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

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

Furthermore, R contains one SCC.

R
DPs
→DP Problem 1
Polynomial Ordering

Dependency Pairs:

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

Rules:

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

The following dependency pairs can be strictly oriented:

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

Additionally, the following usable rules w.r.t. to the implicit AFS can be oriented:

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

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(*'(x1, x2)) =  x2 POL(*(x1, x2)) =  x2 POL(+(x1, x2)) =  1 + x1 + x2 POL(+'(x1, x2)) =  x1 + x2

resulting in one new DP problem.

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

Dependency Pairs:

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

Rules:

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

Using the Dependency Graph the DP problem was split into 1 DP problems.

R
DPs
→DP Problem 1
Polo
→DP Problem 2
DGraph
...
→DP Problem 3
Remaining Obligation(s)

The following remains to be proven:
Dependency Pairs:

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

Rules:

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

Termination of R could not be shown.
Duration:
0:00 minutes