On Choice Principles and Fan Theorems

Veldman proved that the contrapositive of countable binary choice is a theorem of full-fledged intuitionism, to which end he used a principle of continuous choice and the fan theorem. it has turned out that continuous choice is unnecessary in this context, and that a weak form of the fan theorem suffices which holds in the presence of countable choice. in particular, the contrapositive of countable binary choice is valid in bishop-style constructive mathematics. we further discuss a generalisation of this result and link it to ishihara's boundedness principle bd-n.