A theory of recursive definitions has been mechanized in Isabelle's Zermelo-Fraenkel (ZF) set theory. The objective is to support the formalization of particular recursive definitions for use in verification, semantics proofs and other computational reasoning.

Martin-Löf's Intuitionistic Theory of Types is becoming popular for formal reasoning about computer programs. To handle recursion schemes other than primitive recursion, a theory of well-founded relations is presented. Using primitive recursion over higher types, induction and recursion are formally derived for a large class of well-founded relations. Included are < on natural numbers, and relations formed by inverse images, addition, multiplication, and exponentiation of other relations. The constructions are given in full detail to allow their use in theorem provers for Type Theory, such as Nuprl. The theory is compared with work in the field of ordinal recursion over higher types.

This paper describes a tool for use in user-directed synthesis of circuits specified using the relational VLSI description language Ruby. The synthesis method is based on transformational rewriting of Ruby terms in accordance with previously defined term equivalences. The tool permits the introduction of constraints into the specification, thus enhancing the usefulness of the rewrite system in relation to simple rewriting. Keyword Codes: B.7.2, D.1.1. Keywords: Integrated Circuits, Design Aids; Applicative Programming. 1. Introduction Ruby [3] is a language intended for specifying VLSI circuits in terms of relational abstractions of their behaviour. A circuit is described by a binary relation, and the language permits simple relations to be composed into more complex ones by the use of a variety of combining forms which are higher-order functions. Similarly, simple combining forms can be composed into more complex ones, as in conventional languages for functional programming. The basi...

This report describes Version 2.4 of a tool for parsing and typechecking expressions in the relational VLSI description language Ruby, and for rewriting them according to previously defined equivalences as part of a synthesis process. The tool also offers facilities for translating an implementable class of Ruby expressions into VHDL with a view to subsequent synthesis into VLSI circuits, and for simulating the behaviour of the corresponding relations.

This report describes a framework for the manipulation of Ruby expressions, regarded as terms, and their types. Ruby is here considered as a strongly-typed relational language with a type system which permits dependent types in a stratified system of type universes. The type system is closely related to that of Edinburgh LF, and uses the a similar division of entities into terms, types and kinds. The system is first presented in terms of a set of inference rules for deducing types and kinds from well-formed pseudo-terms. Then a more implementation-oriented formal description is given. This has formed the basis for an actual implementation, on which a number of practical tools have been built up.

. This paper describes a tool for use in user-directed synthesis of circuits specified using the relational VLSI description language Ruby. The synthesis method is based on syntactic rewriting of Ruby terms, combined with the introduction of constraints into the specification. The rewriting process is described in a meta-language based on the use of tactics and tacticals, which makes it possible to develop complex specialised strategies for the refinement of specifications. 1 Introduction The relational VLSI specification language Ruby [4] forms a good basis for a transformational style of VLSI synthesis, in which an abstract specification is systematically modified into a description of an implementable circuit by "calculation " [5, 10]. This paper describes some of the principles behind a tool, known as TRuby [12], for handling this style of synthesis. In T-Ruby, the transformation process relies on two elements: Syntactic rewriting, based on (possibly conditional) equivalences betw...