We investigate the problem of translating between different styles of proof systems in higherorder logic: analytic proofs which are well suited for automated theorem proving, and nonanalytic deductions which are well suited for the mathematician. Analytic proofs are represented as expansion proofs, H, a form of the sequent calculus we define, non-analytic proofs are represented by natural deductions. A non-deterministic translation algorithm between expansion proofs and H-deductions is presented and its correctness is proven. We also present an algorithm for translation in the other direction and prove its correctness. A cut-elimination algorithm for expansion proofs is given and its partial correctness is proven. Strong termination of this algorithm remains a conjecture for the full higher-order system, but is proven for the first-order fragment. We extend the translations to a non-analytic proof system which contains a primitive notion of equality, while leaving the notion of expansion proof unaltered. This is possible, since a non-extensional equality is definable in our system of type theory. Next we extend analytic and non-analytic proof systems and the translations between them to include extensionality. Finally, we show how the methods and notions used so far apply to the problem of translating expansion proofs into natural deductions. Much care is taken to specify this translation in a