Initialize the dense-stop counter and exact weighted threshold

Proved

ThreeSum.LightDenseSetup.program_meets

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

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

Let bounded stored arrays $A,B,C$ , valid flagged band records, and distinct auxiliary primes $p_j$ be given, with $g,q>0$ , $q\le M$ , and $qp_j\le M$ . Suppose the complete cover-key data for $h,J$ is retained before free address $f$ . Let $H$ be the aggregate weighted false count over physical owner products at modulus $q$ , and let

C=(\operatorname{scheduleSize}(h)\operatorname{localBucketCap}(h)^2)^J.

Assume all input regions end by $f$ , the weighted-score workspace beginning at $f+4$ and dense-setup word budget fit, space fits in a word, and depth $d+\operatorname{clog}_2M+7$ is available. The program returns $f$ and stores

\mu'[f:f+4]=[0,CH,C,H].

It preserves the cover-key predicate and all memory below $f$ . Its budget is the representation-cap cost plus the aggregate score cost and $400$ steps.

These four cells initialize the dense-stop counter and threshold from actual source records.

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_Bundle05
import Definitions.Def_ThreeSum1962Source_Bundle07
import Definitions.Def_ThreeSum1962Source_Bundle08
import Definitions.Def_ThreeSum1962Source_Bundle09
import Definitions.Def_ThreeSum1962Source_Bundle10
import Definitions.Def_ThreeSum1962Source_Bundle11
import Definitions.Def_ThreeSum1962Source_Bundle12
import Definitions.Def_ThreeSum1962Source_Bundle14
import Definitions.Def_TrulySubcubicAPSP_SourceSpecification
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.GCDMonoid.Nat
import Mathlib.Algebra.Group.Equiv.Basic
import Mathlib.Algebra.Group.Int.Defs
import Mathlib.Algebra.GroupWithZero.Nat
import Mathlib.Algebra.MvPolynomial.Basic
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.BigOperators.GroupWithZero.List
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.Analysis.Complex.ExponentialBounds
import Mathlib.Analysis.Convex.Deriv
import Mathlib.Analysis.SpecialFunctions.BinaryEntropy
import Mathlib.Analysis.SpecialFunctions.Log.Base
import Mathlib.Analysis.SpecialFunctions.Log.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Combinatorics.Colex
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Data.Bool.Basic
import Mathlib.Data.Fin.Embedding
import Mathlib.Data.Fin.VecNotation
import Mathlib.Data.Finset.Basic
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.Fintype.Sets
import Mathlib.Data.Fintype.Sum
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.FinRange
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.ProdSigma
import Mathlib.Data.List.Range
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.GCD.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Nat.Notation
import Mathlib.Data.Nat.Prime.Basic
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.Notation
import Mathlib.LinearAlgebra.Matrix.ToLin
import Mathlib.Logic.Equiv.Fin.Basic
import Mathlib.Logic.Function.Basic
import Mathlib.MeasureTheory.Integral.Bochner.Basic
import Mathlib.NumberTheory.Bertrand
import Mathlib.NumberTheory.Chebyshev
import Mathlib.NumberTheory.PrimeCounting
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.Monotone.Basic
import Mathlib.Probability.Independence.Basic
import Mathlib.RingTheory.Coprime.Lemmas
import Mathlib.Tactic.Abel
import Mathlib.Tactic.ByContra
import Mathlib.Tactic.DeriveFintype
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_BandRawA_mem_aIndices_iff_owner
import Theorems.Thm_ThreeSum_BandRawA_mem_allProducts
import Theorems.Thm_ThreeSum_BandRawA_owner_weightedFalseCount
import Theorems.Thm_ThreeSum_BandWeightedScore_block_bound
import Theorems.Thm_ThreeSum_LightBandWeightedScore_program_meets
import Theorems.Thm_ThreeSum_LightCoverKeyData_Stored_of_kept
import Theorems.Thm_ThreeSum_LightNatPower_meets


