awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.IsSheafFor (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
P : Functor.{v₁, w, u₁, w + 1} Cᵒᵖ (Type w)
X : C
R : Presieve.{v₁, u₁} X
x : FamilyOfElements P R
t : P.obj (op X)
ht : x.IsAmalgamation (R := R) t
Y₁ Y₂ Z : C
g₁ : Z ⟶ Y₁
g₂ : Z ⟶ Y₂
f₁ : Y₁ ⟶ X
f₂ : Y₂ ⟶ X
h₁ : R f₁
h₂ : R f₂
comm : g₁ ≫ f₁ = g₂ ≫ f₂
⊢ Eq (α := (fun (X : Type w) ↦ X) (P.obj (op Z)))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (P.map g₁.op) :
          (a : (fun (X : Type w) ↦ X) (P.obj (op Y₁))) → (fun (X : Type w) ↦ X) (P.obj (op Z)))
        (x f₁ h₁) :
      (fun (X : Type w) ↦ X) (P.obj (op Z)))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (P.map g₂.op) :
          (a : (fun (X : Type w) ↦ X) (P.obj (op Y₂))) → (fun (X : Type w) ↦ X) (P.obj (op Z)))
        (x f₂ h₂) :
      (fun (X : Type w) ↦ X) (P.obj (op Z)))
have rw : Eq (α := (fun (X : Type w) ↦ X) (P.obj (op Z)))
  ((ConcreteCategory.hom (FC := TypeCat.Fun) (P.map (g₂ ≫ f₂).op) :
        (a : (fun (X : Type w) ↦ X) (P.obj (op X))) → P.obj (op Z))
      t :
    P.obj (op Z))
  ((ConcreteCategory.hom (FC := TypeCat.Fun) (P.map g₂.op) :
        (a : (fun (X : Type w) ↦ X) (P.obj (op Y₂))) → (fun (X : Type w) ↦ X) (P.obj (op Z)))
      ((ConcreteCategory.hom (FC := TypeCat.Fun) (P.map f₂.op) :
            (a : (fun (X : Type w) ↦ X) (P.obj (op X))) → (fun (X : Type w) ↦ X) (P.obj (op Y₂)))
          t :
        (fun (X : Type w) ↦ X) (P.obj (op Y₂))) :
    (fun (X : Type w) ↦ X) (P.obj (op Z)))
Tactic: rw [← ht _ h₁, ← ht _ h₂, ← comp_apply, ← Functor.map_comp, ← op_comp, comm]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.IsSheafFor (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
P : Functor.{v₁, w, u₁, w + 1} Cᵒᵖ (Type w)
X Y : C
f : X ⟶ Y
x : FamilyOfElements P (singleton f)
y : P.obj (op Y)
⊢ x.IsAmalgamation (R := singleton f) y ↔
    (ConcreteCategory.hom (FC := TypeCat.Fun) (P.map f.op) :
          (a : (fun (X : Type w) ↦ X) (P.obj (op Y))) → (fun (X : Type w) ↦ X) (P.obj (op X)))
        y =
      x f ⋯
have refine : (ConcreteCategory.hom (FC := TypeCat.Fun) (P.map f.op) :
        (a : (fun (X : Type w) ↦ X) (P.obj (op Y))) → (fun (X : Type w) ↦ X) (P.obj (op X)))
      y =
    x f sorry →
  x.IsAmalgamation (R := singleton f) y
Tactic: refine ⟨fun H ↦ H _ _, ?_⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.IsSheafFor (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
P : Functor.{v₁, w, u₁, w + 1} Cᵒᵖ (Type w)
X : C
R : Presieve.{v₁, u₁} X
⊢ IsSeparatedFor P R ↔ IsSeparatedFor P (generate R).arrows
have mp : IsSeparatedFor P R → IsSeparatedFor P (generate R).arrows
Tactic: constructor
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.IsSheafFor (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
P : Functor.{v₁, w, u₁, w + 1} Cᵒᵖ (Type w)
X : C
R : Presieve.{v₁, u₁} X
mp : IsSeparatedFor P R → IsSeparatedFor P (generate R).arrows
⊢ IsSeparatedFor P R ↔ IsSeparatedFor P (generate R).arrows
