Programs and Proofs Mechanizing Mathematics with Dependent Types Lecture Notes

ing away from the details, we can see that nat rect is indeed a function with three parameters (the keyword fun is similar the lambda notation and is common in the family of ML-like languages). The body of nat rect is implemented as a recursive function (defined via the keyword fix) taking an argument n of type nat. Internally, it proceeds similarly to our implementation of my plus: if the argument n is zero, then the “default” value f of type P 0 is returned. Otherwise, the function proceeds recursively with a smaller argument n0 by applying the “step” function f0 to the n0 and the result of recursive call F n0. Therefore, the summing function can be implemented via the nat’s recursion combinator as follows: Definition my plus1 n m := nat rec (fun ⇒ nat) m (fun n’ m’ ⇒ m’.+1) n. Eval compute in my plus1 16 12. = 28 : (fun : nat ⇒ nat) 16 The result of invoking my plus1 is expectable. Notice, however, that when defining it we didn’t have to use the keyword Fixpoint (or, equivalently, fix), since all recursion has been “sealed” within the definition of the combinator nat rect. 2.2.1 Dependent function types and pattern matching An important thing to notice is the fact that the type of P in the definition of nat rec is a function that maps values of type nat into arbitrary types. This gives us a possibility to define dependently-typed functions, whose return type depends on their input argument value. A simple example of such a function is below: