active1(terms1(N)) -> mark1(cons2(recip1(sqr1(N)), terms1(s1(N))))
active1(sqr1(0)) -> mark1(0)
active1(sqr1(s1(X))) -> mark1(s1(add2(sqr1(X), dbl1(X))))
active1(dbl1(0)) -> mark1(0)
active1(dbl1(s1(X))) -> mark1(s1(s1(dbl1(X))))
active1(add2(0, X)) -> mark1(X)
active1(add2(s1(X), Y)) -> mark1(s1(add2(X, Y)))
active1(first2(0, X)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(terms1(X)) -> terms1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(recip1(X)) -> recip1(active1(X))
active1(sqr1(X)) -> sqr1(active1(X))
active1(s1(X)) -> s1(active1(X))
active1(add2(X1, X2)) -> add2(active1(X1), X2)
active1(add2(X1, X2)) -> add2(X1, active1(X2))
active1(dbl1(X)) -> dbl1(active1(X))
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
terms1(mark1(X)) -> mark1(terms1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
recip1(mark1(X)) -> mark1(recip1(X))
sqr1(mark1(X)) -> mark1(sqr1(X))
s1(mark1(X)) -> mark1(s1(X))
add2(mark1(X1), X2) -> mark1(add2(X1, X2))
add2(X1, mark1(X2)) -> mark1(add2(X1, X2))
dbl1(mark1(X)) -> mark1(dbl1(X))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
proper1(terms1(X)) -> terms1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(recip1(X)) -> recip1(proper1(X))
proper1(sqr1(X)) -> sqr1(proper1(X))
proper1(s1(X)) -> s1(proper1(X))
proper1(0) -> ok1(0)
proper1(add2(X1, X2)) -> add2(proper1(X1), proper1(X2))
proper1(dbl1(X)) -> dbl1(proper1(X))
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
terms1(ok1(X)) -> ok1(terms1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
recip1(ok1(X)) -> ok1(recip1(X))
sqr1(ok1(X)) -> ok1(sqr1(X))
s1(ok1(X)) -> ok1(s1(X))
add2(ok1(X1), ok1(X2)) -> ok1(add2(X1, X2))
dbl1(ok1(X)) -> ok1(dbl1(X))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))
↳ QTRS
↳ DependencyPairsProof
active1(terms1(N)) -> mark1(cons2(recip1(sqr1(N)), terms1(s1(N))))
active1(sqr1(0)) -> mark1(0)
active1(sqr1(s1(X))) -> mark1(s1(add2(sqr1(X), dbl1(X))))
active1(dbl1(0)) -> mark1(0)
active1(dbl1(s1(X))) -> mark1(s1(s1(dbl1(X))))
active1(add2(0, X)) -> mark1(X)
active1(add2(s1(X), Y)) -> mark1(s1(add2(X, Y)))
active1(first2(0, X)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(terms1(X)) -> terms1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(recip1(X)) -> recip1(active1(X))
active1(sqr1(X)) -> sqr1(active1(X))
active1(s1(X)) -> s1(active1(X))
active1(add2(X1, X2)) -> add2(active1(X1), X2)
active1(add2(X1, X2)) -> add2(X1, active1(X2))
active1(dbl1(X)) -> dbl1(active1(X))
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
terms1(mark1(X)) -> mark1(terms1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
recip1(mark1(X)) -> mark1(recip1(X))
sqr1(mark1(X)) -> mark1(sqr1(X))
s1(mark1(X)) -> mark1(s1(X))
add2(mark1(X1), X2) -> mark1(add2(X1, X2))
add2(X1, mark1(X2)) -> mark1(add2(X1, X2))
dbl1(mark1(X)) -> mark1(dbl1(X))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
proper1(terms1(X)) -> terms1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(recip1(X)) -> recip1(proper1(X))
proper1(sqr1(X)) -> sqr1(proper1(X))
proper1(s1(X)) -> s1(proper1(X))
proper1(0) -> ok1(0)
proper1(add2(X1, X2)) -> add2(proper1(X1), proper1(X2))
proper1(dbl1(X)) -> dbl1(proper1(X))
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
terms1(ok1(X)) -> ok1(terms1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
recip1(ok1(X)) -> ok1(recip1(X))
sqr1(ok1(X)) -> ok1(sqr1(X))
s1(ok1(X)) -> ok1(s1(X))
add2(ok1(X1), ok1(X2)) -> ok1(add2(X1, X2))
dbl1(ok1(X)) -> ok1(dbl1(X))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))
FIRST2(X1, mark1(X2)) -> FIRST2(X1, X2)
RECIP1(mark1(X)) -> RECIP1(X)
ACTIVE1(recip1(X)) -> RECIP1(active1(X))
S1(mark1(X)) -> S1(X)
FIRST2(mark1(X1), X2) -> FIRST2(X1, X2)
RECIP1(ok1(X)) -> RECIP1(X)
ACTIVE1(add2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(dbl1(s1(X))) -> DBL1(X)
ACTIVE1(first2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(first2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(terms1(X)) -> ACTIVE1(X)
ACTIVE1(terms1(N)) -> SQR1(N)
ACTIVE1(first2(X1, X2)) -> FIRST2(active1(X1), X2)
TOP1(mark1(X)) -> TOP1(proper1(X))
PROPER1(sqr1(X)) -> PROPER1(X)
ACTIVE1(dbl1(X)) -> ACTIVE1(X)
PROPER1(cons2(X1, X2)) -> PROPER1(X1)
ACTIVE1(add2(X1, X2)) -> ADD2(active1(X1), X2)
SQR1(mark1(X)) -> SQR1(X)
TOP1(ok1(X)) -> ACTIVE1(X)
ACTIVE1(terms1(N)) -> CONS2(recip1(sqr1(N)), terms1(s1(N)))
PROPER1(sqr1(X)) -> SQR1(proper1(X))
ACTIVE1(s1(X)) -> S1(active1(X))
PROPER1(s1(X)) -> PROPER1(X)
ACTIVE1(add2(X1, X2)) -> ADD2(X1, active1(X2))
PROPER1(recip1(X)) -> RECIP1(proper1(X))
ACTIVE1(add2(s1(X), Y)) -> ADD2(X, Y)
ACTIVE1(sqr1(s1(X))) -> ADD2(sqr1(X), dbl1(X))
ACTIVE1(terms1(N)) -> RECIP1(sqr1(N))
ACTIVE1(first2(s1(X), cons2(Y, Z))) -> FIRST2(X, Z)
TOP1(mark1(X)) -> PROPER1(X)
ADD2(mark1(X1), X2) -> ADD2(X1, X2)
ACTIVE1(add2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(sqr1(X)) -> ACTIVE1(X)
PROPER1(terms1(X)) -> PROPER1(X)
DBL1(ok1(X)) -> DBL1(X)
PROPER1(first2(X1, X2)) -> PROPER1(X1)
PROPER1(first2(X1, X2)) -> PROPER1(X2)
ACTIVE1(sqr1(s1(X))) -> DBL1(X)
ACTIVE1(first2(X1, X2)) -> FIRST2(X1, active1(X2))
ACTIVE1(dbl1(X)) -> DBL1(active1(X))
PROPER1(add2(X1, X2)) -> PROPER1(X1)
PROPER1(add2(X1, X2)) -> ADD2(proper1(X1), proper1(X2))
PROPER1(s1(X)) -> S1(proper1(X))
S1(ok1(X)) -> S1(X)
ACTIVE1(first2(s1(X), cons2(Y, Z))) -> CONS2(Y, first2(X, Z))
ACTIVE1(s1(X)) -> ACTIVE1(X)
PROPER1(recip1(X)) -> PROPER1(X)
CONS2(mark1(X1), X2) -> CONS2(X1, X2)
ACTIVE1(cons2(X1, X2)) -> CONS2(active1(X1), X2)
ADD2(ok1(X1), ok1(X2)) -> ADD2(X1, X2)
ACTIVE1(cons2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(sqr1(s1(X))) -> S1(add2(sqr1(X), dbl1(X)))
TERMS1(ok1(X)) -> TERMS1(X)
ACTIVE1(terms1(X)) -> TERMS1(active1(X))
ACTIVE1(sqr1(X)) -> SQR1(active1(X))
ACTIVE1(terms1(N)) -> TERMS1(s1(N))
DBL1(mark1(X)) -> DBL1(X)
PROPER1(first2(X1, X2)) -> FIRST2(proper1(X1), proper1(X2))
ACTIVE1(dbl1(s1(X))) -> S1(dbl1(X))
PROPER1(add2(X1, X2)) -> PROPER1(X2)
ACTIVE1(sqr1(s1(X))) -> SQR1(X)
FIRST2(ok1(X1), ok1(X2)) -> FIRST2(X1, X2)
ADD2(X1, mark1(X2)) -> ADD2(X1, X2)
PROPER1(dbl1(X)) -> DBL1(proper1(X))
PROPER1(cons2(X1, X2)) -> CONS2(proper1(X1), proper1(X2))
SQR1(ok1(X)) -> SQR1(X)
TERMS1(mark1(X)) -> TERMS1(X)
PROPER1(terms1(X)) -> TERMS1(proper1(X))
TOP1(ok1(X)) -> TOP1(active1(X))
PROPER1(cons2(X1, X2)) -> PROPER1(X2)
ACTIVE1(terms1(N)) -> S1(N)
CONS2(ok1(X1), ok1(X2)) -> CONS2(X1, X2)
ACTIVE1(dbl1(s1(X))) -> S1(s1(dbl1(X)))
PROPER1(dbl1(X)) -> PROPER1(X)
ACTIVE1(recip1(X)) -> ACTIVE1(X)
ACTIVE1(add2(s1(X), Y)) -> S1(add2(X, Y))
active1(terms1(N)) -> mark1(cons2(recip1(sqr1(N)), terms1(s1(N))))
active1(sqr1(0)) -> mark1(0)
active1(sqr1(s1(X))) -> mark1(s1(add2(sqr1(X), dbl1(X))))
active1(dbl1(0)) -> mark1(0)
active1(dbl1(s1(X))) -> mark1(s1(s1(dbl1(X))))
active1(add2(0, X)) -> mark1(X)
active1(add2(s1(X), Y)) -> mark1(s1(add2(X, Y)))
active1(first2(0, X)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(terms1(X)) -> terms1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(recip1(X)) -> recip1(active1(X))
active1(sqr1(X)) -> sqr1(active1(X))
active1(s1(X)) -> s1(active1(X))
active1(add2(X1, X2)) -> add2(active1(X1), X2)
active1(add2(X1, X2)) -> add2(X1, active1(X2))
active1(dbl1(X)) -> dbl1(active1(X))
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
terms1(mark1(X)) -> mark1(terms1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
recip1(mark1(X)) -> mark1(recip1(X))
sqr1(mark1(X)) -> mark1(sqr1(X))
s1(mark1(X)) -> mark1(s1(X))
add2(mark1(X1), X2) -> mark1(add2(X1, X2))
add2(X1, mark1(X2)) -> mark1(add2(X1, X2))
dbl1(mark1(X)) -> mark1(dbl1(X))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
proper1(terms1(X)) -> terms1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(recip1(X)) -> recip1(proper1(X))
proper1(sqr1(X)) -> sqr1(proper1(X))
proper1(s1(X)) -> s1(proper1(X))
proper1(0) -> ok1(0)
proper1(add2(X1, X2)) -> add2(proper1(X1), proper1(X2))
proper1(dbl1(X)) -> dbl1(proper1(X))
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
