{ ... }:
rec {
  # Compute the fixed point of the given function `f`, which is usually an
  # attribute set that expects its final, non-recursive representation as an
  # argument:
  #
  #     f = self: { foo = "foo"; bar = "bar"; foobar = self.foo + self.bar; }
  #
  # Nix evaluates this recursion until all references to `self` have been
  # resolved. At that point, the final result is returned and `f x = x` holds:
  #
  #     nix-repl> fix f
  #     { bar = "bar"; foo = "foo"; foobar = "foobar"; }
  #
  #  Type: fix :: (a -> a) -> a
  #
  # See https://en.wikipedia.org/wiki/Fixed-point_combinator for further
  # details.
  fix = f: let x = f x; in x;

  # A variant of `fix` that records the original recursive attribute set in the
  # result. This is useful in combination with the `extends` function to
  # implement deep overriding. See pkgs/development/haskell-modules/default.nix
  # for a concrete example.
  fix' = f: let x = f x // { __unfix__ = f; }; in x;

  # Return the fixpoint that `f` converges to when called recursively, starting
  # with the input `x`.
  #
  #     nix-repl> converge (x: x / 2) 16
  #     0
  converge = f: x:
    let
      x' = f x;
    in
      if x' == x
      then x
      else converge f x';

  # Modify the contents of an explicitly recursive attribute set in a way that
  # honors `self`-references. This is accomplished with a function
  #
  #     g = self: super: { foo = super.foo + " + "; }
  #
  # that has access to the unmodified input (`super`) as well as the final
  # non-recursive representation of the attribute set (`self`). `extends`
  # differs from the native `//` operator insofar as that it's applied *before*
  # references to `self` are resolved:
  #
  #     nix-repl> fix (extends g f)
  #     { bar = "bar"; foo = "foo + "; foobar = "foo + bar"; }
  #
  # The name of the function is inspired by object-oriented inheritance, i.e.
  # think of it as an infix operator `g extends f` that mimics the syntax from
  # Java. It may seem counter-intuitive to have the "base class" as the second
  # argument, but it's nice this way if several uses of `extends` are cascaded.
  #
  # To get a better understanding how `extends` turns a function with a fix
  # point (the package set we start with) into a new function with a different fix
  # point (the desired packages set) lets just see, how `extends g f`
  # unfolds with `g` and `f` defined above:
  #
  # extends g f = self: let super = f self; in super // g self super;
  #             = self: let super = { foo = "foo"; bar = "bar"; foobar = self.foo + self.bar; }; in super // g self super
  #             = self: { foo = "foo"; bar = "bar"; foobar = self.foo + self.bar; } // g self { foo = "foo"; bar = "bar"; foobar = self.foo + self.bar; }
  #             = self: { foo = "foo"; bar = "bar"; foobar = self.foo + self.bar; } // { foo = "foo" + " + "; }
  #             = self: { foo = "foo + "; bar = "bar"; foobar = self.foo + self.bar; }
  #
  extends = f: rattrs: self: let super = rattrs self; in super // f self super;

  # Compose two extending functions of the type expected by 'extends'
  # into one where changes made in the first are available in the
  # 'super' of the second
  composeExtensions =
    f: g: self: super:
      let fApplied = f self super;
          super' = super // fApplied;
      in fApplied // g self super';

  # Create an overridable, recursive attribute set. For example:
  #
  #     nix-repl> obj = makeExtensible (self: { })
  #
  #     nix-repl> obj
  #     { __unfix__ = «lambda»; extend = «lambda»; }
  #
  #     nix-repl> obj = obj.extend (self: super: { foo = "foo"; })
  #
  #     nix-repl> obj
  #     { __unfix__ = «lambda»; extend = «lambda»; foo = "foo"; }
  #
  #     nix-repl> obj = obj.extend (self: super: { foo = super.foo + " + "; bar = "bar"; foobar = self.foo + self.bar; })
  #
  #     nix-repl> obj
  #     { __unfix__ = «lambda»; bar = "bar"; extend = «lambda»; foo = "foo + "; foobar = "foo + bar"; }
  makeExtensible = makeExtensibleWithCustomName "extend";

  # Same as `makeExtensible` but the name of the extending attribute is
  # customized.
  makeExtensibleWithCustomName = extenderName: rattrs:
    fix' rattrs // {
      ${extenderName} = f: makeExtensibleWithCustomName extenderName (extends f rattrs);
   };
}