YES

Problem 1: 

(VAR v_NonEmpty:S N:S X:S X1:S X2:S XS:S Y:S YS:S)
(RULES
active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S)))
active(incr(X:S)) -> incr(active(X:S))
active(pair(X1:S,X2:S)) -> pair(active(X1:S),X2:S)
active(pair(X1:S,X2:S)) -> pair(X1:S,active(X2:S))
active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S))))
active(repItems(nil)) -> mark(nil)
active(repItems(X:S)) -> repItems(active(X:S))
active(s(X:S)) -> s(active(X:S))
active(tail(cons(X:S,XS:S))) -> mark(XS:S)
active(tail(X:S)) -> tail(active(X:S))
active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S)))
active(take(0,XS:S)) -> mark(nil)
active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S)
active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S))
active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S)))
active(zip(nil,XS:S)) -> mark(nil)
active(zip(X:S,nil)) -> mark(nil)
active(zip(X1:S,X2:S)) -> zip(active(X1:S),X2:S)
active(zip(X1:S,X2:S)) -> zip(X1:S,active(X2:S))
active(oddNs) -> mark(incr(pairNs))
active(pairNs) -> mark(cons(0,incr(oddNs)))
cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
incr(mark(X:S)) -> mark(incr(X:S))
incr(ok(X:S)) -> ok(incr(X:S))
pair(mark(X1:S),X2:S) -> mark(pair(X1:S,X2:S))
pair(ok(X1:S),ok(X2:S)) -> ok(pair(X1:S,X2:S))
pair(X1:S,mark(X2:S)) -> mark(pair(X1:S,X2:S))
proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
proper(incr(X:S)) -> incr(proper(X:S))
proper(pair(X1:S,X2:S)) -> pair(proper(X1:S),proper(X2:S))
proper(repItems(X:S)) -> repItems(proper(X:S))
proper(s(X:S)) -> s(proper(X:S))
proper(tail(X:S)) -> tail(proper(X:S))
proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2:S))
proper(zip(X1:S,X2:S)) -> zip(proper(X1:S),proper(X2:S))
proper(0) -> ok(0)
proper(nil) -> ok(nil)
proper(oddNs) -> ok(oddNs)
proper(pairNs) -> ok(pairNs)
repItems(mark(X:S)) -> mark(repItems(X:S))
repItems(ok(X:S)) -> ok(repItems(X:S))
s(mark(X:S)) -> mark(s(X:S))
s(ok(X:S)) -> ok(s(X:S))
tail(mark(X:S)) -> mark(tail(X:S))
tail(ok(X:S)) -> ok(tail(X:S))
take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S))
take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S))
take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S))
top(mark(X:S)) -> top(proper(X:S))
top(ok(X:S)) -> top(active(X:S))
zip(mark(X1:S),X2:S) -> mark(zip(X1:S,X2:S))
zip(ok(X1:S),ok(X2:S)) -> ok(zip(X1:S,X2:S))
zip(X1:S,mark(X2:S)) -> mark(zip(X1:S,X2:S))
) 
(STRATEGY INNERMOST)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 ACTIVE(cons(X1:S,X2:S)) -> ACTIVE(X1:S)
 ACTIVE(cons(X1:S,X2:S)) -> CONS(active(X1:S),X2:S)
 ACTIVE(incr(cons(X:S,XS:S))) -> CONS(s(X:S),incr(XS:S))
 ACTIVE(incr(cons(X:S,XS:S))) -> INCR(XS:S)
 ACTIVE(incr(cons(X:S,XS:S))) -> S(X:S)
 ACTIVE(incr(X:S)) -> ACTIVE(X:S)
 ACTIVE(incr(X:S)) -> INCR(active(X:S))
 ACTIVE(pair(X1:S,X2:S)) -> ACTIVE(X1:S)
 ACTIVE(pair(X1:S,X2:S)) -> ACTIVE(X2:S)
 ACTIVE(pair(X1:S,X2:S)) -> PAIR(active(X1:S),X2:S)
 ACTIVE(pair(X1:S,X2:S)) -> PAIR(X1:S,active(X2:S))
 ACTIVE(repItems(cons(X:S,XS:S))) -> CONS(X:S,cons(X:S,repItems(XS:S)))
 ACTIVE(repItems(cons(X:S,XS:S))) -> CONS(X:S,repItems(XS:S))
 ACTIVE(repItems(cons(X:S,XS:S))) -> REPITEMS(XS:S)
 ACTIVE(repItems(X:S)) -> ACTIVE(X:S)
 ACTIVE(repItems(X:S)) -> REPITEMS(active(X:S))
 ACTIVE(s(X:S)) -> ACTIVE(X:S)
 ACTIVE(s(X:S)) -> S(active(X:S))
 ACTIVE(tail(X:S)) -> ACTIVE(X:S)
 ACTIVE(tail(X:S)) -> TAIL(active(X:S))
 ACTIVE(take(s(N:S),cons(X:S,XS:S))) -> CONS(X:S,take(N:S,XS:S))
 ACTIVE(take(s(N:S),cons(X:S,XS:S))) -> TAKE(N:S,XS:S)
 ACTIVE(take(X1:S,X2:S)) -> ACTIVE(X1:S)
 ACTIVE(take(X1:S,X2:S)) -> ACTIVE(X2:S)
 ACTIVE(take(X1:S,X2:S)) -> TAKE(active(X1:S),X2:S)
 ACTIVE(take(X1:S,X2:S)) -> TAKE(X1:S,active(X2:S))
 ACTIVE(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> CONS(pair(X:S,Y:S),zip(XS:S,YS:S))
 ACTIVE(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> PAIR(X:S,Y:S)
 ACTIVE(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> ZIP(XS:S,YS:S)
 ACTIVE(zip(X1:S,X2:S)) -> ACTIVE(X1:S)
 ACTIVE(zip(X1:S,X2:S)) -> ACTIVE(X2:S)
 ACTIVE(zip(X1:S,X2:S)) -> ZIP(active(X1:S),X2:S)
 ACTIVE(zip(X1:S,X2:S)) -> ZIP(X1:S,active(X2:S))
 CONS(mark(X1:S),X2:S) -> CONS(X1:S,X2:S)
 CONS(ok(X1:S),ok(X2:S)) -> CONS(X1:S,X2:S)
 INCR(mark(X:S)) -> INCR(X:S)
 INCR(ok(X:S)) -> INCR(X:S)
 PAIR(mark(X1:S),X2:S) -> PAIR(X1:S,X2:S)
 PAIR(ok(X1:S),ok(X2:S)) -> PAIR(X1:S,X2:S)
 PAIR(X1:S,mark(X2:S)) -> PAIR(X1:S,X2:S)
 PROPER(cons(X1:S,X2:S)) -> CONS(proper(X1:S),proper(X2:S))
 PROPER(cons(X1:S,X2:S)) -> PROPER(X1:S)
 PROPER(cons(X1:S,X2:S)) -> PROPER(X2:S)
 PROPER(incr(X:S)) -> INCR(proper(X:S))
 PROPER(incr(X:S)) -> PROPER(X:S)
 PROPER(pair(X1:S,X2:S)) -> PAIR(proper(X1:S),proper(X2:S))
 PROPER(pair(X1:S,X2:S)) -> PROPER(X1:S)
 PROPER(pair(X1:S,X2:S)) -> PROPER(X2:S)
 PROPER(repItems(X:S)) -> PROPER(X:S)
 PROPER(repItems(X:S)) -> REPITEMS(proper(X:S))
 PROPER(s(X:S)) -> PROPER(X:S)
 PROPER(s(X:S)) -> S(proper(X:S))
 PROPER(tail(X:S)) -> PROPER(X:S)
 PROPER(tail(X:S)) -> TAIL(proper(X:S))
 PROPER(take(X1:S,X2:S)) -> PROPER(X1:S)
 PROPER(take(X1:S,X2:S)) -> PROPER(X2:S)
 PROPER(take(X1:S,X2:S)) -> TAKE(proper(X1:S),proper(X2:S))
 PROPER(zip(X1:S,X2:S)) -> PROPER(X1:S)
 PROPER(zip(X1:S,X2:S)) -> PROPER(X2:S)
 PROPER(zip(X1:S,X2:S)) -> ZIP(proper(X1:S),proper(X2:S))
 REPITEMS(mark(X:S)) -> REPITEMS(X:S)
 REPITEMS(ok(X:S)) -> REPITEMS(X:S)
 S(mark(X:S)) -> S(X:S)
 S(ok(X:S)) -> S(X:S)
 TAIL(mark(X:S)) -> TAIL(X:S)
 TAIL(ok(X:S)) -> TAIL(X:S)
 TAKE(mark(X1:S),X2:S) -> TAKE(X1:S,X2:S)
 TAKE(ok(X1:S),ok(X2:S)) -> TAKE(X1:S,X2:S)
 TAKE(X1:S,mark(X2:S)) -> TAKE(X1:S,X2:S)
 TOP(mark(X:S)) -> PROPER(X:S)
 TOP(mark(X:S)) -> TOP(proper(X:S))
 TOP(ok(X:S)) -> ACTIVE(X:S)
 TOP(ok(X:S)) -> TOP(active(X:S))
 ZIP(mark(X1:S),X2:S) -> ZIP(X1:S,X2:S)
 ZIP(ok(X1:S),ok(X2:S)) -> ZIP(X1:S,X2:S)
 ZIP(X1:S,mark(X2:S)) -> ZIP(X1:S,X2:S)
-> Rules:
 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
 active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S)))
 active(incr(X:S)) -> incr(active(X:S))
 active(pair(X1:S,X2:S)) -> pair(active(X1:S),X2:S)
 active(pair(X1:S,X2:S)) -> pair(X1:S,active(X2:S))
 active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S))))
 active(repItems(nil)) -> mark(nil)
 active(repItems(X:S)) -> repItems(active(X:S))
 active(s(X:S)) -> s(active(X:S))
 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
 active(tail(X:S)) -> tail(active(X:S))
 active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S)))
 active(take(0,XS:S)) -> mark(nil)
 active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S)
 active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S))
 active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S)))
 active(zip(nil,XS:S)) -> mark(nil)
 active(zip(X:S,nil)) -> mark(nil)
 active(zip(X1:S,X2:S)) -> zip(active(X1:S),X2:S)
 active(zip(X1:S,X2:S)) -> zip(X1:S,active(X2:S))
 active(oddNs) -> mark(incr(pairNs))
 active(pairNs) -> mark(cons(0,incr(oddNs)))
 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
 incr(mark(X:S)) -> mark(incr(X:S))
 incr(ok(X:S)) -> ok(incr(X:S))
 pair(mark(X1:S),X2:S) -> mark(pair(X1:S,X2:S))
 pair(ok(X1:S),ok(X2:S)) -> ok(pair(X1:S,X2:S))
 pair(X1:S,mark(X2:S)) -> mark(pair(X1:S,X2:S))
 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
 proper(incr(X:S)) -> incr(proper(X:S))
 proper(pair(X1:S,X2:S)) -> pair(proper(X1:S),proper(X2:S))
 proper(repItems(X:S)) -> repItems(proper(X:S))
 proper(s(X:S)) -> s(proper(X:S))
 proper(tail(X:S)) -> tail(proper(X:S))
 proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2:S))
 proper(zip(X1:S,X2:S)) -> zip(proper(X1:S),proper(X2:S))
 proper(0) -> ok(0)
 proper(nil) -> ok(nil)
 proper(oddNs) -> ok(oddNs)
 proper(pairNs) -> ok(pairNs)
 repItems(mark(X:S)) -> mark(repItems(X:S))
 repItems(ok(X:S)) -> ok(repItems(X:S))
 s(mark(X:S)) -> mark(s(X:S))
 s(ok(X:S)) -> ok(s(X:S))
 tail(mark(X:S)) -> mark(tail(X:S))
 tail(ok(X:S)) -> ok(tail(X:S))
 take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S))
 take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S))
 take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S))
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))
 zip(mark(X1:S),X2:S) -> mark(zip(X1:S,X2:S))
 zip(ok(X1:S),ok(X2:S)) -> ok(zip(X1:S,X2:S))
 zip(X1:S,mark(X2:S)) -> mark(zip(X1:S,X2:S))

Problem 1: 

