Certification Problem
Input (TPDB TRS_Standard/Transformed_CSR_04/Ex1_Zan97_GM)
The rewrite relation of the following TRS is considered.
|
a__g(X) |
→ |
a__h(X) |
(1) |
| a__c |
→ |
d |
(2) |
|
a__h(d) |
→ |
a__g(c) |
(3) |
|
mark(g(X)) |
→ |
a__g(X) |
(4) |
|
mark(h(X)) |
→ |
a__h(X) |
(5) |
|
mark(c) |
→ |
a__c |
(6) |
|
mark(d) |
→ |
d |
(7) |
|
a__g(X) |
→ |
g(X) |
(8) |
|
a__h(X) |
→ |
h(X) |
(9) |
| a__c |
→ |
c |
(10) |
Property / Task
Prove or disprove termination.Answer / Result
Yes.Proof (by ttt2 @ termCOMP 2023)
1 Rule Removal
Using the
Knuth Bendix order with w0 = 1 and the following precedence and weight functions
| prec(h) |
= |
5 |
|
weight(h) |
= |
3 |
|
|
|
| prec(mark) |
= |
6 |
|
weight(mark) |
= |
1 |
|
|
|
| prec(g) |
= |
7 |
|
weight(g) |
= |
4 |
|
|
|
| prec(c) |
= |
3 |
|
weight(c) |
= |
4 |
|
|
|
| prec(d) |
= |
2 |
|
weight(d) |
= |
5 |
|
|
|
| prec(a__c) |
= |
4 |
|
weight(a__c) |
= |
5 |
|
|
|
| prec(a__h) |
= |
1 |
|
weight(a__h) |
= |
4 |
|
|
|
| prec(a__g) |
= |
0 |
|
weight(a__g) |
= |
5 |
|
|
|
all of the following rules can be deleted.
|
a__g(X) |
→ |
a__h(X) |
(1) |
| a__c |
→ |
d |
(2) |
|
a__h(d) |
→ |
a__g(c) |
(3) |
|
mark(g(X)) |
→ |
a__g(X) |
(4) |
|
mark(h(X)) |
→ |
a__h(X) |
(5) |
|
mark(c) |
→ |
a__c |
(6) |
|
mark(d) |
→ |
d |
(7) |
|
a__g(X) |
→ |
g(X) |
(8) |
|
a__h(X) |
→ |
h(X) |
(9) |
| a__c |
→ |
c |
(10) |
1.1 R is empty
There are no rules in the TRS. Hence, it is terminating.