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

Goal state before:
pp: case a
X : SSet
x y : X.N
h : x.subcomplex = y.subcomplex
⊢ x.subcomplex ≤ y.subcomplex

Tactic:
h

Goal state after:
case a
X : SSet
x y : X.N
h : x.subcomplex = y.subcomplex
⊢ y.subcomplex ≤ y.subcomplex
Module: Mathlib.AlgebraicTopology.SimplicialSet.NonDegenerateSimplices

Goal state before:
pp: case a
X : SSet
x y : X.N
h : x.subcomplex = y.subcomplex
⊢ y.subcomplex ≤ x.subcomplex

Tactic:
h

Goal state after:
case a
X : SSet
x y : X.N
h : x.subcomplex = y.subcomplex
⊢ y.subcomplex ≤ y.subcomplex
Module: Mathlib.AlgebraicTopology.SimplicialSet.Subcomplex

Goal state before:
pp: X Y : SSet
A : X.Subcomplex
f : X ⟶ Y
⊢ ⊤ ≤ range (A.toImage f)

Tactic:
rintro m ⟨_, ⟨y, hy, rfl⟩⟩ _

Goal state after:
X Y : SSet
A : X.Subcomplex
f : X ⟶ Y
m : SimplexCategoryᵒᵖ
y : X.obj m
hy : y ∈ A.obj m
a✝ : ⟨(ConcreteCategory.hom (f.app m)) y, ⋯⟩ ∈ ⊤.obj m
⊢ ⟨(ConcreteCategory.hom (f.app m)) y, ⋯⟩ ∈ (range (A.toImage f)).obj m
Module: Mathlib.AlgebraicTopology.SimplicialSet.Subcomplex

Goal state before:
pp: X Y : SSet
f : X ⟶ Y
S T : X.Subcomplex
⊢ S ≤ T → (fun S ↦ S.image f) S ≤ (fun S ↦ S.image f) T

Tactic:
h

Goal state after:
X Y : SSet
f : X ⟶ Y
S T : X.Subcomplex
h : S ≤ T
⊢ (fun S ↦ S.image f) S ≤ (fun S ↦ S.image f) T
Module: Mathlib.AlgebraicTopology.SimplicialSet.NonDegenerateSimplices

Goal state before:
pp: case a
X : SSet
x✝¹ x✝ : X.N
h : x✝¹.subcomplex = x✝.subcomplex
⊢ x✝¹.subcomplex ≤ x✝.subcomplex

Tactic:
h

Goal state after:
case a
X : SSet
x✝¹ x✝ : X.N
h : x✝¹.subcomplex = x✝.subcomplex
⊢ x✝.subcomplex ≤ x✝.subcomplex
Module: Mathlib.AlgebraicTopology.SimplicialSet.NonDegenerateSimplices

Goal state before:
pp: case a
X : SSet
x✝¹ x✝ : X.N
h : x✝¹.subcomplex = x✝.subcomplex
⊢ x✝.subcomplex ≤ x✝¹.subcomplex

Tactic:
h

Goal state after:
case a
X : SSet
x✝¹ x✝ : X.N
h : x✝¹.subcomplex = x✝.subcomplex
⊢ x✝.subcomplex ≤ x✝.subcomplex
Module: Mathlib.AlgebraicTopology.SimplicialSet.NonDegenerateSimplices

Goal state before:
pp: X : SSet
x : X.S
y : X.N
⊢ x.toN = y → y.subcomplex = x.subcomplex

Tactic:
rintro rfl; simp

Goal state after:
Module: Mathlib.Order.Fin.SuccAboveOrderIso

Goal state before:
pp: case left
n : ℕ
i : Fin (n + 2)
a b : Fin (n + 1)
⊢ (fun a ↦ ⟨i.succAboveOrderEmb a, ⋯⟩) a = (fun a ↦ ⟨i.succAboveOrderEmb a, ⋯⟩) b → a = b

Tactic:
h

Goal state after:
case left
n : ℕ
i : Fin (n + 2)
a b : Fin (n + 1)
h : (fun a ↦ ⟨i.succAboveOrderEmb a, ⋯⟩) a = (fun a ↦ ⟨i.succAboveOrderEmb a, ⋯⟩) b
⊢ a = b
Module: Mathlib.AlgebraicTopology.SimplicialSet.Nerve

