Computing signed local keys from coefficients outside each mask

Proved

ThreeSum.LightLocalKeyRow.Procs.meets

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

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

Let memory store $n$ masks $q_i<2^L$ and two natural coefficient rows $w_F,w_T$ of length $L$ , each bounded by $U$ . All three input arrays end at or before a free address $f$ . Assume the specified fill, hash, powers, and linear-array procedures, with

f+2L+2n+1\le S\le B,\quad2\bigl(2^{L+1}+(L+1)U\bigr)+2\le B,\quad d+L+2\le D.

The routine returns $f+L$ and, within $200nL+223n+47L+462$ steps, writes

\nu(f+L+i)=\sum_{\substack{a<L\\\operatorname{bit}_a(q_i)=0}}\bigl(w_T(a)-w_F(a)\bigr)\qquad(i<n).

Every cell below $f$ retains its initial value. The differences and output entries are integers, so negative keys are included.

This constructs the signed output-key row associated with the stored masks and one local coefficient map.

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 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.Field.ZMod
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.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.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.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.Function.Basic
import Mathlib.NumberTheory.Bertrand
import Mathlib.Order.ConditionallyCompleteLattice.Basic
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 Theorems.Thm_Light_Ends_pass
import Theorems.Thm_ThreeSum_LightArray_linear_bounded_meets
import Theorems.Thm_ThreeSum_LightMaskHashes_meets
import Theorems.Thm_ThreeSum_OneHotMask_eval_le


set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option Elab.async false
namespace Light
end Light
namespace ThreeSum.LightSubsetAddress
end ThreeSum.LightSubsetAddress

Formal statement

theorem ThreeSum.LightLocalKeyRow.Procs.meets : ∀ {lim : Light.Limits} {P : Light.Program} {d pMain : Nat} (S : ThreeSum.LightLocalKeyRow.Procs P pMain) {μ : Nat → Int}
  {L masks n srcFalse srcTrue free U : Nat} (mask wFalse wTrue : 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 L)
        (@OfNat.ofNat.{0} Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))
      lim.depth)
  (hspace : @LE.le.{0} Nat instLENat (ThreeSum.LightLocalKeyRow.space free L n) lim.space)
  (hmasks : @LE.le.{0} Nat instLENat (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) masks n) free)
  (hfalse : @LE.le.{0} Nat instLENat (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) srcFalse L) free)
  (htrue : @LE.le.{0} Nat instLENat (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) srcTrue L) free)
  (hword :
    @LE.le.{0} Int Int.instLEInt (@Nat.cast.{0} Int instNatCastInt (ThreeSum.LightLocalKeyRow.word L U)) lim.word)
  (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)))
  (hmaskLt :
    ∀ (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))) L))
  (hreadFalse :
    ∀ (i : Nat),
      @LT.lt.{0} Nat instLTNat i L →
        @Eq.{1} Int (μ (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) srcFalse i))
          (@Nat.cast.{0} Int instNatCastInt (wFalse i)))
  (hreadTrue :
    ∀ (i : Nat),
      @LT.lt.{0} Nat instLTNat i L →
        @Eq.{1} Int (μ (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) srcTrue i))
          (@Nat.cast.{0} Int instNatCastInt (wTrue i)))
  (hboundFalse : ∀ (i : Nat), @LT.lt.{0} Nat instLTNat i L → @LE.le.{0} Nat instLENat (wFalse i) U)
  (hboundTrue : ∀ (i : Nat), @LT.lt.{0} Nat instLTNat i L → @LE.le.{0} Nat instLENat (wTrue i) U),
  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 masks)
        (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt n)
          (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt srcFalse)
            (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt srcTrue)
              (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt free) (@List.nil.{0} Int)))))))
    μ (ThreeSum.LightLocalKeyRow.time L n) fun (r : Int) (ν : Nat → Int) =>
    And
      (@Eq.{1} Int r
        (@Nat.cast.{0} Int instNatCastInt (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) free L)))
      (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) free L) i))
              (@HSub.hSub.{0, 0, 0} Int Int Int (@instHSub.{0} Int Int.instSub)
                (@Nat.cast.{0} Int instNatCastInt
                  (ThreeSum.OneHotMask.eval wTrue
                    (fun (x : Nat) => @OfNat.ofNat.{0} Nat (nat_lit 0) (instOfNatNat (nat_lit 0))) L (mask i)))
                (@Nat.cast.{0} Int instNatCastInt
                  (ThreeSum.OneHotMask.eval wFalse
                    (fun (x : Nat) => @OfNat.ofNat.{0} Nat (nat_lit 0) (instOfNatNat (nat_lit 0))) L (mask i)))))
        (Light.Kept μ ν free)) := by
  sorry

Source

https://github.com/c0ldzy17/three-sum-lean/blob/2b639655e07421b9ac743de9090459bab473a7c0/Submission/ThreeSum/LightLocalKeyRow.lean#L50-L139

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