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

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasDerivedCategory C
n : ℤ
x✝¹ x✝ : C
f₁ f₂ : x✝¹ ⟶ x✝
h : (singleFunctor C n).map f₁ = (singleFunctor C n).map f₂
⊢ f₁ = f₂

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasDerivedCategory C
n : ℤ
A B : C
f : Q.obj ((CochainComplex.singleFunctor C n).obj A) ⟶ Q.obj ((CochainComplex.singleFunctor C n).obj B)
this : ∃ f', f = Q.map f'
⊢ ∃ a, (singleFunctor C n).map a = f

Tactic:
by

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 L : CochainComplex C ℤ
φ : K ⟶ L
e : K ≅ L
inst✝ : HasZeroObject C
A : C
n i : ℤ
hi : autoParam (i < n) isStrictlyGE_iff._auto_1
⊢ i ≠ n

Tactic:
by

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 L : CochainComplex C ℤ
φ : K ⟶ L
e : K ≅ L
inst✝ : HasZeroObject C
A : C
n : ℤ
⊢ IsStrictlyGE ((single C (up ℤ) n).obj A) n

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasDerivedCategory C
n : ℤ
A B : C
X : CochainComplex C ℤ
w✝² : X.IsStrictlyGE n
w✝¹ : X.IsStrictlyLE n
s : X ⟶ (CochainComplex.singleFunctor C n).obj A
w✝ : IsIso (Q.map s)
g : X ⟶ (CochainComplex.singleFunctor C n).obj B
A₀ : C
e : X ≅ (HomologicalComplex.single C (ComplexShape.up ℤ) n).obj A₀
φ : A₀ ⟶ A
hφ : (CochainComplex.singleFunctor C n).map φ = e.inv ≫ s
⊢ IsIso ((singleFunctor C n).map φ)

Tactic:
by

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 L : CochainComplex C ℤ
φ : K ⟶ L
e : K ≅ L
inst✝ : HasZeroObject C
A : C
n i : ℤ
hi : n < i
⊢ i ≠ n

Tactic:
by

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 L : CochainComplex C ℤ
φ : K ⟶ L
e : K ≅ L
inst✝ : HasZeroObject C
A : C
n : ℤ
⊢ IsStrictlyLE ((single C (up ℤ) n).obj A) n

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasDerivedCategory C
n : ℤ
A B : C
X : CochainComplex C ℤ
w✝² : X.IsStrictlyGE n
w✝¹ : X.IsStrictlyLE n
s : X ⟶ (CochainComplex.singleFunctor C n).obj A
w✝ : IsIso (Q.map s)
g : X ⟶ (CochainComplex.singleFunctor C n).obj B
A₀ : C
e : X ≅ (HomologicalComplex.single C (ComplexShape.up ℤ) n).obj A₀
φ : A₀ ⟶ A
hφ : (CochainComplex.singleFunctor C n).map φ = e.inv ≫ s
this : IsIso ((singleFunctor C n).map φ)
⊢ IsIso ((singleFunctor C n ⋙ homologyFunctor C n).map φ)

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
inst✝¹ : HasZeroObject C
ι : Type u_1
inst✝ : DecidableEq ι
c : ComplexShape ι
j : ι
A : C
i : ι
⊢ ((single C c j).obj A).HasHomology i

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : HasZeroMorphisms C
K L : CochainComplex C ℤ
φ : K ⟶ L
e : K ≅ L
inst✝³ : HasZeroObject C
inst✝² : ∀ (i : ℤ), HasHomology K i
inst✝¹ : ∀ (i : ℤ), HasHomology L i
n : ℤ
inst✝ : K.IsStrictlyGE n
⊢ IsIso (K.πTruncGE n)

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasDerivedCategory C
X Y : CochainComplex C ℤ
f : Q.obj X ⟶ Q.obj Y
⊢ ∃ Y' g s, ∃ (x : IsIso (Q.map s)), f = Q.map g ≫ inv (Q.map s)

