A Logic for Reasoning about Generic Judgments

This paper presents an extension of a proof system for encoding generic judgments, the logic [email protected]^@D^@? of Miller and Tiu, with an induction principle. The logic [email protected]^@D^@? is itself an extension of intuitionistic logic with fixed points and a ''generic quantifier'', @?, which is used to reason about the dynamics of bindings in object systems encoded in the logic. A previous attempt to extend [email protected]^@D^@? with an induction principle has been unsuccessful in modeling some behaviours of bindings in inductive specifications. It turns out that this problem can be solved by relaxing some restrictions on @?, in particular by adding the axiom [email protected]?x.B, where x is not free in B. We show that by adopting the equivariance principle, the presentation of the extended logic can be much simplified. Cut-elimination for the extended logic is stated, and some applications in reasoning about an object logic and a simply typed @l-calculus are illustrated.

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