Rearranging a symmetric cubic inequality

Proved

WorkbookSource.problem_1096

by wamlart · Sep 12, 2026 · Mathlib 0df444a (Lean v4.33.1)

lean-workbooksource-checked

By am-gm $, \ \ c+ab\le \frac{c^2+1}{2}+\frac{a^2+b^2}{2}$ . Then

$\ (ab+bc+ca)-abc\le \frac{c^2+1}{2}+\frac{a^2+b^2}{2} \ \ \ \iff \ \ \ a^2+b^2+c^2+2abc+1\ge 2(ab+bc+ca)$

Source: InternLM Lean-Workbook, record lean_workbook_1096 (Apache-2.0). The complete source proposition and explicit variable declarations are preserved.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.problem_1096  (a b c : ℝ) :
  a * b + b * c + c * a - a * b * c ≤ (c^2 + 1) / 2 + (a^2 + b^2) / 2 ↔
  a^2 + b^2 + c^2 + 2 * a * b * c + 1 ≥ 2 * (a * b + b * c + c * a)  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_1096; Apache-2.0

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