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