Term Rewriting System R:
[x, f, t]
app(app(fmap, fnil), x) -> nil
app(app(fmap, app(app(fcons, f), t)), x) -> app(app(cons, app(f, x)), app(app(fmap, t), x))

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

APP(app(fmap, app(app(fcons, f), t)), x) -> APP(app(cons, app(f, x)), app(app(fmap, t), x))
APP(app(fmap, app(app(fcons, f), t)), x) -> APP(cons, app(f, x))
APP(app(fmap, app(app(fcons, f), t)), x) -> APP(f, x)
APP(app(fmap, app(app(fcons, f), t)), x) -> APP(app(fmap, t), x)
APP(app(fmap, app(app(fcons, f), t)), x) -> APP(fmap, t)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Narrowing Transformation


Dependency Pairs:

APP(app(fmap, app(app(fcons, f), t)), x) -> APP(app(fmap, t), x)
APP(app(fmap, app(app(fcons, f), t)), x) -> APP(f, x)
APP(app(fmap, app(app(fcons, f), t)), x) -> APP(app(cons, app(f, x)), app(app(fmap, t), x))


Rules:


app(app(fmap, fnil), x) -> nil
app(app(fmap, app(app(fcons, f), t)), x) -> app(app(cons, app(f, x)), app(app(fmap, t), 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(fcons, f), t)), x) -> APP(app(cons, app(f, x)), app(app(fmap, t), x))
four new Dependency Pairs are created:

APP(app(fmap, app(app(fcons, app(fmap, fnil)), t)), x'') -> APP(app(cons, nil), app(app(fmap, t), x''))
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''), t''))), t)), x'') -> APP(app(cons, app(app(cons, app(f'', x'')), app(app(fmap, t''), x''))), app(app(fmap, t), x''))
APP(app(fmap, app(app(fcons, f), fnil)), x'') -> APP(app(cons, app(f, x'')), nil)
APP(app(fmap, app(app(fcons, f), app(app(fcons, f''), t''))), x'') -> APP(app(cons, app(f, x'')), app(app(cons, app(f'', x'')), app(app(fmap, t''), x'')))

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(fcons, f), app(app(fcons, f''), t''))), x'') -> APP(app(cons, app(f, x'')), app(app(cons, app(f'', x'')), app(app(fmap, t''), x'')))
APP(app(fmap, app(app(fcons, f), fnil)), x'') -> APP(app(cons, app(f, x'')), nil)
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''), t''))), t)), x'') -> APP(app(cons, app(app(cons, app(f'', x'')), app(app(fmap, t''), x''))), app(app(fmap, t), x''))
APP(app(fmap, app(app(fcons, app(fmap, fnil)), t)), x'') -> APP(app(cons, nil), app(app(fmap, t), x''))
APP(app(fmap, app(app(fcons, f), t)), x) -> APP(f, x)
APP(app(fmap, app(app(fcons, f), t)), x) -> APP(app(fmap, t), x)


Rules:


app(app(fmap, fnil), x) -> nil
app(app(fmap, app(app(fcons, f), t)), x) -> app(app(cons, app(f, x)), app(app(fmap, t), 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(fcons, app(fmap, fnil)), t)), x'') -> APP(app(cons, nil), app(app(fmap, t), x''))
two new Dependency Pairs are created:

APP(app(fmap, app(app(fcons, app(fmap, fnil)), fnil)), x''') -> APP(app(cons, nil), nil)
APP(app(fmap, app(app(fcons, app(fmap, fnil)), app(app(fcons, f'), t''))), x''') -> APP(app(cons, nil), app(app(cons, app(f', x''')), app(app(fmap, t''), x''')))

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(fmap, app(app(fcons, app(fmap, fnil)), app(app(fcons, f'), t''))), x''') -> APP(app(cons, nil), app(app(cons, app(f', x''')), app(app(fmap, t''), x''')))
APP(app(fmap, app(app(fcons, app(fmap, fnil)), fnil)), x''') -> APP(app(cons, nil), nil)
APP(app(fmap, app(app(fcons, f), fnil)), x'') -> APP(app(cons, app(f, x'')), nil)
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''), t''))), t)), x'') -> APP(app(cons, app(app(cons, app(f'', x'')), app(app(fmap, t''), x''))), app(app(fmap, t), x''))
APP(app(fmap, app(app(fcons, f), t)), x) -> APP(app(fmap, t), x)
APP(app(fmap, app(app(fcons, f), t)), x) -> APP(f, x)
APP(app(fmap, app(app(fcons, f), app(app(fcons, f''), t''))), x'') -> APP(app(cons, app(f, x'')), app(app(cons, app(f'', x'')), app(app(fmap, t''), x'')))


Rules:


app(app(fmap, fnil), x) -> nil
app(app(fmap, app(app(fcons, f), t)), x) -> app(app(cons, app(f, x)), app(app(fmap, t), 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(fcons, app(fmap, app(app(fcons, f''), t''))), t)), x'') -> APP(app(cons, app(app(cons, app(f'', x'')), app(app(fmap, t''), x''))), app(app(fmap, t), x''))
six new Dependency Pairs are created:

APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, app(fmap, fnil)), t''))), t)), x''') -> APP(app(cons, app(app(cons, nil), app(app(fmap, t''), x'''))), app(app(fmap, t), x'''))
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, app(fmap, app(app(fcons, f'), t'''))), t''))), t)), x''') -> APP(app(cons, app(app(cons, app(app(cons, app(f', x''')), app(app(fmap, t'''), x'''))), app(app(fmap, t''), x'''))), app(app(fmap, t), x'''))
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''), fnil))), t)), x''') -> APP(app(cons, app(app(cons, app(f'', x''')), nil)), app(app(fmap, t), x'''))
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''), app(app(fcons, f'), t''')))), t)), x''') -> APP(app(cons, app(app(cons, app(f'', x''')), app(app(cons, app(f', x''')), app(app(fmap, t'''), x''')))), app(app(fmap, t), x'''))
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''), t''))), fnil)), x''') -> APP(app(cons, app(app(cons, app(f'', x''')), app(app(fmap, t''), x'''))), nil)
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''), t''))), app(app(fcons, f'), t'''))), x''') -> APP(app(cons, app(app(cons, app(f'', x''')), app(app(fmap, t''), x'''))), app(app(cons, app(f', x''')), app(app(fmap, t'''), 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(fcons, app(fmap, app(app(fcons, f''), t''))), app(app(fcons, f'), t'''))), x''') -> APP(app(cons, app(app(cons, app(f'', x''')), app(app(fmap, t''), x'''))), app(app(cons, app(f', x''')), app(app(fmap, t'''), x''')))
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''), t''))), fnil)), x''') -> APP(app(cons, app(app(cons, app(f'', x''')), app(app(fmap, t''), x'''))), nil)
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''), app(app(fcons, f'), t''')))), t)), x''') -> APP(app(cons, app(app(cons, app(f'', x''')), app(app(cons, app(f', x''')), app(app(fmap, t'''), x''')))), app(app(fmap, t), x'''))
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''), fnil))), t)), x''') -> APP(app(cons, app(app(cons, app(f'', x''')), nil)), app(app(fmap, t), x'''))
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, app(fmap, app(app(fcons, f'), t'''))), t''))), t)), x''') -> APP(app(cons, app(app(cons, app(app(cons, app(f', x''')), app(app(fmap, t'''), x'''))), app(app(fmap, t''), x'''))), app(app(fmap, t), x'''))
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, app(fmap, fnil)), t''))), t)), x''') -> APP(app(cons, app(app(cons, nil), app(app(fmap, t''), x'''))), app(app(fmap, t), x'''))
APP(app(fmap, app(app(fcons, app(fmap, fnil)), fnil)), x''') -> APP(app(cons, nil), nil)
APP(app(fmap, app(app(fcons, f), app(app(fcons, f''), t''))), x'') -> APP(app(cons, app(f, x'')), app(app(cons, app(f'', x'')), app(app(fmap, t''), x'')))
APP(app(fmap, app(app(fcons, f), fnil)), x'') -> APP(app(cons, app(f, x'')), nil)
APP(app(fmap, app(app(fcons, f), t)), x) -> APP(app(fmap, t), x)
APP(app(fmap, app(app(fcons, f), t)), x) -> APP(f, x)
APP(app(fmap, app(app(fcons, app(fmap, fnil)), app(app(fcons, f'), t''))), x''') -> APP(app(cons, nil), app(app(cons, app(f', x''')), app(app(fmap, t''), x''')))


Rules:


app(app(fmap, fnil), x) -> nil
app(app(fmap, app(app(fcons, f), t)), x) -> app(app(cons, app(f, x)), app(app(fmap, t), 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(fcons, f), fnil)), x'') -> APP(app(cons, app(f, x'')), nil)
two new Dependency Pairs are created:

APP(app(fmap, app(app(fcons, app(fmap, fnil)), fnil)), x''') -> APP(app(cons, nil), nil)
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''), t'))), fnil)), x''') -> APP(app(cons, app(app(cons, app(f'', x''')), app(app(fmap, t'), x'''))), nil)

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(fcons, app(fmap, app(app(fcons, f''), t'))), fnil)), x''') -> APP(app(cons, app(app(cons, app(f'', x''')), app(app(fmap, t'), x'''))), nil)
APP(app(fmap, app(app(fcons, app(fmap, fnil)), fnil)), x''') -> APP(app(cons, nil), nil)
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''), t''))), fnil)), x''') -> APP(app(cons, app(app(cons, app(f'', x''')), app(app(fmap, t''), x'''))), nil)
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''), app(app(fcons, f'), t''')))), t)), x''') -> APP(app(cons, app(app(cons, app(f'', x''')), app(app(cons, app(f', x''')), app(app(fmap, t'''), x''')))), app(app(fmap, t), x'''))
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''), fnil))), t)), x''') -> APP(app(cons, app(app(cons, app(f'', x''')), nil)), app(app(fmap, t), x'''))
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, app(fmap, app(app(fcons, f'), t'''))), t''))), t)), x''') -> APP(app(cons, app(app(cons, app(app(cons, app(f', x''')), app(app(fmap, t'''), x'''))), app(app(fmap, t''), x'''))), app(app(fmap, t), x'''))
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, app(fmap, fnil)), t''))), t)), x''') -> APP(app(cons, app(app(cons, nil), app(app(fmap, t''), x'''))), app(app(fmap, t), x'''))
APP(app(fmap, app(app(fcons, app(fmap, fnil)), app(app(fcons, f'), t''))), x''') -> APP(app(cons, nil), app(app(cons, app(f', x''')), app(app(fmap, t''), x''')))
APP(app(fmap, app(app(fcons, app(fmap, fnil)), fnil)), x''') -> APP(app(cons, nil), nil)
APP(app(fmap, app(app(fcons, f), app(app(fcons, f''), t''))), x'') -> APP(app(cons, app(f, x'')), app(app(cons, app(f'', x'')), app(app(fmap, t''), x'')))
APP(app(fmap, app(app(fcons, f), t)), x) -> APP(app(fmap, t), x)
APP(app(fmap, app(app(fcons, f), t)), x) -> APP(f, x)
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''), t''))), app(app(fcons, f'), t'''))), x''') -> APP(app(cons, app(app(cons, app(f'', x''')), app(app(fmap, t''), x'''))), app(app(cons, app(f', x''')), app(app(fmap, t'''), x''')))


Rules:


app(app(fmap, fnil), x) -> nil
app(app(fmap, app(app(fcons, f), t)), x) -> app(app(cons, app(f, x)), app(app(fmap, t), 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(fcons, f), app(app(fcons, f''), t''))), x'') -> APP(app(cons, app(f, x'')), app(app(cons, app(f'', x'')), app(app(fmap, t''), x'')))
six new Dependency Pairs are created:

APP(app(fmap, app(app(fcons, app(fmap, fnil)), app(app(fcons, f''), t''))), x''') -> APP(app(cons, nil), app(app(cons, app(f'', x''')), app(app(fmap, t''), x''')))
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f'''), t'))), app(app(fcons, f''), t''))), x''') -> APP(app(cons, app(app(cons, app(f''', x''')), app(app(fmap, t'), x'''))), app(app(cons, app(f'', x''')), app(app(fmap, t''), x''')))
APP(app(fmap, app(app(fcons, f), app(app(fcons, app(fmap, fnil)), t''))), x''') -> APP(app(cons, app(f, x''')), app(app(cons, nil), app(app(fmap, t''), x''')))
APP(app(fmap, app(app(fcons, f), app(app(fcons, app(fmap, app(app(fcons, f'''), t'))), t''))), x''') -> APP(app(cons, app(f, x''')), app(app(cons, app(app(cons, app(f''', x''')), app(app(fmap, t'), x'''))), app(app(fmap, t''), x''')))
APP(app(fmap, app(app(fcons, f), app(app(fcons, f''), fnil))), x''') -> APP(app(cons, app(f, x''')), app(app(cons, app(f'', x''')), nil))
APP(app(fmap, app(app(fcons, f), app(app(fcons, f''), app(app(fcons, f'''), t')))), x''') -> APP(app(cons, app(f, x''')), app(app(cons, app(f'', x''')), app(app(cons, app(f''', x''')), app(app(fmap, t'), x'''))))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 6
                 ↳Narrowing Transformation


Dependency Pairs:

APP(app(fmap, app(app(fcons, f), app(app(fcons, f''), app(app(fcons, f'''), t')))), x''') -> APP(app(cons, app(f, x''')), app(app(cons, app(f'', x''')), app(app(cons, app(f''', x''')), app(app(fmap, t'), x'''))))
APP(app(fmap, app(app(fcons, f), app(app(fcons, f''), fnil))), x''') -> APP(app(cons, app(f, x''')), app(app(cons, app(f'', x''')), nil))
APP(app(fmap, app(app(fcons, f), app(app(fcons, app(fmap, app(app(fcons, f'''), t'))), t''))), x''') -> APP(app(cons, app(f, x''')), app(app(cons, app(app(cons, app(f''', x''')), app(app(fmap, t'), x'''))), app(app(fmap, t''), x''')))
APP(app(fmap, app(app(fcons, f), app(app(fcons, app(fmap, fnil)), t''))), x''') -> APP(app(cons, app(f, x''')), app(app(cons, nil), app(app(fmap, t''), x''')))
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f'''), t'))), app(app(fcons, f''), t''))), x''') -> APP(app(cons, app(app(cons, app(f''', x''')), app(app(fmap, t'), x'''))), app(app(cons, app(f'', x''')), app(app(fmap, t''), x''')))
APP(app(fmap, app(app(fcons, app(fmap, fnil)), app(app(fcons, f''), t''))), x''') -> APP(app(cons, nil), app(app(cons, app(f'', x''')), app(app(fmap, t''), x''')))
APP(app(fmap, app(app(fcons, app(fmap, fnil)), fnil)), x''') -> APP(app(cons, nil), nil)
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''), t''))), app(app(fcons, f'), t'''))), x''') -> APP(app(cons, app(app(cons, app(f'', x''')), app(app(fmap, t''), x'''))), app(app(cons, app(f', x''')), app(app(fmap, t'''), x''')))
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''), t''))), fnil)), x''') -> APP(app(cons, app(app(cons, app(f'', x''')), app(app(fmap, t''), x'''))), nil)
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''), app(app(fcons, f'), t''')))), t)), x''') -> APP(app(cons, app(app(cons, app(f'', x''')), app(app(cons, app(f', x''')), app(app(fmap, t'''), x''')))), app(app(fmap, t), x'''))
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''), fnil))), t)), x''') -> APP(app(cons, app(app(cons, app(f'', x''')), nil)), app(app(fmap, t), x'''))
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, app(fmap, app(app(fcons, f'), t'''))), t''))), t)), x''') -> APP(app(cons, app(app(cons, app(app(cons, app(f', x''')), app(app(fmap, t'''), x'''))), app(app(fmap, t''), x'''))), app(app(fmap, t), x'''))
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, app(fmap, fnil)), t''))), t)), x''') -> APP(app(cons, app(app(cons, nil), app(app(fmap, t''), x'''))), app(app(fmap, t), x'''))
APP(app(fmap, app(app(fcons, app(fmap, fnil)), app(app(fcons, f'), t''))), x''') -> APP(app(cons, nil), app(app(cons, app(f', x''')), app(app(fmap, t''), x''')))
APP(app(fmap, app(app(fcons, app(fmap, fnil)), fnil)), x''') -> APP(app(cons, nil), nil)
APP(app(fmap, app(app(fcons, f), t)), x) -> APP(app(fmap, t), x)
APP(app(fmap, app(app(fcons, f), t)), x) -> APP(f, x)
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''), t'))), fnil)), x''') -> APP(app(cons, app(app(cons, app(f'', x''')), app(app(fmap, t'), x'''))), nil)


