mathlib-initiative/mathlib-tactics huggingface dataset:
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
i j k : A
h : i + j = k
X : t.Category
⊢ (shiftFunctorAdd' t.Category i j k h).hom.app X =
    ↑(t.z i j) •
      (t.shiftIso k).hom.app X ≫
        (shiftFunctorAdd' C i j k h).hom.app X ≫
          (shiftFunctor C j).map ((t.shiftIso i).inv.app X) ≫ (t.shiftIso j).inv.app ((shiftFunctor t.Category i).obj X)

Tactic:
have : (shiftFunctorAdd' t.Category i j k h).hom.app X =
      (t.z i j).val • (shiftFunctorAdd' C i j k h).hom.app X := by
    dsimp [shiftFunctorAdd']
    cat_disch

Goal state after:
C : Type u
inst✝² : Category.{v, u} C
A : Type w
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
t : TwistShiftData C A
i j k : A
h : i + j = k
X : t.Category
this : (shiftFunctorAdd' t.Category i j k h).hom.app X = ↑(t.z i j) • (shiftFunctorAdd' C i j k h).hom.app X
⊢ (shiftFunctorAdd' t.Category i j k h).hom.app X =
    ↑(t.z i j) •
      (t.shiftIso k).hom.app X ≫
        (shiftFunctorAdd' C i j k h).hom.app X ≫
          (shiftFunctor C j).map ((t.shiftIso i).inv.app X) ≫ (t.shiftIso j).inv.app ((shiftFunctor t.Category i).obj X)
Module: Mathlib.Algebra.Homology.TotalComplexShift

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
K : HomologicalComplex₂ C (up ℤ) (up ℤ)
x y : ℤ
inst✝ : K.HasTotal (up ℤ)
n n₁ n₂ : ℤ
h : n₁ + n₂ = n
⊢ (((shiftFunctor₂ C y).obj K).shiftFunctor₁XXIso n₁ x (n₁ + x) ⋯ n₂).hom ≫
      ((shiftFunctor₂ C y).obj K).ιTotal (up ℤ) (n₁ + x) n₂ (n + x) ⋯ ≫
        (CochainComplex.shiftFunctorObjXIso (((shiftFunctor₂ C y).obj K).total (up ℤ)) x n (n + x) ⋯).inv ≫
          (K.totalShift₂Iso y).hom.f (n + x) =
    (x * y).negOnePow •
      (n₁ * y).negOnePow •
        (((shiftFunctor₁₂CommIso C x y).hom.app K).f n₁).f n₂ ≫
          ((((shiftFunctor₁ C x).obj K).shiftFunctor₂XXIso n₁ n₂ y (n₂ + y) ⋯).hom ≫
              ((shiftFunctor₁ C x).obj K).ιTotal (up ℤ) n₁ (n₂ + y) (n + y) ⋯ ≫
                (CochainComplex.shiftFunctorObjXIso (((shiftFunctor₁ C x).obj K).total (up ℤ)) y n (n + y) ⋯).inv) ≫
            (K.totalShift₁Iso x).hom.f (n + y) ≫
              ((shiftFunctorComm (CochainComplex C ℤ) x y).hom.app (K.total (up ℤ))).f n

Tactic:
dsimp [shiftFunctor₁₂CommIso]

