A cyclic shifted linear ratio sum is at least three

Proved

WorkbookSource.base_13239

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: $\frac{2z+1}{x+y+1}+\frac{1+2x}{y+z+1}+\frac{2y+1}{z+x+1}\geq 3$

Source: InternLM Lean-Workbook, record lean_workbook_13239 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_13239 (x y z : ℝ) (hx : 0 < x) (hy : 0 < y) (hz : 0 < z) : (2 * z + 1) / (x + y + 1) + (1 + 2 * x) / (y + z + 1) + (2 * y + 1) / (z + x + 1) ≥ 3  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_13239; Apache-2.0

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