Confluence of HoTT with univalence

Prove confluence for homotopy type theory incorporating the univalence axiom, thereby addressing the claimed obstacle to establishing decidability of identity in that setting.

Background

The paper contrasts its computationally generated two-dimensional structure with homotopy type theory (HoTT), where univalence is characterized as a postulate without computational content. It states that the labelled beta- and eta-conversion system preserves a nontrivial parity distinction without invoking univalence.

In discussing the relationship with HoTT, the paper identifies confluence as an unresolved metatheoretic issue when univalence is incorporated into Martin-Löf type theory. Establishing confluence would be relevant to the decidability of identity, which the authors describe as being hindered by the current lack of a confluence proof.

References

This would represent an advantage over HoTT, since incorporating the univalence axiom into MLTT complicates the proof of confluence in HoTT (which remains unproven) and thereby hinders the decidability of the identity.

— A Theory of a Two-Dimensional Typed Lambda Calculus  (2609.19479 - Martínez-Rivillas et al., 16 Sep 2026) in Section 6, “Equality evidence and the identity of proofs,” subsection “Relation to homotopy type theory”