Well, as far as the RAIM and baro-aiding, Here's how RAIM works on a conceptual, mathematical level. A GPS receiver computes a position by solving for 4 unknowns: Coordinates in 3 Dimensions (X,Y,Z) and receiver time offset from the GPS satellite system. The latitude and Longitude you see displayed on the receiver are translated from the GPS systems native geocentric (earth centered) coordinate system. As you may remember from algebra, to solve a system of equations for 4 unknowns, you have to have 4 equations. Each measurement to a different satellite adds one equation to the system, so if you have measurements to 4 satellites, the receiver can solve for X,Y,Z and dT. However, with only 4 satellites you have no way of checking that the 4 signals are all correct. However if you have measurements to 5 satellites you have a redundant, or over-determined solution, and with a redundant solution you can tell if the measurement from one satellite doesn't fit the solution. But you have no way of telling which measurement is bad, just that the 5 measurements don't all fit the solution within a certain tolerance. If you have measurements to 6 satellites, that gives you a tie breaker, which allows you to determine which satellite signal is bad, and drop it from the solution.
Now, how does baro-aiding fit into that? Baro-aiding can be used to add one more observation to the solution. If you have baro-aiding, you know what your altitude is, and that essentially reduces the number of unknowns that you have.
If it helps, you can think of it as the receiver solving for lat, long, altitude, and time offset, requiring 4 satellites for a solution, but with baroaiding, altitude is known, so you can solve for lat, long, and time offset with measurements from 3 satellites. That's not quite accurate, like I said the receiver is calculating in Geocentric Cartesian coordinates, but conceptually it's close enough that it's not incorrect, exactly. This was common in the early days of GPS, especially for marine applications, when the GPS satellite system was not yet complete and coverage was spotty. GPS receivers had an option of the user entering the elevation if known (sea level plus antenna height for a boat on the ocean ) which allowed navigation on only 3 GPS satellites. Baro-aiding does the same thing, it automatically enters the altitude derived from the barometric altimeter to the GPS solution and reduces the number of satellites required by 1; 3 satellites fro a minimum position solution, 4 satellites for an over determined position that checks for a good solution, and 5 satellites for the minimum satellites to catch an error, and identify the satellite signal in error.
So, that's the basic concept. I would expect that the TSOs and other standards under whcih the systems are designed and approved would require a higher level of redundancy, so it's likely that "RAIM Availability" from a regulatory standpoint requires more satellites than in my description, but at the algebra/geometry level, that's what's required.