Term Rewriting System R:
[x, y, z]
*(x, +(y, z)) -> +(*(x, y), *(x, z))
*(+(x, y), z) -> +(*(x, z), *(y, z))
*(x, 1) -> x
*(1, y) -> y

Termination of R to be shown.

R
Dependency Pair Analysis

R contains the following Dependency Pairs:

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

Rules:

*(x, +(y, z)) -> +(*(x, y), *(x, z))
*(+(x, y), z) -> +(*(x, z), *(y, z))
*(x, 1) -> x
*(1, y) -> y

The following dependency pairs can be strictly oriented:

*'(+(x, y), z) -> *'(y, z)
*'(+(x, y), z) -> *'(x, z)
*'(x, +(y, z)) -> *'(x, z)
*'(x, +(y, z)) -> *'(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(*'(x1, x2)) =  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
Dependency Graph

Dependency Pair:

Rules:

*(x, +(y, z)) -> +(*(x, y), *(x, z))
*(+(x, y), z) -> +(*(x, z), *(y, z))
*(x, 1) -> x
*(1, y) -> y

Using the Dependency Graph resulted in no new DP problems.

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