state after:
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'' : I₁₂
⊢ K.D₁ c₁₂ i₁₂ i₁₂' ≫ K.D₂ c₁₂ i₁₂' i₁₂'' = -K.D₂ c₁₂ i₁₂ i₁₂' ≫ K.D₁ c₁₂ i₁₂' i₁₂''

Tactic:
by simp

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✝⁸ : Category.{v_1, u_1} C
inst✝⁷ : Preadditive C
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
K : HomologicalComplex₂ C c₁ c₂
c : ComplexShape J
inst✝⁶ : TotalComplexShape c₁ c₂ c
inst✝⁵ : TotalComplexShape c₂ c₁ c
inst✝⁴ : TotalComplexShapeSymmetry c₁ c₂ c
inst✝³ : K.HasTotal c
inst✝² : DecidableEq J
inst✝¹ : TotalComplexShapeSymmetry c₂ c₁ c
inst✝ : TotalComplexShapeSymmetrySymmetry c₁ c₂ c
⊢ K.flip.totalFlipIso c = (K.totalFlipIso c).symm

Tactic:
ext j i₁ i₂ h
  rw [Iso.symm_hom, ιTotal_totalFlipIso_f_hom]
  dsimp only [flip_flip]
  rw [ιTotal_totalFlipIso_f_inv, ComplexShape.σ_symm]

Goal state after:
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 M : HomologicalComplex₂ C c₁ c₂
φ : K ⟶ L
e : K ≅ L
ψ : L ⟶ M
c₁₂ : ComplexShape I₁₂
inst✝³ : TotalComplexShape c₁ c₂ c₁₂
inst✝² : DecidableEq I₁₂
inst✝¹ : K.HasTotal c₁₂
inst✝ : L.HasTotal c₁₂
i₁₂ i₁₂' : I₁₂
x✝ : c₁₂.Rel i₁₂ i₁₂'
⊢ GradedObject.mapMap (toGradedObjectMap φ) (c₁.π c₂ c₁₂) i₁₂ ≫ L.D₁ c₁₂ i₁₂ i₁₂' +
      GradedObject.mapMap (toGradedObjectMap φ) (c₁.π c₂ c₁₂) i₁₂ ≫ L.D₂ c₁₂ i₁₂ i₁₂' =
    (K.D₁ c₁₂ i₁₂ i₁₂' + K.D₂ c₁₂ i₁₂ i₁₂') ≫ GradedObject.mapMap (toGradedObjectMap φ) (c₁.π c₂ c₁₂) i₁₂'

Tactic:
add_comp,

Goal state after:
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 M : HomologicalComplex₂ C c₁ c₂
φ : K ⟶ L
e : K ≅ L
ψ : L ⟶ M
c₁₂ : ComplexShape I₁₂
inst✝³ : TotalComplexShape c₁ c₂ c₁₂
inst✝² : DecidableEq I₁₂
inst✝¹ : K.HasTotal c₁₂
inst✝ : L.HasTotal c₁₂
i₁₂ i₁₂' : I₁₂
x✝ : c₁₂.Rel i₁₂ i₁₂'
⊢ GradedObject.mapMap (toGradedObjectMap φ) (c₁.π c₂ c₁₂) i₁₂ ≫ L.D₁ c₁₂ i₁₂ i₁₂' +
      GradedObject.mapMap (toGradedObjectMap φ) (c₁.π c₂ c₁₂) i₁₂ ≫ L.D₂ c₁₂ i₁₂ i₁₂' =
    K.D₁ c₁₂ i₁₂ i₁₂' ≫ GradedObject.mapMap (toGradedObjectMap φ) (c₁.π c₂ c₁₂) i₁₂' +
      K.D₂ c₁₂ i₁₂ i₁₂' ≫ GradedObject.mapMap (toGradedObjectMap φ) (c₁.π c₂ c₁₂) i₁₂'
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
i₁ i₁' : I₁
hi₁ : c₁.Rel i₁ i₁'
i₂ i₂' : I₂
hi₂ : c₂.Rel i₂ i₂'
j : J
hj : c₁.π c₂ c (i₁', i₂) = j
hj' : c₁.π c₂ c (i₁, i₂') = j
⊢ (-c₁.ε₂ c₂ c (i₁', i₂) * c₁.ε₁ c₂ c (i₁, i₂')) •
      (F.map (h₁.hom i₁' i₁)).app (K₂.X i₂) ≫
        (F.obj (L₁.X i₁)).map (f₂.f i₂) ≫
          (F.obj (L₁.X i₁)).map (L₂.d i₂ i₂') ≫
            (((F.mapBifunctorHomologicalComplex c₁ c₂).obj L₁).obj L₂).ιTotal c i₁ i₂' j hj' =
    -((c₁.ε₂ c₂ c (i₁', i₂) * c₁.ε₁ c₂ c (i₁, i₂')) •
        (F.map (h₁.hom i₁' i₁)).app (K₂.X i₂) ≫
          (F.obj (L₁.X i₁)).map (K₂.d i₂ i₂') ≫ (F.obj (L₁.X i₁)).map (f₂.f i₂') ≫ L₁.ιMapBifunctor L₂ F c i₁ i₂' j hj')

