Maximal satisfying element above a given element in a finite preorder

Proved

finite_predicate_has_maximal_above__p2m_a7e40797_bbb7_4ed5_a4c8_71639bc08809_a1

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

approximation-algorithmsplanar-graphstreewidth

For a predicate on a finite preorder, every satisfying element lies below a satisfying element that has no strictly greater satisfying element.

Preamble

import Mathlib

Formal statement

theorem finite_predicate_has_maximal_above__p2m_a7e40797_bbb7_4ed5_a4c8_71639bc08809_a1
    {α : Type} [Preorder α] [Finite α]
    (S : α → Prop) (x : α) (hx : S x) :
    ∃ y, S y ∧ x ≤ y ∧ ∀ z, S z → y ≤ z → z ≤ y := 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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