Term Rewriting System R:
[x]
f(g(a)) -> f(s(g(b)))
f(f(x)) -> b
g(x) -> f(g(x))
Termination of R to be shown.
R
↳Dependency Pair Analysis
R contains the following Dependency Pairs:
F(g(a)) -> F(s(g(b)))
F(g(a)) -> G(b)
G(x) -> F(g(x))
G(x) -> G(x)
Furthermore, R contains one SCC.
R
↳DPs
→DP Problem 1
↳Argument Filtering and Ordering
Dependency Pairs:
G(x) -> G(x)
G(x) -> F(g(x))
F(g(a)) -> G(b)
Rules:
f(g(a)) -> f(s(g(b)))
f(f(x)) -> b
g(x) -> f(g(x))
The following dependency pairs can be strictly oriented:
G(x) -> F(g(x))
F(g(a)) -> G(b)
The following usable rules using the Ce-refinement can be oriented:
g(x) -> f(g(x))
f(g(a)) -> f(s(g(b)))
f(f(x)) -> b
Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
a > G > F > b
a > G > g > b
a > s > b
f > b
resulting in one new DP problem.
Used Argument Filtering System: G(x1) -> G(x1)
F(x1) -> F(x1)
g(x1) -> g(x1)
f(x1) -> x1
s(x1) -> s(x1)
R
↳DPs
→DP Problem 1
↳AFS
→DP Problem 2
↳Remaining Obligation(s)
The following remains to be proven:
Dependency Pair:
G(x) -> G(x)
Rules:
f(g(a)) -> f(s(g(b)))
f(f(x)) -> b
g(x) -> f(g(x))
Termination of R could not be shown.
Duration:
0:00 minutes