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 Q U : Functor.{v₁, w, u₁, w + 1} Cᵒᵖ (Type w)
X_1 Y_1 : C
S : Sieve.{v₁, u₁} X_1
R : Presieve.{v₁, u₁} X_1
X Y : C
f : X ⟶ Y
x : FamilyOfElements P (singleton f)
⊢ (fun (x : P.obj (op X)) (Z : C) (g : Z ⟶ Y) (hg : singleton f g) ↦
        ((ConcreteCategory.hom (FC := TypeCat.Fun) (P.map (Quiver.Hom.op (eqToHom ⋯))) :
              (a : (fun (X : Type w) ↦ X) (P.obj (op X))) → (fun (X : Type w) ↦ X) (P.obj (op Z)))
            x :
          (fun (X : Type w) ↦ X) (P.obj (op Z))))
      ((fun (x : FamilyOfElements P (singleton f)) ↦ x f ⋯) x) =
    x
have H : ∀ ⦃Y_2 : C⦄,
  (fun (x : P.obj (op X)) (Z : C) (g : Z ⟶ Y) (hg : singleton f g) ↦
        ((ConcreteCategory.hom (FC := TypeCat.Fun) (P.map (Quiver.Hom.op (eqToHom sorry))) :
              (a : (fun (X : Type w) ↦ X) (P.obj (op X))) → (fun (X : Type w) ↦ X) (P.obj (op Z)))
            x :
          (fun (X : Type w) ↦ X) (P.obj (op Z))))
      ((fun (x : FamilyOfElements P (singleton f)) ↦ x f sorry) x) X f sorry =
    x f sorry
Tactic: ext _ _ ⟨rfl⟩
-- 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
inst_1 : R.HasPairwisePullbacks
⊢ x.Compatible (R := R) ↔ x.PullbackCompatible (R := R)
have mp : x.Compatible (R := R) → x.PullbackCompatible (R := 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
x : FamilyOfElements P R
inst_1 : R.HasPairwisePullbacks
mp : x.Compatible (R := R) → x.PullbackCompatible (R := R)
⊢ x.Compatible (R := R) ↔ x.PullbackCompatible (R := R)
have mpr : x.PullbackCompatible (R := R) → x.Compatible (R := 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
x : FamilyOfElements P R
inst_1 : R.HasPairwisePullbacks
t : x.Compatible (R := R)
Y₁ Y₂ : C
f₁ : Y₁ ⟶ X
f₂ : Y₂ ⟶ X
hf₁ : R f₁
hf₂ : R f₂
⊢ Eq (α := (fun (X : Type w) ↦ X) (P.obj (op (Limits.pullback f₁ f₂))))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (P.map (Quiver.Hom.op (pullback.fst f₁ f₂))) :
          (a : (fun (X : Type w) ↦ X) (P.obj (op Y₁))) → (fun (X : Type w) ↦ X) (P.obj (op (Limits.pullback f₁ f₂))))
        (x f₁ hf₁) :
      (fun (X : Type w) ↦ X) (P.obj (op (Limits.pullback f₁ f₂))))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (P.map (Quiver.Hom.op (pullback.snd f₁ f₂))) :
          (a : (fun (X : Type w) ↦ X) (P.obj (op Y₂))) → (fun (X : Type w) ↦ X) (P.obj (op (Limits.pullback f₁ f₂))))
        (x f₂ hf₂) :
      (fun (X : Type w) ↦ X) (P.obj (op (Limits.pullback f₁ f₂))))
have mp : Eq (α := Limits.pullback f₁ f₂ ⟶ X) (pullback.fst f₁ f₂ ≫ f₁) (pullback.snd f₁ f₂ ≫ f₂)
Tactic: apply t
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.IsSheafFor (hypothesis)
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
