A quadratic and triple-product sum has a product lower bound at fixed total four

Proved

WorkbookSource.plus_61764

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

lean-workbooksource-checked

Prove that for $a, b, c, d > 0$ and $a + b + c + d = 4$ , $a^2 + b^2 + c^2 + d^2 + abc + bcd + cda + dab \geq \frac{1}{2}(abcd + 15)$ .

Source: InternLM Lean-Workbook, record lean_workbook_plus_61764 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_61764 (a b c d : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hd : 0 < d) (hab : a + b + c + d = 4) : a^2 + b^2 + c^2 + d^2 + a * b * c + b * c * d + c * d * a + d * a * b ≥ 1 / 2 * (a * b * c * d + 15)   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_61764; Apache-2.0

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