I'm working my way through the Cutland text on computability and I'm having a little trouble understanding exactly what he's saying in regards to substituting functions into other functions (if you have the text, the passage I'm referring to is at the bottom of page 31, but you might have to read the previous two pages to get the context).
Firstly, Cutland shows that if $f(y_1,..., y_k)$ and $g_1(x_1, ..., x_n), ...,g_k(x_1, ..., x_n)$ are all computable functions, then it follows that the function $h(x_1, ..., x_n)$ must also be computable if $h(x_1, ..., x_n) \simeq f(g_1(x_1, ..., x_n), ...,g_k(x_1, ..., x_n))$. Let's refer to this theorem as "Theorem 3.1", because Cutland refers to it as such later (believe me, this will simplify what I fear will be an already convoluted question).
Secondly, Cutland demonstrates that if the function $f(y_1,..., y_k)$ is computable and $x_{i_1},..., x_{i_k}$ is a sequence of $k$ of the variables $x_1, ..., x_n$, then the function $h(x_1, ..., x_n)$ must also be computable if $h(x_1, ..., x_n) \simeq f(y_1,..., y_k)$. Let's call this theorem "Theorem 3.2". Now, from Theorem 3.2, it follows that, given a function $f(y_1, y_2)$, you can get:
$\bullet h(x_1, x_2) \simeq f(x_2,x_1)$ (rearrangement)
$\bullet h(x) \simeq f(x,x)$ (identification)
$\bullet h(x_1, x_2, x_3) \simeq f(x_2,x_3)$ (adding dummy variables)
Ok, I get all of that, or at least I'm pretty sure that I do. What I don't get is the very next part. Here's the direct quote:
"Using this result, we can see that theorem 3.1 also holds when the functions $g_1, ...,g_k$ substituted into $f$ are not necessarily functions of all of the variables $x_1, ..., x_n$"
Is Cutland saying that, for example, if the functions $f(y_1, y_2)$, $g_1(x_1)$ and $g_2(x_1)$ are all computable, then the function $h(x_1, x_2, x_3)$ must also be computable if $h(x_1, x_2, x_3) \simeq f(g_1(x_1), g_2(x_1))$? If so, how does this relate to Theorem 3.2 and the operations of rearrangement, identification and the adding of dummy variables?
It says that combining 3.1 and 3.2, if $f$ and $g_i$ are computable, then (for example) $$h(x_1,x_2,\dots,x_n)\approx f(g_1(x_2),g_2(x_4,x_3),g_3(x_1,x_1))$$ is also computable