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

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Instantiation Transformation


Dependency Pairs:

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


Rules:


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


Strategy:

innermost




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

F(a(x), y) -> F(x, a(y))
three new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Inst
           →DP Problem 2
             ↳Instantiation Transformation


Dependency Pairs:

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


Rules:


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


Strategy:

innermost




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

F(b(x), y) -> F(x, b(y))
four new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Inst
           →DP Problem 2
             ↳Inst
             ...
               →DP Problem 3
                 ↳Instantiation Transformation


Dependency Pairs:

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


Rules:


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


Strategy:

innermost




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

F(x, b(a(y))) -> F(b(b(x)), y)
five new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem: 



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


Dependency Pairs:

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


Rules:


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


Strategy:

innermost




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

F(a(x''), a(y'')) -> F(x'', a(a(y'')))
four new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem: 



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


Dependency Pairs:

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


Rules:


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


Strategy:

innermost




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

F(a(a(x'')), y'') -> F(a(x''), a(y''))
six new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Inst
           →DP Problem 2
             ↳Inst
             ...
               →DP Problem 6
                 ↳Forward Instantiation Transformation


Dependency Pairs:

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


Rules:


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


Strategy:

innermost




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

F(x, a(b(y))) -> F(a(a(x)), y)
13 new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Inst
           →DP Problem 2
             ↳Inst
             ...
               →DP Problem 7
                 ↳Forward Instantiation Transformation


Dependency Pairs:

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


Rules:


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


Strategy:

innermost




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

F(a(x''), b(y'')) -> F(x'', a(b(y'')))
23 new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Inst
           →DP Problem 2
             ↳Inst
             ...
               →DP Problem 8
                 ↳Forward Instantiation Transformation


Dependency Pairs:

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


Rules:


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


Strategy:

innermost




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

F(b(x''), b(y'')) -> F(x'', b(b(y'')))
23 new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Inst
           →DP Problem 2
             ↳Inst
             ...
               →DP Problem 9
                 ↳Forward Instantiation Transformation


Dependency Pairs:

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


Rules:


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


Strategy:

innermost




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

F(b(b(x'')), y'') -> F(b(x''), b(y''))
28 new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Inst
           →DP Problem 2
             ↳Inst
             ...
               →DP Problem 10
                 ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

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


Rules:


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


Strategy:

innermost



Innermost Termination of R could not be shown.
Duration:
0:48 minutes