Definition map := fpm A (lifted Share.Join_ba * B).Instance Join_map : Join map := Join_fpm _.Instance pa_map : Perm_alg map := Perm_fpm _ _.Instance sa_map : Sep_alg map := Sep_fpm _ _.Instance ca_map {CA: Canc_alg B} : Canc_alg map := Canc_fpm _.Instance da_map {DA: Disj_alg B} : Disj_alg map := @Disj_fpm _ _ _ _ _ _.Definition map_share (a:A) (m:map) : share :=
    match lookup_fpm m a with
    | Some (sh,_) => lifted_obj sh
    | None => Share.bot
    end.Definition map_val (a:A) (m:map) : option B :=
    match lookup_fpm m a with
    | Some (_,b) => Some b
    | None => None
    end.Definition empty_map : map := empty_fpm _ _.Definition map_upd (a:A) (b:B) (m:map) : option map :=
    match lookup_fpm m a with
    | Some (sh,_) =>
        if eq_dec (lifted_obj sh) fullshare
           then Some (insert_fpm _ a (sh,b) m)
           else None
    | None => None
    end.

Lemma map_val_bot : forall a m, map_val a m = None <-> map_share a m = Share.bot.