awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Topology.Sheaves.LocalPredicate (goal)
Goal:
X : TopCat
T : (↑X : Type u_2) → Type u_1
P : LocalPredicate T
ι : Type u_2
U : ι → Opens (↑X : Type u_2)
sf :
  (i : ι) → ToType (FC := TypeCat.Fun) ((subpresheafToTypes (T := T) (P.toPrelocalPredicate (T := T))).obj (op (U i)))
sf_comp : (subpresheafToTypes (T := T) (P.toPrelocalPredicate (T := T))).IsCompatible (FC := TypeCat.Fun) U sf
sf' : (i : ι) → (X.presheafToTypes T).obj (op (U i)) :=
  fun (i : ι) ↦ (↑(sf i) : (x : ↥(unop (op (U i)))) → T (↑x : (↑X : Type u_2)))
sf'_comp : (X.presheafToTypes T).IsCompatible (FC := TypeCat.Fun) U sf'
⊢ ∃! s : ToType (FC := TypeCat.Fun) ((subpresheafToTypes (T := T) (P.toPrelocalPredicate (T := T))).obj (op (iSup U))),
    (subpresheafToTypes (T := T) (P.toPrelocalPredicate (T := T))).IsGluing (FC := TypeCat.Fun) U sf s
have obtain : ∀
  (gl :
    ToType (FC := TypeCat.Fun)
      ((X.sheafToTypes T).obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u_2))).obj
        (op (iSup U))))
  (gl_spec :
    IsGluing (FC := TypeCat.Fun)
      ((X.sheafToTypes T).obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u_2)))) U sf'
      gl)
  (gl_uniq :
    ∀
      (y :
        ToType (FC := TypeCat.Fun)
          ((X.sheafToTypes T).obj (P :=
                CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u_2))).obj
            (op (iSup U)))),
      (fun
              (s :
                ToType (FC := TypeCat.Fun)
                  ((X.sheafToTypes T).obj (P :=
                        CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u_2))).obj
                    (op (iSup U)))) ↦
            IsGluing (FC := TypeCat.Fun)
              ((X.sheafToTypes T).obj (P :=
                CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u_2))))
              U sf' s)
          y →
        y = gl),
  ∃! s : ToType (FC := TypeCat.Fun) ((subpresheafToTypes (T := T) (P.toPrelocalPredicate (T := T))).obj (op (iSup U))),
    (subpresheafToTypes (T := T) (P.toPrelocalPredicate (T := T))).IsGluing (FC := TypeCat.Fun) U sf s
Tactic: obtain ⟨gl, gl_spec, gl_uniq⟩ := (sheafToTypes X T).existsUnique_gluing U sf'
      -- `by exact` to help Lean infer the `ConcreteCategory` instance
      (by exact sf'_comp)
-- Lean 4 proof step from Mathlib.Topology.Sheaves.LocalPredicate (goal)
Goal:
X : TopCat
T : (↑X : Type u_2) → Type u_1
P : LocalPredicate T
ι : Type u_2
U : ι → Opens (↑X : Type u_2)
sf :
  (i : ι) → ToType (FC := TypeCat.Fun) ((subpresheafToTypes (T := T) (P.toPrelocalPredicate (T := T))).obj (op (U i)))
sf_comp : (subpresheafToTypes (T := T) (P.toPrelocalPredicate (T := T))).IsCompatible (FC := TypeCat.Fun) U sf
sf' : (i : ι) → (X.presheafToTypes T).obj (op (U i)) :=
  fun (i : ι) ↦ (↑(sf i) : (x : ↥(unop (op (U i)))) → T (↑x : (↑X : Type u_2)))
sf'_comp : (X.presheafToTypes T).IsCompatible (FC := TypeCat.Fun) U sf'
gl :
  ToType (FC := TypeCat.Fun)
    ((X.sheafToTypes T).obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u_2))).obj
      (op (iSup U)))
gl_spec :
  IsGluing (FC := TypeCat.Fun)
    ((X.sheafToTypes T).obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u_2)))) U sf'
    gl
gl_uniq :
  ∀
    (y :
      ToType (FC := TypeCat.Fun)
