Prove2Me

Construct raw source endpoints and both independent trim flags

Proved
ThreeSum.LightSourcePreparation.Procs.meets

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

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

Fix JJ blocks of width LL, required row size h≤Lh\le L, modulus p>0p>0, and nn masks below 2JL2^{JL}. Memory stores valid block selections into MM pairs of weight rows, a valid map-to-candidate table into KK histogram pairs, weights bounded by UU, and histogram entries bounded by VV. All inputs precede two coefficient arrays of length JLJL, then two raw arrays of length nn, a flag area of length 2n+22n+2, JLJL bit cells, two row-result cells, and scratch zz, in that order.

Assume the specified procedures, S≤WS\le W, d+JL+5≤Dd+JL+5\le D, and the defined source-preparation space and word budgets fit S,WS,W. For mask mim_i, let Hℓ(mi)H_\ell(m_i) be the one-hot sum with coefficients pjwbit(sel(j),a)p^jw_{\mathrm{bit}}(\mathrm{sel}(j),a), and let HrH_r interchange the two weight rows. The output satisfies

rawℓ[i]=Hℓ(mi),rawr[i]=Hr(mi),flagsℓ[i]=Fℓ(mi),flagsr[i]=Fr(mi).\mathrm{raw}_\ell[i]=H_\ell(m_i),\quad\mathrm{raw}_r[i]=H_r(m_i),\quad\mathrm{flags}_\ell[i]=F_\ell(m_i),\quad\mathrm{flags}_r[i]=F_r(m_i).

Each flag requires row cardinality hh and the corresponding bucket cap in every block. Execution meets the defined preparation time budget and preserves memory below the first coefficient array.

This constructs endpoint arithmetic and trimming decisions together.

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_Bundle05
import Definitions.Def_ThreeSum1962Source_Bundle06
import Definitions.Def_ThreeSum1962Source_Bundle07
import Definitions.Def_ThreeSum1962Source_Bundle09
import Definitions.Def_ThreeSum1962Source_Bundle10
import Definitions.Def_ThreeSum1962Source_Bundle11
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.Nat
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.Card
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.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.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
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_LightBlockSourceHashes_Procs_meets
import Theorems.Thm_ThreeSum_LightSourceFlagArray_Procs_meets


