Prove2Me

Construct and retain all frontiers through a requested depth

Proved
ThreeSum.LightReportingLevels.program_meets

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

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

Let ww be a key-consistent wanted item for the raw incidence set II of finite integer-valued sets. Fix k≤K≤Jk\le K\le J, and let C=(m+1)(2hh)C=(m+1)\binom{2h}{h}, where mm is the root mass. Memory stores the exact root frontier, capacity, packed output rows, source stages, and the metadata and payload of every catalog through level KK.

Assume all per-level source bounds and query-scratch contracts, the ordered frontier and child buffers, the scalar space condition, S≤WS\le W, and entry-depth allowance k+2(512hJ)+19k+2(512hJ)+19. Execution returns

r=∣Fk∣,∀j≤k, FrontierAt⁡(ν,I,w,r0,C,j),ν(r0+3)=C.r=|\mathcal F_k|,\qquad\forall j\le k,\ \operatorname{FrontierAt}(\nu,I,w,r_0,C,j),\qquad\nu(r_0+3)=C.

Memory below frontier base r0r_0 is preserved. The time budget starts at 3030 and adds each stored-step budget and 200200 per level.

This provides the full retained sequence of positive frontiers needed for terminal reporting.

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_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_Bundle14
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.Range
import Mathlib.Data.Finset.Sigma
import Mathlib.Data.Finset.Sort
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Fintype.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.ProdSigma
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_LightReportingStoredStep_program_meets
import Theorems.Thm_ThreeSum_RawReporting_partitionWord_length
import Theorems.Thm_ThreeSum_RawReporting_word_children
import Theorems.Thm_ThreeSum_ReportingHostFrontierLayout_At_of_kept
import Theorems.Thm_ThreeSum_ReportingHostLevelData_Stored_of_kept
import Theorems.Thm_ThreeSum_ReportingHostMasks_blockSet_localRow
import Theorems.Thm_ThreeSum_ReportingHostMasks_localChildren_mem
import Theorems.Thm_ThreeSum_Reporting_frontier_length_le_mass
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.LightBlockQuery
end ThreeSum.LightBlockQuery
namespace ThreeSum.LightPascal
end ThreeSum.LightPascal
namespace ThreeSum.LightReportingPrefixCatalog
end ThreeSum.LightReportingPrefixCatalog
namespace ThreeSum.LightReportingSuffixCatalog
end ThreeSum.LightReportingSuffixCatalog
namespace ThreeSum.LightZetaCell
end ThreeSum.LightZetaCell
namespace ThreeSum.LocalCoverBounds
end ThreeSum.LocalCoverBounds
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.ReportingHostFrontierLayout
end ThreeSum.ReportingHostFrontierLayout
namespace ThreeSum.ReportingHostLevelData
end ThreeSum.ReportingHostLevelData
namespace ThreeSum.ReportingHostMasks
end ThreeSum.ReportingHostMasks
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.LightReportingLevels.program_meets : ∀ {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] {lim : Light.Limits}
  {μ : Nat → Int} {h J K k baseL baseR catalogs rows root masks childL childR counts selected maskBase work depth : Nat}
  (D : Nat → ThreeSum.ReportingHostLevelData.Catalog) (hk : @LE.le.{0} Nat instLENat k K)
  (hK : @LE.le.{0} Nat instLENat K J) (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)
  (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)
  (HS :
    @ThreeSum.ReportingHostLevelData.Sources.{u_2, u_3} β γ h J μ pMod shift B C b 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))
      baseL baseR)
  (HC :
    ∀ (j : Nat) (hj : @LE.le.{0} Nat instLENat j K),
      @ThreeSum.ReportingHostLevelData.Stored.{u_2, u_3} β γ h J j μ (D j)
        (@LE.le.trans.{0} Nat Nat.instPreorder j K J hj hK) pMod shift B C b 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)))
  (HB : ∀ (j : Nat), @LE.le.{0} Nat instLENat j K → ThreeSum.ReportingHostLevelData.Bounds h J j (D j) baseL baseR root)
  (HW :
    ∀ (j : Nat),
      @LE.le.{0} Nat instLENat j K →
        ThreeSum.ReportingHostLevelData.Scratch lim h J j (D j) (@Finset.card.{u_2} β B) (@Finset.card.{u_3} γ C)
          maskBase work)
  (hmeta :
    ∀ (j : Nat),
      @LE.le.{0} Nat instLENat j K →
        Light.SegN μ
          (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) catalogs
            (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
              (@OfNat.ofNat.{0} Nat (nat_lit 5) (instOfNatNat (nat_lit 5))) j))
          ((D j).record h j))
  (hmetaEnd :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) catalogs
        (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
          (@OfNat.ofNat.{0} Nat (nat_lit 5) (instOfNatNat (nat_lit 5)))
          (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) K
            (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))))
      root)
  (hrows : @LE.le.{0} Nat instLENat (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) rows J) root)
