Definition restrict_env_comp  {key: Type}{A: Type}{KE: EqDec key} (ids: list key) (rho:env key A) : env key A:=
  mk_env _ (restrict_env_comp'_finite ids rho).

Lemma restrict_env_app {key: Type}{A: Type}{KE: EqDec key} : forall ids1 ids2 (rho: env key A), list_disjoint ids1 ids2 -> join (restrict_env ids1 rho) (restrict_env ids2 rho) (restrict_env (ids1++ids2) rho).