update merkle stdlib functions docs

docs/standard_library/merkle_trees.md

@@ -1,8 +1,7 @@
 ---
 title: Merkle Trees
 description:
-  Learn about Merkle Trees in Noir with this tutorial. Explore the basics of check membership and
-  computing root from leaf, and implement them in your code with the help of code
+  Learn about Merkle Trees in Noir with this tutorial. Explore the basics of computing a merkle root using a proof, with examples.
 keywords:
   [
     Merkle trees in Noir,
@@ -17,62 +16,15 @@ keywords:
   ]
 ---
 
-## check_membership
-
-Returns 1 if the specified leaf is at the given index on a tree.
-
-```rust
-fn check_membership(_root : Field, _leaf : Field, _index : Field, _hash_path: [Field]) -> Field
-```
-
-example:
-
-```rust
-/**
- *
-    index = "0"
-    priv_key = "0x000000000000000000000000000000000000000000000000000000616c696365"
-    secret = "0x1929ea3ab8d9106a899386883d9428f8256cfedb3c4f6b66bf4aa4d28a79988f"
-    root = "0x2f36d4404719a30512af45be47c9732e916cb131933102b04ba6432602db209c"
-    hash_path = [
-    "0x1e61bdae0f027b1b2159e1f9d3f8d00fa668a952dddd822fda80dc745d6f65cc",
-    "0x0e4223f3925f98934393c74975142bd73079ab0621f4ee133cee050a3c194f1a",
-    "0x2fd7bb412155bf8693a3bd2a3e7581a679c95c68a052f835dddca85fa1569a40"
-    ]
- */
-
-fn main(root : Field, index : Field, hash_path : [Field; 3], secret: Field, priv_key: Field) {
-    constrain index == index;
-
-    let pubkey = std::scalar_mul::fixed_base(priv_key);
-    let pubkey_x = pubkey[0];
-    let pubkey_y = pubkey[1];
-    let note_commitment = std::hash::pedersen([pubkey_x, pubkey_y, secret]);
-
-    let root = std::merkle::check_membership(root, note_commitment[0], index, hash_path);
-    std::println(root);
-}
-```
-
-## check_membership_in_noir
-
-Behaves exactly the same as above, but it's computed in Noir in order to accept many backends.
-
-```rust
-fn check_membership_in_noir(root : Field, leaf : Field, index : Field, hash_path: [Field]) -> Field
-```
-
-For examples, you can literally replace `check_membership` for this method, in the above example.
-
-## compute_root_from_leaf
+## compute_merkle_root
 
-Returns the root of the tree from the provided leaf and its hash path, using a pedersen hash.
+Returns the root of the tree from the provided leaf and its hash path, using a [Pedersen hash](cryptographic_primitives/00_hashes.mdx#pedersen).
 
 ```rust
-fn compute_root_from_leaf(leaf : Field, index : Field, hash_path: [Field]) -> Field
+fn compute_merkle_root(leaf : Field, index : Field, hash_path: [Field]) -> Field
 ```
 
-example:
+### Example:
 
 ```rust
 /**
@@ -85,30 +37,14 @@ example:
     "0x2fd7bb412155bf8693a3bd2a3e7581a679c95c68a052f835dddca85fa1569a40"
     ]
  */
-fn main(index : Field, priv_key : Field, secret : Field, note_hash_path : [Field; 3]) {
-    constrain index == index;
+fn main(index: Field, priv_key: Field, secret: Field, note_hash_path: [Field; 3]) {
 
     let pubkey = std::scalar_mul::fixed_base(priv_key);
     let pubkey_x = pubkey[0];
     let pubkey_y = pubkey[1];
     let note_commitment = std::hash::pedersen([pubkey_x, pubkey_y, secret]);
 
-    let root = std::merkle::compute_root_from_leaf(note_commitment[0], index, note_hash_path);
+    let root = std::merkle::compute_merkle_root(note_commitment[0], index, note_hash_path);
     std::println(root);
 }
 ```
-
-## compute_merkle_root
-
-Returns the root of the tree from the provided leaf and its hash path, using a pedersen hash.
-
-```rust
-#[foreign(compute_merkle_root)]
-fn compute_merkle_root(_leaf : Field, _index : Field, _hash_path: [Field]) -> Field {}
-```
-
-:::info
-
-This is a black box function. Read [this section](black_box_fns) to learn more about black box functions in Noir.
-
-:::