# Reference Request – Cases where it is proven that multiple introductory steps are required

I'm looking for references for theorems of the form:

1) Any proof of the theorem $$X$$ requires $$n$$ Applications of induction axioms

and specially

2) Any proof of the theorem $$X$$ requires $$n$$ nested applications of induction axioms.

I have seen similar propositions for the use of axioms other than induction axioms (and I would also be pleased with references, but I am particularly interested in induction)