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

Goal state before:
pp: case inr
X : SSet
y : X.N
d : ℕ
x : X _⦋d⦌
hx : x ∈ X.nonDegenerate d
h : mk ↑⟨x, hx⟩ ⋯ < y
h' : (mk ↑⟨x, hx⟩ ⋯).dim = y.dim
f : ⦋(mk ↑⟨x, hx⟩ ⋯).dim⦌ ⟶ ⦋y.dim⦌
w✝ : Mono f
hf : (ConcreteCategory.hom (X.map f.op)) y.simplex = (mk ↑⟨x, hx⟩ ⋯).simplex
⊢ (mk ↑⟨x, hx⟩ ⋯).dim < y.dim

Tactic:
obtain ⟨d', ⟨y, hy⟩, rfl⟩ := y.mk_surjective

Goal state after:
case inr
X : SSet
d : ℕ
x : X _⦋d⦌
hx : x ∈ X.nonDegenerate d
d' : ℕ
y : X _⦋d'⦌
hy : y ∈ X.nonDegenerate d'
h : mk ↑⟨x, hx⟩ ⋯ < mk ↑⟨y, hy⟩ ⋯
h' : (mk ↑⟨x, hx⟩ ⋯).dim = (mk ↑⟨y, hy⟩ ⋯).dim
f : ⦋(mk ↑⟨x, hx⟩ ⋯).dim⦌ ⟶ ⦋(mk ↑⟨y, hy⟩ ⋯).dim⦌
w✝ : Mono f
hf : (ConcreteCategory.hom (X.map f.op)) (mk ↑⟨y, hy⟩ ⋯).simplex = (mk ↑⟨x, hx⟩ ⋯).simplex
⊢ (mk ↑⟨x, hx⟩ ⋯).dim < (mk ↑⟨y, hy⟩ ⋯).dim
Module: Mathlib.AlgebraicTopology.SimplicialSet.Subcomplex

Goal state before:
pp: 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

Tactic:
rw [image_le_iff]

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

Goal state before:
pp: X Y : SSet
B : X.Subcomplex
f : Y ⟶ X
⊢ (B.preimage f).image f ≤ B

Tactic:
rw [image_le_iff]

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

Goal state before:
pp: X : SSet
inst✝ : X.Finite
n : ℕ
x y : ↑(X.nonDegenerate n)
h : (fun x ↦ N.mk ↑x ⋯) x = (fun x ↦ N.mk ↑x ⋯) y
⊢ x = y

Tactic:
by
    rw [N.ext_iff, S.ext_iff'] at h
    aesop

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⦌
i : Fin (n + 1)
this : s.obj i.castSucc ≤ s.obj i.succ
⊢ s.obj i.castSucc = s.obj i.succ ↔ ¬s.obj i.castSucc < s.obj i.succ

Tactic:
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 0 Edge._proof_2 _proof_2 ≫ σ₂ 0 _proof_2 Edge._proof_2).op)) e.edge = e.edge

Tactic:
δ₂_zero_comp_σ₂_zero

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 := ⋯ }).op)) e.edge = e.edge
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
h' : (edgeMk f₀₁).CompStruct (edgeMk f₁₂) (edgeMk (f₀₁ ≫ f₁₂)) :=
  Edge.CompStruct.mk (ComposableArrows.mk₂ f₀₁ f₁₂) ⋯ ⋯ ⋯
x✝ : Nonempty ((edgeMk f₀₁).CompStruct (edgeMk f₁₂) (edgeMk f₀₂))
h : (edgeMk f₀₁).CompStruct (edgeMk f₁₂) (edgeMk f₀₂)
⊢ f₀₁ ≫ f₁₂ = f₀₂

Tactic:
rw [← Arrow.mk_inj]

