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
P₁ : Functor.{v₁, w, u₁, w + 1} Cᵒᵖ (Type w)
P₂ : Functor.{v₁, w', u₁, w' + 1} Cᵒᵖ (Type w')
e : ⦃X : C⦄ → Equiv.{w + 1, w' + 1} (P₁.obj (op X)) (P₂.obj (op X))
he :
  ∀ ⦃X Y : C⦄ (f : X ⟶ Y) (x : P₁.obj (op Y)),
    Eq.{w' + 1} (α := P₂.obj (op X))
      ((e : (a : P₁.obj (op X)) → P₂.obj (op X))
          ((ConcreteCategory.hom (FC := TypeCat.Fun) (P₁.map f.op) :
              (a : (fun (X : Type w) ↦ X) (P₁.obj (op Y))) → (fun (X : Type w) ↦ X) (P₁.obj (op X)))
            x) :
        P₂.obj (op X))
      ((ConcreteCategory.hom (FC := TypeCat.Fun) (P₂.map f.op) :
            (a : (fun (X : Type w') ↦ X) (P₂.obj (op Y))) → (fun (X : Type w') ↦ X) (P₂.obj (op X)))
          ((e : (a : P₁.obj (op Y)) → P₂.obj (op Y)) x : P₂.obj (op Y)) :
        (fun (X : Type w') ↦ X) (P₂.obj (op X)))
X_1 : C
R : Presieve.{v₁, u₁} X_1
h : IsSheafFor P₂ R
X Y : C
f : X ⟶ Y
x : P₂.obj (op Y)
⊢ Eq.{w + 1} (α := P₁.obj (op X))
    ((((fun (x : C) ↦ (Equiv.symm e : Equiv.{w' + 1, w + 1} (P₂.obj (op x)) (P₁.obj (op x)))) X :
            Equiv.{w' + 1, w + 1} (P₂.obj (op X)) (P₁.obj (op X))) :
          (a : P₂.obj (op X)) → P₁.obj (op X))
        ((ConcreteCategory.hom (FC := TypeCat.Fun) (P₂.map f.op) :
            (a : (fun (X : Type w') ↦ X) (P₂.obj (op Y))) → (fun (X : Type w') ↦ X) (P₂.obj (op X)))
          x) :
      P₁.obj (op X))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (P₁.map f.op) :
          (a : (fun (X : Type w) ↦ X) (P₁.obj (op Y))) → (fun (X : Type w) ↦ X) (P₁.obj (op X)))
        ((((fun (x : C) ↦ (Equiv.symm e : Equiv.{w' + 1, w + 1} (P₂.obj (op x)) (P₁.obj (op x)))) Y :
                Equiv.{w' + 1, w + 1} (P₂.obj (op Y)) (P₁.obj (op Y))) :
              (a : P₂.obj (op Y)) → P₁.obj (op Y))
            x :
          P₁.obj (op Y)) :
      (fun (X : Type w) ↦ X) (P₁.obj (op X)))
have obtain : ∀ (y : P₁.obj (op Y)),
  Eq.{w + 1} (α := P₁.obj (op X))
    ((((fun (x : C) ↦ (Equiv.symm e : Equiv.{w' + 1, w + 1} (P₂.obj (op x)) (P₁.obj (op x)))) X :
            Equiv.{w' + 1, w + 1} (P₂.obj (op X)) (P₁.obj (op X))) :
          (a : P₂.obj (op X)) → P₁.obj (op X))
        ((ConcreteCategory.hom (FC := TypeCat.Fun) (P₂.map f.op) :
            (a : (fun (X : Type w') ↦ X) (P₂.obj (op Y))) → (fun (X : Type w') ↦ X) (P₂.obj (op X)))
          ((e : (a : P₁.obj (op Y)) → P₂.obj (op Y)) y : P₂.obj (op Y))) :
      P₁.obj (op X))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (P₁.map f.op) :
          (a : (fun (X : Type w) ↦ X) (P₁.obj (op Y))) → (fun (X : Type w) ↦ X) (P₁.obj (op X)))
        ((((fun (x : C) ↦ (Equiv.symm e : Equiv.{w' + 1, w + 1} (P₂.obj (op x)) (P₁.obj (op x)))) Y :
                Equiv.{w' + 1, w + 1} (P₂.obj (op Y)) (P₁.obj (op Y))) :
              (a : P₂.obj (op Y)) → P₁.obj (op Y))
