Non-null pointers in C/C++?

It's been some time since I started wondering why the C/C++ languages
do not have a non-null pointer (something like a copyable C++
reference)? In my opinion, using non-null pointers instead of the
"traditional" ones (wherever appropriate) would provide the following
benefits:
-allows restrictive programming techniques to be apllied at a lower
cost (e.g. if a pointer passed to a function may not be null, there
would be no need for a null-check before dereferencing it);
-allow for improvements in programming practices (having two types of
pointers will bring to attention the difference between them, namely
the possibility that "traditional" pointers can be null, hopefully
thus making it more obvious that "traditional" pointers should be
checked for null before thay are dereferenced);
-allow for more reliable code (having a "traditional" pointer where a
non-null pointer would do increases the risk of undefined behavior; on
the other hand, using a non-null pointer instead would be safer);
-provide compilers with further semantic information, thus enabling
them to perform more optimizations;
-make source code more concise (as a result of having fewer null-tests
when replacing "traditional" pointers with non-null ones).

What do you think?
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