A quadratic reciprocal sum bounds two symmetric reciprocals

Proved

WorkbookSource.base_48460

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

lean-workbooksource-checked

Let $a,b,c>0$ . Prove that $\sum_{cyc}\frac{1}{a^2+b^2}\geq\frac{1}{2(ab+ac+bc)}+\frac{4}{a^2+b^2+c^2}$

Source: InternLM Lean-Workbook, record lean_workbook_48460 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_48460 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (1 / (a ^ 2 + b ^ 2) + 1 / (b ^ 2 + c ^ 2) + 1 / (c ^ 2 + a ^ 2)) ≥ 1 / (2 * (a * b + a * c + b * c)) + 4 / (a ^ 2 + b ^ 2 + c ^ 2)  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_48460; Apache-2.0

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