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https://github.com/affeldt-aist/monae

Monadic effects and equational reasonig in Coq
https://github.com/affeldt-aist/monae

math-comp mathcomp monad-transformers monadic-effects monads nondeterminism probabilistic-programming ssreflect

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Monadic effects and equational reasonig in Coq

Host: GitHub
URL: https://github.com/affeldt-aist/monae
Owner: affeldt-aist
License: lgpl-2.1
Created: 2018-08-06T00:36:18.000Z (about 6 years ago)
Default Branch: master
Last Pushed: 2024-07-12T12:05:29.000Z (about 2 months ago)
Last Synced: 2024-07-12T14:55:02.415Z (about 2 months ago)
Topics: math-comp, mathcomp, monad-transformers, monadic-effects, monads, nondeterminism, probabilistic-programming, ssreflect
Language: Coq
Homepage:
Size: 5.01 MB
Stars: 68
Watchers: 6
Forks: 11
Open Issues: 16
Metadata Files:
- Readme: README.md
- License: LICENSE.txt

Awesome Lists containing this project

README

        
# Monadic effects and equational reasoning in Coq

[![Docker CI][docker-action-shield]][docker-action-link]

[docker-action-shield]: https://github.com/affeldt-aist/monae/workflows/Docker%20CI/badge.svg?branch=master

[docker-action-link]: https://github.com/affeldt-aist/monae/actions?query=workflow:"Docker%20CI"

This Coq library contains a hierarchy of monads with their laws used

in several examples of monadic equational reasoning.

## Meta

- Author(s):

  - Reynald Affeldt (initial)

  - David Nowak (initial)

  - Takafumi Saikawa (initial)

  - Jacques Garrigue

  - Ayumu Saito

  - Celestine Sauvage

  - Kazunari Tanaka

- License: [LGPL-2.1-or-later](LICENSE)

- Compatible Coq versions: Coq 8.18-8.19

- Additional dependencies:

  - [MathComp ssreflect](https://math-comp.github.io)

  - [MathComp fingroup](https://math-comp.github.io)

  - [MathComp algebra](https://math-comp.github.io)

  - [MathComp solvable](https://math-comp.github.io)

  - [MathComp field](https://math-comp.github.io)

  - [MathComp analysis](https://github.com/math-comp/analysis)

  - [Infotheo](https://github.com/affeldt-aist/infotheo)

  - [Paramcoq](https://github.com/coq-community/paramcoq)

  - [Hierarchy Builder](https://github.com/math-comp/hierarchy-builder)

  - [Coq-Equations](https://github.com/mattam82/Coq-Equations)

- Coq namespace: `monae`

- Related publication(s):

  - [A hierarchy of monadic effects for program verification using equational reasoning](https://staff.aist.go.jp/reynald.affeldt/documents/monae.pdf) doi:[10.1007/978-3-030-33636-3_9](https://doi.org/10.1007/978-3-030-33636-3_9)

  - [Extending Equational Monadic Reasoning with Monad Transformers](https://drops.dagstuhl.de/opus/volltexte/2021/13881/) doi:[10.4230/LIPIcs.TYPES.2020.2](https://doi.org/10.4230/LIPIcs.TYPES.2020.2)

  - [A Trustful Monad for Axiomatic Reasoning with Probability and Nondeterminism](https://arxiv.org/abs/2003.09993) doi:[10.1017/S0956796821000137](https://doi.org/10.1017/S0956796821000137)

  - [Towards a practical library for monadic equational reasoning in Coq](https://staff.aist.go.jp/reynald.affeldt/documents/monae-mpc2022.pdf) doi:[10.1007/978-3-031-16912-0_6](https://doi.org/10.1007/978-3-031-16912-0_6)

  - [Environment-friendly monadic equational reasoning for OCaml](https://coq-workshop.gitlab.io/2023/abstracts/coq2023_monadic-reasoning.pdf) 

## Building and installation instructions

The easiest way to install the latest released version of Monadic effects and equational reasoning in Coq

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-monae

```

It installs two directories in `coq/user-contrib`: `monae` and

`monaeImpredicativeSet`.

To instead build and install manually (with GNU `make`), do:

