Prove2Me

Dispatch one retained wanted slot with an exact mass update

Proved
ThreeSum.LightReportingWantedDispatchStored.program_meets

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

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

Suppose the encoded color payload and shared reporting catalog for bounded input ∣xi∣≤U|x_i|\le U are retained. A stored wanted slot contains valid raw/output indices matching a label w=(i,ℓ,q)w=(i,\ell,q), where ii names a physical AA incidence, ℓ∈{−1,0,1}\ell\in\{-1,0,1\} is its arithmetic lift, and qq its decoded global output. Require the key congruence M∣(ai mod p)+ℓp+σ−globalKey⁡(q)M\mid(a_i\bmod p)+\ell p+\sigma-\operatorname{globalKey}(q), with selected p>0p>0 and 4p≤M4p\le M. Let mwm_w be the label's root reporting mass and HwH_w its terminal original-value hit predicate. The retained counter, threshold, and representation cap are correct, with o,mw≤3E7o,m_w\le3E^7 and θ≤9E8\theta\le9E^8, where E=22048(inputBits⁡(n)+1)E=2^{2048(\operatorname{inputBits}(n)+1)}. Under the stated wanted-resource need at the layout header and depth d+1d+1, the dispatcher returns

r∈{0,1},r=1  ⟺  θ<o+mw ∨ Hw,μ′(counter⁡)=o+mw.r\in\{0,1\},\qquad r=1\iff\theta<o+m_w\ \lor\ H_w,\qquad\mu'(\operatorname{counter})=o+m_w.

It meets the decode-and-record time budget; below the layout header only the counter cell may change.

This connects a physical wanted slot to the cumulative reporting rule.

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_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 Definitions.Def_ThreeSum1962Source_Bundle15
import Definitions.Def_ThreeSum1962Source_Bundle16
import Definitions.Def_ThreeSum1962Source_Bundle17
import Definitions.Def_ThreeSum1962Source_Bundle18
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.Image
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.Basic
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.Dedup
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.Bitwise
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.Choose.Vandermonde
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_BandCandidateRows_fromOwner_sound
import Theorems.Thm_ThreeSum_BandCandidateRows_mem_records_owner
import Theorems.Thm_ThreeSum_BandRawA_mem_row
import Theorems.Thm_ThreeSum_EncodedColorPipelineData_Stored_of_kept
import Theorems.Thm_ThreeSum_EncodedColorWantedConsumerData_incidence_length_le_magnitude
import Theorems.Thm_ThreeSum_EncodedColorWantedConsumerData_persistent_ends
import Theorems.Thm_ThreeSum_LightReportingWantedDecodeAccess_program_meets_fits
import Theorems.Thm_ThreeSum_LightReportingWantedDispatch_program_meets
import Theorems.Thm_ThreeSum_LightReportingWantedLayout_program_meets_relative
import Theorems.Thm_ThreeSum_LightReportingWantedRecordNeed_program_meets
import Theorems.Thm_ThreeSum_ReportingHostWantedRecordData_cover_scalars
import Theorems.Thm_ThreeSum_SharedReportingBankHeader_Stored_of_kept
import Theorems.Thm_ThreeSum_SharedReportingBankResourceBounds_WantedSourceEnds_values
import Theorems.Thm_ThreeSum_SharedReportingBankResourceBounds_components
import Theorems.Thm_ThreeSum_SharedReportingBankResourceBounds_output_count_le_magnitude
import Theorems.Thm_ThreeSum_SharedReportingBankResourceBounds_query_word_le
import Theorems.Thm_ThreeSum_SharedReportingBankResourceBounds_source_span_le_magnitude
import Theorems.Thm_ThreeSum_SharedReportingBankResourceBounds_wanted_buffers
import Theorems.Thm_ThreeSum_SharedReportingBankResourceBounds_wanted_decoder_word_le
import Theorems.Thm_ThreeSum_SharedReportingBankResourceBounds_wanted_depth_le
import Theorems.Thm_ThreeSum_SharedReportingBankResourceBounds_wanted_numeric_le
import Theorems.Thm_ThreeSum_SharedReportingBankResourceBounds_wanted_scratch_le
import Theorems.Thm_ThreeSum_SharedReportingBankResourceBounds_wanted_selected_ends
import Theorems.Thm_ThreeSum_SharedReportingBankResourceBounds_wanted_terminal_word_le
import Theorems.Thm_ThreeSum_SharedReportingBankResourceBounds_wanted_work_le


