Rubrication

The following is some SML code that implements the natural numbers using peano arithmetic in the SML type system. These numbers can be used to protect access to list functions. I haven’t yet figured out if addition is possible, but I’m hoping that it is. It could be really handy!

signature SLIST =
sig
    type 'a nat
    type Z
    type 'a S
    val S : 'a nat -> 'a S nat
    val P : 'a S nat -> 'a nat
    val Z : unit -> Z nat
    val zerop : 'a nat -> bool
    val toInt : 'a nat -> int 

    type ('elt,'n) slist

    val snil : unit -> ('elt, Z nat) slist
    val scons : 'elt -> ('elt, 'n nat) slist -> ('elt, 'n S nat) slist
    val :+: : 'elt * ('elt, 'n nat) slist -> ('elt, 'n S nat) slist
    val shd : ('elt, 'n S nat) slist -> 'elt
    val stl : ('elt, 'n S nat) slist -> ('elt, 'n nat) slist
    val slen : ('elt, 'n S nat) slist -> int

end

structure SList :> SLIST =
struct
    (* encode integer types *)
    type 'a nat = int
    type Z = unit
    type 'a S = unit
    fun S i = i+1;
    fun P i = i-1;
    fun Z () = 0
    fun zerop 0 = true
      | zerop _ = false
    fun toInt d = d

    type ('elt, 'n) slist = 'elt list * 'n

    fun snil () = ([],0)

    fun scons elt sl =
	let val (l,i) = sl
	in ((elt::l),S i)
	end    

    infixr :+:
    fun x :+: y = scons x y

    fun shd sl =
	let val (h::t,i) = sl
	in h
	end

    fun stl sl  =
	let val (h::t,i) = sl
	in (t,P i)
	end

    fun slen sl =
	let val (_,i) = sl
	in i
	end

end 

open SList
infixr :+:

val mylist = 1 :+: 2 :+: 3 :+: snil();
val the_head = shd mylist;
val the_tail = stl mylist;
val the_next_head = shd the_tail;
(*
This doesn't even compile!
val head_of_nil = shd (snil())
*)