Prove2Me

Building the guarded wanted-pair list with a balanced shift

Proved
ThreeSum.LightGuardedWanted.program_meets

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

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

Write S,Ω,ΔS,\Omega,\Delta for machine space, word magnitude, and depth limits. Let x0,…,xn−1x_0,\ldots,x_{n-1} be stored integers with ∣xi∣≤U|x_i|\le U, and let P,M>0P,M>0 and h,J∈Nh,J\in\mathbb N. The input ends before free address ff. Assume the guarded-wanted space and word bounds, S≤ΩS\le\Omega, and

d+512h+J+⌈log⁡2M⌉+3n+12≤Δ.d+512h+J+\lceil\log_2 M\rceil+3n+12\le\Delta.

Let LL be the length-3n3n list of lifts xi mod P+ℓPx_i\bmod P+\ell P, for ℓ=−1,0,1\ell=-1,0,1, and let K0,…,KN−1K_0,\ldots,K_{N-1} be the enumerated global keys. The routine chooses the specified first minimizing shift δ<M\delta<M and returns oo, the address immediately after the retained cover, keys, and lift list. At oo it stores the count HH, pair-array bounds, and δ\delta, with a duplicate-free list satisfying

(i,j) occurs  ⟺  i<3n, j<N, M∣Li+δ−Kj,MH≤3nN.(i,j)\text{ occurs}\iff i<3n,\ j<N,\ M\mid L_i+\delta-K_j, \qquad MH\le3nN.

The exact cover/key data and lift list remain stored, and cells below ff are preserved. The guarded-wanted time budget applies.

This produces the guarded physical pairs consumed by reporting.

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_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_ThreeSum1962Source_Bundle15
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.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.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.Pow.Asymptotics
import Mathlib.Combinatorics.Colex
import Mathlib.Data.Bool.Basic
import Mathlib.Data.Fin.Embedding
import Mathlib.Data.Fin.VecNotation
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.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.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.Abel
import Mathlib.Tactic.ByContra
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_GlobalKeyEnumeration_rawOutput_abs_le
import Theorems.Thm_ThreeSum_LightBalancedWanted_program_meets
import Theorems.Thm_ThreeSum_LightCoverKeyData_Stored_of_kept
import Theorems.Thm_ThreeSum_LightCoverKeys_Procs_meets_with_masks
import Theorems.Thm_ThreeSum_LightRawAKeys_meets
import Theorems.Thm_ThreeSum_RadixKeyLists_values_bounded


set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option Elab.async false
namespace Light
end Light
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.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.RawAKeyLists
end ThreeSum.RawAKeyLists
namespace ThreeSum.RawIncidences
end ThreeSum.RawIncidences
namespace ThreeSumApsp
end ThreeSumApsp
Formal statement
theorem ThreeSum.LightGuardedWanted.program_meets : ∀ {lim : Light.Limits} {μ : Nat → Int} {a free P M h J U d : Nat}
  [inst : @NeZero.{0} Nat (@MulZeroClass.toZero.{0} Nat Nat.instMulZeroClass) M] (xs : List.{0} Int)
  (hw : @LE.le.{0} Int Int.instLEInt (@Nat.cast.{0} Int instNatCastInt lim.space) lim.word)
  (hP : @LT.lt.{0} Nat instLTNat (@OfNat.ofNat.{0} Nat (nat_lit 0) (instOfNatNat (nat_lit 0))) P)
  (ha :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) a (@List.length.{0} Int xs)) free)
  (hspace : @LE.le.{0} Nat instLENat (@ThreeSum.LightGuardedWanted.space free P M h J U inst xs) lim.space)
  (hword :
    @LE.le.{0} Int Int.instLEInt
      (@Nat.cast.{0} Int instNatCastInt (ThreeSum.LightGuardedWanted.word P M h J U (@List.length.{0} Int xs)))
      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)
          (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
            (@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)
                  (@OfNat.ofNat.{0} Nat (nat_lit 512) (instOfNatNat (nat_lit 512))) h))
              J)
            (Nat.clog (@OfNat.ofNat.{0} Nat (nat_lit 2) (instOfNatNat (nat_lit 2))) M))
          (@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)))
        (@OfNat.ofNat.{0} Nat (nat_lit 12) (instOfNatNat (nat_lit 12))))
      lim.depth)
  (hread : Light.Seg μ a xs)
  (hvalues :
    ∀ (x : Int),
      @Membership.mem.{0, 0} Int (List.{0} Int) (@List.instMembership.{0} Int) xs x →
        @LE.le.{0} Int Int.instLEInt (@abs.{0} Int instLatticeInt Int.instAddGroup x)
          (@Nat.cast.{0} Int instNatCastInt U)),
  Light.Meets lim ThreeSum.LightGuardedWanted.program (@OfNat.ofNat.{0} Nat (nat_lit 98) (instOfNatNat (nat_lit 98))) 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 a)
          (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt (@List.length.{0} Int xs))
            (@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 free) (@List.nil.{0} Int))))))))
    μ (@ThreeSum.LightGuardedWanted.time P M h J U inst xs) fun (r : Int) (ν : Nat → Int) =>
    And
      (@Eq.{1} Int r
        (@Nat.cast.{0} Int instNatCastInt (ThreeSum.LightGuardedWanted.wantedFree free h J (@List.length.{0} Int xs))))
      (And
        (ThreeSum.LightWantedSetup.Output ν (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)))
        (And
          (@LE.le.{0} Nat instLENat
            (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat) M
              (ν (ThreeSum.LightGuardedWanted.wantedFree free h J (@List.length.{0} Int xs))).toNat)
            (@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 3) (instOfNatNat (nat_lit 3))) (@List.length.{0} Int xs))
              (ThreeSum.LightGuardedWanted.totalKeys h J)))
          (And (ThreeSum.LightCoverKeyData.Stored ν free h J)
            (And
              (Light.Seg ν (ThreeSum.LightCoverKeyData.keyEnd free h J)
                (ThreeSum.RawAKeyLists.keys (@Nat.cast.{0} Int instNatCastInt P) xs))
              (Light.Kept μ ν free))))) := by
  sorry
Source
https://github.com/c0ldzy17/three-sum-lean/blob/2b639655e07421b9ac743de9090459bab473a7c0/Submission/ThreeSum/LightGuardedWanted.lean#L91-L174

View graph