mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Algebra.Homology.Localization

Goal state before:
pp: C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
ι : Type u_2
c : ComplexShape ι
inst✝³ : Preadditive C
inst✝² : CategoryWithHomology C
inst✝¹ : (HomologicalComplex.quasiIso C c).HasLocalization
inst✝ : c.QFactorsThroughHomotopy C
K : HomologicalComplex C c
i : ι
⊢ (((Quotient.functor (homotopic C c)).associator Qh (homologyFunctor C c i)).symm ≪≫
            Functor.isoWhiskerRight (quotientCompQhIso C c) (homologyFunctor C c i) ≪≫
              homologyFunctorFactors C c i ≪≫ (HomotopyCategory.homologyFunctorFactors C c i).symm).hom.app
      K =
    (homologyFunctor C c i).map ((quotientCompQhIso C c).hom.app K) ≫
      (homologyFunctorFactors C c i).hom.app K ≫ (HomotopyCategory.homologyFunctorFactors C c i).inv.app K

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : CategoryWithHomology C
K : CochainComplex C ℤ
A : C
n i : ℤ
f : A ⟶ ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) n).obj K).X i
j : ℤ
hj : (up ℤ).next i = j
hf : f ≫ ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) n).obj K).d i j = 0
i' : ℤ
hi' : n + i = i'
j' : ℤ
hj' : (up ℤ).next i' = j'
⊢ i' = i + n

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
ι : Type u_2
c : ComplexShape ι
inst✝³ : Preadditive C
inst✝² : CategoryWithHomology C
inst✝¹ : (HomologicalComplex.quasiIso C c).HasLocalization
inst✝ : c.QFactorsThroughHomotopy C
K : HomologicalComplex C c
i : ι
⊢ (((Quotient.functor (homotopic C c)).associator Qh (homologyFunctor C c i)).symm ≪≫
            Functor.isoWhiskerRight (quotientCompQhIso C c) (homologyFunctor C c i) ≪≫
              homologyFunctorFactors C c i ≪≫ (HomotopyCategory.homologyFunctorFactors C c i).symm).inv.app
      K =
    (HomotopyCategory.homologyFunctorFactors C c i).hom.app K ≫
      (homologyFunctorFactors C c i).inv.app K ≫ (homologyFunctor C c i).map ((quotientCompQhIso C c).inv.app K)

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : CategoryWithHomology C
K : CochainComplex C ℤ
A : C
n i : ℤ
f : A ⟶ ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) n).obj K).X i
j : ℤ
hf : f ≫ ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) n).obj K).d i j = 0
i' : ℤ
hi' : n + i = i'
j' : ℤ
hj : i + 1 = j
hj' : i' + 1 = j'
⊢ i' = i + n

Tactic:
by

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

Goal state before:
pp: 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 : ObjectProperty C
inst✝ : P.IsStableUnderShift ℤ
X₁ X₂ : C
f : X₁ ⟶ X₂
hf : P.trW f
n : ℤ
⊢ P.trW ((shiftFunctor C n).map f)

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : CategoryWithHomology C
K : CochainComplex C ℤ
A : C
n i : ℤ
f : A ⟶ ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) n).obj K).X i
j : ℤ
hf : f ≫ ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) n).obj K).d i j = 0
j' : ℤ
hj : i + 1 = j
hi' : n + i = i + n
hj' : i + n + 1 = j'
⊢ j' = j + n

Tactic:
by

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

Goal state before:
pp: 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 : ObjectProperty C
inst✝ : P.IsStableUnderShift ℤ
n : ℤ
⊢ P.trW.inverseImage (shiftFunctor C n) = P.trW

Tactic:
by

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

Goal state before:
pp: ι : Type u_1
c : ComplexShape ι
hc : ∀ (j : ι), ∃ i, c.Rel i j
C : Type u_2
inst✝³ : Category.{v_1, u_2} C
inst✝² : Preadditive C
inst✝¹ : HasBinaryBiproducts C
E : Type u_3
inst✝ : Category.{v_2, u_3} E
F : HomologicalComplex C c ⥤ E
hF : (HomologicalComplex.homotopyEquivalences C c).IsInvertedBy F
K L : HomologicalComplex C c
f g : K ⟶ L
h : Homotopy f g
⊢ DecidableRel c.Rel

