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

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

Furthermore, R contains three SCCs.

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

Dependency Pair:

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

Rules:

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

On this DP problem, an 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(x''), s(y''))) -> F(c(x'', s(s(y''))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Inst`
`           →DP Problem 4`
`             ↳Instantiation Transformation`
`       →DP Problem 2`
`         ↳Inst`
`       →DP Problem 3`
`         ↳Nar`

Dependency Pair:

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

Rules:

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

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

F(c(s(x''), s(y''))) -> F(c(x'', s(s(y''))))
one new Dependency Pair is created:

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

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Inst`
`           →DP Problem 4`
`             ↳Inst`
`             ...`
`               →DP Problem 5`
`                 ↳Polynomial Ordering`
`       →DP Problem 2`
`         ↳Inst`
`       →DP Problem 3`
`         ↳Nar`

Dependency Pair:

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

Rules:

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

The following dependency pair can be strictly oriented:

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

There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

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

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Inst`
`           →DP Problem 4`
`             ↳Inst`
`             ...`
`               →DP Problem 6`
`                 ↳Dependency Graph`
`       →DP Problem 2`
`         ↳Inst`
`       →DP Problem 3`
`         ↳Nar`

Dependency Pair:

Rules:

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

Using the Dependency Graph resulted in no new DP problems.

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

Dependency Pair:

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

Rules:

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

On this DP problem, an 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(s(x''), s(y''))) -> G(c(s(s(x'')), y''))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Inst`
`       →DP Problem 2`
`         ↳Inst`
`           →DP Problem 7`
`             ↳Instantiation Transformation`
`       →DP Problem 3`
`         ↳Nar`

Dependency Pair:

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

Rules:

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

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

G(c(s(x''), s(y''))) -> G(c(s(s(x'')), y''))
one new Dependency Pair is created:

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

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Inst`
`       →DP Problem 2`
`         ↳Inst`
`           →DP Problem 7`
`             ↳Inst`
`             ...`
`               →DP Problem 8`
`                 ↳Polynomial Ordering`
`       →DP Problem 3`
`         ↳Nar`

Dependency Pair:

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

Rules:

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

The following dependency pair can be strictly oriented:

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

There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

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

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Inst`
`       →DP Problem 2`
`         ↳Inst`
`           →DP Problem 7`
`             ↳Inst`
`             ...`
`               →DP Problem 9`
`                 ↳Dependency Graph`
`       →DP Problem 3`
`         ↳Nar`

Dependency Pair:

Rules:

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

Using the Dependency Graph resulted in no new DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Inst`
`       →DP Problem 2`
`         ↳Inst`
`       →DP Problem 3`
`         ↳Narrowing Transformation`

Dependency Pair:

G(s(f(x))) -> G(f(x))

Rules:

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

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

G(s(f(x))) -> G(f(x))
one new Dependency Pair is created:

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

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Inst`
`       →DP Problem 2`
`         ↳Inst`
`       →DP Problem 3`
`         ↳Nar`
`           →DP Problem 10`
`             ↳Narrowing Transformation`

Dependency Pair:

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

Rules:

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

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

G(s(f(c(s(x''), y')))) -> G(f(c(x'', s(y'))))
one new Dependency Pair is created:

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

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Inst`
`       →DP Problem 2`
`         ↳Inst`
`       →DP Problem 3`
`         ↳Nar`
`           →DP Problem 10`
`             ↳Nar`
`             ...`
`               →DP Problem 11`
`                 ↳Forward Instantiation Transformation`

Dependency Pair:

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

Rules:

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

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

G(s(f(c(s(s(x')), y'')))) -> G(f(c(x', s(s(y'')))))
no new Dependency Pairs are created.
The transformation is resulting in no new DP problems.

Termination of R successfully shown.
Duration:
0:00 minutes