diff --git a/examples/circuit-layout.rs b/examples/circuit-layout.rs
index 6020686..93ad054 100644
--- a/examples/circuit-layout.rs
+++ b/examples/circuit-layout.rs
@@ -8,6 +8,7 @@ use halo2::{
 use plotters::prelude::*;
 use std::marker::PhantomData;
 
+#[allow(clippy::many_single_char_names)]
 fn main() {
     /// This represents an advice column at a certain row in the ConstraintSystem
     #[derive(Copy, Clone, Debug)]
@@ -372,7 +373,7 @@ fn main() {
     }
 
     let a = Fp::rand();
-    let a_squared = a * &a;
+    let a_squared = a * a;
     let aux = Fp::one() + Fp::one();
     let lookup_table = vec![aux, a, a, Fp::zero()];
     let lookup_table_2 = vec![Fp::zero(), a, a_squared, Fp::zero()];
diff --git a/src/circuit.rs b/src/circuit.rs
index 562246f..e9ad299 100644
--- a/src/circuit.rs
+++ b/src/circuit.rs
@@ -154,9 +154,9 @@ pub trait Layouter<C: Chip> {
     ///     region.assign_advice(self.config.a, offset, || { Some(value)});
     /// });
     /// ```
-    fn assign_region<A, N, NR>(&mut self, name: N, assignment: A) -> Result<(), Error>
+    fn assign_region<A, AR, N, NR>(&mut self, name: N, assignment: A) -> Result<AR, Error>
     where
-        A: FnMut(Region<'_, C>) -> Result<(), Error>,
+        A: FnMut(Region<'_, C>) -> Result<AR, Error>,
         N: Fn() -> NR,
         NR: Into<String>;
 
@@ -202,9 +202,9 @@ impl<'a, C: Chip, L: Layouter<C> + 'a> Layouter<C> for NamespacedLayouter<'a, C,
         self.0.config()
     }
 
-    fn assign_region<A, N, NR>(&mut self, name: N, assignment: A) -> Result<(), Error>
+    fn assign_region<A, AR, N, NR>(&mut self, name: N, assignment: A) -> Result<AR, Error>
     where
-        A: FnMut(Region<'_, C>) -> Result<(), Error>,
+        A: FnMut(Region<'_, C>) -> Result<AR, Error>,
         N: Fn() -> NR,
         NR: Into<String>,
     {
diff --git a/src/circuit/layouter.rs b/src/circuit/layouter.rs
index 6f1a348..88ad9cc 100644
--- a/src/circuit/layouter.rs
+++ b/src/circuit/layouter.rs
@@ -106,9 +106,9 @@ impl<'a, C: Chip, CS: Assignment<C::Field> + 'a> Layouter<C> for SingleChip<'a,
         &self.config
     }
 
-    fn assign_region<A, N, NR>(&mut self, name: N, mut assignment: A) -> Result<(), Error>
+    fn assign_region<A, AR, N, NR>(&mut self, name: N, mut assignment: A) -> Result<AR, Error>
     where
-        A: FnMut(Region<'_, C>) -> Result<(), Error>,
+        A: FnMut(Region<'_, C>) -> Result<AR, Error>,
         N: Fn() -> NR,
         NR: Into<String>,
     {
@@ -136,13 +136,13 @@ impl<'a, C: Chip, CS: Assignment<C::Field> + 'a> Layouter<C> for SingleChip<'a,
 
         self.cs.enter_region(name);
         let mut region = SingleChipRegion::new(self, region_index);
-        {
+        let result = {
             let region: &mut dyn RegionLayouter<C> = &mut region;
-            assignment(region.into())?;
-        }
+            assignment(region.into())
+        }?;
         self.cs.exit_region();
 
-        Ok(())
+        Ok(result)
     }
 
     fn get_root(&mut self) -> &mut Self::Root {
diff --git a/src/dev/graph/layout.rs b/src/dev/graph/layout.rs
index 7131680..78c8657 100644
--- a/src/dev/graph/layout.rs
+++ b/src/dev/graph/layout.rs
@@ -107,7 +107,7 @@ pub fn circuit_layout<F: Field, ConcreteCircuit: Circuit<F>, DB: DrawingBackend>
         if let Some(offset) = region.offset {
             // Sort the region's columns according to the defined ordering.
             let mut columns: Vec<_> = region.columns.into_iter().collect();
-            columns.sort_unstable_by(|a, b| column_index(a).cmp(&column_index(b)));
+            columns.sort_unstable_by_key(|a| column_index(a));
 
             // Render contiguous parts of the same region as a single box.
             let mut width = None;
diff --git a/src/plonk/circuit.rs b/src/plonk/circuit.rs
index 21879cb..3e8c71f 100644
--- a/src/plonk/circuit.rs
+++ b/src/plonk/circuit.rs
@@ -1,7 +1,10 @@
 use core::cmp::max;
 use core::ops::{Add, Mul};
 use ff::Field;
-use std::convert::TryFrom;
+use std::{
+    convert::TryFrom,
+    ops::{Neg, Sub},
+};
 
 use super::{lookup, permutation, Error};
 use crate::poly::Rotation;
@@ -316,6 +319,13 @@ impl<F: Field> Expression<F> {
     }
 }
 
+impl<F: Field> Neg for Expression<F> {
+    type Output = Expression<F>;
+    fn neg(self) -> Self::Output {
+        Expression::Scaled(Box::new(self), -F::one())
+    }
+}
+
 impl<F> Add for Expression<F> {
     type Output = Expression<F>;
     fn add(self, rhs: Expression<F>) -> Expression<F> {
@@ -323,6 +333,13 @@ impl<F> Add for Expression<F> {
     }
 }
 
+impl<F: Field> Sub for Expression<F> {
+    type Output = Expression<F>;
+    fn sub(self, rhs: Expression<F>) -> Expression<F> {
+        Expression::Sum(Box::new(self), Box::new(-rhs))
+    }
+}
+
 impl<F> Mul for Expression<F> {
     type Output = Expression<F>;
     fn mul(self, rhs: Expression<F>) -> Expression<F> {