awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.DenseSubsite.OneHypercoverDense (goal)
Goal:
C₀ : Type u₀
C : Type u
inst : Category.{v₀, u₀} C₀
inst_1 : Category.{v, u} C
F : Functor.{v₀, v, u₀, u} C₀ C
J₀ : GrothendieckTopology.{v₀, u₀} C₀
J : GrothendieckTopology.{v, u} C
A : Type u'
inst_2 : Category.{v', u'} A
inst_3 : IsDenseSubsite J₀ J F
data : (X : C) → OneHypercoverDenseData.{w, v₀, v, u₀, u} F J₀ J X
inst_4 : HasLimitsOfSize.{w, w, v', u'} A
G₀ : Sheaf.{v₀, v', u₀, u'} J₀ A
X : C
i : (data X).I₀
j : PreOneHypercoverDenseData.I₀ (S := F.obj ((data X).X i)) (toPreOneHypercoverDenseData (data (F.obj ((data X).X i))))
⊢ Eq (α :=
    presheafObj data G₀ X ⟶
      G₀.obj (P := Presheaf.IsSheaf J₀).obj
        (op (PreOneHypercoverDenseData.X (toPreOneHypercoverDenseData (data (F.obj ((data X).X i)))) j)))
    (presheafMap data G₀ ((data X).f i) ≫ presheafObjπ data G₀ (F.obj ((data X).X i)) j)
    (CategoryStruct.comp (Y := presheafObj data G₀ (F.obj ((data X).X i)))
      (presheafObjπ data G₀ X i ≫ Iso.inv (presheafObjObjIso data G₀ ((data X).X i)) :
        presheafObj data G₀ X ⟶ (presheaf data G₀).obj (op (F.obj ((data X).X i))))
      (presheafObjπ data G₀ (F.obj ((data X).X i)) j))
have h : Eq (α :=
  presheafObj data G₀ X ⟶
    G₀.obj (P := Presheaf.IsSheaf J₀).obj
      (op (PreOneHypercoverDenseData.X (toPreOneHypercoverDenseData (data (F.obj ((data X).X i)))) j)))
  (restriction data G₀
    (PreOneHypercoverDenseData.f (toPreOneHypercoverDenseData (data (F.obj ((data X).X i)))) j ≫ (data X).f i))
  (presheafObjπ data G₀ X i ≫
    IsDenseSubsite.mapPreimage J F G₀
      (PreOneHypercoverDenseData.f (toPreOneHypercoverDenseData (data (F.obj ((data X).X i)))) j))
Tactic: rw [assoc, presheafMap_π, presheafObjObjIso, presheafObjObjIso.inv_π data G₀]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.DenseSubsite.OneHypercoverDense (goal)
Goal:
C₀ : Type u₀
C : Type u
inst : Category.{v₀, u₀} C₀
inst_1 : Category.{v, u} C
F : Functor.{v₀, v, u₀, u} C₀ C
J₀ : GrothendieckTopology.{v₀, u₀} C₀
J : GrothendieckTopology.{v, u} C
A : Type u'
inst_2 : Category.{v', u'} A
inst_3 : IsDenseSubsite J₀ J F
data : (X : C) → OneHypercoverDenseData.{w, v₀, v, u₀, u} F J₀ J X
inst_4 : HasLimitsOfSize.{w, w, v', u'} A
G₀ : Sheaf.{v₀, v', u₀, u'} J₀ A
X : C
i : (data X).I₀
j : PreOneHypercoverDenseData.I₀ (S := F.obj ((data X).X i)) (toPreOneHypercoverDenseData (data (F.obj ((data X).X i))))
⊢ Eq (α :=
    presheafObj data G₀ X ⟶
      G₀.obj (P := Presheaf.IsSheaf J₀).obj
        (op (PreOneHypercoverDenseData.X (toPreOneHypercoverDenseData (data (F.obj ((data X).X i)))) j)))
    (restriction data G₀
      (PreOneHypercoverDenseData.f (toPreOneHypercoverDenseData (data (F.obj ((data X).X i)))) j ≫ (data X).f i))
    (presheafObjπ data G₀ X i ≫
      IsDenseSubsite.mapPreimage J F G₀
        (PreOneHypercoverDenseData.f (toPreOneHypercoverDenseData (data (F.obj ((data X).X i)))) j))
