mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.AlgebraicTopology.SimplexCategory.Basic

Goal state before:
pp: case inr
n : ℕ
i j : Fin (n + 1)
H : i ≤ j
k : Fin (n + 1)
h : i ≤ k
hkj : j < k
⊢ i ≤ j.predAbove k.castSucc

Tactic:
· rw [Fin.predAbove_of_castSucc_lt _ _ (Fin.castSucc_lt_castSucc_iff.mpr hkj),
          Fin.le_pred_iff,
          Fin.succ_le_castSucc_iff]
          exact H.trans_lt hkj

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

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
X : CosimplicialObject C
D : Type u_1
inst✝ : Category.{v_1, u_1} D
F : C ⥤ D
X✝ Y✝ : Augmented C
η : X✝ ⟶ Y✝
⊢ (const D).map (F.map η.left) ≫
      { left := F.obj (point.obj Y✝), right := ((whiskering C D).obj F).obj (drop.obj Y✝),
          hom := (constComp SimplexCategory (point.obj Y✝) F).inv ≫ whiskerRight Y✝.hom F }.hom =
    { left := F.obj (point.obj X✝), right := ((whiskering C D).obj F).obj (drop.obj X✝),
          hom := (constComp SimplexCategory (point.obj X✝) F).inv ≫ whiskerRight X✝.hom F }.hom ≫
      (𝟭 (CosimplicialObject D)).map (whiskerRight η.right F)

Tactic:
ext
        dsimp
        rw [Category.id_comp, Category.id_comp, ← F.map_comp, ← F.map_comp]
        simp [w_app, map_comp]

Goal state after:
Module: Mathlib.CategoryTheory.Subfunctor.OfSection

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
F : Cᵒᵖ ⥤ Type (max v w)
X : Cᵒᵖ
x : F.obj X
⊢ ofSection x = range (uliftYonedaEquiv.symm x)

Tactic:
ext U y
  dsimp [uliftYonedaEquiv]
  simp only [Set.mem_range, ULift.exists]
  constructor
  · rintro ⟨f, rfl⟩
    exact ⟨f.unop, rfl⟩
  · rintro ⟨f, rfl⟩
    exact ⟨f.op, rfl⟩

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

Goal state before:
pp: X : SSet
n : ℕ
x : X _⦋n⦌
m₁ m₂ : ℕ
f₁ : ⦋n⦌ ⟶ ⦋m₁⦌
hf₁ : SplitEpi f₁
y₁ : ↑(X.nonDegenerate m₁)
hy₁ : x = (ConcreteCategory.hom (X.map f₁.op)) ↑y₁
f₂ : ⦋n⦌ ⟶ ⦋m₂⦌
y₂ : X _⦋m₂⦌
hy₂ : x = (ConcreteCategory.hom (X.map f₂.op)) y₂
⊢ IsIso (factorThruImage (g hf₁ f₂))

Tactic:
have := map_g_op_y₂ hf₁ hy₁ hy₂

Goal state after:
X : SSet
n : ℕ
x : X _⦋n⦌
m₁ m₂ : ℕ
f₁ : ⦋n⦌ ⟶ ⦋m₁⦌
hf₁ : SplitEpi f₁
y₁ : ↑(X.nonDegenerate m₁)
hy₁ : x = (ConcreteCategory.hom (X.map f₁.op)) ↑y₁
f₂ : ⦋n⦌ ⟶ ⦋m₂⦌
y₂ : X _⦋m₂⦌
hy₂ : x = (ConcreteCategory.hom (X.map f₂.op)) y₂
this : (ConcreteCategory.hom (X.map (g hf₁ f₂).op)) y₂ = ↑y₁
⊢ IsIso (factorThruImage (g hf₁ f₂))
Module: Mathlib.AlgebraicTopology.SimplicialSet.Subcomplex