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) ⋯ ⋯ ⋯
x✝ : Nonempty ((edgeMk f₀₁).CompStruct (edgeMk f₁₂) (edgeMk f₀₂))
h : (edgeMk f₀₁).CompStruct (edgeMk f₁₂) (edgeMk f₀₂)
⊢ Arrow.mk (f₀₁ ≫ f₁₂) = Arrow.mk f₀₂
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)
⊢ K.d₂ c₁₂ i₁ i₂ i₁₂' ≫ K.D₁ c₁₂ i₁₂' i₁₂'' =
    -(c₁.ε₁ c₂ c₁₂ (i₁, i₂) • (K.d i₁ (c₁.next i₁)).f i₂ ≫ K.d₂ c₁₂ (c₁.next i₁) i₂ i₁₂'')

Tactic:
by_cases h₄ : c₂.Rel i₂ (c₂.next i₂)

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)
⊢ K.d₂ c₁₂ i₁ i₂ i₁₂' ≫ K.D₁ c₁₂ i₁₂' i₁₂'' =
    -(c₁.ε₁ c₂ c₁₂ (i₁, i₂) • (K.d i₁ (c₁.next i₁)).f i₂ ≫ K.d₂ c₁₂ (c₁.next i₁) i₂ i₁₂'')

case neg
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)
⊢ K.d₂ c₁₂ i₁ i₂ i₁₂' ≫ K.D₁ c₁₂ i₁₂' i₁₂'' =
    -(c₁.ε₁ c₂ c₁₂ (i₁, i₂) • (K.d i₁ (c₁.next i₁)).f i₂ ≫ K.d₂ c₁₂ (c₁.next i₁) i₂ i₁₂'')
Module: Mathlib.Algebra.Homology.TotalComplex

Goal state before:
pp: case neg
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)
⊢ K.d₂ c₁₂ i₁ i₂ i₁₂' ≫ K.D₁ c₁₂ i₁₂' i₁₂'' = 0

Tactic:
by_cases h₄ : c₂.Rel i₂ (c₂.next i₂)

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)
⊢ K.d₂ c₁₂ i₁ i₂ i₁₂' ≫ K.D₁ c₁₂ i₁₂' i₁₂'' = 0

case neg
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)
⊢ K.d₂ c₁₂ i₁ i₂ i₁₂' ≫ K.D₁ c₁₂ i₁₂' i₁₂'' = 0
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
h₀ : c.Rel j j'
i₂ : I₂
i₁ : I₁
h₁ : c₂.π c₁ c (i₂, i₁) = j
h₂ : c₂.Rel i₂ (c₂.next i₂)
⊢ c₂.π c₁ c (c₂.next i₂, i₁) = j'

Tactic:
← ComplexShape.next_π₁ c₁ c h₂ i₁,

Goal state after:
C : Type u_1
I₁ : Type u_2
I₂ : Type u_3
J : Type u_4
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : Preadditive C
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
K : HomologicalComplex₂ C c₁ c₂
c : ComplexShape J
inst✝⁴ : TotalComplexShape c₁ c₂ c
inst✝³ : TotalComplexShape c₂ c₁ c
inst✝² : TotalComplexShapeSymmetry c₁ c₂ c
inst✝¹ : K.HasTotal c
inst✝ : DecidableEq J
j j' : J
h₀ : c.Rel j j'
i₂ : I₂
i₁ : I₁
h₁ : c₂.π c₁ c (i₂, i₁) = j
h₂ : c₂.Rel i₂ (c₂.next i₂)
⊢ c.next (c₂.π c₁ c (i₂, i₁)) = j'
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
⊢ (((F.mapBifunctorHomologicalComplex c₁ c₂).obj K₁).obj K₂).d₂ c i₁ i₂ (c.next j) ≫ hom₁ h₁ f₂ F c (c.next j) j = 0

Tactic:
by_cases h₄ : c₂.Rel i₂ (c₂.next i₂)

Goal state after:
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)
⊢ (((F.mapBifunctorHomologicalComplex c₁ c₂).obj K₁).obj K₂).d₂ c i₁ i₂ (c.next j) ≫ hom₁ h₁ f₂ F c (c.next j) j = 0

