 This gives rises to the familiar sequence 0, 1, 1, 2, 5, 8, 13, 21, 34, 55, 89, 144,… wherein \(F_0 =0,\) \(F_1 = 1,\) and \(F_{i+2} = F_{i+1} + F_i.\) Note, however, the definition (\ref{fibdefn}) cannot be directly assimilated to the primitive recursion scheme (\ref{prscheme}) since the third clause defines the value of \(\fib(y+1)\) in terms of both \(\fib(y)\) and \(\fib(y-1)\). It is, however, still possible to show that \(\fib \in \mathbf{PR}\). One means of doing this is to again make use of the binary coding and projection functions to first define an auxiliary function \(g(0) = \langle 0,1 \rangle\) and