SCC Processor:
-> Pairs:
 ACTIVE(cons(X1:S,X2:S)) -> ACTIVE(X1:S)
 ACTIVE(cons(X1:S,X2:S)) -> CONS(active(X1:S),X2:S)
 ACTIVE(incr(cons(X:S,XS:S))) -> CONS(s(X:S),incr(XS:S))
 ACTIVE(incr(cons(X:S,XS:S))) -> INCR(XS:S)
 ACTIVE(incr(cons(X:S,XS:S))) -> S(X:S)
 ACTIVE(incr(X:S)) -> ACTIVE(X:S)
 ACTIVE(incr(X:S)) -> INCR(active(X:S))
 ACTIVE(pair(X1:S,X2:S)) -> ACTIVE(X1:S)
 ACTIVE(pair(X1:S,X2:S)) -> ACTIVE(X2:S)
 ACTIVE(pair(X1:S,X2:S)) -> PAIR(active(X1:S),X2:S)
 ACTIVE(pair(X1:S,X2:S)) -> PAIR(X1:S,active(X2:S))
 ACTIVE(repItems(cons(X:S,XS:S))) -> CONS(X:S,cons(X:S,repItems(XS:S)))
 ACTIVE(repItems(cons(X:S,XS:S))) -> CONS(X:S,repItems(XS:S))
 ACTIVE(repItems(cons(X:S,XS:S))) -> REPITEMS(XS:S)
 ACTIVE(repItems(X:S)) -> ACTIVE(X:S)
 ACTIVE(repItems(X:S)) -> REPITEMS(active(X:S))
 ACTIVE(s(X:S)) -> ACTIVE(X:S)
 ACTIVE(s(X:S)) -> S(active(X:S))
 ACTIVE(tail(X:S)) -> ACTIVE(X:S)
 ACTIVE(tail(X:S)) -> TAIL(active(X:S))
 ACTIVE(take(s(N:S),cons(X:S,XS:S))) -> CONS(X:S,take(N:S,XS:S))
 ACTIVE(take(s(N:S),cons(X:S,XS:S))) -> TAKE(N:S,XS:S)
 ACTIVE(take(X1:S,X2:S)) -> ACTIVE(X1:S)
 ACTIVE(take(X1:S,X2:S)) -> ACTIVE(X2:S)
 ACTIVE(take(X1:S,X2:S)) -> TAKE(active(X1:S),X2:S)
 ACTIVE(take(X1:S,X2:S)) -> TAKE(X1:S,active(X2:S))
 ACTIVE(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> CONS(pair(X:S,Y:S),zip(XS:S,YS:S))
 ACTIVE(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> PAIR(X:S,Y:S)
 ACTIVE(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> ZIP(XS:S,YS:S)
 ACTIVE(zip(X1:S,X2:S)) -> ACTIVE(X1:S)
 ACTIVE(zip(X1:S,X2:S)) -> ACTIVE(X2:S)
 ACTIVE(zip(X1:S,X2:S)) -> ZIP(active(X1:S),X2:S)
 ACTIVE(zip(X1:S,X2:S)) -> ZIP(X1:S,active(X2:S))
 CONS(mark(X1:S),X2:S) -> CONS(X1:S,X2:S)
 CONS(ok(X1:S),ok(X2:S)) -> CONS(X1:S,X2:S)
 INCR(mark(X:S)) -> INCR(X:S)
 INCR(ok(X:S)) -> INCR(X:S)
 PAIR(mark(X1:S),X2:S) -> PAIR(X1:S,X2:S)
 PAIR(ok(X1:S),ok(X2:S)) -> PAIR(X1:S,X2:S)
 PAIR(X1:S,mark(X2:S)) -> PAIR(X1:S,X2:S)
 PROPER(cons(X1:S,X2:S)) -> CONS(proper(X1:S),proper(X2:S))
 PROPER(cons(X1:S,X2:S)) -> PROPER(X1:S)
 PROPER(cons(X1:S,X2:S)) -> PROPER(X2:S)
 PROPER(incr(X:S)) -> INCR(proper(X:S))
 PROPER(incr(X:S)) -> PROPER(X:S)
 PROPER(pair(X1:S,X2:S)) -> PAIR(proper(X1:S),proper(X2:S))
 PROPER(pair(X1:S,X2:S)) -> PROPER(X1:S)
 PROPER(pair(X1:S,X2:S)) -> PROPER(X2:S)
 PROPER(repItems(X:S)) -> PROPER(X:S)
 PROPER(repItems(X:S)) -> REPITEMS(proper(X:S))
 PROPER(s(X:S)) -> PROPER(X:S)
 PROPER(s(X:S)) -> S(proper(X:S))
 PROPER(tail(X:S)) -> PROPER(X:S)
 PROPER(tail(X:S)) -> TAIL(proper(X:S))
 PROPER(take(X1:S,X2:S)) -> PROPER(X1:S)
 PROPER(take(X1:S,X2:S)) -> PROPER(X2:S)
 PROPER(take(X1:S,X2:S)) -> TAKE(proper(X1:S),proper(X2:S))
 PROPER(zip(X1:S,X2:S)) -> PROPER(X1:S)
 PROPER(zip(X1:S,X2:S)) -> PROPER(X2:S)
 PROPER(zip(X1:S,X2:S)) -> ZIP(proper(X1:S),proper(X2:S))
 REPITEMS(mark(X:S)) -> REPITEMS(X:S)
 REPITEMS(ok(X:S)) -> REPITEMS(X:S)
 S(mark(X:S)) -> S(X:S)
 S(ok(X:S)) -> S(X:S)
 TAIL(mark(X:S)) -> TAIL(X:S)
 TAIL(ok(X:S)) -> TAIL(X:S)
 TAKE(mark(X1:S),X2:S) -> TAKE(X1:S,X2:S)
 TAKE(ok(X1:S),ok(X2:S)) -> TAKE(X1:S,X2:S)
 TAKE(X1:S,mark(X2:S)) -> TAKE(X1:S,X2:S)
 TOP(mark(X:S)) -> PROPER(X:S)
 TOP(mark(X:S)) -> TOP(proper(X:S))
 TOP(ok(X:S)) -> ACTIVE(X:S)
 TOP(ok(X:S)) -> TOP(active(X:S))
 ZIP(mark(X1:S),X2:S) -> ZIP(X1:S,X2:S)
 ZIP(ok(X1:S),ok(X2:S)) -> ZIP(X1:S,X2:S)
 ZIP(X1:S,mark(X2:S)) -> ZIP(X1:S,X2:S)
-> Rules:
 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
 active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S)))
 active(incr(X:S)) -> incr(active(X:S))
 active(pair(X1:S,X2:S)) -> pair(active(X1:S),X2:S)
 active(pair(X1:S,X2:S)) -> pair(X1:S,active(X2:S))
 active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S))))
 active(repItems(nil)) -> mark(nil)
 active(repItems(X:S)) -> repItems(active(X:S))
 active(s(X:S)) -> s(active(X:S))
 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
 active(tail(X:S)) -> tail(active(X:S))
 active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S)))
 active(take(0,XS:S)) -> mark(nil)
 active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S)
 active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S))
 active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S)))
 active(zip(nil,XS:S)) -> mark(nil)
 active(zip(X:S,nil)) -> mark(nil)
 active(zip(X1:S,X2:S)) -> zip(active(X1:S),X2:S)
 active(zip(X1:S,X2:S)) -> zip(X1:S,active(X2:S))
 active(oddNs) -> mark(incr(pairNs))
 active(pairNs) -> mark(cons(0,incr(oddNs)))
 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
 incr(mark(X:S)) -> mark(incr(X:S))
 incr(ok(X:S)) -> ok(incr(X:S))
 pair(mark(X1:S),X2:S) -> mark(pair(X1:S,X2:S))
 pair(ok(X1:S),ok(X2:S)) -> ok(pair(X1:S,X2:S))
 pair(X1:S,mark(X2:S)) -> mark(pair(X1:S,X2:S))
 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
 proper(incr(X:S)) -> incr(proper(X:S))
 proper(pair(X1:S,X2:S)) -> pair(proper(X1:S),proper(X2:S))
 proper(repItems(X:S)) -> repItems(proper(X:S))
 proper(s(X:S)) -> s(proper(X:S))
 proper(tail(X:S)) -> tail(proper(X:S))
 proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2:S))
 proper(zip(X1:S,X2:S)) -> zip(proper(X1:S),proper(X2:S))
 proper(0) -> ok(0)
 proper(nil) -> ok(nil)
 proper(oddNs) -> ok(oddNs)
 proper(pairNs) -> ok(pairNs)
 repItems(mark(X:S)) -> mark(repItems(X:S))
 repItems(ok(X:S)) -> ok(repItems(X:S))
 s(mark(X:S)) -> mark(s(X:S))
 s(ok(X:S)) -> ok(s(X:S))
 tail(mark(X:S)) -> mark(tail(X:S))
 tail(ok(X:S)) -> ok(tail(X:S))
 take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S))
 take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S))
 take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S))
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))
 zip(mark(X1:S),X2:S) -> mark(zip(X1:S,X2:S))
 zip(ok(X1:S),ok(X2:S)) -> ok(zip(X1:S,X2:S))
 zip(X1:S,mark(X2:S)) -> mark(zip(X1:S,X2:S))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ZIP(mark(X1:S),X2:S) -> ZIP(X1:S,X2:S)
 ZIP(ok(X1:S),ok(X2:S)) -> ZIP(X1:S,X2:S)
 ZIP(X1:S,mark(X2:S)) -> ZIP(X1:S,X2:S)
->->-> Rules:
 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
 active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S)))
 active(incr(X:S)) -> incr(active(X:S))
 active(pair(X1:S,X2:S)) -> pair(active(X1:S),X2:S)
 active(pair(X1:S,X2:S)) -> pair(X1:S,active(X2:S))
 active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S))))
 active(repItems(nil)) -> mark(nil)
 active(repItems(X:S)) -> repItems(active(X:S))
 active(s(X:S)) -> s(active(X:S))
 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
 active(tail(X:S)) -> tail(active(X:S))
 active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S)))
 active(take(0,XS:S)) -> mark(nil)
 active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S)
 active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S))
 active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S)))
 active(zip(nil,XS:S)) -> mark(nil)
 active(zip(X:S,nil)) -> mark(nil)
 active(zip(X1:S,X2:S)) -> zip(active(X1:S),X2:S)
 active(zip(X1:S,X2:S)) -> zip(X1:S,active(X2:S))
 active(oddNs) -> mark(incr(pairNs))
 active(pairNs) -> mark(cons(0,incr(oddNs)))
 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
 incr(mark(X:S)) -> mark(incr(X:S))
 incr(ok(X:S)) -> ok(incr(X:S))
 pair(mark(X1:S),X2:S) -> mark(pair(X1:S,X2:S))
 pair(ok(X1:S),ok(X2:S)) -> ok(pair(X1:S,X2:S))
 pair(X1:S,mark(X2:S)) -> mark(pair(X1:S,X2:S))
 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
 proper(incr(X:S)) -> incr(proper(X:S))
 proper(pair(X1:S,X2:S)) -> pair(proper(X1:S),proper(X2:S))
 proper(repItems(X:S)) -> repItems(proper(X:S))
 proper(s(X:S)) -> s(proper(X:S))
 proper(tail(X:S)) -> tail(proper(X:S))
 proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2:S))
 proper(zip(X1:S,X2:S)) -> zip(proper(X1:S),proper(X2:S))
 proper(0) -> ok(0)
 proper(nil) -> ok(nil)
 proper(oddNs) -> ok(oddNs)
 proper(pairNs) -> ok(pairNs)
 repItems(mark(X:S)) -> mark(repItems(X:S))
 repItems(ok(X:S)) -> ok(repItems(X:S))
 s(mark(X:S)) -> mark(s(X:S))
 s(ok(X:S)) -> ok(s(X:S))
 tail(mark(X:S)) -> mark(tail(X:S))
 tail(ok(X:S)) -> ok(tail(X:S))
 take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S))
 take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S))
 take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S))
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))
 zip(mark(X1:S),X2:S) -> mark(zip(X1:S,X2:S))
 zip(ok(X1:S),ok(X2:S)) -> ok(zip(X1:S,X2:S))
 zip(X1:S,mark(X2:S)) -> mark(zip(X1:S,X2:S))
->->Cycle:
->->-> Pairs:
 TAKE(mark(X1:S),X2:S) -> TAKE(X1:S,X2:S)
 TAKE(ok(X1:S),ok(X2:S)) -> TAKE(X1:S,X2:S)
 TAKE(X1:S,mark(X2:S)) -> TAKE(X1:S,X2:S)
->->-> Rules:
 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
 active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S)))
 active(incr(X:S)) -> incr(active(X:S))
 active(pair(X1:S,X2:S)) -> pair(active(X1:S),X2:S)
 active(pair(X1:S,X2:S)) -> pair(X1:S,active(X2:S))
 active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S))))
 active(repItems(nil)) -> mark(nil)
 active(repItems(X:S)) -> repItems(active(X:S))
 active(s(X:S)) -> s(active(X:S))
 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
 active(tail(X:S)) -> tail(active(X:S))
 active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S)))
 active(take(0,XS:S)) -> mark(nil)
 active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S)
 active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S))
 active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S)))
 active(zip(nil,XS:S)) -> mark(nil)
 active(zip(X:S,nil)) -> mark(nil)
 active(zip(X1:S,X2:S)) -> zip(active(X1:S),X2:S)
 active(zip(X1:S,X2:S)) -> zip(X1:S,active(X2:S))
 active(oddNs) -> mark(incr(pairNs))
 active(pairNs) -> mark(cons(0,incr(oddNs)))
 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
 incr(mark(X:S)) -> mark(incr(X:S))
 incr(ok(X:S)) -> ok(incr(X:S))
 pair(mark(X1:S),X2:S) -> mark(pair(X1:S,X2:S))
 pair(ok(X1:S),ok(X2:S)) -> ok(pair(X1:S,X2:S))
 pair(X1:S,mark(X2:S)) -> mark(pair(X1:S,X2:S))
 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
 proper(incr(X:S)) -> incr(proper(X:S))
 proper(pair(X1:S,X2:S)) -> pair(proper(X1:S),proper(X2:S))
 proper(repItems(X:S)) -> repItems(proper(X:S))
 proper(s(X:S)) -> s(proper(X:S))
 proper(tail(X:S)) -> tail(proper(X:S))
 proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2:S))
 proper(zip(X1:S,X2:S)) -> zip(proper(X1:S),proper(X2:S))
 proper(0) -> ok(0)
 proper(nil) -> ok(nil)
 proper(oddNs) -> ok(oddNs)
 proper(pairNs) -> ok(pairNs)
 repItems(mark(X:S)) -> mark(repItems(X:S))
 repItems(ok(X:S)) -> ok(repItems(X:S))
 s(mark(X:S)) -> mark(s(X:S))
 s(ok(X:S)) -> ok(s(X:S))
 tail(mark(X:S)) -> mark(tail(X:S))
 tail(ok(X:S)) -> ok(tail(X:S))
 take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S))
 take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S))
 take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S))
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))
 zip(mark(X1:S),X2:S) -> mark(zip(X1:S,X2:S))
 zip(ok(X1:S),ok(X2:S)) -> ok(zip(X1:S,X2:S))
 zip(X1:S,mark(X2:S)) -> mark(zip(X1:S,X2:S))
->->Cycle:
->->-> Pairs:
 TAIL(mark(X:S)) -> TAIL(X:S)
 TAIL(ok(X:S)) -> TAIL(X:S)
->->-> Rules:
 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
 active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S)))
 active(incr(X:S)) -> incr(active(X:S))
 active(pair(X1:S,X2:S)) -> pair(active(X1:S),X2:S)
 active(pair(X1:S,X2:S)) -> pair(X1:S,active(X2:S))
 active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S))))
 active(repItems(nil)) -> mark(nil)
 active(repItems(X:S)) -> repItems(active(X:S))
 active(s(X:S)) -> s(active(X:S))
 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
 active(tail(X:S)) -> tail(active(X:S))
 active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S)))
 active(take(0,XS:S)) -> mark(nil)
 active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S)
 active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S))
 active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S)))
 active(zip(nil,XS:S)) -> mark(nil)
 active(zip(X:S,nil)) -> mark(nil)
 active(zip(X1:S,X2:S)) -> zip(active(X1:S),X2:S)
 active(zip(X1:S,X2:S)) -> zip(X1:S,active(X2:S))
 active(oddNs) -> mark(incr(pairNs))
 active(pairNs) -> mark(cons(0,incr(oddNs)))
 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
 incr(mark(X:S)) -> mark(incr(X:S))
 incr(ok(X:S)) -> ok(incr(X:S))
 pair(mark(X1:S),X2:S) -> mark(pair(X1:S,X2:S))
 pair(ok(X1:S),ok(X2:S)) -> ok(pair(X1:S,X2:S))
 pair(X1:S,mark(X2:S)) -> mark(pair(X1:S,X2:S))
 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
 proper(incr(X:S)) -> incr(proper(X:S))
 proper(pair(X1:S,X2:S)) -> pair(proper(X1:S),proper(X2:S))
 proper(repItems(X:S)) -> repItems(proper(X:S))
 proper(s(X:S)) -> s(proper(X:S))
 proper(tail(X:S)) -> tail(proper(X:S))
 proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2:S))
 proper(zip(X1:S,X2:S)) -> zip(proper(X1:S),proper(X2:S))
 proper(0) -> ok(0)
 proper(nil) -> ok(nil)
 proper(oddNs) -> ok(oddNs)
 proper(pairNs) -> ok(pairNs)
 repItems(mark(X:S)) -> mark(repItems(X:S))
 repItems(ok(X:S)) -> ok(repItems(X:S))
 s(mark(X:S)) -> mark(s(X:S))
 s(ok(X:S)) -> ok(s(X:S))
 tail(mark(X:S)) -> mark(tail(X:S))
 tail(ok(X:S)) -> ok(tail(X:S))
 take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S))
 take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S))
 take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S))
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))
 zip(mark(X1:S),X2:S) -> mark(zip(X1:S,X2:S))
 zip(ok(X1:S),ok(X2:S)) -> ok(zip(X1:S,X2:S))
 zip(X1:S,mark(X2:S)) -> mark(zip(X1:S,X2:S))
->->Cycle:
->->-> Pairs:
 S(mark(X:S)) -> S(X:S)
 S(ok(X:S)) -> S(X:S)
->->-> Rules:
 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
 active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S)))
 active(incr(X:S)) -> incr(active(X:S))
 active(pair(X1:S,X2:S)) -> pair(active(X1:S),X2:S)
 active(pair(X1:S,X2:S)) -> pair(X1:S,active(X2:S))
 active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S))))
 active(repItems(nil)) -> mark(nil)
 active(repItems(X:S)) -> repItems(active(X:S))
 active(s(X:S)) -> s(active(X:S))
 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
 active(tail(X:S)) -> tail(active(X:S))
 active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S)))
 active(take(0,XS:S)) -> mark(nil)
 active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S)
 active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S))
 active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S)))
 active(zip(nil,XS:S)) -> mark(nil)
 active(zip(X:S,nil)) -> mark(nil)
 active(zip(X1:S,X2:S)) -> zip(active(X1:S),X2:S)
 active(zip(X1:S,X2:S)) -> zip(X1:S,active(X2:S))
 active(oddNs) -> mark(incr(pairNs))
 active(pairNs) -> mark(cons(0,incr(oddNs)))
 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
 incr(mark(X:S)) -> mark(incr(X:S))
 incr(ok(X:S)) -> ok(incr(X:S))
 pair(mark(X1:S),X2:S) -> mark(pair(X1:S,X2:S))
 pair(ok(X1:S),ok(X2:S)) -> ok(pair(X1:S,X2:S))
 pair(X1:S,mark(X2:S)) -> mark(pair(X1:S,X2:S))
 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
 proper(incr(X:S)) -> incr(proper(X:S))
 proper(pair(X1:S,X2:S)) -> pair(proper(X1:S),proper(X2:S))
 proper(repItems(X:S)) -> repItems(proper(X:S))
 proper(s(X:S)) -> s(proper(X:S))
 proper(tail(X:S)) -> tail(proper(X:S))
 proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2:S))
 proper(zip(X1:S,X2:S)) -> zip(proper(X1:S),proper(X2:S))
 proper(0) -> ok(0)
 proper(nil) -> ok(nil)
 proper(oddNs) -> ok(oddNs)
 proper(pairNs) -> ok(pairNs)
 repItems(mark(X:S)) -> mark(repItems(X:S))
 repItems(ok(X:S)) -> ok(repItems(X:S))
 s(mark(X:S)) -> mark(s(X:S))
 s(ok(X:S)) -> ok(s(X:S))
 tail(mark(X:S)) -> mark(tail(X:S))
 tail(ok(X:S)) -> ok(tail(X:S))
 take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S))
 take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S))
 take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S))
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))
 zip(mark(X1:S),X2:S) -> mark(zip(X1:S,X2:S))
 zip(ok(X1:S),ok(X2:S)) -> ok(zip(X1:S,X2:S))
 zip(X1:S,mark(X2:S)) -> mark(zip(X1:S,X2:S))
->->Cycle:
->->-> Pairs:
 REPITEMS(mark(X:S)) -> REPITEMS(X:S)
 REPITEMS(ok(X:S)) -> REPITEMS(X:S)
->->-> Rules:
 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
 active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S)))
 active(incr(X:S)) -> incr(active(X:S))
 active(pair(X1:S,X2:S)) -> pair(active(X1:S),X2:S)
 active(pair(X1:S,X2:S)) -> pair(X1:S,active(X2:S))
 active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S))))
 active(repItems(nil)) -> mark(nil)
 active(repItems(X:S)) -> repItems(active(X:S))
 active(s(X:S)) -> s(active(X:S))
 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
 active(tail(X:S)) -> tail(active(X:S))
 active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S)))
 active(take(0,XS:S)) -> mark(nil)
 active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S)
 active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S))
 active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S)))
 active(zip(nil,XS:S)) -> mark(nil)
 active(zip(X:S,nil)) -> mark(nil)
 active(zip(X1:S,X2:S)) -> zip(active(X1:S),X2:S)
 active(zip(X1:S,X2:S)) -> zip(X1:S,active(X2:S))
 active(oddNs) -> mark(incr(pairNs))
 active(pairNs) -> mark(cons(0,incr(oddNs)))
 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
 incr(mark(X:S)) -> mark(incr(X:S))
 incr(ok(X:S)) -> ok(incr(X:S))
 pair(mark(X1:S),X2:S) -> mark(pair(X1:S,X2:S))
 pair(ok(X1:S),ok(X2:S)) -> ok(pair(X1:S,X2:S))
 pair(X1:S,mark(X2:S)) -> mark(pair(X1:S,X2:S))
 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
 proper(incr(X:S)) -> incr(proper(X:S))
 proper(pair(X1:S,X2:S)) -> pair(proper(X1:S),proper(X2:S))
 proper(repItems(X:S)) -> repItems(proper(X:S))
 proper(s(X:S)) -> s(proper(X:S))
 proper(tail(X:S)) -> tail(proper(X:S))
 proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2:S))
 proper(zip(X1:S,X2:S)) -> zip(proper(X1:S),proper(X2:S))
 proper(0) -> ok(0)
 proper(nil) -> ok(nil)
 proper(oddNs) -> ok(oddNs)
 proper(pairNs) -> ok(pairNs)
 repItems(mark(X:S)) -> mark(repItems(X:S))
 repItems(ok(X:S)) -> ok(repItems(X:S))
 s(mark(X:S)) -> mark(s(X:S))
 s(ok(X:S)) -> ok(s(X:S))
 tail(mark(X:S)) -> mark(tail(X:S))
 tail(ok(X:S)) -> ok(tail(X:S))
 take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S))
 take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S))
 take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S))
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))
 zip(mark(X1:S),X2:S) -> mark(zip(X1:S,X2:S))
 zip(ok(X1:S),ok(X2:S)) -> ok(zip(X1:S,X2:S))
 zip(X1:S,mark(X2:S)) -> mark(zip(X1:S,X2:S))
->->Cycle:
->->-> Pairs:
 PAIR(mark(X1:S),X2:S) -> PAIR(X1:S,X2:S)
 PAIR(ok(X1:S),ok(X2:S)) -> PAIR(X1:S,X2:S)
 PAIR(X1:S,mark(X2:S)) -> PAIR(X1:S,X2:S)
->->-> Rules:
 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
 active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S)))
 active(incr(X:S)) -> incr(active(X:S))
 active(pair(X1:S,X2:S)) -> pair(active(X1:S),X2:S)
 active(pair(X1:S,X2:S)) -> pair(X1:S,active(X2:S))
 active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S))))
 active(repItems(nil)) -> mark(nil)
 active(repItems(X:S)) -> repItems(active(X:S))
 active(s(X:S)) -> s(active(X:S))
 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
 active(tail(X:S)) -> tail(active(X:S))
 active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S)))
 active(take(0,XS:S)) -> mark(nil)
 active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S)
 active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S))
 active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S)))
 active(zip(nil,XS:S)) -> mark(nil)
 active(zip(X:S,nil)) -> mark(nil)
 active(zip(X1:S,X2:S)) -> zip(active(X1:S),X2:S)
 active(zip(X1:S,X2:S)) -> zip(X1:S,active(X2:S))
 active(oddNs) -> mark(incr(pairNs))
 active(pairNs) -> mark(cons(0,incr(oddNs)))
 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
 incr(mark(X:S)) -> mark(incr(X:S))
 incr(ok(X:S)) -> ok(incr(X:S))
 pair(mark(X1:S),X2:S) -> mark(pair(X1:S,X2:S))
 pair(ok(X1:S),ok(X2:S)) -> ok(pair(X1:S,X2:S))
 pair(X1:S,mark(X2:S)) -> mark(pair(X1:S,X2:S))
 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
 proper(incr(X:S)) -> incr(proper(X:S))
 proper(pair(X1:S,X2:S)) -> pair(proper(X1:S),proper(X2:S))
 proper(repItems(X:S)) -> repItems(proper(X:S))
 proper(s(X:S)) -> s(proper(X:S))
 proper(tail(X:S)) -> tail(proper(X:S))
 proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2:S))
 proper(zip(X1:S,X2:S)) -> zip(proper(X1:S),proper(X2:S))
 proper(0) -> ok(0)
 proper(nil) -> ok(nil)
 proper(oddNs) -> ok(oddNs)
 proper(pairNs) -> ok(pairNs)
 repItems(mark(X:S)) -> mark(repItems(X:S))
 repItems(ok(X:S)) -> ok(repItems(X:S))
 s(mark(X:S)) -> mark(s(X:S))
 s(ok(X:S)) -> ok(s(X:S))
 tail(mark(X:S)) -> mark(tail(X:S))
 tail(ok(X:S)) -> ok(tail(X:S))
 take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S))
 take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S))
 take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S))
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))
 zip(mark(X1:S),X2:S) -> mark(zip(X1:S,X2:S))
 zip(ok(X1:S),ok(X2:S)) -> ok(zip(X1:S,X2:S))
 zip(X1:S,mark(X2:S)) -> mark(zip(X1:S,X2:S))
->->Cycle:
->->-> Pairs:
 INCR(mark(X:S)) -> INCR(X:S)
 INCR(ok(X:S)) -> INCR(X:S)
->->-> Rules:
 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
 active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S)))
 active(incr(X:S)) -> incr(active(X:S))
 active(pair(X1:S,X2:S)) -> pair(active(X1:S),X2:S)
 active(pair(X1:S,X2:S)) -> pair(X1:S,active(X2:S))
 active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S))))
 active(repItems(nil)) -> mark(nil)
 active(repItems(X:S)) -> repItems(active(X:S))
 active(s(X:S)) -> s(active(X:S))
 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
 active(tail(X:S)) -> tail(active(X:S))
 active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S)))
 active(take(0,XS:S)) -> mark(nil)
 active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S)
 active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S))
 active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S)))
 active(zip(nil,XS:S)) -> mark(nil)
 active(zip(X:S,nil)) -> mark(nil)
 active(zip(X1:S,X2:S)) -> zip(active(X1:S),X2:S)
 active(zip(X1:S,X2:S)) -> zip(X1:S,active(X2:S))
 active(oddNs) -> mark(incr(pairNs))
 active(pairNs) -> mark(cons(0,incr(oddNs)))
 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
 incr(mark(X:S)) -> mark(incr(X:S))
 incr(ok(X:S)) -> ok(incr(X:S))
 pair(mark(X1:S),X2:S) -> mark(pair(X1:S,X2:S))
 pair(ok(X1:S),ok(X2:S)) -> ok(pair(X1:S,X2:S))
 pair(X1:S,mark(X2:S)) -> mark(pair(X1:S,X2:S))
 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
 proper(incr(X:S)) -> incr(proper(X:S))
 proper(pair(X1:S,X2:S)) -> pair(proper(X1:S),proper(X2:S))
 proper(repItems(X:S)) -> repItems(proper(X:S))
 proper(s(X:S)) -> s(proper(X:S))
 proper(tail(X:S)) -> tail(proper(X:S))
 proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2:S))
 proper(zip(X1:S,X2:S)) -> zip(proper(X1:S),proper(X2:S))
 proper(0) -> ok(0)
 proper(nil) -> ok(nil)
 proper(oddNs) -> ok(oddNs)
 proper(pairNs) -> ok(pairNs)
 repItems(mark(X:S)) -> mark(repItems(X:S))
 repItems(ok(X:S)) -> ok(repItems(X:S))
 s(mark(X:S)) -> mark(s(X:S))
 s(ok(X:S)) -> ok(s(X:S))
 tail(mark(X:S)) -> mark(tail(X:S))
 tail(ok(X:S)) -> ok(tail(X:S))
 take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S))
 take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S))
 take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S))
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))
 zip(mark(X1:S),X2:S) -> mark(zip(X1:S,X2:S))
 zip(ok(X1:S),ok(X2:S)) -> ok(zip(X1:S,X2:S))
 zip(X1:S,mark(X2:S)) -> mark(zip(X1:S,X2:S))
->->Cycle:
->->-> Pairs:
 CONS(mark(X1:S),X2:S) -> CONS(X1:S,X2:S)
 CONS(ok(X1:S),ok(X2:S)) -> CONS(X1:S,X2:S)
->->-> Rules:
 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
 active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S)))
 active(incr(X:S)) -> incr(active(X:S))
 active(pair(X1:S,X2:S)) -> pair(active(X1:S),X2:S)
 active(pair(X1:S,X2:S)) -> pair(X1:S,active(X2:S))
 active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S))))
 active(repItems(nil)) -> mark(nil)
 active(repItems(X:S)) -> repItems(active(X:S))
 active(s(X:S)) -> s(active(X:S))
 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
 active(tail(X:S)) -> tail(active(X:S))
 active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S)))
 active(take(0,XS:S)) -> mark(nil)
 active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S)
 active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S))
 active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S)))
 active(zip(nil,XS:S)) -> mark(nil)
 active(zip(X:S,nil)) -> mark(nil)
 active(zip(X1:S,X2:S)) -> zip(active(X1:S),X2:S)
 active(zip(X1:S,X2:S)) -> zip(X1:S,active(X2:S))
 active(oddNs) -> mark(incr(pairNs))
 active(pairNs) -> mark(cons(0,incr(oddNs)))
 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
 incr(mark(X:S)) -> mark(incr(X:S))
 incr(ok(X:S)) -> ok(incr(X:S))
 pair(mark(X1:S),X2:S) -> mark(pair(X1:S,X2:S))
 pair(ok(X1:S),ok(X2:S)) -> ok(pair(X1:S,X2:S))
 pair(X1:S,mark(X2:S)) -> mark(pair(X1:S,X2:S))
 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
 proper(incr(X:S)) -> incr(proper(X:S))
 proper(pair(X1:S,X2:S)) -> pair(proper(X1:S),proper(X2:S))
 proper(repItems(X:S)) -> repItems(proper(X:S))
 proper(s(X:S)) -> s(proper(X:S))
 proper(tail(X:S)) -> tail(proper(X:S))
 proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2:S))
 proper(zip(X1:S,X2:S)) -> zip(proper(X1:S),proper(X2:S))
 proper(0) -> ok(0)
 proper(nil) -> ok(nil)
 proper(oddNs) -> ok(oddNs)
 proper(pairNs) -> ok(pairNs)
 repItems(mark(X:S)) -> mark(repItems(X:S))
 repItems(ok(X:S)) -> ok(repItems(X:S))
 s(mark(X:S)) -> mark(s(X:S))
 s(ok(X:S)) -> ok(s(X:S))
 tail(mark(X:S)) -> mark(tail(X:S))
 tail(ok(X:S)) -> ok(tail(X:S))
 take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S))
 take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S))
 take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S))
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))
 zip(mark(X1:S),X2:S) -> mark(zip(X1:S,X2:S))
 zip(ok(X1:S),ok(X2:S)) -> ok(zip(X1:S,X2:S))
 zip(X1:S,mark(X2:S)) -> mark(zip(X1:S,X2:S))
->->Cycle:
->->-> Pairs:
 PROPER(cons(X1:S,X2:S)) -> PROPER(X1:S)
 PROPER(cons(X1:S,X2:S)) -> PROPER(X2:S)
 PROPER(incr(X:S)) -> PROPER(X:S)
 PROPER(pair(X1:S,X2:S)) -> PROPER(X1:S)
 PROPER(pair(X1:S,X2:S)) -> PROPER(X2:S)
 PROPER(repItems(X:S)) -> PROPER(X:S)
 PROPER(s(X:S)) -> PROPER(X:S)
 PROPER(tail(X:S)) -> PROPER(X:S)
 PROPER(take(X1:S,X2:S)) -> PROPER(X1:S)
 PROPER(take(X1:S,X2:S)) -> PROPER(X2:S)
 PROPER(zip(X1:S,X2:S)) -> PROPER(X1:S)
 PROPER(zip(X1:S,X2:S)) -> PROPER(X2:S)
->->-> Rules:
 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
 active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S)))
 active(incr(X:S)) -> incr(active(X:S))
 active(pair(X1:S,X2:S)) -> pair(active(X1:S),X2:S)
 active(pair(X1:S,X2:S)) -> pair(X1:S,active(X2:S))
 active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S))))
 active(repItems(nil)) -> mark(nil)
 active(repItems(X:S)) -> repItems(active(X:S))
 active(s(X:S)) -> s(active(X:S))
 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
 active(tail(X:S)) -> tail(active(X:S))
 active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S)))
 active(take(0,XS:S)) -> mark(nil)
 active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S)
 active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S))
 active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S)))
 active(zip(nil,XS:S)) -> mark(nil)
 active(zip(X:S,nil)) -> mark(nil)
 active(zip(X1:S,X2:S)) -> zip(active(X1:S),X2:S)
 active(zip(X1:S,X2:S)) -> zip(X1:S,active(X2:S))
 active(oddNs) -> mark(incr(pairNs))
 active(pairNs) -> mark(cons(0,incr(oddNs)))
 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
 incr(mark(X:S)) -> mark(incr(X:S))
 incr(ok(X:S)) -> ok(incr(X:S))
 pair(mark(X1:S),X2:S) -> mark(pair(X1:S,X2:S))
 pair(ok(X1:S),ok(X2:S)) -> ok(pair(X1:S,X2:S))
 pair(X1:S,mark(X2:S)) -> mark(pair(X1:S,X2:S))
 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
 proper(incr(X:S)) -> incr(proper(X:S))
 proper(pair(X1:S,X2:S)) -> pair(proper(X1:S),proper(X2:S))
 proper(repItems(X:S)) -> repItems(proper(X:S))
 proper(s(X:S)) -> s(proper(X:S))
 proper(tail(X:S)) -> tail(proper(X:S))
 proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2:S))
 proper(zip(X1:S,X2:S)) -> zip(proper(X1:S),proper(X2:S))
 proper(0) -> ok(0)
 proper(nil) -> ok(nil)
 proper(oddNs) -> ok(oddNs)
 proper(pairNs) -> ok(pairNs)
 repItems(mark(X:S)) -> mark(repItems(X:S))
 repItems(ok(X:S)) -> ok(repItems(X:S))
 s(mark(X:S)) -> mark(s(X:S))
 s(ok(X:S)) -> ok(s(X:S))
 tail(mark(X:S)) -> mark(tail(X:S))
 tail(ok(X:S)) -> ok(tail(X:S))
 take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S))
 take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S))
 take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S))
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))
 zip(mark(X1:S),X2:S) -> mark(zip(X1:S,X2:S))
 zip(ok(X1:S),ok(X2:S)) -> ok(zip(X1:S,X2:S))
 zip(X1:S,mark(X2:S)) -> mark(zip(X1:S,X2:S))
->->Cycle:
->->-> Pairs:
 ACTIVE(cons(X1:S,X2:S)) -> ACTIVE(X1:S)
 ACTIVE(incr(X:S)) -> ACTIVE(X:S)
 ACTIVE(pair(X1:S,X2:S)) -> ACTIVE(X1:S)
 ACTIVE(pair(X1:S,X2:S)) -> ACTIVE(X2:S)
 ACTIVE(repItems(X:S)) -> ACTIVE(X:S)
 ACTIVE(s(X:S)) -> ACTIVE(X:S)
 ACTIVE(tail(X:S)) -> ACTIVE(X:S)
 ACTIVE(take(X1:S,X2:S)) -> ACTIVE(X1:S)
 ACTIVE(take(X1:S,X2:S)) -> ACTIVE(X2:S)
 ACTIVE(zip(X1:S,X2:S)) -> ACTIVE(X1:S)
 ACTIVE(zip(X1:S,X2:S)) -> ACTIVE(X2:S)
->->-> Rules:
 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
 active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S)))
 active(incr(X:S)) -> incr(active(X:S))
 active(pair(X1:S,X2:S)) -> pair(active(X1:S),X2:S)
 active(pair(X1:S,X2:S)) -> pair(X1:S,active(X2:S))
 active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S))))
 active(repItems(nil)) -> mark(nil)
 active(repItems(X:S)) -> repItems(active(X:S))
 active(s(X:S)) -> s(active(X:S))
 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
 active(tail(X:S)) -> tail(active(X:S))
 active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S)))
 active(take(0,XS:S)) -> mark(nil)
 active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S)
 active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S))
 active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S)))
 active(zip(nil,XS:S)) -> mark(nil)
 active(zip(X:S,nil)) -> mark(nil)
 active(zip(X1:S,X2:S)) -> zip(active(X1:S),X2:S)
 active(zip(X1:S,X2:S)) -> zip(X1:S,active(X2:S))
 active(oddNs) -> mark(incr(pairNs))
 active(pairNs) -> mark(cons(0,incr(oddNs)))
 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
 incr(mark(X:S)) -> mark(incr(X:S))
 incr(ok(X:S)) -> ok(incr(X:S))
 pair(mark(X1:S),X2:S) -> mark(pair(X1:S,X2:S))
 pair(ok(X1:S),ok(X2:S)) -> ok(pair(X1:S,X2:S))
 pair(X1:S,mark(X2:S)) -> mark(pair(X1:S,X2:S))
 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
 proper(incr(X:S)) -> incr(proper(X:S))
 proper(pair(X1:S,X2:S)) -> pair(proper(X1:S),proper(X2:S))
 proper(repItems(X:S)) -> repItems(proper(X:S))
 proper(s(X:S)) -> s(proper(X:S))
 proper(tail(X:S)) -> tail(proper(X:S))
 proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2:S))
 proper(zip(X1:S,X2:S)) -> zip(proper(X1:S),proper(X2:S))
 proper(0) -> ok(0)
 proper(nil) -> ok(nil)
 proper(oddNs) -> ok(oddNs)
 proper(pairNs) -> ok(pairNs)
 repItems(mark(X:S)) -> mark(repItems(X:S))
 repItems(ok(X:S)) -> ok(repItems(X:S))
 s(mark(X:S)) -> mark(s(X:S))
 s(ok(X:S)) -> ok(s(X:S))
 tail(mark(X:S)) -> mark(tail(X:S))
 tail(ok(X:S)) -> ok(tail(X:S))
 take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S))
 take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S))
 take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S))
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))
 zip(mark(X1:S),X2:S) -> mark(zip(X1:S,X2:S))
 zip(ok(X1:S),ok(X2:S)) -> ok(zip(X1:S,X2:S))
 zip(X1:S,mark(X2:S)) -> mark(zip(X1:S,X2:S))
->->Cycle:
->->-> Pairs:
 TOP(mark(X:S)) -> TOP(proper(X:S))
 TOP(ok(X:S)) -> TOP(active(X:S))
->->-> Rules:
 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
 active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S)))
 active(incr(X:S)) -> incr(active(X:S))
 active(pair(X1:S,X2:S)) -> pair(active(X1:S),X2:S)
 active(pair(X1:S,X2:S)) -> pair(X1:S,active(X2:S))
 active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S))))
 active(repItems(nil)) -> mark(nil)
 active(repItems(X:S)) -> repItems(active(X:S))
 active(s(X:S)) -> s(active(X:S))
 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
 active(tail(X:S)) -> tail(active(X:S))
 active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S)))
 active(take(0,XS:S)) -> mark(nil)
 active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S)
 active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S))
 active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S)))
 active(zip(nil,XS:S)) -> mark(nil)
 active(zip(X:S,nil)) -> mark(nil)
 active(zip(X1:S,X2:S)) -> zip(active(X1:S),X2:S)
 active(zip(X1:S,X2:S)) -> zip(X1:S,active(X2:S))
 active(oddNs) -> mark(incr(pairNs))
 active(pairNs) -> mark(cons(0,incr(oddNs)))
 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
 incr(mark(X:S)) -> mark(incr(X:S))
 incr(ok(X:S)) -> ok(incr(X:S))
 pair(mark(X1:S),X2:S) -> mark(pair(X1:S,X2:S))
 pair(ok(X1:S),ok(X2:S)) -> ok(pair(X1:S,X2:S))
 pair(X1:S,mark(X2:S)) -> mark(pair(X1:S,X2:S))
 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
 proper(incr(X:S)) -> incr(proper(X:S))
 proper(pair(X1:S,X2:S)) -> pair(proper(X1:S),proper(X2:S))
 proper(repItems(X:S)) -> repItems(proper(X:S))
 proper(s(X:S)) -> s(proper(X:S))
 proper(tail(X:S)) -> tail(proper(X:S))
 proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2:S))
 proper(zip(X1:S,X2:S)) -> zip(proper(X1:S),proper(X2:S))
 proper(0) -> ok(0)
 proper(nil) -> ok(nil)
 proper(oddNs) -> ok(oddNs)
 proper(pairNs) -> ok(pairNs)
 repItems(mark(X:S)) -> mark(repItems(X:S))
 repItems(ok(X:S)) -> ok(repItems(X:S))
 s(mark(X:S)) -> mark(s(X:S))
 s(ok(X:S)) -> ok(s(X:S))
 tail(mark(X:S)) -> mark(tail(X:S))
 tail(ok(X:S)) -> ok(tail(X:S))
 take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S))
 take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S))
 take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S))
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))
 zip(mark(X1:S),X2:S) -> mark(zip(X1:S,X2:S))
 zip(ok(X1:S),ok(X2:S)) -> ok(zip(X1:S,X2:S))
 zip(X1:S,mark(X2:S)) -> mark(zip(X1:S,X2:S))


