In this dissertation, we exhibit work revolving around the FO meta-logic. After going over some of the background material on FO , we discuss work about induction, and give induction principles that are richer than mathematical induction, in that they yield shorter, simpler and more elegant proofs. We then define and study notions of entailment and equivalence between two FO definitions. We make use of these notions by defining a framework, closely related to FO , for performing program transformation on definitions, and proving it correct. Finally, we discuss implementations in Prolog of Iris, a theorem prover based on FO tool based on the framework we built.