|DownValues[f]||give the list of downvalues of f|
|UpValues[f]||give the list of upvalues of f|
|DownValues[f]=rules||set the downvalues of f|
|UpValues[f]=rules||set the upvalues of f|
Under most circumstances, you do not need direct access to the actual transformation rules associated with definitions you have given. Instead, you can simply use and lhs=. to add and remove rules. In some cases, however, you may find it useful to have direct access to the actual rules.
Notice that the rules returned by DownValues and UpValues are set up so that neither their left‐ nor right‐hand sides get evaluated. The left‐hand sides are wrapped in HoldPattern, and the rules are delayed, so that the right‐hand sides are not immediately evaluated.
As discussed in "Making Definitions for Functions", the Wolfram Language tries to order definitions so that more specific ones appear before more general ones. In general, however, there is no unique way to make this ordering, and you may want to choose a different ordering from the one that the Wolfram Language chooses by default. You can do this by reordering the list of rules obtained from DownValues or UpValues.