set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option Elab.async false
namespace Light
end Light
namespace ThreeSum.BlockCoefficientRows
end ThreeSum.BlockCoefficientRows
Formal statement
theorem ThreeSum.LightSourcePreparation.Procs.meets : ∀ {lim : Light.Limits} {P : Light.Program} {d pMain : Nat} (C : ThreeSum.LightSourcePreparation.Procs P pMain)
  {μ : Nat → Int}
  {J L h p cap occ maps candidates histL histR masks n coeff0 coeff1 rawLeft rawRight flags bitBase rowOut work U V M
    K : Nat}
  (sel candidate mask : Nat → Nat) (w0 w1 leftCount rightCount : Nat → Nat → Nat)
  (hw : @LE.le.{0} Int Int.instLEInt (@Nat.cast.{0} Int instNatCastInt lim.space) lim.word)
  (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 L))
        (@OfNat.ofNat.{0} Nat (nat_lit 5) (instOfNatNat (nat_lit 5))))
      lim.depth)
  (hp : @LT.lt.{0} Nat instLTNat (@OfNat.ofNat.{0} Nat (nat_lit 0) (instOfNatNat (nat_lit 0))) p)
  (hh : @LE.le.{0} Nat instLENat h L)
  (hocc : @LE.le.{0} Nat instLENat (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) occ J) coeff0)
  (hmaps :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) maps
        (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
          (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
            (@OfNat.ofNat.{0} Nat (nat_lit 2) (instOfNatNat (nat_lit 2))) L)
          M))
      coeff0)
  (hcandidates :
    @LE.le.{0} Nat instLENat (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) candidates M) coeff0)
  (hleft :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) histL
        (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat) K p))
      coeff0)
  (hright :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) histR
        (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat) K p))
      coeff0)
  (hmasks : @LE.le.{0} Nat instLENat (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) masks n) coeff0)
  (hc0 :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) coeff0
        (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat) J L))
      coeff1)
  (hc1 :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) coeff1
        (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat) J L))
      rawLeft)
  (hrawLeft :
    @LE.le.{0} Nat instLENat (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) rawLeft n) rawRight)
  (hrawRight :
    @LE.le.{0} Nat instLENat (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) rawRight n) flags)
  (hflags :
    @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) flags
          (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
            (@OfNat.ofNat.{0} Nat (nat_lit 2) (instOfNatNat (nat_lit 2))) n))
        (@OfNat.ofNat.{0} Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))
      bitBase)
  (hbits :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) bitBase
        (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat) J L))
      rowOut)
  (hrowOut :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) rowOut
        (@OfNat.ofNat.{0} Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))
      work)
  (hspace : @LE.le.{0} Nat instLENat (ThreeSum.LightSourcePreparation.space work J L U) lim.space)
  (hword :
    @LE.le.{0} Int Int.instLEInt (@Nat.cast.{0} Int instNatCastInt (ThreeSum.LightSourcePreparation.word J L p U V cap))
      lim.word)
  (hsel : ∀ (j : Nat), @LT.lt.{0} Nat instLTNat j J → @LT.lt.{0} Nat instLTNat (sel j) M)
  (hcandidate : ∀ (s : Nat), @LT.lt.{0} Nat instLTNat s M → @LT.lt.{0} Nat instLTNat (candidate s) K)
  (hoccRead :
    ∀ (j : Nat),
      @LT.lt.{0} Nat instLTNat j J →
        @Eq.{1} Int (μ (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) occ j))
          (@Nat.cast.{0} Int instNatCastInt (sel j)))
  (hcandidateRead :
    ∀ (s : Nat),
      @LT.lt.{0} Nat instLTNat s M →
        @Eq.{1} Int (μ (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) candidates s))
          (@Nat.cast.{0} Int instNatCastInt (candidate s)))
  (hread0 :
    ∀ (s : Nat),
      @LT.lt.{0} Nat instLTNat s M →
        ∀ (a : Nat),
          @LT.lt.{0} Nat instLTNat a L →
            @Eq.{1} Int
              (μ
                (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
                  (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) maps
                    (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
                      (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
                        (@OfNat.ofNat.{0} Nat (nat_lit 2) (instOfNatNat (nat_lit 2))) L)
                      s))
                  a))
              (@Nat.cast.{0} Int instNatCastInt (w0 s a)))
  (hread1 :
    ∀ (s : Nat),
      @LT.lt.{0} Nat instLTNat s M →
        ∀ (a : Nat),
          @LT.lt.{0} Nat instLTNat a L →
            @Eq.{1} Int
              (μ
                (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
                  (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
                    (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) maps
                      (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
                        (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
                          (@OfNat.ofNat.{0} Nat (nat_lit 2) (instOfNatNat (nat_lit 2))) L)
                        s))
                    L)
                  a))
              (@Nat.cast.{0} Int instNatCastInt (w1 s a)))
  (hbound0 :
    ∀ (s : Nat),
      @LT.lt.{0} Nat instLTNat s M → ∀ (a : Nat), @LT.lt.{0} Nat instLTNat a L → @LE.le.{0} Nat instLENat (w0 s a) U)
  (hbound1 :
    ∀ (s : Nat),
