Term Rewriting System R:
[y, n, x]
app(nil, y) -> y
reverse(nil) -> nil
shuffle(nil) -> nil

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

APP(add(n, x), y) -> APP(x, y)

Furthermore, R contains three SCCs.

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

Dependency Pair:

APP(add(n, x), y) -> APP(x, y)

Rules:

app(nil, y) -> y
reverse(nil) -> nil
shuffle(nil) -> nil

Strategy:

innermost

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

APP(add(n, x), y) -> APP(x, y)
one new Dependency Pair is created:

The transformation is resulting in one new DP problem:

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

Dependency Pair:

Rules:

app(nil, y) -> y
reverse(nil) -> nil
shuffle(nil) -> nil

Strategy:

innermost

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

one new Dependency Pair is created:

The transformation is resulting in one new DP problem:

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

Dependency Pair:

Rules:

app(nil, y) -> y
reverse(nil) -> nil
shuffle(nil) -> nil

Strategy:

innermost

The following dependency pair can be strictly oriented:

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(APP(x1, x2)) =  1 + x1 POL(add(x1, x2)) =  1 + x2

resulting in one new DP problem.

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

Dependency Pair:

Rules:

app(nil, y) -> y
reverse(nil) -> nil
shuffle(nil) -> nil

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

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

Dependency Pair:

Rules:

app(nil, y) -> y
reverse(nil) -> nil
shuffle(nil) -> nil

Strategy:

innermost

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

one new Dependency Pair is created:

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`
`       →DP Problem 3`
`         ↳Nar`

Dependency Pair:

Rules:

app(nil, y) -> y
reverse(nil) -> nil
shuffle(nil) -> nil

Strategy:

innermost

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

one new Dependency Pair is created:

The transformation is resulting in one new DP problem:

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

Dependency Pair:

Rules:

app(nil, y) -> y
reverse(nil) -> nil
shuffle(nil) -> nil

Strategy:

innermost

The following dependency pair can be strictly oriented:

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(REVERSE(x1)) =  1 + x1 POL(add(x1, x2)) =  1 + x2

resulting in one new DP problem.

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

Dependency Pair:

Rules:

app(nil, y) -> y
reverse(nil) -> nil
shuffle(nil) -> nil

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

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

Dependency Pair:

Rules:

app(nil, y) -> y
reverse(nil) -> nil
shuffle(nil) -> nil

Strategy:

innermost

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

two new Dependency Pairs are created:

The transformation is resulting in one new DP problem:

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

Dependency Pair:

Rules:

app(nil, y) -> y
reverse(nil) -> nil
shuffle(nil) -> nil

Strategy:

innermost

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

two new Dependency Pairs are created:

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳Nar`
`           →DP Problem 10`
`             ↳Nar`
`             ...`
`               →DP Problem 11`
`                 ↳Rewriting Transformation`

Dependency Pairs:

Rules:

app(nil, y) -> y
reverse(nil) -> nil
shuffle(nil) -> nil

Strategy:

innermost

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

one new Dependency Pair is created:

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳Nar`
`           →DP Problem 10`
`             ↳Nar`
`             ...`
`               →DP Problem 12`
`                 ↳Polynomial Ordering`

Dependency Pair:

Rules:

app(nil, y) -> y
reverse(nil) -> nil
shuffle(nil) -> nil

Strategy:

innermost

The following dependency pair can be strictly oriented:

Additionally, the following usable rules for innermost w.r.t. to the implicit AFS can be oriented:

reverse(nil) -> nil
app(nil, y) -> y

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(reverse(x1)) =  x1 POL(SHUFFLE(x1)) =  1 + x1 POL(nil) =  0 POL(app(x1, x2)) =  x1 + x2 POL(add(x1, x2)) =  1 + x2

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳Nar`
`           →DP Problem 10`
`             ↳Nar`
`             ...`
`               →DP Problem 13`
`                 ↳Dependency Graph`

Dependency Pair:

Rules:

app(nil, y) -> y
reverse(nil) -> nil
shuffle(nil) -> nil

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

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