Primitive Recursive Function is Total Recursive Function

Theorem

Every primitive recursive function is a total recursive function.

Proof

A primitive recursive function is a total function, which is apparent from its method of definition.

As the processes for generate a primitive recursive function are a subset of those to generate a recursive function, it follows that a primitive recursive function is also a recursive function.

The result follows from the definition of a total recursive function.

$\blacksquare$