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Superposition: Types and Induction


Wand,  Daniel
Automation of Logic, MPI for Informatics, Max Planck Society;
International Max Planck Research School, MPI for Informatics, Max Planck Society;

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Wand, D. (2017). Superposition: Types and Induction. PhD Thesis, Universität des Saarlandes, Saarbrücken. doi:10.22028/D291-26789.

Cite as: https://hdl.handle.net/11858/00-001M-0000-002D-E99C-5
Proof assistants are becoming widespread for formalization of theories both in computer science and mathematics. They provide rich logics with powerful type systems and machine-checked proofs which increase the confidence in the correctness in complicated and detailed proofs.
However, they incur a significant overhead compared to pen-and-paper proofs.
This thesis describes work on bridging the gap between high-order proof assistants and first-order automated theorem provers by extending the capabilities of the automated theorem provers to provide features usually found in proof assistants.
My first contribution is the development and implementation of a first-order superposition calculus with a polymorphic type system that supports type classes and the accompanying refutational completeness proof for that calculus. The inclusion of the type system into the superposition calculus and solvers completely removes the type encoding overhead when encoding problems from many proof assistants.
My second contribution is the development of SupInd, an extension of the typed superposition calculus that supports data types and structural induction over those data types. It includes heuristics that guide the induction and conjecture strengthening techniques, which can be applied independently of the underlying calculus.
I have implemented the contributions in a tool called Pirate. The evaluations of both contributions show promising results.