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 H : Functor.{v₁, v₂, u₁, u₂} C V
inst_5 : DayConvolution F H
ℌ : DayConvolutionInternalHom F G H
x y : C × C
f : x ⟶ y
this :
  Eq (α :=
    tensorObj.{v₂, u₂} (𝒞 := inst_1) (F.obj x.1) (H.obj x.2) ⟶
      tensorObj.{v₂, u₂} (𝒞 := inst_1) (F.obj x.1)
        (Functor.obj.{v₂, v₂, u₂, u₂} (ihom (F.obj x.1)) (G.obj (tensorObj.{v₁, u₁} (𝒞 := inst) y.1 x.2))))
    (F.obj x.1 ◁ ℌ.π x.2 x.1 ≫ F.obj x.1 ◁ Functor.map (ihom (F.obj x.1)) (G.map (f.1 ▷ x.2)))
    (F.obj x.1 ◁ ℌ.π x.2 y.1 ≫
      F.obj x.1 ◁ NatTrans.app (MonoidalClosed.pre (F.map f.1)) (G.obj (tensorObj.{v₁, u₁} (𝒞 := inst) y.1 x.2)))
⊢ (F.map f.1 ▷ H.obj x.2 ≫ F.obj y.1 ◁ H.map f.2) ≫ MonoidalClosed.uncurry (ℌ.π y.2 y.1) =
    MonoidalClosed.uncurry (ℌ.π x.2 x.1) ≫ G.map (f.1 ⊗ₘ f.2)
have simp : F.map f.1 ▷ H.obj x.2 ≫
    F.obj y.1 ◁ (ℌ.π x.2 y.1 ≫ Functor.map (ihom (F.obj y.1)) (G.map (y.1 ◁ f.2))) ≫
      NatTrans.app (ihom.ev (F.obj y.1)) (G.obj (tensorObj.{v₁, u₁} (𝒞 := inst) y.1 y.2)) =
  F.obj x.1 ◁ ℌ.π x.2 x.1 ≫
    NatTrans.app (ihom.ev (F.obj x.1)) (G.obj (tensorObj.{v₁, u₁} (𝒞 := inst) x.1 x.2)) ≫ G.map (f.1 ⊗ₘ f.2)
Tactic: simp only [Category.assoc, MonoidalClosed.uncurry_eq, Functor.id_obj,
          ← whiskerLeft_comp_assoc, map_comp_π]
-- 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 H : Functor.{v₁, v₂, u₁, u₂} C V
inst_5 : DayConvolution F H
ℌ : DayConvolutionInternalHom F G H
x y : C × C
f : x ⟶ y
this :
  Eq (α :=
    tensorObj.{v₂, u₂} (𝒞 := inst_1) (F.obj x.1) (H.obj x.2) ⟶
      tensorObj.{v₂, u₂} (𝒞 := inst_1) (F.obj x.1)
        (Functor.obj.{v₂, v₂, u₂, u₂} (ihom (F.obj x.1)) (G.obj (tensorObj.{v₁, u₁} (𝒞 := inst) y.1 x.2))))
    (F.obj x.1 ◁ ℌ.π x.2 x.1 ≫ F.obj x.1 ◁ Functor.map (ihom (F.obj x.1)) (G.map (f.1 ▷ x.2)))
    (F.obj x.1 ◁ ℌ.π x.2 y.1 ≫
      F.obj x.1 ◁ NatTrans.app (MonoidalClosed.pre (F.map f.1)) (G.obj (tensorObj.{v₁, u₁} (𝒞 := inst) y.1 x.2)))
⊢ F.map f.1 ▷ H.obj x.2 ≫
      F.obj y.1 ◁ (ℌ.π x.2 y.1 ≫ Functor.map (ihom (F.obj y.1)) (G.map (y.1 ◁ f.2))) ≫
        NatTrans.app (ihom.ev (F.obj y.1)) (G.obj (tensorObj.{v₁, u₁} (𝒞 := inst) y.1 y.2)) =
    F.obj x.1 ◁ ℌ.π x.2 x.1 ≫
      NatTrans.app (ihom.ev (F.obj x.1)) (G.obj (tensorObj.{v₁, u₁} (𝒞 := inst) x.1 x.2)) ≫ G.map (f.1 ⊗ₘ f.2)
have simp : F.obj x.1 ◁ ℌ.π x.2 y.1 ≫
    F.map f.1 ▷ Functor.obj.{v₂, v₂, u₂, u₂} (ihom (F.obj y.1)) (G.obj (tensorObj.{v₁, u₁} (𝒞 := inst) y.1 x.2)) ≫
      NatTrans.app (ihom.ev (F.obj y.1)) (G.obj (tensorObj.{v₁, u₁} (𝒞 := inst) y.1 x.2)) ≫ G.map (y.1 ◁ f.2) =
  F.obj x.1 ◁ ℌ.π x.2 x.1 ≫
    F.obj x.1 ◁ Functor.map (ihom (F.obj x.1)) (G.map (f.1 ▷ x.2)) ≫
      NatTrans.app (ihom.ev (F.obj x.1)) (G.obj (tensorObj.{v₁, u₁} (𝒞 := inst) y.1 x.2)) ≫ G.map (y.1 ◁ f.2)
Tactic: simp only [whiskerLeft_comp, Category.assoc, ihom.ev_naturality,
          Functor.comp_obj, curriedTensor_obj_obj, Functor.id_obj,
          ← whisker_exchange_assoc, tensorHom_def, Functor.map_comp,
          ← ihom.ev_naturality_assoc]
-- 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 H : Functor.{v₁, v₂, u₁, u₂} C V
inst_5 : DayConvolution F H
ℌ : DayConvolutionInternalHom F G H
