Prove2Me

A three-term symmetric rational lower bound

Proved
WorkbookSource.base_5609

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

lean-workbooksource-checked

Prove that: x,y,z>0⇒3(x2+y2+z2)xy+yz+zx+24xyzxyz+(x+y)(y+z)(z+x)+4(xy+yz+zx)(x+y+z)2≥7x,y,z>0\Rightarrow\frac{3(x^2+y^2+z^2)}{xy+yz+zx}+\frac{24xyz}{xyz+(x+y)(y+z)(z+x)}+\frac{4(xy+yz+zx)}{(x+y+z)^2}\ge 7

Source: InternLM Lean-Workbook, record lean_workbook_5609 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble
import Mathlib
open Real Nat
Formal statement
theorem WorkbookSource.base_5609 (x y z : ℝ) (hx : 0 < x) (hy : 0 < y) (hz : 0 < z) :  3 * (x ^ 2 + y ^ 2 + z ^ 2) / (x * y + y * z + z * x) + 24 * x * y * z / (x * y * z + (x + y) * (y + z) * (z + x)) + 4 * (x * y + y * z + z * x) / (x + y + z) ^ 2 ≥ 7  :=  by sorry
Source
https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_5609; Apache-2.0

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