Five axioms of alpha-conversion

By Andrew D. Gordon and Tom Melham, from TPHOLs 1996, available from Andrew D. Gordon’s website:

We present five axioms of name-carrying lambda-terms identified 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 defined 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 five 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.

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