== Summary ==

We explored modular design in some detail; developing interfaces for numbers
that support various operations.

 - we designed an "Addable" interface
 - we brainstormed different implementations:
     ints, floats, fractions, complex numbers, matrices, vectors

 - we separated the interface into a "Core" interface that must be implemented
   and an "Extension" interface that can be derived from the Core.  We will
   revisit this in more detail next lecture

 - we extended the interface to create a "Multipliable" interface using the
   "include" statement

 - we used the "MT with type t := t'" syntax, used to modify a module type by
   replacing it with another type throughout.  For example, by writing

     module Int : Addable with type t := int

   we exposed the fact that Int.t = int, allowing Int functions to be used with
   regular ints.

== Additional reading ==

Real World OCaml, chapter 9, discusses many of the techniques we covered in this
lecture (and the next).

== Provided code ==

I have extended the file algebra.mli that we developed in class.  Just as we
added multiplication to the addable interface to create the multipliable
interface, I have separately added ordering and division.

I have taken out the concrete implementations (Ints, Floats, etc.)  We will
re-add them (and more!) next time.  You are encouraged to add some on your own!

I have also added implementations of all the Extension interfaces in the .ml
file; I will discuss these further next time.

== Foreshadowing the payoff ==

The payoff of designing the code this way may not be apparent until after the
next lecture.  You can start to see it by looking at the amount of effort we
have saved an implementor of the Number type.  There are 18 functions required
in the Number interface:

utop # #use "algebra.ml";;
utop # module Int : Number = struct end;;
Error: Signature mismatch:                                                                                                                                                     Modules do not match: sig  end is not included in Number                                                                                                                The field `>=' is required but not provided                                                                                                                      
       The field `<=' is required but not provided
       The field `>' is required but not provided
       The field `<' is required but not provided
       The field `compare' is required but not provided
       The field `from_int' is required but not provided
       The field `<>' is required but not provided
       The field `-' is required but not provided
       The field `/' is required but not provided
       The field `to_float' is required but not provided
       The field `isPositive' is required but not provided
       The field `one' is required but not provided
       The field `*' is required but not provided
       The field `=' is required but not provided
       The field `zero' is required but not provided
       The field `~-' is required but not provided
       The field `+' is required but not provided
       The field `inv' is required but not provided
       The field `t' is required but not provided

but only 9 required in the NumberCore interface:

utop # module IntCore : NumberCore = struct end;;
Error: Signature mismatch:                                                                                                                                                     Modules do not match: sig  end is not included in NumberCore                                                                                                            The field `to_float' is required but not provided                                                                                                                
       The field `isPositive' is required but not provided
       The field `one' is required but not provided
       The field `*' is required but not provided
       The field `=' is required but not provided
       The field `zero' is required but not provided
       The field `~-' is required but not provided
       The field `+' is required but not provided
       The field `inv' is required but not provided
       The field `t' is required but not provided

Since there will be many number implementations (some more complex than others!)
reducing the amount of work (and potential for bugs) in half is a big win.