have mpr : IsSeparatedFor P (generate R).arrows → IsSeparatedFor P R
Tactic: constructor
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.IsSheafFor (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
P : Functor.{v₁, w, u₁, w + 1} Cᵒᵖ (Type w)
X : C
R : Presieve.{v₁, u₁} X
h : IsSeparatedFor P R
x : FamilyOfElements P (generate R).arrows
t₁ t₂ : P.obj (op X)
ht₁ : x.IsAmalgamation (R := (generate R).arrows) t₁
ht₂ : x.IsAmalgamation (R := (generate R).arrows) t₂
⊢ t₁ = t₂
have apply : (FamilyOfElements.restrict (R₁ := R) (R₂ := (generate R).arrows) sorry x).IsAmalgamation (R := R) t₁
Tactic: apply h (x.restrict (le_generate R)) t₁ t₂ _ _
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.IsSheafFor (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
P : Functor.{v₁, w, u₁, w + 1} Cᵒᵖ (Type w)
X : C
R : Presieve.{v₁, u₁} X
h : IsSeparatedFor P R
x : FamilyOfElements P (generate R).arrows
t₁ t₂ : P.obj (op X)
ht₁ : x.IsAmalgamation (R := (generate R).arrows) t₁
ht₂ : x.IsAmalgamation (R := (generate R).arrows) t₂
apply : (FamilyOfElements.restrict (R₁ := R) (R₂ := (generate R).arrows) ⋯ x).IsAmalgamation (R := R) t₁
⊢ t₁ = t₂
have apply₁ : (FamilyOfElements.restrict (R₁ := R) (R₂ := (generate R).arrows) sorry x).IsAmalgamation (R := R) t₂
Tactic: apply h (x.restrict (le_generate R)) t₁ t₂ _ _
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.IsSheafFor (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
P : Functor.{v₁, w, u₁, w + 1} Cᵒᵖ (Type w)
X : C
R : Presieve.{v₁, u₁} X
h : IsSeparatedFor P (generate R).arrows
x : FamilyOfElements P R
t₁ t₂ : P.obj (op X)
ht₁ : x.IsAmalgamation (R := R) t₁
ht₂ : x.IsAmalgamation (R := R) t₂
⊢ t₁ = t₂
have mpr : x.sieveExtend (R := R).IsAmalgamation (R := (generate R).arrows) t₁
Tactic: apply h x.sieveExtend
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.IsSheafFor (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
P : Functor.{v₁, w, u₁, w + 1} Cᵒᵖ (Type w)
X : C
R : Presieve.{v₁, u₁} X
h : IsSeparatedFor P (generate R).arrows
x : FamilyOfElements P R
t₁ t₂ : P.obj (op X)
ht₁ : x.IsAmalgamation (R := R) t₁
ht₂ : x.IsAmalgamation (R := R) t₂
mpr : x.sieveExtend (R := R).IsAmalgamation (R := (generate R).arrows) t₁
⊢ t₁ = t₂
have mpr₁ : x.sieveExtend (R := R).IsAmalgamation (R := (generate R).arrows) t₂
Tactic: apply h x.sieveExtend
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.IsSheafFor (hypothesis)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
X : C
P : Functor.{v₁, w, u₁, w + 1} Cᵒᵖ (Type w)
x : FamilyOfElements (X := X) P ⊤
t₁ t₂ : P.obj (op X)
h₁ : x.IsAmalgamation (R := ⊤) t₁
h₂ : x.IsAmalgamation (R := ⊤) t₂
⊢ t₁ = t₂
have q₁ : (ConcreteCategory.hom (FC := TypeCat.Fun) (P.map (Quiver.Hom.op (𝟙 X))) :
      (a : (fun (X : Type w) ↦ X) (P.obj (op X))) → (fun (X : Type w) ↦ X) (P.obj (op X)))
    t₁ =
  x (𝟙 X) trivial
Tactic: have q₁ := h₁ (𝟙 X) (by tauto)
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.IsSheafFor (hypothesis)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
X : C
