MapFusion.hs

{-@ LIQUID "--reflection" @-}
{-@ LIQUID "--ple"        @-}

{-# LANGUAGE IncoherentInstances #-}
{-# LANGUAGE FlexibleContexts    #-}


module MapFusion where

import Prelude hiding (map)

import Language.Haskell.Liquid.ProofCombinators

{-@ axiomatize compose @-}
compose :: (b -> c) -> (a -> b) -> a -> c
compose f g x = f (g x)

{-@ axiomatize map @-}
map :: (a -> b) -> L a -> L b
map f N = N 
map f (C x xs) = f x `C` map f xs 


{-@ map_fusion :: f:(a -> a) -> g:(a -> a) -> xs:L a
      -> {map (compose f g) xs == compose (map f) (map g) xs } @-}
map_fusion :: (a -> a) -> (a -> a) -> L a -> Proof
map_fusion _ _ N        = trivial 
map_fusion f g (C x xs) =  map_fusion f g xs


data L a = N | C a (L a)
{-@ data L [llen] a = N | C {headlist :: a, taillist :: L a }@-}



{-@ measure llen @-}
llen :: L a -> Int
{-@ llen :: L a -> Nat @-}
llen N        = 0
llen (C _ xs) = 1 + llen xs