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1 (* Chapter 2: Type-Level Computation *)
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2
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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>. *)
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4
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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. *)
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6
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7 (* * Names and Records *)
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8
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9 (* Last chapter, we met Ur's basic record features, including record construction and field projection. *)
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10
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11 val r = { A = 0, B = 1.2, C = "hi"}
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12
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13 (* begin eval *)
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14 r.B
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15 (* end *)
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16
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17 (* Our first taste of Ur's novel expressive power is with the following function, which implements record field projection in a completely generic way. *)
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18
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19 fun project [nm :: Name] [t ::: Type] [ts ::: {Type}] [[nm] ~ ts] (r : $([nm = t] ++ ts)) : t = r.nm
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20
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21 (* begin eval *)
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22 project [#B] r
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23 (* end *)
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24
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25 (* This function introduces a slew of essential features. First, we see type parameters with explicit kind annotations. Formal parameter syntax like <tt>[a :: K]</tt> declares an <b>explicit</b> parameter <tt>a</tt> of kind <tt>K</tt>. Explicit parameters must be passed explicitly at call sites. In contrast, implicit parameters, declared like <tt>[a ::: K]</tt>, are inferred in the usual way.<br>
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26 <br>
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27 Two new kinds appear in this example. We met the basic kind <tt>Type</tt> in a previous example. Here we meet <tt>Name</tt>, the kind of record field names; and <tt>{Type}</tt> the type of finite maps from field names to types, where we'll generally refer to this notion of "finite map" by the name <b>record</b>, as it will be clear from context whether we're discussing type-level or value-level records. That is, in this case, we are referring to names and records <b>at the level of types</b> that <b>exist only at compile time</b>! By the way, the kind <tt>{Type}</tt> is one example of the general <tt>{K}</tt> kind form, which refers to records with fields of kind <tt>K</tt>.<br>
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28 <br>
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29 The English description of <tt>project</tt> is that it projects a field with name <tt>nm</tt> and type <tt>t</tt> out of a record <tt>r</tt> whose other fields are described by type-level record <tt>ts</tt>. We make all this formal by assigning <tt>r</tt> a type that first builds the singleton record <tt>[nm = t]</tt> that maps <tt>nm</tt> to <tt>t</tt>, and then concatenating this record with the remaining field information in <tt>ts</tt>. The <tt>$</tt> operator translates a type-level record (of kind <tt>{Type}</tt>) into a record type (of kind <tt>Type</tt>).<br>
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30 <br>
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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>
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32 <br>
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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>
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34 <br>
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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>). *)
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