A Constructive Proof of Dependent Choice, Compatible with Classical Logic

Hugo Herbelin
2012 2012 27th Annual IEEE Symposium on Logic in Computer Science  
Martin-Löf's type theory has strong existential elimination (dependent sum type) that allows to prove the full axiom of choice. However the theory is intuitionistic. We give a condition on strong existential elimination that makes it computationally compatible with classical logic. With this restriction, we lose the full axiom of choice but, thanks to a lazily-evaluated coinductive representation of quantification, we are still able to constructively prove the axiom of countable choice, the
more » ... m of dependent choice, and a form of bar induction in ways that make each of them computationally compatible with classical logic. Keywords-Dependent choice; classical logic; constructive logic; strong existential 3 Failure of subject reduction when combining strong existential quantification and computational classical logic was also observed by P. Blain Levy (private communication).
doi:10.1109/lics.2012.47 dblp:conf/lics/Herbelin12 fatcat:nesmi636cregfia2nfimv4zeze