Expand a valid prefix into exactly its positive children

Proved

ThreeSum.LightReportingPositiveExpand.program_meets

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

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

Let $w$ be a key-consistent wanted item for the raw incidences of finite sets $A,B,C$ with integer values. Fix a valid prefix $u$ at depth $k<J$ , and let $\mathcal P(u)$ be its children whose next-prefix incidence count is positive, in the machine's enumeration order. Set $\rho=2h(J-k-1)$ and $G=\binom{2h}{h}$ .

Assume exact source zeta stages and the level- $(k+1)$ sorted catalog, pointer row, and box stages. Its threshold $t$ and cutoff $b$ satisfy $t\le\rho+1$ , $b\le\rho$ , and $\rho-t\le b$ . The prescribed separated buffers, query scratch, lookup and value word bounds hold, with $S\le W$ and depth allowance $2(512hJ)+16$ . Then

r=|\mathcal P(u)|,\qquad\nu(\mathrm{out}_\ell+i)=\operatorname{mask}(P_\ell(\mathcal P(u)_i)),\quad\nu(\mathrm{out}_r+i)=\operatorname{mask}(P_r(\mathcal P(u)_i)).

Time is the catalog-expansion budget plus the pair-gather budget for $G$ entries and $120$ . Below candidate scratch, memory outside the two populated output segments is unchanged.

This materializes exactly the surviving children for a reporting step.

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_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 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_LightReportingCatalogExpand_program_meets
import Theorems.Thm_ThreeSum_LightReportingPositiveMasks_program_meets


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.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.LightBlockChildrenIndexed
end ThreeSum.LightBlockChildrenIndexed
namespace ThreeSum.LightBlockQuery
end ThreeSum.LightBlockQuery
namespace ThreeSum.LightPascal
end ThreeSum.LightPascal
namespace ThreeSum.LightReportingPositiveMasks
end ThreeSum.LightReportingPositiveMasks
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.ReportingHostMasks
end ThreeSum.ReportingHostMasks
namespace ThreeSum.ReportingHostPositive
end ThreeSum.ReportingHostPositive
namespace ThreeSum.ReportingHostPrefixOrder
end ThreeSum.ReportingHostPrefixOrder
namespace ThreeSum.ReportingHostSuffix
end ThreeSum.ReportingHostSuffix
namespace ThreeSum.StoredReporting
end ThreeSum.StoredReporting
namespace ThreeSum.SubsetAddress
end ThreeSum.SubsetAddress

