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

Goal state before:
pp: n : ℕ
Δ : SimplexCategory
θ : Δ ⟶ ⦋n + 1⦌
i : Fin (n + 2)
hi : ∀ (x : Fin (Δ.len + 1)), (Hom.toOrderHom θ) x ≠ i
x : Fin (Δ.len + 1)
this : ∀ (j : Fin (n + 2)), j ≠ i → i.succAbove (((Fin.last n).predAbove i).predAbove j) = j
⊢ (Hom.toOrderHom θ) x = (Hom.toOrderHom ((θ ≫ σ ((Fin.last n).predAbove i)) ≫ δ i)) x

Tactic:
dsimp [δ, σ]
    exact .symm <| this _ (hi _)

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

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
F : Cᵒᵖ ⥤ Type w
X : Cᵒᵖ
x : F.obj X
F' : Cᵒᵖ ⥤ Type w
f : F ⟶ F'
⊢ (ofSection x).image f = ofSection ((ConcreteCategory.hom (f.app X)) x)

Tactic:
apply le_antisymm

Goal state after:
case a
C : Type u
inst✝ : Category.{v, u} C
F : Cᵒᵖ ⥤ Type w
X : Cᵒᵖ
x : F.obj X
F' : Cᵒᵖ ⥤ Type w
f : F ⟶ F'
⊢ (ofSection x).image f ≤ ofSection ((ConcreteCategory.hom (f.app X)) x)

case a
C : Type u
inst✝ : Category.{v, u} C
F : Cᵒᵖ ⥤ Type w
X : Cᵒᵖ
x : F.obj X
F' : Cᵒᵖ ⥤ Type w
f : F ⟶ F'
⊢ ofSection ((ConcreteCategory.hom (f.app X)) x) ≤ (ofSection x).image f
Module: Mathlib.AlgebraicTopology.SimplicialSet.Dimension

Goal state before:
pp: X : SSet
n : ℕ
x : ↑(X.nonDegenerate n)
d : ℕ
inst✝ : X.HasDimensionLT d
this : d ≤ n
⊢ False

Tactic:
obtain ⟨x, hx⟩ := x

Goal state after:
X : SSet
n d : ℕ
inst✝ : X.HasDimensionLT d
this : d ≤ n
x : X _⦋n⦌
hx : x ∈ X.nonDegenerate n
⊢ False
Module: Mathlib.AlgebraicTopology.SimplicialSet.Degenerate

Goal state before:
pp: case mp
X : SSet
n : ℕ
x : X _⦋n + 1⦌
⊢ x ∈ X.degenerate (n + 1) → x ∈ ⋃ i, Set.range ⇑(ConcreteCategory.hom (SimplicialObject.σ X i))

pp: case mpr
X : SSet
n : ℕ
x : X _⦋n + 1⦌
⊢ x ∈ ⋃ i, Set.range ⇑(ConcreteCategory.hom (SimplicialObject.σ X i)) → x ∈ X.degenerate (n + 1)

Tactic:
· intro hx
    rw [mem_degenerate_iff] at hx
    obtain ⟨m, hm, f, hf, y, rfl⟩ := hx
    obtain ⟨i, θ, rfl⟩ := SimplexCategory.eq_σ_comp_of_not_injective f (fun hf ↦ by
      rw [← SimplexCategory.mono_iff_injective] at hf
      have := SimplexCategory.le_of_mono f
      lia)
    aesop

Goal state after:
case mpr
X : SSet
n : ℕ
x : X _⦋n + 1⦌
⊢ x ∈ ⋃ i, Set.range ⇑(ConcreteCategory.hom (SimplicialObject.σ X i)) → x ∈ X.degenerate (n + 1)
Module: Mathlib.AlgebraicTopology.SimplicialSet.Simplices

Goal state before:
pp: case mpr
X : SSet
x y : X.S
⊢ Nonempty (equivElements y ⟶ equivElements x) → ∃ f, (ConcreteCategory.hom (X.map f.op)) y.simplex = x.simplex

Tactic:
· rintro ⟨f, hf⟩
    exact ⟨f.unop, hf⟩

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

Goal state before:
pp: X : SSet
n : ℕ
x : ↑(X.nonDegenerate n)
m : SimplexCategory
f : ⦋n⦌ ⟶ m
inst✝ : Epi f
y : X.obj (op m)
hy : (ConcreteCategory.hom (X.map f.op)) y = ↑x
⊢ IsIso f

