Termination Proof Script
Consider the TRS R consisting of the rewrite rules
|
| 1: |
|
half(0) |
→ 0 |
| 2: |
|
half(s(s(x))) |
→ s(half(x)) |
| 3: |
|
log(s(0)) |
→ 0 |
| 4: |
|
log(s(s(x))) |
→ s(log(s(half(x)))) |
|
There are 3 dependency pairs:
|
| 5: |
|
HALF(s(s(x))) |
→ HALF(x) |
| 6: |
|
LOG(s(s(x))) |
→ LOG(s(half(x))) |
| 7: |
|
LOG(s(s(x))) |
→ HALF(x) |
|
The approximated dependency graph contains 2 SCCs:
{5}
and {6}.
-
Consider the SCC {5}.
There are no usable rules.
By taking the AF π with
π(HALF) = 1 together with
the lexicographic path order with
empty precedence,
rule 5
is strictly decreasing.
-
Consider the SCC {6}.
The usable rules are {1,2}.
By taking the AF π with
π(half) = π(LOG) = 1 together with
the lexicographic path order with
empty precedence,
rule 1
is weakly decreasing and
the rules in {2,6}
are strictly decreasing.
Hence the TRS is terminating.
Tyrolean Termination Tool (0.01 seconds)
--- May 3, 2006