Goal state after:
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
K : HomologicalComplex₂ C (up ℤ) (up ℤ)
x y : ℤ
inst✝ : K.HasTotal (up ℤ)
n n₁ n₂ : ℤ
h : n₁ + n₂ = n
⊢ 𝟙 ((K.X (n₁ + x)).X (n₂ + y)) ≫
      ((shiftFunctor₂ C y).obj K).ιTotal (up ℤ) (n₁ + x) n₂ (n + x) ⋯ ≫
        𝟙 ((((shiftFunctor₂ C y).obj K).total (up ℤ)).X (n + x)) ≫ (K.totalShift₂Iso y).hom.f (n + x) =
    (x * y).negOnePow •
      (n₁ * y).negOnePow •
        𝟙 ((K.X (n₁ + x)).X (n₂ + y)) ≫
          (𝟙 ((K.X (n₁ + x)).X (n₂ + y)) ≫
              ((shiftFunctor₁ C x).obj K).ιTotal (up ℤ) n₁ (n₂ + y) (n + y) ⋯ ≫
                𝟙 ((((shiftFunctor₁ C x).obj K).total (up ℤ)).X (n + y))) ≫
            (K.totalShift₁Iso x).hom.f (n + y) ≫
              ((shiftFunctorComm (CochainComplex C ℤ) x y).hom.app (K.total (up ℤ))).f n
Module: Mathlib.Algebra.Homology.CochainComplexPlus

Goal state before:
pp: C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : HasZeroMorphisms C
J : Type u_2
inst✝² : SmallCategory J
inst✝¹ : FinCategory J
inst✝ : HasLimitsOfShape J C
K : CochainComplex C ℤ
p : (CochainComplex.plus C).LimitOfShape J K
n : J → ℤ
hn : ∀ (j : J), (p.diag.obj j).IsStrictlyGE (n j)
⊢ 0 ∈ Finset.image n ⊤ ∪ {0}

Tactic:
by grind

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

Goal state before:
pp: C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : HasZeroMorphisms C
J : Type u_2
inst✝² : SmallCategory J
inst✝¹ : FinCategory J
inst✝ : HasColimitsOfShape J C
K : CochainComplex C ℤ
p : (CochainComplex.plus C).ColimitOfShape J K
n : J → ℤ
hn : ∀ (j : J), (p.diag.obj j).IsStrictlyGE (n j)
⊢ 0 ∈ Finset.image n ⊤ ∪ {0}

Tactic:
by grind

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

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
K L : CochainComplex C ℤ
n : ℤ
x : Cocycle K L n
⊢ toHom (mk x) = 0 ↔ x ∈ coboundaries K L n

Tactic:
by
  refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩
  · simp only [coboundaries, exists_prop, AddSubgroup.mem_mk, AddSubmonoid.mem_mk,
      AddSubsemigroup.mem_mk, Set.mem_ofPred_eq]
    rw [toHom_mk, HomotopyCategory.quotient_map_eq_zero_iff] at h
    obtain ⟨γ, h⟩ := Cochain.equivHomotopy _ _ h.some
    simp only [Cochain.ofHom_zero, add_zero, Cocycle.equivHomShift_symm_apply,
      Cocycle.cochain_ofHom_homOf_eq_coe, Cocycle.rightShift_coe] at h
    exact ⟨n - 1, by simp, n.negOnePow • γ.rightUnshift _ (by lia),
      by simp [Cochain.δ_rightUnshift _ _ _ _ _ (zero_add n), smul_smul, ← h]⟩
  · rw [← mk_eq_zero_iff] at h
    rw [h, map_zero]

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
j' : ι'
i : ι
hij' : ¬c'.Rel j' (e.f i)
hj' : ∃ j, e.f j = j'
⊢ extend.hom e h.hom (e.f i) j' = 0

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
j' : ι'
i : ι
hij' : ¬c'.Rel j' (e.f i)
hj' : ¬∃ j, e.f j = j'
⊢ extend.hom e h.hom (e.f i) j' = 0

Tactic:
· obtain ⟨j, rfl⟩ := hj'
        rw [extend.hom_eq _ _ rfl rfl, h.zero _ _ (by rwa [← e.rel_iff]),
          zero_comp, comp_zero]

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
h : Homotopy f g
e : c.Embedding c'
inst✝ : e.IsRelIff
j' : ι'
i : ι
hij' : ¬c'.Rel j' (e.f i)
hj' : ¬∃ j, e.f j = j'
⊢ extend.hom e h.hom (e.f i) j' = 0
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
H_one : H 1
a b c : ℕ
h : a + b = c
ha : H a
hb : H b
n : ℤ
⊢ t.le n ≤ t.le (n + ↑(a + b))

