YES
Termination Proof
Termination Proof
by ttt2 (version ttt2 1.15)
Input
The rewrite relation of the following TRS is considered.
|
a12(a12(x0)) |
→ |
x0 |
|
a13(a13(x0)) |
→ |
x0 |
|
a14(a14(x0)) |
→ |
x0 |
|
a15(a15(x0)) |
→ |
x0 |
|
a16(a16(x0)) |
→ |
x0 |
|
a23(a23(x0)) |
→ |
x0 |
|
a24(a24(x0)) |
→ |
x0 |
|
a25(a25(x0)) |
→ |
x0 |
|
a26(a26(x0)) |
→ |
x0 |
|
a34(a34(x0)) |
→ |
x0 |
|
a35(a35(x0)) |
→ |
x0 |
|
a36(a36(x0)) |
→ |
x0 |
|
a45(a45(x0)) |
→ |
x0 |
|
a46(a46(x0)) |
→ |
x0 |
|
a56(a56(x0)) |
→ |
x0 |
|
a13(x0) |
→ |
a12(a23(a12(x0))) |
|
a14(x0) |
→ |
a12(a23(a34(a23(a12(x0))))) |
|
a15(x0) |
→ |
a12(a23(a34(a45(a34(a23(a12(x0))))))) |
|
a16(x0) |
→ |
a12(a23(a34(a45(a56(a45(a34(a23(a12(x0))))))))) |
|
a24(x0) |
→ |
a23(a34(a23(x0))) |
|
a25(x0) |
→ |
a23(a34(a45(a34(a23(x0))))) |
|
a26(x0) |
→ |
a23(a34(a45(a56(a45(a34(a23(x0))))))) |
|
a35(x0) |
→ |
a34(a45(a34(x0))) |
|
a36(x0) |
→ |
a34(a45(a56(a45(a34(x0))))) |
|
a46(x0) |
→ |
a45(a56(a45(x0))) |
|
a12(a23(a12(a23(a12(a23(x0)))))) |
→ |
x0 |
|
a23(a34(a23(a34(a23(a34(x0)))))) |
→ |
x0 |
|
a34(a45(a34(a45(a34(a45(x0)))))) |
→ |
x0 |
|
a45(a56(a45(a56(a45(a56(x0)))))) |
→ |
x0 |
|
a12(a34(x0)) |
→ |
a34(a12(x0)) |
|
a12(a45(x0)) |
→ |
a45(a12(x0)) |
|
a12(a56(x0)) |
→ |
a56(a12(x0)) |
|
a23(a45(x0)) |
→ |
a45(a23(x0)) |
|
a23(a56(x0)) |
→ |
a56(a23(x0)) |
|
a34(a56(x0)) |
→ |
a56(a34(x0)) |
Proof
1 Rule Removal
Using the
linear polynomial interpretation over the arctic semiring over the integers
| [a16(x1)] |
= |
0 ·
x1 +
-∞
|
| [a26(x1)] |
= |
0 ·
x1 +
-∞
|
| [a13(x1)] |
= |
0 ·
x1 +
-∞
|
| [a15(x1)] |
= |
4 ·
x1 +
-∞
|
| [a23(x1)] |
= |
0 ·
x1 +
-∞
|
| [a35(x1)] |
= |
5 ·
x1 +
-∞
|
| [a12(x1)] |
= |
0 ·
x1 +
-∞
|
| [a25(x1)] |
= |
0 ·
x1 +
-∞
|
| [a56(x1)] |
= |
0 ·
x1 +
-∞
|
| [a45(x1)] |
= |
0 ·
x1 +
-∞
|
| [a36(x1)] |
= |
0 ·
x1 +
-∞
|
| [a34(x1)] |
= |
0 ·
x1 +
-∞
|
| [a46(x1)] |
= |
0 ·
x1 +
-∞
|
| [a24(x1)] |
= |
0 ·
x1 +
-∞
|
| [a14(x1)] |
= |
0 ·
x1 +
-∞
|
the
rules
|
a12(a12(x0)) |
→ |
x0 |
|
a13(a13(x0)) |
→ |
x0 |
|
a14(a14(x0)) |
→ |
x0 |
|
a16(a16(x0)) |
→ |
x0 |
|
a23(a23(x0)) |
→ |
x0 |
|
a24(a24(x0)) |
→ |
x0 |
|
a25(a25(x0)) |
→ |
x0 |
|
a26(a26(x0)) |
→ |
x0 |
|
a34(a34(x0)) |
→ |
x0 |
|
a36(a36(x0)) |
→ |
x0 |
|
a45(a45(x0)) |
→ |
x0 |
