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Tags, Upset, UpValues ...

  • To: mathgroup at
  • Subject: [mg8673] Tags, Upset, UpValues ...
  • From: Allan Hayes <hay at>
  • Date: Mon, 15 Sep 1997 02:49:01 -0400
  • Sender: owner-wri-mathgroup at

Here are some notes on Tags, Upset, UpValues ...that I hope may be  
of help:

We are dealing with the following kinds of situations
In[1]:=Sin[s] = .3;
Set::"write": "Tag \!\(Sin\) in \!\(Sin[s]\) is Protected."
Explanation: Mma tried to store the rule under tag Sin, which is  

Here are three way out

1)  Unprotect Sin; make  the definition and then Protect Sin again.

Out[5]= 0.3

2) Store the rule under tag s using UpSet
In[6]:= Sin[s]^=.5;  (*UpSet[Sin[s],.3] *)

In[7]:=Sin[s]   (*uses new definition*)
Out[7]= 0.5

This is usually better because it does not cause the new rule to be  
looked up every time Sin is used.

3) Store the rule under tag s using TagSet
In[8]:= s/:Sin[s]=.7;  (*TagSet[s,Sin[s],.7]*)

In[9]:= Sin[s]		(*new definition replaces old one*)
Out[9]= 0.7

The rules are stored in different ways according to how they are  
In[10]:= DownValues[Sin]
Out[10]= {{HoldPattern[Sin[a]] :> 0.3, HoldPattern[Sin[s]] :> 0.3}

In[11]:= UpValues[s]  (* Sin[s]^=.5 has been replaced*)
Out[11]= {HoldPattern[Sin[s]] :> 0.7}

They are stored as replacement rules because ultimately Mathematica  
works by pattern matching and replacement.

You can see the more usual forms by using ? and ??

"Sin[z] gives the sine of z."
Attributes[Sin] = {Listable, NumericFunction, Protected}

Sin[s] = 0.3

Sin[s] ^= 0.7

There are also OwnValues, SubValues and NValues.Summary


Here is a short rundown on making and storing rules, and on tags,  
UpValues , Down Values etc.
   r* means the value of r;
   a --> b means that in the evaluation process a is changed to b;
   s will always mean an unprotected symbol.

There are some additional points, for example to do with Conditions  
that I shall not include.

I show the full forms on the left and the special forms on the right.

Set[s,r] --> Set[s, r*] ; 		          (s=r)
   add HoldPattern[s ]:> r* to the list OwnValues[s] (tag is s);
   output r**

Set[f[g,h,..] ,r]--> Set[f*[g*,h*,..], r*] ;       (f[g,h,..] = r)
   add HoldPattern[f*[g*,h*,..]]:> r* to DownValues[f*] (tag is f*)
   if f* = s[..]  then
   add HoldPattern[f*[g*,h*,..]]:> r* to SubValues[s] (tag is s )
   output r**

UpSet[f[g,h,..] ,r]--> Set[f*[g*,h*,..], r*] ; 	    (f[g,h,..]  ^= r)
   for each s that is a leftmost part of g* or h*
   add HoldPattern[f*[g*,h*,..]]:> r* to UpValuesValues[s] (tag is s)
   output r**

TagSet[s,f[g,h,..],r]-->TagSet[s,f*[g*,h*,..],r*] (a/:f[g,h,..]=r)
   if f* =s then
   add HoldPattern[f*[g*,h*,..]]:> r* to DownValues[s](tag is s);
   if  f* =s[..]  then
   add HoldPattern[f*[g*,h*,..]]:> r* to SubValues[s] (tag is s);
   if s the leftmost part of at least one of g*,h* then
   add HoldPattern[f*[g*,h*,..]]:> r* to UpValues[s] (tag is s);
   output r**

If a rule cannot be stored because it has no suitably placed  
unprotected symbol to use as tag then the evaluation proceeds except  
the no rule is stored and a message is displayed; in particular,  
the output is still generated.
The corresponding delayed versions SetDelayed, UpSetDelayed and  
TagSetDelayed work similarly but do not evaluate r and they output  

The lists DownValues, UpValues or SubValues are each ordered  
according to the left sides of the rules, first by generality, then  
by default ordering and then by order of introduction; if a rule is  
added that has the same left side as an existing one then it  
replaces the existing one

Entering ?f and ??f  caused the the rules that  have been stored  
with the tag f to be printed out , but they use the familiar forms   
like f[x_,y_] = r  instead of HoldPattern[f[x_,y_] ]:> r.

Stored rules will be normally be used in the order
	user defined OwnValues,  inbuilt OwnValues
	user defined UpValues,  inbuilt UpValues
	user defined DownValues,  inbuilt DownValues
	user defined SubValues,  inbuilt SubValues
(please look up Appendix A.4.1 for more on this)
This is usually, but not always,  the order is the same as shown by  
? and ??

Allan Hayes
hay at
voice:+44 (0)116 2714198
fax: +44 (0)116 2718642
Leicester,  UK


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