define $.Function and $.UnaryTypeVariable

Author: davidchambers

README.md

@@ -23,7 +23,7 @@ const Integer = ...;
 //    NonZeroInteger :: Type
 const NonZeroInteger = ...;
 
-//    env :: [Type]
+//    env :: Array Type
 const env = $.env.concat([Integer, NonZeroInteger]);
 ```
 
@@ -165,6 +165,14 @@ $.Any :: Type
 
 Type comprising every JavaScript value.
 
+#### `AnyFunction`
+
+```haskell
+$.AnyFunction :: Type
+```
+
+Type comprising every Function value.
+
 #### `Arguments`
 
 ```haskell
@@ -218,10 +226,15 @@ Type comprising every [`ValidNumber`](#validnumber) value except `Infinity` and
 #### `Function`
 
 ```haskell
-$.Function :: Type
+$.Function :: [Type] -> Type
 ```
 
-Type comprising every Function value.
+Constructor for Function types.
+
+Examples:
+
+  - `$.Function([$.Date, $.String])` represents the `Date -> String` type; and
+  - `$.Function([a, b, a])` represents the `(a, b) -> a` type.
 
 #### `Integer`
 
@@ -474,63 +487,6 @@ bodies of incoming POST requests against these types.
 
 sanctuary-def provides several functions for defining types.
 
-#### `TypeVariable`
-
-Polymorphism is powerful. Not being able to define a function for all types
-would be very limiting indeed: one couldn't even define the identity function!
-
-```haskell
-TypeVariable :: String -> Type
-```
-
-Before defining a polymorphic function one must define one or more type
-variables:
-
-```javascript
-const a = $.TypeVariable('a');
-const b = $.TypeVariable('b');
-
-//    id :: a -> a
-const id = def('id', {}, [a, a], x => x);
-
-id(42);
-// => 42
-
-id(null);
-// => null
-```
-
-The same type variable may be used in multiple positions, creating a
-constraint:
-
-```javascript
-//    cmp :: a -> a -> Number
-const cmp =
-def('cmp', {}, [a, a, $.Number], (x, y) => x < y ? -1 : x > y ? 1 : 0);
-
-cmp(42, 42);
-// => 0
-
-cmp('a', 'z');
-// => -1
-
-cmp('z', 'a');
-// => 1
-
-cmp(0, '1');
-// ! TypeError: Type-variable constraint violation
-//
-//   cmp :: a -> a -> Number
-//          ^    ^
-//          1    2
-//
-//   1)  0 :: Number
-//
-//   2)  "1" :: String
-//
-//   Since there is no type of which all the above values are members, the type-variable constraint has been violated.
-```
-
 #### `NullaryType`
 
 `NullaryType` is used to construct types with no type variables. `$.Number` is
@@ -602,7 +558,7 @@ rem(42, 0);
 defined via `UnaryType`.
 
 ```javascript
-//    sum :: [Number] -> Number
+//    sum :: Array Number -> Number
 const sum =
 def('sum', {}, [$.Array($.Number), $.Number], xs => xs.reduce((x, y) => x + y, 0));
 
@@ -629,9 +585,10 @@ To define a unary type `t a` one must provide:
     if (and only if) the value is a member of `t x` for some type `x`;
 
   - a function which takes any value of type `t a` and returns an array
-    of the values of type `a` contained in the `t` (exposed as `t._1`); and
+    of the values of type `a` contained in the `t` (exposed as
+    `t.types.$1.extractor`); and
 
-  - the type of `a` (exposed as `t.$1`).
+  - the type of `a` (exposed as `t.types.$1.type`).
 
