mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.CategoryTheory.Enriched.FunctorCategory

Goal state before:
pp: V : Type u₁
inst✝⁸ : Category.{v₁, u₁} V
inst✝⁷ : MonoidalCategory V
C : Type u₂
inst✝⁶ : Category.{v₂, u₂} C
J : Type u₃
inst✝⁵ : Category.{v₃, u₃} J
K : Type u₄
inst✝⁴ : Category.{v₄, u₄} K
inst✝³ : EnrichedOrdinaryCategory V C
F₁ F₂ F₃ F₄ : J ⥤ C
inst✝² : HasFunctorEnrichedHom V F₁ F₂
inst✝¹ : HasFunctorEnrichedHom V F₂ F₃
inst✝ : HasFunctorEnrichedHom V F₁ F₃
j j' : J
f : j ⟶ j'
k : Under j'
⊢ (precompEnrichedHom' V (Under.map f) (Iso.refl (Under.map f ⋙ Under.forget j ⋙ F₁))
            (Iso.refl (Under.map f ⋙ Under.forget j ⋙ F₂)) ≫
          end_.π (diagram V (Under.forget j' ⋙ F₁) (Under.forget j' ⋙ F₂)) k ⊗ₘ
        precompEnrichedHom' V (Under.map f) (Iso.refl (Under.map f ⋙ Under.forget j ⋙ F₂))
            (Iso.refl (Under.map f ⋙ Under.forget j ⋙ F₃)) ≫
          end_.π (diagram V (Under.forget j' ⋙ F₂) (Under.forget j' ⋙ F₃)) k) ≫
      eComp V (F₁.obj k.right) (F₂.obj k.right) (F₃.obj k.right) =
    enrichedComp V (Under.forget j ⋙ F₁) (Under.forget j ⋙ F₂) (Under.forget j ⋙ F₃) ≫
      precompEnrichedHom' V (Under.map f) (Iso.refl (Under.map f ⋙ Under.forget j ⋙ F₁))
          (Iso.refl (Under.map f ⋙ Under.forget j ⋙ F₃)) ≫
        end_.π (diagram V (Under.forget j' ⋙ F₁) (Under.forget j' ⋙ F₃)) k

Tactic:
]

Goal state after:
V : Type u₁
inst✝⁸ : Category.{v₁, u₁} V
inst✝⁷ : MonoidalCategory V
C : Type u₂
inst✝⁶ : Category.{v₂, u₂} C
J : Type u₃
inst✝⁵ : Category.{v₃, u₃} J
K : Type u₄
inst✝⁴ : Category.{v₄, u₄} K
inst✝³ : EnrichedOrdinaryCategory V C
F₁ F₂ F₃ F₄ : J ⥤ C
inst✝² : HasFunctorEnrichedHom V F₁ F₂
inst✝¹ : HasFunctorEnrichedHom V F₂ F₃
inst✝ : HasFunctorEnrichedHom V F₁ F₃
j j' : J
f : j ⟶ j'
k : Under j'
⊢ (precompEnrichedHom' V (Under.map f) (Iso.refl (Under.map f ⋙ Under.forget j ⋙ F₁))
            (Iso.refl (Under.map f ⋙ Under.forget j ⋙ F₂)) ≫
          end_.π (diagram V (Under.forget j' ⋙ F₁) (Under.forget j' ⋙ F₂)) k ⊗ₘ
        precompEnrichedHom' V (Under.map f) (Iso.refl (Under.map f ⋙ Under.forget j ⋙ F₂))
            (Iso.refl (Under.map f ⋙ Under.forget j ⋙ F₃)) ≫
          end_.π (diagram V (Under.forget j' ⋙ F₂) (Under.forget j' ⋙ F₃)) k) ≫
      eComp V (F₁.obj k.right) (F₂.obj k.right) (F₃.obj k.right) =
    enrichedComp V (Under.forget j ⋙ F₁) (Under.forget j ⋙ F₂) (Under.forget j ⋙ F₃) ≫
      precompEnrichedHom' V (Under.map f) (Iso.refl (Under.map f ⋙ Under.forget j ⋙ F₁))
          (Iso.refl (Under.map f ⋙ Under.forget j ⋙ F₃)) ≫
        end_.π (diagram V (Under.forget j' ⋙ F₁) (Under.forget j' ⋙ F₃)) k
Module: Mathlib.CategoryTheory.Enriched.FunctorCategory

