awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.CategoryTheory.Monoidal.Preadditive
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
X : C
J : Type
inst_4 : Finite J
val : Fintype J
f : J → C
b : Bicone.{0, v_1, u_1} f
i : b.IsBilimit
⊢ Eq.{v_1 + 1} (α := tensorObj.{v_1, u_1} (𝒞 := inst) b.pt X ⟶ tensorObj.{v_1, u_1} (𝒞 := inst) b.pt X)
    (∑ x : J, (b.π x ⊗ₘ 𝟙 X) ≫ (b.ι x ⊗ₘ 𝟙 X)) (𝟙 (tensorObj.{v_1, u_1} (𝒞 := inst) b.pt X))

Tactic:
simp only [tensorHom_comp_tensorHom, Category.comp_id, ← sum_tensor, ← id_tensorHom_id,
               IsBilimit.total i]

Premises used: CategoryTheory.sum_tensor, CategoryTheory.Limits.IsBilimit.total, CategoryTheory.MonoidalCategory.tensorHom_comp_tensorHom, CategoryTheory.Category.comp_id, CategoryTheory.MonoidalCategory.id_tensorHom_id
-- Module: Mathlib.CategoryTheory.Monoidal.Preadditive
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
X : C
f : J → C
w :
  ∀ (j : J),
    Eq.{v_1 + 1} (α :=
      tensorObj.{v_1, u_1} (𝒞 := inst) X (biproduct.{0, v_1, u_1} f) ⟶ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j))
      (CategoryStruct.comp (Y := biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j))
          (Iso.hom (leftDistributor X f) :
            tensorObj.{v_1, u_1} (𝒞 := inst) X (biproduct.{0, v_1, u_1} f) ⟶
              biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j))
          (biproduct.π (fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) j :
            (biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) ⟶
              tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) :
        tensorObj.{v_1, u_1} (𝒞 := inst) X (biproduct.{0, v_1, u_1} f) ⟶ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j))
      (CategoryStruct.comp (Y := biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j))
          (∑ j : J,
            X ◁ biproduct.π f j ≫
              biproduct.ι.{0, v_1, u_1} (C := C) (fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) j)
          (biproduct.π (fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) j) :
        tensorObj.{v_1, u_1} (𝒞 := inst) X (biproduct.{0, v_1, u_1} f) ⟶ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j))
⊢ Eq.{v_1 + 1} (α :=
    tensorObj.{v_1, u_1} (𝒞 := inst) X (biproduct.{0, v_1, u_1} f) ⟶
      biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j))
    (Iso.hom (leftDistributor X f) :
      tensorObj.{v_1, u_1} (𝒞 := inst) X (biproduct.{0, v_1, u_1} f) ⟶
        biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j))
    (∑ j : J,
        X ◁ biproduct.π f j ≫
          biproduct.ι.{0, v_1, u_1} (C := C) (fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) j :
      tensorObj.{v_1, u_1} (𝒞 := inst) X (biproduct.{0, v_1, u_1} f) ⟶
        biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j))

Tactic:
ext
-- Module: Mathlib.CategoryTheory.Monoidal.Preadditive
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
X : C
f : J → C
j : J
w :
  Eq.{v_1 + 1} (α :=
    tensorObj.{v_1, u_1} (𝒞 := inst) X (biproduct.{0, v_1, u_1} f) ⟶ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j))
    (CategoryStruct.comp (Y := biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j))
        (biproduct.lift fun (j : J) ↦ X ◁ biproduct.π f j)
        (biproduct.π (fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) j) :
      tensorObj.{v_1, u_1} (𝒞 := inst) X (biproduct.{0, v_1, u_1} f) ⟶ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j))
    (CategoryStruct.comp (Y := biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j))
        (∑ j : J,
          X ◁ biproduct.π f j ≫
            biproduct.ι.{0, v_1, u_1} (C := C) (fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) j)
        (biproduct.π (fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) j) :
      tensorObj.{v_1, u_1} (𝒞 := inst) X (biproduct.{0, v_1, u_1} f) ⟶ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j))
⊢ Eq.{v_1 + 1} (α :=
    tensorObj.{v_1, u_1} (𝒞 := inst) X (biproduct.{0, v_1, u_1} f) ⟶ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j))
    (CategoryStruct.comp (Y := biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j))
        (Iso.hom (leftDistributor X f) :
          tensorObj.{v_1, u_1} (𝒞 := inst) X (biproduct.{0, v_1, u_1} f) ⟶
            biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j))
        (biproduct.π (fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) j :
          (biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) ⟶
            tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) :
      tensorObj.{v_1, u_1} (𝒞 := inst) X (biproduct.{0, v_1, u_1} f) ⟶ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j))
    (CategoryStruct.comp (Y := biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j))
        (∑ j : J,
          X ◁ biproduct.π f j ≫
            biproduct.ι.{0, v_1, u_1} (C := C) (fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) j)
        (biproduct.π (fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) j) :
      tensorObj.{v_1, u_1} (𝒞 := inst) X (biproduct.{0, v_1, u_1} f) ⟶ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j))