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
x₀ x₁ x₂ : C
f₀₁ : x₀ ⟶ x₁
f₁₂ : x₁ ⟶ x₂
f₀₂ : x₀ ⟶ x₂
⊢ Nonempty ((edgeMk f₀₁).CompStruct (edgeMk f₁₂) (edgeMk f₀₂)) ↔ f₀₁ ≫ f₁₂ = f₀₂

Tactic:
let h' : Edge.CompStruct (edgeMk f₀₁) (edgeMk f₁₂) (edgeMk (f₀₁ ≫ f₁₂)) :=
      Edge.CompStruct.mk (ComposableArrows.mk₂ f₀₁ f₁₂)
        (by cat_disch) (by cat_disch) (by cat_disch)

Goal state after:
C : Type u
inst✝ : Category.{v, u} C
x₀ x₁ x₂ : C
f₀₁ : x₀ ⟶ x₁
f₁₂ : x₁ ⟶ x₂
f₀₂ : x₀ ⟶ x₂
h' : (edgeMk f₀₁).CompStruct (edgeMk f₁₂) (edgeMk (f₀₁ ≫ f₁₂)) :=
  Edge.CompStruct.mk (ComposableArrows.mk₂ f₀₁ f₁₂) ⋯ ⋯ ⋯
⊢ Nonempty ((edgeMk f₀₁).CompStruct (edgeMk f₁₂) (edgeMk f₀₂)) ↔ f₀₁ ≫ f₁₂ = f₀₂
Module: Mathlib.AlgebraicTopology.SimplicialSet.StdSimplex

Goal state before:
pp: case right.refine_1
n m : ℕ
S : Finset (Fin (n + 1))
hS : #S = m + 1
e : Fin (m + 1) ≃o ↥S := monoEquivOfFin ↥S ⋯
a b : Fin ((unop (op ⦋m⦌)).len + 1)
⊢ (Hom.toOrderHom (objEquiv (objMk ((OrderHom.Subtype.val fun x ↦ x ∈ S).comp e.toOrderEmbedding.toOrderHom)))) a =
      (Hom.toOrderHom (objEquiv (objMk ((OrderHom.Subtype.val fun x ↦ x ∈ S).comp e.toOrderEmbedding.toOrderHom)))) b →
    a = b

Tactic:
h

Goal state after:
case right.refine_1
n m : ℕ
S : Finset (Fin (n + 1))
hS : #S = m + 1
e : Fin (m + 1) ≃o ↥S := monoEquivOfFin ↥S ⋯
a b : Fin ((unop (op ⦋m⦌)).len + 1)
h :
  (Hom.toOrderHom (objEquiv (objMk ((OrderHom.Subtype.val fun x ↦ x ∈ S).comp e.toOrderEmbedding.toOrderHom)))) a =
    (Hom.toOrderHom (objEquiv (objMk ((OrderHom.Subtype.val fun x ↦ x ∈ S).comp e.toOrderEmbedding.toOrderHom)))) b
⊢ a = b
Module: Mathlib.AlgebraicTopology.SimplicialSet.StdSimplex

Goal state before:
pp: n d : ℕ
A : Δ[n].Subcomplex
x : ↑(Δ[n].nonDegenerate d)
⊢ ↑(nonDegenerateEquiv'.symm (nonDegenerateEquiv' x)) ∈ A.obj (op ⦋d⦌) ↔ Subcomplex.ofSimplex ↑x ≤ A

Tactic:
Equiv.symm_apply_apply,

Goal state after:
n d : ℕ
A : Δ[n].Subcomplex
x : ↑(Δ[n].nonDegenerate d)
⊢ ↑x ∈ A.obj (op ⦋d⦌) ↔ Subcomplex.ofSimplex ↑x ≤ A
Module: Mathlib.Algebra.Homology.TotalComplex

