Agreement on one memory window preserves a shared catalog

Proved

ThreeSum.SharedReportingBankData.Tables.of_window

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

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

Let $h,J,k\in\mathbb N$ with $k\le J$ , let $D$ describe a shared reporting catalog, and let $f,g$ be integer-valued functions on subsets of the $512hJ$ global coordinates. Suppose $\mu$ stores the catalog tables for $f,g$ . Put $N=\operatorname{count}(h,k)$ , $V=(J-k)512h$ , and $s=(V+1)\sum_{r=0}^{\min(D.\mathrm{cut},V)}\binom Vr$ , the suffix-box stride. Assume $\nu=\mu$ on the address window $[\ell,b)$ and

\ell\le D.\mathrm{index},D.\mathrm{ptr},D.\mathrm{box},

D.\mathrm{index}+N\le b,\quad D.\mathrm{ptr}+N\le b,\quad D.\mathrm{box}+Ns\le b.

Then the same table specification holds in $\nu$ . This window form permits changes on either side of the catalog while retaining its contents and 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_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 Definitions.Def_ThreeSum1962Source_Bundle15
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.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.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.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.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_SubsetAddress_layerRank_lt


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.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.LightReportingPrefixCatalog
end ThreeSum.LightReportingPrefixCatalog
namespace ThreeSum.LightReportingSuffixCatalog
end ThreeSum.LightReportingSuffixCatalog
namespace ThreeSum.LocalCoverBounds
end ThreeSum.LocalCoverBounds
namespace ThreeSum.RawIncidences
end ThreeSum.RawIncidences
namespace ThreeSum.RawPrefixCounts
end ThreeSum.RawPrefixCounts
namespace ThreeSum.RawReporting
end ThreeSum.RawReporting
namespace ThreeSum.ReportingHostLevelData
end ThreeSum.ReportingHostLevelData
namespace ThreeSum.ReportingHostPrefixCatalog
end ThreeSum.ReportingHostPrefixCatalog
namespace ThreeSum.ReportingHostPrefixOrder
end ThreeSum.ReportingHostPrefixOrder
namespace ThreeSum.ReportingHostSuffix
end ThreeSum.ReportingHostSuffix
namespace ThreeSum.StoredReporting
end ThreeSum.StoredReporting
namespace ThreeSum.SubsetAddress
end ThreeSum.SubsetAddress

Formal statement

theorem ThreeSum.SharedReportingBankData.Tables.of_window : ∀ {h J k low bound : Nat} {μ ν : Nat → Int} {D : ThreeSum.ReportingHostLevelData.Catalog}
  {hk : @LE.le.{0} Nat instLENat k J}
  {f g :
    Finset.{0}
        (Fin
          (@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))) →
      Int}
  (H : @ThreeSum.SharedReportingBankData.Tables h J k μ D hk f g)
  (heq : ∀ (a : Nat), @LE.le.{0} Nat instLENat low a → @LT.lt.{0} Nat instLTNat a bound → @Eq.{1} Int (ν a) (μ a))
  (hindexLow : @LE.le.{0} Nat instLENat low D.indexBase) (hptrLow : @LE.le.{0} Nat instLENat low D.ptrBase)
  (hboxLow : @LE.le.{0} Nat instLENat low D.boxBase)
  (hindex :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) D.indexBase
        (ThreeSum.LightReportingPrefixCatalog.count h k))
      bound)
  (hptr :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) D.ptrBase
        (ThreeSum.LightReportingPrefixCatalog.count h k))
      bound)
  (hbox :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) D.boxBase
        (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
          (ThreeSum.LightReportingPrefixCatalog.count h k)
          (ThreeSum.LightReportingSuffixCatalog.stride
            (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
              (@HSub.hSub.{0, 0, 0} Nat Nat Nat (@instHSub.{0} Nat instSubNat) J k)
              (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
                (@OfNat.ofNat.{0} Nat (nat_lit 512) (instOfNatNat (nat_lit 512))) h))
            D.boxCut)))
      bound),
  @ThreeSum.SharedReportingBankData.Tables h J k ν D hk f g := by
  sorry

Source

https://github.com/c0ldzy17/three-sum-lean/blob/2b639655e07421b9ac743de9090459bab473a7c0/Submission/ThreeSum/SharedReportingBankMemory.lean#L10-L31

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