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))}.