Term Rewriting System R:
[x, y]
f(f(0, x), 1) -> f(g(f(x, x)), x)
f(g(x), y) -> g(f(x, y))

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

F(f(0, x), 1) -> F(g(f(x, x)), x)
F(f(0, x), 1) -> F(x, x)
F(g(x), y) -> F(x, y)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Forward Instantiation Transformation


Dependency Pairs:

F(f(0, x), 1) -> F(x, x)
F(g(x), y) -> F(x, y)
F(f(0, x), 1) -> F(g(f(x, x)), x)


Rules:


f(f(0, x), 1) -> f(g(f(x, x)), x)
f(g(x), y) -> g(f(x, y))


Strategy:

innermost




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

F(f(0, x), 1) -> F(x, x)
one new Dependency Pair is created:

F(f(0, g(x'')), 1) -> F(g(x''), g(x''))

The transformation is resulting in one new DP problem: 



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


Dependency Pairs:

F(f(0, g(x'')), 1) -> F(g(x''), g(x''))
F(f(0, x), 1) -> F(g(f(x, x)), x)
F(g(x), y) -> F(x, y)


Rules:


f(f(0, x), 1) -> f(g(f(x, x)), x)
f(g(x), y) -> g(f(x, y))


Strategy:

innermost




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

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

F(g(g(x'')), y'') -> F(g(x''), y'')
F(g(f(0, x'')), 1) -> F(f(0, x''), 1)
F(g(f(0, g(x''''))), 1) -> F(f(0, g(x'''')), 1)

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
           →DP Problem 2
             ↳FwdInst
             ...
               →DP Problem 3
                 ↳Narrowing Transformation


Dependency Pairs:

F(g(f(0, g(x''''))), 1) -> F(f(0, g(x'''')), 1)
F(f(0, x), 1) -> F(g(f(x, x)), x)
F(g(f(0, x'')), 1) -> F(f(0, x''), 1)
F(g(g(x'')), y'') -> F(g(x''), y'')
F(f(0, g(x'')), 1) -> F(g(x''), g(x''))


Rules:


f(f(0, x), 1) -> f(g(f(x, x)), x)
f(g(x), y) -> g(f(x, y))


Strategy:

innermost




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

F(f(0, x), 1) -> F(g(f(x, x)), x)
one new Dependency Pair is created:

F(f(0, g(x'')), 1) -> F(g(g(f(x'', g(x'')))), g(x''))

The transformation is resulting in one new DP problem: 



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


Dependency Pairs:

F(f(0, g(x'')), 1) -> F(g(g(f(x'', g(x'')))), g(x''))
F(g(f(0, x'')), 1) -> F(f(0, x''), 1)
F(g(g(x'')), y'') -> F(g(x''), y'')
F(f(0, g(x'')), 1) -> F(g(x''), g(x''))
F(g(f(0, g(x''''))), 1) -> F(f(0, g(x'''')), 1)


Rules:


f(f(0, x), 1) -> f(g(f(x, x)), x)
f(g(x), y) -> g(f(x, y))


Strategy:

innermost




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

F(f(0, g(x'')), 1) -> F(g(x''), g(x''))
one new Dependency Pair is created:

F(f(0, g(g(x''''))), 1) -> F(g(g(x'''')), g(g(x'''')))

The transformation is resulting in one new DP problem: 



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


Dependency Pairs:

F(g(f(0, g(x''''))), 1) -> F(f(0, g(x'''')), 1)
F(f(0, g(g(x''''))), 1) -> F(g(g(x'''')), g(g(x'''')))
F(g(f(0, x'')), 1) -> F(f(0, x''), 1)
F(g(g(x'')), y'') -> F(g(x''), y'')
F(f(0, g(x'')), 1) -> F(g(g(f(x'', g(x'')))), g(x''))


Rules:


f(f(0, x), 1) -> f(g(f(x, x)), x)
f(g(x), y) -> g(f(x, y))


Strategy:

innermost




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

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

F(g(g(g(x''''))), y'''') -> F(g(g(x'''')), y'''')
F(g(g(f(0, x''''))), 1) -> F(g(f(0, x'''')), 1)
F(g(g(f(0, g(x'''''')))), 1) -> F(g(f(0, g(x''''''))), 1)

The transformation is resulting in one new DP problem: 



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


Dependency Pairs:

F(g(g(f(0, g(x'''''')))), 1) -> F(g(f(0, g(x''''''))), 1)
F(f(0, g(g(x''''))), 1) -> F(g(g(x'''')), g(g(x'''')))
F(g(f(0, x'')), 1) -> F(f(0, x''), 1)
F(g(g(f(0, x''''))), 1) -> F(g(f(0, x'''')), 1)
F(g(g(g(x''''))), y'''') -> F(g(g(x'''')), y'''')
F(f(0, g(x'')), 1) -> F(g(g(f(x'', g(x'')))), g(x''))
F(g(f(0, g(x''''))), 1) -> F(f(0, g(x'''')), 1)


Rules:


f(f(0, x), 1) -> f(g(f(x, x)), x)
f(g(x), y) -> g(f(x, y))


Strategy:

innermost




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

F(f(0, g(x'')), 1) -> F(g(g(f(x'', g(x'')))), g(x''))
one new Dependency Pair is created:

F(f(0, g(g(x'))), 1) -> F(g(g(g(f(x', g(g(x')))))), g(g(x')))

The transformation is resulting in one new DP problem: 



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


Dependency Pairs:

F(f(0, g(g(x'))), 1) -> F(g(g(g(f(x', g(g(x')))))), g(g(x')))
F(g(f(0, g(x''''))), 1) -> F(f(0, g(x'''')), 1)
F(g(g(f(0, x''''))), 1) -> F(g(f(0, x'''')), 1)
F(g(g(g(x''''))), y'''') -> F(g(g(x'''')), y'''')
F(f(0, g(g(x''''))), 1) -> F(g(g(x'''')), g(g(x'''')))
F(g(f(0, x'')), 1) -> F(f(0, x''), 1)
F(g(g(f(0, g(x'''''')))), 1) -> F(g(f(0, g(x''''''))), 1)


Rules:


f(f(0, x), 1) -> f(g(f(x, x)), x)
f(g(x), y) -> g(f(x, y))


Strategy:

innermost




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

F(f(0, g(g(x''''))), 1) -> F(g(g(x'''')), g(g(x'''')))
one new Dependency Pair is created:

F(f(0, g(g(g(x'''''')))), 1) -> F(g(g(g(x''''''))), g(g(g(x''''''))))

The transformation is resulting in one new DP problem: 



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


Dependency Pairs:

F(g(g(f(0, g(x'''''')))), 1) -> F(g(f(0, g(x''''''))), 1)
F(g(f(0, g(x''''))), 1) -> F(f(0, g(x'''')), 1)
F(f(0, g(g(g(x'''''')))), 1) -> F(g(g(g(x''''''))), g(g(g(x''''''))))
F(g(f(0, x'')), 1) -> F(f(0, x''), 1)
F(g(g(f(0, x''''))), 1) -> F(g(f(0, x'''')), 1)
F(g(g(g(x''''))), y'''') -> F(g(g(x'''')), y'''')
F(f(0, g(g(x'))), 1) -> F(g(g(g(f(x', g(g(x')))))), g(g(x')))


Rules:


f(f(0, x), 1) -> f(g(f(x, x)), x)
f(g(x), y) -> g(f(x, y))


Strategy:

innermost




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

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

F(g(g(g(g(x'''''')))), y'''''') -> F(g(g(g(x''''''))), y'''''')
F(g(g(g(f(0, x'''''')))), 1) -> F(g(g(f(0, x''''''))), 1)
F(g(g(g(f(0, g(x''''''''))))), 1) -> F(g(g(f(0, g(x'''''''')))), 1)