Tactic:
neg_mul,

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 : 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
i₁ i₁' : I₁
hi₁ : c₁.Rel i₁ i₁'
i₂ i₂' : I₂
hi₂ : c₂.Rel i₂ i₂'
j : J
hj : c₁.π c₂ c (i₁', i₂) = j
hj' : c₁.π c₂ c (i₁, i₂') = j
⊢ -(c₁.ε₂ c₂ c (i₁', i₂) * c₁.ε₁ c₂ c (i₁, i₂')) •
      (F.map (h₁.hom i₁' i₁)).app (K₂.X i₂) ≫
        (F.obj (L₁.X i₁)).map (f₂.f i₂) ≫
          (F.obj (L₁.X i₁)).map (L₂.d i₂ i₂') ≫
            (((F.mapBifunctorHomologicalComplex c₁ c₂).obj L₁).obj L₂).ιTotal c i₁ i₂' j hj' =
    -((c₁.ε₂ c₂ c (i₁', i₂) * c₁.ε₁ c₂ c (i₁, i₂')) •
        (F.map (h₁.hom i₁' i₁)).app (K₂.X i₂) ≫
          (F.obj (L₁.X i₁)).map (K₂.d i₂ i₂') ≫ (F.obj (L₁.X i₁)).map (f₂.f i₂') ≫ L₁.ιMapBifunctor L₂ F c i₁ i₂' j hj')
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 : ℤ
x✝ : ↑((up ℤ).π (up ℤ) (up ℤ) ⁻¹' {n})
⊢ (fun x ↦
        match x with
        | ⟨(a, b), h⟩ => ⟨(a, b - y), ⋯⟩)
      ((fun x ↦
          match x with
          | ⟨(a, b), h⟩ => ⟨(a, b + y), ⋯⟩)
        x✝) =
    x✝

Tactic:
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 L : HomologicalComplex₂ C (up ℤ) (up ℤ)
f : K ⟶ L
x y : ℤ
inst✝ : K.HasTotal (up ℤ)
n : ℤ
x✝ : ↑((up ℤ).π (up ℤ) (up ℤ) ⁻¹' {n + y})
⊢ (fun x ↦
        match x with
        | ⟨(a, b), h⟩ => ⟨(a, b + y), ⋯⟩)
      ((fun x ↦
          match x with
          | ⟨(a, b), h⟩ => ⟨(a, b - y), ⋯⟩)
        x✝) =
    x✝

Tactic:
by simp

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

Goal state before:
pp: case pos
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:
exact comm₁_aux h₁ f₂ F c h₃ h₄ j h

Goal state after:
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
i₁ : ι₁
i₂ : ι₂
i₃ : ι₃
j j' : ι₄
h : c₁.r c₂ c₃ c₁₂ c₄ (i₁, i₂, i₃) = j
inst✝ : HasGoodTrifunctor₁₂Obj F₁₂ G K₁ K₂ K₃ c₁₂ c₄
⊢ ι F₁₂ G K₁ K₂ K₃ c₁₂ c₄ i₁ i₂ i₃ j h ≫ D₂ F₁₂ G K₁ K₂ K₃ c₁₂ c₄ j j' = d₂ F₁₂ G K₁ K₂ K₃ c₁₂ c₄ i₁ i₂ i₃ j'

Tactic:
by
  simp [D₂]

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 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
⊢ mapBifunctorMap f₁ f₂ F c =
    (K₁.mapBifunctorFlipIso K₂ F c).inv ≫ mapBifunctorMap f₂ f₁ F.flip c ≫ (L₁.mapBifunctorFlipIso L₂ F c).hom

Tactic:
by simp

Goal state after: