How to prove that two classes of simple primitive recursive functions are distinct

Let $X$, $Y$ be classes of primitive recursive functions (for short PRF) and let it be required to prove the following statement: (I) It is not true that every function of class $X$ belongs to class $Y$. Such an assertion is usually proved by exhibiting a PRF of class $X$ which grows faster than any PRF of class $Y$, or by constructing some simple PRF's, e.g., $\lambda x(x+1)$, $\lambda xy(x+y)$, from class X, which do not belong to class $Y$. A method is proposed which gives a proof of an assertion of type (1) for some classes of PRF's $X$ and $Y$ such that first, functions of class $X$ do not increase faster than functions of class $Y$, and second, the class $Y$ contains simple PRF's such as $\lambda xy(x+y)$, $\lambda xy(x-y)$, etc. The proposed method is as follows. We choose some class of PRF's $Z$ and for each PRF $f$ we construct an operation $\delta_f$ on the functions of the class $Z$ such that for every $f$ in $Y$, the class $Z$ is closed with respect to the operation $\delta_f$, but on the other hand it is not true that for every f of $X$ the class $Z$ is closed with respect to $\delta_f$. We describe one of the possible applications of the method. We shall not distinguish between word and number PRF's and predicates, bearing in mind the following one-to-one correspondence between words in the alphabet $\{1,2\}$ and natural numbers: the word $a_na_{n-1}\dots a_1a_0$ in the alphabet $\{1,2\}$ corresponds to the number $\sum^n_{i=o}a_i2^1$. Let $\nu(x,i)$ equal the $i$-th letter in the word $x$ (the last letter – is taken to be the $0$-th sign), $|x|$ is the length of $x$; $\operatorname{con}(x,y)$ is the result of concatenating the word $y$ to the right of the word $x$; $\theta$, $\overline\theta$ are the characteristic functions of equality and inequality. Let $RF$ be a class of PRF's obtained from the PRF's $\theta$, $\overline\theta$, $\operatorname{con}$, $\lambda x0$, $\sigma$, $I^n_k$, by the operation of bounded minimalization and composition of functions. We note that the class of relations of the class $RF$ is equal to the class of rudimentary relations. Let $R_sF(f_1,\dots,f_l)$ be a class of PRF's obtained from the PRF's $f_1,\dots,f_l$ by the operation of composition and the operation of putting into correspondence with the function $g$ an $f$ such that $f(\overline x,y)=\mu t_{\leqslant y}[tPy\& g(\overline x,t)=0]$, where $tPy\leftrightharpoons t$ is the subword $y$. We note that the characteristic function of every $s$-rudimentary predicate belongs to $R_sF_\alpha(\theta,\overline\theta,\operatorname{con},\lambda x0,\sigma,I^n_k)$. We take as $Z$ the class of such $\alpha$ in $RF$ which have the values $0,1,2$ and if $\alpha(\overline z,i)=0$ then $\forall k[\alpha(\overline z,i+k)=0]$. The operation $\delta_f$ is such that $[\delta_f(\alpha_1,\dots,\alpha_n)](\overline x,\overline z,j)=\nu(f(\sum^{x_1}_{i=0}\alpha_1(\overline z,i),2^i),j)$. Assertions of type (1) can be proved by the proposed method if for $X$ we take $RF$, and as $Y$ we take $Y-R_sF_\alpha(\theta,\overline\theta,\operatorname{con},\lambda x0,\sigma,I^n_k,\lambda xy(x+y),\lambda xy(x-y),\lambda xf(|x|),\lambda x2^{g(|x|)})$, where $f$ and $g$ belong to the class $RF$.