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(X) -> sqr(X)
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(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)) -> ASQR(mark(X))
ASQR(s(X)) -> MARK(X)
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(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:

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)
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))
ATERMS(N) -> ASQR(mark(N))

Rules:

aterms(N) -> cons(recip(asqr(mark(N))), terms(s(N)))
aterms(X) -> terms(X)
asqr(0) -> 0
asqr(X) -> sqr(X)
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(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 could not be shown.
Duration:
0:01 minutes