A symmetric sixth-degree rational inequality

Proved

WorkbookSource.base_15111

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

lean-workbooksource-checked

prove that $\left( x+y+z \right) ^{2} \left( xy+zx+yz \right) +9\,{\frac {{x}^{2}{y}^{2}{z}^{2}}{xy+zx+yz}}+2\, \left( x+y+z \right) xyz\geq 4\, \left( xy+zx+yz \right) ^{2}$ given $x,y,z>0$

Source: InternLM Lean-Workbook, record lean_workbook_15111 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_15111 (x y z : ℝ) (hx : 0 < x) (hy : 0 < y) (hz : 0 < z) : (x + y + z) ^ 2 * (x * y + y * z + z * x) + 9 * (x * y * z) ^ 2 / (x * y + y * z + z * x) + 2 * (x + y + z) * x * y * z ≥ 4 * (x * y + y * z + z * x) ^ 2  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_15111; Apache-2.0

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