# Mutual Recursion¶

Agda offers multiple ways to write mutually-defined data types, record types and functions.

The last two are more expressive than the first one as they allow the interleaving of declarations and definitions thus making it possible for some types to refere to the constructors of a mutually-defined datatype.

## Interleaved mutual blocks¶

Mutual recursive functions can be written by placing them inside an `interleaved mutual` block. The type signature of each function must come before its defining clauses and its usage sites on the right-hand side of other functions. The clauses for different functions can be interleaved e.g. for pedagogical purposes:

```interleaved mutual

-- Declarations:
even : Nat → Bool
odd  : Nat → Bool

-- zero is even, not odd
even zero = true
odd  zero = false

-- suc case: switch evenness on the predecessor
even (suc n) = odd n
odd  (suc n) = even n
```

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 introduction of constructors in one or many `data ... where` blocks:

```interleaved mutual

-- Declaration of a product record, a universe of codes, and a decoding function
record _×_ (A B : Set) : Set
data U : Set
El : U → Set

-- We have a code for the type of natural numbers in our universe
data U where `Nat : U
El `Nat = Nat

-- Btw we know how to pair values in a record
record _×_ A B where
inductive; constructor _,_
field fst : A; snd : B

-- And we have a code for pairs in our universe
data _ where
_`×_ : (A B : U) → U
El (A `× B) = El A × El B

-- we can now build types of nested pairs of natural numbers
ty-example : U
ty-example = `Nat `× ((`Nat `× `Nat) `× `Nat)

-- and their values
val-example : El ty-example
val-example = 0 , ((1 , 2) , 3)
```

You can mix constructors for different data types in a `data _ where` block (underscore instead of name).

The `interleaved mutual` blocks get desugared into the Forward declaration blocks described below by:

• leaving the signatures where they are,
• grouping the clauses for a function together with the first of them, and
• grouping the constructors for a datatype together with the first of them.

## Forward declaration¶

Mutual recursive functions can be written by placing the type signatures of all mutually recursive function before their definitions. The span of the mutual block will be automatically inferred by Agda:

```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                -- Note the absence of a type signature.
[]   : 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)
```

Note

In contrast to Interleaved mutual blocks, in forward-declaration style we can only have one `data ... where` block per data type.

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-style `mutual` blocks¶

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 by sorting the content of the mutual block into a declaration and a definition part and placing the declarations before the definitions.

Declarations comprise:

• Type signatures of functions, `data` and `record` declarations, `unquoteDecl`. (Function includes here `postulate` and `primitive` etc.)
• Module statements, such as `module` aliases, `import` and `open` statements.
• Pragmas that only need the name, but not the definition of the thing they affect (e.g. `INJECTIVE`).

Definitions comprise:

• Function clauses, `data` constructors and `record` definitions, `unquoteDef`.
• `pattern` synonym definitions.
• Pragmas that need the definition, e.g. `INLINE`, `ETA`, etc.
• Pragmas that are not needed for type checking, like compiler pragmas.

Module definitions with `module ... where` are not supported in old-style `mutual` blocks.