Tactic:
Nat.cast_add,

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.le n ≤ t.le (n + ↑a)
H_zero : H 0
H_one : H 1
a b c : ℕ
h : a + b = c
ha : H a
hb : H b
n : ℤ
⊢ t.le n ≤ t.le (n + (↑a + ↑b))
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
H_one : H 1
a b c : ℕ
h : a + b = c
ha : H a
hb : H b
n : ℤ
⊢ t.ge (n + ↑(a + b)) ≤ t.ge n

Tactic:
Nat.cast_add,

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
H_one : H 1
a b c : ℕ
h : a + b = c
ha : H a
hb : H b
n : ℤ
⊢ t.ge (n + (↑a + ↑b)) ≤ t.ge n
Module: Mathlib.Algebra.Homology.HomotopyCategory.KInjective

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
K L : CochainComplex C ℤ
n : ℤ
z : Cocycle K L n
inst✝ : L.IsKInjective
hL : HomologicalComplex.Acyclic L
m : ℤ
hm : m + 1 = n
⊢ ∃ α, δ m n α = ↑z

Tactic:
obtain ⟨β, hβ⟩ :=
    IsKInjective.eq_δ_of_cocycle (Cocycle.ofHom (𝟙 L)) hL (-1) (by simp)
  exact ⟨z.1.comp β (by lia), by simp [δ_comp z.1 β _ _ 0 _ hm rfl (by simp), hβ]⟩

Goal state after:
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 : HasFiniteLimits C
inst✝³ : HasFiniteColimits C
inst✝² : (weakEquivalences C).HasTwoOutOfThreeProperty
inst✝¹ : (cofibrations C).IsWeakFactorizationSystem (trivialFibrations C)
inst✝ : (trivialCofibrations C).IsWeakFactorizationSystem (fibrations C)
A B X Y : C
f : A ⟶ B
w : X ⟶ Y
h : RetractArrow f w
hw : WeakEquivalence w
hf : (trivialCofibrations C).MapFactorizationData (fibrations C) f
this✝³ : Cofibration hf.i
this✝² : WeakEquivalence hf.i
φ : pushout hf.i (Arrow.Hom.left h.i) ⟶ Y := pushout.desc (hf.p ≫ Arrow.Hom.right h.i) w ⋯
this✝¹ : Fibration hf.p
this✝ : WeakEquivalence (pushout.inr hf.i (Arrow.Hom.left h.i) ≫ φ)
this : WeakEquivalence φ
⊢ weakEquivalences C f

Tactic:
have hp : RetractArrow hf.p φ :=
      { i := Arrow.homMk (pushout.inl _ _) h.i.right
        r := Arrow.homMk (pushout.desc (𝟙 _) (h.r.left ≫ hf.i) (by simp)) h.r.right }

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 : HasFiniteLimits C
inst✝³ : HasFiniteColimits C
inst✝² : (weakEquivalences C).HasTwoOutOfThreeProperty
inst✝¹ : (cofibrations C).IsWeakFactorizationSystem (trivialFibrations C)
inst✝ : (trivialCofibrations C).IsWeakFactorizationSystem (fibrations C)
A B X Y : C
f : A ⟶ B
w : X ⟶ Y
h : RetractArrow f w
hw : WeakEquivalence w
hf : (trivialCofibrations C).MapFactorizationData (fibrations C) f
this✝³ : Cofibration hf.i
this✝² : WeakEquivalence hf.i
φ : pushout hf.i (Arrow.Hom.left h.i) ⟶ Y := pushout.desc (hf.p ≫ Arrow.Hom.right h.i) w ⋯
this✝¹ : Fibration hf.p
this✝ : WeakEquivalence (pushout.inr hf.i (Arrow.Hom.left h.i) ≫ φ)
this : WeakEquivalence φ
hp : RetractArrow hf.p φ
⊢ weakEquivalences C f
Module: Mathlib.CategoryTheory.Triangulated.Opposite.Subcategory