Rules:


app(app(fmap, fnil), x) -> nil
app(app(fmap, app(app(fcons, f), t)), x) -> app(app(cons, app(f, x)), app(app(fmap, t), 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(fcons, app(fmap, fnil)), fnil)), x''') -> APP(app(cons, nil), 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 7
                 ↳Narrowing Transformation


Dependency Pairs:

APP(app(fmap, app(app(fcons, f), app(app(fcons, f''), fnil))), x''') -> APP(app(cons, app(f, x''')), app(app(cons, app(f'', x''')), nil))
APP(app(fmap, app(app(fcons, f), app(app(fcons, app(fmap, app(app(fcons, f'''), t'))), t''))), x''') -> APP(app(cons, app(f, x''')), app(app(cons, app(app(cons, app(f''', x''')), app(app(fmap, t'), x'''))), app(app(fmap, t''), x''')))
APP(app(fmap, app(app(fcons, f), app(app(fcons, app(fmap, fnil)), t''))), x''') -> APP(app(cons, app(f, x''')), app(app(cons, nil), app(app(fmap, t''), x''')))
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f'''), t'))), app(app(fcons, f''), t''))), x''') -> APP(app(cons, app(app(cons, app(f''', x''')), app(app(fmap, t'), x'''))), app(app(cons, app(f'', x''')), app(app(fmap, t''), x''')))
APP(app(fmap, app(app(fcons, app(fmap, fnil)), app(app(fcons, f''), t''))), x''') -> APP(app(cons, nil), app(app(cons, app(f'', x''')), app(app(fmap, t''), x''')))
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''), t'))), fnil)), x''') -> APP(app(cons, app(app(cons, app(f'', x''')), app(app(fmap, t'), x'''))), nil)
APP(app(fmap, app(app(fcons, app(fmap, fnil)), fnil)), x''') -> APP(app(cons, nil), nil)
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''), t''))), app(app(fcons, f'), t'''))), x''') -> APP(app(cons, app(app(cons, app(f'', x''')), app(app(fmap, t''), x'''))), app(app(cons, app(f', x''')), app(app(fmap, t'''), x''')))
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''), t''))), fnil)), x''') -> APP(app(cons, app(app(cons, app(f'', x''')), app(app(fmap, t''), x'''))), nil)
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''), app(app(fcons, f'), t''')))), t)), x''') -> APP(app(cons, app(app(cons, app(f'', x''')), app(app(cons, app(f', x''')), app(app(fmap, t'''), x''')))), app(app(fmap, t), x'''))
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''), fnil))), t)), x''') -> APP(app(cons, app(app(cons, app(f'', x''')), nil)), app(app(fmap, t), x'''))
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, app(fmap, app(app(fcons, f'), t'''))), t''))), t)), x''') -> APP(app(cons, app(app(cons, app(app(cons, app(f', x''')), app(app(fmap, t'''), x'''))), app(app(fmap, t''), x'''))), app(app(fmap, t), x'''))
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, app(fmap, fnil)), t''))), t)), x''') -> APP(app(cons, app(app(cons, nil), app(app(fmap, t''), x'''))), app(app(fmap, t), x'''))
APP(app(fmap, app(app(fcons, app(fmap, fnil)), app(app(fcons, f'), t''))), x''') -> APP(app(cons, nil), app(app(cons, app(f', x''')), app(app(fmap, t''), x''')))
APP(app(fmap, app(app(fcons, f), t)), x) -> APP(app(fmap, t), x)
APP(app(fmap, app(app(fcons, f), t)), x) -> APP(f, x)
APP(app(fmap, app(app(fcons, f), app(app(fcons, f''), app(app(fcons, f'''), t')))), x''') -> APP(app(cons, app(f, x''')), app(app(cons, app(f'', x''')), app(app(cons, app(f''', x''')), app(app(fmap, t'), x'''))))


Rules:


app(app(fmap, fnil), x) -> nil
app(app(fmap, app(app(fcons, f), t)), x) -> app(app(cons, app(f, x)), app(app(fmap, t), 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(fcons, app(fmap, fnil)), fnil)), x''') -> APP(app(cons, nil), 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 8
                 ↳Forward Instantiation Transformation


Dependency Pairs:

APP(app(fmap, app(app(fcons, f), app(app(fcons, f''), app(app(fcons, f'''), t')))), x''') -> APP(app(cons, app(f, x''')), app(app(cons, app(f'', x''')), app(app(cons, app(f''', x''')), app(app(fmap, t'), x'''))))
APP(app(fmap, app(app(fcons, f), app(app(fcons, app(fmap, app(app(fcons, f'''), t'))), t''))), x''') -> APP(app(cons, app(f, x''')), app(app(cons, app(app(cons, app(f''', x''')), app(app(fmap, t'), x'''))), app(app(fmap, t''), x''')))
APP(app(fmap, app(app(fcons, f), app(app(fcons, app(fmap, fnil)), t''))), x''') -> APP(app(cons, app(f, x''')), app(app(cons, nil), app(app(fmap, t''), x''')))
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f'''), t'))), app(app(fcons, f''), t''))), x''') -> APP(app(cons, app(app(cons, app(f''', x''')), app(app(fmap, t'), x'''))), app(app(cons, app(f'', x''')), app(app(fmap, t''), x''')))
APP(app(fmap, app(app(fcons, app(fmap, fnil)), app(app(fcons, f''), t''))), x''') -> APP(app(cons, nil), app(app(cons, app(f'', x''')), app(app(fmap, t''), x''')))
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''), t'))), fnil)), x''') -> APP(app(cons, app(app(cons, app(f'', x''')), app(app(fmap, t'), x'''))), nil)
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''), t''))), app(app(fcons, f'), t'''))), x''') -> APP(app(cons, app(app(cons, app(f'', x''')), app(app(fmap, t''), x'''))), app(app(cons, app(f', x''')), app(app(fmap, t'''), x''')))
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''), t''))), fnil)), x''') -> APP(app(cons, app(app(cons, app(f'', x''')), app(app(fmap, t''), x'''))), nil)
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''), app(app(fcons, f'), t''')))), t)), x''') -> APP(app(cons, app(app(cons, app(f'', x''')), app(app(cons, app(f', x''')), app(app(fmap, t'''), x''')))), app(app(fmap, t), x'''))
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''), fnil))), t)), x''') -> APP(app(cons, app(app(cons, app(f'', x''')), nil)), app(app(fmap, t), x'''))
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, app(fmap, app(app(fcons, f'), t'''))), t''))), t)), x''') -> APP(app(cons, app(app(cons, app(app(cons, app(f', x''')), app(app(fmap, t'''), x'''))), app(app(fmap, t''), x'''))), app(app(fmap, t), x'''))
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, app(fmap, fnil)), t''))), t)), x''') -> APP(app(cons, app(app(cons, nil), app(app(fmap, t''), x'''))), app(app(fmap, t), x'''))
APP(app(fmap, app(app(fcons, app(fmap, fnil)), app(app(fcons, f'), t''))), x''') -> APP(app(cons, nil), app(app(cons, app(f', x''')), app(app(fmap, t''), x''')))
APP(app(fmap, app(app(fcons, f), t)), x) -> APP(app(fmap, t), x)
APP(app(fmap, app(app(fcons, f), t)), x) -> APP(f, x)
APP(app(fmap, app(app(fcons, f), app(app(fcons, f''), fnil))), x''') -> APP(app(cons, app(f, x''')), app(app(cons, app(f'', x''')), nil))


Rules:


app(app(fmap, fnil), x) -> nil
app(app(fmap, app(app(fcons, f), t)), x) -> app(app(cons, app(f, x)), app(app(fmap, t), x))


Strategy:

innermost




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

APP(app(fmap, app(app(fcons, f), t)), x) -> APP(f, x)
15 new Dependency Pairs are created:

APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''), t''))), t)), x'') -> APP(app(fmap, app(app(fcons, f''), t'')), x'')
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, app(fmap, fnil)), app(app(fcons, f'''), t'''')))), t)), x') -> APP(app(fmap, app(app(fcons, app(fmap, fnil)), app(app(fcons, f'''), t''''))), x')
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, app(fmap, app(app(fcons, app(fmap, fnil)), t''''))), t''))), t)), x') -> APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, app(fmap, fnil)), t''''))), t'')), x')
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, app(fmap, app(app(fcons, app(fmap, app(app(fcons, f'''), t'''''))), t''0'))), t''))), t)), x') -> APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, app(fmap, app(app(fcons, f'''), t'''''))), t''0'))), t'')), x')
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''''), fnil))), t''))), t)), x') -> APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''''), fnil))), t'')), x')
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''''), app(app(fcons, f'0'), t''''')))), t''))), t)), x') -> APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''''), app(app(fcons, f'0'), t''''')))), t'')), x')
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''''), t''''))), fnil))), t)), x') -> APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''''), t''''))), fnil)), x')
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''''), t''''0))), app(app(fcons, f'0'), t'''''')))), t)), x') -> APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''''), t''''0))), app(app(fcons, f'0'), t''''''))), x')
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''''), t'''))), fnil))), t)), x') -> APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''''), t'''))), fnil)), x')
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, app(fmap, fnil)), app(app(fcons, f''''), t'''')))), t)), x') -> APP(app(fmap, app(app(fcons, app(fmap, fnil)), app(app(fcons, f''''), t''''))), x')
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, app(fmap, app(app(fcons, f'''''), t'''''))), app(app(fcons, f''0'), t'''''')))), t)), x') -> APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f'''''), t'''''))), app(app(fcons, f''0'), t''''''))), x')
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''), app(app(fcons, app(fmap, fnil)), t'''')))), t)), x') -> APP(app(fmap, app(app(fcons, f''), app(app(fcons, app(fmap, fnil)), t''''))), x')
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''), app(app(fcons, app(fmap, app(app(fcons, f'''''), t'''''))), t'''''')))), t)), x') -> APP(app(fmap, app(app(fcons, f''), app(app(fcons, app(fmap, app(app(fcons, f'''''), t'''''))), t''''''))), x')
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''), app(app(fcons, f''''), fnil)))), t)), x') -> APP(app(fmap, app(app(fcons, f''), app(app(fcons, f''''), fnil))), x')
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''), app(app(fcons, f''''0), app(app(fcons, f''''''), t'''))))), t)), x') -> APP(app(fmap, app(app(fcons, f''), app(app(fcons, f''''0), app(app(fcons, f''''''), t''')))), x')

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 9
                 ↳Argument Filtering and Ordering


Dependency Pairs:

APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''), app(app(fcons, f''''0), app(app(fcons, f''''''), t'''))))), t)), x') -> APP(app(fmap, app(app(fcons, f''), app(app(fcons, f''''0), app(app(fcons, f''''''), t''')))), x')
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''), app(app(fcons, f''''), fnil)))), t)), x') -> APP(app(fmap, app(app(fcons, f''), app(app(fcons, f''''), fnil))), x')
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''), app(app(fcons, app(fmap, app(app(fcons, f'''''), t'''''))), t'''''')))), t)), x') -> APP(app(fmap, app(app(fcons, f''), app(app(fcons, app(fmap, app(app(fcons, f'''''), t'''''))), t''''''))), x')
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, app(fmap, app(app(fcons, f'''''), t'''''))), app(app(fcons, f''0'), t'''''')))), t)), x') -> APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f'''''), t'''''))), app(app(fcons, f''0'), t''''''))), x')
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''''), t'''))), fnil))), t)), x') -> APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''''), t'''))), fnil)), x')
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''''), t''''))), fnil))), t)), x') -> APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''''), t''''))), fnil)), x')
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''''), t''''0))), app(app(fcons, f'0'), t'''''')))), t)), x') -> APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''''), t''''0))), app(app(fcons, f'0'), t''''''))), x')
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''''), app(app(fcons, f'0'), t''''')))), t''))), t)), x') -> APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''''), app(app(fcons, f'0'), t''''')))), t'')), x')
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''''), fnil))), t''))), t)), x') -> APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''''), fnil))), t'')), x')
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, app(fmap, app(app(fcons, app(fmap, app(app(fcons, f'''), t'''''))), t''0'))), t''))), t)), x') -> APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, app(fmap, app(app(fcons, f'''), t'''''))), t''0'))), t'')), x')
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''), app(app(fcons, app(fmap, fnil)), t'''')))), t)), x') -> APP(app(fmap, app(app(fcons, f''), app(app(fcons, app(fmap, fnil)), t''''))), x')
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, app(fmap, fnil)), app(app(fcons, f''''), t'''')))), t)), x') -> APP(app(fmap, app(app(fcons, app(fmap, fnil)), app(app(fcons, f''''), t''''))), x')
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, app(fmap, app(app(fcons, app(fmap, fnil)), t''''))), t''))), t)), x') -> APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, app(fmap, fnil)), t''''))), t'')), x')
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, app(fmap, fnil)), app(app(fcons, f'''), t'''')))), t)), x') -> APP(app(fmap, app(app(fcons, app(fmap, fnil)), app(app(fcons, f'''), t''''))), x')
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''), t''))), t)), x'') -> APP(app(fmap, app(app(fcons, f''), t'')), x'')
APP(app(fmap, app(app(fcons, f), app(app(fcons, f''), fnil))), x''') -> APP(app(cons, app(f, x''')), app(app(cons, app(f'', x''')), nil))
APP(app(fmap, app(app(fcons, f), app(app(fcons, app(fmap, app(app(fcons, f'''), t'))), t''))), x''') -> APP(app(cons, app(f, x''')), app(app(cons, app(app(cons, app(f''', x''')), app(app(fmap, t'), x'''))), app(app(fmap, t''), x''')))
APP(app(fmap, app(app(fcons, f), app(app(fcons, app(fmap, fnil)), t''))), x''') -> APP(app(cons, app(f, x''')), app(app(cons, nil), app(app(fmap, t''), x''')))
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f'''), t'))), app(app(fcons, f''), t''))), x''') -> APP(app(cons, app(app(cons, app(f''', x''')), app(app(fmap, t'), x'''))), app(app(cons, app(f'', x''')), app(app(fmap, t''), x''')))
APP(app(fmap, app(app(fcons, app(fmap, fnil)), app(app(fcons, f''), t''))), x''') -> APP(app(cons, nil), app(app(cons, app(f'', x''')), app(app(fmap, t''), x''')))
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''), t'))), fnil)), x''') -> APP(app(cons, app(app(cons, app(f'', x''')), app(app(fmap, t'), x'''))), nil)
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''), t''))), app(app(fcons, f'), t'''))), x''') -> APP(app(cons, app(app(cons, app(f'', x''')), app(app(fmap, t''), x'''))), app(app(cons, app(f', x''')), app(app(fmap, t'''), x''')))
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''), t''))), fnil)), x''') -> APP(app(cons, app(app(cons, app(f'', x''')), app(app(fmap, t''), x'''))), nil)
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''), app(app(fcons, f'), t''')))), t)), x''') -> APP(app(cons, app(app(cons, app(f'', x''')), app(app(cons, app(f', x''')), app(app(fmap, t'''), x''')))), app(app(fmap, t), x'''))
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''), fnil))), t)), x''') -> APP(app(cons, app(app(cons, app(f'', x''')), nil)), app(app(fmap, t), x'''))
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, app(fmap, app(app(fcons, f'), t'''))), t''))), t)), x''') -> APP(app(cons, app(app(cons, app(app(cons, app(f', x''')), app(app(fmap, t'''), x'''))), app(app(fmap, t''), x'''))), app(app(fmap, t), x'''))
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, app(fmap, fnil)), t''))), t)), x''') -> APP(app(cons, app(app(cons, nil), app(app(fmap, t''), x'''))), app(app(fmap, t), x'''))
APP(app(fmap, app(app(fcons, app(fmap, fnil)), app(app(fcons, f'), t''))), x''') -> APP(app(cons, nil), app(app(cons, app(f', x''')), app(app(fmap, t''), x''')))
APP(app(fmap, app(app(fcons, f), t)), x) -> APP(app(fmap, t), x)
APP(app(fmap, app(app(fcons, f), app(app(fcons, f''), app(app(fcons, f'''), t')))), x''') -> APP(app(cons, app(f, x''')), app(app(cons, app(f'', x''')), app(app(cons, app(f''', x''')), app(app(fmap, t'), x'''))))


Rules:


app(app(fmap, fnil), x) -> nil
app(app(fmap, app(app(fcons, f), t)), x) -> app(app(cons, app(f, x)), app(app(fmap, t), x))


