awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Subcanonical (goal)
Goal:
C : Type u
inst : Category.{v, u} C
D : Type u_1
inst_1 : Category.{v', u_1} D
F : Functor.{v, v', u, u_1} C D
J : GrothendieckTopology.{v, u} C
K : GrothendieckTopology.{v', u_1} D
inst_2 : F.Full
inst_3 : F.Faithful
inst_4 : F.IsContinuous J K
inst_5 : K.Subcanonical
Y : C
h : Presieve.IsSheaf J (obj CategoryTheory.uliftYoneda.{v', v, u} Y)
⊢ Presieve.IsSheaf J (obj CategoryTheory.yoneda Y)
have e : ⦃X : C⦄ → obj (obj CategoryTheory.yoneda Y) (op X) ≃ obj (obj CategoryTheory.uliftYoneda.{v', v, u} Y) (op X)
Tactic: rwa [Presieve.isSheaf_iff_of_nat_equiv]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Subcanonical (goal)
Goal:
C : Type u
inst : Category.{v, u} C
D : Type u_1
inst_1 : Category.{v', u_1} D
F : Functor.{v, v', u, u_1} C D
J : GrothendieckTopology.{v, u} C
K : GrothendieckTopology.{v', u_1} D
inst_2 : F.Full
inst_3 : F.Faithful
inst_4 : F.IsContinuous J K
inst_5 : K.Subcanonical
Y : C
h : Presieve.IsSheaf J (obj CategoryTheory.uliftYoneda.{v', v, u} Y)
e : ⦃X : C⦄ → obj (obj CategoryTheory.yoneda Y) (op X) ≃ obj (obj CategoryTheory.uliftYoneda.{v', v, u} Y) (op X)
⊢ Presieve.IsSheaf J (obj CategoryTheory.yoneda Y)
have he : ∀ ⦃X Y_1 : C⦄ (f : X ⟶ Y_1) (x : obj (obj CategoryTheory.yoneda Y) (op Y_1)),
  Eq (α := obj (obj CategoryTheory.uliftYoneda.{v', v, u} Y) (op X))
    ((Equiv.symm Equiv.ulift :
          (a : (CategoryTheory.yoneda.obj Y).obj (op X)) → (CategoryTheory.uliftYoneda.{v', v, u}.obj Y).obj (op X))
        ((ConcreteCategory.hom (FC := TypeCat.Fun) (map (obj CategoryTheory.yoneda Y) f.op) :
            (a : (fun (X : Type v) ↦ X) ((CategoryTheory.yoneda.obj Y).obj (op Y_1))) →
              (fun (X : Type v) ↦ X) ((CategoryTheory.yoneda.obj Y).obj (op X)))
          x) :
      obj (obj CategoryTheory.uliftYoneda.{v', v, u} Y) (op X))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (map (obj CategoryTheory.uliftYoneda.{v', v, u} Y) f.op) :
          (a : (fun (X : Type (max v v')) ↦ X) ((CategoryTheory.uliftYoneda.{v', v, u}.obj Y).obj (op Y_1))) →
            (fun (X : Type (max v v')) ↦ X) ((CategoryTheory.uliftYoneda.{v', v, u}.obj Y).obj (op X)))
        ((Equiv.symm Equiv.ulift :
            (a : (CategoryTheory.yoneda.obj Y).obj (op Y_1)) →
              (CategoryTheory.uliftYoneda.{v', v, u}.obj Y).obj (op Y_1))
          x) :
      (fun (X : Type (max v v')) ↦ X) (obj (obj CategoryTheory.uliftYoneda.{v', v, u} Y) (op X)))
Tactic: rwa [Presieve.isSheaf_iff_of_nat_equiv]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Subcanonical (goal)
Goal:
C : Type u
inst : Category.{v, u} C
D : Type u_1
inst_1 : Category.{v', u_1} D
F : Functor.{v, v', u, u_1} C D
J : GrothendieckTopology.{v, u} C
K : GrothendieckTopology.{v', u_1} D
inst_2 : F.Full
