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
↳Argument Filtering and 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 pair can be strictly oriented:
F(s(s(x))) -> F(s(x))
The following usable rules using the Ce-refinement can be oriented:
f(0) -> s(0)
f(s(0)) -> s(0)
f(s(s(x))) -> f(f(s(x)))
Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
{f, s}
resulting in one new DP problem.
Used Argument Filtering System: F(x1) -> F(x1)
s(x1) -> s(x1)
f(x1) -> f(x1)
R
↳DPs
→DP Problem 1
↳AFS
→DP Problem 2
↳Remaining Obligation(s)
The following remains to be proven:
Dependency Pair:
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)))
Termination of R could not be shown.
Duration:
0:00 minutes