> proven that it is possible to take an Int and return an Array of Integers. proving that something is POSSIBLE is very easy: just write this function in an un

> proven that it is possible to take an Int and return an Array of Integers. proving that something is POSSIBLE is very easy: just write this function in an untyped language and call it with an int, and show that you get back an array. proven: it's possible. static type systems give you something much stronger: the guarantee that, always, Int -> [Int]. sometimes languages weaken this to exceptions, sometimes, certain kinds of exceptions. these systems are pretty straightforward, we've had them for a while, and they're even in environments like c++ and java. the "dependent types" area of the world that you talk about is more exciting: a desire to push more information about the program, as a specification, into the type system. so for example session types are a technology by which you can encode the state machine of a network protocol into the types of data values and functions. then, if your state machine is correct, and the process of translating the state machine into a session types implementation is correct, you can prove that your implementation is a correct implementation of a given protocol. the dynamic typing half of this, as I understand it, is contracts. you write executable guards around the beginnings of functions that check, at runtime (usually), that the environment the function is in is something the function expects. both of these technologies come from, I think, the same place: the desire to control the steps that a program makes during runtime. static type systems don't permit programs to be compiled that could take invalid steps, dynamic type systems don't permit programs that run to advance past an invalid step.