Looking up a prefix catalog entry and returning its exact count

Proved

ThreeSum.LightReportingCatalogQuery.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 $A,B,C$ be finite labeled integer inputs with maps $a,b,c$ . Fix integer $p,\delta$ and wanted label $w=(x,\ell,q)$ satisfying $x\in A$ and the global key congruence

M\mid a(x)\bmod p+\ell p+\delta-\operatorname{key}(q),\qquad \ell\in\{-1,0,1\}.

Let $P$ be a valid prefix of length $k\le J$ . Put $W=512hJ$ , $V=512h(J-k)$ , $r=2h(J-k)$ , and

N=\left(\binom{512h}{2h}\binom{2h}h\right)^k.

Assume exact source transforms, a sorted packed index for all $N$ prefix pairs, their pointers and suffix-table families, with $t\le r+1$ , $b\le r$ , and $r-t\le b$ . All retained regions precede the query mask buffer. The contract's scratch, lookup, address, and value bounds hold, with $S\le\Omega$ and $d+2W+12\le\Delta$ . Here $M$ is the global block modulus.

For the induced raw representation set $I$ , the returned value is

r_{\rm out}=\#\{u\in I:\operatorname{wanted}(u)=w,\ \operatorname{word}(u)|_k=P\}.

It takes $T_{\rm lookup}(W,N)+T_{\rm suffix}(W,V,r,t)+100$ steps and preserves cells below scratch outside the query-mask buffer.

This makes exact prefix counting accessible through the retained catalog index.

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 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_LightReportingCatalogLookup_program_meets
import Theorems.Thm_ThreeSum_LightReportingSuffixQueryIndexed_program_meets
import Theorems.Thm_ThreeSum_RadixKeyLists_tuples_nodup
import Theorems.Thm_ThreeSum_ReportingHostPairEnumeration_pairs_length
import Theorems.Thm_ThreeSum_ReportingHostPrefixMasks_maskPair_injective
import Theorems.Thm_ThreeSum_SubsetAddress_enumerateLayer_nodup
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.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.LayerCounts
end ThreeSum.LayerCounts
namespace ThreeSum.LightBlockQuery
end ThreeSum.LightBlockQuery
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.LocalCoverBounds
end ThreeSum.LocalCoverBounds
namespace ThreeSum.LocalGraph
end ThreeSum.LocalGraph
namespace ThreeSum.LocalHashMoments
end ThreeSum.LocalHashMoments
namespace ThreeSum.MaskDecode
end ThreeSum.MaskDecode
namespace ThreeSum.RawIncidences
end ThreeSum.RawIncidences
namespace ThreeSum.RawPrefixCounts
end ThreeSum.RawPrefixCounts
namespace ThreeSum.RawReporting
end ThreeSum.RawReporting
namespace ThreeSum.ReportingHostCatalogSelection
end ThreeSum.ReportingHostCatalogSelection
namespace ThreeSum.ReportingHostPairEnumeration
end ThreeSum.ReportingHostPairEnumeration
namespace ThreeSum.ReportingHostPairSegments
end ThreeSum.ReportingHostPairSegments
namespace ThreeSum.ReportingHostPrefixCatalog
end ThreeSum.ReportingHostPrefixCatalog
namespace ThreeSum.ReportingHostPrefixMasks
end ThreeSum.ReportingHostPrefixMasks
namespace ThreeSum.ReportingHostPrefixOrder
end ThreeSum.ReportingHostPrefixOrder
namespace ThreeSum.ReportingHostSuffix
end ThreeSum.ReportingHostSuffix
namespace ThreeSum.StoredReporting
end ThreeSum.StoredReporting
namespace ThreeSum.SubsetAddress
end ThreeSum.SubsetAddress
namespace ThreeSumApsp
end ThreeSumApsp

universe u_1 u_2 u_3

Formal statement

theorem ThreeSum.LightReportingCatalogQuery.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 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 : @LE.le.{0} Nat instLENat 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))))
      maskBase)
  (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))))
      maskBase)
  (hindex :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) indexBase
        (ThreeSum.LightReportingPrefixCatalog.count h k))
      maskBase)
  (hptr :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) ptrBase
        (ThreeSum.LightReportingPrefixCatalog.count h k))
      maskBase)
  (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 k)
          (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)
              k)
            boxCut)))
      maskBase)
  (hmasks :
    @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 k) 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 k))
              (@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 k)))
      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)
            k)
          (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)
            k)
          (ThreeSum.LightBlockQuery.residualSize h J k) 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 k)
        (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))
  (hboxCut : @LE.le.{0} Nat instLENat boxCut (ThreeSum.LightBlockQuery.residualSize h J k))
  (hflags :
    @LE.le.{0} Nat instLENat
      (@HSub.hSub.{0, 0, 0} Nat Nat Nat (@instHSub.{0} Nat instSubNat) (ThreeSum.LightBlockQuery.residualSize h J k) 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 k)
              (ThreeSum.LightReportingPrefixCatalog.count h k)))
          (@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 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
              (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)
                k)
              boxCut))))
  (hboxes :
    ∀ (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
                (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)
                  k)
                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)
            (@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)
              k)
            (ThreeSum.ReportingHostPrefixOrder.preL h J k i) (ThreeSum.ReportingHostPrefixOrder.preR h J k 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)
            k))
  (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 12) (instOfNatNat (nat_lit 12))))
      lim.depth),
  Light.Meets lim ThreeSum.LightReportingCatalogQuery.program
    (@OfNat.ofNat.{0} Nat (nat_lit 35) (instOfNatNat (nat_lit 35))) depth
    (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt (ThreeSum.LightBlockQuery.width h J))
      (@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)
              k))
          (@List.cons.{0} Int
            (@HMul.hMul.{0, 0, 0} Int Int Int (@instHMul.{0} Int Int.instMul) (@Nat.cast.{0} Int instNatCastInt k)
              (@Nat.cast.{0} Int instNatCastInt h))
            (@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
                (@HMul.hMul.{0, 0, 0} Int Int Int (@instHMul.{0} Int Int.instMul) (@Nat.cast.{0} Int instNatCastInt k)
                  (@Nat.cast.{0} Int instNatCastInt h))
                (@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.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)
                        k))
                    (@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 (ThreeSum.LightBlockQuery.residualSize h J k))
                        (@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 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 k))
                                        (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt maskBase)
                                          (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt work)
                                            (@List.nil.{0} Int)))))))))))))))))))))
    μ (ThreeSum.LightReportingCatalogQuery.time h J k t) fun (r : Int) (μ' : Nat → Int) =>
    And
      (@Eq.{1} Int r
        (@Nat.cast.{0} Int instNatCastInt
          (@ThreeSum.RawReporting.prefixQuery.{u_1, u_2, u_3} α β γ inst h J
            (@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 k pre)))
      (Light.KeptBut μ μ' work maskBase
        (ThreeSum.LightSparseQuery.queryItems (ThreeSum.LightBlockQuery.residualSize h J k) t)) := by
  sorry

Source

https://github.com/c0ldzy17/three-sum-lean/blob/2b639655e07421b9ac743de9090459bab473a7c0/Submission/ThreeSum/LightReportingCatalogQuery.lean#L41-L129

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