Definition bool_fun_impl (bf1 bf2 : bool_fun) : bool_fun :=
  fun vb : var_binding => implb (bf1 vb) (bf2 vb).

(* Specifications: *)
Lemma bool_fun_impl_is_neg_or_bf2 : forall bf1 bf2 : bool_fun, bool_fun_eq (bool_fun_impl bf1 bf2) (bool_fun_or (bool_fun_neg bf1) bf2).
Lemma BDDimpl_is_impl : 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 -> bool_fun_eq (bool_fun_of_BDD (fst (BDDimpl cfg negm orm node1 node2)) (fst (snd (BDDimpl cfg negm orm node1 node2)))) (bool_fun_impl (bool_fun_of_BDD cfg node1) (bool_fun_of_BDD cfg node2)).
Lemma BDDimpl_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 (BDDimpl cfg negm orm node1 node2)).
Lemma BDDimpl_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 (BDDimpl cfg negm orm node1 node2)) node.
Lemma BDDimpl_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 (BDDimpl cfg negm orm node1 node2)) node) (bool_fun_of_BDD cfg node).
Lemma BDDimpl_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 (BDDimpl cfg negm orm node1 node2)) (fst (snd (snd (BDDimpl cfg negm orm node1 node2)))).
Lemma BDDimpl_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 (BDDimpl cfg negm orm node1 node2)) (snd (snd (snd (BDDimpl cfg negm orm node1 node2)))).