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

Goal state before:
pp: f : ⦋1⦌ ⟶ ⦋2⦌
this : (Hom.toOrderHom f) 0 ≤ (Hom.toOrderHom f) 1
e0 : (Hom.toOrderHom f) 0 = 1
e1 : (Hom.toOrderHom f) 1 = 2
⊢ (∃ i, f = δ i) ∨ ∃ a, f = ⦋1⦌.const ⦋2⦌ a

Tactic:
refine .inl ⟨0, ?_⟩
    ext i : 3
    match i with
    | 0 => exact e0
    | 1 => exact e1

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

Goal state before:
pp: f : ⦋1⦌ ⟶ ⦋2⦌
this : (Hom.toOrderHom f) 0 ≤ (Hom.toOrderHom f) 1
e0 : (Hom.toOrderHom f) 0 = 0
e1 : (Hom.toOrderHom f) 1 = 0
⊢ (∃ i, f = δ i) ∨ ∃ a, f = ⦋1⦌.const ⦋2⦌ a

Tactic:
refine .inr ⟨f.toOrderHom 0, ?_⟩

Goal state after:
f : ⦋1⦌ ⟶ ⦋2⦌
this : (Hom.toOrderHom f) 0 ≤ (Hom.toOrderHom f) 1
e0 : (Hom.toOrderHom f) 0 = 0
e1 : (Hom.toOrderHom f) 1 = 0
⊢ f = ⦋1⦌.const ⦋2⦌ ((Hom.toOrderHom f) 0)
Module: Mathlib.AlgebraicTopology.SimplexCategory.Basic

Goal state before:
pp: f : ⦋1⦌ ⟶ ⦋2⦌
this : (Hom.toOrderHom f) 0 ≤ (Hom.toOrderHom f) 1
e0 : (Hom.toOrderHom f) 0 = 1
e1 : (Hom.toOrderHom f) 1 = 1
⊢ (∃ i, f = δ i) ∨ ∃ a, f = ⦋1⦌.const ⦋2⦌ a

Tactic:
refine .inr ⟨f.toOrderHom 0, ?_⟩

Goal state after:
f : ⦋1⦌ ⟶ ⦋2⦌
this : (Hom.toOrderHom f) 0 ≤ (Hom.toOrderHom f) 1
e0 : (Hom.toOrderHom f) 0 = 1
e1 : (Hom.toOrderHom f) 1 = 1
⊢ f = ⦋1⦌.const ⦋2⦌ ((Hom.toOrderHom f) 0)
Module: Mathlib.AlgebraicTopology.SimplexCategory.Basic

Goal state before:
pp: f : ⦋1⦌ ⟶ ⦋2⦌
this : (Hom.toOrderHom f) 0 ≤ (Hom.toOrderHom f) 1
e0 : (Hom.toOrderHom f) 0 = 2
e1 : (Hom.toOrderHom f) 1 = 2
⊢ (∃ i, f = δ i) ∨ ∃ a, f = ⦋1⦌.const ⦋2⦌ a

Tactic:
refine .inr ⟨f.toOrderHom 0, ?_⟩

Goal state after:
f : ⦋1⦌ ⟶ ⦋2⦌
this : (Hom.toOrderHom f) 0 ≤ (Hom.toOrderHom f) 1
e0 : (Hom.toOrderHom f) 0 = 2
e1 : (Hom.toOrderHom f) 1 = 2
⊢ f = ⦋1⦌.const ⦋2⦌ ((Hom.toOrderHom f) 0)
Module: Mathlib.AlgebraicTopology.SimplicialObject.DeltaZeroIter

Goal state before:
pp: C : Type u_1
inst✝ : Category.{v_1, u_1} C
X : SimplicialObject C
i n m : ℕ
h : n + i = m
⊢ X.δ₀Iter (i + 1) ⋯ = X.δ 0 ≫ X.δ₀Iter i h

