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

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

Tactic:
rcases le_or_gt k j with (hkj | hkj)

Goal state after:
case inl
n : ℕ
i j : Fin (n + 1)
H : i ≤ j
k : Fin (n + 1)
h : i ≤ k
hkj : k ≤ j
⊢ i ≤ j.predAbove k.castSucc

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
Module: Mathlib.AlgebraicTopology.SimplicialObject.Basic

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
X : SimplicialObject C
X✝ Y✝ : Augmented C
η : X✝ ⟶ Y✝
⊢ (𝟭 C).map ((drop.map η).app (op ⦋0⦌)) ≫
      { left := drop.obj Y✝ _⦋0⦌, right := point.obj Y✝, hom := Y✝.hom.app (op ⦋0⦌) }.hom =
    { left := drop.obj X✝ _⦋0⦌, right := point.obj X✝, hom := X✝.hom.app (op ⦋0⦌) }.hom ≫ (𝟭 C).map (point.map η)

Tactic:
simp [w_app]

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

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
X : CosimplicialObject C
X✝ Y✝ : Augmented C
η : X✝ ⟶ Y✝
⊢ (𝟭 C).map (point.map η) ≫ { left := point.obj Y✝, right := drop.obj Y✝ ^⦋0⦌, hom := Y✝.hom.app ⦋0⦌ }.hom =
    { left := point.obj X✝, right := drop.obj X✝ ^⦋0⦌, hom := X✝.hom.app ⦋0⦌ }.hom ≫ (𝟭 C).map ((drop.map η).app ⦋0⦌)

Tactic:
simp [w_app]

Goal state after:
Module: Mathlib.AlgebraicTopology.SimplexCategory.DeltaZeroIter

Goal state before:
pp: i : ℕ
hi :
  ∀ {n m : ℕ} (j : Fin (m + 1)) (j' : Fin (n + 1)) (hi' : n + i = m),
    ↑j = ↑j' + i → δ₀Iter i ⋯ ≫ σ j = σ j' ≫ δ₀Iter i hi'
n : ℕ
j' : Fin (n + 1)
m : ℕ
j : Fin (m + 1 + 1)
hi' : n + (i + 1) = m + 1
hj' : ↑j = ↑j' + (i + 1)
⊢ j ≠ 0

Tactic:
rintro rfl

Goal state after:
i : ℕ
hi :
  ∀ {n m : ℕ} (j : Fin (m + 1)) (j' : Fin (n + 1)) (hi' : n + i = m),
    ↑j = ↑j' + i → δ₀Iter i ⋯ ≫ σ j = σ j' ≫ δ₀Iter i hi'
n : ℕ
j' : Fin (n + 1)
m : ℕ
hi' : n + (i + 1) = m + 1
hj' : ↑0 = ↑j' + (i + 1)
⊢ False
Module: Mathlib.AlgebraicTopology.SimplicialObject.Homotopy

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
X Y : SimplicialObject C
f g : X ⟶ Y
H : Homotopy f g
Y' : SimplicialObject C
p : Y ⟶ Y'
n✝ : ℕ
i j : Fin (n✝ + 1)
hij : i ≤ j
⊢ (H.h j ≫ p.app (op ⦋n✝ + 1⦌)) ≫ Y'.σ i.castSucc = X.σ i ≫ H.h j.succ ≫ p.app (op ⦋n✝ + 1 + 1⦌)

Tactic:
by
    simpa using H.h_comp_σ_castSucc_of_le i j hij =≫ p.app _

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

Goal state before:
pp: case h
n : ℕ
θ : ⦋n + 1⦌ ⟶ ⦋n⦌
inst✝ : Epi θ
i : Fin (n + 1)
θ' : ⦋n⦌ ⟶ ⦋n⦌
h : θ = σ i ≫ θ'
⊢ θ = σ i

Tactic:
have : Epi (σ i ≫ θ') := by
    rw [← h]
    infer_instance

Goal state after:
case h
n : ℕ
θ : ⦋n + 1⦌ ⟶ ⦋n⦌
inst✝ : Epi θ
i : Fin (n + 1)
θ' : ⦋n⦌ ⟶ ⦋n⦌
h : θ = σ i ≫ θ'
this : Epi (σ i ≫ θ')
⊢ θ = σ i
Module: Mathlib.AlgebraicTopology.SimplicialSet.Degenerate

