Term Rewriting System R:
[x, y]
f(f(x)) -> f(g(f(x), x))
f(f(x)) -> f(h(f(x), f(x)))
g(x, y) -> y
h(x, x) -> g(x, 0)

Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

F(f(x)) -> F(g(f(x), x))
F(f(x)) -> G(f(x), x)
F(f(x)) -> F(h(f(x), f(x)))
F(f(x)) -> H(f(x), f(x))
H(x, x) -> G(x, 0)

Furthermore, R contains one SCC.

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

Dependency Pairs:

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

Rules:

f(f(x)) -> f(g(f(x), x))
f(f(x)) -> f(h(f(x), f(x)))
g(x, y) -> y
h(x, x) -> g(x, 0)

The following dependency pairs can be strictly oriented:

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

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

h(x, x) -> g(x, 0)
f(f(x)) -> f(g(f(x), x))
f(f(x)) -> f(h(f(x), f(x)))
g(x, y) -> y

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

resulting in one new DP problem.

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

Dependency Pair:

Rules:

f(f(x)) -> f(g(f(x), x))
f(f(x)) -> f(h(f(x), f(x)))
g(x, y) -> y
h(x, x) -> g(x, 0)

Using the Dependency Graph resulted in no new DP problems.

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