The rewrite relation of the following TRS is considered.
a(x1) | → | x1 | (1) |
a(b(x1)) | → | c(x1) | (2) |
b(x1) | → | x1 | (3) |
c(c(x1)) | → | b(b(a(a(c(x1))))) | (4) |
{a(☐), b(☐), c(☐)}
We obtain the transformed TRSa(a(x1)) | → | a(x1) | (5) |
b(a(x1)) | → | b(x1) | (6) |
c(a(x1)) | → | c(x1) | (7) |
a(a(b(x1))) | → | a(c(x1)) | (8) |
b(a(b(x1))) | → | b(c(x1)) | (9) |
c(a(b(x1))) | → | c(c(x1)) | (10) |
a(b(x1)) | → | a(x1) | (11) |
b(b(x1)) | → | b(x1) | (12) |
c(b(x1)) | → | c(x1) | (13) |
a(c(c(x1))) | → | a(b(b(a(a(c(x1)))))) | (14) |
b(c(c(x1))) | → | b(b(b(a(a(c(x1)))))) | (15) |
c(c(c(x1))) | → | c(b(b(a(a(c(x1)))))) | (16) |
Root-labeling is applied.
We obtain the labeled TRSaa(aa(x1)) | → | aa(x1) | (17) |
aa(ab(x1)) | → | ab(x1) | (18) |
aa(ac(x1)) | → | ac(x1) | (19) |
ba(aa(x1)) | → | ba(x1) | (20) |
ba(ab(x1)) | → | bb(x1) | (21) |
ba(ac(x1)) | → | bc(x1) | (22) |
ca(aa(x1)) | → | ca(x1) | (23) |
ca(ab(x1)) | → | cb(x1) | (24) |
ca(ac(x1)) | → | cc(x1) | (25) |
aa(ab(ba(x1))) | → | ac(ca(x1)) | (26) |
aa(ab(bb(x1))) | → | ac(cb(x1)) | (27) |
aa(ab(bc(x1))) | → | ac(cc(x1)) | (28) |
ba(ab(ba(x1))) | → | bc(ca(x1)) | (29) |
ba(ab(bb(x1))) | → | bc(cb(x1)) | (30) |
ba(ab(bc(x1))) | → | bc(cc(x1)) | (31) |
ca(ab(ba(x1))) | → | cc(ca(x1)) | (32) |
ca(ab(bb(x1))) | → | cc(cb(x1)) | (33) |
ca(ab(bc(x1))) | → | cc(cc(x1)) | (34) |
ab(ba(x1)) | → | aa(x1) | (35) |
ab(bb(x1)) | → | ab(x1) | (36) |
ab(bc(x1)) | → | ac(x1) | (37) |
bb(ba(x1)) | → | ba(x1) | (38) |
bb(bb(x1)) | → | bb(x1) | (39) |
bb(bc(x1)) | → | bc(x1) | (40) |
cb(ba(x1)) | → | ca(x1) | (41) |
cb(bb(x1)) | → | cb(x1) | (42) |
cb(bc(x1)) | → | cc(x1) | (43) |
ac(cc(ca(x1))) | → | ab(bb(ba(aa(ac(ca(x1)))))) | (44) |
ac(cc(cb(x1))) | → | ab(bb(ba(aa(ac(cb(x1)))))) | (45) |
ac(cc(cc(x1))) | → | ab(bb(ba(aa(ac(cc(x1)))))) | (46) |
bc(cc(ca(x1))) | → | bb(bb(ba(aa(ac(ca(x1)))))) | (47) |
bc(cc(cb(x1))) | → | bb(bb(ba(aa(ac(cb(x1)))))) | (48) |
bc(cc(cc(x1))) | → | bb(bb(ba(aa(ac(cc(x1)))))) | (49) |
cc(cc(ca(x1))) | → | cb(bb(ba(aa(ac(ca(x1)))))) | (50) |
cc(cc(cb(x1))) | → | cb(bb(ba(aa(ac(cb(x1)))))) | (51) |
