Definition var_env_or (ve1 ve2 : var_env) (x : ad) := ve1 x || ve2 x.
Definition var_env'_or (ve1 ve2 : var_env') (x : nat) := ve1 x || ve2 x.

Lemma bool_fun_of_be_ext1 :
forall (be : bool_expr) (ve ve' : var_env'),
(forall x : ad,
be_x_free x be = true -> ve (nat_of_N x) = ve' (nat_of_N x)) ->
eval_be' be ve = eval_be' be ve'.