inst_1 : R.HasPairwisePullbacks
t : x.Compatible (R := R)
Y₁ Y₂ : C
f₁ : Y₁ ⟶ X
f₂ : Y₂ ⟶ X
hf₁ : R f₁
hf₂ : R f₂
⊢ Eq (α := Limits.pullback f₁ f₂ ⟶ X) (pullback.fst f₁ f₂ ≫ f₁) (pullback.snd f₁ f₂ ≫ f₂)
have this : HasPullback f₁ f₂
Tactic: haveI := HasPairwisePullbacks.has_pullbacks hf₁ hf₂
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.IsSheafFor (hypothesis)
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
inst_1 : R.HasPairwisePullbacks
t : x.PullbackCompatible (R := R)
Y₁ Y₂ Z : C
g₁ : Z ⟶ Y₁
g₂ : Z ⟶ Y₂
f₁ : Y₁ ⟶ X
f₂ : Y₂ ⟶ X
hf₁ : R f₁
hf₂ : 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 hf₁) :
      (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 hf₂) :
      (fun (X : Type w) ↦ X) (P.obj (op Z)))
have this : HasPullback f₁ f₂
Tactic: haveI := HasPairwisePullbacks.has_pullbacks hf₁ hf₂
-- 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
inst_1 : R.HasPairwisePullbacks
t : x.PullbackCompatible (R := R)
Y₁ Y₂ Z : C
g₁ : Z ⟶ Y₁
g₂ : Z ⟶ Y₂
f₁ : Y₁ ⟶ X
f₂ : Y₂ ⟶ X
hf₁ : R f₁
hf₂ : R f₂
comm : g₁ ≫ f₁ = g₂ ≫ f₂
this : HasPullback f₁ 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 hf₁) :
      (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 hf₂) :
      (fun (X : Type w) ↦ X) (P.obj (op Z)))
have mpr : Eq (α := (fun (X : Type w) ↦ X) (P.obj (op Z)))
  ((ConcreteCategory.hom (FC := TypeCat.Fun)
          (P.map
            (CategoryStruct.comp (Y := Limits.pullback f₁ f₂) (pullback.lift g₁ g₂ comm) (pullback.fst f₁ f₂)).op) :
        (a : (fun (X : Type w) ↦ X) (P.obj (op Y₁))) → (fun (X : Type w) ↦ X) (P.obj (op Z)))
      (x f₁ hf₁) :
    (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 hf₂) :
    (fun (X : Type w) ↦ X) (P.obj (op Z)))
Tactic: rw [← pullback.lift_fst _ _ comm, op_comp, Functor.map_comp, comp_apply,
      t hf₁ hf₂, ← comp_apply, ← Functor.map_comp, ← op_comp, pullback.lift_snd]
-- 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
inst_1 : R.HasPairwisePullbacks
t : x.PullbackCompatible (R := R)
Y₁ Y₂ Z : C
g₁ : Z ⟶ Y₁
g₂ : Z ⟶ Y₂
f₁ : Y₁ ⟶ X
f₂ : Y₂ ⟶ X
hf₁ : R f₁
hf₂ : R f₂
comm : g₁ ≫ f₁ = g₂ ≫ f₂
this : HasPullback f₁ f₂
mpr :
  Eq (α := (fun (X : Type w) ↦ X) (P.obj (op Z)))
    ((ConcreteCategory.hom (FC := TypeCat.Fun)
            (P.map
              (CategoryStruct.comp (Y := Limits.pullback f₁ f₂) (pullback.lift g₁ g₂ comm) (pullback.fst f₁ f₂)).op) :
          (a : (fun (X : Type w) ↦ X) (P.obj (op Y₁))) → (fun (X : Type w) ↦ X) (P.obj (op Z)))
        (x f₁ hf₁) :
      (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 hf₂) :
      (fun (X : Type w) ↦ X) (P.obj (op Z)))
⊢ 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 hf₁) :
