feat(UnionFind): union by rank with verified O(log n) find#695
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feat(UnionFind): union by rank with verified O(log n) find#695barni120400 wants to merge 1 commit into
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Formalize a union-by-rank union-find over `Fin n` in `Cslib/Algorithms/Lean/UnionFind/`, reusing the `TimeM` cost monad like `MergeSort`. Main results (all sorry-free): * `rank_le_log`: every rank is at most log2 n. * `findDepth_le_log`: `find` dereferences at most log2 n parent pointers. * `link_wellFormed`: union by rank preserves the well-formedness invariant, so the bound holds for any forest built from singletons by unions. * `find_time_le_log` / `find_ret`: a timed `find` in `TimeM` costs at most log2 n ticks and returns the correct root. No path compression yet (worst-case union-by-rank bound). AI usage: developed with a mix of Claude Code and manual proving/review; proofs iterated against the Lean compiler. Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>
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Formalizes union-find (union by rank) in
Cslib/Algorithms/Lean/UnionFind/, the "Union find" item under Verified data structures with time complexity (Batteries + Time Monad) in CONTRIBUTING. Placed next toMergeSortand reusing theTimeMcost monad the same way.Main results, all
sorry-free (lake build+lake testpass):rank_le_log: every rank is at most⌊log₂ n⌋.findDepth_le_log:finddereferences at most⌊log₂ n⌋parent pointers.link_wellFormed: union by rank preserves the well-formedness invariant, so the bound holds for any forest built from singletons by unions.find_time_le_log/find_ret: a timedfindinTimeMcosts at most⌊log₂ n⌋ticks and returns the correct root.No path compression yet (worst-case union-by-rank bound; the inverse-Ackermann amortized result would be a larger follow-up).
This is a single PR for the whole feature (I had first drafted it as a stack; consolidated here). Supersedes #692.
AI usage: developed with a mix of Claude Code and manual proving/review; the proofs were iterated against the Lean compiler.