Section Invariants.
Instance unit_PCM : Ghost := { valid a := True; Join_G a b c := True }.
Inductive list_join {P : Ghost} : Join (list (option G)) :=
| list_join_nil_l m: list_join nil m m
| list_join_nil_r m: list_join m nil m
| list_join_cons a1 a2 m1 m2 a3 m3: join a1 a2 a3 -> list_join m1 m2 m3 ->
list_join (a1 :: m1) (a2 :: m2) (a3 :: m3).
Existing Instance list_join.
Instance list_PCM (P : Ghost) : Ghost := { valid a := True; Join_G := list_join }.
Instance list_order A : @PCM_order (list_PCM (discrete_PCM A)) list_incl.
Instance set_PCM A : Ghost := { valid := fun _ : Ensemble A => True;
Join_G a b c := Disjoint a b /\ c = Union a b }.

Corollary ghost_set_remove : forall {A} g a (s : Ensemble A) (Hdec : forall b, b = a \/ b <> a),
In s a -> ghost_set g s = ghost_set g (Singleton a) * ghost_set g (Subtract s a).