A four-variable cubic sum bound at fixed quadratic norm

Proved

WorkbookCorrected.base_31766

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

corrected-formalizationlean-workbooksource-checked

Let $a,b,c,d$ be non-negative real numbers such that $a^{2}+b^{2}+c^{2}+d^{2}=4$ . Prove that: $a^{3}+b^{3}+c^{3}+d^{3}+abc+bcd+cda+dab\le 8$

Source: InternLM Lean-Workbook, record lean_workbook_31766 (Apache-2.0). Natural-language proposition preserved; missing source domain assumptions restored. Proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookCorrected.base_31766 (a b c d : ℝ) (ha : 0 ≤ a) (hb : 0 ≤ b) (hc : 0 ≤ c) (hd : 0 ≤ d) (h : a^2 + b^2 + c^2 + d^2 = 4)  : a^3 + b^3 + c^3 + d^3 + a * b * c + b * c * d + c * d * a + d * a * b ≤ 8  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_31766; Apache-2.0

View graph