A fourth-power sum bounds twice the cubic sum minus two at fixed sum two

Proved

WorkbookSource.base_18235

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

lean-workbooksource-checked

Let a,b,c be positive real numbers such that a+b+c=2 Prove the inequality $\frac{1}{2}(a^4+b^4+c^4)+1\geq a^3+b^3+c^3$

Source: InternLM Lean-Workbook, record lean_workbook_18235 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_18235 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (habc : a + b + c = 2) : 1 / 2 * (a ^ 4 + b ^ 4 + c ^ 4) + 1 ≥ a ^ 3 + b ^ 3 + c ^ 3  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_18235; Apache-2.0

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