Termination Proof using AProVE

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

Innermost Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

F(c(s(x), y)) -> F(c(x, s(y)))
F(c(s(x), s(y))) -> G(c(x, y))
G(c(x, s(y))) -> G(c(s(x), y))
G(c(s(x), s(y))) -> F(c(x, y))

Furthermore, R contains one SCC.

     ↳DPs

       →DP Problem 1

         ↳Forward Instantiation Transformation

Dependency Pairs:

G(c(s(x), s(y))) -> F(c(x, y))
G(c(x, s(y))) -> G(c(s(x), y))
F(c(s(x), s(y))) -> G(c(x, y))
F(c(s(x), y)) -> F(c(x, s(y)))

Rules:

f(c(s(x), y)) -> f(c(x, s(y)))
f(c(s(x), s(y))) -> g(c(x, y))
g(c(x, s(y))) -> g(c(s(x), y))
g(c(s(x), s(y))) -> f(c(x, y))

Strategy:

innermost

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

F(c(s(x), y)) -> F(c(x, s(y)))

one new Dependency Pair is created:

F(c(s(s(x'')), y'')) -> F(c(s(x''), s(y'')))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 2

             ↳Forward Instantiation Transformation

Dependency Pairs:

F(c(s(s(x'')), y'')) -> F(c(s(x''), s(y'')))
G(c(x, s(y))) -> G(c(s(x), y))
F(c(s(x), s(y))) -> G(c(x, y))
G(c(s(x), s(y))) -> F(c(x, y))

Rules:

f(c(s(x), y)) -> f(c(x, s(y)))
f(c(s(x), s(y))) -> g(c(x, y))
g(c(x, s(y))) -> g(c(s(x), y))
g(c(s(x), s(y))) -> f(c(x, y))

Strategy:

innermost

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

F(c(s(x), s(y))) -> G(c(x, y))

two new Dependency Pairs are created:

F(c(s(x''), s(s(y'')))) -> G(c(x'', s(y'')))
F(c(s(s(x'')), s(s(y'')))) -> G(c(s(x''), s(y'')))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 2

             ↳FwdInst

...

               →DP Problem 3

                 ↳Forward Instantiation Transformation

Dependency Pairs:

F(c(s(s(x'')), s(s(y'')))) -> G(c(s(x''), s(y'')))
G(c(s(x), s(y))) -> F(c(x, y))
G(c(x, s(y))) -> G(c(s(x), y))
F(c(s(x''), s(s(y'')))) -> G(c(x'', s(y'')))
F(c(s(s(x'')), y'')) -> F(c(s(x''), s(y'')))

Rules:

f(c(s(x), y)) -> f(c(x, s(y)))
f(c(s(x), s(y))) -> g(c(x, y))
g(c(x, s(y))) -> g(c(s(x), y))
g(c(s(x), s(y))) -> f(c(x, y))

Strategy:

innermost

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

G(c(x, s(y))) -> G(c(s(x), y))

one new Dependency Pair is created:

G(c(x'', s(s(y'')))) -> G(c(s(x''), s(y'')))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 2

             ↳FwdInst

...

               →DP Problem 4

                 ↳Forward Instantiation Transformation

Dependency Pairs:

G(c(x'', s(s(y'')))) -> G(c(s(x''), s(y'')))
F(c(s(x''), s(s(y'')))) -> G(c(x'', s(y'')))
F(c(s(s(x'')), y'')) -> F(c(s(x''), s(y'')))
G(c(s(x), s(y))) -> F(c(x, y))
F(c(s(s(x'')), s(s(y'')))) -> G(c(s(x''), s(y'')))

Rules:

f(c(s(x), y)) -> f(c(x, s(y)))
f(c(s(x), s(y))) -> g(c(x, y))
g(c(x, s(y))) -> g(c(s(x), y))
g(c(s(x), s(y))) -> f(c(x, y))

Strategy:

innermost

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

G(c(s(x), s(y))) -> F(c(x, y))

three new Dependency Pairs are created:

G(c(s(s(s(x''''))), s(y'))) -> F(c(s(s(x'''')), y'))
G(c(s(s(x'''')), s(s(s(y''''))))) -> F(c(s(x''''), s(s(y''''))))
G(c(s(s(s(x''''))), s(s(s(y''''))))) -> F(c(s(s(x'''')), s(s(y''''))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 2

             ↳FwdInst

...

