Term Rewriting System R:
[x, y, z]
f(f(a, a), x) -> f(a, f(b, f(a, x)))
f(x, f(y, z)) -> f(f(x, y), z)

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

F(f(a, a), x) -> F(a, f(b, f(a, x)))
F(f(a, a), x) -> F(b, f(a, x))
F(f(a, a), x) -> F(a, x)
F(x, f(y, z)) -> F(f(x, y), z)
F(x, f(y, z)) -> F(x, y)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Narrowing Transformation


Dependency Pairs:

F(f(a, a), x) -> F(a, x)
F(x, f(y, z)) -> F(x, y)
F(f(a, a), x) -> F(b, f(a, x))
F(x, f(y, z)) -> F(f(x, y), z)
F(f(a, a), x) -> F(a, f(b, f(a, x)))


Rules:


f(f(a, a), x) -> f(a, f(b, f(a, x)))
f(x, f(y, z)) -> f(f(x, y), z)


Strategy:

innermost




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

F(f(a, a), x) -> F(a, f(b, f(a, x)))
two new Dependency Pairs are created:

F(f(a, a), x'') -> F(a, f(f(b, a), x''))
F(f(a, a), f(y', z')) -> F(a, f(b, f(f(a, y'), z')))

The transformation is resulting in one new DP problem: 



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


Dependency Pairs:

F(f(a, a), f(y', z')) -> F(a, f(b, f(f(a, y'), z')))
F(f(a, a), x'') -> F(a, f(f(b, a), x''))
F(x, f(y, z)) -> F(x, y)
F(f(a, a), x) -> F(b, f(a, x))
F(x, f(y, z)) -> F(f(x, y), z)
F(f(a, a), x) -> F(a, x)


Rules:


f(f(a, a), x) -> f(a, f(b, f(a, x)))
f(x, f(y, z)) -> f(f(x, y), z)


Strategy:

innermost




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

F(f(a, a), f(y', z')) -> F(a, f(b, f(f(a, y'), z')))
three new Dependency Pairs are created:

F(f(a, a), f(y'', z'')) -> F(a, f(f(b, f(a, y'')), z''))
F(f(a, a), f(a, z'')) -> F(a, f(b, f(a, f(b, f(a, z'')))))
F(f(a, a), f(f(y'', z''), z')) -> F(a, f(b, f(f(f(a, y''), z''), 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:

F(f(a, a), f(f(y'', z''), z')) -> F(a, f(b, f(f(f(a, y''), z''), z')))
F(f(a, a), f(a, z'')) -> F(a, f(b, f(a, f(b, f(a, z'')))))
F(f(a, a), f(y'', z'')) -> F(a, f(f(b, f(a, y'')), z''))
F(f(a, a), x) -> F(a, x)
F(x, f(y, z)) -> F(x, y)
F(f(a, a), x) -> F(b, f(a, x))
F(x, f(y, z)) -> F(f(x, y), z)
F(f(a, a), x'') -> F(a, f(f(b, a), x''))


Rules:


f(f(a, a), x) -> f(a, f(b, f(a, x)))
f(x, f(y, z)) -> f(f(x, y), z)


Strategy:

innermost




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

F(f(a, a), x) -> F(a, x)
one new Dependency Pair is created:

F(f(a, a), f(y'', z'')) -> F(a, f(y'', z''))

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:

F(f(a, a), f(y'', z'')) -> F(a, f(y'', z''))
F(f(a, a), f(a, z'')) -> F(a, f(b, f(a, f(b, f(a, z'')))))
F(f(a, a), f(y'', z'')) -> F(a, f(f(b, f(a, y'')), z''))
F(f(a, a), x'') -> F(a, f(f(b, a), x''))
F(x, f(y, z)) -> F(x, y)
F(f(a, a), x) -> F(b, f(a, x))
F(x, f(y, z)) -> F(f(x, y), z)
F(f(a, a), f(f(y'', z''), z')) -> F(a, f(b, f(f(f(a, y''), z''), z')))


Rules:


f(f(a, a), x) -> f(a, f(b, f(a, x)))
f(x, f(y, z)) -> f(f(x, y), z)


Strategy:

innermost




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

F(x, f(y, z)) -> F(x, y)
five new Dependency Pairs are created:

F(x'', f(f(y'', z''), z)) -> F(x'', f(y'', z''))
F(f(a, a), f(y', z)) -> F(f(a, a), y')
F(f(a, a), f(f(y'''', z''''), z)) -> F(f(a, a), f(y'''', z''''))
F(f(a, a), f(f(a, z''''), z)) -> F(f(a, a), f(a, z''''))
F(f(a, a), f(f(f(y'''', z''''), z'0'), z)) -> F(f(a, a), f(f(y'''', z''''), z'0'))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 5
                 ↳Argument Filtering and Ordering


Dependency Pairs:

F(f(a, a), f(f(f(y'''', z''''), z'0'), z)) -> F(f(a, a), f(f(y'''', z''''), z'0'))
F(f(a, a), f(f(a, z''''), z)) -> F(f(a, a), f(a, z''''))
F(f(a, a), f(f(y'''', z''''), z)) -> F(f(a, a), f(y'''', z''''))
F(f(a, a), f(y', z)) -> F(f(a, a), y')
F(f(a, a), f(f(y'', z''), z')) -> F(a, f(b, f(f(f(a, y''), z''), z')))
F(f(a, a), f(a, z'')) -> F(a, f(b, f(a, f(b, f(a, z'')))))
F(f(a, a), f(y'', z'')) -> F(a, f(f(b, f(a, y'')), z''))
F(f(a, a), x'') -> F(a, f(f(b, a), x''))
F(x'', f(f(y'', z''), z)) -> F(x'', f(y'', z''))
F(f(a, a), x) -> F(b, f(a, x))
F(x, f(y, z)) -> F(f(x, y), z)
F(f(a, a), f(y'', z'')) -> F(a, f(y'', z''))


Rules:


f(f(a, a), x) -> f(a, f(b, f(a, x)))
f(x, f(y, z)) -> f(f(x, y), z)


Strategy:

innermost




The following dependency pairs can be strictly oriented:

F(f(a, a), x) -> F(b, f(a, x))
F(f(a, a), f(y'', z'')) -> F(a, f(y'', z''))


The following usable rules for innermost can be oriented:

f(f(a, a), x) -> f(a, f(b, f(a, x)))
f(x, f(y, z)) -> f(f(x, y), z)


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(b)=  0  
  POL(a)=  1  
  POL(F(x1, x2))=  1 + x1 + x2  
  POL(f(x1, x2))=  x1 + x2  

resulting in one new DP problem.
Used Argument Filtering System:
F(x1, x2) -> F(x1, x2)
f(x1, x2) -> f(x1, x2)


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


Dependency Pairs:

F(f(a, a), f(f(f(y'''', z''''), z'0'), z)) -> F(f(a, a), f(f(y'''', z''''), z'0'))
F(f(a, a), f(f(a, z''''), z)) -> F(f(a, a), f(a, z''''))
F(f(a, a), f(f(y'''', z''''), z)) -> F(f(a, a), f(y'''', z''''))
F(f(a, a), f(y', z)) -> F(f(a, a), y')
F(f(a, a), f(f(y'', z''), z')) -> F(a, f(b, f(f(f(a, y''), z''), z')))
F(f(a, a), f(a, z'')) -> F(a, f(b, f(a, f(b, f(a, z'')))))
F(f(a, a), f(y'', z'')) -> F(a, f(f(b, f(a, y'')), z''))
F(f(a, a), x'') -> F(a, f(f(b, a), x''))
F(x'', f(f(y'', z''), z)) -> F(x'', f(y'', z''))
F(x, f(y, z)) -> F(f(x, y), z)


Rules:


f(f(a, a), x) -> f(a, f(b, f(a, x)))
f(x, f(y, z)) -> f(f(x, y), z)


Strategy:

innermost




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

F(f(a, a), f(y'', z'')) -> F(a, f(f(b, f(a, y'')), z''))
two new Dependency Pairs are created:

F(f(a, a), f(y''', z'')) -> F(a, f(f(f(b, a), y'''), z''))
F(f(a, a), f(f(y', z'), z'')) -> F(a, f(f(b, f(f(a, y'), z')), z''))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 7
                 ↳Narrowing Transformation


Dependency Pairs:

F(f(a, a), f(f(y', z'), z'')) -> F(a, f(f(b, f(f(a, y'), z')), z''))
F(f(a, a), f(y''', z'')) -> F(a, f(f(f(b, a), y'''), z''))
F(f(a, a), f(f(a, z''''), z)) -> F(f(a, a), f(a, z''''))
F(f(a, a), f(f(y'''', z''''), z)) -> F(f(a, a), f(y'''', z''''))
F(f(a, a), f(y', z)) -> F(f(a, a), y')
F(f(a, a), f(f(y'', z''), z')) -> F(a, f(b, f(f(f(a, y''), z''), z')))
F(f(a, a), f(a, z'')) -> F(a, f(b, f(a, f(b, f(a, z'')))))
F(x'', f(f(y'', z''), z)) -> F(x'', f(y'', z''))
F(f(a, a), x'') -> F(a, f(f(b, a), x''))
F(x, f(y, z)) -> F(f(x, y), z)
F(f(a, a), f(f(f(y'''', z''''), z'0'), z)) -> F(f(a, a), f(f(y'''', z''''), z'0'))


Rules:


f(f(a, a), x) -> f(a, f(b, f(a, x)))
f(x, f(y, z)) -> f(f(x, y), z)


Strategy:

innermost




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

F(f(a, a), f(a, z'')) -> F(a, f(b, f(a, f(b, f(a, z'')))))
three new Dependency Pairs are created:

F(f(a, a), f(a, z''')) -> F(a, f(f(b, a), f(b, f(a, z'''))))
F(f(a, a), f(a, z''')) -> F(a, f(b, f(f(a, b), f(a, z'''))))
F(f(a, a), f(a, z''')) -> F(a, f(b, f(a, f(f(b, a), z'''))))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 8
                 ↳Narrowing Transformation


Dependency Pairs:

F(f(a, a), f(f(f(y'''', z''''), z'0'), z)) -> F(f(a, a), f(f(y'''', z''''), z'0'))
F(f(a, a), f(a, z''')) -> F(a, f(b, f(a, f(f(b, a), z'''))))
F(f(a, a), f(a, z''')) -> F(a, f(b, f(f(a, b), f(a, z'''))))
F(f(a, a), f(a, z''')) -> F(a, f(f(b, a), f(b, f(a, z'''))))
F(f(a, a), f(y''', z'')) -> F(a, f(f(f(b, a), y'''), z''))
F(f(a, a), f(f(a, z''''), z)) -> F(f(a, a), f(a, z''''))
F(f(a, a), f(f(y'''', z''''), z)) -> F(f(a, a), f(y'''', z''''))
F(f(a, a), f(y', z)) -> F(f(a, a), y')
F(f(a, a), f(f(y'', z''), z')) -> F(a, f(b, f(f(f(a, y''), z''), z')))
F(x'', f(f(y'', z''), z)) -> F(x'', f(y'', z''))
F(f(a, a), x'') -> F(a, f(f(b, a), x''))
F(x, f(y, z)) -> F(f(x, y), z)
F(f(a, a), f(f(y', z'), z'')) -> F(a, f(f(b, f(f(a, y'), z')), z''))


Rules:


f(f(a, a), x) -> f(a, f(b, f(a, x)))
f(x, f(y, z)) -> f(f(x, y), z)


Strategy:

innermost




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

F(f(a, a), f(f(y'', z''), z')) -> F(a, f(b, f(f(f(a, y''), z''), z')))
three new Dependency Pairs are created:

F(f(a, a), f(f(y''', z'''), z'0)) -> F(a, f(f(b, f(f(a, y'''), z''')), z'0))
F(f(a, a), f(f(a, z'''), z')) -> F(a, f(b, f(f(a, f(b, f(a, z'''))), z')))
F(f(a, a), f(f(f(y', z0), z''), z')) -> F(a, f(b, f(f(f(f(a, y'), z0), z''), z')))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 9
                 ↳Narrowing Transformation


Dependency Pairs:

F(f(a, a), f(f(f(y', z0), z''), z')) -> F(a, f(b, f(f(f(f(a, y'), z0), z''), z')))
F(f(a, a), f(f(a, z'''), z')) -> F(a, f(b, f(f(a, f(b, f(a, z'''))), z')))
F(f(a, a), f(f(y''', z'''), z'0)) -> F(a, f(f(b, f(f(a, y'''), z''')), z'0))
F(f(a, a), f(f(y', z'), z'')) -> F(a, f(f(b, f(f(a, y'), z')), z''))
F(f(a, a), f(a, z''')) -> F(a, f(b, f(a, f(f(b, a), z'''))))
F(f(a, a), f(a, z''')) -> F(a, f(b, f(f(a, b), f(a, z'''))))
F(f(a, a), f(a, z''')) -> F(a, f(f(b, a), f(b, f(a, z'''))))
F(f(a, a), f(y''', z'')) -> F(a, f(f(f(b, a), y'''), z''))
F(f(a, a), f(f(a, z''''), z)) -> F(f(a, a), f(a, z''''))
F(f(a, a), f(f(y'''', z''''), z)) -> F(f(a, a), f(y'''', z''''))
F(f(a, a), f(y', z)) -> F(f(a, a), y')
F(x'', f(f(y'', z''), z)) -> F(x'', f(y'', z''))
F(f(a, a), x'') -> F(a, f(f(b, a), x''))
F(x, f(y, z)) -> F(f(x, y), z)
F(f(a, a), f(f(f(y'''', z''''), z'0'), z)) -> F(f(a, a), f(f(y'''', z''''), z'0'))


Rules:


f(f(a, a), x) -> f(a, f(b, f(a, x)))
f(x, f(y, z)) -> f(f(x, y), z)


Strategy:

innermost




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

F(f(a, a), f(f(y', z'), z'')) -> F(a, f(f(b, f(f(a, y'), z')), z''))
three new Dependency Pairs are created:

F(f(a, a), f(f(y'', z'''), z'')) -> F(a, f(f(f(b, f(a, y'')), z'''), z''))
F(f(a, a), f(f(a, z'''), z'')) -> F(a, f(f(b, f(a, f(b, f(a, z''')))), z''))
F(f(a, a), f(f(f(y'', z0), z'), z'')) -> F(a, f(f(b, f(f(f(a, y''), z0), z')), z''))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 10
                 ↳Narrowing Transformation


Dependency Pairs:

