mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.CategoryTheory.Sums.Products

Goal state before:
pp: A : Type u_1
inst✝³ : Category.{v_1, u_1} A
A' : Type u_2
inst✝² : Category.{v_2, u_2} A'
B : Type u
inst✝¹ : Category.{v, u} B
T : Type u_3
inst✝ : Category.{v_3, u_3} T
⊢ ∀ {X Y : (A ⊕ A') ⊕ T ⥤ B} (f : X ⟶ Y),
    ((sum.associativity A A' T).congrLeft.trans
                ((functorEquiv A (A' ⊕ T) B).trans (Equivalence.refl.prod (functorEquiv A' T B)))).functor.map
          f ≫
        ((((inl_ A (A' ⊕ T)).associator (sum.associativity A A' T).inverse Y).symm ≪≫
                isoWhiskerRight (sum.inlCompInverseAssociator A A' T) Y ≪≫
                  (inl_ A A').associator (inl_ (A ⊕ A') T) Y).prod
            (((inl_ A' T).isoWhiskerLeft ((inr_ A (A' ⊕ T)).associator (sum.associativity A A' T).inverse Y).symm ≪≫
                  ((inl_ A' T).associator (inr_ A (A' ⊕ T) ⋙ (sum.associativity A A' T).inverse) Y).symm ≪≫
                    isoWhiskerRight (sum.inlCompInrCompInverseAssociator A A' T) Y ≪≫
                      (inr_ A A').associator (inl_ (A ⊕ A') T) Y).prod
              ((inr_ A' T).isoWhiskerLeft ((inr_ A (A' ⊕ T)).associator (sum.associativity A A' T).inverse Y).symm ≪≫
                ((inr_ A' T).associator (inr_ A (A' ⊕ T) ⋙ (sum.associativity A A' T).inverse) Y).symm ≪≫
                  isoWhiskerRight (sum.inrCompInrCompInverseAssociator A A' T) Y))).hom =
      ((((inl_ A (A' ⊕ T)).associator (sum.associativity A A' T).inverse X).symm ≪≫
                isoWhiskerRight (sum.inlCompInverseAssociator A A' T) X ≪≫
                  (inl_ A A').associator (inl_ (A ⊕ A') T) X).prod
            (((inl_ A' T).isoWhiskerLeft ((inr_ A (A' ⊕ T)).associator (sum.associativity A A' T).inverse X).symm ≪≫
                  ((inl_ A' T).associator (inr_ A (A' ⊕ T) ⋙ (sum.associativity A A' T).inverse) X).symm ≪≫
                    isoWhiskerRight (sum.inlCompInrCompInverseAssociator A A' T) X ≪≫
                      (inr_ A A').associator (inl_ (A ⊕ A') T) X).prod
              ((inr_ A' T).isoWhiskerLeft ((inr_ A (A' ⊕ T)).associator (sum.associativity A A' T).inverse X).symm ≪≫
                ((inr_ A' T).associator (inr_ A (A' ⊕ T) ⋙ (sum.associativity A A' T).inverse) X).symm ≪≫
                  isoWhiskerRight (sum.inrCompInrCompInverseAssociator A A' T) X))).hom ≫
        (((functorEquiv (A ⊕ A') T B).trans ((functorEquiv A A' B).prod Equivalence.refl)).trans
                (prod.associativity (A ⥤ B) (A' ⥤ B) (T ⥤ B))).functor.map
          f

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Topos.Sheaf

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
F G : Cᵒᵖ ⥤ Type (max u v)
m : F ⟶ G
χ' : G ⟶ Functor.sieves C
hχ' : IsPullback m ((Functor.isTerminalConst Cᵒᵖ Types.isTerminalPUnit).from F) χ' (truth C)
⊢ χ' = χ m

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.SheafCohomology.MayerVietoris

