Adapt Frontend/Circom for FCircuit Trait (#71)

* initial

* improved z_i

* improved

* Redefined the .circom and the test code for CircomWrapper

* added test function for wrapper_circuit which has circom_fcircuit, but incompatibility error of Pairing and Pallas

* changed the path in the gitignore

* Remove circom generated path

* improved variable vector

* Pallas::Fr

* improved to the primefield

* mentioned the issues:1. extract z_i1(only public output) 2.constraintsSystem and its Ref

* modified

* public input in circom

* generalized; removed the hardcorded value

* Generalization using FpVar::<Fr>::new_input

* initial

* improved z_i

* improved

* Redefined the .circom and the test code for CircomWrapper

* added test function for wrapper_circuit which has circom_fcircuit, but incompatibility error of Pairing and Pallas

* Remove circom generated path

* improved variable vector

* Pallas::Fr

* improved to the primefield

* mentioned the issues:1. extract z_i1(only public output) 2.constraintsSystem and its Ref

* modified

* Small updates:

- update cubic_circuit.circom: remove extra constraint, remove public
  inputs
- remove allocations of inputs in arkworks
- add return of z_{i+1} at the end of CircomtoFCircuit::generate_step_constraints

With this tmp fix the `test_circom_step_constraints` passes, but needs to be iterated and polished.

* Update circom-compat to re-allocate inputs

* update after rebase to latest main: add usage of self.state_len() to circom frontend

* move circom frontend related structs into frontend/circom dir

* clippy lints

* extract_witness

* add comments

* clean

* fmt, lint, and spell

* CI Check trigger

* fmt

* applied the feedback

---------

Co-authored-by: Y5 <76672645+yugonsan@users.noreply.github.com>
Co-authored-by: arnaucube <root@arnaucube.com>

folding-schemes/src/frontend/mod.rs

@@ -1,11 +1,11 @@
-pub mod circom;
-
 use crate::Error;
 use ark_ff::PrimeField;
 use ark_r1cs_std::fields::fp::FpVar;
 use ark_relations::r1cs::{ConstraintSystemRef, SynthesisError};
 use ark_std::fmt::Debug;
 
+pub mod circom;
+
 /// FCircuit defines the trait of the circuit of the F function, which is the one being folded (ie.
 /// inside the agmented F' function).
 /// The parameter z_i denotes the current state, and z_{i+1} denotes the next state after applying
@@ -44,7 +44,7 @@ pub trait FCircuit<F: PrimeField>: Clone + Debug {
 #[cfg(test)]
 pub mod tests {
     use super::*;
-    use ark_pallas::Fr;
+    use ark_bn254::Fr;
     use ark_r1cs_std::{alloc::AllocVar, eq::EqGadget};
     use ark_relations::r1cs::{
         ConstraintSynthesizer, ConstraintSystem, ConstraintSystemRef, SynthesisError,