mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.CategoryTheory.Triangulated.Pretriangulated

Goal state before:
pp: C : Type u
inst✝⁸ : Category.{v, u} C
inst✝⁷ : HasZeroObject C
inst✝⁶ : HasShift C ℤ
inst✝⁵ : Preadditive C
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
J : Type u_1
T : J → Triangle C
hT : ∀ (j : J), T j ∈ distinguishedTriangles
inst✝³ : HasProduct fun j ↦ (T j).obj₁
inst✝² : HasProduct fun j ↦ (T j).obj₂
inst✝¹ : HasProduct fun j ↦ (T j).obj₃
inst✝ : HasProduct fun j ↦ (shiftFunctor C 1).obj (T j).obj₁
f₁ : (∏ᶜ fun j ↦ (T j).obj₁) ⟶ ∏ᶜ fun j ↦ (T j).obj₂ := Limits.Pi.map fun j ↦ (T j).mor₁
Z : C
f₂ : (∏ᶜ fun j ↦ (T j).obj₂) ⟶ Z
f₃ : Z ⟶ (shiftFunctor C 1).obj (∏ᶜ fun j ↦ (T j).obj₁)
T' : Triangle C := Triangle.mk f₁ f₂ f₃
hT' : T' ∈ distinguishedTriangles
φ : (j : J) → T' ⟶ T j :=
  fun j ↦
    completeDistinguishedTriangleMorphism T' (T j) hT' ⋯ (Pi.π (fun j ↦ (T j).obj₁) j) (Pi.π (fun j ↦ (T j).obj₂) j) ⋯
φ' : T' ⟶ productTriangle T := productTriangle.lift T φ
h₁ : φ'.hom₁ = 𝟙 T'.obj₁
h₂ : φ'.hom₂ = 𝟙 T'.obj₂
this✝ : IsIso φ'.hom₁
this : IsIso φ'.hom₂
A : C
a : A ⟶ (productTriangle T).obj₃
a' : A ⟶ Z
ha' : a' ≫ T'.mor₃ = a ≫ (productTriangle T).mor₃
j : J
⊢ ((a - a' ≫ φ'.hom₃) ≫ Pi.π (fun j ↦ (T j).obj₃) j) ≫ (T j).mor₃ = 0

Tactic:
simp only [sub_comp, assoc]
      erw [← (productTriangle.π T j).comm₃]
      rw [← φ'.comm₃_assoc]
      rw [reassoc_of% ha', sub_eq_zero, h₁, Functor.map_id, id_comp]

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

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
X₁ X₂ X₃ Z₁₂ Z₂₃ Z₁₃ : C
u₁₂ : X₁ ⟶ X₂
u₂₃ : X₂ ⟶ X₃
u₁₃ : X₁ ⟶ X₃
comm : u₁₂ ≫ u₂₃ = u₁₃
v₁₂ : Z₁₂ ⟶ X₁
w₁₂ : X₂ ⟶ (shiftFunctor C 1).obj Z₁₂
h₁₂ : Triangle.mk v₁₂ u₁₂ w₁₂ ∈ distinguishedTriangles
v₂₃ : Z₂₃ ⟶ X₂
w₂₃ : X₃ ⟶ (shiftFunctor C 1).obj Z₂₃
h₂₃ : Triangle.mk v₂₃ u₂₃ w₂₃ ∈ distinguishedTriangles
v₁₃ : Z₁₃ ⟶ X₁
w₁₃ : X₃ ⟶ (shiftFunctor C 1).obj Z₁₃
h₁₃ : Triangle.mk v₁₃ u₁₃ w₁₃ ∈ distinguishedTriangles
h : Octahedron' comm h₁₂ h₂₃ h₁₃
⊢ w₁₃ ≫ (shiftFunctor C 1).map h.m₃ = w₂₃

Tactic:
h.comm₃

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
X₁ X₂ X₃ Z₁₂ Z₂₃ Z₁₃ : C
u₁₂ : X₁ ⟶ X₂
u₂₃ : X₂ ⟶ X₃
u₁₃ : X₁ ⟶ X₃
comm : u₁₂ ≫ u₂₃ = u₁₃
v₁₂ : Z₁₂ ⟶ X₁
w₁₂ : X₂ ⟶ (shiftFunctor C 1).obj Z₁₂
h₁₂ : Triangle.mk v₁₂ u₁₂ w₁₂ ∈ distinguishedTriangles
v₂₃ : Z₂₃ ⟶ X₂
w₂₃ : X₃ ⟶ (shiftFunctor C 1).obj Z₂₃
h₂₃ : Triangle.mk v₂₃ u₂₃ w₂₃ ∈ distinguishedTriangles
v₁₃ : Z₁₃ ⟶ X₁
w₁₃ : X₃ ⟶ (shiftFunctor C 1).obj Z₁₃
h₁₃ : Triangle.mk v₁₃ u₁₃ w₁₃ ∈ distinguishedTriangles
h : Octahedron' comm h₁₂ h₂₃ h₁₃
⊢ w₂₃ = w₂₃
Module: Mathlib.CategoryTheory.Localization.CalculusOfFractions.Preadditive

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_2, u_2} D
inst✝² : Preadditive C
L : C ⥤ D
W : MorphismProperty C
inst✝¹ : L.IsLocalization W
inst✝ : W.HasLeftCalculusOfFractions
X Y : C
f₁ f₂ : L.obj X ⟶ L.obj Y
φ : W.LeftFraction₂ X Y
hφ₁ : f₁ = φ.fst.map L ⋯
hφ₂ : f₂ = φ.snd.map L ⋯
⊢ add' W f₁ f₂ = φ.add.map L ⋯

Tactic:
by
  obtain ⟨φ₀, rfl, rfl, hφ₀⟩ : ∃ (φ₀ : W.LeftFraction₂ X Y)
    (_ : f₁ = φ₀.fst.map L (inverts L W))
    (_ : f₂ = φ₀.snd.map L (inverts L W)),
    add' W f₁ f₂ = φ₀.add.map L (inverts L W) :=
    ⟨(exists_leftFraction₂ L W f₁ f₂).choose,
      (exists_leftFraction₂ L W f₁ f₂).choose_spec.1,
      (exists_leftFraction₂ L W f₁ f₂).choose_spec.2, rfl⟩
  obtain ⟨Z, t₁, t₂, hst, hft, hft', ht⟩ := (LeftFraction₂.map_eq_iff L W φ₀ φ).1 ⟨hφ₁, hφ₂⟩
  have := inverts L W _ ht
  rw [hφ₀, ← cancel_mono (L.map (φ₀.s ≫ t₁))]
  nth_rw 2 [hst]
  rw [L.map_comp, L.map_comp, LeftFraction.map_comp_map_s_assoc,
    LeftFraction.map_comp_map_s_assoc, ← L.map_comp, ← L.map_comp,
    add_comp, add_comp, hft, hft']

Goal state after:
Module: Mathlib.CategoryTheory.Localization.CalculusOfFractions.Preadditive

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_2, u_2} D
inst✝² : Preadditive C
L : C ⥤ D
W : MorphismProperty C
inst✝¹ : L.IsLocalization W
inst✝ : W.HasLeftCalculusOfFractions
X Y : C
f₁ f₂ : L.obj X ⟶ L.obj Y
α : W.LeftFraction₂ X Y
h₁ : f₁ = α.fst.map L ⋯
h₂ : f₂ = α.snd.map L ⋯
⊢ add' W f₁ f₂ = add' W f₂ f₁

Tactic:
add'_eq W f₁ f₂ α h₁ h₂,

Goal state after:
C : Type u_1
D : Type u_2
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_2, u_2} D
inst✝² : Preadditive C
L : C ⥤ D
W : MorphismProperty C
inst✝¹ : L.IsLocalization W
inst✝ : W.HasLeftCalculusOfFractions
X Y : C
f₁ f₂ : L.obj X ⟶ L.obj Y
α : W.LeftFraction₂ X Y
h₁ : f₁ = α.fst.map L ⋯
h₂ : f₂ = α.snd.map L ⋯
⊢ α.add.map L ⋯ = add' W f₂ f₁
Module: Mathlib.CategoryTheory.Localization.CalculusOfFractions.Preadditive

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_2, u_2} D
inst✝² : Preadditive C
L : C ⥤ D
W : MorphismProperty C
inst✝¹ : L.IsLocalization W
inst✝ : W.HasLeftCalculusOfFractions
X Y Z : C
f₁ f₂ : L.obj X ⟶ L.obj Y
g : L.obj Y ⟶ L.obj Z
α : W.LeftFraction₂ X Y
h₁ : f₁ = α.fst.map L ⋯
h₂ : f₂ = α.snd.map L ⋯
β : W.LeftFraction Y Z
hβ : g = β.map L ⋯
γ : W.LeftFraction α.Y' β.Y'
hγ : β.f ≫ γ.s = α.s ≫ γ.f
⊢ add' W f₁ f₂ ≫ g = add' W (f₁ ≫ g) (f₂ ≫ g)

Tactic:
add'_eq W f₁ f₂ α h₁ h₂,

Goal state after:
C : Type u_1
D : Type u_2
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_2, u_2} D
inst✝² : Preadditive C
L : C ⥤ D
W : MorphismProperty C
inst✝¹ : L.IsLocalization W
inst✝ : W.HasLeftCalculusOfFractions
X Y Z : C
f₁ f₂ : L.obj X ⟶ L.obj Y
g : L.obj Y ⟶ L.obj Z
α : W.LeftFraction₂ X Y
h₁ : f₁ = α.fst.map L ⋯
h₂ : f₂ = α.snd.map L ⋯
β : W.LeftFraction Y Z
hβ : g = β.map L ⋯
γ : W.LeftFraction α.Y' β.Y'
hγ : β.f ≫ γ.s = α.s ≫ γ.f
⊢ α.add.map L ⋯ ≫ g = add' W (f₁ ≫ g) (f₂ ≫ g)
Module: Mathlib.CategoryTheory.ObjectProperty.LimitsClosure

Goal state before:
pp: case succ
C : Type u
inst✝¹⁰ : Category.{v, u} C
P : ObjectProperty C
α : Type t
J : α → Type u'
inst✝⁹ : (a : α) → Category.{v', u'} (J a)
β : Type w'
inst✝⁸ : LinearOrder β
inst✝⁷ : OrderBot β
inst✝⁶ : SuccOrder β
inst✝⁵ : WellFoundedLT β
inst✝⁴ : ObjectProperty.Small.{w, v, u} P
inst✝³ : LocallySmall.{w, v, u} C
inst✝² : Small.{w, t} α
inst✝¹ : ∀ (a : α), Small.{w, u'} (J a)
inst✝ : ∀ (a : α), LocallySmall.{w, v', u'} (J a)
H : ∀ {b c : β}, b ≤ c → ∀ [Small.{w, w'} ↑(Set.Iio c)], Small.{w, w'} ↑(Set.Iio b)
b : β
hb : ¬IsMax b
hb' : ∀ [hb₀ : Small.{w, w'} ↑(Set.Iio b)], ObjectProperty.Small.{w, v, u} (P.strictLimitsClosureIter J b)
hb₀ : Small.{w, w'} ↑(Set.Iio (Order.succ b))
⊢ ObjectProperty.Small.{w, v, u} (P.strictLimitsClosureIter J (Order.succ b))

Tactic:
have := H (Order.le_succ b)

Goal state after:
case succ
C : Type u
inst✝¹⁰ : Category.{v, u} C
P : ObjectProperty C
α : Type t
J : α → Type u'
inst✝⁹ : (a : α) → Category.{v', u'} (J a)
β : Type w'
inst✝⁸ : LinearOrder β
inst✝⁷ : OrderBot β
inst✝⁶ : SuccOrder β
inst✝⁵ : WellFoundedLT β
inst✝⁴ : ObjectProperty.Small.{w, v, u} P
inst✝³ : LocallySmall.{w, v, u} C
inst✝² : Small.{w, t} α
inst✝¹ : ∀ (a : α), Small.{w, u'} (J a)
inst✝ : ∀ (a : α), LocallySmall.{w, v', u'} (J a)
H : ∀ {b c : β}, b ≤ c → ∀ [Small.{w, w'} ↑(Set.Iio c)], Small.{w, w'} ↑(Set.Iio b)
b : β
hb : ¬IsMax b
hb' : ∀ [hb₀ : Small.{w, w'} ↑(Set.Iio b)], ObjectProperty.Small.{w, v, u} (P.strictLimitsClosureIter J b)
hb₀ : Small.{w, w'} ↑(Set.Iio (Order.succ b))
this : Small.{w, w'} ↑(Set.Iio b)
⊢ ObjectProperty.Small.{w, v, u} (P.strictLimitsClosureIter J (Order.succ b))
Module: Mathlib.CategoryTheory.Localization.Triangulated

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝¹¹ : Category.{v_1, u_1} C
inst✝¹⁰ : Category.{v_2, u_2} D
L : C ⥤ D
inst✝⁹ : HasShift C ℤ
inst✝⁸ : Preadditive C
inst✝⁷ : HasZeroObject C
inst✝⁶ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁵ : Pretriangulated C
inst✝⁴ : HasShift D ℤ
inst✝³ : L.CommShift ℤ
W : MorphismProperty C
inst✝² : L.IsLocalization W
inst✝¹ : W.HasLeftCalculusOfFractions
inst✝ : W.IsCompatibleWithTriangulation
T₁ T₂ : Triangle C
hT₁ : T₁ ∈ distinguishedTriangles
hT₂ : T₂ ∈ distinguishedTriangles
a : L.obj T₁.obj₁ ⟶ L.obj T₂.obj₁
b : L.obj T₁.obj₂ ⟶ L.obj T₂.obj₂
fac : L.map T₁.mor₁ ≫ b = a ≫ L.map T₂.mor₁
α : W.LeftFraction T₁.obj₁ T₂.obj₁
hα : a = α.map L ⋯
β : W.LeftFraction α.Y' T₂.obj₂
hβ :
  { X' := T₂.obj₁, s := α.s, hs := ⋯, f := T₂.mor₁ }.f ≫ β.s =
    { X' := T₂.obj₁, s := α.s, hs := ⋯, f := T₂.mor₁ }.s ≫ β.f
γ : W.LeftFraction T₁.obj₂ β.Y'
hγ : b ≫ L.map β.s = γ.map L ⋯
⊢ ∃ φ, φ.hom₁ = a ∧ φ.hom₂ = b

Tactic:
dsimp at hβ

Goal state after:
C : Type u_1
D : Type u_2
inst✝¹¹ : Category.{v_1, u_1} C
inst✝¹⁰ : Category.{v_2, u_2} D
L : C ⥤ D
inst✝⁹ : HasShift C ℤ
inst✝⁸ : Preadditive C
inst✝⁷ : HasZeroObject C
inst✝⁶ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁵ : Pretriangulated C
inst✝⁴ : HasShift D ℤ
inst✝³ : L.CommShift ℤ
W : MorphismProperty C
inst✝² : L.IsLocalization W
inst✝¹ : W.HasLeftCalculusOfFractions
inst✝ : W.IsCompatibleWithTriangulation
T₁ T₂ : Triangle C
hT₁ : T₁ ∈ distinguishedTriangles
hT₂ : T₂ ∈ distinguishedTriangles
a : L.obj T₁.obj₁ ⟶ L.obj T₂.obj₁
b : L.obj T₁.obj₂ ⟶ L.obj T₂.obj₂
fac : L.map T₁.mor₁ ≫ b = a ≫ L.map T₂.mor₁
α : W.LeftFraction T₁.obj₁ T₂.obj₁
hα : a = α.map L ⋯
β : W.LeftFraction α.Y' T₂.obj₂
hβ : T₂.mor₁ ≫ β.s = α.s ≫ β.f
γ : W.LeftFraction T₁.obj₂ β.Y'
hγ : b ≫ L.map β.s = γ.map L ⋯
⊢ ∃ φ, φ.hom₁ = a ∧ φ.hom₂ = b
Module: Mathlib.CategoryTheory.Triangulated.Subcategory

Goal state before:
pp: C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : HasZeroObject C
inst✝³ : HasShift C ℤ
inst✝² : Preadditive C
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
n : ℕ
⊢ ⊥.extensionProductIter n = ⊥

Tactic:
cases n
  case zero => rw [extensionProductIter_zero]
  case succ n => rw [extensionProductIter_succ, extensionProduct_bot_left]

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

Goal state before:
pp: C : Type u_1
inst✝¹⁵ : Category.{v_1, u_1} C
inst✝¹⁴ : HasZeroObject C
inst✝¹³ : HasShift C ℤ
inst✝¹² : Preadditive C
inst✝¹¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹⁰ : Pretriangulated C
D : Type u_2
inst✝⁹ : Category.{v_2, u_2} D
inst✝⁸ : Preadditive D
inst✝⁷ : HasZeroObject D
inst✝⁶ : HasShift D ℤ
inst✝⁵ : ∀ (n : ℤ), (shiftFunctor D n).Additive
inst✝⁴ : Pretriangulated D
E : Type u_3
inst✝³ : Category.{v_3, u_3} E
inst✝² : HasShift E ℤ
P : ObjectProperty C
inst✝¹ : IsTriangulated C
inst✝ : P.IsTriangulated
X₁ X₂ X₃ : C
u₁₂ : X₁ ⟶ X₂
u₂₃ : X₂ ⟶ X₃
Z₁₂ : C
v₁₂ : X₂ ⟶ Z₁₂
w₁₂✝ : Z₁₂ ⟶ (shiftFunctor C 1).obj X₁
H₁₂ : Triangle.mk u₁₂ v₁₂ w₁₂✝ ∈ distinguishedTriangles
mem₁₂ : P.isoClosure Z₁₂
Z₂₃ : C
v₂₃ : X₃ ⟶ Z₂₃
w₂₃ : Z₂₃ ⟶ (shiftFunctor C 1).obj X₂
H₂₃ : Triangle.mk u₂₃ v₂₃ w₂₃ ∈ distinguishedTriangles
mem₂₃ : P.isoClosure Z₂₃
Z₁₃ : C
v₁₃ : X₃ ⟶ Z₁₃
w₁₂ : Z₁₃ ⟶ (shiftFunctor C 1).obj X₁
H₁₃ : Triangle.mk (u₁₂ ≫ u₂₃) v₁₃ w₁₂ ∈ distinguishedTriangles
⊢ P.isoClosure.trW (u₁₂ ≫ u₂₃)