P : Functor.{v₁, w, u₁, w + 1} Cᵒᵖ (Type w)
x : FamilyOfElements (X := X) P ⊤
t₁ t₂ : P.obj (op X)
h₁ : x.IsAmalgamation (R := ⊤) t₁
h₂ : x.IsAmalgamation (R := ⊤) t₂
q₁ :
  (ConcreteCategory.hom (FC := TypeCat.Fun) (P.map (Quiver.Hom.op (𝟙 X))) :
        (a : (fun (X : Type w) ↦ X) (P.obj (op X))) → (fun (X : Type w) ↦ X) (P.obj (op X)))
      t₁ =
    x (𝟙 X) trivial
⊢ t₁ = t₂
have q₂ : (ConcreteCategory.hom (FC := TypeCat.Fun) (P.map (Quiver.Hom.op (𝟙 X))) :
      (a : (fun (X : Type w) ↦ X) (P.obj (op X))) → (fun (X : Type w) ↦ X) (P.obj (op X)))
    t₂ =
  x (𝟙 X) trivial
Tactic: have q₂ := h₂ (𝟙 X) (by tauto)
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.IsSheafFor (hypothesis)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
X : C
P : Functor.{v₁, w, u₁, w + 1} Cᵒᵖ (Type w)
x : FamilyOfElements (X := X) P ⊤
t₁ t₂ : P.obj (op X)
h₁ : x.IsAmalgamation (R := ⊤) t₁
h₂ : x.IsAmalgamation (R := ⊤) t₂
q₁ :
  (ConcreteCategory.hom (FC := TypeCat.Fun) (P.map (Quiver.Hom.op (𝟙 X))) :
        (a : (fun (X : Type w) ↦ X) (P.obj (op X))) → (fun (X : Type w) ↦ X) (P.obj (op X)))
      t₁ =
    x (𝟙 X) trivial
q₂ :
  (ConcreteCategory.hom (FC := TypeCat.Fun) (P.map (Quiver.Hom.op (𝟙 X))) :
        (a : (fun (X : Type w) ↦ X) (P.obj (op X))) → (fun (X : Type w) ↦ X) (P.obj (op X)))
      t₂ =
    x (𝟙 X) trivial
⊢ t₁ = t₂
have q₃ : t₁ = x (𝟙 X) trivial
Tactic: simp only [op_id, Functor.map_id, id_apply] at q₁ q₂
Cleared hypotheses: q₁, q₂
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.IsSheafFor (hypothesis)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
X : C
P : Functor.{v₁, w, u₁, w + 1} Cᵒᵖ (Type w)
x : FamilyOfElements (X := X) P ⊤
t₁ t₂ : P.obj (op X)
h₁ : x.IsAmalgamation (R := ⊤) t₁
h₂ : x.IsAmalgamation (R := ⊤) t₂
q₁ :
  (ConcreteCategory.hom (FC := TypeCat.Fun) (P.map (Quiver.Hom.op (𝟙 X))) :
        (a : (fun (X : Type w) ↦ X) (P.obj (op X))) → (fun (X : Type w) ↦ X) (P.obj (op X)))
      t₁ =
    x (𝟙 X) trivial
q₂ :
  (ConcreteCategory.hom (FC := TypeCat.Fun) (P.map (Quiver.Hom.op (𝟙 X))) :
        (a : (fun (X : Type w) ↦ X) (P.obj (op X))) → (fun (X : Type w) ↦ X) (P.obj (op X)))
      t₂ =
    x (𝟙 X) trivial
q₃ : t₁ = x (𝟙 X) trivial
⊢ t₁ = t₂
have q₄ : t₂ = x (𝟙 X) trivial
Tactic: simp only [op_id, Functor.map_id, id_apply] at q₁ q₂
Cleared hypotheses: q₁, q₂
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.IsSheafFor (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
X : C
P : Functor.{v₁, w, u₁, w + 1} Cᵒᵖ (Type w)
x : FamilyOfElements (X := X) P ⊤
t₁ t₂ : P.obj (op X)
h₁ : x.IsAmalgamation (R := ⊤) t₁
h₂ : x.IsAmalgamation (R := ⊤) t₂
q₁ : t₁ = x (𝟙 X) trivial
q₂ : t₂ = x (𝟙 X) trivial
⊢ t₁ = t₂
have rw : x (𝟙 X) trivial = t₂
Tactic: rw [q₁, q₂]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.IsSheafFor (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
X : C
P : Functor.{v₁, w, u₁, w + 1} Cᵒᵖ (Type w)
x : FamilyOfElements (X := X) P ⊤
t₁ t₂ : P.obj (op X)
h₁ : x.IsAmalgamation (R := ⊤) t₁
