awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Shapes.Biproducts (goal)
Goal:
J : Type w
C : Type uC
inst : Category.{uC', uC} C
inst_1 : HasZeroMorphisms.{uC', uC} C
D : Type uD
inst_2 : Category.{uD', uD} D
inst_3 : HasZeroMorphisms.{uD', uD} D
F : J → C
K : Type w'
f : J → C
c : Bicone.{w, uC', uC} f
g : K ≃ J
refine_1 : (hc : (c.whisker g).IsBilimit) → IsLimit c.toCone
⊢ (c.whisker g).IsBilimit ≃ c.IsBilimit
have refine_2 : (hc : (c.whisker g).IsBilimit) → IsColimit c.toCocone
Tactic: refine equivOfSubsingletonOfSubsingleton (fun hc => ⟨?_, ?_⟩) fun hc => ⟨?_, ?_⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Shapes.Biproducts (goal)
Goal:
J : Type w
C : Type uC
inst : Category.{uC', uC} C
inst_1 : HasZeroMorphisms.{uC', uC} C
D : Type uD
inst_2 : Category.{uD', uD} D
inst_3 : HasZeroMorphisms.{uD', uD} D
F : J → C
K : Type w'
f : J → C
c : Bicone.{w, uC', uC} f
g : K ≃ J
refine_1 : (hc : (c.whisker g).IsBilimit) → IsLimit c.toCone
refine_2 : (hc : (c.whisker g).IsBilimit) → IsColimit c.toCocone
⊢ (c.whisker g).IsBilimit ≃ c.IsBilimit
have refine_3 : (hc : c.IsBilimit) → IsLimit (c.whisker g).toCone
Tactic: refine equivOfSubsingletonOfSubsingleton (fun hc => ⟨?_, ?_⟩) fun hc => ⟨?_, ?_⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Shapes.Biproducts (goal)
Goal:
J : Type w
C : Type uC
inst : Category.{uC', uC} C
inst_1 : HasZeroMorphisms.{uC', uC} C
D : Type uD
inst_2 : Category.{uD', uD} D
inst_3 : HasZeroMorphisms.{uD', uD} D
F : J → C
K : Type w'
f : J → C
c : Bicone.{w, uC', uC} f
g : K ≃ J
refine_1 : (hc : (c.whisker g).IsBilimit) → IsLimit c.toCone
refine_2 : (hc : (c.whisker g).IsBilimit) → IsColimit c.toCocone
refine_3 : (hc : c.IsBilimit) → IsLimit (c.whisker g).toCone
⊢ (c.whisker g).IsBilimit ≃ c.IsBilimit
have refine_4 : (hc : c.IsBilimit) → IsColimit (c.whisker g).toCocone
Tactic: refine equivOfSubsingletonOfSubsingleton (fun hc => ⟨?_, ?_⟩) fun hc => ⟨?_, ?_⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Shapes.Biproducts (goal)
Goal:
J : Type w
C : Type uC
inst : Category.{uC', uC} C
inst_1 : HasZeroMorphisms.{uC', uC} C
D : Type uD
inst_2 : Category.{uD', uD} D
inst_3 : HasZeroMorphisms.{uD', uD} D
F : J → C
K : Type w'
f : J → C
c : Bicone.{w, uC', uC} f
g : K ≃ J
hc : c.IsBilimit
⊢ IsLimit (c.whisker g).toCone
have apply : IsLimit.{w', uC', w', uC} (J := Discrete K) (F := Discrete.functor (f ∘ (⇑g : (a : K) → J)))
  (obj (Cone.postcompose (Iso.inv (Discrete.functorComp f (⇑g : (a : K) → J))))
    (Cone.whisker (Discrete.functor (Discrete.mk ∘ (⇑g : (a : K) → J))) c.toCone))
Tactic: apply IsLimit.ofIsoLimit _ (Bicone.whiskerToCone c g).symm
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Shapes.Biproducts (goal)
Goal:
J : Type w
C : Type uC
inst : Category.{uC', uC} C
inst_1 : HasZeroMorphisms.{uC', uC} C
D : Type uD
inst_2 : Category.{uD', uD} D
inst_3 : HasZeroMorphisms.{uD', uD} D
F : J → C
K : Type w'
f : J → C
c : Bicone.{w, uC', uC} f
g : K ≃ J
hc : c.IsBilimit
