A fourth-power and squared-sum bound on a sphere

Proved

WorkbookSource.base_4129

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

lean-workbooksource-checked

If a, b, c, d are real numbers such that $a^2+b^2+c^2+d^2=4$ show that $a^4+b^4+c^4+d^4+4(a+b+c+d)^2 \leq 68.$

Source: InternLM Lean-Workbook, record lean_workbook_4129 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_4129 (a b c d : ℝ) (h : a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2 = 4) :
  a ^ 4 + b ^ 4 + c ^ 4 + d ^ 4 + 4 * (a + b + c + d) ^ 2 ≤ 68  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_4129; Apache-2.0

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