SN and CR for Free-Style LKtq: Linear Decorations and Simulation of Normalization

The present report is a, somewhat lengthy, addendum to Danos et al.(1997), where the elimination of cuts from derivations in sequent calculus for classical logic was studied`from the point of view of linear logic'. To that purpose a formulation of classical logic was used, that-as in linear logic-distinguishes between multiplicative and additive versions of the binary connectives. The main novelty here is the observation that this type-distinction is not essential: we can allow classical sequent derivations to use any combination of additive and multiplicative introduction rules for each of the connectives, and still have strong normalization and connuence of tq-reductions. We give a detailed description of the simulation of tq-reductions by means of reductions of the interpretation of any given classical proof as a proof net of PN2 (the system of second order proof nets for linear logic), in which moreover all connectives can be taken to be of one type, e.g. multiplicative. We nally observe that dynamically the diierent logical cuts, as determined by the four possible combinations of introduction rules, are independent: it is not possible to simulate them internally, i.e. by only one speciic combination, and structural rules.