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

Goal state before:
pp: 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 : C⦄ (s : X ⟶ Y), A X → B.trW s → ∃ Z s' b, A Z ∧ (A ⊓ B).trW s' ∧ s ≫ b = s'
X Y : C
f : X ⟶ Y
hX : B X
hY : A Y
Z : C
s : Y ⟶ Z
b : Z ⟶ (shiftFunctor C 1).obj X
hT : Triangle.mk f s b ∈ Pretriangulated.distinguishedTriangles
hs : B.trW s
W : C
s' : Y ⟶ W
g : Z ⟶ W
hW : A W
hs' : (A ⊓ B).trW s'
fac : s ≫ g = s'
U : C
hU : A U
a : U ⟶ Y
c : W ⟶ (shiftFunctor C 1).obj U
hT' : Triangle.mk a s' c ∈ Pretriangulated.distinguishedTriangles
t : (Triangle.mk f s b).obj₁ ⟶ (Triangle.mk a s' c).obj₁
ht : (Triangle.mk f s b).mor₁ ≫ 𝟙 Y = t ≫ (Triangle.mk a s' c).mor₁
ht' : (Triangle.mk f s b).mor₃ ≫ (shiftFunctor C 1).map t = g ≫ (Triangle.mk a s' c).mor₃
⊢ A ⊓ B ≤ B

Tactic:
by

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

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : Preadditive C
inst✝⁴ : HasZeroObject C
inst✝³ : HasShift C ℤ
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
t : TStructure C
inst✝ : IsTriangulated C
T : Triangle C
hT : T ∈ distinguishedTriangles
n₀ n₁ : ℤ
h : n₀ + 1 = n₁
h₃ : t.IsGE T.obj₃ n₁
⊢ IsIso ((t.truncLE n₀).map T.mor₁)

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝⁷ : Category.{v_1, u_1} C
inst✝⁶ : Preadditive C
inst✝⁵ : HasZeroObject C
inst✝⁴ : HasShift C ℤ
inst✝³ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝² : Pretriangulated C
t : TStructure C
inst✝¹ : IsTriangulated C
X : C
a b : ℤ
inst✝ : t.IsGE X a
⊢ t.IsGE ((t.truncLE b).obj X) a

Tactic:
by

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

Goal state before:
pp: 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 : C⦄ (s : X ⟶ Y), A X → B.trW s → ∃ Z s' b, A Z ∧ (A ⊓ B).trW s' ∧ s ≫ b = s'
X Y : C
f : X ⟶ Y
hX : B X
hY : A Y
Z : C
s : Y ⟶ Z
b : Z ⟶ (shiftFunctor C 1).obj X
hT : Triangle.mk f s b ∈ Pretriangulated.distinguishedTriangles
hs : B.trW s
W : C
s' : Y ⟶ W
g : Z ⟶ W
hW : A W
hs' : (A ⊓ B).trW s'
fac : s ≫ g = s'
U : C
hU : A U
a : U ⟶ Y
c : W ⟶ (shiftFunctor C 1).obj U
hT' : Triangle.mk a s' c ∈ Pretriangulated.distinguishedTriangles
t : (Triangle.mk f s b).obj₁ ⟶ (Triangle.mk a s' c).obj₁
ht : (Triangle.mk f s b).mor₁ ≫ 𝟙 Y = t ≫ (Triangle.mk a s' c).mor₁
ht' : (Triangle.mk f s b).mor₃ ≫ (shiftFunctor C 1).map t = g ≫ (Triangle.mk a s' c).mor₃
⊢ t ≫ a = f

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝⁷ : Category.{v_1, u_1} C
inst✝⁶ : Preadditive C
inst✝⁵ : HasZeroObject C
inst✝⁴ : HasShift C ℤ
inst✝³ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝² : Pretriangulated C
t : TStructure C
inst✝¹ : IsTriangulated C
X : C
a b : ℤ
inst✝ : t.IsLE X b
⊢ t.IsLE ((t.truncGT a).obj X) b