Tactic:
obtain ⟨x, hx⟩ := x

Goal state after:
X : SSet
n : ℕ
m : SimplexCategory
f : ⦋n⦌ ⟶ m
inst✝ : Epi f
y : X.obj (op m)
x : X _⦋n⦌
hx : x ∈ X.nonDegenerate n
hy : (ConcreteCategory.hom (X.map f.op)) y = ↑⟨x, hx⟩
⊢ IsIso f
Module: Mathlib.AlgebraicTopology.SimplicialSet.Subcomplex

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

Tactic:
apply le_antisymm

Goal state after:
case a
X Y : SSet
n : ℕ
x : X _⦋n⦌
f : X ⟶ Y
⊢ (ofSimplex x).image f ≤ ofSimplex ((ConcreteCategory.hom (f.app (Opposite.op ⦋n⦌))) x)

case a
X Y : SSet
n : ℕ
x : X _⦋n⦌
f : X ⟶ Y
⊢ ofSimplex ((ConcreteCategory.hom (f.app (Opposite.op ⦋n⦌))) x) ≤ (ofSimplex x).image f
Module: Mathlib.AlgebraicTopology.SimplicialSet.NonDegenerateSimplices

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

Tactic:
apply le_antisymm

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

case a
X : SSet
x y : X.N
h : x.subcomplex = y.subcomplex
⊢ y ≤ x
Module: Mathlib.AlgebraicTopology.CechNerve

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasTerminal C
ι : Type w
inst✝ : HasFiniteProducts C
X : C
m n : SimplexCategoryᵒᵖ
f : m ⟶ n
⊢ (Arrow.mk (terminal.from X)).cechNerve.map f ≫
      (wideCospan.limitIsoPi (Fin ((Opposite.unop n).len + 1)) (Arrow.mk (terminal.from X)).left).hom =
    (wideCospan.limitIsoPi (Fin ((Opposite.unop m).len + 1)) (Arrow.mk (terminal.from X)).left).hom ≫
      (cechNerveTerminalFrom X).map f

Tactic:
by
    dsimp only [cechNerveTerminalFrom, Arrow.cechNerve]
    ext ⟨j⟩
    simp

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

Goal state before:
pp: X : SSet
inst✝ : X.Finite
⊢ ∃ d, X.HasDimensionLT d

Tactic:
have : Fintype X.N := Fintype.ofFinite _

Goal state after:
X : SSet
inst✝ : X.Finite
this : Fintype X.N
⊢ ∃ d, X.HasDimensionLT d
Module: Mathlib.AlgebraicTopology.SimplicialSet.NonDegenerateSimplices

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

Tactic:
apply le_antisymm

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

case a
X : SSet
x✝¹ x✝ : X.N
h : x✝¹.subcomplex = x✝.subcomplex
⊢ x✝ ≤ x✝¹
Module: Mathlib.AlgebraicTopology.SimplicialSet.Degenerate

Goal state before:
pp: case mpr
X : SSet
A : X.Subcomplex
n : ℕ
x : ↑(A.obj (op ⦋n⦌))
⊢ (∃ m, ∃ (_ : m < n), ∃ f, ∃ (_ : Epi f), ↑x ∈ Set.range ⇑(ConcreteCategory.hom (X.map f.op))) →
    ∃ m, ∃ (_ : m < n), ∃ f, ∃ (_ : Epi f), x ∈ Set.range ⇑(ConcreteCategory.hom (A.toSSet.map f.op))

Tactic:
obtain ⟨x, hx⟩ := x

Goal state after:
case mpr
X : SSet
A : X.Subcomplex
n : ℕ
x : X _⦋n⦌
hx : x ∈ A.obj (op ⦋n⦌)
⊢ (∃ m, ∃ (_ : m < n), ∃ f, ∃ (_ : Epi f), ↑⟨x, hx⟩ ∈ Set.range ⇑(ConcreteCategory.hom (X.map f.op))) →
    ∃ m, ∃ (_ : m < n), ∃ f, ∃ (_ : Epi f), ⟨x, hx⟩ ∈ Set.range ⇑(ConcreteCategory.hom (A.toSSet.map f.op))
Module: Mathlib.AlgebraicTopology.SimplicialSet.CompStruct

Goal state before:
pp: X Y : SSet
x₀ x₁ x₂ y₀ y₁ : X _⦋0⦌
hom : Edge x₀ x₁
hom' : Edge y₀ y₁
I : hom.InvStruct
hhom : hom.edge = hom'.edge
⊢ (ConcreteCategory.hom (SimplicialObject.δ X 0)) hom'.edge = y₁

Tactic:
hom'.tgt_eq

Goal state after:
X Y : SSet
x₀ x₁ x₂ y₀ y₁ : X _⦋0⦌
hom : Edge x₀ x₁
hom' : Edge y₀ y₁
I : hom.InvStruct
hhom : hom.edge = hom'.edge
⊢ y₁ = y₁
Module: Mathlib.AlgebraicTopology.SimplicialSet.CompStruct

