A product of shifted quadratics at fixed sum four

Proved

WorkbookSource.base_19033

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

lean-workbooksource-checked

Prove that $({a^2} - a + 1)({b^2} - b + 1)({c^2} - c + 1) \le 13$ for non-negative reals $a$ , $b$ and $c$ such that $a+b+c=4$ .

Source: InternLM Lean-Workbook, record lean_workbook_19033 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_19033 (a b c : ℝ) (ha : 0 ≤ a) (hb : 0 ≤ b) (hc : 0 ≤ c) (habc : a + b + c = 4) : (a^2 - a + 1) * (b^2 - b + 1) * (c^2 - c + 1) ≤ 13  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_19033; Apache-2.0

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