Termination Proof using AProVE

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

Innermost Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC.

     ↳DPs

       →DP Problem 1

         ↳Instantiation Transformation

Dependency Pairs:

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

Rules:

f(x, a(b(y))) -> f(c(d(x)), y)
f(c(x), y) -> f(x, a(y))
f(d(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(c(x), y) -> F(x, a(y))

three new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Inst

           →DP Problem 2

             ↳Instantiation Transformation

Dependency Pairs:

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

Rules:

f(x, a(b(y))) -> f(c(d(x)), y)
f(c(x), y) -> f(x, a(y))
f(d(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(d(x), y) -> F(x, b(y))

four new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Inst

           →DP Problem 2

             ↳Inst

...

               →DP Problem 3

                 ↳Forward Instantiation Transformation

Dependency Pairs:

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

Rules:

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

three new Dependency Pairs are created:

F(c(c(x'''')), a(y'''')) -> F(c(x''''), a(a(y'''')))
F(c(c(d(x''''))), a(y'''')) -> F(c(d(x'''')), a(a(y'''')))
F(c(d(x'''')), a(y''')) -> F(d(x''''), a(a(y''')))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Inst

           →DP Problem 2

             ↳Inst

...

               →DP Problem 4

                 ↳Forward Instantiation Transformation

Dependency Pairs:

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

Rules:

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

three new Dependency Pairs are created:

F(d(c(d(x''''))), b(y'''')) -> F(c(d(x'''')), b(b(y'''')))
F(d(c(x'''')), b(y'''')) -> F(c(x''''), b(b(y'''')))
F(d(d(x'''')), b(y'''')) -> F(d(x''''), b(b(y'''')))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Inst

           →DP Problem 2

             ↳Inst

...

               →DP Problem 5

                 ↳Forward Instantiation Transformation

Dependency Pairs:

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

Rules:

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

five new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Inst

           →DP Problem 2

             ↳Inst

...

               →DP Problem 6

                 ↳Forward Instantiation Transformation

Dependency Pairs:

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

Rules:

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

five new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Inst

           →DP Problem 2

             ↳Inst

...

               →DP Problem 7

                 ↳Forward Instantiation Transformation

Dependency Pairs:

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

Rules:

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

12 new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Inst

           →DP Problem 2

             ↳Inst

...

               →DP Problem 8

                 ↳Forward Instantiation Transformation

Dependency Pairs:

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

Rules:

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

15 new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Inst

           →DP Problem 2

             ↳Inst

...

               →DP Problem 9

                 ↳Forward Instantiation Transformation

Dependency Pairs:

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

Rules:

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

29 new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Inst

           →DP Problem 2

             ↳Inst

...

               →DP Problem 10

                 ↳Forward Instantiation Transformation

Dependency Pairs:

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

Rules:

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

31 new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Inst

           →DP Problem 2

             ↳Inst

...

               →DP Problem 11

                 ↳Remaining Obligation(s)

The following remains to be proven:
Dependency Pairs:

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

Rules:

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

Strategy:

innermost

Innermost Termination of R could not be shown.
Duration:
0:55 minutes