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

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

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Forward Instantiation Transformation


Dependency Pairs:

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


Rules:


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


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)
two new Dependency Pairs are created:

+'(x'', +(+(y'', z''), z)) -> +'(x'', +(y'', z''))
+'(*(x'0, y''), +(+(*(x''', z''), u''), z)) -> +'(*(x'0, y''), +(*(x''', z''), u''))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
           →DP Problem 2
             ↳Forward Instantiation Transformation


Dependency Pairs:

+'(*(x'0, y''), +(+(*(x''', z''), u''), z)) -> +'(*(x'0, y''), +(*(x''', z''), u''))
+'(x'', +(+(y'', z''), z)) -> +'(x'', +(y'', z''))
+'(*(x, y), +(*(x, z), u)) -> +'(y, z)


Rules:


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


Strategy:

innermost




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

+'(*(x, y), +(*(x, z), u)) -> +'(y, z)
three new Dependency Pairs are created:

+'(*(x, *(x'0, y'')), +(*(x, +(*(x''', z''), u'')), u)) -> +'(*(x'0, y''), +(*(x''', z''), u''))
+'(*(x, y'), +(*(x, +(+(y'''', z''''), z'')), u)) -> +'(y', +(+(y'''', z''''), z''))
+'(*(x, *(x'0'', y'''')), +(*(x, +(+(*(x''''', z''''), u''''), z'')), u)) -> +'(*(x'0'', y''''), +(+(*(x''''', z''''), u''''), z''))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
           →DP Problem 2
             ↳FwdInst
             ...
               →DP Problem 3
                 ↳Polynomial Ordering


Dependency Pairs:

+'(*(x, *(x'0'', y'''')), +(*(x, +(+(*(x''''', z''''), u''''), z'')), u)) -> +'(*(x'0'', y''''), +(+(*(x''''', z''''), u''''), z''))
+'(x'', +(+(y'', z''), z)) -> +'(x'', +(y'', z''))
+'(*(x, y'), +(*(x, +(+(y'''', z''''), z'')), u)) -> +'(y', +(+(y'''', z''''), z''))
+'(*(x, *(x'0, y'')), +(*(x, +(*(x''', z''), u'')), u)) -> +'(*(x'0, y''), +(*(x''', z''), u''))
+'(*(x'0, y''), +(+(*(x''', z''), u''), z)) -> +'(*(x'0, y''), +(*(x''', z''), u''))


Rules:


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


Strategy:

innermost




The following dependency pairs can be strictly oriented:

+'(*(x, *(x'0'', y'''')), +(*(x, +(+(*(x''''', z''''), u''''), z'')), u)) -> +'(*(x'0'', y''''), +(+(*(x''''', z''''), u''''), z''))
+'(*(x, y'), +(*(x, +(+(y'''', z''''), z'')), u)) -> +'(y', +(+(y'''', z''''), z''))
+'(*(x, *(x'0, y'')), +(*(x, +(*(x''', z''), u'')), u)) -> +'(*(x'0, y''), +(*(x''', z''), u''))


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

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

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
           →DP Problem 2
             ↳FwdInst
             ...
               →DP Problem 4
                 ↳Dependency Graph


Dependency Pairs:

+'(x'', +(+(y'', z''), z)) -> +'(x'', +(y'', z''))
+'(*(x'0, y''), +(+(*(x''', z''), u''), z)) -> +'(*(x'0, y''), +(*(x''', z''), u''))


Rules:


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


Strategy:

innermost




Using the Dependency Graph the DP problem was split into 1 DP problems.


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


Dependency Pair:

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


Rules:


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


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


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(*(x1, x2))=  0  
  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
             ↳FwdInst
             ...
               →DP Problem 6
                 ↳Dependency Graph


Dependency Pair:


Rules:


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


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

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