Tactic:
dsimp [leftDistributor, Functor.mapBiproduct, Functor.mapBicone]

Premises used: CategoryTheory.Functor.mapBicone, CategoryTheory.Functor.mapBiproduct, CategoryTheory.leftDistributor
-- Module: Mathlib.CategoryTheory.Monoidal.Preadditive
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
X : C
f : J → C
j : J
w :
  Eq.{v_1 + 1} (α :=
    tensorObj.{v_1, u_1} (𝒞 := inst) X (biproduct.{0, v_1, u_1} f) ⟶ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j))
    (X ◁ biproduct.π f j)
    (CategoryStruct.comp (Y := biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j))
      (∑ j : J,
        X ◁ biproduct.π f j ≫
          biproduct.ι.{0, v_1, u_1} (C := C) (fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) j)
      (biproduct.π (fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) j))
⊢ Eq.{v_1 + 1} (α :=
    tensorObj.{v_1, u_1} (𝒞 := inst) X (biproduct.{0, v_1, u_1} f) ⟶ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j))
    (CategoryStruct.comp (Y := biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j))
        (biproduct.lift fun (j : J) ↦ X ◁ biproduct.π f j)
        (biproduct.π (fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) j) :
      tensorObj.{v_1, u_1} (𝒞 := inst) X (biproduct.{0, v_1, u_1} f) ⟶ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j))
    (CategoryStruct.comp (Y := biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j))
        (∑ j : J,
          X ◁ biproduct.π f j ≫
            biproduct.ι.{0, v_1, u_1} (C := C) (fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) j)
        (biproduct.π (fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) j) :
      tensorObj.{v_1, u_1} (𝒞 := inst) X (biproduct.{0, v_1, u_1} f) ⟶ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j))

Tactic:
erw [biproduct.lift_π]

Premises used: CategoryTheory.Limits.biproduct.lift_π
-- Module: Mathlib.CategoryTheory.Monoidal.Preadditive
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
X : C
f : J → C
j : J
⊢ Eq.{v_1 + 1} (α :=
    tensorObj.{v_1, u_1} (𝒞 := inst) X (biproduct.{0, v_1, u_1} f) ⟶ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j))
    (X ◁ biproduct.π f j)
    (CategoryStruct.comp (Y := biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j))
      (∑ j : J,
        X ◁ biproduct.π f j ≫
          biproduct.ι.{0, v_1, u_1} (C := C) (fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) j)
      (biproduct.π (fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) j))

Tactic:
simp only [Preadditive.sum_comp, Category.assoc, biproduct.ι_π, comp_dite, comp_zero,
    Finset.sum_dite_eq', Finset.mem_univ, ite_true, eqToHom_refl, Category.comp_id]

