awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.Topology.Sheaves.Stalks
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasColimits.{v, u} C
X : TopCat
FC : C → C → Type u_1
CC : C → Type v
inst_2 : (X Y : C) → FunLike.{u_1 + 1, v + 1, v + 1} (FC X Y) (CC X) (CC Y)
instCC : ConcreteCategory C FC (CC := CC)
inst_3 : PreservesFilteredColimits (forget C (FC := FC) (CC := CC))
F : Presheaf C X
x : (↑X : Type v)
t : ToType (FC := FC) (CC := CC) (F.stalk x)
U : (OpenNhds x)ᵒᵖ
s :
  (obj (obj (whiskeringLeft (OpenNhds x)ᵒᵖ (Opens (↑X : Type v))ᵒᵖ C) (OpenNhds.inclusion x).op) F ⋙
        forget C (FC := FC) (CC := CC)).obj
    U
e :
  (ConcreteCategory.hom (FC := TypeCat.Fun)
          (NatTrans.app
            (Cocone.ι
              (mapCocone (forget C (FC := FC) (CC := CC))
                (colimit.cocone
                  (obj (obj (whiskeringLeft (OpenNhds x)ᵒᵖ (Opens (↑X : Type v))ᵒᵖ C) (OpenNhds.inclusion x).op) F))))
            U) :
        (a :
            (fun (X : Type v) ↦ X)
              ((((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens (↑X : Type v))ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj F ⋙
                    forget C).obj
                U)) →
          (fun (X : Type v) ↦ X)
            (((Functor.const (OpenNhds x)ᵒᵖ).obj
                  ((forget C).mapCocone
                      (colimit.cocone
                        (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens (↑X : Type v))ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj
                          F))).pt).obj
              U))
      s =
    t
revert :
  ∀
    (s :
      (obj (obj (whiskeringLeft (OpenNhds x)ᵒᵖ (Opens (↑X : Type v))ᵒᵖ C) (OpenNhds.inclusion x).op) F ⋙
            forget C (FC := FC) (CC := CC)).obj
        U)
    (e :
      (ConcreteCategory.hom (FC := TypeCat.Fun)
              (NatTrans.app
                (Cocone.ι
                  (mapCocone (forget C (FC := FC) (CC := CC))
                    (colimit.cocone
                      (obj (obj (whiskeringLeft (OpenNhds x)ᵒᵖ (Opens (↑X : Type v))ᵒᵖ C) (OpenNhds.inclusion x).op)
                        F))))
                U) :
            (a :
                (fun (X : Type v) ↦ X)
                  ((((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens (↑X : Type v))ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj F ⋙
                        forget C).obj
                    U)) →
              (fun (X : Type v) ↦ X)
                (((Functor.const (OpenNhds x)ᵒᵖ).obj
                      ((forget C).mapCocone
                          (colimit.cocone
                            (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens (↑X : Type v))ᵒᵖ C).obj
                                  (OpenNhds.inclusion x).op).obj
                              F))).pt).obj
                  U))
          s =
        t),
    ∃ (U : Opens (↑X : Type v)) (m : x ∈ U) (s : ToType (FC := FC) (CC := CC) (F.obj (op U))),
      (ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ U x m) : (a : CC (F.obj (op U))) → CC (F.stalk x)) s = t
⊢ ∃ (U : Opens (↑X : Type v)) (m : x ∈ U) (s : ToType (FC := FC) (CC := CC) (F.obj (op U))),
    (ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ U x m) : (a : CC (F.obj (op U))) → CC (F.stalk x)) s = t

