mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.CategoryTheory.Localization.Construction

Goal state before:
pp: case refine_2.inr.refine_2
C : Type uC
inst✝¹ : Category.{uC', uC} C
W : MorphismProperty C
D : Type uD
inst✝ : Category.{uD', uD} D
G₁ G₂ : W.Localization ⥤ D
h : W.Q ⋙ G₁ = W.Q ⋙ G₂
X Y : C
w : { obj := Y }.obj ⟶ { obj := X }.obj
hw✝ : W w
hw : W.Q.map w = (wIso w hw✝).hom
hw' : G₁.map (wIso w hw✝).hom = eqToHom ⋯ ≫ G₂.map (wIso w hw✝).hom ≫ eqToHom ⋯
⊢ G₁.obj (W.Q.obj X) = G₂.obj (W.Q.obj X)

Tactic:
apply Functor.congr_obj h

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

Goal state before:
pp: case mpr
C : Type uC
inst✝¹ : Category.{uC', uC} C
W : MorphismProperty C
P : MorphismProperty W.Localization
inst✝ : P.IsStableUnderComposition
hP₁ : ∀ ⦃X Y : C⦄ (f : X ⟶ Y), P (W.Q.map f)
hP₂ : ∀ ⦃X Y : C⦄ (w : X ⟶ Y) (hw : W w), P (wInv w hw)
X Y : W.Localization
f : X ⟶ Y
a✝ : ⊤ f
G : Paths (LocQuiver W) ⥤ W.Localization := Quotient.functor (relations W)
this : G.Full
⊢ ∀ (X₁ X₂ : Paths (LocQuiver W)) (f : X₁ ⟶ X₂), P (G.map f)

Tactic:
intro X₁

Goal state after:
case mpr
C : Type uC
inst✝¹ : Category.{uC', uC} C
W : MorphismProperty C
P : MorphismProperty W.Localization
inst✝ : P.IsStableUnderComposition
hP₁ : ∀ ⦃X Y : C⦄ (f : X ⟶ Y), P (W.Q.map f)
hP₂ : ∀ ⦃X Y : C⦄ (w : X ⟶ Y) (hw : W w), P (wInv w hw)
X Y : W.Localization
f : X ⟶ Y
a✝ : ⊤ f
G : Paths (LocQuiver W) ⥤ W.Localization := Quotient.functor (relations W)
this : G.Full
X₁ : Paths (LocQuiver W)
⊢ ∀ (X₂ : Paths (LocQuiver W)) (f : X₁ ⟶ X₂), P (G.map f)
Module: Mathlib.CategoryTheory.Localization.Construction

Goal state before:
pp: case mpr
C : Type uC
inst✝¹ : Category.{uC', uC} C
W : MorphismProperty C
P : MorphismProperty W.Localization
inst✝ : P.IsStableUnderComposition
hP₁ : ∀ ⦃X Y : C⦄ (f : X ⟶ Y), P (W.Q.map f)
hP₂ : ∀ ⦃X Y : C⦄ (w : X ⟶ Y) (hw : W w), P (wInv w hw)
X Y : W.Localization
f : X ⟶ Y
a✝ : ⊤ f
G : Paths (LocQuiver W) ⥤ W.Localization := Quotient.functor (relations W)
this : G.Full
⊢ ∀ (X₁ X₂ : Paths (LocQuiver W)) (f : X₁ ⟶ X₂), P (G.map f)

Tactic:
intro X₁ X₂ p

Goal state after:
case mpr
C : Type uC
inst✝¹ : Category.{uC', uC} C
W : MorphismProperty C
P : MorphismProperty W.Localization
inst✝ : P.IsStableUnderComposition
hP₁ : ∀ ⦃X Y : C⦄ (f : X ⟶ Y), P (W.Q.map f)
hP₂ : ∀ ⦃X Y : C⦄ (w : X ⟶ Y) (hw : W w), P (wInv w hw)
X Y : W.Localization
f : X ⟶ Y
a✝ : ⊤ f
G : Paths (LocQuiver W) ⥤ W.Localization := Quotient.functor (relations W)
this : G.Full
X₁ X₂ : Paths (LocQuiver W)
p : X₁ ⟶ X₂
⊢ P (G.map p)
Module: Mathlib.CategoryTheory.Localization.Predicate

