awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.CategoryTheory.Monoidal.DayConvolution
Goal:
C_1 : Type u₁
inst : Category.{v₁, u₁} C_1
V_1 : Type u₂
inst_1 : Category.{v₂, u₂} V_1
inst_2 : MonoidalCategory.{v₁, u₁} C_1
inst_3 : MonoidalCategory.{v₂, u₂} V_1
C : Type u₁
inst_4 : Category.{v₁, u₁} C
V : Type u₂
inst_5 : Category.{v₂, u₂} V
inst_6 : MonoidalCategory.{v₁, u₁} C
inst_7 : MonoidalCategory.{v₂, u₂} V
D : Type u₃
inst_8 : Category.{v₃, u₃} D
inst_9 : MonoidalCategoryStruct.{v₃, u₃} D
inst_10 : LawfulDayConvolutionMonoidalCategoryStruct.{v₁, v₂, v₃, u₁, u₂, u₃} C V D
inst_11 :
  ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow.{v₁, v₁, u₁, u₁} (tensor C) d) (tensorLeft v)
inst_12 :
  ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow.{v₁, v₁, u₁, u₁} (tensor C) d) (tensorRight v)
inst_13 :
  ∀ (v : V) (d : C),
    Limits.PreservesColimitsOfShape.{0, v₁, v₂, v₂, u₂, u₂}
      (CostructuredArrow (Functor.fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst_4))) d) (tensorLeft v)
inst_14 :
  ∀ (v : V) (d : C),
    Limits.PreservesColimitsOfShape.{0, v₁, v₂, v₂, u₂, u₂}
      (CostructuredArrow (Functor.fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst_4))) d) (tensorRight v)
inst_15 :
  ∀ (v : V) (d : C × C),
    Limits.PreservesColimitsOfShape
      (CostructuredArrow.{v₁, v₁, u₁, u₁} (Functor.prod (𝟭 C) (Functor.fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst_4))))
        d)
      (tensorRight v)
inst_16 :
  ∀ (v : V) (d : C × C),
    Limits.PreservesColimitsOfShape (CostructuredArrow.{v₁, v₁, u₁, u₁} (Functor.prod (tensor C) (𝟭 C)) d)
      (tensorRight v)
a b : D
a₁ :
  Eq (α :=
    Quiver.Hom (V := Functor.{v₁, v₂, u₁, u₂} C V)
      (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _)
        (tensorObj.{v₃, u₃} (𝒞 := inst_8) (tensorObj.{v₃, u₃} (𝒞 := inst_8) a (tensorUnit.{v₃, u₃} D (𝒞 := inst_8))) b))
      (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) (tensorObj.{v₃, u₃} (𝒞 := inst_8) a b)))
    (Functor.map (@ι C _ V _ inst_6 inst_7 D _ inst_9 _)
      (Iso.hom (α_ a (tensorUnit.{v₃, u₃} D (𝒞 := inst_8)) b) ≫ (𝟙 a ⊗ₘ Iso.hom (λ_ b))))
    (Functor.map (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) (Iso.hom (ρ_ a) ⊗ₘ 𝟙 b))
⊢ Eq.{v₃ + 1} (α :=
    tensorObj.{v₃, u₃} (𝒞 := inst_8) (tensorObj.{v₃, u₃} (𝒞 := inst_8) a (tensorUnit.{v₃, u₃} D (𝒞 := inst_8))) b ⟶
      tensorObj.{v₃, u₃} (𝒞 := inst_8) a b)
    (Iso.hom (α_ a (tensorUnit.{v₃, u₃} D (𝒞 := inst_8)) b) ≫ (𝟙 a ⊗ₘ Iso.hom (λ_ b))) (Iso.hom (ρ_ a) ⊗ₘ 𝟙 b)

Tactic:
apply Functor.Faithful.map_injective (F := ι C V D)

Premises used: CategoryTheory.Functor.Faithful.map_injective
-- Module: Mathlib.CategoryTheory.Monoidal.DayConvolution
Goal:
C_1 : Type u₁
inst : Category.{v₁, u₁} C_1
V_1 : Type u₂
inst_1 : Category.{v₂, u₂} V_1
inst_2 : MonoidalCategory.{v₁, u₁} C_1
inst_3 : MonoidalCategory.{v₂, u₂} V_1
C : Type u₁
inst_4 : Category.{v₁, u₁} C
V : Type u₂
inst_5 : Category.{v₂, u₂} V
inst_6 : MonoidalCategory.{v₁, u₁} C
inst_7 : MonoidalCategory.{v₂, u₂} V
D : Type u₃
inst_8 : Category.{v₃, u₃} D
inst_9 : InducedLawfulDayConvolutionMonoidalCategoryStructCore.{v₁, v₂, v₃, u₁, u₂, u₃} C V D
d d' : D
X Y : C × C
f : X ⟶ Y
⊢ Eq.{v₂ + 1} (α :=
    Functor.obj.{v₁, v₂, u₁, u₂}
        (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) d ⊠ Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) d') X ⟶
      Functor.obj.{v₁, v₂, u₁, u₂}
        (tensor C ⋙
          Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _)
            (@tensorObj.{v₁, v₂, v₃, u₁, u₂, u₃} C inst_4 V inst_5 inst_6 inst_7 _ inst_8 _ d d'))
        Y)
    (CategoryStruct.comp (Y :=
      Functor.obj.{v₁, v₂, u₁, u₂}
        (tensor C ⋙
          @DayConvolution.convolution _ _ _ _ inst_6 inst_7 (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) d)
            (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) d') _)
        X)
      (NatTrans.app
        (DayConvolution.unit (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) d)
          (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) d'))
        X)
      (Functor.map
          (tensor C ⋙
            @DayConvolution.convolution _ _ _ _ inst_6 inst_7 (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) d)
              (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) d') _)
          f ≫
        NatTrans.app (Iso.inv (tensorObjIsoConvolution C V d d'))
          (MonoidalCategoryStruct.tensorObj.{v₁, u₁} (𝒞 := inst_4) Y.1 Y.2)))
    (NatTrans.app
        (DayConvolution.unit (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) d)
          (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) d'))
        (X.1, X.2) ≫
      NatTrans.app (Iso.inv (tensorObjIsoConvolution C V d d'))
          (MonoidalCategoryStruct.tensorObj.{v₁, u₁} (𝒞 := inst_4) X.1 X.2) ≫
        Functor.map
          (tensor C ⋙
            Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _)
              (@tensorObj.{v₁, v₂, v₃, u₁, u₂, u₃} C inst_4 V inst_5 inst_6 inst_7 _ inst_8 _ d d'))
          f)

Tactic:
simp
-- Module: Mathlib.CategoryTheory.Monoidal.DayConvolution
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
D : Type u₃
inst_4 : Category.{v₃, u₃} D
inst_5 : InducedLawfulDayConvolutionMonoidalCategoryStructCore.{v₁, v₂, v₃, u₁, u₂, u₃} C V D
inst_6 :
  ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow.{v₁, v₁, u₁, u₁} (tensor C) d) (tensorLeft v)
inst_7 :
  ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow.{v₁, v₁, u₁, u₁} (tensor C) d) (tensorRight v)
inst_8 :
  ∀ (v : V) (d : C),
    Limits.PreservesColimitsOfShape.{0, v₁, v₂, v₂, u₂, u₂}
      (CostructuredArrow (Functor.fromPUnit (MonoidalCategoryStruct.tensorUnit.{v₁, u₁} C (𝒞 := inst))) d)
      (tensorRight v)
inst_9 :
  ∀ (v : V) (d : C),
    Limits.PreservesColimitsOfShape.{0, v₁, v₂, v₂, u₂, u₂}
      (CostructuredArrow (Functor.fromPUnit (MonoidalCategoryStruct.tensorUnit.{v₁, u₁} C (𝒞 := inst))) d)
      (tensorLeft v)
x y y' : D
f : y ⟶ y'
⊢ Eq (α :=
    MonoidalCategoryStruct.tensorObj.{v₃, u₃} (𝒞 := inst_4) x y ⟶
      MonoidalCategoryStruct.tensorObj.{v₃, u₃} (𝒞 := inst_4) x y')
    (𝟙 x ⊗ₘ f) (x ◁ f)

