awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.DayConvolution.Closed (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
V : Type u₂
inst_1 : Category.{v₂, u₂} V
inst_2 : MonoidalCategory.{v₁, u₁} C
inst_3 : MonoidalCategory.{v₂, u₂} V
inst_4 : MonoidalClosed.{v₂, u₂} V
F G_1 H G : Functor.{v₁, v₂, u₁, u₂} C V
inst_5 : DayConvolution F G
ℌ : DayConvolutionInternalHom F (@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _) H
c c' : C
f : c ⟶ c'
j : C
⊢ G.map f ≫
      Wedge.IsLimit.lift (ℌ.isLimitWedge c')
          (fun (c'_1 : C) ↦ MonoidalClosed.curry (NatTrans.app (DayConvolution.unit F G) (c'_1, c'))) ⋯ ≫
        ℌ.π c' j =
    Wedge.IsLimit.lift (ℌ.isLimitWedge c)
        (fun (c' : C) ↦ MonoidalClosed.curry (NatTrans.app (DayConvolution.unit F G) (c', c))) ⋯ ≫
      ℌ.π c j ≫ Functor.map (ihom (F.obj j)) ((@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _).map (j ◁ f))
have rw : Eq (α :=
  G.obj c ⟶
    Functor.obj.{v₂, v₂, u₂, u₂} (ihom (F.obj j))
      ((@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _).obj (tensorObj.{v₁, u₁} (𝒞 := inst) j c')))
  (G.map f ≫ MonoidalClosed.curry (NatTrans.app (DayConvolution.unit F G) (j, c')))
  (MonoidalClosed.curry (NatTrans.app (DayConvolution.unit F G) (j, c)) ≫
    Functor.map (ihom (F.obj j)) ((@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _).map (j ◁ f)))
Tactic: rw [← Wedge.mk_ι
        (F := dayConvolutionInternalHomDiagramFunctor F |>.obj _ |>.obj c)
        (H.obj c) (ℌ.π c) (ℌ.hπ c),
      ← Wedge.mk_ι
        (F := dayConvolutionInternalHomDiagramFunctor F |>.obj _ |>.obj c')
        (H.obj c') (ℌ.π c') (ℌ.hπ c'),
      Wedge.IsLimit.lift_ι_assoc, Wedge.IsLimit.lift_ι]
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.DayConvolution.Closed (hypothesis)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
V : Type u₂
inst_1 : Category.{v₂, u₂} V
inst_2 : MonoidalCategory.{v₁, u₁} C
inst_3 : MonoidalCategory.{v₂, u₂} V
inst_4 : MonoidalClosed.{v₂, u₂} V
F G_1 H G : Functor.{v₁, v₂, u₁, u₂} C V
inst_5 : DayConvolution F G
ℌ : DayConvolutionInternalHom F (@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _) H
c c' : C
f : c ⟶ c'
j : C
⊢ Eq (α :=
    G.obj c ⟶
      Functor.obj.{v₂, v₂, u₂, u₂} (ihom (F.obj j))
        ((@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _).obj (tensorObj.{v₁, u₁} (𝒞 := inst) j c')))
    (G.map f ≫ MonoidalClosed.curry (NatTrans.app (DayConvolution.unit F G) (j, c')))
    (MonoidalClosed.curry (NatTrans.app (DayConvolution.unit F G) (j, c)) ≫
      Functor.map (ihom (F.obj j)) ((@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _).map (j ◁ f)))
have this : Eq (α :=
  tensorObj.{v₂, u₂} (𝒞 := inst_1) (F.obj j) (G.obj c) ⟶
    (tensor C ⋙ @DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _).obj (j, c'))
  ((F.map (𝟙 j) ⊗ₘ G.map f) ≫ NatTrans.app (DayConvolution.unit F G) (j, c'))
  (NatTrans.app (DayConvolution.unit F G) (j, c) ≫
    (@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _).map (𝟙 j ⊗ₘ f))
Tactic: have := DayConvolution.unit_naturality F G (𝟙 j) f
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.DayConvolution.Closed (hypothesis)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
V : Type u₂
inst_1 : Category.{v₂, u₂} V
