This Coq plugin provides tactics for rewriting and proving universally quantified equations modulo associativity and commutativity of some operator, with idempotent commutative operators enabling additional simplifications. The tactics can be applied for custom operators by registering the operators and their properties as type class instances. Instances for many commonly used operators, such as for binary integer arithmetic and booleans, are provided with the plugin.

• Author(s):
  • Thomas Braibant (initial)
  • Damien Pous (initial)
  • Fabian Kunze
• Coq-community maintainer(s):
  • Karl Palmskog (@palmskog)
• License: GNU Lesser General Public License v3.0 or later
• Compatible Coq versions: master (use the corresponding branch or release for other Coq versions)
• Compatible OCaml versions: 4.09.0 or later
• Additional dependencies: none
• Coq namespace: AAC_tactics
• Related publication(s):
  • Tactics for Reasoning modulo AC in Coq doi:10.1007/978-3-642-25379-9_14

The easiest way to install the latest released version of AAC Tactics 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.git
cd aac-tactics
make   # or make -j <number-of-cores-on-your-machine> 
make install

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

From AAC_tactics Require Import AAC.
From AAC_tactics Require Instances.
From Coq 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.

  Goal Z.max (b + c) (c + b) + a + Z.opp (c + b) = a.
    aac_normalise.
    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.)
• Lists (Import Instances.Lists.)
• N binary numbers (Import Instances.N.)
• Positive 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.

See also the latest coqdoc documentation and the latest ocamldoc documentation.

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.