h₂ : x.IsAmalgamation (R := ⊤) t₂
q₁ : t₁ = x (𝟙 X) trivial
q₂ : t₂ = x (𝟙 X) trivial
rw : x (𝟙 X) trivial = t₂
⊢ t₁ = t₂
have rw₁ : x (𝟙 X) trivial = x (𝟙 X) trivial
Tactic: rw [q₁, q₂]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.IsSheafFor (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
P Q U : Functor.{v₁, w, u₁, w + 1} Cᵒᵖ (Type w)
X Y : C
S : Sieve.{v₁, u₁} X
R : Presieve.{v₁, u₁} X
inst_1 : LocallySmall.{w, v₁, u₁} C
F : Functor.{v₁, w, u₁, w + 1} Cᵒᵖ (Type w)
t : Subfunctor.toFunctor (F := Functor.obj shrinkYoneda.{w, v₁, u₁} X) (shrinkFunctor.{w, v₁, u₁} S) ⟶ F
⊢ FamilyOfElements.Compatible.{w, v₁, u₁} (P := F) (R := S.arrows) fun (Y : C) (f : Y ⟶ X) (hf : S.arrows f) ↦
    (ConcreteCategory.hom (FC := TypeCat.Fun) (t.app (op Y)) :
        (a : (fun (X : Type w) ↦ X) ((shrinkFunctor.{w, v₁, u₁} S).toFunctor.obj (op Y))) →
          (fun (X : Type w) ↦ X) (F.obj (op Y)))
      (Subtype.mk (p := fun (x : Functor.obj.{v₁, w, u₁, w + 1} (Functor.obj shrinkYoneda.{w, v₁, u₁} X) (op Y)) ↦
          Membership.mem (γ := Set (Functor.obj.{v₁, w, u₁, w + 1} (Functor.obj shrinkYoneda.{w, v₁, u₁} X) (op Y)))
            (Subfunctor.obj (shrinkFunctor.{w, v₁, u₁} S) (op Y)) x)
          ((Equiv.symm shrinkYonedaObjObjEquiv : (a : unop (op Y) ⟶ X) → (shrinkYoneda.{w, v₁, u₁}.obj X).obj (op Y)) f)
          ⋯ :
        { x : Functor.obj.{v₁, w, u₁, w + 1} (Functor.obj shrinkYoneda.{w, v₁, u₁} X) (op Y) //
          Membership.mem (γ := Set (Functor.obj.{v₁, w, u₁, w + 1} (Functor.obj shrinkYoneda.{w, v₁, u₁} X) (op Y)))
            (Subfunctor.obj (shrinkFunctor.{w, v₁, u₁} S) (op Y)) x })
have rw : FamilyOfElements.SieveCompatible.{w, v₁, u₁} (C := C) (P := F) (X := X) (S := S)
  fun (Y : C) (f : Y ⟶ X) (hf : S.arrows f) ↦
  (ConcreteCategory.hom (FC := TypeCat.Fun) (t.app (op Y)) :
      (a : (fun (X : Type w) ↦ X) ((shrinkFunctor.{w, v₁, u₁} S).toFunctor.obj (op Y))) →
        (fun (X : Type w) ↦ X) (F.obj (op Y)))
    (Subtype.mk (p := fun (x : Functor.obj.{v₁, w, u₁, w + 1} (Functor.obj shrinkYoneda.{w, v₁, u₁} X) (op Y)) ↦
        Membership.mem (γ := Set (Functor.obj.{v₁, w, u₁, w + 1} (Functor.obj shrinkYoneda.{w, v₁, u₁} X) (op Y)))
          (Subfunctor.obj (shrinkFunctor.{w, v₁, u₁} S) (op Y)) x)
        ((Equiv.symm shrinkYonedaObjObjEquiv : (a : unop (op Y) ⟶ X) → (shrinkYoneda.{w, v₁, u₁}.obj X).obj (op Y)) f)
        sorry :
      { x : Functor.obj.{v₁, w, u₁, w + 1} (Functor.obj shrinkYoneda.{w, v₁, u₁} X) (op Y) //
        Membership.mem (γ := Set (Functor.obj.{v₁, w, u₁, w + 1} (Functor.obj shrinkYoneda.{w, v₁, u₁} X) (op Y)))
          (Subfunctor.obj (shrinkFunctor.{w, v₁, u₁} S) (op Y)) x })
Tactic: rw [Presieve.compatible_iff_sieveCompatible]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.IsSheafFor (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
P Q U : Functor.{v₁, w, u₁, w + 1} Cᵒᵖ (Type w)
X Y_1 : C
S : Sieve.{v₁, u₁} X
R : Presieve.{v₁, u₁} X
inst_1 : LocallySmall.{w, v₁, u₁} C
F : Functor.{v₁, w, u₁, w + 1} Cᵒᵖ (Type w)
