dhil

Operations are structural typed in Links, therefore operation names are not declared before use.
In OCaml operations are nomial. This difference is not really an issue as we can simply collect all operations from the Links IR and declare them globally in OCaml.
However, Links' presence polymorphism introduces a small problem as operation signatures can change throughout a computation!
As an example of this consider the following:

sig fooInt : (() {Foo:(Int) {}-> Int|e}~> Int) -> 
              () {Foo{_}            |e}~> Int 
fun fooInt(m)() { 
  open handle(m) { 
    case Foo(42,k) -> k(1) 

dhil

make_cache=false
use_cache=false
cache_directory=$LINKSHOME/cache
prelude=$LINKSHOME/prelude.links
jslibdir=$LINKSHOME/lib/js
jsliburl=/lib/
database_args=:links:
database_driver=postgresql
port=10088
js_pretty_print=true

dhil

// Typing the Y combinator in Dart.
class Fix<A> {
  A Function(Fix<A>) f;
  Fix.inject(this.f);

  A Function(Fix<A>) get project => f;
}

// fix : (Fix<A> -> a) -> Fix<A>.
Fix<A> fix<A>(A Function(Fix<A>) f) => Fix.inject(f);

dhil

# Open files in Windows from within WSL.
function _windows_open()
{
    # Absolute, inside WSL, path to the file.
    local file=$(readlink -e "$1")
    if [[ ! (-f "$file" || -d "$file") ]]; then
        echo "error: no such file '$1'"
        return 1
    else
        # Convert slashes into backslashes.

dhil

// Typing the Y combinator in Java (tested with OpenJDK 11.0.4).
// $ javac Y.java
// $ java Y
// 720
// Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))

import java.util.function.Function;

// A typed implementation of the Y-combinator.
public class Y {

dhil

# Typing the Y combinator in Links.
# links> examples();
# 720
# Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))
# () : ()

sig y : (((a) ~e~> b) ~e~> (a) ~e~> b) -> (a) ~e~> b
fun y(f) {
  fun g(x : (mu f . (((f) ~e~> (a) ~e~> b))))(a) {
    f(x(x))(a)

dhil

(* Typing the Y combinator in OCaml.
 * $ ocaml y.ml
   720
   Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))
 *)

type 'a fix = Fix of ('a fix -> 'a)

let fix x = Fix x
let unfix (Fix x) = x

dhil

{- Typing the Y combinator in Haskell.
 - $ runghc Y.hs
   720
   Succ (Succ (Succ (Succ (Succ (Succ Zero)))))
 -}

newtype Fix a = Fix { unfix :: Fix a -> a }

fix :: (Fix a -> a) -> Fix a
fix x = Fix x

dhil

(* Typing the Y combinator in SML/NJ.
 * $ sml y.sml
   720
   Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))
*)

datatype 'a fix = Fix of ('a fix -> 'a)

fun fix f = Fix f
fun unfix (Fix f) = f

dhil

// Typing the Y combinator in Scala 2.
// $ scalac Y.scala
// $ scala Y
// 720
// Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))

object Y {

  case class Fix[A,B](f : Fix[A,B] => A => B) {
    def apply(x : Fix[A,B]) = f(x)