An implementation of MTT with modal dependent products (pi), modal types, dependent sums (sigma), natural numbers, and a cumulative hierarchy. This implementation correctly handles eta for both pi and sigma.

This implementation only permits pre-order mode theories so that there is at most one 2-cell between any pair of modalities.

This implementation has also been extended to include a type checker based on Coquand's semantic type checker. In order to interact with the normalizer, therefore, one can write a file containing a list of definitions and commands to normalize various terms.

Tested under ocaml 4.07.1 and dune 2.8.5. Sexplib, menhir, ppx_compare and cmdliner libraries need to be installed.

Building mitten with make build or dune build. Execute mitten with dune exec mitten PATH/TO/FILE. If there is no output, everything type checked. The commands normalize and normalize def print the normalized term.

let plus : (x : {idm | Nat}) -> (y : {idm | Nat}) -> Nat @ s =
    fun m ->
    fun n ->
    rec n at x -> Nat with
    | zero -> m
    | suc _, p -> suc p
normalize plus {idm, 2} {idm, 2} at Nat @ s
let fib : (x : {idm | Nat}) -> Nat @ s =
    fun n ->
    let worker : Nat * Nat =
      rec n at _ -> Nat * Nat with
      | zero -> pair (1, 0)
      | suc _, p -> pair (plus {idm, (fst p)} {idm, (snd p)}, fst p) in
    snd worker
normalize fib {idm, 25} at Nat @ s

A list of other examples may be found in test/.

The implementation is derived from nbe-for-mltt.