A seventh-degree inequality involving squared symmetric sums

Proved

WorkbookSource.plus_75942

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

lean-workbooksource-checked

For any three positive real numbers $a,b,c$ , show that: $(a+b)(b+c)(c+a)[(a+b+c)^2+ab+bc+ca]^2\ge 32abc(a^2+b^2+c^2+ab+bc+ca)^2$

Source: InternLM Lean-Workbook, record lean_workbook_plus_75942 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_75942 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a + b) * (b + c) * (c + a) * ((a + b + c) ^ 2 + a * b + b * c + c * a) ^ 2 ≥ 32 * a * b * c * (a ^ 2 + b ^ 2 + c ^ 2 + a * b + b * c + c * a) ^ 2   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_75942; Apache-2.0

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