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

Furthermore, R contains one SCC.


   R
DPs
       →DP Problem 1
Narrowing Transformation


Dependency Pairs:

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


Rules:


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


Strategy:

innermost




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

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

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

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
Nar
           →DP Problem 2
Narrowing Transformation


Dependency Pairs:

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


Rules:


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


Strategy:

innermost




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

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

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

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
Nar
           →DP Problem 2
Nar
             ...
               →DP Problem 3
Forward Instantiation Transformation


Dependency Pairs:

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


Rules:


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

+'(x', +(y0, *(y'', u''))) -> +'(+(x', y0), *(y'', u''))
+'(x', +(y', *(y'''', u'''))) -> +'(+(x', y'), *(y'''', u'''))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
Nar
           →DP Problem 2
Nar
             ...
               →DP Problem 4
Forward Instantiation Transformation


Dependency Pairs:

+'(x', +(y', *(y'''', u'''))) -> +'(+(x', y'), *(y'''', u'''))
+'(x', +(y0, *(y'', u''))) -> +'(+(x', y0), *(y'', u''))
*'(x'', +(+(y'', z''), z)) -> +'(+(*(x'', y''), *(x'', z'')), *(x'', z))
*'(x, +(y, z)) -> *'(x, y)
+'(+(x, *(y, z)), *(y, u)) -> *'(y, +(z, u))
+'(x, +(y, z)) -> +'(x, y)
+'(+(x, *(y'', z'')), *(y'', u')) -> +'(x, +(*(y'', z''), *(y'', u')))


Rules:


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

+'(x'', +(+(y'', z''), z)) -> +'(x'', +(y'', z''))
+'(+(x'', *(y''', z'')), +(*(y'''', u''), z)) -> +'(+(x'', *(y''', z'')), *(y'''', u''))
+'(+(x'', *(y''''', z'''')), +(*(y'''''', u'''), z)) -> +'(+(x'', *(y''''', z'''')), *(y'''''', u'''))
+'(x', +(+(y0'', *(y'''', u'''')), z)) -> +'(x', +(y0'', *(y'''', u'''')))
+'(x', +(+(y''', *(y'''''', u''''')), z)) -> +'(x', +(y''', *(y'''''', u''''')))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
Nar
           →DP Problem 2
Nar
             ...
               →DP Problem 5
Forward Instantiation Transformation


Dependency Pairs:

+'(x', +(+(y''', *(y'''''', u''''')), z)) -> +'(x', +(y''', *(y'''''', u''''')))
+'(x', +(+(y0'', *(y'''', u'''')), z)) -> +'(x', +(y0'', *(y'''', u'''')))
+'(+(x'', *(y''''', z'''')), +(*(y'''''', u'''), z)) -> +'(+(x'', *(y''''', z'''')), *(y'''''', u'''))
+'(+(x'', *(y''', z'')), +(*(y'''', u''), z)) -> +'(+(x'', *(y''', z'')), *(y'''', u''))
+'(x'', +(+(y'', z''), z)) -> +'(x'', +(y'', z''))
+'(x', +(y0, *(y'', u''))) -> +'(+(x', y0), *(y'', u''))
+'(+(x, *(y'', z'')), *(y'', u')) -> +'(x, +(*(y'', z''), *(y'', u')))
*'(x'', +(+(y'', z''), z)) -> +'(+(*(x'', y''), *(x'', z'')), *(x'', z))
*'(x, +(y, z)) -> *'(x, y)
+'(+(x, *(y, z)), *(y, u)) -> *'(y, +(z, u))
+'(x', +(y', *(y'''', u'''))) -> +'(+(x', y'), *(y'''', u'''))


Rules:


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

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

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
Nar
           →DP Problem 2
Nar
             ...
               →DP Problem 6
Narrowing Transformation


Dependency Pairs:

