Apply one zeta coordinate update to a stored list of masks

Proved

ThreeSum.LightZetaBatch.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 $[L]$ , with $|f(T)|\le U$ for $|T|\le c$ , and fix a coordinate $a\in[L]$ . A list $\mathcal M$ contains valid masks of one cardinality $k\le c$ , with repetitions permitted. Set $C=\sum_{j=0}^{\min(c,L)}\binom Lj$ . The source table, destination table, and mask list end below scratch $z$ ; the destination interval is disjoint from both source and masks. Memory stores $f$ at the source and $\mathcal M$ at the mask base.

For the specified cell and setup procedures, require $z+(L+1)^2+L+1\le S\le W$ , $(L+1)2^{L+1},2U\le W$ , and $d+L+5\le D$ . The resulting destination table represents

g(T)=\begin{cases}f(T)+f(T\setminus\{a\}),&\operatorname{mask}(T)\in\mathcal M,\ a\in T,\\f(T),&\operatorname{mask}(T)\in\mathcal M,\ a\notin T,\\\mu(\mathrm{dst}+\operatorname{rank}(T)),&\operatorname{mask}(T)\notin\mathcal M.\end{cases}

Execution takes $|\mathcal M|(T_{\rm cell}(L)+40)+10$ steps, where $T_{\rm cell}$ is the defined zeta-cell budget, and preserves memory below $z$ outside the destination.

This certifies partial batches of a compact zeta pass.

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 Definitions.Def_ThreeSum1962Source_Bundle08
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_Light_Ends_for
import Theorems.Thm_ThreeSum_LightZetaCell_meets
import Theorems.Thm_ThreeSum_SubsetAddress_layerRank_injective
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.LightZetaCell
end ThreeSum.LightZetaCell
namespace ThreeSum.MaskDecode
end ThreeSum.MaskDecode
namespace ThreeSum.SubsetAddress
end ThreeSum.SubsetAddress
namespace ThreeSumApsp
end ThreeSumApsp

Formal statement

theorem ThreeSum.LightZetaBatch.meets : ∀ {lim : Light.Limits} {P : Light.Program}
  {pMain pCell pSetup pBuild pFill pRow pLinear pPowers pRank pOffset pErase : 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.LightZetaBatch.body pCell)))
  (hcell :
    @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 pCell)
      (@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 cardinal maskBase src dst work U d : Nat} (masks : List.{0} Nat) (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)
  (hmasks :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) maskBase (@List.length.{0} Nat masks)) work)
  (hsepSrc : Light.Apart src (ThreeSum.SubsetAddress.downsetCap L cut) dst (ThreeSum.SubsetAddress.downsetCap L cut))
  (hsepMasks : Light.Apart maskBase (@List.length.{0} Nat masks) dst (ThreeSum.SubsetAddress.downsetCap L cut))
  (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))
  (hseg : Light.SegN μ maskBase masks)
  (hvalid :
    ∀ (m : Nat),
      @Membership.mem.{0, 0} Nat (List.{0} Nat) (@List.instMembership.{0} Nat) masks m →
        ThreeSum.SubsetAddress.IsMask L cardinal m)
  (hcardinal : @LE.le.{0} Nat instLENat cardinal 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 5) (instOfNatNat (nat_lit 5))))
      lim.depth),
  Light.Meets lim P pMain d
    (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt L)
      (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt cardinal)
        (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt (@List.length.{0} Nat masks))
          (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt maskBase)
            (@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.LightZetaBatch.time L (@List.length.{0} Nat masks)) fun (x : Int) (μ' : Nat → Int) =>
    And (@ThreeSum.LightZetaCell.TableAt L μ' dst cut (@ThreeSum.LightZetaBatch.values L μ dst masks a f))
      (Light.KeptBut μ μ' work dst (ThreeSum.SubsetAddress.downsetCap L cut)) := by
  sorry

Source

https://github.com/c0ldzy17/three-sum-lean/blob/2b639655e07421b9ac743de9090459bab473a7c0/Submission/ThreeSum/LightZetaBatch.lean#L41-L126

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