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

Tactic:
h

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

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

Tactic:
h

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

Goal state before:
pp: C₁ : Type u_1
C₂ : Type u_2
C₁₂ : Type u_3
C₂₃ : Type u_4
C₃ : Type u_5
C₄ : Type u_6
inst✝⁹ : Category.{v_1, u_1} C₁
inst✝⁸ : Category.{v_2, u_2} C₂
inst✝⁷ : Category.{v_3, u_5} C₃
inst✝⁶ : Category.{v_4, u_6} C₄
inst✝⁵ : Category.{v_5, u_3} C₁₂
inst✝⁴ : Category.{v_6, u_4} C₂₃
F₁₂ : C₁ ⥤ C₂ ⥤ C₁₂
G : C₁₂ ⥤ C₃ ⥤ C₄
F : C₁ ⥤ C₂₃ ⥤ C₄
G₂₃ : C₂ ⥤ C₃ ⥤ C₂₃
associator : bifunctorComp₁₂ F₁₂ G ≅ bifunctorComp₂₃ F G₂₃
I₁ : Type u_7
I₂ : Type u_8
I₃ : Type u_9
J : Type u_10
r : I₁ × I₂ × I₃ → J
ρ₁₂ : BifunctorComp₁₂IndexData r
ρ₂₃ : BifunctorComp₂₃IndexData r
X₁ : GradedObject I₁ C₁
X₂ : GradedObject I₂ C₂
X₃ : GradedObject I₃ C₃
inst✝³ : (((mapBifunctor F₁₂ I₁ I₂).obj X₁).obj X₂).HasMap ρ₁₂.p
inst✝² : (((mapBifunctor G ρ₁₂.I₁₂ I₃).obj (mapBifunctorMapObj F₁₂ ρ₁₂.p X₁ X₂)).obj X₃).HasMap ρ₁₂.q
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
H₁₂ : HasGoodTrifunctor₁₂Obj F₁₂ G ρ₁₂ X₁ X₂ X₃
H₂₃ : HasGoodTrifunctor₂₃Obj F G₂₃ ρ₂₃ X₁ X₂ X₃
i₁ : I₁
i₂ : I₂
i₃ : I₃
j : J
h : r (i₁, i₂, i₃) = j
⊢ ιMapBifunctorBifunctor₂₃MapObj F G₂₃ ρ₂₃ X₁ X₂ X₃ i₁ i₂ i₃ j h ≫
      (mapBifunctorAssociator associator ρ₁₂ ρ₂₃ X₁ X₂ X₃).inv j =
    ((associator.inv.app (X₁ i₁)).app (X₂ i₂)).app (X₃ i₃) ≫
      ιMapBifunctor₁₂BifunctorMapObj F₁₂ G ρ₁₂ X₁ X₂ X₃ i₁ i₂ i₃ j h

Tactic:
by
  rw [← cancel_mono ((mapBifunctorAssociator associator ρ₁₂ ρ₂₃ X₁ X₂ X₃).hom j),
    assoc, assoc, Iso.inv_hom_id_eval, comp_id, ι_mapBifunctorAssociator_hom,
    ← NatTrans.comp_app_assoc, ← NatTrans.comp_app, Iso.inv_hom_id_app,
    NatTrans.id_app, NatTrans.id_app, id_comp]

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
i₂ : I₂
i₁ : I₁
h : c₂.π c₁ c (i₂, i₁) = j
⊢ c₂.π c₁ c (i₂, i₁) = j

Tactic:
h

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
i₂ : I₂
i₁ : I₁
h : c₂.π c₁ c (i₂, i₁) = j
⊢ j = 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
⊢ c₁.π c₂ c (i₁, i₂) = j

Tactic:
h

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
i₁ : I₁
i₂ : I₂
h : c₁.π c₂ c (i₁, i₂) = j
⊢ j = j
Module: Mathlib.Algebra.Homology.TotalComplex

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

Tactic:
h

Goal state after:
case pos
C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Preadditive C
I₁ : Type u_2
I₂ : Type u_3
I₁₂ : Type u_4
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
K : HomologicalComplex₂ C c₁ c₂
c₁₂ : ComplexShape I₁₂
inst✝² : TotalComplexShape c₁ c₂ c₁₂
inst✝¹ : DecidableEq I₁₂
inst✝ : K.HasTotal c₁₂
i₁₂ i₁₂' i₁₂'' : I₁₂
h₁ : c₁₂.Rel i₁₂ i₁₂'
h₂ : c₁₂.Rel i₁₂' i₁₂''
i₁ : I₁
i₂ : I₂
h : c₁.π c₂ c₁₂ (i₁, i₂) = i₁₂
h₃ : ¬c₁.Rel i₁ (c₁.next i₁)
h₄ : c₂.Rel i₂ (c₂.next i₂)
⊢ c₁₂.next i₁₂ = c₁₂.next i₁₂
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 L M : HomologicalComplex₂ C c₁ c₂
φ : K ⟶ L
e : K ≅ L
ψ : L ⟶ M
c₁₂ : ComplexShape I₁₂
inst✝² : TotalComplexShape c₁ c₂ c₁₂
inst✝¹ : DecidableEq I₁₂
inst✝ : K.HasTotal c₁₂
x₁ y₁ : I₁
h₁ : x₁ = y₁
x₂ y₂ : I₂
h₂ : x₂ = y₂
i₁₂ : I₁₂
h : c₁.π c₂ c₁₂ (y₁, y₂) = i₁₂
⊢ c₁.π c₂ c₁₂ (y₁, y₂) = i₁₂

Tactic:
h

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 L M : HomologicalComplex₂ C c₁ c₂
φ : K ⟶ L
e : K ≅ L
ψ : L ⟶ M
c₁₂ : ComplexShape I₁₂
inst✝² : TotalComplexShape c₁ c₂ c₁₂
inst✝¹ : DecidableEq I₁₂
inst✝ : K.HasTotal c₁₂
x₁ y₁ : I₁
h₁ : x₁ = y₁
x₂ y₂ : I₂
h₂ : x₂ = y₂
i₁₂ : I₁₂
h : c₁.π c₂ c₁₂ (y₁, y₂) = i₁₂
⊢ i₁₂ = i₁₂
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]
    rw [add_comm]

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
i₁ : I₁
i₂ : I₂
j : J
h : c₂.π c₁ c (i₂, i₁) = j
⊢ c₂.π c₁ c (i₂, i₁) = j

Tactic:
h

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
i₁ : I₁
i₂ : I₂
j : J
h : c₂.π c₁ c (i₂, i₁) = j
⊢ j = 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
i₁ : I₁
i₂ : I₂
j : J
h : c₁.π c₂ c (i₁, i₂) = j
⊢ c₁.π c₂ c (i₁, i₂) = j

Tactic:
h

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
i₁ : I₁
i₂ : I₂
j : J
h : c₁.π c₂ c (i₁, i₂) = j
⊢ j = j
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
⊢ c.prev (c₁.π c₂ c (i₁, i₂)) = c.prev j

Tactic:
h

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
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
⊢ c.prev j = c.prev j
Module: Mathlib.Algebra.Homology.TotalComplexShift

Goal state before:
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)
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
⊢ 0 = x.negOnePow • (K.totalShift₁XIso x n₀ n₀' h₀).hom ≫ K.D₁ (up ℤ) n₀' n₁'

Tactic:
D₁_shape,

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)
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
⊢ 0 = x.negOnePow • (K.totalShift₁XIso x n₀ n₀' h₀).hom ≫ 0

case neg.h₁₂
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
⊢ ¬(up ℤ).Rel 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 : 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)
⊢ c.next (c₁.π c₂ c (i₁, i₂)) = c.next j

Tactic:
h

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
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)
⊢ c.next j = c.next j
Module: Mathlib.Algebra.Homology.TotalComplexShift

Goal state before:
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
⊢ 0 = y.negOnePow • (K.totalShift₂XIso y n₀ n₀' h₀).hom ≫ K.D₁ (up ℤ) n₀' n₁'

Tactic:
D₁_shape,

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
⊢ 0 = y.negOnePow • (K.totalShift₂XIso y n₀ n₀' h₀).hom ≫ 0

case neg.h₁₂
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
⊢ ¬(up ℤ).Rel n₀' n₁'
Module: Mathlib.Algebra.Homology.CommSq