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
h₄ : ¬c₂.Rel i₂ (c₂.next i₂)
⊢ (((F.mapBifunctorHomologicalComplex c₁ c₂).obj K₁).obj K₂).d₂ c i₁ i₂ (c.next j) ≫ hom₁ h₁ f₂ F c (c.next j) j = 0
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
⊢ f ≫ biprod.inl ≫ s.π = g ≫ biprod.inr ≫ s.π

Tactic:
by
          rw [← sub_eq_zero, ← assoc, ← assoc, ← Preadditive.sub_comp]
          convert! s.condition <;> cat_disch

Goal state after:
Module: Mathlib.Algebra.Homology.HomotopyCategory.HomComplexCohomology

Goal state before:
pp: case refine_1
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
K L : CochainComplex C ℤ
n : ℤ
x : Cocycle K L n
h✝ : Nonempty (Homotopy (Cocycle.equivHomShift.symm x) 0)
γ : Cochain K ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) n).obj L) (-1)
h : (↑x).rightShift n 0 ⋯ = δ (-1) 0 γ
⊢ ∃ m, m + 1 = n ∧ ∃ β, δ m n β = ↑x

Tactic:
exact ⟨n - 1, by simp, n.negOnePow • γ.rightUnshift _ (by lia),
      by simp [Cochain.δ_rightUnshift _ _ _ _ _ (zero_add n), smul_smul, ← h]⟩

Goal state after:
Module: Mathlib.Algebra.Homology.Embedding.ExtendHomotopy

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝³ : Category.{v_1, u_3} C
inst✝² : HasZeroObject C
inst✝¹ : Preadditive C
K L : HomologicalComplex C c
f g : K ⟶ L
h : Homotopy f g
e : c.Embedding c'
inst✝ : e.IsRelIff
i j : ι
hij' : ¬c'.Rel (e.f j) (e.f i)
⊢ ¬c.Rel j i

Tactic:
rwa [← e.rel_iff]

Goal state after:
Module: Mathlib.Algebra.Homology.Embedding.ExtendHomotopy

Goal state before:
pp: case pos
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝³ : Category.{v_1, u_3} C
inst✝² : HasZeroObject C
inst✝¹ : Preadditive C
K L : HomologicalComplex C c
f g : K ⟶ L
e : c.Embedding c'
inst✝ : e.IsRelIff
h : Homotopy (extendMap f e) (extendMap g e)
i : ι
this :
  (K.extendXIso e ⋯).hom ≫ f.f i ≫ (L.extendXIso e ⋯).inv =
    (dNext (e.f i)) h.hom + (prevD (e.f i)) h.hom + (K.extendXIso e ⋯).hom ≫ g.f i ≫ (L.extendXIso e ⋯).inv
hi : c.Rel i (c.next i)
⊢ (dNext (e.f i)) h.hom =
    (K.extendXIso e ⋯).hom ≫
      ((dNext i) fun i j ↦ (K.extendXIso e ⋯).inv ≫ h.hom (e.f i) (e.f j) ≫ (L.extendXIso e ⋯).hom) ≫
        (L.extendXIso e ⋯).inv

pp: case neg
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝³ : Category.{v_1, u_3} C
inst✝² : HasZeroObject C
inst✝¹ : Preadditive C
K L : HomologicalComplex C c
f g : K ⟶ L
e : c.Embedding c'
inst✝ : e.IsRelIff
h : Homotopy (extendMap f e) (extendMap g e)
i : ι
this :
  (K.extendXIso e ⋯).hom ≫ f.f i ≫ (L.extendXIso e ⋯).inv =
    (dNext (e.f i)) h.hom + (prevD (e.f i)) h.hom + (K.extendXIso e ⋯).hom ≫ g.f i ≫ (L.extendXIso e ⋯).inv
hi : ¬c.Rel i (c.next i)
⊢ (dNext (e.f i)) h.hom =
    (K.extendXIso e ⋯).hom ≫
      ((dNext i) fun i j ↦ (K.extendXIso e ⋯).inv ≫ h.hom (e.f i) (e.f j) ≫ (L.extendXIso e ⋯).hom) ≫
        (L.extendXIso e ⋯).inv