Goal state before:
pp: X Y : SSet
f : X ⟶ Y
⊢ Monotone fun S ↦ S.image f

Tactic:
intro S

Goal state after:
X Y : SSet
f : X ⟶ Y
S : X.Subcomplex
⊢ ∀ ⦃b : X.Subcomplex⦄, S ≤ b → (fun S ↦ S.image f) S ≤ (fun S ↦ S.image f) b
Module: Mathlib.AlgebraicTopology.SimplicialSet.Finite

Goal state before:
pp: X : SSet
inst✝ : X.Finite
this : Fintype X.N
φ : X.N → ℕ := fun x ↦ x.dim
h : (Finset.image φ ⊤).Nonempty
d : ℕ
hd : (Finset.image φ ⊤).max = ↑d
x✝¹ : ℕ
x✝ : x✝¹ ∈ Finset.image φ ⊤
⊢ x✝¹ < d + 1

Tactic:
by grind [WithBot.coe_le_coe, → Finset.le_max]

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

Goal state before:
pp: case refine_2
X : SSet
A B : X.Subcomplex
h : ∀ (s : X.N), s.subcomplex ≤ A → s.subcomplex ≤ B
n : ℕ
x : ↑(X.nonDegenerate n)
hx : ↑x ∈ A.obj (Opposite.op ⦋n⦌)
⊢ ↑x ∈ B.obj (Opposite.op ⦋n⦌)

Tactic:
· simpa using! h (N.mk _ x.prop) (by simpa)

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

Goal state before:
pp: X : SSet
n : ℕ
x✝ : X _⦋n⦌
m : ℕ
f₁ : ⦋n⦌ ⟶ ⦋m⦌
inst✝ : Epi f₁
y₁ : ↑(X.nonDegenerate m)
hy₁ : x✝ = (ConcreteCategory.hom (X.map f₁.op)) ↑y₁
f₂ : ⦋n⦌ ⟶ ⦋m⦌
y₂ : ↑(X.nonDegenerate m)
hy₂ : x✝ = (ConcreteCategory.hom (X.map f₂.op)) ↑y₂
x : Fin (⦋n⦌.len + 1)
this : ∃ hf₁, (SimplexCategory.Hom.toOrderHom hf₁.section_) ((SimplexCategory.Hom.toOrderHom f₁) x) = x
⊢ (SimplexCategory.Hom.toOrderHom f₁) x = (SimplexCategory.Hom.toOrderHom f₂) x

Tactic:
obtain ⟨hf₁, hf₁'⟩ := this

Goal state after:
X : SSet
n : ℕ
x✝ : X _⦋n⦌
m : ℕ
f₁ : ⦋n⦌ ⟶ ⦋m⦌
inst✝ : Epi f₁
y₁ : ↑(X.nonDegenerate m)
hy₁ : x✝ = (ConcreteCategory.hom (X.map f₁.op)) ↑y₁
f₂ : ⦋n⦌ ⟶ ⦋m⦌
y₂ : ↑(X.nonDegenerate m)
hy₂ : x✝ = (ConcreteCategory.hom (X.map f₂.op)) ↑y₂
x : Fin (⦋n⦌.len + 1)
hf₁ : SplitEpi f₁
hf₁' : (SimplexCategory.Hom.toOrderHom hf₁.section_) ((SimplexCategory.Hom.toOrderHom f₁) x) = x
⊢ (SimplexCategory.Hom.toOrderHom f₁) x = (SimplexCategory.Hom.toOrderHom f₂) x
Module: Mathlib.AlgebraicTopology.SimplicialSet.NerveNondegenerate

Goal state before:
pp: X : Type u_1
inst✝ : PartialOrder X
n : ℕ
s : nerve X _⦋n + 1⦌
i : Fin (n + 1)
hi : s.obj i.castSucc = s.obj i.succ
⊢ s ∈ Set.range ⇑(ConcreteCategory.hom (SimplicialObject.σ (nerve X) i))

