Term Rewriting System R:
[y, u, v, 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(x) | = 0 |
| POL(false) | = 0 |
| POL(true) | = 0 |
| POL(less_leaves(x1, x2)) | = x1 + x2 |
| 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(less_leaves(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