Goal state before:
pp: X Y : SSet
x₀ x₁ x₂ y₀ y₁ : X _⦋0⦌
hom : Edge x₀ x₁
hom' : Edge y₀ y₁
I : hom.InvStruct
hhom : hom.edge = hom'.edge
⊢ (id ((ConcreteCategory.hom (SimplicialObject.δ X 0)) hom'.edge)).edge = (id y₁).edge

Tactic:
hom'.tgt_eq

Goal state after:
X Y : SSet
x₀ x₁ x₂ y₀ y₁ : X _⦋0⦌
hom : Edge x₀ x₁
hom' : Edge y₀ y₁
I : hom.InvStruct
hhom : hom.edge = hom'.edge
⊢ (id y₁).edge = (id y₁).edge
Module: Mathlib.Algebra.Homology.AlternatingConst

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
X Y : C
f : X ⟶ Y
i j : ℕ
hij : (ComplexShape.down ℕ).Rel i j
⊢ f ≫ (HomologicalComplex.alternatingConst Y ⋯ ⋯ ⋯).d i j = (HomologicalComplex.alternatingConst X ⋯ ⋯ ⋯).d i j ≫ f

Tactic:
by_cases Even i

Goal state after:
case pos
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
X Y : C
f : X ⟶ Y
i j : ℕ
hij : (ComplexShape.down ℕ).Rel i j
h✝ : Even i
⊢ f ≫ (HomologicalComplex.alternatingConst Y ⋯ ⋯ ⋯).d i j = (HomologicalComplex.alternatingConst X ⋯ ⋯ ⋯).d i j ≫ f

case neg
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
X Y : C
f : X ⟶ Y
i j : ℕ
hij : (ComplexShape.down ℕ).Rel i j
h✝ : ¬Even i
⊢ f ≫ (HomologicalComplex.alternatingConst Y ⋯ ⋯ ⋯).d i j = (HomologicalComplex.alternatingConst X ⋯ ⋯ ⋯).d i j ≫ f
Module: Mathlib.AlgebraicTopology.SimplicialSet.StdSimplex

Goal state before:
pp: n : ℕ
x✝¹ : SimplexCategoryᵒᵖ
x✝ : Δ[n].obj x✝¹
⊢ x✝ ∈ (face univ).obj x✝¹ ↔ x✝ ∈ ⊤.obj x✝¹

Tactic:
simp only [Subfunctor.top_obj, Set.top_eq_univ, Set.mem_univ, iff_true]

Goal state after:
n : ℕ
x✝¹ : SimplexCategoryᵒᵖ
x✝ : Δ[n].obj x✝¹
⊢ x✝ ∈ (face univ).obj x✝¹
Module: Mathlib.AlgebraicTopology.SimplicialSet.StdSimplex

Goal state before:
pp: n : ℕ
S : Finset (Fin (n + 1))
m : ℕ
e : Fin (m + 1) ≃o ↥S
⊢ face S = Subcomplex.ofSimplex (objMk ((OrderHom.Subtype.val fun x ↦ x ∈ S).comp e.toOrderEmbedding.toOrderHom))

Tactic:
apply le_antisymm

Goal state after:
case a
n : ℕ
S : Finset (Fin (n + 1))
m : ℕ
e : Fin (m + 1) ≃o ↥S
⊢ face S ≤ Subcomplex.ofSimplex (objMk ((OrderHom.Subtype.val fun x ↦ x ∈ S).comp e.toOrderEmbedding.toOrderHom))

case a
n : ℕ
S : Finset (Fin (n + 1))
m : ℕ
e : Fin (m + 1) ≃o ↥S
⊢ Subcomplex.ofSimplex (objMk ((OrderHom.Subtype.val fun x ↦ x ∈ S).comp e.toOrderEmbedding.toOrderHom)) ≤ face S
Module: Mathlib.AlgebraicTopology.SimplicialSet.StdSimplex

Goal state before:
pp: n d : ℕ
x : ↑(Δ[n].nonDegenerate d)
j : Fin (n + 1)
⊢ j ∈ ↑(nonDegenerateEquiv' x) ↔ ∃ i, ↑x i = j

Tactic:
by
  unfold nonDegenerateEquiv'
  simp

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

Goal state before:
pp: n : ℕ
S : Finset (Fin (n + 1))
d : ℕ
hd : #S ≤ d
m : ℕ
hm : #S = m
⊢ (face S).toSSet.HasDimensionLT d