⊢ IsLimit.{w', uC', w', uC} (J := Discrete K) (F := Discrete.functor (f ∘ (⇑g : (a : K) → J)))
    (obj (Cone.postcompose (Iso.inv (Discrete.functorComp f (⇑g : (a : K) → J))))
      (Cone.whisker (Discrete.functor (Discrete.mk ∘ (⇑g : (a : K) → J))) c.toCone))
have apply : IsLimit.{w', uC', w', uC} (J := Discrete K) (F :=
  Discrete.functor (Discrete.mk ∘ (⇑g : (a : K) → J)) ⋙ Discrete.functor f)
  (Cone.whisker (Discrete.functor (Discrete.mk ∘ (⇑g : (a : K) → J))) c.toCone)
Tactic: apply (IsLimit.postcomposeHomEquiv (Discrete.functorComp f g).symm _).symm _
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Shapes.Biproducts (goal)
Goal:
J : Type w
C : Type uC
inst : Category.{uC', uC} C
inst_1 : HasZeroMorphisms.{uC', uC} C
D : Type uD
inst_2 : Category.{uD', uD} D
inst_3 : HasZeroMorphisms.{uD', uD} D
F : J → C
K : Type w'
f : J → C
c : Bicone.{w, uC', uC} f
g : K ≃ J
hc : c.IsBilimit
⊢ IsColimit (c.whisker g).toCocone
have apply : IsColimit.{w', uC', w', uC} (J := Discrete K) (F := Discrete.functor (f ∘ (⇑g : (a : K) → J)))
  (obj (Cocone.precompose (Iso.hom (Discrete.functorComp f (⇑g : (a : K) → J))))
    (Cocone.whisker (Discrete.functor (Discrete.mk ∘ (⇑g : (a : K) → J))) c.toCocone))
Tactic: apply IsColimit.ofIsoColimit _ (Bicone.whiskerToCocone c g).symm
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Shapes.Biproducts (goal)
Goal:
J : Type w
C : Type uC
inst : Category.{uC', uC} C
inst_1 : HasZeroMorphisms.{uC', uC} C
D : Type uD
inst_2 : Category.{uD', uD} D
inst_3 : HasZeroMorphisms.{uD', uD} D
F : J → C
K : Type w'
f : J → C
c : Bicone.{w, uC', uC} f
g : K ≃ J
hc : c.IsBilimit
⊢ IsColimit.{w', uC', w', uC} (J := Discrete K) (F := Discrete.functor (f ∘ (⇑g : (a : K) → J)))
    (obj (Cocone.precompose (Iso.hom (Discrete.functorComp f (⇑g : (a : K) → J))))
      (Cocone.whisker (Discrete.functor (Discrete.mk ∘ (⇑g : (a : K) → J))) c.toCocone))
have apply : IsColimit.{w', uC', w', uC} (J := Discrete K) (F :=
  Discrete.functor (Discrete.mk ∘ (⇑g : (a : K) → J)) ⋙ Discrete.functor f)
  (Cocone.whisker (Discrete.functor (Discrete.mk ∘ (⇑g : (a : K) → J))) c.toCocone)
Tactic: apply (IsColimit.precomposeHomEquiv (Discrete.functorComp f g) _).symm _
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Shapes.Biproducts (goal)
Goal:
J : Type w
C : Type uC
inst : Category.{uC', uC} C
inst_1 : HasZeroMorphisms.{uC', uC} C
D : Type uD
inst_2 : Category.{uD', uD} D
inst_3 : HasZeroMorphisms.{uD', uD} D
F : J → C
inst_4 : HasFiniteBiproducts.{uC', uC} C
inst_5 : Finite J
⊢ HasBiproductsOfShape.{w, uC', uC} J C
have rcases : ∀ (n : ℕ) (e : J ≃ Fin n), HasBiproductsOfShape.{w, uC', uC} J C
Tactic: rcases Finite.exists_equiv_fin J with ⟨n, ⟨e⟩⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Shapes.Biproducts (hypothesis)
Goal:
J : Type w
C : Type uC
inst : Category.{uC', uC} C
inst_1 : HasZeroMorphisms.{uC', uC} C
D : Type uD
inst_2 : Category.{uD', uD} D
inst_3 : HasZeroMorphisms.{uD', uD} D
F : J → C
inst_4 : HasFiniteBiproducts.{uC', uC} C
inst_5 : Finite J
n : ℕ
e : J ≃ Fin n
⊢ HasBiproductsOfShape.{w, uC', uC} J C
