A sixth-power bound for symmetric pairwise products

Proved

WorkbookSource.plus_68139

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

lean-workbooksource-checked

Prove that for all $a,b,c$ positive real number holds:

$5(a+b+c)^6\geq27(ab+ac+bc)(11(a^2b^2+a^2c^2+b^2c^2)+4abc(a+b+c))$

Source: InternLM Lean-Workbook, record lean_workbook_plus_68139 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_68139 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : 5 * (a + b + c) ^ 6 ≥ 27 * (a * b + a * c + b * c) * (11 * (a ^ 2 * b ^ 2 + a ^ 2 * c ^ 2 + b ^ 2 * c ^ 2) + 4 * a * b * c * (a + b + c))   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_68139; Apache-2.0

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