Goal state before:
pp: case mp
X : SSet
n : ℕ
x : X _⦋n⦌
m : ℕ
hm : m < n
f : ⦋n⦌ ⟶ ⦋m⦌
y : X _⦋m⦌
hy : (ConcreteCategory.hom (X.map f.op)) y = x
⊢ ∃ m, ∃ (_ : m < n), ∃ f, ∃ (_ : Epi f), x ∈ Set.range ⇑(ConcreteCategory.hom (X.map f.op))

Tactic:
← image.fac f,

Goal state after:
case mp
X : SSet
n : ℕ
x : X _⦋n⦌
m : ℕ
hm : m < n
f : ⦋n⦌ ⟶ ⦋m⦌
y : X _⦋m⦌
hy : (ConcreteCategory.hom (X.map (factorThruImage f ≫ image.ι f).op)) y = x
⊢ ∃ m, ∃ (_ : m < n), ∃ f, ∃ (_ : Epi f), x ∈ Set.range ⇑(ConcreteCategory.hom (X.map f.op))
Module: Mathlib.AlgebraicTopology.SimplicialSet.Simplices

Goal state before:
pp: X : SSet
s t : X.S
⊢ s = t → ∃ (h : s.dim = t.dim), (s.cast h).simplex = t.simplex

Tactic:
rintro rfl

Goal state after:
X : SSet
s : X.S
⊢ ∃ (h : s.dim = s.dim), (s.cast h).simplex = s.simplex
Module: Mathlib.AlgebraicTopology.SimplicialSet.Simplices

Goal state before:
pp: X : SSet
⊢ Functor.Elements X → X.S

Tactic:
rintro ⟨⟨⟨n⟩⟩, x⟩; exact S.mk x

Goal state after:
Module: Mathlib.AlgebraicTopology.CechNerve

Goal state before:
pp: case a
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : ∀ (n : ℕ) (f : Arrow C), HasWidePushout f.left (fun x ↦ f.right) fun x ↦ f.hom
F : Arrow C
X : Augmented C
G : F ⟶ Augmented.toArrow.obj X
x y : SimplexCategory
f : x ⟶ y
j✝ : Fin (x.len + 1)
⊢ WidePushout.ι (fun x ↦ F.hom) j✝ ≫
      WidePushout.desc (WidePushout.head fun x ↦ F.hom)
          (fun i ↦ WidePushout.ι (fun x ↦ F.hom) ((SimplexCategory.Hom.toOrderHom f) i)) ⋯ ≫
        WidePushout.desc (Arrow.Hom.left G ≫ X.hom.app y) (fun i ↦ Arrow.Hom.right G ≫ X.right.map (⦋0⦌.const y i)) ⋯ =
    WidePushout.ι (fun x ↦ F.hom) j✝ ≫
      WidePushout.desc (Arrow.Hom.left G ≫ X.hom.app x) (fun i ↦ Arrow.Hom.right G ≫ X.right.map (⦋0⦌.const x i)) ⋯ ≫
        X.right.map f

pp: case a
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : ∀ (n : ℕ) (f : Arrow C), HasWidePushout f.left (fun x ↦ f.right) fun x ↦ f.hom
F : Arrow C
X : Augmented C
G : F ⟶ Augmented.toArrow.obj X
x y : SimplexCategory
f : x ⟶ y
⊢ (WidePushout.head fun x ↦ F.hom) ≫
      WidePushout.desc (WidePushout.head fun x ↦ F.hom)
          (fun i ↦ WidePushout.ι (fun x ↦ F.hom) ((SimplexCategory.Hom.toOrderHom f) i)) ⋯ ≫
        WidePushout.desc (Arrow.Hom.left G ≫ X.hom.app y) (fun i ↦ Arrow.Hom.right G ≫ X.right.map (⦋0⦌.const y i)) ⋯ =
    (WidePushout.head fun x ↦ F.hom) ≫
      WidePushout.desc (Arrow.Hom.left G ≫ X.hom.app x) (fun i ↦ Arrow.Hom.right G ≫ X.right.map (⦋0⦌.const x i)) ⋯ ≫
        X.right.map f

Tactic:
· simp only [WidePushout.ι_desc_assoc, WidePushout.ι_desc]
          rw [Category.assoc, ← X.right.map_comp]
          rfl