Goal state before:
pp: 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
P : ObjectProperty C
inst✝ : P.IsTriangulatedClosed₂
T : Triangle Cᵒᵖ
hT : T ∈ Pretriangulated.distinguishedTriangles
h₁ : P.op T.obj₁
h₃ : P.op T.obj₃
⊢ P.op.isoClosure T.obj₂

Tactic:
mem_distTriang_op_iff

Goal state after:
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
P : ObjectProperty C
inst✝ : P.IsTriangulatedClosed₂
T : Triangle Cᵒᵖ
hT : Opposite.unop ((triangleOpEquivalence C).inverse.obj T) ∈ Pretriangulated.distinguishedTriangles
h₁ : P.op T.obj₁
h₃ : P.op T.obj₃
⊢ P.op.isoClosure T.obj₂
Module: Mathlib.CategoryTheory.Triangulated.TStructure.TruncLEGT

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
X Y : C
f : X ⟶ Y
a : ℤ
⊢ IsIso ((t.truncLE a).map f) ↔
    ∃ Z g h, ∃ (_ : Triangle.mk ((t.truncLEι a).app X ≫ f) g h ∈ distinguishedTriangles), t.IsGE Z (a + 1)

Tactic:
apply isIso_truncLT_map_iff

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

Goal state before:
pp: case refine_1
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 : Cᵒᵖ⦄ (s : X ⟶ Y), A.op X → B.op.trW s → ∃ Z s' b, A.op Z ∧ (A.op ⊓ B.op).trW s' ∧ s ≫ b = s'
X Y : C
s : X ⟶ Y
hY : A Y
hs : B.trW s
Z' : Cᵒᵖ
s' : op Y ⟶ Z'
b : op X ⟶ Z'
hZ' : A.op Z'
hs' : (A.op ⊓ B.op).trW s'
fac : s.op ≫ b = s'
⊢ ∃ Z s' a, A Z ∧ (A ⊓ B).trW s' ∧ a ≫ s = s'

Tactic:
exact ⟨Z'.unop, s'.unop, b.unop, hZ', trW_of_op _ hs', by cat_disch⟩

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✝¹⁵ : Category.{v_1, u_1} C
inst✝¹⁴ : Category.{v_2, u_2} D
inst✝¹³ : Category.{v_3, u_3} D₁
inst✝¹² : Category.{v_4, u_4} D₂
A B : ObjectProperty C
inst✝¹¹ : HasZeroObject C
inst✝¹⁰ : HasShift C ℤ
inst✝⁹ : Preadditive C
inst✝⁸ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁷ : Pretriangulated C
inst✝⁶ : IsTriangulated C
inst✝⁵ : A.IsTriangulated
inst✝⁴ : B.IsTriangulated
inst✝³ : B.IsClosedUnderIsomorphisms
inst✝² : A.IsVerdierRightLocalizing B
L₁ : A.FullSubcategory ⥤ D₁
L₂ : C ⥤ D₂
inst✝¹ : L₁.IsLocalization (B.inverseImage A.ι).trW
inst✝ : L₂.IsLocalization B.trW
this✝² : Preadditive D₁ := Localization.preadditive L₁ (B.inverseImage A.ι).trW
this✝¹ : Preadditive D₂ := Localization.preadditive L₂ B.trW
this✝ : L₁.Additive
this : L₂.Additive
F : D₁ ⥤ D₂ := (A.triangulatedLocalizerMorphism B).localizedFunctor L₁ L₂
e : A.ι ⋙ L₂ ≅ L₁ ⋙ F := CatCommSq.iso (A.triangulatedLocalizerMorphism B).functor L₁ L₂ F
X₁ X₂ : A.FullSubcategory
f : X₁ ⟶ X₂
X₃ : C
s : X₂.obj ⟶ X₃
hs : B.trW s
fac : f.hom ≫ s = 0 ≫ s
⊢ L₁.map f = 0

