Term Rewriting System R:
[a, x, k, d]
f(a, empty) -> g(a, empty)
f(a, cons(x, k)) -> f(cons(x, a), k)
g(empty, d) -> d
g(cons(x, k), d) -> g(k, cons(x, d))
Termination of R to be shown.
R
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
f(a, empty) -> g(a, empty)
where the Polynomial interpretation:
| POL(g(x1, x2)) | = x1 + x2 |
| POL(cons(x1, x2)) | = x1 + x2 |
| POL(f(x1, x2)) | = 1 + x1 + x2 |
| POL(empty) | = 0 |
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:
g(empty, d) -> d
where the Polynomial interpretation:
| POL(g(x1, x2)) | = 1 + x1 + x2 |
| POL(cons(x1, x2)) | = x1 + x2 |
| POL(f(x1, x2)) | = x1 + x2 |
| POL(empty) | = 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:
f(a, cons(x, k)) -> f(cons(x, a), k)
where the Polynomial interpretation:
| POL(g(x1, x2)) | = x1 + x2 |
| POL(cons(x1, x2)) | = 1 + x1 + x2 |
| POL(f(x1, x2)) | = x1 + 2·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
↳RRRPolo
...
→TRS4
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
g(cons(x, k), d) -> g(k, cons(x, d))
where the Polynomial interpretation:
| POL(g(x1, x2)) | = 2·x1 + x2 |
| POL(cons(x1, x2)) | = 1 + x1 + x2 |
was used.
All Rules of R can be deleted.
R
↳RRRPolo
→TRS2
↳RRRPolo
→TRS3
↳RRRPolo
...
→TRS5
↳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
...
→TRS6
↳Dependency Pair Analysis
R contains no Dependency Pairs and therefore no SCCs.
Termination of R successfully shown.
Duration:
0:00 minutes