      (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 hf₂) :
      (fun (X : Type w) ↦ X) (P.obj (op Z)))
have mpr₁ : Eq (α := (fun (X : Type w) ↦ X) (P.obj (op Z)))
  ((ConcreteCategory.hom (FC := TypeCat.Fun)
          (P.map
            (CategoryStruct.comp (Y := op (Limits.pullback f₁ f₂)) (Quiver.Hom.op (pullback.fst f₁ f₂))
              (Quiver.Hom.op (pullback.lift g₁ g₂ comm)))) :
        (a : (fun (X : Type w) ↦ X) (P.obj (op Y₁))) → (fun (X : Type w) ↦ X) (P.obj (op Z)))
      (x f₁ hf₁) :
    (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 hf₂) :
    (fun (X : Type w) ↦ X) (P.obj (op Z)))
Tactic: rw [← pullback.lift_fst _ _ comm, op_comp, Functor.map_comp, comp_apply,
      t hf₁ hf₂, ← comp_apply, ← Functor.map_comp, ← op_comp, pullback.lift_snd]
-- 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
inst_1 : R.HasPairwisePullbacks
t : x.PullbackCompatible (R := R)
Y₁ Y₂ Z : C
g₁ : Z ⟶ Y₁
g₂ : Z ⟶ Y₂
f₁ : Y₁ ⟶ X
f₂ : Y₂ ⟶ X
hf₁ : R f₁
hf₂ : R f₂
comm : g₁ ≫ f₁ = g₂ ≫ f₂
this : HasPullback f₁ f₂
mpr :
  Eq (α := (fun (X : Type w) ↦ X) (P.obj (op Z)))
    ((ConcreteCategory.hom (FC := TypeCat.Fun)
            (P.map
              (CategoryStruct.comp (Y := Limits.pullback f₁ f₂) (pullback.lift g₁ g₂ comm) (pullback.fst f₁ f₂)).op) :
          (a : (fun (X : Type w) ↦ X) (P.obj (op Y₁))) → (fun (X : Type w) ↦ X) (P.obj (op Z)))
        (x f₁ hf₁) :
      (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 hf₂) :
      (fun (X : Type w) ↦ X) (P.obj (op Z)))
mpr₁ :
  Eq (α := (fun (X : Type w) ↦ X) (P.obj (op Z)))
    ((ConcreteCategory.hom (FC := TypeCat.Fun)
            (P.map
              (CategoryStruct.comp (Y := op (Limits.pullback f₁ f₂)) (Quiver.Hom.op (pullback.fst f₁ f₂))
                (Quiver.Hom.op (pullback.lift g₁ g₂ comm)))) :
          (a : (fun (X : Type w) ↦ X) (P.obj (op Y₁))) → (fun (X : Type w) ↦ X) (P.obj (op Z)))
        (x f₁ hf₁) :
      (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 hf₂) :
      (fun (X : Type w) ↦ X) (P.obj (op Z)))
⊢ 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 hf₁) :
      (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 hf₂) :
      (fun (X : Type w) ↦ X) (P.obj (op Z)))
have mpr₂ : Eq (α := (fun (X : Type w) ↦ X) (P.obj (op Z)))
  ((ConcreteCategory.hom (FC := TypeCat.Fun)
          (CategoryStruct.comp (Y := P.obj (op (Limits.pullback f₁ f₂))) (P.map (Quiver.Hom.op (pullback.fst f₁ f₂)))
            (P.map (Quiver.Hom.op (pullback.lift g₁ g₂ comm)))) :
        (a : (fun (X : Type w) ↦ X) (P.obj (op Y₁))) → (fun (X : Type w) ↦ X) (P.obj (op Z)))
      (x f₁ hf₁) :
    (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 hf₂) :
