Prove2Me

A squared-difference bound under a mixed sum relation

Proved
WorkbookSource.base_14624

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

lean-workbooksource-checked

Given a,b,ca, b, c positive real numbers, proof that

(a−b)2+(b−c)2+(c−a)2≥6(a+b+c−3)(a-b)^2+(b-c)^2+(c-a)^2 \geq 6(a+b+c-3)

Given that

a+b+c=ab+bc+caa+b+c=ab+bc+ca

Source: InternLM Lean-Workbook, record lean_workbook_14624 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble
import Mathlib
open Real Nat
Formal statement
theorem WorkbookSource.base_14624 {a b c : ℝ} (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (habc : a + b + c = a * b + b * c + c * a) : (a - b) ^ 2 + (b - c) ^ 2 + (c - a) ^ 2 ≥ 6 * (a + b + c - 3)  :=  by sorry
Source
https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_14624; Apache-2.0

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