Termination Proof using AProVE

Term Rewriting System R:
[x, y, z]
rev(nil) -> nil
rev(++(x, y)) -> ++(rev1(x, y), rev2(x, y))
rev1(x, nil) -> x
rev1(x, ++(y, z)) -> rev1(y, z)
rev2(x, nil) -> nil
rev2(x, ++(y, z)) -> rev(++(x, rev(rev2(y, z))))

Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

REV(++(x, y)) -> REV1(x, y)
REV(++(x, y)) -> REV2(x, y)
REV1(x, ++(y, z)) -> REV1(y, z)
REV2(x, ++(y, z)) -> REV(++(x, rev(rev2(y, z))))
REV2(x, ++(y, z)) -> REV(rev2(y, z))
REV2(x, ++(y, z)) -> REV2(y, z)

Furthermore, R contains two SCCs.

     ↳DPs

       →DP Problem 1

         ↳Polynomial Ordering

       →DP Problem 2

         ↳Nar

Dependency Pair:

REV1(x, ++(y, z)) -> REV1(y, z)

Rules:

rev(nil) -> nil
rev(++(x, y)) -> ++(rev1(x, y), rev2(x, y))
rev1(x, nil) -> x
rev1(x, ++(y, z)) -> rev1(y, z)
rev2(x, nil) -> nil
rev2(x, ++(y, z)) -> rev(++(x, rev(rev2(y, z))))

The following dependency pair can be strictly oriented:

REV1(x, ++(y, z)) -> REV1(y, z)

There are no usable rules using the Ce-refinement that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(REV1(x₁, x₂)) = x₂

POL(++(x₁, x₂)) = 1 + x₂

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

           →DP Problem 3

             ↳Dependency Graph

       →DP Problem 2

         ↳Nar

Dependency Pair:

Rules:

rev(nil) -> nil
rev(++(x, y)) -> ++(rev1(x, y), rev2(x, y))
rev1(x, nil) -> x
rev1(x, ++(y, z)) -> rev1(y, z)
rev2(x, nil) -> nil
rev2(x, ++(y, z)) -> rev(++(x, rev(rev2(y, z))))

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Narrowing Transformation

Dependency Pairs:

REV2(x, ++(y, z)) -> REV2(y, z)
REV2(x, ++(y, z)) -> REV(rev2(y, z))
REV2(x, ++(y, z)) -> REV(++(x, rev(rev2(y, z))))
REV(++(x, y)) -> REV2(x, y)

Rules:

rev(nil) -> nil
rev(++(x, y)) -> ++(rev1(x, y), rev2(x, y))
rev1(x, nil) -> x
rev1(x, ++(y, z)) -> rev1(y, z)
rev2(x, nil) -> nil
rev2(x, ++(y, z)) -> rev(++(x, rev(rev2(y, z))))

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

REV2(x, ++(y, z)) -> REV(rev2(y, z))

two new Dependency Pairs are created:

REV2(x, ++(y', nil)) -> REV(nil)
REV2(x, ++(y0, ++(y'', z''))) -> REV(rev(++(y0, rev(rev2(y'', z'')))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Nar

           →DP Problem 4

             ↳Narrowing Transformation

Dependency Pairs:

REV2(x, ++(y0, ++(y'', z''))) -> REV(rev(++(y0, rev(rev2(y'', z'')))))
REV(++(x, y)) -> REV2(x, y)
REV2(x, ++(y, z)) -> REV(++(x, rev(rev2(y, z))))
REV2(x, ++(y, z)) -> REV2(y, z)

Rules:

rev(nil) -> nil
rev(++(x, y)) -> ++(rev1(x, y), rev2(x, y))
rev1(x, nil) -> x
rev1(x, ++(y, z)) -> rev1(y, z)
rev2(x, nil) -> nil
rev2(x, ++(y, z)) -> rev(++(x, rev(rev2(y, z))))

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

REV2(x, ++(y0, ++(y'', z''))) -> REV(rev(++(y0, rev(rev2(y'', z'')))))

three new Dependency Pairs are created:

REV2(x, ++(y0', ++(y''', z'''))) -> REV(++(rev1(y0', rev(rev2(y''', z'''))), rev2(y0', rev(rev2(y''', z''')))))
REV2(x, ++(y0, ++(y''', nil))) -> REV(rev(++(y0, rev(nil))))
REV2(x, ++(y0, ++(y''', ++(y', z')))) -> REV(rev(++(y0, rev(rev(++(y''', rev(rev2(y', z'))))))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Nar

           →DP Problem 4

             ↳Nar

...

