Select a least-correlated residue offset from original arrays

Proved

ThreeSum.LightOriginalOffset.Procs.meets

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

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

Let two integer arrays $A,B$ of lengths $n_A,n_B\le N$ lie below scratch base $f$ , with entry magnitudes at most $U$ . Fix $p>0$ , put $q=2^{\lceil\log_2p\rceil}$ , and assume the specified capacity, signed-histogram, and balanced-offset procedures. Require $f+13q+N+\lfloor\log_2(U+1)\rfloor+2\le S\le W$ , $128q^3(N+1)^2+4U+4p+16\le W$ , and $d+\lceil\log_2p\rceil+4\le D$ .

For residues $z$ , let $H_A(z),H_B(z)$ count the original entries modulo $p$ . The routine returns

r=\min\operatorname*{arg\,min}_{0\le t<p}\sum_z H_B(z)H_A(z-t).

It preserves all memory below $f$ and meets the capacity, two histogram, and balanced-offset time budgets plus $300$ .

This constructs the deterministic balancing offset directly from signed input values, with multiplicities retained.

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_Bundle02
import Definitions.Def_ThreeSum1962Source_Bundle03
import Definitions.Def_ThreeSum1962Source_Bundle05
import Definitions.Def_ThreeSum1962Source_Bundle07
import Definitions.Def_ThreeSum1962Source_Bundle09
import Lean.Elab.Tactic.Omega
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Algebra.Group.Equiv.Basic
import Mathlib.Algebra.Group.Int.Defs
import Mathlib.Algebra.GroupWithZero.Nat
import Mathlib.Algebra.Order.BigOperators.Group.Finset
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.Algebra.Polynomial.Basic
import Mathlib.Algebra.Polynomial.Coeff
import Mathlib.Data.Finset.Card
import Mathlib.Data.Finset.Prod
import Mathlib.Data.Finset.Range
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Fintype.EquivFin
import Mathlib.Data.Fintype.Prod
import Mathlib.Data.Int.Cast.Lemmas
import Mathlib.Data.Int.ModEq
import Mathlib.Data.Int.Notation
import Mathlib.Data.List.GetD
import Mathlib.Data.List.MinMax
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.Size
import Mathlib.Data.Nat.SuccPred
import Mathlib.Data.ZMod.Basic
import Mathlib.LinearAlgebra.Matrix.Defs
import Mathlib.Logic.Function.Basic
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.RingTheory.Coprime.Lemmas
import Mathlib.Tactic.Abel
import Mathlib.Tactic.ByContra
import Mathlib.Tactic.GCongr
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.NormNum
import Mathlib.Tactic.Positivity
import Mathlib.Tactic.Push
import Mathlib.Tactic.Ring
import Theorems.Thm_ThreeSum_LightBalancedOffset_meets
import Theorems.Thm_ThreeSum_LightCapacity_meets
import Theorems.Thm_ThreeSum_LightResidueHistogram_meets


set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option Elab.async false
namespace Light
end Light
namespace ThreeSum.CyclicFold
end ThreeSum.CyclicFold
namespace ThreeSum.Karatsuba
end ThreeSum.Karatsuba
namespace ThreeSum.KaratsubaInterface
end ThreeSum.KaratsubaInterface
namespace ThreeSum.LightHistogram
end ThreeSum.LightHistogram
namespace ThreeSum.LightKaratsuba
end ThreeSum.LightKaratsuba
namespace ThreeSum.LightResidueHistogram
end ThreeSum.LightResidueHistogram
namespace ThreeSum.LightResidues
end ThreeSum.LightResidues

Formal statement

theorem ThreeSum.LightOriginalOffset.Procs.meets : ∀ {lim : Light.Limits} {P : Light.Program} {d pMain a b nLeft nRight p N free U : Nat}
  [inst : @NeZero.{0} Nat (@MulZeroClass.toZero.{0} Nat Nat.instMulZeroClass) p] {μ : Nat → Int}
  (C : ThreeSum.LightOriginalOffset.Procs P pMain)
  (hw : @LE.le.{0} Int Int.instLEInt (@Nat.cast.{0} Int instNatCastInt lim.space) lim.word)
  (ha : @LE.le.{0} Nat instLENat (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) a nLeft) free)
  (hb : @LE.le.{0} Nat instLENat (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) b nRight) free)
  (hnLeft : @LE.le.{0} Nat instLENat nLeft N) (hnRight : @LE.le.{0} Nat instLENat nRight N)
  (hspace : @LE.le.{0} Nat instLENat (ThreeSum.LightOriginalOffset.space free p N U) lim.space)
  (hword :
    @LE.le.{0} Int Int.instLEInt (@Nat.cast.{0} Int instNatCastInt (ThreeSum.LightOriginalOffset.word p N U)) lim.word)
  (hd :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
        (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) d
          (Nat.clog (@OfNat.ofNat.{0} Nat (nat_lit 2) (instOfNatNat (nat_lit 2))) p))
        (@OfNat.ofNat.{0} Nat (nat_lit 4) (instOfNatNat (nat_lit 4))))
      lim.depth)
  (hLeft :
    ∀ (i : Nat),
      @LT.lt.{0} Nat instLTNat i nLeft →
        @LE.le.{0} Int Int.instLEInt
          (@abs.{0} Int instLatticeInt Int.instAddGroup
            (μ (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) a i)))
          (@Nat.cast.{0} Int instNatCastInt U))
  (hRight :
    ∀ (i : Nat),
      @LT.lt.{0} Nat instLTNat i nRight →
        @LE.le.{0} Int Int.instLEInt
          (@abs.{0} Int instLatticeInt Int.instAddGroup
            (μ (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) b i)))
          (@Nat.cast.{0} Int instNatCastInt U)),
  Light.Meets lim P pMain d
    (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt a)
      (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt b)
        (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt nLeft)
          (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt nRight)
            (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt p)
              (@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.LightOriginalOffset.time p nLeft nRight U) fun (r : Int) (ν : Nat → Int) =>
    And
      (@Eq.{1} Int r
        (@Nat.cast.{0} Int instNatCastInt
          (@ThreeSum.LightOriginalOffset.selected p inst
            (fun (i : Nat) => μ (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) a i))
            (fun (i : Nat) => μ (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) b i)) nLeft nRight)))
      (Light.Kept μ ν free) := by
  sorry

Source

https://github.com/c0ldzy17/three-sum-lean/blob/2b639655e07421b9ac743de9090459bab473a7c0/Submission/ThreeSum/LightOriginalOffset.lean#L91-L182

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