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. 



   R
     ↳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. 


   R
     ↳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: 



   R
     ↳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: 



   R
     ↳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: 



   R
     ↳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: 



   R
     ↳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: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 6
                 ↳Forward Instantiation 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(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 Forward Instantiation SCC transformation can be performed.
As a result of transforming the rule

APP(app(filter, f), app(app(cons, x), xs)) -> APP(f, x)
six new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 7
                 ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

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



Innermost Termination of R could not be shown.
Duration:
0:12 minutes