      @LT.lt.{0} Nat instLTNat s M → ∀ (a : Nat), @LT.lt.{0} Nat instLTNat a L → @LE.le.{0} Nat instLENat (w1 s a) U)
  (hcountL :
    ∀ (k : Nat),
      @LT.lt.{0} Nat instLTNat k K →
        ∀ (r : Nat),
          @LT.lt.{0} Nat instLTNat r p →
            @Eq.{1} Int
              (μ
                (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
                  (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) histL
                    (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat) k p))
                  r))
              (@Nat.cast.{0} Int instNatCastInt (leftCount k r)))
  (hcountR :
    ∀ (k : Nat),
      @LT.lt.{0} Nat instLTNat k K →
        ∀ (r : Nat),
          @LT.lt.{0} Nat instLTNat r p →
            @Eq.{1} Int
              (μ
                (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
                  (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) histR
                    (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat) k p))
                  r))
              (@Nat.cast.{0} Int instNatCastInt (rightCount k r)))
  (hcountBoundL :
    ∀ (k : Nat),
      @LT.lt.{0} Nat instLTNat k K →
        ∀ (r : Nat), @LT.lt.{0} Nat instLTNat r p → @LE.le.{0} Nat instLENat (leftCount k r) V)
  (hcountBoundR :
    ∀ (k : Nat),
      @LT.lt.{0} Nat instLTNat k K →
        ∀ (r : Nat), @LT.lt.{0} Nat instLTNat r p → @LE.le.{0} Nat instLENat (rightCount k r) V)
  (hmaskRead :
    ∀ (i : Nat),
      @LT.lt.{0} Nat instLTNat i n →
        @Eq.{1} Int (μ (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) masks i))
          (@Nat.cast.{0} Int instNatCastInt (mask i)))
  (hmaskBound :
    ∀ (i : Nat),
      @LT.lt.{0} Nat instLTNat i n →
        @LT.lt.{0} Nat instLTNat (mask i)
          (@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) J L))),
  Light.Meets lim P pMain d
    (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt J)
      (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt L)
        (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt h)
          (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt p)
            (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt cap)
              (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt occ)
                (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt maps)
                  (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt candidates)
                    (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt histL)
                      (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt histR)
                        (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt masks)
                          (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt n)
                            (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt coeff0)
                              (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt coeff1)
                                (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt rawLeft)
                                  (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt rawRight)
                                    (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt flags)
                                      (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt bitBase)
                                        (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt rowOut)
                                          (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt work)
                                            (@List.nil.{0} Int)))))))))))))))))))))
    μ (ThreeSum.LightSourcePreparation.time n J L U) fun (x : Int) (ν : Nat → Int) =>
    And
      (∀ (i : Nat),
        @LT.lt.{0} Nat instLTNat i n →
          @Eq.{1} Int (ν (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) rawLeft i))
            (@Nat.cast.{0} Int instNatCastInt
              (ThreeSum.OneHotMask.eval (ThreeSum.BlockCoefficientRows.flat J L p sel w0)
                (ThreeSum.BlockCoefficientRows.flat J L p sel w1)
                (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat) J L) (mask i))))
      (And
        (∀ (i : Nat),
          @LT.lt.{0} Nat instLTNat i n →
            @Eq.{1} Int (ν (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) rawRight i))
              (@Nat.cast.{0} Int instNatCastInt
                (ThreeSum.OneHotMask.eval (ThreeSum.BlockCoefficientRows.flat J L p sel w1)
                  (ThreeSum.BlockCoefficientRows.flat J L p sel w0)
                  (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat) J L) (mask i))))
        (And
          (∀ (i : Nat),
            @LT.lt.{0} Nat instLTNat i n →
              @Eq.{1} Int (ν (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) flags i))
                (@Nat.cast.{0} Int instNatCastInt
                  (ThreeSum.LightSourceFlagArray.flag J L h p cap sel candidate w0 w1 leftCount (mask i))))
          (And
            (∀ (i : Nat),
              @LT.lt.{0} Nat instLTNat i n →
                @Eq.{1} Int
                  (ν
                    (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
                      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) flags n) i))
                  (@Nat.cast.{0} Int instNatCastInt
                    (ThreeSum.LightSourceFlagArray.flag J L h p cap sel candidate w1 w0 rightCount (mask i))))
            (Light.Kept μ ν coeff0)))) := by
  sorry
Source
https://github.com/c0ldzy17/three-sum-lean/blob/2b639655e07421b9ac743de9090459bab473a7c0/Submission/ThreeSum/LightSourcePreparation.lean#L44-L149

View graph