Here's a direct translation of the original code.  What is missing is an instance for Number for Prelude.Float.  That is because it is using 4.0 + 0.5 * (t :: Time), where Time must be a P.Float, or else if it is a data type it must having instances for Fractional, Num, and Floating, so that it can be represented as a floating point literal (ie. 4.0).

It might be easier in the long run if you got rid of the Number class and just used Prelude's Num class instead.  You would have to do some slightly different handling to deal with sqrt.  Admittedly that might be difficult.

{-# LANGUAGE NoImplicitPrelude, MultiParamTypeClasses, FlexibleInstances #-}

import qualified Prelude as P ((+), (-), (*), sqrt, Float)

class Number a where
  (+), (-), (*) :: a -> a -> a
  sqr, sqrt :: a -> a
  sqr a = a * a

type Time = P.Float

type Moving v = Time -> v

instance Number P.Float where
  (+) = (P.+)
  (-) = (P.-)
  (*) = (P.*)
  sqrt = P.sqrt


instance Number v => Number (Moving v) where
 (+) a b = \t -> (a t) + (b t)
 (-) a b = \t -> (a t) - (b t)
 (*) a b = \t -> (a t) * (b t)
 sqrt a = \t -> sqrt (a t)

class Number s => Points p s where
  x, y :: p s -> s
  xy :: s -> s -> p s
  dist :: p s -> p s -> s
  dist a b = sqrt (sqr ((x a) - (x b)) + sqr ((y a) - (y b)))

data Point f = Point f f

instance Number v => Points Point v where
  x (Point x1 y1) = x1
  y (Point x1 y1) = y1
  xy x1 y1 = Point x1 y1

instance Number v => Number (Point v) where
  a + b = xy (x a + x b) (y a + y b)
  a - b = xy (x a - x b) (y a - y b)


np1 :: Point (Moving P.Float)
np1 = xy (\t -> 4.0 + 0.5 * t) (\t -> 4.0 - 0.5 * t)
np2 = xy (\t -> 0.0 + 1.0 * t) (\t -> 0.0 - 1.0 * t)
movingDist_1_2 = dist np1 np2
dist_at_1 = movingDist_1_2 1.0


On Sun, Oct 8, 2017 at 1:24 AM, PATRICK BROWNE <patrick.browne@dit.ie> wrote:
Thomas,
Thanks for your response. I agree that using (+) in this manner leads to name clashes. 
My question is prompted by a research paper [1] where a form of lifting is attempted  using  type classes. 
I think that the original code in paper (below) is intended to be illustrative rather than practical. 
I have edited the code to compile and run (after a fashion, see below). But I cannot get the functions to return the lifted type.
Also I have a problem compiling the lines with two lambdas, e.g. (np1 = xy (\t -> 4.0 + 0.5 * t) (\t -> 4.0 - 0.5 * t))
Regards,
Pat
---------------------------------------------------------------------------------------------
Original code [1]:
------------------------------------------------------------------------------------------
class Number a where
(+), (-), (*) :: a -> a -> a
sqr, sqrt :: a -> a
sqr a = a * a

type Moving v = Time -> v

instance Number v => Number (Moving v) where
 (+) a b = \t -> (a t) + (b t)
 (-) a b = \t -> (a t) - (b t)
 (*) a b = \t -> (a t) * (b t)
 sqrt a = \t -> sqrt (a t)

class Number s => Points p s where
 x, y :: p s -> s
 xy :: s -> s -> p s
 dist :: p s -> p s -> s
 dist a b = sqrt (sqr ((x a) - (x b)) +
 sqr ((y a) - (y b)))

data Point f = Point f f

instance Number v => Points Point v where
 x (Point x1 y1) = x1
 y (Point x1 y1) = y1
 xy x1 y1 = Point x1 y1

instance Number v => (Point v) where
 (+) a b = xy (x a + x b) (y a + y b)
 (-) a b = xy (x a - x b) (y a - y b)


np1, np2 :: Point (Moving Float)
np1 = xy (\t -> 4.0 + 0.5 * t) (\t -> 4.0 - 0.5 * t)
np2 = xy (\t -> 0.0 + 1.0 * t) (\t -> 0.0 - 1.0 * t)
movingDist_1_2 = dist np1 np2
dist_at_1 = movingDist_1_2 1.0


