Term Rewriting System R:
[x]
g(s(x)) -> f(x)
g(0) -> 0
f(0) -> s(0)
f(s(x)) -> s(s(g(x)))
Termination of R to be shown.
R
↳Dependency Pair Analysis
R contains the following Dependency Pairs:
G(s(x)) -> F(x)
F(s(x)) -> G(x)
Furthermore, R contains one SCC.
R
↳DPs
→DP Problem 1
↳Polynomial Ordering
Dependency Pairs:
F(s(x)) -> G(x)
G(s(x)) -> F(x)
Rules:
g(s(x)) -> f(x)
g(0) -> 0
f(0) -> s(0)
f(s(x)) -> s(s(g(x)))
The following dependency pairs can be strictly oriented:
F(s(x)) -> G(x)
G(s(x)) -> F(x)
Additionally, the following rules can be oriented:
g(s(x)) -> f(x)
g(0) -> 0
f(0) -> s(0)
f(s(x)) -> s(s(g(x)))
Used ordering: Polynomial ordering with Polynomial interpretation:
| POL(0) | = 0 |
| POL(g(x1)) | = x1 |
| POL(G(x1)) | = x1 |
| POL(s(x1)) | = 1 + x1 |
| POL(f(x1)) | = 1 + x1 |
| POL(F(x1)) | = x1 |
resulting in one new DP problem.
R
↳DPs
→DP Problem 1
↳Polo
→DP Problem 2
↳Dependency Graph
Dependency Pair:
Rules:
g(s(x)) -> f(x)
g(0) -> 0
f(0) -> s(0)
f(s(x)) -> s(s(g(x)))
Using the Dependency Graph resulted in no new DP problems.
Termination of R successfully shown.
Duration:
0:00 minutes