               →DP Problem 5

                 ↳Narrowing Transformation

Dependency Pairs:

REV2(x, ++(y0, ++(y''', ++(y', z')))) -> REV(rev(++(y0, rev(rev(++(y''', rev(rev2(y', z'))))))))
REV2(x, ++(y0, ++(y''', nil))) -> REV(rev(++(y0, rev(nil))))
REV2(x, ++(y0', ++(y''', z'''))) -> REV(++(rev1(y0', rev(rev2(y''', z'''))), rev2(y0', rev(rev2(y''', z''')))))
REV2(x, ++(y, z)) -> REV2(y, z)
REV2(x, ++(y, z)) -> REV(++(x, rev(rev2(y, z))))
REV(++(x, y)) -> REV2(x, y)

Rules:

rev(nil) -> nil
rev(++(x, y)) -> ++(rev1(x, y), rev2(x, y))
rev1(x, nil) -> x
rev1(x, ++(y, z)) -> rev1(y, z)
rev2(x, nil) -> nil
rev2(x, ++(y, z)) -> rev(++(x, rev(rev2(y, z))))

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

REV2(x, ++(y0, ++(y''', nil))) -> REV(rev(++(y0, rev(nil))))

two new Dependency Pairs are created:

REV2(x, ++(y0', ++(y''', nil))) -> REV(++(rev1(y0', rev(nil)), rev2(y0', rev(nil))))
REV2(x, ++(y0, ++(y''', nil))) -> REV(rev(++(y0, nil)))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Nar

           →DP Problem 4

             ↳Nar

...

               →DP Problem 6

                 ↳Narrowing Transformation

Dependency Pairs:

REV2(x, ++(y0, ++(y''', nil))) -> REV(rev(++(y0, nil)))
REV2(x, ++(y0', ++(y''', nil))) -> REV(++(rev1(y0', rev(nil)), rev2(y0', rev(nil))))
REV2(x, ++(y0', ++(y''', z'''))) -> REV(++(rev1(y0', rev(rev2(y''', z'''))), rev2(y0', rev(rev2(y''', z''')))))
REV2(x, ++(y, z)) -> REV2(y, z)
REV2(x, ++(y, z)) -> REV(++(x, rev(rev2(y, z))))
REV(++(x, y)) -> REV2(x, y)
REV2(x, ++(y0, ++(y''', ++(y', z')))) -> REV(rev(++(y0, rev(rev(++(y''', rev(rev2(y', z'))))))))

Rules:

rev(nil) -> nil
rev(++(x, y)) -> ++(rev1(x, y), rev2(x, y))
rev1(x, nil) -> x
rev1(x, ++(y, z)) -> rev1(y, z)
rev2(x, nil) -> nil
rev2(x, ++(y, z)) -> rev(++(x, rev(rev2(y, z))))

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

REV2(x, ++(y0, ++(y''', ++(y', z')))) -> REV(rev(++(y0, rev(rev(++(y''', rev(rev2(y', z'))))))))

four new Dependency Pairs are created:

REV2(x, ++(y0', ++(y'''', ++(y'', z'')))) -> REV(++(rev1(y0', rev(rev(++(y'''', rev(rev2(y'', z'')))))), rev2(y0', rev(rev(++(y'''', rev(rev2(y'', z''))))))))
REV2(x, ++(y0, ++(y'''', ++(y'', z'')))) -> REV(rev(++(y0, rev(++(rev1(y'''', rev(rev2(y'', z''))), rev2(y'''', rev(rev2(y'', z''))))))))
REV2(x, ++(y0, ++(y''', ++(y'', nil)))) -> REV(rev(++(y0, rev(rev(++(y''', rev(nil)))))))
REV2(x, ++(y0, ++(y''', ++(y'', ++(y1, z''))))) -> REV(rev(++(y0, rev(rev(++(y''', rev(rev(++(y'', rev(rev2(y1, z'')))))))))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Nar

           →DP Problem 4

             ↳Nar

...

               →DP Problem 7

                 ↳Narrowing Transformation

Dependency Pairs:

REV2(x, ++(y0, ++(y''', ++(y'', ++(y1, z''))))) -> REV(rev(++(y0, rev(rev(++(y''', rev(rev(++(y'', rev(rev2(y1, z'')))))))))))
REV2(x, ++(y0, ++(y''', ++(y'', nil)))) -> REV(rev(++(y0, rev(rev(++(y''', rev(nil)))))))
REV2(x, ++(y0, ++(y'''', ++(y'', z'')))) -> REV(rev(++(y0, rev(++(rev1(y'''', rev(rev2(y'', z''))), rev2(y'''', rev(rev2(y'', z''))))))))
REV2(x, ++(y0', ++(y'''', ++(y'', z'')))) -> REV(++(rev1(y0', rev(rev(++(y'''', rev(rev2(y'', z'')))))), rev2(y0', rev(rev(++(y'''', rev(rev2(y'', z''))))))))
REV2(x, ++(y0', ++(y''', nil))) -> REV(++(rev1(y0', rev(nil)), rev2(y0', rev(nil))))
REV2(x, ++(y0', ++(y''', z'''))) -> REV(++(rev1(y0', rev(rev2(y''', z'''))), rev2(y0', rev(rev2(y''', z''')))))
REV2(x, ++(y, z)) -> REV2(y, z)
REV2(x, ++(y, z)) -> REV(++(x, rev(rev2(y, z))))
REV(++(x, y)) -> REV2(x, y)
REV2(x, ++(y0, ++(y''', nil))) -> REV(rev(++(y0, nil)))

Rules:

rev(nil) -> nil
rev(++(x, y)) -> ++(rev1(x, y), rev2(x, y))
rev1(x, nil) -> x
rev1(x, ++(y, z)) -> rev1(y, z)
rev2(x, nil) -> nil
rev2(x, ++(y, z)) -> rev(++(x, rev(rev2(y, z))))

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

REV2(x, ++(y0, ++(y''', nil))) -> REV(rev(++(y0, nil)))

one new Dependency Pair is created:

REV2(x, ++(y0', ++(y''', nil))) -> REV(++(rev1(y0', nil), rev2(y0', nil)))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Nar

           →DP Problem 4

             ↳Nar

...

               →DP Problem 8

                 ↳Narrowing Transformation

Dependency Pairs:

REV2(x, ++(y0', ++(y''', nil))) -> REV(++(rev1(y0', nil), rev2(y0', nil)))
REV2(x, ++(y0, ++(y''', ++(y'', nil)))) -> REV(rev(++(y0, rev(rev(++(y''', rev(nil)))))))
REV2(x, ++(y0, ++(y'''', ++(y'', z'')))) -> REV(rev(++(y0, rev(++(rev1(y'''', rev(rev2(y'', z''))), rev2(y'''', rev(rev2(y'', z''))))))))
REV2(x, ++(y0', ++(y'''', ++(y'', z'')))) -> REV(++(rev1(y0', rev(rev(++(y'''', rev(rev2(y'', z'')))))), rev2(y0', rev(rev(++(y'''', rev(rev2(y'', z''))))))))
REV2(x, ++(y0', ++(y''', nil))) -> REV(++(rev1(y0', rev(nil)), rev2(y0', rev(nil))))
REV2(x, ++(y0', ++(y''', z'''))) -> REV(++(rev1(y0', rev(rev2(y''', z'''))), rev2(y0', rev(rev2(y''', z''')))))
REV2(x, ++(y, z)) -> REV2(y, z)
REV2(x, ++(y, z)) -> REV(++(x, rev(rev2(y, z))))
REV(++(x, y)) -> REV2(x, y)
REV2(x, ++(y0, ++(y''', ++(y'', ++(y1, z''))))) -> REV(rev(++(y0, rev(rev(++(y''', rev(rev(++(y'', rev(rev2(y1, z'')))))))))))

Rules:

rev(nil) -> nil
rev(++(x, y)) -> ++(rev1(x, y), rev2(x, y))
rev1(x, nil) -> x
rev1(x, ++(y, z)) -> rev1(y, z)
rev2(x, nil) -> nil
rev2(x, ++(y, z)) -> rev(++(x, rev(rev2(y, z))))

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

REV2(x, ++(y0, ++(y''', ++(y'', nil)))) -> REV(rev(++(y0, rev(rev(++(y''', rev(nil)))))))

three new Dependency Pairs are created:

REV2(x, ++(y0', ++(y'''', ++(y'', nil)))) -> REV(++(rev1(y0', rev(rev(++(y'''', rev(nil))))), rev2(y0', rev(rev(++(y'''', rev(nil)))))))
REV2(x, ++(y0, ++(y'''', ++(y'', nil)))) -> REV(rev(++(y0, rev(++(rev1(y'''', rev(nil)), rev2(y'''', rev(nil)))))))
REV2(x, ++(y0, ++(y''', ++(y'', nil)))) -> REV(rev(++(y0, rev(rev(++(y''', nil))))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Nar

           →DP Problem 4

             ↳Nar

...

               →DP Problem 9

                 ↳Narrowing Transformation

Dependency Pairs:

REV2(x, ++(y0, ++(y''', ++(y'', nil)))) -> REV(rev(++(y0, rev(rev(++(y''', nil))))))
REV2(x, ++(y0, ++(y'''', ++(y'', nil)))) -> REV(rev(++(y0, rev(++(rev1(y'''', rev(nil)), rev2(y'''', rev(nil)))))))
REV2(x, ++(y0', ++(y'''', ++(y'', nil)))) -> REV(++(rev1(y0', rev(rev(++(y'''', rev(nil))))), rev2(y0', rev(rev(++(y'''', rev(nil)))))))
REV2(x, ++(y0, ++(y''', ++(y'', ++(y1, z''))))) -> REV(rev(++(y0, rev(rev(++(y''', rev(rev(++(y'', rev(rev2(y1, z'')))))))))))
REV2(x, ++(y0, ++(y'''', ++(y'', z'')))) -> REV(rev(++(y0, rev(++(rev1(y'''', rev(rev2(y'', z''))), rev2(y'''', rev(rev2(y'', z''))))))))
REV2(x, ++(y0', ++(y'''', ++(y'', z'')))) -> REV(++(rev1(y0', rev(rev(++(y'''', rev(rev2(y'', z'')))))), rev2(y0', rev(rev(++(y'''', rev(rev2(y'', z''))))))))
REV2(x, ++(y0', ++(y''', nil))) -> REV(++(rev1(y0', rev(nil)), rev2(y0', rev(nil))))
REV2(x, ++(y0', ++(y''', z'''))) -> REV(++(rev1(y0', rev(rev2(y''', z'''))), rev2(y0', rev(rev2(y''', z''')))))
REV2(x, ++(y, z)) -> REV2(y, z)
REV2(x, ++(y, z)) -> REV(++(x, rev(rev2(y, z))))
REV(++(x, y)) -> REV2(x, y)
REV2(x, ++(y0', ++(y''', nil))) -> REV(++(rev1(y0', nil), rev2(y0', nil)))

