diff --git a/rustfmt.toml b/rustfmt.toml
index 7b20d96..351c26a 100644
--- a/rustfmt.toml
+++ b/rustfmt.toml
@@ -2,3 +2,4 @@ edition = "2018"
 tab_spaces = 2
 newline_style = "Unix"
 use_try_shorthand = true
+imports_granularity = "crate"
diff --git a/src/bellperson/r1cs.rs b/src/bellperson/r1cs.rs
index afed62f..c3851a8 100644
--- a/src/bellperson/r1cs.rs
+++ b/src/bellperson/r1cs.rs
@@ -2,8 +2,7 @@
 
 #![allow(non_snake_case)]
 
-use super::shape_cs::ShapeCS;
-use super::solver::SatisfyingAssignment;
+use super::{shape_cs::ShapeCS, solver::SatisfyingAssignment};
 use bellperson::{Index, LinearCombination};
 
 use ff::PrimeField;
diff --git a/src/bellperson/shape_cs.rs b/src/bellperson/shape_cs.rs
index 5ca5526..6e9468f 100644
--- a/src/bellperson/shape_cs.rs
+++ b/src/bellperson/shape_cs.rs
@@ -1,7 +1,9 @@
 //! Support for generating R1CS shape using bellperson.
 
-use std::cmp::Ordering;
-use std::collections::{BTreeMap, HashMap};
+use std::{
+  cmp::Ordering,
+  collections::{BTreeMap, HashMap},
+};
 
 use crate::traits::{Group, PrimeField as PF};
 use ff::{Field, PrimeField};
diff --git a/src/circuit.rs b/src/circuit.rs
index e6aa13f..705795b 100644
--- a/src/circuit.rs
+++ b/src/circuit.rs
@@ -10,18 +10,19 @@
 //! otherwise
 //! h1 = H(u2, i) and h2 = H(params = H(shape, gens), u1, i, z0, zi)
 
-use super::commitments::Commitment;
-use super::gadgets::{
-  ecc::AllocatedPoint,
-  utils::{
-    alloc_bignat_constant, alloc_num_equals, alloc_one, alloc_zero, conditionally_select,
-    conditionally_select_bignat, le_bits_to_num,
+use super::{
+  commitments::Commitment,
+  gadgets::{
+    ecc::AllocatedPoint,
+    utils::{
+      alloc_bignat_constant, alloc_num_equals, alloc_one, alloc_zero, conditionally_select,
+      conditionally_select_bignat, le_bits_to_num,
+    },
   },
+  poseidon::{NovaPoseidonConstants, PoseidonROGadget},
+  r1cs::RelaxedR1CSInstance,
+  traits::{Group, PrimeField, StepCircuit},
 };
-use super::poseidon::NovaPoseidonConstants;
-use super::poseidon::PoseidonROGadget;
-use super::r1cs::RelaxedR1CSInstance;
-use super::traits::{Group, InnerCircuit, PrimeField};
 use bellperson::{
   gadgets::{boolean::Boolean, num::AllocatedNum, Assignment},
   Circuit, ConstraintSystem, SynthesisError,
@@ -33,12 +34,12 @@ use bellperson_nonnative::{
 use ff::PrimeFieldBits;
 
 #[derive(Debug, Clone)]
-pub struct VerificationCircuitParams {
+pub struct NIFSVerifierCircuitParams {
   limb_width: usize,
   n_limbs: usize,
 }
 
-impl VerificationCircuitParams {
+impl NIFSVerifierCircuitParams {
   #[allow(dead_code)]
   pub fn new(limb_width: usize, n_limbs: usize) -> Self {
     Self {
@@ -48,7 +49,7 @@ impl VerificationCircuitParams {
   }
 }
 
-pub struct VerificationCircuitInputs<G>
+pub struct NIFSVerifierCircuitInputs<G>
 where
   G: Group,
 {
@@ -63,7 +64,7 @@ where
   w: Commitment<G>, // The commitment to the witness of the fresh r1cs instance
 }
 
-impl<G> VerificationCircuitInputs<G>
+impl<G> NIFSVerifierCircuitInputs<G>
 where
   G: Group,
 {
@@ -95,30 +96,30 @@ where
 }
 
 /// Circuit that encodes only the folding verifier
-pub struct VerificationCircuit<G, IC>
+pub struct NIFSVerifierCircuit<G, SC>
 where
   G: Group,
   <G as Group>::Base: ff::PrimeField,
-  IC: InnerCircuit<G::Base>,
+  SC: StepCircuit<G::Base>,
 {
-  params: VerificationCircuitParams,
-  inputs: Option<VerificationCircuitInputs<G>>,
-  inner_circuit: Option<IC>, // The function that is applied for each step. may be None.
+  params: NIFSVerifierCircuitParams,
+  inputs: Option<NIFSVerifierCircuitInputs<G>>,
+  step_circuit: Option<SC>, // The function that is applied for each step. may be None.
   poseidon_constants: NovaPoseidonConstants<G::Base>,
 }
 
-impl<G, IC> VerificationCircuit<G, IC>
+impl<G, SC> NIFSVerifierCircuit<G, SC>
 where
   G: Group,
   <G as Group>::Base: ff::PrimeField,
-  IC: InnerCircuit<G::Base>,
+  SC: StepCircuit<G::Base>,
 {
   /// Create a new verification circuit for the input relaxed r1cs instances
   #[allow(dead_code)]
   pub fn new(
-    params: VerificationCircuitParams,
-    inputs: Option<VerificationCircuitInputs<G>>,
-    inner_circuit: Option<IC>,
+    params: NIFSVerifierCircuitParams,
+    inputs: Option<NIFSVerifierCircuitInputs<G>>,
+    step_circuit: Option<SC>,
     poseidon_constants: NovaPoseidonConstants<G::Base>,
   ) -> Self
   where
@@ -127,18 +128,18 @@ where
     Self {
       params,
       inputs,
-      inner_circuit,
+      step_circuit,
       poseidon_constants,
     }
   }
 }
 
-impl<G, IC> Circuit<<G as Group>::Base> for VerificationCircuit<G, IC>
+impl<G, SC> Circuit<<G as Group>::Base> for NIFSVerifierCircuit<G, SC>
 where
   G: Group,
   <G as Group>::Base: ff::PrimeField + PrimeField + PrimeFieldBits,
   <G as Group>::Scalar: PrimeFieldBits,
-  IC: InnerCircuit<G::Base>,
+  SC: StepCircuit<G::Base>,
 {
   fn synthesize<CS: ConstraintSystem<<G as Group>::Base>>(
     self,
@@ -304,7 +305,7 @@ where
     // Allocate W
     /***********************************************************************/
 
-    // T = (x, y, infinity)
+    // W = (x, y, infinity)
     let W_x = AllocatedNum::alloc(cs.namespace(|| "W.x"), || {
       Ok(self.inputs.get()?.w.comm.to_coordinates().0)
     })?;
@@ -336,7 +337,7 @@ where
     // Compute default values of U2':
     let zero_commitment = AllocatedPoint::new(zero.clone(), zero.clone(), one);
 
-    //W_default and E_default are a commitment to zero
+    // W_default and E_default are a commitment to zero
     let W_default = zero_commitment.clone();
     let E_default = zero_commitment;
 
@@ -501,7 +502,7 @@ where
       .collect::<Result<Vec<AllocatedNum<G::Base>>, _>>()?;
 
     /***********************************************************************/
-    //Compute i + 1
+    // Compute i + 1
     /***********************************************************************/
 
     let next_i = AllocatedNum::alloc(cs.namespace(|| "i + 1"), || {
@@ -515,7 +516,7 @@ where
       |lc| lc + next_i.get_variable() - CS::one() - i.get_variable(),
     );
 
-    if self.inner_circuit.is_some() {
+    if self.step_circuit.is_some() {
       /***********************************************************************/
       //Allocate z0
       /***********************************************************************/
@@ -590,7 +591,7 @@ where
       /***********************************************************************/
 
       let z_next = self
-        .inner_circuit
+        .step_circuit
         .unwrap()
         .synthesize(&mut cs.namespace(|| "F"), z_i)?;
 
@@ -641,12 +642,12 @@ where
       h1_hash.absorb(u_r);
       h1_hash.absorb(i.clone());
 
-      //absorb each of the limbs of X_r[0]
+      // absorb each of the limbs of X_r[0]
       for limb in Xr0_bn.into_iter() {
         h1_hash.absorb(limb);
       }
 
-      //absorb each of the limbs of X_r[1]
+      // absorb each of the limbs of X_r[1]
       for limb in Xr1_bn.into_iter() {
         h1_hash.absorb(limb);
       }
@@ -698,12 +699,13 @@ where
 #[cfg(test)]
 mod tests {
   use super::*;
-  use crate::bellperson::shape_cs::ShapeCS;
-  use crate::bellperson::solver::SatisfyingAssignment;
+  use crate::bellperson::{shape_cs::ShapeCS, solver::SatisfyingAssignment};
   type G1 = pasta_curves::pallas::Point;
   type G2 = pasta_curves::vesta::Point;
-  use crate::bellperson::r1cs::{NovaShape, NovaWitness};
-  use crate::commitments::CommitTrait;
+  use crate::{
+    bellperson::r1cs::{NovaShape, NovaWitness},
+    commitments::CommitTrait,
+  };
   use std::marker::PhantomData;
 
   struct TestCircuit<F>
@@ -713,7 +715,7 @@ mod tests {
     _p: PhantomData<F>,
   }
 
-  impl<F> InnerCircuit<F> for TestCircuit<F>
+  impl<F> StepCircuit<F> for TestCircuit<F>
   where
     F: PrimeField + ff::PrimeField,
   {
@@ -729,12 +731,12 @@ mod tests {
   #[test]
   fn test_verification_circuit() {
     // We experiment with 8 limbs of 32 bits each
-    let params = VerificationCircuitParams::new(32, 8);
+    let params = NIFSVerifierCircuitParams::new(32, 8);
     // The first circuit that verifies G2
     let poseidon_constants1: NovaPoseidonConstants<<G2 as Group>::Base> =
       NovaPoseidonConstants::new();
-    let circuit1: VerificationCircuit<G2, TestCircuit<<G2 as Group>::Base>> =
-      VerificationCircuit::new(
+    let circuit1: NIFSVerifierCircuit<G2, TestCircuit<<G2 as Group>::Base>> =
+      NIFSVerifierCircuit::new(
         params.clone(),
         None,
         Some(TestCircuit {
@@ -745,8 +747,7 @@ mod tests {
     // First create the shape
     let mut cs: ShapeCS<G1> = ShapeCS::new();
     let _ = circuit1.synthesize(&mut cs);
-    let shape1 = cs.r1cs_shape();
-    let gens1 = cs.r1cs_gens();
+    let (shape1, gens1) = (cs.r1cs_shape(), cs.r1cs_gens());
     println!(
       "Circuit1 -> Number of constraints: {}",
       cs.num_constraints()
@@ -755,28 +756,29 @@ mod tests {
     // The second circuit that verifies G1
     let poseidon_constants2: NovaPoseidonConstants<<G1 as Group>::Base> =
       NovaPoseidonConstants::new();
-    let circuit2: VerificationCircuit<G1, TestCircuit<<G1 as Group>::Base>> =
-      VerificationCircuit::new(params.clone(), None, None, poseidon_constants2);
+    let circuit2: NIFSVerifierCircuit<G1, TestCircuit<<G1 as Group>::Base>> =
+      NIFSVerifierCircuit::new(params.clone(), None, None, poseidon_constants2);
     // First create the shape
     let mut cs: ShapeCS<G2> = ShapeCS::new();
     let _ = circuit2.synthesize(&mut cs);
-    let shape2 = cs.r1cs_shape();
-    let gens2 = cs.r1cs_gens();
+    let (shape2, gens2) = (cs.r1cs_shape(), cs.r1cs_gens());
     println!(
       "Circuit2 -> Number of constraints: {}",
       cs.num_constraints()
     );
 
