  Modern programming frequently requires generalised notions of program
equivalence based on a metric or a similar structure. Previous work addressed
this challenge by introducing the notion of a V-equation, i.e. an equation
labelled by an element of a quantale V, which covers inter alia (ultra-)metric,
classical, and fuzzy (in)equations. It also introduced a V-equational system
for the linear variant of lambda-calculus where any given resource must be used
exactly once.
  In this paper we drop the (often too strict) linearity constraint by adding
graded modal types which allow multiple uses of a resource in a controlled
manner. We show that such a control, whilst providing more expressivity to the
programmer, also interacts more richly with V-equations than the linear or
Cartesian cases. Our main result is the introduction of a sound and complete
V-equational system for a lambda-calculus with graded modal types interpreted
by what we call a Lipschitz exponential comonad. We also show how to build such
comonads canonically via a universal construction, and use our results to
derive graded metric equational systems (and corresponding models) for programs
with timed and probabilistic behaviour.
