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.

The implementation and underlying theory is decribed in the paper Tactics for Reasoning modulo AC in Coq.

• Author(s):
  • Thomas Braibant (initial)
  • Damien Pous (initial)
  • Fabian Kunze
• Coq-community maintainer(s):
  • Fabian Kunze (@fakusb)
  • Karl Palmskog (@palmskog)
• License: GNU Lesser General Public License v3.0 or later
• Compatible Coq versions: Coq master (use the corresponding branch or release for other Coq versions)
• Compatible OCaml versions: all versions supported by Coq
• Additional dependencies: none

The easiest way to install the latest released version is via OPAM:

opam repo add coq-released https://coq.inria.fr/opam/released
opam install coq-aac-tactics

To instead build and install manually, do:

git clone https://github.com/coq-community/aac-tactics
cd aac-tactics
make   # or make -j <number-of-cores-on-your-machine>
make install

After installation, the included modules are available under the AAC_tactics namespace.

The following example shows an application of the tactics for reasoning over Z binary numbers:

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 provides a succinct introduction and more examples of how to use this plugin.

The file 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.

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.