Term Rewriting System R:
[x]
g(s(x)) -> f(x)
g(0) -> 0
f(0) -> s(0)
f(s(x)) -> s(s(g(x)))
Termination of R to be shown.
R
↳Dependency Pair Analysis
R contains the following Dependency Pairs:
G(s(x)) -> F(x)
F(s(x)) -> G(x)
Furthermore, R contains one SCC.
R
↳DPs
→DP Problem 1
↳Argument Filtering and Ordering
Dependency Pairs:
F(s(x)) -> G(x)
G(s(x)) -> F(x)
Rules:
g(s(x)) -> f(x)
g(0) -> 0
f(0) -> s(0)
f(s(x)) -> s(s(g(x)))
The following dependency pairs can be strictly oriented:
F(s(x)) -> G(x)
G(s(x)) -> F(x)
There are no usable rules using the Ce-refinement that need to be oriented.
Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
{G, s} > F
resulting in one new DP problem.
Used Argument Filtering System: F(x1) -> F(x1)
G(x1) -> G(x1)
s(x1) -> s(x1)
R
↳DPs
→DP Problem 1
↳AFS
→DP Problem 2
↳Dependency Graph
Dependency Pair:
Rules:
g(s(x)) -> f(x)
g(0) -> 0
f(0) -> s(0)
f(s(x)) -> s(s(g(x)))
Using the Dependency Graph resulted in no new DP problems.
Termination of R successfully shown.
Duration:
0:00 minutes