Prove that the function computing the product
(12 + 1) * (22 + 1) * (32 + 1) * ... * (n2 + 1)
is primitive recursive. This proof should follow the same pattern that we used in class to prove that addition and multiplication are primitive recursive:
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
(12 + 1) * (22 + 1) * (32 + 1) * ... * (n2 + 1)
that proves that this function is primitive recursive, i.e., that it can be formed from 0, pi ki, and sigma by composition and primitive recursion.