Term Rewriting System R:
[x]
f(a, x) -> f(g(x), x)
h(g(x)) -> h(a)
h(h(x)) -> x
g(h(x)) -> g(x)

Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

F(a, x) -> F(g(x), x)
F(a, x) -> G(x)
H(g(x)) -> H(a)
G(h(x)) -> G(x)

Furthermore, R contains two SCCs.

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

Dependency Pair:

G(h(x)) -> G(x)

Rules:

f(a, x) -> f(g(x), x)
h(g(x)) -> h(a)
h(h(x)) -> x
g(h(x)) -> g(x)

The following dependency pair can be strictly oriented:

G(h(x)) -> G(x)

There are no usable rules w.r.t. to the AFS that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(G(x1)) =  x1 POL(h(x1)) =  1 + x1

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

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

Dependency Pair:

Rules:

f(a, x) -> f(g(x), x)
h(g(x)) -> h(a)
h(h(x)) -> x
g(h(x)) -> g(x)

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(a, x) -> F(g(x), x)

Rules:

f(a, x) -> f(g(x), x)
h(g(x)) -> h(a)
h(h(x)) -> x
g(h(x)) -> g(x)

The following dependency pair can be strictly oriented:

F(a, x) -> F(g(x), x)

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

g(h(x)) -> g(x)

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(g) =  0 POL(a) =  1 POL(F(x1, x2)) =  x1 + x2

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

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

Dependency Pair:

Rules:

f(a, x) -> f(g(x), x)
h(g(x)) -> h(a)
h(h(x)) -> x
g(h(x)) -> g(x)

Using the Dependency Graph resulted in no new DP problems.

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