Expanding a symmetric cubic expression

Proved

WorkbookSource.problem_49786

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

lean-workbooksource-checked

Prove that for a, b, c > 0: $(a + b + c)^3 - \left(\sum_{cyc} a(a^2 + 3b^2 + 3c^2)\right)\ge 0\iff 6abc\ge 0$

Source: InternLM Lean-Workbook, record lean_workbook_49786 (Apache-2.0). The complete source proposition and explicit variable declarations are preserved.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.problem_49786 (a b c : ℝ) :
  (a + b + c) ^ 3 - (a * (a ^ 2 + 3 * b ^ 2 + 3 * c ^ 2) + b * (b ^ 2 + 3 * c ^ 2 + 3 * a ^ 2) + c * (c ^ 2 + 3 * a ^ 2 + 3 * b ^ 2)) ≥ 0 ↔ 6 * a * b * c ≥ 0  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_49786; Apache-2.0

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