Evaluation in Patterns, Rules, and Definitions

There are a number of important interactions in Mathematica between evaluation and pattern matching. The first observation is that pattern matching is usually done on expressions that have already been at least partly evaluated. As a result, it is usually appropriate that the patterns to which these expressions are matched should themselves be evaluated.

The fact that the pattern is evaluated means that it matches the expression given.
In[1]:=
Click for copyable input
Out[1]=
The right-hand side of the condition is not evaluated until it is used during pattern matching.
In[2]:=
Click for copyable input
Out[2]=

There are some cases, however, where you may want to keep all or part of a pattern unevaluated. You can do this by wrapping the parts you do not want to evaluate with HoldPattern. In general, whenever HoldPattern[patt] appears within a pattern, this form is taken to be equivalent to patt for the purpose of pattern matching, but the expression patt is maintained unevaluated.

HoldPattern[patt]equivalent to patt for pattern matching, with patt kept unevaluated

Preventing evaluation in patterns.

One application for HoldPattern is in specifying patterns which can apply to unevaluated expressions, or expressions held in an unevaluated form.

HoldPattern keeps the from being evaluated, and allows it to match the on the left-hand side of the operator.
In[3]:=
Click for copyable input
Out[3]=

Notice that while functions like Hold prevent evaluation of expressions, they do not affect the manipulation of parts of those expressions with and other operators.

This defines values for whenever its argument is not an atomic object.
In[4]:=
Click for copyable input
According to the definition, expressions like are left unchanged.
In[5]:=
Click for copyable input
Out[5]=
However, the pattern is transformed according to the definition for .
In[6]:=
Click for copyable input
Out[6]=
You need to wrap HoldPattern around to prevent it from being evaluated.
In[7]:=
Click for copyable input
Out[7]=

As illustrated above, the left-hand sides of transformation rules such as are usually evaluated immediately, since the rules are usually applied to expressions which have already been evaluated. The right-hand side of is also evaluated immediately. With the delayed rule , however, the expression rhs is not evaluated.

The right-hand side is evaluated immediately in but not rules.
In[8]:=
Click for copyable input
Out[8]=
Here are the results of applying the rules. The right-hand side of the rule gets inserted inside the Hold without evaluation.
In[9]:=
Click for copyable input
Out[9]=
lhs->rhsevaluate both lhs and rhs
lhs:>rhsevaluate lhs but not rhs

Evaluation in transformation rules.

While the left-hand sides of transformation rules are usually evaluated, the left-hand sides of definitions are usually not. The reason for the difference is as follows. Transformation rules are typically applied using to expressions that have already been evaluated. Definitions, however, are used during the evaluation of expressions, and are applied to expressions that have not yet been completely evaluated. To work on such expressions, the left-hand sides of definitions must be maintained in a form that is at least partially unevaluated.

Definitions for symbols are the simplest case. As discussed in "Non-Standard Evaluation", a symbol on the left-hand side of a definition such as is not evaluated. If x had previously been assigned a value y, then if the left-hand side of were evaluated, it would turn into the quite unrelated definition .

Here is a definition. The symbol on the left-hand side is not evaluated.
In[10]:=
Click for copyable input
Out[10]=
This redefines the symbol.
In[11]:=
Click for copyable input
Out[11]=
If you evaluate the left-hand side, then you define not the symbol , but the value of the symbol .
In[12]:=
Click for copyable input
Out[12]=
Now has value .
In[13]:=
Click for copyable input
Out[13]=

Although individual symbols that appear on the left-hand sides of definitions are not evaluated, more complicated expressions are partially evaluated. In an expression such as on the left-hand side of a definition, the args are evaluated.

The is evaluated, so that a value is defined for .
In[14]:=
Click for copyable input
Out[14]=
This shows the value defined for .

You can see why the arguments of a function that appears on the left-hand side of a definition must be evaluated by considering how the definition is used during the evaluation of an expression. As discussed in "Principles of Evaluation", when Mathematica evaluates a function, it first evaluates each of the arguments, then tries to find definitions for the function. As a result, by the time Mathematica applies any definition you have given for a function, the arguments of the function must already have been evaluated. An exception to this occurs when the function in question has attributes which specify that it should hold some of its arguments unevaluated.

symbol=valuesymbol is not evaluated; value is evaluated
symbol:=valueneither symbol nor value is evaluated
f[args]=valueargs are evaluated; left-hand side as a whole is not
f[HoldPattern[arg]]=valuef[arg] is assigned, without evaluating arg
Evaluate[lhs]=valueleft-hand side is evaluated completely

Evaluation in definitions.

While in most cases it is appropriate for the arguments of a function that appears on the left-hand side of a definition to be evaluated, there are some situations in which you do not want this to happen. In such cases, you can wrap HoldPattern around the parts that you do not want to be evaluated.

New to Mathematica? Find your learning path »
Have a question? Ask support »