2.05/2.14 YES 2.05/2.14 2.05/2.14 Problem 1: 2.05/2.14 2.05/2.14 (VAR v_NonEmpty:S N:S X:S X1:S X2:S XS:S Y:S YS:S) 2.05/2.14 (RULES 2.05/2.14 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 2.05/2.14 active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) 2.05/2.14 active(incr(X:S)) -> incr(active(X:S)) 2.05/2.14 active(pair(X1:S,X2:S)) -> pair(active(X1:S),X2:S) 2.05/2.14 active(pair(X1:S,X2:S)) -> pair(X1:S,active(X2:S)) 2.05/2.14 active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) 2.05/2.14 active(repItems(nil)) -> mark(nil) 2.05/2.14 active(repItems(X:S)) -> repItems(active(X:S)) 2.05/2.14 active(s(X:S)) -> s(active(X:S)) 2.05/2.14 active(tail(cons(X:S,XS:S))) -> mark(XS:S) 2.05/2.14 active(tail(X:S)) -> tail(active(X:S)) 2.05/2.14 active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) 2.05/2.14 active(take(0,XS:S)) -> mark(nil) 2.05/2.14 active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S) 2.05/2.14 active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S)) 2.05/2.14 active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) 2.05/2.14 active(zip(nil,XS:S)) -> mark(nil) 2.05/2.14 active(zip(X:S,nil)) -> mark(nil) 2.05/2.14 active(zip(X1:S,X2:S)) -> zip(active(X1:S),X2:S) 2.05/2.14 active(zip(X1:S,X2:S)) -> zip(X1:S,active(X2:S)) 2.05/2.14 active(oddNs) -> mark(incr(pairNs)) 2.05/2.14 active(pairNs) -> mark(cons(0,incr(oddNs))) 2.05/2.14 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 2.05/2.14 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 2.05/2.14 incr(mark(X:S)) -> mark(incr(X:S)) 2.05/2.14 incr(ok(X:S)) -> ok(incr(X:S)) 2.05/2.14 pair(mark(X1:S),X2:S) -> mark(pair(X1:S,X2:S)) 2.05/2.14 pair(ok(X1:S),ok(X2:S)) -> ok(pair(X1:S,X2:S)) 2.05/2.14 pair(X1:S,mark(X2:S)) -> mark(pair(X1:S,X2:S)) 2.05/2.14 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 2.05/2.14 proper(incr(X:S)) -> incr(proper(X:S)) 2.05/2.14 proper(pair(X1:S,X2:S)) -> pair(proper(X1:S),proper(X2:S)) 2.05/2.14 proper(repItems(X:S)) -> repItems(proper(X:S)) 2.05/2.14 proper(s(X:S)) -> s(proper(X:S)) 2.05/2.14 proper(tail(X:S)) -> tail(proper(X:S)) 2.05/2.14 proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2:S)) 2.05/2.14 proper(zip(X1:S,X2:S)) -> zip(proper(X1:S),proper(X2:S)) 2.05/2.14 proper(0) -> ok(0) 2.05/2.14 proper(nil) -> ok(nil) 2.05/2.14 proper(oddNs) -> ok(oddNs) 2.05/2.14 proper(pairNs) -> ok(pairNs) 2.05/2.14 repItems(mark(X:S)) -> mark(repItems(X:S)) 2.05/2.14 repItems(ok(X:S)) -> ok(repItems(X:S)) 2.05/2.14 s(mark(X:S)) -> mark(s(X:S)) 2.05/2.14 s(ok(X:S)) -> ok(s(X:S)) 2.05/2.14 tail(mark(X:S)) -> mark(tail(X:S)) 2.05/2.14 tail(ok(X:S)) -> ok(tail(X:S)) 2.05/2.14 take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S)) 2.05/2.14 take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S)) 2.05/2.14 take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S)) 2.05/2.14 top(mark(X:S)) -> top(proper(X:S)) 2.05/2.14 top(ok(X:S)) -> top(active(X:S)) 2.05/2.14 zip(mark(X1:S),X2:S) -> mark(zip(X1:S,X2:S)) 2.05/2.14 zip(ok(X1:S),ok(X2:S)) -> ok(zip(X1:S,X2:S)) 2.05/2.14 zip(X1:S,mark(X2:S)) -> mark(zip(X1:S,X2:S)) 2.05/2.14 ) 2.05/2.14 (STRATEGY INNERMOST) 2.05/2.14 2.05/2.14 Problem 1: 2.05/2.14 2.05/2.14 Dependency Pairs Processor: 2.05/2.14 -> Pairs: 2.05/2.14 ACTIVE(cons(X1:S,X2:S)) -> ACTIVE(X1:S) 2.05/2.14 ACTIVE(cons(X1:S,X2:S)) -> CONS(active(X1:S),X2:S) 2.05/2.14 ACTIVE(incr(cons(X:S,XS:S))) -> CONS(s(X:S),incr(XS:S)) 2.05/2.14 ACTIVE(incr(cons(X:S,XS:S))) -> INCR(XS:S) 2.05/2.14 ACTIVE(incr(cons(X:S,XS:S))) -> S(X:S) 2.05/2.14 ACTIVE(incr(X:S)) -> ACTIVE(X:S) 2.05/2.14 ACTIVE(incr(X:S)) -> INCR(active(X:S)) 2.05/2.14 ACTIVE(pair(X1:S,X2:S)) -> ACTIVE(X1:S) 2.05/2.14 ACTIVE(pair(X1:S,X2:S)) -> ACTIVE(X2:S) 2.05/2.14 ACTIVE(pair(X1:S,X2:S)) -> PAIR(active(X1:S),X2:S) 2.05/2.14 ACTIVE(pair(X1:S,X2:S)) -> PAIR(X1:S,active(X2:S)) 2.05/2.14 ACTIVE(repItems(cons(X:S,XS:S))) -> CONS(X:S,cons(X:S,repItems(XS:S))) 2.05/2.14 ACTIVE(repItems(cons(X:S,XS:S))) -> CONS(X:S,repItems(XS:S)) 2.05/2.14 ACTIVE(repItems(cons(X:S,XS:S))) -> REPITEMS(XS:S) 2.05/2.14 ACTIVE(repItems(X:S)) -> ACTIVE(X:S) 2.05/2.14 ACTIVE(repItems(X:S)) -> REPITEMS(active(X:S)) 2.05/2.14 ACTIVE(s(X:S)) -> ACTIVE(X:S) 2.05/2.14 ACTIVE(s(X:S)) -> S(active(X:S)) 2.05/2.14 ACTIVE(tail(X:S)) -> ACTIVE(X:S) 2.05/2.14 ACTIVE(tail(X:S)) -> TAIL(active(X:S)) 2.05/2.14 ACTIVE(take(s(N:S),cons(X:S,XS:S))) -> CONS(X:S,take(N:S,XS:S)) 2.05/2.14 ACTIVE(take(s(N:S),cons(X:S,XS:S))) -> TAKE(N:S,XS:S) 2.05/2.14 ACTIVE(take(X1:S,X2:S)) -> ACTIVE(X1:S) 2.05/2.14 ACTIVE(take(X1:S,X2:S)) -> ACTIVE(X2:S) 2.05/2.14 ACTIVE(take(X1:S,X2:S)) -> TAKE(active(X1:S),X2:S) 2.05/2.14 ACTIVE(take(X1:S,X2:S)) -> TAKE(X1:S,active(X2:S)) 2.05/2.14 ACTIVE(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> CONS(pair(X:S,Y:S),zip(XS:S,YS:S)) 2.05/2.14 ACTIVE(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> PAIR(X:S,Y:S) 2.05/2.14 ACTIVE(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> ZIP(XS:S,YS:S) 2.05/2.14 ACTIVE(zip(X1:S,X2:S)) -> ACTIVE(X1:S) 2.05/2.14 ACTIVE(zip(X1:S,X2:S)) -> ACTIVE(X2:S) 2.05/2.14 ACTIVE(zip(X1:S,X2:S)) -> ZIP(active(X1:S),X2:S) 2.05/2.14 ACTIVE(zip(X1:S,X2:S)) -> ZIP(X1:S,active(X2:S)) 2.05/2.14 CONS(mark(X1:S),X2:S) -> CONS(X1:S,X2:S) 2.05/2.14 CONS(ok(X1:S),ok(X2:S)) -> CONS(X1:S,X2:S) 2.05/2.14 INCR(mark(X:S)) -> INCR(X:S) 2.05/2.14 INCR(ok(X:S)) -> INCR(X:S) 2.05/2.14 PAIR(mark(X1:S),X2:S) -> PAIR(X1:S,X2:S) 2.05/2.14 PAIR(ok(X1:S),ok(X2:S)) -> PAIR(X1:S,X2:S) 2.05/2.14 PAIR(X1:S,mark(X2:S)) -> PAIR(X1:S,X2:S) 2.05/2.14 PROPER(cons(X1:S,X2:S)) -> CONS(proper(X1:S),proper(X2:S)) 2.05/2.14 PROPER(cons(X1:S,X2:S)) -> PROPER(X1:S) 2.05/2.14 PROPER(cons(X1:S,X2:S)) -> PROPER(X2:S) 2.05/2.14 PROPER(incr(X:S)) -> INCR(proper(X:S)) 2.05/2.14 PROPER(incr(X:S)) -> PROPER(X:S) 2.05/2.14 PROPER(pair(X1:S,X2:S)) -> PAIR(proper(X1:S),proper(X2:S)) 