The problem is decomposed in 11 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 ZIP(mark(X1:S),X2:S) -> ZIP(X1:S,X2:S)
 ZIP(ok(X1:S),ok(X2:S)) -> ZIP(X1:S,X2:S)
 ZIP(X1:S,mark(X2:S)) -> ZIP(X1:S,X2:S)
-> Rules:
 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
 active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S)))
 active(incr(X:S)) -> incr(active(X:S))
 active(pair(X1:S,X2:S)) -> pair(active(X1:S),X2:S)
 active(pair(X1:S,X2:S)) -> pair(X1:S,active(X2:S))
 active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S))))
 active(repItems(nil)) -> mark(nil)
 active(repItems(X:S)) -> repItems(active(X:S))
 active(s(X:S)) -> s(active(X:S))
 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
 active(tail(X:S)) -> tail(active(X:S))
 active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S)))
 active(take(0,XS:S)) -> mark(nil)
 active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S)
 active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S))
 active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S)))
 active(zip(nil,XS:S)) -> mark(nil)
 active(zip(X:S,nil)) -> mark(nil)
 active(zip(X1:S,X2:S)) -> zip(active(X1:S),X2:S)
 active(zip(X1:S,X2:S)) -> zip(X1:S,active(X2:S))
 active(oddNs) -> mark(incr(pairNs))
 active(pairNs) -> mark(cons(0,incr(oddNs)))
 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
 incr(mark(X:S)) -> mark(incr(X:S))
 incr(ok(X:S)) -> ok(incr(X:S))
 pair(mark(X1:S),X2:S) -> mark(pair(X1:S,X2:S))
 pair(ok(X1:S),ok(X2:S)) -> ok(pair(X1:S,X2:S))
 pair(X1:S,mark(X2:S)) -> mark(pair(X1:S,X2:S))
 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
 proper(incr(X:S)) -> incr(proper(X:S))
 proper(pair(X1:S,X2:S)) -> pair(proper(X1:S),proper(X2:S))
 proper(repItems(X:S)) -> repItems(proper(X:S))
 proper(s(X:S)) -> s(proper(X:S))
 proper(tail(X:S)) -> tail(proper(X:S))
 proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2:S))
 proper(zip(X1:S,X2:S)) -> zip(proper(X1:S),proper(X2:S))
 proper(0) -> ok(0)
 proper(nil) -> ok(nil)
 proper(oddNs) -> ok(oddNs)
 proper(pairNs) -> ok(pairNs)
 repItems(mark(X:S)) -> mark(repItems(X:S))
 repItems(ok(X:S)) -> ok(repItems(X:S))
 s(mark(X:S)) -> mark(s(X:S))
 s(ok(X:S)) -> ok(s(X:S))
 tail(mark(X:S)) -> mark(tail(X:S))
 tail(ok(X:S)) -> ok(tail(X:S))
 take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S))
 take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S))
 take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S))
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))
 zip(mark(X1:S),X2:S) -> mark(zip(X1:S,X2:S))
 zip(ok(X1:S),ok(X2:S)) -> ok(zip(X1:S,X2:S))
 zip(X1:S,mark(X2:S)) -> mark(zip(X1:S,X2:S))
->Projection:
 pi(ZIP) = 1

Problem 1.1: 

