Term Rewriting System R:
[X, XS, X1, X2]
azeros -> cons(0, zeros)
azeros -> zeros
atail(cons(X, XS)) -> mark(XS)
atail(X) -> tail(X)
mark(zeros) -> azeros
mark(tail(X)) -> atail(mark(X))
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(0) -> 0
Termination of R to be shown.
R
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
azeros -> cons(0, zeros)
azeros -> zeros
mark(0) -> 0
where the Polynomial interpretation:
| POL(0) | = 0 |
| POL(a__zeros) | = 1 |
| POL(cons(x1, x2)) | = x1 + x2 |
| POL(tail(x1)) | = 1 + x1 |
| POL(a__tail(x1)) | = 1 + x1 |
| POL(zeros) | = 0 |
| POL(mark(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:
atail(X) -> tail(X)
atail(cons(X, XS)) -> mark(XS)
where the Polynomial interpretation:
| POL(cons(x1, x2)) | = x1 + 2·x2 |
| POL(a__zeros) | = 0 |
| POL(tail(x1)) | = 1 + x1 |
| POL(a__tail(x1)) | = 2 + x1 |
| POL(mark(x1)) | = 2·x1 |
| POL(zeros) | = 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:
mark(tail(X)) -> atail(mark(X))
where the Polynomial interpretation:
| POL(a__zeros) | = 0 |
| POL(cons(x1, x2)) | = x1 + x2 |
| POL(tail(x1)) | = 1 + x1 |
| POL(a__tail(x1)) | = x1 |
| POL(mark(x1)) | = x1 |
| POL(zeros) | = 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
↳RRRPolo
...
→TRS4
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
mark(zeros) -> azeros
where the Polynomial interpretation:
| POL(cons(x1, x2)) | = x1 + x2 |
| POL(a__zeros) | = 0 |
| POL(mark(x1)) | = x1 |
| POL(zeros) | = 1 |
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
↳RRRPolo
...
→TRS5
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
mark(cons(X1, X2)) -> cons(mark(X1), X2)
where the Polynomial interpretation:
| POL(cons(x1, x2)) | = 1 + x1 + x2 |
| POL(mark(x1)) | = 2·x1 |
was used.
All Rules of R can be deleted.
R
↳RRRPolo
→TRS2
↳RRRPolo
→TRS3
↳RRRPolo
...
→TRS6
↳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
...
→TRS7
↳Dependency Pair Analysis
R contains no Dependency Pairs and therefore no SCCs.
Termination of R successfully shown.
Duration:
0:00 minutes