An asymmetric pairwise ratio lower bound

Proved

WorkbookSource.base_37184

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

lean-workbooksource-checked

Prove that for positive real numbers x, y, and z: $\frac{x}{y+z}+\frac{z}{x+y}\geq \frac{z+x}{2y+z+x}+\frac{1}{8}\frac{-2y+7z+7x}{x+y+z}$

Source: InternLM Lean-Workbook, record lean_workbook_37184 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_37184 (x y z : ℝ) (hx : 0 < x) (hy : 0 < y) (hz : 0 < z) : (x / (y + z) + z / (x + y)) ≥ (z + x) / (2 * y + z + x) + 1 / 8 * (-2 * y + 7 * z + 7 * x) / (x + y + z)  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_37184; Apache-2.0

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