In equational theorem proving, convergent term rewriting systems play a crucial role. In order to compute convergent term rewriting systems, the standard completion procedure (KB) was proposed by Knuth and Bendix and has been improved in various ways. The multi-completion system MKB developed by Kurihara and Kondo accepts as input a set of reduction orders in addition to equations and efficiently simulates parallel processes each of which executes the KB procedure with one of the given orderings. Wehrman and Stump also developed a new variant of completion procedure, constraint-based completion, in which reduction orders need not be given by using automated modern termination checkers. As a result, the constraint-based procedures simulate the execution of parallel KB processes in a sequential way, but naive search algorithms sometimes cause serious inefficiency when the number of the potential reduction orders is large. In this paper, we present a new procedure, called a constraint-based multi-completion procedure MKBcs, by augmenting the constraint-based completion with the framework of the multi-completion for suppressing the combinatorial explosion by sharing inferences among the processes. The existing constraint-based system SLOTHROP, which basically employs the best-first search, is more efficient when its built-in heuristics for process selection are appropriate, but when they are not, our system is more efficient. Therefore, both systems have their role to play.

We present an automated lemma generation method for equational, inductive theorem proving based on the term rewriting induction of Reddy and Aoto as well as the divergence critic framework of Walsh. The method effectively works by using the divergence-detection technique to locate differences in diverging sequences, and generates potential lemmas automatically by analyzing these differences. We have incorporated this method in the multi-context inductive theorem prover of Sato and Kurihara to overcome the strategic problems resulting from the unsoundness of the method. The experimental results show that our method is effective especially for some problems diverging with complex differences (i.e., parallel and nested differences).

Inductive theorem proving plays an important role in the field of formal verification of systems. The rewriting induction (RI) is a method for inductive theorem proving proposed by Reddy. In order to obtain successful proofs, it is very important to choose appropriate contexts (such as in which direction each equation should be oriented) when applying RI inference rules. If the choice is not appropriate, the procedure may diverge or the users have to come up with several lemmas to prove together with the main theorem. Therefore we have a good reason to consider parallel execution of several instances of the rewriting induction procedure, each in charge of a distinguished single context in search of a successful proof. In this paper, we propose a new procedure, called multi-context rewriting induction, which efficiently simulates parallel execution of rewriting induction procedures in a single process, based on the idea of the multi-completion procedure. By the experiments with a well-known problem set, we discuss the effectiveness of the proposed procedure when searching along various contexts for a successful inductive proof.