universe u_1 u_2 u_3

Formal statement

theorem ThreeSum.LightReportingPositiveExpand.program_meets : ∀ {lim : Light.Limits} {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] {μ : Nat → Int}
  {h J k t baseL baseR boxCut boxBase indexBase ptrBase outL outR masks dstL dstR dst selected maskBase work depth :
    Nat}
  (pMod 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) (w : ThreeSum.RawReporting.Wanted.{u_1} h J α)
  (pre :
    List.{0}
      (Finset.{0}
        (Fin
          (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
            (@OfNat.ofNat.{0} Nat (nat_lit 512) (instOfNatNat (nat_lit 512))) h))))
  (hk : @LT.lt.{0} Nat instLTNat k J)
  (hp :
    @ThreeSum.StoredReporting.ValidPrefix 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))
      k pre)
  (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))
                pMod)
              (@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)))
                pMod))
            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))))))
  (hw : @LE.le.{0} Int Int.instLEInt (@Nat.cast.{0} Int instNatCastInt lim.space) lim.word)
  (hdataL :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) baseL
        (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
          (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) (ThreeSum.LightBlockQuery.width h J)
            (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))
          (ThreeSum.SubsetAddress.downsetCap (ThreeSum.LightBlockQuery.width h J)
            (ThreeSum.LightBlockQuery.sourceCut h J))))
      outL)
  (hdataR :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) baseR
        (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
          (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) (ThreeSum.LightBlockQuery.width h J)
            (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))
          (ThreeSum.SubsetAddress.downsetCap (ThreeSum.LightBlockQuery.width h J)
            (ThreeSum.LightBlockQuery.sourceCut h J))))
      outL)
  (hindex :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) indexBase
        (ThreeSum.LightReportingPrefixCatalog.count h
          (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) k
            (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))))
      outL)
  (hptr :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) ptrBase
        (ThreeSum.LightReportingPrefixCatalog.count h
          (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) k
            (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))))
      outL)
  (hboxesEnd :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) boxBase
        (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
          (ThreeSum.LightReportingPrefixCatalog.count h
            (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) k
              (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))
          (ThreeSum.LightReportingSuffixCatalog.stride
            (ThreeSum.BlockIndexedQuery.suffixStage 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)
              (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) k
                (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))
            boxCut)))
      outL)
  (houtL :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) outL (ThreeSum.LocalCoverBounds.groupSize h))
      outR)
  (houtR :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) outR (ThreeSum.LocalCoverBounds.groupSize h))
      masks)
  (hmasks :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) masks (ThreeSum.LocalCoverBounds.groupSize h))
      dstL)
  (hleft :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) dstL (ThreeSum.LocalCoverBounds.groupSize h))
      dstR)
  (hright :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) dstR (ThreeSum.LocalCoverBounds.groupSize h))
      dst)
  (hdst :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) dst (ThreeSum.LocalCoverBounds.groupSize h))
      selected)
  (hselected :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) selected (ThreeSum.LocalCoverBounds.groupSize h))
      maskBase)
  (hqueryMasks :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) maskBase
        (ThreeSum.LightSparseQuery.queryItems
          (ThreeSum.LightBlockQuery.residualSize h J
            (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) k
              (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))
          t))
      work)
  (hspace :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
        (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
          (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) work
            (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))
          (ThreeSum.LightPascal.tableSize (ThreeSum.LightBlockQuery.width h J)))
        (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) (ThreeSum.LightBlockQuery.width h J)
          (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))
      lim.space)
  (hpowSpace :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) work
        (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
          (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
            (@OfNat.ofNat.{0} Nat (nat_lit 2) (instOfNatNat (nat_lit 2))) (ThreeSum.LightBlockQuery.width h J))
          (@OfNat.ofNat.{0} Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))))
      lim.space)
  (hemodSpace :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) work
        (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
          (Nat.log (@OfNat.ofNat.{0} Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))
            (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
              (ThreeSum.LightReportingCatalogLookup.valueBound (ThreeSum.LightBlockQuery.width h J)
                (ThreeSum.LightReportingPrefixCatalog.count h
                  (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) k
                    (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))))
              (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))
          (@OfNat.ofNat.{0} Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))))
      lim.space)
  (hword :
    @LE.le.{0} Int Int.instLEInt
      (@Nat.cast.{0} Int instNatCastInt
        (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
          (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) (ThreeSum.LightBlockQuery.width h J)