Tactic:
by

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

Goal state before:
pp: ι : Type u_1
c : ComplexShape ι
hc : ∀ (j : ι), ∃ i, c.Rel i j
C : Type u_2
inst✝³ : Category.{v_1, u_2} C
inst✝² : Preadditive C
inst✝¹ : HasBinaryBiproducts C
E : Type u_3
inst✝ : Category.{v_2, u_3} E
F : HomologicalComplex C c ⥤ E
hF : (HomologicalComplex.homotopyEquivalences C c).IsInvertedBy F
⊢ ∀ (x y : HomologicalComplex C c) (f₁ f₂ : x ⟶ y), homotopic C c f₁ f₂ → F.map f₁ = F.map f₂

Tactic:
by

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

Goal state before:
pp: ι : Type u_1
c : ComplexShape ι
hc : ∀ (j : ι), ∃ i, c.Rel i j
C : Type u_2
inst✝² : Category.{v_1, u_2} C
inst✝¹ : Preadditive C
inst✝ : HasBinaryBiproducts C
⊢ (HomotopyCategory.quotient C c).IsLocalization (HomologicalComplex.homotopyEquivalences C c)

Tactic:
by

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

Goal state before:
pp: ι : Type u_1
c : ComplexShape ι
hc : ∀ (j : ι), ∃ i, c.Rel i j
C : Type u_2
inst✝³ : Category.{v_1, u_2} C
inst✝² : Preadditive C
inst✝¹ : HasBinaryBiproducts C
inst✝ : CategoryWithHomology C
K L : HomologicalComplex C c
f g : K ⟶ L
h : Homotopy f g
⊢ AreEqualizedByLocalization (HomologicalComplex.quasiIso C c) f g

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
ι : Type u_2
inst✝¹ : Preadditive C
inst✝ : HasBinaryBiproducts C
n : ℤ
⊢ (ComplexShape.up ℤ).Rel (n - 1) n

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{v_1, u_1} C
ι : Type u_2
inst✝² : Preadditive C
inst✝¹ : HasBinaryBiproducts C
inst✝ : CategoryWithHomology C
n : ℤ
⊢ (ComplexShape.up ℤ).Rel (n - 1) n

Tactic:
by

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

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
F : C ⥤ D
ι : Type u_3
c : ComplexShape ι
inst✝⁶ : Preadditive C
inst✝⁵ : Preadditive D
inst✝⁴ : CategoryWithHomology C
inst✝³ : CategoryWithHomology D
inst✝² : (HomologicalComplex.quasiIso D c).HasLocalization
inst✝¹ : F.Additive
inst✝ : F.PreservesHomology
⊢ (HomologicalComplex.quasiIso C c).IsInvertedBy (F.mapHomologicalComplex c ⋙ HomologicalComplexUpToQuasiIso.Q)

Tactic:
by

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
⊢ ∀ {X Y Z : C} (f : X ⟶ Y) (g : Y ⟶ Z), P.trW f → P.trW g → P.trW (f ≫ g)

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : CategoryWithHomology C
K : CochainComplex C ℤ
A : C
n i : ℤ
f : A ⟶ ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) n).obj K).X i
j : ℤ
hj : (up ℤ).next i = j
hf : f ≫ ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) n).obj K).d i j = 0
i' : ℤ
hi' : n + i = i'
j' : ℤ
hj' : (up ℤ).next i' = j'
⊢ (f ≫ (K.shiftFunctorObjXIso n i i' ⋯).hom) ≫ K.d i' j' = 0

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : CategoryWithHomology C
K : CochainComplex C ℤ
A : C
n i : ℤ
f : A ⟶ ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) n).obj K).X i
j : ℤ
hj : (up ℤ).next i = j
hf : f ≫ ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) n).obj K).d i j = 0
i' : ℤ
hi' : n + i = i'
j' : ℤ
hj' : (up ℤ).next i' = j'
⊢ liftCycles ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) n).obj K) f j hj hf ≫
      homologyπ ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) n).obj K) i =
    liftCycles K (f ≫ (K.shiftFunctorObjXIso n i i' ⋯).hom) j' hj' ⋯ ≫
      homologyπ K i' ≫ ((homologyFunctor C (up ℤ) 0).shiftIso n i i' hi').inv.app K