terms1(ok1(X)) -> ok1(terms1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
recip1(ok1(X)) -> ok1(recip1(X))
sqr1(ok1(X)) -> ok1(sqr1(X))
s1(ok1(X)) -> ok1(s1(X))
add2(ok1(X1), ok1(X2)) -> ok1(add2(X1, X2))
dbl1(ok1(X)) -> ok1(dbl1(X))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
FIRST2(X1, mark1(X2)) -> FIRST2(X1, X2)
RECIP1(mark1(X)) -> RECIP1(X)
ACTIVE1(recip1(X)) -> RECIP1(active1(X))
S1(mark1(X)) -> S1(X)
FIRST2(mark1(X1), X2) -> FIRST2(X1, X2)
RECIP1(ok1(X)) -> RECIP1(X)
ACTIVE1(add2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(dbl1(s1(X))) -> DBL1(X)
ACTIVE1(first2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(first2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(terms1(X)) -> ACTIVE1(X)
ACTIVE1(terms1(N)) -> SQR1(N)
ACTIVE1(first2(X1, X2)) -> FIRST2(active1(X1), X2)
TOP1(mark1(X)) -> TOP1(proper1(X))
PROPER1(sqr1(X)) -> PROPER1(X)
ACTIVE1(dbl1(X)) -> ACTIVE1(X)
PROPER1(cons2(X1, X2)) -> PROPER1(X1)
ACTIVE1(add2(X1, X2)) -> ADD2(active1(X1), X2)
SQR1(mark1(X)) -> SQR1(X)
TOP1(ok1(X)) -> ACTIVE1(X)
ACTIVE1(terms1(N)) -> CONS2(recip1(sqr1(N)), terms1(s1(N)))
PROPER1(sqr1(X)) -> SQR1(proper1(X))
ACTIVE1(s1(X)) -> S1(active1(X))
PROPER1(s1(X)) -> PROPER1(X)
ACTIVE1(add2(X1, X2)) -> ADD2(X1, active1(X2))
PROPER1(recip1(X)) -> RECIP1(proper1(X))
ACTIVE1(add2(s1(X), Y)) -> ADD2(X, Y)
ACTIVE1(sqr1(s1(X))) -> ADD2(sqr1(X), dbl1(X))
ACTIVE1(terms1(N)) -> RECIP1(sqr1(N))
ACTIVE1(first2(s1(X), cons2(Y, Z))) -> FIRST2(X, Z)
TOP1(mark1(X)) -> PROPER1(X)
ADD2(mark1(X1), X2) -> ADD2(X1, X2)
ACTIVE1(add2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(sqr1(X)) -> ACTIVE1(X)
PROPER1(terms1(X)) -> PROPER1(X)
DBL1(ok1(X)) -> DBL1(X)
PROPER1(first2(X1, X2)) -> PROPER1(X1)
PROPER1(first2(X1, X2)) -> PROPER1(X2)
ACTIVE1(sqr1(s1(X))) -> DBL1(X)
ACTIVE1(first2(X1, X2)) -> FIRST2(X1, active1(X2))
ACTIVE1(dbl1(X)) -> DBL1(active1(X))
PROPER1(add2(X1, X2)) -> PROPER1(X1)
PROPER1(add2(X1, X2)) -> ADD2(proper1(X1), proper1(X2))
PROPER1(s1(X)) -> S1(proper1(X))
S1(ok1(X)) -> S1(X)
ACTIVE1(first2(s1(X), cons2(Y, Z))) -> CONS2(Y, first2(X, Z))
ACTIVE1(s1(X)) -> ACTIVE1(X)
PROPER1(recip1(X)) -> PROPER1(X)
CONS2(mark1(X1), X2) -> CONS2(X1, X2)
ACTIVE1(cons2(X1, X2)) -> CONS2(active1(X1), X2)
ADD2(ok1(X1), ok1(X2)) -> ADD2(X1, X2)
ACTIVE1(cons2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(sqr1(s1(X))) -> S1(add2(sqr1(X), dbl1(X)))
TERMS1(ok1(X)) -> TERMS1(X)
ACTIVE1(terms1(X)) -> TERMS1(active1(X))
ACTIVE1(sqr1(X)) -> SQR1(active1(X))
ACTIVE1(terms1(N)) -> TERMS1(s1(N))
DBL1(mark1(X)) -> DBL1(X)
PROPER1(first2(X1, X2)) -> FIRST2(proper1(X1), proper1(X2))
ACTIVE1(dbl1(s1(X))) -> S1(dbl1(X))
PROPER1(add2(X1, X2)) -> PROPER1(X2)
ACTIVE1(sqr1(s1(X))) -> SQR1(X)
FIRST2(ok1(X1), ok1(X2)) -> FIRST2(X1, X2)
ADD2(X1, mark1(X2)) -> ADD2(X1, X2)
PROPER1(dbl1(X)) -> DBL1(proper1(X))
PROPER1(cons2(X1, X2)) -> CONS2(proper1(X1), proper1(X2))
SQR1(ok1(X)) -> SQR1(X)
TERMS1(mark1(X)) -> TERMS1(X)
PROPER1(terms1(X)) -> TERMS1(proper1(X))
TOP1(ok1(X)) -> TOP1(active1(X))
PROPER1(cons2(X1, X2)) -> PROPER1(X2)
ACTIVE1(terms1(N)) -> S1(N)
CONS2(ok1(X1), ok1(X2)) -> CONS2(X1, X2)
ACTIVE1(dbl1(s1(X))) -> S1(s1(dbl1(X)))
PROPER1(dbl1(X)) -> PROPER1(X)
ACTIVE1(recip1(X)) -> ACTIVE1(X)
ACTIVE1(add2(s1(X), Y)) -> S1(add2(X, Y))
active1(terms1(N)) -> mark1(cons2(recip1(sqr1(N)), terms1(s1(N))))
active1(sqr1(0)) -> mark1(0)
active1(sqr1(s1(X))) -> mark1(s1(add2(sqr1(X), dbl1(X))))
active1(dbl1(0)) -> mark1(0)
active1(dbl1(s1(X))) -> mark1(s1(s1(dbl1(X))))
active1(add2(0, X)) -> mark1(X)
active1(add2(s1(X), Y)) -> mark1(s1(add2(X, Y)))
active1(first2(0, X)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(terms1(X)) -> terms1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(recip1(X)) -> recip1(active1(X))
active1(sqr1(X)) -> sqr1(active1(X))
active1(s1(X)) -> s1(active1(X))
active1(add2(X1, X2)) -> add2(active1(X1), X2)
active1(add2(X1, X2)) -> add2(X1, active1(X2))
active1(dbl1(X)) -> dbl1(active1(X))
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
terms1(mark1(X)) -> mark1(terms1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
recip1(mark1(X)) -> mark1(recip1(X))
sqr1(mark1(X)) -> mark1(sqr1(X))
s1(mark1(X)) -> mark1(s1(X))
add2(mark1(X1), X2) -> mark1(add2(X1, X2))
add2(X1, mark1(X2)) -> mark1(add2(X1, X2))
dbl1(mark1(X)) -> mark1(dbl1(X))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
proper1(terms1(X)) -> terms1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(recip1(X)) -> recip1(proper1(X))
proper1(sqr1(X)) -> sqr1(proper1(X))
proper1(s1(X)) -> s1(proper1(X))
proper1(0) -> ok1(0)
proper1(add2(X1, X2)) -> add2(proper1(X1), proper1(X2))
proper1(dbl1(X)) -> dbl1(proper1(X))
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
terms1(ok1(X)) -> ok1(terms1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
recip1(ok1(X)) -> ok1(recip1(X))
sqr1(ok1(X)) -> ok1(sqr1(X))
s1(ok1(X)) -> ok1(s1(X))
add2(ok1(X1), ok1(X2)) -> ok1(add2(X1, X2))
dbl1(ok1(X)) -> ok1(dbl1(X))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
FIRST2(X1, mark1(X2)) -> FIRST2(X1, X2)
FIRST2(ok1(X1), ok1(X2)) -> FIRST2(X1, X2)
FIRST2(mark1(X1), X2) -> FIRST2(X1, X2)
active1(terms1(N)) -> mark1(cons2(recip1(sqr1(N)), terms1(s1(N))))
active1(sqr1(0)) -> mark1(0)
active1(sqr1(s1(X))) -> mark1(s1(add2(sqr1(X), dbl1(X))))
active1(dbl1(0)) -> mark1(0)
active1(dbl1(s1(X))) -> mark1(s1(s1(dbl1(X))))
active1(add2(0, X)) -> mark1(X)
active1(add2(s1(X), Y)) -> mark1(s1(add2(X, Y)))
active1(first2(0, X)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(terms1(X)) -> terms1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(recip1(X)) -> recip1(active1(X))
active1(sqr1(X)) -> sqr1(active1(X))
active1(s1(X)) -> s1(active1(X))
active1(add2(X1, X2)) -> add2(active1(X1), X2)
active1(add2(X1, X2)) -> add2(X1, active1(X2))
active1(dbl1(X)) -> dbl1(active1(X))
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
terms1(mark1(X)) -> mark1(terms1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
recip1(mark1(X)) -> mark1(recip1(X))
sqr1(mark1(X)) -> mark1(sqr1(X))
s1(mark1(X)) -> mark1(s1(X))
add2(mark1(X1), X2) -> mark1(add2(X1, X2))
add2(X1, mark1(X2)) -> mark1(add2(X1, X2))
dbl1(mark1(X)) -> mark1(dbl1(X))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
proper1(terms1(X)) -> terms1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(recip1(X)) -> recip1(proper1(X))
proper1(sqr1(X)) -> sqr1(proper1(X))
proper1(s1(X)) -> s1(proper1(X))
proper1(0) -> ok1(0)
proper1(add2(X1, X2)) -> add2(proper1(X1), proper1(X2))
proper1(dbl1(X)) -> dbl1(proper1(X))
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
terms1(ok1(X)) -> ok1(terms1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
recip1(ok1(X)) -> ok1(recip1(X))
sqr1(ok1(X)) -> ok1(sqr1(X))
s1(ok1(X)) -> ok1(s1(X))
add2(ok1(X1), ok1(X2)) -> ok1(add2(X1, X2))
dbl1(ok1(X)) -> ok1(dbl1(X))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
FIRST2(ok1(X1), ok1(X2)) -> FIRST2(X1, X2)
FIRST2(mark1(X1), X2) -> FIRST2(X1, X2)
Used ordering: Polynomial interpretation [21]:
FIRST2(X1, mark1(X2)) -> FIRST2(X1, X2)
POL(FIRST2(x1, x2)) = x1
POL(mark1(x1)) = 1 + x1
POL(ok1(x1)) = 1 + x1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
FIRST2(X1, mark1(X2)) -> FIRST2(X1, X2)
active1(terms1(N)) -> mark1(cons2(recip1(sqr1(N)), terms1(s1(N))))
active1(sqr1(0)) -> mark1(0)
active1(sqr1(s1(X))) -> mark1(s1(add2(sqr1(X), dbl1(X))))
active1(dbl1(0)) -> mark1(0)
active1(dbl1(s1(X))) -> mark1(s1(s1(dbl1(X))))
active1(add2(0, X)) -> mark1(X)
active1(add2(s1(X), Y)) -> mark1(s1(add2(X, Y)))
active1(first2(0, X)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(terms1(X)) -> terms1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(recip1(X)) -> recip1(active1(X))
active1(sqr1(X)) -> sqr1(active1(X))
active1(s1(X)) -> s1(active1(X))
active1(add2(X1, X2)) -> add2(active1(X1), X2)
active1(add2(X1, X2)) -> add2(X1, active1(X2))
active1(dbl1(X)) -> dbl1(active1(X))
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
terms1(mark1(X)) -> mark1(terms1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
