Definitional Inversion, Without Normalisation

2026-07-15

AI Summary

This work addresses the challenge of proving definitional inversion properties—such as injectivity and non-confusion—for type constructors in dependent type theories without relying on normalization. To this end, it introduces a novel metatheoretic framework grounded in domain theory, which for the first time establishes the injectivity of type constructors rigorously within a non-normalizing system featuring both η-laws and the type-in-type rule. The approach entirely dispenses with normalization assumptions, has been validated in a minimal type theory, and shows promise for extension to practical non-normalizing systems such as Idris and Lean, thereby opening a new avenue for metatheoretic investigations of dependent type theories.

Abstract

We contribute a new proof technique, based on domain theory, to prove key meta-theoretic properties of dependent type systems: definitional inversion properties, i.e. injectivity and no-confusion of type constructors. This proof technique is independent of normalisation, and indeed applies even for the "type-in-type" rule of Martin-Löf's original type theory. Our proof is the first to establish injectivity of type constructors for such a system in the presence of $η$ laws. More generally, the technique is motivated by, and intended for, the metatheory of systems such as Idris, Lean, or dependent Haskell, whose underlying type theory is known to be non-normalising, as well as projects such as MetaRocq or Lean4Lean, where Gödel's second incompleteness theorem means we cannot show normalisation of the object logic in itself. We showcase the method on a small type theory, then explain how it extends to more ambitious extensions.

Problem

• definitional inversion
• dependent type theory
• injectivity
• no-confusion
• normalisation

Innovation

• definitional inversion
• domain theory
• dependent type theory
• normalisation-free
• injectivity of type constructors