User:IssaRice/Computability and logic/Decrement command in loop programs

(started typing this up as a question for math SE, but I think I got it)

Herbert Enderton’s Computability Theory: An Introduction to Recursion Theory defines a loop program as one made up of the following commands:

The clear command $X_{n} \leftarrow 0$ that sets the value of the variable $X_{n}$ to zero.
The increment command $X_{n} \leftarrow X_{n} + 1$ that increases the value of $X_{n}$ by one.
The copy command $X_{n} \leftarrow X_{m}$ that sets the value of $X_{n}$ to that of $X_{m}$ and leaves the value of $X_{m}$ unchanged.
The loop command $loop X_{n}; P; endloop X_{n}$ that takes a program $P$ and executes it $X_{n}$ times. Here the value of $X_{n}$ may be changed inside the loop, but this will not affect the number of times $P$ is executed. (This ensures the overall program always terminates, assuming $P$ does.)

The output of a program is always stored in $X_{0}$ . A function $f : N^{k} \to N$ is loop-computable if there is a loop program such that for every $x \in N^{k}$ the program started with $(X_{1}, \dots, X_{k}) = x$ (and zero in all the other variables) halts with $f (x)$ in $X_{0}$ .

The book then states (without proof, as far as I can tell) that a function taking values in $N$ is loop-computable if and only if it is primitive recursive.

I am confused that the above definition of a loop program does not have a decrement command. When I look at other definitions I see that they have a decrement command.

I have noticed that I can implement a decrement program given if-else conditionals, as follows:

loop X₁
    if counter != 0
        X₀ ← X₀ + 1
    endif
    counter ← counter + 1
endloop X₁

$n \neq 0 ⟺ n \geq 1$ , so we just need a program that checks if the input is at least 1:

loop X₁
     X₀ ← 1
endloop X₁