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 show the derivations from kbmag):
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 show the derivations from kbmag):
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 show the derivations from kbmag):
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 getshow the right presentation]derivations from kbmag):
#Initial equation number equations1:
#x*xi -> IdWord
#Initial equation number 2:=
[#xi*x -> IdWord
#Initial equation number 3:
#y*yi [x*xi,IdWord],-> IdWord
#Initial equation number 4:
#yi*y [xi*x,IdWord],-> IdWord
#Initial equation number 5:
#y^4 [y*yi,IdWord],-> x^3*yi
#Initial equation number 6:
#y*x*y [yi*y-> x^2
#New equation number 7,IdWord] from overlap 5,
3:
#x^3*yi^2->y^3
#New equation number [y^48,x*y*x] from overlap 4,
5:
#yi*x^3->y^4
#New equation number [y*x*y9,x^2] from overlap 6,
3:
#x^2*yi->y*x
#New equation number [x^2*yi10,y*x] from overlap 4,
6:
#yi*x^2->x*y
#New equation number [yi*x^211,x*y] from overlap 2,
7:
#xi*y^3->y*x*yi
#New equation number [xi*y^312,y*x*yi] from overlap 2,
9:
#xi*y*x->x*yi
#New equation number [xi*y*x13,x*yi] from overlap 10,
1:
#x*y*xi->yi*x
#New equation number [x*y*xi14,yi*x] from overlap 1,
11:
#x*y*x*yi->y^3
#New equation number [x*y*x*yi15,y^3] from overlap 11,
3:
#y*x*yi^2->xi*y^2
#New equation number [y*x*yi^216,xi*y^2] from overlap 12,
1:
#x*yi*xi->xi*y
#New equation number [x*yi*xi17,xi*y] from overlap 2,
13:
#xi*yi*x->y*xi
#New equation number [xi*yi*x18,y*xi] from overlap 4,
15:
#yi*xi*y^2->x*yi^2
#New equation number [yi*xi*y^219,x*yi^2] from overlap 2,
16:
#xi^2*y->yi*xi
#New equation number [xi^2*y20,yi*xi] from overlap 17,
1:
#y*xi^2->xi*yi
#New equation number [y*xi^221,xi*yi] from overlap 18,
3:
#x*yi^3->yi*xi*y
#New equation number [x*yi^322,yi*xi*y] from overlap 19,
3:
#yi*xi*yi->xi^2
#New equation number [yi*xi*yi23,xi^2] from overlap 2,
21:
#xi*yi*xi*y->yi^3
#New equation number [xi*yi*xi*y24,yi^3] from overlap 23,
3:
#yi^4->xi*yi*xi
#New equation number [yi^425,xi*yi*xi] from overlap 3,
24:
#y*xi*yi*xi->yi^3
#New equation number [y*xi*yi*xi26,yi^3] from overlap 25,
2:
#yi^3*x->y*xi*yi
#New equation number [yi^3*x27,y*xi*yi] from overlap 3,
26:
#y^2*xi*yi->yi^2*x
#New equation number [y^2*xi*yi28,yi^2*x] from overlap 27,
4:
#yi^2*x*y->y^2*xi
#New equation number [yi^2*x*y29,y^2*xi] from overlap 3,
28:
#y^3*xi->yi*x*y
#New equation number [y^3*xi30,yi*x*y] from overlap 29,
2:
#yi*x*y*x->y^3
#New equation number [yi*x*y*x31,y^3] from overlap 5,
5:
#y*x^3->x^3*y
#New equation number [y*x^332,xi*yi*xi] from overlap 5,
6:
#y^3*x^2->x*y*x^2*y
#New equation number [y^3*x^233,yi^2*xi] from overlap 8,
7:
#yi*x*y^3->x*y^2*x*yi
#New equation number [yi*x*y^334,x*y^2*x*yi] from overlap 7,
8:
#y*x^2*y*x->x^2*y^3
#New equation number [y*x*yi*x*y35,x^2*y^2*xi] from overlap 11,
6:
#y*x*yi*x*y->xi*y^2*x^2
#New equation number [xi*y^2*x36,x*yi*x*yi] from overlap 12,
9:
#xi*y^2*x->x*yi*x*yi
#New equation number [yi*x*yi*x37,x*y^2*xi] from overlap 10,
13:
#yi*x*yi*x->x*y^2*xi
#New equation number [y*x*yi*x*yi^238,xi*y^2*xi*y^2] from overlap 11,
15:
#y*x*yi*x*yi^2->xi*y^2*xi*y^2
#New equation number [xi*y*xi*y39,x*yi^2*xi] from overlap 12,
16:
#xi*y*xi*y->x*yi^2*xi
#New equation number [y*xi*y*xi40,xi*yi^2*x] from overlap 17,
13:
#y*xi*y*xi->xi*yi^2*x
#New equation number [yi*x*yi^2*xi*y41,x*yi^2*x*yi^2] from overlap 18,
15:
#yi*x*yi^2*xi*y->x*yi^2*x*yi^2
#New equation number [yi*xi*y*xi*yi42,x*y^3*x] from overlap 18,
20:
#yi*xi*y*xi*yi->x*yi^2*xi^2
#New equation number [yi^2*x*yi^2*x43,y*xi*yi^2*x*yi] from overlap 19,
20:
#yi*xi^3->xi^3*yi
#New equation number [y*xi*yi^2*xi*y44,yi^2*x*yi^2] from overlap 17,
21:
#y*xi*yi^3->xi*yi^2*xi*y
#New equation number [yi*x*y^2*xi*y45,y^2*xi*y^2*xi] from overlap 23,
20:
#yi^3*xi^2->xi*yi*xi^2*yi
#New equation number [yi*x*y^2*x46,y^3*x*yi] from overlap 22,
24:
#yi*xi^2*yi*xi->xi^2*yi^3
#New equation number [x*yi*x*yi^2*xi47,xi*y^2*xi*y] from overlap 25,
19:
#yi^3*xi*y->y*xi*yi^2*xi
#New equation number [y*x^2*y48,xi*yi^2*xi] from overlap 22,
26:
#xi^2*yi^2*x->x*yi^2*xi^2
