A sharp sum bound from reciprocal quadratic denominators

Proved

WorkbookCorrected.plus_13466

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

corrected-formalizationlean-workbooksource-checked

Let $a,b,c$ be positive numbers satisfying $\frac{1}{a^2+2}+\frac{1}{b^2+2}+\frac{1}{c^2+2}=\frac{1}{3}$ . Prove that $a+b+c\ge 3\sqrt{7}.$

Formalization Note: The original formalization added abc=1, absent from the source. This correction removes that added hypothesis and proves the source lower bound for all positive real variables satisfying the reciprocal identity.

Source: InternLM Lean-Workbook, record lean_workbook_plus_13466 (Apache-2.0).

Preamble

import Mathlib

Formal statement

theorem WorkbookCorrected.plus_13466 (a b c : ℝ) (ha : 0<a) (hb : 0<b) (hc : 0<c)
    (h : 1/(a^2+2)+1/(b^2+2)+1/(c^2+2)=1/3) : a+b+c ≥ 3*Real.sqrt 7 := by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_13466; Apache-2.0

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