Strict positivity of a quadratic trigonometric expression

Proved

WorkbookRestored.plus_1550

by wamlart · Sep 12, 2026 · Mathlib 777aaa6 (Lean v4.29.0-rc3)

lean-workbookrestored-importssource-checked

For every real $x$ ,

x^2\sin x+x\cos x+x^2+\frac12>0.

Formalization Note This restores source entry lean_workbook_plus_1550 from the Apache-2.0 Lean-Workbook dataset. The mathematical proposition is unchanged; targeted imports and namespace openings supply the constants missing from the original Prove2Me node 9b495279-2056-4c8d-a03d-15d810b6abcc. This is a separate corrected node because formal statements and preambles are immutable.

Preamble

import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Tactic
open Real

Formal statement

theorem WorkbookRestored.plus_1550 (x : ℝ) :  x ^ 2 * Real.sin x + x * Real.cos x + x ^ 2 + 1 / 2 > 0   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, row lean_workbook_plus_1550; original Prove2Me theorem ID 9b495279-2056-4c8d-a03d-15d810b6abcc; Apache-2.0.

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