Tactic:
rwa [← mem_range_nerve_σ_iff] at hi

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

Goal state before:
pp: X : Type u_1
inst✝ : PartialOrder X
n : ℕ
s : nerve X _⦋n⦌
⊢ s ∈ (nerve X).nonDegenerate n ↔ Function.Injective s.obj

Tactic:
by
  rw [mem_nerve_nonDegenerate_iff_strictMono]
  refine ⟨fun h ↦ h.injective, fun h i j hij ↦ ?_⟩
  obtain h' | h' := (s.monotone hij.le).lt_or_eq
  · exact h'
  · exact ((h h').not_lt hij).elim

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

Goal state before:
pp: i j : ℕ
h : (up ℕ).Rel i j
⊢ Odd (i + j)

Tactic:
by
  subst h
  norm_num

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

Goal state before:
pp: i j : ℕ
h : (down ℕ).Rel i j
⊢ Odd (i + j)

Tactic:
by
  subst h
  norm_num

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

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasZeroObject C
⊢ ∀ {X Y : C} (f : X ⟶ Y),
    (Functor.const SimplexCategoryᵒᵖ ⋙ alternatingFaceMapComplex C).map f ≫
        (HomologicalComplex.Hom.isoOfComponents
            (fun x ↦ Iso.refl (((Functor.const SimplexCategoryᵒᵖ ⋙ alternatingFaceMapComplex C).obj Y).X x)) ⋯).hom =
      (HomologicalComplex.Hom.isoOfComponents
            (fun x ↦ Iso.refl (((Functor.const SimplexCategoryᵒᵖ ⋙ alternatingFaceMapComplex C).obj X).X x)) ⋯).hom ≫
        ChainComplex.alternatingConst.map f

Tactic:
intros

Goal state after:
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasZeroObject C
X✝ Y✝ : C
f✝ : X✝ ⟶ Y✝
⊢ (Functor.const SimplexCategoryᵒᵖ ⋙ alternatingFaceMapComplex C).map f✝ ≫
      (HomologicalComplex.Hom.isoOfComponents
          (fun x ↦ Iso.refl (((Functor.const SimplexCategoryᵒᵖ ⋙ alternatingFaceMapComplex C).obj Y✝).X x)) ⋯).hom =
    (HomologicalComplex.Hom.isoOfComponents
          (fun x ↦ Iso.refl (((Functor.const SimplexCategoryᵒᵖ ⋙ alternatingFaceMapComplex C).obj X✝).X x)) ⋯).hom ≫
      ChainComplex.alternatingConst.map f✝
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 : SimplexCategoryᵒᵖ
⊢ ed.s' ≫ X.hom.app (op ⦋0⦌) = 𝟙 X.right

Tactic:
exact ed.s'_comp_ε

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
X : GradedObject (I₁ × I₂) C
d₁ : (i₁ i₁' : I₁) → (i₂ : I₂) → X (i₁, i₂) ⟶ X (i₁', i₂)
d₂ : (i₁ : I₁) → (i₂ i₂' : I₂) → X (i₁, i₂) ⟶ X (i₁, i₂')
shape₁ : ∀ (i₁ i₁' : I₁), ¬c₁.Rel i₁ i₁' → ∀ (i₂ : I₂), d₁ i₁ i₁' i₂ = 0
shape₂ : ∀ (i₁ : I₁) (i₂ i₂' : I₂), ¬c₂.Rel i₂ i₂' → d₂ i₁ i₂ i₂' = 0
d₁_comp_d₁ : ∀ (i₁ i₁' i₁'' : I₁) (i₂ : I₂), d₁ i₁ i₁' i₂ ≫ d₁ i₁' i₁'' i₂ = 0
d₂_comp_d₂ : ∀ (i₁ : I₁) (i₂ i₂' i₂'' : I₂), d₂ i₁ i₂ i₂' ≫ d₂ i₁ i₂' i₂'' = 0
comm : ∀ (i₁ i₁' : I₁) (i₂ i₂' : I₂), d₁ i₁ i₁' i₂ ≫ d₂ i₁' i₂ i₂' = d₂ i₁ i₂ i₂' ≫ d₁ i₁ i₁' i₂'
i₁ : I₁
⊢ ∀ (i j k : I₂), c₂.Rel i j → c₂.Rel j k → d₂ i₁ i j ≫ d₂ i₁ j k = 0