            ((e : (a : P₁.obj (op Y)) → P₂.obj (op Y)) y) :
          P₁.obj (op Y)) :
      (fun (X : Type w) ↦ X) (P₁.obj (op X)))
Tactic: obtain ⟨y, rfl⟩ := e.surjective x
Cleared hypotheses: x
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.IsSheafFor (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
P₁ : Functor.{v₁, w, u₁, w + 1} Cᵒᵖ (Type w)
P₂ : Functor.{v₁, w', u₁, w' + 1} Cᵒᵖ (Type w')
e : ⦃X : C⦄ → Equiv.{w + 1, w' + 1} (P₁.obj (op X)) (P₂.obj (op X))
he :
  ∀ ⦃X Y : C⦄ (f : X ⟶ Y) (x : P₁.obj (op Y)),
    Eq.{w' + 1} (α := P₂.obj (op X))
      ((e : (a : P₁.obj (op X)) → P₂.obj (op X))
          ((ConcreteCategory.hom (FC := TypeCat.Fun) (P₁.map f.op) :
              (a : (fun (X : Type w) ↦ X) (P₁.obj (op Y))) → (fun (X : Type w) ↦ X) (P₁.obj (op X)))
            x) :
        P₂.obj (op X))
      ((ConcreteCategory.hom (FC := TypeCat.Fun) (P₂.map f.op) :
            (a : (fun (X : Type w') ↦ X) (P₂.obj (op Y))) → (fun (X : Type w') ↦ X) (P₂.obj (op X)))
          ((e : (a : P₁.obj (op Y)) → P₂.obj (op Y)) x : P₂.obj (op Y)) :
        (fun (X : Type w') ↦ X) (P₂.obj (op X)))
X_1 : C
R : Presieve.{v₁, u₁} X_1
h : IsSheafFor P₂ R
X Y : C
f : X ⟶ Y
y : P₁.obj (op Y)
⊢ Eq.{w + 1} (α := P₁.obj (op X))
    ((((fun (x : C) ↦ (Equiv.symm e : Equiv.{w' + 1, w + 1} (P₂.obj (op x)) (P₁.obj (op x)))) X :
            Equiv.{w' + 1, w + 1} (P₂.obj (op X)) (P₁.obj (op X))) :
          (a : P₂.obj (op X)) → P₁.obj (op X))
        ((ConcreteCategory.hom (FC := TypeCat.Fun) (P₂.map f.op) :
            (a : (fun (X : Type w') ↦ X) (P₂.obj (op Y))) → (fun (X : Type w') ↦ X) (P₂.obj (op X)))
          ((e : (a : P₁.obj (op Y)) → P₂.obj (op Y)) y : P₂.obj (op Y))) :
      P₁.obj (op X))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (P₁.map f.op) :
          (a : (fun (X : Type w) ↦ X) (P₁.obj (op Y))) → (fun (X : Type w) ↦ X) (P₁.obj (op X)))
        ((((fun (x : C) ↦ (Equiv.symm e : Equiv.{w' + 1, w + 1} (P₂.obj (op x)) (P₁.obj (op x)))) Y :
                Equiv.{w' + 1, w + 1} (P₂.obj (op Y)) (P₁.obj (op Y))) :
              (a : P₂.obj (op Y)) → P₁.obj (op Y))
            ((e : (a : P₁.obj (op Y)) → P₂.obj (op Y)) y) :
          P₁.obj (op Y)) :
      (fun (X : Type w) ↦ X) (P₁.obj (op X)))
have refine : Eq.{w' + 1} (α := P₂.obj (op X))
  ((e : (a : P₁.obj (op X)) → P₂.obj (op X))
      ((((fun (x : C) ↦ (Equiv.symm e : Equiv.{w' + 1, w + 1} (P₂.obj (op x)) (P₁.obj (op x)))) X :
            Equiv.{w' + 1, w + 1} (P₂.obj (op X)) (P₁.obj (op X))) :
          (a : P₂.obj (op X)) → P₁.obj (op X))
        ((ConcreteCategory.hom (FC := TypeCat.Fun) (P₂.map f.op) :
            (a : (fun (X : Type w') ↦ X) (P₂.obj (op Y))) → (fun (X : Type w') ↦ X) (P₂.obj (op X)))
          ((e : (a : P₁.obj (op Y)) → P₂.obj (op Y)) y : P₂.obj (op Y)))) :
