Term Rewriting System R:
[x]
f(f(x)) -> g(f(x))
g(g(x)) -> f(x)
Termination of R to be shown.
R
↳Dependency Pair Analysis
R contains the following Dependency Pairs:
F(f(x)) -> G(f(x))
G(g(x)) -> F(x)
Furthermore, R contains one SCC.
R
↳DPs
→DP Problem 1
↳Narrowing Transformation
Dependency Pairs:
G(g(x)) -> F(x)
F(f(x)) -> G(f(x))
Rules:
f(f(x)) -> g(f(x))
g(g(x)) -> f(x)
On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule
F(f(x)) -> G(f(x))
one new Dependency Pair
is created:
F(f(f(x''))) -> G(g(f(x'')))
The transformation is resulting in one new DP problem:
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 2
↳Argument Filtering and Ordering
Dependency Pairs:
F(f(f(x''))) -> G(g(f(x'')))
G(g(x)) -> F(x)
Rules:
f(f(x)) -> g(f(x))
g(g(x)) -> f(x)
The following dependency pairs can be strictly oriented:
F(f(f(x''))) -> G(g(f(x'')))
G(g(x)) -> F(x)
The following usable rules w.r.t. to the AFS can be oriented:
g(g(x)) -> f(x)
f(f(x)) -> g(f(x))
Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
{F, f, g} > G
resulting in one new DP problem.
Used Argument Filtering System: G(x1) -> G(x1)
F(x1) -> F(x1)
g(x1) -> g(x1)
f(x1) -> f(x1)
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 2
↳AFS
...
→DP Problem 3
↳Dependency Graph
Dependency Pair:
Rules:
f(f(x)) -> g(f(x))
g(g(x)) -> f(x)
Using the Dependency Graph resulted in no new DP problems.
Termination of R successfully shown.
Duration:
0:00 minutes