Termination Proof using AProVE

Term Rewriting System R:
[f, x, xs]
app(app(mapt, f), app(leaf, x)) -> app(leaf, app(f, x))
app(app(mapt, f), app(node, xs)) -> app(node, app(app(maptlist, f), xs))
app(app(maptlist, f), nil) -> nil
app(app(maptlist, f), app(app(cons, x), xs)) -> app(app(cons, app(app(mapt, f), x)), app(app(maptlist, f), xs))

Innermost Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

APP(app(mapt, f), app(leaf, x)) -> APP(leaf, app(f, x))
APP(app(mapt, f), app(leaf, x)) -> APP(f, x)
APP(app(mapt, f), app(node, xs)) -> APP(node, app(app(maptlist, f), xs))
APP(app(mapt, f), app(node, xs)) -> APP(app(maptlist, f), xs)
APP(app(mapt, f), app(node, xs)) -> APP(maptlist, f)
APP(app(maptlist, f), app(app(cons, x), xs)) -> APP(app(cons, app(app(mapt, f), x)), app(app(maptlist, f), xs))
APP(app(maptlist, f), app(app(cons, x), xs)) -> APP(cons, app(app(mapt, f), x))
APP(app(maptlist, f), app(app(cons, x), xs)) -> APP(app(mapt, f), x)
APP(app(maptlist, f), app(app(cons, x), xs)) -> APP(mapt, f)
APP(app(maptlist, f), app(app(cons, x), xs)) -> APP(app(maptlist, f), xs)

Furthermore, R contains one SCC.

     ↳DPs

       →DP Problem 1

         ↳Narrowing Transformation

Dependency Pairs:

APP(app(maptlist, f), app(app(cons, x), xs)) -> APP(app(maptlist, f), xs)
APP(app(maptlist, f), app(app(cons, x), xs)) -> APP(app(mapt, f), x)
APP(app(maptlist, f), app(app(cons, x), xs)) -> APP(app(cons, app(app(mapt, f), x)), app(app(maptlist, f), xs))
APP(app(mapt, f), app(node, xs)) -> APP(app(maptlist, f), xs)
APP(app(mapt, f), app(leaf, x)) -> APP(f, x)

Rules:

app(app(mapt, f), app(leaf, x)) -> app(leaf, app(f, x))
app(app(mapt, f), app(node, xs)) -> app(node, app(app(maptlist, f), xs))
app(app(maptlist, f), nil) -> nil
app(app(maptlist, f), app(app(cons, x), xs)) -> app(app(cons, app(app(mapt, f), x)), app(app(maptlist, f), xs))

Strategy:

innermost

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

APP(app(maptlist, f), app(app(cons, x), xs)) -> APP(app(cons, app(app(mapt, f), x)), app(app(maptlist, f), xs))

four new Dependency Pairs are created:

APP(app(maptlist, f''), app(app(cons, app(leaf, x'')), xs)) -> APP(app(cons, app(leaf, app(f'', x''))), app(app(maptlist, f''), xs))
APP(app(maptlist, f''), app(app(cons, app(node, xs'')), xs)) -> APP(app(cons, app(node, app(app(maptlist, f''), xs''))), app(app(maptlist, f''), xs))
APP(app(maptlist, f''), app(app(cons, x), nil)) -> APP(app(cons, app(app(mapt, f''), x)), nil)
APP(app(maptlist, f''), app(app(cons, x), app(app(cons, x''), xs''))) -> APP(app(cons, app(app(mapt, f''), x)), app(app(cons, app(app(mapt, f''), x'')), app(app(maptlist, f''), xs'')))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Narrowing Transformation

Dependency Pairs:

APP(app(maptlist, f''), app(app(cons, x), app(app(cons, x''), xs''))) -> APP(app(cons, app(app(mapt, f''), x)), app(app(cons, app(app(mapt, f''), x'')), app(app(maptlist, f''), xs'')))
APP(app(maptlist, f''), app(app(cons, app(node, xs'')), xs)) -> APP(app(cons, app(node, app(app(maptlist, f''), xs''))), app(app(maptlist, f''), xs))
APP(app(maptlist, f''), app(app(cons, app(leaf, x'')), xs)) -> APP(app(cons, app(leaf, app(f'', x''))), app(app(maptlist, f''), xs))
APP(app(mapt, f), app(node, xs)) -> APP(app(maptlist, f), xs)
APP(app(mapt, f), app(leaf, x)) -> APP(f, x)
APP(app(maptlist, f), app(app(cons, x), xs)) -> APP(app(mapt, f), x)
APP(app(maptlist, f), app(app(cons, x), xs)) -> APP(app(maptlist, f), xs)

Rules:

app(app(mapt, f), app(leaf, x)) -> app(leaf, app(f, x))
app(app(mapt, f), app(node, xs)) -> app(node, app(app(maptlist, f), xs))
app(app(maptlist, f), nil) -> nil
app(app(maptlist, f), app(app(cons, x), xs)) -> app(app(cons, app(app(mapt, f), x)), app(app(maptlist, f), xs))

Strategy:

innermost

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

APP(app(maptlist, f''), app(app(cons, app(leaf, x'')), xs)) -> APP(app(cons, app(leaf, app(f'', x''))), app(app(maptlist, f''), xs))

six new Dependency Pairs are created:

APP(app(maptlist, app(mapt, f')), app(app(cons, app(leaf, app(leaf, x'))), xs)) -> APP(app(cons, app(leaf, app(leaf, app(f', x')))), app(app(maptlist, app(mapt, f')), xs))
APP(app(maptlist, app(mapt, f')), app(app(cons, app(leaf, app(node, xs''))), xs)) -> APP(app(cons, app(leaf, app(node, app(app(maptlist, f'), xs'')))), app(app(maptlist, app(mapt, f')), xs))
APP(app(maptlist, app(maptlist, f')), app(app(cons, app(leaf, nil)), xs)) -> APP(app(cons, app(leaf, nil)), app(app(maptlist, app(maptlist, f')), xs))
APP(app(maptlist, app(maptlist, f')), app(app(cons, app(leaf, app(app(cons, x'), xs''))), xs)) -> APP(app(cons, app(leaf, app(app(cons, app(app(mapt, f'), x')), app(app(maptlist, f'), xs'')))), app(app(maptlist, app(maptlist, f')), xs))
APP(app(maptlist, f'''), app(app(cons, app(leaf, x'')), nil)) -> APP(app(cons, app(leaf, app(f''', x''))), nil)
APP(app(maptlist, f'''), app(app(cons, app(leaf, x'')), app(app(cons, x'), xs''))) -> APP(app(cons, app(leaf, app(f''', x''))), app(app(cons, app(app(mapt, f'''), x')), app(app(maptlist, 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(maptlist, f'''), app(app(cons, app(leaf, x'')), app(app(cons, x'), xs''))) -> APP(app(cons, app(leaf, app(f''', x''))), app(app(cons, app(app(mapt, f'''), x')), app(app(maptlist, f'''), xs'')))
APP(app(maptlist, app(maptlist, f')), app(app(cons, app(leaf, app(app(cons, x'), xs''))), xs)) -> APP(app(cons, app(leaf, app(app(cons, app(app(mapt, f'), x')), app(app(maptlist, f'), xs'')))), app(app(maptlist, app(maptlist, f')), xs))
APP(app(maptlist, app(mapt, f')), app(app(cons, app(leaf, app(node, xs''))), xs)) -> APP(app(cons, app(leaf, app(node, app(app(maptlist, f'), xs'')))), app(app(maptlist, app(mapt, f')), xs))
APP(app(maptlist, app(mapt, f')), app(app(cons, app(leaf, app(leaf, x'))), xs)) -> APP(app(cons, app(leaf, app(leaf, app(f', x')))), app(app(maptlist, app(mapt, f')), xs))
APP(app(maptlist, f''), app(app(cons, app(node, xs'')), xs)) -> APP(app(cons, app(node, app(app(maptlist, f''), xs''))), app(app(maptlist, f''), xs))
APP(app(maptlist, f), app(app(cons, x), xs)) -> APP(app(maptlist, f), xs)
APP(app(maptlist, f), app(app(cons, x), xs)) -> APP(app(mapt, f), x)
APP(app(mapt, f), app(node, xs)) -> APP(app(maptlist, f), xs)
APP(app(mapt, f), app(leaf, x)) -> APP(f, x)
APP(app(maptlist, f''), app(app(cons, x), app(app(cons, x''), xs''))) -> APP(app(cons, app(app(mapt, f''), x)), app(app(cons, app(app(mapt, f''), x'')), app(app(maptlist, f''), xs'')))

Rules:

app(app(mapt, f), app(leaf, x)) -> app(leaf, app(f, x))
app(app(mapt, f), app(node, xs)) -> app(node, app(app(maptlist, f), xs))
app(app(maptlist, f), nil) -> nil
app(app(maptlist, f), app(app(cons, x), xs)) -> app(app(cons, app(app(mapt, f), x)), app(app(maptlist, f), xs))

Strategy:

innermost

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

APP(app(maptlist, f''), app(app(cons, app(node, xs'')), xs)) -> APP(app(cons, app(node, app(app(maptlist, f''), xs''))), app(app(maptlist, f''), xs))

four new Dependency Pairs are created:

APP(app(maptlist, f'''), app(app(cons, app(node, nil)), xs)) -> APP(app(cons, app(node, nil)), app(app(maptlist, f'''), xs))
APP(app(maptlist, f'''), app(app(cons, app(node, app(app(cons, x'), xs'''))), xs)) -> APP(app(cons, app(node, app(app(cons, app(app(mapt, f'''), x')), app(app(maptlist, f'''), xs''')))), app(app(maptlist, f'''), xs))
APP(app(maptlist, f'''), app(app(cons, app(node, xs'')), nil)) -> APP(app(cons, app(node, app(app(maptlist, f'''), xs''))), nil)
APP(app(maptlist, f'''), app(app(cons, app(node, xs'')), app(app(cons, x'), xs'''))) -> APP(app(cons, app(node, app(app(maptlist, f'''), xs''))), app(app(cons, app(app(mapt, f'''), x')), app(app(maptlist, f'''), 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(maptlist, f'''), app(app(cons, app(node, xs'')), app(app(cons, x'), xs'''))) -> APP(app(cons, app(node, app(app(maptlist, f'''), xs''))), app(app(cons, app(app(mapt, f'''), x')), app(app(maptlist, f'''), xs''')))
APP(app(maptlist, f'''), app(app(cons, app(node, app(app(cons, x'), xs'''))), xs)) -> APP(app(cons, app(node, app(app(cons, app(app(mapt, f'''), x')), app(app(maptlist, f'''), xs''')))), app(app(maptlist, f'''), xs))
APP(app(maptlist, app(maptlist, f')), app(app(cons, app(leaf, app(app(cons, x'), xs''))), xs)) -> APP(app(cons, app(leaf, app(app(cons, app(app(mapt, f'), x')), app(app(maptlist, f'), xs'')))), app(app(maptlist, app(maptlist, f')), xs))
APP(app(maptlist, app(mapt, f')), app(app(cons, app(leaf, app(node, xs''))), xs)) -> APP(app(cons, app(leaf, app(node, app(app(maptlist, f'), xs'')))), app(app(maptlist, app(mapt, f')), xs))
APP(app(maptlist, app(mapt, f')), app(app(cons, app(leaf, app(leaf, x'))), xs)) -> APP(app(cons, app(leaf, app(leaf, app(f', x')))), app(app(maptlist, app(mapt, f')), xs))
APP(app(maptlist, f''), app(app(cons, x), app(app(cons, x''), xs''))) -> APP(app(cons, app(app(mapt, f''), x)), app(app(cons, app(app(mapt, f''), x'')), app(app(maptlist, f''), xs'')))
APP(app(maptlist, f), app(app(cons, x), xs)) -> APP(app(maptlist, f), xs)
APP(app(maptlist, f), app(app(cons, x), xs)) -> APP(app(mapt, f), x)
APP(app(mapt, f), app(node, xs)) -> APP(app(maptlist, f), xs)
APP(app(mapt, f), app(leaf, x)) -> APP(f, x)
APP(app(maptlist, f'''), app(app(cons, app(leaf, x'')), app(app(cons, x'), xs''))) -> APP(app(cons, app(leaf, app(f''', x''))), app(app(cons, app(app(mapt, f'''), x')), app(app(maptlist, f'''), xs'')))