|
a46(a46(x0)) |
→ |
x0 |
|
a56(a56(x0)) |
→ |
x0 |
|
a13(x0) |
→ |
a12(a23(a12(x0))) |
|
a14(x0) |
→ |
a12(a23(a34(a23(a12(x0))))) |
|
a16(x0) |
→ |
a12(a23(a34(a45(a56(a45(a34(a23(a12(x0))))))))) |
|
a24(x0) |
→ |
a23(a34(a23(x0))) |
|
a25(x0) |
→ |
a23(a34(a45(a34(a23(x0))))) |
|
a26(x0) |
→ |
a23(a34(a45(a56(a45(a34(a23(x0))))))) |
|
a36(x0) |
→ |
a34(a45(a56(a45(a34(x0))))) |
|
a46(x0) |
→ |
a45(a56(a45(x0))) |
|
a12(a23(a12(a23(a12(a23(x0)))))) |
→ |
x0 |
|
a23(a34(a23(a34(a23(a34(x0)))))) |
→ |
x0 |
|
a34(a45(a34(a45(a34(a45(x0)))))) |
→ |
x0 |
|
a45(a56(a45(a56(a45(a56(x0)))))) |
→ |
x0 |
|
a12(a34(x0)) |
→ |
a34(a12(x0)) |
|
a12(a45(x0)) |
→ |
a45(a12(x0)) |
|
a12(a56(x0)) |
→ |
a56(a12(x0)) |
|
a23(a45(x0)) |
→ |
a45(a23(x0)) |
|
a23(a56(x0)) |
→ |
a56(a23(x0)) |
|
a34(a56(x0)) |
→ |
a56(a34(x0)) |
remain.
1.1 Rule Removal
Using the
linear polynomial interpretation over the arctic semiring over the integers
| [a16(x1)] |
= |
0 ·
x1 +
-∞
|
| [a26(x1)] |
= |
0 ·
x1 +
-∞
|
| [a13(x1)] |
= |
0 ·
x1 +
-∞
|
| [a23(x1)] |
= |
0 ·
x1 +
-∞
|
| [a12(x1)] |
= |
0 ·
x1 +
-∞
|
| [a25(x1)] |
= |
0 ·
x1 +
-∞
|
| [a56(x1)] |
= |
0 ·
x1 +
-∞
|
| [a45(x1)] |
= |
0 ·
x1 +
-∞
|
| [a36(x1)] |
= |
1 ·
x1 +
-∞
|
| [a34(x1)] |
= |
0 ·
x1 +
-∞
|
| [a46(x1)] |
= |
1 ·
x1 +
-∞
|
| [a24(x1)] |
= |
0 ·
x1 +
-∞
|
| [a14(x1)] |
= |
2 ·
x1 +
-∞
|
the
rules
|
a12(a12(x0)) |
→ |
x0 |
|
a13(a13(x0)) |
→ |
x0 |
|
a16(a16(x0)) |
→ |
x0 |
|
a23(a23(x0)) |
→ |
x0 |
|
a24(a24(x0)) |
→ |
x0 |
|
a25(a25(x0)) |
→ |
x0 |
|
a26(a26(x0)) |
→ |
x0 |
|
a34(a34(x0)) |
→ |
x0 |
|
a45(a45(x0)) |
→ |
x0 |
|
a56(a56(x0)) |
→ |
x0 |
|
a13(x0) |
→ |
a12(a23(a12(x0))) |
|
a16(x0) |
→ |
a12(a23(a34(a45(a56(a45(a34(a23(a12(x0))))))))) |
|
a24(x0) |
→ |
a23(a34(a23(x0))) |
|
a25(x0) |
→ |
a23(a34(a45(a34(a23(x0))))) |
|
a26(x0) |
→ |
a23(a34(a45(a56(a45(a34(a23(x0))))))) |
|
a12(a23(a12(a23(a12(a23(x0)))))) |
→ |
x0 |
|
a23(a34(a23(a34(a23(a34(x0)))))) |
→ |
x0 |
|
a34(a45(a34(a45(a34(a45(x0)))))) |
→ |
x0 |
|
a45(a56(a45(a56(a45(a56(x0)))))) |
→ |
x0 |
|
a12(a34(x0)) |
→ |
a34(a12(x0)) |
|
a12(a45(x0)) |
→ |
a45(a12(x0)) |
|
a12(a56(x0)) |
→ |
a56(a12(x0)) |
|
a23(a45(x0)) |
→ |
a45(a23(x0)) |
|
a23(a56(x0)) |
→ |
a56(a23(x0)) |
|
a34(a56(x0)) |
→ |
a56(a34(x0)) |
remain.
