awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.CategoryTheory.Presentable.ColimitPresentation (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : Type w
I : J → Type w
inst_1 : SmallCategory J
inst_2 : (j : J) → SmallCategory (I j)
D : Functor.{w, v, w, u} J C
P : (j : J) → ColimitPresentation.{w, w, v, u} (I j) (D.obj j)
j j' : J
i : I j
u : j ⟶ j'
inst_3 : IsFiltered.{w, w} (I j')
inst_4 : IsFinitelyPresentable.{w, v, u} ((P j).diag.obj i)
⊢ ∃ (i' : I j') (f : mk (I := I) P j i ⟶ mk (I := I) P j' i'), f.base (I := I) = u
have obtain : ∀ (i' : I j') (q : (P j).diag.obj i ⟶ (P j').diag.obj i')
  (hq : q ≫ { pt := D.obj j', ι := (P j').ι : Cocone (P j').diag }.ι.app i' = (P j).ι.app i ≫ D.map u),
  ∃ (i' : I j') (f : mk (I := I) P j i ⟶ mk (I := I) P j' i'), f.base (I := I) = u
Tactic: obtain ⟨i', q, hq⟩ := IsFinitelyPresentable.exists_hom_of_isColimit (P j').isColimit
    ((P j).ι.app i ≫ D.map u)
-- Lean 4 proof step from Mathlib.CategoryTheory.Presentable.ColimitPresentation (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : Type w
I : J → Type w
inst_1 : SmallCategory J
inst_2 : (j : J) → SmallCategory (I j)
D : Functor.{w, v, w, u} J C
P : (j : J) → ColimitPresentation.{w, w, v, u} (I j) (D.obj j)
j j' : J
i : I j
u : j ⟶ j'
inst_3 : IsFiltered.{w, w} (I j')
inst_4 : IsFinitelyPresentable.{w, v, u} ((P j).diag.obj i)
i' : I j'
q : (P j).diag.obj i ⟶ (P j').diag.obj i'
hq : q ≫ { pt := D.obj j', ι := (P j').ι : Cocone (P j').diag }.ι.app i' = (P j).ι.app i ≫ D.map u
w : I j'
⊢ ∃ (i' : I j') (f : mk (I := I) P j i ⟶ mk (I := I) P j' i'), f.base (I := I) = u
have w₁ : mk (I := I) P j i ⟶ mk (I := I) P j' i'
Tactic: use i', { base := u, hom := q, w := by simp [← hq] }
-- Lean 4 proof step from Mathlib.CategoryTheory.Presentable.ColimitPresentation (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : Type w
I : J → Type w
inst_1 : SmallCategory J
inst_2 : (j : J) → SmallCategory (I j)
D : Functor.{w, v, w, u} J C
P : (j : J) → ColimitPresentation.{w, w, v, u} (I j) (D.obj j)
j j' : J
i : I j
u : j ⟶ j'
inst_3 : IsFiltered.{w, w} (I j')
inst_4 : IsFinitelyPresentable.{w, v, u} ((P j).diag.obj i)
i' : I j'
q : (P j).diag.obj i ⟶ (P j').diag.obj i'
hq : q ≫ { pt := D.obj j', ι := (P j').ι : Cocone (P j').diag }.ι.app i' = (P j).ι.app i ≫ D.map u
w : I j'
w₁ : mk (I := I) P j i ⟶ mk (I := I) P j' i'
⊢ ∃ (i' : I j') (f : mk (I := I) P j i ⟶ mk (I := I) P j' i'), f.base (I := I) = u
have h : Hom.base (I := I) (Hom.mk (I := I) u q sorry) = u
Tactic: use i', { base := u, hom := q, w := by simp [← hq] }
-- Lean 4 proof step from Mathlib.CategoryTheory.Presentable.ColimitPresentation (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J_1 : Type u_1
I_1 : J_1 → Type u_2
