Why does existence of predecessor imply adequacy of a numeral system?
Recall that the set of partial recursive functions $\mathcal{R}$ is defined inductively by the following rules:


(zero) $\dfrac{}{\zeta : n \mapsto 0 \in \mathcal{R}}$
(successor) $\dfrac{}{\sigma : n \mapsto n + 1 \in \mathcal{R}}$
(projection) $\dfrac{}{\Pi_m^i : (n_1, \cdots n_m) \mapsto n_i \in \mathcal{R}}$
(composition) $\dfrac{\chi : \mathbb{N}^m \to \mathbb{N}, \psi_i : \mathbb{N}^k \to \mathbb{N} \in \mathcal{R} }{\phi : (n_1, \cdots n_k) \mapsto \chi(\psi_1(n_1, \cdots, n_k), \cdots \psi_m(n_1, \cdots, n_k)) \in \mathcal{R}}$
(primitive recursion) $\dfrac{\chi : \mathbb{N}^m \to \mathbb{N}, \psi : \mathbb{N}^{n+2} \to \mathbb{N} \in \mathcal{R} }{\phi : (k,n_1, \cdots n_m) \mapsto \begin{cases} 
      \chi(n_1, \cdots n_m) &amp; \text{ if } k=0 \\
       \psi(\phi(k-1, n_1, \cdots, n_m),k-1, n_1 \cdots n_m ) &amp; \text{ if } k &gt; 0\\
   \end{cases}  \in \mathcal{R}}$
(minimalization) $\dfrac{\chi : \mathbb{N}^{m+1} \to \mathbb{N} \in \mathcal{R} }{\phi : (n_1, \cdots n_m) \mapsto k \text{, the least integer with } \chi(k,n_1, \cdots n_m) =0
   \in \mathcal{R}}$


So we will prove the result by structural induction on $\mathcal{R}$.


Initial functions.


It is clear that $\zeta$ is defined by $\lambda x. d_0$, $\sigma$ by $S$, and $\Pi_m^i$ by $\lambda x_1 \cdots x_m . x_i$.


Composition


Suppose $\chi$ is defined by $f$ and $\psi_i$ by $g_i$. 

Let $h$ be the term $\lambda x_1 \cdots x_k.f (g_1x_1 \cdots x_k) \cdots (g_mx_1 \cdots x_k)$. Then $h$ defines the function $\phi : (n_1, \cdots n_k) \mapsto \chi(\psi_1(\vec{n}), \cdots \psi_m(\vec{n}))$


Primitive recursion.


Suppose $\chi$ is defined by $f$ and $\psi$ by $g$. 

Let's define the combinator $\textbf{r} = \lambda nfg.isZ\:n\:f\:(g \:(P\:n))$, with the property that:

$$
\textbf{r} \: d_n \:f \: g = \begin{cases} f &amp; \text{ if } n = 0 \\
g \: (d_{n-1}) &amp; \text{ if } n &gt; 0
\end{cases}
$$

Now let $h$ be the term $\textbf{y} \: (\lambda s y x_1 \cdots x_m . \textbf{r} \: y(f x_1\cdots x_m)(\lambda z.g (s z x_1 \cdots x_m)z x_1 \cdots x_m )$. Then $h$ defines $\phi: (k,n_1, \cdots n_m) \mapsto \begin{cases} 
      \chi(n_1, \cdots n_m) &amp; \text{ if } k=0 \\
       \psi(\phi(k-1, n_1, \cdots, n_m),k-1, n_1 \cdots n_m ) &amp; \text{ if } k &gt; 0\\
   \end{cases}$.


Minimalization


Suppose $\chi$ is defined by $f.

Let $g$ be the term $\textbf{y} (\lambda syx_1 \cdots x_m.isZ \: (f\: y\: x_1\cdots x_m) \: y\:  (s\: (S \: y)\:  x_1 \cdots x_m))$. Then take $h$ to be the term $g \: d_0$, and $h$ defines the function $\phi : (n_1, \cdots n_m) \mapsto k \text{, the least integer with } \chi(k,n_1, \cdots n_m) =0$.

By induction, every function in $\mathcal{R}$ is definable.

(I leave the proving the fact that the terms $h$ define the stated functions as an exercise!)