Can lucidity be dreamed?

Know that I say this having never “experienced” a lucid dream before:
The human brain dreams in such a way that, upon awakening, recollection of a dream is no different from a memory of a physical event. While in a dream, the untrained brain will not differ that dream world from physical reality. One who has trained their mind to recognize a dream can then take control of it, making it a lucid dream. Nearly all non-lucid dreams relate to or are inspired by events or ideas experienced in the physical reality. My theoretical situation/question is: Those who have supposedly trained themselves to recognize a dream and become lucid have obviously thought about lucidity in depth in their waking hours. What if those thoughts in the awakened state inspire non-lucid dreams in which we dream of controlling a dream? So what if there is no such thing as a “trained mind?” Because the “untrained mind” cannot recognize a dream as a dream, we would simply wake up with the recollection of a seemingly lucid dream, as we would the recollection of falling had we fallen in our dream. So what if lucidity is no more than an unachievable concept?