awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Preadditive (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : Preadditive.{v_1, u_1} C
inst_2 : MonoidalCategory.{v_1, u_1} C
inst_3 : MonoidalPreadditive.{v_1, u_1} C
inst_4 : HasFiniteBiproducts.{v_1, u_1} C
J : Type
inst_5 : Fintype J
f : J → C
X : C
⊢ Eq.{v_1 + 1} (α :=
    tensorObj.{v_1, u_1} (𝒞 := inst) (biproduct.{0, v_1, u_1} f) X ⟶
      biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X)
    (Iso.hom (rightDistributor f X) :
      tensorObj.{v_1, u_1} (𝒞 := inst) (biproduct.{0, v_1, u_1} f) X ⟶
        biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X)
    (∑ j : J,
        biproduct.π f j ▷ X ≫
          biproduct.ι.{0, v_1, u_1} (C := C) (fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X) j :
      tensorObj.{v_1, u_1} (𝒞 := inst) (biproduct.{0, v_1, u_1} f) X ⟶
        biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X)
have w : ∀ (j : J),
  Eq.{v_1 + 1} (α :=
    tensorObj.{v_1, u_1} (𝒞 := inst) (biproduct.{0, v_1, u_1} f) X ⟶ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X)
    (CategoryStruct.comp (Y := biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X)
        (Iso.hom (rightDistributor f X) :
          tensorObj.{v_1, u_1} (𝒞 := inst) (biproduct.{0, v_1, u_1} f) X ⟶
            biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X)
        (biproduct.π (fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X) j :
          (biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X) ⟶
            tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X) :
      tensorObj.{v_1, u_1} (𝒞 := inst) (biproduct.{0, v_1, u_1} f) X ⟶ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X)
    (CategoryStruct.comp (Y := biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X)
        (∑ j : J,
          biproduct.π f j ▷ X ≫
            biproduct.ι.{0, v_1, u_1} (C := C) (fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X) j)
        (biproduct.π (fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X) j) :
      tensorObj.{v_1, u_1} (𝒞 := inst) (biproduct.{0, v_1, u_1} f) X ⟶ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X)
Tactic: classical
  ext
  dsimp [rightDistributor, Functor.mapBiproduct, Functor.mapBicone]
  erw [biproduct.lift_π]
  simp only [Preadditive.sum_comp, Category.assoc, biproduct.ι_π, comp_dite, comp_zero,
    Finset.sum_dite_eq', Finset.mem_univ, eqToHom_refl, Category.comp_id, ite_true]
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Preadditive (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : Preadditive.{v_1, u_1} C
inst_2 : MonoidalCategory.{v_1, u_1} C
inst_3 : MonoidalPreadditive.{v_1, u_1} C
inst_4 : HasFiniteBiproducts.{v_1, u_1} C
J : Type
inst_5 : Fintype J
f : J → C
X : C
⊢ Eq.{v_1 + 1} (α :=
    (biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X) ⟶
      tensorObj.{v_1, u_1} (𝒞 := inst) (biproduct.{0, v_1, u_1} f) X)
    (Iso.inv (rightDistributor f X) :
      (biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X) ⟶
        tensorObj.{v_1, u_1} (𝒞 := inst) (biproduct.{0, v_1, u_1} f) X)
    (∑ j : J, biproduct.π (fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X) j ≫ biproduct.ι f j ▷ X :
      (biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X) ⟶
        tensorObj.{v_1, u_1} (𝒞 := inst) (biproduct.{0, v_1, u_1} f) X)
have w : ∀ (j : J),
  Eq.{v_1 + 1} (α :=
    tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X ⟶ tensorObj.{v_1, u_1} (𝒞 := inst) (biproduct.{0, v_1, u_1} f) X)
    (CategoryStruct.comp (Y := biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X)
        (biproduct.ι.{0, v_1, u_1} (C := C) (fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X) j :
          tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X ⟶
            biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X)
        (Iso.inv (rightDistributor f X) :
          (biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X) ⟶
            tensorObj.{v_1, u_1} (𝒞 := inst) (biproduct.{0, v_1, u_1} f) X) :
      tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X ⟶ tensorObj.{v_1, u_1} (𝒞 := inst) (biproduct.{0, v_1, u_1} f) X)
    (CategoryStruct.comp (Y := biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X)
        (biproduct.ι.{0, v_1, u_1} (C := C) (fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X) j)
        (∑ j : J, biproduct.π (fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X) j ≫ biproduct.ι f j ▷ X) :
      tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X ⟶ tensorObj.{v_1, u_1} (𝒞 := inst) (biproduct.{0, v_1, u_1} f) X)
Tactic: ext
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Preadditive (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : Preadditive.{v_1, u_1} C
