Term Rewriting System R:
[N, X, Y]
terms(N) -> cons(recip(sqr(N)))
sqr(0) -> 0
sqr(s) -> s
dbl(0) -> 0
dbl(s) -> s
first(0, X) -> nil
first(s, cons(Y)) -> cons(Y)

Termination of R to be shown.

`   R`
`     ↳Removing Redundant Rules`

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

terms(N) -> cons(recip(sqr(N)))

where the Polynomial interpretation:
 POL(0) =  0 POL(first(x1, x2)) =  x1 + x2 POL(cons(x1)) =  x1 POL(sqr(x1)) =  x1 POL(dbl(x1)) =  x1 POL(nil) =  0 POL(s) =  0 POL(terms(x1)) =  1 + x1 POL(recip(x1)) =  x1 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`
`         ↳Removing Redundant Rules`

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

where the Polynomial interpretation:
 POL(0) =  0 POL(first(x1, x2)) =  x1 + x2 POL(cons(x1)) =  x1 POL(sqr(x1)) =  x1 POL(dbl(x1)) =  x1 POL(nil) =  0 POL(s) =  0 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:

sqr(s) -> s
sqr(0) -> 0

where the Polynomial interpretation:
 POL(0) =  0 POL(first(x1, x2)) =  x1 + x2 POL(cons(x1)) =  x1 POL(sqr(x1)) =  1 + x1 POL(dbl(x1)) =  x1 POL(nil) =  0 POL(s) =  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`
`             ↳RRRPolo`
`             ...`
`               →TRS4`
`                 ↳Removing Redundant Rules`

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

dbl(s) -> s
dbl(0) -> 0

where the Polynomial interpretation:
 POL(0) =  0 POL(first(x1, x2)) =  x1 + x2 POL(cons(x1)) =  x1 POL(dbl(x1)) =  1 + x1 POL(nil) =  0 POL(s) =  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`
`             ↳RRRPolo`
`             ...`
`               →TRS5`
`                 ↳Removing Redundant Rules`

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

first(0, X) -> nil

where the Polynomial interpretation:
 POL(first(x1, x2)) =  x1 + x2 POL(0) =  1 POL(cons(x1)) =  x1 POL(nil) =  0 POL(s) =  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`
`             ↳RRRPolo`
`             ...`
`               →TRS6`
`                 ↳Removing Redundant Rules`

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

first(s, cons(Y)) -> cons(Y)

where the Polynomial interpretation:
 POL(first(x1, x2)) =  1 + x1 + x2 POL(cons(x1)) =  x1 POL(s) =  0
was used.

All Rules of R can be deleted.

`   R`
`     ↳RRRPolo`
`       →TRS2`
`         ↳RRRPolo`
`           →TRS3`
`             ↳RRRPolo`
`             ...`
`               →TRS7`
`                 ↳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`
`             ...`
`               →TRS8`
`                 ↳Dependency Pair Analysis`

R contains no Dependency Pairs and therefore no SCCs.

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