Certification Problem
Input (TPDB TRS_Standard/AG01/#3.7)
The rewrite relation of the following TRS is considered.
|
half(0) |
→ |
0 |
(1) |
|
half(s(s(x))) |
→ |
s(half(x)) |
(2) |
|
log(s(0)) |
→ |
0 |
(3) |
|
log(s(s(x))) |
→ |
s(log(s(half(x)))) |
(4) |
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(log) |
= |
1 |
|
weight(log) |
= |
2 |
|
|
|
| prec(s) |
= |
0 |
|
weight(s) |
= |
1 |
|
|
|
| prec(half) |
= |
3 |
|
weight(half) |
= |
0 |
|
|
|
| prec(0) |
= |
2 |
|
weight(0) |
= |
1 |
|
|
|
all of the following rules can be deleted.
|
half(0) |
→ |
0 |
(1) |
|
half(s(s(x))) |
→ |
s(half(x)) |
(2) |
|
log(s(0)) |
→ |
0 |
(3) |
|
log(s(s(x))) |
→ |
s(log(s(half(x)))) |
(4) |
1.1 R is empty
There are no rules in the TRS. Hence, it is terminating.