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

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} 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
n : ℤ
X : C
h : t.IsGE X n
⊢ t.IsGE ((t.truncGE n).obj X) n

Tactic:
;

Goal state after:
C : Type u
inst✝⁵ : Category.{v, u} 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
n : ℤ
X : C
h : t.IsGE X n
⊢ t.IsGE ((t.truncGE n).obj X) n
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✝¹³ : Category.{v_1, u_1} C
inst✝¹² : Category.{v_2, u_2} D
inst✝¹¹ : Category.{v_3, u_3} D₁
inst✝¹⁰ : Category.{v_4, u_4} D₂
A B : ObjectProperty C
inst✝⁹ : HasZeroObject C
inst✝⁸ : HasShift C ℤ
inst✝⁷ : Preadditive C
inst✝⁶ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁵ : Pretriangulated C
inst✝⁴ : IsTriangulated C
inst✝³ : A.IsTriangulated
inst✝² : B.IsTriangulated
inst✝¹ : B.IsClosedUnderIsomorphisms
inst✝ : A.IsVerdierLeftLocalizing B
L₁ : A.FullSubcategory ⥤ (B.inverseImage A.ι).trW.Localization := (B.inverseImage A.ι).trW.Q
L₂ : C ⥤ B.trW.Localization := B.trW.Q
F : (B.inverseImage A.ι).trW.Localization ⥤ B.trW.Localization :=
  (A.triangulatedLocalizerMorphism B).localizedFunctor L₁ L₂
this : CatCommSq (A.op.triangulatedLocalizerMorphism B.op).functor (A.opEquivalence.functor ⋙ L₁.op) L₂.op F.op :=
  {
    iso :=
      A.opEquivalence.functor.isoWhiskerLeft
        (NatIso.op (CatCommSq.iso (A.triangulatedLocalizerMorphism B).functor L₁ L₂ F).symm) }
⊢ L₂.op.IsLocalization B.trW.op

Tactic:
;

Goal state after:
C : Type u_1
D : Type u_2
D₁ : Type u_3
D₂ : Type u_4
inst✝¹³ : Category.{v_1, u_1} C
inst✝¹² : Category.{v_2, u_2} D
inst✝¹¹ : Category.{v_3, u_3} D₁
inst✝¹⁰ : Category.{v_4, u_4} D₂
A B : ObjectProperty C
inst✝⁹ : HasZeroObject C
inst✝⁸ : HasShift C ℤ
inst✝⁷ : Preadditive C
inst✝⁶ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁵ : Pretriangulated C
inst✝⁴ : IsTriangulated C
inst✝³ : A.IsTriangulated
inst✝² : B.IsTriangulated
inst✝¹ : B.IsClosedUnderIsomorphisms
inst✝ : A.IsVerdierLeftLocalizing B
L₁ : A.FullSubcategory ⥤ (B.inverseImage A.ι).trW.Localization := (B.inverseImage A.ι).trW.Q
L₂ : C ⥤ B.trW.Localization := B.trW.Q
F : (B.inverseImage A.ι).trW.Localization ⥤ B.trW.Localization :=
  (A.triangulatedLocalizerMorphism B).localizedFunctor L₁ L₂
this : CatCommSq (A.op.triangulatedLocalizerMorphism B.op).functor (A.opEquivalence.functor ⋙ L₁.op) L₂.op F.op :=
  {
    iso :=
      A.opEquivalence.functor.isoWhiskerLeft
        (NatIso.op (CatCommSq.iso (A.triangulatedLocalizerMorphism B).functor L₁ L₂ F).symm) }
⊢ L₂.op.IsLocalization B.trW.op
Module: Mathlib.CategoryTheory.Abelian.EpiWithInjectiveKernel

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
X' Y' X Y : C
f' : X' ⟶ Y'
f : X ⟶ Y
r : RetractArrow f' f
left✝ : Epi f
hf : Injective (kernel f)
this : Epi f'
⊢ (kernel.map f' f (Arrow.Hom.left r.i) (Arrow.Hom.right r.i) ⋯ ≫
        kernel.map f f' (Arrow.Hom.left r.r) (Arrow.Hom.right r.r) ⋯) ≫
      equalizer.ι f' 0 =
    𝟙 (kernel f') ≫ equalizer.ι f' 0

