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

Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

G(g(x)) -> G(h(g(x)))
G(g(x)) -> H(g(x))
H(h(x)) -> H(f(h(x), x))

Furthermore, R contains one SCC.

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

Dependency Pair:

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

Rules:

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

The following dependency pair can be strictly oriented:

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

The following usable rules using the Ce-refinement can be oriented:

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

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(g) =  1 POL(G(x1)) =  x1 POL(h) =  0

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

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

Dependency Pair:

Rules:

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

Using the Dependency Graph resulted in no new DP problems.

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