Termination Proof using AProVE

Term Rewriting System R:
[x, ys, f, xs]
app(app(app(consif, true), x), ys) -> app(app(cons, x), ys)
app(app(app(consif, false), x), ys) -> ys
app(app(filter, f), nil) -> nil
app(app(filter, f), app(app(cons, x), xs)) -> app(app(app(consif, app(f, x)), x), app(app(filter, f), xs))

Innermost Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

APP(app(app(consif, true), x), ys) -> APP(app(cons, x), ys)
APP(app(app(consif, true), x), ys) -> APP(cons, x)
APP(app(filter, f), app(app(cons, x), xs)) -> APP(app(app(consif, app(f, x)), x), app(app(filter, f), xs))
APP(app(filter, f), app(app(cons, x), xs)) -> APP(app(consif, app(f, x)), x)
APP(app(filter, f), app(app(cons, x), xs)) -> APP(consif, app(f, x))
APP(app(filter, f), app(app(cons, x), xs)) -> APP(f, x)
APP(app(filter, f), app(app(cons, x), xs)) -> APP(app(filter, f), xs)

Furthermore, R contains one SCC.

     ↳DPs

       →DP Problem 1

         ↳Narrowing Transformation

Dependency Pairs:

APP(app(filter, f), app(app(cons, x), xs)) -> APP(app(filter, f), xs)
APP(app(filter, f), app(app(cons, x), xs)) -> APP(f, x)
APP(app(filter, f), app(app(cons, x), xs)) -> APP(app(consif, app(f, x)), x)
APP(app(filter, f), app(app(cons, x), xs)) -> APP(app(app(consif, app(f, x)), x), app(app(filter, f), xs))
APP(app(app(consif, true), x), ys) -> APP(app(cons, x), ys)

Rules:

app(app(app(consif, true), x), ys) -> app(app(cons, x), ys)
app(app(app(consif, false), x), ys) -> ys
app(app(filter, f), nil) -> nil
app(app(filter, f), app(app(cons, x), xs)) -> app(app(app(consif, app(f, x)), x), app(app(filter, f), xs))

Strategy:

innermost

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

APP(app(app(consif, true), x), ys) -> APP(app(cons, x), ys)

no new Dependency Pairs are created.
The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Narrowing Transformation

Dependency Pairs:

APP(app(filter, f), app(app(cons, x), xs)) -> APP(f, x)
APP(app(filter, f), app(app(cons, x), xs)) -> APP(app(consif, app(f, x)), x)
APP(app(filter, f), app(app(cons, x), xs)) -> APP(app(app(consif, app(f, x)), x), app(app(filter, f), xs))
APP(app(filter, f), app(app(cons, x), xs)) -> APP(app(filter, f), xs)

Rules:

app(app(app(consif, true), x), ys) -> app(app(cons, x), ys)
app(app(app(consif, false), x), ys) -> ys
app(app(filter, f), nil) -> nil
app(app(filter, f), app(app(cons, x), xs)) -> app(app(app(consif, app(f, x)), x), app(app(filter, f), xs))

Strategy:

innermost

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

APP(app(filter, f), app(app(cons, x), xs)) -> APP(app(app(consif, app(f, x)), x), app(app(filter, f), xs))

six new Dependency Pairs are created:

APP(app(filter, app(app(consif, true), x'')), app(app(cons, x0), xs)) -> APP(app(app(consif, app(app(cons, x''), x0)), x0), app(app(filter, app(app(consif, true), x'')), xs))
APP(app(filter, app(app(consif, false), x'')), app(app(cons, x0), xs)) -> APP(app(app(consif, x0), x0), app(app(filter, app(app(consif, false), x'')), xs))
APP(app(filter, app(filter, f'')), app(app(cons, nil), xs)) -> APP(app(app(consif, nil), nil), app(app(filter, app(filter, f'')), xs))
APP(app(filter, app(filter, f'')), app(app(cons, app(app(cons, x''), xs'')), xs)) -> APP(app(app(consif, app(app(app(consif, app(f'', x'')), x''), app(app(filter, f''), xs''))), app(app(cons, x''), xs'')), app(app(filter, app(filter, f'')), xs))
APP(app(filter, f''), app(app(cons, x), nil)) -> APP(app(app(consif, app(f'', x)), x), nil)
APP(app(filter, f''), app(app(cons, x), app(app(cons, x''), xs''))) -> APP(app(app(consif, app(f'', x)), x), app(app(app(consif, app(f'', x'')), x''), app(app(filter, f''), xs'')))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 3

                 ↳Narrowing Transformation

Dependency Pairs:

APP(app(filter, f''), app(app(cons, x), app(app(cons, x''), xs''))) -> APP(app(app(consif, app(f'', x)), x), app(app(app(consif, app(f'', x'')), x''), app(app(filter, f''), xs'')))
APP(app(filter, app(filter, f'')), app(app(cons, app(app(cons, x''), xs'')), xs)) -> APP(app(app(consif, app(app(app(consif, app(f'', x'')), x''), app(app(filter, f''), xs''))), app(app(cons, x''), xs'')), app(app(filter, app(filter, f'')), xs))
APP(app(filter, app(filter, f'')), app(app(cons, nil), xs)) -> APP(app(app(consif, nil), nil), app(app(filter, app(filter, f'')), xs))
APP(app(filter, app(app(consif, false), x'')), app(app(cons, x0), xs)) -> APP(app(app(consif, x0), x0), app(app(filter, app(app(consif, false), x'')), xs))
APP(app(filter, app(app(consif, true), x'')), app(app(cons, x0), xs)) -> APP(app(app(consif, app(app(cons, x''), x0)), x0), app(app(filter, app(app(consif, true), x'')), xs))
APP(app(filter, f), app(app(cons, x), xs)) -> APP(app(filter, f), xs)
APP(app(filter, f), app(app(cons, x), xs)) -> APP(app(consif, app(f, x)), x)
APP(app(filter, f), app(app(cons, x), xs)) -> APP(f, x)

Rules:

app(app(app(consif, true), x), ys) -> app(app(cons, x), ys)
app(app(app(consif, false), x), ys) -> ys
app(app(filter, f), nil) -> nil
app(app(filter, f), app(app(cons, x), xs)) -> app(app(app(consif, app(f, x)), x), app(app(filter, f), xs))

Strategy:

innermost

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

APP(app(filter, f), app(app(cons, x), xs)) -> APP(app(consif, app(f, x)), x)

four new Dependency Pairs are created:

APP(app(filter, app(app(consif, true), x'')), app(app(cons, x0), xs)) -> APP(app(consif, app(app(cons, x''), x0)), x0)
APP(app(filter, app(app(consif, false), x'')), app(app(cons, x0), xs)) -> APP(app(consif, x0), x0)
APP(app(filter, app(filter, f'')), app(app(cons, nil), xs)) -> APP(app(consif, nil), nil)
APP(app(filter, app(filter, f'')), app(app(cons, app(app(cons, x''), xs'')), xs)) -> APP(app(consif, app(app(app(consif, app(f'', x'')), x''), app(app(filter, f''), xs''))), app(app(cons, x''), xs''))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 4

                 ↳Narrowing Transformation

Dependency Pairs:

APP(app(filter, app(filter, f'')), app(app(cons, app(app(cons, x''), xs'')), xs)) -> APP(app(consif, app(app(app(consif, app(f'', x'')), x''), app(app(filter, f''), xs''))), app(app(cons, x''), xs''))
APP(app(filter, app(app(consif, false), x'')), app(app(cons, x0), xs)) -> APP(app(consif, x0), x0)
APP(app(filter, app(app(consif, true), x'')), app(app(cons, x0), xs)) -> APP(app(consif, app(app(cons, x''), x0)), x0)
APP(app(filter, app(filter, f'')), app(app(cons, app(app(cons, x''), xs'')), xs)) -> APP(app(app(consif, app(app(app(consif, app(f'', x'')), x''), app(app(filter, f''), xs''))), app(app(cons, x''), xs'')), app(app(filter, app(filter, f'')), xs))
APP(app(filter, app(filter, f'')), app(app(cons, nil), xs)) -> APP(app(app(consif, nil), nil), app(app(filter, app(filter, f'')), xs))
APP(app(filter, app(app(consif, false), x'')), app(app(cons, x0), xs)) -> APP(app(app(consif, x0), x0), app(app(filter, app(app(consif, false), x'')), xs))
APP(app(filter, app(app(consif, true), x'')), app(app(cons, x0), xs)) -> APP(app(app(consif, app(app(cons, x''), x0)), x0), app(app(filter, app(app(consif, true), x'')), xs))
APP(app(filter, f), app(app(cons, x), xs)) -> APP(app(filter, f), xs)
APP(app(filter, f), app(app(cons, x), xs)) -> APP(f, x)
APP(app(filter, f''), app(app(cons, x), app(app(cons, x''), xs''))) -> APP(app(app(consif, app(f'', x)), x), app(app(app(consif, app(f'', x'')), x''), app(app(filter, f''), xs'')))

Rules:

app(app(app(consif, true), x), ys) -> app(app(cons, x), ys)
app(app(app(consif, false), x), ys) -> ys
app(app(filter, f), nil) -> nil
app(app(filter, f), app(app(cons, x), xs)) -> app(app(app(consif, app(f, x)), x), app(app(filter, f), xs))

