Term Rewriting System R:
[x, y]
natsactive -> addactive(zerosactive)
natsactive -> nats
hdactive(x) -> hd(x)
hdactive(cons(x, y)) -> mark(x)
zerosactive -> cons(0, zeros)
zerosactive -> zeros
tlactive(x) -> tl(x)
tlactive(cons(x, y)) -> mark(y)
incractive(cons(x, y)) -> cons(s(x), incr(y))
incractive(x) -> incr(x)
mark(nats) -> natsactive
mark(zeros) -> zerosactive
mark(incr(x)) -> incractive(mark(x))
mark(add(x)) -> addactive(mark(x))
mark(hd(x)) -> hdactive(mark(x))
mark(tl(x)) -> tlactive(mark(x))
mark(0) -> 0
mark(s(x)) -> s(x)
mark(cons(x, y)) -> cons(x, y)
addactive(cons(x, y)) -> incractive(cons(x, add(y)))
addactive(x) -> add(x)

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

NATSACTIVE -> ADDACTIVE(zerosactive)
NATSACTIVE -> ZEROSACTIVE
HDACTIVE(cons(x, y)) -> MARK(x)
TLACTIVE(cons(x, y)) -> MARK(y)
MARK(nats) -> NATSACTIVE
MARK(zeros) -> ZEROSACTIVE
MARK(incr(x)) -> INCRACTIVE(mark(x))
MARK(incr(x)) -> MARK(x)
MARK(add(x)) -> ADDACTIVE(mark(x))
MARK(add(x)) -> MARK(x)
MARK(hd(x)) -> HDACTIVE(mark(x))
MARK(hd(x)) -> MARK(x)
MARK(tl(x)) -> TLACTIVE(mark(x))
MARK(tl(x)) -> MARK(x)
ADDACTIVE(cons(x, y)) -> INCRACTIVE(cons(x, add(y)))

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Narrowing Transformation


Dependency Pairs:

MARK(tl(x)) -> MARK(x)
TLACTIVE(cons(x, y)) -> MARK(y)
MARK(tl(x)) -> TLACTIVE(mark(x))
MARK(hd(x)) -> MARK(x)
MARK(hd(x)) -> HDACTIVE(mark(x))
MARK(add(x)) -> MARK(x)
MARK(incr(x)) -> MARK(x)
HDACTIVE(cons(x, y)) -> MARK(x)


Rules:


natsactive -> addactive(zerosactive)
natsactive -> nats
hdactive(x) -> hd(x)
hdactive(cons(x, y)) -> mark(x)
zerosactive -> cons(0, zeros)
zerosactive -> zeros
tlactive(x) -> tl(x)
tlactive(cons(x, y)) -> mark(y)
incractive(cons(x, y)) -> cons(s(x), incr(y))
incractive(x) -> incr(x)
mark(nats) -> natsactive
mark(zeros) -> zerosactive
mark(incr(x)) -> incractive(mark(x))
mark(add(x)) -> addactive(mark(x))
mark(hd(x)) -> hdactive(mark(x))
mark(tl(x)) -> tlactive(mark(x))
mark(0) -> 0
mark(s(x)) -> s(x)
mark(cons(x, y)) -> cons(x, y)
addactive(cons(x, y)) -> incractive(cons(x, add(y)))
addactive(x) -> add(x)





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

MARK(hd(x)) -> HDACTIVE(mark(x))
nine new Dependency Pairs are created:

MARK(hd(nats)) -> HDACTIVE(natsactive)
MARK(hd(zeros)) -> HDACTIVE(zerosactive)
MARK(hd(incr(x''))) -> HDACTIVE(incractive(mark(x'')))
MARK(hd(add(x''))) -> HDACTIVE(addactive(mark(x'')))
MARK(hd(hd(x''))) -> HDACTIVE(hdactive(mark(x'')))
MARK(hd(tl(x''))) -> HDACTIVE(tlactive(mark(x'')))
MARK(hd(0)) -> HDACTIVE(0)
MARK(hd(s(x''))) -> HDACTIVE(s(x''))
MARK(hd(cons(x'', y'))) -> HDACTIVE(cons(x'', y'))

The transformation is resulting in one new DP problem: 



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


Dependency Pairs:

MARK(hd(cons(x'', y'))) -> HDACTIVE(cons(x'', y'))
MARK(hd(tl(x''))) -> HDACTIVE(tlactive(mark(x'')))
MARK(hd(hd(x''))) -> HDACTIVE(hdactive(mark(x'')))
MARK(hd(add(x''))) -> HDACTIVE(addactive(mark(x'')))
MARK(hd(incr(x''))) -> HDACTIVE(incractive(mark(x'')))
MARK(hd(zeros)) -> HDACTIVE(zerosactive)
HDACTIVE(cons(x, y)) -> MARK(x)
MARK(hd(nats)) -> HDACTIVE(natsactive)
TLACTIVE(cons(x, y)) -> MARK(y)
MARK(tl(x)) -> TLACTIVE(mark(x))
MARK(hd(x)) -> MARK(x)
MARK(add(x)) -> MARK(x)
MARK(incr(x)) -> MARK(x)
MARK(tl(x)) -> MARK(x)


Rules:


natsactive -> addactive(zerosactive)
natsactive -> nats
hdactive(x) -> hd(x)
hdactive(cons(x, y)) -> mark(x)
zerosactive -> cons(0, zeros)
zerosactive -> zeros
tlactive(x) -> tl(x)
tlactive(cons(x, y)) -> mark(y)
incractive(cons(x, y)) -> cons(s(x), incr(y))
incractive(x) -> incr(x)
mark(nats) -> natsactive
mark(zeros) -> zerosactive
mark(incr(x)) -> incractive(mark(x))
mark(add(x)) -> addactive(mark(x))
mark(hd(x)) -> hdactive(mark(x))
mark(tl(x)) -> tlactive(mark(x))
mark(0) -> 0
mark(s(x)) -> s(x)
mark(cons(x, y)) -> cons(x, y)
addactive(cons(x, y)) -> incractive(cons(x, add(y)))
addactive(x) -> add(x)





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

MARK(tl(x)) -> TLACTIVE(mark(x))
nine new Dependency Pairs are created:

MARK(tl(nats)) -> TLACTIVE(natsactive)
MARK(tl(zeros)) -> TLACTIVE(zerosactive)
MARK(tl(incr(x''))) -> TLACTIVE(incractive(mark(x'')))
MARK(tl(add(x''))) -> TLACTIVE(addactive(mark(x'')))
MARK(tl(hd(x''))) -> TLACTIVE(hdactive(mark(x'')))
MARK(tl(tl(x''))) -> TLACTIVE(tlactive(mark(x'')))
MARK(tl(0)) -> TLACTIVE(0)
MARK(tl(s(x''))) -> TLACTIVE(s(x''))
MARK(tl(cons(x'', y'))) -> TLACTIVE(cons(x'', y'))

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:

MARK(tl(cons(x'', y'))) -> TLACTIVE(cons(x'', y'))
MARK(tl(tl(x''))) -> TLACTIVE(tlactive(mark(x'')))
MARK(tl(hd(x''))) -> TLACTIVE(hdactive(mark(x'')))
MARK(tl(add(x''))) -> TLACTIVE(addactive(mark(x'')))
MARK(tl(incr(x''))) -> TLACTIVE(incractive(mark(x'')))
MARK(tl(zeros)) -> TLACTIVE(zerosactive)
TLACTIVE(cons(x, y)) -> MARK(y)
MARK(tl(nats)) -> TLACTIVE(natsactive)
MARK(hd(tl(x''))) -> HDACTIVE(tlactive(mark(x'')))
MARK(hd(hd(x''))) -> HDACTIVE(hdactive(mark(x'')))
MARK(hd(add(x''))) -> HDACTIVE(addactive(mark(x'')))
MARK(hd(incr(x''))) -> HDACTIVE(incractive(mark(x'')))
MARK(hd(zeros)) -> HDACTIVE(zerosactive)
MARK(hd(nats)) -> HDACTIVE(natsactive)
MARK(tl(x)) -> MARK(x)
MARK(hd(x)) -> MARK(x)
MARK(add(x)) -> MARK(x)
MARK(incr(x)) -> MARK(x)
HDACTIVE(cons(x, y)) -> MARK(x)
MARK(hd(cons(x'', y'))) -> HDACTIVE(cons(x'', y'))


Rules:


natsactive -> addactive(zerosactive)
natsactive -> nats
hdactive(x) -> hd(x)
hdactive(cons(x, y)) -> mark(x)
zerosactive -> cons(0, zeros)
zerosactive -> zeros
tlactive(x) -> tl(x)
tlactive(cons(x, y)) -> mark(y)
incractive(cons(x, y)) -> cons(s(x), incr(y))
incractive(x) -> incr(x)
mark(nats) -> natsactive
mark(zeros) -> zerosactive
mark(incr(x)) -> incractive(mark(x))
mark(add(x)) -> addactive(mark(x))
mark(hd(x)) -> hdactive(mark(x))
mark(tl(x)) -> tlactive(mark(x))
mark(0) -> 0
mark(s(x)) -> s(x)
mark(cons(x, y)) -> cons(x, y)
addactive(cons(x, y)) -> incractive(cons(x, add(y)))
addactive(x) -> add(x)





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

MARK(hd(nats)) -> HDACTIVE(natsactive)
two new Dependency Pairs are created:

MARK(hd(nats)) -> HDACTIVE(addactive(zerosactive))
MARK(hd(nats)) -> HDACTIVE(nats)

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:

MARK(hd(nats)) -> HDACTIVE(addactive(zerosactive))
MARK(tl(tl(x''))) -> TLACTIVE(tlactive(mark(x'')))
MARK(tl(hd(x''))) -> TLACTIVE(hdactive(mark(x'')))
MARK(tl(add(x''))) -> TLACTIVE(addactive(mark(x'')))
MARK(tl(incr(x''))) -> TLACTIVE(incractive(mark(x'')))
MARK(tl(zeros)) -> TLACTIVE(zerosactive)
MARK(tl(nats)) -> TLACTIVE(natsactive)
MARK(hd(cons(x'', y'))) -> HDACTIVE(cons(x'', y'))
MARK(hd(tl(x''))) -> HDACTIVE(tlactive(mark(x'')))
MARK(hd(hd(x''))) -> HDACTIVE(hdactive(mark(x'')))
MARK(hd(add(x''))) -> HDACTIVE(addactive(mark(x'')))
MARK(hd(incr(x''))) -> HDACTIVE(incractive(mark(x'')))
HDACTIVE(cons(x, y)) -> MARK(x)
MARK(hd(zeros)) -> HDACTIVE(zerosactive)
MARK(tl(x)) -> MARK(x)
MARK(hd(x)) -> MARK(x)
MARK(add(x)) -> MARK(x)
MARK(incr(x)) -> MARK(x)
TLACTIVE(cons(x, y)) -> MARK(y)
MARK(tl(cons(x'', y'))) -> TLACTIVE(cons(x'', y'))


Rules:


natsactive -> addactive(zerosactive)
natsactive -> nats
hdactive(x) -> hd(x)
hdactive(cons(x, y)) -> mark(x)
zerosactive -> cons(0, zeros)
zerosactive -> zeros
tlactive(x) -> tl(x)
tlactive(cons(x, y)) -> mark(y)
incractive(cons(x, y)) -> cons(s(x), incr(y))
incractive(x) -> incr(x)
mark(nats) -> natsactive
mark(zeros) -> zerosactive
mark(incr(x)) -> incractive(mark(x))
mark(add(x)) -> addactive(mark(x))
mark(hd(x)) -> hdactive(mark(x))
mark(tl(x)) -> tlactive(mark(x))
mark(0) -> 0
mark(s(x)) -> s(x)
mark(cons(x, y)) -> cons(x, y)
addactive(cons(x, y)) -> incractive(cons(x, add(y)))
addactive(x) -> add(x)





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

MARK(hd(zeros)) -> HDACTIVE(zerosactive)
two new Dependency Pairs are created:

MARK(hd(zeros)) -> HDACTIVE(cons(0, zeros))
MARK(hd(zeros)) -> HDACTIVE(zeros)

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:

MARK(hd(zeros)) -> HDACTIVE(cons(0, zeros))
MARK(tl(cons(x'', y'))) -> TLACTIVE(cons(x'', y'))
MARK(tl(tl(x''))) -> TLACTIVE(tlactive(mark(x'')))
MARK(tl(hd(x''))) -> TLACTIVE(hdactive(mark(x'')))
MARK(tl(add(x''))) -> TLACTIVE(addactive(mark(x'')))
MARK(tl(incr(x''))) -> TLACTIVE(incractive(mark(x'')))
MARK(tl(zeros)) -> TLACTIVE(zerosactive)
TLACTIVE(cons(x, y)) -> MARK(y)
MARK(tl(nats)) -> TLACTIVE(natsactive)
MARK(hd(cons(x'', y'))) -> HDACTIVE(cons(x'', y'))
MARK(hd(tl(x''))) -> HDACTIVE(tlactive(mark(x'')))
MARK(hd(hd(x''))) -> HDACTIVE(hdactive(mark(x'')))
MARK(hd(add(x''))) -> HDACTIVE(addactive(mark(x'')))
MARK(hd(incr(x''))) -> HDACTIVE(incractive(mark(x'')))
MARK(tl(x)) -> MARK(x)
MARK(hd(x)) -> MARK(x)
MARK(add(x)) -> MARK(x)
MARK(incr(x)) -> MARK(x)
HDACTIVE(cons(x, y)) -> MARK(x)
MARK(hd(nats)) -> HDACTIVE(addactive(zerosactive))


Rules:


natsactive -> addactive(zerosactive)
natsactive -> nats
hdactive(x) -> hd(x)
hdactive(cons(x, y)) -> mark(x)
zerosactive -> cons(0, zeros)
zerosactive -> zeros
tlactive(x) -> tl(x)
tlactive(cons(x, y)) -> mark(y)
incractive(cons(x, y)) -> cons(s(x), incr(y))
incractive(x) -> incr(x)
mark(nats) -> natsactive
mark(zeros) -> zerosactive
mark(incr(x)) -> incractive(mark(x))
mark(add(x)) -> addactive(mark(x))
mark(hd(x)) -> hdactive(mark(x))
mark(tl(x)) -> tlactive(mark(x))
mark(0) -> 0
mark(s(x)) -> s(x)
mark(cons(x, y)) -> cons(x, y)
addactive(cons(x, y)) -> incractive(cons(x, add(y)))
addactive(x) -> add(x)





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

MARK(hd(incr(x''))) -> HDACTIVE(incractive(mark(x'')))
10 new Dependency Pairs are created:

MARK(hd(incr(x'''))) -> HDACTIVE(incr(mark(x''')))
MARK(hd(incr(nats))) -> HDACTIVE(incractive(natsactive))
MARK(hd(incr(zeros))) -> HDACTIVE(incractive(zerosactive))
MARK(hd(incr(incr(x')))) -> HDACTIVE(incractive(incractive(mark(x'))))
MARK(hd(incr(add(x')))) -> HDACTIVE(incractive(addactive(mark(x'))))
MARK(hd(incr(hd(x')))) -> HDACTIVE(incractive(hdactive(mark(x'))))
MARK(hd(incr(tl(x')))) -> HDACTIVE(incractive(tlactive(mark(x'))))
MARK(hd(incr(0))) -> HDACTIVE(incractive(0))
MARK(hd(incr(s(x')))) -> HDACTIVE(incractive(s(x')))
MARK(hd(incr(cons(x', y')))) -> HDACTIVE(incractive(cons(x', y')))

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:

MARK(hd(incr(cons(x', y')))) -> HDACTIVE(incractive(cons(x', y')))
MARK(hd(incr(s(x')))) -> HDACTIVE(incractive(s(x')))
MARK(hd(incr(0))) -> HDACTIVE(incractive(0))
MARK(hd(incr(tl(x')))) -> HDACTIVE(incractive(tlactive(mark(x'))))
MARK(hd(incr(hd(x')))) -> HDACTIVE(incractive(hdactive(mark(x'))))
MARK(hd(incr(add(x')))) -> HDACTIVE(incractive(addactive(mark(x'))))
MARK(hd(incr(incr(x')))) -> HDACTIVE(incractive(incractive(mark(x'))))
MARK(hd(incr(zeros))) -> HDACTIVE(incractive(zerosactive))
MARK(hd(incr(nats))) -> HDACTIVE(incractive(natsactive))
MARK(hd(nats)) -> HDACTIVE(addactive(zerosactive))
MARK(tl(cons(x'', y'))) -> TLACTIVE(cons(x'', y'))
MARK(tl(tl(x''))) -> TLACTIVE(tlactive(mark(x'')))
MARK(tl(hd(x''))) -> TLACTIVE(hdactive(mark(x'')))
MARK(tl(add(x''))) -> TLACTIVE(addactive(mark(x'')))
MARK(tl(incr(x''))) -> TLACTIVE(incractive(mark(x'')))
MARK(tl(zeros)) -> TLACTIVE(zerosactive)
TLACTIVE(cons(x, y)) -> MARK(y)
MARK(tl(nats)) -> TLACTIVE(natsactive)
MARK(hd(cons(x'', y'))) -> HDACTIVE(cons(x'', y'))
MARK(hd(tl(x''))) -> HDACTIVE(tlactive(mark(x'')))
MARK(hd(hd(x''))) -> HDACTIVE(hdactive(mark(x'')))
MARK(hd(add(x''))) -> HDACTIVE(addactive(mark(x'')))
MARK(tl(x)) -> MARK(x)
MARK(hd(x)) -> MARK(x)
MARK(add(x)) -> MARK(x)
MARK(incr(x)) -> MARK(x)
HDACTIVE(cons(x, y)) -> MARK(x)
MARK(hd(zeros)) -> HDACTIVE(cons(0, zeros))


Rules:


natsactive -> addactive(zerosactive)
natsactive -> nats
hdactive(x) -> hd(x)
hdactive(cons(x, y)) -> mark(x)
zerosactive -> cons(0, zeros)
zerosactive -> zeros
tlactive(x) -> tl(x)
tlactive(cons(x, y)) -> mark(y)
incractive(cons(x, y)) -> cons(s(x), incr(y))
incractive(x) -> incr(x)
mark(nats) -> natsactive
mark(zeros) -> zerosactive
mark(incr(x)) -> incractive(mark(x))
mark(add(x)) -> addactive(mark(x))
mark(hd(x)) -> hdactive(mark(x))
mark(tl(x)) -> tlactive(mark(x))
mark(0) -> 0
mark(s(x)) -> s(x)
mark(cons(x, y)) -> cons(x, y)
addactive(cons(x, y)) -> incractive(cons(x, add(y)))
addactive(x) -> add(x)





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

MARK(hd(add(x''))) -> HDACTIVE(addactive(mark(x'')))
10 new Dependency Pairs are created:

MARK(hd(add(x'''))) -> HDACTIVE(add(mark(x''')))
MARK(hd(add(nats))) -> HDACTIVE(addactive(natsactive))
MARK(hd(add(zeros))) -> HDACTIVE(addactive(zerosactive))
MARK(hd(add(incr(x')))) -> HDACTIVE(addactive(incractive(mark(x'))))
MARK(hd(add(add(x')))) -> HDACTIVE(addactive(addactive(mark(x'))))
MARK(hd(add(hd(x')))) -> HDACTIVE(addactive(hdactive(mark(x'))))
MARK(hd(add(tl(x')))) -> HDACTIVE(addactive(tlactive(mark(x'))))
MARK(hd(add(0))) -> HDACTIVE(addactive(0))
MARK(hd(add(s(x')))) -> HDACTIVE(addactive(s(x')))
MARK(hd(add(cons(x', y')))) -> HDACTIVE(addactive(cons(x', y')))

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:

MARK(hd(add(cons(x', y')))) -> HDACTIVE(addactive(cons(x', y')))
MARK(hd(add(s(x')))) -> HDACTIVE(addactive(s(x')))
MARK(hd(add(0))) -> HDACTIVE(addactive(0))
MARK(hd(add(tl(x')))) -> HDACTIVE(addactive(tlactive(mark(x'))))
MARK(hd(add(hd(x')))) -> HDACTIVE(addactive(hdactive(mark(x'))))
MARK(hd(add(add(x')))) -> HDACTIVE(addactive(addactive(mark(x'))))
MARK(hd(add(incr(x')))) -> HDACTIVE(addactive(incractive(mark(x'))))
MARK(hd(add(zeros))) -> HDACTIVE(addactive(zerosactive))
MARK(hd(add(nats))) -> HDACTIVE(addactive(natsactive))
MARK(hd(incr(s(x')))) -> HDACTIVE(incractive(s(x')))
MARK(hd(incr(0))) -> HDACTIVE(incractive(0))
MARK(hd(incr(tl(x')))) -> HDACTIVE(incractive(tlactive(mark(x'))))
MARK(hd(incr(hd(x')))) -> HDACTIVE(incractive(hdactive(mark(x'))))
MARK(hd(incr(add(x')))) -> HDACTIVE(incractive(addactive(mark(x'))))
MARK(hd(incr(incr(x')))) -> HDACTIVE(incractive(incractive(mark(x'))))
MARK(hd(incr(zeros))) -> HDACTIVE(incractive(zerosactive))
MARK(hd(incr(nats))) -> HDACTIVE(incractive(natsactive))
MARK(hd(zeros)) -> HDACTIVE(cons(0, zeros))
MARK(hd(nats)) -> HDACTIVE(addactive(zerosactive))
MARK(tl(cons(x'', y'))) -> TLACTIVE(cons(x'', y'))
MARK(tl(tl(x''))) -> TLACTIVE(tlactive(mark(x'')))
MARK(tl(hd(x''))) -> TLACTIVE(hdactive(mark(x'')))
MARK(tl(add(x''))) -> TLACTIVE(addactive(mark(x'')))
MARK(tl(incr(x''))) -> TLACTIVE(incractive(mark(x'')))
MARK(tl(zeros)) -> TLACTIVE(zerosactive)
TLACTIVE(cons(x, y)) -> MARK(y)
MARK(tl(nats)) -> TLACTIVE(natsactive)
MARK(hd(cons(x'', y'))) -> HDACTIVE(cons(x'', y'))
MARK(hd(tl(x''))) -> HDACTIVE(tlactive(mark(x'')))
MARK(hd(hd(x''))) -> HDACTIVE(hdactive(mark(x'')))
MARK(tl(x)) -> MARK(x)
MARK(hd(x)) -> MARK(x)
MARK(add(x)) -> MARK(x)
MARK(incr(x)) -> MARK(x)
HDACTIVE(cons(x, y)) -> MARK(x)
MARK(hd(incr(cons(x', y')))) -> HDACTIVE(incractive(cons(x', y')))


Rules:


natsactive -> addactive(zerosactive)
natsactive -> nats
hdactive(x) -> hd(x)
hdactive(cons(x, y)) -> mark(x)
zerosactive -> cons(0, zeros)
zerosactive -> zeros
tlactive(x) -> tl(x)
tlactive(cons(x, y)) -> mark(y)
incractive(cons(x, y)) -> cons(s(x), incr(y))
incractive(x) -> incr(x)
mark(nats) -> natsactive
mark(zeros) -> zerosactive
mark(incr(x)) -> incractive(mark(x))
mark(add(x)) -> addactive(mark(x))
mark(hd(x)) -> hdactive(mark(x))
mark(tl(x)) -> tlactive(mark(x))
mark(0) -> 0
mark(s(x)) -> s(x)
mark(cons(x, y)) -> cons(x, y)
addactive(cons(x, y)) -> incractive(cons(x, add(y)))
addactive(x) -> add(x)





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

MARK(hd(hd(x''))) -> HDACTIVE(hdactive(mark(x'')))
10 new Dependency Pairs are created:

MARK(hd(hd(x'''))) -> HDACTIVE(hd(mark(x''')))
MARK(hd(hd(nats))) -> HDACTIVE(hdactive(natsactive))
MARK(hd(hd(zeros))) -> HDACTIVE(hdactive(zerosactive))
MARK(hd(hd(incr(x')))) -> HDACTIVE(hdactive(incractive(mark(x'))))
MARK(hd(hd(add(x')))) -> HDACTIVE(hdactive(addactive(mark(x'))))
MARK(hd(hd(hd(x')))) -> HDACTIVE(hdactive(hdactive(mark(x'))))
MARK(hd(hd(tl(x')))) -> HDACTIVE(hdactive(tlactive(mark(x'))))
MARK(hd(hd(0))) -> HDACTIVE(hdactive(0))
MARK(hd(hd(s(x')))) -> HDACTIVE(hdactive(s(x')))
MARK(hd(hd(cons(x', y')))) -> HDACTIVE(hdactive(cons(x', y')))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 8
                 ↳Narrowing Transformation


Dependency Pairs:

MARK(hd(hd(cons(x', y')))) -> HDACTIVE(hdactive(cons(x', y')))
MARK(hd(hd(s(x')))) -> HDACTIVE(hdactive(s(x')))
MARK(hd(hd(0))) -> HDACTIVE(hdactive(0))
MARK(hd(hd(tl(x')))) -> HDACTIVE(hdactive(tlactive(mark(x'))))
MARK(hd(hd(hd(x')))) -> HDACTIVE(hdactive(hdactive(mark(x'))))
MARK(hd(hd(add(x')))) -> HDACTIVE(hdactive(addactive(mark(x'))))
MARK(hd(hd(incr(x')))) -> HDACTIVE(hdactive(incractive(mark(x'))))
MARK(hd(hd(zeros))) -> HDACTIVE(hdactive(zerosactive))
MARK(hd(hd(nats))) -> HDACTIVE(hdactive(natsactive))
MARK(hd(add(s(x')))) -> HDACTIVE(addactive(s(x')))
MARK(hd(add(0))) -> HDACTIVE(addactive(0))
MARK(hd(add(tl(x')))) -> HDACTIVE(addactive(tlactive(mark(x'))))
MARK(hd(add(hd(x')))) -> HDACTIVE(addactive(hdactive(mark(x'))))
MARK(hd(add(add(x')))) -> HDACTIVE(addactive(addactive(mark(x'))))
MARK(hd(add(incr(x')))) -> HDACTIVE(addactive(incractive(mark(x'))))
MARK(hd(add(zeros))) -> HDACTIVE(addactive(zerosactive))
MARK(hd(add(nats))) -> HDACTIVE(addactive(natsactive))
MARK(hd(incr(cons(x', y')))) -> HDACTIVE(incractive(cons(x', y')))
MARK(hd(incr(s(x')))) -> HDACTIVE(incractive(s(x')))
MARK(hd(incr(0))) -> HDACTIVE(incractive(0))
MARK(hd(incr(tl(x')))) -> HDACTIVE(incractive(tlactive(mark(x'))))
MARK(hd(incr(hd(x')))) -> HDACTIVE(incractive(hdactive(mark(x'))))
MARK(hd(incr(add(x')))) -> HDACTIVE(incractive(addactive(mark(x'))))
MARK(hd(incr(incr(x')))) -> HDACTIVE(incractive(incractive(mark(x'))))
MARK(hd(incr(zeros))) -> HDACTIVE(incractive(zerosactive))
MARK(hd(incr(nats))) -> HDACTIVE(incractive(natsactive))
MARK(hd(zeros)) -> HDACTIVE(cons(0, zeros))
MARK(hd(nats)) -> HDACTIVE(addactive(zerosactive))
MARK(tl(cons(x'', y'))) -> TLACTIVE(cons(x'', y'))
MARK(tl(tl(x''))) -> TLACTIVE(tlactive(mark(x'')))
MARK(tl(hd(x''))) -> TLACTIVE(hdactive(mark(x'')))
MARK(tl(add(x''))) -> TLACTIVE(addactive(mark(x'')))
MARK(tl(incr(x''))) -> TLACTIVE(incractive(mark(x'')))
MARK(tl(zeros)) -> TLACTIVE(zerosactive)
TLACTIVE(cons(x, y)) -> MARK(y)
MARK(tl(nats)) -> TLACTIVE(natsactive)
MARK(hd(cons(x'', y'))) -> HDACTIVE(cons(x'', y'))
MARK(hd(tl(x''))) -> HDACTIVE(tlactive(mark(x'')))
MARK(tl(x)) -> MARK(x)
MARK(hd(x)) -> MARK(x)
MARK(add(x)) -> MARK(x)
MARK(incr(x)) -> MARK(x)
HDACTIVE(cons(x, y)) -> MARK(x)
MARK(hd(add(cons(x', y')))) -> HDACTIVE(addactive(cons(x', y')))


Rules:


natsactive -> addactive(zerosactive)
natsactive -> nats
hdactive(x) -> hd(x)
hdactive(cons(x, y)) -> mark(x)
zerosactive -> cons(0, zeros)
zerosactive -> zeros
tlactive(x) -> tl(x)
tlactive(cons(x, y)) -> mark(y)
incractive(cons(x, y)) -> cons(s(x), incr(y))
incractive(x) -> incr(x)
mark(nats) -> natsactive
mark(zeros) -> zerosactive
mark(incr(x)) -> incractive(mark(x))
mark(add(x)) -> addactive(mark(x))
mark(hd(x)) -> hdactive(mark(x))
mark(tl(x)) -> tlactive(mark(x))
mark(0) -> 0
mark(s(x)) -> s(x)
mark(cons(x, y)) -> cons(x, y)
addactive(cons(x, y)) -> incractive(cons(x, add(y)))
addactive(x) -> add(x)





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

MARK(hd(tl(x''))) -> HDACTIVE(tlactive(mark(x'')))
10 new Dependency Pairs are created:

MARK(hd(tl(x'''))) -> HDACTIVE(tl(mark(x''')))
MARK(hd(tl(nats))) -> HDACTIVE(tlactive(natsactive))
MARK(hd(tl(zeros))) -> HDACTIVE(tlactive(zerosactive))
MARK(hd(tl(incr(x')))) -> HDACTIVE(tlactive(incractive(mark(x'))))
MARK(hd(tl(add(x')))) -> HDACTIVE(tlactive(addactive(mark(x'))))
MARK(hd(tl(hd(x')))) -> HDACTIVE(tlactive(hdactive(mark(x'))))
MARK(hd(tl(tl(x')))) -> HDACTIVE(tlactive(tlactive(mark(x'))))
MARK(hd(tl(0))) -> HDACTIVE(tlactive(0))
MARK(hd(tl(s(x')))) -> HDACTIVE(tlactive(s(x')))
MARK(hd(tl(cons(x', y')))) -> HDACTIVE(tlactive(cons(x', y')))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 9
                 ↳Narrowing Transformation


Dependency Pairs:

MARK(hd(tl(cons(x', y')))) -> HDACTIVE(tlactive(cons(x', y')))
MARK(hd(tl(s(x')))) -> HDACTIVE(tlactive(s(x')))
MARK(hd(tl(0))) -> HDACTIVE(tlactive(0))
MARK(hd(tl(tl(x')))) -> HDACTIVE(tlactive(tlactive(mark(x'))))
MARK(hd(tl(hd(x')))) -> HDACTIVE(tlactive(hdactive(mark(x'))))
MARK(hd(tl(add(x')))) -> HDACTIVE(tlactive(addactive(mark(x'))))
MARK(hd(tl(incr(x')))) -> HDACTIVE(tlactive(incractive(mark(x'))))
MARK(hd(tl(zeros))) -> HDACTIVE(tlactive(zerosactive))
MARK(hd(tl(nats))) -> HDACTIVE(tlactive(natsactive))
MARK(hd(hd(s(x')))) -> HDACTIVE(hdactive(s(x')))
MARK(hd(hd(0))) -> HDACTIVE(hdactive(0))
MARK(hd(hd(tl(x')))) -> HDACTIVE(hdactive(tlactive(mark(x'))))
MARK(hd(hd(hd(x')))) -> HDACTIVE(hdactive(hdactive(mark(x'))))
MARK(hd(hd(add(x')))) -> HDACTIVE(hdactive(addactive(mark(x'))))
MARK(hd(hd(incr(x')))) -> HDACTIVE(hdactive(incractive(mark(x'))))
MARK(hd(hd(zeros))) -> HDACTIVE(hdactive(zerosactive))
MARK(hd(hd(nats))) -> HDACTIVE(hdactive(natsactive))
MARK(hd(add(cons(x', y')))) -> HDACTIVE(addactive(cons(x', y')))
MARK(hd(add(s(x')))) -> HDACTIVE(addactive(s(x')))
MARK(hd(add(0))) -> HDACTIVE(addactive(0))
MARK(hd(add(tl(x')))) -> HDACTIVE(addactive(tlactive(mark(x'))))
MARK(hd(add(hd(x')))) -> HDACTIVE(addactive(hdactive(mark(x'))))
MARK(hd(add(add(x')))) -> HDACTIVE(addactive(addactive(mark(x'))))
MARK(hd(add(incr(x')))) -> HDACTIVE(addactive(incractive(mark(x'))))
MARK(hd(add(zeros))) -> HDACTIVE(addactive(zerosactive))
MARK(hd(add(nats))) -> HDACTIVE(addactive(natsactive))
MARK(hd(incr(cons(x', y')))) -> HDACTIVE(incractive(cons(x', y')))
MARK(hd(incr(s(x')))) -> HDACTIVE(incractive(s(x')))
MARK(hd(incr(0))) -> HDACTIVE(incractive(0))
MARK(hd(incr(tl(x')))) -> HDACTIVE(incractive(tlactive(mark(x'))))
MARK(hd(incr(hd(x')))) -> HDACTIVE(incractive(hdactive(mark(x'))))
MARK(hd(incr(add(x')))) -> HDACTIVE(incractive(addactive(mark(x'))))
MARK(hd(incr(incr(x')))) -> HDACTIVE(incractive(incractive(mark(x'))))
MARK(hd(incr(zeros))) -> HDACTIVE(incractive(zerosactive))
MARK(hd(incr(nats))) -> HDACTIVE(incractive(natsactive))
MARK(hd(zeros)) -> HDACTIVE(cons(0, zeros))
MARK(hd(nats)) -> HDACTIVE(addactive(zerosactive))
MARK(tl(cons(x'', y'))) -> TLACTIVE(cons(x'', y'))
MARK(tl(tl(x''))) -> TLACTIVE(tlactive(mark(x'')))
MARK(tl(hd(x''))) -> TLACTIVE(hdactive(mark(x'')))
MARK(tl(add(x''))) -> TLACTIVE(addactive(mark(x'')))
MARK(tl(incr(x''))) -> TLACTIVE(incractive(mark(x'')))
MARK(tl(zeros)) -> TLACTIVE(zerosactive)
TLACTIVE(cons(x, y)) -> MARK(y)
MARK(tl(nats)) -> TLACTIVE(natsactive)
MARK(hd(cons(x'', y'))) -> HDACTIVE(cons(x'', y'))
MARK(tl(x)) -> MARK(x)
MARK(hd(x)) -> MARK(x)
MARK(add(x)) -> MARK(x)
MARK(incr(x)) -> MARK(x)
HDACTIVE(cons(x, y)) -> MARK(x)
MARK(hd(hd(cons(x', y')))) -> HDACTIVE(hdactive(cons(x', y')))


Rules:


natsactive -> addactive(zerosactive)
natsactive -> nats
hdactive(x) -> hd(x)
hdactive(cons(x, y)) -> mark(x)
zerosactive -> cons(0, zeros)
zerosactive -> zeros
tlactive(x) -> tl(x)
tlactive(cons(x, y)) -> mark(y)
incractive(cons(x, y)) -> cons(s(x), incr(y))
incractive(x) -> incr(x)
mark(nats) -> natsactive
mark(zeros) -> zerosactive
mark(incr(x)) -> incractive(mark(x))
mark(add(x)) -> addactive(mark(x))
mark(hd(x)) -> hdactive(mark(x))
mark(tl(x)) -> tlactive(mark(x))
mark(0) -> 0
mark(s(x)) -> s(x)
mark(cons(x, y)) -> cons(x, y)
addactive(cons(x, y)) -> incractive(cons(x, add(y)))
addactive(x) -> add(x)





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

MARK(tl(nats)) -> TLACTIVE(natsactive)
two new Dependency Pairs are created:

MARK(tl(nats)) -> TLACTIVE(addactive(zerosactive))
MARK(tl(nats)) -> TLACTIVE(nats)

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 10
                 ↳Narrowing Transformation


Dependency Pairs:

MARK(tl(nats)) -> TLACTIVE(addactive(zerosactive))
MARK(hd(tl(s(x')))) -> HDACTIVE(tlactive(s(x')))
MARK(hd(tl(0))) -> HDACTIVE(tlactive(0))
MARK(hd(tl(tl(x')))) -> HDACTIVE(tlactive(tlactive(mark(x'))))
MARK(hd(tl(hd(x')))) -> HDACTIVE(tlactive(hdactive(mark(x'))))
MARK(hd(tl(add(x')))) -> HDACTIVE(tlactive(addactive(mark(x'))))
MARK(hd(tl(incr(x')))) -> HDACTIVE(tlactive(incractive(mark(x'))))
MARK(hd(tl(zeros))) -> HDACTIVE(tlactive(zerosactive))
MARK(hd(tl(nats))) -> HDACTIVE(tlactive(natsactive))
MARK(hd(hd(cons(x', y')))) -> HDACTIVE(hdactive(cons(x', y')))
MARK(hd(hd(s(x')))) -> HDACTIVE(hdactive(s(x')))
MARK(hd(hd(0))) -> HDACTIVE(hdactive(0))
MARK(hd(hd(tl(x')))) -> HDACTIVE(hdactive(tlactive(mark(x'))))
MARK(hd(hd(hd(x')))) -> HDACTIVE(hdactive(hdactive(mark(x'))))
MARK(hd(hd(add(x')))) -> HDACTIVE(hdactive(addactive(mark(x'))))
MARK(hd(hd(incr(x')))) -> HDACTIVE(hdactive(incractive(mark(x'))))
MARK(hd(hd(zeros))) -> HDACTIVE(hdactive(zerosactive))
MARK(hd(hd(nats))) -> HDACTIVE(hdactive(natsactive))
MARK(hd(add(cons(x', y')))) -> HDACTIVE(addactive(cons(x', y')))
MARK(hd(add(s(x')))) -> HDACTIVE(addactive(s(x')))
MARK(hd(add(0))) -> HDACTIVE(addactive(0))
MARK(hd(add(tl(x')))) -> HDACTIVE(addactive(tlactive(mark(x'))))
MARK(hd(add(hd(x')))) -> HDACTIVE(addactive(hdactive(mark(x'))))
MARK(hd(add(add(x')))) -> HDACTIVE(addactive(addactive(mark(x'))))
MARK(hd(add(incr(x')))) -> HDACTIVE(addactive(incractive(mark(x'))))
MARK(hd(add(zeros))) -> HDACTIVE(addactive(zerosactive))
MARK(hd(add(nats))) -> HDACTIVE(addactive(natsactive))
MARK(hd(incr(cons(x', y')))) -> HDACTIVE(incractive(cons(x', y')))
MARK(hd(incr(s(x')))) -> HDACTIVE(incractive(s(x')))
MARK(hd(incr(0))) -> HDACTIVE(incractive(0))
MARK(hd(incr(tl(x')))) -> HDACTIVE(incractive(tlactive(mark(x'))))
MARK(hd(incr(hd(x')))) -> HDACTIVE(incractive(hdactive(mark(x'))))
MARK(hd(incr(add(x')))) -> HDACTIVE(incractive(addactive(mark(x'))))
MARK(hd(incr(incr(x')))) -> HDACTIVE(incractive(incractive(mark(x'))))
MARK(hd(incr(zeros))) -> HDACTIVE(incractive(zerosactive))
MARK(hd(incr(nats))) -> HDACTIVE(incractive(natsactive))
MARK(hd(zeros)) -> HDACTIVE(cons(0, zeros))
MARK(hd(nats)) -> HDACTIVE(addactive(zerosactive))
MARK(tl(cons(x'', y'))) -> TLACTIVE(cons(x'', y'))
MARK(tl(tl(x''))) -> TLACTIVE(tlactive(mark(x'')))
MARK(tl(hd(x''))) -> TLACTIVE(hdactive(mark(x'')))
MARK(tl(add(x''))) -> TLACTIVE(addactive(mark(x'')))
MARK(tl(incr(x''))) -> TLACTIVE(incractive(mark(x'')))
TLACTIVE(cons(x, y)) -> MARK(y)
MARK(tl(zeros)) -> TLACTIVE(zerosactive)
MARK(hd(cons(x'', y'))) -> HDACTIVE(cons(x'', y'))
MARK(tl(x)) -> MARK(x)
MARK(hd(x)) -> MARK(x)
MARK(add(x)) -> MARK(x)
MARK(incr(x)) -> MARK(x)
HDACTIVE(cons(x, y)) -> MARK(x)
MARK(hd(tl(cons(x', y')))) -> HDACTIVE(tlactive(cons(x', y')))


Rules:


natsactive -> addactive(zerosactive)
natsactive -> nats
hdactive(x) -> hd(x)
hdactive(cons(x, y)) -> mark(x)
zerosactive -> cons(0, zeros)
zerosactive -> zeros
tlactive(x) -> tl(x)
tlactive(cons(x, y)) -> mark(y)
incractive(cons(x, y)) -> cons(s(x), incr(y))
incractive(x) -> incr(x)
mark(nats) -> natsactive
mark(zeros) -> zerosactive
mark(incr(x)) -> incractive(mark(x))
mark(add(x)) -> addactive(mark(x))
mark(hd(x)) -> hdactive(mark(x))
mark(tl(x)) -> tlactive(mark(x))
mark(0) -> 0
mark(s(x)) -> s(x)
mark(cons(x, y)) -> cons(x, y)
addactive(cons(x, y)) -> incractive(cons(x, add(y)))
addactive(x) -> add(x)





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

MARK(tl(zeros)) -> TLACTIVE(zerosactive)
two new Dependency Pairs are created:

MARK(tl(zeros)) -> TLACTIVE(cons(0, zeros))
MARK(tl(zeros)) -> TLACTIVE(zeros)

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 11
                 ↳Narrowing Transformation


Dependency Pairs:

MARK(tl(zeros)) -> TLACTIVE(cons(0, zeros))
MARK(hd(tl(cons(x', y')))) -> HDACTIVE(tlactive(cons(x', y')))
MARK(hd(tl(s(x')))) -> HDACTIVE(tlactive(s(x')))
MARK(hd(tl(0))) -> HDACTIVE(tlactive(0))
MARK(hd(tl(tl(x')))) -> HDACTIVE(tlactive(tlactive(mark(x'))))
MARK(hd(tl(hd(x')))) -> HDACTIVE(tlactive(hdactive(mark(x'))))
MARK(hd(tl(add(x')))) -> HDACTIVE(tlactive(addactive(mark(x'))))
MARK(hd(tl(incr(x')))) -> HDACTIVE(tlactive(incractive(mark(x'))))
MARK(hd(tl(zeros))) -> HDACTIVE(tlactive(zerosactive))
MARK(hd(tl(nats))) -> HDACTIVE(tlactive(natsactive))
MARK(hd(hd(cons(x', y')))) -> HDACTIVE(hdactive(cons(x', y')))
MARK(hd(hd(s(x')))) -> HDACTIVE(hdactive(s(x')))
MARK(hd(hd(0))) -> HDACTIVE(hdactive(0))
MARK(hd(hd(tl(x')))) -> HDACTIVE(hdactive(tlactive(mark(x'))))
MARK(hd(hd(hd(x')))) -> HDACTIVE(hdactive(hdactive(mark(x'))))
MARK(hd(hd(add(x')))) -> HDACTIVE(hdactive(addactive(mark(x'))))
MARK(hd(hd(incr(x')))) -> HDACTIVE(hdactive(incractive(mark(x'))))
MARK(hd(hd(zeros))) -> HDACTIVE(hdactive(zerosactive))
MARK(hd(hd(nats))) -> HDACTIVE(hdactive(natsactive))
MARK(hd(add(cons(x', y')))) -> HDACTIVE(addactive(cons(x', y')))
MARK(hd(add(s(x')))) -> HDACTIVE(addactive(s(x')))
MARK(hd(add(0))) -> HDACTIVE(addactive(0))
MARK(hd(add(tl(x')))) -> HDACTIVE(addactive(tlactive(mark(x'))))
MARK(hd(add(hd(x')))) -> HDACTIVE(addactive(hdactive(mark(x'))))
MARK(hd(add(add(x')))) -> HDACTIVE(addactive(addactive(mark(x'))))
MARK(hd(add(incr(x')))) -> HDACTIVE(addactive(incractive(mark(x'))))
MARK(hd(add(zeros))) -> HDACTIVE(addactive(zerosactive))
MARK(hd(add(nats))) -> HDACTIVE(addactive(natsactive))
MARK(hd(incr(cons(x', y')))) -> HDACTIVE(incractive(cons(x', y')))
MARK(hd(incr(s(x')))) -> HDACTIVE(incractive(s(x')))
MARK(hd(incr(0))) -> HDACTIVE(incractive(0))
MARK(hd(incr(tl(x')))) -> HDACTIVE(incractive(tlactive(mark(x'))))
MARK(hd(incr(hd(x')))) -> HDACTIVE(incractive(hdactive(mark(x'))))
MARK(hd(incr(add(x')))) -> HDACTIVE(incractive(addactive(mark(x'))))
MARK(hd(incr(incr(x')))) -> HDACTIVE(incractive(incractive(mark(x'))))
MARK(hd(incr(zeros))) -> HDACTIVE(incractive(zerosactive))
MARK(hd(incr(nats))) -> HDACTIVE(incractive(natsactive))
MARK(hd(zeros)) -> HDACTIVE(cons(0, zeros))
MARK(hd(nats)) -> HDACTIVE(addactive(zerosactive))
MARK(tl(cons(x'', y'))) -> TLACTIVE(cons(x'', y'))
MARK(tl(tl(x''))) -> TLACTIVE(tlactive(mark(x'')))
MARK(tl(hd(x''))) -> TLACTIVE(hdactive(mark(x'')))
MARK(tl(add(x''))) -> TLACTIVE(addactive(mark(x'')))
MARK(tl(incr(x''))) -> TLACTIVE(incractive(mark(x'')))
HDACTIVE(cons(x, y)) -> MARK(x)
MARK(hd(cons(x'', y'))) -> HDACTIVE(cons(x'', y'))
MARK(tl(x)) -> MARK(x)
MARK(hd(x)) -> MARK(x)
MARK(add(x)) -> MARK(x)
MARK(incr(x)) -> MARK(x)
TLACTIVE(cons(x, y)) -> MARK(y)
MARK(tl(nats)) -> TLACTIVE(addactive(zerosactive))


Rules:


natsactive -> addactive(zerosactive)
natsactive -> nats
hdactive(x) -> hd(x)
hdactive(cons(x, y)) -> mark(x)
zerosactive -> cons(0, zeros)
zerosactive -> zeros
tlactive(x) -> tl(x)
tlactive(cons(x, y)) -> mark(y)
incractive(cons(x, y)) -> cons(s(x), incr(y))
incractive(x) -> incr(x)
mark(nats) -> natsactive
mark(zeros) -> zerosactive
mark(incr(x)) -> incractive(mark(x))
mark(add(x)) -> addactive(mark(x))
mark(hd(x)) -> hdactive(mark(x))
mark(tl(x)) -> tlactive(mark(x))
mark(0) -> 0
mark(s(x)) -> s(x)
mark(cons(x, y)) -> cons(x, y)
addactive(cons(x, y)) -> incractive(cons(x, add(y)))
addactive(x) -> add(x)