x y : C × C
f : x ⟶ y
this :
  Eq (α :=
    tensorObj.{v₂, u₂} (𝒞 := inst_1) (F.obj x.1) (H.obj x.2) ⟶
      tensorObj.{v₂, u₂} (𝒞 := inst_1) (F.obj x.1)
        (Functor.obj.{v₂, v₂, u₂, u₂} (ihom (F.obj x.1)) (G.obj (tensorObj.{v₁, u₁} (𝒞 := inst) y.1 x.2))))
    (F.obj x.1 ◁ ℌ.π x.2 x.1 ≫ F.obj x.1 ◁ Functor.map (ihom (F.obj x.1)) (G.map (f.1 ▷ x.2)))
    (F.obj x.1 ◁ ℌ.π x.2 y.1 ≫
      F.obj x.1 ◁ NatTrans.app (MonoidalClosed.pre (F.map f.1)) (G.obj (tensorObj.{v₁, u₁} (𝒞 := inst) y.1 x.2)))
⊢ F.obj x.1 ◁ ℌ.π x.2 y.1 ≫
      F.map f.1 ▷ Functor.obj.{v₂, v₂, u₂, u₂} (ihom (F.obj y.1)) (G.obj (tensorObj.{v₁, u₁} (𝒞 := inst) y.1 x.2)) ≫
        NatTrans.app (ihom.ev (F.obj y.1)) (G.obj (tensorObj.{v₁, u₁} (𝒞 := inst) y.1 x.2)) ≫ G.map (y.1 ◁ f.2) =
    F.obj x.1 ◁ ℌ.π x.2 x.1 ≫
      F.obj x.1 ◁ Functor.map (ihom (F.obj x.1)) (G.map (f.1 ▷ x.2)) ≫
        NatTrans.app (ihom.ev (F.obj x.1)) (G.obj (tensorObj.{v₁, u₁} (𝒞 := inst) y.1 x.2)) ≫ G.map (y.1 ◁ f.2)
have rw : F.obj x.1 ◁ ℌ.π x.2 y.1 ≫
    F.map f.1 ▷ Functor.obj.{v₂, v₂, u₂, u₂} (ihom (F.obj y.1)) (G.obj (tensorObj.{v₁, u₁} (𝒞 := inst) y.1 x.2)) ≫
      NatTrans.app (ihom.ev (F.obj y.1)) (G.obj (tensorObj.{v₁, u₁} (𝒞 := inst) y.1 x.2)) ≫ G.map (y.1 ◁ f.2) =
  F.obj x.1 ◁ ℌ.π x.2 y.1 ≫
    CategoryStruct.comp (Y :=
      tensorObj.{v₂, u₂} (𝒞 := inst_1) (F.obj x.1)
        (Functor.obj.{v₂, v₂, u₂, u₂} (ihom (F.obj x.1)) (G.obj (tensorObj.{v₁, u₁} (𝒞 := inst) y.1 x.2))))
      (F.obj x.1 ◁ NatTrans.app (MonoidalClosed.pre (F.map f.1)) (G.obj (tensorObj.{v₁, u₁} (𝒞 := inst) y.1 x.2)))
      (NatTrans.app (ihom.ev (F.obj x.1)) (G.obj (tensorObj.{v₁, u₁} (𝒞 := inst) y.1 x.2)) ≫ G.map (y.1 ◁ f.2))
Tactic: rw [reassoc_of% 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 H : Functor.{v₁, v₂, u₁, u₂} C V
inst_5 : DayConvolution F H
ℌ : DayConvolutionInternalHom F G H
x y : C
⊢ CategoryStruct.comp (Y := (tensor C ⋙ @DayConvolution.convolution _ _ _ _ inst_2 inst_3 F H _).obj (x, y))
      (NatTrans.app (DayConvolution.unit F H) (x, y)) (NatTrans.app ℌ.ev_app (tensorObj.{v₁, u₁} (𝒞 := inst) x y)) =
    MonoidalClosed.uncurry (ℌ.π y x)
have this : ∀ (β : F ⊠ H ⟶ tensor C ⋙ G) (X : C × C),
  NatTrans.app (DayConvolution.unit F H) X ≫
      ((@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F H _).descOfIsLeftKanExtension (DayConvolution.unit F H) G
            β).app
        (Functor.obj.{v₁, v₁, u₁, u₁} (tensor C) X) =
    β.app X
Tactic: have := Functor.descOfIsLeftKanExtension_fac_app (F ⊛ H)
    (DayConvolution.unit F H) G
-- 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 H : Functor.{v₁, v₂, u₁, u₂} C V
inst_5 : DayConvolution F H
ℌ : DayConvolutionInternalHom F G H
G' H' : Functor.{v₁, v₂, u₁, u₂} C V
ℌ' : DayConvolutionInternalHom F G' H'
inst_6 : DayConvolution F H'
η : G ⟶ G'
⊢ Eq (α := @DayConvolution.convolution _ _ _ _ inst_2 inst_3 F H _ ⟶ G')
    (CategoryStruct.comp (Y := @DayConvolution.convolution _ _ _ _ inst_2 inst_3 F H' _)
      (DayConvolution.map (𝟙 F) (ℌ.map η ℌ')) ℌ'.ev_app)
    (ℌ.ev_app ≫ η)
have a : Eq (α := (Functor.obj (Functor.whiskeringLeft (C × C) C V) (tensor C) ⋙ Functor.obj coyoneda (op (F ⊠ H))).obj G')