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
J : GrothendieckTopology C
inst✝² : HasWeakSheafify J (Type v)
inst✝¹ : HasSheafify J AddCommGrpCat
inst✝ : HasExt (Sheaf J AddCommGrpCat)
S : J.MayerVietorisSquare
F : Sheaf J AddCommGrpCat
n : ℕ
y : ↑(F.H' n S.X₄)
⊢ (ConcreteCategory.hom (S.toBiprod F n)) y =
    (ConcreteCategory.hom
        (((F.cohomologyPresheaf n).obj (op S.X₂)).biprodIsoProd ((F.cohomologyPresheaf n).obj (op S.X₃))).inv)
      ((ConcreteCategory.hom ((F.cohomologyPresheaf n).map S.f₂₄.op)) y,
        (ConcreteCategory.hom ((F.cohomologyPresheaf n).map S.f₃₄.op)) y)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.SheafCohomology.MayerVietoris

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
J : GrothendieckTopology C
inst✝² : HasWeakSheafify J (Type v)
inst✝¹ : HasSheafify J AddCommGrpCat
inst✝ : HasExt (Sheaf J AddCommGrpCat)
S : J.MayerVietorisSquare
F : Sheaf J AddCommGrpCat
n : ℕ
⊢ S.toBiprod F n ≫ S.fromBiprod F n = 0

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.SheafCohomology.MayerVietoris

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
J : GrothendieckTopology C
inst✝² : HasWeakSheafify J (Type v)
inst✝¹ : HasSheafify J AddCommGrpCat
inst✝ : HasExt (Sheaf J AddCommGrpCat)
S : J.MayerVietorisSquare
F : Sheaf J AddCommGrpCat
n : ℕ
y₁ : ↑(F.H' n S.X₂)
y₂ : ↑(F.H' n S.X₃)
⊢ (ConcreteCategory.hom (S.fromBiprod F n))
      ((ConcreteCategory.hom ((F.H' n S.X₂).biprodIsoProd (F.H' n S.X₃)).inv) (y₁, y₂)) =
    (ConcreteCategory.hom ((F.cohomologyPresheaf n).map S.f₁₂.op)) y₁ -
      (ConcreteCategory.hom ((F.cohomologyPresheaf n).map S.f₁₃.op)) y₂

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.SheafCohomology.MayerVietoris

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
J : GrothendieckTopology C
inst✝² : HasWeakSheafify J (Type v)
inst✝¹ : HasSheafify J AddCommGrpCat
inst✝ : HasExt (Sheaf J AddCommGrpCat)
S : J.MayerVietorisSquare
F : Sheaf J AddCommGrpCat
n : ℕ
x : ↑(F.H' n S.X₄)
⊢ Ext.biprodAddEquiv
      (Ext.biprodAddEquiv.symm
        ((ConcreteCategory.hom ((F.H' n S.X₂).biprodIsoProd (F.H' n S.X₃)).hom)
          ((ConcreteCategory.hom (S.toBiprod F n)) x))) =
    Ext.biprodAddEquiv ((Ext.mk₀ S.shortComplex.g).comp x ⋯)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Topos.Sheaf

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
J : GrothendieckTopology C
F G : Cᵒᵖ ⥤ Type (max u v)
m : F ⟶ G
inst✝ : Mono m
hF : Presieve.IsSheaf J F
hG : Presieve.IsSeparated J G
X : Cᵒᵖ
x : G.obj X
⊢ J.IsClosed ((ConcreteCategory.hom ((χ m).app X)) x)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.SheafCohomology.MayerVietoris

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
J : GrothendieckTopology C
inst✝² : HasWeakSheafify J (Type v)
inst✝¹ : HasSheafify J AddCommGrpCat
inst✝ : HasExt (Sheaf J AddCommGrpCat)
S : J.MayerVietorisSquare
F : Sheaf J AddCommGrpCat
n : ℕ
x : ↑(F.H' n S.X₂ ⊞ F.H' n S.X₃)
⊢ (Ext.mk₀ S.shortComplex.f).comp
      (Ext.biprodAddEquiv.symm ((ConcreteCategory.hom ((F.H' n S.X₂).biprodIsoProd (F.H' n S.X₃)).hom) x)) ⋯ =
    (ConcreteCategory.hom (S.fromBiprod F n)) x

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Topos.Sheaf

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
J✝ : GrothendieckTopology C
F G : Sheaf J✝ (Type (max u v))
J : GrothendieckTopology C
⊢ Presheaf.IsSheaf J (closedSieves J).toFunctor

