Maximal extension in a finite partial order

Proved

WilliamsonShmoys.finite_preorder_maximal_extension__p2m_8fa7b5e9_9ae2_4561_bc62_ceb5c7ac1a09_a1

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

approximation-algorithmsplanar-graphstreewidth

In a finite partially ordered type, if a predicate holds at an element x, there is an element y above x satisfying the predicate that is maximal among predicate-satisfying elements above x.

Preamble

import Mathlib
set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.finite_preorder_maximal_extension__p2m_8fa7b5e9_9ae2_4561_bc62_ceb5c7ac1a09_a1
    {α : Type} [Finite α] [PartialOrder α]
    (P : α → Prop) (x : α) (hx : P x) :
    Exists fun y : α => And (x ≤ y) (And (P y)
      (forall z : α, y ≤ z → P 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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