1.1.1 Rule Removal
Using the
linear polynomial interpretation over the arctic semiring over the integers
| [a16(x1)] |
= |
0 ·
x1 +
-∞
|
| [a26(x1)] |
= |
8 ·
x1 +
-∞
|
| [a13(x1)] |
= |
0 ·
x1 +
-∞
|
| [a23(x1)] |
= |
0 ·
x1 +
-∞
|
| [a12(x1)] |
= |
0 ·
x1 +
-∞
|
| [a25(x1)] |
= |
0 ·
x1 +
-∞
|
| [a56(x1)] |
= |
0 ·
x1 +
-∞
|
| [a45(x1)] |
= |
0 ·
x1 +
-∞
|
| [a34(x1)] |
= |
0 ·
x1 +
-∞
|
| [a24(x1)] |
= |
8 ·
x1 +
-∞
|
the
rules
|
a12(a12(x0)) |
→ |
x0 |
|
a13(a13(x0)) |
→ |
x0 |
|
a16(a16(x0)) |
→ |
x0 |
|
a23(a23(x0)) |
→ |
x0 |
|
a25(a25(x0)) |
→ |
x0 |
|
a34(a34(x0)) |
→ |
x0 |
|
a45(a45(x0)) |
→ |
x0 |
|
a56(a56(x0)) |
→ |
x0 |
|
a13(x0) |
→ |
a12(a23(a12(x0))) |
|
a16(x0) |
→ |
a12(a23(a34(a45(a56(a45(a34(a23(a12(x0))))))))) |
|
a25(x0) |
→ |
a23(a34(a45(a34(a23(x0))))) |
|
a12(a23(a12(a23(a12(a23(x0)))))) |
→ |
x0 |
|
a23(a34(a23(a34(a23(a34(x0)))))) |
→ |
x0 |
|
a34(a45(a34(a45(a34(a45(x0)))))) |
→ |
x0 |
|
a45(a56(a45(a56(a45(a56(x0)))))) |
→ |
x0 |
|
a12(a34(x0)) |
→ |
a34(a12(x0)) |
|
a12(a45(x0)) |
→ |
a45(a12(x0)) |
|
a12(a56(x0)) |
→ |
a56(a12(x0)) |
|
a23(a45(x0)) |
→ |
a45(a23(x0)) |
|
a23(a56(x0)) |
→ |
a56(a23(x0)) |
|
a34(a56(x0)) |
→ |
a56(a34(x0)) |
remain.
1.1.1.1 Rule Removal
Using the
linear polynomial interpretation over the arctic semiring over the integers
| [a16(x1)] |
= |
0 ·
x1 +
-∞
|
| [a13(x1)] |
= |
10 ·
x1 +
-∞
|
| [a23(x1)] |
= |
0 ·
x1 +
-∞
|
| [a12(x1)] |
= |
0 ·
x1 +
-∞
|
| [a25(x1)] |
= |
0 ·
x1 +
-∞
|
| [a56(x1)] |
= |
0 ·
x1 +
-∞
|
| [a45(x1)] |
= |
0 ·
x1 +
-∞
|
| [a34(x1)] |
= |
0 ·
x1 +
-∞
|
the
rules
|
a12(a12(x0)) |
→ |
x0 |
|
a16(a16(x0)) |
→ |
x0 |
|
a23(a23(x0)) |
→ |
x0 |
|
a25(a25(x0)) |
→ |
x0 |
|
a34(a34(x0)) |
→ |
x0 |
|
a45(a45(x0)) |
→ |
x0 |
|
a56(a56(x0)) |
→ |
x0 |
|
a16(x0) |
→ |
a12(a23(a34(a45(a56(a45(a34(a23(a12(x0))))))))) |
|
a25(x0) |
→ |
a23(a34(a45(a34(a23(x0))))) |
|
a12(a23(a12(a23(a12(a23(x0)))))) |
→ |
x0 |
|
a23(a34(a23(a34(a23(a34(x0)))))) |
→ |
x0 |
|
a34(a45(a34(a45(a34(a45(x0)))))) |
→ |
x0 |
|
a45(a56(a45(a56(a45(a56(x0)))))) |
→ |
x0 |
|
a12(a34(x0)) |
→ |
a34(a12(x0)) |
|
a12(a45(x0)) |
→ |
a45(a12(x0)) |
|
a12(a56(x0)) |
→ |
a56(a12(x0)) |
|
a23(a45(x0)) |
→ |
a45(a23(x0)) |
|
a23(a56(x0)) |
→ |
a56(a23(x0)) |
|
a34(a56(x0)) |
→ |
a56(a34(x0)) |
remain.
