A cyclic harmonic product sum has a normalized alternating-product upper bound

Proved

WorkbookSource.base_52794

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

lean-workbooksource-checked

For $a, b, c,d>0$ prove that $\frac{ab}{a+b}+\frac{bc}{b+c}+\frac{cd}{c+d}+\frac{da}{d+a}\le\frac{2(ab+bc+cd+da)}{a+b+c+d}$

Source: InternLM Lean-Workbook, record lean_workbook_52794 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_52794 (a b c d : ℝ) (hab : 0 < a) (hbc : 0 < b) (hcd : 0 < c) (hda : 0 < d) : (a * b / (a + b) + b * c / (b + c) + c * d / (c + d) + d * a / (d + a)) ≤ (2 * (a * b + b * c + c * d + d * a)) / (a + b + c + d)  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_52794; Apache-2.0

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