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Topos.Sheaf

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
J✝ : GrothendieckTopology C
F G : Sheaf J✝ (Type (max u v))
J : GrothendieckTopology C
X : Cᵒᵖ
x : (sieves C).obj X
⊢ x ∈ (Subfunctor.range (Presheaf.truth C)).obj X → x ∈ (closedSieves J).obj X

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.SheafCohomology.MayerVietoris

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
J : GrothendieckTopology C
inst✝² : HasWeakSheafify J (Type v)
inst✝¹ : HasSheafify J AddCommGrpCat
inst✝ : HasExt (Sheaf J AddCommGrpCat)
S : J.MayerVietorisSquare
F : Sheaf J AddCommGrpCat
n₀ n₁ : ℕ
h : n₀ + 1 = n₁
⊢ 1 + n₀ = n₁

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Topos.Sheaf

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
J : GrothendieckTopology C
F G : Sheaf J (Type (max u v))
m : F ⟶ G
inst✝ : Mono m
⊢ Subfunctor.range (Presheaf.χ m.hom) ≤ closedSieves J

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.SheafCohomology.MayerVietoris

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
J : GrothendieckTopology C
inst✝² : HasWeakSheafify J (Type v)
inst✝¹ : HasSheafify J AddCommGrpCat
inst✝ : HasExt (Sheaf J AddCommGrpCat)
S : J.MayerVietorisSquare
F : Sheaf J AddCommGrpCat
n₀ n₁ : ℕ
h : n₀ + 1 = n₁
⊢ (S.sequence F n₀ n₁ h).map' 0 1 homMk₁._proof_4 _proof_450✝ ≫
      ((F.H' n₀ S.X₂).biprodIsoProd (F.H' n₀ S.X₃) ≪≫ Ext.biprodAddEquiv.symm.toAddCommGrpIso).hom =
    (Iso.refl ((S.sequence F n₀ n₁ h).obj' 0 _proof_449✝)).hom ≫
      (Ext.contravariantSequence ⋯ F n₀ n₁ ⋯).map' 0 1 homMk₁._proof_4 _proof_450✝

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.SheafCohomology.MayerVietoris

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
J : GrothendieckTopology C
inst✝² : HasWeakSheafify J (Type v)
inst✝¹ : HasSheafify J AddCommGrpCat
inst✝ : HasExt (Sheaf J AddCommGrpCat)
S : J.MayerVietorisSquare
F : Sheaf J AddCommGrpCat
n₀ n₁ : ℕ
h : n₀ + 1 = n₁
⊢ (S.sequence F n₀ n₁ h).map' 1 2 _proof_246✝ _proof_451✝ ≫ (Iso.refl ((S.sequence F n₀ n₁ h).obj' 2 _proof_451✝)).hom =
    ((F.H' n₀ S.X₂).biprodIsoProd (F.H' n₀ S.X₃) ≪≫ Ext.biprodAddEquiv.symm.toAddCommGrpIso).hom ≫
      (Ext.contravariantSequence ⋯ F n₀ n₁ ⋯).map' 1 2 _proof_246✝ _proof_451✝

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Topos.Sheaf

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
J : GrothendieckTopology C
F G : Sheaf J (Type (max u v))
m : F ⟶ G
inst✝ : Mono m
⊢ (m.hom ≫ Subfunctor.lift (Presheaf.χ m.hom) ⋯) ≫ (closedSieves J).ι =
    (((isTerminalTerminal J Types.isTerminalPUnit).from F).hom ≫ Subfunctor.lift (Presheaf.truth C) ⋯) ≫
      (closedSieves J).ι

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.SheafCohomology.MayerVietoris

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
J : GrothendieckTopology C
inst✝² : HasWeakSheafify J (Type v)
inst✝¹ : HasSheafify J AddCommGrpCat
inst✝ : HasExt (Sheaf J AddCommGrpCat)
S : J.MayerVietorisSquare
F : Sheaf J AddCommGrpCat
n₀ n₁ : ℕ
h : n₀ + 1 = n₁
⊢ (S.sequence F n₀ n₁ h).map' 2 3 _proof_299✝ _proof_452✝ ≫ (Iso.refl ((S.sequence F n₀ n₁ h).obj' 3 _proof_452✝)).hom =
    (Iso.refl ((S.sequence F n₀ n₁ h).obj' 2 _proof_451✝)).hom ≫
      (Ext.contravariantSequence ⋯ F n₀ n₁ ⋯).map' 2 3 _proof_299✝ _proof_452✝