Goal state after:
case a
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : ∀ (n : ℕ) (f : Arrow C), HasWidePushout f.left (fun x ↦ f.right) fun x ↦ f.hom
F : Arrow C
X : Augmented C
G : F ⟶ Augmented.toArrow.obj X
x y : SimplexCategory
f : x ⟶ y
⊢ (WidePushout.head fun x ↦ F.hom) ≫
      WidePushout.desc (WidePushout.head fun x ↦ F.hom)
          (fun i ↦ WidePushout.ι (fun x ↦ F.hom) ((SimplexCategory.Hom.toOrderHom f) i)) ⋯ ≫
        WidePushout.desc (Arrow.Hom.left G ≫ X.hom.app y) (fun i ↦ Arrow.Hom.right G ≫ X.right.map (⦋0⦌.const y i)) ⋯ =
    (WidePushout.head fun x ↦ F.hom) ≫
      WidePushout.desc (Arrow.Hom.left G ≫ X.hom.app x) (fun i ↦ Arrow.Hom.right G ≫ X.right.map (⦋0⦌.const x i)) ⋯ ≫
        X.right.map f
Module: Mathlib.AlgebraicTopology.SimplicialSet.Degenerate

Goal state before:
pp: case mk
X : SSet
n : ℕ
x : X _⦋n⦌
hx✝ : x ∈ X.nonDegenerate n
hx : x ∉ X.degenerate n
m : ℕ
f : ⦋n⦌ ⟶ ⦋m⦌
inst✝ : Epi f
y : X _⦋m⦌
hy : (ConcreteCategory.hom (X.map f.op)) y = ↑⟨x, hx✝⟩
hf : ¬IsIso f
h : n ≤ m
⊢ IsIso f

Tactic:
SimplexCategory.isIso_iff_of_epi

Goal state after:
case mk
X : SSet
n : ℕ
x : X _⦋n⦌
hx✝ : x ∈ X.nonDegenerate n
hx : x ∉ X.degenerate n
m : ℕ
f : ⦋n⦌ ⟶ ⦋m⦌
inst✝ : Epi f
y : X _⦋m⦌
hy : (ConcreteCategory.hom (X.map f.op)) y = ↑⟨x, hx✝⟩
hf : ¬IsIso f
h : n ≤ m
⊢ ⦋n⦌.len = ⦋m⦌.len
Module: Mathlib.AlgebraicTopology.SimplicialSet.Subcomplex

Goal state before:
pp: case a
X Y : SSet
n : ℕ
x : X _⦋n⦌
f : X ⟶ Y
⊢ (ConcreteCategory.hom (f.app (Opposite.op ⦋n⦌))) x ∈
    (ofSimplex ((ConcreteCategory.hom (f.app (Opposite.op ⦋n⦌))) x)).obj (Opposite.op ⦋n⦌)

Tactic:
apply mem_ofSimplex_obj

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
⊢ (ConcreteCategory.hom (X.map (g hf₁ f₂).op)) y₂ = ↑y₁

Tactic:
dsimp [g]
  rw [Functor.map_comp, comp_apply, ← hy₂, hy₁, ← comp_apply, ← Functor.map_comp, ← op_comp,
    SplitEpi.id, op_id, CategoryTheory.Functor.map_id, id_apply]

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

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

Tactic:
rintro rfl

Goal state after:
X : SSet
x : X.N
⊢ x.subcomplex = x.subcomplex
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
this : (ConcreteCategory.hom (X.map (image.ι (g hf₁ f₂)).op ≫ X.map (factorThruImage (g hf₁ f₂)).op)) y₂ = ↑y₁
⊢ IsIso (factorThruImage (g hf₁ f₂))

Tactic:
comp_apply

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 (factorThruImage (g hf₁ f₂)).op))
      ((ConcreteCategory.hom (X.map (image.ι (g hf₁ f₂)).op)) y₂) =
    ↑y₁
⊢ IsIso (factorThruImage (g hf₁ f₂))
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

Goal state after:
X : SSet
x : X.S
⊢ x.toN.subcomplex = x.subcomplex
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)
⊢ s ∈ Set.range ⇑(ConcreteCategory.hom (SimplicialObject.σ (nerve X) i)) ↔ s.obj i.castSucc = s.obj i.succ

