(* This program is free software; you can redistribute it and/or      *)
(* modify it under the terms of the GNU Lesser General Public License *)
(* as published by the Free Software Foundation; either version 2.1   *)
(* of the License, or (at your option) any later version.             *)
(*                                                                    *)
(* This program is distributed in the hope that it will be useful,    *)
(* but WITHOUT ANY WARRANTY; without even the implied warranty of     *)
(* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the      *)
(* GNU General Public License for more details.                       *)
(*                                                                    *)
(* You should have received a copy of the GNU Lesser General Public   *)
(* License along with this program; if not, write to the Free         *)
(* Software Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA *)
(* 02110-1301 USA                                                     *)
Definition BDDneg (cfg : BDDconfig) (memo : BDDneg_memo)
(node : ad) :=
match BDDneg_1_1 cfg memo node (S (nat_of_N (var cfg node))) with
| ((cfg', node'), memo') => (cfg', (node', memo'))
end.
Definition BDDor (cfg : BDDconfig) (memo : BDDor_memo)
(node1 node2 : ad) :=
BDDor_1_1 cfg memo node1 node2
(S (max (nat_of_N (var cfg node1)) (nat_of_N (var cfg node2)))).
Definition BDDand (cfg : BDDconfig) (negm : BDDneg_memo)
(orm : BDDor_memo) (node1 node2 : ad) :=
match BDDneg cfg negm node1 with
| (cfg', (node1', negm')) =>
match BDDneg cfg' negm' node2 with
| (cfg'', (node2', negm'')) =>
match BDDor cfg'' orm node1' node2' with
| (cfg''', (node, orm')) =>
match BDDneg cfg''' negm'' node with
| (cfg'''', (node', negm''')) =>
(cfg'''', (node', (negm''', orm')))
end
end
end
end.

(* Specifications: *)
Lemma nodes_preserved_orm_OK :
forall (cfg cfg' : BDDconfig) (orm : BDDor_memo),
BDDconfig_OK cfg ->
BDDconfig_OK cfg' ->
nodes_preserved cfg cfg' -> BDDor_memo_OK cfg orm -> BDDor_memo_OK cfg' orm.
Lemma nodes_preserved_negm_OK :
forall (cfg cfg' : BDDconfig) (negm : BDDneg_memo),
BDDconfig_OK cfg ->
BDDconfig_OK cfg' ->
nodes_preserved cfg cfg' ->
BDDneg_memo_OK_2 cfg negm -> BDDneg_memo_OK_2 cfg' negm.
Lemma BDDneg_keeps_config_OK :
forall (cfg : BDDconfig) (negm : BDDneg_memo) (node : ad),
BDDconfig_OK cfg ->
BDDneg_memo_OK_2 cfg negm ->
config_node_OK cfg node -> BDDconfig_OK (fst (BDDneg cfg negm node)).
Lemma BDDneg_node_OK :
forall (cfg : BDDconfig) (negm : BDDneg_memo) (node : ad),
BDDconfig_OK cfg ->
BDDneg_memo_OK_2 cfg negm ->
config_node_OK cfg node ->
config_node_OK (fst (BDDneg cfg negm node))
(fst (snd (BDDneg cfg negm node))).
Lemma BDDneg_preserves_nodes :
forall (cfg : BDDconfig) (negm : BDDneg_memo) (node : ad),
BDDconfig_OK cfg ->
BDDneg_memo_OK_2 cfg negm ->
config_node_OK cfg node -> nodes_preserved cfg (fst (BDDneg cfg negm node)).
Lemma BDDneg_keeps_neg_memo_OK :
forall (cfg : BDDconfig) (negm : BDDneg_memo) (node : ad),
BDDconfig_OK cfg ->
BDDneg_memo_OK_2 cfg negm ->
config_node_OK cfg node ->
BDDneg_memo_OK_2 (fst (BDDneg cfg negm node))
(snd (snd (BDDneg cfg negm node))).
Lemma BDDneg_keeps_or_memo_OK :
forall (cfg : BDDconfig) (negm : BDDneg_memo) (orm : BDDor_memo) (node : ad),
BDDconfig_OK cfg ->
BDDneg_memo_OK_2 cfg negm ->
config_node_OK cfg node ->
BDDor_memo_OK cfg orm -> BDDor_memo_OK (fst (BDDneg cfg negm node)) orm.
