Prove2Me

Building a pointer catalog of signed suffix tables

Proved
ThreeSum.LightReportingSuffixCatalog.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 bounded integer set functions f,gf,g on WW coordinates have retained source-zeta tables through cutoff c≤Wc\le W. Fix stage s≤Ws\le W, offset oo and width VV with o+V≤Wo+V\le W, box cutoff b≤Vb\le V, and nn stored prefix pairs (Ai,Bi)(A_i,B_i) with constant cardinalities cL,cRc_L,c_R. Assume cL+b,cR+b≤cc_L+b,c_R+b\le c. For ι(T)=o+T\iota(T)=o+T, define

Fi(T)=(−1)∣T∣Zsf(Ai∪ι(T))Zsg(Bi∪ι(T)).F_i(T)=(-1)^{|T|}Z_sf(A_i\cup\iota(T))Z_sg(B_i\cup\iota(T)).

Let K=∑j≤b(Vj)K=\sum_{j\le b}\binom Vj and σ=(V+1)K\sigma=(V+1)K. Under the contract's ordered source, input, pointer, box, and scratch regions; space and word bounds, including 2V(2sB0)(2sD0)2^V(2^sB_0)(2^sD_0) for source bounds B0,D0B_0,D_0; and d+W+12≤Δd+W+12\le\Delta, the routine stores

ν(p+i)=q+iσ(i<n)\nu(p+i)=q+i\sigma\quad(i<n)

and every forward zeta stage of FiF_i in the indicated family. It retains both source families and preserves all cells below pp. Its time is n(Tsetup+200)+30n(T_{\rm setup}+200)+30.

This constructs a reusable suffix-table catalog, including the empty-catalog case.

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_Bundle04
import Definitions.Def_ThreeSum1962Source_Bundle10
import Definitions.Def_ThreeSum1962Source_Bundle12
import Definitions.Def_ThreeSum1962Source_Bundle13
import Definitions.Def_ThreeSum1962Source_Bundle14
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_LightReportingSuffixSetup_program_meets
import Theorems.Thm_ThreeSum_SubsetAddress_layerRank_lt