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
inst✝¹ : HasZeroObject C
ι : Type u_1
inst✝ : DecidableEq ι
c : ComplexShape ι
j : ι
A : C
i : ι
hi : i ≠ j
⊢ IsZero (((single C c j).obj A).homology i)

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
inst✝¹ : HasZeroObject C
ι : Type u_1
inst✝ : DecidableEq ι
c : ComplexShape ι
j : ι
A : C
⊢ (singleObjCyclesSelfIso c j A).inv ≫ ((single C c j).obj A).iCycles j = (singleObjXSelf c j A).inv

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
inst✝¹ : HasZeroObject C
ι : Type u_1
inst✝ : DecidableEq ι
c : ComplexShape ι
j : ι
A : C
⊢ ((single C c j).obj A).homologyπ j ≫ (singleObjHomologySelfIso c j A).hom = (singleObjCyclesSelfIso c j A).hom

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : HasZeroMorphisms C
K L : CochainComplex C ℤ
φ : K ⟶ L
e : K ≅ L
inst✝³ : HasZeroObject C
inst✝² : ∀ (i : ℤ), HasHomology K i
inst✝¹ : ∀ (i : ℤ), HasHomology L i
n : ℤ
inst✝ : K.IsStrictlyLE n
⊢ IsIso (K.ιTruncLE n)

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasDerivedCategory C
n : ℤ
A B : C
X : CochainComplex C ℤ
w✝² : X.IsStrictlyGE n
w✝¹ : X.IsStrictlyLE n
s : X ⟶ (CochainComplex.singleFunctor C n).obj A
w✝ : IsIso (Q.map s)
g : X ⟶ (CochainComplex.singleFunctor C n).obj B
A₀ : C
e : X ≅ (HomologicalComplex.single C (ComplexShape.up ℤ) n).obj A₀
φ : A₀ ⟶ A
hφ : (CochainComplex.singleFunctor C n).map φ = e.inv ≫ s
this✝ : IsIso ((singleFunctor C n).map φ)
this : IsIso φ
⊢ IsIso (e.inv ≫ s)

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : HasZeroMorphisms C
K : CochainComplex C ℤ
inst✝¹ : HasZeroObject C
inst✝ : ∀ (i : ℤ), HasHomology K i
n : ℤ
⊢ IsIso (K.πTruncGE n) ↔ K.IsStrictlyGE n

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
inst✝¹ : HasZeroObject C
ι : Type u_1
inst✝ : DecidableEq ι
c : ComplexShape ι
j : ι
A : C
⊢ (singleObjCyclesSelfIso c j A).hom ≫ (singleObjHomologySelfIso c j A).inv = ((single C c j).obj A).homologyπ j

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : HasZeroMorphisms C
K : CochainComplex C ℤ
inst✝¹ : HasZeroObject C
inst✝ : ∀ (i : ℤ), HasHomology K i
n : ℤ
⊢ IsIso (K.ιTruncLE n) ↔ K.IsStrictlyLE n

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
inst✝¹ : HasZeroObject C
ι : Type u_1
inst✝ : DecidableEq ι
c : ComplexShape ι
j : ι
A : C
⊢ (singleObjCyclesSelfIso c j A).hom ≫ (singleObjOpcyclesSelfIso c j A).hom =
    ((single C c j).obj A).iCycles j ≫ ((single C c j).obj A).pOpcycles j

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasDerivedCategory C
n : ℤ
A B : C
X : CochainComplex C ℤ
w✝² : X.IsStrictlyGE n
w✝¹ : X.IsStrictlyLE n
s : X ⟶ (CochainComplex.singleFunctor C n).obj A
w✝ : IsIso (Q.map s)
g : X ⟶ (CochainComplex.singleFunctor C n).obj B
A₀ : C
e : X ≅ (HomologicalComplex.single C (ComplexShape.up ℤ) n).obj A₀
φ : A₀ ⟶ A
hφ : (CochainComplex.singleFunctor C n).map φ = e.inv ≫ s
this✝² : IsIso ((singleFunctor C n).map φ)
this✝¹ : IsIso φ
this✝ : IsIso (e.inv ≫ s)
this : IsIso s
⊢ inv (Q.map s) ≫ Q.map g = Q.map (inv s ≫ g)

