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Programming Languages, Martials Arts and Computers. The Weblog of Chris Double. datatypes where the constructors for the datatype are explicitly defined by the programmer. This has a number of uses and I'll go through some examples in this post. GADTs are sometimes referred to as Guarded Recursive Datatypes. Some useful resources for reading up on GADTs that I used to write this post are: This is probably the most common demonstration of GADT usage and it's useful to see how to do it in ATS. The example is taken from the First we'll create datatype to represent a simple expression language, and write an evaluation function for it, without using GADTs: implement main0() = let   val term = Mul(I(2), I(4)) Expr is a datatype with three constructors. I represents an integer, Add adds two expressions together and Mul multiples two expressions. The eval function pattern matches on these and evaluates them. The example can be compiled with the following if placed in a file Now we extend the expression language with another type, booleans, and add an Expr constructor to compare for equality: This code fails to typecheck - the eval function is defined to return an int but the B b pattern match returns a boolean. We can work around this by making the result value of eval return a datatype that can represent either an int or a bool but then we have to add code throughout eval to detect the invalid addition or multiplication of a boolean and raise a runtime error. Ideally we'd like to have make it impossible to construct invalid expressions such that they error out at compile time. We can imagine the type constructors for Expr as if they were functions with a type signature. The type signature of the fun Mul(t1:Expr, t1:Expr): Expr fun Eq(t1:Expr, t2:Expr): Expr The problem here is that Add and Mul both take Expr as arguments but we want to restrict them to integer expressions only - we need to differentiate between integer and boolean expressions. We'd like to add a type index to the Expr datatype that indicates if it is a