Termination of Hippogriff type-equality normalization

Prove that the algorithm for deciding type equality in Hippogriff terminates, rather than relying solely on implementation behavior as evidence of termination.

Background

Hippogriff combines dependent typing with general recursion by using a synthetic phase distinction and opaque values during type checking. The type system has a formal specification and mathematical semantics, and the implementation includes an algorithm for deciding type equality.

Despite this formal setup, the paper explicitly acknowledges that termination of the type-equality algorithm has not been proved. The proposed route is to extend relative-induction techniques for normalization by evaluation to incorporate both the open modality and guarded recursion, ultimately proving the required normalization and termination properties.