Term Rewriting System R:
[x, y]
merge(x, nil) -> x
merge(nil, y) -> y
merge(++(x, y), ++(u, v)) -> ++(x, merge(y, ++(u, v)))
merge(++(x, y), ++(u, v)) -> ++(u, merge(++(x, y), v))
Termination of R to be shown.
R
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
merge(x, nil) -> x
merge(nil, y) -> y
where the Polynomial interpretation:
| POL(v) | = 0 |
| POL(merge(x1, x2)) | = x1 + x2 |
| POL(++(x1, x2)) | = x1 + x2 |
| POL(nil) | = 1 |
| POL(u) | = 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:
merge(++(x, y), ++(u, v)) -> ++(x, merge(y, ++(u, v)))
where the Polynomial interpretation:
| POL(v) | = 0 |
| POL(merge(x1, x2)) | = 2·x1 + x2 |
| POL(++(x1, x2)) | = 1 + x1 + x2 |
| POL(u) | = 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:
merge(++(x, y), ++(u, v)) -> ++(u, merge(++(x, y), v))
where the Polynomial interpretation:
| POL(v) | = 1 |
| POL(merge(x1, x2)) | = x1 + 2·x2 |
| POL(++(x1, x2)) | = x1 + x2 |
| POL(u) | = 1 |
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