A finite partially ordered type has a maximal element above any satisfying point
Provedfinite_poset_maximal_above__p2m_cf1fad93_82bb_45b4_92ca_a77558fa3813_a1approximation-algorithmsplanar-graphstreewidth
In a finite partially ordered type, any point satisfying a predicate lies below a predicate-satisfying maximal point.
Preamble
import Mathlib
Formal statement
theorem finite_poset_maximal_above__p2m_cf1fad93_82bb_45b4_92ca_a77558fa3813_a1
{α : Type} [Fintype α] [PartialOrder α]
(P : α → Prop) {a : α} (ha : P a) :
∃ m, a ≤ m ∧ P m ∧ ∀ b, m ≤ b → P b → b ≤ m := by sorrySource
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).