Termination Proof using AProVE

Term Rewriting System R:
[x, l, y]
app(rev, nil) -> nil
app(rev, app(app(cons, x), l)) -> app(app(cons, app(app(rev1, x), l)), app(app(rev2, x), l))
app(app(rev1, 0), nil) -> 0
app(app(rev1, app(s, x)), nil) -> app(s, x)
app(app(rev1, x), app(app(cons, y), l)) -> app(app(rev1, y), l)
app(app(rev2, x), nil) -> nil
app(app(rev2, x), app(app(cons, y), l)) -> app(rev, app(app(cons, x), app(app(rev2, y), l)))

Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

APP(rev, app(app(cons, x), l)) -> APP(app(cons, app(app(rev1, x), l)), app(app(rev2, x), l))
APP(rev, app(app(cons, x), l)) -> APP(cons, app(app(rev1, x), l))
APP(rev, app(app(cons, x), l)) -> APP(app(rev1, x), l)
APP(rev, app(app(cons, x), l)) -> APP(rev1, x)
APP(rev, app(app(cons, x), l)) -> APP(app(rev2, x), l)
APP(rev, app(app(cons, x), l)) -> APP(rev2, x)
APP(app(rev1, x), app(app(cons, y), l)) -> APP(app(rev1, y), l)
APP(app(rev1, x), app(app(cons, y), l)) -> APP(rev1, y)
APP(app(rev2, x), app(app(cons, y), l)) -> APP(rev, app(app(cons, x), app(app(rev2, y), l)))
APP(app(rev2, x), app(app(cons, y), l)) -> APP(app(cons, x), app(app(rev2, y), l))
APP(app(rev2, x), app(app(cons, y), l)) -> APP(cons, x)
APP(app(rev2, x), app(app(cons, y), l)) -> APP(app(rev2, y), l)
APP(app(rev2, x), app(app(cons, y), l)) -> APP(rev2, y)

Furthermore, R contains one SCC.

     ↳DPs

       →DP Problem 1

         ↳Narrowing Transformation

Dependency Pairs:

APP(app(rev2, x), app(app(cons, y), l)) -> APP(app(rev2, y), l)
APP(app(rev2, x), app(app(cons, y), l)) -> APP(app(cons, x), app(app(rev2, y), l))
APP(app(rev2, x), app(app(cons, y), l)) -> APP(rev, app(app(cons, x), app(app(rev2, y), l)))
APP(app(rev1, x), app(app(cons, y), l)) -> APP(app(rev1, y), l)
APP(rev, app(app(cons, x), l)) -> APP(app(rev2, x), l)
APP(rev, app(app(cons, x), l)) -> APP(app(rev1, x), l)
APP(rev, app(app(cons, x), l)) -> APP(app(cons, app(app(rev1, x), l)), app(app(rev2, x), l))

Rules:

app(rev, nil) -> nil
app(rev, app(app(cons, x), l)) -> app(app(cons, app(app(rev1, x), l)), app(app(rev2, x), l))
app(app(rev1, 0), nil) -> 0
app(app(rev1, app(s, x)), nil) -> app(s, x)
app(app(rev1, x), app(app(cons, y), l)) -> app(app(rev1, y), l)
app(app(rev2, x), nil) -> nil
app(app(rev2, x), app(app(cons, y), l)) -> app(rev, app(app(cons, x), app(app(rev2, y), l)))

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

APP(app(rev2, x), app(app(cons, y), l)) -> APP(app(cons, x), app(app(rev2, y), l))

two new Dependency Pairs are created:

APP(app(rev2, x), app(app(cons, y'), nil)) -> APP(app(cons, x), nil)
APP(app(rev2, x), app(app(cons, y0), app(app(cons, y''), l''))) -> APP(app(cons, x), app(rev, app(app(cons, y0), app(app(rev2, y''), l''))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Narrowing Transformation

Dependency Pairs:

APP(app(rev2, x), app(app(cons, y0), app(app(cons, y''), l''))) -> APP(app(cons, x), app(rev, app(app(cons, y0), app(app(rev2, y''), l''))))
APP(app(rev2, x), app(app(cons, y), l)) -> APP(rev, app(app(cons, x), app(app(rev2, y), l)))
APP(app(rev1, x), app(app(cons, y), l)) -> APP(app(rev1, y), l)
APP(rev, app(app(cons, x), l)) -> APP(app(rev2, x), l)
APP(rev, app(app(cons, x), l)) -> APP(app(rev1, x), l)
APP(rev, app(app(cons, x), l)) -> APP(app(cons, app(app(rev1, x), l)), app(app(rev2, x), l))
APP(app(rev2, x), app(app(cons, y), l)) -> APP(app(rev2, y), l)

Rules:

app(rev, nil) -> nil
app(rev, app(app(cons, x), l)) -> app(app(cons, app(app(rev1, x), l)), app(app(rev2, x), l))
app(app(rev1, 0), nil) -> 0
app(app(rev1, app(s, x)), nil) -> app(s, x)
app(app(rev1, x), app(app(cons, y), l)) -> app(app(rev1, y), l)
app(app(rev2, x), nil) -> nil
app(app(rev2, x), app(app(cons, y), l)) -> app(rev, app(app(cons, x), app(app(rev2, y), l)))

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

APP(app(rev2, x), app(app(cons, y0), app(app(cons, y''), l''))) -> APP(app(cons, x), app(rev, app(app(cons, y0), app(app(rev2, y''), l''))))

three new Dependency Pairs are created:

APP(app(rev2, x), app(app(cons, y0'), app(app(cons, y'''), l'''))) -> APP(app(cons, x), app(app(cons, app(app(rev1, y0'), app(app(rev2, y'''), l'''))), app(app(rev2, y0'), app(app(rev2, y'''), l'''))))
APP(app(rev2, x), app(app(cons, y0), app(app(cons, y'''), nil))) -> APP(app(cons, x), app(rev, app(app(cons, y0), nil)))
APP(app(rev2, x), app(app(cons, y0), app(app(cons, y'''), app(app(cons, y'), l')))) -> APP(app(cons, x), app(rev, app(app(cons, y0), app(rev, app(app(cons, y'''), app(app(rev2, y'), l'))))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 3

                 ↳Polynomial Ordering

Dependency Pairs:

APP(app(rev2, x), app(app(cons, y0), app(app(cons, y'''), app(app(cons, y'), l')))) -> APP(app(cons, x), app(rev, app(app(cons, y0), app(rev, app(app(cons, y'''), app(app(rev2, y'), l'))))))
APP(app(rev2, x), app(app(cons, y0), app(app(cons, y'''), nil))) -> APP(app(cons, x), app(rev, app(app(cons, y0), nil)))
APP(app(rev2, x), app(app(cons, y0'), app(app(cons, y'''), l'''))) -> APP(app(cons, x), app(app(cons, app(app(rev1, y0'), app(app(rev2, y'''), l'''))), app(app(rev2, y0'), app(app(rev2, y'''), l'''))))
APP(app(rev2, x), app(app(cons, y), l)) -> APP(app(rev2, y), l)
APP(app(rev1, x), app(app(cons, y), l)) -> APP(app(rev1, y), l)
APP(rev, app(app(cons, x), l)) -> APP(app(rev2, x), l)
APP(rev, app(app(cons, x), l)) -> APP(app(rev1, x), l)
APP(rev, app(app(cons, x), l)) -> APP(app(cons, app(app(rev1, x), l)), app(app(rev2, x), l))
APP(app(rev2, x), app(app(cons, y), l)) -> APP(rev, app(app(cons, x), app(app(rev2, y), l)))

Rules:

app(rev, nil) -> nil
app(rev, app(app(cons, x), l)) -> app(app(cons, app(app(rev1, x), l)), app(app(rev2, x), l))
app(app(rev1, 0), nil) -> 0
app(app(rev1, app(s, x)), nil) -> app(s, x)
app(app(rev1, x), app(app(cons, y), l)) -> app(app(rev1, y), l)
app(app(rev2, x), nil) -> nil
app(app(rev2, x), app(app(cons, y), l)) -> app(rev, app(app(cons, x), app(app(rev2, y), l)))

The following dependency pairs can be strictly oriented:

APP(app(rev2, x), app(app(cons, y0), app(app(cons, y'''), app(app(cons, y'), l')))) -> APP(app(cons, x), app(rev, app(app(cons, y0), app(rev, app(app(cons, y'''), app(app(rev2, y'), l'))))))
APP(app(rev2, x), app(app(cons, y0), app(app(cons, y'''), nil))) -> APP(app(cons, x), app(rev, app(app(cons, y0), nil)))
APP(app(rev2, x), app(app(cons, y0'), app(app(cons, y'''), l'''))) -> APP(app(cons, x), app(app(cons, app(app(rev1, y0'), app(app(rev2, y'''), l'''))), app(app(rev2, y0'), app(app(rev2, y'''), l'''))))
APP(rev, app(app(cons, x), l)) -> APP(app(rev1, x), l)
APP(rev, app(app(cons, x), l)) -> APP(app(cons, app(app(rev1, x), l)), app(app(rev2, x), l))

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

app(rev, nil) -> nil
app(rev, app(app(cons, x), l)) -> app(app(cons, app(app(rev1, x), l)), app(app(rev2, x), l))
app(app(rev1, 0), nil) -> 0
app(app(rev1, app(s, x)), nil) -> app(s, x)
app(app(rev1, x), app(app(cons, y), l)) -> app(app(rev1, y), l)
app(app(rev2, x), nil) -> nil
app(app(rev2, x), app(app(cons, y), l)) -> app(rev, app(app(cons, x), app(app(rev2, y), l)))

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(rev2) = 1

POL(rev) = 1

POL(0) = 0

POL(cons) = 0

POL(rev1) = 0

POL(nil) = 0

POL(s) = 0

POL(app(x₁, x₂)) = x₁

POL(APP(x₁, x₂)) = 1 + x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 4

                 ↳Remaining Obligation(s)

The following remains to be proven:
Dependency Pairs:

APP(app(rev2, x), app(app(cons, y), l)) -> APP(app(rev2, y), l)
APP(app(rev1, x), app(app(cons, y), l)) -> APP(app(rev1, y), l)
APP(rev, app(app(cons, x), l)) -> APP(app(rev2, x), l)
APP(app(rev2, x), app(app(cons, y), l)) -> APP(rev, app(app(cons, x), app(app(rev2, y), l)))

Rules:

app(rev, nil) -> nil
app(rev, app(app(cons, x), l)) -> app(app(cons, app(app(rev1, x), l)), app(app(rev2, x), l))
app(app(rev1, 0), nil) -> 0
app(app(rev1, app(s, x)), nil) -> app(s, x)
app(app(rev1, x), app(app(cons, y), l)) -> app(app(rev1, y), l)
app(app(rev2, x), nil) -> nil
app(app(rev2, x), app(app(cons, y), l)) -> app(rev, app(app(cons, x), app(app(rev2, y), l)))

Termination of R could not be shown.
Duration:
0:00 minutes

POL(rev2)	= 1
POL(rev)	= 1
POL(0)	= 0
POL(cons)	= 0
POL(rev1)	= 0
POL(nil)	= 0
POL(s)	= 0
POL(app(x₁, x₂))	= x₁
POL(APP(x₁, x₂))	= 1 + x₁