Term Rewriting System R:
[x]
f(x) -> s(x)
f(s(s(x))) -> s(f(f(x)))
Termination of R to be shown.
R
↳Dependency Pair Analysis
R contains the following Dependency Pairs:
F(s(s(x))) -> F(f(x))
F(s(s(x))) -> F(x)
Furthermore, R contains one SCC.
R
↳DPs
→DP Problem 1
↳Polynomial Ordering
Dependency Pairs:
F(s(s(x))) -> F(x)
F(s(s(x))) -> F(f(x))
Rules:
f(x) -> s(x)
f(s(s(x))) -> s(f(f(x)))
The following dependency pairs can be strictly oriented:
F(s(s(x))) -> F(x)
F(s(s(x))) -> F(f(x))
Additionally, the following usable rules using the Ce-refinement can be oriented:
f(x) -> s(x)
f(s(s(x))) -> s(f(f(x)))
Used ordering: Polynomial ordering with Polynomial interpretation:
| POL(s(x1)) | = 1 + x1 |
| POL(f(x1)) | = 1 + x1 |
| POL(F(x1)) | = 1 + x1 |
resulting in one new DP problem.
R
↳DPs
→DP Problem 1
↳Polo
→DP Problem 2
↳Dependency Graph
Dependency Pair:
Rules:
f(x) -> s(x)
f(s(s(x))) -> s(f(f(x)))
Using the Dependency Graph resulted in no new DP problems.
Termination of R successfully shown.
Duration:
0:00 minutes