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))
There are no usable rules 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
↳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