Tactic:
constructor
  · rintro ⟨s, rfl⟩
    simp [-nerve_obj, nerve.σ_obj]
  · intro h
    refine ⟨(nerve X).δ i.castSucc s, ?_⟩
    ext j
    rw [nerve.σ_obj, nerve.δ_obj]
    by_cases h₁ : i.castSucc < j
    · obtain ⟨j, rfl⟩ := Fin.eq_succ_of_ne_zero (Fin.ne_zero_of_lt h₁)
      rw [Fin.predAbove_of_castSucc_lt _ _ h₁, Fin.pred_succ,
        Fin.succAbove_of_le_castSucc _ _ (Fin.le_castSucc_iff.2 h₁)]
    · simp only [not_lt] at h₁
      grind [→ Fin.succAbove_of_castSucc_lt,
        → Fin.predAbove_of_le_castSucc, Fin.castSucc_castPred, Fin.castPred_castSucc,
        Fin.succAbove_castSucc_self, → LE.le.lt_or_eq]

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

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

Tactic:
rintro rfl

Goal state after:
case refine_2
n : ℕ
⊢ objEquiv.symm (𝟙 ⦋n⦌) ∈ Δ[n].nonDegenerate n
Module: Mathlib.AlgebraicTopology.ExtraDegeneracy

Goal state before:
pp: n : ℕ
Δ : SimplexCategory
f : ⦋n⦌ ⟶ Δ
i₁ i₂ : Fin (⦋n + 1⦌.len + 1)
hi : i₁ ≤ i₂
h₁ : ¬i₁ = 0
⊢ i₂ ≠ 0

Tactic:
rintro rfl

Goal state after:
n : ℕ
Δ : SimplexCategory
f : ⦋n⦌ ⟶ Δ
i₁ : Fin (⦋n + 1⦌.len + 1)
h₁ : ¬i₁ = 0
hi : i₁ ≤ 0
⊢ False
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)
⊢ K.d₂ c₁₂ i₁ i₂ i₁₂' ≫ K.D₁ c₁₂ i₁₂' i₁₂'' = 0

Tactic:
rw [totalAux.d₂_eq K c₁₂ i₁ h₄ i₁₂']; swap
          · rw [← ComplexShape.next_π₂ c₁ c₁₂ i₁ h₄, ← c₁₂.next_eq' h₁, h]
          simp only [Linear.units_smul_comp, assoc, totalAux.ιMapObj_D₁]
          rw [K.d₁_eq_zero c₁₂ _ _ _ h₃, comp_zero, smul_zero]

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

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
K L : CochainComplex C ℤ
f g : K ⟶ L
h : Homotopy ((opEquivalence C).functor.map f.op) ((opEquivalence C).functor.map g.op)
H :
  ∀ (p q p' q' : ℤ) (hp : p = p') (hq : q = q'),
    h.hom p q = (HomologicalComplex.XIsoOfEq L ⋯).hom.op ≫ h.hom p' q' ≫ (HomologicalComplex.XIsoOfEq K ⋯).hom.op
n : ℤ
this : (f.f (-n)).op = (dNext n) h.hom + (prevD n) h.hom + (g.f (-n)).op
⊢ (ComplexShape.up ℤ).Rel n (n + 1)

Tactic:
by dsimp

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

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
P : ObjectProperty C
inst✝⁵ : HasZeroObject C
inst✝⁴ : HasShift C ℤ
inst✝³ : Preadditive C
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
inst✝ : P.IsTriangulated
Y : C
hY : P.rightOrthogonal Y
X₁ X₂ : C
f : X₁ ⟶ X₂
x✝ : P.trW f
X₃ : C
g : X₂ ⟶ X₃
h : X₃ ⟶ (shiftFunctor C 1).obj X₁
hT : Triangle.mk f g h ∈ distinguishedTriangles
hX₃ : P X₃
this : ∀ (α : X₂ ⟶ Y), f ≫ α = 0 → α = 0
α₁ α₂ : X₂ ⟶ Y
hα : (fun g ↦ f ≫ g) α₁ = (fun g ↦ f ≫ g) α₂
⊢ α₁ = α₂

Tactic:
by
      simpa [sub_eq_zero] using this (α₁ - α₂) (by simpa [sub_eq_zero] using hα)

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
h : Homotopy f g
e : c.Embedding c'
inst✝ : e.IsRelIff
i : ι
⊢ (K.extendXIso e ⋯).hom ≫ f.f i ≫ (L.extendXIso e ⋯).inv =
    (dNext (e.f i)) (extend.hom e h.hom) + (prevD (e.f i)) (extend.hom e h.hom) +
      (K.extendXIso e ⋯).hom ≫ g.f i ≫ (L.extendXIso e ⋯).inv

