The actor wanting to reanimate the coins on the Bitcoin network provides a proof to the Bitcoin network that 1) the coins on the sidechain have been moved to a particular address that puts them out of reach, 2) that this transaction has been buried by sufficient blocks so that it's unlikely to be reversed, 3) that it was this actor that did it.

This is why the sidechains paper suggests a one-off change to Bitcoin might be needed: there's no out-of-the-box Bitcoin feature to validate this proof. The authors propose an interim measure where n-of-m "functionaries" are trusted to validate the proof.

With a small change to bitcoin protocol it should be possible in some cases. But right now the only viable option is to use M of N oracles:

https://github.com/orisi/wiki/wiki/Orisi-White-Paper http://gavintech.blogspot.com/2014/06/bit-thereum.html

Basically, you choose a panel of up to 15 independent entities to verify that the funds were locked indeed. The panel can be modified by a majority of votes if some of the entities prove to be unreliable - kind of like the Supreme Court works.

This obviously introduces a new vulnerability to the system, but is the best solution available right now.