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