set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option Elab.async false
namespace Light
end Light
namespace ThreeSum.BandCandidateRows
end ThreeSum.BandCandidateRows
namespace ThreeSum.BandEnumeration
end ThreeSum.BandEnumeration
namespace ThreeSum.BandPartition
end ThreeSum.BandPartition
namespace ThreeSum.BandRawA
end ThreeSum.BandRawA
namespace ThreeSum.BandScales
end ThreeSum.BandScales
namespace ThreeSum.BlockSourceEncoding
end ThreeSum.BlockSourceEncoding
namespace ThreeSum.ColorClassLists
end ThreeSum.ColorClassLists
namespace ThreeSum.CostLedger
end ThreeSum.CostLedger
namespace ThreeSum.EncodedColorNormalizedData
end ThreeSum.EncodedColorNormalizedData
namespace ThreeSum.EncodedColorWantedConsumerData
end ThreeSum.EncodedColorWantedConsumerData
namespace ThreeSum.GlobalEncoding
end ThreeSum.GlobalEncoding
namespace ThreeSum.GlobalKeyEnumeration
end ThreeSum.GlobalKeyEnumeration
namespace ThreeSum.LightBlockQuery
end ThreeSum.LightBlockQuery
namespace ThreeSum.LightClassFilter
end ThreeSum.LightClassFilter
namespace ThreeSum.LightLocalCoverLayout
end ThreeSum.LightLocalCoverLayout
namespace ThreeSum.LightLocalCoverMetadata
end ThreeSum.LightLocalCoverMetadata
namespace ThreeSum.LightLocalCoverSchedule
end ThreeSum.LightLocalCoverSchedule
namespace ThreeSum.LightLocalCoverTables
end ThreeSum.LightLocalCoverTables
namespace ThreeSum.LocalCoverBounds
end ThreeSum.LocalCoverBounds
namespace ThreeSum.OutputDecodeOrder
end ThreeSum.OutputDecodeOrder
namespace ThreeSum.OutputDecodeRecords
end ThreeSum.OutputDecodeRecords
namespace ThreeSum.RawIncidences
end ThreeSum.RawIncidences
namespace ThreeSum.RawReporting
end ThreeSum.RawReporting
namespace ThreeSum.ReportingHostFrontierLayout
end ThreeSum.ReportingHostFrontierLayout
namespace ThreeSum.ReportingHostSourceSelection
end ThreeSum.ReportingHostSourceSelection
namespace ThreeSum.ReportingHostSourceSelectionStored
end ThreeSum.ReportingHostSourceSelectionStored
namespace ThreeSum.ReportingHostWantedConsumerData
end ThreeSum.ReportingHostWantedConsumerData
namespace ThreeSum.ReportingHostWantedLayout
end ThreeSum.ReportingHostWantedLayout
namespace ThreeSum.ReportingHostWantedOriginal
end ThreeSum.ReportingHostWantedOriginal
namespace ThreeSum.ReportingHostWantedPhysical
end ThreeSum.ReportingHostWantedPhysical
namespace ThreeSum.Scales
end ThreeSum.Scales
namespace ThreeSum.SharedReportingBankData
end ThreeSum.SharedReportingBankData
namespace ThreeSum.SharedReportingBankHeader
end ThreeSum.SharedReportingBankHeader
namespace ThreeSum.SharedReportingBankLayout
end ThreeSum.SharedReportingBankLayout
namespace ThreeSum.SharedReportingBankLevels
end ThreeSum.SharedReportingBankLevels
namespace ThreeSum.SharedReportingBankResourceBounds
end ThreeSum.SharedReportingBankResourceBounds
namespace ThreeSum.SharedReportingBankRows
end ThreeSum.SharedReportingBankRows
namespace ThreeSum.StageBounds
end ThreeSum.StageBounds
namespace ThreeSum.SubsetAddress
end ThreeSum.SubsetAddress
namespace ThreeSum.ThreeSumWantedHostData
end ThreeSum.ThreeSumWantedHostData
namespace ThreeSumApsp
end ThreeSumApsp
Formal statement
theorem ThreeSum.LightReportingWantedDispatchStored.program_meets : ∀ {lim : Light.Limits} {μ : Nat → Int} {free n U d s raw output old threshold : Nat}
  {color :
    ThreeSum.Coloring.Parameter