Tactic:
exact ⟨_, _, _, H₁₃, P.isoClosure.ext_of_isTriangulatedClosed₂
      _ (someOctahedron rfl H₁₂ H₂₃ H₁₃).mem mem₁₂ mem₂₃⟩

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

Goal state before:
pp: C : Type u_1
D : Type u_2
E : Type u_3
E' : Type u_4
inst✝⁸ : Category.{v_1, u_1} C
inst✝⁷ : Category.{v_2, u_2} D
inst✝⁶ : Category.{v_3, u_3} E
inst✝⁵ : Category.{v_4, u_4} E'
A : Type u_5
inst✝⁴ : AddMonoid A
inst✝³ : HasShift D A
inst✝² : HasShift E A
inst✝¹ : HasShift E' A
F G✝ H : SingleFunctors C D A
G : D ⥤ E
inst✝ : G.CommShift A
F₁ F₂ : SingleFunctors C D A
φ : F₁ ⟶ F₂
n a a' : A
ha' : n + a = a'
⊢ ((F₁.postcomp G).shiftIso n a a' ha').hom ≫ whiskerRight (φ.hom a) G =
    whiskerRight (whiskerRight (φ.hom a') G) (shiftFunctor E n) ≫ ((F₂.postcomp G).shiftIso n a a' ha').hom

Tactic:
by
        ext X
        simpa using G.congr_map (congr_app (φ.comm n a a' ha') X)

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

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasBinaryBiproducts C
X₁ X₂ X₃ : CochainComplex C ℤ
f : X₁ ⟶ X₂
g : X₂ ⟶ X₃
⊢ δ (-1) 0 ((snd f).comp (inl g) ⋯) =
    Cochain.ofHom ((mappingConeCompTriangle f g).mor₁ ≫ desc (f ≫ g) ((Cochain.ofHom f).comp (inl g) ⋯) (inr g) ⋯)

Tactic:
by
      ext p
      rw [ext_from_iff _ (p + 1) _ rfl, ext_to_iff _ _ (p + 1) rfl]
      simp [map, δ_zero_cochain_comp,
        Cochain.comp_v _ _ (add_neg_cancel 1) p (p + 1) p (by lia) (by lia)]

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

Goal state before:
pp: case mpr
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝¹ : Category.{v_1, u_3} C
inst✝ : HasZeroMorphisms C
K : HomologicalComplex C c'
e : c.Embedding c'
h₁ : K.IsStrictlySupported e
h₂ : K.IsStrictlySupportedOutside e
n : ι'
⊢ IsZero (K.X n)

Tactic:
by_cases hn : ∃ i, e.f i = n

Goal state after:
case pos
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝¹ : Category.{v_1, u_3} C
inst✝ : HasZeroMorphisms C
K : HomologicalComplex C c'
e : c.Embedding c'
h₁ : K.IsStrictlySupported e
h₂ : K.IsStrictlySupportedOutside e
n : ι'
hn : ∃ i, e.f i = n
⊢ IsZero (K.X n)

case neg
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝¹ : Category.{v_1, u_3} C
inst✝ : HasZeroMorphisms C
K : HomologicalComplex C c'
e : c.Embedding c'
h₁ : K.IsStrictlySupported e
h₂ : K.IsStrictlySupportedOutside e
n : ι'
hn : ¬∃ i, e.f i = n
⊢ IsZero (K.X n)
Module: Mathlib.Algebra.Homology.Embedding.IsSupported

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝¹ : Category.{v_1, u_3} C
inst✝ : HasZeroMorphisms C
K : HomologicalComplex C c'
e : c.Embedding c'
h₁ : K.IsStrictlySupported e
h₂ : K.IsStrictlySupportedOutside e
n : ι'
hn : ¬∃ i, e.f i = n
⊢ ∀ (i : ι), e.f i ≠ n

Tactic:
simpa using hn

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

Goal state before:
pp: case mp
p : ℤ
n : ℕ
⊢ (embeddingUpIntGE p).BoundaryGE n → n = 0

Tactic:
intro h
    obtain _ | n := n
    · rfl
    · have := h.2 n
      dsimp at this
      lia

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

Goal state before:
pp: case mp
p : ℤ
n : ℕ
⊢ (embeddingUpIntLE p).BoundaryLE n → n = 0

Tactic:
intro h
    obtain _ | n := n
    · rfl
    · have := h.2 n
      dsimp at this
      lia

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

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝³ : Category.{v_1, u_3} C
inst✝² : HasZeroMorphisms C
inst✝¹ : HasZeroObject C
K L M : HomologicalComplex C c
φ : K ⟶ L
φ' : L ⟶ M
e : c.Embedding c'
inst✝ : ∀ (j : ι), K.HasHomology j
j' : ι'
⊢ (K.extend e).HasHomology j'

Tactic:
by
  by_cases h : ∃ j, e.f j = j'
  · obtain ⟨j, rfl⟩ := h
    infer_instance
  · have hj := isZero_extend_X K e j' (by tauto)
    exact ShortComplex.HasHomology.mk'
      (ShortComplex.HomologyData.ofZeros _ (hj.eq_of_tgt _ _) (hj.eq_of_src _ _))

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

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✝⁷ : Category.{v_1, u_3} C
inst✝⁶ : HasZeroMorphisms C
K L M : HomologicalComplex C c'
φ : K ⟶ L
φ' : L ⟶ M
e : c.Embedding c'
inst✝⁵ : e.IsTruncGE
inst✝⁴ : ∀ (i' : ι'), K.HasHomology i'
inst✝³ : ∀ (i' : ι'), L.HasHomology i'
inst✝² : ∀ (i' : ι'), M.HasHomology i'
inst✝¹ : HasZeroObject C
ι'' : Type u_4
c'' : ComplexShape ι''
e' : c''.Embedding c'
inst✝ : K.IsStrictlySupported e'
i' : ι'
hi' : ∀ (i : ι''), e'.f i ≠ i'
hi'' : ∃ i, e.f i = i'
⊢ IsZero ((K.truncGE e).X i')

Tactic:
obtain ⟨i, hi⟩ := hi''

Goal state after:
case pos
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝⁷ : Category.{v_1, u_3} C
inst✝⁶ : HasZeroMorphisms C
K L M : HomologicalComplex C c'
φ : K ⟶ L
φ' : L ⟶ M
e : c.Embedding c'
inst✝⁵ : e.IsTruncGE
inst✝⁴ : ∀ (i' : ι'), K.HasHomology i'
inst✝³ : ∀ (i' : ι'), L.HasHomology i'
inst✝² : ∀ (i' : ι'), M.HasHomology i'
inst✝¹ : HasZeroObject C
ι'' : Type u_4
c'' : ComplexShape ι''
e' : c''.Embedding c'
inst✝ : K.IsStrictlySupported e'
i' : ι'
hi' : ∀ (i : ι''), e'.f i ≠ i'
i : ι
hi : e.f i = i'
⊢ IsZero ((K.truncGE e).X i')
Module: Mathlib.Algebra.Homology.Embedding.TruncLEHomology

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✝² : Category.{v_1, u_3} C
inst✝¹ : Abelian C
K : HomologicalComplex C c'
e : c.Embedding c'
inst✝ : e.IsTruncLE
i' : ι'
hi' : ∃ j', c'.Rel i' j'
⊢ Epi (homologyMap (K.shortComplexTruncLE e).g i')

Tactic:
obtain ⟨j', hj'⟩ := hi'
    exact ((K.shortComplexTruncLE_shortExact e).homology_exact₃ i' j' hj').epi_f (by simp)

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

Goal state before:
pp: case neg
n₀ n₁ : ℤ
h : n₀ + 1 = n₁
k : ℕ
hi : ¬n₁ + ↑k ≤ n₀
⊢ (∃ i₁, (embeddingUpIntLE n₀).f i₁ = n₁ + ↑k) ∨ ∃ i₂, (embeddingUpIntGE n₁).f i₂ = n₁ + ↑k

