Update public examples

[qwang98]

Depends on #2567.

Did a blanket search of public declarations and added public references in all such cases. This will be required once public reference values instead of witness is solely used for proving.

To make all such updated public examples work, also make the following changes:

1. Updated mock backend to use public reference values (this is technically the first backend that's edited to use public reference witgen).
2. Updated pipeline functions for soundness tests (set_witness --> set_witness_and_publics).

[Schaeff]

Out of curiosity: do I understand correctly that we still have a RHS on public declarations, but it is not enforced in the circuit? Is it used for witgen? Can it be omitted?