Prove2Me

A squared cyclic ratio sum bounds a refined reciprocal product

Proved
WorkbookSource.plus_8538

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

lean-workbooksource-checked

Suppose a,b,c∈R+a,b,c\in \mathbb R^+ . Prove that : (ab+bc+ca)2≥(a+b+c)(1a+1b+1c)+2⋅(b−c)2+(c−a)2+(a−b)2bc+ca+ab.\left(\frac ab+\frac bc+\frac ca\right)^2\geq (a+b+c)\left(\frac1a+\frac1b+\frac1c\right)+2\cdot \frac{(b-c)^2+(c-a)^2+(a-b)^2}{bc+ca+ab} .

Source: InternLM Lean-Workbook, record lean_workbook_plus_8538 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble
import Mathlib
open Real Nat
Formal statement
theorem WorkbookSource.plus_8538 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a / b + b / c + c / a) ^ 2 ≥ (a + b + c) * (1 / a + 1 / b + 1 / c) + 2 * ((b - c) ^ 2 + (c - a) ^ 2 + (a - b) ^ 2) / (b * c + c * a + a * b)   :=  by sorry
Source
https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_8538; Apache-2.0

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