``` shell

git clone https://github.com/affeldt-aist/monae.git

cd monae

make -j 4

make install

```

## Overview

This repository contains a formalization of monads including examples

of monadic equational reasoning and several models. This includes for

example the formalization of the following papers:

- [Gibbons and Hinze, Just do It: Simple Monadic Equational Reasoning, ICFP 2011] (except Sect. 10.2)

- [Gibbons, Unifying Theories of Programming with Monads, UTP 2012] (up to Sect. 7.2)

- [Mu, Equational Reasoning for Non-determinism Monad: A Case study of Spark Aggregation, TR-IIS-19-002, Academia Sinica]

- [Mu, Calculating a Backtracking Algorithm: An exercise in Monadic Program Derivation, TR-IIS-29-003, Academia Sinica]

- [Mu, Functional Pearls: Reasoning and Derivation of Monadic Programs, A case study of non-determinism and state, 2017]

  + This is a draft paper. In the first release, we formalized this draft up to Sect. 5.

    The contents have been since superseded by [mu2019tr2] and [mu2019tr3].

- [Mu and Chiang, Deriving Monadic Quicksort (Declarative Pearl), 2020]

This library has been applied to other formalizations:

- application to program semantics (see file `smallstep.v`)

- formalization of monad composition [Jones and Duponcheel, Composing Monads, Yale RR 1993] (Sections 2 and 3)

- formalization of monad transformers [Jaskelioff, Modular Monad Transformers, ESOP 2009] (up to Sect. 4)

  + completed with details from [Jaskelioff, Lifting of Operations in Modular Monadic Semantics, PhD 2009]

  + see directory `impredicative_set` for the formalization of [Jaskelioff, Modular Monad Transformers, ESOP 2009] (from Sect. 5)

- formalization of the geometrically convex monad (main reference:

  [Cheung, Distributive Interaction of Algebraic Effects, PhD Thesis, U. Oxford, 2017])

![Available monads](./hier.png "Available Monads")

## Files

- core:

  + [monae_lib.v](./theories/core/monae_lib.v): simple additions to base libraries

  + [hierarchy.v](./theories/core/hierarchy.v): hierarchy of monadic effects

  + [category.v](./theories/core/category.v): formalization of concrete categories (generalization of the bottom layers of `hierarchy.v`)

  + [monad_transformer.v](./theories/core/monad_transformer.v): monad transformers

    * completed by `ifmt_lifting.v` and `iparametricty_codensity.v` in the directory `impredicative_set`

      - the directory `impredicative_set` contains a lighter version of Monae where monads live in `Set` and that compiles with `impredicative-set`

- libraries for each monad theory:  

  + [monad_lib.v](./theories/lib/monad_lib.v): basic lemmas about monads

  + [fail_lib.v](./theories/lib/fail_lib.v): lemmas about the fail monad and related monads

  + [state_lib.v](./theories/lib/state_lib.v): lemmas about state-related monads

  + [array_lib.v](./theories/lib/array_lib.v): lemmas about the array monad

  + [trace_lib.v](./theories/lib/trace_lib.v): lemmas about about the state-trace monad

  + [proba_lib.v](./theories/lib/proba_lib.v): lemmas about the probability monad

  + [typed_store_lib.v](./theories/lib/typed_store_lib.v): derived definitions and lemmas about about the typed store monad

- models of monads:

  + [monad_model.v](./theories/models/monad_model.v): concrete models of monads

  + [proba_monad_model.v](./theories/models//proba_monad_model.v): model of the probability monad

  + [gcm_model.v](./theories/models//gcm_model.v): model of the geometrically convex monad

  + [altprob_model.v](./theories/models//altprob_model.v): model of a monad that mixes non-deterministic choice and probabilistic choice

  + [typed_store_model.v](./theories/models//typed_store_model.v): alternative model of the typed store monad

- applications:

  + [example_relabeling.v](./theories/applications/example_relabeling.v): tree relabeling

  + [example_monty.v](./theories/applications/example_monty.v): Monty Hall problem

  + [example_spark.v](./theories/applications/example_spark.v): Spark aggregation

  + [example_iquicksort.v](./theories/applications/example_iquicksort.v): in-place quicksort

  + [example_quicksort.v](./theories/applications/example_quicksort.v): functional quicksort

  + [example_nqueens.v](./theories/applications/example_nqueens.v): the n-queens puzzle

  + [example_typed_store.v](./theories/applications/example_typed_store.v): ML programs with references

  + [example_transformer.v](./theories/applications/example_transformer.v): monad transformers

  + [smallstep.v](./theories/applications/smallstep.v): semantics for an imperative language, with equivalence operational semantics/denotation and sample imperative programs

  + [monad_composition.v](./theories/applications/monad_composition.v): composing monads

  + [category_ext.v](./theories/applications/category_ext.v): experimental library about categories

## About Installation with Windows 11

Installation of monae on Windows is less simple.

First install infotheo following the [instructions for Windows 11](https://github.com/affeldt-aist/infotheo).

Once infotheo is installed (with opam), do:

- `opam install coq-monae` or `git clone [email protected]:affeldt-aist/monae.git; opam install .`

## Original License

Before version 0.2, monae was distributed under the terms of the `GPL-3.0-or-later` license