2.05/2.14 PROPER(pair(X1:S,X2:S)) -> PROPER(X1:S) 2.05/2.14 PROPER(pair(X1:S,X2:S)) -> PROPER(X2:S) 2.05/2.14 PROPER(repItems(X:S)) -> PROPER(X:S) 2.05/2.14 PROPER(repItems(X:S)) -> REPITEMS(proper(X:S)) 2.05/2.14 PROPER(s(X:S)) -> PROPER(X:S) 2.05/2.14 PROPER(s(X:S)) -> S(proper(X:S)) 2.05/2.14 PROPER(tail(X:S)) -> PROPER(X:S) 2.05/2.14 PROPER(tail(X:S)) -> TAIL(proper(X:S)) 2.05/2.14 PROPER(take(X1:S,X2:S)) -> PROPER(X1:S) 2.05/2.14 PROPER(take(X1:S,X2:S)) -> PROPER(X2:S) 2.05/2.14 PROPER(take(X1:S,X2:S)) -> TAKE(proper(X1:S),proper(X2:S)) 2.05/2.14 PROPER(zip(X1:S,X2:S)) -> PROPER(X1:S) 2.05/2.14 PROPER(zip(X1:S,X2:S)) -> PROPER(X2:S) 2.05/2.14 PROPER(zip(X1:S,X2:S)) -> ZIP(proper(X1:S),proper(X2:S)) 2.05/2.14 REPITEMS(mark(X:S)) -> REPITEMS(X:S) 2.05/2.14 REPITEMS(ok(X:S)) -> REPITEMS(X:S) 2.05/2.14 S(mark(X:S)) -> S(X:S) 2.05/2.14 S(ok(X:S)) -> S(X:S) 2.05/2.14 TAIL(mark(X:S)) -> TAIL(X:S) 2.05/2.14 TAIL(ok(X:S)) -> TAIL(X:S) 2.05/2.14 TAKE(mark(X1:S),X2:S) -> TAKE(X1:S,X2:S) 2.05/2.14 TAKE(ok(X1:S),ok(X2:S)) -> TAKE(X1:S,X2:S) 2.05/2.14 TAKE(X1:S,mark(X2:S)) -> TAKE(X1:S,X2:S) 2.05/2.14 TOP(mark(X:S)) -> PROPER(X:S) 2.05/2.14 TOP(mark(X:S)) -> TOP(proper(X:S)) 2.05/2.14 TOP(ok(X:S)) -> ACTIVE(X:S) 2.05/2.14 TOP(ok(X:S)) -> TOP(active(X:S)) 2.05/2.14 ZIP(mark(X1:S),X2:S) -> ZIP(X1:S,X2:S) 2.05/2.14 ZIP(ok(X1:S),ok(X2:S)) -> ZIP(X1:S,X2:S) 2.05/2.14 ZIP(X1:S,mark(X2:S)) -> ZIP(X1:S,X2:S) 2.05/2.14 -> Rules: 2.05/2.14 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 2.05/2.14 active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) 2.05/2.14 active(incr(X:S)) -> incr(active(X:S)) 2.05/2.14 active(pair(X1:S,X2:S)) -> pair(active(X1:S),X2:S) 2.05/2.14 active(pair(X1:S,X2:S)) -> pair(X1:S,active(X2:S)) 2.05/2.14 active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) 2.05/2.14 active(repItems(nil)) -> mark(nil) 2.05/2.14 active(repItems(X:S)) -> repItems(active(X:S)) 2.05/2.14 active(s(X:S)) -> s(active(X:S)) 2.05/2.14 active(tail(cons(X:S,XS:S))) -> mark(XS:S) 2.05/2.14 active(tail(X:S)) -> tail(active(X:S)) 2.05/2.14 active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) 2.05/2.14 active(take(0,XS:S)) -> mark(nil) 2.05/2.14 active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S) 2.05/2.14 active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S)) 2.05/2.14 active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) 2.05/2.14 active(zip(nil,XS:S)) -> mark(nil) 2.05/2.14 active(zip(X:S,nil)) -> mark(nil) 2.05/2.14 active(zip(X1:S,X2:S)) -> zip(active(X1:S),X2:S) 2.05/2.14 active(zip(X1:S,X2:S)) -> zip(X1:S,active(X2:S)) 2.05/2.14 active(oddNs) -> mark(incr(pairNs)) 2.05/2.14 active(pairNs) -> mark(cons(0,incr(oddNs))) 2.05/2.14 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 2.05/2.14 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 2.05/2.14 incr(mark(X:S)) -> mark(incr(X:S)) 2.05/2.14 incr(ok(X:S)) -> ok(incr(X:S)) 2.05/2.14 pair(mark(X1:S),X2:S) -> mark(pair(X1:S,X2:S)) 2.05/2.14 pair(ok(X1:S),ok(X2:S)) -> ok(pair(X1:S,X2:S)) 2.05/2.14 pair(X1:S,mark(X2:S)) -> mark(pair(X1:S,X2:S)) 2.05/2.14 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 2.05/2.14 proper(incr(X:S)) -> incr(proper(X:S)) 2.05/2.14 proper(pair(X1:S,X2:S)) -> pair(proper(X1:S),proper(X2:S)) 2.05/2.14 proper(repItems(X:S)) -> repItems(proper(X:S)) 2.05/2.14 proper(s(X:S)) -> s(proper(X:S)) 2.05/2.14 proper(tail(X:S)) -> tail(proper(X:S)) 2.05/2.14 proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2:S)) 2.05/2.14 proper(zip(X1:S,X2:S)) -> zip(proper(X1:S),proper(X2:S)) 2.05/2.14 proper(0) -> ok(0) 2.05/2.14 proper(nil) -> ok(nil) 2.05/2.14 proper(oddNs) -> ok(oddNs) 2.05/2.14 proper(pairNs) -> ok(pairNs) 2.05/2.14 repItems(mark(X:S)) -> mark(repItems(X:S)) 2.05/2.14 repItems(ok(X:S)) -> ok(repItems(X:S)) 2.05/2.14 s(mark(X:S)) -> mark(s(X:S)) 2.05/2.14 s(ok(X:S)) -> ok(s(X:S)) 2.05/2.14 tail(mark(X:S)) -> mark(tail(X:S)) 2.05/2.14 tail(ok(X:S)) -> ok(tail(X:S)) 2.05/2.14 take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S)) 2.05/2.14 take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S)) 2.05/2.14 take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S)) 2.05/2.14 top(mark(X:S)) -> top(proper(X:S)) 2.05/2.14 top(ok(X:S)) -> top(active(X:S)) 2.05/2.14 zip(mark(X1:S),X2:S) -> mark(zip(X1:S,X2:S)) 2.05/2.14 zip(ok(X1:S),ok(X2:S)) -> ok(zip(X1:S,X2:S)) 2.05/2.14 zip(X1:S,mark(X2:S)) -> mark(zip(X1:S,X2:S)) 2.05/2.14 2.05/2.14 Problem 1: 2.05/2.14 2.05/2.14 SCC Processor: 2.05/2.14 -> Pairs: 2.05/2.14 ACTIVE(cons(X1:S,X2:S)) -> ACTIVE(X1:S) 2.05/2.14 ACTIVE(cons(X1:S,X2:S)) -> CONS(active(X1:S),X2:S) 2.05/2.14 ACTIVE(incr(cons(X:S,XS:S))) -> CONS(s(X:S),incr(XS:S)) 2.05/2.14 ACTIVE(incr(cons(X:S,XS:S))) -> INCR(XS:S) 2.05/2.14 ACTIVE(incr(cons(X:S,XS:S))) -> S(X:S) 2.05/2.14 ACTIVE(incr(X:S)) -> ACTIVE(X:S) 2.05/2.14 ACTIVE(incr(X:S)) -> INCR(active(X:S)) 2.05/2.14 ACTIVE(pair(X1:S,X2:S)) -> ACTIVE(X1:S) 2.05/2.14 ACTIVE(pair(X1:S,X2:S)) -> ACTIVE(X2:S) 2.05/2.14 ACTIVE(pair(X1:S,X2:S)) -> PAIR(active(X1:S),X2:S) 2.05/2.14 ACTIVE(pair(X1:S,X2:S)) -> PAIR(X1:S,active(X2:S)) 2.05/2.14 ACTIVE(repItems(cons(X:S,XS:S))) -> CONS(X:S,cons(X:S,repItems(XS:S))) 2.05/2.14 ACTIVE(repItems(cons(X:S,XS:S))) -> CONS(X:S,repItems(XS:S)) 2.05/2.14 ACTIVE(repItems(cons(X:S,XS:S))) -> REPITEMS(XS:S) 2.05/2.14 ACTIVE(repItems(X:S)) -> ACTIVE(X:S) 2.05/2.14 ACTIVE(repItems(X:S)) -> REPITEMS(active(X:S)) 2.05/2.14 ACTIVE(s(X:S)) -> ACTIVE(X:S) 2.05/2.14 ACTIVE(s(X:S)) -> S(active(X:S)) 2.05/2.14 ACTIVE(tail(X:S)) -> ACTIVE(X:S) 2.05/2.14 ACTIVE(tail(X:S)) -> TAIL(active(X:S)) 2.05/2.14 ACTIVE(take(s(N:S),cons(X:S,XS:S))) -> CONS(X:S,take(N:S,XS:S)) 2.05/2.14 ACTIVE(take(s(N:S),cons(X:S,XS:S))) -> TAKE(N:S,XS:S) 2.05/2.14 ACTIVE(take(X1:S,X2:S)) -> ACTIVE(X1:S) 2.05/2.14 ACTIVE(take(X1:S,X2:S)) -> ACTIVE(X2:S) 2.05/2.14 ACTIVE(take(X1:S,X2:S)) -> TAKE(active(X1:S),X2:S) 2.05/2.14 ACTIVE(take(X1:S,X2:S)) -> TAKE(X1:S,active(X2:S)) 2.05/2.14 ACTIVE(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> CONS(pair(X:S,Y:S),zip(XS:S,YS:S)) 2.05/2.14 ACTIVE(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> PAIR(X:S,Y:S) 2.05/2.14 ACTIVE(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> ZIP(XS:S,YS:S) 2.05/2.14 ACTIVE(zip(X1:S,X2:S)) -> ACTIVE(X1:S) 2.05/2.14 ACTIVE(zip(X1:S,X2:S)) -> ACTIVE(X2:S) 2.05/2.14 ACTIVE(zip(X1:S,X2:S)) -> ZIP(active(X1:S),X2:S) 2.05/2.14 ACTIVE(zip(X1:S,X2:S)) -> ZIP(X1:S,active(X2:S)) 2.05/2.14 CONS(mark(X1:S),X2:S) -> CONS(X1:S,X2:S) 2.05/2.14 CONS(ok(X1:S),ok(X2:S)) -> CONS(X1:S,X2:S) 2.05/2.14 INCR(mark(X:S)) -> INCR(X:S) 2.05/2.14 INCR(ok(X:S)) -> INCR(X:S) 2.05/2.14 PAIR(mark(X1:S),X2:S) -> PAIR(X1:S,X2:S) 2.05/2.14 PAIR(ok(X1:S),ok(X2:S)) -> PAIR(X1:S,X2:S) 2.05/2.14 PAIR(X1:S,mark(X2:S)) -> PAIR(X1:S,X2:S) 2.05/2.14 PROPER(cons(X1:S,X2:S)) -> CONS(proper(X1:S),proper(X2:S)) 2.05/2.14 PROPER(cons(X1:S,X2:S)) -> PROPER(X1:S) 2.05/2.14 PROPER(cons(X1:S,X2:S)) -> PROPER(X2:S) 2.05/2.14 PROPER(incr(X:S)) -> INCR(proper(X:S)) 2.05/2.14 PROPER(incr(X:S)) -> PROPER(X:S) 2.05/2.14 PROPER(pair(X1:S,X2:S)) -> PAIR(proper(X1:S),proper(X2:S)) 2.05/2.14 PROPER(pair(X1:S,X2:S)) -> PROPER(X1:S) 2.05/2.14 PROPER(pair(X1:S,X2:S)) -> PROPER(X2:S) 2.05/2.14 PROPER(repItems(X:S)) -> PROPER(X:S) 2.05/2.14 PROPER(repItems(X:S)) -> REPITEMS(proper(X:S)) 2.05/2.14 PROPER(s(X:S)) -> PROPER(X:S) 2.05/2.14 PROPER(s(X:S)) -> S(proper(X:S)) 2.05/2.14 PROPER(tail(X:S)) -> PROPER(X:S) 2.05/2.14 PROPER(tail(X:S)) -> TAIL(proper(X:S)) 2.05/2.14 PROPER(take(X1:S,X2:S)) -> PROPER(X1:S) 2.05/2.14 PROPER(take(X1:S,X2:S)) -> PROPER(X2:S) 2.05/2.14 PROPER(take(X1:S,X2:S)) -> TAKE(proper(X1:S),proper(X2:S)) 2.05/2.14 PROPER(zip(X1:S,X2:S)) -> PROPER(X1:S) 2.05/2.14 PROPER(zip(X1:S,X2:S)) -> PROPER(X2:S) 2.05/2.14 PROPER(zip(X1:S,X2:S)) -> ZIP(proper(X1:S),proper(X2:S)) 2.05/2.14 REPITEMS(mark(X:S)) -> REPITEMS(X:S) 2.05/2.14 REPITEMS(ok(X:S)) -> REPITEMS(X:S) 2.05/2.14 S(mark(X:S)) -> S(X:S) 2.05/2.14 S(ok(X:S)) -> S(X:S) 2.05/2.14 TAIL(mark(X:S)) -> TAIL(X:S) 2.05/2.14 TAIL(ok(X:S)) -> TAIL(X:S) 2.05/2.14 TAKE(mark(X1:S),X2:S) -> TAKE(X1:S,X2:S) 2.05/2.14 TAKE(ok(X1:S),ok(X2:S)) -> TAKE(X1:S,X2:S) 2.05/2.14 TAKE(X1:S,mark(X2:S)) -> TAKE(X1:S,X2:S) 2.05/2.14 TOP(mark(X:S)) -> PROPER(X:S) 2.05/2.14 TOP(mark(X:S)) -> TOP(proper(X:S)) 2.05/2.14 TOP(ok(X:S)) -> ACTIVE(X:S) 2.05/2.14 TOP(ok(X:S)) -> TOP(active(X:S)) 2.05/2.14 ZIP(mark(X1:S),X2:S) -> ZIP(X1:S,X2:S) 2.05/2.14 ZIP(ok(X1:S),ok(X2:S)) -> ZIP(X1:S,X2:S) 2.05/2.14 ZIP(X1:S,mark(X2:S)) -> ZIP(X1:S,X2:S) 2.05/2.14 -> Rules: 2.05/2.14 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 2.05/2.14 active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) 2.05/2.14 active(incr(X:S)) -> incr(active(X:S)) 2.05/2.14 active(pair(X1:S,X2:S)) -> pair(active(X1:S),X2:S) 2.05/2.14 active(pair(X1:S,X2:S)) -> pair(X1:S,active(X2:S)) 2.05/2.14 active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) 2.05/2.14 active(repItems(nil)) -> mark(nil) 2.05/2.14 active(repItems(X:S)) -> repItems(active(X:S)) 2.05/2.14 active(s(X:S)) -> s(active(X:S)) 2.05/2.14 active(tail(cons(X:S,XS:S))) -> mark(XS:S) 2.05/2.14 active(tail(X:S)) -> tail(active(X:S)) 2.05/2.14 active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) 2.05/2.14 active(take(0,XS:S)) -> mark(nil) 2.05/2.14 active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S) 2.05/2.14 active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S)) 2.05/2.14 active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) 2.05/2.14 active(zip(nil,XS:S)) -> mark(nil) 2.05/2.14 active(zip(X:S,nil)) -> mark(nil) 2.05/2.14 active(zip(X1:S,X2:S)) -> zip(active(X1:S),X2:S) 2.05/2.14 active(zip(X1:S,X2:S)) -> zip(X1:S,active(X2:S)) 2.05/2.14 active(oddNs) -> mark(incr(pairNs)) 2.05/2.14 active(pairNs) -> mark(cons(0,incr(oddNs))) 2.05/2.14 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 2.05/2.14 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 2.05/2.14 incr(mark(X:S)) -> mark(incr(X:S)) 2.05/2.14 incr(ok(X:S)) -> ok(incr(X:S)) 2.05/2.14 pair(mark(X1:S),X2:S) -> mark(pair(X1:S,X2:S)) 2.05/2.14 pair(ok(X1:S),ok(X2:S)) -> ok(pair(X1:S,X2:S)) 2.05/2.14 pair(X1:S,mark(X2:S)) -> mark(pair(X1:S,X2:S)) 2.05/2.14 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 2.05/2.14 proper(incr(X:S)) -> incr(proper(X:S)) 2.05/2.14 proper(pair(X1:S,X2:S)) -> pair(proper(X1:S),proper(X2:S)) 2.05/2.14 proper(repItems(X:S)) -> repItems(proper(X:S)) 2.05/2.14 proper(s(X:S)) -> s(proper(X:S)) 2.05/2.14 proper(tail(X:S)) -> tail(proper(X:S)) 2.05/2.14 proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2:S)) 2.05/2.14 proper(zip(X1:S,X2:S)) -> zip(proper(X1:S),proper(X2:S)) 2.05/2.14 proper(0) -> ok(0) 2.05/2.14 proper(nil) -> ok(nil) 2.05/2.14 proper(oddNs) -> ok(oddNs) 2.05/2.14 proper(pairNs) -> ok(pairNs) 2.05/2.14 repItems(mark(X:S)) -> mark(repItems(X:S)) 2.05/2.14 repItems(ok(X:S)) -> ok(repItems(X:S)) 2.05/2.14 s(mark(X:S)) -> mark(s(X:S)) 2.05/2.14 s(ok(X:S)) -> ok(s(X:S)) 2.05/2.14 tail(mark(X:S)) -> mark(tail(X:S)) 2.05/2.14 tail(ok(X:S)) -> ok(tail(X:S)) 2.05/2.14 take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S)) 2.05/2.14 take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S)) 2.05/2.14 take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S)) 2.05/2.14 top(mark(X:S)) -> top(proper(X:S)) 2.05/2.14 top(ok(X:S)) -> top(active(X:S)) 2.05/2.14 zip(mark(X1:S),X2:S) -> mark(zip(X1:S,X2:S)) 2.05/2.14 zip(ok(X1:S),ok(X2:S)) -> ok(zip(X1:S,X2:S)) 2.05/2.14 zip(X1:S,mark(X2:S)) -> mark(zip(X1:S,X2:S)) 2.05/2.14 ->Strongly Connected Components: 2.05/2.14 ->->Cycle: 2.05/2.14 ->->-> Pairs: 2.05/2.14 ZIP(mark(X1:S),X2:S) -> ZIP(X1:S,X2:S) 