# Canonical Equational Proofs by Bachmair

By Bachmair

Equations ensue in lots of laptop purposes, corresponding to symbolic compu­ tation, practical programming, summary info kind requirements, software verification, software synthesis, and automatic theorem proving. Rewrite structures are directed equations used to compute by means of changing subterms in a given formulation by means of equivalent phrases until eventually a least difficult shape attainable, referred to as a regular shape, is acquired. the speculation of rewriting is anxious with the compu­ tation of ordinary types. we will research using rewrite options for reasoning approximately equations. Reasoning approximately equations may well, for example, contain figuring out even if an equation is a logical outcome of a given set of equational axioms. Convergent rewrite platforms are these for which the rewriting strategy de­ fines designated general kinds. they are often regarded as non-deterministic useful courses and supply kind of effective determination approaches for the underlying equational theories. The Knuth-Bendix final touch technique offers a way of checking out for convergence and will usually be used to con­ struct convergent rewrite platforms from non-convergent ones. We enhance a proof-theoretic framework for learning finishing touch and comparable rewrite­ dependent facts tactics. we will view theorem provers as evidence transformation strategies, in an effort to convey their crucial houses as evidence normalization theorems.

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Example text

By the induction hypothesis, the proof Q (and hence P) can be transformed into a rewrite proof modulo A. D. =>! =>! =>! Let us emphasize that this theorem applies to arbitrary rewrite systems R, not just left-linear systems. Left-linearity will be crucial, however, in deriving effective conditions under which all cliffs can be transformed to rewrite proofs modulo A. By N,£ we denote the set of all rewrite proofs modulo A. We can now formulate a sufficient condition for fairness. 6. A non-failing derivation Eo; Ro I- c E 1 ; Rl I- c ...

By the induction hypothesis, the proof Q (and hence P) can be transformed into a rewrite proof modulo A. D. =>! =>! =>! Let us emphasize that this theorem applies to arbitrary rewrite systems R, not just left-linear systems. Left-linearity will be crucial, however, in deriving effective conditions under which all cliffs can be transformed to rewrite proofs modulo A. By N,£ we denote the set of all rewrite proofs modulo A. We can now formulate a sufficient condition for fairness. 6. A non-failing derivation Eo; Ro I- c E 1 ; Rl I- c ...

Thus the subterm ordering modulo A is not well-founded. D. Note that if the subterm ordering modulo A is well-founded, then all congruence classes of A have to be finite. 12. (Jouannaud and Kirchner 1986) Let A be a set of equations for which minimal complete sets of A-unifiers can be computed, and let R be a rewrite system such that RIA is terminating.

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