Certification Problem
Input (TPDB TRS_Standard/Transformed_CSR_04/Ex4_7_77_Bor03_FR)
The rewrite relation of the following TRS is considered.
| zeros |
→ |
cons(0,n__zeros) |
(1) |
|
tail(cons(X,XS)) |
→ |
activate(XS) |
(2) |
| zeros |
→ |
n__zeros |
(3) |
|
activate(n__zeros) |
→ |
zeros |
(4) |
|
activate(X) |
→ |
X |
(5) |
Property / Task
Prove or disprove termination.Answer / Result
Yes.Proof (by AProVE @ termCOMP 2023)
1 Rule Removal
Using the
Knuth Bendix order with w0 = 1 and the following precedence and weight functions
| prec(zeros) |
= |
1 |
|
weight(zeros) |
= |
3 |
|
|
|
| prec(0) |
= |
2 |
|
weight(0) |
= |
2 |
|
|
|
| prec(n__zeros) |
= |
3 |
|
weight(n__zeros) |
= |
1 |
|
|
|
| prec(tail) |
= |
5 |
|
weight(tail) |
= |
1 |
|
|
|
| prec(activate) |
= |
4 |
|
weight(activate) |
= |
2 |
|
|
|
| prec(cons) |
= |
0 |
|
weight(cons) |
= |
0 |
|
|
|
all of the following rules can be deleted.
| zeros |
→ |
cons(0,n__zeros) |
(1) |
|
tail(cons(X,XS)) |
→ |
activate(XS) |
(2) |
| zeros |
→ |
n__zeros |
(3) |
|
activate(n__zeros) |
→ |
zeros |
(4) |
|
activate(X) |
→ |
X |
(5) |
1.1 R is empty
There are no rules in the TRS. Hence, it is terminating.