Constructing a two-coordinate coloring array

Proved

ThreeSum.LightColorArray.meets

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

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

Let $S,B,D$ be the machine's space, word, and call-depth limits. Choose coordinates $a,b<W$ and a map $\phi:\{0,1\}^2\to\{0,1,2\}$ . Suppose $n\le2^W$ , the destination satisfies $\mathrm{dst}+n\le\mathrm{work}$ , and

\mathrm{work}+W+1\le S\le B,\qquad 2^{W+1}\le B,\qquad d+W+2\le D.

For a program containing the specified coloring, powers-table, and mask-erasure procedures, the array constructor terminates from any initial memory $\mu$ within $n(200W+310)+17W+125$ steps. Its final memory $\nu$ satisfies

\nu(\mathrm{dst}+i)=\phi(\operatorname{bit}_a(i),\operatorname{bit}_b(i))\quad(i<n),\qquad \nu(\mathrm{work}+j)=2^j\quad(j\le W).

Every address below $\mathrm{work}$ outside the destination interval retains its initial value.

This implements a coloring parameter as a literal array while retaining the powers table used to read its two coordinates.

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 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_Light_Ends_for
import Theorems.Thm_Light_powTable_ends
import Theorems.Thm_ThreeSum_LightColorIndex_meets


set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option Elab.async false
namespace Light
end Light
namespace Light.PowTable
end Light.PowTable
namespace ThreeSum.LightPascal
end ThreeSum.LightPascal
namespace ThreeSum.LightSubsetAddress
end ThreeSum.LightSubsetAddress
namespace ThreeSum.SubsetAddress
end ThreeSum.SubsetAddress

Formal statement

theorem ThreeSum.LightColorArray.meets : ∀ {lim : Light.Limits} {P : Light.Program} {d pMain pPowers pColor pErase W n dst work : 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.LightColorArray.body pPowers pColor)))
  (hpowers :
    @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 pPowers)
      (@Option.some.{0} Light.Stmt Light.powTableBody))
  (hcolor :
    @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 pColor)
      (@Option.some.{0} Light.Stmt (ThreeSum.LightColorIndex.body pErase)))
  (herase :
    @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 pErase)
      (@Option.some.{0} Light.Stmt (ThreeSum.LightMaskErase.body pErase)))
  (μ : Nat → Int) (p : ThreeSum.Coloring.Parameter W)
  (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))
  (hdst : @LE.le.{0} Nat instLENat (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) dst n) 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)
  (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 2) (instOfNatNat (nat_lit 2))))
      lim.depth),
  Light.Meets lim P pMain d
    (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt W)
      (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt n)
        (@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 dst)
                      (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt work) (@List.nil.{0} Int)))))))))))
    μ (ThreeSum.LightColorArray.time W n) fun (x : Int) (ν : Nat → Int) =>
    And
      (∀ (i : Nat),
        @LT.lt.{0} Nat instLTNat i n →
          @Eq.{1} Int (ν (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) dst i))
            (@Nat.cast.{0} Int instNatCastInt (@ThreeSum.LightColorIndex.color W p i)))
      (And (ThreeSum.LightSubsetAddress.Powers ν work W) (Light.KeptBut μ ν work dst n)) := by
  sorry

Source

https://github.com/c0ldzy17/three-sum-lean/blob/2b639655e07421b9ac743de9090459bab473a7c0/Submission/ThreeSum/LightColorArray.lean#L37-L99

View graph