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\todo[inline]{Discuss associativity.}


== Structural Recursion (Optional)

Here is a copy of the definition of addition:

```idris
plus' : Nat -> Nat -> Nat
plus'  Z       right = right
plus' (S left) r = S (!'  rt)
```

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(0 ==) . snd export length : HasIO io => IORingQueue a -> io Nat length q = readIORef q.loc >>= pure . snd export push : HasIO io => a -> IORingQueue a -> io Bool push x q = do (n, m) <- readIORef q.loc if m >= capacity q.content then pure False else do writeIOArray q.content (restrict q.content (cast $ n + m)) (Just x) writeIORef q.loc (n, m + 1) pure True export pull : HasIO io => IORingQueue a -> io (Maybe a) pull q = do (n, (S m)) <- readIORef q.loc | (_, Z) => pure Nothing writeIORef q.loc ((n + 1) `mod` (capacity q.content), m) let i = restrict q.content $ cast n r <- readIOArray q.content i writeIOArray q.content i Nothing pure r {- export HasIO io => MutableQueue io IORingQueue where null q = readIORef q.loc >>= pure . (0 ==) . snd length q = readIORef q.loc >>= pure . snd push x q = do (n, m) <- readIORef q.loc if m >= capacity q.content then pure False else do writeIOArray q.content (restrict q.content (cast $ n + m)) (Just x) writeIORef q.loc (n, m + 1) pure True pull q = do (n, (S m)) <- readIORef q.loc | (_, Z) => pure Nothing writeIORef q.loc ((n + 1) `mod` (capacity q.content), m) let i = restrict q.content $ cast n r <- readIOArray q.content i writeIOArray q.content i Nothing pure r -} -- -------------------------------------------------------------------------- -- vim: tw=80 sw=2 expandtab : � import Data.List -- Taken from http://nicolas.biri.name/posts/2016-07-26-union-type-in-idris-part-1.html data Union : List Type -> Type where MemberHere : ty -> Union (ty::ts) MemberThere : Union ts -> Union (ty::ts) member : ty -> {auto e: Elem ty ts} -> Union ts member x {e = Here} = MemberHere x member x {e = There later} = MemberThere (member x {e = later}) x : Union [String, Nat, List String] x = member "Ahoy!" y : Union [String, Nat, List String] y = member (S Z) ������0����.��{-�� �Equivlance relation rules mod n. �j�-} |||� ModEq n x y is (aweful) prefix not }Hfor a type x == y (�).G ThisDPa singleton dependent;, and it&<inhabited solelyC�by its constructor. 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