Term Rewriting System R:
[y, n, x]
app(nil, y) -> y
reverse(nil) -> nil
shuffle(nil) -> nil

Innermost Termination of R to be shown.

`   R`
`     ↳Removing Redundant Rules`

Removing the following rules from R which fullfill a polynomial ordering:

reverse(nil) -> nil

where the Polynomial interpretation:
 POL(reverse(x1)) =  1 + x1 POL(shuffle(x1)) =  2·x1 POL(nil) =  0 POL(app(x1, x2)) =  x1 + x2 POL(add(x1, x2)) =  2 + 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:

where the Polynomial interpretation:
 POL(reverse(x1)) =  x1 POL(shuffle(x1)) =  2·x1 POL(nil) =  0 POL(app(x1, x2)) =  x1 + x2 POL(add(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:

app(nil, y) -> y

where the Polynomial interpretation:
 POL(reverse(x1)) =  2·x1 POL(shuffle(x1)) =  x1 POL(nil) =  1 POL(app(x1, x2)) =  x1 + x2 POL(add(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:

shuffle(nil) -> nil

where the Polynomial interpretation:
 POL(reverse(x1)) =  x1 POL(shuffle(x1)) =  1 + x1 POL(nil) =  0 POL(app(x1, x2)) =  x1 + x2 POL(add(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`
`             ↳RRRPolo`
`             ...`
`               →TRS5`
`                 ↳Removing Redundant Rules`

Removing the following rules from R which fullfill a polynomial ordering:

where the Polynomial interpretation:
 POL(reverse(x1)) =  2·x1 POL(nil) =  0 POL(app(x1, x2)) =  x1 + x2 POL(add(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`
`             ...`
`               →TRS6`
`                 ↳Removing Redundant Rules`

Removing the following rules from R which fullfill a polynomial ordering:

where the Polynomial interpretation:
 POL(app(x1, x2)) =  2·x1 + x2 POL(add(x1, x2)) =  1 + x1 + x2
was used.

All Rules of R can be deleted.

`   R`
`     ↳RRRPolo`
`       →TRS2`
`         ↳RRRPolo`
`           →TRS3`
`             ↳RRRPolo`
`             ...`
`               →TRS7`
`                 ↳Dependency Pair Analysis`

R contains no Dependency Pairs and therefore no SCCs.

Innermost Termination of R successfully shown.
Duration:
0:00 minutes