inst_2 : MonoidalCategory.{v₁, u₁} C
inst_3 : MonoidalCategory.{v₂, u₂} V
inst_4 : MonoidalClosed.{v₂, u₂} V
F G_1 H G : Functor.{v₁, v₂, u₁, u₂} C V
inst_5 : DayConvolution F G
ℌ : DayConvolutionInternalHom F (@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _) H
c c' : C
f : c ⟶ c'
j : C
this :
  Eq (α :=
    tensorObj.{v₂, u₂} (𝒞 := inst_1) (F.obj j) (G.obj c) ⟶
      (tensor C ⋙ @DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _).obj (j, c'))
    ((F.map (𝟙 j) ⊗ₘ G.map f) ≫ NatTrans.app (DayConvolution.unit F G) (j, c'))
    (NatTrans.app (DayConvolution.unit F G) (j, c) ≫
      (@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _).map (𝟙 j ⊗ₘ f))
⊢ Eq (α :=
    G.obj c ⟶
      Functor.obj.{v₂, v₂, u₂, u₂} (ihom (F.obj j))
        ((@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _).obj (tensorObj.{v₁, u₁} (𝒞 := inst) j c')))
    (G.map f ≫ MonoidalClosed.curry (NatTrans.app (DayConvolution.unit F G) (j, c')))
    (MonoidalClosed.curry (NatTrans.app (DayConvolution.unit F G) (j, c)) ≫
      Functor.map (ihom (F.obj j)) ((@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _).map (j ◁ f)))
have this₁ : Eq (α :=
  tensorObj.{v₂, u₂} (𝒞 := inst_1) (F.obj j) (G.obj c) ⟶
    (tensor C ⋙ @DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _).obj (j, c'))
  (F.obj j ◁ G.map f ≫ NatTrans.app (DayConvolution.unit F G) (j, c'))
  (NatTrans.app (DayConvolution.unit F G) (j, c) ≫
    (@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _).map (j ◁ f))
Tactic: simp only [Functor.map_id, id_tensorHom] at this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.DayConvolution.Closed (hypothesis)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
V : Type u₂
inst_1 : Category.{v₂, u₂} V
inst_2 : MonoidalCategory.{v₁, u₁} C
inst_3 : MonoidalCategory.{v₂, u₂} V
inst_4 : MonoidalClosed.{v₂, u₂} V
F G_1 H G : Functor.{v₁, v₂, u₁, u₂} C V
inst_5 : DayConvolution F G
ℌ : DayConvolutionInternalHom F (@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _) H
c c' : C
f : c ⟶ c'
j : C
this :
  Eq (α :=
    tensorObj.{v₂, u₂} (𝒞 := inst_1) (F.obj j) (G.obj c) ⟶
      (tensor C ⋙ @DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _).obj (j, c'))
    (F.obj j ◁ G.map f ≫ NatTrans.app (DayConvolution.unit F G) (j, c'))
    (NatTrans.app (DayConvolution.unit F G) (j, c) ≫
      (@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _).map (j ◁ f))
⊢ Eq (α :=
    G.obj c ⟶
      Functor.obj.{v₂, v₂, u₂, u₂} (ihom (F.obj j))
        ((@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _).obj (tensorObj.{v₁, u₁} (𝒞 := inst) j c')))
    (G.map f ≫ MonoidalClosed.curry (NatTrans.app (DayConvolution.unit F G) (j, c')))
    (MonoidalClosed.curry (NatTrans.app (DayConvolution.unit F G) (j, c)) ≫
      Functor.map (ihom (F.obj j)) ((@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _).map (j ◁ f)))
have this₁ : Eq (α :=
  G.obj c ⟶
    Functor.obj.{v₂, v₂, u₂, u₂} (ihom (F.obj j))
      ((tensor C ⋙ @DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _).obj (j, c')))
  (MonoidalClosed.curry (F.obj j ◁ G.map f ≫ NatTrans.app (DayConvolution.unit F G) (j, c')))
  (MonoidalClosed.curry
    (NatTrans.app (DayConvolution.unit F G) (j, c) ≫
