A pairwise weighted-square inequality at unit sum

Proved

WorkbookSource.base_12795

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

lean-workbooksource-checked

Let $a,b,c$ be positive real numbers such that $a+b+c=1$ . Prove that

ab(a+b)^2+bc(b+c)^2+ca(c+a)^2\geq 4abc

Source: InternLM Lean-Workbook, record lean_workbook_12795 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_12795 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hab : a + b + c = 1) : a * b * (a + b) ^ 2 + b * c * (b + c) ^ 2 + c * a * (c + a) ^ 2 ≥ 4 * a * b * c  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_12795; Apache-2.0

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