A cyclic quadratic difference over shifted pair products

Proved

WorkbookSource.base_12997

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

lean-workbooksource-checked

Let a,b,c be positive real numbers. Prove that: $\frac{a(a-2b+c)}{ab+1} + \frac{b(b-2c+a)}{bc+1} + \frac{c(c-2a+b)}{ac+1} \geq 0$

Source: InternLM Lean-Workbook, record lean_workbook_12997 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_12997 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a * (a - 2 * b + c) / (a * b + 1) + b * (b - 2 * c + a) / (b * c + 1) + c * (c - 2 * a + b) / (c * a + 1)) ≥ 0  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_12997; Apache-2.0

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