Term Rewriting System R:
[x]
f(s(x)) -> s(f(f(p(s(x)))))
f(0) -> 0
p(s(x)) -> x

Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

F(s(x)) -> F(f(p(s(x))))
F(s(x)) -> F(p(s(x)))
F(s(x)) -> P(s(x))

Furthermore, R contains one SCC.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Narrowing Transformation`

Dependency Pairs:

F(s(x)) -> F(p(s(x)))
F(s(x)) -> F(f(p(s(x))))

Rules:

f(s(x)) -> s(f(f(p(s(x)))))
f(0) -> 0
p(s(x)) -> x

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

F(s(x)) -> F(f(p(s(x))))
one new Dependency Pair is created:

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

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳Narrowing Transformation`

Dependency Pairs:

F(s(x'')) -> F(f(x''))
F(s(x)) -> F(p(s(x)))

Rules:

f(s(x)) -> s(f(f(p(s(x)))))
f(0) -> 0
p(s(x)) -> x

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

F(s(x)) -> F(p(s(x)))
one new Dependency Pair is created:

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

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳Nar`
`             ...`
`               →DP Problem 3`
`                 ↳Narrowing Transformation`

Dependency Pairs:

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

Rules:

f(s(x)) -> s(f(f(p(s(x)))))
f(0) -> 0
p(s(x)) -> x

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

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

F(s(s(x'))) -> F(s(f(f(p(s(x'))))))
F(s(0)) -> F(0)

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳Nar`
`             ...`
`               →DP Problem 4`
`                 ↳Remaining Obligation(s)`

The following remains to be proven:
Dependency Pairs:

F(s(s(x'))) -> F(s(f(f(p(s(x'))))))
F(s(x'')) -> F(x'')

Rules:

f(s(x)) -> s(f(f(p(s(x)))))
f(0) -> 0
p(s(x)) -> x

Termination of R could not be shown.
Duration:
0:00 minutes