A cyclic shifted-square inequality at unit product

Proved

WorkbookSource.plus_18705

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

lean-workbooksource-checked

Let $x$ , $y$ , and $z$ be positive reals such that $xyz=1$ . Prove that $\sum x^2(2y-1)^2+5\ge(1+x)(1+y)(1+z)+3(1-x)(1-y)(1-z)$ .

Source: InternLM Lean-Workbook, record lean_workbook_plus_18705 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_18705 (x y z : ℝ) (hx : 0 < x) (hy : 0 < y) (hz : 0 < z) (habc : x*y*z = 1) : x^2 * (2*y - 1)^2 + y^2 * (2*z - 1)^2 + z^2 * (2*x - 1)^2 + 5 ≥ (1 + x) * (1 + y) * (1 + z) + 3 * (1 - x) * (1 - y) * (1 - z)   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_18705; Apache-2.0

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