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.

In this paper, we extend the Gnaedig's results on termination of order-sorted rewriting. Gnaedig required a condition for order-sorted signatures, called minimality, for the termination proof. We get rid of this restriction by introducing a transformation from a TRS with an arbitrary order-sorted signature to another TRS with a minimal signature, and proving that this transformation preserves termination.

Dynamic Term Rewriting Calculus is a new computation model proposed by the authors for the purpose of formal description and verification of algorithms treating Term Rewriting Systems. The computation of DTRC is basically term rewriting. The characteristic features of DTRC are dynamic change of rewriting rules during computation and hierarchical declaration of not only function symbols and variables but also rewriting rules. These features allow us to program metacomputation of TRSs in DTRC, that is , we can implement in DTRC in a natural way those algorithms which manipulate term rewriting systems as well as those procedures which verify such algorithms. In this paper, we give a formal description of DTRC. We then show some results on confluence property of DTRC.

The currying of term rewriting systems (TRSs) is a transformation of TRSs from a functional form to an applicative form. We have already introduced an order-sorted version of currying and proved that the compatibility and confluence of order-sorted TRSs were preserved by currying. In this paper, we focus on a key property of TRSs, completeness. We first show some proofs omitted in Ref. [3]. Then, we prove that the SN (strongly normalizing) property, which corresponds to termination of a program, is preserved by currying. Finally, we prove that the completeness of compatible order-sorted TRSs is preserved by currying.

Dynamic Term Rewriting Calculus (DTRC) is a new computation model aiming at formal description and verification of algorithms treating Term Rewriting Systems (TRSs). In this paper, we show that we can use DTRC to mechanize explicit induction for proving an inductive theorem, that is, we can translate the statements of base and induction steps for proving by induction into a DTRC term. The translation reduces the proof of the statements into the evaluation of the corresponding DTRC term.