inst_3 : F.Faithful
inst_4 : F.IsContinuous J K
inst_5 : K.Subcanonical
Y : C
⊢ Presieve.IsSheaf J (obj CategoryTheory.yoneda Y)
have suffices : Presieve.IsSheaf J (obj CategoryTheory.uliftYoneda.{v', v, u} Y)
Tactic: suffices h : Presieve.IsSheaf J (CategoryTheory.uliftYoneda.{v'}.obj Y) by
    rwa [Presieve.isSheaf_iff_of_nat_equiv]
    · intro
      exact Equiv.ulift.symm
    · intros
      rfl
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Subcanonical (goal)
Goal:
C : Type u
inst : Category.{v, u} C
D : Type u_1
inst_1 : Category.{v', u_1} D
F : Functor.{v, v', u, u_1} C D
J : GrothendieckTopology.{v, u} C
K : GrothendieckTopology.{v', u_1} D
inst_2 : F.Full
inst_3 : F.Faithful
inst_4 : F.IsContinuous J K
inst_5 : K.Subcanonical
Y : C
⊢ Presieve.IsSheaf J (obj CategoryTheory.uliftYoneda.{v', v, u} Y)
have rw : Presheaf.IsSheaf J
  ((F ⋙ CategoryTheory.uliftYoneda.{v, v', u_1} ⋙ obj (whiskeringLeft Cᵒᵖ Dᵒᵖ (Type (max v v'))) F.op).obj Y)
Tactic: rw [← isSheaf_iff_isSheaf_of_type, Presheaf.isSheaf_of_iso_iff
    ((Functor.FullyFaithful.ofFullyFaithful F).compUliftYonedaCompWhiskeringLeft.app Y).symm]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Subcanonical (goal)
Goal:
C : Type u
inst : Category.{v, u} C
D : Type u_1
inst_1 : Category.{v', u_1} D
F : Functor.{v, v', u, u_1} C D
J : GrothendieckTopology.{v, u} C
K : GrothendieckTopology.{v', u_1} D
inst_2 : F.Full
inst_3 : F.Faithful
inst_4 : F.IsContinuous J K
inst_5 : K.Subcanonical
Y : C
⊢ Presheaf.IsSheaf J
    ((F ⋙ CategoryTheory.uliftYoneda.{v, v', u_1} ⋙ obj (whiskeringLeft Cᵒᵖ Dᵒᵖ (Type (max v v'))) F.op).obj Y)
have refine : Presheaf.IsSheaf K (obj CategoryTheory.uliftYoneda.{v, v', u_1} (F.obj Y))
Tactic: refine F.op_comp_isSheaf_of_isSheaf J K _ ?_
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Subcanonical (goal)
Goal:
C : Type u
inst : Category.{v, u} C
D : Type u_1
inst_1 : Category.{v', u_1} D
F : Functor.{v, v', u, u_1} C D
J : GrothendieckTopology.{v, u} C
K : GrothendieckTopology.{v', u_1} D
inst_2 : F.Full
inst_3 : F.Faithful
inst_4 : F.IsContinuous J K
inst_5 : K.Subcanonical
Y : C
⊢ Presheaf.IsSheaf K (obj CategoryTheory.uliftYoneda.{v, v', u_1} (F.obj Y))
have rw : Presieve.IsSheaf K (obj CategoryTheory.uliftYoneda.{v, v', u_1} (F.obj Y))
Tactic: rw [isSheaf_iff_isSheaf_of_type]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.SubcanonicalOver (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
J : GrothendieckTopology.{v_1, u_1} C
inst_1 : J.Subcanonical
X : C
⊢ (J.over X).Subcanonical
have refine : ∀ (E Z : Over.{v_1, u_1} X) (R : Sieve Z) (hR : R ∈ (J.over X : (a : Over X) → Set (Sieve a)) Z)
  (t : Presieve.FamilyOfElements (Functor.obj CategoryTheory.yoneda E) R.arrows) (ht : t.Compatible (R := R.arrows)),
