Prove2Me

Decide the original zero-sum witness for one selected color

Proved
ThreeSum.LightThreeSumColor.program_meets

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

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

Let x:[n]→Zx:[n]\to\mathbb Z satisfy ∣xi∣≤nk|x_i|\le n^k, and choose a color parameter cc at width inputBits⁡(n)+1\operatorname{inputBits}(n)+1. Assume Ready⁡(k,n)\operatorname{Ready}(k,n): positive size and micro-half, radix at least four, the scheduled divisor-budget inequality, and an auxiliary modulus failing to divide every nonzero integer of magnitude at most 3nk3n^k. Store the input, computed nineteen-cell parameter header, and six color fields below free address ff. If the color resource need at magnitude nkn^k fits the limits from f,df,d, the fixed program returns

r∈{0,1},r=1  ⟺  ∃i,j,ℓ: c(i)=0, c(j)=1, c(ℓ)=2, xi+xj+xℓ=0.r\in\{0,1\},\qquad r=1\iff\exists i,j,\ell:\ c(i)=0,\ c(j)=1,\ c(\ell)=2,\ x_i+x_j+x_\ell=0.

The three positions are distinct because their colors differ. Memory below ff is preserved, and the runtime is bounded by colorBudget⁡(k,n)\operatorname{colorBudget}(k,n), comprising the reporting prelude, dispatch budget, and 50005000 steps.

This closes the connection from the executed dense/reporting decision to the original selected-color witness.

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 Definitions.Def_ThreeSum1962Source_Bundle08
import Definitions.Def_ThreeSum1962Source_Bundle09
import Definitions.Def_ThreeSum1962Source_Bundle10
import Definitions.Def_ThreeSum1962Source_Bundle11
import Definitions.Def_ThreeSum1962Source_Bundle12
import Definitions.Def_ThreeSum1962Source_Bundle13
import Definitions.Def_ThreeSum1962Source_Bundle14
import Definitions.Def_ThreeSum1962Source_Bundle15
import Definitions.Def_ThreeSum1962Source_Bundle16
import Definitions.Def_ThreeSum1962Source_Bundle17
import Definitions.Def_ThreeSum1962Source_Bundle18
import Definitions.Def_TrulySubcubicAPSP_SourceSpecification
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.GCDMonoid.Nat
import Mathlib.Algebra.Group.Equiv.Basic
import Mathlib.Algebra.Group.Int.Defs
import Mathlib.Algebra.GroupWithZero.Nat
import Mathlib.Algebra.MvPolynomial.Basic
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.BigOperators.GroupWithZero.List
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.Nat
import Mathlib.Algebra.Order.Ring.Pow
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.Algebra.Polynomial.Coeff
import Mathlib.Algebra.Ring.Nat
import Mathlib.Analysis.Complex.ExponentialBounds
import Mathlib.Analysis.Convex.Deriv
import Mathlib.Analysis.SpecialFunctions.BinaryEntropy
import Mathlib.Analysis.SpecialFunctions.Log.Base
import Mathlib.Analysis.SpecialFunctions.Log.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics
import Mathlib.Analysis.SpecialFunctions.Pow.NthRootLemmas
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.Combinatorics.Colex
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Data.Bool.Basic
import Mathlib.Data.Fin.Embedding
import Mathlib.Data.Fin.VecNotation
import Mathlib.Data.Finset.Basic
import Mathlib.Data.Finset.Card
import Mathlib.Data.Finset.Image
import Mathlib.Data.Finset.Max
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.Fin
import Mathlib.Data.Fintype.Pi
import Mathlib.Data.Fintype.Prod
import Mathlib.Data.Fintype.Sets
import Mathlib.Data.Fintype.Sum
import Mathlib.Data.Int.Basic
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.Dedup
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.Range
import Mathlib.Data.List.Sublists
import Mathlib.Data.List.TakeDrop
import Mathlib.Data.Nat.Bitwise
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.Choose.Vandermonde
import Mathlib.Data.Nat.Count
import Mathlib.Data.Nat.GCD.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Nat.Notation
import Mathlib.Data.Nat.Prime.Basic
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.Notation
import Mathlib.LinearAlgebra.Matrix.ToLin
import Mathlib.Logic.Equiv.Fin.Basic
import Mathlib.Logic.Function.Basic
import Mathlib.MeasureTheory.Integral.Bochner.Basic
import Mathlib.NumberTheory.Bertrand
import Mathlib.NumberTheory.Chebyshev
import Mathlib.NumberTheory.PrimeCounting
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.Monotone.Basic
import Mathlib.Probability.Independence.Basic
import Mathlib.RingTheory.Coprime.Lemmas
import Mathlib.Tactic
import Mathlib.Tactic.Abel
import Mathlib.Tactic.ByContra
import Mathlib.Tactic.DeriveFintype
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.FinCases
import Mathlib.Tactic.GCongr
import Mathlib.Tactic.IntervalCases
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.LinearCombination
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_ThreeSum_BandPrimeChoice_modulus_score_bound
import Theorems.Thm_ThreeSum_BandRawA_mem_aIndices_iff_owner
import Theorems.Thm_ThreeSum_BandRawA_mem_allProducts
import Theorems.Thm_ThreeSum_BandRawA_mem_row
import Theorems.Thm_ThreeSum_BandRawA_owner_weightedFalseCount
import Theorems.Thm_ThreeSum_BandWeightedScore_block_bound
import Theorems.Thm_ThreeSum_EncodedColorPipelineData_ordered
import Theorems.Thm_ThreeSum_EncodedColorReportingData_report_le_source_allocation
import Theorems.Thm_ThreeSum_LightEncodedColorReportingPipeline_program_meets
import Theorems.Thm_ThreeSum_LightLocalCoverMagnitude_rounds_bound
import Theorems.Thm_ThreeSum_LightPrimePool_prefix_increasing
import Theorems.Thm_ThreeSum_LightReportingWantedScanStored_program_meets
import Theorems.Thm_ThreeSum_LightThreeSumWantedHost_program_meets
import Theorems.Thm_ThreeSum_Scales_primeDepth_loss_bound
import Theorems.Thm_ThreeSum_Scales_scoreLoss_finite_bound
import Theorems.Thm_ThreeSum_SharedReportingBankResourceBounds_components
import Theorems.Thm_ThreeSum_SharedReportingBankResourceBounds_offset_le
import Theorems.Thm_ThreeSum_SharedReportingBankResourceBounds_wanted_threshold_le
import Theorems.Thm_ThreeSum_SharedTraversalCosts_prefix_time_le_dispatch_budget
import Theorems.Thm_ThreeSum_ThreeSumColorBudgets_wanted_count_le
import Theorems.Thm_ThreeSum_ThreeSumWantedHostCosts_physical_step_le
import Theorems.Thm_ThreeSum_ThreeSumWantedHostDecision_accepted_iff_witness


set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option Elab.async false
namespace Light
end Light
namespace ThreeSum.ArithmeticStageBounds
end ThreeSum.ArithmeticStageBounds
namespace ThreeSum.BandCandidateRows
end ThreeSum.BandCandidateRows
namespace ThreeSum.BandEnumeration
end ThreeSum.BandEnumeration
namespace ThreeSum.BandPartition
end ThreeSum.BandPartition
namespace ThreeSum.BandRawA
end ThreeSum.BandRawA
namespace ThreeSum.BandScales
end ThreeSum.BandScales
namespace ThreeSum.BandWeightedScore
end ThreeSum.BandWeightedScore
namespace ThreeSum.BlockEncoding
end ThreeSum.BlockEncoding
namespace ThreeSum.BlockIndexedQuery
end ThreeSum.BlockIndexedQuery
namespace ThreeSum.ColorClassLists
end ThreeSum.ColorClassLists
namespace ThreeSum.CostLedger
end ThreeSum.CostLedger
namespace ThreeSum.EncodedColorLayout
end ThreeSum.EncodedColorLayout
namespace ThreeSum.EncodedColorNormalizedData
end ThreeSum.EncodedColorNormalizedData
namespace ThreeSum.EncodedColorWantedConsumerData
end ThreeSum.EncodedColorWantedConsumerData
namespace ThreeSum.GlobalEncoding
end ThreeSum.GlobalEncoding
namespace ThreeSum.HierarchyScales
end ThreeSum.HierarchyScales
namespace ThreeSum.LightClassFilter
end ThreeSum.LightClassFilter
namespace ThreeSum.LightLocalCoverLayout
end ThreeSum.LightLocalCoverLayout
namespace ThreeSum.LightLocalCoverSchedule
end ThreeSum.LightLocalCoverSchedule
namespace ThreeSum.LightReportingPrefixCatalog
end ThreeSum.LightReportingPrefixCatalog
namespace ThreeSum.LightReportingSuffixCatalog
end ThreeSum.LightReportingSuffixCatalog
namespace ThreeSum.LocalCoverBounds
end ThreeSum.LocalCoverBounds
namespace ThreeSum.MassScales
end ThreeSum.MassScales
namespace ThreeSum.OutputDecodeRecords
end ThreeSum.OutputDecodeRecords
namespace ThreeSum.Parameters
end ThreeSum.Parameters
namespace ThreeSum.RawIncidences
end ThreeSum.RawIncidences
namespace ThreeSum.RawReporting
end ThreeSum.RawReporting
namespace ThreeSum.ReportingHostFrontierLayout
end ThreeSum.ReportingHostFrontierLayout
namespace ThreeSum.ReportingHostLevelData
end ThreeSum.ReportingHostLevelData
namespace ThreeSum.ReportingHostWantedConsumerData
end ThreeSum.ReportingHostWantedConsumerData
namespace ThreeSum.ReportingHostWantedLayout
end ThreeSum.ReportingHostWantedLayout
namespace ThreeSum.ReportingHostWantedOriginal
end ThreeSum.ReportingHostWantedOriginal
namespace ThreeSum.ReportingHostWantedPhysical
end ThreeSum.ReportingHostWantedPhysical
namespace ThreeSum.Scales
end ThreeSum.Scales
namespace ThreeSum.SharedReportingBankData
end ThreeSum.SharedReportingBankData
namespace ThreeSum.SharedReportingBankHeader
end ThreeSum.SharedReportingBankHeader
namespace ThreeSum.SharedReportingBankLayout
end ThreeSum.SharedReportingBankLayout
namespace ThreeSum.SharedReportingBankLevels
end ThreeSum.SharedReportingBankLevels
namespace ThreeSum.SharedReportingBankResourceBounds
end ThreeSum.SharedReportingBankResourceBounds
namespace ThreeSum.SharedReportingBankRows
end ThreeSum.SharedReportingBankRows
namespace ThreeSum.SharedTraversalCosts
end ThreeSum.SharedTraversalCosts
namespace ThreeSum.StageBounds
end ThreeSum.StageBounds
namespace ThreeSum.SubsetAddress
end ThreeSum.SubsetAddress
namespace ThreeSum.Switch
end ThreeSum.Switch
namespace ThreeSum.ThreeSumReportingSetupData
end ThreeSum.ThreeSumReportingSetupData
