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Show have P!v. = (vii U o-)!V. = '~i!V = .,,i' I I I where Vi can be dropped simply because D(v,i) ~ V 1 • Now, let v i+l = (Vi u IC~ i) ) - D(tJi) Y/·+1 = '1/,IV . I i+l We then have O{v,i+l) ~ Vi+l' proving 1. By the definition of Vi+l and the assumption that D{1:.) n l(1:.i) have (*) .,,. = fJIV = (v/tJ -)IV = ((vJ'Iv )1:. -)IV = (vi -+11:. -)IV. I i I i (i+l) I i I I i Now assume that 2 holds for i, i.e., 11iv = ( 11i 9i )IV and show that it holds for i+ 1. From (*) above, we get 0, we From the definition of ei, we have 1(9) ~ Vi and V ~ Vi u 0(9). The righthand side of the above expression then simplifies to ((vli+1 ""i9i) lv)!v = (vli+1""i9i)IV = (v1i+1 9i+1)IV' I proving 2. Finally, it is easy to see that V(C) ~ Vi, from which we get .,,i+,(ci+1) = .,,i+1 1:.i(ci[ui <- 13k.n = .,,i+1pi(Ci[ui <- v/IJi (fjk_}]) = Y/i+11:1 i(Ci[U i <- v/ p i1/i+; pi({j k_)]) = .,,.(C.[u. <- o-(fjk }]) I I I I . = (v,.C.)[u. <- o-(fjk1 )] I I I . = B.[u. <- o-(fjk )] I I I . = 8 i+1' I proving 3. 0 Theorem 4-17: A term rewriting system is confluent if it is linear and closed. 0 68 A term rewriting system defines a subtree replacement system. See [86] for a detailed study of subtree replacement systems and a proof of the above theorem. A proof of the correctness of the narrowing method, i.e., satisfaction of part (ii} of Definition 2-5 is provided in [51]; this proof does not rely on the ter- |