               →DP Problem 5

                 ↳Forward Instantiation Transformation

Dependency Pairs:

G(c(s(s(s(x''''))), s(s(s(y''''))))) -> F(c(s(s(x'''')), s(s(y''''))))
F(c(s(s(x'')), s(s(y'')))) -> G(c(s(x''), s(y'')))
G(c(s(s(x'''')), s(s(s(y''''))))) -> F(c(s(x''''), s(s(y''''))))
F(c(s(x''), s(s(y'')))) -> G(c(x'', s(y'')))
F(c(s(s(x'')), y'')) -> F(c(s(x''), s(y'')))
G(c(s(s(s(x''''))), s(y'))) -> F(c(s(s(x'''')), y'))
G(c(x'', s(s(y'')))) -> G(c(s(x''), s(y'')))

Rules:

f(c(s(x), y)) -> f(c(x, s(y)))
f(c(s(x), s(y))) -> g(c(x, y))
g(c(x, s(y))) -> g(c(s(x), y))
g(c(s(x), s(y))) -> f(c(x, y))

Strategy:

innermost

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

F(c(s(s(x'')), y'')) -> F(c(s(x''), s(y'')))

three new Dependency Pairs are created:

F(c(s(s(s(x''''))), y'''')) -> F(c(s(s(x'''')), s(y'''')))
F(c(s(s(x'''')), s(y''''))) -> F(c(s(x''''), s(s(y''''))))
F(c(s(s(s(x''''))), s(y''''))) -> F(c(s(s(x'''')), s(s(y''''))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 2

             ↳FwdInst

...

               →DP Problem 6

                 ↳Forward Instantiation Transformation

Dependency Pairs:

F(c(s(s(s(x''''))), s(y''''))) -> F(c(s(s(x'''')), s(s(y''''))))
F(c(s(s(x'''')), s(y''''))) -> F(c(s(x''''), s(s(y''''))))
F(c(s(s(s(x''''))), y'''')) -> F(c(s(s(x'''')), s(y'''')))
G(c(s(s(x'''')), s(s(s(y''''))))) -> F(c(s(x''''), s(s(y''''))))
F(c(s(s(x'')), s(s(y'')))) -> G(c(s(x''), s(y'')))
G(c(s(s(s(x''''))), s(y'))) -> F(c(s(s(x'''')), y'))
G(c(x'', s(s(y'')))) -> G(c(s(x''), s(y'')))
F(c(s(x''), s(s(y'')))) -> G(c(x'', s(y'')))
G(c(s(s(s(x''''))), s(s(s(y''''))))) -> F(c(s(s(x'''')), s(s(y''''))))

Rules:

f(c(s(x), y)) -> f(c(x, s(y)))
f(c(s(x), s(y))) -> g(c(x, y))
g(c(x, s(y))) -> g(c(s(x), y))
g(c(s(x), s(y))) -> f(c(x, y))

Strategy:

innermost

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

F(c(s(x''), s(s(y'')))) -> G(c(x'', s(y'')))

four new Dependency Pairs are created:

F(c(s(x''''), s(s(s(y''''))))) -> G(c(x'''', s(s(y''''))))
F(c(s(s(s(s(x'''''')))), s(s(y'''')))) -> G(c(s(s(s(x''''''))), s(y'''')))
F(c(s(s(s(x''''''))), s(s(s(s(y'''''')))))) -> G(c(s(s(x'''''')), s(s(s(y'''''')))))
F(c(s(s(s(s(x'''''')))), s(s(s(s(y'''''')))))) -> G(c(s(s(s(x''''''))), s(s(s(y'''''')))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 2

             ↳FwdInst

...

               →DP Problem 7

                 ↳Forward Instantiation Transformation

Dependency Pairs:

F(c(s(s(s(s(x'''''')))), s(s(s(s(y'''''')))))) -> G(c(s(s(s(x''''''))), s(s(s(y'''''')))))
F(c(s(s(s(x''''''))), s(s(s(s(y'''''')))))) -> G(c(s(s(x'''''')), s(s(s(y'''''')))))
G(c(s(s(s(x''''))), s(s(s(y''''))))) -> F(c(s(s(x'''')), s(s(y''''))))
F(c(s(s(s(s(x'''''')))), s(s(y'''')))) -> G(c(s(s(s(x''''''))), s(y'''')))
G(c(s(s(x'''')), s(s(s(y''''))))) -> F(c(s(x''''), s(s(y''''))))
F(c(s(x''''), s(s(s(y''''))))) -> G(c(x'''', s(s(y''''))))
F(c(s(s(x'''')), s(y''''))) -> F(c(s(x''''), s(s(y''''))))
F(c(s(s(s(x''''))), y'''')) -> F(c(s(s(x'''')), s(y'''')))
G(c(s(s(s(x''''))), s(y'))) -> F(c(s(s(x'''')), y'))
G(c(x'', s(s(y'')))) -> G(c(s(x''), s(y'')))
F(c(s(s(x'')), s(s(y'')))) -> G(c(s(x''), s(y'')))
F(c(s(s(s(x''''))), s(y''''))) -> F(c(s(s(x'''')), s(s(y''''))))

