Templated Class Friend Operator Member Function

I'm trying to get a friend function inside a templated class to compile, but the error message and warning I do not understand. I've made a demonstration of the issue. The error I'm getting is:

prog.cpp:8:57: error: non-class, non-variable partial specialization C operator+(const B& lhs, const C& rhs);

prog.cpp:15:59: warning: friend declaration 'C operator+(const B&, const C&)' declares a non-template function [-Wnon-template-friend] friend C operator+(const B& lhs, const C& rhs);

prog.cpp:15:59: note: (if this is not what you intended, make sure the function template has already been declared and add <> after the function name here)

#include <iostream>
using namespace std;

template<typename A, typename B>
class C;

template<typename A, typename B>
C<A, B> operator+<A, B>(const B& lhs, const C<A, B>& rhs);

template<typename A, typename B>
struct C
{
    A val_;
    C operator+(const C& other) const;
    friend C<A, B> operator+(const B& lhs, const C<A, B>& rhs);
};

template<typename A, typename B>
C<A, B> C<A, B>::operator+(const C<A, B>& other) const
{
    C<A, B> c;
    c.val_ = this->val_ + other.val_;
    return c;
}

template<typename A, typename B>
 C<A, B> operator+(const B& lhs, const C<A, B>& rhs)
{
    C<A, B> c;
    c.val_ = lhs + rhs.val_;
    return c;
}

int main() 
{
    C<string, char> c0,c1;
    c0.val_ = " C0 ";
    c1.val_ = " C1 ";
    cout << "Stuct:" << (c0 + c1).val_ << '\n';
    cout << "Friend:" << ('~' + c1).val_ << endl;
    return 0;
}