A modular construction of type theories
Fr\'ed\'eric Blanqui, Gilles Dowek, Emilie Grienenberger and, Gabriel Hondet, Fran\c{c}ois Thir\'e
TL;DR
This paper introduces a modular framework for constructing various type theories within the lambda-Pi-calculus modulo theory, enabling flexible expression and proof translation across systems.
Contribution
It presents the theory U, a unified framework that encapsulates multiple logical systems as sub-theories, with proofs in U corresponding to proofs in these systems.
Findings
Defined the theory U for expressing multiple logical systems
Proved that proofs using only sub-theory symbols are valid in the sub-theories
Established a modular approach to constructing and relating type theories
Abstract
The lambda-Pi-calculus modulo theory is a logical framework in which many type systems can be expressed as theories. We present such a theory, the theory U, where proofs of several logical systems can be expressed. Moreover, we identify a sub-theory of U corresponding to each of these systems, and prove that, when a proof in U uses only symbols of a sub-theory, then it is a proof in that sub-theory.
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Taxonomy
TopicsLogic, programming, and type systems · Logic, Reasoning, and Knowledge · Advanced Algebra and Logic