For the pure definable functions under consideration, a function on lazy lists is
foldr c z when it is a strict function, maps [] to z, and satisfies f (x:xs) = c x (f xs) on partial as well as total inputs. Necessity follows from the fold equations. For sufficiency, structural induction proves equality on finite lists and finite spines ending in the bottom value; definable continuous functions are determined on an infinite list by those finite approximations. The important information constraint is that a step receives the tail only through its folded result. Equal folded tails must yield equal extended results when the new head is the same. Articles by others on the same topic
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