Viewing [formula omitted]-terms through maps

Abstract

In this paper we introduce the notion of map, which is a notation for the set of occurrences of a symbol in a syntactic expression such as a formula or a λ term. We use binary trees over 0 and 1 as maps, but some well-formedness conditions are required. We develop a representation of lambda terms using maps. The representation is concrete (inductively definable in HOL or Constructive Type Theory) and canonical (one representative per λ term). We define substitution for our map representation, and prove the representation is in substitution preserving isomorphism with both nominal logic λ terms and de Bruijn nameless terms. These proofs are mechanically checked in Isabelle/HOL and Minlog respectively.The map representation has good properties. Substitution does not require adjustment of binding information: neither α conversion of the body being substituted into, nor de Bruijn lifting of the term being implanted. We have a natural proof of the substitution lemma of λ calculus that requires no fresh names, or index manipulation.Using the notion of map we study conventional raw λ syntax. E.g. we give, and prove correct, a decision procedure for α equivalence of raw λ terms that does not require fresh names.We conclude with a definition of β reduction for map terms, some discussion on the limitations of our current work, and suggestions for future work.