have h : PreOneHypercoverDenseData.f (toPreOneHypercoverDenseData (data (F.obj ((data X).X i)))) j ≫ (data X).f i =
  PreOneHypercoverDenseData.f (toPreOneHypercoverDenseData (data (F.obj ((data X).X i)))) j ≫ (data X).f i
Tactic: apply restriction_eq_of_fac
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.DenseSubsite.OneHypercoverDense (goal)
Goal:
C₀ : Type u₀
C : Type u
inst : Category.{v₀, u₀} C₀
inst_1 : Category.{v, u} C
F : Functor.{v₀, v, u₀, u} C₀ C
J₀ : GrothendieckTopology.{v₀, u₀} C₀
J : GrothendieckTopology.{v, u} C
A : Type u'
inst_2 : Category.{v', u'} A
inst_3 : IsDenseSubsite J₀ J F
data : (X : C) → OneHypercoverDenseData.{w, v₀, v, u₀, u} F J₀ J X
inst_4 : HasLimitsOfSize.{w, w, v', u'} A
G₀ : Sheaf.{v₀, v', u₀, u'} J₀ A
X₀ Y₀ : C₀
f : X₀ ⟶ Y₀
⊢ Eq (α := G₀.obj (P := Presheaf.IsSheaf J₀).obj (op Y₀) ⟶ (presheaf data G₀).obj (op (F.obj X₀)))
    (G₀.obj (P := Presheaf.IsSheaf J₀).map f.op ≫ Iso.inv (presheafObjObjIso data G₀ X₀))
    (CategoryStruct.comp (Y := (presheaf data G₀).obj (op (F.obj Y₀))) (Iso.inv (presheafObjObjIso data G₀ Y₀))
      (presheafMap data G₀ (F.map f)))
have h : ∀
  (i :
    PreOneHypercoverDenseData.I₀ (S := unop (op (F.obj X₀)))
      (toPreOneHypercoverDenseData (data (unop (op (F.obj X₀)))))),
  CategoryStruct.comp (Y := presheafObj data G₀ (unop (op (F.obj X₀))))
      (G₀.obj (P := Presheaf.IsSheaf J₀).map f.op ≫ Iso.inv (presheafObjObjIso data G₀ X₀) :
        G₀.obj (P := Presheaf.IsSheaf J₀).obj (op Y₀) ⟶ (presheaf data G₀).obj (op (F.obj X₀)))
      (presheafObjπ data G₀ (unop (op (F.obj X₀))) i) =
    CategoryStruct.comp (Y := presheafObj data G₀ (unop (op (F.obj X₀))))
      (CategoryStruct.comp (Y := (presheaf data G₀).obj (op (F.obj Y₀))) (Iso.inv (presheafObjObjIso data G₀ Y₀))
        (presheafMap data G₀ (F.map f)))
      (presheafObjπ data G₀ (unop (op (F.obj X₀))) i)
Tactic: apply presheafObj_hom_ext
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.DenseSubsite.OneHypercoverDense (goal)
Goal:
C₀ : Type u₀
C : Type u
inst : Category.{v₀, u₀} C₀
inst_1 : Category.{v, u} C
F : Functor.{v₀, v, u₀, u} C₀ C
J₀ : GrothendieckTopology.{v₀, u₀} C₀
J : GrothendieckTopology.{v, u} C
A : Type u'
inst_2 : Category.{v', u'} A
inst_3 : IsDenseSubsite J₀ J F
data : (X : C) → OneHypercoverDenseData.{w, v₀, v, u₀, u} F J₀ J X
inst_4 : HasLimitsOfSize.{w, w, v', u'} A
G₀ : Sheaf.{v₀, v', u₀, u'} J₀ A
⊢ Presheaf.IsSheaf J (presheaf data G₀)
have rw : Presheaf.IsSheaf J₀ (F.op ⋙ presheaf data G₀) ∧
  ∀ (X : C), Nonempty (IsLimit ((data X).toOneHypercover.multifork (presheaf data G₀)))
Tactic: rw [isSheaf_iff data]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.DenseSubsite.OneHypercoverDense (goal)
Goal:
C₀ : Type u₀
C : Type u
inst : Category.{v₀, u₀} C₀
inst_1 : Category.{v, u} C
F : Functor.{v₀, v, u₀, u} C₀ C
J₀ : GrothendieckTopology.{v₀, u₀} C₀
J : GrothendieckTopology.{v, u} C
A : Type u'
inst_2 : Category.{v', u'} A
inst_3 : IsDenseSubsite J₀ J F