-    //TODO: We need to hardwire default hash or give it as input
+    // TODO: We need to hardwire default hash or give it as input
     let default_hash = <<G2 as Group>::Base as ff::PrimeField>::from_str_vartime(
       "332553638888022689042501686561503049809",
     )
     .unwrap();
+
     let T = vec![<G2 as Group>::Scalar::zero()].commit(&gens2.gens_E);
     let w = vec![<G2 as Group>::Scalar::zero()].commit(&gens2.gens_E);
+
     // Now get an assignment
     let mut cs: SatisfyingAssignment<G1> = SatisfyingAssignment::new();
-    let inputs: VerificationCircuitInputs<G2> = VerificationCircuitInputs::new(
+    let inputs: NIFSVerifierCircuitInputs<G2> = NIFSVerifierCircuitInputs::new(
       default_hash,
       RelaxedR1CSInstance::default(&gens2, &shape2),
       <<G2 as Group>::Base as PrimeField>::zero(), // TODO: provide real inputs
@@ -787,8 +789,9 @@ mod tests {
       T,                                           // TODO: provide real inputs
       w,
     );
-    let circuit: VerificationCircuit<G2, TestCircuit<<G2 as Group>::Base>> =
-      VerificationCircuit::new(
+
+    let circuit: NIFSVerifierCircuit<G2, TestCircuit<<G2 as Group>::Base>> =
+      NIFSVerifierCircuit::new(
         params,
         Some(inputs),
         Some(TestCircuit {
diff --git a/src/commitments.rs b/src/commitments.rs
index 9ac322c..1e1c2e5 100644
--- a/src/commitments.rs
+++ b/src/commitments.rs
@@ -1,7 +1,11 @@
-use super::errors::NovaError;
-use super::traits::{CompressedGroup, Group};
-use core::fmt::Debug;
-use core::ops::{Add, AddAssign, Mul, MulAssign};
+use super::{
+  errors::NovaError,
+  traits::{CompressedGroup, Group},
+};
+use core::{
+  fmt::Debug,
+  ops::{Add, AddAssign, Mul, MulAssign},
+};
 use digest::{ExtendableOutput, Input};
 use merlin::Transcript;
 use sha3::Shake256;
diff --git a/src/gadgets/ecc.rs b/src/gadgets/ecc.rs
index e101e5e..4e8d7a4 100644
--- a/src/gadgets/ecc.rs
+++ b/src/gadgets/ecc.rs
@@ -619,12 +619,9 @@ mod tests {
     assert_eq!(e_pasta, e_pasta_2);
   }
 
-  use crate::bellperson::shape_cs::ShapeCS;
-  use crate::bellperson::solver::SatisfyingAssignment;
+  use crate::bellperson::{shape_cs::ShapeCS, solver::SatisfyingAssignment};
   use ff::{Field, PrimeFieldBits};
-  use pasta_curves::arithmetic::CurveAffine;
-  use pasta_curves::group::Curve;
-  use pasta_curves::EpAffine;
+  use pasta_curves::{arithmetic::CurveAffine, group::Curve, EpAffine};
   use std::ops::Mul;
   type G = pasta_curves::pallas::Point;
   type Fp = pasta_curves::pallas::Scalar;
diff --git a/src/pasta.rs b/src/pasta.rs
index dd5b98e..aaa7089 100644
--- a/src/pasta.rs
+++ b/src/pasta.rs
@@ -1,13 +1,15 @@
 //! This module implements the Nova traits for pallas::Point, pallas::Scalar, vesta::Point, vesta::Scalar.
 use crate::traits::{ChallengeTrait, CompressedGroup, Group, PrimeField};
 use merlin::Transcript;
-use pasta_curves::arithmetic::{CurveAffine, CurveExt, FieldExt, Group as Grp};
-use pasta_curves::group::{Curve, GroupEncoding};
-use pasta_curves::{self, pallas, vesta, Ep, Eq, Fp, Fq};
+use pasta_curves::{
+  self,
+  arithmetic::{CurveAffine, CurveExt, FieldExt, Group as Grp},
+  group::{Curve, GroupEncoding},
+  pallas, vesta, Ep, Eq, Fp, Fq,
+};
 use rand::{CryptoRng, RngCore};
 use rug::Integer;
-use std::borrow::Borrow;
-use std::ops::Mul;
+use std::{borrow::Borrow, ops::Mul};
 
