🤖 AI Summary
This work presents the first complete formal verification in Lean 4 of the informal Euclidean domain algorithms originally described in the 1986 ICON language. By separating concerns into mathematical definitions, computable implementations, and output formatting, the project constructs a computable mirror atop Mathlib’s `EuclideanDomain` hierarchy and integrates a regression testing infrastructure to reproduce the original outputs. All 14 algorithms are formally specified, with core procedures such as integer GCD and the extended Euclidean algorithm accompanied by machine-checked correctness proofs. The formalization precisely delineates the boundaries between computability and mathematical correctness while fully replicating the benchmark results reported in Ericson’s technical report.
📝 Abstract
We describe a Lean 4 formalization of the algorithms and domain types from NYU Computer Science Technical Report \#232, \emph{An ICON Package for Experimenting with Euclidean Domains} (Ericson, 1986). The original system implemented Lipson's catalog of procedures over integers, rationals, modular rings, polynomial rings, and truncated power series via a custom runtime dispatch mechanism in Icon. The present work separates three concerns: mathematical definitions grounded in Mathlib's \texttt{EuclideanDomain} hierarchy, computable mirrors suitable for evaluation and regression testing, and report-formatting infrastructure that reproduces the 1986 benchmark output line-for-line. All fourteen application algorithms from Section 3 of the report are defined and typecheck without \texttt{sorry}; those grounded in Mathlib -- chiefly integer gcd and extended Euclid -- additionally carry machine-checked proofs. We classify each procedure by its epistemic status relative to Mathlib, enumerate the coherence obligations between the proof and computable layers, and state precisely what is theorem-backed versus regression-trusted. The formalization makes explicit the verification boundary that the 1986 package crossed only informally.