Rules:

rev(nil) -> nil
rev(++(x, y)) -> ++(rev1(x, y), rev2(x, y))
rev1(x, nil) -> x
rev1(x, ++(y, z)) -> rev1(y, z)
rev2(x, nil) -> nil
rev2(x, ++(y, z)) -> rev(++(x, rev(rev2(y, z))))

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

REV2(x, ++(y0, ++(y''', ++(y'', ++(y1, z''))))) -> REV(rev(++(y0, rev(rev(++(y''', rev(rev(++(y'', rev(rev2(y1, z'')))))))))))

five new Dependency Pairs are created:

REV2(x, ++(y0', ++(y'''', ++(y''0, ++(y1', z'''))))) -> REV(++(rev1(y0', rev(rev(++(y'''', rev(rev(++(y''0, rev(rev2(y1', z'''))))))))), rev2(y0', rev(rev(++(y'''', rev(rev(++(y''0, rev(rev2(y1', z''')))))))))))
REV2(x, ++(y0, ++(y'''', ++(y''0, ++(y1', z'''))))) -> REV(rev(++(y0, rev(++(rev1(y'''', rev(rev(++(y''0, rev(rev2(y1', z''')))))), rev2(y'''', rev(rev(++(y''0, rev(rev2(y1', z''')))))))))))
REV2(x, ++(y0, ++(y''', ++(y'''', ++(y1', z'''))))) -> REV(rev(++(y0, rev(rev(++(y''', rev(++(rev1(y'''', rev(rev2(y1', z'''))), rev2(y'''', rev(rev2(y1', z''')))))))))))
REV2(x, ++(y0, ++(y''', ++(y'', ++(y1', nil))))) -> REV(rev(++(y0, rev(rev(++(y''', rev(rev(++(y'', rev(nil))))))))))
REV2(x, ++(y0, ++(y''', ++(y'', ++(y1', ++(y', z')))))) -> REV(rev(++(y0, rev(rev(++(y''', rev(rev(++(y'', rev(rev(++(y1', rev(rev2(y', z'))))))))))))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Nar

           →DP Problem 4

             ↳Nar

...

               →DP Problem 10

                 ↳Narrowing Transformation

Dependency Pairs:

REV2(x, ++(y0, ++(y''', ++(y'', ++(y1', ++(y', z')))))) -> REV(rev(++(y0, rev(rev(++(y''', rev(rev(++(y'', rev(rev(++(y1', rev(rev2(y', z'))))))))))))))
REV2(x, ++(y0, ++(y''', ++(y'', ++(y1', nil))))) -> REV(rev(++(y0, rev(rev(++(y''', rev(rev(++(y'', rev(nil))))))))))
REV2(x, ++(y0, ++(y''', ++(y'''', ++(y1', z'''))))) -> REV(rev(++(y0, rev(rev(++(y''', rev(++(rev1(y'''', rev(rev2(y1', z'''))), rev2(y'''', rev(rev2(y1', z''')))))))))))
REV2(x, ++(y0, ++(y'''', ++(y''0, ++(y1', z'''))))) -> REV(rev(++(y0, rev(++(rev1(y'''', rev(rev(++(y''0, rev(rev2(y1', z''')))))), rev2(y'''', rev(rev(++(y''0, rev(rev2(y1', z''')))))))))))
REV2(x, ++(y0', ++(y'''', ++(y''0, ++(y1', z'''))))) -> REV(++(rev1(y0', rev(rev(++(y'''', rev(rev(++(y''0, rev(rev2(y1', z'''))))))))), rev2(y0', rev(rev(++(y'''', rev(rev(++(y''0, rev(rev2(y1', z''')))))))))))
REV2(x, ++(y0, ++(y'''', ++(y'', nil)))) -> REV(rev(++(y0, rev(++(rev1(y'''', rev(nil)), rev2(y'''', rev(nil)))))))
REV2(x, ++(y0', ++(y'''', ++(y'', nil)))) -> REV(++(rev1(y0', rev(rev(++(y'''', rev(nil))))), rev2(y0', rev(rev(++(y'''', rev(nil)))))))
REV2(x, ++(y0', ++(y''', nil))) -> REV(++(rev1(y0', nil), rev2(y0', nil)))
REV2(x, ++(y0, ++(y'''', ++(y'', z'')))) -> REV(rev(++(y0, rev(++(rev1(y'''', rev(rev2(y'', z''))), rev2(y'''', rev(rev2(y'', z''))))))))
REV2(x, ++(y0', ++(y'''', ++(y'', z'')))) -> REV(++(rev1(y0', rev(rev(++(y'''', rev(rev2(y'', z'')))))), rev2(y0', rev(rev(++(y'''', rev(rev2(y'', z''))))))))
REV2(x, ++(y0', ++(y''', nil))) -> REV(++(rev1(y0', rev(nil)), rev2(y0', rev(nil))))
REV2(x, ++(y0', ++(y''', z'''))) -> REV(++(rev1(y0', rev(rev2(y''', z'''))), rev2(y0', rev(rev2(y''', z''')))))
REV2(x, ++(y, z)) -> REV2(y, z)
REV2(x, ++(y, z)) -> REV(++(x, rev(rev2(y, z))))
REV(++(x, y)) -> REV2(x, y)
REV2(x, ++(y0, ++(y''', ++(y'', nil)))) -> REV(rev(++(y0, rev(rev(++(y''', nil))))))

