A packed-record wrapper preserves the reporter contract with fixed overhead

Proved

ThreeSum.LightReportingWantedRecordCall.program_meets

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

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

Let integer memory $\mu$ store a twenty-cell selection record $S$ at $s$ and a fourteen-cell layout record $L$ at $\ell$ . Let $h,J,K,d,T,c,q\in\mathbb N$ be the local half-size, block count, reporting parameter, call depth, time budget, cover address, and counter address. The cover cells $c+6,c+7$ must contain the local prime $p_h$ and bucket cap $b_h$ .

For machine limits $M$ (word magnitude), $A$ (memory), and $D$ (depth), assume $A\le M$ , $d<D$ , every selection entry has magnitude at most $M$ , and every layout entry is at most $A$ . Also require $\max\{s+20,\ell+38+3J,c+8,q+2,J+512h+K+p_h^J+2hJ+p_h+b_h+1024\}\le A$ .

If reporter procedure $137$ , called at depth $d+1$ with the forty-four arguments encoded by these records, meets a postcondition $Q$ within $T$ steps, then wrapper procedure $138$ satisfies

\operatorname{Wrapper}(s,\ell,h,J,K,p_h^J,c,q;\mu) \text{ meets }Q\text{ within }T+4000\text{ steps}.

This connects stored records to the existing reporter without changing its postcondition.

Preamble

/-
Copyright (c) 2026 Anthropic, PBC. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
SPDX-License-Identifier: Apache-2.0
-/

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_Bundle07
import Definitions.Def_ThreeSum1962Source_Bundle13
import Definitions.Def_ThreeSum1962Source_Bundle17
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


set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option Elab.async true
namespace Light
end Light
namespace ThreeSum.BlockEncoding
end ThreeSum.BlockEncoding
namespace ThreeSum.GlobalEncoding
end ThreeSum.GlobalEncoding
namespace ThreeSum.LightBlockQuery
end ThreeSum.LightBlockQuery
namespace ThreeSum.LightSourceResidues
end ThreeSum.LightSourceResidues
namespace ThreeSum.LocalCoverBounds
end ThreeSum.LocalCoverBounds
namespace ThreeSum.ReportingHostWantedControls
end ThreeSum.ReportingHostWantedControls
namespace ThreeSum.ReportingHostWantedLayout
end ThreeSum.ReportingHostWantedLayout

Formal statement

theorem ThreeSum.LightReportingWantedRecordCall.program_meets : ∀ {lim : Light.Limits} {μ : Nat → Int} {S : ThreeSum.ReportingHostWantedControls.Selection}
  {L : ThreeSum.ReportingHostWantedLayout.Layout} {selectionHeader layoutHeader cover counter h J K d T : Nat}
  {Q : Int → (Nat → Int) → Prop} (H : Light.Seg μ selectionHeader (ThreeSum.ReportingHostWantedControls.record S))
  (HL : Light.SegN μ layoutHeader (ThreeSum.ReportingHostWantedLayout.values L))
  (hprime :
    @Eq.{1} Int
      (μ
        (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) cover
          (@OfNat.ofNat.{0} Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))))
      (@Nat.cast.{0} Int instNatCastInt (ThreeSum.LocalCoverBounds.localPrime h)))
  (hcap :
    @Eq.{1} Int
      (μ
        (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) cover
          (@OfNat.ofNat.{0} Nat (nat_lit 7) (instOfNatNat (nat_lit 7)))))
      (@Nat.cast.{0} Int instNatCastInt (ThreeSum.LocalCoverBounds.localBucketCap h)))
  (hw : @LE.le.{0} Int Int.instLEInt (@Nat.cast.{0} Int instNatCastInt lim.space) lim.word)
  (hd : @LT.lt.{0} Nat instLTNat d lim.depth)
  (hs :
    @LE.le.{0} Nat instLENat
      (ThreeSum.LightReportingWantedRecordCall.space selectionHeader layoutHeader cover counter h J K) lim.space)
  (hselected :
    ∀ (x : Int),
      @Membership.mem.{0, 0} Int (List.{0} Int) (@List.instMembership.{0} Int)
          (ThreeSum.ReportingHostWantedControls.record S) x →
        @LE.le.{0} Int Int.instLEInt (@abs.{0} Int instLatticeInt Int.instAddGroup x) lim.word)
  (hlayout :
    ∀ (x : Nat),
      @Membership.mem.{0, 0} Nat (List.{0} Nat) (@List.instMembership.{0} Nat)
          (ThreeSum.ReportingHostWantedLayout.values L) x →
        @LE.le.{0} Nat instLENat x lim.space)
  (hreport :
    Light.Meets lim ThreeSum.LightReportingWanted.program
      (@OfNat.ofNat.{0} Nat (nat_lit 137) (instOfNatNat (nat_lit 137)))
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) d
        (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))
      (ThreeSum.ReportingHostWantedControls.arguments S L layoutHeader cover counter h J K) μ T Q),
  Light.Meets lim ThreeSum.LightReportingWantedRecordCall.program
    (@OfNat.ofNat.{0} Nat (nat_lit 138) (instOfNatNat (nat_lit 138))) d
    (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt selectionHeader)
      (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt layoutHeader)
        (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt h)
          (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt J)
            (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt K)
              (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt (ThreeSum.BlockSourceEncoding.modulus h J))
                (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt cover)
                  (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt counter) (@List.nil.{0} Int)))))))))
    μ
    (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) T
      (@OfNat.ofNat.{0} Nat (nat_lit 4000) (instOfNatNat (nat_lit 4000))))
    Q := by
  sorry

Source

https://github.com/c0ldzy17/three-sum-lean/blob/2b639655e07421b9ac743de9090459bab473a7c0/Submission/ThreeSum/LightReportingWantedRecordCall.lean#L75-L354

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