recip1(mark1(X)) -> mark1(recip1(X))
sqr1(mark1(X)) -> mark1(sqr1(X))
s1(mark1(X)) -> mark1(s1(X))
add2(mark1(X1), X2) -> mark1(add2(X1, X2))
add2(X1, mark1(X2)) -> mark1(add2(X1, X2))
dbl1(mark1(X)) -> mark1(dbl1(X))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
proper1(terms1(X)) -> terms1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(recip1(X)) -> recip1(proper1(X))
proper1(sqr1(X)) -> sqr1(proper1(X))
proper1(s1(X)) -> s1(proper1(X))
proper1(0) -> ok1(0)
proper1(add2(X1, X2)) -> add2(proper1(X1), proper1(X2))
proper1(dbl1(X)) -> dbl1(proper1(X))
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
terms1(ok1(X)) -> ok1(terms1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
recip1(ok1(X)) -> ok1(recip1(X))
sqr1(ok1(X)) -> ok1(sqr1(X))
s1(ok1(X)) -> ok1(s1(X))
add2(ok1(X1), ok1(X2)) -> ok1(add2(X1, X2))
dbl1(ok1(X)) -> ok1(dbl1(X))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
FIRST2(X1, mark1(X2)) -> FIRST2(X1, X2)
POL(FIRST2(x1, x2)) = x2
POL(mark1(x1)) = 1 + x1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
active1(terms1(N)) -> mark1(cons2(recip1(sqr1(N)), terms1(s1(N))))
active1(sqr1(0)) -> mark1(0)
active1(sqr1(s1(X))) -> mark1(s1(add2(sqr1(X), dbl1(X))))
active1(dbl1(0)) -> mark1(0)
active1(dbl1(s1(X))) -> mark1(s1(s1(dbl1(X))))
active1(add2(0, X)) -> mark1(X)
active1(add2(s1(X), Y)) -> mark1(s1(add2(X, Y)))
active1(first2(0, X)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(terms1(X)) -> terms1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(recip1(X)) -> recip1(active1(X))
active1(sqr1(X)) -> sqr1(active1(X))
active1(s1(X)) -> s1(active1(X))
active1(add2(X1, X2)) -> add2(active1(X1), X2)
active1(add2(X1, X2)) -> add2(X1, active1(X2))
active1(dbl1(X)) -> dbl1(active1(X))
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
terms1(mark1(X)) -> mark1(terms1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
recip1(mark1(X)) -> mark1(recip1(X))
sqr1(mark1(X)) -> mark1(sqr1(X))
s1(mark1(X)) -> mark1(s1(X))
add2(mark1(X1), X2) -> mark1(add2(X1, X2))
add2(X1, mark1(X2)) -> mark1(add2(X1, X2))
dbl1(mark1(X)) -> mark1(dbl1(X))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
proper1(terms1(X)) -> terms1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(recip1(X)) -> recip1(proper1(X))
proper1(sqr1(X)) -> sqr1(proper1(X))
proper1(s1(X)) -> s1(proper1(X))
proper1(0) -> ok1(0)
proper1(add2(X1, X2)) -> add2(proper1(X1), proper1(X2))
proper1(dbl1(X)) -> dbl1(proper1(X))
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
terms1(ok1(X)) -> ok1(terms1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
recip1(ok1(X)) -> ok1(recip1(X))
sqr1(ok1(X)) -> ok1(sqr1(X))
s1(ok1(X)) -> ok1(s1(X))
add2(ok1(X1), ok1(X2)) -> ok1(add2(X1, X2))
dbl1(ok1(X)) -> ok1(dbl1(X))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
DBL1(mark1(X)) -> DBL1(X)
DBL1(ok1(X)) -> DBL1(X)
active1(terms1(N)) -> mark1(cons2(recip1(sqr1(N)), terms1(s1(N))))
active1(sqr1(0)) -> mark1(0)
active1(sqr1(s1(X))) -> mark1(s1(add2(sqr1(X), dbl1(X))))
active1(dbl1(0)) -> mark1(0)
active1(dbl1(s1(X))) -> mark1(s1(s1(dbl1(X))))
active1(add2(0, X)) -> mark1(X)
active1(add2(s1(X), Y)) -> mark1(s1(add2(X, Y)))
active1(first2(0, X)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(terms1(X)) -> terms1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(recip1(X)) -> recip1(active1(X))
active1(sqr1(X)) -> sqr1(active1(X))
active1(s1(X)) -> s1(active1(X))
active1(add2(X1, X2)) -> add2(active1(X1), X2)
active1(add2(X1, X2)) -> add2(X1, active1(X2))
active1(dbl1(X)) -> dbl1(active1(X))
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
terms1(mark1(X)) -> mark1(terms1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
recip1(mark1(X)) -> mark1(recip1(X))
sqr1(mark1(X)) -> mark1(sqr1(X))
s1(mark1(X)) -> mark1(s1(X))
add2(mark1(X1), X2) -> mark1(add2(X1, X2))
add2(X1, mark1(X2)) -> mark1(add2(X1, X2))
dbl1(mark1(X)) -> mark1(dbl1(X))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
proper1(terms1(X)) -> terms1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(recip1(X)) -> recip1(proper1(X))
proper1(sqr1(X)) -> sqr1(proper1(X))
proper1(s1(X)) -> s1(proper1(X))
proper1(0) -> ok1(0)
proper1(add2(X1, X2)) -> add2(proper1(X1), proper1(X2))
proper1(dbl1(X)) -> dbl1(proper1(X))
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
terms1(ok1(X)) -> ok1(terms1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
recip1(ok1(X)) -> ok1(recip1(X))
sqr1(ok1(X)) -> ok1(sqr1(X))
s1(ok1(X)) -> ok1(s1(X))
add2(ok1(X1), ok1(X2)) -> ok1(add2(X1, X2))
dbl1(ok1(X)) -> ok1(dbl1(X))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
DBL1(mark1(X)) -> DBL1(X)
DBL1(ok1(X)) -> DBL1(X)
POL(DBL1(x1)) = x1
POL(mark1(x1)) = 1 + x1
POL(ok1(x1)) = 1 + x1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
active1(terms1(N)) -> mark1(cons2(recip1(sqr1(N)), terms1(s1(N))))
active1(sqr1(0)) -> mark1(0)
active1(sqr1(s1(X))) -> mark1(s1(add2(sqr1(X), dbl1(X))))
active1(dbl1(0)) -> mark1(0)
active1(dbl1(s1(X))) -> mark1(s1(s1(dbl1(X))))
active1(add2(0, X)) -> mark1(X)
active1(add2(s1(X), Y)) -> mark1(s1(add2(X, Y)))
active1(first2(0, X)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(terms1(X)) -> terms1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(recip1(X)) -> recip1(active1(X))
active1(sqr1(X)) -> sqr1(active1(X))
active1(s1(X)) -> s1(active1(X))
active1(add2(X1, X2)) -> add2(active1(X1), X2)
active1(add2(X1, X2)) -> add2(X1, active1(X2))
active1(dbl1(X)) -> dbl1(active1(X))
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
terms1(mark1(X)) -> mark1(terms1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
recip1(mark1(X)) -> mark1(recip1(X))
sqr1(mark1(X)) -> mark1(sqr1(X))
s1(mark1(X)) -> mark1(s1(X))
add2(mark1(X1), X2) -> mark1(add2(X1, X2))
add2(X1, mark1(X2)) -> mark1(add2(X1, X2))
dbl1(mark1(X)) -> mark1(dbl1(X))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
proper1(terms1(X)) -> terms1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(recip1(X)) -> recip1(proper1(X))
proper1(sqr1(X)) -> sqr1(proper1(X))
proper1(s1(X)) -> s1(proper1(X))
proper1(0) -> ok1(0)
proper1(add2(X1, X2)) -> add2(proper1(X1), proper1(X2))
proper1(dbl1(X)) -> dbl1(proper1(X))
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
terms1(ok1(X)) -> ok1(terms1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
recip1(ok1(X)) -> ok1(recip1(X))
sqr1(ok1(X)) -> ok1(sqr1(X))
s1(ok1(X)) -> ok1(s1(X))
add2(ok1(X1), ok1(X2)) -> ok1(add2(X1, X2))
dbl1(ok1(X)) -> ok1(dbl1(X))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
ADD2(X1, mark1(X2)) -> ADD2(X1, X2)
ADD2(mark1(X1), X2) -> ADD2(X1, X2)
ADD2(ok1(X1), ok1(X2)) -> ADD2(X1, X2)
active1(terms1(N)) -> mark1(cons2(recip1(sqr1(N)), terms1(s1(N))))
active1(sqr1(0)) -> mark1(0)
active1(sqr1(s1(X))) -> mark1(s1(add2(sqr1(X), dbl1(X))))
active1(dbl1(0)) -> mark1(0)
active1(dbl1(s1(X))) -> mark1(s1(s1(dbl1(X))))
active1(add2(0, X)) -> mark1(X)
active1(add2(s1(X), Y)) -> mark1(s1(add2(X, Y)))
active1(first2(0, X)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(terms1(X)) -> terms1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(recip1(X)) -> recip1(active1(X))
active1(sqr1(X)) -> sqr1(active1(X))
active1(s1(X)) -> s1(active1(X))
active1(add2(X1, X2)) -> add2(active1(X1), X2)
active1(add2(X1, X2)) -> add2(X1, active1(X2))
active1(dbl1(X)) -> dbl1(active1(X))
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
terms1(mark1(X)) -> mark1(terms1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
recip1(mark1(X)) -> mark1(recip1(X))
sqr1(mark1(X)) -> mark1(sqr1(X))
s1(mark1(X)) -> mark1(s1(X))
add2(mark1(X1), X2) -> mark1(add2(X1, X2))
add2(X1, mark1(X2)) -> mark1(add2(X1, X2))
dbl1(mark1(X)) -> mark1(dbl1(X))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
proper1(terms1(X)) -> terms1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(recip1(X)) -> recip1(proper1(X))
proper1(sqr1(X)) -> sqr1(proper1(X))
proper1(s1(X)) -> s1(proper1(X))
proper1(0) -> ok1(0)
