mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.CategoryTheory.ObjectProperty.LimitsClosure

Goal state before:
pp: case of_isoClosure
C : Type u
inst✝³ : Category.{v, u} C
P : ObjectProperty C
α : Type t
J : α → Type u'
inst✝² : (a : α) → Category.{v', u'} (J a)
Q : ObjectProperty C
inst✝¹ : Q.IsClosedUnderIsomorphisms
inst✝ : ∀ (a : α), Q.IsClosedUnderLimitsOfShape (J a)
h : P ≤ Q
X X✝ Y✝ : C
e : X✝ ≅ Y✝
hX : P.limitsClosure J X✝
hX' : Q X✝
⊢ Q Y✝

Tactic:
| of_isoClosure e hX hX' =>

Goal state after:
Module: Mathlib.SetTheory.Cardinal.HasCardinalLT

Goal state before:
pp: ι : Type u_1
X : Type u_2
P : ι → X → Prop
κ : Cardinal.{u_3}
inst✝ : Fact κ.IsRegular
hι : HasCardinalLT ι κ
hP : ∀ (i : ι), HasCardinalLT (Subtype (P i)) κ
val✝ : X
h : (⨆ i, P i) val✝
⊢ ∃ a,
    (fun x ↦
          match x with
          | ⟨i, ⟨x, hx⟩⟩ => ⟨x, ⋯⟩)
        a =
      ⟨val✝, h⟩

Tactic:
simp only [iSup_apply, iSup_Prop_eq] at h

Goal state after:
ι : Type u_1
X : Type u_2
P : ι → X → Prop
κ : Cardinal.{u_3}
inst✝ : Fact κ.IsRegular
hι : HasCardinalLT ι κ
hP : ∀ (i : ι), HasCardinalLT (Subtype (P i)) κ
val✝ : X
h✝ : (⨆ i, P i) val✝
h : ∃ i, P i val✝
⊢ ∃ a,
    (fun x ↦
          match x with
          | ⟨i, ⟨x, hx⟩⟩ => ⟨x, ⋯⟩)
        a =
      ⟨val✝, h✝⟩
Module: Mathlib.CategoryTheory.ObjectProperty.ColimitsClosure

Goal state before:
pp: case of_isoClosure
C : Type u
inst✝³ : Category.{v, u} C
P : ObjectProperty C
α : Type t
J : α → Type u'
inst✝² : (a : α) → Category.{v', u'} (J a)
Q : ObjectProperty C
inst✝¹ : Q.IsClosedUnderIsomorphisms
inst✝ : ∀ (a : α), Q.IsClosedUnderColimitsOfShape (J a)
h : P ≤ Q
X X✝ Y✝ : C
e : X✝ ≅ Y✝
hX : P.colimitsClosure J X✝
hX' : Q X✝
⊢ Q Y✝

Tactic:
| of_isoClosure e hX hX' =>

Goal state after:
Module: Mathlib.CategoryTheory.ObjectProperty.ColimitsClosure

Goal state before:
pp: case refine_1
C : Type u
inst✝¹ : Category.{v, u} C
P : ObjectProperty C
α : Type t
J : α → Type u'
inst✝ : (a : α) → Category.{v', u'} (J a)
⊢ P.op ≤ P.op.limitsClosure fun a ↦ (J a)ᵒᵖ

Tactic:
apply le_limitsClosure

Goal state after:
Module: Mathlib.CategoryTheory.ObjectProperty.LimitsClosure

Goal state before:
pp: case inr
C : Type u
inst✝³ : Category.{v, u} C
P : ObjectProperty C
α : Type t
J : α → Type u'
inst✝² : (a : α) → Category.{v', u'} (J a)
κ : Cardinal.{w}
inst✝¹ : Fact κ.IsRegular
h : ∀ (a : α), HasCardinalLT (J a) κ
hκ : κ.IsRegular
this : ∀ (a : α), Small.{w, u'} (J a)
a : α
F : J a ⥤ C
inst✝ : HasLimit F
hF : ∀ (j : J a), P.strictLimitsClosureIter J κ.ord (F.obj j)
⊢ P.strictLimitsClosureIter J κ.ord (limit F)

Tactic:
· simp only [strictLimitsClosureIter, transfiniteIterate_limit _ _ _
      (Cardinal.isSuccLimit_ord hκ.aleph0_le), prop_iSup_iff,
      Subtype.exists, Set.mem_Iio, exists_prop] at hF
    choose o ho ho' using hF
    obtain ⟨m, hm, hm'⟩ : ∃ (m : Ordinal.{w}) (hm : m < κ.ord), ∀ (j : J a), o j ≤ m := by
      refine ⟨⨆ j, o ((equivShrink.{w} (J a)).symm j),
        Ordinal.iSup_lt_of_lt_cof ?_ (fun _ ↦ ho _), fun j ↦ ?_⟩
      · rw [hκ.cof_ord, ← hasCardinalLT_iff_cardinal_mk_lt _ κ,
          ← hasCardinalLT_iff_of_equiv (equivShrink.{w} (J a))]
        exact h a
      · obtain ⟨j, rfl⟩ := (equivShrink.{w} (J a)).symm.surjective j
        exact le_ciSup Ordinal.bddAbove_of_small _
    refine monotone_transfiniteIterate _ _
      (fun (Q : ObjectProperty C) ↦ Q.le_strictLimitsClosureStep J) (Order.succ_le_iff.2 hm) _ ?_
    dsimp
    rw [transfiniteIterate_succ _ _ _ (by simp)]
    simp only [strictLimitsClosureStep, prop_sup_iff, prop_iSup_iff]
    exact Or.inr ⟨a, ⟨_, fun j ↦ monotone_transfiniteIterate _ _
      (fun (Q : ObjectProperty C) ↦ Q.le_strictLimitsClosureStep J)  (hm' j) _ (ho' j)⟩⟩

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

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} D
W : MorphismProperty C
inst✝ : W.HasLeftCalculusOfFractions
X Y : C
z : W.LeftFraction X Y
⊢ Hom.mk { Y' := z.Y', f := 𝟙 X ≫ z.f, s := z.s ≫ 𝟙 z.Y', hs := ⋯ } = Hom.mk z

Tactic:
simp only [id_comp, comp_id]

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

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} D
W : MorphismProperty C
inst✝ : W.HasLeftCalculusOfFractions
⊢ ∀ {X Y : Localization W} (f : Hom W X Y), (Hom.mk (ofHom W (𝟙 X))).comp f = f

Tactic:
rintro (X Y : C) f
    obtain ⟨z, rfl⟩ := Hom.mk_surjective f
    rw [Hom.comp_eq, comp_eq (ofHom W (𝟙 X)) z (ofHom W z.f) (by simp)]
    dsimp
    simp only [id_comp, comp_id]

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

Goal state before:
pp: C : Type u_1
D : Type u_2
A : Type u_3
inst✝¹⁴ : Category.{v_1, u_1} C
inst✝¹³ : HasShift C ℤ
inst✝¹² : Category.{v_2, u_2} D
inst✝¹¹ : HasZeroObject D
inst✝¹⁰ : HasShift D ℤ
inst✝⁹ : Preadditive D
inst✝⁸ : ∀ (n : ℤ), (shiftFunctor D n).Additive
inst✝⁷ : Pretriangulated D
inst✝⁶ : Category.{v_3, u_3} A
F : C ⥤ A
inst✝⁵ : HasZeroObject C
inst✝⁴ : Preadditive C
inst✝³ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝² : Pretriangulated C
inst✝¹ : Abelian A
inst✝ : F.IsHomological
X₁ X₂ : C
this : HasBinaryBiproduct (F.obj X₁) (F.obj X₂)
Z : A
f : Z ⟶ F.obj (X₁ ⊞ X₂)
hf : f ≫ F.biprodComparison X₁ X₂ = 0
S : ShortComplex A := { X₁ := X₁, X₂ := X₁ ⊞ X₂, X₃ := X₂, f := biprod.inl, g := biprod.snd, zero := ⋯ }.map F
⊢ f = 0

Tactic:
have : Mono S.f := by dsimp [S]; infer_instance

Goal state after:
C : Type u_1
D : Type u_2
A : Type u_3
inst✝¹⁴ : Category.{v_1, u_1} C
inst✝¹³ : HasShift C ℤ
inst✝¹² : Category.{v_2, u_2} D
inst✝¹¹ : HasZeroObject D
inst✝¹⁰ : HasShift D ℤ
inst✝⁹ : Preadditive D
inst✝⁸ : ∀ (n : ℤ), (shiftFunctor D n).Additive
inst✝⁷ : Pretriangulated D
inst✝⁶ : Category.{v_3, u_3} A
F : C ⥤ A
inst✝⁵ : HasZeroObject C
inst✝⁴ : Preadditive C
inst✝³ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝² : Pretriangulated C
inst✝¹ : Abelian A
inst✝ : F.IsHomological
X₁ X₂ : C
this✝ : HasBinaryBiproduct (F.obj X₁) (F.obj X₂)
Z : A
f : Z ⟶ F.obj (X₁ ⊞ X₂)
hf : f ≫ F.biprodComparison X₁ X₂ = 0
S : ShortComplex A := { X₁ := X₁, X₂ := X₁ ⊞ X₂, X₃ := X₂, f := biprod.inl, g := biprod.snd, zero := ⋯ }.map F
this : Mono S.f
⊢ f = 0
Module: Mathlib.CategoryTheory.Triangulated.HomologicalFunctor

Goal state before:
pp: C : Type u_1
A : Type u_3
inst✝⁹ : Category.{v_1, u_1} C
inst✝⁸ : HasShift C ℤ
inst✝⁷ : Category.{v_3, u_3} A
F : C ⥤ A
inst✝⁶ : HasZeroObject C
inst✝⁵ : Preadditive C
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝³ : Pretriangulated C
inst✝² : Abelian A
inst✝¹ : F.IsHomological
inst✝ : F.ShiftSequence ℤ
T : Triangle C
hT : T ∈ distinguishedTriangles
n₀ n₁ : ℤ
h : n₀ + 1 = n₁
⊢ (Iso.refl
          { X₁ := (F.shift n₀).obj T.rotate.obj₁, X₂ := (F.shift n₀).obj T.rotate.obj₂,
              X₃ := (F.shift n₀).obj T.rotate.obj₃, f := (F.shift n₀).map T.rotate.mor₁,
              g := (F.shift n₀).map T.rotate.mor₂, zero := ⋯ }.X₂).hom ≫
      { X₁ := (F.shift n₀).obj T.obj₂, X₂ := (F.shift n₀).obj T.obj₃, X₃ := (F.shift n₁).obj T.obj₁,
          f := (F.shift n₀).map T.mor₂, g := F.homologySequenceδ T n₀ n₁ h, zero := ⋯ }.g =
    { X₁ := (F.shift n₀).obj T.rotate.obj₁, X₂ := (F.shift n₀).obj T.rotate.obj₂, X₃ := (F.shift n₀).obj T.rotate.obj₃,
          f := (F.shift n₀).map T.rotate.mor₁, g := (F.shift n₀).map T.rotate.mor₂, zero := ⋯ }.g ≫
      ((F.shiftIso 1 n₀ n₁ ⋯).app T.obj₁).hom

Tactic:
by simp [homologySequenceδ, shiftMap]

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

Goal state before:
pp: C : Type u_1
inst✝⁶ : Category.{v_1, u_1} C
D : Type u_2
inst✝⁵ : Category.{v_2, u_2} D
E : Type u_3
inst✝⁴ : Category.{v_3, u_3} E
M : Type u_4
inst✝³ : AddMonoid M
inst✝² : HasShift C M
inst✝¹ : HasShift D M
inst✝ : HasShift E M
X Y Z T : C
a₁ a₂ a₃ a₁₂ a₂₃ a : M
α : ShiftedHom X Y a₁
β : ShiftedHom Y Z a₂
γ : ShiftedHom Z T a₃
h₁₂ : a₂ + a₁ = a₁₂
h₂₃ : a₃ + a₂ = a₂₃
h : a₃ + a₂ + a₁ = a
⊢ a₃ + a₁₂ = a

Tactic:
← h₁₂,

Goal state after:
C : Type u_1
inst✝⁶ : Category.{v_1, u_1} C
D : Type u_2
inst✝⁵ : Category.{v_2, u_2} D
E : Type u_3
inst✝⁴ : Category.{v_3, u_3} E
M : Type u_4
inst✝³ : AddMonoid M
inst✝² : HasShift C M
inst✝¹ : HasShift D M
inst✝ : HasShift E M
X Y Z T : C
a₁ a₂ a₃ a₁₂ a₂₃ a : M
α : ShiftedHom X Y a₁
β : ShiftedHom Y Z a₂
γ : ShiftedHom Z T a₃
h₁₂ : a₂ + a₁ = a₁₂
h₂₃ : a₃ + a₂ = a₂₃
h : a₃ + a₂ + a₁ = a
⊢ a₃ + (a₂ + a₁) = a
Module: Mathlib.CategoryTheory.Shift.ShiftedHom

Goal state before:
pp: C : Type u_1
inst✝⁸ : Category.{v_1, u_1} C
D : Type u_2
inst✝⁷ : Category.{v_2, u_2} D
M : Type u_4
inst✝⁶ : AddMonoid M
inst✝⁵ : HasShift C M
inst✝⁴ : HasShift D M
X Y : C
inst✝³ : Preadditive C
inst✝² : Preadditive D
a : M
α₁ α₂ : ShiftedHom X Y a
F : C ⥤ D
inst✝¹ : F.CommShift M
inst✝ : F.Additive
⊢ (α₁ + α₂).map F = α₁.map F + α₂.map F

Tactic:
simp [ShiftedHom.map, F.map_add]

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

Goal state before:
pp: C : Type u_1
inst✝¹³ : Category.{v_1, u_1} C
inst✝¹² : HasZeroObject C
inst✝¹¹ : HasShift C ℤ
inst✝¹⁰ : Preadditive C
inst✝⁹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁸ : Pretriangulated C
D : Type u_2
inst✝⁷ : Category.{v_2, u_2} D
inst✝⁶ : Preadditive D
inst✝⁵ : HasZeroObject D
inst✝⁴ : HasShift D ℤ
inst✝³ : ∀ (n : ℤ), (shiftFunctor D n).Additive
inst✝² : Pretriangulated D
E : Type u_3
inst✝¹ : Category.{v_3, u_3} E
inst✝ : HasShift E ℤ
P Q R : ObjectProperty C
⊢ ∀ {X Y : C} (x : X ≅ Y), P.extensionProduct Q X → P.extensionProduct Q Y

Tactic:
intro X X' i ⟨Y, Z, f, g, h, hT, hP, hQ⟩

Goal state after:
C : Type u_1
inst✝¹³ : Category.{v_1, u_1} C
inst✝¹² : HasZeroObject C
inst✝¹¹ : HasShift C ℤ
inst✝¹⁰ : Preadditive C
inst✝⁹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁸ : Pretriangulated C
D : Type u_2
inst✝⁷ : Category.{v_2, u_2} D
inst✝⁶ : Preadditive D
inst✝⁵ : HasZeroObject D
inst✝⁴ : HasShift D ℤ
inst✝³ : ∀ (n : ℤ), (shiftFunctor D n).Additive
inst✝² : Pretriangulated D
E : Type u_3
inst✝¹ : Category.{v_3, u_3} E
inst✝ : HasShift E ℤ
P Q R : ObjectProperty C
X X' : C
i : X ≅ X'
Y Z : C
f : Y ⟶ X
g : X ⟶ Z
h : Z ⟶ (shiftFunctor C 1).obj Y
hT : Triangle.mk f g h ∈ distinguishedTriangles
hP : P Y
hQ : Q Z
⊢ P.extensionProduct Q X'
Module: Mathlib.CategoryTheory.Triangulated.Subcategory

Goal state before:
pp: case succ
C : Type u_1
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : HasZeroObject C
inst✝⁴ : HasShift C ℤ
inst✝³ : Preadditive C
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
P Q : ObjectProperty C
inst✝ : Q.IsTriangulatedClosed₂
h : P ≤ Q
n : ℕ
H : P.extensionProductIter n ≤ Q.isoClosure
⊢ P.extensionProductIter (n + 1) ≤ Q.isoClosure

Tactic:
| succ n H =>

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

Goal state before:
pp: case succ
C : Type u_1
inst✝¹⁴ : Category.{v_1, u_1} C
inst✝¹³ : HasZeroObject C
inst✝¹² : HasShift C ℤ
inst✝¹¹ : Preadditive C
inst✝¹⁰ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁹ : Pretriangulated C
D : Type u_2
inst✝⁸ : Category.{v_2, u_2} D
inst✝⁷ : Preadditive D
inst✝⁶ : HasZeroObject D
inst✝⁵ : HasShift D ℤ
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor D n).Additive
inst✝³ : Pretriangulated D
E : Type u_3
inst✝² : Category.{v_3, u_3} E
inst✝¹ : HasShift E ℤ
P Q R : ObjectProperty C
inst✝ : P.IsStableUnderShift ℤ
n : ℕ
H : (P.extensionProductIter n).IsStableUnderShift ℤ
⊢ (P.extensionProductIter (n + 1)).IsStableUnderShift ℤ

Tactic:
| succ n H =>

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

Goal state before:
pp: case succ
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
P : ObjectProperty C
n : ℕ
H : P.retractClosure.extensionProductIter n ≤ (P.extensionProductIter n).retractClosure
⊢ P.retractClosure.extensionProductIter (n + 1) ≤ (P.extensionProductIter (n + 1)).retractClosure

Tactic:
| succ n H =>

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

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasDerivedCategory C
n : ℤ
⊢ (shiftFunctor (DerivedCategory C) n).Additive

Tactic:
by
  rw [Localization.functor_additive_iff
    Qh (HomotopyCategory.subcategoryAcyclic C).trW]
  exact Functor.additive_of_iso (Qh.commShiftIso n)

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
T₁ T₃ : Triangle C
mem₁ : T₁ ∈ distinguishedTriangles
mem₃ : T₃ ∈ distinguishedTriangles
a : T₁.obj₁ ⟶ T₃.obj₁
b : T₁.obj₂ ⟶ T₃.obj₂
Z₅ : C
g₅ : T₃.obj₁ ⟶ Z₅
h₅ : Z₅ ⟶ (shiftFunctor C 1).obj T₁.obj₁
mem₅ : Triangle.mk a g₅ h₅ ∈ distinguishedTriangles
mem₅' : P Z₅
Z₄ : C
g₄ : T₃.obj₂ ⟶ Z₄
h₄ : Z₄ ⟶ (shiftFunctor C 1).obj T₁.obj₂
mem₄ : Triangle.mk b g₄ h₄ ∈ distinguishedTriangles
mem₄' : P Z₄
comm : T₁.mor₁ ≫ b = a ≫ T₃.mor₁
Z₂ : C
g₂ : T₃.obj₂ ⟶ Z₂
h₂ : Z₂ ⟶ (shiftFunctor C 1).obj T₁.obj₁
mem₂ : Triangle.mk (T₁.mor₁ ≫ b) g₂ h₂ ∈ distinguishedTriangles
⊢ ∃ c, ∃ (_ : P.trW c), T₁.mor₂ ≫ c = b ≫ T₃.mor₂ ∧ T₁.mor₃ ≫ (shiftFunctor C 1).map a = c ≫ T₃.mor₃

Tactic:
have H := someOctahedron rfl mem₁ mem₄ mem₂

Goal state after:
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
T₁ T₃ : Triangle C
mem₁ : T₁ ∈ distinguishedTriangles
mem₃ : T₃ ∈ distinguishedTriangles
a : T₁.obj₁ ⟶ T₃.obj₁
b : T₁.obj₂ ⟶ T₃.obj₂
Z₅ : C
g₅ : T₃.obj₁ ⟶ Z₅
h₅ : Z₅ ⟶ (shiftFunctor C 1).obj T₁.obj₁
mem₅ : Triangle.mk a g₅ h₅ ∈ distinguishedTriangles
mem₅' : P Z₅
Z₄ : C
g₄ : T₃.obj₂ ⟶ Z₄
h₄ : Z₄ ⟶ (shiftFunctor C 1).obj T₁.obj₂
mem₄ : Triangle.mk b g₄ h₄ ∈ distinguishedTriangles
mem₄' : P Z₄
comm : T₁.mor₁ ≫ b = a ≫ T₃.mor₁
Z₂ : C
g₂ : T₃.obj₂ ⟶ Z₂
h₂ : Z₂ ⟶ (shiftFunctor C 1).obj T₁.obj₁
mem₂ : Triangle.mk (T₁.mor₁ ≫ b) g₂ h₂ ∈ distinguishedTriangles
H : Octahedron ⋯ mem₁ mem₄ mem₂
⊢ ∃ c, ∃ (_ : P.trW c), T₁.mor₂ ≫ c = b ≫ T₃.mor₂ ∧ T₁.mor₃ ≫ (shiftFunctor C 1).map a = c ≫ T₃.mor₃
Module: Mathlib.Algebra.Homology.Embedding.Extend

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 : HasZeroObject C
inst✝ : HasZeroMorphisms C
K L M : HomologicalComplex C c
φ : K ⟶ L
φ' : L ⟶ M
e : c.Embedding c'
i' j' : ι'
x✝ : c'.Rel i' j'
i : ι
hi : e.f i = i'
hj : ∃ j, e.f j = j'
⊢ extend.mapX φ (e.r i') ≫ (L.extend e).d i' j' = (K.extend e).d i' j' ≫ extend.mapX φ (e.r j')

Tactic:
obtain ⟨j, hj⟩ := hj
        rw [K.extend_d_eq e hi hj, L.extend_d_eq e hi hj,
          extend.mapX_some φ (e.r_eq_some hi),
          extend.mapX_some φ (e.r_eq_some hj)]
        simp only [extendXIso, assoc, Iso.inv_hom_id_assoc, Hom.comm_assoc]

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

Goal state before:
pp: case neg
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝² : Category.{v_1, u_3} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasZeroObject C
K : HomologicalComplex C c
e : c.Embedding c'
i j : ι
i' j' : ι'
hj' : e.f j = j'
hi : c.prev j = i
hi' : c'.prev j' = i'
W : C
φ : K.X j ⟶ W
hij : ¬c.Rel i j
⊢ K.d i j ≫ φ = 0 ↔ (K.extend e).d i' j' ≫ (K.extendXIso e hj').hom ≫ φ = 0

Tactic:
simp only [K.shape _ _ hij, zero_comp, true_iff]
    rw [K.extend_d_to_eq_zero e i' j' j hj', zero_comp]
    rw [hi]
    exact hij

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

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 : 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'
h : ∃ j, e.f j = j'
⊢ (K.extend e).HasHomology j'

Tactic:
obtain ⟨j, rfl⟩ := h

Goal state after:
case pos
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝³ : Category.{v_1, u_3} C
inst✝² : 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 (e.f j)
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
f : (singleFunctor C n).obj A ⟶ (singleFunctor C n).obj B
⊢ ∃ a, (singleFunctor C n).map a = f

Tactic:
change Q.obj ((CochainComplex.singleFunctor C n).obj A) ⟶
      Q.obj ((CochainComplex.singleFunctor C n).obj B) at f

Goal state after:
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasDerivedCategory C
n : ℤ
A B : C
f : Q.obj ((CochainComplex.singleFunctor C n).obj A) ⟶ Q.obj ((CochainComplex.singleFunctor C n).obj B)
⊢ ∃ a, (singleFunctor C n).map a = f
Module: Mathlib.Algebra.Homology.Embedding.CochainComplex

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
K : CochainComplex C ℤ
n : ℤ
⊢ (K.shortComplexTruncLE n).ShortExact

Tactic:
by
  apply HomologicalComplex.shortComplexTruncLE_shortExact

Goal state after:
Module: Mathlib.CategoryTheory.Localization.SmallShiftedHom

Goal state before:
pp: C : Type u₁
inst✝¹³ : Category.{v₁, u₁} C
D : Type u₂
inst✝¹² : Category.{v₂, u₂} D
W : MorphismProperty C
M : Type w'
inst✝¹¹ : AddMonoid M
inst✝¹⁰ : HasShift C M
inst✝⁹ : HasShift D M
inst✝⁸ : W.IsCompatibleWithShift M
X Y Z T : C
a₁ a₂ a₃ a₁₂ a₂₃ a : M
inst✝⁷ : HasSmallLocalizedShiftedHom W M X Y
inst✝⁶ : HasSmallLocalizedShiftedHom W M X Z
inst✝⁵ : HasSmallLocalizedShiftedHom W M X T
inst✝⁴ : HasSmallLocalizedShiftedHom W M Y Z
inst✝³ : HasSmallLocalizedShiftedHom W M Y T
inst✝² : HasSmallLocalizedShiftedHom W M Z T
inst✝¹ : HasSmallLocalizedShiftedHom W M Z Z
inst✝ : HasSmallLocalizedShiftedHom W M T T
α : SmallShiftedHom W X Y a₁
β : SmallShiftedHom W Y Z a₂
γ : SmallShiftedHom W Z T a₃
h₁₂ : a₂ + a₁ = a₁₂
h₂₃ : a₃ + a₂ = a₂₃
h : a₃ + a₂ + a₁ = a
⊢ a₃ + a₁₂ = a

