Constructing common-modulus data from normalized finite inputs

Proved

ThreeSum.LightEncodedColorPrime.program_meets

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

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

Write $S,\Omega,\Delta$ for machine space, word magnitude, and depth limits. Let original integer lists have absolute values at most $U$ , and suppose their normalization is stored as strictly increasing lists $A,B,C$ , with prepared band headers, tables, and the canonical owner records for band size $g>0$ . Fix $y>0$ and natural target scale $M$ . Let $L$ be the specified prime-chain depth, and $Q_j$ the common modulus chain for these normalized lists, using the increasing prime pool $y<p\le4y$ and first-minimum weighted-score selection.

Assume the value arrays, length headers, and owner-record cells end before $f$ , $(4y)^{L+1}\le M_{\max}$ , the core scratch and wrapper word bounds, $S\le\Omega$ , and $d+\lceil\log_2 M_{\max}\rceil+10\le\Delta$ . The routine returns

r=Q_L

and stores the exact prime-data record at $f$ : its header, prime pool, and all moduli $Q_0,\ldots,Q_L$ . Every cell below $f$ is preserved. The time bound is the prime-core budget for the actual normalized lengths and record count, plus $400$ .

This adapts common-modulus construction to the normalized finite-input representation.

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_Bundle08
import Definitions.Def_ThreeSum1962Source_Bundle09
import Definitions.Def_ThreeSum1962Source_Bundle10
import Definitions.Def_ThreeSum1962Source_Bundle12
import Definitions.Def_ThreeSum1962Source_Bundle13
import Definitions.Def_ThreeSum1962Source_Bundle15
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.Nat
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.NthRootLemmas
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Analysis.SpecificLimits.Normed
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.Image
import Mathlib.Data.Finset.Max
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.Fin
import Mathlib.Data.Fintype.Pi
import Mathlib.Data.Fintype.Prod
import Mathlib.Data.Fintype.Sets
import Mathlib.Data.Fintype.Sum
import Mathlib.Data.Int.Basic
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.Dedup
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.Sublists
import Mathlib.Data.List.TakeDrop
import Mathlib.Data.Nat.Bitwise
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.Choose.Vandermonde
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
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.LinearCombination
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_BandCandidateRows_fromOwner_sound
import Theorems.Thm_ThreeSum_BandCandidateRows_mem_records_owner
import Theorems.Thm_ThreeSum_LightEncodedColorPrimeCore_program_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.BandEnumeration
end ThreeSum.BandEnumeration
namespace ThreeSum.BandPartition
end ThreeSum.BandPartition
namespace ThreeSum.BandRawA
end ThreeSum.BandRawA
namespace ThreeSum.BandScales
end ThreeSum.BandScales
namespace ThreeSum.BandWeightedScore
end ThreeSum.BandWeightedScore
namespace ThreeSum.CostLedger
end ThreeSum.CostLedger
namespace ThreeSum.GlobalEncoding
end ThreeSum.GlobalEncoding
namespace ThreeSum.LightLocalCoverLayout
end ThreeSum.LightLocalCoverLayout
namespace ThreeSum.LightLocalCoverSchedule
end ThreeSum.LightLocalCoverSchedule
namespace ThreeSum.LocalCoverBounds
end ThreeSum.LocalCoverBounds
namespace ThreeSum.Scales
end ThreeSum.Scales
namespace ThreeSum.SourceFiniteInputs
end ThreeSum.SourceFiniteInputs
namespace ThreeSum.WeightedScore
end ThreeSum.WeightedScore

