A (very experimental) attempt at formalizing type theory internally in cubical agda such that:

1. Only well-typed terms are permitted

2. Equality of terms is exactly βη-equality.

The first condition can be acheived in plain agda (e.g. https://github.com/AndrasKovacs/stlc-nbe) but the second can only be acheived with higher inductive types.

This work was inspired by Type Theory in Type Theory using Quotient Inductive Types by Thorsten Altenkirch and Ambrus Kaposi, and an adaptation of the core of their paper can be found in DTLC.agda.

Structure

• STLC.Base gives a higher inductive presentation of a simply typed lambda calculus with a unit type.

• STLC+N.Base gives an analgous HIT for simply typed lambda calculus with a natural numbers type and recursor. This is older work that will be returned to once everything is finished in the simpler case above.

  Note: If you're wondering why we bother with apˡ and apʳ in the first place, see Cubical.HITs.Ints.BiInvInt and Cubical.HITs.Ints.IsoInt in agda/cubical. That being said, this is no longer relevant since sub needs a set truncation constructor anyhow.

Currently normalization for STLC (and STLC+N) is a mess and incomplete.

• STLC+N.NormShort was my first pass at trying to do this directly, and it got very difficult to work with.

• After doing some research, I began trying normalization by evaluation in STLC+N.NormLong and STLC.Norm -- inspired by Andras Kovacs' above repository, among others. Finishing up the proofs in these files is ongoing work.