A normalized triple product bounds a symmetric quadratic ratio

Proved

WorkbookSource.base_35760

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

lean-workbooksource-checked

Prove that $\frac{3abc}{a^3+b^3+c^3+3abc}+\frac{1}{2}\ge \frac{ab+bc+ca}{a^2+b^2+c^2}$ given $a,b,c>0$ using only classical inequalities.

Source: InternLM Lean-Workbook, record lean_workbook_35760 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_35760 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (3 * a * b * c) / (a ^ 3 + b ^ 3 + c ^ 3 + 3 * a * b * c) + 1 / 2 ≥ (a * b + b * c + c * a) / (a ^ 2 + b ^ 2 + c ^ 2)  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_35760; Apache-2.0

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