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bash - What does "plus colon" ("+:") mean in shell script expressions?

What does this mean?

if ${ac_cv_lib_lept_pixCreate+:} false; then :
  $as_echo_n "(cached) " >&6
else
  ac_check_lib_save_LIBS=$LIBS

Looks like ac_cv_lib_lept_pixCreate is some variable, so what does +: mean?

Where to learn complete syntax of curly bracket expressions? What is the name of this syntax?

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In the “plus colon” ${...+:} expression, only the + has special meaning in the shell. The colon is just a string value in this case, so we could write that snippet as ${...+":"}.

For convenience, let's pretend the variable is called var, and consider the expression:

if ${var+:} false; then ...

If the shell variable $var exists, the entire expression is replaced with :, if not, it returns an empty string.

Therefore the entire expression ${var+:} false becomes either : false (returning true) or false (returning false).

This comes down to a test for existence, which can be true even if the variable has no value assigned.

It is very cryptic, but as it happens, is one of the few tests for the existence of a variable that actually works in most, if not all, shells of Bourne descent.

Possible equivalents: (substitute any variable name here for var)

if [[ ${var+"is_set"} == is_set ]]; then ...

Or, probably more portable:

case ${var+"IS_SET"} in IS_SET) ...;; esac

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