I want to solve the following problem :
Let $f_1(x), f_2(x), g_1(x_1, x_2, x_3, x_4)$ and $g_2(x_1, x_2, x_3, x_4)$ be primitive recursive functions. Also, let $h_1, h_2$ be defined as :
$h_1(x, 0) = f_1(x)$
$h_2(x, 0) = f_2(x)$
$h_1(x, t + 1) = g_1(x, t, h_1(x, t), h_2(x, t))$
$h_2(x, t+1) = g_2(x, t, h_1(x, t), h_2(x, t))$.
Show that $h_1, h_2$ are primitive recursive.
The problem also notes that we could use the function ${\lambda}(x, t) = 2^{h_1(x, t)}\cdot 3^{h_2(x, t)}$.
I do not have much knowledge on this area and it is now that I am trying to be familiar. My thoughts is to :
Define $K(x,y) = \langle h_1(x, y), h_2(x, y)\rangle$ and show that $K$ is primitive recursive function. So, if we define $K$ using primitive recursion, we get that : $K(x, 0) = \langle f_1(x), f_2(x)\rangle$ and $K(x, t+1) = j (t, K(x, t), x)$ and
$j(x_1, x_2, x_3, x_4) = \langle g_1(x_4, l(x_3), r(x_3), p(x_3)), g_2(x_4, l(x_3), r(x_3), p(x_3))\rangle$.
Now, we have that $j$ is obtained by recursion and composition of $g_1, g_2$ and initial functions and so, $j$ is primitive recursive. Hence, $K$ is primitive recursive. Therefore, $h_1, h_2$ are defined to be $l(g(x, y))$ and $r(g(x, y))$ respectively, which are both primitive recursive.
Does all these make sense and complete the proof? Otherwise, is there another way that someone can show me on how to solve this?
Thank you in advance.