Term Rewriting System R:
[x, y, z]
norm(nil) -> 0
norm(g(x, y)) -> s(norm(x))
f(x, nil) -> g(nil, x)
f(x, g(y, z)) -> g(f(x, y), z)
rem(nil, y) -> nil
rem(g(x, y), 0) -> g(x, y)
rem(g(x, y), s(z)) -> rem(x, z)

Innermost Termination of R to be shown.

`   R`
`     ↳Removing Redundant Rules`

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

norm(nil) -> 0

where the Polynomial interpretation:
 POL(0) =  0 POL(g(x1, x2)) =  x1 + x2 POL(nil) =  0 POL(s(x1)) =  x1 POL(rem(x1, x2)) =  x1 + x2 POL(f(x1, x2)) =  x1 + x2 POL(norm(x1)) =  1 + x1
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:

norm(g(x, y)) -> s(norm(x))
rem(g(x, y), s(z)) -> rem(x, z)

where the Polynomial interpretation:
 POL(0) =  0 POL(g(x1, x2)) =  1 + x1 + x2 POL(nil) =  0 POL(rem(x1, x2)) =  x1 + x2 POL(s(x1)) =  x1 POL(f(x1, x2)) =  1 + x1 + x2 POL(norm(x1)) =  x1
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:

rem(nil, y) -> nil
rem(g(x, y), 0) -> g(x, y)

where the Polynomial interpretation:
 POL(0) =  0 POL(g(x1, x2)) =  x1 + x2 POL(nil) =  0 POL(rem(x1, x2)) =  1 + x1 + x2 POL(f(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`
`         ↳RRRPolo`
`           →TRS3`
`             ↳RRRPolo`
`             ...`
`               →TRS4`
`                 ↳Removing Redundant Rules`

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

f(x, nil) -> g(nil, x)

where the Polynomial interpretation:
 POL(g(x1, x2)) =  x1 + x2 POL(nil) =  1 POL(f(x1, x2)) =  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`
`             ↳RRRPolo`
`             ...`
`               →TRS5`
`                 ↳Removing Redundant Rules`

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

f(x, g(y, z)) -> g(f(x, y), z)

where the Polynomial interpretation:
 POL(g(x1, x2)) =  1 + x1 + x2 POL(f(x1, x2)) =  x1 + 2·x2
was used.

All Rules of R can be deleted.

`   R`
`     ↳RRRPolo`
`       →TRS2`
`         ↳RRRPolo`
`           →TRS3`
`             ↳RRRPolo`
`             ...`
`               →TRS6`
`                 ↳Dependency Pair Analysis`

R contains no Dependency Pairs and therefore no SCCs.

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