F(f(a, a), f(f(f(y'', z0), z'), z'')) -> F(a, f(f(b, f(f(f(a, y''), z0), z')), z''))
F(f(a, a), f(f(a, z'''), z'')) -> F(a, f(f(b, f(a, f(b, f(a, z''')))), z''))
F(f(a, a), f(f(y'', z'''), z'')) -> F(a, f(f(f(b, f(a, y'')), z'''), z''))
F(f(a, a), f(f(a, z'''), z')) -> F(a, f(b, f(f(a, f(b, f(a, z'''))), z')))
F(f(a, a), f(f(y''', z'''), z'0)) -> F(a, f(f(b, f(f(a, y'''), z''')), z'0))
F(f(a, a), f(f(f(y'''', z''''), z'0'), z)) -> F(f(a, a), f(f(y'''', z''''), z'0'))
F(f(a, a), f(a, z''')) -> F(a, f(b, f(a, f(f(b, a), z'''))))
F(f(a, a), f(a, z''')) -> F(a, f(b, f(f(a, b), f(a, z'''))))
F(f(a, a), f(a, z''')) -> F(a, f(f(b, a), f(b, f(a, z'''))))
F(f(a, a), f(y''', z'')) -> F(a, f(f(f(b, a), y'''), z''))
F(f(a, a), f(f(a, z''''), z)) -> F(f(a, a), f(a, z''''))
F(f(a, a), f(f(y'''', z''''), z)) -> F(f(a, a), f(y'''', z''''))
F(f(a, a), f(y', z)) -> F(f(a, a), y')
F(x'', f(f(y'', z''), z)) -> F(x'', f(y'', z''))
F(f(a, a), x'') -> F(a, f(f(b, a), x''))
F(x, f(y, z)) -> F(f(x, y), z)
F(f(a, a), f(f(f(y', z0), z''), z')) -> F(a, f(b, f(f(f(f(a, y'), z0), z''), z')))


Rules:


f(f(a, a), x) -> f(a, f(b, f(a, x)))
f(x, f(y, z)) -> f(f(x, y), z)


Strategy:

innermost




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

F(f(a, a), f(a, z''')) -> F(a, f(f(b, a), f(b, f(a, z'''))))
two new Dependency Pairs are created:

F(f(a, a), f(a, z'''')) -> F(a, f(f(f(b, a), b), f(a, z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(f(b, a), f(f(b, a), z'''')))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 11
                 ↳Narrowing Transformation


Dependency Pairs:

F(f(a, a), f(f(a, z'''), z'')) -> F(a, f(f(b, f(a, f(b, f(a, z''')))), z''))
F(f(a, a), f(f(y'', z'''), z'')) -> F(a, f(f(f(b, f(a, y'')), z'''), z''))
F(f(a, a), f(f(f(y', z0), z''), z')) -> F(a, f(b, f(f(f(f(a, y'), z0), z''), z')))
F(f(a, a), f(f(a, z'''), z')) -> F(a, f(b, f(f(a, f(b, f(a, z'''))), z')))
F(f(a, a), f(f(y''', z'''), z'0)) -> F(a, f(f(b, f(f(a, y'''), z''')), z'0))
F(f(a, a), f(f(f(y'''', z''''), z'0'), z)) -> F(f(a, a), f(f(y'''', z''''), z'0'))
F(f(a, a), f(a, z'''')) -> F(a, f(f(b, a), f(f(b, a), z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(f(f(b, a), b), f(a, z'''')))
F(f(a, a), f(a, z''')) -> F(a, f(b, f(a, f(f(b, a), z'''))))
F(f(a, a), f(a, z''')) -> F(a, f(b, f(f(a, b), f(a, z'''))))
F(f(a, a), f(y''', z'')) -> F(a, f(f(f(b, a), y'''), z''))
F(f(a, a), f(f(a, z''''), z)) -> F(f(a, a), f(a, z''''))
F(f(a, a), f(f(y'''', z''''), z)) -> F(f(a, a), f(y'''', z''''))
F(f(a, a), f(y', z)) -> F(f(a, a), y')
F(x'', f(f(y'', z''), z)) -> F(x'', f(y'', z''))
F(f(a, a), x'') -> F(a, f(f(b, a), x''))
F(x, f(y, z)) -> F(f(x, y), z)
F(f(a, a), f(f(f(y'', z0), z'), z'')) -> F(a, f(f(b, f(f(f(a, y''), z0), z')), z''))


Rules:


f(f(a, a), x) -> f(a, f(b, f(a, x)))
f(x, f(y, z)) -> f(f(x, y), z)


Strategy:

innermost




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

F(f(a, a), f(a, z''')) -> F(a, f(b, f(f(a, b), f(a, z'''))))
two new Dependency Pairs are created:

F(f(a, a), f(a, z'''')) -> F(a, f(f(b, f(a, b)), f(a, z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(b, f(f(f(a, b), a), z'''')))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 12
                 ↳Narrowing Transformation


Dependency Pairs:

F(f(a, a), f(f(f(y'', z0), z'), z'')) -> F(a, f(f(b, f(f(f(a, y''), z0), z')), z''))
F(f(a, a), f(f(y'', z'''), z'')) -> F(a, f(f(f(b, f(a, y'')), z'''), z''))
F(f(a, a), f(f(f(y', z0), z''), z')) -> F(a, f(b, f(f(f(f(a, y'), z0), z''), z')))
F(f(a, a), f(f(a, z'''), z')) -> F(a, f(b, f(f(a, f(b, f(a, z'''))), z')))
F(f(a, a), f(f(y''', z'''), z'0)) -> F(a, f(f(b, f(f(a, y'''), z''')), z'0))
F(f(a, a), f(f(f(y'''', z''''), z'0'), z)) -> F(f(a, a), f(f(y'''', z''''), z'0'))
F(f(a, a), f(a, z'''')) -> F(a, f(b, f(f(f(a, b), a), z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(f(b, f(a, b)), f(a, z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(f(b, a), f(f(b, a), z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(f(f(b, a), b), f(a, z'''')))
F(f(a, a), f(a, z''')) -> F(a, f(b, f(a, f(f(b, a), z'''))))
F(f(a, a), f(y''', z'')) -> F(a, f(f(f(b, a), y'''), z''))
F(f(a, a), f(f(a, z''''), z)) -> F(f(a, a), f(a, z''''))
F(f(a, a), f(f(y'''', z''''), z)) -> F(f(a, a), f(y'''', z''''))
F(f(a, a), f(y', z)) -> F(f(a, a), y')
F(x'', f(f(y'', z''), z)) -> F(x'', f(y'', z''))
F(f(a, a), x'') -> F(a, f(f(b, a), x''))
F(x, f(y, z)) -> F(f(x, y), z)
F(f(a, a), f(f(a, z'''), z'')) -> F(a, f(f(b, f(a, f(b, f(a, z''')))), z''))


Rules:


f(f(a, a), x) -> f(a, f(b, f(a, x)))
f(x, f(y, z)) -> f(f(x, y), z)


Strategy:

innermost




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

F(f(a, a), f(a, z''')) -> F(a, f(b, f(a, f(f(b, a), z'''))))
two new Dependency Pairs are created:

F(f(a, a), f(a, z'''')) -> F(a, f(f(b, a), f(f(b, a), z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(b, f(f(a, f(b, a)), z'''')))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 13
                 ↳Narrowing Transformation


Dependency Pairs:

F(f(a, a), f(f(a, z'''), z'')) -> F(a, f(f(b, f(a, f(b, f(a, z''')))), z''))
F(f(a, a), f(f(y'', z'''), z'')) -> F(a, f(f(f(b, f(a, y'')), z'''), z''))
F(f(a, a), f(f(f(y', z0), z''), z')) -> F(a, f(b, f(f(f(f(a, y'), z0), z''), z')))
F(f(a, a), f(f(a, z'''), z')) -> F(a, f(b, f(f(a, f(b, f(a, z'''))), z')))
F(f(a, a), f(f(y''', z'''), z'0)) -> F(a, f(f(b, f(f(a, y'''), z''')), z'0))
F(f(a, a), f(f(f(y'''', z''''), z'0'), z)) -> F(f(a, a), f(f(y'''', z''''), z'0'))
F(f(a, a), f(a, z'''')) -> F(a, f(b, f(f(a, f(b, a)), z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(f(b, a), f(f(b, a), z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(b, f(f(f(a, b), a), z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(f(b, f(a, b)), f(a, z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(f(b, a), f(f(b, a), z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(f(f(b, a), b), f(a, z'''')))
F(f(a, a), f(y''', z'')) -> F(a, f(f(f(b, a), y'''), z''))
F(f(a, a), f(f(a, z''''), z)) -> F(f(a, a), f(a, z''''))
F(f(a, a), f(f(y'''', z''''), z)) -> F(f(a, a), f(y'''', z''''))
F(f(a, a), f(y', z)) -> F(f(a, a), y')
F(x'', f(f(y'', z''), z)) -> F(x'', f(y'', z''))
F(f(a, a), x'') -> F(a, f(f(b, a), x''))
F(x, f(y, z)) -> F(f(x, y), z)
F(f(a, a), f(f(f(y'', z0), z'), z'')) -> F(a, f(f(b, f(f(f(a, y''), z0), z')), z''))


Rules:


f(f(a, a), x) -> f(a, f(b, f(a, x)))
f(x, f(y, z)) -> f(f(x, y), z)


Strategy:

innermost




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

F(f(a, a), f(f(y''', z'''), z'0)) -> F(a, f(f(b, f(f(a, y'''), z''')), z'0))
three new Dependency Pairs are created:

F(f(a, a), f(f(y'''', z''''), z'0)) -> F(a, f(f(f(b, f(a, y'''')), z''''), z'0))
F(f(a, a), f(f(a, z''''), z'0)) -> F(a, f(f(b, f(a, f(b, f(a, z'''')))), z'0))
F(f(a, a), f(f(f(y', z'), z'''), z'0)) -> F(a, f(f(b, f(f(f(a, y'), z'), z''')), z'0))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 14
                 ↳Narrowing Transformation


Dependency Pairs:

F(f(a, a), f(f(f(y', z'), z'''), z'0)) -> F(a, f(f(b, f(f(f(a, y'), z'), z''')), z'0))
F(f(a, a), f(f(a, z''''), z'0)) -> F(a, f(f(b, f(a, f(b, f(a, z'''')))), z'0))
F(f(a, a), f(f(y'''', z''''), z'0)) -> F(a, f(f(f(b, f(a, y'''')), z''''), z'0))
F(f(a, a), f(f(f(y'', z0), z'), z'')) -> F(a, f(f(b, f(f(f(a, y''), z0), z')), z''))
F(f(a, a), f(f(y'', z'''), z'')) -> F(a, f(f(f(b, f(a, y'')), z'''), z''))
F(f(a, a), f(f(f(y', z0), z''), z')) -> F(a, f(b, f(f(f(f(a, y'), z0), z''), z')))
F(f(a, a), f(f(a, z'''), z')) -> F(a, f(b, f(f(a, f(b, f(a, z'''))), z')))
F(f(a, a), f(f(f(y'''', z''''), z'0'), z)) -> F(f(a, a), f(f(y'''', z''''), z'0'))
F(f(a, a), f(a, z'''')) -> F(a, f(b, f(f(a, f(b, a)), z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(f(b, a), f(f(b, a), z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(b, f(f(f(a, b), a), z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(f(b, f(a, b)), f(a, z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(f(b, a), f(f(b, a), z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(f(f(b, a), b), f(a, z'''')))
F(f(a, a), f(y''', z'')) -> F(a, f(f(f(b, a), y'''), z''))
F(f(a, a), f(f(a, z''''), z)) -> F(f(a, a), f(a, z''''))
F(f(a, a), f(f(y'''', z''''), z)) -> F(f(a, a), f(y'''', z''''))
F(f(a, a), f(y', z)) -> F(f(a, a), y')
F(x'', f(f(y'', z''), z)) -> F(x'', f(y'', z''))
F(f(a, a), x'') -> F(a, f(f(b, a), x''))
F(x, f(y, z)) -> F(f(x, y), z)
F(f(a, a), f(f(a, z'''), z'')) -> F(a, f(f(b, f(a, f(b, f(a, z''')))), z''))


Rules:


f(f(a, a), x) -> f(a, f(b, f(a, x)))
f(x, f(y, z)) -> f(f(x, y), z)


Strategy:

innermost




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

F(f(a, a), f(f(a, z'''), z')) -> F(a, f(b, f(f(a, f(b, f(a, z'''))), z')))
three new Dependency Pairs are created:

F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(b, f(a, f(b, f(a, z'''')))), z''))
F(f(a, a), f(f(a, z''''), z')) -> F(a, f(b, f(f(f(a, b), f(a, z'''')), z')))
F(f(a, a), f(f(a, z''''), z')) -> F(a, f(b, f(f(a, f(f(b, a), z'''')), z')))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 15
                 ↳Narrowing Transformation


Dependency Pairs:

F(f(a, a), f(f(a, z''''), z')) -> F(a, f(b, f(f(a, f(f(b, a), z'''')), z')))
F(f(a, a), f(f(a, z''''), z')) -> F(a, f(b, f(f(f(a, b), f(a, z'''')), z')))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(b, f(a, f(b, f(a, z'''')))), z''))
F(f(a, a), f(f(a, z''''), z'0)) -> F(a, f(f(b, f(a, f(b, f(a, z'''')))), z'0))
F(f(a, a), f(f(y'''', z''''), z'0)) -> F(a, f(f(f(b, f(a, y'''')), z''''), z'0))
F(f(a, a), f(f(f(y'', z0), z'), z'')) -> F(a, f(f(b, f(f(f(a, y''), z0), z')), z''))
F(f(a, a), f(f(a, z'''), z'')) -> F(a, f(f(b, f(a, f(b, f(a, z''')))), z''))
F(f(a, a), f(f(y'', z'''), z'')) -> F(a, f(f(f(b, f(a, y'')), z'''), z''))
F(f(a, a), f(f(f(y', z0), z''), z')) -> F(a, f(b, f(f(f(f(a, y'), z0), z''), z')))
F(f(a, a), f(f(f(y'''', z''''), z'0'), z)) -> F(f(a, a), f(f(y'''', z''''), z'0'))
F(f(a, a), f(a, z'''')) -> F(a, f(b, f(f(a, f(b, a)), z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(f(b, a), f(f(b, a), z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(b, f(f(f(a, b), a), z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(f(b, f(a, b)), f(a, z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(f(b, a), f(f(b, a), z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(f(f(b, a), b), f(a, z'''')))
F(f(a, a), f(y''', z'')) -> F(a, f(f(f(b, a), y'''), z''))
F(f(a, a), f(f(a, z''''), z)) -> F(f(a, a), f(a, z''''))
F(f(a, a), f(f(y'''', z''''), z)) -> F(f(a, a), f(y'''', z''''))
F(f(a, a), f(y', z)) -> F(f(a, a), y')
F(x'', f(f(y'', z''), z)) -> F(x'', f(y'', z''))
F(f(a, a), x'') -> F(a, f(f(b, a), x''))
F(x, f(y, z)) -> F(f(x, y), z)
F(f(a, a), f(f(f(y', z'), z'''), z'0)) -> F(a, f(f(b, f(f(f(a, y'), z'), z''')), z'0))


