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

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

F(g(x), s(0), y) -> F(y, y, g(x))
G(s(x)) -> G(x)

Furthermore, R contains one SCC.

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

Dependency Pair:

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

Rules:

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

Strategy:

innermost

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

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

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

The transformation is resulting in one new DP problem:

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

Dependency Pair:

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

Rules:

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

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

G(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`
`                 ↳Argument Filtering and Ordering`

Dependency Pair:

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

Rules:

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost 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)
s(x1) -> s(x1)

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 2`
`             ↳FwdInst`
`             ...`
`               →DP Problem 4`
`                 ↳Dependency Graph`

Dependency Pair:

Rules:

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

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

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