Rules:

app(app(mapt, f), app(leaf, x)) -> app(leaf, app(f, x))
app(app(mapt, f), app(node, xs)) -> app(node, app(app(maptlist, f), xs))
app(app(maptlist, f), nil) -> nil
app(app(maptlist, f), app(app(cons, x), xs)) -> app(app(cons, app(app(mapt, f), x)), app(app(maptlist, f), xs))

Strategy:

innermost

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

APP(app(maptlist, f''), app(app(cons, x), app(app(cons, x''), xs''))) -> APP(app(cons, app(app(mapt, f''), x)), app(app(cons, app(app(mapt, f''), x'')), app(app(maptlist, f''), xs'')))

six new Dependency Pairs are created:

APP(app(maptlist, f'''), app(app(cons, app(leaf, x''')), app(app(cons, x''), xs''))) -> APP(app(cons, app(leaf, app(f''', x'''))), app(app(cons, app(app(mapt, f'''), x'')), app(app(maptlist, f'''), xs'')))
APP(app(maptlist, f'''), app(app(cons, app(node, xs')), app(app(cons, x''), xs''))) -> APP(app(cons, app(node, app(app(maptlist, f'''), xs'))), app(app(cons, app(app(mapt, f'''), x'')), app(app(maptlist, f'''), xs'')))
APP(app(maptlist, f'''), app(app(cons, x), app(app(cons, app(leaf, x''')), xs''))) -> APP(app(cons, app(app(mapt, f'''), x)), app(app(cons, app(leaf, app(f''', x'''))), app(app(maptlist, f'''), xs'')))
APP(app(maptlist, f'''), app(app(cons, x), app(app(cons, app(node, xs')), xs''))) -> APP(app(cons, app(app(mapt, f'''), x)), app(app(cons, app(node, app(app(maptlist, f'''), xs'))), app(app(maptlist, f'''), xs'')))
APP(app(maptlist, f'''), app(app(cons, x), app(app(cons, x''), nil))) -> APP(app(cons, app(app(mapt, f'''), x)), app(app(cons, app(app(mapt, f'''), x'')), nil))
APP(app(maptlist, f'''), app(app(cons, x), app(app(cons, x''), app(app(cons, x'''), xs')))) -> APP(app(cons, app(app(mapt, f'''), x)), app(app(cons, app(app(mapt, f'''), x'')), app(app(cons, app(app(mapt, f'''), x''')), app(app(maptlist, f'''), xs'))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 5

                 ↳Forward Instantiation Transformation

Dependency Pairs:

APP(app(maptlist, f'''), app(app(cons, x), app(app(cons, x''), app(app(cons, x'''), xs')))) -> APP(app(cons, app(app(mapt, f'''), x)), app(app(cons, app(app(mapt, f'''), x'')), app(app(cons, app(app(mapt, f'''), x''')), app(app(maptlist, f'''), xs'))))
APP(app(maptlist, f'''), app(app(cons, x), app(app(cons, x''), nil))) -> APP(app(cons, app(app(mapt, f'''), x)), app(app(cons, app(app(mapt, f'''), x'')), nil))
APP(app(maptlist, f'''), app(app(cons, x), app(app(cons, app(node, xs')), xs''))) -> APP(app(cons, app(app(mapt, f'''), x)), app(app(cons, app(node, app(app(maptlist, f'''), xs'))), app(app(maptlist, f'''), xs'')))
APP(app(maptlist, f'''), app(app(cons, x), app(app(cons, app(leaf, x''')), xs''))) -> APP(app(cons, app(app(mapt, f'''), x)), app(app(cons, app(leaf, app(f''', x'''))), app(app(maptlist, f'''), xs'')))
APP(app(maptlist, f'''), app(app(cons, app(node, xs')), app(app(cons, x''), xs''))) -> APP(app(cons, app(node, app(app(maptlist, f'''), xs'))), app(app(cons, app(app(mapt, f'''), x'')), app(app(maptlist, f'''), xs'')))
APP(app(maptlist, f'''), app(app(cons, app(leaf, x''')), app(app(cons, x''), xs''))) -> APP(app(cons, app(leaf, app(f''', x'''))), app(app(cons, app(app(mapt, f'''), x'')), app(app(maptlist, f'''), xs'')))
APP(app(maptlist, f'''), app(app(cons, app(node, app(app(cons, x'), xs'''))), xs)) -> APP(app(cons, app(node, app(app(cons, app(app(mapt, f'''), x')), app(app(maptlist, f'''), xs''')))), app(app(maptlist, f'''), xs))
APP(app(maptlist, f'''), app(app(cons, app(leaf, x'')), app(app(cons, x'), xs''))) -> APP(app(cons, app(leaf, app(f''', x''))), app(app(cons, app(app(mapt, f'''), x')), app(app(maptlist, f'''), xs'')))
APP(app(maptlist, app(maptlist, f')), app(app(cons, app(leaf, app(app(cons, x'), xs''))), xs)) -> APP(app(cons, app(leaf, app(app(cons, app(app(mapt, f'), x')), app(app(maptlist, f'), xs'')))), app(app(maptlist, app(maptlist, f')), xs))
APP(app(maptlist, app(mapt, f')), app(app(cons, app(leaf, app(node, xs''))), xs)) -> APP(app(cons, app(leaf, app(node, app(app(maptlist, f'), xs'')))), app(app(maptlist, app(mapt, f')), xs))
APP(app(maptlist, app(mapt, f')), app(app(cons, app(leaf, app(leaf, x'))), xs)) -> APP(app(cons, app(leaf, app(leaf, app(f', x')))), app(app(maptlist, app(mapt, f')), xs))
APP(app(maptlist, f), app(app(cons, x), xs)) -> APP(app(maptlist, f), xs)
APP(app(maptlist, f), app(app(cons, x), xs)) -> APP(app(mapt, f), x)
APP(app(mapt, f), app(node, xs)) -> APP(app(maptlist, f), xs)
APP(app(mapt, f), app(leaf, x)) -> APP(f, x)
APP(app(maptlist, f'''), app(app(cons, app(node, xs'')), app(app(cons, x'), xs'''))) -> APP(app(cons, app(node, app(app(maptlist, f'''), xs''))), app(app(cons, app(app(mapt, f'''), x')), app(app(maptlist, f'''), xs''')))

Rules:

app(app(mapt, f), app(leaf, x)) -> app(leaf, app(f, x))
app(app(mapt, f), app(node, xs)) -> app(node, app(app(maptlist, f), xs))
app(app(maptlist, f), nil) -> nil
app(app(maptlist, f), app(app(cons, x), xs)) -> app(app(cons, app(app(mapt, f), x)), app(app(maptlist, f), xs))

Strategy:

innermost

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

APP(app(mapt, f), app(leaf, x)) -> APP(f, x)

15 new Dependency Pairs are created:

APP(app(mapt, app(mapt, f'')), app(leaf, app(leaf, x''))) -> APP(app(mapt, f''), app(leaf, x''))
APP(app(mapt, app(mapt, f'')), app(leaf, app(node, xs''))) -> APP(app(mapt, f''), app(node, xs''))
APP(app(mapt, app(maptlist, f'')), app(leaf, app(app(cons, x''), xs''))) -> APP(app(maptlist, f''), app(app(cons, x''), xs''))
APP(app(mapt, app(maptlist, app(mapt, f'''))), app(leaf, app(app(cons, app(leaf, app(leaf, x'''))), xs''))) -> APP(app(maptlist, app(mapt, f''')), app(app(cons, app(leaf, app(leaf, x'''))), xs''))
APP(app(mapt, app(maptlist, app(mapt, f'''))), app(leaf, app(app(cons, app(leaf, app(node, xs''''))), xs''))) -> APP(app(maptlist, app(mapt, f''')), app(app(cons, app(leaf, app(node, xs''''))), xs''))
APP(app(mapt, app(maptlist, app(maptlist, f'''))), app(leaf, app(app(cons, app(leaf, app(app(cons, x'''), xs''''))), xs''))) -> APP(app(maptlist, app(maptlist, f''')), app(app(cons, app(leaf, app(app(cons, x'''), xs''''))), xs''))
APP(app(mapt, app(maptlist, f''''')), app(leaf, app(app(cons, app(leaf, x'''')), app(app(cons, x'0'), xs'''')))) -> APP(app(maptlist, f'''''), app(app(cons, app(leaf, x'''')), app(app(cons, x'0'), xs'''')))
APP(app(mapt, app(maptlist, f''''')), app(leaf, app(app(cons, app(node, app(app(cons, x'''), xs'''''))), xs''))) -> APP(app(maptlist, f'''''), app(app(cons, app(node, app(app(cons, x'''), xs'''''))), xs''))
APP(app(mapt, app(maptlist, f''''')), app(leaf, app(app(cons, app(node, xs''''0)), app(app(cons, x'''), xs'''''')))) -> APP(app(maptlist, f'''''), app(app(cons, app(node, xs''''0)), app(app(cons, x'''), xs'''''')))
APP(app(mapt, app(maptlist, f''''')), app(leaf, app(app(cons, app(leaf, x''''')), app(app(cons, x''0'), xs'''')))) -> APP(app(maptlist, f'''''), app(app(cons, app(leaf, x''''')), app(app(cons, x''0'), xs'''')))
APP(app(mapt, app(maptlist, f''''')), app(leaf, app(app(cons, app(node, xs''''')), app(app(cons, x''''), xs'''''')))) -> APP(app(maptlist, f'''''), app(app(cons, app(node, xs''''')), app(app(cons, x''''), xs'''''')))
APP(app(mapt, app(maptlist, f''''')), app(leaf, app(app(cons, x''), app(app(cons, app(leaf, x''''')), xs'''')))) -> APP(app(maptlist, f'''''), app(app(cons, x''), app(app(cons, app(leaf, x''''')), xs'''')))
APP(app(mapt, app(maptlist, f''''')), app(leaf, app(app(cons, x''), app(app(cons, app(node, xs''''')), xs'''''')))) -> APP(app(maptlist, f'''''), app(app(cons, x''), app(app(cons, app(node, xs''''')), xs'''''')))
APP(app(mapt, app(maptlist, f''''')), app(leaf, app(app(cons, x''), app(app(cons, x''''), nil)))) -> APP(app(maptlist, f'''''), app(app(cons, x''), app(app(cons, x''''), nil)))
APP(app(mapt, app(maptlist, f''''')), app(leaf, app(app(cons, x''), app(app(cons, x''''0), app(app(cons, x''''''), xs'''))))) -> APP(app(maptlist, f'''''), app(app(cons, x''), app(app(cons, x''''0), 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 6

                 ↳Polynomial Ordering

Dependency Pairs:

APP(app(mapt, app(maptlist, f''''')), app(leaf, app(app(cons, x''), app(app(cons, x''''0), app(app(cons, x''''''), xs'''))))) -> APP(app(maptlist, f'''''), app(app(cons, x''), app(app(cons, x''''0), app(app(cons, x''''''), xs'''))))
APP(app(mapt, app(maptlist, f''''')), app(leaf, app(app(cons, x''), app(app(cons, x''''), nil)))) -> APP(app(maptlist, f'''''), app(app(cons, x''), app(app(cons, x''''), nil)))
APP(app(mapt, app(maptlist, f''''')), app(leaf, app(app(cons, x''), app(app(cons, app(node, xs''''')), xs'''''')))) -> APP(app(maptlist, f'''''), app(app(cons, x''), app(app(cons, app(node, xs''''')), xs'''''')))
APP(app(mapt, app(maptlist, f''''')), app(leaf, app(app(cons, x''), app(app(cons, app(leaf, x''''')), xs'''')))) -> APP(app(maptlist, f'''''), app(app(cons, x''), app(app(cons, app(leaf, x''''')), xs'''')))
APP(app(mapt, app(maptlist, f''''')), app(leaf, app(app(cons, app(node, xs''''')), app(app(cons, x''''), xs'''''')))) -> APP(app(maptlist, f'''''), app(app(cons, app(node, xs''''')), app(app(cons, x''''), xs'''''')))
APP(app(mapt, app(maptlist, f''''')), app(leaf, app(app(cons, app(leaf, x''''')), app(app(cons, x''0'), xs'''')))) -> APP(app(maptlist, f'''''), app(app(cons, app(leaf, x''''')), app(app(cons, x''0'), xs'''')))
APP(app(mapt, app(maptlist, f''''')), app(leaf, app(app(cons, app(node, xs''''0)), app(app(cons, x'''), xs'''''')))) -> APP(app(maptlist, f'''''), app(app(cons, app(node, xs''''0)), app(app(cons, x'''), xs'''''')))
APP(app(mapt, app(maptlist, f''''')), app(leaf, app(app(cons, app(node, app(app(cons, x'''), xs'''''))), xs''))) -> APP(app(maptlist, f'''''), app(app(cons, app(node, app(app(cons, x'''), xs'''''))), xs''))
APP(app(mapt, app(maptlist, f''''')), app(leaf, app(app(cons, app(leaf, x'''')), app(app(cons, x'0'), xs'''')))) -> APP(app(maptlist, f'''''), app(app(cons, app(leaf, x'''')), app(app(cons, x'0'), xs'''')))
APP(app(mapt, app(maptlist, app(maptlist, f'''))), app(leaf, app(app(cons, app(leaf, app(app(cons, x'''), xs''''))), xs''))) -> APP(app(maptlist, app(maptlist, f''')), app(app(cons, app(leaf, app(app(cons, x'''), xs''''))), xs''))
APP(app(mapt, app(maptlist, app(mapt, f'''))), app(leaf, app(app(cons, app(leaf, app(node, xs''''))), xs''))) -> APP(app(maptlist, app(mapt, f''')), app(app(cons, app(leaf, app(node, xs''''))), xs''))
APP(app(mapt, app(maptlist, app(mapt, f'''))), app(leaf, app(app(cons, app(leaf, app(leaf, x'''))), xs''))) -> APP(app(maptlist, app(mapt, f''')), app(app(cons, app(leaf, app(leaf, x'''))), xs''))
APP(app(mapt, app(maptlist, f'')), app(leaf, app(app(cons, x''), xs''))) -> APP(app(maptlist, f''), app(app(cons, x''), xs''))
APP(app(mapt, app(mapt, f'')), app(leaf, app(node, xs''))) -> APP(app(mapt, f''), app(node, xs''))
APP(app(mapt, app(mapt, f'')), app(leaf, app(leaf, x''))) -> APP(app(mapt, f''), app(leaf, x''))
APP(app(maptlist, f'''), app(app(cons, x), app(app(cons, x''), nil))) -> APP(app(cons, app(app(mapt, f'''), x)), app(app(cons, app(app(mapt, f'''), x'')), nil))
APP(app(maptlist, f'''), app(app(cons, x), app(app(cons, app(node, xs')), xs''))) -> APP(app(cons, app(app(mapt, f'''), x)), app(app(cons, app(node, app(app(maptlist, f'''), xs'))), app(app(maptlist, f'''), xs'')))
APP(app(maptlist, f'''), app(app(cons, x), app(app(cons, app(leaf, x''')), xs''))) -> APP(app(cons, app(app(mapt, f'''), x)), app(app(cons, app(leaf, app(f''', x'''))), app(app(maptlist, f'''), xs'')))
APP(app(maptlist, f'''), app(app(cons, app(node, xs')), app(app(cons, x''), xs''))) -> APP(app(cons, app(node, app(app(maptlist, f'''), xs'))), app(app(cons, app(app(mapt, f'''), x'')), app(app(maptlist, f'''), xs'')))
APP(app(maptlist, f'''), app(app(cons, app(leaf, x''')), app(app(cons, x''), xs''))) -> APP(app(cons, app(leaf, app(f''', x'''))), app(app(cons, app(app(mapt, f'''), x'')), app(app(maptlist, f'''), xs'')))
APP(app(maptlist, f'''), app(app(cons, app(node, xs'')), app(app(cons, x'), xs'''))) -> APP(app(cons, app(node, app(app(maptlist, f'''), xs''))), app(app(cons, app(app(mapt, f'''), x')), app(app(maptlist, f'''), xs''')))
APP(app(maptlist, f'''), app(app(cons, app(node, app(app(cons, x'), xs'''))), xs)) -> APP(app(cons, app(node, app(app(cons, app(app(mapt, f'''), x')), app(app(maptlist, f'''), xs''')))), app(app(maptlist, f'''), xs))
APP(app(maptlist, f'''), app(app(cons, app(leaf, x'')), app(app(cons, x'), xs''))) -> APP(app(cons, app(leaf, app(f''', x''))), app(app(cons, app(app(mapt, f'''), x')), app(app(maptlist, f'''), xs'')))
APP(app(maptlist, app(maptlist, f')), app(app(cons, app(leaf, app(app(cons, x'), xs''))), xs)) -> APP(app(cons, app(leaf, app(app(cons, app(app(mapt, f'), x')), app(app(maptlist, f'), xs'')))), app(app(maptlist, app(maptlist, f')), xs))
APP(app(maptlist, app(mapt, f')), app(app(cons, app(leaf, app(node, xs''))), xs)) -> APP(app(cons, app(leaf, app(node, app(app(maptlist, f'), xs'')))), app(app(maptlist, app(mapt, f')), xs))
APP(app(maptlist, app(mapt, f')), app(app(cons, app(leaf, app(leaf, x'))), xs)) -> APP(app(cons, app(leaf, app(leaf, app(f', x')))), app(app(maptlist, app(mapt, f')), xs))
APP(app(maptlist, f), app(app(cons, x), xs)) -> APP(app(maptlist, f), xs)
APP(app(maptlist, f), app(app(cons, x), xs)) -> APP(app(mapt, f), x)
APP(app(mapt, f), app(node, xs)) -> APP(app(maptlist, f), xs)
APP(app(maptlist, f'''), app(app(cons, x), app(app(cons, x''), app(app(cons, x'''), xs')))) -> APP(app(cons, app(app(mapt, f'''), x)), app(app(cons, app(app(mapt, f'''), x'')), app(app(cons, app(app(mapt, f'''), x''')), app(app(maptlist, f'''), xs'))))

