A refined cyclic rational comparison with squared differences

Proved

WorkbookSource.plus_11955

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

lean-workbooksource-checked

For positive reals $a,b,c$ , prove the inequality

\frac{(a-b)^2}{ab} + \frac{(b-c)^2}{bc} + \frac{(c-a)^2}{ca} + \frac ca + \frac ab + \frac bc \geq \frac{3bc+ac-ab}{2ab+ac} + \frac{3ac+ab-bc}{2bc+ab} + \frac{3ab+bc-ac}{2ac+bc}.

Proposed by Danylo Hilko

Source: InternLM Lean-Workbook, record lean_workbook_plus_11955 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_11955 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a - b) ^ 2 / (a * b) + (b - c) ^ 2 / (b * c) + (c - a) ^ 2 / (c * a) + c / a + a / b + b / c ≥ (3 * b * c + a * c - a * b) / (2 * a * b + a * c) + (3 * a * c + a * b - b * c) / (2 * b * c + a * b) + (3 * a * b + b * c - a * c) / (2 * a * c + b * c)   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_11955; Apache-2.0

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