Tactic:
← h₁₂,

Goal state after:
C : Type u₁
inst✝¹³ : Category.{v₁, u₁} C
D : Type u₂
inst✝¹² : Category.{v₂, u₂} D
W : MorphismProperty C
M : Type w'
inst✝¹¹ : AddMonoid M
inst✝¹⁰ : HasShift C M
inst✝⁹ : HasShift D M
inst✝⁸ : W.IsCompatibleWithShift M
X Y Z T : C
a₁ a₂ a₃ a₁₂ a₂₃ a : M
inst✝⁷ : HasSmallLocalizedShiftedHom W M X Y
inst✝⁶ : HasSmallLocalizedShiftedHom W M X Z
inst✝⁵ : HasSmallLocalizedShiftedHom W M X T
inst✝⁴ : HasSmallLocalizedShiftedHom W M Y Z
inst✝³ : HasSmallLocalizedShiftedHom W M Y T
inst✝² : HasSmallLocalizedShiftedHom W M Z T
inst✝¹ : HasSmallLocalizedShiftedHom W M Z Z
inst✝ : HasSmallLocalizedShiftedHom W M T T
α : SmallShiftedHom W X Y a₁
β : SmallShiftedHom W Y Z a₂
γ : SmallShiftedHom W Z T a₃
h₁₂ : a₂ + a₁ = a₁₂
h₂₃ : a₃ + a₂ = a₂₃
h : a₃ + a₂ + a₁ = a
⊢ a₃ + (a₂ + a₁) = a
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 : shiftFunctor D a ⋙ G ≅ G ⋙ shiftFunctor C a
h : CompatibilityCounit adj e₁ e₂
X : C
this :
  F.map (e₂.inv.app (F.obj X)) ≫ adj.counit.app ((shiftFunctor D a).obj (F.obj X)) =
    e₁.hom.app (G.obj (F.obj X)) ≫ (shiftFunctor D a).map (adj.counit.app (F.obj X))
⊢ e₁.hom.app X =
    F.map ((shiftFunctor C a).map (adj.unit.app X)) ≫
      F.map (e₂.inv.app (F.obj X)) ≫ adj.counit.app ((shiftFunctor D a).obj (F.obj X))

Tactic:
dsimp only [Functor.comp_obj, Functor.id_obj]

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 : shiftFunctor D a ⋙ G ≅ G ⋙ shiftFunctor C a
h : CompatibilityCounit adj e₁ e₂
X : C
this :
  F.map (e₂.inv.app (F.obj X)) ≫ adj.counit.app ((shiftFunctor D a).obj (F.obj X)) =
    e₁.hom.app (G.obj (F.obj X)) ≫ (shiftFunctor D a).map (adj.counit.app (F.obj X))
⊢ e₁.hom.app X =
    F.map ((shiftFunctor C a).map (adj.unit.app X)) ≫
      F.map (e₂.inv.app (F.obj X)) ≫ adj.counit.app ((shiftFunctor D a).obj (F.obj X))
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 e₁' : shiftFunctor C a ⋙ F ≅ F ⋙ shiftFunctor D a
e₂ : shiftFunctor D a ⋙ G ≅ G ⋙ shiftFunctor C a
h : CompatibilityUnit adj e₁ e₂
h' : CompatibilityUnit adj e₁' e₂
x✝ : C
⊢ F.map ((shiftFunctor C a).map (adj.unit.app x✝)) ≫
      F.map (e₂.inv.app (F.obj x✝)) ≫ adj.counit.app ((shiftFunctor D a).obj (F.obj x✝)) =
    e₁'.hom.app x✝

Tactic:
compatibilityCounit_left adj e₁' e₂ (compatibilityCounit_of_compatibilityUnit adj _ _ 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 e₁' : shiftFunctor C a ⋙ F ≅ F ⋙ shiftFunctor D a
e₂ : shiftFunctor D a ⋙ G ≅ G ⋙ shiftFunctor C a
h : CompatibilityUnit adj e₁ e₂
h' : CompatibilityUnit adj e₁' e₂
x✝ : C
⊢ F.map ((shiftFunctor C a).map (adj.unit.app x✝)) ≫
      F.map (e₂.inv.app (F.obj x✝)) ≫ adj.counit.app ((shiftFunctor D a).obj (F.obj x✝)) =
    F.map ((shiftFunctor C a).map (adj.unit.app x✝)) ≫
      F.map (e₂.inv.app (F.obj x✝)) ≫ adj.counit.app ((shiftFunctor D a).obj (F.obj x✝))