inst_1 : Category.{v_1, u_1} J_1
inst_2 : (j : J_1) → Category.{?u.29355, u_2} (I_1 j)
D_1 : Functor.{v_1, v, u_1, u} J_1 C
P_1 : (j : J_1) → ColimitPresentation.{?u.29355, u_2, v, u} (I_1 j) (D_1.obj j)
J : Type w
I : J → Type w
inst_3 : SmallCategory J
inst_4 : (j : J) → SmallCategory (I j)
D : Functor.{w, v, w, u} J C
P : (j : J) → ColimitPresentation.{w, w, v, u} (I j) (D.obj j)
inst_5 : IsFiltered J
inst_6 : ∀ (j : J), IsFiltered.{w, w} (I j)
⊢ Nonempty.{w + 1} (Total (I := I) P)
have obtain : ∀ (j : J), Nonempty.{w + 1} (Total (I := I) P)
Tactic: obtain ⟨j⟩ : Nonempty J := IsFiltered.nonempty
-- Lean 4 proof step from Mathlib.CategoryTheory.Presentable.ColimitPresentation (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J_1 : Type u_1
I_1 : J_1 → Type u_2
inst_1 : Category.{v_1, u_1} J_1
inst_2 : (j : J_1) → Category.{?u.29355, u_2} (I_1 j)
D_1 : Functor.{v_1, v, u_1, u} J_1 C
P_1 : (j : J_1) → ColimitPresentation.{?u.29355, u_2, v, u} (I_1 j) (D_1.obj j)
J : Type w
I : J → Type w
inst_3 : SmallCategory J
inst_4 : (j : J) → SmallCategory (I j)
D : Functor.{w, v, w, u} J C
P : (j : J) → ColimitPresentation.{w, w, v, u} (I j) (D.obj j)
inst_5 : IsFiltered J
inst_6 : ∀ (j : J), IsFiltered.{w, w} (I j)
j : J
⊢ Nonempty.{w + 1} (Total (I := I) P)
have obtain : ∀ (i : I j), Nonempty.{w + 1} (Total (I := I) P)
Tactic: obtain ⟨i⟩ : Nonempty (I j) := IsFiltered.nonempty
-- Lean 4 proof step from Mathlib.CategoryTheory.Presentable.ColimitPresentation (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J_1 : Type u_1
I_1 : J_1 → Type u_2
inst_1 : Category.{v_1, u_1} J_1
inst_2 : (j : J_1) → Category.{?u.30679, u_2} (I_1 j)
D_1 : Functor.{v_1, v, u_1, u} J_1 C
P_1 : (j : J_1) → ColimitPresentation.{?u.30679, u_2, v, u} (I_1 j) (D_1.obj j)
J : Type w
I : J → Type w
inst_3 : SmallCategory J
inst_4 : (j : J) → SmallCategory (I j)
D : Functor.{w, v, w, u} J C
P : (j : J) → ColimitPresentation.{w, w, v, u} (I j) (D.obj j)
inst_5 : IsFiltered J
inst_6 : ∀ (j : J), IsFiltered.{w, w} (I j)
inst_7 : ∀ (j : J) (i : I j), IsFinitelyPresentable.{w, v, u} ((P j).diag.obj i)
k l : Total (I := I) P
a : J := IsFiltered.max (k.fst (β := fun (j : J) ↦ I j)) (l.fst (β := fun (j : J) ↦ I j))
⊢ ∃ (Z : Total (I := I) P) (x : k ⟶ Z) (x : l ⟶ Z), True
have obtain : ∀ (a' : I (IsFiltered.max (k.fst (β := fun (j : J) ↦ I j)) (l.fst (β := fun (j : J) ↦ I j))))
  (f :
    Total.mk (I := I) P (k.fst (β := fun (j : J) ↦ I j)) (k.snd (β := fun (j : J) ↦ I j)) ⟶
      Total.mk (I := I) P (IsFiltered.max (k.fst (β := fun (j : J) ↦ I j)) (l.fst (β := fun (j : J) ↦ I j))) a')
  (hf : f.base (I := I) = IsFiltered.leftToMax (k.fst (β := fun (j : J) ↦ I j)) (l.fst (β := fun (j : J) ↦ I j))),
  ∃ (Z : Total (I := I) P) (x : k ⟶ Z) (x : l ⟶ Z), True