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
inst✝¹ : HasZeroObject C
ι : Type u_1
inst✝ : DecidableEq ι
c : ComplexShape ι
j : ι
A : C
⊢ (singleObjCyclesSelfIso c j A).inv ≫ ((single C c j).obj A).homologyπ j = (singleObjHomologySelfIso c j A).inv

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
inst✝¹ : HasDerivedCategory C
X Y : CochainComplex C ℤ
n : ℤ
inst✝ : X.IsStrictlyLE n
X' : CochainComplex C ℤ
s : X' ⟶ X
hs : IsIso (Q.map s)
g : X' ⟶ Y
⊢ IsIso (Q.map (CochainComplex.truncLEMap s n))

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasDerivedCategory C
n : ℤ
A B : C
f : (singleFunctor C n).obj A ⟶ (singleFunctor C n).obj B
⊢ ∃ a, (singleFunctor C n).map a = f

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : HasZeroMorphisms C
K L : CochainComplex C ℤ
φ : K ⟶ L
e : K ≅ L
inst✝³ : HasZeroObject C
inst✝² : ∀ (i : ℤ), HasHomology K i
inst✝¹ : ∀ (i : ℤ), HasHomology L i
n : ℤ
inst✝ : K.IsGE n
⊢ QuasiIso (K.πTruncGE n)

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : HasZeroMorphisms C
K L : CochainComplex C ℤ
φ : K ⟶ L
e : K ≅ L
inst✝³ : HasZeroObject C
inst✝² : ∀ (i : ℤ), HasHomology K i
inst✝¹ : ∀ (i : ℤ), HasHomology L i
n : ℤ
inst✝ : K.IsLE n
⊢ QuasiIso (K.ιTruncLE n)

Tactic:
by

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

Goal state before:
pp: C₁ : Type u_2
C₂ : Type u_3
D₁ : Type u_5
D₂ : Type u_6
inst✝⁵ : Category.{v_2, u_2} C₁
inst✝⁴ : Category.{v_3, u_3} C₂
inst✝³ : Category.{v_5, u_5} D₁
inst✝² : Category.{v_6, u_6} D₂
W₁ : MorphismProperty C₁
W₂ : MorphismProperty C₂
Φ : LocalizerMorphism W₁ W₂
L₁ : C₁ ⥤ D₁
inst✝¹ : L₁.IsLocalization W₁
L₂ : C₂ ⥤ D₂
inst✝ : L₂.IsLocalization W₂
X Y : C₁
f : X ⟶ Y
⊢ Φ.homMap L₁ L₂ (L₁.map f) = L₂.map (Φ.functor.map f)

Tactic:
by

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

Goal state before:
pp: C₁ : Type u_2
C₂ : Type u_3
D₁ : Type u_5
D₂ : Type u_6
inst✝⁵ : Category.{v_2, u_2} C₁
inst✝⁴ : Category.{v_3, u_3} C₂
inst✝³ : Category.{v_5, u_5} D₁
inst✝² : Category.{v_6, u_6} D₂
W₁ : MorphismProperty C₁
W₂ : MorphismProperty C₂
Φ : LocalizerMorphism W₁ W₂
L₁ : C₁ ⥤ D₁
inst✝¹ : L₁.IsLocalization W₁
L₂ : C₂ ⥤ D₂
inst✝ : L₂.IsLocalization W₂
X : C₁
⊢ Φ.homMap L₁ L₂ (𝟙 (L₁.obj X)) = 𝟙 (L₂.obj (Φ.functor.obj X))