proper1(add2(X1, X2)) -> add2(proper1(X1), proper1(X2))
proper1(dbl1(X)) -> dbl1(proper1(X))
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
terms1(ok1(X)) -> ok1(terms1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
recip1(ok1(X)) -> ok1(recip1(X))
sqr1(ok1(X)) -> ok1(sqr1(X))
s1(ok1(X)) -> ok1(s1(X))
add2(ok1(X1), ok1(X2)) -> ok1(add2(X1, X2))
dbl1(ok1(X)) -> ok1(dbl1(X))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
ADD2(mark1(X1), X2) -> ADD2(X1, X2)
ADD2(ok1(X1), ok1(X2)) -> ADD2(X1, X2)
Used ordering: Polynomial interpretation [21]:
ADD2(X1, mark1(X2)) -> ADD2(X1, X2)
POL(ADD2(x1, x2)) = x1
POL(mark1(x1)) = 1 + x1
POL(ok1(x1)) = 1 + x1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
ADD2(X1, mark1(X2)) -> ADD2(X1, X2)
active1(terms1(N)) -> mark1(cons2(recip1(sqr1(N)), terms1(s1(N))))
active1(sqr1(0)) -> mark1(0)
active1(sqr1(s1(X))) -> mark1(s1(add2(sqr1(X), dbl1(X))))
active1(dbl1(0)) -> mark1(0)
active1(dbl1(s1(X))) -> mark1(s1(s1(dbl1(X))))
active1(add2(0, X)) -> mark1(X)
active1(add2(s1(X), Y)) -> mark1(s1(add2(X, Y)))
active1(first2(0, X)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(terms1(X)) -> terms1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(recip1(X)) -> recip1(active1(X))
active1(sqr1(X)) -> sqr1(active1(X))
active1(s1(X)) -> s1(active1(X))
active1(add2(X1, X2)) -> add2(active1(X1), X2)
active1(add2(X1, X2)) -> add2(X1, active1(X2))
active1(dbl1(X)) -> dbl1(active1(X))
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
terms1(mark1(X)) -> mark1(terms1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
recip1(mark1(X)) -> mark1(recip1(X))
sqr1(mark1(X)) -> mark1(sqr1(X))
s1(mark1(X)) -> mark1(s1(X))
add2(mark1(X1), X2) -> mark1(add2(X1, X2))
add2(X1, mark1(X2)) -> mark1(add2(X1, X2))
dbl1(mark1(X)) -> mark1(dbl1(X))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
proper1(terms1(X)) -> terms1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(recip1(X)) -> recip1(proper1(X))
proper1(sqr1(X)) -> sqr1(proper1(X))
proper1(s1(X)) -> s1(proper1(X))
proper1(0) -> ok1(0)
proper1(add2(X1, X2)) -> add2(proper1(X1), proper1(X2))
proper1(dbl1(X)) -> dbl1(proper1(X))
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
terms1(ok1(X)) -> ok1(terms1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
recip1(ok1(X)) -> ok1(recip1(X))
sqr1(ok1(X)) -> ok1(sqr1(X))
s1(ok1(X)) -> ok1(s1(X))
add2(ok1(X1), ok1(X2)) -> ok1(add2(X1, X2))
dbl1(ok1(X)) -> ok1(dbl1(X))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
ADD2(X1, mark1(X2)) -> ADD2(X1, X2)
POL(ADD2(x1, x2)) = x2
POL(mark1(x1)) = 1 + x1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
active1(terms1(N)) -> mark1(cons2(recip1(sqr1(N)), terms1(s1(N))))
active1(sqr1(0)) -> mark1(0)
active1(sqr1(s1(X))) -> mark1(s1(add2(sqr1(X), dbl1(X))))
active1(dbl1(0)) -> mark1(0)
active1(dbl1(s1(X))) -> mark1(s1(s1(dbl1(X))))
active1(add2(0, X)) -> mark1(X)
active1(add2(s1(X), Y)) -> mark1(s1(add2(X, Y)))
active1(first2(0, X)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(terms1(X)) -> terms1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(recip1(X)) -> recip1(active1(X))
active1(sqr1(X)) -> sqr1(active1(X))
active1(s1(X)) -> s1(active1(X))
active1(add2(X1, X2)) -> add2(active1(X1), X2)
active1(add2(X1, X2)) -> add2(X1, active1(X2))
active1(dbl1(X)) -> dbl1(active1(X))
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
terms1(mark1(X)) -> mark1(terms1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
recip1(mark1(X)) -> mark1(recip1(X))
sqr1(mark1(X)) -> mark1(sqr1(X))
s1(mark1(X)) -> mark1(s1(X))
add2(mark1(X1), X2) -> mark1(add2(X1, X2))
add2(X1, mark1(X2)) -> mark1(add2(X1, X2))
dbl1(mark1(X)) -> mark1(dbl1(X))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
proper1(terms1(X)) -> terms1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(recip1(X)) -> recip1(proper1(X))
proper1(sqr1(X)) -> sqr1(proper1(X))
proper1(s1(X)) -> s1(proper1(X))
proper1(0) -> ok1(0)
proper1(add2(X1, X2)) -> add2(proper1(X1), proper1(X2))
proper1(dbl1(X)) -> dbl1(proper1(X))
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
terms1(ok1(X)) -> ok1(terms1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
recip1(ok1(X)) -> ok1(recip1(X))
sqr1(ok1(X)) -> ok1(sqr1(X))
s1(ok1(X)) -> ok1(s1(X))
add2(ok1(X1), ok1(X2)) -> ok1(add2(X1, X2))
dbl1(ok1(X)) -> ok1(dbl1(X))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
S1(ok1(X)) -> S1(X)
S1(mark1(X)) -> S1(X)
active1(terms1(N)) -> mark1(cons2(recip1(sqr1(N)), terms1(s1(N))))
active1(sqr1(0)) -> mark1(0)
active1(sqr1(s1(X))) -> mark1(s1(add2(sqr1(X), dbl1(X))))
active1(dbl1(0)) -> mark1(0)
active1(dbl1(s1(X))) -> mark1(s1(s1(dbl1(X))))
active1(add2(0, X)) -> mark1(X)
active1(add2(s1(X), Y)) -> mark1(s1(add2(X, Y)))
active1(first2(0, X)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(terms1(X)) -> terms1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(recip1(X)) -> recip1(active1(X))
active1(sqr1(X)) -> sqr1(active1(X))
active1(s1(X)) -> s1(active1(X))
active1(add2(X1, X2)) -> add2(active1(X1), X2)
active1(add2(X1, X2)) -> add2(X1, active1(X2))
active1(dbl1(X)) -> dbl1(active1(X))
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
terms1(mark1(X)) -> mark1(terms1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
recip1(mark1(X)) -> mark1(recip1(X))
sqr1(mark1(X)) -> mark1(sqr1(X))
s1(mark1(X)) -> mark1(s1(X))
add2(mark1(X1), X2) -> mark1(add2(X1, X2))
add2(X1, mark1(X2)) -> mark1(add2(X1, X2))
dbl1(mark1(X)) -> mark1(dbl1(X))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
proper1(terms1(X)) -> terms1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(recip1(X)) -> recip1(proper1(X))
proper1(sqr1(X)) -> sqr1(proper1(X))
proper1(s1(X)) -> s1(proper1(X))
proper1(0) -> ok1(0)
proper1(add2(X1, X2)) -> add2(proper1(X1), proper1(X2))
proper1(dbl1(X)) -> dbl1(proper1(X))
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
terms1(ok1(X)) -> ok1(terms1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
recip1(ok1(X)) -> ok1(recip1(X))
sqr1(ok1(X)) -> ok1(sqr1(X))
s1(ok1(X)) -> ok1(s1(X))
add2(ok1(X1), ok1(X2)) -> ok1(add2(X1, X2))
dbl1(ok1(X)) -> ok1(dbl1(X))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
S1(ok1(X)) -> S1(X)
S1(mark1(X)) -> S1(X)
POL(S1(x1)) = x1
POL(mark1(x1)) = 1 + x1
POL(ok1(x1)) = 1 + x1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
active1(terms1(N)) -> mark1(cons2(recip1(sqr1(N)), terms1(s1(N))))
active1(sqr1(0)) -> mark1(0)
active1(sqr1(s1(X))) -> mark1(s1(add2(sqr1(X), dbl1(X))))
active1(dbl1(0)) -> mark1(0)
active1(dbl1(s1(X))) -> mark1(s1(s1(dbl1(X))))
active1(add2(0, X)) -> mark1(X)
active1(add2(s1(X), Y)) -> mark1(s1(add2(X, Y)))
active1(first2(0, X)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(terms1(X)) -> terms1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(recip1(X)) -> recip1(active1(X))
active1(sqr1(X)) -> sqr1(active1(X))
active1(s1(X)) -> s1(active1(X))
active1(add2(X1, X2)) -> add2(active1(X1), X2)
active1(add2(X1, X2)) -> add2(X1, active1(X2))
active1(dbl1(X)) -> dbl1(active1(X))
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
terms1(mark1(X)) -> mark1(terms1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
recip1(mark1(X)) -> mark1(recip1(X))
sqr1(mark1(X)) -> mark1(sqr1(X))
s1(mark1(X)) -> mark1(s1(X))
add2(mark1(X1), X2) -> mark1(add2(X1, X2))
add2(X1, mark1(X2)) -> mark1(add2(X1, X2))
dbl1(mark1(X)) -> mark1(dbl1(X))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
proper1(terms1(X)) -> terms1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(recip1(X)) -> recip1(proper1(X))
proper1(sqr1(X)) -> sqr1(proper1(X))
proper1(s1(X)) -> s1(proper1(X))
proper1(0) -> ok1(0)
proper1(add2(X1, X2)) -> add2(proper1(X1), proper1(X2))
proper1(dbl1(X)) -> dbl1(proper1(X))
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
terms1(ok1(X)) -> ok1(terms1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
recip1(ok1(X)) -> ok1(recip1(X))
sqr1(ok1(X)) -> ok1(sqr1(X))
s1(ok1(X)) -> ok1(s1(X))
add2(ok1(X1), ok1(X2)) -> ok1(add2(X1, X2))
