Definition node_OK (bs : BDDstate) (node : ad) :=
    node = BDDzero \/ node = BDDone \/ in_dom _ node bs = true.

Lemma BDDalloc_semantics : forall (cfg : BDDconfig) (l r : ad) (x : BDDvar), BDDconfig_OK cfg -> node_OK (fst cfg) l -> node_OK (fst cfg) r -> Neqb l r <> true -> (forall (xl : BDDvar) (ll rl : ad), MapGet _ (fst cfg) l = Some (xl, (ll, rl)) -> BDDcompare xl x = Datatypes.Lt) -> (forall (xr : BDDvar) (lr rr : ad), MapGet _ (fst cfg) r = Some (xr, (lr, rr)) -> BDDcompare xr x = Datatypes.Lt) -> (forall (x' : BDDvar) (l' r' a : ad), MapGet _ (fst cfg) a = Some (x', (l', r')) -> (x, (l, r)) <> (x', (l', r'))) -> BDDconfig_OK (fst (BDDalloc cfg x l r)) /\ MapGet _ (fst (fst (BDDalloc cfg x l r))) (snd (BDDalloc cfg x l r)) = Some (x, (l, r)) /\ (forall (a l' r' : ad) (x' : BDDvar), (MapGet _ (fst (fst (BDDalloc cfg x l r))) a = Some (x', (l', r')) -> MapGet _ (fst cfg) a = Some (x', (l', r')) \/ snd (BDDalloc cfg x l r) = a) /\ (MapGet _ (fst cfg) a = Some (x', (l', r')) -> MapGet _ (fst (fst (BDDalloc cfg x l r))) a = Some (x', (l', r')))) /\ node_OK (fst (fst (BDDalloc cfg x l r))) (snd (BDDalloc cfg x l r)).