Split base rollup into public base and private base.

[LeilaWang]

Public Base

• Verify vm proof
• Verify output from private kernel vs inputs to the vm (part of the output from the vm)
• Verify transaction fee (or the values used to compute the transaction fee)

Private Base

• Update nullifier tree root
• Update note hash tree root
• Silo l2_to_l1_message
• Compute transaction fee
• Inject protocol update requests for deducting tx_fee from fee_payer's balance

For both

• Verify tube
• Archive membership check