Formalização da lógica linear em Coq

Abstract

In proof theory, the cut-elimination theorem (or Hauptsatz, which means main result) is of paramount importance since it implies the consistency and the subformula property for the given system. This theorem states that any proof in the sequent calculus that makes use of the cut rule can be replaced by other that does not make use of it. The proof of cut-elimination proceeds by induction on the lexicographical order (formula weight, cut height) and generates multiple cases, considering for instance, when the formula generated by the cut rule is, or is not, principal. In general, one must consider the last rule applied in the two premises immediately after applying the cut rule (seeing the proof bottom-up). This thus generates a considerable amount of cases. For this reason, the proof of cut-elimination includes several cases and it could be error prone if we use an informal proof. Linear Logic (LL) is one of the most significant substructural logics and the cut rule is admissible in its sequent calculus. LL is a refinement of the classical and the intuitionistic model. As a resource sensible logic, LL has been widely used in the specification and verification of computer systems. In view of this, it becomes relevant the study of this logic in this work. In this dissertation we formalize three sequent calculus for linear logic in Coq and prove all of them equivalent. Additionally, we formalize meta-theorems such as admissibility of cut, generalization of initial rule, bang and copy and invertibility of the rules for the connectives par, bot, with and quest. Regarding the invertibility, we demonstrate this theorem in two different ways: a version by induction on the height of the derivation and by using the cut rule. This allows us to show how the cut rule greatly simplifies the proofs in the sequent calculus. In order to mitigate the number of several cases in the proofs, we develop several tactics in Coq that allow us to perform semi-automatic reasoning.