Term Rewriting System R:
[X]
f(g(X)) -> g(f(f(X)))
f(h(X)) -> h(g(X))
Termination of R to be shown.
R
↳Dependency Pair Analysis
R contains the following Dependency Pairs:
F(g(X)) -> F(f(X))
F(g(X)) -> F(X)
Furthermore, R contains one SCC.
R
↳DPs
→DP Problem 1
↳Argument Filtering and Ordering
Dependency Pairs:
F(g(X)) -> F(X)
F(g(X)) -> F(f(X))
Rules:
f(g(X)) -> g(f(f(X)))
f(h(X)) -> h(g(X))
The following dependency pairs can be strictly oriented:
F(g(X)) -> F(X)
F(g(X)) -> F(f(X))
The following usable rules using the Ce-refinement can be oriented:
f(g(X)) -> g(f(f(X)))
f(h(X)) -> h(g(X))
Used ordering: Homeomorphic Embedding Order with EMB
resulting in one new DP problem.
Used Argument Filtering System: F(x1) -> F(x1)
g(x1) -> g(x1)
f(x1) -> x1
h(x1) -> h
R
↳DPs
→DP Problem 1
↳AFS
→DP Problem 2
↳Dependency Graph
Dependency Pair:
Rules:
f(g(X)) -> g(f(f(X)))
f(h(X)) -> h(g(X))
Using the Dependency Graph resulted in no new DP problems.
Termination of R successfully shown.
Duration:
0:00 minutes