      (@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _).map (j ◁ f)))
Tactic: replace this := congrArg MonoidalClosed.curry this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.DayConvolution.Closed (hypothesis)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
V : Type u₂
inst_1 : Category.{v₂, u₂} V
inst_2 : MonoidalCategory.{v₁, u₁} C
inst_3 : MonoidalCategory.{v₂, u₂} V
inst_4 : MonoidalClosed.{v₂, u₂} V
F G_1 H G : Functor.{v₁, v₂, u₁, u₂} C V
inst_5 : DayConvolution F G
ℌ : DayConvolutionInternalHom F (@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _) H
c c' : C
f : c ⟶ c'
j : C
this :
  Eq (α :=
    G.obj c ⟶
      Functor.obj.{v₂, v₂, u₂, u₂} (ihom (F.obj j))
        ((tensor C ⋙ @DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _).obj (j, c')))
    (MonoidalClosed.curry (F.obj j ◁ G.map f ≫ NatTrans.app (DayConvolution.unit F G) (j, c')))
    (MonoidalClosed.curry
      (NatTrans.app (DayConvolution.unit F G) (j, c) ≫
        (@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _).map (j ◁ f)))
⊢ Eq (α :=
    G.obj c ⟶
      Functor.obj.{v₂, v₂, u₂, u₂} (ihom (F.obj j))
        ((@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _).obj (tensorObj.{v₁, u₁} (𝒞 := inst) j c')))
    (G.map f ≫ MonoidalClosed.curry (NatTrans.app (DayConvolution.unit F G) (j, c')))
    (MonoidalClosed.curry (NatTrans.app (DayConvolution.unit F G) (j, c)) ≫
      Functor.map (ihom (F.obj j)) ((@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _).map (j ◁ f)))
have this₁ : Eq (α :=
  G.obj c ⟶
    Functor.obj.{v₂, v₂, u₂, u₂} (ihom (F.obj j))
      ((tensor C ⋙ @DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _).obj (j, c')))
  (CategoryStruct.comp (Y :=
    Functor.obj.{v₂, v₂, u₂, u₂} (ihom (F.obj j)) (tensorObj.{v₂, u₂} (𝒞 := inst_1) (F.obj j) (G.obj c')))
    (MonoidalClosed.curry (F.obj j ◁ G.map f))
    (Functor.map (ihom (F.obj j)) (NatTrans.app (DayConvolution.unit F G) (j, c'))))
  (MonoidalClosed.curry (NatTrans.app (DayConvolution.unit F G) (j, c)) ≫
    Functor.map (ihom (F.obj j)) ((@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _).map (j ◁ f)))
Tactic: simp only [MonoidalClosed.curry_natural_right] at this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.DayConvolution.Closed (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
V : Type u₂
inst_1 : Category.{v₂, u₂} V
inst_2 : MonoidalCategory.{v₁, u₁} C
inst_3 : MonoidalCategory.{v₂, u₂} V
inst_4 : MonoidalClosed.{v₂, u₂} V
F G_1 H G : Functor.{v₁, v₂, u₁, u₂} C V
inst_5 : DayConvolution F G
ℌ : DayConvolutionInternalHom F (@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _) H
c c' : C
f : c ⟶ c'
j : C
this :
  Eq (α :=
    G.obj c ⟶
      Functor.obj.{v₂, v₂, u₂, u₂} (ihom (F.obj j))
        ((tensor C ⋙ @DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _).obj (j, c')))
    (CategoryStruct.comp (Y :=
      Functor.obj.{v₂, v₂, u₂, u₂} (ihom (F.obj j)) (tensorObj.{v₂, u₂} (𝒞 := inst_1) (F.obj j) (G.obj c')))
      (MonoidalClosed.curry (F.obj j ◁ G.map f))
      (Functor.map (ihom (F.obj j)) (NatTrans.app (DayConvolution.unit F G) (j, c'))))
    (MonoidalClosed.curry (NatTrans.app (DayConvolution.unit F G) (j, c)) ≫
      Functor.map (ihom (F.obj j)) ((@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _).map (j ◁ f)))
