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 : Finite J
X : C
f : J → C
j : J
⊢ 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 (f j))
    (CategoryStruct.comp (Y := 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))
        (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)) :
      (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))
    (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))
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) X (f j)) ⟶
      tensorObj.{v_1, u_1} (𝒞 := inst) X (f j))
    (CategoryStruct.comp (Y := 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))
        (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)) :
      (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))
    (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))
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
X : C
f : J → C
j : J
⊢ 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 (f j))
    (CategoryStruct.comp (Y := 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))
        (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)) :
      (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))
    (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))
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) X (f j)) ⟶
      tensorObj.{v_1, u_1} (𝒞 := inst) X (f j))
    (CategoryStruct.comp (Y := 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))
        (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)) :
      (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))
    (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))
Tactic: classical
  cases nonempty_fintype J
  simp [leftDistributor_inv, Preadditive.sum_comp, ← whiskerLeft_comp,
    biproduct.ι_π, whiskerLeft_dite, 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
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 :
          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))
    (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))
have intro : ∀ (val : Fintype 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))
    (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: 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
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 :
          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))
    (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))
have intro : ∀ (val : Fintype 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))
    (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: classical
  cases nonempty_fintype J
  simp [leftDistributor_inv, Preadditive.comp_sum, biproduct.ι_π_assoc, 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
X Y : C
f : J → C
val : Fintype J
j : J
⊢ Eq (α :=
    tensorObj.{v_1, u_1} (𝒞 := inst) X (tensorObj.{v_1, u_1} (𝒞 := inst) Y (biproduct.{0, v_1, u_1} f)) ⟶
      tensorObj.{v_1, u_1} (𝒞 := inst) X (tensorObj.{v_1, u_1} (𝒞 := inst) Y (f j)))
    (if Membership.mem (γ := Finset J) Finset.univ j then
      ∑ x : J,
        (𝟙 X ⊗ₘ Y ◁ biproduct.π f x) ≫
          CategoryStruct.comp (Y :=
            tensorObj.{v_1, u_1} (𝒞 := inst) X
              (biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) Y (f j)))
            (𝟙 X ⊗ₘ biproduct.ι.{0, v_1, u_1} (C := C) (fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) Y (f j)) x)
            (X ◁ biproduct.π (fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) Y (f j)) j)
    else 0)
    (if Membership.mem (γ := Finset J) Finset.univ j then
      Iso.inv (α_ X Y (biproduct.{0, v_1, u_1} f)) ≫
        tensorObj.{v_1, u_1} (𝒞 := inst) X Y ◁ biproduct.π f j ≫ Iso.hom (α_ X Y (f j))
    else 0)
have intro : Eq (α :=
  tensorObj.{v_1, u_1} (𝒞 := inst) X (tensorObj.{v_1, u_1} (𝒞 := inst) Y (biproduct.{0, v_1, u_1} f)) ⟶
    tensorObj.{v_1, u_1} (𝒞 := inst) X (tensorObj.{v_1, u_1} (𝒞 := inst) Y (f j)))
  (if Membership.mem (γ := Finset J) Finset.univ j then
    ∑ x : J,
      (𝟙 X ⊗ₘ 𝟙 Y ⊗ₘ biproduct.π f x) ≫
        CategoryStruct.comp (Y :=
          tensorObj.{v_1, u_1} (𝒞 := inst) X
            (biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) Y (f j)))
          (𝟙 X ⊗ₘ biproduct.ι.{0, v_1, u_1} (C := C) (fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) Y (f j)) x)