Tactic:
by

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

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝¹⁵ : Category.{v_1, u_1} C
inst✝¹⁴ : Category.{v_2, u_2} D
inst✝¹³ : HasZeroObject C
inst✝¹² : HasZeroObject D
inst✝¹¹ : Preadditive C
inst✝¹⁰ : Preadditive D
inst✝⁹ : HasShift C ℤ
inst✝⁸ : HasShift D ℤ
inst✝⁷ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁶ : ∀ (n : ℤ), (shiftFunctor D n).Additive
inst✝⁵ : Pretriangulated C
inst✝⁴ : Pretriangulated D
F : C ⥤ D
G : D ⥤ C
adj : F ⊣ G
inst✝³ : F.CommShift ℤ
inst✝² : G.CommShift ℤ
inst✝¹ : adj.CommShift ℤ
inst✝ : F.IsTriangulated
T : Triangle D
hT : T ∈ distinguishedTriangles
this : G.Additive
Z : C
f : G.obj T.obj₂ ⟶ Z
g : Z ⟶ (shiftFunctor C 1).obj (G.obj T.obj₁)
mem : Triangle.mk (G.map T.mor₁) f g ∈ distinguishedTriangles
⊢ (F.mapTriangle.obj (Triangle.mk (G.map T.mor₁) f g)).mor₁ ≫ adj.counit.app T.obj₂ = adj.counit.app T.obj₁ ≫ T.mor₁

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.SheafCohomology.MayerVietoris

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
J : GrothendieckTopology C
inst✝² : HasWeakSheafify J (Type v)
inst✝¹ : HasSheafify J AddCommGrpCat
inst✝ : HasExt (Sheaf J AddCommGrpCat)
S : J.MayerVietorisSquare
F : Sheaf J AddCommGrpCat
n₀ n₁ : ℕ
h : n₀ + 1 = n₁
⊢ (S.sequence F n₀ n₁ h).map' 3 4 _proof_364✝ _proof_453✝ ≫
      ((F.H' n₁ S.X₂).biprodIsoProd (F.H' n₁ S.X₃) ≪≫ Ext.biprodAddEquiv.symm.toAddCommGrpIso).hom =
    (Iso.refl ((S.sequence F n₀ n₁ h).obj' 3 _proof_452✝)).hom ≫
      (Ext.contravariantSequence ⋯ F n₀ n₁ ⋯).map' 3 4 _proof_364✝ _proof_453✝

Tactic:
by

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

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝¹⁵ : Category.{v_1, u_1} C
inst✝¹⁴ : Category.{v_2, u_2} D
inst✝¹³ : HasZeroObject C
inst✝¹² : HasZeroObject D
inst✝¹¹ : Preadditive C
inst✝¹⁰ : Preadditive D
inst✝⁹ : HasShift C ℤ
inst✝⁸ : HasShift D ℤ
inst✝⁷ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁶ : ∀ (n : ℤ), (shiftFunctor D n).Additive
inst✝⁵ : Pretriangulated C
inst✝⁴ : Pretriangulated D
F : C ⥤ D
G : D ⥤ C
adj : F ⊣ G
inst✝³ : F.CommShift ℤ
inst✝² : G.CommShift ℤ
inst✝¹ : adj.CommShift ℤ
inst✝ : F.IsTriangulated
T : Triangle D
hT : T ∈ distinguishedTriangles
this : G.Additive
Z : C
f : G.obj T.obj₂ ⟶ Z
g : Z ⟶ (shiftFunctor C 1).obj (G.obj T.obj₁)
mem : Triangle.mk (G.map T.mor₁) f g ∈ distinguishedTriangles
h : F.obj Z ⟶ T.obj₃
h₁ : F.map f ≫ h = adj.counit.app T.obj₂ ≫ T.mor₂
h₂ :
  (F.map g ≫ (Functor.commShiftIso F 1).hom.app (G.obj T.obj₁)) ≫ (shiftFunctor D 1).map (adj.counit.app T.obj₁) =
    h ≫ T.mor₃
⊢ f ≫ adj.unit.app Z ≫ G.map h = G.map T.mor₂