have this : HasBiproductsOfShape.{0, uC', uC} (Fin n) C
Tactic: haveI : HasBiproductsOfShape (Fin n) C := HasFiniteBiproducts.out n
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Shapes.Biproducts (goal)
Goal:
J : Type w
C : Type u
inst : Category.{v, u} C
inst_1 : HasZeroMorphisms.{v, u} C
f : J → C
inst_2 : HasBiproduct.{w, v, u} f
j j' : J
w : j = j'
⊢ Eq.{v + 1} (α := f j ⟶ biproduct.{w, v, u} f) (eqToHom ⋯ ≫ ι f j') (ι f j)
have refl : Eq.{v + 1} (α := f j ⟶ biproduct.{w, v, u} f) (eqToHom sorry ≫ ι f j) (ι f j)
Tactic: cases w
Cleared hypotheses: j', w
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Shapes.Biproducts (goal)
Goal:
J : Type w
C : Type u
inst : Category.{v, u} C
inst_1 : HasZeroMorphisms.{v, u} C
f g : J → C
inst_2 : HasBiproduct.{w, v, u} f
inst_3 : HasBiproduct.{w, v, u} g
p : (b : J) → f b ⟶ g b
⊢ Eq (α := biproduct.{w, v, u} f ⟶ biproduct.{w, v, u} g) (map p) (map' p)
have w : ∀ (j j_1 : J),
  CategoryStruct.comp (Y := biproduct.{w, v, u} f) (ι f j_1)
      (CategoryStruct.comp (Y := biproduct.{w, v, u} g) (map p) (π g j)) =
    CategoryStruct.comp (Y := biproduct.{w, v, u} f) (ι f j_1)
      (CategoryStruct.comp (Y := biproduct.{w, v, u} g) (map' p) (π g j))
Tactic: ext
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Shapes.Biproducts (goal)
Goal:
J : Type w
C : Type u
inst : Category.{v, u} C
inst_1 : HasZeroMorphisms.{v, u} C
f g : J → C
inst_2 : HasBiproduct.{w, v, u} f
inst_3 : HasBiproduct.{w, v, u} g
p : (b : J) → f b ⟶ g b
j j_1 : J
⊢ CategoryStruct.comp (Y := biproduct.{w, v, u} f) (ι f j_1)
      (CategoryStruct.comp (Y := biproduct.{w, v, u} g) (map p) (π g j)) =
    CategoryStruct.comp (Y := biproduct.{w, v, u} f) (ι f j_1)
      (CategoryStruct.comp (Y := biproduct.{w, v, u} g) (map' p) (π g j))
have w : Eq (α := f j_1 ⟶ g j)
  (CategoryStruct.comp (Y := biproduct.{w, v, u} f)
    (NatTrans.app (Cocone.ι (Bicone.toCocone (bicone f))) { as := j_1 } :
      obj.{w, v, w, u} (Discrete.functor f) { as := j_1 : Discrete J } ⟶
        obj.{w, v, w, u} (obj (const (Discrete J)) (Cocone.pt.{w, v, w, u} (Bicone.toCocone (bicone f))))
          { as := j_1 : Discrete J })
    (NatTrans.app (Cone.π (Bicone.toCone (bicone f))) { as := j } ≫ p j))
  (p j_1 ≫
    CategoryStruct.comp (Y := Cocone.pt.{w, v, w, u} (Bicone.toCocone (bicone g)))
      (NatTrans.app (Cocone.ι (Bicone.toCocone (bicone g))) { as := j_1 } :
        obj.{w, v, w, u} (Discrete.functor g) { as := j_1 : Discrete J } ⟶
          obj.{w, v, w, u} (obj (const (Discrete J)) (Cocone.pt.{w, v, w, u} (Bicone.toCocone (bicone g))))
            { as := j_1 : Discrete J })
      (NatTrans.app (Cone.π (Bicone.toCone (bicone g))) { as := j }))
Tactic: simp only [Discrete.natTrans_app, Limits.IsColimit.ι_map_assoc, Limits.IsLimit.map_π,
    ← Bicone.toCone_π_app_mk, ← Bicone.toCocone_ι_app_mk]
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Shapes.Biproducts (goal)
Goal:
J : Type w
C : Type u
inst : Category.{v, u} C
inst_1 : HasZeroMorphisms.{v, u} C