Tactic:
by

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

Goal state before:
pp: C₁ : Type u_2
C₂ : Type u_3
D₁ : Type u_5
D₂ : Type u_6
inst✝⁵ : Category.{v_2, u_2} C₁
inst✝⁴ : Category.{v_3, u_3} C₂
inst✝³ : Category.{v_5, u_5} D₁
inst✝² : Category.{v_6, u_6} D₂
W₁ : MorphismProperty C₁
W₂ : MorphismProperty C₂
Φ : LocalizerMorphism W₁ W₂
L₁ : C₁ ⥤ D₁
inst✝¹ : L₁.IsLocalization W₁
L₂ : C₂ ⥤ D₂
inst✝ : L₂.IsLocalization W₂
X Y Z : C₁
f : L₁.obj X ⟶ L₁.obj Y
g : L₁.obj Y ⟶ L₁.obj Z
⊢ Φ.homMap L₁ L₂ (f ≫ g) = Φ.homMap L₁ L₂ f ≫ Φ.homMap L₁ L₂ g

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
inst✝¹ : HasZeroObject C
ι : Type u_1
inst✝ : DecidableEq ι
c : ComplexShape ι
j : ι
A : C
⊢ (singleObjHomologySelfIso c j A).inv ≫ ((single C c j).obj A).homologyι j = (singleObjOpcyclesSelfIso c j A).hom

Tactic:
by

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

Goal state before:
pp: C₁ : Type u_2
C₂ : Type u_3
D₁ : Type u_5
D₂ : Type u_6
inst✝⁵ : Category.{v_2, u_2} C₁
inst✝⁴ : Category.{v_3, u_3} C₂
inst✝³ : Category.{v_5, u_5} D₁
inst✝² : Category.{v_6, u_6} D₂
W₁ : MorphismProperty C₁
W₂ : MorphismProperty C₂
Φ : LocalizerMorphism W₁ W₂
L₁ : C₁ ⥤ D₁
inst✝¹ : L₁.IsLocalization W₁
L₂ : C₂ ⥤ D₂
inst✝ : L₂.IsLocalization W₂
X Y : C₁
G : D₁ ⥤ D₂
e : Φ.functor ⋙ L₂ ≅ L₁ ⋙ G
f : L₁.obj X ⟶ L₁.obj Y
G' : D₁ ⥤ D₂ := Φ.localizedFunctor L₁ L₂
e' : Φ.functor ⋙ L₂ ≅ L₁ ⋙ G' := CatCommSq.iso Φ.functor L₁ L₂ G'
this : Localization.Lifting L₁ W₁ (Φ.functor ⋙ L₂) G := { iso := e.symm }
α : G' ≅ G := Localization.liftNatIso L₁ W₁ (L₁ ⋙ G') (Φ.functor ⋙ L₂) G' G e'.symm
⊢ e = e' ≪≫ L₁.isoWhiskerLeft α

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
inst✝¹ : HasZeroObject C
ι : Type u_1
inst✝ : DecidableEq ι
c : ComplexShape ι
j : ι
A : C
⊢ ((single C c j).obj A).homologyι j ≫ (singleObjOpcyclesSelfIso c j A).inv = (singleObjHomologySelfIso c j A).hom

Tactic:
by

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

Goal state before:
pp: C₁ : Type u_2
C₂ : Type u_3
D₁ : Type u_5
D₂ : Type u_6
inst✝⁵ : Category.{v_2, u_2} C₁
inst✝⁴ : Category.{v_3, u_3} C₂
inst✝³ : Category.{v_5, u_5} D₁
inst✝² : Category.{v_6, u_6} D₂
W₁ : MorphismProperty C₁
W₂ : MorphismProperty C₂
Φ : LocalizerMorphism W₁ W₂
L₁ : C₁ ⥤ D₁
inst✝¹ : L₁.IsLocalization W₁
L₂ : C₂ ⥤ D₂
inst✝ : L₂.IsLocalization W₂
X Y : C₁
G : D₁ ⥤ D₂
e : Φ.functor ⋙ L₂ ≅ L₁ ⋙ G
f : L₁.obj X ⟶ L₁.obj Y
⊢ Φ.homMap L₁ L₂ f = e.hom.app X ≫ G.map f ≫ e.inv.app Y

Tactic:
by

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

Goal state before:
pp: C₁ : Type u_2
D₁ : Type u_5
inst✝² : Category.{v_2, u_2} C₁
inst✝¹ : Category.{v_5, u_5} D₁
W₁ : MorphismProperty C₁
L₁ : C₁ ⥤ D₁
inst✝ : L₁.IsLocalization W₁
X Y : C₁
f : L₁.obj X ⟶ L₁.obj Y
⊢ (id W₁).homMap L₁ L₁ f = f

