Prove2Me

Constructing a retained signed suffix table and all its box transforms

Proved
ThreeSum.LightReportingSuffixSetup.program_meets

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

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

Write S,Ω,ΔS,\Omega,\Delta for machine space, word magnitude, and depth limits. Let f,gf,g be integer functions on subsets of WW coordinates, bounded respectively by B0,D0B_0,D_0 on sets of size at most c≤Wc\le W. Their complete partial suffix-zeta tables are stored. Fix s≤Ws\le W, an interval of VV coordinates starting at oo with o+V≤Wo+V\le W, prefixes A0,B0′A_0,B_0' and cutoff b≤Vb\le V, with ∣A0∣+b,∣B0′∣+b≤c|A_0|+b,|B_0'|+b\le c. Write ι(T)=o+T\iota(T)=o+T and

F(T)=(−1)∣T∣Zsf(A0∪ι(T)) Zsg(B0′∪ι(T)).F(T)=(-1)^{|T|}Z_sf(A_0\cup\iota(T))\,Z_sg(B_0'\cup\iota(T)).

Here ZsZ_s sums over deletions in the last ss ambient coordinates.

Assume the ordered source, box, mask, and scratch regions, the stated address-space and address-word bounds, S≤ΩS\le\Omega, 2V(2sB0)(2sD0)≤Ω2^V(2^sB_0)(2^sD_0)\le\Omega, and d+W+11≤Δd+W+11\le\Delta. The routine stores every forward partial zeta transform of FF, for all V+1V+1 stages and ∣T∣≤b|T|\le b, in the box family. It retains both source families and preserves cells below the mask base outside the box family, within the specified setup time.

This prepares the signed suffix data used by sparse reporting queries.

