Directed proof-relevant logical relations in simplicial HoTT

2026-07-09

AI Summary

Traditional logical relations struggle to accommodate directed reductions in type theory, impeding normalization proofs in dependent type systems. This work presents the first logical relation model within simplicial homotopy type theory that integrates contravariant computability predicates with directed quotient inductive types. Reduction is internalized as an inequality type, and a comonadic flat modality cleanly separates vertical reductions from horizontal parametricity. Leveraging built-in functoriality and universal properties, the approach supports computability reasoning under directed reduction, enabling a successful proof of directed Boolean normalization. The method extends to systems featuring dependent types and universes, yielding the first formalization of representation independence with proof relevance.

Abstract

Intrinsically-typed presentations of type theory often use equality in the meta-language to represent object-language judgmental equality. In such equational syntax, proof-relevant logical relations define computability predicates on judgmental equivalence classes of types and terms. This approach, however, does not directly account for reduction, which is directed and plays a central role in many logical-relations arguments. This paper develops a directed version of proof-relevant logical relations in simplicial homotopy type theory, where reductions are internalized as \emph{inequality types}. We construct object syntax as a directed quotient inductive type. The central observation is that contravariant families in simplicial type theory provide exactly the proof-relevant form of closure under expansion for logical relations: computability evidence can be transported backward along reductions, with the required functoriality and universal property built in. Using this observation, we construct a unary logical relations model with contravariant computability predicates and prove directed Boolean canonicity: every closed Boolean term reduces to either true or false. We then extend the construction to dependent types and universes, where a comonadic flat modality provides the discreteness needed for type conversion and universe predicates. Finally, we adapt the method to binary logical relations, separating vertical reduction from horizontal parametricity and obtaining a proof-relevant account of representation independence.

Problem

• directed logical relations
• proof-relevance
• simplicial homotopy type theory
• reduction
• canonicity

Innovation

• directed logical relations
• simplicial homotopy type theory
• contravariant families
• inequality types
• proof-relevant semantics