Nice. But the implementation has some subtleties. I use OCaml. Fist, `fix f` must explicitly be defined as a function. If we replace the expression `fun n -> f

Nice. But the implementation has some subtleties. I use OCaml. Fist, `fix f` must explicitly be defined as a function. If we replace the expression `fun n -> f (fix f) n` by `f (fix f)` then we got "Stack overflow during evaluation" of fix(f). let rec fix f = fun n -> f (fix f) n Then fix can be applied to some generator : let fibgen(fib) = fun n -> if n = 0 then 0 else if n = 1 then 1 else fib(n-1) + fib(n-2) let fib = fix(fibgen) assert (fib 6 = 8) The compose and trace functions pose no issue. let (>>) f g x = f(g(x)) let trace f_name f = fun n -> Format.printf "%s(%d)\n%!" f_name n; f n let trace_fib = fix (trace "fib" >> fibgen) But the memoize function cannot simply be defined as `memoize f = fun n -> ...`, otherwise a fresh cache will be used for each call ! Indeed, `fix f` is called for each recursive call in the generator. In the case of `fix (memoize >> fibgen)` memoize is recursively called with a new function to be cached. Hence, the use of an extra step to explicitly create once the cache. let memoize size = let cache = Hashtbl.create size in fun f n -> try Hashtbl.find cache n with Not_found -> ( let r = f n in Hashtbl.add cache n r; r ) let memoize_fib = fix (memoize 16 >> trace "fib" >> fibgen) Last subtilty: the order of application of the misc transformers. When we write `memoize 16 >> trace "fib" >> fibgen`, the resulting function `f` adds , in order, memoization, trace and fib recursive computation to itself in disguise `fix f`. The latter applied transformation takes the control. Here the fib computation only calls the chain `fix f` if n is greater than 1. So, a call to `trace_fib(0)` leads to no trace; and a call to `memoize_fib(6)` caches the results for 4 and 5 but not for 6. Hence, to memoize all calls and trace only actual call to fib, we must reorder the transformations: let memoize_fib = fix (fibgen >> trace "fib" >> memoize 16)