Term Rewriting System R:
[x, y, z]
rev(nil) -> nil
rev(rev(x)) -> x
rev(++(x, y)) -> ++(rev(y), rev(x))
++(nil, y) -> y
++(x, nil) -> x
++(.(x, y), z) -> .(x, ++(y, z))
++(x, ++(y, z)) -> ++(++(x, y), z)
make(x) -> .(x, nil)

Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

REV(++(x, y)) -> ++'(rev(y), rev(x))
REV(++(x, y)) -> REV(y)
REV(++(x, y)) -> REV(x)
++'(.(x, y), z) -> ++'(y, z)
++'(x, ++(y, z)) -> ++'(++(x, y), z)
++'(x, ++(y, z)) -> ++'(x, y)

Furthermore, R contains two SCCs.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Argument Filtering and Ordering`
`       →DP Problem 2`
`         ↳AFS`

Dependency Pairs:

++'(x, ++(y, z)) -> ++'(x, y)
++'(x, ++(y, z)) -> ++'(++(x, y), z)
++'(.(x, y), z) -> ++'(y, z)

Rules:

rev(nil) -> nil
rev(rev(x)) -> x
rev(++(x, y)) -> ++(rev(y), rev(x))
++(nil, y) -> y
++(x, nil) -> x
++(.(x, y), z) -> .(x, ++(y, z))
++(x, ++(y, z)) -> ++(++(x, y), z)
make(x) -> .(x, nil)

The following dependency pair can be strictly oriented:

++'(.(x, y), z) -> ++'(y, z)

The following usable rules w.r.t. to the AFS can be oriented:

++(nil, y) -> y
++(x, nil) -> x
++(.(x, y), z) -> .(x, ++(y, z))
++(x, ++(y, z)) -> ++(++(x, y), z)

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(++'(x1, x2)) =  1 + x1 + x2 POL(++(x1, x2)) =  x1 + x2 POL(nil) =  0 POL(.(x1, x2)) =  1 + x1 + x2

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

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`           →DP Problem 3`
`             ↳Argument Filtering and Ordering`
`       →DP Problem 2`
`         ↳AFS`

Dependency Pairs:

++'(x, ++(y, z)) -> ++'(x, y)
++'(x, ++(y, z)) -> ++'(++(x, y), z)

Rules:

rev(nil) -> nil
rev(rev(x)) -> x
rev(++(x, y)) -> ++(rev(y), rev(x))
++(nil, y) -> y
++(x, nil) -> x
++(.(x, y), z) -> .(x, ++(y, z))
++(x, ++(y, z)) -> ++(++(x, y), z)
make(x) -> .(x, nil)

The following dependency pair can be strictly oriented:

++'(x, ++(y, z)) -> ++'(x, y)

The following usable rules w.r.t. to the AFS can be oriented:

++(nil, y) -> y
++(x, nil) -> x
++(.(x, y), z) -> .(x, ++(y, z))
++(x, ++(y, z)) -> ++(++(x, y), z)

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(++'(x1, x2)) =  1 + x1 + x2 POL(++(x1, x2)) =  1 + x1 + x2 POL(nil) =  0 POL(.(x1, x2)) =  x1 + x2

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

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`           →DP Problem 3`
`             ↳AFS`
`             ...`
`               →DP Problem 4`
`                 ↳Argument Filtering and Ordering`
`       →DP Problem 2`
`         ↳AFS`

Dependency Pair:

++'(x, ++(y, z)) -> ++'(++(x, y), z)

Rules:

rev(nil) -> nil
rev(rev(x)) -> x
rev(++(x, y)) -> ++(rev(y), rev(x))
++(nil, y) -> y
++(x, nil) -> x
++(.(x, y), z) -> .(x, ++(y, z))
++(x, ++(y, z)) -> ++(++(x, y), z)
make(x) -> .(x, nil)

The following dependency pair can be strictly oriented:

++'(x, ++(y, z)) -> ++'(++(x, y), z)

There are no usable rules w.r.t. to the AFS that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(++(x1, x2)) =  1 + x1 + x2

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

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

Dependency Pair:

Rules:

rev(nil) -> nil
rev(rev(x)) -> x
rev(++(x, y)) -> ++(rev(y), rev(x))
++(nil, y) -> y
++(x, nil) -> x
++(.(x, y), z) -> .(x, ++(y, z))
++(x, ++(y, z)) -> ++(++(x, y), z)
make(x) -> .(x, nil)

Using the Dependency Graph resulted in no new DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳Argument Filtering and Ordering`

Dependency Pairs:

REV(++(x, y)) -> REV(x)
REV(++(x, y)) -> REV(y)

Rules:

rev(nil) -> nil
rev(rev(x)) -> x
rev(++(x, y)) -> ++(rev(y), rev(x))
++(nil, y) -> y
++(x, nil) -> x
++(.(x, y), z) -> .(x, ++(y, z))
++(x, ++(y, z)) -> ++(++(x, y), z)
make(x) -> .(x, nil)

The following dependency pairs can be strictly oriented:

REV(++(x, y)) -> REV(x)
REV(++(x, y)) -> REV(y)

There are no usable rules w.r.t. to the AFS that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(REV(x1)) =  x1 POL(++(x1, x2)) =  1 + x1 + x2

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

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳AFS`
`           →DP Problem 6`
`             ↳Dependency Graph`

Dependency Pair:

Rules:

rev(nil) -> nil
rev(rev(x)) -> x
rev(++(x, y)) -> ++(rev(y), rev(x))
++(nil, y) -> y
++(x, nil) -> x
++(.(x, y), z) -> .(x, ++(y, z))
++(x, ++(y, z)) -> ++(++(x, y), z)
make(x) -> .(x, nil)

Using the Dependency Graph resulted in no new DP problems.

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