inst_2 : MonoidalCategory.{v_1, u_1} C
inst_3 : MonoidalPreadditive.{v_1, u_1} C
inst_4 : HasFiniteBiproducts.{v_1, u_1} C
J : Type
inst_5 : Fintype J
f : J → C
X : C
j : J
⊢ Eq.{v_1 + 1} (α :=
    tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X ⟶ tensorObj.{v_1, u_1} (𝒞 := inst) (biproduct.{0, v_1, u_1} f) X)
    (CategoryStruct.comp (Y := biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X)
        (biproduct.ι.{0, v_1, u_1} (C := C) (fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X) j :
          tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X ⟶
            biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X)
        (Iso.inv (rightDistributor f X) :
          (biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X) ⟶
            tensorObj.{v_1, u_1} (𝒞 := inst) (biproduct.{0, v_1, u_1} f) X) :
      tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X ⟶ tensorObj.{v_1, u_1} (𝒞 := inst) (biproduct.{0, v_1, u_1} f) X)
    (CategoryStruct.comp (Y := biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X)
        (biproduct.ι.{0, v_1, u_1} (C := C) (fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X) j)
        (∑ j : J, biproduct.π (fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X) j ≫ biproduct.ι f j ▷ X) :
      tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X ⟶ tensorObj.{v_1, u_1} (𝒞 := inst) (biproduct.{0, v_1, u_1} f) X)
have w : Eq.{v_1 + 1} (α :=
  tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X ⟶ tensorObj.{v_1, u_1} (𝒞 := inst) (biproduct.{0, v_1, u_1} f) X)
  (CategoryStruct.comp (Y := biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X)
      (biproduct.ι.{0, v_1, u_1} (C := C) (fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X) j)
      (biproduct.desc fun (j : J) ↦ biproduct.ι f j ▷ X) :
    tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X ⟶ tensorObj.{v_1, u_1} (𝒞 := inst) (biproduct.{0, v_1, u_1} f) X)
  (CategoryStruct.comp (Y := biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X)
      (biproduct.ι.{0, v_1, u_1} (C := C) (fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X) j)
      (∑ j : J, biproduct.π (fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X) j ≫ biproduct.ι f j ▷ X) :
    tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X ⟶ tensorObj.{v_1, u_1} (𝒞 := inst) (biproduct.{0, v_1, u_1} f) X)
Tactic: dsimp [rightDistributor, Functor.mapBiproduct, Functor.mapBicone]
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Preadditive (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : Preadditive.{v_1, u_1} C
inst_2 : MonoidalCategory.{v_1, u_1} C
inst_3 : MonoidalPreadditive.{v_1, u_1} C
inst_4 : HasFiniteBiproducts.{v_1, u_1} C
J : Type
inst_5 : Fintype J
f : J → C
X : C
⊢ Eq.{v_1 + 1} (α :=
    (biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X) ⟶
      tensorObj.{v_1, u_1} (𝒞 := inst) (biproduct.{0, v_1, u_1} f) X)
    (Iso.inv (rightDistributor f X) :
      (biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X) ⟶
        tensorObj.{v_1, u_1} (𝒞 := inst) (biproduct.{0, v_1, u_1} f) X)
    (∑ j : J, biproduct.π (fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X) j ≫ biproduct.ι f j ▷ X :
      (biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X) ⟶
        tensorObj.{v_1, u_1} (𝒞 := inst) (biproduct.{0, v_1, u_1} f) X)
have w : ∀ (j : J),
  Eq.{v_1 + 1} (α :=
    tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X ⟶ tensorObj.{v_1, u_1} (𝒞 := inst) (biproduct.{0, v_1, u_1} f) X)
    (CategoryStruct.comp (Y := biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X)
        (biproduct.ι.{0, v_1, u_1} (C := C) (fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X) j :
          tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X ⟶
            biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X)
        (Iso.inv (rightDistributor f X) :
          (biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X) ⟶
            tensorObj.{v_1, u_1} (𝒞 := inst) (biproduct.{0, v_1, u_1} f) X) :
      tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X ⟶ tensorObj.{v_1, u_1} (𝒞 := inst) (biproduct.{0, v_1, u_1} f) X)
    (CategoryStruct.comp (Y := biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X)
        (biproduct.ι.{0, v_1, u_1} (C := C) (fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X) j)
        (∑ j : J, biproduct.π (fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X) j ≫ biproduct.ι f j ▷ X) :
      tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X ⟶ tensorObj.{v_1, u_1} (𝒞 := inst) (biproduct.{0, v_1, u_1} f) X)
Tactic: classical
  ext
  dsimp [rightDistributor, Functor.mapBiproduct, Functor.mapBicone]
