diff --git a/src/fri/recursive_verifier.rs b/src/fri/recursive_verifier.rs
index a424dee0..4219aefc 100644
--- a/src/fri/recursive_verifier.rs
+++ b/src/fri/recursive_verifier.rs
@@ -1,22 +1,16 @@
-use anyhow::{ensure, Result};
 use itertools::izip;
 
 use crate::circuit_builder::CircuitBuilder;
 use crate::field::extension_field::target::{flatten_target, ExtensionTarget};
-use crate::field::extension_field::{flatten, Extendable, FieldExtension, OEF};
+use crate::field::extension_field::Extendable;
 use crate::field::field::Field;
-use crate::field::lagrange::{barycentric_weights, interpolant, interpolate};
 use crate::fri::FriConfig;
-use crate::hash::hash_n_to_1;
-use crate::merkle_proofs::verify_merkle_proof;
-use crate::plonk_challenger::{Challenger, RecursiveChallenger};
-use crate::plonk_common::reduce_with_iter;
+use crate::plonk_challenger::RecursiveChallenger;
 use crate::proof::{
-    FriInitialTreeProof, FriInitialTreeProofTarget, FriProof, FriProofTarget, FriQueryRound,
-    FriQueryRoundTarget, Hash, HashTarget, OpeningSet, OpeningSetTarget,
+    FriInitialTreeProofTarget, FriProofTarget, FriQueryRoundTarget, HashTarget, OpeningSetTarget,
 };
 use crate::target::Target;
-use crate::util::{log2_strict, reverse_bits, reverse_index_bits_in_place};
+use crate::util::{log2_strict, reverse_index_bits_in_place};
 
 impl<F: Extendable<D>, const D: usize> CircuitBuilder<F, D> {
     /// Computes P'(x^arity) from {P(x*g^i)}_(i=0..arity), where g is a `arity`-th root of unity
@@ -180,7 +174,7 @@ impl<F: Extendable<D>, const D: usize> CircuitBuilder<F, D> {
         }
         let denominator = self.sub_extension(subgroup_x, zeta);
         let quotient = self.div_unsafe_extension(numerator, denominator);
-        let sum = self.add_extension(sum, quotient);
+        sum = self.add_extension(sum, quotient);
 
         let evs = proof
             .unsalted_evals(3, config)
@@ -213,7 +207,7 @@ impl<F: Extendable<D>, const D: usize> CircuitBuilder<F, D> {
         let vanish_right = self.sub_extension(subgroup_x, zeta_right);
         let denominator = self.mul_extension(vanish, vanish_right);
         let quotient = self.div_unsafe_extension(numerator, denominator);
-        let sum = self.add_extension(sum, quotient);
+        sum = self.add_extension(sum, quotient);
 
         let evs = proof
             .unsalted_evals(2, config)
@@ -245,7 +239,7 @@ impl<F: Extendable<D>, const D: usize> CircuitBuilder<F, D> {
         let vanish_frob = self.sub_extension(subgroup_x, zeta_frob);
         let denominator = self.mul_extension(vanish, vanish_frob);
         let quotient = self.div_unsafe_extension(numerator, denominator);
-        let sum = self.add_extension(sum, quotient);
+        sum = self.add_extension(sum, quotient);
 
         sum
     }
@@ -285,7 +279,6 @@ impl<F: Extendable<D>, const D: usize> CircuitBuilder<F, D> {
         let mut subgroup_x = self.mul(g, phi);
 
         for (i, &arity_bits) in config.reduction_arity_bits.iter().enumerate() {
-            let arity = 1 << arity_bits;
             let next_domain_size = domain_size >> arity_bits;
             let e_x = if i == 0 {
                 self.fri_combine_initial(
@@ -309,8 +302,7 @@ impl<F: Extendable<D>, const D: usize> CircuitBuilder<F, D> {
             let mut evals = round_proof.steps[i].evals.clone();
             // Insert P(y) into the evaluation vector, since it wasn't included by the prover.
             let (low_x_index, high_x_index) = self.split_low_high(x_index, arity_bits);
-            // TODO: Uncomment this.
-            // evals.insert(x_index & (arity - 1), e_x);
+            evals = self.insert(low_x_index, e_x, evals);
             evaluations.push(evals);
             self.verify_merkle_proof(
                 flatten_target(&evaluations[i]),
diff --git a/src/gadgets/insert.rs b/src/gadgets/insert.rs
new file mode 100644
index 00000000..e8a1f6a4
--- /dev/null
+++ b/src/gadgets/insert.rs
@@ -0,0 +1,29 @@
+use crate::circuit_builder::CircuitBuilder;
+use crate::field::extension_field::target::ExtensionTarget;
+use crate::field::extension_field::Extendable;
+use crate::target::Target;
+use crate::util::log2_strict;
+
+impl<F: Extendable<D>, const D: usize> CircuitBuilder<F, D> {
+    /// Inserts a `Target` in a vector at a non-deterministic index. This is done by rotating to the
+    /// left, inserting at 0 and then rotating to the right.
+    /// Note: `index` is not range-checked.
+    pub fn insert(
+        &mut self,
+        index: Target,
+        element: ExtensionTarget<D>,
+        mut v: Vec<ExtensionTarget<D>>,
+    ) -> Vec<ExtensionTarget<D>> {
+        let n = v.len();
+        debug_assert!(n.is_power_of_two());
+        let n_log = log2_strict(n);
+
+        v.push(self.zero_extension());
+        let mut v = self.rotate_left(index, &v, n_log);
+
+        v.insert(0, element);
+        v.pop().unwrap();
+
+        self.rotate_right(index, &v, n_log)
+    }
+}
diff --git a/src/gadgets/mod.rs b/src/gadgets/mod.rs
index 137b991e..a1e041fc 100644
--- a/src/gadgets/mod.rs
+++ b/src/gadgets/mod.rs
@@ -1,5 +1,6 @@
 pub mod arithmetic;
 pub mod hash;
+pub mod insert;
 pub mod interpolation;
 pub mod polynomial;
 pub mod range_check;
diff --git a/src/gadgets/rotate.rs b/src/gadgets/rotate.rs
index 12bf3bc0..cc3c7fce 100644
--- a/src/gadgets/rotate.rs
+++ b/src/gadgets/rotate.rs
@@ -37,6 +37,22 @@ impl<F: Extendable<D>, const D: usize> CircuitBuilder<F, D> {
         res
     }
 
