A natural-valued function attains a maximum on a nonempty finite predicate

Proved

finite_maximum_on_nonempty_predicate__p2m_16fb1a0d_e68c_4595_a206_2f4554b0589b_a2

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

approximation-algorithmsplanar-graphstreewidth

On a finite type, every natural-valued function attains a maximum among the elements satisfying an inhabited predicate.

Preamble

import Mathlib

Formal statement

theorem finite_maximum_on_nonempty_predicate__p2m_16fb1a0d_e68c_4595_a206_2f4554b0589b_a2
    {α : Type} [Fintype α] (P : α → Prop) (f : α → Nat)
    (hP : ∃ x, P x) :
    ∃ x, P x ∧ ∀ y, P y → f y ≤ f x := 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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