Term Rewriting System R:
[X, Z, Y]
h(X, Z) -> f(X, s(X), Z)
f(X, Y, g(X, Y)) -> h(0, g(X, Y))
g(0, Y) -> 0
g(X, s(Y)) -> g(X, Y)

Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

H(X, Z) -> F(X, s(X), Z)
F(X, Y, g(X, Y)) -> H(0, g(X, Y))
G(X, s(Y)) -> G(X, Y)

Furthermore, R contains two SCCs.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polynomial Ordering`
`       →DP Problem 2`
`         ↳Inst`

Dependency Pair:

G(X, s(Y)) -> G(X, Y)

Rules:

h(X, Z) -> f(X, s(X), Z)
f(X, Y, g(X, Y)) -> h(0, g(X, Y))
g(0, Y) -> 0
g(X, s(Y)) -> g(X, Y)

The following dependency pair can be strictly oriented:

G(X, s(Y)) -> G(X, Y)

There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(G(x1, x2)) =  x2 POL(s(x1)) =  1 + x1

resulting in one new DP problem.

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

Dependency Pair:

Rules:

h(X, Z) -> f(X, s(X), Z)
f(X, Y, g(X, Y)) -> h(0, g(X, Y))
g(0, Y) -> 0
g(X, s(Y)) -> g(X, Y)

Using the Dependency Graph resulted in no new DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Instantiation Transformation`

Dependency Pairs:

F(X, Y, g(X, Y)) -> H(0, g(X, Y))
H(X, Z) -> F(X, s(X), Z)

Rules:

h(X, Z) -> f(X, s(X), Z)
f(X, Y, g(X, Y)) -> h(0, g(X, Y))
g(0, Y) -> 0
g(X, s(Y)) -> g(X, Y)

On this DP problem, an Instantiation SCC transformation can be performed.
As a result of transforming the rule

H(X, Z) -> F(X, s(X), Z)
one new Dependency Pair is created:

H(0, Z') -> F(0, s(0), Z')

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Inst`
`           →DP Problem 4`
`             ↳Instantiation Transformation`

Dependency Pairs:

H(0, Z') -> F(0, s(0), Z')
F(X, Y, g(X, Y)) -> H(0, g(X, Y))

Rules:

h(X, Z) -> f(X, s(X), Z)
f(X, Y, g(X, Y)) -> h(0, g(X, Y))
g(0, Y) -> 0
g(X, s(Y)) -> g(X, Y)

On this DP problem, an Instantiation SCC transformation can be performed.
As a result of transforming the rule

F(X, Y, g(X, Y)) -> H(0, g(X, Y))
one new Dependency Pair is created:

F(0, s(0), g(0, s(0))) -> H(0, g(0, s(0)))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Inst`
`           →DP Problem 4`
`             ↳Inst`
`             ...`
`               →DP Problem 5`
`                 ↳Remaining Obligation(s)`

The following remains to be proven:
Dependency Pairs:

F(0, s(0), g(0, s(0))) -> H(0, g(0, s(0)))
H(0, Z') -> F(0, s(0), Z')

Rules:

h(X, Z) -> f(X, s(X), Z)
f(X, Y, g(X, Y)) -> h(0, g(X, Y))
g(0, Y) -> 0
g(X, s(Y)) -> g(X, Y)

Termination of R could not be shown.
Duration:
0:00 minutes