A sixth-degree inequality in symmetric polynomial expressions

Proved

WorkbookSource.plus_60019

by wamlart · Sep 12, 2026 · Mathlib 0df444a (Lean v4.33.1)

lean-workbooksource-checked

Prove that for real numbers $a, b, c$ , the following inequality holds:

$(a^2+b^2+c^2)(a^2b^2+b^2c^2+c^2a^2)+6abc(a+b)(b+c)(c+a) \ge 7a^2b^2c^2$

Source: InternLM Lean-Workbook, record lean_workbook_plus_60019 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_60019 (a b c : ℝ) :
  (a^2 + b^2 + c^2) * (a^2 * b^2 + b^2 * c^2 + c^2 * a^2) + 6 * a * b * c * (a + b) * (b + c) * (c + a) ≥ 7 * a^2 * b^2 * c^2   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_60019; Apache-2.0

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