YES
Termination Proof
Termination Proof
by ttt2 (version ttt2 1.15)
Input
The rewrite relation of the following TRS is considered.
|
a(a(x0)) |
→ |
b(b(b(x0))) |
|
b(b(x0)) |
→ |
c(c(c(x0))) |
|
c(c(c(c(x0)))) |
→ |
a(b(x0)) |
Proof
1 Rule Removal
Using the
linear polynomial interpretation over the arctic semiring over the integers
| [b(x1)] |
= |
6 ·
x1 +
-∞
|
| [a(x1)] |
= |
10 ·
x1 +
-∞
|
| [c(x1)] |
= |
4 ·
x1 +
-∞
|
the
rules
|
b(b(x0)) |
→ |
c(c(c(x0))) |
|
c(c(c(c(x0)))) |
→ |
a(b(x0)) |
remain.
1.1 Rule Removal
Using the
linear polynomial interpretation over the arctic semiring over the integers
| [b(x1)] |
= |
5 ·
x1 +
-∞
|
| [a(x1)] |
= |
7 ·
x1 +
-∞
|
| [c(x1)] |
= |
3 ·
x1 +
-∞
|
the
rule
|
c(c(c(c(x0)))) |
→ |
a(b(x0)) |
remains.
1.1.1 Rule Removal
Using the
linear polynomial interpretation over (2 x 2)-matrices with strict dimension 1
over the arctic semiring over the integers
| [b(x1)] |
= |
·
x1 +
|
| [a(x1)] |
= |
·
x1 +
|
| [c(x1)] |
= |
·
x1 +
|
all rules could be removed.
1.1.1.1 R is empty
There are no rules in the TRS. Hence, it is terminating.