Tactic: obtain ⟨a', f, hf⟩ := Total.exists_hom_of_hom (P := P) k.2 (IsFiltered.leftToMax k.1 l.1)
-- Lean 4 proof step from Mathlib.CategoryTheory.Presentable.ColimitPresentation (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J_1 : Type u_1
I_1 : J_1 → Type u_2
inst_1 : Category.{v_1, u_1} J_1
inst_2 : (j : J_1) → Category.{?u.30679, u_2} (I_1 j)
D_1 : Functor.{v_1, v, u_1, u} J_1 C
P_1 : (j : J_1) → ColimitPresentation.{?u.30679, u_2, v, u} (I_1 j) (D_1.obj j)
J : Type w
I : J → Type w
inst_3 : SmallCategory J
inst_4 : (j : J) → SmallCategory (I j)
D : Functor.{w, v, w, u} J C
P : (j : J) → ColimitPresentation.{w, w, v, u} (I j) (D.obj j)
inst_5 : IsFiltered J
inst_6 : ∀ (j : J), IsFiltered.{w, w} (I j)
inst_7 : ∀ (j : J) (i : I j), IsFinitelyPresentable.{w, v, u} ((P j).diag.obj i)
k l : Total (I := I) P
a : J := IsFiltered.max (k.fst (β := fun (j : J) ↦ I j)) (l.fst (β := fun (j : J) ↦ I j))
a' : I (IsFiltered.max (k.fst (β := fun (j : J) ↦ I j)) (l.fst (β := fun (j : J) ↦ I j)))
f :
  Total.mk (I := I) P (k.fst (β := fun (j : J) ↦ I j)) (k.snd (β := fun (j : J) ↦ I j)) ⟶
    Total.mk (I := I) P (IsFiltered.max (k.fst (β := fun (j : J) ↦ I j)) (l.fst (β := fun (j : J) ↦ I j))) a'
hf : f.base (I := I) = IsFiltered.leftToMax (k.fst (β := fun (j : J) ↦ I j)) (l.fst (β := fun (j : J) ↦ I j))
⊢ ∃ (Z : Total (I := I) P) (x : k ⟶ Z) (x : l ⟶ Z), True
have obtain : ∀ (b' : I (IsFiltered.max (k.fst (β := fun (j : J) ↦ I j)) (l.fst (β := fun (j : J) ↦ I j))))
  (g :
    Total.mk (I := I) P (l.fst (β := fun (j : J) ↦ I j)) (l.snd (β := fun (j : J) ↦ I j)) ⟶
      Total.mk (I := I) P (IsFiltered.max (k.fst (β := fun (j : J) ↦ I j)) (l.fst (β := fun (j : J) ↦ I j))) b')
  (hg : g.base (I := I) = IsFiltered.rightToMax (k.fst (β := fun (j : J) ↦ I j)) (l.fst (β := fun (j : J) ↦ I j))),
  ∃ (Z : Total (I := I) P) (x : k ⟶ Z) (x : l ⟶ Z), True
Tactic: obtain ⟨b', g, hg⟩ := Total.exists_hom_of_hom (P := P) l.2 (IsFiltered.rightToMax k.1 l.1)
-- Lean 4 proof step from Mathlib.CategoryTheory.Presentable.ColimitPresentation (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J_1 : Type u_1
I_1 : J_1 → Type u_2
inst_1 : Category.{v_1, u_1} J_1
inst_2 : (j : J_1) → Category.{?u.30679, u_2} (I_1 j)
D_1 : Functor.{v_1, v, u_1, u} J_1 C
P_1 : (j : J_1) → ColimitPresentation.{?u.30679, u_2, v, u} (I_1 j) (D_1.obj j)
J : Type w
I : J → Type w
inst_3 : SmallCategory J
inst_4 : (j : J) → SmallCategory (I j)
D : Functor.{w, v, w, u} J C
P : (j : J) → ColimitPresentation.{w, w, v, u} (I j) (D.obj j)
inst_5 : IsFiltered J
inst_6 : ∀ (j : J), IsFiltered.{w, w} (I j)
inst_7 : ∀ (j : J) (i : I j), IsFinitelyPresentable.{w, v, u} ((P j).diag.obj i)
k l : Total (I := I) P
a : J := IsFiltered.max (k.fst (β := fun (j : J) ↦ I j)) (l.fst (β := fun (j : J) ↦ I j))
