Termination Proof Script
Consider the TRS R consisting of the rewrite rules
|
| 1: |
|
sum(0) |
→ 0 |
| 2: |
|
sum(s(x)) |
→ sum(x) + s(x) |
| 3: |
|
sum1(0) |
→ 0 |
| 4: |
|
sum1(s(x)) |
→ s(sum1(x) + (x + x)) |
|
There are 2 dependency pairs:
|
| 5: |
|
SUM(s(x)) |
→ SUM(x) |
| 6: |
|
SUM1(s(x)) |
→ SUM1(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(SUM) = 1 together with
the lexicographic path order with
empty precedence,
rule 5
is strictly decreasing.
-
Consider the SCC {6}.
There are no usable rules.
By taking the AF π with
π(SUM1) = 1 together with
the lexicographic path order with
empty precedence,
rule 6
is strictly decreasing.
Hence the TRS is terminating.
Tyrolean Termination Tool (0.01 seconds)
--- May 4, 2006