The relative rewrite relation R/S is considered where R is the following TRS
a(b(a(b(a(x1))))) | → | x1 | (1) |
and S is the following TRS.
a(b(x1)) | → | b(b(a(a(x1)))) | (2) |
{a(☐), b(☐)}
We obtain the transformed TRSa(a(b(a(b(a(x1)))))) | → | a(x1) | (3) |
b(a(b(a(b(a(x1)))))) | → | b(x1) | (4) |
a(a(b(x1))) | → | a(b(b(a(a(x1))))) | (5) |
b(a(b(x1))) | → | b(b(b(a(a(x1))))) | (6) |
Root-labeling is applied.
We obtain the labeled TRSaa(ab(ba(ab(ba(aa(x1)))))) | → | aa(x1) | (7) |
aa(ab(ba(ab(ba(ab(x1)))))) | → | ab(x1) | (8) |
ba(ab(ba(ab(ba(aa(x1)))))) | → | ba(x1) | (9) |
ba(ab(ba(ab(ba(ab(x1)))))) | → | bb(x1) | (10) |
aa(ab(ba(x1))) | → | ab(bb(ba(aa(aa(x1))))) | (11) |
aa(ab(bb(x1))) | → | ab(bb(ba(aa(ab(x1))))) | (12) |
ba(ab(ba(x1))) | → | bb(bb(ba(aa(aa(x1))))) | (13) |
ba(ab(bb(x1))) | → | bb(bb(ba(aa(ab(x1))))) | (14) |
{aa(☐), ab(☐), ba(☐), bb(☐)}
We obtain the transformed TRSaa(ab(ba(ab(ba(aa(x1)))))) | → | aa(x1) | (7) |
ba(ab(ba(ab(ba(aa(x1)))))) | → | ba(x1) | (9) |
aa(aa(ab(ba(ab(ba(ab(x1))))))) | → | aa(ab(x1)) | (15) |
ab(aa(ab(ba(ab(ba(ab(x1))))))) | → | ab(ab(x1)) | (16) |
ba(aa(ab(ba(ab(ba(ab(x1))))))) | → | ba(ab(x1)) | (17) |
bb(aa(ab(ba(ab(ba(ab(x1))))))) | → | bb(ab(x1)) | (18) |
aa(ba(ab(ba(ab(ba(ab(x1))))))) | → | aa(bb(x1)) | (19) |
ab(ba(ab(ba(ab(ba(ab(x1))))))) | → | ab(bb(x1)) | (20) |
ba(ba(ab(ba(ab(ba(ab(x1))))))) | → | ba(bb(x1)) | (21) |
bb(ba(ab(ba(ab(ba(ab(x1))))))) | → | bb(bb(x1)) | (22) |
aa(aa(ab(ba(x1)))) | → | aa(ab(bb(ba(aa(aa(x1)))))) | (23) |
ab(aa(ab(ba(x1)))) | → | ab(ab(bb(ba(aa(aa(x1)))))) | (24) |
ba(aa(ab(ba(x1)))) | → | ba(ab(bb(ba(aa(aa(x1)))))) | (25) |
bb(aa(ab(ba(x1)))) | → | bb(ab(bb(ba(aa(aa(x1)))))) | (26) |
aa(aa(ab(bb(x1)))) | → | aa(ab(bb(ba(aa(ab(x1)))))) | (27) |
ab(aa(ab(bb(x1)))) | → | ab(ab(bb(ba(aa(ab(x1)))))) | (28) |
ba(aa(ab(bb(x1)))) | → | ba(ab(bb(ba(aa(ab(x1)))))) | (29) |
bb(aa(ab(bb(x1)))) | → | bb(ab(bb(ba(aa(ab(x1)))))) | (30) |
aa(ba(ab(ba(x1)))) | → | aa(bb(bb(ba(aa(aa(x1)))))) | (31) |
ab(ba(ab(ba(x1)))) | → | ab(bb(bb(ba(aa(aa(x1)))))) | (32) |
ba(ba(ab(ba(x1)))) | → | ba(bb(bb(ba(aa(aa(x1)))))) | (33) |
bb(ba(ab(ba(x1)))) | → | bb(bb(bb(ba(aa(aa(x1)))))) | (34) |
aa(ba(ab(bb(x1)))) | → | aa(bb(bb(ba(aa(ab(x1)))))) | (35) |
ab(ba(ab(bb(x1)))) | → | ab(bb(bb(ba(aa(ab(x1)))))) | (36) |
ba(ba(ab(bb(x1)))) | → | ba(bb(bb(ba(aa(ab(x1)))))) | (37) |
bb(ba(ab(bb(x1)))) | → | bb(bb(bb(ba(aa(ab(x1)))))) | (38) |
Root-labeling is applied.
