Building a sorted packed index for the complete prefix catalog

Proved

ThreeSum.LightReportingCatalogIndex.program_meets

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

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

Write $S,\Omega,\Delta$ for machine space, word magnitude, and depth limits. Let $k\le J$ , $W=512hJ$ , $V=512h(J-k)$ , and $N=(\binom{512h}{2h}\binom{2h}h)^k$ . Let $f,g$ be integer set functions bounded by $B_0,D_0$ through source cutoff $c\le W$ , with exact retained source transforms. Let $b\le V$ and $kh+b\le c$ . Enumerate all disjoint blockwise $h$ -element prefix pairs $(A_i,B_i)$ .

Assume the prefix-catalog layout and value bounds, a subsequent index region and sorting scratch, the packed-index word bound, $S\le\Omega$ , and depth bounds $d+512h+k+6\le\Delta$ , $d+W+15\le\Delta$ , $d+\lceil\log_2N\rceil+3\le\Delta$ . The routine retains the complete suffix catalog, its pointer array and source transforms, and stores a nondecreasing permutation of

\left[(\operatorname{mask}(A_i)+2^W\operatorname{mask}(B_i))(N+1)+i\right]_{i<N}.

It returns the left prefix-array header, whose value remains $N$ , and preserves memory before prefix enumeration. The time is prefix-catalog construction plus sorted-index construction plus $250$ .

This adds a searchable mask-pair index while retaining each entry's original catalog position.

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_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_Bundle10
import Definitions.Def_ThreeSum1962Source_Bundle12
import Definitions.Def_ThreeSum1962Source_Bundle13
import Definitions.Def_ThreeSum1962Source_Bundle14
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.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_LightReportingPrefixCatalog_program_meets
import Theorems.Thm_ThreeSum_LightReportingSortedCatalog_program_meets
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.LightPascal
end ThreeSum.LightPascal
namespace ThreeSum.LightReportingPrefixCatalog
end ThreeSum.LightReportingPrefixCatalog
namespace ThreeSum.LightReportingSuffixCatalog
end ThreeSum.LightReportingSuffixCatalog
namespace ThreeSum.LightSubsetAddress
end ThreeSum.LightSubsetAddress
namespace ThreeSum.LightZetaCell
end ThreeSum.LightZetaCell
namespace ThreeSum.MaskDecode
end ThreeSum.MaskDecode
namespace ThreeSum.ReportingHostPrefixCatalog
end ThreeSum.ReportingHostPrefixCatalog
namespace ThreeSum.ReportingHostPrefixOrder
end ThreeSum.ReportingHostPrefixOrder
namespace ThreeSum.ReportingHostSuffix
end ThreeSum.ReportingHostSuffix
namespace ThreeSum.SubsetAddress
end ThreeSum.SubsetAddress

