Prove2Me

A cyclic linear-over-quadratic sum bounded by reciprocals

Proved
WorkbookSource.base_14171

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

lean-workbooksource-checked

For x,y,z>0 x,y,z > 0 , prove that: 2xyz+x2+2yzx+y2+2zxy+z2≤1x+1y+1z\frac{2x}{yz+x^2}+ \frac{2y}{zx+y^2}+\frac{ 2z}{xy+z^2} \le \frac{1}{x}+ \frac{1}{y}+ \frac{1}{z}

Source: InternLM Lean-Workbook, record lean_workbook_14171 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble
import Mathlib
open Real Nat
Formal statement
theorem WorkbookSource.base_14171 (x y z : ℝ) (hx : 0 < x) (hy : 0 < y) (hz : 0 < z) : (2 * x / (y * z + x ^ 2) + 2 * y / (z * x + y ^ 2) + 2 * z / (x * y + z ^ 2)) ≤ 1 / x + 1 / y + 1 / z  :=  by sorry
Source
https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_14171; Apache-2.0

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