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) (ThreeSum.Scales.inputBits n)
        (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))}
  {x : Fin n → Int} {A B C : List.{0} Int}
  {hsA : @List.Pairwise.{0} Int (fun (x1 x2 : Int) => @LT.lt.{0} Int Int.instLTInt x1 x2) A}
  {hsB : @List.Pairwise.{0} Int (fun (x1 x2 : Int) => @LT.lt.{0} Int Int.instLTInt x1 x2) B}
  {hsC : @List.Pairwise.{0} Int (fun (x1 x2 : Int) => @LT.lt.{0} Int Int.instLTInt x1 x2) C}
  (H : ThreeSum.EncodedColorPipelineData.Stored μ free n color x A B C hsA hsB hsC)
  (HQ : ThreeSum.ThreeSumReportingSetupData.Catalog μ free n A B C hsA hsB hsC)
  (hx :
    ∀ (i : Fin n),
      @LE.le.{0} Int Int.instLEInt (@abs.{0} Int instLatticeInt Int.instAddGroup (x i))
        (@Nat.cast.{0} Int instNatCastInt U))
  (w : ThreeSum.ReportingHostWantedConsumerData.Label n A B C hsA hsB hsC)
  (hs : @LT.lt.{0} Nat instLTNat s (ThreeSum.ThreeSumWantedHostData.count n A B C hsA hsB hsC))
  (hraw :
    @LT.lt.{0} Nat instLTNat raw
      (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
        (@OfNat.ofNat.{0} Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))
        (ThreeSum.ThreeSumWantedHostData.config free n A B C hsA hsB hsC).N))
  (hout :
    @LT.lt.{0} Nat instLTNat output
      (@HPow.hPow.{0, 0, 0} Nat Nat Nat
        (@instHPow.{0, 0} Nat Nat (@NPow.toPow.{0} Nat (@Monoid.toNPow.{0} Nat Nat.instMonoid)))
        (ThreeSum.OutputDecodeRecords.localCount (ThreeSum.SharedReportingBankResourceBounds.half n))
        (ThreeSum.SharedReportingBankResourceBounds.blocks n)))
  (hpos :
    @Eq.{1} Nat
      (@Fin.val
        (@List.length.{0} ThreeSum.BandRawA.Label
          (ThreeSum.BandRawA.items (@List.length.{0} Int A) (ThreeSum.BandScales.bandSize n)
            (ThreeSum.EncodedColorWantedConsumerData.records n A B C hsA hsB hsC)))
        (@Prod.fst.{0, 0}
          (ThreeSum.ReportingHostWantedOriginal.AType (@List.length.{0} Int A) (ThreeSum.BandScales.bandSize n)
            (ThreeSum.EncodedColorWantedConsumerData.records n A B C hsA hsB hsC))
          (Prod.{0, 0} (Fin (@OfNat.ofNat.{0} Nat (nat_lit 3) (instOfNatNat (nat_lit 3))))
            (ThreeSum.GlobalEncoding.GlobalOutput (ThreeSum.SharedReportingBankResourceBounds.half n)
              (ThreeSum.SharedReportingBankResourceBounds.blocks n)))
          w))
      (@HMod.hMod.{0, 0, 0} Nat Nat Nat (@instHMod.{0} Nat Nat.instMod) raw
        (ThreeSum.ThreeSumWantedHostData.config free n A B C hsA hsB hsC).N))
  (he :
    @Eq.{1}
      (ThreeSum.GlobalEncoding.GlobalOutput (ThreeSum.SharedReportingBankResourceBounds.half n)
        (ThreeSum.SharedReportingBankResourceBounds.blocks n))
      (@Prod.snd.{0, 0} (Fin (@OfNat.ofNat.{0} Nat (nat_lit 3) (instOfNatNat (nat_lit 3))))
        (ThreeSum.GlobalEncoding.GlobalOutput (ThreeSum.SharedReportingBankResourceBounds.half n)
          (ThreeSum.SharedReportingBankResourceBounds.blocks n))
        (@Prod.snd.{0, 0}
          (ThreeSum.ReportingHostWantedOriginal.AType (@List.length.{0} Int A) (ThreeSum.BandScales.bandSize n)
            (ThreeSum.EncodedColorWantedConsumerData.records n A B C hsA hsB hsC))
          (Prod.{0, 0} (Fin (@OfNat.ofNat.{0} Nat (nat_lit 3) (instOfNatNat (nat_lit 3))))
            (ThreeSum.GlobalEncoding.GlobalOutput (ThreeSum.SharedReportingBankResourceBounds.half n)
