Prove2Me

Constructing paired prefix-mask product arrays in matching order

Proved
ThreeSum.LightReportingPrefixArrays.program_meets

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

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

Let 2h≤L2h\le L and k∈Nk\in\mathbb N. Enumerate all ordered disjoint pairs (A,B)(A,B) of hh-subsets of an LL-element set, first by AA and then by BB. Their number is m=(L2h)(2hh)m=\binom L{2h}\binom{2h}h. Let X,YX,Y be the resulting left and right mask lists, and define V0(Z)=[0]V_0(Z)=[0], Vj+1(Z)=[z+2Ly:z∈Z, y∈Vj(Z)]V_{j+1}(Z)=[z+2^Ly:z\in Z,\ y\in V_j(Z)] in nested list order.

The fixed program uses nonoverlapping ordered regions for the initial hh-subset list, both mm-cell pair arrays, both complete radix-stage constructions, and scratch. Assume their stated endpoints fit SS, S≤WS\le W, 2L+1≤W2^{L+1}\le W, the radix word budget for (2L,2L,k)(2^L,2^L,k) fits WW, and d+L+k+4≤Dd+L+k+4\le D. It writes headers mkm^k followed by

Vk(X)andVk(Y),V_k(X)\quad\text{and}\quad V_k(Y),

returns the left header address, preserves cells below the initial subset-list base, and meets the defined prefix-array budget.

Both outputs retain the same ordered pair-occurrence indexing.

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_Bundle05
import Definitions.Def_ThreeSum1962Source_Bundle07
import Definitions.Def_ThreeSum1962Source_Bundle08
import Definitions.Def_ThreeSum1962Source_Bundle09
import Definitions.Def_ThreeSum1962Source_Bundle10
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.List.TakeDrop
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_Light_seg_append
import Theorems.Thm_ThreeSum_LightRadixKeys_meets
import Theorems.Thm_ThreeSum_LightReportingPairRows_program_meets
import Theorems.Thm_ThreeSum_ReportingHostPairEnumeration_leftAt_card
import Theorems.Thm_ThreeSum_ReportingHostPairEnumeration_lefts_length
import Theorems.Thm_ThreeSum_ReportingHostPairEnumeration_pairs_length
import Theorems.Thm_ThreeSum_ReportingHostPairEnumeration_rights_length


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.LayerCounts
end ThreeSum.LayerCounts
namespace ThreeSum.LightPascal
end ThreeSum.LightPascal
namespace ThreeSum.LightSubsetAddress
end ThreeSum.LightSubsetAddress
namespace ThreeSum.LocalGraph
end ThreeSum.LocalGraph
namespace ThreeSum.ReportingHostPairEnumeration
end ThreeSum.ReportingHostPairEnumeration
namespace ThreeSum.ReportingHostPairSegments
end ThreeSum.ReportingHostPairSegments
namespace ThreeSum.SubsetAddress
end ThreeSum.SubsetAddress
Formal statement
theorem ThreeSum.LightReportingPrefixArrays.program_meets : ∀ {lim : Light.Limits} {μ : Nat → Int} {L h k srcS pairL pairR stageL stageR work d : Nat}
  (hw : @LE.le.{0} Int Int.instLEInt (@Nat.cast.{0} Int instNatCastInt lim.space) lim.word)
  (hh :
    @LE.le.{0} Nat instLENat
      (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
        (@OfNat.ofNat.{0} Nat (nat_lit 2) (instOfNatNat (nat_lit 2))) h)
      L)
  (hS :
    @LE.le.{0} Nat instLENat (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) srcS (L.choose h)) pairL)
  (hL :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) pairL
        (ThreeSum.LightReportingPrefixArrays.localCount L h))
      pairR)
  (hR :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) pairR
        (ThreeSum.LightReportingPrefixArrays.localCount L h))
      stageL)
  (hstageL :
    @LE.le.{0} Nat instLENat
      (ThreeSum.LightRadixKeys.space stageL (ThreeSum.LightReportingPrefixArrays.localCount L h) k) stageR)
  (hstageR :
    @LE.le.{0} Nat instLENat
      (ThreeSum.LightRadixKeys.space stageR (ThreeSum.LightReportingPrefixArrays.localCount L h) k) 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
          (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
            (@OfNat.ofNat.{0} Nat (nat_lit 2) (instOfNatNat (nat_lit 2))) L))
        (@OfNat.ofNat.{0} Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))
      lim.space)
  (hpowWord :
    @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) L
            (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))))
      lim.word)
  (hkeysWord :
    @LE.le.{0} Int Int.instLEInt
      (@Nat.cast.{0} Int instNatCastInt
        (ThreeSum.LightRadixKeys.word
          (@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))) L)
          (@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))) L)
          k))
      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)
          (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) d L) k)
        (@OfNat.ofNat.{0} Nat (nat_lit 4) (instOfNatNat (nat_lit 4))))
      lim.depth),
  Light.Meets lim ThreeSum.LightReportingPrefixArrays.program
    (@OfNat.ofNat.{0} Nat (nat_lit 10) (instOfNatNat (nat_lit 10))) d
    (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt L)
      (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt h)
        (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt k)
          (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt srcS)
            (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt pairL)
              (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt pairR)
                (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt stageL)
                  (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt stageR)
                    (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt work) (@List.nil.{0} Int))))))))))
    μ (ThreeSum.LightReportingPrefixArrays.time L h k) fun (r : Int) (μ' : Nat → Int) =>
    And
      (@Eq.{1} Int r
        (@Nat.cast.{0} Int instNatCastInt
          (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) stageL
            (ThreeSum.LightRadixKeys.offset (ThreeSum.LightReportingPrefixArrays.localCount L h) k))))
      (And
        (@Eq.{1} Int
          (μ'
            (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) stageL
              (ThreeSum.LightRadixKeys.offset (ThreeSum.LightReportingPrefixArrays.localCount L h) k)))
          (@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)))
              (ThreeSum.LightReportingPrefixArrays.localCount L h) k)))
        (And
          (@Eq.{1} Int
            (μ'
              (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) stageR
                (ThreeSum.LightRadixKeys.offset (ThreeSum.LightReportingPrefixArrays.localCount L h) k)))
            (@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)))
                (ThreeSum.LightReportingPrefixArrays.localCount L h) k)))
          (And
            (Light.Seg μ'
              (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
                (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) stageL
                  (ThreeSum.LightRadixKeys.offset (ThreeSum.LightReportingPrefixArrays.localCount L h) k))
                (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))
              (ThreeSum.RadixKeyLists.values
                (@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))) L))
                (ThreeSum.ReportingHostPairSegments.leftMasks L h) k))
            (And
              (Light.Seg μ'
                (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
                  (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) stageR
                    (ThreeSum.LightRadixKeys.offset (ThreeSum.LightReportingPrefixArrays.localCount L h) k))
                  (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))
                (ThreeSum.RadixKeyLists.values
                  (@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))) L))
                  (ThreeSum.ReportingHostPairSegments.rightMasks L h) k))
              (Light.Kept μ μ' srcS))))) := by
  sorry
Source
https://github.com/c0ldzy17/three-sum-lean/blob/2b639655e07421b9ac743de9090459bab473a7c0/Submission/ThreeSum/LightReportingPrefixArrays.lean#L38-L125

View graph