Prove2Me

Wanted traversal time is charged to its actual root mass

Proved
ThreeSum.SharedTraversalCosts.wanted_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 index sets with integer value maps a,b,ca,b,c. Suppose arrays of lengths NL,NRN_L,N_R enumerate B,CB,C through bijections and store their exact values. Fix natural modulus pp, integer shift ss, local parameters h,Jh,J, and depth K≤JK\le J. Let ww be a wanted item whose source index lies in AA and whose lifted source residue agrees with its global output key modulo the global encoding modulus.

Let II be the raw incidence set of A×B×CA\times B\times C, let mm be its root-prefix mass for ww, and let Dk\mathcal D_k be the supplied catalog at level kk. For the old count oo and threshold θ\theta, the concrete wanted routine's time function obeys

T≤I0(h,J,D,K)+{0,θ<o+m,m Q(h,J,D,K),θ≥o+m.T\le I_0(h,J,\mathcal D,K)+\begin{cases}0,&\theta<o+m,\\m\,Q(h,J,\mathcal D,K),&\theta\ge o+m.\end{cases}

Here I0I_0 includes wanted initialization and fixed loop overhead; QQ sums positive-expansion costs and the terminal-frontier charge.

This bounds traversal work by actual incidence multiplicity whenever the dense guard permits searching.

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_Bundle04
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 Definitions.Def_ThreeSum1962Source_Bundle16
import Definitions.Def_ThreeSum1962Source_Bundle17
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.Nat
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.Log.Base
import Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics
import Mathlib.Analysis.SpecialFunctions.Pow.NthRootLemmas
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Analysis.SpecificLimits.Normed
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.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
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_RawReporting_partitionWord_length
import Theorems.Thm_ThreeSum_RawReporting_word_children
import Theorems.Thm_ThreeSum_ReportingHostMasks_blockSet_localRow
import Theorems.Thm_ThreeSum_ReportingHostMasks_localChildren_mem
import Theorems.Thm_ThreeSum_ReportingHostTerminalFrontier_frontier_time_le
import Theorems.Thm_ThreeSum_Reporting_frontier_length_le_mass


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.BlockMaskLayout
end ThreeSum.BlockMaskLayout
namespace ThreeSum.BlockRows
end ThreeSum.BlockRows
namespace ThreeSum.GlobalEncoding
end ThreeSum.GlobalEncoding
namespace ThreeSum.LightBlockQuery
end ThreeSum.LightBlockQuery
namespace ThreeSum.LocalCoverBounds
end ThreeSum.LocalCoverBounds
namespace ThreeSum.RawIncidences
end ThreeSum.RawIncidences
namespace ThreeSum.RawReporting
end ThreeSum.RawReporting
namespace ThreeSum.ReportingHostFrontierLayout
end ThreeSum.ReportingHostFrontierLayout
namespace ThreeSum.ReportingHostLevelData
end ThreeSum.ReportingHostLevelData
namespace ThreeSum.ReportingHostMasks
end ThreeSum.ReportingHostMasks
namespace ThreeSum.StoredReporting
end ThreeSum.StoredReporting
namespace ThreeSum.SubsetAddress
end ThreeSum.SubsetAddress

universe u_1 u_2 u_3
Formal statement
theorem ThreeSum.SharedTraversalCosts.wanted_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 old threshold : 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 :
    ∀ (i : Fin NR),
      @Eq.{1} Int (right (@Fin.val NR i))
        (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 i))))
  (w : ThreeSum.RawReporting.Wanted.{u_1} h J α) (D : Nat → ThreeSum.ReportingHostLevelData.Catalog)
  (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;
  @LE.le.{0} Nat instLENat
    (@ThreeSum.LightReportingWanted.time.{u_1, u_2, u_3} α β γ inst h J p shift left right NL NR old threshold I w D K)
    (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) (ThreeSum.SharedTraversalCosts.initialUnit h J D K)
      (@ite.{1} Nat
        (@LT.lt.{0} Nat instLTNat threshold
          (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) old
            (@ThreeSum.ReportingHostFrontierLayout.rootMass.{u_1, u_2, u_3} α β γ inst h J I w)))
        (threshold.decLt
          (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) old
            (@ThreeSum.ReportingHostFrontierLayout.rootMass.{u_1, u_2, u_3} α β γ inst h J I w)))
        (@OfNat.ofNat.{0} Nat (nat_lit 0) (instOfNatNat (nat_lit 0)))
        (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
          (@ThreeSum.ReportingHostFrontierLayout.rootMass.{u_1, u_2, u_3} α β γ inst h J I w)
          (ThreeSum.SharedTraversalCosts.searchUnit h J D K)))) := by
  sorry
Source
https://github.com/c0ldzy17/three-sum-lean/blob/2b639655e07421b9ac743de9090459bab473a7c0/Submission/ThreeSum/SharedTraversalCosts.lean#L110-L137

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