The rewrite relation of the following TRS is considered.
c(b(a(a(x1)))) | → | a(a(b(a(a(b(c(b(a(c(x1)))))))))) | (1) |
{c(☐), b(☐), a(☐)}
We obtain the transformed TRSc(c(b(a(a(x1))))) | → | c(a(a(b(a(a(b(c(b(a(c(x1))))))))))) | (2) |
b(c(b(a(a(x1))))) | → | b(a(a(b(a(a(b(c(b(a(c(x1))))))))))) | (3) |
a(c(b(a(a(x1))))) | → | a(a(a(b(a(a(b(c(b(a(c(x1))))))))))) | (4) |
Root-labeling is applied.
We obtain the labeled TRScc(cb(ba(aa(ac(x1))))) | → | ca(aa(ab(ba(aa(ab(bc(cb(ba(ac(cc(x1))))))))))) | (5) |
cc(cb(ba(aa(ab(x1))))) | → | ca(aa(ab(ba(aa(ab(bc(cb(ba(ac(cb(x1))))))))))) | (6) |
cc(cb(ba(aa(aa(x1))))) | → | ca(aa(ab(ba(aa(ab(bc(cb(ba(ac(ca(x1))))))))))) | (7) |
bc(cb(ba(aa(ac(x1))))) | → | ba(aa(ab(ba(aa(ab(bc(cb(ba(ac(cc(x1))))))))))) | (8) |
bc(cb(ba(aa(ab(x1))))) | → | ba(aa(ab(ba(aa(ab(bc(cb(ba(ac(cb(x1))))))))))) | (9) |
bc(cb(ba(aa(aa(x1))))) | → | ba(aa(ab(ba(aa(ab(bc(cb(ba(ac(ca(x1))))))))))) | (10) |
ac(cb(ba(aa(ac(x1))))) | → | aa(aa(ab(ba(aa(ab(bc(cb(ba(ac(cc(x1))))))))))) | (11) |
ac(cb(ba(aa(ab(x1))))) | → | aa(aa(ab(ba(aa(ab(bc(cb(ba(ac(cb(x1))))))))))) | (12) |
ac(cb(ba(aa(aa(x1))))) | → | aa(aa(ab(ba(aa(ab(bc(cb(ba(ac(ca(x1))))))))))) | (13) |
cc#(cb(ba(aa(ac(x1))))) | → | bc#(cb(ba(ac(cc(x1))))) | (14) |
cc#(cb(ba(aa(ac(x1))))) | → | ac#(cc(x1)) | (15) |
cc#(cb(ba(aa(ac(x1))))) | → | cc#(x1) | (16) |
cc#(cb(ba(aa(ab(x1))))) | → | bc#(cb(ba(ac(cb(x1))))) | (17) |
cc#(cb(ba(aa(ab(x1))))) | → | ac#(cb(x1)) | (18) |
cc#(cb(ba(aa(aa(x1))))) | → | bc#(cb(ba(ac(ca(x1))))) | (19) |
cc#(cb(ba(aa(aa(x1))))) | → | ac#(ca(x1)) | (20) |
bc#(cb(ba(aa(ac(x1))))) | → | bc#(cb(ba(ac(cc(x1))))) | (21) |
bc#(cb(ba(aa(ac(x1))))) | → | ac#(cc(x1)) | (22) |
bc#(cb(ba(aa(ac(x1))))) | → | cc#(x1) | (23) |
bc#(cb(ba(aa(ab(x1))))) | → | bc#(cb(ba(ac(cb(x1))))) | (24) |
bc#(cb(ba(aa(ab(x1))))) | → | ac#(cb(x1)) | (25) |
bc#(cb(ba(aa(aa(x1))))) | → | bc#(cb(ba(ac(ca(x1))))) | (26) |
bc#(cb(ba(aa(aa(x1))))) | → | ac#(ca(x1)) | (27) |
ac#(cb(ba(aa(ac(x1))))) | → | bc#(cb(ba(ac(cc(x1))))) | (28) |
ac#(cb(ba(aa(ac(x1))))) | → | ac#(cc(x1)) | (29) |
ac#(cb(ba(aa(ac(x1))))) | → | cc#(x1) | (30) |
ac#(cb(ba(aa(ab(x1))))) | → | bc#(cb(ba(ac(cb(x1))))) | (31) |
ac#(cb(ba(aa(ab(x1))))) | → | ac#(cb(x1)) | (32) |
ac#(cb(ba(aa(aa(x1))))) | → | bc#(cb(ba(ac(ca(x1))))) | (33) |
ac#(cb(ba(aa(aa(x1))))) | → | ac#(ca(x1)) | (34) |
The dependency pairs are split into 1 component.
