A cyclic squared-difference rational inequality

Proved

WorkbookSource.base_35848

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

lean-workbooksource-checked

Let $a,b,c>0$ . Prove that

\sum\limits_{cyc} (a-b)^2\Big[ \frac{12a^2+12b^2-3c^2+3ab+9ca+9cb}{3(2a^2+(b+c)^2)(2b^2+(c+a)^2)} - \frac{1}{(a+b+c)^2}\Big] \geqq 0

Source: InternLM Lean-Workbook, record lean_workbook_35848 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_35848 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a - b) ^ 2 * ((12 * a ^ 2 + 12 * b ^ 2 - 3 * c ^ 2 + 3 * a * b + 9 * c * a + 9 * c * b) / (3 * (2 * a ^ 2 + (b + c) ^ 2) * (2 * b ^ 2 + (c + a) ^ 2)) - 1 / (a + b + c) ^ 2) + (b - c) ^ 2 * ((12 * b ^ 2 + 12 * c ^ 2 - 3 * a ^ 2 + 3 * b * c + 9 * a * b + 9 * a * c) / (3 * (2 * b ^ 2 + (c + a) ^ 2) * (2 * c ^ 2 + (a + b) ^ 2)) - 1 / (a + b + c) ^ 2) + (c - a) ^ 2 * ((12 * c ^ 2 + 12 * a ^ 2 - 3 * b ^ 2 + 3 * c * a + 9 * b * c + 9 * b * a) / (3 * (2 * c ^ 2 + (a + b) ^ 2) * (2 * a ^ 2 + (b + c) ^ 2)) - 1 / (a + b + c) ^ 2) ≥ 0  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_35848; Apache-2.0

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