Formal statement

theorem ThreeSum.LightEncodedColorPrime.program_meets : ∀ {lim : Light.Limits} {d : Nat} {μ : Nat → Int} {av ae bv be cv ce tables rec vals g M y free maxMod U : Nat}
  {hg : @LT.lt.{0} Nat instLTNat (@OfNat.ofNat.{0} Nat (nat_lit 0) (instOfNatNat (nat_lit 0))) g}
  {originalA originalB originalC A B C : List.{0} Int}
  {hsA : @List.Pairwise.{0} Int (fun (x1 x2 : Int) => @LT.lt.{0} Int Int.instLTInt x1 x2) A}
  {hsB : @List.Pairwise.{0} Int (fun (x1 x2 : Int) => @LT.lt.{0} Int Int.instLTInt x1 x2) B}
  {hsC : @List.Pairwise.{0} Int (fun (x1 x2 : Int) => @LT.lt.{0} Int Int.instLTInt x1 x2) C} {rA : Int}
  (hw : @LE.le.{0} Int Int.instLEInt (@Nat.cast.{0} Int instNatCastInt lim.space) lim.word)
  (hy : @LT.lt.{0} Nat instLTNat (@OfNat.ofNat.{0} Nat (nat_lit 0) (instOfNatNat (nat_lit 0))) y)
  (hav :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) av (@List.length.{0} Int A)) free)
  (hbv :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) bv (@List.length.{0} Int B)) free)
  (hcv :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) cv (@List.length.{0} Int C)) free)
  (hae :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) ae
        (@OfNat.ofNat.{0} Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))
      free)
  (hbe :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) be
        (@OfNat.ofNat.{0} Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))
      free)
  (hce :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) ce
        (@OfNat.ofNat.{0} Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))
      free)
  (hrec :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) rec
        (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
          (@OfNat.ofNat.{0} Nat (nat_lit 4) (instOfNatNat (nat_lit 4)))
          (ThreeSum.LightEncodedColorPrime.count (@List.length.{0} Int A) (@List.length.{0} Int B)
            (@List.length.{0} Int C) g)))
      free)
  (hspace : @LE.le.{0} Nat instLENat (ThreeSum.LightEncodedColorPrimeCore.space free maxMod g M y U) lim.space)
  (hword :
    @LE.le.{0} Int Int.instLEInt
      (@Nat.cast.{0} Int instNatCastInt
        (ThreeSum.LightEncodedColorPrime.word maxMod (@List.length.{0} Int A) (@List.length.{0} Int B)
          (@List.length.{0} Int C) g M y U))
      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 10) (instOfNatNat (nat_lit 10))))
      lim.depth)
  (hA :
    ∀ (z : Int),
      @Membership.mem.{0, 0} Int (List.{0} Int) (@List.instMembership.{0} Int) originalA z →
        @LE.le.{0} Int Int.instLEInt (@abs.{0} Int instLatticeInt Int.instAddGroup z)
          (@Nat.cast.{0} Int instNatCastInt U))
  (hB :
    ∀ (z : Int),
      @Membership.mem.{0, 0} Int (List.{0} Int) (@List.instMembership.{0} Int) originalB z →
        @LE.le.{0} Int Int.instLEInt (@abs.{0} Int instLatticeInt Int.instAddGroup z)
          (@Nat.cast.{0} Int instNatCastInt U))
  (hC :
    ∀ (z : Int),
      @Membership.mem.{0, 0} Int (List.{0} Int) (@List.instMembership.{0} Int) originalC z →
        @LE.le.{0} Int Int.instLEInt (@abs.{0} Int instLatticeInt Int.instAddGroup z)
          (@Nat.cast.{0} Int instNatCastInt U))
  (H :
    ThreeSum.SourceFiniteData.Normalized μ av ae bv be cv ce tables rec vals g hg originalA originalB originalC A B C
      hsA hsB hsC rA)
  (hmax :
    @LE.le.{0} Nat instLENat
      (@HPow.hPow.{0, 0, 0} Nat Nat Nat
        (@instHPow.{0, 0} Nat Nat (@NPow.toPow.{0} Nat (@Monoid.toNPow.{0} Nat Nat.instMonoid)))
        (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
          (@OfNat.ofNat.{0} Nat (nat_lit 4) (instOfNatNat (nat_lit 4))) y)
        (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) (ThreeSum.LightBandPrimeDepth.depth M y)
          (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))
      maxMod),
  Light.Meets lim ThreeSum.LightEncodedColorPrime.program
    (@OfNat.ofNat.{0} Nat (nat_lit 24) (instOfNatNat (nat_lit 24))) d
    (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt av)
      (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt ae)
        (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt bv)
          (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt be)
            (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt cv)
              (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt ce)
                (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt g)
                  (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt rec)
                    (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt M)
                      (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt y)
                        (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt free) (@List.nil.{0} Int))))))))))))
    μ
    (ThreeSum.LightEncodedColorPrime.time maxMod (@List.length.{0} Int A) (@List.length.{0} Int B)
      (@List.length.{0} Int C) g M y U)
    fun (result : Int) (ν : Nat → Int) =>
    And
      (@Eq.{1} Int result
        (@Nat.cast.{0} Int instNatCastInt
          (ThreeSum.LightEncodedColorPrime.moduli A B C g hg hsA hsB hsC y (ThreeSum.LightBandPrimeDepth.depth M y))))
      (And
        (ThreeSum.EncodedColorPrimeData.Stored ν free M y
          (ThreeSum.LightEncodedColorPrime.moduli A B C g hg hsA hsB hsC y))
        (Light.Kept μ ν free)) := by
  sorry

Source

https://github.com/c0ldzy17/three-sum-lean/blob/2b639655e07421b9ac743de9090459bab473a7c0/Submission/ThreeSum/LightEncodedColorPrime.lean#L46-L134

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