Term Rewriting System R:
[x, y]
f(0) -> s(0)
f(s(0)) -> s(s(0))
f(s(0)) -> *(s(s(0)), f(0))
f(+(x, s(0))) -> +(s(s(0)), f(x))
f(+(x, y)) -> *(f(x), f(y))

Termination of R to be shown.

`   R`
`     ↳Removing Redundant Rules`

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

f(0) -> s(0)
f(s(0)) -> s(s(0))

where the Polynomial interpretation:
 POL(0) =  0 POL(*(x1, x2)) =  x1 + x2 POL(s(x1)) =  x1 POL(f(x1)) =  1 + x1 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`
`         ↳Removing Redundant Rules`

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

f(+(x, y)) -> *(f(x), f(y))

where the Polynomial interpretation:
 POL(0) =  0 POL(*(x1, x2)) =  x1 + x2 POL(s(x1)) =  x1 POL(f(x1)) =  x1 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:

f(+(x, s(0))) -> +(s(s(0)), f(x))

where the Polynomial interpretation:
 POL(0) =  0 POL(*(x1, x2)) =  x1 + x2 POL(s(x1)) =  x1 POL(f(x1)) =  2·x1 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`
`                 ↳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`
`             ...`
`               →TRS5`
`                 ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

F(s(0)) -> F(0)

R contains no SCCs.

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