Prove2Me

Build shared reporting levels for a selected source-bank pair

Proved
ThreeSum.LightSharedReportingPair.program_meets

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

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

Suppose memory stores complete source banks for arrays fB,fCf_B,f_C of lengths NB,NCN_B,N_C, band parameter gg, local parameters h,Jh,J, source cutoff 2hJ2hJ, integer modulus pp, and shift δ\delta. Select an occurrence code ss below the occurrence count, and common-band indices i,ji,j below the maximum of the two defined band counts. Let K≤⌊13J/20⌋+1K\le\lfloor13J/20\rfloor+1.

The source banks end before record base r0r_0; its 5K5K records end before payload base aa. Assume S≤WS\le W and the level resource contract at entry depth plus one, using the source magnitude bounds g,gg,g. The routine returns aKa_K and constructs

all K reporting catalogs for the left source of band i and right source of band j at occurrence s.\text{all }K\text{ reporting catalogs for the left source of band }i\text{ and right source of band }j\text{ at occurrence }s.

Time is the level-construction budget plus 500500. Memory below r0r_0, and below aa outside the records, is preserved.

This selects reusable source tables from the bank before constructing their shared reporting levels.

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_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_Bundle13
import Definitions.Def_ThreeSum1962Source_Bundle14
import Definitions.Def_ThreeSum1962Source_Bundle15
import Definitions.Def_ThreeSum1962Source_Bundle16
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.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.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.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.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_LightSharedReportingLevels_program_meets
import Theorems.Thm_ThreeSum_SharedReportingBankSources_source_bounds


