Term Rewriting System R:
[t, u, v, x, f]
app(app(app(app(rec, t), u), v), 0) -> t
app(app(app(app(rec, t), u), v), app(s, x)) -> app(app(u, x), app(app(app(app(rec, t), u), v), x))
app(app(app(app(rec, t), u), v), app(lim, f)) -> app(app(v, f), app(app(app(app(rectuv, t), u), v), app(f, n)))
app(app(app(app(rectuv, t), u), v), n) -> app(app(app(app(rec, t), u), v), n)

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

APP(app(app(app(rec, t), u), v), app(s, x)) -> APP(app(u, x), app(app(app(app(rec, t), u), v), x))
APP(app(app(app(rec, t), u), v), app(s, x)) -> APP(u, x)
APP(app(app(app(rec, t), u), v), app(s, x)) -> APP(app(app(app(rec, t), u), v), x)
APP(app(app(app(rec, t), u), v), app(lim, f)) -> APP(app(v, f), app(app(app(app(rectuv, t), u), v), app(f, n)))
APP(app(app(app(rec, t), u), v), app(lim, f)) -> APP(v, f)
APP(app(app(app(rec, t), u), v), app(lim, f)) -> APP(app(app(app(rectuv, t), u), v), app(f, n))
APP(app(app(app(rec, t), u), v), app(lim, f)) -> APP(app(app(rectuv, t), u), v)
APP(app(app(app(rec, t), u), v), app(lim, f)) -> APP(app(rectuv, t), u)
APP(app(app(app(rec, t), u), v), app(lim, f)) -> APP(rectuv, t)
APP(app(app(app(rec, t), u), v), app(lim, f)) -> APP(f, n)
APP(app(app(app(rectuv, t), u), v), n) -> APP(app(app(app(rec, t), u), v), n)
APP(app(app(app(rectuv, t), u), v), n) -> APP(app(app(rec, t), u), v)
APP(app(app(app(rectuv, t), u), v), n) -> APP(app(rec, t), u)
APP(app(app(app(rectuv, t), u), v), n) -> APP(rec, t)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Narrowing Transformation


Dependency Pairs:

APP(app(app(app(rectuv, t), u), v), n) -> APP(app(rec, t), u)
APP(app(app(app(rectuv, t), u), v), n) -> APP(app(app(rec, t), u), v)
APP(app(app(app(rectuv, t), u), v), n) -> APP(app(app(app(rec, t), u), v), n)
APP(app(app(app(rec, t), u), v), app(lim, f)) -> APP(f, n)
APP(app(app(app(rec, t), u), v), app(lim, f)) -> APP(app(rectuv, t), u)
APP(app(app(app(rec, t), u), v), app(lim, f)) -> APP(app(app(rectuv, t), u), v)
APP(app(app(app(rec, t), u), v), app(lim, f)) -> APP(app(app(app(rectuv, t), u), v), app(f, n))
APP(app(app(app(rec, t), u), v), app(lim, f)) -> APP(v, f)
APP(app(app(app(rec, t), u), v), app(lim, f)) -> APP(app(v, f), app(app(app(app(rectuv, t), u), v), app(f, n)))
APP(app(app(app(rec, t), u), v), app(s, x)) -> APP(app(app(app(rec, t), u), v), x)
APP(app(app(app(rec, t), u), v), app(s, x)) -> APP(u, x)
APP(app(app(app(rec, t), u), v), app(s, x)) -> APP(app(u, x), app(app(app(app(rec, t), u), v), x))


Rules:


app(app(app(app(rec, t), u), v), 0) -> t
app(app(app(app(rec, t), u), v), app(s, x)) -> app(app(u, x), app(app(app(app(rec, t), u), v), x))
app(app(app(app(rec, t), u), v), app(lim, f)) -> app(app(v, f), app(app(app(app(rectuv, t), u), v), app(f, n)))
app(app(app(app(rectuv, t), u), v), n) -> app(app(app(app(rec, t), u), v), n)





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

APP(app(app(app(rec, t), u), v), app(lim, f)) -> APP(app(app(rectuv, t), u), v)
no new Dependency Pairs are created.
The transformation is resulting in one new DP problem: 



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


Dependency Pairs:

APP(app(app(app(rectuv, t), u), v), n) -> APP(app(app(rec, t), u), v)
APP(app(app(app(rectuv, t), u), v), n) -> APP(app(app(app(rec, t), u), v), n)
APP(app(app(app(rec, t), u), v), app(lim, f)) -> APP(f, n)
APP(app(app(app(rec, t), u), v), app(lim, f)) -> APP(app(rectuv, t), u)
APP(app(app(app(rec, t), u), v), app(lim, f)) -> APP(app(app(app(rectuv, t), u), v), app(f, n))
APP(app(app(app(rec, t), u), v), app(lim, f)) -> APP(v, f)
APP(app(app(app(rec, t), u), v), app(lim, f)) -> APP(app(v, f), app(app(app(app(rectuv, t), u), v), app(f, n)))
APP(app(app(app(rec, t), u), v), app(s, x)) -> APP(app(app(app(rec, t), u), v), x)
APP(app(app(app(rec, t), u), v), app(s, x)) -> APP(u, x)
APP(app(app(app(rec, t), u), v), app(s, x)) -> APP(app(u, x), app(app(app(app(rec, t), u), v), x))
APP(app(app(app(rectuv, t), u), v), n) -> APP(app(rec, t), u)