  ∃! t_1 : Functor.obj (Functor.obj CategoryTheory.yoneda E) (Opposite.op Z), t.IsAmalgamation (R := R.arrows) t_1
Tactic: refine .of_isSheaf_yoneda_obj _ fun E Z R hR t ht ↦ ?_
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.SubcanonicalOver (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
J : GrothendieckTopology.{v_1, u_1} C
inst_1 : J.Subcanonical
X : C
E Z : Over.{v_1, u_1} X
R : Sieve Z
hR : R ∈ (J.over X : (a : Over X) → Set (Sieve a)) Z
t : Presieve.FamilyOfElements (Functor.obj CategoryTheory.yoneda E) R.arrows
ht : t.Compatible (R := R.arrows)
⊢ ∃! t_1 : Functor.obj (Functor.obj CategoryTheory.yoneda E) (Opposite.op Z), t.IsAmalgamation (R := R.arrows) t_1
have obtain : ∀ (ι : Type (max u_1 v_1)) (T : ι → Over.{v_1, u_1} X) (g : (i : ι) → T i ⟶ Z)
  (hR : Sieve.ofArrows T g ∈ (J.over X : (a : Over X) → Set (Sieve a)) Z)
  (t : Presieve.FamilyOfElements (Functor.obj CategoryTheory.yoneda E) (Sieve.ofArrows T g).arrows)
  (ht : t.Compatible (R := (Sieve.ofArrows T g).arrows)),
  ∃! t_1 : Functor.obj (Functor.obj CategoryTheory.yoneda E) (Opposite.op Z),
    t.IsAmalgamation (R := (Sieve.ofArrows T g).arrows) t_1
Tactic: obtain ⟨ι, T, g, rfl⟩ := R.exists_eq_ofArrows
Cleared hypotheses: R, hR, t, ht
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.SubcanonicalOver (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
J : GrothendieckTopology.{v_1, u_1} C
inst_1 : J.Subcanonical
X : C
E Z_1 : Over.{v_1, u_1} X
ι : Type (max u_1 v_1)
T : ι → Over.{v_1, u_1} X
g : (i : ι) → T i ⟶ Z_1
hR : Sieve.ofArrows T g ∈ (J.over X : (a : Over X) → Set (Sieve a)) Z_1
t : Presieve.FamilyOfElements (Functor.obj CategoryTheory.yoneda E) (Sieve.ofArrows T g).arrows
ht : t.Compatible (R := (Sieve.ofArrows T g).arrows)
i j : ι
Z : C
gi : Z ⟶ (T i).left
gj : Z ⟶ (T j).left
hgij : Eq (α := Z ⟶ Z_1.left) (gi ≫ (fun (i : ι) ↦ Over.Hom.left (g i)) i) (gj ≫ (fun (i : ι) ↦ Over.Hom.left (g i)) j)
⊢ gj ≫ (T j).hom = (Over.mk (gi ≫ (T i).hom)).hom
have dsimp : gj ≫ (T j).hom = gi ≫ (T i).hom
Tactic: dsimp
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.SubcanonicalOver (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
J : GrothendieckTopology.{v_1, u_1} C
inst_1 : J.Subcanonical
X : C
E Z_1 : Over.{v_1, u_1} X
ι : Type (max u_1 v_1)
T : ι → Over.{v_1, u_1} X
g : (i : ι) → T i ⟶ Z_1
hR : Sieve.ofArrows T g ∈ (J.over X : (a : Over X) → Set (Sieve a)) Z_1
t : Presieve.FamilyOfElements (Functor.obj CategoryTheory.yoneda E) (Sieve.ofArrows T g).arrows
ht : t.Compatible (R := (Sieve.ofArrows T g).arrows)
i j : ι
Z : C
gi : Z ⟶ (T i).left
gj : Z ⟶ (T j).left
hgij : Eq (α := Z ⟶ Z_1.left) (gi ≫ (fun (i : ι) ↦ Over.Hom.left (g i)) i) (gj ≫ (fun (i : ι) ↦ Over.Hom.left (g i)) j)
⊢ gj ≫ (T j).hom = gi ≫ (T i).hom
have rw : gj ≫ (T j).hom = gi ≫ Over.Hom.left (g i) ≫ Z_1.hom
