Prove2Me

Terminal frontier cost is charged to raw incidence mass

Proved
ThreeSum.ReportingHostTerminalFrontier.frontier_time_le

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

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

Let A,B,CA,B,C be finite subsets of finite index types, with integer value maps a,b,ca,b,c. Arrays L,RL,R enumerate B,CB,C bijectively, preserving their values. Fix h,J,k∈Nh,J,k\in\mathbb N with k≤Jk\le J, a natural modulus parameter pp, and an integer shift ss. Let w=(x,ℓ,q)w=(x,\ell,q) be a wanted label: an AA index, a lift ℓ∈{0,1,2}\ell\in\{0,1,2\}, and a global output qq. Assume x∈Ax\in A and

Mh,J∣a(x) mod p+(ℓ−1)p+s−K(q),M_{h,J}\mid a(x)\bmod p+(\ell-1)p+s-K(q),

where Mh,JM_{h,J} and KK are the global encoding modulus and key. Let II be the raw incidences over A×B×CA\times B\times C, IwI_w its fiber at ww, and FF the positive frontier at depth kk. With C(h,J,k)C(h,J,k) the defined terminal overhead charge,

Tfrontier(F)≤240∣Iw∣+C(h,J,k)∣F∣+30.T_{\mathrm{frontier}}(F)\le240|I_w|+C(h,J,k)|F|+30.

The charge includes binomial-table, mask, preparation, and residue-processing budgets.

This sums terminal work over the retained frontier using the original indexed incidence mass.

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_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_Bundle09
import Definitions.Def_ThreeSum1962Source_Bundle10
import Definitions.Def_ThreeSum1962Source_Bundle11
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.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.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.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.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.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
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_BlockForced_completed_bucket_sum
import Theorems.Thm_ThreeSum_RawPrefixCounts_prefixQuery_eq_bucketPrefixMass
import Theorems.Thm_ThreeSum_ReportingHostFrontier_mem_frontier_iff
import Theorems.Thm_ThreeSum_ReportingHostMasks_blockSet_localRow
import Theorems.Thm_ThreeSum_ReportingHostTerminalChoices_mem_choices
import Theorems.Thm_ThreeSum_Reporting_frontier_nodup
import Theorems.Thm_ThreeSum_StoredReporting_positivePrefix_valid


set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option Elab.async false
namespace Light
end Light
namespace ThreeSum.BlockEncoding
end ThreeSum.BlockEncoding
namespace ThreeSum.BlockForced
end ThreeSum.BlockForced
namespace ThreeSum.BlockIndex
end ThreeSum.BlockIndex
namespace ThreeSum.BlockIndexedQuery
end ThreeSum.BlockIndexedQuery
namespace ThreeSum.BlockMaskLayout
end ThreeSum.BlockMaskLayout
namespace ThreeSum.BlockRows
end ThreeSum.BlockRows
namespace ThreeSum.BlockSourceEncoding
end ThreeSum.BlockSourceEncoding
namespace ThreeSum.GlobalEncoding
end ThreeSum.GlobalEncoding
namespace ThreeSum.LightBlockChildrenIndexed
end ThreeSum.LightBlockChildrenIndexed
namespace ThreeSum.LightBlockSourceBuild
end ThreeSum.LightBlockSourceBuild
namespace ThreeSum.LightHistogram
end ThreeSum.LightHistogram
namespace ThreeSum.LightReportingTerminalMasks
end ThreeSum.LightReportingTerminalMasks
namespace ThreeSum.LightResidues
end ThreeSum.LightResidues
namespace ThreeSum.LightSourceBucketSelect
end ThreeSum.LightSourceBucketSelect
namespace ThreeSum.LightSourceResidues
end ThreeSum.LightSourceResidues
namespace ThreeSum.LocalCoverBounds
end ThreeSum.LocalCoverBounds
namespace ThreeSum.LocalGraph
end ThreeSum.LocalGraph
namespace ThreeSum.LocalHashMoments
end ThreeSum.LocalHashMoments
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.ReportingHostBlockBucketTest
end ThreeSum.ReportingHostBlockBucketTest
namespace ThreeSum.ReportingHostBucketTest
end ThreeSum.ReportingHostBucketTest
namespace ThreeSum.ReportingHostFrontier
end ThreeSum.ReportingHostFrontier
namespace ThreeSum.ReportingHostMasks
end ThreeSum.ReportingHostMasks
namespace ThreeSum.ReportingHostTerminalChoices
end ThreeSum.ReportingHostTerminalChoices
namespace ThreeSum.SourceBuckets
end ThreeSum.SourceBuckets
namespace ThreeSum.StoredReporting
end ThreeSum.StoredReporting
namespace ThreeSum.SubsetAddress
end ThreeSum.SubsetAddress