dbl1(ok1(X)) -> ok1(dbl1(X))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
SQR1(ok1(X)) -> SQR1(X)
SQR1(mark1(X)) -> SQR1(X)
active1(terms1(N)) -> mark1(cons2(recip1(sqr1(N)), terms1(s1(N))))
active1(sqr1(0)) -> mark1(0)
active1(sqr1(s1(X))) -> mark1(s1(add2(sqr1(X), dbl1(X))))
active1(dbl1(0)) -> mark1(0)
active1(dbl1(s1(X))) -> mark1(s1(s1(dbl1(X))))
active1(add2(0, X)) -> mark1(X)
active1(add2(s1(X), Y)) -> mark1(s1(add2(X, Y)))
active1(first2(0, X)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(terms1(X)) -> terms1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(recip1(X)) -> recip1(active1(X))
active1(sqr1(X)) -> sqr1(active1(X))
active1(s1(X)) -> s1(active1(X))
active1(add2(X1, X2)) -> add2(active1(X1), X2)
active1(add2(X1, X2)) -> add2(X1, active1(X2))
active1(dbl1(X)) -> dbl1(active1(X))
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
terms1(mark1(X)) -> mark1(terms1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
recip1(mark1(X)) -> mark1(recip1(X))
sqr1(mark1(X)) -> mark1(sqr1(X))
s1(mark1(X)) -> mark1(s1(X))
add2(mark1(X1), X2) -> mark1(add2(X1, X2))
add2(X1, mark1(X2)) -> mark1(add2(X1, X2))
dbl1(mark1(X)) -> mark1(dbl1(X))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
proper1(terms1(X)) -> terms1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(recip1(X)) -> recip1(proper1(X))
proper1(sqr1(X)) -> sqr1(proper1(X))
proper1(s1(X)) -> s1(proper1(X))
proper1(0) -> ok1(0)
proper1(add2(X1, X2)) -> add2(proper1(X1), proper1(X2))
proper1(dbl1(X)) -> dbl1(proper1(X))
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
terms1(ok1(X)) -> ok1(terms1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
recip1(ok1(X)) -> ok1(recip1(X))
sqr1(ok1(X)) -> ok1(sqr1(X))
s1(ok1(X)) -> ok1(s1(X))
add2(ok1(X1), ok1(X2)) -> ok1(add2(X1, X2))
dbl1(ok1(X)) -> ok1(dbl1(X))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
SQR1(ok1(X)) -> SQR1(X)
SQR1(mark1(X)) -> SQR1(X)
POL(SQR1(x1)) = x1
POL(mark1(x1)) = 1 + x1
POL(ok1(x1)) = 1 + x1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
active1(terms1(N)) -> mark1(cons2(recip1(sqr1(N)), terms1(s1(N))))
active1(sqr1(0)) -> mark1(0)
active1(sqr1(s1(X))) -> mark1(s1(add2(sqr1(X), dbl1(X))))
active1(dbl1(0)) -> mark1(0)
active1(dbl1(s1(X))) -> mark1(s1(s1(dbl1(X))))
active1(add2(0, X)) -> mark1(X)
active1(add2(s1(X), Y)) -> mark1(s1(add2(X, Y)))
active1(first2(0, X)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(terms1(X)) -> terms1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(recip1(X)) -> recip1(active1(X))
active1(sqr1(X)) -> sqr1(active1(X))
active1(s1(X)) -> s1(active1(X))
active1(add2(X1, X2)) -> add2(active1(X1), X2)
active1(add2(X1, X2)) -> add2(X1, active1(X2))
active1(dbl1(X)) -> dbl1(active1(X))
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
terms1(mark1(X)) -> mark1(terms1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
recip1(mark1(X)) -> mark1(recip1(X))
sqr1(mark1(X)) -> mark1(sqr1(X))
s1(mark1(X)) -> mark1(s1(X))
add2(mark1(X1), X2) -> mark1(add2(X1, X2))
add2(X1, mark1(X2)) -> mark1(add2(X1, X2))
dbl1(mark1(X)) -> mark1(dbl1(X))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
proper1(terms1(X)) -> terms1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(recip1(X)) -> recip1(proper1(X))
proper1(sqr1(X)) -> sqr1(proper1(X))
proper1(s1(X)) -> s1(proper1(X))
proper1(0) -> ok1(0)
proper1(add2(X1, X2)) -> add2(proper1(X1), proper1(X2))
proper1(dbl1(X)) -> dbl1(proper1(X))
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
terms1(ok1(X)) -> ok1(terms1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
recip1(ok1(X)) -> ok1(recip1(X))
sqr1(ok1(X)) -> ok1(sqr1(X))
s1(ok1(X)) -> ok1(s1(X))
add2(ok1(X1), ok1(X2)) -> ok1(add2(X1, X2))
dbl1(ok1(X)) -> ok1(dbl1(X))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
RECIP1(mark1(X)) -> RECIP1(X)
RECIP1(ok1(X)) -> RECIP1(X)
active1(terms1(N)) -> mark1(cons2(recip1(sqr1(N)), terms1(s1(N))))
active1(sqr1(0)) -> mark1(0)
active1(sqr1(s1(X))) -> mark1(s1(add2(sqr1(X), dbl1(X))))
active1(dbl1(0)) -> mark1(0)
active1(dbl1(s1(X))) -> mark1(s1(s1(dbl1(X))))
active1(add2(0, X)) -> mark1(X)
active1(add2(s1(X), Y)) -> mark1(s1(add2(X, Y)))
active1(first2(0, X)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(terms1(X)) -> terms1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(recip1(X)) -> recip1(active1(X))
active1(sqr1(X)) -> sqr1(active1(X))
active1(s1(X)) -> s1(active1(X))
active1(add2(X1, X2)) -> add2(active1(X1), X2)
active1(add2(X1, X2)) -> add2(X1, active1(X2))
active1(dbl1(X)) -> dbl1(active1(X))
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
terms1(mark1(X)) -> mark1(terms1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
recip1(mark1(X)) -> mark1(recip1(X))
sqr1(mark1(X)) -> mark1(sqr1(X))
s1(mark1(X)) -> mark1(s1(X))
add2(mark1(X1), X2) -> mark1(add2(X1, X2))
add2(X1, mark1(X2)) -> mark1(add2(X1, X2))
dbl1(mark1(X)) -> mark1(dbl1(X))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
proper1(terms1(X)) -> terms1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(recip1(X)) -> recip1(proper1(X))
proper1(sqr1(X)) -> sqr1(proper1(X))
proper1(s1(X)) -> s1(proper1(X))
proper1(0) -> ok1(0)
proper1(add2(X1, X2)) -> add2(proper1(X1), proper1(X2))
proper1(dbl1(X)) -> dbl1(proper1(X))
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
terms1(ok1(X)) -> ok1(terms1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
recip1(ok1(X)) -> ok1(recip1(X))
sqr1(ok1(X)) -> ok1(sqr1(X))
s1(ok1(X)) -> ok1(s1(X))
add2(ok1(X1), ok1(X2)) -> ok1(add2(X1, X2))
dbl1(ok1(X)) -> ok1(dbl1(X))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
RECIP1(mark1(X)) -> RECIP1(X)
RECIP1(ok1(X)) -> RECIP1(X)
POL(RECIP1(x1)) = x1
POL(mark1(x1)) = 1 + x1
POL(ok1(x1)) = 1 + x1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
active1(terms1(N)) -> mark1(cons2(recip1(sqr1(N)), terms1(s1(N))))
active1(sqr1(0)) -> mark1(0)
active1(sqr1(s1(X))) -> mark1(s1(add2(sqr1(X), dbl1(X))))
active1(dbl1(0)) -> mark1(0)
active1(dbl1(s1(X))) -> mark1(s1(s1(dbl1(X))))
active1(add2(0, X)) -> mark1(X)
active1(add2(s1(X), Y)) -> mark1(s1(add2(X, Y)))
active1(first2(0, X)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(terms1(X)) -> terms1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(recip1(X)) -> recip1(active1(X))
active1(sqr1(X)) -> sqr1(active1(X))
active1(s1(X)) -> s1(active1(X))
active1(add2(X1, X2)) -> add2(active1(X1), X2)
active1(add2(X1, X2)) -> add2(X1, active1(X2))
active1(dbl1(X)) -> dbl1(active1(X))
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
terms1(mark1(X)) -> mark1(terms1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
recip1(mark1(X)) -> mark1(recip1(X))
sqr1(mark1(X)) -> mark1(sqr1(X))
s1(mark1(X)) -> mark1(s1(X))
add2(mark1(X1), X2) -> mark1(add2(X1, X2))
add2(X1, mark1(X2)) -> mark1(add2(X1, X2))
dbl1(mark1(X)) -> mark1(dbl1(X))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
proper1(terms1(X)) -> terms1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(recip1(X)) -> recip1(proper1(X))
proper1(sqr1(X)) -> sqr1(proper1(X))
proper1(s1(X)) -> s1(proper1(X))
proper1(0) -> ok1(0)
proper1(add2(X1, X2)) -> add2(proper1(X1), proper1(X2))
proper1(dbl1(X)) -> dbl1(proper1(X))
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
terms1(ok1(X)) -> ok1(terms1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
recip1(ok1(X)) -> ok1(recip1(X))
sqr1(ok1(X)) -> ok1(sqr1(X))
s1(ok1(X)) -> ok1(s1(X))
add2(ok1(X1), ok1(X2)) -> ok1(add2(X1, X2))
dbl1(ok1(X)) -> ok1(dbl1(X))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
CONS2(mark1(X1), X2) -> CONS2(X1, X2)
CONS2(ok1(X1), ok1(X2)) -> CONS2(X1, X2)
active1(terms1(N)) -> mark1(cons2(recip1(sqr1(N)), terms1(s1(N))))
active1(sqr1(0)) -> mark1(0)
active1(sqr1(s1(X))) -> mark1(s1(add2(sqr1(X), dbl1(X))))
active1(dbl1(0)) -> mark1(0)
active1(dbl1(s1(X))) -> mark1(s1(s1(dbl1(X))))
active1(add2(0, X)) -> mark1(X)
active1(add2(s1(X), Y)) -> mark1(s1(add2(X, Y)))
