awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.DayConvolution (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
U : Functor.{v₁, v₂, u₁, u₂} C V
inst_4 : DayConvolutionUnit U
F : Functor.{v₁, v₂, u₁, u₂} C V
inst_5 :
  ∀ (v : V) (d : C),
    Limits.PreservesColimitsOfShape.{0, v₁, v₂, v₂, u₂, u₂}
      (CostructuredArrow (fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst))) d) (tensorRight v)
inst_6 : DayConvolution U F
G : Functor.{v₁, v₂, u₁, u₂} C V
inst_7 : DayConvolution U G
f : F ⟶ G
⊢ Eq (α := @DayConvolution.convolution _ _ _ _ inst_2 inst_3 U F _ ⟶ G)
    (CategoryStruct.comp (Y := @DayConvolution.convolution _ _ _ _ inst_2 inst_3 U G _) (DayConvolution.map (𝟙 U) f)
      (Iso.hom (leftUnitor U G)))
    (Iso.hom (leftUnitor U F) ≫ f)
have hγ : Eq (α := U ⊠ F ⟶ tensor C ⋙ G)
  (CategoryStruct.comp (Y := tensor C ⋙ @DayConvolution.convolution _ _ _ _ inst_2 inst_3 U F _)
    (DayConvolution.unit U F)
    (Functor.whiskerLeft (tensor C)
      (CategoryStruct.comp (Y := @DayConvolution.convolution _ _ _ _ inst_2 inst_3 U G _) (DayConvolution.map (𝟙 U) f)
        (Iso.hom (leftUnitor U G)))))
  (CategoryStruct.comp (Y := tensor C ⋙ @DayConvolution.convolution _ _ _ _ inst_2 inst_3 U F _)
    (DayConvolution.unit U F) (Functor.whiskerLeft (tensor C) (Iso.hom (leftUnitor U F) ≫ f)))
Tactic: apply Functor.hom_ext_of_isLeftKanExtension _ (DayConvolution.unit _ _) _
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.DayConvolution (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
U : Functor.{v₁, v₂, u₁, u₂} C V
inst_4 : DayConvolutionUnit U
F : Functor.{v₁, v₂, u₁, u₂} C V
inst_5 :
  ∀ (v : V) (d : C),
    Limits.PreservesColimitsOfShape.{0, v₁, v₂, v₂, u₂, u₂}
      (CostructuredArrow (fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst))) d) (tensorRight v)
inst_6 : DayConvolution U F
G : Functor.{v₁, v₂, u₁, u₂} C V
inst_7 : DayConvolution U G
f : F ⟶ G
⊢ Eq (α := U ⊠ F ⟶ tensor C ⋙ G)
    (CategoryStruct.comp (Y := tensor C ⋙ @DayConvolution.convolution _ _ _ _ inst_2 inst_3 U F _)
      (DayConvolution.unit U F)
      (Functor.whiskerLeft (tensor C)
        (CategoryStruct.comp (Y := @DayConvolution.convolution _ _ _ _ inst_2 inst_3 U G _) (DayConvolution.map (𝟙 U) f)
          (Iso.hom (leftUnitor U G)))))
    (CategoryStruct.comp (Y := tensor C ⋙ @DayConvolution.convolution _ _ _ _ inst_2 inst_3 U F _)
      (DayConvolution.unit U F) (Functor.whiskerLeft (tensor C) (Iso.hom (leftUnitor U F) ≫ f)))
have hγ : Eq (α :=
  fromPUnit (tensorUnit.{v₂, u₂} V (𝒞 := inst_1)) ⊠ F ⟶
    Functor.prod (fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst))) (𝟭 C) ⋙ tensor C ⋙ G)
  (extensionUnitLeft U (φ U) F ≫
    Functor.whiskerLeft (Functor.prod (fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst))) (𝟭 C))
      (CategoryStruct.comp (Y := tensor C ⋙ @DayConvolution.convolution _ _ _ _ inst_2 inst_3 U F _)
        (DayConvolution.unit U F)
        (Functor.whiskerLeft (tensor C)
          (CategoryStruct.comp (Y := @DayConvolution.convolution _ _ _ _ inst_2 inst_3 U G _)
            (DayConvolution.map (𝟙 U) f) (Iso.hom (leftUnitor U G))))))
  (extensionUnitLeft U (φ U) F ≫
    Functor.whiskerLeft (Functor.prod (fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst))) (𝟭 C))
      (CategoryStruct.comp (Y := tensor C ⋙ @DayConvolution.convolution _ _ _ _ inst_2 inst_3 U F _)
        (DayConvolution.unit U F) (Functor.whiskerLeft (tensor C) (Iso.hom (leftUnitor U F) ≫ f))))
Tactic: apply Functor.hom_ext_of_isLeftKanExtension _ (extensionUnitLeft U (φ U) F) _
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.DayConvolution (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
U : Functor.{v₁, v₂, u₁, u₂} C V
inst_4 : DayConvolutionUnit U
F : Functor.{v₁, v₂, u₁, u₂} C V
inst_5 :
  ∀ (v : V) (d : C),
    Limits.PreservesColimitsOfShape.{0, v₁, v₂, v₂, u₂, u₂}
      (CostructuredArrow (fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst))) d) (tensorRight v)
inst_6 : DayConvolution U F
G : Functor.{v₁, v₂, u₁, u₂} C V
inst_7 : DayConvolution U G
f : F ⟶ G
⊢ Eq (α :=
    fromPUnit (tensorUnit.{v₂, u₂} V (𝒞 := inst_1)) ⊠ F ⟶
      Functor.prod (fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst))) (𝟭 C) ⋙ tensor C ⋙ G)
    (extensionUnitLeft U (φ U) F ≫
      Functor.whiskerLeft (Functor.prod (fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst))) (𝟭 C))
        (CategoryStruct.comp (Y := tensor C ⋙ @DayConvolution.convolution _ _ _ _ inst_2 inst_3 U F _)
          (DayConvolution.unit U F)
          (Functor.whiskerLeft (tensor C)
            (CategoryStruct.comp (Y := @DayConvolution.convolution _ _ _ _ inst_2 inst_3 U G _)
              (DayConvolution.map (𝟙 U) f) (Iso.hom (leftUnitor U G))))))
    (extensionUnitLeft U (φ U) F ≫
      Functor.whiskerLeft (Functor.prod (fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst))) (𝟭 C))
        (CategoryStruct.comp (Y := tensor C ⋙ @DayConvolution.convolution _ _ _ _ inst_2 inst_3 U F _)
          (DayConvolution.unit U F) (Functor.whiskerLeft (tensor C) (Iso.hom (leftUnitor U F) ≫ f))))
have hγ : ∀ (x : Discrete PUnit.{1} × C),
  Eq (α :=
    (fromPUnit (tensorUnit.{v₂, u₂} V (𝒞 := inst_1)) ⊠ F).obj x ⟶
      (Functor.prod (fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst))) (𝟭 C) ⋙ tensor C ⋙ G).obj x)
    (NatTrans.app
      (extensionUnitLeft U (φ U) F ≫
        Functor.whiskerLeft (Functor.prod (fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst))) (𝟭 C))
          (CategoryStruct.comp (Y := tensor C ⋙ @DayConvolution.convolution _ _ _ _ inst_2 inst_3 U F _)
            (DayConvolution.unit U F)
            (Functor.whiskerLeft (tensor C)
              (CategoryStruct.comp (Y := @DayConvolution.convolution _ _ _ _ inst_2 inst_3 U G _)
                (DayConvolution.map (𝟙 U) f) (Iso.hom (leftUnitor U G))))))
      x)
    (NatTrans.app
      (extensionUnitLeft U (φ U) F ≫
        Functor.whiskerLeft (Functor.prod (fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst))) (𝟭 C))
          (CategoryStruct.comp (Y := tensor C ⋙ @DayConvolution.convolution _ _ _ _ inst_2 inst_3 U F _)
            (DayConvolution.unit U F) (Functor.whiskerLeft (tensor C) (Iso.hom (leftUnitor U F) ≫ f))))
      x)
Tactic: ext
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.DayConvolution (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
U : Functor.{v₁, v₂, u₁, u₂} C V
inst_4 : DayConvolutionUnit U
F : Functor.{v₁, v₂, u₁, u₂} C V
inst_5 :
  ∀ (v : V) (d : C),
    Limits.PreservesColimitsOfShape.{0, v₁, v₂, v₂, u₂, u₂}
      (CostructuredArrow (fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst))) d) (tensorLeft v)
inst_6 : DayConvolution F U
G : Functor.{v₁, v₂, u₁, u₂} C V
inst_7 : DayConvolution G U
f : F ⟶ G
⊢ Eq (α := @DayConvolution.convolution _ _ _ _ inst_2 inst_3 F U _ ⟶ G)
    (CategoryStruct.comp (Y := @DayConvolution.convolution _ _ _ _ inst_2 inst_3 G U _) (DayConvolution.map f (𝟙 U))
      (Iso.hom (rightUnitor U G)))
    (Iso.hom (rightUnitor U F) ≫ f)
have hγ : Eq (α := F ⊠ U ⟶ tensor C ⋙ G)
  (CategoryStruct.comp (Y := tensor C ⋙ @DayConvolution.convolution _ _ _ _ inst_2 inst_3 F U _)