Goal state before:
pp: V : Type u₁
inst✝¹⁰ : Category.{v₁, u₁} V
inst✝⁹ : MonoidalCategory V
C : Type u₂
inst✝⁸ : Category.{v₂, u₂} C
J : Type u₃
inst✝⁷ : Category.{v₃, u₃} J
inst✝⁶ : EnrichedOrdinaryCategory V C
F₁ F₂ F₃ F₄ : J ⥤ C
inst✝⁵ : HasFunctorEnrichedHom V F₁ F₂
inst✝⁴ : HasFunctorEnrichedHom V F₂ F₃
inst✝³ : HasFunctorEnrichedHom V F₃ F₄
inst✝² : HasFunctorEnrichedHom V F₁ F₃
inst✝¹ : HasFunctorEnrichedHom V F₂ F₄
inst✝ : HasFunctorEnrichedHom V F₁ F₄
j : J
⊢ enrichedHom V (Under.forget j ⋙ F₁) (Under.forget j ⋙ F₂) ◁
        enrichedComp V (Under.forget j ⋙ F₂) (Under.forget j ⋙ F₃) (Under.forget j ⋙ F₄) ≫
      enrichedComp V (Under.forget j ⋙ F₁) (Under.forget j ⋙ F₂) (Under.forget j ⋙ F₄) =
    enrichedHom V (Under.forget j ⋙ F₁) (Under.forget j ⋙ F₂) ◁
        enrichedComp V (Under.forget j ⋙ F₂) (Under.forget j ⋙ F₃) (Under.forget j ⋙ F₄) ≫
      enrichedComp V (Under.forget j ⋙ F₁) (Under.forget j ⋙ F₂) (Under.forget j ⋙ F₄)

Tactic:
]

Goal state after:
Module: Mathlib.CategoryTheory.FiberedCategory.Grothendieck

Goal state before:
pp: case h.base
𝒮 : Type u_1
inst✝ : Category.{v_1, u_1} 𝒮
F : LocallyDiscrete 𝒮ᵒᵖ ⥤ᵖ Cat
S : 𝒮
Y : (forget F).Fiber S
hYS : (fiberInclusion.obj Y).base = S
⊢ (fiberInclusion.obj ((Fiber.inducedFunctor ⋯).obj (hYS ▸ (fiberInclusion.obj Y).fiber))).base =
    (fiberInclusion.obj Y).base

Tactic:
simp [hYS]

Goal state after:
Module: Mathlib.CategoryTheory.FiberedCategory.Grothendieck

Goal state before:
pp: case h.fiber
𝒮 : Type u_1
inst✝ : Category.{v_1, u_1} 𝒮
F : LocallyDiscrete 𝒮ᵒᵖ ⥤ᵖ Cat
S : 𝒮
Y : (forget F).Fiber S
hYS : (fiberInclusion.obj Y).base = S
⊢ (fiberInclusion.obj ((Fiber.inducedFunctor ⋯).obj (hYS ▸ (fiberInclusion.obj Y).fiber))).fiber ≍
    (fiberInclusion.obj Y).fiber

Tactic:
simp [hYS]

Goal state after:
Module: Mathlib.CategoryTheory.Enriched.FunctorCategory

Goal state before:
pp: V : Type u₁
inst✝¹⁰ : Category.{v₁, u₁} V
inst✝⁹ : MonoidalCategory V
C : Type u₂
inst✝⁸ : Category.{v₂, u₂} C
J : Type u₃
inst✝⁷ : Category.{v₃, u₃} J
inst✝⁶ : EnrichedOrdinaryCategory V C
F₁ F₂ F₃ : J ⥤ C
inst✝⁵ : HasFunctorEnrichedHom V F₁ F₂
inst✝⁴ : HasEnrichedHom V F₁ F₂
inst✝³ : HasFunctorEnrichedHom V F₂ F₃
inst✝² : HasEnrichedHom V F₂ F₃
inst✝¹ : HasFunctorEnrichedHom V F₁ F₃
inst✝ : HasEnrichedHom V F₁ F₃
f : F₁ ⟶ F₂
g : F₂ ⟶ F₃
j : J
k : Under j
⊢ (λ_ (𝟙_ V)).inv ≫
      ((homEquiv V) f ⊗ₘ (homEquiv V) g) ≫
        enrichedComp V F₁ F₂ F₃ ≫
          enrichedHomπ V F₁ F₃ ((Under.forget j).obj k) ≫
            eHomWhiskerRight V ((Iso.refl (Under.forget j ⋙ F₁)).inv.app k) (F₃.obj ((Under.forget j).obj k)) ≫
              eHomWhiskerLeft V ((Under.forget j ⋙ F₁).obj k) ((Iso.refl (Under.forget j ⋙ F₃)).hom.app k) =
    (λ_ (𝟙_ V)).inv ≫
      ((homEquiv V) f ≫ precompEnrichedHom V F₁ F₂ (Under.forget j) ⊗ₘ
          (homEquiv V) g ≫ precompEnrichedHom V F₂ F₃ (Under.forget j)) ≫
        (end_.π (diagram V (Under.forget j ⋙ F₁) (Under.forget j ⋙ F₂)) k ⊗ₘ
            end_.π (diagram V (Under.forget j ⋙ F₂) (Under.forget j ⋙ F₃)) k) ≫
          eComp V (Opposite.unop ((Under.forget j ⋙ F₁).op.obj (Opposite.op k))) ((Under.forget j ⋙ F₂).obj k)
            ((Under.forget j ⋙ F₃).obj k)

Tactic:
]

