Term Rewriting System R:
[x, y, z]
rev(nil) -> nil
rev(.(x, y)) -> ++(rev(y), .(x, nil))
car(.(x, y)) -> x
cdr(.(x, y)) -> y
null(nil) -> true
null(.(x, y)) -> false
++(nil, y) -> y
++(.(x, y), z) -> .(x, ++(y, z))
Termination of R to be shown.
R
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
rev(nil) -> nil
car(.(x, y)) -> x
where the Polynomial interpretation:
| POL(cdr(x1)) | = x1 |
| POL(rev(x1)) | = 1 + x1 |
| POL(null(x1)) | = x1 |
| POL(false) | = 0 |
| POL(++(x1, x2)) | = x1 + x2 |
| POL(nil) | = 0 |
| POL(car(x1)) | = 1 + x1 |
| POL(true) | = 0 |
| POL(.(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:
++(nil, y) -> y
null(nil) -> true
null(.(x, y)) -> false
cdr(.(x, y)) -> y
where the Polynomial interpretation:
| POL(cdr(x1)) | = 1 + x1 |
| POL(rev(x1)) | = 2·x1 |
| POL(null(x1)) | = x1 |
| POL(false) | = 0 |
| POL(++(x1, x2)) | = x1 + x2 |
| POL(nil) | = 1 |
| POL(true) | = 0 |
| POL(.(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:
rev(.(x, y)) -> ++(rev(y), .(x, nil))
where the Polynomial interpretation:
| POL(rev(x1)) | = 2·x1 |
| POL(++(x1, x2)) | = x1 + x2 |
| POL(nil) | = 0 |
| POL(.(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
↳RRRPolo
...
→TRS4
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
++(.(x, y), z) -> .(x, ++(y, z))
where the Polynomial interpretation:
| POL(++(x1, x2)) | = 2·x1 + x2 |
| POL(.(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