SCC Processor:
-> Pairs:
 ZIP(X1:S,mark(X2:S)) -> ZIP(X1:S,X2:S)
-> Rules:
 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
 active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S)))
 active(incr(X:S)) -> incr(active(X:S))
 active(pair(X1:S,X2:S)) -> pair(active(X1:S),X2:S)
 active(pair(X1:S,X2:S)) -> pair(X1:S,active(X2:S))
 active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S))))
 active(repItems(nil)) -> mark(nil)
 active(repItems(X:S)) -> repItems(active(X:S))
 active(s(X:S)) -> s(active(X:S))
 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
 active(tail(X:S)) -> tail(active(X:S))
 active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S)))
 active(take(0,XS:S)) -> mark(nil)
 active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S)
 active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S))
 active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S)))
 active(zip(nil,XS:S)) -> mark(nil)
 active(zip(X:S,nil)) -> mark(nil)
 active(zip(X1:S,X2:S)) -> zip(active(X1:S),X2:S)
 active(zip(X1:S,X2:S)) -> zip(X1:S,active(X2:S))
 active(oddNs) -> mark(incr(pairNs))
 active(pairNs) -> mark(cons(0,incr(oddNs)))
 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
 incr(mark(X:S)) -> mark(incr(X:S))
 incr(ok(X:S)) -> ok(incr(X:S))
 pair(mark(X1:S),X2:S) -> mark(pair(X1:S,X2:S))
 pair(ok(X1:S),ok(X2:S)) -> ok(pair(X1:S,X2:S))
 pair(X1:S,mark(X2:S)) -> mark(pair(X1:S,X2:S))
 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
 proper(incr(X:S)) -> incr(proper(X:S))
 proper(pair(X1:S,X2:S)) -> pair(proper(X1:S),proper(X2:S))
 proper(repItems(X:S)) -> repItems(proper(X:S))
 proper(s(X:S)) -> s(proper(X:S))
 proper(tail(X:S)) -> tail(proper(X:S))
 proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2:S))
 proper(zip(X1:S,X2:S)) -> zip(proper(X1:S),proper(X2:S))
 proper(0) -> ok(0)
 proper(nil) -> ok(nil)
 proper(oddNs) -> ok(oddNs)
 proper(pairNs) -> ok(pairNs)
 repItems(mark(X:S)) -> mark(repItems(X:S))
 repItems(ok(X:S)) -> ok(repItems(X:S))
 s(mark(X:S)) -> mark(s(X:S))
 s(ok(X:S)) -> ok(s(X:S))
 tail(mark(X:S)) -> mark(tail(X:S))
 tail(ok(X:S)) -> ok(tail(X:S))
 take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S))
 take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S))
 take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S))
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))
 zip(mark(X1:S),X2:S) -> mark(zip(X1:S,X2:S))
 zip(ok(X1:S),ok(X2:S)) -> ok(zip(X1:S,X2:S))
 zip(X1:S,mark(X2:S)) -> mark(zip(X1:S,X2:S))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ZIP(X1:S,mark(X2:S)) -> ZIP(X1:S,X2:S)
->->-> Rules:
 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
 active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S)))
 active(incr(X:S)) -> incr(active(X:S))
 active(pair(X1:S,X2:S)) -> pair(active(X1:S),X2:S)
 active(pair(X1:S,X2:S)) -> pair(X1:S,active(X2:S))
 active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S))))
 active(repItems(nil)) -> mark(nil)
 active(repItems(X:S)) -> repItems(active(X:S))
 active(s(X:S)) -> s(active(X:S))
 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
 active(tail(X:S)) -> tail(active(X:S))
 active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S)))
 active(take(0,XS:S)) -> mark(nil)
 active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S)
 active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S))
 active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S)))
 active(zip(nil,XS:S)) -> mark(nil)
 active(zip(X:S,nil)) -> mark(nil)
 active(zip(X1:S,X2:S)) -> zip(active(X1:S),X2:S)
 active(zip(X1:S,X2:S)) -> zip(X1:S,active(X2:S))
 active(oddNs) -> mark(incr(pairNs))
 active(pairNs) -> mark(cons(0,incr(oddNs)))
 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
 incr(mark(X:S)) -> mark(incr(X:S))
 incr(ok(X:S)) -> ok(incr(X:S))
 pair(mark(X1:S),X2:S) -> mark(pair(X1:S,X2:S))
 pair(ok(X1:S),ok(X2:S)) -> ok(pair(X1:S,X2:S))
 pair(X1:S,mark(X2:S)) -> mark(pair(X1:S,X2:S))
 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
 proper(incr(X:S)) -> incr(proper(X:S))
 proper(pair(X1:S,X2:S)) -> pair(proper(X1:S),proper(X2:S))
 proper(repItems(X:S)) -> repItems(proper(X:S))
 proper(s(X:S)) -> s(proper(X:S))
 proper(tail(X:S)) -> tail(proper(X:S))
 proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2:S))
 proper(zip(X1:S,X2:S)) -> zip(proper(X1:S),proper(X2:S))
 proper(0) -> ok(0)
 proper(nil) -> ok(nil)
 proper(oddNs) -> ok(oddNs)
 proper(pairNs) -> ok(pairNs)
 repItems(mark(X:S)) -> mark(repItems(X:S))
 repItems(ok(X:S)) -> ok(repItems(X:S))
 s(mark(X:S)) -> mark(s(X:S))
 s(ok(X:S)) -> ok(s(X:S))
 tail(mark(X:S)) -> mark(tail(X:S))
 tail(ok(X:S)) -> ok(tail(X:S))
 take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S))
 take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S))
 take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S))
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))
 zip(mark(X1:S),X2:S) -> mark(zip(X1:S,X2:S))
 zip(ok(X1:S),ok(X2:S)) -> ok(zip(X1:S,X2:S))
 zip(X1:S,mark(X2:S)) -> mark(zip(X1:S,X2:S))

Problem 1.1: 

Subterm Processor:
-> Pairs:
 ZIP(X1:S,mark(X2:S)) -> ZIP(X1:S,X2:S)
-> Rules:
 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
 active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S)))
 active(incr(X:S)) -> incr(active(X:S))
 active(pair(X1:S,X2:S)) -> pair(active(X1:S),X2:S)
 active(pair(X1:S,X2:S)) -> pair(X1:S,active(X2:S))
 active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S))))
 active(repItems(nil)) -> mark(nil)
 active(repItems(X:S)) -> repItems(active(X:S))
 active(s(X:S)) -> s(active(X:S))
 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
 active(tail(X:S)) -> tail(active(X:S))
 active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S)))
 active(take(0,XS:S)) -> mark(nil)
 active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S)
 active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S))
 active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S)))
 active(zip(nil,XS:S)) -> mark(nil)
 active(zip(X:S,nil)) -> mark(nil)
 active(zip(X1:S,X2:S)) -> zip(active(X1:S),X2:S)
 active(zip(X1:S,X2:S)) -> zip(X1:S,active(X2:S))
 active(oddNs) -> mark(incr(pairNs))
 active(pairNs) -> mark(cons(0,incr(oddNs)))
 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
 incr(mark(X:S)) -> mark(incr(X:S))
 incr(ok(X:S)) -> ok(incr(X:S))
 pair(mark(X1:S),X2:S) -> mark(pair(X1:S,X2:S))
 pair(ok(X1:S),ok(X2:S)) -> ok(pair(X1:S,X2:S))
 pair(X1:S,mark(X2:S)) -> mark(pair(X1:S,X2:S))
 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
 proper(incr(X:S)) -> incr(proper(X:S))
 proper(pair(X1:S,X2:S)) -> pair(proper(X1:S),proper(X2:S))
 proper(repItems(X:S)) -> repItems(proper(X:S))
 proper(s(X:S)) -> s(proper(X:S))
 proper(tail(X:S)) -> tail(proper(X:S))
 proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2:S))
 proper(zip(X1:S,X2:S)) -> zip(proper(X1:S),proper(X2:S))
 proper(0) -> ok(0)
 proper(nil) -> ok(nil)
 proper(oddNs) -> ok(oddNs)
 proper(pairNs) -> ok(pairNs)
 repItems(mark(X:S)) -> mark(repItems(X:S))
 repItems(ok(X:S)) -> ok(repItems(X:S))
 s(mark(X:S)) -> mark(s(X:S))
 s(ok(X:S)) -> ok(s(X:S))
 tail(mark(X:S)) -> mark(tail(X:S))
 tail(ok(X:S)) -> ok(tail(X:S))
 take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S))
 take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S))
 take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S))
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))
 zip(mark(X1:S),X2:S) -> mark(zip(X1:S,X2:S))
 zip(ok(X1:S),ok(X2:S)) -> ok(zip(X1:S,X2:S))
 zip(X1:S,mark(X2:S)) -> mark(zip(X1:S,X2:S))
->Projection:
 pi(ZIP) = 2

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
 active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S)))
 active(incr(X:S)) -> incr(active(X:S))
 active(pair(X1:S,X2:S)) -> pair(active(X1:S),X2:S)
 active(pair(X1:S,X2:S)) -> pair(X1:S,active(X2:S))
 active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S))))
 active(repItems(nil)) -> mark(nil)
 active(repItems(X:S)) -> repItems(active(X:S))
 active(s(X:S)) -> s(active(X:S))
 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
 active(tail(X:S)) -> tail(active(X:S))
 active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S)))
 active(take(0,XS:S)) -> mark(nil)
 active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S)
 active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S))
 active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S)))
 active(zip(nil,XS:S)) -> mark(nil)
 active(zip(X:S,nil)) -> mark(nil)
 active(zip(X1:S,X2:S)) -> zip(active(X1:S),X2:S)
 active(zip(X1:S,X2:S)) -> zip(X1:S,active(X2:S))
 active(oddNs) -> mark(incr(pairNs))
 active(pairNs) -> mark(cons(0,incr(oddNs)))
 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
 incr(mark(X:S)) -> mark(incr(X:S))
 incr(ok(X:S)) -> ok(incr(X:S))
 pair(mark(X1:S),X2:S) -> mark(pair(X1:S,X2:S))
 pair(ok(X1:S),ok(X2:S)) -> ok(pair(X1:S,X2:S))
 pair(X1:S,mark(X2:S)) -> mark(pair(X1:S,X2:S))
 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
 proper(incr(X:S)) -> incr(proper(X:S))
 proper(pair(X1:S,X2:S)) -> pair(proper(X1:S),proper(X2:S))
 proper(repItems(X:S)) -> repItems(proper(X:S))
 proper(s(X:S)) -> s(proper(X:S))
 proper(tail(X:S)) -> tail(proper(X:S))
 proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2:S))
 proper(zip(X1:S,X2:S)) -> zip(proper(X1:S),proper(X2:S))
 proper(0) -> ok(0)
 proper(nil) -> ok(nil)
 proper(oddNs) -> ok(oddNs)
 proper(pairNs) -> ok(pairNs)
 repItems(mark(X:S)) -> mark(repItems(X:S))
 repItems(ok(X:S)) -> ok(repItems(X:S))
 s(mark(X:S)) -> mark(s(X:S))
 s(ok(X:S)) -> ok(s(X:S))
 tail(mark(X:S)) -> mark(tail(X:S))
 tail(ok(X:S)) -> ok(tail(X:S))
 take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S))
 take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S))
 take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S))
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))
 zip(mark(X1:S),X2:S) -> mark(zip(X1:S,X2:S))
 zip(ok(X1:S),ok(X2:S)) -> ok(zip(X1:S,X2:S))
 zip(X1:S,mark(X2:S)) -> mark(zip(X1:S,X2:S))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Subterm Processor:
-> Pairs:
 TAKE(mark(X1:S),X2:S) -> TAKE(X1:S,X2:S)
 TAKE(ok(X1:S),ok(X2:S)) -> TAKE(X1:S,X2:S)
 TAKE(X1:S,mark(X2:S)) -> TAKE(X1:S,X2:S)
-> Rules:
 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
 active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S)))
 active(incr(X:S)) -> incr(active(X:S))
 active(pair(X1:S,X2:S)) -> pair(active(X1:S),X2:S)
 active(pair(X1:S,X2:S)) -> pair(X1:S,active(X2:S))
 active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S))))
 active(repItems(nil)) -> mark(nil)
 active(repItems(X:S)) -> repItems(active(X:S))
 active(s(X:S)) -> s(active(X:S))
 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
 active(tail(X:S)) -> tail(active(X:S))
 active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S)))
 active(take(0,XS:S)) -> mark(nil)
 active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S)
 active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S))
 active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S)))
 active(zip(nil,XS:S)) -> mark(nil)
 active(zip(X:S,nil)) -> mark(nil)
 active(zip(X1:S,X2:S)) -> zip(active(X1:S),X2:S)
 active(zip(X1:S,X2:S)) -> zip(X1:S,active(X2:S))
 active(oddNs) -> mark(incr(pairNs))
 active(pairNs) -> mark(cons(0,incr(oddNs)))
 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
 incr(mark(X:S)) -> mark(incr(X:S))
 incr(ok(X:S)) -> ok(incr(X:S))
 pair(mark(X1:S),X2:S) -> mark(pair(X1:S,X2:S))
 pair(ok(X1:S),ok(X2:S)) -> ok(pair(X1:S,X2:S))
 pair(X1:S,mark(X2:S)) -> mark(pair(X1:S,X2:S))
 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
 proper(incr(X:S)) -> incr(proper(X:S))
 proper(pair(X1:S,X2:S)) -> pair(proper(X1:S),proper(X2:S))
 proper(repItems(X:S)) -> repItems(proper(X:S))
 proper(s(X:S)) -> s(proper(X:S))
 proper(tail(X:S)) -> tail(proper(X:S))
 proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2:S))
 proper(zip(X1:S,X2:S)) -> zip(proper(X1:S),proper(X2:S))
 proper(0) -> ok(0)
 proper(nil) -> ok(nil)
 proper(oddNs) -> ok(oddNs)
 proper(pairNs) -> ok(pairNs)
 repItems(mark(X:S)) -> mark(repItems(X:S))
 repItems(ok(X:S)) -> ok(repItems(X:S))
 s(mark(X:S)) -> mark(s(X:S))
 s(ok(X:S)) -> ok(s(X:S))
 tail(mark(X:S)) -> mark(tail(X:S))
 tail(ok(X:S)) -> ok(tail(X:S))
 take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S))
 take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S))
 take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S))
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))
 zip(mark(X1:S),X2:S) -> mark(zip(X1:S,X2:S))
 zip(ok(X1:S),ok(X2:S)) -> ok(zip(X1:S,X2:S))
 zip(X1:S,mark(X2:S)) -> mark(zip(X1:S,X2:S))
->Projection:
 pi(TAKE) = 1

Problem 1.2: 

SCC Processor:
-> Pairs:
 TAKE(X1:S,mark(X2:S)) -> TAKE(X1:S,X2:S)
-> Rules:
 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
 active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S)))
 active(incr(X:S)) -> incr(active(X:S))
 active(pair(X1:S,X2:S)) -> pair(active(X1:S),X2:S)
 active(pair(X1:S,X2:S)) -> pair(X1:S,active(X2:S))
 active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S))))
 active(repItems(nil)) -> mark(nil)
 active(repItems(X:S)) -> repItems(active(X:S))
 active(s(X:S)) -> s(active(X:S))
 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
 active(tail(X:S)) -> tail(active(X:S))
 active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S)))
 active(take(0,XS:S)) -> mark(nil)
 active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S)
 active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S))
 active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S)))
 active(zip(nil,XS:S)) -> mark(nil)
 active(zip(X:S,nil)) -> mark(nil)
 active(zip(X1:S,X2:S)) -> zip(active(X1:S),X2:S)
 active(zip(X1:S,X2:S)) -> zip(X1:S,active(X2:S))
 active(oddNs) -> mark(incr(pairNs))
 active(pairNs) -> mark(cons(0,incr(oddNs)))
 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
 incr(mark(X:S)) -> mark(incr(X:S))
 incr(ok(X:S)) -> ok(incr(X:S))
 pair(mark(X1:S),X2:S) -> mark(pair(X1:S,X2:S))
 pair(ok(X1:S),ok(X2:S)) -> ok(pair(X1:S,X2:S))
 pair(X1:S,mark(X2:S)) -> mark(pair(X1:S,X2:S))
 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
 proper(incr(X:S)) -> incr(proper(X:S))
 proper(pair(X1:S,X2:S)) -> pair(proper(X1:S),proper(X2:S))
 proper(repItems(X:S)) -> repItems(proper(X:S))
 proper(s(X:S)) -> s(proper(X:S))
 proper(tail(X:S)) -> tail(proper(X:S))
 proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2:S))
 proper(zip(X1:S,X2:S)) -> zip(proper(X1:S),proper(X2:S))
 proper(0) -> ok(0)
 proper(nil) -> ok(nil)
 proper(oddNs) -> ok(oddNs)
 proper(pairNs) -> ok(pairNs)
 repItems(mark(X:S)) -> mark(repItems(X:S))
 repItems(ok(X:S)) -> ok(repItems(X:S))
 s(mark(X:S)) -> mark(s(X:S))
 s(ok(X:S)) -> ok(s(X:S))
 tail(mark(X:S)) -> mark(tail(X:S))
 tail(ok(X:S)) -> ok(tail(X:S))
 take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S))
 take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S))
 take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S))
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))
 zip(mark(X1:S),X2:S) -> mark(zip(X1:S,X2:S))
 zip(ok(X1:S),ok(X2:S)) -> ok(zip(X1:S,X2:S))
 zip(X1:S,mark(X2:S)) -> mark(zip(X1:S,X2:S))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 TAKE(X1:S,mark(X2:S)) -> TAKE(X1:S,X2:S)
->->-> Rules:
 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
 active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S)))
 active(incr(X:S)) -> incr(active(X:S))
 active(pair(X1:S,X2:S)) -> pair(active(X1:S),X2:S)
 active(pair(X1:S,X2:S)) -> pair(X1:S,active(X2:S))
 active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S))))
 active(repItems(nil)) -> mark(nil)
 active(repItems(X:S)) -> repItems(active(X:S))
 active(s(X:S)) -> s(active(X:S))
 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
 active(tail(X:S)) -> tail(active(X:S))
 active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S)))
 active(take(0,XS:S)) -> mark(nil)
 active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S)
 active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S))
 active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S)))
 active(zip(nil,XS:S)) -> mark(nil)
 active(zip(X:S,nil)) -> mark(nil)
 active(zip(X1:S,X2:S)) -> zip(active(X1:S),X2:S)
 active(zip(X1:S,X2:S)) -> zip(X1:S,active(X2:S))
 active(oddNs) -> mark(incr(pairNs))
 active(pairNs) -> mark(cons(0,incr(oddNs)))
 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
 incr(mark(X:S)) -> mark(incr(X:S))
 incr(ok(X:S)) -> ok(incr(X:S))
 pair(mark(X1:S),X2:S) -> mark(pair(X1:S,X2:S))
 pair(ok(X1:S),ok(X2:S)) -> ok(pair(X1:S,X2:S))
 pair(X1:S,mark(X2:S)) -> mark(pair(X1:S,X2:S))
 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
 proper(incr(X:S)) -> incr(proper(X:S))
 proper(pair(X1:S,X2:S)) -> pair(proper(X1:S),proper(X2:S))
 proper(repItems(X:S)) -> repItems(proper(X:S))
 proper(s(X:S)) -> s(proper(X:S))
 proper(tail(X:S)) -> tail(proper(X:S))
 proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2:S))
 proper(zip(X1:S,X2:S)) -> zip(proper(X1:S),proper(X2:S))
 proper(0) -> ok(0)
 proper(nil) -> ok(nil)
 proper(oddNs) -> ok(oddNs)
 proper(pairNs) -> ok(pairNs)
 repItems(mark(X:S)) -> mark(repItems(X:S))
 repItems(ok(X:S)) -> ok(repItems(X:S))
 s(mark(X:S)) -> mark(s(X:S))
 s(ok(X:S)) -> ok(s(X:S))
 tail(mark(X:S)) -> mark(tail(X:S))
 tail(ok(X:S)) -> ok(tail(X:S))
 take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S))
 take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S))
 take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S))
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))
 zip(mark(X1:S),X2:S) -> mark(zip(X1:S,X2:S))
 zip(ok(X1:S),ok(X2:S)) -> ok(zip(X1:S,X2:S))
 zip(X1:S,mark(X2:S)) -> mark(zip(X1:S,X2:S))

Problem 1.2: 

Subterm Processor:
-> Pairs:
 TAKE(X1:S,mark(X2:S)) -> TAKE(X1:S,X2:S)
-> Rules:
 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
 active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S)))
 active(incr(X:S)) -> incr(active(X:S))
 active(pair(X1:S,X2:S)) -> pair(active(X1:S),X2:S)
 active(pair(X1:S,X2:S)) -> pair(X1:S,active(X2:S))
 active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S))))
 active(repItems(nil)) -> mark(nil)
 active(repItems(X:S)) -> repItems(active(X:S))
 active(s(X:S)) -> s(active(X:S))
 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
 active(tail(X:S)) -> tail(active(X:S))
 active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S)))
 active(take(0,XS:S)) -> mark(nil)
 active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S)
 active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S))
 active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S)))
 active(zip(nil,XS:S)) -> mark(nil)
 active(zip(X:S,nil)) -> mark(nil)
 active(zip(X1:S,X2:S)) -> zip(active(X1:S),X2:S)
 active(zip(X1:S,X2:S)) -> zip(X1:S,active(X2:S))
 active(oddNs) -> mark(incr(pairNs))
 active(pairNs) -> mark(cons(0,incr(oddNs)))
 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
 incr(mark(X:S)) -> mark(incr(X:S))
 incr(ok(X:S)) -> ok(incr(X:S))
 pair(mark(X1:S),X2:S) -> mark(pair(X1:S,X2:S))
 pair(ok(X1:S),ok(X2:S)) -> ok(pair(X1:S,X2:S))
 pair(X1:S,mark(X2:S)) -> mark(pair(X1:S,X2:S))
 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
 proper(incr(X:S)) -> incr(proper(X:S))
 proper(pair(X1:S,X2:S)) -> pair(proper(X1:S),proper(X2:S))
 proper(repItems(X:S)) -> repItems(proper(X:S))
 proper(s(X:S)) -> s(proper(X:S))
 proper(tail(X:S)) -> tail(proper(X:S))
 proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2:S))
 proper(zip(X1:S,X2:S)) -> zip(proper(X1:S),proper(X2:S))
 proper(0) -> ok(0)
 proper(nil) -> ok(nil)
 proper(oddNs) -> ok(oddNs)
 proper(pairNs) -> ok(pairNs)
 repItems(mark(X:S)) -> mark(repItems(X:S))
 repItems(ok(X:S)) -> ok(repItems(X:S))
 s(mark(X:S)) -> mark(s(X:S))
 s(ok(X:S)) -> ok(s(X:S))
 tail(mark(X:S)) -> mark(tail(X:S))
 tail(ok(X:S)) -> ok(tail(X:S))
 take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S))
 take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S))
 take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S))
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))
 zip(mark(X1:S),X2:S) -> mark(zip(X1:S,X2:S))
 zip(ok(X1:S),ok(X2:S)) -> ok(zip(X1:S,X2:S))
 zip(X1:S,mark(X2:S)) -> mark(zip(X1:S,X2:S))
->Projection:
 pi(TAKE) = 2