Tactic:
by
  simp [δ₀Iter, SimplexCategory.δ₀Iter_succ _ h, δ_def]

Goal state after:
Module: Mathlib.AlgebraicTopology.CechNerve

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : ∀ (n : ℕ) (f : Arrow C), HasWidePullback f.right (fun x ↦ f.left) fun x ↦ f.hom
X : Augmented C
F : Arrow C
G : Augmented.toArrow.obj X ⟶ F
x y : SimplexCategoryᵒᵖ
⊢ ∀ (f : x ⟶ y),
    X.left.map f ≫
        WidePullback.lift (X.hom.app y ≫ Arrow.Hom.right G)
          (fun i ↦ X.left.map (⦋0⦌.const (Opposite.unop y) i).op ≫ Arrow.Hom.left G) ⋯ =
      WidePullback.lift (X.hom.app x ≫ Arrow.Hom.right G)
          (fun i ↦ X.left.map (⦋0⦌.const (Opposite.unop x) i).op ≫ Arrow.Hom.left G) ⋯ ≫
        F.augmentedCechNerve.left.map f

Tactic:
f

Goal state after:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : ∀ (n : ℕ) (f : Arrow C), HasWidePullback f.right (fun x ↦ f.left) fun x ↦ f.hom
X : Augmented C
F : Arrow C
G : Augmented.toArrow.obj X ⟶ F
x y : SimplexCategoryᵒᵖ
f : x ⟶ y
⊢ X.left.map f ≫
      WidePullback.lift (X.hom.app y ≫ Arrow.Hom.right G)
        (fun i ↦ X.left.map (⦋0⦌.const (Opposite.unop y) i).op ≫ Arrow.Hom.left G) ⋯ =
    WidePullback.lift (X.hom.app x ≫ Arrow.Hom.right G)
        (fun i ↦ X.left.map (⦋0⦌.const (Opposite.unop x) i).op ≫ Arrow.Hom.left G) ⋯ ≫
      F.augmentedCechNerve.left.map f
Module: Mathlib.AlgebraicTopology.CechNerve

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : ∀ (n : ℕ) (f : Arrow C), HasWidePullback f.right (fun x ↦ f.left) fun x ↦ f.hom
X : Augmented C
Y Y' : Arrow C
⊢ ∀ (f : Augmented.toArrow.obj X ⟶ Y) (g : Y ⟶ Y'),
    { toFun := equivalenceLeftToRight X Y', invFun := equivalenceRightToLeft X Y', left_inv := ⋯, right_inv := ⋯ }
        (f ≫ g) =
      { toFun := equivalenceLeftToRight X Y, invFun := equivalenceRightToLeft X Y, left_inv := ⋯, right_inv := ⋯ } f ≫
        augmentedCechNerve.map g

Tactic:
f

Goal state after:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : ∀ (n : ℕ) (f : Arrow C), HasWidePullback f.right (fun x ↦ f.left) fun x ↦ f.hom
X : Augmented C
Y Y' : Arrow C
f : Augmented.toArrow.obj X ⟶ Y
⊢ ∀ (g : Y ⟶ Y'),
    { toFun := equivalenceLeftToRight X Y', invFun := equivalenceRightToLeft X Y', left_inv := ⋯, right_inv := ⋯ }
        (f ≫ g) =
      { toFun := equivalenceLeftToRight X Y, invFun := equivalenceRightToLeft X Y, left_inv := ⋯, right_inv := ⋯ } f ≫
        augmentedCechNerve.map g
Module: Mathlib.AlgebraicTopology.CechNerve

Goal state before:
pp: 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),
    F.augmentedCechConerve.right.map f ≫
        WidePushout.desc (Arrow.Hom.left G ≫ X.hom.app y) (fun i ↦ Arrow.Hom.right G ≫ X.right.map (⦋0⦌.const y i)) ⋯ =
      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:
f