set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option Elab.async false
namespace Light
end Light
namespace ThreeSum.BandCandidateRows
end ThreeSum.BandCandidateRows
namespace ThreeSum.GlobalEncoding
end ThreeSum.GlobalEncoding
namespace ThreeSum.GlobalKeyEnumeration
end ThreeSum.GlobalKeyEnumeration
namespace ThreeSum.LightLocalCoverLayout
end ThreeSum.LightLocalCoverLayout
namespace ThreeSum.LightLocalCoverMetadata
end ThreeSum.LightLocalCoverMetadata
namespace ThreeSum.LightLocalCoverSchedule
end ThreeSum.LightLocalCoverSchedule
namespace ThreeSum.LightLocalCoverTables
end ThreeSum.LightLocalCoverTables
namespace ThreeSum.LocalCoverBounds
end ThreeSum.LocalCoverBounds

Formal statement

theorem ThreeSum.LightDenseSetup.program_meets : ∀ {lim : Light.Limits} {d : Nat} {μ : Nat → Int} {a b c nA nB nC g recs q aux s cover free maxMod U h J : Nat}
  (primes : Nat → Nat) (rs : List.{0} ThreeSum.BandCandidateRows.Record)
  (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
          (Nat.clog (@OfNat.ofNat.{0} Nat (nat_lit 2) (instOfNatNat (nat_lit 2))) maxMod))
        (@OfNat.ofNat.{0} Nat (nat_lit 7) (instOfNatNat (nat_lit 7))))
      lim.depth)
  (hg : @LT.lt.{0} Nat instLTNat (@OfNat.ofNat.{0} Nat (nat_lit 0) (instOfNatNat (nat_lit 0))) g)
  (hq : @LT.lt.{0} Nat instLTNat (@OfNat.ofNat.{0} Nat (nat_lit 0) (instOfNatNat (nat_lit 0))) q)
  (hqmax : @LE.le.{0} Nat instLENat q maxMod)
  (ha : @LE.le.{0} Nat instLENat (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) a nA) free)
  (hb : @LE.le.{0} Nat instLENat (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) b nB) free)
  (hc : @LE.le.{0} Nat instLENat (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) c nC) free)
  (hrecs :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) recs
        (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
          (@OfNat.ofNat.{0} Nat (nat_lit 4) (instOfNatNat (nat_lit 4)))
          (@List.length.{0} ThreeSum.BandCandidateRows.Record rs)))
      free)
  (haux : @LE.le.{0} Nat instLENat (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) aux s) free)
  (hcover : @LE.le.{0} Nat instLENat (ThreeSum.LightCoverKeyData.keyEnd cover h J) free)
  (hspace :
    @LE.le.{0} Nat instLENat
      (ThreeSum.LightWeightedScore.space
        (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) free
          (@OfNat.ofNat.{0} Nat (nat_lit 4) (instOfNatNat (nat_lit 4))))
        maxMod g U)
      lim.space)
  (hword :
    @LE.le.{0} Int Int.instLEInt
      (@Nat.cast.{0} Int instNatCastInt
        (ThreeSum.LightDenseSetup.word maxMod nA nB nC g (@List.length.{0} ThreeSum.BandCandidateRows.Record rs) s U h
          J))
      lim.word)
  (H : ThreeSum.LightCoverKeyData.Stored μ cover h J)
  (hrecords : Light.Seg μ recs (ThreeSum.BandCandidateRows.cells rs))
  (hflag :
    ∀ (r : ThreeSum.BandCandidateRows.Record),
      @Membership.mem.{0, 0} ThreeSum.BandCandidateRows.Record (List.{0} ThreeSum.BandCandidateRows.Record)
          (@List.instMembership.{0} ThreeSum.BandCandidateRows.Record) rs r →
        Or (@Eq.{1} Nat r.flag (@OfNat.ofNat.{0} Nat (nat_lit 0) (instOfNatNat (nat_lit 0))))
          (@Eq.{1} Nat r.flag (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))
  (hvalid :
    ∀ (r : ThreeSum.BandCandidateRows.Record),
      @Membership.mem.{0, 0} ThreeSum.BandCandidateRows.Record (List.{0} ThreeSum.BandCandidateRows.Record)
          (@List.instMembership.{0} ThreeSum.BandCandidateRows.Record) rs r →
        @Eq.{1} Nat r.flag (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1))) →
          And (@LE.le.{0} Nat instLENat (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat) r.a g) nA)
            (And (@LE.le.{0} Nat instLENat (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat) r.b g) nB)
              (@LE.le.{0} Nat instLENat (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat) r.c g) nC)))
