A Sequent Formulation of a Logic of Predicates in HOL

By a predicate we mean a term in the HOL logic of type *-> bool, where * can be any type. Boolean connectives, quantiiers and sequents in the HOL logic can all be lifted to operate on predicates. The lifted logical operators and sequents form a Logic of Predicates (LP) whose behavior resembles closely that of the unlifted HOL logic. Of the applications of LP we describe two in some detail: (1) a semantic embedding of Lamport's Temporal Logic of Actions, and (2) an alternative formulation of set theory. The main contribution of this paper is a systematic approach for lifting tactics that works in the unlifted HOL logic to ones that works in LP, so that one can enjoy the rich proof infrastructure of HOL when reasoning in LP. The power of this approach is illustrated by examples from modal and temporal logics. The implementation technique is brieey described. 1 A Logic of Predicates By a predicate we mean a term in the HOL 3] logic of type *-> bool, where * is called the domain of the predicate and can be any type. Boolean connectives and quantiiers in the HOL logic can all be lifted to operate on predicates with the following deenitions: (TT)(x) = T (FF)(x) = F (~~ P)(x) = ~ P(x) (P //\\ Q)(x) = P(x) /\ Q(x) (P \\// Q)(x) = P(x) \/ Q(x) (P ==>> Q)(x) = P(x) ==> Q(x) (P == Q)(x) = (P(x) = Q(x)) (!! R)(x) = ! i. (R i)(x)