Fuss-free cumulative universes: theory and practice

2026-07-13

AI Summary

This work addresses the tension between correctness and usability in polymorphic cumulative universes within dependent type theory by proposing a concise generalized algebraic formulation that avoids intricate coherence coercion mechanisms through a “no-frills” universe hierarchy. The approach supports judgmental descriptions of datatypes and subsumes existing cumulative inductive types as derivable constructs. We present an abstract specification of a bidirectional elaboration algorithm together with a practical Haskell implementation, and establish their semantic equivalence via a normalization theorem, thereby achieving both theoretical rigor and engineering feasibility.

Abstract

Universes are central to dependent type theory, and they are notoriously difficult to handle in a way that is both correct and usable. We propose a new "fuss-free" generalised algebraic presentation for polymorphic cumulative universes that dispenses with the intricate theory of coherent universe coercions in favour of a simpler formulation, which we prove equivalent by means of a normalisation theorem for the former. Evidence for the utility of the fuss-free formulation is provided in the form of (1) an abstract specification of its bidirectional elaboration algorithm, and (2) a concrete implementation in Haskell. We also describe and implement an extension of the fuss-free universe hierarchy with a judgemental notion of datatype description from which prior notions of cumulative inductive type may be derived.

Problem

• cumulative universes
• dependent type theory
• universe coercions
• type theory
• polymorphic universes

Innovation

• cumulative universes
• dependent type theory
• normalisation theorem
• bidirectional elaboration
• inductive types