A quadratic-factor product bound with an attaining example

Proved

WorkbookCorrected.base_47199

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

corrected-formalizationlean-workbooksource-checked

Prove that for all real numbers $x, y, z$ satisfying $x+y+z+2=xyz$ , $(x^2+1)(y^2+1)(z^2+1) \geq 4$ and equality holds for example when $x=0, \; y=z=-1$ .

Source: InternLM Lean-Workbook, record lean_workbook_47199 (Apache-2.0). Natural-language proposition preserved; the source bound is completed with an exact attainment witness. Proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookCorrected.base_47199 : (∀ (x y z : ℝ) (h : x + y + z + 2 = x * y * z), (x ^ 2 + 1) * (y ^ 2 + 1) * (z ^ 2 + 1) ≥ 4) ∧ ((((0) : ℝ) + ((-1) : ℝ) + ((-1) : ℝ) + 2 = ((0) : ℝ) * ((-1) : ℝ) * ((-1) : ℝ)) ∧ ( (((0) : ℝ) ^ 2 + 1) * (((-1) : ℝ) ^ 2 + 1) * (((-1) : ℝ) ^ 2 + 1)  =  4  )) := by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_47199; Apache-2.0

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