Termination Proof Script
Consider the TRS R consisting of the rewrite rules
|
| 1: |
|
fib(0) |
→ 0 |
| 2: |
|
fib(s(0)) |
→ s(0) |
| 3: |
|
fib(s(s(x))) |
→ fib(s(x)) + fib(x) |
| 4: |
|
x + 0 |
→ x |
| 5: |
|
x + s(y) |
→ s(x + y) |
|
There are 4 dependency pairs:
|
| 6: |
|
FIB(s(s(x))) |
→ fib(s(x)) +# fib(x) |
| 7: |
|
FIB(s(s(x))) |
→ FIB(s(x)) |
| 8: |
|
FIB(s(s(x))) |
→ FIB(x) |
| 9: |
|
x +# s(y) |
→ x +# y |
|
The approximated dependency graph contains 2 SCCs:
{9}
and {7,8}.
-
Consider the SCC {9}.
There are no usable rules.
By taking the AF π with
π(+#) = 2 together with
the lexicographic path order with
empty precedence,
rule 9
is strictly decreasing.
-
Consider the SCC {7,8}.
There are no usable rules.
By taking the AF π with
π(FIB) = 1 together with
the lexicographic path order with
empty precedence,
the rules in {7,8}
are strictly decreasing.
Hence the TRS is terminating.
Tyrolean Termination Tool (0.01 seconds)
--- May 3, 2006