Polymorphic Type Checking and Monadic Evaluation

f

(λ x. (λ y. x + y) :: (Int -> (Int -> Int))

g

10

h

1

expression

f g

(f g) h

f (g h)

>>> let f = {Conc[Int], {Conc[Int], Conc[Int]}}[x, {Conc[Int], Conc[Int]}[y, Conc[Int][x + y]]](λ x. (λ y. x + y) :: (Int -> (Int -> Int)) >>> let g = Conc[Int][10]10 :: Int >>> let h = Conc[Int][1]1 :: Int >>> f g h11 :: Int >>> ▮

This Demonstration implements an extremely primitive type system in Mathematica. All functions applications are type checked at every stage of evaluation, guaranteeing that any type variables are replaced with new concrete referents and that no evaluation takes place if the types do not match up. Any type of kind * can be supported, as can a select few types of kind * -> * (lists specifically, though that is not shown here). Types are stored in the heads of data, and in order to avoid boilerplate, functions are written monadically. There are also facilities for translating terms (written in the ugly style of the input lines) into strings (the output lines), alongside their type signatures.