Tactic:
by

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

Goal state before:
pp: C₁ : Type u_1
C₂ : Type u_2
C₁₂ : Type u_3
C₂₃ : Type u_4
C₃ : Type u_5
C₄ : Type u_6
inst✝³³ : Category.{v_1, u_1} C₁
inst✝³² : Category.{v_2, u_2} C₂
inst✝³¹ : Category.{v_3, u_5} C₃
inst✝³⁰ : Category.{v_4, u_6} C₄
inst✝²⁹ : Category.{v_5, u_3} C₁₂
inst✝²⁸ : Category.{v_6, u_4} C₂₃
inst✝²⁷ : HasZeroMorphisms C₁
inst✝²⁶ : HasZeroMorphisms C₂
inst✝²⁵ : HasZeroMorphisms C₃
inst✝²⁴ : Preadditive C₁₂
inst✝²³ : Preadditive C₂₃
inst✝²² : Preadditive C₄
F₁₂ : C₁ ⥤ C₂ ⥤ C₁₂
G : C₁₂ ⥤ C₃ ⥤ C₄
F : C₁ ⥤ C₂₃ ⥤ C₄
G₂₃ : C₂ ⥤ C₃ ⥤ C₂₃
inst✝²¹ : F₁₂.PreservesZeroMorphisms
inst✝²⁰ : ∀ (X₁ : C₁), (F₁₂.obj X₁).PreservesZeroMorphisms
inst✝¹⁹ : G.Additive
inst✝¹⁸ : ∀ (X₁₂ : C₁₂), (G.obj X₁₂).PreservesZeroMorphisms
inst✝¹⁷ : G₂₃.PreservesZeroMorphisms
inst✝¹⁶ : ∀ (X₂ : C₂), (G₂₃.obj X₂).PreservesZeroMorphisms
inst✝¹⁵ : F.PreservesZeroMorphisms
inst✝¹⁴ : ∀ (X₁ : C₁), (F.obj X₁).Additive
associator : bifunctorComp₁₂ F₁₂ G ≅ bifunctorComp₂₃ F G₂₃
ι₁ : Type u_7
ι₂ : Type u_8
ι₃ : Type u_9
ι₁₂ : Type u_10
ι₂₃ : Type u_11
ι₄ : Type u_12
inst✝¹³ : DecidableEq ι₄
c₁ : ComplexShape ι₁
c₂ : ComplexShape ι₂
c₃ : ComplexShape ι₃
K₁ : HomologicalComplex C₁ c₁
K₂ : HomologicalComplex C₂ c₂
K₃ : HomologicalComplex C₃ c₃
c₁₂ : ComplexShape ι₁₂
c₂₃ : ComplexShape ι₂₃
c₄ : ComplexShape ι₄
inst✝¹² : TotalComplexShape c₁ c₂ c₁₂
inst✝¹¹ : TotalComplexShape c₁₂ c₃ c₄
inst✝¹⁰ : TotalComplexShape c₂ c₃ c₂₃
inst✝⁹ : TotalComplexShape c₁ c₂₃ c₄
inst✝⁸ : K₁.HasMapBifunctor K₂ F₁₂ c₁₂
inst✝⁷ : K₂.HasMapBifunctor K₃ G₂₃ c₂₃
inst✝⁶ : c₁.Associative c₂ c₃ c₁₂ c₂₃ c₄
inst✝⁵ : DecidableEq ι₁₂
inst✝⁴ : DecidableEq ι₂₃
inst✝³ : (K₁.mapBifunctor K₂ F₁₂ c₁₂).HasMapBifunctor K₃ G c₄
inst✝² : K₁.HasMapBifunctor (K₂.mapBifunctor K₃ G₂₃ c₂₃) F c₄
inst✝¹ : HasGoodTrifunctor₁₂Obj F₁₂ G K₁ K₂ K₃ c₁₂ c₄
inst✝ : HasGoodTrifunctor₂₃Obj F G₂₃ K₁ K₂ K₃ c₁₂ c₂₃ c₄
j j' : ι₄
⊢ (mapBifunctorAssociatorX associator K₁ K₂ K₃ c₁₂ c₂₃ c₄ j).hom ≫ mapBifunctor₂₃.D₂ F G₂₃ K₁ K₂ K₃ c₁₂ c₂₃ c₄ j j' =
    mapBifunctor₁₂.D₂ F₁₂ G K₁ K₂ K₃ c₁₂ c₄ j j' ≫ (mapBifunctorAssociatorX associator K₁ K₂ K₃ c₁₂ c₂₃ c₄ j').hom

