 Example 3 We now provide a mathematical example. Mathematical ideas can be expressed in type theory without introducing any new constants. An iterate of a function f from a set to itself is a function which applies f one or more times. For example, if \(g(x) = f(f(f(x)))\), then g is an iterate of f. \([\text{ITERATE+}_{{o}(\imath\imath)(\imath\imath)}f_{\imath\imath}g_{\imath\imath}]\) means that \(g_{\imath\imath}\) is an iterate of \(f_{\imath\imath}\). \(\text{ITERATE+}_{{o}(\imath\imath)(\imath\imath)}\) is defined (inductively) as 