a' : I (IsFiltered.max (k.fst (β := fun (j : J) ↦ I j)) (l.fst (β := fun (j : J) ↦ I j)))
f :
  Total.mk (I := I) P (k.fst (β := fun (j : J) ↦ I j)) (k.snd (β := fun (j : J) ↦ I j)) ⟶
    Total.mk (I := I) P (IsFiltered.max (k.fst (β := fun (j : J) ↦ I j)) (l.fst (β := fun (j : J) ↦ I j))) a'
hf : f.base (I := I) = IsFiltered.leftToMax (k.fst (β := fun (j : J) ↦ I j)) (l.fst (β := fun (j : J) ↦ I j))
b' : I (IsFiltered.max (k.fst (β := fun (j : J) ↦ I j)) (l.fst (β := fun (j : J) ↦ I j)))
g :
  Total.mk (I := I) P (l.fst (β := fun (j : J) ↦ I j)) (l.snd (β := fun (j : J) ↦ I j)) ⟶
    Total.mk (I := I) P (IsFiltered.max (k.fst (β := fun (j : J) ↦ I j)) (l.fst (β := fun (j : J) ↦ I j))) b'
hg : g.base (I := I) = IsFiltered.rightToMax (k.fst (β := fun (j : J) ↦ I j)) (l.fst (β := fun (j : J) ↦ I j))
⊢ ∃ (Z : Total (I := I) P) (x : k ⟶ Z) (x : l ⟶ Z), True
have refine_1 : k ⟶ Sigma.mk (β := fun (j : J) ↦ I j) a (IsFiltered.max a' b')
Tactic: refine ⟨⟨a, IsFiltered.max a' b'⟩, ?_, ?_, trivial⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Presentable.ColimitPresentation (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J_1 : Type u_1
I_1 : J_1 → Type u_2
inst_1 : Category.{v_1, u_1} J_1
inst_2 : (j : J_1) → Category.{?u.30679, u_2} (I_1 j)
D_1 : Functor.{v_1, v, u_1, u} J_1 C
P_1 : (j : J_1) → ColimitPresentation.{?u.30679, u_2, v, u} (I_1 j) (D_1.obj j)
J : Type w
I : J → Type w
inst_3 : SmallCategory J
inst_4 : (j : J) → SmallCategory (I j)
D : Functor.{w, v, w, u} J C
P : (j : J) → ColimitPresentation.{w, w, v, u} (I j) (D.obj j)
inst_5 : IsFiltered J
inst_6 : ∀ (j : J), IsFiltered.{w, w} (I j)
inst_7 : ∀ (j : J) (i : I j), IsFinitelyPresentable.{w, v, u} ((P j).diag.obj i)
k l : Total (I := I) P
a : J := IsFiltered.max (k.fst (β := fun (j : J) ↦ I j)) (l.fst (β := fun (j : J) ↦ I j))
a' : I (IsFiltered.max (k.fst (β := fun (j : J) ↦ I j)) (l.fst (β := fun (j : J) ↦ I j)))
f :
  Total.mk (I := I) P (k.fst (β := fun (j : J) ↦ I j)) (k.snd (β := fun (j : J) ↦ I j)) ⟶
    Total.mk (I := I) P (IsFiltered.max (k.fst (β := fun (j : J) ↦ I j)) (l.fst (β := fun (j : J) ↦ I j))) a'
hf : f.base (I := I) = IsFiltered.leftToMax (k.fst (β := fun (j : J) ↦ I j)) (l.fst (β := fun (j : J) ↦ I j))
b' : I (IsFiltered.max (k.fst (β := fun (j : J) ↦ I j)) (l.fst (β := fun (j : J) ↦ I j)))
g :
  Total.mk (I := I) P (l.fst (β := fun (j : J) ↦ I j)) (l.snd (β := fun (j : J) ↦ I j)) ⟶
    Total.mk (I := I) P (IsFiltered.max (k.fst (β := fun (j : J) ↦ I j)) (l.fst (β := fun (j : J) ↦ I j))) b'
hg : g.base (I := I) = IsFiltered.rightToMax (k.fst (β := fun (j : J) ↦ I j)) (l.fst (β := fun (j : J) ↦ I j))
refine_1 : k ⟶ Sigma.mk (β := fun (j : J) ↦ I j) a (IsFiltered.max a' b')
