A shifted cyclic triple-product ratio is at most four fifths at fixed total four

Proved

WorkbookSource.base_52908

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

lean-workbooksource-checked

Let $a,b,c,d>0,a+b+c+d=4$ ,prove that: $\frac{abc}{d+4}+\frac{bcd}{a+4}+\frac{cda}{b+4}+\frac{dab}{c+4}\le \frac{4}{5}$

Source: InternLM Lean-Workbook, record lean_workbook_52908 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_52908 (a b c d : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hd : 0 < d) (habc : a + b + c + d = 4) : (a * b * c) / (d + 4) + (b * c * d) / (a + 4) + (c * d * a) / (b + 4) + (d * a * b) / (c + 4) ≤ 4 / 5  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_52908; Apache-2.0

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