mirror of
https://github.com/idris-lang/Idris2.git
synced 2024-12-21 02:31:50 +03:00
165 lines
5.2 KiB
ReStructuredText
165 lines
5.2 KiB
ReStructuredText
.. _sect-misc:
|
||
|
||
**********
|
||
Miscellany
|
||
**********
|
||
|
||
In this section we discuss a variety of additional features:
|
||
|
||
+ auto, implicit, and default arguments;
|
||
+ literate programming; and
|
||
+ the universe hierarchy.
|
||
|
||
Implicit arguments
|
||
==================
|
||
|
||
We have already seen implicit arguments, which allows arguments to be
|
||
omitted when they can be inferred by the type checker [#IdrisType]_, e.g.
|
||
|
||
.. code-block:: idris
|
||
|
||
index : forall a, n . Fin n -> Vect n a -> a
|
||
|
||
Auto implicit arguments
|
||
-----------------------
|
||
|
||
In other situations, it may be possible to infer arguments not by type
|
||
checking but by searching the context for an appropriate value, or
|
||
constructing a proof. For example, the following definition of ``head``
|
||
which requires a proof that the list is non-empty:
|
||
|
||
.. code-block:: idris
|
||
|
||
isCons : List a -> Bool
|
||
isCons [] = False
|
||
isCons (x :: xs) = True
|
||
|
||
head : (xs : List a) -> (isCons xs = True) -> a
|
||
head (x :: xs) _ = x
|
||
|
||
If the list is statically known to be non-empty, either because its
|
||
value is known or because a proof already exists in the context, the
|
||
proof can be constructed automatically. Auto implicit arguments allow
|
||
this to happen silently. We define ``head`` as follows:
|
||
|
||
.. code-block:: idris
|
||
|
||
head : (xs : List a) -> {auto p : isCons xs = True} -> a
|
||
head (x :: xs) = x
|
||
|
||
The ``auto`` annotation on the implicit argument means that Idris
|
||
will attempt to fill in the implicit argument by searching for a value
|
||
of the appropriate type. In fact, internally, this is exactly how interface
|
||
resolution works. It will try the following, in order:
|
||
|
||
- Local variables, i.e. names bound in pattern matches or ``let`` bindings,
|
||
with exactly the right type.
|
||
- The constructors of the required type. If they have arguments, it will
|
||
search recursively up to a maximum depth of 100.
|
||
- Local variables with function types, searching recursively for the
|
||
arguments.
|
||
- Any function with the appropriate return type which is marked with the
|
||
``%hint`` annotation.
|
||
|
||
In the case that a proof is not found, it can be provided explicitly as normal:
|
||
|
||
.. code-block:: idris
|
||
|
||
head xs {p = ?headProof}
|
||
|
||
Default implicit arguments
|
||
---------------------------
|
||
|
||
Besides having Idris automatically find a value of a given type, sometimes we
|
||
want to have an implicit argument with a specific default value. In Idris, we can
|
||
do this using the ``default`` annotation. While this is primarily intended to assist
|
||
in automatically constructing a proof where auto fails, or finds an unhelpful value,
|
||
it might be easier to first consider a simpler case, not involving proofs.
|
||
|
||
If we want to compute the n'th fibonacci number (and defining the 0th fibonacci
|
||
number as 0), we could write:
|
||
|
||
.. code-block:: idris
|
||
|
||
fibonacci : {default 0 lag : Nat} -> {default 1 lead : Nat} -> (n : Nat) -> Nat
|
||
fibonacci {lag} Z = lag
|
||
fibonacci {lag} {lead} (S n) = fibonacci {lag=lead} {lead=lag+lead} n
|
||
|
||
After this definition, ``fibonacci 5`` is equivalent to ``fibonacci {lag=0} {lead=1} 5``,
|
||
and will return the 5th fibonacci number. Note that while this works, this is not the
|
||
intended use of the ``default`` annotation. It is included here for illustrative purposes
|
||
only. Usually, ``default`` is used to provide things like a custom proof search script.
|
||
|
||
Literate programming
|
||
====================
|
||
|
||
Like Haskell, Idris supports *literate* programming. If a file has
|
||
an extension of ``.lidr`` then it is assumed to be a literate file. In
|
||
literate programs, everything is assumed to be a comment unless the line
|
||
begins with a greater than sign ``>``, for example:
|
||
|
||
::
|
||
|
||
> module literate
|
||
|
||
This is a comment. The main program is below
|
||
|
||
> main : IO ()
|
||
> main = putStrLn "Hello literate world!\n"
|
||
|
||
An additional restriction is that there must be a blank line between a
|
||
program line (beginning with ``>``) and a comment line (beginning with
|
||
any other character).
|
||
|
||
Cumulativity
|
||
============
|
||
|
||
.. warning::
|
||
|
||
NOT YET IN IDRIS 2
|
||
|
||
Since values can appear in types and *vice versa*, it is natural that
|
||
types themselves have types. For example:
|
||
|
||
::
|
||
|
||
*universe> :t Nat
|
||
Nat : Type
|
||
*universe> :t Vect
|
||
Vect : Nat -> Type -> Type
|
||
|
||
But what about the type of ``Type``? If we ask Idris it reports:
|
||
|
||
::
|
||
|
||
*universe> :t Type
|
||
Type : Type 1
|
||
|
||
If ``Type`` were its own type, it would lead to an inconsistency due to
|
||
`Girard’s paradox <https://www.cs.cmu.edu/afs/cs.cmu.edu/user/kw/www/scans/girard72thesis.pdf>`_,
|
||
so internally there is a *hierarchy* of types (or *universes*):
|
||
|
||
.. code-block:: idris
|
||
|
||
Type : Type 1 : Type 2 : Type 3 : ...
|
||
|
||
Universes are *cumulative*, that is, if ``x : Type n`` we can also have
|
||
that ``x : Type m``, as long as ``n < m``. The typechecker generates
|
||
such universe constraints and reports an error if any inconsistencies
|
||
are found. Ordinarily, a programmer does not need to worry about this,
|
||
but it does prevent (contrived) programs such as the following:
|
||
|
||
.. code-block:: idris
|
||
|
||
myid : (a : Type) -> a -> a
|
||
myid _ x = x
|
||
|
||
idid : (a : Type) -> a -> a
|
||
idid = myid _ myid
|
||
|
||
The application of ``myid`` to itself leads to a cycle in the universe
|
||
hierarchy — ``myid``\ ’s first argument is a ``Type``, which cannot be
|
||
at a lower level than required if it is applied to itself.
|
||
|
||
.. [#IdrisType] https://github.com/david-christiansen/idris-type-providers
|