PRF proof and length-preserving
A PRF does not have to be length-preserving. We sometimes define it that way for simplicity. In general, $F:\{0,1\}^n \times \{0,1\}^{\ell_1(n)} \rightarrow \{0,1\}^{\ell_2(n)}$ for polynomials $\ell_1,\ell_2$.