Term Rewriting System R:
[Y, U, V, X, W, Z]
concat(leaf, Y) -> Y
concat(cons(U, V), Y) -> cons(U, concat(V, Y))
lessleaves(X, leaf) -> false
lessleaves(leaf, cons(W, Z)) -> true
lessleaves(cons(U, V), cons(W, Z)) -> lessleaves(concat(U, V), concat(W, Z))
Termination of R to be shown.
R
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
concat(leaf, Y) -> Y
lessleaves(X, leaf) -> false
lessleaves(leaf, cons(W, Z)) -> true
where the Polynomial interpretation:
| POL(cons(x1, x2)) | = x1 + x2 |
| POL(false) | = 0 |
| POL(lessleaves(x1, x2)) | = x1 + x2 |
| POL(true) | = 0 |
| POL(leaf) | = 1 |
| POL(concat(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:
lessleaves(cons(U, V), cons(W, Z)) -> lessleaves(concat(U, V), concat(W, Z))
where the Polynomial interpretation:
| POL(cons(x1, x2)) | = 1 + x1 + x2 |
| POL(lessleaves(x1, x2)) | = x1 + x2 |
| POL(concat(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
↳RRRPolo
→TRS3
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
concat(cons(U, V), Y) -> cons(U, concat(V, Y))
where the Polynomial interpretation:
| POL(cons(x1, x2)) | = 1 + x1 + x2 |
| POL(concat(x1, x2)) | = 2·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