[Haskell-beginners] Lifting Numbers

[PATRICK BROWNE]

Hi,
I am trying to compile, run, and understand the following code from [1].

type Moving v = Time -> v

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

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)

I followed the compiler advice to produce the following version which
compiles:

{-# LANGUAGE FlexibleInstances #-}
{-# LANGUAGE TypeSynonymInstances #-}
module MovingPoint where
type Time  = Float -- Type synonym assumed, could it be data type??
type Moving v  = Time -> v

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

instance (Floating v) => Number (Moving v) 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)
 sqr a =  \t -> (a t)  Prelude.* (a t)
 sqrt a =  \t -> Prelude.sqrt (a t)

I do not know how to invoke any of the operations. In general I do know how
to execute lambdas.
I do not understand the bracketed pairs e.g. (a t).
Any help on understanding and running the program would be appreciated.
Thanks,
Pat


[1] Ontology for Spatio-temporal Databases
http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.113.9804&rep=rep1&type=pdf

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