Tactic:
rfl

Premises used: rfl
-- Module: Mathlib.CategoryTheory.Monoidal.DayConvolution
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
D : Type u₃
inst_4 : Category.{v₃, u₃} D
inst_5 : InducedLawfulDayConvolutionMonoidalCategoryStructCore.{v₁, v₂, v₃, u₁, u₂, u₃} C V D
inst_6 :
  ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow.{v₁, v₁, u₁, u₁} (tensor C) d) (tensorLeft v)
inst_7 :
  ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow.{v₁, v₁, u₁, u₁} (tensor C) d) (tensorRight v)
inst_8 :
  ∀ (v : V) (d : C),
    Limits.PreservesColimitsOfShape.{0, v₁, v₂, v₂, u₂, u₂}
      (CostructuredArrow (Functor.fromPUnit (MonoidalCategoryStruct.tensorUnit.{v₁, u₁} C (𝒞 := inst))) d)
      (tensorRight v)
inst_9 :
  ∀ (v : V) (d : C),
    Limits.PreservesColimitsOfShape.{0, v₁, v₂, v₂, u₂, u₂}
      (CostructuredArrow (Functor.fromPUnit (MonoidalCategoryStruct.tensorUnit.{v₁, u₁} C (𝒞 := inst))) d)
      (tensorLeft v)
x x' : D
f : x ⟶ x'
y : D
⊢ Eq (α :=
    MonoidalCategoryStruct.tensorObj.{v₃, u₃} (𝒞 := inst_4) x y ⟶
      MonoidalCategoryStruct.tensorObj.{v₃, u₃} (𝒞 := inst_4) x' y)
    (f ⊗ₘ 𝟙 y) (f ▷ y)

Tactic:
rfl

Premises used: rfl
-- Module: Mathlib.CategoryTheory.Monoidal.DayConvolution
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
D : Type u₃
inst_4 : Category.{v₃, u₃} D
inst_5 : InducedLawfulDayConvolutionMonoidalCategoryStructCore.{v₁, v₂, v₃, u₁, u₂, u₃} C V D
inst_6 :
  ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow.{v₁, v₁, u₁, u₁} (tensor C) d) (tensorLeft v)
inst_7 :
  ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow.{v₁, v₁, u₁, u₁} (tensor C) d) (tensorRight v)
inst_8 :
  ∀ (v : V) (d : C),
    Limits.PreservesColimitsOfShape.{0, v₁, v₂, v₂, u₂, u₂}
      (CostructuredArrow (Functor.fromPUnit (MonoidalCategoryStruct.tensorUnit.{v₁, u₁} C (𝒞 := inst))) d)
      (tensorRight v)
inst_9 :
  ∀ (v : V) (d : C),
    Limits.PreservesColimitsOfShape.{0, v₁, v₂, v₂, u₂, u₂}
      (CostructuredArrow (Functor.fromPUnit (MonoidalCategoryStruct.tensorUnit.{v₁, u₁} C (𝒞 := inst))) d)
      (tensorLeft v)
d₁ d₁' d₂ d₂' : D
f : d₁ ⟶ d₂
f' : d₁' ⟶ d₂'
dsimp :
  Eq (α :=
    Quiver.Hom (V := Functor.{v₁, v₂, u₁, u₂} C V)
      (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ _)
        (@tensorObj.{v₁, v₂, v₃, u₁, u₂, u₃} C inst V inst_1 inst_2 inst_3 _ inst_4 _ d₁ d₁'))
      (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ _)
        (@tensorObj.{v₁, v₂, v₃, u₁, u₂, u₃} C inst V inst_1 inst_2 inst_3 _ inst_4 _ d₂ d₂')))
    (Functor.map (@ι C _ V _ inst_2 inst_3 D _ _) (tensorHom f f'))
    (DayConvolution.map (Functor.map (@ι C _ V _ inst_2 inst_3 D _ _) f)
      (Functor.map (@ι C _ V _ inst_2 inst_3 D _ _) f'))
⊢ Eq (α :=
    Quiver.Hom (V := Functor.{v₁, v₂, u₁, u₂} C V)
      (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ _) (MonoidalCategoryStruct.tensorObj.{v₃, u₃} (𝒞 := inst_4) d₁ d₁'))
      (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ _) (MonoidalCategoryStruct.tensorObj.{v₃, u₃} (𝒞 := inst_4) d₂ d₂')))
    (Functor.map (@ι C _ V _ inst_2 inst_3 D _ _) (f ⊗ₘ f'))
    (DayConvolution.map (Functor.map (@ι C _ V _ inst_2 inst_3 D _ _) f)
      (Functor.map (@ι C _ V _ inst_2 inst_3 D _ _) f'))