Tactic:
revert s e
-- Module: Mathlib.Topology.Sheaves.Stalks
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasColimits.{v, u} C
X : TopCat
FC : C → C → Type u_1
CC : C → Type v
inst_2 : (X Y : C) → FunLike.{u_1 + 1, v + 1, v + 1} (FC X Y) (CC X) (CC Y)
instCC : ConcreteCategory C FC (CC := CC)
inst_3 : PreservesFilteredColimits (forget C (FC := FC) (CC := CC))
F : Presheaf C X
x : (↑X : Type v)
t : ToType (FC := FC) (CC := CC) (F.stalk x)
U : OpenNhds x
⊢ ∀
    (s :
      (obj (obj (whiskeringLeft (OpenNhds x)ᵒᵖ (Opens (↑X : Type v))ᵒᵖ C) (OpenNhds.inclusion x).op) F ⋙
            forget C (FC := FC) (CC := CC)).obj
        (op U))
    (e :
      (ConcreteCategory.hom (FC := TypeCat.Fun)
              (NatTrans.app
                (Cocone.ι
                  (mapCocone (forget C (FC := FC) (CC := CC))
                    (colimit.cocone
                      (obj (obj (whiskeringLeft (OpenNhds x)ᵒᵖ (Opens (↑X : Type v))ᵒᵖ C) (OpenNhds.inclusion x).op)
                        F))))
                (op U)) :
            (a :
                (fun (X : Type v) ↦ X)
                  ((((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens (↑X : Type v))ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj F ⋙
                        forget C).obj
                    (op U))) →
              (fun (X : Type v) ↦ X)
                (((Functor.const (OpenNhds x)ᵒᵖ).obj
                      ((forget C).mapCocone
                          (colimit.cocone
                            (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens (↑X : Type v))ᵒᵖ C).obj
                                  (OpenNhds.inclusion x).op).obj
                              F))).pt).obj
                  (op U)))
          s =
        t),
    ∃ (U : Opens (↑X : Type v)) (m : x ∈ U) (s : ToType (FC := FC) (CC := CC) (F.obj (op U))),
      (ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ U x m) : (a : CC (F.obj (op U))) → CC (F.stalk x)) s = t

Tactic:
?_
-- Module: Mathlib.Topology.Sheaves.Stalks
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasColimits.{v, u} C
X : TopCat
FC : C → C → Type u_1
CC : C → Type v
inst_2 : (X Y : C) → FunLike.{u_1 + 1, v + 1, v + 1} (FC X Y) (CC X) (CC Y)
instCC : ConcreteCategory C FC (CC := CC)
inst_3 : PreservesFilteredColimits (forget C (FC := FC) (CC := CC))
F : Presheaf C X
x : (↑X : Type v)
t : ToType (FC := FC) (CC := CC) (F.stalk x)
V : Opens (↑X : Type v)
m : x ∈ V
s :
  (obj (obj (whiskeringLeft (OpenNhds x)ᵒᵖ (Opens (↑X : Type v))ᵒᵖ C) (OpenNhds.inclusion x).op) F ⋙
        forget C (FC := FC) (CC := CC)).obj
    (op (Subtype.mk (p := fun (U : Opens (↑X : Type v)) ↦ x ∈ U) V m))
e :
  (ConcreteCategory.hom (FC := TypeCat.Fun)
          (NatTrans.app
            (Cocone.ι
              (mapCocone (forget C (FC := FC) (CC := CC))
                (colimit.cocone
                  (obj (obj (whiskeringLeft (OpenNhds x)ᵒᵖ (Opens (↑X : Type v))ᵒᵖ C) (OpenNhds.inclusion x).op) F))))
            (op (Subtype.mk (p := fun (U : Opens (↑X : Type v)) ↦ x ∈ U) V m))) :
        (a :
            (fun (X : Type v) ↦ X)
              ((((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens (↑X : Type v))ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj F ⋙
                    forget C).obj
                (op ⟨V, m⟩))) →
          (fun (X : Type v) ↦ X)
            (((Functor.const (OpenNhds x)ᵒᵖ).obj
                  ((forget C).mapCocone
                      (colimit.cocone
                        (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens (↑X : Type v))ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj
                          F))).pt).obj
              (op ⟨V, m⟩)))
      s =
    t
⊢ ∃ (U : Opens (↑X : Type v)) (m : x ∈ U) (s : ToType (FC := FC) (CC := CC) (F.obj (op U))),
    (ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ U x m) : (a : CC (F.obj (op U))) → CC (F.stalk x)) s = t

