Prove2Me

A quadratic ratio with a normalized triple-product correction

Proved
WorkbookSource.base_12221

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

lean-workbooksource-checked

If x,y,z>0x,y,z>0 prove that: 3(x2+y2+z2)xy+yz+zx+9xyz(x+y+z)(x2+y2+z2)≥4\frac{3(x^2+y^2+z^2)}{xy+yz+zx}+\frac{9xyz}{(x+y+z)(x^2+y^2+z^2)}\ge 4

Source: InternLM Lean-Workbook, record lean_workbook_12221 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble
import Mathlib
open Real Nat
Formal statement
theorem WorkbookSource.base_12221 (x y z : ℝ) (hx : 0 < x) (hy : 0 < y) (hz : 0 < z) : (3 * (x ^ 2 + y ^ 2 + z ^ 2) / (x * y + x * z + y * z) + 9 * x * y * z / ((x + y + z) * (x ^ 2 + y ^ 2 + z ^ 2))) ≥ 4  :=  by sorry
Source
https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_12221; Apache-2.0

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