  simp only [biproduct.ι_desc, Preadditive.comp_sum, biproduct.ι_π_assoc, dite_comp,
    zero_comp, Finset.sum_dite_eq, Finset.mem_univ, eqToHom_refl, Category.id_comp, ite_true]
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Preadditive (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : Preadditive.{v_1, u_1} C
inst_2 : MonoidalCategory.{v_1, u_1} C
inst_3 : MonoidalPreadditive.{v_1, u_1} C
inst_4 : HasFiniteBiproducts.{v_1, u_1} C
J : Type
inst_5 : Finite J
f : J → C
X : C
j : J
⊢ Eq.{v_1 + 1} (α :=
    tensorObj.{v_1, u_1} (𝒞 := inst) (biproduct.{0, v_1, u_1} f) X ⟶ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X)
    (CategoryStruct.comp (Y := biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X)
        (Iso.hom (rightDistributor f X) :
          tensorObj.{v_1, u_1} (𝒞 := inst) (biproduct.{0, v_1, u_1} f) X ⟶
            biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X)
        (biproduct.π (fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X) j :
          (biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X) ⟶
            tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X) :
      tensorObj.{v_1, u_1} (𝒞 := inst) (biproduct.{0, v_1, u_1} f) X ⟶ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X)
    (biproduct.π f j ▷ X :
      tensorObj.{v_1, u_1} (𝒞 := inst) (biproduct.{0, v_1, u_1} f) X ⟶ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X)
have intro : ∀ (val : Fintype J),
  Eq.{v_1 + 1} (α :=
    tensorObj.{v_1, u_1} (𝒞 := inst) (biproduct.{0, v_1, u_1} f) X ⟶ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X)
    (CategoryStruct.comp (Y := biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X)
        (Iso.hom (rightDistributor f X) :
          tensorObj.{v_1, u_1} (𝒞 := inst) (biproduct.{0, v_1, u_1} f) X ⟶
            biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X)
        (biproduct.π (fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X) j :
          (biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X) ⟶
            tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X) :
      tensorObj.{v_1, u_1} (𝒞 := inst) (biproduct.{0, v_1, u_1} f) X ⟶ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X)
    (biproduct.π f j ▷ X :
      tensorObj.{v_1, u_1} (𝒞 := inst) (biproduct.{0, v_1, u_1} f) X ⟶ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X)
Tactic: cases nonempty_fintype J
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Preadditive (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : Preadditive.{v_1, u_1} C
inst_2 : MonoidalCategory.{v_1, u_1} C
inst_3 : MonoidalPreadditive.{v_1, u_1} C
inst_4 : HasFiniteBiproducts.{v_1, u_1} C
J : Type
inst_5 : Finite J
f : J → C
X : C
j : J
⊢ Eq.{v_1 + 1} (α :=
    tensorObj.{v_1, u_1} (𝒞 := inst) (biproduct.{0, v_1, u_1} f) X ⟶ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X)
    (CategoryStruct.comp (Y := biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X)
        (Iso.hom (rightDistributor f X) :
          tensorObj.{v_1, u_1} (𝒞 := inst) (biproduct.{0, v_1, u_1} f) X ⟶
            biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X)
        (biproduct.π (fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X) j :
          (biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X) ⟶
            tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X) :
      tensorObj.{v_1, u_1} (𝒞 := inst) (biproduct.{0, v_1, u_1} f) X ⟶ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X)
    (biproduct.π f j ▷ X :
      tensorObj.{v_1, u_1} (𝒞 := inst) (biproduct.{0, v_1, u_1} f) X ⟶ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X)
have intro : ∀ (val : Fintype J),
  Eq.{v_1 + 1} (α :=
    tensorObj.{v_1, u_1} (𝒞 := inst) (biproduct.{0, v_1, u_1} f) X ⟶ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X)
    (CategoryStruct.comp (Y := biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X)
        (Iso.hom (rightDistributor f X) :
          tensorObj.{v_1, u_1} (𝒞 := inst) (biproduct.{0, v_1, u_1} f) X ⟶
            biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X)
        (biproduct.π (fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X) j :
          (biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X) ⟶
            tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X) :
      tensorObj.{v_1, u_1} (𝒞 := inst) (biproduct.{0, v_1, u_1} f) X ⟶ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X)
    (biproduct.π f j ▷ X :
      tensorObj.{v_1, u_1} (𝒞 := inst) (biproduct.{0, v_1, u_1} f) X ⟶ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X)
Tactic: classical
  cases nonempty_fintype J
