Normalizing a color class and constructing its retained prime data

Proved

ThreeSum.LightEncodedColorPrimeHost.program_meets_input

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

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

Write $S,\Omega,\Delta$ for machine space, word magnitude, and depth limits. Let $n>0$ and $x:\{0,\ldots,n-1\}\to\mathbb Z$ satisfy $|x_i|\le U$ . Fix a coloring parameter of width $\operatorname{inputBits}(n)+1$ . Memory stores the input array, exact 19-word parameter record, and six coloring fields, all ending before free address $f$ .

Assume $\operatorname{radix}(n)\ge4$ , the input-host space and word bounds, $S\le\Omega$ , and the specified input-host depth bound. The routine returns

r=\operatorname{prime}(f,n,\operatorname{bandSize}(n)),

the address of the resulting prime-data record. It retains strictly increasing normalized lists for color classes $0,1,2$ , with the third class's values negated, together with their band structures, canonical owner records, raw- $A$ incidence count, prime pool, and full common-modulus chain. The normalized header and prime-data record satisfy the exact combined output predicate. Every cell below $f$ is preserved, within the stated input-host time budget.

This prepares the normalized sources and modulus data for one encoded coloring.

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_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_EncodedColorParameterData_reads
import Theorems.Thm_ThreeSum_LightEncodedColorNormalize_program_meets
import Theorems.Thm_ThreeSum_LightEncodedColorPrime_program_meets
import Theorems.Thm_ThreeSum_SourceFiniteData_Normalized_keep


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.BandRawA
end ThreeSum.BandRawA
namespace ThreeSum.BandScales
end ThreeSum.BandScales
namespace ThreeSum.ColorClassLists
end ThreeSum.ColorClassLists
namespace ThreeSum.CostLedger
end ThreeSum.CostLedger
namespace ThreeSum.EncodedColorLayout
end ThreeSum.EncodedColorLayout
namespace ThreeSum.LightClassFilter
end ThreeSum.LightClassFilter
namespace ThreeSum.Scales
end ThreeSum.Scales
namespace ThreeSumApsp
end ThreeSumApsp

Formal statement

theorem ThreeSum.LightEncodedColorPrimeHost.program_meets_input : ∀ {lim : Light.Limits} {d : Nat} {μ : Nat → Int} {src n params row free U : 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) (hw : @LE.le.{0} Int Int.instLEInt (@Nat.cast.{0} Int instNatCastInt lim.space) lim.word)
  (hn : @LT.lt.{0} Nat instLTNat (@OfNat.ofNat.{0} Nat (nat_lit 0) (instOfNatNat (nat_lit 0))) n)
  (hR :
    @LE.le.{0} Nat instLENat (@OfNat.ofNat.{0} Nat (nat_lit 4) (instOfNatNat (nat_lit 4))) (ThreeSum.Scales.radix 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)
  (hspace : @LE.le.{0} Nat instLENat (ThreeSum.LightEncodedColorPrimeHost.inputSpace free n U) lim.space)
  (hword :
    @LE.le.{0} Int Int.instLEInt (@Nat.cast.{0} Int instNatCastInt (ThreeSum.LightEncodedColorPrimeHost.inputWord n U))
      lim.word)
  (hd :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) d (ThreeSum.LightEncodedColorPrimeHost.depth n))
      lim.depth)
  (hread : Light.Seg μ src (@List.ofFn.{0} Int n x))
  (hbound :
    ∀ (i : Fin n),
      @LE.le.{0} Int Int.instLEInt (@abs.{0} Int instLatticeInt Int.instAddGroup (x i))
        (@Nat.cast.{0} Int instNatCastInt U))
  (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.LightEncodedColorPrimeHost.program
    (@OfNat.ofNat.{0} Nat (nat_lit 71) (instOfNatNat (nat_lit 71))) 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.LightEncodedColorPrimeHost.inputTime n U) fun (r : Int) (ν : Nat → Int) =>
    And
      (@Eq.{1} Int r
        (@Nat.cast.{0} Int instNatCastInt (ThreeSum.EncodedColorLayout.prime free n (ThreeSum.BandScales.bandSize n))))
      (And (ThreeSum.EncodedColorPrimeReadyData.Output ν free n color x) (Light.Kept μ ν free)) := by
  sorry

Source

https://github.com/c0ldzy17/three-sum-lean/blob/2b639655e07421b9ac743de9090459bab473a7c0/Submission/ThreeSum/LightEncodedColorPrimeHost.lean#L61-L158

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