Miscellaneous¶

Things we have yet to classify, or are two small to justify their own page.

The Unifier Log¶

If you’re having a hard time debugging why the unifier won’t accept something (often while debugging the compiler itself), try applying the special operator %unifyLog to the expression in question. This will cause the type checker to spit out all sorts of informative messages.

Namespaces and type-directed disambiguation¶

Names can be defined in separate namespaces, and disambiguated by type. An expression with NAME EXPR will privilege the namespace NAME in the expression EXPR. For example:

Idris> with List [[1,2],[3,4],[5,6]]
[[1, 2], [3, 4], [5, 6]] : List (List Integer)

Idris> with Vect [[1,2],[3,4],[5,6]]
[[1, 2], [3, 4], [5, 6]] : Vect 3 (Vect 2 Integer)

Idris> [[1,2],[3,4],[5,6]]
Can't disambiguate name: Prelude.List.::, Prelude.Stream.::, Prelude.Vect.::

Alternatives¶

The syntax (| option1, option2, option3, ... |) type checks each of the options in turn until one of them works. This is used, for example, when translating integer literals.

Idris> the Nat (| "foo", Z, (-3) |)
0 : Nat

This can also be used to give simple automated proofs, for example: trying some constructors of proofs.

syntax Trivial = (| Oh, Refl |)

Totality checking assertions¶

All definitions are checked for coverage (i.e. all well-typed applications are handled) and either for termination (i.e. all well-typed applications will eventually produce an answer) or, if returning codata, for productivity (in practice, all recursive calls are constructor guarded).

Obviously, termination checking is undecidable. In practice, the termination checker looks for size change - every cycle of recursive calls must have a decreasing argument, such as a recursive argument of a strictly positive data type.

There are two built-in functions which can be used to give the totality checker a hint:

assert_total x asserts that the expression x is terminating and covering, even if the totality checker cannot tell. This can be used for example if x uses a function which does not cover all inputs, but the caller knows that the specific input is covered.
assert_smaller p x asserts that the expression x is structurally smaller than the pattern p.

For example, the following function is not checked as total:

qsort : Ord a => List a -> List a
qsort [] = []
qsort (x :: xs) = qsort (filter (<= x) xs) ++ (x :: qsort (filter (>= x) xs)))

This is because the checker cannot tell that filter will always produce a value smaller than the pattern x :: xs for the recursive call to qsort. We can assert that this will always be true as follows:

total
qsort : Ord a => List a -> List a
qsort [] = []
qsort (x :: xs) = qsort (assert_smaller (x :: xs) (filter (<= x) xs)) ++
                  (x :: qsort (assert_smaller (x :: xs) (filter (>= x) xs))))

Preorder reasoning¶

This syntax is defined in the module Syntax.PreorderReasoning in the base package. It provides a syntax for composing proofs of reflexive-transitive relations, using overloadable functions called step and qed. This module also defines step and qed functions allowing the syntax to be used for demonstrating equality. Here is an example:

import Syntax.PreorderReasoning
multThree : (a, b, c : Nat) -> a * b * c = c * a * b
multThree a b c =
  (a * b * c) ={ sym (multAssociative a b c) }=
  (a * (b * c)) ={ cong (multCommutative b c) }=
  (a * (c * b)) ={ multAssociative a c b }=
  (a * c * b) ={ cong {f = (* b)} (multCommutative a c) }=
  (c * a * b) QED

Note that the parentheses are required – only a simple expression can be on the left of ={ }= or QED. Also, when using preorder reasoning syntax to prove things about equality, remember that you can only relate the entire expression, not subexpressions. This might occasionally require the use of cong.

Finally, although equality is the most obvious application of preorder reasoning, it can be used for any reflexive-transitive relation. Something like step1 ={ just1 }= step2 ={ just2 }= end QED is translated to (step step1 just1 (step step2 just2 (qed end))), selecting the appropriate definitions of step and qed through the normal disambiguation process. The standard library, for example, also contains an implementation of preorder reasoning on isomorphisms.

Pattern matching on Implicit Arguments¶

Pattern matching is only allowed on implicit arguments when they are referred by name, e.g.

foo : {n : Nat} -> Nat
foo {n = Z} = Z
foo {n = S k} = k

or

foo : {n : Nat} -> Nat
foo {n = n} = n

The latter could be shortened to the following:

foo : {n : Nat} -> Nat
foo {n} = n

That is, {x} behaves like {x=x}.

Existence of an implementation¶

In order to show that an implementation of some interface is defined for some type, one could use the %implementation keyword:

foo : Num Nat
foo = %implementation

‘match’ application¶

ty <== name applies the function name in such a way that it has the type ty, by matching ty against the function’s type. This can be used in proofs, for example:

plus_comm : (n : Nat) -> (m : Nat) -> (n + m = m + n)
-- Base case
(Z + m = m + Z) <== plus_comm =
    rewrite ((m + Z = m) <== plusZeroRightNeutral) ==>
            (Z + m = m) in Refl

-- Step case
(S k + m = m + S k) <== plus_comm =
    rewrite ((k + m = m + k) <== plus_comm) in
    rewrite ((S (m + k) = m + S k) <== plusSuccRightSucc) in
        Refl

Reflection¶

Including %reflection functions and quoteGoal x by fn in t, which applies fn to the expected type of the current expression, and puts the result in x which is in scope when elaborating t.

Bash Completion¶

Use of optparse-applicative allows Idris to support Bash completion. You can obtain the completion script for Idris using the following command:

idris --bash-completion-script `which idris`

To enable completion for the lifetime of your current session, run the following command:

source <(idris --bash-completion-script `which idris`)

To enable completion permenatly you must either:

Modify your bash init script with the above command.
Add the completion script to the appropriate bash_completion.d/ folder on your machine.