Rules:

app(app(mapt, f), app(leaf, x)) -> app(leaf, app(f, x))
app(app(mapt, f), app(node, xs)) -> app(node, app(app(maptlist, f), xs))
app(app(maptlist, f), nil) -> nil
app(app(maptlist, f), app(app(cons, x), xs)) -> app(app(cons, app(app(mapt, f), x)), app(app(maptlist, f), xs))

Strategy:

innermost

The following dependency pairs can be strictly oriented:

APP(app(maptlist, f'''), app(app(cons, x), app(app(cons, x''), nil))) -> APP(app(cons, app(app(mapt, f'''), x)), app(app(cons, app(app(mapt, f'''), x'')), nil))
APP(app(maptlist, f'''), app(app(cons, x), app(app(cons, app(node, xs')), xs''))) -> APP(app(cons, app(app(mapt, f'''), x)), app(app(cons, app(node, app(app(maptlist, f'''), xs'))), app(app(maptlist, f'''), xs'')))
APP(app(maptlist, f'''), app(app(cons, x), app(app(cons, app(leaf, x''')), xs''))) -> APP(app(cons, app(app(mapt, f'''), x)), app(app(cons, app(leaf, app(f''', x'''))), app(app(maptlist, f'''), xs'')))
APP(app(maptlist, f'''), app(app(cons, app(node, xs')), app(app(cons, x''), xs''))) -> APP(app(cons, app(node, app(app(maptlist, f'''), xs'))), app(app(cons, app(app(mapt, f'''), x'')), app(app(maptlist, f'''), xs'')))
APP(app(maptlist, f'''), app(app(cons, app(leaf, x''')), app(app(cons, x''), xs''))) -> APP(app(cons, app(leaf, app(f''', x'''))), app(app(cons, app(app(mapt, f'''), x'')), app(app(maptlist, f'''), xs'')))
APP(app(maptlist, f'''), app(app(cons, app(node, xs'')), app(app(cons, x'), xs'''))) -> APP(app(cons, app(node, app(app(maptlist, f'''), xs''))), app(app(cons, app(app(mapt, f'''), x')), app(app(maptlist, f'''), xs''')))
APP(app(maptlist, f'''), app(app(cons, app(node, app(app(cons, x'), xs'''))), xs)) -> APP(app(cons, app(node, app(app(cons, app(app(mapt, f'''), x')), app(app(maptlist, f'''), xs''')))), app(app(maptlist, f'''), xs))
APP(app(maptlist, f'''), app(app(cons, app(leaf, x'')), app(app(cons, x'), xs''))) -> APP(app(cons, app(leaf, app(f''', x''))), app(app(cons, app(app(mapt, f'''), x')), app(app(maptlist, f'''), xs'')))
APP(app(maptlist, app(maptlist, f')), app(app(cons, app(leaf, app(app(cons, x'), xs''))), xs)) -> APP(app(cons, app(leaf, app(app(cons, app(app(mapt, f'), x')), app(app(maptlist, f'), xs'')))), app(app(maptlist, app(maptlist, f')), xs))
APP(app(maptlist, app(mapt, f')), app(app(cons, app(leaf, app(node, xs''))), xs)) -> APP(app(cons, app(leaf, app(node, app(app(maptlist, f'), xs'')))), app(app(maptlist, app(mapt, f')), xs))
APP(app(maptlist, app(mapt, f')), app(app(cons, app(leaf, app(leaf, x'))), xs)) -> APP(app(cons, app(leaf, app(leaf, app(f', x')))), app(app(maptlist, app(mapt, f')), xs))
APP(app(maptlist, f'''), app(app(cons, x), app(app(cons, x''), app(app(cons, x'''), xs')))) -> APP(app(cons, app(app(mapt, f'''), x)), app(app(cons, app(app(mapt, f'''), x'')), app(app(cons, app(app(mapt, f'''), x''')), app(app(maptlist, f'''), xs'))))

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

app(app(mapt, f), app(leaf, x)) -> app(leaf, app(f, x))
app(app(mapt, f), app(node, xs)) -> app(node, app(app(maptlist, f), xs))
app(app(maptlist, f), nil) -> nil
app(app(maptlist, f), app(app(cons, x), xs)) -> app(app(cons, app(app(mapt, f), x)), app(app(maptlist, f), xs))

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(cons) = 0

POL(mapt) = 1

POL(maptlist) = 1

POL(nil) = 0

POL(node) = 0

POL(leaf) = 0

POL(app(x₁, x₂)) = x₁

POL(APP(x₁, x₂)) = x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 7

                 ↳Polynomial Ordering

Dependency Pairs:

Rules:

app(app(mapt, f), app(leaf, x)) -> app(leaf, app(f, x))
app(app(mapt, f), app(node, xs)) -> app(node, app(app(maptlist, f), xs))
app(app(maptlist, f), nil) -> nil
app(app(maptlist, f), app(app(cons, x), xs)) -> app(app(cons, app(app(mapt, f), x)), app(app(maptlist, f), xs))

Strategy:

innermost

The following dependency pairs can be strictly oriented:

APP(app(mapt, app(maptlist, f''''')), app(leaf, app(app(cons, x''), app(app(cons, x''''0), app(app(cons, x''''''), xs'''))))) -> APP(app(maptlist, f'''''), app(app(cons, x''), app(app(cons, x''''0), app(app(cons, x''''''), xs'''))))
APP(app(mapt, app(maptlist, f''''')), app(leaf, app(app(cons, x''), app(app(cons, x''''), nil)))) -> APP(app(maptlist, f'''''), app(app(cons, x''), app(app(cons, x''''), nil)))
APP(app(mapt, app(maptlist, f''''')), app(leaf, app(app(cons, x''), app(app(cons, app(node, xs''''')), xs'''''')))) -> APP(app(maptlist, f'''''), app(app(cons, x''), app(app(cons, app(node, xs''''')), xs'''''')))
APP(app(mapt, app(maptlist, f''''')), app(leaf, app(app(cons, x''), app(app(cons, app(leaf, x''''')), xs'''')))) -> APP(app(maptlist, f'''''), app(app(cons, x''), app(app(cons, app(leaf, x''''')), xs'''')))
APP(app(mapt, app(maptlist, f''''')), app(leaf, app(app(cons, app(node, xs''''')), app(app(cons, x''''), xs'''''')))) -> APP(app(maptlist, f'''''), app(app(cons, app(node, xs''''')), app(app(cons, x''''), xs'''''')))
APP(app(mapt, app(maptlist, f''''')), app(leaf, app(app(cons, app(leaf, x''''')), app(app(cons, x''0'), xs'''')))) -> APP(app(maptlist, f'''''), app(app(cons, app(leaf, x''''')), app(app(cons, x''0'), xs'''')))
APP(app(mapt, app(maptlist, f''''')), app(leaf, app(app(cons, app(node, xs''''0)), app(app(cons, x'''), xs'''''')))) -> APP(app(maptlist, f'''''), app(app(cons, app(node, xs''''0)), app(app(cons, x'''), xs'''''')))
APP(app(mapt, app(maptlist, f''''')), app(leaf, app(app(cons, app(node, app(app(cons, x'''), xs'''''))), xs''))) -> APP(app(maptlist, f'''''), app(app(cons, app(node, app(app(cons, x'''), xs'''''))), xs''))
APP(app(mapt, app(maptlist, f''''')), app(leaf, app(app(cons, app(leaf, x'''')), app(app(cons, x'0'), xs'''')))) -> APP(app(maptlist, f'''''), app(app(cons, app(leaf, x'''')), app(app(cons, x'0'), xs'''')))
APP(app(mapt, app(maptlist, app(maptlist, f'''))), app(leaf, app(app(cons, app(leaf, app(app(cons, x'''), xs''''))), xs''))) -> APP(app(maptlist, app(maptlist, f''')), app(app(cons, app(leaf, app(app(cons, x'''), xs''''))), xs''))
APP(app(mapt, app(maptlist, app(mapt, f'''))), app(leaf, app(app(cons, app(leaf, app(node, xs''''))), xs''))) -> APP(app(maptlist, app(mapt, f''')), app(app(cons, app(leaf, app(node, xs''''))), xs''))
APP(app(mapt, app(maptlist, app(mapt, f'''))), app(leaf, app(app(cons, app(leaf, app(leaf, x'''))), xs''))) -> APP(app(maptlist, app(mapt, f''')), app(app(cons, app(leaf, app(leaf, x'''))), xs''))
APP(app(mapt, app(maptlist, f'')), app(leaf, app(app(cons, x''), xs''))) -> APP(app(maptlist, f''), app(app(cons, x''), xs''))
APP(app(mapt, app(mapt, f'')), app(leaf, app(node, xs''))) -> APP(app(mapt, f''), app(node, xs''))
APP(app(mapt, app(mapt, f'')), app(leaf, app(leaf, x''))) -> APP(app(mapt, f''), app(leaf, x''))