namespace ThreeSum.ThreeSumWantedHostConditions
end ThreeSum.ThreeSumWantedHostConditions
namespace ThreeSum.ThreeSumWantedHostData
end ThreeSum.ThreeSumWantedHostData
namespace ThreeSum.TrimScales
end ThreeSum.TrimScales
namespace ThreeSum.WantedRecordOrder
end ThreeSum.WantedRecordOrder
namespace ThreeSumApsp
end ThreeSumApsp
Formal statement
theorem ThreeSum.LightThreeSumColor.program_meets : ∀ {lim : Light.Limits} {μ : Nat → Int} {src n params row free k d : Nat}
  (color :
    ThreeSum.Coloring.Parameter
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) (ThreeSum.Scales.inputBits n)
        (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))
  (x : Fin n → Int) (hn : ThreeSum.ThreeSumWantedHostConditions.Ready k n)
  (hsrc : @LE.le.{0} Nat instLENat (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) src n) free)
  (hparams :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) params
        (@OfNat.ofNat.{0} Nat (nat_lit 19) (instOfNatNat (nat_lit 19))))
      free)
  (hrow :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) row
        (@OfNat.ofNat.{0} Nat (nat_lit 6) (instOfNatNat (nat_lit 6))))
      free)
  (hok :
    (ThreeSum.ThreeSumFastResourcesColor.need n
          (@HPow.hPow.{0, 0, 0} Nat Nat Nat
            (@instHPow.{0, 0} Nat Nat (@NPow.toPow.{0} Nat (@Monoid.toNPow.{0} Nat Nat.instMonoid))) n k)).Ok
      lim free d)
  (hread : Light.Seg μ src (@List.ofFn.{0} Int n x))
  (hx :
    ∀ (i : Fin n),
      @LE.le.{0} Int Int.instLEInt (@abs.{0} Int instLatticeInt Int.instAddGroup (x i))
        (@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))) n k)))
  (hparameters : Light.Seg μ params (ThreeSum.LightParameterInit.values n))
  (hcolor :
    Light.Seg μ row
      (@ThreeSum.ColorParameters.fields
        (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) (ThreeSum.Scales.inputBits n)
          (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))
        color)),
  Light.Meets lim ThreeSum.LightThreeSumColor.program (@OfNat.ofNat.{0} Nat (nat_lit 509) (instOfNatNat (nat_lit 509)))
    d
    (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt src)
      (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt n)
        (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt params)
          (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt row)
            (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt free) (@List.nil.{0} Int))))))
    μ (ThreeSum.SharedTraversalCosts.colorBudget k n) fun (r : Int) (ξ : Nat → Int) =>
    And
      (Or (@Eq.{1} Int r (@OfNat.ofNat.{0} Int (nat_lit 0) (@instOfNat (nat_lit 0))))
        (@Eq.{1} Int r (@OfNat.ofNat.{0} Int (nat_lit 1) (@instOfNat (nat_lit 1)))))
      (And
        (Iff (@Eq.{1} Int r (@OfNat.ofNat.{0} Int (nat_lit 1) (@instOfNat (nat_lit 1))))
          (@ThreeSum.ColorClassLists.witness n
            (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) (ThreeSum.Scales.inputBits n)
              (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))
            color x))
        (Light.Kept μ ξ free)) := by
  sorry
Source
https://github.com/c0ldzy17/three-sum-lean/blob/2b639655e07421b9ac743de9090459bab473a7c0/Submission/ThreeSum/LightThreeSumColor.lean#L35-L85

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