+    pub fn rotate_right_fixed(
+        &mut self,
+        b: Target,
+        k: usize,
+        v: &[ExtensionTarget<D>],
+        len: usize,
+    ) -> Vec<ExtensionTarget<D>> {
+        let mut res = Vec::new();
+
+        for i in 0..len {
+            res.push(self.select(b, v[(i - k) % len], v[i]));
+        }
+
+        res
+    }
+
     /// Left-rotates an array by `num_rotation`. Assumes that `num_rotation` is range-checked to be
     /// less than `len`.
     /// Note: We assume `len` is less than 8 since we won't use any arity greater than 8 in FRI (maybe?).
@@ -58,6 +74,24 @@ impl<F: Extendable<D>, const D: usize> CircuitBuilder<F, D> {
         v
     }
 
+    pub fn rotate_right_from_bits(
+        &mut self,
+        num_rotation_bits: &[Target],
+        v: &[ExtensionTarget<D>],
+        len_log: usize,
+    ) -> Vec<ExtensionTarget<D>> {
+        debug_assert_eq!(num_rotation_bits.len(), len_log);
+        let len = 1 << len_log;
+        debug_assert_eq!(v.len(), len);
+        let mut v = v.to_vec();
+
+        for i in 0..len_log {
+            v = self.rotate_right_fixed(num_rotation_bits[i], 1 << i, &v, len);
+        }
+
+        v
+    }
+
     /// Left-rotates an array by `num_rotation`. Assumes that `num_rotation` is range-checked to be
     /// less than `len`.
     /// Note: We assume `len` is a power of two less than or equal to 8, since we won't use any
@@ -74,6 +108,19 @@ impl<F: Extendable<D>, const D: usize> CircuitBuilder<F, D> {
 
         self.rotate_left_from_bits(&bits, v, len_log)
     }
+
+    pub fn rotate_right(
+        &mut self,
+        num_rotation: Target,
+        v: &[ExtensionTarget<D>],
+        len_log: usize,
+    ) -> Vec<ExtensionTarget<D>> {
+        let len = 1 << len_log;
+        debug_assert_eq!(v.len(), len);
+        let bits = self.split_le(num_rotation, len_log);
+
+        self.rotate_right_from_bits(&bits, v, len_log)
+    }
 }
 
 #[cfg(test)]
diff --git a/src/gates/base_sum.rs b/src/gates/base_sum.rs
index e3b69618..150a492e 100644
--- a/src/gates/base_sum.rs
+++ b/src/gates/base_sum.rs
@@ -1,5 +1,4 @@
 use crate::circuit_builder::CircuitBuilder;
-use crate::circuit_data::CircuitConfig;
 use crate::field::extension_field::target::ExtensionTarget;
 use crate::field::extension_field::Extendable;
 use crate::field::field::Field;
@@ -8,7 +7,6 @@ use crate::generator::{SimpleGenerator, WitnessGenerator};
 use crate::plonk_common::{reduce_with_powers, reduce_with_powers_recursive};
 use crate::target::Target;
 use crate::vars::{EvaluationTargets, EvaluationVars};
-use crate::wire::Wire;
 use crate::witness::PartialWitness;
 use std::ops::Range;
 
diff --git a/src/gates/noop.rs b/src/gates/noop.rs
index eddd0361..a12df932 100644
--- a/src/gates/noop.rs
+++ b/src/gates/noop.rs
@@ -5,7 +5,7 @@ use crate::gates::gate::{Gate, GateRef};
 use crate::generator::WitnessGenerator;
 use crate::vars::{EvaluationTargets, EvaluationVars};
 
-/// A gate which takes a single constant parameter and outputs that value.
+/// A gate which does nothing.
 pub struct NoopGate;
 
 impl NoopGate {