            (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))
          (@HPow.hPow.{0, 0, 0} Nat Nat Nat
            (@instHPow.{0, 0} Nat Nat (@NPow.toPow.{0} Nat (@Monoid.toNPow.{0} Nat Nat.instMonoid)))
            (@OfNat.ofNat.{0} Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))
            (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) (ThreeSum.LightBlockQuery.width h J)
              (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))))
      lim.word)
  (hlookupWord :
    @LE.le.{0} Int Int.instLEInt
      (@Nat.cast.{0} Int instNatCastInt
        (ThreeSum.LightReportingCatalogLookup.word (ThreeSum.LightBlockQuery.width h J)
          (ThreeSum.LightReportingPrefixCatalog.count h
            (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) k
              (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))))
      lim.word)
  (hvalueWord :
    @LE.le.{0} Int Int.instLEInt
      (@Nat.cast.{0} Int instNatCastInt
        (ThreeSum.LightSparseQuery.magnitudeBound
          (ThreeSum.BlockIndexedQuery.suffixStage 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)
            (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) k
              (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))
          (ThreeSum.BlockIndexedQuery.suffixStage 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)
            (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) k
              (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))
          (ThreeSum.LightBlockQuery.residualSize h J
            (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) k
              (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))
          t (@Finset.card.{u_2} β B) (@Finset.card.{u_3} γ C)))
      lim.word)
  (ht :
    @LE.le.{0} Nat instLENat t
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
        (ThreeSum.LightBlockQuery.residualSize h J
          (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) k
            (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))
        (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))
  (hboxCut :
    @LE.le.{0} Nat instLENat boxCut
      (ThreeSum.LightBlockQuery.residualSize h J
        (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) k
          (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))))
  (hflags :
    @LE.le.{0} Nat instLENat
      (@HSub.hSub.{0, 0, 0} Nat Nat Nat (@instHSub.{0} Nat instSubNat)
        (ThreeSum.LightBlockQuery.residualSize h J
          (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) k
            (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))
        t)
      boxCut)
  (hstoredL :
    @ThreeSum.LightZetaStages.StoredThrough
      (@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))
      μ baseL (ThreeSum.LightBlockQuery.sourceCut h J)
      (@ThreeSum.BlockIndexedQuery.leftSource.{u_2} β h J pMod B b
        (@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)))
      (ThreeSum.LightBlockQuery.width h J))
  (hstoredR :
    @ThreeSum.LightZetaStages.StoredThrough
      (@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))
      μ baseR (ThreeSum.LightBlockQuery.sourceCut h J)
      (@ThreeSum.BlockIndexedQuery.rightSource.{u_3} γ h J pMod shift C c
        (@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)))
      (ThreeSum.LightBlockQuery.width h J))
  (hindexData :
    ∃ (S : List.{0} Int),
      And (Light.Seg μ indexBase S)
        (And
          (@List.Perm.{0} Int S
            (ThreeSum.ReportingHostPackedCatalog.codes
              (ThreeSum.ReportingHostPackedCatalog.prefixKey h J
                (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) k
                  (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))
              (ThreeSum.LightReportingPrefixCatalog.count h
                (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) k
                  (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))))
          (@List.Pairwise.{0} Int (fun (x1 x2 : Int) => @LE.le.{0} Int Int.instLEInt x1 x2) S)))
  (hptrs :
    Light.SegN μ ptrBase
      (@List.ofFn.{0} Nat
        (ThreeSum.LightReportingPrefixCatalog.count h
          (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) k
            (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))
        fun
          (i :
            Fin
              (ThreeSum.LightReportingPrefixCatalog.count h
                (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) k
                  (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))) =>
        @HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) boxBase
          (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
            (@Fin.val
              (ThreeSum.LightReportingPrefixCatalog.count h
                (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) k
                  (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))
              i)
            (ThreeSum.LightReportingSuffixCatalog.stride
              (ThreeSum.BlockIndexedQuery.suffixStage 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)
                (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) k
                  (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))
              boxCut))))
  (hboxes :
    ∀ (i : Nat),
      @LT.lt.{0} Nat instLTNat i
          (ThreeSum.LightReportingPrefixCatalog.count h
            (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) k
              (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))) →
        @ThreeSum.LightBoxStages.StoredThrough
          (@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.succ)
            (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
              (@OfNat.ofNat.{0} Nat (nat_lit 512) (instOfNatNat (nat_lit 512))) h))
          μ
          (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) boxBase
            (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat) i
              (ThreeSum.LightReportingSuffixCatalog.stride
                (ThreeSum.BlockIndexedQuery.suffixStage 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)
                  (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) k
                    (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))
                boxCut)))
          boxCut
          (@ThreeSum.ReportingHostSuffix.leafWeight
            (@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))
