Retain every suffix stage of the actual left block source

Proved

ThreeSum.LightBlockSourceRetainedStages.meets_left

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

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

Let $B$ be a finite set of source indices with values $b_x$ , and let a length- $N$ physical array enumerate $B$ bijectively with matching values. Fix $h,J$ , a selected global output $q$ , any natural modulus parameter $p$ , $W=512hJ$ , and cutoff $c\le W$ . Set $m=\operatorname{modulus}(h,J)>0$ . Assume the retained arrays contain the actual left endpoints and independent allowed flags for $q$ , and the histogram counts the physical keys $((b_x\bmod p)\bmod m)$ . Under the prescribed destination, mask and scratch layout, retained-source space and word budgets using endpoint bound $E=\operatorname{endpointBound}(h,J)$ and histogram bound $N$ , space fitting in a word, and depth $d+W+8$ , the installed routine retains all suffix stages of

F_L(S)=\mathbf1_{\operatorname{LeftAllowed}(q,S)}\#\{x\in B:((b_x\bmod p)\bmod m)=\operatorname{endpointLeft}(q,S)\bmod m\}.

Coordinate sets are interpreted through the block-index bijection. Memory below the destination is preserved, with the retained-stage construction budget.

This identifies executable source tables with the finite left-source semantics.

Preamble

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_Bundle05
import Definitions.Def_ThreeSum1962Source_Bundle06
import Definitions.Def_ThreeSum1962Source_Bundle07
import Definitions.Def_ThreeSum1962Source_Bundle09
import Definitions.Def_ThreeSum1962Source_Bundle10
import Definitions.Def_ThreeSum1962Source_Bundle11
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.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.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.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.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.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_ThreeSum_BlockIndex_maskHash_left
import Theorems.Thm_ThreeSum_LightSourceRetainedStages_Procs_meets
import Theorems.Thm_ThreeSum_OneHotMask_eval_le


set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option Elab.async false
namespace Light
end Light
namespace ThreeSum.BlockCoefficientRows
end ThreeSum.BlockCoefficientRows
namespace ThreeSum.BlockEncoding
end ThreeSum.BlockEncoding
namespace ThreeSum.BlockIndex
end ThreeSum.BlockIndex
namespace ThreeSum.BlockRows
end ThreeSum.BlockRows
namespace ThreeSum.BlockSourceEncoding
end ThreeSum.BlockSourceEncoding
namespace ThreeSum.GlobalEncoding
end ThreeSum.GlobalEncoding
namespace ThreeSum.LightBlockSourceBuild
end ThreeSum.LightBlockSourceBuild
namespace ThreeSum.LightSourceResidues
end ThreeSum.LightSourceResidues
namespace ThreeSum.LocalCoverBounds
end ThreeSum.LocalCoverBounds
namespace ThreeSum.LocalHashMoments
end ThreeSum.LocalHashMoments
namespace ThreeSum.SubsetAddress
end ThreeSum.SubsetAddress