Tactic:
obtain ⟨X₄, t, a, hX₄, ht, fac'⟩ :=
    IsVerdierRightLocalizing.fac' s X₂.property hs

Goal state after:
C : Type u_1
D : Type u_2
D₁ : Type u_3
D₂ : Type u_4
inst✝¹⁵ : Category.{v_1, u_1} C
inst✝¹⁴ : Category.{v_2, u_2} D
inst✝¹³ : Category.{v_3, u_3} D₁
inst✝¹² : Category.{v_4, u_4} D₂
A B : ObjectProperty C
inst✝¹¹ : HasZeroObject C
inst✝¹⁰ : HasShift C ℤ
inst✝⁹ : Preadditive C
inst✝⁸ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁷ : Pretriangulated C
inst✝⁶ : IsTriangulated C
inst✝⁵ : A.IsTriangulated
inst✝⁴ : B.IsTriangulated
inst✝³ : B.IsClosedUnderIsomorphisms
inst✝² : A.IsVerdierRightLocalizing B
L₁ : A.FullSubcategory ⥤ D₁
L₂ : C ⥤ D₂
inst✝¹ : L₁.IsLocalization (B.inverseImage A.ι).trW
inst✝ : L₂.IsLocalization B.trW
this✝² : Preadditive D₁ := Localization.preadditive L₁ (B.inverseImage A.ι).trW
this✝¹ : Preadditive D₂ := Localization.preadditive L₂ B.trW
this✝ : L₁.Additive
this : L₂.Additive
F : D₁ ⥤ D₂ := (A.triangulatedLocalizerMorphism B).localizedFunctor L₁ L₂
e : A.ι ⋙ L₂ ≅ L₁ ⋙ F := CatCommSq.iso (A.triangulatedLocalizerMorphism B).functor L₁ L₂ F
X₁ X₂ : A.FullSubcategory
f : X₁ ⟶ X₂
X₃ : C
s : X₂.obj ⟶ X₃
hs : B.trW s
fac : f.hom ≫ s = 0 ≫ s
X₄ : C
t : X₂.obj ⟶ X₄
a : X₃ ⟶ X₄
hX₄ : A X₄
ht : (A ⊓ B).trW t
fac' : s ≫ a = t
⊢ L₁.map f = 0
Module: Mathlib.CategoryTheory.Abelian.EpiWithInjectiveKernel

Goal state before:
pp: case h₁.h₀
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
X Y Z : C
g₁ : X ⟶ Y
g₂ : Y ⟶ Z
I₁ : C
w✝¹ : Injective I₁
f₁ : I₁ ⟶ X
w₁ : f₁ ≫ g₁ = 0
σ₁ : { X₁ := I₁, X₂ := X, X₃ := Y, f := f₁, g := g₁, zero := w₁ }.Splitting
I₂ : C
w✝ : Injective I₂
f₂ : I₂ ⟶ Y
w₂ : f₂ ≫ g₂ = 0
σ₂ : { X₁ := I₂, X₂ := Y, X₃ := Z, f := f₂, g := g₂, zero := w₂ }.Splitting
⊢ (biprod.inr ≫
        { X₁ := I₁ ⊞ I₂, X₂ := X, X₃ := Z, f := biprod.fst ≫ f₁ + biprod.snd ≫ f₂ ≫ σ₁.s, g := g₁ ≫ g₂, zero := ⋯ }.f ≫
          (σ₁.r ≫ biprod.inl + g₁ ≫ σ₂.r ≫ biprod.inr)) ≫
      biprod.fst =
    (biprod.inr ≫
        𝟙
          { X₁ := I₁ ⊞ I₂, X₂ := X, X₃ := Z, f := biprod.fst ≫ f₁ + biprod.snd ≫ f₂ ≫ σ₁.s, g := g₁ ≫ g₂,
              zero := ⋯ }.X₁) ≫
      biprod.fst