Goal state after:
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
⊢ F.augmentedCechConerve.right.map f ≫
      WidePushout.desc (Arrow.Hom.left G ≫ X.hom.app y) (fun i ↦ Arrow.Hom.right G ≫ X.right.map (⦋0⦌.const y i)) ⋯ =
    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.NonDegenerateSimplices

Goal state before:
pp: X : SSet
⊢ ⊤ ≤ ⨆ s, s.subcomplex

Tactic:
← Subcomplex.iSup_ofSimplex_nonDegenerate_eq_top X,

Goal state after:
X : SSet
⊢ ⨆ x, Subcomplex.ofSimplex ↑x.snd ≤ ⨆ s, s.subcomplex
Module: Mathlib.AlgebraicTopology.SimplicialSet.NonDegenerateSimplices

Goal state before:
pp: X : SSet
⊢ ⨆ x, Subcomplex.ofSimplex ↑x.snd ≤ ⨆ s, s.subcomplex

Tactic:
iSup_le_iff

Goal state after:
X : SSet
⊢ ∀ (i : (p : ℕ) × ↑(X.nonDegenerate p)), Subcomplex.ofSimplex ↑i.snd ≤ ⨆ s, s.subcomplex
Module: Mathlib.AlgebraicTopology.SimplicialSet.Finite

Goal state before:
pp: X : SSet
ι : Type u_1
inst✝ : Finite ι
A : ι → X.Subcomplex
⊢ (⨆ i, A i).toSSet.Finite ↔ ∀ (i : ι), (A i).toSSet.Finite

Tactic:
refine ⟨fun h i ↦ finite_of_mono (Subcomplex.homOfLE (le_iSup A i)), fun h ↦ ⟨?_⟩⟩

Goal state after:
X : SSet
ι : Type u_1
inst✝ : Finite ι
A : ι → X.Subcomplex
h : ∀ (i : ι), (A i).toSSet.Finite
⊢ Finite (⨆ i, A i).toSSet.N
Module: Mathlib.AlgebraicTopology.SimplicialSet.Degenerate

Goal state before:
pp: X : SSet
A B : X.Subcomplex
⊢ A ≤ B ↔ ∀ (n : ℕ) (x : ↑(X.nonDegenerate n)), ↑x ∈ A.obj (op ⦋n⦌) → ↑x ∈ B.obj (op ⦋n⦌)

Tactic:
by
  constructor
  · aesop
  · rintro h ⟨n⟩ x hx
    induction n using SimplexCategory.rec with | _ n =>
    obtain ⟨m, f, _, ⟨a, ha⟩, ha'⟩ := exists_nonDegenerate A ⟨x, hx⟩
    simp only [Subfunctor.toFunctor_obj, Subtype.ext_iff,
      Subfunctor.toFunctor_map] at ha'
    subst ha'
    rw [mem_nonDegenerate_iff] at ha
    exact B.map f.op (h _ ⟨_, ha⟩ a.prop)

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

Goal state before:
pp: V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : HasZeroMorphisms V
C : ChainComplex V ℕ
i j : ℕ
⊢ (ComplexShape.down ℕ).Rel i j →
    (match i with
        | 0 => 𝟙 (((truncate.obj C).augment (C.d 1 0) ⋯).X 0)
        | n.succ => 𝟙 (((truncate.obj C).augment (C.d 1 0) ⋯).X (n + 1))) ≫
        C.d i j =
      ((truncate.obj C).augment (C.d 1 0) ⋯).d i j ≫
        match j with
        | 0 => 𝟙 (((truncate.obj C).augment (C.d 1 0) ⋯).X 0)
        | n.succ => 𝟙 (((truncate.obj C).augment (C.d 1 0) ⋯).X (n + 1))

Tactic:
match i with
        | 0 | 1 | n + 2 =>
          rcases j with - | j <;> dsimp [augment, truncate] <;> simp

