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InductiveDataType Systems
, 2002
"... In a previous work ("Abstract Data Type Systems", TCS 173(2), 1997), the leI two authors presented a combined lmbined made of a (strongl normal3zG9 alrmal rewrite system and a typed #calA#Ik enriched by patternmatching definitions folnitio a certain format,calat the "General Schem ..."
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Cited by 821 (23 self)
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In a previous work ("Abstract Data Type Systems", TCS 173(2), 1997), the leI two authors presented a combined lmbined made of a (strongl normal3zG9 alrmal rewrite system and a typed #calA#Ik enriched by patternmatching definitions folnitio a certain format,calat the "General Schema", whichgeneral39I theusual recursor definitions fornatural numbers and simil9 "basic inductive types". This combined lmbined was shown to bestrongl normalIk39f The purpose of this paper is toreformul33 and extend theGeneral Schema in order to make it easil extensibl3 to capture a more general cler of inductive types, cals, "strictly positive", and to ease the strong normalgAg9Ik proof of theresulGGg system. Thisresul provides a computation model for the combination of anal"DAfGI specification language based on abstract data types and of astrongl typed functional language with strictly positive inductive types.
Equational Inference, Canonical Proofs, And Proof Orderings
 Journal of the ACM
, 1992
"... We describe the application of proof orderingsa technique for reasoning about inference systemsto various rewritebased theoremproving methods, including re#nements of the standard KnuthBendix completion procedure based on critical pair criteria; Huet's procedure for rewriting modulo a ..."
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Cited by 30 (10 self)
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We describe the application of proof orderingsa technique for reasoning about inference systemsto various rewritebased theoremproving methods, including re#nements of the standard KnuthBendix completion procedure based on critical pair criteria; Huet's procedure for rewriting modulo a congruence; ordered completion #a refutationally complete extension of standard completion#; and a proof by consistency procedure for proving inductive theorems. # This is a substantially revised version of the paper, #Orderings for equational proofs," coauthored with J. Hsiang and presented at the Symp. on Logic in Computer Science #Boston, Massachusetts, June 1986#. It includes material from the paper #Proof by consistency in equational theories," by the #rst author, presented at the ThirdAnnual Symp. on Logic in Computer Science #Edinburgh, Scotland, July 1988#. This researchwas supported in part by the National Science Foundation under grants CCR8901322, CCR9007195, and CCR9024271. 1 ...
33 Examples of Termination
 In Proc. French Spring School of Theoretical Computer Science, LNCS 909
, 1995
"... . A graded sequence of examplespresented in a uniform frameworkspotlights stages in the development of methods for proving termination of rewrite systems. Let T be the set of all terms over some vocabulary.Arewrite system over T is a ##nite or in#nite# set of rules, eachoftheforml ! r, where ..."
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Cited by 23 (0 self)
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. A graded sequence of examplespresented in a uniform frameworkspotlights stages in the development of methods for proving termination of rewrite systems. Let T be the set of all terms over some vocabulary.Arewrite system over T is a ##nite or in#nite# set of rules, eachoftheforml ! r, where l and r are terms containing variables ranging over T . A rule l ! r applies to a term t in T if a subterm s of t matches the lefthand side l with some substitution # of terms in T for variables appearing in l. The rule is applied by replacing the redex s in t with the corresponding righthand side r# of the rule, to which the same substitution # of terms for variables has been applied. We write t !u to indicate that the term t in T rewrites in this waytothetermu in T by a single application of some rule. Note that more than one rule can apply to t and rules can apply at more than one subterm s. Rewrite systems have long been used as decision procedures for validity in equational theories,...
Rewriting Methods for Word Problems
 Words, Languages & Combinatorics (Proceedings of the International Colloquium, Kyoto
, 1992
"... This paper outlines various recent approaches to solving word problems. Term orderings are used to define a terminating rewrite relation. When confluent, that relation defines unique normal forms that can be used to decide word problems. Some results obtained by these methods are summarized. 1. Intr ..."
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Cited by 2 (1 self)
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This paper outlines various recent approaches to solving word problems. Term orderings are used to define a terminating rewrite relation. When confluent, that relation defines unique normal forms that can be used to decide word problems. Some results obtained by these methods are summarized. 1. Introduction The central idea of rewriting is to impose directionality on the use of equations in proofs. A rewrite rule is an ordered pair of terms, written l ! r. Like equations, rules are used to replace instances of l by corresponding instances of r; unlike equations, rules are not used to replace instances of the righthand side r. For any given set R of rules, the rewrite relation !R is the closure of R (viewed as a binary relation) under the "replacement" property (within any context) and "fully invariant property" (under any substitution). In other words, s !R t if s contains a subterm that is an instance loe of l, for some rule l ! r in R, and t is s with that subterm replaced by roe...