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

Innermost 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
         ↳Forward Instantiation Transformation


Dependency Pair:

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


Rules:


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


Strategy:

innermost




On this DP problem, a Forward Instantiation SCC transformation can be performed.
As a result of transforming the rule

*'(*(x, y), z) -> *'(y, z)
one new Dependency Pair is created:

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

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
           →DP Problem 2
             ↳Polynomial Ordering


Dependency Pair:

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


Rules:


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


Strategy:

innermost




The following dependency pair can be strictly oriented:

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


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

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


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. 



   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
           →DP Problem 2
             ↳Polo
             ...
               →DP Problem 3
                 ↳Dependency Graph


Dependency Pair:


Rules:


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


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

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