Term Rewriting System R:
[x, y, z]
*(i(x), x) -> 1
*(1, y) -> y
*(x, 0) -> 0
*(*(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), 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:


*(i(x), x) -> 1
*(1, y) -> y
*(x, 0) -> 0
*(*(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
             ↳Argument Filtering and Ordering


Dependency Pair:

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


Rules:


*(i(x), x) -> 1
*(1, y) -> y
*(x, 0) -> 0
*(*(x, y), z) -> *(x, *(y, z))


Strategy:

innermost




The following dependency pair can be strictly oriented:

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


The following usable rules for innermost can be oriented:

*(i(x), x) -> 1
*(1, y) -> y
*(x, 0) -> 0
*(*(x, y), z) -> *(x, *(y, z))


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

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


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


Dependency Pair:


Rules:


*(i(x), x) -> 1
*(1, y) -> y
*(x, 0) -> 0
*(*(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