Term Rewriting System R:
[f, x, xs, ys]
app(app(map, f), nil) -> nil
app(app(map, f), app(app(cons, x), xs)) -> app(app(cons, app(f, x)), app(app(map, f), xs))
app(flatten, app(app(node, x), xs)) -> app(app(cons, x), app(concat, app(app(map, flatten), xs)))
app(concat, nil) -> nil
app(concat, app(app(cons, x), xs)) -> app(app(append, x), app(concat, xs))
app(app(append, nil), xs) -> xs
app(app(append, app(app(cons, x), xs)), ys) -> app(app(cons, x), app(app(append, xs), ys))

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

APP(app(map, f), app(app(cons, x), xs)) -> APP(app(cons, app(f, x)), app(app(map, f), xs))
APP(app(map, f), app(app(cons, x), xs)) -> APP(cons, app(f, x))
APP(app(map, f), app(app(cons, x), xs)) -> APP(f, x)
APP(app(map, f), app(app(cons, x), xs)) -> APP(app(map, f), xs)
APP(flatten, app(app(node, x), xs)) -> APP(app(cons, x), app(concat, app(app(map, flatten), xs)))
APP(flatten, app(app(node, x), xs)) -> APP(cons, x)
APP(flatten, app(app(node, x), xs)) -> APP(concat, app(app(map, flatten), xs))
APP(flatten, app(app(node, x), xs)) -> APP(app(map, flatten), xs)
APP(flatten, app(app(node, x), xs)) -> APP(map, flatten)
APP(concat, app(app(cons, x), xs)) -> APP(app(append, x), app(concat, xs))
APP(concat, app(app(cons, x), xs)) -> APP(append, x)
APP(concat, app(app(cons, x), xs)) -> APP(concat, xs)
APP(app(append, app(app(cons, x), xs)), ys) -> APP(app(cons, x), app(app(append, xs), ys))
APP(app(append, app(app(cons, x), xs)), ys) -> APP(app(append, xs), ys)
APP(app(append, app(app(cons, x), xs)), ys) -> APP(append, xs)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Narrowing Transformation


Dependency Pairs:

APP(app(append, app(app(cons, x), xs)), ys) -> APP(app(append, xs), ys)
APP(concat, app(app(cons, x), xs)) -> APP(concat, xs)
APP(flatten, app(app(node, x), xs)) -> APP(app(map, flatten), xs)
APP(concat, app(app(cons, x), xs)) -> APP(app(append, x), app(concat, xs))
APP(flatten, app(app(node, x), xs)) -> APP(concat, app(app(map, flatten), xs))
APP(flatten, app(app(node, x), xs)) -> APP(app(cons, x), app(concat, app(app(map, flatten), xs)))
APP(app(map, f), app(app(cons, x), xs)) -> APP(app(map, f), xs)
APP(app(map, f), app(app(cons, x), xs)) -> APP(f, x)
APP(app(map, f), app(app(cons, x), xs)) -> APP(app(cons, app(f, x)), app(app(map, f), xs))


Rules:


app(app(map, f), nil) -> nil
app(app(map, f), app(app(cons, x), xs)) -> app(app(cons, app(f, x)), app(app(map, f), xs))
app(flatten, app(app(node, x), xs)) -> app(app(cons, x), app(concat, app(app(map, flatten), xs)))
app(concat, nil) -> nil
app(concat, app(app(cons, x), xs)) -> app(app(append, x), app(concat, xs))
app(app(append, nil), xs) -> xs
app(app(append, app(app(cons, x), xs)), ys) -> app(app(cons, x), app(app(append, xs), ys))


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

APP(app(map, app(map, f'')), app(app(cons, nil), xs)) -> APP(app(cons, nil), app(app(map, app(map, f'')), xs))
APP(app(map, app(map, f'')), app(app(cons, app(app(cons, x''), xs'')), xs)) -> APP(app(cons, app(app(cons, app(f'', x'')), app(app(map, f''), xs''))), app(app(map, app(map, f'')), xs))
APP(app(map, flatten), app(app(cons, app(app(node, x''), xs'')), xs)) -> APP(app(cons, app(app(cons, x''), app(concat, app(app(map, flatten), xs'')))), app(app(map, flatten), xs))
APP(app(map, concat), app(app(cons, nil), xs)) -> APP(app(cons, nil), app(app(map, concat), xs))
APP(app(map, concat), app(app(cons, app(app(cons, x''), xs'')), xs)) -> APP(app(cons, app(app(append, x''), app(concat, xs''))), app(app(map, concat), xs))
APP(app(map, app(append, nil)), app(app(cons, x'), xs)) -> APP(app(cons, x'), app(app(map, app(append, nil)), xs))
APP(app(map, app(append, app(app(cons, x''), xs''))), app(app(cons, x0), xs)) -> APP(app(cons, app(app(cons, x''), app(app(append, xs''), x0))), app(app(map, app(append, app(app(cons, x''), xs''))), xs))
APP(app(map, f''), app(app(cons, x), nil)) -> APP(app(cons, app(f'', x)), nil)
APP(app(map, f''), app(app(cons, x), app(app(cons, x''), xs''))) -> APP(app(cons, app(f'', x)), app(app(cons, app(f'', x'')), app(app(map, f''), xs'')))

