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`
`         ↳Narrowing Transformation`
`       →DP Problem 2`
`         ↳Inst`

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))

On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

F(g(x)) -> F(f(x))
two new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 3`
`             ↳Polynomial Ordering`
`       →DP Problem 2`
`         ↳Inst`

Dependency Pairs:

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

Additionally, the following usable rules w.r.t. to the implicit AFS can be oriented:

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

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

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 3`
`             ↳Polo`
`             ...`
`               →DP Problem 4`
`                 ↳Dependency Graph`
`       →DP Problem 2`
`         ↳Inst`

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`
`         ↳Nar`
`       →DP Problem 2`
`         ↳Instantiation Transformation`

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))

On this DP problem, an Instantiation SCC transformation can be performed.
As a result of transforming the rule

F'(s(x), y, y) -> F'(y, x, s(x))
one new Dependency Pair is created:

F'(s(x0), s(x'''), s(x''')) -> F'(s(x'''), x0, s(x0))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`       →DP Problem 2`
`         ↳Inst`
`           →DP Problem 5`
`             ↳Forward Instantiation Transformation`

Dependency Pair:

F'(s(x0), s(x'''), s(x''')) -> F'(s(x'''), x0, s(x0))

Rules:

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

On this DP problem, a Forward Instantiation SCC transformation can be performed.
As a result of transforming the rule

F'(s(x0), s(x'''), s(x''')) -> F'(s(x'''), x0, s(x0))
one new Dependency Pair is created:

F'(s(s(x'''''')), s(x'''0), s(x'''0)) -> F'(s(x'''0), s(x''''''), s(s(x'''''')))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`       →DP Problem 2`
`         ↳Inst`
`           →DP Problem 5`
`             ↳FwdInst`
`             ...`
`               →DP Problem 6`
`                 ↳Polynomial Ordering`

Dependency Pair:

F'(s(s(x'''''')), s(x'''0), s(x'''0)) -> F'(s(x'''0), s(x''''''), s(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(s(x'''''')), s(x'''0), s(x'''0)) -> F'(s(x'''0), s(x''''''), s(s(x'''''')))

There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(F'(x1, x2, x3)) =  1 + x1 + x2 POL(s(x1)) =  1 + x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`       →DP Problem 2`
`         ↳Inst`
`           →DP Problem 5`
`             ↳FwdInst`
`             ...`
`               →DP Problem 7`
`                 ↳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