bc#(cb(ba(aa(ab(x1))))) | → | bc#(cb(ba(ac(cb(x1))))) | (24) |
bc#(cb(ba(aa(ab(x1))))) | → | ac#(cb(x1)) | (25) |
ac#(cb(ba(aa(ac(x1))))) | → | bc#(cb(ba(ac(cc(x1))))) | (28) |
ac#(cb(ba(aa(ac(x1))))) | → | cc#(x1) | (30) |
cc#(cb(ba(aa(ac(x1))))) | → | bc#(cb(ba(ac(cc(x1))))) | (14) |
cc#(cb(ba(aa(ac(x1))))) | → | cc#(x1) | (16) |
cc#(cb(ba(aa(ab(x1))))) | → | bc#(cb(ba(ac(cb(x1))))) | (17) |
cc#(cb(ba(aa(ab(x1))))) | → | ac#(cb(x1)) | (18) |
ac#(cb(ba(aa(ab(x1))))) | → | bc#(cb(ba(ac(cb(x1))))) | (31) |
ac#(cb(ba(aa(ab(x1))))) | → | ac#(cb(x1)) | (32) |
[bc#(x1)] | = | -1 + 2 · x1 |
[ba(x1)] | = | 2 · x1 |
[cb(x1)] | = | -2 + 2 · x1 |
[ac(x1)] | = | 0 |
[aa(x1)] | = | -1 + x1 |
[ab(x1)] | = | 2 |
[bc(x1)] | = | -2 |
[cc(x1)] | = | x1 |
[ca(x1)] | = | 1 |
[ac#(x1)] | = | 0 |
[cc#(x1)] | = | 0 |
ac(cb(ba(aa(ac(x1))))) | → | aa(aa(ab(ba(aa(ab(bc(cb(ba(ac(cc(x1))))))))))) | (11) |
ac(cb(ba(aa(ab(x1))))) | → | aa(aa(ab(ba(aa(ab(bc(cb(ba(ac(cb(x1))))))))))) | (12) |
ac(cb(ba(aa(aa(x1))))) | → | aa(aa(ab(ba(aa(ab(bc(cb(ba(ac(ca(x1))))))))))) | (13) |
bc#(cb(ba(aa(ab(x1))))) | → | bc#(cb(ba(ac(cb(x1))))) | (24) |
bc#(cb(ba(aa(ab(x1))))) | → | ac#(cb(x1)) | (25) |
The dependency pairs are split into 1 component.
ac#(cb(ba(aa(ac(x1))))) | → | cc#(x1) | (30) |
cc#(cb(ba(aa(ac(x1))))) | → | cc#(x1) | (16) |
cc#(cb(ba(aa(ab(x1))))) | → | ac#(cb(x1)) | (18) |
ac#(cb(ba(aa(ab(x1))))) | → | ac#(cb(x1)) | (32) |
[cb(x1)] | = | 1 · x1 |
[ba(x1)] | = | 1 · x1 |
[aa(x1)] | = | 1 · x1 |
[ac(x1)] | = | 1 · x1 |
[ab(x1)] | = | 1 · x1 |
[cc#(x1)] | = | 1 · x1 |
[ac#(x1)] | = | 1 · x1 |
[ac#(x1)] | = | 1 · x1 |
[cb(x1)] | = | 1 · x1 |
[ba(x1)] | = | 1 · x1 |
[aa(x1)] | = | 1 · x1 |
[ac(x1)] | = | 1 + 1 · x1 |
[cc#(x1)] | = | 1 · x1 |
[ab(x1)] | = | 1 · x1 |
ac#(cb(ba(aa(ac(x1))))) | → | cc#(x1) | (30) |
cc#(cb(ba(aa(ac(x1))))) | → | cc#(x1) | (16) |
The dependency pairs are split into 1 component.
ac#(cb(ba(aa(ab(x1))))) | → | ac#(cb(x1)) | (32) |
[ac#(x1)] | = | -2 + x1 |
[cb(x1)] | = | 2 · x1 |
[ba(x1)] | = | 2 · x1 |
[aa(x1)] | = | 2 · x1 |
[ab(x1)] | = | 2 + 2 · x1 |
ac#(cb(ba(aa(ab(x1))))) | → | ac#(cb(x1)) | (32) |
There are no pairs anymore.