Lean-Workbook Plus 65632: Exponential inequality

Proved

WorkbookRestored.plus_65632

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

lean-workbookrestored-importssource-checked

$\frac{v}{1+e^{-v}}<0$ for all $v<0$ .

Source: Lean-Workbook row lean_workbook_plus_65632 (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_65632 (v : ℝ) (h : v < 0) : v / (1 + exp (- v)) < 0   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, row lean_workbook_plus_65632; immutable original Prove2Me node ee22a04f-d199-41bf-91a8-aa5560942d98

View graph