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The most important guideline to bear in mind when checking for
features is to mimic as much as possible the intended use. 
Unfortunately, old versions of AC_CHECK_HEADER and
AC_CHECK_HEADERS failed to follow this idea, and called
the preprocessor, instead of the compiler, to check for headers.  As a
result, incompatibilities between headers went unnoticed during
configuration, and maintainers finally had to deal with this issue
elsewhere.
   
As of Autoconf 2.56 both checks are performed, and configure
complains loudly if the compiler and the preprocessor do not agree. 
For the time being the result used is that of the preprocessor, to give
maintainers time to adjust their configure.ac, but in the
future, only the compiler will be considered.
   
Consider the following example:
     $ cat number.h
     typedef int number;
     $ cat pi.h
     const number pi = 3;
     $ cat configure.ac
     AC_INIT([Example], [1.0], [bug-example@example.org])
     AC_CHECK_HEADERS([pi.h])
     $ autoconf -Wall
     $ ./configure
     checking for gcc... gcc
     checking for C compiler default output file name... a.out
     checking whether the C compiler works... yes
     checking whether we are cross compiling... no
     checking for suffix of executables...
     checking for suffix of object files... o
     checking whether we are using the GNU C compiler... yes
     checking whether gcc accepts -g... yes
     checking for gcc option to accept ISO C89... none needed
     checking how to run the C preprocessor... gcc -E
     checking for grep that handles long lines and -e... grep
     checking for egrep... grep -E
     checking for ANSI C header files... yes
     checking for sys/types.h... yes
     checking for sys/stat.h... yes
     checking for stdlib.h... yes
     checking for string.h... yes
     checking for memory.h... yes
     checking for strings.h... yes
     checking for inttypes.h... yes
     checking for stdint.h... yes
     checking for unistd.h... yes
     checking pi.h usability... no
     checking pi.h presence... yes
     configure: WARNING: pi.h: present but cannot be compiled
     configure: WARNING: pi.h:     check for missing prerequisite headers?
     configure: WARNING: pi.h: see the Autoconf documentation
     configure: WARNING: pi.h:     section "Present But Cannot Be Compiled"
     configure: WARNING: pi.h: proceeding with the preprocessor's result
     configure: WARNING: pi.h: in the future, the compiler will take precedence
     configure: WARNING:     ## -------------------------------------- ##
     configure: WARNING:     ## Report this to bug-example@example.org ##
     configure: WARNING:     ## -------------------------------------- ##
     checking for pi.h... yes
   The proper way the handle this case is using the fourth argument (see Generic Headers):
     $ cat configure.ac
     AC_INIT([Example], [1.0], [bug-example@example.org])
     AC_CHECK_HEADERS([number.h pi.h], [], [],
     [[#ifdef HAVE_NUMBER_H
     # include <number.h>
     #endif
     ]])
     $ autoconf -Wall
     $ ./configure
     checking for gcc... gcc
     checking for C compiler default output... a.out
     checking whether the C compiler works... yes
     checking whether we are cross compiling... no
     checking for suffix of executables...
     checking for suffix of object files... o
     checking whether we are using the GNU C compiler... yes
     checking whether gcc accepts -g... yes
     checking for gcc option to accept ANSI C... none needed
     checking for number.h... yes
     checking for pi.h... yes
   See Particular Headers, for a list of headers with their prerequisite.