 The appropriate class \(\Phi\) is revealed in the process of reduction itself, so that in the statement of theorems we simply say that \(\bT_1\) is proof-theoretically reducible to \(\bT_2\) (written \(\bT_1\leq \bT_2\)) and \(\bT_1\) and \(\bT_2\) are proof-theoretically equivalent (written \(\bT_1 \equiv \bT_2\)), respectively. Alternatively, we shall say that \(\bT_1\) and \(\bT_2\) have the same proof-theoretic strength when \(\bT_1\equiv \bT_2\). In practice, if \(\bT_1\equiv \bT_2\) is shown via proof-theoretic  means[11]  this always entails that the two theories prove at least the same \(\Pi^0_2\) sentences (those of the complexity of the twin prime conjecture). The complexity of formulae of PRA is stratified as follows. The \(\Sigma^0_0\) and \(\Pi^0_0\) formulae are of the form \(R(t_1,\ldots,t_n)\) where R is a predicate symbol for an n-ary primitive recursive predicate. A formula is \(\Sigma^0_{k+1}\) (\(\Pi^0_{k+1}\)) if it is of the form 