Prove2Me

Construct the three color-class lists with their exact contents

Proved
ThreeSum.LightColorClasses.Procs.meets

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

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

Let pp be a two-bit coloring parameter of width ww, and let integers xix_i (i<ni<n) satisfy n≤2wn\le2^w and ∣xi∣≤U|x_i|\le U. Memory stores xx at ss. Place the nn color cells at bb, the three class buffers at a0,a1,a2a_0,a_1,a_2, and scratch at ff, with s+n≤bs+n\le b, b+n≤a0b+n\le a_0, a0+n+1≤a1a_0+n+1\le a_1, a1+n+1≤a2a_1+n+1\le a_2, and a2+n+1≤fa_2+n+1\le f. Assume the specified procedures and f+w+1≤S≤Wf+w+1\le S\le W, 2w+1,U≤W2^{w+1},U\le W, and d+w+3≤Dd+w+3\le D.

The routine writes each index's actual color and length-prefixed class lists

L0=[xi:p(i)=0],L1=[xi:p(i)=1],L2=[−xi:p(i)=2],L_0=[x_i:p(i)=0],\quad L_1=[x_i:p(i)=1],\quad L_2=[-x_i:p(i)=2],

retaining original index order and repetitions. It takes at most 17w+625+n(200w+610)17w+625+n(200w+610) steps and preserves all memory below bb.

This produces the concrete lists whose two-sum relation represents a zero sum across the three colors.

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_Bundle04
import Definitions.Def_ThreeSum1962Source_Bundle06
import Definitions.Def_ThreeSum1962Source_Bundle07
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.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.Combinatorics.Colex
import Mathlib.Data.Fin.Embedding
import Mathlib.Data.Finset.Image
import Mathlib.Data.Finset.Powerset
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Fintype.Pi
import Mathlib.Data.Fintype.Prod
import Mathlib.Data.Int.Basic
import Mathlib.Data.Int.Cast.Lemmas
import Mathlib.Data.Int.Notation
import Mathlib.Data.Int.SuccPred
import Mathlib.Data.List.Basic
import Mathlib.Data.List.GetD
import Mathlib.Data.List.Nodup
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.Cast.Order.Ring
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Data.Nat.Choose.Bounds
import Mathlib.Data.Nat.Count
import Mathlib.Data.Nat.Notation
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.Push
import Mathlib.Tactic.Ring
import Theorems.Thm_ThreeSum_LightClassFilter_meets
import Theorems.Thm_ThreeSum_LightColorArray_meets


