Term Rewriting System R:
[l, h, t, f, l1, l2, l3]
app(app(append, nil), l) -> l
app(app(append, app(app(cons, h), t)), l) -> app(app(cons, h), app(app(append, t), l))
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(append, app(app(append, l1), l2)), l3) -> app(app(append, l1), app(app(append, l2), l3))
app(app(map, f), app(app(append, l1), l2)) -> app(app(append, app(app(map, f), l1)), app(app(map, f), l2))

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

APP(app(append, app(app(cons, h), t)), l) -> APP(app(cons, h), app(app(append, t), l))
APP(app(append, app(app(cons, h), t)), l) -> APP(app(append, t), l)
APP(app(append, app(app(cons, h), t)), l) -> APP(append, t)
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(append, app(app(append, l1), l2)), l3) -> APP(app(append, l1), app(app(append, l2), l3))
APP(app(append, app(app(append, l1), l2)), l3) -> APP(app(append, l2), l3)
APP(app(append, app(app(append, l1), l2)), l3) -> APP(append, l2)
APP(app(map, f), app(app(append, l1), l2)) -> APP(app(append, app(app(map, f), l1)), app(app(map, f), l2))
APP(app(map, f), app(app(append, l1), l2)) -> APP(append, app(app(map, f), l1))
APP(app(map, f), app(app(append, l1), l2)) -> APP(app(map, f), l1)
APP(app(map, f), app(app(append, l1), l2)) -> APP(app(map, f), l2)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Narrowing Transformation


Dependency Pairs:

APP(app(map, f), app(app(append, l1), l2)) -> APP(app(map, f), l2)
APP(app(append, app(app(append, l1), l2)), l3) -> APP(app(append, l2), l3)
APP(app(map, f), app(app(append, l1), l2)) -> APP(app(append, app(app(map, f), l1)), app(app(map, f), l2))
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(append, app(app(cons, h), t)), l) -> APP(app(append, t), l)


Rules:


app(app(append, nil), l) -> l
app(app(append, app(app(cons, h), t)), l) -> app(app(cons, h), app(app(append, t), l))
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(append, app(app(append, l1), l2)), l3) -> app(app(append, l1), app(app(append, l2), l3))
app(app(map, f), app(app(append, l1), l2)) -> app(app(append, app(app(map, f), l1)), app(app(map, f), l2))


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))
nine new Dependency Pairs are created:

APP(app(map, app(append, nil)), app(app(cons, h'), t)) -> APP(app(cons, h'), app(app(map, app(append, nil)), t))
APP(app(map, app(append, app(app(cons, h'''), t''))), app(app(cons, h''), t)) -> APP(app(cons, app(app(cons, h'''), app(app(append, t''), h''))), app(app(map, app(append, app(app(cons, h'''), t''))), 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(append, app(app(append, l1'), l2'))), app(app(cons, h'), t)) -> APP(app(cons, app(app(append, l1'), app(app(append, l2'), h'))), app(app(map, app(append, app(app(append, l1'), l2'))), t))
APP(app(map, app(map, f'')), app(app(cons, app(app(append, l1'), l2')), t)) -> APP(app(cons, app(app(append, app(app(map, f''), l1')), app(app(map, f''), l2'))), app(app(map, app(map, f'')), 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'')))
APP(app(map, f''), app(app(cons, h), app(app(append, l1'), l2'))) -> APP(app(cons, app(f'', h)), app(app(append, app(app(map, f''), l1')), app(app(map, f''), l2')))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Narrowing Transformation


Dependency Pairs:

APP(app(map, f''), app(app(cons, h), app(app(append, l1'), l2'))) -> APP(app(cons, app(f'', h)), app(app(append, app(app(map, f''), l1')), app(app(map, f''), l2')))
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(map, f'')), app(app(cons, app(app(append, l1'), l2')), t)) -> APP(app(cons, app(app(append, app(app(map, f''), l1')), app(app(map, f''), l2'))), app(app(map, app(map, f'')), t))
APP(app(map, app(append, app(app(append, l1'), l2'))), app(app(cons, h'), t)) -> APP(app(cons, app(app(append, l1'), app(app(append, l2'), h'))), app(app(map, app(append, app(app(append, l1'), l2'))), 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(append, app(app(cons, h'''), t''))), app(app(cons, h''), t)) -> APP(app(cons, app(app(cons, h'''), app(app(append, t''), h''))), app(app(map, app(append, app(app(cons, h'''), t''))), t))
APP(app(append, app(app(append, l1), l2)), l3) -> APP(app(append, l2), l3)
APP(app(map, f), app(app(append, l1), l2)) -> APP(app(append, app(app(map, f), l1)), app(app(map, f), l2))
APP(app(map, f), app(app(cons, h), t)) -> APP(app(map, f), t)
APP(app(append, app(app(cons, h), t)), l) -> APP(app(append, t), l)
APP(app(map, f), app(app(cons, h), t)) -> APP(f, h)
APP(app(map, f), app(app(append, l1), l2)) -> APP(app(map, f), l2)


Rules:


app(app(append, nil), l) -> l
app(app(append, app(app(cons, h), t)), l) -> app(app(cons, h), app(app(append, t), l))
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(append, app(app(append, l1), l2)), l3) -> app(app(append, l1), app(app(append, l2), l3))
app(app(map, f), app(app(append, l1), l2)) -> app(app(append, app(app(map, f), l1)), app(app(map, f), l2))


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(append, l1), l2)) -> APP(app(append, app(app(map, f), l1)), app(app(map, f), l2))
three new Dependency Pairs are created:

APP(app(map, f''), app(app(append, l1), nil)) -> APP(app(append, app(app(map, f''), l1)), nil)
APP(app(map, f''), app(app(append, l1), app(app(cons, h'), t'))) -> APP(app(append, app(app(map, f''), l1)), app(app(cons, app(f'', h')), app(app(map, f''), t')))
APP(app(map, f''), app(app(append, l1), app(app(append, l1''), l2''))) -> APP(app(append, app(app(map, f''), l1)), app(app(append, app(app(map, f''), l1'')), app(app(map, f''), l2'')))

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(append, l1), app(app(append, l1''), l2''))) -> APP(app(append, app(app(map, f''), l1)), app(app(append, app(app(map, f''), l1'')), app(app(map, f''), l2'')))
APP(app(map, f''), app(app(append, l1), app(app(cons, h'), t'))) -> APP(app(append, app(app(map, f''), l1)), app(app(cons, app(f'', h')), app(app(map, f''), t')))
APP(app(map, f''), app(app(append, l1), nil)) -> APP(app(append, app(app(map, f''), l1)), 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'')))
APP(app(map, f''), app(app(cons, h), nil)) -> APP(app(cons, app(f'', h)), nil)
APP(app(map, app(map, f'')), app(app(cons, app(app(append, l1'), l2')), t)) -> APP(app(cons, app(app(append, app(app(map, f''), l1')), app(app(map, f''), l2'))), app(app(map, app(map, f'')), t))
APP(app(map, app(append, app(app(append, l1'), l2'))), app(app(cons, h'), t)) -> APP(app(cons, app(app(append, l1'), app(app(append, l2'), h'))), app(app(map, app(append, app(app(append, l1'), l2'))), 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(append, app(app(append, l1), l2)), l3) -> APP(app(append, l2), l3)
APP(app(map, app(append, app(app(cons, h'''), t''))), app(app(cons, h''), t)) -> APP(app(cons, app(app(cons, h'''), app(app(append, t''), h''))), app(app(map, app(append, app(app(cons, h'''), t''))), t))
APP(app(map, f), app(app(append, l1), l2)) -> APP(app(map, f), l2)
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(append, app(app(cons, h), t)), l) -> APP(app(append, t), l)
APP(app(map, f''), app(app(cons, h), app(app(append, l1'), l2'))) -> APP(app(cons, app(f'', h)), app(app(append, app(app(map, f''), l1')), app(app(map, f''), l2')))


Rules:


app(app(append, nil), l) -> l
app(app(append, app(app(cons, h), t)), l) -> app(app(cons, h), app(app(append, t), l))
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(append, app(app(append, l1), l2)), l3) -> app(app(append, l1), app(app(append, l2), l3))
app(app(map, f), app(app(append, l1), l2)) -> app(app(append, app(app(map, f), l1)), app(app(map, f), l2))


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(append, l1), nil)) -> APP(app(append, app(app(map, f''), l1)), 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 4
                 ↳Forward Instantiation Transformation


Dependency Pairs:

APP(app(map, f''), app(app(append, l1), app(app(cons, h'), t'))) -> APP(app(append, app(app(map, f''), l1)), app(app(cons, app(f'', h')), app(app(map, f''), t')))
APP(app(map, f''), app(app(cons, h), app(app(append, l1'), l2'))) -> APP(app(cons, app(f'', h)), app(app(append, app(app(map, f''), l1')), app(app(map, f''), l2')))
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(map, f'')), app(app(cons, app(app(append, l1'), l2')), t)) -> APP(app(cons, app(app(append, app(app(map, f''), l1')), app(app(map, f''), l2'))), app(app(map, app(map, f'')), t))
APP(app(map, app(append, app(app(append, l1'), l2'))), app(app(cons, h'), t)) -> APP(app(cons, app(app(append, l1'), app(app(append, l2'), h'))), app(app(map, app(append, app(app(append, l1'), l2'))), 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(append, app(app(append, l1), l2)), l3) -> APP(app(append, l2), l3)
APP(app(map, app(append, app(app(cons, h'''), t''))), app(app(cons, h''), t)) -> APP(app(cons, app(app(cons, h'''), app(app(append, t''), h''))), app(app(map, app(append, app(app(cons, h'''), t''))), t))
APP(app(map, f), app(app(append, l1), l2)) -> APP(app(map, f), l2)
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(append, app(app(cons, h), t)), l) -> APP(app(append, t), l)
APP(app(map, f''), app(app(append, l1), app(app(append, l1''), l2''))) -> APP(app(append, app(app(map, f''), l1)), app(app(append, app(app(map, f''), l1'')), app(app(map, f''), l2'')))


Rules:


app(app(append, nil), l) -> l
app(app(append, app(app(cons, h), t)), l) -> app(app(cons, h), app(app(append, t), l))
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(append, app(app(append, l1), l2)), l3) -> app(app(append, l1), app(app(append, l2), l3))
app(app(map, f), app(app(append, l1), l2)) -> app(app(append, app(app(map, f), l1)), app(app(map, f), l2))


Strategy:

innermost




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

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

APP(app(map, app(map, f'')), app(app(cons, app(app(cons, h''), t'')), t)) -> APP(app(map, f''), app(app(cons, h''), t''))
APP(app(map, app(append, app(app(cons, h'''), t''))), app(app(cons, h''), t)) -> APP(app(append, app(app(cons, h'''), t'')), h'')
APP(app(map, app(append, app(app(append, l1''), l2''))), app(app(cons, h'), t)) -> APP(app(append, app(app(append, l1''), l2'')), h')
APP(app(map, app(map, f'')), app(app(cons, app(app(append, l1''), l2'')), t)) -> APP(app(map, f''), app(app(append, l1''), l2''))
APP(app(map, app(map, app(append, app(app(cons, h'''''), t'''')))), app(app(cons, app(app(cons, h''0'), t'')), t)) -> APP(app(map, app(append, app(app(cons, h'''''), t''''))), app(app(cons, h''0'), t''))
APP(app(map, app(map, app(map, f''''))), app(app(cons, app(app(cons, app(app(cons, h''''), t'''')), t'')), t)) -> APP(app(map, app(map, f'''')), app(app(cons, app(app(cons, h''''), t'''')), t''))
APP(app(map, app(map, app(append, app(app(append, l1'''), l2''')))), app(app(cons, app(app(cons, h'''), t'')), t)) -> APP(app(map, app(append, app(app(append, l1'''), l2'''))), app(app(cons, h'''), t''))
APP(app(map, app(map, app(map, f''''))), app(app(cons, app(app(cons, app(app(append, l1'''), l2''')), t'')), t)) -> APP(app(map, app(map, f'''')), app(app(cons, app(app(append, l1'''), l2''')), t''))
APP(app(map, app(map, f'''')), app(app(cons, app(app(cons, h''), nil)), t)) -> APP(app(map, f''''), app(app(cons, h''), nil))
APP(app(map, app(map, f'''')), app(app(cons, app(app(cons, h''), app(app(cons, h''''), t''''))), t)) -> APP(app(map, f''''), app(app(cons, h''), app(app(cons, h''''), t'''')))
APP(app(map, app(map, f'''')), app(app(cons, app(app(cons, h''), app(app(append, l1'''), l2'''))), t)) -> APP(app(map, f''''), app(app(cons, h''), app(app(append, l1'''), l2''')))
APP(app(map, app(map, f'''')), app(app(cons, app(app(append, l1''), app(app(cons, h'''), t'''))), t)) -> APP(app(map, f''''), app(app(append, l1''), app(app(cons, h'''), t''')))
APP(app(map, app(map, f'''')), app(app(cons, app(app(append, l1''), app(app(append, l1''''), l2''''))), t)) -> APP(app(map, f''''), app(app(append, l1''), app(app(append, l1''''), l2'''')))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 5
                 ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

