Term Rewriting System R:
[X, XS, N, Y, YS, X1, X2]
from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(X)
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
minus(X, 0) -> 0
minus(s(X), s(Y)) -> minus(X, Y)
quot(0, s(Y)) -> 0
quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
zWquot(XS, nil) -> nil
zWquot(nil, XS) -> nil
zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), nzWquot(activate(XS), activate(YS)))
zWquot(X1, X2) -> nzWquot(X1, X2)
s(X) -> ns(X)
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(activate(X))
activate(nzWquot(X1, X2)) -> zWquot(activate(X1), activate(X2))
activate(X) -> X

Innermost Termination of R to be shown.



   R
Dependency Pair Analysis



R contains the following Dependency Pairs:

SEL(s(N), cons(X, XS)) -> SEL(N, activate(XS))
SEL(s(N), cons(X, XS)) -> ACTIVATE(XS)
MINUS(s(X), s(Y)) -> MINUS(X, Y)
QUOT(s(X), s(Y)) -> S(quot(minus(X, Y), s(Y)))
QUOT(s(X), s(Y)) -> QUOT(minus(X, Y), s(Y))
QUOT(s(X), s(Y)) -> MINUS(X, Y)
ZWQUOT(cons(X, XS), cons(Y, YS)) -> QUOT(X, Y)
ZWQUOT(cons(X, XS), cons(Y, YS)) -> ACTIVATE(XS)
ZWQUOT(cons(X, XS), cons(Y, YS)) -> ACTIVATE(YS)
ACTIVATE(nfrom(X)) -> FROM(activate(X))
ACTIVATE(nfrom(X)) -> ACTIVATE(X)
ACTIVATE(ns(X)) -> S(activate(X))
ACTIVATE(ns(X)) -> ACTIVATE(X)
ACTIVATE(nzWquot(X1, X2)) -> ZWQUOT(activate(X1), activate(X2))
ACTIVATE(nzWquot(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(nzWquot(X1, X2)) -> ACTIVATE(X2)

Furthermore, R contains one SCC.


   R
DPs
       →DP Problem 1
Narrowing Transformation


Dependency Pairs:

ACTIVATE(nzWquot(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(nzWquot(X1, X2)) -> ACTIVATE(X1)
ZWQUOT(cons(X, XS), cons(Y, YS)) -> ACTIVATE(YS)
ACTIVATE(nzWquot(X1, X2)) -> ZWQUOT(activate(X1), activate(X2))
ACTIVATE(ns(X)) -> ACTIVATE(X)
ACTIVATE(nfrom(X)) -> ACTIVATE(X)
ZWQUOT(cons(X, XS), cons(Y, YS)) -> ACTIVATE(XS)


Rules:


from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(X)
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
minus(X, 0) -> 0
minus(s(X), s(Y)) -> minus(X, Y)
quot(0, s(Y)) -> 0
quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
zWquot(XS, nil) -> nil
zWquot(nil, XS) -> nil
zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), nzWquot(activate(XS), activate(YS)))
zWquot(X1, X2) -> nzWquot(X1, X2)
s(X) -> ns(X)
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(activate(X))
activate(nzWquot(X1, X2)) -> zWquot(activate(X1), activate(X2))
activate(X) -> X


Strategy:

innermost




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

ACTIVATE(nzWquot(X1, X2)) -> ZWQUOT(activate(X1), activate(X2))
eight new Dependency Pairs are created:

ACTIVATE(nzWquot(nfrom(X'), X2)) -> ZWQUOT(from(activate(X')), activate(X2))
ACTIVATE(nzWquot(ns(X'), X2)) -> ZWQUOT(s(activate(X')), activate(X2))
ACTIVATE(nzWquot(nzWquot(X1'', X2''), X2)) -> ZWQUOT(zWquot(activate(X1''), activate(X2'')), activate(X2))
ACTIVATE(nzWquot(X1', X2)) -> ZWQUOT(X1', activate(X2))
ACTIVATE(nzWquot(X1, nfrom(X'))) -> ZWQUOT(activate(X1), from(activate(X')))
ACTIVATE(nzWquot(X1, ns(X'))) -> ZWQUOT(activate(X1), s(activate(X')))
ACTIVATE(nzWquot(X1, nzWquot(X1'', X2''))) -> ZWQUOT(activate(X1), zWquot(activate(X1''), activate(X2'')))
ACTIVATE(nzWquot(X1, X2')) -> ZWQUOT(activate(X1), X2')

The transformation is resulting in one new DP problem:



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




The following remains to be proven:
Dependency Pairs:

ACTIVATE(nzWquot(X1, X2')) -> ZWQUOT(activate(X1), X2')
ACTIVATE(nzWquot(X1, nzWquot(X1'', X2''))) -> ZWQUOT(activate(X1), zWquot(activate(X1''), activate(X2'')))
ACTIVATE(nzWquot(X1, ns(X'))) -> ZWQUOT(activate(X1), s(activate(X')))
ACTIVATE(nzWquot(X1, nfrom(X'))) -> ZWQUOT(activate(X1), from(activate(X')))
ACTIVATE(nzWquot(X1', X2)) -> ZWQUOT(X1', activate(X2))
ACTIVATE(nzWquot(nzWquot(X1'', X2''), X2)) -> ZWQUOT(zWquot(activate(X1''), activate(X2'')), activate(X2))
ZWQUOT(cons(X, XS), cons(Y, YS)) -> ACTIVATE(YS)
ACTIVATE(nzWquot(ns(X'), X2)) -> ZWQUOT(s(activate(X')), activate(X2))
ZWQUOT(cons(X, XS), cons(Y, YS)) -> ACTIVATE(XS)
ACTIVATE(nzWquot(nfrom(X'), X2)) -> ZWQUOT(from(activate(X')), activate(X2))
ACTIVATE(nzWquot(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(ns(X)) -> ACTIVATE(X)
ACTIVATE(nfrom(X)) -> ACTIVATE(X)
ACTIVATE(nzWquot(X1, X2)) -> ACTIVATE(X2)


Rules:


from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(X)
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
minus(X, 0) -> 0
minus(s(X), s(Y)) -> minus(X, Y)
quot(0, s(Y)) -> 0
quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
zWquot(XS, nil) -> nil
zWquot(nil, XS) -> nil
zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), nzWquot(activate(XS), activate(YS)))
zWquot(X1, X2) -> nzWquot(X1, X2)
s(X) -> ns(X)
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(activate(X))
activate(nzWquot(X1, X2)) -> zWquot(activate(X1), activate(X2))
activate(X) -> X


Strategy:

innermost



Innermost Termination of R could not be shown.
Duration:
0:06 minutes