#New equation number [xi*y*xi*yi^2*xi49,x^2*y^2*xi*y] from overlap 27, 26:
#yi^2*x*yi^2*x->y*xi*yi^2*x*yi
#New equation number 50, from [xi*yi*xi^2overlap 49,y*x^2] 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, [xi^2*yi*xifrom overlap 27,x^2*y] 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, [x^2*y^2*xfrom overlap 31,xi*y*xi*yi] 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 [x*y*x^232,yi*xi^2] 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, [x^3*yfrom overlap 56,xi*yi*xi] 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, [x^2*y*xfrom overlap 33,xi^2*yi] 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, [xi*yi^3from overlap 35,x^2*y^2] 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 [yi*xi^2*yioverlap 33,x*y^2*x] 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, [xi^3from overlap 4,x^3] 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, [x^2*y^3from overlap 4,xi*yi^2] 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, [x^4from overlap 4,xi^2] 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 [xi^2*yi^2overlap 1,x*y^3] 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 [y^3*x*yi*x70,yi^2*xi*y*xi] 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, [yi^2*xi^2from overlap 2,y^3*x] 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 [yi^3*xi73,y^2*x^2] 3:
#xi^3->x^3
#New equation number 79, from overlap 75, 4:
#x^2*y^3->xi*yi^2
#New equation number 80, [x^2*y^2*xi*y^2from overlap 1,xi*yi^2*x*yi^2] 78:
#x^4->xi^2
#New equation number 81, from overlap 2, 79:
#xi^2*yi^2->x*y^3
#New equation number 82, [xi*y^2*xi*y^2*xifrom overlap 29,y*xi*y^2*xi*y] 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 [y*xi*yi^2*x*yi^2*xi4,yi*x*yi^2*x*yi^2] 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 [x*y^2*x^2overlap 9,yi*xi*y*xi] 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 [yi*x*yi^2*x*yi^2*xi11,x*yi*x*yi^2*x*yi^2] 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 [xi*y*xi*yi^2*x*yi^2overlap 38,x*y^2*xi*y^2*xi*y] 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 [y^2*xi*y^2*xi*y^2overlap 41,x*y^2*x*yi*x*yi] 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 = 299, 246; 77 states; 0 secs. max len: lhs, rhs = 8, 8.
#System is confluent.
#Halting with 68 equations. #Exit status is 0
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 the right presentation]):
equations := [
[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]
]
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 show the derivations from kbmag):
#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 = 299, 246; 77 states; 0 secs. max len: lhs, rhs = 8, 8.
#System is confluent.
#Halting with 68 equations. #Exit status is 0
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 resepctivelyrespectively, idWord is 1 [edited to get the right presentation]):
equations := [
[x*xi,IdWord],
[xi*x,IdWord],
[y*yi,IdWord],
[yi*y,IdWord],
[x^2[y^4,xi]x*y*x],
[y^3[y*x*y,yi^2]x^2],
[y*x[x^2*yi,xi*yi]y*x],
[xi^2[yi*x^2,x]x*y],
[yi^3[xi*y^3,y^2]y*x*yi],
[yi*xi[xi*y*x,x*y]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],
[x*y*xi[yi*xi*y^2,yi*x]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],
[xi*y*xi[y^3*xi,x*yi*x]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],
[yi^2*x*yi^2[y*x*yi*x*yi^2,y^2*xi*y^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],
[y^2*xi*y^2*xi*y^2[yi*x*y^2*x,yi*x*yi^2*x*yi]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]
]
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 resepctively, idWord is 1):
equations := [
[x*xi,IdWord],
[xi*x,IdWord],
[y*yi,IdWord],
[yi*y,IdWord],
[x^2,xi],
[y^3,yi^2],
[y*x,xi*yi],
[xi^2,x],
[yi^3,y^2],
[yi*xi,x*y],
[xi*yi*x,y*xi],
[x*y*xi,yi*x],
[y^2*xi*yi,yi^2*x],
[yi^2*x*y,y^2*xi],
[xi*y*xi,x*yi*x],
[yi*x*yi*x,x*y^2*xi],
[yi^2*x*yi^2,y^2*xi*y^2],
[yi*x*y^2*xi*y,y^2*xi*y^2*xi],
[y^2*xi*y^2*xi*y^2,yi*x*yi^2*x*yi],
[xi*y^2*xi*y^2*xi,y*xi*y^2*xi*y]
]
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 the right presentation]):
equations := [
[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]
]
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