Multiple definition of inline functions when linking static libs

I have a C++ program that I compile with mingw (gcc for Windows). Using the TDM release of mingw which includes gcc 4.4.1. The executable links to two static library (.a) files: On of them is a third-party library written in C; the other is a C++ library, written by me, that uses the C library provides my own C++ API on top.

An (in my view, excessive) portion of the C library's functionality is implemented in inline functions. You can't avoid including the inline functions when you use the C library's API, but when I try to link it all together, I'm getting link errors saying there is a multiple definition of all of the inline functions - both ones I have called in my C++ wrapper library and ones which I have not, basically anything defined inline in the headers has gotten a function created for it in both the C library and the C++ library.

It doesn't cause multiple definition errors when the include files are used multiple times in different .c or .cpp files in the same project; the problem is just that it generates one definition per library.

How/why is the compiler generating functions and symbols for these inline functions in both libraries? How can I force it to stop generating them in my code? Is there a tool I can run to strip the duplicate functions from the .a file, or a way to make the linker ignore multiple definitions?

(FYI, the third party library does include #ifdef __cplusplus and extern "C" guards in all its headers; anyway if that were the problem, it would not cause a multiple definition of symbol, it would cause the opposite problem because the symbol would be undefined or at least different.)

Notably, the link errors do NOT occur if I link to the third party C library's DLL; however then I get strange runtime failures that seem to have to do with my code having its own version of functions it should be calling from the DLL. (As if the compiler is creating local versions of functions I didn't ask for.)

Similar versions of this question have been asked before, however, I didn't find the answer to my situation in any of these:

The answer to this question was that the poster was multiply defining variables, my problem is multiple definition of inline functions: Repeated Multiple Definition Errors from including same header in multiple cpps

This was an MSVC program, but I'm using mingw; also, the poster's problem in this question was the definition of a C++ class constructor outside of the class body in a header, while my problem is with C functions that are inline: Static Lib Multiple Definition Problem

This fool renamed all his C code as C++ files and his C code wasn't C++-safe: Multiple definition of lots of std:: functions when linking

This one was just wanting to know why violating the one definition rule was not an error: unpredictable behavior of Inline functions with different definitions

First you have to understand the C99 inline model - perhaps there is something wrong with your headers. There are two kind of definitions for inline functions with external (non-static) linkage

External definition
This definition of a function can only appear once in the whole program, in a designated TU. It provides an exported function that can be used from other TUs.
Inline definition
These appear in every TU where declared as a separate definition. The definitions do not need to be identical to one another or to the external definition. If used internal in a library, they can do omit checking on function arguments that would otherwise be done in the external definition.

Each definition of the function has its own local static variables, because their local declarations have no linkage (they are not shared like in C++). A definition of a non-static inline function will be an inline definition if

Every function declaration in a TU includes the specifier inline, and
No function declaration in a TU includes the specifier extern.

Otherwise, the definition that must appear in that TU (because inline functions must be defined in the same TU where declared) is an external definition. In a call to an inline function it's unspecified whether the external or the inline definition is used. However, because the function defined in all cases is still the same (because it has external linkage), the address of it compares equal in all cases, no matter how many inline definitions appear. So if you take the function's address, it's likely the compiler resolves to the external definition (especially if optimizations are disabled).

An example that demonstrates a wrong use of inline, because it includes an external definition of a function twice in two TUs, causing a multiple definition error

// included into two TUs
void f(void); // no inline specifier
inline void f(void) { }

The following program is dangerous, because the compiler is free to use the external definition, but the program does not provide one

// main.c, only TU of the program
inline void g(void) {
  printf("inline definition\n");
}

int main(void) {
  g(); // could use external definition!
}

I have made some test cases using GCC that demonstrate the mechanism further:

main.c

#include <stdio.h>

inline void f(void);

// inline definition of 'f'
inline void f(void) {
  printf("inline def main.c\n");
}

// defined in TU of second inline definition
void g(void);

// defined in TU of external definition
void h(void);

int main(void) {
  // unspecified whether external definition is used!
  f();
  g();
  h();

  // will probably use external definition. But since we won't compare
  // the address taken, the compiler can still use the inline definition.
  // To prevent it, i tried and succeeded using "volatile". 
  void (*volatile fp)() = &f;
  fp();
  return 0;
}

main1.c

#include <stdio.h>

inline void f(void);

// inline definition of 'f'
inline void f(void) {
  printf("inline def main1.c\n");
}

void g(void) {
  f();
}

main2.c

#include <stdio.h>

// external definition!
extern inline void f(void);

inline void f(void) {
  printf("external def\n");
}


void h(void) {
  f(); // calls external def
}

Now, the program outputs what we expected!

$ gcc -std=c99 -O2 main.c main1.c main2.c
inline def main.c
inline def main1.c
external def
external def

Looking at the symbol table, we will see that the symbol of an inline definition is not exported (from main1.o), while an external definition is exported (from main2.o).

Now, if your static libraries each have an external definition of their inline functions (as they should), they will naturally conflict with each other. The solution is to make the inline functions static or just to rename them. These will always provide external definitions (so they are full fledged definitions), but they are not exported because they have internal linkage, thus not conflicting

static inline void f(void) {
  printf("i'm unique in every TU\n");
}

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