Premises used: Finset.mem_univ, CategoryTheory.Category.assoc, CategoryTheory.Limits.comp_zero, CategoryTheory.Preadditive.sum_comp, CategoryTheory.eqToHom_refl, ite_true, Finset.sum_dite_eq', CategoryTheory.Category.comp_id, CategoryTheory.Limits.biproduct.ι_π, CategoryTheory.comp_dite
-- Module: Mathlib.CategoryTheory.Monoidal.Preadditive
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
X : C
f : J → C
w :
  ∀ (j : J),
    Eq.{v_1 + 1} (α :=
      tensorObj.{v_1, u_1} (𝒞 := inst) X (f j) ⟶ tensorObj.{v_1, u_1} (𝒞 := inst) X (biproduct.{0, v_1, u_1} f))
      (CategoryStruct.comp (Y := biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j))
          (biproduct.ι.{0, v_1, u_1} (C := C) (fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) j :
            tensorObj.{v_1, u_1} (𝒞 := inst) X (f j) ⟶
              biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j))
          (Iso.inv (leftDistributor X f) :
            (biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) ⟶
              tensorObj.{v_1, u_1} (𝒞 := inst) X (biproduct.{0, v_1, u_1} f)) :
        tensorObj.{v_1, u_1} (𝒞 := inst) X (f j) ⟶ tensorObj.{v_1, u_1} (𝒞 := inst) X (biproduct.{0, v_1, u_1} f))
      (CategoryStruct.comp (Y := biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j))
          (biproduct.ι.{0, v_1, u_1} (C := C) (fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) j)
          (∑ j : J, biproduct.π (fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) j ≫ X ◁ biproduct.ι f j) :
        tensorObj.{v_1, u_1} (𝒞 := inst) X (f j) ⟶ tensorObj.{v_1, u_1} (𝒞 := inst) X (biproduct.{0, v_1, u_1} f))
⊢ Eq.{v_1 + 1} (α :=
    (biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) ⟶
      tensorObj.{v_1, u_1} (𝒞 := inst) X (biproduct.{0, v_1, u_1} f))
    (Iso.inv (leftDistributor X f) :
      (biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) ⟶
        tensorObj.{v_1, u_1} (𝒞 := inst) X (biproduct.{0, v_1, u_1} f))
    (∑ j : J, biproduct.π (fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) j ≫ X ◁ biproduct.ι f j :
      (biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) ⟶
        tensorObj.{v_1, u_1} (𝒞 := inst) X (biproduct.{0, v_1, u_1} f))

Tactic:
ext
-- Module: Mathlib.CategoryTheory.Monoidal.Preadditive
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
X : C
f : J → C
j : J
w :
  Eq.{v_1 + 1} (α :=
    tensorObj.{v_1, u_1} (𝒞 := inst) X (f j) ⟶ tensorObj.{v_1, u_1} (𝒞 := inst) X (biproduct.{0, v_1, u_1} f))
    (CategoryStruct.comp (Y := biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j))
        (biproduct.ι.{0, v_1, u_1} (C := C) (fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) j)
        (biproduct.desc fun (j : J) ↦ X ◁ biproduct.ι f j) :
      tensorObj.{v_1, u_1} (𝒞 := inst) X (f j) ⟶ tensorObj.{v_1, u_1} (𝒞 := inst) X (biproduct.{0, v_1, u_1} f))
    (CategoryStruct.comp (Y := biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j))
        (biproduct.ι.{0, v_1, u_1} (C := C) (fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) j)
        (∑ j : J, biproduct.π (fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) j ≫ X ◁ biproduct.ι f j) :
      tensorObj.{v_1, u_1} (𝒞 := inst) X (f j) ⟶ tensorObj.{v_1, u_1} (𝒞 := inst) X (biproduct.{0, v_1, u_1} f))
⊢ Eq.{v_1 + 1} (α :=
    tensorObj.{v_1, u_1} (𝒞 := inst) X (f j) ⟶ tensorObj.{v_1, u_1} (𝒞 := inst) X (biproduct.{0, v_1, u_1} f))
    (CategoryStruct.comp (Y := biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j))
        (biproduct.ι.{0, v_1, u_1} (C := C) (fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) j :
          tensorObj.{v_1, u_1} (𝒞 := inst) X (f j) ⟶
            biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j))
        (Iso.inv (leftDistributor X f) :
          (biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) ⟶
            tensorObj.{v_1, u_1} (𝒞 := inst) X (biproduct.{0, v_1, u_1} f)) :
      tensorObj.{v_1, u_1} (𝒞 := inst) X (f j) ⟶ tensorObj.{v_1, u_1} (𝒞 := inst) X (biproduct.{0, v_1, u_1} f))
    (CategoryStruct.comp (Y := biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j))
        (biproduct.ι.{0, v_1, u_1} (C := C) (fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) j)
        (∑ j : J, biproduct.π (fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) j ≫ X ◁ biproduct.ι f j) :
      tensorObj.{v_1, u_1} (𝒞 := inst) X (f j) ⟶ tensorObj.{v_1, u_1} (𝒞 := inst) X (biproduct.{0, v_1, u_1} f))