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

MARK(tl(incr(x''))) -> TLACTIVE(incractive(mark(x'')))
10 new Dependency Pairs are created:

MARK(tl(incr(x'''))) -> TLACTIVE(incr(mark(x''')))
MARK(tl(incr(nats))) -> TLACTIVE(incractive(natsactive))
MARK(tl(incr(zeros))) -> TLACTIVE(incractive(zerosactive))
MARK(tl(incr(incr(x')))) -> TLACTIVE(incractive(incractive(mark(x'))))
MARK(tl(incr(add(x')))) -> TLACTIVE(incractive(addactive(mark(x'))))
MARK(tl(incr(hd(x')))) -> TLACTIVE(incractive(hdactive(mark(x'))))
MARK(tl(incr(tl(x')))) -> TLACTIVE(incractive(tlactive(mark(x'))))
MARK(tl(incr(0))) -> TLACTIVE(incractive(0))
MARK(tl(incr(s(x')))) -> TLACTIVE(incractive(s(x')))
MARK(tl(incr(cons(x', y')))) -> TLACTIVE(incractive(cons(x', y')))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 12
                 ↳Narrowing Transformation


Dependency Pairs:

MARK(tl(incr(cons(x', y')))) -> TLACTIVE(incractive(cons(x', y')))
MARK(tl(incr(s(x')))) -> TLACTIVE(incractive(s(x')))
MARK(tl(incr(0))) -> TLACTIVE(incractive(0))
MARK(tl(incr(tl(x')))) -> TLACTIVE(incractive(tlactive(mark(x'))))
MARK(tl(incr(hd(x')))) -> TLACTIVE(incractive(hdactive(mark(x'))))
MARK(tl(incr(add(x')))) -> TLACTIVE(incractive(addactive(mark(x'))))
MARK(tl(incr(incr(x')))) -> TLACTIVE(incractive(incractive(mark(x'))))
MARK(tl(incr(zeros))) -> TLACTIVE(incractive(zerosactive))
MARK(tl(incr(nats))) -> TLACTIVE(incractive(natsactive))
MARK(tl(nats)) -> TLACTIVE(addactive(zerosactive))
MARK(hd(tl(cons(x', y')))) -> HDACTIVE(tlactive(cons(x', y')))
MARK(hd(tl(s(x')))) -> HDACTIVE(tlactive(s(x')))
MARK(hd(tl(0))) -> HDACTIVE(tlactive(0))
MARK(hd(tl(tl(x')))) -> HDACTIVE(tlactive(tlactive(mark(x'))))
MARK(hd(tl(hd(x')))) -> HDACTIVE(tlactive(hdactive(mark(x'))))
MARK(hd(tl(add(x')))) -> HDACTIVE(tlactive(addactive(mark(x'))))
MARK(hd(tl(incr(x')))) -> HDACTIVE(tlactive(incractive(mark(x'))))
MARK(hd(tl(zeros))) -> HDACTIVE(tlactive(zerosactive))
MARK(hd(tl(nats))) -> HDACTIVE(tlactive(natsactive))
MARK(hd(hd(cons(x', y')))) -> HDACTIVE(hdactive(cons(x', y')))
MARK(hd(hd(s(x')))) -> HDACTIVE(hdactive(s(x')))
MARK(hd(hd(0))) -> HDACTIVE(hdactive(0))
MARK(hd(hd(tl(x')))) -> HDACTIVE(hdactive(tlactive(mark(x'))))
MARK(hd(hd(hd(x')))) -> HDACTIVE(hdactive(hdactive(mark(x'))))
MARK(hd(hd(add(x')))) -> HDACTIVE(hdactive(addactive(mark(x'))))
MARK(hd(hd(incr(x')))) -> HDACTIVE(hdactive(incractive(mark(x'))))
MARK(hd(hd(zeros))) -> HDACTIVE(hdactive(zerosactive))
MARK(hd(hd(nats))) -> HDACTIVE(hdactive(natsactive))
MARK(hd(add(cons(x', y')))) -> HDACTIVE(addactive(cons(x', y')))
MARK(hd(add(s(x')))) -> HDACTIVE(addactive(s(x')))
MARK(hd(add(0))) -> HDACTIVE(addactive(0))
MARK(hd(add(tl(x')))) -> HDACTIVE(addactive(tlactive(mark(x'))))
MARK(hd(add(hd(x')))) -> HDACTIVE(addactive(hdactive(mark(x'))))
MARK(hd(add(add(x')))) -> HDACTIVE(addactive(addactive(mark(x'))))
MARK(hd(add(incr(x')))) -> HDACTIVE(addactive(incractive(mark(x'))))
MARK(hd(add(zeros))) -> HDACTIVE(addactive(zerosactive))
MARK(hd(add(nats))) -> HDACTIVE(addactive(natsactive))
MARK(hd(incr(cons(x', y')))) -> HDACTIVE(incractive(cons(x', y')))
MARK(hd(incr(s(x')))) -> HDACTIVE(incractive(s(x')))
MARK(hd(incr(0))) -> HDACTIVE(incractive(0))
MARK(hd(incr(tl(x')))) -> HDACTIVE(incractive(tlactive(mark(x'))))
MARK(hd(incr(hd(x')))) -> HDACTIVE(incractive(hdactive(mark(x'))))
MARK(hd(incr(add(x')))) -> HDACTIVE(incractive(addactive(mark(x'))))
MARK(hd(incr(incr(x')))) -> HDACTIVE(incractive(incractive(mark(x'))))
MARK(hd(incr(zeros))) -> HDACTIVE(incractive(zerosactive))
MARK(hd(incr(nats))) -> HDACTIVE(incractive(natsactive))
MARK(hd(zeros)) -> HDACTIVE(cons(0, zeros))
MARK(hd(nats)) -> HDACTIVE(addactive(zerosactive))
MARK(tl(cons(x'', y'))) -> TLACTIVE(cons(x'', y'))
MARK(tl(tl(x''))) -> TLACTIVE(tlactive(mark(x'')))
MARK(tl(hd(x''))) -> TLACTIVE(hdactive(mark(x'')))
MARK(tl(add(x''))) -> TLACTIVE(addactive(mark(x'')))
HDACTIVE(cons(x, y)) -> MARK(x)
MARK(hd(cons(x'', y'))) -> HDACTIVE(cons(x'', y'))
MARK(tl(x)) -> MARK(x)
MARK(hd(x)) -> MARK(x)
MARK(add(x)) -> MARK(x)
MARK(incr(x)) -> MARK(x)
TLACTIVE(cons(x, y)) -> MARK(y)
MARK(tl(zeros)) -> TLACTIVE(cons(0, zeros))


Rules:


natsactive -> addactive(zerosactive)
natsactive -> nats
hdactive(x) -> hd(x)
hdactive(cons(x, y)) -> mark(x)
zerosactive -> cons(0, zeros)
zerosactive -> zeros
tlactive(x) -> tl(x)
tlactive(cons(x, y)) -> mark(y)
incractive(cons(x, y)) -> cons(s(x), incr(y))
incractive(x) -> incr(x)
mark(nats) -> natsactive
mark(zeros) -> zerosactive
mark(incr(x)) -> incractive(mark(x))
mark(add(x)) -> addactive(mark(x))
mark(hd(x)) -> hdactive(mark(x))
mark(tl(x)) -> tlactive(mark(x))
mark(0) -> 0
mark(s(x)) -> s(x)
mark(cons(x, y)) -> cons(x, y)
addactive(cons(x, y)) -> incractive(cons(x, add(y)))
addactive(x) -> add(x)





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

MARK(tl(add(x''))) -> TLACTIVE(addactive(mark(x'')))
10 new Dependency Pairs are created:

MARK(tl(add(x'''))) -> TLACTIVE(add(mark(x''')))
MARK(tl(add(nats))) -> TLACTIVE(addactive(natsactive))
MARK(tl(add(zeros))) -> TLACTIVE(addactive(zerosactive))
MARK(tl(add(incr(x')))) -> TLACTIVE(addactive(incractive(mark(x'))))
MARK(tl(add(add(x')))) -> TLACTIVE(addactive(addactive(mark(x'))))
MARK(tl(add(hd(x')))) -> TLACTIVE(addactive(hdactive(mark(x'))))
MARK(tl(add(tl(x')))) -> TLACTIVE(addactive(tlactive(mark(x'))))
MARK(tl(add(0))) -> TLACTIVE(addactive(0))
MARK(tl(add(s(x')))) -> TLACTIVE(addactive(s(x')))
MARK(tl(add(cons(x', y')))) -> TLACTIVE(addactive(cons(x', y')))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 13
                 ↳Narrowing Transformation


Dependency Pairs:

MARK(tl(add(cons(x', y')))) -> TLACTIVE(addactive(cons(x', y')))
MARK(tl(add(s(x')))) -> TLACTIVE(addactive(s(x')))
MARK(tl(add(0))) -> TLACTIVE(addactive(0))
MARK(tl(add(tl(x')))) -> TLACTIVE(addactive(tlactive(mark(x'))))
MARK(tl(add(hd(x')))) -> TLACTIVE(addactive(hdactive(mark(x'))))
MARK(tl(add(add(x')))) -> TLACTIVE(addactive(addactive(mark(x'))))
MARK(tl(add(incr(x')))) -> TLACTIVE(addactive(incractive(mark(x'))))
MARK(tl(add(zeros))) -> TLACTIVE(addactive(zerosactive))
MARK(tl(add(nats))) -> TLACTIVE(addactive(natsactive))
MARK(tl(incr(s(x')))) -> TLACTIVE(incractive(s(x')))
MARK(tl(incr(0))) -> TLACTIVE(incractive(0))
MARK(tl(incr(tl(x')))) -> TLACTIVE(incractive(tlactive(mark(x'))))
MARK(tl(incr(hd(x')))) -> TLACTIVE(incractive(hdactive(mark(x'))))
MARK(tl(incr(add(x')))) -> TLACTIVE(incractive(addactive(mark(x'))))
MARK(tl(incr(incr(x')))) -> TLACTIVE(incractive(incractive(mark(x'))))
MARK(tl(incr(zeros))) -> TLACTIVE(incractive(zerosactive))
MARK(tl(incr(nats))) -> TLACTIVE(incractive(natsactive))
MARK(tl(zeros)) -> TLACTIVE(cons(0, zeros))
MARK(tl(nats)) -> TLACTIVE(addactive(zerosactive))
MARK(hd(tl(cons(x', y')))) -> HDACTIVE(tlactive(cons(x', y')))
MARK(hd(tl(s(x')))) -> HDACTIVE(tlactive(s(x')))
MARK(hd(tl(0))) -> HDACTIVE(tlactive(0))
MARK(hd(tl(tl(x')))) -> HDACTIVE(tlactive(tlactive(mark(x'))))
MARK(hd(tl(hd(x')))) -> HDACTIVE(tlactive(hdactive(mark(x'))))
MARK(hd(tl(add(x')))) -> HDACTIVE(tlactive(addactive(mark(x'))))
MARK(hd(tl(incr(x')))) -> HDACTIVE(tlactive(incractive(mark(x'))))
MARK(hd(tl(zeros))) -> HDACTIVE(tlactive(zerosactive))
MARK(hd(tl(nats))) -> HDACTIVE(tlactive(natsactive))
MARK(hd(hd(cons(x', y')))) -> HDACTIVE(hdactive(cons(x', y')))
MARK(hd(hd(s(x')))) -> HDACTIVE(hdactive(s(x')))
MARK(hd(hd(0))) -> HDACTIVE(hdactive(0))
MARK(hd(hd(tl(x')))) -> HDACTIVE(hdactive(tlactive(mark(x'))))
MARK(hd(hd(hd(x')))) -> HDACTIVE(hdactive(hdactive(mark(x'))))
MARK(hd(hd(add(x')))) -> HDACTIVE(hdactive(addactive(mark(x'))))
MARK(hd(hd(incr(x')))) -> HDACTIVE(hdactive(incractive(mark(x'))))
MARK(hd(hd(zeros))) -> HDACTIVE(hdactive(zerosactive))
MARK(hd(hd(nats))) -> HDACTIVE(hdactive(natsactive))
MARK(hd(add(cons(x', y')))) -> HDACTIVE(addactive(cons(x', y')))
MARK(hd(add(s(x')))) -> HDACTIVE(addactive(s(x')))
MARK(hd(add(0))) -> HDACTIVE(addactive(0))
MARK(hd(add(tl(x')))) -> HDACTIVE(addactive(tlactive(mark(x'))))
MARK(hd(add(hd(x')))) -> HDACTIVE(addactive(hdactive(mark(x'))))
MARK(hd(add(add(x')))) -> HDACTIVE(addactive(addactive(mark(x'))))
MARK(hd(add(incr(x')))) -> HDACTIVE(addactive(incractive(mark(x'))))
MARK(hd(add(zeros))) -> HDACTIVE(addactive(zerosactive))
MARK(hd(add(nats))) -> HDACTIVE(addactive(natsactive))
MARK(hd(incr(cons(x', y')))) -> HDACTIVE(incractive(cons(x', y')))
MARK(hd(incr(s(x')))) -> HDACTIVE(incractive(s(x')))
MARK(hd(incr(0))) -> HDACTIVE(incractive(0))
MARK(hd(incr(tl(x')))) -> HDACTIVE(incractive(tlactive(mark(x'))))
MARK(hd(incr(hd(x')))) -> HDACTIVE(incractive(hdactive(mark(x'))))
MARK(hd(incr(add(x')))) -> HDACTIVE(incractive(addactive(mark(x'))))
MARK(hd(incr(incr(x')))) -> HDACTIVE(incractive(incractive(mark(x'))))
MARK(hd(incr(zeros))) -> HDACTIVE(incractive(zerosactive))
MARK(hd(incr(nats))) -> HDACTIVE(incractive(natsactive))
MARK(hd(zeros)) -> HDACTIVE(cons(0, zeros))
MARK(hd(nats)) -> HDACTIVE(addactive(zerosactive))
MARK(tl(cons(x'', y'))) -> TLACTIVE(cons(x'', y'))
MARK(tl(tl(x''))) -> TLACTIVE(tlactive(mark(x'')))
MARK(tl(hd(x''))) -> TLACTIVE(hdactive(mark(x'')))
HDACTIVE(cons(x, y)) -> MARK(x)
MARK(hd(cons(x'', y'))) -> HDACTIVE(cons(x'', y'))
MARK(tl(x)) -> MARK(x)
MARK(hd(x)) -> MARK(x)
MARK(add(x)) -> MARK(x)
MARK(incr(x)) -> MARK(x)
TLACTIVE(cons(x, y)) -> MARK(y)
MARK(tl(incr(cons(x', y')))) -> TLACTIVE(incractive(cons(x', y')))


Rules:


natsactive -> addactive(zerosactive)
natsactive -> nats
hdactive(x) -> hd(x)
hdactive(cons(x, y)) -> mark(x)
zerosactive -> cons(0, zeros)
zerosactive -> zeros
tlactive(x) -> tl(x)
tlactive(cons(x, y)) -> mark(y)
incractive(cons(x, y)) -> cons(s(x), incr(y))
incractive(x) -> incr(x)
mark(nats) -> natsactive
mark(zeros) -> zerosactive
mark(incr(x)) -> incractive(mark(x))
mark(add(x)) -> addactive(mark(x))
mark(hd(x)) -> hdactive(mark(x))
mark(tl(x)) -> tlactive(mark(x))
mark(0) -> 0
mark(s(x)) -> s(x)
mark(cons(x, y)) -> cons(x, y)
addactive(cons(x, y)) -> incractive(cons(x, add(y)))
addactive(x) -> add(x)





