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

Innermost 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`
`         ↳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:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Rw`
`           →DP Problem 2`
`             ↳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:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Rw`
`           →DP Problem 2`
`             ↳Rw`
`             ...`
`               →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

Strategy:

innermost

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`
`         ↳Rw`
`           →DP Problem 2`
`             ↳Rw`
`             ...`
`               →DP Problem 4`
`                 ↳Rewriting Transformation`

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

Strategy:

innermost

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

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

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

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Rw`
`           →DP Problem 2`
`             ↳Rw`
`             ...`
`               →DP Problem 5`
`                 ↳Forward Instantiation Transformation`

Dependency Pairs:

F(s(s(x''))) -> F(s(f(f(x''))))
F(s(x)) -> F(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 Forward Instantiation SCC transformation can be performed.
As a result of transforming the rule

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

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

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Rw`
`           →DP Problem 2`
`             ↳Rw`
`             ...`
`               →DP Problem 6`
`                 ↳Narrowing Transformation`

Dependency Pairs:

F(s(s(s(x'''')))) -> F(s(s(x'''')))
F(s(s(x''))) -> F(s(x''))
F(s(s(x''))) -> F(s(f(f(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 Narrowing SCC transformation can be performed.
As a result of transforming the rule

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

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

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Rw`
`           →DP Problem 2`
`             ↳Rw`
`             ...`
`               →DP Problem 7`
`                 ↳Rewriting Transformation`

Dependency Pairs:

F(s(s(0))) -> F(s(f(0)))
F(s(s(s(x')))) -> F(s(f(s(f(f(p(s(x'))))))))
F(s(s(x''))) -> F(s(x''))
F(s(s(s(x'''')))) -> F(s(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(s(s(x')))) -> F(s(f(s(f(f(p(s(x'))))))))
one new Dependency Pair is created:

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

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Rw`
`           →DP Problem 2`
`             ↳Rw`
`             ...`
`               →DP Problem 8`
`                 ↳Rewriting Transformation`

Dependency Pairs:

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

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(s(0))) -> F(s(f(0)))
one new Dependency Pair is created:

F(s(s(0))) -> F(s(0))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Rw`
`           →DP Problem 2`
`             ↳Rw`
`             ...`
`               →DP Problem 9`
`                 ↳Rewriting Transformation`

Dependency Pairs:

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

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

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Rw`
`           →DP Problem 2`
`             ↳Rw`
`             ...`
`               →DP Problem 10`
`                 ↳Rewriting Transformation`

Dependency Pairs:

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

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

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Rw`
`           →DP Problem 2`
`             ↳Rw`
`             ...`
`               →DP Problem 11`
`                 ↳Forward Instantiation Transformation`

Dependency Pairs:

F(s(s(s(x')))) -> F(s(s(f(f(f(f(x')))))))
F(s(s(s(x'''')))) -> F(s(s(x'''')))
F(s(s(x''))) -> F(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 Forward Instantiation SCC transformation can be performed.
As a result of transforming the rule

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

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

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Rw`
`           →DP Problem 2`
`             ↳Rw`
`             ...`
`               →DP Problem 12`
`                 ↳Forward Instantiation Transformation`

Dependency Pairs:

F(s(s(s(s(x''''))))) -> F(s(s(s(x''''))))
F(s(s(s(s(x''''''))))) -> F(s(s(s(x''''''))))
F(s(s(s(x'''')))) -> F(s(s(x'''')))
F(s(s(s(x'''')))) -> F(s(s(x'''')))
F(s(s(s(x')))) -> F(s(s(f(f(f(f(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 Forward Instantiation SCC transformation can be performed.
As a result of transforming the rule

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

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

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Rw`
`           →DP Problem 2`
`             ↳Rw`
`             ...`
`               →DP Problem 13`
`                 ↳Forward Instantiation Transformation`

Dependency Pairs:

F(s(s(s(s(s(x'''''')))))) -> F(s(s(s(s(x'''''')))))
F(s(s(s(s(s(x'''''''')))))) -> F(s(s(s(s(x'''''''')))))
F(s(s(s(s(x'''))))) -> F(s(s(s(x'''))))
F(s(s(s(s(x''''''))))) -> F(s(s(s(x''''''))))
F(s(s(s(s(x''''''))))) -> F(s(s(s(x''''''))))
F(s(s(s(x'''')))) -> F(s(s(x'''')))
F(s(s(s(x')))) -> F(s(s(f(f(f(f(x')))))))
F(s(s(s(s(x''''))))) -> F(s(s(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 Forward Instantiation SCC transformation can be performed.
As a result of transforming the rule

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

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

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Rw`
`           →DP Problem 2`
`             ↳Rw`
`             ...`
`               →DP Problem 14`
`                 ↳Remaining Obligation(s)`

The following remains to be proven:
Dependency Pairs:

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

Rules:

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

Strategy:

innermost

Innermost Termination of R could not be shown.
Duration:
0:01 minutes