We discuss the Paris-Wilkie translation from bounded arithmeticproofs to bounded depth propositional proofs in both relativized and non-relativized forms. We describe normal forms for proofs in boundedarithmetic, and a definition of \Sigma 0-depth for PK-proofs that makes the translation from bounded arithmetic to propositional logic particularlytransparent. Using this, we give new proofs of the witnessing theorems for S12and T 12; namely, new proofs that the \Sigma b1-definable functions of S12are polynomial time computable and that the \Sigma b1-definable functions of T 12 are in Polynomial Local Search (PLS). Both proofs generalize to \Sigma