Certification Problem
Input (TPDB SRS_Standard/Secret_06_SRS/multum6)
The rewrite relation of the following TRS is considered.
|
c(c(b(x1))) |
→ |
a(c(b(x1))) |
(1) |
|
a(c(b(a(x1)))) |
→ |
b(c(c(x1))) |
(2) |
|
b(a(c(x1))) |
→ |
a(b(c(a(x1)))) |
(3) |
|
b(c(a(x1))) |
→ |
c(a(b(x1))) |
(4) |
Property / Task
Prove or disprove termination.Answer / Result
Yes.Proof (by ttt2 @ termCOMP 2023)
1 Rule Removal
Using the
linear polynomial interpretation over (5 x 5)-matrices with strict dimension 1
over the naturals
| [c(x1)] |
= |
|
| 1 |
1 |
0 |
0 |
0 |
| 0 |
0 |
1 |
1 |
0 |
| 0 |
0 |
0 |
1 |
0 |
| 0 |
0 |
1 |
0 |
0 |
| 0 |
0 |
0 |
1 |
0 |
|
|
· x1 +
|
| 0 |
0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
0 |
|
|
|
| [b(x1)] |
= |
|
| 1 |
0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
1 |
| 0 |
0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
1 |
0 |
|
|
· x1 +
|
| 0 |
0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
0 |
| 1 |
0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
0 |
| 1 |
0 |
0 |
0 |
0 |
|
|
|
| [a(x1)] |
= |
|
| 1 |
1 |
0 |
0 |
0 |
| 0 |
0 |
0 |
1 |
0 |
| 0 |
1 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
0 |
| 0 |
0 |
1 |
1 |
0 |
|
|
· x1 +
|
| 0 |
0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
0 |
|
|
|
all of the following rules can be deleted.
|
a(c(b(a(x1)))) |
→ |
b(c(c(x1))) |
(2) |
1.1 Rule Removal
Using the
linear polynomial interpretation over the arctic semiring over the integers
| [c(x1)] |
= |
1 · x1 +
-∞ |
| [b(x1)] |
= |
0 · x1 +
-∞ |
| [a(x1)] |
= |
0 · x1 +
-∞ |
all of the following rules can be deleted.
|
c(c(b(x1))) |
→ |
a(c(b(x1))) |
(1) |
1.1.1 Rule Removal
Using the
linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1
over the naturals
| [c(x1)] |
= |
· x1 +
|
| [b(x1)] |
= |
· x1 +
|
| [a(x1)] |
= |
· x1 +
|
all of the following rules can be deleted.
|
b(c(a(x1))) |
→ |
c(a(b(x1))) |
(4) |
1.1.1.1 Rule Removal
Using the
linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1
over the naturals
| [c(x1)] |
= |
· x1 +
|
| [b(x1)] |
= |
· x1 +
|
| [a(x1)] |
= |
· x1 +
|
all of the following rules can be deleted.
|
b(a(c(x1))) |
→ |
a(b(c(a(x1)))) |
(3) |
1.1.1.1.1 R is empty
There are no rules in the TRS. Hence, it is terminating.