  ((Functor.CorepresentableBy.homEquiv (DayConvolution.corepresentableBy F H) :
        (a : F ⊛ H ⟶ G') → ((Functor.whiskeringLeft (C × C) C V).obj (tensor C) ⋙ coyoneda.obj (op (F ⊠ H))).obj G')
      (CategoryStruct.comp (Y := @DayConvolution.convolution _ _ _ _ inst_2 inst_3 F H' _)
        (DayConvolution.map (𝟙 F) (ℌ.map η ℌ')) ℌ'.ev_app) :
    (Functor.obj (Functor.whiskeringLeft (C × C) C V) (tensor C) ⋙ Functor.obj coyoneda (op (F ⊠ H))).obj G')
  ((Functor.CorepresentableBy.homEquiv (DayConvolution.corepresentableBy F H) :
        (a : F ⊛ H ⟶ G') → ((Functor.whiskeringLeft (C × C) C V).obj (tensor C) ⋙ coyoneda.obj (op (F ⊠ H))).obj G')
      (ℌ.ev_app ≫ η) :
    (Functor.obj (Functor.whiskeringLeft (C × C) C V) (tensor C) ⋙ Functor.obj coyoneda (op (F ⊠ H))).obj G')
Tactic: apply DayConvolution.corepresentableBy F H |>.homEquiv.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 G H : Functor.{v₁, v₂, u₁, u₂} C V
inst_5 : DayConvolution F H
ℌ : DayConvolutionInternalHom F G H
G' H' : Functor.{v₁, v₂, u₁, u₂} C V
ℌ' : DayConvolutionInternalHom F G' H'
inst_6 : DayConvolution F H'
η : G ⟶ G'
⊢ Eq (α := F ⊠ H ⟶ tensor C ⋙ G')
    ((Functor.CorepresentableBy.homEquiv (DayConvolution.corepresentableBy F H) :
          (a : F ⊛ H ⟶ G') → F ⊠ H ⟶ tensor C ⋙ G')
        (CategoryStruct.comp (Y := @DayConvolution.convolution _ _ _ _ inst_2 inst_3 F H' _)
          (DayConvolution.map (𝟙 F) (ℌ.map η ℌ')) ℌ'.ev_app) :
      F ⊠ H ⟶ tensor C ⋙ G')
    ((Functor.CorepresentableBy.homEquiv (DayConvolution.corepresentableBy F H) :
          (a : F ⊛ H ⟶ G') → F ⊠ H ⟶ tensor C ⋙ G')
        (ℌ.ev_app ≫ η) :
      F ⊠ H ⟶ tensor C ⋙ G')
have a : ∀ (x y : C),
  Eq.{v₂ + 1} (α := (F ⊠ H).obj (x, y) ⟶ (tensor C ⋙ G').obj (x, y))
    (NatTrans.app
      ((Functor.CorepresentableBy.homEquiv (DayConvolution.corepresentableBy F H) :
          (a : F ⊛ H ⟶ G') → F ⊠ H ⟶ tensor C ⋙ G')
        (CategoryStruct.comp (Y := @DayConvolution.convolution _ _ _ _ inst_2 inst_3 F H' _)
          (DayConvolution.map (𝟙 F) (ℌ.map η ℌ')) ℌ'.ev_app))
      (x, y))
    (NatTrans.app
      ((Functor.CorepresentableBy.homEquiv (DayConvolution.corepresentableBy F H) :
          (a : F ⊛ H ⟶ G') → F ⊠ H ⟶ tensor C ⋙ G')
        (ℌ.ev_app ≫ η))
      (x, y))
Tactic: ext ⟨x, y⟩
-- 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'' : C
f : c' ⟶ c''
⊢ Eq (α :=
    G.obj c ⟶
      ((((dayConvolutionInternalHomDiagramFunctor F).obj (@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _)).obj
                c).obj
            (op c')).obj
        c'')
    ((fun (c' : C) ↦
          (MonoidalClosed.curry (NatTrans.app (DayConvolution.unit F G) (c', c)) :
            G.obj c ⟶
              Functor.obj.{v₂, v₂, u₂, u₂} (ihom (unop (F.op.obj (op c'))))
                ((tensorRight c ⋙ @DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _).obj c')))
        c' ≫
      ((((dayConvolutionInternalHomDiagramFunctor F).obj (@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _)).obj
                c).obj
            (op c')).map
        f)
    ((fun (c' : C) ↦