data : (X : C) → OneHypercoverDenseData.{w, v₀, v, u₀, u} F J₀ J X
inst_4 : HasLimitsOfSize.{w, w, v', u'} A
G₀ : Sheaf.{v₀, v', u₀, u'} J₀ A
⊢ Presheaf.IsSheaf J₀ (F.op ⋙ presheaf data G₀) ∧
    ∀ (X : C), Nonempty (IsLimit ((data X).toOneHypercover.multifork (presheaf data G₀)))
have left : Presheaf.IsSheaf J₀ (F.op ⋙ presheaf data G₀)
Tactic: constructor
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.DenseSubsite.OneHypercoverDense (goal)
Goal:
C₀ : Type u₀
C : Type u
inst : Category.{v₀, u₀} C₀
inst_1 : Category.{v, u} C
F : Functor.{v₀, v, u₀, u} C₀ C
J₀ : GrothendieckTopology.{v₀, u₀} C₀
J : GrothendieckTopology.{v, u} C
A : Type u'
inst_2 : Category.{v', u'} A
inst_3 : IsDenseSubsite J₀ J F
data : (X : C) → OneHypercoverDenseData.{w, v₀, v, u₀, u} F J₀ J X
inst_4 : HasLimitsOfSize.{w, w, v', u'} A
G₀ : Sheaf.{v₀, v', u₀, u'} J₀ A
left : Presheaf.IsSheaf J₀ (F.op ⋙ presheaf data G₀)
⊢ Presheaf.IsSheaf J₀ (F.op ⋙ presheaf data G₀) ∧
    ∀ (X : C), Nonempty (IsLimit ((data X).toOneHypercover.multifork (presheaf data G₀)))
have right : ∀ (X : C), Nonempty (IsLimit ((data X).toOneHypercover.multifork (presheaf data G₀)))
Tactic: constructor
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.DenseSubsite.OneHypercoverDense (goal)
Goal:
C₀ : Type u₀
C : Type u
inst : Category.{v₀, u₀} C₀
inst_1 : Category.{v, u} C
F : Functor.{v₀, v, u₀, u} C₀ C
J₀ : GrothendieckTopology.{v₀, u₀} C₀
J : GrothendieckTopology.{v, u} C
A : Type u'
inst_2 : Category.{v', u'} A
inst_3 : IsDenseSubsite J₀ J F
data : (X : C) → OneHypercoverDenseData.{w, v₀, v, u₀, u} F J₀ J X
inst_4 : HasLimitsOfSize.{w, w, v', u'} A
G₀ : Sheaf.{v₀, v', u₀, u'} J₀ A
X : C
⊢ Nonempty (IsLimit ((data X).toOneHypercover.multifork (presheaf data G₀)))
have right : ∀ (i : (data X).multicospanShape.L),
  Iso.hom (Iso.refl (presheafObjMultifork data G₀ X).pt) ≫
      Multifork.ι
        (obj
          (Cone.postcompose
            (Iso.hom
              (WalkingMulticospan.functorExt
                (fun (x : (data X).toOneHypercover.multicospanShape.L) ↦ presheafObjObjIso data G₀ ((data X).X x))
                (fun (x : (data X).toOneHypercover.multicospanShape.R) ↦
                  presheafObjObjIso data G₀
                    ((data X).Y
                      (x.snd (β := fun (i : (data X).toOneHypercover.I₀ × (data X).toOneHypercover.I₀) ↦
                        (data X).toOneHypercover.I₁ i.1 i.2))))
                sorry sorry)))
          ((data X).toOneHypercover.multifork (presheaf data G₀)))
        i =
    (presheafObjMultifork data G₀ X).ι i
Tactic: refine ⟨(IsLimit.postcomposeHomEquiv
      (WalkingMulticospan.functorExt
          (fun _ ↦ presheafObjObjIso _ _ _) (fun _ ↦ presheafObjObjIso _ _ _)
          (fun _ ↦ (compPresheafIso _ _).hom.naturality _)
          (fun _ ↦ (compPresheafIso _ _).hom.naturality _)) _).1
      (IsLimit.ofIsoLimit (presheafObjIsLimit data G₀ X)
