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 forall_lemma1 :
forall (be : bool_expr) (ve : var_env) (a : ad),
bool_fun_of_bool_expr (forall_ a be) ve = true ->
bool_fun_of_bool_expr be ve = true.