Implement a proof assistant for OHTT

Develop a proof assistant for Open Horn Type Theory that supports both coherence (⊢⁺) and gap (⊢⁻) judgments, enforces the Exclusion Law, treats horn types as first-class objects, and provides mechanisms or tactics for constructing and manipulating gap witnesses.

Background

The paper outlines requirements for an OHTT proof assistant—dual judgment forms, Exclusion, horn types, and gap‑oriented tactics—but no implementation exists.

They explicitly state that building such a system remains an open task.