We obtain the labeled TRSThere are 104 ruless (increase limit for explicit display).
[aaab(x1)] | = | 1 · x1 |
[abba(x1)] | = | 1 · x1 |
[baab(x1)] | = | 1 · x1 |
[baaa(x1)] | = | 1 · x1 |
[aaaa(x1)] | = | 1 · x1 |
[aaba(x1)] | = | 1 + 1 · x1 |
[aabb(x1)] | = | 1 · x1 |
[baba(x1)] | = | 1 + 1 · x1 |
[babb(x1)] | = | 1 · x1 |
[abaa(x1)] | = | 1 + 1 · x1 |
[abab(x1)] | = | 1 + 1 · x1 |
[abbb(x1)] | = | 1 · x1 |
[bbaa(x1)] | = | 1 + 1 · x1 |
[bbab(x1)] | = | 1 + 1 · x1 |
[bbba(x1)] | = | 1 · x1 |
[bbbb(x1)] | = | 1 · x1 |
aaba(baab(abba(baab(abba(baab(abaa(x1))))))) | → | aabb(bbaa(x1)) | (63) |
aaba(baab(abba(baab(abba(baab(abab(x1))))))) | → | aabb(bbab(x1)) | (64) |
aaba(baab(abba(baab(abba(baab(abba(x1))))))) | → | aabb(bbba(x1)) | (65) |
aaba(baab(abba(baab(abba(baab(abbb(x1))))))) | → | aabb(bbbb(x1)) | (66) |
baba(baab(abba(baab(abba(baab(abaa(x1))))))) | → | babb(bbaa(x1)) | (71) |
baba(baab(abba(baab(abba(baab(abab(x1))))))) | → | babb(bbab(x1)) | (72) |
baba(baab(abba(baab(abba(baab(abba(x1))))))) | → | babb(bbba(x1)) | (73) |
baba(baab(abba(baab(abba(baab(abbb(x1))))))) | → | babb(bbbb(x1)) | (74) |
aaba(baab(abba(baaa(x1)))) | → | aabb(bbbb(bbba(baaa(aaaa(aaaa(x1)))))) | (111) |
aaba(baab(abba(baab(x1)))) | → | aabb(bbbb(bbba(baaa(aaaa(aaab(x1)))))) | (112) |
aaba(baab(abba(baba(x1)))) | → | aabb(bbbb(bbba(baaa(aaaa(aaba(x1)))))) | (113) |
aaba(baab(abba(babb(x1)))) | → | aabb(bbbb(bbba(baaa(aaaa(aabb(x1)))))) | (114) |
baba(baab(abba(baaa(x1)))) | → | babb(bbbb(bbba(baaa(aaaa(aaaa(x1)))))) | (119) |
baba(baab(abba(baab(x1)))) | → | babb(bbbb(bbba(baaa(aaaa(aaab(x1)))))) | (120) |
baba(baab(abba(baba(x1)))) | → | babb(bbbb(bbba(baaa(aaaa(aaba(x1)))))) | (121) |
baba(baab(abba(babb(x1)))) | → | babb(bbbb(bbba(baaa(aaaa(aabb(x1)))))) | (122) |
aaba(baab(abbb(bbaa(x1)))) | → | aabb(bbbb(bbba(baaa(aaab(abaa(x1)))))) | (127) |
aaba(baab(abbb(bbab(x1)))) | → | aabb(bbbb(bbba(baaa(aaab(abab(x1)))))) | (128) |
aaba(baab(abbb(bbba(x1)))) | → | aabb(bbbb(bbba(baaa(aaab(abba(x1)))))) | (129) |
aaba(baab(abbb(bbbb(x1)))) | → | aabb(bbbb(bbba(baaa(aaab(abbb(x1)))))) | (130) |
baba(baab(abbb(bbaa(x1)))) | → | babb(bbbb(bbba(baaa(aaab(abaa(x1)))))) | (135) |
baba(baab(abbb(bbab(x1)))) | → | babb(bbbb(bbba(baaa(aaab(abab(x1)))))) | (136) |
baba(baab(abbb(bbba(x1)))) | → | babb(bbbb(bbba(baaa(aaab(abba(x1)))))) | (137) |
baba(baab(abbb(bbbb(x1)))) | → | babb(bbbb(bbba(baaa(aaab(abbb(x1)))))) | (138) |
[aaab(x1)] | = | 1 · x1 |
[abba(x1)] | = | 1 · x1 |
[baab(x1)] | = | 1 · x1 |
[baaa(x1)] | = | 1 · x1 |
[aaaa(x1)] | = | 1 · x1 |
[aaba(x1)] | = | 1 · x1 |
[aabb(x1)] | = | 1 · x1 |
[baba(x1)] | = | 1 · x1 |
[babb(x1)] | = | 1 · x1 |
[abaa(x1)] | = | 1 + 1 · x1 |
[abab(x1)] | = | 1 · x1 |
[abbb(x1)] | = | 1 · x1 |
[bbaa(x1)] | = | 1 + 1 · x1 |
[bbab(x1)] | = | 1 · x1 |
[bbba(x1)] | = | 1 · x1 |
[bbbb(x1)] | = | 1 · x1 |
abaa(aaab(abba(baab(abba(baab(abaa(x1))))))) | → | abab(abaa(x1)) | (51) |
abaa(aaab(abba(baab(abba(baab(abab(x1))))))) | → | abab(abab(x1)) | (52) |
abaa(aaab(abba(baab(abba(baab(abba(x1))))))) | → | abab(abba(x1)) | (53) |
abaa(aaab(abba(baab(abba(baab(abbb(x1))))))) | → | abab(abbb(x1)) | (54) |
