Termination Proof using AProVE

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

Innermost Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

APP(app(filter, f), app(app(cons, y), ys)) -> APP(app(app(filtersub, app(f, y)), f), app(app(cons, y), ys))
APP(app(filter, f), app(app(cons, y), ys)) -> APP(app(filtersub, app(f, y)), f)
APP(app(filter, f), app(app(cons, y), ys)) -> APP(filtersub, app(f, y))
APP(app(filter, f), app(app(cons, y), ys)) -> APP(f, y)
APP(app(app(filtersub, true), f), app(app(cons, y), ys)) -> APP(app(cons, y), app(app(filter, f), ys))
APP(app(app(filtersub, true), f), app(app(cons, y), ys)) -> APP(app(filter, f), ys)
APP(app(app(filtersub, true), f), app(app(cons, y), ys)) -> APP(filter, f)
APP(app(app(filtersub, false), f), app(app(cons, y), ys)) -> APP(app(filter, f), ys)
APP(app(app(filtersub, false), f), app(app(cons, y), ys)) -> APP(filter, f)

Furthermore, R contains one SCC.

     ↳DPs

       →DP Problem 1

         ↳Narrowing Transformation

Dependency Pairs:

APP(app(app(filtersub, false), f), app(app(cons, y), ys)) -> APP(app(filter, f), ys)
APP(app(app(filtersub, true), f), app(app(cons, y), ys)) -> APP(app(filter, f), ys)
APP(app(filter, f), app(app(cons, y), ys)) -> APP(f, y)
APP(app(filter, f), app(app(cons, y), ys)) -> APP(app(filtersub, app(f, y)), f)
APP(app(filter, f), app(app(cons, y), ys)) -> APP(app(app(filtersub, app(f, y)), f), app(app(cons, y), ys))

Rules:

app(app(filter, f), nil) -> nil
app(app(filter, f), app(app(cons, y), ys)) -> app(app(app(filtersub, app(f, y)), f), app(app(cons, y), ys))
app(app(app(filtersub, true), f), app(app(cons, y), ys)) -> app(app(cons, y), app(app(filter, f), ys))
app(app(app(filtersub, false), f), app(app(cons, y), ys)) -> app(app(filter, f), ys)

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, y), ys)) -> APP(app(app(filtersub, app(f, y)), f), app(app(cons, y), ys))

four new Dependency Pairs are created:

APP(app(filter, app(filter, f'')), app(app(cons, nil), ys)) -> APP(app(app(filtersub, nil), app(filter, f'')), app(app(cons, nil), ys))
APP(app(filter, app(filter, f'')), app(app(cons, app(app(cons, y''), ys'')), ys)) -> APP(app(app(filtersub, app(app(app(filtersub, app(f'', y'')), f''), app(app(cons, y''), ys''))), app(filter, f'')), app(app(cons, app(app(cons, y''), ys'')), ys))
APP(app(filter, app(app(filtersub, true), f'')), app(app(cons, app(app(cons, y''), ys'')), ys)) -> APP(app(app(filtersub, app(app(cons, y''), app(app(filter, f''), ys''))), app(app(filtersub, true), f'')), app(app(cons, app(app(cons, y''), ys'')), ys))
APP(app(filter, app(app(filtersub, false), f'')), app(app(cons, app(app(cons, y''), ys'')), ys)) -> APP(app(app(filtersub, app(app(filter, f''), ys'')), app(app(filtersub, false), f'')), app(app(cons, app(app(cons, y''), ys'')), ys))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Narrowing Transformation

Dependency Pairs:

APP(app(filter, app(app(filtersub, false), f'')), app(app(cons, app(app(cons, y''), ys'')), ys)) -> APP(app(app(filtersub, app(app(filter, f''), ys'')), app(app(filtersub, false), f'')), app(app(cons, app(app(cons, y''), ys'')), ys))
APP(app(filter, app(app(filtersub, true), f'')), app(app(cons, app(app(cons, y''), ys'')), ys)) -> APP(app(app(filtersub, app(app(cons, y''), app(app(filter, f''), ys''))), app(app(filtersub, true), f'')), app(app(cons, app(app(cons, y''), ys'')), ys))
APP(app(filter, app(filter, f'')), app(app(cons, app(app(cons, y''), ys'')), ys)) -> APP(app(app(filtersub, app(app(app(filtersub, app(f'', y'')), f''), app(app(cons, y''), ys''))), app(filter, f'')), app(app(cons, app(app(cons, y''), ys'')), ys))
APP(app(filter, app(filter, f'')), app(app(cons, nil), ys)) -> APP(app(app(filtersub, nil), app(filter, f'')), app(app(cons, nil), ys))
APP(app(app(filtersub, true), f), app(app(cons, y), ys)) -> APP(app(filter, f), ys)
APP(app(filter, f), app(app(cons, y), ys)) -> APP(f, y)
APP(app(filter, f), app(app(cons, y), ys)) -> APP(app(filtersub, app(f, y)), f)
APP(app(app(filtersub, false), f), app(app(cons, y), ys)) -> APP(app(filter, f), ys)

Rules:

app(app(filter, f), nil) -> nil
app(app(filter, f), app(app(cons, y), ys)) -> app(app(app(filtersub, app(f, y)), f), app(app(cons, y), ys))
app(app(app(filtersub, true), f), app(app(cons, y), ys)) -> app(app(cons, y), app(app(filter, f), ys))
app(app(app(filtersub, false), f), app(app(cons, y), ys)) -> app(app(filter, f), ys)

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, y), ys)) -> APP(app(filtersub, app(f, y)), f)

four new Dependency Pairs are created:

APP(app(filter, app(filter, f'')), app(app(cons, nil), ys)) -> APP(app(filtersub, nil), app(filter, f''))
APP(app(filter, app(filter, f'')), app(app(cons, app(app(cons, y''), ys'')), ys)) -> APP(app(filtersub, app(app(app(filtersub, app(f'', y'')), f''), app(app(cons, y''), ys''))), app(filter, f''))
APP(app(filter, app(app(filtersub, true), f'')), app(app(cons, app(app(cons, y''), ys'')), ys)) -> APP(app(filtersub, app(app(cons, y''), app(app(filter, f''), ys''))), app(app(filtersub, true), f''))
APP(app(filter, app(app(filtersub, false), f'')), app(app(cons, app(app(cons, y''), ys'')), ys)) -> APP(app(filtersub, app(app(filter, f''), ys'')), app(app(filtersub, false), f''))

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, app(app(filtersub, true), f'')), app(app(cons, app(app(cons, y''), ys'')), ys)) -> APP(app(app(filtersub, app(app(cons, y''), app(app(filter, f''), ys''))), app(app(filtersub, true), f'')), app(app(cons, app(app(cons, y''), ys'')), ys))
APP(app(filter, app(filter, f'')), app(app(cons, app(app(cons, y''), ys'')), ys)) -> APP(app(app(filtersub, app(app(app(filtersub, app(f'', y'')), f''), app(app(cons, y''), ys''))), app(filter, f'')), app(app(cons, app(app(cons, y''), ys'')), ys))
APP(app(filter, app(filter, f'')), app(app(cons, nil), ys)) -> APP(app(app(filtersub, nil), app(filter, f'')), app(app(cons, nil), ys))
APP(app(app(filtersub, false), f), app(app(cons, y), ys)) -> APP(app(filter, f), ys)
APP(app(app(filtersub, true), f), app(app(cons, y), ys)) -> APP(app(filter, f), ys)
APP(app(filter, f), app(app(cons, y), ys)) -> APP(f, y)
APP(app(filter, app(app(filtersub, false), f'')), app(app(cons, app(app(cons, y''), ys'')), ys)) -> APP(app(app(filtersub, app(app(filter, f''), ys'')), app(app(filtersub, false), f'')), app(app(cons, app(app(cons, y''), ys'')), ys))

Rules:

app(app(filter, f), nil) -> nil
app(app(filter, f), app(app(cons, y), ys)) -> app(app(app(filtersub, app(f, y)), f), app(app(cons, y), ys))
app(app(app(filtersub, true), f), app(app(cons, y), ys)) -> app(app(cons, y), app(app(filter, f), ys))
app(app(app(filtersub, false), f), app(app(cons, y), ys)) -> app(app(filter, f), ys)

Strategy:

innermost

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

APP(app(filter, app(filter, f'')), app(app(cons, nil), ys)) -> APP(app(app(filtersub, nil), app(filter, f'')), app(app(cons, nil), ys))

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 4

                 ↳Forward Instantiation Transformation

Dependency Pairs:

APP(app(filter, app(app(filtersub, false), f'')), app(app(cons, app(app(cons, y''), ys'')), ys)) -> APP(app(app(filtersub, app(app(filter, f''), ys'')), app(app(filtersub, false), f'')), app(app(cons, app(app(cons, y''), ys'')), ys))
APP(app(filter, app(filter, f'')), app(app(cons, app(app(cons, y''), ys'')), ys)) -> APP(app(app(filtersub, app(app(app(filtersub, app(f'', y'')), f''), app(app(cons, y''), ys''))), app(filter, f'')), app(app(cons, app(app(cons, y''), ys'')), ys))
APP(app(app(filtersub, false), f), app(app(cons, y), ys)) -> APP(app(filter, f), ys)
APP(app(app(filtersub, true), f), app(app(cons, y), ys)) -> APP(app(filter, f), ys)
APP(app(filter, f), app(app(cons, y), ys)) -> APP(f, y)
APP(app(filter, app(app(filtersub, true), f'')), app(app(cons, app(app(cons, y''), ys'')), ys)) -> APP(app(app(filtersub, app(app(cons, y''), app(app(filter, f''), ys''))), app(app(filtersub, true), f'')), app(app(cons, app(app(cons, y''), ys'')), ys))

Rules:

app(app(filter, f), nil) -> nil
app(app(filter, f), app(app(cons, y), ys)) -> app(app(app(filtersub, app(f, y)), f), app(app(cons, y), ys))
app(app(app(filtersub, true), f), app(app(cons, y), ys)) -> app(app(cons, y), app(app(filter, f), ys))
app(app(app(filtersub, false), f), app(app(cons, y), ys)) -> app(app(filter, f), ys)

Strategy:

innermost

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

APP(app(filter, f), app(app(cons, y), ys)) -> APP(f, y)

six new Dependency Pairs are created:

APP(app(filter, app(filter, f'')), app(app(cons, app(app(cons, y''), ys'')), ys)) -> APP(app(filter, f''), app(app(cons, y''), ys''))
APP(app(filter, app(app(filtersub, true), f'')), app(app(cons, app(app(cons, y''), ys'')), ys)) -> APP(app(app(filtersub, true), f''), app(app(cons, y''), ys''))
APP(app(filter, app(app(filtersub, false), f'')), app(app(cons, app(app(cons, y''), ys'')), ys)) -> APP(app(app(filtersub, false), f''), app(app(cons, y''), ys''))
APP(app(filter, app(filter, app(filter, f''''))), app(app(cons, app(app(cons, app(app(cons, y''''), ys'''')), ys'')), ys)) -> APP(app(filter, app(filter, f'''')), app(app(cons, app(app(cons, y''''), ys'''')), ys''))
APP(app(filter, app(filter, app(app(filtersub, true), f''''))), app(app(cons, app(app(cons, app(app(cons, y''''), ys'''')), ys'')), ys)) -> APP(app(filter, app(app(filtersub, true), f'''')), app(app(cons, app(app(cons, y''''), ys'''')), ys''))
APP(app(filter, app(filter, app(app(filtersub, false), f''''))), app(app(cons, app(app(cons, app(app(cons, y''''), ys'''')), ys'')), ys)) -> APP(app(filter, app(app(filtersub, false), f'''')), app(app(cons, app(app(cons, y''''), ys'''')), ys''))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 5

                 ↳Argument Filtering and Ordering

Dependency Pairs:

APP(app(filter, app(filter, app(app(filtersub, false), f''''))), app(app(cons, app(app(cons, app(app(cons, y''''), ys'''')), ys'')), ys)) -> APP(app(filter, app(app(filtersub, false), f'''')), app(app(cons, app(app(cons, y''''), ys'''')), ys''))
APP(app(filter, app(filter, app(app(filtersub, true), f''''))), app(app(cons, app(app(cons, app(app(cons, y''''), ys'''')), ys'')), ys)) -> APP(app(filter, app(app(filtersub, true), f'''')), app(app(cons, app(app(cons, y''''), ys'''')), ys''))
APP(app(filter, app(filter, app(filter, f''''))), app(app(cons, app(app(cons, app(app(cons, y''''), ys'''')), ys'')), ys)) -> APP(app(filter, app(filter, f'''')), app(app(cons, app(app(cons, y''''), ys'''')), ys''))
APP(app(filter, app(app(filtersub, false), f'')), app(app(cons, app(app(cons, y''), ys'')), ys)) -> APP(app(app(filtersub, false), f''), app(app(cons, y''), ys''))
APP(app(filter, app(app(filtersub, true), f'')), app(app(cons, app(app(cons, y''), ys'')), ys)) -> APP(app(app(filtersub, true), f''), app(app(cons, y''), ys''))
APP(app(filter, app(filter, f'')), app(app(cons, app(app(cons, y''), ys'')), ys)) -> APP(app(filter, f''), app(app(cons, y''), ys''))
APP(app(filter, app(app(filtersub, true), f'')), app(app(cons, app(app(cons, y''), ys'')), ys)) -> APP(app(app(filtersub, app(app(cons, y''), app(app(filter, f''), ys''))), app(app(filtersub, true), f'')), app(app(cons, app(app(cons, y''), ys'')), ys))
APP(app(filter, app(filter, f'')), app(app(cons, app(app(cons, y''), ys'')), ys)) -> APP(app(app(filtersub, app(app(app(filtersub, app(f'', y'')), f''), app(app(cons, y''), ys''))), app(filter, f'')), app(app(cons, app(app(cons, y''), ys'')), ys))
APP(app(app(filtersub, false), f), app(app(cons, y), ys)) -> APP(app(filter, f), ys)
APP(app(app(filtersub, true), f), app(app(cons, y), ys)) -> APP(app(filter, f), ys)
APP(app(filter, app(app(filtersub, false), f'')), app(app(cons, app(app(cons, y''), ys'')), ys)) -> APP(app(app(filtersub, app(app(filter, f''), ys'')), app(app(filtersub, false), f'')), app(app(cons, app(app(cons, y''), ys'')), ys))

Rules:

app(app(filter, f), nil) -> nil
app(app(filter, f), app(app(cons, y), ys)) -> app(app(app(filtersub, app(f, y)), f), app(app(cons, y), ys))
app(app(app(filtersub, true), f), app(app(cons, y), ys)) -> app(app(cons, y), app(app(filter, f), ys))
app(app(app(filtersub, false), f), app(app(cons, y), ys)) -> app(app(filter, f), ys)

Strategy:

innermost

The following dependency pairs can be strictly oriented:

APP(app(filter, app(filter, app(app(filtersub, false), f''''))), app(app(cons, app(app(cons, app(app(cons, y''''), ys'''')), ys'')), ys)) -> APP(app(filter, app(app(filtersub, false), f'''')), app(app(cons, app(app(cons, y''''), ys'''')), ys''))
APP(app(filter, app(filter, app(app(filtersub, true), f''''))), app(app(cons, app(app(cons, app(app(cons, y''''), ys'''')), ys'')), ys)) -> APP(app(filter, app(app(filtersub, true), f'''')), app(app(cons, app(app(cons, y''''), ys'''')), ys''))
APP(app(filter, app(filter, app(filter, f''''))), app(app(cons, app(app(cons, app(app(cons, y''''), ys'''')), ys'')), ys)) -> APP(app(filter, app(filter, f'''')), app(app(cons, app(app(cons, y''''), ys'''')), ys''))
APP(app(filter, app(app(filtersub, false), f'')), app(app(cons, app(app(cons, y''), ys'')), ys)) -> APP(app(app(filtersub, false), f''), app(app(cons, y''), ys''))
APP(app(filter, app(app(filtersub, true), f'')), app(app(cons, app(app(cons, y''), ys'')), ys)) -> APP(app(app(filtersub, true), f''), app(app(cons, y''), ys''))
APP(app(filter, app(filter, f'')), app(app(cons, app(app(cons, y''), ys'')), ys)) -> APP(app(filter, f''), app(app(cons, y''), ys''))