              (ThreeSum.SharedReportingBankResourceBounds.blocks n)))
          w))
      (ThreeSum.LightOutputDecodeOriginal.output (ThreeSum.SharedReportingBankResourceBounds.half n)
        (ThreeSum.SharedReportingBankResourceBounds.blocks n) output))
  (hleft :
    @Eq.{1} Int
      (μ
        (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
          (ThreeSum.ThreeSumWantedHostData.config free n A B C hsA hsB hsC).pairs
          (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
            (@OfNat.ofNat.{0} Nat (nat_lit 2) (instOfNatNat (nat_lit 2))) s)))
      (@Nat.cast.{0} Int instNatCastInt raw))
  (hright :
    @Eq.{1} Int
      (μ
        (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
          (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
            (ThreeSum.ThreeSumWantedHostData.config free n A B C hsA hsB hsC).pairs
            (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
              (@OfNat.ofNat.{0} Nat (nat_lit 2) (instOfNatNat (nat_lit 2))) s))
          (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))
      (@Nat.cast.{0} Int instNatCastInt output))
  (hp :
    @LT.lt.{0} Nat instLTNat (@OfNat.ofNat.{0} Nat (nat_lit 0) (instOfNatNat (nat_lit 0)))
      (ThreeSum.EncodedColorPrimeReadyData.selected n A B C hsA hsB hsC))
  (hguard :
    @LE.le.{0} Int Int.instLEInt
      (@HMul.hMul.{0, 0, 0} Int Int Int (@instHMul.{0} Int Int.instMul)
        (@OfNat.ofNat.{0} Int (nat_lit 4) (@instOfNat (nat_lit 4)))
        (@Nat.cast.{0} Int instNatCastInt (ThreeSum.EncodedColorPrimeReadyData.selected n A B C hsA hsB hsC)))
      (ThreeSum.GlobalEncoding.globalModulus (ThreeSum.SharedReportingBankResourceBounds.half n)
        (ThreeSum.SharedReportingBankResourceBounds.blocks n)))
  (hkey :
    @Dvd.dvd.{0} Int Int.instDvd
      (ThreeSum.GlobalEncoding.globalModulus (ThreeSum.SharedReportingBankResourceBounds.half n)
        (ThreeSum.SharedReportingBankResourceBounds.blocks n))
      (@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)
              (ThreeSum.BandRawA.valueAt (@List.length.{0} Int A) (ThreeSum.BandScales.bandSize n)
                (ThreeSum.EncodedColorWantedConsumerData.records n A B C hsA hsB hsC)
                (fun (j : Nat) => @List.getD.{0} Int A j (@OfNat.ofNat.{0} Int (nat_lit 0) (@instOfNat (nat_lit 0))))
                (@Fin.val
                  (@List.length.{0} ThreeSum.BandRawA.Label
                    (ThreeSum.BandRawA.items (@List.length.{0} Int A) (ThreeSum.BandScales.bandSize n)
                      (ThreeSum.EncodedColorWantedConsumerData.records n A B C hsA hsB hsC)))
                  (@Prod.fst.{0, 0}
                    (ThreeSum.ReportingHostWantedOriginal.AType (@List.length.{0} Int A)
                      (ThreeSum.BandScales.bandSize n)
                      (ThreeSum.EncodedColorWantedConsumerData.records n A B C hsA hsB hsC))
                    (Prod.{0, 0} (Fin (@OfNat.ofNat.{0} Nat (nat_lit 3) (instOfNatNat (nat_lit 3))))
                      (ThreeSum.GlobalEncoding.GlobalOutput (ThreeSum.SharedReportingBankResourceBounds.half n)
                        (ThreeSum.SharedReportingBankResourceBounds.blocks n)))
                    w)))
              (@Nat.cast.{0} Int instNatCastInt (ThreeSum.EncodedColorPrimeReadyData.selected n A B C hsA hsB hsC)))
