  We study weighted programming, a programming paradigm for specifying
mathematical models. More specifically, the weighted programs we investigate
are like usual imperative programs with two additional features: (1)
nondeterministic branching and (2) weighting execution traces. Weights can be
numbers but also other objects like words from an alphabet, polynomials, formal
power series, or cardinal numbers. We argue that weighted programming as a
paradigm can be used to specify mathematical models beyond probability
distributions (as is done in probabilistic programming).
  We develop weakest-precondition- and weakest-liberal-precondition-style
calculi \`{a} la Dijkstra for reasoning about mathematical models specified by
weighted programs. We present several case studies. For instance, we use
weighted programming to model the ski rental problem - an optimization problem.
We model not only the optimization problem itself, but also the best
deterministic online algorithm for solving this problem as weighted programs.
By means of weakest-precondition-style reasoning, we can determine the
competitive ratio of the online algorithm on source code level.
