On proving inductive properties of abstract data types

Abstract

The eq~ational,axioms of an algebraic speciflcatlort of a data type (such as flnlte sequences) often can be formed into a convergent sot of rewrite rules; Le. such that all sequences of rewrites are finite and unlque{y termhtatlng, If one adds [

Equational inference, canonical proofs, and proof orderings

“…Finally, in Section 7, we present, within the same proof-transformation framework, a method for proving inductive theorems, due originally to Musser [49], based on the concept of "proof by consistency." With this method, it is easy to prove automatically from the definition of multiplication given at the outset that 0 X x R:: O.…”

“…Musser [49] was the first to describe an inductive completion procedure. His procedure applies to abstract data type specifications, where an equality predicate eq is associated with each data type and the specification is "sufficiently complete," so that each ground expression eq( s, t) can be reduced to the Boolean constant true or false.…”

Rewriting Induction + Linear Arithmetic = Decision Procedure

“…Here, an induction scheme explicitly gives the base cases and the step cases, where a step case consists of an obligation and one or more hypotheses. In implicit induction (see, e.g., [33,21,26,24,16,27,34,7,1,37]), no concrete induction scheme is constructed a priori. Instead, an induction scheme is implicitly constructed during the proof attempt.…”

Completion and Its Applications

“…Were a convergent system R to include rules that reduce any valid ground equation eq (g, d) to T and invalid ones to F (for some equality symbol eq), then letting (Eo; Ra) be ({ S -t}; R) and completing fairly, would generate the contradiction F -T whenever S = t is not an identity in the initial model of R. This is the method of Musser (1980) (see also Goguen (1980), Huet and Oppen (1980), and Kapur (1980)); its correctness follows directly from the fairness of completion, since, if S # I(R) t, then…”