Conference Paper

A Dependent Type Theory with Names and Binding.

DOI: 10.1007/b100120 Conference: Computer Science Logic, 18th International Workshop, CSL 2004, 13th Annual Conference of the EACSL, Karpacz, Poland, September 20-24, 2004, Proceedings
Source: DBLP

ABSTRACT We consider the problem of providing formal support for working with abstract syntax involving variable binders. Gabbay and Pitts have shown in their work on Fraenkel-Mostowski (FM) set theory how to address this through first-class names: in this paper we present a dependent type theory for programming and reasoning with such names. Our development is based on a categorical axiomatisation of names, with freshness as its central notion. An associated adjunction captures constructions known from FM theory: the freshness quantifier, name-binding, and unique choice of fresh names. The Schanuel topos – the category underlying FM set theory – is an instance of this axiomatisation. Working from the categorical structure, we define a dependent type theory which it models. This uses bunches to integrate the monoidal structure corresponding to freshness, from which we define novel multiplicative dependent products Π * and sums Σ * , as well as a propositions-as-types generalisation of the freshness quantifier.

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