Considering the enormous amount of brawn involved in demolition + transport + disposal and then the formwork + transport + mixing of 800kg of cement, moving the slab is far less work, far quicker, and far cheaper. Regardless of the labor and expense of demolition approach, the manufacture of cement carries a very large carbon footprint. R&R would be a -1 for the environment.
The underside is likely very rough. If you are willing to risk breaking the slab, it could be pried up, rails slipped underneath, then slid over. 3" pipe or some such could be used for rails. If you want to move it over N feet, the rails could be as short as 3+N. Pry up one side at a time, using something between the lever and slab to distribute the pressure. Lifting one side with a single 1:10 lever is: 0.5 * 800kg * 0.1 = 40kg = 90lbs ... that's completely doable by one person for anyone weighing more than 90lbs.
Alternatively, an off-center tripod could be built over the slab, and a come-along used to hoist the slab.
Alternatively, if it is to be moved it over N feet, then rails that are about N+1 feet long would suffice. This method combines the previous two. Pry up the slab and place the rails under the slab aligned with the far edge. Build an A-frame and stand it a foot away from the near edge. Run a strap around the slab at the near edge, attach a 30ft rope to the strap and route the rope over the A frame. Lean the A-frame towards the slab, then heave on the loose end of the rope while standing 20-25ft away. With enormous leverage, this will cause 1) the A-frame to become perpendicular, 2) lift the near edge of slab, and 3) pull the slab towards its new position by, say, 6", the far edge of the slab sliding along the 3" pipe rails. Reposition the A-frame and repeat.
Note: created this answer out of my several comments, added the 2nd alternative, then deleted the redundancies.