Rules:


f(f(a, a), x) -> f(a, f(b, f(a, x)))
f(x, f(y, z)) -> f(f(x, y), z)


Strategy:

innermost




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

F(f(a, a), f(f(f(y', z0), z''), z')) -> F(a, f(b, f(f(f(f(a, y'), z0), z''), z')))
three new Dependency Pairs are created:

F(f(a, a), f(f(f(y'', z0'), z'''), z'0)) -> F(a, f(f(b, f(f(f(a, y''), z0'), z''')), z'0))
F(f(a, a), f(f(f(a, z0'), z''), z')) -> F(a, f(b, f(f(f(a, f(b, f(a, z0'))), z''), z')))
F(f(a, a), f(f(f(f(y'', z1), z0), z''), z')) -> F(a, f(b, f(f(f(f(f(a, y''), z1), z0), z''), z')))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 16
                 ↳Narrowing Transformation


Dependency Pairs:

F(f(a, a), f(f(f(f(y'', z1), z0), z''), z')) -> F(a, f(b, f(f(f(f(f(a, y''), z1), z0), z''), z')))
F(f(a, a), f(f(f(a, z0'), z''), z')) -> F(a, f(b, f(f(f(a, f(b, f(a, z0'))), z''), z')))
F(f(a, a), f(f(f(y'', z0'), z'''), z'0)) -> F(a, f(f(b, f(f(f(a, y''), z0'), z''')), z'0))
F(f(a, a), f(f(a, z''''), z')) -> F(a, f(b, f(f(f(a, b), f(a, z'''')), z')))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(b, f(a, f(b, f(a, z'''')))), z''))
F(f(a, a), f(f(f(y', z'), z'''), z'0)) -> F(a, f(f(b, f(f(f(a, y'), z'), z''')), z'0))
F(f(a, a), f(f(a, z''''), z'0)) -> F(a, f(f(b, f(a, f(b, f(a, z'''')))), z'0))
F(f(a, a), f(f(y'''', z''''), z'0)) -> F(a, f(f(f(b, f(a, y'''')), z''''), z'0))
F(f(a, a), f(f(f(y'', z0), z'), z'')) -> F(a, f(f(b, f(f(f(a, y''), z0), z')), z''))
F(f(a, a), f(f(a, z'''), z'')) -> F(a, f(f(b, f(a, f(b, f(a, z''')))), z''))
F(f(a, a), f(f(y'', z'''), z'')) -> F(a, f(f(f(b, f(a, y'')), z'''), z''))
F(f(a, a), f(f(f(y'''', z''''), z'0'), z)) -> F(f(a, a), f(f(y'''', z''''), z'0'))
F(f(a, a), f(a, z'''')) -> F(a, f(b, f(f(a, f(b, a)), z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(f(b, a), f(f(b, a), z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(b, f(f(f(a, b), a), z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(f(b, f(a, b)), f(a, z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(f(b, a), f(f(b, a), z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(f(f(b, a), b), f(a, z'''')))
F(f(a, a), f(y''', z'')) -> F(a, f(f(f(b, a), y'''), z''))
F(f(a, a), f(f(a, z''''), z)) -> F(f(a, a), f(a, z''''))
F(f(a, a), f(f(y'''', z''''), z)) -> F(f(a, a), f(y'''', z''''))
F(f(a, a), f(y', z)) -> F(f(a, a), y')
F(x'', f(f(y'', z''), z)) -> F(x'', f(y'', z''))
F(f(a, a), x'') -> F(a, f(f(b, a), x''))
F(x, f(y, z)) -> F(f(x, y), z)
F(f(a, a), f(f(a, z''''), z')) -> F(a, f(b, f(f(a, f(f(b, a), z'''')), z')))


Rules:


f(f(a, a), x) -> f(a, f(b, f(a, x)))
f(x, f(y, z)) -> f(f(x, y), z)


Strategy:

innermost




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

F(f(a, a), f(f(y'', z'''), z'')) -> F(a, f(f(f(b, f(a, y'')), z'''), z''))
two new Dependency Pairs are created:

F(f(a, a), f(f(y''', z'''), z'')) -> F(a, f(f(f(f(b, a), y'''), z'''), z''))
F(f(a, a), f(f(f(y', z'), z'''), z'')) -> F(a, f(f(f(b, f(f(a, y'), z')), z'''), z''))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 17
                 ↳Narrowing Transformation


Dependency Pairs:

F(f(a, a), f(f(f(y', z'), z'''), z'')) -> F(a, f(f(f(b, f(f(a, y'), z')), z'''), z''))
F(f(a, a), f(f(y''', z'''), z'')) -> F(a, f(f(f(f(b, a), y'''), z'''), z''))
F(f(a, a), f(f(f(a, z0'), z''), z')) -> F(a, f(b, f(f(f(a, f(b, f(a, z0'))), z''), z')))
F(f(a, a), f(f(f(y'', z0'), z'''), z'0)) -> F(a, f(f(b, f(f(f(a, y''), z0'), z''')), z'0))
F(f(a, a), f(f(a, z''''), z')) -> F(a, f(b, f(f(a, f(f(b, a), z'''')), z')))
F(f(a, a), f(f(a, z''''), z')) -> F(a, f(b, f(f(f(a, b), f(a, z'''')), z')))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(b, f(a, f(b, f(a, z'''')))), z''))
F(f(a, a), f(f(f(y', z'), z'''), z'0)) -> F(a, f(f(b, f(f(f(a, y'), z'), z''')), z'0))
F(f(a, a), f(f(a, z''''), z'0)) -> F(a, f(f(b, f(a, f(b, f(a, z'''')))), z'0))
F(f(a, a), f(f(y'''', z''''), z'0)) -> F(a, f(f(f(b, f(a, y'''')), z''''), z'0))
F(f(a, a), f(f(f(y'', z0), z'), z'')) -> F(a, f(f(b, f(f(f(a, y''), z0), z')), z''))
F(f(a, a), f(f(a, z'''), z'')) -> F(a, f(f(b, f(a, f(b, f(a, z''')))), z''))
F(f(a, a), f(f(f(y'''', z''''), z'0'), z)) -> F(f(a, a), f(f(y'''', z''''), z'0'))
F(f(a, a), f(a, z'''')) -> F(a, f(b, f(f(a, f(b, a)), z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(f(b, a), f(f(b, a), z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(b, f(f(f(a, b), a), z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(f(b, f(a, b)), f(a, z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(f(b, a), f(f(b, a), z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(f(f(b, a), b), f(a, z'''')))
F(f(a, a), f(y''', z'')) -> F(a, f(f(f(b, a), y'''), z''))
F(f(a, a), f(f(a, z''''), z)) -> F(f(a, a), f(a, z''''))
F(f(a, a), f(f(y'''', z''''), z)) -> F(f(a, a), f(y'''', z''''))
F(f(a, a), f(y', z)) -> F(f(a, a), y')
F(x'', f(f(y'', z''), z)) -> F(x'', f(y'', z''))
F(f(a, a), x'') -> F(a, f(f(b, a), x''))
F(x, f(y, z)) -> F(f(x, y), z)
F(f(a, a), f(f(f(f(y'', z1), z0), z''), z')) -> F(a, f(b, f(f(f(f(f(a, y''), z1), z0), z''), z')))


Rules:


f(f(a, a), x) -> f(a, f(b, f(a, x)))
f(x, f(y, z)) -> f(f(x, y), z)


Strategy:

innermost




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

F(f(a, a), f(f(a, z'''), z'')) -> F(a, f(f(b, f(a, f(b, f(a, z''')))), z''))
three new Dependency Pairs are created:

F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(f(b, a), f(b, f(a, z''''))), z''))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(b, f(f(a, b), f(a, z''''))), z''))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(b, f(a, f(f(b, a), z''''))), z''))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 18
                 ↳Narrowing Transformation


Dependency Pairs:

F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(b, f(a, f(f(b, a), z''''))), z''))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(b, f(f(a, b), f(a, z''''))), z''))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(f(b, a), f(b, f(a, z''''))), z''))
F(f(a, a), f(f(y''', z'''), z'')) -> F(a, f(f(f(f(b, a), y'''), z'''), z''))
F(f(a, a), f(f(f(f(y'', z1), z0), z''), z')) -> F(a, f(b, f(f(f(f(f(a, y''), z1), z0), z''), z')))
F(f(a, a), f(f(f(a, z0'), z''), z')) -> F(a, f(b, f(f(f(a, f(b, f(a, z0'))), z''), z')))
F(f(a, a), f(f(f(y'', z0'), z'''), z'0)) -> F(a, f(f(b, f(f(f(a, y''), z0'), z''')), z'0))
F(f(a, a), f(f(a, z''''), z')) -> F(a, f(b, f(f(a, f(f(b, a), z'''')), z')))
F(f(a, a), f(f(a, z''''), z')) -> F(a, f(b, f(f(f(a, b), f(a, z'''')), z')))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(b, f(a, f(b, f(a, z'''')))), z''))
F(f(a, a), f(f(f(y', z'), z'''), z'0)) -> F(a, f(f(b, f(f(f(a, y'), z'), z''')), z'0))
F(f(a, a), f(f(a, z''''), z'0)) -> F(a, f(f(b, f(a, f(b, f(a, z'''')))), z'0))
F(f(a, a), f(f(y'''', z''''), z'0)) -> F(a, f(f(f(b, f(a, y'''')), z''''), z'0))
F(f(a, a), f(f(f(y'', z0), z'), z'')) -> F(a, f(f(b, f(f(f(a, y''), z0), z')), z''))
F(f(a, a), f(f(f(y'''', z''''), z'0'), z)) -> F(f(a, a), f(f(y'''', z''''), z'0'))
F(f(a, a), f(a, z'''')) -> F(a, f(b, f(f(a, f(b, a)), z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(f(b, a), f(f(b, a), z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(b, f(f(f(a, b), a), z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(f(b, f(a, b)), f(a, z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(f(b, a), f(f(b, a), z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(f(f(b, a), b), f(a, z'''')))
F(f(a, a), f(y''', z'')) -> F(a, f(f(f(b, a), y'''), z''))
F(f(a, a), f(f(a, z''''), z)) -> F(f(a, a), f(a, z''''))
F(f(a, a), f(f(y'''', z''''), z)) -> F(f(a, a), f(y'''', z''''))
F(f(a, a), f(y', z)) -> F(f(a, a), y')
F(x'', f(f(y'', z''), z)) -> F(x'', f(y'', z''))
F(f(a, a), x'') -> F(a, f(f(b, a), x''))
F(x, f(y, z)) -> F(f(x, y), z)
F(f(a, a), f(f(f(y', z'), z'''), z'')) -> F(a, f(f(f(b, f(f(a, y'), z')), z'''), z''))


Rules:


f(f(a, a), x) -> f(a, f(b, f(a, x)))
f(x, f(y, z)) -> f(f(x, y), z)


Strategy:

innermost




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

F(f(a, a), f(f(f(y'', z0), z'), z'')) -> F(a, f(f(b, f(f(f(a, y''), z0), z')), z''))
three new Dependency Pairs are created:

F(f(a, a), f(f(f(y''', z0'), z'''), z'')) -> F(a, f(f(f(b, f(f(a, y'''), z0')), z'''), z''))
F(f(a, a), f(f(f(a, z0'), z'), z'')) -> F(a, f(f(b, f(f(a, f(b, f(a, z0'))), z')), z''))
F(f(a, a), f(f(f(f(y', z1), z0), z'), z'')) -> F(a, f(f(b, f(f(f(f(a, y'), z1), z0), z')), z''))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 19
                 ↳Narrowing Transformation


Dependency Pairs:

F(f(a, a), f(f(f(f(y', z1), z0), z'), z'')) -> F(a, f(f(b, f(f(f(f(a, y'), z1), z0), z')), z''))
F(f(a, a), f(f(f(a, z0'), z'), z'')) -> F(a, f(f(b, f(f(a, f(b, f(a, z0'))), z')), z''))
F(f(a, a), f(f(f(y''', z0'), z'''), z'')) -> F(a, f(f(f(b, f(f(a, y'''), z0')), z'''), z''))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(b, f(f(a, b), f(a, z''''))), z''))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(f(b, a), f(b, f(a, z''''))), z''))
F(f(a, a), f(f(f(y', z'), z'''), z'')) -> F(a, f(f(f(b, f(f(a, y'), z')), z'''), z''))
F(f(a, a), f(f(y''', z'''), z'')) -> F(a, f(f(f(f(b, a), y'''), z'''), z''))
F(f(a, a), f(f(f(f(y'', z1), z0), z''), z')) -> F(a, f(b, f(f(f(f(f(a, y''), z1), z0), z''), z')))
F(f(a, a), f(f(f(a, z0'), z''), z')) -> F(a, f(b, f(f(f(a, f(b, f(a, z0'))), z''), z')))
F(f(a, a), f(f(f(y'', z0'), z'''), z'0)) -> F(a, f(f(b, f(f(f(a, y''), z0'), z''')), z'0))
F(f(a, a), f(f(a, z''''), z')) -> F(a, f(b, f(f(a, f(f(b, a), z'''')), z')))
F(f(a, a), f(f(a, z''''), z')) -> F(a, f(b, f(f(f(a, b), f(a, z'''')), z')))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(b, f(a, f(b, f(a, z'''')))), z''))
F(f(a, a), f(f(f(y', z'), z'''), z'0)) -> F(a, f(f(b, f(f(f(a, y'), z'), z''')), z'0))
F(f(a, a), f(f(a, z''''), z'0)) -> F(a, f(f(b, f(a, f(b, f(a, z'''')))), z'0))
F(f(a, a), f(f(y'''', z''''), z'0)) -> F(a, f(f(f(b, f(a, y'''')), z''''), z'0))
F(f(a, a), f(f(f(y'''', z''''), z'0'), z)) -> F(f(a, a), f(f(y'''', z''''), z'0'))
F(f(a, a), f(a, z'''')) -> F(a, f(b, f(f(a, f(b, a)), z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(f(b, a), f(f(b, a), z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(b, f(f(f(a, b), a), z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(f(b, f(a, b)), f(a, z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(f(b, a), f(f(b, a), z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(f(f(b, a), b), f(a, z'''')))
F(f(a, a), f(y''', z'')) -> F(a, f(f(f(b, a), y'''), z''))
F(f(a, a), f(f(a, z''''), z)) -> F(f(a, a), f(a, z''''))
F(f(a, a), f(f(y'''', z''''), z)) -> F(f(a, a), f(y'''', z''''))
F(f(a, a), f(y', z)) -> F(f(a, a), y')
F(x'', f(f(y'', z''), z)) -> F(x'', f(y'', z''))
F(f(a, a), x'') -> F(a, f(f(b, a), x''))
F(x, f(y, z)) -> F(f(x, y), z)
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(b, f(a, f(f(b, a), z''''))), z''))


Rules:


f(f(a, a), x) -> f(a, f(b, f(a, x)))
f(x, f(y, z)) -> f(f(x, y), z)


