Certification Problem
Input (TPDB SRS_Relative/Waldmann_19/random-105)
The relative rewrite relation R/S is considered where R is the following TRS
|
c(c(c(x1))) |
→ |
b(b(b(x1))) |
(1) |
and S is the following TRS.
|
b(c(a(x1))) |
→ |
a(c(a(x1))) |
(2) |
|
b(b(b(x1))) |
→ |
a(a(b(x1))) |
(3) |
|
b(a(a(x1))) |
→ |
a(b(c(x1))) |
(4) |
|
a(c(b(x1))) |
→ |
c(a(a(x1))) |
(5) |
Property / Task
Prove or disprove termination.Answer / Result
Yes.Proof (by AProVE @ termCOMP 2023)
1 Rule Removal
Using the
matrix interpretations of dimension 6 with strict dimension 1 over the integers
| [c(x1)] |
= |
+
|
| 1 |
0 |
0 |
0 |
0 |
0 |
| 0 |
0 |
1 |
0 |
0 |
0 |
| 0 |
1 |
0 |
0 |
0 |
0 |
| 0 |
1 |
0 |
0 |
1 |
1 |
| 0 |
0 |
0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
0 |
0 |
|
|
· x1
|
| [b(x1)] |
= |
+
|
| 1 |
0 |
0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
0 |
0 |
| 0 |
1 |
0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
1 |
0 |
0 |
| 0 |
0 |
0 |
1 |
0 |
0 |
|
|
· x1
|
| [a(x1)] |
= |
+
|
| 1 |
0 |
0 |
1 |
0 |
0 |
| 0 |
0 |
0 |
0 |
0 |
0 |
| 0 |
1 |
0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
1 |
0 |
0 |
| 0 |
0 |
0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
0 |
0 |
|
|
· x1
|
all of the following rules can be deleted.
|
a(c(b(x1))) |
→ |
c(a(a(x1))) |
(5) |
1.1 Rule Removal
Using the
Knuth Bendix order with w0 = 1 and the following precedence and weight functions
| prec(c) |
= |
2 |
|
weight(c) |
= |
1 |
|
|
|
| prec(b) |
= |
1 |
|
weight(b) |
= |
1 |
|
|
|
| prec(a) |
= |
0 |
|
weight(a) |
= |
1 |
|
|
|
all of the following rules can be deleted.
|
c(c(c(x1))) |
→ |
b(b(b(x1))) |
(1) |
|
b(c(a(x1))) |
→ |
a(c(a(x1))) |
(2) |
|
b(b(b(x1))) |
→ |
a(a(b(x1))) |
(3) |
|
b(a(a(x1))) |
→ |
a(b(c(x1))) |
(4) |
1.1.1 R is empty
There are no rules in the TRS. Hence, it is terminating.