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@@ -1,5 +1,9 @@ (* Chapter 2: Type-Level Computation *) +(* 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>. *) (* The last chapter reviewed some Ur features imported from ML and Haskell. This chapter explores uncharted territory, introducing the features that make Ur unique. *) @@ -33,3 +37,85 @@ The type annotation on <tt>r</tt> uses the record concatenation operator <tt>++< For a polymorphic function like <tt>project</tt>, the compiler doesn't know which fields a type-level record variable like <tt>ts</tt> contains. To enable self-contained type-checking, we need to declare some constraints about field disjointness. That's exactly the meaning of syntax like <tt>[r1 ~ r2]</tt>, which asserts disjointness of two type-level records. The disjointness clause for <tt>project</tt> asserts that the name <tt>nm</tt> is not used by <tt>ts</tt>. The syntax <tt>[nm]</tt> is shorthand for <tt>[nm = ()]</tt>, which defines a singleton record of kind <tt>{Unit}</tt>, where <tt>Unit</tt> is the degenerate kind inhabited only by the constructor <tt>()</tt>.<br> <br> The last piece of this puzzle is the easiest. In the example call to <tt>project</tt>, we see that the only parameters passed are the one explicit constructor parameter <tt>nm</tt> and the value-level parameter <tt>r</tt>. The rest are inferred, and the disjointness proof obligation is discharged automatically. The syntax <tt>#A</tt> denotes the constructor standing for first-class field name <tt>A</tt>, and we pass all constructor parameters to value-level functions within square brackets (which bear no formal relation to the syntax for type-level record literals <tt>[A = c, ..., A = c]</tt>). *) + + +(* * Basic Type-Level Programming *) + +(* To help us express more interesting operations over records, we will need to do some type-level programming. Ur makes that fairly congenial, since Ur's constructor level includes an embedded copy of the simply-typed lambda calculus. Here are a few examples. *) + +con id = fn t :: Type => t + +val x : id int = 0 +val x : id float = 1.2 + +con pair = fn t :: Type => t * t + +val x : pair int = (0, 1) +val x : pair float = (1.2, 2.3) + +con compose = fn (f :: Type -> Type) (g :: Type -> Type) (t :: Type) => f (g t) + +val x : compose pair pair int = ((0, 1), (2, 3)) + +con fst = fn t :: (Type * Type) => t.1 +con snd = fn t :: (Type * Type) => t.2 + +con p = (int, float) +val x : fst p = 0 +val x : snd p = 1.2 + +con mp = fn (f :: Type -> Type) (t1 :: Type, t2 :: Type) => (f t1, f t2) + +val x : fst (mp pair p) = (1, 2) + +(* Actually, Ur's constructor level goes further than merely including a copy of the simply-typed lambda calculus with tuples. We also effectively import classic <b>let-polymorphism</b>, via <b>kind polymorphism</b>, which we can use to make some of the definitions above more generic. *) + +con fst = K1 ==> K2 ==> fn t :: (K1 * K2) => t.1 +con snd = K1 ==> K2 ==> fn t :: (K1 * K2) => t.2 + +con twoFuncs :: ((Type -> Type) * (Type -> Type)) = (id, compose pair pair) + +val x : fst twoFuncs int = 0 +val x : snd twoFuncs int = ((1, 2), (3, 4)) + + +(* * Type-Level Map *) + +(* The examples from the same section may seem cute but not especially useful. In this section, we meet <tt>map</tt>, the real workhorse of Ur's type-level computation. We'll use it to type some useful operations over value-level records. A few more pieces will be necessary before getting there, so we'll start just by showing how interesting type-level operations on records may be built from <tt>map</tt>. *) + +con r = [A = int, B = float, C = string] + +con optionify = map option +val x : $(optionify r) = {A = Some 1, B = None, C = Some "hi"} + +con pairify = map pair +val x : $(pairify r) = {A = (1, 2), B = (3.0, 4.0), C = ("5", "6")} + +con stringify = map (fn _ => string) +val x : $(stringify r) = {A = "1", B = "2", C = "3"} + +(* We'll also give our first hint at the cleverness within Ur's type inference engine. The following definition type-checks, despite the fact that doing so requires applying several algebraic identities about <tt>map</tt> and <tt>++</tt>. This is the first point where we see a clear advantage of Ur over the type-level computation facilities that have become popular in GHC Haskell. *) + +fun concat [f :: Type -> Type] [r1 :: {Type}] [r2 :: {Type}] [r1 ~ r2] + (r1 : $(map f r1)) (r2 : $(map f r2)) : $(map f (r1 ++ r2)) = r1 ++ r2 + + +(* * First-Class Polymorphism *) + +(* The idea of <b>first-class polymorphism</b> or <b>impredicative polymorphism</b> has also become popular in GHC Haskell. This feature, which has a long history in type theory, is also central to Ur's metaprogramming facilities. First-class polymorphism goes beyond Hindley-Milner's let-polymorphism to allow arguments to functions to themselves be polymorphic. Among other things, this enables the classic example of Church encodings, as for the natural numbers in this example. *) + +type nat = t :: Type -> t -> (t -> t) -> t +val zero : nat = fn [t :: Type] (z : t) (s : t -> t) => z +fun succ (n : nat) : nat = fn [t :: Type] (z : t) (s : t -> t) => s (n [t] z s) + +val one = succ zero +val two = succ one +val three = succ two + +(* begin eval *) +three [int] 0 (plus 1) +(* end *) + +(* begin eval *) +three [string] "" (strcat "!") +(* end *) |