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

app(app(mapt, f), app(leaf, x)) -> app(leaf, app(f, x))
app(app(mapt, f), app(node, xs)) -> app(node, app(app(maptlist, f), xs))
app(app(maptlist, f), nil) -> nil
app(app(maptlist, f), app(app(cons, x), xs)) -> app(app(cons, app(app(mapt, f), x)), app(app(maptlist, f), xs))

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(cons) = 0

POL(mapt) = 0

POL(maptlist) = 0

POL(nil) = 0

POL(node) = 0

POL(leaf) = 0

POL(app(x₁, x₂)) = 1 + x₂

POL(APP(x₁, x₂)) = x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 8

                 ↳Polynomial Ordering

Dependency Pairs:

APP(app(maptlist, f), app(app(cons, x), xs)) -> APP(app(maptlist, f), xs)
APP(app(maptlist, f), app(app(cons, x), xs)) -> APP(app(mapt, f), x)
APP(app(mapt, f), app(node, xs)) -> APP(app(maptlist, f), xs)

Rules:

app(app(mapt, f), app(leaf, x)) -> app(leaf, app(f, x))
app(app(mapt, f), app(node, xs)) -> app(node, app(app(maptlist, f), xs))
app(app(maptlist, f), nil) -> nil
app(app(maptlist, f), app(app(cons, x), xs)) -> app(app(cons, app(app(mapt, f), x)), app(app(maptlist, f), xs))

Strategy:

innermost

The following dependency pair can be strictly oriented:

APP(app(mapt, f), app(node, xs)) -> APP(app(maptlist, f), xs)

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(cons) = 0

POL(mapt) = 0

POL(maptlist) = 0

POL(nil) = 0

POL(node) = 1

POL(leaf) = 0

POL(app(x₁, x₂)) = x₁ + x₂

POL(APP(x₁, x₂)) = x₂

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 9

                 ↳Narrowing Transformation

Dependency Pairs:

APP(app(maptlist, f), app(app(cons, x), xs)) -> APP(app(maptlist, f), xs)
APP(app(maptlist, f), app(app(cons, x), xs)) -> APP(app(mapt, f), x)

Rules:

app(app(mapt, f), app(leaf, x)) -> app(leaf, app(f, x))
app(app(mapt, f), app(node, xs)) -> app(node, app(app(maptlist, f), xs))
app(app(maptlist, f), nil) -> nil
app(app(maptlist, f), app(app(cons, x), xs)) -> app(app(cons, app(app(mapt, f), x)), app(app(maptlist, f), xs))

Strategy:

innermost

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

APP(app(maptlist, f), app(app(cons, x), xs)) -> APP(app(mapt, f), x)

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 10

                 ↳Polynomial Ordering

Dependency Pair:

APP(app(maptlist, f), app(app(cons, x), xs)) -> APP(app(maptlist, f), xs)

Rules:

app(app(mapt, f), app(leaf, x)) -> app(leaf, app(f, x))
app(app(mapt, f), app(node, xs)) -> app(node, app(app(maptlist, f), xs))
app(app(maptlist, f), nil) -> nil
app(app(maptlist, f), app(app(cons, x), xs)) -> app(app(cons, app(app(mapt, f), x)), app(app(maptlist, f), xs))

Strategy:

innermost

The following dependency pair can be strictly oriented:

APP(app(maptlist, f), app(app(cons, x), xs)) -> APP(app(maptlist, f), xs)

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(cons) = 0

POL(mapt) = 0

POL(maptlist) = 0

POL(nil) = 0

POL(node) = 0

POL(leaf) = 0

POL(app(x₁, x₂)) = 1 + x₂

POL(APP(x₁, x₂)) = x₂

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 11

                 ↳Dependency Graph

Dependency Pair:

Rules:

app(app(mapt, f), app(leaf, x)) -> app(leaf, app(f, x))
app(app(mapt, f), app(node, xs)) -> app(node, app(app(maptlist, f), xs))
app(app(maptlist, f), nil) -> nil
app(app(maptlist, f), app(app(cons, x), xs)) -> app(app(cons, app(app(mapt, f), x)), app(app(maptlist, f), xs))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

Innermost Termination of R successfully shown.
Duration:
0:33 minutes

POL(cons)	= 0
POL(mapt)	= 1
POL(maptlist)	= 1
POL(nil)	= 0
POL(node)	= 0
POL(leaf)	= 0
POL(app(x₁, x₂))	= x₁
POL(APP(x₁, x₂))	= x₁