 //////////////////////////////////////Pallas///////////////////////////////////////////////
 
diff --git a/src/poseidon.rs b/src/poseidon.rs
index 48ff399..57e8b18 100644
--- a/src/poseidon.rs
+++ b/src/poseidon.rs
@@ -189,9 +189,9 @@ mod tests {
   use super::*;
   type S = pasta_curves::pallas::Scalar;
   type G = pasta_curves::pallas::Point;
-  use crate::bellperson::solver::SatisfyingAssignment;
-  use crate::gadgets::utils::le_bits_to_num;
-  use crate::traits::PrimeField;
+  use crate::{
+    bellperson::solver::SatisfyingAssignment, gadgets::utils::le_bits_to_num, traits::PrimeField,
+  };
   use rand::rngs::OsRng;
 
   #[test]
diff --git a/src/r1cs.rs b/src/r1cs.rs
index fed9356..a2f3081 100644
--- a/src/r1cs.rs
+++ b/src/r1cs.rs
@@ -1,8 +1,10 @@
 //! This module defines R1CS related types and a folding scheme for Relaxed R1CS
 #![allow(clippy::type_complexity)]
-use super::commitments::{CommitGens, CommitTrait, Commitment, CompressedCommitment};
-use super::errors::NovaError;
-use super::traits::{Group, PrimeField};
+use super::{
+  commitments::{CommitGens, CommitTrait, Commitment, CompressedCommitment},
+  errors::NovaError,
+  traits::{Group, PrimeField},
+};
 use itertools::concat;
 use rayon::prelude::*;
 
diff --git a/src/traits.rs b/src/traits.rs
index d98c9d8..b4f89fb 100644
--- a/src/traits.rs
+++ b/src/traits.rs
@@ -1,8 +1,10 @@
 //! This module defines various traits required by the users of the library to implement.
 use bellperson::{gadgets::num::AllocatedNum, ConstraintSystem, SynthesisError};
-use core::borrow::Borrow;
-use core::fmt::Debug;
-use core::ops::{Add, AddAssign, Mul, MulAssign, Neg, Sub, SubAssign};
+use core::{
+  borrow::Borrow,
+  fmt::Debug,
+  ops::{Add, AddAssign, Mul, MulAssign, Neg, Sub, SubAssign},
+};
 use merlin::Transcript;
 use rand::{CryptoRng, RngCore};
 use rug::Integer;
@@ -131,9 +133,10 @@ impl<T, Rhs, Output> ScalarMul<Rhs, Output> for T where T: Mul<Rhs, Output = Out
 pub trait ScalarMulOwned<Rhs, Output = Self>: for<'r> ScalarMul<&'r Rhs, Output> {}
 impl<T, Rhs, Output> ScalarMulOwned<Rhs, Output> for T where T: for<'r> ScalarMul<&'r Rhs, Output> {}
 
-///A helper trait for the inner circuit F
-pub trait InnerCircuit<F: PrimeField + ff::PrimeField> {
-  ///Sythesize the circuit for a computation step and return variable that corresponds to z_{i+1}
+/// A helper trait for a step of the incremental computation (i.e., circuit for F)
+pub trait StepCircuit<F: PrimeField + ff::PrimeField> {
+  /// Sythesize the circuit for a computation step and return variable
+  /// that corresponds to the output of the step z_{i+1}
   fn synthesize<CS: ConstraintSystem<F>>(
     &self,
     cs: &mut CS,