Term Rewriting System R:
[x, y]
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))
+(0, s(y)) -> s(y)
s(+(0, y)) -> s(y)
Termination of R to be shown.
R
↳Dependency Pair Analysis
R contains the following Dependency Pairs:
+'(x, s(y)) -> S(+(x, y))
+'(x, s(y)) -> +'(x, y)
S(+(0, y)) -> S(y)
Furthermore, R contains two SCCs.
R
↳DPs
→DP Problem 1
↳Argument Filtering and Ordering
→DP Problem 2
↳AFS
Dependency Pair:
S(+(0, y)) -> S(y)
Rules:
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))
+(0, s(y)) -> s(y)
s(+(0, y)) -> s(y)
The following dependency pair can be strictly oriented:
S(+(0, y)) -> S(y)
There are no usable rules w.r.t. to the AFS that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:
| POL(0) | = 0 |
| POL(S(x1)) | = x1 |
| POL(+(x1, x2)) | = 1 + x1 + x2 |
resulting in one new DP problem.
Used Argument Filtering System: S(x1) -> S(x1)
+(x1, x2) -> +(x1, x2)
R
↳DPs
→DP Problem 1
↳AFS
→DP Problem 3
↳Dependency Graph
→DP Problem 2
↳AFS
Dependency Pair:
Rules:
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))
+(0, s(y)) -> s(y)
s(+(0, y)) -> s(y)
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:
+'(x, s(y)) -> +'(x, y)
Rules:
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))
+(0, s(y)) -> s(y)
s(+(0, y)) -> s(y)
The following dependency pair can be strictly oriented:
+'(x, s(y)) -> +'(x, y)
There are no usable rules w.r.t. to the AFS that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:
| POL(s(x1)) | = 1 + x1 |
| POL(+'(x1, x2)) | = x1 + x2 |
resulting in one new DP problem.
Used Argument Filtering System: +'(x1, x2) -> +'(x1, x2)
s(x1) -> s(x1)
R
↳DPs
→DP Problem 1
↳AFS
→DP Problem 2
↳AFS
→DP Problem 4
↳Dependency Graph
Dependency Pair:
Rules:
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))
+(0, s(y)) -> s(y)
s(+(0, y)) -> s(y)
Using the Dependency Graph resulted in no new DP problems.
Termination of R successfully shown.
Duration:
0:00 minutes