The rewrite relation of the following TRS is considered.
a(x1) | → | x1 | (1) |
a(b(x1)) | → | b(a(c(b(a(c(x1)))))) | (2) |
b(x1) | → | x1 | (3) |
c(c(x1)) | → | x1 | (4) |
a(x1) | → | x1 | (1) |
b(a(x1)) | → | c(a(b(c(a(b(x1)))))) | (5) |
b(x1) | → | x1 | (3) |
c(c(x1)) | → | x1 | (4) |
{a(☐), b(☐), c(☐)}
We obtain the transformed TRSa(a(x1)) | → | a(x1) | (6) |
b(a(x1)) | → | b(x1) | (7) |
c(a(x1)) | → | c(x1) | (8) |
a(b(a(x1))) | → | a(c(a(b(c(a(b(x1))))))) | (9) |
b(b(a(x1))) | → | b(c(a(b(c(a(b(x1))))))) | (10) |
c(b(a(x1))) | → | c(c(a(b(c(a(b(x1))))))) | (11) |
a(b(x1)) | → | a(x1) | (12) |
b(b(x1)) | → | b(x1) | (13) |
c(b(x1)) | → | c(x1) | (14) |
a(c(c(x1))) | → | a(x1) | (15) |
b(c(c(x1))) | → | b(x1) | (16) |
c(c(c(x1))) | → | c(x1) | (17) |
Root-labeling is applied.
We obtain the labeled TRSaa(aa(x1)) | → | aa(x1) | (18) |
aa(ab(x1)) | → | ab(x1) | (19) |
aa(ac(x1)) | → | ac(x1) | (20) |
ba(aa(x1)) | → | ba(x1) | (21) |
ba(ab(x1)) | → | bb(x1) | (22) |
ba(ac(x1)) | → | bc(x1) | (23) |
ca(aa(x1)) | → | ca(x1) | (24) |
ca(ab(x1)) | → | cb(x1) | (25) |
ca(ac(x1)) | → | cc(x1) | (26) |
ab(ba(aa(x1))) | → | ac(ca(ab(bc(ca(ab(ba(x1))))))) | (27) |
ab(ba(ab(x1))) | → | ac(ca(ab(bc(ca(ab(bb(x1))))))) | (28) |
ab(ba(ac(x1))) | → | ac(ca(ab(bc(ca(ab(bc(x1))))))) | (29) |
bb(ba(aa(x1))) | → | bc(ca(ab(bc(ca(ab(ba(x1))))))) | (30) |
bb(ba(ab(x1))) | → | bc(ca(ab(bc(ca(ab(bb(x1))))))) | (31) |
bb(ba(ac(x1))) | → | bc(ca(ab(bc(ca(ab(bc(x1))))))) | (32) |
cb(ba(aa(x1))) | → | cc(ca(ab(bc(ca(ab(ba(x1))))))) | (33) |
cb(ba(ab(x1))) | → | cc(ca(ab(bc(ca(ab(bb(x1))))))) | (34) |
cb(ba(ac(x1))) | → | cc(ca(ab(bc(ca(ab(bc(x1))))))) | (35) |
ab(ba(x1)) | → | aa(x1) | (36) |
ab(bb(x1)) | → | ab(x1) | (37) |
ab(bc(x1)) | → | ac(x1) | (38) |
bb(ba(x1)) | → | ba(x1) | (39) |
bb(bb(x1)) | → | bb(x1) | (40) |
bb(bc(x1)) | → | bc(x1) | (41) |
cb(ba(x1)) | → | ca(x1) | (42) |
cb(bb(x1)) | → | cb(x1) | (43) |
cb(bc(x1)) | → | cc(x1) | (44) |
ac(cc(ca(x1))) | → | aa(x1) | (45) |
ac(cc(cb(x1))) | → | ab(x1) | (46) |
ac(cc(cc(x1))) | → | ac(x1) | (47) |
