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`
`         ↳Argument Filtering and Ordering`
`       →DP Problem 2`
`         ↳Remaining`

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 using the Ce-refinement that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(G(x1, x2)) =  x1 + x2 POL(s(x1)) =  1 + x1

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

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

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`
`         ↳AFS`
`       →DP Problem 2`
`         ↳Remaining Obligation(s)`

The following remains to be proven:
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)

Termination of R could not be shown.
Duration:
0:01 minutes