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`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

F(j(x, y), y) -> G(f(x, k(y)))
F(j(x, y), y) -> F(x, k(y))
F(j(x, y), y) -> 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))

Furthermore, R contains two SCCs.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Argument Filtering and Ordering`
`       →DP Problem 2`
`         ↳AFS`

Dependency Pair:

H2(x, j(y, h1(z, u)), h1(z, u)) -> H2(s(x), y, h1(s(z), u))

Rules:

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)

The following dependency pair can be strictly oriented:

H2(x, j(y, h1(z, u)), h1(z, u)) -> H2(s(x), y, h1(s(z), u))

There are no usable rules w.r.t. to the AFS that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(h1(x1, x2)) =  x1 + x2 POL(s(x1)) =  x1 POL(j(x1, x2)) =  1 + x1 + x2 POL(H2(x1, x2, x3)) =  1 + x1 + x2 + x3

resulting in one new DP problem.
Used Argument Filtering System:
H2(x1, x2, x3) -> H2(x1, x2, x3)
j(x1, x2) -> j(x1, x2)
h1(x1, x2) -> h1(x1, x2)
s(x1) -> s(x1)

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`           →DP Problem 3`
`             ↳Dependency Graph`
`       →DP Problem 2`
`         ↳AFS`

Dependency Pair:

Rules:

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)

Using the Dependency Graph resulted in no new DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳Argument Filtering and Ordering`

Dependency Pair:

F(j(x, y), y) -> F(x, k(y))

Rules:

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)

The following dependency pair can be strictly oriented:

F(j(x, y), y) -> F(x, k(y))

The following usable rules w.r.t. to the AFS can be oriented:

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

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(0) =  0 POL(h1(x1, x2)) =  x1 + x2 POL(h(x1)) =  x1 POL(s(x1)) =  x1 POL(j(x1, x2)) =  1 + x1 + x2 POL(F(x1, x2)) =  x1 + x2 POL(k(x1)) =  x1

resulting in one new DP problem.
Used Argument Filtering System:
F(x1, x2) -> F(x1, x2)
j(x1, x2) -> j(x1, x2)
k(x1) -> k(x1)
h(x1) -> h(x1)
h1(x1, x2) -> h1(x1, x2)
s(x1) -> s(x1)

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳AFS`
`           →DP Problem 4`
`             ↳Dependency Graph`

Dependency Pair:

Rules:

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)

Using the Dependency Graph resulted in no new DP problems.

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