          (MonoidalClosed.curry (NatTrans.app (DayConvolution.unit F G) (c', c)) :
            G.obj c ⟶
              Functor.obj.{v₂, v₂, u₂, u₂} (ihom (unop (F.op.obj (op c'))))
                ((tensorRight c ⋙ @DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _).obj c')))
        c'' ≫
      ((((dayConvolutionInternalHomDiagramFunctor F).obj (@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _)).obj
                c).map
            f.op).app
        c'')
have this : Eq (α :=
  tensorObj.{v₂, u₂} (𝒞 := inst_1) (F.obj c') (G.obj c) ⟶
    (tensor C ⋙ @DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _).obj (c'', c))
  ((F.map f ⊗ₘ G.map (𝟙 c)) ≫ NatTrans.app (DayConvolution.unit F G) (c'', c))
  (NatTrans.app (DayConvolution.unit F G) (c', c) ≫
    (@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _).map (f ⊗ₘ 𝟙 c))
Tactic: have := DayConvolution.unit_naturality F G f (𝟙 c)
-- 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'' : C
f : c' ⟶ c''
this :
  Eq (α :=
    tensorObj.{v₂, u₂} (𝒞 := inst_1) (F.obj c') (G.obj c) ⟶
      (tensor C ⋙ @DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _).obj (c'', c))
    ((F.map f ⊗ₘ G.map (𝟙 c)) ≫ NatTrans.app (DayConvolution.unit F G) (c'', c))
    (NatTrans.app (DayConvolution.unit F G) (c', c) ≫
      (@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _).map (f ⊗ₘ 𝟙 c))
⊢ Eq (α :=
    G.obj c ⟶
      ((((dayConvolutionInternalHomDiagramFunctor F).obj (@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _)).obj
                c).obj
            (op c')).obj
        c'')
    ((fun (c' : C) ↦
          (MonoidalClosed.curry (NatTrans.app (DayConvolution.unit F G) (c', c)) :
            G.obj c ⟶
              Functor.obj.{v₂, v₂, u₂, u₂} (ihom (unop (F.op.obj (op c'))))
                ((tensorRight c ⋙ @DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _).obj c')))
        c' ≫
      ((((dayConvolutionInternalHomDiagramFunctor F).obj (@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _)).obj
                c).obj
            (op c')).map
        f)
    ((fun (c' : C) ↦
          (MonoidalClosed.curry (NatTrans.app (DayConvolution.unit F G) (c', c)) :
            G.obj c ⟶
              Functor.obj.{v₂, v₂, u₂, u₂} (ihom (unop (F.op.obj (op c'))))
                ((tensorRight c ⋙ @DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _).obj c')))
        c'' ≫
      ((((dayConvolutionInternalHomDiagramFunctor F).obj (@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _)).obj
                c).map
            f.op).app
        c'')
have this₁ : Eq (α :=
  tensorObj.{v₂, u₂} (𝒞 := inst_1) (F.obj c') (G.obj c) ⟶
    (tensor C ⋙ @DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _).obj (c'', c))
  (F.map f ▷ G.obj c ≫ NatTrans.app (DayConvolution.unit F G) (c'', c))
  (NatTrans.app (DayConvolution.unit F G) (c', c) ≫
    (@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _).map (f ▷ c))
Tactic: simp only [Functor.map_id, tensorHom_id] 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'' : C
f : c' ⟶ c''
this :
  Eq (α :=
    tensorObj.{v₂, u₂} (𝒞 := inst_1) (F.obj c') (G.obj c) ⟶
