A previous answerer (Steve Huntsman) has given a specific derivation for your example. But I made up a short input file for kbmag, and it immediately came back with a "confluent" system of relations (a particular system which has a technical property that when you just do a series of string substitutions you always get the same answer no matter what order you do them in). For your edification, here they are (xi and yi are x^-1 and yi^-1 respectively, idWord is 1 [edited to get show the right presentation])derivations from kbmag):
equations #Initial equation number 1: #x*xi -> IdWord#Initial equation number 2: #xi*x -> IdWord#Initial equation number 3: #y*yi -> IdWord#Initial equation number 4: #yi*y -> IdWord#Initial equation number 5: #y^4 -> x^3*yi#Initial equation number 6: #y*x*y -> x^2#New equation number 7, from overlap 5, 3: #x^3*yi^2->y^3#New equation number 8, from overlap 4, 5: #yi*x^3->y^4#New equation number 9, from overlap 6, 3: #x^2*yi->y*x#New equation number 10, from overlap 4, 6: #yi*x^2->x*y#New equation number 11, from overlap 2, 7: #xi*y^3->y*x*yi#New equation number 12, from overlap 2, 9: #xi*y*x->x*yi#New equation number 13, from overlap 10, 1: #x*y*xi->yi*x#New equation number 14, from overlap 1, 11: #x*y*x*yi->y^3#New equation number 15, from overlap 11, 3: #y*x*yi^2->xi*y^2#New equation number 16, from overlap 12, 1: #x*yi*xi->xi*y#New equation number 17, from overlap 2, 13: #xi*yi*x->y*xi#New equation number 18, from overlap 4, 15: #yi*xi*y^2->x*yi^2#New equation number 19, from overlap 2, 16: #xi^2*y->yi*xi#New equation number 20, from overlap 17, 1: #y*xi^2->xi*yi#New equation number 21, from overlap 18, 3: #x*yi^3->yi*xi*y#New equation number 22, from overlap 19, 3: #yi*xi*yi->xi^2#New equation number 23, from overlap 2, 21: #xi*yi*xi*y->yi^3#New equation number 24, from overlap 23, 3: #yi^4->xi*yi*xi#New equation number 25, from overlap 3, 24: #y*xi*yi*xi->yi^3#New equation number 26, from overlap 25, 2: #yi^3*x->y*xi*yi#New equation number 27, from overlap 3, 26: #y^2*xi*yi->yi^2*x#New equation number 28, from overlap 27, 4: #yi^2*x*y->y^2*xi#New equation number 29, from overlap 3, 28: #y^3*xi->yi*x*y#New equation number 30, from overlap 29, 2: #yi*x*y*x->y^3#New equation number 31, from overlap 5, 5: #y*x^3->x^3*y#New equation number 32, from overlap 5, 6: #y^3*x^2->x*y*x^2*y#New equation number 33, from overlap 8, 7: #yi*x*y^3->x*y^2*x*yi#New equation number 34, from overlap 7, 8: #y*x^2*y*x->x^2*y^3#New equation number 35, from overlap 11, 6: #y*x*yi*x*y->xi*y^2*x^2#New equation number 36, from overlap 12, 9: #xi*y^2*x->x*yi*x*yi#New equation number 37, from overlap 10, 13: #yi*x*yi*x->x*y^2*xi#New equation number 38, from overlap 11, 15: #y*x*yi*x*yi^2->xi*y^2*xi*y^2#New equation number 39, from overlap 12, 16: #xi*y*xi*y->x*yi^2*xi#New equation number 40, from overlap 17, 13: #y*xi*y*xi->xi*yi^2*x#New equation number 41, from overlap 18, 15: #yi*x*yi^2*xi*y->x*yi^2*x*yi^2#New equation number 42, from overlap 18, 20: #yi*xi*y*xi*yi->x*yi^2*xi^2#New equation number 43, from overlap 19, 20: #yi*xi^3->xi^3*yi#New equation number 44, from overlap 17, 21: #y*xi*yi^3->xi*yi^2*xi*y#New equation number 45, from overlap 23, 20: #yi^3*xi^2->xi*yi*xi^2*yi#New equation number 46, from overlap 22, 24: #yi*xi^2*yi*xi->xi^2*yi^3#New equation number 47, from overlap 25, 19: #yi^3*xi*y->y*xi*yi^2*xi#New equation number 48, from overlap 22, 26: #xi^2*yi^2*x->x*yi^2*xi^2#New equation number 49, from overlap 27, 26: #yi^2*x*yi^2*x->y*xi*yi^2*x*yi#New equation number 50, from overlap 49, 1: #y*xi*yi^2*xi*y->yi^2*x*yi^2#New equation number 51, from overlap 7, 28: #x^3*y^2*xi->y^2*x^2#New equation number 52, from overlap 27, 28: #yi*x*y^2*xi*y->y^2*xi*y^2*xi#New equation number 53, from overlap 29, 12: #yi*x*y^2*x->y^3*x*yi#New equation number 54, from overlap 31, 7: #x^3*y^2*x*yi->y*x^2*y^3#New equation number 55, from overlap 32, 13: #x*y*x^2*y^2*xi->y^3*x*yi*x#New equation number 56, from overlap 32, 14: #x*y*x^2*y^2*x*yi->y^2*x^2*y^2#New equation number 57, from overlap 2, 56: #y*x^2*y^2*x*yi->x^2*y^2*xi*y^2#New equation number 58, from overlap 56, 4: #y^2*x^2*y^3->x*y*x^2*y^2*x#New equation number 59, from overlap 57, 4: #x^2*y^3*x*yi->y*x^2*y^2*x#New equation number 60, from overlap 33, 34: #y^2*x^2*y^2*xi*y^2->x*y^2*x^2*y^2*x#New equation number 61, from overlap 11, 35: #x^2*y^2*xi*y^2*xi->y*x^2*y^2*xi*y#New equation number 62, from overlap 35, 25: #x*yi*x*yi^2*xi->xi*y^2*xi*y#New equation number 63, from overlap 35, 32: #x^2*y^2*x^2*y^2*xi->y*x^2*y^2*x^2*y#New equation number 64, from overlap 33, 35: #y^2*x^2*y^2*xi*y->yi*x*yi^2*x*yi^2#New equation number 65, from overlap 64, 3: #y^2*x^2*y^2->yi^2*x*yi^2#New equation number 66, from overlap 4, 64: #y*x^2*y^2*xi*y->xi*yi^2*x*yi^2*xi#New equation number 67, from overlap 65, 3: #y*x^2*y->xi*yi^2*xi#New equation number 68, from overlap 4, 66: #xi*y*xi*yi^2*xi->x^2*y^2*xi*y#New equation number 69, from overlap 67, 3: #xi*yi*xi^2->y*x^2#New equation number 70, from overlap 4, 67: #xi^2*yi*xi->x^2*y#New equation number 71, from overlap 68, 2: #x^2*y^2*x->xi*y*xi*yi#New equation number 72, from overlap 1, 69: #x*y*x^2->yi*xi^2#New equation number 73, from overlap 69, 2: #x^3*y->xi*yi*xi#New equation number 74, from overlap 70, 2: #x^2*y*x->xi^2*yi#New equation number 75, from overlap 71, 1: #xi*yi^3->x^2*y^2#New equation number 76, from overlap 2, 71: #yi*xi^2*yi->x*y^2*x#New equation number 77, from overlap 72, 1: #xi^3*yi->x*y*x#New equation number 78, from overlap 73, 3: #xi^3->x^3#New equation number 79, from overlap 75, 4: #x^2*y^3->xi*yi^2#New equation number 80, from overlap 1, 78: #x^4->xi^2#New equation number 81, from overlap 2, 79: #xi^2*yi^2->x*y^3#New equation number 82, from overlap 29, 36: #y^3*x*yi*x*yi->x*y^2*x*yi*x#New equation number 83, from overlap 7, 37: #y^3*x*yi*x->yi^2*xi*y*xi#New equation number 84, from overlap 4, 83: #xi*yi^2*xi^2->y*x*yi*x#New equation number 85, from overlap 1, 84: #yi^2*xi^2->y^3*x#New equation number 86, from overlap 9, 37: #yi^3*xi->y^2*x^2#New equation number 87, from overlap 38, 8: #x^2*y^2*xi*y^2->xi*yi^2*x*yi^2#New equation number 88, from overlap 11, 38: #xi*y^2*xi*y^2*xi*y->y^2*x*yi*x*yi^2#New equation number 89, from overlap 88, 3: #xi*y^2*xi*y^2*xi->y*xi*y^2*xi*y#New equation number 90, from overlap 38, 37: #y*xi*yi^2*x*yi^2*xi->yi*x*yi^2*x*yi^2#New equation number 91, from overlap 39, 39: #x*y^2*x^2->yi*xi*y*xi#New equation number 92, from overlap 41, 39: #yi*x*yi^2*x*yi^2*xi->x*yi*x*yi^2*x*yi^2#New equation number 93, from overlap 42, 47: #xi*y*xi*yi^2*x*yi^2->x*y^2*xi*y^2*xi*y#New equation number 94, from overlap 93, 4: #x*y^2*xi*y^2*xi*y^2->xi*y*xi*yi^2*x*yi#New equation number 95, from overlap 2, 94: #y^2*xi*y^2*xi*y^2->x*y^2*x*yi*x*yi#68 eqns; total len: lhs, rhs = [ [x*xi,IdWord], [xi*x,IdWord], [y*yi,IdWord], [yi*y,IdWord], [y^4,x*y*x], [y*x*y,x^2], [x^2*yi,y*x], [yi*x^2,x*y], [xi*y^3,y*x*yi], [xi*y*x,x*yi], [x*y*xi,yi*x], [x*y*x*yi,y^3], [y*x*yi^2,xi*y^2], [x*yi*xi,xi*y], [xi*yi*x,y*xi], [yi*xi*y^2,x*yi^2], [xi^2*y,yi*xi], [y*xi^2,xi*yi], [x*yi^3,yi*xi*y], [yi*xi*yi,xi^2], [xi*yi*xi*y,yi^3], [yi^4,xi*yi*xi], [y*xi*yi*xi,yi^3], [yi^3*x,y*xi*yi], [y^2*xi*yi,yi^2*x], [yi^2*x*y,y^2*xi], [y^3*xi,yi*x*y], [yi*x*y*x,y^3], [y*x^3,xi*yi*xi], [y^3*x^2,yi^2*xi], [yi*x*y^3,x*y^2*x*yi], [y*x*yi*x*y,x^2*y^2*xi], [xi*y^2*x,x*yi*x*yi], [yi*x*yi*x,x*y^2*xi], [y*x*yi*x*yi^2,xi*y^2*xi*y^2], [xi*y*xi*y,x*yi^2*xi], [y*xi*y*xi,xi*yi^2*x], [yi*x*yi^2*xi*y,x*yi^2*x*yi^2], [yi*xi*y*xi*yi,x*y^3*x], [yi^2*x*yi^2*x,y*xi*yi^2*x*yi], [y*xi*yi^2*xi*y,yi^2*x*yi^2], [yi*x*y^2*xi*y,y^2*xi*y^2*xi], [yi*x*y^2*x,y^3*x*yi], [x*yi*x*yi^2*xi,xi*y^2*xi*y], [y*x^2*y,xi*yi^2*xi], [xi*y*xi*yi^2*xi,x^2*y^2*xi*y], [xi*yi*xi^2,y*x^2], [xi^2*yi*xi,x^2*y], [x^2*y^2*x,xi*y*xi*yi], [x*y*x^2,yi*xi^2], [x^3*y,xi*yi*xi], [x^2*y*x,xi^2*yi], [xi*yi^3,x^2*y^2], [yi*xi^2*yi,x*y^2*x], [xi^3,x^3], [x^2*y^3,xi*yi^2], [x^4,xi^2], [xi^2*yi^2,x*y^3], [y^3*x*yi*x,yi^2*xi*y*xi], [yi^2*xi^2,y^3*x], [yi^3*xi,y^2*x^2], [x^2*y^2*xi*y^2,xi*yi^2*x*yi^2], [xi*y^2*xi*y^2*xi,y*xi*y^2*xi*y], [y*xi*yi^2*x*yi^2*xi,yi*x*yi^2*x*yi^2], [x*y^2*x^2,yi*xi*y*xi], [yi*x*yi^2*x*yi^2*xi,x*yi*x*yi^2*x*yi^2], [xi*y*xi*yi^2*x*yi^2,x*y^2*xi*y^2*xi*y], [y^2*xi*y^2*xi*y^2,x*y^2*x*yi*x*yi]299, 246; 77 states; 0 secs. max len: lhs, rhs = 8, 8.
#System is confluent.
#Halting with 68 equations. #Exit status is 0
