Continuations may be unreasonable

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Abstract
We show that two lambda calculus terms can be observationally congruent (i.e., agree in all contexts) but their continuation-passing transforms may not be. We also show that two terms may be congruent in all untyped contexts but fail to be congruent in a calculus with call/cc operators. Hence, familiar reasoning about functional terms may be unsound if the terms use continuations explicitly or access them implicitly through new operators. We then examine one method of connecting terms with their continuized form, extending the work of Meyer and Wand [8].

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