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A proof theory for generic judgments

The operational semantics of a computation system is often presented as inference rules or, equivalently, as logical theories. Specifications can be made more declarative and high level if syntactic details concerning bound variables and substitutions are encoded directly into the logic using term-level abstractions (λ-abstraction) and proof-level abstractions (eigenvariables). When one wishes to use such logical theories to support reasoning about properties of computation, the usual quantifiers and proof-level abstractions do not seem adequate: proof-level abstraction of variables with scope over sequents (global scope) as well as over only formulas (local scope) seem required for many examples. We will present a sequent calculus that provides this local notion of proof-level abstraction via generic judgment and a new quantifier, ∇, which explicitly manipulates such local scope. Intuitionistic logic extended with ∇ satisfies cut-elimination even when the logic is additionally strengthened with a proof theoretic notion of definitions. The resulting logic can be used to encode naturally a number of examples involving abstractions, and we illustrate the uses of ∇ with the π-calculus and an encoding of provability of an object-logic.