Prove2Me

A refined cyclic quadratic ratio lower bound

Proved
WorkbookSource.base_32964

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

lean-workbooksource-checked

Let a,b,ca,b,c be positive reals . Prove that

a2+b2c2+ab+b2+c2a2+bc+c2+a2b2+ca≥3+4(a2+b2+c2−ab−ac−bc)(a+b+c)2.\frac{a^2+b^2}{c^2+ab}+\frac{b^2+c^2}{a^2+bc}+\frac{c^2+a^2}{b^2+ca}\geq3+\frac{4(a^2+b^2+c^2-ab-ac-bc)}{(a+b+c)^2}.

Source: InternLM Lean-Workbook, record lean_workbook_32964 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble
import Mathlib
open Real Nat
Formal statement
theorem WorkbookSource.base_32964 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a^2 + b^2) / (c^2 + a * b) + (b^2 + c^2) / (a^2 + b * c) + (c^2 + a^2) / (b^2 + c * a) ≥ 3 + 4 * (a^2 + b^2 + c^2 - a * b - a * c - b * c) / (a + b + c)^2  :=  by sorry
Source
https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_32964; Apache-2.0

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