Kripke models and the (in)equational logic of the second-order lambda-calculus

We define a new class of Kripke structures for the second-order lambda-calculus, and investigate the soundness and completeness of some proof systems for proving inequalities (rewrite rules) as well as equations. The Kripke structures under consideration are equipped with preorders that correspond to an abstract form of reduction, and they are not necessarily extensional. A novelty of our approach is that we define these structures directly as functors A: W --> Preor equipped with certain natural transformations corresponding to application and abstraction (where W is a preorder, the set of worlds, and Preor is the category of preorders). We make use of an explicit construction of the exponential of functors in the Cartesian-closed category Preor(W), and we also define a kind of exponential Pi(Phi)(A(s))(s epsilon T) to take care of type abstraction. However, we strive for simplicity, and we only use very elementary categorical concepts. Consequently, we believe that the models described in this paper are more palatable than abstract categorical models which require much more sophisticated machinery (and are not models of rewrite rules anyway). We obtain soundness and completeness theorems that generalize some results of Mitchell and Moggi to the second-order lambda-calculus, and to sets of inequalities (rewrite rules).