Strategy:

innermost

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

APP(app(filter, app(app(consif, true), x'')), app(app(cons, x0), xs)) -> APP(app(consif, app(app(cons, x''), x0)), x0)

no new Dependency Pairs are created.
The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 5

                 ↳Narrowing Transformation

Dependency Pairs:

APP(app(filter, app(app(consif, false), x'')), app(app(cons, x0), xs)) -> APP(app(consif, x0), x0)
APP(app(filter, f''), app(app(cons, x), app(app(cons, x''), xs''))) -> APP(app(app(consif, app(f'', x)), x), app(app(app(consif, app(f'', x'')), x''), app(app(filter, f''), xs'')))
APP(app(filter, app(filter, f'')), app(app(cons, app(app(cons, x''), xs'')), xs)) -> APP(app(app(consif, app(app(app(consif, app(f'', x'')), x''), app(app(filter, f''), xs''))), app(app(cons, x''), xs'')), app(app(filter, app(filter, f'')), xs))
APP(app(filter, app(filter, f'')), app(app(cons, nil), xs)) -> APP(app(app(consif, nil), nil), app(app(filter, app(filter, f'')), xs))
APP(app(filter, app(app(consif, false), x'')), app(app(cons, x0), xs)) -> APP(app(app(consif, x0), x0), app(app(filter, app(app(consif, false), x'')), xs))
APP(app(filter, app(app(consif, true), x'')), app(app(cons, x0), xs)) -> APP(app(app(consif, app(app(cons, x''), x0)), x0), app(app(filter, app(app(consif, true), x'')), xs))
APP(app(filter, f), app(app(cons, x), xs)) -> APP(app(filter, f), xs)
APP(app(filter, f), app(app(cons, x), xs)) -> APP(f, x)
APP(app(filter, app(filter, f'')), app(app(cons, app(app(cons, x''), xs'')), xs)) -> APP(app(consif, app(app(app(consif, app(f'', x'')), x''), app(app(filter, f''), xs''))), app(app(cons, x''), xs''))

Rules:

app(app(app(consif, true), x), ys) -> app(app(cons, x), ys)
app(app(app(consif, false), x), ys) -> ys
app(app(filter, f), nil) -> nil
app(app(filter, f), app(app(cons, x), xs)) -> app(app(app(consif, app(f, x)), x), app(app(filter, f), xs))

Strategy:

innermost

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

APP(app(filter, app(app(consif, false), x'')), app(app(cons, x0), xs)) -> APP(app(consif, x0), x0)

no new Dependency Pairs are created.
The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 6

                 ↳Remaining Obligation(s)

The following remains to be proven:
Dependency Pairs:

APP(app(filter, app(filter, f'')), app(app(cons, app(app(cons, x''), xs'')), xs)) -> APP(app(consif, app(app(app(consif, app(f'', x'')), x''), app(app(filter, f''), xs''))), app(app(cons, x''), xs''))
APP(app(filter, app(filter, f'')), app(app(cons, app(app(cons, x''), xs'')), xs)) -> APP(app(app(consif, app(app(app(consif, app(f'', x'')), x''), app(app(filter, f''), xs''))), app(app(cons, x''), xs'')), app(app(filter, app(filter, f'')), xs))
APP(app(filter, app(filter, f'')), app(app(cons, nil), xs)) -> APP(app(app(consif, nil), nil), app(app(filter, app(filter, f'')), xs))
APP(app(filter, app(app(consif, false), x'')), app(app(cons, x0), xs)) -> APP(app(app(consif, x0), x0), app(app(filter, app(app(consif, false), x'')), xs))
APP(app(filter, app(app(consif, true), x'')), app(app(cons, x0), xs)) -> APP(app(app(consif, app(app(cons, x''), x0)), x0), app(app(filter, app(app(consif, true), x'')), xs))
APP(app(filter, f), app(app(cons, x), xs)) -> APP(app(filter, f), xs)
APP(app(filter, f), app(app(cons, x), xs)) -> APP(f, x)
APP(app(filter, f''), app(app(cons, x), app(app(cons, x''), xs''))) -> APP(app(app(consif, app(f'', x)), x), app(app(app(consif, app(f'', x'')), x''), app(app(filter, f''), xs'')))

Rules:

app(app(app(consif, true), x), ys) -> app(app(cons, x), ys)
app(app(app(consif, false), x), ys) -> ys
app(app(filter, f), nil) -> nil
app(app(filter, f), app(app(cons, x), xs)) -> app(app(app(consif, app(f, x)), x), app(app(filter, f), xs))

Strategy:

innermost

Innermost Termination of R could not be shown.
Duration:
0:09 minutes