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : CategoryWithHomology C
n a a' : ℤ
ha' : n + a = a'
K : CochainComplex C ℤ
⊢ ((homologyFunctor C (up ℤ) 0).shiftIso n a a' ha').hom.app ((quotient C (up ℤ)).obj K) =
    (homologyFunctor C (up ℤ) a).map ((Functor.commShiftIso (quotient C (up ℤ)) n).inv.app K) ≫
      (homologyFunctorFactors C (up ℤ) a).hom.app ((shiftFunctor (HomologicalComplex C (up ℤ)) n).obj K) ≫
        ((HomologicalComplex.homologyFunctor C (up ℤ) 0).shiftIso n a a' ha').hom.app K ≫
          (homologyFunctorFactors C (up ℤ) a').inv.app K

Tactic:
by

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

Goal state before:
pp: 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 : ObjectProperty C
inst✝ : P.IsClosedUnderIsomorphisms
T : Triangle C
hT : T ∈ distinguishedTriangles
⊢ P.trW T.mor₁ ↔ P T.obj₃

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝⁷ : Category.{v_1, u_1} C
inst✝⁶ : HasZeroObject C
inst✝⁵ : HasShift C ℤ
inst✝⁴ : Preadditive C
inst✝³ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝² : Pretriangulated C
P : ObjectProperty C
inst✝¹ : P.IsStableUnderShift ℤ
inst✝ : P.IsClosedUnderIsomorphisms
T : Triangle C
hT : T ∈ distinguishedTriangles
⊢ P.trW T.mor₂ ↔ P T.obj₁

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : CategoryWithHomology C
K L : CochainComplex C ℤ
n : ℤ
f : K ⟶ (shiftFunctor (CochainComplex C ℤ) n).obj L
a a' : ℤ
h : n + a = a'
⊢ (homologyFunctor C (up ℤ) 0).shiftMap (ShiftedHom.map f (quotient C (up ℤ))) a a' h =
    (homologyFunctorFactors C (up ℤ) a).hom.app K ≫
      (HomologicalComplex.homologyFunctor C (up ℤ) 0).shiftMap f a a' h ≫ (homologyFunctorFactors C (up ℤ) a').inv.app L

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preadditive C
inst✝ : HasZeroObject C
n a a' : ℤ
ha' : n + a = a'
X : C
i : ℤ
⊢ a' = a + n

Tactic:
by

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

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
F : C ⥤ D
ι : Type u_3
c : ComplexShape ι
inst✝¹⁰ : Preadditive C
inst✝⁹ : Preadditive D
inst✝⁸ : CategoryWithHomology C
inst✝⁷ : CategoryWithHomology D
inst✝⁶ : (HomologicalComplex.quasiIso D c).HasLocalization
inst✝⁵ : F.Additive
inst✝⁴ : F.PreservesHomology
inst✝³ : (HomologicalComplex.quasiIso C c).HasLocalization
inst✝² : c.QFactorsThroughHomotopy C
inst✝¹ : c.QFactorsThroughHomotopy D
inst✝ : (HomotopyCategory.quotient C c).IsLocalization (HomologicalComplex.homotopyEquivalences C c)
K : HomologicalComplex C c
⊢ (F.mapHomologicalComplexUpToQuasiIsoFactorsh c).hom.app ((HomotopyCategory.quotient C c).obj K) =
    (F.mapHomologicalComplexUpToQuasiIso c).map ((HomologicalComplexUpToQuasiIso.quotientCompQhIso C c).hom.app K) ≫
      (F.mapHomologicalComplexUpToQuasiIsoFactors c).hom.app K ≫
        (HomologicalComplexUpToQuasiIso.quotientCompQhIso D c).inv.app ((F.mapHomologicalComplex c).obj K) ≫
          HomologicalComplexUpToQuasiIso.Qh.map ((F.mapHomotopyCategoryFactors c).inv.app K)

Tactic:
by

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

Goal state before:
pp: 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
P : ObjectProperty C
F : D ⥤ C
inst✝² : F.CommShift ℤ
inst✝¹ : F.IsTriangulated
inst✝ : P.IsClosedUnderIsomorphisms
X Y : D
s : X ⟶ Y
⊢ (P.inverseImage F).trW s ↔ P.trW (F.map s)

