Normalize a value list and construct its band endpoints

Proved

ThreeSum.LightSortedBands.Procs.meets

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

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

Let an integer list $X$ of length $n$ be stored at source $s$ , and let $g>0$ be the band width. Reserve, in order, the source, an $n$ -cell values buffer, an endpoint region of $2+2n$ cells, and $n$ scratch cells. Assume space fits in a word, $n+g+2$ fits in a word, and $d+n+\operatorname{clog}_2n+3$ fits in depth. The installed program constructs a strictly increasing list $L$ with

|L|\le n,\qquad z\in L\iff z\in X,

stores $L$ , and writes its length and band count in the endpoint header. For each nonempty consecutive $g$ -band it stores the first and last values, returning the band count. Time is $\operatorname{Normalize.time}(n)+200n+300$ .

Below scratch, memory outside the source, values, and endpoint regions is preserved. This prepares an original color class for interval-based owner enumeration.

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_Bundle03
import Definitions.Def_ThreeSum1962Source_Bundle05
import Definitions.Def_ThreeSum1962Source_Bundle06
import Definitions.Def_ThreeSum1962Source_Bundle07
import Definitions.Def_ThreeSum1962Source_Bundle08
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.Finset
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.Algebra.Polynomial.Basic
import Mathlib.Algebra.Polynomial.Coeff
import Mathlib.Combinatorics.Colex
import Mathlib.Data.Fin.Embedding
import Mathlib.Data.Finset.Basic
import Mathlib.Data.Finset.Powerset
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Fintype.Prod
import Mathlib.Data.Fintype.Sets
import Mathlib.Data.Fintype.Sum
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.Dedup
import Mathlib.Data.List.FinRange
import Mathlib.Data.List.GetD
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.ProdSigma
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.Log
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 Mathlib.Tactic.SplitIfs
import Theorems.Thm_ThreeSum_LightBandPartition_meets
import Theorems.Thm_ThreeSum_LightNormalize_Procs_meets


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

Formal statement

theorem ThreeSum.LightSortedBands.Procs.meets : ∀ {lim : Light.Limits} {P : Light.Program} {pMain src values endpoints work g d : Nat} {μ : Nat → Int}
  {original : List.{0} Int} (R : ThreeSum.LightSortedBands.Procs P pMain)
  (hw : @LE.le.{0} Int Int.instLEInt (@Nat.cast.{0} Int instNatCastInt lim.space) lim.word)
  (hg : @LT.lt.{0} Nat instLTNat (@OfNat.ofNat.{0} Nat (nat_lit 0) (instOfNatNat (nat_lit 0))) g)
  (hsrc :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) src (@List.length.{0} Int original)) values)
  (hvalues :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) values (@List.length.{0} Int original))
      endpoints)
  (hendpoints :
    @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) endpoints
          (@OfNat.ofNat.{0} Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))
        (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
          (@OfNat.ofNat.{0} Nat (nat_lit 2) (instOfNatNat (nat_lit 2))) (@List.length.{0} Int original)))
      work)
  (hspace : @LE.le.{0} Nat instLENat (ThreeSum.LightSortedBands.space work (@List.length.{0} Int original)) lim.space)
  (hword :
    @LE.le.{0} Int Int.instLEInt
      (@Nat.cast.{0} Int instNatCastInt
        (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
          (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) (@List.length.{0} Int original) g)
          (@OfNat.ofNat.{0} Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))))
      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)
          (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) d (@List.length.{0} Int original))
          (Nat.clog (@OfNat.ofNat.{0} Nat (nat_lit 2) (instOfNatNat (nat_lit 2))) (@List.length.{0} Int original)))
        (@OfNat.ofNat.{0} Nat (nat_lit 3) (instOfNatNat (nat_lit 3))))
      lim.depth)
  (hread : Light.Seg μ src original),
  Light.Meets lim P pMain d
    (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt src)
      (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt (@List.length.{0} Int original))
        (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt g)
          (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt values)
            (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt endpoints)
              (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt work) (@List.nil.{0} Int)))))))
    μ (ThreeSum.LightSortedBands.time (@List.length.{0} Int original)) fun (r : Int) (μ' : Nat → Int) =>
    And
      (@Eq.{1} Int r
        (μ'
          (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) endpoints
            (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))))
      (And (ThreeSum.LightSortedBands.Prepared μ' values endpoints g original)
        (∀ (a : Nat),
          @LT.lt.{0} Nat instLTNat a work →
            Or (@LT.lt.{0} Nat instLTNat a src)
                (@LE.le.{0} Nat instLENat
                  (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) src (@List.length.{0} Int original))
                  a) →
              Or (@LT.lt.{0} Nat instLTNat a values)
                  (@LE.le.{0} Nat instLENat
                    (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) values
                      (@List.length.{0} Int original))
                    a) →
                Or (@LT.lt.{0} Nat instLTNat a endpoints)
                    (@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) endpoints
                          (@OfNat.ofNat.{0} Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))
                        (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
                          (@OfNat.ofNat.{0} Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))
                          (@List.length.{0} Int original)))
                      a) →
                  @Eq.{1} Int (μ' a) (μ a))) := by
  sorry

Source

https://github.com/c0ldzy17/three-sum-lean/blob/2b639655e07421b9ac743de9090459bab473a7c0/Submission/ThreeSum/LightSortedBands.lean#L63-L130

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