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
 active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S)))
 active(incr(X:S)) -> incr(active(X:S))
 active(pair(X1:S,X2:S)) -> pair(active(X1:S),X2:S)
 active(pair(X1:S,X2:S)) -> pair(X1:S,active(X2:S))
 active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S))))
 active(repItems(nil)) -> mark(nil)
 active(repItems(X:S)) -> repItems(active(X:S))
 active(s(X:S)) -> s(active(X:S))
 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
 active(tail(X:S)) -> tail(active(X:S))
 active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S)))
 active(take(0,XS:S)) -> mark(nil)
 active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S)
 active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S))
 active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S)))
 active(zip(nil,XS:S)) -> mark(nil)
 active(zip(X:S,nil)) -> mark(nil)
 active(zip(X1:S,X2:S)) -> zip(active(X1:S),X2:S)
 active(zip(X1:S,X2:S)) -> zip(X1:S,active(X2:S))
 active(oddNs) -> mark(incr(pairNs))
 active(pairNs) -> mark(cons(0,incr(oddNs)))
 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
 incr(mark(X:S)) -> mark(incr(X:S))
 incr(ok(X:S)) -> ok(incr(X:S))
 pair(mark(X1:S),X2:S) -> mark(pair(X1:S,X2:S))
 pair(ok(X1:S),ok(X2:S)) -> ok(pair(X1:S,X2:S))
 pair(X1:S,mark(X2:S)) -> mark(pair(X1:S,X2:S))
 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
 proper(incr(X:S)) -> incr(proper(X:S))
 proper(pair(X1:S,X2:S)) -> pair(proper(X1:S),proper(X2:S))
 proper(repItems(X:S)) -> repItems(proper(X:S))
 proper(s(X:S)) -> s(proper(X:S))
 proper(tail(X:S)) -> tail(proper(X:S))
 proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2:S))
 proper(zip(X1:S,X2:S)) -> zip(proper(X1:S),proper(X2:S))
 proper(0) -> ok(0)
 proper(nil) -> ok(nil)
 proper(oddNs) -> ok(oddNs)
 proper(pairNs) -> ok(pairNs)
 repItems(mark(X:S)) -> mark(repItems(X:S))
 repItems(ok(X:S)) -> ok(repItems(X:S))
 s(mark(X:S)) -> mark(s(X:S))
 s(ok(X:S)) -> ok(s(X:S))
 tail(mark(X:S)) -> mark(tail(X:S))
 tail(ok(X:S)) -> ok(tail(X:S))
 take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S))
 take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S))
 take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S))
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))
 zip(mark(X1:S),X2:S) -> mark(zip(X1:S,X2:S))
 zip(ok(X1:S),ok(X2:S)) -> ok(zip(X1:S,X2:S))
 zip(X1:S,mark(X2:S)) -> mark(zip(X1:S,X2:S))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.3: 

Subterm Processor:
-> Pairs:
 TAIL(mark(X:S)) -> TAIL(X:S)
 TAIL(ok(X:S)) -> TAIL(X:S)
-> Rules:
 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
 active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S)))
 active(incr(X:S)) -> incr(active(X:S))
 active(pair(X1:S,X2:S)) -> pair(active(X1:S),X2:S)
 active(pair(X1:S,X2:S)) -> pair(X1:S,active(X2:S))
 active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S))))
 active(repItems(nil)) -> mark(nil)
 active(repItems(X:S)) -> repItems(active(X:S))
 active(s(X:S)) -> s(active(X:S))
 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
 active(tail(X:S)) -> tail(active(X:S))
 active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S)))
 active(take(0,XS:S)) -> mark(nil)
 active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S)
 active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S))
 active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S)))
 active(zip(nil,XS:S)) -> mark(nil)
 active(zip(X:S,nil)) -> mark(nil)
 active(zip(X1:S,X2:S)) -> zip(active(X1:S),X2:S)
 active(zip(X1:S,X2:S)) -> zip(X1:S,active(X2:S))
 active(oddNs) -> mark(incr(pairNs))
 active(pairNs) -> mark(cons(0,incr(oddNs)))
 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
 incr(mark(X:S)) -> mark(incr(X:S))
 incr(ok(X:S)) -> ok(incr(X:S))
 pair(mark(X1:S),X2:S) -> mark(pair(X1:S,X2:S))
 pair(ok(X1:S),ok(X2:S)) -> ok(pair(X1:S,X2:S))
 pair(X1:S,mark(X2:S)) -> mark(pair(X1:S,X2:S))
 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
 proper(incr(X:S)) -> incr(proper(X:S))
 proper(pair(X1:S,X2:S)) -> pair(proper(X1:S),proper(X2:S))
 proper(repItems(X:S)) -> repItems(proper(X:S))
 proper(s(X:S)) -> s(proper(X:S))
 proper(tail(X:S)) -> tail(proper(X:S))
 proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2:S))
 proper(zip(X1:S,X2:S)) -> zip(proper(X1:S),proper(X2:S))
 proper(0) -> ok(0)
 proper(nil) -> ok(nil)
 proper(oddNs) -> ok(oddNs)
 proper(pairNs) -> ok(pairNs)
 repItems(mark(X:S)) -> mark(repItems(X:S))
 repItems(ok(X:S)) -> ok(repItems(X:S))
 s(mark(X:S)) -> mark(s(X:S))
 s(ok(X:S)) -> ok(s(X:S))
 tail(mark(X:S)) -> mark(tail(X:S))
 tail(ok(X:S)) -> ok(tail(X:S))
 take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S))
 take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S))
 take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S))
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))
 zip(mark(X1:S),X2:S) -> mark(zip(X1:S,X2:S))
 zip(ok(X1:S),ok(X2:S)) -> ok(zip(X1:S,X2:S))
 zip(X1:S,mark(X2:S)) -> mark(zip(X1:S,X2:S))
->Projection:
 pi(TAIL) = 1

Problem 1.3: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
 active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S)))
 active(incr(X:S)) -> incr(active(X:S))
 active(pair(X1:S,X2:S)) -> pair(active(X1:S),X2:S)
 active(pair(X1:S,X2:S)) -> pair(X1:S,active(X2:S))
 active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S))))
 active(repItems(nil)) -> mark(nil)
 active(repItems(X:S)) -> repItems(active(X:S))
 active(s(X:S)) -> s(active(X:S))
 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
 active(tail(X:S)) -> tail(active(X:S))
 active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S)))
 active(take(0,XS:S)) -> mark(nil)
 active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S)
 active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S))
 active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S)))
 active(zip(nil,XS:S)) -> mark(nil)
 active(zip(X:S,nil)) -> mark(nil)
 active(zip(X1:S,X2:S)) -> zip(active(X1:S),X2:S)
 active(zip(X1:S,X2:S)) -> zip(X1:S,active(X2:S))
 active(oddNs) -> mark(incr(pairNs))
 active(pairNs) -> mark(cons(0,incr(oddNs)))
 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
 incr(mark(X:S)) -> mark(incr(X:S))
 incr(ok(X:S)) -> ok(incr(X:S))
 pair(mark(X1:S),X2:S) -> mark(pair(X1:S,X2:S))
 pair(ok(X1:S),ok(X2:S)) -> ok(pair(X1:S,X2:S))
 pair(X1:S,mark(X2:S)) -> mark(pair(X1:S,X2:S))
 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
 proper(incr(X:S)) -> incr(proper(X:S))
 proper(pair(X1:S,X2:S)) -> pair(proper(X1:S),proper(X2:S))
 proper(repItems(X:S)) -> repItems(proper(X:S))
 proper(s(X:S)) -> s(proper(X:S))
 proper(tail(X:S)) -> tail(proper(X:S))
 proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2:S))
 proper(zip(X1:S,X2:S)) -> zip(proper(X1:S),proper(X2:S))
 proper(0) -> ok(0)
 proper(nil) -> ok(nil)
 proper(oddNs) -> ok(oddNs)
 proper(pairNs) -> ok(pairNs)
 repItems(mark(X:S)) -> mark(repItems(X:S))
 repItems(ok(X:S)) -> ok(repItems(X:S))
 s(mark(X:S)) -> mark(s(X:S))
 s(ok(X:S)) -> ok(s(X:S))
 tail(mark(X:S)) -> mark(tail(X:S))
 tail(ok(X:S)) -> ok(tail(X:S))
 take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S))
 take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S))
 take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S))
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))
 zip(mark(X1:S),X2:S) -> mark(zip(X1:S,X2:S))
 zip(ok(X1:S),ok(X2:S)) -> ok(zip(X1:S,X2:S))
 zip(X1:S,mark(X2:S)) -> mark(zip(X1:S,X2:S))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.4: 

Subterm Processor:
-> Pairs:
 S(mark(X:S)) -> S(X:S)
 S(ok(X:S)) -> S(X:S)
-> Rules:
 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
 active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S)))
 active(incr(X:S)) -> incr(active(X:S))
 active(pair(X1:S,X2:S)) -> pair(active(X1:S),X2:S)
 active(pair(X1:S,X2:S)) -> pair(X1:S,active(X2:S))
 active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S))))
 active(repItems(nil)) -> mark(nil)
 active(repItems(X:S)) -> repItems(active(X:S))
 active(s(X:S)) -> s(active(X:S))
 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
 active(tail(X:S)) -> tail(active(X:S))
 active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S)))
 active(take(0,XS:S)) -> mark(nil)
 active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S)
 active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S))
 active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S)))
 active(zip(nil,XS:S)) -> mark(nil)
 active(zip(X:S,nil)) -> mark(nil)
 active(zip(X1:S,X2:S)) -> zip(active(X1:S),X2:S)
 active(zip(X1:S,X2:S)) -> zip(X1:S,active(X2:S))
 active(oddNs) -> mark(incr(pairNs))
 active(pairNs) -> mark(cons(0,incr(oddNs)))
 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
 incr(mark(X:S)) -> mark(incr(X:S))
 incr(ok(X:S)) -> ok(incr(X:S))
 pair(mark(X1:S),X2:S) -> mark(pair(X1:S,X2:S))
 pair(ok(X1:S),ok(X2:S)) -> ok(pair(X1:S,X2:S))
 pair(X1:S,mark(X2:S)) -> mark(pair(X1:S,X2:S))
 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
 proper(incr(X:S)) -> incr(proper(X:S))
 proper(pair(X1:S,X2:S)) -> pair(proper(X1:S),proper(X2:S))
 proper(repItems(X:S)) -> repItems(proper(X:S))
 proper(s(X:S)) -> s(proper(X:S))
 proper(tail(X:S)) -> tail(proper(X:S))
 proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2:S))
 proper(zip(X1:S,X2:S)) -> zip(proper(X1:S),proper(X2:S))
 proper(0) -> ok(0)
 proper(nil) -> ok(nil)
 proper(oddNs) -> ok(oddNs)
 proper(pairNs) -> ok(pairNs)
 repItems(mark(X:S)) -> mark(repItems(X:S))
 repItems(ok(X:S)) -> ok(repItems(X:S))
 s(mark(X:S)) -> mark(s(X:S))
 s(ok(X:S)) -> ok(s(X:S))
 tail(mark(X:S)) -> mark(tail(X:S))
 tail(ok(X:S)) -> ok(tail(X:S))
 take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S))
 take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S))
 take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S))
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))
 zip(mark(X1:S),X2:S) -> mark(zip(X1:S,X2:S))
 zip(ok(X1:S),ok(X2:S)) -> ok(zip(X1:S,X2:S))
 zip(X1:S,mark(X2:S)) -> mark(zip(X1:S,X2:S))
->Projection:
 pi(S) = 1

Problem 1.4: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
 active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S)))
 active(incr(X:S)) -> incr(active(X:S))
 active(pair(X1:S,X2:S)) -> pair(active(X1:S),X2:S)
 active(pair(X1:S,X2:S)) -> pair(X1:S,active(X2:S))
 active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S))))
 active(repItems(nil)) -> mark(nil)
 active(repItems(X:S)) -> repItems(active(X:S))
 active(s(X:S)) -> s(active(X:S))
 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
 active(tail(X:S)) -> tail(active(X:S))
 active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S)))
 active(take(0,XS:S)) -> mark(nil)
 active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S)
 active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S))
 active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S)))
 active(zip(nil,XS:S)) -> mark(nil)
 active(zip(X:S,nil)) -> mark(nil)
 active(zip(X1:S,X2:S)) -> zip(active(X1:S),X2:S)
 active(zip(X1:S,X2:S)) -> zip(X1:S,active(X2:S))
 active(oddNs) -> mark(incr(pairNs))
 active(pairNs) -> mark(cons(0,incr(oddNs)))
 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
 incr(mark(X:S)) -> mark(incr(X:S))
 incr(ok(X:S)) -> ok(incr(X:S))
 pair(mark(X1:S),X2:S) -> mark(pair(X1:S,X2:S))
 pair(ok(X1:S),ok(X2:S)) -> ok(pair(X1:S,X2:S))
 pair(X1:S,mark(X2:S)) -> mark(pair(X1:S,X2:S))
 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
 proper(incr(X:S)) -> incr(proper(X:S))
 proper(pair(X1:S,X2:S)) -> pair(proper(X1:S),proper(X2:S))
 proper(repItems(X:S)) -> repItems(proper(X:S))
 proper(s(X:S)) -> s(proper(X:S))
 proper(tail(X:S)) -> tail(proper(X:S))
 proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2:S))
 proper(zip(X1:S,X2:S)) -> zip(proper(X1:S),proper(X2:S))
 proper(0) -> ok(0)
 proper(nil) -> ok(nil)
 proper(oddNs) -> ok(oddNs)
 proper(pairNs) -> ok(pairNs)
 repItems(mark(X:S)) -> mark(repItems(X:S))
 repItems(ok(X:S)) -> ok(repItems(X:S))
 s(mark(X:S)) -> mark(s(X:S))
 s(ok(X:S)) -> ok(s(X:S))
 tail(mark(X:S)) -> mark(tail(X:S))
 tail(ok(X:S)) -> ok(tail(X:S))
 take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S))
 take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S))
 take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S))
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))
 zip(mark(X1:S),X2:S) -> mark(zip(X1:S,X2:S))
 zip(ok(X1:S),ok(X2:S)) -> ok(zip(X1:S,X2:S))
 zip(X1:S,mark(X2:S)) -> mark(zip(X1:S,X2:S))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.5: 

Subterm Processor:
-> Pairs:
 REPITEMS(mark(X:S)) -> REPITEMS(X:S)
 REPITEMS(ok(X:S)) -> REPITEMS(X:S)
-> Rules:
 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
 active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S)))
 active(incr(X:S)) -> incr(active(X:S))
 active(pair(X1:S,X2:S)) -> pair(active(X1:S),X2:S)
 active(pair(X1:S,X2:S)) -> pair(X1:S,active(X2:S))
 active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S))))
 active(repItems(nil)) -> mark(nil)
 active(repItems(X:S)) -> repItems(active(X:S))
 active(s(X:S)) -> s(active(X:S))
 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
 active(tail(X:S)) -> tail(active(X:S))
 active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S)))
 active(take(0,XS:S)) -> mark(nil)
 active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S)
 active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S))
 active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S)))
 active(zip(nil,XS:S)) -> mark(nil)
 active(zip(X:S,nil)) -> mark(nil)
 active(zip(X1:S,X2:S)) -> zip(active(X1:S),X2:S)
 active(zip(X1:S,X2:S)) -> zip(X1:S,active(X2:S))
 active(oddNs) -> mark(incr(pairNs))
 active(pairNs) -> mark(cons(0,incr(oddNs)))
 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
 incr(mark(X:S)) -> mark(incr(X:S))
 incr(ok(X:S)) -> ok(incr(X:S))
 pair(mark(X1:S),X2:S) -> mark(pair(X1:S,X2:S))
 pair(ok(X1:S),ok(X2:S)) -> ok(pair(X1:S,X2:S))
 pair(X1:S,mark(X2:S)) -> mark(pair(X1:S,X2:S))
 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
 proper(incr(X:S)) -> incr(proper(X:S))
 proper(pair(X1:S,X2:S)) -> pair(proper(X1:S),proper(X2:S))
 proper(repItems(X:S)) -> repItems(proper(X:S))
 proper(s(X:S)) -> s(proper(X:S))
 proper(tail(X:S)) -> tail(proper(X:S))
 proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2:S))
 proper(zip(X1:S,X2:S)) -> zip(proper(X1:S),proper(X2:S))
 proper(0) -> ok(0)
 proper(nil) -> ok(nil)
 proper(oddNs) -> ok(oddNs)
 proper(pairNs) -> ok(pairNs)
 repItems(mark(X:S)) -> mark(repItems(X:S))
 repItems(ok(X:S)) -> ok(repItems(X:S))
 s(mark(X:S)) -> mark(s(X:S))
 s(ok(X:S)) -> ok(s(X:S))
 tail(mark(X:S)) -> mark(tail(X:S))
 tail(ok(X:S)) -> ok(tail(X:S))
 take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S))
 take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S))
 take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S))
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))
 zip(mark(X1:S),X2:S) -> mark(zip(X1:S,X2:S))
 zip(ok(X1:S),ok(X2:S)) -> ok(zip(X1:S,X2:S))
 zip(X1:S,mark(X2:S)) -> mark(zip(X1:S,X2:S))
->Projection:
 pi(REPITEMS) = 1

Problem 1.5: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
 active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S)))
 active(incr(X:S)) -> incr(active(X:S))
 active(pair(X1:S,X2:S)) -> pair(active(X1:S),X2:S)
 active(pair(X1:S,X2:S)) -> pair(X1:S,active(X2:S))
 active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S))))
 active(repItems(nil)) -> mark(nil)
 active(repItems(X:S)) -> repItems(active(X:S))
 active(s(X:S)) -> s(active(X:S))
 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
 active(tail(X:S)) -> tail(active(X:S))
 active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S)))
 active(take(0,XS:S)) -> mark(nil)
 active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S)
 active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S))
 active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S)))
 active(zip(nil,XS:S)) -> mark(nil)
 active(zip(X:S,nil)) -> mark(nil)
 active(zip(X1:S,X2:S)) -> zip(active(X1:S),X2:S)
 active(zip(X1:S,X2:S)) -> zip(X1:S,active(X2:S))
 active(oddNs) -> mark(incr(pairNs))
 active(pairNs) -> mark(cons(0,incr(oddNs)))
 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
 incr(mark(X:S)) -> mark(incr(X:S))
 incr(ok(X:S)) -> ok(incr(X:S))
 pair(mark(X1:S),X2:S) -> mark(pair(X1:S,X2:S))
 pair(ok(X1:S),ok(X2:S)) -> ok(pair(X1:S,X2:S))
 pair(X1:S,mark(X2:S)) -> mark(pair(X1:S,X2:S))
 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
 proper(incr(X:S)) -> incr(proper(X:S))
 proper(pair(X1:S,X2:S)) -> pair(proper(X1:S),proper(X2:S))
 proper(repItems(X:S)) -> repItems(proper(X:S))
 proper(s(X:S)) -> s(proper(X:S))
 proper(tail(X:S)) -> tail(proper(X:S))
 proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2:S))
 proper(zip(X1:S,X2:S)) -> zip(proper(X1:S),proper(X2:S))
 proper(0) -> ok(0)
 proper(nil) -> ok(nil)
 proper(oddNs) -> ok(oddNs)
 proper(pairNs) -> ok(pairNs)
 repItems(mark(X:S)) -> mark(repItems(X:S))
 repItems(ok(X:S)) -> ok(repItems(X:S))
 s(mark(X:S)) -> mark(s(X:S))
 s(ok(X:S)) -> ok(s(X:S))
 tail(mark(X:S)) -> mark(tail(X:S))
 tail(ok(X:S)) -> ok(tail(X:S))
 take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S))
 take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S))
 take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S))
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))
 zip(mark(X1:S),X2:S) -> mark(zip(X1:S,X2:S))
 zip(ok(X1:S),ok(X2:S)) -> ok(zip(X1:S,X2:S))
 zip(X1:S,mark(X2:S)) -> mark(zip(X1:S,X2:S))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.6: 

