A nonempty predicate on a finite partial order has a maximal element

Proved

WilliamsonShmoys.finite_predicate_has_maximal_element__p2m_c0ec6b8f_fcbf_4b2c_8dba_71231ff2f6b6_a1

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

approximation-algorithmsplanar-graphstreewidth

Every nonempty predicate on a finite partially ordered type contains an element that has no strictly larger element satisfying the predicate.

Preamble

import Mathlib

set_option autoImplicit false

universe u

Formal statement

theorem WilliamsonShmoys.finite_predicate_has_maximal_element__p2m_c0ec6b8f_fcbf_4b2c_8dba_71231ff2f6b6_a1
    {α : Type u} [Fintype α] [PartialOrder α]
    (P : α → Prop) (hP : ∃ x, P x) :
    ∃ x, P x ∧ ∀ y, P y → x ≤ y → y ≤ x := 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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