Just to have an high level idea in a quick/lazy way ;-) , any diagram/note/schema about who plays the role of public STATEMENT & private WITNESS (in the SNARK meaning of those keywords) in each of the 3 stages of your chainproof (header/assumedvalid/full)?
The statement are the bitcoin consensus rules, basically expressing "I know a chain of blocks that is valid and results in the chain state X". The (private) witness is the chain of blocks.
The chain state contains data like the block height, the total work, etc, but also a UTXO set commitment.