A fourth-power and cubic lower bound at unit sum

Proved

WorkbookSource.base_13751

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

lean-workbooksource-checked

Let $a,b,c,d>0$ and $a+b+c+d=1$ . Then, $16(a^4+b^4+c^4+d^4)+8(a^3+b^3+c^3+d^3)-3(a^2+b^2+c^2+d^2)\ge 0$ .

Source: InternLM Lean-Workbook, record lean_workbook_13751 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_13751 (a b c d : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hd : 0 < d) (habc : a + b + c + d = 1) :(16 * (a ^ 4 + b ^ 4 + c ^ 4 + d ^ 4) + 8 * (a ^ 3 + b ^ 3 + c ^ 3 + d ^ 3) - 3 * (a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2)) ≥ 0  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_13751; Apache-2.0

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