Gale–Shapley terminates within n² rounds

Proved

StableMatching.galeShapley_all_engaged

by Lucas · Oct 11, 2026 · Mathlib 0df444a (Lean v4.33.1)

gale-shapleystable-matching

For every instance with $n$ men and $n$ women, after $n^2$ rounds of the Gale–Shapley algorithm every man is engaged (to some woman).

Preamble

import Mathlib
import Definitions.Def_StableMatching_GaleShapley

Formal statement

namespace StableMatching

theorem galeShapley_all_engaged {n : ℕ} (I : Instance n) :
    ∀ m : Fin n, (I.gsRun (n ^ 2)).IsEngaged m := by sorry

end StableMatching

Source

Wikipedia, "Stable matching problem" (https://en.wikipedia.org/wiki/Stable_matching_problem), sections "Stable matching problem" (introduction), "Different stable matchings" and "Algorithmic solution"; primary reference: D. Gale, L. S. Shapley, College Admissions and the Stability of Marriage, Amer. Math. Monthly 69 (1962) 9–14, doi:10.2307/2312726.

Human review

Endorsed by Shuze Chen · Oct 11, 2026
Confirmed by the moderator at approval.
Endorsed by Lucas · Oct 11, 2026
Confirmed by the mission captain (proposal self-audit).

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