Tactic:
h.comm i,

Goal state after:
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
h : Homotopy f g
e : c.Embedding c'
inst✝ : e.IsRelIff
i : ι
⊢ (K.extendXIso e ⋯).hom ≫ ((dNext i) h.hom + (prevD i) h.hom + g.f i) ≫ (L.extendXIso e ⋯).inv =
    (dNext (e.f i)) (extend.hom e h.hom) + (prevD (e.f i)) (extend.hom e h.hom) +
      (K.extendXIso e ⋯).hom ≫ g.f i ≫ (L.extendXIso e ⋯).inv
Module: Mathlib.Algebra.Homology.Embedding.ExtendHomotopy

Goal state before:
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
h : Homotopy f g
e : c.Embedding c'
inst✝ : e.IsRelIff
i : ι
hi : ¬c.Rel i (c.next i)
⊢ (K.extendXIso e ⋯).hom ≫ 0 ≫ (L.extendXIso e ⋯).inv = (dNext (e.f i)) (extend.hom e h.hom)

Tactic:
by_cases hi' : c'.Rel (e.f i) (c'.next (e.f i))

Goal state after:
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
h : Homotopy f g
e : c.Embedding c'
inst✝ : e.IsRelIff
i : ι
hi : ¬c.Rel i (c.next i)
hi' : c'.Rel (e.f i) (c'.next (e.f i))
⊢ (K.extendXIso e ⋯).hom ≫ 0 ≫ (L.extendXIso e ⋯).inv = (dNext (e.f i)) (extend.hom e h.hom)

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
h : Homotopy f g
e : c.Embedding c'
inst✝ : e.IsRelIff
i : ι
hi : ¬c.Rel i (c.next i)
hi' : ¬c'.Rel (e.f i) (c'.next (e.f i))
⊢ (K.extendXIso e ⋯).hom ≫ 0 ≫ (L.extendXIso e ⋯).inv = (dNext (e.f i)) (extend.hom e h.hom)
Module: Mathlib.Algebra.Homology.Embedding.ExtendHomotopy

Goal state before:
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 ≫ 0 ≫ (L.extendXIso e ⋯).inv

Tactic:
by_cases hi' : c'.Rel (e.f i) (c'.next (e.f i))

Goal state after:
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)
hi' : c'.Rel (e.f i) (c'.next (e.f i))
⊢ (dNext (e.f i)) h.hom = (K.extendXIso e ⋯).hom ≫ 0 ≫ (L.extendXIso e ⋯).inv

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)
hi' : ¬c'.Rel (e.f i) (c'.next (e.f i))
⊢ (dNext (e.f i)) h.hom = (K.extendXIso e ⋯).hom ≫ 0 ≫ (L.extendXIso e ⋯).inv
Module: Mathlib.CategoryTheory.MorphismProperty.LiftingProperty

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
D : Type u_1
inst✝¹ : Category.{v_1, u_1} D
G : C ⥤ D
inst✝ : G.IsEquivalence
A B X Y : C
i : A ⟶ B
p : X ⟶ Y
⊢ HasLiftingProperty (G.asEquivalence.functor.map i) (G.asEquivalence.functor.map p) ↔ HasLiftingProperty i p

Tactic:
G.asEquivalence.toAdjunction.hasLiftingProperty_iff

Goal state after:
C : Type u
inst✝² : Category.{v, u} C
D : Type u_1
inst✝¹ : Category.{v_1, u_1} D
G : C ⥤ D
inst✝ : G.IsEquivalence
A B X Y : C
i : A ⟶ B
p : X ⟶ Y
⊢ HasLiftingProperty i (G.asEquivalence.inverse.map (G.asEquivalence.functor.map p)) ↔ HasLiftingProperty i p
Module: Mathlib.AlgebraicTopology.ModelCategory.Instances

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : CategoryWithWeakEquivalences C
inst✝² : CategoryWithCofibrations C
inst✝¹ : CategoryWithFibrations C
inst✝ : (trivialCofibrations C).IsWeakFactorizationSystem (fibrations C)
J : Type w
⊢ (fibrations C).IsStableUnderProductsOfShape J

Tactic:
by
  rw [← trivialCofibrations_rlp]
  apply MorphismProperty.rlp_isStableUnderProductsOfShape