Tactic:
⟨V, m, s, e⟩

Premises used: Exists.intro
-- Module: Mathlib.Topology.Sheaves.Stalks
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasColimits.{v, u} C
X : TopCat
FC : C → C → Type u_1
CC : C → Type v
inst_2 : (X Y : C) → FunLike.{u_1 + 1, v + 1, v + 1} (FC X Y) (CC X) (CC Y)
instCC : ConcreteCategory C FC (CC := CC)
inst_3 : PreservesFilteredColimits (forget C (FC := FC) (CC := CC))
F : Presheaf C X
x : (↑X : Type v)
t : ToType (FC := FC) (CC := CC) (F.stalk x)
V : Opens (↑X : Type v)
m : x ∈ V
s :
  (obj (obj (whiskeringLeft (OpenNhds x)ᵒᵖ (Opens (↑X : Type v))ᵒᵖ C) (OpenNhds.inclusion x).op) F ⋙
        forget C (FC := FC) (CC := CC)).obj
    (op (Subtype.mk (p := fun (U : Opens (↑X : Type v)) ↦ x ∈ U) V m))
e :
  (ConcreteCategory.hom (FC := TypeCat.Fun)
          (NatTrans.app
            (Cocone.ι
              (mapCocone (forget C (FC := FC) (CC := CC))
                (colimit.cocone
                  (obj (obj (whiskeringLeft (OpenNhds x)ᵒᵖ (Opens (↑X : Type v))ᵒᵖ C) (OpenNhds.inclusion x).op) F))))
            (op (Subtype.mk (p := fun (U : Opens (↑X : Type v)) ↦ x ∈ U) V m))) :
        (a :
            (fun (X : Type v) ↦ X)
              ((((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens (↑X : Type v))ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj F ⋙
                    forget C).obj
                (op ⟨V, m⟩))) →
          (fun (X : Type v) ↦ X)
            (((Functor.const (OpenNhds x)ᵒᵖ).obj
                  ((forget C).mapCocone
                      (colimit.cocone
                        (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens (↑X : Type v))ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj
                          F))).pt).obj
              (op ⟨V, m⟩)))
      s =
    t
⊢ ∃ (U : Opens (↑X : Type v)) (m : x ∈ U) (s : ToType (FC := FC) (CC := CC) (F.obj (op U))),
    (ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ U x m) : (a : CC (F.obj (op U))) → CC (F.stalk x)) s = t

Tactic:
exact ⟨V, m, s, e⟩
-- Module: Mathlib.Topology.Sheaves.Stalks
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasColimits.{v, u} C
X : TopCat
FC : C → C → Type u_1
CC : C → Type v
inst_2 : (X Y : C) → FunLike.{u_1 + 1, v + 1, v + 1} (FC X Y) (CC X) (CC Y)
instCC : ConcreteCategory C FC (CC := CC)
inst_3 : PreservesFilteredColimits (forget C (FC := FC) (CC := CC))
F : Presheaf C X
x : (↑X : Type v)
t : ToType (FC := FC) (CC := CC) (F.stalk x)
V : Opens (↑X : Type v)
hV : x ∈ V
rcases :
  ∀ (U : Opens (↑X : Type v)) (hxU : x ∈ U) (s : ToType (FC := FC) (CC := CC) (F.obj (op U))),
    ∃ U_1 ≤ V,
      ∃ (m : x ∈ U_1) (s_1 : ToType (FC := FC) (CC := CC) (F.obj (op U_1))),
        Eq (α := CC (F.stalk x))
          ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ U_1 x m) : (a : CC (F.obj (op U_1))) → CC (F.stalk x))
              s_1 :
            CC (F.stalk x))
          ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ U x hxU) : (a : CC (F.obj (op U))) → CC (F.stalk x)) s :
            CC (F.stalk x))
⊢ ∃ U ≤ V,
    ∃ (m : x ∈ U) (s : ToType (FC := FC) (CC := CC) (F.obj (op U))),
      (ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ U x m) : (a : CC (F.obj (op U))) → CC (F.stalk x)) s = t

Tactic:
rcases F.exists_germ_eq t with ⟨U, hxU, s, rfl⟩

Premises used: TopCat.Presheaf.exists_germ_eq, rfl
-- Module: Mathlib.Topology.Sheaves.Stalks
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasColimits.{v, u} C
X : TopCat
FC : C → C → Type u_1
CC : C → Type v
inst_2 : (X Y : C) → FunLike.{u_1 + 1, v + 1, v + 1} (FC X Y) (CC X) (CC Y)
instCC : ConcreteCategory C FC (CC := CC)
inst_3 : PreservesFilteredColimits (forget C (FC := FC) (CC := CC))
F : Presheaf C X
x : (↑X : Type v)
V : Opens (↑X : Type v)
hV : x ∈ V
U : Opens (↑X : Type v)
hxU : x ∈ U
s : ToType (FC := FC) (CC := CC) (F.obj (op U))
⊢ x ∈ U ⊓ V

