Prove that the factorial function n! is primitive recursive. This proof should follow the following pattern:

You start with a 3-dot expression

First you write a for-loop corresponding to this function

Then you describe this for-loop in mathematical terms

Then, to prepare for a match with the general expression for primitive recursion, you rename the function to f and the parameters to n₁, ..., m

Then you write down the general expression of primitive recursion for the corresponding k

Then you match: find g and h for which the specific case of primitive recursion will be exactly the functions corresponding to initialization and to what is happening inside the loop

Then, you get a final expression for the function n! that proves that this function is primitive recursive, i.e., that it can be formed from 0, ^k_i, and by composition and primitive recursion.