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
L : C ⥤ D
W : MorphismProperty C
hW : W ≤ MorphismProperty.isomorphisms C
inst✝ : L.IsEquivalence
⊢ L.IsLocalization W

Tactic:
have : (𝟭 C).IsLocalization W := for_id W hW
  exact of_equivalence_target (𝟭 C) W L L.asEquivalence L.leftUnitor

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

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝¹² : Category.{v_1, u_1} C
inst✝¹¹ : Category.{v_2, u_2} D
L : C ⥤ D
inst✝¹⁰ : HasShift C ℤ
inst✝⁹ : Preadditive C
inst✝⁸ : HasZeroObject C
inst✝⁷ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁶ : Pretriangulated C
inst✝⁵ : HasShift D ℤ
inst✝⁴ : L.CommShift ℤ
inst✝³ : L.EssSurj
inst✝² : HasZeroObject D
inst✝¹ : HasZeroMorphisms D
inst✝ : L.PreservesZeroMorphisms
X : D
⊢ contractibleTriangle X ∈ L.essImageDistTriang

Tactic:
by
  refine ⟨contractibleTriangle (L.objPreimage X), ?_, contractible_distinguished _⟩
  exact ((contractibleTriangleFunctor D).mapIso (L.objObjPreimageIso X)).symm ≪≫
    Triangle.isoMk _ _ (Iso.refl _) (Iso.refl _) L.mapZeroObject.symm (by simp) (by simp) (by simp)

Goal state after:
Module: Mathlib.Order.TransfiniteIteration

Goal state before:
pp: I : Type u
inst✝⁴ : CompleteLattice I
φ : I → I
i₀ : I
J : Type w
inst✝³ : LinearOrder J
inst✝² : OrderBot J
inst✝¹ : SuccOrder J
inst✝ : WellFoundedLT J
hφ : ∀ (i : I), i ≤ φ i
⊢ Monotone fun j ↦ transfiniteIterate φ j i₀

Tactic:
by
  intro k j hkj
  induction j using SuccOrder.limitRecOn with
  | isMin k hk =>
    obtain rfl := hk.eq_bot
    obtain rfl : k = ⊥ := by simpa using hkj
    rfl
  | succ k' hk' hkk' =>
    obtain hkj | rfl := hkj.lt_or_eq
    · refine (hkk' ((Order.lt_succ_iff_of_not_isMax hk').mp hkj)).trans ?_
      dsimp
      rw [transfiniteIterate_succ _ _ _ hk']
      apply hφ
    · rfl
  | isSuccLimit k' hk' _ =>
    obtain hkj | rfl := hkj.lt_or_eq
    · dsimp
      rw [transfiniteIterate_limit _ _ _ hk']
      exact le_iSup (fun (⟨l, hl⟩ : Set.Iio k') ↦ transfiniteIterate φ l i₀) ⟨k, hkj⟩
    · rfl

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

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁸ : Category.{v_1, u_1} C
inst✝⁷ : Category.{v_2, u_2} D
L : C ⥤ D
inst✝⁶ : HasShift C ℤ
inst✝⁵ : Preadditive C
inst✝⁴ : HasZeroObject C
inst✝³ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝² : Pretriangulated C
inst✝¹ : HasShift D ℤ
inst✝ : L.CommShift ℤ
H :
  ∀ (T₁' T₂' : Triangle C),
    T₁' ∈ distinguishedTriangles →
      T₂' ∈ distinguishedTriangles →
        ∀ (a : L.obj T₁'.obj₁ ⟶ L.obj T₂'.obj₁) (b : L.obj T₁'.obj₂ ⟶ L.obj T₂'.obj₂),
          L.map T₁'.mor₁ ≫ b = a ≫ L.map T₂'.mor₁ → ∃ φ, φ.hom₁ = a ∧ φ.hom₂ = b
T₁ T₂ : Triangle D
a : T₁.obj₁ ⟶ T₂.obj₁
b : T₁.obj₂ ⟶ T₂.obj₂
fac : T₁.mor₁ ≫ b = a ≫ T₂.mor₁
T₁' : Triangle C
e₁ : T₁ ≅ L.mapTriangle.obj T₁'
hT₁' : T₁' ∈ distinguishedTriangles
T₂' : Triangle C
e₂ : T₂ ≅ L.mapTriangle.obj T₂'
hT₂' : T₂' ∈ distinguishedTriangles
comm₁ : L.map T₁'.mor₁ ≫ e₁.inv.hom₂ = e₁.inv.hom₁ ≫ T₁.mor₁
comm₁' : T₂.mor₁ ≫ e₂.hom.hom₂ = e₂.hom.hom₁ ≫ L.map T₂'.mor₁
comm₂ : T₁.mor₂ ≫ e₁.hom.hom₃ = e₁.hom.hom₂ ≫ L.map T₁'.mor₂
comm₂' : T₂.mor₂ ≫ e₂.hom.hom₃ = e₂.hom.hom₂ ≫ L.map T₂'.mor₂
comm₃ :
  (L.map T₁'.mor₃ ≫ (commShiftIso L 1).hom.app T₁'.obj₁) ≫ (shiftFunctor D 1).map e₁.inv.hom₁ = e₁.inv.hom₃ ≫ T₁.mor₃
comm₃' :
  T₂.mor₃ ≫ (shiftFunctor D 1).map e₂.hom.hom₁ = e₂.hom.hom₃ ≫ L.map T₂'.mor₃ ≫ (commShiftIso L 1).hom.app T₂'.obj₁
⊢ ∃ c, T₁.mor₂ ≫ c = b ≫ T₂.mor₂ ∧ T₁.mor₃ ≫ (shiftFunctor D 1).map a = c ≫ T₂.mor₃

