Prove2Me

Build paired source-band structures and their shared zero bucket

Proved
ThreeSum.LightSourceBandPair.Procs.meets

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

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

Let stored arrays L,RL,R have lengths NL,NRN_L,N_R and magnitudes at most UU. Fix g,p,m>0g,p,m>0 and ∣σ∣≤D|\sigma|\le D, with consistent stored length/band-count headers and shift cell before destination bb. Assume the paired-source space and word budgets, space fitting in a word, depth strictly above d+6d+6, and the installed band-setup procedures. The routine constructs every left and right band using respectively

kL(x)=((x mod p) mod m),kR(x)=((x mod p+σ) mod m).k_L(x)=((x\bmod p)\bmod m),\qquad k_R(x)=((x\bmod p+\sigma)\bmod m).

Each band retains its header, codes, histogram, and physical-position fibers. An eight-cell pair header records the two band bases and counts, shared zero-bucket base, next address, and original lengths. The routine also stores m+1m+1 zero cells, returns the next address, and preserves memory below bb.

Its time is the two setup costs plus zero filling and 20002000 steps. This provides both source orientations for bandwise queries.

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_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.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.Pow.Asymptotics
import Mathlib.Analysis.SpecialFunctions.Pow.NthRootLemmas
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.Combinatorics.Colex
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.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.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.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.NumberTheory.Chebyshev
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.Monotone.Basic
import Mathlib.RingTheory.Coprime.Lemmas
import Mathlib.Tactic
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.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_Light_Ends_pass
import Theorems.Thm_ThreeSum_LightSourceBandSetup_Procs_meets
import Theorems.Thm_ThreeSum_SourceBandData_At_transfer
import Theorems.Thm_ThreeSum_SourceBandPairData_Data_transfer


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.LightHistogram
end ThreeSum.LightHistogram
namespace ThreeSum.SourceBandLayout
end ThreeSum.SourceBandLayout
namespace ThreeSum.SourceBandPairData
end ThreeSum.SourceBandPairData
namespace ThreeSum.SourceBuckets
end ThreeSum.SourceBuckets
Formal statement
theorem ThreeSum.LightSourceBandPair.Procs.meets : ∀ {lim : Light.Limits} {P : Light.Program} {d pMain : Nat} (C : ThreeSum.LightSourceBandPair.Procs P pMain)
  {μ : Nat → Int} {srcL headL srcR headR shiftCell base NL NR g p m U D : Nat} {shift : Int} (fL fR : Nat → Int)
  (hw : @LE.le.{0} Int Int.instLEInt (@Nat.cast.{0} Int instNatCastInt lim.space) lim.word)
  (hd :
    @LT.lt.{0} Nat instLTNat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) d
        (@OfNat.ofNat.{0} Nat (nat_lit 6) (instOfNatNat (nat_lit 6))))
      lim.depth)
  (hg : @LT.lt.{0} Nat instLTNat (@OfNat.ofNat.{0} Nat (nat_lit 0) (instOfNatNat (nat_lit 0))) g)
  (hp : @LT.lt.{0} Nat instLTNat (@OfNat.ofNat.{0} Nat (nat_lit 0) (instOfNatNat (nat_lit 0))) p)
  (hm : @LT.lt.{0} Nat instLTNat (@OfNat.ofNat.{0} Nat (nat_lit 0) (instOfNatNat (nat_lit 0))) m)
  (hsrcL : @LE.le.{0} Nat instLENat (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) srcL NL) base)
  (hsrcR : @LE.le.{0} Nat instLENat (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) srcR NR) base)
  (hheadL :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) headL
        (@OfNat.ofNat.{0} Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))
      base)
  (hheadR :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) headR
        (@OfNat.ofNat.{0} Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))
      base)
  (hshiftCell : @LT.lt.{0} Nat instLTNat shiftCell base)
  (hspace : @LE.le.{0} Nat instLENat (ThreeSum.LightSourceBandPair.space base NL NR g p m U D) lim.space)
  (hword :
    @LE.le.{0} Int Int.instLEInt (@Nat.cast.{0} Int instNatCastInt (ThreeSum.LightSourceBandPair.word NL NR g p m U D))
      lim.word)
  (hshift :
    @LE.le.{0} Int Int.instLEInt (@abs.{0} Int instLatticeInt Int.instAddGroup shift)
      (@Nat.cast.{0} Int instNatCastInt D))
  (hlenL : @Eq.{1} Int (μ headL) (@Nat.cast.{0} Int instNatCastInt NL))
  (hlenR : @Eq.{1} Int (μ headR) (@Nat.cast.{0} Int instNatCastInt NR))
  (hcountL :
    @Eq.{1} Int
      (μ
        (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) headL
          (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))
      (@Nat.cast.{0} Int instNatCastInt (ThreeSum.BandPartition.size NL g)))
  (hcountR :
    @Eq.{1} Int
      (μ
        (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) headR
          (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))
      (@Nat.cast.{0} Int instNatCastInt (ThreeSum.BandPartition.size NR g)))
  (hshiftRead : @Eq.{1} Int (μ shiftCell) shift)
  (hreadL :
    ∀ (j : Nat),
      @LT.lt.{0} Nat instLTNat j NL →
        @Eq.{1} Int (μ (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) srcL j)) (fL j))
  (hreadR :
    ∀ (j : Nat),
      @LT.lt.{0} Nat instLTNat j NR →
        @Eq.{1} Int (μ (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) srcR j)) (fR j))
  (hvaluesL :
    ∀ (j : Nat),
      @LT.lt.{0} Nat instLTNat j NL →
        @LE.le.{0} Int Int.instLEInt (@abs.{0} Int instLatticeInt Int.instAddGroup (fL j))
          (@Nat.cast.{0} Int instNatCastInt U))
  (hvaluesR :
    ∀ (j : Nat),
      @LT.lt.{0} Nat instLTNat j NR →
        @LE.le.{0} Int Int.instLEInt (@abs.{0} Int instLatticeInt Int.instAddGroup (fR j))
          (@Nat.cast.{0} Int instNatCastInt U)),
  Light.Meets lim P pMain d
    (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt srcL)
      (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt headL)
        (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt srcR)
          (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt headR)
            (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt g)
              (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt p)
                (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt m)
                  (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt shiftCell)
                    (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt base) (@List.nil.{0} Int))))))))))
    μ (ThreeSum.LightSourceBandPair.time NL NR g p m U D) fun (r : Int) (ν : Nat → Int) =>
    And (@Eq.{1} Int r (@Nat.cast.{0} Int instNatCastInt (ThreeSum.SourceBandPairData.next base NL NR g m)))
      (And (ThreeSum.SourceBandPairData.Stored ν fL fR base NL NR g p m shift) (Light.Kept μ ν base)) := by
  sorry
Source
https://github.com/c0ldzy17/three-sum-lean/blob/2b639655e07421b9ac743de9090459bab473a7c0/Submission/ThreeSum/LightSourceBandPair.lean#L64-L177

View graph