Tactic:
dsimp [leftDistributor, Functor.mapBiproduct, Functor.mapBicone]

Premises used: CategoryTheory.Functor.mapBicone, CategoryTheory.Functor.mapBiproduct, CategoryTheory.leftDistributor
-- Module: Mathlib.CategoryTheory.Monoidal.Preadditive
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
X : C
f : J → C
j : J
⊢ Eq.{v_1 + 1} (α :=
    tensorObj.{v_1, u_1} (𝒞 := inst) X (f j) ⟶ tensorObj.{v_1, u_1} (𝒞 := inst) X (biproduct.{0, v_1, u_1} f))
    (CategoryStruct.comp (Y := biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j))
        (biproduct.ι.{0, v_1, u_1} (C := C) (fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) j)
        (biproduct.desc fun (j : J) ↦ X ◁ biproduct.ι f j) :
      tensorObj.{v_1, u_1} (𝒞 := inst) X (f j) ⟶ tensorObj.{v_1, u_1} (𝒞 := inst) X (biproduct.{0, v_1, u_1} f))
    (CategoryStruct.comp (Y := biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j))
        (biproduct.ι.{0, v_1, u_1} (C := C) (fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) j)
        (∑ j : J, biproduct.π (fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) j ≫ X ◁ biproduct.ι f j) :
      tensorObj.{v_1, u_1} (𝒞 := inst) X (f j) ⟶ tensorObj.{v_1, u_1} (𝒞 := inst) X (biproduct.{0, v_1, u_1} f))

Tactic:
simp only [Preadditive.comp_sum, biproduct.ι_π_assoc, dite_comp, zero_comp,
    Finset.sum_dite_eq, Finset.mem_univ, ite_true, eqToHom_refl, Category.id_comp,
    biproduct.ι_desc]

Premises used: Finset.mem_univ, CategoryTheory.Limits.biproduct.ι_π_assoc, CategoryTheory.dite_comp, CategoryTheory.eqToHom_refl, ite_true, CategoryTheory.Limits.biproduct.ι_desc, CategoryTheory.Preadditive.comp_sum, CategoryTheory.Limits.zero_comp, Finset.sum_dite_eq, CategoryTheory.Category.id_comp
-- Module: Mathlib.CategoryTheory.Monoidal.Preadditive
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
X : C
f : J → C
j : J
intro :
  ∀ (val : Fintype J),
    Eq.{v_1 + 1} (α :=
      tensorObj.{v_1, u_1} (𝒞 := inst) X (biproduct.{0, v_1, u_1} f) ⟶ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j))
      (CategoryStruct.comp (Y := biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j))
          (Iso.hom (leftDistributor X f) :
            tensorObj.{v_1, u_1} (𝒞 := inst) X (biproduct.{0, v_1, u_1} f) ⟶
              biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j))
          (biproduct.π (fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) j :
            (biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) ⟶
              tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) :
        tensorObj.{v_1, u_1} (𝒞 := inst) X (biproduct.{0, v_1, u_1} f) ⟶ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j))
      (X ◁ biproduct.π f j :
        tensorObj.{v_1, u_1} (𝒞 := inst) X (biproduct.{0, v_1, u_1} f) ⟶ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j))
⊢ Eq.{v_1 + 1} (α :=
    tensorObj.{v_1, u_1} (𝒞 := inst) X (biproduct.{0, v_1, u_1} f) ⟶ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j))
    (CategoryStruct.comp (Y := biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j))
        (Iso.hom (leftDistributor X f) :
          tensorObj.{v_1, u_1} (𝒞 := inst) X (biproduct.{0, v_1, u_1} f) ⟶
            biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j))
        (biproduct.π (fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) j :
          (biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) ⟶
            tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) :
      tensorObj.{v_1, u_1} (𝒞 := inst) X (biproduct.{0, v_1, u_1} f) ⟶ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j))
    (X ◁ biproduct.π f j :
      tensorObj.{v_1, u_1} (𝒞 := inst) X (biproduct.{0, v_1, u_1} f) ⟶ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j))

