Term Rewriting System R:
[x]
sum(0) -> 0
sum(s(x)) -> +(sum(x), s(x))
sum1(0) -> 0
sum1(s(x)) -> s(+(sum1(x), +(x, x)))
Termination of R to be shown.
R
↳Dependency Pair Analysis
R contains the following Dependency Pairs:
SUM(s(x)) -> SUM(x)
SUM1(s(x)) -> SUM1(x)
Furthermore, R contains two SCCs.
R
↳DPs
→DP Problem 1
↳Argument Filtering and Ordering
→DP Problem 2
↳AFS
Dependency Pair:
SUM(s(x)) -> SUM(x)
Rules:
sum(0) -> 0
sum(s(x)) -> +(sum(x), s(x))
sum1(0) -> 0
sum1(s(x)) -> s(+(sum1(x), +(x, x)))
The following dependency pair can be strictly oriented:
SUM(s(x)) -> SUM(x)
There are no usable rules using the Ce-refinement that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:
| POL(SUM(x1)) | = x1 |
| POL(s(x1)) | = 1 + x1 |
resulting in one new DP problem.
Used Argument Filtering System: SUM(x1) -> SUM(x1)
s(x1) -> s(x1)
R
↳DPs
→DP Problem 1
↳AFS
→DP Problem 3
↳Dependency Graph
→DP Problem 2
↳AFS
Dependency Pair:
Rules:
sum(0) -> 0
sum(s(x)) -> +(sum(x), s(x))
sum1(0) -> 0
sum1(s(x)) -> s(+(sum1(x), +(x, 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:
SUM1(s(x)) -> SUM1(x)
Rules:
sum(0) -> 0
sum(s(x)) -> +(sum(x), s(x))
sum1(0) -> 0
sum1(s(x)) -> s(+(sum1(x), +(x, x)))
The following dependency pair can be strictly oriented:
SUM1(s(x)) -> SUM1(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(SUM1(x1)) | = x1 |
resulting in one new DP problem.
Used Argument Filtering System: SUM1(x1) -> SUM1(x1)
s(x1) -> s(x1)
R
↳DPs
→DP Problem 1
↳AFS
→DP Problem 2
↳AFS
→DP Problem 4
↳Dependency Graph
Dependency Pair:
Rules:
sum(0) -> 0
sum(s(x)) -> +(sum(x), s(x))
sum1(0) -> 0
sum1(s(x)) -> s(+(sum1(x), +(x, x)))
Using the Dependency Graph resulted in no new DP problems.
Termination of R successfully shown.
Duration:
0:00 minutes