A product of four shifted cyclic ratios is at least nine

Proved

WorkbookSource.plus_21539

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

lean-workbooksource-checked

Let $a, b, c, d$ be positive real number. Prove that $(1+\frac{2a}{b+c})(1+\frac{2b}{c+d})(1+\frac{2c}{d+a})(1+\frac{2d}{a+b}) \ge 9$ .

Source: InternLM Lean-Workbook, record lean_workbook_plus_21539 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_21539 (a b c d : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hd : 0 < d) : (1 + 2 * a / (b + c)) * (1 + 2 * b / (c + d)) * (1 + 2 * c / (d + a)) * (1 + 2 * d / (a + b)) ≥ 9   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_21539; Apache-2.0

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