Prove2Me

Computing a union mask and its cardinality

Proved
ThreeSum.LightMaskUnion.meets

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

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

Let A,B⊆{0,…,W−1}A,B\subseteq\{0,\ldots,W-1\}, with masks m(S)=∑i∈S2im(S)=\sum_{i\in S}2^i. Suppose a Light program contains the union body, power-table body and recursive mask-clearing body. For result cell DD and scratch address FF, assume D+1≤FD+1\le F, F+W+1≤space≤wordF+W+1\le\mathrm{space}\le\mathrm{word}, 2W+1≤word2^{W+1}\le\mathrm{word}, and d+W+1≤depthd+W+1\le\mathrm{depth}.

The call with (W,∣A∣,m(A),m(B),D,F)(W,|A|,m(A),m(B),D,F) terminates within 200W2+257W+225200W^2+257W+225 steps and yields

r=m(A∪B),ν(D)=∣A∪B∣,ν(F+i)=2i(i≤W).r=m(A\cup B),\qquad \nu(D)=|A\cup B|,\qquad \nu(F+i)=2^i\quad(i\le W).

Every cell below FF, except DD, is preserved.

This implements union for arbitrary, possibly overlapping masks while tracking the exact cardinality.

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_Bundle04
import Definitions.Def_ThreeSum1962Source_Bundle05
import Definitions.Def_ThreeSum1962Source_Bundle07
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.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.Combinatorics.Colex
import Mathlib.Data.Fin.Embedding
import Mathlib.Data.Finset.Card
import Mathlib.Data.Finset.Max
import Mathlib.Data.Finset.Powerset
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.GetD
import Mathlib.Data.List.Nodup
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.Notation
import Mathlib.Data.Nat.SuccPred
import Mathlib.LinearAlgebra.Matrix.Defs
import Mathlib.Logic.Function.Basic
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Tactic.ByContra
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_for
import Theorems.Thm_Light_powTable_ends
import Theorems.Thm_ThreeSum_LightMaskErase_meets
import Theorems.Thm_ThreeSum_MaskErase_eraseMask_le
import Theorems.Thm_ThreeSum_MaskErase_eraseMask_maskOf


set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option Elab.async false
namespace Light
end Light
namespace Light.PowTable
end Light.PowTable
namespace ThreeSum.LightPascal
end ThreeSum.LightPascal
namespace ThreeSum.LightSubsetAddress
end ThreeSum.LightSubsetAddress
namespace ThreeSum.MaskErase
end ThreeSum.MaskErase
namespace ThreeSum.MaskUnion
end ThreeSum.MaskUnion
namespace ThreeSum.SubsetAddress
end ThreeSum.SubsetAddress
Formal statement
theorem ThreeSum.LightMaskUnion.meets : ∀ {lim : Light.Limits} {P : Light.Program} {pMain pPowers pErase : 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.LightMaskUnion.body pPowers pErase)))
  (hPowers :
    @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 pPowers)
      (@Option.some.{0} Light.Stmt Light.powTableBody))
  (hErase :
    @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 pErase)
      (@Option.some.{0} Light.Stmt (ThreeSum.LightMaskErase.body pErase)))
  {μ : Nat → Int} {W dst work d : Nat} (A B : Finset.{0} (Fin W))
  (hw : @LE.le.{0} Int Int.instLEInt (@Nat.cast.{0} Int instNatCastInt lim.space) lim.word)
  (hdst :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) dst
        (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))
      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) W
          (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))
      lim.space)
  (hword :
    @LE.le.{0} Int Int.instLEInt
      (@Nat.cast.{0} Int instNatCastInt
        (@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)))
          (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) W
            (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))))
      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 W)
        (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))
      lim.depth),
  Light.Meets lim P pMain d
    (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt W)
      (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt (@Finset.card.{0} (Fin W) A))
        (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt (@ThreeSum.SubsetAddress.subsetMask W A))
          (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt (@ThreeSum.SubsetAddress.subsetMask W B))
            (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt dst)
              (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt work) (@List.nil.{0} Int)))))))
    μ (ThreeSum.LightMaskUnion.time W) fun (r : Int) (μ' : Nat → Int) =>
    And
      (@Eq.{1} Int r
        (@Nat.cast.{0} Int instNatCastInt
          (@ThreeSum.SubsetAddress.subsetMask W
            (@Union.union.{0} (Finset.{0} (Fin W)) (@Finset.instUnion.{0} (Fin W) (instDecidableEqFin W)) A B))))
      (And
        (@Eq.{1} Int (μ' dst)
          (@Nat.cast.{0} Int instNatCastInt
            (@Finset.card.{0} (Fin W)
              (@Union.union.{0} (Finset.{0} (Fin W)) (@Finset.instUnion.{0} (Fin W) (instDecidableEqFin W)) A B))))
        (And (ThreeSum.LightSubsetAddress.Powers μ' work W)
          (Light.KeptBut μ μ' work dst (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))) := by
  sorry
Source
https://github.com/c0ldzy17/three-sum-lean/blob/2b639655e07421b9ac743de9090459bab473a7c0/Submission/ThreeSum/LightMaskUnion.lean#L46-L161

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