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nbe-for-mltt

An implementation of Normalization by Evaluation for Martin-Löf Type Theory with dependent products (pi), dependent sums (sigma), natural numbers, and a cumulative hierarchy. This implementation correctly handles eta for both pi and sigma.

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.

For example:

let plus : Nat -> Nat -> Nat =
  fun m ->
  fun n ->
  rec n at x -> Nat with
  | zero -> m
  | suc _, p -> suc p

let fib : Nat -> Nat =
  fun n ->
  let worker : Nat * Nat =
    rec n at _ -> Nat * Nat with
    | zero -> <1, 0>
    | suc _, p -> <plus (fst p) (snd p), fst p> in
  snd worker

normalize fib 25 at Nat

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

The algorithm for normalization is heavily based on the description provided in "Normalization by Evaluation Dependent Types and Impredicativity" by Andreas Abel. The type checker is loosely based on "A simple type-theoretic language: Mini-TT" by Thierry Coquand, Yoshiki Kinoshita, Bengt Nordström, and Makoto Takeyama.

An explanation of the algorithm may be found in nbe-explanation.md. An explanation of the type checker may be found (eventually) in check-explanation.md.

Compiling the code

In order to compile the code, it is necessary to install the dependent libraries:

opam install --deps-only .

Afterwards run make all or dune build @install to build the code. Specific files can be executed with dune exec -- mltt test/somefile.tt.