Term Rewriting System R:
[x, y]
minus(minus(x)) -> x
minux(+(x, y)) -> +(minus(y), minus(x))
+(minus(x), +(x, y)) -> y
+(+(x, y), minus(y)) -> x

Termination of R to be shown.

`   R`
`     ↳Removing Redundant Rules`

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

minus(minus(x)) -> x
+(minus(x), +(x, y)) -> y
+(+(x, y), minus(y)) -> x

where the Polynomial interpretation:
 POL(minus(x1)) =  1 + x1 POL(minux(x1)) =  2 + x1 POL(+(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:

minux(+(x, y)) -> +(minus(y), minus(x))

where the Polynomial interpretation:
 POL(minus(x1)) =  x1 POL(minux(x1)) =  1 + x1 POL(+(x1, x2)) =  x1 + x2
was used.

All Rules of R can be deleted.

`   R`
`     ↳RRRPolo`
`       →TRS2`
`         ↳RRRPolo`
`           →TRS3`
`             ↳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`
`             ↳OC`
`             ...`
`               →TRS4`
`                 ↳Dependency Pair Analysis`

R contains no Dependency Pairs and therefore no SCCs.

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