std::is_sorted and strictly less comparison?

Question

I do not understand well the std::is_sorted algorithm and its default behaviour. If we look to cppreference, it says that by default std::is_sorted use

Answer 1

Even if you only have the < operator you can figure out if two numbers are equivalent not necessarily equal.

if !(first < second) and !(second < first)
then first equivalent to second

In addition, as paxdiablo's solution actually mentioned first, you could implement is_sorted as going up the list and continually checking for < not to be true, if it is ever true you stop.

Here is the correct behavior of the function from cplusplus.com

template 
  bool is_sorted (ForwardIterator first, ForwardIterator last)
{
  if (first==last) return true;
  ForwardIterator next = first;
  while (++next!=last) {
    if (*next<*first)     // or, if (comp(*next,*first)) for version (2)
      return false;
    ++first;
  }
  return true;
}