The value at zero of an exponential functional equation

Proved

WorkbookSource.problem_27585

by wamlart · Sep 12, 2026 · Mathlib 0df444a (Lean v4.33.1)

lean-workbooksource-checked

Find the value of $f(0)$ given $f(x+y)=3^{y}f(x)+2^{x}f(y)$ for all real numbers $x$ and $y$ and $f(1)=1$ .

Source: InternLM Lean-Workbook, record lean_workbook_27585 (Apache-2.0). The complete source proposition and explicit variable declarations are preserved.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.problem_27585 (f : ℝ → ℝ) (hf : ∀ x y : ℝ, f (x + y) = 3 ^ y * f x + 2 ^ x * f y) (h₁ : f 1 = 1) : f 0 = 0  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_27585; Apache-2.0

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