    (fun (X : Type w) ↦ X) (P.obj (op Z)))
Tactic: rw [← pullback.lift_fst _ _ comm, op_comp, Functor.map_comp, comp_apply,
      t hf₁ hf₂, ← comp_apply, ← Functor.map_comp, ← op_comp, pullback.lift_snd]
-- 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
hx : x.Compatible (R := R)
Y₁ Y₂ Z : C
g₁ : Z ⟶ Y₁
g₂ : Z ⟶ Y₂
f₁ : Y₁ ⟶ X
f₂ : Y₂ ⟶ X
h₁ : (generate R).arrows f₁
h₂ : (generate R).arrows 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.sieveExtend (R := R) 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.sieveExtend (R := R) f₂ h₂) :
      (fun (X : Type w) ↦ X) (P.obj (op Z)))
have simp : Eq (α := P.obj (op Z))
  ((ConcreteCategory.hom (FC := TypeCat.Fun)
          (P.map
            (g₁ ≫
                sorry.choose (p :=
                  fun (h : Y₁ ⟶ Exists.choose (p := fun (Y : C) ↦ ∃ (h : Y₁ ⟶ Y) (g : Y ⟶ X), R g ∧ h ≫ g = f₁) h₁) ↦
                  ∃ (g : Exists.choose (p := fun (Y : C) ↦ ∃ (h : Y₁ ⟶ Y) (g : Y ⟶ X), R g ∧ h ≫ g = f₁) h₁ ⟶ X),
                    R g ∧ h ≫ g = f₁)).op) :
        (a : P.obj (op (Exists.choose h₁))) → P.obj (op Z))
      (x
        (sorry.choose (p :=
          fun (g : Exists.choose (p := fun (Y : C) ↦ ∃ (h : Y₁ ⟶ Y) (g : Y ⟶ X), R g ∧ h ≫ g = f₁) h₁ ⟶ X) ↦
          R g ∧
            sorry.choose (p :=
                  fun (h : Y₁ ⟶ Exists.choose (p := fun (Y : C) ↦ ∃ (h : Y₁ ⟶ Y) (g : Y ⟶ X), R g ∧ h ≫ g = f₁) h₁) ↦
                  ∃ (g : Exists.choose (p := fun (Y : C) ↦ ∃ (h : Y₁ ⟶ Y) (g : Y ⟶ X), R g ∧ h ≫ g = f₁) h₁ ⟶ X),
                    R g ∧ h ≫ g = f₁) ≫
                g =
              f₁))
        sorry) :
    P.obj (op Z))
  ((ConcreteCategory.hom (FC := TypeCat.Fun)
          (P.map
            (g₂ ≫
                sorry.choose (p :=
                  fun (h : Y₂ ⟶ Exists.choose (p := fun (Y : C) ↦ ∃ (h : Y₂ ⟶ Y) (g : Y ⟶ X), R g ∧ h ≫ g = f₂) h₂) ↦
                  ∃ (g : Exists.choose (p := fun (Y : C) ↦ ∃ (h : Y₂ ⟶ Y) (g : Y ⟶ X), R g ∧ h ≫ g = f₂) h₂ ⟶ X),
                    R g ∧ h ≫ g = f₂)).op) :
        (a : P.obj (op (Exists.choose h₂))) → P.obj (op Z))
      (x
        (sorry.choose (p :=
          fun (g : Exists.choose (p := fun (Y : C) ↦ ∃ (h : Y₂ ⟶ Y) (g : Y ⟶ X), R g ∧ h ≫ g = f₂) h₂ ⟶ X) ↦
          R g ∧
            sorry.choose (p :=
                  fun (h : Y₂ ⟶ Exists.choose (p := fun (Y : C) ↦ ∃ (h : Y₂ ⟶ Y) (g : Y ⟶ X), R g ∧ h ≫ g = f₂) h₂) ↦
                  ∃ (g : Exists.choose (p := fun (Y : C) ↦ ∃ (h : Y₂ ⟶ Y) (g : Y ⟶ X), R g ∧ h ≫ g = f₂) h₂ ⟶ X),
                    R g ∧ h ≫ g = f₂) ≫
                g =
              f₂))
        sorry) :
    P.obj (op Z))
