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So if I understand correctly, these are examples of free variables: (all occurrences of $x$ are free) $$ x*0 $$ $$ 0+x*0 $$ $$ f: y \mapsto x*0 $$ $$ x*12345*(1-1) $$ $$ x*12345*(5-(10/2)) $$

What is an example of a variable that is not free (expensive variable?)

Is this $x$ here one? $$ f: x \mapsto x^{x^{x^{x^x}}} $$

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    $x$ in $f: x \to x*0$ is not free. May be [reading this](http://en.wikipedia.org/wiki/Free_variables_and_bound_variables) will help.2012-07-04
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    @J.D. sorry, it was a typo2012-07-04
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    Then you got the example: $f : x \to x * 0$ is not free.2012-07-04
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    @J.D. but $x*0 =0$, isn't it free if it's zero?2012-07-04
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    Few comments: the opposite of "free" is "bound", not "expensive". A variable is bound if it's within the "scope" of a quantifier. Think of it as an argument of a function "$f(x) =\text{ .. something something }x\text{ ..}$", here $x$ is bound by $f(x).$ Also, the expressions $x \mapsto x*0$ and $x \mapsto 0$ are 2 *different* expressions. It's true they're "equivalent" under arithmetic reductions, but as far as syntax goes, these are 2 different expressions. The bottom line, when we say $x$ is free in a particular expression $E$, we really mean in $E$; not in some other $E'$ equivalent to $E.$2012-07-04

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