Term Rewriting System R:
[X]
f(s(X), X) -> f(X, a(X))
f(X, c(X)) -> f(s(X), X)
f(X, X) -> c(X)
Termination of R to be shown.
R
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
f(s(X), X) -> f(X, a(X))
where the Polynomial interpretation:
| POL(c(x1)) | = 1 + x1 |
| POL(s(x1)) | = 1 + x1 |
| POL(a(x1)) | = x1 |
| POL(f(x1, x2)) | = 1 + 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:
f(X, c(X)) -> f(s(X), X)
where the Polynomial interpretation:
| POL(c(x1)) | = 1 + x1 |
| POL(s(x1)) | = x1 |
| POL(f(x1, x2)) | = 1 + 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
↳RRRPolo
→TRS3
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
f(X, X) -> c(X)
where the Polynomial interpretation:
| POL(c(x1)) | = x1 |
| POL(f(x1, x2)) | = 1 + 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