⊢ Eq (α :=
    G.obj c ⟶
      Functor.obj.{v₂, v₂, u₂, u₂} (ihom (F.obj j))
        ((@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _).obj (tensorObj.{v₁, u₁} (𝒞 := inst) j c')))
    (G.map f ≫ MonoidalClosed.curry (NatTrans.app (DayConvolution.unit F G) (j, c')))
    (MonoidalClosed.curry (NatTrans.app (DayConvolution.unit F G) (j, c)) ≫
      Functor.map (ihom (F.obj j)) ((@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _).map (j ◁ f)))
have rw : Eq (α :=
  G.obj c ⟶
    Functor.obj.{v₂, v₂, u₂, u₂} (ihom (F.obj j))
      ((@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _).obj (tensorObj.{v₁, u₁} (𝒞 := inst) j c')))
  (G.map f ≫ MonoidalClosed.curry (NatTrans.app (DayConvolution.unit F G) (j, c')))
  (CategoryStruct.comp (Y :=
    Functor.obj.{v₂, v₂, u₂, u₂} (ihom (F.obj j)) (tensorObj.{v₂, u₂} (𝒞 := inst_1) (F.obj j) (G.obj c')))
    (MonoidalClosed.curry (F.obj j ◁ G.map f))
    (Functor.map (ihom (F.obj j)) (NatTrans.app (DayConvolution.unit F G) (j, c'))))
Tactic: rw [← this]
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.DayConvolution.Closed (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
V : Type u₂
inst_1 : Category.{v₂, u₂} V
inst_2 : MonoidalCategory.{v₁, u₁} C
inst_3 : MonoidalCategory.{v₂, u₂} V
inst_4 : MonoidalClosed.{v₂, u₂} V
F H G : Functor.{v₁, v₂, u₁, u₂} C V
inst_5 : DayConvolution F G
ℌ : DayConvolutionInternalHom F (@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _) H
c j : C
⊢ ℌ.coev_app.app c ≫ ℌ.π c j = MonoidalClosed.curry (NatTrans.app (DayConvolution.unit F G) (j, c))
have dsimp : Eq (α :=
  G.obj c ⟶
    Functor.obj.{v₂, v₂, u₂, u₂} (ihom (F.obj j))
      ((@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _).obj (tensorObj.{v₁, u₁} (𝒞 := inst) j c)))
  (Wedge.IsLimit.lift (ℌ.isLimitWedge c)
      (fun (c' : C) ↦ MonoidalClosed.curry (NatTrans.app (DayConvolution.unit F G) (c', c))) sorry ≫
    ℌ.π c j)
  (MonoidalClosed.curry (NatTrans.app (DayConvolution.unit F G) (j, c)))
Tactic: dsimp [coev_app]
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.DayConvolution.Closed (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
V : Type u₂
inst_1 : Category.{v₂, u₂} V
inst_2 : MonoidalCategory.{v₁, u₁} C
inst_3 : MonoidalCategory.{v₂, u₂} V
inst_4 : MonoidalClosed.{v₂, u₂} V
F H G : Functor.{v₁, v₂, u₁, u₂} C V
inst_5 : DayConvolution F G
ℌ : DayConvolutionInternalHom F (@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _) H
c j : C
⊢ Eq (α :=
    G.obj c ⟶
      Functor.obj.{v₂, v₂, u₂, u₂} (ihom (F.obj j))
        ((@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _).obj (tensorObj.{v₁, u₁} (𝒞 := inst) j c)))
    (Wedge.IsLimit.lift (ℌ.isLimitWedge c)
        (fun (c' : C) ↦ MonoidalClosed.curry (NatTrans.app (DayConvolution.unit F G) (c', c))) ⋯ ≫
      ℌ.π c j)
    (MonoidalClosed.curry (NatTrans.app (DayConvolution.unit F G) (j, c)))
have rw : Eq (α :=
  G.obj c ⟶
    Functor.obj.{v₂, v₂, u₂, u₂} (ihom (F.obj j))
      ((@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _).obj (tensorObj.{v₁, u₁} (𝒞 := inst) j c)))
  (Wedge.IsLimit.lift (ℌ.isLimitWedge c)
      (fun (c' : C) ↦ MonoidalClosed.curry (NatTrans.app (DayConvolution.unit F G) (c', c))) sorry ≫