Tactic:
simp only [assoc] at comm₃

Goal state after:
C : Type u_1
D : Type u_2
inst✝⁸ : Category.{v_1, u_1} C
inst✝⁷ : Category.{v_2, u_2} D
L : C ⥤ D
inst✝⁶ : HasShift C ℤ
inst✝⁵ : Preadditive C
inst✝⁴ : HasZeroObject C
inst✝³ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝² : Pretriangulated C
inst✝¹ : HasShift D ℤ
inst✝ : L.CommShift ℤ
H :
  ∀ (T₁' T₂' : Triangle C),
    T₁' ∈ distinguishedTriangles →
      T₂' ∈ distinguishedTriangles →
        ∀ (a : L.obj T₁'.obj₁ ⟶ L.obj T₂'.obj₁) (b : L.obj T₁'.obj₂ ⟶ L.obj T₂'.obj₂),
          L.map T₁'.mor₁ ≫ b = a ≫ L.map T₂'.mor₁ → ∃ φ, φ.hom₁ = a ∧ φ.hom₂ = b
T₁ T₂ : Triangle D
a : T₁.obj₁ ⟶ T₂.obj₁
b : T₁.obj₂ ⟶ T₂.obj₂
fac : T₁.mor₁ ≫ b = a ≫ T₂.mor₁
T₁' : Triangle C
e₁ : T₁ ≅ L.mapTriangle.obj T₁'
hT₁' : T₁' ∈ distinguishedTriangles
T₂' : Triangle C
e₂ : T₂ ≅ L.mapTriangle.obj T₂'
hT₂' : T₂' ∈ distinguishedTriangles
comm₁ : L.map T₁'.mor₁ ≫ e₁.inv.hom₂ = e₁.inv.hom₁ ≫ T₁.mor₁
comm₁' : T₂.mor₁ ≫ e₂.hom.hom₂ = e₂.hom.hom₁ ≫ L.map T₂'.mor₁
comm₂ : T₁.mor₂ ≫ e₁.hom.hom₃ = e₁.hom.hom₂ ≫ L.map T₁'.mor₂
comm₂' : T₂.mor₂ ≫ e₂.hom.hom₃ = e₂.hom.hom₂ ≫ L.map T₂'.mor₂
comm₃' :
  T₂.mor₃ ≫ (shiftFunctor D 1).map e₂.hom.hom₁ = e₂.hom.hom₃ ≫ L.map T₂'.mor₃ ≫ (commShiftIso L 1).hom.app T₂'.obj₁
comm₃ :
  L.map T₁'.mor₃ ≫ (commShiftIso L 1).hom.app T₁'.obj₁ ≫ (shiftFunctor D 1).map e₁.inv.hom₁ = e₁.inv.hom₃ ≫ T₁.mor₃
⊢ ∃ c, T₁.mor₂ ≫ c = b ≫ T₂.mor₂ ∧ T₁.mor₃ ≫ (shiftFunctor D 1).map a = c ≫ T₂.mor₃
Module: Mathlib.CategoryTheory.ObjectProperty.LimitsClosure