Rules:

f(c(s(x), y)) -> f(c(x, s(y)))
f(c(s(x), s(y))) -> g(c(x, y))
g(c(x, s(y))) -> g(c(s(x), y))
g(c(s(x), s(y))) -> f(c(x, y))

Strategy:

innermost

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

F(c(s(s(x'')), s(s(y'')))) -> G(c(s(x''), s(y'')))

four new Dependency Pairs are created:

F(c(s(s(x'''')), s(s(s(y''''))))) -> G(c(s(x''''), s(s(y''''))))
F(c(s(s(s(s(x'''''')))), s(s(y'''')))) -> G(c(s(s(s(x''''''))), s(y'''')))
F(c(s(s(s(x''''''))), s(s(s(s(y'''''')))))) -> G(c(s(s(x'''''')), s(s(s(y'''''')))))
F(c(s(s(s(s(x'''''')))), s(s(s(s(y'''''')))))) -> G(c(s(s(s(x''''''))), s(s(s(y'''''')))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 2

             ↳FwdInst

...

               →DP Problem 8

                 ↳Forward Instantiation Transformation

Dependency Pairs:

F(c(s(s(s(s(x'''''')))), s(s(s(s(y'''''')))))) -> G(c(s(s(s(x''''''))), s(s(s(y'''''')))))
F(c(s(s(s(x''''''))), s(s(s(s(y'''''')))))) -> G(c(s(s(x'''''')), s(s(s(y'''''')))))
F(c(s(s(s(s(x'''''')))), s(s(y'''')))) -> G(c(s(s(s(x''''''))), s(y'''')))
F(c(s(s(x'''')), s(s(s(y''''))))) -> G(c(s(x''''), s(s(y''''))))
F(c(s(s(s(x''''''))), s(s(s(s(y'''''')))))) -> G(c(s(s(x'''''')), s(s(s(y'''''')))))
G(c(s(s(s(x''''))), s(s(s(y''''))))) -> F(c(s(s(x'''')), s(s(y''''))))
F(c(s(s(s(s(x'''''')))), s(s(y'''')))) -> G(c(s(s(s(x''''''))), s(y'''')))
G(c(s(s(x'''')), s(s(s(y''''))))) -> F(c(s(x''''), s(s(y''''))))
F(c(s(x''''), s(s(s(y''''))))) -> G(c(x'''', s(s(y''''))))
F(c(s(s(s(x''''))), s(y''''))) -> F(c(s(s(x'''')), s(s(y''''))))
F(c(s(s(x'''')), s(y''''))) -> F(c(s(x''''), s(s(y''''))))
F(c(s(s(s(x''''))), y'''')) -> F(c(s(s(x'''')), s(y'''')))
G(c(s(s(s(x''''))), s(y'))) -> F(c(s(s(x'''')), y'))
G(c(x'', s(s(y'')))) -> G(c(s(x''), s(y'')))
F(c(s(s(s(s(x'''''')))), s(s(s(s(y'''''')))))) -> G(c(s(s(s(x''''''))), s(s(s(y'''''')))))

Rules:

f(c(s(x), y)) -> f(c(x, s(y)))
f(c(s(x), s(y))) -> g(c(x, y))
g(c(x, s(y))) -> g(c(s(x), y))
g(c(s(x), s(y))) -> f(c(x, y))

Strategy:

innermost

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

G(c(x'', s(s(y'')))) -> G(c(s(x''), s(y'')))

four new Dependency Pairs are created:

G(c(x'''', s(s(s(y''''))))) -> G(c(s(x''''), s(s(y''''))))
G(c(s(s(x'''''')), s(s(y'''')))) -> G(c(s(s(s(x''''''))), s(y'''')))
G(c(s(x''''''), s(s(s(s(y'''''')))))) -> G(c(s(s(x'''''')), s(s(s(y'''''')))))
G(c(s(s(x'''''')), s(s(s(s(y'''''')))))) -> G(c(s(s(s(x''''''))), s(s(s(y'''''')))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 2

             ↳FwdInst

...

