Prove2Me

Reading the initialized global parameter block

Proved
ThreeSum.EncodedColorParameterData.reads

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

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

Let n,q∈Nn,q\in\mathbb N, and suppose memory μ:N→Z\mu:\mathbb N\to\mathbb Z stores the parameter initialization list for input size nn consecutively at base qq. Write b=inputBits⁡(n)b=\operatorname{inputBits}(n), h=microHalf⁡(b)h=\operatorname{microHalf}(b), J=blockCount⁡(b)J=\operatorname{blockCount}(b), u=mu⁡(n)u=\operatorname{mu}(n), R=radix⁡(n)R=\operatorname{radix}(n), g=bandSize⁡(n)g=\operatorname{bandSize}(n), and y=primeWindow⁡(n)y=\operatorname{primeWindow}(n). Then

(μ(q),μ(q+1),μ(q+3),μ(q+4))=(b,h,J,u),(μ(q+14),μ(q+17),μ(q+18))=(R,g,y).\begin{aligned} (\mu(q),\mu(q+1),\mu(q+3),\mu(q+4))&=(b,h,J,u),\\ (\mu(q+14),\mu(q+17),\mu(q+18))&=(R,g,y). \end{aligned}

The natural parameter values are embedded in the integers. This provides the exact memory reads used to pass the initialized scales to later stages of the color pipeline.

Preamble
import Aesop
import Definitions.Def_APSPSource_RemainingDefinitions
import Definitions.Def_APSPSource_ThreeSumApsp_Lang_Logic
import Definitions.Def_APSPSource_ThreeSumApsp_Lang_Regions
import Definitions.Def_ThreeSum1962Source_Bundle01
import Definitions.Def_ThreeSum1962Source_Bundle07
import Definitions.Def_ThreeSum1962Source_Bundle08
import Definitions.Def_ThreeSum1962Source_Bundle11
import Definitions.Def_ThreeSum1962Source_Bundle12
import Lean.Elab.Tactic.Omega
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
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.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.Analysis.Complex.ExponentialBounds
import Mathlib.Analysis.Convex.Deriv
import Mathlib.Analysis.SpecialFunctions.BinaryEntropy
import Mathlib.Analysis.SpecialFunctions.Log.Base
import Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics
import Mathlib.Analysis.SpecialFunctions.Pow.NthRootLemmas
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Analysis.SpecificLimits.Normed
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.Image
import Mathlib.Data.Finset.Powerset
import Mathlib.Data.Finset.Prod
import Mathlib.Data.Finset.Range
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.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.FinRange
import Mathlib.Data.List.GetD
import Mathlib.Data.List.MinMax
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Pi
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.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.ToLin
import Mathlib.Logic.Function.Basic
import Mathlib.NumberTheory.Bertrand
import Mathlib.NumberTheory.Chebyshev
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.Monotone.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


set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option Elab.async false
namespace Light
end Light
namespace ThreeSum.BandScales
end ThreeSum.BandScales
namespace ThreeSum.LocalCoverBounds
end ThreeSum.LocalCoverBounds
namespace ThreeSum.Scales
end ThreeSum.Scales
Formal statement
theorem ThreeSum.EncodedColorParameterData.reads : ∀ {μ : Nat → Int} {params n : Nat} (H : Light.Seg μ params (ThreeSum.LightParameterInit.values n)),
  And (@Eq.{1} Int (μ params) (@Nat.cast.{0} Int instNatCastInt (ThreeSum.Scales.inputBits n)))
    (And
      (@Eq.{1} Int
        (μ
          (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) params
            (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))
        (@Nat.cast.{0} Int instNatCastInt (ThreeSum.Scales.microHalf (ThreeSum.Scales.inputBits n))))
      (And
        (@Eq.{1} Int
          (μ
            (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) params
              (@OfNat.ofNat.{0} Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))))
          (@Nat.cast.{0} Int instNatCastInt (ThreeSum.Scales.blockCount (ThreeSum.Scales.inputBits n))))
        (And
          (@Eq.{1} Int
            (μ
              (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) params
                (@OfNat.ofNat.{0} Nat (nat_lit 4) (instOfNatNat (nat_lit 4)))))
            (@Nat.cast.{0} Int instNatCastInt (ThreeSum.Scales.mu n)))
          (And
            (@Eq.{1} Int
              (μ
                (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) params
                  (@OfNat.ofNat.{0} Nat (nat_lit 14) (instOfNatNat (nat_lit 14)))))
              (@Nat.cast.{0} Int instNatCastInt (ThreeSum.Scales.radix n)))
            (And
              (@Eq.{1} Int
                (μ
                  (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) params
                    (@OfNat.ofNat.{0} Nat (nat_lit 17) (instOfNatNat (nat_lit 17)))))
                (@Nat.cast.{0} Int instNatCastInt (ThreeSum.BandScales.bandSize n)))
              (@Eq.{1} Int
                (μ
                  (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) params
                    (@OfNat.ofNat.{0} Nat (nat_lit 18) (instOfNatNat (nat_lit 18)))))
                (@Nat.cast.{0} Int instNatCastInt (ThreeSum.Scales.primeWindow n)))))))) := by
  sorry
Source
https://github.com/c0ldzy17/three-sum-lean/blob/2b639655e07421b9ac743de9090459bab473a7c0/Submission/ThreeSum/EncodedColorParameterData.lean#L10-L28

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