Prove2Me

Evaluate the explicit missing-set layers of a suffix query

Proved
ThreeSum.LightReportingSuffixExplicit.meets

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

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

Let Q⊆[V]Q\subseteq[V] be embedded into [W][W] by a coordinate shift ι\iota, and let t≤∣Q∣+1t\le|Q|+1. Suppose suffix stages s≤Ws\le W of f,gf,g are retained on the cc-downset, c≤Wc\le W, with magnitude bounds B,DB,D, and both PL∪ιQP_L\cup\iota Q and PR∪ιQP_R\cup\iota Q have size at most cc. Set λ(S)=(−1)∣S∣Zsf(PL∪ιS)Zsg(PR∪ιS)\lambda(S)=(-1)^{|S|}Z_sf(P_L\cup\iota S)Z_sg(P_R\cup\iota S) and A=∑j<t(∣Q∣j)A=\sum_{j<t}\binom{|Q|}{j}. With source banks before an AA-cell mask buffer, sufficient scratch, space fitting in a word, the control and (A+1)(2sB)(2sD)(A+1)(2^sB)(2^sD) word bounds, and depth d+W+8d+W+8, the program returns

∑U⊆Q∣U∣<tλ(Q∖U).\sum_{\substack{U\subseteq Q\\|U|<t}}\lambda(Q\setminus U).

Its budget is t(10500(W+1)2(A+1)+30)+40t(10500(W+1)^2(A+1)+30)+40. Below scratch only the mask buffer may change.

The cutoff counts missing coordinates, providing the explicit part of the reporting query.

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_Bundle10
import Definitions.Def_ThreeSum1962Source_Bundle11
import Definitions.Def_ThreeSum1962Source_Bundle12
import Definitions.Def_ThreeSum1962Source_Bundle13
import Lean.Elab.Tactic.Omega
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
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.Pow
import Mathlib.Algebra.Ring.Nat
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.Max
import Mathlib.Data.Finset.Powerset
import Mathlib.Data.Finset.Prod
import Mathlib.Data.Finset.Sigma
import Mathlib.Data.Finset.Sort
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Fintype.Fin
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.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.Sublists
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.Notation
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.Order.ConditionallyCompleteLattice.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_Light_Ends_for
import Theorems.Thm_ThreeSum_LightQueryEnumeration_meets
import Theorems.Thm_ThreeSum_LightReportingSuffixSum_meets
import Theorems.Thm_ThreeSum_SubsetAddress_enumerationWork_bound
import Theorems.Thm_ThreeSum_SubsetAddress_layerRank_lt
import Theorems.Thm_ThreeSum_SubsetStages_stageValue_abs_le