            (@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 (ThreeSum.SharedReportingBankResourceBounds.half n)
                    (ThreeSum.SharedReportingBankResourceBounds.blocks n))
                  (@Prod.snd.{0, 0}
                    (ThreeSum.ReportingHostWantedOriginal.AType (@List.length.{0} Int A)
                      (ThreeSum.BandScales.bandSize n)
                      (ThreeSum.EncodedColorWantedConsumerData.records n A B C hsA hsB hsC))
                    (Prod.{0, 0} (Fin (@OfNat.ofNat.{0} Nat (nat_lit 3) (instOfNatNat (nat_lit 3))))
                      (ThreeSum.GlobalEncoding.GlobalOutput (ThreeSum.SharedReportingBankResourceBounds.half n)
                        (ThreeSum.SharedReportingBankResourceBounds.blocks n)))
                    w)))
              (@Nat.cast.{0} Int instNatCastInt (ThreeSum.EncodedColorPrimeReadyData.selected n A B C hsA hsB hsC))))
          (ThreeSum.ThreeSumReportingSetupData.shift n A B C hsA hsB hsC))
        (@ThreeSum.GlobalEncoding.globalKey (ThreeSum.SharedReportingBankResourceBounds.half n)
          (ThreeSum.SharedReportingBankResourceBounds.blocks n)
          (@Prod.snd.{0, 0} (Fin (@OfNat.ofNat.{0} Nat (nat_lit 3) (instOfNatNat (nat_lit 3))))
            (ThreeSum.GlobalEncoding.GlobalOutput (ThreeSum.SharedReportingBankResourceBounds.half n)
              (ThreeSum.SharedReportingBankResourceBounds.blocks n))
            (@Prod.snd.{0, 0}
              (ThreeSum.ReportingHostWantedOriginal.AType (@List.length.{0} Int A) (ThreeSum.BandScales.bandSize n)
                (ThreeSum.EncodedColorWantedConsumerData.records n A B C hsA hsB hsC))
              (Prod.{0, 0} (Fin (@OfNat.ofNat.{0} Nat (nat_lit 3) (instOfNatNat (nat_lit 3))))
                (ThreeSum.GlobalEncoding.GlobalOutput (ThreeSum.SharedReportingBankResourceBounds.half n)
                  (ThreeSum.SharedReportingBankResourceBounds.blocks n)))
              w)))))
  (hcounter :
    @Eq.{1} Int (μ (ThreeSum.ThreeSumWantedHostData.config free n A B C hsA hsB hsC).counter)
      (@Nat.cast.{0} Int instNatCastInt old))
  (hthreshold :
    @Eq.{1} Int
      (μ
        (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
          (ThreeSum.ThreeSumWantedHostData.config free n A B C hsA hsB hsC).counter
          (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))
      (@Nat.cast.{0} Int instNatCastInt threshold))
  (hcap :
    @Eq.{1} Int
      (μ
        (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
          (ThreeSum.ThreeSumWantedHostData.config free n A B C hsA hsB hsC).counter
          (@OfNat.ofNat.{0} Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))))
      (@Nat.cast.{0} Int instNatCastInt
        (ThreeSum.GlobalEncoding.representationCap (ThreeSum.SharedReportingBankResourceBounds.half n)
          (ThreeSum.SharedReportingBankResourceBounds.blocks n))))
  (hold :
    @LE.le.{0} Nat instLENat old
      (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
        (@OfNat.ofNat.{0} Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))
        (@HPow.hPow.{0, 0, 0} Nat Nat Nat
          (@instHPow.{0, 0} Nat Nat (@NPow.toPow.{0} Nat (@Monoid.toNPow.{0} Nat Nat.instMonoid)))
          (ThreeSum.SharedReportingBankResourceBounds.magnitude n)
          (@OfNat.ofNat.{0} Nat (nat_lit 7) (instOfNatNat (nat_lit 7))))))