2.05/2.14 ZIP(ok(X1:S),ok(X2:S)) -> ZIP(X1:S,X2:S) 2.05/2.14 ZIP(X1:S,mark(X2:S)) -> ZIP(X1:S,X2:S) 2.05/2.14 ->->-> Rules: 2.05/2.14 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 2.05/2.14 active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) 2.05/2.14 active(incr(X:S)) -> incr(active(X:S)) 2.05/2.14 active(pair(X1:S,X2:S)) -> pair(active(X1:S),X2:S) 2.05/2.14 active(pair(X1:S,X2:S)) -> pair(X1:S,active(X2:S)) 2.05/2.14 active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) 2.05/2.14 active(repItems(nil)) -> mark(nil) 2.05/2.14 active(repItems(X:S)) -> repItems(active(X:S)) 2.05/2.14 active(s(X:S)) -> s(active(X:S)) 2.05/2.14 active(tail(cons(X:S,XS:S))) -> mark(XS:S) 2.05/2.14 active(tail(X:S)) -> tail(active(X:S)) 2.05/2.14 active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) 2.05/2.14 active(take(0,XS:S)) -> mark(nil) 2.05/2.14 active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S) 2.05/2.14 active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S)) 2.05/2.14 active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) 2.05/2.14 active(zip(nil,XS:S)) -> mark(nil) 2.05/2.14 active(zip(X:S,nil)) -> mark(nil) 2.05/2.14 active(zip(X1:S,X2:S)) -> zip(active(X1:S),X2:S) 2.05/2.14 active(zip(X1:S,X2:S)) -> zip(X1:S,active(X2:S)) 2.05/2.14 active(oddNs) -> mark(incr(pairNs)) 2.05/2.14 active(pairNs) -> mark(cons(0,incr(oddNs))) 2.05/2.14 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 2.05/2.14 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 2.05/2.14 incr(mark(X:S)) -> mark(incr(X:S)) 2.05/2.14 incr(ok(X:S)) -> ok(incr(X:S)) 2.05/2.14 pair(mark(X1:S),X2:S) -> mark(pair(X1:S,X2:S)) 2.05/2.14 pair(ok(X1:S),ok(X2:S)) -> ok(pair(X1:S,X2:S)) 2.05/2.14 pair(X1:S,mark(X2:S)) -> mark(pair(X1:S,X2:S)) 2.05/2.14 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 2.05/2.14 proper(incr(X:S)) -> incr(proper(X:S)) 2.05/2.14 proper(pair(X1:S,X2:S)) -> pair(proper(X1:S),proper(X2:S)) 2.05/2.14 proper(repItems(X:S)) -> repItems(proper(X:S)) 2.05/2.14 proper(s(X:S)) -> s(proper(X:S)) 2.05/2.14 proper(tail(X:S)) -> tail(proper(X:S)) 2.05/2.14 proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2:S)) 2.05/2.14 proper(zip(X1:S,X2:S)) -> zip(proper(X1:S),proper(X2:S)) 2.05/2.14 proper(0) -> ok(0) 2.05/2.14 proper(nil) -> ok(nil) 2.05/2.14 proper(oddNs) -> ok(oddNs) 2.05/2.14 proper(pairNs) -> ok(pairNs) 2.05/2.14 repItems(mark(X:S)) -> mark(repItems(X:S)) 2.05/2.14 repItems(ok(X:S)) -> ok(repItems(X:S)) 2.05/2.14 s(mark(X:S)) -> mark(s(X:S)) 2.05/2.14 s(ok(X:S)) -> ok(s(X:S)) 2.05/2.14 tail(mark(X:S)) -> mark(tail(X:S)) 2.05/2.14 tail(ok(X:S)) -> ok(tail(X:S)) 2.05/2.14 take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S)) 2.05/2.14 take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S)) 2.05/2.14 take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S)) 2.05/2.14 top(mark(X:S)) -> top(proper(X:S)) 2.05/2.14 top(ok(X:S)) -> top(active(X:S)) 2.05/2.14 zip(mark(X1:S),X2:S) -> mark(zip(X1:S,X2:S)) 2.05/2.14 zip(ok(X1:S),ok(X2:S)) -> ok(zip(X1:S,X2:S)) 2.05/2.14 zip(X1:S,mark(X2:S)) -> mark(zip(X1:S,X2:S)) 2.05/2.14 ->->Cycle: 2.05/2.14 ->->-> Pairs: 2.05/2.14 TAKE(mark(X1:S),X2:S) -> TAKE(X1:S,X2:S) 2.05/2.14 TAKE(ok(X1:S),ok(X2:S)) -> TAKE(X1:S,X2:S) 2.05/2.14 TAKE(X1:S,mark(X2:S)) -> TAKE(X1:S,X2:S) 2.05/2.14 ->->-> Rules: 2.05/2.14 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 2.05/2.14 active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) 2.05/2.14 active(incr(X:S)) -> incr(active(X:S)) 2.05/2.14 active(pair(X1:S,X2:S)) -> pair(active(X1:S),X2:S) 2.05/2.14 active(pair(X1:S,X2:S)) -> pair(X1:S,active(X2:S)) 2.05/2.14 active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) 2.05/2.14 active(repItems(nil)) -> mark(nil) 2.05/2.14 active(repItems(X:S)) -> repItems(active(X:S)) 2.05/2.14 active(s(X:S)) -> s(active(X:S)) 2.05/2.14 active(tail(cons(X:S,XS:S))) -> mark(XS:S) 2.05/2.14 active(tail(X:S)) -> tail(active(X:S)) 2.05/2.14 active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) 2.05/2.14 active(take(0,XS:S)) -> mark(nil) 2.05/2.14 active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S) 2.05/2.14 active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S)) 2.05/2.14 active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) 2.05/2.14 active(zip(nil,XS:S)) -> mark(nil) 2.05/2.14 active(zip(X:S,nil)) -> mark(nil) 2.05/2.14 active(zip(X1:S,X2:S)) -> zip(active(X1:S),X2:S) 2.05/2.14 active(zip(X1:S,X2:S)) -> zip(X1:S,active(X2:S)) 2.05/2.14 active(oddNs) -> mark(incr(pairNs)) 2.05/2.14 active(pairNs) -> mark(cons(0,incr(oddNs))) 2.05/2.14 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 2.05/2.14 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 2.05/2.14 incr(mark(X:S)) -> mark(incr(X:S)) 2.05/2.14 incr(ok(X:S)) -> ok(incr(X:S)) 2.05/2.14 pair(mark(X1:S),X2:S) -> mark(pair(X1:S,X2:S)) 2.05/2.14 pair(ok(X1:S),ok(X2:S)) -> ok(pair(X1:S,X2:S)) 2.05/2.14 pair(X1:S,mark(X2:S)) -> mark(pair(X1:S,X2:S)) 2.05/2.14 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 2.05/2.14 proper(incr(X:S)) -> incr(proper(X:S)) 2.05/2.14 proper(pair(X1:S,X2:S)) -> pair(proper(X1:S),proper(X2:S)) 2.05/2.14 proper(repItems(X:S)) -> repItems(proper(X:S)) 2.05/2.14 proper(s(X:S)) -> s(proper(X:S)) 2.05/2.14 proper(tail(X:S)) -> tail(proper(X:S)) 2.05/2.14 proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2:S)) 2.05/2.14 proper(zip(X1:S,X2:S)) -> zip(proper(X1:S),proper(X2:S)) 2.05/2.14 proper(0) -> ok(0) 2.05/2.14 proper(nil) -> ok(nil) 2.05/2.14 proper(oddNs) -> ok(oddNs) 2.05/2.14 proper(pairNs) -> ok(pairNs) 2.05/2.14 repItems(mark(X:S)) -> mark(repItems(X:S)) 2.05/2.14 repItems(ok(X:S)) -> ok(repItems(X:S)) 2.05/2.14 s(mark(X:S)) -> mark(s(X:S)) 2.05/2.14 s(ok(X:S)) -> ok(s(X:S)) 2.05/2.14 tail(mark(X:S)) -> mark(tail(X:S)) 2.05/2.14 