Term Rewriting System R:
[x]
f(f(x)) -> f(x)
f(s(x)) -> f(x)
g(s(0)) -> g(f(s(0)))
Termination of R to be shown.
R
↳Dependency Pair Analysis
R contains the following Dependency Pairs:
F(s(x)) -> F(x)
G(s(0)) -> G(f(s(0)))
G(s(0)) -> F(s(0))
Furthermore, R contains two SCCs.
R
↳DPs
→DP Problem 1
↳Argument Filtering and Ordering
→DP Problem 2
↳AFS
Dependency Pair:
F(s(x)) -> F(x)
Rules:
f(f(x)) -> f(x)
f(s(x)) -> f(x)
g(s(0)) -> g(f(s(0)))
The following dependency pair can be strictly oriented:
F(s(x)) -> F(x)
There are no usable rules w.r.t. to the AFS that need to be oriented.
Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
trivial
resulting in one new DP problem.
Used Argument Filtering System: F(x1) -> F(x1)
s(x1) -> s(x1)
R
↳DPs
→DP Problem 1
↳AFS
→DP Problem 3
↳Dependency Graph
→DP Problem 2
↳AFS
Dependency Pair:
Rules:
f(f(x)) -> f(x)
f(s(x)) -> f(x)
g(s(0)) -> g(f(s(0)))
Using the Dependency Graph resulted in no new DP problems.
R
↳DPs
→DP Problem 1
↳AFS
→DP Problem 2
↳Argument Filtering and Ordering
Dependency Pair:
G(s(0)) -> G(f(s(0)))
Rules:
f(f(x)) -> f(x)
f(s(x)) -> f(x)
g(s(0)) -> g(f(s(0)))
The following dependency pair can be strictly oriented:
G(s(0)) -> G(f(s(0)))
The following usable rules w.r.t. to the AFS can be oriented:
f(f(x)) -> f(x)
f(s(x)) -> f(x)
Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
{f, 0}
resulting in one new DP problem.
Used Argument Filtering System: G(x1) -> G(x1)
s(x1) -> s(x1)
f(x1) -> f
R
↳DPs
→DP Problem 1
↳AFS
→DP Problem 2
↳AFS
→DP Problem 4
↳Dependency Graph
Dependency Pair:
Rules:
f(f(x)) -> f(x)
f(s(x)) -> f(x)
g(s(0)) -> g(f(s(0)))
Using the Dependency Graph resulted in no new DP problems.
Termination of R successfully shown.
Duration:
0:00 minutes