Goal state before:
pp: case isSuccLimit
C : Type u
inst✝⁵ : Category.{v, u} C
P : ObjectProperty C
α : Type t
J : α → Type u'
inst✝⁴ : (a : α) → Category.{v', u'} (J a)
β : Type w'
inst✝³ : LinearOrder β
inst✝² : OrderBot β
inst✝¹ : SuccOrder β
inst✝ : WellFoundedLT β
b : β
hb : Order.IsSuccLimit b
hb' : ∀ b_1 < b, P.strictLimitsClosureIter J b_1 ≤ P.limitsClosure J
⊢ P.strictLimitsClosureIter J b ≤ P.limitsClosure J

Tactic:
simp only [transfiniteIterate_limit _ _ _ hb,
      iSup_le_iff, Subtype.forall, Set.mem_Iio]

Goal state after:
case isSuccLimit
C : Type u
inst✝⁵ : Category.{v, u} C
P : ObjectProperty C
α : Type t
J : α → Type u'
inst✝⁴ : (a : α) → Category.{v', u'} (J a)
β : Type w'
inst✝³ : LinearOrder β
inst✝² : OrderBot β
inst✝¹ : SuccOrder β
inst✝ : WellFoundedLT β
b : β
hb : Order.IsSuccLimit b
hb' : ∀ b_1 < b, P.strictLimitsClosureIter J b_1 ≤ P.limitsClosure J
⊢ ∀ a < b, transfiniteIterate (fun Q ↦ Q.strictLimitsClosureStep J) a P ≤ P.limitsClosure J
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.colimitsClosure J ≤ (P.op.limitsClosure fun a ↦ (J a)ᵒᵖ).unop

pp: case refine_2
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.limitsClosure fun a ↦ (J a)ᵒᵖ).unop ≤ P.colimitsClosure J

Tactic:
· apply colimitsClosure_le
    rw [← op_monotone_iff, op_unop]
    apply le_limitsClosure

Goal state after:
case refine_2
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.limitsClosure fun a ↦ (J a)ᵒᵖ).unop ≤ P.colimitsClosure J
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), PreservesLimit (pair X₁ X₂) F

Tactic:
intro X₁

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 : C
⊢ ∀ (X₂ : C), PreservesLimit (pair X₁ X₂) F
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 : ℤ
⊢ (F.shift n₀).map T.mor₁ ≫ (F.shift n₀).map T.mor₂ = 0

Tactic:
rw [← Functor.map_comp, comp_distTriang_mor_zero₁₂ _ hT, Functor.map_zero]

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

Goal state before:
pp: case mpr
C : Type u_1
D : Type u_2
inst✝¹⁴ : Category.{v_1, u_1} C
inst✝¹³ : Category.{v_2, u_2} D
L : C ⥤ D
inst✝¹² : HasShift C ℤ
inst✝¹¹ : Preadditive C
inst✝¹⁰ : HasZeroObject C
inst✝⁹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁸ : Pretriangulated C
inst✝⁷ : HasShift D ℤ
inst✝⁶ : L.CommShift ℤ
inst✝⁵ : HasZeroObject D
inst✝⁴ : Preadditive D
inst✝³ : ∀ (n : ℤ), (shiftFunctor D n).Additive
inst✝² : Pretriangulated D
inst✝¹ : L.mapArrow.EssSurj
inst✝ : L.IsTriangulated
T : Triangle D
⊢ T ∈ L.essImageDistTriang → T ∈ distinguishedTriangles

Tactic:
· rintro ⟨T₀, e, hT₀⟩
    exact isomorphic_distinguished _ (L.map_distinguished _ hT₀) _ e

Goal state after:
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
P Q : ObjectProperty C
hPQ : P ≤ Q
n : ℕ
⊢ P.extensionProductIter n ≤ Q.extensionProductIter n

Tactic:
by
  induction n with
  | zero => exact hPQ
  | succ n h => grw [extensionProductIter_succ, extensionProductIter_succ,
    monotone_extensionProduct_left _ hPQ, monotone_extensionProduct_right _ h]