universe u_1 u_2 u_3
Formal statement
theorem ThreeSum.ReportingHostTerminalFrontier.frontier_time_le : ∀ {α : Type u_1} {β : Type u_2} {γ : Type u_3} [Fintype.{u_1} α] [Fintype.{u_2} β] [Fintype.{u_3} γ]
  [inst : DecidableEq.{u_1 + 1} α] [DecidableEq.{u_2 + 1} β] [DecidableEq.{u_3 + 1} γ] {h J k NL NR : Nat} (p : Nat)
  (shift : Int) (A : Finset.{u_1} α) (B : Finset.{u_2} β) (C : Finset.{u_3} γ) (a : α → Int) (b : β → Int) (c : γ → Int)
  (left right : Nat → Int) (eL : Equiv.{1, u_2 + 1} (Fin NL) ↥B) (eR : Equiv.{1, u_3 + 1} (Fin NR) ↥C)
  (hvL :
    ∀ (i : Fin NL),
      @Eq.{1} Int (left (@Fin.val NL i))
        (b
          (@Subtype.val.{u_2 + 1} β
            (fun (x : β) =>
              @Membership.mem.{u_2, u_2} β (Finset.{u_2} β)
                (@SetLike.instMembership.{u_2, u_2} (Finset.{u_2} β) β (@Finset.instSetLike.{u_2} β)) B x)
            (@DFunLike.coe.{u_2 + 1, 1, u_2 + 1} (Equiv.{1, u_2 + 1} (Fin NL) ↥B) (Fin NL) (fun (x : Fin NL) => ↥B)
              (@EquivLike.toFunLike.{u_2 + 1, 1, u_2 + 1} (Equiv.{1, u_2 + 1} (Fin NL) ↥B) (Fin NL) (↥B)
                (@Equiv.instEquivLike.{1, u_2 + 1} (Fin NL) ↥B))
              eL i))))
  (hvR :
    ∀ (j : Fin NR),
      @Eq.{1} Int (right (@Fin.val NR j))
        (c
          (@Subtype.val.{u_3 + 1} γ
            (fun (x : γ) =>
              @Membership.mem.{u_3, u_3} γ (Finset.{u_3} γ)
                (@SetLike.instMembership.{u_3, u_3} (Finset.{u_3} γ) γ (@Finset.instSetLike.{u_3} γ)) C x)
            (@DFunLike.coe.{u_3 + 1, 1, u_3 + 1} (Equiv.{1, u_3 + 1} (Fin NR) ↥C) (Fin NR) (fun (x : Fin NR) => ↥C)
              (@EquivLike.toFunLike.{u_3 + 1, 1, u_3 + 1} (Equiv.{1, u_3 + 1} (Fin NR) ↥C) (Fin NR) (↥C)
                (@Equiv.instEquivLike.{1, u_3 + 1} (Fin NR) ↥C))
              eR j))))
  (w : ThreeSum.RawReporting.Wanted.{u_1} h J α) (hk : @LE.le.{0} Nat instLENat k J)
  (hkey :
    And
      (@Membership.mem.{u_1, u_1} α (Finset.{u_1} α)
        (@SetLike.instMembership.{u_1, u_1} (Finset.{u_1} α) α (@Finset.instSetLike.{u_1} α)) A
        (@Prod.fst.{u_1, 0} α
          (Prod.{0, 0} (Fin (@OfNat.ofNat.{0} Nat (nat_lit 3) (instOfNatNat (nat_lit 3))))
            (ThreeSum.GlobalEncoding.GlobalOutput h J))
          w))
      (@Dvd.dvd.{0} Int Int.instDvd (ThreeSum.GlobalEncoding.globalModulus h J)
        (@HSub.hSub.{0, 0, 0} Int Int Int (@instHSub.{0} Int Int.instSub)
          (@HAdd.hAdd.{0, 0, 0} Int Int Int (@instHAdd.{0} Int Int.instAdd)
            (@HAdd.hAdd.{0, 0, 0} Int Int Int (@instHAdd.{0} Int Int.instAdd)
              (@HMod.hMod.{0, 0, 0} Int Int Int (@instHMod.{0} Int Int.instMod)
                (a
                  (@Prod.fst.{u_1, 0} α
                    (Prod.{0, 0} (Fin (@OfNat.ofNat.{0} Nat (nat_lit 3) (instOfNatNat (nat_lit 3))))
                      (ThreeSum.GlobalEncoding.GlobalOutput h J))
                    w))
                (@Nat.cast.{0} Int instNatCastInt p))
              (@HMul.hMul.{0, 0, 0} Int Int Int (@instHMul.{0} Int Int.instMul)
                (ThreeSum.RawIncidences.lift
                  (@Prod.fst.{0, 0} (Fin (@OfNat.ofNat.{0} Nat (nat_lit 3) (instOfNatNat (nat_lit 3))))
                    (ThreeSum.GlobalEncoding.GlobalOutput h J)