----------------------------------------------------------------------------------------------
My attempt at getting above code to run:
-----------------------------------------------------------------------------------------------
{-# LANGUAGE MultiParamTypeClasses #-}
{-# LANGUAGE FlexibleInstances #-}
{-# LANGUAGE TypeSynonymInstances #-}
module Moving where
data  Time  = Time Double
type Moving v  = Time -> v
data Point v  = Point v v deriving Show

class  Number a where
 (+),(-),(*)  ::  a -> a  -> a
 sqrt :: a -> a


instance (Fractional a,Floating a) => Number  (Moving a) where
 (+) a b = \t -> ((a t) Prelude.+ (b t))
 (-) a b = \t -> ((a t) Prelude.- (b t))
 (*) a b = \t -> ((a t) Prelude.* (b t))
 sqrt a =  \t -> Prelude.sqrt (a t)

a,b ::  Moving Double
a (Time x) = 4.0
b (Time x) = 4.0
testPlus ::(Moving Double)
testPlus = (a Moving.+ b)
testPlusArg = (a Moving.+ b) (Time 2.0)
testSqrt  = (Moving.sqrt a) (Time 2.0)               



class (Number s) =>  Points p s  where
 x, y :: p s -> s
 xy :: s -> s -> p s
 dist :: p s -> p s -> s
 dist a b = Moving.sqrt (sqr ((x a) Moving.- (x b))  Moving.+ sqr ((y a) Moving.- (y b)))
               where sqr z = z Moving.* z



-- instance (Floating v,Number v) => Points Point v  where
instance  (Number s) => Points Point s where
 x (Point x1 y1) = x1
 y (Point x1 y1) = y1
 xy x1 y1 = Point x1 y1

instance Number v =>  Number (Point v) where
 (+) a b = xy (x a Moving.+ x b) (y a Moving.+ y b)
 (-) a b = xy (x a Moving.- x b) (y a Moving.- y b)


md1,md2,md3,md4 ::  Moving Double
md1  (Time x) = 0.0
md2  (Time x) = 0.0
md3  (Time x) = 10.0
md4  (Time x) = 10.0
testMD1 = (md1  (Time 2.0))
testX =  x (Point md1 md2)  (Time 2.0)
testY =  y (Point md1 md2) (Time 2.0)
testXY =  (xy md1 md2)::(Point (Moving Double))
testX' =  x testXY  (Time 1.0)
testD = dist (Point md1 md2) (Point md3 md4)  (Time 1.0)

--- I cannot get the rest to work

[1] Ontology for Spatio-temporal Databases

On 7 October 2017 at 23:30, Thomas Jakway <tjakway@nyu.edu> wrote:

You could hide Prelude and define it yourself in a different module but that would be a pretty bad idea.  Everyone who wanted to use it would have to import your module qualified and refer to it as MyModule.+, which defeats the point of making it (+) and not `myAdditionFunction` in the first place.

Haskell deliberately doesn't allow overloading.  Having (+) return something other than Num would be extremely confusing.


On 10/07/2017 05:07 AM, PATRICK BROWNE wrote:
Hi,
Is there a way rewriting the definition of (+) so that testPlusArg returns a (Moving Double). My current intuition is that the signature [(+)  ::  a -> a  -> a] says that the type should be the same as the arguments. And indeed (:t testPlus) confirms this. But the type of  testPlusArg is a Double.
 Can I make it (Moving Double) ?
Thanks,
Pat
 

{-# LANGUAGE FlexibleInstances #-}
{-# LANGUAGE TypeSynonymInstances #-}
module Moving where
data  Time  = Time Double
type Moving v  = Time -> v

class  Number a where
 (+)  ::  a -> a  -> a

instance Number  (Moving Double) where
 (+) a b = \t -> ((a t) Prelude.+ (b t))

a,b ::  Moving Double
a (Time x) = 2.0
b (Time x) = 2.0
testPlus ::(Moving Double)
testPlus = (a Moving.+ b)
testPlusArg = (a Moving.+ b) (Time 2.0)

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