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

Goal state before:
pp: case refine_2
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
X' X Y : C
f₁ f₂ : X ⟶ Y
s : X' ⟶ X
Z : C
g : X ⟶ Z
h : Z ⟶ (shiftFunctor C 1).obj X'
H : Triangle.mk s g h ∈ distinguishedTriangles
mem : P Z
hf₁ : s ≫ f₁ = s ≫ f₂
hf₂ : s ≫ (f₁ - f₂) = 0
q : (Triangle.mk s g h).obj₃ ⟶ Y
hq : f₁ - f₂ = (Triangle.mk s g h).mor₂ ≫ q
Y' : C
r : Y ⟶ Y'
t : Y' ⟶ (shiftFunctor C 1).obj (Triangle.mk s g h).obj₃
mem' : Triangle.mk q r t ∈ distinguishedTriangles
⊢ f₁ ≫ r = f₂ ≫ r

Tactic:
· have eq := comp_distTriang_mor_zero₁₂ _ mem'
      dsimp at eq
      rw [← sub_eq_zero, ← sub_comp, hq, assoc, eq, comp_zero]

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
Y Z Z' : C
f₁ f₂ : Y ⟶ Z
s : Z ⟶ Z'
hs : P.trW s
hf₁ : f₁ ≫ s = f₂ ≫ s
⊢ ∃ X' t, ∃ (_ : P.trW t), t ≫ f₁ = t ≫ f₂

Tactic:
rw [P.trW_iff'] at hs

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
Y Z Z' : C
f₁ f₂ : Y ⟶ Z
s : Z ⟶ Z'
hs : ∃ X f h, ∃ (_ : Triangle.mk f s h ∈ distinguishedTriangles), P X
hf₁ : f₁ ≫ s = f₂ ≫ s
⊢ ∃ X' t, ∃ (_ : P.trW t), t ≫ f₁ = t ≫ f₂
Module: Mathlib.Algebra.Homology.Embedding.Extend

Goal state before:
pp: case inr.inr.a
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
hi : e.r (e.f i) = some i
hj : e.r (e.f j) = some j
h : ¬c'.Rel (e.f i) (e.f j)
hij : c.Rel i j
⊢ False

Tactic:
exact h (e.rel hij)

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

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
e : c.Embedding c'
inst✝ : e.IsRelIff
j k : ι
hjk : c.next j = k
hj : ¬e.BoundaryLE j
hjk' : ¬c.Rel j k
⊢ ¬c.Rel j (c.next j)

Tactic:
by simpa only [hjk] using hjk'

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✝³ : Category.{v_1, u_3} C
inst✝² : HasZeroMorphisms C
K : HomologicalComplex C c'
e : c.Embedding c'
inst✝¹ : e.IsTruncGE
inst✝ : ∀ (i' : ι'), K.HasHomology i'
i j k : ι
hij : c.Rel i j
hjk : c.Rel j k
⊢ (if hi : e.BoundaryGE i then (XIsoOpcycles K e hi).hom ≫ K.fromOpcycles (e.f i) (e.f j) ≫ (XIso K e ⋯).inv
      else (XIso K e hi).hom ≫ K.d (e.f i) (e.f j) ≫ (XIso K e ⋯).inv) ≫
      (XIso K e ⋯).hom ≫ K.d (e.f j) (e.f k) ≫ (XIso K e ⋯).inv =
    0

Tactic:
split_ifs

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
K : HomologicalComplex C c'
e : c.Embedding c'
inst✝¹ : e.IsTruncGE
inst✝ : ∀ (i' : ι'), K.HasHomology i'
i j k : ι
hij : c.Rel i j
hjk : c.Rel j k
h✝ : e.BoundaryGE i
⊢ ((XIsoOpcycles K e h✝).hom ≫ K.fromOpcycles (e.f i) (e.f j) ≫ (XIso K e ⋯).inv) ≫
      (XIso K e ⋯).hom ≫ K.d (e.f j) (e.f k) ≫ (XIso K e ⋯).inv =
    0

