According to the first sidechain article (Link), the cross chain transaction process can be briefly describe as follows:
Alice has a wallet on sidechain S1. She wants to send some coins to sidechain S2. So she makes a transaction containing some coins, send it to a certain locked-output on S1.
Wait for confirmation period to end.
After the transaction is confirmed, she submits a SPV proof to S2.
Again, she has to wait for a contest period.
After the contest period ends, an equivalent amount of coins is created and sent to her wallet on S2.
I assume that both the sidechains are secure, otherwise there is not much left to discuss. If they are secure, then after the confirmation period in step 2 above, the sidechain S2 already knows that the transaction is confirmed on sidechain S1. For example, if Alice purchases something using Bitcoins, step 3 is where she can get the product she purchased from a normal store.
In my opinion, after the confirmation period, the transaction is already in a stable part of S1. Thus, there cannot be any valid SPV proof that contradicts the SPV proof Alice sent, unless S1 does not satisfy the persistence property.
Therfore, I would like to ask, why is the contest period necessary? Is there something I am missing in my argument? Thank you.