A reflection identity for an exponential quotient

Proved

WorkbookRestored.plus_32500

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

lean-workbookrestored-importssource-checked

If $f(x)=9^x/(9^x+3)$ for every real $x$ , then $f(x)+f(1-x)=1$ for $0<x<1$ .

This restores the missing imports or namespaces of the original node, retaining its formal proposition. Source: Lean-Workbook row lean_workbook_plus_32500 (Apache-2.0).

Preamble

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

Formal statement

theorem WorkbookRestored.plus_32500  (x : ℝ)
  (f : ℝ → ℝ)
  (h₀ : ∀ x, f x = 9^x / (9^x + 3))
  (h₁ : 0 < x)
  (h₂ : x < 1) :
  f x + f (1 - x) = 1   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, row lean_workbook_plus_32500; original Prove2Me node 01067a07-b06d-4e35-992f-64bfbbb06511; Apache-2.0

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