 ```haskell
 UnaryType :: String -> (Any -> Boolean) -> (t a -> [a]) -> Type -> Type
@@ -701,14 +658,16 @@ To define a binary type `t a b` one must provide:
     `x` and `y`;
 
   - a function which takes any value of type `t a b` and returns an array
-    of the values of type `a` contained in the `t` (exposed as `t._1`);
+    of the values of type `a` contained in the `t` (exposed as
+    `t.types.$1.extractor`);
 
   - a function which takes any value of type `t a b` and returns an array
-    of the values of type `b` contained in the `t` (exposed as `t._2`);
+    of the values of type `b` contained in the `t` (exposed as
+    `t.types.$2.extractor`);
 
-  - the type of `a` (exposed as `t.$1`); and
+  - the type of `a` (exposed as `t.types.$1.type`); and
 
-  - the type of `b` (exposed as `t.$2`).
+  - the type of `b` (exposed as `t.types.$2.type`).
 
 ```haskell
 BinaryType :: String -> (Any -> Boolean) -> (t a b -> [a]) -> (t a b -> [b]) -> Type -> Type -> Type
@@ -870,6 +829,115 @@ dist(0);
 //   The value at position 1 is not a member of ‘{ x :: FiniteNumber, y :: FiniteNumber }’.
 ```
 
+#### `TypeVariable`
+
+Polymorphism is powerful. Not being able to define a function for all types
+would be very limiting indeed: one couldn't even define the identity function!
+
+```haskell
+TypeVariable :: String -> Type
+```
+
+Before defining a polymorphic function one must define one or more type
+variables:
+
+```javascript
+const a = $.TypeVariable('a');
+const b = $.TypeVariable('b');
+
+//    id :: a -> a
+const id = def('id', {}, [a, a], x => x);
+
+id(42);
+// => 42
+
+id(null);
+// => null
+```
+
+The same type variable may be used in multiple positions, creating a
+constraint:
+
+```javascript
+//    cmp :: a -> a -> Number
+const cmp =
+def('cmp', {}, [a, a, $.Number], (x, y) => x < y ? -1 : x > y ? 1 : 0);
+
+cmp(42, 42);
+// => 0
+
+cmp('a', 'z');
+// => -1
+
+cmp('z', 'a');
+// => 1
+
+cmp(0, '1');
+// ! TypeError: Type-variable constraint violation
+//
+//   cmp :: a -> a -> Number
+//          ^    ^
+//          1    2
+//
+//   1)  0 :: Number
+//
+//   2)  "1" :: String
+//
+//   Since there is no type of which all the above values are members, the type-variable constraint has been violated.
+```
+
+#### `UnaryTypeVariable`
+
+As its name suggests, `UnaryTypeVariable` combines [`UnaryType`](#unarytype)
+and [`TypeVariable`](#typevariable).
+
+To define a unary type variable `t a` one must provide:
+
+  - a name (conventionally matching `^[a-z]$`); and
+
+  - the type of `a` (exposed as `t.types.$1.type`).
+
+```haskell
+UnaryTypeVariable :: String -> Type -> Type
+```
+
+Consider the type of a generalized `map`:
+
+```haskell
+map :: Functor f => (a -> b) -> f a -> f b
+```
+
+`f` is a unary type variable. With two (regular) type variables, one unary
+type variable, and one [type class](#type-classes) it's possible to define
+a fully polymorphic `map` function:
+
+```javascript
+const a = $.TypeVariable('a');
+const b = $.TypeVariable('b');
+const f = $.UnaryTypeVariable('f');
+
+//    Functor :: TypeClass
+const Functor = ...;
+
+//    map :: Functor f => (a -> b) -> f a -> f b
+const map =
+def('map',
+    {f: [Functor]},
+    [$.Function([a, b]), f(a), f(b)],
+    (fn, functor) => functor.map(fn));
+```
+
+Whereas a regular type variable is fully resolved (`a` might become
+`Array (Array String)`, for example), a unary type variable defers to
+its type argument, which may itself be a type variable. The type argument
+corresponds to the type argument of a unary type or the *second* type
+argument of a binary type. The second type argument of `Map k v`, for
+example, is `v`. One could replace `Functor => f` with `Map k` or with
+`Map Integer`, but not with `Map`.
+
+This shallow inspection makes it possible to constrain a value's "outer"
+and "inner" types independently.
+
 ### Type classes
 
 `concatS`, defined earlier, is a function which concatenates two strings.