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
inst✝¹ : HasZeroObject C
ι : Type u_1
inst✝ : DecidableEq ι
c : ComplexShape ι
j : ι
A : C
⊢ (singleObjHomologySelfIso c j A).hom ≫ (singleObjOpcyclesSelfIso c j A).hom = ((single C c j).obj A).homologyι j

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
inst✝¹ : HasZeroObject C
ι : Type u_1
inst✝ : DecidableEq ι
c : ComplexShape ι
j : ι
A : C
⊢ ((single C c j).obj A).d j (c.next j) = 0

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
inst✝¹ : HasDerivedCategory C
X Y : CochainComplex C ℤ
f : Q.obj X ⟶ Q.obj Y
n : ℤ
inst✝ : X.IsStrictlyLE n
⊢ ∃ X', ∃ (_ : X'.IsStrictlyLE n), ∃ s, ∃ (x : IsIso (Q.map s)), ∃ g, f = inv (Q.map s) ≫ Q.map g

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
inst✝¹ : HasZeroObject C
ι : Type u_1
inst✝ : DecidableEq ι
c : ComplexShape ι
j : ι
A : C
⊢ ((single C c j).obj A).pOpcycles j ≫ (singleObjOpcyclesSelfIso c j A).inv = (singleObjXSelf c j A).hom

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : HasZeroMorphisms C
K L : CochainComplex C ℤ
φ : K ⟶ L
inst✝² : HasZeroObject C
inst✝¹ : ∀ (i : ℤ), HasHomology K i
inst✝ : ∀ (i : ℤ), HasHomology L i
n : ℤ
h : ∀ (i : ℤ), n ≤ i → QuasiIsoAt φ i
i : ℕ
⊢ n ≤ (embeddingUpIntGE n).f i

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : HasZeroMorphisms C
K L : CochainComplex C ℤ
φ : K ⟶ L
inst✝² : HasZeroObject C
inst✝¹ : ∀ (i : ℤ), HasHomology K i
inst✝ : ∀ (i : ℤ), HasHomology L i
n : ℤ
⊢ QuasiIso (truncGEMap φ n) ↔ ∀ (i : ℤ), n ≤ i → QuasiIsoAt φ i

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
inst✝¹ : HasDerivedCategory C
X Y : CochainComplex C ℤ
n : ℤ
inst✝ : Y.IsStrictlyGE n
Y' : CochainComplex C ℤ
g : X ⟶ Y'
s : Y ⟶ Y'
hs : IsIso (Q.map s)
⊢ IsIso (Q.map (CochainComplex.truncGEMap s n))

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
inst✝¹ : HasZeroObject C
ι : Type u_1
inst✝ : DecidableEq ι
c : ComplexShape ι
j : ι
A B : C
f : A ⟶ B
⊢ cyclesMap ((single C c j).map f) j ≫ (singleObjCyclesSelfIso c j B).hom = (singleObjCyclesSelfIso c j A).hom ≫ f

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
inst✝¹ : HasZeroObject C
ι : Type u_1
inst✝ : DecidableEq ι
c : ComplexShape ι
j : ι
A B : C
f : A ⟶ B
⊢ (singleObjCyclesSelfIso c j A).inv ≫ cyclesMap ((single C c j).map f) j = f ≫ (singleObjCyclesSelfIso c j B).inv

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
inst✝¹ : HasZeroObject C
ι : Type u_1
inst✝ : DecidableEq ι
c : ComplexShape ι
j : ι
A B : C
f : A ⟶ B
⊢ homologyMap ((single C c j).map f) j ≫ (singleObjHomologySelfIso c j B).hom = (singleObjHomologySelfIso c j A).hom ≫ f

