Rewrote set tutorial from scratch.

This is intended to address the issue mentioned in
ocaml/ocaml.org#598

This version of the tutorial now demonstrates how to use Set with
arbitrary types. It also provides a demonstration on how to reason
about the behavior of functions based on their type signatures.

Author: NebuPookins

site/learn/tutorials/set.md

@@ -3,74 +3,252 @@
 
 # Set
 
-## Module Set
-To make a set of strings:
+`Set` is a functor, which means that it is a module that is parameterized
+by another module. More concretely, this means you cannot directly create
+a set; instead, you must first specify what type of elements your set will
+contain.
+
+The `Set` functor provides a function `Make` which accepts a module as a
+parameter, and returns a new module representing a set whose elements have
+the type that you passed in. For example, if you want to work with sets of
+strings, you can invoke `Set.Make(String)` which will return you a new module
+which you can assign the name `SS` (short for "String Set"). Note: Be sure to
+pay attention to the case; you need to type `Set.Make(String)` and not
+`Set.Make(string)`. The reason behind this is explained in the
+"Technical Details" section at the bottom.
+
+Doing this in the OCaml's top level will yield a lot of output:
 
 ```ocamltop
 module SS = Set.Make(String);;
 ```
-To create a set you need to start somewhere so here is the empty set:
+
+What happened here is that after assigning your newly created module to the name
+`SS`, OCaml's top level then displayed the module, which in this case contains
+a large number of convenience functions for working with sets (for example `is_empty`
+for checking if you set is empty, `add` to add an element to your set, `remove` to
+remove an element from your set, and so on).
+
+Note also that this module defines two types: `type elt = String.t` representing
+the type of the elements, and `type t = Set.Make(String).t` representing the type of
+the set itself. It's important to note this, because these types are used in the
+signatures of many of the functions defined in this module.
+
+For example, the `add` function has the signature `elt -> t -> t`, which means
+that it expects an element (a String), and a set of strings, and will return to you
+a set of strings. As you gain more experience in OCaml and other function languages,
+the type signature of functions are often the most convenient form of documentation
+on how to use those functions.
+
+## Creating a Set
+
+You've created your module representing a set of strings, but now you actually want
+to create an instance of a set of strings. So how do we go about doing this? Well, you
+could search through the documentation for the original `Set` functor to try and
+find what function or value you should use to do this, but this is an excellent
+opportunity to practice reading the type signatures and inferring the answer from them.
+
+You want to create a new set (as opposed to modifying an existing set). So you should
+look for functions whose return result has type `t` (the type representing the set),
+and which *does not* require a parameter of type `t`.
+
+Skimming through the list of functions in the module, there's only a handful of functions
+that match that criteria: `empty: t`, `singleton : elt -> t`, `of_list : elt list -> t`
+and `of_seq : elt Seq.t -> t`.
+
+Perhaps you already know how to work with lists and sequences in OCaml or
+perhaps you don't. For now, let's assume you don't know, and so we'll focus
+our attention on the first two functions in that list: `empty` and `singleton`.
+
+The type signature for `empty` says that it simply returns `t`, i.e. an instance
+of our set, without requiring any parameters at all. By intuition, you might
+guess that the only reasonable set that a library function could return when
+given zero parameters is the empty set. And the fact that the function is named
+`empty` reinforces this theory.
+
+Is there a way to test this theory? Perhaps if we had a function which
+could print out the size of a set, then we could check if the set we get
+from `empty` has a size of zero. In other words, we want a function which
+receives a set as a parameter, and returns an integer as a result. Again,
+skimming through the list of functions in the module, we see there is a
+function which matches this signature: `cardinal : t -> int`. If you're
+not familiar with the word "cardinal", you can look it up on Wikipedia
+and notice that it basically refers to the size of sets, so this reinforces
+the idea that this is exactly the function we want.
+
+So let's test our hypothesis:
 
 ```ocamltop
 let s = SS.empty;;
+SS.cardinal s;;
 ```
-Alternatively if we know an element to start with we can create a set
-like
+
+Excellent, it looks like `SS.empty` does indeed create an empty set,
+and `SS.cardinal` does indeed print out the size of a set.
+
+What about that other function we saw, `singleton : elt -> t`? Again,
+using our intuition, if we provide the function with a single element,
+and the function returns a set, then probably the function will return
+a set containing that element (or else what else would it do with the
+parameter we gave it?). The name of the function is `singleton`, and
+again if you're unfamiliar with what word, you can look it up on
+Wikipedia and see that the word means "a set with exactly one element".
+It sounds like we're on the right track again. Let's test our theory.
 
 ```ocamltop
 let s = SS.singleton "hello";;
+SS.cardinal s;;
 ```
-To add some elements to the set we can do.
 
-```ocamltop
-let s =
-  List.fold_right SS.add ["hello"; "world"; "community"; "manager";
-                          "stuff"; "blue"; "green"] s;;
-```
-Now if we are playing around with sets we will probably want to see what
-is in the set that we have created. To do this we can write a function
-that will print the set out.
+It looks like we were right again!
+
+## Working with Sets
+
+Now let's say we want to build bigger and more complex sets. Specifically,
+let's say we want to add another element to our existing set. So we're
+looking for a function with two parameters: One of the parameters should
+be the element we wish to add, and the other parameter should be the set
+that we're adding to. For the return value, we would expect it to either
+return unit (if the function modifies the set in place), or it returns a
+new set representing the result of adding the new element. So we're
+looking for signatures that look something like `elt -> t -> unit` or
+`t -> elt -> unit` (since we don't know what order the two parameters
+should appear in), or `elt -> t -> t` or `t -> elt -> t`.
+
+Skimming through the list, we see 2 functions with matching signatures:
+`add : elt -> t -> t` and `remove : elt -> t -> t`. Based on their names,
+`add` is probably the function we're looking for. `remove` probably removes
+an element from a set, and using our intuition again, it does seem like
+the type signature makes sense: To remove an element from a set, you need
+to tell it what set you want to perform the removal on and what element
+you want to remove; and the return result will be the resulting set after
+the removal.
+
+Furthermore, because we see that these functions return `t` and not `unit`,
+we can infer that these functions do not modify the set in place, but
+instead return a new set. Again, we can test this theory:
 
 ```ocamltop
-(* Prints a new line "\n" after each string is printed *)
-let print_set s = 
-   SS.iter print_endline s;;
+let firstSet = SS.singleton "hello";;
+let secondSet = SS.add "world" firstSet;;
+SS.cardinal firstSet;;
+SS.cardinal secondSet;;
 ```
-If we want to remove a specific element of a set there is a remove
-function. However if we want to remove several elements at once we could
-think of it as doing a 'filter'. Let's filter out all words that are
-longer than 5 characters.
 