t : { x : FamilyOfElements F S.arrows // x.Compatible (R := S.arrows) }
Y Z : Cᵒᵖ
g : Y ⟶ Z
f : Functor.obj.{v₁, w, u₁, w + 1} (Functor.obj shrinkYoneda.{w, v₁, u₁} X) Y
hf :
  Membership.mem (γ := Set (Functor.obj.{v₁, w, u₁, w + 1} (Functor.obj shrinkYoneda.{w, v₁, u₁} X) Y))
    (Subfunctor.obj (shrinkFunctor.{w, v₁, u₁} S) Y) f
⊢ TypeCat.Fun.toFun
      (ConcreteCategory.hom (FC := TypeCat.Fun)
        ((Subfunctor.toFunctor (F := Functor.obj shrinkYoneda.{w, v₁, u₁} X) (shrinkFunctor.{w, v₁, u₁} S)).map g ≫
          ↾fun
              (f :
                (Subfunctor.toFunctor (F := Functor.obj shrinkYoneda.{w, v₁, u₁} X) (shrinkFunctor.{w, v₁, u₁} S)).obj
                  Z) ↦
            (↑t : FamilyOfElements F S.arrows)
              ((shrinkYonedaObjObjEquiv : (a : (shrinkYoneda.{w, v₁, u₁}.obj X).obj Z) → unop Z ⟶ X)
                (↑f : (shrinkYoneda.{w, v₁, u₁}.obj X).obj Z))
              ⋯))
      (Subtype.mk (p := fun (x : Functor.obj.{v₁, w, u₁, w + 1} (Functor.obj shrinkYoneda.{w, v₁, u₁} X) Y) ↦
        Membership.mem (γ := Set (Functor.obj.{v₁, w, u₁, w + 1} (Functor.obj shrinkYoneda.{w, v₁, u₁} X) Y))
          (Subfunctor.obj (shrinkFunctor.{w, v₁, u₁} S) Y) x)
        f hf) =
    TypeCat.Fun.toFun
      (ConcreteCategory.hom (FC := TypeCat.Fun)
        ((↾fun
              (f :
                (Subfunctor.toFunctor (F := Functor.obj shrinkYoneda.{w, v₁, u₁} X) (shrinkFunctor.{w, v₁, u₁} S)).obj
                  Y) ↦
            (↑t : FamilyOfElements F S.arrows)
              ((shrinkYonedaObjObjEquiv : (a : (shrinkYoneda.{w, v₁, u₁}.obj X).obj Y) → unop Y ⟶ X)
                (↑f : (shrinkYoneda.{w, v₁, u₁}.obj X).obj Y))
              ⋯) ≫
          F.map g))
      (Subtype.mk (p := fun (x : Functor.obj.{v₁, w, u₁, w + 1} (Functor.obj shrinkYoneda.{w, v₁, u₁} X) Y) ↦
        Membership.mem (γ := Set (Functor.obj.{v₁, w, u₁, w + 1} (Functor.obj shrinkYoneda.{w, v₁, u₁} X) Y))
          (Subfunctor.obj (shrinkFunctor.{w, v₁, u₁} S) Y) x)
        f hf)
have h : (↑t : FamilyOfElements F S.arrows)
    ((shrinkYonedaObjObjEquiv : (a : (shrinkYoneda.{w, v₁, u₁}.obj X).obj Z) → unop Z ⟶ X)
      ((ConcreteCategory.hom (FC := TypeCat.Fun) (Functor.map (Functor.obj shrinkYoneda.{w, v₁, u₁} X) g) :
          (a : (shrinkYoneda.{w, v₁, u₁}.obj X).obj Y) → (shrinkYoneda.{w, v₁, u₁}.obj X).obj Z)
        f))
    sorry =
  (ConcreteCategory.hom (FC := TypeCat.Fun) (F.map g) : (a : F.obj Y) → F.obj Z)
    ((↑t : FamilyOfElements F S.arrows)
      ((shrinkYonedaObjObjEquiv : (a : (shrinkYoneda.{w, v₁, u₁}.obj X).obj Y) → unop Y ⟶ X) f) sorry)
Tactic: dsimp
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.IsSheafFor (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
P Q U : Functor.{v₁, w, u₁, w + 1} Cᵒᵖ (Type w)
X Y_1 : C
S : Sieve.{v₁, u₁} X
R : Presieve.{v₁, u₁} X
inst_1 : LocallySmall.{w, v₁, u₁} C
F : Functor.{v₁, w, u₁, w + 1} Cᵒᵖ (Type w)
t : { x : FamilyOfElements F S.arrows // x.Compatible (R := S.arrows) }
Y Z : Cᵒᵖ
g : Y ⟶ Z
f : Functor.obj.{v₁, w, u₁, w + 1} (Functor.obj shrinkYoneda.{w, v₁, u₁} X) Y
hf :
  Membership.mem (γ := Set (Functor.obj.{v₁, w, u₁, w + 1} (Functor.obj shrinkYoneda.{w, v₁, u₁} X) Y))
    (Subfunctor.obj (shrinkFunctor.{w, v₁, u₁} S) Y) f
⊢ (↑t : FamilyOfElements F S.arrows)
      ((shrinkYonedaObjObjEquiv : (a : (shrinkYoneda.{w, v₁, u₁}.obj X).obj Z) → unop Z ⟶ X)
        ((ConcreteCategory.hom (FC := TypeCat.Fun) (Functor.map (Functor.obj shrinkYoneda.{w, v₁, u₁} X) g) :
            (a : (shrinkYoneda.{w, v₁, u₁}.obj X).obj Y) → (shrinkYoneda.{w, v₁, u₁}.obj X).obj Z)
          f))
      ⋯ =
    (ConcreteCategory.hom (FC := TypeCat.Fun) (F.map g) : (a : F.obj Y) → F.obj Z)
      ((↑t : FamilyOfElements F S.arrows)
        ((shrinkYonedaObjObjEquiv : (a : (shrinkYoneda.{w, v₁, u₁}.obj X).obj Y) → unop Y ⟶ X) f) ⋯)
have h : F.obj Z = F.obj (op (unop Z)) →
  Eq.{v₁ + 1} (α := unop Z ⟶ X)
    ((shrinkYonedaObjObjEquiv : (a : (shrinkYoneda.{w, v₁, u₁}.obj X).obj Z) → unop Z ⟶ X)
        ((ConcreteCategory.hom (FC := TypeCat.Fun) (Functor.map (Functor.obj shrinkYoneda.{w, v₁, u₁} X) g) :
            (a : (shrinkYoneda.{w, v₁, u₁}.obj X).obj Y) → (shrinkYoneda.{w, v₁, u₁}.obj X).obj Z)
          f) :
      unop Z ⟶ X)
    (unop g ≫ (shrinkYonedaObjObjEquiv : (a : (shrinkYoneda.{w, v₁, u₁}.obj X).obj Y) → unop Y ⟶ X) f : unop Z ⟶ X)
Tactic: convert! t.2.to_sieveCompatible _ _ _
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.IsSheafFor (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
P Q U : Functor.{v₁, w, u₁, w + 1} Cᵒᵖ (Type w)
X Y_1 : C
S : Sieve.{v₁, u₁} X
R : Presieve.{v₁, u₁} X
inst_1 : LocallySmall.{w, v₁, u₁} C
F : Functor.{v₁, w, u₁, w + 1} Cᵒᵖ (Type w)
t : { x : FamilyOfElements F S.arrows // x.Compatible (R := S.arrows) }
Y Z : Cᵒᵖ
g : Y ⟶ Z
f : Functor.obj.{v₁, w, u₁, w + 1} (Functor.obj shrinkYoneda.{w, v₁, u₁} X) Y
hf :
  Membership.mem (γ := Set (Functor.obj.{v₁, w, u₁, w + 1} (Functor.obj shrinkYoneda.{w, v₁, u₁} X) Y))
    (Subfunctor.obj (shrinkFunctor.{w, v₁, u₁} S) Y) f
e_1 : F.obj Z = F.obj (op (unop Z))
⊢ Eq.{v₁ + 1} (α := unop Z ⟶ X)
    ((shrinkYonedaObjObjEquiv : (a : (shrinkYoneda.{w, v₁, u₁}.obj X).obj Z) → unop Z ⟶ X)
        ((ConcreteCategory.hom (FC := TypeCat.Fun) (Functor.map (Functor.obj shrinkYoneda.{w, v₁, u₁} X) g) :
            (a : (shrinkYoneda.{w, v₁, u₁}.obj X).obj Y) → (shrinkYoneda.{w, v₁, u₁}.obj X).obj Z)
          f) :
      unop Z ⟶ X)
    (unop g ≫ (shrinkYonedaObjObjEquiv : (a : (shrinkYoneda.{w, v₁, u₁}.obj X).obj Y) → unop Y ⟶ X) f : unop Z ⟶ X)
have h : Eq (α := unop Z ⟶ X) (g.unop ≫ (shrinkYonedaObjObjEquiv : (a : (shrinkYoneda.{w, v₁, u₁}.obj X).obj Y) → unop Y ⟶ X) f)
  (unop g ≫ (shrinkYonedaObjObjEquiv : (a : (shrinkYoneda.{w, v₁, u₁}.obj X).obj Y) → unop Y ⟶ X) f)
Tactic: simp only [Opposite.op_unop, shrinkYonedaObjObjEquiv_obj_map]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.IsSheafFor (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
P Q U : Functor.{v₁, w, u₁, w + 1} Cᵒᵖ (Type w)