    P₂.obj (op X))
  ((e : (a : P₁.obj (op X)) → P₂.obj (op X))
      ((ConcreteCategory.hom (FC := TypeCat.Fun) (P₁.map f.op) :
          (a : (fun (X : Type w) ↦ X) (P₁.obj (op Y))) → (fun (X : Type w) ↦ X) (P₁.obj (op X)))
        ((((fun (x : C) ↦ (Equiv.symm e : Equiv.{w' + 1, w + 1} (P₂.obj (op x)) (P₁.obj (op x)))) Y :
                Equiv.{w' + 1, w + 1} (P₂.obj (op Y)) (P₁.obj (op Y))) :
              (a : P₂.obj (op Y)) → P₁.obj (op Y))
            ((e : (a : P₁.obj (op Y)) → P₂.obj (op Y)) y) :
          P₁.obj (op Y))) :
    P₂.obj (op X))
Tactic: refine e.injective ?_
-- 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)
S : Sieve.{v₁, u₁} X
R : Presieve.{v₁, u₁} X
h : S.arrows ≤ R
hS : IsSheafFor P S.arrows
trans : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R f → IsSeparatedFor P (Sieve.pullback f S).arrows
⊢ IsSheafFor P R
have rw : IsSeparatedFor P R ∧
  ∀ (x : FamilyOfElements P R), x.Compatible (R := R) → ∃ (t : P.obj (op X)), x.IsAmalgamation (R := R) t
Tactic: rw [← isSeparatedFor_and_exists_isAmalgamation_iff_isSheafFor]
-- 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)
S : Sieve.{v₁, u₁} X
R : Presieve.{v₁, u₁} X
h : S.arrows ≤ R
hS : IsSheafFor P S.arrows
trans : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R f → IsSeparatedFor P (Sieve.pullback f S).arrows
⊢ IsSeparatedFor P R ∧
    ∀ (x : FamilyOfElements P R), x.Compatible (R := R) → ∃ (t : P.obj (op X)), x.IsAmalgamation (R := R) t
have left : IsSeparatedFor P R
Tactic: constructor
-- 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)
S : Sieve.{v₁, u₁} X
R : Presieve.{v₁, u₁} X
h : S.arrows ≤ R
hS : IsSheafFor P S.arrows
trans : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R f → IsSeparatedFor P (Sieve.pullback f S).arrows
left : IsSeparatedFor P R
⊢ IsSeparatedFor P R ∧
    ∀ (x : FamilyOfElements P R), x.Compatible (R := R) → ∃ (t : P.obj (op X)), x.IsAmalgamation (R := R) t
have right : ∀ (x : FamilyOfElements P R), x.Compatible (R := R) → ∃ (t : P.obj (op X)), x.IsAmalgamation (R := R) t
Tactic: constructor
-- 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)
S : Sieve.{v₁, u₁} X
R : Presieve.{v₁, u₁} X
h : S.arrows ≤ R
hS : IsSheafFor P S.arrows
trans : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R f → IsSeparatedFor P (Sieve.pullback f S).arrows
x : FamilyOfElements P R
hx : x.Compatible (R := R)
⊢ ∃ (t : P.obj (op X)), x.IsAmalgamation (R := R) t
have h₁ : x.IsAmalgamation (R := R) (hS.amalgamate (R := S.arrows) (FamilyOfElements.restrict (R₁ := S.arrows) (R₂ := R) h x) sorry)
Tactic: use hS.amalgamate _ (hx.restrict h)
-- 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)
S : Sieve.{v₁, u₁} X
R : Presieve.{v₁, u₁} X
h : S.arrows ≤ R
hS : IsSheafFor P S.arrows