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 : HasZeroMorphisms C
K : HomologicalComplex C c'
e : c.Embedding c'
inst✝¹ : e.IsTruncGE
inst✝ : ∀ (i' : ι'), K.HasHomology i'
i j k : ι
hij : c.Rel i j
hjk : c.Rel j k
h✝ : ¬e.BoundaryGE i
⊢ ((XIso K e h✝).hom ≫ K.d (e.f i) (e.f j) ≫ (XIso K e ⋯).inv) ≫
      (XIso K e ⋯).hom ≫ K.d (e.f j) (e.f k) ≫ (XIso K e ⋯).inv =
    0
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✝⁴ : Category.{v_1, u_3} C
inst✝³ : HasZeroMorphisms C
inst✝² : HasZeroObject C
K L : HomologicalComplex C c
φ : K ⟶ L
e : c.Embedding c'
inst✝¹ : ∀ (j : ι), K.HasHomology j
inst✝ : ∀ (j : ι), L.HasHomology j
h : ∀ (i : ι), QuasiIsoAt (extendMap φ e) (e.f i)
j' : ι'
hj' : ¬∃ j, e.f j = j'
⊢ (L.extend e).ExactAt j'

Tactic:
exact extend_exactAt _ _ _ (by simpa using hj')

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

Goal state before:
pp: case neg.hK
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝⁴ : Category.{v_1, u_3} C
inst✝³ : HasZeroMorphisms C
inst✝² : HasZeroObject C
K L : HomologicalComplex C c
φ : K ⟶ L
e : c.Embedding c'
inst✝¹ : ∀ (j : ι), K.HasHomology j
inst✝ : ∀ (j : ι), L.HasHomology j
h : ∀ (i : ι), QuasiIsoAt (extendMap φ e) (e.f i)
j' : ι'
hj' : ¬∃ j, e.f j = j'
⊢ (K.extend e).ExactAt j'

Tactic:
exact extend_exactAt _ _ _ (by simpa using hj')

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

Goal state before:
pp: case right.inr
ι : Type u_1
ι₁ : Type u_2
ι₂ : Type u_3
c : ComplexShape ι
c₁ : ComplexShape ι₁
c₂ : ComplexShape ι₂
e₁ : c₁.Embedding c
e₂ : c₂.Embedding c
ac : e₁.AreComplementary e₂
i₂ : ι₂
⊢ ∃ a, fromSum e₁ e₂ a = e₂.f i₂

Tactic:
exact ⟨Sum.inr i₂, rfl⟩

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

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : HasZeroMorphisms C
K : CochainComplex C ℤ
p q : ℤ
hpq : p ≤ q
inst✝ : K.IsGE q
i : ℤ
hi : i < p
⊢ ExactAt K i

Tactic:
exact K.exactAt_of_isGE q _

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

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasDerivedCategory C
S₁ S₂ : ShortComplex (CochainComplex C ℤ)
hS₁ : S₁.ShortExact
hS₂ : S₂.ShortExact
f : S₁ ⟶ S₂
⊢ (CochainComplex.mappingCone.triangle S₁.f).mor₃ ≫ (shiftFunctor (CochainComplex C ℤ) 1).map f.τ₁ =
    CochainComplex.mappingCone.map S₁.f S₂.f f.τ₁ f.τ₂ ⋯ ≫ (CochainComplex.mappingCone.triangle S₂.f).mor₃

Tactic:
exact (CochainComplex.mappingCone.triangleMap S₁.f S₂.f f.τ₁ f.τ₂ f.comm₁₂.symm).comm₃

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

Goal state before:
pp: case e_a
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasDerivedCategory C
S : ShortComplex (CochainComplex C ℤ)
hS : S.ShortExact
S₁ S₂ : ShortComplex (CochainComplex C ℤ)
h₁ : S₁.ShortExact
h₂ : S₂.ShortExact
f : S₁ ⟶ S₂
⊢ (CochainComplex.mappingCone.triangle S₁.f).mor₃ ≫ (shiftFunctor (CochainComplex C ℤ) 1).map f.τ₁ =
    CochainComplex.mappingCone.map S₁.f S₂.f f.τ₁ f.τ₂ ⋯ ≫ (CochainComplex.mappingCone.triangle S₂.f).mor₃

Tactic:
exact (CochainComplex.mappingCone.triangleMap S₁.f S₂.f f.τ₁ f.τ₂ f.comm₁₂.symm).comm₃

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

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasExt C
X : C
n : ℕ
X✝ Y✝ Z✝ : C
f : X✝ ⟶ Y✝
f' : Y✝ ⟶ Z✝
⊢ AddCommGrpCat.ofHom ((Ext.mk₀ (f ≫ f')).postcomp X ⋯) =
    AddCommGrpCat.ofHom ((Ext.mk₀ f).postcomp X ⋯) ≫ AddCommGrpCat.ofHom ((Ext.mk₀ f').postcomp X ⋯)