tail(ok(X:S)) -> ok(tail(X:S)) 2.05/2.14 take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S)) 2.05/2.14 take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S)) 2.05/2.14 take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S)) 2.05/2.14 top(mark(X:S)) -> top(proper(X:S)) 2.05/2.14 top(ok(X:S)) -> top(active(X:S)) 2.05/2.14 zip(mark(X1:S),X2:S) -> mark(zip(X1:S,X2:S)) 2.05/2.14 zip(ok(X1:S),ok(X2:S)) -> ok(zip(X1:S,X2:S)) 2.05/2.14 zip(X1:S,mark(X2:S)) -> mark(zip(X1:S,X2:S)) 2.05/2.14 ->->Cycle: 2.05/2.14 ->->-> Pairs: 2.05/2.14 TAIL(mark(X:S)) -> TAIL(X:S) 2.05/2.14 TAIL(ok(X:S)) -> TAIL(X:S) 2.05/2.14 ->->-> Rules: 2.05/2.14 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 2.05/2.14 active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) 2.05/2.14 active(incr(X:S)) -> incr(active(X:S)) 2.05/2.14 active(pair(X1:S,X2:S)) -> pair(active(X1:S),X2:S) 2.05/2.14 active(pair(X1:S,X2:S)) -> pair(X1:S,active(X2:S)) 2.05/2.14 active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) 2.05/2.14 active(repItems(nil)) -> mark(nil) 2.05/2.14 active(repItems(X:S)) -> repItems(active(X:S)) 2.05/2.14 active(s(X:S)) -> s(active(X:S)) 2.05/2.14 active(tail(cons(X:S,XS:S))) -> mark(XS:S) 2.05/2.14 active(tail(X:S)) -> tail(active(X:S)) 2.05/2.14 active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) 2.05/2.14 active(take(0,XS:S)) -> mark(nil) 2.05/2.14 active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S) 2.05/2.14 active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S)) 2.05/2.14 active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) 2.05/2.14 active(zip(nil,XS:S)) -> mark(nil) 2.05/2.14 active(zip(X:S,nil)) -> mark(nil) 2.05/2.14 active(zip(X1:S,X2:S)) -> zip(active(X1:S),X2:S) 2.05/2.14 active(zip(X1:S,X2:S)) -> zip(X1:S,active(X2:S)) 2.05/2.14 active(oddNs) -> mark(incr(pairNs)) 2.05/2.14 active(pairNs) -> mark(cons(0,incr(oddNs))) 2.05/2.14 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 2.05/2.14 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 2.05/2.14 incr(mark(X:S)) -> mark(incr(X:S)) 2.05/2.14 incr(ok(X:S)) -> ok(incr(X:S)) 2.05/2.14 pair(mark(X1:S),X2:S) -> mark(pair(X1:S,X2:S)) 2.05/2.14 pair(ok(X1:S),ok(X2:S)) -> ok(pair(X1:S,X2:S)) 2.05/2.14 pair(X1:S,mark(X2:S)) -> mark(pair(X1:S,X2:S)) 2.05/2.14 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 2.05/2.14 proper(incr(X:S)) -> incr(proper(X:S)) 2.05/2.14 proper(pair(X1:S,X2:S)) -> pair(proper(X1:S),proper(X2:S)) 2.05/2.14 proper(repItems(X:S)) -> repItems(proper(X:S)) 2.05/2.14 proper(s(X:S)) -> s(proper(X:S)) 2.05/2.14 proper(tail(X:S)) -> tail(proper(X:S)) 2.05/2.14 proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2:S)) 2.05/2.14 proper(zip(X1:S,X2:S)) -> zip(proper(X1:S),proper(X2:S)) 2.05/2.14 proper(0) -> ok(0) 2.05/2.14 proper(nil) -> ok(nil) 2.05/2.14 proper(oddNs) -> ok(oddNs) 2.05/2.14 proper(pairNs) -> ok(pairNs) 2.05/2.14 repItems(mark(X:S)) -> mark(repItems(X:S)) 2.05/2.14 repItems(ok(X:S)) -> ok(repItems(X:S)) 2.05/2.14 s(mark(X:S)) -> mark(s(X:S)) 2.05/2.14 s(ok(X:S)) -> ok(s(X:S)) 2.05/2.14 tail(mark(X:S)) -> mark(tail(X:S)) 2.05/2.14 tail(ok(X:S)) -> ok(tail(X:S)) 2.05/2.14 take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S)) 2.05/2.14 take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S)) 2.05/2.14 take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S)) 2.05/2.14 top(mark(X:S)) -> top(proper(X:S)) 2.05/2.14 top(ok(X:S)) -> top(active(X:S)) 2.05/2.14 zip(mark(X1:S),X2:S) -> mark(zip(X1:S,X2:S)) 2.05/2.14 zip(ok(X1:S),ok(X2:S)) -> ok(zip(X1:S,X2:S)) 2.05/2.14 zip(X1:S,mark(X2:S)) -> mark(zip(X1:S,X2:S)) 2.05/2.14 ->->Cycle: 2.05/2.14 ->->-> Pairs: 2.05/2.14 S(mark(X:S)) -> S(X:S) 2.05/2.14 S(ok(X:S)) -> S(X:S) 2.05/2.14 ->->-> Rules: 2.05/2.14 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 2.05/2.14 active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) 2.05/2.14 active(incr(X:S)) -> incr(active(X:S)) 2.05/2.14 active(pair(X1:S,X2:S)) -> pair(active(X1:S),X2:S) 2.05/2.14 active(pair(X1:S,X2:S)) -> pair(X1:S,active(X2:S)) 2.05/2.14 active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) 2.05/2.14 active(repItems(nil)) -> mark(nil) 2.05/2.14 active(repItems(X:S)) -> repItems(active(X:S)) 2.05/2.14 active(s(X:S)) -> s(active(X:S)) 2.05/2.14 active(tail(cons(X:S,XS:S))) -> mark(XS:S) 2.05/2.14 active(tail(X:S)) -> tail(active(X:S)) 2.05/2.14 active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) 2.05/2.14 active(take(0,XS:S)) -> mark(nil) 2.05/2.14 active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S) 2.05/2.14 active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S)) 2.05/2.14 active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) 2.05/2.14 active(zip(nil,XS:S)) -> mark(nil) 2.05/2.14 active(zip(X:S,nil)) -> mark(nil) 2.05/2.14 active(zip(X1:S,X2:S)) -> zip(active(X1:S),X2:S) 2.05/2.14 active(zip(X1:S,X2:S)) -> zip(X1:S,active(X2:S)) 2.05/2.14 active(oddNs) -> mark(incr(pairNs)) 2.05/2.14 active(pairNs) -> mark(cons(0,incr(oddNs))) 2.05/2.14 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 2.05/2.14 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 2.05/2.14 incr(mark(X:S)) -> mark(incr(X:S)) 2.05/2.14 incr(ok(X:S)) -> ok(incr(X:S)) 2.05/2.14 pair(mark(X1:S),X2:S) -> mark(pair(X1:S,X2:S)) 2.05/2.14 pair(ok(X1:S),ok(X2:S)) -> ok(pair(X1:S,X2:S)) 2.05/2.14 pair(X1:S,mark(X2:S)) -> mark(pair(X1:S,X2:S)) 2.05/2.14 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 2.05/2.14 proper(incr(X:S)) -> incr(proper(X:S)) 2.05/2.14 proper(pair(X1:S,X2:S)) -> pair(proper(X1:S),proper(X2:S)) 2.05/2.14 proper(repItems(X:S)) -> repItems(proper(X:S)) 2.05/2.14 proper(s(X:S)) -> s(proper(X:S)) 2.05/2.14 proper(tail(X:S)) -> tail(proper(X:S)) 2.05/2.14 proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2:S)) 2.05/2.14 proper(zip(X1:S,X2:S)) -> zip(proper(X1:S),proper(X2:S)) 