How to get all definitions associated with other symbols?

Question

How to get all definitions for a symbol associated with other symbols by TagSet, TagSetDelayed, UpSet or UpSetDelayed?

For example, if one has defined

area[square] ^= s^2
area[cube] ^= 6*s^2

how to obtain these definitions, not knowing the names square, cube but knowing only the name area?

---

I just have found that UpValues does not return definitions for MakeBoxes and N since they are stored in FormatValues and NValues correspondingly:

In[1]:= rotate /: MakeBoxes[expr_rotate, "StandardForm"] := x
UpValues[rotate]
FormatValues[rotate]

Out[2]= {}

Out[3]= {HoldPattern[MakeBoxes[expr_rotate, "StandardForm"]] :> x}

In[4]:= pi /: N[pi] = 3.14
UpValues[pi]
NValues[pi]

Out[4]= 3.14

Out[5]= {}

Out[6]= {HoldPattern[N[pi, {MachinePrecision, MachinePrecision}]] :> 
  3.14}

In this way instead of UpValues we should use a combination of UpValues, FormatValues and NValues.

---

When trying to output a list of FormatValues one can face problems with MakeBoxes since FormatValues gives definitions for MakeBoxes those are further processed by MakeBoxes on creating the output for the FrontEnd. This problem can be solved by switching FormatType temporarily to OutputForm or by converting these definitions to strings.

In[1]:= SetOptions[$Output,FormatType->OutputForm];
FormatValues[DialogNotebook]
Out[2]= {HoldPattern[MakeBoxes[BoxForm`apat$:HoldPattern[DialogNotebook[___]], BoxForm`fpat$_]] :> 

   BoxForm`BoxFormAutoLoad[MakeBoxes, BoxForm`apat$, BoxForm`fpat$, Typeset`CellNotebook`, 

    {{CellGroup, _}, {DocumentNotebook, _}, {PaletteNotebook, _}, {DialogNotebook, _}, {ExpressionCell, _}, {Text, _}, 

     {TextCell, _}, {Cell, HoldPattern[MakeExpression[_Cell, _]]}, {Notebook, HoldPattern[MakeExpression[_Notebook, _]]}}]}

In[1]:= ToString@FormatValues[DialogNotebook]
Out[1]= {HoldPattern[MakeBoxes[BoxForm`apat$:HoldPattern[DialogNotebook[___]], BoxForm`fpat$_]] :> BoxForm`BoxFormAutoLoad[MakeBoxes, BoxForm`apat$, BoxForm`fpat$, Typeset`CellNotebook`, {{CellGroup, _}, {DocumentNotebook, _}, {PaletteNotebook, _}, {DialogNotebook, _}, {ExpressionCell, _}, {Text, _}, {TextCell, _}, {Cell, HoldPattern[MakeExpression[_Cell, _]]}, {Notebook, HoldPattern[MakeExpression[_Notebook, _]]}}]}

Answer 1

Attempting to address Alexey's concerns with Howard's answer, I came up with this:

Cases[
   UpValues @@@ MakeExpression /@ Names["Global`*"],
   HoldPattern[_@_area :> _],
   {2}
]

---

In response to your updated requirements, here is the advanced version:

SetAttributes[otherValues, HoldFirst]

otherValues[sym_] :=
  With[{names = MakeExpression /@ Names["Global`*"]},
    Join[
      Cases[UpValues @@@ names, HoldPattern[_@_sym :> _], {2}],
      Cases[NValues @@@ names, HoldPattern[_@N[sym, ___] :> _], {2}],
      Select[Join @@ FormatValues @@@ names, ! FreeQ[#, HoldPattern@sym] &]
    ]
  ]

Answer 2

You can try an exhaustive search via

Select[UpValues /@ Cases[ToExpression[Names["*"]], _Symbol], ! FreeQ[#, area] &]

which in your example will yield

{{HoldPattern[area[cube]] :> 6 s^2}, {HoldPattern[area[square]] :> s^2}}

Answer 3

The following version

Cases[
  Flatten@Map[
    ToExpression[#, InputForm, Function[sym, UpValues[sym], HoldAllComplete]] &,
    Names["Global`*"]],
  Verbatim[RuleDelayed][Verbatim[HoldPattern][_area], _]
]

will not evaluate symbols as well. It is similar in spirit to @Mr. Wizard's answer, but I prefer ToExpression to MakeExpression since the latter is tied to the FrontEnd and boxes ( at least conceptually) , while the former is a general-purpose command (although it is mentioned in the documentation that it will use rules for MakeExpression).

If you have an access to the full Mathematica session from the start, another solution would be to overload TagSet, TagSetDelayed, UpSet and UpSetDelayed so that they will record the symbol dependencies in some kind of hash. Here is an example for UpSet:

Unprotect[UpSet];
Module[{tried, upsetHash},
  upsetHash[_] = {};
  getUpsetHash[] := upsetHash;
  UpSet[f_[args___], rhs_] :=
    Block[{tried = True},
       AppendTo[upsetHash[f], 
          Select[HoldComplete[args], 
            Function[Null, Head[Unevaluated[#]] === Symbol, HoldAll]]];
       UpSet[f[args], rhs]] /; ! TrueQ[tried]
];
Protect[UpSet];

All assignments made with UpSet after this redefinition will be recorded. For example, after executing your example above, you can call

In[6]:= getUpsetHash[][area]

Out[6]= {HoldComplete[square], HoldComplete[cube]} 

This way you get your information much faster, especially if you want to make such inquiries frequently, and/or you have lots of packages loaded. You can also automate the process further, to switch to the standard definitions for assignments once you load the functionality of interest.