Prove2Me

Lean-Workbook Plus 81331: Exponential identity

Proved
WorkbookRestored.plus_81331

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

lean-workbookrestored-importssource-checked

For a function f:R→Rf:\mathbb R\to\mathbb R and real xx, 1−f(x)=(9ex+2)/(12ex+3)1-f(x)=(9e^x+2)/(12e^x+3) if and only if f(x)=(3ex+1)/(12ex+3)f(x)=(3e^x+1)/(12e^x+3).

Source: Lean-Workbook row lean_workbook_plus_81331 (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_81331 (f : ℝ → ℝ) (x : ℝ) : 1 - f x = (9 * exp x + 2) / (12 * exp x + 3) ↔ f x = (3 * exp x + 1) / (12 * exp x + 3)   :=  by sorry
Source
https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, row lean_workbook_plus_81331; immutable original Prove2Me node 6bfe64bb-31c2-4adc-b2b2-72c1c791b28c

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