Execution contract for a padded signed residue histogram

Proved

ThreeSum.LightResidueHistogram.meets

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

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

Let $x_i=\mu(s+i)$ for $i<n$ , with $|x_i|\le U$ , and let $c\in\mathbb Z$ satisfy $|c|\le1$ . Let $0<p\le q$ , where $q$ is the output capacity. The program contains the specified residue, remainder, histogram, and clearing routines. For space, word, and depth limits $S,W,D$ , assume $S\le W$ , $d+1<D$ , $s+n\le b$ , $a+q\le b$ , $b+n\le f$ , $f+\lfloor\log_2(U+1)\rfloor+2\le S$ , and $4U+4p+16\le W$ . Here $a,b,f$ are the histogram, code-array, and scratch bases.

The routine finishes within $n(R_U+54)+13q+53$ steps, where $R_U=43(\lfloor\log_2(U+1)\rfloor+1)+34$ , and writes

\nu(a+r)=\begin{cases}\#\{i<n:cx_i\bmod p=r\},&r<p,\\0,&p\le r<q.\end{cases}

Memory below $b$ outside $[a,a+q)$ is preserved.

This supplies the padded integer histogram used by subsequent modular counting routines.

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_Bundle05
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.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.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.Data.Finset.Card
import Mathlib.Data.Finset.Prod
import Mathlib.Data.Finset.Range
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Fintype.EquivFin
import Mathlib.Data.Fintype.Prod
import Mathlib.Data.Int.Cast.Lemmas
import Mathlib.Data.Int.Notation
import Mathlib.Data.List.GetD
import Mathlib.Data.List.MinMax
import Mathlib.Data.Nat.Cast.Order.Ring
import Mathlib.Data.Nat.Count
import Mathlib.Data.Nat.Log
import Mathlib.Data.Nat.Notation
import Mathlib.Data.Nat.Size
import Mathlib.Data.Nat.SuccPred
import Mathlib.Data.ZMod.Basic
import Mathlib.LinearAlgebra.Matrix.Defs
import Mathlib.Logic.Function.Basic
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Tactic.Abel
import Mathlib.Tactic.ByContra
import Mathlib.Tactic.GCongr
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.NormNum
import Mathlib.Tactic.Positivity
import Mathlib.Tactic.Ring
import Theorems.Thm_ThreeSum_LightHistogram_histogram_meets
import Theorems.Thm_ThreeSum_LightResidues_residues_ends


set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option Elab.async false
namespace Light
end Light
namespace ThreeSum.CyclicFold
end ThreeSum.CyclicFold
namespace ThreeSum.Karatsuba
end ThreeSum.Karatsuba
namespace ThreeSum.LightHistogram
end ThreeSum.LightHistogram
namespace ThreeSum.LightResidues
end ThreeSum.LightResidues

Formal statement

theorem ThreeSum.LightResidueHistogram.meets : ∀ {lim : Light.Limits} {P : Light.Program} {d pMain pResidues pEmod pHistogram pFill : 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.LightResidueHistogram.body pResidues pHistogram)))
  (hres :
    @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 pResidues)
      (@Option.some.{0} Light.Stmt (ThreeSum.LightResidues.residuesBody pEmod)))
  (hem :
    @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 pEmod)
      (@Option.some.{0} Light.Stmt Light.emodBody))
  (hhist :
    @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 pHistogram)
      (@Option.some.{0} Light.Stmt (ThreeSum.LightHistogram.histogramBody pFill)))
  (hfill :
    @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 pFill)
      (@Option.some.{0} Light.Stmt Light.fillBody))
  {μ : Nat → Int} {src dst n p cap codes work U : Nat} {c : Int}
  (hw : @LE.le.{0} Int Int.instLEInt (@Nat.cast.{0} Int instNatCastInt lim.space) lim.word)
  (hd :
    @LT.lt.{0} Nat instLTNat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) d
        (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))
      lim.depth)
  (hp : @LT.lt.{0} Nat instLTNat (@OfNat.ofNat.{0} Nat (nat_lit 0) (instOfNatNat (nat_lit 0))) p)
  (hcap : @LE.le.{0} Nat instLENat p cap)
  (hsrc : @LE.le.{0} Nat instLENat (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) src n) codes)
  (hdst : @LE.le.{0} Nat instLENat (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) dst cap) codes)
  (hcodes : @LE.le.{0} Nat instLENat (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) codes n) work)
  (hspace :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) work
        (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
          (Nat.log (@OfNat.ofNat.{0} Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))
            (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) U
              (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))
          (@OfNat.ofNat.{0} Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))))
      lim.space)
  (hc :
    @LE.le.{0} Int Int.instLEInt (@abs.{0} Int instLatticeInt Int.instAddGroup c)
      (@OfNat.ofNat.{0} Int (nat_lit 1) (@instOfNat (nat_lit 1))))
  (hword :
    @LE.le.{0} Int Int.instLEInt
      (@HAdd.hAdd.{0, 0, 0} Int Int Int (@instHAdd.{0} Int Int.instAdd)
        (@HAdd.hAdd.{0, 0, 0} Int Int Int (@instHAdd.{0} Int Int.instAdd)
          (@HMul.hMul.{0, 0, 0} Int Int Int (@instHMul.{0} Int Int.instMul)
            (@OfNat.ofNat.{0} Int (nat_lit 4) (@instOfNat (nat_lit 4))) (@Nat.cast.{0} Int instNatCastInt U))
          (@HMul.hMul.{0, 0, 0} Int Int Int (@instHMul.{0} Int Int.instMul)
            (@OfNat.ofNat.{0} Int (nat_lit 4) (@instOfNat (nat_lit 4))) (@Nat.cast.{0} Int instNatCastInt p)))
        (@OfNat.ofNat.{0} Int (nat_lit 16) (@instOfNat (nat_lit 16))))
      lim.word)
  (hvalues :
    ∀ (i : Nat),
      @LT.lt.{0} Nat instLTNat i n →
        @LE.le.{0} Int Int.instLEInt
          (@abs.{0} Int instLatticeInt Int.instAddGroup
            (μ (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) src i)))
          (@Nat.cast.{0} Int instNatCastInt U)),
  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 dst)
        (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt n)
          (@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 codes)
                (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt work)
                  (@List.cons.{0} Int c (@List.nil.{0} Int)))))))))
    μ (ThreeSum.LightResidueHistogram.time n cap U) fun (x : Int) (μ' : Nat → Int) =>
    And
      (Light.Seg μ' dst
        (ThreeSum.CyclicFold.padTo cap
          (ThreeSum.CyclicFold.vector p
            (ThreeSum.LightResidueHistogram.signedHistogram p c
              (fun (i : Nat) => μ (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) src i)) n))))
      (Light.KeptBut μ μ' codes dst cap) := by
  sorry

Source

https://github.com/c0ldzy17/three-sum-lean/blob/2b639655e07421b9ac743de9090459bab473a7c0/Submission/ThreeSum/LightResidueHistogram.lean#L121-L156

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