/// For a function, f, define fix(f) as the "fixed point" of f:
/// A value, z, such that f(z) = z.
///
/// Substituting fix(f) for z, gives f(fix(f)) = fix(f), which
/// we flip to fix(f) = f(fix(f)).
///
/// This fixed point, z, is itself a function that takes an
/// argument, x. We have to make x explicit in the definition
/// in order to avoid infinite recursion when fix is called.
///
/// (In a typed language like F#, fix has to be defined recursively.
/// However, in an untyped system, such as the pure lambda calculus,
/// fix can be defined without recursion. That version of fix is
/// called the "Y" combinator.)
///
/// https://en.wikipedia.org/wiki/Fixed-point_combinator
let rec fix f =
let z x = (f (fix f)) x
z
/// As an example, we create a factorial "generator", which, when
/// given a function that implements factorial correctly for inputs
/// up to some number, n-1, answers a slightly better function that
/// implements factorial correctly for n as well. Note that this
/// function is not recursive.
let factorialGen (factorialWeak : int -> int) : (int -> int) =
fun n ->
if n = 0 then 1
else n * factorialWeak (n - 1)
/// We could feed such a generated function back into the generator
/// to produce an even better version that works for all numbers
/// up to n + 1. Instead, applying fix to the generator itself
/// returns the generator's fixed point, as though we had iterated
/// the feedback infinitely. The result is a function that implements
/// factorial correctly for *all* inputs n.
let factorial = fix factorialGen
/// Another example: Fibonacci numbers. Again, fibGen itself isn't
/// recursive. It only behaves recursively once we apply fix.
let fib =
let fibGen fibWeak =
fun n ->
if n <= 1 then 1
else fibWeak (n - 1) + fibWeak (n - 2)
fix fibGen
[]
let main argv =
printfn ""
for i = 0 to 10 do
printfn "%A!: %A" i (factorial i)
printfn ""
for i = 0 to 10 do
printfn "fib(%A): %A" i (fib i)
0