A product of pairwise ratio sums bounds a cyclic ratio sum

Proved

WorkbookSource.base_34321

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

lean-workbooksource-checked

Prove that for $x,y,z>0$ , $\frac{1}{6}\, \left( {\frac {x+y}{y+z}}+{\frac {y+z}{z+x}}+{\frac {z+x}{x+y}} \right) \left( {\frac {x+y}{z+x}}+{\frac {y+z}{x+y}}+{\frac {z+x}{y+z}} \right) \geq {\frac {x}{y+z}}+{\frac {y}{z+x}}+{\frac {z}{x+y}}$

Source: InternLM Lean-Workbook, record lean_workbook_34321 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_34321 (x y z : ℝ) (hx : 0 < x) (hy : 0 < y) (hz : 0 < z) : (1 / 6) * ((x + y) / (y + z) + (y + z) / (z + x) + (z + x) / (x + y)) * ((x + y) / (z + x) + (y + z) / (x + y) + (z + x) / (y + z)) ≥ x / (y + z) + y / (z + x) + z / (x + y)  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_34321; Apache-2.0

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