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))

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(app(append, app(app(cons, x), xs)), ys) -> APP(app(cons, x), 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))





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
             ↳Narrowing 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(append, app(app(cons, x), xs)), ys) -> APP(app(cons, x), 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, 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))
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))





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 3
                 ↳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), app(concat, nil))
APP(app(append, app(app(cons, x), xs)), ys) -> APP(app(append, xs), ys)
APP(app(append, app(app(cons, x), xs)), ys) -> APP(app(cons, x), 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, 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))
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))





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

APP(app(append, app(app(cons, x), xs)), ys) -> APP(app(cons, x), app(app(append, xs), ys))
two new Dependency Pairs are created:

APP(app(append, app(app(cons, x), nil)), ys') -> APP(app(cons, x), ys')
APP(app(append, app(app(cons, x), app(app(cons, x''), xs''))), ys'') -> APP(app(cons, x), app(app(cons, x''), app(app(append, xs''), ys'')))

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(append, app(app(cons, x), app(app(cons, x''), xs''))), ys'') -> APP(app(cons, x), app(app(cons, x''), app(app(append, xs''), ys'')))
APP(app(append, app(app(cons, x), nil)), ys') -> APP(app(cons, x), ys')
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(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, 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))
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))





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

APP(app(append, app(app(cons, x), nil)), ys') -> APP(app(cons, x), ys')
no new Dependency Pairs are created.
The transformation is resulting in one new DP problem: 



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




The following remains to be proven:
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), 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(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, 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))
APP(app(append, app(app(cons, x), app(app(cons, x''), xs''))), ys'') -> APP(app(cons, x), app(app(cons, x''), 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))




Termination of R could not be shown.
Duration:
0:01 minutes