A quadratic pairwise-sum bound with a cubic correction

Proved

WorkbookSource.plus_3249

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

lean-workbooksource-checked

Let a, b, c are positive real numbers such that $a+b+c=1$ . Prove that $4\left(ab+bc+ac\right)^2+6abc+1\ge 5\left(ab+bc+ac\right)$

Source: InternLM Lean-Workbook, record lean_workbook_plus_3249 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_3249 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hab : a + b + c = 1) : 4 * (a * b + b * c + a * c) ^ 2 + 6 * a * b * c + 1 ≥ 5 * (a * b + b * c + a * c)   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_3249; Apache-2.0

View graph