A finite nonempty set in a preorder has a maximal element

Proved

exists_maximal_element_of_finite_set__p2m_5ad4dfa2_7b03_4649_bf47_2550ad7ee6b8_a1

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

approximation-algorithmsplanar-graphstreewidth

Every finite nonempty subset of a preorder contains an element with no strictly larger member of that subset.

Preamble

import Mathlib

set_option autoImplicit false

Formal statement

theorem exists_maximal_element_of_finite_set__p2m_5ad4dfa2_7b03_4649_bf47_2550ad7ee6b8_a1
    {α : Type} [Finite α] [Preorder α]
    (S : Set α) (hS : S.Finite) (hSne : S.Nonempty) :
    ∃ a ∈ S, ∀ b ∈ S, a ≤ b → b ≤ a := 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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