               →DP Problem 9

                 ↳Forward Instantiation Transformation

Dependency Pairs:

F(c(s(s(s(x''''''))), s(s(s(s(y'''''')))))) -> G(c(s(s(x'''''')), s(s(s(y'''''')))))
F(c(s(s(s(s(x'''''')))), s(s(y'''')))) -> G(c(s(s(s(x''''''))), s(y'''')))
F(c(s(s(x'''')), s(s(s(y''''))))) -> G(c(s(x''''), s(s(y''''))))
F(c(s(s(s(s(x'''''')))), s(s(s(s(y'''''')))))) -> G(c(s(s(s(x''''''))), s(s(s(y'''''')))))
G(c(s(s(x'''''')), s(s(s(s(y'''''')))))) -> G(c(s(s(s(x''''''))), s(s(s(y'''''')))))
G(c(s(x''''''), s(s(s(s(y'''''')))))) -> G(c(s(s(x'''''')), s(s(s(y'''''')))))
G(c(s(s(x'''''')), s(s(y'''')))) -> G(c(s(s(s(x''''''))), s(y'''')))
G(c(x'''', s(s(s(y''''))))) -> G(c(s(x''''), s(s(y''''))))
F(c(s(s(s(x''''''))), s(s(s(s(y'''''')))))) -> G(c(s(s(x'''''')), s(s(s(y'''''')))))
G(c(s(s(s(x''''))), s(s(s(y''''))))) -> F(c(s(s(x'''')), s(s(y''''))))
F(c(s(s(s(s(x'''''')))), s(s(y'''')))) -> G(c(s(s(s(x''''''))), s(y'''')))
G(c(s(s(x'''')), s(s(s(y''''))))) -> F(c(s(x''''), s(s(y''''))))
F(c(s(x''''), s(s(s(y''''))))) -> G(c(x'''', s(s(y''''))))
F(c(s(s(s(x''''))), s(y''''))) -> F(c(s(s(x'''')), s(s(y''''))))
F(c(s(s(x'''')), s(y''''))) -> F(c(s(x''''), s(s(y''''))))
F(c(s(s(s(x''''))), y'''')) -> F(c(s(s(x'''')), s(y'''')))
G(c(s(s(s(x''''))), s(y'))) -> F(c(s(s(x'''')), y'))
F(c(s(s(s(s(x'''''')))), s(s(s(s(y'''''')))))) -> G(c(s(s(s(x''''''))), s(s(s(y'''''')))))

Rules:

f(c(s(x), y)) -> f(c(x, s(y)))
f(c(s(x), s(y))) -> g(c(x, y))
g(c(x, s(y))) -> g(c(s(x), y))
g(c(s(x), s(y))) -> f(c(x, y))

Strategy:

innermost

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

G(c(s(s(s(x''''))), s(y'))) -> F(c(s(s(x'''')), y'))

seven new Dependency Pairs are created:

G(c(s(s(s(s(x'''''')))), s(y''))) -> F(c(s(s(s(x''''''))), y''))
G(c(s(s(s(x''''''))), s(s(y'''''')))) -> F(c(s(s(x'''''')), s(y'''''')))
G(c(s(s(s(s(x'''''')))), s(s(y'''''')))) -> F(c(s(s(s(x''''''))), s(y'''''')))
G(c(s(s(s(x''''''))), s(s(s(s(y'''''')))))) -> F(c(s(s(x'''''')), s(s(s(y'''''')))))
G(c(s(s(s(s(s(x''''''''))))), s(s(s(y''''''))))) -> F(c(s(s(s(s(x'''''''')))), s(s(y''''''))))
G(c(s(s(s(s(x'''''''')))), s(s(s(s(s(y''''''''))))))) -> F(c(s(s(s(x''''''''))), s(s(s(s(y''''''''))))))
G(c(s(s(s(s(s(x''''''''))))), s(s(s(s(s(y''''''''))))))) -> F(c(s(s(s(s(x'''''''')))), s(s(s(s(y''''''''))))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 2

             ↳FwdInst

...