Formal statement

theorem ThreeSum.LightReportingCatalogIndex.program_meets : ∀ {lim : Light.Limits} {μ : Nat → Int}
  {h J k cut boxCut baseL baseR srcS pairL pairR stageL stageR ptrBase boxBase maskBase indexBase work B D d : Nat}
  (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)
  (hw : @LE.le.{0} Int Int.instLEInt (@Nat.cast.{0} Int instNatCastInt lim.space) lim.word)
  (hcut :
    @LE.le.{0} Nat instLENat cut
      (@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)))
  (hboxCut :
    @LE.le.{0} Nat instLENat boxCut
      (@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)))
  (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)
            (@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))
            (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))
          (ThreeSum.SubsetAddress.downsetCap
            (@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))
            cut)))
      srcS)
  (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)
            (@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))
            (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))
          (ThreeSum.SubsetAddress.downsetCap
            (@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))
            cut)))
      srcS)
  (hS :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) srcS
        ((@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
              (@OfNat.ofNat.{0} Nat (nat_lit 512) (instOfNatNat (nat_lit 512))) h).choose
          h))
      pairL)
  (hL :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) pairL
        (ThreeSum.LightReportingPrefixArrays.localCount
          (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
            (@OfNat.ofNat.{0} Nat (nat_lit 512) (instOfNatNat (nat_lit 512))) h)
          h))
      pairR)
  (hR :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) pairR
        (ThreeSum.LightReportingPrefixArrays.localCount
          (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
            (@OfNat.ofNat.{0} Nat (nat_lit 512) (instOfNatNat (nat_lit 512))) h)
          h))
      stageL)
  (hstageL :
    @LE.le.{0} Nat instLENat
      (ThreeSum.LightRadixKeys.space stageL
        (ThreeSum.LightReportingPrefixArrays.localCount
          (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
            (@OfNat.ofNat.{0} Nat (nat_lit 512) (instOfNatNat (nat_lit 512))) h)
          h)
        k)
      stageR)
  (hstageR :
    @LE.le.{0} Nat instLENat
      (ThreeSum.LightRadixKeys.space stageR
        (ThreeSum.LightReportingPrefixArrays.localCount
          (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
            (@OfNat.ofNat.{0} Nat (nat_lit 512) (instOfNatNat (nat_lit 512))) h)
          h)
        k)
      ptrBase)
  (hptr :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) ptrBase
        (ThreeSum.LightReportingPrefixCatalog.count h k))
      boxBase)
  (hbox :
    @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 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))
            boxCut)))
      maskBase)
  (hmasks :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) maskBase
        (ThreeSum.SubsetAddress.downsetCap
          (@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))
          boxCut))
      work)
  (hindex : @LE.le.{0} Nat instLENat maskBase indexBase)
  (hindexEnd :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) indexBase
        (ThreeSum.LightReportingPrefixCatalog.count h k))
      work)
  (hpairSpace :
    @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) work
          (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
            (@OfNat.ofNat.{0} Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))
            (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
              (@OfNat.ofNat.{0} Nat (nat_lit 512) (instOfNatNat (nat_lit 512))) h)))
        (@OfNat.ofNat.{0} Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))
      lim.space)
  (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
            (@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))))
        (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
          (@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))
          (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))
      lim.space)
  (hsortSpace :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) work
        (ThreeSum.LightReportingPrefixCatalog.count h k))
      lim.space)
  (hpowWord :
    @LE.le.{0} Int Int.instLEInt
      (@Nat.cast.{0} Int instNatCastInt
        (@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)
            (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
              (@OfNat.ofNat.{0} Nat (nat_lit 512) (instOfNatNat (nat_lit 512))) h)
            (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))))
      lim.word)
  (hkeysWord :
    @LE.le.{0} Int Int.instLEInt
      (@Nat.cast.{0} Int instNatCastInt
        (ThreeSum.LightRadixKeys.word
          (@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)))
            (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
              (@OfNat.ofNat.{0} Nat (nat_lit 512) (instOfNatNat (nat_lit 512))) h))
          (@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)))
            (@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))
      lim.word)
  (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)
            (@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))
            (@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)
              (@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))
              (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))))
      lim.word)
  (hindexWord :
    @LE.le.{0} Int Int.instLEInt
      (@Nat.cast.{0} Int instNatCastInt
        (ThreeSum.LightReportingSortedCatalog.word
          (@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.LightReportingPrefixCatalog.count h k)))
      lim.word)
  (hvalueWord :
    @LE.le.{0} Int Int.instLEInt
      (@Nat.cast.{0} Int instNatCastInt
        (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
          (@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)))
            (@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)))
          (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
            (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
              (@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)))
                (@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)))
              B)
            (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
              (@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)))
                (@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))))
      lim.word)
  (hpre :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
        (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat) k h) boxCut)
      cut)
  (hf :
    ∀
      (S :
        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)))),
      @LE.le.{0} Nat instLENat
          (@Finset.card.{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)))
            S)
          cut →
        @LE.le.{0} Int Int.instLEInt (@abs.{0} Int instLatticeInt Int.instAddGroup (f S))
          (@Nat.cast.{0} Int instNatCastInt B))
  (hg :
    ∀
      (S :
        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)))),
