Definition bool_expr_to_var_env'' (L U : nat) (be : bool_expr) :
  Ensemble var_env'' :=
  fun ve =>
  eval_be' be (var_env''_to_env' ve) = true /\ In _ (Evar_env'' L U) ve.Definition be_le1 (L U : nat) (be1 be2 : bool_expr) :=
  forall ve : var_env'',
  In _ (bool_expr_to_var_env'' L U be1) ve ->
  In _ (bool_expr_to_var_env'' L U be2) ve.

(* Specifications: *)
Lemma var_env'_to_var_env''_lemma1 : forall (n L U : nat) (ve : var_env'), n = U - L -> exists ve'' : var_env'', In _ (Evar_env'' L U) ve'' /\ (forall x : ad, var_lu L U x = true -> in_dom _ x ve'' = ve (nat_of_N x)).
Lemma var_env'_to_var_env''_lemma2 : forall (n L U : nat) (ve : var_env'), n = U - L -> {ve'' : var_env'' | In _ (Evar_env'' L U) ve'' /\ (forall x : ad, var_lu L U x = true -> in_dom _ x ve'' = ve (nat_of_N x))}.