trans : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R f → IsSeparatedFor P (Sieve.pullback f S).arrows
x : FamilyOfElements P R
hx : x.Compatible (R := R)
W : C
j : W ⟶ X
hj : R j
⊢ (ConcreteCategory.hom (FC := TypeCat.Fun) (P.map j.op) :
        (a : (fun (X : Type w) ↦ X) (P.obj (op X))) → (fun (X : Type w) ↦ X) (P.obj (op W)))
      (hS.amalgamate (R := S.arrows) (FamilyOfElements.restrict (R₁ := S.arrows) (R₂ := R) h x) ⋯) =
    x j hj
have h₁ : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ W⦄,
  (Sieve.pullback j S).arrows f →
    Eq (α := (fun (X : Type w) ↦ X) (P.obj (op Y)))
      ((ConcreteCategory.hom (FC := TypeCat.Fun) (P.map f.op) :
            (a : (fun (X : Type w) ↦ X) (P.obj (op W))) → (fun (X : Type w) ↦ X) (P.obj (op Y)))
          ((ConcreteCategory.hom (FC := TypeCat.Fun) (P.map j.op) :
                (a : (fun (X : Type w) ↦ X) (P.obj (op X))) → (fun (X : Type w) ↦ X) (P.obj (op W)))
              (hS.amalgamate (R := S.arrows) (FamilyOfElements.restrict (R₁ := S.arrows) (R₂ := R) h x) sorry) :
            (fun (X : Type w) ↦ X) (P.obj (op W))) :
        (fun (X : Type w) ↦ X) (P.obj (op Y)))
      ((ConcreteCategory.hom (FC := TypeCat.Fun) (P.map f.op) :
            (a : (fun (X : Type w) ↦ X) (P.obj (op W))) → (fun (X : Type w) ↦ X) (P.obj (op Y)))
          (x j hj) :
        (fun (X : Type w) ↦ X) (P.obj (op Y)))
Tactic: apply (trans hj).ext
-- 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)
S : Sieve.{v₁, u₁} X
R : Presieve.{v₁, u₁} X
h : S.arrows ≤ R
hS : IsSheafFor P S.arrows
trans : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R f → IsSeparatedFor P (Sieve.pullback f S).arrows
x : FamilyOfElements P R
hx : x.Compatible (R := R)
W : C
j : W ⟶ X
hj : R j
Y : C
f : Y ⟶ W
hf : (Sieve.pullback j S).arrows f
⊢ Eq (α := (fun (X : Type w) ↦ X) (P.obj (op Y)))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (P.map f.op) :
          (a : (fun (X : Type w) ↦ X) (P.obj (op W))) → (fun (X : Type w) ↦ X) (P.obj (op Y)))
        ((ConcreteCategory.hom (FC := TypeCat.Fun) (P.map j.op) :
              (a : (fun (X : Type w) ↦ X) (P.obj (op X))) → (fun (X : Type w) ↦ X) (P.obj (op W)))
            (hS.amalgamate (R := S.arrows) (FamilyOfElements.restrict (R₁ := S.arrows) (R₂ := R) h x) ⋯) :
          (fun (X : Type w) ↦ X) (P.obj (op W))) :
      (fun (X : Type w) ↦ X) (P.obj (op Y)))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (P.map f.op) :
          (a : (fun (X : Type w) ↦ X) (P.obj (op W))) → (fun (X : Type w) ↦ X) (P.obj (op Y)))
        (x j hj) :
      (fun (X : Type w) ↦ X) (P.obj (op Y)))
have h₁ : x (f ≫ j) sorry =
  (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 ≫ j) sorry)
Tactic: rw [← comp_apply, ← Functor.map_comp, ← op_comp, hS.valid_glue (hx.restrict h) _ hf,
