Term Rewriting System R:
[x, y, z, u]
f(j(x, y), y) -> g(f(x, k(y)))
f(x, h1(y, z)) -> h2(0, x, h1(y, z))
g(h2(x, y, h1(z, u))) -> h2(s(x), y, h1(z, u))
h2(x, j(y, h1(z, u)), h1(z, u)) -> h2(s(x), y, h1(s(z), u))
i(f(x, h(y))) -> y
i(h2(s(x), y, h1(x, z))) -> z
k(h(x)) -> h1(0, x)
k(h1(x, y)) -> h1(s(x), y)

Termination of R to be shown.

`   R`
`     ↳Removing Redundant Rules`

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

f(j(x, y), y) -> g(f(x, k(y)))
h2(x, j(y, h1(z, u)), h1(z, u)) -> h2(s(x), y, h1(s(z), u))
i(f(x, h(y))) -> y
i(h2(s(x), y, h1(x, z))) -> z

where the Polynomial interpretation:
 POL(0) =  0 POL(i(x1)) =  1 + x1 POL(g(x1)) =  x1 POL(h1(x1, x2)) =  x1 + x2 POL(s(x1)) =  x1 POL(h(x1)) =  x1 POL(j(x1, x2)) =  1 + x1 + x2 POL(f(x1, x2)) =  x1 + x2 POL(h2(x1, x2, x3)) =  x1 + x2 + x3 POL(k(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`
`         ↳Removing Redundant Rules`

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

k(h1(x, y)) -> h1(s(x), y)
k(h(x)) -> h1(0, x)

where the Polynomial interpretation:
 POL(0) =  0 POL(g(x1)) =  x1 POL(h1(x1, x2)) =  x1 + x2 POL(h(x1)) =  x1 POL(s(x1)) =  x1 POL(f(x1, x2)) =  x1 + x2 POL(h2(x1, x2, x3)) =  x1 + x2 + x3 POL(k(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`
`         ↳RRRPolo`
`           →TRS3`
`             ↳Removing Redundant Rules`

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

f(x, h1(y, z)) -> h2(0, x, h1(y, z))

where the Polynomial interpretation:
 POL(0) =  0 POL(g(x1)) =  x1 POL(h1(x1, x2)) =  x1 + x2 POL(s(x1)) =  x1 POL(h2(x1, x2, x3)) =  x1 + x2 + x3 POL(f(x1, x2)) =  1 + 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:

g(h2(x, y, h1(z, u))) -> h2(s(x), y, h1(z, u))

where the Polynomial interpretation:
 POL(g(x1)) =  1 + x1 POL(h1(x1, x2)) =  x1 + x2 POL(s(x1)) =  x1 POL(h2(x1, x2, x3)) =  x1 + x2 + x3
was used.

All Rules of R can be deleted.

`   R`
`     ↳RRRPolo`
`       →TRS2`
`         ↳RRRPolo`
`           →TRS3`
`             ↳RRRPolo`
`             ...`
`               →TRS5`
`                 ↳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`
`             ...`
`               →TRS6`
`                 ↳Dependency Pair Analysis`

R contains no Dependency Pairs and therefore no SCCs.

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