Thanks for looking that up, Dadpad. Very interesting. No doubt in my mind that what I've seen my dogs and cats do is dreaming. All my dogs are gone now, but I still have four of my cats -- all well over 10 years young and still dreaming.
I used to laugh at myself when I'd pretend that my cats were saying one thing or another, but over the years I've decided that my anthropomorphisizing (gosh, is that a word?) was not only OK but not that unrealistic. So I don't believe my assumptions about my cats are a fantasy. I'll never know exactly what's going on inside their heads, of course, but what I see and hear outside, "in" their bodies, tells me I'm not, in many ways, that different from them.
Their behavior also tells me that they remember things for years, not just minutes, days or months. Yes, they seem to live in the here and now more than humans do, but more than once I've experienced them remembering something from the past that even I had forgotten. They probably don't "think" about the future as we do, but it isn't much of a stretch, at least for me, to interpret some of the things I see them do as anticipating/planning for something not in the present moment. Stalking a gopher, for example, implies something they are waiting for in the future. Well, maybe I am reaching too far for that one, and if so, please don't hesitate to laugh at me...