    (DayConvolution.unit F U)
    (Functor.whiskerLeft (tensor C)
      (CategoryStruct.comp (Y := @DayConvolution.convolution _ _ _ _ inst_2 inst_3 G U _) (DayConvolution.map f (𝟙 U))
        (Iso.hom (rightUnitor U G)))))
  (CategoryStruct.comp (Y := tensor C ⋙ @DayConvolution.convolution _ _ _ _ inst_2 inst_3 F U _)
    (DayConvolution.unit F U) (Functor.whiskerLeft (tensor C) (Iso.hom (rightUnitor U F) ≫ f)))
Tactic: apply Functor.hom_ext_of_isLeftKanExtension _ (DayConvolution.unit _ _) _
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.DayConvolution (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
U : Functor.{v₁, v₂, u₁, u₂} C V
inst_4 : DayConvolutionUnit U
F : Functor.{v₁, v₂, u₁, u₂} C V
inst_5 :
  ∀ (v : V) (d : C),
    Limits.PreservesColimitsOfShape.{0, v₁, v₂, v₂, u₂, u₂}
      (CostructuredArrow (fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst))) d) (tensorLeft v)
inst_6 : DayConvolution F U
G : Functor.{v₁, v₂, u₁, u₂} C V
inst_7 : DayConvolution G U
f : F ⟶ G
⊢ Eq (α := F ⊠ U ⟶ tensor C ⋙ G)
    (CategoryStruct.comp (Y := tensor C ⋙ @DayConvolution.convolution _ _ _ _ inst_2 inst_3 F U _)
      (DayConvolution.unit F U)
      (Functor.whiskerLeft (tensor C)
        (CategoryStruct.comp (Y := @DayConvolution.convolution _ _ _ _ inst_2 inst_3 G U _) (DayConvolution.map f (𝟙 U))
          (Iso.hom (rightUnitor U G)))))
    (CategoryStruct.comp (Y := tensor C ⋙ @DayConvolution.convolution _ _ _ _ inst_2 inst_3 F U _)
      (DayConvolution.unit F U) (Functor.whiskerLeft (tensor C) (Iso.hom (rightUnitor U F) ≫ f)))
have hγ : Eq (α :=
  F ⊠ fromPUnit (tensorUnit.{v₂, u₂} V (𝒞 := inst_1)) ⟶
    Functor.prod (𝟭 C) (fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst))) ⋙ tensor C ⋙ G)
  (extensionUnitRight U (φ U) F ≫
    Functor.whiskerLeft (Functor.prod (𝟭 C) (fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst))))
      (CategoryStruct.comp (Y := tensor C ⋙ @DayConvolution.convolution _ _ _ _ inst_2 inst_3 F U _)
        (DayConvolution.unit F U)
        (Functor.whiskerLeft (tensor C)
          (CategoryStruct.comp (Y := @DayConvolution.convolution _ _ _ _ inst_2 inst_3 G U _)
            (DayConvolution.map f (𝟙 U)) (Iso.hom (rightUnitor U G))))))
  (extensionUnitRight U (φ U) F ≫
    Functor.whiskerLeft (Functor.prod (𝟭 C) (fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst))))
      (CategoryStruct.comp (Y := tensor C ⋙ @DayConvolution.convolution _ _ _ _ inst_2 inst_3 F U _)
        (DayConvolution.unit F U) (Functor.whiskerLeft (tensor C) (Iso.hom (rightUnitor U F) ≫ f))))
Tactic: apply Functor.hom_ext_of_isLeftKanExtension _ (extensionUnitRight U (φ U) F) _
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.DayConvolution (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
U : Functor.{v₁, v₂, u₁, u₂} C V
inst_4 : DayConvolutionUnit U
F : Functor.{v₁, v₂, u₁, u₂} C V
inst_5 :
  ∀ (v : V) (d : C),
    Limits.PreservesColimitsOfShape.{0, v₁, v₂, v₂, u₂, u₂}
      (CostructuredArrow (fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst))) d) (tensorLeft v)
inst_6 : DayConvolution F U
G : Functor.{v₁, v₂, u₁, u₂} C V
inst_7 : DayConvolution G U
f : F ⟶ G
⊢ Eq (α :=
    F ⊠ fromPUnit (tensorUnit.{v₂, u₂} V (𝒞 := inst_1)) ⟶
      Functor.prod (𝟭 C) (fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst))) ⋙ tensor C ⋙ G)
    (extensionUnitRight U (φ U) F ≫
      Functor.whiskerLeft (Functor.prod (𝟭 C) (fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst))))
