Assumes that we have 3 signature algorithms, $S^A$ with key pair $(sk^A,pk^A)$, $S^B$ with key pair $(sk^B,pk^B)$,$S^C$ with key pair $(sk^C,pk^C)$. We denote by $\epsilon$, $\epsilon'$ and $\epsilon''$ the advantages for breaking $S^A$, $S^B$ and $S^C$ respectively, in the sense of weak unforgeability.
I have a composed signature algorithm which works as follows : First we sign a message with $S^A$, then the result is signed with $S^B$, and finally the result of $S^B$ is signed with $S^C$. An adversary can plays to the game in which he can make queries to a composed signing oracle for the signature of messages of his choices. At the end, the adversay has to find a composed signature for a message which was never queried at the oracle.
Is the adversary advantage bounded by $\epsilon+\epsilon'+\epsilon''+\epsilon \epsilon'+\epsilon \epsilon''+\epsilon' \epsilon''+\epsilon \epsilon'\epsilon''$ ?
I try to construct the proof.. This could be a nice example for me to understand well provable security.