YES
Termination Proof
Termination Proof
by ttt2 (version ttt2 1.15)
Input
The rewrite relation of the following TRS is considered.
|
a(x0) |
→ |
b(c(x0)) |
|
a(b(x0)) |
→ |
b(a(x0)) |
|
d(c(x0)) |
→ |
d(a(x0)) |
|
a(c(x0)) |
→ |
c(a(x0)) |
Proof
1 Rule Removal
Using the
linear polynomial interpretation over (2 x 2)-matrices with strict dimension 1
over the arctic semiring over the integers
| [c(x1)] |
= |
·
x1 +
|
| [d(x1)] |
= |
·
x1 +
|
| [a(x1)] |
= |
·
x1 +
|
| [b(x1)] |
= |
·
x1 +
|
the
rules
|
a(x0) |
→ |
b(c(x0)) |
|
a(b(x0)) |
→ |
b(a(x0)) |
|
a(c(x0)) |
→ |
c(a(x0)) |
remain.
1.1 Rule Removal
Using the
linear polynomial interpretation over the arctic semiring over the integers
| [c(x1)] |
= |
0 ·
x1 +
-∞
|
| [a(x1)] |
= |
8 ·
x1 +
-∞
|
| [b(x1)] |
= |
5 ·
x1 +
-∞
|
the
rules
|
a(b(x0)) |
→ |
b(a(x0)) |
|
a(c(x0)) |
→ |
c(a(x0)) |
remain.
1.1.1 Rule Removal
Using the
Knuth Bendix order with w0 = 1 and the following precedence and weight function
| prec(b) |
= |
0 |
|
weight(b) |
= |
1 |
|
|
|
| prec(c) |
= |
0 |
|
weight(c) |
= |
1 |
|
|
|
| prec(a) |
= |
1 |
|
weight(a) |
= |
0 |
|
|
|
all rules could be removed.
1.1.1.1 R is empty
There are no rules in the TRS. Hence, it is terminating.