Lemma BDDneg_keeps_node_OK :
forall (cfg : BDDconfig) (negm : BDDneg_memo) (node : ad),
BDDconfig_OK cfg ->
BDDneg_memo_OK_2 cfg negm ->
config_node_OK cfg node ->
forall node' : ad,
config_node_OK cfg node' ->
config_node_OK (fst (BDDneg cfg negm node)) node'.
Lemma BDDneg_preserves_bool_fun :
forall (cfg : BDDconfig) (negm : BDDneg_memo) (node : ad),
BDDconfig_OK cfg ->
BDDneg_memo_OK_2 cfg negm ->
config_node_OK cfg node ->
forall node' : ad,
config_node_OK cfg node' ->
bool_fun_eq (bool_fun_of_BDD (fst (BDDneg cfg negm node)) node')
(bool_fun_of_BDD cfg node').
Lemma BDDneg_is_neg :
forall (cfg : BDDconfig) (negm : BDDneg_memo) (node : ad),
BDDconfig_OK cfg ->
BDDneg_memo_OK_2 cfg negm ->
config_node_OK cfg node ->
bool_fun_eq
(bool_fun_of_BDD (fst (BDDneg cfg negm node))
(fst (snd (BDDneg cfg negm node))))
(bool_fun_neg (bool_fun_of_BDD cfg node)).
Lemma BDDor_keeps_config_OK :
forall (cfg : BDDconfig) (orm : BDDor_memo) (node1 node2 : ad),
BDDconfig_OK cfg ->
BDDor_memo_OK cfg orm ->
config_node_OK cfg node1 ->
config_node_OK cfg node2 -> BDDconfig_OK (fst (BDDor cfg orm node1 node2)).
Lemma BDDor_node_OK :
forall (cfg : BDDconfig) (orm : BDDor_memo) (node1 node2 : ad),
BDDconfig_OK cfg ->
BDDor_memo_OK cfg orm ->
config_node_OK cfg node1 ->
config_node_OK cfg node2 ->
config_node_OK (fst (BDDor cfg orm node1 node2))
(fst (snd (BDDor cfg orm node1 node2))).
Lemma BDDor_keeps_or_memo_OK :
forall (cfg : BDDconfig) (orm : BDDor_memo) (node1 node2 : ad),
BDDconfig_OK cfg ->
BDDor_memo_OK cfg orm ->
config_node_OK cfg node1 ->
config_node_OK cfg node2 ->
BDDor_memo_OK (fst (BDDor cfg orm node1 node2))
(snd (snd (BDDor cfg orm node1 node2))).
Lemma BDDor_preserves_nodes :
forall (cfg : BDDconfig) (orm : BDDor_memo) (node1 node2 : ad),
BDDconfig_OK cfg ->
BDDor_memo_OK cfg orm ->
config_node_OK cfg node1 ->
config_node_OK cfg node2 ->
nodes_preserved cfg (fst (BDDor cfg orm node1 node2)).
Lemma BDDor_keeps_node_OK :
forall (cfg : BDDconfig) (orm : BDDor_memo) (node1 node2 : ad),
BDDconfig_OK cfg ->
BDDor_memo_OK cfg orm ->
config_node_OK cfg node1 ->
config_node_OK cfg node2 ->
forall node : ad,
config_node_OK cfg node ->
config_node_OK (fst (BDDor cfg orm node1 node2)) node.
Lemma BDDor_preserves_bool_fun :
forall (cfg : BDDconfig) (orm : BDDor_memo) (node1 node2 : ad),
BDDconfig_OK cfg ->
BDDor_memo_OK cfg orm ->
config_node_OK cfg node1 ->
config_node_OK cfg node2 ->
forall node : ad,
config_node_OK cfg node ->
bool_fun_eq (bool_fun_of_BDD (fst (BDDor cfg orm node1 node2)) node)
(bool_fun_of_BDD cfg node).
Lemma BDDor_keeps_neg_memo_OK :
forall (cfg : BDDconfig) (negm : BDDneg_memo) (orm : BDDor_memo)
(node1 node2 : ad),
BDDconfig_OK cfg ->
BDDor_memo_OK cfg orm ->
config_node_OK cfg node1 ->
config_node_OK cfg node2 ->
BDDneg_memo_OK_2 cfg negm ->
BDDneg_memo_OK_2 (fst (BDDor cfg orm node1 node2)) negm.