Tactic:
;

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
X' Y' X Y : C
f' : X' ⟶ Y'
f : X ⟶ Y
r : RetractArrow f' f
left✝ : Epi f
hf : Injective (kernel f)
this : Epi f'
⊢ (kernel.map f' f (Arrow.Hom.left r.i) (Arrow.Hom.right r.i) ⋯ ≫
        kernel.map f f' (Arrow.Hom.left r.r) (Arrow.Hom.right r.r) ⋯) ≫
      equalizer.ι f' 0 =
    𝟙 (kernel f') ≫ equalizer.ι f' 0
Module: Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : Preadditive C
inst✝⁴ : HasZeroObject C
inst✝³ : HasShift C ℤ
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
t : TStructure C
n : ℤ
Y X : C
f₁ f₂ : Y ⟶ (t.truncLT n).obj X
h : f₁ ≫ (t.truncLTι n).app X = f₂ ≫ (t.truncLTι n).app X
inst✝ : t.IsLE Y (n - 1)
f : Y ⟶ (t.truncLT n).obj X
hf : f ≫ (t.truncLTι n).app X = 0
g : Y ⟶ ((t.triangleLTGE n).obj X).invRotate.obj₁
hg : f = g ≫ ((t.triangleLTGE n).obj X).invRotate.mor₁
⊢ t.IsGE ((shiftFunctor C (-1)).obj ((t.truncGE n).obj X)) (n + 1)

Tactic:
;

Goal state after:
C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : Preadditive C
inst✝⁴ : HasZeroObject C
inst✝³ : HasShift C ℤ
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
t : TStructure C
n : ℤ
Y X : C
f₁ f₂ : Y ⟶ (t.truncLT n).obj X
h : f₁ ≫ (t.truncLTι n).app X = f₂ ≫ (t.truncLTι n).app X
inst✝ : t.IsLE Y (n - 1)
f : Y ⟶ (t.truncLT n).obj X
hf : f ≫ (t.truncLTι n).app X = 0
g : Y ⟶ ((t.triangleLTGE n).obj X).invRotate.obj₁
hg : f = g ≫ ((t.triangleLTGE n).obj X).invRotate.mor₁
⊢ t.IsGE ((shiftFunctor C (-1)).obj ((t.truncGE n).obj X)) (n + 1)
Module: Mathlib.CategoryTheory.Category.Factorisation

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

Tactic:
;

Goal state after:
C : Type u
inst✝ : Category.{v, u} C
X Y : C
f✝ : X ⟶ Y
d : Factorisation f✝
f : Factorisation.initial ⟶ d
⊢ f.h = d.initialHom.h
Module: Mathlib.CategoryTheory.Category.Factorisation

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

Tactic:
;

Goal state after:
C : Type u
inst✝ : Category.{v, u} C
X Y : C
f✝ : X ⟶ Y
d : Factorisation f✝
f : d ⟶ Factorisation.terminal
⊢ f.h = d.terminalHom.h
Module: Mathlib.Algebra.Homology.Factorizations.CM5b

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
inst✝ : EnoughInjectives C
K L : CochainComplex C ℤ
⊢ degreewiseEpiWithInjectiveKernel (p K L)

Tactic:
intro n
  rw [epiWithInjectiveKernel_iff]
  refine ⟨(mappingCone (𝟙 (I K))).X n, inferInstance,
    (biprod.inl :_ ⟶ (mappingCone (𝟙 (I K))) ⊞ L).f n, ?_,
    (biprod.fst : (mappingCone (𝟙 (I K))) ⊞ L ⟶ _).f n,
    (biprod.inr :_ ⟶ (mappingCone (𝟙 (I K))) ⊞ L).f n, ?_, ?_, ?_⟩
  all_goals simp [← HomologicalComplex.comp_f, ← HomologicalComplex.add_f_apply]

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

Goal state before:
pp: case zero
C : Type u_1
inst✝ : Category.{v_1, u_1} C
X✝ : ℕ → C
f✝ : (n : ℕ) → X✝ n ⟶ X✝ (n + 1)
⊢ map f✝ 0 0 ⋯ = 𝟙 (X✝ 0)

