[Haskell-beginners] Lifting Numbers

[David McBride]

You can get some intuition for how this works by replacing "Moving v" with
its definition "Time -> v".  Let's look at the + operation.

class Number a where
  (+) :: a -> a -> a
instance Number v => Number (Moving v)
instance Number v => Number (Time -> v)
(+) :: Number v => (Time -> v) -> (Time -> v) -> (Time -> v)

So each argument of + must take a Time, the end result must also take a
Time, and whatever each argument returns must be a Number (and thus has +
defined for it).  So you can sort of see how it works.  + for a Moving v
takes a time, then passes that time to each of its arguments, then adds the
result.

(+) a b = \t -> (a t)  Prelude.+ (b t)

data Time = Time Double -- For example.

Then you can make formulas that are rooted in time.  For example
(contrived) if you are throwing a ball, the distance of the ball from you
at time f could be something like the following:

balldistance ::  Moving Double
balldistance (Time f) = f * 1.2

ball1 :: Moving Double
ball1 = balldistance

ball2 :: Moving Double
ball2 = balldistance

-- the combined distance of both balls at time f
bothballs :: Moving Double
bothballs = ball1 + ball2

Then you can get the combined distance of both balls after 12 seconds, for
example.

test :: Double
test = bothballs (Time 12.0)





On Tue, Oct 3, 2017 at 9:07 AM, PATRICK BROWNE <patrick.browne at dit.ie>
wrote:

> 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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