chore(docs): Update proving backend related docs (#5601)

# Description

## Summary\*

Update docs related to the installation and use of proving backends
following #4960.

Focusing mainly on improving:
1. Noir's proving backend agnosticity
2. Formatting/readability

Best to merge this PR in before stable releasing v0.31.0 and v0.32.0.

## Documentation\*

Check one:
- [ ] No documentation needed.
- [x] Documentation included in this PR.
- [ ] **[For Experimental Features]** Documentation to be submitted in a
separate PR.

# PR Checklist\*

- [x] I have tested the changes locally.
- [x] I have formatted the changes with [Prettier](https://prettier.io/)
and/or `cargo fmt` on default settings.

---------

Co-authored-by: James Zaki <[email protected]>

docs/docs/getting_started/barretenberg/_category_.json → docs/docs/getting_started/backend/_category_.json

@@ -1,6 +1,6 @@
 {
     "position": 1,
-    "label": "Install Barretenberg",
+    "label": "Install Proving Backend",
     "collapsible": true,
     "collapsed": true
 }

docs/docs/getting_started/backend/index.md

@@ -0,0 +1,31 @@
+---
+title: Proving Backend Installation
+description: Proving backends offer command line tools for proving and verifying Noir programs. This page describes how to install `bb` as an example.
+keywords: [
+  Proving
+  Backend
+  Barretenberg
+  bb
+  bbup
+  Installation
+  Terminal
+  Command
+  CLI
+  Version
+]
+pagination_next: getting_started/hello_noir/index
+---
+
+Proving backends each provide their own tools for working with Noir programs, providing functionality like proof generation, proof verification, and verifier smart contract generation.
+
+For the latest information on tooling provided by each proving backend, installation instructions, Noir version compatibility... you may refer to the proving backends' own documentation.
+
+You can find the full list of proving backends compatible with Noir in [Awesome Noir](https://github.com/noir-lang/awesome-noir/?tab=readme-ov-file#proving-backends).
+
+## Example: Installing `bb`
+
+`bb` is the CLI tool provided by the [Barretenberg proving backend](https://github.com/AztecProtocol/barretenberg) developed by Aztec Labs.
+
+You can find the instructions for installation in [`bb`'s documentation](https://github.com/AztecProtocol/aztec-packages/blob/master/barretenberg/cpp/src/barretenberg/bb/readme.md#installation).
+
+Once installed, we are ready to start working on [our first Noir program](../hello_noir/index.md).

docs/docs/getting_started/hello_noir/index.md

@@ -17,41 +17,41 @@ sidebar_position: 1
 
 ---
 
-Now that we have installed Nargo, it is time to make our first hello world program!
+Now that we have installed Nargo and a proving backend, it is time to make our first hello world program!
 
-## Create a Project Directory
+### 1. Create a new project directory
 
 Noir code can live anywhere on your computer. Let us create a _projects_ folder in the home
-directory to house our Noir programs.
+directory to house our first Noir program.
 
-For Linux, macOS, and Windows PowerShell, create the directory and change directory into it by
-running:
+Create the directory and change directory into it by running:
 
 ```sh
 mkdir ~/projects
 cd ~/projects
 ```
 
-## Create Our First Nargo Project
+## Nargo
+
+Nargo provides the ability to initiate and execute Noir projects. Read the [Nargo installation](../installation/index.md) section to learn more about Nargo and how to install it.
+
+### 2. Create a new Noir project
 
 Now that we are in the projects directory, create a new Nargo project by running:
 
 ```sh
 nargo new hello_world
 ```
 
-> **Note:** `hello_world` can be any arbitrary project name, we are simply using `hello_world` for
-> demonstration.
->
-> In production, the common practice is to name the project folder as `circuits` for better
-> identifiability when sitting alongside other folders in the codebase (e.g. `contracts`, `scripts`,
-> `test`).
+`hello_world` can be any arbitrary project name, we are simply using `hello_world` for demonstration.
+
+In production, it is common practice to name the project folder, `circuits`, for clarity amongst other folders in the codebase (like: `contracts`, `scripts`, and `test`).
 
 A `hello_world` folder would be created. Similar to Rust, the folder houses _src/main.nr_ and
 _Nargo.toml_ which contain the source code and environmental options of your Noir program
 respectively.
 
-### Intro to Noir Syntax
+#### Intro to Noir Syntax
 
 Let us take a closer look at _main.nr_. The default _main.nr_ generated should look like this:
 
@@ -81,7 +81,7 @@ The Noir syntax `assert` can be interpreted as something similar to constraints
 
 For more Noir syntax, check the [Language Concepts](../../noir/concepts/comments.md) chapter.
 
-## Build In/Output Files
+### 3. Build in/output files
 
 Change directory into _hello_world_ and build in/output files for your Noir program by running:
 
@@ -92,7 +92,7 @@ nargo check
 
 A _Prover.toml_ file will be generated in your project directory, to allow specifying input values to the program.
 
-## Execute Our Noir Program
+### 4. Execute the Noir program
 
 Now that the project is set up, we can execute our Noir program.
 
@@ -111,34 +111,41 @@ nargo execute witness-name
 
 The witness corresponding to this execution will then be written to the file `./target/witness-name.gz`.
 
-## Prove Our Noir Program
+The command also automatically compiles your Noir program if it was not already / was edited, which you may notice the compiled artifacts being written to the file `./target/hello_world.json`.
+
+## Proving Backend
 
-:::info
+Proving backends provide the ability to generate and verify proofs of executing Noir programs, following Noir's tooling that compiles and executes the programs. Read the [proving backend installation](../backend/index.md) section to learn more about proving backends and how to install them.
 
-Nargo no longer handles communicating with backends in order to generate proofs. In order to prove/verify your Noir programs, you'll need an installation of [bb](../barretenberg/index.md).
+Barretenberg is used as an example here to demonstrate how proving and verifying could be implemented and used. Read the [`bb` installation](../backend/index.md#example-installing-bb) section for how to install Barretenberg's CLI tool; refer to [`bb`'s documentation](https://github.com/AztecProtocol/aztec-packages/blob/master/barretenberg/cpp/src/barretenberg/bb/readme.md) for full details about the tool.
 
-:::
+### 5. Prove an execution of the Noir program
 
-Prove the valid execution of your Noir program using `bb`:
+Using Barretenberg as an example, prove the valid execution of your Noir program running:
 
 ```sh
-bb prove -b ./target/hello_world.json -w ./target/witness-name.gz -o ./proof
+bb prove -b ./target/hello_world.json -w ./target/witness-name.gz -o ./target/proof
 ```
 
-A new file called `proof` will be generated in your project directory, containing the generated proof for your program.
+The proof generated will then be written to the file `./target/proof`.
 
-## Verify Our Noir Program
+### 6. Verify the execution proof
 
 Once a proof is generated, we can verify correct execution of our Noir program by verifying the proof file.
 
-Verify your proof by running:
+Using Barretenberg as an example, compute the verification key for the Noir program by running:
 
 ```sh
 bb write_vk -b ./target/hello_world.json -o ./target/vk
-bb verify -k ./target/vk -p ./proof
 ```
 
-The verification will complete in silence if it is successful. If it fails, it will log the corresponding error instead.
+And verify your proof by running:
+
+```sh
+bb verify -k ./target/vk -p ./target/proof
+```
+
+If successful, the verification will complete in silence; if unsuccessful, the command will trigger logging of the corresponding error.
 
 Congratulations, you have now created and verified a proof for your very first Noir program!
 

docs/docs/getting_started/hello_noir/project_breakdown.md

@@ -8,8 +8,7 @@ keywords:
 sidebar_position: 2 
 ---
 
-This section breaks down our hello world program from the previous section. We elaborate on the project
-structure and what the `prove` and `verify` commands did.
+This section breaks down our hello world program from the previous section.
 
 ## Anatomy of a Nargo Project
 

docs/docs/getting_started/installation/index.md

@@ -19,11 +19,9 @@ keywords: [
 pagination_next: getting_started/hello_noir/index
 ---
 
-`nargo` is the one-stop-shop for almost everything related with Noir. The name comes from our love for Rust and its package manager `cargo`.
+`nargo` is a tool for working with Noir programs on the CLI, providing you with the ability to start new projects, compile, execute and test Noir programs from the terminal.
 
-With `nargo`, you can start new projects, compile, execute, prove, verify, test, generate solidity contracts, and do pretty much all that is available in Noir.
-
-Similarly to `rustup`, we also maintain an easy installation method that covers most machines: `noirup`.
+The name is inspired by Rust's package manager `cargo`; and similar to Rust's `rustup`, Noir also has an easy installation script `noirup`.
 
 ## Installing Noirup
 

docs/versioned_docs/version-v0.19.4/getting_started/01_hello_world.md

@@ -84,7 +84,7 @@ assert(x != y);
 
 The Noir syntax `assert` can be interpreted as something similar to constraints in other zk-contract languages.
 
-For more Noir syntax, check the [Language Concepts](../language_concepts/comments.md) chapter.
+For more Noir syntax, check the [Language Concepts](../language_concepts/09_comments.md) chapter.
 
 ## Build In/Output Files
 

docs/versioned_docs/version-v0.19.4/language_concepts/data_types/02_booleans.md

@@ -26,5 +26,5 @@ fn main() {
 > `false` in _Verifier.toml_.
 
 The boolean type is most commonly used in conditionals like `if` expressions and `assert`
-statements. More about conditionals is covered in the [Control Flow](../control_flow.md) and
-[Assert Function](../assert.md) sections.
+statements. More about conditionals is covered in the [Control Flow](../02_control_flow.md) and
+[Assert Function](../04_assert.md) sections.

docs/versioned_docs/version-v0.19.4/language_concepts/data_types/04_arrays.md

@@ -56,7 +56,7 @@ You can instantiate a new array of a fixed size with the same value repeated for
 let array: [Field; 32] = [0; 32];
 ```
 
-Like in Rust, arrays in Noir are a fixed size. However, if you wish to convert an array to a [slice](./slices.mdx), you can just call `as_slice` on your array:
+Like in Rust, arrays in Noir are a fixed size. However, if you wish to convert an array to a [slice](./05_slices.mdx), you can just call `as_slice` on your array:
 
 ```rust
 let array: [Field; 32] = [0; 32];

docs/versioned_docs/version-v0.19.4/modules_packages_crates/dependencies.md

@@ -81,7 +81,7 @@ use dep::std::scalar_mul::fixed_base_embedded_curve;
 ```
 
 Lastly, as demonstrated in the
-[elliptic curve example](../standard_library/cryptographic_primitives/ec_primitives.md#examples), you
+[elliptic curve example](../standard_library/cryptographic_primitives/04_ec_primitives.md#examples), you
 can import multiple items in the same line by enclosing them in curly braces:
 
 ```rust

docs/versioned_docs/version-v0.19.4/nargo/01_commands.md

@@ -213,7 +213,7 @@ you run `nargo test`. To print `println` statements in tests, use the `--show-ou
 
 Takes an optional `--exact` flag which allows you to select tests based on an exact name.
 
-See an example on the [testing page](./testing.md).
+See an example on the [testing page](./02_testing.md).
 
 ### Options
 

docs/versioned_docs/version-v0.19.4/standard_library/black_box_fns.md

@@ -26,19 +26,19 @@ fn sha256<N>(_input : [u8; N]) -> [u8; 32] {}
 Here is a list of the current black box functions that are supported by UltraPlonk:
 
 - AES
-- [SHA256](./cryptographic_primitives/hashes.mdx#sha256)
-- [Schnorr signature verification](./cryptographic_primitives/schnorr.mdx)
-- [Blake2s](./cryptographic_primitives/hashes.mdx#blake2s)
-- [Pedersen Hash](./cryptographic_primitives/hashes.mdx#pedersen_hash)
-- [Pedersen Commitment](./cryptographic_primitives/hashes.mdx#pedersen_commitment)
-- [HashToField128Security](./cryptographic_primitives/hashes.mdx#hash_to_field)
-- [ECDSA signature verification](./cryptographic_primitives/ecdsa_sig_verification.mdx)
-- [Fixed base scalar multiplication](./cryptographic_primitives/scalar.mdx)
+- [SHA256](./cryptographic_primitives/00_hashes.mdx#sha256)
+- [Schnorr signature verification](./cryptographic_primitives/02_schnorr.mdx)
+- [Blake2s](./cryptographic_primitives/00_hashes.mdx#blake2s)
+- [Pedersen Hash](./cryptographic_primitives/00_hashes.mdx#pedersen_hash)
+- [Pedersen Commitment](./cryptographic_primitives/00_hashes.mdx#pedersen_commitment)
+- [HashToField128Security](./cryptographic_primitives/00_hashes.mdx#hash_to_field)
+- [ECDSA signature verification](./cryptographic_primitives/03_ecdsa_sig_verification.mdx)
+- [Fixed base scalar multiplication](./cryptographic_primitives/01_scalar.mdx)
 - [Compute merkle root](./merkle_trees.md#compute_merkle_root)
 - AND
 - XOR
 - RANGE
-- [Keccak256](./cryptographic_primitives/hashes.mdx#keccak256)
+- [Keccak256](./cryptographic_primitives/00_hashes.mdx#keccak256)
 - [Recursive proof verification](./recursion.md)
 
 Most black box functions are included as part of the Noir standard library, however `AND`, `XOR` and `RANGE` are used as part of the Noir language syntax. For instance, using the bitwise operator `&` will invoke the `AND` black box function. To ensure compatibility across backends, the ACVM has fallback implementations of `AND`, `XOR` and `RANGE` defined in its standard library which it can seamlessly fallback to if the backend doesn't support them.

docs/versioned_docs/version-v0.32.0/getting_started/barretenberg/_category_.json → docs/versioned_docs/version-v0.31.0/getting_started/backend/_category_.json

@@ -1,6 +1,6 @@
 {
     "position": 1,
-    "label": "Install Barretenberg",
+    "label": "Install Proving Backend",
     "collapsible": true,
     "collapsed": true
 }

docs/versioned_docs/version-v0.31.0/getting_started/backend/index.md

@@ -0,0 +1,31 @@
+---
+title: Proving Backend Installation
+description: Proving backends offer command line tools for proving and verifying Noir programs. This page describes how to install `bb` as an example.
+keywords: [
+  Proving
+  Backend
+  Barretenberg
+  bb
+  bbup
+  Installation
+  Terminal
+  Command
+  CLI
+  Version
+]
+pagination_next: getting_started/hello_noir/index
+---
+
+Proving backends each provide their own tools for working with Noir programs, providing functionality like proof generation, proof verification, and verifier smart contract generation.
+
+For the latest information on tooling provided by each proving backend, installation instructions, Noir version compatibility... you may refer to the proving backends' own documentation.
+
+You can find the full list of proving backends compatible with Noir in [Awesome Noir](https://github.com/noir-lang/awesome-noir/?tab=readme-ov-file#proving-backends).
+
+## Example: Installing `bb`
+
+`bb` is the CLI tool provided by the [Barretenberg proving backend](https://github.com/AztecProtocol/barretenberg) developed by Aztec Labs.
+
+You can find the instructions for installation in [`bb`'s documentation](https://github.com/AztecProtocol/aztec-packages/blob/master/barretenberg/cpp/src/barretenberg/bb/readme.md#installation).
+
+Once installed, we are ready to start working on [our first Noir program](../hello_noir/index.md).