Tactic:
by
    ext α
    dsimp [AddCommGrpCat.ofHom]
    rw [← Ext.mk₀_comp_mk₀]
    symm
    apply Ext.comp_assoc
    lia

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

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : Category.{v_2, u_2} D
F : C ⥤ D
G : D ⥤ C
adj : F ⊣ G
A : Type u_3
inst✝⁴ : AddMonoid A
inst✝³ : HasShift C A
inst✝² : HasShift D A
inst✝¹ : F.CommShift A
inst✝ : G.CommShift A
x✝ : NatTrans.CommShift adj.unit A
a : A
⊢ (Functor.commShiftIso (G ⋙ F) a).hom ≫ Functor.whiskerRight adj.counit (shiftFunctor D a) =
    (shiftFunctor D a).whiskerLeft adj.counit ≫ (Functor.commShiftIso (𝟭 D) a).hom

Tactic:
by
    ext
    simp only [Functor.comp_obj, Functor.id_obj, NatTrans.comp_app,
      Functor.commShiftIso_comp_hom_app, Functor.whiskerRight_app, assoc, Functor.whiskerLeft_app,
      Functor.commShiftIso_id_hom_app, comp_id]
    refine (compatibilityCounit_of_compatibilityUnit adj _ _ (fun X ↦ ?_) _).symm
    simpa [Functor.commShiftIso_comp_hom_app] using NatTrans.shift_app_comm adj.unit a X

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

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : Abelian C
P : C
inst✝² : Projective P
K : CochainComplex C ℤ
i : ℤ
π : K ⟶ (singleFunctor C i).obj P
inst✝¹ : K.IsStrictlyLE i
inst✝ : QuasiIsoAt π i
e : cycles K i ≅ K.X i := iCyclesIso K i (i + 1) ⋯ ⋯
α : K.X i ⟶ P := e.inv ≫ homologyπ K i ≫ homologyMap π i ≫ (singleObjHomologySelfIso (ComplexShape.up ℤ) i P).hom
this : π.f i = α ≫ (singleObjXSelf (ComplexShape.up ℤ) i P).inv
⊢ Projective.factorThru (𝟙 P) α ≫ K.d i (i + 1) = 0

Tactic:
apply (K.isZero_of_isStrictlyLE i (i + 1) (by simp)).eq_of_tgt

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

Goal state before:
pp: R : Type u_1
inst✝¹³ : Ring R
C : Type u_4
inst✝¹² : Category.{v_1, u_4} C
inst✝¹¹ : Preadditive C
inst✝¹⁰ : Linear R C
D : Type u_2
inst✝⁹ : Category.{v_2, u_2} D
inst✝⁸ : Preadditive D
inst✝⁷ : Linear R D
M : Type u_3
inst✝⁶ : AddMonoid M
inst✝⁵ : HasShift C M
inst✝⁴ : ∀ (n : M), Functor.Linear R (shiftFunctor C n)
L : C ⥤ D
W : MorphismProperty C
inst✝³ : L.IsLocalization W
inst✝² : Functor.Linear R L
inst✝¹ : HasShift D M
inst✝ : L.CommShift M
n : M
⊢ Functor.Linear R (shiftFunctor D n)

Tactic:
have : Localization.Lifting L W (shiftFunctor C n ⋙ L) (shiftFunctor D n) :=
    ⟨(L.commShiftIso n).symm⟩

Goal state after:
R : Type u_1
inst✝¹³ : Ring R
C : Type u_4
inst✝¹² : Category.{v_1, u_4} C
inst✝¹¹ : Preadditive C
inst✝¹⁰ : Linear R C
D : Type u_2
inst✝⁹ : Category.{v_2, u_2} D
inst✝⁸ : Preadditive D
inst✝⁷ : Linear R D
M : Type u_3
inst✝⁶ : AddMonoid M
inst✝⁵ : HasShift C M
inst✝⁴ : ∀ (n : M), Functor.Linear R (shiftFunctor C n)
L : C ⥤ D
W : MorphismProperty C
inst✝³ : L.IsLocalization W
inst✝² : Functor.Linear R L
inst✝¹ : HasShift D M
inst✝ : L.CommShift M
n : M
this : Localization.Lifting L W (shiftFunctor C n ⋙ L) (shiftFunctor D n)
⊢ Functor.Linear R (shiftFunctor D n)