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

MARK(tl(hd(x''))) -> TLACTIVE(hdactive(mark(x'')))
10 new Dependency Pairs are created:

MARK(tl(hd(x'''))) -> TLACTIVE(hd(mark(x''')))
MARK(tl(hd(nats))) -> TLACTIVE(hdactive(natsactive))
MARK(tl(hd(zeros))) -> TLACTIVE(hdactive(zerosactive))
MARK(tl(hd(incr(x')))) -> TLACTIVE(hdactive(incractive(mark(x'))))
MARK(tl(hd(add(x')))) -> TLACTIVE(hdactive(addactive(mark(x'))))
MARK(tl(hd(hd(x')))) -> TLACTIVE(hdactive(hdactive(mark(x'))))
MARK(tl(hd(tl(x')))) -> TLACTIVE(hdactive(tlactive(mark(x'))))
MARK(tl(hd(0))) -> TLACTIVE(hdactive(0))
MARK(tl(hd(s(x')))) -> TLACTIVE(hdactive(s(x')))
MARK(tl(hd(cons(x', y')))) -> TLACTIVE(hdactive(cons(x', y')))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 14
                 ↳Narrowing Transformation


Dependency Pairs:

MARK(tl(hd(cons(x', y')))) -> TLACTIVE(hdactive(cons(x', y')))
MARK(tl(hd(s(x')))) -> TLACTIVE(hdactive(s(x')))
MARK(tl(hd(0))) -> TLACTIVE(hdactive(0))
MARK(tl(hd(tl(x')))) -> TLACTIVE(hdactive(tlactive(mark(x'))))
MARK(tl(hd(hd(x')))) -> TLACTIVE(hdactive(hdactive(mark(x'))))
MARK(tl(hd(add(x')))) -> TLACTIVE(hdactive(addactive(mark(x'))))
MARK(tl(hd(incr(x')))) -> TLACTIVE(hdactive(incractive(mark(x'))))
MARK(tl(hd(zeros))) -> TLACTIVE(hdactive(zerosactive))
MARK(tl(hd(nats))) -> TLACTIVE(hdactive(natsactive))
MARK(tl(add(s(x')))) -> TLACTIVE(addactive(s(x')))
MARK(tl(add(0))) -> TLACTIVE(addactive(0))
MARK(tl(add(tl(x')))) -> TLACTIVE(addactive(tlactive(mark(x'))))
MARK(tl(add(hd(x')))) -> TLACTIVE(addactive(hdactive(mark(x'))))
MARK(tl(add(add(x')))) -> TLACTIVE(addactive(addactive(mark(x'))))
MARK(tl(add(incr(x')))) -> TLACTIVE(addactive(incractive(mark(x'))))
MARK(tl(add(zeros))) -> TLACTIVE(addactive(zerosactive))
MARK(tl(add(nats))) -> TLACTIVE(addactive(natsactive))
MARK(tl(incr(cons(x', y')))) -> TLACTIVE(incractive(cons(x', y')))
MARK(tl(incr(s(x')))) -> TLACTIVE(incractive(s(x')))
MARK(tl(incr(0))) -> TLACTIVE(incractive(0))
MARK(tl(incr(tl(x')))) -> TLACTIVE(incractive(tlactive(mark(x'))))
MARK(tl(incr(hd(x')))) -> TLACTIVE(incractive(hdactive(mark(x'))))
MARK(tl(incr(add(x')))) -> TLACTIVE(incractive(addactive(mark(x'))))
MARK(tl(incr(incr(x')))) -> TLACTIVE(incractive(incractive(mark(x'))))
MARK(tl(incr(zeros))) -> TLACTIVE(incractive(zerosactive))
MARK(tl(incr(nats))) -> TLACTIVE(incractive(natsactive))
MARK(tl(zeros)) -> TLACTIVE(cons(0, zeros))
MARK(tl(nats)) -> TLACTIVE(addactive(zerosactive))
MARK(hd(tl(cons(x', y')))) -> HDACTIVE(tlactive(cons(x', y')))
MARK(hd(tl(s(x')))) -> HDACTIVE(tlactive(s(x')))
MARK(hd(tl(0))) -> HDACTIVE(tlactive(0))
MARK(hd(tl(tl(x')))) -> HDACTIVE(tlactive(tlactive(mark(x'))))
MARK(hd(tl(hd(x')))) -> HDACTIVE(tlactive(hdactive(mark(x'))))
MARK(hd(tl(add(x')))) -> HDACTIVE(tlactive(addactive(mark(x'))))
MARK(hd(tl(incr(x')))) -> HDACTIVE(tlactive(incractive(mark(x'))))
MARK(hd(tl(zeros))) -> HDACTIVE(tlactive(zerosactive))
MARK(hd(tl(nats))) -> HDACTIVE(tlactive(natsactive))
MARK(hd(hd(cons(x', y')))) -> HDACTIVE(hdactive(cons(x', y')))
MARK(hd(hd(s(x')))) -> HDACTIVE(hdactive(s(x')))
MARK(hd(hd(0))) -> HDACTIVE(hdactive(0))
MARK(hd(hd(tl(x')))) -> HDACTIVE(hdactive(tlactive(mark(x'))))
MARK(hd(hd(hd(x')))) -> HDACTIVE(hdactive(hdactive(mark(x'))))
MARK(hd(hd(add(x')))) -> HDACTIVE(hdactive(addactive(mark(x'))))
MARK(hd(hd(incr(x')))) -> HDACTIVE(hdactive(incractive(mark(x'))))
MARK(hd(hd(zeros))) -> HDACTIVE(hdactive(zerosactive))
MARK(hd(hd(nats))) -> HDACTIVE(hdactive(natsactive))
MARK(hd(add(cons(x', y')))) -> HDACTIVE(addactive(cons(x', y')))
MARK(hd(add(s(x')))) -> HDACTIVE(addactive(s(x')))
MARK(hd(add(0))) -> HDACTIVE(addactive(0))
MARK(hd(add(tl(x')))) -> HDACTIVE(addactive(tlactive(mark(x'))))
MARK(hd(add(hd(x')))) -> HDACTIVE(addactive(hdactive(mark(x'))))
MARK(hd(add(add(x')))) -> HDACTIVE(addactive(addactive(mark(x'))))
MARK(hd(add(incr(x')))) -> HDACTIVE(addactive(incractive(mark(x'))))
MARK(hd(add(zeros))) -> HDACTIVE(addactive(zerosactive))
MARK(hd(add(nats))) -> HDACTIVE(addactive(natsactive))
MARK(hd(incr(cons(x', y')))) -> HDACTIVE(incractive(cons(x', y')))
MARK(hd(incr(s(x')))) -> HDACTIVE(incractive(s(x')))
MARK(hd(incr(0))) -> HDACTIVE(incractive(0))
MARK(hd(incr(tl(x')))) -> HDACTIVE(incractive(tlactive(mark(x'))))
MARK(hd(incr(hd(x')))) -> HDACTIVE(incractive(hdactive(mark(x'))))
MARK(hd(incr(add(x')))) -> HDACTIVE(incractive(addactive(mark(x'))))
MARK(hd(incr(incr(x')))) -> HDACTIVE(incractive(incractive(mark(x'))))
MARK(hd(incr(zeros))) -> HDACTIVE(incractive(zerosactive))
MARK(hd(incr(nats))) -> HDACTIVE(incractive(natsactive))
MARK(hd(zeros)) -> HDACTIVE(cons(0, zeros))
MARK(hd(nats)) -> HDACTIVE(addactive(zerosactive))
MARK(tl(cons(x'', y'))) -> TLACTIVE(cons(x'', y'))
MARK(tl(tl(x''))) -> TLACTIVE(tlactive(mark(x'')))
HDACTIVE(cons(x, y)) -> MARK(x)
MARK(hd(cons(x'', y'))) -> HDACTIVE(cons(x'', y'))
MARK(tl(x)) -> MARK(x)
MARK(hd(x)) -> MARK(x)
MARK(add(x)) -> MARK(x)
MARK(incr(x)) -> MARK(x)
TLACTIVE(cons(x, y)) -> MARK(y)
MARK(tl(add(cons(x', y')))) -> TLACTIVE(addactive(cons(x', y')))


Rules:


natsactive -> addactive(zerosactive)
natsactive -> nats
hdactive(x) -> hd(x)
hdactive(cons(x, y)) -> mark(x)
zerosactive -> cons(0, zeros)
zerosactive -> zeros
tlactive(x) -> tl(x)
tlactive(cons(x, y)) -> mark(y)
incractive(cons(x, y)) -> cons(s(x), incr(y))
incractive(x) -> incr(x)
mark(nats) -> natsactive
mark(zeros) -> zerosactive
mark(incr(x)) -> incractive(mark(x))
mark(add(x)) -> addactive(mark(x))
mark(hd(x)) -> hdactive(mark(x))
mark(tl(x)) -> tlactive(mark(x))
mark(0) -> 0
mark(s(x)) -> s(x)
mark(cons(x, y)) -> cons(x, y)
addactive(cons(x, y)) -> incractive(cons(x, add(y)))
addactive(x) -> add(x)





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

MARK(tl(tl(x''))) -> TLACTIVE(tlactive(mark(x'')))
10 new Dependency Pairs are created:

MARK(tl(tl(x'''))) -> TLACTIVE(tl(mark(x''')))
MARK(tl(tl(nats))) -> TLACTIVE(tlactive(natsactive))
MARK(tl(tl(zeros))) -> TLACTIVE(tlactive(zerosactive))
MARK(tl(tl(incr(x')))) -> TLACTIVE(tlactive(incractive(mark(x'))))
MARK(tl(tl(add(x')))) -> TLACTIVE(tlactive(addactive(mark(x'))))
MARK(tl(tl(hd(x')))) -> TLACTIVE(tlactive(hdactive(mark(x'))))
MARK(tl(tl(tl(x')))) -> TLACTIVE(tlactive(tlactive(mark(x'))))
MARK(tl(tl(0))) -> TLACTIVE(tlactive(0))
MARK(tl(tl(s(x')))) -> TLACTIVE(tlactive(s(x')))
MARK(tl(tl(cons(x', y')))) -> TLACTIVE(tlactive(cons(x', y')))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 15
                 ↳Polynomial Ordering


Dependency Pairs:

MARK(tl(tl(cons(x', y')))) -> TLACTIVE(tlactive(cons(x', y')))
MARK(tl(tl(s(x')))) -> TLACTIVE(tlactive(s(x')))
MARK(tl(tl(0))) -> TLACTIVE(tlactive(0))
MARK(tl(tl(tl(x')))) -> TLACTIVE(tlactive(tlactive(mark(x'))))
MARK(tl(tl(hd(x')))) -> TLACTIVE(tlactive(hdactive(mark(x'))))
MARK(tl(tl(add(x')))) -> TLACTIVE(tlactive(addactive(mark(x'))))
MARK(tl(tl(incr(x')))) -> TLACTIVE(tlactive(incractive(mark(x'))))
MARK(tl(tl(zeros))) -> TLACTIVE(tlactive(zerosactive))
MARK(tl(tl(nats))) -> TLACTIVE(tlactive(natsactive))
MARK(tl(hd(s(x')))) -> TLACTIVE(hdactive(s(x')))
MARK(tl(hd(0))) -> TLACTIVE(hdactive(0))
MARK(tl(hd(tl(x')))) -> TLACTIVE(hdactive(tlactive(mark(x'))))
MARK(tl(hd(hd(x')))) -> TLACTIVE(hdactive(hdactive(mark(x'))))
MARK(tl(hd(add(x')))) -> TLACTIVE(hdactive(addactive(mark(x'))))
MARK(tl(hd(incr(x')))) -> TLACTIVE(hdactive(incractive(mark(x'))))
MARK(tl(hd(zeros))) -> TLACTIVE(hdactive(zerosactive))
MARK(tl(hd(nats))) -> TLACTIVE(hdactive(natsactive))
MARK(tl(add(cons(x', y')))) -> TLACTIVE(addactive(cons(x', y')))
MARK(tl(add(s(x')))) -> TLACTIVE(addactive(s(x')))
MARK(tl(add(0))) -> TLACTIVE(addactive(0))
MARK(tl(add(tl(x')))) -> TLACTIVE(addactive(tlactive(mark(x'))))
MARK(tl(add(hd(x')))) -> TLACTIVE(addactive(hdactive(mark(x'))))
MARK(tl(add(add(x')))) -> TLACTIVE(addactive(addactive(mark(x'))))
MARK(tl(add(incr(x')))) -> TLACTIVE(addactive(incractive(mark(x'))))
MARK(tl(add(zeros))) -> TLACTIVE(addactive(zerosactive))
MARK(tl(add(nats))) -> TLACTIVE(addactive(natsactive))
MARK(tl(incr(cons(x', y')))) -> TLACTIVE(incractive(cons(x', y')))
MARK(tl(incr(s(x')))) -> TLACTIVE(incractive(s(x')))
MARK(tl(incr(0))) -> TLACTIVE(incractive(0))
MARK(tl(incr(tl(x')))) -> TLACTIVE(incractive(tlactive(mark(x'))))
MARK(tl(incr(hd(x')))) -> TLACTIVE(incractive(hdactive(mark(x'))))
MARK(tl(incr(add(x')))) -> TLACTIVE(incractive(addactive(mark(x'))))
MARK(tl(incr(incr(x')))) -> TLACTIVE(incractive(incractive(mark(x'))))
MARK(tl(incr(zeros))) -> TLACTIVE(incractive(zerosactive))
MARK(tl(incr(nats))) -> TLACTIVE(incractive(natsactive))
MARK(tl(zeros)) -> TLACTIVE(cons(0, zeros))
MARK(tl(nats)) -> TLACTIVE(addactive(zerosactive))
MARK(hd(tl(cons(x', y')))) -> HDACTIVE(tlactive(cons(x', y')))
MARK(hd(tl(s(x')))) -> HDACTIVE(tlactive(s(x')))
MARK(hd(tl(0))) -> HDACTIVE(tlactive(0))
MARK(hd(tl(tl(x')))) -> HDACTIVE(tlactive(tlactive(mark(x'))))
MARK(hd(tl(hd(x')))) -> HDACTIVE(tlactive(hdactive(mark(x'))))
MARK(hd(tl(add(x')))) -> HDACTIVE(tlactive(addactive(mark(x'))))
MARK(hd(tl(incr(x')))) -> HDACTIVE(tlactive(incractive(mark(x'))))
MARK(hd(tl(zeros))) -> HDACTIVE(tlactive(zerosactive))
MARK(hd(tl(nats))) -> HDACTIVE(tlactive(natsactive))
MARK(hd(hd(cons(x', y')))) -> HDACTIVE(hdactive(cons(x', y')))
MARK(hd(hd(s(x')))) -> HDACTIVE(hdactive(s(x')))
MARK(hd(hd(0))) -> HDACTIVE(hdactive(0))
MARK(hd(hd(tl(x')))) -> HDACTIVE(hdactive(tlactive(mark(x'))))
MARK(hd(hd(hd(x')))) -> HDACTIVE(hdactive(hdactive(mark(x'))))
MARK(hd(hd(add(x')))) -> HDACTIVE(hdactive(addactive(mark(x'))))
MARK(hd(hd(incr(x')))) -> HDACTIVE(hdactive(incractive(mark(x'))))
MARK(hd(hd(zeros))) -> HDACTIVE(hdactive(zerosactive))
MARK(hd(hd(nats))) -> HDACTIVE(hdactive(natsactive))
MARK(hd(add(cons(x', y')))) -> HDACTIVE(addactive(cons(x', y')))
MARK(hd(add(s(x')))) -> HDACTIVE(addactive(s(x')))
MARK(hd(add(0))) -> HDACTIVE(addactive(0))
MARK(hd(add(tl(x')))) -> HDACTIVE(addactive(tlactive(mark(x'))))
MARK(hd(add(hd(x')))) -> HDACTIVE(addactive(hdactive(mark(x'))))
MARK(hd(add(add(x')))) -> HDACTIVE(addactive(addactive(mark(x'))))
MARK(hd(add(incr(x')))) -> HDACTIVE(addactive(incractive(mark(x'))))
MARK(hd(add(zeros))) -> HDACTIVE(addactive(zerosactive))
MARK(hd(add(nats))) -> HDACTIVE(addactive(natsactive))
MARK(hd(incr(cons(x', y')))) -> HDACTIVE(incractive(cons(x', y')))
MARK(hd(incr(s(x')))) -> HDACTIVE(incractive(s(x')))
MARK(hd(incr(0))) -> HDACTIVE(incractive(0))
MARK(hd(incr(tl(x')))) -> HDACTIVE(incractive(tlactive(mark(x'))))
MARK(hd(incr(hd(x')))) -> HDACTIVE(incractive(hdactive(mark(x'))))
MARK(hd(incr(add(x')))) -> HDACTIVE(incractive(addactive(mark(x'))))
MARK(hd(incr(incr(x')))) -> HDACTIVE(incractive(incractive(mark(x'))))
MARK(hd(incr(zeros))) -> HDACTIVE(incractive(zerosactive))
MARK(hd(incr(nats))) -> HDACTIVE(incractive(natsactive))
MARK(hd(zeros)) -> HDACTIVE(cons(0, zeros))
MARK(hd(nats)) -> HDACTIVE(addactive(zerosactive))
MARK(tl(cons(x'', y'))) -> TLACTIVE(cons(x'', y'))
HDACTIVE(cons(x, y)) -> MARK(x)
MARK(hd(cons(x'', y'))) -> HDACTIVE(cons(x'', y'))
MARK(tl(x)) -> MARK(x)
MARK(hd(x)) -> MARK(x)
MARK(add(x)) -> MARK(x)
MARK(incr(x)) -> MARK(x)
TLACTIVE(cons(x, y)) -> MARK(y)
MARK(tl(hd(cons(x', y')))) -> TLACTIVE(hdactive(cons(x', y')))


