Five Axioms of Alpha -

We present ve axioms of name-carrying lambda-terms iden-tiied up to alpha-conversion|that is, up to renaming of bound variables. We assume constructors for constants, variables, application and lambda-abstraction. Other constants represent a function Fv that returns the set of free variables in a term and a function that substitutes a term for a variable free in another term. Our axioms are (1) equations relating Fv and each constructor, (2) equations relating substitution and each constructor, (3) alpha-conversion itself, (4) unique existence of functions on lambda-terms deened by structural iteration, and (5) construction of lambda-abstractions given certain functions from variables to terms. By building a model from de Bruijn's nameless lambda-terms, we show that our ve axioms are a conservative extension of HOL. Theorems provable from the axioms include distinctness, injectivity and an exhaustion principle for the constructors, principles of structural induction and primitive recursion on lambda-terms, Hindley and Seldin's substitution lemmas and the existence of their length function. These theorems and the model have been mechanically checked in the Cambridge HOL system. The axioms presented in this paper are intended to give a simple, abstract characterisation of untyped lambda-terms, with constants, identiied up to alpha-conversion, that is, renaming of bound variables. We were led to develop these axioms because we are interested in representing the syntax of programming languages with binding operators within a theorem prover. The diiculty of correctly deening substitution on lambda-terms is notorious. Previous experience with the pi-calculus (Milner, Parrow, and Walker 1992) in HOL (Melham 1994) suggests that developing substitution and binding operators directly is a tedious and error-prone business. Instead, to avoid error and repetition, we advocate rst developing a metatheory of untyped lambda-terms, and secondly deriving syntax for a particular programming language as abbreviations for un-typed lambda-terms. We will show in section 4 how to do this for a nitary pi-calculus. Given higher-order logic, as implemented in the Cambridge HOL system (Gor-don and Melham 1993), what we are after is a logical type ()term that stands for the set of lambda-terms, where is the type of constants. Terms are generated by the four constructors: