Maximal feasible element in a finite partial order

Proved

finite_maximal_extension__p2m_03bfe837_ddc7_49ff_ba57_41dcefd0c29f_a1

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

approximation-algorithmsplanar-graphstreewidth

In a finite partially ordered type, any element satisfying a downward-closed predicate lies below a maximal element satisfying that predicate.

Preamble

import Mathlib

Formal statement

theorem finite_maximal_extension__p2m_03bfe837_ddc7_49ff_ba57_41dcefd0c29f_a1
    {α : Type} [Finite α] [PartialOrder α]
    (P : α → Prop)
    (hdown : ∀ {a b : α}, a ≤ b → P b → P a)
    (a : α) (ha : P a) :
    ∃ b : α, a ≤ b ∧ P b ∧ ∀ c : α, b ≤ c → P c → c = b := 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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