Tactic: rw [← Over.w (g i), reassoc_of% hgij, ← Over.w (g j)]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.SubcanonicalOver (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
J : GrothendieckTopology.{v_1, u_1} C
inst_1 : J.Subcanonical
X : C
E Z_1 : Over.{v_1, u_1} X
ι : Type (max u_1 v_1)
T : ι → Over.{v_1, u_1} X
g : (i : ι) → T i ⟶ Z_1
hR : Sieve.ofArrows T g ∈ (J.over X : (a : Over X) → Set (Sieve a)) Z_1
t : Presieve.FamilyOfElements (Functor.obj CategoryTheory.yoneda E) (Sieve.ofArrows T g).arrows
ht : t.Compatible (R := (Sieve.ofArrows T g).arrows)
i j : ι
Z : C
gi : Z ⟶ (T i).left
gj : Z ⟶ (T j).left
hgij : Eq (α := Z ⟶ Z_1.left) (gi ≫ (fun (i : ι) ↦ Over.Hom.left (g i)) i) (gj ≫ (fun (i : ι) ↦ Over.Hom.left (g i)) j)
rw : gj ≫ (T j).hom = gi ≫ Over.Hom.left (g i) ≫ Z_1.hom
⊢ gj ≫ (T j).hom = gi ≫ (T i).hom
have rw₁ : gj ≫ (T j).hom = gj ≫ Over.Hom.left (g j) ≫ Z_1.hom
Tactic: rw [← Over.w (g i), reassoc_of% hgij, ← Over.w (g j)]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.SubcanonicalOver (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
J : GrothendieckTopology.{v_1, u_1} C
inst_1 : J.Subcanonical
X : C
E Z_1 : Over.{v_1, u_1} X
ι : Type (max u_1 v_1)
T : ι → Over.{v_1, u_1} X
g : (i : ι) → T i ⟶ Z_1
hR : Sieve.ofArrows T g ∈ (J.over X : (a : Over X) → Set (Sieve a)) Z_1
t : Presieve.FamilyOfElements (Functor.obj CategoryTheory.yoneda E) (Sieve.ofArrows T g).arrows
ht : t.Compatible (R := (Sieve.ofArrows T g).arrows)
i j : ι
Z : C
gi : Z ⟶ (T i).left
gj : Z ⟶ (T j).left
hgij : Eq (α := Z ⟶ Z_1.left) (gi ≫ (fun (i : ι) ↦ Over.Hom.left (g i)) i) (gj ≫ (fun (i : ι) ↦ Over.Hom.left (g i)) j)
rw : gj ≫ (T j).hom = gi ≫ Over.Hom.left (g i) ≫ Z_1.hom
rw₁ : gj ≫ (T j).hom = gj ≫ Over.Hom.left (g j) ≫ Z_1.hom
⊢ gj ≫ (T j).hom = gi ≫ (T i).hom
have rw₂ : gj ≫ Over.Hom.left (g j) ≫ Z_1.hom = gj ≫ Over.Hom.left (g j) ≫ Z_1.hom
Tactic: rw [← Over.w (g i), reassoc_of% hgij, ← Over.w (g j)]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.SubcanonicalOver (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
J : GrothendieckTopology.{v_1, u_1} C
inst_1 : J.Subcanonical
X : C
E Z_1 : Over.{v_1, u_1} X
ι : Type (max u_1 v_1)
T : ι → Over.{v_1, u_1} X
g : (i : ι) → T i ⟶ Z_1
hR : Sieve.ofArrows T g ∈ (J.over X : (a : Over X) → Set (Sieve a)) Z_1
t : Presieve.FamilyOfElements (Functor.obj CategoryTheory.yoneda E) (Sieve.ofArrows T g).arrows
ht : t.Compatible (R := (Sieve.ofArrows T g).arrows)
i j : ι
Z : C
gi : Z ⟶ (T i).left
gj : Z ⟶ (T j).left
hgij : Eq (α := Z ⟶ Z_1.left) (gi ≫ (fun (i : ι) ↦ Over.Hom.left (g i)) i) (gj ≫ (fun (i : ι) ↦ Over.Hom.left (g i)) j)
⊢ Over.homMk gi ⋯ ≫ g i = Over.homMk gj ⋯ ≫ g j
have h : Over.Hom.left (Over.homMk gi sorry ≫ g i) = Over.Hom.left (Over.homMk gj sorry ≫ g j)
Tactic: ext