Goal state after:
Module: Mathlib.AlgebraicTopology.ModelCategory.IsCofibrant

Goal state before:
pp: C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : CategoryWithFibrations C
inst✝² : HasTerminal C
X Y : C
inst✝¹ : (fibrations C).IsStableUnderBaseChange
inst✝ : HasBinaryProduct X Y
hY : IsFibrant Y
⊢ Fibration prod.fst

Tactic:
rw [isFibrant_iff] at hY

Goal state after:
C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : CategoryWithFibrations C
inst✝² : HasTerminal C
X Y : C
inst✝¹ : (fibrations C).IsStableUnderBaseChange
inst✝ : HasBinaryProduct X Y
hY : Fibration (terminal.from Y)
⊢ Fibration prod.fst
Module: Mathlib.Algebra.Homology.BifunctorAssociator

Goal state before:
pp: case pos
C₁ : Type u_1
C₂ : Type u_2
C₂₃ : Type u_4
C₃ : Type u_5
C₄ : Type u_6
inst✝²³ : Category.{v_1, u_1} C₁
inst✝²² : Category.{v_2, u_2} C₂
inst✝²¹ : Category.{v_3, u_5} C₃
inst✝²⁰ : Category.{v_4, u_6} C₄
inst✝¹⁹ : Category.{v_6, u_4} C₂₃
inst✝¹⁸ : HasZeroMorphisms C₁
inst✝¹⁷ : HasZeroMorphisms C₂
inst✝¹⁶ : HasZeroMorphisms C₃
inst✝¹⁵ : Preadditive C₂₃
inst✝¹⁴ : Preadditive C₄
F : C₁ ⥤ C₂₃ ⥤ C₄
G₂₃ : C₂ ⥤ C₃ ⥤ C₂₃
inst✝¹³ : G₂₃.PreservesZeroMorphisms
inst✝¹² : ∀ (X₂ : C₂), (G₂₃.obj X₂).PreservesZeroMorphisms
inst✝¹¹ : F.PreservesZeroMorphisms
inst✝¹⁰ : ∀ (X₁ : C₁), (F.obj X₁).Additive
ι₁ : Type u_7
ι₂ : Type u_8
ι₃ : Type u_9
ι₁₂ : Type u_10
ι₂₃ : Type u_11
ι₄ : Type u_12
inst✝⁹ : 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
c₂₃ : ComplexShape ι₂₃
c₄ : ComplexShape ι₄
inst✝⁸ : TotalComplexShape c₁ c₂ c₁₂
inst✝⁷ : TotalComplexShape c₁₂ c₃ c₄
inst✝⁶ : TotalComplexShape c₂ c₃ c₂₃
inst✝⁵ : TotalComplexShape c₁ c₂₃ c₄
inst✝⁴ : K₂.HasMapBifunctor K₃ G₂₃ c₂₃
inst✝³ : c₁.Associative c₂ c₃ c₁₂ c₂₃ c₄
inst✝² : DecidableEq ι₂₃
inst✝¹ : K₁.HasMapBifunctor (K₂.mapBifunctor K₃ G₂₃ c₂₃) F c₄
inst✝ : HasGoodTrifunctor₂₃Obj F G₂₃ K₁ K₂ K₃ c₁₂ c₂₃ c₄
j j' : ι₄
i₁ : ι₁
i₂ : ι₂
i₃ : ι₃
h : c₁.r c₂ c₃ c₁₂ c₄ (i₁, i₂, i₃) = j
i₂₃ : ι₂₃ := c₂.π c₃ c₂₃ (i₂, i₃)
h₁ : c₂₃.Rel i₂₃ (c₂₃.next i₂₃)
h₂ : ¬c₁.π c₂₃ c₄ (i₁, c₂₃.next i₂₃) = j'
h₃ : c₂.Rel i₂ (c₂.next i₂)
⊢ 0 = d₂ F G₂₃ K₁ K₂ K₃ c₁₂ c₂₃ c₄ i₁ i₂ i₃ j'

Tactic:
rw [d₂_eq _ _ _ _ _ _ _ _ _ h₃, ιOrZero_eq_zero, comp_zero, smul_zero]
            intro h₄
            apply h₂
            rw [← h₄]
            dsimp [ComplexShape.r]
            rw [ComplexShape.assoc c₁ c₂ c₃ c₁₂ c₂₃ c₄, ComplexShape.next_π₁ c₃ c₂₃ h₃ i₃]

