Non-Standard Evaluation

While most built-in Mathematica functions follow the standard evaluation procedure, some important ones do not. For example, most of the Mathematica functions associated with the construction and execution of programs use non-standard evaluation procedures. In typical cases, the functions either never evaluate some of their arguments, or do so in a special way under their own control.

x=ydo not evaluate the left-hand side
If[p,a,b]evaluate a if p is True, and b if it is False
Do[expr,{n}]evaluate expr n times
Plot[f,{x,...}]evaluate f with a sequence of numerical values for x
Function[{x},body]do not evaluate until the function is applied

Some functions that use non-standard evaluation procedures.

When you give a definition such as , Mathematica does not evaluate the that appears on the left-hand side. You can see that there would be trouble if the was evaluated. The reason is that if you had previously set , then evaluating in the definition would put the definition into the nonsensical form .

In the standard evaluation procedure, each argument of a function is evaluated in turn. This is prevented by setting the attributes HoldFirst, HoldRest and HoldAll. These attributes make Mathematica "hold" particular arguments in an unevaluated form.

HoldFirstdo not evaluate the first argument
HoldRestevaluate only the first argument
HoldAllevaluate none of the arguments

Attributes for holding function arguments in unevaluated form.

With the standard evaluation procedure, all arguments to a function are evaluated.
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This assigns the attribute HoldFirst to .
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The first argument to is now held in an unevaluated form.
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When you use the first argument to like this, it will get evaluated.
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Built-in functions like Set carry attributes such as HoldFirst.
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Even though a function may have attributes which specify that it should hold certain arguments unevaluated, you can always explicitly tell Mathematica to evaluate those arguments by giving the arguments in the form Evaluate[arg].

Evaluate effectively overrides the HoldFirst attribute, and causes the first argument to be evaluated.
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f[Evaluate[arg]]evaluate arg immediately, even though attributes of f may specify that it should be held

Forcing the evaluation of function arguments.

By holding its arguments, a function can control when those arguments are evaluated. By using Evaluate, you can force the arguments to be evaluated immediately, rather than being evaluated under the control of the function. This capability is useful in a number of circumstances.

The Mathematica Set function holds its first argument, so the symbol is not evaluated in this case.
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You can make Set evaluate its first argument using Evaluate. In this case, the result is the object which is the value of , namely is set to .
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has now been set to .
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In most cases, you want all expressions you give to Mathematica to be evaluated. Sometimes, however, you may want to prevent the evaluation of certain expressions. For example, if you want to manipulate pieces of a Mathematica program symbolically, then you must prevent those pieces from being evaluated while you are manipulating them.

You can use the functions Hold and HoldForm to keep expressions unevaluated. These functions work simply by carrying the attribute HoldAll, which prevents their arguments from being evaluated. The functions provide "wrappers" inside which expressions remain unevaluated.

The difference between Hold[expr] and HoldForm[expr] is that in standard Mathematica output format, Hold is printed explicitly, while HoldForm is not. If you look at the full internal Mathematica form, you can however see both functions.

Hold maintains expressions in an unevaluated form.
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HoldForm also keeps expressions unevaluated, but is invisible in standard Mathematica output format.
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HoldForm is still present internally.
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The function ReleaseHold removes Hold and HoldForm, so the expressions they contain get evaluated.
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Hold[expr]keep expr unevaluated
HoldComplete[expr]keep expr unevaluated and prevent upvalues associated with expr from being used
HoldForm[expr]keep expr unevaluated, and print without HoldForm
ReleaseHold[expr]remove Hold and HoldForm in expr
Extract[expr,index,Hold]get a part of expr, wrapping it with Hold to prevent evaluation

Functions for handling unevaluated expressions.

Parts of expressions are usually evaluated as soon as you extract them.
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This extracts a part and immediately wraps it with Hold, so it does not get evaluated.
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f[...,Unevaluated[expr],...]give expr unevaluated as an argument to f

Temporary prevention of argument evaluation.

evaluates to , and Length[2] gives .
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This gives the unevaluated form as the argument of Length.
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Unevaluated[expr] effectively works by temporarily giving a function an attribute like HoldFirst, and then supplying expr as an argument to the function.

SequenceHolddo not flatten out Sequence objects that appear as arguments
HoldAllCompletetreat all arguments as completely inert

Attributes for preventing other aspects of evaluation.

By setting the attribute HoldAll, you can prevent Mathematica from evaluating the arguments of a function. But even with this attribute set, Mathematica will still do some transformations on the arguments. By setting SequenceHold you can prevent it from flattening out Sequence objects that appear in the arguments. And by setting HoldAllComplete you can also inhibit the stripping of Unevaluated, and prevent Mathematica from using any upvalues it finds associated with the arguments.

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