bbaa(aaab(abba(baab(abba(baab(abaa(x1))))))) | → | bbab(abaa(x1)) | (59) |
bbaa(aaab(abba(baab(abba(baab(abab(x1))))))) | → | bbab(abab(x1)) | (60) |
bbaa(aaab(abba(baab(abba(baab(abba(x1))))))) | → | bbab(abba(x1)) | (61) |
bbaa(aaab(abba(baab(abba(baab(abbb(x1))))))) | → | bbab(abbb(x1)) | (62) |
abaa(aaab(abba(baaa(x1)))) | → | abab(abbb(bbba(baaa(aaaa(aaaa(x1)))))) | (83) |
abaa(aaab(abba(baab(x1)))) | → | abab(abbb(bbba(baaa(aaaa(aaab(x1)))))) | (84) |
abaa(aaab(abba(baba(x1)))) | → | abab(abbb(bbba(baaa(aaaa(aaba(x1)))))) | (85) |
abaa(aaab(abba(babb(x1)))) | → | abab(abbb(bbba(baaa(aaaa(aabb(x1)))))) | (86) |
bbaa(aaab(abba(baaa(x1)))) | → | bbab(abbb(bbba(baaa(aaaa(aaaa(x1)))))) | (91) |
bbaa(aaab(abba(baab(x1)))) | → | bbab(abbb(bbba(baaa(aaaa(aaab(x1)))))) | (92) |
bbaa(aaab(abba(baba(x1)))) | → | bbab(abbb(bbba(baaa(aaaa(aaba(x1)))))) | (93) |
bbaa(aaab(abba(babb(x1)))) | → | bbab(abbb(bbba(baaa(aaaa(aabb(x1)))))) | (94) |
abaa(aaab(abbb(bbaa(x1)))) | → | abab(abbb(bbba(baaa(aaab(abaa(x1)))))) | (99) |
abaa(aaab(abbb(bbab(x1)))) | → | abab(abbb(bbba(baaa(aaab(abab(x1)))))) | (100) |
abaa(aaab(abbb(bbba(x1)))) | → | abab(abbb(bbba(baaa(aaab(abba(x1)))))) | (101) |
abaa(aaab(abbb(bbbb(x1)))) | → | abab(abbb(bbba(baaa(aaab(abbb(x1)))))) | (102) |
bbaa(aaab(abbb(bbaa(x1)))) | → | bbab(abbb(bbba(baaa(aaab(abaa(x1)))))) | (107) |
bbaa(aaab(abbb(bbab(x1)))) | → | bbab(abbb(bbba(baaa(aaab(abab(x1)))))) | (108) |
bbaa(aaab(abbb(bbba(x1)))) | → | bbab(abbb(bbba(baaa(aaab(abba(x1)))))) | (109) |
bbaa(aaab(abbb(bbbb(x1)))) | → | bbab(abbb(bbba(baaa(aaab(abbb(x1)))))) | (110) |
[aaab(x1)] | = |
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[abba(x1)] | = |
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[baab(x1)] | = |
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[baaa(x1)] | = |
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[aaaa(x1)] | = |
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[aaba(x1)] | = |
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[aabb(x1)] | = |
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[baba(x1)] | = |
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[babb(x1)] | = |
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[abaa(x1)] | = |
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[abab(x1)] | = |
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[abbb(x1)] | = |
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[bbaa(x1)] | = |
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[bbab(x1)] | = |
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[bbba(x1)] | = |
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[bbbb(x1)] | = |
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aaab(abba(baab(abba(baaa(aaaa(x1)))))) | → | aaaa(x1) | (39) |
aaab(abba(baab(abba(baaa(aaab(x1)))))) | → | aaab(x1) | (40) |
aaab(abba(baab(abba(baaa(aaba(x1)))))) | → | aaba(x1) | (41) |
aaab(abba(baab(abba(baaa(aabb(x1)))))) | → | aabb(x1) | (42) |
baab(abba(baab(abba(baaa(aaaa(x1)))))) | → | baaa(x1) | (43) |
baab(abba(baab(abba(baaa(aaab(x1)))))) | → | baab(x1) | (44) |
baab(abba(baab(abba(baaa(aaba(x1)))))) | → | baba(x1) | (45) |
baab(abba(baab(abba(baaa(aabb(x1)))))) | → | babb(x1) | (46) |