Strategy:

innermost




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

F(f(a, a), f(a, z'''')) -> F(a, f(f(f(b, a), b), f(a, z'''')))
one new Dependency Pair is created:

F(f(a, a), f(a, z''''')) -> F(a, f(f(f(f(b, a), b), a), z'''''))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 20
                 ↳Narrowing Transformation


Dependency Pairs:

F(f(a, a), f(f(f(a, z0'), z'), z'')) -> F(a, f(f(b, f(f(a, f(b, f(a, z0'))), z')), z''))
F(f(a, a), f(f(f(y''', z0'), z'''), z'')) -> F(a, f(f(f(b, f(f(a, y'''), z0')), z'''), z''))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(b, f(a, f(f(b, a), z''''))), z''))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(b, f(f(a, b), f(a, z''''))), z''))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(f(b, a), f(b, f(a, z''''))), z''))
F(f(a, a), f(f(f(y', z'), z'''), z'')) -> F(a, f(f(f(b, f(f(a, y'), z')), z'''), z''))
F(f(a, a), f(f(y''', z'''), z'')) -> F(a, f(f(f(f(b, a), y'''), z'''), z''))
F(f(a, a), f(f(f(f(y'', z1), z0), z''), z')) -> F(a, f(b, f(f(f(f(f(a, y''), z1), z0), z''), z')))
F(f(a, a), f(f(f(a, z0'), z''), z')) -> F(a, f(b, f(f(f(a, f(b, f(a, z0'))), z''), z')))
F(f(a, a), f(f(f(y'', z0'), z'''), z'0)) -> F(a, f(f(b, f(f(f(a, y''), z0'), z''')), z'0))
F(f(a, a), f(f(a, z''''), z')) -> F(a, f(b, f(f(a, f(f(b, a), z'''')), z')))
F(f(a, a), f(f(a, z''''), z')) -> F(a, f(b, f(f(f(a, b), f(a, z'''')), z')))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(b, f(a, f(b, f(a, z'''')))), z''))
F(f(a, a), f(f(f(y', z'), z'''), z'0)) -> F(a, f(f(b, f(f(f(a, y'), z'), z''')), z'0))
F(f(a, a), f(f(a, z''''), z'0)) -> F(a, f(f(b, f(a, f(b, f(a, z'''')))), z'0))
F(f(a, a), f(f(y'''', z''''), z'0)) -> F(a, f(f(f(b, f(a, y'''')), z''''), z'0))
F(f(a, a), f(f(f(y'''', z''''), z'0'), z)) -> F(f(a, a), f(f(y'''', z''''), z'0'))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(f(b, a), b), a), z'''''))
F(f(a, a), f(a, z'''')) -> F(a, f(b, f(f(a, f(b, a)), z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(f(b, a), f(f(b, a), z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(b, f(f(f(a, b), a), z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(f(b, f(a, b)), f(a, z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(f(b, a), f(f(b, a), z'''')))
F(f(a, a), f(y''', z'')) -> F(a, f(f(f(b, a), y'''), z''))
F(f(a, a), f(f(a, z''''), z)) -> F(f(a, a), f(a, z''''))
F(f(a, a), f(f(y'''', z''''), z)) -> F(f(a, a), f(y'''', z''''))
F(f(a, a), f(y', z)) -> F(f(a, a), y')
F(x'', f(f(y'', z''), z)) -> F(x'', f(y'', z''))
F(f(a, a), x'') -> F(a, f(f(b, a), x''))
F(x, f(y, z)) -> F(f(x, y), z)
F(f(a, a), f(f(f(f(y', z1), z0), z'), z'')) -> F(a, f(f(b, f(f(f(f(a, y'), z1), z0), z')), z''))


Rules:


f(f(a, a), x) -> f(a, f(b, f(a, x)))
f(x, f(y, z)) -> f(f(x, y), z)


Strategy:

innermost




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

F(f(a, a), f(a, z'''')) -> F(a, f(f(b, a), f(f(b, a), z'''')))
one new Dependency Pair is created:

F(f(a, a), f(a, z''''')) -> F(a, f(f(f(b, a), f(b, a)), z'''''))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 21
                 ↳Narrowing Transformation


Dependency Pairs:

F(f(a, a), f(f(f(f(y', z1), z0), z'), z'')) -> F(a, f(f(b, f(f(f(f(a, y'), z1), z0), z')), z''))
F(f(a, a), f(f(f(y''', z0'), z'''), z'')) -> F(a, f(f(f(b, f(f(a, y'''), z0')), z'''), z''))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(b, f(a, f(f(b, a), z''''))), z''))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(b, f(f(a, b), f(a, z''''))), z''))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(f(b, a), f(b, f(a, z''''))), z''))
F(f(a, a), f(f(f(y', z'), z'''), z'')) -> F(a, f(f(f(b, f(f(a, y'), z')), z'''), z''))
F(f(a, a), f(f(y''', z'''), z'')) -> F(a, f(f(f(f(b, a), y'''), z'''), z''))
F(f(a, a), f(f(f(f(y'', z1), z0), z''), z')) -> F(a, f(b, f(f(f(f(f(a, y''), z1), z0), z''), z')))
F(f(a, a), f(f(f(a, z0'), z''), z')) -> F(a, f(b, f(f(f(a, f(b, f(a, z0'))), z''), z')))
F(f(a, a), f(f(f(y'', z0'), z'''), z'0)) -> F(a, f(f(b, f(f(f(a, y''), z0'), z''')), z'0))
F(f(a, a), f(f(a, z''''), z')) -> F(a, f(b, f(f(a, f(f(b, a), z'''')), z')))
F(f(a, a), f(f(a, z''''), z')) -> F(a, f(b, f(f(f(a, b), f(a, z'''')), z')))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(b, f(a, f(b, f(a, z'''')))), z''))
F(f(a, a), f(f(f(y', z'), z'''), z'0)) -> F(a, f(f(b, f(f(f(a, y'), z'), z''')), z'0))
F(f(a, a), f(f(a, z''''), z'0)) -> F(a, f(f(b, f(a, f(b, f(a, z'''')))), z'0))
F(f(a, a), f(f(y'''', z''''), z'0)) -> F(a, f(f(f(b, f(a, y'''')), z''''), z'0))
F(f(a, a), f(f(f(y'''', z''''), z'0'), z)) -> F(f(a, a), f(f(y'''', z''''), z'0'))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(b, a), f(b, a)), z'''''))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(f(b, a), b), a), z'''''))
F(f(a, a), f(a, z'''')) -> F(a, f(b, f(f(a, f(b, a)), z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(f(b, a), f(f(b, a), z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(b, f(f(f(a, b), a), z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(f(b, f(a, b)), f(a, z'''')))
F(f(a, a), f(y''', z'')) -> F(a, f(f(f(b, a), y'''), z''))
F(f(a, a), f(f(a, z''''), z)) -> F(f(a, a), f(a, z''''))
F(f(a, a), f(f(y'''', z''''), z)) -> F(f(a, a), f(y'''', z''''))
F(f(a, a), f(y', z)) -> F(f(a, a), y')
F(x'', f(f(y'', z''), z)) -> F(x'', f(y'', z''))
F(f(a, a), x'') -> F(a, f(f(b, a), x''))
F(x, f(y, z)) -> F(f(x, y), z)
F(f(a, a), f(f(f(a, z0'), z'), z'')) -> F(a, f(f(b, f(f(a, f(b, f(a, z0'))), z')), z''))


Rules:


f(f(a, a), x) -> f(a, f(b, f(a, x)))
f(x, f(y, z)) -> f(f(x, y), z)


Strategy:

innermost




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

F(f(a, a), f(a, z'''')) -> F(a, f(f(b, f(a, b)), f(a, z'''')))
two new Dependency Pairs are created:

F(f(a, a), f(a, z''''')) -> F(a, f(f(f(b, f(a, b)), a), z'''''))
F(f(a, a), f(a, z'''')) -> F(a, f(f(f(b, a), b), f(a, z'''')))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 22
                 ↳Narrowing Transformation


Dependency Pairs:

F(f(a, a), f(f(f(a, z0'), z'), z'')) -> F(a, f(f(b, f(f(a, f(b, f(a, z0'))), z')), z''))
F(f(a, a), f(f(f(y''', z0'), z'''), z'')) -> F(a, f(f(f(b, f(f(a, y'''), z0')), z'''), z''))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(b, f(a, f(f(b, a), z''''))), z''))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(b, f(f(a, b), f(a, z''''))), z''))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(f(b, a), f(b, f(a, z''''))), z''))
F(f(a, a), f(f(f(y', z'), z'''), z'')) -> F(a, f(f(f(b, f(f(a, y'), z')), z'''), z''))
F(f(a, a), f(f(y''', z'''), z'')) -> F(a, f(f(f(f(b, a), y'''), z'''), z''))
F(f(a, a), f(f(f(f(y'', z1), z0), z''), z')) -> F(a, f(b, f(f(f(f(f(a, y''), z1), z0), z''), z')))
F(f(a, a), f(f(f(a, z0'), z''), z')) -> F(a, f(b, f(f(f(a, f(b, f(a, z0'))), z''), z')))
F(f(a, a), f(f(f(y'', z0'), z'''), z'0)) -> F(a, f(f(b, f(f(f(a, y''), z0'), z''')), z'0))
F(f(a, a), f(f(a, z''''), z')) -> F(a, f(b, f(f(a, f(f(b, a), z'''')), z')))
F(f(a, a), f(f(a, z''''), z')) -> F(a, f(b, f(f(f(a, b), f(a, z'''')), z')))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(b, f(a, f(b, f(a, z'''')))), z''))
F(f(a, a), f(f(f(y', z'), z'''), z'0)) -> F(a, f(f(b, f(f(f(a, y'), z'), z''')), z'0))
F(f(a, a), f(f(a, z''''), z'0)) -> F(a, f(f(b, f(a, f(b, f(a, z'''')))), z'0))
F(f(a, a), f(f(y'''', z''''), z'0)) -> F(a, f(f(f(b, f(a, y'''')), z''''), z'0))
F(f(a, a), f(f(f(y'''', z''''), z'0'), z)) -> F(f(a, a), f(f(y'''', z''''), z'0'))
F(f(a, a), f(a, z'''')) -> F(a, f(f(f(b, a), b), f(a, z'''')))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(b, f(a, b)), a), z'''''))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(b, a), f(b, a)), z'''''))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(f(b, a), b), a), z'''''))
F(f(a, a), f(a, z'''')) -> F(a, f(b, f(f(a, f(b, a)), z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(f(b, a), f(f(b, a), z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(b, f(f(f(a, b), a), z'''')))
F(f(a, a), f(y''', z'')) -> F(a, f(f(f(b, a), y'''), z''))
F(f(a, a), f(f(a, z''''), z)) -> F(f(a, a), f(a, z''''))
F(f(a, a), f(f(y'''', z''''), z)) -> F(f(a, a), f(y'''', z''''))
F(f(a, a), f(y', z)) -> F(f(a, a), y')
F(x'', f(f(y'', z''), z)) -> F(x'', f(y'', z''))
F(f(a, a), x'') -> F(a, f(f(b, a), x''))
F(x, f(y, z)) -> F(f(x, y), z)
F(f(a, a), f(f(f(f(y', z1), z0), z'), z'')) -> F(a, f(f(b, f(f(f(f(a, y'), z1), z0), z')), z''))


Rules:


f(f(a, a), x) -> f(a, f(b, f(a, x)))
f(x, f(y, z)) -> f(f(x, y), z)


Strategy:

innermost




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

F(f(a, a), f(a, z'''')) -> F(a, f(b, f(f(f(a, b), a), z'''')))
one new Dependency Pair is created:

F(f(a, a), f(a, z''''')) -> F(a, f(f(b, f(f(a, b), a)), z'''''))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 23
                 ↳Narrowing Transformation


Dependency Pairs:

F(f(a, a), f(f(f(f(y', z1), z0), z'), z'')) -> F(a, f(f(b, f(f(f(f(a, y'), z1), z0), z')), z''))
F(f(a, a), f(f(f(y''', z0'), z'''), z'')) -> F(a, f(f(f(b, f(f(a, y'''), z0')), z'''), z''))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(b, f(a, f(f(b, a), z''''))), z''))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(b, f(f(a, b), f(a, z''''))), z''))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(f(b, a), f(b, f(a, z''''))), z''))
F(f(a, a), f(f(f(y', z'), z'''), z'')) -> F(a, f(f(f(b, f(f(a, y'), z')), z'''), z''))
F(f(a, a), f(f(y''', z'''), z'')) -> F(a, f(f(f(f(b, a), y'''), z'''), z''))
F(f(a, a), f(f(f(f(y'', z1), z0), z''), z')) -> F(a, f(b, f(f(f(f(f(a, y''), z1), z0), z''), z')))
F(f(a, a), f(f(f(a, z0'), z''), z')) -> F(a, f(b, f(f(f(a, f(b, f(a, z0'))), z''), z')))
F(f(a, a), f(f(f(y'', z0'), z'''), z'0)) -> F(a, f(f(b, f(f(f(a, y''), z0'), z''')), z'0))
F(f(a, a), f(f(a, z''''), z')) -> F(a, f(b, f(f(a, f(f(b, a), z'''')), z')))
F(f(a, a), f(f(a, z''''), z')) -> F(a, f(b, f(f(f(a, b), f(a, z'''')), z')))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(b, f(a, f(b, f(a, z'''')))), z''))
F(f(a, a), f(f(f(y', z'), z'''), z'0)) -> F(a, f(f(b, f(f(f(a, y'), z'), z''')), z'0))
F(f(a, a), f(f(a, z''''), z'0)) -> F(a, f(f(b, f(a, f(b, f(a, z'''')))), z'0))
F(f(a, a), f(f(y'''', z''''), z'0)) -> F(a, f(f(f(b, f(a, y'''')), z''''), z'0))
F(f(a, a), f(f(f(y'''', z''''), z'0'), z)) -> F(f(a, a), f(f(y'''', z''''), z'0'))
F(f(a, a), f(a, z''''')) -> F(a, f(f(b, f(f(a, b), a)), z'''''))
F(f(a, a), f(a, z'''')) -> F(a, f(f(f(b, a), b), f(a, z'''')))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(b, f(a, b)), a), z'''''))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(b, a), f(b, a)), z'''''))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(f(b, a), b), a), z'''''))
F(f(a, a), f(a, z'''')) -> F(a, f(b, f(f(a, f(b, a)), z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(f(b, a), f(f(b, a), z'''')))
F(f(a, a), f(y''', z'')) -> F(a, f(f(f(b, a), y'''), z''))
F(f(a, a), f(f(a, z''''), z)) -> F(f(a, a), f(a, z''''))
F(f(a, a), f(f(y'''', z''''), z)) -> F(f(a, a), f(y'''', z''''))
F(f(a, a), f(y', z)) -> F(f(a, a), y')
F(x'', f(f(y'', z''), z)) -> F(x'', f(y'', z''))
F(f(a, a), x'') -> F(a, f(f(b, a), x''))
F(x, f(y, z)) -> F(f(x, y), z)
F(f(a, a), f(f(f(a, z0'), z'), z'')) -> F(a, f(f(b, f(f(a, f(b, f(a, z0'))), z')), z''))