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, x), app(app(cons, x''), xs''))) -> APP(app(cons, app(f'', x)), app(app(cons, app(f'', x'')), app(app(map, f''), xs'')))
APP(app(map, f''), app(app(cons, x), nil)) -> APP(app(cons, app(f'', x)), nil)
APP(app(map, app(append, app(app(cons, x''), xs''))), app(app(cons, x0), xs)) -> APP(app(cons, app(app(cons, x''), app(app(append, xs''), x0))), app(app(map, app(append, app(app(cons, x''), xs''))), xs))
APP(app(map, concat), app(app(cons, app(app(cons, x''), xs'')), xs)) -> APP(app(cons, app(app(append, x''), app(concat, xs''))), app(app(map, concat), xs))
APP(app(map, flatten), app(app(cons, app(app(node, x''), xs'')), xs)) -> APP(app(cons, app(app(cons, x''), app(concat, app(app(map, flatten), xs'')))), app(app(map, flatten), xs))
APP(concat, app(app(cons, x), xs)) -> APP(concat, xs)
APP(flatten, app(app(node, x), xs)) -> APP(app(map, flatten), xs)
APP(concat, app(app(cons, x), xs)) -> APP(app(append, x), app(concat, xs))
APP(flatten, app(app(node, x), xs)) -> APP(concat, app(app(map, flatten), xs))
APP(flatten, app(app(node, x), xs)) -> APP(app(cons, x), app(concat, app(app(map, flatten), xs)))
APP(app(map, app(map, f'')), app(app(cons, app(app(cons, x''), xs'')), xs)) -> APP(app(cons, app(app(cons, app(f'', x'')), app(app(map, f''), xs''))), app(app(map, app(map, f'')), xs))
APP(app(map, f), app(app(cons, x), xs)) -> APP(app(map, f), xs)
APP(app(map, f), app(app(cons, x), xs)) -> APP(f, x)
APP(app(append, app(app(cons, x), xs)), ys) -> APP(app(append, xs), ys)


Rules:


app(app(map, f), nil) -> nil
app(app(map, f), app(app(cons, x), xs)) -> app(app(cons, app(f, x)), app(app(map, f), xs))
app(flatten, app(app(node, x), xs)) -> app(app(cons, x), app(concat, app(app(map, flatten), xs)))
app(concat, nil) -> nil
app(concat, app(app(cons, x), xs)) -> app(app(append, x), app(concat, xs))
app(app(append, nil), xs) -> xs
app(app(append, app(app(cons, x), xs)), ys) -> app(app(cons, x), app(app(append, xs), ys))


Strategy:

innermost




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

APP(flatten, app(app(node, x), xs)) -> APP(app(cons, x), app(concat, app(app(map, flatten), xs)))
two new Dependency Pairs are created:

APP(flatten, app(app(node, x), nil)) -> APP(app(cons, x), app(concat, nil))
APP(flatten, app(app(node, x), app(app(cons, x''), xs''))) -> APP(app(cons, x), app(concat, app(app(cons, app(flatten, x'')), app(app(map, flatten), xs''))))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 3
                 ↳Rewriting Transformation


Dependency Pairs:

APP(flatten, app(app(node, x), app(app(cons, x''), xs''))) -> APP(app(cons, x), app(concat, app(app(cons, app(flatten, x'')), app(app(map, flatten), xs''))))
APP(flatten, app(app(node, x), nil)) -> APP(app(cons, x), app(concat, nil))
APP(app(map, f''), app(app(cons, x), nil)) -> APP(app(cons, app(f'', x)), nil)
APP(app(map, app(append, app(app(cons, x''), xs''))), app(app(cons, x0), xs)) -> APP(app(cons, app(app(cons, x''), app(app(append, xs''), x0))), app(app(map, app(append, app(app(cons, x''), xs''))), xs))
APP(app(map, concat), app(app(cons, app(app(cons, x''), xs'')), xs)) -> APP(app(cons, app(app(append, x''), app(concat, xs''))), app(app(map, concat), xs))
APP(app(map, flatten), app(app(cons, app(app(node, x''), xs'')), xs)) -> APP(app(cons, app(app(cons, x''), app(concat, app(app(map, flatten), xs'')))), app(app(map, flatten), xs))
APP(app(append, app(app(cons, x), xs)), ys) -> APP(app(append, xs), ys)
APP(concat, app(app(cons, x), xs)) -> APP(concat, xs)
APP(flatten, app(app(node, x), xs)) -> APP(app(map, flatten), xs)
APP(concat, app(app(cons, x), xs)) -> APP(app(append, x), app(concat, xs))
APP(flatten, app(app(node, x), xs)) -> APP(concat, app(app(map, flatten), xs))
APP(app(map, app(map, f'')), app(app(cons, app(app(cons, x''), xs'')), xs)) -> APP(app(cons, app(app(cons, app(f'', x'')), app(app(map, f''), xs''))), app(app(map, app(map, f'')), xs))
APP(app(map, f), app(app(cons, x), xs)) -> APP(app(map, f), xs)
APP(app(map, f), app(app(cons, x), xs)) -> APP(f, x)
APP(app(map, f''), app(app(cons, x), app(app(cons, x''), xs''))) -> APP(app(cons, app(f'', x)), app(app(cons, app(f'', x'')), app(app(map, f''), xs'')))


Rules:


app(app(map, f), nil) -> nil
app(app(map, f), app(app(cons, x), xs)) -> app(app(cons, app(f, x)), app(app(map, f), xs))
app(flatten, app(app(node, x), xs)) -> app(app(cons, x), app(concat, app(app(map, flatten), xs)))
app(concat, nil) -> nil
app(concat, app(app(cons, x), xs)) -> app(app(append, x), app(concat, xs))
app(app(append, nil), xs) -> xs
app(app(append, app(app(cons, x), xs)), ys) -> app(app(cons, x), app(app(append, xs), ys))


Strategy:

innermost




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

APP(flatten, app(app(node, x), nil)) -> APP(app(cons, x), app(concat, nil))
one new Dependency Pair is created:

APP(flatten, app(app(node, x), nil)) -> APP(app(cons, x), nil)

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 4
                 ↳Rewriting Transformation


Dependency Pairs:

APP(flatten, app(app(node, x), nil)) -> APP(app(cons, x), nil)
APP(app(map, f''), app(app(cons, x), app(app(cons, x''), xs''))) -> APP(app(cons, app(f'', x)), app(app(cons, app(f'', x'')), app(app(map, f''), xs'')))
APP(app(map, f''), app(app(cons, x), nil)) -> APP(app(cons, app(f'', x)), nil)
APP(app(map, app(append, app(app(cons, x''), xs''))), app(app(cons, x0), xs)) -> APP(app(cons, app(app(cons, x''), app(app(append, xs''), x0))), app(app(map, app(append, app(app(cons, x''), xs''))), xs))
APP(app(map, concat), app(app(cons, app(app(cons, x''), xs'')), xs)) -> APP(app(cons, app(app(append, x''), app(concat, xs''))), app(app(map, concat), xs))
APP(app(map, flatten), app(app(cons, app(app(node, x''), xs'')), xs)) -> APP(app(cons, app(app(cons, x''), app(concat, app(app(map, flatten), xs'')))), app(app(map, flatten), xs))
APP(app(append, app(app(cons, x), xs)), ys) -> APP(app(append, xs), ys)
APP(concat, app(app(cons, x), xs)) -> APP(concat, xs)
APP(flatten, app(app(node, x), xs)) -> APP(app(map, flatten), xs)
APP(concat, app(app(cons, x), xs)) -> APP(app(append, x), app(concat, xs))
APP(flatten, app(app(node, x), xs)) -> APP(concat, app(app(map, flatten), xs))
APP(app(map, app(map, f'')), app(app(cons, app(app(cons, x''), xs'')), xs)) -> APP(app(cons, app(app(cons, app(f'', x'')), app(app(map, f''), xs''))), app(app(map, app(map, f'')), xs))
APP(app(map, f), app(app(cons, x), xs)) -> APP(app(map, f), xs)
APP(app(map, f), app(app(cons, x), xs)) -> APP(f, x)
APP(flatten, app(app(node, x), app(app(cons, x''), xs''))) -> APP(app(cons, x), app(concat, app(app(cons, app(flatten, x'')), app(app(map, flatten), xs''))))