        ((X.sheafToTypes T).obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u_2))).obj
          (op (iSup U)))),
    (fun
            (s :
              ToType (FC := TypeCat.Fun)
                ((X.sheafToTypes T).obj (P :=
                      CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u_2))).obj
                  (op (iSup U)))) ↦
          IsGluing (FC := TypeCat.Fun)
            ((X.sheafToTypes T).obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u_2))))
            U sf' s)
        y →
      y = gl
⊢ ∃! s : ToType (FC := TypeCat.Fun) ((subpresheafToTypes (T := T) (P.toPrelocalPredicate (T := T))).obj (op (iSup U))),
    (subpresheafToTypes (T := T) (P.toPrelocalPredicate (T := T))).IsGluing (FC := TypeCat.Fun) U sf s
have refine_1 : P.toPrelocalPredicate (T := T).pred (T := T) gl
Tactic: refine ⟨⟨gl, ?_⟩, ?_, ?_⟩
-- Lean 4 proof step from Mathlib.Topology.Sheaves.LocalPredicate (goal)
Goal:
X : TopCat
T : (↑X : Type u_2) → Type u_1
P : LocalPredicate T
ι : Type u_2
U : ι → Opens (↑X : Type u_2)
sf :
  (i : ι) → ToType (FC := TypeCat.Fun) ((subpresheafToTypes (T := T) (P.toPrelocalPredicate (T := T))).obj (op (U i)))
sf_comp : (subpresheafToTypes (T := T) (P.toPrelocalPredicate (T := T))).IsCompatible (FC := TypeCat.Fun) U sf
sf' : (i : ι) → (X.presheafToTypes T).obj (op (U i)) :=
  fun (i : ι) ↦ (↑(sf i) : (x : ↥(unop (op (U i)))) → T (↑x : (↑X : Type u_2)))
sf'_comp : (X.presheafToTypes T).IsCompatible (FC := TypeCat.Fun) U sf'
gl :
  ToType (FC := TypeCat.Fun)
    ((X.sheafToTypes T).obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u_2))).obj
      (op (iSup U)))
gl_spec :
  IsGluing (FC := TypeCat.Fun)
    ((X.sheafToTypes T).obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u_2)))) U sf'
    gl
gl_uniq :
  ∀
    (y :
      ToType (FC := TypeCat.Fun)
        ((X.sheafToTypes T).obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u_2))).obj
          (op (iSup U)))),
    (fun
            (s :
              ToType (FC := TypeCat.Fun)
                ((X.sheafToTypes T).obj (P :=
                      CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u_2))).obj
                  (op (iSup U)))) ↦
          IsGluing (FC := TypeCat.Fun)
            ((X.sheafToTypes T).obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u_2))))
            U sf' s)
        y →
      y = gl
refine_1 : P.toPrelocalPredicate (T := T).pred (T := T) gl
⊢ ∃! s : ToType (FC := TypeCat.Fun) ((subpresheafToTypes (T := T) (P.toPrelocalPredicate (T := T))).obj (op (iSup U))),
    (subpresheafToTypes (T := T) (P.toPrelocalPredicate (T := T))).IsGluing (FC := TypeCat.Fun) U sf s
have refine_2 : (fun
      (s :
        ToType (FC := TypeCat.Fun) ((subpresheafToTypes (T := T) (P.toPrelocalPredicate (T := T))).obj (op (iSup U)))) ↦
    (subpresheafToTypes (T := T) (P.toPrelocalPredicate (T := T))).IsGluing (FC := TypeCat.Fun) U sf s)
  (Subtype.mk (p := fun (f : (x : ↥(unop (op (iSup U)))) → T (↑x : (↑X : Type u_2))) ↦
    P.toPrelocalPredicate (T := T).pred (T := T) f) gl sorry)
Tactic: refine ⟨⟨gl, ?_⟩, ?_, ?_⟩
-- Lean 4 proof step from Mathlib.Topology.Sheaves.LocalPredicate (goal)
Goal:
X : TopCat
T : (↑X : Type u_2) → Type u_1
P : LocalPredicate T
ι : Type u_2
U : ι → Opens (↑X : Type u_2)
sf :
  (i : ι) → ToType (FC := TypeCat.Fun) ((subpresheafToTypes (T := T) (P.toPrelocalPredicate (T := T))).obj (op (U i)))
sf_comp : (subpresheafToTypes (T := T) (P.toPrelocalPredicate (T := T))).IsCompatible (FC := TypeCat.Fun) U sf
sf' : (i : ι) → (X.presheafToTypes T).obj (op (U i)) :=
  fun (i : ι) ↦ (↑(sf i) : (x : ↥(unop (op (U i)))) → T (↑x : (↑X : Type u_2)))
sf'_comp : (X.presheafToTypes T).IsCompatible (FC := TypeCat.Fun) U sf'
gl :
  ToType (FC := TypeCat.Fun)
    ((X.sheafToTypes T).obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u_2))).obj
      (op (iSup U)))
gl_spec :
  IsGluing (FC := TypeCat.Fun)
    ((X.sheafToTypes T).obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u_2)))) U sf'
    gl
gl_uniq :
  ∀
    (y :
      ToType (FC := TypeCat.Fun)
        ((X.sheafToTypes T).obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u_2))).obj
          (op (iSup U)))),
    (fun
            (s :
              ToType (FC := TypeCat.Fun)
                ((X.sheafToTypes T).obj (P :=
                      CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u_2))).obj
                  (op (iSup U)))) ↦
          IsGluing (FC := TypeCat.Fun)
            ((X.sheafToTypes T).obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u_2))))
            U sf' s)
        y →
      y = gl
refine_1 : P.toPrelocalPredicate (T := T).pred (T := T) gl
refine_2 :
  (fun
        (s :
          ToType (FC := TypeCat.Fun)
            ((subpresheafToTypes (T := T) (P.toPrelocalPredicate (T := T))).obj (op (iSup U)))) ↦
      (subpresheafToTypes (T := T) (P.toPrelocalPredicate (T := T))).IsGluing (FC := TypeCat.Fun) U sf s)
    (Subtype.mk (p := fun (f : (x : ↥(unop (op (iSup U)))) → T (↑x : (↑X : Type u_2))) ↦
      P.toPrelocalPredicate (T := T).pred (T := T) f) gl ⋯)
⊢ ∃! s : ToType (FC := TypeCat.Fun) ((subpresheafToTypes (T := T) (P.toPrelocalPredicate (T := T))).obj (op (iSup U))),
    (subpresheafToTypes (T := T) (P.toPrelocalPredicate (T := T))).IsGluing (FC := TypeCat.Fun) U sf s
have refine_3 : ∀ (y : ToType (FC := TypeCat.Fun) ((subpresheafToTypes (T := T) (P.toPrelocalPredicate (T := T))).obj (op (iSup U)))),
  (fun
          (s :
            ToType (FC := TypeCat.Fun)
              ((subpresheafToTypes (T := T) (P.toPrelocalPredicate (T := T))).obj (op (iSup U)))) ↦
        (subpresheafToTypes (T := T) (P.toPrelocalPredicate (T := T))).IsGluing (FC := TypeCat.Fun) U sf s)
      y →
    y =
      Subtype.mk (p := fun (f : (x : ↥(unop (op (iSup U)))) → T (↑x : (↑X : Type u_2))) ↦
        P.toPrelocalPredicate (T := T).pred (T := T) f) gl sorry
Tactic: refine ⟨⟨gl, ?_⟩, ?_, ?_⟩
-- Lean 4 proof step from Mathlib.Topology.Sheaves.LocalPredicate (goal)
Goal:
X : TopCat
T : (↑X : Type u_2) → Type u_1
P : LocalPredicate T
ι : Type u_2
U : ι → Opens (↑X : Type u_2)
sf :
  (i : ι) → ToType (FC := TypeCat.Fun) ((subpresheafToTypes (T := T) (P.toPrelocalPredicate (T := T))).obj (op (U i)))
sf_comp : (subpresheafToTypes (T := T) (P.toPrelocalPredicate (T := T))).IsCompatible (FC := TypeCat.Fun) U sf
sf' : (i : ι) → (X.presheafToTypes T).obj (op (U i)) :=
  fun (i : ι) ↦ (↑(sf i) : (x : ↥(unop (op (U i)))) → T (↑x : (↑X : Type u_2)))
sf'_comp : (X.presheafToTypes T).IsCompatible (FC := TypeCat.Fun) U sf'
gl :
  ToType (FC := TypeCat.Fun)
    ((X.sheafToTypes T).obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u_2))).obj
      (op (iSup U)))
gl_spec :
  IsGluing (FC := TypeCat.Fun)
    ((X.sheafToTypes T).obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u_2)))) U sf'
    gl
gl_uniq :
  ∀
    (y :
      ToType (FC := TypeCat.Fun)
        ((X.sheafToTypes T).obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u_2))).obj
          (op (iSup U)))),
    (fun
            (s :
              ToType (FC := TypeCat.Fun)
                ((X.sheafToTypes T).obj (P :=
                      CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u_2))).obj
                  (op (iSup U)))) ↦
          IsGluing (FC := TypeCat.Fun)
            ((X.sheafToTypes T).obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u_2))))
            U sf' s)
        y →
      y = gl
⊢ P.toPrelocalPredicate (T := T).pred (T := T) gl
have refine_1 : ∀ (x : ↥(unop (op (iSup U)))),
  ∃ (V : Opens (↑X : Type u_2)) (_ : (↑x : (↑X : Type u_2)) ∈ V) (i : V ⟶ unop (op (iSup U))),
    P.toPrelocalPredicate (T := T).pred (T := T) (U := V) fun (x : ↥V) ↦ gl ((i : (a : ↥V) → ↥(unop (op (iSup U)))) x)
Tactic: apply P.locality
-- Lean 4 proof step from Mathlib.Topology.Sheaves.LocalPredicate (goal)
Goal:
X : TopCat
T : (↑X : Type u_2) → Type u_1
P : LocalPredicate T
ι : Type u_2
U : ι → Opens (↑X : Type u_2)
sf :
  (i : ι) → ToType (FC := TypeCat.Fun) ((subpresheafToTypes (T := T) (P.toPrelocalPredicate (T := T))).obj (op (U i)))
sf_comp : (subpresheafToTypes (T := T) (P.toPrelocalPredicate (T := T))).IsCompatible (FC := TypeCat.Fun) U sf
sf' : (i : ι) → (X.presheafToTypes T).obj (op (U i)) :=
  fun (i : ι) ↦ (↑(sf i) : (x : ↥(unop (op (U i)))) → T (↑x : (↑X : Type u_2)))
sf'_comp : (X.presheafToTypes T).IsCompatible (FC := TypeCat.Fun) U sf'
gl :
  ToType (FC := TypeCat.Fun)
    ((X.sheafToTypes T).obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u_2))).obj
      (op (iSup U)))
gl_spec :
  IsGluing (FC := TypeCat.Fun)
    ((X.sheafToTypes T).obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u_2)))) U sf'
    gl
gl_uniq :
  ∀
    (y :
      ToType (FC := TypeCat.Fun)
        ((X.sheafToTypes T).obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u_2))).obj
          (op (iSup U)))),
    (fun
            (s :
              ToType (FC := TypeCat.Fun)
                ((X.sheafToTypes T).obj (P :=
                      CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u_2))).obj
                  (op (iSup U)))) ↦
          IsGluing (FC := TypeCat.Fun)
            ((X.sheafToTypes T).obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u_2))))
            U sf' s)
        y →
      y = gl
x : (↑X : Type u_2)
mem : x ∈ unop (op (iSup U))
⊢ ∃ (V : Opens (↑X : Type u_2)) (_ :
    (↑(Subtype.mk (p := fun (x : (↑X : Type u_2)) ↦ x ∈ unop (op (iSup U))) x mem) : (↑X : Type u_2)) ∈ V) (i :
    V ⟶ unop (op (iSup U))),
    P.toPrelocalPredicate (T := T).pred (T := T) (U := V) fun (x : ↥V) ↦ gl ((i : (a : ↥V) → ↥(unop (op (iSup U)))) x)
have refine_1 : ∀ (i : ι) (hi : x ∈ U i),
  ∃ (V : Opens (↑X : Type u_2)) (_ :
    (↑(Subtype.mk (p := fun (x : (↑X : Type u_2)) ↦ x ∈ unop (op (iSup U))) x mem) : (↑X : Type u_2)) ∈ V) (i :
    V ⟶ unop (op (iSup U))),
    P.toPrelocalPredicate (T := T).pred (T := T) (U := V) fun (x : ↥V) ↦ gl ((i : (a : ↥V) → ↥(unop (op (iSup U)))) x)
Tactic: choose i hi using Opens.mem_iSup.mp mem
-- Lean 4 proof step from Mathlib.Topology.Sheaves.LocalPredicate (goal)
Goal:
X : TopCat
T : (↑X : Type u_2) → Type u_1
P : LocalPredicate T
ι : Type u_2
U : ι → Opens (↑X : Type u_2)
sf :
  (i : ι) → ToType (FC := TypeCat.Fun) ((subpresheafToTypes (T := T) (P.toPrelocalPredicate (T := T))).obj (op (U i)))
sf_comp : (subpresheafToTypes (T := T) (P.toPrelocalPredicate (T := T))).IsCompatible (FC := TypeCat.Fun) U sf
sf' : (i : ι) → (X.presheafToTypes T).obj (op (U i)) :=
  fun (i : ι) ↦ (↑(sf i) : (x : ↥(unop (op (U i)))) → T (↑x : (↑X : Type u_2)))
sf'_comp : (X.presheafToTypes T).IsCompatible (FC := TypeCat.Fun) U sf'
gl :
  ToType (FC := TypeCat.Fun)
    ((X.sheafToTypes T).obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u_2))).obj
      (op (iSup U)))
gl_spec :
  IsGluing (FC := TypeCat.Fun)
    ((X.sheafToTypes T).obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u_2)))) U sf'
    gl
gl_uniq :
  ∀
    (y :
      ToType (FC := TypeCat.Fun)
        ((X.sheafToTypes T).obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u_2))).obj
          (op (iSup U)))),
    (fun
            (s :
              ToType (FC := TypeCat.Fun)
                ((X.sheafToTypes T).obj (P :=
                      CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u_2))).obj
                  (op (iSup U)))) ↦
          IsGluing (FC := TypeCat.Fun)
            ((X.sheafToTypes T).obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u_2))))
            U sf' s)
        y →
      y = gl
x : (↑X : Type u_2)
mem : x ∈ unop (op (iSup U))
i : ι
hi : x ∈ U i
⊢ ∃ (V : Opens (↑X : Type u_2)) (_ :
    (↑(Subtype.mk (p := fun (x : (↑X : Type u_2)) ↦ x ∈ unop (op (iSup U))) x mem) : (↑X : Type u_2)) ∈ V) (i :
    V ⟶ unop (op (iSup U))),
    P.toPrelocalPredicate (T := T).pred (T := T) (U := V) fun (x : ↥V) ↦ gl ((i : (a : ↥V) → ↥(unop (op (iSup U)))) x)
have h : P.toPrelocalPredicate (T := T).pred (T := T) (U := U i) fun (x : ↥(U i)) ↦
  gl ((Opens.leSupr U i : (a : ↥(U i)) → ↥(unop (op (iSup U)))) x)
Tactic: use U i, hi, Opens.leSupr U i
-- Lean 4 proof step from Mathlib.Topology.Sheaves.LocalPredicate (goal)
Goal:
X : TopCat
T : (↑X : Type u_2) → Type u_1
P : LocalPredicate T
ι : Type u_2
U : ι → Opens (↑X : Type u_2)
sf :
  (i : ι) → ToType (FC := TypeCat.Fun) ((subpresheafToTypes (T := T) (P.toPrelocalPredicate (T := T))).obj (op (U i)))
sf_comp : (subpresheafToTypes (T := T) (P.toPrelocalPredicate (T := T))).IsCompatible (FC := TypeCat.Fun) U sf
sf' : (i : ι) → (X.presheafToTypes T).obj (op (U i)) :=
  fun (i : ι) ↦ (↑(sf i) : (x : ↥(unop (op (U i)))) → T (↑x : (↑X : Type u_2)))
sf'_comp : (X.presheafToTypes T).IsCompatible (FC := TypeCat.Fun) U sf'
gl :
  ToType (FC := TypeCat.Fun)
    ((X.sheafToTypes T).obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u_2))).obj
      (op (iSup U)))
gl_spec :
  IsGluing (FC := TypeCat.Fun)
    ((X.sheafToTypes T).obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u_2)))) U sf'
    gl
gl_uniq :
  ∀
    (y :
      ToType (FC := TypeCat.Fun)
        ((X.sheafToTypes T).obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u_2))).obj
          (op (iSup U)))),
    (fun
            (s :
              ToType (FC := TypeCat.Fun)
                ((X.sheafToTypes T).obj (P :=
                      CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u_2))).obj
                  (op (iSup U)))) ↦
          IsGluing (FC := TypeCat.Fun)
            ((X.sheafToTypes T).obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u_2))))
            U sf' s)
        y →
      y = gl
x : (↑X : Type u_2)
mem : x ∈ unop (op (iSup U))
i : ι
hi : x ∈ U i
⊢ P.toPrelocalPredicate (T := T).pred (T := T) (U := U i) fun (x : ↥(U i)) ↦
    gl ((Opens.leSupr U i : (a : ↥(U i)) → ↥(unop (op (iSup U)))) x)
have h : (fun (x : ↥(U i)) ↦ gl ((Opens.leSupr U i : (a : ↥(U i)) → ↥(unop (op (iSup U)))) x)) =
  (↑(sf i) : (x : ↥(unop (op (U i)))) → T (↑x : (↑X : Type u_2)))
Tactic: convert! (sf i).property using 1
-- Lean 4 proof step from Mathlib.Topology.Sheaves.LocalPredicate (goal)
Goal:
X : TopCat
T : (↑X : Type u_2) → Type u_1
P : LocalPredicate T
ι : Type u_2
U : ι → Opens (↑X : Type u_2)
sf :
  (i : ι) → ToType (FC := TypeCat.Fun) ((subpresheafToTypes (T := T) (P.toPrelocalPredicate (T := T))).obj (op (U i)))
sf_comp : (subpresheafToTypes (T := T) (P.toPrelocalPredicate (T := T))).IsCompatible (FC := TypeCat.Fun) U sf
sf' : (i : ι) → (X.presheafToTypes T).obj (op (U i)) :=
  fun (i : ι) ↦ (↑(sf i) : (x : ↥(unop (op (U i)))) → T (↑x : (↑X : Type u_2)))
sf'_comp : (X.presheafToTypes T).IsCompatible (FC := TypeCat.Fun) U sf'
gl :
  ToType (FC := TypeCat.Fun)
    ((X.sheafToTypes T).obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u_2))).obj
      (op (iSup U)))
gl_spec :
  IsGluing (FC := TypeCat.Fun)
    ((X.sheafToTypes T).obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u_2)))) U sf'
    gl
gl_uniq :
  ∀
    (y :
      ToType (FC := TypeCat.Fun)
        ((X.sheafToTypes T).obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u_2))).obj
          (op (iSup U)))),
    (fun
            (s :
              ToType (FC := TypeCat.Fun)
                ((X.sheafToTypes T).obj (P :=
                      CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u_2))).obj
                  (op (iSup U)))) ↦
          IsGluing (FC := TypeCat.Fun)
            ((X.sheafToTypes T).obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u_2))))
            U sf' s)
        y →
      y = gl
gl' : ToType (FC := TypeCat.Fun) ((subpresheafToTypes (T := T) (P.toPrelocalPredicate (T := T))).obj (op (iSup U)))
hgl' : (subpresheafToTypes (T := T) (P.toPrelocalPredicate (T := T))).IsGluing (FC := TypeCat.Fun) U sf gl'
⊢ gl' =
    Subtype.mk (p := fun (f : (x : ↥(unop (op (iSup U)))) → T (↑x : (↑X : Type u_2))) ↦
      P.toPrelocalPredicate (T := T).pred (T := T) f) gl ⋯
have refine_3 : (↑gl' : (x : ↥(unop (op (iSup U)))) → T (↑x : (↑X : Type u_2))) =
  (↑(Subtype.mk (p := fun (f : (x : ↥(unop (op (iSup U)))) → T (↑x : (↑X : Type u_2))) ↦
        P.toPrelocalPredicate (T := T).pred (T := T) f) gl sorry) :
    (x : ↥(unop (op (iSup U)))) → T (↑x : (↑X : Type u_2)))
Tactic: refine Subtype.ext ?_
-- Lean 4 proof step from Mathlib.Topology.Sheaves.LocalPredicate (goal)
Goal:
X : TopCat
T : (↑X : Type u_1) → Type u_1
P : LocalPredicate T
U : Opens (↑X : Type u_1)
x : (↑X : Type u_1)
hx : x ∈ U
f : (fun (X : Type u_1) ↦ X) ((subsheafToTypes (T := T) P).presheaf.obj (op U))
⊢ (ConcreteCategory.hom (FC := TypeCat.Fun) (stalkToFiber (T := T) P x) :
