A Pascal-table program returns an exact binomial coefficient

Proved

ThreeSum.LightReportingBinomial.program_meets

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

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

Let $0\le k\le n$ be natural numbers and let $f$ be a scratch address. For machine limits $(S,V,L)$ , assume $S\le V$ , $f+(n+1)^2\le S$ , $2^{n+1}\le V$ , and $d+3\le L$ . The reporting-binomial program, called with $(n,k,f)$ at depth $d$ , returns

r=\binom nk

within $\operatorname{buildTime}(n)+100$ steps. Its final memory agrees with the initial memory outside the scratch interval $[f,f+(n+1)^2)$ .

This supplies an actual arithmetic procedure for the combinatorial counts used by reporting children and terminal candidates.

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 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_Bundle04
import Definitions.Def_ThreeSum1962Source_Bundle06
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
import Mathlib.Algebra.Group.Int.Defs
import Mathlib.Algebra.GroupWithZero.Nat
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.Data.Int.Cast.Lemmas
import Mathlib.Data.Int.Notation
import Mathlib.Data.List.GetD
import Mathlib.Data.Nat.Cast.Order.Ring
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.GCongr
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.NormNum
import Mathlib.Tactic.Positivity
import Mathlib.Tactic.Ring
import Theorems.Thm_ThreeSum_LightPascal_build_meets


set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option Elab.async false
namespace Light
end Light
namespace ThreeSum.LightPascal
end ThreeSum.LightPascal

Formal statement

theorem ThreeSum.LightReportingBinomial.program_meets : ∀ {lim : Light.Limits} {μ : Nat → Int} {n k work d : Nat} (hk : @LE.le.{0} Nat instLENat k n)
  (hw : @LE.le.{0} Int Int.instLEInt (@Nat.cast.{0} Int instNatCastInt lim.space) lim.word)
  (hspace :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) work (ThreeSum.LightPascal.tableSize n))
      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) n
            (@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) d
        (@OfNat.ofNat.{0} Nat (nat_lit 3) (instOfNatNat (nat_lit 3))))
      lim.depth),
  Light.Meets lim ThreeSum.LightReportingBinomial.program (@OfNat.ofNat.{0} Nat (nat_lit 4) (instOfNatNat (nat_lit 4)))
    d
    (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt n)
      (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt k)
        (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt work) (@List.nil.{0} Int))))
    μ (ThreeSum.LightReportingBinomial.time n) fun (r : Int) (ν : Nat → Int) =>
    And (@Eq.{1} Int r (@Nat.cast.{0} Int instNatCastInt (n.choose k)))
      (Light.SameOutside μ ν work (ThreeSum.LightPascal.tableSize n)) := by
  sorry

Source

https://github.com/c0ldzy17/three-sum-lean/blob/2b639655e07421b9ac743de9090459bab473a7c0/Submission/ThreeSum/LightReportingBinomial.lean#L23-L46

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