    Multifork.ι (Wedge.mk (H.obj c) (ℌ.π c) sorry) j)
  (MonoidalClosed.curry (NatTrans.app (DayConvolution.unit F G) (j, c)))
Tactic: rw [← Wedge.mk_ι
      (F := dayConvolutionInternalHomDiagramFunctor F |>.obj _ |>.obj c)
      (H.obj c) (ℌ.π c) (ℌ.hπ c),
    Wedge.IsLimit.lift_ι]
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.DayConvolution.Closed (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
V : Type u₂
inst_1 : Category.{v₂, u₂} V
inst_2 : MonoidalCategory.{v₁, u₁} C
inst_3 : MonoidalCategory.{v₂, u₂} V
inst_4 : MonoidalClosed.{v₂, u₂} V
F H G : Functor.{v₁, v₂, u₁, u₂} C V
inst_5 : DayConvolution F G
ℌ : DayConvolutionInternalHom F (@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _) H
c j : C
rw :
  Eq (α :=
    G.obj c ⟶
      Functor.obj.{v₂, v₂, u₂, u₂} (ihom (F.obj j))
        ((@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _).obj (tensorObj.{v₁, u₁} (𝒞 := inst) j c)))
    (Wedge.IsLimit.lift (ℌ.isLimitWedge c)
        (fun (c' : C) ↦ MonoidalClosed.curry (NatTrans.app (DayConvolution.unit F G) (c', c))) ⋯ ≫
      Multifork.ι (Wedge.mk (H.obj c) (ℌ.π c) ⋯) j)
    (MonoidalClosed.curry (NatTrans.app (DayConvolution.unit F G) (j, c)))
⊢ Eq (α :=
    G.obj c ⟶
      Functor.obj.{v₂, v₂, u₂, u₂} (ihom (F.obj j))
        ((@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _).obj (tensorObj.{v₁, u₁} (𝒞 := inst) j c)))
    (Wedge.IsLimit.lift (ℌ.isLimitWedge c)
        (fun (c' : C) ↦ MonoidalClosed.curry (NatTrans.app (DayConvolution.unit F G) (c', c))) ⋯ ≫
      ℌ.π c j)
    (MonoidalClosed.curry (NatTrans.app (DayConvolution.unit F G) (j, c)))
have rw₁ : Eq (α :=
  G.obj c ⟶
    Functor.obj.{v₂, v₂, u₂, u₂} (ihom (F.obj j))
      ((@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _).obj (tensorObj.{v₁, u₁} (𝒞 := inst) j c)))
  (MonoidalClosed.curry (NatTrans.app (DayConvolution.unit F G) (j, c)) :
    G.obj c ⟶
      Functor.obj.{v₂, v₂, u₂, u₂} (ihom (F.obj j))
        ((@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _).obj (tensorObj.{v₁, u₁} (𝒞 := inst) j c)))
  (MonoidalClosed.curry (NatTrans.app (DayConvolution.unit F G) (j, c)) :
    G.obj c ⟶
      Functor.obj.{v₂, v₂, u₂, u₂} (ihom (F.obj j))
        ((@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _).obj (tensorObj.{v₁, u₁} (𝒞 := inst) j c)))
Tactic: rw [← Wedge.mk_ι
      (F := dayConvolutionInternalHomDiagramFunctor F |>.obj _ |>.obj c)
      (H.obj c) (ℌ.π c) (ℌ.hπ c),
    Wedge.IsLimit.lift_ι]
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.DayConvolution.Closed (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
V : Type u₂
inst_1 : Category.{v₂, u₂} V
inst_2 : MonoidalCategory.{v₁, u₁} C
inst_3 : MonoidalCategory.{v₂, u₂} V
inst_4 : MonoidalClosed.{v₂, u₂} V
F H G : Functor.{v₁, v₂, u₁, u₂} C V
inst_5 : DayConvolution F G
ℌ : DayConvolutionInternalHom F (@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _) H
G' H' : Functor.{v₁, v₂, u₁, u₂} C V
inst_6 : DayConvolution F G'
η : G ⟶ G'
ℌ' : DayConvolutionInternalHom F (@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G' _) H'
⊢ η ≫ ℌ'.coev_app = ℌ.coev_app ≫ ℌ.map (DayConvolution.map (𝟙 F) η) ℌ'
have w : ∀ (c : C), (η ≫ ℌ'.coev_app).app c = (ℌ.coev_app ≫ ℌ.map (DayConvolution.map (𝟙 F) η) ℌ').app c