Rules:


f(f(a, a), x) -> f(a, f(b, f(a, x)))
f(x, f(y, z)) -> f(f(x, y), z)


Strategy:

innermost




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

F(f(a, a), f(a, z'''')) -> F(a, f(f(b, a), f(f(b, a), z'''')))
one new Dependency Pair is created:

F(f(a, a), f(a, z''''')) -> F(a, f(f(f(b, a), f(b, a)), z'''''))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 24
                 ↳Narrowing Transformation


Dependency Pairs:

F(f(a, a), f(f(f(a, z0'), z'), z'')) -> F(a, f(f(b, f(f(a, f(b, f(a, z0'))), z')), z''))
F(f(a, a), f(f(f(y''', z0'), z'''), z'')) -> F(a, f(f(f(b, f(f(a, y'''), z0')), z'''), z''))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(b, f(a, f(f(b, a), z''''))), z''))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(b, f(f(a, b), f(a, z''''))), z''))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(f(b, a), f(b, f(a, z''''))), z''))
F(f(a, a), f(f(f(y', z'), z'''), z'')) -> F(a, f(f(f(b, f(f(a, y'), z')), z'''), z''))
F(f(a, a), f(f(y''', z'''), z'')) -> F(a, f(f(f(f(b, a), y'''), z'''), z''))
F(f(a, a), f(f(f(f(y'', z1), z0), z''), z')) -> F(a, f(b, f(f(f(f(f(a, y''), z1), z0), z''), z')))
F(f(a, a), f(f(f(a, z0'), z''), z')) -> F(a, f(b, f(f(f(a, f(b, f(a, z0'))), z''), z')))
F(f(a, a), f(f(f(y'', z0'), z'''), z'0)) -> F(a, f(f(b, f(f(f(a, y''), z0'), z''')), z'0))
F(f(a, a), f(f(a, z''''), z')) -> F(a, f(b, f(f(a, f(f(b, a), z'''')), z')))
F(f(a, a), f(f(a, z''''), z')) -> F(a, f(b, f(f(f(a, b), f(a, z'''')), z')))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(b, f(a, f(b, f(a, z'''')))), z''))
F(f(a, a), f(f(f(y', z'), z'''), z'0)) -> F(a, f(f(b, f(f(f(a, y'), z'), z''')), z'0))
F(f(a, a), f(f(a, z''''), z'0)) -> F(a, f(f(b, f(a, f(b, f(a, z'''')))), z'0))
F(f(a, a), f(f(y'''', z''''), z'0)) -> F(a, f(f(f(b, f(a, y'''')), z''''), z'0))
F(f(a, a), f(f(f(y'''', z''''), z'0'), z)) -> F(f(a, a), f(f(y'''', z''''), z'0'))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(b, a), f(b, a)), z'''''))
F(f(a, a), f(a, z''''')) -> F(a, f(f(b, f(f(a, b), a)), z'''''))
F(f(a, a), f(a, z'''')) -> F(a, f(f(f(b, a), b), f(a, z'''')))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(b, f(a, b)), a), z'''''))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(b, a), f(b, a)), z'''''))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(f(b, a), b), a), z'''''))
F(f(a, a), f(a, z'''')) -> F(a, f(b, f(f(a, f(b, a)), z'''')))
F(f(a, a), f(y''', z'')) -> F(a, f(f(f(b, a), y'''), z''))
F(f(a, a), f(f(a, z''''), z)) -> F(f(a, a), f(a, z''''))
F(f(a, a), f(f(y'''', z''''), z)) -> F(f(a, a), f(y'''', z''''))
F(f(a, a), f(y', z)) -> F(f(a, a), y')
F(x'', f(f(y'', z''), z)) -> F(x'', f(y'', z''))
F(f(a, a), x'') -> F(a, f(f(b, a), x''))
F(x, f(y, z)) -> F(f(x, y), z)
F(f(a, a), f(f(f(f(y', z1), z0), z'), z'')) -> F(a, f(f(b, f(f(f(f(a, y'), z1), z0), z')), z''))


Rules:


f(f(a, a), x) -> f(a, f(b, f(a, x)))
f(x, f(y, z)) -> f(f(x, y), z)


Strategy:

innermost




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

F(f(a, a), f(a, z'''')) -> F(a, f(b, f(f(a, f(b, a)), z'''')))
two new Dependency Pairs are created:

F(f(a, a), f(a, z''''')) -> F(a, f(f(b, f(a, f(b, a))), z'''''))
F(f(a, a), f(a, z'''')) -> F(a, f(b, f(f(f(a, b), a), z'''')))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 25
                 ↳Narrowing Transformation


Dependency Pairs:

F(f(a, a), f(f(f(f(y', z1), z0), z'), z'')) -> F(a, f(f(b, f(f(f(f(a, y'), z1), z0), z')), z''))
F(f(a, a), f(f(f(y''', z0'), z'''), z'')) -> F(a, f(f(f(b, f(f(a, y'''), z0')), z'''), z''))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(b, f(a, f(f(b, a), z''''))), z''))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(b, f(f(a, b), f(a, z''''))), z''))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(f(b, a), f(b, f(a, z''''))), z''))
F(f(a, a), f(f(f(y', z'), z'''), z'')) -> F(a, f(f(f(b, f(f(a, y'), z')), z'''), z''))
F(f(a, a), f(f(y''', z'''), z'')) -> F(a, f(f(f(f(b, a), y'''), z'''), z''))
F(f(a, a), f(f(f(f(y'', z1), z0), z''), z')) -> F(a, f(b, f(f(f(f(f(a, y''), z1), z0), z''), z')))
F(f(a, a), f(f(f(a, z0'), z''), z')) -> F(a, f(b, f(f(f(a, f(b, f(a, z0'))), z''), z')))
F(f(a, a), f(f(f(y'', z0'), z'''), z'0)) -> F(a, f(f(b, f(f(f(a, y''), z0'), z''')), z'0))
F(f(a, a), f(f(a, z''''), z')) -> F(a, f(b, f(f(a, f(f(b, a), z'''')), z')))
F(f(a, a), f(f(a, z''''), z')) -> F(a, f(b, f(f(f(a, b), f(a, z'''')), z')))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(b, f(a, f(b, f(a, z'''')))), z''))
F(f(a, a), f(f(f(y', z'), z'''), z'0)) -> F(a, f(f(b, f(f(f(a, y'), z'), z''')), z'0))
F(f(a, a), f(f(a, z''''), z'0)) -> F(a, f(f(b, f(a, f(b, f(a, z'''')))), z'0))
F(f(a, a), f(f(y'''', z''''), z'0)) -> F(a, f(f(f(b, f(a, y'''')), z''''), z'0))
F(f(a, a), f(f(f(y'''', z''''), z'0'), z)) -> F(f(a, a), f(f(y'''', z''''), z'0'))
F(f(a, a), f(a, z'''')) -> F(a, f(b, f(f(f(a, b), a), z'''')))
F(f(a, a), f(a, z''''')) -> F(a, f(f(b, f(a, f(b, a))), z'''''))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(b, a), f(b, a)), z'''''))
F(f(a, a), f(a, z''''')) -> F(a, f(f(b, f(f(a, b), a)), z'''''))
F(f(a, a), f(a, z'''')) -> F(a, f(f(f(b, a), b), f(a, z'''')))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(b, f(a, b)), a), z'''''))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(b, a), f(b, a)), z'''''))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(f(b, a), b), a), z'''''))
F(f(a, a), f(y''', z'')) -> F(a, f(f(f(b, a), y'''), z''))
F(f(a, a), f(f(a, z''''), z)) -> F(f(a, a), f(a, z''''))
F(f(a, a), f(f(y'''', z''''), z)) -> F(f(a, a), f(y'''', z''''))
F(f(a, a), f(y', z)) -> F(f(a, a), y')
F(x'', f(f(y'', z''), z)) -> F(x'', f(y'', z''))
F(f(a, a), x'') -> F(a, f(f(b, a), x''))
F(x, f(y, z)) -> F(f(x, y), z)
F(f(a, a), f(f(f(a, z0'), z'), z'')) -> F(a, f(f(b, f(f(a, f(b, f(a, z0'))), z')), z''))


Rules:


f(f(a, a), x) -> f(a, f(b, f(a, x)))
f(x, f(y, z)) -> f(f(x, y), z)


Strategy:

innermost




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

F(f(a, a), f(f(y'''', z''''), z'0)) -> F(a, f(f(f(b, f(a, y'''')), z''''), z'0))
two new Dependency Pairs are created:

F(f(a, a), f(f(y''''', z''''), z'0)) -> F(a, f(f(f(f(b, a), y'''''), z''''), z'0))
F(f(a, a), f(f(f(y', z'), z''''), z'0)) -> F(a, f(f(f(b, f(f(a, y'), z')), z''''), z'0))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 26
                 ↳Narrowing Transformation


Dependency Pairs:

F(f(a, a), f(f(f(y', z'), z''''), z'0)) -> F(a, f(f(f(b, f(f(a, y'), z')), z''''), z'0))
F(f(a, a), f(f(y''''', z''''), z'0)) -> F(a, f(f(f(f(b, a), y'''''), z''''), z'0))
F(f(a, a), f(f(f(a, z0'), z'), z'')) -> F(a, f(f(b, f(f(a, f(b, f(a, z0'))), z')), z''))
F(f(a, a), f(f(f(y''', z0'), z'''), z'')) -> F(a, f(f(f(b, f(f(a, y'''), z0')), z'''), z''))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(b, f(a, f(f(b, a), z''''))), z''))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(b, f(f(a, b), f(a, z''''))), z''))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(f(b, a), f(b, f(a, z''''))), z''))
F(f(a, a), f(f(f(y', z'), z'''), z'')) -> F(a, f(f(f(b, f(f(a, y'), z')), z'''), z''))
F(f(a, a), f(f(y''', z'''), z'')) -> F(a, f(f(f(f(b, a), y'''), z'''), z''))
F(f(a, a), f(f(f(f(y'', z1), z0), z''), z')) -> F(a, f(b, f(f(f(f(f(a, y''), z1), z0), z''), z')))
F(f(a, a), f(f(f(a, z0'), z''), z')) -> F(a, f(b, f(f(f(a, f(b, f(a, z0'))), z''), z')))
F(f(a, a), f(f(f(y'', z0'), z'''), z'0)) -> F(a, f(f(b, f(f(f(a, y''), z0'), z''')), z'0))
F(f(a, a), f(f(a, z''''), z')) -> F(a, f(b, f(f(a, f(f(b, a), z'''')), z')))
F(f(a, a), f(f(a, z''''), z')) -> F(a, f(b, f(f(f(a, b), f(a, z'''')), z')))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(b, f(a, f(b, f(a, z'''')))), z''))
F(f(a, a), f(f(f(y', z'), z'''), z'0)) -> F(a, f(f(b, f(f(f(a, y'), z'), z''')), z'0))
F(f(a, a), f(f(a, z''''), z'0)) -> F(a, f(f(b, f(a, f(b, f(a, z'''')))), z'0))
F(f(a, a), f(f(f(y'''', z''''), z'0'), z)) -> F(f(a, a), f(f(y'''', z''''), z'0'))
F(f(a, a), f(a, z'''')) -> F(a, f(b, f(f(f(a, b), a), z'''')))
F(f(a, a), f(a, z''''')) -> F(a, f(f(b, f(a, f(b, a))), z'''''))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(b, a), f(b, a)), z'''''))
F(f(a, a), f(a, z''''')) -> F(a, f(f(b, f(f(a, b), a)), z'''''))
F(f(a, a), f(a, z'''')) -> F(a, f(f(f(b, a), b), f(a, z'''')))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(b, f(a, b)), a), z'''''))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(b, a), f(b, a)), z'''''))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(f(b, a), b), a), z'''''))
F(f(a, a), f(y''', z'')) -> F(a, f(f(f(b, a), y'''), z''))
F(f(a, a), f(f(a, z''''), z)) -> F(f(a, a), f(a, z''''))
F(f(a, a), f(f(y'''', z''''), z)) -> F(f(a, a), f(y'''', z''''))
F(f(a, a), f(y', z)) -> F(f(a, a), y')
F(x'', f(f(y'', z''), z)) -> F(x'', f(y'', z''))
F(f(a, a), x'') -> F(a, f(f(b, a), x''))
F(x, f(y, z)) -> F(f(x, y), z)
F(f(a, a), f(f(f(f(y', z1), z0), z'), z'')) -> F(a, f(f(b, f(f(f(f(a, y'), z1), z0), z')), z''))


Rules:


f(f(a, a), x) -> f(a, f(b, f(a, x)))
f(x, f(y, z)) -> f(f(x, y), z)


Strategy:

innermost




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

F(f(a, a), f(f(a, z''''), z'0)) -> F(a, f(f(b, f(a, f(b, f(a, z'''')))), z'0))
three new Dependency Pairs are created:

F(f(a, a), f(f(a, z'''''), z'0)) -> F(a, f(f(f(b, a), f(b, f(a, z'''''))), z'0))
F(f(a, a), f(f(a, z'''''), z'0)) -> F(a, f(f(b, f(f(a, b), f(a, z'''''))), z'0))
F(f(a, a), f(f(a, z'''''), z'0)) -> F(a, f(f(b, f(a, f(f(b, a), z'''''))), z'0))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 27
                 ↳Narrowing Transformation


Dependency Pairs:

F(f(a, a), f(f(a, z'''''), z'0)) -> F(a, f(f(b, f(a, f(f(b, a), z'''''))), z'0))
F(f(a, a), f(f(a, z'''''), z'0)) -> F(a, f(f(b, f(f(a, b), f(a, z'''''))), z'0))
F(f(a, a), f(f(a, z'''''), z'0)) -> F(a, f(f(f(b, a), f(b, f(a, z'''''))), z'0))
F(f(a, a), f(f(y''''', z''''), z'0)) -> F(a, f(f(f(f(b, a), y'''''), z''''), z'0))
F(f(a, a), f(f(f(f(y', z1), z0), z'), z'')) -> F(a, f(f(b, f(f(f(f(a, y'), z1), z0), z')), z''))
F(f(a, a), f(f(f(a, z0'), z'), z'')) -> F(a, f(f(b, f(f(a, f(b, f(a, z0'))), z')), z''))
F(f(a, a), f(f(f(y''', z0'), z'''), z'')) -> F(a, f(f(f(b, f(f(a, y'''), z0')), z'''), z''))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(b, f(a, f(f(b, a), z''''))), z''))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(b, f(f(a, b), f(a, z''''))), z''))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(f(b, a), f(b, f(a, z''''))), z''))
F(f(a, a), f(f(f(y', z'), z'''), z'')) -> F(a, f(f(f(b, f(f(a, y'), z')), z'''), z''))
F(f(a, a), f(f(y''', z'''), z'')) -> F(a, f(f(f(f(b, a), y'''), z'''), z''))
F(f(a, a), f(f(f(f(y'', z1), z0), z''), z')) -> F(a, f(b, f(f(f(f(f(a, y''), z1), z0), z''), z')))
F(f(a, a), f(f(f(a, z0'), z''), z')) -> F(a, f(b, f(f(f(a, f(b, f(a, z0'))), z''), z')))
F(f(a, a), f(f(f(y'', z0'), z'''), z'0)) -> F(a, f(f(b, f(f(f(a, y''), z0'), z''')), z'0))
F(f(a, a), f(f(a, z''''), z')) -> F(a, f(b, f(f(a, f(f(b, a), z'''')), z')))
F(f(a, a), f(f(a, z''''), z')) -> F(a, f(b, f(f(f(a, b), f(a, z'''')), z')))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(b, f(a, f(b, f(a, z'''')))), z''))
F(f(a, a), f(f(f(y', z'), z'''), z'0)) -> F(a, f(f(b, f(f(f(a, y'), z'), z''')), z'0))
F(f(a, a), f(f(f(y'''', z''''), z'0'), z)) -> F(f(a, a), f(f(y'''', z''''), z'0'))
F(f(a, a), f(a, z'''')) -> F(a, f(b, f(f(f(a, b), a), z'''')))
F(f(a, a), f(a, z''''')) -> F(a, f(f(b, f(a, f(b, a))), z'''''))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(b, a), f(b, a)), z'''''))
F(f(a, a), f(a, z''''')) -> F(a, f(f(b, f(f(a, b), a)), z'''''))
F(f(a, a), f(a, z'''')) -> F(a, f(f(f(b, a), b), f(a, z'''')))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(b, f(a, b)), a), z'''''))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(b, a), f(b, a)), z'''''))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(f(b, a), b), a), z'''''))
F(f(a, a), f(y''', z'')) -> F(a, f(f(f(b, a), y'''), z''))
F(f(a, a), f(f(a, z''''), z)) -> F(f(a, a), f(a, z''''))
F(f(a, a), f(f(y'''', z''''), z)) -> F(f(a, a), f(y'''', z''''))
F(f(a, a), f(y', z)) -> F(f(a, a), y')
F(x'', f(f(y'', z''), z)) -> F(x'', f(y'', z''))
F(f(a, a), x'') -> F(a, f(f(b, a), x''))
F(x, f(y, z)) -> F(f(x, y), z)
F(f(a, a), f(f(f(y', z'), z''''), z'0)) -> F(a, f(f(f(b, f(f(a, y'), z')), z''''), z'0))


Rules:


f(f(a, a), x) -> f(a, f(b, f(a, x)))
f(x, f(y, z)) -> f(f(x, y), z)


Strategy:

innermost




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

F(f(a, a), f(f(f(y', z'), z'''), z'0)) -> F(a, f(f(b, f(f(f(a, y'), z'), z''')), z'0))
three new Dependency Pairs are created:

F(f(a, a), f(f(f(y'', z''), z''''), z'0)) -> F(a, f(f(f(b, f(f(a, y''), z'')), z''''), z'0))
F(f(a, a), f(f(f(a, z''), z'''), z'0)) -> F(a, f(f(b, f(f(a, f(b, f(a, z''))), z''')), z'0))
F(f(a, a), f(f(f(f(y'', z''), z'), z'''), z'0)) -> F(a, f(f(b, f(f(f(f(a, y''), z''), z'), z''')), z'0))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 28
                 ↳Narrowing Transformation


Dependency Pairs:

F(f(a, a), f(f(f(f(y'', z''), z'), z'''), z'0)) -> F(a, f(f(b, f(f(f(f(a, y''), z''), z'), z''')), z'0))
F(f(a, a), f(f(f(a, z''), z'''), z'0)) -> F(a, f(f(b, f(f(a, f(b, f(a, z''))), z''')), z'0))
F(f(a, a), f(f(f(y'', z''), z''''), z'0)) -> F(a, f(f(f(b, f(f(a, y''), z'')), z''''), z'0))
F(f(a, a), f(f(a, z'''''), z'0)) -> F(a, f(f(b, f(f(a, b), f(a, z'''''))), z'0))
F(f(a, a), f(f(a, z'''''), z'0)) -> F(a, f(f(f(b, a), f(b, f(a, z'''''))), z'0))
F(f(a, a), f(f(f(y', z'), z''''), z'0)) -> F(a, f(f(f(b, f(f(a, y'), z')), z''''), z'0))
F(f(a, a), f(f(y''''', z''''), z'0)) -> F(a, f(f(f(f(b, a), y'''''), z''''), z'0))
F(f(a, a), f(f(f(f(y', z1), z0), z'), z'')) -> F(a, f(f(b, f(f(f(f(a, y'), z1), z0), z')), z''))
F(f(a, a), f(f(f(a, z0'), z'), z'')) -> F(a, f(f(b, f(f(a, f(b, f(a, z0'))), z')), z''))
F(f(a, a), f(f(f(y''', z0'), z'''), z'')) -> F(a, f(f(f(b, f(f(a, y'''), z0')), z'''), z''))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(b, f(a, f(f(b, a), z''''))), z''))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(b, f(f(a, b), f(a, z''''))), z''))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(f(b, a), f(b, f(a, z''''))), z''))
F(f(a, a), f(f(f(y', z'), z'''), z'')) -> F(a, f(f(f(b, f(f(a, y'), z')), z'''), z''))
F(f(a, a), f(f(y''', z'''), z'')) -> F(a, f(f(f(f(b, a), y'''), z'''), z''))
F(f(a, a), f(f(f(f(y'', z1), z0), z''), z')) -> F(a, f(b, f(f(f(f(f(a, y''), z1), z0), z''), z')))
F(f(a, a), f(f(f(a, z0'), z''), z')) -> F(a, f(b, f(f(f(a, f(b, f(a, z0'))), z''), z')))
F(f(a, a), f(f(f(y'', z0'), z'''), z'0)) -> F(a, f(f(b, f(f(f(a, y''), z0'), z''')), z'0))
F(f(a, a), f(f(a, z''''), z')) -> F(a, f(b, f(f(a, f(f(b, a), z'''')), z')))
F(f(a, a), f(f(a, z''''), z')) -> F(a, f(b, f(f(f(a, b), f(a, z'''')), z')))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(b, f(a, f(b, f(a, z'''')))), z''))
F(f(a, a), f(f(f(y'''', z''''), z'0'), z)) -> F(f(a, a), f(f(y'''', z''''), z'0'))
F(f(a, a), f(a, z'''')) -> F(a, f(b, f(f(f(a, b), a), z'''')))
F(f(a, a), f(a, z''''')) -> F(a, f(f(b, f(a, f(b, a))), z'''''))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(b, a), f(b, a)), z'''''))
F(f(a, a), f(a, z''''')) -> F(a, f(f(b, f(f(a, b), a)), z'''''))
F(f(a, a), f(a, z'''')) -> F(a, f(f(f(b, a), b), f(a, z'''')))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(b, f(a, b)), a), z'''''))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(b, a), f(b, a)), z'''''))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(f(b, a), b), a), z'''''))
F(f(a, a), f(y''', z'')) -> F(a, f(f(f(b, a), y'''), z''))
F(f(a, a), f(f(a, z''''), z)) -> F(f(a, a), f(a, z''''))
F(f(a, a), f(f(y'''', z''''), z)) -> F(f(a, a), f(y'''', z''''))
F(f(a, a), f(y', z)) -> F(f(a, a), y')
F(x'', f(f(y'', z''), z)) -> F(x'', f(y'', z''))
F(f(a, a), x'') -> F(a, f(f(b, a), x''))
F(x, f(y, z)) -> F(f(x, y), z)
F(f(a, a), f(f(a, z'''''), z'0)) -> F(a, f(f(b, f(a, f(f(b, a), z'''''))), z'0))


Rules:


f(f(a, a), x) -> f(a, f(b, f(a, x)))
f(x, f(y, z)) -> f(f(x, y), z)


Strategy:

innermost




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

F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(b, f(a, f(b, f(a, z'''')))), z''))
three new Dependency Pairs are created:

F(f(a, a), f(f(a, z'''''), z'')) -> F(a, f(f(f(b, a), f(b, f(a, z'''''))), z''))
F(f(a, a), f(f(a, z'''''), z'')) -> F(a, f(f(b, f(f(a, b), f(a, z'''''))), z''))
F(f(a, a), f(f(a, z'''''), z'')) -> F(a, f(f(b, f(a, f(f(b, a), z'''''))), z''))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 29
                 ↳Narrowing Transformation


Dependency Pairs:

F(f(a, a), f(f(a, z'''''), z'')) -> F(a, f(f(b, f(a, f(f(b, a), z'''''))), z''))
F(f(a, a), f(f(a, z'''''), z'')) -> F(a, f(f(b, f(f(a, b), f(a, z'''''))), z''))
F(f(a, a), f(f(a, z'''''), z'')) -> F(a, f(f(f(b, a), f(b, f(a, z'''''))), z''))
F(f(a, a), f(f(f(a, z''), z'''), z'0)) -> F(a, f(f(b, f(f(a, f(b, f(a, z''))), z''')), z'0))
F(f(a, a), f(f(f(y'', z''), z''''), z'0)) -> F(a, f(f(f(b, f(f(a, y''), z'')), z''''), z'0))
F(f(a, a), f(f(a, z'''''), z'0)) -> F(a, f(f(b, f(a, f(f(b, a), z'''''))), z'0))
F(f(a, a), f(f(a, z'''''), z'0)) -> F(a, f(f(b, f(f(a, b), f(a, z'''''))), z'0))
F(f(a, a), f(f(a, z'''''), z'0)) -> F(a, f(f(f(b, a), f(b, f(a, z'''''))), z'0))
F(f(a, a), f(f(f(y', z'), z''''), z'0)) -> F(a, f(f(f(b, f(f(a, y'), z')), z''''), z'0))
F(f(a, a), f(f(y''''', z''''), z'0)) -> F(a, f(f(f(f(b, a), y'''''), z''''), z'0))
F(f(a, a), f(f(f(f(y', z1), z0), z'), z'')) -> F(a, f(f(b, f(f(f(f(a, y'), z1), z0), z')), z''))
F(f(a, a), f(f(f(a, z0'), z'), z'')) -> F(a, f(f(b, f(f(a, f(b, f(a, z0'))), z')), z''))
F(f(a, a), f(f(f(y''', z0'), z'''), z'')) -> F(a, f(f(f(b, f(f(a, y'''), z0')), z'''), z''))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(b, f(a, f(f(b, a), z''''))), z''))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(b, f(f(a, b), f(a, z''''))), z''))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(f(b, a), f(b, f(a, z''''))), z''))
F(f(a, a), f(f(f(y', z'), z'''), z'')) -> F(a, f(f(f(b, f(f(a, y'), z')), z'''), z''))
F(f(a, a), f(f(y''', z'''), z'')) -> F(a, f(f(f(f(b, a), y'''), z'''), z''))
F(f(a, a), f(f(f(f(y'', z1), z0), z''), z')) -> F(a, f(b, f(f(f(f(f(a, y''), z1), z0), z''), z')))
F(f(a, a), f(f(f(a, z0'), z''), z')) -> F(a, f(b, f(f(f(a, f(b, f(a, z0'))), z''), z')))
F(f(a, a), f(f(f(y'', z0'), z'''), z'0)) -> F(a, f(f(b, f(f(f(a, y''), z0'), z''')), z'0))
F(f(a, a), f(f(a, z''''), z')) -> F(a, f(b, f(f(a, f(f(b, a), z'''')), z')))
F(f(a, a), f(f(a, z''''), z')) -> F(a, f(b, f(f(f(a, b), f(a, z'''')), z')))
F(f(a, a), f(f(f(y'''', z''''), z'0'), z)) -> F(f(a, a), f(f(y'''', z''''), z'0'))
F(f(a, a), f(a, z'''')) -> F(a, f(b, f(f(f(a, b), a), z'''')))
F(f(a, a), f(a, z''''')) -> F(a, f(f(b, f(a, f(b, a))), z'''''))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(b, a), f(b, a)), z'''''))
F(f(a, a), f(a, z''''')) -> F(a, f(f(b, f(f(a, b), a)), z'''''))
F(f(a, a), f(a, z'''')) -> F(a, f(f(f(b, a), b), f(a, z'''')))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(b, f(a, b)), a), z'''''))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(b, a), f(b, a)), z'''''))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(f(b, a), b), a), z'''''))
F(f(a, a), f(y''', z'')) -> F(a, f(f(f(b, a), y'''), z''))
F(f(a, a), f(f(a, z''''), z)) -> F(f(a, a), f(a, z''''))
F(f(a, a), f(f(y'''', z''''), z)) -> F(f(a, a), f(y'''', z''''))
F(f(a, a), f(y', z)) -> F(f(a, a), y')
F(x'', f(f(y'', z''), z)) -> F(x'', f(y'', z''))
F(f(a, a), x'') -> F(a, f(f(b, a), x''))
F(x, f(y, z)) -> F(f(x, y), z)
F(f(a, a), f(f(f(f(y'', z''), z'), z'''), z'0)) -> F(a, f(f(b, f(f(f(f(a, y''), z''), z'), z''')), z'0))


Rules:


f(f(a, a), x) -> f(a, f(b, f(a, x)))
f(x, f(y, z)) -> f(f(x, y), z)


Strategy:

innermost




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

F(f(a, a), f(f(a, z''''), z')) -> F(a, f(b, f(f(f(a, b), f(a, z'''')), z')))
two new Dependency Pairs are created:

F(f(a, a), f(f(a, z'''''), z'')) -> F(a, f(f(b, f(f(a, b), f(a, z'''''))), z''))
F(f(a, a), f(f(a, z'''''), z')) -> F(a, f(b, f(f(f(f(a, b), a), z'''''), z')))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 30
                 ↳Narrowing Transformation


Dependency Pairs:

F(f(a, a), f(f(a, z'''''), z')) -> F(a, f(b, f(f(f(f(a, b), a), z'''''), z')))
F(f(a, a), f(f(a, z'''''), z'')) -> F(a, f(f(b, f(f(a, b), f(a, z'''''))), z''))
F(f(a, a), f(f(a, z'''''), z'')) -> F(a, f(f(b, f(f(a, b), f(a, z'''''))), z''))
F(f(a, a), f(f(a, z'''''), z'')) -> F(a, f(f(f(b, a), f(b, f(a, z'''''))), z''))
F(f(a, a), f(f(f(f(y'', z''), z'), z'''), z'0)) -> F(a, f(f(b, f(f(f(f(a, y''), z''), z'), z''')), z'0))
F(f(a, a), f(f(f(a, z''), z'''), z'0)) -> F(a, f(f(b, f(f(a, f(b, f(a, z''))), z''')), z'0))
F(f(a, a), f(f(f(y'', z''), z''''), z'0)) -> F(a, f(f(f(b, f(f(a, y''), z'')), z''''), z'0))
F(f(a, a), f(f(a, z'''''), z'0)) -> F(a, f(f(b, f(a, f(f(b, a), z'''''))), z'0))
F(f(a, a), f(f(a, z'''''), z'0)) -> F(a, f(f(b, f(f(a, b), f(a, z'''''))), z'0))
F(f(a, a), f(f(a, z'''''), z'0)) -> F(a, f(f(f(b, a), f(b, f(a, z'''''))), z'0))
F(f(a, a), f(f(f(y', z'), z''''), z'0)) -> F(a, f(f(f(b, f(f(a, y'), z')), z''''), z'0))
F(f(a, a), f(f(y''''', z''''), z'0)) -> F(a, f(f(f(f(b, a), y'''''), z''''), z'0))
F(f(a, a), f(f(f(f(y', z1), z0), z'), z'')) -> F(a, f(f(b, f(f(f(f(a, y'), z1), z0), z')), z''))
F(f(a, a), f(f(f(a, z0'), z'), z'')) -> F(a, f(f(b, f(f(a, f(b, f(a, z0'))), z')), z''))
F(f(a, a), f(f(f(y''', z0'), z'''), z'')) -> F(a, f(f(f(b, f(f(a, y'''), z0')), z'''), z''))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(b, f(a, f(f(b, a), z''''))), z''))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(b, f(f(a, b), f(a, z''''))), z''))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(f(b, a), f(b, f(a, z''''))), z''))
F(f(a, a), f(f(f(y', z'), z'''), z'')) -> F(a, f(f(f(b, f(f(a, y'), z')), z'''), z''))
F(f(a, a), f(f(y''', z'''), z'')) -> F(a, f(f(f(f(b, a), y'''), z'''), z''))
F(f(a, a), f(f(f(f(y'', z1), z0), z''), z')) -> F(a, f(b, f(f(f(f(f(a, y''), z1), z0), z''), z')))
F(f(a, a), f(f(f(a, z0'), z''), z')) -> F(a, f(b, f(f(f(a, f(b, f(a, z0'))), z''), z')))
F(f(a, a), f(f(f(y'', z0'), z'''), z'0)) -> F(a, f(f(b, f(f(f(a, y''), z0'), z''')), z'0))
F(f(a, a), f(f(a, z''''), z')) -> F(a, f(b, f(f(a, f(f(b, a), z'''')), z')))
F(f(a, a), f(f(f(y'''', z''''), z'0'), z)) -> F(f(a, a), f(f(y'''', z''''), z'0'))
F(f(a, a), f(a, z'''')) -> F(a, f(b, f(f(f(a, b), a), z'''')))
F(f(a, a), f(a, z''''')) -> F(a, f(f(b, f(a, f(b, a))), z'''''))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(b, a), f(b, a)), z'''''))
F(f(a, a), f(a, z''''')) -> F(a, f(f(b, f(f(a, b), a)), z'''''))
F(f(a, a), f(a, z'''')) -> F(a, f(f(f(b, a), b), f(a, z'''')))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(b, f(a, b)), a), z'''''))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(b, a), f(b, a)), z'''''))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(f(b, a), b), a), z'''''))
F(f(a, a), f(y''', z'')) -> F(a, f(f(f(b, a), y'''), z''))
F(f(a, a), f(f(a, z''''), z)) -> F(f(a, a), f(a, z''''))
F(f(a, a), f(f(y'''', z''''), z)) -> F(f(a, a), f(y'''', z''''))
F(f(a, a), f(y', z)) -> F(f(a, a), y')
F(x'', f(f(y'', z''), z)) -> F(x'', f(y'', z''))
F(f(a, a), x'') -> F(a, f(f(b, a), x''))
F(x, f(y, z)) -> F(f(x, y), z)
F(f(a, a), f(f(a, z'''''), z'')) -> F(a, f(f(b, f(a, f(f(b, a), z'''''))), z''))


Rules:


f(f(a, a), x) -> f(a, f(b, f(a, x)))
f(x, f(y, z)) -> f(f(x, y), z)


Strategy:

innermost




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

F(f(a, a), f(f(a, z''''), z')) -> F(a, f(b, f(f(a, f(f(b, a), z'''')), z')))
two new Dependency Pairs are created:

F(f(a, a), f(f(a, z'''''), z'')) -> F(a, f(f(b, f(a, f(f(b, a), z'''''))), z''))
F(f(a, a), f(f(a, z'''''), z')) -> F(a, f(b, f(f(f(a, f(b, a)), z'''''), z')))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 31
                 ↳Narrowing Transformation


Dependency Pairs:

F(f(a, a), f(f(a, z'''''), z')) -> F(a, f(b, f(f(f(a, f(b, a)), z'''''), z')))
F(f(a, a), f(f(a, z'''''), z'')) -> F(a, f(f(b, f(a, f(f(b, a), z'''''))), z''))
F(f(a, a), f(f(a, z'''''), z'')) -> F(a, f(f(b, f(f(a, b), f(a, z'''''))), z''))
F(f(a, a), f(f(a, z'''''), z'')) -> F(a, f(f(b, f(a, f(f(b, a), z'''''))), z''))
F(f(a, a), f(f(a, z'''''), z'')) -> F(a, f(f(b, f(f(a, b), f(a, z'''''))), z''))
F(f(a, a), f(f(a, z'''''), z'')) -> F(a, f(f(f(b, a), f(b, f(a, z'''''))), z''))
F(f(a, a), f(f(f(f(y'', z''), z'), z'''), z'0)) -> F(a, f(f(b, f(f(f(f(a, y''), z''), z'), z''')), z'0))
F(f(a, a), f(f(f(a, z''), z'''), z'0)) -> F(a, f(f(b, f(f(a, f(b, f(a, z''))), z''')), z'0))
F(f(a, a), f(f(f(y'', z''), z''''), z'0)) -> F(a, f(f(f(b, f(f(a, y''), z'')), z''''), z'0))
F(f(a, a), f(f(a, z'''''), z'0)) -> F(a, f(f(b, f(a, f(f(b, a), z'''''))), z'0))
F(f(a, a), f(f(a, z'''''), z'0)) -> F(a, f(f(b, f(f(a, b), f(a, z'''''))), z'0))
F(f(a, a), f(f(a, z'''''), z'0)) -> F(a, f(f(f(b, a), f(b, f(a, z'''''))), z'0))
F(f(a, a), f(f(f(y', z'), z''''), z'0)) -> F(a, f(f(f(b, f(f(a, y'), z')), z''''), z'0))
F(f(a, a), f(f(y''''', z''''), z'0)) -> F(a, f(f(f(f(b, a), y'''''), z''''), z'0))
F(f(a, a), f(f(f(f(y', z1), z0), z'), z'')) -> F(a, f(f(b, f(f(f(f(a, y'), z1), z0), z')), z''))
F(f(a, a), f(f(f(a, z0'), z'), z'')) -> F(a, f(f(b, f(f(a, f(b, f(a, z0'))), z')), z''))
F(f(a, a), f(f(f(y''', z0'), z'''), z'')) -> F(a, f(f(f(b, f(f(a, y'''), z0')), z'''), z''))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(b, f(a, f(f(b, a), z''''))), z''))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(b, f(f(a, b), f(a, z''''))), z''))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(f(b, a), f(b, f(a, z''''))), z''))
F(f(a, a), f(f(f(y', z'), z'''), z'')) -> F(a, f(f(f(b, f(f(a, y'), z')), z'''), z''))
F(f(a, a), f(f(y''', z'''), z'')) -> F(a, f(f(f(f(b, a), y'''), z'''), z''))
F(f(a, a), f(f(f(f(y'', z1), z0), z''), z')) -> F(a, f(b, f(f(f(f(f(a, y''), z1), z0), z''), z')))
F(f(a, a), f(f(f(a, z0'), z''), z')) -> F(a, f(b, f(f(f(a, f(b, f(a, z0'))), z''), z')))
F(f(a, a), f(f(f(y'', z0'), z'''), z'0)) -> F(a, f(f(b, f(f(f(a, y''), z0'), z''')), z'0))
F(f(a, a), f(f(f(y'''', z''''), z'0'), z)) -> F(f(a, a), f(f(y'''', z''''), z'0'))
F(f(a, a), f(a, z'''')) -> F(a, f(b, f(f(f(a, b), a), z'''')))
F(f(a, a), f(a, z''''')) -> F(a, f(f(b, f(a, f(b, a))), z'''''))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(b, a), f(b, a)), z'''''))
F(f(a, a), f(a, z''''')) -> F(a, f(f(b, f(f(a, b), a)), z'''''))
F(f(a, a), f(a, z'''')) -> F(a, f(f(f(b, a), b), f(a, z'''')))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(b, f(a, b)), a), z'''''))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(b, a), f(b, a)), z'''''))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(f(b, a), b), a), z'''''))
F(f(a, a), f(y''', z'')) -> F(a, f(f(f(b, a), y'''), z''))
F(f(a, a), f(f(a, z''''), z)) -> F(f(a, a), f(a, z''''))
F(f(a, a), f(f(y'''', z''''), z)) -> F(f(a, a), f(y'''', z''''))
F(f(a, a), f(y', z)) -> F(f(a, a), y')
F(x'', f(f(y'', z''), z)) -> F(x'', f(y'', z''))
F(f(a, a), x'') -> F(a, f(f(b, a), x''))
F(x, f(y, z)) -> F(f(x, y), z)
F(f(a, a), f(f(a, z'''''), z')) -> F(a, f(b, f(f(f(f(a, b), a), z'''''), z')))


Rules:


f(f(a, a), x) -> f(a, f(b, f(a, x)))
f(x, f(y, z)) -> f(f(x, y), z)


Strategy:

innermost




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

F(f(a, a), f(f(f(y'', z0'), z'''), z'0)) -> F(a, f(f(b, f(f(f(a, y''), z0'), z''')), z'0))
three new Dependency Pairs are created:

F(f(a, a), f(f(f(y''', z0''), z''''), z'0)) -> F(a, f(f(f(b, f(f(a, y'''), z0'')), z''''), z'0))
F(f(a, a), f(f(f(a, z0''), z'''), z'0)) -> F(a, f(f(b, f(f(a, f(b, f(a, z0''))), z''')), z'0))
F(f(a, a), f(f(f(f(y', z'), z0'), z'''), z'0)) -> F(a, f(f(b, f(f(f(f(a, y'), z'), z0'), z''')), z'0))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 32
                 ↳Narrowing Transformation


Dependency Pairs:

F(f(a, a), f(f(f(f(y', z'), z0'), z'''), z'0)) -> F(a, f(f(b, f(f(f(f(a, y'), z'), z0'), z''')), z'0))
F(f(a, a), f(f(f(a, z0''), z'''), z'0)) -> F(a, f(f(b, f(f(a, f(b, f(a, z0''))), z''')), z'0))
F(f(a, a), f(f(f(y''', z0''), z''''), z'0)) -> F(a, f(f(f(b, f(f(a, y'''), z0'')), z''''), z'0))
F(f(a, a), f(f(a, z'''''), z'')) -> F(a, f(f(b, f(a, f(f(b, a), z'''''))), z''))
F(f(a, a), f(f(a, z'''''), z')) -> F(a, f(b, f(f(f(f(a, b), a), z'''''), z')))
F(f(a, a), f(f(a, z'''''), z'')) -> F(a, f(f(b, f(f(a, b), f(a, z'''''))), z''))
F(f(a, a), f(f(a, z'''''), z'')) -> F(a, f(f(b, f(a, f(f(b, a), z'''''))), z''))
F(f(a, a), f(f(a, z'''''), z'')) -> F(a, f(f(b, f(f(a, b), f(a, z'''''))), z''))
F(f(a, a), f(f(a, z'''''), z'')) -> F(a, f(f(f(b, a), f(b, f(a, z'''''))), z''))
F(f(a, a), f(f(f(f(y'', z''), z'), z'''), z'0)) -> F(a, f(f(b, f(f(f(f(a, y''), z''), z'), z''')), z'0))
F(f(a, a), f(f(f(a, z''), z'''), z'0)) -> F(a, f(f(b, f(f(a, f(b, f(a, z''))), z''')), z'0))
F(f(a, a), f(f(f(y'', z''), z''''), z'0)) -> F(a, f(f(f(b, f(f(a, y''), z'')), z''''), z'0))
F(f(a, a), f(f(a, z'''''), z'0)) -> F(a, f(f(b, f(a, f(f(b, a), z'''''))), z'0))
F(f(a, a), f(f(a, z'''''), z'0)) -> F(a, f(f(b, f(f(a, b), f(a, z'''''))), z'0))
F(f(a, a), f(f(a, z'''''), z'0)) -> F(a, f(f(f(b, a), f(b, f(a, z'''''))), z'0))
F(f(a, a), f(f(f(y', z'), z''''), z'0)) -> F(a, f(f(f(b, f(f(a, y'), z')), z''''), z'0))
F(f(a, a), f(f(y''''', z''''), z'0)) -> F(a, f(f(f(f(b, a), y'''''), z''''), z'0))
F(f(a, a), f(f(f(f(y', z1), z0), z'), z'')) -> F(a, f(f(b, f(f(f(f(a, y'), z1), z0), z')), z''))
F(f(a, a), f(f(f(a, z0'), z'), z'')) -> F(a, f(f(b, f(f(a, f(b, f(a, z0'))), z')), z''))
F(f(a, a), f(f(f(y''', z0'), z'''), z'')) -> F(a, f(f(f(b, f(f(a, y'''), z0')), z'''), z''))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(b, f(a, f(f(b, a), z''''))), z''))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(b, f(f(a, b), f(a, z''''))), z''))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(f(b, a), f(b, f(a, z''''))), z''))
F(f(a, a), f(f(f(y', z'), z'''), z'')) -> F(a, f(f(f(b, f(f(a, y'), z')), z'''), z''))
F(f(a, a), f(f(y''', z'''), z'')) -> F(a, f(f(f(f(b, a), y'''), z'''), z''))
F(f(a, a), f(f(f(f(y'', z1), z0), z''), z')) -> F(a, f(b, f(f(f(f(f(a, y''), z1), z0), z''), z')))
F(f(a, a), f(f(f(a, z0'), z''), z')) -> F(a, f(b, f(f(f(a, f(b, f(a, z0'))), z''), z')))
F(f(a, a), f(f(f(y'''', z''''), z'0'), z)) -> F(f(a, a), f(f(y'''', z''''), z'0'))
F(f(a, a), f(a, z'''')) -> F(a, f(b, f(f(f(a, b), a), z'''')))
F(f(a, a), f(a, z''''')) -> F(a, f(f(b, f(a, f(b, a))), z'''''))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(b, a), f(b, a)), z'''''))
F(f(a, a), f(a, z''''')) -> F(a, f(f(b, f(f(a, b), a)), z'''''))
F(f(a, a), f(a, z'''')) -> F(a, f(f(f(b, a), b), f(a, z'''')))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(b, f(a, b)), a), z'''''))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(b, a), f(b, a)), z'''''))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(f(b, a), b), a), z'''''))
F(f(a, a), f(y''', z'')) -> F(a, f(f(f(b, a), y'''), z''))
F(f(a, a), f(f(a, z''''), z)) -> F(f(a, a), f(a, z''''))
F(f(a, a), f(f(y'''', z''''), z)) -> F(f(a, a), f(y'''', z''''))
F(f(a, a), f(y', z)) -> F(f(a, a), y')
F(x'', f(f(y'', z''), z)) -> F(x'', f(y'', z''))
F(f(a, a), x'') -> F(a, f(f(b, a), x''))
F(x, f(y, z)) -> F(f(x, y), z)
F(f(a, a), f(f(a, z'''''), z')) -> F(a, f(b, f(f(f(a, f(b, a)), z'''''), z')))