Tactic:
exact Or.inr ⟨k, rfl⟩

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

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasDerivedCategory C
n : ℤ
A B : C
X : CochainComplex C ℤ
w✝² : X.IsStrictlyGE n
w✝¹ : X.IsStrictlyLE n
s : X ⟶ (CochainComplex.singleFunctor C n).obj A
w✝ : IsIso (Q.map s)
g : X ⟶ (CochainComplex.singleFunctor C n).obj B
A₀ : C
e : X ≅ (HomologicalComplex.single C (ComplexShape.up ℤ) n).obj A₀
φ : A₀ ⟶ A
hφ : (CochainComplex.singleFunctor C n).map φ = e.inv ≫ s
⊢ ∃ f', inv (Q.map s) ≫ Q.map g = Q.map f'

Tactic:
have : IsIso ((singleFunctor C n).map φ) := by
      change IsIso (Q.map ((CochainComplex.singleFunctor C n).map φ))
      rw [hφ, Functor.map_comp]
      infer_instance

Goal state after:
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasDerivedCategory C
n : ℤ
A B : C
X : CochainComplex C ℤ
w✝² : X.IsStrictlyGE n
w✝¹ : X.IsStrictlyLE n
s : X ⟶ (CochainComplex.singleFunctor C n).obj A
w✝ : IsIso (Q.map s)
g : X ⟶ (CochainComplex.singleFunctor C n).obj B
A₀ : C
e : X ≅ (HomologicalComplex.single C (ComplexShape.up ℤ) n).obj A₀
φ : A₀ ⟶ A
hφ : (CochainComplex.singleFunctor C n).map φ = e.inv ≫ s
this : IsIso ((singleFunctor C n).map φ)
⊢ ∃ f', inv (Q.map s) ≫ Q.map g = Q.map f'
Module: Mathlib.CategoryTheory.Localization.HomEquiv

Goal state before:
pp: C : Type u_1
D₁ : Type u_5
D₂ : Type u_6
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_5, u_5} D₁
inst✝² : Category.{v_6, u_6} D₂
W : MorphismProperty C
L₁ : C ⥤ D₁
inst✝¹ : L₁.IsLocalization W
L₂ : C ⥤ D₂
inst✝ : L₂.IsLocalization W
X Y : C
G : D₁ ⥤ D₂
e : L₁ ⋙ G ≅ L₂
f : L₁.obj X ⟶ L₁.obj Y
⊢ e.inv.app X ≫ G.map f ≫ e.symm.inv.app Y = e.inv.app X ≫ G.map f ≫ e.hom.app Y

Tactic:
Iso.symm_inv

Goal state after:
C : Type u_1
D₁ : Type u_5
D₂ : Type u_6
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_5, u_5} D₁
inst✝² : Category.{v_6, u_6} D₂
W : MorphismProperty C
L₁ : C ⥤ D₁
inst✝¹ : L₁.IsLocalization W
L₂ : C ⥤ D₂
inst✝ : L₂.IsLocalization W
X Y : C
G : D₁ ⥤ D₂
e : L₁ ⋙ G ≅ L₂
f : L₁.obj X ⟶ L₁.obj Y
⊢ e.inv.app X ≫ G.map f ≫ e.hom.app Y = e.inv.app X ≫ G.map f ≫ e.hom.app Y
Module: Mathlib.CategoryTheory.Localization.SmallShiftedHom

Goal state before:
pp: C : Type u₁
inst✝⁸ : Category.{v₁, u₁} C
D : Type u₂
inst✝⁷ : Category.{v₂, u₂} D
W : MorphismProperty C
M : Type w'
inst✝⁶ : AddMonoid M
inst✝⁵ : HasShift C M
inst✝⁴ : HasShift D M
L : C ⥤ D
inst✝³ : L.IsLocalization W
inst✝² : L.CommShift M
X Y : C
inst✝¹ : HasSmallLocalizedShiftedHom W M Y X
inst✝ : W.RespectsIso
m₀ : M
hm₀ : m₀ = 0
f : X ⟶ Y
hf : W f
hf' : W ((shiftFunctorZero' C m₀ hm₀).hom.app X ≫ f)
⊢ ((Functor.commShiftIso L m₀).app X).hom =
    (isoOfHom L W ((shiftFunctorZero' C m₀ hm₀).hom.app X ≫ f) hf').hom ≫ ShiftedHom.mk₀ m₀ hm₀ (isoOfHom L W f hf).inv

Tactic:
simp [ShiftedHom.mk₀, Functor.commShiftIso_zero' _ _ m₀ hm₀]

