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https://github.com/xsnow/distributingtypes


https://github.com/xsnow/distributingtypes

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# Distributing Intersection and Union Types with Splits and Duality (Artifact)



DistributingTypes | Xuejing Huang 2021 | Distributed under the terms of [the GPL-v3 license](./LICENSE)



- The companion paper can be found at .

- [impl/](./impl) for the Haskell implementation

- [coq/](./coq) for the Coq formalization

- [spec/](./spec) for the Ott specification (that is used to generate the syntax

definition in Coq)



## Haskell Implementation



- Fig. 3 is in [Algo_bcd.hs](./impl/Algo_bcd.hs)

- Fig. 9 and 10 are in [Algo_duo.hs](./impl/Algo_duo.hs)

- [Algo_sub.hs](./impl/Algo_sub.hs) implements the algorithmic subtyping defined

in Fig. 6.

- [Algo_alt.hs](./impl/Algo_alt.hs) implements the algorithmic duotyping in an alternative way (discussed in Section 5.4).



### Building Instructions



1. Install GHCi.

2. Enter [impl/](./impl) directory.

3. Run ghci with the code loaded. There are some example defined in each .hs

that can be used to test.



   ```sh

   ghci Algo2.hs

   *Main> test1

   "(Int -> (Int /\\ Int)) <: (Int -> (Int /\\ Int))  Result: True"

   ```



## Coq Formalization



### Third Party Materials



We use [LibTactics.v](./coq/LibTactics.v) from [the TLC Coq library](https://www.chargueraud.org/softs/tlc/)

by Arthur Chargueraud.



### Building Instructions



Our Coq proofs are verified in the latest version of Coq: **8.13.2**.



#### Prerequisites



1. Install Coq 8.13.2.

   The recommended way to install Coq is via `OPAM`. Refer to

   [here](https://coq.inria.fr/opam/www/using.html) for detailed steps. Or one could

   download the pre-built packages for Windows and MacOS via

   [here](https://github.com/coq/coq/releases/tag/V8.13.2). Choose a suitable installer

   according to your platform.



2. Make sure `Coq` is installed (type `coqc` in the terminal, if you see "command

   not found" this means you have not properly installed Coq).



#### Build and Compile the Proofs



1. Enter [coq/](./coq) directory.



2. Type `make` in the terminal to build and compile the proofs.



3. You should see something like the following (suppose `>` is the prompt):

   ```sh

   coq_makefile -arg '-w -variable-collision,-meta-collision,-require-in-module' -f _CoqProject -o CoqSrc.mk

   COQDEP VFILES

   COQC LibTactics.v

   COQC Definitions.v

   ```

4. `Definitions.v` is generated by Ott. To reproduce it (some comments will be

lost), please remove it and run `make` (with Ott installed).



### Proof Structure



- [Definitions.v](./coq/Definitions.v) contains all definitions. It is generated

by [spec/rules.ott](spec/rules.ott).



- [TypeSize.v](./coq/TypeSize.v) defines the size of type and proves some

straightforward lemmas in it. It helps us to do induction on the size of type.



- [LibTactics.v](./coq/LibTactics.v) is a Coq library by Arthur Chargueraud.

We downloaded it from [here](http://gallium.inria.fr/~fpottier/ssphs/LibTactics.html).



- [Duotyping.v](./coq/Duotyping.v) is the main file. It contains most theorems

and lemmas in Sec. 4: Lemma 4.1, Theorem 4.5, Theorem 4.6, Lemma 4.7, Lemma 4.8,

Lemma 4.9, Theorem 4.10, and Theorem 4.11.



- [Equivalence.v](./coq/Equivalence.v) relates the two declarative systems and

the two algorithmic systems, respectively.

It contains Theorem 4.2, Lemma 4.3, and Theorem 4.4.



- [Subtyping.v](./coq/Subtyping.v) contains some lemmas about the two subtyping

systems. It is not discussed in the paper. Three of them are used in the

proof of Theorem 4.4 in [Equivalence.v](./coq/Equivalence.v).



- [DistAnd.v](./coq/DistAnd.v) justifies one statement in the paper. It shows

that the rule DS-distAnd (Fig. 4) is omittable.



- [DistSubtyping.v](./coq/DistSubtyping.v) is a stand-alone file which contains

subtyping definitions and related lemmas and theorems.