      FamilyOfElements.restrict, ← hx (𝟙 _) f (h _ _ hf) _ (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)
B : C
I : Type u_1
X : I → C
π : (i : I) → X i ⟶ B
x : (i : I) → P.obj (op (X i))
hx : Compatible P π x
⊢ ∃ (x' : FamilyOfElements P (ofArrows X π)), ∀ (i : I), x' (π i) ⋯ = x i
have choose : ∀ (i : {Z : C} → (g : Z ⟶ B) → (hg : ofArrows X π g) → I)
  (h : ∀ {Z : C} (g : Z ⟶ B) (hg : ofArrows X π g), X (i g hg) = Z)
  (h' : ∀ {Z : C} (g : Z ⟶ B) (hg : ofArrows X π g), g = eqToHom sorry ≫ π (i g hg)),
  ∃ (x' : FamilyOfElements P (ofArrows X π)), ∀ (i : I), x' (π i) sorry = x i
Tactic: choose i h h' using @ofArrows_surj _ _ _ _ _ π
-- 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)
B : C
I : Type u_1
X : I → C
π : (i : I) → X i ⟶ B
x : (i : I) → P.obj (op (X i))
hx : Compatible P π x
i : {Z : C} → (g : Z ⟶ B) → (hg : ofArrows X π g) → I
h : ∀ {Z : C} (g : Z ⟶ B) (hg : ofArrows X π g), X (i g hg) = Z
h' : ∀ {Z : C} (g : Z ⟶ B) (hg : ofArrows X π g), g = eqToHom ⋯ ≫ π (i g hg)
j : I
⊢ CategoryStruct.comp (Y := X (i (π j) ⋯)) (eqToHom ⋯) (π (i (π j) ⋯)) = 𝟙 (X j) ≫ π j
have rw : π j = 𝟙 (X j) ≫ π j
Tactic: rw [← h', 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)
B : C
I : Type u_1
X : I → C
π : (i : I) → X i ⟶ B
x : (i : I) → P.obj (op (X i))
hx : Compatible P π x
i : {Z : C} → (g : Z ⟶ B) → (hg : ofArrows X π g) → I
h : ∀ {Z : C} (g : Z ⟶ B) (hg : ofArrows X π g), X (i g hg) = Z
h' : ∀ {Z : C} (g : Z ⟶ B) (hg : ofArrows X π g), g = eqToHom ⋯ ≫ π (i g hg)
j : I
rw : π j = 𝟙 (X j) ≫ π j
⊢ CategoryStruct.comp (Y := X (i (π j) ⋯)) (eqToHom ⋯) (π (i (π j) ⋯)) = 𝟙 (X j) ≫ π j
have rw₁ : π j = π j
Tactic: rw [← h', 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)
B : C
I : Type u_1
X : I → C
π : (i : I) → X i ⟶ B
⊢ IsSheafFor P (ofArrows X π) ↔
    ∀ (x : (i : I) → P.obj (op (X i))),
      Arrows.Compatible P π x →
        ∃! t : (fun (X : Type w) ↦ X) (P.obj (op B)),
          ∀ (i : I),
            (ConcreteCategory.hom (FC := TypeCat.Fun) (P.map (π i).op) :
                  (a : (fun (X : Type w) ↦ X) (P.obj (op B))) → (fun (X : Type w) ↦ X) (P.obj (op (X i))))
                t =
              x i
have refine_1 : ∀ (h : IsSheafFor P (ofArrows X π)) (x : (i : I) → P.obj (op (X i))) (hx : Arrows.Compatible P π x),
  ∃! t : (fun (X : Type w) ↦ X) (P.obj (op B)),
    ∀ (i : I),
      (ConcreteCategory.hom (FC := TypeCat.Fun) (P.map (π i).op) :
            (a : (fun (X : Type w) ↦ X) (P.obj (op B))) → (fun (X : Type w) ↦ X) (P.obj (op (X i))))
          t =
        x i
Tactic: refine ⟨fun h x hx ↦ ?_, fun h x hx ↦ ?_⟩
-- 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)
B : C
I : Type u_1
X : I → C
π : (i : I) → X i ⟶ B
refine_1 :
  ∀ (h : IsSheafFor P (ofArrows X π)) (x : (i : I) → P.obj (op (X i))) (hx : Arrows.Compatible P π x),
    ∃! t : (fun (X : Type w) ↦ X) (P.obj (op B)),
