Certification Problem
Input (TPDB SRS_Relative/Waldmann_19/random-202)
The relative rewrite relation R/S is considered where R is the following TRS
|
a(a(a(x1))) |
→ |
b(b(c(x1))) |
(1) |
|
b(b(b(x1))) |
→ |
c(c(b(x1))) |
(2) |
and S is the following TRS.
|
b(c(b(x1))) |
→ |
a(c(c(x1))) |
(3) |
|
a(c(a(x1))) |
→ |
b(c(b(x1))) |
(4) |
|
c(c(a(x1))) |
→ |
b(b(a(x1))) |
(5) |
|
c(a(a(x1))) |
→ |
b(b(c(x1))) |
(6) |
Property / Task
Prove or disprove termination.Answer / Result
Yes.Proof (by AProVE @ termCOMP 2023)
1 String Reversal
Since only unary symbols occur, one can reverse all terms and obtains the TRS
|
a(a(a(x1))) |
→ |
c(b(b(x1))) |
(7) |
|
b(b(b(x1))) |
→ |
b(c(c(x1))) |
(8) |
|
b(c(b(x1))) |
→ |
c(c(a(x1))) |
(9) |
|
a(c(a(x1))) |
→ |
b(c(b(x1))) |
(4) |
|
a(c(c(x1))) |
→ |
a(b(b(x1))) |
(10) |
|
a(a(c(x1))) |
→ |
c(b(b(x1))) |
(11) |
1.1 Rule Removal
Using the
matrix interpretations of dimension 6 with strict dimension 1 over the integers
| [a(x1)] |
= |
+
|
| 1 |
1 |
0 |
0 |
0 |
1 |
| 0 |
1 |
1 |
0 |
0 |
0 |
| 0 |
1 |
0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
0 |
0 |
|
|
· x1
|
| [c(x1)] |
= |
+
|
| 1 |
0 |
0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
0 |
1 |
| 0 |
0 |
0 |
0 |
0 |
1 |
| 0 |
0 |
0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
1 |
0 |
0 |
|
|
· x1
|
| [b(x1)] |
= |
+
|
| 1 |
0 |
0 |
0 |
0 |
1 |
| 0 |
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 |
0 |
0 |
0 |
|
|
· x1
|
all of the following rules can be deleted.
|
a(a(a(x1))) |
→ |
c(b(b(x1))) |
(7) |
1.1.1 Rule Removal
Using the
matrix interpretations of dimension 6 with strict dimension 1 over the integers
| [b(x1)] |
= |
+
|
| 1 |
0 |
0 |
1 |
0 |
0 |
| 0 |
0 |
0 |
0 |
1 |
0 |
| 0 |
0 |
0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
1 |
0 |
|
|
· x1
|
| [c(x1)] |
= |
+
|
| 1 |
0 |
0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
0 |
0 |
| 0 |
1 |
0 |
0 |
0 |
1 |
| 0 |
0 |
1 |
0 |
0 |
0 |
| 0 |
0 |
0 |
1 |
0 |
0 |
|
|
· x1
|
| [a(x1)] |
= |
+
|
| 1 |
0 |
0 |
1 |
1 |
0 |
| 0 |
0 |
0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
1 |
0 |
| 0 |
0 |
0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
0 |
1 |
| 0 |
0 |
0 |
0 |
0 |
0 |
|
|
· x1
|
all of the following rules can be deleted.
|
a(c(a(x1))) |
→ |
b(c(b(x1))) |
(4) |
1.1.1.1 Rule Removal
Using the
matrix interpretations of dimension 6 with strict dimension 1 over the integers
| [b(x1)] |
= |
+
|
| 1 |
0 |
1 |
0 |
0 |
0 |
| 0 |
0 |
1 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
0 |
0 |
| 0 |
1 |
0 |
0 |
1 |
0 |
| 0 |
0 |
0 |
0 |
0 |
0 |
|
|
· x1
|
| [c(x1)] |
= |
+
|
| 1 |
0 |
0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
1 |
0 |
0 |
| 0 |
0 |
0 |
0 |
1 |
0 |
| 0 |
0 |
1 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
0 |
0 |
| 0 |
1 |
0 |
0 |
1 |
0 |
|
|
· x1
|
| [a(x1)] |
= |
+
|
| 1 |
1 |
0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
1 |
0 |
1 |
| 0 |
1 |
0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
0 |
0 |
| 0 |
1 |
0 |
0 |
0 |
0 |
|
|
· x1
|
all of the following rules can be deleted.
|
a(a(c(x1))) |
→ |
c(b(b(x1))) |
(11) |
1.1.1.1.1 Rule Removal
Using the
linear polynomial interpretation over the naturals
| [b(x1)] |
= |
1 + 1 · x1
|
| [c(x1)] |
= |
1 + 1 · x1
|
| [a(x1)] |
= |
1 · x1
|
all of the following rules can be deleted.
|
b(c(b(x1))) |
→ |
c(c(a(x1))) |
(9) |
1.1.1.1.1.1 Rule Removal
Using the
matrix interpretations of dimension 2 with strict dimension 1 over the integers
| [b(x1)] |
= |
+ · x1
|
| [c(x1)] |
= |
+ · x1
|
| [a(x1)] |
= |
+ · x1
|
all of the following rules can be deleted.
|
b(b(b(x1))) |
→ |
b(c(c(x1))) |
(8) |
|
a(c(c(x1))) |
→ |
a(b(b(x1))) |
(10) |
1.1.1.1.1.1.1 R is empty
There are no rules in the TRS. Hence, it is terminating.