Rules:


natsactive -> addactive(zerosactive)
natsactive -> nats
hdactive(x) -> hd(x)
hdactive(cons(x, y)) -> mark(x)
zerosactive -> cons(0, zeros)
zerosactive -> zeros
tlactive(x) -> tl(x)
tlactive(cons(x, y)) -> mark(y)
incractive(cons(x, y)) -> cons(s(x), incr(y))
incractive(x) -> incr(x)
mark(nats) -> natsactive
mark(zeros) -> zerosactive
mark(incr(x)) -> incractive(mark(x))
mark(add(x)) -> addactive(mark(x))
mark(hd(x)) -> hdactive(mark(x))
mark(tl(x)) -> tlactive(mark(x))
mark(0) -> 0
mark(s(x)) -> s(x)
mark(cons(x, y)) -> cons(x, y)
addactive(cons(x, y)) -> incractive(cons(x, add(y)))
addactive(x) -> add(x)





The following dependency pairs can be strictly oriented:

MARK(tl(nats)) -> TLACTIVE(addactive(zerosactive))
MARK(hd(nats)) -> HDACTIVE(addactive(zerosactive))


Additionally, the following usable rules w.r.t. to the implicit AFS can be oriented:

hdactive(x) -> hd(x)
hdactive(cons(x, y)) -> mark(x)
mark(nats) -> natsactive
mark(zeros) -> zerosactive
mark(incr(x)) -> incractive(mark(x))
mark(add(x)) -> addactive(mark(x))
mark(hd(x)) -> hdactive(mark(x))
mark(tl(x)) -> tlactive(mark(x))
mark(0) -> 0
mark(s(x)) -> s(x)
mark(cons(x, y)) -> cons(x, y)
tlactive(x) -> tl(x)
tlactive(cons(x, y)) -> mark(y)
addactive(cons(x, y)) -> incractive(cons(x, add(y)))
addactive(x) -> add(x)
incractive(cons(x, y)) -> cons(s(x), incr(y))
incractive(x) -> incr(x)
zerosactive -> cons(0, zeros)
zerosactive -> zeros
natsactive -> addactive(zerosactive)
natsactive -> nats


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(zeros_active)=  0  
  POL(hd_active(x1))=  x1  
  POL(incr_active(x1))=  x1  
  POL(MARK(x1))=  x1  
  POL(TL_ACTIVE(x1))=  x1  
  POL(incr(x1))=  x1  
  POL(mark(x1))=  x1  
  POL(tl(x1))=  x1  
  POL(HD_ACTIVE(x1))=  x1  
  POL(add(x1))=  x1  
  POL(add_active(x1))=  x1  
  POL(0)=  0  
  POL(cons(x1, x2))=  x1 + x2  
  POL(hd(x1))=  x1  
  POL(nats)=  1  
  POL(s(x1))=  0  
  POL(zeros)=  0  
  POL(nats_active)=  1  
  POL(tl_active(x1))=  x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 16
                 ↳Polynomial Ordering


Dependency Pairs:

MARK(tl(tl(cons(x', y')))) -> TLACTIVE(tlactive(cons(x', y')))
MARK(tl(tl(s(x')))) -> TLACTIVE(tlactive(s(x')))
MARK(tl(tl(0))) -> TLACTIVE(tlactive(0))
MARK(tl(tl(tl(x')))) -> TLACTIVE(tlactive(tlactive(mark(x'))))
MARK(tl(tl(hd(x')))) -> TLACTIVE(tlactive(hdactive(mark(x'))))
MARK(tl(tl(add(x')))) -> TLACTIVE(tlactive(addactive(mark(x'))))
MARK(tl(tl(incr(x')))) -> TLACTIVE(tlactive(incractive(mark(x'))))
MARK(tl(tl(zeros))) -> TLACTIVE(tlactive(zerosactive))
MARK(tl(tl(nats))) -> TLACTIVE(tlactive(natsactive))
MARK(tl(hd(s(x')))) -> TLACTIVE(hdactive(s(x')))
MARK(tl(hd(0))) -> TLACTIVE(hdactive(0))
MARK(tl(hd(tl(x')))) -> TLACTIVE(hdactive(tlactive(mark(x'))))
MARK(tl(hd(hd(x')))) -> TLACTIVE(hdactive(hdactive(mark(x'))))
MARK(tl(hd(add(x')))) -> TLACTIVE(hdactive(addactive(mark(x'))))
MARK(tl(hd(incr(x')))) -> TLACTIVE(hdactive(incractive(mark(x'))))
MARK(tl(hd(zeros))) -> TLACTIVE(hdactive(zerosactive))
MARK(tl(hd(nats))) -> TLACTIVE(hdactive(natsactive))
MARK(tl(add(cons(x', y')))) -> TLACTIVE(addactive(cons(x', y')))
MARK(tl(add(s(x')))) -> TLACTIVE(addactive(s(x')))
MARK(tl(add(0))) -> TLACTIVE(addactive(0))
MARK(tl(add(tl(x')))) -> TLACTIVE(addactive(tlactive(mark(x'))))
MARK(tl(add(hd(x')))) -> TLACTIVE(addactive(hdactive(mark(x'))))
MARK(tl(add(add(x')))) -> TLACTIVE(addactive(addactive(mark(x'))))
MARK(tl(add(incr(x')))) -> TLACTIVE(addactive(incractive(mark(x'))))
MARK(tl(add(zeros))) -> TLACTIVE(addactive(zerosactive))
MARK(tl(add(nats))) -> TLACTIVE(addactive(natsactive))
MARK(tl(incr(cons(x', y')))) -> TLACTIVE(incractive(cons(x', y')))
MARK(tl(incr(s(x')))) -> TLACTIVE(incractive(s(x')))
MARK(tl(incr(0))) -> TLACTIVE(incractive(0))
MARK(tl(incr(tl(x')))) -> TLACTIVE(incractive(tlactive(mark(x'))))
MARK(tl(incr(hd(x')))) -> TLACTIVE(incractive(hdactive(mark(x'))))
MARK(tl(incr(add(x')))) -> TLACTIVE(incractive(addactive(mark(x'))))
MARK(tl(incr(incr(x')))) -> TLACTIVE(incractive(incractive(mark(x'))))
MARK(tl(incr(zeros))) -> TLACTIVE(incractive(zerosactive))
MARK(tl(incr(nats))) -> TLACTIVE(incractive(natsactive))
MARK(tl(zeros)) -> TLACTIVE(cons(0, zeros))
MARK(hd(tl(cons(x', y')))) -> HDACTIVE(tlactive(cons(x', y')))
MARK(hd(tl(s(x')))) -> HDACTIVE(tlactive(s(x')))
MARK(hd(tl(0))) -> HDACTIVE(tlactive(0))
MARK(hd(tl(tl(x')))) -> HDACTIVE(tlactive(tlactive(mark(x'))))
MARK(hd(tl(hd(x')))) -> HDACTIVE(tlactive(hdactive(mark(x'))))
MARK(hd(tl(add(x')))) -> HDACTIVE(tlactive(addactive(mark(x'))))
MARK(hd(tl(incr(x')))) -> HDACTIVE(tlactive(incractive(mark(x'))))
MARK(hd(tl(zeros))) -> HDACTIVE(tlactive(zerosactive))
MARK(hd(tl(nats))) -> HDACTIVE(tlactive(natsactive))
MARK(hd(hd(cons(x', y')))) -> HDACTIVE(hdactive(cons(x', y')))
MARK(hd(hd(s(x')))) -> HDACTIVE(hdactive(s(x')))
MARK(hd(hd(0))) -> HDACTIVE(hdactive(0))
MARK(hd(hd(tl(x')))) -> HDACTIVE(hdactive(tlactive(mark(x'))))
MARK(hd(hd(hd(x')))) -> HDACTIVE(hdactive(hdactive(mark(x'))))
MARK(hd(hd(add(x')))) -> HDACTIVE(hdactive(addactive(mark(x'))))
MARK(hd(hd(incr(x')))) -> HDACTIVE(hdactive(incractive(mark(x'))))
MARK(hd(hd(zeros))) -> HDACTIVE(hdactive(zerosactive))
MARK(hd(hd(nats))) -> HDACTIVE(hdactive(natsactive))
MARK(hd(add(cons(x', y')))) -> HDACTIVE(addactive(cons(x', y')))
MARK(hd(add(s(x')))) -> HDACTIVE(addactive(s(x')))
MARK(hd(add(0))) -> HDACTIVE(addactive(0))
MARK(hd(add(tl(x')))) -> HDACTIVE(addactive(tlactive(mark(x'))))
MARK(hd(add(hd(x')))) -> HDACTIVE(addactive(hdactive(mark(x'))))
MARK(hd(add(add(x')))) -> HDACTIVE(addactive(addactive(mark(x'))))
MARK(hd(add(incr(x')))) -> HDACTIVE(addactive(incractive(mark(x'))))
MARK(hd(add(zeros))) -> HDACTIVE(addactive(zerosactive))
MARK(hd(add(nats))) -> HDACTIVE(addactive(natsactive))
MARK(hd(incr(cons(x', y')))) -> HDACTIVE(incractive(cons(x', y')))
MARK(hd(incr(s(x')))) -> HDACTIVE(incractive(s(x')))
MARK(hd(incr(0))) -> HDACTIVE(incractive(0))
MARK(hd(incr(tl(x')))) -> HDACTIVE(incractive(tlactive(mark(x'))))
MARK(hd(incr(hd(x')))) -> HDACTIVE(incractive(hdactive(mark(x'))))
MARK(hd(incr(add(x')))) -> HDACTIVE(incractive(addactive(mark(x'))))
MARK(hd(incr(incr(x')))) -> HDACTIVE(incractive(incractive(mark(x'))))
MARK(hd(incr(zeros))) -> HDACTIVE(incractive(zerosactive))
MARK(hd(incr(nats))) -> HDACTIVE(incractive(natsactive))
MARK(hd(zeros)) -> HDACTIVE(cons(0, zeros))
MARK(tl(cons(x'', y'))) -> TLACTIVE(cons(x'', y'))
HDACTIVE(cons(x, y)) -> MARK(x)
MARK(hd(cons(x'', y'))) -> HDACTIVE(cons(x'', y'))
MARK(tl(x)) -> MARK(x)
MARK(hd(x)) -> MARK(x)
MARK(add(x)) -> MARK(x)
MARK(incr(x)) -> MARK(x)
TLACTIVE(cons(x, y)) -> MARK(y)
MARK(tl(hd(cons(x', y')))) -> TLACTIVE(hdactive(cons(x', y')))


Rules:


natsactive -> addactive(zerosactive)
natsactive -> nats
hdactive(x) -> hd(x)
hdactive(cons(x, y)) -> mark(x)
zerosactive -> cons(0, zeros)
zerosactive -> zeros
tlactive(x) -> tl(x)
tlactive(cons(x, y)) -> mark(y)
incractive(cons(x, y)) -> cons(s(x), incr(y))
incractive(x) -> incr(x)
mark(nats) -> natsactive
mark(zeros) -> zerosactive
mark(incr(x)) -> incractive(mark(x))
mark(add(x)) -> addactive(mark(x))
mark(hd(x)) -> hdactive(mark(x))
mark(tl(x)) -> tlactive(mark(x))
mark(0) -> 0
mark(s(x)) -> s(x)
mark(cons(x, y)) -> cons(x, y)
addactive(cons(x, y)) -> incractive(cons(x, add(y)))
addactive(x) -> add(x)





The following dependency pair can be strictly oriented:

MARK(add(x)) -> MARK(x)


Additionally, the following usable rules w.r.t. to the implicit AFS can be oriented:

hdactive(x) -> hd(x)
hdactive(cons(x, y)) -> mark(x)
mark(nats) -> natsactive
mark(zeros) -> zerosactive
mark(incr(x)) -> incractive(mark(x))
mark(add(x)) -> addactive(mark(x))
mark(hd(x)) -> hdactive(mark(x))
mark(tl(x)) -> tlactive(mark(x))
mark(0) -> 0
mark(s(x)) -> s(x)
mark(cons(x, y)) -> cons(x, y)
tlactive(x) -> tl(x)
tlactive(cons(x, y)) -> mark(y)
addactive(cons(x, y)) -> incractive(cons(x, add(y)))
addactive(x) -> add(x)
incractive(cons(x, y)) -> cons(s(x), incr(y))
incractive(x) -> incr(x)
zerosactive -> cons(0, zeros)
zerosactive -> zeros
natsactive -> addactive(zerosactive)
natsactive -> nats


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(zeros_active)=  0  
  POL(hd_active(x1))=  x1  
  POL(incr_active(x1))=  x1  
  POL(MARK(x1))=  x1  
  POL(TL_ACTIVE(x1))=  x1  
  POL(incr(x1))=  x1  
  POL(mark(x1))=  x1  
  POL(tl(x1))=  x1  
  POL(HD_ACTIVE(x1))=  x1  
  POL(add(x1))=  1 + x1  
  POL(add_active(x1))=  1 + x1  
  POL(0)=  0  
  POL(cons(x1, x2))=  x1 + x2  
  POL(hd(x1))=  x1  
  POL(nats)=  1  
  POL(s(x1))=  0  
  POL(zeros)=  0  
  POL(nats_active)=  1  
  POL(tl_active(x1))=  x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 17
                 ↳Polynomial Ordering


Dependency Pairs:

