  Programs with a continuous state space or that interact with physical
processes often require notions of equivalence going beyond the standard binary
setting in which equivalence either holds or does not hold. In this paper we
explore the idea of equivalence taking values in a quantale V, which covers the
cases of (in)equations and (ultra)metric equations among others. Our main
result is the introduction of a V-equational deductive system for linear
{\lambda}-calculus together with a proof that it is sound and complete (in
fact, an internal language) for a class of enriched autonomous categories. In
the case of inequations, we get an internal language for autonomous categories
enriched over partial orders. In the case of (ultra)metric equations, we get an
internal language for autonomous categories enriched over (ultra)metric spaces.
We use our results to obtain examples of inequational and metric equational
systems for higher-order programs that contain real-time and probabilistic
behaviour
