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
         ↳Remaining Obligation(s)




The following remains to be proven:
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)




Termination of R could not be shown.
Duration:
0:00 minutes