Definition be_le2 (be1 be2 : bool_expr) :=
forall ve : var_env,
bool_fun_of_bool_expr be1 ve = true -> bool_fun_of_bool_expr be2 ve = true.

Lemma and_le2 :
forall be1 be2 be1' be2' : bool_expr,
be_le2 be1 be2 -> be_le2 be1' be2' -> be_le2 (ANd be1 be1') (ANd be2 be2').