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In the cross-compilation case, setup.py incorrectly adds /usr/include to self.compiler.include_dirs, and results in the following invalid compilation line:
/home/thomas/projets/buildroot/output/host/usr/bin/arm-none-linux-gnueabi-gcc
 -fPIC -Wno-unused-result -Werror=declaration-after-statement -DNDEBUG -g
 -O3 -Wall -Wstrict-prototypes -D_LARGEFILE_SOURCE -D_LARGEFILE64_SOURCE
 -D_FILE_OFFSET_BITS=64 -pipe -Os
 -I./Include -I/usr/include -I. -IInclude
 -I/home/thomas/projets/buildroot/output/host/usr/arm-buildroot-linux-gnueabi/sysroot/usr/include
 -I/home/thomas/projets/buildroot/output/build/python3-3.4.0b1/Include
 -I/home/thomas/projets/buildroot/output/build/python3-3.4.0b1
 -c /home/thomas/projets/buildroot/output/build/python3-3.4.0b1/Modules/_struct.c
 -o build/temp.linux-arm-3.4/home/thomas/projets/buildroot/output/build/python3-3.4.0b1/Modules/_struct.o
cc1: warning: include location "/usr/include" is unsafe for cross-compilation [-Wpoison-system-directories]
The -I/usr/include is wrong when cross compiling, so we disable adding INCLUDEDIR and LIBDIR from the host when cross compiling.

Can we have a patch review on this issue please.

The patch looks good to me, and works in my setup.

Got the same issue here on building Python for Android. See https://circleci.com/gh/yan12125/python3-android/11 for an example build log. /usr/include is erroneously included and leads to build failures:
building 'xxlimited' extension
/home/ubuntu/android-ndk-r13/toolchains/llvm/prebuilt/linux-x86_64/bin/clang -target aarch64-none-linux-android -gcc-toolchain /home/ubuntu/android-ndk-r13/toolchains/aarch64-linux-android-4.9/prebuilt/linux-x86_64 --sysroot=/home/ubuntu/android-ndk-r13/platforms/android-21/arch-arm64/usr -fPIC -Wno-unused-result -Wsign-compare -Wunreachable-code -DNDEBUG -g -fwrapv -O3 -Wall -Wstrict-prototypes -fPIE -fno-integrated-as -fPIE -fno-integrated-as -std=c99 -Wextra -Wno-unused-result -Wno-unused-parameter -Wno-missing-field-initializers -DPy_LIMITED_API=0x03050000 -I../Include -I/usr/include -IObjects -IPython -I. -I/home/ubuntu/python3-android/build/21-aarch64-linux-android-4.9/usr/include -I/home/ubuntu/python3-android/src/cpython/Include -I/home/ubuntu/python3-android/src/cpython/build-target -c /home/ubuntu/python3-android/src/cpython/Modules/xxlimited.c -o build/temp.linux-aarch64-3.7/home/ubuntu/python3-android/src/cpython/Modules/xxlimited.o
In file included from /home/ubuntu/python3-android/src/cpython/Modules/xxlimited.c:17:
In file included from ../Include/Python.h:11:
In file included from /usr/include/limits.h:25:
/usr/include/features.h:398:10: fatal error: 'gnu/stubs.h' file not found
#include <gnu/stubs.h>
 ^
1 error generated.
The patch looks reasonable and still applies to the default branch. With this patch the build is fine: https://circleci.com/gh/yan12125/python3-android/12 

Modified this issue a bit and add some experts on the build system

When cross compiling, setup.py must not look for headers in the directory for installing C header files and must not look for libraries in the directory for installing object code libraries of the _native_ interpreter which is being used to run setup.py.
Thomas patch updated to the tip of the default branch.

this assumption is wrong for the multiarch case.

xdegaye's explanation is incorrect. The actual scene is much more complicated. First, let me explain why things may be broken. The wrong directory is included only if all of the following conditions are met:
1. PYTHON_FOR_BUILD is not installed with --prefix=/usr. A common example is those which are installed with pyenv. This condition is here simply because of `os.path.normpath(sys.base_prefix) != '/usr'`
2. The build script uses $DESTDIR convention to compile CPython
3. Header files in $DESTDIR$PREFIX/include are not equivalent to $PREFIX/include. Here "equivalent" means they are for the same set of architecture/kernel/... This is a common case when cross compiling, while a famous exception is the multiarch framework on Debian-based systems.
For example, if I use --prefix=/usr and DESTDIR=/tmp/python3-android/build, INCLUDEDIR will be /usr/include, which should not be included. On the other hand, if I use --prefix=/tmp/python3-android/build/usr and omit DESTDIR, INCLUDEDIR is correct /tmp/python3-android/build/usr/include. With $DESTDIR specified, the build script has the responsibility to specify $DESTDIR$PREFIX/include in CPPFLAGS, or CPython interpreter core won't even build, so skipping adding INCLUDEDIR in setup.py is safe. However, for the latter case (not using DESTDIR), the build script still needs to specify correct CPPFLAGS. I see it as a CPython limitation and worth documentation.
For Debian's multiarch framework, paths are already handled by add_multiarch_paths(). I'm not sure whether $PREFIX/include should still be included or not. (I don't use Debian-based systems)

> this assumption is wrong for the multiarch case.
Please explain why it is wrong.

New changeset d1b400943483 by Xavier de Gaye in branch '3.6':
Issue #20211: Do not add the directory for installing C header files and
https://hg.python.org/cpython/rev/d1b400943483
New changeset fcc9f19fcc13 by Xavier de Gaye in branch 'default':
Issue #20211: Merge 3.6.
https://hg.python.org/cpython/rev/fcc9f19fcc13 

>> this assumption is wrong for the multiarch case.
>
> Please explain why it is wrong.
Sadly no explanations have been given, we have to take Matthias word for it.
This new patch updates the add_multiarch_paths() method and includes LIBDIR and INCLUDEDIR in their corresponding search paths for the multiarch case.

Same patch, but this one is properly set to be handled by Rietveld.