Goal state after:
Module: Mathlib.AlgebraicTopology.ModelCategory.LeftHomotopy

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
X✝ : C
P✝ : Precylinder X✝
Y✝ : C
P : ObjectProperty C
X Y : P.FullSubcategory
Q : Precylinder X
f g : X ⟶ Y
h : (Q.map P.ι).LeftHomotopy f.hom g.hom
⊢ Q.i₀ ≫ ObjectProperty.homMk h.h = f

Tactic:
ext; exact h.h₀

Goal state after:
Module: Mathlib.AlgebraicTopology.ModelCategory.LeftHomotopy

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
X Y : C
inst✝¹ : ModelCategory C
f₀ f₁ f₂ : X ⟶ Y
inst✝ : IsCofibrant X
h : LeftHomotopyRel f₀ f₁
h' : LeftHomotopyRel f₁ f₂
⊢ LeftHomotopyRel f₀ f₂

Tactic:
by
  obtain ⟨P, ⟨h⟩⟩ := h
  obtain ⟨P', _, ⟨h'⟩⟩ := h'.exists_good_cylinder
  exact (h.trans h').leftHomotopyRel

Goal state after:
Module: Mathlib.AlgebraicTopology.ModelCategory.RightHomotopy

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
Y✝ : C
P✝ : PrepathObject Y✝
X✝ : C
P : ObjectProperty C
X Y : P.FullSubcategory
Q : PrepathObject Y
f g : X ⟶ Y
h : (Q.map P.ι).RightHomotopy f.hom g.hom
⊢ ObjectProperty.homMk h.h ≫ Q.p₀ = f

Tactic:
ext; exact h.h₀

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

Goal state before:
pp: C : Type u_1
D : Type u_2
D₁ : Type u_3
D₂ : Type u_4
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_2, u_2} D
inst✝² : Category.{v_3, u_3} D₁
inst✝¹ : Category.{v_4, u_4} D₂
A B : ObjectProperty C
inst✝ : A.IsVerdierRightLocalizing B
X✝ Y✝ : Cᵒᵖ
f : X✝ ⟶ Y✝
hX : A.op X✝
hY : B.op Y✝
⊢ ∃ Z a b, A.op Z ∧ B.op Z ∧ a ≫ b = f

Tactic:
by
    obtain ⟨Z, a, b, h₁, h₂, fac⟩ := IsVerdierRightLocalizing.fac f.unop hY hX
    exact ⟨_, b.op, a.op, h₁, h₂, Quiver.Hom.unop_inj fac⟩

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

Goal state before:
pp: case mpr
C : Type u_1
inst✝⁶ : Category.{v_1, u_1} C
A B : ObjectProperty C
inst✝⁵ : HasZeroObject C
inst✝⁴ : HasShift C ℤ
inst✝³ : Preadditive C
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
inst✝ : A.IsTriangulated
X Y : A.FullSubcategory
f : X ⟶ Y
Z : C
a : Y.obj ⟶ Z
b : Z ⟶ (shiftFunctor C 1).obj X.obj
h : Triangle.mk f.hom a b ∈ Pretriangulated.distinguishedTriangles
hZ : (A ⊓ B) Z
⊢ Triangle.mk f (homMk a) (homMk (b ≫ (Functor.commShiftIso A.ι 1).inv.app X)) ∈ Pretriangulated.distinguishedTriangles

Tactic:
rw [← A.ι.map_distinguished_iff]

Goal state after:
case mpr
C : Type u_1
inst✝⁶ : Category.{v_1, u_1} C
A B : ObjectProperty C
inst✝⁵ : HasZeroObject C
inst✝⁴ : HasShift C ℤ
inst✝³ : Preadditive C
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
inst✝ : A.IsTriangulated
X Y : A.FullSubcategory
f : X ⟶ Y
Z : C
a : Y.obj ⟶ Z
b : Z ⟶ (shiftFunctor C 1).obj X.obj
h : Triangle.mk f.hom a b ∈ Pretriangulated.distinguishedTriangles
hZ : (A ⊓ B) Z
⊢ A.ι.mapTriangle.obj (Triangle.mk f (homMk a) (homMk (b ≫ (Functor.commShiftIso A.ι 1).inv.app X))) ∈
    Pretriangulated.distinguishedTriangles