A unique shared second-fastest repair forces strictness

Proved

LonelyRunner.CollectiveSecondFastest.strict_of_shared_second_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 $n,r,s,m,k,Q,\ell\in\mathbb N$ , $P=mr=ks$ , with $\ell\ge3$ , $|W|\le\ell$ , $n\le2r$ , $r<s<n$ , $r\in R$ , $m\ge2$ , and $k>0$ . Assume $W$ is positive, $P,Q\in W$ , $P<Q$ , every member of $W\setminus\{P,Q\}$ is less than $P$ , and $P$ is the unique member of $W$ divisible by $r$ . If $\varphi(r)\ge2\ell-2$ , 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.CollectiveSecondFastest
open BadCover

/-- A unique repair shared with a larger baseline divisor cannot be the
second-fastest insertion when the primitive-unit capacity is large enough.
The conclusion is an actual strict time, with arbitrary additional deletions. -/
theorem LonelyRunner.CollectiveSecondFastest.strict_of_shared_second_fastest {n r s m k Q ell : ℕ} {R W : Finset ℕ}
    (hell : 3 ≤ ell) (hcard : W.card ≤ ell)
    (hlarge : n ≤ 2*r) (hrs : r < s) (hsn : s < n)
    (hrR : r ∈ R) (hm : 2 ≤ m) (hk : 0 < k) (hshared : m*r=k*s)
    (hP : m*r ∈ W) (hQ : Q ∈ W) (hpq : m*r < Q)
    (hWpos : ∀p ∈ W, 0 < p)
    (hslow : ∀p ∈ (W.erase (m*r)).erase Q, p < m*r)
    (hunique : ∀p ∈ W.erase (m*r), ¬r ∣ p)
    (hphi : 2*ell-2 ≤ r.totient) :
    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/CollectiveSecondFastest.lean, lines 327–360, declaration LonelyRunner.CollectiveSecondFastest.strict_of_shared_second_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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