Universes for generic programs and proofs in dependent type theory

  • Authors:

  • Marcin Benke;Peter Dybjer;Patrik Jansson

  • Affiliations:

  • Department of Computing Science, Chalmers University of Technology, SE-412 96 Göteborg, Sweden;Department of Computing Science, Chalmers University of Technology, SE-412 96 Göteborg, Sweden;Department of Computing Science, Chalmers University of Technology, SE-412 96 Göteborg, Sweden

  • Venue:
  • Nordic Journal of Computing
  • Year:
  • 2003

Quantified Score

Hi-index 0.00

Visualization

Abstract

We show how to write generic programs and proofs in Martin-Löf type theory. To this end we consider several extensions of Martin-Löf's logical framework for dependent types. Each extension has a universe of codes (signatures) for inductively defined sets with generic formation, introduction, elimination, and equality rules. These extensions are modeled on Dybjer and Setzer's finitely axiomatized theories of inductive-recursive definitions, which also have universes of codes for sets, and generic formation, introduction, elimination, and equality rules. Here we consider several smaller universes of interest for generic programming and universal algebra. We formalize one-sorted and many-sorted term algebras, as well as iterated, generalized, parameterized, and indexed inductive definitions. We also show how to extend the techniques of generic programming to these universes. Furthermore, we give generic proofs of reflexivity and substitutivity of a generic equality test: Most of the definitions in the paper have been implemented using the proof assistant Alfa for dependent type theory.