Bennett positive series mul has Sum

Proved

D1Bound5.bennett_positive_series_mul_hasSum

by tp · Oct 23, 2026 · Mathlib 0df444a (Lean v4.33.1)

diophantine-tuplesnumber-theory

Weighted Cauchy multiplication for positive real power series. The two series are actually summable at the chosen radius; no formal evaluation is silently treated as an analytic sum.

Preamble

import Lean.Elab.Tactic.Omega
import Mathlib.Analysis.Analytic.Binomial
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.Normed.Ring.Basic
import Mathlib.Analysis.Normed.Ring.InfiniteSum
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.Data.Sym.Basic
import Mathlib.Data.Sym.Card
import Mathlib.RingTheory.MvPolynomial.Symmetric.Defs
import Mathlib.RingTheory.PowerSeries.Binomial
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.NormNum
import Mathlib.Tactic.Positivity
import Mathlib.Tactic.Ring
import Mathlib.Topology.Algebra.InfiniteSum.Order

Formal statement

section
open Finset

namespace D1Bound5
















theorem _root_.D1Bound5.bennett_positive_series_mul_hasSum
    (P Q : PowerSeries ℝ) (r s t : ℝ) (hr : 0 ≤ r)
    (hP : ∀ n, 0 ≤ PowerSeries.coeff n P)
    (hQ : ∀ n, 0 ≤ PowerSeries.coeff n Q)
    (hs : HasSum (fun n : ℕ => PowerSeries.coeff n P * r^n) s)
    (ht : HasSum (fun n : ℕ => PowerSeries.coeff n Q * r^n) t) :
    HasSum (fun n : ℕ => PowerSeries.coeff n (P*Q) * r^n) (s*t) := by sorry





end D1Bound5
end

Source

D1Bound5/BennettPositiveSeries.lean, lines 105-138, SHA-256 97c8e1703cfe77445d9777396cd5b2920aa185f7c75649ef58d66b03bb6b026f. Project: https://github.com/tomislav-math/d1-bound5-lean

View graph