set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option Elab.async false
namespace Light
end Light
namespace ThreeSum.LightPascal
end ThreeSum.LightPascal
namespace ThreeSum.LightSubsetAddress
end ThreeSum.LightSubsetAddress
namespace ThreeSum.LightZetaCell
end ThreeSum.LightZetaCell
namespace ThreeSum.MaskDecode
end ThreeSum.MaskDecode
namespace ThreeSum.ReportingHostQueryBounds
end ThreeSum.ReportingHostQueryBounds
namespace ThreeSum.ReportingHostSuffix
end ThreeSum.ReportingHostSuffix
namespace ThreeSum.SubsetAddress
end ThreeSum.SubsetAddress
Formal statement
theorem ThreeSum.LightReportingSuffixCatalog.program_meets : ∀ {lim : Light.Limits} {μ : Nat → Int}
  {W V offset cut boxCut stage cL cR baseL baseR inL inR n ptrBase boxBase maskBase 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)
  (preL preR : Nat → 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) (hboxCut : @LE.le.{0} Nat instLENat boxCut V)
  (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)))
      ptrBase)
  (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)))
      ptrBase)
  (hinL : @LE.le.{0} Nat instLENat (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) inL n) ptrBase)
  (hinR : @LE.le.{0} Nat instLENat (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) inR n) ptrBase)
  (hptr : @LE.le.{0} Nat instLENat (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) ptrBase n) boxBase)
  (hbox :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) boxBase
        (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat) n
          (ThreeSum.LightReportingSuffixCatalog.stride V boxCut)))
      maskBase)
  (hmasks :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) maskBase
        (ThreeSum.SubsetAddress.downsetCap V boxCut))
      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)
  (hvalueWord :
    @LE.le.{0} Int Int.instLEInt
      (@Nat.cast.{0} Int instNatCastInt
        (@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))) V)
          (@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)
  (hcardL : ∀ (i : Nat), @LT.lt.{0} Nat instLTNat i n → @Eq.{1} Nat (@Finset.card.{0} (Fin W) (preL i)) cL)
  (hcardR : ∀ (i : Nat), @LT.lt.{0} Nat instLTNat i n → @Eq.{1} Nat (@Finset.card.{0} (Fin W) (preR i)) cR)
  (hpreL : @LE.le.{0} Nat instLENat (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) cL boxCut) cut)
  (hpreR : @LE.le.{0} Nat instLENat (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) cR boxCut) cut)
  (hf :
    ∀ (S : Finset.{0} (Fin W)),
      @LE.le.{0} Nat instLENat (@Finset.card.{0} (Fin W) S) cut →
        @LE.le.{0} Int Int.instLEInt (@abs.{0} Int instLatticeInt Int.instAddGroup (f S))
          (@Nat.cast.{0} Int instNatCastInt B))
  (hg :
    ∀ (S : Finset.{0} (Fin W)),
      @LE.le.{0} Nat instLENat (@Finset.card.{0} (Fin W) S) cut →
        @LE.le.{0} Int Int.instLEInt (@abs.{0} Int instLatticeInt Int.instAddGroup (g S))
          (@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)
  (hreadL :
    ∀ (i : Nat),
      @LT.lt.{0} Nat instLTNat i n →
        @Eq.{1} Int (μ (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) inL i))
          (@Nat.cast.{0} Int instNatCastInt (@ThreeSum.SubsetAddress.subsetMask W (preL i))))
  (hreadR :
    ∀ (i : Nat),
      @LT.lt.{0} Nat instLTNat i n →
        @Eq.{1} Int (μ (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) inR i))
          (@Nat.cast.{0} Int instNatCastInt (@ThreeSum.SubsetAddress.subsetMask W (preR i))))
  (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 12) (instOfNatNat (nat_lit 12))))
      lim.depth),
  Light.Meets lim ThreeSum.LightReportingSuffixCatalog.program
    (@OfNat.ofNat.{0} Nat (nat_lit 30) (instOfNatNat (nat_lit 30))) 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 boxCut)
          (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt stage)
            (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt cL)
              (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt cR)
                (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt V)
                  (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt offset)
                    (@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 inL)
                          (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt inR)
                            (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt n)
                              (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt ptrBase)
                                (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt boxBase)
                                  (@List.cons.{0} Int
                                    (@Nat.cast.{0} Int instNatCastInt
                                      (ThreeSum.LightReportingSuffixCatalog.stride V boxCut))
                                    (@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.LightReportingSuffixCatalog.time W V boxCut offset n) fun (x : Int) (μ' : Nat → Int) =>
    And
      (Light.SegN μ' ptrBase
        (@List.ofFn.{0} Nat n fun (i : Fin n) =>
          @HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) boxBase
            (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat) (@Fin.val n i)
              (ThreeSum.LightReportingSuffixCatalog.stride V boxCut))))
      (And
        (∀ (i : Nat),
          @LT.lt.{0} Nat instLTNat i n →
            @ThreeSum.LightBoxStages.StoredThrough V μ'
              (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) boxBase
                (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat) i
                  (ThreeSum.LightReportingSuffixCatalog.stride V boxCut)))
              boxCut (@ThreeSum.ReportingHostSuffix.leafWeight W V offset hfit f g stage (preL i) (preR i)) V)
        (And (@ThreeSum.LightZetaStages.StoredThrough W μ' baseL cut f W)
          (And (@ThreeSum.LightZetaStages.StoredThrough W μ' baseR cut g W) (Light.Kept μ μ' ptrBase)))) := by
  sorry
Source
https://github.com/c0ldzy17/three-sum-lean/blob/2b639655e07421b9ac743de9090459bab473a7c0/Submission/ThreeSum/LightReportingSuffixCatalog.lean#L49-L166

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