Tactic:
;

Goal state after:
case zero
C : Type u_1
inst✝ : Category.{v_1, u_1} C
X✝ : ℕ → C
f✝ : (n : ℕ) → X✝ n ⟶ X✝ (n + 1)
⊢ map f✝ 0 0 ⋯ = 𝟙 (X✝ 0)
Module: Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : Preadditive C
inst✝⁴ : HasZeroObject C
inst✝³ : HasShift C ℤ
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
t : TStructure C
X Y : C
f : X ⟶ Y
n₀ n₁ : ℤ
h : n₀ + 1 = n₁
inst✝ : t.IsLE X n₀
⊢ t.IsGE ((t.truncGE n₁).obj Y) n₁

Tactic:
;

Goal state after:
C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : Preadditive C
inst✝⁴ : HasZeroObject C
inst✝³ : HasShift C ℤ
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
t : TStructure C
X Y : C
f : X ⟶ Y
n₀ n₁ : ℤ
h : n₀ + 1 = n₁
inst✝ : t.IsLE X n₀
⊢ t.IsGE ((t.truncGE n₁).obj Y) n₁
Module: Mathlib.CategoryTheory.Functor.OfSequence

Goal state before:
pp: case zero
C : Type u_1
inst✝ : Category.{v_1, u_1} C
X✝ : ℕ → C
f✝ : (n : ℕ) → X✝ n ⟶ X✝ (n + 1)
⊢ map f✝ 0 (0 + 1) ⋯ = f✝ 0

Tactic:
;

Goal state after:
case zero
C : Type u_1
inst✝ : Category.{v_1, u_1} C
X✝ : ℕ → C
f✝ : (n : ℕ) → X✝ n ⟶ X✝ (n + 1)
⊢ map f✝ 0 (0 + 1) ⋯ = f✝ 0
Module: Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : Preadditive C
inst✝⁴ : HasZeroObject C
inst✝³ : HasShift C ℤ
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
t : TStructure C
X Y : C
f : X ⟶ Y
n : ℤ
inst✝ : t.IsGE Y n
⊢ t.IsLE ((t.truncLT n).obj X) (n - 1)

Tactic:
;

Goal state after:
C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : Preadditive C
inst✝⁴ : HasZeroObject C
inst✝³ : HasShift C ℤ
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
t : TStructure C
X Y : C
f : X ⟶ Y
n : ℤ
inst✝ : t.IsGE Y n
⊢ t.IsLE ((t.truncLT n).obj X) (n - 1)
Module: Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} 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
n₀ n₁ : ℤ
h : n₀ + 1 = n₁
X : C
hX : ∀ (Y : C) (f : Y ⟶ X), t.IsLE Y n₀ → f = 0
⊢ t.IsLE ((t.truncLT (n₀ + 1)).obj X) n₀

Tactic:
;

Goal state after:
C : Type u
inst✝⁵ : Category.{v, u} 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
n₀ n₁ : ℤ
h : n₀ + 1 = n₁
X : C
hX : ∀ (Y : C) (f : Y ⟶ X), t.IsLE Y n₀ → f = 0
⊢ t.IsLE ((t.truncLT (n₀ + 1)).obj X) n₀
Module: Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
t : TStructure C
X Y : C
f : X ⟶ Y
n : ℤ
x✝ : ∃ Z g h, ∃ (_ : Triangle.mk ((t.truncLTι n).app X ≫ f) g h ∈ distinguishedTriangles), t.IsGE Z n
Z : C
g : Y ⟶ Z
h : Z ⟶ (shiftFunctor C 1).obj ((t.truncLT n).obj X)
mem : Triangle.mk ((t.truncLTι n).app X ≫ f) g h ∈ distinguishedTriangles
h✝ : t.IsGE Z n
⊢ t.IsLE ((t.truncLT n).obj X) (n - 1)

Tactic:
;

Goal state after:
C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
t : TStructure C
X Y : C
f : X ⟶ Y
n : ℤ
x✝ : ∃ Z g h, ∃ (_ : Triangle.mk ((t.truncLTι n).app X ≫ f) g h ∈ distinguishedTriangles), t.IsGE Z n
Z : C
g : Y ⟶ Z
h : Z ⟶ (shiftFunctor C 1).obj ((t.truncLT n).obj X)
mem : Triangle.mk ((t.truncLTι n).app X ≫ f) g h ∈ distinguishedTriangles
h✝ : t.IsGE Z n
⊢ t.IsLE ((t.truncLT n).obj X) (n - 1)
Module: Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
t : TStructure C
X Y : C
f : X ⟶ Y
n : ℤ
x✝ : ∃ Z g h, ∃ (_ : Triangle.mk ((t.truncLTι n).app X ≫ f) g h ∈ distinguishedTriangles), t.IsGE Z n
Z : C
g : Y ⟶ Z
h : Z ⟶ (shiftFunctor C 1).obj ((t.truncLT n).obj X)
mem : Triangle.mk ((t.truncLTι n).app X ≫ f) g h ∈ distinguishedTriangles
h✝ : t.IsGE Z n
⊢ t.IsGE Z n

Tactic:
;

Goal state after:
C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
t : TStructure C
X Y : C
f : X ⟶ Y
n : ℤ
x✝ : ∃ Z g h, ∃ (_ : Triangle.mk ((t.truncLTι n).app X ≫ f) g h ∈ distinguishedTriangles), t.IsGE Z n
Z : C
g : Y ⟶ Z
h : Z ⟶ (shiftFunctor C 1).obj ((t.truncLT n).obj X)
mem : Triangle.mk ((t.truncLTι n).app X ≫ f) g h ∈ distinguishedTriangles
h✝ : t.IsGE Z n
⊢ t.IsGE Z n
Module: Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
t : TStructure C
X Y : C
f : X ⟶ Y
n : ℤ
x✝ : ∃ Z g h, ∃ (_ : Triangle.mk ((t.truncLTι n).app X ≫ f) g h ∈ distinguishedTriangles), t.IsGE Z n
Z : C
g : Y ⟶ Z
h : Z ⟶ (shiftFunctor C 1).obj ((t.truncLT n).obj X)
mem : Triangle.mk ((t.truncLTι n).app X ≫ f) g h ∈ distinguishedTriangles
h✝ : t.IsGE Z n
⊢ t.IsLE ((t.truncLT n).obj Y) (n - 1)

Tactic:
;

Goal state after:
C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
t : TStructure C
X Y : C
f : X ⟶ Y
n : ℤ
x✝ : ∃ Z g h, ∃ (_ : Triangle.mk ((t.truncLTι n).app X ≫ f) g h ∈ distinguishedTriangles), t.IsGE Z n
Z : C
g : Y ⟶ Z
h : Z ⟶ (shiftFunctor C 1).obj ((t.truncLT n).obj X)
mem : Triangle.mk ((t.truncLTι n).app X ≫ f) g h ∈ distinguishedTriangles
h✝ : t.IsGE Z n
⊢ t.IsLE ((t.truncLT n).obj Y) (n - 1)
Module: Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
t : TStructure C
X Y : C
f : X ⟶ Y
n : ℤ
x✝ : ∃ Z g h, ∃ (_ : Triangle.mk ((t.truncLTι n).app X ≫ f) g h ∈ distinguishedTriangles), t.IsGE Z n
Z : C
g : Y ⟶ Z
h : Z ⟶ (shiftFunctor C 1).obj ((t.truncLT n).obj X)
mem : Triangle.mk ((t.truncLTι n).app X ≫ f) g h ∈ distinguishedTriangles
h✝ : t.IsGE Z n
⊢ t.IsGE ((t.truncGE n).obj Y) n

Tactic:
;

Goal state after:
C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
t : TStructure C
X Y : C
f : X ⟶ Y
n : ℤ
x✝ : ∃ Z g h, ∃ (_ : Triangle.mk ((t.truncLTι n).app X ≫ f) g h ∈ distinguishedTriangles), t.IsGE Z n
Z : C
g : Y ⟶ Z
h : Z ⟶ (shiftFunctor C 1).obj ((t.truncLT n).obj X)
mem : Triangle.mk ((t.truncLTι n).app X ≫ f) g h ∈ distinguishedTriangles
h✝ : t.IsGE Z n
⊢ t.IsGE ((t.truncGE n).obj Y) n
Module: Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
t : TStructure C
X Y : C
f : X ⟶ Y
n : ℤ
x✝ : ∃ Z g h, ∃ (_ : Triangle.mk ((t.truncLTι n).app X ≫ f) g h ∈ distinguishedTriangles), t.IsGE Z n
Z : C
g : Y ⟶ Z
h : Z ⟶ (shiftFunctor C 1).obj ((t.truncLT n).obj X)
mem : Triangle.mk ((t.truncLTι n).app X ≫ f) g h ∈ distinguishedTriangles
h✝ : t.IsGE Z n
e : Triangle.mk ((t.truncLTι n).app X ≫ f) g h ≅ (t.triangleLTGE n).obj Y
he : e.hom.hom₂ = (Iso.refl (Triangle.mk ((t.truncLTι n).app X ≫ f) g h).obj₂).hom
this : (t.truncLT n).map f = e.hom.hom₁
⊢ IsIso e.hom.hom₁

Tactic:
;

Goal state after:
C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
t : TStructure C
X Y : C
f : X ⟶ Y
n : ℤ
x✝ : ∃ Z g h, ∃ (_ : Triangle.mk ((t.truncLTι n).app X ≫ f) g h ∈ distinguishedTriangles), t.IsGE Z n
Z : C
g : Y ⟶ Z
h : Z ⟶ (shiftFunctor C 1).obj ((t.truncLT n).obj X)
mem : Triangle.mk ((t.truncLTι n).app X ≫ f) g h ∈ distinguishedTriangles
h✝ : t.IsGE Z n
e : Triangle.mk ((t.truncLTι n).app X ≫ f) g h ≅ (t.triangleLTGE n).obj Y
he : e.hom.hom₂ = (Iso.refl (Triangle.mk ((t.truncLTι n).app X ≫ f) g h).obj₂).hom
this : (t.truncLT n).map f = e.hom.hom₁
⊢ IsIso e.hom.hom₁
Module: Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} 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
Y Z : C
g : Y ⟶ Z
n₀ : ℤ
hf : IsIso ((t.truncGE (n₀ + 1)).map g)
⊢ t.IsLE ((t.truncLT (n₀ + 1)).obj Y) n₀

Tactic:
;

Goal state after:
C : Type u
inst✝⁵ : Category.{v, u} 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
Y Z : C
g : Y ⟶ Z
n₀ : ℤ
hf : IsIso ((t.truncGE (n₀ + 1)).map g)
⊢ t.IsLE ((t.truncLT (n₀ + 1)).obj Y) n₀
Module: Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} 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
Y Z : C
g : Y ⟶ Z
n₀ n₁ : ℤ
hn : n₀ + 1 = n₁
x✝ : ∃ X f h, ∃ (_ : Triangle.mk f (g ≫ (t.truncGEπ n₁).app Z) h ∈ distinguishedTriangles), t.IsLE X n₀
X : C
f : X ⟶ Y
h : (t.truncGE n₁).obj Z ⟶ (shiftFunctor C 1).obj X
mem : Triangle.mk f (g ≫ (t.truncGEπ n₁).app Z) h ∈ distinguishedTriangles
h✝ : t.IsLE X n₀
⊢ t.IsLE ((t.truncLT n₁).obj Y) n₀

Tactic:
;

Goal state after:
C : Type u
inst✝⁵ : Category.{v, u} 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
Y Z : C
g : Y ⟶ Z
n₀ n₁ : ℤ
hn : n₀ + 1 = n₁
x✝ : ∃ X f h, ∃ (_ : Triangle.mk f (g ≫ (t.truncGEπ n₁).app Z) h ∈ distinguishedTriangles), t.IsLE X n₀
X : C
f : X ⟶ Y
h : (t.truncGE n₁).obj Z ⟶ (shiftFunctor C 1).obj X
mem : Triangle.mk f (g ≫ (t.truncGEπ n₁).app Z) h ∈ distinguishedTriangles
h✝ : t.IsLE X n₀
⊢ t.IsLE ((t.truncLT n₁).obj Y) n₀
Module: Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} 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
Y Z : C
g : Y ⟶ Z
n₀ n₁ : ℤ
hn : n₀ + 1 = n₁
x✝ : ∃ X f h, ∃ (_ : Triangle.mk f (g ≫ (t.truncGEπ n₁).app Z) h ∈ distinguishedTriangles), t.IsLE X n₀
X : C
f : X ⟶ Y
h : (t.truncGE n₁).obj Z ⟶ (shiftFunctor C 1).obj X
mem : Triangle.mk f (g ≫ (t.truncGEπ n₁).app Z) h ∈ distinguishedTriangles
h✝ : t.IsLE X n₀
⊢ t.IsLE ((t.truncLT (n₀ + 1)).obj Y) n₀