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.SubcanonicalOver (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
J : GrothendieckTopology.{v_1, u_1} C
inst_1 : J.Subcanonical
X : C
E Z : Over.{v_1, u_1} X
ι : Type (max u_1 v_1)
T : ι → Over.{v_1, u_1} X
g : (i : ι) → T i ⟶ Z
hR : Sieve.ofArrows T g ∈ (J.over X : (a : Over X) → Set (Sieve a)) Z
t : Presieve.FamilyOfElements (Functor.obj CategoryTheory.yoneda E) (Sieve.ofArrows T g).arrows
ht : t.Compatible (R := (Sieve.ofArrows T g).arrows)
hg : Presieve.Arrows.Compatible (Functor.obj CategoryTheory.yoneda E.left) (B := Z.left) (X := fun (i : ι) ↦ (T i).left)
  (fun (i : ι) ↦ Over.Hom.left (g i)) fun (i : ι) ↦ Over.Hom.left (t (g i) ⋯) :=
  ⋯
⊢ ∃! t_1 : Functor.obj (Functor.obj CategoryTheory.yoneda E) (Opposite.op Z),
    t.IsAmalgamation (R := (Sieve.ofArrows T g).arrows) t_1
have rw : ∀
  (hR :
    Membership.mem (γ := Set (Sieve.{v_1, u_1} Z.left)) ((J : (a : C) → Set (Sieve a)) Z.left)
      (Sieve.generate
        (Presieve.ofArrows (fun (i : ι) ↦ Functor.obj (Over.forget X) (T i)) fun (i : ι) ↦
          Functor.map (Over.forget X) (g i))))
  (t : Presieve.FamilyOfElements (Functor.obj CategoryTheory.yoneda E) (Sieve.ofArrows T g).arrows)
  (ht : t.Compatible (R := (Sieve.ofArrows T g).arrows)),
  ∃! t_1 : Functor.obj (Functor.obj CategoryTheory.yoneda E) (Opposite.op Z),
    t.IsAmalgamation (R := (Sieve.ofArrows T g).arrows) t_1
Tactic: rw [J.mem_over_iff, Sieve.ofArrows, Sieve.overEquiv_generate,
    ← Sieve.arrows_generate_map_eq_functorPushforward, Sieve.generate_sieve,
    Presieve.map_ofArrows] at hR
Cleared hypotheses: hR, t, ht, hg
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.SubcanonicalOver (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
J : GrothendieckTopology.{v_1, u_1} C
inst_1 : J.Subcanonical
X : C
E Z : Over.{v_1, u_1} X
ι : Type (max u_1 v_1)
T : ι → Over.{v_1, u_1} X
g : (i : ι) → T i ⟶ Z
hR :
  Membership.mem (γ := Set (Sieve.{v_1, u_1} Z.left)) ((J : (a : C) → Set (Sieve a)) Z.left)
    (Sieve.generate
      (Presieve.ofArrows (fun (i : ι) ↦ Functor.obj (Over.forget X) (T i)) fun (i : ι) ↦
        Functor.map (Over.forget X) (g i)))
t : Presieve.FamilyOfElements (Functor.obj CategoryTheory.yoneda E) (Sieve.ofArrows T g).arrows
ht : t.Compatible (R := (Sieve.ofArrows T g).arrows)
hg : Presieve.Arrows.Compatible (Functor.obj CategoryTheory.yoneda E.left) (B := Z.left) (X := fun (i : ι) ↦ (T i).left)
  (fun (i : ι) ↦ Over.Hom.left (g i)) fun (i : ι) ↦ Over.Hom.left (t (g i) ⋯) :=
  ⋯
⊢ ∃! t_1 : Functor.obj (Functor.obj CategoryTheory.yoneda E) (Opposite.op Z),
    t.IsAmalgamation (R := (Sieve.ofArrows T g).arrows) t_1
have obtain : ∀ (a : Functor.obj (Functor.obj CategoryTheory.yoneda E.left) (Opposite.op Z.left))
  (ha :
    hg.familyOfElements.sieveExtend (R :=
