Operational and Goal-Independent Denotational Semantics for Prolog with Cut: Proofs

This paper contains the proofs of the results of 1] not already contained there. We refer to 1] for notation. We deene a stronger ordering on trees than. We say that T2 is a blooming of T1, and we write T1 T2, if and only if T2 is obtained with a strict expansion of the divergent leaves of T1. More formally we deene: (G; ") (G; ") if not div(G) (G; ") (G; ~ t) if div(G) The interest in the blooming relation is strictly related to the notion of P-directed tree. Namely, a chain of strongly P-directed trees for G increasing w.r.t. must eventually reach every strongly P-directed tree for G. This is formalized by the lemma below: Lemma P.1. Given a program P, let fTig i0 be a sequence of strongly P-directed trees for R such that Ti Ti+1. Let T be any strongly P-directed tree for R. Then there exists j such that T Ti for every i j. Proof. The proof will be carried by induction on the height of T. Suppose T = (R; "). Since every Ti is for R, we have T Ti for every i and the thesis follows. If T = (R; ~ t) with ~ t = t1 :: :: tn, let R k be the root of t k. By deenition of tree, we have div(R). Since T0 Ti for i 1, we must have Ti = (R; t i 1 :: :: t i m) for i 1. Since both T and Ti are strongly P-directed, we have n = m and t i k is for R k for k = 1; : : : ; n. Moreover, since T1 T2 T3 we conclude that t 1 k t 2 k t 3 k for any k = 1 ft i+1 k g i0 is a sequence of strongly P-directed trees for R k such that t i+1 k t i+2 k for every i 0. But t k is a strongly P-directed tree for R k. By inductive hypothesis we conclude that for every k = 1; : : : ; n there exists an index j k such that t k t i k for all i j k. Letting j = max j k , we have T Ti for every i j. We now show that sequences (6) and (7) are formed by …