Tactic:
;

Goal state after:
C : Type u
inst✝⁵ : Category.{v, u} 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
Y Z : C
g : Y ⟶ Z
n₀ n₁ : ℤ
hn : n₀ + 1 = n₁
x✝ : ∃ X f h, ∃ (_ : Triangle.mk f (g ≫ (t.truncGEπ n₁).app Z) h ∈ distinguishedTriangles), t.IsLE X n₀
X : C
f : X ⟶ Y
h : (t.truncGE n₁).obj Z ⟶ (shiftFunctor C 1).obj X
mem : Triangle.mk f (g ≫ (t.truncGEπ n₁).app Z) h ∈ distinguishedTriangles
h✝ : t.IsLE X n₀
⊢ t.IsLE ((t.truncLT (n₀ + 1)).obj Y) n₀
Module: Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} 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
Y Z : C
g : Y ⟶ Z
n₀ n₁ : ℤ
hn : n₀ + 1 = n₁
x✝ : ∃ X f h, ∃ (_ : Triangle.mk f (g ≫ (t.truncGEπ n₁).app Z) h ∈ distinguishedTriangles), t.IsLE X n₀
X : C
f : X ⟶ Y
h : (t.truncGE n₁).obj Z ⟶ (shiftFunctor C 1).obj X
mem : Triangle.mk f (g ≫ (t.truncGEπ n₁).app Z) h ∈ distinguishedTriangles
h✝ : t.IsLE X n₀
⊢ t.IsGE ((t.truncGE n₁).obj Y) n₁

Tactic:
;

Goal state after:
C : Type u
inst✝⁵ : Category.{v, u} 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
Y Z : C
g : Y ⟶ Z
n₀ n₁ : ℤ
hn : n₀ + 1 = n₁
x✝ : ∃ X f h, ∃ (_ : Triangle.mk f (g ≫ (t.truncGEπ n₁).app Z) h ∈ distinguishedTriangles), t.IsLE X n₀
X : C
f : X ⟶ Y
h : (t.truncGE n₁).obj Z ⟶ (shiftFunctor C 1).obj X
mem : Triangle.mk f (g ≫ (t.truncGEπ n₁).app Z) h ∈ distinguishedTriangles
h✝ : t.IsLE X n₀
⊢ t.IsGE ((t.truncGE n₁).obj Y) n₁
Module: Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} 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
Y Z : C
g : Y ⟶ Z
n₀ n₁ : ℤ
hn : n₀ + 1 = n₁
x✝ : ∃ X f h, ∃ (_ : Triangle.mk f (g ≫ (t.truncGEπ n₁).app Z) h ∈ distinguishedTriangles), t.IsLE X n₀
X : C
f : X ⟶ Y
h : (t.truncGE n₁).obj Z ⟶ (shiftFunctor C 1).obj X
mem : Triangle.mk f (g ≫ (t.truncGEπ n₁).app Z) h ∈ distinguishedTriangles
h✝ : t.IsLE X n₀
⊢ t.IsLE X n₀

Tactic:
;

