A parametric pairwise-product bound at unit sum

Proved

WorkbookSource.plus_59988

by wamlart · Sep 13, 2026 · Mathlib 0df444a (Lean v4.33.1)

lean-workbooksource-checked

Let $a,b,c,d\geq 0$ and $a+b +c +d =1.$ Prove that $ab +kbc +cd \leq \dfrac{k}{4}$ Where $k\geq 1.$

Source: InternLM Lean-Workbook, record lean_workbook_plus_59988 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_59988 (a b c d : ℝ) (ha : 0 ≤ a) (hb : 0 ≤ b) (hc : 0 ≤ c) (hd : 0 ≤ d) (hab : a + b + c + d = 1) (k : ℝ) (hk : 1 ≤ k) : a * b + k * b * c + c * d ≤ k / 4   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_59988; Apache-2.0

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