Construct the local covering matrix and gather its greedy schedule

Proved

ThreeSum.LightLocalCoverHost.Procs.meets

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

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

Fix $h\in\mathbb N$ , width $L=512h$ , local prime $p$ , $V=\binom Lh$ vertices, $R=p^{2L}$ candidates, and schedule length $T$ . Memory initially stores the exact $h$ -subset masks and their $V\times L$ Boolean incidence table. These inputs end before the coefficient area. Reserve, in order, $2LR$ coefficient cells, two reusable raw areas of $2V$ cells each, two $Rp$ histogram areas, and an $Rp^2$ covering matrix, followed by scratch $z$ . The $2LT$ gathered output cells begin at $a\ge z+p^2+R+T$ .

Assume the specified matrix, greedy, and gather procedures; their exact host word budget fits $W$ , both the candidate workspace and output fit $S\le W$ , and $d+L+6\le D$ . The routine returns $a$ and establishes

\text{all candidate tables and cover entries are correct},\qquad\nu[a,a+2LT)=\text{the actual scheduled weight rows}.

The greedy output indices equal the actual candidate choices, and memory below the coefficient area is preserved. Time is the sum of matrix, greedy, and gather budgets plus $200$ .

This connects the concrete covering computation to the canonical greedy schedule.

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_Bundle05
import Definitions.Def_ThreeSum1962Source_Bundle06
import Definitions.Def_ThreeSum1962Source_Bundle07
import Definitions.Def_ThreeSum1962Source_Bundle09
import Definitions.Def_ThreeSum1962Source_Bundle10
import Definitions.Def_ThreeSum1962Source_Bundle11
import Definitions.Def_ThreeSum1962Source_Bundle12
import Lean.Elab.Tactic.Omega
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
import Mathlib.Algebra.BigOperators.Group.Finset.Piecewise
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.Polynomial.Basic
import Mathlib.Algebra.Polynomial.Coeff
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.Powerset
import Mathlib.Data.Finset.Prod
import Mathlib.Data.Finset.Range
import Mathlib.Data.Finset.Sigma
import Mathlib.Data.Finset.Sort
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Fintype.EquivFin
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.Range
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.Log
import Mathlib.Data.Nat.Notation
import Mathlib.Data.Nat.Size
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_ThreeSum_LightCoverGreedy_Procs_meets
import Theorems.Thm_ThreeSum_LightLocalCoverMatrix_Procs_meets
import Theorems.Thm_ThreeSum_LightLocalCoverSchedule_indices_lt
import Theorems.Thm_ThreeSum_LightLocalCoverTables_Data_gathered_schedule
import Theorems.Thm_ThreeSum_LightLocalCoverTables_Data_of_kept
import Theorems.Thm_ThreeSum_LightWeightGather_Procs_meets


set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option Elab.async false
namespace Light
end Light
namespace ThreeSum.GlobalEncoding
end ThreeSum.GlobalEncoding
namespace ThreeSum.LightLocalCoverSchedule
end ThreeSum.LightLocalCoverSchedule
namespace ThreeSum.LightLocalCoverTables
end ThreeSum.LightLocalCoverTables
namespace ThreeSum.LightSubsetAddress
end ThreeSum.LightSubsetAddress
namespace ThreeSum.LocalCoverBounds
end ThreeSum.LocalCoverBounds
namespace ThreeSum.LocalHashMoments
end ThreeSum.LocalHashMoments
namespace ThreeSum.LocalMaskGraph
end ThreeSum.LocalMaskGraph
namespace ThreeSum.LocalMaskHashes
end ThreeSum.LocalMaskHashes
namespace ThreeSum.MaskDecode
end ThreeSum.MaskDecode
namespace ThreeSum.MaskErase
end ThreeSum.MaskErase
namespace ThreeSum.SubsetAddress
end ThreeSum.SubsetAddress

