Computing a compact downset address from a mask

Proved

ThreeSum.LightDownsetAddress.meets

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

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

Let $m<2^L$ be a mask with exactly $k$ set bits. Its compact downset address is the number of smaller cardinality subsets plus its position among the $k$ -subsets ordered by increasing binary mask:

\rho_L(m)=\sum_{i<k}\binom Li+\operatorname{rank}_{L,k}(m).

Assume the specified subset-setup and layer-offset procedures are present. With scratch base $w$ and machine limits $S,W,D$ , require

w+(L+1)^2+(L+1)\le S\le W,\quad(L+1)2^{L+1}\le W,\quad d+L+4\le D.

Starting from arbitrary memory, the procedure returns $\rho_L(m)$ within $23L^2+286L+284$ steps and preserves every address below $w$ .

This supplies a compact subset address while constructing its own Pascal and powers tables.

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 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_Bundle06
import Definitions.Def_ThreeSum1962Source_Bundle07
import Lean.Elab.Tactic.Omega
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Algebra.Group.Int.Defs
import Mathlib.Algebra.GroupWithZero.Nat
import Mathlib.Algebra.Order.BigOperators.Group.List
import Mathlib.Algebra.Order.BigOperators.Group.Multiset
import Mathlib.Algebra.Order.Field.Basic
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.Combinatorics.Colex
import Mathlib.Data.Fin.Embedding
import Mathlib.Data.Finset.Powerset
import Mathlib.Data.Int.Cast.Lemmas
import Mathlib.Data.Int.Notation
import Mathlib.Data.Int.SuccPred
import Mathlib.Data.List.Basic
import Mathlib.Data.List.GetD
import Mathlib.Data.List.Nodup
import Mathlib.Data.Nat.Cast.Order.Ring
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Data.Nat.Choose.Bounds
import Mathlib.Data.Nat.Count
import Mathlib.Data.Nat.Notation
import Mathlib.Data.Nat.SuccPred
import Mathlib.LinearAlgebra.Matrix.Defs
import Mathlib.Logic.Function.Basic
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Tactic.GCongr
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.NormNum
import Mathlib.Tactic.Positivity
import Mathlib.Tactic.Ring
import Theorems.Thm_ThreeSum_LightSubsetOffset_meets
import Theorems.Thm_ThreeSum_LightSubsetSetup_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.LightSubsetAddress
end ThreeSum.LightSubsetAddress
namespace ThreeSum.SubsetAddress
end ThreeSum.SubsetAddress

Formal statement

theorem ThreeSum.LightDownsetAddress.meets : ∀ {lim : Light.Limits} {P : Light.Program} {pMain pFresh pBuild pFill pRow pLinear pPowers pRank pOffset : Nat}
  (hmain :
    @Eq.{1} (Option.{0} Light.Stmt)
      (@GetElem?.getElem?.{0, 0, 0} Light.Program Nat Light.Stmt
        (fun (as : List.{0} Light.Stmt) (i : Nat) => @LT.lt.{0} Nat instLTNat i (@List.length.{0} Light.Stmt as))
        (@List.instGetElem?NatLtLength.{0} Light.Stmt) P pMain)
      (@Option.some.{0} Light.Stmt (ThreeSum.LightDownsetAddress.body pFresh pOffset)))
  (C : ThreeSum.LightSubsetSetup.Procs P pFresh pBuild pFill pRow pLinear pPowers pRank)
  (hoff :
    @Eq.{1} (Option.{0} Light.Stmt)
      (@GetElem?.getElem?.{0, 0, 0} Light.Program Nat Light.Stmt
        (fun (as : List.{0} Light.Stmt) (i : Nat) => @LT.lt.{0} Nat instLTNat i (@List.length.{0} Light.Stmt as))
        (@List.instGetElem?NatLtLength.{0} Light.Stmt) P pOffset)
      (@Option.some.{0} Light.Stmt ThreeSum.LightSubsetOffset.body))
  {μ : Nat → Int} {L k m work d : Nat}
  (hw : @LE.le.{0} Int Int.instLEInt (@Nat.cast.{0} Int instNatCastInt lim.space) lim.word)
  (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) work (ThreeSum.LightPascal.tableSize L))
        (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) L
          (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))
      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) L
            (@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) L
              (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))))
      lim.word)
  (hm : ThreeSum.SubsetAddress.IsMask L k m)
  (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 L)
        (@OfNat.ofNat.{0} Nat (nat_lit 4) (instOfNatNat (nat_lit 4))))
      lim.depth),
  Light.Meets lim P pMain d
    (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt L)
      (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt k)
        (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt m)
          (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt work) (@List.nil.{0} Int)))))
    μ (ThreeSum.LightDownsetAddress.time L) fun (r : Int) (μ' : Nat → Int) =>
    And (@Eq.{1} Int r (@Nat.cast.{0} Int instNatCastInt (ThreeSum.SubsetAddress.downsetRank L m)))
      (Light.Kept μ μ' work) := by
  sorry

Source

https://github.com/c0ldzy17/three-sum-lean/blob/2b639655e07421b9ac743de9090459bab473a7c0/Submission/ThreeSum/LightDownsetAddress.lean#L26-L71

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