Prove2Me

A cyclic linear ratio sum with a symmetric product correction

Proved
WorkbookSource.base_54981

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

lean-workbooksource-checked

Prove that for positive reals x, y, and z, the following inequality holds: y+z2 x+z+y+z+x2 y+x+z+x+y2 z+y+x≥16+32 (x+y)(z+x)(y+z)(x+y+z)(xy+xz+yz){\frac {y+z}{2\,x+z+y}}+{\frac {z+x}{2\,y+x+z}}+{\frac {x+y}{2\,z+y+x}}\geq \frac{1}{6}+\frac{3}{2}\,{\frac { \left( x+y \right) \left( z+x \right) \left( y+z \right) }{ \left( x+y+z \right) \left( xy+xz+yz \right) }}

Source: InternLM Lean-Workbook, record lean_workbook_54981 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble
import Mathlib
open Real Nat
Formal statement
theorem WorkbookSource.base_54981 (x y z : ℝ) (hx : 0 < x) (hy : 0 < y) (hz : 0 < z) : (y + z) / (2 * x + z + y) + (z + x) / (2 * y + x + z) + (x + y) / (2 * z + y + x) ≥ 1 / 6 + 3 / 2 * ((x + y) * (z + x) * (y + z) / ((x + y + z) * (x * y + x * z + y * z)))  :=  by sorry
Source
https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_54981; Apache-2.0

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