The following usable rules for innermost can be oriented:

app(app(filter, f), nil) -> nil
app(app(filter, f), app(app(cons, y), ys)) -> app(app(app(filtersub, app(f, y)), f), app(app(cons, y), ys))
app(app(app(filtersub, true), f), app(app(cons, y), ys)) -> app(app(cons, y), app(app(filter, f), ys))
app(app(app(filtersub, false), f), app(app(cons, y), ys)) -> app(app(filter, f), ys)

Used ordering: Homeomorphic Embedding Order with EMB
resulting in one new DP problem.
Used Argument Filtering System:

APP(x₁, x₂) -> x₁
app(x₁, x₂) -> app(x₂)

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 6

                 ↳Argument Filtering and Ordering

Dependency Pairs:

APP(app(filter, app(app(filtersub, true), f'')), app(app(cons, app(app(cons, y''), ys'')), ys)) -> APP(app(app(filtersub, app(app(cons, y''), app(app(filter, f''), ys''))), app(app(filtersub, true), f'')), app(app(cons, app(app(cons, y''), ys'')), ys))
APP(app(filter, app(filter, f'')), app(app(cons, app(app(cons, y''), ys'')), ys)) -> APP(app(app(filtersub, app(app(app(filtersub, app(f'', y'')), f''), app(app(cons, y''), ys''))), app(filter, f'')), app(app(cons, app(app(cons, y''), ys'')), ys))
APP(app(app(filtersub, false), f), app(app(cons, y), ys)) -> APP(app(filter, f), ys)
APP(app(app(filtersub, true), f), app(app(cons, y), ys)) -> APP(app(filter, f), ys)
APP(app(filter, app(app(filtersub, false), f'')), app(app(cons, app(app(cons, y''), ys'')), ys)) -> APP(app(app(filtersub, app(app(filter, f''), ys'')), app(app(filtersub, false), f'')), app(app(cons, app(app(cons, y''), ys'')), ys))

Rules:

app(app(filter, f), nil) -> nil
app(app(filter, f), app(app(cons, y), ys)) -> app(app(app(filtersub, app(f, y)), f), app(app(cons, y), ys))
app(app(app(filtersub, true), f), app(app(cons, y), ys)) -> app(app(cons, y), app(app(filter, f), ys))
app(app(app(filtersub, false), f), app(app(cons, y), ys)) -> app(app(filter, f), ys)

Strategy:

innermost

The following dependency pairs can be strictly oriented:

APP(app(app(filtersub, false), f), app(app(cons, y), ys)) -> APP(app(filter, f), ys)
APP(app(app(filtersub, true), f), app(app(cons, y), ys)) -> APP(app(filter, f), ys)

The following usable rules for innermost can be oriented:

app(app(filter, f), nil) -> nil
app(app(filter, f), app(app(cons, y), ys)) -> app(app(app(filtersub, app(f, y)), f), app(app(cons, y), ys))
app(app(app(filtersub, true), f), app(app(cons, y), ys)) -> app(app(cons, y), app(app(filter, f), ys))
app(app(app(filtersub, false), f), app(app(cons, y), ys)) -> app(app(filter, f), ys)

Used ordering: Homeomorphic Embedding Order with EMB
resulting in one new DP problem.
Used Argument Filtering System:

APP(x₁, x₂) -> APP(x₁, x₂)
app(x₁, x₂) -> app(x₂)

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 7

                 ↳Remaining Obligation(s)

The following remains to be proven:
Dependency Pairs:

Rules:

app(app(filter, f), nil) -> nil
app(app(filter, f), app(app(cons, y), ys)) -> app(app(app(filtersub, app(f, y)), f), app(app(cons, y), ys))
app(app(app(filtersub, true), f), app(app(cons, y), ys)) -> app(app(cons, y), app(app(filter, f), ys))
app(app(app(filtersub, false), f), app(app(cons, y), ys)) -> app(app(filter, f), ys)

Strategy:

innermost

Innermost Termination of R could not be shown.
Duration:
0:03 minutes