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Formalizing Implicative Algebras in Coq

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Abstract

We present a Coq formalization of Alexandre Miquel’s implicative algebras, which aim at providing a general algebraic framework for the study of classical realizability models. We first give a self-contained presentation of the underlying implicative structures, which roughly consists of a complete lattice equipped with a binary law representing the implication. We then explain how these structures can beturned into models by adding separators, giving rise to the so-called implicative algebras. Additionally, we show how they generalize Boolean and Heyting algebras as well as the usual algebraic structures used in the analysis of classical realizability.
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hal-01703524 , version 1 (07-02-2018)

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Étienne Miquey. Formalizing Implicative Algebras in Coq. ITP 2018 - 9th International Conference on Interactive Theorem Proving, Jul 2018, Oxford, United Kingdom. pp.459-476, ⟨10.1007/978-3-319-94821-8_27⟩. ⟨hal-01703524⟩
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