thon is a small programming language. Here’s an example program that verifies the singleton list is not empty.

data list = nil unit | cons nat * list in
fun isempty : list -> nat =
  \ natlist : list ->
  case exposelist natlist of
      empty -> S Z
    | not -> Z
in (isempty (cons (Z, (nil unit))))

thon has natural numbers, functions, recursion, binary product and sum types, polymorphism, existential packages (a formalization of interfaces), recursive types, and binary algebraic datatypes.

natural numbers

Z is the natural number 0. S Z is 1 (the succesor of zero). S S Z is 2, and so on.


In thon, functions are expressions just like numbers are. thon supports anonymous functions and named, recursive functions.

Here are some example anonymous functions.

\ x : nat -> x
\ x : nat -> (\ y : nat -> y)

Functions are applied to their arguments by juxtaposition.

((\ x : nat -> x) Z)

Here’s a divide-by-two function:

fun divbytwo : nat -> nat =
  \ n : nat ->
    ifz n of
      Z -> Z
    | S p -> ifz p of Z -> Z | S p' -> (S (divbytwo p'))
in divbytwo (S S S S Z)

If the number is zero, we’re done. Otherwise, it has some predecessor number p. If p is zero, then return zero (taking the floor). Otherwise, recurse on the predecessor of the predecessor n-2 and add one to whatever that gave us.

Under the hood, recursive functions are implemented as a fixed point expression that substitutes itself in for itself. It’s like a recursive function, but it doesn’t have to be a function, it can be any expression. Here’s an amusing way to loop forever:

fix loop : nat in loop


let x : nat = Z in x

binds the name x in the expression following the in keyword.


Polymorphism lets us reuse code you wrote for many different types, with the guarantee that the code will behave the same for all types.

poly t -> \ x : t -> x

is the polymorphic identity function. Feed it a type to get the identity function on that type. e.g.

(poly t -> \ x : t -> x) nat

evaluates to the identity function on natural numbers.

existential packages hide types

They let us write a piece of code with a private implementation type. Clients that use this implementation don’t know what type was used. This property is enforced by the type system.

Ok, so how do we use them in thon? Let’s consider a sort-of-silly example.

The interface is just “set” and “get.” We feed in a number, get a number back. However the implementation stores the number is up to them.

We have two implementations with two separate implementation types. The first just holds on to the number.

((*set*) \ x : nat -> x,
 (*get*) \ x : nat -> x)

The second stores in the number in a tuple (for no real good reason - you didn’t write this code, it’s not your fault it does it this way).

((*set*) \ x : nat -> (x, Z),
 (*get*) \ tup : (nat * nat) -> fst tup)

Now each of these implementations can be packed away with the syntax

impl some t. ((nat -> t) * (t -> nat)) with nat as
    ((*set*) \ x : nat -> x,
    (*get*) \ x : nat -> x)


impl some t. ((nat -> t) * (t -> nat)) with (nat, nat) as
    ((*set*) \ x : nat -> (x, Z),
    (*get*) \ tup : (nat * nat) -> fst tup)

Both of these expression have type ((nat -> T) * (T -> nat)) for some type T. Note this is an existential claim, hence the name existential packages.

An implementation can be used as follows:

let setget : some t. ((nat -> t) * (t -> nat)) =
    (impl some t. ((nat -> t) * (t -> nat)) with nat as
        ((*set*) \ x : nat -> x,
        (*get*) \ x : nat -> x)
in use setget as (sg, t) in
let set : (nat -> t) = fst sg in
let get : (t -> nat) = snd sg in
let s : t = set (S S Z) in
let g : nat = get s in

Note that since the type variable t declared in the use clause is abstract, we can equivalently use the other implementation.

recursive types

u nats. (unit | (nat * nats)) is the type of lists natural numbers.

fold u nats.  (unit | (nat * nats))
with left unit : (unit | (nat * (u nats. (unit | (nat * nats)))))

is the empty list of natural numbers.

\ (nat * (u nats. (unit | nat * nats))) ->
   fold u nats. (unit | nat * nats) with
   right 0 : (unit | nat * (u nats. (unit | nat * nats)))

is a function that takes a pair (nat, natlist) and prepends nat to natlist.


I’ve mostly been working out of Bob Harper’s “Practical Foundations for Programming Languages,” though Pierce’s “Types and Programming Languages” has been a useful source of examples and exposition as well. I am also grateful to Rob Simmons and every other contributor to the SML starter code for CMU’s Fall 2016 compilers course.

install (ubuntu 20)

Wow, you read this far! (or scrolled this far, at least) If you’d like to program in thon, the code is publicly available.

$ git clone
$ sudo apt install smlnj ml-yaxx ml-lex ml-lpt
$ sml
- CM.make "path/to/your/git/clone/";
- "some thon program here";

If you figure out install instructions on mac or windows or have any other questions or comments, please email me at I would love to hear from you!

collatz conjecture

A fun program I wrote after adding recursion. Pretty much all the code I’ve written in thon is available through the git repo.

let isone : nat -> nat =
  \ n : nat ->
    ifz n of
      Z -> Z (*false*)
    | S p -> ifz p of Z -> S Z | S p -> Z
in fun iseven : nat -> nat =
  \ n : nat ->
    ifz n of
      Z -> S Z (*true*)
    | S p -> ifz (iseven p) of Z -> S Z | S p -> Z
in fun divbytwo : nat -> nat =
  \ n : nat ->
    ifz n of
      Z -> Z
    | S p -> ifz p of Z -> Z | S p' -> (S (divbytwo p'))
in fun multbythree : nat -> nat =
  \ n : nat ->
    ifz n of
      Z -> Z
   | S nminusone -> S S S (multbythree nminusone)
in fun collatz : nat -> nat =
  \ n : nat ->
    ifz (isone n) of
      Z -> (
        ifz (iseven n) of
          Z -> collatz (S (multbythree n))
        | S p -> (collatz (divbytwo n))
    | S p -> (S Z)
in (collatz (S S Z))

relevant xkcd