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : HasZeroMorphisms C
K L : CochainComplex C ℤ
φ : K ⟶ L
inst✝² : HasZeroObject C
inst✝¹ : ∀ (i : ℤ), HasHomology K i
inst✝ : ∀ (i : ℤ), HasHomology L i
i : ℤ
k : ℕ
h : ∀ (i_1 : ℕ) (i' : ℤ), (embeddingUpIntLE (i + ↑k)).f i_1 = i' → QuasiIsoAt φ i'
hi : i ≤ i + ↑k
⊢ (embeddingUpIntLE (i + ↑k)).f k = i

Tactic:
by

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

Goal state before:
pp: C₁ : Type u_2
C₂ : Type u_3
C₃ : Type u_4
D₁ : Type u_5
D₂ : Type u_6
D₃ : Type u_7
inst✝⁸ : Category.{v_2, u_2} C₁
inst✝⁷ : Category.{v_3, u_3} C₂
inst✝⁶ : Category.{v_4, u_4} C₃
inst✝⁵ : Category.{v_5, u_5} D₁
inst✝⁴ : Category.{v_6, u_6} D₂
inst✝³ : Category.{v_7, u_7} D₃
W₁ : MorphismProperty C₁
W₂ : MorphismProperty C₂
W₃ : MorphismProperty C₃
Φ : LocalizerMorphism W₁ W₂
Ψ : LocalizerMorphism W₂ W₃
L₁ : C₁ ⥤ D₁
inst✝² : L₁.IsLocalization W₁
L₂ : C₂ ⥤ D₂
inst✝¹ : L₂.IsLocalization W₂
L₃ : C₃ ⥤ D₃
inst✝ : L₃.IsLocalization W₃
X Y : C₁
f : L₁.obj X ⟶ L₁.obj Y
⊢ Ψ.homMap L₂ L₃ (Φ.homMap L₁ L₂ f) = (Φ.comp Ψ).homMap L₁ L₃ f

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
inst✝¹ : HasZeroObject C
ι : Type u_1
inst✝ : DecidableEq ι
c : ComplexShape ι
j : ι
A B : C
f : A ⟶ B
⊢ (singleObjHomologySelfIso c j A).inv ≫ homologyMap ((single C c j).map f) j = f ≫ (singleObjHomologySelfIso c j B).inv

Tactic:
by

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

Goal state before:
pp: C : Type u_1
C₁ : Type u_2
C₂ : Type u_3
C₃ : Type u_4
D₁ : Type u_5
D₂ : Type u_6
D₃ : Type u_7
inst✝⁹ : Category.{v_1, u_1} C
inst✝⁸ : Category.{v_2, u_2} C₁
inst✝⁷ : Category.{v_3, u_3} C₂
inst✝⁶ : Category.{v_4, u_4} C₃
inst✝⁵ : Category.{v_5, u_5} D₁
inst✝⁴ : Category.{v_6, u_6} D₂
inst✝³ : Category.{v_7, u_7} D₃
W : MorphismProperty C
L₁ : C ⥤ D₁
inst✝² : L₁.IsLocalization W
L₂ : C ⥤ D₂
inst✝¹ : L₂.IsLocalization W
L₃ : C ⥤ D₃
inst✝ : L₃.IsLocalization W
X Y Z : C
f : L₁.obj X ⟶ L₁.obj Y
⊢ (LocalizerMorphism.id W).homMap L₂ L₁ ((LocalizerMorphism.id W).homMap L₁ L₂ f) = f

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : HasZeroMorphisms C
K L : CochainComplex C ℤ
φ : K ⟶ L
inst✝² : HasZeroObject C
inst✝¹ : ∀ (i : ℤ), HasHomology K i
inst✝ : ∀ (i : ℤ), HasHomology L i
n : ℤ
h : ∀ i ≤ n, QuasiIsoAt φ i
i : ℕ
⊢ (embeddingUpIntLE n).f i ≤ n

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : HasZeroMorphisms C
K L : CochainComplex C ℤ
φ : K ⟶ L
inst✝² : HasZeroObject C
inst✝¹ : ∀ (i : ℤ), HasHomology K i
inst✝ : ∀ (i : ℤ), HasHomology L i
n : ℤ
⊢ QuasiIso (truncLEMap φ n) ↔ ∀ i ≤ n, QuasiIsoAt φ i