f g : J → C
inst_2 : HasBiproduct.{w, v, u} f
inst_3 : HasBiproduct.{w, v, u} g
p : (b : J) → f b ⟶ g b
j j_1 : J
⊢ Eq (α := f j_1 ⟶ g j)
    (CategoryStruct.comp (Y := biproduct.{w, v, u} f)
      (NatTrans.app (Cocone.ι (Bicone.toCocone (bicone f))) { as := j_1 } :
        obj.{w, v, w, u} (Discrete.functor f) { as := j_1 : Discrete J } ⟶
          obj.{w, v, w, u} (obj (const (Discrete J)) (Cocone.pt.{w, v, w, u} (Bicone.toCocone (bicone f))))
            { as := j_1 : Discrete J })
      (NatTrans.app (Cone.π (Bicone.toCone (bicone f))) { as := j } ≫ p j))
    (p j_1 ≫
      CategoryStruct.comp (Y := Cocone.pt.{w, v, w, u} (Bicone.toCocone (bicone g)))
        (NatTrans.app (Cocone.ι (Bicone.toCocone (bicone g))) { as := j_1 } :
          obj.{w, v, w, u} (Discrete.functor g) { as := j_1 : Discrete J } ⟶
            obj.{w, v, w, u} (obj (const (Discrete J)) (Cocone.pt.{w, v, w, u} (Bicone.toCocone (bicone g))))
              { as := j_1 : Discrete J })
        (NatTrans.app (Cone.π (Bicone.toCone (bicone g))) { as := j }))
have w : CategoryStruct.comp (Y := biproduct.{w, v, u} f) (ι f j_1) (π f j ≫ p j) =
  p j_1 ≫ CategoryStruct.comp (Y := Bicone.pt.{w, v, u} (bicone g)) (ι g j_1) (π g j)
Tactic: dsimp
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Shapes.Biproducts (goal)
Goal:
J : Type w
C : Type u
inst : Category.{v, u} C
inst_1 : HasZeroMorphisms.{v, u} C
f g : J → C
inst_2 : HasBiproduct.{w, v, u} f
inst_3 : HasBiproduct.{w, v, u} g
p : (b : J) → f b ⟶ g b
j j_1 : J
⊢ CategoryStruct.comp (Y := biproduct.{w, v, u} f) (ι f j_1) (π f j ≫ p j) =
    p j_1 ≫ CategoryStruct.comp (Y := Bicone.pt.{w, v, u} (bicone g)) (ι g j_1) (π g j)
have w : Eq.{v + 1} ((if h : j_1 = j then eqToHom sorry else 0) ≫ p j) (p j_1 ≫ if h : j_1 = j then eqToHom sorry else 0)
Tactic: rw [biproduct.ι_π_assoc, biproduct.ι_π]
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Shapes.Biproducts (goal)
Goal:
J : Type w
C : Type u
inst : Category.{v, u} C
inst_1 : HasZeroMorphisms.{v, u} C
f g : J → C
inst_2 : HasBiproduct.{w, v, u} f
inst_3 : HasBiproduct.{w, v, u} g
p : (b : J) → f b ⟶ g b
j j_1 : J
⊢ Eq.{v + 1} ((if h : j_1 = j then eqToHom ⋯ else 0) ≫ p j) (p j_1 ≫ if h : j_1 = j then eqToHom ⋯ else 0)
have pos : ∀ (h : j_1 = j), eqToHom sorry ≫ p j = p j_1 ≫ eqToHom sorry
Tactic: split_ifs with h
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Shapes.Biproducts (goal)
Goal:
J : Type w
C : Type u
inst : Category.{v, u} C
inst_1 : HasZeroMorphisms.{v, u} C
f g : J → C
inst_2 : HasBiproduct.{w, v, u} f
inst_3 : HasBiproduct.{w, v, u} g
p : (b : J) → f b ⟶ g b
j j_1 : J
pos : ∀ (h : j_1 = j), eqToHom ⋯ ≫ p j = p j_1 ≫ eqToHom ⋯
⊢ Eq.{v + 1} ((if h : j_1 = j then eqToHom ⋯ else 0) ≫ p j) (p j_1 ≫ if h : j_1 = j then eqToHom ⋯ else 0)
have neg : ∀ (h : ¬j_1 = j), Eq.{v + 1} (0 ≫ p j) (p j_1 ≫ 0)
Tactic: split_ifs with h
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Shapes.Biproducts (goal)
Goal:
J : Type w
C : Type u
inst : Category.{v, u} C
inst_1 : HasZeroMorphisms.{v, u} C
