A four-variable triple quadratic ratio sum is at least four

Proved

WorkbookSource.base_17392

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

lean-workbooksource-checked

Let $a,b,c,d$ -reals positive numbers. Prove inequality: $\frac{a^2+b^2+c^2}{ab+bc+cd}+\frac{b^2+c^2+d^2}{bc+cd+ad}+\frac{a^2+c^2+d^2}{ab+ad+cd}+\frac{a^2+b^2+d^2}{ab+ad+bc} \geq 4$

Source: InternLM Lean-Workbook, record lean_workbook_17392 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_17392 (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 + b * c + c * d) + (b^2 + c^2 + d^2) / (b * c + c * d + d * a) + (a^2 + c^2 + d^2) / (a * b + a * d + c * d) + (a^2 + b^2 + d^2) / (a * b + a * d + b * c) ≥ 4  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_17392; Apache-2.0

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