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---
fullname: AAC tactics
shortname: aac-tactics
description: |
This Coq plugin provides tactics for rewriting universally quantified
equations, modulo associativity and commutativity of some operator.
The tactics can be applied for custom operators by registering the
operators and their properties as type class instances. Many common
operator instances, such as for Z binary arithmetic and booleans, are
provided with the plugin.
paper:
doi: 10.1007/978-3-642-25379-9_14
url: https://arxiv.org/abs/1106.4448
title: Tactics for Reasoning modulo AC in Coq
authors:
- name: Thomas Braibant
initial: true
- name: Damien Pous
initial: true
- name: Fabian Kunze
initial: false
maintainers:
- name: Fabian Kunze
nickname: fakusb
- name: Karl Palmskog
nickname: palmskog
opam-file-maintainer: palmskog@gmail.com
license:
fullname: GNU Lesser General Public License v3.0 or later
identifier: LGPL-3.0-or-later
plugin: true
supported_coq_versions:
text: Coq 8.9 (use the corresponding branch or release for other Coq versions)
opam: '{>= "8.9" & < "8.10~"}'
tested_coq_versions:
- version_or_url: 8.9
tested_coq_opam_version: 8.9
namespace: AAC_tactics
keywords:
- name: reflexive tactic
- name: rewriting
- name: rewriting modulo associativity and commutativity
- name: rewriting modulo ac
- name: decision procedure
categories:
- name: Miscellaneous/Coq Extensions
- name: Computer Science/Decision Procedures and Certified Algorithms/Decision procedures
documentation: |
## Documentation
The following example shows an application of the tactics for reasoning over Z binary numbers:
```coq
Require Import AAC_tactics.AAC.
Require AAC_tactics.Instances.
Require Import ZArith.
Section ZOpp.
Import Instances.Z.
Variables a b c : Z.
Hypothesis H: forall x, x + Z.opp x = 0.
Goal a + b + c + Z.opp (c + a) = b.
aac_rewrite H.
aac_reflexivity.
Qed.
End ZOpp.
```
The file [Tutorial.v](theories/Tutorial.v) provides a succinct introduction
and more examples of how to use this plugin.
The file [Instances.v](theories/Instances.v) defines several type class instances
for frequent use-cases of this plugin, that should allow you to use it off-the-shelf.
Namely, it contains instances for:
- Peano naturals (`Import Instances.Peano.`)
- Z binary numbers (`Import Instances.Z.`)
- N binary numbers (`Import Instances.N.`)
- P binary numbers (`Import Instances.P.`)
- Rational numbers (`Import Instances.Q.`)
- Prop (`Import Instances.Prop_ops.`)
- Booleans (`Import Instances.Bool.`)
- Relations (`Import Instances.Relations.`)
- all of the above (`Import Instances.All.`)
To understand the inner workings of the tactics, please refer to
the `.mli` files as the main source of information on each `.ml` file.
## Acknowledgements
The initial authors are grateful to Evelyne Contejean, Hugo Herbelin,
Assia Mahboubi, and Matthieu Sozeau for highly instructive discussions.
The plugin took inspiration from the plugin tutorial "constructors" by
Matthieu Sozeau, distributed under the LGPL 2.1.
---
|