A fourth power of the pairwise sum bounds a mixed symmetric product

Proved

WorkbookSource.base_40061

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

lean-workbooksource-checked

The following statement is true. Let $a,b,c \ge 0$ . Prove that : $(ab+ac+bc)^4 \geq a^2b^2c^2(16(a^2+b^2+c^2)+11(ab+ac+bc))$ . Equivalently, $(a+b+c)^4 \ge 16(a^2b^2+b^2c^2+c^2a^2)+11abc(a+b+c)$ .

Source: InternLM Lean-Workbook, record lean_workbook_40061 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_40061 (a b c : ℝ) (ha : 0 ≤ a) (hb : 0 ≤ b) (hc : 0 ≤ c) : (a * b + b * c + c * a) ^ 4 ≥ a ^ 2 * b ^ 2 * c ^ 2 * (16 * (a ^ 2 + b ^ 2 + c ^ 2) + 11 * (a * b + b * c + c * a))  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_40061; Apache-2.0

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