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

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Argument Filtering and Ordering


Dependency Pairs:

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


Rules:


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


Strategy:

innermost




The following dependency pairs can be strictly oriented:

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


There are no usable rules for innermost that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(*'(x1, x2))=  x1 + x2  
  POL(oplus(x1, x2))=  1 + x1 + x2  
  POL(+(x1, x2))=  x1 + x2  

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


   R
     ↳DPs
       →DP Problem 1
         ↳AFS
           →DP Problem 2
             ↳Argument Filtering and Ordering


Dependency Pairs:

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


Rules:


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


Strategy:

innermost




The following dependency pairs can be strictly oriented:

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


There are no usable rules for innermost 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
             ↳AFS
             ...
               →DP Problem 3
                 ↳Dependency Graph


Dependency Pair:


Rules:


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


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

Innermost Termination of R successfully shown.
Duration:
0:01 minutes