Maximal element in a finite preorder subset

Proved

finite_preorder_has_maximal__p2m_d50bd5f9_430a_4ac6_90e0_7716421734c3_a1

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

approximation-algorithmsplanar-graphstreewidth

Every nonempty predicate-defined subset of a finite preorder contains an element with no strictly larger member in that subset: whenever the chosen element is below another member, that other member is also below the chosen one.

Preamble

import Mathlib

set_option autoImplicit false

Formal statement

theorem finite_preorder_has_maximal__p2m_d50bd5f9_430a_4ac6_90e0_7716421734c3_a1
    {α : Type} [Preorder α] [Finite α]
    (P : α → Prop) (hne : ∃ 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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