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Fix termination bug in tutorial's isEven/isOdd

author | Adam Chlipala <adam@chlipala.net> |
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date | Sun, 09 Aug 2015 11:14:09 -0400 |

parents | 2d9f831d45c9 |

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(* Chapter 1: Introduction *) (* begin hide *) val show_string = mkShow (fn s => "\"" ^ s ^ "\"") (* end *) (* This tutorial by <a href="http://adam.chlipala.net/">Adam Chlipala</a> is licensed under a <a href="http://creativecommons.org/licenses/by-nc-nd/3.0/">Creative Commons Attribution-Noncommercial-No Derivative Works 3.0 Unported License</a>. *) (* This is a tutorial for the <a href="http://www.impredicative.com/ur/">Ur/Web</a> programming language.<br> <br> Briefly, <b>Ur</b> is a programming language in the tradition of <a target="_top" href="http://en.wikipedia.org/wiki/ML_(programming_language)">ML</a> and <a target="_top" href="http://en.wikipedia.org/wiki/Haskell_(programming_language)">Haskell</a>, but featuring a significantly richer type system. Ur is <a target="_top" href="http://en.wikipedia.org/wiki/Functional_programming">functional</a>, <a target="_top" href="http://en.wikipedia.org/wiki/Purely_functional">pure</a>, <a target="_top" href="http://en.wikipedia.org/wiki/Statically-typed">statically-typed</a>, and <a target="_top" href="http://en.wikipedia.org/wiki/Strict_programming_language">strict</a>. Ur supports a powerful kind of <b><a target="_top" href="http://en.wikipedia.org/wiki/Metaprogramming">metaprogramming</a></b> based on <b><a target=_"top" href="http://citeseer.ist.psu.edu/viewdoc/summary?doi=10.1.1.44.6387">row types</a></b>.<br> <br> <b>Ur/Web</b> is Ur plus a special standard library and associated rules for parsing and optimization. Ur/Web supports construction of dynamic web applications backed by SQL databases, with mixed server-side and client-side applications generated from source code in one language.<br> <br> Ur inherits its foundation from ML and Haskell, then going further to add fancier stuff. This first chapter of the tutorial reviews the key ML and Haskell features, giving their syntax in Ur. I do assume reading familiarity with ML and Haskell and won't dwell too much on explaining the imported features.<br> <br> For information on compiling applications (and for some full example applications), see the intro page of <a href="http://www.impredicative.com/ur/demo/">the online demo</a>, with further detail available in <a href="http://www.impredicative.com/ur/manual.pdf">the reference manual</a>. *) (* * Basics *) (* Let's start with features shared with both ML and Haskell. First, we have the basic numeric, string, and Boolean stuff. (In the following examples, <tt>==</tt> is used to indicate the result of evaluating an expression. It's not valid Ur syntax!) *) (* begin eval *) 1 + 1 (* end *) (* begin eval *) 1.2 + 3.4 (* end *) (* begin eval *) "Hello " ^ "world!" (* end *) (* begin eval *) 1 + 1 < 6 (* end *) (* begin eval *) 0.0 < -3.2 (* end *) (* begin eval *) "Hello" = "Goodbye" || (1 * 2 <> 8 && True <> False) (* end *) (* We also have function definitions with type inference for parameter and return types. *) fun double n = 2 * n (* begin eval *) double 8 (* end *) fun fact n = if n = 0 then 1 else n * fact (n - 1) (* begin eval *) fact 5 (* end *) fun isEven n = n = 0 || (n > 1 && isOdd (n - 1)) and isOdd n = n = 1 || (n > 1 && isEven (n - 1)) (* begin eval *) isEven 32 (* end *) (* begin eval *) isEven 31 (* end *) (* Of course we have anonymous functions, too. *) val inc = fn x => x + 1 (* begin eval *) inc 3 (* end *) (* Then there's parametric polymorphism. Unlike in ML and Haskell, polymorphic functions in Ur/Web often require full type annotations. That is because more advanced features (which we'll get to in the next chapter) make Ur type inference undecidable. *) fun id [a] (x : a) : a = x (* begin eval *) id "hi" (* end *) fun compose [a] [b] [c] (f : b -> c) (g : a -> b) (x : a) : c = f (g x) (* begin eval *) compose inc inc 3 (* end *) (* The <tt>option</tt> type family is like ML's <tt>option</tt> or Haskell's <tt>Maybe</tt>. We also have a <tt>case</tt> expression form lifted directly from ML. Note that, while Ur follows most syntactic conventions of ML, one key difference is that type family names appear before their arguments, as in Haskell. *) fun predecessor (n : int) : option int = if n >= 1 then Some (n - 1) else None (* begin hide *) fun show_option [t] (_ : show t) : show (option t) = mkShow (fn x => case x of None => "None" | Some x => "Some(" ^ show x ^ ")") (* end *) (* begin eval *) predecessor 6 (* end *) (* begin eval *) predecessor 0 (* end *) (* Naturally, there are lists, too! *) val numbers : list int = 1 :: 2 :: 3 :: [] val strings : list string = "a" :: "bc" :: [] fun length [a] (ls : list a) : int = case ls of [] => 0 | _ :: ls' => 1 + length ls' (* begin eval *) length numbers (* end *) (* begin eval *) length strings (* end *) (* And lists make a good setting for demonstrating higher-order functions and local functions. (This example also introduces one idiosyncrasy of Ur, which is that <tt>map</tt> is a keyword, so we name our "map" function <tt>mp</tt>.) *) (* begin hide *) fun show_list [t] (_ : show t) : show (list t) = mkShow (let fun shower (ls : list t) = case ls of [] => "[]" | x :: ls' => show x ^ " :: " ^ shower ls' in shower end) (* end *) fun mp [a] [b] (f : a -> b) : list a -> list b = let fun loop (ls : list a) = case ls of [] => [] | x :: ls' => f x :: loop ls' in loop end (* begin eval *) mp inc numbers (* end *) (* begin eval *) mp (fn s => s ^ "!") strings (* end *) (* We can define our own polymorphic datatypes and write higher-order functions over them. *) datatype tree a = Leaf of a | Node of tree a * tree a (* begin hide *) fun show_tree [t] (_ : show t) : show (tree t) = mkShow (let fun shower (t : tree t) = case t of Leaf x => "Leaf(" ^ show x ^ ")" | Node (t1, t2) => "Node(" ^ shower t1 ^ ", " ^ shower t2 ^ ")" in shower end) (* end *) fun size [a] (t : tree a) : int = case t of Leaf _ => 1 | Node (t1, t2) => size t1 + size t2 (* begin eval *) size (Node (Leaf 0, Leaf 1)) (* end *) (* begin eval *) size (Node (Leaf 1.2, Node (Leaf 3.4, Leaf 4.5))) (* end *) fun tmap [a] [b] (f : a -> b) : tree a -> tree b = let fun loop (t : tree a) : tree b = case t of Leaf x => Leaf (f x) | Node (t1, t2) => Node (loop t1, loop t2) in loop end (* begin eval *) tmap inc (Node (Leaf 0, Leaf 1)) (* end *) (* We also have anonymous record types, as in Standard ML. The next chapter will show that there is quite a lot more going on here with records than in SML or OCaml, but we'll stick to the basics in this chapter. We will add one tantalizing hint of what's to come by demonstrating the record concatention operator <tt>++</tt> and the record field removal operator <tt>--</tt>. *) val x = { A = 0, B = 1.2, C = "hi", D = True } (* begin eval *) x.A (* end *) (* begin eval *) x.C (* end *) type myRecord = { A : int, B : float, C : string, D : bool } fun getA (r : myRecord) = r.A (* begin eval *) getA x (* end *) (* begin eval *) getA (x -- #A ++ {A = 4}) (* end *) val y = { A = "uhoh", B = 2.3, C = "bye", D = False } (* begin eval *) getA (y -- #A ++ {A = 5}) (* end *) (* * Borrowed from ML *) (* Ur includes an ML-style <b>module system</b>. The most basic use case involves packaging abstract types with their "methods." *) signature COUNTER = sig type t val zero : t val increment : t -> t val toInt : t -> int end structure Counter : COUNTER = struct type t = int val zero = 0 val increment = plus 1 fun toInt x = x end (* begin eval *) Counter.toInt (Counter.increment Counter.zero) (* end *) (* We may package not just abstract types, but also abstract type families. Here we see our first use of the <tt>con</tt> keyword, which stands for <b>constructor</b>. Constructors are a generalization of types to include other "compile-time things"; for instance, basic type families, which are assigned the <b>kind</b> <tt>Type -> Type</tt>. Kinds are to constructors as types are to normal values. We also see how to write the type of a polymorphic function, using the <tt>:::</tt> syntax for type variable binding. This <tt>:::</tt> differs from the <tt>::</tt> used with the <tt>con</tt> keyword because it marks a type parameter as implicit, so that it need not be supplied explicitly at call sites. Such an option is the only one available in ML and Haskell, but, in the next chapter, we'll meet cases where it is appropriate to use explicit constructor parameters. *) signature STACK = sig con t :: Type -> Type val empty : a ::: Type -> t a val push : a ::: Type -> t a -> a -> t a val peek : a ::: Type -> t a -> option a val pop : a ::: Type -> t a -> option (t a) end structure Stack : STACK = struct con t = list val empty [a] = [] fun push [a] (t : t a) (x : a) = x :: t fun peek [a] (t : t a) = case t of [] => None | x :: _ => Some x fun pop [a] (t : t a) = case t of [] => None | _ :: t' => Some t' end (* begin eval *) Stack.peek (Stack.push (Stack.push Stack.empty "A") "B") (* end *) (* Ur also inherits the ML concept of <b>functors</b>, which are functions from modules to modules. *) datatype order = Less | Equal | Greater signature COMPARABLE = sig type t val compare : t -> t -> order end signature DICTIONARY = sig type key con t :: Type -> Type val empty : a ::: Type -> t a val insert : a ::: Type -> t a -> key -> a -> t a val lookup : a ::: Type -> t a -> key -> option a end functor BinarySearchTree(M : COMPARABLE) : DICTIONARY where type key = M.t = struct type key = M.t datatype t a = Leaf | Node of t a * key * a * t a val empty [a] = Leaf fun insert [a] (t : t a) (k : key) (v : a) : t a = case t of Leaf => Node (Leaf, k, v, Leaf) | Node (left, k', v', right) => case M.compare k k' of Equal => Node (left, k, v, right) | Less => Node (insert left k v, k', v', right) | Greater => Node (left, k', v', insert right k v) fun lookup [a] (t : t a) (k : key) : option a = case t of Leaf => None | Node (left, k', v, right) => case M.compare k k' of Equal => Some v | Less => lookup left k | Greater => lookup right k end structure IntTree = BinarySearchTree(struct type t = int fun compare n1 n2 = if n1 = n2 then Equal else if n1 < n2 then Less else Greater end) (* begin eval *) IntTree.lookup (IntTree.insert (IntTree.insert IntTree.empty 0 "A") 1 "B") 1 (* end *) (* It is sometimes handy to rebind modules to shorter names. *) structure IT = IntTree (* begin eval *) IT.lookup (IT.insert (IT.insert IT.empty 0 "A") 1 "B") 0 (* end *) (* One can even use the <tt>open</tt> command to import a module's namespace wholesale, though this can make it harder for someone reading code to tell which identifiers come from which modules. *) open IT (* begin eval *) lookup (insert (insert empty 0 "A") 1 "B") 2 (* end *) (* Ur adopts OCaml's approach to splitting projects across source files. When a project contains files <tt>foo.ur</tt> and <tt>foo.urs</tt>, these are taken as defining a module named <tt>Foo</tt> whose signature is drawn from <tt>foo.urs</tt> and whose implementation is drawn from <tt>foo.ur</tt>. If <tt>foo.ur</tt> exists without <tt>foo.urs</tt>, then module <tt>Foo</tt> is defined without an explicit signature, so that it is assigned its <b>principal signature</b>, which exposes all typing details without abstraction. *) (* * Borrowed from Haskell *) (* Ur includes a take on <b>type classes</b>. For instance, here is a generic "max" function that relies on a type class <tt>ord</tt>. Notice that the type class membership witness is treated like an ordinary function parameter, though we don't assign it a name here, because type inference figures out where it should be used. The more advanced examples of the next chapter will include cases where we manipulate type class witnesses explicitly. *) fun max [a] (_ : ord a) (x : a) (y : a) : a = if x < y then y else x (* begin eval *) max 1 2 (* end *) (* begin eval *) max "ABC" "ABA" (* end *) (* The idiomatic way to define a new type class is to stash it inside a module, like in this example: *) signature DOUBLE = sig class double val double : a ::: Type -> double a -> a -> a val mkDouble : a ::: Type -> (a -> a) -> double a val double_int : double int val double_string : double string end structure Double : DOUBLE = struct con double a = a -> a fun double [a] (f : double a) (x : a) : a = f x fun mkDouble [a] (f : a -> a) : double a = f val double_int = mkDouble (times 2) val double_string = mkDouble (fn s => s ^ s) end open Double (* begin eval *) double 13 (* end *) (* begin eval *) double "ho" (* end *) val double_float = mkDouble (times 2.0) (* begin eval *) double 2.3 (* end *) (* That example had a mix of instances defined with a class and instances defined outside its module. Its possible to create <b>closed type classes</b> simply by omitting from the module an instance creation function like <tt>mkDouble</tt>. This way, only the instances you decide on may be allowed, which enables you to enforce program-wide invariants over instances. *) signature OK_TYPE = sig class ok val importantOperation : a ::: Type -> ok a -> a -> string val ok_int : ok int val ok_float : ok float end structure OkType : OK_TYPE = struct con ok a = unit fun importantOperation [a] (_ : ok a) (_ : a) = "You found an OK value!" val ok_int = () val ok_float = () end open OkType (* begin eval *) importantOperation 13 (* end *) (* Like Haskell, Ur supports the more general notion of <b>constructor classes</b>, whose instances may be parameterized over constructors with kinds beside <tt>Type</tt>. Also like in Haskell, the flagship constructor class is <tt>monad</tt>. Ur/Web's counterpart of Haskell's <tt>IO</tt> monad is <tt>transaction</tt>, which indicates the tight coupling with transactional execution in server-side code. Just as in Haskell, <tt>transaction</tt> must be used to create side-effecting actions, since Ur is purely functional (but has eager evaluation). Here is a quick example transaction, showcasing Ur's variation on Haskell <tt>do</tt> notation. *) val readBack : transaction int = src <- source 0; set src 1; n <- get src; return (n + 1) (* We get ahead of ourselves a bit here, as this example uses functions associated with client-side code to create and manipulate a mutable data source. *)