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