One active half-phase blocker and one non-half-phase blocker escape

Proved

LonelyRunner.SharedSlowestHalf.strict_of_half_and_nonhalf

by Whunt003 · Oct 7, 2026 · Mathlib 0df444a (Lean v4.33.1)

formalizationlonely-runnerlrc-shared-repair-20261007number-theory

Write $d(x)=|x-\operatorname{round}(x)|$ for distance to the nearest integer. For finite sets $R,W\subseteq\mathbb N$ , let $S(n,R,W)=\{v\in\mathbb N:0<v<n,\ v\notin R\}\cup W$ . A strict time means a real $t$ such that $d(vt)>1/n$ for every $v\in S(n,R,W)$ . Let $n,r,m,Q,w\in\mathbb N$ with $r\ge7$ , $r+2\le n\le2r$ , $m>0$ , $r\in R$ , $n<m(n-r)$ , and $mr<Q$ . Assume $r\mid2Q$ , $r\nmid Q$ , $r\nmid2w$ , and $d(1/2-Q/(nmr))\le1/n$ . Then $S(n,R,\{mr,Q,w\})$ has a strict time. No upper bound or ordering is imposed on $w$ .

Preamble

import Mathlib
import Mathlib.Data.ZMod.Units
import Mathlib.GroupTheory.Index
import Mathlib.RingTheory.Coprime.Lemmas
import Mathlib.RingTheory.Int.Basic
import Definitions.Def_LRC_SharedRepair_Defs

Formal statement

open LonelyRunner
open LonelyRunner.SharedSlowestHalf
open CollectiveSecondFastest CollectiveUnits BadCover

/-- An active half-phase blocker and any non-half-phase blocker leave a strict
time; the latter's speed may be arbitrarily large. -/
theorem LonelyRunner.SharedSlowestHalf.strict_of_half_and_nonhalf {n r m Q w : ℕ} {R : Finset ℕ}
    (hr : 7 ≤ r) (hrmax : r+2 ≤ n) (hm : 0 < m)
    (hrR : r ∈ R) (hlarge : n ≤ 2*r) (hfull : n < m*(n-r))
    (hpQ : m*r < Q) (hd : r ∣ 2*Q) (hnQ : ¬r ∣ Q)
    (hnw : ¬r ∣ 2*w)
    (hband : ndist (1/2-((Q:ℝ)/((m:ℝ)*r))/(n:ℝ)) ≤ 1/(n:ℝ)) :
    SeparatedMulti.HasStrictTime n R {m*r,Q,w} := by sorry

Source

Wade Hunter, lonely-runner research repository, https://github.com/huntrontrakkr/lonely-runner; local source snapshot 9b5f51aa235e321a3ec794c9bf60358933a8e6dc, lean/LonelyRunner/SharedSlowestHalf.lean, lines 113–197, declaration LonelyRunner.SharedSlowestHalf.strict_of_half_and_nonhalf. AI-assisted formalization. The snapshot identifier records the submitted local source; it is not a claim that this commit is publicly hosted.

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