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_comp_join {key: Type}{A: Type}{KE: EqDec key}: forall ids (ge: env key A), join (restrict_env ids ge) (restrict_env_comp ids ge) ge.