Rules:

rev(nil) -> nil
rev(++(x, y)) -> ++(rev1(x, y), rev2(x, y))
rev1(x, nil) -> x
rev1(x, ++(y, z)) -> rev1(y, z)
rev2(x, nil) -> nil
rev2(x, ++(y, z)) -> rev(++(x, rev(rev2(y, z))))

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

REV2(x, ++(y0, ++(y''', ++(y'', nil)))) -> REV(rev(++(y0, rev(rev(++(y''', nil))))))

two new Dependency Pairs are created:

REV2(x, ++(y0', ++(y'''', ++(y'', nil)))) -> REV(++(rev1(y0', rev(rev(++(y'''', nil)))), rev2(y0', rev(rev(++(y'''', nil))))))
REV2(x, ++(y0, ++(y'''', ++(y'', nil)))) -> REV(rev(++(y0, rev(++(rev1(y'''', nil), rev2(y'''', nil))))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Nar

           →DP Problem 4

             ↳Nar

...

               →DP Problem 11

                 ↳Narrowing Transformation

Dependency Pairs:

REV2(x, ++(y0, ++(y'''', ++(y'', nil)))) -> REV(rev(++(y0, rev(++(rev1(y'''', nil), rev2(y'''', nil))))))
REV2(x, ++(y0', ++(y'''', ++(y'', nil)))) -> REV(++(rev1(y0', rev(rev(++(y'''', nil)))), rev2(y0', rev(rev(++(y'''', nil))))))
REV2(x, ++(y0, ++(y''', ++(y'', ++(y1', nil))))) -> REV(rev(++(y0, rev(rev(++(y''', rev(rev(++(y'', rev(nil))))))))))
REV2(x, ++(y0, ++(y''', ++(y'''', ++(y1', z'''))))) -> REV(rev(++(y0, rev(rev(++(y''', rev(++(rev1(y'''', rev(rev2(y1', z'''))), rev2(y'''', rev(rev2(y1', z''')))))))))))
REV2(x, ++(y0, ++(y'''', ++(y''0, ++(y1', z'''))))) -> REV(rev(++(y0, rev(++(rev1(y'''', rev(rev(++(y''0, rev(rev2(y1', z''')))))), rev2(y'''', rev(rev(++(y''0, rev(rev2(y1', z''')))))))))))
REV2(x, ++(y0', ++(y'''', ++(y''0, ++(y1', z'''))))) -> REV(++(rev1(y0', rev(rev(++(y'''', rev(rev(++(y''0, rev(rev2(y1', z'''))))))))), rev2(y0', rev(rev(++(y'''', rev(rev(++(y''0, rev(rev2(y1', z''')))))))))))
REV2(x, ++(y0, ++(y'''', ++(y'', nil)))) -> REV(rev(++(y0, rev(++(rev1(y'''', rev(nil)), rev2(y'''', rev(nil)))))))
REV2(x, ++(y0', ++(y'''', ++(y'', nil)))) -> REV(++(rev1(y0', rev(rev(++(y'''', rev(nil))))), rev2(y0', rev(rev(++(y'''', rev(nil)))))))
REV2(x, ++(y0', ++(y''', nil))) -> REV(++(rev1(y0', nil), rev2(y0', nil)))
REV2(x, ++(y0, ++(y'''', ++(y'', z'')))) -> REV(rev(++(y0, rev(++(rev1(y'''', rev(rev2(y'', z''))), rev2(y'''', rev(rev2(y'', z''))))))))
REV2(x, ++(y0', ++(y'''', ++(y'', z'')))) -> REV(++(rev1(y0', rev(rev(++(y'''', rev(rev2(y'', z'')))))), rev2(y0', rev(rev(++(y'''', rev(rev2(y'', z''))))))))
REV2(x, ++(y0', ++(y''', nil))) -> REV(++(rev1(y0', rev(nil)), rev2(y0', rev(nil))))
REV2(x, ++(y0', ++(y''', z'''))) -> REV(++(rev1(y0', rev(rev2(y''', z'''))), rev2(y0', rev(rev2(y''', z''')))))
REV2(x, ++(y, z)) -> REV2(y, z)
REV2(x, ++(y, z)) -> REV(++(x, rev(rev2(y, z))))
REV(++(x, y)) -> REV2(x, y)
REV2(x, ++(y0, ++(y''', ++(y'', ++(y1', ++(y', z')))))) -> REV(rev(++(y0, rev(rev(++(y''', rev(rev(++(y'', rev(rev(++(y1', rev(rev2(y', z'))))))))))))))

Rules:

rev(nil) -> nil
rev(++(x, y)) -> ++(rev1(x, y), rev2(x, y))
rev1(x, nil) -> x
rev1(x, ++(y, z)) -> rev1(y, z)
rev2(x, nil) -> nil
rev2(x, ++(y, z)) -> rev(++(x, rev(rev2(y, z))))

