Does CCA-security implies CPA-security
My Sketch on the reduction proof:

Basic Assumption: I got a Cryptosystem #1 that is CCA secure

Based on this Cryptosystem I can build a CPA secure Cryptosystem #2, and I assume, that an adversary can break it with non negligible probability

Therefore I can build a Reduction: R can simulate the CPA game with A and just gives the messages of A to his challenger and the response back to A. R then outputs the result of A.

Because A has a non negligible probability in breaking #2, R also has a non negligible probability in breaking #1, which gives a contradiction to the basic assumption

Therefore CCA implies CPA


I hope the basic idea is correct.