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

Innermost Termination of R to be shown.

`   R`
`     ↳Removing Redundant Rules`

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

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

where the Polynomial interpretation:
 POL(0) =  1 POL(s(x1)) =  x1 POL(-(x1, x2)) =  x1 + x2 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:

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

where the Polynomial interpretation:
 POL(0) =  0 POL(s(x1)) =  1 + x1 POL(-(x1, x2)) =  x1 + x2 POL(+(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`
`         ↳RRRPolo`
`           →TRS3`
`             ↳Removing Redundant Rules`

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

-(0, y) -> 0

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

All Rules of R can be deleted.

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

R contains no Dependency Pairs and therefore no SCCs.

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