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

Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

F(g(x)) -> F(f(x))
F(g(x)) -> F(x)
F'(s(x), y, y) -> F'(y, x, s(x))

Furthermore, R contains two SCCs.

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

Dependency Pairs:

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

Rules:

f(g(x)) -> g(f(f(x)))
f(h(x)) -> h(g(x))
f'(s(x), y, y) -> f'(y, x, s(x))

The following dependency pairs can be strictly oriented:

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

The following rules can be oriented:

f(g(x)) -> g(f(f(x)))
f(h(x)) -> h(g(x))
f'(s(x), y, y) -> f'(y, x, s(x))

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

resulting in one new DP problem.
Used Argument Filtering System:
F(x1) -> F(x1)
g(x1) -> g(x1)
f(x1) -> f(x1)
h(x1) -> h
f'(x1, x2, x3) -> x3
s(x1) -> s

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

Dependency Pair:

Rules:

f(g(x)) -> g(f(f(x)))
f(h(x)) -> h(g(x))
f'(s(x), y, y) -> f'(y, x, s(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'(s(x), y, y) -> F'(y, x, s(x))

Rules:

f(g(x)) -> g(f(f(x)))
f(h(x)) -> h(g(x))
f'(s(x), y, y) -> f'(y, x, s(x))

The following dependency pair can be strictly oriented:

F'(s(x), y, y) -> F'(y, x, s(x))

The following rules can be oriented:

f(g(x)) -> g(f(f(x)))
f(h(x)) -> h(g(x))
f'(s(x), y, y) -> f'(y, x, s(x))

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

resulting in one new DP problem.
Used Argument Filtering System:
F'(x1, x2, x3) -> F'(x1, x2)
s(x1) -> s(x1)
f(x1) -> f(x1)
g(x1) -> g(x1)
h(x1) -> h(x1)
f'(x1, x2, x3) -> f'

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

Dependency Pair:

Rules:

f(g(x)) -> g(f(f(x)))
f(h(x)) -> h(g(x))
f'(s(x), y, y) -> f'(y, x, s(x))

Using the Dependency Graph resulted in no new DP problems.

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