Prove2Me

Verified output-sensitive join of left keys with stored right buckets

Proved
ThreeSum.LightWantedJoin.meets

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

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

Let natural right keys rj<Mr_j<M for j<NRj<N_R and left keys ℓi<M\ell_i<M for i<ni<n. Suppose integer memory μ\mu stores the left-key array at aa and the exact right-index bucket representation at offset and flat bases o,fo,f. Let zz be the pair-output base, with a+n,o+M+1,f+NR≤za+n,o+M+1,f+N_R\le z. Put C=∑i<n∣{j<NR:rj=ℓi}∣C=\sum_{i<n}|\{j<N_R:r_j=\ell_i\}|. Assume z+2C≤S≤Wz+2C\le S\le W, d+n+2≤Dd+n+2\le D, and that PP contains the recursive join and wanted-row copier. Execution on (a,o,f,n,NR,M,z)(a,o,f,n,N_R,M,z) finishes within 120C+270n+50120C+270n+50 steps and yields

r=C,ν[z:z+2C]=flatten⁡([(i,j):i<n, j in stored bucket ℓi]).r=C,\qquad \nu[z:z+2C]=\operatorname{flatten}\bigl([(i,j):i<n,\ j\text{ in stored bucket }\ell_i]\bigr).

Rows follow left-index order and each row follows bucket order. Memory below zz is preserved. This enumerates exactly the matching physical index pairs.

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_Bundle06
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.Piecewise
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.Polynomial.Basic
import Mathlib.Algebra.Polynomial.Coeff
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.EquivFin
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.Int.SuccPred
import Mathlib.Data.List.Basic
import Mathlib.Data.List.FinRange
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.List.ProdSigma
import Mathlib.Data.List.TakeDrop
import Mathlib.Data.Nat.Cast.Order.Ring
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Data.Nat.Choose.Bounds
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_seg_append
import Theorems.Thm_ThreeSum_LightWantedRow_meets
import Theorems.Thm_ThreeSum_SourceBuckets_position_bucket_iff
import Theorems.Thm_ThreeSum_SourceBuckets_position_injective
import Theorems.Thm_ThreeSum_WantedPairs_row_segment


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
namespace ThreeSum.WantedPairs
end ThreeSum.WantedPairs
namespace ThreeSumApsp
end ThreeSumApsp
Formal statement
theorem ThreeSum.LightWantedJoin.meets : ∀ {lim : Light.Limits} {P : Light.Program} {pMain pRow keys offsets flat NR M dst : 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.LightWantedJoin.body pMain pRow)))
  (hrow :
    @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 pRow)
      (@Option.some.{0} Light.Stmt ThreeSum.LightWantedRow.body))
  (left right : Nat → Nat) (hright : ∀ (j : Nat), @LT.lt.{0} Nat instLTNat j NR → @LT.lt.{0} Nat instLTNat (right j) M)
  (n d : Nat) (μ : Nat → Int),
  @LE.le.{0} Int Int.instLEInt (@Nat.cast.{0} Int instNatCastInt lim.space) lim.word →
    @LE.le.{0} Nat instLENat (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) keys n) dst →
      @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))))
          dst →
        @LE.le.{0} Nat instLENat (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) flat NR) dst →
          @LE.le.{0} Nat instLENat
              (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) dst
                (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
                  (@OfNat.ofNat.{0} Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))
                  (ThreeSum.WantedPairs.count left right NR n)))
              lim.space →
            @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 n)
                  (@OfNat.ofNat.{0} Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))
                lim.depth →
              (∀ (i : Nat), @LT.lt.{0} Nat instLTNat i n → @LT.lt.{0} Nat instLTNat (left i) M) →
                (∀ (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 (left i))) →
                  ThreeSum.SourceBuckets.Stored μ right NR M offsets flat →
                    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 offsets)
                          (@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 NR)
                                (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt M)
                                  (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt dst) (@List.nil.{0} Int))))))))
                      μ (ThreeSum.LightWantedJoin.time left right NR n) fun (r : Int) (ν : Nat → Int) =>
                      And
                        (@Eq.{1} Int r (@Nat.cast.{0} Int instNatCastInt (ThreeSum.WantedPairs.count left right NR n)))
                        (And
                          (Light.Seg ν dst
                            (ThreeSum.WantedPairs.cells (ThreeSum.WantedPairs.pairs μ offsets flat left n)))
                          (Light.Kept μ ν dst)) := by
  sorry
Source
https://github.com/c0ldzy17/three-sum-lean/blob/2b639655e07421b9ac743de9090459bab473a7c0/Submission/ThreeSum/LightWantedJoin.lean#L45-L136

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