* Add external inputs logic to F function/circuit. Add an example of usage with external inputs too.

* Add examples run into CI

* Change CycleFold approach:

Instead of having a single CycleFold circuit that checks the 2 forign
scalarmul of the main circuit instances, now there are 2 separated
CycleFold circuits each of them checking a single foreign scalarmul.
Increasing the number of constraints of the AugmentedFCircuit, but
reducing the number of constraints in the CycleFold circuit, which will
translate into reducing the number of constraints in the Decider
circuit.

* CycleFold circuits checks in AugmentedFCircuit:

- update NonNativeAffineVar to work with NonNativeFieldVar directly
  instead of FpVar comming from NonNativeFieldVar.to_constraint_field()
- include in AugmentedFCircuit intermediate steps inbetween CycleFold
  circuits, and update the internal checks of the CycleFold circuits

Pending to document the new CycleFold circuits approach and better
variable namings, rm unwraps, etc

* matrix_vec_mul_sparse gadget: skip value * v[col_i] mul when value==1

Saves a notable amount of constraints since there is a notable amount of
1 values in R1CS matrices.

* Reuse computed vector of U_i

Reuse computed vector of U_i, saving 4k constraints in AugmentedFCircuit.

* fixes post last rebase to main

* rm test_augmentedfcircuit since it is already tested in test_ivc (and is a slow computation)

* rm dbg!()

* small fixes after last main rebase

* Add IPA commitment native implementation

* Add IPA Gadget verifier

* polish Pedersen & IPA, add blind bool param to IPA

* Optimize IPA gadget constraints (and native):

- optimize <s,b> computation from linear to log time
- optimize s computation from k*2^k to k*(2^k)/2

* add small optimization: delegate u_i^-1 to prover and just check u_i*u_i^-1==1 in verifier circuit

* IPA polish and document

* Add 'BLIND' parameter to CommitmentProver trait (and to Pedersen and KZG impls). Fit IPA into CommitmentProver trait.

* rename 'BLIND' to 'H' (hiding) in commitment

* IPA: rm u_invs from Proof and compute them incircuit

* Update IPA's build_s & gadget to use Halo2 approach following @han0110 's suggestion.

This reduced further the amount of constraints needed.
- for k=4: -9k constraints (-7%)
- for k=8: -473k constr (-31%)
- for k=9: -1123k constr (-35%)
- for k=10: -2578k constr (-39%)
And now IPA verification (without amortizing) is very close to Pedersen
verification (in-circuits).

* rm dbg!(cs.num_constraints()) from multiple tests

* IPA::prove remove intermediate v_lo,v_hi vectors, add doc to build_s_gadget

* move powers_of into utils/mod.rs, update iters to cfg_iter

* Add typos to CI

* Apply typos suggestions

* missing typos

* Fix Nova multi-elements state

In the AugmentedFCircuit the default value for the state when no input
is provided was `vec![F::zero()]`, which defaults to length `1`. So when
having more than 1 element in the state, before even starting to fold,
the circuit was already already failing.

Additionally this commit adds an example for a circuit with a state of 5
elements.

* abstract 'nova_setup' helper to avoid code duplication in examples

* update example naming to 'MultiInputs'

* rename nova_setup -> test_nova_setup to make it more explicit

* change: Refactor structure into workspace

* chore: Add empty readme

* change: Transform repo into workspace

* add: Create folding-verifier-solidity crate

* add: Include askama.toml for `sol` extension escaper

* add: Jordi's old Groth16 verifier .sol template and adapt it

* tmp: create simple template struct to test

* Update FoldingSchemes trait, fit Nova+CycleFold

- update lib.rs's `FoldingScheme` trait interface
- fit Nova+CycleFold into the `FoldingScheme` trait
- refactor `src/nova/*`

* chore: add serialization assets for testing

Now we include an `assets` folder with a serialized proof & vk for tests

* Add `examples` dir, with Nova's `FoldingScheme` example

* polishing

* expose poseidon_test_config outside tests

* change: Refactor structure into workspace

* chore: Add empty readme

* change: Transform repo into workspace

* add: Create folding-verifier-solidity crate

* add: Include askama.toml for `sol` extension escaper

* add: Jordi's old Groth16 verifier .sol template and adapt it

* tmp: create simple template struct to test

* feat: templating kzg working

* chore: add emv and revm

* feat: start evm file

* chore: add ark-poly-commit

* chore: move `commitment` to `folding-schemes`

* chore: update `.gitignore` to ignore generated contracts

* chore: update template with bn254 lib on it (avoids import), update for loop to account for whitespaces

* refactor: update template with no lib

* feat: add evm deploy code, compile and create kzg verifier

* chore: update `Cargo.toml` to have `folding-schemes` available with verifiers

* feat: start kzg prove and verify with sol

* chore: compute crs from kzg prover

* feat: evm kzg verification passing

* tmp

* change: Swap order of G2 coordinates within the template

* Update way to serialize proof with correct order

* chore: update `Cargo.toml`

* chore: add revm

* chore: add `save_solidity`

* refactor: verifiers in dedicated mod

* refactor: have dedicated `utils` module

* chore: expose modules

* chore: update verifier for kzg

* chore: rename templates

* fix: look for binary using also name of contract

* refactor: generate groth16 proof for sha256 pre-image, generate groth16 template with verifying key

* chore: template renaming

* fix: switch circuit for circuit that simply adds

* feat: generates test data on the fly

* feat: update to latest groth16 verifier

* refactor: rename folder, update `.gitignore`

* chore: update `Cargo.toml`

* chore: update templates extension to indicate that they are templates

* chore: rename templates, both files and structs

* fix: template inheritance working

* feat: template spdx and pragma statements

* feat: decider verifier compiles, update test for kzg10 and groth16 templates

* feat: parameterize which size of the crs should be stored on the contract

* chore: add comment on how the groth16 and kzg10 proofs will be linked together

* chore: cargo clippy run

* chore: cargo clippy tests

* chore: cargo fmt

* refactor: remove unused lifetime parameter

* chore: end merge

* chore: move examples to `folding-schemes` workspace

* get latest main changes

* fix: temp fix clippy warnings, will remove lints once not used in tests only

* fix: cargo clippy lint added on `code_size`

* fix: update path to test circuit and add step for installing solc

* chore: remove `save_solidity` steps

* fix: the borrowed expression implements the required traits

* chore: update `Cargo.toml`

* chore: remove extra `[patch.crates-io]`

* fix: update to patch at the workspace level and add comment explaining this

* refactor: correct `staticcall` with valid input/output sizes and change return syntax for pairing

* refactor: expose modules and remove `dead_code` calls

* chore: update `README.md`, add additional comments on `kzg10` template and update `groth16` template comments

* chore: be clearer on attributions on `kzg10`

---------

Co-authored-by: CPerezz <c.perezbaro@gmail.com>
Co-authored-by: arnaucube <root@arnaucube.com>