Verified stable source buckets from retained keys and histogram counts

Proved

ThreeSum.LightSourceBucketFromHistogram.Procs.meets

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

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

Let $k_i<M$ for $i<N$ . Suppose integer memory $\mu$ stores keys at $a$ and their exact histogram at $h$ . Let $o,c,f$ be offset, cursor, and flat-index bases, with $a+N,h+M\le o$ , $o+M+1\le c$ , $c+M\le f$ , and $f+N\le S\le W$ . Assume $d<D$ and that $P$ contains the bucket constructor with its offset and scatter helpers. Define $H_r=|\{i<N:k_i=r\}|$ and $O(r)=\sum_{u<r}H_u$ . Execution on $(a,h,o,c,f,N,M)$ takes at most $120N+120M+220$ steps and constructs

\nu(o+r)=O(r)\ (r\le M),\qquad \nu(f+O(k_i)+|\{j<i:k_j=k_i\}|)=i\ (i<N).

Memory below $o$ and outside $[o,f+N)$ is preserved. These equations give the exact stored-bucket representation used by source scans and joins.

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_Bundle04
import Definitions.Def_ThreeSum1962Source_Bundle06
import Definitions.Def_ThreeSum1962Source_Bundle07
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
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.Data.Finset.Card
import Mathlib.Data.Finset.Range
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Int.Cast.Lemmas
import Mathlib.Data.Int.Notation
import Mathlib.Data.List.GetD
import Mathlib.Data.List.OfFn
import Mathlib.Data.Nat.Cast.Order.Ring
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_LightSourceBucketOffsets_meets
import Theorems.Thm_ThreeSum_LightSourceBucketScatter_meets


set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option Elab.async false
namespace Light
end Light
namespace ThreeSum.LightHistogram
end ThreeSum.LightHistogram
namespace ThreeSum.SourceBuckets
end ThreeSum.SourceBuckets

Formal statement

theorem ThreeSum.LightSourceBucketFromHistogram.Procs.meets : ∀ {lim : Light.Limits} {P : Light.Program} {d pMain : Nat} (C : ThreeSum.LightSourceBucketFromHistogram.Procs P pMain)
  {μ : Nat → Int} {keys hist offsets cursor flat N M : Nat} (key : Nat → Nat)
  (hw : @LE.le.{0} Int Int.instLEInt (@Nat.cast.{0} Int instNatCastInt lim.space) lim.word)
  (hd : @LT.lt.{0} Nat instLTNat d lim.depth)
  (hkeys : @LE.le.{0} Nat instLENat (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) keys N) offsets)
  (hhist : @LE.le.{0} Nat instLENat (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) hist M) offsets)
  (hoffsets :
    @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) offsets M)
        (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))
      cursor)
  (hcursor : @LE.le.{0} Nat instLENat (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) cursor M) flat)
  (hspace : @LE.le.{0} Nat instLENat (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) flat N) lim.space)
  (hkey : ∀ (i : Nat), @LT.lt.{0} Nat instLTNat i N → @LT.lt.{0} Nat instLTNat (key i) M)
  (hread :
    ∀ (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) keys i))
          (@Nat.cast.{0} Int instNatCastInt (key i)))
  (hhistRead :
    ∀ (r : Nat),
      @LT.lt.{0} Nat instLTNat r M →
        @Eq.{1} Int (μ (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) hist r))
          (@Nat.cast.{0} Int instNatCastInt (ThreeSum.LightHistogram.count key r N))),
  Light.Meets lim P pMain d
    (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt keys)
      (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt hist)
        (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt offsets)
          (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt cursor)
            (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt flat)
              (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt N)
                (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt M) (@List.nil.{0} Int))))))))
    μ (ThreeSum.LightSourceBucketFromHistogram.time N M) fun (x : Int) (ν : Nat → Int) =>
    And (ThreeSum.SourceBuckets.Stored ν key N M offsets flat)
      (And (Light.Kept μ ν offsets)
        (Light.SameOutside μ ν offsets
          (@HSub.hSub.{0, 0, 0} Nat Nat Nat (@instHSub.{0} Nat instSubNat)
            (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) flat N) offsets))) := by
  sorry

Source

https://github.com/c0ldzy17/three-sum-lean/blob/2b639655e07421b9ac743de9090459bab473a7c0/Submission/ThreeSum/LightSourceBucketFromHistogram.lean#L37-L72

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