-This can be written as:
+It looks like our theories were correct!
+
+## Sets of With Custom Comparators
+
+The `SS` module we created uses the built-in comparison function provided
+by the `String` module, which performs a case-sensitive comparison. We
+can test that with the following code:
 
 ```ocamltop
-let my_filter str =
-  String.length str <= 5;;
-let s2 = SS.filter my_filter s;;
+let firstSet = SS.singleton "hello";;
+let secondSet = SS.add "HELLO" firstSet;;
+SS.cardinal firstSet;;
+SS.cardinal secondSet;;
 ```
-or using an anonymous function:
+
+As we can see, the `secondSet` has a cardinality of 2, indicating that
+`"hello"` and `"HELLO"` are considered two distinct elements.
+
+Let's say we want to create a set which performs a case-insensitive
+comparison instead. To do this, we simply have to change the parameter
+that we pass to the `Set.Make` function.
+
+The `Set.Make` function expects a struct with two fields: a type `t`
+that represents the type of the element, and a function `compare`
+whose signature is `t -> t -> int` and essentially returns 0 if two
+values are equal, and non-zero if they are non-equal. It just so happens
+that the `String` module matches that structure, which is why we could
+directly pass `String` as a parameter to `Set.Make`. Incidentally, many
+other modules also have that structure, including `Int` and `Float`,
+and so they too can be directly passed into `Set.Make` to construct a
+set of integers, or a set of floating point numbers.
+
+For our use case, we still want our elements to be of type string, but
+we want to change the comparison function to ignore the case of the
+strings. We can accomplish this by directly passing in a literal struct
+to the `Set.Make` function:
 
 ```ocamltop
-let s2 = SS.filter (fun str -> String.length str <= 5) s;;
+module CISS = Set.Make(struct
+  type t = string
+  let compare a b = compare (String.lowercase_ascii a) (String.lowercase_ascii b)
+end);;
 ```
-If we want to check and see if an element is in the set it might look
-like this.
+
+We name the resulting module CISS (short for "Case Insensitive String Set").
+We can now test whether this module has the desired behavior:
+
 
 ```ocamltop
-SS.mem "hello" s2;;
+let firstSet = CISS.singleton "hello";;
+let secondSet = CISS.add "HELLO" firstSet;;
+CISS.cardinal firstSet;;
+CISS.cardinal secondSet;;
 ```
 
-The Set module also provides the set theoretic operations union,
-intersection and difference. For example, the difference of the original
-set and the set with short strings (≤ 5 characters) is the set of long
-strings:
+Success! `secondSet` has a cardinality of 1, showing that `"hello"`
+and `"HELLO"` are now considered to be the same element in this set.
+We now have a set of strings whose compare function performs a case
+insensitive comparison.
+
+Note that this technique can also be used to allow arbitrary types
+to be used as the element type for set, as long as you can define a
+meaningful compare operation:
 
 ```ocamltop
-print_set (SS.diff s s2);;
+type color = Red | Green | Blue;;
+
+module SC = Set.Make(struct
+  type t = color
+  let compare a b =
+    match (a, b) with
+    | (Red, Red) -> 0
+    | (Red, Green) -> 1
+    | (Red, Blue) -> 1
+    | (Green, Red) -> -1
+    | (Green, Green) -> 0
+    | (Green, Blue) -> 1
+    | (Blue, Red) -> -1
+    | (Blue, Green) -> -1
+    | (Blue, Blue) -> 0
+end);;
 ```
-Note that the Set module provides a purely functional data structure:
-removing an element from a set does not alter that set but, rather,
-returns a new set that is very similar to (and shares much of its
-internals with) the original set.
+
+## Technical Details
+
+### Set.Make, types and modules
+
+As mentioned in a previous section, the `Set.Make` function accepts a structure
+with two specific fields, `t` and `compare`. Modules have structure, and thus
+it's possible (but not guaranteed) for a module to have the structure that
+`Set.Make` expects. On the other hand, types do not have structure, and so you
+can never pass a type to the `Set.Make` function. In OCaml, modules start with
+an upper case letter and types start with a lower case letter. This is why
+when creating a set of strings, you have to use `Set.Make(String)` (passing in
+the module named `String`), and not `Set.Make(string)` (which would be attempting
+to pass in the type named `string`, which will not work).
+
+### Purely Functional Data Structures
+
+The data structure implemented by the Set functor is a purely functional one.
+What exactly that means is a big topic in itself (feel free to search for
+"Purely Functional Data Structure" in Google or Wikipedia to learn more). As a
+short oversimplification, this means that all instances of the data structure
+that you create are immutable. The functions like `add` and `remove` do not
+actually modify the set you pass in, but instead return a new set representing
+the results of having performed the corresponding operation.
+
+### Full API documentation
+
+This tutorial focused on teaching how to quickly find a function that does what
+you want by looking at the type signature. This is often the quickest and most
+convenient way to discover useful functions. However, sometimes you do want to
+see the formal documentation for the API provided by a module. For sets, the
+API documentation you probably want to look at is at
+https://ocaml.org/api/Set.Make.html