Tactic:
simp [*]
-- Module: Mathlib.Topology.Sheaves.Stalks
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasColimits.{v, u} C
X : TopCat
FC : C → C → Type u_1
CC : C → Type v
inst_2 : (X Y : C) → FunLike.{u_1 + 1, v + 1, v + 1} (FC X Y) (CC X) (CC Y)
instCC : ConcreteCategory C FC (CC := CC)
inst_3 : PreservesFilteredColimits (forget C (FC := FC) (CC := CC))
F : Presheaf C X
x : (↑X : Type v)
V : Opens (↑X : Type v)
hV : x ∈ V
U : Opens (↑X : Type v)
hxU : x ∈ U
s : ToType (FC := FC) (CC := CC) (F.obj (op U))
refine :
  Eq (α := CC (F.stalk x))
    ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ (U ⊓ V) x ⋯) : (a : CC (F.obj (op (U ⊓ V)))) → CC (F.stalk x))
        ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.map (Quiver.Hom.op (homOfLE ⋯))) :
              (a : CC (F.obj (op U))) → CC (F.obj (op (U ⊓ V))))
            s :
          CC (F.obj (op (U ⊓ V)))) :
      CC (F.stalk x))
    ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ U x hxU) : (a : CC (F.obj (op U))) → CC (F.stalk x)) s :
      CC (F.stalk x))
⊢ ∃ U_1 ≤ V,
    ∃ (m : x ∈ U_1) (s_1 : ToType (FC := FC) (CC := CC) (F.obj (op U_1))),
      Eq (α := CC (F.stalk x))
        ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ U_1 x m) : (a : CC (F.obj (op U_1))) → CC (F.stalk x))
            s_1 :
          CC (F.stalk x))
        ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ U x hxU) : (a : CC (F.obj (op U))) → CC (F.stalk x)) s :
          CC (F.stalk x))

Tactic:
refine ⟨U ⊓ V, inf_le_right, by simp [*], F.map (homOfLE inf_le_left).op s, ?_⟩
-- Module: Mathlib.Topology.Sheaves.Stalks
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasColimits.{v, u} C
X : TopCat
FC : C → C → Type u_1
CC : C → Type v
inst_2 : (X Y : C) → FunLike.{u_1 + 1, v + 1, v + 1} (FC X Y) (CC X) (CC Y)
instCC : ConcreteCategory C FC (CC := CC)
inst_3 : PreservesFilteredColimits (forget C (FC := FC) (CC := CC))
F : Presheaf C X
x : (↑X : Type v)
V : Opens (↑X : Type v)
hV : x ∈ V
U : Opens (↑X : Type v)
hxU : x ∈ U
s : ToType (FC := FC) (CC := CC) (F.obj (op U))
⊢ Eq (α := CC (F.stalk x))
    ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ (U ⊓ V) x ⋯) : (a : CC (F.obj (op (U ⊓ V)))) → CC (F.stalk x))
        ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.map (Quiver.Hom.op (homOfLE ⋯))) :
              (a : CC (F.obj (op U))) → CC (F.obj (op (U ⊓ V))))
            s :
          CC (F.obj (op (U ⊓ V)))) :
      CC (F.stalk x))
    ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ U x hxU) : (a : CC (F.obj (op U))) → CC (F.stalk x)) s :
      CC (F.stalk x))

Tactic:
germ_res_apply ..

Premises used: TopCat.Presheaf.germ_res_apply
-- Module: Mathlib.Topology.Sheaves.Stalks
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasColimits.{v, u} C
X : TopCat
FC : C → C → Type u_1
CC : C → Type v
inst_2 : (X Y : C) → FunLike.{u_1 + 1, v + 1, v + 1} (FC X Y) (CC X) (CC Y)
instCC : ConcreteCategory C FC (CC := CC)
inst_3 : PreservesFilteredColimits (forget C (FC := FC) (CC := CC))
F : Presheaf C X
x : (↑X : Type v)
V : Opens (↑X : Type v)
hV : x ∈ V
U : Opens (↑X : Type v)
hxU : x ∈ U
s : ToType (FC := FC) (CC := CC) (F.obj (op U))
⊢ Eq (α := CC (F.stalk x))
    ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ (U ⊓ V) x ⋯) : (a : CC (F.obj (op (U ⊓ V)))) → CC (F.stalk x))
        ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.map (Quiver.Hom.op (homOfLE ⋯))) :
              (a : CC (F.obj (op U))) → CC (F.obj (op (U ⊓ V))))
            s :
          CC (F.obj (op (U ⊓ V)))) :
      CC (F.stalk x))
    ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ U x hxU) : (a : CC (F.obj (op U))) → CC (F.stalk x)) s :
      CC (F.stalk x))

Tactic:
exact germ_res_apply ..