⊢ ∃ (Z : Total (I := I) P) (x : k ⟶ Z) (x : l ⟶ Z), True
have refine_2 : l ⟶ Sigma.mk (β := fun (j : J) ↦ I j) a (IsFiltered.max a' b')
Tactic: refine ⟨⟨a, IsFiltered.max a' b'⟩, ?_, ?_, trivial⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Presentable.ColimitPresentation (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J_1 : Type u_1
I_1 : J_1 → Type u_2
inst_1 : Category.{v_1, u_1} J_1
inst_2 : (j : J_1) → Category.{?u.30679, u_2} (I_1 j)
D_1 : Functor.{v_1, v, u_1, u} J_1 C
P_1 : (j : J_1) → ColimitPresentation.{?u.30679, u_2, v, u} (I_1 j) (D_1.obj j)
J : Type w
I : J → Type w
inst_3 : SmallCategory J
inst_4 : (j : J) → SmallCategory (I j)
D : Functor.{w, v, w, u} J C
P : (j : J) → ColimitPresentation.{w, w, v, u} (I j) (D.obj j)
inst_5 : IsFiltered J
inst_6 : ∀ (j : J), IsFiltered.{w, w} (I j)
inst_7 : ∀ (j : J) (i : I j), IsFinitelyPresentable.{w, v, u} ((P j).diag.obj i)
k l : Total (I := I) P
f g : k ⟶ l
a : J := IsFiltered.coeq (f.base (I := I)) (g.base (I := I))
⊢ ∃ (Z : Total (I := I) P) (h : l ⟶ Z), f ≫ h = g ≫ h
have obtain : ∀ (a' : I (IsFiltered.coeq (f.base (I := I)) (g.base (I := I))))
  (u :
    Total.mk (I := I) P (l.fst (β := fun (j : J) ↦ I j)) (l.snd (β := fun (j : J) ↦ I j)) ⟶
      Total.mk (I := I) P (IsFiltered.coeq (f.base (I := I)) (g.base (I := I))) a')
  (hu : u.base (I := I) = IsFiltered.coeqHom (f.base (I := I)) (g.base (I := I))),
  ∃ (Z : Total (I := I) P) (h : l ⟶ Z), f ≫ h = g ≫ h
Tactic: obtain ⟨a', u, hu⟩ := Total.exists_hom_of_hom (P := P) l.2 (IsFiltered.coeqHom f.base g.base)
-- Lean 4 proof step from Mathlib.CategoryTheory.Presentable.ColimitPresentation (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J_1 : Type u_1
I_1 : J_1 → Type u_2
inst_1 : Category.{v_1, u_1} J_1
inst_2 : (j : J_1) → Category.{?u.30679, u_2} (I_1 j)
D_1 : Functor.{v_1, v, u_1, u} J_1 C
P_1 : (j : J_1) → ColimitPresentation.{?u.30679, u_2, v, u} (I_1 j) (D_1.obj j)
J : Type w
I : J → Type w
inst_3 : SmallCategory J
inst_4 : (j : J) → SmallCategory (I j)
D : Functor.{w, v, w, u} J C
P : (j : J) → ColimitPresentation.{w, w, v, u} (I j) (D.obj j)
inst_5 : IsFiltered J
inst_6 : ∀ (j : J), IsFiltered.{w, w} (I j)
inst_7 : ∀ (j : J) (i : I j), IsFinitelyPresentable.{w, v, u} ((P j).diag.obj i)
k l : Total (I := I) P
f g : k ⟶ l
a : J := IsFiltered.coeq (f.base (I := I)) (g.base (I := I))
a' : I (IsFiltered.coeq (f.base (I := I)) (g.base (I := I)))
u :
  Total.mk (I := I) P (l.fst (β := fun (j : J) ↦ I j)) (l.snd (β := fun (j : J) ↦ I j)) ⟶
    Total.mk (I := I) P (IsFiltered.coeq (f.base (I := I)) (g.base (I := I))) a'
hu : u.base (I := I) = IsFiltered.coeqHom (f.base (I := I)) (g.base (I := I))
⊢ (f.hom (I := I) ≫ u.hom (I := I)) ≫
      NatTrans.app
        (ι
          (P
