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.

`   R`
`     ↳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.

`   R`
`     ↳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:

`   R`
`     ↳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:

`   R`
`     ↳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:

`   R`
`     ↳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:

`   R`
`     ↳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:

`   R`
`     ↳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:

`   R`
`     ↳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:

`   R`
`     ↳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:

`   R`
`     ↳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:

`   R`
`     ↳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:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 2`
`             ↳FwdInst`
`             ...`
`               →DP Problem 11`
`                 ↳Polynomial 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 implicit 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.

`   R`
`     ↳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.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 2`
`             ↳FwdInst`
`             ...`
`               →DP Problem 13`
`                 ↳Polynomial 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 implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(c(x1, x2)) =  1 + x2 POL(G(x1)) =  1 + x1 POL(s(x1)) =  1 + x1

resulting in one new DP problem.

`   R`
`     ↳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.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 2`
`             ↳FwdInst`
`             ...`
`               →DP Problem 14`
`                 ↳Polynomial 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 implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(c(x1, x2)) =  1 + x1 POL(s(x1)) =  1 + x1 POL(F(x1)) =  1 + x1

resulting in one new DP problem.

Innermost Termination of R successfully shown.
Duration:
0:13 minutes