          Presieve.ofArrows (fun (i : ι) ↦ (T i).left) fun (i : ι) ↦ Over.Hom.left (g i)).IsAmalgamation
      (R :=
      Sieve.arrows
        (Sieve.generate
          (Presieve.ofArrows (fun (i : ι) ↦ Functor.obj (Over.forget X) (T i)) fun (i : ι) ↦
            Functor.map (Over.forget X) (g i))))
      a)
  (huniq :
    ∀ (y : Functor.obj (Functor.obj CategoryTheory.yoneda E.left) (Opposite.op Z.left)),
      (fun (t_1 : Functor.obj (Functor.obj CategoryTheory.yoneda E.left) (Opposite.op Z.left)) ↦
            hg.familyOfElements.sieveExtend (R :=
                  Presieve.ofArrows (fun (i : ι) ↦ (T i).left) fun (i : ι) ↦ Over.Hom.left (g i)).IsAmalgamation
              (R :=
              Sieve.arrows
                (Sieve.generate
                  (Presieve.ofArrows (fun (i : ι) ↦ Functor.obj (Over.forget X) (T i)) fun (i : ι) ↦
                    Functor.map (Over.forget X) (g i))))
              t_1)
          y →
        y = a),
  ∃! t_1 : Functor.obj (Functor.obj CategoryTheory.yoneda E) (Opposite.op Z),
    t.IsAmalgamation (R := (Sieve.ofArrows T g).arrows) t_1
Tactic: obtain ⟨a, ha, huniq⟩ := Subcanonical.isSheaf_of_isRepresentable
    (CategoryTheory.yoneda.obj E.left) _ hR _ hg.familyOfElements_compatible.sieveExtend
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.SubcanonicalOver (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
J : GrothendieckTopology.{v_1, u_1} C
inst_1 : J.Subcanonical
X : C
E Z : Over.{v_1, u_1} X
ι : Type (max u_1 v_1)
T : ι → Over.{v_1, u_1} X
g : (i : ι) → T i ⟶ Z
hR :
  Membership.mem (γ := Set (Sieve.{v_1, u_1} Z.left)) ((J : (a : C) → Set (Sieve a)) Z.left)
    (Sieve.generate
      (Presieve.ofArrows (fun (i : ι) ↦ Functor.obj (Over.forget X) (T i)) fun (i : ι) ↦
        Functor.map (Over.forget X) (g i)))
t : Presieve.FamilyOfElements (Functor.obj CategoryTheory.yoneda E) (Sieve.ofArrows T g).arrows
ht : t.Compatible (R := (Sieve.ofArrows T g).arrows)
hg : Presieve.Arrows.Compatible (Functor.obj CategoryTheory.yoneda E.left) (B := Z.left) (X := fun (i : ι) ↦ (T i).left)
  (fun (i : ι) ↦ Over.Hom.left (g i)) fun (i : ι) ↦ Over.Hom.left (t (g i) ⋯) :=
  ⋯
a : Functor.obj (Functor.obj CategoryTheory.yoneda E.left) (Opposite.op Z.left)
ha :
  hg.familyOfElements.sieveExtend (R :=
        Presieve.ofArrows (fun (i : ι) ↦ (T i).left) fun (i : ι) ↦ Over.Hom.left (g i)).IsAmalgamation
    (R :=
    Sieve.arrows
      (Sieve.generate
        (Presieve.ofArrows (fun (i : ι) ↦ Functor.obj (Over.forget X) (T i)) fun (i : ι) ↦
          Functor.map (Over.forget X) (g i))))
    a
huniq :
  ∀ (y : Functor.obj (Functor.obj CategoryTheory.yoneda E.left) (Opposite.op Z.left)),
    (fun (t_1 : Functor.obj (Functor.obj CategoryTheory.yoneda E.left) (Opposite.op Z.left)) ↦
          hg.familyOfElements.sieveExtend (R :=