Rules:


f(f(a, a), x) -> f(a, f(b, f(a, x)))
f(x, f(y, z)) -> f(f(x, y), z)


Strategy:

innermost




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

F(f(a, a), f(f(f(a, z0'), z''), z')) -> F(a, f(b, f(f(f(a, f(b, f(a, z0'))), z''), z')))
three new Dependency Pairs are created:

F(f(a, a), f(f(f(a, z0''), z'''), z'0)) -> F(a, f(f(b, f(f(a, f(b, f(a, z0''))), z''')), z'0))
F(f(a, a), f(f(f(a, z0''), z''), z')) -> F(a, f(b, f(f(f(f(a, b), f(a, z0'')), z''), z')))
F(f(a, a), f(f(f(a, z0''), z''), z')) -> F(a, f(b, f(f(f(a, f(f(b, a), z0'')), z''), z')))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 33
                 ↳Narrowing Transformation


Dependency Pairs:

F(f(a, a), f(f(f(a, z0''), z''), z')) -> F(a, f(b, f(f(f(a, f(f(b, a), z0'')), z''), z')))
F(f(a, a), f(f(f(a, z0''), z''), z')) -> F(a, f(b, f(f(f(f(a, b), f(a, z0'')), z''), z')))
F(f(a, a), f(f(f(a, z0''), z'''), z'0)) -> F(a, f(f(b, f(f(a, f(b, f(a, z0''))), z''')), z'0))
F(f(a, a), f(f(f(a, z0''), z'''), z'0)) -> F(a, f(f(b, f(f(a, f(b, f(a, z0''))), z''')), z'0))
F(f(a, a), f(f(f(y''', z0''), z''''), z'0)) -> F(a, f(f(f(b, f(f(a, y'''), z0'')), z''''), z'0))
F(f(a, a), f(f(a, z'''''), z')) -> F(a, f(b, f(f(f(a, f(b, a)), z'''''), z')))
F(f(a, a), f(f(a, z'''''), z'')) -> F(a, f(f(b, f(a, f(f(b, a), z'''''))), z''))
F(f(a, a), f(f(a, z'''''), z')) -> F(a, f(b, f(f(f(f(a, b), a), z'''''), z')))
F(f(a, a), f(f(a, z'''''), z'')) -> F(a, f(f(b, f(f(a, b), f(a, z'''''))), z''))
F(f(a, a), f(f(a, z'''''), z'')) -> F(a, f(f(b, f(a, f(f(b, a), z'''''))), z''))
F(f(a, a), f(f(a, z'''''), z'')) -> F(a, f(f(b, f(f(a, b), f(a, z'''''))), z''))
F(f(a, a), f(f(a, z'''''), z'')) -> F(a, f(f(f(b, a), f(b, f(a, z'''''))), z''))
F(f(a, a), f(f(f(f(y'', z''), z'), z'''), z'0)) -> F(a, f(f(b, f(f(f(f(a, y''), z''), z'), z''')), z'0))
F(f(a, a), f(f(f(a, z''), z'''), z'0)) -> F(a, f(f(b, f(f(a, f(b, f(a, z''))), z''')), z'0))
F(f(a, a), f(f(f(y'', z''), z''''), z'0)) -> F(a, f(f(f(b, f(f(a, y''), z'')), z''''), z'0))
F(f(a, a), f(f(a, z'''''), z'0)) -> F(a, f(f(b, f(a, f(f(b, a), z'''''))), z'0))
F(f(a, a), f(f(a, z'''''), z'0)) -> F(a, f(f(b, f(f(a, b), f(a, z'''''))), z'0))
F(f(a, a), f(f(a, z'''''), z'0)) -> F(a, f(f(f(b, a), f(b, f(a, z'''''))), z'0))
F(f(a, a), f(f(f(y', z'), z''''), z'0)) -> F(a, f(f(f(b, f(f(a, y'), z')), z''''), z'0))
F(f(a, a), f(f(y''''', z''''), z'0)) -> F(a, f(f(f(f(b, a), y'''''), z''''), z'0))
F(f(a, a), f(f(f(f(y', z1), z0), z'), z'')) -> F(a, f(f(b, f(f(f(f(a, y'), z1), z0), z')), z''))
F(f(a, a), f(f(f(a, z0'), z'), z'')) -> F(a, f(f(b, f(f(a, f(b, f(a, z0'))), z')), z''))
F(f(a, a), f(f(f(y''', z0'), z'''), z'')) -> F(a, f(f(f(b, f(f(a, y'''), z0')), z'''), z''))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(b, f(a, f(f(b, a), z''''))), z''))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(b, f(f(a, b), f(a, z''''))), z''))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(f(b, a), f(b, f(a, z''''))), z''))
F(f(a, a), f(f(f(y', z'), z'''), z'')) -> F(a, f(f(f(b, f(f(a, y'), z')), z'''), z''))
F(f(a, a), f(f(y''', z'''), z'')) -> F(a, f(f(f(f(b, a), y'''), z'''), z''))
F(f(a, a), f(f(f(f(y'', z1), z0), z''), z')) -> F(a, f(b, f(f(f(f(f(a, y''), z1), z0), z''), z')))
F(f(a, a), f(f(f(y'''', z''''), z'0'), z)) -> F(f(a, a), f(f(y'''', z''''), z'0'))
F(f(a, a), f(a, z'''')) -> F(a, f(b, f(f(f(a, b), a), z'''')))
F(f(a, a), f(a, z''''')) -> F(a, f(f(b, f(a, f(b, a))), z'''''))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(b, a), f(b, a)), z'''''))
F(f(a, a), f(a, z''''')) -> F(a, f(f(b, f(f(a, b), a)), z'''''))
F(f(a, a), f(a, z'''')) -> F(a, f(f(f(b, a), b), f(a, z'''')))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(b, f(a, b)), a), z'''''))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(b, a), f(b, a)), z'''''))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(f(b, a), b), a), z'''''))
F(f(a, a), f(y''', z'')) -> F(a, f(f(f(b, a), y'''), z''))
F(f(a, a), f(f(a, z''''), z)) -> F(f(a, a), f(a, z''''))
F(f(a, a), f(f(y'''', z''''), z)) -> F(f(a, a), f(y'''', z''''))
F(f(a, a), f(y', z)) -> F(f(a, a), y')
F(x'', f(f(y'', z''), z)) -> F(x'', f(y'', z''))
F(f(a, a), x'') -> F(a, f(f(b, a), x''))
F(x, f(y, z)) -> F(f(x, y), z)
F(f(a, a), f(f(f(f(y', z'), z0'), z'''), z'0)) -> F(a, f(f(b, f(f(f(f(a, y'), z'), z0'), z''')), z'0))


Rules:


f(f(a, a), x) -> f(a, f(b, f(a, x)))
f(x, f(y, z)) -> f(f(x, y), z)


Strategy:

innermost




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

F(f(a, a), f(f(f(f(y'', z1), z0), z''), z')) -> F(a, f(b, f(f(f(f(f(a, y''), z1), z0), z''), z')))
three new Dependency Pairs are created:

F(f(a, a), f(f(f(f(y''', z1'), z0'), z'''), z'0)) -> F(a, f(f(b, f(f(f(f(a, y'''), z1'), z0'), z''')), z'0))
F(f(a, a), f(f(f(f(a, z1'), z0), z''), z')) -> F(a, f(b, f(f(f(f(a, f(b, f(a, z1'))), z0), z''), z')))
F(f(a, a), f(f(f(f(f(y', z2), z1), z0), z''), z')) -> F(a, f(b, f(f(f(f(f(f(a, y'), z2), z1), z0), z''), z')))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 34
                 ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

F(f(a, a), f(f(f(f(f(y', z2), z1), z0), z''), z')) -> F(a, f(b, f(f(f(f(f(f(a, y'), z2), z1), z0), z''), z')))
F(f(a, a), f(f(f(f(a, z1'), z0), z''), z')) -> F(a, f(b, f(f(f(f(a, f(b, f(a, z1'))), z0), z''), z')))
F(f(a, a), f(f(f(f(y''', z1'), z0'), z'''), z'0)) -> F(a, f(f(b, f(f(f(f(a, y'''), z1'), z0'), z''')), z'0))
F(f(a, a), f(f(f(a, z0''), z''), z')) -> F(a, f(b, f(f(f(f(a, b), f(a, z0'')), z''), z')))
F(f(a, a), f(f(f(a, z0''), z'''), z'0)) -> F(a, f(f(b, f(f(a, f(b, f(a, z0''))), z''')), z'0))
F(f(a, a), f(f(f(f(y', z'), z0'), z'''), z'0)) -> F(a, f(f(b, f(f(f(f(a, y'), z'), z0'), z''')), z'0))
F(f(a, a), f(f(f(a, z0''), z'''), z'0)) -> F(a, f(f(b, f(f(a, f(b, f(a, z0''))), z''')), z'0))
F(f(a, a), f(f(f(y''', z0''), z''''), z'0)) -> F(a, f(f(f(b, f(f(a, y'''), z0'')), z''''), z'0))
F(f(a, a), f(f(a, z'''''), z')) -> F(a, f(b, f(f(f(a, f(b, a)), z'''''), z')))
F(f(a, a), f(f(a, z'''''), z'')) -> F(a, f(f(b, f(a, f(f(b, a), z'''''))), z''))
F(f(a, a), f(f(a, z'''''), z')) -> F(a, f(b, f(f(f(f(a, b), a), z'''''), z')))
F(f(a, a), f(f(a, z'''''), z'')) -> F(a, f(f(b, f(f(a, b), f(a, z'''''))), z''))
F(f(a, a), f(f(a, z'''''), z'')) -> F(a, f(f(b, f(a, f(f(b, a), z'''''))), z''))
F(f(a, a), f(f(a, z'''''), z'')) -> F(a, f(f(b, f(f(a, b), f(a, z'''''))), z''))
F(f(a, a), f(f(a, z'''''), z'')) -> F(a, f(f(f(b, a), f(b, f(a, z'''''))), z''))
F(f(a, a), f(f(f(f(y'', z''), z'), z'''), z'0)) -> F(a, f(f(b, f(f(f(f(a, y''), z''), z'), z''')), z'0))
F(f(a, a), f(f(f(a, z''), z'''), z'0)) -> F(a, f(f(b, f(f(a, f(b, f(a, z''))), z''')), z'0))
F(f(a, a), f(f(f(y'', z''), z''''), z'0)) -> F(a, f(f(f(b, f(f(a, y''), z'')), z''''), z'0))
F(f(a, a), f(f(a, z'''''), z'0)) -> F(a, f(f(b, f(a, f(f(b, a), z'''''))), z'0))
F(f(a, a), f(f(a, z'''''), z'0)) -> F(a, f(f(b, f(f(a, b), f(a, z'''''))), z'0))
F(f(a, a), f(f(a, z'''''), z'0)) -> F(a, f(f(f(b, a), f(b, f(a, z'''''))), z'0))
F(f(a, a), f(f(f(y', z'), z''''), z'0)) -> F(a, f(f(f(b, f(f(a, y'), z')), z''''), z'0))
F(f(a, a), f(f(y''''', z''''), z'0)) -> F(a, f(f(f(f(b, a), y'''''), z''''), z'0))
F(f(a, a), f(f(f(f(y', z1), z0), z'), z'')) -> F(a, f(f(b, f(f(f(f(a, y'), z1), z0), z')), z''))
F(f(a, a), f(f(f(a, z0'), z'), z'')) -> F(a, f(f(b, f(f(a, f(b, f(a, z0'))), z')), z''))
F(f(a, a), f(f(f(y''', z0'), z'''), z'')) -> F(a, f(f(f(b, f(f(a, y'''), z0')), z'''), z''))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(b, f(a, f(f(b, a), z''''))), z''))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(b, f(f(a, b), f(a, z''''))), z''))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(f(b, a), f(b, f(a, z''''))), z''))
F(f(a, a), f(f(f(y', z'), z'''), z'')) -> F(a, f(f(f(b, f(f(a, y'), z')), z'''), z''))
F(f(a, a), f(f(y''', z'''), z'')) -> F(a, f(f(f(f(b, a), y'''), z'''), z''))
F(f(a, a), f(f(f(y'''', z''''), z'0'), z)) -> F(f(a, a), f(f(y'''', z''''), z'0'))
F(f(a, a), f(a, z'''')) -> F(a, f(b, f(f(f(a, b), a), z'''')))
F(f(a, a), f(a, z''''')) -> F(a, f(f(b, f(a, f(b, a))), z'''''))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(b, a), f(b, a)), z'''''))
F(f(a, a), f(a, z''''')) -> F(a, f(f(b, f(f(a, b), a)), z'''''))
F(f(a, a), f(a, z'''')) -> F(a, f(f(f(b, a), b), f(a, z'''')))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(b, f(a, b)), a), z'''''))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(b, a), f(b, a)), z'''''))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(f(b, a), b), a), z'''''))
F(f(a, a), f(y''', z'')) -> F(a, f(f(f(b, a), y'''), z''))
F(f(a, a), f(f(a, z''''), z)) -> F(f(a, a), f(a, z''''))
F(f(a, a), f(f(y'''', z''''), z)) -> F(f(a, a), f(y'''', z''''))
F(f(a, a), f(y', z)) -> F(f(a, a), y')
F(x'', f(f(y'', z''), z)) -> F(x'', f(y'', z''))
F(f(a, a), x'') -> F(a, f(f(b, a), x''))
F(x, f(y, z)) -> F(f(x, y), z)
F(f(a, a), f(f(f(a, z0''), z''), z')) -> F(a, f(b, f(f(f(a, f(f(b, a), z0'')), z''), z')))


Rules:


f(f(a, a), x) -> f(a, f(b, f(a, x)))
f(x, f(y, z)) -> f(f(x, y), z)


Strategy:

innermost



Innermost Termination of R could not be shown.
Duration:
0:21 minutes