Formal statement

theorem ThreeSum.LightLocalCoverHost.Procs.meets : ∀ {lim : Light.Limits} {P : Light.Program} {d pMain : Nat} (S : ThreeSum.LightLocalCoverHost.Procs P pMain)
  {μ : Nat → Int} {h masks bitPtr maps raw0 raw1 hist0 hist1 matrixPtr work scheduleMaps : Nat}
  (hw : @LE.le.{0} Int Int.instLEInt (@Nat.cast.{0} Int instNatCastInt lim.space) 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
          (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
            (@OfNat.ofNat.{0} Nat (nat_lit 512) (instOfNatNat (nat_lit 512))) h))
        (@OfNat.ofNat.{0} Nat (nat_lit 6) (instOfNatNat (nat_lit 6))))
      lim.depth)
  (hspace :
    @LE.le.{0} Nat instLENat
      (ThreeSum.LightLocalCoverCandidate.space work
        (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
          (@OfNat.ofNat.{0} Nat (nat_lit 512) (instOfNatNat (nat_lit 512))) h)
        (ThreeSum.LocalMaskGraph.vertexCount
          (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
            (@OfNat.ofNat.{0} Nat (nat_lit 512) (instOfNatNat (nat_lit 512))) h)
          h)
        (ThreeSum.LocalCoverBounds.localPrime h))
      lim.space)
  (hmasks :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) masks
        (ThreeSum.LocalMaskGraph.vertexCount
          (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
            (@OfNat.ofNat.{0} Nat (nat_lit 512) (instOfNatNat (nat_lit 512))) h)
          h))
      maps)
  (hbits :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) bitPtr
        (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
          (ThreeSum.LocalMaskGraph.vertexCount
            (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
              (@OfNat.ofNat.{0} Nat (nat_lit 512) (instOfNatNat (nat_lit 512))) h)
            h)
          (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
            (@OfNat.ofNat.{0} Nat (nat_lit 512) (instOfNatNat (nat_lit 512))) h)))
      maps)
  (hmaps :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) maps
        (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat) (ThreeSum.LightLocalCoverSchedule.rows h)
          (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
            (@OfNat.ofNat.{0} Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))
            (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
              (@OfNat.ofNat.{0} Nat (nat_lit 512) (instOfNatNat (nat_lit 512))) h))))
      raw0)
  (hraw0 :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) raw0
        (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
          (@OfNat.ofNat.{0} Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))
          (ThreeSum.LocalMaskGraph.vertexCount
            (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
              (@OfNat.ofNat.{0} Nat (nat_lit 512) (instOfNatNat (nat_lit 512))) h)
            h)))
      raw1)
  (hraw1 :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) raw1
        (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
          (@OfNat.ofNat.{0} Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))
          (ThreeSum.LocalMaskGraph.vertexCount
            (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
              (@OfNat.ofNat.{0} Nat (nat_lit 512) (instOfNatNat (nat_lit 512))) h)
            h)))
      hist0)
  (hhist0 :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) hist0
        (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat) (ThreeSum.LightLocalCoverSchedule.rows h)
          (ThreeSum.LocalCoverBounds.localPrime h)))
      hist1)
  (hhist1 :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) hist1
        (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat) (ThreeSum.LightLocalCoverSchedule.rows h)
          (ThreeSum.LocalCoverBounds.localPrime h)))
      matrixPtr)
  (hmatrix :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) matrixPtr
        (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat) (ThreeSum.LightLocalCoverSchedule.rows h)
          (ThreeSum.LightLocalCoverSchedule.cols h)))
      work)
  (hschedule :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) work
        (ThreeSum.LightCoverGreedy.total (ThreeSum.LightLocalCoverSchedule.rows h)
          (ThreeSum.LightLocalCoverSchedule.cols h) (ThreeSum.GlobalEncoding.scheduleSize h)))
      scheduleMaps)
  (houtSpace :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) scheduleMaps
        (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat) (ThreeSum.GlobalEncoding.scheduleSize h)
          (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
            (@OfNat.ofNat.{0} Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))
            (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
              (@OfNat.ofNat.{0} Nat (nat_lit 512) (instOfNatNat (nat_lit 512))) h))))
      lim.space)
  (hword :