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_Bundle03
import Definitions.Def_ThreeSum1962Source_Bundle04
import Definitions.Def_ThreeSum1962Source_Bundle10
import Definitions.Def_ThreeSum1962Source_Bundle12
import Definitions.Def_ThreeSum1962Source_Bundle13
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.Field.ZMod
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.Chebyshev
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Floor.Div
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.Order.Ring.Pow
import Mathlib.Algebra.Ring.Nat
import Mathlib.Combinatorics.Colex
import Mathlib.Data.Bool.Basic
import Mathlib.Data.Fin.Embedding
import Mathlib.Data.Fin.VecNotation
import Mathlib.Data.Finset.Card
import Mathlib.Data.Finset.Max
import Mathlib.Data.Finset.Powerset
import Mathlib.Data.Finset.Prod
import Mathlib.Data.Finset.Sigma
import Mathlib.Data.Finset.Sort
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Fintype.Fin
import Mathlib.Data.Fintype.Pi
import Mathlib.Data.Fintype.Prod
import Mathlib.Data.Int.Cast.Lemmas
import Mathlib.Data.Int.ModEq
import Mathlib.Data.Int.Notation
import Mathlib.Data.Int.SuccPred
import Mathlib.Data.List.Basic
import Mathlib.Data.List.GetD
import Mathlib.Data.List.MinMax
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.OfFn
import Mathlib.Data.List.Pi
import Mathlib.Data.List.Sublists
import Mathlib.Data.Nat.Cast.Order.Ring
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Data.Nat.Choose.Bounds
import Mathlib.Data.Nat.Choose.Central
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.Data.Nat.Count
import Mathlib.Data.Nat.Notation
import Mathlib.Data.Nat.Sqrt
import Mathlib.Data.Nat.SuccPred
import Mathlib.Data.Real.Basic
import Mathlib.Data.ZMod.Basic
import Mathlib.GroupTheory.Index
import Mathlib.LinearAlgebra.Dual.Lemmas
import Mathlib.LinearAlgebra.Matrix.Defs
import Mathlib.LinearAlgebra.Matrix.Dual
import Mathlib.LinearAlgebra.Matrix.ToLin
import Mathlib.Logic.Equiv.Fin.Basic
import Mathlib.Logic.Function.Basic
import Mathlib.NumberTheory.Bertrand
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.RingTheory.Coprime.Lemmas
import Mathlib.Tactic.Abel
import Mathlib.Tactic.ByContra
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.FinCases
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 Mathlib.Tactic.SplitIfs
import Mathlib.Tactic.Tauto
import Theorems.Thm_Light_powTable_ends
import Theorems.Thm_ThreeSum_LightReportingSuffixSetup_core_meets
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 Light.PowTable
end Light.PowTable
namespace ThreeSum.LightPascal
end ThreeSum.LightPascal
namespace ThreeSum.LightSubsetAddress
end ThreeSum.LightSubsetAddress
namespace ThreeSum.LightZetaCell
end ThreeSum.LightZetaCell
namespace ThreeSum.MaskDecode
end ThreeSum.MaskDecode
namespace ThreeSum.ReportingHostSuffix
end ThreeSum.ReportingHostSuffix
namespace ThreeSum.SubsetAddress
end ThreeSum.SubsetAddress
Formal statement
theorem ThreeSum.LightReportingSuffixSetup.program_meets : ∀ {lim : Light.Limits} {μ : Nat → Int} {W V offset sourceCut boxCut stage baseL baseR boxBase maskBase work B D d : Nat}
  (hfit : @LE.le.{0} Nat instLENat (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) offset V) W)
  (preL preR : Finset.{0} (Fin W)) (f g : Finset.{0} (Fin W) → Int)
  (hw : @LE.le.{0} Int Int.instLEInt (@Nat.cast.{0} Int instNatCastInt lim.space) lim.word)
  (hcut : @LE.le.{0} Nat instLENat sourceCut W) (hboxCut : @LE.le.{0} Nat instLENat boxCut V)
  (hstage : @LE.le.{0} Nat instLENat stage W)
  (hdataL :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) baseL
        (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
          (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) W
            (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))
          (ThreeSum.SubsetAddress.downsetCap W sourceCut)))
      boxBase)
  (hdataR :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) baseR
        (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
          (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) W
            (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))
          (ThreeSum.SubsetAddress.downsetCap W sourceCut)))
      boxBase)
  (hbox :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) boxBase
        (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
          (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) V
            (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))
          (ThreeSum.SubsetAddress.downsetCap V boxCut)))
      maskBase)
  (hmasks :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) maskBase
        (ThreeSum.SubsetAddress.downsetCap V boxCut))
      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)
          (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) work
            (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))
          (ThreeSum.LightPascal.tableSize W))
        (@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
        (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
          (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) W
            (@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) W
              (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))))
      lim.word)
  (hvalueWord :
    @LE.le.{0} Int Int.instLEInt
      (@Nat.cast.{0} Int instNatCastInt
        (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
          (@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))) V)
          (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
            (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
              (@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))) stage)
              B)
            (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
              (@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))) stage)
              D))))
      lim.word)
  (hpreL :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) (@Finset.card.{0} (Fin W) preL) boxCut)
      sourceCut)
  (hpreR :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) (@Finset.card.{0} (Fin W) preR) boxCut)
      sourceCut)
  (hf :
    ∀ (T : Finset.{0} (Fin W)),
      @LE.le.{0} Nat instLENat (@Finset.card.{0} (Fin W) T) sourceCut →
        @LE.le.{0} Int Int.instLEInt (@abs.{0} Int instLatticeInt Int.instAddGroup (f T))
          (@Nat.cast.{0} Int instNatCastInt B))
  (hg :
    ∀ (T : Finset.{0} (Fin W)),
      @LE.le.{0} Nat instLENat (@Finset.card.{0} (Fin W) T) sourceCut →
        @LE.le.{0} Int Int.instLEInt (@abs.{0} Int instLatticeInt Int.instAddGroup (g T))
          (@Nat.cast.{0} Int instNatCastInt D))
  (hstoredL : @ThreeSum.LightZetaStages.StoredThrough W μ baseL sourceCut f W)
  (hstoredR : @ThreeSum.LightZetaStages.StoredThrough W μ baseR sourceCut g W)
  (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 11) (instOfNatNat (nat_lit 11))))
      lim.depth),
  Light.Meets lim ThreeSum.LightReportingSuffixSetup.program
    (@OfNat.ofNat.{0} Nat (nat_lit 29) (instOfNatNat (nat_lit 29))) d
    (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt W)
      (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt sourceCut)
        (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt boxCut)
          (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt stage)
            (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt (@Finset.card.{0} (Fin W) preL))
              (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt (@ThreeSum.SubsetAddress.subsetMask W preL))
                (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt (@Finset.card.{0} (Fin W) preR))
                  (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt (@ThreeSum.SubsetAddress.subsetMask W preR))
                    (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt V)
                      (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt offset)
                        (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt baseL)
                          (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt baseR)
                            (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt boxBase)
                              (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt maskBase)
                                (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt work)
                                  (@List.nil.{0} Int))))))))))))))))
    μ (ThreeSum.LightReportingSuffixSetup.time W V boxCut offset) fun (x : Int) (μ' : Nat → Int) =>
    And
      (@ThreeSum.LightBoxStages.StoredThrough V μ' boxBase boxCut
        (@ThreeSum.ReportingHostSuffix.leafWeight W V offset hfit f g stage preL preR) V)
      (And (@ThreeSum.LightZetaStages.StoredThrough W μ' baseL sourceCut f W)
        (And (@ThreeSum.LightZetaStages.StoredThrough W μ' baseR sourceCut g W)
          (Light.KeptBut μ μ' maskBase boxBase
            (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
              (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) V
                (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))
              (ThreeSum.SubsetAddress.downsetCap V boxCut))))) := by
  sorry
Source
https://github.com/c0ldzy17/three-sum-lean/blob/2b639655e07421b9ac743de9090459bab473a7c0/Submission/ThreeSum/LightReportingSuffixSetup.lean#L129-L179

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