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : Preadditive C
inst✝⁴ : HasZeroObject C
inst✝³ : HasShift C ℤ
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
t : TStructure C
inst✝ : IsTriangulated C
X : C
a b : ℤ
⊢ t.IsLE ((t.truncGELE a b).obj X) b

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : Preadditive C
inst✝⁴ : HasZeroObject C
inst✝³ : HasShift C ℤ
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
t : TStructure C
inst✝ : IsTriangulated C
a b : ℤ
h : a ≤ b
X : C
⊢ a + 1 ≤ b + 1

Tactic:
by

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

Goal state before:
pp: 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
⊢ A.IsVerdierRightLocalizing B ↔ ∀ ⦃X Y : C⦄ (s : X ⟶ Y), A X → B.trW s → ∃ Z s' b, A Z ∧ (A ⊓ B).trW s' ∧ s ≫ b = s'

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : Preadditive C
inst✝⁴ : HasZeroObject C
inst✝³ : HasShift C ℤ
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
t : TStructure C
inst✝ : IsTriangulated C
a b : ℤ
h : b ≤ a
X : C
⊢ b + 1 ≤ a + 1

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : Preadditive C
inst✝⁴ : HasZeroObject C
inst✝³ : HasShift C ℤ
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
t : TStructure C
inst✝ : IsTriangulated C
X : C
n : ℤ
⊢ n ≤ n

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : Preadditive C
inst✝⁴ : HasZeroObject C
inst✝³ : HasShift C ℤ
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
P : ObjectProperty C
t : TStructure C
inst✝ : P.IsTriangulated
h : P.HasInducedTStructure t
X Y : P.FullSubcategory
f : X ⟶ Y
hX : t.le 0 X.obj
hY : t.ge 1 Y.obj
⊢ P.ι.map f = P.ι.map 0

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : Preadditive C
inst✝⁴ : HasZeroObject C
inst✝³ : HasShift C ℤ
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
P : ObjectProperty C
t : TStructure C
inst✝ : P.IsTriangulated
h : P.HasInducedTStructure t
A : P.FullSubcategory
⊢ ∃ X Y, ∃ (_ : t.le 0 X.obj) (_ : t.ge 1 Y.obj), ∃ f g h, Triangle.mk f g h ∈ distinguishedTriangles

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : Preadditive C
inst✝⁴ : HasZeroObject C
inst✝³ : HasShift C ℤ
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
P : ObjectProperty C
inst✝ : P.IsTriangulated
t : TStructure C
h : ∀ (X : C), P X → ∀ (n : ℤ), P ((t.truncLE n).obj X) ∧ P ((t.truncGE n).obj X)
X : C
hX : P X
⊢ 0 + 1 = 1

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝⁸ : Category.{v_1, u_1} C
inst✝⁷ : Preadditive C
inst✝⁶ : HasZeroObject C
inst✝⁵ : HasShift C ℤ
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝³ : Pretriangulated C
P : ObjectProperty C
inst✝² : P.IsTriangulated
t : TStructure C
inst✝¹ : P.IsClosedUnderIsomorphisms
inst✝ : P.HasInducedTStructure t
T : Triangle C
hT : T ∈ distinguishedTriangles
n₀ n₁ : ℤ
h : n₀ + 1 = n₁
h₁ : t.IsLE T.obj₁ n₀
h₂ : P T.obj₂
h₃ : t.IsGE T.obj₃ n₁
⊢ t.IsLE (P.ι.mapTriangle.obj (((P.tStructure t).triangleLEGE n₀ n₁ h).obj { obj := T.obj₂, property := h₂ })).obj₁ n₀

Tactic:
by

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

Goal state before:
pp: 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
⊢ B.op.trW s.op

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝⁸ : Category.{v_1, u_1} C
inst✝⁷ : Preadditive C
inst✝⁶ : HasZeroObject C
inst✝⁵ : HasShift C ℤ
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝³ : Pretriangulated C
P : ObjectProperty C
inst✝² : P.IsTriangulated
t : TStructure C
inst✝¹ : P.IsClosedUnderIsomorphisms
inst✝ : P.HasInducedTStructure t
T : Triangle C
hT : T ∈ distinguishedTriangles
n₀ n₁ : ℤ
h : n₀ + 1 = n₁
h₁ : t.IsLE T.obj₁ n₀
h₂ : P T.obj₂
h₃ : t.IsGE T.obj₃ n₁
⊢ t.IsGE (P.ι.mapTriangle.obj (((P.tStructure t).triangleLEGE n₀ n₁ h).obj { obj := T.obj₂, property := h₂ })).obj₃ n₁

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝⁸ : Category.{v_1, u_1} C
inst✝⁷ : Preadditive C
inst✝⁶ : HasZeroObject C
inst✝⁵ : HasShift C ℤ
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝³ : Pretriangulated C
P : ObjectProperty C
inst✝² : P.IsTriangulated
t : TStructure C
inst✝¹ : P.IsClosedUnderIsomorphisms
inst✝ : P.HasInducedTStructure t
T : Triangle C
hT : T ∈ distinguishedTriangles
n₀ n₁ : ℤ
h : n₀ + 1 = n₁
h₁ : t.IsLE T.obj₁ n₀
h₂ : P T.obj₂
h₃ : t.IsGE T.obj₃ n₁
⊢ P T.obj₁ ∧ P T.obj₃

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝¹² : Category.{v_1, u_1} C
inst✝¹¹ : Preadditive C
inst✝¹⁰ : HasZeroObject C
inst✝⁹ : HasShift C ℤ
inst✝⁸ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁷ : Pretriangulated C
P✝ : ObjectProperty C
t✝ : TStructure C
inst✝⁶ : P✝.IsTriangulated
h : P✝.HasInducedTStructure t✝
P P' : ObjectProperty C
inst✝⁵ : P.IsTriangulated
inst✝⁴ : P'.IsTriangulated
t : TStructure C
inst✝³ : P.HasInducedTStructure t
inst✝² : P'.HasInducedTStructure t
inst✝¹ : P.IsClosedUnderIsomorphisms
inst✝ : P'.IsClosedUnderIsomorphisms
X : C
hX : P X
hX' : P' X
n : ℤ
⊢ t.IsLE ((t.triangleLEGE n (n + 1) ⋯).obj X).obj₁ n

Tactic:
by

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

Goal state before:
pp: 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'
⊢ b.unop ≫ s = s'.unop

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝¹² : Category.{v_1, u_1} C
inst✝¹¹ : Preadditive C
inst✝¹⁰ : HasZeroObject C
inst✝⁹ : HasShift C ℤ
inst✝⁸ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁷ : Pretriangulated C
P✝ : ObjectProperty C
t✝ : TStructure C
inst✝⁶ : P✝.IsTriangulated
h : P✝.HasInducedTStructure t✝
P P' : ObjectProperty C
inst✝⁵ : P.IsTriangulated
inst✝⁴ : P'.IsTriangulated
t : TStructure C
inst✝³ : P.HasInducedTStructure t
inst✝² : P'.HasInducedTStructure t
inst✝¹ : P.IsClosedUnderIsomorphisms
inst✝ : P'.IsClosedUnderIsomorphisms
X : C
hX : P X
hX' : P' X
n : ℤ
⊢ t.IsGE ((t.triangleLEGE n (n + 1) ⋯).obj X).obj₃ (n + 1)

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝¹² : Category.{v_1, u_1} C
inst✝¹¹ : Preadditive C
inst✝¹⁰ : HasZeroObject C
inst✝⁹ : HasShift C ℤ
inst✝⁸ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁷ : Pretriangulated C
P✝ : ObjectProperty C
t✝ : TStructure C
inst✝⁶ : P✝.IsTriangulated
h : P✝.HasInducedTStructure t✝
P P' : ObjectProperty C
inst✝⁵ : P.IsTriangulated
inst✝⁴ : P'.IsTriangulated
t : TStructure C
inst✝³ : P.HasInducedTStructure t
inst✝² : P'.HasInducedTStructure t
inst✝¹ : P.IsClosedUnderIsomorphisms
inst✝ : P'.IsClosedUnderIsomorphisms
X : C
hX : P X
hX' : P' X
n : ℤ
⊢ n - 1 + 1 = n

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝¹² : Category.{v_1, u_1} C
inst✝¹¹ : Preadditive C
inst✝¹⁰ : HasZeroObject C
inst✝⁹ : HasShift C ℤ
inst✝⁸ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁷ : Pretriangulated C
P✝ : ObjectProperty C
t✝ : TStructure C
inst✝⁶ : P✝.IsTriangulated
h : P✝.HasInducedTStructure t✝
P P' : ObjectProperty C
inst✝⁵ : P.IsTriangulated
inst✝⁴ : P'.IsTriangulated
t : TStructure C
inst✝³ : P.HasInducedTStructure t
inst✝² : P'.HasInducedTStructure t
inst✝¹ : P.IsClosedUnderIsomorphisms
inst✝ : P'.IsClosedUnderIsomorphisms
X : C
hX : P X
hX' : P' X
n : ℤ
⊢ t.IsLE ((t.triangleLEGE (n - 1) n ⋯).obj X).obj₁ (n - 1)

Tactic:
by

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

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

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝¹² : Category.{v_1, u_1} C
inst✝¹¹ : Preadditive C
inst✝¹⁰ : HasZeroObject C
inst✝⁹ : HasShift C ℤ
inst✝⁸ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁷ : Pretriangulated C
P✝ : ObjectProperty C
t✝ : TStructure C
inst✝⁶ : P✝.IsTriangulated
h : P✝.HasInducedTStructure t✝
P P' : ObjectProperty C
inst✝⁵ : P.IsTriangulated
inst✝⁴ : P'.IsTriangulated
t : TStructure C
inst✝³ : P.HasInducedTStructure t
inst✝² : P'.HasInducedTStructure t
inst✝¹ : P.IsClosedUnderIsomorphisms
inst✝ : P'.IsClosedUnderIsomorphisms
X : C
hX : P X
hX' : P' X
n : ℤ
⊢ t.IsGE ((t.triangleLEGE (n - 1) n ⋯).obj X).obj₃ n

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝¹² : Category.{v_1, u_1} C
inst✝¹¹ : Preadditive C
inst✝¹⁰ : HasZeroObject C
inst✝⁹ : HasShift C ℤ
inst✝⁸ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁷ : Pretriangulated C
P✝ : ObjectProperty C
t✝ : TStructure C
inst✝⁶ : P✝.IsTriangulated
h : P✝.HasInducedTStructure t✝
P P' : ObjectProperty C
inst✝⁵ : P.IsTriangulated
inst✝⁴ : P'.IsTriangulated
t : TStructure C
inst✝³ : P.HasInducedTStructure t
inst✝² : P'.HasInducedTStructure t
inst✝¹ : P.IsClosedUnderIsomorphisms
inst✝ : P'.IsClosedUnderIsomorphisms
⊢ ∀ (X : C), (P ⊓ P') X → ∀ (n : ℤ), (P ⊓ P') ((t.truncLE n).obj X) ∧ (P ⊓ P') ((t.truncGE n).obj X)

