A sum of triple quadratic averages bounds the four-variable total

Proved

WorkbookSource.base_17800

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

lean-workbooksource-checked

Let be a,b,c,d real positive numbers. Show that

$\frac{a^{2}+b^{2}+c^{2}}{a+b+c}+\frac{b^{2}+c^{2}+d^{2}}{b+c+d}+\frac{c^{2}+d^{2}+a^{2}}{c+d+a}+\frac{d^{2}+a^{2}+b^{2}}{d+a+b}\geq a+b+c+d$

Source: InternLM Lean-Workbook, record lean_workbook_17800 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_17800 (a b c d : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hd : 0 < d) : (a^2 + b^2 + c^2)/(a + b + c) + (b^2 + c^2 + d^2)/(b + c + d) + (c^2 + d^2 + a^2)/(c + d + a) + (d^2 + a^2 + b^2)/(d + a + b) ≥ a + b + c + d  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_17800; Apache-2.0

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