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Topos.Sheaf

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
J : GrothendieckTopology C
F G : Sheaf J (Type (max u v))
m : F ⟶ G
inst✝ : Mono m
⊢ IsPullback m ((isTerminalTerminal J Types.isTerminalPUnit).from F) (χ m) (truth J)

Tactic:
by

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

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝¹⁵ : Category.{v_1, u_1} C
inst✝¹⁴ : Category.{v_2, u_2} D
inst✝¹³ : HasZeroObject C
inst✝¹² : HasZeroObject D
inst✝¹¹ : Preadditive C
inst✝¹⁰ : Preadditive D
inst✝⁹ : HasShift C ℤ
inst✝⁸ : HasShift D ℤ
inst✝⁷ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁶ : ∀ (n : ℤ), (shiftFunctor D n).Additive
inst✝⁵ : Pretriangulated C
inst✝⁴ : Pretriangulated D
F : C ⥤ D
G : D ⥤ C
adj : F ⊣ G
inst✝³ : F.CommShift ℤ
inst✝² : G.CommShift ℤ
inst✝¹ : adj.CommShift ℤ
inst✝ : F.IsTriangulated
T : Triangle D
hT : T ∈ distinguishedTriangles
this : G.Additive
Z : C
f : G.obj T.obj₂ ⟶ Z
g : Z ⟶ (shiftFunctor C 1).obj (G.obj T.obj₁)
mem : Triangle.mk (G.map T.mor₁) f g ∈ distinguishedTriangles
h : F.obj Z ⟶ T.obj₃
h₁ : F.map f ≫ h = adj.counit.app T.obj₂ ≫ T.mor₂
h₂ :
  (F.map g ≫ (Functor.commShiftIso F 1).hom.app (G.obj T.obj₁)) ≫ (shiftFunctor D 1).map (adj.counit.app T.obj₁) =
    h ≫ T.mor₃
h₁' : f ≫ adj.unit.app Z ≫ G.map h = G.map T.mor₂
⊢ g ≫ (Functor.commShiftIso G 1).inv.app T.obj₁ = (adj.homEquiv Z T.obj₃) h ≫ G.map T.mor₃

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.SheafCohomology.MayerVietoris

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
J : GrothendieckTopology C
inst✝² : HasWeakSheafify J (Type v)
inst✝¹ : HasSheafify J AddCommGrpCat
inst✝ : HasExt (Sheaf J AddCommGrpCat)
S : J.MayerVietorisSquare
F : Sheaf J AddCommGrpCat
n₀ n₁ : ℕ
h : n₀ + 1 = n₁
⊢ (S.sequence F n₀ n₁ h).map' 4 5 _proof_453✝ _proof_454✝ ≫ (Iso.refl ((S.sequence F n₀ n₁ h).obj' 5 _proof_454✝)).hom =
    ((F.H' n₁ S.X₂).biprodIsoProd (F.H' n₁ S.X₃) ≪≫ Ext.biprodAddEquiv.symm.toAddCommGrpIso).hom ≫
      (Ext.contravariantSequence ⋯ F n₀ n₁ ⋯).map' 4 5 _proof_453✝ _proof_454✝

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Topos.Sheaf

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
J : GrothendieckTopology C
F G : Sheaf J (Type (max u v))
m : F ⟶ G
inst✝ : Mono m
χ' : G ⟶ Ω J
hχ' : IsPullback m ((isTerminalTerminal J Types.isTerminalPUnit).from F) χ' (truth J)
⊢ CommSq (𝟙 G.obj) χ'.hom (χ'.hom ≫ (closedSieves J).ι) (closedSieves J).ι

