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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, thedoi:10.1109/lics.2012.47 dblp:conf/lics/Herbelin12 fatcat:nesmi636cregfia2nfimv4zeze