A fourth-power lower bound at unit positive sum

Proved

WorkbookSource.base_34623

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

lean-workbooksource-checked

Prove that $x^{4}+y^{4}+z^{4}+1\geq 2(x^{2}+y^{2}+z^{2})$ given $x+y+z=1$ and $x,y,z$ are positive real numbers.

Source: InternLM Lean-Workbook, record lean_workbook_34623 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_34623 (x y z : ℝ) (h1 : x + y + z = 1) (hx : 0 < x) (hy : 0 < y) (hz : 0 < z) : x^4 + y^4 + z^4 + 1 ≥ 2 * (x^2 + y^2 + z^2)  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_34623; Apache-2.0

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