cc(cc(cc(x1))) | → | cb(bb(ba(aa(ac(cc(x1)))))) | (52) |
[aa(x1)] | = | 1 · x1 |
[ab(x1)] | = | 1 · x1 + 1 |
[ac(x1)] | = | 1 · x1 |
[ba(x1)] | = | 1 · x1 |
[bb(x1)] | = | 1 · x1 |
[bc(x1)] | = | 1 · x1 |
[ca(x1)] | = | 1 · x1 + 1 |
[cb(x1)] | = | 1 · x1 + 1 |
[cc(x1)] | = | 1 · x1 + 1 |
ba(ab(x1)) | → | bb(x1) | (21) |
ca(ab(x1)) | → | cb(x1) | (24) |
ab(ba(x1)) | → | aa(x1) | (35) |
ab(bc(x1)) | → | ac(x1) | (37) |
bc(cc(ca(x1))) | → | bb(bb(ba(aa(ac(ca(x1)))))) | (47) |
bc(cc(cb(x1))) | → | bb(bb(ba(aa(ac(cb(x1)))))) | (48) |
bc(cc(cc(x1))) | → | bb(bb(ba(aa(ac(cc(x1)))))) | (49) |
cc(cc(ca(x1))) | → | cb(bb(ba(aa(ac(ca(x1)))))) | (50) |
cc(cc(cb(x1))) | → | cb(bb(ba(aa(ac(cb(x1)))))) | (51) |
cc(cc(cc(x1))) | → | cb(bb(ba(aa(ac(cc(x1)))))) | (52) |
ba#(aa(x1)) | → | ba#(x1) | (53) |
ca#(aa(x1)) | → | ca#(x1) | (54) |
aa#(ab(ba(x1))) | → | ac#(ca(x1)) | (55) |
aa#(ab(ba(x1))) | → | ca#(x1) | (56) |
aa#(ab(bb(x1))) | → | ac#(cb(x1)) | (57) |
aa#(ab(bb(x1))) | → | cb#(x1) | (58) |
aa#(ab(bc(x1))) | → | ac#(cc(x1)) | (59) |
ba#(ab(ba(x1))) | → | ca#(x1) | (60) |
ba#(ab(bb(x1))) | → | cb#(x1) | (61) |
ca#(ab(ba(x1))) | → | ca#(x1) | (62) |
ca#(ab(bb(x1))) | → | cb#(x1) | (63) |
ab#(bb(x1)) | → | ab#(x1) | (64) |
cb#(ba(x1)) | → | ca#(x1) | (65) |
cb#(bb(x1)) | → | cb#(x1) | (66) |
ac#(cc(ca(x1))) | → | ab#(bb(ba(aa(ac(ca(x1)))))) | (67) |
ac#(cc(ca(x1))) | → | bb#(ba(aa(ac(ca(x1))))) | (68) |
ac#(cc(ca(x1))) | → | ba#(aa(ac(ca(x1)))) | (69) |
ac#(cc(ca(x1))) | → | aa#(ac(ca(x1))) | (70) |
ac#(cc(ca(x1))) | → | ac#(ca(x1)) | (71) |
ac#(cc(cb(x1))) | → | ab#(bb(ba(aa(ac(cb(x1)))))) | (72) |
ac#(cc(cb(x1))) | → | bb#(ba(aa(ac(cb(x1))))) | (73) |
ac#(cc(cb(x1))) | → | ba#(aa(ac(cb(x1)))) | (74) |
ac#(cc(cb(x1))) | → | aa#(ac(cb(x1))) | (75) |
ac#(cc(cb(x1))) | → | ac#(cb(x1)) | (76) |
ac#(cc(cc(x1))) | → | ab#(bb(ba(aa(ac(cc(x1)))))) | (77) |
ac#(cc(cc(x1))) | → | bb#(ba(aa(ac(cc(x1))))) | (78) |
ac#(cc(cc(x1))) | → | ba#(aa(ac(cc(x1)))) | (79) |
ac#(cc(cc(x1))) | → | aa#(ac(cc(x1))) | (80) |
ac#(cc(cc(x1))) | → | ac#(cc(x1)) | (81) |
The dependency pairs are split into 4 components.