        (Multifork.ext (Iso.refl _) (fun i ↦ ?_)))⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.DenseSubsite.SheafEquiv (goal)
Goal:
C : Type u_1
D : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} D
G : Functor.{v_1, v_2, u_1, u_2} C D
J : GrothendieckTopology.{v_1, u_1} C
K : GrothendieckTopology.{v_2, u_2} D
A : Type w
inst_2 : Category.{w', w} A
inst_3 : ∀ (X : Dᵒᵖ), Limits.HasLimitsOfShape (StructuredArrow X G.op) A
inst_4 : IsDenseSubsite J K G
Y : Sheaf.{v_1, w', u_1, w} J A
U : C
X : A
⊢ IsIso (X :=
    obj
      (obj yoneda ((obj (G.op.ran ⋙ obj (whiskeringLeft Cᵒᵖ Dᵒᵖ A) G.op) (Y.obj (P := Presheaf.IsSheaf J))).obj (op U)))
      (op X))
    (NatTrans.app (map yoneda (NatTrans.app (NatTrans.app G.op.ranCounit (Y.obj (P := Presheaf.IsSheaf J))) (op U)))
      (op X))
have rw : Function.Bijective (α :=
  (fun (X : Type w') ↦ X)
    (obj
      (obj yoneda ((obj (G.op.ran ⋙ obj (whiskeringLeft Cᵒᵖ Dᵒᵖ A) G.op) (Y.obj (P := Presheaf.IsSheaf J))).obj (op U)))
      (op X)))
  (β :=
  (fun (X : Type w') ↦ X)
    (obj (obj yoneda ((obj (𝟭 (Functor.{v_1, w', u_1, w} Cᵒᵖ A)) (Y.obj (P := Presheaf.IsSheaf J))).obj (op U)))
      (op X)))
  (⇑(ConcreteCategory.hom (FC := TypeCat.Fun)
        (NatTrans.app (map yoneda (NatTrans.app (NatTrans.app G.op.ranCounit (Y.obj (P := Presheaf.IsSheaf J))) (op U)))
          (op X))) :
    (a :
        (fun (X : Type w') ↦ X)
          ((yoneda.obj (((G.op.ran ⋙ (whiskeringLeft Cᵒᵖ Dᵒᵖ A).obj G.op).obj Y.obj).obj (op U))).obj (op X))) →
      (fun (X : Type w') ↦ X) ((yoneda.obj (((𝟭 (Cᵒᵖ ⥤ A)).obj Y.obj).obj (op U))).obj (op X)))
Tactic: rw [isIso_iff_bijective]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.DenseSubsite.SheafEquiv (goal)
Goal:
C : Type u_1
D : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} D
G : Functor.{v_1, v_2, u_1, u_2} C D
J : GrothendieckTopology.{v_1, u_1} C
K : GrothendieckTopology.{v_2, u_2} D
A : Type w
inst_2 : Category.{w', w} A
inst_3 : ∀ (X : Dᵒᵖ), Limits.HasLimitsOfShape (StructuredArrow X G.op) A
inst_4 : IsDenseSubsite J K G
Y : Sheaf.{v_1, w', u_1, w} J A
U : C
X : A
⊢ Function.Bijective (α :=
    (fun (X : Type w') ↦ X)
      (obj
        (obj yoneda
          ((obj (G.op.ran ⋙ obj (whiskeringLeft Cᵒᵖ Dᵒᵖ A) G.op) (Y.obj (P := Presheaf.IsSheaf J))).obj (op U)))
        (op X)))
    (β :=
    (fun (X : Type w') ↦ X)
      (obj (obj yoneda ((obj (𝟭 (Functor.{v_1, w', u_1, w} Cᵒᵖ A)) (Y.obj (P := Presheaf.IsSheaf J))).obj (op U)))
        (op X)))
    (⇑(ConcreteCategory.hom (FC := TypeCat.Fun)
          (NatTrans.app
            (map yoneda (NatTrans.app (NatTrans.app G.op.ranCounit (Y.obj (P := Presheaf.IsSheaf J))) (op U)))
            (op X))) :
      (a :
          (fun (X : Type w') ↦ X)
            ((yoneda.obj (((G.op.ran ⋙ (whiskeringLeft Cᵒᵖ Dᵒᵖ A).obj G.op).obj Y.obj).obj (op U))).obj (op X))) →
        (fun (X : Type w') ↦ X) ((yoneda.obj (((𝟭 (Cᵒᵖ ⥤ A)).obj Y.obj).obj (op U))).obj (op X)))
have left : Function.Injective (α :=
  (fun (X : Type w') ↦ X)