pp: case h₁
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
X Y Z : C
g₁ : X ⟶ Y
g₂ : Y ⟶ Z
I₁ : C
w✝¹ : Injective I₁
f₁ : I₁ ⟶ X
w₁ : f₁ ≫ g₁ = 0
σ₁ : { X₁ := I₁, X₂ := X, X₃ := Y, f := f₁, g := g₁, zero := w₁ }.Splitting
I₂ : C
w✝ : Injective I₂
f₂ : I₂ ⟶ Y
w₂ : f₂ ≫ g₂ = 0
σ₂ : { X₁ := I₂, X₂ := Y, X₃ := Z, f := f₂, g := g₂, zero := w₂ }.Splitting
⊢ (biprod.inr ≫
        { X₁ := I₁ ⊞ I₂, X₂ := X, X₃ := Z, f := biprod.fst ≫ f₁ + biprod.snd ≫ f₂ ≫ σ₁.s, g := g₁ ≫ g₂, zero := ⋯ }.f ≫
          (σ₁.r ≫ biprod.inl + g₁ ≫ σ₂.r ≫ biprod.inr)) ≫
      biprod.snd =
    (biprod.inr ≫
        𝟙
          { X₁ := I₁ ⊞ I₂, X₂ := X, X₃ := Z, f := biprod.fst ≫ f₁ + biprod.snd ≫ f₂ ≫ σ₁.s, g := g₁ ≫ g₂,
              zero := ⋯ }.X₁) ≫
      biprod.snd

Tactic:
· simp

Goal state after:
case h₁
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
X Y Z : C
g₁ : X ⟶ Y
g₂ : Y ⟶ Z
I₁ : C
w✝¹ : Injective I₁
f₁ : I₁ ⟶ X
w₁ : f₁ ≫ g₁ = 0
σ₁ : { X₁ := I₁, X₂ := X, X₃ := Y, f := f₁, g := g₁, zero := w₁ }.Splitting
I₂ : C
w✝ : Injective I₂
f₂ : I₂ ⟶ Y
w₂ : f₂ ≫ g₂ = 0
σ₂ : { X₁ := I₂, X₂ := Y, X₃ := Z, f := f₂, g := g₂, zero := w₂ }.Splitting
⊢ (biprod.inr ≫
        { X₁ := I₁ ⊞ I₂, X₂ := X, X₃ := Z, f := biprod.fst ≫ f₁ + biprod.snd ≫ f₂ ≫ σ₁.s, g := g₁ ≫ g₂, zero := ⋯ }.f ≫
          (σ₁.r ≫ biprod.inl + g₁ ≫ σ₂.r ≫ biprod.inr)) ≫
      biprod.snd =
    (biprod.inr ≫
        𝟙
          { X₁ := I₁ ⊞ I₂, X₂ := X, X₃ := Z, f := biprod.fst ≫ f₁ + biprod.snd ≫ f₂ ≫ σ₁.s, g := g₁ ≫ g₂,
              zero := ⋯ }.X₁) ≫
      biprod.snd
Module: Mathlib.CategoryTheory.Category.Factorisation

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
X Y : C
f✝ : X ⟶ Y
d : Factorisation f✝
f : Factorisation.initial ⟶ d
⊢ f = d.initialHom

Tactic:
apply Factorisation.Hom.ext; simp [← f.ι_h]

