Term Rewriting System R:
[x]
g(s(x)) -> f(x)
g(0) -> 0
f(0) -> s(0)
f(s(x)) -> s(s(g(x)))

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

G(s(x)) -> F(x)
F(s(x)) -> G(x)

Furthermore, R contains one SCC.

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

Dependency Pairs:

F(s(x)) -> G(x)
G(s(x)) -> F(x)

Rules:

g(s(x)) -> f(x)
g(0) -> 0
f(0) -> s(0)
f(s(x)) -> s(s(g(x)))

Strategy:

innermost

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

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

G(s(s(x''))) -> F(s(x''))

The transformation is resulting in one new DP problem:

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

Dependency Pairs:

G(s(s(x''))) -> F(s(x''))
F(s(x)) -> G(x)

Rules:

g(s(x)) -> f(x)
g(0) -> 0
f(0) -> s(0)
f(s(x)) -> s(s(g(x)))

Strategy:

innermost

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

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

F(s(s(s(x'''')))) -> G(s(s(x'''')))

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(s(s(s(x'''')))) -> G(s(s(x'''')))
G(s(s(x''))) -> F(s(x''))

Rules:

g(s(x)) -> f(x)
g(0) -> 0
f(0) -> s(0)
f(s(x)) -> s(s(g(x)))

Strategy:

innermost

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

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

G(s(s(s(s(x''''''))))) -> F(s(s(s(x''''''))))

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(s(s(s(s(x''''''))))) -> F(s(s(s(x''''''))))
F(s(s(s(x'''')))) -> G(s(s(x'''')))

Rules:

g(s(x)) -> f(x)
g(0) -> 0
f(0) -> s(0)
f(s(x)) -> s(s(g(x)))

Strategy:

innermost

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

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

F(s(s(s(s(s(x'''''''')))))) -> G(s(s(s(s(x'''''''')))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 2`
`             ↳FwdInst`
`             ...`
`               →DP Problem 5`
`                 ↳Polynomial Ordering`

Dependency Pairs:

F(s(s(s(s(s(x'''''''')))))) -> G(s(s(s(s(x'''''''')))))
G(s(s(s(s(x''''''))))) -> F(s(s(s(x''''''))))

Rules:

g(s(x)) -> f(x)
g(0) -> 0
f(0) -> s(0)
f(s(x)) -> s(s(g(x)))

Strategy:

innermost

The following dependency pairs can be strictly oriented:

F(s(s(s(s(s(x'''''''')))))) -> G(s(s(s(s(x'''''''')))))
G(s(s(s(s(x''''''))))) -> F(s(s(s(x''''''))))

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(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 6`
`                 ↳Dependency Graph`

Dependency Pair:

Rules:

g(s(x)) -> f(x)
g(0) -> 0
f(0) -> s(0)
f(s(x)) -> s(s(g(x)))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

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