Here is a link to the original paper where C.A.R. Hoare introduces assertions. Yes "assert()" in C is the same. The D spec claims to provide C-style assertions. That means it should abide to what is described in this paper:http://sunnyday.mit.edu/16.355/Hoare-CACM-69.pdf If not, call it something else.
So you'll be happy if we call them D-style assertions?