continued support for noir -> r1cs

[dolores-park]

• add r1cs optimization, 1 poseidon row count reduced from 644->428

[recmo]

There is a more general result: The last mul term never needs an intermediate constraint (even if there are multiple mul-terms).

[dolores-park]

fixed witness generation

[dolores-park]

@recmo think this is a good place to merge into main, would love to get feedback / approval from you
for the updates:

• fixed & optimized the -> r1cs process
• added witness population from .gz witness files
• added bunch of IO functions

to test,
run commands like
cargo run --bin noir-r1cs -- noir /Users/ende.shen/pork/ProveKit/noir-r1cs/noir-examples/basic-3/target/basic.json /Users/ende.shen/pork/ProveKit/noir-r1cs/noir-examples/basic-3/target/basic.gz
after that,
there should be a r1cs.json file generated,
and that file can be copied into ssh://git@github.com/reilabs/ProveKit.git, and proven with their prover/src/main.rs

On a separate note,
I've found the std impl of sha256, in the history of noir-lang repo,
https://github.com/noir-lang/noir/blob/e895feb4e7b25530a22668bca597dfc78be92584/noir_stdlib/src/hash/sha256.nr
In all its history, it always had

#[foreign(sha256_compression)]
pub fn sha256_compression(_input: [u32; 16], _state: [u32; 8]) -> [u32; 8] {}

as a blackbox function anyways, so will try to
rip the r1cs constraints from circom / gnark

[dolores-park]

Added some scripts from @RyanCao7 , to run use

cd noir-r1cs
bash scripts/run_all.sh

see the outputs at scripts/all_outputs.txt

[dolores-park]

To have a full end to end run,
try picking a noir program
say cd noir-r1cs/noir-examples/poseidon-rounds
provide the correct Prover.toml file under that dir (inputs to noir programs)
and run
nargo execute
to generate
.json file -- represent the circuit represented in ACIR
and .gz file -- represent the witness

after having the noir intermediate representations, compile file at noir-r1cs/src/main.rs
and run
target/debug/noir-r1cs /path-to/*.json /path-to/*.gz
the two args are path to .json file and path to .gz file

after that, there should be a r1cs.json at this ProveKit, this is the ABC matrices that representing the r1cs

==========================================================================================
(For these below, also checkout https://github.com/reilabs/gnark-whir/blob/main/README.md)

copy that to the repo at
https://github.com/reilabs/ProveKit.git
as prover/r1cs.json
and run
rustup run beta cargo run --bin prover -- --input_file_path prover/r1cs.json
to get the two files
prover/params and prover/proof

checkout the repo at
https://github.com/reilabs/gnark-whir
cd into the repo, and run
go run .
note that it assumes
https://github.com/reilabs/gnark-whir and https://github.com/reilabs/ProveKit.git live side by side, and the two files prover/params and prover/proof live in https://github.com/reilabs/ProveKit.git.

==========================================================================================

[recmo]

For above instructions to work it is important that the noirc version matches the noirc_artifacts crate.

noirup -C 03b58fa2