Prove2Me

A squared quadratic expression bounds four fourth powers

Proved
WorkbookSource.plus_76314

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

lean-workbooksource-checked

Prove that (4∑cyca2+43∑cyc(a−c)2+83∑cyc(a−b)2)2≥64(a4+b4+c4+d4)(4\sum_{\mathrm{cyc}} a^2 + \frac{4}{3}\sum_{\mathrm{cyc}}(a - c)^2 + \frac{8}{3}\sum_{\mathrm{cyc}} (a - b)^2)^2 \ge 64(a^4+b^4+c^4+d^4) for a,b,c,d≥0a, b, c, d \ge 0.

Source: InternLM Lean-Workbook, record lean_workbook_plus_76314 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble
import Mathlib
open Real Nat
Formal statement
theorem WorkbookSource.plus_76314 (a b c d : ℝ) : (4 * (a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2) + (4 / 3) * ((a - c) ^ 2 + (b - d) ^ 2 + (c - a) ^ 2 + (d - b) ^ 2) + (8 / 3) * ((a - b) ^ 2 + (b - c) ^ 2 + (c - d) ^ 2 + (d - a) ^ 2)) ^ 2 ≥ 64 * (a ^ 4 + b ^ 4 + c ^ 4 + d ^ 4)   :=  by sorry
Source
https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_76314; Apache-2.0

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