Term Rewriting System R:
[f, x, h, t]
app(app(twice, f), x) -> app(f, app(f, x))
app(app(map, f), nil) -> nil
app(app(map, f), app(app(cons, h), t)) -> app(app(cons, app(f, h)), app(app(map, f), t))
app(app(fmap, nil), x) -> nil
app(app(fmap, app(app(cons, f), tf)), x) -> app(app(cons, app(f, x)), app(app(fmap, tf), x))

Innermost Termination of R to be shown.



   R
Dependency Pair Analysis



R contains the following Dependency Pairs:

APP(app(twice, f), x) -> APP(f, app(f, x))
APP(app(twice, f), x) -> APP(f, x)
APP(app(map, f), app(app(cons, h), t)) -> APP(app(cons, app(f, h)), app(app(map, f), t))
APP(app(map, f), app(app(cons, h), t)) -> APP(cons, app(f, h))
APP(app(map, f), app(app(cons, h), t)) -> APP(f, h)
APP(app(map, f), app(app(cons, h), t)) -> APP(app(map, f), t)
APP(app(fmap, app(app(cons, f), tf)), x) -> APP(app(cons, app(f, x)), app(app(fmap, tf), x))
APP(app(fmap, app(app(cons, f), tf)), x) -> APP(cons, app(f, x))
APP(app(fmap, app(app(cons, f), tf)), x) -> APP(f, x)
APP(app(fmap, app(app(cons, f), tf)), x) -> APP(app(fmap, tf), x)
APP(app(fmap, app(app(cons, f), tf)), x) -> APP(fmap, tf)

Furthermore, R contains one SCC.


   R
DPs
       →DP Problem 1
Narrowing Transformation


Dependency Pairs:

APP(app(fmap, app(app(cons, f), tf)), x) -> APP(app(fmap, tf), x)
APP(app(fmap, app(app(cons, f), tf)), x) -> APP(f, x)
APP(app(fmap, app(app(cons, f), tf)), x) -> APP(app(cons, app(f, x)), app(app(fmap, tf), x))
APP(app(map, f), app(app(cons, h), t)) -> APP(app(map, f), t)
APP(app(map, f), app(app(cons, h), t)) -> APP(f, h)
APP(app(map, f), app(app(cons, h), t)) -> APP(app(cons, app(f, h)), app(app(map, f), t))
APP(app(twice, f), x) -> APP(f, x)
APP(app(twice, f), x) -> APP(f, app(f, x))


Rules:


app(app(twice, f), x) -> app(f, app(f, x))
app(app(map, f), nil) -> nil
app(app(map, f), app(app(cons, h), t)) -> app(app(cons, app(f, h)), app(app(map, f), t))
app(app(fmap, nil), x) -> nil
app(app(fmap, app(app(cons, f), tf)), x) -> app(app(cons, app(f, x)), app(app(fmap, tf), x))


Strategy:

innermost




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

APP(app(fmap, app(app(cons, f), tf)), x) -> APP(app(fmap, tf), x)
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(fmap, app(app(cons, f), tf)), x) -> APP(app(cons, app(f, x)), app(app(fmap, tf), x))
APP(app(map, f), app(app(cons, h), t)) -> APP(app(map, f), t)
APP(app(map, f), app(app(cons, h), t)) -> APP(f, h)
APP(app(map, f), app(app(cons, h), t)) -> APP(app(cons, app(f, h)), app(app(map, f), t))
APP(app(twice, f), x) -> APP(f, x)
APP(app(twice, f), x) -> APP(f, app(f, x))
APP(app(fmap, app(app(cons, f), tf)), x) -> APP(f, x)


Rules:


app(app(twice, f), x) -> app(f, app(f, x))
app(app(map, f), nil) -> nil
app(app(map, f), app(app(cons, h), t)) -> app(app(cons, app(f, h)), app(app(map, f), t))
app(app(fmap, nil), x) -> nil
app(app(fmap, app(app(cons, f), tf)), x) -> app(app(cons, app(f, x)), app(app(fmap, tf), x))


Strategy:

innermost




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

APP(app(map, f), app(app(cons, h), t)) -> APP(app(cons, app(f, h)), app(app(map, f), t))
seven new Dependency Pairs are created:

