Certification Problem
Input (TPDB TRS_Standard/AG01/#3.15)
The rewrite relation of the following TRS is considered.
|
average(s(x),y) |
→ |
average(x,s(y)) |
(1) |
|
average(x,s(s(s(y)))) |
→ |
s(average(s(x),y)) |
(2) |
|
average(0,0) |
→ |
0 |
(3) |
|
average(0,s(0)) |
→ |
0 |
(4) |
|
average(0,s(s(0))) |
→ |
s(0) |
(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(0) |
= |
0 |
|
weight(0) |
= |
1 |
|
|
|
| prec(s) |
= |
1 |
|
weight(s) |
= |
1 |
|
|
|
| prec(average) |
= |
2 |
|
weight(average) |
= |
0 |
|
|
|
all of the following rules can be deleted.
|
average(s(x),y) |
→ |
average(x,s(y)) |
(1) |
|
average(x,s(s(s(y)))) |
→ |
s(average(s(x),y)) |
(2) |
|
average(0,0) |
→ |
0 |
(3) |
|
average(0,s(0)) |
→ |
0 |
(4) |
|
average(0,s(s(0))) |
→ |
s(0) |
(5) |
1.1 R is empty
There are no rules in the TRS. Hence, it is terminating.