set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option Elab.async false
namespace Light
end Light
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.GlobalEncoding
end ThreeSum.GlobalEncoding
namespace ThreeSum.LightBlockQuery
end ThreeSum.LightBlockQuery
namespace ThreeSum.LightPascal
end ThreeSum.LightPascal
namespace ThreeSum.LightSubsetAddress
end ThreeSum.LightSubsetAddress
namespace ThreeSum.LightZetaCell
end ThreeSum.LightZetaCell
namespace ThreeSum.LocalCoverBounds
end ThreeSum.LocalCoverBounds
namespace ThreeSum.MaskDecode
end ThreeSum.MaskDecode
namespace ThreeSum.MaskErase
end ThreeSum.MaskErase
namespace ThreeSum.MaskUnion
end ThreeSum.MaskUnion
namespace ThreeSum.QueryCoordinates
end ThreeSum.QueryCoordinates
namespace ThreeSum.RawIncidences
end ThreeSum.RawIncidences
namespace ThreeSum.RawPrefixCounts
end ThreeSum.RawPrefixCounts
namespace ThreeSum.RawReporting
end ThreeSum.RawReporting
namespace ThreeSum.ReportingHostSuffix
end ThreeSum.ReportingHostSuffix
namespace ThreeSum.StoredReporting
end ThreeSum.StoredReporting
namespace ThreeSum.SubsetAddress
end ThreeSum.SubsetAddress
Formal statement
theorem ThreeSum.LightReportingSuffixExplicit.meets : ∀ {lim : Light.Limits} {P : Light.Program} {pMain pEnum pSum : Nat}
  (hmain :
    @Eq.{1} (Option.{0} Light.Stmt)
      (@GetElem?.getElem?.{0, 0, 0} Light.Program Nat Light.Stmt
        (fun (as : List.{0} Light.Stmt) (i : Nat) => @LT.lt.{0} Nat instLTNat i (@List.length.{0} Light.Stmt as))
        (@List.instGetElem?NatLtLength.{0} Light.Stmt) P pMain)
      (@Option.some.{0} Light.Stmt (ThreeSum.LightReportingSuffixExplicit.body pEnum pSum)))
  (C : ThreeSum.LightReportingSuffixExplicit.Procs P pEnum pSum) {μ : Nat → Int}
  {W V offset cut stage t maskBase baseL baseR work B D d : Nat}
  (hfit : @LE.le.{0} Nat instLENat (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) offset V) W)
  (Q : Finset.{0} (Fin V)) (preL preR : Finset.{0} (Fin W)) (f g : Finset.{0} (Fin W) → Int)
  (hw : @LE.le.{0} Int Int.instLEInt (@Nat.cast.{0} Int instNatCastInt lim.space) lim.word)
  (hcut : @LE.le.{0} Nat instLENat cut W) (hstage : @LE.le.{0} Nat instLENat stage W)
  (hdataL :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) baseL
        (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
          (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) W
            (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))
          (ThreeSum.SubsetAddress.downsetCap W cut)))
      maskBase)
  (hdataR :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) baseR
        (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
          (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) W
            (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))
          (ThreeSum.SubsetAddress.downsetCap W cut)))
      maskBase)
  (hmasks :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) maskBase
        (ThreeSum.SubsetAddress.layerOffset (@Finset.card.{0} (Fin V) Q) t))
      work)
  (hspace :
    @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) work
            (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))
          (ThreeSum.LightPascal.tableSize W))
        (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) W
          (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))
      lim.space)
  (hword :
    @LE.le.{0} Int Int.instLEInt
      (@Nat.cast.{0} Int instNatCastInt
        (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
          (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) W
            (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))
          (@HPow.hPow.{0, 0, 0} Nat Nat Nat
            (@instHPow.{0, 0} Nat Nat (@NPow.toPow.{0} Nat (@Monoid.toNPow.{0} Nat Nat.instMonoid)))
            (@OfNat.ofNat.{0} Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))
            (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) W
              (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))))
      lim.word)
  (haccWord :
    @LE.le.{0} Int Int.instLEInt
      (@Nat.cast.{0} Int instNatCastInt
        (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
          (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
            (ThreeSum.SubsetAddress.layerOffset (@Finset.card.{0} (Fin V) Q) t)
            (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))
          (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
            (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