abba(baab(abba(baaa(x1)))) | → | abbb(bbbb(bbba(baaa(aaaa(aaaa(x1)))))) | (115) |
bbba(baab(abba(baaa(x1)))) | → | bbbb(bbbb(bbba(baaa(aaaa(aaaa(x1)))))) | (123) |
[aaaa(x1)] | = | 1 · x1 |
[aaab(x1)] | = | 1 · x1 |
[abba(x1)] | = | 1 · x1 |
[baab(x1)] | = | 1 · x1 |
[abaa(x1)] | = | 1 · x1 |
[abab(x1)] | = | 1 · x1 |
[abbb(x1)] | = | 1 · x1 |
[baaa(x1)] | = | 1 · x1 |
[bbaa(x1)] | = | 1 · x1 |
[bbab(x1)] | = | 1 · x1 |
[bbba(x1)] | = | 1 · x1 |
[bbbb(x1)] | = | 1 · x1 |
[baba(x1)] | = | 1 + 1 · x1 |
[aaba(x1)] | = | 1 + 1 · x1 |
[babb(x1)] | = | 1 + 1 · x1 |
[aabb(x1)] | = | 1 · x1 |
aaaa(aaab(abba(babb(x1)))) | → | aaab(abbb(bbba(baaa(aaaa(aabb(x1)))))) | (82) |
baaa(aaab(abba(babb(x1)))) | → | baab(abbb(bbba(baaa(aaaa(aabb(x1)))))) | (90) |
abba(baab(abba(babb(x1)))) | → | abbb(bbbb(bbba(baaa(aaaa(aabb(x1)))))) | (118) |
bbba(baab(abba(babb(x1)))) | → | bbbb(bbbb(bbba(baaa(aaaa(aabb(x1)))))) | (126) |
[aaaa(x1)] | = | 1 · x1 |
[aaab(x1)] | = | 1 · x1 |
[abba(x1)] | = | 1 · x1 |
[baab(x1)] | = | 1 · x1 |
[abaa(x1)] | = | 1 · x1 |
[abab(x1)] | = | 1 · x1 |
[abbb(x1)] | = | 1 · x1 |
[baaa(x1)] | = | 1 · x1 |
[bbaa(x1)] | = | 1 · x1 |
[bbab(x1)] | = | 1 · x1 |
[bbba(x1)] | = | 1 · x1 |
[bbbb(x1)] | = | 1 · x1 |
[baba(x1)] | = | 1 + 1 · x1 |
[aaba(x1)] | = | 1 · x1 |
aaaa(aaab(abba(baba(x1)))) | → | aaab(abbb(bbba(baaa(aaaa(aaba(x1)))))) | (81) |
baaa(aaab(abba(baba(x1)))) | → | baab(abbb(bbba(baaa(aaaa(aaba(x1)))))) | (89) |
abba(baab(abba(baba(x1)))) | → | abbb(bbbb(bbba(baaa(aaaa(aaba(x1)))))) | (117) |
bbba(baab(abba(baba(x1)))) | → | bbbb(bbbb(bbba(baaa(aaaa(aaba(x1)))))) | (125) |
[aaaa(x1)] | = |
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[aaab(x1)] | = |
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[abba(x1)] | = |
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[baab(x1)] | = |
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[abaa(x1)] | = |
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[abab(x1)] | = |
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[abbb(x1)] | = |
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[baaa(x1)] | = |
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[bbaa(x1)] | = |
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[bbab(x1)] | = |
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[bbba(x1)] | = |
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[bbbb(x1)] | = |
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abba(baab(abba(baab(abba(baab(abab(x1))))))) | → | abbb(bbab(x1)) | (68) |
abba(baab(abba(baab(abba(baab(abba(x1))))))) | → | abbb(bbba(x1)) | (69) |
abba(baab(abba(baab(abba(baab(abbb(x1))))))) | → | abbb(bbbb(x1)) | (70) |
bbba(baab(abba(baab(abba(baab(abab(x1))))))) | → | bbbb(bbab(x1)) | (76) |
bbba(baab(abba(baab(abba(baab(abba(x1))))))) | → | bbbb(bbba(x1)) | (77) |
bbba(baab(abba(baab(abba(baab(abbb(x1))))))) | → | bbbb(bbbb(x1)) | (78) |
aaaa(aaab(abbb(bbaa(x1)))) | → | aaab(abbb(bbba(baaa(aaab(abaa(x1)))))) | (95) |
baaa(aaab(abbb(bbaa(x1)))) | → | baab(abbb(bbba(baaa(aaab(abaa(x1)))))) | (103) |
abba(baab(abbb(bbaa(x1)))) | → | abbb(bbbb(bbba(baaa(aaab(abaa(x1)))))) | (131) |