Tactic:
· have hi' : c'.Rel (e.f i) (e.f (c.next i)) := by rwa [e.rel_iff]
        simp [dNext_eq _ hi, dNext_eq _ hi', K.extend_d_eq _ rfl rfl]

Goal state after:
case neg
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝³ : Category.{v_1, u_3} C
inst✝² : HasZeroObject C
inst✝¹ : Preadditive C
K L : HomologicalComplex C c
f g : K ⟶ L
e : c.Embedding c'
inst✝ : e.IsRelIff
h : Homotopy (extendMap f e) (extendMap g e)
i : ι
this :
  (K.extendXIso e ⋯).hom ≫ f.f i ≫ (L.extendXIso e ⋯).inv =
    (dNext (e.f i)) h.hom + (prevD (e.f i)) h.hom + (K.extendXIso e ⋯).hom ≫ g.f i ≫ (L.extendXIso e ⋯).inv
hi : ¬c.Rel i (c.next i)
⊢ (dNext (e.f i)) h.hom =
    (K.extendXIso e ⋯).hom ≫
      ((dNext i) fun i j ↦ (K.extendXIso e ⋯).inv ≫ h.hom (e.f i) (e.f j) ≫ (L.extendXIso e ⋯).hom) ≫
        (L.extendXIso e ⋯).inv
Module: Mathlib.CategoryTheory.Triangulated.TStructure.Basic

Goal state before:
pp: C : Type u_1
inst✝⁵ : Category.{v_1, u_1} 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
⊢ Monotone t.le

Tactic:
let H := fun (a : ℕ) => ∀ (n : ℤ), t.le n ≤ t.le (n + a)
  suffices ∀ (a : ℕ), H a by
    intro n₀ n₁ h
    obtain ⟨a, ha⟩ := Int.nonneg_def.1 h
    obtain rfl : n₁ = n₀ + a := by lia
    apply this
  have H_zero : H 0 := fun n => by
    simp only [Nat.cast_zero, add_zero]
    rfl
  have H_one : H 1 := fun n X hX => by
    rw [← t.shift_le n 1 (n + (1 : ℕ)) rfl, ObjectProperty.prop_shift_iff]
    rw [← t.shift_le n 0 n (add_zero n), ObjectProperty.prop_shift_iff] at hX
    exact t.le_zero_le _ hX
  have H_add : ∀ (a b c : ℕ) (_ : a + b = c) (_ : H a) (_ : H b), H c := by
    intro a b c h ha hb n
    rw [← h, Nat.cast_add, ← add_assoc]
    exact (ha n).trans (hb (n + a))
  intro a
  induction a with
  | zero => exact H_zero
  | succ a ha => exact H_add a 1 _ rfl ha H_one

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.TStructure.Basic

Goal state before:
pp: C : Type u_1
inst✝⁷ : Category.{v_1, u_1} C
inst✝⁶ : Preadditive C
inst✝⁵ : HasZeroObject C
inst✝⁴ : HasShift C ℤ
inst✝³ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝² : Pretriangulated C
t : TStructure C
X Y : C
f : X ⟶ Y
n₀ n₁ : ℤ
h : n₀ < n₁
inst✝¹ : t.IsLE X n₀
inst✝ : t.IsGE Y n₁
this✝¹ : t.IsLE ((shiftFunctor C n₀).obj X) 0
this✝ : t.IsGE ((shiftFunctor C n₀).obj Y) (n₁ - n₀)
this : t.IsGE ((shiftFunctor C n₀).obj Y) 1
⊢ f = 0

Tactic:
apply (shiftFunctor C n₀).map_injective

Goal state after:
C : Type u_1
inst✝⁷ : Category.{v_1, u_1} C
inst✝⁶ : Preadditive C
inst✝⁵ : HasZeroObject C
inst✝⁴ : HasShift C ℤ
inst✝³ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝² : Pretriangulated C
t : TStructure C
X Y : C
f : X ⟶ Y
n₀ n₁ : ℤ
h : n₀ < n₁
inst✝¹ : t.IsLE X n₀
inst✝ : t.IsGE Y n₁
this✝¹ : t.IsLE ((shiftFunctor C n₀).obj X) 0
this✝ : t.IsGE ((shiftFunctor C n₀).obj Y) (n₁ - n₀)
this : t.IsGE ((shiftFunctor C n₀).obj Y) 1
⊢ (shiftFunctor C n₀).map f = (shiftFunctor C n₀).map 0
Module: Mathlib.AlgebraicTopology.ModelCategory.LeftHomotopy

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : CategoryWithWeakEquivalences C
X Y : C
f g : X ⟶ Y
P : Cylinder X
h : P.LeftHomotopy f g
L : C ⥤ (weakEquivalences C).Localization := (weakEquivalences C).Q
⊢ AreEqualizedByLocalization (weakEquivalences C) f g

Tactic:
areEqualizedByLocalization_iff L

Goal state after:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : CategoryWithWeakEquivalences C
X Y : C
f g : X ⟶ Y
P : Cylinder X
h : P.LeftHomotopy f g
L : C ⥤ (weakEquivalences C).Localization := (weakEquivalences C).Q
⊢ L.map f = L.map g
Module: Mathlib.AlgebraicTopology.ModelCategory.RightHomotopy

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : CategoryWithWeakEquivalences C
X Y : C
f g : X ⟶ Y
P : PathObject Y
h : P.RightHomotopy f g
L : C ⥤ (weakEquivalences C).Localization := (weakEquivalences C).Q
⊢ AreEqualizedByLocalization (weakEquivalences C) f g

Tactic:
areEqualizedByLocalization_iff L

Goal state after:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : CategoryWithWeakEquivalences C
X Y : C
f g : X ⟶ Y
P : PathObject Y
h : P.RightHomotopy f g
L : C ⥤ (weakEquivalences C).Localization := (weakEquivalences C).Q
⊢ L.map f = L.map g
Module: Mathlib.CategoryTheory.Triangulated.LocalizingSubcategory

Goal state before:
pp: case refine_2
C : Type u_1
inst✝⁸ : Category.{v_1, u_1} C
A B : 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 : A.IsTriangulated
inst✝¹ : B.IsTriangulated
inst✝ : B.IsClosedUnderIsomorphisms
hA : ∀ ⦃X Y : C⦄ (s : X ⟶ Y), A Y → B.trW s → ∃ Z s' a, A Z ∧ (A ⊓ B).trW s' ∧ a ≫ s = s'
X Y : Cᵒᵖ
s : X ⟶ Y
hX : A.op X
hs : B.op.trW s
Z' : C
s' : Z' ⟶ unop X
b : Z' ⟶ unop Y
hZ' : A Z'
hs' : (A ⊓ B).trW s'
fac : b ≫ s.unop = s'
⊢ ∃ Z s' b, A.op Z ∧ (A.op ⊓ B.op).trW s' ∧ s ≫ b = s'

Tactic:
exact ⟨_, s'.op, b.op, hZ', trW_of_unop _ hs', by cat_disch⟩

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.Injective.LiftingProperties

Goal state before:
pp: case mp
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasZeroMorphisms C
I Z : C
hZ : IsZero Z
a✝ : Injective I
A B : C
i : A ⟶ B
⊢ MorphismProperty.monomorphisms C i → HasLiftingProperty i 0

Tactic:
(_ : Mono i)

Goal state after:
case mp
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasZeroMorphisms C
I Z : C
hZ : IsZero Z
a✝ : Injective I
A B : C
i : A ⟶ B
x✝ : Mono i
⊢ HasLiftingProperty i 0
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
n : ℤ
x✝³ x✝² : C
x✝¹ x✝ : x✝³ ⟶ x✝²
⊢ Triangle.π₁.map ((TruncAux.triangleFunctor t n).map (x✝¹ + x✝)) =
    Triangle.π₁.map ((TruncAux.triangleFunctor t n).map x✝¹) + Triangle.π₁.map ((TruncAux.triangleFunctor t n).map x✝)

Tactic:
rw [Functor.map_add]