    (obj
      (obj yoneda ((obj (G.op.ran ⋙ obj (whiskeringLeft Cᵒᵖ Dᵒᵖ A) G.op) (Y.obj (P := Presheaf.IsSheaf J))).obj (op U)))
      (op X)))
  (β :=
  (fun (X : Type w') ↦ X)
    (obj (obj yoneda ((obj (𝟭 (Functor.{v_1, w', u_1, w} Cᵒᵖ A)) (Y.obj (P := Presheaf.IsSheaf J))).obj (op U)))
      (op X)))
  (⇑(ConcreteCategory.hom (FC := TypeCat.Fun)
        (NatTrans.app (map yoneda (NatTrans.app (NatTrans.app G.op.ranCounit (Y.obj (P := Presheaf.IsSheaf J))) (op U)))
          (op X))) :
    (a :
        (fun (X : Type w') ↦ X)
          ((yoneda.obj (((G.op.ran ⋙ (whiskeringLeft Cᵒᵖ Dᵒᵖ A).obj G.op).obj Y.obj).obj (op U))).obj (op X))) →
      (fun (X : Type w') ↦ X) ((yoneda.obj (((𝟭 (Cᵒᵖ ⥤ A)).obj Y.obj).obj (op U))).obj (op X)))
Tactic: constructor
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.DenseSubsite.SheafEquiv (goal)
Goal:
C : Type u_1
D : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} D
G : Functor.{v_1, v_2, u_1, u_2} C D
J : GrothendieckTopology.{v_1, u_1} C
K : GrothendieckTopology.{v_2, u_2} D
A : Type w
inst_2 : Category.{w', w} A
inst_3 : ∀ (X : Dᵒᵖ), Limits.HasLimitsOfShape (StructuredArrow X G.op) A
inst_4 : IsDenseSubsite J K G
Y : Sheaf.{v_1, w', u_1, w} J A
U : C
X : A
left :
  Function.Injective (α :=
    (fun (X : Type w') ↦ X)
      (obj
        (obj yoneda
          ((obj (G.op.ran ⋙ obj (whiskeringLeft Cᵒᵖ Dᵒᵖ A) G.op) (Y.obj (P := Presheaf.IsSheaf J))).obj (op U)))
        (op X)))
    (β :=
    (fun (X : Type w') ↦ X)
      (obj (obj yoneda ((obj (𝟭 (Functor.{v_1, w', u_1, w} Cᵒᵖ A)) (Y.obj (P := Presheaf.IsSheaf J))).obj (op U)))
        (op X)))
    (⇑(ConcreteCategory.hom (FC := TypeCat.Fun)
          (NatTrans.app
            (map yoneda (NatTrans.app (NatTrans.app G.op.ranCounit (Y.obj (P := Presheaf.IsSheaf J))) (op U)))
            (op X))) :
      (a :
          (fun (X : Type w') ↦ X)
            ((yoneda.obj (((G.op.ran ⋙ (whiskeringLeft Cᵒᵖ Dᵒᵖ A).obj G.op).obj Y.obj).obj (op U))).obj (op X))) →
        (fun (X : Type w') ↦ X) ((yoneda.obj (((𝟭 (Cᵒᵖ ⥤ A)).obj Y.obj).obj (op U))).obj (op X)))
⊢ Function.Bijective (α :=
    (fun (X : Type w') ↦ X)
      (obj
        (obj yoneda
          ((obj (G.op.ran ⋙ obj (whiskeringLeft Cᵒᵖ Dᵒᵖ A) G.op) (Y.obj (P := Presheaf.IsSheaf J))).obj (op U)))
        (op X)))
    (β :=
    (fun (X : Type w') ↦ X)
      (obj (obj yoneda ((obj (𝟭 (Functor.{v_1, w', u_1, w} Cᵒᵖ A)) (Y.obj (P := Presheaf.IsSheaf J))).obj (op U)))
        (op X)))
    (⇑(ConcreteCategory.hom (FC := TypeCat.Fun)
          (NatTrans.app
            (map yoneda (NatTrans.app (NatTrans.app G.op.ranCounit (Y.obj (P := Presheaf.IsSheaf J))) (op U)))
            (op X))) :
      (a :
          (fun (X : Type w') ↦ X)
            ((yoneda.obj (((G.op.ran ⋙ (whiskeringLeft Cᵒᵖ Dᵒᵖ A).obj G.op).obj Y.obj).obj (op U))).obj (op X))) →
        (fun (X : Type w') ↦ X) ((yoneda.obj (((𝟭 (Cᵒᵖ ⥤ A)).obj Y.obj).obj (op U))).obj (op X)))
have right : Function.Surjective (α :=