               →DP Problem 10

                 ↳Forward Instantiation Transformation

Dependency Pairs:

G(c(s(s(s(s(s(x''''''''))))), s(s(s(s(s(y''''''''))))))) -> F(c(s(s(s(s(x'''''''')))), s(s(s(s(y''''''''))))))
G(c(s(s(s(s(x'''''''')))), s(s(s(s(s(y''''''''))))))) -> F(c(s(s(s(x''''''''))), s(s(s(s(y''''''''))))))
F(c(s(s(s(s(x'''''')))), s(s(s(s(y'''''')))))) -> G(c(s(s(s(x''''''))), s(s(s(y'''''')))))
G(c(s(s(s(s(s(x''''''''))))), s(s(s(y''''''))))) -> F(c(s(s(s(s(x'''''''')))), s(s(y''''''))))
F(c(s(s(s(s(x'''''')))), s(s(y'''')))) -> G(c(s(s(s(x''''''))), s(y'''')))
G(c(s(s(s(x''''''))), s(s(s(s(y'''''')))))) -> F(c(s(s(x'''''')), s(s(s(y'''''')))))
F(c(s(s(x'''')), s(s(s(y''''))))) -> G(c(s(x''''), s(s(y''''))))
G(c(s(s(s(s(x'''''')))), s(s(y'''''')))) -> F(c(s(s(s(x''''''))), s(y'''''')))
F(c(s(s(s(s(x'''''')))), s(s(s(s(y'''''')))))) -> G(c(s(s(s(x''''''))), s(s(s(y'''''')))))
G(c(s(s(s(x''''''))), s(s(y'''''')))) -> F(c(s(s(x'''''')), s(y'''''')))
F(c(s(s(s(x''''''))), s(s(s(s(y'''''')))))) -> G(c(s(s(x'''''')), s(s(s(y'''''')))))
G(c(s(s(s(s(x'''''')))), s(y''))) -> F(c(s(s(s(x''''''))), y''))
G(c(s(s(x'''''')), s(s(s(s(y'''''')))))) -> G(c(s(s(s(x''''''))), s(s(s(y'''''')))))
G(c(s(x''''''), s(s(s(s(y'''''')))))) -> G(c(s(s(x'''''')), s(s(s(y'''''')))))
G(c(s(s(x'''''')), s(s(y'''')))) -> G(c(s(s(s(x''''''))), s(y'''')))
G(c(x'''', s(s(s(y''''))))) -> G(c(s(x''''), s(s(y''''))))
F(c(s(s(s(s(x'''''')))), s(s(y'''')))) -> G(c(s(s(s(x''''''))), s(y'''')))
G(c(s(s(s(x''''))), s(s(s(y''''))))) -> F(c(s(s(x'''')), s(s(y''''))))
F(c(s(x''''), s(s(s(y''''))))) -> G(c(x'''', s(s(y''''))))
F(c(s(s(s(x''''))), s(y''''))) -> F(c(s(s(x'''')), s(s(y''''))))
F(c(s(s(x'''')), s(y''''))) -> F(c(s(x''''), s(s(y''''))))
F(c(s(s(s(x''''))), y'''')) -> F(c(s(s(x'''')), s(y'''')))
G(c(s(s(x'''')), s(s(s(y''''))))) -> F(c(s(x''''), s(s(y''''))))
F(c(s(s(s(x''''''))), s(s(s(s(y'''''')))))) -> G(c(s(s(x'''''')), s(s(s(y'''''')))))

Rules:

f(c(s(x), y)) -> f(c(x, s(y)))
f(c(s(x), s(y))) -> g(c(x, y))
g(c(x, s(y))) -> g(c(s(x), y))
g(c(s(x), s(y))) -> f(c(x, y))

Strategy:

innermost

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

G(c(s(s(x'''')), s(s(s(y''''))))) -> F(c(s(x''''), s(s(y''''))))

seven new Dependency Pairs are created:

G(c(s(s(s(s(x'''''')))), s(s(s(y''''''))))) -> F(c(s(s(s(x''''''))), s(s(y''''''))))
G(c(s(s(s(x''''''))), s(s(s(y''''''))))) -> F(c(s(s(x'''''')), s(s(y''''''))))
G(c(s(s(x'''''')), s(s(s(s(y'''''')))))) -> F(c(s(x''''''), s(s(s(y'''''')))))
G(c(s(s(s(s(s(x''''''''))))), s(s(s(y''''''))))) -> F(c(s(s(s(s(x'''''''')))), s(s(y''''''))))
G(c(s(s(s(s(x'''''''')))), s(s(s(s(s(y''''''''))))))) -> F(c(s(s(s(x''''''''))), s(s(s(s(y''''''''))))))
G(c(s(s(s(s(s(x''''''''))))), s(s(s(s(s(y''''''''))))))) -> F(c(s(s(s(s(x'''''''')))), s(s(s(s(y''''''''))))))
G(c(s(s(s(x''''''))), s(s(s(s(y'''''')))))) -> F(c(s(s(x'''''')), s(s(s(y'''''')))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 2

             ↳FwdInst

...