APP(app(map, app(map, f'''')), app(app(cons, app(app(cons, h''), app(app(append, l1'''), l2'''))), t)) -> APP(app(map, f''''), app(app(cons, h''), app(app(append, l1'''), l2''')))
APP(app(map, app(map, f'''')), app(app(cons, app(app(cons, h''), app(app(cons, h''''), t''''))), t)) -> APP(app(map, f''''), app(app(cons, h''), app(app(cons, h''''), t'''')))
APP(app(map, app(map, f'''')), app(app(cons, app(app(cons, h''), nil)), t)) -> APP(app(map, f''''), app(app(cons, h''), nil))
APP(app(map, app(map, f'''')), app(app(cons, app(app(append, l1''), app(app(append, l1''''), l2''''))), t)) -> APP(app(map, f''''), app(app(append, l1''), app(app(append, l1''''), l2'''')))
APP(app(map, app(map, f'''')), app(app(cons, app(app(append, l1''), app(app(cons, h'''), t'''))), t)) -> APP(app(map, f''''), app(app(append, l1''), app(app(cons, h'''), t''')))
APP(app(map, app(map, app(map, f''''))), app(app(cons, app(app(cons, app(app(append, l1'''), l2''')), t'')), t)) -> APP(app(map, app(map, f'''')), app(app(cons, app(app(append, l1'''), l2''')), t''))
APP(app(map, app(map, app(append, app(app(append, l1'''), l2''')))), app(app(cons, app(app(cons, h'''), t'')), t)) -> APP(app(map, app(append, app(app(append, l1'''), l2'''))), app(app(cons, h'''), t''))
APP(app(map, app(map, app(map, f''''))), app(app(cons, app(app(cons, app(app(cons, h''''), t'''')), t'')), t)) -> APP(app(map, app(map, f'''')), app(app(cons, app(app(cons, h''''), t'''')), t''))
APP(app(map, app(map, app(append, app(app(cons, h'''''), t'''')))), app(app(cons, app(app(cons, h''0'), t'')), t)) -> APP(app(map, app(append, app(app(cons, h'''''), t''''))), app(app(cons, h''0'), t''))
APP(app(map, app(map, f'')), app(app(cons, app(app(append, l1''), l2'')), t)) -> APP(app(map, f''), app(app(append, l1''), l2''))
APP(app(map, app(append, app(app(append, l1''), l2''))), app(app(cons, h'), t)) -> APP(app(append, app(app(append, l1''), l2'')), h')
APP(app(map, app(append, app(app(cons, h'''), t''))), app(app(cons, h''), t)) -> APP(app(append, app(app(cons, h'''), t'')), h'')
APP(app(map, app(map, f'')), app(app(cons, app(app(cons, h''), t'')), t)) -> APP(app(map, f''), app(app(cons, h''), t''))
APP(app(map, f''), app(app(append, l1), app(app(append, l1''), l2''))) -> APP(app(append, app(app(map, f''), l1)), app(app(append, app(app(map, f''), l1'')), app(app(map, f''), l2'')))
APP(app(map, f''), app(app(cons, h), app(app(append, l1'), l2'))) -> APP(app(cons, app(f'', h)), app(app(append, app(app(map, f''), l1')), app(app(map, f''), l2')))
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(map, f'')), app(app(cons, app(app(append, l1'), l2')), t)) -> APP(app(cons, app(app(append, app(app(map, f''), l1')), app(app(map, f''), l2'))), app(app(map, app(map, f'')), t))
APP(app(map, app(append, app(app(append, l1'), l2'))), app(app(cons, h'), t)) -> APP(app(cons, app(app(append, l1'), app(app(append, l2'), h'))), app(app(map, app(append, app(app(append, l1'), l2'))), 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(append, app(app(append, l1), l2)), l3) -> APP(app(append, l2), l3)
APP(app(map, app(append, app(app(cons, h'''), t''))), app(app(cons, h''), t)) -> APP(app(cons, app(app(cons, h'''), app(app(append, t''), h''))), app(app(map, app(append, app(app(cons, h'''), t''))), t))
APP(app(map, f), app(app(append, l1), l2)) -> APP(app(map, f), l2)
APP(app(map, f), app(app(cons, h), t)) -> APP(app(map, f), t)
APP(app(append, app(app(cons, h), t)), l) -> APP(app(append, t), l)
APP(app(map, f''), app(app(append, l1), app(app(cons, h'), t'))) -> APP(app(append, app(app(map, f''), l1)), app(app(cons, app(f'', h')), app(app(map, f''), t')))


Rules:


app(app(append, nil), l) -> l
app(app(append, app(app(cons, h), t)), l) -> app(app(cons, h), app(app(append, t), l))
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(append, app(app(append, l1), l2)), l3) -> app(app(append, l1), app(app(append, l2), l3))
app(app(map, f), app(app(append, l1), l2)) -> app(app(append, app(app(map, f), l1)), app(app(map, f), l2))


Strategy:

innermost



Innermost Termination of R could not be shown.
Duration:
0:47 minutes