A symmetric reciprocal-product lower bound

Proved

WorkbookSource.base_7101

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

lean-workbooksource-checked

Prove that for all positive real numbers $a,b,c$ the following inequality $\frac{1}{a+b+c} (\frac{1}{a+b} + \frac{1}{b+c} + \frac{1}{a+c}) \ge \frac{1}{ab+bc+ca} + \frac{1}{2(a^2+b^2+c^2}$ .

Source: InternLM Lean-Workbook, record lean_workbook_7101 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_7101 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (1 / (a + b + c)) * (1 / (a + b) + 1 / (b + c) + 1 / (a + c)) ≥ 1 / (a * b + b * c + c * a) + 1 / (2 * (a ^ 2 + b ^ 2 + c ^ 2))  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_7101; Apache-2.0

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