Term Rewriting System R:
[N, X, Y, X1, X2, Z]
aterms(N) -> cons(recip(asqr(mark(N))), terms(s(N)))
aterms(X) -> terms(X)
asqr(0) -> 0
asqr(s(X)) -> s(aadd(asqr(mark(X)), adbl(mark(X))))
asqr(X) -> sqr(X)
adbl(0) -> 0
adbl(s(X)) -> s(s(adbl(mark(X))))
adbl(X) -> dbl(X)
aadd(0, X) -> mark(X)
aadd(s(X), Y) -> s(aadd(mark(X), mark(Y)))
aadd(X1, X2) -> add(X1, X2)
afirst(0, X) -> nil
afirst(s(X), cons(Y, Z)) -> cons(mark(Y), first(X, Z))
afirst(X1, X2) -> first(X1, X2)
ahalf(0) -> 0
ahalf(s(0)) -> 0
ahalf(s(s(X))) -> s(ahalf(mark(X)))
ahalf(dbl(X)) -> mark(X)
ahalf(X) -> half(X)
mark(terms(X)) -> aterms(mark(X))
mark(sqr(X)) -> asqr(mark(X))
mark(add(X1, X2)) -> aadd(mark(X1), mark(X2))
mark(dbl(X)) -> adbl(mark(X))
mark(first(X1, X2)) -> afirst(mark(X1), mark(X2))
mark(half(X)) -> ahalf(mark(X))
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(recip(X)) -> recip(mark(X))
mark(s(X)) -> s(mark(X))
mark(0) -> 0
mark(nil) -> nil

Termination of R to be shown.



   R
Dependency Pair Analysis



R contains the following Dependency Pairs:

ATERMS(N) -> ASQR(mark(N))
ATERMS(N) -> MARK(N)
ASQR(s(X)) -> AADD(asqr(mark(X)), adbl(mark(X)))
ASQR(s(X)) -> ASQR(mark(X))
ASQR(s(X)) -> MARK(X)
ASQR(s(X)) -> ADBL(mark(X))
ADBL(s(X)) -> ADBL(mark(X))
ADBL(s(X)) -> MARK(X)
AADD(0, X) -> MARK(X)
AADD(s(X), Y) -> AADD(mark(X), mark(Y))
AADD(s(X), Y) -> MARK(X)
AADD(s(X), Y) -> MARK(Y)
AFIRST(s(X), cons(Y, Z)) -> MARK(Y)
AHALF(s(s(X))) -> AHALF(mark(X))
AHALF(s(s(X))) -> MARK(X)
AHALF(dbl(X)) -> MARK(X)
MARK(terms(X)) -> ATERMS(mark(X))
MARK(terms(X)) -> MARK(X)
MARK(sqr(X)) -> ASQR(mark(X))
MARK(sqr(X)) -> MARK(X)
MARK(add(X1, X2)) -> AADD(mark(X1), mark(X2))
MARK(add(X1, X2)) -> MARK(X1)
MARK(add(X1, X2)) -> MARK(X2)
MARK(dbl(X)) -> ADBL(mark(X))
MARK(dbl(X)) -> MARK(X)
MARK(first(X1, X2)) -> AFIRST(mark(X1), mark(X2))
MARK(first(X1, X2)) -> MARK(X1)
MARK(first(X1, X2)) -> MARK(X2)
MARK(half(X)) -> AHALF(mark(X))
MARK(half(X)) -> MARK(X)
MARK(cons(X1, X2)) -> MARK(X1)
MARK(recip(X)) -> MARK(X)
MARK(s(X)) -> MARK(X)

Furthermore, R contains one SCC.


   R
DPs
       →DP Problem 1
Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

ASQR(s(X)) -> ADBL(mark(X))
AADD(s(X), Y) -> MARK(Y)
AHALF(dbl(X)) -> MARK(X)
MARK(s(X)) -> MARK(X)
MARK(recip(X)) -> MARK(X)
MARK(cons(X1, X2)) -> MARK(X1)
MARK(half(X)) -> MARK(X)
AHALF(s(s(X))) -> MARK(X)
AHALF(s(s(X))) -> AHALF(mark(X))
MARK(half(X)) -> AHALF(mark(X))
MARK(first(X1, X2)) -> MARK(X2)
MARK(first(X1, X2)) -> MARK(X1)
AFIRST(s(X), cons(Y, Z)) -> MARK(Y)
MARK(first(X1, X2)) -> AFIRST(mark(X1), mark(X2))
MARK(dbl(X)) -> MARK(X)
ADBL(s(X)) -> MARK(X)
ADBL(s(X)) -> ADBL(mark(X))
MARK(dbl(X)) -> ADBL(mark(X))
MARK(add(X1, X2)) -> MARK(X2)
MARK(add(X1, X2)) -> MARK(X1)
AADD(s(X), Y) -> MARK(X)
AADD(s(X), Y) -> AADD(mark(X), mark(Y))
MARK(add(X1, X2)) -> AADD(mark(X1), mark(X2))
MARK(sqr(X)) -> MARK(X)
ASQR(s(X)) -> MARK(X)
ASQR(s(X)) -> ASQR(mark(X))
MARK(sqr(X)) -> ASQR(mark(X))
MARK(terms(X)) -> MARK(X)
ATERMS(N) -> MARK(N)
MARK(terms(X)) -> ATERMS(mark(X))
AADD(0, X) -> MARK(X)
ASQR(s(X)) -> AADD(asqr(mark(X)), adbl(mark(X)))
ATERMS(N) -> ASQR(mark(N))


Rules:


aterms(N) -> cons(recip(asqr(mark(N))), terms(s(N)))
aterms(X) -> terms(X)
asqr(0) -> 0
asqr(s(X)) -> s(aadd(asqr(mark(X)), adbl(mark(X))))
asqr(X) -> sqr(X)
adbl(0) -> 0
adbl(s(X)) -> s(s(adbl(mark(X))))
adbl(X) -> dbl(X)
aadd(0, X) -> mark(X)
aadd(s(X), Y) -> s(aadd(mark(X), mark(Y)))
aadd(X1, X2) -> add(X1, X2)
afirst(0, X) -> nil
afirst(s(X), cons(Y, Z)) -> cons(mark(Y), first(X, Z))
afirst(X1, X2) -> first(X1, X2)
ahalf(0) -> 0
ahalf(s(0)) -> 0
ahalf(s(s(X))) -> s(ahalf(mark(X)))
ahalf(dbl(X)) -> mark(X)
ahalf(X) -> half(X)
mark(terms(X)) -> aterms(mark(X))
mark(sqr(X)) -> asqr(mark(X))
mark(add(X1, X2)) -> aadd(mark(X1), mark(X2))
mark(dbl(X)) -> adbl(mark(X))
mark(first(X1, X2)) -> afirst(mark(X1), mark(X2))
mark(half(X)) -> ahalf(mark(X))
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(recip(X)) -> recip(mark(X))
mark(s(X)) -> s(mark(X))
mark(0) -> 0
mark(nil) -> nil




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