Order Maintenance in Worst-Case Constant Time, Verified

Abstract

Order maintenance asks for a data structure that keeps a sequence of elements under insertions and deletions and tells which of two elements comes first. We give an algorithm, assembled from published components, that performs each operation in worst-case constant time, and verify it in Lean 4. The algorithm is a program for a random-access machine deeply embedded in Lean, whose interpreter counts every instruction. We prove that the program answers every query as a simple reference specification does, that each operation takes at most a fixed number of steps, independent of the number of elements and of the history, and that after $t$ operations no value it has computed has more than $O(\log t)$ bits, so it runs unchanged on a machine with words of that size. A verified translation that preserves step counts exactly connects the high-level proofs to the machine program, so the interpreter is the only cost model to trust. An unverified native implementation shows the algorithm's practical cost.