f g : J → C
inst_2 : HasBiproduct.{w, v, u} f
inst_3 : HasBiproduct.{w, v, u} g
p : (b : J) → f b ⟶ g b
j j_1 : J
h : j_1 = j
⊢ eqToHom ⋯ ≫ p j = p j_1 ≫ eqToHom ⋯
have pos : eqToHom sorry ≫ p j_1 = p j_1 ≫ eqToHom sorry
Tactic: subst h
Cleared hypotheses: j, h
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Shapes.Biproducts (goal)
Goal:
J : Type w
C : Type u
inst : Category.{v, u} C
inst_1 : HasZeroMorphisms.{v, u} C
f g : J → C
inst_2 : HasBiproduct.{w, v, u} f
inst_3 : HasBiproduct.{w, v, u} g
p : (b : J) → f b ⟶ g b
⊢ Eq (α := biproduct.{w, v, u} f ⟶ biproduct.{w, v, u} g) (map p) (map' p)
have w : ∀ (j j_1 : J),
  CategoryStruct.comp (Y := biproduct.{w, v, u} f) (ι f j_1)
      (CategoryStruct.comp (Y := biproduct.{w, v, u} g) (map p) (π g j)) =
    CategoryStruct.comp (Y := biproduct.{w, v, u} f) (ι f j_1)
      (CategoryStruct.comp (Y := biproduct.{w, v, u} g) (map' p) (π g j))
Tactic: classical
  ext
  dsimp
  simp only [Discrete.natTrans_app, Limits.IsColimit.ι_map_assoc, Limits.IsLimit.map_π,
    ← Bicone.toCone_π_app_mk, ← Bicone.toCocone_ι_app_mk]
  dsimp
  rw [biproduct.ι_π_assoc, biproduct.ι_π]
  split_ifs with h
  · subst h; simp
  · simp
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Shapes.Biproducts (goal)
Goal:
J : Type w
C : Type u
inst : Category.{v, u} C
inst_1 : HasZeroMorphisms.{v, u} C
f g : J → C
inst_2 : HasBiproduct.{w, v, u} f
inst_3 : HasBiproduct.{w, v, u} g
p : (b : J) → f b ⟶ g b
w w₁ :
  ∀ (j j_1 : J),
    CategoryStruct.comp (Y := biproduct.{w, v, u} f) (ι f j_1)
        (CategoryStruct.comp (Y := biproduct.{w, v, u} g) (map p) (π g j)) =
      CategoryStruct.comp (Y := biproduct.{w, v, u} f) (ι f j_1)
        (CategoryStruct.comp (Y := biproduct.{w, v, u} g) (map' p) (π g j))
⊢ Eq (α := biproduct.{w, v, u} f ⟶ biproduct.{w, v, u} g) (map p) (map' p)
have w₂ : ∀ (j j_1 : J),
  Eq (α := f j_1 ⟶ g j)
    (CategoryStruct.comp (Y := biproduct.{w, v, u} f)
      (NatTrans.app (Cocone.ι (Bicone.toCocone (bicone f))) { as := j_1 } :
        obj.{w, v, w, u} (Discrete.functor f) { as := j_1 : Discrete J } ⟶
          obj.{w, v, w, u} (obj (const (Discrete J)) (Cocone.pt.{w, v, w, u} (Bicone.toCocone (bicone f))))
            { as := j_1 : Discrete J })
      (NatTrans.app (Cone.π (Bicone.toCone (bicone f))) { as := j } ≫ p j))
    (p j_1 ≫
      CategoryStruct.comp (Y := Cocone.pt.{w, v, w, u} (Bicone.toCocone (bicone g)))
        (NatTrans.app (Cocone.ι (Bicone.toCocone (bicone g))) { as := j_1 } :
          obj.{w, v, w, u} (Discrete.functor g) { as := j_1 : Discrete J } ⟶
            obj.{w, v, w, u} (obj (const (Discrete J)) (Cocone.pt.{w, v, w, u} (Bicone.toCocone (bicone g))))
              { as := j_1 : Discrete J })
        (NatTrans.app (Cone.π (Bicone.toCone (bicone g))) { as := j }))
Tactic: classical
  ext
  dsimp
  simp only [Discrete.natTrans_app, Limits.IsColimit.ι_map_assoc, Limits.IsLimit.map_π,
    ← Bicone.toCone_π_app_mk, ← Bicone.toCocone_ι_app_mk]
  dsimp
  rw [biproduct.ι_π_assoc, biproduct.ι_π]