Tactic:
intros

Goal state after:
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
X : GradedObject (I₁ × I₂) C
d₁ : (i₁ i₁' : I₁) → (i₂ : I₂) → X (i₁, i₂) ⟶ X (i₁', i₂)
d₂ : (i₁ : I₁) → (i₂ i₂' : I₂) → X (i₁, i₂) ⟶ X (i₁, i₂')
shape₁ : ∀ (i₁ i₁' : I₁), ¬c₁.Rel i₁ i₁' → ∀ (i₂ : I₂), d₁ i₁ i₁' i₂ = 0
shape₂ : ∀ (i₁ : I₁) (i₂ i₂' : I₂), ¬c₂.Rel i₂ i₂' → d₂ i₁ i₂ i₂' = 0
d₁_comp_d₁ : ∀ (i₁ i₁' i₁'' : I₁) (i₂ : I₂), d₁ i₁ i₁' i₂ ≫ d₁ i₁' i₁'' i₂ = 0
d₂_comp_d₂ : ∀ (i₁ : I₁) (i₂ i₂' i₂'' : I₂), d₂ i₁ i₂ i₂' ≫ d₂ i₁ i₂' i₂'' = 0
comm : ∀ (i₁ i₁' : I₁) (i₂ i₂' : I₂), d₁ i₁ i₁' i₂ ≫ d₂ i₁' i₂ i₂' = d₂ i₁ i₂ i₂' ≫ d₁ i₁ i₁' i₂'
i₁ : I₁
i✝ j✝ k✝ : I₂
a✝¹ : c₂.Rel i✝ j✝
a✝ : c₂.Rel j✝ k✝
⊢ d₂ i₁ i✝ j✝ ≫ d₂ i₁ j✝ k✝ = 0
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
X : GradedObject (I₁ × I₂) C
d₁ : (i₁ i₁' : I₁) → (i₂ : I₂) → X (i₁, i₂) ⟶ X (i₁', i₂)
d₂ : (i₁ : I₁) → (i₂ i₂' : I₂) → X (i₁, i₂) ⟶ X (i₁, i₂')
shape₁ : ∀ (i₁ i₁' : I₁), ¬c₁.Rel i₁ i₁' → ∀ (i₂ : I₂), d₁ i₁ i₁' i₂ = 0
shape₂ : ∀ (i₁ : I₁) (i₂ i₂' : I₂), ¬c₂.Rel i₂ i₂' → d₂ i₁ i₂ i₂' = 0
d₁_comp_d₁ : ∀ (i₁ i₁' i₁'' : I₁) (i₂ : I₂), d₁ i₁ i₁' i₂ ≫ d₁ i₁' i₁'' i₂ = 0
d₂_comp_d₂ : ∀ (i₁ : I₁) (i₂ i₂' i₂'' : I₂), d₂ i₁ i₂ i₂' ≫ d₂ i₁ i₂' i₂'' = 0
comm : ∀ (i₁ i₁' : I₁) (i₂ i₂' : I₂), d₁ i₁ i₁' i₂ ≫ d₂ i₁' i₂ i₂' = d₂ i₁ i₂ i₂' ≫ d₁ i₁ i₁' i₂'
i₁ i₁' : I₁
⊢ ∀ (i j : I₂),
    c₂.Rel i j →
      d₁ i₁ i₁' i ≫ { X := fun i₂ ↦ X (i₁', i₂), d := fun i₂ i₂' ↦ d₂ i₁' i₂ i₂', shape := ⋯, d_comp_d' := ⋯ }.d i j =
        { X := fun i₂ ↦ X (i₁, i₂), d := fun i₂ i₂' ↦ d₂ i₁ i₂ i₂', shape := ⋯, d_comp_d' := ⋯ }.d i j ≫ d₁ i₁ i₁' j