Strategy:

innermost




The following dependency pairs can be strictly oriented:

APP(app(fmap, app(app(fcons, f), app(app(fcons, f''), fnil))), x''') -> APP(app(cons, app(f, x''')), app(app(cons, app(f'', x''')), nil))
APP(app(fmap, app(app(fcons, f), app(app(fcons, app(fmap, app(app(fcons, f'''), t'))), t''))), x''') -> APP(app(cons, app(f, x''')), app(app(cons, app(app(cons, app(f''', x''')), app(app(fmap, t'), x'''))), app(app(fmap, t''), x''')))
APP(app(fmap, app(app(fcons, f), app(app(fcons, app(fmap, fnil)), t''))), x''') -> APP(app(cons, app(f, x''')), app(app(cons, nil), app(app(fmap, t''), x''')))
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f'''), t'))), app(app(fcons, f''), t''))), x''') -> APP(app(cons, app(app(cons, app(f''', x''')), app(app(fmap, t'), x'''))), app(app(cons, app(f'', x''')), app(app(fmap, t''), x''')))
APP(app(fmap, app(app(fcons, app(fmap, fnil)), app(app(fcons, f''), t''))), x''') -> APP(app(cons, nil), app(app(cons, app(f'', x''')), app(app(fmap, t''), x''')))
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''), t'))), fnil)), x''') -> APP(app(cons, app(app(cons, app(f'', x''')), app(app(fmap, t'), x'''))), nil)
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''), t''))), app(app(fcons, f'), t'''))), x''') -> APP(app(cons, app(app(cons, app(f'', x''')), app(app(fmap, t''), x'''))), app(app(cons, app(f', x''')), app(app(fmap, t'''), x''')))
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''), t''))), fnil)), x''') -> APP(app(cons, app(app(cons, app(f'', x''')), app(app(fmap, t''), x'''))), nil)
APP(app(fmap, app(app(fcons, app(fmap, fnil)), app(app(fcons, f'), t''))), x''') -> APP(app(cons, nil), app(app(cons, app(f', x''')), app(app(fmap, t''), x''')))
APP(app(fmap, app(app(fcons, f), app(app(fcons, f''), app(app(fcons, f'''), t')))), x''') -> APP(app(cons, app(f, x''')), app(app(cons, app(f'', x''')), app(app(cons, app(f''', x''')), app(app(fmap, t'), x'''))))


The following usable rules for innermost can be oriented:

app(app(fmap, fnil), x) -> nil
app(app(fmap, app(app(fcons, f), t)), x) -> app(app(cons, app(f, x)), app(app(fmap, t), x))


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(cons)=  0  
  POL(fmap)=  1  
  POL(nil)=  0  
  POL(APP(x1, x2))=  1 + x1 + x2  

resulting in one new DP problem.
Used Argument Filtering System:
APP(x1, x2) -> APP(x1, x2)
app(x1, x2) -> x1


   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 10
                 ↳Argument Filtering and Ordering


Dependency Pairs:

APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''), app(app(fcons, f''''0), app(app(fcons, f''''''), t'''))))), t)), x') -> APP(app(fmap, app(app(fcons, f''), app(app(fcons, f''''0), app(app(fcons, f''''''), t''')))), x')
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''), app(app(fcons, f''''), fnil)))), t)), x') -> APP(app(fmap, app(app(fcons, f''), app(app(fcons, f''''), fnil))), x')
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''), app(app(fcons, app(fmap, app(app(fcons, f'''''), t'''''))), t'''''')))), t)), x') -> APP(app(fmap, app(app(fcons, f''), app(app(fcons, app(fmap, app(app(fcons, f'''''), t'''''))), t''''''))), x')
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, app(fmap, app(app(fcons, f'''''), t'''''))), app(app(fcons, f''0'), t'''''')))), t)), x') -> APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f'''''), t'''''))), app(app(fcons, f''0'), t''''''))), x')
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''''), t'''))), fnil))), t)), x') -> APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''''), t'''))), fnil)), x')
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''''), t''''))), fnil))), t)), x') -> APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''''), t''''))), fnil)), x')
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''''), t''''0))), app(app(fcons, f'0'), t'''''')))), t)), x') -> APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''''), t''''0))), app(app(fcons, f'0'), t''''''))), x')
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''''), app(app(fcons, f'0'), t''''')))), t''))), t)), x') -> APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''''), app(app(fcons, f'0'), t''''')))), t'')), x')
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''''), fnil))), t''))), t)), x') -> APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''''), fnil))), t'')), x')
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, app(fmap, app(app(fcons, app(fmap, app(app(fcons, f'''), t'''''))), t''0'))), t''))), t)), x') -> APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, app(fmap, app(app(fcons, f'''), t'''''))), t''0'))), t'')), x')
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''), app(app(fcons, app(fmap, fnil)), t'''')))), t)), x') -> APP(app(fmap, app(app(fcons, f''), app(app(fcons, app(fmap, fnil)), t''''))), x')
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, app(fmap, fnil)), app(app(fcons, f''''), t'''')))), t)), x') -> APP(app(fmap, app(app(fcons, app(fmap, fnil)), app(app(fcons, f''''), t''''))), x')
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, app(fmap, app(app(fcons, app(fmap, fnil)), t''''))), t''))), t)), x') -> APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, app(fmap, fnil)), t''''))), t'')), x')
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, app(fmap, fnil)), app(app(fcons, f'''), t'''')))), t)), x') -> APP(app(fmap, app(app(fcons, app(fmap, fnil)), app(app(fcons, f'''), t''''))), x')
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''), t''))), t)), x'') -> APP(app(fmap, app(app(fcons, f''), t'')), x'')
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''), app(app(fcons, f'), t''')))), t)), x''') -> APP(app(cons, app(app(cons, app(f'', x''')), app(app(cons, app(f', x''')), app(app(fmap, t'''), x''')))), app(app(fmap, t), x'''))
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''), fnil))), t)), x''') -> APP(app(cons, app(app(cons, app(f'', x''')), nil)), app(app(fmap, t), x'''))
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, app(fmap, app(app(fcons, f'), t'''))), t''))), t)), x''') -> APP(app(cons, app(app(cons, app(app(cons, app(f', x''')), app(app(fmap, t'''), x'''))), app(app(fmap, t''), x'''))), app(app(fmap, t), x'''))
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, app(fmap, fnil)), t''))), t)), x''') -> APP(app(cons, app(app(cons, nil), app(app(fmap, t''), x'''))), app(app(fmap, t), x'''))
APP(app(fmap, app(app(fcons, f), t)), x) -> APP(app(fmap, t), x)


Rules:


app(app(fmap, fnil), x) -> nil
app(app(fmap, app(app(fcons, f), t)), x) -> app(app(cons, app(f, x)), app(app(fmap, t), x))


Strategy:

innermost




The following dependency pairs can be strictly oriented:

APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''), app(app(fcons, f'), t''')))), t)), x''') -> APP(app(cons, app(app(cons, app(f'', x''')), app(app(cons, app(f', x''')), app(app(fmap, t'''), x''')))), app(app(fmap, t), x'''))
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''), fnil))), t)), x''') -> APP(app(cons, app(app(cons, app(f'', x''')), nil)), app(app(fmap, t), x'''))
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, app(fmap, app(app(fcons, f'), t'''))), t''))), t)), x''') -> APP(app(cons, app(app(cons, app(app(cons, app(f', x''')), app(app(fmap, t'''), x'''))), app(app(fmap, t''), x'''))), app(app(fmap, t), x'''))
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, app(fmap, fnil)), t''))), t)), x''') -> APP(app(cons, app(app(cons, nil), app(app(fmap, t''), x'''))), app(app(fmap, t), x'''))


The following usable rules for innermost can be oriented:

app(app(fmap, fnil), x) -> nil
app(app(fmap, app(app(fcons, f), t)), x) -> app(app(cons, app(f, x)), app(app(fmap, t), x))


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(cons)=  0  
  POL(fmap)=  1  
  POL(nil)=  0  

resulting in one new DP problem.
Used Argument Filtering System:
APP(x1, x2) -> x1
app(x1, x2) -> x1


   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 11
                 ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''), app(app(fcons, f''''0), app(app(fcons, f''''''), t'''))))), t)), x') -> APP(app(fmap, app(app(fcons, f''), app(app(fcons, f''''0), app(app(fcons, f''''''), t''')))), x')
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''), app(app(fcons, f''''), fnil)))), t)), x') -> APP(app(fmap, app(app(fcons, f''), app(app(fcons, f''''), fnil))), x')
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''), app(app(fcons, app(fmap, app(app(fcons, f'''''), t'''''))), t'''''')))), t)), x') -> APP(app(fmap, app(app(fcons, f''), app(app(fcons, app(fmap, app(app(fcons, f'''''), t'''''))), t''''''))), x')
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, app(fmap, app(app(fcons, f'''''), t'''''))), app(app(fcons, f''0'), t'''''')))), t)), x') -> APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f'''''), t'''''))), app(app(fcons, f''0'), t''''''))), x')
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''''), t'''))), fnil))), t)), x') -> APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''''), t'''))), fnil)), x')
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''''), t''''))), fnil))), t)), x') -> APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''''), t''''))), fnil)), x')
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''''), t''''0))), app(app(fcons, f'0'), t'''''')))), t)), x') -> APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''''), t''''0))), app(app(fcons, f'0'), t''''''))), x')
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''''), app(app(fcons, f'0'), t''''')))), t''))), t)), x') -> APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''''), app(app(fcons, f'0'), t''''')))), t'')), x')
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''''), fnil))), t''))), t)), x') -> APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''''), fnil))), t'')), x')
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, app(fmap, app(app(fcons, app(fmap, app(app(fcons, f'''), t'''''))), t''0'))), t''))), t)), x') -> APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, app(fmap, app(app(fcons, f'''), t'''''))), t''0'))), t'')), x')
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''), app(app(fcons, app(fmap, fnil)), t'''')))), t)), x') -> APP(app(fmap, app(app(fcons, f''), app(app(fcons, app(fmap, fnil)), t''''))), x')
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, app(fmap, fnil)), app(app(fcons, f''''), t'''')))), t)), x') -> APP(app(fmap, app(app(fcons, app(fmap, fnil)), app(app(fcons, f''''), t''''))), x')
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, app(fmap, app(app(fcons, app(fmap, fnil)), t''''))), t''))), t)), x') -> APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, app(fmap, fnil)), t''''))), t'')), x')
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, app(fmap, fnil)), app(app(fcons, f'''), t'''')))), t)), x') -> APP(app(fmap, app(app(fcons, app(fmap, fnil)), app(app(fcons, f'''), t''''))), x')
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''), t''))), t)), x'') -> APP(app(fmap, app(app(fcons, f''), t'')), x'')
APP(app(fmap, app(app(fcons, f), t)), x) -> APP(app(fmap, t), x)


Rules:


app(app(fmap, fnil), x) -> nil
app(app(fmap, app(app(fcons, f), t)), x) -> app(app(cons, app(f, x)), app(app(fmap, t), x))


Strategy:

innermost



Innermost Termination of R could not be shown.
Duration:
0:44 minutes