active1(first2(0, X)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(terms1(X)) -> terms1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(recip1(X)) -> recip1(active1(X))
active1(sqr1(X)) -> sqr1(active1(X))
active1(s1(X)) -> s1(active1(X))
active1(add2(X1, X2)) -> add2(active1(X1), X2)
active1(add2(X1, X2)) -> add2(X1, active1(X2))
active1(dbl1(X)) -> dbl1(active1(X))
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
terms1(mark1(X)) -> mark1(terms1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
recip1(mark1(X)) -> mark1(recip1(X))
sqr1(mark1(X)) -> mark1(sqr1(X))
s1(mark1(X)) -> mark1(s1(X))
add2(mark1(X1), X2) -> mark1(add2(X1, X2))
add2(X1, mark1(X2)) -> mark1(add2(X1, X2))
dbl1(mark1(X)) -> mark1(dbl1(X))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
proper1(terms1(X)) -> terms1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(recip1(X)) -> recip1(proper1(X))
proper1(sqr1(X)) -> sqr1(proper1(X))
proper1(s1(X)) -> s1(proper1(X))
proper1(0) -> ok1(0)
proper1(add2(X1, X2)) -> add2(proper1(X1), proper1(X2))
proper1(dbl1(X)) -> dbl1(proper1(X))
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
terms1(ok1(X)) -> ok1(terms1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
recip1(ok1(X)) -> ok1(recip1(X))
sqr1(ok1(X)) -> ok1(sqr1(X))
s1(ok1(X)) -> ok1(s1(X))
add2(ok1(X1), ok1(X2)) -> ok1(add2(X1, X2))
dbl1(ok1(X)) -> ok1(dbl1(X))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
CONS2(ok1(X1), ok1(X2)) -> CONS2(X1, X2)
Used ordering: Polynomial interpretation [21]:
CONS2(mark1(X1), X2) -> CONS2(X1, X2)
POL(CONS2(x1, x2)) = x2
POL(mark1(x1)) = 0
POL(ok1(x1)) = 1 + x1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
CONS2(mark1(X1), X2) -> CONS2(X1, X2)
active1(terms1(N)) -> mark1(cons2(recip1(sqr1(N)), terms1(s1(N))))
active1(sqr1(0)) -> mark1(0)
active1(sqr1(s1(X))) -> mark1(s1(add2(sqr1(X), dbl1(X))))
active1(dbl1(0)) -> mark1(0)
active1(dbl1(s1(X))) -> mark1(s1(s1(dbl1(X))))
active1(add2(0, X)) -> mark1(X)
active1(add2(s1(X), Y)) -> mark1(s1(add2(X, Y)))
active1(first2(0, X)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(terms1(X)) -> terms1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(recip1(X)) -> recip1(active1(X))
active1(sqr1(X)) -> sqr1(active1(X))
active1(s1(X)) -> s1(active1(X))
active1(add2(X1, X2)) -> add2(active1(X1), X2)
active1(add2(X1, X2)) -> add2(X1, active1(X2))
active1(dbl1(X)) -> dbl1(active1(X))
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
terms1(mark1(X)) -> mark1(terms1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
recip1(mark1(X)) -> mark1(recip1(X))
sqr1(mark1(X)) -> mark1(sqr1(X))
s1(mark1(X)) -> mark1(s1(X))
add2(mark1(X1), X2) -> mark1(add2(X1, X2))
add2(X1, mark1(X2)) -> mark1(add2(X1, X2))
dbl1(mark1(X)) -> mark1(dbl1(X))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
proper1(terms1(X)) -> terms1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(recip1(X)) -> recip1(proper1(X))
proper1(sqr1(X)) -> sqr1(proper1(X))
proper1(s1(X)) -> s1(proper1(X))
proper1(0) -> ok1(0)
proper1(add2(X1, X2)) -> add2(proper1(X1), proper1(X2))
proper1(dbl1(X)) -> dbl1(proper1(X))
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
terms1(ok1(X)) -> ok1(terms1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
recip1(ok1(X)) -> ok1(recip1(X))
sqr1(ok1(X)) -> ok1(sqr1(X))
s1(ok1(X)) -> ok1(s1(X))
add2(ok1(X1), ok1(X2)) -> ok1(add2(X1, X2))
dbl1(ok1(X)) -> ok1(dbl1(X))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
CONS2(mark1(X1), X2) -> CONS2(X1, X2)
POL(CONS2(x1, x2)) = x1
POL(mark1(x1)) = 1 + x1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
active1(terms1(N)) -> mark1(cons2(recip1(sqr1(N)), terms1(s1(N))))
active1(sqr1(0)) -> mark1(0)
active1(sqr1(s1(X))) -> mark1(s1(add2(sqr1(X), dbl1(X))))
active1(dbl1(0)) -> mark1(0)
active1(dbl1(s1(X))) -> mark1(s1(s1(dbl1(X))))
active1(add2(0, X)) -> mark1(X)
active1(add2(s1(X), Y)) -> mark1(s1(add2(X, Y)))
active1(first2(0, X)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(terms1(X)) -> terms1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(recip1(X)) -> recip1(active1(X))
active1(sqr1(X)) -> sqr1(active1(X))
active1(s1(X)) -> s1(active1(X))
active1(add2(X1, X2)) -> add2(active1(X1), X2)
active1(add2(X1, X2)) -> add2(X1, active1(X2))
active1(dbl1(X)) -> dbl1(active1(X))
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
terms1(mark1(X)) -> mark1(terms1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
recip1(mark1(X)) -> mark1(recip1(X))
sqr1(mark1(X)) -> mark1(sqr1(X))
s1(mark1(X)) -> mark1(s1(X))
add2(mark1(X1), X2) -> mark1(add2(X1, X2))
add2(X1, mark1(X2)) -> mark1(add2(X1, X2))
dbl1(mark1(X)) -> mark1(dbl1(X))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
proper1(terms1(X)) -> terms1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(recip1(X)) -> recip1(proper1(X))
proper1(sqr1(X)) -> sqr1(proper1(X))
proper1(s1(X)) -> s1(proper1(X))
proper1(0) -> ok1(0)
proper1(add2(X1, X2)) -> add2(proper1(X1), proper1(X2))
proper1(dbl1(X)) -> dbl1(proper1(X))
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
terms1(ok1(X)) -> ok1(terms1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
recip1(ok1(X)) -> ok1(recip1(X))
sqr1(ok1(X)) -> ok1(sqr1(X))
s1(ok1(X)) -> ok1(s1(X))
add2(ok1(X1), ok1(X2)) -> ok1(add2(X1, X2))
dbl1(ok1(X)) -> ok1(dbl1(X))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
↳ QDP
TERMS1(mark1(X)) -> TERMS1(X)
TERMS1(ok1(X)) -> TERMS1(X)
active1(terms1(N)) -> mark1(cons2(recip1(sqr1(N)), terms1(s1(N))))
active1(sqr1(0)) -> mark1(0)
active1(sqr1(s1(X))) -> mark1(s1(add2(sqr1(X), dbl1(X))))
active1(dbl1(0)) -> mark1(0)
active1(dbl1(s1(X))) -> mark1(s1(s1(dbl1(X))))
active1(add2(0, X)) -> mark1(X)
active1(add2(s1(X), Y)) -> mark1(s1(add2(X, Y)))
active1(first2(0, X)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(terms1(X)) -> terms1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(recip1(X)) -> recip1(active1(X))
active1(sqr1(X)) -> sqr1(active1(X))
active1(s1(X)) -> s1(active1(X))
active1(add2(X1, X2)) -> add2(active1(X1), X2)
active1(add2(X1, X2)) -> add2(X1, active1(X2))
active1(dbl1(X)) -> dbl1(active1(X))
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
terms1(mark1(X)) -> mark1(terms1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
recip1(mark1(X)) -> mark1(recip1(X))
sqr1(mark1(X)) -> mark1(sqr1(X))
s1(mark1(X)) -> mark1(s1(X))
add2(mark1(X1), X2) -> mark1(add2(X1, X2))
add2(X1, mark1(X2)) -> mark1(add2(X1, X2))
dbl1(mark1(X)) -> mark1(dbl1(X))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
proper1(terms1(X)) -> terms1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(recip1(X)) -> recip1(proper1(X))
proper1(sqr1(X)) -> sqr1(proper1(X))
proper1(s1(X)) -> s1(proper1(X))
proper1(0) -> ok1(0)
proper1(add2(X1, X2)) -> add2(proper1(X1), proper1(X2))
proper1(dbl1(X)) -> dbl1(proper1(X))
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
terms1(ok1(X)) -> ok1(terms1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
recip1(ok1(X)) -> ok1(recip1(X))
sqr1(ok1(X)) -> ok1(sqr1(X))
s1(ok1(X)) -> ok1(s1(X))
add2(ok1(X1), ok1(X2)) -> ok1(add2(X1, X2))
dbl1(ok1(X)) -> ok1(dbl1(X))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
TERMS1(mark1(X)) -> TERMS1(X)
TERMS1(ok1(X)) -> TERMS1(X)
POL(TERMS1(x1)) = x1
POL(mark1(x1)) = 1 + x1
POL(ok1(x1)) = 1 + x1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
↳ QDP
↳ QDP
↳ QDP
active1(terms1(N)) -> mark1(cons2(recip1(sqr1(N)), terms1(s1(N))))
active1(sqr1(0)) -> mark1(0)
active1(sqr1(s1(X))) -> mark1(s1(add2(sqr1(X), dbl1(X))))
active1(dbl1(0)) -> mark1(0)
active1(dbl1(s1(X))) -> mark1(s1(s1(dbl1(X))))
active1(add2(0, X)) -> mark1(X)
active1(add2(s1(X), Y)) -> mark1(s1(add2(X, Y)))
active1(first2(0, X)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(terms1(X)) -> terms1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(recip1(X)) -> recip1(active1(X))
active1(sqr1(X)) -> sqr1(active1(X))
active1(s1(X)) -> s1(active1(X))
active1(add2(X1, X2)) -> add2(active1(X1), X2)
active1(add2(X1, X2)) -> add2(X1, active1(X2))
active1(dbl1(X)) -> dbl1(active1(X))