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

Goal state before:
pp: C : Type u_1
ι : Type u_2
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
c : ComplexShape ι
K : HomologicalComplex C c
j : ι
A : C
z₂ : A ⟶ K.cycles j
⊢ z₂ ≫ K.homologyπ j = 0 ↔ ∃ A' π, ∃ (_ : Epi π), ∃ x₁, π ≫ z₂ = x₁ ≫ K.toCycles (c.prev j) j

Tactic:
exact (K.sc j).comp_homologyπ_eq_zero_iff_up_to_refinements z₂

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

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Category.{v_2, u_2} D
A : Type u_3
inst✝³ : AddMonoid A
inst✝² : HasShift C A
inst✝¹ : HasShift D A
F : C ⥤ D
inst✝ : F.CommShift A
a b : A
⊢ (NatIso.op (commShiftIso F (a + b))).symm =
    CommShift.isoAdd (NatIso.op (commShiftIso F a)).symm (NatIso.op (commShiftIso F b)).symm

Tactic:
commShiftIso_add

Goal state after:
C : Type u_1
D : Type u_2
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Category.{v_2, u_2} D
A : Type u_3
inst✝³ : AddMonoid A
inst✝² : HasShift C A
inst✝¹ : HasShift D A
F : C ⥤ D
inst✝ : F.CommShift A
a b : A
⊢ (NatIso.op (CommShift.isoAdd (commShiftIso F a) (commShiftIso F b))).symm =
    CommShift.isoAdd (NatIso.op (commShiftIso F a)).symm (NatIso.op (commShiftIso F b)).symm
Module: Mathlib.CategoryTheory.Shift.Opposite

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Category.{v_2, u_2} D
A : Type u_3
inst✝³ : AddMonoid A
inst✝² : HasShift C A
inst✝¹ : HasShift D A
F : C ⥤ D
inst✝ : (OppositeShift.functor A F).CommShift A
a b : A
⊢ NatIso.removeOp (commShiftIso (OppositeShift.functor A F) (a + b)).symm =
    CommShift.isoAdd (NatIso.removeOp (commShiftIso (OppositeShift.functor A F) a).symm)
      (NatIso.removeOp (commShiftIso (OppositeShift.functor A F) b).symm)

Tactic:
commShiftIso_add

Goal state after:
C : Type u_1
D : Type u_2
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Category.{v_2, u_2} D
A : Type u_3
inst✝³ : AddMonoid A
inst✝² : HasShift C A
inst✝¹ : HasShift D A
F : C ⥤ D
inst✝ : (OppositeShift.functor A F).CommShift A
a b : A
⊢ NatIso.removeOp
      (CommShift.isoAdd (commShiftIso (OppositeShift.functor A F) a)
          (commShiftIso (OppositeShift.functor A F) b)).symm =
    CommShift.isoAdd (NatIso.removeOp (commShiftIso (OppositeShift.functor A F) a).symm)
      (NatIso.removeOp (commShiftIso (OppositeShift.functor A F) b).symm)
Module: Mathlib.CategoryTheory.Shift.Adjunction

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_2, u_2} D
F : C ⥤ D
G : D ⥤ C
adj : F ⊣ G
A : Type u_3
inst✝² : AddMonoid A
inst✝¹ : HasShift C A
inst✝ : HasShift D A
a : A
e₁ : shiftFunctor C a ⋙ F ≅ F ⋙ shiftFunctor D a
e₂ e₂' : shiftFunctor D a ⋙ G ≅ G ⋙ shiftFunctor C a
h : CompatibilityUnit adj e₁ e₂
h' : CompatibilityUnit adj e₁ e₂'
x✝ : D
⊢ e₂.inv.app x✝ = e₂'.inv.app x✝

Tactic:
compatibilityUnit_right adj e₁ e₂ h,

Goal state after:
C : Type u_1
D : Type u_2
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_2, u_2} D
F : C ⥤ D
G : D ⥤ C
adj : F ⊣ G
A : Type u_3
inst✝² : AddMonoid A
inst✝¹ : HasShift C A
inst✝ : HasShift D A
a : A
e₁ : shiftFunctor C a ⋙ F ≅ F ⋙ shiftFunctor D a
e₂ e₂' : shiftFunctor D a ⋙ G ≅ G ⋙ shiftFunctor C a
h : CompatibilityUnit adj e₁ e₂
h' : CompatibilityUnit adj e₁ e₂'
x✝ : D
⊢ adj.unit.app ((shiftFunctor C a).obj (G.obj x✝)) ≫
      G.map (e₁.hom.app (G.obj x✝)) ≫ G.map ((shiftFunctor D a).map (adj.counit.app x✝)) =
    e₂'.inv.app x✝