Write easy examples to compare PIL1 and PIL2 resources

[hecmas]

We have to study and understand better how to compare programs made in PIL1 vs their equivalents in PIL2. For this, we're going to go through a series of "small" examples that are intended to cover most frequent scenarios.

Examples

• Basic arithmetic

Parameters to compare

• Area, i.e., the number of columns times the number of rows.
• Number of intended program's operations with such area. For example, for the binary program, this would be the number of binary operations it can perform with particular area.
• Number of constraints.
• Constraint degree
• Constraint "window" (some people refer to this as the number of openings).

Consequences to consider

• Proving time.
• Proof size.
• Verification time.

Note 1: The main objective should be achieving the least proving time, without incurring in a prohibitive proof size and/or verification time.
Note 2: Verification time is, at least, linearly dependent on the proof size. Therefore, proof size is the parameter to worry about between these two. Anyways, these can always be reduced using proof composition.

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Started to work on this in this note.