A Higher-Order Colon Translation

A lambda-encoding such as the CPS transformation gives rise to administrative redexes. In his seminal article "Call-by-name, call-by-value and the lambda-calculus", 25 years ago, Plotkin tackled administrative reductions using a so-called colon translation. In "Representing control,a study of the CPS transformation", 15 years later, Danvy and Filinski integrated administrative reductions in the CPS transformation, making it operate in one pass. This one-pass transformation is higher-order, and can be used for other lambda-encodings, but we do not see its associated proof technique used in practice - instead, Plotkin's colon translation appears to be favored. Therefore, in an attempt to link the higher-order transformation and Plotkin's proof technique, we recast Plotkin's proof of Indifference and Simulation in a higher-order setting. To this end, we extend the colon translation from first order to higher order.Keywords: Call by name, call by value, lambda-calculus, continuation-passing style(CPS), CPS transformation, administrative reductions, colon translation, one-pass CPS transformation, Indifference, Simulation.