Premises used: TopCat.Presheaf.germ_res_apply
-- Module: Mathlib.Topology.Sheaves.Stalks
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasColimits.{v, u} C
X : TopCat
FC : C → C → Type u_1
CC : C → Type v
inst_2 : (X Y : C) → FunLike.{u_1 + 1, v + 1, v + 1} (FC X Y) (CC X) (CC Y)
instCC : ConcreteCategory C FC (CC := CC)
inst_3 : PreservesFilteredColimits (forget C (FC := FC) (CC := CC))
F : Presheaf C X
U V : Opens (↑X : Type v)
x : (↑X : Type v)
mU : x ∈ U
mV : x ∈ V
s : ToType (FC := FC) (CC := CC) (F.obj (op U))
t : ToType (FC := FC) (CC := CC) (F.obj (op V))
h :
  Eq (α := CC (F.stalk x))
    ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ U x mU) : (a : CC (F.obj (op U))) → CC (F.stalk x)) s :
      CC (F.stalk x))
    ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ V x mV) : (a : CC (F.obj (op V))) → CC (F.stalk x)) t :
      CC (F.stalk x))
obtain :
  ∀ (W : (OpenNhds x)ᵒᵖ) (iU : op (Subtype.mk (p := fun (U : Opens (↑X : Type v)) ↦ x ∈ U) U mU) ⟶ W)
    (iV : op (Subtype.mk (p := fun (U : Opens (↑X : Type v)) ↦ x ∈ U) V mV) ⟶ W)
    (e :
      Eq (α := CC (((OpenNhds.inclusion x).op ⋙ F).obj W))
        ((ConcreteCategory.hom (FC := FC) (CC := CC) (((OpenNhds.inclusion x).op ⋙ F).map iU) :
              (a : CC (((OpenNhds.inclusion x).op ⋙ F).obj (op ⟨U, mU⟩))) → CC (((OpenNhds.inclusion x).op ⋙ F).obj W))
            s :
          CC (((OpenNhds.inclusion x).op ⋙ F).obj W))
        ((ConcreteCategory.hom (FC := FC) (CC := CC) (((OpenNhds.inclusion x).op ⋙ F).map iV) :
              (a : CC (((OpenNhds.inclusion x).op ⋙ F).obj (op ⟨V, mV⟩))) → CC (((OpenNhds.inclusion x).op ⋙ F).obj W))
            t :
          CC (((OpenNhds.inclusion x).op ⋙ F).obj W))),
    ∃ (W : Opens (↑X : Type v)) (_ : x ∈ W) (iU : W ⟶ U) (iV : W ⟶ V),
      Eq (α := CC (F.obj (op W)))
        ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.map iU.op) : (a : CC (F.obj (op U))) → CC (F.obj (op W))) s :
          CC (F.obj (op W)))
        ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.map iV.op) : (a : CC (F.obj (op V))) → CC (F.obj (op W))) t :
          CC (F.obj (op W)))
⊢ ∃ (W : Opens (↑X : Type v)) (_ : x ∈ W) (iU : W ⟶ U) (iV : W ⟶ V),
    Eq (α := CC (F.obj (op W)))
      ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.map iU.op) : (a : CC (F.obj (op U))) → CC (F.obj (op W))) s :
        CC (F.obj (op W)))
      ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.map iV.op) : (a : CC (F.obj (op V))) → CC (F.obj (op W))) t :
        CC (F.obj (op W)))

Tactic:
obtain ⟨W, iU, iV, e⟩ := (colimit.isColimit ((OpenNhds.inclusion x).op ⋙ F)).eq_iff.mp h