              (@HPow.hPow.{0, 0, 0} Nat Nat Nat
                (@instHPow.{0, 0} Nat Nat (@NPow.toPow.{0} Nat (@Monoid.toNPow.{0} Nat Nat.instMonoid)))
                (@OfNat.ofNat.{0} Nat (nat_lit 2) (instOfNatNat (nat_lit 2))) stage)
              B)
            (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
              (@HPow.hPow.{0, 0, 0} Nat Nat Nat
                (@instHPow.{0, 0} Nat Nat (@NPow.toPow.{0} Nat (@Monoid.toNPow.{0} Nat Nat.instMonoid)))
                (@OfNat.ofNat.{0} Nat (nat_lit 2) (instOfNatNat (nat_lit 2))) stage)
              D))))
      lim.word)
  (ht :
    @LE.le.{0} Nat instLENat t
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) (@Finset.card.{0} (Fin V) Q)
        (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))
  (hcardL :
    @LE.le.{0} Nat instLENat
      (@Finset.card.{0} (Fin W)
        (@Union.union.{0} (Finset.{0} (Fin W)) (@Finset.instUnion.{0} (Fin W) (instDecidableEqFin W)) preL
          (@ThreeSum.ReportingHostSuffix.liftSet W V offset hfit Q)))
      cut)
  (hcardR :
    @LE.le.{0} Nat instLENat
      (@Finset.card.{0} (Fin W)
        (@Union.union.{0} (Finset.{0} (Fin W)) (@Finset.instUnion.{0} (Fin W) (instDecidableEqFin W)) preR
          (@ThreeSum.ReportingHostSuffix.liftSet W V offset hfit Q)))
      cut)
  (hf :
    ∀ (T : Finset.{0} (Fin W)),
      @LE.le.{0} Nat instLENat (@Finset.card.{0} (Fin W) T) cut →
        @LE.le.{0} Int Int.instLEInt (@abs.{0} Int instLatticeInt Int.instAddGroup (f T))
          (@Nat.cast.{0} Int instNatCastInt B))
  (hg :
    ∀ (T : Finset.{0} (Fin W)),
      @LE.le.{0} Nat instLENat (@Finset.card.{0} (Fin W) T) cut →
        @LE.le.{0} Int Int.instLEInt (@abs.{0} Int instLatticeInt Int.instAddGroup (g T))
          (@Nat.cast.{0} Int instNatCastInt D))
  (hstoredL : @ThreeSum.LightZetaStages.StoredThrough W μ baseL cut f W)
  (hstoredR : @ThreeSum.LightZetaStages.StoredThrough W μ baseR cut g W)
  (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 W)
        (@OfNat.ofNat.{0} Nat (nat_lit 8) (instOfNatNat (nat_lit 8))))
      lim.depth),
  Light.Meets lim P pMain d
    (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt W)
      (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt cut)
        (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt stage)
          (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt (@Finset.card.{0} (Fin W) preL))
            (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt (@ThreeSum.SubsetAddress.subsetMask W preL))
              (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt (@Finset.card.{0} (Fin W) preR))
                (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt (@ThreeSum.SubsetAddress.subsetMask W preR))
                  (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt V)
                    (@List.cons.{0} Int
                      (@HPow.hPow.{0, 0, 0} Int Nat Int
                        (@instHPow.{0, 0} Int Nat (@NPow.toPow.{0} Int (@Monoid.toNPow.{0} Int Int.instMonoid)))
                        (@OfNat.ofNat.{0} Int (nat_lit 2) (@instOfNat (nat_lit 2))) offset)
                      (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt (@Finset.card.{0} (Fin V) Q))
                        (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt (@ThreeSum.SubsetAddress.subsetMask V Q))
                          (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt t)
                            (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt baseL)
                              (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt baseR)
                                (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt maskBase)
                                  (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt work)
                                    (@List.nil.{0} Int)))))))))))))))))
    μ (ThreeSum.LightReportingSuffixExplicit.time W (@Finset.card.{0} (Fin V) Q) t) fun (r : Int) (μ' : Nat → Int) =>
    And
      (@Eq.{1} Int r
        (@ThreeSum.ExplicitLayers.valueSum V Q
          (@ThreeSum.ReportingHostSuffix.leafWeight W V offset hfit f g stage preL preR) t))
      (Light.KeptBut μ μ' work maskBase (ThreeSum.SubsetAddress.layerOffset (@Finset.card.{0} (Fin V) Q) t)) := by
  sorry
Source
https://github.com/c0ldzy17/three-sum-lean/blob/2b639655e07421b9ac743de9090459bab473a7c0/Submission/ThreeSum/LightReportingSuffixExplicit.lean#L43-L175

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