  (fun (X : Type w') ↦ X)
    (obj
      (obj yoneda ((obj (G.op.ran ⋙ obj (whiskeringLeft Cᵒᵖ Dᵒᵖ A) G.op) (Y.obj (P := Presheaf.IsSheaf J))).obj (op U)))
      (op X)))
  (β :=
  (fun (X : Type w') ↦ X)
    (obj (obj yoneda ((obj (𝟭 (Functor.{v_1, w', u_1, w} Cᵒᵖ A)) (Y.obj (P := Presheaf.IsSheaf J))).obj (op U)))
      (op X)))
  (⇑(ConcreteCategory.hom (FC := TypeCat.Fun)
        (NatTrans.app (map yoneda (NatTrans.app (NatTrans.app G.op.ranCounit (Y.obj (P := Presheaf.IsSheaf J))) (op U)))
          (op X))) :
    (a :
        (fun (X : Type w') ↦ X)
          ((yoneda.obj (((G.op.ran ⋙ (whiskeringLeft Cᵒᵖ Dᵒᵖ A).obj G.op).obj Y.obj).obj (op U))).obj (op X))) →
      (fun (X : Type w') ↦ X) ((yoneda.obj (((𝟭 (Cᵒᵖ ⥤ A)).obj Y.obj).obj (op U))).obj (op X)))
Tactic: constructor
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.DenseSubsite.SheafEquiv (goal)
Goal:
C : Type u_1
D : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} D
G : Functor.{v_1, v_2, u_1, u_2} C D
J : GrothendieckTopology.{v_1, u_1} C
K : GrothendieckTopology.{v_2, u_2} D
A : Type w
inst_2 : Category.{w', w} A
inst_3 : ∀ (X : Dᵒᵖ), Limits.HasLimitsOfShape (StructuredArrow X G.op) A
inst_4 : IsDenseSubsite J K G
Y : Sheaf.{v_1, w', u_1, w} J A
U : C
X : A
f₁ f₂ :
  obj (obj yoneda ((obj (G.op.ran ⋙ obj (whiskeringLeft Cᵒᵖ Dᵒᵖ A) G.op) (Y.obj (P := Presheaf.IsSheaf J))).obj (op U)))
    (op X)
e :
  Eq (α :=
    (fun (X : Type w') ↦ X)
      (obj (obj yoneda ((obj (𝟭 (Functor.{v_1, w', u_1, w} Cᵒᵖ A)) (Y.obj (P := Presheaf.IsSheaf J))).obj (op U)))
        (op X)))
    ((ConcreteCategory.hom (FC := TypeCat.Fun)
            (NatTrans.app
              (map yoneda (NatTrans.app (NatTrans.app G.op.ranCounit (Y.obj (P := Presheaf.IsSheaf J))) (op U)))
              (op X)) :
          (a :
              (fun (X : Type w') ↦ X)
                ((yoneda.obj (((G.op.ran ⋙ (whiskeringLeft Cᵒᵖ Dᵒᵖ A).obj G.op).obj Y.obj).obj (op U))).obj (op X))) →
            (fun (X : Type w') ↦ X) ((yoneda.obj (((𝟭 (Cᵒᵖ ⥤ A)).obj Y.obj).obj (op U))).obj (op X)))
        f₁ :
      (fun (X : Type w') ↦ X)
        (obj (obj yoneda ((obj (𝟭 (Functor.{v_1, w', u_1, w} Cᵒᵖ A)) (Y.obj (P := Presheaf.IsSheaf J))).obj (op U)))
          (op X)))
    ((ConcreteCategory.hom (FC := TypeCat.Fun)
            (NatTrans.app
              (map yoneda (NatTrans.app (NatTrans.app G.op.ranCounit (Y.obj (P := Presheaf.IsSheaf J))) (op U)))
              (op X)) :
          (a :
              (fun (X : Type w') ↦ X)
                ((yoneda.obj (((G.op.ran ⋙ (whiskeringLeft Cᵒᵖ Dᵒᵖ A).obj G.op).obj Y.obj).obj (op U))).obj (op X))) →
            (fun (X : Type w') ↦ X) ((yoneda.obj (((𝟭 (Cᵒᵖ ⥤ A)).obj Y.obj).obj (op U))).obj (op X)))
        f₂ :
      (fun (X : Type w') ↦ X)
        (obj (obj yoneda ((obj (𝟭 (Functor.{v_1, w', u_1, w} Cᵒᵖ A)) (Y.obj (P := Presheaf.IsSheaf J))).obj (op U)))
          (op X)))
⊢ f₁ = f₂
have left : ∀ (g : StructuredArrow (op (G.obj U)) G.op),