Goal state after:
V : Type u₁
inst✝¹⁰ : Category.{v₁, u₁} V
inst✝⁹ : MonoidalCategory V
C : Type u₂
inst✝⁸ : Category.{v₂, u₂} C
J : Type u₃
inst✝⁷ : Category.{v₃, u₃} J
inst✝⁶ : EnrichedOrdinaryCategory V C
F₁ F₂ F₃ : J ⥤ C
inst✝⁵ : HasFunctorEnrichedHom V F₁ F₂
inst✝⁴ : HasEnrichedHom V F₁ F₂
inst✝³ : HasFunctorEnrichedHom V F₂ F₃
inst✝² : HasEnrichedHom V F₂ F₃
inst✝¹ : HasFunctorEnrichedHom V F₁ F₃
inst✝ : HasEnrichedHom V F₁ F₃
f : F₁ ⟶ F₂
g : F₂ ⟶ F₃
j : J
k : Under j
⊢ (λ_ (𝟙_ V)).inv ≫
      ((homEquiv V) f ⊗ₘ (homEquiv V) g) ≫
        enrichedComp V F₁ F₂ F₃ ≫
          enrichedHomπ V F₁ F₃ ((Under.forget j).obj k) ≫
            eHomWhiskerRight V ((Iso.refl (Under.forget j ⋙ F₁)).inv.app k) (F₃.obj ((Under.forget j).obj k)) ≫
              eHomWhiskerLeft V ((Under.forget j ⋙ F₁).obj k) ((Iso.refl (Under.forget j ⋙ F₃)).hom.app k) =
    (λ_ (𝟙_ V)).inv ≫
      ((homEquiv V) f ≫ precompEnrichedHom V F₁ F₂ (Under.forget j) ⊗ₘ
          (homEquiv V) g ≫ precompEnrichedHom V F₂ F₃ (Under.forget j)) ≫
        (end_.π (diagram V (Under.forget j ⋙ F₁) (Under.forget j ⋙ F₂)) k ⊗ₘ
            end_.π (diagram V (Under.forget j ⋙ F₂) (Under.forget j ⋙ F₃)) k) ≫
          eComp V (Opposite.unop ((Under.forget j ⋙ F₁).op.obj (Opposite.op k))) ((Under.forget j ⋙ F₂).obj k)
            ((Under.forget j ⋙ F₃).obj k)
Module: Mathlib.CategoryTheory.Enriched.FunctorCategory

Goal state before:
pp: V : Type u₁
inst✝¹⁰ : Category.{v₁, u₁} V
inst✝⁹ : MonoidalCategory V
C : Type u₂
inst✝⁸ : Category.{v₂, u₂} C
J : Type u₃
inst✝⁷ : Category.{v₃, u₃} J
inst✝⁶ : EnrichedOrdinaryCategory V C
F₁ F₂ F₃ : J ⥤ C
inst✝⁵ : HasFunctorEnrichedHom V F₁ F₂
inst✝⁴ : HasEnrichedHom V F₁ F₂
inst✝³ : HasFunctorEnrichedHom V F₂ F₃
inst✝² : HasEnrichedHom V F₂ F₃
inst✝¹ : HasFunctorEnrichedHom V F₁ F₃
inst✝ : HasEnrichedHom V F₁ F₃
f : F₁ ⟶ F₂
g : F₂ ⟶ F₃
j : J
k : Under j
⊢ ((homEquiv V) (f ≫ g) ≫ precompEnrichedHom V F₁ F₃ (Under.forget j)) ≫
      end_.π (diagram V (Under.forget j ⋙ F₁) (Under.forget j ⋙ F₃)) k =
    ((λ_ (𝟙_ V)).inv ≫
        ((homEquiv V) f ≫ precompEnrichedHom V F₁ F₂ (Under.forget j) ⊗ₘ
            (homEquiv V) g ≫ precompEnrichedHom V F₂ F₃ (Under.forget j)) ≫
          enrichedComp V (Under.forget j ⋙ F₁) (Under.forget j ⋙ F₂) (Under.forget j ⋙ F₃)) ≫
      end_.π (diagram V (Under.forget j ⋙ F₁) (Under.forget j ⋙ F₃)) k

Tactic:
rw [homEquiv_comp, assoc, assoc, assoc, assoc, assoc, end_.lift_π, enrichedComp_π]