        (CategoryStruct.comp (Y := tensor C ⋙ @DayConvolution.convolution _ _ _ _ inst_2 inst_3 F U _)
          (DayConvolution.unit F U)
          (Functor.whiskerLeft (tensor C)
            (CategoryStruct.comp (Y := @DayConvolution.convolution _ _ _ _ inst_2 inst_3 G U _)
              (DayConvolution.map f (𝟙 U)) (Iso.hom (rightUnitor U G))))))
    (extensionUnitRight U (φ U) F ≫
      Functor.whiskerLeft (Functor.prod (𝟭 C) (fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst))))
        (CategoryStruct.comp (Y := tensor C ⋙ @DayConvolution.convolution _ _ _ _ inst_2 inst_3 F U _)
          (DayConvolution.unit F U) (Functor.whiskerLeft (tensor C) (Iso.hom (rightUnitor U F) ≫ f))))
have hγ : ∀ (x : C × Discrete PUnit.{1}),
  Eq (α :=
    (F ⊠ fromPUnit (tensorUnit.{v₂, u₂} V (𝒞 := inst_1))).obj x ⟶
      (Functor.prod (𝟭 C) (fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst))) ⋙ tensor C ⋙ G).obj x)
    (NatTrans.app
      (extensionUnitRight U (φ U) F ≫
        Functor.whiskerLeft (Functor.prod (𝟭 C) (fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst))))
          (CategoryStruct.comp (Y := tensor C ⋙ @DayConvolution.convolution _ _ _ _ inst_2 inst_3 F U _)
            (DayConvolution.unit F U)
            (Functor.whiskerLeft (tensor C)
              (CategoryStruct.comp (Y := @DayConvolution.convolution _ _ _ _ inst_2 inst_3 G U _)
                (DayConvolution.map f (𝟙 U)) (Iso.hom (rightUnitor U G))))))
      x)
    (NatTrans.app
      (extensionUnitRight U (φ U) F ≫
        Functor.whiskerLeft (Functor.prod (𝟭 C) (fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst))))
          (CategoryStruct.comp (Y := tensor C ⋙ @DayConvolution.convolution _ _ _ _ inst_2 inst_3 F U _)
            (DayConvolution.unit F U) (Functor.whiskerLeft (tensor C) (Iso.hom (rightUnitor U F) ≫ f))))
      x)
Tactic: ext
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.DayConvolution (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 :
  ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow.{v₁, v₁, u₁, u₁} (tensor C) d) (tensorLeft v)
inst_5 :
  ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow.{v₁, v₁, u₁, u₁} (tensor C) d) (tensorRight v)
inst_6 :
  ∀ (v : V) (d : C),
    Limits.PreservesColimitsOfShape.{0, v₁, v₂, v₂, u₂, u₂}
      (CostructuredArrow (Functor.fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst))) d) (tensorLeft v)
inst_7 :
  ∀ (v : V) (d : C),
    Limits.PreservesColimitsOfShape.{0, v₁, v₂, v₂, u₂, u₂}
      (CostructuredArrow (Functor.fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst))) d) (tensorRight v)
inst_8 :
  ∀ (v : V) (d : C × C),
    Limits.PreservesColimitsOfShape
      (CostructuredArrow.{v₁, v₁, u₁, u₁} (Functor.prod (𝟭 C) (Functor.fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst))))
        d)
      (tensorRight v)
F G U : Functor.{v₁, v₂, u₁, u₂} C V
inst_9 : DayConvolutionUnit U
inst_10 : DayConvolution F U
inst_11 : DayConvolution U G
inst_12 : DayConvolution F (@convolution _ _ _ _ inst_2 inst_3 U G _)
inst_13 : DayConvolution (@convolution _ _ _ _ inst_2 inst_3 F U _) G
inst_14 : DayConvolution F G
⊢ Eq (α :=
    @convolution _ _ _ _ inst_2 inst_3 (@convolution _ _ _ _ inst_2 inst_3 F U _) G _ ⟶
      @convolution _ _ _ _ inst_2 inst_3 F G _)
    (CategoryStruct.comp (Y := @convolution _ _ _ _ inst_2 inst_3 F (@convolution _ _ _ _ inst_2 inst_3 U G _) _)
      (Iso.hom (associator F U G)) (map (𝟙 F) (Iso.hom (DayConvolutionUnit.leftUnitor U G))))
    (map (Iso.hom (DayConvolutionUnit.rightUnitor U F)) (𝟙 G))