                    (@Prod.snd.{u_1, 0} α
                      (Prod.{0, 0} (Fin (@OfNat.ofNat.{0} Nat (nat_lit 3) (instOfNatNat (nat_lit 3))))
                        (ThreeSum.GlobalEncoding.GlobalOutput h J))
                      w)))
                (@Nat.cast.{0} Int instNatCastInt p)))
            shift)
          (@ThreeSum.GlobalEncoding.globalKey h J
            (@Prod.snd.{0, 0} (Fin (@OfNat.ofNat.{0} Nat (nat_lit 3) (instOfNatNat (nat_lit 3))))
              (ThreeSum.GlobalEncoding.GlobalOutput h J)
              (@Prod.snd.{u_1, 0} α
                (Prod.{0, 0} (Fin (@OfNat.ofNat.{0} Nat (nat_lit 3) (instOfNatNat (nat_lit 3))))
                  (ThreeSum.GlobalEncoding.GlobalOutput h J))
                w)))))),
  have I :=
    @ThreeSum.RawIncidences.rawOn.{u_1, u_2, u_3} α β γ h J (@Nat.cast.{0} Int instNatCastInt p) shift
      (@SProd.sprod.{u_1, max u_2 u_3, max (max u_1 u_2) u_3} (Finset.{u_1} α)
        (Finset.{max u_3 u_2} (Prod.{u_2, u_3} β γ))
        (Finset.{max (max u_2 u_3) u_1} (Prod.{u_1, max u_2 u_3} α (Prod.{u_2, u_3} β γ)))
        (@Finset.instSProd.{u_1, max u_2 u_3} α (Prod.{u_2, u_3} β γ)) A
        (@SProd.sprod.{u_2, u_3, max u_2 u_3} (Finset.{u_2} β) (Finset.{u_3} γ)
          (Finset.{max u_3 u_2} (Prod.{u_2, u_3} β γ)) (@Finset.instSProd.{u_2, u_3} β γ) B C))
      a b c;
  have pres := @ThreeSum.ReportingHostFrontier.frontier.{u_1, u_2, u_3} α β γ inst h J I w k;
  @LE.le.{0} Nat instLENat
    (@ThreeSum.ReportingHostTerminalFrontier.time h J p shift left right NL NR
      (@Prod.snd.{0, 0} (Fin (@OfNat.ofNat.{0} Nat (nat_lit 3) (instOfNatNat (nat_lit 3))))
        (ThreeSum.GlobalEncoding.GlobalOutput h J)
        (@Prod.snd.{u_1, 0} α
          (Prod.{0, 0} (Fin (@OfNat.ofNat.{0} Nat (nat_lit 3) (instOfNatNat (nat_lit 3))))
            (ThreeSum.GlobalEncoding.GlobalOutput h J))
          w))
      k pres
      (@List.length.{0}
        (List.{0}
          (Finset.{0}
            (Fin
              (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
                (@OfNat.ofNat.{0} Nat (nat_lit 512) (instOfNatNat (nat_lit 512))) h))))
        pres))
    (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
        (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
          (@OfNat.ofNat.{0} Nat (nat_lit 240) (instOfNatNat (nat_lit 240)))
          (@Finset.card.{max (max u_1 u_2) u_3} (ThreeSum.RawIncidences.RawItem.{u_1, u_2, u_3} h J α β γ)
            (@ThreeSum.RawReporting.domain.{u_1, u_2, u_3} α β γ inst h J I w)))
        (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
          (ThreeSum.ReportingHostTerminalFrontier.charge h J k)
          (@List.length.{0}
            (List.{0}
              (Finset.{0}
                (Fin
                  (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
                    (@OfNat.ofNat.{0} Nat (nat_lit 512) (instOfNatNat (nat_lit 512))) h))))
            pres)))
      (@OfNat.ofNat.{0} Nat (nat_lit 30) (instOfNatNat (nat_lit 30)))) := by
  sorry
Source
https://github.com/c0ldzy17/three-sum-lean/blob/2b639655e07421b9ac743de9090459bab473a7c0/Submission/ThreeSum/ReportingHostTerminalFrontier.lean#L101-L123

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