Term Rewriting System R:
[x]
fac(s(x)) -> *(fac(p(s(x))), s(x))
p(s(0)) -> 0
p(s(s(x))) -> s(p(s(x)))

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

FAC(s(x)) -> FAC(p(s(x)))
FAC(s(x)) -> P(s(x))
P(s(s(x))) -> P(s(x))

Furthermore, R contains two SCCs.

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

Dependency Pair:

P(s(s(x))) -> P(s(x))

Rules:

fac(s(x)) -> *(fac(p(s(x))), s(x))
p(s(0)) -> 0
p(s(s(x))) -> s(p(s(x)))

Strategy:

innermost

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

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

P(s(s(s(x'')))) -> P(s(s(x'')))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 3`
`             ↳Forward Instantiation Transformation`
`       →DP Problem 2`
`         ↳Nar`

Dependency Pair:

P(s(s(s(x'')))) -> P(s(s(x'')))

Rules:

fac(s(x)) -> *(fac(p(s(x))), s(x))
p(s(0)) -> 0
p(s(s(x))) -> s(p(s(x)))

Strategy:

innermost

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

P(s(s(s(x'')))) -> P(s(s(x'')))
one new Dependency Pair is created:

P(s(s(s(s(x''''))))) -> P(s(s(s(x''''))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 3`
`             ↳FwdInst`
`             ...`
`               →DP Problem 4`
`                 ↳Polynomial Ordering`
`       →DP Problem 2`
`         ↳Nar`

Dependency Pair:

P(s(s(s(s(x''''))))) -> P(s(s(s(x''''))))

Rules:

fac(s(x)) -> *(fac(p(s(x))), s(x))
p(s(0)) -> 0
p(s(s(x))) -> s(p(s(x)))

Strategy:

innermost

The following dependency pair can be strictly oriented:

P(s(s(s(s(x''''))))) -> P(s(s(s(x''''))))

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

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(P(x1)) =  1 + x1 POL(s(x1)) =  1 + x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 3`
`             ↳FwdInst`
`             ...`
`               →DP Problem 5`
`                 ↳Dependency Graph`
`       →DP Problem 2`
`         ↳Nar`

Dependency Pair:

Rules:

fac(s(x)) -> *(fac(p(s(x))), s(x))
p(s(0)) -> 0
p(s(s(x))) -> s(p(s(x)))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

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

Dependency Pair:

FAC(s(x)) -> FAC(p(s(x)))

Rules:

fac(s(x)) -> *(fac(p(s(x))), s(x))
p(s(0)) -> 0
p(s(s(x))) -> s(p(s(x)))

Strategy:

innermost

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

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

FAC(s(0)) -> FAC(0)
FAC(s(s(x''))) -> FAC(s(p(s(x''))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳Nar`
`           →DP Problem 6`
`             ↳Narrowing Transformation`

Dependency Pair:

FAC(s(s(x''))) -> FAC(s(p(s(x''))))

Rules:

fac(s(x)) -> *(fac(p(s(x))), s(x))
p(s(0)) -> 0
p(s(s(x))) -> s(p(s(x)))

Strategy:

innermost

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

FAC(s(s(x''))) -> FAC(s(p(s(x''))))
two new Dependency Pairs are created:

FAC(s(s(0))) -> FAC(s(0))
FAC(s(s(s(x')))) -> FAC(s(s(p(s(x')))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳Nar`
`           →DP Problem 6`
`             ↳Nar`
`             ...`
`               →DP Problem 7`
`                 ↳Narrowing Transformation`

Dependency Pair:

FAC(s(s(s(x')))) -> FAC(s(s(p(s(x')))))

Rules:

fac(s(x)) -> *(fac(p(s(x))), s(x))
p(s(0)) -> 0
p(s(s(x))) -> s(p(s(x)))

Strategy:

innermost

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

FAC(s(s(s(x')))) -> FAC(s(s(p(s(x')))))
two new Dependency Pairs are created:

FAC(s(s(s(0)))) -> FAC(s(s(0)))
FAC(s(s(s(s(x''))))) -> FAC(s(s(s(p(s(x''))))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳Nar`
`           →DP Problem 6`
`             ↳Nar`
`             ...`
`               →DP Problem 8`
`                 ↳Narrowing Transformation`

Dependency Pair:

FAC(s(s(s(s(x''))))) -> FAC(s(s(s(p(s(x''))))))

Rules:

fac(s(x)) -> *(fac(p(s(x))), s(x))
p(s(0)) -> 0
p(s(s(x))) -> s(p(s(x)))

Strategy:

innermost

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

FAC(s(s(s(s(x''))))) -> FAC(s(s(s(p(s(x''))))))
two new Dependency Pairs are created:

FAC(s(s(s(s(0))))) -> FAC(s(s(s(0))))
FAC(s(s(s(s(s(x')))))) -> FAC(s(s(s(s(p(s(x')))))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳Nar`
`           →DP Problem 6`
`             ↳Nar`
`             ...`
`               →DP Problem 9`
`                 ↳Narrowing Transformation`

Dependency Pair:

FAC(s(s(s(s(s(x')))))) -> FAC(s(s(s(s(p(s(x')))))))

Rules:

fac(s(x)) -> *(fac(p(s(x))), s(x))
p(s(0)) -> 0
p(s(s(x))) -> s(p(s(x)))

Strategy:

innermost

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

FAC(s(s(s(s(s(x')))))) -> FAC(s(s(s(s(p(s(x')))))))
two new Dependency Pairs are created:

FAC(s(s(s(s(s(0)))))) -> FAC(s(s(s(s(0)))))
FAC(s(s(s(s(s(s(x''))))))) -> FAC(s(s(s(s(s(p(s(x''))))))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳Nar`
`           →DP Problem 6`
`             ↳Nar`
`             ...`
`               →DP Problem 10`
`                 ↳Remaining Obligation(s)`

The following remains to be proven:
Dependency Pair:

FAC(s(s(s(s(s(s(x''))))))) -> FAC(s(s(s(s(s(p(s(x''))))))))

Rules:

fac(s(x)) -> *(fac(p(s(x))), s(x))
p(s(0)) -> 0
p(s(s(x))) -> s(p(s(x)))

Strategy:

innermost

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