Goal state after:
Module: Mathlib.CategoryTheory.Functor.OfSequence

Goal state before:
pp: case zero.succ
C : Type u_1
inst✝ : Category.{v_1, u_1} C
j : ℕ
hj :
  ∀ {X : ℕ → C} (f : (n : ℕ) → X n ⟶ X (n + 1)) (k : ℕ) (hij : 0 ≤ j) (hjk : j ≤ k),
    map f 0 k ⋯ = map f 0 j hij ≫ map f j k hjk
X : ℕ → C
f : (n : ℕ) → X n ⟶ X (n + 1)
k : ℕ
hij : 0 ≤ j + 1
hjk : j + 1 ≤ k
⊢ map f 0 k ⋯ = map f 0 (j + 1) hij ≫ map f (j + 1) k hjk

Tactic:
| succ j hj =>

Goal state after:
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
X : C
a b : ℤ
inst✝ : t.IsGE X a
⊢ t.IsGE ((t.truncGE b).obj X) a

Tactic:
by_cases h : a ≤ b

Goal state after:
case pos
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
X : C
a b : ℤ
inst✝ : t.IsGE X a
h : a ≤ b
⊢ t.IsGE ((t.truncGE b).obj X) a

case neg
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
X : C
a b : ℤ
inst✝ : t.IsGE X a
h : ¬a ≤ b
⊢ t.IsGE ((t.truncGE b).obj X) a
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
inst✝ : IsTriangulated C
a b : ℤ
X : C
⊢ IsIso ((t.truncGELTToLTGE a b).app X)

Tactic:
by_cases h : a ≤ b

Goal state after:
case pos
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
inst✝ : IsTriangulated C
a b : ℤ
X : C
h : a ≤ b
⊢ IsIso ((t.truncGELTToLTGE a b).app X)

case neg
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
inst✝ : IsTriangulated C
a b : ℤ
X : C
h : ¬a ≤ b
⊢ IsIso ((t.truncGELTToLTGE a b).app X)
Module: Mathlib.AlgebraicTopology.ModelCategory.Homotopy

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : ModelCategory C
X Y Z : C
f g : X ⟶ Y
inst✝¹ : IsFibrant Y
h✝ : RightHomotopyRel f g
Q : Cylinder X
inst✝ : Q.IsGood
P : PathObject Y := ⋯.choose
h : ⋯.choose.RightHomotopy f g
h' : ⋯.choose.IsGood
sq : CommSq (coprod.desc (f ≫ P.ι) h.h) Q.i P.p₀ (Q.π ≫ f)
this : coprod.inr ≫ Q.i ≫ sq.lift = coprod.inr ≫ coprod.desc (f ≫ P.ι) h.h
⊢ Q.i₁ ≫ sq.lift ≫ P.p₁ = g

Tactic:
rw [Q.inr_i_assoc, coprod.inr_desc] at this

Goal state after:
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : ModelCategory C
X Y Z : C
f g : X ⟶ Y
inst✝¹ : IsFibrant Y
h✝ : RightHomotopyRel f g
Q : Cylinder X
inst✝ : Q.IsGood
P : PathObject Y := ⋯.choose
h : ⋯.choose.RightHomotopy f g
h' : ⋯.choose.IsGood
sq : CommSq (coprod.desc (f ≫ P.ι) h.h) Q.i P.p₀ (Q.π ≫ f)
this : Q.i₁ ≫ sq.lift = h.h
⊢ Q.i₁ ≫ sq.lift ≫ P.p₁ = g
Module: Mathlib.Algebra.Homology.Factorizations.CM5a

Goal state before:
pp: case right
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
K L : CochainComplex C ℤ
inst✝ : EnoughInjectives C
n₁ i : ℤ
hi : i ≠ n₁
⊢ ((π K L n₁).f i ≫ biprod.inr.f i) ≫ biprod.snd.f i = 𝟙 ((mid K L n₁).X i) ≫ biprod.snd.f i

Tactic:
· simp

Goal state after:
Module: Mathlib.Algebra.Homology.ModelCategory.Injective

Goal state before:
pp: C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Abelian C
inst✝ : EnoughInjectives C
K : CochainComplex C ℤ
n : ℤ
hn : K.IsStrictlyGE n
L : CochainComplex C ℤ
m : ℤ
hm : L.IsStrictlyGE m
f : { obj := K, property := ⋯ }.obj ⟶ { obj := L, property := ⋯ }.obj
⊢ min (n - 1) m + 1 ≤ n