APP(app(map, app(twice, f'')), app(app(cons, h'), t)) -> APP(app(cons, app(f'', app(f'', h'))), app(app(map, app(twice, f'')), t))
APP(app(map, app(map, f'')), app(app(cons, nil), t)) -> APP(app(cons, nil), app(app(map, app(map, f'')), t))
APP(app(map, app(map, f'')), app(app(cons, app(app(cons, h''), t'')), t)) -> APP(app(cons, app(app(cons, app(f'', h'')), app(app(map, f''), t''))), app(app(map, app(map, f'')), t))
APP(app(map, app(fmap, nil)), app(app(cons, h'), t)) -> APP(app(cons, nil), app(app(map, app(fmap, nil)), t))
APP(app(map, app(fmap, app(app(cons, f''), tf))), app(app(cons, h'), t)) -> APP(app(cons, app(app(cons, app(f'', h')), app(app(fmap, tf), h'))), app(app(map, app(fmap, app(app(cons, f''), tf))), t))
APP(app(map, f''), app(app(cons, h), nil)) -> APP(app(cons, app(f'', h)), nil)
APP(app(map, f''), app(app(cons, h), app(app(cons, h''), t''))) -> APP(app(cons, app(f'', h)), app(app(cons, app(f'', h'')), app(app(map, f''), t'')))

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(map, f''), app(app(cons, h), app(app(cons, h''), t''))) -> APP(app(cons, app(f'', h)), app(app(cons, app(f'', h'')), app(app(map, f''), t'')))
APP(app(map, f''), app(app(cons, h), nil)) -> APP(app(cons, app(f'', h)), nil)
APP(app(map, app(fmap, app(app(cons, f''), tf))), app(app(cons, h'), t)) -> APP(app(cons, app(app(cons, app(f'', h')), app(app(fmap, tf), h'))), app(app(map, app(fmap, app(app(cons, f''), tf))), t))
APP(app(map, app(fmap, nil)), app(app(cons, h'), t)) -> APP(app(cons, nil), app(app(map, app(fmap, nil)), t))
APP(app(map, app(map, f'')), app(app(cons, app(app(cons, h''), t'')), t)) -> APP(app(cons, app(app(cons, app(f'', h'')), app(app(map, f''), t''))), app(app(map, app(map, f'')), t))
APP(app(fmap, app(app(cons, f), tf)), x) -> APP(f, x)
APP(app(map, app(twice, f'')), app(app(cons, h'), t)) -> APP(app(cons, app(f'', app(f'', h'))), app(app(map, app(twice, f'')), t))
APP(app(map, f), app(app(cons, h), t)) -> APP(app(map, f), t)
APP(app(map, f), app(app(cons, h), t)) -> APP(f, h)
APP(app(twice, f), x) -> APP(f, x)
APP(app(twice, f), x) -> APP(f, app(f, x))
APP(app(fmap, app(app(cons, f), tf)), x) -> APP(app(cons, app(f, x)), app(app(fmap, tf), x))


Rules:


app(app(twice, f), x) -> app(f, app(f, x))
app(app(map, f), nil) -> nil
app(app(map, f), app(app(cons, h), t)) -> app(app(cons, app(f, h)), app(app(map, f), t))
app(app(fmap, nil), x) -> nil
app(app(fmap, app(app(cons, f), tf)), x) -> app(app(cons, app(f, x)), app(app(fmap, tf), x))


Strategy:

innermost




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

APP(app(fmap, app(app(cons, f), tf)), x) -> APP(app(cons, app(f, x)), app(app(fmap, tf), x))
five new Dependency Pairs are created:

APP(app(fmap, app(app(cons, app(twice, f'')), tf)), x'') -> APP(app(cons, app(f'', app(f'', x''))), app(app(fmap, tf), x''))
APP(app(fmap, app(app(cons, app(map, f'')), tf)), nil) -> APP(app(cons, nil), app(app(fmap, tf), nil))
APP(app(fmap, app(app(cons, app(map, f'')), tf)), app(app(cons, h'), t')) -> APP(app(cons, app(app(cons, app(f'', h')), app(app(map, f''), t'))), app(app(fmap, tf), app(app(cons, h'), t')))
APP(app(fmap, app(app(cons, app(fmap, nil)), tf)), x'') -> APP(app(cons, nil), app(app(fmap, tf), x''))
APP(app(fmap, app(app(cons, app(fmap, app(app(cons, f''), tf))), tf)), x'') -> APP(app(cons, app(app(cons, app(f'', x'')), app(app(fmap, tf), x''))), app(app(fmap, tf), x''))