               →DP Problem 11

                 ↳Argument Filtering and Ordering

Dependency Pairs:

G(c(s(s(s(x''''''))), s(s(s(s(y'''''')))))) -> F(c(s(s(x'''''')), s(s(s(y'''''')))))
G(c(s(s(s(s(s(x''''''''))))), s(s(s(s(s(y''''''''))))))) -> F(c(s(s(s(s(x'''''''')))), s(s(s(s(y''''''''))))))
G(c(s(s(s(s(x'''''''')))), s(s(s(s(s(y''''''''))))))) -> F(c(s(s(s(x''''''''))), s(s(s(s(y''''''''))))))
G(c(s(s(s(s(s(x''''''''))))), s(s(s(y''''''))))) -> F(c(s(s(s(s(x'''''''')))), s(s(y''''''))))
G(c(s(s(x'''''')), s(s(s(s(y'''''')))))) -> F(c(s(x''''''), s(s(s(y'''''')))))
G(c(s(s(s(x''''''))), s(s(s(y''''''))))) -> F(c(s(s(x'''''')), s(s(y''''''))))
G(c(s(s(s(s(x'''''')))), s(s(s(y''''''))))) -> F(c(s(s(s(x''''''))), s(s(y''''''))))
F(c(s(s(s(s(x'''''')))), s(s(s(s(y'''''')))))) -> G(c(s(s(s(x''''''))), s(s(s(y'''''')))))
G(c(s(s(s(s(x'''''''')))), s(s(s(s(s(y''''''''))))))) -> F(c(s(s(s(x''''''''))), s(s(s(s(y''''''''))))))
F(c(s(s(s(x''''''))), s(s(s(s(y'''''')))))) -> G(c(s(s(x'''''')), s(s(s(y'''''')))))
G(c(s(s(s(s(s(x''''''''))))), s(s(s(y''''''))))) -> F(c(s(s(s(s(x'''''''')))), s(s(y''''''))))
F(c(s(s(s(s(x'''''')))), s(s(y'''')))) -> G(c(s(s(s(x''''''))), s(y'''')))
G(c(s(s(s(x''''''))), s(s(s(s(y'''''')))))) -> F(c(s(s(x'''''')), s(s(s(y'''''')))))
F(c(s(s(x'''')), s(s(s(y''''))))) -> G(c(s(x''''), s(s(y''''))))
G(c(s(s(s(s(x'''''')))), s(s(y'''''')))) -> F(c(s(s(s(x''''''))), s(y'''''')))
F(c(s(s(s(s(x'''''')))), s(s(s(s(y'''''')))))) -> G(c(s(s(s(x''''''))), s(s(s(y'''''')))))
G(c(s(s(s(x''''''))), s(s(y'''''')))) -> F(c(s(s(x'''''')), s(y'''''')))
F(c(s(s(s(x''''''))), s(s(s(s(y'''''')))))) -> G(c(s(s(x'''''')), s(s(s(y'''''')))))
G(c(s(s(s(s(x'''''')))), s(y''))) -> F(c(s(s(s(x''''''))), y''))
G(c(s(s(x'''''')), s(s(s(s(y'''''')))))) -> G(c(s(s(s(x''''''))), s(s(s(y'''''')))))
G(c(s(x''''''), s(s(s(s(y'''''')))))) -> G(c(s(s(x'''''')), s(s(s(y'''''')))))
G(c(s(s(x'''''')), s(s(y'''')))) -> G(c(s(s(s(x''''''))), s(y'''')))
G(c(x'''', s(s(s(y''''))))) -> G(c(s(x''''), s(s(y''''))))
F(c(s(s(s(s(x'''''')))), s(s(y'''')))) -> G(c(s(s(s(x''''''))), s(y'''')))
G(c(s(s(s(x''''))), s(s(s(y''''))))) -> F(c(s(s(x'''')), s(s(y''''))))
F(c(s(x''''), s(s(s(y''''))))) -> G(c(x'''', s(s(y''''))))
F(c(s(s(s(x''''))), s(y''''))) -> F(c(s(s(x'''')), s(s(y''''))))
F(c(s(s(x'''')), s(y''''))) -> F(c(s(x''''), s(s(y''''))))
F(c(s(s(s(x''''))), y'''')) -> F(c(s(s(x'''')), s(y'''')))
G(c(s(s(s(s(s(x''''''''))))), s(s(s(s(s(y''''''''))))))) -> F(c(s(s(s(s(x'''''''')))), s(s(s(s(y''''''''))))))

