feat(Algorithms): comparison sort lower bound via decision-tree induction - #770
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…tion A comparison program is a free monad over a single comparison query, i.e. a binary decision tree. card_image_run_le_two_pow_of_cost_le bounds its distinct results over a finite comparator family by 2 ^ t whenever every comparator in the family costs at most t comparisons, by structural induction: the family splits at the root comparison. A program that sorts under every hidden permutation order on Fin n therefore takes at least log2(n!) comparisons in the worst case; (n/2) * log2(n/2) follows as a corollary.
SamuelSchlesinger
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August 3, 2026 06:41
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An attempt to prove the comparison sort lower bound of #685 taking, in my view, a more natural proof approach for this model. The original proof analyzes traces of the computation, while this approach takes direct advantage of the inductive structure of the free monad and essentially does induction on the program. At a high level, #685 formalizes the encoding argument whereas we formalize the decision tree argument. The arithmetic estimate of log_2 n! is adapted from #685.
Claude Fable 5 authored this code under my direction and review.