  split_ifs with h
  · subst h; simp
  · simp
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Shapes.Biproducts (goal)
Goal:
J : Type w
C : Type u
inst : Category.{v, u} C
inst_1 : HasZeroMorphisms.{v, u} C
f g : J → C
inst_2 : HasBiproduct.{w, v, u} f
inst_3 : HasBiproduct.{w, v, u} g
p : (b : J) → f b ⟶ g b
w w₁ :
  ∀ (j j_1 : J),
    CategoryStruct.comp (Y := biproduct.{w, v, u} f) (ι f j_1)
        (CategoryStruct.comp (Y := biproduct.{w, v, u} g) (map p) (π g j)) =
      CategoryStruct.comp (Y := biproduct.{w, v, u} f) (ι f j_1)
        (CategoryStruct.comp (Y := biproduct.{w, v, u} g) (map' p) (π g j))
w₂ :
  ∀ (j j_1 : J),
    Eq (α := f j_1 ⟶ g j)
      (CategoryStruct.comp (Y := biproduct.{w, v, u} f)
        (NatTrans.app (Cocone.ι (Bicone.toCocone (bicone f))) { as := j_1 } :
          obj.{w, v, w, u} (Discrete.functor f) { as := j_1 : Discrete J } ⟶
            obj.{w, v, w, u} (obj (const (Discrete J)) (Cocone.pt.{w, v, w, u} (Bicone.toCocone (bicone f))))
              { as := j_1 : Discrete J })
        (NatTrans.app (Cone.π (Bicone.toCone (bicone f))) { as := j } ≫ p j))
      (p j_1 ≫
        CategoryStruct.comp (Y := Cocone.pt.{w, v, w, u} (Bicone.toCocone (bicone g)))
          (NatTrans.app (Cocone.ι (Bicone.toCocone (bicone g))) { as := j_1 } :
            obj.{w, v, w, u} (Discrete.functor g) { as := j_1 : Discrete J } ⟶
              obj.{w, v, w, u} (obj (const (Discrete J)) (Cocone.pt.{w, v, w, u} (Bicone.toCocone (bicone g))))
                { as := j_1 : Discrete J })
          (NatTrans.app (Cone.π (Bicone.toCone (bicone g))) { as := j }))
⊢ Eq (α := biproduct.{w, v, u} f ⟶ biproduct.{w, v, u} g) (map p) (map' p)
have w₃ : ∀ (j j_1 : J),
  CategoryStruct.comp (Y := biproduct.{w, v, u} f) (ι f j_1) (π f j ≫ p j) =
    p j_1 ≫ CategoryStruct.comp (Y := Bicone.pt.{w, v, u} (bicone g)) (ι g j_1) (π g j)
Tactic: classical
  ext
  dsimp
  simp only [Discrete.natTrans_app, Limits.IsColimit.ι_map_assoc, Limits.IsLimit.map_π,
    ← Bicone.toCone_π_app_mk, ← Bicone.toCocone_ι_app_mk]
  dsimp
  rw [biproduct.ι_π_assoc, biproduct.ι_π]
  split_ifs with h
  · subst h; simp
  · simp
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Shapes.Biproducts (goal)
Goal:
J : Type w
C : Type u
inst : Category.{v, u} C
inst_1 : HasZeroMorphisms.{v, u} C
f g : J → C
inst_2 : HasBiproduct.{w, v, u} f
inst_3 : HasBiproduct.{w, v, u} g
p : (b : J) → f b ⟶ g b
w w₁ :
  ∀ (j j_1 : J),
    CategoryStruct.comp (Y := biproduct.{w, v, u} f) (ι f j_1)
        (CategoryStruct.comp (Y := biproduct.{w, v, u} g) (map p) (π g j)) =
      CategoryStruct.comp (Y := biproduct.{w, v, u} f) (ι f j_1)
        (CategoryStruct.comp (Y := biproduct.{w, v, u} g) (map' p) (π g j))
w₂ :
  ∀ (j j_1 : J),
    Eq (α := f j_1 ⟶ g j)
      (CategoryStruct.comp (Y := biproduct.{w, v, u} f)
        (NatTrans.app (Cocone.ι (Bicone.toCocone (bicone f))) { as := j_1 } :
          obj.{w, v, w, u} (Discrete.functor f) { as := j_1 : Discrete J } ⟶
