BiSikkel: A Multimode Logical Framework in Agda

Embedding Multimode Type Theory (MTT) as a library enables the usage of additional reasoning principles in off-the-shelf proof assistants without risking soundness or compatibility. Moreover, by interpreting embedded MTT terms in an internally constructed model of MTT, we can extract programs and proofs to the meta language and obtain interoperability between the embedded language and the metalanguage. The existing Sikkel library for Agda achieves this for Multimode Simple Type Theory (MSTT) with an internal presheaf model of dependent MTT. In this work, we add, on top of the simply-typed layer, a logical framework in which users can write multimode proofs about multimode Sikkel programs, still in an off-the-shelf proof assistant. To this end, we carve out of MTT a new multimode logical framework µLF over MSTT and implement it on top of Sikkel, interpreting both in the existing internal model. In the process, we further extend and improve the original codebase for each of the three layers (syntax, semantics and extraction) of Sikkel. We demonstrate the use of µLF by proving some properties about functions manipulating guarded streams and by implementing an example involving parametricity predicates.