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Tutorial: up to First-Class Polymorphism
author Adam Chlipala <adam@chlipala.net>
date Sun, 17 Jul 2011 11:00:04 -0400
parents 71fdaef3b5dd
children 44fda91f5fa0
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1504:71fdaef3b5dd 1505:8c851e5508a7
1 (* Chapter 2: Type-Level Computation *) 1 (* Chapter 2: Type-Level Computation *)
2
3 (* begin hide *)
4 val show_string = mkShow (fn s => "\"" ^ s ^ "\"")
5 (* end *)
2 6
3 (* 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>. *) 7 (* 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>. *)
4 8
5 (* The last chapter reviewed some Ur features imported from ML and Haskell. This chapter explores uncharted territory, introducing the features that make Ur unique. *) 9 (* The last chapter reviewed some Ur features imported from ML and Haskell. This chapter explores uncharted territory, introducing the features that make Ur unique. *)
6 10
31 The type annotation on <tt>r</tt> uses the record concatenation operator <tt>++</tt>. Ur enforces that any concatenation happens between records that share no field names. Otherwise, we'd need to resolve field name ambiguity in some predictable way, which would force us to treat <tt>++</tt> as non-commutative, if we are to maintain the nice modularity properties of polymorphism. However, treating <tt>++</tt> as commutative, and treating records as equal up to field permutation in general, are very convenient for type inference and general programmer experience. Thus, we enforce disjointness to keep things simple.<br> 35 The type annotation on <tt>r</tt> uses the record concatenation operator <tt>++</tt>. Ur enforces that any concatenation happens between records that share no field names. Otherwise, we'd need to resolve field name ambiguity in some predictable way, which would force us to treat <tt>++</tt> as non-commutative, if we are to maintain the nice modularity properties of polymorphism. However, treating <tt>++</tt> as commutative, and treating records as equal up to field permutation in general, are very convenient for type inference and general programmer experience. Thus, we enforce disjointness to keep things simple.<br>
32 <br> 36 <br>
33 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> 37 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>
34 <br> 38 <br>
35 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>). *) 39 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>). *)
40
41
42 (* * Basic Type-Level Programming *)
43
44 (* 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. *)
45
46 con id = fn t :: Type => t
47
48 val x : id int = 0
49 val x : id float = 1.2
50
51 con pair = fn t :: Type => t * t
52
53 val x : pair int = (0, 1)
54 val x : pair float = (1.2, 2.3)
55
56 con compose = fn (f :: Type -> Type) (g :: Type -> Type) (t :: Type) => f (g t)
57
58 val x : compose pair pair int = ((0, 1), (2, 3))
59
60 con fst = fn t :: (Type * Type) => t.1
61 con snd = fn t :: (Type * Type) => t.2
62
63 con p = (int, float)
64 val x : fst p = 0
65 val x : snd p = 1.2
66
67 con mp = fn (f :: Type -> Type) (t1 :: Type, t2 :: Type) => (f t1, f t2)
68
69 val x : fst (mp pair p) = (1, 2)
70
71 (* 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. *)
72
73 con fst = K1 ==> K2 ==> fn t :: (K1 * K2) => t.1
74 con snd = K1 ==> K2 ==> fn t :: (K1 * K2) => t.2
75
76 con twoFuncs :: ((Type -> Type) * (Type -> Type)) = (id, compose pair pair)
77
78 val x : fst twoFuncs int = 0
79 val x : snd twoFuncs int = ((1, 2), (3, 4))
80
81
82 (* * Type-Level Map *)
83
84 (* 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>. *)
85
86 con r = [A = int, B = float, C = string]
87
88 con optionify = map option
89 val x : $(optionify r) = {A = Some 1, B = None, C = Some "hi"}
90
91 con pairify = map pair
92 val x : $(pairify r) = {A = (1, 2), B = (3.0, 4.0), C = ("5", "6")}
93
94 con stringify = map (fn _ => string)
95 val x : $(stringify r) = {A = "1", B = "2", C = "3"}
96
97 (* 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. *)
98
99 fun concat [f :: Type -> Type] [r1 :: {Type}] [r2 :: {Type}] [r1 ~ r2]
100 (r1 : $(map f r1)) (r2 : $(map f r2)) : $(map f (r1 ++ r2)) = r1 ++ r2
101
102
103 (* * First-Class Polymorphism *)
104
105 (* 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. *)
106
107 type nat = t :: Type -> t -> (t -> t) -> t
108 val zero : nat = fn [t :: Type] (z : t) (s : t -> t) => z
109 fun succ (n : nat) : nat = fn [t :: Type] (z : t) (s : t -> t) => s (n [t] z s)
110
111 val one = succ zero
112 val two = succ one
113 val three = succ two
114
115 (* begin eval *)
116 three [int] 0 (plus 1)
117 (* end *)
118
119 (* begin eval *)
120 three [string] "" (strcat "!")
121 (* end *)