            obj.{w, v, w, u} (obj (const (Discrete J)) (Cocone.pt.{w, v, w, u} (Bicone.toCocone (bicone f))))
              { as := j_1 : Discrete J })
        (NatTrans.app (Cone.π (Bicone.toCone (bicone f))) { as := j } ≫ p j))
      (p j_1 ≫
        CategoryStruct.comp (Y := Cocone.pt.{w, v, w, u} (Bicone.toCocone (bicone g)))
          (NatTrans.app (Cocone.ι (Bicone.toCocone (bicone g))) { as := j_1 } :
            obj.{w, v, w, u} (Discrete.functor g) { as := j_1 : Discrete J } ⟶
              obj.{w, v, w, u} (obj (const (Discrete J)) (Cocone.pt.{w, v, w, u} (Bicone.toCocone (bicone g))))
                { as := j_1 : Discrete J })
          (NatTrans.app (Cone.π (Bicone.toCone (bicone g))) { as := j }))
w₃ :
  ∀ (j j_1 : J),
    CategoryStruct.comp (Y := biproduct.{w, v, u} f) (ι f j_1) (π f j ≫ p j) =
      p j_1 ≫ CategoryStruct.comp (Y := Bicone.pt.{w, v, u} (bicone g)) (ι g j_1) (π g j)
⊢ Eq (α := biproduct.{w, v, u} f ⟶ biproduct.{w, v, u} g) (map p) (map' p)
have w₄ : ∀ (j j_1 : J), Eq.{v + 1} ((if h : j_1 = j then eqToHom sorry else 0) ≫ p j) (p j_1 ≫ if h : j_1 = j then eqToHom sorry else 0)
Tactic: classical
  ext
  dsimp
  simp only [Discrete.natTrans_app, Limits.IsColimit.ι_map_assoc, Limits.IsLimit.map_π,
    ← Bicone.toCone_π_app_mk, ← Bicone.toCocone_ι_app_mk]
  dsimp
  rw [biproduct.ι_π_assoc, biproduct.ι_π]
  split_ifs with h
  · subst h; simp
  · simp
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Shapes.Biproducts (goal)
Goal:
J : Type w
C : Type u
inst : Category.{v, u} C
inst_1 : HasZeroMorphisms.{v, u} C
f g : J → C
inst_2 : HasBiproduct.{w, v, u} f
inst_3 : HasBiproduct.{w, v, u} g
p : (b : J) → f b ⟶ g b
w w₁ :
  ∀ (j j_1 : J),
    CategoryStruct.comp (Y := biproduct.{w, v, u} f) (ι f j_1)
        (CategoryStruct.comp (Y := biproduct.{w, v, u} g) (map p) (π g j)) =
      CategoryStruct.comp (Y := biproduct.{w, v, u} f) (ι f j_1)
        (CategoryStruct.comp (Y := biproduct.{w, v, u} g) (map' p) (π g j))
w₂ :
  ∀ (j j_1 : J),
    Eq (α := f j_1 ⟶ g j)
      (CategoryStruct.comp (Y := biproduct.{w, v, u} f)
        (NatTrans.app (Cocone.ι (Bicone.toCocone (bicone f))) { as := j_1 } :
          obj.{w, v, w, u} (Discrete.functor f) { as := j_1 : Discrete J } ⟶
            obj.{w, v, w, u} (obj (const (Discrete J)) (Cocone.pt.{w, v, w, u} (Bicone.toCocone (bicone f))))
              { as := j_1 : Discrete J })
        (NatTrans.app (Cone.π (Bicone.toCone (bicone f))) { as := j } ≫ p j))
      (p j_1 ≫
        CategoryStruct.comp (Y := Cocone.pt.{w, v, w, u} (Bicone.toCocone (bicone g)))
          (NatTrans.app (Cocone.ι (Bicone.toCocone (bicone g))) { as := j_1 } :
            obj.{w, v, w, u} (Discrete.functor g) { as := j_1 : Discrete J } ⟶
              obj.{w, v, w, u} (obj (const (Discrete J)) (Cocone.pt.{w, v, w, u} (Bicone.toCocone (bicone g))))
                { as := j_1 : Discrete J })
          (NatTrans.app (Cone.π (Bicone.toCone (bicone g))) { as := j }))
w₃ :
  ∀ (j j_1 : J),
    CategoryStruct.comp (Y := biproduct.{w, v, u} f) (ι f j_1) (π f j ≫ p j) =