have hγ : Eq (α := @convolution _ _ _ _ inst_2 inst_3 F U _ ⊠ G ⟶ tensor C ⋙ @convolution _ _ _ _ inst_2 inst_3 F G _)
  (CategoryStruct.comp (Y :=
    tensor C ⋙ @convolution _ _ _ _ inst_2 inst_3 (@convolution _ _ _ _ inst_2 inst_3 F U _) G _)
    (unit (@convolution _ _ _ _ inst_2 inst_3 F U _) G)
    (Functor.whiskerLeft (tensor C)
      (CategoryStruct.comp (Y := @convolution _ _ _ _ inst_2 inst_3 F (@convolution _ _ _ _ inst_2 inst_3 U G _) _)
        (Iso.hom (associator F U G)) (map (𝟙 F) (Iso.hom (DayConvolutionUnit.leftUnitor U G))))))
  (CategoryStruct.comp (Y :=
    tensor C ⋙ @convolution _ _ _ _ inst_2 inst_3 (@convolution _ _ _ _ inst_2 inst_3 F U _) G _)
    (unit (@convolution _ _ _ _ inst_2 inst_3 F U _) G)
    (Functor.whiskerLeft (tensor C) (map (Iso.hom (DayConvolutionUnit.rightUnitor U F)) (𝟙 G))))
Tactic: apply Functor.hom_ext_of_isLeftKanExtension _ (DayConvolution.unit _ _) _
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.DayConvolution (goal)
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)
x y : D
⊢ 𝟙 x ⊗ₘ 𝟙 y = 𝟙 (tensorObj.{v₃, u₃} (𝒞 := inst_8) x y)
have a : 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) x y))
    (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) (tensorObj.{v₃, u₃} (𝒞 := inst_8) x y)))
  (Functor.map (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) (𝟙 x ⊗ₘ 𝟙 y))
  (Functor.map (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) (𝟙 (tensorObj.{v₃, u₃} (𝒞 := inst_8) x y)))
Tactic: apply Functor.Faithful.map_injective (F := ι C V D)
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.DayConvolution (goal)
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)
x y : D
⊢ 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) x y))
      (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) (tensorObj.{v₃, u₃} (𝒞 := inst_8) x y)))
    (Functor.map (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) (𝟙 x ⊗ₘ 𝟙 y))
    (Functor.map (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) (𝟙 (tensorObj.{v₃, u₃} (𝒞 := inst_8) x y)))
have a : 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) x y))
    (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) (tensorObj.{v₃, u₃} (𝒞 := inst_8) x y)))
  (map (𝟙 (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) x))
      (𝟙 (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) y)) :
    Quiver.Hom (V := Functor.{v₁, v₂, u₁, u₂} C V)
      (@DayConvolution.convolution _ _ _ _ inst_6 inst_7 (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) x)
        (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) y) _)
      (@DayConvolution.convolution _ _ _ _ inst_6 inst_7 (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) x)
        (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) y) _))
  (𝟙 (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) (tensorObj.{v₃, u₃} (𝒞 := inst_8) x y)))
Tactic: simp only [ι_map_tensorHom_hom_eq_tensorHom, Functor.map_id]
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.DayConvolution (goal)
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)
x y₁ y₂ : D
f : y₁ ⟶ y₂
⊢ Eq (α := tensorObj.{v₃, u₃} (𝒞 := inst_8) x y₁ ⟶ tensorObj.{v₃, u₃} (𝒞 := inst_8) x y₂) (𝟙 x ⊗ₘ f) (x ◁ f)
have a : 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) x y₁))
    (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) (tensorObj.{v₃, u₃} (𝒞 := inst_8) x y₂)))
  (Functor.map (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) (𝟙 x ⊗ₘ f))
  (Functor.map (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) (x ◁ f))
Tactic: apply Functor.Faithful.map_injective (F := ι C V D)