bc(cc(ca(x1))) | → | ba(x1) | (48) |
bc(cc(cb(x1))) | → | bb(x1) | (49) |
bc(cc(cc(x1))) | → | bc(x1) | (50) |
cc(cc(ca(x1))) | → | ca(x1) | (51) |
cc(cc(cb(x1))) | → | cb(x1) | (52) |
cc(cc(cc(x1))) | → | cc(x1) | (53) |
[aa(x1)] | = | 1 · x1 + 2 |
[ab(x1)] | = | 1 · x1 |
[ac(x1)] | = | 1 · x1 + 1 |
[ba(x1)] | = | 1 · x1 + 2 |
[bb(x1)] | = | 1 · x1 |
[bc(x1)] | = | 1 · x1 + 1 |
[ca(x1)] | = | 1 · x1 |
[cb(x1)] | = | 1 · x1 |
[cc(x1)] | = | 1 · x1 + 1 |
aa(aa(x1)) | → | aa(x1) | (18) |
aa(ab(x1)) | → | ab(x1) | (19) |
aa(ac(x1)) | → | ac(x1) | (20) |
ba(aa(x1)) | → | ba(x1) | (21) |
ba(ab(x1)) | → | bb(x1) | (22) |
ba(ac(x1)) | → | bc(x1) | (23) |
ca(aa(x1)) | → | ca(x1) | (24) |
cb(ba(x1)) | → | ca(x1) | (42) |
ac(cc(cb(x1))) | → | ab(x1) | (46) |
ac(cc(cc(x1))) | → | ac(x1) | (47) |
bc(cc(cb(x1))) | → | bb(x1) | (49) |
bc(cc(cc(x1))) | → | bc(x1) | (50) |
cc(cc(ca(x1))) | → | ca(x1) | (51) |
cc(cc(cb(x1))) | → | cb(x1) | (52) |
cc(cc(cc(x1))) | → | cc(x1) | (53) |
ca#(ab(x1)) | → | cb#(x1) | (54) |
ab#(ba(aa(x1))) | → | ac#(ca(ab(bc(ca(ab(ba(x1))))))) | (55) |
ab#(ba(aa(x1))) | → | ca#(ab(bc(ca(ab(ba(x1)))))) | (56) |
ab#(ba(aa(x1))) | → | ab#(bc(ca(ab(ba(x1))))) | (57) |
ab#(ba(aa(x1))) | → | bc#(ca(ab(ba(x1)))) | (58) |
ab#(ba(aa(x1))) | → | ca#(ab(ba(x1))) | (59) |
ab#(ba(aa(x1))) | → | ab#(ba(x1)) | (60) |
ab#(ba(ab(x1))) | → | ac#(ca(ab(bc(ca(ab(bb(x1))))))) | (61) |
ab#(ba(ab(x1))) | → | ca#(ab(bc(ca(ab(bb(x1)))))) | (62) |
ab#(ba(ab(x1))) | → | ab#(bc(ca(ab(bb(x1))))) | (63) |
ab#(ba(ab(x1))) | → | bc#(ca(ab(bb(x1)))) | (64) |
ab#(ba(ab(x1))) | → | ca#(ab(bb(x1))) | (65) |
ab#(ba(ab(x1))) | → | ab#(bb(x1)) | (66) |
ab#(ba(ab(x1))) | → | bb#(x1) | (67) |
ab#(ba(ac(x1))) | → | ac#(ca(ab(bc(ca(ab(bc(x1))))))) | (68) |
ab#(ba(ac(x1))) | → | ca#(ab(bc(ca(ab(bc(x1)))))) | (69) |
ab#(ba(ac(x1))) | → | ab#(bc(ca(ab(bc(x1))))) | (70) |
ab#(ba(ac(x1))) | → | bc#(ca(ab(bc(x1)))) | (71) |
ab#(ba(ac(x1))) | → | ca#(ab(bc(x1))) | (72) |
ab#(ba(ac(x1))) | → | ab#(bc(x1)) | (73) |
ab#(ba(ac(x1))) | → | bc#(x1) | (74) |
