Prove2Me

Construct finite source data from normalized arrays and a selected shift

Proved
ThreeSum.LightSourceFinitePrepared.meets

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

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

Let B,CB,C be strictly increasing lists represented by stored normalized values and band headers for band size g>0g>0. Their original lists have magnitudes at most UU. Fix p>0p>0, positive shift-selection modulus MM, local parameters h,Jh,J, and cutoff c≤512hJc\le512hJ. Suppose the exact cover data and wanted-setup output are already stored for a list XX; let δ\delta be that setup's selected offset.

All normalized input reservations and the selected-shift cell precede source base bb. Assume the source-controls procedures, their exact space and word budgets with shift bound MM, S≤WS\le W, and d+512hJ+16≤Dd+512hJ+16\le D. The routine returns the source finish address and establishes

FiniteSources⁡(ν,b,g,p,h,J,c,B,C,δ).\operatorname{FiniteSources}(\nu,b,g,p,h,J,c,B,C,\delta).

This includes controls, original-band residue buckets, and retained source banks. The cover data remain valid and memory below bb is unchanged; execution meets the source-controls time budget.

This specializes the generic source host to the actual normalized finite lists and computed shift.

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_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_Bundle11
import Definitions.Def_ThreeSum1962Source_Bundle12
import Definitions.Def_ThreeSum1962Source_Bundle14
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_LightCoverKeyData_Stored_of_kept
import Theorems.Thm_ThreeSum_LightSourceControlsHost_Procs_meets


set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option Elab.async false
namespace Light
end Light
namespace ThreeSum.BlockCoefficientRows
end ThreeSum.BlockCoefficientRows
namespace ThreeSum.BlockSourceEncoding
end ThreeSum.BlockSourceEncoding
namespace ThreeSum.BlockSourceFlags
end ThreeSum.BlockSourceFlags
namespace ThreeSum.CyclicFold
end ThreeSum.CyclicFold
namespace ThreeSum.GlobalEncoding
end ThreeSum.GlobalEncoding
namespace ThreeSum.GlobalKeyEnumeration
end ThreeSum.GlobalKeyEnumeration
namespace ThreeSum.Karatsuba
end ThreeSum.Karatsuba
namespace ThreeSum.KaratsubaInterface
end ThreeSum.KaratsubaInterface
namespace ThreeSum.LightKaratsuba
end ThreeSum.LightKaratsuba
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.LightResidueHistogram
end ThreeSum.LightResidueHistogram
namespace ThreeSum.LightResidues
end ThreeSum.LightResidues
namespace ThreeSum.LocalCoverBounds
end ThreeSum.LocalCoverBounds
namespace ThreeSum.LocalGraph
end ThreeSum.LocalGraph
namespace ThreeSum.LocalHashMoments
end ThreeSum.LocalHashMoments
namespace ThreeSum.RadixKeyLists
end ThreeSum.RadixKeyLists
namespace ThreeSum.RawIncidences
end ThreeSum.RawIncidences
namespace ThreeSum.SourceFiniteInputs
end ThreeSum.SourceFiniteInputs
Formal statement
theorem ThreeSum.LightSourceFinitePrepared.meets : ∀ {lim : Light.Limits} {P : Light.Program} {d pMain : Nat} (R : ThreeSum.LightSourceControlsHost.Procs P pMain)
  {μ : Nat → Int} {free base bv be cv ce g p h J cut U M : Nat}
  [inst : @NeZero.{0} Nat (@MulZeroClass.toZero.{0} Nat Nat.instMulZeroClass) M]
  {hg : @LT.lt.{0} Nat instLTNat (@OfNat.ofNat.{0} Nat (nat_lit 0) (instOfNatNat (nat_lit 0))) g}
  {originalB originalC B C xs : List.{0} Int}
  {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}
  (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) 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)))
        (@OfNat.ofNat.{0} Nat (nat_lit 16) (instOfNatNat (nat_lit 16))))
      lim.depth)
  (hp : @LT.lt.{0} Nat instLTNat (@OfNat.ofNat.{0} Nat (nat_lit 0) (instOfNatNat (nat_lit 0))) p)
  (hcut :
    @LE.le.{0} Nat instLENat cut
      (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat) 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)))
  (hbv :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) bv (@List.length.{0} Int originalB)) base)
  (hcv :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) cv (@List.length.{0} Int originalC)) base)
  (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))))
      base)
  (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))))
      base)
  (hshiftCell :
    @LT.lt.{0} Nat instLTNat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
        (ThreeSum.LightGuardedWanted.wantedFree free h J (@List.length.{0} Int xs))
        (@OfNat.ofNat.{0} Nat (nat_lit 3) (instOfNatNat (nat_lit 3))))
      base)
  (hspace :
    @LE.le.{0} Nat instLENat
      (ThreeSum.LightSourceControlsHost.space base (@List.length.{0} Int B) (@List.length.{0} Int C) g p h J cut U M)
      lim.space)
  (hword :
    @LE.le.{0} Int Int.instLEInt
      (@Nat.cast.{0} Int instNatCastInt
        (ThreeSum.LightSourceControlsHost.word (@List.length.{0} Int B) (@List.length.{0} Int C) g p h J cut U M))
      lim.word)
  (hB : ThreeSum.LightBandPreprocessing.Represents μ bv be g hg originalB B hsB)
  (hC : ThreeSum.LightBandPreprocessing.Represents μ cv ce g hg originalC C hsC)
  (hboundB :
    ∀ (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))
  (hboundC :
    ∀ (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))
  (hcover : ThreeSum.LightCoverKeyData.Stored μ free h J)
  (hwanted :
    ThreeSum.LightWantedSetup.Output μ (ThreeSum.LightGuardedWanted.wantedFree free h J (@List.length.{0} Int xs)) M
      (ThreeSum.LightGuardedWanted.left p xs) (ThreeSum.LightGuardedWanted.right h J)
      (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
        (@OfNat.ofNat.{0} Nat (nat_lit 3) (instOfNatNat (nat_lit 3))) (@List.length.{0} Int xs))
      (ThreeSum.LightGuardedWanted.totalKeys h J)
      (@Nat.cast.{0} Int instNatCastInt
        (@ThreeSum.RawAKeyLists.selected M inst (@Nat.cast.{0} Int instNatCastInt p) xs h J))),
  Light.Meets lim P pMain d
    (ThreeSum.LightSourceFinitePrepared.args free base bv be cv ce g p h J cut (@List.length.{0} Int xs)) μ
    (ThreeSum.LightSourceControlsHost.time (@List.length.{0} Int B) (@List.length.{0} Int C) g p h J cut U M)
    fun (r : Int) (ν : Nat → Int) =>
    And
      (ThreeSum.SourceFiniteData.Stored ν base g p h J cut B C
        (@Nat.cast.{0} Int instNatCastInt
          (@ThreeSum.RawAKeyLists.selected M inst (@Nat.cast.{0} Int instNatCastInt p) xs h J))
        r)
      (And (ThreeSum.LightCoverKeyData.Stored ν free h J) (Light.Kept μ ν base)) := by
  sorry
Source
https://github.com/c0ldzy17/three-sum-lean/blob/2b639655e07421b9ac743de9090459bab473a7c0/Submission/ThreeSum/LightSourceFinitePrepared.lean#L16-L56

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