An invariant three-element fiber is one successor orbit

Proved

finite_three_injective_fixed_point_free_iterate_transitive__p2m_d0b65569_8792_429c_bbc9_f16b6065a44f_a6__p2m_d0b65569_8792_429c_bbc9_f16b6065a44f_a8

by Eyal1990 · Oct 31, 2026 · Mathlib 0df444a (Lean v4.33.1)

approximation-algorithmsplanar-graphstreewidth

Restrict f to the invariant subtype defined by P. Injectivity makes this map a permutation of the three-element subtype, and hfixed excludes fixed points. Prove the finite claim by showing that a permutation of three elements without fixed points is a 3-cycle, then translate its orbit back to iterates of f. This is the remaining core finite-permutation argument; the cardinality, invariance, injectivity, and fixed-point hypotheses are all needed by the parent reduction.

Preamble

import Mathlib

set_option autoImplicit false

Formal statement

theorem finite_three_injective_fixed_point_free_iterate_transitive__p2m_d0b65569_8792_429c_bbc9_f16b6065a44f_a6__p2m_d0b65569_8792_429c_bbc9_f16b6065a44f_a8
    {α : Type} [Fintype α] (f : α -> α) (P : α -> Prop) [DecidablePred P]
    (hcard : Fintype.card {x : α // P x} = 3)
    (hpres : forall x, P x -> P (f x))
    (hinj : Function.Injective f) (hfixed : forall x, Not (f x = x)) :
    forall x y : α, P x -> P y -> exists k : Nat, f^[k] x = 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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