In the past two months I was in exams for my Lic. but now I have the time to continue with my post’s.
Back to my blog I find out that anyone can copy (CTRL+C) the images that are here and what in fact you are copying is the LaTeX code that I write to produce the images. That’s great! Now you can see my Xy-pic skills (lolz)…
In the last post of this series I have talk about Products of functions, in this post we will see more about point-free calculation using co-product of functions.
Because of the length of what I intend to explain I talk about these theme in three more post’s. Consider this part II.
- Products (Part I)
- Constants and conditionals (Part III)
- Pointfree – Pointwise conversion & Program calculation (Part IV)
At the time I finish the other posts I will put their link here.
In this post I will talk about co-products of functions and some of their rules.
I will use that means ‘def’ from ‘definition’. I can’t use ‘def’ in this LaTeX package. And I confess that didn’t investigate a lot about that, in matter of fact I like the output of .
I will also use Commutative diagrams to explain the rules, Because I think that is the easy way to get it.
I will use combinator defined in Part I.
Like we sow for the structs (split combinator, Part I), we have also a combinator for unions (eithers).
The is often called ’either’. This is the combinator responsible to take care of union’s (eithers in Haskell).
in Haskell became:
either :: (a -> c) -> (b -> c) -> Either a b -> c either f g (Left x) = f x either f g (Right y) = g y
As we will see with this combinator we can combine functions with the same counterdomain.
‘Left‘ and ‘Right‘ are constructors of data type Either (in Haskell) and we can see them as a functions with the folowing type:
As you can see with both functions we can inject information in our functions, in the oposite side we have ´fst´ and ´snd´ functions that we use to project information (Part I).
Let A, B, C, D be Types and
f : A -> C, g : B -> C and we want the function
either : A + B -> C
Like diagram is commutative we can infer two rules:
Now, if type C = A + B, we will have
To have a commutative diagram we must have and
And we can infer the rule:
Let A, B, C, D be Types and g : A -> C, h : B -> C and f : C -> D.
When we want to defin a function that takes a co-product and goes to a co-product we may use the function (\bigoplus)
in Haskell became:
infixr 4 -|- (-|-) :: (a -> b) -> (c -> d) -> Either a c -> Either b d f -|- g = either (Left . f) (Right . g)
here is the diagram of it definition:
We also have the equalty of eithers;
And, to conclude, we have a rule that relationate ‘eithers’ with ‘splits’,
To show you some of the power of program calculation here is the prove of
(please open the image to see it in full size)
The combinator have more precedence than the combinator.
I would like to thank dysfunctor to the corrections made to this post.