Rules:

f(c(s(x), y)) -> f(c(x, s(y)))
f(c(s(x), s(y))) -> g(c(x, y))
g(c(x, s(y))) -> g(c(s(x), y))
g(c(s(x), s(y))) -> f(c(x, y))

Strategy:

innermost

The following dependency pairs can be strictly oriented:

G(c(s(s(s(x''''''))), s(s(s(s(y'''''')))))) -> F(c(s(s(x'''''')), s(s(s(y'''''')))))
G(c(s(s(s(s(s(x''''''''))))), s(s(s(s(s(y''''''''))))))) -> F(c(s(s(s(s(x'''''''')))), s(s(s(s(y''''''''))))))
G(c(s(s(s(s(x'''''''')))), s(s(s(s(s(y''''''''))))))) -> F(c(s(s(s(x''''''''))), s(s(s(s(y''''''''))))))
G(c(s(s(s(s(s(x''''''''))))), s(s(s(y''''''))))) -> F(c(s(s(s(s(x'''''''')))), s(s(y''''''))))
G(c(s(s(x'''''')), s(s(s(s(y'''''')))))) -> F(c(s(x''''''), s(s(s(y'''''')))))
G(c(s(s(s(x''''''))), s(s(s(y''''''))))) -> F(c(s(s(x'''''')), s(s(y''''''))))
G(c(s(s(s(s(x'''''')))), s(s(s(y''''''))))) -> F(c(s(s(s(x''''''))), s(s(y''''''))))
F(c(s(s(s(s(x'''''')))), s(s(s(s(y'''''')))))) -> G(c(s(s(s(x''''''))), s(s(s(y'''''')))))
F(c(s(s(s(x''''''))), s(s(s(s(y'''''')))))) -> G(c(s(s(x'''''')), s(s(s(y'''''')))))
F(c(s(s(s(s(x'''''')))), s(s(y'''')))) -> G(c(s(s(s(x''''''))), s(y'''')))
F(c(s(s(x'''')), s(s(s(y''''))))) -> G(c(s(x''''), s(s(y''''))))
G(c(s(s(s(s(x'''''')))), s(s(y'''''')))) -> F(c(s(s(s(x''''''))), s(y'''''')))
G(c(s(s(s(x''''''))), s(s(y'''''')))) -> F(c(s(s(x'''''')), s(y'''''')))
G(c(s(s(s(s(x'''''')))), s(y''))) -> F(c(s(s(s(x''''''))), y''))
G(c(s(s(s(x''''))), s(s(s(y''''))))) -> F(c(s(s(x'''')), s(s(y''''))))
F(c(s(x''''), s(s(s(y''''))))) -> G(c(x'''', s(s(y''''))))

There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:

POL(c(x₁, x₂)) = x₁ + x₂

POL(G(x₁)) = x₁

POL(s(x₁)) = 1 + x₁

POL(F(x₁)) = x₁

resulting in one new DP problem.
Used Argument Filtering System:

F(x₁) -> F(x₁)
G(x₁) -> G(x₁)
c(x₁, x₂) -> c(x₁, x₂)
s(x₁) -> s(x₁)

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 2

             ↳FwdInst

...

               →DP Problem 12

                 ↳Dependency Graph

Dependency Pairs:

G(c(s(s(x'''''')), s(s(s(s(y'''''')))))) -> G(c(s(s(s(x''''''))), s(s(s(y'''''')))))
G(c(s(x''''''), s(s(s(s(y'''''')))))) -> G(c(s(s(x'''''')), s(s(s(y'''''')))))
G(c(s(s(x'''''')), s(s(y'''')))) -> G(c(s(s(s(x''''''))), s(y'''')))
G(c(x'''', s(s(s(y''''))))) -> G(c(s(x''''), s(s(y''''))))
F(c(s(s(s(x''''))), s(y''''))) -> F(c(s(s(x'''')), s(s(y''''))))
F(c(s(s(x'''')), s(y''''))) -> F(c(s(x''''), s(s(y''''))))
F(c(s(s(s(x''''))), y'''')) -> F(c(s(s(x'''')), s(y'''')))

Rules:

f(c(s(x), y)) -> f(c(x, s(y)))
f(c(s(x), s(y))) -> g(c(x, y))
g(c(x, s(y))) -> g(c(s(x), y))
g(c(s(x), s(y))) -> f(c(x, y))

Strategy:

innermost

Using the Dependency Graph the DP problem was split into 2 DP problems.