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Topos.Sheaf

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
J : GrothendieckTopology C
F G : Sheaf J (Type (max u v))
m : F ⟶ G
inst✝ : Mono m
χ' : G ⟶ Ω J
hχ' : IsPullback m ((isTerminalTerminal J Types.isTerminalPUnit).from F) χ' (truth J)
pb : IsPullback (𝟙 G.obj) χ'.hom (χ'.hom ≫ (closedSieves J).ι) (closedSieves J).ι
⊢ IsPullback m.hom ?refine_1 χ'.hom (truth J).hom

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Topos.Sheaf

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
J : GrothendieckTopology C
F G : Sheaf J (Type (max u v))
m : F ⟶ G
inst✝ : Mono m
χ' : G ⟶ Ω J
hχ' : IsPullback m ((isTerminalTerminal J Types.isTerminalPUnit).from F) χ' (truth J)
⊢ χ' = χ m

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.Opposite.Functor

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_2, u_2} D
inst✝² : HasShift C ℤ
inst✝¹ : HasShift D ℤ
F : C ⥤ D
inst✝ : F.CommShift ℤ
⊢ ∀ (a b : ℤ), -(a + b) = -a + -b

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.Opposite.Functor

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : Category.{v_2, u_2} D
inst✝⁴ : HasShift C ℤ
inst✝³ : HasShift D ℤ
F : C ⥤ D
inst✝² : F.CommShift ℤ
G : C ⥤ D
inst✝¹ : G.CommShift ℤ
τ : F ⟶ G
inst✝ : NatTrans.CommShift τ ℤ
⊢ ∀ (a b : ℤ), -(a + b) = -a + -b

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.Opposite.Functor

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : Category.{v_2, u_2} D
inst✝⁴ : HasShift C ℤ
inst✝³ : HasShift D ℤ
F : C ⥤ D
inst✝² : F.CommShift ℤ
G : D ⥤ C
inst✝¹ : G.CommShift ℤ
adj : F ⊣ G
inst✝ : adj.CommShift ℤ
⊢ ∀ (a b : ℤ), -(a + b) = -a + -b

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.Opposite.Functor

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : Category.{v_2, u_2} D
inst✝⁴ : HasShift C ℤ
inst✝³ : HasShift D ℤ
F : C ⥤ D
inst✝² : F.CommShift ℤ
G : D ⥤ C
inst✝¹ : G.CommShift ℤ
adj : F ⊣ G
inst✝ : adj.CommShift ℤ
⊢ adj.op = PullbackShift.adjunction (AddMonoidHom.mk' (fun n ↦ -n) ⋯) (OppositeShift.adjunction ℤ adj)

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : HasZeroObject C
inst✝³ : HasShift C ℤ
inst✝² : Preadditive C
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
P : ObjectProperty C
⊢ P.triangEnvelopeIter 0 = (P.shiftClosure ℤ).binaryProductsClosure.retractClosure

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : HasZeroObject C
inst✝³ : HasShift C ℤ
inst✝² : Preadditive C
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
P : ObjectProperty C
n : ℕ
⊢ P.triangEnvelopeIter (n + 1) =
    ((P.shiftClosure ℤ).binaryProductsClosure.retractClosure.extensionProduct (P.triangEnvelopeIter n)).retractClosure

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.Opposite.Functor

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : Category.{v_2, u_2} D
inst✝⁴ : HasShift C ℤ
inst✝³ : HasShift D ℤ
F : C ⥤ D
inst✝² : F.CommShift ℤ
G : D ⥤ C
inst✝¹ : G.CommShift ℤ
adj : F ⊣ G
inst✝ : adj.CommShift ℤ
eq : adj.op = PullbackShift.adjunction (AddMonoidHom.mk' (fun n ↦ -n) ⋯) (OppositeShift.adjunction ℤ adj)
⊢ ∀ (a b : ℤ), -(a + b) = -a + -b

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.Opposite.Functor

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : Category.{v_2, u_2} D
inst✝⁴ : HasShift C ℤ
inst✝³ : HasShift D ℤ
F : C ⥤ D
inst✝² : F.CommShift ℤ
G : D ⥤ C
inst✝¹ : G.CommShift ℤ
adj : F ⊣ G
inst✝ : adj.CommShift ℤ
⊢ adj.op.CommShift ℤ

