 Note first that although the familiar recursive definitions of addition (\ref{defnadd}) and multiplication (\ref{defnmult}) fit the format of (\ref{prex3}), they do not fit the format of (\ref{prscheme}) which in this case requires that the argument \(g\) to the primitive recursion scheme be a \(3\)-ary function. It is, however, possible to provide a definition of \(\textit{add}(x,y)\) in the official form by taking \(f(x_0) = \pi^1_0(x_0)\)—i.e., the identity function—and \(g(x_0,x_1,x_2) = \mathcal{Comp}^1_3[s,\pi^3_1]\)—i.e., the function which results from composing the successor function with the \(3\)-ary projection function on to its second argument. The expression \(\mathcal{PrimRec}_1[\pi^1_0,\mathcal{Comp}^1_3[s,\pi^3_1]]\) may then be understood as a term which encodes the definition we have provided for addition. Multiplication can then be defined via (\ref{prscheme}) with \(f = \mathbf{0}\) and \(g(x_0,x_1,x_2) = \mathcal{Comp}^2_3[add,\pi^3_0,\pi^3_2]\). Thus  \[\mathcal{PrimRec}_1[\mathbf{0},\mathcal{Comp}^2_3[add,\pi^3_0,\pi^3_2]]\]  —or in explicit form   \[\mathcal{PrimRec}_1[\mathbf{0},\mathcal{Comp}^2_3[\mathcal{PrimRec}_1[\pi^1_0,\mathcal{Comp}^1_3[s,\pi^3_1]],\pi^3_0,\pi^3_2]]\]  —can be taken as a similar term encoding the definition of multiplication we have abbreviated by \(\textit{mult}(x,y)\).