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : Preadditive C
inst✝⁴ : HasZeroObject C
inst✝³ : HasShift C ℤ
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
P : ObjectProperty C
t : TStructure C
inst✝ : IsTriangulated C
⊢ ∀ (X : C), t.plus X → ∀ (n : ℤ), t.plus ((t.truncLE n).obj X) ∧ t.plus ((t.truncGE n).obj X)

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : Preadditive C
inst✝⁴ : HasZeroObject C
inst✝³ : HasShift C ℤ
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
P : ObjectProperty C
t : TStructure C
inst✝ : IsTriangulated C
⊢ ∀ (X : C), t.minus X → ∀ (n : ℤ), t.minus ((t.truncLE n).obj X) ∧ t.minus ((t.truncGE n).obj X)

Tactic:
by

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

Goal state before:
pp: 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
⊢ A.IsVerdierLeftLocalizing B ↔ ∀ ⦃X Y : C⦄ (s : X ⟶ Y), A Y → B.trW s → ∃ Z s' a, A Z ∧ (A ⊓ B).trW s' ∧ a ≫ s = s'

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : Preadditive C
inst✝⁴ : HasZeroObject C
inst✝³ : HasShift C ℤ
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
P : ObjectProperty C
t : TStructure C
inst✝ : IsTriangulated C
⊢ t.bounded.HasInducedTStructure t

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
⊢ quasiIso C = (subcategoryAcyclic C).trW

Tactic:
by

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✝⁹ : Category.{v_1, u_1} C
inst✝⁸ : Category.{v_2, u_2} D
inst✝⁷ : Category.{v_3, u_3} D₁
inst✝⁶ : Category.{v_4, u_4} D₂
A B : ObjectProperty 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
inst✝ : A.IsTriangulated
X Y : A.FullSubcategory
f : X ⟶ Y
hf : (B.inverseImage A.ι).trW f
⊢ B.trW.inverseImage A.ι f

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
inst✝ : HasDerivedCategory C
⊢ ∀ (X : HomotopyCategory.Plus C), t.plus ((HomotopyCategory.Plus.ι C ⋙ DerivedCategory.Qh).obj X)

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
inst✝ : HasDerivedCategory C
⊢ Qh.CommShift ℤ

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
inst✝ : HasDerivedCategory C
⊢ Qh.IsTriangulated

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
inst✝ : HasDerivedCategory C
K L : HomotopyCategory.Plus C
x✝ : CochainComplex.IsKInjective L.obj.as
⊢ Function.Bijective Qh.map

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
inst✝ : HasDerivedCategory C
K : CochainComplex C ℤ
hK : HomologicalComplex.Acyclic K
L : CochainComplex C ℤ
n : ℤ
hn : L.IsStrictlyGE n
φ : K ⟶ L
⊢ (HomotopyCategory.quotient C (ComplexShape.up ℤ)).map (K.πTruncGE n) ≫
      (HomotopyCategory.quotient C (ComplexShape.up ℤ)).map (CochainComplex.truncGEMap φ n ≫ inv (L.πTruncGE n)) =
    (HomotopyCategory.quotient C (ComplexShape.up ℤ)).map φ

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
inst✝ : HasDerivedCategory C
K L : HomotopyCategory C (ComplexShape.up ℤ)
φ : K ⟶ L
hK : HomotopyCategory.subcategoryAcyclic C K
hL : HomotopyCategory.plus C L
⊢ ∃ Z a b, HomotopyCategory.plus C Z ∧ HomotopyCategory.subcategoryAcyclic C Z ∧ a ≫ b = φ

Tactic:
by

Goal state after: