Sequent Calculus 3.1 Cut-free Sequent Calculus

In the previous chapter we developed linear logic in the form of natural deduction , which is appropriate for many applications of linear logic. It is also highly economical, in that we only needed one basic judgment (A true) and two judgment forms (linear and unrestricted hypothetical judgments) to explain the meaning of all connectives we have encountered so far. However, it is not immediately well-suited for proof search, because it involves mixing forward and backward reasoning even if we restrict ourselves to searching for normal deductions. In this chapter we develop a sequent calculus as a calculus of proof search for normal natural deductions. We then extend it with a rule of cut that allows us to model arbitrary natural deductions. The central theorem of this chapter is cut elimination which shows that the cut rule is admissible. We obtain the normalization theorem for natural deduction as a direct consequence of this theorem. It was this latter application which led to the original discovery of the sequent calculus by Gentzen [Gen35]. There are many useful immediate corollaries of the cut elimination theorem, such as consistency of the logic, or the disjunction property. In this section we transcribe the process of searching for normal natural deductions into an inference system. In the context of sequent calculus, proof search is seen entirely as the bottom-up construction of a derivation. This means that elimination rules must be turned " upside-down " so they can also be applied bottom-up rather than top-down. In terms of judgments we develop the sequent calculus via a splitting of the judgment " A is true " into two judgments: " A is a resource " (A res) and " A is a goal " (A goal). Ignoring unrestricted hypothesis for the moment, the main judgment w