      p j_1 ≫ CategoryStruct.comp (Y := Bicone.pt.{w, v, u} (bicone g)) (ι g j_1) (π g j)
w₄ :
  ∀ (j j_1 : J),
    Eq.{v + 1} ((if h : j_1 = j then eqToHom ⋯ else 0) ≫ p j) (p j_1 ≫ if h : j_1 = j then eqToHom ⋯ else 0)
⊢ Eq (α := biproduct.{w, v, u} f ⟶ biproduct.{w, v, u} g) (map p) (map' p)
have pos : ∀ (j j_1 : J) (h : j_1 = j), eqToHom sorry ≫ p j = p j_1 ≫ eqToHom sorry
Tactic: classical
  ext
  dsimp
  simp only [Discrete.natTrans_app, Limits.IsColimit.ι_map_assoc, Limits.IsLimit.map_π,
    ← Bicone.toCone_π_app_mk, ← Bicone.toCocone_ι_app_mk]
  dsimp
  rw [biproduct.ι_π_assoc, biproduct.ι_π]
  split_ifs with h
  · subst h; simp
  · simp
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Shapes.Biproducts (goal)
Goal:
J : Type w
C : Type u
inst : Category.{v, u} C
inst_1 : HasZeroMorphisms.{v, u} C
f g : J → C
inst_2 : HasBiproduct.{w, v, u} f
inst_3 : HasBiproduct.{w, v, u} g
p : (b : J) → f b ⟶ g b
w w₁ :
  ∀ (j j_1 : J),
    CategoryStruct.comp (Y := biproduct.{w, v, u} f) (ι f j_1)
        (CategoryStruct.comp (Y := biproduct.{w, v, u} g) (map p) (π g j)) =
      CategoryStruct.comp (Y := biproduct.{w, v, u} f) (ι f j_1)
        (CategoryStruct.comp (Y := biproduct.{w, v, u} g) (map' p) (π g j))
w₂ :
  ∀ (j j_1 : J),
    Eq (α := f j_1 ⟶ g j)
      (CategoryStruct.comp (Y := biproduct.{w, v, u} f)
        (NatTrans.app (Cocone.ι (Bicone.toCocone (bicone f))) { as := j_1 } :
          obj.{w, v, w, u} (Discrete.functor f) { as := j_1 : Discrete J } ⟶
            obj.{w, v, w, u} (obj (const (Discrete J)) (Cocone.pt.{w, v, w, u} (Bicone.toCocone (bicone f))))
              { as := j_1 : Discrete J })
        (NatTrans.app (Cone.π (Bicone.toCone (bicone f))) { as := j } ≫ p j))
      (p j_1 ≫
        CategoryStruct.comp (Y := Cocone.pt.{w, v, w, u} (Bicone.toCocone (bicone g)))
          (NatTrans.app (Cocone.ι (Bicone.toCocone (bicone g))) { as := j_1 } :
            obj.{w, v, w, u} (Discrete.functor g) { as := j_1 : Discrete J } ⟶
              obj.{w, v, w, u} (obj (const (Discrete J)) (Cocone.pt.{w, v, w, u} (Bicone.toCocone (bicone g))))
                { as := j_1 : Discrete J })
          (NatTrans.app (Cone.π (Bicone.toCone (bicone g))) { as := j }))
w₃ :
  ∀ (j j_1 : J),
    CategoryStruct.comp (Y := biproduct.{w, v, u} f) (ι f j_1) (π f j ≫ p j) =
      p j_1 ≫ CategoryStruct.comp (Y := Bicone.pt.{w, v, u} (bicone g)) (ι g j_1) (π g j)
w₄ :
  ∀ (j j_1 : J),
    Eq.{v + 1} ((if h : j_1 = j then eqToHom ⋯ else 0) ≫ p j) (p j_1 ≫ if h : j_1 = j then eqToHom ⋯ else 0)
pos : ∀ (j j_1 : J) (h : j_1 = j), eqToHom ⋯ ≫ p j = p j_1 ≫ eqToHom ⋯
⊢ Eq (α := biproduct.{w, v, u} f ⟶ biproduct.{w, v, u} g) (map p) (map' p)
have neg : ∀ (j j_1 : J) (h : ¬j_1 = j), Eq.{v + 1} (0 ≫ p j) (p j_1 ≫ 0)
Tactic: classical
  ext
  dsimp
  simp only [Discrete.natTrans_app, Limits.IsColimit.ι_map_assoc, Limits.IsLimit.map_π,
    ← Bicone.toCone_π_app_mk, ← Bicone.toCocone_ι_app_mk]
  dsimp
  rw [biproduct.ι_π_assoc, biproduct.ι_π]
  split_ifs with h
  · subst h; simp
  · simp