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))

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
         ↳Polynomial Ordering


Dependency Pairs:

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





The following dependency pairs can be strictly oriented:

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


There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(*'(x1, x2))=  x1  
  POL(otimes(x1, x2))=  0  
  POL(*(x1, x2))=  0  
  POL(oplus(x1, x2))=  0  
  POL(+(x1, x2))=  1 + x1 + x2  

resulting in one new DP problem. 



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


Dependency Pairs:

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





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 w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(*'(x1, x2))=  x2  
  POL(otimes(x1, x2))=  0  
  POL(*(x1, x2))=  x2  
  POL(oplus(x1, x2))=  1 + x1 + x2  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
           →DP Problem 2
             ↳Polo
             ...
               →DP Problem 3
                 ↳Instantiation Transformation


Dependency Pair:

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





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

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

*'(otimes(x'', y''), *(y0, z'')) -> *'(otimes(otimes(x'', y''), y0), z'')

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
           →DP Problem 2
             ↳Polo
             ...
               →DP Problem 4
                 ↳Instantiation Transformation


Dependency Pair:

*'(otimes(x'', y''), *(y0, z'')) -> *'(otimes(otimes(x'', y''), y0), 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))





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

*'(otimes(x'', y''), *(y0, z'')) -> *'(otimes(otimes(x'', y''), y0), z'')
one new Dependency Pair is created:

*'(otimes(otimes(x'''', y''''), y''0), *(y0'', z'''')) -> *'(otimes(otimes(otimes(x'''', y''''), y''0), y0''), z'''')

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
           →DP Problem 2
             ↳Polo
             ...
               →DP Problem 5
                 ↳Polynomial Ordering


Dependency Pair:

*'(otimes(otimes(x'''', y''''), y''0), *(y0'', z'''')) -> *'(otimes(otimes(otimes(x'''', y''''), y''0), y0''), 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))





The following dependency pair can be strictly oriented:

*'(otimes(otimes(x'''', y''''), y''0), *(y0'', z'''')) -> *'(otimes(otimes(otimes(x'''', y''''), y''0), y0''), z'''')


There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(*'(x1, x2))=  x2  
  POL(otimes(x1, x2))=  0  
  POL(*(x1, x2))=  1 + x2  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
           →DP Problem 2
             ↳Polo
             ...
               →DP Problem 6
                 ↳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))





Using the Dependency Graph resulted in no new DP problems.

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