      (tensor C ⋙ @DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _).obj (c'', c))
    (F.map f ▷ G.obj c ≫ NatTrans.app (DayConvolution.unit F G) (c'', c))
    (NatTrans.app (DayConvolution.unit F G) (c', c) ≫
      (@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _).map (f ▷ c))
⊢ Eq (α :=
    G.obj c ⟶
      ((((dayConvolutionInternalHomDiagramFunctor F).obj (@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _)).obj
                c).obj
            (op c')).obj
        c'')
    ((fun (c' : C) ↦
          (MonoidalClosed.curry (NatTrans.app (DayConvolution.unit F G) (c', c)) :
            G.obj c ⟶
              Functor.obj.{v₂, v₂, u₂, u₂} (ihom (unop (F.op.obj (op c'))))
                ((tensorRight c ⋙ @DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _).obj c')))
        c' ≫
      ((((dayConvolutionInternalHomDiagramFunctor F).obj (@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _)).obj
                c).obj
            (op c')).map
        f)
    ((fun (c' : C) ↦
          (MonoidalClosed.curry (NatTrans.app (DayConvolution.unit F G) (c', c)) :
            G.obj c ⟶
              Functor.obj.{v₂, v₂, u₂, u₂} (ihom (unop (F.op.obj (op c'))))
                ((tensorRight c ⋙ @DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _).obj c')))
        c'' ≫
      ((((dayConvolutionInternalHomDiagramFunctor F).obj (@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _)).obj
                c).map
            f.op).app
        c'')
have this₁ : Eq (α :=
  G.obj c ⟶
    Functor.obj.{v₂, v₂, u₂, u₂} (ihom (F.obj c'))
      ((tensor C ⋙ @DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _).obj (c'', c)))
  (MonoidalClosed.curry (F.map f ▷ G.obj c ≫ NatTrans.app (DayConvolution.unit F G) (c'', c)))
  (MonoidalClosed.curry
    (NatTrans.app (DayConvolution.unit F G) (c', c) ≫
      (@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _).map (f ▷ c)))
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'' : C
f : c' ⟶ c''
this :
  Eq (α :=
    G.obj c ⟶
      Functor.obj.{v₂, v₂, u₂, u₂} (ihom (F.obj c'))
        ((tensor C ⋙ @DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _).obj (c'', c)))
    (MonoidalClosed.curry (F.map f ▷ G.obj c ≫ NatTrans.app (DayConvolution.unit F G) (c'', c)))
    (MonoidalClosed.curry
      (NatTrans.app (DayConvolution.unit F G) (c', c) ≫
        (@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _).map (f ▷ c)))
⊢ Eq (α :=
    G.obj c ⟶
      ((((dayConvolutionInternalHomDiagramFunctor F).obj (@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _)).obj
                c).obj
            (op c')).obj
        c'')
    ((fun (c' : C) ↦
          (MonoidalClosed.curry (NatTrans.app (DayConvolution.unit F G) (c', c)) :
            G.obj c ⟶
              Functor.obj.{v₂, v₂, u₂, u₂} (ihom (unop (F.op.obj (op c'))))
                ((tensorRight c ⋙ @DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _).obj c')))
        c' ≫
      ((((dayConvolutionInternalHomDiagramFunctor F).obj (@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _)).obj
                c).obj
            (op c')).map
        f)
    ((fun (c' : C) ↦
          (MonoidalClosed.curry (NatTrans.app (DayConvolution.unit F G) (c', c)) :
            G.obj c ⟶
              Functor.obj.{v₂, v₂, u₂, u₂} (ihom (unop (F.op.obj (op c'))))
                ((tensorRight c ⋙ @DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _).obj c')))
        c'' ≫
      ((((dayConvolutionInternalHomDiagramFunctor F).obj (@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _)).obj
                c).map
            f.op).app
        c'')
have this₁ : Eq (α :=
  G.obj c ⟶
    Functor.obj.{v₂, v₂, u₂, u₂} (ihom (F.obj c'))
      ((tensor C ⋙ @DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _).obj (c'', c)))
  (CategoryStruct.comp (Y :=
      Functor.obj.{v₂, v₂, u₂, u₂} (ihom (F.obj c')) (tensorObj.{v₂, u₂} (𝒞 := inst_1) (F.obj c'') (G.obj c)))
      (MonoidalClosed.curry (F.map f ▷ G.obj c))
      (Functor.map (ihom (F.obj c')) (NatTrans.app (DayConvolution.unit F G) (c'', c))) :
    G.obj c ⟶
      Functor.obj.{v₂, v₂, u₂, u₂} (ihom (F.obj c'))
        ((tensor C ⋙ @DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _).obj (c'', c)))
  (MonoidalClosed.curry (NatTrans.app (DayConvolution.unit F G) (c', c)) ≫
      Functor.map (ihom (F.obj c')) ((@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _).map (f ▷ c)) :
    G.obj c ⟶
      Functor.obj.{v₂, v₂, u₂, u₂} (ihom (F.obj c'))
        ((tensor C ⋙ @DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _).obj (c'', c)))
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'
⊢ G.map f ≫
      Wedge.IsLimit.lift (ℌ.isLimitWedge c')
        (fun (c'_1 : C) ↦ MonoidalClosed.curry (NatTrans.app (DayConvolution.unit F G) (c'_1, c'))) ⋯ =
    Wedge.IsLimit.lift (ℌ.isLimitWedge c)
        (fun (c' : C) ↦ MonoidalClosed.curry (NatTrans.app (DayConvolution.unit F G) (c', c))) ⋯ ≫
      H.map f
have apply : ∀ (j : (multicospanShapeEnd.{v₁, u₁} C).L),
  (G.map f ≫
        Wedge.IsLimit.lift (ℌ.isLimitWedge c')
          (fun (c'_1 : C) ↦ MonoidalClosed.curry (NatTrans.app (DayConvolution.unit F G) (c'_1, c'))) sorry) ≫
      Multifork.ι (Wedge.mk (H.obj c') (ℌ.π c') sorry) j =
    (Wedge.IsLimit.lift (ℌ.isLimitWedge c)
          (fun (c' : C) ↦ MonoidalClosed.curry (NatTrans.app (DayConvolution.unit F G) (c', c))) sorry ≫
        H.map f) ≫
      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 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'))) ⋯) ≫
      Multifork.ι (Wedge.mk (H.obj 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.map f) ≫
      Multifork.ι (Wedge.mk (H.obj c') (ℌ.π c') ⋯) j
have simp : G.map f ≫
    Wedge.IsLimit.lift (ℌ.isLimitWedge c')
        (fun (c'_1 : C) ↦ MonoidalClosed.curry (NatTrans.app (DayConvolution.unit F G) (c'_1, c'))) sorry ≫
      ℌ.π c' j =
  Wedge.IsLimit.lift (ℌ.isLimitWedge c)
      (fun (c' : C) ↦ MonoidalClosed.curry (NatTrans.app (DayConvolution.unit F G) (c', c))) sorry ≫
    ℌ.π c j ≫ Functor.map (ihom (F.obj j)) ((@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _).map (j ◁ f))
Tactic: simp only [multicospanIndexEnd_left,
      dayConvolutionInternalHomDiagramFunctor_obj_obj_obj_obj, Multifork.ofι_pt,
      Wedge.mk_ι, Category.assoc, map_comp_π]