Running example: Addition of Natural Numbers

Throughout this tutorial, we will be working with the following function, defined in the Idris prelude, which defines addition on natural numbers:

plus : Nat -> Nat -> Nat
plus Z     m = m
plus (S k) m = S (plus k m)

It is defined by the above equations, meaning that we have for free the properties that adding m to zero always results in m, and that adding m to any non-zero number S k always results in S (plus k m). We can see this by evaluation at the Idris REPL (i.e. the prompt, the read-eval-print loop):

Main> \m => plus Z m
\m => m

Idris> \k,m => plus (S k) m
\k => \m => S (plus k m)

Note that unlike many other language REPLs, the Idris REPL performs evaluation on open terms, meaning that it can reduce terms which appear inside lambda bindings, like those above. Therefore, we can introduce unknowns k and m as lambda bindings and see how plus reduces.

The plus function has a number of other useful properties, for example:

  • It is commutative, that is for all Nat inputs n and m, we know that plus n m = plus m n.
  • It is associative, that is for all Nat inputs n, m and p, we know that plus n (plus m p) = plus (plus m n) p.

We can use these properties in an Idris program, but in order to do so we must prove them.

Equality Proofs

Idris defines a propositional equality type as follows:

data Equal : a -> b -> Type where
   Refl : Equal x x

As syntactic sugar, Equal x y can be written as x = y.

It is propositional equality, where the type states that any two values in different types a and b may be proposed to be equal. There is only one way to prove equality, however, which is by reflexivity (Refl).

We have a type for propositional equality here, and correspondingly a program inhabiting an instance of this type can be seen as a proof of the corresponding proposition [1]. So, trivially, we can prove that 4 equals 4:

four_eq : 4 = 4
four_eq = Refl

However, trying to prove that 4 = 5 results in failure:

four_eq_five : 4 = 5
four_eq_five = Refl

The type 4 = 5 is a perfectly valid type, but is uninhabited, so when trying to type check this definition, Idris gives the following error:

When unifying 4 = 4 and (fromInteger 4) = (fromInteger 5)
Mismatch between:

Type checking equality proofs

An important step in type checking Idris programs is unification, which attempts to resolve implicit arguments such as the implicit argument x in Refl. As far as our understanding of type checking proofs is concerned, it suffices to know that unifying two terms involves reducing both to normal form then trying to find an assignment to implicit arguments which will make those normal forms equal.

When type checking Refl, Idris requires that the type is of the form x = x, as we see from the type of Refl. In the case of four_eq_five, Idris will try to unify the expected type 4 = 5 with the type of Refl, x = x, notice that a solution requires that x be both 4 and 5, and therefore fail.

Since type checking involves reduction to normal form, we can write the following equalities directly:

twoplustwo_eq_four : 2 + 2 = 4
twoplustwo_eq_four = Refl

plus_reduces_Z : (m : Nat) -> plus Z m = m
plus_reduces_Z m = Refl

plus_reduces_Sk : (k, m : Nat) -> plus (S k) m = S (plus k m)
plus_reduces_Sk k m = Refl

Heterogeneous Equality

Equality in Idris is heterogeneous, meaning that we can even propose equalities between values in different types:

idris_not_php : Z = "Z"

The type Z = "Z" is uninhabited, and one might wonder why it is useful to be able to propose equalities between values in different types. However, with dependent types, such equalities can arise naturally. For example, if two vectors are equal, their lengths must be equal:

vect_eq_length : (xs : Vect n a) -> (ys : Vect m a) ->
                 (xs = ys) -> n = m

In the above declaration, xs and ys have different types because their lengths are different, but we would still like to draw a conclusion about the lengths if they happen to be equal. We can define vect_eq_length as follows:

vect_eq_length xs xs Refl = Refl

By matching on Refl for the third argument, we know that the only valid value for ys is xs, because they must be equal, and therefore their types must be equal, so the lengths must be equal.

Alternatively, we can put an underscore for the second xs, since there is only one value which will type check:

vect_eq_length xs _ Refl = Refl

Properties of plus

Using the (=) type, we can now state the properties of plus given above as Idris type declarations:

plus_commutes : (n, m : Nat) -> plus n m = plus m n
plus_assoc : (n, m, p : Nat) -> plus n (plus m p) = plus (plus n m) p

Both of these properties (and many others) are proved for natural number addition in the Idris standard library, using (+) from the Num interface rather than using plus directly. They have the names plusCommutative and plusAssociative respectively.

In the remainder of this tutorial, we will explore several different ways of proving plus_commutes (or, to put it another way, writing the function.) We will also discuss how to use such equality proofs, and see where the need for them arises in practice.

[1]This is known as the Curry-Howard correspondence.