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

REV2(x, ++(y0, ++(y''', ++(y'', ++(y1', nil))))) -> REV(rev(++(y0, rev(rev(++(y''', rev(rev(++(y'', rev(nil))))))))))

four new Dependency Pairs are created:

REV2(x, ++(y0', ++(y'''', ++(y''0, ++(y1', nil))))) -> REV(++(rev1(y0', rev(rev(++(y'''', rev(rev(++(y''0, rev(nil)))))))), rev2(y0', rev(rev(++(y'''', rev(rev(++(y''0, rev(nil))))))))))
REV2(x, ++(y0, ++(y'''', ++(y''0, ++(y1', nil))))) -> REV(rev(++(y0, rev(++(rev1(y'''', rev(rev(++(y''0, rev(nil))))), rev2(y'''', rev(rev(++(y''0, rev(nil))))))))))
REV2(x, ++(y0, ++(y''', ++(y'''', ++(y1', nil))))) -> REV(rev(++(y0, rev(rev(++(y''', rev(++(rev1(y'''', rev(nil)), rev2(y'''', rev(nil))))))))))
REV2(x, ++(y0, ++(y''', ++(y'', ++(y1', nil))))) -> REV(rev(++(y0, rev(rev(++(y''', rev(rev(++(y'', nil)))))))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Nar

           →DP Problem 4

             ↳Nar

...

               →DP Problem 12

                 ↳Narrowing Transformation

Dependency Pairs:

REV2(x, ++(y0, ++(y''', ++(y'', ++(y1', nil))))) -> REV(rev(++(y0, rev(rev(++(y''', rev(rev(++(y'', nil)))))))))
REV2(x, ++(y0, ++(y''', ++(y'''', ++(y1', nil))))) -> REV(rev(++(y0, rev(rev(++(y''', rev(++(rev1(y'''', rev(nil)), rev2(y'''', rev(nil))))))))))
REV2(x, ++(y0, ++(y'''', ++(y''0, ++(y1', nil))))) -> REV(rev(++(y0, rev(++(rev1(y'''', rev(rev(++(y''0, rev(nil))))), rev2(y'''', rev(rev(++(y''0, rev(nil))))))))))
REV2(x, ++(y0', ++(y'''', ++(y''0, ++(y1', nil))))) -> REV(++(rev1(y0', rev(rev(++(y'''', rev(rev(++(y''0, rev(nil)))))))), rev2(y0', rev(rev(++(y'''', rev(rev(++(y''0, rev(nil))))))))))
REV2(x, ++(y0', ++(y'''', ++(y'', nil)))) -> REV(++(rev1(y0', rev(rev(++(y'''', nil)))), rev2(y0', rev(rev(++(y'''', nil))))))
REV2(x, ++(y0, ++(y''', ++(y'', ++(y1', ++(y', z')))))) -> REV(rev(++(y0, rev(rev(++(y''', rev(rev(++(y'', rev(rev(++(y1', rev(rev2(y', z'))))))))))))))
REV2(x, ++(y0, ++(y''', ++(y'''', ++(y1', z'''))))) -> REV(rev(++(y0, rev(rev(++(y''', rev(++(rev1(y'''', rev(rev2(y1', z'''))), rev2(y'''', rev(rev2(y1', z''')))))))))))
REV2(x, ++(y0, ++(y'''', ++(y''0, ++(y1', z'''))))) -> REV(rev(++(y0, rev(++(rev1(y'''', rev(rev(++(y''0, rev(rev2(y1', z''')))))), rev2(y'''', rev(rev(++(y''0, rev(rev2(y1', z''')))))))))))
REV2(x, ++(y0', ++(y'''', ++(y''0, ++(y1', z'''))))) -> REV(++(rev1(y0', rev(rev(++(y'''', rev(rev(++(y''0, rev(rev2(y1', z'''))))))))), rev2(y0', rev(rev(++(y'''', rev(rev(++(y''0, rev(rev2(y1', z''')))))))))))
REV2(x, ++(y0, ++(y'''', ++(y'', nil)))) -> REV(rev(++(y0, rev(++(rev1(y'''', rev(nil)), rev2(y'''', rev(nil)))))))
REV2(x, ++(y0', ++(y'''', ++(y'', nil)))) -> REV(++(rev1(y0', rev(rev(++(y'''', rev(nil))))), rev2(y0', rev(rev(++(y'''', rev(nil)))))))
REV2(x, ++(y0', ++(y''', nil))) -> REV(++(rev1(y0', nil), rev2(y0', nil)))
REV2(x, ++(y0, ++(y'''', ++(y'', z'')))) -> REV(rev(++(y0, rev(++(rev1(y'''', rev(rev2(y'', z''))), rev2(y'''', rev(rev2(y'', z''))))))))
REV2(x, ++(y0', ++(y'''', ++(y'', z'')))) -> REV(++(rev1(y0', rev(rev(++(y'''', rev(rev2(y'', z'')))))), rev2(y0', rev(rev(++(y'''', rev(rev2(y'', z''))))))))
REV2(x, ++(y0', ++(y''', nil))) -> REV(++(rev1(y0', rev(nil)), rev2(y0', rev(nil))))
REV2(x, ++(y0', ++(y''', z'''))) -> REV(++(rev1(y0', rev(rev2(y''', z'''))), rev2(y0', rev(rev2(y''', z''')))))
REV2(x, ++(y, z)) -> REV2(y, z)
REV2(x, ++(y, z)) -> REV(++(x, rev(rev2(y, z))))
REV(++(x, y)) -> REV2(x, y)
REV2(x, ++(y0, ++(y'''', ++(y'', nil)))) -> REV(rev(++(y0, rev(++(rev1(y'''', nil), rev2(y'''', nil))))))