Tactic:
cases nonempty_fintype J

Premises used: nonempty_fintype
-- Module: Mathlib.CategoryTheory.Monoidal.Preadditive
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
X : C
f : J → C
j : J
val : Fintype J
⊢ Eq.{v_1 + 1} (α :=
    tensorObj.{v_1, u_1} (𝒞 := inst) X (biproduct.{0, v_1, u_1} f) ⟶ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j))
    (CategoryStruct.comp (Y := biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j))
        (Iso.hom (leftDistributor X f) :
          tensorObj.{v_1, u_1} (𝒞 := inst) X (biproduct.{0, v_1, u_1} f) ⟶
            biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j))
        (biproduct.π (fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) j :
          (biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) ⟶
            tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) :
      tensorObj.{v_1, u_1} (𝒞 := inst) X (biproduct.{0, v_1, u_1} f) ⟶ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j))
    (X ◁ biproduct.π f j :
      tensorObj.{v_1, u_1} (𝒞 := inst) X (biproduct.{0, v_1, u_1} f) ⟶ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j))

Tactic:
simp [leftDistributor_hom, Preadditive.sum_comp, biproduct.ι_π, comp_dite]

Premises used: CategoryTheory.leftDistributor_hom, CategoryTheory.Preadditive.sum_comp, CategoryTheory.Limits.biproduct.ι_π, CategoryTheory.comp_dite
-- Module: Mathlib.CategoryTheory.Monoidal.Preadditive
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
X : C
f : J → C
j : J
intro :
  ∀ (val : Fintype J),
    Eq.{v_1 + 1} (α :=
      tensorObj.{v_1, u_1} (𝒞 := inst) X (biproduct.{0, v_1, u_1} f) ⟶ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j))
      (CategoryStruct.comp (Y := biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j))
          (Iso.hom (leftDistributor X f) :
            tensorObj.{v_1, u_1} (𝒞 := inst) X (biproduct.{0, v_1, u_1} f) ⟶
              biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j))
          (biproduct.π (fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) j :
            (biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) ⟶
              tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) :
        tensorObj.{v_1, u_1} (𝒞 := inst) X (biproduct.{0, v_1, u_1} f) ⟶ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j))
      (X ◁ biproduct.π f j :
        tensorObj.{v_1, u_1} (𝒞 := inst) X (biproduct.{0, v_1, u_1} f) ⟶ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j))
⊢ Eq.{v_1 + 1} (α :=
    tensorObj.{v_1, u_1} (𝒞 := inst) X (biproduct.{0, v_1, u_1} f) ⟶ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j))
    (CategoryStruct.comp (Y := biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j))
        (Iso.hom (leftDistributor X f) :
          tensorObj.{v_1, u_1} (𝒞 := inst) X (biproduct.{0, v_1, u_1} f) ⟶
            biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j))
        (biproduct.π (fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) j :
          (biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) ⟶
            tensorObj.{v_1, u_1} (𝒞 := inst) X (f j)) :
      tensorObj.{v_1, u_1} (𝒞 := inst) X (biproduct.{0, v_1, u_1} f) ⟶ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j))
    (X ◁ biproduct.π f j :
      tensorObj.{v_1, u_1} (𝒞 := inst) X (biproduct.{0, v_1, u_1} f) ⟶ tensorObj.{v_1, u_1} (𝒞 := inst) X (f j))

Tactic:
classical
  cases nonempty_fintype J
  simp [leftDistributor_hom, Preadditive.sum_comp, biproduct.ι_π, comp_dite]

Premises used: CategoryTheory.leftDistributor_hom, CategoryTheory.Preadditive.sum_comp, nonempty_fintype, CategoryTheory.Limits.biproduct.ι_π, CategoryTheory.comp_dite