Term Rewriting System R:
[x, y, z]
+(*(x, y), *(a, y)) -> *(+(x, a), y)
*(*(x, y), z) -> *(x, *(y, z))

Termination of R to be shown.



   R
Dependency Pair Analysis



R contains the following Dependency Pairs:

+'(*(x, y), *(a, y)) -> *'(+(x, a), y)
+'(*(x, y), *(a, y)) -> +'(x, a)
*'(*(x, y), z) -> *'(x, *(y, z))
*'(*(x, y), z) -> *'(y, z)

Furthermore, R contains one SCC.


   R
DPs
       →DP Problem 1
Argument Filtering and Ordering


Dependency Pairs:

*'(*(x, y), z) -> *'(y, z)
*'(*(x, y), z) -> *'(x, *(y, z))


Rules:


+(*(x, y), *(a, y)) -> *(+(x, a), y)
*(*(x, y), z) -> *(x, *(y, z))





The following dependency pairs can be strictly oriented:

*'(*(x, y), z) -> *'(y, z)
*'(*(x, y), z) -> *'(x, *(y, z))


The following rules can be oriented:

*(*(x, y), z) -> *(x, *(y, z))
+(*(x, y), *(a, y)) -> *(+(x, a), y)


Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
*' > *
+ > *

resulting in one new DP problem.
Used Argument Filtering System:
*'(x1, x2) -> *'(x1, x2)
*(x1, x2) -> *(x1, x2)
+(x1, x2) -> +(x1, x2)


   R
DPs
       →DP Problem 1
AFS
           →DP Problem 2
Dependency Graph


Dependency Pair:


Rules:


+(*(x, y), *(a, y)) -> *(+(x, a), y)
*(*(x, y), z) -> *(x, *(y, z))





Using the Dependency Graph resulted in no new DP problems.

Termination of R successfully shown.
Duration:
0:00 minutes