Definition restrict_env_comp' {key: Type}{A: Type}{KE: EqDec key} (ids: list key) (rho: env key A) (id: key) : option (pshare * A) :=
if  In_dec eq_dec id ids
then None
else env_get rho id.

Lemma restrict_env_comp'_finite {key: Type}{A: Type}{KE: EqDec key}:
forall ids (rho: env key A), finite_idfun (restrict_env_comp' ids rho).