Goal state after:
Module: Mathlib.AlgebraicTopology.ExtraDegeneracy

Goal state before:
pp: C : Type u_1
inst✝ : Category.{v_1, u_1} C
X : Augmented C
ed : X.ExtraDegeneracy
n : ℕ
i : Fin (n + 2)
j : Fin (n + 1)
hij : i ≤ j.castSucc
k : Fin (n + 1 + 1)
hk : j.succ.rev = k
⊢ X.left.δ₀Iter (↑j + 1) ⋯ ≫ ed.s ↑k ≫ X.left.σ₀Iter (↑j + 1) ⋯ ≫ X.left.δ i.castSucc =
    X.left.δ i ≫ X.left.δ₀Iter ↑j ⋯ ≫ ed.s ↑k ≫ X.left.σ₀Iter ↑j ⋯

Tactic:
rw [X.left.σ₀Iter_δ _ _ (by grind), X.left.δ_δ₀Iter_assoc _ _ (by grind)]

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

Goal state before:
pp: case left
n m : ℕ
f₁ : unop (op ⦋m⦌) ⟶ ⦋n⦌
h₁ : objEquiv.symm f₁ ∈ Δ[n].nonDegenerate m
f₂ : unop (op ⦋m⦌) ⟶ ⦋n⦌
h₂ : objEquiv.symm f₂ ∈ Δ[n].nonDegenerate m
h₃ :
  (fun x ↦
        match x with
        | ⟨x, hx⟩ => ⟨Finset.image (⇑(Hom.toOrderHom (objEquiv x))) univ, ⋯⟩)
      ⟨objEquiv.symm f₁, h₁⟩ =
    (fun x ↦
        match x with
        | ⟨x, hx⟩ => ⟨Finset.image (⇑(Hom.toOrderHom (objEquiv x))) univ, ⋯⟩)
      ⟨objEquiv.symm f₂, h₂⟩
⊢ ⟨objEquiv.symm f₁, h₁⟩ = ⟨objEquiv.symm f₂, h₂⟩

Tactic:
simp only [mem_nonDegenerate_iff_mono, Equiv.apply_symm_apply,
      SimplexCategory.mono_iff_injective, SimplexCategory.len_mk] at h₁ h₂

Goal state after:
case left
n m : ℕ
f₁ : unop (op ⦋m⦌) ⟶ ⦋n⦌
h₁✝ : objEquiv.symm f₁ ∈ Δ[n].nonDegenerate m
f₂ : unop (op ⦋m⦌) ⟶ ⦋n⦌
h₂✝ : objEquiv.symm f₂ ∈ Δ[n].nonDegenerate m
h₃ :
  (fun x ↦
        match x with
        | ⟨x, hx⟩ => ⟨Finset.image (⇑(Hom.toOrderHom (objEquiv x))) univ, ⋯⟩)
      ⟨objEquiv.symm f₁, h₁✝⟩ =
    (fun x ↦
        match x with
        | ⟨x, hx⟩ => ⟨Finset.image (⇑(Hom.toOrderHom (objEquiv x))) univ, ⋯⟩)
      ⟨objEquiv.symm f₂, h₂✝⟩
h₁ : Function.Injective ⇑(Hom.toOrderHom f₁)
h₂ : Function.Injective ⇑(Hom.toOrderHom f₂)
⊢ ⟨objEquiv.symm f₁, h₁✝⟩ = ⟨objEquiv.symm f₂, h₂✝⟩
Module: Mathlib.Algebra.Homology.Bifunctor