      ∀ (i : I),
        (ConcreteCategory.hom (FC := TypeCat.Fun) (P.map (π i).op) :
              (a : (fun (X : Type w) ↦ X) (P.obj (op B))) → (fun (X : Type w) ↦ X) (P.obj (op (X i))))
            t =
          x i
⊢ IsSheafFor P (ofArrows X π) ↔
    ∀ (x : (i : I) → P.obj (op (X i))),
      Arrows.Compatible P π x →
        ∃! t : (fun (X : Type w) ↦ X) (P.obj (op B)),
          ∀ (i : I),
            (ConcreteCategory.hom (FC := TypeCat.Fun) (P.map (π i).op) :
                  (a : (fun (X : Type w) ↦ X) (P.obj (op B))) → (fun (X : Type w) ↦ X) (P.obj (op (X i))))
                t =
              x i
have refine_2 : ∀
  (h :
    ∀ (x : (i : I) → P.obj (op (X i))),
      Arrows.Compatible P π x →
        ∃! t : (fun (X : Type w) ↦ X) (P.obj (op B)),
          ∀ (i : I),
            (ConcreteCategory.hom (FC := TypeCat.Fun) (P.map (π i).op) :
                  (a : (fun (X : Type w) ↦ X) (P.obj (op B))) → (fun (X : Type w) ↦ X) (P.obj (op (X i))))
                t =
              x i)
  (x : FamilyOfElements P (ofArrows X π)) (hx : x.Compatible (R := ofArrows X π)),
  ∃! t : P.obj (op B), x.IsAmalgamation (R := ofArrows X π) t
Tactic: refine ⟨fun h x hx ↦ ?_, fun h x hx ↦ ?_⟩
-- 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)
B : C
I : Type u_1
X : I → C
π : (i : I) → X i ⟶ B
h : IsSheafFor P (ofArrows X π)
x : (i : I) → P.obj (op (X i))
hx : Arrows.Compatible P π x
⊢ ∃! t : (fun (X : Type w) ↦ X) (P.obj (op B)),
    ∀ (i : I),
      (ConcreteCategory.hom (FC := TypeCat.Fun) (P.map (π i).op) :
            (a : (fun (X : Type w) ↦ X) (P.obj (op B))) → (fun (X : Type w) ↦ X) (P.obj (op (X i))))
          t =
        x i
have refine_1 : ∀ (t : P.obj (op B)) (ht₁ : hx.familyOfElements.IsAmalgamation (R := ofArrows X π) t)
  (ht₂ :
    ∀ (y : P.obj (op B)),
      (fun (t : P.obj (op B)) ↦ hx.familyOfElements.IsAmalgamation (R := ofArrows X π) t) y → y = t),
  ∃! t : (fun (X : Type w) ↦ X) (P.obj (op B)),
    ∀ (i : I),
      (ConcreteCategory.hom (FC := TypeCat.Fun) (P.map (π i).op) :
            (a : (fun (X : Type w) ↦ X) (P.obj (op B))) → (fun (X : Type w) ↦ X) (P.obj (op (X i))))
          t =
        x i
Tactic: obtain ⟨t, ht₁, ht₂⟩ := h _ hx.familyOfElements_compatible
-- 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)
B : C
I : Type u_1
X : I → C
π : (i : I) → X i ⟶ B
h : IsSheafFor P (ofArrows X π)
x : (i : I) → P.obj (op (X i))
hx : Arrows.Compatible P π x
t : P.obj (op B)
ht₁ : hx.familyOfElements.IsAmalgamation (R := ofArrows X π) t
ht₂ :
  ∀ (y : P.obj (op B)), (fun (t : P.obj (op B)) ↦ hx.familyOfElements.IsAmalgamation (R := ofArrows X π) t) y → y = t
⊢ ∃! t : (fun (X : Type w) ↦ X) (P.obj (op B)),
    ∀ (i : I),
      (ConcreteCategory.hom (FC := TypeCat.Fun) (P.map (π i).op) :