Tactic:
by grind

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

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : ModelCategory C
X Y : C
inst✝⁴ : IsCofibrant X
inst✝³ : IsCofibrant Y
inst✝² : IsFibrant X
inst✝¹ : IsFibrant Y
f : X ⟶ Y
inst✝ : WeakEquivalence f
g : Y ⟶ X
hg : mk (f ≫ g) = mk (𝟙 X)
⊢ mk (f ≫ g ≫ f) = mk f

Tactic:
rw [mk_eq_mk_iff, ← leftHomotopyRel_iff_rightHomotopyRel] at hg ⊢

Goal state after:
C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : ModelCategory C
X Y : C
inst✝⁴ : IsCofibrant X
inst✝³ : IsCofibrant Y
inst✝² : IsFibrant X
inst✝¹ : IsFibrant Y
f : X ⟶ Y
inst✝ : WeakEquivalence f
g : Y ⟶ X
hg : LeftHomotopyRel (f ≫ g) (𝟙 X)
⊢ LeftHomotopyRel (f ≫ g ≫ f) f
Module: Mathlib.CategoryTheory.Localization.DerivabilityStructure.Basic

Goal state before:
pp: C₁ : Type u₁
C₂ : Type u₂
inst✝⁶ : Category.{v₁, u₁} C₁
inst✝⁵ : Category.{v₂, u₂} C₂
W₁ : MorphismProperty C₁
W₂ : MorphismProperty C₂
Φ : LocalizerMorphism W₁ W₂
D₁ : Type u_1
D₂ : Type u_2
inst✝⁴ : Category.{v_1, u_1} D₁
inst✝³ : Category.{v_2, u_2} D₂
L₁ : C₁ ⥤ D₁
L₂ : C₂ ⥤ D₂
inst✝² : L₁.IsLocalization W₁
inst✝¹ : L₂.IsLocalization W₂
F : D₁ ⥤ D₂
inst✝ : Φ.HasRightResolutions
e : Φ.functor ⋙ L₂ ≅ L₁ ⋙ F
⊢ Φ.IsRightDerivabilityStructure ↔ TwoSquare.GuitartExact e.hom

Tactic:
have : Φ.IsRightDerivabilityStructure ↔
      TwoSquare.GuitartExact ((Φ.catCommSq W₁.Q W₂.Q).iso).hom :=
    ⟨fun h => h.guitartExact', fun h => ⟨inferInstance, h⟩⟩

Goal state after:
C₁ : Type u₁
C₂ : Type u₂
inst✝⁶ : Category.{v₁, u₁} C₁
inst✝⁵ : Category.{v₂, u₂} C₂
W₁ : MorphismProperty C₁
W₂ : MorphismProperty C₂
Φ : LocalizerMorphism W₁ W₂
D₁ : Type u_1
D₂ : Type u_2
inst✝⁴ : Category.{v_1, u_1} D₁
inst✝³ : Category.{v_2, u_2} D₂
L₁ : C₁ ⥤ D₁
L₂ : C₂ ⥤ D₂
inst✝² : L₁.IsLocalization W₁
inst✝¹ : L₂.IsLocalization W₂
F : D₁ ⥤ D₂
inst✝ : Φ.HasRightResolutions
e : Φ.functor ⋙ L₂ ≅ L₁ ⋙ F
this :
  Φ.IsRightDerivabilityStructure ↔
    TwoSquare.GuitartExact (CatCommSq.iso Φ.functor W₁.Q W₂.Q (Φ.localizedFunctor W₁.Q W₂.Q)).hom
⊢ Φ.IsRightDerivabilityStructure ↔ TwoSquare.GuitartExact e.hom