Dealing with Non-orientable Equations in Rewriting Induction

Rewriting Induction + Linear Arithmetic = Decision Procedure

“…This principle is quite similar to an abstract principle of Koike and Toyama [32] as reported in [1,2] but differs from that principle by incorporating E LIA . With this principle, the statement of the theorem can be shown.…”

“…The presentation follows [1,2]. The main idea of this method is to expand certain subterms of an atomic conjecture using narrowing with the rewrite rules of R.…”

“…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.…”

Multi-Context Rewriting Induction with Termination Checkers

“…A TRS R is quasi-reducible (also called ground-reducible) if every ground basic term is reducible in R. An equation s = t is an inductive theorem of R if all its ground instances sσ = tσ are equational consequences of the equational axioms R (regarded as a set of equations), i.e., sσ ↔ * R tσ. Given a set R of rewrite rules representing equational axioms and a reduction order containing R, RI is represented as an inference system working on a pair of a set of equations E and a set of rewrite rules H. Intuitively, E represents conjectures (i.e., theorems and lemmas) to be proved and H represents inductive hypotheses applicable to E. Figure 1 shows the inference rules of RI proposed in [1].…”

“…Moreover, they can implicitly exploit modern termination proving methods more powerful than the classical, simply parameterized reduction orders (such as recursive path orders and polynomial orders). We should say that RIt, which solves the strategic issue shown above as (1), has another issue instead: (1') in which direction hypothetical equations should be oriented. From the viewpoint of strategy, the use of termination checkers gives us more flexibility in the orientation strategy, because they increase the possibility of success in the orientation and we can decide the direction of the equations dynamically.…”