Rules:


app(app(app(app(rec, t), u), v), 0) -> t
app(app(app(app(rec, t), u), v), app(s, x)) -> app(app(u, x), app(app(app(app(rec, t), u), v), x))
app(app(app(app(rec, t), u), v), app(lim, f)) -> app(app(v, f), app(app(app(app(rectuv, t), u), v), app(f, n)))
app(app(app(app(rectuv, t), u), v), n) -> app(app(app(app(rec, t), u), v), n)





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

APP(app(app(app(rec, t), u), v), app(lim, f)) -> APP(app(rectuv, t), u)
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 3
                 ↳Narrowing Transformation


Dependency Pairs:

APP(app(app(app(rectuv, t), u), v), n) -> APP(app(rec, t), u)
APP(app(app(app(rectuv, t), u), v), n) -> APP(app(app(app(rec, t), u), v), n)
APP(app(app(app(rec, t), u), v), app(lim, f)) -> APP(f, n)
APP(app(app(app(rec, t), u), v), app(lim, f)) -> APP(app(app(app(rectuv, t), u), v), app(f, n))
APP(app(app(app(rec, t), u), v), app(lim, f)) -> APP(v, f)
APP(app(app(app(rec, t), u), v), app(lim, f)) -> APP(app(v, f), app(app(app(app(rectuv, t), u), v), app(f, n)))
APP(app(app(app(rec, t), u), v), app(s, x)) -> APP(app(app(app(rec, t), u), v), x)
APP(app(app(app(rec, t), u), v), app(s, x)) -> APP(u, x)
APP(app(app(app(rec, t), u), v), app(s, x)) -> APP(app(u, x), app(app(app(app(rec, t), u), v), x))
APP(app(app(app(rectuv, t), u), v), n) -> APP(app(app(rec, t), u), v)


Rules:


app(app(app(app(rec, t), u), v), 0) -> t
app(app(app(app(rec, t), u), v), app(s, x)) -> app(app(u, x), app(app(app(app(rec, t), u), v), x))
app(app(app(app(rec, t), u), v), app(lim, f)) -> app(app(v, f), app(app(app(app(rectuv, t), u), v), app(f, n)))
app(app(app(app(rectuv, t), u), v), n) -> app(app(app(app(rec, t), u), v), n)





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

APP(app(app(app(rectuv, t), u), v), n) -> APP(app(app(app(rec, t), u), v), n)
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
                 ↳Narrowing Transformation


Dependency Pairs:

APP(app(app(app(rectuv, t), u), v), n) -> APP(app(app(rec, t), u), v)
APP(app(app(app(rec, t), u), v), app(lim, f)) -> APP(f, n)
APP(app(app(app(rec, t), u), v), app(lim, f)) -> APP(app(app(app(rectuv, t), u), v), app(f, n))
APP(app(app(app(rec, t), u), v), app(lim, f)) -> APP(v, f)
APP(app(app(app(rec, t), u), v), app(lim, f)) -> APP(app(v, f), app(app(app(app(rectuv, t), u), v), app(f, n)))
APP(app(app(app(rec, t), u), v), app(s, x)) -> APP(app(app(app(rec, t), u), v), x)
APP(app(app(app(rec, t), u), v), app(s, x)) -> APP(u, x)
APP(app(app(app(rec, t), u), v), app(s, x)) -> APP(app(u, x), app(app(app(app(rec, t), u), v), x))
APP(app(app(app(rectuv, t), u), v), n) -> APP(app(rec, t), u)


Rules:


app(app(app(app(rec, t), u), v), 0) -> t
app(app(app(app(rec, t), u), v), app(s, x)) -> app(app(u, x), app(app(app(app(rec, t), u), v), x))
app(app(app(app(rec, t), u), v), app(lim, f)) -> app(app(v, f), app(app(app(app(rectuv, t), u), v), app(f, n)))
app(app(app(app(rectuv, t), u), v), n) -> app(app(app(app(rec, t), u), v), n)





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

APP(app(app(app(rectuv, t), u), v), n) -> APP(app(app(rec, t), u), v)
no new Dependency Pairs are created.
The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 5
                 ↳Narrowing Transformation


Dependency Pairs:

APP(app(app(app(rec, t), u), v), app(lim, f)) -> APP(app(app(app(rectuv, t), u), v), app(f, n))
APP(app(app(app(rec, t), u), v), app(lim, f)) -> APP(v, f)
APP(app(app(app(rec, t), u), v), app(lim, f)) -> APP(app(v, f), app(app(app(app(rectuv, t), u), v), app(f, n)))
APP(app(app(app(rec, t), u), v), app(s, x)) -> APP(app(app(app(rec, t), u), v), x)
APP(app(app(app(rec, t), u), v), app(s, x)) -> APP(u, x)
APP(app(app(app(rec, t), u), v), app(s, x)) -> APP(app(u, x), app(app(app(app(rec, t), u), v), x))
APP(app(app(app(rectuv, t), u), v), n) -> APP(app(rec, t), u)
APP(app(app(app(rec, t), u), v), app(lim, f)) -> APP(f, n)