            (@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.succ)
              (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
                (@OfNat.ofNat.{0} Nat (nat_lit 512) (instOfNatNat (nat_lit 512))) h))
            (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat) k.succ
              (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
                (@OfNat.ofNat.{0} Nat (nat_lit 512) (instOfNatNat (nat_lit 512))) h))
            (@ThreeSum.ReportingHostSuffix.block_fit 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)
              k.succ (@Nat.succ_le_of_lt k J hk))
            (@ThreeSum.BlockIndexedQuery.leftSource.{u_2} β h J pMod B b
              (@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)))
            (@ThreeSum.BlockIndexedQuery.rightSource.{u_3} γ h J pMod shift C c
              (@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)))
            (ThreeSum.BlockIndexedQuery.suffixStage 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)
              (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) k
                (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))
            (ThreeSum.ReportingHostPrefixOrder.preL h J
              (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) k
                (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))
              i)
            (ThreeSum.ReportingHostPrefixOrder.preR h J
              (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) k
                (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))
              i))
          (ThreeSum.BlockIndexedQuery.suffixStage 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)
            (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) k
              (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))))
  (hd :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
        (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) depth
          (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
            (@OfNat.ofNat.{0} Nat (nat_lit 2) (instOfNatNat (nat_lit 2))) (ThreeSum.LightBlockQuery.width h J)))
        (@OfNat.ofNat.{0} Nat (nat_lit 16) (instOfNatNat (nat_lit 16))))
      lim.depth),
  Light.Meets lim ThreeSum.LightReportingPositiveExpand.program
    (@OfNat.ofNat.{0} Nat (nat_lit 49) (instOfNatNat (nat_lit 49))) depth
    (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt (ThreeSum.LightBlockQuery.width h J))
      (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt h)
        (@List.cons.{0} Int
          (@Nat.cast.{0} Int instNatCastInt
            (@ThreeSum.BlockMaskLayout.rowMask 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)
              (@ThreeSum.BlockForced.outputRows 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)))
              (@Fin.mk J k hk)))
          (@List.cons.{0} Int
            (@Nat.cast.{0} Int instNatCastInt
              (@ThreeSum.SubsetAddress.subsetMask
                (@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))
                (@ThreeSum.BlockIndexedQuery.leftPrefix 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)
                  k
                  (@ThreeSum.StoredReporting.prefixRows 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)
                    pre))))
            (@List.cons.{0} Int
              (@Nat.cast.{0} Int instNatCastInt
                (@ThreeSum.SubsetAddress.subsetMask
                  (@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))
                  (@ThreeSum.BlockIndexedQuery.rightPrefix 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)
                    k
                    (@ThreeSum.BlockForced.outputRows 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)))
                    (@ThreeSum.StoredReporting.prefixRows 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)
                      pre))))
              (@List.cons.{0} Int
                (@Nat.cast.{0} Int instNatCastInt
                  (@ThreeSum.SubsetAddress.subsetMask
                    (@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))
                    (@ThreeSum.BlockIndexedQuery.querySet 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)
                      k
                      (@ThreeSum.BlockForced.outputRows 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))))))
                (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt masks)
                  (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt dstL)
                    (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt dstR)
                      (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt work)
                        (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt (ThreeSum.LightBlockQuery.sourceCut h J))
                          (@List.cons.{0} Int
                            (@Nat.cast.{0} Int instNatCastInt
                              (ThreeSum.BlockIndexedQuery.suffixStage 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)
                                (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) k
                                  (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))))
                            (@List.cons.{0} Int
                              (@HMul.hMul.{0, 0, 0} Int Int Int (@instHMul.{0} Int Int.instMul)
                                (@HAdd.hAdd.{0, 0, 0} Int Int Int (@instHAdd.{0} Int Int.instAdd)
                                  (@Nat.cast.{0} Int instNatCastInt k)
                                  (@OfNat.ofNat.{0} Int (nat_lit 1) (@instOfNat (nat_lit 1))))
                                (@Nat.cast.{0} Int instNatCastInt h))
                              (@List.cons.{0} Int
                                (@Nat.cast.{0} Int instNatCastInt
                                  (ThreeSum.LightBlockQuery.residualSize h J
                                    (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) k
                                      (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))))
                                (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt t)
                                  (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt baseL)
                                    (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt baseR)
                                      (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt boxCut)