Tactic:
intros

Goal state after:
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
X : GradedObject (I₁ × I₂) C
d₁ : (i₁ i₁' : I₁) → (i₂ : I₂) → X (i₁, i₂) ⟶ X (i₁', i₂)
d₂ : (i₁ : I₁) → (i₂ i₂' : I₂) → X (i₁, i₂) ⟶ X (i₁, i₂')
shape₁ : ∀ (i₁ i₁' : I₁), ¬c₁.Rel i₁ i₁' → ∀ (i₂ : I₂), d₁ i₁ i₁' i₂ = 0
shape₂ : ∀ (i₁ : I₁) (i₂ i₂' : I₂), ¬c₂.Rel i₂ i₂' → d₂ i₁ i₂ i₂' = 0
d₁_comp_d₁ : ∀ (i₁ i₁' i₁'' : I₁) (i₂ : I₂), d₁ i₁ i₁' i₂ ≫ d₁ i₁' i₁'' i₂ = 0
d₂_comp_d₂ : ∀ (i₁ : I₁) (i₂ i₂' i₂'' : I₂), d₂ i₁ i₂ i₂' ≫ d₂ i₁ i₂' i₂'' = 0
comm : ∀ (i₁ i₁' : I₁) (i₂ i₂' : I₂), d₁ i₁ i₁' i₂ ≫ d₂ i₁' i₂ i₂' = d₂ i₁ i₂ i₂' ≫ d₁ i₁ i₁' i₂'
i₁ i₁' : I₁
i✝ j✝ : I₂
a✝ : c₂.Rel i✝ j✝
⊢ d₁ i₁ i₁' i✝ ≫ { X := fun i₂ ↦ X (i₁', i₂), d := fun i₂ i₂' ↦ d₂ i₁' i₂ i₂', shape := ⋯, d_comp_d' := ⋯ }.d i✝ j✝ =
    { X := fun i₂ ↦ X (i₁, i₂), d := fun i₂ i₂' ↦ d₂ i₁ i₂ i₂', shape := ⋯, d_comp_d' := ⋯ }.d i✝ j✝ ≫ d₁ i₁ i₁' j✝
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 L : HomologicalComplex₂ C c₁ c₂
f : K ⟶ L
i : I₂
⊢ ∀ (i_1 j : I₁), c₁.Rel i_1 j → (f.f i_1).f i ≫ (L.flip.X i).d i_1 j = (K.flip.X i).d i_1 j ≫ (f.f j).f i

Tactic:
intros

Goal state after:
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 L : HomologicalComplex₂ C c₁ c₂
f : K ⟶ L
i : I₂
i✝ j✝ : I₁
a✝ : c₁.Rel i✝ j✝
⊢ (f.f i✝).f i ≫ (L.flip.X i).d i✝ j✝ = (K.flip.X i).d i✝ j✝ ≫ (f.f j✝).f i
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 L : HomologicalComplex₂ C c₁ c₂
f : K ⟶ L
⊢ ∀ (i j : I₂),
    c₂.Rel i j →
      { f := fun j ↦ (f.f j).f i, comm' := ⋯ } ≫ L.flip.d i j =
        K.flip.d i j ≫ { f := fun j_1 ↦ (f.f j_1).f j, comm' := ⋯ }

