{-# LANGUAGE RankNTypes      #-}
{-# LANGUAGE GADTs           #-}
{-# LANGUAGE KindSignatures  #-}
{-@ LIQUID "--reflection" @-} 
-- Z3 encoding cannot be used until this is fixed: 
-- https://github.com/Z3Prover/z3/issues/3930
{-@ LIQUID "--no-adt"         @-} 
{-@ LIQUID "--ple-local"      @-} 
{-@ LIQUID "--prune-unsorted" @-} 

module Properties where 



{-@ assume injectiveEqRTFun :: x:(a->b) -> y:(a->b) -> d:{EqRT (a->b) {x} {y} | isEqFun d} 
                         -> {x = eqFunX d && y = eqFunY d} @-}
injectiveEqRTFun :: (a->b) -> (a->b) -> EqT (a->b) -> () 
injectiveEqRTFun _ _ _ = () 


{-@ measure isEqFun @-}
isEqFun :: EqT a -> Bool 
isEqFun (EqFun _ _ _) = True 
isEqFun _             = False 

{-@ reflect eqFunX @-}
{-@ eqFunX :: {d:EqT (a -> b) | isEqFun d} -> (a -> b) @-}
eqFunX :: EqT (a -> b) ->  (a -> b) 
eqFunX (EqFun x _ _) = x 

{-@ reflect eqFunY @-}
{-@ eqFunY :: {d:EqT (a -> b) | isEqFun d} -> (a -> b) @-}
eqFunY :: EqT (a -> b) ->  (a -> b) 
eqFunY (EqFun _ y _) = y 


{-@ ple     eqFunP @-}
{-@ reflect eqFunP @-}
{-@ eqFunP :: d:{EqT (a -> b) | isEqFun d} -> (x:a -> EqRT b {eqFunX d x} {eqFunY d x}) @-}
eqFunP :: EqT (a -> b) ->  (a -> EqT b) 
eqFunP (EqFun _ _ p) = p


data EqT  :: * -> *  where 
   EqFun  :: (a -> b) -> (a -> b) -> (a -> EqT b) -> EqT (a -> b)

{-@ data EqT  :: * -> *  where 
     EqFun  :: ff:(a -> b) -> gg:(a -> b) -> (x:a -> EqRT b {ff x} {gg x}) -> EqRT (a -> b) {ff} {gg}
@-}   


{-@ type EqRT a E1 E2 = {v:EqT a | E1 = E2} @-}
