Term Rewriting System R:
[X, Y, Z, X1, X2]
dbl(0) -> 0
dbl(s(X)) -> s(ns(ndbl(activate(X))))
dbl(X) -> ndbl(X)
dbls(nil) -> nil
dbls(cons(X, Y)) -> cons(ndbl(activate(X)), ndbls(activate(Y)))
dbls(X) -> ndbls(X)
sel(0, cons(X, Y)) -> activate(X)
sel(s(X), cons(Y, Z)) -> sel(activate(X), activate(Z))
sel(X1, X2) -> nsel(X1, X2)
indx(nil, X) -> nil
indx(cons(X, Y), Z) -> cons(nsel(activate(X), activate(Z)), nindx(activate(Y), activate(Z)))
indx(X1, X2) -> nindx(X1, X2)
from(X) -> cons(activate(X), nfrom(ns(activate(X))))
from(X) -> nfrom(X)
s(X) -> ns(X)
activate(ns(X)) -> s(X)
activate(ndbl(X)) -> dbl(activate(X))
activate(ndbls(X)) -> dbls(activate(X))
activate(nsel(X1, X2)) -> sel(activate(X1), activate(X2))
activate(nindx(X1, X2)) -> indx(activate(X1), X2)
activate(nfrom(X)) -> from(X)
activate(X) -> X

Innermost Termination of R to be shown. 



   R
     ↳Removing Redundant Rules for Innermost Termination



Removing the following rules from R which left hand sides contain non normal subterms

dbl(s(X)) -> s(ns(ndbl(activate(X))))
sel(s(X), cons(Y, Z)) -> sel(activate(X), activate(Z))


   R
     ↳RRRI
       →TRS2
         ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

DBLS(cons(X, Y)) -> ACTIVATE(X)
DBLS(cons(X, Y)) -> ACTIVATE(Y)
ACTIVATE(ns(X)) -> S(X)
ACTIVATE(ndbl(X)) -> DBL(activate(X))
ACTIVATE(ndbl(X)) -> ACTIVATE(X)
ACTIVATE(ndbls(X)) -> DBLS(activate(X))
ACTIVATE(ndbls(X)) -> ACTIVATE(X)
ACTIVATE(nsel(X1, X2)) -> SEL(activate(X1), activate(X2))
ACTIVATE(nsel(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(nsel(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(nindx(X1, X2)) -> INDX(activate(X1), X2)
ACTIVATE(nindx(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(nfrom(X)) -> FROM(X)
SEL(0, cons(X, Y)) -> ACTIVATE(X)
INDX(cons(X, Y), Z) -> ACTIVATE(X)
INDX(cons(X, Y), Z) -> ACTIVATE(Z)
INDX(cons(X, Y), Z) -> ACTIVATE(Y)
FROM(X) -> ACTIVATE(X)

Furthermore, R contains one SCC. 


   R
     ↳RRRI
       →TRS2
         ↳DPs
           →DP Problem 1
             ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

INDX(cons(X, Y), Z) -> ACTIVATE(Y)
INDX(cons(X, Y), Z) -> ACTIVATE(Z)
FROM(X) -> ACTIVATE(X)
ACTIVATE(nfrom(X)) -> FROM(X)
ACTIVATE(nindx(X1, X2)) -> ACTIVATE(X1)
INDX(cons(X, Y), Z) -> ACTIVATE(X)
ACTIVATE(nindx(X1, X2)) -> INDX(activate(X1), X2)
ACTIVATE(nsel(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(nsel(X1, X2)) -> ACTIVATE(X1)
SEL(0, cons(X, Y)) -> ACTIVATE(X)
ACTIVATE(nsel(X1, X2)) -> SEL(activate(X1), activate(X2))
ACTIVATE(ndbls(X)) -> ACTIVATE(X)
DBLS(cons(X, Y)) -> ACTIVATE(Y)
ACTIVATE(ndbls(X)) -> DBLS(activate(X))
ACTIVATE(ndbl(X)) -> ACTIVATE(X)
DBLS(cons(X, Y)) -> ACTIVATE(X)


Rules:


dbl(0) -> 0
dbl(X) -> ndbl(X)
dbls(nil) -> nil
dbls(cons(X, Y)) -> cons(ndbl(activate(X)), ndbls(activate(Y)))
dbls(X) -> ndbls(X)
activate(ns(X)) -> s(X)
activate(ndbl(X)) -> dbl(activate(X))
activate(ndbls(X)) -> dbls(activate(X))
activate(nsel(X1, X2)) -> sel(activate(X1), activate(X2))
activate(nindx(X1, X2)) -> indx(activate(X1), X2)
activate(nfrom(X)) -> from(X)
activate(X) -> X
sel(0, cons(X, Y)) -> activate(X)
sel(X1, X2) -> nsel(X1, X2)
indx(nil, X) -> nil
indx(cons(X, Y), Z) -> cons(nsel(activate(X), activate(Z)), nindx(activate(Y), activate(Z)))
indx(X1, X2) -> nindx(X1, X2)
from(X) -> cons(activate(X), nfrom(ns(activate(X))))
from(X) -> nfrom(X)
s(X) -> ns(X)




The Proof could not be continued due to a Timeout.
Innermost Termination of R could not be shown.
Duration:
1:00 minutes