Tactic:
intros

Goal state after:
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 L : HomologicalComplex₂ C c₁ c₂
f : K ⟶ L
i✝ j✝ : I₂
a✝ : c₂.Rel i✝ j✝
⊢ { f := fun j ↦ (f.f j).f i✝, comm' := ⋯ } ≫ L.flip.d i✝ j✝ =
    K.flip.d i✝ j✝ ≫ { f := fun j ↦ (f.f j).f j✝, comm' := ⋯ }
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 : HomologicalComplex C₁ c₁
K₂ K₂' : HomologicalComplex C₂ c₂
φ : K₂ ⟶ K₂'
⊢ ∀ (i₁ i₁' : I₁) (i₂ : I₂),
    c₁.Rel i₁ i₁' →
      (F.obj (K₁.X i₁)).map (φ.f i₂) ≫ (F.map (K₁.d i₁ i₁')).app (K₂'.X i₂) =
        (F.map (K₁.d i₁ i₁')).app (K₂.X i₂) ≫ (F.obj (K₁.X i₁')).map (φ.f i₂)

Tactic:
intros

Goal state after:
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 : HomologicalComplex C₁ c₁
K₂ K₂' : HomologicalComplex C₂ c₂
φ : K₂ ⟶ K₂'
i₁✝ i₁'✝ : I₁
i₂✝ : I₂
a✝ : c₁.Rel i₁✝ i₁'✝
⊢ (F.obj (K₁.X i₁✝)).map (φ.f i₂✝) ≫ (F.map (K₁.d i₁✝ i₁'✝)).app (K₂'.X i₂✝) =
    (F.map (K₁.d i₁✝ i₁'✝)).app (K₂.X i₂✝) ≫ (F.obj (K₁.X i₁'✝)).map (φ.f i₂✝)
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 : HomologicalComplex C₁ c₁
K₂ K₂' : HomologicalComplex C₂ c₂
φ : K₂ ⟶ K₂'
⊢ ∀ (i₁ : I₁) (i₂ i₂' : I₂),
    c₂.Rel i₂ i₂' →
      (F.obj (K₁.X i₁)).map (φ.f i₂) ≫ (F.obj (K₁.X i₁)).map (K₂'.d i₂ i₂') =
        (F.obj (K₁.X i₁)).map (K₂.d i₂ i₂') ≫ (F.obj (K₁.X i₁)).map (φ.f i₂')

Tactic:
intros

Goal state after:
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 : HomologicalComplex C₁ c₁
K₂ K₂' : HomologicalComplex C₂ c₂
φ : K₂ ⟶ K₂'
i₁✝ : I₁
i₂✝ i₂'✝ : I₂
a✝ : c₂.Rel i₂✝ i₂'✝
⊢ (F.obj (K₁.X i₁✝)).map (φ.f i₂✝) ≫ (F.obj (K₁.X i₁✝)).map (K₂'.d i₂✝ i₂'✝) =
    (F.obj (K₁.X i₁✝)).map (K₂.d i₂✝ i₂'✝) ≫ (F.obj (K₁.X i₁✝)).map (φ.f i₂'✝)
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:
intros

Goal state after:
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
a✝ : 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✝)
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
A : C
i₁₂ : I₁₂
f g : (K.total c₁₂).X i₁₂ ⟶ A
h :
  ∀ (i₁ : I₁) (i₂ : I₂) (hi : c₁.π c₂ c₁₂ (i₁, i₂) = i₁₂), K.ιTotal c₁₂ i₁ i₂ i₁₂ hi ≫ f = K.ιTotal c₁₂ i₁ i₂ i₁₂ hi ≫ g
⊢ f = g