Tactic: ext c
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.DayConvolution.Closed (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
V : Type u₂
inst_1 : Category.{v₂, u₂} V
inst_2 : MonoidalCategory.{v₁, u₁} C
inst_3 : MonoidalCategory.{v₂, u₂} V
inst_4 : MonoidalClosed.{v₂, u₂} V
F H G : Functor.{v₁, v₂, u₁, u₂} C V
inst_5 : DayConvolution F G
ℌ : DayConvolutionInternalHom F (@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _) H
G' H' : Functor.{v₁, v₂, u₁, u₂} C V
inst_6 : DayConvolution F G'
η : G ⟶ G'
ℌ' : DayConvolutionInternalHom F (@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G' _) H'
c : C
⊢ (η ≫ ℌ'.coev_app).app c = (ℌ.coev_app ≫ ℌ.map (DayConvolution.map (𝟙 F) η) ℌ').app c
have w : η.app c ≫ ℌ'.coev_app.app c = ℌ.coev_app.app c ≫ (ℌ.map (DayConvolution.map (𝟙 F) η) ℌ').app c
Tactic: dsimp
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.DayConvolution.Closed (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
V : Type u₂
inst_1 : Category.{v₂, u₂} V
inst_2 : MonoidalCategory.{v₁, u₁} C
inst_3 : MonoidalCategory.{v₂, u₂} V
inst_4 : MonoidalClosed.{v₂, u₂} V
F H G : Functor.{v₁, v₂, u₁, u₂} C V
inst_5 : DayConvolution F G
ℌ : DayConvolutionInternalHom F (@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _) H
G' H' : Functor.{v₁, v₂, u₁, u₂} C V
inst_6 : DayConvolution F G'
η : G ⟶ G'
ℌ' : DayConvolutionInternalHom F (@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G' _) H'
c : C
⊢ η.app c ≫ ℌ'.coev_app.app c = ℌ.coev_app.app c ≫ (ℌ.map (DayConvolution.map (𝟙 F) η) ℌ').app c
have w : ∀ (j : (multicospanShapeEnd.{v₁, u₁} C).L),
  Eq (α :=
    G.obj c ⟶
      (multicospanIndexEnd
            (((dayConvolutionInternalHomDiagramFunctor F).obj
                  (@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G' _)).obj
              c)).left
        j)
    ((η.app c ≫ ℌ'.coev_app.app c) ≫ Multifork.ι (Wedge.mk (H'.obj c) (ℌ'.π c) sorry) j)
    ((ℌ.coev_app.app c ≫ (ℌ.map (DayConvolution.map (𝟙 F) η) ℌ').app c) ≫
      Multifork.ι (Wedge.mk (H'.obj c) (ℌ'.π c) sorry) j)
Tactic: apply Wedge.IsLimit.hom_ext <| ℌ'.isLimitWedge c
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.DayConvolution.Closed (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
V : Type u₂
inst_1 : Category.{v₂, u₂} V
inst_2 : MonoidalCategory.{v₁, u₁} C
inst_3 : MonoidalCategory.{v₂, u₂} V
inst_4 : MonoidalClosed.{v₂, u₂} V
F H G : Functor.{v₁, v₂, u₁, u₂} C V
inst_5 : DayConvolution F G
ℌ : DayConvolutionInternalHom F (@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _) H
G' H' : Functor.{v₁, v₂, u₁, u₂} C V
inst_6 : DayConvolution F G'
η : G ⟶ G'
ℌ' : DayConvolutionInternalHom F (@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G' _) H'
c : C
j : (multicospanShapeEnd.{v₁, u₁} C).L
⊢ Eq (α :=
    G.obj c ⟶
      (multicospanIndexEnd
            (((dayConvolutionInternalHomDiagramFunctor F).obj
                  (@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G' _)).obj
              c)).left
        j)
    ((η.app c ≫ ℌ'.coev_app.app c) ≫ Multifork.ι (Wedge.mk (H'.obj c) (ℌ'.π c) ⋯) j)
    ((ℌ.coev_app.app c ≫ (ℌ.map (DayConvolution.map (𝟙 F) η) ℌ').app c) ≫
      Multifork.ι (Wedge.mk (H'.obj c) (ℌ'.π c) ⋯) j)
have w : Eq (α :=