Rules:


app(app(map, f), nil) -> nil
app(app(map, f), app(app(cons, x), xs)) -> app(app(cons, app(f, x)), app(app(map, f), xs))
app(flatten, app(app(node, x), xs)) -> app(app(cons, x), app(concat, app(app(map, flatten), xs)))
app(concat, nil) -> nil
app(concat, app(app(cons, x), xs)) -> app(app(append, x), app(concat, xs))
app(app(append, nil), xs) -> xs
app(app(append, app(app(cons, x), xs)), ys) -> app(app(cons, x), app(app(append, xs), ys))


Strategy:

innermost




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

APP(flatten, app(app(node, x), app(app(cons, x''), xs''))) -> APP(app(cons, x), app(concat, app(app(cons, app(flatten, x'')), app(app(map, flatten), xs''))))
one new Dependency Pair is created:

APP(flatten, app(app(node, x), app(app(cons, x''), xs''))) -> APP(app(cons, x), app(app(append, app(flatten, x'')), app(concat, app(app(map, flatten), xs''))))

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(flatten, app(app(node, x), app(app(cons, x''), xs''))) -> APP(app(cons, x), app(app(append, app(flatten, x'')), app(concat, app(app(map, flatten), xs''))))
APP(app(map, f''), app(app(cons, x), app(app(cons, x''), xs''))) -> APP(app(cons, app(f'', x)), app(app(cons, app(f'', x'')), app(app(map, f''), xs'')))
APP(app(map, f''), app(app(cons, x), nil)) -> APP(app(cons, app(f'', x)), nil)
APP(app(map, app(append, app(app(cons, x''), xs''))), app(app(cons, x0), xs)) -> APP(app(cons, app(app(cons, x''), app(app(append, xs''), x0))), app(app(map, app(append, app(app(cons, x''), xs''))), xs))
APP(app(map, concat), app(app(cons, app(app(cons, x''), xs'')), xs)) -> APP(app(cons, app(app(append, x''), app(concat, xs''))), app(app(map, concat), xs))
APP(app(map, flatten), app(app(cons, app(app(node, x''), xs'')), xs)) -> APP(app(cons, app(app(cons, x''), app(concat, app(app(map, flatten), xs'')))), app(app(map, flatten), xs))
APP(concat, app(app(cons, x), xs)) -> APP(concat, xs)
APP(flatten, app(app(node, x), xs)) -> APP(app(map, flatten), xs)
APP(concat, app(app(cons, x), xs)) -> APP(app(append, x), app(concat, xs))
APP(flatten, app(app(node, x), xs)) -> APP(concat, app(app(map, flatten), xs))
APP(app(map, app(map, f'')), app(app(cons, app(app(cons, x''), xs'')), xs)) -> APP(app(cons, app(app(cons, app(f'', x'')), app(app(map, f''), xs''))), app(app(map, app(map, f'')), xs))
APP(app(map, f), app(app(cons, x), xs)) -> APP(app(map, f), xs)
APP(app(map, f), app(app(cons, x), xs)) -> APP(f, x)
APP(app(append, app(app(cons, x), xs)), ys) -> APP(app(append, xs), ys)
APP(flatten, app(app(node, x), nil)) -> APP(app(cons, x), nil)


Rules:


app(app(map, f), nil) -> nil
app(app(map, f), app(app(cons, x), xs)) -> app(app(cons, app(f, x)), app(app(map, f), xs))
app(flatten, app(app(node, x), xs)) -> app(app(cons, x), app(concat, app(app(map, flatten), xs)))
app(concat, nil) -> nil
app(concat, app(app(cons, x), xs)) -> app(app(append, x), app(concat, xs))
app(app(append, nil), xs) -> xs
app(app(append, app(app(cons, x), xs)), ys) -> app(app(cons, x), app(app(append, xs), ys))


Strategy:

innermost




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

APP(flatten, app(app(node, x), xs)) -> APP(concat, app(app(map, flatten), xs))
two new Dependency Pairs are created:

APP(flatten, app(app(node, x), nil)) -> APP(concat, nil)
APP(flatten, app(app(node, x), app(app(cons, x''), xs''))) -> APP(concat, app(app(cons, app(flatten, x'')), app(app(map, flatten), xs'')))

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(flatten, app(app(node, x), app(app(cons, x''), xs''))) -> APP(concat, app(app(cons, app(flatten, x'')), app(app(map, flatten), xs'')))
APP(flatten, app(app(node, x), nil)) -> APP(app(cons, x), nil)
APP(app(map, f''), app(app(cons, x), app(app(cons, x''), xs''))) -> APP(app(cons, app(f'', x)), app(app(cons, app(f'', x'')), app(app(map, f''), xs'')))
APP(app(map, f''), app(app(cons, x), nil)) -> APP(app(cons, app(f'', x)), nil)
APP(app(map, app(append, app(app(cons, x''), xs''))), app(app(cons, x0), xs)) -> APP(app(cons, app(app(cons, x''), app(app(append, xs''), x0))), app(app(map, app(append, app(app(cons, x''), xs''))), xs))
APP(app(map, concat), app(app(cons, app(app(cons, x''), xs'')), xs)) -> APP(app(cons, app(app(append, x''), app(concat, xs''))), app(app(map, concat), xs))
APP(app(map, flatten), app(app(cons, app(app(node, x''), xs'')), xs)) -> APP(app(cons, app(app(cons, x''), app(concat, app(app(map, flatten), xs'')))), app(app(map, flatten), xs))
APP(app(append, app(app(cons, x), xs)), ys) -> APP(app(append, xs), ys)
APP(concat, app(app(cons, x), xs)) -> APP(concat, xs)
APP(concat, app(app(cons, x), xs)) -> APP(app(append, x), app(concat, xs))
APP(flatten, app(app(node, x), xs)) -> APP(app(map, flatten), xs)
APP(app(map, app(map, f'')), app(app(cons, app(app(cons, x''), xs'')), xs)) -> APP(app(cons, app(app(cons, app(f'', x'')), app(app(map, f''), xs''))), app(app(map, app(map, f'')), xs))
APP(app(map, f), app(app(cons, x), xs)) -> APP(app(map, f), xs)
APP(app(map, f), app(app(cons, x), xs)) -> APP(f, x)
APP(flatten, app(app(node, x), app(app(cons, x''), xs''))) -> APP(app(cons, x), app(app(append, app(flatten, x'')), app(concat, app(app(map, flatten), xs''))))


Rules:


app(app(map, f), nil) -> nil
app(app(map, f), app(app(cons, x), xs)) -> app(app(cons, app(f, x)), app(app(map, f), xs))
app(flatten, app(app(node, x), xs)) -> app(app(cons, x), app(concat, app(app(map, flatten), xs)))
app(concat, nil) -> nil
app(concat, app(app(cons, x), xs)) -> app(app(append, x), app(concat, xs))
app(app(append, nil), xs) -> xs
app(app(append, app(app(cons, x), xs)), ys) -> app(app(cons, x), app(app(append, xs), ys))


Strategy:

innermost




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

APP(flatten, app(app(node, x), nil)) -> APP(app(cons, x), 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
                 ↳Forward Instantiation Transformation


Dependency Pairs:

APP(flatten, app(app(node, x), app(app(cons, x''), xs''))) -> APP(app(cons, x), app(app(append, app(flatten, x'')), app(concat, app(app(map, flatten), xs''))))
APP(app(map, f''), app(app(cons, x), app(app(cons, x''), xs''))) -> APP(app(cons, app(f'', x)), app(app(cons, app(f'', x'')), app(app(map, f''), xs'')))
APP(app(map, f''), app(app(cons, x), nil)) -> APP(app(cons, app(f'', x)), nil)
APP(app(map, app(append, app(app(cons, x''), xs''))), app(app(cons, x0), xs)) -> APP(app(cons, app(app(cons, x''), app(app(append, xs''), x0))), app(app(map, app(append, app(app(cons, x''), xs''))), xs))
APP(app(map, concat), app(app(cons, app(app(cons, x''), xs'')), xs)) -> APP(app(cons, app(app(append, x''), app(concat, xs''))), app(app(map, concat), xs))
APP(app(map, flatten), app(app(cons, app(app(node, x''), xs'')), xs)) -> APP(app(cons, app(app(cons, x''), app(concat, app(app(map, flatten), xs'')))), app(app(map, flatten), xs))
APP(app(append, app(app(cons, x), xs)), ys) -> APP(app(append, xs), ys)
APP(concat, app(app(cons, x), xs)) -> APP(concat, xs)
APP(flatten, app(app(node, x), xs)) -> APP(app(map, flatten), xs)
APP(app(map, app(map, f'')), app(app(cons, app(app(cons, x''), xs'')), xs)) -> APP(app(cons, app(app(cons, app(f'', x'')), app(app(map, f''), xs''))), app(app(map, app(map, f'')), xs))
APP(app(map, f), app(app(cons, x), xs)) -> APP(app(map, f), xs)
APP(app(map, f), app(app(cons, x), xs)) -> APP(f, x)
APP(concat, app(app(cons, x), xs)) -> APP(app(append, x), app(concat, xs))
APP(flatten, app(app(node, x), app(app(cons, x''), xs''))) -> APP(concat, app(app(cons, app(flatten, x'')), app(app(map, flatten), xs'')))


Rules:


app(app(map, f), nil) -> nil
app(app(map, f), app(app(cons, x), xs)) -> app(app(cons, app(f, x)), app(app(map, f), xs))
app(flatten, app(app(node, x), xs)) -> app(app(cons, x), app(concat, app(app(map, flatten), xs)))
app(concat, nil) -> nil
app(concat, app(app(cons, x), xs)) -> app(app(append, x), app(concat, xs))
app(app(append, nil), xs) -> xs
app(app(append, app(app(cons, x), xs)), ys) -> app(app(cons, x), app(app(append, xs), ys))


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, x), xs)) -> APP(f, x)
11 new Dependency Pairs are created:

APP(app(map, app(map, f'')), app(app(cons, app(app(cons, x''), xs'')), xs)) -> APP(app(map, f''), app(app(cons, x''), xs''))
APP(app(map, flatten), app(app(cons, app(app(node, x''), xs'')), xs)) -> APP(flatten, app(app(node, x''), xs''))
APP(app(map, concat), app(app(cons, app(app(cons, x''), xs'')), xs)) -> APP(concat, app(app(cons, x''), xs''))
APP(app(map, app(append, app(app(cons, x''), xs''))), app(app(cons, x0), xs)) -> APP(app(append, app(app(cons, x''), xs'')), x0)
APP(app(map, app(map, app(map, f''''))), app(app(cons, app(app(cons, app(app(cons, x''''), xs'''')), xs'')), xs)) -> APP(app(map, app(map, f'''')), app(app(cons, app(app(cons, x''''), xs'''')), xs''))
APP(app(map, app(map, flatten)), app(app(cons, app(app(cons, app(app(node, x''''), xs'''')), xs'')), xs)) -> APP(app(map, flatten), app(app(cons, app(app(node, x''''), xs'''')), xs''))
APP(app(map, app(map, concat)), app(app(cons, app(app(cons, app(app(cons, x''''), xs'''')), xs'')), xs)) -> APP(app(map, concat), app(app(cons, app(app(cons, x''''), xs'''')), xs''))
APP(app(map, app(map, app(append, app(app(cons, x''''), xs'''')))), app(app(cons, app(app(cons, x0''), xs'')), xs)) -> APP(app(map, app(append, app(app(cons, x''''), xs''''))), app(app(cons, x0''), xs''))
APP(app(map, app(map, f'''')), app(app(cons, app(app(cons, x''), nil)), xs)) -> APP(app(map, f''''), app(app(cons, x''), nil))
APP(app(map, app(map, f'''')), app(app(cons, app(app(cons, x''), app(app(cons, x''''), xs''''))), xs)) -> APP(app(map, f''''), app(app(cons, x''), app(app(cons, x''''), xs'''')))
APP(app(map, flatten), app(app(cons, app(app(node, x''), app(app(cons, x''''), xs''''))), xs)) -> APP(flatten, app(app(node, 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 8
                 ↳Forward Instantiation Transformation


Dependency Pairs:

APP(app(map, app(map, f'''')), app(app(cons, app(app(cons, x''), app(app(cons, x''''), xs''''))), xs)) -> APP(app(map, f''''), app(app(cons, x''), app(app(cons, x''''), xs'''')))
APP(app(map, app(map, f'''')), app(app(cons, app(app(cons, x''), nil)), xs)) -> APP(app(map, f''''), app(app(cons, x''), nil))
APP(app(map, app(map, app(append, app(app(cons, x''''), xs'''')))), app(app(cons, app(app(cons, x0''), xs'')), xs)) -> APP(app(map, app(append, app(app(cons, x''''), xs''''))), app(app(cons, x0''), xs''))
APP(app(map, app(map, concat)), app(app(cons, app(app(cons, app(app(cons, x''''), xs'''')), xs'')), xs)) -> APP(app(map, concat), app(app(cons, app(app(cons, x''''), xs'''')), xs''))
APP(app(map, flatten), app(app(cons, app(app(node, x''), app(app(cons, x''''), xs''''))), xs)) -> APP(flatten, app(app(node, x''), app(app(cons, x''''), xs'''')))
APP(app(map, app(map, flatten)), app(app(cons, app(app(cons, app(app(node, x''''), xs'''')), xs'')), xs)) -> APP(app(map, flatten), app(app(cons, app(app(node, x''''), xs'''')), xs''))
APP(app(map, app(map, app(map, f''''))), app(app(cons, app(app(cons, app(app(cons, x''''), xs'''')), xs'')), xs)) -> APP(app(map, app(map, f'''')), app(app(cons, app(app(cons, x''''), xs'''')), xs''))
APP(app(map, app(append, app(app(cons, x''), xs''))), app(app(cons, x0), xs)) -> APP(app(append, app(app(cons, x''), xs'')), x0)
APP(app(map, concat), app(app(cons, app(app(cons, x''), xs'')), xs)) -> APP(concat, app(app(cons, x''), xs''))
APP(app(map, flatten), app(app(cons, app(app(node, x''), xs'')), xs)) -> APP(flatten, app(app(node, x''), xs''))
APP(app(map, app(map, f'')), app(app(cons, app(app(cons, x''), xs'')), xs)) -> APP(app(map, f''), app(app(cons, x''), xs''))
APP(flatten, app(app(node, x), app(app(cons, x''), xs''))) -> APP(concat, app(app(cons, app(flatten, x'')), app(app(map, flatten), xs'')))
APP(app(map, f''), app(app(cons, x), app(app(cons, x''), xs''))) -> APP(app(cons, app(f'', x)), app(app(cons, app(f'', x'')), app(app(map, f''), xs'')))
APP(app(map, f''), app(app(cons, x), nil)) -> APP(app(cons, app(f'', x)), nil)
APP(app(map, app(append, app(app(cons, x''), xs''))), app(app(cons, x0), xs)) -> APP(app(cons, app(app(cons, x''), app(app(append, xs''), x0))), app(app(map, app(append, app(app(cons, x''), xs''))), xs))
APP(app(map, concat), app(app(cons, app(app(cons, x''), xs'')), xs)) -> APP(app(cons, app(app(append, x''), app(concat, xs''))), app(app(map, concat), xs))
APP(app(map, flatten), app(app(cons, app(app(node, x''), xs'')), xs)) -> APP(app(cons, app(app(cons, x''), app(concat, app(app(map, flatten), xs'')))), app(app(map, flatten), xs))
APP(app(append, app(app(cons, x), xs)), ys) -> APP(app(append, xs), ys)
APP(concat, app(app(cons, x), xs)) -> APP(concat, xs)
APP(concat, app(app(cons, x), xs)) -> APP(app(append, x), app(concat, xs))
APP(flatten, app(app(node, x), xs)) -> APP(app(map, flatten), xs)
APP(app(map, app(map, f'')), app(app(cons, app(app(cons, x''), xs'')), xs)) -> APP(app(cons, app(app(cons, app(f'', x'')), app(app(map, f''), xs''))), app(app(map, app(map, f'')), xs))
APP(app(map, f), app(app(cons, x), xs)) -> APP(app(map, f), xs)
APP(flatten, app(app(node, x), app(app(cons, x''), xs''))) -> APP(app(cons, x), app(app(append, app(flatten, x'')), app(concat, app(app(map, flatten), xs''))))


Rules:


app(app(map, f), nil) -> nil
app(app(map, f), app(app(cons, x), xs)) -> app(app(cons, app(f, x)), app(app(map, f), xs))
app(flatten, app(app(node, x), xs)) -> app(app(cons, x), app(concat, app(app(map, flatten), xs)))
app(concat, nil) -> nil
app(concat, app(app(cons, x), xs)) -> app(app(append, x), app(concat, xs))
app(app(append, nil), xs) -> xs
app(app(append, app(app(cons, x), xs)), ys) -> app(app(cons, x), app(app(append, xs), ys))


Strategy:

innermost




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

APP(concat, app(app(cons, x), xs)) -> APP(concat, xs)
one new Dependency Pair is created:

APP(concat, app(app(cons, x), app(app(cons, x''), xs''))) -> APP(concat, 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 9
                 ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