            ((Total.mk (I := I) P (IsFiltered.coeq (f.base (I := I)) (g.base (I := I))) a').fst (β := fun (j : J) ↦
              I j))))
        ((Total.mk (I := I) P (IsFiltered.coeq (f.base (I := I)) (g.base (I := I))) a').snd (β := fun (j : J) ↦ I j)) =
    (g.hom (I := I) ≫ u.hom (I := I)) ≫
      NatTrans.app
        (ι
          (P
            ((Total.mk (I := I) P (IsFiltered.coeq (f.base (I := I)) (g.base (I := I))) a').fst (β := fun (j : J) ↦
              I j))))
        ((Total.mk (I := I) P (IsFiltered.coeq (f.base (I := I)) (g.base (I := I))) a').snd (β := fun (j : J) ↦ I j))
have simp : Eq (α :=
  Functor.obj (diag (P (k.fst (β := fun (j : J) ↦ I j)))) (k.snd (β := fun (j : J) ↦ I j)) ⟶
    D.obj (IsFiltered.coeq (f.base (I := I)) (g.base (I := I))))
  (NatTrans.app (ι (P (k.fst (β := fun (j : J) ↦ I j)))) (k.snd (β := fun (j : J) ↦ I j)) ≫
    D.map (f.base (I := I)) ≫ D.map (u.base (I := I)))
  (NatTrans.app (ι (P (k.fst (β := fun (j : J) ↦ I j)))) (k.snd (β := fun (j : J) ↦ I j)) ≫
    D.map (g.base (I := I)) ≫ D.map (u.base (I := I)))
Tactic: simp only [Category.assoc, Functor.const_obj_obj, ← u.w, ← f.w_assoc, ← g.w_assoc]
-- Lean 4 proof step from Mathlib.CategoryTheory.Presentable.ColimitPresentation (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J_1 : Type u_1
I_1 : J_1 → Type u_2
inst_1 : Category.{v_1, u_1} J_1
inst_2 : (j : J_1) → Category.{?u.30679, u_2} (I_1 j)
D_1 : Functor.{v_1, v, u_1, u} J_1 C
P_1 : (j : J_1) → ColimitPresentation.{?u.30679, u_2, v, u} (I_1 j) (D_1.obj j)
J : Type w
I : J → Type w
inst_3 : SmallCategory J
inst_4 : (j : J) → SmallCategory (I j)
D : Functor.{w, v, w, u} J C
P : (j : J) → ColimitPresentation.{w, w, v, u} (I j) (D.obj j)
inst_5 : IsFiltered J
inst_6 : ∀ (j : J), IsFiltered.{w, w} (I j)
inst_7 : ∀ (j : J) (i : I j), IsFinitelyPresentable.{w, v, u} ((P j).diag.obj i)
k l : Total (I := I) P
f g : k ⟶ l
a : J := IsFiltered.coeq (f.base (I := I)) (g.base (I := I))
a' : I (IsFiltered.coeq (f.base (I := I)) (g.base (I := I)))
u :
  Total.mk (I := I) P (l.fst (β := fun (j : J) ↦ I j)) (l.snd (β := fun (j : J) ↦ I j)) ⟶
    Total.mk (I := I) P (IsFiltered.coeq (f.base (I := I)) (g.base (I := I))) a'
hu : u.base (I := I) = IsFiltered.coeqHom (f.base (I := I)) (g.base (I := I))
⊢ Eq (α :=
    Functor.obj (diag (P (k.fst (β := fun (j : J) ↦ I j)))) (k.snd (β := fun (j : J) ↦ I j)) ⟶
      D.obj (IsFiltered.coeq (f.base (I := I)) (g.base (I := I))))
    (NatTrans.app (ι (P (k.fst (β := fun (j : J) ↦ I j)))) (k.snd (β := fun (j : J) ↦ I j)) ≫
      D.map (f.base (I := I)) ≫ D.map (u.base (I := I)))
    (NatTrans.app (ι (P (k.fst (β := fun (j : J) ↦ I j)))) (k.snd (β := fun (j : J) ↦ I j)) ≫
      D.map (g.base (I := I)) ≫ D.map (u.base (I := I)))
