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`
`         ↳Argument Filtering and Ordering`
`       →DP Problem 2`
`         ↳Remaining`

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

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost w.r.t. to the 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.
Used Argument Filtering System:
P(x1) -> P(x1)
s(x1) -> s(x1)

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`           →DP Problem 3`
`             ↳Dependency Graph`
`       →DP Problem 2`
`         ↳Remaining`

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`
`         ↳AFS`
`       →DP Problem 2`
`         ↳Remaining Obligation(s)`

The following remains to be proven:
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

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