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.

`   TRS`
`     ↳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.

`   TRS`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Non-Overlappingness Check`

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

R does not overlap into P. Moreover, R is locally confluent (all critical pairs are trivially joinable).Hence we can switch to innermost.
The transformation is resulting in one subcycle:

`   TRS`
`     ↳DPs`
`       →DP Problem 1`
`         ↳NOC`
`           →DP Problem 2`
`             ↳Rewriting 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

Strategy:

innermost

On this DP problem, a Rewriting 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:

`   TRS`
`     ↳DPs`
`       →DP Problem 1`
`         ↳NOC`
`           →DP Problem 2`
`             ↳Rw`
`             ...`
`               →DP Problem 3`
`                 ↳Rewriting 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

Strategy:

innermost

On this DP problem, a Rewriting 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:

`   TRS`
`     ↳DPs`
`       →DP Problem 1`
`         ↳NOC`
`           →DP Problem 2`
`             ↳Rw`
`             ...`
`               →DP Problem 4`
`                 ↳Negative Polynomial Order`

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

Strategy:

innermost

The following Dependency Pairs can be strictly oriented using the given order.

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

Moreover, the following usable rules (regarding the implicit AFS) are oriented.

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

Used ordering:
Polynomial Order with Interpretation:

POL( F(x1) ) = x1

POL( s(x1) ) = x1 + 1

POL( f(x1) ) = x1

POL( 0 ) = 0

POL( p(x1) ) = max{0, x1 - 1}

This results in one new DP problem.

`   TRS`
`     ↳DPs`
`       →DP Problem 1`
`         ↳NOC`
`           →DP Problem 2`
`             ↳Rw`
`             ...`
`               →DP Problem 5`
`                 ↳Dependency Graph`

Dependency Pair:

Rules:

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

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

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