Explain the purpose of the training order leg-yield shoulder-in haunches-in half-pass?

The sequence of leg-yield, shoulder-in, haunches-in, and half-pass represents one of the most intelligently organized progressions in all of classical horsemanship — a training order in which each movement builds on what the preceding one has developed and introduces one additional dimension of difficulty that prepares the horse for the movement that follows. Leg-yield is appropriately the first movement in the sequence because it asks for the simplest version of lateral movement — sideways and forward simultaneously — without requiring the horse to maintain a specific bend through his body. The horse moves sideways away from the rider's leg while remaining approximately parallel to the long side, crossing his inside legs in front of his outside legs in a movement that develops the horse's understanding of the lateral leg aid, his willingness to move his whole body sideways in response to a specific leg, and the crossing and adduction of the hind legs that all subsequent lateral work requires at progressively greater intensity. Shoulder-in is the second movement and the one that most classical trainers regard as the most important single gymnastic exercise available. Shoulder-in introduces the element that leg-yield deliberately excluded: a uniform body bend from poll to tail in the direction of the movement, combined with the horse traveling on three tracks rather than two. The specific gymnastic demand that shoulder-in places on the inside hind leg — stepping under the body and carrying rather than simply crossing — develops the carrying capacity of that hind leg in a way that leg-yield alone cannot, and it is that carrying capacity that collection specifically requires. Haunches-in introduces the third dimension: the displacement of the hindquarters to the inside rather than the displacement of the forehand. In haunches-in the horse maintains a bend in the direction of travel while his hindquarters are brought in off the rail so that he travels on three tracks with his inside hind following the track of the outside fore. This reverses the lateral orientation of shoulder-in in a way that develops both the horse's lateral suppleness in the opposite direction of bend and the inside hind leg's ability to cross in front of the outside hind — the specific movement that half-pass requires. Half-pass is the culminating movement of the sequence and synthesizes all of the qualities that the preceding three movements have developed in combination rather than separately. The horse moves forward and sideways simultaneously while maintaining a clear bend in the direction of travel across a diagonal rather than along a rail — requiring the inside hind leg to step deeply under and across with each stride in a movement that combines the adduction of leg-yield, the carrying of shoulder-in, and the crossing of haunches-in in a single sophisticated gymnastic demand. A horse that reaches the half-pass through this systematic preparation performs it with the physical capacity and the communicative understanding that allow the movement to be genuinely gymnastic rather than forced.