set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option Elab.async false
namespace Light
end Light
namespace ThreeSum.BandPartition
end ThreeSum.BandPartition
namespace ThreeSum.BlockEncoding
end ThreeSum.BlockEncoding
namespace ThreeSum.BlockForced
end ThreeSum.BlockForced
namespace ThreeSum.BlockIndex
end ThreeSum.BlockIndex
namespace ThreeSum.BlockIndexedQuery
end ThreeSum.BlockIndexedQuery
namespace ThreeSum.BlockRows
end ThreeSum.BlockRows
namespace ThreeSum.BlockSourceOccurrenceOrder
end ThreeSum.BlockSourceOccurrenceOrder
namespace ThreeSum.BlockSourceSharing
end ThreeSum.BlockSourceSharing
namespace ThreeSum.GlobalEncoding
end ThreeSum.GlobalEncoding
namespace ThreeSum.LocalCoverBounds
end ThreeSum.LocalCoverBounds
namespace ThreeSum.OutputDecodeRecords
end ThreeSum.OutputDecodeRecords
namespace ThreeSum.RawIncidences
end ThreeSum.RawIncidences
namespace ThreeSum.RawPrefixCounts
end ThreeSum.RawPrefixCounts
namespace ThreeSum.RawReporting
end ThreeSum.RawReporting
namespace ThreeSum.SharedReportingBankData
end ThreeSum.SharedReportingBankData
namespace ThreeSum.SharedReportingBankLayout
end ThreeSum.SharedReportingBankLayout
namespace ThreeSum.SharedReportingBankLevels
end ThreeSum.SharedReportingBankLevels
namespace ThreeSum.SharedReportingBankSources
end ThreeSum.SharedReportingBankSources
namespace ThreeSum.SourceBandLayout
end ThreeSum.SourceBandLayout
namespace ThreeSum.SubsetAddress
end ThreeSum.SubsetAddress
Formal statement
theorem ThreeSum.LightSharedReportingPair.program_meets : ∀ {lim : Light.Limits} {μ fB fC : Nat → Int} {header NB NC g h J occ s i j K record base d : Nat} {p shift : Int}
  (H :
    ThreeSum.SourceBandBankData.Stored μ fB fC header NB NC g h J
      (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
        (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
          (@OfNat.ofNat.{0} Nat (nat_lit 2) (instOfNatNat (nat_lit 2))) h)
        J)
      occ p shift)
  (hs : @LT.lt.{0} Nat instLTNat s (ThreeSum.LightBlockSourceCatalog.count h J))
  (hi : @LT.lt.{0} Nat instLTNat i (ThreeSum.SourceBandPairSlices.count NB NC g))
  (hj : @LT.lt.{0} Nat instLTNat j (ThreeSum.SourceBandPairSlices.count NB NC g))
  (hw : @LE.le.{0} Int Int.instLEInt (@Nat.cast.{0} Int instNatCastInt lim.space) lim.word)
  (hK :
    @LE.le.{0} Nat instLENat K
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) (ThreeSum.LightSwitchDepth.depth J)
        (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))
  (hsource :
    @LE.le.{0} Nat instLENat
      (ThreeSum.SourceBandBankData.finish header NB NC g h J
        (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
          (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
            (@OfNat.ofNat.{0} Nat (nat_lit 2) (instOfNatNat (nat_lit 2))) h)
          J))
      record)
  (hrecords :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) record
        (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
          (@OfNat.ofNat.{0} Nat (nat_lit 5) (instOfNatNat (nat_lit 5))) K))
      base)
  (R :
    ThreeSum.LightSharedReportingLevels.Resources lim base h J K g g
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) d
        (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))),
  Light.Meets lim ThreeSum.LightSharedReportingPair.program
    (@OfNat.ofNat.{0} Nat (nat_lit 65) (instOfNatNat (nat_lit 65))) d
    (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt h)
      (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt J)
        (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt K)
          (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt (ThreeSum.SourceBandBankData.first header))
            (@List.cons.{0} Int
              (@Nat.cast.{0} Int instNatCastInt
                (ThreeSum.SourceBandBankLayout.span h J
                  (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
                    (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
                      (@OfNat.ofNat.{0} Nat (nat_lit 2) (instOfNatNat (nat_lit 2))) h)
                    J)))
              (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt s)
                (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt i)
                  (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt j)
                    (@List.cons.{0} Int
                      (@Nat.cast.{0} Int instNatCastInt (ThreeSum.SharedReportingBankLayout.localChoose h))
                      (@List.cons.{0} Int
                        (@Nat.cast.{0} Int instNatCastInt (ThreeSum.SharedReportingBankLayout.localPairs h))
                        (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt record)
                          (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt base) (@List.nil.{0} Int)))))))))))))
    μ (ThreeSum.LightSharedReportingPair.time h J K) fun (result : Int) (ν : Nat → Int) =>
    And (@Eq.{1} Int result (@Nat.cast.{0} Int instNatCastInt (ThreeSum.SharedReportingBankLevels.baseAt base h J K)))
      (And
        (ThreeSum.SharedReportingBankLevels.Built ν record base h J K
          (@ThreeSum.BlockIndexedQuery.leftSource.{0} (Fin NB) h J p (ThreeSum.SourceBandPartition.source NB g i)
            (fun (t : Fin NB) => fB (@Fin.val NB t)) (ThreeSum.BlockSourceOccurrenceOrder.representative h J s))
          (@ThreeSum.BlockIndexedQuery.rightSource.{0} (Fin NC) h J p shift (ThreeSum.SourceBandPartition.source NC g j)
            (fun (t : Fin NC) => fC (@Fin.val NC t)) (ThreeSum.BlockSourceOccurrenceOrder.representative h J s)))
        (And (Light.Kept μ ν record)
          (Light.KeptBut μ ν base record
            (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
              (@OfNat.ofNat.{0} Nat (nat_lit 5) (instOfNatNat (nat_lit 5))) K)))) := by
  sorry
Source
https://github.com/c0ldzy17/three-sum-lean/blob/2b639655e07421b9ac743de9090459bab473a7c0/Submission/ThreeSum/LightSharedReportingPair.lean#L26-L77

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