Instance Perm_env {PA: Perm_alg A}: @Perm_alg env Join_env.Instance Sep_env {SA: Sep_alg A}: @Sep_alg env Join_env.Instance Sing_env {SA: Sep_alg A}: @Sing_alg env Join_env Sep_env.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_splittable {KE: EqDec key}: forall id v (sh sh1 sh2: pshare) rho, join (proj1_sig sh1) (proj1_sig sh2) (proj1_sig sh) -> (_env_mapsto id (pshare_sh sh) v rho <-> (_env_mapsto id (pshare_sh sh1) v * _env_mapsto id (pshare_sh sh2) v)%pred rho).