bbba(baab(abbb(bbaa(x1)))) | → | bbbb(bbbb(bbba(baaa(aaab(abaa(x1)))))) | (139) |
[aaaa(x1)] | = | 1 · x1 |
[aaab(x1)] | = | 1 · x1 |
[abba(x1)] | = | 1 · x1 |
[baab(x1)] | = | 1 · x1 |
[abaa(x1)] | = | 1 + 1 · x1 |
[abab(x1)] | = | 1 · x1 |
[abbb(x1)] | = | 1 · x1 |
[baaa(x1)] | = | 1 · x1 |
[bbaa(x1)] | = | 1 + 1 · x1 |
[bbba(x1)] | = | 1 · x1 |
[bbbb(x1)] | = | 1 · x1 |
[bbab(x1)] | = | 1 + 1 · x1 |
aaaa(aaab(abbb(bbab(x1)))) | → | aaab(abbb(bbba(baaa(aaab(abab(x1)))))) | (96) |
baaa(aaab(abbb(bbab(x1)))) | → | baab(abbb(bbba(baaa(aaab(abab(x1)))))) | (104) |
abba(baab(abbb(bbab(x1)))) | → | abbb(bbbb(bbba(baaa(aaab(abab(x1)))))) | (132) |
bbba(baab(abbb(bbab(x1)))) | → | bbbb(bbbb(bbba(baaa(aaab(abab(x1)))))) | (140) |
[aaaa(x1)] | = | 1 · x1 |
[aaab(x1)] | = | 1 · x1 |
[abba(x1)] | = | 1 · x1 |
[baab(x1)] | = | 1 · x1 |
[abaa(x1)] | = | 1 + 1 · x1 |
[abab(x1)] | = | 1 · x1 |
[abbb(x1)] | = | 1 · x1 |
[baaa(x1)] | = | 1 · x1 |
[bbaa(x1)] | = | 1 · x1 |
[bbba(x1)] | = | 1 · x1 |
[bbbb(x1)] | = | 1 · x1 |
abba(baab(abba(baab(abba(baab(abaa(x1))))))) | → | abbb(bbaa(x1)) | (67) |
bbba(baab(abba(baab(abba(baab(abaa(x1))))))) | → | bbbb(bbaa(x1)) | (75) |
[aaaa(x1)] | = |
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[aaab(x1)] | = |
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[abba(x1)] | = |
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[baab(x1)] | = |
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[abaa(x1)] | = |
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[abab(x1)] | = |
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[abbb(x1)] | = |
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[baaa(x1)] | = |
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[bbba(x1)] | = |
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[bbbb(x1)] | = |
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aaaa(aaab(abba(baab(abba(baab(abaa(x1))))))) | → | aaab(abaa(x1)) | (47) |
[aaaa(x1)] | = |
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[aaab(x1)] | = |
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[abba(x1)] | = |
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[baab(x1)] | = |
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[abab(x1)] | = |
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[abbb(x1)] | = |
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[baaa(x1)] | = |
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[abaa(x1)] | = |
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[bbba(x1)] | = |
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[bbbb(x1)] | = |
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baaa(aaab(abba(baab(abba(baab(abaa(x1))))))) | → | baab(abaa(x1)) | (55) |
[aaaa(x1)] | = |
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[aaab(x1)] | = |
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[abba(x1)] | = |
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[baab(x1)] | = |
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[abab(x1)] | = |
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[abbb(x1)] | = |
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[baaa(x1)] | = |
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[bbba(x1)] | = |