Subterm Processor:
-> Pairs:
 PAIR(mark(X1:S),X2:S) -> PAIR(X1:S,X2:S)
 PAIR(ok(X1:S),ok(X2:S)) -> PAIR(X1:S,X2:S)
 PAIR(X1:S,mark(X2:S)) -> PAIR(X1:S,X2:S)
-> Rules:
 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
 active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S)))
 active(incr(X:S)) -> incr(active(X:S))
 active(pair(X1:S,X2:S)) -> pair(active(X1:S),X2:S)
 active(pair(X1:S,X2:S)) -> pair(X1:S,active(X2:S))
 active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S))))
 active(repItems(nil)) -> mark(nil)
 active(repItems(X:S)) -> repItems(active(X:S))
 active(s(X:S)) -> s(active(X:S))
 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
 active(tail(X:S)) -> tail(active(X:S))
 active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S)))
 active(take(0,XS:S)) -> mark(nil)
 active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S)
 active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S))
 active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S)))
 active(zip(nil,XS:S)) -> mark(nil)
 active(zip(X:S,nil)) -> mark(nil)
 active(zip(X1:S,X2:S)) -> zip(active(X1:S),X2:S)
 active(zip(X1:S,X2:S)) -> zip(X1:S,active(X2:S))
 active(oddNs) -> mark(incr(pairNs))
 active(pairNs) -> mark(cons(0,incr(oddNs)))
 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
 incr(mark(X:S)) -> mark(incr(X:S))
 incr(ok(X:S)) -> ok(incr(X:S))
 pair(mark(X1:S),X2:S) -> mark(pair(X1:S,X2:S))
 pair(ok(X1:S),ok(X2:S)) -> ok(pair(X1:S,X2:S))
 pair(X1:S,mark(X2:S)) -> mark(pair(X1:S,X2:S))
 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
 proper(incr(X:S)) -> incr(proper(X:S))
 proper(pair(X1:S,X2:S)) -> pair(proper(X1:S),proper(X2:S))
 proper(repItems(X:S)) -> repItems(proper(X:S))
 proper(s(X:S)) -> s(proper(X:S))
 proper(tail(X:S)) -> tail(proper(X:S))
 proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2:S))
 proper(zip(X1:S,X2:S)) -> zip(proper(X1:S),proper(X2:S))
 proper(0) -> ok(0)
 proper(nil) -> ok(nil)
 proper(oddNs) -> ok(oddNs)
 proper(pairNs) -> ok(pairNs)
 repItems(mark(X:S)) -> mark(repItems(X:S))
 repItems(ok(X:S)) -> ok(repItems(X:S))
 s(mark(X:S)) -> mark(s(X:S))
 s(ok(X:S)) -> ok(s(X:S))
 tail(mark(X:S)) -> mark(tail(X:S))
 tail(ok(X:S)) -> ok(tail(X:S))
 take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S))
 take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S))
 take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S))
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))
 zip(mark(X1:S),X2:S) -> mark(zip(X1:S,X2:S))
 zip(ok(X1:S),ok(X2:S)) -> ok(zip(X1:S,X2:S))
 zip(X1:S,mark(X2:S)) -> mark(zip(X1:S,X2:S))
->Projection:
 pi(PAIR) = 1

Problem 1.6: 

SCC Processor:
-> Pairs:
 PAIR(X1:S,mark(X2:S)) -> PAIR(X1:S,X2:S)
-> Rules:
 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
 active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S)))
 active(incr(X:S)) -> incr(active(X:S))
 active(pair(X1:S,X2:S)) -> pair(active(X1:S),X2:S)
 active(pair(X1:S,X2:S)) -> pair(X1:S,active(X2:S))
 active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S))))
 active(repItems(nil)) -> mark(nil)
 active(repItems(X:S)) -> repItems(active(X:S))
 active(s(X:S)) -> s(active(X:S))
 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
 active(tail(X:S)) -> tail(active(X:S))
 active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S)))
 active(take(0,XS:S)) -> mark(nil)
 active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S)
 active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S))
 active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S)))
 active(zip(nil,XS:S)) -> mark(nil)
 active(zip(X:S,nil)) -> mark(nil)
 active(zip(X1:S,X2:S)) -> zip(active(X1:S),X2:S)
 active(zip(X1:S,X2:S)) -> zip(X1:S,active(X2:S))
 active(oddNs) -> mark(incr(pairNs))
 active(pairNs) -> mark(cons(0,incr(oddNs)))
 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
 incr(mark(X:S)) -> mark(incr(X:S))
 incr(ok(X:S)) -> ok(incr(X:S))
 pair(mark(X1:S),X2:S) -> mark(pair(X1:S,X2:S))
 pair(ok(X1:S),ok(X2:S)) -> ok(pair(X1:S,X2:S))
 pair(X1:S,mark(X2:S)) -> mark(pair(X1:S,X2:S))
 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
 proper(incr(X:S)) -> incr(proper(X:S))
 proper(pair(X1:S,X2:S)) -> pair(proper(X1:S),proper(X2:S))
 proper(repItems(X:S)) -> repItems(proper(X:S))
 proper(s(X:S)) -> s(proper(X:S))
 proper(tail(X:S)) -> tail(proper(X:S))
 proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2:S))
 proper(zip(X1:S,X2:S)) -> zip(proper(X1:S),proper(X2:S))
 proper(0) -> ok(0)
 proper(nil) -> ok(nil)
 proper(oddNs) -> ok(oddNs)
 proper(pairNs) -> ok(pairNs)
 repItems(mark(X:S)) -> mark(repItems(X:S))
 repItems(ok(X:S)) -> ok(repItems(X:S))
 s(mark(X:S)) -> mark(s(X:S))
 s(ok(X:S)) -> ok(s(X:S))
 tail(mark(X:S)) -> mark(tail(X:S))
 tail(ok(X:S)) -> ok(tail(X:S))
 take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S))
 take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S))
 take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S))
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))
 zip(mark(X1:S),X2:S) -> mark(zip(X1:S,X2:S))
 zip(ok(X1:S),ok(X2:S)) -> ok(zip(X1:S,X2:S))
 zip(X1:S,mark(X2:S)) -> mark(zip(X1:S,X2:S))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 PAIR(X1:S,mark(X2:S)) -> PAIR(X1:S,X2:S)
->->-> Rules:
 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
 active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S)))
 active(incr(X:S)) -> incr(active(X:S))
 active(pair(X1:S,X2:S)) -> pair(active(X1:S),X2:S)
 active(pair(X1:S,X2:S)) -> pair(X1:S,active(X2:S))
 active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S))))
 active(repItems(nil)) -> mark(nil)
 active(repItems(X:S)) -> repItems(active(X:S))
 active(s(X:S)) -> s(active(X:S))
 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
 active(tail(X:S)) -> tail(active(X:S))
 active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S)))
 active(take(0,XS:S)) -> mark(nil)
 active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S)
 active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S))
 active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S)))
 active(zip(nil,XS:S)) -> mark(nil)
 active(zip(X:S,nil)) -> mark(nil)
 active(zip(X1:S,X2:S)) -> zip(active(X1:S),X2:S)
 active(zip(X1:S,X2:S)) -> zip(X1:S,active(X2:S))
 active(oddNs) -> mark(incr(pairNs))
 active(pairNs) -> mark(cons(0,incr(oddNs)))
 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
 incr(mark(X:S)) -> mark(incr(X:S))
 incr(ok(X:S)) -> ok(incr(X:S))
 pair(mark(X1:S),X2:S) -> mark(pair(X1:S,X2:S))
 pair(ok(X1:S),ok(X2:S)) -> ok(pair(X1:S,X2:S))
 pair(X1:S,mark(X2:S)) -> mark(pair(X1:S,X2:S))
 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
 proper(incr(X:S)) -> incr(proper(X:S))
 proper(pair(X1:S,X2:S)) -> pair(proper(X1:S),proper(X2:S))
 proper(repItems(X:S)) -> repItems(proper(X:S))
 proper(s(X:S)) -> s(proper(X:S))
 proper(tail(X:S)) -> tail(proper(X:S))
 proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2:S))
 proper(zip(X1:S,X2:S)) -> zip(proper(X1:S),proper(X2:S))
 proper(0) -> ok(0)
 proper(nil) -> ok(nil)
 proper(oddNs) -> ok(oddNs)
 proper(pairNs) -> ok(pairNs)
 repItems(mark(X:S)) -> mark(repItems(X:S))
 repItems(ok(X:S)) -> ok(repItems(X:S))
 s(mark(X:S)) -> mark(s(X:S))
 s(ok(X:S)) -> ok(s(X:S))
 tail(mark(X:S)) -> mark(tail(X:S))
 tail(ok(X:S)) -> ok(tail(X:S))
 take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S))
 take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S))
 take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S))
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))
 zip(mark(X1:S),X2:S) -> mark(zip(X1:S,X2:S))
 zip(ok(X1:S),ok(X2:S)) -> ok(zip(X1:S,X2:S))
 zip(X1:S,mark(X2:S)) -> mark(zip(X1:S,X2:S))

Problem 1.6: 

Subterm Processor:
-> Pairs:
 PAIR(X1:S,mark(X2:S)) -> PAIR(X1:S,X2:S)
-> Rules:
 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
 active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S)))
 active(incr(X:S)) -> incr(active(X:S))
 active(pair(X1:S,X2:S)) -> pair(active(X1:S),X2:S)
 active(pair(X1:S,X2:S)) -> pair(X1:S,active(X2:S))
 active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S))))
 active(repItems(nil)) -> mark(nil)
 active(repItems(X:S)) -> repItems(active(X:S))
 active(s(X:S)) -> s(active(X:S))
 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
 active(tail(X:S)) -> tail(active(X:S))
 active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S)))
 active(take(0,XS:S)) -> mark(nil)
 active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S)
 active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S))
 active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S)))
 active(zip(nil,XS:S)) -> mark(nil)
 active(zip(X:S,nil)) -> mark(nil)
 active(zip(X1:S,X2:S)) -> zip(active(X1:S),X2:S)
 active(zip(X1:S,X2:S)) -> zip(X1:S,active(X2:S))
 active(oddNs) -> mark(incr(pairNs))
 active(pairNs) -> mark(cons(0,incr(oddNs)))
 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
 incr(mark(X:S)) -> mark(incr(X:S))
 incr(ok(X:S)) -> ok(incr(X:S))
 pair(mark(X1:S),X2:S) -> mark(pair(X1:S,X2:S))
 pair(ok(X1:S),ok(X2:S)) -> ok(pair(X1:S,X2:S))
 pair(X1:S,mark(X2:S)) -> mark(pair(X1:S,X2:S))
 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
 proper(incr(X:S)) -> incr(proper(X:S))
 proper(pair(X1:S,X2:S)) -> pair(proper(X1:S),proper(X2:S))
 proper(repItems(X:S)) -> repItems(proper(X:S))
 proper(s(X:S)) -> s(proper(X:S))
 proper(tail(X:S)) -> tail(proper(X:S))
 proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2:S))
 proper(zip(X1:S,X2:S)) -> zip(proper(X1:S),proper(X2:S))
 proper(0) -> ok(0)
 proper(nil) -> ok(nil)
 proper(oddNs) -> ok(oddNs)
 proper(pairNs) -> ok(pairNs)
 repItems(mark(X:S)) -> mark(repItems(X:S))
 repItems(ok(X:S)) -> ok(repItems(X:S))
 s(mark(X:S)) -> mark(s(X:S))
 s(ok(X:S)) -> ok(s(X:S))
 tail(mark(X:S)) -> mark(tail(X:S))
 tail(ok(X:S)) -> ok(tail(X:S))
 take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S))
 take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S))
 take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S))
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))
 zip(mark(X1:S),X2:S) -> mark(zip(X1:S,X2:S))
 zip(ok(X1:S),ok(X2:S)) -> ok(zip(X1:S,X2:S))
 zip(X1:S,mark(X2:S)) -> mark(zip(X1:S,X2:S))
->Projection:
 pi(PAIR) = 2

Problem 1.6: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
 active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S)))
 active(incr(X:S)) -> incr(active(X:S))
 active(pair(X1:S,X2:S)) -> pair(active(X1:S),X2:S)
 active(pair(X1:S,X2:S)) -> pair(X1:S,active(X2:S))
 active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S))))
 active(repItems(nil)) -> mark(nil)
 active(repItems(X:S)) -> repItems(active(X:S))
 active(s(X:S)) -> s(active(X:S))
 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
 active(tail(X:S)) -> tail(active(X:S))
 active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S)))
 active(take(0,XS:S)) -> mark(nil)
 active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S)
 active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S))
 active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S)))
 active(zip(nil,XS:S)) -> mark(nil)
 active(zip(X:S,nil)) -> mark(nil)
 active(zip(X1:S,X2:S)) -> zip(active(X1:S),X2:S)
 active(zip(X1:S,X2:S)) -> zip(X1:S,active(X2:S))
 active(oddNs) -> mark(incr(pairNs))
 active(pairNs) -> mark(cons(0,incr(oddNs)))
 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
 incr(mark(X:S)) -> mark(incr(X:S))
 incr(ok(X:S)) -> ok(incr(X:S))
 pair(mark(X1:S),X2:S) -> mark(pair(X1:S,X2:S))
 pair(ok(X1:S),ok(X2:S)) -> ok(pair(X1:S,X2:S))
 pair(X1:S,mark(X2:S)) -> mark(pair(X1:S,X2:S))
 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
 proper(incr(X:S)) -> incr(proper(X:S))
 proper(pair(X1:S,X2:S)) -> pair(proper(X1:S),proper(X2:S))
 proper(repItems(X:S)) -> repItems(proper(X:S))
 proper(s(X:S)) -> s(proper(X:S))
 proper(tail(X:S)) -> tail(proper(X:S))
 proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2:S))
 proper(zip(X1:S,X2:S)) -> zip(proper(X1:S),proper(X2:S))
 proper(0) -> ok(0)
 proper(nil) -> ok(nil)
 proper(oddNs) -> ok(oddNs)
 proper(pairNs) -> ok(pairNs)
 repItems(mark(X:S)) -> mark(repItems(X:S))
 repItems(ok(X:S)) -> ok(repItems(X:S))
 s(mark(X:S)) -> mark(s(X:S))
 s(ok(X:S)) -> ok(s(X:S))
 tail(mark(X:S)) -> mark(tail(X:S))
 tail(ok(X:S)) -> ok(tail(X:S))
 take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S))
 take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S))
 take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S))
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))
 zip(mark(X1:S),X2:S) -> mark(zip(X1:S,X2:S))
 zip(ok(X1:S),ok(X2:S)) -> ok(zip(X1:S,X2:S))
 zip(X1:S,mark(X2:S)) -> mark(zip(X1:S,X2:S))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.7: 

Subterm Processor:
-> Pairs:
 INCR(mark(X:S)) -> INCR(X:S)
 INCR(ok(X:S)) -> INCR(X:S)
-> Rules:
 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
 active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S)))
 active(incr(X:S)) -> incr(active(X:S))
 active(pair(X1:S,X2:S)) -> pair(active(X1:S),X2:S)
 active(pair(X1:S,X2:S)) -> pair(X1:S,active(X2:S))
 active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S))))
 active(repItems(nil)) -> mark(nil)
 active(repItems(X:S)) -> repItems(active(X:S))
 active(s(X:S)) -> s(active(X:S))
 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
 active(tail(X:S)) -> tail(active(X:S))
 active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S)))
 active(take(0,XS:S)) -> mark(nil)
 active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S)
 active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S))
 active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S)))
 active(zip(nil,XS:S)) -> mark(nil)
 active(zip(X:S,nil)) -> mark(nil)
 active(zip(X1:S,X2:S)) -> zip(active(X1:S),X2:S)
 active(zip(X1:S,X2:S)) -> zip(X1:S,active(X2:S))
 active(oddNs) -> mark(incr(pairNs))
 active(pairNs) -> mark(cons(0,incr(oddNs)))
 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
 incr(mark(X:S)) -> mark(incr(X:S))
 incr(ok(X:S)) -> ok(incr(X:S))
 pair(mark(X1:S),X2:S) -> mark(pair(X1:S,X2:S))
 pair(ok(X1:S),ok(X2:S)) -> ok(pair(X1:S,X2:S))
 pair(X1:S,mark(X2:S)) -> mark(pair(X1:S,X2:S))
 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
 proper(incr(X:S)) -> incr(proper(X:S))
 proper(pair(X1:S,X2:S)) -> pair(proper(X1:S),proper(X2:S))
 proper(repItems(X:S)) -> repItems(proper(X:S))
 proper(s(X:S)) -> s(proper(X:S))
 proper(tail(X:S)) -> tail(proper(X:S))
 proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2:S))
 proper(zip(X1:S,X2:S)) -> zip(proper(X1:S),proper(X2:S))
 proper(0) -> ok(0)
 proper(nil) -> ok(nil)
 proper(oddNs) -> ok(oddNs)
 proper(pairNs) -> ok(pairNs)
 repItems(mark(X:S)) -> mark(repItems(X:S))
 repItems(ok(X:S)) -> ok(repItems(X:S))
 s(mark(X:S)) -> mark(s(X:S))
 s(ok(X:S)) -> ok(s(X:S))
 tail(mark(X:S)) -> mark(tail(X:S))
 tail(ok(X:S)) -> ok(tail(X:S))
 take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S))
 take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S))
 take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S))
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))
 zip(mark(X1:S),X2:S) -> mark(zip(X1:S,X2:S))
 zip(ok(X1:S),ok(X2:S)) -> ok(zip(X1:S,X2:S))
 zip(X1:S,mark(X2:S)) -> mark(zip(X1:S,X2:S))
->Projection:
 pi(INCR) = 1

Problem 1.7: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
 active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S)))
 active(incr(X:S)) -> incr(active(X:S))
 active(pair(X1:S,X2:S)) -> pair(active(X1:S),X2:S)
 active(pair(X1:S,X2:S)) -> pair(X1:S,active(X2:S))
 active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S))))
 active(repItems(nil)) -> mark(nil)
 active(repItems(X:S)) -> repItems(active(X:S))
 active(s(X:S)) -> s(active(X:S))
 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
 active(tail(X:S)) -> tail(active(X:S))
 active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S)))
 active(take(0,XS:S)) -> mark(nil)
 active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S)
 active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S))
 active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S)))
 active(zip(nil,XS:S)) -> mark(nil)
 active(zip(X:S,nil)) -> mark(nil)
 active(zip(X1:S,X2:S)) -> zip(active(X1:S),X2:S)
 active(zip(X1:S,X2:S)) -> zip(X1:S,active(X2:S))
 active(oddNs) -> mark(incr(pairNs))
 active(pairNs) -> mark(cons(0,incr(oddNs)))
 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
 incr(mark(X:S)) -> mark(incr(X:S))
 incr(ok(X:S)) -> ok(incr(X:S))
 pair(mark(X1:S),X2:S) -> mark(pair(X1:S,X2:S))
 pair(ok(X1:S),ok(X2:S)) -> ok(pair(X1:S,X2:S))
 pair(X1:S,mark(X2:S)) -> mark(pair(X1:S,X2:S))
 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
 proper(incr(X:S)) -> incr(proper(X:S))
 proper(pair(X1:S,X2:S)) -> pair(proper(X1:S),proper(X2:S))
 proper(repItems(X:S)) -> repItems(proper(X:S))
 proper(s(X:S)) -> s(proper(X:S))
 proper(tail(X:S)) -> tail(proper(X:S))
 proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2:S))
 proper(zip(X1:S,X2:S)) -> zip(proper(X1:S),proper(X2:S))
 proper(0) -> ok(0)
 proper(nil) -> ok(nil)
 proper(oddNs) -> ok(oddNs)
 proper(pairNs) -> ok(pairNs)
 repItems(mark(X:S)) -> mark(repItems(X:S))
 repItems(ok(X:S)) -> ok(repItems(X:S))
 s(mark(X:S)) -> mark(s(X:S))
 s(ok(X:S)) -> ok(s(X:S))
 tail(mark(X:S)) -> mark(tail(X:S))
 tail(ok(X:S)) -> ok(tail(X:S))
 take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S))
 take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S))
 take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S))
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))
 zip(mark(X1:S),X2:S) -> mark(zip(X1:S,X2:S))
 zip(ok(X1:S),ok(X2:S)) -> ok(zip(X1:S,X2:S))
 zip(X1:S,mark(X2:S)) -> mark(zip(X1:S,X2:S))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.8: 

Subterm Processor:
-> Pairs:
 CONS(mark(X1:S),X2:S) -> CONS(X1:S,X2:S)
 CONS(ok(X1:S),ok(X2:S)) -> CONS(X1:S,X2:S)
-> Rules:
 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
 active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S)))
 active(incr(X:S)) -> incr(active(X:S))
 active(pair(X1:S,X2:S)) -> pair(active(X1:S),X2:S)
 active(pair(X1:S,X2:S)) -> pair(X1:S,active(X2:S))
 active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S))))
 active(repItems(nil)) -> mark(nil)
 active(repItems(X:S)) -> repItems(active(X:S))
 active(s(X:S)) -> s(active(X:S))
 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
 active(tail(X:S)) -> tail(active(X:S))
 active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S)))
 active(take(0,XS:S)) -> mark(nil)
 active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S)
 active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S))
 active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S)))
 active(zip(nil,XS:S)) -> mark(nil)
 active(zip(X:S,nil)) -> mark(nil)
 active(zip(X1:S,X2:S)) -> zip(active(X1:S),X2:S)
 active(zip(X1:S,X2:S)) -> zip(X1:S,active(X2:S))
 active(oddNs) -> mark(incr(pairNs))
 active(pairNs) -> mark(cons(0,incr(oddNs)))
 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
 incr(mark(X:S)) -> mark(incr(X:S))
 incr(ok(X:S)) -> ok(incr(X:S))
 pair(mark(X1:S),X2:S) -> mark(pair(X1:S,X2:S))
 pair(ok(X1:S),ok(X2:S)) -> ok(pair(X1:S,X2:S))
 pair(X1:S,mark(X2:S)) -> mark(pair(X1:S,X2:S))
 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
 proper(incr(X:S)) -> incr(proper(X:S))
 proper(pair(X1:S,X2:S)) -> pair(proper(X1:S),proper(X2:S))
 proper(repItems(X:S)) -> repItems(proper(X:S))
 proper(s(X:S)) -> s(proper(X:S))
 proper(tail(X:S)) -> tail(proper(X:S))
 proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2:S))
 proper(zip(X1:S,X2:S)) -> zip(proper(X1:S),proper(X2:S))
 proper(0) -> ok(0)
 proper(nil) -> ok(nil)
 proper(oddNs) -> ok(oddNs)
 proper(pairNs) -> ok(pairNs)
 repItems(mark(X:S)) -> mark(repItems(X:S))
 repItems(ok(X:S)) -> ok(repItems(X:S))
 s(mark(X:S)) -> mark(s(X:S))
 s(ok(X:S)) -> ok(s(X:S))
 tail(mark(X:S)) -> mark(tail(X:S))
 tail(ok(X:S)) -> ok(tail(X:S))
 take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S))
 take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S))
 take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S))
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))
 zip(mark(X1:S),X2:S) -> mark(zip(X1:S,X2:S))
 zip(ok(X1:S),ok(X2:S)) -> ok(zip(X1:S,X2:S))
 zip(X1:S,mark(X2:S)) -> mark(zip(X1:S,X2:S))
->Projection:
 pi(CONS) = 1

Problem 1.8: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
 active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S)))
 active(incr(X:S)) -> incr(active(X:S))
 active(pair(X1:S,X2:S)) -> pair(active(X1:S),X2:S)
 active(pair(X1:S,X2:S)) -> pair(X1:S,active(X2:S))
 active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S))))
 active(repItems(nil)) -> mark(nil)
 active(repItems(X:S)) -> repItems(active(X:S))
 active(s(X:S)) -> s(active(X:S))
 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
 active(tail(X:S)) -> tail(active(X:S))
 active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S)))
 active(take(0,XS:S)) -> mark(nil)
 active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S)
 active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S))
 active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S)))
 active(zip(nil,XS:S)) -> mark(nil)
 active(zip(X:S,nil)) -> mark(nil)
 active(zip(X1:S,X2:S)) -> zip(active(X1:S),X2:S)
 active(zip(X1:S,X2:S)) -> zip(X1:S,active(X2:S))
 active(oddNs) -> mark(incr(pairNs))
 active(pairNs) -> mark(cons(0,incr(oddNs)))
 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
 incr(mark(X:S)) -> mark(incr(X:S))
 incr(ok(X:S)) -> ok(incr(X:S))
 pair(mark(X1:S),X2:S) -> mark(pair(X1:S,X2:S))
 pair(ok(X1:S),ok(X2:S)) -> ok(pair(X1:S,X2:S))
 pair(X1:S,mark(X2:S)) -> mark(pair(X1:S,X2:S))
 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
 proper(incr(X:S)) -> incr(proper(X:S))
 proper(pair(X1:S,X2:S)) -> pair(proper(X1:S),proper(X2:S))
 proper(repItems(X:S)) -> repItems(proper(X:S))
 proper(s(X:S)) -> s(proper(X:S))
 proper(tail(X:S)) -> tail(proper(X:S))
 proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2:S))
 proper(zip(X1:S,X2:S)) -> zip(proper(X1:S),proper(X2:S))
 proper(0) -> ok(0)
 proper(nil) -> ok(nil)
 proper(oddNs) -> ok(oddNs)
 proper(pairNs) -> ok(pairNs)
 repItems(mark(X:S)) -> mark(repItems(X:S))
 repItems(ok(X:S)) -> ok(repItems(X:S))
 s(mark(X:S)) -> mark(s(X:S))
 s(ok(X:S)) -> ok(s(X:S))
 tail(mark(X:S)) -> mark(tail(X:S))
 tail(ok(X:S)) -> ok(tail(X:S))
 take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S))
 take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S))
 take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S))
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))
 zip(mark(X1:S),X2:S) -> mark(zip(X1:S,X2:S))
 zip(ok(X1:S),ok(X2:S)) -> ok(zip(X1:S,X2:S))
 zip(X1:S,mark(X2:S)) -> mark(zip(X1:S,X2:S))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.9: 

Subterm Processor:
-> Pairs:
 PROPER(cons(X1:S,X2:S)) -> PROPER(X1:S)
 PROPER(cons(X1:S,X2:S)) -> PROPER(X2:S)
 PROPER(incr(X:S)) -> PROPER(X:S)
 PROPER(pair(X1:S,X2:S)) -> PROPER(X1:S)
 PROPER(pair(X1:S,X2:S)) -> PROPER(X2:S)
 PROPER(repItems(X:S)) -> PROPER(X:S)
 PROPER(s(X:S)) -> PROPER(X:S)
 PROPER(tail(X:S)) -> PROPER(X:S)
 PROPER(take(X1:S,X2:S)) -> PROPER(X1:S)
 PROPER(take(X1:S,X2:S)) -> PROPER(X2:S)
 PROPER(zip(X1:S,X2:S)) -> PROPER(X1:S)
 PROPER(zip(X1:S,X2:S)) -> PROPER(X2:S)
-> Rules:
 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
 active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S)))
 active(incr(X:S)) -> incr(active(X:S))
 active(pair(X1:S,X2:S)) -> pair(active(X1:S),X2:S)
 active(pair(X1:S,X2:S)) -> pair(X1:S,active(X2:S))
 active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S))))
 active(repItems(nil)) -> mark(nil)
 active(repItems(X:S)) -> repItems(active(X:S))
 active(s(X:S)) -> s(active(X:S))
 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
 active(tail(X:S)) -> tail(active(X:S))
 active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S)))
 active(take(0,XS:S)) -> mark(nil)
 active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S)
 active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S))
 active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S)))
 active(zip(nil,XS:S)) -> mark(nil)
 active(zip(X:S,nil)) -> mark(nil)
 active(zip(X1:S,X2:S)) -> zip(active(X1:S),X2:S)
 active(zip(X1:S,X2:S)) -> zip(X1:S,active(X2:S))
 active(oddNs) -> mark(incr(pairNs))
 active(pairNs) -> mark(cons(0,incr(oddNs)))
 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
 incr(mark(X:S)) -> mark(incr(X:S))
 incr(ok(X:S)) -> ok(incr(X:S))
 pair(mark(X1:S),X2:S) -> mark(pair(X1:S,X2:S))
 pair(ok(X1:S),ok(X2:S)) -> ok(pair(X1:S,X2:S))
 pair(X1:S,mark(X2:S)) -> mark(pair(X1:S,X2:S))
 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
 proper(incr(X:S)) -> incr(proper(X:S))
 proper(pair(X1:S,X2:S)) -> pair(proper(X1:S),proper(X2:S))
 proper(repItems(X:S)) -> repItems(proper(X:S))
 proper(s(X:S)) -> s(proper(X:S))
 proper(tail(X:S)) -> tail(proper(X:S))
 proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2:S))
 proper(zip(X1:S,X2:S)) -> zip(proper(X1:S),proper(X2:S))
 proper(0) -> ok(0)
 proper(nil) -> ok(nil)
 proper(oddNs) -> ok(oddNs)
 proper(pairNs) -> ok(pairNs)
 repItems(mark(X:S)) -> mark(repItems(X:S))
 repItems(ok(X:S)) -> ok(repItems(X:S))
 s(mark(X:S)) -> mark(s(X:S))
 s(ok(X:S)) -> ok(s(X:S))
 tail(mark(X:S)) -> mark(tail(X:S))
 tail(ok(X:S)) -> ok(tail(X:S))
 take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S))
 take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S))
 take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S))
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))
 zip(mark(X1:S),X2:S) -> mark(zip(X1:S,X2:S))
 zip(ok(X1:S),ok(X2:S)) -> ok(zip(X1:S,X2:S))
 zip(X1:S,mark(X2:S)) -> mark(zip(X1:S,X2:S))
->Projection:
 pi(PROPER) = 1

Problem 1.9: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
 active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S)))
 active(incr(X:S)) -> incr(active(X:S))
 active(pair(X1:S,X2:S)) -> pair(active(X1:S),X2:S)
 active(pair(X1:S,X2:S)) -> pair(X1:S,active(X2:S))
 active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S))))
 active(repItems(nil)) -> mark(nil)
 active(repItems(X:S)) -> repItems(active(X:S))
 active(s(X:S)) -> s(active(X:S))
 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
 active(tail(X:S)) -> tail(active(X:S))
 active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S)))
 active(take(0,XS:S)) -> mark(nil)
 active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S)
 active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S))
 active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S)))
 active(zip(nil,XS:S)) -> mark(nil)
 active(zip(X:S,nil)) -> mark(nil)
 active(zip(X1:S,X2:S)) -> zip(active(X1:S),X2:S)
 active(zip(X1:S,X2:S)) -> zip(X1:S,active(X2:S))
 active(oddNs) -> mark(incr(pairNs))
 active(pairNs) -> mark(cons(0,incr(oddNs)))
 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
 incr(mark(X:S)) -> mark(incr(X:S))
 incr(ok(X:S)) -> ok(incr(X:S))
 pair(mark(X1:S),X2:S) -> mark(pair(X1:S,X2:S))
 pair(ok(X1:S),ok(X2:S)) -> ok(pair(X1:S,X2:S))
 pair(X1:S,mark(X2:S)) -> mark(pair(X1:S,X2:S))
 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
 proper(incr(X:S)) -> incr(proper(X:S))
 proper(pair(X1:S,X2:S)) -> pair(proper(X1:S),proper(X2:S))
 proper(repItems(X:S)) -> repItems(proper(X:S))
 proper(s(X:S)) -> s(proper(X:S))
 proper(tail(X:S)) -> tail(proper(X:S))
 proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2:S))
 proper(zip(X1:S,X2:S)) -> zip(proper(X1:S),proper(X2:S))
 proper(0) -> ok(0)
 proper(nil) -> ok(nil)
 proper(oddNs) -> ok(oddNs)
 proper(pairNs) -> ok(pairNs)
 repItems(mark(X:S)) -> mark(repItems(X:S))
 repItems(ok(X:S)) -> ok(repItems(X:S))
 s(mark(X:S)) -> mark(s(X:S))
 s(ok(X:S)) -> ok(s(X:S))
 tail(mark(X:S)) -> mark(tail(X:S))
 tail(ok(X:S)) -> ok(tail(X:S))
 take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S))
 take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S))
 take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S))
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))
 zip(mark(X1:S),X2:S) -> mark(zip(X1:S,X2:S))
 zip(ok(X1:S),ok(X2:S)) -> ok(zip(X1:S,X2:S))
 zip(X1:S,mark(X2:S)) -> mark(zip(X1:S,X2:S))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.10: 

Subterm Processor:
-> Pairs:
 ACTIVE(cons(X1:S,X2:S)) -> ACTIVE(X1:S)
 ACTIVE(incr(X:S)) -> ACTIVE(X:S)
 ACTIVE(pair(X1:S,X2:S)) -> ACTIVE(X1:S)
 ACTIVE(pair(X1:S,X2:S)) -> ACTIVE(X2:S)
 ACTIVE(repItems(X:S)) -> ACTIVE(X:S)
 ACTIVE(s(X:S)) -> ACTIVE(X:S)
 ACTIVE(tail(X:S)) -> ACTIVE(X:S)
 ACTIVE(take(X1:S,X2:S)) -> ACTIVE(X1:S)
 ACTIVE(take(X1:S,X2:S)) -> ACTIVE(X2:S)
 ACTIVE(zip(X1:S,X2:S)) -> ACTIVE(X1:S)
 ACTIVE(zip(X1:S,X2:S)) -> ACTIVE(X2:S)
-> Rules:
 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
 active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S)))
 active(incr(X:S)) -> incr(active(X:S))
 active(pair(X1:S,X2:S)) -> pair(active(X1:S),X2:S)
 active(pair(X1:S,X2:S)) -> pair(X1:S,active(X2:S))
 active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S))))
 active(repItems(nil)) -> mark(nil)
 active(repItems(X:S)) -> repItems(active(X:S))
 active(s(X:S)) -> s(active(X:S))
 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
 active(tail(X:S)) -> tail(active(X:S))
 active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S)))
 active(take(0,XS:S)) -> mark(nil)
 active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S)
 active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S))
 active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S)))
 active(zip(nil,XS:S)) -> mark(nil)
 active(zip(X:S,nil)) -> mark(nil)
 active(zip(X1:S,X2:S)) -> zip(active(X1:S),X2:S)
 active(zip(X1:S,X2:S)) -> zip(X1:S,active(X2:S))
 active(oddNs) -> mark(incr(pairNs))
 active(pairNs) -> mark(cons(0,incr(oddNs)))
 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
 incr(mark(X:S)) -> mark(incr(X:S))
 incr(ok(X:S)) -> ok(incr(X:S))
 pair(mark(X1:S),X2:S) -> mark(pair(X1:S,X2:S))
 pair(ok(X1:S),ok(X2:S)) -> ok(pair(X1:S,X2:S))
 pair(X1:S,mark(X2:S)) -> mark(pair(X1:S,X2:S))
 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
 proper(incr(X:S)) -> incr(proper(X:S))
 proper(pair(X1:S,X2:S)) -> pair(proper(X1:S),proper(X2:S))
 proper(repItems(X:S)) -> repItems(proper(X:S))
 proper(s(X:S)) -> s(proper(X:S))
 proper(tail(X:S)) -> tail(proper(X:S))
 proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2:S))
 proper(zip(X1:S,X2:S)) -> zip(proper(X1:S),proper(X2:S))
 proper(0) -> ok(0)
 proper(nil) -> ok(nil)
 proper(oddNs) -> ok(oddNs)
 proper(pairNs) -> ok(pairNs)
 repItems(mark(X:S)) -> mark(repItems(X:S))
 repItems(ok(X:S)) -> ok(repItems(X:S))
 s(mark(X:S)) -> mark(s(X:S))
 s(ok(X:S)) -> ok(s(X:S))
 tail(mark(X:S)) -> mark(tail(X:S))
 tail(ok(X:S)) -> ok(tail(X:S))
 take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S))
 take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S))
 take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S))
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))
 zip(mark(X1:S),X2:S) -> mark(zip(X1:S,X2:S))
 zip(ok(X1:S),ok(X2:S)) -> ok(zip(X1:S,X2:S))
 zip(X1:S,mark(X2:S)) -> mark(zip(X1:S,X2:S))
->Projection:
 pi(ACTIVE) = 1

Problem 1.10: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
 active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S)))
 active(incr(X:S)) -> incr(active(X:S))
 active(pair(X1:S,X2:S)) -> pair(active(X1:S),X2:S)
 active(pair(X1:S,X2:S)) -> pair(X1:S,active(X2:S))
 active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S))))
 active(repItems(nil)) -> mark(nil)
 active(repItems(X:S)) -> repItems(active(X:S))
 active(s(X:S)) -> s(active(X:S))
 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
 active(tail(X:S)) -> tail(active(X:S))
 active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S)))
 active(take(0,XS:S)) -> mark(nil)
 active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S)
 active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S))
 active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S)))
 active(zip(nil,XS:S)) -> mark(nil)
 active(zip(X:S,nil)) -> mark(nil)
 active(zip(X1:S,X2:S)) -> zip(active(X1:S),X2:S)
 active(zip(X1:S,X2:S)) -> zip(X1:S,active(X2:S))
 active(oddNs) -> mark(incr(pairNs))
 active(pairNs) -> mark(cons(0,incr(oddNs)))
 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
 incr(mark(X:S)) -> mark(incr(X:S))
 incr(ok(X:S)) -> ok(incr(X:S))
 pair(mark(X1:S),X2:S) -> mark(pair(X1:S,X2:S))
 pair(ok(X1:S),ok(X2:S)) -> ok(pair(X1:S,X2:S))
 pair(X1:S,mark(X2:S)) -> mark(pair(X1:S,X2:S))
 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
 proper(incr(X:S)) -> incr(proper(X:S))
 proper(pair(X1:S,X2:S)) -> pair(proper(X1:S),proper(X2:S))
 proper(repItems(X:S)) -> repItems(proper(X:S))
 proper(s(X:S)) -> s(proper(X:S))
 proper(tail(X:S)) -> tail(proper(X:S))
 proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2:S))
 proper(zip(X1:S,X2:S)) -> zip(proper(X1:S),proper(X2:S))
 proper(0) -> ok(0)
 proper(nil) -> ok(nil)
 proper(oddNs) -> ok(oddNs)
 proper(pairNs) -> ok(pairNs)
 repItems(mark(X:S)) -> mark(repItems(X:S))
 repItems(ok(X:S)) -> ok(repItems(X:S))
 s(mark(X:S)) -> mark(s(X:S))
 s(ok(X:S)) -> ok(s(X:S))
 tail(mark(X:S)) -> mark(tail(X:S))
 tail(ok(X:S)) -> ok(tail(X:S))
 take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S))
 take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S))
 take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S))
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))
 zip(mark(X1:S),X2:S) -> mark(zip(X1:S,X2:S))
 zip(ok(X1:S),ok(X2:S)) -> ok(zip(X1:S,X2:S))
 zip(X1:S,mark(X2:S)) -> mark(zip(X1:S,X2:S))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.11: 

