### Schematization of infinite sets of rewrite rules generated by divergent completion processes

Hélène Kirchner
1989 Theoretical Computer Science
Infinite sets of rewrite rules may be generated for example by completion of termrewriting systems or by a narrowing process for solving equations in equational theories. This is a severe limitation to the practical use of these processes. We propose in this paper a notion of schematization for an infinite set of rewrite rules. We show how to use a schematization for deciding the validity or satisfiability of an equation in the equational theory defined by the infinite set of rules, provided
more » ... schematization is sound and complete. A process to build a complete schematization from a sound one is proposed. Proof. From the proof of the order-sorted equational completion procedure  , the returned set of order-sorted rules MR' is convergent modulo r" and ~~~~I,MRU,.~, coincides with ti~RZU,.~~. So by Theorem 6.3, MR' is a convergent set of metarules modulo -on terms. Let us first show that t ej*".t' implies t ++g= t' for any terms t and t'. From Proposition 5.13,9(t) &R.vl.'> 9( t'), so 9(t) ti~~~iJMRv,.~~ 9( t'). First, \$a([) eT\cx 9(t') implies t -2 t', then t ++*,-t'. Second, 9(t) -* -MR 9( t') implies t esR t'. Since the schematization Y is sound, @LR is included in ++:I. Finally, 9(t) k+ 9( t') implies t -t', so t @LR t'. But since the schematization Y is sound, -is included in ++gr. Conversely, if t -\$a t' then t -2 t', which implies 9(t) eT\c, 9(t'). So ,a( t) +,R',J" 9( t'). Thus the schematization of R" by Y' is complete. 0 Example 6.5 (Example 2.11 continued). The system of order-sorted rules reduced to the only rule (VX : TOP, f(g(X)) + g(X)) can be proved convergent because there is no superposition. Example 6.6 (Example 2.12 continued). The order-sorted completion process is applied to the order-sorted equational system: ro: vx:l~l, --x=x, vx, Y:(T/, -f(X Y) =f(-Y -W, ' > is a I'"-reduction ordering if > is well-founded, compatible with operators in I0 and moreover T S\$' T' implies 2-s T' and T'S T. 4 A" is defined from A as in Definition 5.8 and rules in MR are given as equalities to the order-sorted completion procedure.