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 exl_semantics :
forall (lx : list ad) (be : bool_expr) (ve : var_env'),
(forall n : nat, ve n = true -> ~ In (N_of_nat n) lx) ->
no_dup_list _ lx ->
(eval_be' (exl be lx) ve = true <->
(exists ve' : var_env',
(forall n : nat, ve' n = true -> In (N_of_nat n) lx) /\
eval_be' be (var_env'_or ve ve') = true)).