APP(app(map, app(map, f'''')), app(app(cons, app(app(cons, x''), nil)), xs)) -> APP(app(map, f''''), app(app(cons, x''), nil))
APP(app(map, app(map, app(append, app(app(cons, x''''), xs'''')))), app(app(cons, app(app(cons, x0''), xs'')), xs)) -> APP(app(map, app(append, app(app(cons, x''''), xs''''))), app(app(cons, x0''), xs''))
APP(app(map, app(map, concat)), app(app(cons, app(app(cons, app(app(cons, x''''), xs'''')), xs'')), xs)) -> APP(app(map, concat), app(app(cons, app(app(cons, x''''), xs'''')), xs''))
APP(app(map, flatten), app(app(cons, app(app(node, x''), app(app(cons, x''''), xs''''))), xs)) -> APP(flatten, app(app(node, x''), app(app(cons, x''''), xs'''')))
APP(app(map, app(map, flatten)), app(app(cons, app(app(cons, app(app(node, x''''), xs'''')), xs'')), xs)) -> APP(app(map, flatten), app(app(cons, app(app(node, x''''), xs'''')), xs''))
APP(app(map, app(map, app(map, f''''))), app(app(cons, app(app(cons, app(app(cons, x''''), xs'''')), xs'')), xs)) -> APP(app(map, app(map, f'''')), app(app(cons, app(app(cons, x''''), xs'''')), xs''))
APP(app(map, app(append, app(app(cons, x''), xs''))), app(app(cons, x0), xs)) -> APP(app(append, app(app(cons, x''), xs'')), x0)
APP(app(map, concat), app(app(cons, app(app(cons, x''), xs'')), xs)) -> APP(concat, app(app(cons, x''), xs''))
APP(app(map, flatten), app(app(cons, app(app(node, x''), xs'')), xs)) -> APP(flatten, app(app(node, x''), xs''))
APP(app(map, app(map, f'')), app(app(cons, app(app(cons, x''), xs'')), xs)) -> APP(app(map, f''), app(app(cons, x''), xs''))
APP(concat, app(app(cons, x), app(app(cons, x''), xs''))) -> APP(concat, app(app(cons, x''), xs''))
APP(flatten, app(app(node, x), app(app(cons, x''), xs''))) -> APP(concat, app(app(cons, app(flatten, x'')), app(app(map, flatten), xs'')))
APP(flatten, app(app(node, x), app(app(cons, x''), xs''))) -> APP(app(cons, x), app(app(append, app(flatten, x'')), app(concat, app(app(map, flatten), xs''))))
APP(app(map, f''), app(app(cons, x), app(app(cons, x''), xs''))) -> APP(app(cons, app(f'', x)), app(app(cons, app(f'', x'')), app(app(map, f''), xs'')))
APP(app(map, f''), app(app(cons, x), nil)) -> APP(app(cons, app(f'', x)), nil)
APP(app(map, app(append, app(app(cons, x''), xs''))), app(app(cons, x0), xs)) -> APP(app(cons, app(app(cons, x''), app(app(append, xs''), x0))), app(app(map, app(append, app(app(cons, x''), xs''))), xs))
APP(app(map, concat), app(app(cons, app(app(cons, x''), xs'')), xs)) -> APP(app(cons, app(app(append, x''), app(concat, xs''))), app(app(map, concat), xs))
APP(app(map, flatten), app(app(cons, app(app(node, x''), xs'')), xs)) -> APP(app(cons, app(app(cons, x''), app(concat, app(app(map, flatten), xs'')))), app(app(map, flatten), xs))
APP(app(append, app(app(cons, x), xs)), ys) -> APP(app(append, xs), ys)
APP(concat, app(app(cons, x), xs)) -> APP(app(append, x), app(concat, xs))
APP(flatten, app(app(node, x), xs)) -> APP(app(map, flatten), xs)
APP(app(map, app(map, f'')), app(app(cons, app(app(cons, x''), xs'')), xs)) -> APP(app(cons, app(app(cons, app(f'', x'')), app(app(map, f''), xs''))), app(app(map, app(map, f'')), xs))
APP(app(map, f), app(app(cons, x), xs)) -> APP(app(map, f), xs)
APP(app(map, app(map, f'''')), app(app(cons, app(app(cons, x''), app(app(cons, x''''), xs''''))), xs)) -> APP(app(map, f''''), app(app(cons, x''), app(app(cons, x''''), xs'''')))


Rules:


app(app(map, f), nil) -> nil
app(app(map, f), app(app(cons, x), xs)) -> app(app(cons, app(f, x)), app(app(map, f), xs))
app(flatten, app(app(node, x), xs)) -> app(app(cons, x), app(concat, app(app(map, flatten), xs)))
app(concat, nil) -> nil
app(concat, app(app(cons, x), xs)) -> app(app(append, x), app(concat, xs))
app(app(append, nil), xs) -> xs
app(app(append, app(app(cons, x), xs)), ys) -> app(app(cons, x), app(app(append, xs), ys))


Strategy:

innermost



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