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 identity_empty_env {key: Type}{A: Type}{KE: EqDec key}: forall rho: env key A, identity rho <-> rho = empty_env.