Tactic:
obtain _ | m := m

Goal state after:
case zero
n : ℕ
S : Finset (Fin (n + 1))
d : ℕ
hd : #S ≤ d
hm : #S = 0
⊢ (face S).toSSet.HasDimensionLT d

case succ
n : ℕ
S : Finset (Fin (n + 1))
d : ℕ
hd : #S ≤ d
m : ℕ
hm : #S = m + 1
⊢ (face S).toSSet.HasDimensionLT d
Module: Mathlib.AlgebraicTopology.ExtraDegeneracy

Goal state before:
pp: case neg
n : ℕ
Δ : SimplexCategory
f : ⦋n⦌ ⟶ Δ
j₁ j₂ : Fin ⦋n + 1⦌.len
h₁ : ¬j₁.succ = 0
h₂ : j₂.succ ≠ 0
hi : j₁.succ ≤ j₂.succ
⊢ shiftFun (⇑(SimplexCategory.Hom.toOrderHom f)) j₁.succ ≤ shiftFun (⇑(SimplexCategory.Hom.toOrderHom f)) j₂.succ

Tactic:
simpa only [shiftFun_succ] using f.toOrderHom.monotone (Fin.succ_le_succ_iff.mp hi)

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

Goal state before:
pp: C₁ : Type u_1
C₂ : Type u_2
D : Type u_3
inst✝⁷ : Category.{v_1, u_1} C₁
inst✝⁶ : Category.{v_2, u_2} C₂
inst✝⁵ : Category.{v_3, u_3} D
inst✝⁴ : HasZeroMorphisms C₁
inst✝³ : HasZeroMorphisms C₂
inst✝² : HasZeroMorphisms D
F : C₁ ⥤ C₂ ⥤ D
I₁ : Type u_4
I₂ : Type u_5
J : Type u_6
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
inst✝¹ : F.PreservesZeroMorphisms
inst✝ : ∀ (X₁ : C₁), (F.obj X₁).PreservesZeroMorphisms
K₁ : HomologicalComplex C₁ c₁
K₂ : HomologicalComplex C₂ c₂
i✝¹ : I₁
i✝ : I₂
⊢ (F.obj (K₁.X i✝¹)).map (𝟙 (K₂.X i✝)) = 𝟙 ((F.obj (K₁.X i✝¹)).obj (K₂.X i✝))

Tactic:
rw [Functor.map_id]

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

Goal state before:
pp: C₁ : Type u_1
C₂ : Type u_2
D : Type u_3
I₁ : Type u_4
I₂ : Type u_5
J : Type u_6
inst✝¹¹ : Category.{v_1, u_1} C₁
inst✝¹⁰ : Category.{v_2, u_2} C₂
inst✝⁹ : Category.{v_3, u_3} D
inst✝⁸ : Preadditive C₁
inst✝⁷ : Preadditive C₂
inst✝⁶ : Preadditive D
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
K₁ L₁ : HomologicalComplex C₁ c₁
f₁ : K₁ ⟶ L₁
K₂ L₂ : HomologicalComplex C₂ c₂
f₂ f₂' : K₂ ⟶ L₂
h₂ : Homotopy f₂ f₂'
F : C₁ ⥤ C₂ ⥤ D
inst✝⁵ : F.Additive
inst✝⁴ : ∀ (X₁ : C₁), (F.obj X₁).Additive
c : ComplexShape J
inst✝³ : DecidableEq J
inst✝² : TotalComplexShape c₁ c₂ c
inst✝¹ : K₁.HasMapBifunctor K₂ F c
inst✝ : L₁.HasMapBifunctor L₂ F c
i₁ : I₁
i₂ i₂' : I₂
j j' : J
h : c₁.π c₂ c (i₁, i₂') = j
h' : c₂.prev i₂' = i₂
⊢ K₁.ιMapBifunctor K₂ F c i₁ i₂' j h ≫ hom₂ f₁ h₂ F c j j' =
    c₁.ε₂ c₂ c (i₁, i₂) •
      (F.map (f₁.f i₁)).app (K₂.X i₂') ≫ (F.obj (L₁.X i₁)).map (h₂.hom i₂' i₂) ≫ L₁.ιMapBifunctorOrZero L₂ F c i₁ i₂ j'

Tactic:
by
  subst h'
  simp [hom₂]