Tactic:
dsimp +instances [mkMonoidalCategoryStruct]

Premises used: CategoryTheory.MonoidalCategory.InducedLawfulDayConvolutionMonoidalCategoryStructCore.mkMonoidalCategoryStruct
-- Module: Mathlib.CategoryTheory.Monoidal.DayConvolution
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
D : Type u₃
inst_4 : Category.{v₃, u₃} D
inst_5 : InducedLawfulDayConvolutionMonoidalCategoryStructCore.{v₁, v₂, v₃, u₁, u₂, u₃} C V D
inst_6 :
  ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow.{v₁, v₁, u₁, u₁} (tensor C) d) (tensorLeft v)
inst_7 :
  ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow.{v₁, v₁, u₁, u₁} (tensor C) d) (tensorRight v)
inst_8 :
  ∀ (v : V) (d : C),
    Limits.PreservesColimitsOfShape.{0, v₁, v₂, v₂, u₂, u₂}
      (CostructuredArrow (Functor.fromPUnit (MonoidalCategoryStruct.tensorUnit.{v₁, u₁} C (𝒞 := inst))) d)
      (tensorRight v)
inst_9 :
  ∀ (v : V) (d : C),
    Limits.PreservesColimitsOfShape.{0, v₁, v₂, v₂, u₂, u₂}
      (CostructuredArrow (Functor.fromPUnit (MonoidalCategoryStruct.tensorUnit.{v₁, u₁} C (𝒞 := inst))) d)
      (tensorLeft v)
d₁ d₁' d₂ d₂' : D
f : d₁ ⟶ d₂
f' : d₁' ⟶ d₂'
rw :
  Eq (α :=
    Quiver.Hom (V := Functor.{v₁, v₂, u₁, u₂} C V)
      (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ _)
        (@tensorObj.{v₁, v₂, v₃, u₁, u₂, u₃} C inst V inst_1 inst_2 inst_3 _ inst_4 _ d₁ d₁'))
      (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ _)
        (@tensorObj.{v₁, v₂, v₃, u₁, u₂, u₃} C inst V inst_1 inst_2 inst_3 _ inst_4 _ d₂ d₂')))
    (Iso.hom (tensorObjIsoConvolution C V d₁ d₁') ≫
      DayConvolution.map (Functor.map (@ι C _ V _ inst_2 inst_3 D _ _) f)
          (Functor.map (@ι C _ V _ inst_2 inst_3 D _ _) f') ≫
        Iso.inv (tensorObjIsoConvolution C V d₂ d₂'))
    (DayConvolution.map (Functor.map (@ι C _ V _ inst_2 inst_3 D _ _) f)
      (Functor.map (@ι C _ V _ inst_2 inst_3 D _ _) f'))
⊢ Eq (α :=
    Quiver.Hom (V := Functor.{v₁, v₂, u₁, u₂} C V)
      (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ _)
        (@tensorObj.{v₁, v₂, v₃, u₁, u₂, u₃} C inst V inst_1 inst_2 inst_3 _ inst_4 _ d₁ d₁'))
      (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ _)
        (@tensorObj.{v₁, v₂, v₃, u₁, u₂, u₃} C inst V inst_1 inst_2 inst_3 _ inst_4 _ d₂ d₂')))
    (Functor.map (@ι C _ V _ inst_2 inst_3 D _ _) (tensorHom f f'))
    (DayConvolution.map (Functor.map (@ι C _ V _ inst_2 inst_3 D _ _) f)
      (Functor.map (@ι C _ V _ inst_2 inst_3 D _ _) f'))

Tactic:
rw [tensorHom_eq]