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
n : ℤ
x✝³ x✝² : C
x✝¹ x✝ : x✝³ ⟶ x✝²
⊢ Triangle.π₁.map ((TruncAux.triangleFunctor t n).map x✝¹ + (TruncAux.triangleFunctor t n).map x✝) =
    Triangle.π₁.map ((TruncAux.triangleFunctor t n).map x✝¹) + Triangle.π₁.map ((TruncAux.triangleFunctor t n).map x✝)
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
n : ℤ
x✝³ x✝² : C
x✝¹ x✝ : x✝³ ⟶ x✝²
⊢ Triangle.π₃.map ((TruncAux.triangleFunctor t n).map (x✝¹ + x✝)) =
    Triangle.π₃.map ((TruncAux.triangleFunctor t n).map x✝¹) + Triangle.π₃.map ((TruncAux.triangleFunctor t n).map x✝)

Tactic:
rw [Functor.map_add]

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
n : ℤ
x✝³ x✝² : C
x✝¹ x✝ : x✝³ ⟶ x✝²
⊢ Triangle.π₃.map ((TruncAux.triangleFunctor t n).map x✝¹ + (TruncAux.triangleFunctor t n).map x✝) =
    Triangle.π₃.map ((TruncAux.triangleFunctor t n).map x✝¹) + Triangle.π₃.map ((TruncAux.triangleFunctor t n).map x✝)
Module: Mathlib.Algebra.Homology.ModelCategory.Lifting

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
A B X Y : CochainComplex C ℤ
t : A ⟶ X
i : A ⟶ B
p : X ⟶ Y
b : B ⟶ Y
sq : CommSq t i p b
hsq : (n : ℤ) → ⋯.LiftStruct
Q : CochainComplex C ℤ
π : B ⟶ Q
hπ : i ≫ π = 0
hQ : IsColimit (CokernelCofork.ofπ π hπ)
K : CochainComplex C ℤ
ι : K ⟶ X
hι : ι ≫ p = 0
hK : IsLimit (KernelFork.ofι ι hι)
this✝¹ : Epi π
this✝ : Mono ι
n : ℤ
this :
  (↑(cocycle₁' sq hsq)).v n (n + 1) ⋯ ≫ X.d (n + 1) (n + 2) +
      Int.negOnePow 2 • B.d n (n + 1) ≫ (↑(cocycle₁' sq hsq)).v (n + 1) (n + 2) ⋯ =
    0
⊢ (cochain₁ sq hsq hQ hK).v n (n + 1) ⋯ ≫ K.d (n + 1) (n + 2) +
      Int.negOnePow 2 • Q.d n (n + 1) ≫ (cochain₁ sq hsq hQ hK).v (n + 1) (n + 2) ⋯ =
    0

Tactic:
Int.negOnePow_even 2 ⟨1, by simp⟩,

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
A B X Y : CochainComplex C ℤ
t : A ⟶ X
i : A ⟶ B
p : X ⟶ Y
b : B ⟶ Y
sq : CommSq t i p b
hsq : (n : ℤ) → ⋯.LiftStruct
Q : CochainComplex C ℤ
π : B ⟶ Q
hπ : i ≫ π = 0
hQ : IsColimit (CokernelCofork.ofπ π hπ)
K : CochainComplex C ℤ
ι : K ⟶ X
hι : ι ≫ p = 0
hK : IsLimit (KernelFork.ofι ι hι)
this✝¹ : Epi π
this✝ : Mono ι
n : ℤ
this :
  (↑(cocycle₁' sq hsq)).v n (n + 1) ⋯ ≫ X.d (n + 1) (n + 2) +
      1 • B.d n (n + 1) ≫ (↑(cocycle₁' sq hsq)).v (n + 1) (n + 2) ⋯ =
    0
⊢ (cochain₁ sq hsq hQ hK).v n (n + 1) ⋯ ≫ K.d (n + 1) (n + 2) +
      1 • Q.d n (n + 1) ≫ (cochain₁ sq hsq hQ hK).v (n + 1) (n + 2) ⋯ =
    0