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

Goal state before:
pp: case pos
C₁ : Type u_1
C₂ : Type u_2
C₁₂ : Type u_3
C₃ : Type u_5
C₄ : Type u_6
inst✝¹⁹ : Category.{v_1, u_1} C₁
inst✝¹⁸ : Category.{v_2, u_2} C₂
inst✝¹⁷ : Category.{v_3, u_5} C₃
inst✝¹⁶ : Category.{v_4, u_6} C₄
inst✝¹⁵ : Category.{v_5, u_3} C₁₂
inst✝¹⁴ : HasZeroMorphisms C₁
inst✝¹³ : HasZeroMorphisms C₂
inst✝¹² : HasZeroMorphisms C₃
inst✝¹¹ : Preadditive C₁₂
inst✝¹⁰ : Preadditive C₄
F₁₂ : C₁ ⥤ C₂ ⥤ C₁₂
G : C₁₂ ⥤ C₃ ⥤ C₄
inst✝⁹ : F₁₂.PreservesZeroMorphisms
inst✝⁸ : ∀ (X₁ : C₁), (F₁₂.obj X₁).PreservesZeroMorphisms
inst✝⁷ : G.Additive
inst✝⁶ : ∀ (X₁₂ : C₁₂), (G.obj X₁₂).PreservesZeroMorphisms
ι₁ : Type u_7
ι₂ : Type u_8
ι₃ : Type u_9
ι₁₂ : Type u_10
ι₄ : Type u_12
inst✝⁵ : DecidableEq ι₄
c₁ : ComplexShape ι₁
c₂ : ComplexShape ι₂
c₃ : ComplexShape ι₃
K₁ : HomologicalComplex C₁ c₁
K₂ : HomologicalComplex C₂ c₂
K₃ : HomologicalComplex C₃ c₃
c₁₂ : ComplexShape ι₁₂
c₄ : ComplexShape ι₄
inst✝⁴ : TotalComplexShape c₁ c₂ c₁₂
inst✝³ : TotalComplexShape c₁₂ c₃ c₄
inst✝² : K₁.HasMapBifunctor K₂ F₁₂ c₁₂
inst✝¹ : DecidableEq ι₁₂
inst✝ : (K₁.mapBifunctor K₂ F₁₂ c₁₂).HasMapBifunctor K₃ G c₄
i₁ : ι₁
i₂ : ι₂
i₃ : ι₃
j j' : ι₄
h : c₁.r c₂ c₃ c₁₂ c₄ (i₁, i₂, i₃) = j
h₁ : c₃.Rel i₃ (c₃.next i₃)
⊢ (G.map (K₁.ιMapBifunctor K₂ F₁₂ c₁₂ i₁ i₂ (c₁.π c₂ c₁₂ (i₁, i₂)) ⋯)).app (K₃.X i₃) ≫
      mapBifunctor.d₂ (K₁.mapBifunctor K₂ F₁₂ c₁₂) K₃ G c₄ (c₁.π c₂ c₁₂ (i₁, i₂)) i₃ j' =
    d₃ F₁₂ G K₁ K₂ K₃ c₁₂ c₄ i₁ i₂ i₃ j'

Tactic:
rw [d₃_eq _ _ _ _ _ _ _ _ _ h₁]

Goal state after:
case pos
C₁ : Type u_1
C₂ : Type u_2
C₁₂ : Type u_3
C₃ : Type u_5
C₄ : Type u_6
inst✝¹⁹ : Category.{v_1, u_1} C₁
inst✝¹⁸ : Category.{v_2, u_2} C₂
inst✝¹⁷ : Category.{v_3, u_5} C₃
inst✝¹⁶ : Category.{v_4, u_6} C₄
inst✝¹⁵ : Category.{v_5, u_3} C₁₂
inst✝¹⁴ : HasZeroMorphisms C₁
inst✝¹³ : HasZeroMorphisms C₂
inst✝¹² : HasZeroMorphisms C₃
inst✝¹¹ : Preadditive C₁₂
inst✝¹⁰ : Preadditive C₄
F₁₂ : C₁ ⥤ C₂ ⥤ C₁₂
G : C₁₂ ⥤ C₃ ⥤ C₄
inst✝⁹ : F₁₂.PreservesZeroMorphisms
inst✝⁸ : ∀ (X₁ : C₁), (F₁₂.obj X₁).PreservesZeroMorphisms
inst✝⁷ : G.Additive
inst✝⁶ : ∀ (X₁₂ : C₁₂), (G.obj X₁₂).PreservesZeroMorphisms
ι₁ : Type u_7
ι₂ : Type u_8
ι₃ : Type u_9
ι₁₂ : Type u_10
ι₄ : Type u_12
inst✝⁵ : DecidableEq ι₄
c₁ : ComplexShape ι₁
c₂ : ComplexShape ι₂
c₃ : ComplexShape ι₃
K₁ : HomologicalComplex C₁ c₁
K₂ : HomologicalComplex C₂ c₂
K₃ : HomologicalComplex C₃ c₃
c₁₂ : ComplexShape ι₁₂
c₄ : ComplexShape ι₄
inst✝⁴ : TotalComplexShape c₁ c₂ c₁₂
inst✝³ : TotalComplexShape c₁₂ c₃ c₄
inst✝² : K₁.HasMapBifunctor K₂ F₁₂ c₁₂
inst✝¹ : DecidableEq ι₁₂
inst✝ : (K₁.mapBifunctor K₂ F₁₂ c₁₂).HasMapBifunctor K₃ G c₄
i₁ : ι₁
i₂ : ι₂
i₃ : ι₃
j j' : ι₄
h : c₁.r c₂ c₃ c₁₂ c₄ (i₁, i₂, i₃) = j
h₁ : c₃.Rel i₃ (c₃.next i₃)
⊢ (G.map (K₁.ιMapBifunctor K₂ F₁₂ c₁₂ i₁ i₂ (c₁.π c₂ c₁₂ (i₁, i₂)) ⋯)).app (K₃.X i₃) ≫
      mapBifunctor.d₂ (K₁.mapBifunctor K₂ F₁₂ c₁₂) K₃ G c₄ (c₁.π c₂ c₁₂ (i₁, i₂)) i₃ j' =
    c₁₂.ε₂ c₃ c₄ (c₁.π c₂ c₁₂ (i₁, i₂), i₃) •
      (G.obj ((F₁₂.obj (K₁.X i₁)).obj (K₂.X i₂))).map (K₃.d i₃ (c₃.next i₃)) ≫
        ιOrZero F₁₂ G K₁ K₂ K₃ c₁₂ c₄ i₁ i₂ (c₃.next i₃) j'
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 : CochainComplex C ℤ
m : ℤ
hK : HomologicalComplex.ExactAt K (-m)
hm : -m + m = 0
⊢ (ComplexShape.up ℤ).next (-m) = -(m - 1)

