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https://github.com/cdepillabout/coq-equivalence-not-congruence
Coq proof of an equivalence relation that is not congruent on the Imp language from Software Foundations
https://github.com/cdepillabout/coq-equivalence-not-congruence
congruence coq equivalence-relation proof
Last synced: 14 days ago
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Coq proof of an equivalence relation that is not congruent on the Imp language from Software Foundations
- Host: GitHub
- URL: https://github.com/cdepillabout/coq-equivalence-not-congruence
- Owner: cdepillabout
- License: other
- Created: 2022-03-18T01:35:01.000Z (over 2 years ago)
- Default Branch: master
- Last Pushed: 2022-03-20T08:27:04.000Z (over 2 years ago)
- Last Synced: 2024-10-10T21:41:03.138Z (about 1 month ago)
- Topics: congruence, coq, equivalence-relation, proof
- Language: Coq
- Homepage: https://cs.stackexchange.com/a/98873/130503
- Size: 53.7 KB
- Stars: 1
- Watchers: 3
- Forks: 0
- Open Issues: 0
-
Metadata Files:
- Readme: README.md
- License: LICENSE
Awesome Lists containing this project
README
# Equivalence Not Congruence for Imp
[![Docker CI][docker-action-shield]][docker-action-link]
[![Nix CI][nix-action-shield]][nix-action-link]
[![coqdoc][coqdoc-shield]][coqdoc-link]
[docker-action-shield]: https://github.com/cdepillabout/coq-equivalence-not-congruence/workflows/Docker%20CI/badge.svg?branch=master
[docker-action-link]: https://github.com/cdepillabout/coq-equivalence-not-congruence/actions?query=workflow:"Docker%20CI"
[nix-action-shield]: https://github.com/cdepillabout/coq-equivalence-not-congruence/workflows/Nix%20CI/badge.svg?branch=master
[nix-action-link]: https://github.com/cdepillabout/coq-equivalence-not-congruence/actions?query=workflow:"Nix%20CI"
[coqdoc-shield]: https://img.shields.io/badge/docs-coqdoc-blue.svg
[coqdoc-link]: https://cdepillabout.github.io/coq-equivalence-not-congruence
This project contains a Coq proof of an equivalence relation on the Imp
language that is not congruent. This answers a question from the
[Program Equivalence (Equiv)](https://softwarefoundations.cis.upenn.edu/plf-current/Equiv.html)
chapter of
[Programming Language Foundations](https://softwarefoundations.cis.upenn.edu/plf-current/index.html), which is the
second book of [Software Foundations](https://softwarefoundations.cis.upenn.edu/).
This proof is suggested in
this [answer on the Computer Science StackExchange](https://cs.stackexchange.com/a/98873/130503).
## Meta
- Author(s):
- Dennis Gosnell (initial)
- License: [BSD 3-Clause "New" or "Revised" License](LICENSE)
- Compatible Coq versions: 8.12 or later
- Additional dependencies: none
- Related publication(s): none
## Building and installation instructions
The easiest way to install the latest released version of Equivalence Not Congruence for Imp
is via [OPAM](https://opam.ocaml.org/doc/Install.html):
```shell
opam repo add coq-released https://coq.inria.fr/opam/released
opam install coq-coq-equivalence-not-congruence
```
To instead build and install manually, do:
``` shell
git clone https://github.com/cdepillabout/coq-equivalence-not-congruence.git
cd coq-equivalence-not-congruence
make # or make -j
make install
```
## Documentation
### Building
If you're using Nix, you can get into a shell with Coq available by running
`nix develop`:
```console
$ nix develop
```
You can build all the Coq files in this repo with `make`:
```console
$ make
```
After building, you can open up any of the files in
[`theories/`](./theories/) in `coqide` in order to work through the proofs.
You can regenerate the files in this repo (like `README.md`) from the
[`meta.yml`](./meta.yml) file by cloning
[`coq-community/templates`](https://github.com/coq-community/templates) and
running `generate.sh`:
```console
$ /some/path/to/coq-community/templates/generate.sh
```
You can also generate HTML documentation with `coqdoc`:
```console
$ make html
```
### Overview
The [Program Equivalence (Equiv)](https://softwarefoundations.cis.upenn.edu/plf-current/Equiv.html)
chapter of
[Programming Language Foundations](https://softwarefoundations.cis.upenn.edu/plf-current/index.html)
has a question like the following:
> We've shown that the `cequiv` relation is both an equivalence and
> a congruence on commands. Can you think of a relation on commands
> that is an equivalence but _not_ a congruence?
There is an
[answer to this question on the Computer Science StackExchange](https://cs.stackexchange.com/a/98873/130503):
> Let `x`, `y` be two fixed distinct variable names.
>
> Call `P` and `Q` equivalent iff `Q` is obtained from `P` by optionally
> swapping the variable names `x` and `y`. That is, either `Q = P` or
> `Q = P{x/y,y/x}` where the latter uses simultaneous substitution.
>
> It is an equivalence. Reflexivity follows by construction. For symmetry,
> `P == Q` swaps if `Q == P` swaps (where `==` is the equivalence relation).
> For transitivity, we consider the four
> cases: in the swap-swap case we get the same program back.
>
> It is not a congruence since `(x := x + 1) == (y := y + 1)` and
> `(x := 0) == (x := 0)`, but `(x := 0; x := x + 1) =/= (x := 0; y := y + 1)`
The [`theories/RenameVars.v`](./theories/RenameVars.v) file has a
formalization of this equivalence relation on the Imp language, as well as a
proof that there is no congruence in this case.
### Other approaches
This repo contains other examples of equivalence relations that are not
congruences:
- [`theories/CountUniqVars.v`](./theories/CountUniqVars.v)
This file contains an example of an equivalence relation where
two Imp programs are considered equivalent if they have the
same number of unique assignments for a set of variables.
For instance, `(X := X + 1; X := 200)` is equivalent to
`(Y := 3)` (since they both assign to one unique variable).
This file proves this is an equivalence relation, and shows
that it is not a congruence.