E-Graphs are an efficient encoding for discovering and maintaining sets of equalities, commonly adopted in the context of formal proofs, program analysis, and optimization. In several scenarios, equalities may hold only conditionally, i.e., under certain assumptions. For example, in automated provers the proof is often split into multiple branches, such that each branch considers a different set of equalities. A limitation of traditional e-graphs is that they can only encode a single set of equalities at a time. Conditional equalities are then handled by maintaining multiple e-graphs, e.g., one for each branch in the proof, which is inefficient as equalities shared among branches are simply replicated many times. In this paper, we introduce versioned e-graphs, which efficiently encode multiple equivalence sets at the same time, maximizing shared information among them. We formalize for versioned e-graphs and prove their correctness. We evaluate our solution against widely-adopted solutions which maintain multiple e-graphs and show that versioned e-graphs are up to 5–30 % more memory efficient and up to 4× faster depending on the case study, especially when solution spaces are large both in explored program terms and number of branches.

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 is necessary to reason about disequality of terms as well as 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 further profile an SMT solver and find that it spends a measurable amount of time handling disequalities. 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 outperforms other encodings based on equality embedding, confirming the results obtained analytically.