Goal state before:
pp: C₁ : Type u_1
C₂ : Type u_2
D : Type u_3
inst✝¹⁰ : Category.{v_1, u_1} C₁
inst✝⁹ : Category.{v_2, u_2} C₂
inst✝⁸ : Category.{v_3, u_3} D
I₁ : Type u_4
I₂ : Type u_5
J : Type u_6
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
inst✝⁷ : HasZeroMorphisms C₁
inst✝⁶ : HasZeroMorphisms C₂
inst✝⁵ : Preadditive D
K₁ : HomologicalComplex C₁ c₁
K₂ : HomologicalComplex C₂ c₂
F : C₁ ⥤ C₂ ⥤ D
inst✝⁴ : F.PreservesZeroMorphisms
inst✝³ : ∀ (X₁ : C₁), (F.obj X₁).PreservesZeroMorphisms
c : ComplexShape J
inst✝² : TotalComplexShape c₁ c₂ c
inst✝¹ : K₁.HasMapBifunctor K₂ F c
inst✝ : DecidableEq J
j j' : J
i₁ : I₁
i₂ : I₂
h : c₁.π c₂ c (i₁, i₂) = j
⊢ K₁.ιMapBifunctor K₂ F c i₁ i₂ j h ≫ D₁ K₁ K₂ F c j j' = d₁ K₁ K₂ F c i₁ i₂ j'

Tactic:
by
  apply HomologicalComplex₂.ι_D₁

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:
subst h'
  simp [hom₂]

Goal state after:
Module: Mathlib.CategoryTheory.Shift.Twist

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
A : Type w
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
t : TwistShiftData C A
a b : A
⊢ ∀ {X Y : C} (f : X ⟶ Y),
    (shiftFunctor C (a + b)).map f ≫ (t.z a b • (shiftFunctorAdd C a b).app Y).hom =
      (t.z a b • (shiftFunctorAdd C a b).app X).hom ≫ (shiftFunctor C a ⋙ shiftFunctor C b).map f

Tactic:
by
    simp [CatCenter.naturality_assoc, CatCenter.naturality, CatCenter.smul_iso_hom_eq]

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 f₁' : K₁ ⟶ L₁
h₁ : Homotopy f₁ f₁'
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
this✝² : TotalComplexShape c₂ c₁ c := TotalComplexShape.symm c₁ c₂ c
this✝¹ : TotalComplexShapeSymmetry c₁ c₂ c := TotalComplexShape.symmSymmetry c₁ c₂ c
this✝ : F.flip.Additive
this : ∀ (X₁ : C₂), (F.flip.obj X₁).Additive
H : Homotopy (mapBifunctorMap f₁ f₂ F c) (mapBifunctorMap f₁ f₂' F c) :=
  (Homotopy.ofEq ⋯).trans
    ((((mapBifunctorMapHomotopy₁ h₂ f₁ F.flip c).compRight (L₁.mapBifunctorFlipIso L₂ F c).hom).compLeft
          (K₁.mapBifunctorFlipIso K₂ F c).inv).trans
      (Homotopy.ofEq ⋯))
j j' : J
i₁ : I₁
i₂ : I₂
hj : c₁.π c₂ c (i₁, i₂) = j
hj' : c₁.π c₂ c (i₁, c₂.prev i₂) = j'
⊢ c₂.π c₁ c (c₂.prev i₂, i₁) = j'

Tactic:
rwa [ComplexShape.π_symm c₁ c₂ c]

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 f₁' : K₁ ⟶ L₁
h₁ : Homotopy f₁ f₁'
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
this✝² : TotalComplexShape c₂ c₁ c := TotalComplexShape.symm c₁ c₂ c
this✝¹ : TotalComplexShapeSymmetry c₁ c₂ c := TotalComplexShape.symmSymmetry c₁ c₂ c
this✝ : F.flip.Additive
this : ∀ (X₁ : C₂), (F.flip.obj X₁).Additive
H : Homotopy (mapBifunctorMap f₁ f₂ F c) (mapBifunctorMap f₁ f₂' F c) :=
  (Homotopy.ofEq ⋯).trans
    ((((mapBifunctorMapHomotopy₁ h₂ f₁ F.flip c).compRight (L₁.mapBifunctorFlipIso L₂ F c).hom).compLeft
          (K₁.mapBifunctorFlipIso K₂ F c).inv).trans
      (Homotopy.ofEq ⋯))
j j' : J
i₁ : I₁
i₂ : I₂
hj : c₁.π c₂ c (i₁, i₂) = j
hj' : ¬c₁.π c₂ c (i₁, c₂.prev i₂) = j'
⊢ c₂.π c₁ c (c₂.prev i₂, i₁) ≠ j'