                                        (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt indexBase)
                                          (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt ptrBase)
                                            (@List.cons.{0} Int
                                              (@Nat.cast.{0} Int instNatCastInt
                                                (ThreeSum.LightReportingPrefixCatalog.count h
                                                  (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) k
                                                    (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))))
                                              (@List.cons.{0} Int
                                                (@Nat.cast.{0} Int instNatCastInt
                                                  (ThreeSum.LocalCoverBounds.groupSize h))
                                                (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt dst)
                                                  (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt selected)
                                                    (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt maskBase)
                                                      (@List.cons.{0} Int
                                                        (@HMul.hMul.{0, 0, 0} Int Int Int
                                                          (@instHMul.{0} Int Int.instMul)
                                                          (@HAdd.hAdd.{0, 0, 0} Int Int Int
                                                            (@instHAdd.{0} Int Int.instAdd)
                                                            (@Nat.cast.{0} Int instNatCastInt k)
                                                            (@OfNat.ofNat.{0} Int (nat_lit 1) (@instOfNat (nat_lit 1))))
                                                          (@HMul.hMul.{0, 0, 0} Int Int Int
                                                            (@instHMul.{0} Int Int.instMul)
                                                            (@OfNat.ofNat.{0} Int (nat_lit 512)
                                                              (@instOfNat (nat_lit 512)))
                                                            (@Nat.cast.{0} Int instNatCastInt h)))
                                                        (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt outL)
                                                          (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt outR)
                                                            (@List.nil.{0} Int)))))))))))))))))))))))))))))
    μ (ThreeSum.LightReportingPositiveExpand.time h J k t) fun (r : Int) (ν : Nat → Int) =>
    have pos :=
      @ThreeSum.ReportingHostPositive.positiveChildren.{u_1, u_2, u_3} α β γ inst h J k
        (@ThreeSum.RawIncidences.rawOn.{u_1, u_2, u_3} α β γ h J pMod 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)
        w pre hk;
    And
      (@Eq.{1} Int r
        (@Nat.cast.{0} Int instNatCastInt
          (@List.length.{0}
            (List.{0}
              (Finset.{0}
                (Fin
                  (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
                    (@OfNat.ofNat.{0} Nat (nat_lit 512) (instOfNatNat (nat_lit 512))) h))))
            pos)))
      (And
        (Light.SegN ν outL
          (@List.map.{0, 0}
            (List.{0}
              (Finset.{0}
                (Fin
                  (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
                    (@OfNat.ofNat.{0} Nat (nat_lit 512) (instOfNatNat (nat_lit 512))) h))))
            Nat
            (fun
                (next :
                  List.{0}
                    (Finset.{0}
                      (Fin
                        (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
                          (@OfNat.ofNat.{0} Nat (nat_lit 512) (instOfNatNat (nat_lit 512))) h)))) =>
              @ThreeSum.SubsetAddress.subsetMask
                (@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))
                (@ThreeSum.BlockIndexedQuery.leftPrefix 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)
                  (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) k
                    (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))
                  (@ThreeSum.StoredReporting.prefixRows 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)
                    next)))
            pos))
        (And
          (Light.SegN ν outR
            (@List.map.{0, 0}
              (List.{0}
                (Finset.{0}
                  (Fin
                    (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
                      (@OfNat.ofNat.{0} Nat (nat_lit 512) (instOfNatNat (nat_lit 512))) h))))
              Nat
              (fun
                  (next :
                    List.{0}
                      (Finset.{0}
                        (Fin
                          (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
                            (@OfNat.ofNat.{0} Nat (nat_lit 512) (instOfNatNat (nat_lit 512))) h)))) =>
                @ThreeSum.SubsetAddress.subsetMask
                  (@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))
                  (@ThreeSum.BlockIndexedQuery.rightPrefix 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)
                    (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) k
                      (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))
                    (@ThreeSum.BlockForced.outputRows 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)))
                    (@ThreeSum.StoredReporting.prefixRows 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)
                      next)))
              pos))
          (ThreeSum.LightReportingPositiveExpand.KeptOutputs μ ν masks outL outR
            (@List.length.{0}
              (List.{0}
                (Finset.{0}
                  (Fin
                    (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
                      (@OfNat.ofNat.{0} Nat (nat_lit 512) (instOfNatNat (nat_lit 512))) h))))
              pos)))) := by
  sorry

Source

https://github.com/c0ldzy17/three-sum-lean/blob/2b639655e07421b9ac743de9090459bab473a7c0/Submission/ThreeSum/LightReportingPositiveExpand.lean#L42-L135

View graph