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
X₁ X₂ X₃ X₄ : C
inst✝ : HasBinaryBiproduct X₂ X₃
f : X₁ ⟶ X₂
g : X₁ ⟶ X₃
inl : X₂ ⟶ X₄
inr : X₃ ⟶ X₄
sq : CommSq f g inl inr
h : IsColimit (PushoutCocone.mk inl inr ⋯)
s : Cofork (biprod.lift f (-g)) 0
m : sq.cokernelCofork.pt ⟶ s.pt
hm : Cofork.π sq.cokernelCofork ≫ m = s.π
⊢ m = PushoutCocone.IsColimit.desc h (biprod.inl ≫ s.π) (biprod.inr ≫ s.π) ⋯

Tactic:
apply PushoutCocone.IsColimit.hom_ext h

Goal state after:
case h₀
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
X₁ X₂ X₃ X₄ : C
inst✝ : HasBinaryBiproduct X₂ X₃
f : X₁ ⟶ X₂
g : X₁ ⟶ X₃
inl : X₂ ⟶ X₄
inr : X₃ ⟶ X₄
sq : CommSq f g inl inr
h : IsColimit (PushoutCocone.mk inl inr ⋯)
s : Cofork (biprod.lift f (-g)) 0
m : sq.cokernelCofork.pt ⟶ s.pt
hm : Cofork.π sq.cokernelCofork ≫ m = s.π
⊢ (PushoutCocone.mk inl inr ⋯).inl ≫ m =
    (PushoutCocone.mk inl inr ⋯).inl ≫ PushoutCocone.IsColimit.desc h (biprod.inl ≫ s.π) (biprod.inr ≫ s.π) ⋯

case h₁
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
X₁ X₂ X₃ X₄ : C
inst✝ : HasBinaryBiproduct X₂ X₃
f : X₁ ⟶ X₂
g : X₁ ⟶ X₃
inl : X₂ ⟶ X₄
inr : X₃ ⟶ X₄
sq : CommSq f g inl inr
h : IsColimit (PushoutCocone.mk inl inr ⋯)
s : Cofork (biprod.lift f (-g)) 0
m : sq.cokernelCofork.pt ⟶ s.pt
hm : Cofork.π sq.cokernelCofork ≫ m = s.π
⊢ (PushoutCocone.mk inl inr ⋯).inr ≫ m =
    (PushoutCocone.mk inl inr ⋯).inr ≫ PushoutCocone.IsColimit.desc h (biprod.inl ≫ s.π) (biprod.inr ≫ s.π) ⋯
Module: Mathlib.Algebra.Homology.BifunctorShift

Goal state before:
pp: C₁ : Type u_1
C₂ : Type u_2
D : Type u_3
inst✝⁸ : Category.{v_1, u_1} C₁
inst✝⁷ : Category.{v_2, u_2} C₂
inst✝⁶ : Category.{v_3, u_3} D
inst✝⁵ : Preadditive C₁
inst✝⁴ : Preadditive C₂
inst✝³ : Preadditive D
F : C₁ ⥤ C₂ ⥤ D
inst✝² : F.Additive
inst✝¹ : ∀ (X₁ : C₁), (F.obj X₁).Additive
inst✝ : ∀ (K₁ : CochainComplex C₁ ℤ) (K₂ : CochainComplex C₂ ℤ), K₁.HasMapBifunctor K₂ F
K₁ L₁ : CochainComplex C₁ ℤ
f : K₁ ⟶ L₁
n : ℤ
⊢ (commShiftIso (F.map₂CochainComplex.obj K₁) n).hom ≫
      whiskerRight (F.map₂CochainComplex.map f) (shiftFunctor (CochainComplex D ℤ) n) =
    (shiftFunctor (CochainComplex C₂ ℤ) n).whiskerLeft (F.map₂CochainComplex.map f) ≫
      (commShiftIso (F.map₂CochainComplex.obj L₁) n).hom

Tactic:
by
    ext K₂ d
    dsimp
    ext p q h
    simp [commShiftIso_map₂CochainComplex_hom_app,
      CochainComplex.ι_mapBifunctorShift₂Iso_hom_f _ _ _ _ _ _ _ _ (q + n) (d + n) rfl rfl,
      CochainComplex.ι_mapBifunctorShift₂Iso_hom_f_assoc _ _ _ _ _ _ _ _ (q + n) (d + n) rfl rfl]

Goal state after: