Term Rewriting System R:
[a, x, k, d]
f(a, empty) -> g(a, empty)
f(a, cons(x, k)) -> f(cons(x, a), k)
g(empty, d) -> d
g(cons(x, k), d) -> g(k, cons(x, d))

Innermost Termination of R to be shown.

`   R`
`     ↳Removing Redundant Rules`

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

f(a, empty) -> g(a, empty)

where the Polynomial interpretation:
 POL(g(x1, x2)) =  x1 + x2 POL(cons(x1, x2)) =  x1 + x2 POL(f(x1, x2)) =  1 + x1 + x2 POL(empty) =  0
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:

g(empty, d) -> d

where the Polynomial interpretation:
 POL(g(x1, x2)) =  1 + x1 + x2 POL(cons(x1, x2)) =  x1 + x2 POL(f(x1, x2)) =  x1 + x2 POL(empty) =  0
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(a, cons(x, k)) -> f(cons(x, a), k)

where the Polynomial interpretation:
 POL(g(x1, x2)) =  x1 + x2 POL(cons(x1, x2)) =  1 + x1 + x2 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`
`             ...`
`               →TRS4`
`                 ↳Removing Redundant Rules`

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

g(cons(x, k), d) -> g(k, cons(x, d))

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

All Rules of R can be deleted.

`   R`
`     ↳RRRPolo`
`       →TRS2`
`         ↳RRRPolo`
`           →TRS3`
`             ↳RRRPolo`
`             ...`
`               →TRS5`
`                 ↳Dependency Pair Analysis`

R contains no Dependency Pairs and therefore no SCCs.

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