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

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)) -> *'(x, y)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Forward Instantiation Transformation


Dependency Pair:

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


Rules:


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


Strategy:

innermost




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

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

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


Rules:


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


Strategy:

innermost




The following dependency pair can be strictly oriented:

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


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

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


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


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

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