Term Rewriting System R:
[X]
f(s(X), X) -> f(X, a(X))
f(X, c(X)) -> f(s(X), X)
f(X, X) -> c(X)
Termination of R to be shown.
R
↳Dependency Pair Analysis
R contains the following Dependency Pairs:
F(s(X), X) -> F(X, a(X))
F(X, c(X)) -> F(s(X), X)
Furthermore, R contains two SCCs.
R
↳DPs
→DP Problem 1
↳Polynomial Ordering
→DP Problem 2
↳Polo
Dependency Pair:
F(s(X), X) -> F(X, a(X))
Rules:
f(s(X), X) -> f(X, a(X))
f(X, c(X)) -> f(s(X), X)
f(X, X) -> c(X)
The following dependency pair can be strictly oriented:
F(s(X), X) -> F(X, a(X))
There are no usable rules using the Ce-refinement that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:
| POL(s(x1)) | = 1 + x1 |
| POL(a(x1)) | = 0 |
| POL(F(x1, x2)) | = x1 |
resulting in one new DP problem.
R
↳DPs
→DP Problem 1
↳Polo
→DP Problem 3
↳Dependency Graph
→DP Problem 2
↳Polo
Dependency Pair:
Rules:
f(s(X), X) -> f(X, a(X))
f(X, c(X)) -> f(s(X), X)
f(X, X) -> c(X)
Using the Dependency Graph resulted in no new DP problems.
R
↳DPs
→DP Problem 1
↳Polo
→DP Problem 2
↳Polynomial Ordering
Dependency Pair:
F(X, c(X)) -> F(s(X), X)
Rules:
f(s(X), X) -> f(X, a(X))
f(X, c(X)) -> f(s(X), X)
f(X, X) -> c(X)
The following dependency pair can be strictly oriented:
F(X, c(X)) -> F(s(X), X)
There are no usable rules using the Ce-refinement that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:
| POL(c(x1)) | = 1 + x1 |
| POL(s(x1)) | = 0 |
| POL(F(x1, x2)) | = x2 |
resulting in one new DP problem.
R
↳DPs
→DP Problem 1
↳Polo
→DP Problem 2
↳Polo
→DP Problem 4
↳Dependency Graph
Dependency Pair:
Rules:
f(s(X), X) -> f(X, a(X))
f(X, c(X)) -> f(s(X), X)
f(X, X) -> c(X)
Using the Dependency Graph resulted in no new DP problems.
Termination of R successfully shown.
Duration:
0:00 minutes