  (hrow :
    ∀ (j : Nat) (hj : @LT.lt.{0} Nat instLTNat j J),
      @Eq.{1} Int (μ (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) rows j))
        (@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 j hj))))
  (hcap :
    @Eq.{1} Int
      (μ
        (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) root
          (@OfNat.ofNat.{0} Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))))
      (@Nat.cast.{0} Int instNatCastInt
        (@ThreeSum.ReportingHostFrontierLayout.capacity.{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)))
  (HA :
    @ThreeSum.ReportingHostFrontierLayout.At.{u_1, u_2, u_3} α β γ inst h J μ
      (@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 root
      (@ThreeSum.ReportingHostFrontierLayout.capacity.{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)
      (@OfNat.ofNat.{0} Nat (nat_lit 0) (instOfNatNat (nat_lit 0))))
  (hout :
    @LE.le.{0} Nat instLENat
      (ThreeSum.ReportingHostFrontierLayout.slot root
        (@ThreeSum.ReportingHostFrontierLayout.capacity.{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)
        (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) K
          (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))
      masks)
  (hmasks :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) masks (ThreeSum.LocalCoverBounds.groupSize h))
      childL)
  (hchildL :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) childL (ThreeSum.LocalCoverBounds.groupSize h))
      childR)
  (hchildR :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) childR (ThreeSum.LocalCoverBounds.groupSize h))
      counts)
  (hcounts :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) counts (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)
  (hparams :
    @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) 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.LocalCoverBounds.groupSize h))
        (ThreeSum.LightBlockQuery.sourceCut h J))
      lim.space)
  (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)
          (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) depth k)
          (@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 19) (instOfNatNat (nat_lit 19))))
      lim.depth),
  Light.Meets lim ThreeSum.LightReportingLevels.program (@OfNat.ofNat.{0} Nat (nat_lit 52) (instOfNatNat (nat_lit 52)))
    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 J)
          (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt k)
            (@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 catalogs)
                  (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt rows)
                    (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt root)
                      (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt masks)
                        (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt childL)
                          (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt childR)
                            (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt counts)
                              (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt selected)
                                (@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 (ThreeSum.LocalCoverBounds.groupSize h))
                                      (@List.cons.{0} Int
                                        (@Nat.cast.{0} Int instNatCastInt (ThreeSum.LightBlockQuery.sourceCut h J))
                                        (@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.nil.{0} Int))))))))))))))))))))
    μ
    (@ThreeSum.LightReportingLevels.time.{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 D k)
    fun (r : Int) (ν : Nat → Int) =>
    have I :=
      @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;
    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))))
            (@ThreeSum.ReportingHostFrontier.frontier.{u_1, u_2, u_3} α β γ inst h J I w k))))
      (And
        (∀ (j : Nat),
          @LE.le.{0} Nat instLENat j k →
            @ThreeSum.ReportingHostFrontierLayout.At.{u_1, u_2, u_3} α β γ inst h J ν I w root
              (@ThreeSum.ReportingHostFrontierLayout.capacity.{u_1, u_2, u_3} α β γ inst h J I w) j)
        (And
          (@Eq.{1} Int
            (ν
              (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) root
                (@OfNat.ofNat.{0} Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))))
            (@Nat.cast.{0} Int instNatCastInt
              (@ThreeSum.ReportingHostFrontierLayout.capacity.{u_1, u_2, u_3} α β γ inst h J I w)))
          (Light.Kept μ ν root))) := by
  sorry
Source
https://github.com/c0ldzy17/three-sum-lean/blob/2b639655e07421b9ac743de9090459bab473a7c0/Submission/ThreeSum/LightReportingLevels.lean#L42-L146

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