  tensorObj.{v₂, u₂} (𝒞 := inst_1) (unop (F.op.obj (op j))) (G.obj c) ⟶
    (tensorRight c ⋙ @DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G' _).obj j)
  (MonoidalClosed.uncurry ((η.app c ≫ ℌ'.coev_app.app c) ≫ Multifork.ι (Wedge.mk (H'.obj c) (ℌ'.π c) sorry) j))
  (MonoidalClosed.uncurry
    ((ℌ.coev_app.app c ≫ (ℌ.map (DayConvolution.map (𝟙 F) η) ℌ').app c) ≫
      Multifork.ι (Wedge.mk (H'.obj c) (ℌ'.π c) sorry) j))
Tactic: apply MonoidalClosed.uncurry_injective
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.DayConvolution.Closed (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
V : Type u₂
inst_1 : Category.{v₂, u₂} V
inst_2 : MonoidalCategory.{v₁, u₁} C
inst_3 : MonoidalCategory.{v₂, u₂} V
inst_4 : MonoidalClosed.{v₂, u₂} V
F H G : Functor.{v₁, v₂, u₁, u₂} C V
inst_5 : DayConvolution F G
ℌ : DayConvolutionInternalHom F (@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _) H
G' H' : Functor.{v₁, v₂, u₁, u₂} C V
inst_6 : DayConvolution F G'
η : G ⟶ G'
ℌ' : DayConvolutionInternalHom F (@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G' _) H'
c : C
j : (multicospanShapeEnd.{v₁, u₁} C).L
⊢ Eq (α :=
    tensorObj.{v₂, u₂} (𝒞 := inst_1) (unop (F.op.obj (op j))) (G.obj c) ⟶
      (tensorRight c ⋙ @DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G' _).obj j)
    (MonoidalClosed.uncurry ((η.app c ≫ ℌ'.coev_app.app c) ≫ Multifork.ι (Wedge.mk (H'.obj c) (ℌ'.π c) ⋯) j))
    (MonoidalClosed.uncurry
      ((ℌ.coev_app.app c ≫ (ℌ.map (DayConvolution.map (𝟙 F) η) ℌ').app c) ≫
        Multifork.ι (Wedge.mk (H'.obj c) (ℌ'.π c) ⋯) j))
have w : MonoidalClosed.uncurry ((η.app c ≫ ℌ'.coev_app.app c) ≫ ℌ'.π c j) =
  MonoidalClosed.uncurry ((ℌ.coev_app.app c ≫ (ℌ.map (DayConvolution.map (𝟙 F) η) ℌ').app c) ≫ ℌ'.π c j)
Tactic: dsimp
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.DayConvolution.Closed (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
V : Type u₂
inst_1 : Category.{v₂, u₂} V
inst_2 : MonoidalCategory.{v₁, u₁} C
inst_3 : MonoidalCategory.{v₂, u₂} V
inst_4 : MonoidalClosed.{v₂, u₂} V
F H G : Functor.{v₁, v₂, u₁, u₂} C V
inst_5 : DayConvolution F G
ℌ : DayConvolutionInternalHom F (@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _) H
G' H' : Functor.{v₁, v₂, u₁, u₂} C V
inst_6 : DayConvolution F G'
η : G ⟶ G'
ℌ' : DayConvolutionInternalHom F (@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G' _) H'
c : C
j : (multicospanShapeEnd.{v₁, u₁} C).L
⊢ MonoidalClosed.uncurry ((η.app c ≫ ℌ'.coev_app.app c) ≫ ℌ'.π c j) =
    MonoidalClosed.uncurry ((ℌ.coev_app.app c ≫ (ℌ.map (DayConvolution.map (𝟙 F) η) ℌ').app c) ≫ ℌ'.π c j)
have w : Eq (α :=
  tensorObj.{v₂, u₂} (𝒞 := inst_1) (F.obj j) (G.obj c) ⟶
    (@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G' _).obj (tensorObj.{v₁, u₁} (𝒞 := inst) j c))
  (MonoidalClosed.uncurry (η.app c ≫ MonoidalClosed.curry (NatTrans.app (DayConvolution.unit F G') (j, c))))
  (F.obj j ◁ η.app c ≫ NatTrans.app (DayConvolution.unit F G') (j, c))
Tactic: simp only [Category.assoc, coev_app_π, Functor.comp_obj, tensor_obj,
    map_app_comp_π, coev_app_π_assoc, MonoidalClosed.uncurry_natural_right,
    MonoidalClosed.uncurry_curry, DayConvolution.unit_app_map_app,
    NatTrans.id_app, id_tensorHom]