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_all_eval_semantics1 : 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) -> eval_be' (mu_all_eval N t be) ve = true -> forall 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.