  Eq (α :=
    unop (op X) ⟶
      (StructuredArrow.proj (op (G.obj U)) G.op ⋙
            obj (𝟭 (Functor.{v_1, w', u_1, w} Cᵒᵖ A)) (Y.obj (P := Presheaf.IsSheaf J))).obj
        g)
    (f₁ ≫
      NatTrans.app
        (Limits.Cone.π
          ((RightExtension.mk (obj G.op.ran (Y.obj (P := Presheaf.IsSheaf J)))
                (NatTrans.app G.op.ranCounit (Y.obj (P := Presheaf.IsSheaf J)))).coneAt
            (op (G.obj U))))
        g)
    (f₂ ≫
      NatTrans.app
        (Limits.Cone.π
          ((RightExtension.mk (obj G.op.ran (Y.obj (P := Presheaf.IsSheaf J)))
                (NatTrans.app G.op.ranCounit (Y.obj (P := Presheaf.IsSheaf J)))).coneAt
            (op (G.obj U))))
        g)
Tactic: refine (isPointwiseRightKanExtensionRanCounit G.op Y.1 (.op (G.obj U))).hom_ext (fun g ↦ ?_)
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.DenseSubsite.SheafEquiv (hypothesis)
Goal:
C : Type u_1
D : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} D
G : Functor.{v_1, v_2, u_1, u_2} C D
J : GrothendieckTopology.{v_1, u_1} C
K : GrothendieckTopology.{v_2, u_2} D
A : Type w
inst_2 : Category.{w', w} A
inst_3 : ∀ (X : Dᵒᵖ), Limits.HasLimitsOfShape (StructuredArrow X G.op) A
inst_4 : IsDenseSubsite J K G
Y : Sheaf.{v_1, w', u_1, w} J A
U : C
X : A
f : obj (obj yoneda ((obj (𝟭 (Functor.{v_1, w', u_1, w} Cᵒᵖ A)) (Y.obj (P := Presheaf.IsSheaf J))).obj (op U))) (op X)
⊢ ∃ (a :
    (fun (X : Type w') ↦ X)
      (obj
        (obj yoneda
          ((obj (G.op.ran ⋙ obj (whiskeringLeft Cᵒᵖ Dᵒᵖ A) G.op) (Y.obj (P := Presheaf.IsSheaf J))).obj (op U)))
        (op X))),
    (ConcreteCategory.hom (FC := TypeCat.Fun)
            (NatTrans.app
              (map yoneda (NatTrans.app (NatTrans.app G.op.ranCounit (Y.obj (P := Presheaf.IsSheaf J))) (op U)))
              (op X)) :
          (a :
              (fun (X : Type w') ↦ X)
                ((yoneda.obj (((G.op.ran ⋙ (whiskeringLeft Cᵒᵖ Dᵒᵖ A).obj G.op).obj Y.obj).obj (op U))).obj (op X))) →
            (fun (X : Type w') ↦ X) ((yoneda.obj (((𝟭 (Cᵒᵖ ⥤ A)).obj Y.obj).obj (op U))).obj (op X)))
        a =
      f
have this : ∀ (X Y Z : C) (f : X ⟶ Y) (g : G.obj Y ⟶ G.obj Z) (hf : (G.imageSieve g).arrows f), ∃ (l : X ⟶ Z), G.map l = G.map f ≫ g
Tactic: have (X Y Z) (f : X ⟶ Y) (g : G.obj Y ⟶ G.obj Z) (hf : G.imageSieve g f) : Exists _ := hf
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.DenseSubsite.SheafEquiv (goal)
Goal:
C : Type u_1
D : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} D
G : Functor.{v_1, v_2, u_1, u_2} C D
J : GrothendieckTopology.{v_1, u_1} C
K : GrothendieckTopology.{v_2, u_2} D
A : Type w
inst_2 : Category.{w', w} A
inst_3 : ∀ (X : Dᵒᵖ), Limits.HasLimitsOfShape (StructuredArrow X G.op) A
inst_4 : IsDenseSubsite J K G
Y : Sheaf.{v_1, w', u_1, w} J A
U : C
X : A
f : obj (obj yoneda ((obj (𝟭 (Functor.{v_1, w', u_1, w} Cᵒᵖ A)) (Y.obj (P := Presheaf.IsSheaf J))).obj (op U))) (op X)
this :
