A four-variable mixed quadratic ratio sum bounds the total

Proved

WorkbookSource.base_5049

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

lean-workbooksource-checked

Let $a,b,c,d>0$ ,prove that: ${\frac {{a}^{2}+bc}{d+a}}+{\frac {{b}^{2}+cd}{a+b}}+{\frac {{c}^{2}+ad}{b+c}}+{\frac {{d}^{2}+ab}{c+d}}\geq a+b+c+d$

Source: InternLM Lean-Workbook, record lean_workbook_5049 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_5049 (a b c d : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hd : 0 < d) : (a^2 + b * c) / (d + a) + (b^2 + c * d) / (a + b) + (c^2 + d * a) / (b + c) + (d^2 + a * b) / (c + d) ≥ a + b + c + d  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_5049; Apache-2.0

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