The question is not clear, but there are 2 things which happen before a program gets created,
Compiling (requires prototypes / declarations)
Linking (requires definitions).
Header information is needed for knowing prototypes. Even this would compile fine:
int printf ( const char * format, ... );
int main()
{
printf("Hello world");
}
On linking there will be no issues because the printf function is found in the C standard library, so on linking it will look into the standard directories (of the compiler where the library is kept - bin/lib folder) and link the function.
The source only needs to know the prototype. The problem a programmer will have in this case:
int my_printf ( const char * format, ... );
int main()
{
my_printf("Hello world");
}
The above will compile, but when linking my_printf your code will not have a definition so it will give an error on linking.