Finite maximal element by a natural-valued measure
Provedfinite_maximal_by_nat_measure__p2m_6d710174_22d6_4830_a446_d998b14d2565_a1approximation-algorithmsplanar-graphstreewidth
On a finite type, if a predicate is nonempty and a natural-valued measure strictly increases along every related pair of distinct predicate elements, then some predicate element has no distinct related successor.
Preamble
import Mathlib set_option autoImplicit false
Formal statement
theorem finite_maximal_by_nat_measure__p2m_6d710174_22d6_4830_a446_d998b14d2565_a1
{α : Type} [Fintype α] (P : α → Prop) (R : α → α → Prop) (measure : α → Nat)
(hP : ∃ a, P a)
(hstrict : ∀ a b, P a → P b → R a b → a ≠ b → measure a < measure b) :
∃ a, P a ∧ ∀ b, P b → R a b → a = b := 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).