set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option Elab.async false
namespace Light
end Light
namespace ThreeSum.ColorClassLists
end ThreeSum.ColorClassLists
namespace ThreeSum.LightClassFilter
end ThreeSum.LightClassFilter
namespace ThreeSum.LightSubsetAddress
end ThreeSum.LightSubsetAddress
namespace ThreeSumApsp
end ThreeSumApsp
Formal statement
theorem ThreeSum.LightColorClasses.Procs.meets : ∀ {lim : Light.Limits} {P : Light.Program} {d pMain src n W colors dst0 dst1 dst2 work U : Nat} {μ : Nat → Int}
  (C : ThreeSum.LightColorClasses.Procs P pMain) (p : ThreeSum.Coloring.Parameter W) (x : Fin n → Int)
  (hw : @LE.le.{0} Int Int.instLEInt (@Nat.cast.{0} Int instNatCastInt lim.space) lim.word)
  (hn :
    @LE.le.{0} Nat instLENat n
      (@HPow.hPow.{0, 0, 0} Nat Nat Nat
        (@instHPow.{0, 0} Nat Nat (@NPow.toPow.{0} Nat (@Monoid.toNPow.{0} Nat Nat.instMonoid)))
        (@OfNat.ofNat.{0} Nat (nat_lit 2) (instOfNatNat (nat_lit 2))) W))
  (hsrc : @LE.le.{0} Nat instLENat (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) src n) colors)
  (hcolors : @LE.le.{0} Nat instLENat (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) colors n) dst0)
  (h0 :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) dst0
        (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) n
          (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))
      dst1)
  (h1 :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) dst1
        (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) n
          (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))
      dst2)
  (h2 :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) dst2
        (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) n
          (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))
      work)
  (hspace :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) work
        (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) W
          (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))
      lim.space)
  (hword :
    @LE.le.{0} Int Int.instLEInt
      (@Nat.cast.{0} Int instNatCastInt
        (@HPow.hPow.{0, 0, 0} Nat Nat Nat
          (@instHPow.{0, 0} Nat Nat (@NPow.toPow.{0} Nat (@Monoid.toNPow.{0} Nat Nat.instMonoid)))
          (@OfNat.ofNat.{0} Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))
          (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) W
            (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))))
      lim.word)
  (hU : @LE.le.{0} Int Int.instLEInt (@Nat.cast.{0} Int instNatCastInt 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 W)
        (@OfNat.ofNat.{0} Nat (nat_lit 3) (instOfNatNat (nat_lit 3))))
      lim.depth)
  (hread : Light.Seg μ src (@List.ofFn.{0} Int n x))
  (hbound :
    ∀ (i : Fin n),
      @LE.le.{0} Int Int.instLEInt (@abs.{0} Int instLatticeInt Int.instAddGroup (x i))
        (@Nat.cast.{0} Int instNatCastInt U)),
  Light.Meets lim P pMain d
    (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt src)
      (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt n)
        (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt W)
          (@List.cons.{0} Int
            (@Nat.cast.{0} Int instNatCastInt
              (@Fin.val W
                (@Prod.fst.{0, 0} (Fin W) (Fin W)
                  (@Prod.fst.{0, 0} (Prod.{0, 0} (Fin W) (Fin W))
                    (Prod.{0, 0} Bool Bool → Fin (@OfNat.ofNat.{0} Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) p))))
            (@List.cons.{0} Int
              (@Nat.cast.{0} Int instNatCastInt
                (@Fin.val W
                  (@Prod.snd.{0, 0} (Fin W) (Fin W)
                    (@Prod.fst.{0, 0} (Prod.{0, 0} (Fin W) (Fin W))
                      (Prod.{0, 0} Bool Bool → Fin (@OfNat.ofNat.{0} Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) p))))
              (@List.cons.{0} Int
                (@Nat.cast.{0} Int instNatCastInt
                  (@Fin.val (@OfNat.ofNat.{0} Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))
                    (@Prod.snd.{0, 0} (Prod.{0, 0} (Fin W) (Fin W))
                      (Prod.{0, 0} Bool Bool → Fin (@OfNat.ofNat.{0} Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) p
                      (@Prod.mk.{0, 0} Bool Bool Bool.false Bool.false))))
                (@List.cons.{0} Int
                  (@Nat.cast.{0} Int instNatCastInt
                    (@Fin.val (@OfNat.ofNat.{0} Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))
                      (@Prod.snd.{0, 0} (Prod.{0, 0} (Fin W) (Fin W))
                        (Prod.{0, 0} Bool Bool → Fin (@OfNat.ofNat.{0} Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) p
                        (@Prod.mk.{0, 0} Bool Bool Bool.false Bool.true))))
                  (@List.cons.{0} Int
                    (@Nat.cast.{0} Int instNatCastInt
                      (@Fin.val (@OfNat.ofNat.{0} Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))
                        (@Prod.snd.{0, 0} (Prod.{0, 0} (Fin W) (Fin W))
                          (Prod.{0, 0} Bool Bool → Fin (@OfNat.ofNat.{0} Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) p
                          (@Prod.mk.{0, 0} Bool Bool Bool.true Bool.false))))
                    (@List.cons.{0} Int
                      (@Nat.cast.{0} Int instNatCastInt
                        (@Fin.val (@OfNat.ofNat.{0} Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))
                          (@Prod.snd.{0, 0} (Prod.{0, 0} (Fin W) (Fin W))
                            (Prod.{0, 0} Bool Bool → Fin (@OfNat.ofNat.{0} Nat (nat_lit 3) (instOfNatNat (nat_lit 3))))
                            p (@Prod.mk.{0, 0} Bool Bool Bool.true Bool.true))))
                      (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt colors)
                        (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt dst0)
                          (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt dst1)
                            (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt dst2)
                              (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt work)
                                (@List.nil.{0} Int)))))))))))))))
    μ (ThreeSum.LightColorClasses.time W n) fun (x_1 : Int) (ν : Nat → Int) =>
    @ThreeSum.LightColorClasses.Output n W p x μ ν colors dst0 dst1 dst2 := by
  sorry
Source
https://github.com/c0ldzy17/three-sum-lean/blob/2b639655e07421b9ac743de9090459bab473a7c0/Submission/ThreeSum/LightColorClasses.lean#L55-L133

View graph