            (a : (fun (X : Type w) ↦ X) (P.obj (op B))) → (fun (X : Type w) ↦ X) (P.obj (op (X i))))
          t =
        x i
have refine_1 : ∀ (i : I),
  (ConcreteCategory.hom (FC := TypeCat.Fun) (P.map (π i).op) :
        (a : (fun (X : Type w) ↦ X) (P.obj (op B))) → (fun (X : Type w) ↦ X) (P.obj (op (X i))))
      t =
    x i
Tactic: refine ⟨t, fun i ↦ ?_, fun t' ht' ↦ ht₂ _ fun _ _ ⟨i⟩ ↦ ?_⟩
-- 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)
B : C
I : Type u_1
X : I → C
π : (i : I) → X i ⟶ B
h : IsSheafFor P (ofArrows X π)
x : (i : I) → P.obj (op (X i))
hx : Arrows.Compatible P π x
t : P.obj (op B)
ht₁ : hx.familyOfElements.IsAmalgamation (R := ofArrows X π) t
ht₂ :
  ∀ (y : P.obj (op B)), (fun (t : P.obj (op B)) ↦ hx.familyOfElements.IsAmalgamation (R := ofArrows X π) t) y → y = t
refine_1 :
  ∀ (i : I),
    (ConcreteCategory.hom (FC := TypeCat.Fun) (P.map (π i).op) :
          (a : (fun (X : Type w) ↦ X) (P.obj (op B))) → (fun (X : Type w) ↦ X) (P.obj (op (X i))))
        t =
      x i
⊢ ∃! t : (fun (X : Type w) ↦ X) (P.obj (op B)),
    ∀ (i : I),
      (ConcreteCategory.hom (FC := TypeCat.Fun) (P.map (π i).op) :
            (a : (fun (X : Type w) ↦ X) (P.obj (op B))) → (fun (X : Type w) ↦ X) (P.obj (op (X i))))
          t =
        x i
have refine_1₁ : ∀ (t' : (fun (X : Type w) ↦ X) (P.obj (op B)))
  (ht' :
    (fun (t : (fun (X : Type w) ↦ X) (P.obj (op B))) ↦
        ∀ (i : I),
          (ConcreteCategory.hom (FC := TypeCat.Fun) (P.map (π i).op) :
                (a : (fun (X : Type w) ↦ X) (P.obj (op B))) → (fun (X : Type w) ↦ X) (P.obj (op (X i))))
              t =
            x i)
      t')
  (x_1 : C) (x_2 : x_1 ⟶ B),
  ofArrows X π x_2 →
    ∀ (i : I),
      (ConcreteCategory.hom (FC := TypeCat.Fun) (P.map (π i).op) :
            (a : (fun (X : Type w) ↦ X) (P.obj (op B))) → (fun (X : Type w) ↦ X) (P.obj (op (X i))))
          t' =
        hx.familyOfElements (π i) sorry
Tactic: refine ⟨t, fun i ↦ ?_, fun t' ht' ↦ ht₂ _ fun _ _ ⟨i⟩ ↦ ?_⟩
-- 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)
B : C
I : Type u_1
X : I → C
π : (i : I) → X i ⟶ B
h : IsSheafFor P (ofArrows X π)
x : (i : I) → P.obj (op (X i))
hx : Arrows.Compatible P π x
t : P.obj (op B)
ht₁ : hx.familyOfElements.IsAmalgamation (R := ofArrows X π) t
ht₂ :
  ∀ (y : P.obj (op B)), (fun (t : P.obj (op B)) ↦ hx.familyOfElements.IsAmalgamation (R := ofArrows X π) t) y → y = t
i : I
⊢ (ConcreteCategory.hom (FC := TypeCat.Fun) (P.map (π i).op) :
        (a : (fun (X : Type w) ↦ X) (P.obj (op B))) → (fun (X : Type w) ↦ X) (P.obj (op (X i))))
      t =
    x i
have refine_1 : hx.familyOfElements (π i) sorry = x i
Tactic: rw [ht₁ _ (ofArrows.mk i), hx.familyOfElements_ofArrows_mk]
-- 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)
B : C
I : Type u_1
X : I → C
π : (i : I) → X i ⟶ B
h : IsSheafFor P (ofArrows X π)