  ∀ (X Y Z : C) (f : X ⟶ Y) (g : G.obj Y ⟶ G.obj Z) (hf : (G.imageSieve g).arrows f),
    ∃ (l : X ⟶ Z), G.map l = G.map f ≫ g
⊢ ∃ (a :
    (fun (X : Type w') ↦ X)
      (obj
        (obj yoneda
          ((obj (G.op.ran ⋙ obj (whiskeringLeft Cᵒᵖ Dᵒᵖ A) G.op) (Y.obj (P := Presheaf.IsSheaf J))).obj (op U)))
        (op X))),
    (ConcreteCategory.hom (FC := TypeCat.Fun)
            (NatTrans.app
              (map yoneda (NatTrans.app (NatTrans.app G.op.ranCounit (Y.obj (P := Presheaf.IsSheaf J))) (op U)))
              (op X)) :
          (a :
              (fun (X : Type w') ↦ X)
                ((yoneda.obj (((G.op.ran ⋙ (whiskeringLeft Cᵒᵖ Dᵒᵖ A).obj G.op).obj Y.obj).obj (op U))).obj (op X))) →
            (fun (X : Type w') ↦ X) ((yoneda.obj (((𝟭 (Cᵒᵖ ⥤ A)).obj Y.obj).obj (op U))).obj (op X)))
        a =
      f
have right : ∀ (l : (X Y Z : C) → (f : X ⟶ Y) → (g : G.obj Y ⟶ G.obj Z) → (hf : (G.imageSieve g).arrows f) → X ⟶ Z)
  (hl :
    ∀ (X Y Z : C) (f : X ⟶ Y) (g : G.obj Y ⟶ G.obj Z) (hf : (G.imageSieve g).arrows f),
      G.map (l X Y Z f g hf) = G.map f ≫ g),
  ∃ (a :
    (fun (X : Type w') ↦ X)
      (obj
        (obj yoneda
          ((obj (G.op.ran ⋙ obj (whiskeringLeft Cᵒᵖ Dᵒᵖ A) G.op) (Y.obj (P := Presheaf.IsSheaf J))).obj (op U)))
        (op X))),
    (ConcreteCategory.hom (FC := TypeCat.Fun)
            (NatTrans.app
              (map yoneda (NatTrans.app (NatTrans.app G.op.ranCounit (Y.obj (P := Presheaf.IsSheaf J))) (op U)))
              (op X)) :
          (a :
              (fun (X : Type w') ↦ X)
                ((yoneda.obj (((G.op.ran ⋙ (whiskeringLeft Cᵒᵖ Dᵒᵖ A).obj G.op).obj Y.obj).obj (op U))).obj (op X))) →
            (fun (X : Type w') ↦ X) ((yoneda.obj (((𝟭 (Cᵒᵖ ⥤ A)).obj Y.obj).obj (op U))).obj (op X)))
        a =
      f
Tactic: choose l hl using this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.DenseSubsite.SheafEquiv (goal)
Goal:
C : Type u_1
D : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} D
G : Functor.{v_1, v_2, u_1, u_2} C D
J : GrothendieckTopology.{v_1, u_1} C
K : GrothendieckTopology.{v_2, u_2} D
A : Type w
inst_2 : Category.{w', w} A
inst_3 : ∀ (X : Dᵒᵖ), Limits.HasLimitsOfShape (StructuredArrow X G.op) A
inst_4 : IsDenseSubsite J K G
Y : Sheaf.{v_1, w', u_1, w} J A
U : C
X : A
f : obj (obj yoneda ((obj (𝟭 (Functor.{v_1, w', u_1, w} Cᵒᵖ A)) (Y.obj (P := Presheaf.IsSheaf J))).obj (op U))) (op X)
l : (X Y Z : C) → (f : X ⟶ Y) → (g : G.obj Y ⟶ G.obj Z) → (hf : (G.imageSieve g).arrows f) → X ⟶ Z
hl :
  ∀ (X Y Z : C) (f : X ⟶ Y) (g : G.obj Y ⟶ G.obj Z) (hf : (G.imageSieve g).arrows f),
    G.map (l X Y Z f g hf) = G.map f ≫ g
⊢ Limits.Cone (StructuredArrow.proj (op (G.obj U)) G.op ⋙ Y.obj (P := Presheaf.IsSheaf J))
have refine_1 : (g : StructuredArrow (op (G.obj U)) G.op) →
  obj (obj (const (StructuredArrow (op (G.obj U)) G.op)) X) g ⟶
    (StructuredArrow.proj (op (G.obj U)) G.op ⋙ Y.obj (P := Presheaf.IsSheaf J)).obj g
Tactic: refine ⟨X, ⟨fun g ↦ ?_, fun g₁ g₂ i ↦ ?_⟩⟩