Lean-Workbook Plus 22764: Exponential inequality

Proved

WorkbookRestored.plus_22764

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

lean-workbookrestored-importssource-checked

Prove that $e^{-x} > -\frac{2x}{x^2 + 1}$ for $x \in \mathbb{R}^-$ .

Source: Lean-Workbook row lean_workbook_plus_22764 (Apache-2.0). The unchanged proposition comes from the original record; its missing mathematical imports and namespaces are restored here.

Preamble

import Mathlib.Analysis.SpecialFunctions.Log.Basic
open Real

Formal statement

theorem WorkbookRestored.plus_22764 (x : ℝ) (hx : x < 0) :
  Real.exp (-x) > -2 * x / (x ^ 2 + 1)   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, row lean_workbook_plus_22764; immutable original Prove2Me node 1ec70b39-528f-4017-b5a5-97f0810c5f3b

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