Equality Saturation Tactic for Lean

This repository contains a (work-in-progress) equality saturation tactic for Lean based on egg. This egg tactic is useful for automated equational reasoning based on given equational theorems.

Setup

The egg tactic requires Rust and its package manager Cargo. They are easily installed following the official guide.

To use egg in your Lean project, add the following line to your lakefile.toml:

[[require]]
name = "egg"
git  = "https://github.com/marcusrossel/lean-egg"
rev  = "main"

... or the following line to your lakefile.lean:

require "marcusrossel" / "egg" @ git "main"

Then, add import Egg to the files that require egg.

Basic Usage

The syntax of egg is very similar to that of simp or rw:

import Egg

example : 0 = 0 := by
  egg

example (a b c : Nat) (h₁ : a = b) (h₂ : b = c) : a = c := by
  egg [h₁, h₂]

open List in
example (as bs : List α) : reverse (as ++ bs) = (reverse bs) ++ (reverse as) := by
  induction as generalizing bs with
  | nil         => egg [reverse_nil, append_nil, append]
  | cons _ _ ih => egg [ih, append_assoc, reverse_cons, append]

But you can use it to solve some equations which simp cannot:

import Egg

variable (a b c d : Int)

example : ((a * b) - (2 * c)) * d - (a * b) = (d - 1) * (a * b) - (2 * c * d) := by
  egg [Int.sub_mul, Int.sub_sub, Int.add_comm, Int.mul_comm, Int.one_mul]

Sometimes, egg needs a little help with finding the correct sequence of rewrites. You can help out by providing guide terms which nudge egg in the right direction:

-- From Lean/Egg/Tests/Groups.lean
import Egg

example [Group G] (a : G) : a⁻¹⁻¹ = a := by
  egg [/- group axioms -/] using a⁻¹ * a

And if you want to prove your goal based on an equivalent hypothesis, you can use the from syntax:

import Egg

example (h : p ∧ q ∧ r) : r ∧ r ∧ q ∧ p ∧ q ∧ r ∧ p := by
  egg [and_comm, and_assoc, and_self] from h

For conditional rewriting, hypotheses can be provided after the rewrites:

import Egg

example {p q r : Prop} (h₁ : p) (h₂ : p ↔ q) (h₃ : q → (p ↔ r)) : p ↔ r := by
  egg [h₂, h₃; h₁]

Note that rewrites are also applied to hypotheses.

Related Work

• An Equality Saturation Tactic for Lean contains a detailed description of the tactic's inner workings as of June 2024.
• Guided Equality Saturation introduces the original prototype of the egg tactic.
• Bridging Syntax and Semantics of Lean Expressions in E-Graphs contains a high-level description of how the tactic handles binders and definitional equality.