{-@ LIQUID "--pruneunsorted" @-}
-- | A somewhat fancier example demonstrating the use of Abstract Predicates and exist-types
module Ex (llen) where
-------------------------------------------------------------------------
-- | Data types ---------------------------------------------------------
-------------------------------------------------------------------------
data Vec a = Nil | Cons a (Vec a)
{-@ data Vec [llen] a = Nil | Cons { vx::a, vxs :: Vec a } @-}
-- | We can encode the notion of length as an inductive measure @llen@
{-@ measure llen @-}
llen :: Vec a -> Int
llen (Nil) = 0
llen (Cons x xs) = 1 + llen xs
{-@ invariant {v:Vec a | (llen v) >= 0} @-}
-- | As a warmup, lets check that a /real/ length function indeed computes
-- the length of the list.
{-@ sizeOf :: xs:Vec a -> {v: Int | v = llen xs} @-}
sizeOf :: Vec a -> Int
sizeOf Nil = 0
sizeOf (Cons _ xs) = 1 + sizeOf xs
-------------------------------------------------------------------------
-- | Higher-order fold --------------------------------------------------
-------------------------------------------------------------------------
-- | Time to roll up the sleeves. Here's a a higher-order @foldr@ function
-- for our `Vec` type. Note that the `op` argument takes an extra /ghost/
-- parameter that will let us properly describe the type of `efoldr`
{-@ efoldr :: forall a b <p :: x0:Vec a -> x1:b -> Bool>.
(xs:Vec a -> x:a -> b <p xs> -> b <p (Ex.Cons x xs)>)
-> b <p Ex.Nil>
-> ys: Vec a
-> b <p ys>
@-}
efoldr :: (Vec a -> a -> b -> b) -> b -> Vec a -> b
efoldr op b Nil = b
efoldr op b (Cons x xs) = op xs x (efoldr op b xs)
-------------------------------------------------------------------------
-- | Clients of `efold` -------------------------------------------------
-------------------------------------------------------------------------
-- | Finally, lets write a few /client/ functions that use `efoldr` to
-- operate on the `Vec`s.
-- | First: Computing the length using `efoldr`
{-@ size :: xs:Vec a -> {v: Int | v = llen xs} @-}
size :: Vec a -> Int
size = efoldr (\_ _ n -> n + 1) 0
-- | Second: Appending two lists using `efoldr`
{-@ app :: xs: Vec Int -> ys: Vec Int -> {v: Vec Int | llen v = llen xs + llen ys } @-}
app :: Vec Int -> Vec Int -> Vec Int
app xs ys = efoldr (\_ z zs -> Cons z zs) ys xs