The elegant theory of the call-by-value lambda-calculus relies on
weak evaluation and closed terms, that are natural hypotheses in
the study of programming languages. To model proof assistants,
however, strong evaluation and open terms are required, and it is
well known that the operational semantics of call-by-value becomes
problematic in this case, as first pointed out by Paolini and Ronchi
della Rocca. Here we study the intermediate setting—that we call
Open Call-by-Value—of weak evaluation with open terms, on top
of which Grégoire and Leroy designed the abstract machine of Coq.
Various calculi for Open Call-by-Value already exist, coming from
logical, semantical, or implementative points of view, each one with
its pros and cons. This paper presents a detailed comparative study
of their operational semantics. First, we show that all calculi are
equivalent from a termination point of view, justifying the slogan
Open Call-by-Value. Second, we compare their equational theories.
Third, we present a detailed quantitative analysis of the time cost
model. Four, we introduce a new simple abstract machine, and
prove it a reasonable implementation of Open Call-by-Value with
respect to its cost model. Along the way, there emerges a sharp
deconstruction of call-by-value evaluation and of the complexity of
its implementations.