Instance Sep_env {SA: Sep_alg A}: @Sep_alg env Join_env.
refine (mkSep Join_env (fun _ => empty_env) _ _).
repeat intro; rewrite env_get_empty; constructor.
auto.
Defined.
Instance Sing_env {SA: Sep_alg A}: @Sing_alg env Join_env Sep_env.
refine (mkSing empty_env _). reflexivity.
Defined.
Instance Canc_env {PA: Perm_alg A}{CA: Canc_alg A}: @Canc_alg env Join_env.
Instance Disj_env {PA: Perm_alg A}{DA: Disj_alg A}: @Disj_alg env Join_env.
Instance Cross_env: Cross_alg env.
Definition _env_mapsto {KE: EqDec key} (id: key) (sh: Share.t) (v: A) : pred env :=
fun rho => exists p,
forall id', env_get rho id' = if eq_dec id id' then Some (exist _ sh p,v) else None.

Lemma _env_mapsto_set_sh{KE: EqDec key}: forall id (sh: pshare) v,
_env_mapsto id (pshare_sh sh) v (env_set_sh id (Some (sh,v)) empty_env).