A unique shared fastest repair forces strictness

Proved

LonelyRunner.SharedFastest.strict_of_shared_fastest

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 $n,r,s,m,k\in\mathbb N$ and $P=mr=ks$ . Assume $n\ge5$ , $r\ge7$ , $n\le2r$ , $r<s<n$ , $r\in R$ , $m\ge2$ , $k>0$ , $P\in W$ , $W$ is positive, $|W|\le3$ , every member of $W\setminus\{P\}$ is less than $P$ , and none is divisible by $r$ . Then $S(n,R,W)$ has a strict time.

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.SharedFastest
open FastestBoundary

/-- If the fastest insertion is shared by r<s and uniquely supplies r>=7,
at most three insertions cannot prevent a strictly lonely time. -/
theorem LonelyRunner.SharedFastest.strict_of_shared_fastest {n r s m k : ℕ} {R W : Finset ℕ}
    (hn : 5 ≤ n) (hr : 7 ≤ r) (hrR : r ∈ R)
    (hlarge : n ≤ 2*r) (hrs : r < s) (hsn : s < n)
    (hm : 2 ≤ m) (hk : 0 < k) (heq : m*r=k*s)
    (hpW : m*r ∈ W) (hW : ∀ q ∈ W, 0 < q)
    (hmax : ∀ q ∈ W.erase (m*r), q < m*r)
    (hunique : ∀ q ∈ W.erase (m*r), ¬ r ∣ q)
    (hcard : W.card ≤ 3) : SeparatedMulti.HasStrictTime n R W := by sorry

Source

Wade Hunter, lonely-runner research repository, https://github.com/huntrontrakkr/lonely-runner; local source snapshot 9b5f51aa235e321a3ec794c9bf60358933a8e6dc, lean/LonelyRunner/SharedFastest.lean, lines 113–125, declaration LonelyRunner.SharedFastest.strict_of_shared_fastest. 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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