Lifting E-Graphs: A Function Isn't a Constant

Lifting E-Graphs: A Function Isn’t a Constant proposes a lifting-aware extension of e-graphs that treats context as part of term identity, avoiding unsound variable capture while increasing sharing across differently scoped expressions. Built on ideas from slotted e-graphs and Co-de Bruijn syntax, the paper represents lifted terms with fat identifiers that pair an integer class id with a thinning bitvector, then uses lift-pulling smart constructors and a thinning-aware union-find to canonicalize equalities under α-equivalence-like context changes. The key technical move is to make lifting a first-class combinator: lift compositions collapse by thinning composition, lift homomorphisms push through function symbols, and union can peel common thinnings to recover ordinary equality, or synthesize a fresh representative when two lifted forms are not directly reconcilable. The result is an e-graph modulo theories style design that can compact expressions such as x, y, z, w 7→ 42 + 42 into a single pulled form, while preserving distinctions like sin1 and sin2 only up to their shared lifted structure. The paper also generalizes the same machinery to annotated union-finds, including offset, monus, and factor variants, to show that group structure is not required for relational union-find annotations.