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

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Forward Instantiation Transformation


Dependency Pair:

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


Rules:


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

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


Rules:


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


Strategy:

innermost




The following dependency pair can be strictly oriented:

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


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

-(+(x, y), y) -> x


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(-(x1, x2))=  1 + x1 + x2  
  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)
+(x1, x2) -> +(x1, x2)


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


Dependency Pair:


Rules:


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


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

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