Goal state after:
V : Type u₁
inst✝¹⁰ : Category.{v₁, u₁} V
inst✝⁹ : MonoidalCategory V
C : Type u₂
inst✝⁸ : Category.{v₂, u₂} C
J : Type u₃
inst✝⁷ : Category.{v₃, u₃} J
inst✝⁶ : EnrichedOrdinaryCategory V C
F₁ F₂ F₃ : J ⥤ C
inst✝⁵ : HasFunctorEnrichedHom V F₁ F₂
inst✝⁴ : HasEnrichedHom V F₁ F₂
inst✝³ : HasFunctorEnrichedHom V F₂ F₃
inst✝² : HasEnrichedHom V F₂ F₃
inst✝¹ : HasFunctorEnrichedHom V F₁ F₃
inst✝ : HasEnrichedHom V F₁ F₃
f : F₁ ⟶ F₂
g : F₂ ⟶ F₃
j : J
k : Under j
⊢ (λ_ (𝟙_ V)).inv ≫
      ((homEquiv V) f ⊗ₘ (homEquiv V) g) ≫
        enrichedComp V F₁ F₂ F₃ ≫
          enrichedHomπ V F₁ F₃ ((Under.forget j).obj k) ≫
            eHomWhiskerRight V ((Iso.refl (Under.forget j ⋙ F₁)).inv.app k) (F₃.obj ((Under.forget j).obj k)) ≫
              eHomWhiskerLeft V ((Under.forget j ⋙ F₁).obj k) ((Iso.refl (Under.forget j ⋙ F₃)).hom.app k) =
    (λ_ (𝟙_ V)).inv ≫
      ((homEquiv V) f ≫ precompEnrichedHom V F₁ F₂ (Under.forget j) ⊗ₘ
          (homEquiv V) g ≫ precompEnrichedHom V F₂ F₃ (Under.forget j)) ≫
        (end_.π (diagram V (Under.forget j ⋙ F₁) (Under.forget j ⋙ F₂)) k ⊗ₘ
            end_.π (diagram V (Under.forget j ⋙ F₂) (Under.forget j ⋙ F₃)) k) ≫
          eComp V (Opposite.unop ((Under.forget j ⋙ F₁).op.obj (Opposite.op k))) ((Under.forget j ⋙ F₂).obj k)
            ((Under.forget j ⋙ F₃).obj k)
Module: Mathlib.CategoryTheory.Functor.Derived.LeftDerived

Goal state before:
pp: C : Type u_3
D : Type u_1
H : Type u_2
inst✝⁴ : Category.{v_1, u_3} C
inst✝³ : Category.{v_3, u_1} D
inst✝² : Category.{v_5, u_2} H
LF : D ⥤ H
F : C ⥤ H
L : C ⥤ D
α : L ⋙ LF ⟶ F
W : MorphismProperty C
inst✝¹ : L.IsLocalization W
inst✝ : LF.IsLeftDerivedFunctor α W
G : D ⥤ H
β : L ⋙ G ⟶ F
⊢ G.IsRightKanExtension β ↔ IsIso (LF.leftDerivedLift α W G β)

Tactic:
]

Goal state after:
C : Type u_3
D : Type u_1
H : Type u_2
inst✝⁴ : Category.{v_1, u_3} C
inst✝³ : Category.{v_3, u_1} D
inst✝² : Category.{v_5, u_2} H
LF : D ⥤ H
F : C ⥤ H
L : C ⥤ D
α : L ⋙ LF ⟶ F
W : MorphismProperty C
inst✝¹ : L.IsLocalization W
inst✝ : LF.IsLeftDerivedFunctor α W
G : D ⥤ H
β : L ⋙ G ⟶ F
⊢ G.IsRightKanExtension β ↔ IsIso (LF.leftDerivedLift α W G β)
Module: Mathlib.CategoryTheory.Functor.Derived.PointwiseLeftDerived

Goal state before:
pp: C : Type u₁
D : Type u₂
H : Type u₃
inst✝³ : Category.{v₁, u₁} C
inst✝² : Category.{v₂, u₂} D
inst✝¹ : Category.{v₃, u₃} H
F : C ⥤ H
L : C ⥤ D
W : MorphismProperty C
inst✝ : L.IsLocalization W
X : C
⊢ F.HasPointwiseLeftDerivedFunctorAt W X ↔ W.Q.HasPointwiseRightKanExtensionAt F (W.Q.obj X)

Tactic:
]

Goal state after:
C : Type u₁
D : Type u₂
H : Type u₃
inst✝³ : Category.{v₁, u₁} C
inst✝² : Category.{v₂, u₂} D
inst✝¹ : Category.{v₃, u₃} H
F : C ⥤ H
L : C ⥤ D
W : MorphismProperty C
inst✝ : L.IsLocalization W
X : C
⊢ F.HasPointwiseLeftDerivedFunctorAt W X ↔ W.Q.HasPointwiseRightKanExtensionAt F (W.Q.obj X)
Module: Mathlib.CategoryTheory.Functor.Derived.PointwiseLeftDerived

Goal state before:
pp: C : Type u₁
D : Type u₂
H : Type u₃
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : Category.{v₂, u₂} D
inst✝² : Category.{v₃, u₃} H
F : C ⥤ H
L : C ⥤ D
W : MorphismProperty C
inst✝¹ : L.IsLocalization W
X : C
inst✝ : F.HasPointwiseLeftDerivedFunctorAt W X
⊢ F.HasPointwiseLeftDerivedFunctorAt W X

Tactic:
]

Goal state after:
C : Type u₁
D : Type u₂
H : Type u₃
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : Category.{v₂, u₂} D
inst✝² : Category.{v₃, u₃} H
F : C ⥤ H
L : C ⥤ D
W : MorphismProperty C
inst✝¹ : L.IsLocalization W
X : C
inst✝ : F.HasPointwiseLeftDerivedFunctorAt W X
⊢ F.HasPointwiseLeftDerivedFunctorAt W X
Module: Mathlib.CategoryTheory.Functor.Derived.LeftDerived

