[Fix] Declare public variable 1field once

[vicsn]

Motivation

Closes https://github.com/AleoHQ/snarkVM/issues/2266

We were declaring two hardcoded public One variables (in R1CS and ConstraintSystem), and adding them twice (in to_transition_verifier_inputs and in verify_batch. This PR:

• skips adding the hardcoded variable from R1CS onto the ConstraintSystem
• removes adding the hardcoded input in verify_batch
• adds logic to remove LinearCombination terms with value 0. Similar to https://github.com/AleoHQ/snarkVM/pull/1796 but for a different object doing the same thing.

Test Plan

No new tests were added.

Requires resampling credits and inclusion.

Related PRs

Previous attempt: https://github.com/AleoHQ/snarkVM/pull/2110/

[Pratyush]

This looks good to me, modulo an additional check in generate_constraints that ConstraintSystem allocates a one variable.

[vicsn]

This looks good to me, modulo an additional check in generate_constraints that ConstraintSystem allocates a one variable.

Thank you for the suggestion! I looked into it again and think we have a good setup already (avoiding further refactors):

1. Verifier: Most importantly, the verifier already checks that the first public variable is a one
2. Prover: Indeed clearly pushes the one input variable into the converter. Meanwhile the cs: ConstraintSystem trait doesn't give access to its variables barring other checks on that.

There is a lot of complexity, one day, I hope we can cut out some parts from the R1CS->Assignment->Converter->ConstraintSystem->[Matrix;3] chain.