Comparing two powers of a base between zero and one

Proved

WorkbookRestored.plus_3527

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

lean-workbookrestored-importssource-checked

If $0<x<1$ , then

x^{1/x}\le x^x.

This restores the missing imports or namespaces of the original node, retaining its formal proposition. Source: Lean-Workbook row lean_workbook_plus_3527 (Apache-2.0).

Preamble

import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Analysis.SpecialFunctions.Log.Base
open Real

Formal statement

theorem WorkbookRestored.plus_3527 (x : ℝ) (hx : 0 < x ∧ x < 1) :
  x^(1/x) ≤ x^x   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, row lean_workbook_plus_3527; original Prove2Me node 5662f2e0-c4af-44cd-9059-d84d6ebccdbb; Apache-2.0

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