Tactic: simp only [FamilyOfElements.sieveExtend, ← comp_apply, ← Functor.map_comp, ← op_comp]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.IsSheafFor (hypothesis)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
P : Functor.{v₁, w, u₁, w + 1} Cᵒᵖ (Type w)
X Y : C
R : Presieve.{v₁, u₁} X
x : FamilyOfElements P R
t : x.Compatible (R := R)
f : Y ⟶ X
hf : R f
⊢ x.sieveExtend (R := R) f ⋯ = x f hf
have h : ∃ (h : Y ⟶ Exists.choose (p := fun (Y_1 : C) ↦ ∃ (h : Y ⟶ Y_1) (g : Y_1 ⟶ X), R g ∧ h ≫ g = f) sorry) (g :
  Exists.choose (p := fun (Y_1 : C) ↦ ∃ (h : Y ⟶ Y_1) (g : Y_1 ⟶ X), R g ∧ h ≫ g = f) sorry ⟶ X), R g ∧ h ≫ g = f
Tactic: have h := (le_generate R Y _ hf).choose_spec
-- 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
R : Presieve.{v₁, u₁} X
x : FamilyOfElements P R
t : x.Compatible (R := R)
f : Y ⟶ X
hf : R f
h :
  ∃ (h : Y ⟶ Exists.choose (p := fun (Y_1 : C) ↦ ∃ (h : Y ⟶ Y_1) (g : Y_1 ⟶ X), R g ∧ h ≫ g = f) ⋯) (g :
    Exists.choose (p := fun (Y_1 : C) ↦ ∃ (h : Y ⟶ Y_1) (g : Y_1 ⟶ X), R g ∧ h ≫ g = f) ⋯ ⟶ X), R g ∧ h ≫ g = f
⊢ (ConcreteCategory.hom (FC := TypeCat.Fun)
          (P.map
            ⋯.choose (p :=
                fun (h : Y ⟶ Exists.choose (p := fun (Y_1 : C) ↦ ∃ (h : Y ⟶ Y_1) (g : Y_1 ⟶ X), R g ∧ h ≫ g = f) ⋯) ↦
                ∃ (g : Exists.choose (p := fun (Y_1 : C) ↦ ∃ (h : Y ⟶ Y_1) (g : Y_1 ⟶ X), R g ∧ h ≫ g = f) ⋯ ⟶ X),
                  R g ∧ h ≫ g = f).op) :
        (a : (fun (X : Type w) ↦ X) (P.obj (op (Exists.choose ⋯)))) → (fun (X : Type w) ↦ X) (P.obj (op Y)))
      (x
        (⋯.choose (p :=
          fun (g : Exists.choose (p := fun (Y_1 : C) ↦ ∃ (h : Y ⟶ Y_1) (g : Y_1 ⟶ X), R g ∧ h ≫ g = f) ⋯ ⟶ X) ↦
          R g ∧
            ⋯.choose (p :=
                  fun (h : Y ⟶ Exists.choose (p := fun (Y_1 : C) ↦ ∃ (h : Y ⟶ Y_1) (g : Y_1 ⟶ X), R g ∧ h ≫ g = f) ⋯) ↦
                  ∃ (g : Exists.choose (p := fun (Y_1 : C) ↦ ∃ (h : Y ⟶ Y_1) (g : Y_1 ⟶ X), R g ∧ h ≫ g = f) ⋯ ⟶ X),
                    R g ∧ h ≫ g = f) ≫
                g =
              f))
        ⋯) =
    x f hf
have rw : (ConcreteCategory.hom (FC := TypeCat.Fun) (P.map (Quiver.Hom.op (𝟙 Y))) :
      (a : (fun (X : Type w) ↦ X) (P.obj (op Y))) → (fun (X : Type w) ↦ X) (P.obj (op Y)))
    (x f hf) =
  x f hf
Tactic: rw [t h.choose (𝟙 _) _ hf _]
-- 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
R : Presieve.{v₁, u₁} X
x : FamilyOfElements P R
t : x.Compatible (R := R)
f : Y ⟶ X
hf : R f
h :
  ∃ (h : Y ⟶ Exists.choose (p := fun (Y_1 : C) ↦ ∃ (h : Y ⟶ Y_1) (g : Y_1 ⟶ X), R g ∧ h ≫ g = f) ⋯) (g :
    Exists.choose (p := fun (Y_1 : C) ↦ ∃ (h : Y ⟶ Y_1) (g : Y_1 ⟶ X), R g ∧ h ≫ g = f) ⋯ ⟶ X), R g ∧ h ≫ g = f
rw :
  (ConcreteCategory.hom (FC := TypeCat.Fun) (P.map (Quiver.Hom.op (𝟙 Y))) :
