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Download Computational Aspects of an Order-Sorted Logic with Term by Manfred Schmidt-Schauß PDF

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By Manfred Schmidt-Schauß

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Ii) {s=t} ~- t=s iii) {r=s, s=t} ~- r=t. iv) If f(s 1..... Sn) and f(t 1..... tn) are well-sorted, then {Sl = tl ..... Sn = tn} ~- f(sm ..... Sn)= f(tl ..... tn). v) { s=t} ~- ~s=~t for every well-sorted substitution ~. • W e write E~- s=t if there exists a finite proof of s=t starting with equations from E using the rules (i)- (v). The following completeness theorem is the well-known Birkhoff-Theorem extended to the order-sorted case. 2 T h e o r e m . E ~ s=t iff P r o o f . i) E ~- s=t ~ E~- s=t for all well-sorted terms s,t and all sets of axioms E.

Ii) Completeness: Let z ~ UE,E(F l& 1-'2) with DOM(x) c UV(F l& F2). Then there exists a cy ~ U, such that cy -

Ii) The proof is trivially extended to vectors of finite length and hence to substitutions. I 38 11. Unification and Theory-Matching Let ~: = (X;, E) be an axiomatization of an equational theory. t. In this case we say t is more general than s or s is an E-instance o f t. Obviously the relation >X is a quasi-ordering on Ty.. Note that sometimes the reversed ordering is used, (cf. [Si84, Si86, Sz82, Sch85]). , s ~X,E t i f f s >E,E t and t ->E,E s We extend the subsumption relation to substitutions: Let (r,'~ E SUB x and let V c_ V x.

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