Term Rewriting System R:
[ls, a, x, k]
rev(ls) -> r1(ls, empty)
r1(empty, a) -> a
r1(cons(x, k), a) -> r1(k, cons(x, a))
Termination of R to be shown.
R
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
rev(ls) -> r1(ls, empty)
where the Polynomial interpretation:
| POL(rev(x1)) | = 1 + x1 |
| POL(cons(x1, x2)) | = x1 + x2 |
| POL(r1(x1, x2)) | = x1 + x2 |
| POL(empty) | = 0 |
was used.
Not all Rules of R can be deleted, so we still have to regard a part of R.
R
↳RRRPolo
→TRS2
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
r1(empty, a) -> a
where the Polynomial interpretation:
| POL(cons(x1, x2)) | = x1 + x2 |
| POL(r1(x1, x2)) | = 1 + x1 + x2 |
| POL(empty) | = 0 |
was used.
Not all Rules of R can be deleted, so we still have to regard a part of R.
R
↳RRRPolo
→TRS2
↳RRRPolo
→TRS3
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
r1(cons(x, k), a) -> r1(k, cons(x, a))
where the Polynomial interpretation:
| POL(cons(x1, x2)) | = 1 + x1 + x2 |
| POL(r1(x1, x2)) | = 2·x1 + x2 |
was used.
All Rules of R can be deleted.
R
↳RRRPolo
→TRS2
↳RRRPolo
→TRS3
↳RRRPolo
...
→TRS4
↳Overlay and local confluence Check
The TRS is overlay and locally confluent (all critical pairs are trivially joinable).Hence, we can switch to innermost.
R
↳RRRPolo
→TRS2
↳RRRPolo
→TRS3
↳RRRPolo
...
→TRS5
↳Dependency Pair Analysis
R contains no Dependency Pairs and therefore no SCCs.
Termination of R successfully shown.
Duration:
0:00 minutes