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[bbbb(x1)] | = |
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aaaa(aaab(abba(baab(abba(baab(abab(x1))))))) | → | aaab(abab(x1)) | (48) |
[aaaa(x1)] | = |
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[aaab(x1)] | = |
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[abba(x1)] | = |
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[baab(x1)] | = |
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[abbb(x1)] | = |
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[baaa(x1)] | = |
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[abab(x1)] | = |
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[bbba(x1)] | = |
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[bbbb(x1)] | = |
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aaaa(aaab(abba(baab(abba(baab(abba(x1))))))) | → | aaab(abba(x1)) | (49) |
baaa(aaab(abba(baab(abba(baab(abba(x1))))))) | → | baab(abba(x1)) | (57) |
[aaaa(x1)] | = |
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[aaab(x1)] | = |
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[abba(x1)] | = |
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[baab(x1)] | = |
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[abbb(x1)] | = |
|
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[baaa(x1)] | = |
|
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[abab(x1)] | = |
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[bbba(x1)] | = |
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[bbbb(x1)] | = |
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baaa(aaab(abba(baab(abba(baab(abab(x1))))))) | → | baab(abab(x1)) | (56) |
[aaaa(x1)] | = |
|
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[aaab(x1)] | = |
|
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[abba(x1)] | = |
|
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[baab(x1)] | = |
|
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[abbb(x1)] | = |
|
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[baaa(x1)] | = |
|
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[bbba(x1)] | = |
|
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[bbbb(x1)] | = |
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bbba(baab(abba(baab(x1)))) | → | bbbb(bbbb(bbba(baaa(aaaa(aaab(x1)))))) | (124) |
[aaaa(x1)] | = |
|
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[aaab(x1)] | = |
|
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[abba(x1)] | = |
|
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[baab(x1)] | = |
|
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[abbb(x1)] | = |
|
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[baaa(x1)] | = |
|
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[bbba(x1)] | = |
|
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[bbbb(x1)] | = |
|
aaaa(aaab(abba(baab(abba(baab(abbb(x1))))))) | → | aaab(abbb(x1)) | (50) |
baaa(aaab(abba(baab(abba(baab(abbb(x1))))))) | → | baab(abbb(x1)) | (58) |
abba(baab(abba(baab(x1)))) | → | abbb(bbbb(bbba(baaa(aaaa(aaab(x1)))))) | (116) |
There are no rules in the TRS. Hence, it is terminating.