bb#(ba(aa(x1))) | → | bc#(ca(ab(bc(ca(ab(ba(x1))))))) | (75) |
bb#(ba(aa(x1))) | → | ca#(ab(bc(ca(ab(ba(x1)))))) | (76) |
bb#(ba(aa(x1))) | → | ab#(bc(ca(ab(ba(x1))))) | (77) |
bb#(ba(aa(x1))) | → | bc#(ca(ab(ba(x1)))) | (78) |
bb#(ba(aa(x1))) | → | ca#(ab(ba(x1))) | (79) |
bb#(ba(aa(x1))) | → | ab#(ba(x1)) | (80) |
bb#(ba(ab(x1))) | → | bc#(ca(ab(bc(ca(ab(bb(x1))))))) | (81) |
bb#(ba(ab(x1))) | → | ca#(ab(bc(ca(ab(bb(x1)))))) | (82) |
bb#(ba(ab(x1))) | → | ab#(bc(ca(ab(bb(x1))))) | (83) |
bb#(ba(ab(x1))) | → | bc#(ca(ab(bb(x1)))) | (84) |
bb#(ba(ab(x1))) | → | ca#(ab(bb(x1))) | (85) |
bb#(ba(ab(x1))) | → | ab#(bb(x1)) | (86) |
bb#(ba(ab(x1))) | → | bb#(x1) | (87) |
bb#(ba(ac(x1))) | → | bc#(ca(ab(bc(ca(ab(bc(x1))))))) | (88) |
bb#(ba(ac(x1))) | → | ca#(ab(bc(ca(ab(bc(x1)))))) | (89) |
bb#(ba(ac(x1))) | → | ab#(bc(ca(ab(bc(x1))))) | (90) |
bb#(ba(ac(x1))) | → | bc#(ca(ab(bc(x1)))) | (91) |
bb#(ba(ac(x1))) | → | ca#(ab(bc(x1))) | (92) |
bb#(ba(ac(x1))) | → | ab#(bc(x1)) | (93) |
bb#(ba(ac(x1))) | → | bc#(x1) | (94) |
cb#(ba(aa(x1))) | → | ca#(ab(bc(ca(ab(ba(x1)))))) | (95) |
cb#(ba(aa(x1))) | → | ab#(bc(ca(ab(ba(x1))))) | (96) |
cb#(ba(aa(x1))) | → | bc#(ca(ab(ba(x1)))) | (97) |
cb#(ba(aa(x1))) | → | ca#(ab(ba(x1))) | (98) |
cb#(ba(aa(x1))) | → | ab#(ba(x1)) | (99) |
cb#(ba(ab(x1))) | → | ca#(ab(bc(ca(ab(bb(x1)))))) | (100) |
cb#(ba(ab(x1))) | → | ab#(bc(ca(ab(bb(x1))))) | (101) |
cb#(ba(ab(x1))) | → | bc#(ca(ab(bb(x1)))) | (102) |
cb#(ba(ab(x1))) | → | ca#(ab(bb(x1))) | (103) |
cb#(ba(ab(x1))) | → | ab#(bb(x1)) | (104) |
cb#(ba(ab(x1))) | → | bb#(x1) | (105) |
cb#(ba(ac(x1))) | → | ca#(ab(bc(ca(ab(bc(x1)))))) | (106) |
cb#(ba(ac(x1))) | → | ab#(bc(ca(ab(bc(x1))))) | (107) |
cb#(ba(ac(x1))) | → | bc#(ca(ab(bc(x1)))) | (108) |
cb#(ba(ac(x1))) | → | ca#(ab(bc(x1))) | (109) |
cb#(ba(ac(x1))) | → | ab#(bc(x1)) | (110) |
cb#(ba(ac(x1))) | → | bc#(x1) | (111) |
ab#(bb(x1)) | → | ab#(x1) | (112) |
ab#(bc(x1)) | → | ac#(x1) | (113) |
cb#(bb(x1)) | → | cb#(x1) | (114) |
The dependency pairs are split into 1 component.
