# Mechanizations of Type Theories

This is a collection of mechanizations of type theories during Jason Hu’s PhD at McGill University. This library mechanizes the normalization proofs of many type theories using different methods. A notable one is untyped domain models as per Abel. Please read README for a list of available type theories.

This library also demonstrates that, at least for mechanizations of normalization proofs, Agda does not necessarily require more lines of code than proof assistants with very powerful proof automation, e.g. Coq. For example, the whole normalization proof with completeness and soundness theorems for Martin-Löf type theory is around 8000 LoC. This is the smallest proof in size among all similar mechanizations in 2024, while there is still room to further shorten the mechanization.

This library current works with Agda 2.6.4.3 and agda-stdlib 2.0.

You are welcome to use it anywhere, for teaching or for your own research. Please also feel free to contribute.

Jason Z. S. Hu and Brigitte Pientka. A Categorical Normalization Proof for the Modal Lambda-Calculus (**MFPS 22**)

See the code

Jason Z. S. Hu, Junyoung Jang and Brigitte Pientka. Normalization by Evaluation for Modal Dependent Type Theory (**JFP 23**)

See the code

Please also see here.

Jason Z. S. Hu and Brigitte Pientka. Layered Modal Type Theory: Where Meta-programming Meets Intensional Analysis (**ESOP 24**)

See the code; also

Audience interested in this library might also find another project that Jason Hu is involved in interesting too.

## Installing Agda

It is recommended to build Agda from source. To do so, one would need to install `stack`

. This can be done via

`curl -sSL https://get.haskellstack.org/ | sh`

Then the following script will use `stack`

to install Agda in `~/.local/bin/`

.

```
git clone https://github.com/agda/agda
cd agda
git checkout release-2.6.4.3
cp stack-8.8.4.yaml stack.yaml # choose your favourite Haskell version
stack install # it is going to take a while
cp ~/.local/bin/agda ~/.local/bin/agda-2.6.4.3
cp ~/.local/bin/agda-mode ~/.local/bin/agda-mode-2.6.4.3
```

If Agda does not run, please check to make sure it is in your PATH.

## Foundation and Assumptions

This library is implemented in safe Agda without K as much as possible. For dependent type theories, however, it is necessary to include the extension of induction-recursion. Functional extensionality is sometimes used for dependent type theories as well. These extensions are completely standard, as done in many other works.