                Presieve.ofArrows (fun (i : ι) ↦ (T i).left) fun (i : ι) ↦ Over.Hom.left (g i)).IsAmalgamation
            (R :=
            Sieve.arrows
              (Sieve.generate
                (Presieve.ofArrows (fun (i : ι) ↦ Functor.obj (Over.forget X) (T i)) fun (i : ι) ↦
                  Functor.map (Over.forget X) (g i))))
            t_1)
        y →
      y = a
⊢ ∃! t_1 : Functor.obj (Functor.obj CategoryTheory.yoneda E) (Opposite.op Z),
    t.IsAmalgamation (R := (Sieve.ofArrows T g).arrows) t_1
have refine_1 : Functor.obj (Functor.obj CategoryTheory.yoneda E) (Opposite.op Z)
Tactic: refine ⟨?_, ?_, fun y hty ↦ ?_⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.SubcanonicalOver (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
J : GrothendieckTopology.{v_1, u_1} C
inst_1 : J.Subcanonical
X : C
E Z : Over.{v_1, u_1} X
ι : Type (max u_1 v_1)
T : ι → Over.{v_1, u_1} X
g : (i : ι) → T i ⟶ Z
hR :
  Membership.mem (γ := Set (Sieve.{v_1, u_1} Z.left)) ((J : (a : C) → Set (Sieve a)) Z.left)
    (Sieve.generate
      (Presieve.ofArrows (fun (i : ι) ↦ Functor.obj (Over.forget X) (T i)) fun (i : ι) ↦
        Functor.map (Over.forget X) (g i)))
t : Presieve.FamilyOfElements (Functor.obj CategoryTheory.yoneda E) (Sieve.ofArrows T g).arrows
ht : t.Compatible (R := (Sieve.ofArrows T g).arrows)
hg : Presieve.Arrows.Compatible (Functor.obj CategoryTheory.yoneda E.left) (B := Z.left) (X := fun (i : ι) ↦ (T i).left)
  (fun (i : ι) ↦ Over.Hom.left (g i)) fun (i : ι) ↦ Over.Hom.left (t (g i) ⋯) :=
  ⋯
a : Functor.obj (Functor.obj CategoryTheory.yoneda E.left) (Opposite.op Z.left)
ha :
  hg.familyOfElements.sieveExtend (R :=
        Presieve.ofArrows (fun (i : ι) ↦ (T i).left) fun (i : ι) ↦ Over.Hom.left (g i)).IsAmalgamation
    (R :=
    Sieve.arrows
      (Sieve.generate
        (Presieve.ofArrows (fun (i : ι) ↦ Functor.obj (Over.forget X) (T i)) fun (i : ι) ↦
          Functor.map (Over.forget X) (g i))))
    a
huniq :
  ∀ (y : Functor.obj (Functor.obj CategoryTheory.yoneda E.left) (Opposite.op Z.left)),
    (fun (t_1 : Functor.obj (Functor.obj CategoryTheory.yoneda E.left) (Opposite.op Z.left)) ↦
          hg.familyOfElements.sieveExtend (R :=
                Presieve.ofArrows (fun (i : ι) ↦ (T i).left) fun (i : ι) ↦ Over.Hom.left (g i)).IsAmalgamation
            (R :=
            Sieve.arrows
              (Sieve.generate
                (Presieve.ofArrows (fun (i : ι) ↦ Functor.obj (Over.forget X) (T i)) fun (i : ι) ↦
                  Functor.map (Over.forget X) (g i))))
            t_1)
        y →
      y = a
refine_1 : Functor.obj (Functor.obj CategoryTheory.yoneda E) (Opposite.op Z)
⊢ ∃! t_1 : Functor.obj (Functor.obj CategoryTheory.yoneda E) (Opposite.op Z),
    t.IsAmalgamation (R := (Sieve.ofArrows T g).arrows) t_1
have refine_2 : (fun (t_1 : Functor.obj (Functor.obj CategoryTheory.yoneda E) (Opposite.op Z)) ↦
