Term Rewriting System R:
[x]
f(f(x)) -> f(c(f(x)))
f(f(x)) -> f(d(f(x)))
g(c(x)) -> x
g(d(x)) -> x
g(c(h(0))) -> g(d(1))
g(c(1)) -> g(d(h(0)))
g(h(x)) -> g(x)

Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

F(f(x)) -> F(c(f(x)))
F(f(x)) -> F(d(f(x)))
G(c(h(0))) -> G(d(1))
G(c(1)) -> G(d(h(0)))
G(h(x)) -> G(x)

Furthermore, R contains one SCC.

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

Dependency Pair:

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

Rules:

f(f(x)) -> f(c(f(x)))
f(f(x)) -> f(d(f(x)))
g(c(x)) -> x
g(d(x)) -> x
g(c(h(0))) -> g(d(1))
g(c(1)) -> g(d(h(0)))
g(h(x)) -> g(x)

The following dependency pair can be strictly oriented:

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

There are no usable rules using the Ce-refinement 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 2`
`             ↳Dependency Graph`

Dependency Pair:

Rules:

f(f(x)) -> f(c(f(x)))
f(f(x)) -> f(d(f(x)))
g(c(x)) -> x
g(d(x)) -> x
g(c(h(0))) -> g(d(1))
g(c(1)) -> g(d(h(0)))
g(h(x)) -> g(x)

Using the Dependency Graph resulted in no new DP problems.

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