is an     node.   gain, we start with showing that valid.

et None of the ethyle be the child of that corresponds tail se uence to the binder . . . . the inductive h pothesis, . . . . ( H ) is strongl valid.

(  ). Therefore,   is ei
( H ), since otherwise , N
ther contradicting, or blockwise uivalent to some
(H)
If    is     contradicting,   N is     contradicting too (proof b  contradic tion, as for  N   nodes). If   is blockwise e uivalent to   , then   N is blockwise e uivalent to 
( ) (the proof is eas ).   ence,
( ) is strongl valid. The onl orderings of cluded in the conting the dace algorithm have cheaper permutations of their leading ma  blocks, and therefore are     contradicting, b   emma 7.   ence,
( ) is strongl  valid.

□
Corollary 7 The candidate set found by Algorithm 6 for the root node is valid.

Corollary 8 Algorithm 6 finds a minimal ordering of the given set of subgoals.

## References

ho, V., opcroft, V.,
dison esle .

odd ,   .,     ean, T. (19 9).  olving time dependent planning problems.  In  ridha ran, N. . ( d.), Proceedings of the 11th International Joint Conference on Artificial Intelligence, pp. 979 9 , etroit, I,         organ Kaufmann.

ol, N., pt, K. N., Klop, . . (1991). n anal sis of loop checking mechanisms for logic programs. Theoretical Computer Science, 86 (1),    79.
raem, ., e harlier, ., odar, ., Van entenr ck, . (199). ardinalit nal sis of  rolog. In   ru  nooghe,      (  d.), Logic Programming - Proceedings of the 1994 International Symposium, pp. 7 71, assachusetts Institute of Technolog . The IT ress.

reiman,    .,    ried man, Regression Trees. adsworth International roup, elmont,

ru nooghe,      e   chre e, Logic Programming, 6, 1    1   .
lark, K. ., c abe, . (1979). The control facilities of I rolog. In richie, . ( d.),
Expert Systems in The Microelectronic Age., pp. 1    1 9.   niversit  of  dinburgh, cotland.

locksin, . . ., . ellish, . . (197). Programming in Prolog (Third edition). pringer Verlag, New ork.