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

Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

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

Furthermore, R contains two SCCs.

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

Dependency Pair:

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

Rules:

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

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

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

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

The transformation is resulting in one new DP problem:

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

Dependency Pair:

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

Rules:

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

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

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

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

The transformation is resulting in one new DP problem:

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

Dependency Pair:

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

Rules:

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

The following dependency pair can be strictly oriented:

F(s(x''''), s(c(s(c(s(y'''')))))) -> F(x'''', s(c(s(c(s(c(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)) =  0 POL(s(x1)) =  1 + x1 POL(F(x1, x2)) =  x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Inst`
`           →DP Problem 3`
`             ↳Inst`
`             ...`
`               →DP Problem 5`
`                 ↳Dependency Graph`
`       →DP Problem 2`
`         ↳FwdInst`

Dependency Pair:

Rules:

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

Using the Dependency Graph resulted in no new DP problems.

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

Dependency Pair:

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

Rules:

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

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

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

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

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Inst`
`       →DP Problem 2`
`         ↳FwdInst`
`           →DP Problem 6`
`             ↳Instantiation Transformation`

Dependency Pair:

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

Rules:

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

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

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

F(c(y'''''), c(c(y''0))) -> F(c(y''0), c(y''0))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Inst`
`       →DP Problem 2`
`         ↳FwdInst`
`           →DP Problem 6`
`             ↳Inst`
`             ...`
`               →DP Problem 7`
`                 ↳Polynomial Ordering`

Dependency Pair:

F(c(y'''''), c(c(y''0))) -> F(c(y''0), c(y''0))

Rules:

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

The following dependency pair can be strictly oriented:

F(c(y'''''), c(c(y''0))) -> F(c(y''0), c(y''0))

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)) =  1 + x1 POL(F(x1, x2)) =  1 + x2

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Inst`
`       →DP Problem 2`
`         ↳FwdInst`
`           →DP Problem 6`
`             ↳Inst`
`             ...`
`               →DP Problem 8`
`                 ↳Dependency Graph`

Dependency Pair:

Rules:

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

Using the Dependency Graph resulted in no new DP problems.

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