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.Opposite.Functor

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_2, u_2} D
inst✝² : HasShift C ℤ
inst✝¹ : HasShift D ℤ
F : C ⥤ D
inst✝ : F.CommShift ℤ
X : Cᵒᵖ
n m : ℤ
h : n + m = 0
⊢ m = -n

Tactic:
by

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

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✝ : IsTriangulated C
n : ℕ
⊢ P.triangEnvelopeIter (n + 1) =
    ((P.triangEnvelopeIter n).extensionProduct (P.shiftClosure ℤ).binaryProductsClosure.retractClosure).retractClosure

Tactic:
by

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

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✝ : IsTriangulated C
n m n' : ℕ
h : n = n' + 1
⊢ P.triangEnvelopeIter (n + m) = ((P.triangEnvelopeIter n').extensionProduct (P.triangEnvelopeIter m)).retractClosure

Tactic:
by

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

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✝ : IsTriangulated C
n m m' : ℕ
h : m = m' + 1
⊢ P.triangEnvelopeIter (n + m) = ((P.triangEnvelopeIter n).extensionProduct (P.triangEnvelopeIter m')).retractClosure

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.Opposite.Functor

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_2, u_2} D
inst✝² : HasShift C ℤ
inst✝¹ : HasShift D ℤ
F : C ⥤ D
inst✝ : F.CommShift ℤ
X : Cᵒᵖ
n m : ℤ
h : n + m = 0
⊢ (commShiftIso F.op n).hom.app X =
    (F.map ((shiftFunctorOpIso C n m h).hom.app X).unop).op ≫
      ((commShiftIso F m).inv.app (unop X)).op ≫ (shiftFunctorOpIso D n m h).inv.app (op (F.obj (unop X)))

Tactic:
obtain rfl : m = -n := by lia
  rfl

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.Opposite.Functor

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_2, u_2} D
inst✝² : HasShift C ℤ
inst✝¹ : HasShift D ℤ
F : C ⥤ D
inst✝ : F.CommShift ℤ
X : Cᵒᵖ
n m : ℤ
h : n + m = 0
⊢ (commShiftIso F.op n).hom.app X =
    (F.map ((shiftFunctorOpIso C n m h).hom.app X).unop).op ≫
      ((commShiftIso F m).inv.app (unop X)).op ≫ (shiftFunctorOpIso D n m h).inv.app (op (F.obj (unop X)))

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : HasZeroObject C
inst✝³ : HasShift C ℤ
inst✝² : Preadditive C
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
P : ObjectProperty C
n m : ℕ
h : n ≤ m
⊢ P.triangEnvelopeIter n ≤ P.triangEnvelopeIter m

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : HasZeroObject C
inst✝³ : HasShift C ℤ
inst✝² : Preadditive C
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
P : ObjectProperty C
n : ℕ
⊢ (P.shiftClosure ℤ).binaryProductsClosure.retractClosure ≤ P.triangEnvelopeIter n

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : HasZeroObject C
inst✝³ : HasShift C ℤ
inst✝² : Preadditive C
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
P : ObjectProperty C
⊢ ∀ {X Y : C} (x : Retract X Y), P.triangEnvelope Y → P.triangEnvelope X

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : HasZeroObject C
inst✝³ : HasShift C ℤ
inst✝² : Preadditive C
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
P : ObjectProperty C
a : ℤ
⊢ P.triangEnvelope ≤ P.triangEnvelope.shift a

Tactic:
by

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

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✝ : IsTriangulated C
⊢ P.triangEnvelope.IsTriangulatedClosed₂

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.Opposite.Functor

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_2, u_2} D
inst✝² : HasShift C ℤ
inst✝¹ : HasShift D ℤ
F : C ⥤ D
inst✝ : F.CommShift ℤ
X : Cᵒᵖ
n m : ℤ
h : n + m = 0
⊢ (commShiftIso F.op n).inv.app X =
    (shiftFunctorOpIso D n m h).hom.app (op (F.obj (unop X))) ≫
      ((commShiftIso F m).hom.app (unop X)).op ≫ (F.map ((shiftFunctorOpIso C n m h).inv.app X).unop).op

Tactic:
by

Goal state after: