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)

Innermost 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)


Strategy:

innermost




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)


Strategy:

innermost




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
Remaining Obligation(s)




The following remains to be proven:
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)


Strategy:

innermost



Innermost Termination of R could not be shown.
Duration:
0:07 minutes