Verified gathering of independently flagged source histogram values

Proved

ThreeSum.LightSourceGather.meets

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

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

Let $m>0$ , let endpoints $x_i\in\mathbb Z$ satisfy $|x_i|\le U$ for $i<n$ , let $b_i\in\{0,1\}$ , and let $H:[m]\to\mathbb N$ satisfy $H(r)\le B$ . Suppose memory $\mu$ stores these arrays in segments ending at or before output base $z$ . Let $z+n\le f$ , $f+\lfloor\log_2(U+1)\rfloor+2\le S\le W$ , $4U+4m+16\le W$ , $B\le W$ , and $d<D$ . If $P$ contains the source-gather body and Euclidean-remainder helper, execution finishes within $n(R_U+80)+6$ steps, with $R_U=43(\lfloor\log_2(U+1)\rfloor+1)+34$ , and stores

\nu(z+i)=b_i H(x_i\bmod m)\quad(i<n).

All addresses below $f$ outside $[z,z+n)$ are preserved. This gathers retained source weights from a precomputed histogram without recounting its original inputs.

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_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.Group.Finset.Powerset
import Mathlib.Algebra.BigOperators.Group.Finset.Sigma
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Algebra.Field.ZMod
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.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.Algebra.Ring.Nat
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.Finset.Range
import Mathlib.Data.Finset.Sigma
import Mathlib.Data.Finset.Sort
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Fintype.Fin
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.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.Central
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.Data.Nat.Count
import Mathlib.Data.Nat.Log
import Mathlib.Data.Nat.Notation
import Mathlib.Data.Nat.Size
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.Equiv.Fin.Basic
import Mathlib.Logic.Function.Basic
import Mathlib.NumberTheory.Bertrand
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.RingTheory.Coprime.Lemmas
import Mathlib.Tactic.Abel
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 Mathlib.Tactic.Tauto
import Theorems.Thm_Light_Ends_for
import Theorems.Thm_Light_emod_meets


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

Formal statement

theorem ThreeSum.LightSourceGather.meets : ∀ {lim : Light.Limits} {P : Light.Program} {d pMain pEmod : 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.LightSourceGather.body 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))
  {μ : Nat → Int} {raw flags table dst n m work U B : Nat} (endpoint : Nat → Int) (flag hist : 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)
  (hm : @LT.lt.{0} Nat instLTNat (@OfNat.ofNat.{0} Nat (nat_lit 0) (instOfNatNat (nat_lit 0))) m)
  (hraw : @LE.le.{0} Nat instLENat (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) raw n) dst)
  (hflags : @LE.le.{0} Nat instLENat (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) flags n) dst)
  (htable : @LE.le.{0} Nat instLENat (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) table m) dst)
  (hdst : @LE.le.{0} Nat instLENat (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) dst 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)
  (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 m)))
        (@OfNat.ofNat.{0} Int (nat_lit 16) (@instOfNat (nat_lit 16))))
      lim.word)
  (hcountWord : @LE.le.{0} Int Int.instLEInt (@Nat.cast.{0} Int instNatCastInt B) lim.word)
  (hrawRead :
    ∀ (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) raw i)) (endpoint i))
  (hrawBound :
    ∀ (i : Nat),
      @LT.lt.{0} Nat instLTNat i n →
        @LE.le.{0} Int Int.instLEInt (@abs.{0} Int instLatticeInt Int.instAddGroup (endpoint i))
          (@Nat.cast.{0} Int instNatCastInt U))
  (hflagRead :
    ∀ (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) flags i))
          (@Nat.cast.{0} Int instNatCastInt (flag i)))
  (hflagBound :
    ∀ (i : Nat),
      @LT.lt.{0} Nat instLTNat i n →
        @LE.le.{0} Nat instLENat (flag i) (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))
  (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) table r))
          (@Nat.cast.{0} Int instNatCastInt (hist r)))
  (hhistBound : ∀ (r : Nat), @LT.lt.{0} Nat instLTNat r m → @LE.le.{0} Nat instLENat (hist r) B),
  Light.Meets lim P pMain d
    (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt raw)
      (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt flags)
        (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt table)
          (@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 m)
                (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt work) (@List.nil.{0} Int))))))))
    μ (ThreeSum.LightSourceGather.time n U) fun (x : Int) (ν : Nat → Int) =>
    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) dst i))
            (@Nat.cast.{0} Int instNatCastInt (ThreeSum.LightSourceGather.value m endpoint flag hist i)))
      (Light.KeptBut μ ν work dst n) := by
  sorry

Source

https://github.com/c0ldzy17/three-sum-lean/blob/2b639655e07421b9ac743de9090459bab473a7c0/Submission/ThreeSum/LightSourceGather.lean#L48-L133

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