2.05/2.14 proper(0) -> ok(0) 2.05/2.14 proper(nil) -> ok(nil) 2.05/2.14 proper(oddNs) -> ok(oddNs) 2.05/2.14 proper(pairNs) -> ok(pairNs) 2.05/2.14 repItems(mark(X:S)) -> mark(repItems(X:S)) 2.05/2.14 repItems(ok(X:S)) -> ok(repItems(X:S)) 2.05/2.14 s(mark(X:S)) -> mark(s(X:S)) 2.05/2.14 s(ok(X:S)) -> ok(s(X:S)) 2.05/2.14 tail(mark(X:S)) -> mark(tail(X:S)) 2.05/2.14 tail(ok(X:S)) -> ok(tail(X:S)) 2.05/2.14 take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S)) 2.05/2.14 take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S)) 2.05/2.14 take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S)) 2.05/2.14 top(mark(X:S)) -> top(proper(X:S)) 2.05/2.14 top(ok(X:S)) -> top(active(X:S)) 2.05/2.14 zip(mark(X1:S),X2:S) -> mark(zip(X1:S,X2:S)) 2.05/2.14 zip(ok(X1:S),ok(X2:S)) -> ok(zip(X1:S,X2:S)) 2.05/2.14 zip(X1:S,mark(X2:S)) -> mark(zip(X1:S,X2:S)) 2.05/2.14 ->->Cycle: 2.05/2.14 ->->-> Pairs: 2.05/2.14 REPITEMS(mark(X:S)) -> REPITEMS(X:S) 2.05/2.14 REPITEMS(ok(X:S)) -> REPITEMS(X:S) 2.05/2.15 ->->-> Rules: 2.05/2.15 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 2.05/2.15 active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) 2.05/2.15 active(incr(X:S)) -> incr(active(X:S)) 2.05/2.15 active(pair(X1:S,X2:S)) -> pair(active(X1:S),X2:S) 2.05/2.15 active(pair(X1:S,X2:S)) -> pair(X1:S,active(X2:S)) 2.05/2.15 active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) 2.05/2.15 active(repItems(nil)) -> mark(nil) 2.05/2.15 active(repItems(X:S)) -> repItems(active(X:S)) 2.05/2.15 active(s(X:S)) -> s(active(X:S)) 2.05/2.15 active(tail(cons(X:S,XS:S))) -> mark(XS:S) 2.05/2.15 active(tail(X:S)) -> tail(active(X:S)) 2.05/2.15 active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) 2.05/2.15 active(take(0,XS:S)) -> mark(nil) 2.05/2.15 active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S) 2.05/2.15 active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S)) 2.05/2.15 active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) 2.05/2.15 active(zip(nil,XS:S)) -> mark(nil) 2.05/2.15 active(zip(X:S,nil)) -> mark(nil) 2.05/2.15 active(zip(X1:S,X2:S)) -> zip(active(X1:S),X2:S) 2.05/2.15 active(zip(X1:S,X2:S)) -> zip(X1:S,active(X2:S)) 2.05/2.15 active(oddNs) -> mark(incr(pairNs)) 2.05/2.15 active(pairNs) -> mark(cons(0,incr(oddNs))) 2.05/2.15 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 2.05/2.15 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 2.05/2.15 incr(mark(X:S)) -> mark(incr(X:S)) 2.05/2.15 incr(ok(X:S)) -> ok(incr(X:S)) 2.05/2.15 pair(mark(X1:S),X2:S) -> mark(pair(X1:S,X2:S)) 2.05/2.15 pair(ok(X1:S),ok(X2:S)) -> ok(pair(X1:S,X2:S)) 2.05/2.15 pair(X1:S,mark(X2:S)) -> mark(pair(X1:S,X2:S)) 2.05/2.15 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(incr(X:S)) -> incr(proper(X:S)) 2.05/2.15 proper(pair(X1:S,X2:S)) -> pair(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(repItems(X:S)) -> repItems(proper(X:S)) 2.05/2.15 proper(s(X:S)) -> s(proper(X:S)) 2.05/2.15 proper(tail(X:S)) -> tail(proper(X:S)) 2.05/2.15 proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(zip(X1:S,X2:S)) -> zip(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(0) -> ok(0) 2.05/2.15 proper(nil) -> ok(nil) 2.05/2.15 proper(oddNs) -> ok(oddNs) 2.05/2.15 proper(pairNs) -> ok(pairNs) 2.05/2.15 repItems(mark(X:S)) -> mark(repItems(X:S)) 2.05/2.15 repItems(ok(X:S)) -> ok(repItems(X:S)) 2.05/2.15 s(mark(X:S)) -> mark(s(X:S)) 2.05/2.15 s(ok(X:S)) -> ok(s(X:S)) 2.05/2.15 tail(mark(X:S)) -> mark(tail(X:S)) 2.05/2.15 tail(ok(X:S)) -> ok(tail(X:S)) 2.05/2.15 take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S)) 2.05/2.15 take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S)) 2.05/2.15 take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S)) 2.05/2.15 top(mark(X:S)) -> top(proper(X:S)) 2.05/2.15 top(ok(X:S)) -> top(active(X:S)) 2.05/2.15 zip(mark(X1:S),X2:S) -> mark(zip(X1:S,X2:S)) 2.05/2.15 zip(ok(X1:S),ok(X2:S)) -> ok(zip(X1:S,X2:S)) 2.05/2.15 zip(X1:S,mark(X2:S)) -> mark(zip(X1:S,X2:S)) 2.05/2.15 ->->Cycle: 2.05/2.15 ->->-> Pairs: 2.05/2.15 PAIR(mark(X1:S),X2:S) -> PAIR(X1:S,X2:S) 2.05/2.15 PAIR(ok(X1:S),ok(X2:S)) -> PAIR(X1:S,X2:S) 2.05/2.15 PAIR(X1:S,mark(X2:S)) -> PAIR(X1:S,X2:S) 2.05/2.15 ->->-> Rules: 2.05/2.15 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 2.05/2.15 active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) 2.05/2.15 active(incr(X:S)) -> incr(active(X:S)) 2.05/2.15 active(pair(X1:S,X2:S)) -> pair(active(X1:S),X2:S) 2.05/2.15 active(pair(X1:S,X2:S)) -> pair(X1:S,active(X2:S)) 2.05/2.15 active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) 2.05/2.15 active(repItems(nil)) -> mark(nil) 2.05/2.15 active(repItems(X:S)) -> repItems(active(X:S)) 2.05/2.15 active(s(X:S)) -> s(active(X:S)) 2.05/2.15 active(tail(cons(X:S,XS:S))) -> mark(XS:S) 2.05/2.15 active(tail(X:S)) -> tail(active(X:S)) 2.05/2.15 active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) 2.05/2.15 active(take(0,XS:S)) -> mark(nil) 2.05/2.15 active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S) 2.05/2.15 active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S)) 2.05/2.15 active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) 2.05/2.15 active(zip(nil,XS:S)) -> mark(nil) 2.05/2.15 active(zip(X:S,nil)) -> mark(nil) 2.05/2.15 active(zip(X1:S,X2:S)) -> zip(active(X1:S),X2:S) 2.05/2.15 active(zip(X1:S,X2:S)) -> zip(X1:S,active(X2:S)) 2.05/2.15 active(oddNs) -> mark(incr(pairNs)) 2.05/2.15 active(pairNs) -> mark(cons(0,incr(oddNs))) 2.05/2.15 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 2.05/2.15 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 2.05/2.15 incr(mark(X:S)) -> mark(incr(X:S)) 2.05/2.15 incr(ok(X:S)) -> ok(incr(X:S)) 2.05/2.15 pair(mark(X1:S),X2:S) -> mark(pair(X1:S,X2:S)) 2.05/2.15 pair(ok(X1:S),ok(X2:S)) -> ok(pair(X1:S,X2:S)) 2.05/2.15 