Goal state before:
pp: C : Type u_1
D : Type u_3
H : Type u_2
inst✝³ : Category.{v_1, u_1} C
inst✝² : Category.{v_3, u_3} D
inst✝¹ : Category.{v_5, u_2} H
F : C ⥤ H
L : C ⥤ D
W : MorphismProperty C
inst✝ : L.IsLocalization W
this : F.HasLeftDerivedFunctor W ↔ W.Q.HasRightKanExtension F
⊢ W.Q.HasRightKanExtension F ↔ W.Q.HasRightKanExtension F

Tactic:
]

Goal state after:
Module: Mathlib.CategoryTheory.Functor.Derived.LeftDerived

Goal state before:
pp: C : Type u_1
H : Type u_2
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Category.{v_5, u_2} H
F F' : C ⥤ H
e : F ≅ F'
W : MorphismProperty C
⊢ W.Q.HasRightKanExtension F' ↔ W.Q.HasRightKanExtension F'

Tactic:
]

Goal state after:
Module: Mathlib.CategoryTheory.Functor.Derived.PointwiseLeftDerived

Goal state before:
pp: C : Type u₁
D : Type u₂
H : Type u₃
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : Category.{v₂, u₂} D
inst✝² : Category.{v₃, u₃} H
F : C ⥤ H
L : C ⥤ D
W : MorphismProperty C
inst✝¹ : F.HasPointwiseLeftDerivedFunctor W
inst✝ : L.IsLocalization W
Y : D
this : L.EssSurj
⊢ F.HasPointwiseLeftDerivedFunctorAt W (L.objPreimage Y)

Tactic:
]

Goal state after:
C : Type u₁
D : Type u₂
H : Type u₃
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : Category.{v₂, u₂} D
inst✝² : Category.{v₃, u₃} H
F : C ⥤ H
L : C ⥤ D
W : MorphismProperty C
inst✝¹ : F.HasPointwiseLeftDerivedFunctor W
inst✝ : L.IsLocalization W
Y : D
this : L.EssSurj
⊢ F.HasPointwiseLeftDerivedFunctorAt W (L.objPreimage Y)
Module: Mathlib.CategoryTheory.Functor.Derived.PointwiseLeftDerived

Goal state before:
pp: case refine_1
C : Type u₁
D : Type u₂
H : Type u₃
inst✝³ : Category.{v₁, u₁} C
inst✝² : Category.{v₂, u₂} D
inst✝¹ : Category.{v₃, u₃} H
F' : D ⥤ H
F : C ⥤ H
L : C ⥤ D
α : L ⋙ F' ⟶ F
W : MorphismProperty C
G : D ⥤ H
e : F ≅ L ⋙ G
inst✝ : L.IsLocalization W
Y : C
s : Cone (StructuredArrow.proj (L.obj Y) L ⋙ F)
j : StructuredArrow (L.obj Y) L
X₁ X₂ : C
f : X₁ ⟶ X₂
φ : L.obj Y ⟶ L.obj X₁
hφ :
  (s.π.app (StructuredArrow.mk (𝟙 (L.obj Y))) ≫ e.hom.app Y) ≫ G.map φ ≫ e.inv.app X₁ = s.π.app (StructuredArrow.mk φ)
eq : s.π.app (StructuredArrow.mk (φ ≫ L.map f)) = s.π.app (StructuredArrow.mk φ) ≫ F.map f
⊢ (s.π.app (StructuredArrow.mk (𝟙 (L.obj Y))) ≫ e.hom.app Y) ≫ G.map (φ ≫ L.map f) ≫ e.inv.app X₂ =
    s.π.app (StructuredArrow.mk (φ ≫ L.map f))

Tactic:
]

Goal state after:
case refine_1
C : Type u₁
D : Type u₂
H : Type u₃
inst✝³ : Category.{v₁, u₁} C
inst✝² : Category.{v₂, u₂} D
inst✝¹ : Category.{v₃, u₃} H
F' : D ⥤ H
F : C ⥤ H
L : C ⥤ D
α : L ⋙ F' ⟶ F
W : MorphismProperty C
G : D ⥤ H
e : F ≅ L ⋙ G
inst✝ : L.IsLocalization W
Y : C
s : Cone (StructuredArrow.proj (L.obj Y) L ⋙ F)
j : StructuredArrow (L.obj Y) L
X₁ X₂ : C
f : X₁ ⟶ X₂
φ : L.obj Y ⟶ L.obj X₁
hφ :
  (s.π.app (StructuredArrow.mk (𝟙 (L.obj Y))) ≫ e.hom.app Y) ≫ G.map φ ≫ e.inv.app X₁ = s.π.app (StructuredArrow.mk φ)
eq : s.π.app (StructuredArrow.mk (φ ≫ L.map f)) = s.π.app (StructuredArrow.mk φ) ≫ F.map f
⊢ (s.π.app (StructuredArrow.mk (𝟙 (L.obj Y))) ≫ e.hom.app Y) ≫ G.map (φ ≫ L.map f) ≫ e.inv.app X₂ =
    s.π.app (StructuredArrow.mk (φ ≫ L.map f))
Module: Mathlib.CategoryTheory.Functor.Derived.PointwiseLeftDerived

