Term Rewriting System R:
[f, g, x, l, xs]
app(app(app(compose, f), g), x) -> app(g, app(f, x))
app(reverse, l) -> app(app(reverse2, l), nil)
app(app(reverse2, nil), l) -> l
app(app(reverse2, app(app(cons, x), xs)), l) -> app(app(reverse2, xs), app(app(cons, x), l))
app(hd, app(app(cons, x), xs)) -> x
app(tl, app(app(cons, x), xs)) -> xs
last -> app(app(compose, hd), reverse)
init -> app(app(compose, reverse), app(app(compose, tl), reverse))

Termination of R to be shown.

`   R`
`     ↳Removing Redundant Rules`

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

app(app(app(compose, f), g), x) -> app(g, app(f, x))

where the Polynomial interpretation:
 POL(last) =  1 POL(reverse) =  0 POL(cons) =  0 POL(hd) =  0 POL(nil) =  0 POL(tl) =  0 POL(compose) =  1 POL(init) =  2 POL(app(x1, x2)) =  x1 + x2 POL(reverse2) =  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:

last -> app(app(compose, hd), reverse)

where the Polynomial interpretation:
 POL(last) =  1 POL(reverse) =  0 POL(cons) =  0 POL(hd) =  0 POL(nil) =  0 POL(tl) =  0 POL(init) =  0 POL(compose) =  0 POL(app(x1, x2)) =  x1 + x2 POL(reverse2) =  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:

app(reverse, l) -> app(app(reverse2, l), nil)
app(hd, app(app(cons, x), xs)) -> x

where the Polynomial interpretation:
 POL(reverse) =  1 POL(cons) =  0 POL(hd) =  1 POL(nil) =  0 POL(tl) =  0 POL(init) =  2 POL(compose) =  0 POL(app(x1, x2)) =  x1 + x2 POL(reverse2) =  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`
`             ↳RRRPolo`
`             ...`
`               →TRS4`
`                 ↳Removing Redundant Rules`

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

init -> app(app(compose, reverse), app(app(compose, tl), reverse))
app(app(reverse2, nil), l) -> l

where the Polynomial interpretation:
 POL(reverse) =  0 POL(cons) =  0 POL(nil) =  0 POL(tl) =  0 POL(init) =  1 POL(compose) =  0 POL(app(x1, x2)) =  x1 + x2 POL(reverse2) =  1
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:

app(app(reverse2, app(app(cons, x), xs)), l) -> app(app(reverse2, xs), app(app(cons, x), l))
app(tl, app(app(cons, x), xs)) -> xs

where the Polynomial interpretation:
 POL(cons) =  1 POL(tl) =  2 POL(app(x1, x2)) =  2·x1 + x2 POL(reverse2) =  2
was used.

All Rules of R can be deleted.

`   R`
`     ↳RRRPolo`
`       →TRS2`
`         ↳RRRPolo`
`           →TRS3`
`             ↳RRRPolo`
`             ...`
`               →TRS6`
`                 ↳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`
`             ...`
`               →TRS7`
`                 ↳Dependency Pair Analysis`

R contains no Dependency Pairs and therefore no SCCs.

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