Executing one zeta-transform update at a compact subset address

Proved

ThreeSum.LightZetaCell.meets

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

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

Let $f$ be an integer function on subsets of $\{0,\ldots,L-1\}$ , with $|f(T)|\le U$ whenever $|T|\le c$ . A source table stores $f(T)$ at compact address $s+\rho(T)$ . Fix $A$ with $|A|\le c$ and a coordinate $a<L$ . Both source and destination tables of capacity $K=\sum_{i\le\min(c,L)}\binom Li$ end before scratch $w$ .

Assume the specified setup, offset, mask-erasure, and rank procedures, and

w+(L+1)^2+(L+1)\le S\le W,\quad(L+1)2^{L+1}\le W,\quad2U\le W,\quad d+L+4\le D.

Within $23L^2+516L+534$ steps, the procedure writes

\nu(t+\rho(A))=\begin{cases}f(A)+f(A\setminus\{a\}),&a\in A,\\f(A),&a\notin A.\end{cases}

Every other cell below $w$ is preserved.

This realizes a single subset-zeta addition using computed compact addresses.

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 Aesop
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.Group.Finset.Powerset
import Mathlib.Algebra.BigOperators.Group.Finset.Sigma
import Mathlib.Algebra.BigOperators.Ring.Finset
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.Combinatorics.Colex
import Mathlib.Data.Fin.Embedding
import Mathlib.Data.Finset.Card
import Mathlib.Data.Finset.Powerset
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.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.SuccPred
import Mathlib.LinearAlgebra.Matrix.Defs
import Mathlib.Logic.Function.Basic
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Tactic.ByContra
import Mathlib.Tactic.GCongr
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.NormNum
import Mathlib.Tactic.Positivity
import Mathlib.Tactic.Ring
import Mathlib.Tactic.SplitIfs
import Theorems.Thm_ThreeSum_LightMaskErase_meets
import Theorems.Thm_ThreeSum_LightSubsetAddress_rank_meets
import Theorems.Thm_ThreeSum_LightSubsetOffset_meets
import Theorems.Thm_ThreeSum_LightSubsetSetup_meets
import Theorems.Thm_ThreeSum_MaskErase_eraseMask_le
import Theorems.Thm_ThreeSum_MaskErase_eraseMask_maskOf
import Theorems.Thm_ThreeSum_SubsetAddress_layerRank_lt


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

Formal statement

theorem ThreeSum.LightZetaCell.meets : ∀ {lim : Light.Limits} {P : Light.Program} {pThis pSetup pBuild pFill pRow pLinear pPowers pRank pOffset pErase : Nat}
  (hThis :
    @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 pThis)
      (@Option.some.{0} Light.Stmt (ThreeSum.LightZetaCell.body pSetup pOffset pErase pRank)))
  (C : ThreeSum.LightSubsetSetup.Procs P pSetup pBuild pFill pRow pLinear pPowers pRank)
  (hOffset :
    @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 pOffset)
      (@Option.some.{0} Light.Stmt ThreeSum.LightSubsetOffset.body))
  (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} {L cut src dst work U d : Nat} (S : Finset.{0} (Fin L)) (a : Fin L) (f : Finset.{0} (Fin L) → Int)
  (hw : @LE.le.{0} Int Int.instLEInt (@Nat.cast.{0} Int instNatCastInt lim.space) lim.word)
  (hsrc :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) src (ThreeSum.SubsetAddress.downsetCap L cut))
      work)
  (hdst :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) dst (ThreeSum.SubsetAddress.downsetCap L cut))
      work)
  (hspace :
    @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) work (ThreeSum.LightPascal.tableSize L))
        (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) L
          (@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
        (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
          (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) L
            (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))
          (@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) L
              (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))))
      lim.word)
  (hvalueWord :
    @LE.le.{0} Int Int.instLEInt
      (@HMul.hMul.{0, 0, 0} Int Int Int (@instHMul.{0} Int Int.instMul)
        (@OfNat.ofNat.{0} Int (nat_lit 2) (@instOfNat (nat_lit 2))) (@Nat.cast.{0} Int instNatCastInt U))
      lim.word)
  (hread : @ThreeSum.LightZetaCell.TableAt L μ src cut f)
  (hvalues :
    ∀ (T : Finset.{0} (Fin L)),
      @LE.le.{0} Nat instLENat (@Finset.card.{0} (Fin L) T) cut →
        @LE.le.{0} Int Int.instLEInt (@abs.{0} Int instLatticeInt Int.instAddGroup (f T))
          (@Nat.cast.{0} Int instNatCastInt U))
  (hS : @LE.le.{0} Nat instLENat (@Finset.card.{0} (Fin L) S) cut)
  (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 L)
        (@OfNat.ofNat.{0} Nat (nat_lit 4) (instOfNatNat (nat_lit 4))))
      lim.depth),
  Light.Meets lim P pThis d
    (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt L)
      (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt (@Finset.card.{0} (Fin L) S))
        (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt (@ThreeSum.SubsetAddress.subsetMask L S))
          (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt (@Fin.val L a))
            (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt src)
              (@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.LightZetaCell.time L) fun (x : Int) (μ' : Nat → Int) =>
    And
      (@Eq.{1} Int
        (μ'
          (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) dst
            (@ThreeSum.SubsetAddress.subsetRank L S)))
        (@ThreeSum.Subset.zetaStep.{0} (Fin L) (instDecidableEqFin L) a f S))
      (ThreeSum.LightZetaCell.KeptExcept μ μ' work
        (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) dst (@ThreeSum.SubsetAddress.subsetRank L S))) := by
  sorry

Source

https://github.com/c0ldzy17/three-sum-lean/blob/2b639655e07421b9ac743de9090459bab473a7c0/Submission/ThreeSum/LightZetaCell.lean#L52-L162

View graph