IIUC, when proving / disproving arbitrary propositions (the terms of a contract) in an interactive proof, you need as many game steps as there are alternating exists/forall quantifiers, but each step can otherwise be kept constant size using a zkSNARK.
Thus you can say "This sideledger has an entry that says my address has so many tokens" by exhibiting just the hash L of the entire ledger, the hash E of the entry, your address A (hash of your key), the number of tokens T, and a zkSNARK proving there is a path from L to E.
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Then the sideledger manager can either concede and send T units to your address A with a reference to having paid said claim, or he can show you have already spent from address A on the side-chain since E was current, or closed your account on the main chain.
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