1.1.1.1.1 Rule Removal
Using the
linear polynomial interpretation over the arctic semiring over the integers
| [a16(x1)] |
= |
2 ·
x1 +
-∞
|
| [a23(x1)] |
= |
0 ·
x1 +
-∞
|
| [a12(x1)] |
= |
0 ·
x1 +
-∞
|
| [a25(x1)] |
= |
0 ·
x1 +
-∞
|
| [a56(x1)] |
= |
2 ·
x1 +
-∞
|
| [a45(x1)] |
= |
0 ·
x1 +
-∞
|
| [a34(x1)] |
= |
0 ·
x1 +
-∞
|
the
rules
|
a12(a12(x0)) |
→ |
x0 |
|
a23(a23(x0)) |
→ |
x0 |
|
a25(a25(x0)) |
→ |
x0 |
|
a34(a34(x0)) |
→ |
x0 |
|
a45(a45(x0)) |
→ |
x0 |
|
a16(x0) |
→ |
a12(a23(a34(a45(a56(a45(a34(a23(a12(x0))))))))) |
|
a25(x0) |
→ |
a23(a34(a45(a34(a23(x0))))) |
|
a12(a23(a12(a23(a12(a23(x0)))))) |
→ |
x0 |
|
a23(a34(a23(a34(a23(a34(x0)))))) |
→ |
x0 |
|
a34(a45(a34(a45(a34(a45(x0)))))) |
→ |
x0 |
|
a12(a34(x0)) |
→ |
a34(a12(x0)) |
|
a12(a45(x0)) |
→ |
a45(a12(x0)) |
|
a12(a56(x0)) |
→ |
a56(a12(x0)) |
|
a23(a45(x0)) |
→ |
a45(a23(x0)) |
|
a23(a56(x0)) |
→ |
a56(a23(x0)) |
|
a34(a56(x0)) |
→ |
a56(a34(x0)) |
remain.
1.1.1.1.1.1 Rule Removal
Using the
linear polynomial interpretation over the arctic semiring over the integers
| [a16(x1)] |
= |
12 ·
x1 +
-∞
|
| [a23(x1)] |
= |
0 ·
x1 +
-∞
|
| [a12(x1)] |
= |
0 ·
x1 +
-∞
|
| [a25(x1)] |
= |
1 ·
x1 +
-∞
|
| [a56(x1)] |
= |
8 ·
x1 +
-∞
|
| [a45(x1)] |
= |
1 ·
x1 +
-∞
|
| [a34(x1)] |
= |
0 ·
x1 +
-∞
|
the
rules
|
a12(a12(x0)) |
→ |
x0 |
|
a23(a23(x0)) |
→ |
x0 |
|
a34(a34(x0)) |
→ |
x0 |
|
a25(x0) |
→ |
a23(a34(a45(a34(a23(x0))))) |
|
a12(a23(a12(a23(a12(a23(x0)))))) |
→ |
x0 |
|
a23(a34(a23(a34(a23(a34(x0)))))) |
→ |
x0 |
|
a12(a34(x0)) |
→ |
a34(a12(x0)) |
|
a12(a45(x0)) |
→ |
a45(a12(x0)) |
|
a12(a56(x0)) |
→ |
a56(a12(x0)) |
|
a23(a45(x0)) |
→ |
a45(a23(x0)) |
|
a23(a56(x0)) |
→ |
a56(a23(x0)) |
|
a34(a56(x0)) |
→ |
a56(a34(x0)) |
remain.
1.1.1.1.1.1.1 Rule Removal
Using the
linear polynomial interpretation over the arctic semiring over the integers
| [a23(x1)] |
= |
2 ·
x1 +
-∞
|
| [a12(x1)] |
= |
0 ·
x1 +
-∞
|
| [a25(x1)] |
= |
6 ·
x1 +
-∞
|
| [a56(x1)] |
= |
2 ·
x1 +
-∞
|
| [a45(x1)] |
= |
0 ·
x1 +
-∞
|
| [a34(x1)] |
= |
1 ·
x1 +
-∞
|
the
rules
|
a12(a12(x0)) |
→ |
x0 |
|
a25(x0) |
→ |
a23(a34(a45(a34(a23(x0))))) |
|
a12(a34(x0)) |
→ |
a34(a12(x0)) |
|
a12(a45(x0)) |
→ |
a45(a12(x0)) |
|
a12(a56(x0)) |
→ |
a56(a12(x0)) |
|
a23(a45(x0)) |
→ |
a45(a23(x0)) |
|
a23(a56(x0)) |
→ |
a56(a23(x0)) |
|
a34(a56(x0)) |
→ |
a56(a34(x0)) |
remain.
1.1.1.1.1.1.1.1 Rule Removal
Using the
linear polynomial interpretation over the arctic semiring over the integers
| [a23(x1)] |
= |
3 ·
x1 +
-∞
|
| [a12(x1)] |
= |
6 ·
x1 +
-∞
|
| [a25(x1)] |
= |
8 ·
x1 +
-∞
|
| [a56(x1)] |
= |
0 ·
x1 +
-∞
|
| [a45(x1)] |
= |
0 ·
x1 +
-∞
|
| [a34(x1)] |
= |
1 ·
x1 +
-∞
|
the
rules
|
a25(x0) |
→ |
a23(a34(a45(a34(a23(x0))))) |
|
a12(a34(x0)) |
→ |
a34(a12(x0)) |
|
a12(a45(x0)) |
→ |
a45(a12(x0)) |
|
a12(a56(x0)) |
→ |
a56(a12(x0)) |
|
a23(a45(x0)) |
→ |
a45(a23(x0)) |
|
a23(a56(x0)) |
→ |
a56(a23(x0)) |
|
a34(a56(x0)) |
→ |
a56(a34(x0)) |
remain.
1.1.1.1.1.1.1.1.1 Rule Removal
Using the
linear polynomial interpretation over the arctic semiring over the integers
| [a23(x1)] |
= |
4 ·
x1 +
-∞
|
| [a12(x1)] |
= |
15 ·
x1 +
-∞
|
| [a25(x1)] |
= |
13 ·
x1 +
-∞
|
| [a56(x1)] |
= |
1 ·
x1 +
-∞
|
| [a45(x1)] |
= |
4 ·
x1 +
-∞
|
| [a34(x1)] |
= |
0 ·
x1 +
-∞
|
the
rules
|
a12(a34(x0)) |
→ |
a34(a12(x0)) |
|
a12(a45(x0)) |
→ |
a45(a12(x0)) |
|
a12(a56(x0)) |
→ |
a56(a12(x0)) |
|
a23(a45(x0)) |
→ |
a45(a23(x0)) |
|
a23(a56(x0)) |
→ |
a56(a23(x0)) |
|
a34(a56(x0)) |
→ |
a56(a34(x0)) |
remain.
1.1.1.1.1.1.1.1.1.1 Rule Removal
Using the
Knuth Bendix order with w0 = 1 and the following precedence and weight function
| prec(a56) |
= |
0 |
|
weight(a56) |
= |
1 |
|
|
|
| prec(a45) |
= |
0 |
|
weight(a45) |
= |
1 |
|
|
|
| prec(a34) |
= |
2 |
|
weight(a34) |
= |
1 |
|
|
|
| prec(a23) |
= |
1 |
|
weight(a23) |
= |
1 |
|
|
|
| prec(a12) |
= |
3 |
|
weight(a12) |
= |
0 |
|
|
|
all rules could be removed.
1.1.1.1.1.1.1.1.1.1.1 R is empty
There are no rules in the TRS. Hence, it is terminating.