Term Rewriting System R:
[x]
fib(0) -> 0
fib(s(0)) -> s(0)
fib(s(s(x))) -> +(fib(s(x)), fib(x))
Termination of R to be shown.
R
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
fib(0) -> 0
fib(s(0)) -> s(0)
where the Polynomial interpretation:
| POL(0) | = 1 |
| POL(fib(x1)) | = 2·x1 |
| POL(s(x1)) | = 2·x1 |
| POL(+(x1, x2)) | = x1 + x2 |
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:
fib(s(s(x))) -> +(fib(s(x)), fib(x))
where the Polynomial interpretation:
| POL(fib(x1)) | = x1 |
| POL(s(x1)) | = 1 + 2·x1 |
| POL(+(x1, x2)) | = x1 + x2 |
was used.
All Rules of R can be deleted.
R
↳RRRPolo
→TRS2
↳RRRPolo
→TRS3
↳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
↳OC
...
→TRS4
↳Dependency Pair Analysis
R contains no Dependency Pairs and therefore no SCCs.
Termination of R successfully shown.
Duration:
0:00 minutes