    t.IsAmalgamation (R := (Sieve.ofArrows T g).arrows) t_1)
  (Over.homMk a sorry)
Tactic: refine ⟨?_, ?_, fun y hty ↦ ?_⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.SubcanonicalOver (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
J : GrothendieckTopology.{v_1, u_1} C
inst_1 : J.Subcanonical
X : C
E Z : Over.{v_1, u_1} X
ι : Type (max u_1 v_1)
T : ι → Over.{v_1, u_1} X
g : (i : ι) → T i ⟶ Z
hR :
  Membership.mem (γ := Set (Sieve.{v_1, u_1} Z.left)) ((J : (a : C) → Set (Sieve a)) Z.left)
    (Sieve.generate
      (Presieve.ofArrows (fun (i : ι) ↦ Functor.obj (Over.forget X) (T i)) fun (i : ι) ↦
        Functor.map (Over.forget X) (g i)))
t : Presieve.FamilyOfElements (Functor.obj CategoryTheory.yoneda E) (Sieve.ofArrows T g).arrows
ht : t.Compatible (R := (Sieve.ofArrows T g).arrows)
hg : Presieve.Arrows.Compatible (Functor.obj CategoryTheory.yoneda E.left) (B := Z.left) (X := fun (i : ι) ↦ (T i).left)
  (fun (i : ι) ↦ Over.Hom.left (g i)) fun (i : ι) ↦ Over.Hom.left (t (g i) ⋯) :=
  ⋯
a : Functor.obj (Functor.obj CategoryTheory.yoneda E.left) (Opposite.op Z.left)
ha :
  hg.familyOfElements.sieveExtend (R :=
        Presieve.ofArrows (fun (i : ι) ↦ (T i).left) fun (i : ι) ↦ Over.Hom.left (g i)).IsAmalgamation
    (R :=
    Sieve.arrows
      (Sieve.generate
        (Presieve.ofArrows (fun (i : ι) ↦ Functor.obj (Over.forget X) (T i)) fun (i : ι) ↦
          Functor.map (Over.forget X) (g i))))
    a
huniq :
  ∀ (y : Functor.obj (Functor.obj CategoryTheory.yoneda E.left) (Opposite.op Z.left)),
    (fun (t_1 : Functor.obj (Functor.obj CategoryTheory.yoneda E.left) (Opposite.op Z.left)) ↦
          hg.familyOfElements.sieveExtend (R :=
                Presieve.ofArrows (fun (i : ι) ↦ (T i).left) fun (i : ι) ↦ Over.Hom.left (g i)).IsAmalgamation
            (R :=
            Sieve.arrows
              (Sieve.generate
                (Presieve.ofArrows (fun (i : ι) ↦ Functor.obj (Over.forget X) (T i)) fun (i : ι) ↦
                  Functor.map (Over.forget X) (g i))))
            t_1)
        y →
      y = a
refine_1 : Functor.obj (Functor.obj CategoryTheory.yoneda E) (Opposite.op Z)
refine_2 :
  (fun (t_1 : Functor.obj (Functor.obj CategoryTheory.yoneda E) (Opposite.op Z)) ↦
      t.IsAmalgamation (R := (Sieve.ofArrows T g).arrows) t_1)
    (Over.homMk a ⋯)
⊢ ∃! t_1 : Functor.obj (Functor.obj CategoryTheory.yoneda E) (Opposite.op Z),
    t.IsAmalgamation (R := (Sieve.ofArrows T g).arrows) t_1
have refine_3 : ∀ (y : Functor.obj (Functor.obj CategoryTheory.yoneda E) (Opposite.op Z))
  (hty :
    (fun (t_1 : Functor.obj (Functor.obj CategoryTheory.yoneda E) (Opposite.op Z)) ↦
        t.IsAmalgamation (R := (Sieve.ofArrows T g).arrows) t_1)
      y),
  y = Over.homMk a sorry
Tactic: refine ⟨?_, ?_, fun y hty ↦ ?_⟩