YES
Termination Proof
Termination Proof
by ttt2 (version ttt2 1.15)
Input
The rewrite relation of the following TRS is considered.
|
r(e(x0)) |
→ |
w(r(x0)) |
|
i(t(x0)) |
→ |
e(r(x0)) |
|
e(w(x0)) |
→ |
r(i(x0)) |
|
t(e(x0)) |
→ |
r(e(x0)) |
|
w(r(x0)) |
→ |
i(t(x0)) |
|
e(r(x0)) |
→ |
e(w(x0)) |
|
r(i(t(e(r(x0))))) |
→ |
e(w(r(i(t(e(x0)))))) |
Proof
1 Rule Removal
Using the
linear polynomial interpretation over the arctic semiring over the integers
| [i(x1)] |
= |
0 ·
x1 +
-∞
|
| [r(x1)] |
= |
2 ·
x1 +
-∞
|
| [t(x1)] |
= |
3 ·
x1 +
-∞
|
| [e(x1)] |
= |
1 ·
x1 +
-∞
|
| [w(x1)] |
= |
1 ·
x1 +
-∞
|
the
rules
|
r(e(x0)) |
→ |
w(r(x0)) |
|
i(t(x0)) |
→ |
e(r(x0)) |
|
e(w(x0)) |
→ |
r(i(x0)) |
|
w(r(x0)) |
→ |
i(t(x0)) |
|
r(i(t(e(r(x0))))) |
→ |
e(w(r(i(t(e(x0)))))) |
remain.
1.1 Rule Removal
Using the
linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1
over the naturals
| [i(x1)] |
= |
·
x1 +
|
| [r(x1)] |
= |
·
x1 +
|
| [t(x1)] |
= |
·
x1 +
|
| [e(x1)] |
= |
·
x1 +
|
| [w(x1)] |
= |
·
x1 +
|
the
rules
|
r(e(x0)) |
→ |
w(r(x0)) |
|
i(t(x0)) |
→ |
e(r(x0)) |
|
e(w(x0)) |
→ |
r(i(x0)) |
|
w(r(x0)) |
→ |
i(t(x0)) |
remain.
1.1.1 Rule Removal
Using the
linear polynomial interpretation over the arctic semiring over the integers
| [i(x1)] |
= |
0 ·
x1 +
-∞
|
| [r(x1)] |
= |
0 ·
x1 +
-∞
|
| [t(x1)] |
= |
4 ·
x1 +
-∞
|
| [e(x1)] |
= |
4 ·
x1 +
-∞
|
| [w(x1)] |
= |
4 ·
x1 +
-∞
|
the
rules
|
r(e(x0)) |
→ |
w(r(x0)) |
|
i(t(x0)) |
→ |
e(r(x0)) |
|
w(r(x0)) |
→ |
i(t(x0)) |
remain.
1.1.1.1 Rule Removal
Using the
Knuth Bendix order with w0 = 1 and the following precedence and weight function
| prec(i) |
= |
1 |
|
weight(i) |
= |
1 |
|
|
|
| prec(t) |
= |
0 |
|
weight(t) |
= |
1 |
|
|
|
| prec(w) |
= |
2 |
|
weight(w) |
= |
1 |
|
|
|
| prec(r) |
= |
3 |
|
weight(r) |
= |
1 |
|
|
|
| prec(e) |
= |
0 |
|
weight(e) |
= |
1 |
|
|
|
all rules could be removed.
1.1.1.1.1 R is empty
There are no rules in the TRS. Hence, it is terminating.