A brief tutorial on "display-form" inertness in Maple
Maple offers several forms of inertness, i.e., ways to control when, if ever, expressions or particular subexpressions evaluate. I'm not aware of names for these different forms of inertness. So, for now at least, I'm going to call the topic of this thread and this article "display-form" inertness because it's primarily (although not exclusively) used to manipuate the prettyprinted displayed form of expressions while keeping them within manifestly mathematical data structures that--unlike for example strings--are only slight variations of their noninert (or active) counterparts. This form of inertness is mostly controlled by using the command value, by prefixing symbols and operators with %, and by the package InertForm. The package Typesetting and procedures whose names begin `print/ are also involved, although this aspect is usually transparent to the end user and often also transparent to the programmer who's one step removed from the end user.
The option inert=true vs. inert=false in package InertForm: The inert=true and inert=false options to some commands in InertForm do not change whether the expression is inert; they only change whether the inert operators are displayed as inert (i.e., in gray) rather than as normal (i.e., in blue).[*1] And it's only the inert operators, not the entire expression, whose color changes.
So now that I've pointed out the subtlety of this difference, do you notice it? It's clearly visible in the above Answers by @acer and @Kitonum .
If neither of these options is used, that's equivalent to inert=true.
Making symbols inert with %: Any Maple symbol (e.g., command name (if it's a procedure), function name, variable, named constant (e.g., Pi, infinity)) can be made inert by prefixing an %. Although these inert forms usually "do nothing" (i.e., they are truly "inert" as in the common usage of that word), that is not actually a requirement. Executable code in procedures can be supplied for them, and those procedures can return an inert form as an unevaluated function. One possible use for that is to do syntax checking of arguments without doing the computation. An example of this is shown in the Reply below.
Special-character quoting (e.g., `%Pi`) is usually not needed (in 1D input (I don't know about 2D Input)) when prefixing %, but it is allowed. Multiple levels of inertness can be applied by prefixing multiple %. In that case, the quoting is required.
Inert arithmetic infix operators: The five basic[*2] infix arithmetic operators as well as noncommutative multiplication are available in inert forms for infix use: %^, %., %*, %/, %+, %-. These can be used infix in 1D input. I don't know about 2D Input, but your experience mentioned above suggests that they can't be. As Kitonum showed, they can always be used in prefix (aka functional) form, in which case they're just symbols, and the previous paragraph applies. You cannot create multiple levels of inertness for the infix forms.
You cannot use quotes with any infix operators (regardless of whether they're inert) if they're being used in infix form. The quotes ` ` (sometimes called "back quotes") turn anything into a symbol. Infix operators cannot be symbols, although almost all of them have an equivalent symbol form that allows them to be used in functional form (just like Kitonum used `%*` above).
Like any "multiple-punctuation-mark" infix operator in almost any computer language[*3], you cannot separate the characters with any white space. And of course ab or `ab` (those are identical) and `a b` are distinct symbols.
Other historically inert commands: The commands Eval, Diff, Int, Intat, Limit, Sum, Product, Normal, and Hypergeom (and perhaps I've missed some) are inert forms of the corresponding uncapitalized commands. These are much older than is using % for inertness; however, their form of inertness is essentially the same as with %.
There are many other cases of two Maple commands whose names only differ by capitalization and the capitalized form is inert; yet the lowercase form is not an active form for it. In these cases, the capitalized form is usually a data container for input in some specialized domain of computation such as mod, modp1, and evala. See their help pages for examples of these inert functions.
Removing inertness with value: The value command applied to an expression removes one level of inertness from every inert symbol (including the historical ones from the previous paragraph) and inert infix operator in the expression. If special techniques need to be used to remove the inertness, value has a facility for supplying procedures[*4] for that. Likewise, any otherwise unused symbol, regardless of any special characters, can be made into an inert form of any other symbol (i.e., there need not be any connection between the spellings of the inert and active forms) by teaching[*4] value about it. The previous sentence applies to symbols only, not to infix operators. It is not necessary that any noninert form produced by value actually does anything, i.e., it could also be "inert" for all practical purposes.
The InertForm package: In summary, there are numeous ways to use inertness without using the InertForm package. However, one way (among several) that the package is quite useful is for working with expressions that are so volatile that they must be entered as strings before being Parsed by the package into mathematical expressions. Examples of such volatile expressions are those that would otherwise be changed by Maple's numerous forms of automatic simplification.
The command Typeset produces displayable expressions whose detailed internal structure can be examined with lprint. This is useful for learning how the Typesetting package is used (essentially as a mark-up language).
`print/...` procedures: A procedure named `print/...` where ... is a function name (usually an essentially inert function, although its inertness need not be formally imposed by % or by any other means) is one of the easier ways to create customized prettyprinted displays for inert functions. For example, the inert function can be broken down into simpler inert functions whose prettyprinted display is already handled by the system. An example of this is in the Reply below.
These procedures could manipulate the output of InertForm:-Typeset. An example of this---performing a small "surgery" to correct the blue exclamation points---is also in the Reply below.
Relevant help pages: The most-important help page for more details on what I've presented here is ?value. Also relevant are ?operator,precedence, ?print, and of course ?InertForm (in particular, ?InertForm,Display and ?InertForm,Typeset). Some more details about using "punctuation mark" operators as function symbols can be found at ?use.
The commands FromInert and ToInert are not related to the "display-form" inertness being discussed in this thread and this article. They are oriented towards debugging, automatic code generation, and what I call "code surgery".
I've used several technical terms in this article without providing definitions. These appear in upright bold if they're available verbatim as Maple commands, keywords, etc., or in italics if they're being used in a technical sense (such as specific to formal computer science) rather than as common English. Feel free to ask me for specific definitions of any of these.
[*1] These styles, such as the colors gray and blue, could be changed if you really wanted. I've never seen it done, and I'd recommend against it unless you were a master craftsman at typography and document layout, lest you significantly reduce the readability of the output, especially its readability to experienced Maple users who are expecting certain styles.
An example in the Reply below corrects a bug that causes the color of the exclamation points to be default blue rather than inert gray in inert factorials.
[*2] Personally, I'd also consider mod to be a "basic" infix arithmetic operator, but it doesn't have an infix inert form. Just like most of the infix "punctuation-mark" operators, its inert infix effect can be achieved by quoting, i.e., by using `%mod` as a function symbol.
[*3] Fortran, at least in some of its older forms, is unlike most languages in that it has some exceptions to these white space restrictions.
[*4] Specifically, this is done by defining a procedure named `value/...` where the ... is the inert function symbol. The details are given at ?value. An example is shown in the Reply immediately below.