  (hmass :
    @LE.le.{0} Nat instLENat (ThreeSum.ReportingHostWantedConsumerData.labelMass n A B C hsA hsB hsC w)
      (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
        (@OfNat.ofNat.{0} Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))
        (@HPow.hPow.{0, 0, 0} Nat Nat Nat
          (@instHPow.{0, 0} Nat Nat (@NPow.toPow.{0} Nat (@Monoid.toNPow.{0} Nat Nat.instMonoid)))
          (ThreeSum.SharedReportingBankResourceBounds.magnitude n)
          (@OfNat.ofNat.{0} Nat (nat_lit 7) (instOfNatNat (nat_lit 7))))))
  (hthresholdBound :
    @LE.le.{0} Nat instLENat threshold
      (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
        (@OfNat.ofNat.{0} Nat (nat_lit 9) (instOfNatNat (nat_lit 9)))
        (@HPow.hPow.{0, 0, 0} Nat Nat Nat
          (@instHPow.{0, 0} Nat Nat (@NPow.toPow.{0} Nat (@Monoid.toNPow.{0} Nat Nat.instMonoid)))
          (ThreeSum.SharedReportingBankResourceBounds.magnitude n)
          (@OfNat.ofNat.{0} Nat (nat_lit 8) (instOfNatNat (nat_lit 8))))))
  (hok :
    (ThreeSum.SharedReportingBankResourceBounds.wantedNeed n U).Ok lim
      (ThreeSum.ThreeSumWantedHostData.config free n A B C hsA hsB hsC).layoutHeader
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) d
        (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))),
  Light.Meets lim ThreeSum.LightReportingWantedDispatch.program
    (@OfNat.ofNat.{0} Nat (nat_lit 151) (instOfNatNat (nat_lit 151))) d
    (ThreeSum.LightReportingWantedDispatch.arguments (ThreeSum.ThreeSumWantedHostData.config free n A B C hsA hsB hsC)
      s)
    μ
    (ThreeSum.LightReportingWantedDispatch.time (ThreeSum.ThreeSumWantedHostData.config free n A B C hsA hsB hsC)
      (ThreeSum.SharedReportingBankResourceBounds.wantedDecodeV n)
      (ThreeSum.ReportingHostWantedConsumerData.recordTime free n A B C hsA hsB hsC w output old threshold))
    fun (r : Int) (ν : Nat → Int) =>
    And
      (Or (@Eq.{1} Int r (@OfNat.ofNat.{0} Int (nat_lit 0) (@instOfNat (nat_lit 0))))
        (@Eq.{1} Int r (@OfNat.ofNat.{0} Int (nat_lit 1) (@instOfNat (nat_lit 1)))))
      (And
        (Iff (@Eq.{1} Int r (@OfNat.ofNat.{0} Int (nat_lit 1) (@instOfNat (nat_lit 1))))
          (Or
            (@LT.lt.{0} Nat instLTNat threshold
              (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) old
                (ThreeSum.ReportingHostWantedConsumerData.labelMass n A B C hsA hsB hsC w)))
            (ThreeSum.ReportingHostWantedConsumerData.labelHit n A B C hsA hsB hsC w)))
        (And
          (@Eq.{1} Int (ν (ThreeSum.ThreeSumWantedHostData.config free n A B C hsA hsB hsC).counter)
            (@Nat.cast.{0} Int instNatCastInt
              (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) old
                (ThreeSum.ReportingHostWantedConsumerData.labelMass n A B C hsA hsB hsC w))))
          (Light.KeptBut μ ν (ThreeSum.ThreeSumWantedHostData.config free n A B C hsA hsB hsC).layoutHeader
            (ThreeSum.ThreeSumWantedHostData.config free n A B C hsA hsB hsC).counter
            (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))) := by
  sorry
Source
https://github.com/c0ldzy17/three-sum-lean/blob/2b639655e07421b9ac743de9090459bab473a7c0/Submission/ThreeSum/LightReportingWantedDispatchStored.lean#L22-L212

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