content: module Termination where
  
  import Prelude hiding (gcd, mod, map, repeat, take)
  import Language.Haskell.Liquid.Prelude
  
  fac   :: Int -> Int
  tfac  :: Int -> Int -> Int
  map   :: (a -> b) -> List a -> List b
  merge :: (Ord a) => List a -> List a -> List a
  
  
  -------------------------------------------------------------------------
  -- | Simple Termination
  -------------------------------------------------------------------------
  
  {-@ fac :: Nat -> Nat @-}
  fac 0 = 1
  fac 1 = 1
  fac n = n * fac (n-1)
  
  
  
  
  -------------------------------------------------------------------------
  -- | Semantic Termination
  -------------------------------------------------------------------------
  
  {-@ gcd :: a:Nat -> b:{v:Nat | v < a} -> Int @-}
  gcd :: Int -> Int -> Int
  gcd a 0 = a
  gcd a b = gcd b (a `mod` b)
  
  
  {-@ mod :: a:Nat -> b:{v:Nat| ((v < a) && (v > 0))} -> {v:Nat | v < b} @-}
  mod :: Int -> Int -> Int
  mod a b | a - b >  b = mod (a - b) b
          | a - b <  b = a - b
          | a - b == b = 0
  
  -------------------------------------------------------------------------
  -- Explicit Metrics #1 
  -------------------------------------------------------------------------
  
  {-@ tfac :: Nat -> n:Nat -> Nat / [n] @-}
  tfac acc 0 = acc
  tfac acc n = tfac (n * acc) (n-1)
  
  
  
  
  -------------------------------------------------------------------------
  -- Explicit Metrics #2 
  -------------------------------------------------------------------------
  
  {-@ range :: lo:Nat -> hi:Nat -> [Nat] / [hi-lo] @-}
  range :: Int -> Int -> [Int]
  range lo hi
    | lo < hi   = lo : range (lo + 1) hi
    | otherwise = []
  
  
  
  -------------------------------------------------------------------------
  -- | Structural Recursion 
  -------------------------------------------------------------------------
  
  data List a = N | C a (List a)
  
  {-@ measure sz  :: List a -> Int 
      sz (C x xs) = 1 + (sz xs)
      sz (N)      = 0
    @-}
  
  {-@ map :: (a -> b) -> xs:List a -> (List b) / [sz xs] @-}
  map _ N        = N
  map f (C x xs) = f x `C` map f xs
  
  
  
  -------------------------------------------------------------------------
  -- | Default Metrics
  -------------------------------------------------------------------------
  
  {-@ data List [sz] a = N | C {x :: a, xs :: List a } @-}
  
  map' _ N        = N
  map' f (C x xs) = f x `C` map' f xs
  
  
  
  -------------------------------------------------------------------------
  -- | Termination Expressions Metrics
  -------------------------------------------------------------------------
  
  {-@ merge :: xs:_ -> ys:_ -> _ / [sz xs + sz ys] @-}
  
  merge (C x xs) (C y ys)
    | x < y      = x `C` merge xs (y `C` ys)
    | otherwise  = y `C` merge (x `C` xs) ys
  merge _   ys   = ys
  
  
  
  
  
  
  
  
  
  -------------------------------------------------------------------------
  -- | Infinite Streams
  -------------------------------------------------------------------------
  
  {- data List [sz] a <p :: List a -> Prop>
        = N | C { x  :: a
                , xs :: List <p> a <<p>>
                }
    -}
  
  
  {-@ measure emp  :: (List a) -> Prop
      emp (N)      = true
      emp (C x xs) = false
    @-}
  
  {- type Stream a = {xs: List <{\v -> not (emp v)}> a | not (emp xs)} @-}
  
  {- Lazy repeat @-}
                   
  {- repeat :: a -> Stream a @-}
  -- repeat   :: a -> List a
  -- repeat x = x `C` repeat x
  
  
  {- take :: Nat -> Stream a -> List a @-}
  -- take  :: Int -> List a -> List a
  -- take 0 _        = N
  -- take n (C x xs) = x `C` take (n-1) xs
  -- take _ N        = liquidError "never happens"
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  -----------------------------------------------------
  
  
  {-@ invariant {v : List a | 0 <= sz v} @-}
  
  
  
  
  
  
  
  
  