A shifted cyclic quadratic ratio sum bounds the total

Proved

WorkbookSource.base_41461

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

lean-workbooksource-checked

Prove that for positive real numbers a, b, and c, the following inequality holds:

\frac{a^2+b}{a+1}+\frac{b^2+c}{b+1}+\frac{c^2+a}{c+1} \geq a+b+c

Source: InternLM Lean-Workbook, record lean_workbook_41461 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_41461 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a^2 + b) / (a + 1) + (b^2 + c) / (b + 1) + (c^2 + a) / (c + 1) ≥ a + b + c  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_41461; Apache-2.0

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