The transformation is resulting in one new DP problem: 



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


Dependency Pairs:

F(g(g(g(f(0, g(x''''''''))))), 1) -> F(g(g(f(0, g(x'''''''')))), 1)
F(f(0, g(g(g(x'''''')))), 1) -> F(g(g(g(x''''''))), g(g(g(x''''''))))
F(g(f(0, g(x''''))), 1) -> F(f(0, g(x'''')), 1)
F(g(g(f(0, x''''))), 1) -> F(g(f(0, x'''')), 1)
F(g(g(g(f(0, x'''''')))), 1) -> F(g(g(f(0, x''''''))), 1)
F(g(g(g(g(x'''''')))), y'''''') -> F(g(g(g(x''''''))), y'''''')
F(f(0, g(g(x'))), 1) -> F(g(g(g(f(x', g(g(x')))))), g(g(x')))
F(g(f(0, x'')), 1) -> F(f(0, x''), 1)
F(g(g(f(0, g(x'''''')))), 1) -> F(g(f(0, g(x''''''))), 1)


Rules:


f(f(0, x), 1) -> f(g(f(x, x)), x)
f(g(x), y) -> g(f(x, y))


Strategy:

innermost




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

F(f(0, g(g(x'))), 1) -> F(g(g(g(f(x', g(g(x')))))), g(g(x')))
one new Dependency Pair is created:

F(f(0, g(g(g(x'')))), 1) -> F(g(g(g(g(f(x'', g(g(g(x'')))))))), g(g(g(x''))))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
           →DP Problem 2
             ↳FwdInst
             ...
               →DP Problem 10
                 ↳Forward Instantiation Transformation


Dependency Pairs:

F(f(0, g(g(g(x'')))), 1) -> F(g(g(g(g(f(x'', g(g(g(x'')))))))), g(g(g(x''))))
F(g(f(0, g(x''''))), 1) -> F(f(0, g(x'''')), 1)
F(g(g(f(0, g(x'''''')))), 1) -> F(g(f(0, g(x''''''))), 1)
F(g(g(g(f(0, x'''''')))), 1) -> F(g(g(f(0, x''''''))), 1)
F(g(g(g(g(x'''''')))), y'''''') -> F(g(g(g(x''''''))), y'''''')
F(f(0, g(g(g(x'''''')))), 1) -> F(g(g(g(x''''''))), g(g(g(x''''''))))
F(g(f(0, x'')), 1) -> F(f(0, x''), 1)
F(g(g(f(0, x''''))), 1) -> F(g(f(0, x'''')), 1)
F(g(g(g(f(0, g(x''''''''))))), 1) -> F(g(g(f(0, g(x'''''''')))), 1)


Rules:


f(f(0, x), 1) -> f(g(f(x, x)), x)
f(g(x), y) -> g(f(x, y))


Strategy:

innermost




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

F(f(0, g(g(g(x'''''')))), 1) -> F(g(g(g(x''''''))), g(g(g(x''''''))))
one new Dependency Pair is created:

F(f(0, g(g(g(g(x''''''''))))), 1) -> F(g(g(g(g(x'''''''')))), g(g(g(g(x'''''''')))))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
           →DP Problem 2
             ↳FwdInst
             ...
               →DP Problem 11
                 ↳Forward Instantiation Transformation


Dependency Pairs:

