Pattern Matching Proofs

In this section, we will provide a proof of plus_commutes directly, by writing a pattern matching definition. We will use interactive editing features extensively, since it is significantly easier to produce a proof when the machine can give the types of intermediate values and construct components of the proof itself. The commands we will use are summarised below. Where we refer to commands directly, we will use the Vim version, but these commands have a direct mapping to Emacs commands.

Command Vim binding Emacs binding Explanation
Check type \t C-c C-t Show type of identifier or hole under the cursor.
Proof search \s C-c C-a Attempt to solve hole under the cursor by applying simple proof search.
Make new definition \a C-c C-s Add a template definition for the type defined under the cursor.
Make lemma \l C-c C-e Add a top level function with a type which solves the hole under the cursor.
Split cases \c C-c C-c Create new constructor patterns for each possible case of the variable under the cursor.

Creating a Definition

To begin, create a file pluscomm.idr containing the following type declaration:

plus_commutes : (n : Nat) -> (m : Nat) -> n + m = m + n

To create a template definition for the proof, press \a (or the equivalent in your editor of choice) on the line with the type declaration. You should see:

plus_commutes : (n : Nat) -> (m : Nat) -> n + m = m + n
plus_commutes n m = ?plus_commutes_rhs

To prove this by induction on n, as we sketched in Section Inductive Proofs, we begin with a case split on n (press \c with the cursor over the n in the definition.) You should see:

plus_commutes : (n : Nat) -> (m : Nat) -> n + m = m + n
plus_commutes Z m = ?plus_commutes_rhs_1
plus_commutes (S k) m = ?plus_commutes_rhs_2

If we inspect the types of the newly created holes, plus_commutes_rhs_1 and plus_commutes_rhs_2, we see that the type of each reflects that n has been refined to Z and S k in each respective case. Pressing \t over plus_commutes_rhs_1 shows:

   m : Nat
-------------------------------------
plus_commutes_rhs_1 : m = plus m Z

Similarly, for plus_commutes_rhs_2:

  k : Nat
  m : Nat
--------------------------------------
plus_commutes_rhs_2 : (S (plus k m)) = (plus m (S k))

It is a good idea to give these slightly more meaningful names:

plus_commutes : (n : Nat) -> (m : Nat) -> n + m = m + n
plus_commutes Z m = ?plus_commutes_Z
plus_commutes (S k) m = ?plus_commutes_S

Base Case

We can create a separate lemma for the base case interactively, by pressing \l with the cursor over plus_commutes_Z. This yields:

plus_commutes_Z : (m : Nat) -> m = plus m Z

plus_commutes : (n : Nat) -> (m : Nat) -> n + m = m + n
plus_commutes Z m = plus_commutes_Z m
plus_commutes (S k) m = ?plus_commutes_S

That is, the hole has been filled with a call to a top level function plus_commutes_Z, applied to the variable in scope m.

Unfortunately, we cannot prove this lemma directly, since plus is defined by matching on its first argument, and here plus m Z has a concrete value for its second argument (in fact, the left hand side of the equality has been reduced from plus Z m.) Again, we can prove this by induction, this time on m.

First, create a template definition with \d:

plus_commutes_Z : (m : Nat) -> m = plus m Z
plus_commutes_Z m = ?plus_commutes_Z_rhs

Now, case split on m with \c:

plus_commutes_Z : (m : Nat) -> m = plus m Z
plus_commutes_Z Z = ?plus_commutes_Z_rhs_1
plus_commutes_Z (S k) = ?plus_commutes_Z_rhs_2

Checking the type of plus_commutes_Z_rhs_1 shows the following, which is provable by Refl:

--------------------------------------
plus_commutes_Z_rhs_1 : Z = Z

For such immediate proofs, we can let write the proof automatically by pressing \s with the cursor over plus_commutes_Z_rhs_1. This yields:

plus_commutes_Z : (m : Nat) -> m = plus m Z
plus_commutes_Z Z = Refl
plus_commutes_Z (S k) = ?plus_commutes_Z_rhs_2

For plus_commutes_Z_rhs_2, we are not so lucky:

   k : Nat
-------------------------------------
plus_commutes_Z_rhs_2 : S k = S (plus k Z)

Inductively, we should know that k = plus k Z, and we can get access to this inductive hypothesis by making a recursive call on k, as follows:

plus_commutes_Z : (m : Nat) -> m = plus m Z
plus_commutes_Z Z = Refl
plus_commutes_Z (S k)
   = let rec = plus_commutes_Z k in
         ?plus_commutes_Z_rhs_2

For plus_commutes_Z_rhs_2, we now see:

   k : Nat
   rec : k = plus k Z
-------------------------------------
plus_commutes_Z_rhs_2 : S k = S (plus k Z)

So we know that k = plus k Z, but how do we use this to update the goal to S k = S k?

