Certification Problem
Input (TPDB SRS_Standard/Gebhardt_06/16)
The rewrite relation of the following TRS is considered.
|
0(0(0(0(x1)))) |
→ |
1(0(0(1(x1)))) |
(1) |
|
0(1(0(1(x1)))) |
→ |
0(0(1(0(x1)))) |
(2) |
Property / Task
Prove or disprove termination.Answer / Result
Yes.Proof (by AProVE @ termCOMP 2023)
1 Dependency Pair Transformation
The following set of initial dependency pairs has been identified.
|
0#(0(0(0(x1)))) |
→ |
0#(0(1(x1))) |
(3) |
|
0#(0(0(0(x1)))) |
→ |
0#(1(x1)) |
(4) |
|
0#(1(0(1(x1)))) |
→ |
0#(0(1(0(x1)))) |
(5) |
|
0#(1(0(1(x1)))) |
→ |
0#(1(0(x1))) |
(6) |
|
0#(1(0(1(x1)))) |
→ |
0#(x1) |
(7) |
1.1 Reduction Pair Processor
Using the linear polynomial interpretation over the naturals
| [0#(x1)] |
= |
1 · x1
|
| [0(x1)] |
= |
1 + 1 · x1
|
| [1(x1)] |
= |
1 + 1 · x1
|
the
pairs
|
0#(0(0(0(x1)))) |
→ |
0#(0(1(x1))) |
(3) |
|
0#(0(0(0(x1)))) |
→ |
0#(1(x1)) |
(4) |
|
0#(1(0(1(x1)))) |
→ |
0#(1(0(x1))) |
(6) |
|
0#(1(0(1(x1)))) |
→ |
0#(x1) |
(7) |
could be deleted.
1.1.1 Reduction Pair Processor
Using the matrix interpretations of dimension 3 with strict dimension 1 over the arctic semiring over the naturals
| [0#(x1)] |
= |
+
|
| 0 |
-∞ |
-∞ |
|
-∞ |
-∞ |
-∞ |
|
-∞ |
-∞ |
-∞ |
|
|
· x1
|
| [1(x1)] |
= |
+ · x1
|
| [0(x1)] |
= |
+ · x1
|
the
pair
|
0#(1(0(1(x1)))) |
→ |
0#(0(1(0(x1)))) |
(5) |
could be deleted.
1.1.1.1 P is empty
There are no pairs anymore.