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 2

             ↳FwdInst

...

               →DP Problem 13

                 ↳Argument Filtering and Ordering

Dependency Pairs:

G(c(s(x''''''), s(s(s(s(y'''''')))))) -> G(c(s(s(x'''''')), s(s(s(y'''''')))))
G(c(s(s(x'''''')), s(s(y'''')))) -> G(c(s(s(s(x''''''))), s(y'''')))
G(c(x'''', s(s(s(y''''))))) -> G(c(s(x''''), s(s(y''''))))
G(c(s(s(x'''''')), s(s(s(s(y'''''')))))) -> G(c(s(s(s(x''''''))), s(s(s(y'''''')))))

Rules:

f(c(s(x), y)) -> f(c(x, s(y)))
f(c(s(x), s(y))) -> g(c(x, y))
g(c(x, s(y))) -> g(c(s(x), y))
g(c(s(x), s(y))) -> f(c(x, y))

Strategy:

innermost

The following dependency pairs can be strictly oriented:

G(c(s(x''''''), s(s(s(s(y'''''')))))) -> G(c(s(s(x'''''')), s(s(s(y'''''')))))
G(c(s(s(x'''''')), s(s(y'''')))) -> G(c(s(s(s(x''''''))), s(y'''')))
G(c(x'''', s(s(s(y''''))))) -> G(c(s(x''''), s(s(y''''))))
G(c(s(s(x'''''')), s(s(s(s(y'''''')))))) -> G(c(s(s(s(x''''''))), s(s(s(y'''''')))))

There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:

POL(G(x₁)) = 1 + x₁

POL(s(x₁)) = 1 + x₁

resulting in one new DP problem.
Used Argument Filtering System:

G(x₁) -> G(x₁)
c(x₁, x₂) -> x₂
s(x₁) -> s(x₁)

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 2

             ↳FwdInst

...

               →DP Problem 15

                 ↳Dependency Graph

Dependency Pair:

Rules:

f(c(s(x), y)) -> f(c(x, s(y)))
f(c(s(x), s(y))) -> g(c(x, y))
g(c(x, s(y))) -> g(c(s(x), y))
g(c(s(x), s(y))) -> f(c(x, y))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 2

             ↳FwdInst

...

               →DP Problem 14

                 ↳Argument Filtering and Ordering

Dependency Pairs:

F(c(s(s(x'''')), s(y''''))) -> F(c(s(x''''), s(s(y''''))))
F(c(s(s(s(x''''))), y'''')) -> F(c(s(s(x'''')), s(y'''')))
F(c(s(s(s(x''''))), s(y''''))) -> F(c(s(s(x'''')), s(s(y''''))))

Rules:

f(c(s(x), y)) -> f(c(x, s(y)))
f(c(s(x), s(y))) -> g(c(x, y))
g(c(x, s(y))) -> g(c(s(x), y))
g(c(s(x), s(y))) -> f(c(x, y))

Strategy:

innermost

The following dependency pairs can be strictly oriented:

F(c(s(s(x'''')), s(y''''))) -> F(c(s(x''''), s(s(y''''))))
F(c(s(s(s(x''''))), y'''')) -> F(c(s(s(x'''')), s(y'''')))
F(c(s(s(s(x''''))), s(y''''))) -> F(c(s(s(x'''')), s(s(y''''))))

There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:

POL(s(x₁)) = 1 + x₁

POL(F(x₁)) = 1 + x₁

resulting in one new DP problem.
Used Argument Filtering System:

F(x₁) -> F(x₁)
c(x₁, x₂) -> x₁
s(x₁) -> s(x₁)

Innermost Termination of R successfully shown.
Duration:
0:14 minutes

POL(c(x₁, x₂))	= x₁ + x₂
POL(G(x₁))	= x₁
POL(s(x₁))	= 1 + x₁
POL(F(x₁))	= x₁