Tactic:
by grind [next]

Goal state after:
Module: Mathlib.Algebra.Homology.DerivedCategory.KInjective

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
K L : CochainComplex C ℤ
inst✝¹ : K.IsKInjective
inst✝ : L.IsKInjective
f : K ⟶ L
⊢ QuasiIso f ↔ homotopyEquivalences C (ComplexShape.up ℤ) f

Tactic:
by
  refine ⟨fun _ ↦ ?_, fun hf ↦ homotopyEquivalences_le_quasiIso _ _ _ hf⟩
  rw [← HomotopyCategory.inverseImage_quotient_isomorphisms,
    MorphismProperty.inverseImage_iff, MorphismProperty.isomorphisms.iff]
  obtain ⟨g, hg⟩ := (Qh_map_bijective _ _).surjective
    ((quotientCompQhIso C).hom.app L ≫ inv (Q.map f) ≫ (quotientCompQhIso C).inv.app K)
  refine ⟨g, (Qh_map_bijective _ _).injective ?_, (Qh_map_bijective _ _).injective ?_⟩
  · simp [hg]
  · simp [hg, ← quotientCompQhIso_inv_naturality, -NatTrans.naturality]

Goal state after:
Module: Mathlib.CategoryTheory.MorphismProperty.LiftingProperty

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
T : MorphismProperty C
⊢ T.pushouts.rlp = T.rlp

Tactic:
apply le_antisymm

Goal state after:
case a
C : Type u
inst✝ : Category.{v, u} C
T : MorphismProperty C
⊢ T.pushouts.rlp ≤ T.rlp

case a
C : Type u
inst✝ : Category.{v, u} C
T : MorphismProperty C
⊢ T.rlp ≤ T.pushouts.rlp
Module: Mathlib.CategoryTheory.MorphismProperty.LiftingProperty

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
T : MorphismProperty C
⊢ (coproducts.{w, v, u} T).rlp = T.rlp

Tactic:
apply le_antisymm

Goal state after:
case a
C : Type u
inst✝ : Category.{v, u} C
T : MorphismProperty C
⊢ (coproducts.{w, v, u} T).rlp ≤ T.rlp

case a
C : Type u
inst✝ : Category.{v, u} C
T : MorphismProperty C
⊢ T.rlp ≤ (coproducts.{w, v, u} T).rlp
Module: Mathlib.CategoryTheory.MorphismProperty.LiftingProperty

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
T : MorphismProperty C
⊢ T.retracts.rlp = T.rlp

Tactic:
apply le_antisymm

Goal state after:
case a
C : Type u
inst✝ : Category.{v, u} C
T : MorphismProperty C
⊢ T.retracts.rlp ≤ T.rlp

case a
C : Type u
inst✝ : Category.{v, u} C
T : MorphismProperty C
⊢ T.rlp ≤ T.retracts.rlp
Module: Mathlib.AlgebraicTopology.ModelCategory.Basic