Tactic:
apply GradedObject.mapObj_ext

Goal state after:
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
A : C
i₁₂ : I₁₂
f g : (K.total c₁₂).X i₁₂ ⟶ A
h :
  ∀ (i₁ : I₁) (i₂ : I₂) (hi : c₁.π c₂ c₁₂ (i₁, i₂) = i₁₂), K.ιTotal c₁₂ i₁ i₂ i₁₂ hi ≫ f = K.ιTotal c₁₂ i₁ i₂ i₁₂ hi ≫ g
⊢ ∀ (i : I₁ × I₂) (hij : c₁.π c₂ c₁₂ i = i₁₂),
    K.toGradedObject.ιMapObj (c₁.π c₂ c₁₂) i i₁₂ hij ≫ f = K.toGradedObject.ιMapObj (c₁.π c₂ c₁₂) i i₁₂ hij ≫ g
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
⊢ Function.RightInverse
    (fun x_1 ↦
      match x_1 with
      | ⟨(a, b), h⟩ => ⟨(a - x, b), ⋯⟩)
    fun x_1 ↦
    match x_1 with
    | ⟨(a, b), h⟩ => ⟨(a + x, b), ⋯⟩

Tactic:
intro ⟨⟨a, b⟩, h⟩

Goal state after:
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 a b : ℤ
h : (a, b) ∈ (up ℤ).π (up ℤ) (up ℤ) ⁻¹' {n + x}
⊢ (fun x_1 ↦
        match x_1 with
        | ⟨(a, b), h⟩ => ⟨(a + x, b), ⋯⟩)
      ((fun x_1 ↦
          match x_1 with
          | ⟨(a, b), h⟩ => ⟨(a - x, b), ⋯⟩)
        ⟨(a, b), h⟩) =
    ⟨(a, b), h⟩
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 : n₀ + 1 = n₁
p q : ℤ
hpq : p + q = n₀
⊢ (((shiftFunctor₁ C x).obj K).d₂ (up ℤ) p q n₁ ≫
      ((shiftFunctor₁ C x).obj K).totalDesc fun p q hpq ↦ K.ιTotal (up ℤ) (p + x) q n₁' ⋯) =
    x.negOnePow • K.d₂ (up ℤ) (p + x) q n₁'

Tactic:
((shiftFunctor₁ C x).obj K).d₂_eq _ _ rfl _ (by dsimp; lia),

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 : n₀ + 1 = n₁
p q : ℤ
hpq : p + q = n₀
⊢ (((up ℤ).ε₂ (up ℤ) (up ℤ) (p, q) •
        (((shiftFunctor₁ C x).obj K).X p).d q (q + 1) ≫ ((shiftFunctor₁ C x).obj K).ιTotal (up ℤ) p (q + 1) n₁ ⋯) ≫
      ((shiftFunctor₁ C x).obj K).totalDesc fun p q hpq ↦ K.ιTotal (up ℤ) (p + x) q n₁' ⋯) =
    x.negOnePow • K.d₂ (up ℤ) (p + x) q n₁'
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
⊢ (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) ≫ L₁.ιMapBifunctorOrZero L₂ F c (c₁.prev i₁) i₂ (c.prev j)) ≫
      (((F.mapBifunctorHomologicalComplex c₁ c₂).obj L₁).obj L₂).D₂ c (c.prev j) 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:
· rw [h₁.zero _ _ h₃, Functor.map_zero, zero_app, zero_comp,
        smul_zero, zero_comp, zero_eq_neg]
      by_cases h₄ : c₂.Rel i₂ (c₂.next i₂)
      · by_cases h₅ : c.Rel j (c.next j)
        · rw [HomologicalComplex₂.d₂_eq _ _ _ h₄ _ (by rw [← ComplexShape.next_π₂ c₁ c i₁ h₄, h]),
            Linear.units_smul_comp, assoc, Functor.mapBifunctorHomologicalComplex_obj_obj_X_d,
            ιMapBifunctor_hom₁ _ _ _ _ _ _ _ _ _ _ rfl, h₁.zero _ _ h₃,
            Functor.map_zero, zero_app, zero_comp, smul_zero, comp_zero, smul_zero]
        · rw [zero₁ _ _ _ _ _ _ h₅, comp_zero]
      · rw [HomologicalComplex₂.d₂_eq_zero _ _ _ _ _ h₄, zero_comp]

Goal state after: