# Mutual Recursion¶

Mutual recursive functions can be written by placing the type signatures of all mutually recursive function before their definitions:

```
f : A
g : B[f]
f = a[f, g]
g = b[f, g].
```

You can mix arbitrary declarations, such as modules and postulates, with mutually recursive definitions. For data types and records the following syntax is used to separate the declaration from the definition:

```
-- Declaration.
data Vec (A : Set) : Nat → Set -- Note the absence of ‘where’.
-- Definition.
data Vec A where
[] : Vec A zero
_::_ : {n : Nat} → A → Vec A n → Vec A (suc n)
-- Declaration.
record Sigma (A : Set) (B : A → Set) : Set
-- Definition.
record Sigma A B where
constructor _,_
field fst : A
snd : B fst
```

The parameter lists in the second part of a data or record declaration behave like variables left-hand sides (although infix syntax is not supported). That is, they should have no type signatures, but implicit parameters can be omitted or bound by name.

Such a separation of declaration and definition is for instance needed when defining a set of codes for types and their interpretation as actual types (a so-called *universe*):

```
-- Declarations.
data TypeCode : Set
Interpretation : TypeCode → Set
-- Definitions.
data TypeCode where
nat : TypeCode
pi : (a : TypeCode) (b : Interpretation a → TypeCode) → TypeCode
Interpretation nat = Nat
Interpretation (pi a b) = (x : Interpretation a) → Interpretation (b x)
```

When making separated declarations/definitions private or abstract you should attach the `private`

keyword to the declaration and the `abstract`

keyword to the definition. For instance, a private, abstract function can be defined as

```
private
f : A
abstract
f = e
```

## Old Syntax: Keyword `mutual`

¶

Note

You are advised to avoid using this old syntax if possible, but the old syntax is still supported.

Mutual recursive functions can be written by placing the type signatures of all mutually recursive function before their definitions:

```
mutual
f : A
f = a[f, g]
g : B[f]
g = b[f, g]
```

Using the `mutual`

keyword,
the *universe* example from above is expressed as follows:

```
mutual
data TypeCode : Set where
nat : TypeCode
pi : (a : TypeCode) (b : Interpretation a → TypeCode) → TypeCode
Interpretation : TypeCode → Set
Interpretation nat = Nat
Interpretation (pi a b) = (x : Interpretation a) → Interpretation (b x)
```

This alternative syntax desugars into the new syntax.