Rules:


app(app(app(app(rec, t), u), v), 0) -> t
app(app(app(app(rec, t), u), v), app(s, x)) -> app(app(u, x), app(app(app(app(rec, t), u), v), x))
app(app(app(app(rec, t), u), v), app(lim, f)) -> app(app(v, f), app(app(app(app(rectuv, t), u), v), app(f, n)))
app(app(app(app(rectuv, t), u), v), n) -> app(app(app(app(rec, t), u), v), n)





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

APP(app(app(app(rectuv, t), u), v), n) -> APP(app(rec, t), u)
no new Dependency Pairs are created.
The transformation is resulting in one new DP problem: 



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


Dependency Pairs:

APP(app(app(app(rec, t), u), v), app(lim, f)) -> APP(v, f)
APP(app(app(app(rec, t), u), v), app(lim, f)) -> APP(app(v, f), app(app(app(app(rectuv, t), u), v), app(f, n)))
APP(app(app(app(rec, t), u), v), app(s, x)) -> APP(app(app(app(rec, t), u), v), x)
APP(app(app(app(rec, t), u), v), app(s, x)) -> APP(u, x)
APP(app(app(app(rec, t), u), v), app(s, x)) -> APP(app(u, x), app(app(app(app(rec, t), u), v), x))
APP(app(app(app(rec, t), u), v), app(lim, f)) -> APP(app(app(app(rectuv, t), u), v), app(f, n))


Rules:


app(app(app(app(rec, t), u), v), 0) -> t
app(app(app(app(rec, t), u), v), app(s, x)) -> app(app(u, x), app(app(app(app(rec, t), u), v), x))
app(app(app(app(rec, t), u), v), app(lim, f)) -> app(app(v, f), app(app(app(app(rectuv, t), u), v), app(f, n)))
app(app(app(app(rectuv, t), u), v), n) -> app(app(app(app(rec, t), u), v), n)





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

APP(app(app(app(rec, t), u), v), app(lim, f)) -> APP(app(app(app(rectuv, t), u), v), app(f, n))
one new Dependency Pair is created:

APP(app(app(app(rec, t), u), v), app(lim, app(app(app(rectuv, t''), u''), v''))) -> APP(app(app(app(rectuv, t), u), v), app(app(app(app(rec, t''), u''), v''), n))

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(app(app(rec, t), u), v), app(lim, app(app(app(rectuv, t''), u''), v''))) -> APP(app(app(app(rectuv, t), u), v), app(app(app(app(rec, t''), u''), v''), n))
APP(app(app(app(rec, t), u), v), app(lim, f)) -> APP(app(v, f), app(app(app(app(rectuv, t), u), v), app(f, n)))
APP(app(app(app(rec, t), u), v), app(s, x)) -> APP(app(app(app(rec, t), u), v), x)
APP(app(app(app(rec, t), u), v), app(s, x)) -> APP(u, x)
APP(app(app(app(rec, t), u), v), app(s, x)) -> APP(app(u, x), app(app(app(app(rec, t), u), v), x))
APP(app(app(app(rec, t), u), v), app(lim, f)) -> APP(v, f)


Rules:


app(app(app(app(rec, t), u), v), 0) -> t
app(app(app(app(rec, t), u), v), app(s, x)) -> app(app(u, x), app(app(app(app(rec, t), u), v), x))
app(app(app(app(rec, t), u), v), app(lim, f)) -> app(app(v, f), app(app(app(app(rectuv, t), u), v), app(f, n)))
app(app(app(app(rectuv, t), u), v), n) -> app(app(app(app(rec, t), u), v), n)





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

APP(app(app(app(rec, t), u), v), app(lim, app(app(app(rectuv, t''), u''), v''))) -> APP(app(app(app(rectuv, t), u), v), app(app(app(app(rec, t''), u''), v''), n))
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
                 ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

APP(app(app(app(rec, t), u), v), app(lim, f)) -> APP(v, f)
APP(app(app(app(rec, t), u), v), app(s, x)) -> APP(app(app(app(rec, t), u), v), x)
APP(app(app(app(rec, t), u), v), app(s, x)) -> APP(u, x)
APP(app(app(app(rec, t), u), v), app(s, x)) -> APP(app(u, x), app(app(app(app(rec, t), u), v), x))
APP(app(app(app(rec, t), u), v), app(lim, f)) -> APP(app(v, f), app(app(app(app(rectuv, t), u), v), app(f, n)))


Rules:


app(app(app(app(rec, t), u), v), 0) -> t
app(app(app(app(rec, t), u), v), app(s, x)) -> app(app(u, x), app(app(app(app(rec, t), u), v), x))
app(app(app(app(rec, t), u), v), app(lim, f)) -> app(app(v, f), app(app(app(app(rectuv, t), u), v), app(f, n)))
app(app(app(app(rectuv, t), u), v), n) -> app(app(app(app(rec, t), u), v), n)




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