X Y : C
S : Sieve.{v₁, u₁} X
R : Presieve.{v₁, u₁} X
inst_1 : LocallySmall.{w, v₁, u₁} C
F : Functor.{v₁, w, u₁, w + 1} Cᵒᵖ (Type w)
x : { x : FamilyOfElements F S.arrows // x.Compatible (R := S.arrows) }
Y_1 : C
f : Y_1 ⟶ X
hf : S.arrows f
⊢ (↑x : FamilyOfElements F S.arrows)
      ((shrinkYonedaObjObjEquiv : (a : (shrinkYoneda.{w, v₁, u₁}.obj X).obj (op Y_1)) → Y_1 ⟶ X)
        ((Equiv.symm shrinkYonedaObjObjEquiv : (a : Y_1 ⟶ X) → (shrinkYoneda.{w, v₁, u₁}.obj X).obj (op Y_1)) f))
      ⋯ =
    (↑x : FamilyOfElements F S.arrows) f hf
have a : (↑x : FamilyOfElements F S.arrows) f sorry = (↑x : FamilyOfElements F S.arrows) f hf
Tactic: rw! [Equiv.apply_symm_apply]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.IsSheafFor (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
X : C
S : Sieve.{v₁, u₁} X
inst_1 : LocallySmall.{w, v₁, u₁} C
F : Functor.{v₁, w, u₁, w + 1} Cᵒᵖ (Type w)
f : Subfunctor.toFunctor (F := Functor.obj shrinkYoneda.{w, v₁, u₁} X) (shrinkFunctor.{w, v₁, u₁} S) ⟶ F
g : Functor.obj shrinkYoneda.{w, v₁, u₁} X ⟶ F
⊢ Subfunctor.ι (shrinkFunctor.{w, v₁, u₁} S) ≫ g = f ↔
    FamilyOfElements.IsAmalgamation.{w, v₁, u₁} (P := F) (R := S.arrows)
      (↑((shrinkFunctorHomEquiv :
              (a : (shrinkFunctor.{w, v₁, u₁} S).toFunctor ⟶ F) → { x : FamilyOfElements F S.arrows // x.Compatible })
            f) :
        FamilyOfElements F S.arrows)
      ((shrinkYonedaEquiv : (a : shrinkYoneda.{w, v₁, u₁}.obj X ⟶ F) → F.obj (op X)) g)
have dsimp : Subfunctor.ι (shrinkFunctor.{w, v₁, u₁} S) ≫ g = f ↔
  ∀ ⦃Y : C⦄ (f_1 : Y ⟶ X) (h : S.arrows f_1),
    Eq (α := F.obj (op Y))
      ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map f_1.op) : (a : F.obj (op X)) → F.obj (op Y))
          ((shrinkYonedaEquiv : (a : shrinkYoneda.{w, v₁, u₁}.obj X ⟶ F) → F.obj (op X)) g : F.obj (op X)) :
        F.obj (op Y))
      ((ConcreteCategory.hom (FC := TypeCat.Fun) (f.app (op Y)) :
            (a :
                { x : (shrinkYoneda.{w, v₁, u₁}.obj X).obj (op Y) //
                  S.arrows
                    ((shrinkYonedaObjObjEquiv : (a : (shrinkYoneda.{w, v₁, u₁}.obj X).obj (op Y)) → Y ⟶ X) x) }) →
              F.obj (op Y))
          (Subtype.mk (p := fun (x : Functor.obj.{v₁, w, u₁, w + 1} (Functor.obj shrinkYoneda.{w, v₁, u₁} X) (op Y)) ↦
              S.arrows (Y := Y)
                ((shrinkYonedaObjObjEquiv : (a : (shrinkYoneda.{w, v₁, u₁}.obj X).obj (op Y)) → Y ⟶ X) x))
              ((Equiv.symm shrinkYonedaObjObjEquiv : (a : Y ⟶ X) → (shrinkYoneda.{w, v₁, u₁}.obj X).obj (op Y)) f_1) sorry :
            { x : Functor.obj.{v₁, w, u₁, w + 1} (Functor.obj shrinkYoneda.{w, v₁, u₁} X) (op Y) //
              S.arrows (Y := Y)
                ((shrinkYonedaObjObjEquiv : (a : (shrinkYoneda.{w, v₁, u₁}.obj X).obj (op Y)) → Y ⟶ X) x) }) :
        F.obj (op Y))
Tactic: dsimp [Presieve.FamilyOfElements.IsAmalgamation]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.IsSheafFor (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
X : C
