[libc] Remove integration test target from check libc.

[PaulkaToast]

Summary

This is failing on the asan build because we use -nostdlib.
I also took this opportunity to make the target name match the naming structure we've been using.

[sivachandra]

s/check-libc-public-integration/libc-integration-test

It is cleaner, and easy to remember.

libc/test/src/CMakeLists.txt

COMMAND $<TARGET_FILE:libc-prototype-testgen> -o ${public_test}
           ${LIBC_SOURCE_DIR}/config/${LIBC_TARGET_OS}/api.td
 
   DEPENDS ${LIBC_SOURCE_DIR}/config/${LIBC_TARGET_OS}/api.td
           libc-prototype-testgen ${PUBLIC_HEADERS}
           llvmlibc llvmlibm
 )
 
 add_executable(
-  public_integration_test
+  check-libc-public-integration
   EXCLUDE_FROM_ALL
   ${public_test}
 )
 # Blank out default include directories to prevent accidentally including
 # system headers or our own internal headers.
 set_target_properties(
-  public_integration_test
+  check-libc-public-integration
   PROPERTIES
   INCLUDE_DIRECTORIES ""
 )
 # Only include we need is the include for cpp::IsSame and our generated
 # public headers.
 target_include_directories(
-  public_integration_test BEFORE
+  check-libc-public-integration BEFORE
   PRIVATE
     "${LIBC_SOURCE_DIR}/utils/CPP"
     "${LIBC_BUILD_DIR}/include"
 )
 target_compile_options(
-  public_integration_test
+  check-libc-public-integration
   PRIVATE
   -ffreestanding
 )
 target_link_options(
-  public_integration_test
+  check-libc-public-integration
   PRIVATE "-nostdlib"
 )
 set(library_files)
 foreach(library_name IN LISTS "llvmlibc;llvmlibm")
   get_target_property(library_file ${library_name} "LIBRARY_FILE")
   list(APPEND library_files ${library_file})
 endforeach()
 
 if(COMPILER_RESOURCE_DIR AND LLVM_LIBC_ENABLE_LINTING)
   add_custom_target(
-    public_integration_test-tidy
+    check-libc-public-integration-tidy
     VERBATIM
     COMMAND $<TARGET_FILE:clang-tidy> --system-headers
       --checks=-*,llvmlibc-restrict-system-libc-headers
       "--extra-arg=-resource-dir=${COMPILER_RESOURCE_DIR}"
       --header-filter=.*
       --warnings-as-errors=llvmlibc-*
       "-config={CheckOptions: [{key: llvmlibc-restrict-system-libc-headers.Includes, value: '-*, linux/*, asm/*.h, asm-generic/*.h'}]}"
       --quiet
       -p ${PROJECT_BINARY_DIR}
       ${public_test}
     DEPENDS
       clang-tidy ${public_test}
   )
-  add_dependencies(check-libc public_integration_test-tidy)
+  add_dependencies(check-libc-public-integration check-libc-public-integration-tidy)
 endif()
 
-target_link_libraries(public_integration_test
+target_link_libraries(check-libc-public-integration
   PRIVATE
   ${library_files}
 )
-add_dependencies(check-libc public_integration_test)