Is it safe to "compress" an IV in AES256-CTR?

Alice is using AES256 in CTR mode to send encrypted messages to Bob. The messages are 4 byte values. Along with each encrypted message, Alice sends along a random 16 byte IV. f maps a 4 byte value v to the 16 byte values v|v|v|v (concatenates it with itself 4 times). Would it be secure for Alice to randomly generate 4 bytes s, encrypt her message using IV = f(s) and send Bob the encrypted message along with s (assuming Bob uses the same f to recompute the IV on his end)?