have rw : Eq (α :=
  Functor.obj (diag (P (k.fst (β := fun (j : J) ↦ I j)))) (k.snd (β := fun (j : J) ↦ I j)) ⟶
    D.obj (IsFiltered.coeq (f.base (I := I)) (g.base (I := I))))
  (NatTrans.app (ι (P (k.fst (β := fun (j : J) ↦ I j)))) (k.snd (β := fun (j : J) ↦ I j)) ≫
    D.map (g.base (I := I) ≫ IsFiltered.coeqHom (f.base (I := I)) (g.base (I := I))))
  (NatTrans.app (ι (P (k.fst (β := fun (j : J) ↦ I j)))) (k.snd (β := fun (j : J) ↦ I j)) ≫
    D.map (g.base (I := I)) ≫ D.map (IsFiltered.coeqHom (f.base (I := I)) (g.base (I := I))))
Tactic: rw [← Functor.map_comp, hu, IsFiltered.coeq_condition f.base g.base]
-- Lean 4 proof step from Mathlib.CategoryTheory.Presentable.ColimitPresentation (hypothesis)
Goal:
C : Type u
inst : Category.{v, u} C
J_1 : Type u_1
I_1 : J_1 → Type u_2
inst_1 : Category.{v_1, u_1} J_1
inst_2 : (j : J_1) → Category.{?u.30679, u_2} (I_1 j)
D_1 : Functor.{v_1, v, u_1, u} J_1 C
P_1 : (j : J_1) → ColimitPresentation.{?u.30679, u_2, v, u} (I_1 j) (D_1.obj j)
J : Type w
I : J → Type w
inst_3 : SmallCategory J
inst_4 : (j : J) → SmallCategory (I j)
D : Functor.{w, v, w, u} J C
P : (j : J) → ColimitPresentation.{w, w, v, u} (I j) (D.obj j)
inst_5 : IsFiltered J
inst_6 : ∀ (j : J), IsFiltered.{w, w} (I j)
inst_7 : ∀ (j : J) (i : I j), IsFinitelyPresentable.{w, v, u} ((P j).diag.obj i)
k l : Total (I := I) P
f g : k ⟶ l
a : J := IsFiltered.coeq (f.base (I := I)) (g.base (I := I))
a' : I (IsFiltered.coeq (f.base (I := I)) (g.base (I := I)))
u :
  Total.mk (I := I) P (l.fst (β := fun (j : J) ↦ I j)) (l.snd (β := fun (j : J) ↦ I j)) ⟶
    Total.mk (I := I) P (IsFiltered.coeq (f.base (I := I)) (g.base (I := I))) a'
hu : u.base (I := I) = IsFiltered.coeqHom (f.base (I := I)) (g.base (I := I))
⊢ ∃ (Z : Total (I := I) P) (h : l ⟶ Z), f ≫ h = g ≫ h
have this : (f.hom (I := I) ≫ u.hom (I := I)) ≫
    NatTrans.app
      (ι
        (P
          ((Total.mk (I := I) P (IsFiltered.coeq (f.base (I := I)) (g.base (I := I))) a').fst (β := fun (j : J) ↦
            I j))))
      ((Total.mk (I := I) P (IsFiltered.coeq (f.base (I := I)) (g.base (I := I))) a').snd (β := fun (j : J) ↦ I j)) =
  (g.hom (I := I) ≫ u.hom (I := I)) ≫
    NatTrans.app
      (ι
        (P
          ((Total.mk (I := I) P (IsFiltered.coeq (f.base (I := I)) (g.base (I := I))) a').fst (β := fun (j : J) ↦
            I j))))
      ((Total.mk (I := I) P (IsFiltered.coeq (f.base (I := I)) (g.base (I := I))) a').snd (β := fun (j : J) ↦ I j))
Tactic: have : (f.hom ≫ u.hom) ≫ (P _).ι.app _ = (g.hom ≫ u.hom) ≫ (P _).ι.app _ := by
      simp only [Category.assoc, Functor.const_obj_obj, ← u.w, ← f.w_assoc, ← g.w_assoc]
      rw [← Functor.map_comp, hu, IsFiltered.coeq_condition f.base g.base]
      simp