  simp [rightDistributor_hom, Preadditive.sum_comp, biproduct.ι_π, comp_dite]
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Preadditive (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : Preadditive.{v_1, u_1} C
inst_2 : MonoidalCategory.{v_1, u_1} C
inst_3 : MonoidalPreadditive.{v_1, u_1} C
inst_4 : HasFiniteBiproducts.{v_1, u_1} C
J : Type
inst_5 : Finite J
f : J → C
X : C
j : J
⊢ Eq.{v_1 + 1} (α :=
    tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X ⟶
      biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X)
    (CategoryStruct.comp (Y := tensorObj.{v_1, u_1} (𝒞 := inst) (biproduct.{0, v_1, u_1} f) X)
        (biproduct.ι f j ▷ X :
          tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X ⟶ tensorObj.{v_1, u_1} (𝒞 := inst) (biproduct.{0, v_1, u_1} f) X)
        (Iso.hom (rightDistributor f X) :
          tensorObj.{v_1, u_1} (𝒞 := inst) (biproduct.{0, v_1, u_1} f) X ⟶
            biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X) :
      tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X ⟶
        biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X)
    (biproduct.ι.{0, v_1, u_1} (C := C) (fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X) j :
      tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X ⟶
        biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X)
have intro : ∀ (val : Fintype J),
  Eq.{v_1 + 1} (α :=
    tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X ⟶
      biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X)
    (CategoryStruct.comp (Y := tensorObj.{v_1, u_1} (𝒞 := inst) (biproduct.{0, v_1, u_1} f) X)
        (biproduct.ι f j ▷ X :
          tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X ⟶ tensorObj.{v_1, u_1} (𝒞 := inst) (biproduct.{0, v_1, u_1} f) X)
        (Iso.hom (rightDistributor f X) :
          tensorObj.{v_1, u_1} (𝒞 := inst) (biproduct.{0, v_1, u_1} f) X ⟶
            biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X) :
      tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X ⟶
        biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X)
    (biproduct.ι.{0, v_1, u_1} (C := C) (fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X) j :
      tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X ⟶
        biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X)
Tactic: cases nonempty_fintype J
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Preadditive (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : Preadditive.{v_1, u_1} C
inst_2 : MonoidalCategory.{v_1, u_1} C
inst_3 : MonoidalPreadditive.{v_1, u_1} C
inst_4 : HasFiniteBiproducts.{v_1, u_1} C
J : Type
inst_5 : Finite J
f : J → C
X : C
j : J
⊢ Eq.{v_1 + 1} (α :=
    tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X ⟶
      biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X)
    (CategoryStruct.comp (Y := tensorObj.{v_1, u_1} (𝒞 := inst) (biproduct.{0, v_1, u_1} f) X)
        (biproduct.ι f j ▷ X :
          tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X ⟶ tensorObj.{v_1, u_1} (𝒞 := inst) (biproduct.{0, v_1, u_1} f) X)
        (Iso.hom (rightDistributor f X) :
          tensorObj.{v_1, u_1} (𝒞 := inst) (biproduct.{0, v_1, u_1} f) X ⟶
            biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X) :
      tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X ⟶
        biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X)
    (biproduct.ι.{0, v_1, u_1} (C := C) (fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X) j :
      tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X ⟶
        biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X)
have intro : ∀ (val : Fintype J),
  Eq.{v_1 + 1} (α :=
    tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X ⟶
      biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X)
    (CategoryStruct.comp (Y := tensorObj.{v_1, u_1} (𝒞 := inst) (biproduct.{0, v_1, u_1} f) X)
        (biproduct.ι f j ▷ X :
          tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X ⟶ tensorObj.{v_1, u_1} (𝒞 := inst) (biproduct.{0, v_1, u_1} f) X)
        (Iso.hom (rightDistributor f X) :
          tensorObj.{v_1, u_1} (𝒞 := inst) (biproduct.{0, v_1, u_1} f) X ⟶
            biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X) :
      tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X ⟶
        biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X)
    (biproduct.ι.{0, v_1, u_1} (C := C) (fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X) j :
      tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X ⟶
        biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X)
Tactic: classical
  cases nonempty_fintype J
  simp [rightDistributor_hom, Preadditive.comp_sum, ← comp_whiskerRight_assoc, biproduct.ι_π,