    @LE.le.{0} Int Int.instLEInt (@Nat.cast.{0} Int instNatCastInt (ThreeSum.LightLocalCoverHost.word h)) lim.word)
  (hmaskRead :
    ∀ (i : Nat),
      @LT.lt.{0} Nat instLTNat i
          (ThreeSum.LocalMaskGraph.vertexCount
            (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
              (@OfNat.ofNat.{0} Nat (nat_lit 512) (instOfNatNat (nat_lit 512))) h)
            h) →
        @Eq.{1} Int (μ (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) masks i))
          (@Nat.cast.{0} Int instNatCastInt
            (ThreeSum.LocalMaskGraph.maskAt
              (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
                (@OfNat.ofNat.{0} Nat (nat_lit 512) (instOfNatNat (nat_lit 512))) h)
              h i)))
  (hbitsRead :
    ∀ (i : Nat),
      @LT.lt.{0} Nat instLTNat i
          (ThreeSum.LocalMaskGraph.vertexCount
            (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
              (@OfNat.ofNat.{0} Nat (nat_lit 512) (instOfNatNat (nat_lit 512))) h)
            h) →
        ∀ (j : Nat),
          @LT.lt.{0} Nat instLTNat j
              (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
                (@OfNat.ofNat.{0} Nat (nat_lit 512) (instOfNatNat (nat_lit 512))) h) →
            @Eq.{1} Int
              (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) bitPtr
                    (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat) i
                      (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
                        (@OfNat.ofNat.{0} Nat (nat_lit 512) (instOfNatNat (nat_lit 512))) h)))
                  j))
              (@Nat.cast.{0} Int instNatCastInt
                (ThreeSum.LocalCoverMatrix.bits
                  (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
                    (@OfNat.ofNat.{0} Nat (nat_lit 512) (instOfNatNat (nat_lit 512))) h)
                  h i j))),
  Light.Meets lim P pMain d
    (@List.cons.{0} Int
      (@HMul.hMul.{0, 0, 0} Int Int Int (@instHMul.{0} Int Int.instMul)
        (@OfNat.ofNat.{0} Int (nat_lit 512) (@instOfNat (nat_lit 512))) (@Nat.cast.{0} Int instNatCastInt h))
      (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt (ThreeSum.LocalCoverBounds.localPrime h))
        (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt (ThreeSum.LightLocalCoverSchedule.rows h))
          (@List.cons.{0} Int
            (@Nat.cast.{0} Int instNatCastInt
              (ThreeSum.LocalMaskGraph.vertexCount
                (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
                  (@OfNat.ofNat.{0} Nat (nat_lit 512) (instOfNatNat (nat_lit 512))) h)
                h))
            (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt (ThreeSum.LocalCoverBounds.localBucketCap h))
              (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt masks)
                (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt bitPtr)
                  (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt maps)
                    (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt raw0)
                      (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt raw1)
                        (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt hist0)
                          (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt hist1)
                            (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt matrixPtr)
                              (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt work)
                                (@List.cons.{0} Int
                                  (@Nat.cast.{0} Int instNatCastInt (ThreeSum.GlobalEncoding.scheduleSize h))
                                  (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt scheduleMaps)
                                    (@List.nil.{0} Int)))))))))))))))))
    μ (ThreeSum.LightLocalCoverHost.time h) fun (r : Int) (ν : Nat → Int) =>
    And (@Eq.{1} Int r (@Nat.cast.{0} Int instNatCastInt scheduleMaps))
      (And (ThreeSum.LightLocalCoverTables.Data ν h maps hist0 hist1 matrixPtr)
        (And (ThreeSum.LightLocalCoverTables.ScheduleRows ν h scheduleMaps)
          (And
            (∀ (s : ThreeSum.GlobalEncoding.Occurrence h),
              @Eq.{1} Int
                (ν
                  (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
                    (ThreeSum.LightCoverGreedy.out work (ThreeSum.LightLocalCoverSchedule.rows h)
                      (ThreeSum.LightLocalCoverSchedule.cols h))
                    (@Fin.val (ThreeSum.GlobalEncoding.scheduleSize h) s)))
                (@Nat.cast.{0} Int instNatCastInt (ThreeSum.LightLocalCoverSchedule.candidateAt h s)))
            (Light.Kept μ ν maps)))) := by
  sorry

Source

https://github.com/c0ldzy17/three-sum-lean/blob/2b639655e07421b9ac743de9090459bab473a7c0/Submission/ThreeSum/LightLocalCoverHost.lean#L49-L169

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