Consider the following langugage:$\qquad L = \{ \langle M \rangle \mid M \text{ TM}, w \in L(M) \implies ww \in L(M)\}$.I've been asked to decide whether this language is in R/RE/CO-RE. I've proved that the language doesn't belong to R and RE using Rice theorem, but I can't find a formal proof for CO-RE. Any suggestions? 
Is w  L(M )  ww  L(M) co-semi-decidable?
computability;semi decidability