/*
* The `SelectMany` operation is called _flatMap_ in some functional
* languages. It performs a double `Bind` as it combines a generator for
* `T`'s and a generator for `U`'s, and maps them to a generator for `V`'s.
* Whenever we have more than one select clause in a LINQ expression this
* method is called.
*/
public static Gen <V> SelectMany <T, U, V> (this Gen <T> gen,
Func <T, Gen <U> > project, Func <T, U, V> select)
{
/*
* As for `Select` the correct implementation reveals itself by
* following the types. Note that the types force us to arrange
* the `Bind` operations in a specific order.
*/
return(gen.Bind(a => project(a).Bind(b => select(a, b).ToGen())));
}
public static Gen <float> ToFloat(this Gen <double> gen)
{
return(gen.Bind(x => ((float)x).ToGen()));
}
public static Gen <long> ToLong(this Gen <int> gen)
{
return(gen.Bind(x => ((long)x).ToGen()));
}
/*
## Converting Generators
## We need also to do type conversions between the generators. The
## following methods are defined mostly for convenience, to avoid
## boilerplate code when constructing generators.
*/
public static Gen <U> Cast <T, U> (this Gen <T> gen) where T : U
{
return(gen.Bind(x => ((U)x).ToGen()));
}
/*
* Note that the process was almost mechanical. Think for a while, if you
* could implement the method differently. You will notice that if you try
* to change the implementation in any way, the code will not compile
* anymore; the types do not match.
*
* This is the reason, why we usually do not have to concern ourselves
* with the monad laws. If your monad type is a pure function, it is
* impossible to define the _return_ and _bind_ operation in a way that
* violates the monad laws. There is literally only one way to implement
* them. Be warned that this realization can easily blow your mind...
*
### What Have We Achieved?
###Before going further, let's step back and contemplate what we just
###implemented. We defined two operations: `ToGen` converts a value into
###a generator, and `Bind` turns a generator for `T`'s into generator for
###`U`'s when given a function that maps a value of type `T` to a
###generator of type `Gen<U>`.
###
###Essentially we defined a way to create generators and chain them
###together while hiding the two parameters of the `Gen<T>` delegate.
###This does not sound like a big deal, but it is.
###
###We can construct arbitrary complex generators and implement almost
###all the functionality we need using just these two operations. The
###instance of the Random class and the size parameter are passed along
###transparently in the background. This simplifies our code and makes it
###_composable_. With monads we achieve true code reusability.
###
## Relation to LINQ
##In the introduction there was a vague claim that LINQ and monads are
##somehow related. Now we will show exactly how. We will implement LINQ's
##core operations `Select` and `SelectMany` using `ToGen` and `Bind`. The
##`Select*` methods enable the syntactic sugaring that allows us to write
##generators as LINQ expressions.
##
##When C# compiler is desugaring a LINQ expression, it just rewrites it
##using the `Select` and `SelectMany` extension methods. If these methods
##are defined somewhere and they have the correct signature, then the
##compiler will be happy.
##
##The trick is thus to define those methods for our own monad type, and
##we get a [domain specific language](https://en.wikipedia.org/wiki/Domain-specific_language)
##for our generators. This DSL just happens to have the same syntax as
##LINQ. Haskell provides
##[syntactic sugaring](https://en.wikibooks.org/wiki/Haskell/do_notation)
##for monads too, although its syntax resembles an imperative program
##rather than a SQL query. Nevertheless, the idea is exactly the same.
##
### Implementing Select and SelectMany
###
###The signature of the LINQ's `Select` method is almost the same as for
###`Bind`. The only difference is the type of the function given as the
###second argument. Instead of `Func<T, Gen<U>>` it is `Func<T, U>`.
###Converting the function type to one expected by `Bind` is trivial using
###`ToGen`.
*/
public static Gen <U> Select <T, U> (this Gen <T> gen, Func <T, U> select)
{
return(gen.Bind(a => select(a).ToGen()));
}
/// <summary>
/// SelectMany extension method needed to enable Linq's syntactic sugaring.
/// </summary>
public static Gen <V> SelectMany <T, U, V> (this Gen <T> gen,
Func <T, Gen <U> > project, Func <T, U, V> select)
{
return(gen.Bind(a => project(a).Bind(b => select(a, b).ToGen())));
}