chore(backend): implement more circuit op logic (#173)

* Add backend MerkleProof type

* Add eval_not_contains

* Remove print statement

* Handle some edge cases

* Add test

* Add missing ?

* Optimisation and stylistic changes

* Code review

src/backends/plonky2/primitives/merkletree.rs

@@ -23,13 +23,21 @@ pub struct MerkleTree {
 impl MerkleTree {
     /// builds a new `MerkleTree` where the leaves contain the given key-values
     pub fn new(max_depth: usize, kvs: &HashMap<Value, Value>) -> Result<Self> {
-        let mut root = Node::Intermediate(Intermediate::empty());
+        // Construct leaves.
+        let mut leaves: Vec<_> = kvs
+            .iter()
+            .map(|(k, v)| Leaf::new(max_depth, *k, *v))
+            .collect::<Result<_>>()?;
 
-        for (k, v) in kvs.iter() {
-            let leaf = Leaf::new(max_depth, *k, *v)?;
+        // Start with a leaf or conclude with an empty node as root.
+        let mut root = leaves.pop().map(|l| Node::Leaf(l)).unwrap_or(Node::None);
+
+        // Iterate over remaining leaves (if any) and add them.
+        for leaf in leaves.into_iter() {
             root.add_leaf(0, max_depth, leaf)?;
         }
 
+        // Fill in hashes.
         let _ = root.compute_hash();
         Ok(Self { max_depth, root })
     }

src/backends/plonky2/primitives/merkletree_circuit.rs

@@ -35,8 +35,9 @@ pub struct MerkleProofGadget {
     pub max_depth: usize,
 }
 
-pub struct MerkleProofTarget {
-    max_depth: usize,
+#[derive(Clone)]
+pub struct MerkleClaimAndProofTarget {
+    pub(crate) max_depth: usize,
     // `enabled` determines if the merkleproof verification is enabled
     pub(crate) enabled: BoolTarget,
     pub(crate) root: HashOutTarget,
@@ -51,7 +52,7 @@ pub struct MerkleProofTarget {
 
 impl MerkleProofGadget {
     /// creates the targets and defines the logic of the circuit
-    pub fn eval(&self, builder: &mut CircuitBuilder<F, D>) -> Result<MerkleProofTarget> {
+    pub fn eval(&self, builder: &mut CircuitBuilder<F, D>) -> Result<MerkleClaimAndProofTarget> {
         let enabled = builder.add_virtual_bool_target_safe();
         let root = builder.add_virtual_hash();
         let key = builder.add_virtual_value();
@@ -133,7 +134,7 @@ impl MerkleProofGadget {
             builder.connect(computed_root[j], expected_root[j]);
         }
 
-        Ok(MerkleProofTarget {
+        Ok(MerkleClaimAndProofTarget {
             max_depth: self.max_depth,
             enabled,
             existence,
@@ -148,7 +149,7 @@ impl MerkleProofGadget {
     }
 }
 
-impl MerkleProofTarget {
+impl MerkleClaimAndProofTarget {
     /// assigns the given values to the targets
     pub fn set_targets(
         &self,