Rules:

rev(nil) -> nil
rev(++(x, y)) -> ++(rev1(x, y), rev2(x, y))
rev1(x, nil) -> x
rev1(x, ++(y, z)) -> rev1(y, z)
rev2(x, nil) -> nil
rev2(x, ++(y, z)) -> rev(++(x, rev(rev2(y, z))))

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

REV2(x, ++(y0, ++(y''', ++(y'', ++(y1', ++(y', z')))))) -> REV(rev(++(y0, rev(rev(++(y''', rev(rev(++(y'', rev(rev(++(y1', rev(rev2(y', z'))))))))))))))

six new Dependency Pairs are created:

REV2(x, ++(y0', ++(y'''', ++(y''0, ++(y1'', ++(y'0, z'')))))) -> REV(++(rev1(y0', rev(rev(++(y'''', rev(rev(++(y''0, rev(rev(++(y1'', rev(rev2(y'0, z'')))))))))))), rev2(y0', rev(rev(++(y'''', rev(rev(++(y''0, rev(rev(++(y1'', rev(rev2(y'0, z''))))))))))))))
REV2(x, ++(y0, ++(y'''', ++(y''0, ++(y1'', ++(y'0, z'')))))) -> REV(rev(++(y0, rev(++(rev1(y'''', rev(rev(++(y''0, rev(rev(++(y1'', rev(rev2(y'0, z''))))))))), rev2(y'''', rev(rev(++(y''0, rev(rev(++(y1'', rev(rev2(y'0, z''))))))))))))))
REV2(x, ++(y0, ++(y''', ++(y'''', ++(y1'', ++(y'0, z'')))))) -> REV(rev(++(y0, rev(rev(++(y''', rev(++(rev1(y'''', rev(rev(++(y1'', rev(rev2(y'0, z'')))))), rev2(y'''', rev(rev(++(y1'', rev(rev2(y'0, z''))))))))))))))
REV2(x, ++(y0, ++(y''', ++(y'', ++(y1'', ++(y'0, z'')))))) -> REV(rev(++(y0, rev(rev(++(y''', rev(rev(++(y'', rev(++(rev1(y1'', rev(rev2(y'0, z''))), rev2(y1'', rev(rev2(y'0, z''))))))))))))))
REV2(x, ++(y0, ++(y''', ++(y'', ++(y1', ++(y'0, nil)))))) -> REV(rev(++(y0, rev(rev(++(y''', rev(rev(++(y'', rev(rev(++(y1', rev(nil)))))))))))))
REV2(x, ++(y0, ++(y''', ++(y'', ++(y1', ++(y'0, ++(y1, z''))))))) -> REV(rev(++(y0, rev(rev(++(y''', rev(rev(++(y'', rev(rev(++(y1', rev(rev(++(y'0, rev(rev2(y1, z'')))))))))))))))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Nar

           →DP Problem 4

             ↳Nar

...

               →DP Problem 13

                 ↳Remaining Obligation(s)

The following remains to be proven:
Dependency Pairs:

REV2(x, ++(y0, ++(y''', ++(y'', ++(y1', ++(y'0, ++(y1, z''))))))) -> REV(rev(++(y0, rev(rev(++(y''', rev(rev(++(y'', rev(rev(++(y1', rev(rev(++(y'0, rev(rev2(y1, z'')))))))))))))))))
REV2(x, ++(y0, ++(y''', ++(y'', ++(y1', ++(y'0, nil)))))) -> REV(rev(++(y0, rev(rev(++(y''', rev(rev(++(y'', rev(rev(++(y1', rev(nil)))))))))))))
REV2(x, ++(y0, ++(y''', ++(y'', ++(y1'', ++(y'0, z'')))))) -> REV(rev(++(y0, rev(rev(++(y''', rev(rev(++(y'', rev(++(rev1(y1'', rev(rev2(y'0, z''))), rev2(y1'', rev(rev2(y'0, z''))))))))))))))
REV2(x, ++(y0, ++(y''', ++(y'''', ++(y1'', ++(y'0, z'')))))) -> REV(rev(++(y0, rev(rev(++(y''', rev(++(rev1(y'''', rev(rev(++(y1'', rev(rev2(y'0, z'')))))), rev2(y'''', rev(rev(++(y1'', rev(rev2(y'0, z''))))))))))))))
REV2(x, ++(y0, ++(y'''', ++(y''0, ++(y1'', ++(y'0, z'')))))) -> REV(rev(++(y0, rev(++(rev1(y'''', rev(rev(++(y''0, rev(rev(++(y1'', rev(rev2(y'0, z''))))))))), rev2(y'''', rev(rev(++(y''0, rev(rev(++(y1'', rev(rev2(y'0, z''))))))))))))))
REV2(x, ++(y0', ++(y'''', ++(y''0, ++(y1'', ++(y'0, z'')))))) -> REV(++(rev1(y0', rev(rev(++(y'''', rev(rev(++(y''0, rev(rev(++(y1'', rev(rev2(y'0, z'')))))))))))), rev2(y0', rev(rev(++(y'''', rev(rev(++(y''0, rev(rev(++(y1'', rev(rev2(y'0, z''))))))))))))))
REV2(x, ++(y0, ++(y''', ++(y'''', ++(y1', nil))))) -> REV(rev(++(y0, rev(rev(++(y''', rev(++(rev1(y'''', rev(nil)), rev2(y'''', rev(nil))))))))))
REV2(x, ++(y0, ++(y'''', ++(y''0, ++(y1', nil))))) -> REV(rev(++(y0, rev(++(rev1(y'''', rev(rev(++(y''0, rev(nil))))), rev2(y'''', rev(rev(++(y''0, rev(nil))))))))))
REV2(x, ++(y0', ++(y'''', ++(y''0, ++(y1', nil))))) -> REV(++(rev1(y0', rev(rev(++(y'''', rev(rev(++(y''0, rev(nil)))))))), rev2(y0', rev(rev(++(y'''', rev(rev(++(y''0, rev(nil))))))))))
REV2(x, ++(y0, ++(y'''', ++(y'', nil)))) -> REV(rev(++(y0, rev(++(rev1(y'''', nil), rev2(y'''', nil))))))
REV2(x, ++(y0', ++(y'''', ++(y'', nil)))) -> REV(++(rev1(y0', rev(rev(++(y'''', nil)))), rev2(y0', rev(rev(++(y'''', nil))))))
REV2(x, ++(y0, ++(y''', ++(y'''', ++(y1', z'''))))) -> REV(rev(++(y0, rev(rev(++(y''', rev(++(rev1(y'''', rev(rev2(y1', z'''))), rev2(y'''', rev(rev2(y1', z''')))))))))))
REV2(x, ++(y0, ++(y'''', ++(y''0, ++(y1', z'''))))) -> REV(rev(++(y0, rev(++(rev1(y'''', rev(rev(++(y''0, rev(rev2(y1', z''')))))), rev2(y'''', rev(rev(++(y''0, rev(rev2(y1', z''')))))))))))
REV2(x, ++(y0', ++(y'''', ++(y''0, ++(y1', z'''))))) -> REV(++(rev1(y0', rev(rev(++(y'''', rev(rev(++(y''0, rev(rev2(y1', z'''))))))))), rev2(y0', rev(rev(++(y'''', rev(rev(++(y''0, rev(rev2(y1', z''')))))))))))
REV2(x, ++(y0, ++(y'''', ++(y'', nil)))) -> REV(rev(++(y0, rev(++(rev1(y'''', rev(nil)), rev2(y'''', rev(nil)))))))
REV2(x, ++(y0', ++(y'''', ++(y'', nil)))) -> REV(++(rev1(y0', rev(rev(++(y'''', rev(nil))))), rev2(y0', rev(rev(++(y'''', rev(nil)))))))
REV2(x, ++(y0', ++(y''', nil))) -> REV(++(rev1(y0', nil), rev2(y0', nil)))
REV2(x, ++(y0, ++(y'''', ++(y'', z'')))) -> REV(rev(++(y0, rev(++(rev1(y'''', rev(rev2(y'', z''))), rev2(y'''', rev(rev2(y'', z''))))))))
REV2(x, ++(y0', ++(y'''', ++(y'', z'')))) -> REV(++(rev1(y0', rev(rev(++(y'''', rev(rev2(y'', z'')))))), rev2(y0', rev(rev(++(y'''', rev(rev2(y'', z''))))))))
REV2(x, ++(y0', ++(y''', nil))) -> REV(++(rev1(y0', rev(nil)), rev2(y0', rev(nil))))
REV2(x, ++(y0', ++(y''', z'''))) -> REV(++(rev1(y0', rev(rev2(y''', z'''))), rev2(y0', rev(rev2(y''', z''')))))
REV2(x, ++(y, z)) -> REV2(y, z)
REV2(x, ++(y, z)) -> REV(++(x, rev(rev2(y, z))))
REV(++(x, y)) -> REV2(x, y)
REV2(x, ++(y0, ++(y''', ++(y'', ++(y1', nil))))) -> REV(rev(++(y0, rev(rev(++(y''', rev(rev(++(y'', nil)))))))))

Rules:

rev(nil) -> nil
rev(++(x, y)) -> ++(rev1(x, y), rev2(x, y))
rev1(x, nil) -> x
rev1(x, ++(y, z)) -> rev1(y, z)
rev2(x, nil) -> nil
rev2(x, ++(y, z)) -> rev(++(x, rev(rev2(y, z))))

Termination of R could not be shown.
Duration:
0:06 minutes

POL(REV1(x₁, x₂))	= x₂
POL(++(x₁, x₂))	= 1 + x₂