Term Rewriting System R:
[x]
rec(rec(x)) -> sent(rec(x))
rec(sent(x)) -> sent(rec(x))
rec(no(x)) -> sent(rec(x))
rec(bot) -> up(sent(bot))
rec(up(x)) -> up(rec(x))
sent(up(x)) -> up(sent(x))
no(up(x)) -> up(no(x))
top(rec(up(x))) -> top(check(rec(x)))
top(sent(up(x))) -> top(check(rec(x)))
top(no(up(x))) -> top(check(rec(x)))
check(up(x)) -> up(check(x))
check(sent(x)) -> sent(check(x))
check(rec(x)) -> rec(check(x))
check(no(x)) -> no(check(x))
check(no(x)) -> no(x)
Innermost Termination of R to be shown.
R
↳Removing Redundant Rules for Innermost Termination
Removing the following rules from R which left hand sides contain non normal subterms
top(rec(up(x))) -> top(check(rec(x)))
top(sent(up(x))) -> top(check(rec(x)))
top(no(up(x))) -> top(check(rec(x)))
R
↳RRRI
→TRS2
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
rec(bot) -> up(sent(bot))
where the Polynomial interpretation:
| POL(rec(x1)) | = 2·x1 |
| POL(no(x1)) | = x1 |
| POL(bot) | = 1 |
| POL(up(x1)) | = x1 |
| POL(check(x1)) | = x1 |
| POL(sent(x1)) | = x1 |
was used.
Not all Rules of R can be deleted, so we still have to regard a part of R.
R
↳RRRI
→TRS2
↳RRRPolo
→TRS3
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
rec(no(x)) -> sent(rec(x))
where the Polynomial interpretation:
| POL(rec(x1)) | = x1 |
| POL(no(x1)) | = 1 + x1 |
| POL(up(x1)) | = x1 |
| POL(check(x1)) | = x1 |
| POL(sent(x1)) | = x1 |
was used.
Not all Rules of R can be deleted, so we still have to regard a part of R.
R
↳RRRI
→TRS2
↳RRRPolo
→TRS3
↳RRRPolo
...
→TRS4
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
check(no(x)) -> no(check(x))
check(no(x)) -> no(x)
where the Polynomial interpretation:
| POL(rec(x1)) | = x1 |
| POL(no(x1)) | = 1 + x1 |
| POL(up(x1)) | = x1 |
| POL(check(x1)) | = 2·x1 |
| POL(sent(x1)) | = x1 |
was used.
Not all Rules of R can be deleted, so we still have to regard a part of R.
R
↳RRRI
→TRS2
↳RRRPolo
→TRS3
↳RRRPolo
...
→TRS5
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
no(up(x)) -> up(no(x))
where the Polynomial interpretation:
| POL(rec(x1)) | = x1 |
| POL(no(x1)) | = 2·x1 |
| POL(up(x1)) | = 1 + x1 |
| POL(check(x1)) | = x1 |
| POL(sent(x1)) | = x1 |
was used.
Not all Rules of R can be deleted, so we still have to regard a part of R.
R
↳RRRI
→TRS2
↳RRRPolo
→TRS3
↳RRRPolo
...
→TRS6
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
check(up(x)) -> up(check(x))
where the Polynomial interpretation:
| POL(rec(x1)) | = x1 |
| POL(up(x1)) | = 1 + x1 |
| POL(check(x1)) | = 2·x1 |
| POL(sent(x1)) | = x1 |
was used.
Not all Rules of R can be deleted, so we still have to regard a part of R.
R
↳RRRI
→TRS2
↳RRRPolo
→TRS3
↳RRRPolo
...
→TRS7
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
rec(up(x)) -> up(rec(x))
where the Polynomial interpretation:
| POL(rec(x1)) | = 2·x1 |
| POL(up(x1)) | = 1 + x1 |
| POL(check(x1)) | = x1 |
| POL(sent(x1)) | = x1 |
was used.
Not all Rules of R can be deleted, so we still have to regard a part of R.
R
↳RRRI
→TRS2
↳RRRPolo
→TRS3
↳RRRPolo
...
→TRS8
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
sent(up(x)) -> up(sent(x))
where the Polynomial interpretation:
| POL(rec(x1)) | = 2·x1 |
| POL(up(x1)) | = 1 + x1 |
| POL(check(x1)) | = x1 |
| POL(sent(x1)) | = 2·x1 |
was used.
Not all Rules of R can be deleted, so we still have to regard a part of R.
R
↳RRRI
→TRS2
↳RRRPolo
→TRS3
↳RRRPolo
...
→TRS9
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
rec(rec(x)) -> sent(rec(x))
where the Polynomial interpretation:
| POL(rec(x1)) | = 1 + x1 |
| POL(check(x1)) | = x1 |
| POL(sent(x1)) | = x1 |
was used.
Not all Rules of R can be deleted, so we still have to regard a part of R.
R
↳RRRI
→TRS2
↳RRRPolo
→TRS3
↳RRRPolo
...
→TRS10
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
check(rec(x)) -> rec(check(x))
where the Polynomial interpretation:
| POL(rec(x1)) | = 1 + x1 |
| POL(check(x1)) | = 2·x1 |
| POL(sent(x1)) | = x1 |
was used.
Not all Rules of R can be deleted, so we still have to regard a part of R.
R
↳RRRI
→TRS2
↳RRRPolo
→TRS3
↳RRRPolo
...
→TRS11
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
check(sent(x)) -> sent(check(x))
where the Polynomial interpretation:
| POL(rec(x1)) | = x1 |
| POL(check(x1)) | = 2·x1 |
| POL(sent(x1)) | = 1 + x1 |
was used.
Not all Rules of R can be deleted, so we still have to regard a part of R.
R
↳RRRI
→TRS2
↳RRRPolo
→TRS3
↳RRRPolo
...
→TRS12
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
rec(sent(x)) -> sent(rec(x))
where the Polynomial interpretation:
| POL(rec(x1)) | = 2·x1 |
| POL(sent(x1)) | = 1 + x1 |
was used.
All Rules of R can be deleted.
R
↳RRRI
→TRS2
↳RRRPolo
→TRS3
↳RRRPolo
...
→TRS13
↳Dependency Pair Analysis
R contains no Dependency Pairs and therefore no SCCs.
Innermost Termination of R successfully shown.
Duration:
0:00 minutes