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

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

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


Rules:


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


Strategy:

innermost




The following dependency pairs can be strictly oriented:

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


There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
trivial

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


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


Dependency Pair:


Rules:


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


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

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