YES
Termination Proof
Termination Proof
by ttt2 (version ttt2 1.15)
Input
The rewrite relation of the following TRS is considered.
|
a(a(a(a(x0)))) |
→ |
b(b(b(b(b(b(x0)))))) |
|
b(b(b(b(x0)))) |
→ |
c(c(c(c(c(c(x0)))))) |
|
c(c(c(c(x0)))) |
→ |
d(d(d(d(d(d(x0)))))) |
|
b(b(x0)) |
→ |
d(d(d(d(x0)))) |
|
c(c(d(d(d(d(x0)))))) |
→ |
a(a(x0)) |
Proof
1 Dependency Pair Transformation
The following set of initial dependency pairs has been identified.
|
a#(a(a(a(x0)))) |
→ |
b#(x0) |
|
a#(a(a(a(x0)))) |
→ |
b#(b(x0)) |
|
a#(a(a(a(x0)))) |
→ |
b#(b(b(x0))) |
|
a#(a(a(a(x0)))) |
→ |
b#(b(b(b(x0)))) |
|
a#(a(a(a(x0)))) |
→ |
b#(b(b(b(b(x0))))) |
|
a#(a(a(a(x0)))) |
→ |
b#(b(b(b(b(b(x0)))))) |
|
b#(b(b(b(x0)))) |
→ |
c#(x0) |
|
b#(b(b(b(x0)))) |
→ |
c#(c(x0)) |
|
b#(b(b(b(x0)))) |
→ |
c#(c(c(x0))) |
|
b#(b(b(b(x0)))) |
→ |
c#(c(c(c(x0)))) |
|
b#(b(b(b(x0)))) |
→ |
c#(c(c(c(c(x0))))) |
|
b#(b(b(b(x0)))) |
→ |
c#(c(c(c(c(c(x0)))))) |
|
c#(c(d(d(d(d(x0)))))) |
→ |
a#(x0) |
|
c#(c(d(d(d(d(x0)))))) |
→ |
a#(a(x0)) |
1.1 Reduction Pair Processor
Using the linear polynomial interpretation over the naturals
| [a#(x1)] |
= |
1 ·
x1 + 9 |
| [b(x1)] |
= |
1 ·
x1 + 9 |
| [d(x1)] |
= |
1 ·
x1 + 4 |
| [b#(x1)] |
= |
1 ·
x1 + 6 |
| [a(x1)] |
= |
1 ·
x1 + 14 |
| [c#(x1)] |
= |
1 ·
x1 + 1 |
| [c(x1)] |
= |
1 ·
x1 + 6 |
the
pairs
|
a#(a(a(a(x0)))) |
→ |
b#(b(b(b(b(b(x0)))))) |
|
c#(c(d(d(d(d(x0)))))) |
→ |
a#(a(x0)) |
remain.
1.1.1 Dependency Graph Processor
The dependency pairs are split into 0
components.