(* Chapter 2: Type-Level Computation *)

(* 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. *)

(* * Names and Records *)

(* Last chapter, we met Ur's basic record features, including record construction and field projection. *)

val r = { A = 0, B = 1.2, C = "hi"}

(* begin eval *)
r.B
(* end *)

(* Our first taste of Ur's novel expressive power is with the following function, which implements record field projection in a completely generic way. *)

fun project [nm :: Name] [t ::: Type] [ts ::: {Type}] [[nm] ~ ts] (r : $([nm = t] ++ ts)) : t = r.nm

(* begin eval *)
project [#B] r
(* end *)

(* 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>
<br>
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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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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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>
<br>
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>). *)