The focus of this paper is the technique of recursion analysis. Recursion analysis is used by the Boyer-Moore Theorem Prover to choose an appropriate induction schema and variable to prove theorems by mathematical induction. A rational reconstruction of recursion analysis is outlined, using the technique of proof plans. This rational reconstruction suggests an extension of recursion analysis which frees the induction suggestion from the forms of recursion found in the conjecture. Preliminary results are reported of the automation of this rational reconstruction and extension using the clam- Oyster system. 1 Introduction The work described in this paper is part of a project to explore the use of proof plans for the automatic guidance of mathematical proofs. In particular, we are developing proof plans for the proofs by mathematical induction that are required in the automatic synthesis of computer programs from their specifications. Given a conjecture, the clam plan form...

How can we understand reasoning in general and mathematical proofs in particular? It is argued that a high-level understanding of proofs is needed to complement the low-level understanding provided by Logic. A role for computation is proposed to provide this high-level understanding, namely by the association of proof plans with proofs. Criteria are given for assessing the association of a proof plan with a proof. 1 Motivation: the understanding of mathematical proofs We argue that Logic 1 is not enough to understand reasoning. It provides only a low-level, step by step understanding, whereas a high-level, strategic understanding is also required. Many commonly observed phenomena of reasoning cannot be explained without such a high-level understanding. Furthermore, automatic reasoning is impractical without a high-level understanding. We propose a science of reasoning which provides both a low- and a high-level understanding of reasoning. It combines Logic with the concept of proof plans, [Bundy, 1988]. We illustrate this with examples from mathematical reasoning, but it is intended that the science should eventually apply to all kinds of reasoning. 2 The Need for Higher-Level Explanations A proof in a logic is a partially ordered set of formulae where each formula in the set is either an axiom or is derived from earlier formulae in the set by a rule of inference. Each mathematical theory de nes what it means to be a formula, an axiom or a rule of inference. Thus Logic provides a low-level explanation of a mathematical proof. It explains the proof as a sequence of steps and shows how each step follows from previous ones by a set of rules. Its concerns are limited to the soundness of the proof, and to the truth of proposed conjectures in models of logical theories.

. The close association between higher order functions (HOFs) and algorithmic skeletons is a promising source of automatic parallelisation of programs. An approach to synthesising HOFs from functional programs through proof planning is presented, and its realisation in Clam is discussed. 1. Introduction 1.1. Higher Order Functions Pure functional languages, satisfying the Church-Rosser property of evaluation order independence, have long been proposed as a basis for parallel programming. Thus, Wegner[Weg71] observed in 1971: Note that [the Church-Rosser theorem] essentially states that lambda expressions can be evaluated by asynchronous multiprocesssing applied in arbitrary order to local subexpressions. page 185 Early work on functional parallelism focussed on reduction of Curry combinators [Tur79], lifted automatically from functional programs, but these proved of too low granularity for efficient parallel evaluation [Sto84]. Somewhat more success has obtained from parallel red...

This paper contains a discussion, and reconstruction, of Goad's proof transformation system. Information contained in constructive existence proofs, which goes beyond that needed for simple execution, is exploited in the adaption of algorithms to special situations. Goad's system is reconstructed, and extended, in the Oyster proof refinement environment and subjected to test on a number of examples. Differences in methodology between Goad's system and the reconstruction are discussed. In particular, a more active role is given to the actual proof, as opposed to the extracted algorithm, in the transformation process.