  We study the interpretation of the lambda-calculus in a framework based on
tropical mathematics, and we show that it provides a unifying framework for two
well-developed quantitative approaches to program semantics: on the one hand
program metrics, based on the analysis of program sensitivity via Lipschitz
conditions, on the other hand resource analysis, based on linear logic and
higher-order program differentiation. To do that we focus on the semantic
arising from the relational model weighted over the tropical semiring, and we
discuss its application to the study of "best case" program behavior for
languages with probabilistic and non-deterministic effects. Finally, we show
that a general foundation for this approach is provided by an abstract
correspondence between tropical algebra and Lawvere's theory of generalized
metric spaces.
