(* An interpretation of BDDvar's *)
Definition var_binding := BDDvar -> bool.
(* A boolean function. However not all boolean functions represent a logical formula. *)
Definition bool_fun := var_binding -> bool.
(* Evaluation of a boolean function given a binding of variables *)
Definition bool_fun_eval (bf : bool_fun) (vb : var_binding) := bf vb.
(* The boolean functions representing ZERO and ONE *)
Definition bool_fun_zero (vb : var_binding) := false.
Definition bool_fun_one (vb : var_binding) := true.
Fixpoint bool_fun_of_BDD_1 (cfg : BDDconfig) (node : ad)
(bound : nat) {struct bound} : bool_fun :=
match MapGet _ (fst cfg) node with
| None => if Neqb node BDDzero then bool_fun_zero else bool_fun_one
| Some (x, (l, r)) =>
match bound with
| O => (* Error *)  bool_fun_zero
| S bound' =>
let bfl := bool_fun_of_BDD_1 cfg l bound' in
let bfr := bool_fun_of_BDD_1 cfg r bound' in
fun vb : var_binding => if vb x then bfr vb else bfl vb
end
end.

(* Specifications: *)
Lemma bool_fun_of_BDD_1_semantics :
forall cfg : BDDconfig,
BDDconfig_OK cfg ->
forall bound : nat,
bool_fun_of_BDD_1 cfg BDDzero bound = bool_fun_zero /\
bool_fun_of_BDD_1 cfg BDDone bound = bool_fun_one /\
(forall node : ad,
is_internal_node cfg node ->
nat_of_N (var cfg node) < bound ->
forall vb : var_binding,
(vb (var cfg node) = true ->
bool_fun_eval (bool_fun_of_BDD_1 cfg node bound) vb =
bool_fun_eval (bool_fun_of_BDD_1 cfg (high cfg node) (pred bound)) vb) /\
(vb (var cfg node) = false ->
bool_fun_eval (bool_fun_of_BDD_1 cfg node bound) vb =
bool_fun_eval (bool_fun_of_BDD_1 cfg (low cfg node) (pred bound)) vb)).
Lemma bool_fun_of_BDD_1_semantics_1 :
forall cfg : BDDconfig,
BDDconfig_OK cfg ->
forall bound : nat,
bool_fun_of_BDD_1 cfg BDDzero bound = bool_fun_zero /\
bool_fun_of_BDD_1 cfg BDDone bound = bool_fun_one /\
(forall node : ad,
is_internal_node cfg node ->
nat_of_N (var cfg node) < bound ->
bool_fun_of_BDD_1 cfg node bound =
(fun vb : var_binding =>
if vb (var cfg node)
then bool_fun_of_BDD_1 cfg (high cfg node) (pred bound) vb
else bool_fun_of_BDD_1 cfg (low cfg node) (pred bound) vb)).
Lemma bool_fun_of_BDD_1_semantics_2 :
forall (cfg : BDDconfig) (node : ad) (bound1 bound2 : nat),
MapGet _ (fst cfg) node = None ->
bool_fun_of_BDD_1 cfg node bound1 = bool_fun_of_BDD_1 cfg node bound2.
Lemma bool_fun_of_BDD_1_change_bound :
forall cfg : BDDconfig,
BDDconfig_OK cfg ->
forall (bound : nat) (node : ad),
nat_of_N (var cfg node) < bound ->
bool_fun_of_BDD_1 cfg node bound =
bool_fun_of_BDD_1 cfg node (S (nat_of_N (var cfg node))).