universe u_1

Formal statement

theorem ThreeSum.LightBlockSourceRetainedStages.meets_left : ∀ {lim : Light.Limits} {P : Light.Program} {d : Nat} {β : Type u_1} [Fintype.{u_1} β] [DecidableEq.{u_1 + 1} β]
  {pMain : Nat} (C : ThreeSum.LightSourceRetainedStages.Procs P pMain) {μ : Nat → Int}
  {h J N raw flags p table base work cut maskBase : Nat} (B : Finset.{u_1} β) (b : β → Int)
  (q : ThreeSum.GlobalEncoding.GlobalOutput h J) (v : Nat → Int) (e : Equiv.{1, u_1 + 1} (Fin N) ↥B)
  (hw : @LE.le.{0} Int Int.instLEInt (@Nat.cast.{0} Int instNatCastInt lim.space) 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) d
          (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat) J
            (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
              (@OfNat.ofNat.{0} Nat (nat_lit 512) (instOfNatNat (nat_lit 512))) h)))
        (@OfNat.ofNat.{0} Nat (nat_lit 8) (instOfNatNat (nat_lit 8))))
      lim.depth)
  (hcut :
    @LE.le.{0} Nat instLENat cut
      (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat) J
        (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
          (@OfNat.ofNat.{0} Nat (nat_lit 512) (instOfNatNat (nat_lit 512))) h)))
  (hraw :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) raw
        (ThreeSum.SubsetAddress.downsetCap
          (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat) J
            (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
              (@OfNat.ofNat.{0} Nat (nat_lit 512) (instOfNatNat (nat_lit 512))) h))
          cut))
      base)
  (hflags :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) flags
        (ThreeSum.SubsetAddress.downsetCap
          (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat) J
            (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
              (@OfNat.ofNat.{0} Nat (nat_lit 512) (instOfNatNat (nat_lit 512))) h))
          cut))
      base)
  (htable :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) table (ThreeSum.BlockSourceEncoding.modulus h J))
      base)
  (hdata :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) base
        (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
          (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
            (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat) J
              (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
                (@OfNat.ofNat.{0} Nat (nat_lit 512) (instOfNatNat (nat_lit 512))) h))
            (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))
          (ThreeSum.SubsetAddress.downsetCap
            (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat) J
              (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
                (@OfNat.ofNat.{0} Nat (nat_lit 512) (instOfNatNat (nat_lit 512))) h))
            cut)))
      maskBase)
  (hmasks :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) maskBase
        (ThreeSum.SubsetAddress.downsetCap
          (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat) J
            (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
              (@OfNat.ofNat.{0} Nat (nat_lit 512) (instOfNatNat (nat_lit 512))) h))
          cut))
      work)
  (hspace :
    @LE.le.{0} Nat instLENat
      (ThreeSum.LightSourceRetainedStages.space work
        (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat) J
          (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
            (@OfNat.ofNat.{0} Nat (nat_lit 512) (instOfNatNat (nat_lit 512))) h))
        (ThreeSum.LightBlockSourceBuild.endpointBound h J))
      lim.space)
  (hword :
    @LE.le.{0} Int Int.instLEInt
      (@Nat.cast.{0} Int instNatCastInt
        (ThreeSum.LightSourceRetainedStages.word
          (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat) J
            (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
              (@OfNat.ofNat.{0} Nat (nat_lit 512) (instOfNatNat (nat_lit 512))) h))
          (ThreeSum.BlockSourceEncoding.modulus h J) (ThreeSum.LightBlockSourceBuild.endpointBound h J) N))
      lim.word)
  (hinput :
    ∀ (i : Fin N),
      @Eq.{1} Int (v (@Fin.val N i))
        (b
          (@Subtype.val.{u_1 + 1} β
            (fun (x : β) =>
              @Membership.mem.{u_1, u_1} β (Finset.{u_1} β)
                (@SetLike.instMembership.{u_1, u_1} (Finset.{u_1} β) β (@Finset.instSetLike.{u_1} β)) B x)
            (@DFunLike.coe.{u_1 + 1, 1, u_1 + 1} (Equiv.{1, u_1 + 1} (Fin N) ↥B) (Fin N) (fun (x : Fin N) => ↥B)
              (@EquivLike.toFunLike.{u_1 + 1, 1, u_1 + 1} (Equiv.{1, u_1 + 1} (Fin N) ↥B) (Fin N) (↥B)
                (@Equiv.instEquivLike.{1, u_1 + 1} (Fin N) ↥B))
              e i))))
  (hhistRead :
    ∀ (r : Nat),
      @LT.lt.{0} Nat instLTNat r (ThreeSum.BlockSourceEncoding.modulus h J) →