Premises used: CategoryTheory.Functor.op, CategoryTheory.Functor.comp, CategoryTheory.Limits.colimit.isColimit, CategoryTheory.Limits.IsColimit.eq_iff, Iff.mp, TopologicalSpace.OpenNhds.inclusion
-- Module: Mathlib.Topology.Sheaves.Stalks
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasColimits.{v, u} C
X : TopCat
FC : C → C → Type u_1
CC : C → Type v
inst_2 : (X Y : C) → FunLike.{u_1 + 1, v + 1, v + 1} (FC X Y) (CC X) (CC Y)
instCC : ConcreteCategory C FC (CC := CC)
inst_3 : PreservesFilteredColimits (forget C (FC := FC) (CC := CC))
F : Presheaf C X
U V : Opens (↑X : Type v)
x : (↑X : Type v)
mU : x ∈ U
mV : x ∈ V
s : ToType (FC := FC) (CC := CC) (F.obj (op U))
t : ToType (FC := FC) (CC := CC) (F.obj (op V))
h :
  Eq (α := CC (F.stalk x))
    ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ U x mU) : (a : CC (F.obj (op U))) → CC (F.stalk x)) s :
      CC (F.stalk x))
    ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ V x mV) : (a : CC (F.obj (op V))) → CC (F.stalk x)) t :
      CC (F.stalk x))
W : (OpenNhds x)ᵒᵖ
iU : op (Subtype.mk (p := fun (U : Opens (↑X : Type v)) ↦ x ∈ U) U mU) ⟶ W
iV : op (Subtype.mk (p := fun (U : Opens (↑X : Type v)) ↦ x ∈ U) V mV) ⟶ W
e :
  Eq (α := CC (((OpenNhds.inclusion x).op ⋙ F).obj W))
    ((ConcreteCategory.hom (FC := FC) (CC := CC) (((OpenNhds.inclusion x).op ⋙ F).map iU) :
          (a : CC (((OpenNhds.inclusion x).op ⋙ F).obj (op ⟨U, mU⟩))) → CC (((OpenNhds.inclusion x).op ⋙ F).obj W))
        s :
      CC (((OpenNhds.inclusion x).op ⋙ F).obj W))
    ((ConcreteCategory.hom (FC := FC) (CC := CC) (((OpenNhds.inclusion x).op ⋙ F).map iV) :
          (a : CC (((OpenNhds.inclusion x).op ⋙ F).obj (op ⟨V, mV⟩))) → CC (((OpenNhds.inclusion x).op ⋙ F).obj W))
        t :
      CC (((OpenNhds.inclusion x).op ⋙ F).obj W))
⊢ ∃ (W : Opens (↑X : Type v)) (_ : x ∈ W) (iU : W ⟶ U) (iV : W ⟶ V),
    Eq (α := CC (F.obj (op W)))
      ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.map iU.op) : (a : CC (F.obj (op U))) → CC (F.obj (op W))) s :
        CC (F.obj (op W)))
      ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.map iV.op) : (a : CC (F.obj (op V))) → CC (F.obj (op W))) t :
        CC (F.obj (op W)))

Tactic:
⟨(unop W).1, (unop W).2, iU.unop, iV.unop, e⟩

Premises used: Quiver.Hom.unop, Exists.intro, Subtype.val, Opposite.unop, Subtype.property
-- Module: Mathlib.Topology.Sheaves.Stalks
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasColimits.{v, u} C
X : TopCat
FC : C → C → Type u_1
CC : C → Type v
inst_2 : (X Y : C) → FunLike.{u_1 + 1, v + 1, v + 1} (FC X Y) (CC X) (CC Y)
instCC : ConcreteCategory C FC (CC := CC)
inst_3 : PreservesFilteredColimits (forget C (FC := FC) (CC := CC))
F : Presheaf C X
U V : Opens (↑X : Type v)
x : (↑X : Type v)
mU : x ∈ U
mV : x ∈ V
s : ToType (FC := FC) (CC := CC) (F.obj (op U))
t : ToType (FC := FC) (CC := CC) (F.obj (op V))
h :
  Eq (α := CC (F.stalk x))
    ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ U x mU) : (a : CC (F.obj (op U))) → CC (F.stalk x)) s :
      CC (F.stalk x))
    ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ V x mV) : (a : CC (F.obj (op V))) → CC (F.stalk x)) t :
      CC (F.stalk x))
W : (OpenNhds x)ᵒᵖ
iU : op (Subtype.mk (p := fun (U : Opens (↑X : Type v)) ↦ x ∈ U) U mU) ⟶ W
iV : op (Subtype.mk (p := fun (U : Opens (↑X : Type v)) ↦ x ∈ U) V mV) ⟶ W
e :
  Eq (α := CC (((OpenNhds.inclusion x).op ⋙ F).obj W))
    ((ConcreteCategory.hom (FC := FC) (CC := CC) (((OpenNhds.inclusion x).op ⋙ F).map iU) :
          (a : CC (((OpenNhds.inclusion x).op ⋙ F).obj (op ⟨U, mU⟩))) → CC (((OpenNhds.inclusion x).op ⋙ F).obj W))
        s :
      CC (((OpenNhds.inclusion x).op ⋙ F).obj W))
    ((ConcreteCategory.hom (FC := FC) (CC := CC) (((OpenNhds.inclusion x).op ⋙ F).map iV) :
          (a : CC (((OpenNhds.inclusion x).op ⋙ F).obj (op ⟨V, mV⟩))) → CC (((OpenNhds.inclusion x).op ⋙ F).obj W))
        t :
      CC (((OpenNhds.inclusion x).op ⋙ F).obj W))
⊢ ∃ (W : Opens (↑X : Type v)) (_ : x ∈ W) (iU : W ⟶ U) (iV : W ⟶ V),
    Eq (α := CC (F.obj (op W)))
      ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.map iU.op) : (a : CC (F.obj (op U))) → CC (F.obj (op W))) s :
        CC (F.obj (op W)))
      ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.map iV.op) : (a : CC (F.obj (op V))) → CC (F.obj (op W))) t :
        CC (F.obj (op W)))