      @LE.le.{0} Nat instLENat
          (@Finset.card.{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)))
            S)
          cut →
        @LE.le.{0} Int Int.instLEInt (@abs.{0} Int instLatticeInt Int.instAddGroup (g S))
          (@Nat.cast.{0} Int instNatCastInt D))
  (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 cut f
      (@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)))
  (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 cut g
      (@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)))
  (hdPair :
    @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) d
            (@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)
        (@OfNat.ofNat.{0} Nat (nat_lit 6) (instOfNatNat (nat_lit 6))))
      lim.depth)
  (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) d
          (@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)))
        (@OfNat.ofNat.{0} Nat (nat_lit 15) (instOfNatNat (nat_lit 15))))
      lim.depth)
  (hdSort :
    @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) d
          (Nat.clog (@OfNat.ofNat.{0} Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))
            (ThreeSum.LightReportingPrefixCatalog.count h k)))
        (@OfNat.ofNat.{0} Nat (nat_lit 3) (instOfNatNat (nat_lit 3))))
      lim.depth),
  Light.Meets lim ThreeSum.LightReportingCatalogIndex.program
    (@OfNat.ofNat.{0} Nat (nat_lit 51) (instOfNatNat (nat_lit 51))) d
    (@List.cons.{0} Int
      (@HMul.hMul.{0, 0, 0} Int Int Int (@instHMul.{0} Int Int.instMul) (@Nat.cast.{0} Int instNatCastInt J)
        (@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 cut)
        (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt boxCut)
          (@List.cons.{0} Int
            (@HMul.hMul.{0, 0, 0} Int Int Int (@instHMul.{0} Int Int.instMul)
              (@HSub.hSub.{0, 0, 0} Int Int Int (@instHSub.{0} Int Int.instSub) (@Nat.cast.{0} Int instNatCastInt J)
                (@Nat.cast.{0} Int instNatCastInt k))
              (@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
              (@HMul.hMul.{0, 0, 0} Int Int Int (@instHMul.{0} Int Int.instMul)
                (@HSub.hSub.{0, 0, 0} Int Int Int (@instHSub.{0} Int Int.instSub) (@Nat.cast.{0} Int instNatCastInt J)
                  (@Nat.cast.{0} Int instNatCastInt k))
                (@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
                (@HMul.hMul.{0, 0, 0} Int Int Int (@instHMul.{0} Int Int.instMul) (@Nat.cast.{0} Int instNatCastInt k)
                  (@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 baseL)
                  (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt baseR)
                    (@List.cons.{0} Int
                      (@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 h)
                        (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt k)
                          (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt srcS)
                            (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt pairL)
                              (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt pairR)
                                (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt stageL)
                                  (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt stageR)
                                    (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt ptrBase)
                                      (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt boxBase)
                                        (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt maskBase)
                                          (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt work)
                                            (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt indexBase)
                                              (@List.nil.{0} Int))))))))))))))))))))))
    μ (ThreeSum.LightReportingCatalogIndex.time h J k boxCut) fun (r : Int) (μ' : Nat → Int) =>
    And
      (@Eq.{1} Int r
        (@Nat.cast.{0} Int instNatCastInt
          (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) stageL
            (ThreeSum.LightRadixKeys.offset
              (ThreeSum.LightReportingPrefixArrays.localCount
                (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
                  (@OfNat.ofNat.{0} Nat (nat_lit 512) (instOfNatNat (nat_lit 512))) h)
                h)
              k))))
      (And
        (@Eq.{1} Int
          (μ'
            (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) stageL
              (ThreeSum.LightRadixKeys.offset
                (ThreeSum.LightReportingPrefixArrays.localCount
                  (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
                    (@OfNat.ofNat.{0} Nat (nat_lit 512) (instOfNatNat (nat_lit 512))) h)
                  h)
                k)))
          (@Nat.cast.{0} Int instNatCastInt (ThreeSum.LightReportingPrefixCatalog.count h k)))
        (And
          (Light.SegN μ' ptrBase
            (@List.ofFn.{0} Nat (ThreeSum.LightReportingPrefixCatalog.count h k)
              fun (i : Fin (ThreeSum.LightReportingPrefixCatalog.count h k)) =>
              @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 k) i)
                  (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))
                    boxCut))))
          (And
            (∀ (i : Nat),
              @LT.lt.{0} Nat instLTNat i (ThreeSum.LightReportingPrefixCatalog.count h k) →
                @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)
                    (@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
                        (@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))
                        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)
                      (@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
                      (@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 hk)
                    f g
                    (@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))
                    (ThreeSum.ReportingHostPrefixOrder.preL h J k i) (ThreeSum.ReportingHostPrefixOrder.preR h J k i))
                  (@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)))
            (And
              (@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 cut f
                (@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)))
              (And
                (@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 cut g
                  (@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)))
                (And
                  (∃ (S : List.{0} Int),
                    And (Light.Seg μ' indexBase S)
                      (And
                        (@List.Perm.{0} Int S
                          (ThreeSum.ReportingHostPackedCatalog.codes
                            (ThreeSum.ReportingHostPackedCatalog.prefixKey h J k)
                            (ThreeSum.LightReportingPrefixCatalog.count h k)))
                        (@List.Pairwise.{0} Int (fun (x1 x2 : Int) => @LE.le.{0} Int Int.instLEInt x1 x2) S)))
                  (Light.Kept μ μ' srcS))))))) := by
  sorry

Source

https://github.com/c0ldzy17/three-sum-lean/blob/2b639655e07421b9ac743de9090459bab473a7c0/Submission/ThreeSum/LightReportingCatalogIndex.lean#L40-L150

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