Premises used: CategoryTheory.MonoidalCategory.InducedLawfulDayConvolutionMonoidalCategoryStructCore.tensorHom_eq
-- Module: Mathlib.CategoryTheory.Monoidal.DayConvolution
Goal:
C_1 : Type u₁
inst : Category.{v₁, u₁} C_1
V_1 : Type u₂
inst_1 : Category.{v₂, u₂} V_1
inst_2 : MonoidalCategory.{v₁, u₁} C_1
inst_3 : MonoidalCategory.{v₂, u₂} V_1
C : Type u₁
inst_4 : Category.{v₁, u₁} C
V : Type u₂
inst_5 : Category.{v₂, u₂} V
inst_6 : MonoidalCategory.{v₁, u₁} C
inst_7 : MonoidalCategory.{v₂, u₂} V
D : Type u₃
inst_8 : Category.{v₃, u₃} D
inst_9 : InducedLawfulDayConvolutionMonoidalCategoryStructCore.{v₁, v₂, v₃, u₁, u₂, u₃} C V D
inst_10 :
  ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow.{v₁, v₁, u₁, u₁} (tensor C) d) (tensorLeft v)
inst_11 :
  ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow.{v₁, v₁, u₁, u₁} (tensor C) d) (tensorRight v)
inst_12 :
  ∀ (v : V) (d : C),
    Limits.PreservesColimitsOfShape.{0, v₁, v₂, v₂, u₂, u₂}
      (CostructuredArrow (Functor.fromPUnit (MonoidalCategoryStruct.tensorUnit.{v₁, u₁} C (𝒞 := inst_4))) d)
      (tensorRight v)
inst_13 :
  ∀ (v : V) (d : C),
    Limits.PreservesColimitsOfShape.{0, v₁, v₂, v₂, u₂, u₂}
      (CostructuredArrow (Functor.fromPUnit (MonoidalCategoryStruct.tensorUnit.{v₁, u₁} C (𝒞 := inst_4))) d)
      (tensorLeft v)
this : MonoidalCategoryStruct.{v₃, u₃} D := mkMonoidalCategoryStruct.{v₁, v₂, v₃, u₁, u₂, u₃} C V D
d₁ d₁' d₂ d₂' : D
f : d₁ ⟶ d₂
f' : d₁' ⟶ d₂'
x y : C
⊢ Eq.{v₂ + 1} (α :=
    Functor.obj.{v₁, v₂, u₁, u₂}
        (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) d₁ ⊠ Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) d₁') (x, y) ⟶
      Functor.obj.{v₁, v₂, u₁, u₂}
        (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) (MonoidalCategoryStruct.tensorObj.{v₃, u₃} (𝒞 := inst_8) d₂ d₂'))
        (MonoidalCategoryStruct.tensorObj.{v₁, u₁} (𝒞 := inst_4) x y))
    (CategoryStruct.comp (Y :=
      Functor.obj.{v₁, v₂, u₁, u₂}
        (tensor C ⋙
          Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) (MonoidalCategoryStruct.tensorObj.{v₃, u₃} (𝒞 := inst_8) d₁ d₁'))
        (x, y))
      (NatTrans.app
        (DayConvolution.unit (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) d₁)
          (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) d₁'))
        (x, y))
      (NatTrans.app (Functor.map (@ι C _ V _ inst_6 inst_7 D _ _) (f ⊗ₘ f'))
        (MonoidalCategoryStruct.tensorObj.{v₁, u₁} (𝒞 := inst_4) x y)))
    ((NatTrans.app (Functor.map (@ι C _ V _ inst_6 inst_7 D _ _) f) x ⊗ₘ
        NatTrans.app (Functor.map (@ι C _ V _ inst_6 inst_7 D _ _) f') y) ≫
      NatTrans.app
        (DayConvolution.unit (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) d₂)
          (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) d₂'))
        (x, y))

Tactic:
simp [ι_map_tensorHom_eq C V D f f']