x : (i : I) → P.obj (op (X i))
hx : Arrows.Compatible P π x
t : P.obj (op B)
ht₁ : hx.familyOfElements.IsAmalgamation (R := ofArrows X π) t
ht₂ :
  ∀ (y : P.obj (op B)), (fun (t : P.obj (op B)) ↦ hx.familyOfElements.IsAmalgamation (R := ofArrows X π) t) y → y = t
i : I
refine_1 : hx.familyOfElements (π i) ⋯ = x i
⊢ (ConcreteCategory.hom (FC := TypeCat.Fun) (P.map (π i).op) :
        (a : (fun (X : Type w) ↦ X) (P.obj (op B))) → (fun (X : Type w) ↦ X) (P.obj (op (X i))))
      t =
    x i
have refine_1₁ : x i = x i
Tactic: rw [ht₁ _ (ofArrows.mk i), hx.familyOfElements_ofArrows_mk]
-- 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)
B : C
I : Type u_1
X : I → C
π : (i : I) → X i ⟶ B
h : IsSheafFor P (ofArrows X π)
x : (i : I) → P.obj (op (X i))
hx : Arrows.Compatible P π x
t : P.obj (op B)
ht₁ : hx.familyOfElements.IsAmalgamation (R := ofArrows X π) t
ht₂ :
  ∀ (y : P.obj (op B)), (fun (t : P.obj (op B)) ↦ hx.familyOfElements.IsAmalgamation (R := ofArrows X π) t) y → y = t
t' : (fun (X : Type w) ↦ X) (P.obj (op B))
ht' :
  (fun (t : (fun (X : Type w) ↦ X) (P.obj (op B))) ↦
      ∀ (i : I),
        (ConcreteCategory.hom (FC := TypeCat.Fun) (P.map (π i).op) :
              (a : (fun (X : Type w) ↦ X) (P.obj (op B))) → (fun (X : Type w) ↦ X) (P.obj (op (X i))))
            t =
          x i)
    t'
x_1 : C
x_2 : x_1 ⟶ B
h_1 : ofArrows X π x_2
i : I
⊢ (ConcreteCategory.hom (FC := TypeCat.Fun) (P.map (π i).op) :
        (a : (fun (X : Type w) ↦ X) (P.obj (op B))) → (fun (X : Type w) ↦ X) (P.obj (op (X i))))
      t' =
    hx.familyOfElements (π i) ⋯
have refine_1 : x i = hx.familyOfElements (π i) sorry
Tactic: rw [ht', hx.familyOfElements_ofArrows_mk]
-- 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)
B : C
I : Type u_1
X : I → C
π : (i : I) → X i ⟶ B
h : IsSheafFor P (ofArrows X π)
x : (i : I) → P.obj (op (X i))
hx : Arrows.Compatible P π x
t : P.obj (op B)
ht₁ : hx.familyOfElements.IsAmalgamation (R := ofArrows X π) t
ht₂ :
  ∀ (y : P.obj (op B)), (fun (t : P.obj (op B)) ↦ hx.familyOfElements.IsAmalgamation (R := ofArrows X π) t) y → y = t
t' : (fun (X : Type w) ↦ X) (P.obj (op B))
ht' :
  (fun (t : (fun (X : Type w) ↦ X) (P.obj (op B))) ↦
      ∀ (i : I),
        (ConcreteCategory.hom (FC := TypeCat.Fun) (P.map (π i).op) :
              (a : (fun (X : Type w) ↦ X) (P.obj (op B))) → (fun (X : Type w) ↦ X) (P.obj (op (X i))))
            t =
          x i)
    t'
x_1 : C
x_2 : x_1 ⟶ B
h_1 : ofArrows X π x_2
i : I
refine_1 : x i = hx.familyOfElements (π i) ⋯
⊢ (ConcreteCategory.hom (FC := TypeCat.Fun) (P.map (π i).op) :
        (a : (fun (X : Type w) ↦ X) (P.obj (op B))) → (fun (X : Type w) ↦ X) (P.obj (op (X i))))
      t' =
    hx.familyOfElements (π i) ⋯
have refine_1₁ : x i = x i
Tactic: rw [ht', hx.familyOfElements_ofArrows_mk]