pair(X1:S,mark(X2:S)) -> mark(pair(X1:S,X2:S)) 2.05/2.15 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(incr(X:S)) -> incr(proper(X:S)) 2.05/2.15 proper(pair(X1:S,X2:S)) -> pair(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(repItems(X:S)) -> repItems(proper(X:S)) 2.05/2.15 proper(s(X:S)) -> s(proper(X:S)) 2.05/2.15 proper(tail(X:S)) -> tail(proper(X:S)) 2.05/2.15 proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(zip(X1:S,X2:S)) -> zip(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(0) -> ok(0) 2.05/2.15 proper(nil) -> ok(nil) 2.05/2.15 proper(oddNs) -> ok(oddNs) 2.05/2.15 proper(pairNs) -> ok(pairNs) 2.05/2.15 repItems(mark(X:S)) -> mark(repItems(X:S)) 2.05/2.15 repItems(ok(X:S)) -> ok(repItems(X:S)) 2.05/2.15 s(mark(X:S)) -> mark(s(X:S)) 2.05/2.15 s(ok(X:S)) -> ok(s(X:S)) 2.05/2.15 tail(mark(X:S)) -> mark(tail(X:S)) 2.05/2.15 tail(ok(X:S)) -> ok(tail(X:S)) 2.05/2.15 take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S)) 2.05/2.15 take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S)) 2.05/2.15 take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S)) 2.05/2.15 top(mark(X:S)) -> top(proper(X:S)) 2.05/2.15 top(ok(X:S)) -> top(active(X:S)) 2.05/2.15 zip(mark(X1:S),X2:S) -> mark(zip(X1:S,X2:S)) 2.05/2.15 zip(ok(X1:S),ok(X2:S)) -> ok(zip(X1:S,X2:S)) 2.05/2.15 zip(X1:S,mark(X2:S)) -> mark(zip(X1:S,X2:S)) 2.05/2.15 ->->Cycle: 2.05/2.15 ->->-> Pairs: 2.05/2.15 INCR(mark(X:S)) -> INCR(X:S) 2.05/2.15 INCR(ok(X:S)) -> INCR(X:S) 2.05/2.15 ->->-> Rules: 2.05/2.15 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 2.05/2.15 active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) 2.05/2.15 active(incr(X:S)) -> incr(active(X:S)) 2.05/2.15 active(pair(X1:S,X2:S)) -> pair(active(X1:S),X2:S) 2.05/2.15 active(pair(X1:S,X2:S)) -> pair(X1:S,active(X2:S)) 2.05/2.15 active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) 2.05/2.15 active(repItems(nil)) -> mark(nil) 2.05/2.15 active(repItems(X:S)) -> repItems(active(X:S)) 2.05/2.15 active(s(X:S)) -> s(active(X:S)) 2.05/2.15 active(tail(cons(X:S,XS:S))) -> mark(XS:S) 2.05/2.15 active(tail(X:S)) -> tail(active(X:S)) 2.05/2.15 active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) 2.05/2.15 active(take(0,XS:S)) -> mark(nil) 2.05/2.15 active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S) 2.05/2.15 active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S)) 2.05/2.15 active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) 2.05/2.15 active(zip(nil,XS:S)) -> mark(nil) 2.05/2.15 active(zip(X:S,nil)) -> mark(nil) 2.05/2.15 active(zip(X1:S,X2:S)) -> zip(active(X1:S),X2:S) 2.05/2.15 active(zip(X1:S,X2:S)) -> zip(X1:S,active(X2:S)) 2.05/2.15 active(oddNs) -> mark(incr(pairNs)) 2.05/2.15 active(pairNs) -> mark(cons(0,incr(oddNs))) 2.05/2.15 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 2.05/2.15 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 2.05/2.15 incr(mark(X:S)) -> mark(incr(X:S)) 2.05/2.15 incr(ok(X:S)) -> ok(incr(X:S)) 2.05/2.15 pair(mark(X1:S),X2:S) -> mark(pair(X1:S,X2:S)) 2.05/2.15 pair(ok(X1:S),ok(X2:S)) -> ok(pair(X1:S,X2:S)) 2.05/2.15 pair(X1:S,mark(X2:S)) -> mark(pair(X1:S,X2:S)) 2.05/2.15 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(incr(X:S)) -> incr(proper(X:S)) 2.05/2.15 proper(pair(X1:S,X2:S)) -> pair(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(repItems(X:S)) -> repItems(proper(X:S)) 2.05/2.15 proper(s(X:S)) -> s(proper(X:S)) 2.05/2.15 proper(tail(X:S)) -> tail(proper(X:S)) 2.05/2.15 proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(zip(X1:S,X2:S)) -> zip(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(0) -> ok(0) 2.05/2.15 proper(nil) -> ok(nil) 2.05/2.15 proper(oddNs) -> ok(oddNs) 2.05/2.15 proper(pairNs) -> ok(pairNs) 2.05/2.15 repItems(mark(X:S)) -> mark(repItems(X:S)) 2.05/2.15 repItems(ok(X:S)) -> ok(repItems(X:S)) 2.05/2.15 s(mark(X:S)) -> mark(s(X:S)) 2.05/2.15 s(ok(X:S)) -> ok(s(X:S)) 2.05/2.15 tail(mark(X:S)) -> mark(tail(X:S)) 2.05/2.15 tail(ok(X:S)) -> ok(tail(X:S)) 2.05/2.15 take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S)) 2.05/2.15 take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S)) 2.05/2.15 take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S)) 2.05/2.15 top(mark(X:S)) -> top(proper(X:S)) 2.05/2.15 top(ok(X:S)) -> top(active(X:S)) 2.05/2.15 zip(mark(X1:S),X2:S) -> mark(zip(X1:S,X2:S)) 2.05/2.15 zip(ok(X1:S),ok(X2:S)) -> ok(zip(X1:S,X2:S)) 2.05/2.15 zip(X1:S,mark(X2:S)) -> mark(zip(X1:S,X2:S)) 2.05/2.15 ->->Cycle: 2.05/2.15 ->->-> Pairs: 2.05/2.15 CONS(mark(X1:S),X2:S) -> CONS(X1:S,X2:S) 2.05/2.15 CONS(ok(X1:S),ok(X2:S)) -> CONS(X1:S,X2:S) 2.05/2.15 ->->-> Rules: 2.05/2.15 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 2.05/2.15 active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) 2.05/2.15 active(incr(X:S)) -> incr(active(X:S)) 2.05/2.15 active(pair(X1:S,X2:S)) -> pair(active(X1:S),X2:S) 2.05/2.15 active(pair(X1:S,X2:S)) -> pair(X1:S,active(X2:S)) 2.05/2.15 active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) 2.05/2.15 active(repItems(nil)) -> mark(nil) 2.05/2.15 active(repItems(X:S)) -> repItems(active(X:S)) 2.05/2.15 active(s(X:S)) -> s(active(X:S)) 2.05/2.15 active(tail(cons(X:S,XS:S))) -> mark(XS:S) 2.05/2.15 active(tail(X:S)) -> tail(active(X:S)) 2.05/2.15 active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) 2.05/2.15 active(take(0,XS:S)) -> mark(nil) 2.05/2.15 active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S) 2.05/2.15 active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S)) 2.05/2.15 active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) 2.05/2.15 active(zip(nil,XS:S)) -> mark(nil) 2.05/2.15 active(zip(X:S,nil)) -> mark(nil) 2.05/2.15 active(zip(X1:S,X2:S)) -> zip(active(X1:S),X2:S) 2.05/2.15 active(zip(X1:S,X2:S)) -> zip(X1:S,active(X2:S)) 2.05/2.15 