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Sender: Mailing list for the LaTeX3 project <[log in to unmask]>
Date: Mon, 16 Jan 2017 12:00:20 +0100
Reply-To: Mailing list for the LaTeX3 project <[log in to unmask]>
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From: Frank Mittelbach <[log in to unmask]>
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Am 16.01.2017 um 11:41 schrieb David Carlisle:
> Any apparent correspondence between inner and outer math and
> displaystyle and textstyle is more or less accidental.

\ifinner is described in the TeXbook simply as

True if \TeX\ is in internal vertical mode, or restricted
horizontal mode, or (nondisplay) math mode (see Chapter~13).

which is unfortunately an over-simplification as the "style" can be 
explicitly set via \displaystyle and \textstyle and that is not 
reflected by the \ifinner test.

Most math environment do exactly this, ie typesetting the formulas using 
$...$ = \ifinner=true but changing the style to \displaystyle when 
needed (with still \ifinner=true) which makes \ifinner pretty useless 
inside math

As Joseph said, that is really only reflected when you look at \mathchoice