    dite_whiskerRight, dite_comp]
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Preadditive (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : Preadditive.{v_1, u_1} C
inst_2 : MonoidalCategory.{v_1, u_1} C
inst_3 : MonoidalPreadditive.{v_1, u_1} C
inst_4 : HasFiniteBiproducts.{v_1, u_1} C
J : Type
inst_5 : Finite J
f : J → C
X : C
j : J
⊢ Eq.{v_1 + 1} (α :=
    (biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X) ⟶
      tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X)
    (CategoryStruct.comp (Y := tensorObj.{v_1, u_1} (𝒞 := inst) (biproduct.{0, v_1, u_1} f) X)
        (Iso.inv (rightDistributor f X) :
          (biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X) ⟶
            tensorObj.{v_1, u_1} (𝒞 := inst) (biproduct.{0, v_1, u_1} f) X)
        (biproduct.π f j ▷ X :
          tensorObj.{v_1, u_1} (𝒞 := inst) (biproduct.{0, v_1, u_1} f) X ⟶ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X) :
      (biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X) ⟶
        tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X)
    (biproduct.π (fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X) j :
      (biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X) ⟶
        tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X)
have intro : ∀ (val : Fintype J),
  Eq.{v_1 + 1} (α :=
    (biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X) ⟶
      tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X)
    (CategoryStruct.comp (Y := tensorObj.{v_1, u_1} (𝒞 := inst) (biproduct.{0, v_1, u_1} f) X)
        (Iso.inv (rightDistributor f X) :
          (biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X) ⟶
            tensorObj.{v_1, u_1} (𝒞 := inst) (biproduct.{0, v_1, u_1} f) X)
        (biproduct.π f j ▷ X :
          tensorObj.{v_1, u_1} (𝒞 := inst) (biproduct.{0, v_1, u_1} f) X ⟶ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X) :
      (biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X) ⟶
        tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X)
    (biproduct.π (fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X) j :
      (biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X) ⟶
        tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X)
Tactic: cases nonempty_fintype J
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Preadditive (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : Preadditive.{v_1, u_1} C
inst_2 : MonoidalCategory.{v_1, u_1} C
inst_3 : MonoidalPreadditive.{v_1, u_1} C
inst_4 : HasFiniteBiproducts.{v_1, u_1} C
J : Type
inst_5 : Finite J
f : J → C
X : C
j : J
⊢ Eq.{v_1 + 1} (α :=
    (biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X) ⟶
      tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X)
    (CategoryStruct.comp (Y := tensorObj.{v_1, u_1} (𝒞 := inst) (biproduct.{0, v_1, u_1} f) X)
        (Iso.inv (rightDistributor f X) :
          (biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X) ⟶
            tensorObj.{v_1, u_1} (𝒞 := inst) (biproduct.{0, v_1, u_1} f) X)
        (biproduct.π f j ▷ X :
          tensorObj.{v_1, u_1} (𝒞 := inst) (biproduct.{0, v_1, u_1} f) X ⟶ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X) :
      (biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X) ⟶
        tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X)
    (biproduct.π (fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X) j :
      (biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X) ⟶
        tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X)
have intro : ∀ (val : Fintype J),
  Eq.{v_1 + 1} (α :=
    (biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X) ⟶
      tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X)
    (CategoryStruct.comp (Y := tensorObj.{v_1, u_1} (𝒞 := inst) (biproduct.{0, v_1, u_1} f) X)
        (Iso.inv (rightDistributor f X) :
          (biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X) ⟶
            tensorObj.{v_1, u_1} (𝒞 := inst) (biproduct.{0, v_1, u_1} f) X)
        (biproduct.π f j ▷ X :
          tensorObj.{v_1, u_1} (𝒞 := inst) (biproduct.{0, v_1, u_1} f) X ⟶ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X) :
      (biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X) ⟶
        tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X)
    (biproduct.π (fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X) j :
      (biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X) ⟶
        tensorObj.{v_1, u_1} (𝒞 := inst) (f j) X)
Tactic: classical
  cases nonempty_fintype J
  simp [rightDistributor_inv, Preadditive.sum_comp, ← comp_whiskerRight,
    biproduct.ι_π, dite_whiskerRight, comp_dite]