        (a : (fun (X : Type w) ↦ X) (P.obj (op Y))) → (fun (X : Type w) ↦ X) (P.obj (op Y)))
      (x f hf) =
    x f hf
⊢ (ConcreteCategory.hom (FC := TypeCat.Fun)
          (P.map
            ⋯.choose (p :=
                fun (h : Y ⟶ Exists.choose (p := fun (Y_1 : C) ↦ ∃ (h : Y ⟶ Y_1) (g : Y_1 ⟶ X), R g ∧ h ≫ g = f) ⋯) ↦
                ∃ (g : Exists.choose (p := fun (Y_1 : C) ↦ ∃ (h : Y ⟶ Y_1) (g : Y_1 ⟶ X), R g ∧ h ≫ g = f) ⋯ ⟶ X),
                  R g ∧ h ≫ g = f).op) :
        (a : (fun (X : Type w) ↦ X) (P.obj (op (Exists.choose ⋯)))) → (fun (X : Type w) ↦ X) (P.obj (op Y)))
      (x
        (⋯.choose (p :=
          fun (g : Exists.choose (p := fun (Y_1 : C) ↦ ∃ (h : Y ⟶ Y_1) (g : Y_1 ⟶ X), R g ∧ h ≫ g = f) ⋯ ⟶ X) ↦
          R g ∧
            ⋯.choose (p :=
                  fun (h : Y ⟶ Exists.choose (p := fun (Y_1 : C) ↦ ∃ (h : Y ⟶ Y_1) (g : Y_1 ⟶ X), R g ∧ h ≫ g = f) ⋯) ↦
                  ∃ (g : Exists.choose (p := fun (Y_1 : C) ↦ ∃ (h : Y ⟶ Y_1) (g : Y_1 ⟶ X), R g ∧ h ≫ g = f) ⋯ ⟶ X),
                    R g ∧ h ≫ g = f) ≫
                g =
              f))
        ⋯) =
    x f hf
have rw₁ : h.choose (p := fun (h : Y ⟶ Exists.choose (p := fun (Y_1 : C) ↦ ∃ (h : Y ⟶ Y_1) (g : Y_1 ⟶ X), R g ∧ h ≫ g = f) sorry) ↦
      ∃ (g : Exists.choose (p := fun (Y_1 : C) ↦ ∃ (h : Y ⟶ Y_1) (g : Y_1 ⟶ X), R g ∧ h ≫ g = f) sorry ⟶ X),
        R g ∧ h ≫ g = f) ≫
    sorry.choose (p := fun (g : Exists.choose (p := fun (Y_1 : C) ↦ ∃ (h : Y ⟶ Y_1) (g : Y_1 ⟶ X), R g ∧ h ≫ g = f) sorry ⟶ X) ↦
      R g ∧
        sorry.choose (p :=
              fun (h : Y ⟶ Exists.choose (p := fun (Y_1 : C) ↦ ∃ (h : Y ⟶ Y_1) (g : Y_1 ⟶ X), R g ∧ h ≫ g = f) sorry) ↦
              ∃ (g : Exists.choose (p := fun (Y_1 : C) ↦ ∃ (h : Y ⟶ Y_1) (g : Y_1 ⟶ X), R g ∧ h ≫ g = f) sorry ⟶ X),
                R g ∧ h ≫ g = f) ≫
            g =
          f) =
  𝟙 Y ≫ f
Tactic: rw [t h.choose (𝟙 _) _ hf _]
-- 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
R : Presieve.{v₁, u₁} X
x : FamilyOfElements P R
t : x.Compatible (R := R)
f : Y ⟶ X
hf : R f
h :
  ∃ (h : Y ⟶ Exists.choose (p := fun (Y_1 : C) ↦ ∃ (h : Y ⟶ Y_1) (g : Y_1 ⟶ X), R g ∧ h ≫ g = f) ⋯) (g :
    Exists.choose (p := fun (Y_1 : C) ↦ ∃ (h : Y ⟶ Y_1) (g : Y_1 ⟶ X), R g ∧ h ≫ g = f) ⋯ ⟶ X), R g ∧ h ≫ g = f
⊢ h.choose (p := fun (h : Y ⟶ Exists.choose (p := fun (Y_1 : C) ↦ ∃ (h : Y ⟶ Y_1) (g : Y_1 ⟶ X), R g ∧ h ≫ g = f) ⋯) ↦
        ∃ (g : Exists.choose (p := fun (Y_1 : C) ↦ ∃ (h : Y ⟶ Y_1) (g : Y_1 ⟶ X), R g ∧ h ≫ g = f) ⋯ ⟶ X),
          R g ∧ h ≫ g = f) ≫
      ⋯.choose (p :=
        fun (g : Exists.choose (p := fun (Y_1 : C) ↦ ∃ (h : Y ⟶ Y_1) (g : Y_1 ⟶ X), R g ∧ h ≫ g = f) ⋯ ⟶ X) ↦
        R g ∧
          ⋯.choose (p :=
