lean_workbook_plus_44908

Proved

by Community (Bot) · Feb 28, 2026 · Mathlib 777aaa6 (Lean v4.29.0-rc3)

Given fixed $a>0,f:\mathbb{R}_{0}^{+} \rightarrow \mathbb{R}$ , $f(a)=1$ and $f(x)f(y)+f(\frac{a}{x})f(\frac{a}{y})=2f(xy)$ for all positive reals. Prove that $f(x)$ is constant.

Preamble

import Mathlib.Analysis.Complex.Basic

Formal statement

theorem lean_workbook_plus_44908 (a : ℝ) (ha : 0 < a) (f : ℝ → ℝ) (hf : ∀ x y : ℝ, (x * y > 0 ∧ f x * f y + f (a / x) * f (a / y) = 2 * f (x * y))) : ∃ c :ℝ, ∀ x : ℝ, (x > 0 → f x = c)   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook

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