Verified primality decision by bounded trial division

Proved

ThreeSum.LightPrimality.meets

by wurtle · Oct 14, 2026 · Mathlib 0df444a (Lean v4.33.1)

3sumalgorithmsfine-grained-complexitythree-sum-1962word-ram

Let $2\le n\le V$ and let $f$ be a scratch base. For machine limits $S,W,D$ , assume $f+\lfloor\log_2(V+1)\rfloor+2\le S\le W$ , $8V+16\le W$ , and $d<D$ . Suppose program $P$ contains the primality body and Euclidean-remainder helper. Put $R_V=43(\lfloor\log_2(V+1)\rfloor+1)+34$ . From any integer memory $\mu$ , execution on $(n,f)$ terminates within $(n-2)(R_V+70)+70$ steps and returns

r=\begin{cases}1&n\text{ is prime},\\0&n\text{ is composite},\end{cases} \qquad \mu'(a)=\mu(a)\quad(a<f).

This supplies a concrete primality test for deterministic prime-pool construction.

Preamble

/-
Copyright (c) 2026 Anthropic, PBC. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
SPDX-License-Identifier: Apache-2.0
-/

import Definitions.Def_APSPSource_RemainingDefinitions
import Definitions.Def_APSPSource_ThreeSumApsp_Lang_Logic
import Definitions.Def_APSPSource_ThreeSumApsp_Lang_Regions
import Definitions.Def_APSPSource_ThreeSumApsp_Lang_Syntax
import Definitions.Def_ThreeSum1962Source_Bundle01
import Definitions.Def_ThreeSum1962Source_Bundle03
import Definitions.Def_ThreeSum1962Source_Bundle04
import Definitions.Def_ThreeSum1962Source_Bundle05
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
import Mathlib.Algebra.Group.Int.Defs
import Mathlib.Algebra.GroupWithZero.Nat
import Mathlib.Algebra.Order.BigOperators.Group.List
import Mathlib.Algebra.Order.BigOperators.Group.Multiset
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Group.Abs
import Mathlib.Algebra.Order.Group.Int
import Mathlib.Algebra.Order.Group.Nat
import Mathlib.Algebra.Order.Group.Unbundled.Abs
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Algebra.Order.Ring.Int
import Mathlib.Data.Int.Cast.Lemmas
import Mathlib.Data.Int.Notation
import Mathlib.Data.List.GetD
import Mathlib.Data.Nat.Cast.Order.Ring
import Mathlib.Data.Nat.Count
import Mathlib.Data.Nat.Log
import Mathlib.Data.Nat.Notation
import Mathlib.Data.Nat.Prime.Basic
import Mathlib.Data.Nat.Size
import Mathlib.Data.Nat.SuccPred
import Mathlib.LinearAlgebra.Matrix.Defs
import Mathlib.Logic.Function.Basic
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Tactic.GCongr
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.NormNum
import Mathlib.Tactic.Positivity
import Mathlib.Tactic.Ring
import Theorems.Thm_Light_Ends_for
import Theorems.Thm_Light_emod_meets


set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option Elab.async false
namespace Light
end Light
namespace ThreeSum.LightResidues
end ThreeSum.LightResidues

Formal statement

theorem ThreeSum.LightPrimality.meets : ∀ {lim : Light.Limits} {P : Light.Program} {d pMain pEmod : Nat}
  (hmain :
    @Eq.{1} (Option.{0} Light.Stmt)
      (@GetElem?.getElem?.{0, 0, 0} Light.Program Nat Light.Stmt
        (fun (as : List.{0} Light.Stmt) (i : Nat) => @LT.lt.{0} Nat instLTNat i (@List.length.{0} Light.Stmt as))
        (@List.instGetElem?NatLtLength.{0} Light.Stmt) P pMain)
      (@Option.some.{0} Light.Stmt (ThreeSum.LightPrimality.body pEmod)))
  (hem :
    @Eq.{1} (Option.{0} Light.Stmt)
      (@GetElem?.getElem?.{0, 0, 0} Light.Program Nat Light.Stmt
        (fun (as : List.{0} Light.Stmt) (i : Nat) => @LT.lt.{0} Nat instLTNat i (@List.length.{0} Light.Stmt as))
        (@List.instGetElem?NatLtLength.{0} Light.Stmt) P pEmod)
      (@Option.some.{0} Light.Stmt Light.emodBody))
  {μ : Nat → Int} {n free maxValue : Nat}
  (hw : @LE.le.{0} Int Int.instLEInt (@Nat.cast.{0} Int instNatCastInt lim.space) lim.word)
  (hd : @LT.lt.{0} Nat instLTNat d lim.depth)
  (hn : @LE.le.{0} Nat instLENat (@OfNat.ofNat.{0} Nat (nat_lit 2) (instOfNatNat (nat_lit 2))) n)
  (hmax : @LE.le.{0} Nat instLENat n maxValue)
  (hspace :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) free
        (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
          (Nat.log (@OfNat.ofNat.{0} Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))
            (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) maxValue
              (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))
          (@OfNat.ofNat.{0} Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))))
      lim.space)
  (hword :
    @LE.le.{0} Int Int.instLEInt
      (@Nat.cast.{0} Int instNatCastInt
        (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
          (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
            (@OfNat.ofNat.{0} Nat (nat_lit 8) (instOfNatNat (nat_lit 8))) maxValue)
          (@OfNat.ofNat.{0} Nat (nat_lit 16) (instOfNatNat (nat_lit 16)))))
      lim.word),
  Light.Meets lim P pMain d
    (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt n)
      (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt free) (@List.nil.{0} Int)))
    μ (ThreeSum.LightPrimality.time n maxValue) fun (r : Int) (μ' : Nat → Int) =>
    And (@Eq.{1} Int r (@Nat.cast.{0} Int instNatCastInt (ThreeSum.LightPrimality.answer n))) (Light.Kept μ μ' free) := by
  sorry

Source

https://github.com/c0ldzy17/three-sum-lean/blob/2b639655e07421b9ac743de9090459bab473a7c0/Submission/ThreeSum/LightPrimality.lean#L97-L156

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