Reduction Pairs Processor:
-> Pairs:
 TOP(mark(X:S)) -> TOP(proper(X:S))
 TOP(ok(X:S)) -> TOP(active(X:S))
-> Rules:
 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
 active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S)))
 active(incr(X:S)) -> incr(active(X:S))
 active(pair(X1:S,X2:S)) -> pair(active(X1:S),X2:S)
 active(pair(X1:S,X2:S)) -> pair(X1:S,active(X2:S))
 active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S))))
 active(repItems(nil)) -> mark(nil)
 active(repItems(X:S)) -> repItems(active(X:S))
 active(s(X:S)) -> s(active(X:S))
 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
 active(tail(X:S)) -> tail(active(X:S))
 active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S)))
 active(take(0,XS:S)) -> mark(nil)
 active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S)
 active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S))
 active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S)))
 active(zip(nil,XS:S)) -> mark(nil)
 active(zip(X:S,nil)) -> mark(nil)
 active(zip(X1:S,X2:S)) -> zip(active(X1:S),X2:S)
 active(zip(X1:S,X2:S)) -> zip(X1:S,active(X2:S))
 active(oddNs) -> mark(incr(pairNs))
 active(pairNs) -> mark(cons(0,incr(oddNs)))
 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
 incr(mark(X:S)) -> mark(incr(X:S))
 incr(ok(X:S)) -> ok(incr(X:S))
 pair(mark(X1:S),X2:S) -> mark(pair(X1:S,X2:S))
 pair(ok(X1:S),ok(X2:S)) -> ok(pair(X1:S,X2:S))
 pair(X1:S,mark(X2:S)) -> mark(pair(X1:S,X2:S))
 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
 proper(incr(X:S)) -> incr(proper(X:S))
 proper(pair(X1:S,X2:S)) -> pair(proper(X1:S),proper(X2:S))
 proper(repItems(X:S)) -> repItems(proper(X:S))
 proper(s(X:S)) -> s(proper(X:S))
 proper(tail(X:S)) -> tail(proper(X:S))
 proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2:S))
 proper(zip(X1:S,X2:S)) -> zip(proper(X1:S),proper(X2:S))
 proper(0) -> ok(0)
 proper(nil) -> ok(nil)
 proper(oddNs) -> ok(oddNs)
 proper(pairNs) -> ok(pairNs)
 repItems(mark(X:S)) -> mark(repItems(X:S))
 repItems(ok(X:S)) -> ok(repItems(X:S))
 s(mark(X:S)) -> mark(s(X:S))
 s(ok(X:S)) -> ok(s(X:S))
 tail(mark(X:S)) -> mark(tail(X:S))
 tail(ok(X:S)) -> ok(tail(X:S))
 take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S))
 take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S))
 take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S))
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))
 zip(mark(X1:S),X2:S) -> mark(zip(X1:S,X2:S))
 zip(ok(X1:S),ok(X2:S)) -> ok(zip(X1:S,X2:S))
 zip(X1:S,mark(X2:S)) -> mark(zip(X1:S,X2:S))
-> Usable rules:
 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
 active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S)))
 active(incr(X:S)) -> incr(active(X:S))
 active(pair(X1:S,X2:S)) -> pair(active(X1:S),X2:S)
 active(pair(X1:S,X2:S)) -> pair(X1:S,active(X2:S))
 active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S))))
 active(repItems(nil)) -> mark(nil)
 active(repItems(X:S)) -> repItems(active(X:S))
 active(s(X:S)) -> s(active(X:S))
 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
 active(tail(X:S)) -> tail(active(X:S))
 active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S)))
 active(take(0,XS:S)) -> mark(nil)
 active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S)
 active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S))
 active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S)))
 active(zip(nil,XS:S)) -> mark(nil)
 active(zip(X:S,nil)) -> mark(nil)
 active(zip(X1:S,X2:S)) -> zip(active(X1:S),X2:S)
 active(zip(X1:S,X2:S)) -> zip(X1:S,active(X2:S))
 active(oddNs) -> mark(incr(pairNs))
 active(pairNs) -> mark(cons(0,incr(oddNs)))
 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
 incr(mark(X:S)) -> mark(incr(X:S))
 incr(ok(X:S)) -> ok(incr(X:S))
 pair(mark(X1:S),X2:S) -> mark(pair(X1:S,X2:S))
 pair(ok(X1:S),ok(X2:S)) -> ok(pair(X1:S,X2:S))
 pair(X1:S,mark(X2:S)) -> mark(pair(X1:S,X2:S))
 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
 proper(incr(X:S)) -> incr(proper(X:S))
 proper(pair(X1:S,X2:S)) -> pair(proper(X1:S),proper(X2:S))
 proper(repItems(X:S)) -> repItems(proper(X:S))
 proper(s(X:S)) -> s(proper(X:S))
 proper(tail(X:S)) -> tail(proper(X:S))
 proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2:S))
 proper(zip(X1:S,X2:S)) -> zip(proper(X1:S),proper(X2:S))
 proper(0) -> ok(0)
 proper(nil) -> ok(nil)
 proper(oddNs) -> ok(oddNs)
 proper(pairNs) -> ok(pairNs)
 repItems(mark(X:S)) -> mark(repItems(X:S))
 repItems(ok(X:S)) -> ok(repItems(X:S))
 s(mark(X:S)) -> mark(s(X:S))
 s(ok(X:S)) -> ok(s(X:S))
 tail(mark(X:S)) -> mark(tail(X:S))
 tail(ok(X:S)) -> ok(tail(X:S))
 take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S))
 take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S))
 take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S))
 zip(mark(X1:S),X2:S) -> mark(zip(X1:S,X2:S))
 zip(ok(X1:S),ok(X2:S)) -> ok(zip(X1:S,X2:S))
 zip(X1:S,mark(X2:S)) -> mark(zip(X1:S,X2:S))
->Interpretation type:
 Linear
->Coefficients:
 All rationals
->Dimension:
 1
->Bound:
 4
->Interpretation:
 
[active](X) = X
[cons](X1,X2) = 3/2.X1 + 1/4.X2
[incr](X) = 4/3.X + 1/2
[pair](X1,X2) = X1 + 4/3.X2
[proper](X) = X
[repItems](X) = 3.X + 1
[s](X) = 4/3.X
[tail](X) = 4.X + 2/3
[take](X1,X2) = 4.X1 + 3/2.X2 + 4
[top](X) = 0
[zip](X1,X2) = X1 + 4/3.X2 + 4/3
[0] = 0
[fSNonEmpty] = 0
[mark](X) = X + 1/3
[nil] = 3
[oddNs] = 4
[ok](X) = X
[pairNs] = 2
[ACTIVE](X) = 0
[CONS](X1,X2) = 0
[INCR](X) = 0
[PAIR](X1,X2) = 0
[PROPER](X) = 0
[REPITEMS](X) = 0
[S](X) = 0
[TAIL](X) = 0
[TAKE](X1,X2) = 0
[TOP](X) = 1/3.X
[ZIP](X1,X2) = 0

Problem 1.11: 

SCC Processor:
-> Pairs:
 TOP(ok(X:S)) -> TOP(active(X:S))
-> Rules:
 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
 active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S)))
 active(incr(X:S)) -> incr(active(X:S))
 active(pair(X1:S,X2:S)) -> pair(active(X1:S),X2:S)
 active(pair(X1:S,X2:S)) -> pair(X1:S,active(X2:S))
 active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S))))
 active(repItems(nil)) -> mark(nil)
 active(repItems(X:S)) -> repItems(active(X:S))
 active(s(X:S)) -> s(active(X:S))
 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
 active(tail(X:S)) -> tail(active(X:S))
 active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S)))
 active(take(0,XS:S)) -> mark(nil)
 active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S)
 active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S))
 active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S)))
 active(zip(nil,XS:S)) -> mark(nil)
 active(zip(X:S,nil)) -> mark(nil)
 active(zip(X1:S,X2:S)) -> zip(active(X1:S),X2:S)
 active(zip(X1:S,X2:S)) -> zip(X1:S,active(X2:S))
 active(oddNs) -> mark(incr(pairNs))
 active(pairNs) -> mark(cons(0,incr(oddNs)))
 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
 incr(mark(X:S)) -> mark(incr(X:S))
 incr(ok(X:S)) -> ok(incr(X:S))
 pair(mark(X1:S),X2:S) -> mark(pair(X1:S,X2:S))
 pair(ok(X1:S),ok(X2:S)) -> ok(pair(X1:S,X2:S))
 pair(X1:S,mark(X2:S)) -> mark(pair(X1:S,X2:S))
 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
 proper(incr(X:S)) -> incr(proper(X:S))
 proper(pair(X1:S,X2:S)) -> pair(proper(X1:S),proper(X2:S))
 proper(repItems(X:S)) -> repItems(proper(X:S))
 proper(s(X:S)) -> s(proper(X:S))
 proper(tail(X:S)) -> tail(proper(X:S))
 proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2:S))
 proper(zip(X1:S,X2:S)) -> zip(proper(X1:S),proper(X2:S))
 proper(0) -> ok(0)
 proper(nil) -> ok(nil)
 proper(oddNs) -> ok(oddNs)
 proper(pairNs) -> ok(pairNs)
 repItems(mark(X:S)) -> mark(repItems(X:S))
 repItems(ok(X:S)) -> ok(repItems(X:S))
 s(mark(X:S)) -> mark(s(X:S))
 s(ok(X:S)) -> ok(s(X:S))
 tail(mark(X:S)) -> mark(tail(X:S))
 tail(ok(X:S)) -> ok(tail(X:S))
 take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S))
 take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S))
 take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S))
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))
 zip(mark(X1:S),X2:S) -> mark(zip(X1:S,X2:S))
 zip(ok(X1:S),ok(X2:S)) -> ok(zip(X1:S,X2:S))
 zip(X1:S,mark(X2:S)) -> mark(zip(X1:S,X2:S))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 TOP(ok(X:S)) -> TOP(active(X:S))
->->-> Rules:
 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
 active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S)))
 active(incr(X:S)) -> incr(active(X:S))
 active(pair(X1:S,X2:S)) -> pair(active(X1:S),X2:S)
 active(pair(X1:S,X2:S)) -> pair(X1:S,active(X2:S))
 active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S))))
 active(repItems(nil)) -> mark(nil)
 active(repItems(X:S)) -> repItems(active(X:S))
 active(s(X:S)) -> s(active(X:S))
 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
 active(tail(X:S)) -> tail(active(X:S))
 active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S)))
 active(take(0,XS:S)) -> mark(nil)
 active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S)
 active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S))
 active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S)))
 active(zip(nil,XS:S)) -> mark(nil)
 active(zip(X:S,nil)) -> mark(nil)
 active(zip(X1:S,X2:S)) -> zip(active(X1:S),X2:S)
 active(zip(X1:S,X2:S)) -> zip(X1:S,active(X2:S))
 active(oddNs) -> mark(incr(pairNs))
 active(pairNs) -> mark(cons(0,incr(oddNs)))
 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
 incr(mark(X:S)) -> mark(incr(X:S))
 incr(ok(X:S)) -> ok(incr(X:S))
 pair(mark(X1:S),X2:S) -> mark(pair(X1:S,X2:S))
 pair(ok(X1:S),ok(X2:S)) -> ok(pair(X1:S,X2:S))
 pair(X1:S,mark(X2:S)) -> mark(pair(X1:S,X2:S))
 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
 proper(incr(X:S)) -> incr(proper(X:S))
 proper(pair(X1:S,X2:S)) -> pair(proper(X1:S),proper(X2:S))
 proper(repItems(X:S)) -> repItems(proper(X:S))
 proper(s(X:S)) -> s(proper(X:S))
 proper(tail(X:S)) -> tail(proper(X:S))
 proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2:S))
 proper(zip(X1:S,X2:S)) -> zip(proper(X1:S),proper(X2:S))
 proper(0) -> ok(0)
 proper(nil) -> ok(nil)
 proper(oddNs) -> ok(oddNs)
 proper(pairNs) -> ok(pairNs)
 repItems(mark(X:S)) -> mark(repItems(X:S))
 repItems(ok(X:S)) -> ok(repItems(X:S))
 s(mark(X:S)) -> mark(s(X:S))
 s(ok(X:S)) -> ok(s(X:S))
 tail(mark(X:S)) -> mark(tail(X:S))
 tail(ok(X:S)) -> ok(tail(X:S))
 take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S))
 take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S))
 take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S))
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))
 zip(mark(X1:S),X2:S) -> mark(zip(X1:S,X2:S))
 zip(ok(X1:S),ok(X2:S)) -> ok(zip(X1:S,X2:S))
 zip(X1:S,mark(X2:S)) -> mark(zip(X1:S,X2:S))

Problem 1.11: 

Reduction Pairs Processor:
-> Pairs:
 TOP(ok(X:S)) -> TOP(active(X:S))
-> Rules:
 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
 active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S)))
 active(incr(X:S)) -> incr(active(X:S))
 active(pair(X1:S,X2:S)) -> pair(active(X1:S),X2:S)
 active(pair(X1:S,X2:S)) -> pair(X1:S,active(X2:S))
 active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S))))
 active(repItems(nil)) -> mark(nil)
 active(repItems(X:S)) -> repItems(active(X:S))
 active(s(X:S)) -> s(active(X:S))
 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
 active(tail(X:S)) -> tail(active(X:S))
 active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S)))
 active(take(0,XS:S)) -> mark(nil)
 active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S)
 active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S))
 active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S)))
 active(zip(nil,XS:S)) -> mark(nil)
 active(zip(X:S,nil)) -> mark(nil)
 active(zip(X1:S,X2:S)) -> zip(active(X1:S),X2:S)
 active(zip(X1:S,X2:S)) -> zip(X1:S,active(X2:S))
 active(oddNs) -> mark(incr(pairNs))
 active(pairNs) -> mark(cons(0,incr(oddNs)))
 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
 incr(mark(X:S)) -> mark(incr(X:S))
 incr(ok(X:S)) -> ok(incr(X:S))
 pair(mark(X1:S),X2:S) -> mark(pair(X1:S,X2:S))
 pair(ok(X1:S),ok(X2:S)) -> ok(pair(X1:S,X2:S))
 pair(X1:S,mark(X2:S)) -> mark(pair(X1:S,X2:S))
 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
 proper(incr(X:S)) -> incr(proper(X:S))
 proper(pair(X1:S,X2:S)) -> pair(proper(X1:S),proper(X2:S))
 proper(repItems(X:S)) -> repItems(proper(X:S))
 proper(s(X:S)) -> s(proper(X:S))
 proper(tail(X:S)) -> tail(proper(X:S))
 proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2:S))
 proper(zip(X1:S,X2:S)) -> zip(proper(X1:S),proper(X2:S))
 proper(0) -> ok(0)
 proper(nil) -> ok(nil)
 proper(oddNs) -> ok(oddNs)
 proper(pairNs) -> ok(pairNs)
 repItems(mark(X:S)) -> mark(repItems(X:S))
 repItems(ok(X:S)) -> ok(repItems(X:S))
 s(mark(X:S)) -> mark(s(X:S))
 s(ok(X:S)) -> ok(s(X:S))
 tail(mark(X:S)) -> mark(tail(X:S))
 tail(ok(X:S)) -> ok(tail(X:S))
 take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S))
 take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S))
 take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S))
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))
 zip(mark(X1:S),X2:S) -> mark(zip(X1:S,X2:S))
 zip(ok(X1:S),ok(X2:S)) -> ok(zip(X1:S,X2:S))
 zip(X1:S,mark(X2:S)) -> mark(zip(X1:S,X2:S))
-> Usable rules:
 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
 active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S)))
 active(incr(X:S)) -> incr(active(X:S))
 active(pair(X1:S,X2:S)) -> pair(active(X1:S),X2:S)
 active(pair(X1:S,X2:S)) -> pair(X1:S,active(X2:S))
 active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S))))
 active(repItems(nil)) -> mark(nil)
 active(repItems(X:S)) -> repItems(active(X:S))
 active(s(X:S)) -> s(active(X:S))
 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
 active(tail(X:S)) -> tail(active(X:S))
 active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S)))
 active(take(0,XS:S)) -> mark(nil)
 active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S)
 active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S))
 active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S)))
 active(zip(nil,XS:S)) -> mark(nil)
 active(zip(X:S,nil)) -> mark(nil)
 active(zip(X1:S,X2:S)) -> zip(active(X1:S),X2:S)
 active(zip(X1:S,X2:S)) -> zip(X1:S,active(X2:S))
 active(oddNs) -> mark(incr(pairNs))
 active(pairNs) -> mark(cons(0,incr(oddNs)))
 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
 incr(mark(X:S)) -> mark(incr(X:S))
 incr(ok(X:S)) -> ok(incr(X:S))
 pair(mark(X1:S),X2:S) -> mark(pair(X1:S,X2:S))
 pair(ok(X1:S),ok(X2:S)) -> ok(pair(X1:S,X2:S))
 pair(X1:S,mark(X2:S)) -> mark(pair(X1:S,X2:S))
 repItems(mark(X:S)) -> mark(repItems(X:S))
 repItems(ok(X:S)) -> ok(repItems(X:S))
 s(mark(X:S)) -> mark(s(X:S))
 s(ok(X:S)) -> ok(s(X:S))
 tail(mark(X:S)) -> mark(tail(X:S))
 tail(ok(X:S)) -> ok(tail(X:S))
 take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S))
 take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S))
 take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S))
 zip(mark(X1:S),X2:S) -> mark(zip(X1:S,X2:S))
 zip(ok(X1:S),ok(X2:S)) -> ok(zip(X1:S,X2:S))
 zip(X1:S,mark(X2:S)) -> mark(zip(X1:S,X2:S))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[active](X) = 2.X
[cons](X1,X2) = 2.X1
[incr](X) = 2.X + 2
[pair](X1,X2) = 2.X1 + 2.X2 + 2
[proper](X) = 0
[repItems](X) = 2.X + 1
[s](X) = 2.X
[tail](X) = 2.X + 2
[take](X1,X2) = 2.X1 + 2.X2 + 2
[top](X) = 0
[zip](X1,X2) = 2.X1 + 2.X2 + 1
[0] = 2
[fSNonEmpty] = 0
[mark](X) = 1
[nil] = 2
[oddNs] = 1
[ok](X) = 2.X + 2
[pairNs] = 2
[ACTIVE](X) = 0
[CONS](X1,X2) = 0
[INCR](X) = 0
[PAIR](X1,X2) = 0
[PROPER](X) = 0
[REPITEMS](X) = 0
[S](X) = 0
[TAIL](X) = 0
[TAKE](X1,X2) = 0
[TOP](X) = 2.X
[ZIP](X1,X2) = 0

Problem 1.11: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
 active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S)))
 active(incr(X:S)) -> incr(active(X:S))
 active(pair(X1:S,X2:S)) -> pair(active(X1:S),X2:S)
 active(pair(X1:S,X2:S)) -> pair(X1:S,active(X2:S))
 active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S))))
 active(repItems(nil)) -> mark(nil)
 active(repItems(X:S)) -> repItems(active(X:S))
 active(s(X:S)) -> s(active(X:S))
 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
 active(tail(X:S)) -> tail(active(X:S))
 active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S)))
 active(take(0,XS:S)) -> mark(nil)
 active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S)
 active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S))
 active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S)))
 active(zip(nil,XS:S)) -> mark(nil)
 active(zip(X:S,nil)) -> mark(nil)
 active(zip(X1:S,X2:S)) -> zip(active(X1:S),X2:S)
 active(zip(X1:S,X2:S)) -> zip(X1:S,active(X2:S))
 active(oddNs) -> mark(incr(pairNs))
 active(pairNs) -> mark(cons(0,incr(oddNs)))
 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
 incr(mark(X:S)) -> mark(incr(X:S))
 incr(ok(X:S)) -> ok(incr(X:S))
 pair(mark(X1:S),X2:S) -> mark(pair(X1:S,X2:S))
 pair(ok(X1:S),ok(X2:S)) -> ok(pair(X1:S,X2:S))
 pair(X1:S,mark(X2:S)) -> mark(pair(X1:S,X2:S))
 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
 proper(incr(X:S)) -> incr(proper(X:S))
 proper(pair(X1:S,X2:S)) -> pair(proper(X1:S),proper(X2:S))
 proper(repItems(X:S)) -> repItems(proper(X:S))
 proper(s(X:S)) -> s(proper(X:S))
 proper(tail(X:S)) -> tail(proper(X:S))
 proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2:S))
 proper(zip(X1:S,X2:S)) -> zip(proper(X1:S),proper(X2:S))
 proper(0) -> ok(0)
 proper(nil) -> ok(nil)
 proper(oddNs) -> ok(oddNs)
 proper(pairNs) -> ok(pairNs)
 repItems(mark(X:S)) -> mark(repItems(X:S))
 repItems(ok(X:S)) -> ok(repItems(X:S))
 s(mark(X:S)) -> mark(s(X:S))
 s(ok(X:S)) -> ok(s(X:S))
 tail(mark(X:S)) -> mark(tail(X:S))
 tail(ok(X:S)) -> ok(tail(X:S))
 take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S))
 take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S))
 take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S))
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))
 zip(mark(X1:S),X2:S) -> mark(zip(X1:S,X2:S))
 zip(ok(X1:S),ok(X2:S)) -> ok(zip(X1:S,X2:S))
 zip(X1:S,mark(X2:S)) -> mark(zip(X1:S,X2:S))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.