Cannot access first and second members from an iterator on a templated std::map

Question

I want to write a template function to interpolate missing values within a std::map like this :

template<typename T1, typename T2>
T2 interpolate(const std::map<T1, T2> &data, const T1 at_x);

When I write the function with real type for the map, like std::map <unsigned short, double> , all is fine. But when I want to template the function, after I create an iterator for the map, I cannot access the ->first and ->second membres anymore from the iterator ? why ?

Here's the code of my templated function:

template<typename T1, typename T2>
T2 interpolate(const std::map<T1, T2> &data, const T1 at_x){

    typedef std::map<T1, T2>::const_iterator i_t;

    i_t right = data.upper_bound(at_x);
    i_t left;
    double delta;

    data.begin()->first; // this is ok for the compiler and the auto-completion of eclipse
    right->first; // this is rejected: "Field 'first' could not be resolved"

    // after range of keys in map:
    if (right == data.end()) {
        --right;
        left = right;
        --left;
        delta = ((double)at_x - left->first) / (right->first - left->first);
        return right->second + (delta - 1.0) * (right->second - left->second);
    }

    // before range of keys in map:
    if (right == data.begin()) {
        left = right;
        ++left;
        delta = ((double)at_x - left->first) / (right->first - left->first);
        return right->second + (delta - 1.0) * (right->second - left->second);
    }

    // within range of keys in map:
    left = right;
    --left;
    delta = ((double)at_x - left->first) / (right->first - left->first);
    return delta * right->second + (1.0 - delta) * left->second;
}

Many thanks for any help.

Samuel