Goal state before:
pp: case refine_1
C : Type u₁
D : Type u₂
H : Type u₃
inst✝³ : Category.{v₁, u₁} C
inst✝² : Category.{v₂, u₂} D
inst✝¹ : Category.{v₃, u₃} H
F' : D ⥤ H
F : C ⥤ H
L : C ⥤ D
α : L ⋙ F' ⟶ F
W : MorphismProperty C
G : D ⥤ H
e : F ≅ L ⋙ G
inst✝ : L.IsLocalization W
Y : C
s : Cone (StructuredArrow.proj (L.obj Y) L ⋙ F)
j : StructuredArrow (L.obj Y) L
X₁ X₂ : C
f : X₁ ⟶ X₂
φ : L.obj Y ⟶ L.obj X₁
hφ : s.π.app (StructuredArrow.mk (𝟙 (L.obj Y))) ≫ e.hom.app Y ≫ G.map φ ≫ e.inv.app X₁ = s.π.app (StructuredArrow.mk φ)
eq : s.π.app (StructuredArrow.mk (φ ≫ L.map f)) = s.π.app (StructuredArrow.mk φ) ≫ F.map f
⊢ (s.π.app (StructuredArrow.mk (𝟙 (L.obj Y))) ≫ e.hom.app Y) ≫ G.map (φ ≫ L.map f) ≫ e.inv.app X₂ =
    s.π.app (StructuredArrow.mk (φ ≫ L.map f))

Tactic:
]

Goal state after:
case refine_1
C : Type u₁
D : Type u₂
H : Type u₃
inst✝³ : Category.{v₁, u₁} C
inst✝² : Category.{v₂, u₂} D
inst✝¹ : Category.{v₃, u₃} H
F' : D ⥤ H
F : C ⥤ H
L : C ⥤ D
α : L ⋙ F' ⟶ F
W : MorphismProperty C
G : D ⥤ H
e : F ≅ L ⋙ G
inst✝ : L.IsLocalization W
Y : C
s : Cone (StructuredArrow.proj (L.obj Y) L ⋙ F)
j : StructuredArrow (L.obj Y) L
X₁ X₂ : C
f : X₁ ⟶ X₂
φ : L.obj Y ⟶ L.obj X₁
hφ : s.π.app (StructuredArrow.mk (𝟙 (L.obj Y))) ≫ e.hom.app Y ≫ G.map φ ≫ e.inv.app X₁ = s.π.app (StructuredArrow.mk φ)
eq : s.π.app (StructuredArrow.mk (φ ≫ L.map f)) = s.π.app (StructuredArrow.mk φ) ≫ F.map f
⊢ (s.π.app (StructuredArrow.mk (𝟙 (L.obj Y))) ≫ e.hom.app Y) ≫ G.map (φ ≫ L.map f) ≫ e.inv.app X₂ =
    s.π.app (StructuredArrow.mk (φ ≫ L.map f))
Module: Mathlib.CategoryTheory.Functor.ReflectsIso.Exact

Goal state before:
pp: C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
I : Type u_2
D : I → Type u_3
inst✝⁴ : (i : I) → Category.{v_2, u_3} (D i)
F : (i : I) → C ⥤ D i
hP : JointlyReflectIsomorphisms F
inst✝³ : HasZeroMorphisms C
inst✝² : (i : I) → HasZeroMorphisms (D i)
inst✝¹ : ∀ (i : I), (F i).PreservesZeroMorphisms
inst✝ : HasZeroObject C
X : C
hX : ∀ (i : I), IsZero ((F i).obj X)
φ : 0 ⟶ X := 0
⊢ ∀ (i : I), IsIso ((F i).map φ)

Tactic:
]

Goal state after:
C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
I : Type u_2
D : I → Type u_3
inst✝⁴ : (i : I) → Category.{v_2, u_3} (D i)
F : (i : I) → C ⥤ D i
hP : JointlyReflectIsomorphisms F
inst✝³ : HasZeroMorphisms C
inst✝² : (i : I) → HasZeroMorphisms (D i)
inst✝¹ : ∀ (i : I), (F i).PreservesZeroMorphisms
inst✝ : HasZeroObject C
X : C
hX : ∀ (i : I), IsZero ((F i).obj X)
φ : 0 ⟶ X := 0
⊢ ∀ (i : I), IsIso ((F i).map φ)
Module: Mathlib.CategoryTheory.Functor.Derived.PointwiseLeftDerived