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preadditive C
inst✝ : HasZeroObject C
n a : ℤ
X : C
i : ℤ
ha' : n + a = a + n
h : ¬i = a
h' : i + n = a + n
⊢ i = a

Tactic:
by

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

Goal state before:
pp: C : Type u_1
D : Type u_2
E : Type u_3
E' : Type u_4
inst✝⁷ : Category.{v_1, u_1} C
inst✝⁶ : Category.{v_2, u_2} D
inst✝⁵ : Category.{v_3, u_3} E
inst✝⁴ : Category.{v_4, u_4} E'
A : Type u_5
inst✝³ : AddMonoid A
inst✝² : HasShift D A
inst✝¹ : HasShift E A
inst✝ : HasShift E' A
functor : A → C ⥤ D
shiftIso : (n a a' : A) → n + a = a' → (functor a' ⋙ shiftFunctor D n ≅ functor a)
shiftIso_zero : ∀ (a : A), shiftIso 0 a a ⋯ = (functor a).isoWhiskerLeft (shiftFunctorZero D A)
n m a a' a'' : A
ha' : n + a = a'
ha'' : m + a' = a''
⊢ m + n + a = a''

Tactic:
by

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

Goal state before:
pp: C : Type u_1
D : Type u_2
E : Type u_3
E' : Type u_4
inst✝⁷ : Category.{v_1, u_1} C
inst✝⁶ : Category.{v_2, u_2} D
inst✝⁵ : Category.{v_3, u_3} E
inst✝⁴ : Category.{v_4, u_4} E'
A : Type u_5
inst✝³ : AddMonoid A
inst✝² : HasShift D A
inst✝¹ : HasShift E A
inst✝ : HasShift E' A
F G H : SingleFunctors C D A
n m a a' a'' : A
ha' : n + a = a'
ha'' : m + a' = a''
X : C
⊢ m + n + a = a''

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preadditive C
inst✝ : HasZeroObject C
n a a' : ℤ
ha' : n + a = a'
X : C
i : ℤ
⊢ ((single C (ComplexShape.up ℤ) a' ⋙ CategoryTheory.shiftFunctor (CochainComplex C ℤ) n).obj X).X i =
    ((single C (ComplexShape.up ℤ) a).obj X).X i

Tactic:
by

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

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝³ : Category.{v_1, u_1} C
inst✝² : Category.{v_2, u_2} D
A : Type u_5
inst✝¹ : AddMonoid A
inst✝ : HasShift D A
F : SingleFunctors C D A
n m a a' a'' : A
ha' : n + a = a'
ha'' : m + a' = a''
X : C
⊢ (F.shiftIso (m + n) a a'' ⋯).hom.app X =
    (shiftFunctorAdd D m n).hom.app ((F.functor a'').obj X) ≫
      (shiftFunctor D n).map ((F.shiftIso m a' a'' ha'').hom.app X) ≫ (F.shiftIso n a a' ha').hom.app X

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preadditive C
inst✝ : HasZeroObject C
n a a' : ℤ
ha' : n + a = a'
X Y : C
f : X ⟶ Y
⊢ a' = a + n

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝¹⁶ : Category.{v_1, u_1} C
inst✝¹⁵ : HasZeroObject C
inst✝¹⁴ : HasShift C ℤ
inst✝¹³ : Preadditive C
inst✝¹² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹¹ : Pretriangulated C
D : Type u_2
inst✝¹⁰ : Category.{v_2, u_2} D
inst✝⁹ : Preadditive D
inst✝⁸ : HasZeroObject D
inst✝⁷ : HasShift D ℤ
inst✝⁶ : ∀ (n : ℤ), (shiftFunctor D n).Additive
inst✝⁵ : Pretriangulated D
P : ObjectProperty C
F : D ⥤ C
inst✝⁴ : F.CommShift ℤ
inst✝³ : F.IsTriangulated
inst✝² : P.IsClosedUnderIsomorphisms
E : Type u_4
inst✝¹ : Category.{u_5, u_4} E
L : C ⥤ E
inst✝ : L.IsLocalization P.trW
X Y : D
f : X ⟶ Y
hf : (P.inverseImage F).trW f
⊢ P.trW (F.map f)