Tactic:
rwa [ComplexShape.π_symm c₁ c₂ c]

Goal state after:
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
H : ℕ → Prop := fun a ↦ ∀ (n : ℤ), t.le n ≤ t.le (n + ↑a)
H_zero : H 0
n : ℤ
X : C
hX : t.le 0 ((shiftFunctor C n).obj X)
⊢ t.le 1 ((shiftFunctor C n).obj X)

Tactic:
exact t.le_zero_le _ hX

Goal state after:
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
H : ℕ → Prop := fun a ↦ ∀ (n : ℤ), t.ge (n + ↑a) ≤ t.ge n
H_zero : H 0
n : ℤ
X : C
hX : t.ge (n + ↑1) X
⊢ t.ge n X

Tactic:
← t.shift_ge n 1 (n + (1 : ℕ)) (by simp),

Goal state after:
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
H : ℕ → Prop := fun a ↦ ∀ (n : ℤ), t.ge (n + ↑a) ≤ t.ge n
H_zero : H 0
n : ℤ
X : C
hX : (t.ge 1).shift n X
⊢ t.ge n X
Module: Mathlib.Algebra.Homology.HomotopyCategory.KInjective

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : Abelian C
L : CochainComplex C ℤ
d : ℤ
inst✝¹ : L.IsStrictlyGE d
inst✝ : ∀ (n : ℤ), Injective (L.X n)
K : CochainComplex C ℤ
f : K ⟶ L
hK : HomologicalComplex.Acyclic K
X : ℕ → Set (Cochain K L (-1)) := fun n ↦ {α | (δ (-1) 0 α).EqUpTo (Cochain.ofHom f) (↑n + d - 1)}
x₀ : ↑(X 0) := ⟨0, ⋯⟩
φ : (n : ℕ) → ↑(X n) → ↑(X (n + 1)) := fun n α ↦ ⟨↑α + Cochain.single ⋯.choose (-1), ⋯⟩
hφ : ∀ (k : ℕ) (x : ↑(X k)), (↑(φ k x)).EqUpTo (↑x) (d + ↑k)
⊢ Cochain.ofHom f = δ (-1) 0 (limitSequence φ hφ x₀) + Cochain.ofHom 0

Tactic:
rw [Cochain.ofHom_zero, add_zero]

Goal state after:
C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : Abelian C
L : CochainComplex C ℤ
d : ℤ
inst✝¹ : L.IsStrictlyGE d
inst✝ : ∀ (n : ℤ), Injective (L.X n)
K : CochainComplex C ℤ
f : K ⟶ L
hK : HomologicalComplex.Acyclic K
X : ℕ → Set (Cochain K L (-1)) := fun n ↦ {α | (δ (-1) 0 α).EqUpTo (Cochain.ofHom f) (↑n + d - 1)}
x₀ : ↑(X 0) := ⟨0, ⋯⟩
φ : (n : ℕ) → ↑(X n) → ↑(X (n + 1)) := fun n α ↦ ⟨↑α + Cochain.single ⋯.choose (-1), ⋯⟩
hφ : ∀ (k : ℕ) (x : ↑(X k)), (↑(φ k x)).EqUpTo (↑x) (d + ↑k)
⊢ Cochain.ofHom f = δ (-1) 0 (limitSequence φ hφ x₀)
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
⊢ t.bounded.IsClosedUnderIsomorphisms

Tactic:
dsimp [bounded]
  infer_instance

Goal state after:
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
⊢ t.bounded.IsStableUnderShift ℤ

Tactic:
dsimp [bounded]
  infer_instance

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

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
T : MorphismProperty C
J : Type u_1
A B : J → C
inst✝¹ : Limits.HasCoproduct A
inst✝ : Limits.HasCoproduct B
⊢ ∀ (f : (j : J) → A j ⟶ B j), (∀ (j : J), T.llp (f j)) → T.llp (Limits.Sigma.map f)

Tactic:
f

Goal state after:
C : Type u
inst✝² : Category.{v, u} C
T : MorphismProperty C
J : Type u_1
A B : J → C
inst✝¹ : Limits.HasCoproduct A
inst✝ : Limits.HasCoproduct B
f : (j : J) → A j ⟶ B j
⊢ (∀ (j : J), T.llp (f j)) → T.llp (Limits.Sigma.map f)
Module: Mathlib.CategoryTheory.MorphismProperty.LiftingProperty

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
T : MorphismProperty C
J : Type u_1
A B : J → C
inst✝¹ : Limits.HasProduct A
inst✝ : Limits.HasProduct B
⊢ ∀ (f : (j : J) → A j ⟶ B j), (∀ (j : J), T.rlp (f j)) → T.rlp (Limits.Pi.map f)

Tactic:
f

Goal state after:
C : Type u
inst✝² : Category.{v, u} C
T : MorphismProperty C
J : Type u_1
A B : J → C
inst✝¹ : Limits.HasProduct A
inst✝ : Limits.HasProduct B
f : (j : J) → A j ⟶ B j
⊢ (∀ (j : J), T.rlp (f j)) → T.rlp (Limits.Pi.map f)
Module: Mathlib.CategoryTheory.MorphismProperty.LiftingProperty

Goal state before:
pp: case a
C : Type u
inst✝ : Category.{v, u} C
T : MorphismProperty C
⊢ coproducts.{w, v, u} T ≤ T.rlp.llp

Tactic:
exact T.coproducts_le_llp_rlp

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

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : CategoryWithWeakEquivalences C
inst✝ : (weakEquivalences C).HasTwoOutOfThreeProperty
X Y Z : C
f : X ⟶ Y
g : Y ⟶ Z
hf : WeakEquivalence f
hfg : WeakEquivalence (f ≫ g)
⊢ WeakEquivalence g

Tactic:
by
  rw [weakEquivalence_iff] at hf hfg ⊢
  exact of_precomp _ _ _ hf hfg

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

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : CategoryWithWeakEquivalences C
inst✝³ : CategoryWithCofibrations C
inst✝² : CategoryWithFibrations C
X Y : C
f : X ⟶ Y
inst✝¹ : (cofibrations C).IsWeakFactorizationSystem (trivialFibrations C)
inst✝ : IsIso f
this : trivialFibrations C f
⊢ Fibration f

Tactic:
simpa only [fibration_iff] using this.1

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
f g : X ⟶ Y
inst✝ : IsFibrant Y
h✝ : LeftHomotopyRel f g
P : Cylinder X
left✝ : P.IsGood
h : P.LeftHomotopy f g
fac : (trivialCofibrations C).MapFactorizationData (fibrations C) P.π :=
  (trivialCofibrations C).factorizationData (fibrations C) P.π
P' : Cylinder X :=
  { I := fac.Z, i₀ := P.i₀ ≫ fac.i, i₁ := P.i₁ ≫ fac.i, π := fac.p, i₀_π := ⋯, i₁_π := ⋯, weakEquivalence_π := ⋯ }
⊢ ∃ P, P.IsVeryGood ∧ Nonempty (P.LeftHomotopy f g)

Tactic:
have : Cofibration P'.i := by
    rw [show P'.i = P.i ≫ fac.i by cat_disch]
    infer_instance

Goal state after:
C : Type u
inst✝² : Category.{v, u} C
X Y : C
inst✝¹ : ModelCategory C
f g : X ⟶ Y
inst✝ : IsFibrant Y
h✝ : LeftHomotopyRel f g
P : Cylinder X
left✝ : P.IsGood
h : P.LeftHomotopy f g
fac : (trivialCofibrations C).MapFactorizationData (fibrations C) P.π :=
  (trivialCofibrations C).factorizationData (fibrations C) P.π
P' : Cylinder X :=
  { I := fac.Z, i₀ := P.i₀ ≫ fac.i, i₁ := P.i₁ ≫ fac.i, π := fac.p, i₀_π := ⋯, i₁_π := ⋯, weakEquivalence_π := ⋯ }
this : Cofibration P'.i
⊢ ∃ P, P.IsVeryGood ∧ Nonempty (P.LeftHomotopy f g)
Module: Mathlib.CategoryTheory.Triangulated.Opposite.Triangulated

Goal state before:
pp: case refine_3
C : Type u_1
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : HasShift C ℤ
inst✝⁴ : HasZeroObject C
inst✝³ : Preadditive C
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
inst✝ : IsTriangulated C
X₁ X₂ X₃ Z₁₂ Z₂₃ Z₁₃ : Cᵒᵖ
u₁₂ : X₁ ⟶ X₂
u₂₃ : X₂ ⟶ X₃
u₁₃ : X₁ ⟶ X₃
comm : u₁₂ ≫ u₂₃ = u₁₃
v₁₂ : X₂ ⟶ Z₁₂
w₁₂ : Z₁₂ ⟶ (shiftFunctor Cᵒᵖ 1).obj X₁
h₁₂ : Triangle.mk u₁₂ v₁₂ w₁₂ ∈ Pretriangulated.distinguishedTriangles
v₂₃ : X₃ ⟶ Z₂₃
w₂₃ : Z₂₃ ⟶ (shiftFunctor Cᵒᵖ 1).obj X₂
h₂₃ : Triangle.mk u₂₃ v₂₃ w₂₃ ∈ Pretriangulated.distinguishedTriangles
v₁₃ : X₃ ⟶ Z₁₃
w₁₃ : Z₁₃ ⟶ (shiftFunctor Cᵒᵖ 1).obj X₁
h₁₃ : Triangle.mk u₁₃ v₁₃ w₁₃ ∈ Pretriangulated.distinguishedTriangles
o : Triangulated.Octahedron' ⋯ ⋯ ⋯ ⋯ := Triangulated.someOctahedron' ⋯ ⋯ ⋯ ⋯
⊢ Triangle.mk o.m₃.op o.m₁.op (w₂₃ ≫ (shiftFunctor Cᵒᵖ 1).map v₁₂) ∈ Pretriangulated.distinguishedTriangles

Tactic:
· have := op_distinguished _ o.mem
      dsimp at this
      convert! this using 2
      rw [Category.assoc, Functor.map_comp, Functor.map_comp,
        ← opShiftFunctorEquivalence_counitIso_hom_app_shift,
        ← opShiftFunctorEquivalence_counitIso_inv_naturality_assoc, Iso.inv_hom_id_app_assoc]

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

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
P : ObjectProperty C
t : TStructure C
inst✝ : IsTriangulated C
⊢ t.bounded.HasInducedTStructure t

Tactic:
dsimp [bounded]
  infer_instance

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

Goal state before:
pp: case mpr
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasZeroMorphisms C
I Z : C
hZ : IsZero Z
h : (MorphismProperty.monomorphisms C).rlp 0
A B : C
⊢ ∀ (g : A ⟶ I) (f : A ⟶ B) [Mono f], ∃ h, f ≫ h = g

Tactic:
f

Goal state after:
case mpr
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasZeroMorphisms C
I Z : C
hZ : IsZero Z
h : (MorphismProperty.monomorphisms C).rlp 0
A B : C
f : A ⟶ I
⊢ ∀ (f_1 : A ⟶ B) [Mono f_1], ∃ h, f_1 ≫ h = f