Primitive recursive functions by initial functions and schemes

For each arity $k\ge 0$, primitive recursive functions are total maps ${\mathbb{N}}^k\to \mathbb{N}$ given by finite derivation trees from the following rules. The empty tuple is the unique element of ${\mathbb{N}}^0$.

The initial functions are zero (in every arity), successor $S(x)=x+1$, and projections $P_i^k(x_1,\dots ,x_k)=x_i$ for $1\le i\le k$. Composition forms $h(g_1(\mathbf{a}),\dots ,g_r(\mathbf{a}))$ from an $r$-ary $h$ and $k$-ary $g_i$; $r=0$ gives the constant value of $h$. Primitive recursion forms the $(k+1)$-ary function $f(\mathbf{a},0)=g(\mathbf{a}),f(\mathbf{a},y+1)=h(\mathbf{a},y,f(\mathbf{a},y))$ from a $k$-ary $g$ and a $(k+2)$-ary $h$. In particular $k=0$ is allowed. This finite-tree definition is the least arity-indexed class closed under the rules: every closed class contains the value of each tree, by induction on the tree's finite depth.

A predicate is primitive recursive when its characteristic function, with true represented by $1$ and false by $0$, is primitive recursive. This is a definition by the displayed schemes, not by computability or unbounded search.