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
terms1(mark1(X)) -> mark1(terms1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
recip1(mark1(X)) -> mark1(recip1(X))
sqr1(mark1(X)) -> mark1(sqr1(X))
s1(mark1(X)) -> mark1(s1(X))
add2(mark1(X1), X2) -> mark1(add2(X1, X2))
add2(X1, mark1(X2)) -> mark1(add2(X1, X2))
dbl1(mark1(X)) -> mark1(dbl1(X))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
proper1(terms1(X)) -> terms1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(recip1(X)) -> recip1(proper1(X))
proper1(sqr1(X)) -> sqr1(proper1(X))
proper1(s1(X)) -> s1(proper1(X))
proper1(0) -> ok1(0)
proper1(add2(X1, X2)) -> add2(proper1(X1), proper1(X2))
proper1(dbl1(X)) -> dbl1(proper1(X))
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
terms1(ok1(X)) -> ok1(terms1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
recip1(ok1(X)) -> ok1(recip1(X))
sqr1(ok1(X)) -> ok1(sqr1(X))
s1(ok1(X)) -> ok1(s1(X))
add2(ok1(X1), ok1(X2)) -> ok1(add2(X1, X2))
dbl1(ok1(X)) -> ok1(dbl1(X))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
PROPER1(recip1(X)) -> PROPER1(X)
PROPER1(add2(X1, X2)) -> PROPER1(X1)
PROPER1(s1(X)) -> PROPER1(X)
PROPER1(cons2(X1, X2)) -> PROPER1(X1)
PROPER1(cons2(X1, X2)) -> PROPER1(X2)
PROPER1(terms1(X)) -> PROPER1(X)
PROPER1(first2(X1, X2)) -> PROPER1(X1)
PROPER1(first2(X1, X2)) -> PROPER1(X2)
PROPER1(dbl1(X)) -> PROPER1(X)
PROPER1(add2(X1, X2)) -> PROPER1(X2)
PROPER1(sqr1(X)) -> PROPER1(X)
active1(terms1(N)) -> mark1(cons2(recip1(sqr1(N)), terms1(s1(N))))
active1(sqr1(0)) -> mark1(0)
active1(sqr1(s1(X))) -> mark1(s1(add2(sqr1(X), dbl1(X))))
active1(dbl1(0)) -> mark1(0)
active1(dbl1(s1(X))) -> mark1(s1(s1(dbl1(X))))
active1(add2(0, X)) -> mark1(X)
active1(add2(s1(X), Y)) -> mark1(s1(add2(X, Y)))
active1(first2(0, X)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(terms1(X)) -> terms1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(recip1(X)) -> recip1(active1(X))
active1(sqr1(X)) -> sqr1(active1(X))
active1(s1(X)) -> s1(active1(X))
active1(add2(X1, X2)) -> add2(active1(X1), X2)
active1(add2(X1, X2)) -> add2(X1, active1(X2))
active1(dbl1(X)) -> dbl1(active1(X))
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
terms1(mark1(X)) -> mark1(terms1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
recip1(mark1(X)) -> mark1(recip1(X))
sqr1(mark1(X)) -> mark1(sqr1(X))
s1(mark1(X)) -> mark1(s1(X))
add2(mark1(X1), X2) -> mark1(add2(X1, X2))
add2(X1, mark1(X2)) -> mark1(add2(X1, X2))
dbl1(mark1(X)) -> mark1(dbl1(X))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
proper1(terms1(X)) -> terms1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(recip1(X)) -> recip1(proper1(X))
proper1(sqr1(X)) -> sqr1(proper1(X))
proper1(s1(X)) -> s1(proper1(X))
proper1(0) -> ok1(0)
proper1(add2(X1, X2)) -> add2(proper1(X1), proper1(X2))
proper1(dbl1(X)) -> dbl1(proper1(X))
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
terms1(ok1(X)) -> ok1(terms1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
recip1(ok1(X)) -> ok1(recip1(X))
sqr1(ok1(X)) -> ok1(sqr1(X))
s1(ok1(X)) -> ok1(s1(X))
add2(ok1(X1), ok1(X2)) -> ok1(add2(X1, X2))
dbl1(ok1(X)) -> ok1(dbl1(X))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
PROPER1(recip1(X)) -> PROPER1(X)
PROPER1(add2(X1, X2)) -> PROPER1(X1)
PROPER1(s1(X)) -> PROPER1(X)
PROPER1(cons2(X1, X2)) -> PROPER1(X1)
PROPER1(cons2(X1, X2)) -> PROPER1(X2)
PROPER1(terms1(X)) -> PROPER1(X)
PROPER1(first2(X1, X2)) -> PROPER1(X1)
PROPER1(first2(X1, X2)) -> PROPER1(X2)
PROPER1(dbl1(X)) -> PROPER1(X)
PROPER1(add2(X1, X2)) -> PROPER1(X2)
Used ordering: Polynomial interpretation [21]:
PROPER1(sqr1(X)) -> PROPER1(X)
POL(PROPER1(x1)) = x1
POL(add2(x1, x2)) = 1 + x1 + x2
POL(cons2(x1, x2)) = 1 + x1 + x2
POL(dbl1(x1)) = 1 + x1
POL(first2(x1, x2)) = 1 + x1 + x2
POL(recip1(x1)) = 1 + x1
POL(s1(x1)) = 1 + x1
POL(sqr1(x1)) = x1
POL(terms1(x1)) = 1 + x1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
PROPER1(sqr1(X)) -> PROPER1(X)
active1(terms1(N)) -> mark1(cons2(recip1(sqr1(N)), terms1(s1(N))))
active1(sqr1(0)) -> mark1(0)
active1(sqr1(s1(X))) -> mark1(s1(add2(sqr1(X), dbl1(X))))
active1(dbl1(0)) -> mark1(0)
active1(dbl1(s1(X))) -> mark1(s1(s1(dbl1(X))))
active1(add2(0, X)) -> mark1(X)
active1(add2(s1(X), Y)) -> mark1(s1(add2(X, Y)))
active1(first2(0, X)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(terms1(X)) -> terms1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(recip1(X)) -> recip1(active1(X))
active1(sqr1(X)) -> sqr1(active1(X))
active1(s1(X)) -> s1(active1(X))
active1(add2(X1, X2)) -> add2(active1(X1), X2)
active1(add2(X1, X2)) -> add2(X1, active1(X2))
active1(dbl1(X)) -> dbl1(active1(X))
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
terms1(mark1(X)) -> mark1(terms1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
recip1(mark1(X)) -> mark1(recip1(X))
sqr1(mark1(X)) -> mark1(sqr1(X))
s1(mark1(X)) -> mark1(s1(X))
add2(mark1(X1), X2) -> mark1(add2(X1, X2))
add2(X1, mark1(X2)) -> mark1(add2(X1, X2))
dbl1(mark1(X)) -> mark1(dbl1(X))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
proper1(terms1(X)) -> terms1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(recip1(X)) -> recip1(proper1(X))
proper1(sqr1(X)) -> sqr1(proper1(X))
proper1(s1(X)) -> s1(proper1(X))
proper1(0) -> ok1(0)
proper1(add2(X1, X2)) -> add2(proper1(X1), proper1(X2))
proper1(dbl1(X)) -> dbl1(proper1(X))
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
terms1(ok1(X)) -> ok1(terms1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
recip1(ok1(X)) -> ok1(recip1(X))
sqr1(ok1(X)) -> ok1(sqr1(X))
s1(ok1(X)) -> ok1(s1(X))
add2(ok1(X1), ok1(X2)) -> ok1(add2(X1, X2))
dbl1(ok1(X)) -> ok1(dbl1(X))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
PROPER1(sqr1(X)) -> PROPER1(X)
POL(PROPER1(x1)) = x1
POL(sqr1(x1)) = 1 + x1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
↳ QDP
↳ QDP
active1(terms1(N)) -> mark1(cons2(recip1(sqr1(N)), terms1(s1(N))))
active1(sqr1(0)) -> mark1(0)
active1(sqr1(s1(X))) -> mark1(s1(add2(sqr1(X), dbl1(X))))
active1(dbl1(0)) -> mark1(0)
active1(dbl1(s1(X))) -> mark1(s1(s1(dbl1(X))))
active1(add2(0, X)) -> mark1(X)
active1(add2(s1(X), Y)) -> mark1(s1(add2(X, Y)))
active1(first2(0, X)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(terms1(X)) -> terms1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(recip1(X)) -> recip1(active1(X))
active1(sqr1(X)) -> sqr1(active1(X))
active1(s1(X)) -> s1(active1(X))
active1(add2(X1, X2)) -> add2(active1(X1), X2)
active1(add2(X1, X2)) -> add2(X1, active1(X2))
active1(dbl1(X)) -> dbl1(active1(X))
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
terms1(mark1(X)) -> mark1(terms1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
recip1(mark1(X)) -> mark1(recip1(X))
sqr1(mark1(X)) -> mark1(sqr1(X))
s1(mark1(X)) -> mark1(s1(X))
add2(mark1(X1), X2) -> mark1(add2(X1, X2))
add2(X1, mark1(X2)) -> mark1(add2(X1, X2))
dbl1(mark1(X)) -> mark1(dbl1(X))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
proper1(terms1(X)) -> terms1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(recip1(X)) -> recip1(proper1(X))
proper1(sqr1(X)) -> sqr1(proper1(X))
proper1(s1(X)) -> s1(proper1(X))
proper1(0) -> ok1(0)
proper1(add2(X1, X2)) -> add2(proper1(X1), proper1(X2))
proper1(dbl1(X)) -> dbl1(proper1(X))
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
terms1(ok1(X)) -> ok1(terms1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
recip1(ok1(X)) -> ok1(recip1(X))
sqr1(ok1(X)) -> ok1(sqr1(X))
s1(ok1(X)) -> ok1(s1(X))
add2(ok1(X1), ok1(X2)) -> ok1(add2(X1, X2))
dbl1(ok1(X)) -> ok1(dbl1(X))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
ACTIVE1(dbl1(X)) -> ACTIVE1(X)
ACTIVE1(first2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(first2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(cons2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(terms1(X)) -> ACTIVE1(X)
ACTIVE1(add2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(sqr1(X)) -> ACTIVE1(X)
ACTIVE1(s1(X)) -> ACTIVE1(X)
ACTIVE1(recip1(X)) -> ACTIVE1(X)
ACTIVE1(add2(X1, X2)) -> ACTIVE1(X2)
