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