F(g(g(g(f(0, g(x''''''''))))), 1) -> F(g(g(f(0, g(x'''''''')))), 1)
F(g(g(f(0, g(x'''''')))), 1) -> F(g(f(0, g(x''''''))), 1)
F(g(f(0, g(x''''))), 1) -> F(f(0, g(x'''')), 1)
F(f(0, g(g(g(g(x''''''''))))), 1) -> F(g(g(g(g(x'''''''')))), g(g(g(g(x'''''''')))))
F(g(f(0, x'')), 1) -> F(f(0, x''), 1)
F(g(g(f(0, x''''))), 1) -> F(g(f(0, x'''')), 1)
F(g(g(g(f(0, x'''''')))), 1) -> F(g(g(f(0, x''''''))), 1)
F(g(g(g(g(x'''''')))), y'''''') -> F(g(g(g(x''''''))), y'''''')
F(f(0, g(g(g(x'')))), 1) -> F(g(g(g(g(f(x'', g(g(g(x'')))))))), g(g(g(x''))))


Rules:


f(f(0, x), 1) -> f(g(f(x, x)), x)
f(g(x), y) -> g(f(x, y))


Strategy:

innermost




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

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

F(g(g(g(g(g(x''''''''))))), y'''''''') -> F(g(g(g(g(x'''''''')))), y'''''''')
F(g(g(g(g(f(0, x''''''''))))), 1) -> F(g(g(g(f(0, x'''''''')))), 1)
F(g(g(g(g(f(0, g(x'''''''''')))))), 1) -> F(g(g(g(f(0, g(x''''''''''))))), 1)

The transformation is resulting in one new DP problem: 



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


Dependency Pairs:

F(g(g(g(g(f(0, g(x'''''''''')))))), 1) -> F(g(g(g(f(0, g(x''''''''''))))), 1)
F(f(0, g(g(g(g(x''''''''))))), 1) -> F(g(g(g(g(x'''''''')))), g(g(g(g(x'''''''')))))
F(g(f(0, g(x''''))), 1) -> F(f(0, g(x'''')), 1)
F(g(g(f(0, g(x'''''')))), 1) -> F(g(f(0, g(x''''''))), 1)
F(g(g(g(f(0, x'''''')))), 1) -> F(g(g(f(0, x''''''))), 1)
F(g(g(g(g(f(0, x''''''''))))), 1) -> F(g(g(g(f(0, x'''''''')))), 1)
F(g(g(g(g(g(x''''''''))))), y'''''''') -> F(g(g(g(g(x'''''''')))), y'''''''')
F(f(0, g(g(g(x'')))), 1) -> F(g(g(g(g(f(x'', g(g(g(x'')))))))), g(g(g(x''))))
F(g(f(0, x'')), 1) -> F(f(0, x''), 1)
F(g(g(f(0, x''''))), 1) -> F(g(f(0, x'''')), 1)
F(g(g(g(f(0, g(x''''''''))))), 1) -> F(g(g(f(0, g(x'''''''')))), 1)


Rules:


f(f(0, x), 1) -> f(g(f(x, x)), x)
f(g(x), y) -> g(f(x, y))


Strategy:

innermost




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

F(f(0, g(g(g(x'')))), 1) -> F(g(g(g(g(f(x'', g(g(g(x'')))))))), g(g(g(x''))))
one new Dependency Pair is created:

F(f(0, g(g(g(g(x'))))), 1) -> F(g(g(g(g(g(f(x', g(g(g(g(x')))))))))), g(g(g(g(x')))))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
           →DP Problem 2
             ↳FwdInst
             ...
               →DP Problem 13
                 ↳Forward Instantiation Transformation


Dependency Pairs:

F(f(0, g(g(g(g(x'))))), 1) -> F(g(g(g(g(g(f(x', g(g(g(g(x')))))))))), g(g(g(g(x')))))
F(g(f(0, g(x''''))), 1) -> F(f(0, g(x'''')), 1)
F(g(g(f(0, g(x'''''')))), 1) -> F(g(f(0, g(x''''''))), 1)
F(g(g(g(f(0, g(x''''''''))))), 1) -> F(g(g(f(0, g(x'''''''')))), 1)
F(g(g(g(g(f(0, x''''''''))))), 1) -> F(g(g(g(f(0, x'''''''')))), 1)
F(g(g(g(g(g(x''''''''))))), y'''''''') -> F(g(g(g(g(x'''''''')))), y'''''''')
F(f(0, g(g(g(g(x''''''''))))), 1) -> F(g(g(g(g(x'''''''')))), g(g(g(g(x'''''''')))))
F(g(f(0, x'')), 1) -> F(f(0, x''), 1)
F(g(g(f(0, x''''))), 1) -> F(g(f(0, x'''')), 1)
F(g(g(g(f(0, x'''''')))), 1) -> F(g(g(f(0, x''''''))), 1)
F(g(g(g(g(f(0, g(x'''''''''')))))), 1) -> F(g(g(g(f(0, g(x''''''''''))))), 1)


Rules:


f(f(0, x), 1) -> f(g(f(x, x)), x)
f(g(x), y) -> g(f(x, y))


Strategy:

innermost




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

F(f(0, g(g(g(g(x''''''''))))), 1) -> F(g(g(g(g(x'''''''')))), g(g(g(g(x'''''''')))))
one new Dependency Pair is created:

F(f(0, g(g(g(g(g(x'''''''''')))))), 1) -> F(g(g(g(g(g(x''''''''''))))), g(g(g(g(g(x''''''''''))))))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
           →DP Problem 2
             ↳FwdInst
             ...
               →DP Problem 14
                 ↳Forward Instantiation Transformation


Dependency Pairs:

F(g(g(g(g(f(0, g(x'''''''''')))))), 1) -> F(g(g(g(f(0, g(x''''''''''))))), 1)
F(g(g(g(f(0, g(x''''''''))))), 1) -> F(g(g(f(0, g(x'''''''')))), 1)
F(g(g(f(0, g(x'''''')))), 1) -> F(g(f(0, g(x''''''))), 1)
F(g(f(0, g(x''''))), 1) -> F(f(0, g(x'''')), 1)
F(f(0, g(g(g(g(g(x'''''''''')))))), 1) -> F(g(g(g(g(g(x''''''''''))))), g(g(g(g(g(x''''''''''))))))
F(g(f(0, x'')), 1) -> F(f(0, x''), 1)
F(g(g(f(0, x''''))), 1) -> F(g(f(0, x'''')), 1)
F(g(g(g(f(0, x'''''')))), 1) -> F(g(g(f(0, x''''''))), 1)
F(g(g(g(g(f(0, x''''''''))))), 1) -> F(g(g(g(f(0, x'''''''')))), 1)
F(g(g(g(g(g(x''''''''))))), y'''''''') -> F(g(g(g(g(x'''''''')))), y'''''''')
F(f(0, g(g(g(g(x'))))), 1) -> F(g(g(g(g(g(f(x', g(g(g(g(x')))))))))), g(g(g(g(x')))))


Rules:


f(f(0, x), 1) -> f(g(f(x, x)), x)
f(g(x), y) -> g(f(x, y))


Strategy:

innermost




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

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

F(g(g(g(g(g(g(x'''''''''')))))), y'''''''''') -> F(g(g(g(g(g(x''''''''''))))), y'''''''''')
F(g(g(g(g(g(f(0, x'''''''''')))))), 1) -> F(g(g(g(g(f(0, x''''''''''))))), 1)
F(g(g(g(g(g(f(0, g(x''''''''''''))))))), 1) -> F(g(g(g(g(f(0, g(x'''''''''''')))))), 1)

The transformation is resulting in one new DP problem: 



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


Dependency Pairs:

F(g(g(g(g(g(f(0, g(x''''''''''''))))))), 1) -> F(g(g(g(g(f(0, g(x'''''''''''')))))), 1)
F(f(0, g(g(g(g(g(x'''''''''')))))), 1) -> F(g(g(g(g(g(x''''''''''))))), g(g(g(g(g(x''''''''''))))))
F(g(f(0, g(x''''))), 1) -> F(f(0, g(x'''')), 1)
F(g(g(f(0, g(x'''''')))), 1) -> F(g(f(0, g(x''''''))), 1)
F(g(g(g(f(0, g(x''''''''))))), 1) -> F(g(g(f(0, g(x'''''''')))), 1)
F(g(g(g(g(f(0, x''''''''))))), 1) -> F(g(g(g(f(0, x'''''''')))), 1)
F(g(g(g(g(g(f(0, x'''''''''')))))), 1) -> F(g(g(g(g(f(0, x''''''''''))))), 1)
F(g(g(g(g(g(g(x'''''''''')))))), y'''''''''') -> F(g(g(g(g(g(x''''''''''))))), y'''''''''')
F(f(0, g(g(g(g(x'))))), 1) -> F(g(g(g(g(g(f(x', g(g(g(g(x')))))))))), g(g(g(g(x')))))
F(g(f(0, x'')), 1) -> F(f(0, x''), 1)
F(g(g(f(0, x''''))), 1) -> F(g(f(0, x'''')), 1)
F(g(g(g(f(0, x'''''')))), 1) -> F(g(g(f(0, x''''''))), 1)
F(g(g(g(g(f(0, g(x'''''''''')))))), 1) -> F(g(g(g(f(0, g(x''''''''''))))), 1)


Rules:


f(f(0, x), 1) -> f(g(f(x, x)), x)
f(g(x), y) -> g(f(x, y))


Strategy:

innermost




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

F(f(0, g(g(g(g(x'))))), 1) -> F(g(g(g(g(g(f(x', g(g(g(g(x')))))))))), g(g(g(g(x')))))
one new Dependency Pair is created:

F(f(0, g(g(g(g(g(x'')))))), 1) -> F(g(g(g(g(g(g(f(x'', g(g(g(g(g(x'')))))))))))), g(g(g(g(g(x''))))))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
           →DP Problem 2
             ↳FwdInst
             ...
               →DP Problem 16
                 ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

F(f(0, g(g(g(g(g(x'')))))), 1) -> F(g(g(g(g(g(g(f(x'', g(g(g(g(g(x'')))))))))))), g(g(g(g(g(x''))))))
F(g(f(0, g(x''''))), 1) -> F(f(0, g(x'''')), 1)
F(g(g(f(0, g(x'''''')))), 1) -> F(g(f(0, g(x''''''))), 1)
F(g(g(g(f(0, g(x''''''''))))), 1) -> F(g(g(f(0, g(x'''''''')))), 1)
F(g(g(g(g(f(0, g(x'''''''''')))))), 1) -> F(g(g(g(f(0, g(x''''''''''))))), 1)
F(g(g(g(g(g(f(0, x'''''''''')))))), 1) -> F(g(g(g(g(f(0, x''''''''''))))), 1)
F(g(g(g(g(g(g(x'''''''''')))))), y'''''''''') -> F(g(g(g(g(g(x''''''''''))))), y'''''''''')
F(f(0, g(g(g(g(g(x'''''''''')))))), 1) -> F(g(g(g(g(g(x''''''''''))))), g(g(g(g(g(x''''''''''))))))
F(g(f(0, x'')), 1) -> F(f(0, x''), 1)
F(g(g(f(0, x''''))), 1) -> F(g(f(0, x'''')), 1)
F(g(g(g(f(0, x'''''')))), 1) -> F(g(g(f(0, x''''''))), 1)
F(g(g(g(g(f(0, x''''''''))))), 1) -> F(g(g(g(f(0, x'''''''')))), 1)
F(g(g(g(g(g(f(0, g(x''''''''''''))))))), 1) -> F(g(g(g(g(f(0, g(x'''''''''''')))))), 1)


Rules:


f(f(0, x), 1) -> f(g(f(x, x)), x)
f(g(x), y) -> g(f(x, y))


Strategy:

innermost



Innermost Termination of R could not be shown.
Duration:
0:01 minutes