Certification Problem
Input (TPDB TRS_Standard/Endrullis_06/linear1)
The rewrite relation of the following TRS is considered.
|
a(a(f(b,a(x)))) |
→ |
f(a(a(a(x))),b) |
(1) |
|
a(a(x)) |
→ |
f(b,a(f(a(x),b))) |
(2) |
|
f(a(x),b) |
→ |
f(b,a(x)) |
(3) |
Property / Task
Prove or disprove termination.Answer / Result
Yes.Proof (by ttt2 @ termCOMP 2023)
1 Rule Removal
Using the
linear polynomial interpretation over (4 x 4)-matrices with strict dimension 1
over the naturals
| [a(x1)] |
= |
|
| 1 |
0 |
0 |
1 |
| 0 |
0 |
0 |
1 |
| 0 |
1 |
0 |
0 |
| 0 |
1 |
0 |
0 |
|
|
· x1 +
|
| 0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
| 1 |
0 |
0 |
0 |
|
|
|
| [b] |
= |
|
| 0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
|
|
|
| [f(x1, x2)] |
= |
|
| 1 |
0 |
0 |
0 |
| 0 |
1 |
0 |
0 |
| 0 |
0 |
0 |
0 |
| 0 |
0 |
1 |
0 |
|
|
· x1 +
|
| 1 |
0 |
0 |
0 |
| 0 |
1 |
0 |
0 |
| 0 |
0 |
0 |
0 |
| 0 |
0 |
1 |
0 |
|
|
· x2 +
|
| 0 |
0 |
0 |
0 |
| 1 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
|
|
|
all of the following rules can be deleted.
|
a(a(x)) |
→ |
f(b,a(f(a(x),b))) |
(2) |
1.1 Rule Removal
Using the
Knuth Bendix order with w0 = 1 and the following precedence and weight functions
| prec(f) |
= |
0 |
|
weight(f) |
= |
0 |
|
|
|
| prec(a) |
= |
3 |
|
weight(a) |
= |
4 |
|
|
|
| prec(b) |
= |
2 |
|
weight(b) |
= |
5 |
|
|
|
all of the following rules can be deleted.
|
a(a(f(b,a(x)))) |
→ |
f(a(a(a(x))),b) |
(1) |
|
f(a(x),b) |
→ |
f(b,a(x)) |
(3) |
1.1.1 R is empty
There are no rules in the TRS. Hence, it is terminating.