  (hA :
    ∀ (i : Nat),
      @LT.lt.{0} Nat instLTNat i nA →
        @LE.le.{0} Int Int.instLEInt
          (@abs.{0} Int instLatticeInt Int.instAddGroup
            (μ (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) a i)))
          (@Nat.cast.{0} Int instNatCastInt U))
  (hB :
    ∀ (i : Nat),
      @LT.lt.{0} Nat instLTNat i nB →
        @LE.le.{0} Int Int.instLEInt
          (@abs.{0} Int instLatticeInt Int.instAddGroup
            (μ (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) b i)))
          (@Nat.cast.{0} Int instNatCastInt U))
  (hC :
    ∀ (i : Nat),
      @LT.lt.{0} Nat instLTNat i nC →
        @LE.le.{0} Int Int.instLEInt
          (@abs.{0} Int instLatticeInt Int.instAddGroup
            (μ (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) c i)))
          (@Nat.cast.{0} Int instNatCastInt U))
  (hread :
    ∀ (i : Nat),
      @LT.lt.{0} Nat instLTNat i s →
        @Eq.{1} Int (μ (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) aux i))
          (@Nat.cast.{0} Int instNatCastInt (primes i)))
  (hprime : ∀ (i : Nat), @LT.lt.{0} Nat instLTNat i s → Nat.Prime (primes i))
  (hinj :
    ∀ (i : Nat),
      @LT.lt.{0} Nat instLTNat i s →
        ∀ (j : Nat), @LT.lt.{0} Nat instLTNat j s → @Eq.{1} Nat (primes i) (primes j) → @Eq.{1} Nat i j)
  (hproduct :
    ∀ (i : Nat),
      @LT.lt.{0} Nat instLTNat i s →
        @LE.le.{0} Nat instLENat (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat) q (primes i)) maxMod),
  Light.Meets lim ThreeSum.LightDenseSetup.program (@OfNat.ofNat.{0} Nat (nat_lit 19) (instOfNatNat (nat_lit 19))) d
    (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt a)
      (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt b)
        (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt c)
          (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt nA)
            (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt nB)
              (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt nC)
                (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt g)
                  (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt recs)
                    (@List.cons.{0} Int
                      (@Nat.cast.{0} Int instNatCastInt (@List.length.{0} ThreeSum.BandCandidateRows.Record rs))
                      (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt q)
                        (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt aux)
                          (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt s)
                            (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt cover)
                              (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt J)
                                (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt free)
                                  (@List.nil.{0} Int))))))))))))))))
    μ (ThreeSum.LightDenseSetup.time maxMod g (@List.length.{0} ThreeSum.BandCandidateRows.Record rs) s U J)
    fun (result : Int) (ν : Nat → Int) =>
    And (@Eq.{1} Int result (@Nat.cast.{0} Int instNatCastInt free))
      (And
        (ThreeSum.LightDenseSetup.Stored ν free (ThreeSum.GlobalEncoding.representationCap h J)
          (ThreeSum.LightDenseSetup.falseScore primes s q nA nB nC g
            (fun (i : Nat) => μ (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) a i))
            (fun (i : Nat) => μ (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) b i))
            (fun (i : Nat) => μ (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) c i)) rs))
        (And (ThreeSum.LightCoverKeyData.Stored ν cover h J) (Light.Kept μ ν free))) := by
  sorry

Source

https://github.com/c0ldzy17/three-sum-lean/blob/2b639655e07421b9ac743de9090459bab473a7c0/Submission/ThreeSum/LightDenseSetup.lean#L65-L154

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