Term Rewriting System R:
[x, y, z]
i(0) -> 0
i(i(x)) -> x
i(+(x, y)) -> +(i(x), i(y))
+(0, y) -> y
+(x, 0) -> x
+(i(x), x) -> 0
+(x, i(x)) -> 0
+(x, +(y, z)) -> +(+(x, y), z)
+(+(x, i(y)), y) -> x
+(+(x, y), i(y)) -> x

Termination of R to be shown.

`   R`
`     ↳Removing Redundant Rules`

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

i(0) -> 0
i(+(x, y)) -> +(i(x), i(y))
+(0, y) -> y
+(x, 0) -> x
+(+(x, i(y)), y) -> x
+(+(x, y), i(y)) -> x

where the Polynomial interpretation:
 POL(i(x1)) =  2·x1 POL(0) =  1 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:

+(x, +(y, z)) -> +(+(x, y), z)
+(x, i(x)) -> 0
+(i(x), x) -> 0

where the Polynomial interpretation:
 POL(i(x1)) =  x1 POL(0) =  0 POL(+(x1, x2)) =  1 + x1 + 2·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:

i(i(x)) -> x

where the Polynomial interpretation:
 POL(i(x1)) =  1 + x1
was used.

All Rules of R can be deleted.

`   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 no Dependency Pairs and therefore no SCCs.

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