                fun (h : Y ⟶ Exists.choose (p := fun (Y_1 : C) ↦ ∃ (h : Y ⟶ Y_1) (g : Y_1 ⟶ X), R g ∧ h ≫ g = f) ⋯) ↦
                ∃ (g : Exists.choose (p := fun (Y_1 : C) ↦ ∃ (h : Y ⟶ Y_1) (g : Y_1 ⟶ X), R g ∧ h ≫ g = f) ⋯ ⟶ X),
                  R g ∧ h ≫ g = f) ≫
              g =
            f) =
    𝟙 Y ≫ f
have rw : h.choose (p := fun (h : Y ⟶ Exists.choose (p := fun (Y_1 : C) ↦ ∃ (h : Y ⟶ Y_1) (g : Y_1 ⟶ X), R g ∧ h ≫ g = f) sorry) ↦
      ∃ (g : Exists.choose (p := fun (Y_1 : C) ↦ ∃ (h : Y ⟶ Y_1) (g : Y_1 ⟶ X), R g ∧ h ≫ g = f) sorry ⟶ X),
        R g ∧ h ≫ g = f) ≫
    sorry.choose (p := fun (g : Exists.choose (p := fun (Y_1 : C) ↦ ∃ (h : Y ⟶ Y_1) (g : Y_1 ⟶ X), R g ∧ h ≫ g = f) sorry ⟶ X) ↦
      R g ∧
        sorry.choose (p :=
              fun (h : Y ⟶ Exists.choose (p := fun (Y_1 : C) ↦ ∃ (h : Y ⟶ Y_1) (g : Y_1 ⟶ X), R g ∧ h ≫ g = f) sorry) ↦
              ∃ (g : Exists.choose (p := fun (Y_1 : C) ↦ ∃ (h : Y ⟶ Y_1) (g : Y_1 ⟶ X), R g ∧ h ≫ g = f) sorry ⟶ X),
                R g ∧ h ≫ g = f) ≫
            g =
          f) =
  f
Tactic: rw [id_comp]
-- 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 : x.Compatible (R := R)
⊢ FamilyOfElements.restrict (R₁ := R) (R₂ := (generate R).arrows) ⋯ (x.sieveExtend (R := R)) = x
have h : ∀ (Y : C) (f : Y ⟶ X) (hf : R f),
  FamilyOfElements.restrict (R₁ := R) (R₂ := (generate R).arrows) sorry (x.sieveExtend (R := R)) f hf = x f hf
Tactic: funext Y f hf
-- 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
S : Sieve.{v₁, u₁} X
x : FamilyOfElements P S.arrows
⊢ x.Compatible (R := S.arrows) ↔ x.SieveCompatible
have mp : x.Compatible (R := S.arrows) → x.SieveCompatible
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
S : Sieve.{v₁, u₁} X
x : FamilyOfElements P S.arrows
mp : x.Compatible (R := S.arrows) → x.SieveCompatible
⊢ x.Compatible (R := S.arrows) ↔ x.SieveCompatible
have mpr : x.SieveCompatible → x.Compatible (R := S.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
S : Sieve.{v₁, u₁} X
x : FamilyOfElements P S.arrows
h : x.SieveCompatible
Y₁ Y₂ Z : C
g₁ : Z ⟶ Y₁
g₂ : Z ⟶ Y₂
f₁ : Y₁ ⟶ X
f₂ : Y₂ ⟶ X
h₁ : S.arrows f₁
h₂ : S.arrows f₂
k : 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 mpr : x (g₁ ≫ f₁) sorry = x (g₂ ≫ f₂) sorry
Tactic: simp_rw [← h f₁ g₁ h₁, ← h f₂ g₂ h₂]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.IsSheafFor (hypothesis)
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 (generate R).arrows
t : x.Compatible (R := (generate R).arrows)
⊢ (FamilyOfElements.restrict (R₁ := R) (R₂ := (generate R).arrows) ⋯ x).sieveExtend (R := R) = x
have t₁ : x.SieveCompatible
Tactic: rw [compatible_iff_sieveCompatible] at t
Cleared hypotheses: t