        (a : (fun (X : Type u_1) ↦ X) ((subsheafToTypes P).presheaf.stalk x)) → (fun (X : Type u_1) ↦ X) (T x))
      ((ConcreteCategory.hom (FC := TypeCat.Fun) ((subsheafToTypes (T := T) P).presheaf.germ U x hx) :
          (a : (fun (X : Type u_1) ↦ X) ((subsheafToTypes P).presheaf.obj (op U))) →
            (fun (X : Type u_1) ↦ X) ((subsheafToTypes P).presheaf.stalk x))
        f) =
    (↑f : (x : ↥(unop (op U))) → T (↑x : (↑X : Type u_1)))
      (Subtype.mk (p := fun (x : (↑X : Type u_1)) ↦ x ∈ unop (op U)) x hx)
have dsimp : ((@ConcreteCategory.hom _ types TypeCat.Fun _ _ _ _ _
          (colimit.desc ((OpenNhds.inclusion x).op ⋙ subpresheafToTypes (T := T) (P.toPrelocalPredicate (T := T)))
            (P.cocone (T := T) x)) :
        TypeCat.Fun ((subpresheafToTypes (T := T) (P.toPrelocalPredicate (T := T))).stalk x) (T x)) :
      (a : (subpresheafToTypes P.toPrelocalPredicate).stalk x) → T x)
    (((@ConcreteCategory.hom _ types TypeCat.Fun _ _ _ _ _
            (colimit.ι ((OpenNhds.inclusion x).op ⋙ subpresheafToTypes (T := T) (P.toPrelocalPredicate (T := T)))
              (op (Subtype.mk (p := fun (U : Opens (↑X : Type u_1)) ↦ x ∈ U) U hx))) :
          TypeCat.Fun { f : (x : ↥U) → T (↑x : (↑X : Type u_1)) // P.toPrelocalPredicate (T := T).pred (T := T) f }
            ((subpresheafToTypes (T := T) (P.toPrelocalPredicate (T := T))).stalk x)) :
        (a : { f : (x : ↥U) → T (↑x : (↑X : Type u_1)) // P.pred f }) →
          (subpresheafToTypes P.toPrelocalPredicate).stalk x)
      f) =
  (↑f : (x : ↥U) → T (↑x : (↑X : Type u_1))) (Subtype.mk (p := fun (x : (↑X : Type u_1)) ↦ x ∈ U) x hx)
Tactic: dsimp [stalkToFiber, Presheaf.germ]
-- Lean 4 proof step from Mathlib.Topology.Sheaves.LocalPredicate (goal)
Goal:
X : TopCat
T : (↑X : Type u_1) → Type u_1
P : LocalPredicate T
x : (↑X : Type u_1)
w :
  ∀ (t : T x),
    ∃ (U : OpenNhds x) (f : (y : ↥(↑U : Opens (↑X : Type u_1))) → T (↑y : (↑X : Type u_1))) (_ :
      P.toPrelocalPredicate (T := T).pred (T := T) f),
      f (Subtype.mk (p := fun (x_2 : (↑X : Type u_1)) ↦ x_2 ∈ (↑U : Opens (↑X : Type u_1))) x ⋯) = t
t : (fun (X : Type u_1) ↦ X) (T x)
⊢ ∃ (a : (fun (X : Type u_1) ↦ X) ((subsheafToTypes (T := T) P).presheaf.stalk x)),
    (ConcreteCategory.hom (FC := TypeCat.Fun) (stalkToFiber (T := T) P x) :
          (a : (fun (X : Type u_1) ↦ X) ((subsheafToTypes P).presheaf.stalk x)) → (fun (X : Type u_1) ↦ X) (T x))
        a =
      t
have rcases : ∀ (U : OpenNhds x) (f : (y : ↥(↑U : Opens (↑X : Type u_1))) → T (↑y : (↑X : Type u_1)))
  (h : P.toPrelocalPredicate (T := T).pred (T := T) f),
  ∃ (a : (fun (X : Type u_1) ↦ X) ((subsheafToTypes (T := T) P).presheaf.stalk x)),
    Eq (α := (fun (X : Type u_1) ↦ X) (T x))
      ((ConcreteCategory.hom (FC := TypeCat.Fun) (stalkToFiber (T := T) P x) :
            (a : (fun (X : Type u_1) ↦ X) ((subsheafToTypes P).presheaf.stalk x)) → (fun (X : Type u_1) ↦ X) (T x))
          a :
        (fun (X : Type u_1) ↦ X) (T x))
      (f (Subtype.mk (p := fun (x_1 : (↑X : Type u_1)) ↦ x_1 ∈ (↑U : Opens (↑X : Type u_1))) x sorry) :
        T (↑(Subtype.mk (p := fun (x_1 : (↑X : Type u_1)) ↦ x_1 ∈ (↑U : Opens (↑X : Type u_1))) x sorry) : (↑X : Type u_1)))
Tactic: rcases w t with ⟨U, f, h, rfl⟩
Cleared hypotheses: t