A cubic expression bounds a cyclic rational sum

Proved

WorkbookSource.base_7020

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

lean-workbooksource-checked

Let $x,y,z>0$ . Prove that: $x(y+z-x)^2+y(x+z-y)^2+z(x+y-z)^2 \ge 2xyz(\frac{x}{y+z}+\frac{y}{x+z}+\frac{z}{x+y})$

Source: InternLM Lean-Workbook, record lean_workbook_7020 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_7020 (x y z : ℝ) (hx : 0 < x) (hy : 0 < y) (hz : 0 < z) : x * (y + z - x) ^ 2 + y * (x + z - y) ^ 2 + z * (x + y - z) ^ 2 ≥ 2 * x * y * z * (x/(y+z) + y/(x+z) + z/(x+y))  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_7020; Apache-2.0

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