Goal state before:
pp: C : Type u₁
D : Type u₂
H : Type u₃
inst✝³ : Category.{v₁, u₁} C
inst✝² : Category.{v₂, u₂} D
inst✝¹ : Category.{v₃, u₃} H
F' : D ⥤ H
F : C ⥤ H
L : C ⥤ D
α : L ⋙ F' ⟶ F
W : MorphismProperty C
G : D ⥤ H
e : F ≅ L ⋙ G
inst✝ : L.IsLocalization W
Y : C
s : Cone (StructuredArrow.proj (L.obj Y) L ⋙ F)
j : StructuredArrow (L.obj Y) L
X₁ X₂ : C
w : X₁ ⟶ X₂
hw : W w
φ : L.obj Y ⟶ L.obj X₂
hφ :
  (s.π.app (StructuredArrow.mk (𝟙 (L.obj Y))) ≫ e.hom.app Y) ≫
      ((RightExtension.mk G e.inv).coneAt (L.obj Y)).π.app (StructuredArrow.mk φ) =
    s.π.app (StructuredArrow.mk φ)
this : IsIso (L.map w)
⊢ IsIso (e.hom.app X₁ ≫ (L ⋙ G).map w ≫ e.inv.app X₂)

Tactic:
]

Goal state after:
C : Type u₁
D : Type u₂
H : Type u₃
inst✝³ : Category.{v₁, u₁} C
inst✝² : Category.{v₂, u₂} D
inst✝¹ : Category.{v₃, u₃} H
F' : D ⥤ H
F : C ⥤ H
L : C ⥤ D
α : L ⋙ F' ⟶ F
W : MorphismProperty C
G : D ⥤ H
e : F ≅ L ⋙ G
inst✝ : L.IsLocalization W
Y : C
s : Cone (StructuredArrow.proj (L.obj Y) L ⋙ F)
j : StructuredArrow (L.obj Y) L
X₁ X₂ : C
w : X₁ ⟶ X₂
hw : W w
φ : L.obj Y ⟶ L.obj X₂
hφ :
  (s.π.app (StructuredArrow.mk (𝟙 (L.obj Y))) ≫ e.hom.app Y) ≫
      ((RightExtension.mk G e.inv).coneAt (L.obj Y)).π.app (StructuredArrow.mk φ) =
    s.π.app (StructuredArrow.mk φ)
this : IsIso (L.map w)
⊢ IsIso (e.hom.app X₁ ≫ (L ⋙ G).map w ≫ e.inv.app X₂)
Module: Mathlib.CategoryTheory.Functor.ReflectsIso.Exact

Goal state before:
pp: C : Type u_1
inst✝⁷ : Category.{v_1, u_1} C
I : Type u_2
D : I → Type u_3
inst✝⁶ : (i : I) → Category.{v_2, u_3} (D i)
F : (i : I) → C ⥤ D i
hP : JointlyReflectIsomorphisms F
inst✝⁵ : HasZeroMorphisms C
inst✝⁴ : (i : I) → HasZeroMorphisms (D i)
inst✝³ : ∀ (i : I), (F i).PreservesZeroMorphisms
inst✝² : CategoryWithHomology C
inst✝¹ : ∀ (i : I), (F i).PreservesHomology
inst✝ : HasZeroObject C
S : ShortComplex C
hS : ∀ (i : I), IsZero (S.map (F i)).homology
⊢ ∀ (i : I), IsZero ((F i).obj S.homology)

Tactic:
]

Goal state after:
C : Type u_1
inst✝⁷ : Category.{v_1, u_1} C
I : Type u_2
D : I → Type u_3
inst✝⁶ : (i : I) → Category.{v_2, u_3} (D i)
F : (i : I) → C ⥤ D i
hP : JointlyReflectIsomorphisms F
inst✝⁵ : HasZeroMorphisms C
inst✝⁴ : (i : I) → HasZeroMorphisms (D i)
inst✝³ : ∀ (i : I), (F i).PreservesZeroMorphisms
inst✝² : CategoryWithHomology C
inst✝¹ : ∀ (i : I), (F i).PreservesHomology
inst✝ : HasZeroObject C
S : ShortComplex C
hS : ∀ (i : I), IsZero (S.map (F i)).homology
⊢ ∀ (i : I), IsZero ((F i).obj S.homology)
Module: Mathlib.CategoryTheory.Functor.Derived.PointwiseLeftDerived

Goal state before:
pp: case refine_2
C : Type u₁
D : Type u₂
H : Type u₃
inst✝³ : Category.{v₁, u₁} C
inst✝² : Category.{v₂, u₂} D
inst✝¹ : Category.{v₃, u₃} H
F' : D ⥤ H
F : C ⥤ H
L : C ⥤ D
α : L ⋙ F' ⟶ F
W : MorphismProperty C
G : D ⥤ H
e : F ≅ L ⋙ G
inst✝ : L.IsLocalization W
Y : C
s : Cone (StructuredArrow.proj (L.obj Y) L ⋙ F)
j : StructuredArrow (L.obj Y) L
X₁ X₂ : C
w : X₁ ⟶ X₂
hw : W w
φ : L.obj Y ⟶ L.obj X₂
hφ :
  (s.π.app (StructuredArrow.mk (𝟙 (L.obj Y))) ≫ e.hom.app Y) ≫ G.map φ ≫ e.inv.app X₂ = s.π.app (StructuredArrow.mk φ)
this : IsIso (F.map w)
eq : s.π.app (StructuredArrow.mk φ) = s.π.app (StructuredArrow.mk (φ ≫ (Localization.isoOfHom L W w hw).inv)) ≫ F.map w
⊢ (s.π.app (StructuredArrow.mk (𝟙 (L.obj Y))) ≫ e.hom.app Y) ≫
      G.map (φ ≫ (Localization.isoOfHom L W w hw).inv) ≫ e.inv.app X₁ =
    s.π.app (StructuredArrow.mk (φ ≫ (Localization.isoOfHom L W w hw).inv))

Tactic:
]

Goal state after:
case refine_2
C : Type u₁
D : Type u₂
H : Type u₃
inst✝³ : Category.{v₁, u₁} C
inst✝² : Category.{v₂, u₂} D
inst✝¹ : Category.{v₃, u₃} H
F' : D ⥤ H
F : C ⥤ H
L : C ⥤ D
α : L ⋙ F' ⟶ F
W : MorphismProperty C
G : D ⥤ H
e : F ≅ L ⋙ G
inst✝ : L.IsLocalization W
Y : C
s : Cone (StructuredArrow.proj (L.obj Y) L ⋙ F)
j : StructuredArrow (L.obj Y) L
X₁ X₂ : C
w : X₁ ⟶ X₂
hw : W w
φ : L.obj Y ⟶ L.obj X₂
hφ :
  (s.π.app (StructuredArrow.mk (𝟙 (L.obj Y))) ≫ e.hom.app Y) ≫ G.map φ ≫ e.inv.app X₂ = s.π.app (StructuredArrow.mk φ)
this : IsIso (F.map w)
eq : s.π.app (StructuredArrow.mk φ) = s.π.app (StructuredArrow.mk (φ ≫ (Localization.isoOfHom L W w hw).inv)) ≫ F.map w
⊢ (s.π.app (StructuredArrow.mk (𝟙 (L.obj Y))) ≫ e.hom.app Y) ≫
      G.map (φ ≫ (Localization.isoOfHom L W w hw).inv) ≫ e.inv.app X₁ =
    s.π.app (StructuredArrow.mk (φ ≫ (Localization.isoOfHom L W w hw).inv))
Module: Mathlib.CategoryTheory.Functor.ReflectsIso.Exact

Goal state before:
pp: C : Type u_1
inst✝⁶ : Category.{v_1, u_1} C
I : Type u_2
D : I → Type u_3
inst✝⁵ : (i : I) → Category.{v_2, u_3} (D i)
F : (i : I) → C ⥤ D i
hP : JointlyReflectIsomorphisms F
inst✝⁴ : Abelian C
inst✝³ : (i : I) → Abelian (D i)
inst✝² : CategoryWithHomology C
inst✝¹ : ∀ (i : I), PreservesFiniteLimits (F i)
inst✝ : ∀ (i : I), PreservesFiniteColimits (F i)
S : ShortComplex C
hS : ∀ (i : I), (S.map (F i)).ShortExact
⊢ ∀ (i : I), Mono ((F i).map S.f)

Tactic:
]

Goal state after:
C : Type u_1
inst✝⁶ : Category.{v_1, u_1} C
I : Type u_2
D : I → Type u_3
inst✝⁵ : (i : I) → Category.{v_2, u_3} (D i)
F : (i : I) → C ⥤ D i
hP : JointlyReflectIsomorphisms F
inst✝⁴ : Abelian C
inst✝³ : (i : I) → Abelian (D i)
inst✝² : CategoryWithHomology C
inst✝¹ : ∀ (i : I), PreservesFiniteLimits (F i)
inst✝ : ∀ (i : I), PreservesFiniteColimits (F i)
S : ShortComplex C
hS : ∀ (i : I), (S.map (F i)).ShortExact
⊢ ∀ (i : I), Mono ((F i).map S.f)
Module: Mathlib.CategoryTheory.Functor.ReflectsIso.Limits

Goal state before:
pp: C : Type u_1
inst✝⁴ : Category.{?u.4, u_1} C
I : Type u_2
D : I → Type u_3
inst✝³ : (i : I) → Category.{?u.13, u_3} (D i)
F : (i : I) → C ⥤ D i
hF : JointlyReflectIsomorphisms F
J : Type u_4
inst✝² : Category.{v_1, u_4} J
G : J ⥤ C
c : Cone G
hc : (i : I) → IsLimit ((F i).mapCone c)
inst✝¹ : HasLimit G
inst✝ : ∀ (i : I), PreservesLimit G (F i)
⊢ ∀ (i : I), IsIso ((F i).map (limit.lift G c))

Tactic:
]

Goal state after:
C : Type u_1
inst✝⁴ : Category.{?u.4, u_1} C
I : Type u_2
D : I → Type u_3
inst✝³ : (i : I) → Category.{?u.13, u_3} (D i)
F : (i : I) → C ⥤ D i
hF : JointlyReflectIsomorphisms F
J : Type u_4
inst✝² : Category.{v_1, u_4} J
G : J ⥤ C
c : Cone G
hc : (i : I) → IsLimit ((F i).mapCone c)
inst✝¹ : HasLimit G
inst✝ : ∀ (i : I), PreservesLimit G (F i)
⊢ ∀ (i : I), IsIso ((F i).map (limit.lift G c))