A nominal exploration of intuitionism

Vincent Rahli, Mark Bickford
2016 Proceedings of the 5th ACM SIGPLAN Conference on Certified Programs and Proofs - CPP 2016  
This papers extends the Nuprl proof assistant (a system representative of the class of extensional type theoriesà la Martin-Löf) with named exceptions and handlers, as well as a nominal fresh operator. Using these new features, we prove a version of Brouwer's Continuity Principle for numbers. We also provide a simpler proof of a weaker version of this principle that only uses diverging terms. We prove these two principles in Nuprl's meta-theory using our formalization of Nuprl in Coq and show
more » ... w we can reflect these metatheoretical results in the Nuprl theory as derivation rules. We also show that these additions preserve Nuprl's key meta-theoretical properties, in particular consistency and the congruence of Howe's computational equivalence relation. Using continuity and the fan theorem we prove important results of Intuitionistic Mathematics: Brouwer's continuity theorem and bar induction on monotone bars.
doi:10.1145/2854065.2854077 dblp:conf/cpp/RahliB16 fatcat:55xfsdlzcrc3zoxyhv64kqrz3a