Normalisation in Deep Inference: Cut Elimination in the Calculus of Structures

This book is presenting the calculus of structures, a recently developed proof-theoretic formalism using deep inference. According to this approach, inference rules can apply arbitrarily deep inside formulas. It follows that derivations are now symmetric instead of tree-shape objects. A formal system for classical predicate logic is presented and compared with the corresponding sequent calculus. They are both analytic systems but locality can only be obtained with deep inference, meaning that the effort of applying a rule is always bounded. Then we investigate what normal forms of deductions have been defined. Besides cut elimination, we can adopt two other notions of normalisation that allow cuts inside a derivation. The focus is on common things and differences between normalisation in deep and shallow inference.