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.

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.