Premises used: CategoryTheory.MonoidalCategory.InducedLawfulDayConvolutionMonoidalCategoryStructCore.ι_map_tensorHom_eq
-- Module: Mathlib.CategoryTheory.Monoidal.DayConvolution
Goal:
C_1 : Type u₁
inst : Category.{v₁, u₁} C_1
V_1 : Type u₂
inst_1 : Category.{v₂, u₂} V_1
inst_2 : MonoidalCategory.{v₁, u₁} C_1
inst_3 : MonoidalCategory.{v₂, u₂} V_1
C : Type u₁
inst_4 : Category.{v₁, u₁} C
V : Type u₂
inst_5 : Category.{v₂, u₂} V
inst_6 : MonoidalCategory.{v₁, u₁} C
inst_7 : MonoidalCategory.{v₂, u₂} V
D : Type u₃
inst_8 : Category.{v₃, u₃} D
inst_9 : InducedLawfulDayConvolutionMonoidalCategoryStructCore.{v₁, v₂, v₃, u₁, u₂, u₃} C V D
inst_10 :
  ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow.{v₁, v₁, u₁, u₁} (tensor C) d) (tensorLeft v)
inst_11 :
  ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow.{v₁, v₁, u₁, u₁} (tensor C) d) (tensorRight v)
inst_12 :
  ∀ (v : V) (d : C),
    Limits.PreservesColimitsOfShape.{0, v₁, v₂, v₂, u₂, u₂}
      (CostructuredArrow (Functor.fromPUnit (MonoidalCategoryStruct.tensorUnit.{v₁, u₁} C (𝒞 := inst_4))) d)
      (tensorRight v)
inst_13 :
  ∀ (v : V) (d : C),
    Limits.PreservesColimitsOfShape.{0, v₁, v₂, v₂, u₂, u₂}
      (CostructuredArrow (Functor.fromPUnit (MonoidalCategoryStruct.tensorUnit.{v₁, u₁} C (𝒞 := inst_4))) d)
      (tensorLeft v)
this : MonoidalCategoryStruct.{v₃, u₃} D := mkMonoidalCategoryStruct.{v₁, v₂, v₃, u₁, u₂, u₃} C V D
d₁ d₂ d₂' : D
f : d₂ ⟶ d₂'
x y : C
⊢ Eq.{v₂ + 1} (α :=
    Functor.obj.{v₁, v₂, u₁, u₂}
        (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) d₁ ⊠ Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) d₂) (x, y) ⟶
      Functor.obj.{v₁, v₂, u₁, u₂}
        (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) (MonoidalCategoryStruct.tensorObj.{v₃, u₃} (𝒞 := inst_8) d₁ d₂'))
        (MonoidalCategoryStruct.tensorObj.{v₁, u₁} (𝒞 := inst_4) x y))
    (CategoryStruct.comp (Y :=
      Functor.obj.{v₁, v₂, u₁, u₂}
        (tensor C ⋙
          Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) (MonoidalCategoryStruct.tensorObj.{v₃, u₃} (𝒞 := inst_8) d₁ d₂))
        (x, y))
      (NatTrans.app
        (DayConvolution.unit (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) d₁)
          (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) d₂))
        (x, y))
      (NatTrans.app (Functor.map (@ι C _ V _ inst_6 inst_7 D _ _) (d₁ ◁ f))
        (MonoidalCategoryStruct.tensorObj.{v₁, u₁} (𝒞 := inst_4) x y)))
    (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) d₁) x ◁
        NatTrans.app (Functor.map (@ι C _ V _ inst_6 inst_7 D _ _) f) y ≫
      NatTrans.app
        (DayConvolution.unit (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) d₁)
          (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) d₂'))
        (x, y))

Tactic:
simp [← id_tensorHom, ι_map_tensorHom_eq C V D]

