Helly Property for Subtrees1

One can prove the following proposition (1) For every non empty finite sequence p holds 〈p(1)〉 aa p = p. Let p, q be finite sequences. The functor maxPrefix(p, q) yields a finite sequence and is defined by: (Def. 1) maxPrefix(p, q) p and maxPrefix(p, q) q and for every finite sequence r such that r p and r q holds r maxPrefix(p, q). Let us observe that the functor maxPrefix(p, q) is commutative. Next we state several propositions: (2) For all finite sequences p, q holds p q iff maxPrefix(p, q) = p. (3) For all finite sequences p, q holds lenmaxPrefix(p, q) ≤ len p. (4) For every non empty finite sequence p holds 〈p(1)〉 p. (5) For all non empty finite sequences p, q such that p(1) = q(1) holds 1 ≤ lenmaxPrefix(p, q). This work has been partially supported by the NSERC grant OGP 9207.