A Type-Based Approach to Divide-And-Conquer Recursion in Coq

Artifact for the JFP submission, "AType-BasedApproach to Divide-and-Conquer Recursion". This is an expanded version of the artifact for the POPL'23 paper "A Type-Based Approach to Divide-And-Conquer Recursion in Coq". This README is available in both Markdown and PDF format.

Installation

This artifact may be fully built and installed either manually or via Docker.

Which installation should I use?

Docker installation is faster & easier -- however, we are not entirely sure how a docker installation will integrate with your favorite IDE. Installation via Docker will build the artifact in entirety and allow you to interact with it via shell. You may need to perform a manual installation if you want to interact with proofs incrementally, i.e., with a goal buffer and all of your favorite tools.

In short, we offer Docker installation to guarantee the successful delivery of our artifact, but you may find you want to perform a manual installation in order to interface with the artifact more completely.

Docker Installation

First, install Docker Desktop. Then, run (in this directory):

docker build -t dc-image .
docker run -dt --name dc-container -v $(pwd):/home/coq/dc/ dc-image

The first command builds a docker image with precisely the desired Coq version and dependencies installed. The second command boots up that image as a container with a shared drive between this directory and /home/coq/dc within the container.

We can now build the development in the docker container via:

docker exec -it dc-container sh make-all

This will build all of the coq files & documentation. Documentation is placed in ./html.

Docker Help & FAQ

Once the container is built and running, you can shell into it via

docker exec -ti dc-container sh

If commands such as coqtop do not work, you may need to export all opam environment variables to the shell session via

eval $(opam env)

once inside the container. Instructions for artifact evaluation in this file will be given under the presumption that you have an active shell session in this docker container (or, locally, if you have chosen manual installation).

If you ever need to wipe the whole thing and start over, run

 docker container stop dc-container;
 docker container rm dc-container;
 docker volume rm dc;

Manual Installation

Manual installation instruction can be found here.

Additional Artifact Description

Documentation

Included in ./html/ is html documentation generated by coqdoc. You can view a table of contents by opening ./html/toc.html in the browser of your choice.

Documentation can be regenerated by running make html.

Code Snippets & Figures Index

Below we have indexed most code found in the paper by (sub)section and location in this codebase.

\S3

Section	Figure	Location	Description
\S3.1	Figure 1	WordsBy.hs	The WordsBy function in Haskell.
\S3.2	Figure 2	WordsByWf.v	Using Program to generate a well-founded version of wordsBy.
\S3.2	Figure 3	WordsByEq.v	Using Equations to generate a well-founded version of wordsBy.
\S3.6		SmallerList.v	A custom ordering introduced directly on the list data type.

\S4

Section	Figure	Location	Description
\S4.1		List.v	The List data type as a signature functor & definition of lengthAlg.
\S4.2		Kinds.v	The KAlg kind.
\S4.2		Dc.v	Definition of our divide and conquer interface: the types SAlg, FoldT, Alg, and Dc, and functions inDc, fold, sfold, and out.

\S5

Section	Figure	Location	Description
\S5.1.1	Figure 7	Span.v	The Span (and helper spanr) function.
\S5.1.2	Figure 8	WordsBy.v	The WordsBy function.
\S5.1.2	Figure 9	mapThrough.hs	The WordsBy function.
\S5.2.1	Figure 10	MapThrough.v	The mapThrough combinator.
\S5.2.2	Figure 11	Rle.hs	mapThrough and run-length encoding in Haskell.
\S5.2.2	Figure 12	Rle.v	Run-length encoding.
\S5.2.2	Figure 13	MapThroughWf.v	Using Program to generate a well-founded version of mapThrough.
\S5.3	Figures 14 & 15	Matcher.v	Harper's matcher.
\S5.4	Figures 16 & 17	Mergesort.v	mergesort implemented using split.
\S5.5	Figures 18 & 19	Quicksort.v	quicksort implemented using partition.

\S6

Section	Figure	Location	Description
\S6.1	Figure 20	Mu.v	Derivation of retractive-positive recursive types.
\S6.2	Figure 21	Dc.v	Definition of promote.

\S7

Section	Figure	Location	Description
\S7		Dci.v	Derivation of Divide & Conquer induction.
\S7		Rle.v	The decoding property for rle.
\S7.1	Figure 22	SpanPfs/	Formulations of three lemmas about span.
\S7.2	Figure 23	MotivePres.v	Carrier for proving that Split preserves motives.
\S7.3	Figure 24	Forall.v	Motive used to avoid noncanoicity problems.