Goal state after:
C : Type u
inst✝⁵ : Category.{v, u} 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
Y Z : C
g : Y ⟶ Z
n₀ n₁ : ℤ
hn : n₀ + 1 = n₁
x✝ : ∃ X f h, ∃ (_ : Triangle.mk f (g ≫ (t.truncGEπ n₁).app Z) h ∈ distinguishedTriangles), t.IsLE X n₀
X : C
f : X ⟶ Y
h : (t.truncGE n₁).obj Z ⟶ (shiftFunctor C 1).obj X
mem : Triangle.mk f (g ≫ (t.truncGEπ n₁).app Z) h ∈ distinguishedTriangles
h✝ : t.IsLE X n₀
⊢ t.IsLE X n₀
Module: Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} 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
Y Z : C
g : Y ⟶ Z
n₀ n₁ : ℤ
hn : n₀ + 1 = n₁
x✝ : ∃ X f h, ∃ (_ : Triangle.mk f (g ≫ (t.truncGEπ n₁).app Z) h ∈ distinguishedTriangles), t.IsLE X n₀
X : C
f : X ⟶ Y
h : (t.truncGE n₁).obj Z ⟶ (shiftFunctor C 1).obj X
mem : Triangle.mk f (g ≫ (t.truncGEπ n₁).app Z) h ∈ distinguishedTriangles
h✝ : t.IsLE X n₀
⊢ t.IsGE ((t.truncGE n₁).obj Z) n₁

Tactic:
;

Goal state after:
C : Type u
inst✝⁵ : Category.{v, u} 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
Y Z : C
g : Y ⟶ Z
n₀ n₁ : ℤ
hn : n₀ + 1 = n₁
x✝ : ∃ X f h, ∃ (_ : Triangle.mk f (g ≫ (t.truncGEπ n₁).app Z) h ∈ distinguishedTriangles), t.IsLE X n₀
X : C
f : X ⟶ Y
h : (t.truncGE n₁).obj Z ⟶ (shiftFunctor C 1).obj X
mem : Triangle.mk f (g ≫ (t.truncGEπ n₁).app Z) h ∈ distinguishedTriangles
h✝ : t.IsLE X n₀
⊢ t.IsGE ((t.truncGE n₁).obj Z) n₁
Module: Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} 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
Y Z : C
g : Y ⟶ Z
n₀ n₁ : ℤ
hn : n₀ + 1 = n₁
x✝ : ∃ X f h, ∃ (_ : Triangle.mk f (g ≫ (t.truncGEπ n₁).app Z) h ∈ distinguishedTriangles), t.IsLE X n₀
X : C
f : X ⟶ Y
h : (t.truncGE n₁).obj Z ⟶ (shiftFunctor C 1).obj X
mem : Triangle.mk f (g ≫ (t.truncGEπ n₁).app Z) h ∈ distinguishedTriangles
h✝ : t.IsLE X n₀
e : (t.triangleLTGE n₁).obj Y ≅ Triangle.mk f (g ≫ (t.truncGEπ n₁).app Z) h
he : e.hom.hom₂ = (Iso.refl ((t.triangleLTGE n₁).obj Y).obj₂).hom
this : (t.truncGE n₁).map g = e.hom.hom₃
⊢ IsIso e.hom.hom₃

Tactic:
;

Goal state after:
C : Type u
inst✝⁵ : Category.{v, u} 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
Y Z : C
g : Y ⟶ Z
n₀ n₁ : ℤ
hn : n₀ + 1 = n₁
x✝ : ∃ X f h, ∃ (_ : Triangle.mk f (g ≫ (t.truncGEπ n₁).app Z) h ∈ distinguishedTriangles), t.IsLE X n₀
X : C
f : X ⟶ Y
h : (t.truncGE n₁).obj Z ⟶ (shiftFunctor C 1).obj X
mem : Triangle.mk f (g ≫ (t.truncGEπ n₁).app Z) h ∈ distinguishedTriangles
h✝ : t.IsLE X n₀
e : (t.triangleLTGE n₁).obj Y ≅ Triangle.mk f (g ≫ (t.truncGEπ n₁).app Z) h
he : e.hom.hom₂ = (Iso.refl ((t.triangleLTGE n₁).obj Y).obj₂).hom
this : (t.truncGE n₁).map g = e.hom.hom₃
⊢ IsIso e.hom.hom₃
Module: Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} 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.truncGE a).obj ((t.truncLT b).obj X)) a

Tactic:
;

Goal state after:
C : Type u
inst✝⁵ : Category.{v, u} 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.truncGE a).obj ((t.truncLT b).obj X)) a