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