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(fmap, app(app(cons, app(fmap, app(app(cons, f''), tf))), tf)), x'') -> APP(app(cons, app(app(cons, app(f'', x'')), app(app(fmap, tf), x''))), app(app(fmap, tf), x''))
APP(app(fmap, app(app(cons, app(fmap, nil)), tf)), x'') -> APP(app(cons, nil), app(app(fmap, tf), x''))
APP(app(fmap, app(app(cons, app(map, f'')), tf)), app(app(cons, h'), t')) -> APP(app(cons, app(app(cons, app(f'', h')), app(app(map, f''), t'))), app(app(fmap, tf), app(app(cons, h'), t')))
APP(app(fmap, app(app(cons, app(map, f'')), tf)), nil) -> APP(app(cons, nil), app(app(fmap, tf), nil))
APP(app(fmap, app(app(cons, app(twice, f'')), tf)), x'') -> APP(app(cons, app(f'', app(f'', x''))), app(app(fmap, tf), x''))
APP(app(map, f''), app(app(cons, h), nil)) -> APP(app(cons, app(f'', h)), nil)
APP(app(map, app(fmap, app(app(cons, f''), tf))), app(app(cons, h'), t)) -> APP(app(cons, app(app(cons, app(f'', h')), app(app(fmap, tf), h'))), app(app(map, app(fmap, app(app(cons, f''), tf))), t))
APP(app(map, app(fmap, nil)), app(app(cons, h'), t)) -> APP(app(cons, nil), app(app(map, app(fmap, nil)), t))
APP(app(map, app(map, f'')), app(app(cons, app(app(cons, h''), t'')), t)) -> APP(app(cons, app(app(cons, app(f'', h'')), app(app(map, f''), t''))), app(app(map, app(map, f'')), t))
APP(app(fmap, app(app(cons, f), tf)), x) -> APP(f, x)
APP(app(map, app(twice, f'')), app(app(cons, h'), t)) -> APP(app(cons, app(f'', app(f'', h'))), app(app(map, app(twice, f'')), t))
APP(app(map, f), app(app(cons, h), t)) -> APP(app(map, f), t)
APP(app(map, f), app(app(cons, h), t)) -> APP(f, h)
APP(app(twice, f), x) -> APP(f, x)
APP(app(twice, f), x) -> APP(f, app(f, x))
APP(app(map, f''), app(app(cons, h), app(app(cons, h''), t''))) -> APP(app(cons, app(f'', h)), app(app(cons, app(f'', h'')), app(app(map, f''), t'')))


Rules:


app(app(twice, f), x) -> app(f, app(f, x))
app(app(map, f), nil) -> nil
app(app(map, f), app(app(cons, h), t)) -> app(app(cons, app(f, h)), app(app(map, f), t))
app(app(fmap, nil), x) -> nil
app(app(fmap, app(app(cons, f), tf)), x) -> app(app(cons, app(f, x)), app(app(fmap, tf), x))


Strategy:

innermost




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

APP(app(fmap, app(app(cons, app(map, f'')), tf)), nil) -> APP(app(cons, nil), app(app(fmap, tf), nil))
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(fmap, app(app(cons, app(fmap, nil)), tf)), x'') -> APP(app(cons, nil), app(app(fmap, tf), x''))
APP(app(fmap, app(app(cons, app(map, f'')), tf)), app(app(cons, h'), t')) -> APP(app(cons, app(app(cons, app(f'', h')), app(app(map, f''), t'))), app(app(fmap, tf), app(app(cons, h'), t')))
APP(app(map, f''), app(app(cons, h), app(app(cons, h''), t''))) -> APP(app(cons, app(f'', h)), app(app(cons, app(f'', h'')), app(app(map, f''), t'')))
APP(app(fmap, app(app(cons, app(twice, f'')), tf)), x'') -> APP(app(cons, app(f'', app(f'', x''))), app(app(fmap, tf), x''))
APP(app(map, f''), app(app(cons, h), nil)) -> APP(app(cons, app(f'', h)), nil)
APP(app(map, app(fmap, app(app(cons, f''), tf))), app(app(cons, h'), t)) -> APP(app(cons, app(app(cons, app(f'', h')), app(app(fmap, tf), h'))), app(app(map, app(fmap, app(app(cons, f''), tf))), t))
APP(app(map, app(fmap, nil)), app(app(cons, h'), t)) -> APP(app(cons, nil), app(app(map, app(fmap, nil)), t))
APP(app(map, app(map, f'')), app(app(cons, app(app(cons, h''), t'')), t)) -> APP(app(cons, app(app(cons, app(f'', h'')), app(app(map, f''), t''))), app(app(map, app(map, f'')), t))
APP(app(fmap, app(app(cons, f), tf)), x) -> APP(f, x)
APP(app(map, app(twice, f'')), app(app(cons, h'), t)) -> APP(app(cons, app(f'', app(f'', h'))), app(app(map, app(twice, f'')), t))
APP(app(map, f), app(app(cons, h), t)) -> APP(app(map, f), t)
APP(app(map, f), app(app(cons, h), t)) -> APP(f, h)
APP(app(twice, f), x) -> APP(f, x)
APP(app(twice, f), x) -> APP(f, app(f, x))
APP(app(fmap, app(app(cons, app(fmap, app(app(cons, f''), tf))), tf)), x'') -> APP(app(cons, app(app(cons, app(f'', x'')), app(app(fmap, tf), x''))), app(app(fmap, tf), x''))