          (𝟙 X ⊗ₘ biproduct.π (fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) Y (f j)) j)
  else 0)
  (if Membership.mem (γ := Finset J) Finset.univ j then
    Iso.inv (α_ X Y (biproduct.{0, v_1, u_1} f)) ≫
      (𝟙 (tensorObj.{v_1, u_1} (𝒞 := inst) X Y) ⊗ₘ biproduct.π f j) ≫ Iso.hom (α_ X Y (f j))
  else 0)
Tactic: simp_rw [← id_tensorHom]
-- 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
X Y : C
f : J → C
val : Fintype J
j : J
⊢ Eq (α :=
    tensorObj.{v_1, u_1} (𝒞 := inst) X (tensorObj.{v_1, u_1} (𝒞 := inst) Y (biproduct.{0, v_1, u_1} f)) ⟶
      tensorObj.{v_1, u_1} (𝒞 := inst) X (tensorObj.{v_1, u_1} (𝒞 := inst) Y (f j)))
    (if Membership.mem (γ := Finset J) Finset.univ j then
      ∑ x : J,
        (𝟙 X ⊗ₘ 𝟙 Y ⊗ₘ biproduct.π f x) ≫
          CategoryStruct.comp (Y :=
            tensorObj.{v_1, u_1} (𝒞 := inst) X
              (biproduct.{0, v_1, u_1} fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) Y (f j)))
            (𝟙 X ⊗ₘ biproduct.ι.{0, v_1, u_1} (C := C) (fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) Y (f j)) x)
            (𝟙 X ⊗ₘ biproduct.π (fun (j : J) ↦ tensorObj.{v_1, u_1} (𝒞 := inst) Y (f j)) j)
    else 0)
    (if Membership.mem (γ := Finset J) Finset.univ j then
      Iso.inv (α_ X Y (biproduct.{0, v_1, u_1} f)) ≫
        (𝟙 (tensorObj.{v_1, u_1} (𝒞 := inst) X Y) ⊗ₘ biproduct.π f j) ≫ Iso.hom (α_ X Y (f j))
    else 0)
have intro : Eq (α :=
  tensorObj.{v_1, u_1} (𝒞 := inst) X (tensorObj.{v_1, u_1} (𝒞 := inst) Y (biproduct.{0, v_1, u_1} f)) ⟶
    tensorObj.{v_1, u_1} (𝒞 := inst) X (tensorObj.{v_1, u_1} (𝒞 := inst) Y (f j)))
  (if Membership.mem (γ := Finset J) Finset.univ j then
    ∑ x : J, 𝟙 X ⊗ₘ (𝟙 Y ⊗ₘ biproduct.π f x) ≫ if h : x = j then eqToHom sorry else 0
  else 0)
  (if Membership.mem (γ := Finset J) Finset.univ j then
    Iso.inv (α_ X Y (biproduct.{0, v_1, u_1} f)) ≫
      (𝟙 (tensorObj.{v_1, u_1} (𝒞 := inst) X Y) ⊗ₘ biproduct.π f j) ≫ Iso.hom (α_ X Y (f j))
  else 0)
Tactic: simp only [← id_tensor_comp, biproduct.ι_π]
-- 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
X Y : C
f : J → C
val : Fintype J
j : J
⊢ Eq (α :=
    tensorObj.{v_1, u_1} (𝒞 := inst) X (tensorObj.{v_1, u_1} (𝒞 := inst) Y (biproduct.{0, v_1, u_1} f)) ⟶
      tensorObj.{v_1, u_1} (𝒞 := inst) X (tensorObj.{v_1, u_1} (𝒞 := inst) Y (f j)))
    (if Membership.mem (γ := Finset J) Finset.univ j then
      ∑ x : J, 𝟙 X ⊗ₘ (𝟙 Y ⊗ₘ biproduct.π f x) ≫ if h : x = j then eqToHom ⋯ else 0
    else 0)
    (if Membership.mem (γ := Finset J) Finset.univ j then
      Iso.inv (α_ X Y (biproduct.{0, v_1, u_1} f)) ≫
        (𝟙 (tensorObj.{v_1, u_1} (𝒞 := inst) X Y) ⊗ₘ biproduct.π f j) ≫ Iso.hom (α_ X Y (f j))
    else 0)
have intro : Eq (α :=
  tensorObj.{v_1, u_1} (𝒞 := inst) X (tensorObj.{v_1, u_1} (𝒞 := inst) Y (biproduct.{0, v_1, u_1} f)) ⟶
    tensorObj.{v_1, u_1} (𝒞 := inst) X (tensorObj.{v_1, u_1} (𝒞 := inst) Y (f j)))
  (if Membership.mem (γ := Finset J) Finset.univ j then
    ∑ x : J,
      if h : x = j then (𝟙 X ⊗ₘ 𝟙 Y ⊗ₘ biproduct.π f x) ≫ (𝟙 X ⊗ₘ eqToHom sorry)
      else (𝟙 X ⊗ₘ 𝟙 Y ⊗ₘ biproduct.π f x) ≫ (𝟙 X ⊗ₘ 0)
  else 0)
  (if Membership.mem (γ := Finset J) Finset.univ j then
    Iso.inv (α_ X Y (biproduct.{0, v_1, u_1} f)) ≫
      (𝟙 (tensorObj.{v_1, u_1} (𝒞 := inst) X Y) ⊗ₘ biproduct.π f j) ≫ Iso.hom (α_ X Y (f j))
  else 0)
Tactic: simp only [id_tensor_comp, tensor_dite, 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 : 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: 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) (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)
have w : 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)
      (biproduct.lift fun (j : J) ↦ biproduct.π f j ▷ X)
      (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)
  (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: 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
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)
        (biproduct.lift fun (j : J) ↦ biproduct.π f j ▷ X)
        (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)
    (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)
have w : 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)
  (biproduct.π 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))
Tactic: erw [biproduct.lift_π]