MARK(tl(tl(cons(x', y')))) -> TLACTIVE(tlactive(cons(x', y')))
MARK(tl(tl(s(x')))) -> TLACTIVE(tlactive(s(x')))
MARK(tl(tl(0))) -> TLACTIVE(tlactive(0))
MARK(tl(tl(tl(x')))) -> TLACTIVE(tlactive(tlactive(mark(x'))))
MARK(tl(tl(hd(x')))) -> TLACTIVE(tlactive(hdactive(mark(x'))))
MARK(tl(tl(add(x')))) -> TLACTIVE(tlactive(addactive(mark(x'))))
MARK(tl(tl(incr(x')))) -> TLACTIVE(tlactive(incractive(mark(x'))))
MARK(tl(tl(zeros))) -> TLACTIVE(tlactive(zerosactive))
MARK(tl(tl(nats))) -> TLACTIVE(tlactive(natsactive))
MARK(tl(hd(s(x')))) -> TLACTIVE(hdactive(s(x')))
MARK(tl(hd(0))) -> TLACTIVE(hdactive(0))
MARK(tl(hd(tl(x')))) -> TLACTIVE(hdactive(tlactive(mark(x'))))
MARK(tl(hd(hd(x')))) -> TLACTIVE(hdactive(hdactive(mark(x'))))
MARK(tl(hd(add(x')))) -> TLACTIVE(hdactive(addactive(mark(x'))))
MARK(tl(hd(incr(x')))) -> TLACTIVE(hdactive(incractive(mark(x'))))
MARK(tl(hd(zeros))) -> TLACTIVE(hdactive(zerosactive))
MARK(tl(hd(nats))) -> TLACTIVE(hdactive(natsactive))
MARK(tl(add(cons(x', y')))) -> TLACTIVE(addactive(cons(x', y')))
MARK(tl(add(s(x')))) -> TLACTIVE(addactive(s(x')))
MARK(tl(add(0))) -> TLACTIVE(addactive(0))
MARK(tl(add(tl(x')))) -> TLACTIVE(addactive(tlactive(mark(x'))))
MARK(tl(add(hd(x')))) -> TLACTIVE(addactive(hdactive(mark(x'))))
MARK(tl(add(add(x')))) -> TLACTIVE(addactive(addactive(mark(x'))))
MARK(tl(add(incr(x')))) -> TLACTIVE(addactive(incractive(mark(x'))))
MARK(tl(add(zeros))) -> TLACTIVE(addactive(zerosactive))
MARK(tl(add(nats))) -> TLACTIVE(addactive(natsactive))
MARK(tl(incr(cons(x', y')))) -> TLACTIVE(incractive(cons(x', y')))
MARK(tl(incr(s(x')))) -> TLACTIVE(incractive(s(x')))
MARK(tl(incr(0))) -> TLACTIVE(incractive(0))
MARK(tl(incr(tl(x')))) -> TLACTIVE(incractive(tlactive(mark(x'))))
MARK(tl(incr(hd(x')))) -> TLACTIVE(incractive(hdactive(mark(x'))))
MARK(tl(incr(add(x')))) -> TLACTIVE(incractive(addactive(mark(x'))))
MARK(tl(incr(incr(x')))) -> TLACTIVE(incractive(incractive(mark(x'))))
MARK(tl(incr(zeros))) -> TLACTIVE(incractive(zerosactive))
MARK(tl(incr(nats))) -> TLACTIVE(incractive(natsactive))
MARK(tl(zeros)) -> TLACTIVE(cons(0, zeros))
MARK(hd(tl(cons(x', y')))) -> HDACTIVE(tlactive(cons(x', y')))
MARK(hd(tl(s(x')))) -> HDACTIVE(tlactive(s(x')))
MARK(hd(tl(0))) -> HDACTIVE(tlactive(0))
MARK(hd(tl(tl(x')))) -> HDACTIVE(tlactive(tlactive(mark(x'))))
MARK(hd(tl(hd(x')))) -> HDACTIVE(tlactive(hdactive(mark(x'))))
MARK(hd(tl(add(x')))) -> HDACTIVE(tlactive(addactive(mark(x'))))
MARK(hd(tl(incr(x')))) -> HDACTIVE(tlactive(incractive(mark(x'))))
MARK(hd(tl(zeros))) -> HDACTIVE(tlactive(zerosactive))
MARK(hd(tl(nats))) -> HDACTIVE(tlactive(natsactive))
MARK(hd(hd(cons(x', y')))) -> HDACTIVE(hdactive(cons(x', y')))
MARK(hd(hd(s(x')))) -> HDACTIVE(hdactive(s(x')))
MARK(hd(hd(0))) -> HDACTIVE(hdactive(0))
MARK(hd(hd(tl(x')))) -> HDACTIVE(hdactive(tlactive(mark(x'))))
MARK(hd(hd(hd(x')))) -> HDACTIVE(hdactive(hdactive(mark(x'))))
MARK(hd(hd(add(x')))) -> HDACTIVE(hdactive(addactive(mark(x'))))
MARK(hd(hd(incr(x')))) -> HDACTIVE(hdactive(incractive(mark(x'))))
MARK(hd(hd(zeros))) -> HDACTIVE(hdactive(zerosactive))
MARK(hd(hd(nats))) -> HDACTIVE(hdactive(natsactive))
MARK(hd(add(cons(x', y')))) -> HDACTIVE(addactive(cons(x', y')))
MARK(hd(add(s(x')))) -> HDACTIVE(addactive(s(x')))
MARK(hd(add(0))) -> HDACTIVE(addactive(0))
MARK(hd(add(tl(x')))) -> HDACTIVE(addactive(tlactive(mark(x'))))
MARK(hd(add(hd(x')))) -> HDACTIVE(addactive(hdactive(mark(x'))))
MARK(hd(add(add(x')))) -> HDACTIVE(addactive(addactive(mark(x'))))
MARK(hd(add(incr(x')))) -> HDACTIVE(addactive(incractive(mark(x'))))
MARK(hd(add(zeros))) -> HDACTIVE(addactive(zerosactive))
MARK(hd(add(nats))) -> HDACTIVE(addactive(natsactive))
MARK(hd(incr(cons(x', y')))) -> HDACTIVE(incractive(cons(x', y')))
MARK(hd(incr(s(x')))) -> HDACTIVE(incractive(s(x')))
MARK(hd(incr(0))) -> HDACTIVE(incractive(0))
MARK(hd(incr(tl(x')))) -> HDACTIVE(incractive(tlactive(mark(x'))))
MARK(hd(incr(hd(x')))) -> HDACTIVE(incractive(hdactive(mark(x'))))
MARK(hd(incr(add(x')))) -> HDACTIVE(incractive(addactive(mark(x'))))
MARK(hd(incr(incr(x')))) -> HDACTIVE(incractive(incractive(mark(x'))))
MARK(hd(incr(zeros))) -> HDACTIVE(incractive(zerosactive))
MARK(hd(incr(nats))) -> HDACTIVE(incractive(natsactive))
MARK(hd(zeros)) -> HDACTIVE(cons(0, zeros))
MARK(tl(cons(x'', y'))) -> TLACTIVE(cons(x'', y'))
HDACTIVE(cons(x, y)) -> MARK(x)
MARK(hd(cons(x'', y'))) -> HDACTIVE(cons(x'', y'))
MARK(tl(x)) -> MARK(x)
MARK(hd(x)) -> MARK(x)
MARK(incr(x)) -> MARK(x)
TLACTIVE(cons(x, y)) -> MARK(y)
MARK(tl(hd(cons(x', y')))) -> TLACTIVE(hdactive(cons(x', y')))


Rules:


natsactive -> addactive(zerosactive)
natsactive -> nats
hdactive(x) -> hd(x)
hdactive(cons(x, y)) -> mark(x)
zerosactive -> cons(0, zeros)
zerosactive -> zeros
tlactive(x) -> tl(x)
tlactive(cons(x, y)) -> mark(y)
incractive(cons(x, y)) -> cons(s(x), incr(y))
incractive(x) -> incr(x)
mark(nats) -> natsactive
mark(zeros) -> zerosactive
mark(incr(x)) -> incractive(mark(x))
mark(add(x)) -> addactive(mark(x))
mark(hd(x)) -> hdactive(mark(x))
mark(tl(x)) -> tlactive(mark(x))
mark(0) -> 0
mark(s(x)) -> s(x)
mark(cons(x, y)) -> cons(x, y)
addactive(cons(x, y)) -> incractive(cons(x, add(y)))
addactive(x) -> add(x)





The following dependency pairs can be strictly oriented:

MARK(tl(tl(cons(x', y')))) -> TLACTIVE(tlactive(cons(x', y')))
MARK(tl(tl(s(x')))) -> TLACTIVE(tlactive(s(x')))
MARK(tl(tl(0))) -> TLACTIVE(tlactive(0))
MARK(tl(tl(tl(x')))) -> TLACTIVE(tlactive(tlactive(mark(x'))))
MARK(tl(tl(hd(x')))) -> TLACTIVE(tlactive(hdactive(mark(x'))))
MARK(tl(tl(add(x')))) -> TLACTIVE(tlactive(addactive(mark(x'))))
MARK(tl(tl(incr(x')))) -> TLACTIVE(tlactive(incractive(mark(x'))))
MARK(tl(tl(zeros))) -> TLACTIVE(tlactive(zerosactive))
MARK(tl(tl(nats))) -> TLACTIVE(tlactive(natsactive))
MARK(tl(hd(s(x')))) -> TLACTIVE(hdactive(s(x')))
MARK(tl(hd(0))) -> TLACTIVE(hdactive(0))
MARK(tl(hd(tl(x')))) -> TLACTIVE(hdactive(tlactive(mark(x'))))
MARK(tl(hd(hd(x')))) -> TLACTIVE(hdactive(hdactive(mark(x'))))
MARK(tl(hd(add(x')))) -> TLACTIVE(hdactive(addactive(mark(x'))))
MARK(tl(hd(incr(x')))) -> TLACTIVE(hdactive(incractive(mark(x'))))
MARK(tl(hd(zeros))) -> TLACTIVE(hdactive(zerosactive))
MARK(tl(hd(nats))) -> TLACTIVE(hdactive(natsactive))
MARK(tl(add(cons(x', y')))) -> TLACTIVE(addactive(cons(x', y')))
MARK(tl(add(s(x')))) -> TLACTIVE(addactive(s(x')))
MARK(tl(add(0))) -> TLACTIVE(addactive(0))
MARK(tl(add(tl(x')))) -> TLACTIVE(addactive(tlactive(mark(x'))))
MARK(tl(add(hd(x')))) -> TLACTIVE(addactive(hdactive(mark(x'))))
MARK(tl(add(add(x')))) -> TLACTIVE(addactive(addactive(mark(x'))))
MARK(tl(add(incr(x')))) -> TLACTIVE(addactive(incractive(mark(x'))))
MARK(tl(add(zeros))) -> TLACTIVE(addactive(zerosactive))
MARK(tl(add(nats))) -> TLACTIVE(addactive(natsactive))
MARK(tl(incr(cons(x', y')))) -> TLACTIVE(incractive(cons(x', y')))
MARK(tl(incr(s(x')))) -> TLACTIVE(incractive(s(x')))
MARK(tl(incr(0))) -> TLACTIVE(incractive(0))
MARK(tl(incr(tl(x')))) -> TLACTIVE(incractive(tlactive(mark(x'))))
MARK(tl(incr(hd(x')))) -> TLACTIVE(incractive(hdactive(mark(x'))))
MARK(tl(incr(add(x')))) -> TLACTIVE(incractive(addactive(mark(x'))))
MARK(tl(incr(incr(x')))) -> TLACTIVE(incractive(incractive(mark(x'))))
MARK(tl(incr(zeros))) -> TLACTIVE(incractive(zerosactive))
MARK(tl(incr(nats))) -> TLACTIVE(incractive(natsactive))
MARK(tl(zeros)) -> TLACTIVE(cons(0, zeros))
MARK(tl(cons(x'', y'))) -> TLACTIVE(cons(x'', y'))
MARK(tl(x)) -> MARK(x)
MARK(tl(hd(cons(x', y')))) -> TLACTIVE(hdactive(cons(x', y')))


Additionally, the following usable rules w.r.t. to the implicit AFS can be oriented:

hdactive(x) -> hd(x)
hdactive(cons(x, y)) -> mark(x)
mark(nats) -> natsactive
mark(zeros) -> zerosactive
mark(incr(x)) -> incractive(mark(x))
mark(add(x)) -> addactive(mark(x))
mark(hd(x)) -> hdactive(mark(x))
mark(tl(x)) -> tlactive(mark(x))
mark(0) -> 0
mark(s(x)) -> s(x)
mark(cons(x, y)) -> cons(x, y)
tlactive(x) -> tl(x)
tlactive(cons(x, y)) -> mark(y)
addactive(cons(x, y)) -> incractive(cons(x, add(y)))
addactive(x) -> add(x)
incractive(cons(x, y)) -> cons(s(x), incr(y))
incractive(x) -> incr(x)
zerosactive -> cons(0, zeros)
zerosactive -> zeros
natsactive -> addactive(zerosactive)
natsactive -> nats


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(zeros_active)=  0  
  POL(hd_active(x1))=  x1  
  POL(incr_active(x1))=  x1  
  POL(MARK(x1))=  x1  
  POL(TL_ACTIVE(x1))=  x1  
  POL(incr(x1))=  x1  
  POL(mark(x1))=  x1  
  POL(tl(x1))=  1 + x1  
  POL(HD_ACTIVE(x1))=  x1  
  POL(add(x1))=  x1  
  POL(add_active(x1))=  x1  
  POL(0)=  0  
  POL(cons(x1, x2))=  x1 + x2  
  POL(hd(x1))=  x1  
  POL(nats)=  0  
  POL(s(x1))=  0  
  POL(zeros)=  0  
  POL(nats_active)=  0  
  POL(tl_active(x1))=  1 + x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 18
                 ↳Dependency Graph


Dependency Pairs:

MARK(hd(tl(cons(x', y')))) -> HDACTIVE(tlactive(cons(x', y')))
MARK(hd(tl(s(x')))) -> HDACTIVE(tlactive(s(x')))
MARK(hd(tl(0))) -> HDACTIVE(tlactive(0))
MARK(hd(tl(tl(x')))) -> HDACTIVE(tlactive(tlactive(mark(x'))))
MARK(hd(tl(hd(x')))) -> HDACTIVE(tlactive(hdactive(mark(x'))))
MARK(hd(tl(add(x')))) -> HDACTIVE(tlactive(addactive(mark(x'))))
MARK(hd(tl(incr(x')))) -> HDACTIVE(tlactive(incractive(mark(x'))))
MARK(hd(tl(zeros))) -> HDACTIVE(tlactive(zerosactive))
MARK(hd(tl(nats))) -> HDACTIVE(tlactive(natsactive))
MARK(hd(hd(cons(x', y')))) -> HDACTIVE(hdactive(cons(x', y')))
MARK(hd(hd(s(x')))) -> HDACTIVE(hdactive(s(x')))
MARK(hd(hd(0))) -> HDACTIVE(hdactive(0))
MARK(hd(hd(tl(x')))) -> HDACTIVE(hdactive(tlactive(mark(x'))))
MARK(hd(hd(hd(x')))) -> HDACTIVE(hdactive(hdactive(mark(x'))))
MARK(hd(hd(add(x')))) -> HDACTIVE(hdactive(addactive(mark(x'))))
MARK(hd(hd(incr(x')))) -> HDACTIVE(hdactive(incractive(mark(x'))))
MARK(hd(hd(zeros))) -> HDACTIVE(hdactive(zerosactive))
MARK(hd(hd(nats))) -> HDACTIVE(hdactive(natsactive))
MARK(hd(add(cons(x', y')))) -> HDACTIVE(addactive(cons(x', y')))
MARK(hd(add(s(x')))) -> HDACTIVE(addactive(s(x')))
MARK(hd(add(0))) -> HDACTIVE(addactive(0))
MARK(hd(add(tl(x')))) -> HDACTIVE(addactive(tlactive(mark(x'))))
MARK(hd(add(hd(x')))) -> HDACTIVE(addactive(hdactive(mark(x'))))
MARK(hd(add(add(x')))) -> HDACTIVE(addactive(addactive(mark(x'))))
MARK(hd(add(incr(x')))) -> HDACTIVE(addactive(incractive(mark(x'))))
MARK(hd(add(zeros))) -> HDACTIVE(addactive(zerosactive))
MARK(hd(add(nats))) -> HDACTIVE(addactive(natsactive))
MARK(hd(incr(cons(x', y')))) -> HDACTIVE(incractive(cons(x', y')))
MARK(hd(incr(s(x')))) -> HDACTIVE(incractive(s(x')))
MARK(hd(incr(0))) -> HDACTIVE(incractive(0))
MARK(hd(incr(tl(x')))) -> HDACTIVE(incractive(tlactive(mark(x'))))
MARK(hd(incr(hd(x')))) -> HDACTIVE(incractive(hdactive(mark(x'))))
MARK(hd(incr(add(x')))) -> HDACTIVE(incractive(addactive(mark(x'))))
MARK(hd(incr(incr(x')))) -> HDACTIVE(incractive(incractive(mark(x'))))
MARK(hd(incr(zeros))) -> HDACTIVE(incractive(zerosactive))
MARK(hd(incr(nats))) -> HDACTIVE(incractive(natsactive))
MARK(hd(zeros)) -> HDACTIVE(cons(0, zeros))
HDACTIVE(cons(x, y)) -> MARK(x)
MARK(hd(cons(x'', y'))) -> HDACTIVE(cons(x'', y'))
MARK(hd(x)) -> MARK(x)
MARK(incr(x)) -> MARK(x)
TLACTIVE(cons(x, y)) -> MARK(y)


Rules:


natsactive -> addactive(zerosactive)
natsactive -> nats
hdactive(x) -> hd(x)
hdactive(cons(x, y)) -> mark(x)
zerosactive -> cons(0, zeros)
zerosactive -> zeros
tlactive(x) -> tl(x)
tlactive(cons(x, y)) -> mark(y)
incractive(cons(x, y)) -> cons(s(x), incr(y))
incractive(x) -> incr(x)
mark(nats) -> natsactive
mark(zeros) -> zerosactive
mark(incr(x)) -> incractive(mark(x))
mark(add(x)) -> addactive(mark(x))
mark(hd(x)) -> hdactive(mark(x))
mark(tl(x)) -> tlactive(mark(x))
mark(0) -> 0
mark(s(x)) -> s(x)
mark(cons(x, y)) -> cons(x, y)
addactive(cons(x, y)) -> incractive(cons(x, add(y)))
addactive(x) -> add(x)





Using the Dependency Graph the DP problem was split into 1 DP problems.