Rules:


app(app(twice, f), x) -> app(f, app(f, x))
app(app(map, f), nil) -> nil
app(app(map, f), app(app(cons, h), t)) -> app(app(cons, app(f, h)), app(app(map, f), t))
app(app(fmap, nil), x) -> nil
app(app(fmap, app(app(cons, f), tf)), x) -> app(app(cons, app(f, x)), app(app(fmap, tf), x))


Strategy:

innermost




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

APP(app(fmap, app(app(cons, app(fmap, nil)), tf)), x'') -> APP(app(cons, nil), app(app(fmap, tf), x''))
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
Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

APP(app(fmap, app(app(cons, app(fmap, app(app(cons, f''), tf))), tf)), x'') -> APP(app(cons, app(app(cons, app(f'', x'')), app(app(fmap, tf), x''))), app(app(fmap, tf), x''))
APP(app(map, f''), app(app(cons, h), app(app(cons, h''), t''))) -> APP(app(cons, app(f'', h)), app(app(cons, app(f'', h'')), app(app(map, f''), t'')))
APP(app(fmap, app(app(cons, app(twice, f'')), tf)), x'') -> APP(app(cons, app(f'', app(f'', x''))), app(app(fmap, tf), x''))
APP(app(map, f''), app(app(cons, h), nil)) -> APP(app(cons, app(f'', h)), nil)
APP(app(map, app(fmap, app(app(cons, f''), tf))), app(app(cons, h'), t)) -> APP(app(cons, app(app(cons, app(f'', h')), app(app(fmap, tf), h'))), app(app(map, app(fmap, app(app(cons, f''), tf))), t))
APP(app(map, app(fmap, nil)), app(app(cons, h'), t)) -> APP(app(cons, nil), app(app(map, app(fmap, nil)), t))
APP(app(map, app(map, f'')), app(app(cons, app(app(cons, h''), t'')), t)) -> APP(app(cons, app(app(cons, app(f'', h'')), app(app(map, f''), t''))), app(app(map, app(map, f'')), t))
APP(app(fmap, app(app(cons, f), tf)), x) -> APP(f, x)
APP(app(map, app(twice, f'')), app(app(cons, h'), t)) -> APP(app(cons, app(f'', app(f'', h'))), app(app(map, app(twice, f'')), t))
APP(app(map, f), app(app(cons, h), t)) -> APP(app(map, f), t)
APP(app(map, f), app(app(cons, h), t)) -> APP(f, h)
APP(app(twice, f), x) -> APP(f, x)
APP(app(twice, f), x) -> APP(f, app(f, x))
APP(app(fmap, app(app(cons, app(map, f'')), tf)), app(app(cons, h'), t')) -> APP(app(cons, app(app(cons, app(f'', h')), app(app(map, f''), t'))), app(app(fmap, tf), app(app(cons, h'), t')))


Rules:


app(app(twice, f), x) -> app(f, app(f, x))
app(app(map, f), nil) -> nil
app(app(map, f), app(app(cons, h), t)) -> app(app(cons, app(f, h)), app(app(map, f), t))
app(app(fmap, nil), x) -> nil
app(app(fmap, app(app(cons, f), tf)), x) -> app(app(cons, app(f, x)), app(app(fmap, tf), x))


Strategy:

innermost



Innermost Termination of R could not be shown.
Duration:
0:30 minutes