*'(x', +(+(+(y'''', z''''), z''), z)) -> *'(x', +(+(y'''', z''''), z''))
*'(x'', +(+(y'', z''), z)) -> *'(x'', +(y'', z''))
+'(x', +(+(y0'', *(y'''', u'''')), z)) -> +'(x', +(y0'', *(y'''', u'''')))
+'(+(x'', *(y''''', z'''')), +(*(y'''''', u'''), z)) -> +'(+(x'', *(y''''', z'''')), *(y'''''', u'''))
+'(+(x'', *(y''', z'')), +(*(y'''', u''), z)) -> +'(+(x'', *(y''', z'')), *(y'''', u''))
+'(x'', +(+(y'', z''), z)) -> +'(x'', +(y'', z''))
+'(x', +(y', *(y'''', u'''))) -> +'(+(x', y'), *(y'''', u'''))
+'(+(x, *(y'', z'')), *(y'', u')) -> +'(x, +(*(y'', z''), *(y'', u')))
*'(x'', +(+(y'', z''), z)) -> +'(+(*(x'', y''), *(x'', z'')), *(x'', z))
+'(+(x, *(y, z)), *(y, u)) -> *'(y, +(z, u))
+'(x', +(y0, *(y'', u''))) -> +'(+(x', y0), *(y'', u''))
+'(x', +(+(y''', *(y'''''', u''''')), z)) -> +'(x', +(y''', *(y'''''', u''''')))


Rules:


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


Strategy:

innermost




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

+'(+(x, *(y, z)), *(y, u)) -> *'(y, +(z, u))
no new Dependency Pairs are created.
The transformation is resulting in two new DP problems:



   R
DPs
       →DP Problem 1
Nar
           →DP Problem 2
Nar
             ...
               →DP Problem 7
Polynomial Ordering


Dependency Pairs:

+'(x', +(+(y''', *(y'''''', u''''')), z)) -> +'(x', +(y''', *(y'''''', u''''')))
+'(x', +(+(y0'', *(y'''', u'''')), z)) -> +'(x', +(y0'', *(y'''', u'''')))
+'(+(x'', *(y''''', z'''')), +(*(y'''''', u'''), z)) -> +'(+(x'', *(y''''', z'''')), *(y'''''', u'''))
+'(+(x'', *(y''', z'')), +(*(y'''', u''), z)) -> +'(+(x'', *(y''', z'')), *(y'''', u''))
+'(x'', +(+(y'', z''), z)) -> +'(x'', +(y'', z''))
+'(x', +(y', *(y'''', u'''))) -> +'(+(x', y'), *(y'''', u'''))
+'(x', +(y0, *(y'', u''))) -> +'(+(x', y0), *(y'', u''))
+'(+(x, *(y'', z'')), *(y'', u')) -> +'(x, +(*(y'', z''), *(y'', u')))


Rules:


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


Strategy:

innermost




The following dependency pairs can be strictly oriented:

+'(x', +(+(y''', *(y'''''', u''''')), z)) -> +'(x', +(y''', *(y'''''', u''''')))
+'(x', +(+(y0'', *(y'''', u'''')), z)) -> +'(x', +(y0'', *(y'''', u'''')))
+'(+(x'', *(y''''', z'''')), +(*(y'''''', u'''), z)) -> +'(+(x'', *(y''''', z'''')), *(y'''''', u'''))
+'(+(x'', *(y''', z'')), +(*(y'''', u''), z)) -> +'(+(x'', *(y''', z'')), *(y'''', u''))
+'(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), *(x, z))
+(x, +(y, z)) -> +(+(x, y), z)
+(+(x, *(y, z)), *(y, u)) -> +(x, *(y, +(z, u)))


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

resulting in one new DP problem.



   R
DPs
       →DP Problem 1
Nar
           →DP Problem 2
Nar
             ...
               →DP Problem 9
Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

+'(x', +(y', *(y'''', u'''))) -> +'(+(x', y'), *(y'''', u'''))
+'(x', +(y0, *(y'', u''))) -> +'(+(x', y0), *(y'', u''))
+'(+(x, *(y'', z'')), *(y'', u')) -> +'(x, +(*(y'', z''), *(y'', u')))


Rules:


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


Strategy:

innermost




   R
DPs
       →DP Problem 1
Nar
           →DP Problem 2
Nar
             ...
               →DP Problem 8
Polynomial Ordering


Dependency Pairs:

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


Rules:


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


Strategy:

innermost




The following dependency pairs can be strictly oriented:

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


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

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


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

resulting in one new DP problem.



   R
DPs
       →DP Problem 1
Nar
           →DP Problem 2
Nar
             ...
               →DP Problem 10
Dependency Graph


Dependency Pair:


Rules:


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


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

Innermost Termination of R could not be shown.
Duration:
0:00 minutes