Complete the verification in MainMockPod (#302)

- Update the `RecursivePod` trait to return `vd_set` instead of `vds_root`
  - A native verifier requires the entire set to reason about the circuits that have been used in the recursive tree
- Implement serialization/deserialization for `VDSet`
- Remove `DynError` and use `BackendError` instead for middleware functions that wrap or define trait functions implemented in the backend.  This is based on the fact that we will only have a single backend enabled at a time, so there's no need for a `dyn Error`
  - Move the implementations of `_verify` functions to `verify` and similarly for `_prove`
- Complete the verification of a MockMainPod: the verification of input pods was missing.  The inclusion of these input pods in the serialization was also missing.  With this change a `MockMainPod` will grow after each recursion.  This was expected from the design but was not the case because of the missing recursive native verification implementation.

* apply feedback from @arnaucube

src/frontend/error.rs

@@ -1,6 +1,6 @@
 use std::{backtrace::Backtrace, fmt::Debug};
 
-use crate::middleware::{DynError, Statement, StatementTmpl, Value};
+use crate::middleware::{BackendError, Statement, StatementTmpl, Value};
 
 pub type Result<T, E = Error> = core::result::Result<T, E>;
 
@@ -41,7 +41,7 @@ pub enum Error {
     #[error(transparent)]
     Infallible(#[from] std::convert::Infallible),
     #[error(transparent)]
-    Backend(#[from] Box<DynError>),
+    Backend(#[from] BackendError),
     #[error(transparent)]
     Middleware(#[from] crate::middleware::Error),
 }

src/frontend/mod.rs

@@ -564,7 +564,7 @@ impl MainPodBuilder {
             statements: &statements,
             operations: &operations,
             public_statements: &public_statements,
-            vds_set: self.vd_set.clone(),
+            vd_set: self.vd_set.clone(),
         };
         let pod = prover.prove(&self.params, &self.vd_set, inputs)?;
 

src/frontend/serialization.rs

@@ -4,7 +4,7 @@ use serde::{Deserialize, Serialize};
 use super::Error;
 use crate::{
     frontend::{MainPod, SignedPod},
-    middleware::{deserialize_pod, deserialize_signed_pod, Hash, Params, PodId},
+    middleware::{deserialize_pod, deserialize_signed_pod, Params, PodId, VDSet},
 };
 
 #[derive(Serialize, Deserialize, JsonSchema)]
@@ -29,7 +29,7 @@ pub struct SerializedMainPod {
     params: Params,
     pod_type: (usize, String),
     id: PodId,
-    vds_root: Hash,
+    vd_set: VDSet,
     data: serde_json::Value,
 }
 
@@ -62,7 +62,7 @@ impl From<MainPod> for SerializedMainPod {
         SerializedMainPod {
             pod_type: (pod_type, pod_type_name_str.to_string()),
             id: pod.id(),
-            vds_root: pod.pod.vds_root(),
+            vd_set: pod.pod.vd_set().clone(),
             params: pod.params.clone(),
             data,
         }
@@ -77,7 +77,7 @@ impl TryFrom<SerializedMainPod> for MainPod {
             serialized.pod_type.0,
             serialized.params.clone(),
             serialized.id,
-            serialized.vds_root,
+            serialized.vd_set,
             serialized.data,
         )?;
         let public_statements = pod.pub_statements();