A part of mathematical logic concerned with the notion of proof.
second order arithmetic, elementary function arithmetic
proof management system
Wikipedia, Proof theory.
Richard Statman, Structural Complexity of Proofs, 1974
Jean-Yves Girard, Yves Lafont, and Paul Taylor, Proofs and Types, 1990. (web)
Martin Hyland provides some abstract perspectives on proof theory as a subject in the introduction to his essay, Proof Theory in the Abstract.