Certification Problem
Input (TPDB SRS_Relative/Mixed_relative_SRS/un01)
The relative rewrite relation R/S is considered where R is the following TRS
|
b(b(a(a(b(a(b(x1))))))) |
→ |
b(a(b(b(a(b(b(x1))))))) |
(1) |
|
b(b(b(x1))) |
→ |
b(b(a(b(b(x1))))) |
(2) |
and S is the following TRS.
|
b(a(b(x1))) |
→ |
b(a(b(a(a(b(x1)))))) |
(3) |
Property / Task
Prove or disprove termination.Answer / Result
Yes.Proof (by matchbox @ termCOMP 2023)
1 Rule Removal
Using the
matrix interpretations of dimension 4 with strict dimension 1 over the arctic semiring over the naturals
| [b(x1)] |
= |
|
| 0 |
-∞ |
0 |
0 |
| 0 |
-∞ |
0 |
0 |
| 0 |
-1 |
1 |
1 |
|
-∞ |
-∞ |
-1 |
0 |
|
|
· x1
|
| [a(x1)] |
= |
|
| 0 |
-∞ |
-1 |
-∞ |
| 1 |
1 |
0 |
-∞ |
|
-∞ |
-∞ |
-∞ |
-∞ |
|
-∞ |
-∞ |
-∞ |
0 |
|
|
· x1
|
all of the following rules can be deleted.
|
b(b(b(x1))) |
→ |
b(b(a(b(b(x1))))) |
(2) |
1.1 Rule Removal
Using the
matrix interpretations of dimension 5 with strict dimension 1 over the naturals
| [b(x1)] |
= |
|
| 1 |
0 |
0 |
0 |
1 |
| 0 |
0 |
0 |
1 |
1 |
| 0 |
0 |
1 |
0 |
1 |
| 0 |
0 |
1 |
0 |
0 |
| 0 |
0 |
0 |
0 |
0 |
|
|
· x1 +
|
| [a(x1)] |
= |
|
| 1 |
1 |
0 |
0 |
1 |
| 0 |
0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
2 |
| 0 |
2 |
1 |
0 |
0 |
| 0 |
1 |
0 |
0 |
0 |
|
|
· x1 +
|
all of the following rules can be deleted.
|
b(b(a(a(b(a(b(x1))))))) |
→ |
b(a(b(b(a(b(b(x1))))))) |
(1) |
1.1.1 R is empty
There are no rules in the TRS. Hence, it is terminating.