ac#(cc(ca(x1))) | → | aa#(ac(ca(x1))) | (70) |
aa#(ab(ba(x1))) | → | ac#(ca(x1)) | (55) |
ac#(cc(ca(x1))) | → | ac#(ca(x1)) | (71) |
ac#(cc(cb(x1))) | → | aa#(ac(cb(x1))) | (75) |
aa#(ab(bb(x1))) | → | ac#(cb(x1)) | (57) |
ac#(cc(cb(x1))) | → | ac#(cb(x1)) | (76) |
ac#(cc(cc(x1))) | → | aa#(ac(cc(x1))) | (80) |
aa#(ab(bc(x1))) | → | ac#(cc(x1)) | (59) |
ac#(cc(cc(x1))) | → | ac#(cc(x1)) | (81) |
[ac#(x1)] | = | 1 · x1 |
[cc(x1)] | = | 1 + 1 · x1 |
[ca(x1)] | = | 1 + 1 · x1 |
[aa#(x1)] | = | 1 · x1 |
[ac(x1)] | = | 1 · x1 |
[ab(x1)] | = | 1 + 1 · x1 |
[ba(x1)] | = | 1 · x1 |
[cb(x1)] | = | 1 + 1 · x1 |
[bb(x1)] | = | 1 · x1 |
[bc(x1)] | = | 1 · x1 |
[aa(x1)] | = | 1 · x1 |
ac#(cc(ca(x1))) | → | aa#(ac(ca(x1))) | (70) |
ac#(cc(ca(x1))) | → | ac#(ca(x1)) | (71) |
ac#(cc(cb(x1))) | → | aa#(ac(cb(x1))) | (75) |
ac#(cc(cb(x1))) | → | ac#(cb(x1)) | (76) |
ac#(cc(cc(x1))) | → | aa#(ac(cc(x1))) | (80) |
ac#(cc(cc(x1))) | → | ac#(cc(x1)) | (81) |
The dependency pairs are split into 0 components.
ba#(aa(x1)) | → | ba#(x1) | (53) |
[aa(x1)] | = | 1 · x1 |
[ba#(x1)] | = | 1 · x1 |
Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.
ba#(aa(x1)) | → | ba#(x1) | (53) |
1 | > | 1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
ca#(ab(ba(x1))) | → | ca#(x1) | (62) |
ca#(aa(x1)) | → | ca#(x1) | (54) |
ca#(ab(bb(x1))) | → | cb#(x1) | (63) |
cb#(ba(x1)) | → | ca#(x1) | (65) |
cb#(bb(x1)) | → | cb#(x1) | (66) |
[ab(x1)] | = | 1 · x1 |
[ba(x1)] | = | 1 · x1 |
[aa(x1)] | = | 1 · x1 |
[bb(x1)] | = | 1 · x1 |
[ca#(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.
ca#(ab(bb(x1))) | → | cb#(x1) | (63) |
1 | > | 1 | |
cb#(ba(x1)) | → | ca#(x1) | (65) |
1 | > | 1 | |
cb#(bb(x1)) | → | cb#(x1) | (66) |
1 | > | 1 | |
ca#(ab(ba(x1))) | → | ca#(x1) | (62) |
1 | > | 1 | |
ca#(aa(x1)) | → | ca#(x1) | (54) |
1 | > | 1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
ab#(bb(x1)) | → | ab#(x1) | (64) |
[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) | (64) |
1 | > | 1 |
As there is no critical graph in the transitive closure, there are no infinite chains.