Goal state before:
pp: C : Type u
inst✝⁸ : Category.{v, u} C
inst✝⁷ : CategoryWithFibrations C
inst✝⁶ : CategoryWithCofibrations C
inst✝⁵ : CategoryWithWeakEquivalences C
inst✝⁴ : (weakEquivalences C).HasTwoOutOfThreeProperty
inst✝³ : (trivialCofibrations C).IsWeakFactorizationSystem (fibrations C)
inst✝² : (cofibrations C).IsWeakFactorizationSystem (trivialFibrations C)
A B X Y : C
f : A ⟶ B
w : X ⟶ Y
inst✝¹ : Fibration f
inst✝ : WeakEquivalence w
h : RetractArrow f w
⊢ WeakEquivalence f

Tactic:
have hw := factorizationData (trivialCofibrations C) (fibrations C) w

Goal state after:
C : Type u
inst✝⁸ : Category.{v, u} C
inst✝⁷ : CategoryWithFibrations C
inst✝⁶ : CategoryWithCofibrations C
inst✝⁵ : CategoryWithWeakEquivalences C
inst✝⁴ : (weakEquivalences C).HasTwoOutOfThreeProperty
inst✝³ : (trivialCofibrations C).IsWeakFactorizationSystem (fibrations C)
inst✝² : (cofibrations C).IsWeakFactorizationSystem (trivialFibrations C)
A B X Y : C
f : A ⟶ B
w : X ⟶ Y
inst✝¹ : Fibration f
inst✝ : WeakEquivalence w
h : RetractArrow f w
hw : (trivialCofibrations C).MapFactorizationData (fibrations C) w
⊢ WeakEquivalence f
Module: Mathlib.AlgebraicTopology.ModelCategory.Cylinder

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} C
A : C
inst✝⁴ : CategoryWithWeakEquivalences C
P : Cylinder A
inst✝³ : HasBinaryCoproducts C
inst✝² : CategoryWithCofibrations C
inst✝¹ : P.IsGood
inst✝ : (cofibrations C).RespectsIso
hi : cofibrations C P.i
⊢ Arrow.mk ((coprod.braiding A A).hom ≫ P.i) ≅ Arrow.mk P.i

Tactic:
exact Arrow.isoMk (coprod.braiding A A) (Iso.refl _)

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

Goal state before:
pp: C : Type u_1
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : Preadditive C
inst✝⁴ : HasZeroObject C
inst✝³ : HasShift C ℤ
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
t : TStructure C
n : ℤ
Y X : C
f₁ f₂ : Y ⟶ (t.truncLE n).obj X
h : f₁ ≫ (t.truncLEι n).app X = f₂ ≫ (t.truncLEι n).app X
inst✝ : t.IsLE Y n
⊢ f₁ = f₂

Tactic:
have : t.IsLE Y (n + 1 - 1) := by simpa

Goal state after:
C : Type u_1
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : Preadditive C
inst✝⁴ : HasZeroObject C
inst✝³ : HasShift C ℤ
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
t : TStructure C
n : ℤ
Y X : C
f₁ f₂ : Y ⟶ (t.truncLE n).obj X
h : f₁ ≫ (t.truncLEι n).app X = f₂ ≫ (t.truncLEι n).app X
inst✝ : t.IsLE Y n
this : t.IsLE Y (n + 1 - 1)
⊢ f₁ = f₂
Module: Mathlib.Algebra.Homology.DerivedCategory.Plus

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
inst✝ : HasDerivedCategory C
K L : HomotopyCategory.Plus C
x✝ : CochainComplex.IsKInjective L.obj.as
⊢ Function.Bijective Qh.map

Tactic:
have := CochainComplex.IsKInjective.Qh_map_bijective K.1 L.1.as

Goal state after:
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
inst✝ : HasDerivedCategory C
K L : HomotopyCategory.Plus C
x✝ : CochainComplex.IsKInjective L.obj.as
this : Function.Bijective DerivedCategory.Qh.map
⊢ Function.Bijective Qh.map
Module: Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : Preadditive C
inst✝⁴ : HasZeroObject C
inst✝³ : HasShift C ℤ
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
t : TStructure C
n₀ n₁ : ℤ
h : n₀ + 1 = n₁
X : C
inst✝ : t.IsLE X n₀
⊢ IsZero ((t.truncGE n₁).obj X)