Tactic:
by

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

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝³ : Category.{v_1, u_1} C
inst✝² : Category.{v_2, u_2} D
A : Type u_5
inst✝¹ : AddMonoid A
inst✝ : HasShift D A
F : SingleFunctors C D A
n m a a' a'' : A
ha' : n + a = a'
ha'' : m + a' = a''
X : C
⊢ (F.shiftIso (m + n) a a'' ⋯).inv.app X =
    (F.shiftIso n a a' ha').inv.app X ≫
      (shiftFunctor D n).map ((F.shiftIso m a' a'' ha'').inv.app X) ≫
        (shiftFunctorAdd D m n).inv.app ((F.functor a'').obj X)

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preadditive C
inst✝ : HasZeroObject C
n a a' : ℤ
ha' : n + a = a'
X Y : C
f : X ⟶ Y
⊢ (single C (ComplexShape.up ℤ) a' ⋙ CategoryTheory.shiftFunctor (CochainComplex C ℤ) n).map f ≫
      (Hom.isoOfComponents (fun i ↦ eqToIso ⋯) ⋯).hom =
    (Hom.isoOfComponents (fun i ↦ eqToIso ⋯) ⋯).hom ≫ (single C (ComplexShape.up ℤ) a).map f

Tactic:
by

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

Goal state before:
pp: C : Type u_1
D : Type u_2
E : Type u_3
E' : Type u_4
inst✝⁷ : Category.{v_1, u_1} C
inst✝⁶ : Category.{v_2, u_2} D
inst✝⁵ : Category.{v_3, u_3} E
inst✝⁴ : Category.{v_4, u_4} E'
A : Type u_5
inst✝³ : AddMonoid A
inst✝² : HasShift D A
inst✝¹ : HasShift E A
inst✝ : HasShift E' A
F G H : SingleFunctors C D A
n m mn : A
hnm : m + n = mn
a a' a'' : A
ha' : n + a = a'
ha'' : m + a' = a''
⊢ mn + a = a''

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preadditive C
inst✝ : HasZeroObject C
a : ℤ
⊢ NatIso.ofComponents (fun X ↦ Hom.isoOfComponents (fun i ↦ eqToIso ⋯) ⋯) ⋯ =
    (single C (ComplexShape.up ℤ) a).isoWhiskerLeft (shiftFunctorZero (CochainComplex C ℤ) ℤ)

Tactic:
by

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

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝³ : Category.{v_1, u_1} C
inst✝² : Category.{v_2, u_2} D
A : Type u_5
inst✝¹ : AddMonoid A
inst✝ : HasShift D A
F : SingleFunctors C D A
n m mn : A
hnm : m + n = mn
a a' a'' : A
ha' : n + a = a'
ha'' : m + a' = a''
⊢ F.shiftIso mn a a'' ⋯ =
    (F.functor a'').isoWhiskerLeft (shiftFunctorAdd' D m n mn hnm) ≪≫
      ((F.functor a'').associator (shiftFunctor D m) (shiftFunctor D n)).symm ≪≫
        isoWhiskerRight (F.shiftIso m a' a'' ha'') (shiftFunctor D n) ≪≫ F.shiftIso n a a' ha'

Tactic:
by

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

Goal state before:
pp: C : Type u_1
D : Type u_2
E : Type u_3
E' : Type u_4
inst✝⁷ : Category.{v_1, u_1} C
inst✝⁶ : Category.{v_2, u_2} D
inst✝⁵ : Category.{v_3, u_3} E
inst✝⁴ : Category.{v_4, u_4} E'
A : Type u_5
inst✝³ : AddMonoid A
inst✝² : HasShift D A
inst✝¹ : HasShift E A
inst✝ : HasShift E' A
F G H : SingleFunctors C D A
n m mn : A
hnm : m + n = mn
a a' a'' : A
ha' : n + a = a'
ha'' : m + a' = a''
X : C
⊢ mn + a = a''

Tactic:
by

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

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝³ : Category.{v_1, u_1} C
inst✝² : Category.{v_2, u_2} D
A : Type u_5
inst✝¹ : AddMonoid A
inst✝ : HasShift D A
F : SingleFunctors C D A
n m mn : A
hnm : m + n = mn
a a' a'' : A
ha' : n + a = a'
ha'' : m + a' = a''
X : C
⊢ (F.shiftIso mn a a'' ⋯).hom.app X =
    (shiftFunctorAdd' D m n mn hnm).hom.app ((F.functor a'').obj X) ≫
      (shiftFunctor D n).map ((F.shiftIso m a' a'' ha'').hom.app X) ≫ (F.shiftIso n a a' ha').hom.app X