Premises used: CategoryTheory.MonoidalCategory.InducedLawfulDayConvolutionMonoidalCategoryStructCore.ι_map_tensorHom_eq, CategoryTheory.MonoidalCategory.InducedLawfulDayConvolutionMonoidalCategoryStructCore.id_tensorHom
-- Module: Mathlib.CategoryTheory.Monoidal.DayConvolution
Goal:
C_1 : Type u₁
inst : Category.{v₁, u₁} C_1
V_1 : Type u₂
inst_1 : Category.{v₂, u₂} V_1
inst_2 : MonoidalCategory.{v₁, u₁} C_1
inst_3 : MonoidalCategory.{v₂, u₂} V_1
C : Type u₁
inst_4 : Category.{v₁, u₁} C
V : Type u₂
inst_5 : Category.{v₂, u₂} V
inst_6 : MonoidalCategory.{v₁, u₁} C
inst_7 : MonoidalCategory.{v₂, u₂} V
D : Type u₃
inst_8 : Category.{v₃, u₃} D
inst_9 : InducedLawfulDayConvolutionMonoidalCategoryStructCore.{v₁, v₂, v₃, u₁, u₂, u₃} C V D
inst_10 :
  ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow.{v₁, v₁, u₁, u₁} (tensor C) d) (tensorLeft v)
inst_11 :
  ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow.{v₁, v₁, u₁, u₁} (tensor C) d) (tensorRight v)
inst_12 :
  ∀ (v : V) (d : C),
    Limits.PreservesColimitsOfShape.{0, v₁, v₂, v₂, u₂, u₂}
      (CostructuredArrow (Functor.fromPUnit (MonoidalCategoryStruct.tensorUnit.{v₁, u₁} C (𝒞 := inst_4))) d)
      (tensorRight v)
inst_13 :
  ∀ (v : V) (d : C),
    Limits.PreservesColimitsOfShape.{0, v₁, v₂, v₂, u₂, u₂}
      (CostructuredArrow (Functor.fromPUnit (MonoidalCategoryStruct.tensorUnit.{v₁, u₁} C (𝒞 := inst_4))) d)
      (tensorLeft v)
this : MonoidalCategoryStruct.{v₃, u₃} D := mkMonoidalCategoryStruct.{v₁, v₂, v₃, u₁, u₂, u₃} C V D
d₁ d₁' : D
f₁ : d₁ ⟶ d₁'
d₂ : D
x y : C
⊢ Eq.{v₂ + 1} (α :=
    Functor.obj.{v₁, v₂, u₁, u₂}
        (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) d₁ ⊠ Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) d₂) (x, y) ⟶
      Functor.obj.{v₁, v₂, u₁, u₂}
        (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) (MonoidalCategoryStruct.tensorObj.{v₃, u₃} (𝒞 := inst_8) d₁' d₂))
        (MonoidalCategoryStruct.tensorObj.{v₁, u₁} (𝒞 := inst_4) x y))
    (CategoryStruct.comp (Y :=
      Functor.obj.{v₁, v₂, u₁, u₂}
        (tensor C ⋙
          Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) (MonoidalCategoryStruct.tensorObj.{v₃, u₃} (𝒞 := inst_8) d₁ d₂))
        (x, y))
      (NatTrans.app
        (DayConvolution.unit (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) d₁)
          (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) d₂))
        (x, y))
      (NatTrans.app (Functor.map (@ι C _ V _ inst_6 inst_7 D _ _) (f₁ ▷ d₂))
        (MonoidalCategoryStruct.tensorObj.{v₁, u₁} (𝒞 := inst_4) x y)))
    (NatTrans.app (Functor.map (@ι C _ V _ inst_6 inst_7 D _ _) f₁) x ▷
        Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) d₂) y ≫
      NatTrans.app
        (DayConvolution.unit (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) d₁')
          (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) d₂))
        (x, y))

Tactic:
simp [← tensorHom_id, ι_map_tensorHom_eq C V D]

Premises used: CategoryTheory.MonoidalCategory.InducedLawfulDayConvolutionMonoidalCategoryStructCore.ι_map_tensorHom_eq, CategoryTheory.MonoidalCategory.InducedLawfulDayConvolutionMonoidalCategoryStructCore.tensorHom_id
-- Module: Mathlib.CategoryTheory.Monoidal.Discrete
Goal:
M : Type u
inst : Monoid M
X x x_1 : Discrete M
f : x ⟶ x_1
rw rw₁ : { as := X.as * x_1.as : Discrete M } = { as := X.as * x_1.as }
⊢ { as := X.as * x.as : Discrete M } = { as := X.as * x_1.as }

Tactic:
rw [eq_of_hom f]

Premises used: CategoryTheory.Discrete.eq_of_hom
-- Module: Mathlib.CategoryTheory.Monoidal.Discrete
Goal:
M : Type u
inst : Monoid M
X₁ X₂ : Discrete M
f : X₁ ⟶ X₂
X : Discrete M
rw rw₁ : { as := X₂.as * X.as : Discrete M } = { as := X₂.as * X.as }
⊢ { as := X₁.as * X.as : Discrete M } = { as := X₂.as * X.as }

Tactic:
rw [eq_of_hom f]

Premises used: CategoryTheory.Discrete.eq_of_hom
-- Module: Mathlib.CategoryTheory.Monoidal.Discrete
Goal:
M : Type u
inst : Monoid M
X₁ Y₁ X₂ Y₂ : Discrete M
f : X₁ ⟶ Y₁
g : X₂ ⟶ Y₂
rw : { as := Y₁.as * X₂.as : Discrete M } = { as := Y₁.as * Y₂.as }
rw₁ rw₂ : { as := Y₁.as * Y₂.as : Discrete M } = { as := Y₁.as * Y₂.as }
⊢ { as := X₁.as * X₂.as : Discrete M } = { as := Y₁.as * Y₂.as }

Tactic:
rw [eq_of_hom f, eq_of_hom g]

Premises used: CategoryTheory.Discrete.eq_of_hom
-- Module: Mathlib.CategoryTheory.Monoidal.Discrete
Goal:
M : Type u
inst : Monoid M
⊢ Eq.{u + 1} (α := M) (as ((𝟙_ (Discrete M)) (𝒞 := discreteCategory M))) 1

Tactic:
rfl

Premises used: rfl
-- Module: Mathlib.CategoryTheory.Monoidal.Discrete
Goal:
M : Type u
inst : Monoid M
N : Type u'
inst_1 : Monoid N
F : M →* N
m : M
⊢ (monoidalFunctor F).obj { as := m : Discrete M } = { as := (F : (a : M) → N) m }

Tactic:
rfl

Premises used: rfl
-- Module: Mathlib.CategoryTheory.Monoidal.Discrete
Goal:
M : Type u
inst : Monoid M
N : Type u'
inst_1 : Monoid N
F : M →* N
⊢ ε (monoidalFunctor F) = Discrete.eqToHom ⋯

Tactic:
rfl

Premises used: Eq.mpr, CategoryTheory.eqToHom, rfl
-- Module: Mathlib.CategoryTheory.Monoidal.Discrete
Goal:
M : Type u
inst : Monoid M
N : Type u'
inst_1 : Monoid N
F : M →* N
⊢ η (monoidalFunctor F) = Discrete.eqToHom ⋯

Tactic:
rfl

Premises used: Eq.mpr, CategoryTheory.eqToHom, rfl
-- Module: Mathlib.CategoryTheory.Monoidal.Discrete
Goal:
M : Type u
inst : Monoid M
N : Type u'
inst_1 : Monoid N
F : M →* N
m₁ m₂ : Discrete M
⊢ Eq (α :=
    tensorObj (𝒞 := discreteCategory N) ((monoidalFunctor F).obj m₁) ((monoidalFunctor F).obj m₂) ⟶
      (monoidalFunctor F).obj (tensorObj (𝒞 := discreteCategory M) m₁ m₂))
    (μ (monoidalFunctor F) m₁ m₂) (Discrete.eqToHom ⋯)

Tactic:
rfl

Premises used: Eq.mpr, CategoryTheory.eqToHom, rfl
-- Module: Mathlib.CategoryTheory.Monoidal.Discrete
Goal:
M : Type u
inst : Monoid M
N : Type u'
inst_1 : Monoid N
F : M →* N
m₁ m₂ : Discrete M
⊢ Eq (α :=
    (monoidalFunctor F).obj (tensorObj (𝒞 := discreteCategory M) m₁ m₂) ⟶
      tensorObj (𝒞 := discreteCategory N) ((monoidalFunctor F).obj m₁) ((monoidalFunctor F).obj m₂))
    (δ (monoidalFunctor F) m₁ m₂) (Discrete.eqToHom ⋯)

Tactic:
rfl

Premises used: Eq.mpr, CategoryTheory.eqToHom, rfl