Prove2Me

A squared pairwise ratio expression bounds a symmetric ratio

Proved
WorkbookSource.base_19817

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

lean-workbooksource-checked

For x,y,zx,y, z positive real numbers:

[2(xy+z+yx+z+zx+y)−1]2−1≥2(x+y+z)2x2+y2+z2+xy+yz+zx  ;\left[ 2 \left( \frac{x}{y+z}+\frac{y}{x+z}+\frac{z}{x+y} \right)-1 \right]^2 -1 \ge \frac{2(x+y+z)^2}{x^2+y^2+z^2+xy+yz+zx} \ \ ;

Greetings!

Source: InternLM Lean-Workbook, record lean_workbook_19817 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble
import Mathlib
open Real Nat
Formal statement
theorem WorkbookSource.base_19817 (x y z : ℝ) (hx : 0 < x) (hy : 0 < y) (hz : 0 < z) : (2 * (x / (y + z) + y / (x + z) + z / (x + y)) - 1)^2 - 1 ≥ 2 * (x + y + z)^2 / (x^2 + y^2 + z^2 + x * y + y * z + z * x)  :=  by sorry
Source
https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_19817; Apache-2.0

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