cb#(ba(aa(x1))) | → | ca#(ab(bc(ca(ab(ba(x1)))))) | (95) |
ca#(ab(x1)) | → | cb#(x1) | (54) |
cb#(ba(aa(x1))) | → | ab#(bc(ca(ab(ba(x1))))) | (96) |
ab#(ba(aa(x1))) | → | ca#(ab(bc(ca(ab(ba(x1)))))) | (56) |
ab#(ba(aa(x1))) | → | ab#(bc(ca(ab(ba(x1))))) | (57) |
ab#(ba(aa(x1))) | → | ca#(ab(ba(x1))) | (59) |
ab#(ba(aa(x1))) | → | ab#(ba(x1)) | (60) |
ab#(ba(ab(x1))) | → | ca#(ab(bc(ca(ab(bb(x1)))))) | (62) |
ab#(ba(ab(x1))) | → | ab#(bc(ca(ab(bb(x1))))) | (63) |
ab#(ba(ab(x1))) | → | ca#(ab(bb(x1))) | (65) |
ab#(ba(ab(x1))) | → | ab#(bb(x1)) | (66) |
ab#(ba(ab(x1))) | → | bb#(x1) | (67) |
bb#(ba(aa(x1))) | → | ca#(ab(bc(ca(ab(ba(x1)))))) | (76) |
bb#(ba(aa(x1))) | → | ab#(bc(ca(ab(ba(x1))))) | (77) |
ab#(ba(ac(x1))) | → | ca#(ab(bc(ca(ab(bc(x1)))))) | (69) |
ab#(ba(ac(x1))) | → | ab#(bc(ca(ab(bc(x1))))) | (70) |
ab#(ba(ac(x1))) | → | ca#(ab(bc(x1))) | (72) |
ab#(ba(ac(x1))) | → | ab#(bc(x1)) | (73) |
ab#(bb(x1)) | → | ab#(x1) | (112) |
bb#(ba(aa(x1))) | → | ca#(ab(ba(x1))) | (79) |
bb#(ba(aa(x1))) | → | ab#(ba(x1)) | (80) |
bb#(ba(ab(x1))) | → | ca#(ab(bc(ca(ab(bb(x1)))))) | (82) |
bb#(ba(ab(x1))) | → | ab#(bc(ca(ab(bb(x1))))) | (83) |
bb#(ba(ab(x1))) | → | ca#(ab(bb(x1))) | (85) |
bb#(ba(ab(x1))) | → | ab#(bb(x1)) | (86) |
bb#(ba(ab(x1))) | → | bb#(x1) | (87) |
bb#(ba(ac(x1))) | → | ca#(ab(bc(ca(ab(bc(x1)))))) | (89) |
bb#(ba(ac(x1))) | → | ab#(bc(ca(ab(bc(x1))))) | (90) |
bb#(ba(ac(x1))) | → | ca#(ab(bc(x1))) | (92) |
bb#(ba(ac(x1))) | → | ab#(bc(x1)) | (93) |
cb#(ba(aa(x1))) | → | ca#(ab(ba(x1))) | (98) |
cb#(ba(aa(x1))) | → | ab#(ba(x1)) | (99) |
cb#(ba(ab(x1))) | → | ca#(ab(bc(ca(ab(bb(x1)))))) | (100) |
cb#(ba(ab(x1))) | → | ab#(bc(ca(ab(bb(x1))))) | (101) |
cb#(ba(ab(x1))) | → | ca#(ab(bb(x1))) | (103) |
cb#(ba(ab(x1))) | → | ab#(bb(x1)) | (104) |
cb#(ba(ab(x1))) | → | bb#(x1) | (105) |
cb#(ba(ac(x1))) | → | ca#(ab(bc(ca(ab(bc(x1)))))) | (106) |
cb#(ba(ac(x1))) | → | ab#(bc(ca(ab(bc(x1))))) | (107) |
cb#(ba(ac(x1))) | → | ca#(ab(bc(x1))) | (109) |
cb#(ba(ac(x1))) | → | ab#(bc(x1)) | (110) |
cb#(bb(x1)) | → | cb#(x1) | (114) |
[ca#(x1)] | = | -2 + 2 · x1 |
[ab#(x1)] | = | 2 + 2 · x1 |
[bc(x1)] | = | x1 |
[ca(x1)] | = | -1 + x1 |
[ab(x1)] | = | 2 + x1 |
[ba(x1)] | = | x1 |
[aa(x1)] | = | 2 + x1 |
[ac(x1)] | = | 2 + x1 |
[bb(x1)] | = | x1 |
[cb(x1)] | = | 1 + x1 |
[cc(x1)] | = | 1 + x1 |
[cb#(x1)] | = | 2 · x1 |
[bb#(x1)] | = | 2 · x1 |
ca#(ab(x1)) | → | cb#(x1) | (54) |
ab#(ba(aa(x1))) | → | ca#(ab(bc(ca(ab(ba(x1)))))) | (56) |
ab#(ba(aa(x1))) | → | ab#(bc(ca(ab(ba(x1))))) | (57) |
ab#(ba(aa(x1))) | → | ca#(ab(ba(x1))) | (59) |
ab#(ba(aa(x1))) | → | ab#(ba(x1)) | (60) |
ab#(ba(ab(x1))) | → | ca#(ab(bc(ca(ab(bb(x1)))))) | (62) |
ab#(ba(ab(x1))) | → | ab#(bc(ca(ab(bb(x1))))) | (63) |
ab#(ba(ab(x1))) | → | ca#(ab(bb(x1))) | (65) |
ab#(ba(ab(x1))) | → | ab#(bb(x1)) | (66) |
ab#(ba(ab(x1))) | → | bb#(x1) | (67) |
ab#(ba(ac(x1))) | → | ca#(ab(bc(ca(ab(bc(x1)))))) | (69) |
ab#(ba(ac(x1))) | → | ab#(bc(ca(ab(bc(x1))))) | (70) |
ab#(ba(ac(x1))) | → | ca#(ab(bc(x1))) | (72) |
ab#(ba(ac(x1))) | → | ab#(bc(x1)) | (73) |
bb#(ba(aa(x1))) | → | ca#(ab(ba(x1))) | (79) |
bb#(ba(aa(x1))) | → | ab#(ba(x1)) | (80) |
bb#(ba(ab(x1))) | → | ca#(ab(bb(x1))) | (85) |
bb#(ba(ab(x1))) | → | ab#(bb(x1)) | (86) |
bb#(ba(ab(x1))) | → | bb#(x1) | (87) |
bb#(ba(ac(x1))) | → | ca#(ab(bc(x1))) | (92) |
bb#(ba(ac(x1))) | → | ab#(bc(x1)) | (93) |
cb#(ba(aa(x1))) | → | ca#(ab(ba(x1))) | (98) |
cb#(ba(aa(x1))) | → | ab#(ba(x1)) | (99) |
cb#(ba(ab(x1))) | → | ca#(ab(bb(x1))) | (103) |
cb#(ba(ab(x1))) | → | ab#(bb(x1)) | (104) |
cb#(ba(ab(x1))) | → | bb#(x1) | (105) |
cb#(ba(ac(x1))) | → | ca#(ab(bc(x1))) | (109) |
cb#(ba(ac(x1))) | → | ab#(bc(x1)) | (110) |
The dependency pairs are split into 2 components.
cb#(bb(x1)) | → | cb#(x1) | (114) |
[bb(x1)] | = | 1 · x1 |
[cb#(x1)] | = | 1 · x1 |
Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.
cb#(bb(x1)) | → | cb#(x1) | (114) |
1 | > | 1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
ab#(bb(x1)) | → | ab#(x1) | (112) |
[bb(x1)] | = | 1 · x1 |
[ab#(x1)] | = | 1 · x1 |
Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.
ab#(bb(x1)) | → | ab#(x1) | (112) |
1 | > | 1 |
As there is no critical graph in the transitive closure, there are no infinite chains.