To achieve this, Idris provides a replace function as part of the prelude:

Main> :t replace
Builtin.replace : (0 rule : x = y) -> p x -> p y

Given a proof that x = y, and a property p which holds for x, we can get a proof of the same property for y, because we know x and y must be the same. Note the multiplicity on rule means that it’s guaranteed to be erased at run time. In practice, this function can be a little tricky to use because in general the implicit argument p can be hard to infer by unification, so Idris provides a high level syntax which calculates the property and applies replace:

rewrite prf in expr

If we have prf : x = y, and the required type for expr is some property of x, the rewrite ... in syntax will search for all occurrences of x in the required type of expr and replace them with y. We want to replace plus k Z with k, so we need to apply our rule rec in reverse, which we can do using sym from the Prelude

Main> :t sym
Builtin.sym : (0 rule : x = y) -> y = x

Concretely, in our example, we can say:

plus_commutes_Z (S k)
   = let rec = plus_commutes_Z k in
         rewrite sym rec in ?plus_commutes_Z_rhs_2

Checking the type of plus_commutes_Z_rhs_2 now gives:

   k : Nat
   rec : k = plus k Z
-------------------------------------
plus_commutes_Z_rhs_2 : S k = S k

Using the rewrite rule rec, the goal type has been updated with plus k Z replaced by k.

We can use proof search (\s) to complete the proof, giving:

plus_commutes_Z : (m : Nat) -> m = plus m Z
plus_commutes_Z Z = Refl
plus_commutes_Z (S k)
   = let rec = plus_commutes_Z k in
         rewrite sym rec in Refl

The base case of plus_commutes is now complete.

Inductive Step

Our main theorem, plus_commutes should currently be in the following state:

plus_commutes : (n : Nat) -> (m : Nat) -> n + m = m + n
plus_commutes Z m = plus_commutes_Z m
plus_commutes (S k) m = ?plus_commutes_S

Looking again at the type of plus_commutes_S, we have:

   k : Nat
   m : Nat
-------------------------------------
plus_commutes_S : S (plus k m) = plus m (S k)

Conveniently, by induction we can immediately tell that plus k m = plus m k, so let us rewrite directly by making a recursive call to plus_commutes. We add this directly, by hand, as follows:

plus_commutes : (n : Nat) -> (m : Nat) -> n + m = m + n
plus_commutes Z m = plus_commutes_Z
plus_commutes (S k) m = rewrite plus_commutes k m in ?plus_commutes_S

Checking the type of plus_commutes_S now gives:

   k : Nat
   m : Nat
-------------------------------------
plus_commutes_S : S (plus m k) = plus m (S k)

The good news is that m and k now appear in the correct order. However, we still have to show that the successor symbol S can be moved to the front in the right hand side of this equality. This remaining lemma takes a similar form to the plus_commutes_Z; we begin by making a new top level lemma with \l. This gives:

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

Again, we make a template definition with \a:

plus_commutes_S : (k : Nat) -> (m : Nat) -> S (plus m k) = plus m (S k)
plus_commutes_S k m = ?plus_commutes_S_rhs

Like plus_commutes_Z, we can define this by induction over m, since plus is defined by matching on its first argument. The complete definition is:

total
plus_commutes_S : (k : Nat) -> (m : Nat) -> S (plus m k) = plus m (S k)
plus_commutes_S k Z = Refl
plus_commutes_S k (S j) = rewrite plus_commutes_S k j in Refl

All holes have now been solved.

The total annotation means that we require the final function to pass the totality checker; i.e. it will terminate on all possible well-typed inputs. This is important for proofs, since it provides a guarantee that the proof is valid in all cases, not just those for which it happens to be well-defined.

Now that plus_commutes has a total annotation, we have completed the proof of commutativity of addition on natural numbers.