Automatic signals generation

[leolara]

This is useful for:

• Arbitrary boolean expressions
• Multiplicative inverse elimination
• Automatic witness generation in general
• Conversion to lower degree polynomial identities

[qwang98]

Overall, this PR allows the auto_eq feature which automatically assigns the left hand side queriable of a regular constraint. I'm good with the implementation of this feature and left a couple comments on some potential improvements. Let me know if you agree! @leolara

Looking forward, there are more things we can do:

1. Allow auto witness generation for constraints other than eq.
2. Allow auto witness generation for transition constraints and signals with rotation as well.
3. Similar to PIL, we could eventually enable an auto witness generation "mode" where instead of using the auto_eq API, the user only needs to provide the initial trace input and all signals in all step instances will be inferred, or return an error saying that not enough trace input was provided.

[qwang98]

LGTM with additional features to be implemented in future PRs.

[qwang98]

@leolara Good to go after fixing the conflicts.

[codecov-commenter]

Codecov Report

Attention: 35 lines in your changes are missing coverage. Please review.

Comparison is base (93e0fad) 48.02% compared to head (bb0e74b) 49.81%.

❗ Current head bb0e74b differs from pull request most recent head eb35cca. Consider uploading reports for the commit eb35cca to get more accurate results

Additional details and impacted files

@@            Coverage Diff             @@
##             main     #152      +/-   ##
==========================================
+ Coverage   48.02%   49.81%   +1.79%     
==========================================
  Files          21       21              
  Lines        5483     5729     +246     
==========================================
+ Hits         2633     2854     +221     
- Misses       2850     2875      +25     

Files	Coverage Δ
src/ast/mod.rs	41.41% <100.00%> (+0.18%)	⬆️
src/plonkish/ir/sc.rs	0.00% <ø> (ø)
src/plonkish/compiler/mod.rs	0.00% <0.00%> (ø)
src/poly.rs	86.30% <96.49%> (+12.14%)	⬆️
src/plonkish/backend/halo2.rs	0.00% <0.00%> (ø)
src/plonkish/ir/assignments.rs	16.08% <0.00%> (-0.50%)	⬇️
src/frontend/dsl/mod.rs	16.86% <0.00%> (-0.56%)	⬇️
src/wit_gen.rs	87.46% <90.58%> (+4.59%)	⬆️

... and 1 file with indirect coverage changes

☔ View full report in Codecov by Sentry.
📢 Have feedback on the report? Share it here.