A product of mixed cyclic ratio sums is at most one

Proved

WorkbookSource.base_25667

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

lean-workbooksource-checked

prove that $1\geq \left( {\frac {x}{x+y}}+{\frac {z+x}{2\,x+z+y}} \right) \left( {\frac {y}{y+z}}+{\frac {x+y}{2\,y+x+z}} \right) \left( {\frac {z}{z+x}}+{\frac {y+z}{2\,z+y+x}} \right)$ where $x,y,z>0$

Source: InternLM Lean-Workbook, record lean_workbook_25667 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_25667 (x y z : ℝ) (hx : 0 < x) (hy : 0 < y) (hz : 0 < z) : 1 ≥ (x / (x + y) + (z + x) / (2 * x + z + y)) * (y / (y + z) + (x + y) / (2 * y + x + z)) * (z / (z + x) + (y + z) / (2 * z + y + x))  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_25667; Apache-2.0

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