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preadditive C
inst✝ : HasZeroObject C
n m a a' a'' : ℤ
ha' : n + a = a'
ha'' : m + a' = a''
⊢ NatIso.ofComponents (fun X ↦ Hom.isoOfComponents (fun i ↦ eqToIso ⋯) ⋯) ⋯ =
    (single C (ComplexShape.up ℤ) a'').isoWhiskerLeft (shiftFunctorAdd (CochainComplex C ℤ) m n) ≪≫
      ((single C (ComplexShape.up ℤ) a'').associator (CategoryTheory.shiftFunctor (CochainComplex C ℤ) m)
            (CategoryTheory.shiftFunctor (CochainComplex C ℤ) n)).symm ≪≫
        Functor.isoWhiskerRight (NatIso.ofComponents (fun X ↦ Hom.isoOfComponents (fun i ↦ eqToIso ⋯) ⋯) ⋯)
            (CategoryTheory.shiftFunctor (CochainComplex C ℤ) n) ≪≫
          NatIso.ofComponents (fun X ↦ Hom.isoOfComponents (fun i ↦ eqToIso ⋯) ⋯) ⋯

Tactic:
by

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
X Y : C
φ : P.trW.RightFraction X Y
Z : C
f : X ⟶ Z
g : Z ⟶ (shiftFunctor C 1).obj φ.X'
H : Triangle.mk φ.s f g ∈ distinguishedTriangles
mem : P Z
Y' : C
s' : Y ⟶ Y'
f' : Y' ⟶ Z
mem' : Triangle.mk s' f' (g ≫ (shiftFunctor C 1).map φ.f) ∈ distinguishedTriangles
⊢ (Triangle.mk φ.s f g).mor₃ ≫ (shiftFunctor C 1).map φ.f =
    𝟙 Z ≫ (Triangle.mk s' f' (g ≫ (shiftFunctor C 1).map φ.f)).mor₃

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preadditive C
inst✝ : HasZeroObject C
n : ℤ
⊢ ((singleFunctors C).functor n).Additive

Tactic:
by

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

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝³ : Category.{v_1, u_1} C
inst✝² : Category.{v_2, u_2} D
A : Type u_5
inst✝¹ : AddMonoid A
inst✝ : HasShift D A
F : SingleFunctors C D A
n m mn : A
hnm : m + n = mn
a a' a'' : A
ha' : n + a = a'
ha'' : m + a' = a''
X : C
⊢ (F.shiftIso mn a a'' ⋯).inv.app X =
    (F.shiftIso n a a' ha').inv.app X ≫
      (shiftFunctor D n).map ((F.shiftIso m a' a'' ha'').inv.app X) ≫
        (shiftFunctorAdd' D m n mn hnm).inv.app ((F.functor a'').obj X)

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : Preadditive C
inst✝² : HasZeroObject C
R : Type u_1
inst✝¹ : Ring R
n : ℤ
inst✝ : Linear R C
X✝ Y✝ : C
f : X✝ ⟶ Y✝
r : R
⊢ ((singleFunctors C).functor n).map (r • f) = r • ((singleFunctors C).functor n).map f

Tactic:
by

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

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝³ : Category.{v_1, u_1} C
inst✝² : Category.{v_2, u_2} D
A : Type u_5
inst✝¹ : AddMonoid A
inst✝ : HasShift D A
F : SingleFunctors C D A
a : A
X : C
⊢ (F.shiftIso 0 a a ⋯).hom.app X = (shiftFunctorZero D A).hom.app ((F.functor a).obj X)

Tactic:
by

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

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝³ : Category.{v_1, u_1} C
inst✝² : Category.{v_2, u_2} D
A : Type u_5
inst✝¹ : AddMonoid A
inst✝ : HasShift D A
F : SingleFunctors C D A
a : A
X : C
⊢ (F.shiftIso 0 a a ⋯).inv.app X = (shiftFunctorZero D A).inv.app ((F.functor a).1 X)

Tactic:
by

Goal state after: