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)))

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
Polynomial 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)))

The following dependency pair can be strictly oriented:

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

Additionally, the following rules can be oriented:

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

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(P(x1)) =  x1 POL(0) =  0 POL(fac(x1)) =  0 POL(*(x1, x2)) =  0 POL(s(x1)) =  1 + x1 POL(p(x1)) =  x1

resulting in one new DP problem.

R
DPs
→DP Problem 1
Polo
→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)))

Using the Dependency Graph resulted in no new DP problems.

R
DPs
→DP Problem 1
Polo
→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)))

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