Termination Proof Script
Consider the TRS R consisting of the rewrite rules
|
| 1: |
|
from(X) |
→ cons(X,from(s(X))) |
| 2: |
|
head(cons(X,XS)) |
→ X |
| 3: |
|
2nd(cons(X,XS)) |
→ head(XS) |
| 4: |
|
take(0,XS) |
→ nil |
| 5: |
|
take(s(N),cons(X,XS)) |
→ cons(X,take(N,XS)) |
| 6: |
|
sel(0,cons(X,XS)) |
→ X |
| 7: |
|
sel(s(N),cons(X,XS)) |
→ sel(N,XS) |
|
There are 4 dependency pairs:
|
| 8: |
|
FROM(X) |
→ FROM(s(X)) |
| 9: |
|
2nd#(cons(X,XS)) |
→ HEAD(XS) |
| 10: |
|
TAKE(s(N),cons(X,XS)) |
→ TAKE(N,XS) |
| 11: |
|
SEL(s(N),cons(X,XS)) |
→ SEL(N,XS) |
|
The approximated dependency graph contains 3 SCCs:
{8},
{11}
and {10}.
-
Consider the SCC {8}.
There are no usable rules.
The constraints could not be solved.
-
Consider the SCC {11}.
There are no usable rules.
By taking the AF π with
π(SEL) = 1 together with
the lexicographic path order with
empty precedence,
rule 11
is strictly decreasing.
-
Consider the SCC {10}.
There are no usable rules.
By taking the AF π with
π(TAKE) = 1 together with
the lexicographic path order with
empty precedence,
rule 10
is strictly decreasing.
Tyrolean Termination Tool (0.01 seconds)
--- May 4, 2006