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