Term Rewriting System R:
[x, y]
f(g(x), s(0), y) -> f(g(s(0)), y, 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), y) -> F(g(s(0)), y, g(x))
F(g(x), s(0), y) -> G(s(0))
G(s(x)) -> G(x)
Furthermore, R contains two SCCs.
R
↳DPs
→DP Problem 1
↳Argument Filtering and Ordering
→DP Problem 2
↳Remaining
Dependency Pair:
G(s(x)) -> G(x)
Rules:
f(g(x), s(0), y) -> f(g(s(0)), y, 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: Homeomorphic Embedding Order with EMB
resulting in one new DP problem.
Used Argument Filtering System: G(x1) -> G(x1)
s(x1) -> s(x1)
R
↳DPs
→DP Problem 1
↳AFS
→DP Problem 3
↳Dependency Graph
→DP Problem 2
↳Remaining
Dependency Pair:
Rules:
f(g(x), s(0), y) -> f(g(s(0)), y, 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
↳AFS
→DP Problem 2
↳Remaining Obligation(s)
The following remains to be proven:
Dependency Pair:
F(g(x), s(0), y) -> F(g(s(0)), y, g(x))
Rules:
f(g(x), s(0), y) -> f(g(s(0)), y, g(x))
g(s(x)) -> s(g(x))
g(0) -> 0
Termination of R could not be shown.
Duration:
0:00 minutes