        @Eq.{1} Int (μ (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) table r))
          (@Nat.cast.{0} Int instNatCastInt
            (ThreeSum.LightBlockSourceRetainedStages.histogram p (ThreeSum.BlockSourceEncoding.modulus h J)
              (@OfNat.ofNat.{0} Int (nat_lit 0) (@instOfNat (nat_lit 0))) v N r)))
  (hrawRead :
    ∀ (i : Nat),
      @LT.lt.{0} Nat instLTNat i
          (ThreeSum.SubsetAddress.downsetCap
            (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat) J
              (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
                (@OfNat.ofNat.{0} Nat (nat_lit 512) (instOfNatNat (nat_lit 512))) h))
            cut) →
        @Eq.{1} Int (μ (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) raw i))
          (@ThreeSum.BlockEncoding.endpointLeft h J (@ThreeSum.BlockEncoding.outputOccurrences h J q)
            (ThreeSum.BlockSourceTableLayout.sets h J cut i)))
  (hflagRead :
    ∀ (i : Nat),
      @LT.lt.{0} Nat instLTNat i
          (ThreeSum.SubsetAddress.downsetCap
            (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat) J
              (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
                (@OfNat.ofNat.{0} Nat (nat_lit 512) (instOfNatNat (nat_lit 512))) h))
            cut) →
        @Eq.{1} Int (μ (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) flags i))
          (@Nat.cast.{0} Int instNatCastInt
            (@ite.{1} Nat
              (@ThreeSum.BlockEncoding.LeftAllowed h J (@ThreeSum.BlockEncoding.outputOccurrences h J q)
                (ThreeSum.BlockSourceTableLayout.sets h J cut i))
              (@ThreeSum.BlockEncoding.instDecidableLeftAllowed h J (@ThreeSum.BlockEncoding.outputOccurrences h J q)
                (ThreeSum.BlockSourceTableLayout.sets h J cut i))
              (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))
              (@OfNat.ofNat.{0} Nat (nat_lit 0) (instOfNatNat (nat_lit 0)))))),
  Light.Meets lim P pMain d
    (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt raw)
      (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt flags)
        (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt table)
          (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt base)
            (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt (ThreeSum.BlockSourceEncoding.modulus h J))
              (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt work)
                (@List.cons.{0} Int
                  (@HMul.hMul.{0, 0, 0} Int Int Int (@instHMul.{0} Int Int.instMul) (@Nat.cast.{0} Int instNatCastInt J)
                    (@HMul.hMul.{0, 0, 0} Int Int Int (@instHMul.{0} Int Int.instMul)
                      (@OfNat.ofNat.{0} Int (nat_lit 512) (@instOfNat (nat_lit 512)))
                      (@Nat.cast.{0} Int instNatCastInt h)))
                  (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt cut)
                    (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt maskBase)
                      (@List.cons.{0} Int
                        (@Nat.cast.{0} Int instNatCastInt
                          (ThreeSum.SubsetAddress.downsetCap
                            (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat) J
                              (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
                                (@OfNat.ofNat.{0} Nat (nat_lit 512) (instOfNatNat (nat_lit 512))) h))
                            cut))
                        (@List.nil.{0} Int)))))))))))
    μ
    (ThreeSum.LightSourceRetainedStages.time
      (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat) J
        (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
          (@OfNat.ofNat.{0} Nat (nat_lit 512) (instOfNatNat (nat_lit 512))) h))
      cut (ThreeSum.LightBlockSourceBuild.endpointBound h J))
    fun (x : Int) (ν : Nat → Int) =>
    And
      (@ThreeSum.LightZetaStages.StoredThrough
        (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat) J
          (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
            (@OfNat.ofNat.{0} Nat (nat_lit 512) (instOfNatNat (nat_lit 512))) h))
        ν base cut (@ThreeSum.BlockIndexedQuery.leftSource.{u_1} β h J (@Nat.cast.{0} Int instNatCastInt p) B b q)
        (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat) J
          (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
            (@OfNat.ofNat.{0} Nat (nat_lit 512) (instOfNatNat (nat_lit 512))) h)))
      (Light.Kept μ ν base) := by
  sorry

Source

https://github.com/c0ldzy17/three-sum-lean/blob/2b639655e07421b9ac743de9090459bab473a7c0/Submission/ThreeSum/LightBlockSourceRetainedStages.lean#L53-L88

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