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`
`         ↳Argument Filtering and Ordering`
`       →DP Problem 2`
`         ↳AFS`
`       →DP Problem 3`
`         ↳AFS`

Dependency Pair:

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

Rules:

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(APP(x1, x2)) =  x1 + x2 POL(add(x1, x2)) =  1 + x1 + x2

resulting in one new DP problem.
Used Argument Filtering System:
APP(x1, x2) -> APP(x1, x2)

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`           →DP Problem 4`
`             ↳Dependency Graph`
`       →DP Problem 2`
`         ↳AFS`
`       →DP Problem 3`
`         ↳AFS`

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`
`         ↳AFS`
`       →DP Problem 2`
`         ↳Argument Filtering and Ordering`
`       →DP Problem 3`
`         ↳AFS`

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 that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(REVERSE(x1)) =  x1 POL(add(x1, x2)) =  1 + x1 + x2

resulting in one new DP problem.
Used Argument Filtering System:
REVERSE(x1) -> REVERSE(x1)

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳AFS`
`           →DP Problem 5`
`             ↳Dependency Graph`
`       →DP Problem 3`
`         ↳AFS`

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`
`         ↳AFS`
`       →DP Problem 2`
`         ↳AFS`
`       →DP Problem 3`
`         ↳Argument Filtering and Ordering`

Dependency Pair:

Rules:

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

The following usable rules for innermost 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 + x1 + x2

resulting in one new DP problem.
Used Argument Filtering System:
SHUFFLE(x1) -> SHUFFLE(x1)
reverse(x1) -> reverse(x1)
app(x1, x2) -> app(x1, x2)

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳AFS`
`       →DP Problem 3`
`         ↳AFS`
`           →DP Problem 6`
`             ↳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