Tactic:
by
  rw [← t.isLE_iff_isZero_truncGE_obj _ _ h X]
  infer_instance

Goal state after:
Module: Mathlib.CategoryTheory.GuitartExact.Basic

Goal state before:
pp: C₁ : Type u₁
C₂ : Type u₂
C₃ : Type u₃
C₄ : Type u₄
inst✝³ : Category.{v₁, u₁} C₁
inst✝² : Category.{v₂, u₂} C₂
inst✝¹ : Category.{v₃, u₃} C₃
inst✝ : Category.{v₄, u₄} C₄
T : C₁ ⥤ C₂
L : C₁ ⥤ C₃
R : C₂ ⥤ C₄
B : C₃ ⥤ C₄
w : TwoSquare T L R B
X₂ X₂' : C₂
e : X₂ ≅ X₂'
⊢ (w.structuredArrowDownwards X₂ ⋙ (StructuredArrow.mapIso (R.mapIso e)).functor).Initial ↔
    ((StructuredArrow.mapIso e).functor ⋙ w.structuredArrowDownwards X₂').Initial

Tactic:
exact Functor.initial_natIso_iff
    (NatIso.ofComponents (fun _ ↦ StructuredArrow.isoMk (Iso.refl _)))

Goal state after:
Module: Mathlib.CategoryTheory.Localization.DerivabilityStructure.PointwiseRightDerived

Goal state before:
pp: C₁ : Type u₁
C₂ : Type u₂
H : Type u₃
inst✝³ : Category.{v₁, u₁} C₁
inst✝² : Category.{v₂, u₂} C₂
inst✝¹ : Category.{v₃, u₃} H
W₁ : MorphismProperty C₁
W₂ : MorphismProperty C₂
Φ : LocalizerMorphism W₁ W₂
F : C₂ ⥤ H
inst✝ : Φ.IsRightDerivabilityStructure
⊢ F.HasPointwiseRightDerivedFunctor W₂ ↔ (Φ.functor ⋙ F).HasPointwiseRightDerivedFunctor W₁

Tactic:
constructor
  · intro hF X₁
    rw [hasPointwiseRightDerivedFunctorAt_iff_of_isRightDerivabilityStructure]
    apply hF
  · intro hF X₂
    have R : Φ.RightResolution X₂ := Classical.arbitrary _
    simpa only [hasPointwiseRightDerivedFunctorAt_iff_of_isRightDerivabilityStructure,
      ← F.hasPointwiseRightDerivedFunctorAt_iff_of_mem W₂ R.w R.hw] using hF R.X₁

Goal state after:
Module: Mathlib.Algebra.Homology.DerivedCategory.Ext.Map

Goal state before:
pp: C : Type u
inst✝⁸ : Category.{v, u} C
inst✝⁷ : Abelian C
D : Type u'
inst✝⁶ : Category.{v', u'} D
inst✝⁵ : Abelian D
F : C ⥤ D
inst✝⁴ : F.Additive
inst✝³ : PreservesFiniteLimits F
inst✝² : PreservesFiniteColimits F
inst✝¹ : HasDerivedCategory C
inst✝ : HasDerivedCategory D
S : ShortComplex (CochainComplex C ℤ)
hS : S.ShortExact
this : QuasiIso (CochainComplex.mappingCone.descShortComplex S)
⊢ F.mapDerivedCategory.map (triangleOfSESδ hS) =
    F.mapDerivedCategoryFactors.hom.app S.X₃ ≫
      triangleOfSESδ ⋯ ≫
        (shiftFunctor (DerivedCategory D) 1).map (F.mapDerivedCategoryFactors.inv.app S.X₁) ≫
          (Functor.commShiftIso F.mapDerivedCategory 1).inv.app (Q.obj S.X₁)

Tactic:
← cancel_epi (F.mapDerivedCategory.map
    (Q.map (CochainComplex.mappingCone.descShortComplex S))),

Goal state after:
C : Type u
inst✝⁸ : Category.{v, u} C
inst✝⁷ : Abelian C
D : Type u'
inst✝⁶ : Category.{v', u'} D
inst✝⁵ : Abelian D
F : C ⥤ D
inst✝⁴ : F.Additive
inst✝³ : PreservesFiniteLimits F
inst✝² : PreservesFiniteColimits F
inst✝¹ : HasDerivedCategory C
inst✝ : HasDerivedCategory D
S : ShortComplex (CochainComplex C ℤ)
hS : S.ShortExact
this : QuasiIso (CochainComplex.mappingCone.descShortComplex S)
⊢ F.mapDerivedCategory.map (Q.map (CochainComplex.mappingCone.descShortComplex S)) ≫
      F.mapDerivedCategory.map (triangleOfSESδ hS) =
    F.mapDerivedCategory.map (Q.map (CochainComplex.mappingCone.descShortComplex S)) ≫
      F.mapDerivedCategoryFactors.hom.app S.X₃ ≫
        triangleOfSESδ ⋯ ≫
          (shiftFunctor (DerivedCategory D) 1).map (F.mapDerivedCategoryFactors.inv.app S.X₁) ≫
            (Functor.commShiftIso F.mapDerivedCategory 1).inv.app (Q.obj S.X₁)