E-graphs are a data structure to compactly represent a program space and reason about equality of program terms. E-graphs have been successfully applied to a number of domains, including program optimization and automated theorem proving. In many applications, however, it necessary to reason about disequality of terms, not only about equality. While disequality reasoning can be encoded, direct support for disequalities increases performance and simplifies the metatheory.

In this paper, we develop a framework independent of a specific implementation to formally reason about e-graphs. For the first time, we prove the equivalence of e-graphs to the reflexive, symmetric, transitive, and congruent closure of the equivalence relation they are expected to encode. We use these results to present the first formalization of an extension of e-graphs that directly supports disequalities and prove an analytical result about their superior efficiency compared to embedding techniques that are commonly used in SMT solvers and automated verifiers.

We implement our approach in an extension to egg, a popular e-graph Rust library. We evaluate our solution in an SMT solver and an automated theorem prover using standard benchmarks. The results indicate that direct support for disequalities significantly outperforms other encodings based on disequality embedding confirming the results obtained analitically.