   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 19
                 ↳Polynomial Ordering


Dependency Pairs:

MARK(hd(tl(s(x')))) -> HDACTIVE(tlactive(s(x')))
MARK(hd(tl(0))) -> HDACTIVE(tlactive(0))
MARK(hd(tl(tl(x')))) -> HDACTIVE(tlactive(tlactive(mark(x'))))
MARK(hd(tl(hd(x')))) -> HDACTIVE(tlactive(hdactive(mark(x'))))
MARK(hd(tl(add(x')))) -> HDACTIVE(tlactive(addactive(mark(x'))))
MARK(hd(tl(incr(x')))) -> HDACTIVE(tlactive(incractive(mark(x'))))
MARK(hd(tl(zeros))) -> HDACTIVE(tlactive(zerosactive))
MARK(hd(tl(nats))) -> HDACTIVE(tlactive(natsactive))
MARK(hd(hd(cons(x', y')))) -> HDACTIVE(hdactive(cons(x', y')))
MARK(hd(hd(s(x')))) -> HDACTIVE(hdactive(s(x')))
MARK(hd(hd(0))) -> HDACTIVE(hdactive(0))
MARK(hd(hd(tl(x')))) -> HDACTIVE(hdactive(tlactive(mark(x'))))
MARK(hd(hd(hd(x')))) -> HDACTIVE(hdactive(hdactive(mark(x'))))
MARK(hd(hd(add(x')))) -> HDACTIVE(hdactive(addactive(mark(x'))))
MARK(hd(hd(incr(x')))) -> HDACTIVE(hdactive(incractive(mark(x'))))
MARK(hd(hd(zeros))) -> HDACTIVE(hdactive(zerosactive))
MARK(hd(hd(nats))) -> HDACTIVE(hdactive(natsactive))
MARK(hd(add(cons(x', y')))) -> HDACTIVE(addactive(cons(x', y')))
MARK(hd(add(s(x')))) -> HDACTIVE(addactive(s(x')))
MARK(hd(add(0))) -> HDACTIVE(addactive(0))
MARK(hd(add(tl(x')))) -> HDACTIVE(addactive(tlactive(mark(x'))))
MARK(hd(add(hd(x')))) -> HDACTIVE(addactive(hdactive(mark(x'))))
MARK(hd(add(add(x')))) -> HDACTIVE(addactive(addactive(mark(x'))))
MARK(hd(add(incr(x')))) -> HDACTIVE(addactive(incractive(mark(x'))))
MARK(hd(add(zeros))) -> HDACTIVE(addactive(zerosactive))
MARK(hd(add(nats))) -> HDACTIVE(addactive(natsactive))
MARK(hd(incr(cons(x', y')))) -> HDACTIVE(incractive(cons(x', y')))
MARK(hd(incr(s(x')))) -> HDACTIVE(incractive(s(x')))
MARK(hd(incr(0))) -> HDACTIVE(incractive(0))
MARK(hd(incr(tl(x')))) -> HDACTIVE(incractive(tlactive(mark(x'))))
MARK(hd(incr(hd(x')))) -> HDACTIVE(incractive(hdactive(mark(x'))))
MARK(hd(incr(add(x')))) -> HDACTIVE(incractive(addactive(mark(x'))))
MARK(hd(incr(incr(x')))) -> HDACTIVE(incractive(incractive(mark(x'))))
MARK(hd(incr(zeros))) -> HDACTIVE(incractive(zerosactive))
MARK(hd(incr(nats))) -> HDACTIVE(incractive(natsactive))
MARK(hd(zeros)) -> HDACTIVE(cons(0, zeros))
MARK(hd(cons(x'', y'))) -> HDACTIVE(cons(x'', y'))
MARK(hd(x)) -> MARK(x)
MARK(incr(x)) -> MARK(x)
HDACTIVE(cons(x, y)) -> MARK(x)
MARK(hd(tl(cons(x', y')))) -> HDACTIVE(tlactive(cons(x', y')))


Rules:


natsactive -> addactive(zerosactive)
natsactive -> nats
hdactive(x) -> hd(x)
hdactive(cons(x, y)) -> mark(x)
zerosactive -> cons(0, zeros)
zerosactive -> zeros
tlactive(x) -> tl(x)
tlactive(cons(x, y)) -> mark(y)
incractive(cons(x, y)) -> cons(s(x), incr(y))
incractive(x) -> incr(x)
mark(nats) -> natsactive
mark(zeros) -> zerosactive
mark(incr(x)) -> incractive(mark(x))
mark(add(x)) -> addactive(mark(x))
mark(hd(x)) -> hdactive(mark(x))
mark(tl(x)) -> tlactive(mark(x))
mark(0) -> 0
mark(s(x)) -> s(x)
mark(cons(x, y)) -> cons(x, y)
addactive(cons(x, y)) -> incractive(cons(x, add(y)))
addactive(x) -> add(x)





The following dependency pairs can be strictly oriented:

MARK(hd(tl(s(x')))) -> HDACTIVE(tlactive(s(x')))
MARK(hd(tl(0))) -> HDACTIVE(tlactive(0))
MARK(hd(tl(tl(x')))) -> HDACTIVE(tlactive(tlactive(mark(x'))))
MARK(hd(tl(hd(x')))) -> HDACTIVE(tlactive(hdactive(mark(x'))))
MARK(hd(tl(add(x')))) -> HDACTIVE(tlactive(addactive(mark(x'))))
MARK(hd(tl(incr(x')))) -> HDACTIVE(tlactive(incractive(mark(x'))))
MARK(hd(tl(zeros))) -> HDACTIVE(tlactive(zerosactive))
MARK(hd(tl(nats))) -> HDACTIVE(tlactive(natsactive))
MARK(hd(hd(cons(x', y')))) -> HDACTIVE(hdactive(cons(x', y')))
MARK(hd(hd(s(x')))) -> HDACTIVE(hdactive(s(x')))
MARK(hd(hd(0))) -> HDACTIVE(hdactive(0))
MARK(hd(hd(tl(x')))) -> HDACTIVE(hdactive(tlactive(mark(x'))))
MARK(hd(hd(hd(x')))) -> HDACTIVE(hdactive(hdactive(mark(x'))))
MARK(hd(hd(add(x')))) -> HDACTIVE(hdactive(addactive(mark(x'))))
MARK(hd(hd(incr(x')))) -> HDACTIVE(hdactive(incractive(mark(x'))))
MARK(hd(hd(zeros))) -> HDACTIVE(hdactive(zerosactive))
MARK(hd(hd(nats))) -> HDACTIVE(hdactive(natsactive))
MARK(hd(add(cons(x', y')))) -> HDACTIVE(addactive(cons(x', y')))
MARK(hd(add(s(x')))) -> HDACTIVE(addactive(s(x')))
MARK(hd(add(0))) -> HDACTIVE(addactive(0))
MARK(hd(add(tl(x')))) -> HDACTIVE(addactive(tlactive(mark(x'))))
MARK(hd(add(hd(x')))) -> HDACTIVE(addactive(hdactive(mark(x'))))
MARK(hd(add(add(x')))) -> HDACTIVE(addactive(addactive(mark(x'))))
MARK(hd(add(incr(x')))) -> HDACTIVE(addactive(incractive(mark(x'))))
MARK(hd(add(zeros))) -> HDACTIVE(addactive(zerosactive))
MARK(hd(add(nats))) -> HDACTIVE(addactive(natsactive))
MARK(hd(incr(cons(x', y')))) -> HDACTIVE(incractive(cons(x', y')))
MARK(hd(incr(s(x')))) -> HDACTIVE(incractive(s(x')))
MARK(hd(incr(0))) -> HDACTIVE(incractive(0))
MARK(hd(incr(tl(x')))) -> HDACTIVE(incractive(tlactive(mark(x'))))
MARK(hd(incr(hd(x')))) -> HDACTIVE(incractive(hdactive(mark(x'))))
MARK(hd(incr(add(x')))) -> HDACTIVE(incractive(addactive(mark(x'))))
MARK(hd(incr(incr(x')))) -> HDACTIVE(incractive(incractive(mark(x'))))
MARK(hd(incr(zeros))) -> HDACTIVE(incractive(zerosactive))
MARK(hd(incr(nats))) -> HDACTIVE(incractive(natsactive))
MARK(hd(zeros)) -> HDACTIVE(cons(0, zeros))
MARK(hd(cons(x'', y'))) -> HDACTIVE(cons(x'', y'))
MARK(hd(x)) -> MARK(x)
MARK(hd(tl(cons(x', y')))) -> HDACTIVE(tlactive(cons(x', y')))


Additionally, the following usable rules w.r.t. to the implicit AFS can be oriented:

hdactive(x) -> hd(x)
hdactive(cons(x, y)) -> mark(x)
mark(nats) -> natsactive
mark(zeros) -> zerosactive
mark(incr(x)) -> incractive(mark(x))
mark(add(x)) -> addactive(mark(x))
mark(hd(x)) -> hdactive(mark(x))
mark(tl(x)) -> tlactive(mark(x))
mark(0) -> 0
mark(s(x)) -> s(x)
mark(cons(x, y)) -> cons(x, y)
tlactive(x) -> tl(x)
tlactive(cons(x, y)) -> mark(y)
addactive(cons(x, y)) -> incractive(cons(x, add(y)))
addactive(x) -> add(x)
incractive(cons(x, y)) -> cons(s(x), incr(y))
incractive(x) -> incr(x)
zerosactive -> cons(0, zeros)
zerosactive -> zeros
natsactive -> addactive(zerosactive)
natsactive -> nats


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(zeros_active)=  0  
  POL(hd_active(x1))=  1 + x1  
  POL(incr_active(x1))=  x1  
  POL(MARK(x1))=  x1  
  POL(incr(x1))=  x1  
  POL(mark(x1))=  x1  
  POL(tl(x1))=  x1  
  POL(HD_ACTIVE(x1))=  x1  
  POL(add(x1))=  x1  
  POL(add_active(x1))=  x1  
  POL(0)=  0  
  POL(cons(x1, x2))=  x1 + x2  
  POL(hd(x1))=  1 + x1  
  POL(nats)=  0  
  POL(s(x1))=  0  
  POL(zeros)=  0  
  POL(nats_active)=  0  
  POL(tl_active(x1))=  x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 20
                 ↳Dependency Graph


Dependency Pairs:

MARK(incr(x)) -> MARK(x)
HDACTIVE(cons(x, y)) -> MARK(x)


Rules:


natsactive -> addactive(zerosactive)
natsactive -> nats
hdactive(x) -> hd(x)
hdactive(cons(x, y)) -> mark(x)
zerosactive -> cons(0, zeros)
zerosactive -> zeros
tlactive(x) -> tl(x)
tlactive(cons(x, y)) -> mark(y)
incractive(cons(x, y)) -> cons(s(x), incr(y))
incractive(x) -> incr(x)
mark(nats) -> natsactive
mark(zeros) -> zerosactive
mark(incr(x)) -> incractive(mark(x))
mark(add(x)) -> addactive(mark(x))
mark(hd(x)) -> hdactive(mark(x))
mark(tl(x)) -> tlactive(mark(x))
mark(0) -> 0
mark(s(x)) -> s(x)
mark(cons(x, y)) -> cons(x, y)
addactive(cons(x, y)) -> incractive(cons(x, add(y)))
addactive(x) -> add(x)





Using the Dependency Graph the DP problem was split into 1 DP problems.


   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 21
                 ↳Polynomial Ordering


Dependency Pair:

MARK(incr(x)) -> MARK(x)


Rules:


natsactive -> addactive(zerosactive)
natsactive -> nats
hdactive(x) -> hd(x)
hdactive(cons(x, y)) -> mark(x)
zerosactive -> cons(0, zeros)
zerosactive -> zeros
tlactive(x) -> tl(x)
tlactive(cons(x, y)) -> mark(y)
incractive(cons(x, y)) -> cons(s(x), incr(y))
incractive(x) -> incr(x)
mark(nats) -> natsactive
mark(zeros) -> zerosactive
mark(incr(x)) -> incractive(mark(x))
mark(add(x)) -> addactive(mark(x))
mark(hd(x)) -> hdactive(mark(x))
mark(tl(x)) -> tlactive(mark(x))
mark(0) -> 0
mark(s(x)) -> s(x)
mark(cons(x, y)) -> cons(x, y)
addactive(cons(x, y)) -> incractive(cons(x, add(y)))
addactive(x) -> add(x)





The following dependency pair can be strictly oriented:

MARK(incr(x)) -> MARK(x)


There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(MARK(x1))=  x1  
  POL(incr(x1))=  1 + x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 22
                 ↳Dependency Graph


Dependency Pair:


Rules:


natsactive -> addactive(zerosactive)
natsactive -> nats
hdactive(x) -> hd(x)
hdactive(cons(x, y)) -> mark(x)
zerosactive -> cons(0, zeros)
zerosactive -> zeros
tlactive(x) -> tl(x)
tlactive(cons(x, y)) -> mark(y)
incractive(cons(x, y)) -> cons(s(x), incr(y))
incractive(x) -> incr(x)
mark(nats) -> natsactive
mark(zeros) -> zerosactive
mark(incr(x)) -> incractive(mark(x))
mark(add(x)) -> addactive(mark(x))
mark(hd(x)) -> hdactive(mark(x))
mark(tl(x)) -> tlactive(mark(x))
mark(0) -> 0
mark(s(x)) -> s(x)
mark(cons(x, y)) -> cons(x, y)
addactive(cons(x, y)) -> incractive(cons(x, add(y)))
addactive(x) -> add(x)





Using the Dependency Graph resulted in no new DP problems.

Termination of R successfully shown.
Duration:
0:30 minutes