Function Definitions¶
Introduction¶
A function is defined by first declaring its type followed by a number of
equations called clauses. Each clause consists of the function being defined
applied to a number of patterns, followed by =
and a term called the
right-hand side. For example:
not : Bool → Bool
not true = false
not false = true
Functions are allowed to call themselves recursively, for example:
twice : Nat → Nat
twice zero = zero
twice (suc n) = suc (suc (twice n))
General form¶
The general form for defining a function is
f : (x₁ : A₁) → … → (xₙ : Aₙ) → B
f p₁ … pₙ = d
…
f q₁ … qₙ = e
where f
is a new identifier, pᵢ
and qᵢ
are patterns of type Aᵢ
,
and d
and e
are expressions.
The declaration above gives the identifier f
the type
(x₁ : A₁) → … → (xₙ : Aₙ) → B
and f
is defined by the defining
equations. Patterns are matched from top to bottom, i.e., the first pattern
that matches the actual parameters is the one that is used.
By default, Agda checks the following properties of a function definition:
- The patterns in the left-hand side of each clause should consist only of constructors and variables.
- No variable should occur more than once on the left-hand side of a single clause.
- The patterns of all clauses should together cover all possible inputs of the function, see Coverage Checking.
- The function should be terminating on all possible inputs, see Termination Checking.
Special patterns¶
In addition to constructors consisting of constructors and variables, Agda supports two special kinds of patterns: dot patterns and absurd patterns.
Dot patterns¶
A dot pattern (also called inaccessible pattern) can be used when
the only type-correct value of the argument is determined by the
patterns given for the other arguments.
The syntax for a dot pattern is .t
.
As an example, consider the datatype Square
defined as follows
data Square : Nat → Set where
sq : (m : Nat) → Square (m * m)
Suppose we want to define a function root : (n : Nat) → Square n → Nat
that
takes as its arguments a number n
and a proof that it is a square, and
returns the square root of that number. We can do so as follows:
root : (n : Nat) → Square n → Nat
root .(m * m) (sq m) = m
Notice that by matching on the argument of type Square n
with the constructor
sq : (m : Nat) → Square (m * m)
, n
is forced to be equal to m * m
.
In general, when matching on an argument of type D i₁ … iₙ
with a constructor
c : (x₁ : A₁) → … → (xₘ : Aₘ) → D j₁ … jₙ
, Agda will attempt to unify
i₁ … iₙ
with j₁ … jₙ
. When the unification algorithm instantiates a
variable x
with value t
, the corresponding argument of the function
can be replaced by a dot pattern .t
. Using a dot pattern is optional, but
can help readability. The following are also legal definitions of
root
:
Since Agda 2.4.2.4:
root₁ : (n : Nat) → Square n → Nat
root₁ _ (sq m) = m
Since Agda 2.5.2:
root₂ : (n : Nat) → Square n → Nat
root₂ n (sq m) = m
In the case of root₂
, n
evaluates to m * m
in the body of the
function and is thus equivalent to
root₃ : (n : Nat) → Square n → Nat
root₃ _ (sq m) = let n = m * m in m
Absurd patterns¶
Absurd patterns can be used when none of the constructors for a particular
argument would be valid. The syntax for an absurd pattern is ()
.
As an example, if we have a datatype Even
defined as follows
data Even : Nat → Set where
even-zero : Even zero
even-plus2 : {n : Nat} → Even n → Even (suc (suc n))
then we can define a function one-not-even : Even 1 → ⊥
by using an absurd
pattern:
one-not-even : Even 1 → ⊥
one-not-even ()
Note that if the left-hand side of a clause contains an absurd pattern, its right-hand side must be omitted.
In general, when matching on an argument of type D i₁ … iₙ
with an absurd
pattern, Agda will attempt for each constructor
c : (x₁ : A₁) → … → (xₘ : Aₘ) → D j₁ … jₙ
of the datatype D
to unify
i₁ … iₙ
with j₁ … jₙ
. The absurd pattern will only be accepted if all
of these unifications end in a conflict.
As-patterns¶
As-patterns (or @-patterns
) can be used to name a pattern. The name has the
same scope as normal pattern variables (i.e. the right-hand side, where clause,
and dot patterns). The name reduces to the value of the named pattern. For example:
module _ {A : Set} (_<_ : A → A → Bool) where
merge : List A → List A → List A
merge xs [] = xs
merge [] ys = ys
merge xs@(x ∷ xs₁) ys@(y ∷ ys₁) =
if x < y then x ∷ merge xs₁ ys
else y ∷ merge xs ys₁
As-patterns are properly supported since Agda 2.5.2.
Case trees¶
Internally, Agda represents function definitions as case trees. For example, a function definition
max : Nat → Nat → Nat
max zero n = n
max m zero = m
max (suc m) (suc n) = suc (max m n)
will be represented internally as a case tree that looks like this:
max m n = case m of
zero → n
suc m' → case n of
zero → suc m'
suc n' → suc (max m' n')
Note that because Agda uses this representation of the function
max
, the clause max m zero = m
does not hold definitionally
(i.e. as a reduction rule). If you would try to prove that this
equation holds, you would not be able to write refl
:
data _≡_ {A : Set} (x : A) : A → Set where
refl : x ≡ x
-- Does not work!
lemma : (m : Nat) → max m zero ≡ m
lemma = refl
Clauses which do not hold definitionally are usually (but not always) the result of writing clauses by hand instead of using Agda’s case split tactic. These clauses are highlighted by Emacs.
The --exact-split
flag causes Agda to raise an error whenever a
clause in a definition by pattern matching cannot be made to hold
definitionally. Specific clauses can be excluded from this check by
means of the {-# CATCHALL #-}
pragma.
For instance, the above definition of max
will be rejected when
using the --exact-split
flag because its second clause does not to
hold definitionally.
When using the --exact-split
flag, catch-all clauses have to
be marked as such, for instance:
eq : Nat → Nat → Bool
eq zero zero = true
eq (suc m) (suc n) = eq m n
{-# CATCHALL #-}
eq _ _ = false
The --no-exact-split
flag can be used to override a global
--exact-split
in a file, by adding a pragma
{-# OPTIONS --no-exact-split #-}
. This option is enabled by
default.