Pattern Matching

Introduction

This page explains the reduction behaviour of functions defined by pattern matching. Consider the following two implementations of the less than function on natural numbers.

  data Bool : Set where
    true  : Bool
    false : Bool

  data Nat : Set where
    zero : Nat
    suc  : Nat -> Nat

  -- First implementation
  _<1_ : Nat -> Nat -> Bool
  zero  <1 suc m = true
  suc n <1 suc m = n <1 m
  _     <1 zero  = false

  -- Second implementation
  _<2_ : Nat -> Nat -> Bool
  _     <2 zero  = false
  zero  <2 suc m = true
  suc n <2 suc m = n <2 m

Perhaps surprisingly (since the two implementations have the same non-overlapping clauses), _<1_ and _<2_ does not have the same reduction behaviour. For neutral n we have that n <1 zero does not reduce, whereas n <2 zero reduces to false. The reason for this is explained below.

Case tree equivalence

The set of defining equations for a functions are equivalent to a case tree, and the order of the equations determines which one. For _<1_ the equivalent case tree is

n <1 m = case n of
  zero  -> case m of
    suc m -> true
    zero  -> false
  suc n -> case m of
    suc m -> n < m
    zero  -> false

whereas for _<2_ it's

n <2 m = case m of
  zero  -> false
  suc m -> case n of
    zero  -> true
    suc n -> n <2 m

Looking at the case tree it's clear that in the first case n < zero will not reduce to false for neutral n, but in the second case it will. Pattern matching always starts with the left-most constructor pattern in the first clause. For _<1_ this is the first argument, whereas for _<2_ it's the second argument.

Operational view

An alternative more operational way of looking at what's going on is the following. When reducing an application of a function by pattern matching the clauses will be tried one at a time, from top to bottom. For each clause the arguments will be matched against the corresponding patterns from left to right. If there is a mismatch the next clause is tried. If there is an inconclusive match (e.g. matching a neutral term against a constructor pattern) the application won't reduce. For _<1_ we can't see whether n matches zero or not, so the reduction is suspended.

Berry's majority function

The reason for suspending reduction at the first sign of inconclusive matching is to keep the correspondence to a case tree. Consider Berry's majority function:

data Bool : Set where
  tt : Bool
  ff : Bool

maj tt tt tt = tt
maj tt ff x  = x
maj ff x  tt = x
maj x  tt ff = x
maj ff ff ff = F

Here, all clauses are disjoint, but there is no case tree which delivers all equations as definitional equalities. Following the rules for pattern matching given above you'll discover that with this definition you get all but maj x tt ff = x.