Tactic:
by

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

Goal state before:
pp: C : Type u_1
C₁ : Type u_2
C₂ : Type u_3
C₃ : Type u_4
D₁ : Type u_5
D₂ : Type u_6
D₃ : Type u_7
inst✝⁹ : Category.{v_1, u_1} C
inst✝⁸ : Category.{v_2, u_2} C₁
inst✝⁷ : Category.{v_3, u_3} C₂
inst✝⁶ : Category.{v_4, u_4} C₃
inst✝⁵ : Category.{v_5, u_5} D₁
inst✝⁴ : Category.{v_6, u_6} D₂
inst✝³ : Category.{v_7, u_7} D₃
W : MorphismProperty C
L₁ : C ⥤ D₁
inst✝² : L₁.IsLocalization W
L₂ : C ⥤ D₂
inst✝¹ : L₂.IsLocalization W
L₃ : C ⥤ D₃
inst✝ : L₃.IsLocalization W
X Y Z : C
g : L₂.obj X ⟶ L₂.obj Y
⊢ (LocalizerMorphism.id W).homMap L₁ L₂ ((LocalizerMorphism.id W).homMap L₂ L₁ g) = g

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
inst✝¹ : HasDerivedCategory C
X Y : CochainComplex C ℤ
f : Q.obj X ⟶ Q.obj Y
n : ℤ
inst✝ : Y.IsStrictlyGE n
⊢ ∃ Y', ∃ (_ : Y'.IsStrictlyGE n), ∃ g s, ∃ (x : IsIso (Q.map s)), f = Q.map g ≫ inv (Q.map s)

Tactic:
by

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 : Preadditive C
K : CochainComplex C ℤ
n : ℤ
inst✝ : K.IsStrictlyLE n
a n' : ℤ
h : a + n' = n
i : ℤ
hi : n' < i
⊢ n < i + a

Tactic:
by

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 : Preadditive C
K : CochainComplex C ℤ
n : ℤ
inst✝ : K.IsStrictlyLE n
a n' : ℤ
h : a + n' = n
⊢ ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) a).obj K).IsStrictlyLE n'

Tactic:
by

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

Goal state before:
pp: C : Type u_1
D₁ : Type u_5
D₂ : Type u_6
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_5, u_5} D₁
inst✝² : Category.{v_6, u_6} D₂
W : MorphismProperty C
L₁ : C ⥤ D₁
inst✝¹ : L₁.IsLocalization W
L₂ : C ⥤ D₂
inst✝ : L₂.IsLocalization W
X Y : C
G : D₁ ⥤ D₂
e : L₁ ⋙ G ≅ L₂
f : L₁.obj X ⟶ L₁.obj Y
⊢ (homEquiv W L₁ L₂) f = e.inv.app X ≫ G.map f ≫ e.hom.app Y

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
inst✝¹ : HasZeroObject C
ι : Type u_1
inst✝ : DecidableEq ι
c : ComplexShape ι
j : ι
A B : C
f : A ⟶ B
⊢ (singleObjOpcyclesSelfIso c j A).hom ≫ opcyclesMap ((single C c j).map f) j = f ≫ (singleObjOpcyclesSelfIso c j B).hom

Tactic:
by

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

Goal state before:
pp: C : Type u_1
D₁ : Type u_5
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_5, u_5} D₁
W : MorphismProperty C
L₁ : C ⥤ D₁
inst✝ : L₁.IsLocalization W
X Y : C
f : L₁.obj X ⟶ L₁.obj Y
⊢ (homEquiv W L₁ L₁) f = f

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
inst✝¹ : HasZeroObject C
ι : Type u_1
inst✝ : DecidableEq ι
c : ComplexShape ι
j : ι
A B : C
f : A ⟶ B
⊢ opcyclesMap ((single C c j).map f) j ≫ (singleObjOpcyclesSelfIso c j B).inv = (singleObjOpcyclesSelfIso c j A).inv ≫ f

Tactic:
by

Goal state after: