Cut Elimination for a Logic with Generic Judgments and Induction

This paper presents a cut-elimination proof for the logic LG, which is an extension of a proof system for encoding generic judgments, the logic FOλ of Miller and Tiu, with an induction principle. The logic LG, just as FOλ, features extensions of first-order 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 FOλ 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 B ≡ ∇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. This paper contains the technical proofs for the results stated in [14]; readers are encouraged to consult [14] for motivations and examples for LG.