Appending one catalog row preserves the completed shared-bank prefix

Proved

ThreeSum.SharedReportingBankRows.Done.append

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

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

Fix natural layout parameters $h,J,K,N_B,N_C,g$ , metadata base $r$ , payload base $b$ , and integer source functions $f_B,f_C$ with residue parameters $p,z\in\mathbb Z$ . Let $S$ be the occurrence count, and $n_B,n_C$ the numbers of bands. Rows are ranked by $q=(s n_B+i)n_C+j$ ; row $q$ has metadata base $r_q=r+5Kq$ and payload base $b_q=b+q\Delta$ , where $\Delta$ is the prescribed $K$ -level payload span.

Assume all valid rows of rank below $q$ are built in $\mu$ , and $s<S$ , $i<n_B$ , $j<n_C$ . Suppose row $q$ is built in $\nu$ for these sources and occurrence, $r+5KSn_Bn_C\le b$ , and $\nu$ agrees with $\mu$ below $b_q$ outside $[r_q,r_q+5K)$ . Then

\operatorname{Done}_{\mu}(q)\ \Longrightarrow\ \operatorname{Done}_{\nu}(q+1).

Here a built row contains the correct catalog headers and tables for every level below $K$ .

This advances the shared-bank construction invariant in physical row order.

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_Bundle08
import Definitions.Def_ThreeSum1962Source_Bundle10
import Definitions.Def_ThreeSum1962Source_Bundle11
import Definitions.Def_ThreeSum1962Source_Bundle14
import Definitions.Def_ThreeSum1962Source_Bundle15
import Definitions.Def_ThreeSum1962Source_Bundle16
import Definitions.Def_TrulySubcubicAPSP_SourceSpecification
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.MvPolynomial.Basic
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.BigOperators.GroupWithZero.List
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.Log.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics
import Mathlib.Analysis.SpecialFunctions.Pow.NthRootLemmas
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.Combinatorics.Colex
import Mathlib.Combinatorics.SimpleGraph.Basic
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.Fintype.Sets
import Mathlib.Data.Fintype.Sum
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.FinRange
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.Notation
import Mathlib.LinearAlgebra.Matrix.ToLin
import Mathlib.Logic.Equiv.Fin.Basic
import Mathlib.Logic.Function.Basic
import Mathlib.MeasureTheory.Integral.Bochner.Basic
import Mathlib.NumberTheory.Bertrand
import Mathlib.NumberTheory.Chebyshev
import Mathlib.NumberTheory.PrimeCounting
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.Monotone.Basic
import Mathlib.Probability.Independence.Basic
import Mathlib.RingTheory.Coprime.Lemmas
import Mathlib.Tactic
import Mathlib.Tactic.Abel
import Mathlib.Tactic.ByContra
import Mathlib.Tactic.DeriveFintype
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.FinCases
import Mathlib.Tactic.GCongr
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.LinearCombination
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_SharedReportingBankData_Tables_of_window


set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option Elab.async false
namespace Light
end Light
namespace ThreeSum.BlockIndexedQuery
end ThreeSum.BlockIndexedQuery
namespace ThreeSum.BlockSourceOccurrenceOrder
end ThreeSum.BlockSourceOccurrenceOrder
namespace ThreeSum.GlobalEncoding
end ThreeSum.GlobalEncoding
namespace ThreeSum.LightPascal
end ThreeSum.LightPascal
namespace ThreeSum.LightReportingPrefixCatalog
end ThreeSum.LightReportingPrefixCatalog
namespace ThreeSum.LightReportingSuffixCatalog
end ThreeSum.LightReportingSuffixCatalog
namespace ThreeSum.ReportingHostLevelData
end ThreeSum.ReportingHostLevelData
namespace ThreeSum.SharedReportingBankData
end ThreeSum.SharedReportingBankData
namespace ThreeSum.SharedReportingBankLayout
end ThreeSum.SharedReportingBankLayout
namespace ThreeSum.SharedReportingBankLevels
end ThreeSum.SharedReportingBankLevels
namespace ThreeSum.SubsetAddress
end ThreeSum.SubsetAddress

Formal statement

theorem ThreeSum.SharedReportingBankRows.Done.append : ∀ {μ ν : Nat → Int} {record base h J K NB NC g s i j q : Nat} {fB fC : Nat → Int} {p shift : Int}
  (H : ThreeSum.SharedReportingBankRows.Done μ record base h J K NB NC g q fB fC p shift)
  (hs : @LT.lt.{0} Nat instLTNat s (ThreeSum.LightBlockSourceCatalog.count h J))
  (hi : @LT.lt.{0} Nat instLTNat i (ThreeSum.BandPartition.size NB g))
  (hj : @LT.lt.{0} Nat instLTNat j (ThreeSum.BandPartition.size NC g))
  (hq :
    @Eq.{1} Nat q
      (ThreeSum.SharedReportingBankRows.rank (ThreeSum.BandPartition.size NB g) (ThreeSum.BandPartition.size NC g) s i
        j))
  (Hnew :
    ThreeSum.SharedReportingBankLevels.Built ν (ThreeSum.SharedReportingBankRows.rowRecord record K q)
      (ThreeSum.SharedReportingBankRows.rowBase base h J K q) h J K
      (@ThreeSum.BlockIndexedQuery.leftSource.{0} (Fin NB) h J p (ThreeSum.SourceBandPartition.source NB g i)
        (fun (t : Fin NB) => fB (@Fin.val NB t)) (ThreeSum.BlockSourceOccurrenceOrder.representative h J s))
      (@ThreeSum.BlockIndexedQuery.rightSource.{0} (Fin NC) h J p shift (ThreeSum.SourceBandPartition.source NC g j)
        (fun (t : Fin NC) => fC (@Fin.val NC t)) (ThreeSum.BlockSourceOccurrenceOrder.representative h J s)))
  (heq :
    Light.KeptBut μ ν (ThreeSum.SharedReportingBankRows.rowBase base h J K q)
      (ThreeSum.SharedReportingBankRows.rowRecord record K q)
      (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
        (@OfNat.ofNat.{0} Nat (nat_lit 5) (instOfNatNat (nat_lit 5))) K))
  (hslab :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) record
        (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
          (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
            (@OfNat.ofNat.{0} Nat (nat_lit 5) (instOfNatNat (nat_lit 5))) K)
          (ThreeSum.SharedReportingBankRows.total h J NB NC g)))
      base),
  ThreeSum.SharedReportingBankRows.Done ν record base h J K NB NC g
    (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) q
      (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))
    fB fC p shift := by
  sorry

Source

https://github.com/c0ldzy17/three-sum-lean/blob/2b639655e07421b9ac743de9090459bab473a7c0/Submission/ThreeSum/SharedReportingBankRows.lean#L130-L158

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