-- Lean 4 proof step from Mathlib.CategoryTheory.Presentable.ColimitPresentation (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J_1 : Type u_1
I_1 : J_1 → Type u_2
inst_1 : Category.{v_1, u_1} J_1
inst_2 : (j : J_1) → Category.{?u.30679, u_2} (I_1 j)
D_1 : Functor.{v_1, v, u_1, u} J_1 C
P_1 : (j : J_1) → ColimitPresentation.{?u.30679, u_2, v, u} (I_1 j) (D_1.obj j)
J : Type w
I : J → Type w
inst_3 : SmallCategory J
inst_4 : (j : J) → SmallCategory (I j)
D : Functor.{w, v, w, u} J C
P : (j : J) → ColimitPresentation.{w, w, v, u} (I j) (D.obj j)
inst_5 : IsFiltered J
inst_6 : ∀ (j : J), IsFiltered.{w, w} (I j)
inst_7 : ∀ (j : J) (i : I j), IsFinitelyPresentable.{w, v, u} ((P j).diag.obj i)
k l : Total (I := I) P
f g : k ⟶ l
a : J := IsFiltered.coeq (f.base (I := I)) (g.base (I := I))
a' : I (IsFiltered.coeq (f.base (I := I)) (g.base (I := I)))
u :
  Total.mk (I := I) P (l.fst (β := fun (j : J) ↦ I j)) (l.snd (β := fun (j : J) ↦ I j)) ⟶
    Total.mk (I := I) P (IsFiltered.coeq (f.base (I := I)) (g.base (I := I))) a'
hu : u.base (I := I) = IsFiltered.coeqHom (f.base (I := I)) (g.base (I := I))
this :
  (f.hom (I := I) ≫ u.hom (I := I)) ≫
      NatTrans.app
        (ι
          (P
            ((Total.mk (I := I) P (IsFiltered.coeq (f.base (I := I)) (g.base (I := I))) a').fst (β := fun (j : J) ↦
              I j))))
        ((Total.mk (I := I) P (IsFiltered.coeq (f.base (I := I)) (g.base (I := I))) a').snd (β := fun (j : J) ↦ I j)) =
    (g.hom (I := I) ≫ u.hom (I := I)) ≫
      NatTrans.app
        (ι
          (P
            ((Total.mk (I := I) P (IsFiltered.coeq (f.base (I := I)) (g.base (I := I))) a').fst (β := fun (j : J) ↦
              I j))))
        ((Total.mk (I := I) P (IsFiltered.coeq (f.base (I := I)) (g.base (I := I))) a').snd (β := fun (j : J) ↦ I j))
⊢ ∃ (Z : Total (I := I) P) (h : l ⟶ Z), f ≫ h = g ≫ h
have obtain : ∀ (j : I ((Total.mk (I := I) P (IsFiltered.coeq (f.base (I := I)) (g.base (I := I))) a').fst (β := fun (j : J) ↦ I j)))
  (p q :
    (Total.mk (I := I) P (IsFiltered.coeq (f.base (I := I)) (g.base (I := I))) a').snd (β := fun (j : J) ↦ I j) ⟶ j)
  (hpq :
    (f.hom (I := I) ≫ u.hom (I := I)) ≫
        Functor.map
          (diag
            (P
              ((Total.mk (I := I) P (IsFiltered.coeq (f.base (I := I)) (g.base (I := I))) a').fst (β := fun (j : J) ↦
                I j))))
          p =
      (g.hom (I := I) ≫ u.hom (I := I)) ≫
        Functor.map
          (diag
            (P
              ((Total.mk (I := I) P (IsFiltered.coeq (f.base (I := I)) (g.base (I := I))) a').fst (β := fun (j : J) ↦
                I j))))
          q),
  ∃ (Z : Total (I := I) P) (h : l ⟶ Z), f ≫ h = g ≫ h
Tactic: obtain ⟨j, p, q, hpq⟩ := IsFinitelyPresentable.exists_eq_of_isColimit (P _).isColimit _ _ this