Tactic:
exact ⟨(unop W).1, (unop W).2, iU.unop, iV.unop, e⟩
-- Module: Mathlib.Topology.Sheaves.Stalks
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasColimits.{v, u} C
X : TopCat
FC : C → C → Type u_1
CC : C → Type v
inst_2 : (X Y : C) → FunLike.{u_1 + 1, v + 1, v + 1} (FC X Y) (CC X) (CC Y)
instCC : ConcreteCategory C FC (CC := CC)
inst_3 : PreservesFilteredColimits (forget C (FC := FC) (CC := CC))
F G : Presheaf C X
f : F ⟶ G
h :
  ∀ (U : Opens (↑X : Type v)),
    Function.Injective.{v + 1, v + 1} (α := CC (F.obj (op U))) (β := CC (G.obj (op U)))
      (⇑(ConcreteCategory.hom (FC := FC) (CC := CC) (f.app (op U))) : (a : CC (F.obj (op U))) → CC (G.obj (op U)))
x : (↑X : Type v)
s t : CC (obj (stalkFunctor C x) F)
hst :
  Eq (α := CC (obj (stalkFunctor C x) G))
    ((ConcreteCategory.hom (FC := FC) (CC := CC) (map (stalkFunctor C x) f) :
          (a : CC ((stalkFunctor C x).obj F)) → CC ((stalkFunctor C x).obj G))
        s :
      CC (obj (stalkFunctor C x) G))
    ((ConcreteCategory.hom (FC := FC) (CC := CC) (map (stalkFunctor C x) f) :
          (a : CC ((stalkFunctor C x).obj F)) → CC ((stalkFunctor C x).obj G))
        t :
      CC (obj (stalkFunctor C x) G))
rcases :
  ∀ (U₁ : Opens (↑X : Type v)) (hxU₁ : x ∈ U₁) (s : ToType (FC := FC) (CC := CC) (F.obj (op U₁)))
    (hst :
      Eq (α := CC (obj (stalkFunctor C x) G))
        ((ConcreteCategory.hom (FC := FC) (CC := CC) (map (stalkFunctor C x) f) :
              (a : CC ((stalkFunctor C x).obj F)) → CC ((stalkFunctor C x).obj G))
            ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ U₁ x hxU₁) : (a : CC (F.obj (op U₁))) → CC (F.stalk x))
                s :
              CC (F.stalk x)) :
          CC (obj (stalkFunctor C x) G))
        ((ConcreteCategory.hom (FC := FC) (CC := CC) (map (stalkFunctor C x) f) :
              (a : CC ((stalkFunctor C x).obj F)) → CC ((stalkFunctor C x).obj G))
            t :
          CC (obj (stalkFunctor C x) G))),
    (ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ U₁ x hxU₁) : (a : CC (F.obj (op U₁))) → CC (F.stalk x)) s = t
⊢ s = t

Tactic:
rcases exists_germ_eq F s with ⟨U₁, hxU₁, s, rfl⟩

Premises used: TopCat.Presheaf.exists_germ_eq, rfl