Term Rewriting System R:
[x]
f(g(x), s(0)) -> f(g(x), g(x))
g(s(x)) -> s(g(x))
g(0) -> 0
Termination of R to be shown.
R
↳Dependency Pair Analysis
R contains the following Dependency Pairs:
F(g(x), s(0)) -> F(g(x), g(x))
G(s(x)) -> G(x)
Furthermore, R contains two SCCs.
R
↳DPs
→DP Problem 1
↳Polynomial Ordering
→DP Problem 2
↳Remaining
Dependency Pair:
G(s(x)) -> G(x)
Rules:
f(g(x), s(0)) -> f(g(x), g(x))
g(s(x)) -> s(g(x))
g(0) -> 0
The following dependency pair can be strictly oriented:
G(s(x)) -> G(x)
There are no usable rules using the Ce-refinement that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:
| POL(G(x1)) | = 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:
f(g(x), s(0)) -> f(g(x), g(x))
g(s(x)) -> s(g(x))
g(0) -> 0
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:
F(g(x), s(0)) -> F(g(x), g(x))
Rules:
f(g(x), s(0)) -> f(g(x), g(x))
g(s(x)) -> s(g(x))
g(0) -> 0
Termination of R could not be shown.
Duration:
0:00 minutes