On the formalization of the modal µ-calculus in the Calculus of Inductive Constructions

This paper is part of an ongoing research programme at the Computer Science Department of the University of Udine on proof editors, started in 1992, based on HOAS encodings in dependent typed #-calculus for program logics [15, 21, 16]. In this paper, we investigate the applicability of this approach to the modal -calculus. Due to its expressive power, we adopt the Calculus of Inductive Constructions (CIC), implemented in the system Coq. Beside its importance in the theory and verification of processes, the modal -calculus is interesting also for its syntactic and proof theoretic peculiarities. These idiosyncrasies are mainly due to a) the negative arity of "" (i.e., the bound variable x ranges over the same syntactic class of x#); b) a context-sensitive grammar due the condition on x#; c) rules with complex side conditions (sequent-style "proof " rules). These anomalies escape the "standard" representation paradigm of CIC; hence, we need to accommodate special techniques for enforcing these peculiarities. Moreover, since generated editors allow the user to reason "under assumptions", the designer of a proof editor for a given logic is urged to look for a Natural Deduction formulation of the system. Hence, we introduce a new proof system N # K in Natural Deduction style for K. This system should be more natural to use than traditional Hilbert-style systems; moreover, it takes best advantage of the possibility of manipulating assumptions o#ered by CIC in order to implement the problematic substitution of formul for variables. In fact, substitutions are delayed as much as possible, and are kept in the derivation context by means of assumptions. This mechanism fits perfectly the stack discipline of assumptions of Natural Deduction, and it is neatly formalized in CIC. Bes..