A cyclic quadratic difference ratio sum is nonnegative

Proved

WorkbookSource.base_9765

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

lean-workbooksource-checked

Let $x,y,z>0$ ,prove that: ${\frac {y \left( y-x \right) }{x+y+{x}^{2}}}+{\frac {z \left( z-y \right) }{y+z+{y}^{2}}}+{\frac {x \left( x-z \right) }{z+x+{z}^{2}}}\geq 0$

Source: InternLM Lean-Workbook, record lean_workbook_9765 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_9765 (x y z : ℝ) (hx : 0 < x) (hy : 0 < y) (hz : 0 < z) : (y * (y - x) / (x + y + x ^ 2) + z * (z - y) / (y + z + y ^ 2) + x * (x - z) / (z + x + z ^ 2)) ≥ 0  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_9765; Apache-2.0

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