S : Sieve.{v₁, u₁} X
inst_1 : LocallySmall.{w, v₁, u₁} C
F : Functor.{v₁, w, u₁, w + 1} Cᵒᵖ (Type w)
f : Subfunctor.toFunctor (F := Functor.obj shrinkYoneda.{w, v₁, u₁} X) (shrinkFunctor.{w, v₁, u₁} S) ⟶ F
g : Functor.obj shrinkYoneda.{w, v₁, u₁} X ⟶ F
⊢ Subfunctor.ι (shrinkFunctor.{w, v₁, u₁} S) ≫ g = f ↔
    ∀ ⦃Y : C⦄ (f_1 : Y ⟶ X) (h : S.arrows f_1),
      Eq (α := F.obj (op Y))
        ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map f_1.op) : (a : F.obj (op X)) → F.obj (op Y))
            ((shrinkYonedaEquiv : (a : shrinkYoneda.{w, v₁, u₁}.obj X ⟶ F) → F.obj (op X)) g : F.obj (op X)) :
          F.obj (op Y))
        ((ConcreteCategory.hom (FC := TypeCat.Fun) (f.app (op Y)) :
              (a :
                  { x : (shrinkYoneda.{w, v₁, u₁}.obj X).obj (op Y) //
                    S.arrows
                      ((shrinkYonedaObjObjEquiv : (a : (shrinkYoneda.{w, v₁, u₁}.obj X).obj (op Y)) → Y ⟶ X) x) }) →
                F.obj (op Y))
            (Subtype.mk (p := fun (x : Functor.obj.{v₁, w, u₁, w + 1} (Functor.obj shrinkYoneda.{w, v₁, u₁} X) (op Y)) ↦
                S.arrows (Y := Y)
                  ((shrinkYonedaObjObjEquiv : (a : (shrinkYoneda.{w, v₁, u₁}.obj X).obj (op Y)) → Y ⟶ X) x))
                ((Equiv.symm shrinkYonedaObjObjEquiv : (a : Y ⟶ X) → (shrinkYoneda.{w, v₁, u₁}.obj X).obj (op Y)) f_1)
                ⋯ :
              { x : Functor.obj.{v₁, w, u₁, w + 1} (Functor.obj shrinkYoneda.{w, v₁, u₁} X) (op Y) //
                S.arrows (Y := Y)
                  ((shrinkYonedaObjObjEquiv : (a : (shrinkYoneda.{w, v₁, u₁}.obj X).obj (op Y)) → Y ⟶ X) x) }) :
          F.obj (op Y))
have refine_1 : Subfunctor.ι (shrinkFunctor.{w, v₁, u₁} S) ≫ g = f →
  ∀ ⦃Y : C⦄ (f_1 : Y ⟶ X) (h : S.arrows f_1),
    Eq (α := F.obj (op Y))
      ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map f_1.op) : (a : F.obj (op X)) → F.obj (op Y))
          ((shrinkYonedaEquiv : (a : shrinkYoneda.{w, v₁, u₁}.obj X ⟶ F) → F.obj (op X)) g : F.obj (op X)) :
        F.obj (op Y))
      ((ConcreteCategory.hom (FC := TypeCat.Fun) (f.app (op Y)) :
            (a :
                { x : (shrinkYoneda.{w, v₁, u₁}.obj X).obj (op Y) //
                  S.arrows
                    ((shrinkYonedaObjObjEquiv : (a : (shrinkYoneda.{w, v₁, u₁}.obj X).obj (op Y)) → Y ⟶ X) x) }) →
              F.obj (op Y))
          (Subtype.mk (p := fun (x : Functor.obj.{v₁, w, u₁, w + 1} (Functor.obj shrinkYoneda.{w, v₁, u₁} X) (op Y)) ↦
              S.arrows (Y := Y)
                ((shrinkYonedaObjObjEquiv : (a : (shrinkYoneda.{w, v₁, u₁}.obj X).obj (op Y)) → Y ⟶ X) x))
              ((Equiv.symm shrinkYonedaObjObjEquiv : (a : Y ⟶ X) → (shrinkYoneda.{w, v₁, u₁}.obj X).obj (op Y)) f_1) sorry :
            { x : Functor.obj.{v₁, w, u₁, w + 1} (Functor.obj shrinkYoneda.{w, v₁, u₁} X) (op Y) //
              S.arrows (Y := Y)
                ((shrinkYonedaObjObjEquiv : (a : (shrinkYoneda.{w, v₁, u₁}.obj X).obj (op Y)) → Y ⟶ X) x) }) :
        F.obj (op Y))
Tactic: refine ⟨?_, fun h ↦ ?_⟩