A finite subset with at least as many elements is equal

Proved

WilliamsonShmoys.finset_subset_of_le_card_eq__p2m_15de7f60_3f94_44ed_9413_540d275804dc_a3

by Eyal1990 · Nov 1, 2026 · Mathlib 0df444a (Lean v4.33.1)

approximation-algorithmsplanar-graphstreewidth

If one finite set is contained in another and has at least as many elements, the two sets are equal.

Preamble

import Mathlib

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.finset_subset_of_le_card_eq__p2m_15de7f60_3f94_44ed_9413_540d275804dc_a3
    {α : Type} [DecidableEq α] (s t : Finset α)
    (hst : s ⊆ t) (hcard : t.card ≤ s.card) : s = t := by sorry

Source

David P. Williamson and David B. Shmoys, The Design of Approximation Algorithms, Cambridge University Press, 2011, author electronic manuscript, Section 10.2, proof of Theorem 10.11, pp. 270-272. https://doi.org/10.1017/CBO9780511921735. Bounded tree-width of the pieces: Robertson–Seymour (planar graphs of radius r have tree-width at most 3r+1) / Bodlaender (k-outerplanar graphs have tree-width at most 3k-1).

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