active1(terms1(N)) -> mark1(cons2(recip1(sqr1(N)), terms1(s1(N))))
active1(sqr1(0)) -> mark1(0)
active1(sqr1(s1(X))) -> mark1(s1(add2(sqr1(X), dbl1(X))))
active1(dbl1(0)) -> mark1(0)
active1(dbl1(s1(X))) -> mark1(s1(s1(dbl1(X))))
active1(add2(0, X)) -> mark1(X)
active1(add2(s1(X), Y)) -> mark1(s1(add2(X, Y)))
active1(first2(0, X)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(terms1(X)) -> terms1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(recip1(X)) -> recip1(active1(X))
active1(sqr1(X)) -> sqr1(active1(X))
active1(s1(X)) -> s1(active1(X))
active1(add2(X1, X2)) -> add2(active1(X1), X2)
active1(add2(X1, X2)) -> add2(X1, active1(X2))
active1(dbl1(X)) -> dbl1(active1(X))
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
terms1(mark1(X)) -> mark1(terms1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
recip1(mark1(X)) -> mark1(recip1(X))
sqr1(mark1(X)) -> mark1(sqr1(X))
s1(mark1(X)) -> mark1(s1(X))
add2(mark1(X1), X2) -> mark1(add2(X1, X2))
add2(X1, mark1(X2)) -> mark1(add2(X1, X2))
dbl1(mark1(X)) -> mark1(dbl1(X))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
proper1(terms1(X)) -> terms1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(recip1(X)) -> recip1(proper1(X))
proper1(sqr1(X)) -> sqr1(proper1(X))
proper1(s1(X)) -> s1(proper1(X))
proper1(0) -> ok1(0)
proper1(add2(X1, X2)) -> add2(proper1(X1), proper1(X2))
proper1(dbl1(X)) -> dbl1(proper1(X))
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
terms1(ok1(X)) -> ok1(terms1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
recip1(ok1(X)) -> ok1(recip1(X))
sqr1(ok1(X)) -> ok1(sqr1(X))
s1(ok1(X)) -> ok1(s1(X))
add2(ok1(X1), ok1(X2)) -> ok1(add2(X1, X2))
dbl1(ok1(X)) -> ok1(dbl1(X))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
ACTIVE1(dbl1(X)) -> ACTIVE1(X)
ACTIVE1(first2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(first2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(cons2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(terms1(X)) -> ACTIVE1(X)
ACTIVE1(add2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(sqr1(X)) -> ACTIVE1(X)
ACTIVE1(s1(X)) -> ACTIVE1(X)
ACTIVE1(add2(X1, X2)) -> ACTIVE1(X2)
Used ordering: Polynomial interpretation [21]:
ACTIVE1(recip1(X)) -> ACTIVE1(X)
POL(ACTIVE1(x1)) = x1
POL(add2(x1, x2)) = 1 + x1 + x2
POL(cons2(x1, x2)) = 1 + x1
POL(dbl1(x1)) = 1 + x1
POL(first2(x1, x2)) = 1 + x1 + x2
POL(recip1(x1)) = x1
POL(s1(x1)) = 1 + x1
POL(sqr1(x1)) = 1 + x1
POL(terms1(x1)) = 1 + x1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
ACTIVE1(recip1(X)) -> ACTIVE1(X)
active1(terms1(N)) -> mark1(cons2(recip1(sqr1(N)), terms1(s1(N))))
active1(sqr1(0)) -> mark1(0)
active1(sqr1(s1(X))) -> mark1(s1(add2(sqr1(X), dbl1(X))))
active1(dbl1(0)) -> mark1(0)
active1(dbl1(s1(X))) -> mark1(s1(s1(dbl1(X))))
active1(add2(0, X)) -> mark1(X)
active1(add2(s1(X), Y)) -> mark1(s1(add2(X, Y)))
active1(first2(0, X)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(terms1(X)) -> terms1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(recip1(X)) -> recip1(active1(X))
active1(sqr1(X)) -> sqr1(active1(X))
active1(s1(X)) -> s1(active1(X))
active1(add2(X1, X2)) -> add2(active1(X1), X2)
active1(add2(X1, X2)) -> add2(X1, active1(X2))
active1(dbl1(X)) -> dbl1(active1(X))
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
terms1(mark1(X)) -> mark1(terms1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
recip1(mark1(X)) -> mark1(recip1(X))
sqr1(mark1(X)) -> mark1(sqr1(X))
s1(mark1(X)) -> mark1(s1(X))
add2(mark1(X1), X2) -> mark1(add2(X1, X2))
add2(X1, mark1(X2)) -> mark1(add2(X1, X2))
dbl1(mark1(X)) -> mark1(dbl1(X))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
proper1(terms1(X)) -> terms1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(recip1(X)) -> recip1(proper1(X))
proper1(sqr1(X)) -> sqr1(proper1(X))
proper1(s1(X)) -> s1(proper1(X))
proper1(0) -> ok1(0)
proper1(add2(X1, X2)) -> add2(proper1(X1), proper1(X2))
proper1(dbl1(X)) -> dbl1(proper1(X))
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
terms1(ok1(X)) -> ok1(terms1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
recip1(ok1(X)) -> ok1(recip1(X))
sqr1(ok1(X)) -> ok1(sqr1(X))
s1(ok1(X)) -> ok1(s1(X))
add2(ok1(X1), ok1(X2)) -> ok1(add2(X1, X2))
dbl1(ok1(X)) -> ok1(dbl1(X))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
ACTIVE1(recip1(X)) -> ACTIVE1(X)
POL(ACTIVE1(x1)) = x1
POL(recip1(x1)) = 1 + x1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
↳ QDP
active1(terms1(N)) -> mark1(cons2(recip1(sqr1(N)), terms1(s1(N))))
active1(sqr1(0)) -> mark1(0)
active1(sqr1(s1(X))) -> mark1(s1(add2(sqr1(X), dbl1(X))))
active1(dbl1(0)) -> mark1(0)
active1(dbl1(s1(X))) -> mark1(s1(s1(dbl1(X))))
active1(add2(0, X)) -> mark1(X)
active1(add2(s1(X), Y)) -> mark1(s1(add2(X, Y)))
active1(first2(0, X)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(terms1(X)) -> terms1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(recip1(X)) -> recip1(active1(X))
active1(sqr1(X)) -> sqr1(active1(X))
active1(s1(X)) -> s1(active1(X))
active1(add2(X1, X2)) -> add2(active1(X1), X2)
active1(add2(X1, X2)) -> add2(X1, active1(X2))
active1(dbl1(X)) -> dbl1(active1(X))
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
terms1(mark1(X)) -> mark1(terms1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
recip1(mark1(X)) -> mark1(recip1(X))
sqr1(mark1(X)) -> mark1(sqr1(X))
s1(mark1(X)) -> mark1(s1(X))
add2(mark1(X1), X2) -> mark1(add2(X1, X2))
add2(X1, mark1(X2)) -> mark1(add2(X1, X2))
dbl1(mark1(X)) -> mark1(dbl1(X))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
proper1(terms1(X)) -> terms1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(recip1(X)) -> recip1(proper1(X))
proper1(sqr1(X)) -> sqr1(proper1(X))
proper1(s1(X)) -> s1(proper1(X))
proper1(0) -> ok1(0)
proper1(add2(X1, X2)) -> add2(proper1(X1), proper1(X2))
proper1(dbl1(X)) -> dbl1(proper1(X))
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
terms1(ok1(X)) -> ok1(terms1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
recip1(ok1(X)) -> ok1(recip1(X))
sqr1(ok1(X)) -> ok1(sqr1(X))
s1(ok1(X)) -> ok1(s1(X))
add2(ok1(X1), ok1(X2)) -> ok1(add2(X1, X2))
dbl1(ok1(X)) -> ok1(dbl1(X))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
TOP1(ok1(X)) -> TOP1(active1(X))
TOP1(mark1(X)) -> TOP1(proper1(X))
active1(terms1(N)) -> mark1(cons2(recip1(sqr1(N)), terms1(s1(N))))
active1(sqr1(0)) -> mark1(0)
active1(sqr1(s1(X))) -> mark1(s1(add2(sqr1(X), dbl1(X))))
active1(dbl1(0)) -> mark1(0)
active1(dbl1(s1(X))) -> mark1(s1(s1(dbl1(X))))
active1(add2(0, X)) -> mark1(X)
active1(add2(s1(X), Y)) -> mark1(s1(add2(X, Y)))
active1(first2(0, X)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(terms1(X)) -> terms1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(recip1(X)) -> recip1(active1(X))
active1(sqr1(X)) -> sqr1(active1(X))
active1(s1(X)) -> s1(active1(X))
active1(add2(X1, X2)) -> add2(active1(X1), X2)
active1(add2(X1, X2)) -> add2(X1, active1(X2))
active1(dbl1(X)) -> dbl1(active1(X))
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
terms1(mark1(X)) -> mark1(terms1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
recip1(mark1(X)) -> mark1(recip1(X))
sqr1(mark1(X)) -> mark1(sqr1(X))
s1(mark1(X)) -> mark1(s1(X))
add2(mark1(X1), X2) -> mark1(add2(X1, X2))
add2(X1, mark1(X2)) -> mark1(add2(X1, X2))
dbl1(mark1(X)) -> mark1(dbl1(X))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
proper1(terms1(X)) -> terms1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(recip1(X)) -> recip1(proper1(X))
proper1(sqr1(X)) -> sqr1(proper1(X))
proper1(s1(X)) -> s1(proper1(X))
proper1(0) -> ok1(0)
proper1(add2(X1, X2)) -> add2(proper1(X1), proper1(X2))
proper1(dbl1(X)) -> dbl1(proper1(X))
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
terms1(ok1(X)) -> ok1(terms1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
recip1(ok1(X)) -> ok1(recip1(X))
sqr1(ok1(X)) -> ok1(sqr1(X))
s1(ok1(X)) -> ok1(s1(X))
add2(ok1(X1), ok1(X2)) -> ok1(add2(X1, X2))
dbl1(ok1(X)) -> ok1(dbl1(X))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))