Automating the meta theory of deductive systems

This thesis describes the design of a meta-logical framework that supports the representation and veri cation of deductive systems, its implementation as an automated theorem prover, and experimental results related to the areas of programming languages, type theory, and logics. Design: The meta-logical framework extends the logical framework LF [HHP93] by a meta-logic M+2 . This design is novel and unique since it allows higher-order encodings of deductive systems and induction principles to coexist. On the one hand, higher-order representation techniques lead to concise and direct encodings of programming languages and logic calculi. Inductive de nitions on the other hand allow the formalization of properties about deductive systems, such as the proof that an operational semantics preserves types or the proof that a logic is consistent. M+2 is a proof calculus whose proof terms are recursive functions that may be de ned by cases and range over dependent higher-order types. The soundness of M+2 follows from a realizability interpretation of proof terms as total recursive functions. Implementation: A proof search algorithm for proof terms inM+2 is implemented in the metatheorem prover that is part of the Twelf system [PS99b]. Its takes full advantage of higher-order encodings while using inductive reasoning. Experiments: Twelf has been used for many experiments. Among others, it proved automatically the Church-Rosser theorem for the simply-typed -calculus and the cut-elimination theorem for intuitionistic rst-order logic. In programming languages, it proved various type preservation theorems for di erent operational semantics and compiler correctness theorems. In logics, it was able to derive the equivalence of various logic calculi, such as the natural deduction calculus, the sequent calculus, and the Hilbert calculus. Twelf also proved that Cartesian closed categories can be embedded into the simply-typed -calculus. In the special domains of programming languages, type theory, and logics, Twelf's reasoning power far exceeds that of any other theorem prover.

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