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 mu_ex_eval_semantics2 :
forall t be : bool_expr,
be_ok (var_lu 0 N) be ->
forall ve : var_env',
(forall n : nat, ve n = true -> var_lu 0 N (N_of_nat n) = true) ->
(exists ve' : var_env',
(forall n : nat, ve' n = true -> var_lu 0 N (N_of_nat n) = true) /\
eval_be' t (var_env'_or ve (var_env'_dash ve')) = true /\
eval_be' be ve' = true) -> eval_be' (mu_ex_eval N t be) ve = true.