A quartic and cubic comparison on the four-variable simplex

Proved

WorkbookSource.base_14166

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

lean-workbooksource-checked

Let $a,b,c,d\ge 0$ such that $a+b+c+d=1$ . Prove that

5(a^2+b^2+c^2+d^2)+2(a^2+b^2+c^2+d^2)^2\ge 1 +6(a^3+b^3+c^3+d^3).

Source: InternLM Lean-Workbook, record lean_workbook_14166 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_14166 (a b c d : ℝ) (ha : a ≥ 0) (hb : b ≥ 0) (hc : c ≥ 0) (hd : d ≥ 0) (hab : a + b + c + d = 1) : 5 * (a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2) + 2 * (a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2) ^ 2 ≥ 1 + 6 * (a ^ 3 + b ^ 3 + c ^ 3 + d ^ 3)  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_14166; Apache-2.0

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