Lemma BDDor_is_or :
forall (cfg : BDDconfig) (orm : BDDor_memo) (node1 node2 : ad),
BDDconfig_OK cfg ->
BDDor_memo_OK cfg orm ->
config_node_OK cfg node1 ->
config_node_OK cfg node2 ->
bool_fun_eq
(bool_fun_of_BDD (fst (BDDor cfg orm node1 node2))
(fst (snd (BDDor cfg orm node1 node2))))
(bool_fun_or (bool_fun_of_BDD cfg node1) (bool_fun_of_BDD cfg node2)).
Lemma BDDand_keeps_config_OK :
forall (cfg : BDDconfig) (negm : BDDneg_memo) (orm : BDDor_memo)
(node1 node2 : ad),
BDDconfig_OK cfg ->
BDDneg_memo_OK_2 cfg negm ->
BDDor_memo_OK cfg orm ->
config_node_OK cfg node1 ->
config_node_OK cfg node2 ->
BDDconfig_OK (fst (BDDand cfg negm orm node1 node2)).
Lemma BDDand_node_OK :
forall (cfg : BDDconfig) (negm : BDDneg_memo) (orm : BDDor_memo)
(node1 node2 : ad),
BDDconfig_OK cfg ->
BDDneg_memo_OK_2 cfg negm ->
BDDor_memo_OK cfg orm ->
config_node_OK cfg node1 ->
config_node_OK cfg node2 ->
config_node_OK (fst (BDDand cfg negm orm node1 node2))
(fst (snd (BDDand cfg negm orm node1 node2))).
Lemma BDDand_preserves_nodes :
forall (cfg : BDDconfig) (negm : BDDneg_memo) (orm : BDDor_memo)
(node1 node2 : ad),
BDDconfig_OK cfg ->
BDDneg_memo_OK_2 cfg negm ->
BDDor_memo_OK cfg orm ->
config_node_OK cfg node1 ->
config_node_OK cfg node2 ->
nodes_preserved cfg (fst (BDDand cfg negm orm node1 node2)).
Lemma BDDand_keeps_node_OK :
forall (cfg : BDDconfig) (negm : BDDneg_memo) (orm : BDDor_memo)
(node1 node2 : ad),
BDDconfig_OK cfg ->
BDDneg_memo_OK_2 cfg negm ->
BDDor_memo_OK cfg orm ->
config_node_OK cfg node1 ->
config_node_OK cfg node2 ->
forall node : ad,
config_node_OK cfg node ->
config_node_OK (fst (BDDand cfg negm orm node1 node2)) node.
Lemma BDDand_preserves_bool_fun :
forall (cfg : BDDconfig) (negm : BDDneg_memo) (orm : BDDor_memo)
(node1 node2 : ad),
BDDconfig_OK cfg ->
BDDneg_memo_OK_2 cfg negm ->
BDDor_memo_OK cfg orm ->
config_node_OK cfg node1 ->
config_node_OK cfg node2 ->
forall node : ad,
config_node_OK cfg node ->
bool_fun_eq (bool_fun_of_BDD (fst (BDDand cfg negm orm node1 node2)) node)
(bool_fun_of_BDD cfg node).
Lemma BDDand_keeps_neg_memo_OK :
forall (cfg : BDDconfig) (negm : BDDneg_memo) (orm : BDDor_memo)
(node1 node2 : ad),
BDDconfig_OK cfg ->
BDDneg_memo_OK_2 cfg negm ->
BDDor_memo_OK cfg orm ->
config_node_OK cfg node1 ->
config_node_OK cfg node2 ->
BDDneg_memo_OK_2 (fst (BDDand cfg negm orm node1 node2))
(fst (snd (snd (BDDand cfg negm orm node1 node2)))).
Lemma BDDand_keeps_or_memo_OK :
forall (cfg : BDDconfig) (negm : BDDneg_memo) (orm : BDDor_memo)
(node1 node2 : ad),
BDDconfig_OK cfg ->
BDDneg_memo_OK_2 cfg negm ->
BDDor_memo_OK cfg orm ->
config_node_OK cfg node1 ->
config_node_OK cfg node2 ->
BDDor_memo_OK (fst (BDDand cfg negm orm node1 node2))
(snd (snd (snd (BDDand cfg negm orm node1 node2)))).