
What about this:

 (local
  (defthm try-me-8a
    (implies (and (integerp x)
		  (<= 0 x)
		  (integerp y)
		  (<= 0 y)
		  )
	     (equal (floor x (* 2 y))
		    (if (logbitp 0 (floor x y))
			(+ -1/2 (* 1/2 (floor x y)))
		      (* 1/2 (floor x y)))))
    :hints (("Goal" :in-theory (enable logbitp)))))

