Is the abbreviated pointer comparison "if(p)" to test for non-null pointers valid? What if the internal representation for null pointers is nonzero?

When C requires the boolean value of an expression (in the if, while, for, and do statements, and with the &&, ,!, and ?: operators), a false value is produced when the expression compares equal to zero, and a true value otherwise. That is, whenever one writes
if(expr)
where "expr" is any expression at all, the compiler essentially acts as if it had been written as
if(expr != 0)
Substituting the trivial pointer expression "p" for "expr," we have
if(p) is equivalent to if(p != 0)
and this is a comparison context, so the compiler can tell that the (implicit) 0 is a null pointer, and use the correct value. There is no trickery involved here; compilers do work this way, and generate identical code for both statements. The internal representation of a pointer does _not_ matter.
The Boolean negation operator, !, can be described as follows:
!expr is essentially equivalent to expr?0:1
It is left as an exercise for the reader to show that
if(!p) is equivalent to if(p == 0)