active(oddNs) -> mark(incr(pairNs)) 2.05/2.15 active(pairNs) -> mark(cons(0,incr(oddNs))) 2.05/2.15 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 2.05/2.15 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 2.05/2.15 incr(mark(X:S)) -> mark(incr(X:S)) 2.05/2.15 incr(ok(X:S)) -> ok(incr(X:S)) 2.05/2.15 pair(mark(X1:S),X2:S) -> mark(pair(X1:S,X2:S)) 2.05/2.15 pair(ok(X1:S),ok(X2:S)) -> ok(pair(X1:S,X2:S)) 2.05/2.15 pair(X1:S,mark(X2:S)) -> mark(pair(X1:S,X2:S)) 2.05/2.15 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(incr(X:S)) -> incr(proper(X:S)) 2.05/2.15 proper(pair(X1:S,X2:S)) -> pair(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(repItems(X:S)) -> repItems(proper(X:S)) 2.05/2.15 proper(s(X:S)) -> s(proper(X:S)) 2.05/2.15 proper(tail(X:S)) -> tail(proper(X:S)) 2.05/2.15 proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(zip(X1:S,X2:S)) -> zip(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(0) -> ok(0) 2.05/2.15 proper(nil) -> ok(nil) 2.05/2.15 proper(oddNs) -> ok(oddNs) 2.05/2.15 proper(pairNs) -> ok(pairNs) 2.05/2.15 repItems(mark(X:S)) -> mark(repItems(X:S)) 2.05/2.15 repItems(ok(X:S)) -> ok(repItems(X:S)) 2.05/2.15 s(mark(X:S)) -> mark(s(X:S)) 2.05/2.15 s(ok(X:S)) -> ok(s(X:S)) 2.05/2.15 tail(mark(X:S)) -> mark(tail(X:S)) 2.05/2.15 tail(ok(X:S)) -> ok(tail(X:S)) 2.05/2.15 take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S)) 2.05/2.15 take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S)) 2.05/2.15 take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S)) 2.05/2.15 top(mark(X:S)) -> top(proper(X:S)) 2.05/2.15 top(ok(X:S)) -> top(active(X:S)) 2.05/2.15 zip(mark(X1:S),X2:S) -> mark(zip(X1:S,X2:S)) 2.05/2.15 zip(ok(X1:S),ok(X2:S)) -> ok(zip(X1:S,X2:S)) 2.05/2.15 zip(X1:S,mark(X2:S)) -> mark(zip(X1:S,X2:S)) 2.05/2.15 ->->Cycle: 2.05/2.15 ->->-> Pairs: 2.05/2.15 PROPER(cons(X1:S,X2:S)) -> PROPER(X1:S) 2.05/2.15 PROPER(cons(X1:S,X2:S)) -> PROPER(X2:S) 2.05/2.15 PROPER(incr(X:S)) -> PROPER(X:S) 2.05/2.15 PROPER(pair(X1:S,X2:S)) -> PROPER(X1:S) 2.05/2.15 PROPER(pair(X1:S,X2:S)) -> PROPER(X2:S) 2.05/2.15 PROPER(repItems(X:S)) -> PROPER(X:S) 2.05/2.15 PROPER(s(X:S)) -> PROPER(X:S) 2.05/2.15 PROPER(tail(X:S)) -> PROPER(X:S) 2.05/2.15 PROPER(take(X1:S,X2:S)) -> PROPER(X1:S) 2.05/2.15 PROPER(take(X1:S,X2:S)) -> PROPER(X2:S) 2.05/2.15 PROPER(zip(X1:S,X2:S)) -> PROPER(X1:S) 2.05/2.15 PROPER(zip(X1:S,X2:S)) -> PROPER(X2:S) 2.05/2.15 ->->-> Rules: 2.05/2.15 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 2.05/2.15 active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) 2.05/2.15 active(incr(X:S)) -> incr(active(X:S)) 2.05/2.15 active(pair(X1:S,X2:S)) -> pair(active(X1:S),X2:S) 2.05/2.15 active(pair(X1:S,X2:S)) -> pair(X1:S,active(X2:S)) 2.05/2.15 active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) 2.05/2.15 active(repItems(nil)) -> mark(nil) 2.05/2.15 active(repItems(X:S)) -> repItems(active(X:S)) 2.05/2.15 active(s(X:S)) -> s(active(X:S)) 2.05/2.15 active(tail(cons(X:S,XS:S))) -> mark(XS:S) 2.05/2.15 active(tail(X:S)) -> tail(active(X:S)) 2.05/2.15 active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) 2.05/2.15 active(take(0,XS:S)) -> mark(nil) 2.05/2.15 active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S) 2.05/2.15 active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S)) 2.05/2.15 active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) 2.05/2.15 active(zip(nil,XS:S)) -> mark(nil) 2.05/2.15 active(zip(X:S,nil)) -> mark(nil) 2.05/2.15 active(zip(X1:S,X2:S)) -> zip(active(X1:S),X2:S) 2.05/2.15 active(zip(X1:S,X2:S)) -> zip(X1:S,active(X2:S)) 2.05/2.15 active(oddNs) -> mark(incr(pairNs)) 2.05/2.15 active(pairNs) -> mark(cons(0,incr(oddNs))) 2.05/2.15 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 2.05/2.15 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 2.05/2.15 incr(mark(X:S)) -> mark(incr(X:S)) 2.05/2.15 incr(ok(X:S)) -> ok(incr(X:S)) 2.05/2.15 pair(mark(X1:S),X2:S) -> mark(pair(X1:S,X2:S)) 2.05/2.15 pair(ok(X1:S),ok(X2:S)) -> ok(pair(X1:S,X2:S)) 2.05/2.15 pair(X1:S,mark(X2:S)) -> mark(pair(X1:S,X2:S)) 2.05/2.15 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(incr(X:S)) -> incr(proper(X:S)) 2.05/2.15 proper(pair(X1:S,X2:S)) -> pair(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(repItems(X:S)) -> repItems(proper(X:S)) 2.05/2.15 proper(s(X:S)) -> s(proper(X:S)) 2.05/2.15 proper(tail(X:S)) -> tail(proper(X:S)) 2.05/2.15 proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(zip(X1:S,X2:S)) -> zip(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(0) -> ok(0) 2.05/2.15 proper(nil) -> ok(nil) 2.05/2.15 proper(oddNs) -> ok(oddNs) 2.05/2.15 proper(pairNs) -> ok(pairNs) 2.05/2.15 repItems(mark(X:S)) -> mark(repItems(X:S)) 2.05/2.15 repItems(ok(X:S)) -> ok(repItems(X:S)) 2.05/2.15 s(mark(X:S)) -> mark(s(X:S)) 2.05/2.15 s(ok(X:S)) -> ok(s(X:S)) 2.05/2.15 tail(mark(X:S)) -> mark(tail(X:S)) 2.05/2.15 tail(ok(X:S)) -> ok(tail(X:S)) 2.05/2.15 take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S)) 2.05/2.15 take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S)) 2.05/2.15 take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S)) 2.05/2.15 top(mark(X:S)) -> top(proper(X:S)) 2.05/2.15 top(ok(X:S)) -> top(active(X:S)) 2.05/2.15 zip(mark(X1:S),X2:S) -> mark(zip(X1:S,X2:S)) 2.05/2.15 zip(ok(X1:S),ok(X2:S)) -> ok(zip(X1:S,X2:S)) 2.05/2.15 zip(X1:S,mark(X2:S)) -> mark(zip(X1:S,X2:S)) 2.05/2.15 ->->Cycle: 2.05/2.15 ->->-> Pairs: 2.05/2.15 ACTIVE(cons(X1:S,X2:S)) -> ACTIVE(X1:S) 2.05/2.15 ACTIVE(incr(X:S)) -> ACTIVE(X:S) 2.05/2.15 ACTIVE(pair(X1:S,X2:S)) -> ACTIVE(X1:S) 2.05/2.15 ACTIVE(pair(X1:S,X2:S)) -> ACTIVE(X2:S) 2.05/2.15 ACTIVE(repItems(X:S)) -> ACTIVE(X:S) 2.05/2.15 ACTIVE(s(X:S)) -> ACTIVE(X:S) 2.05/2.15 ACTIVE(tail(X:S)) -> ACTIVE(X:S) 2.05/2.15 ACTIVE(take(X1:S,X2:S)) -> ACTIVE(X1:S) 2.05/2.15 ACTIVE(take(X1:S,X2:S)) -> ACTIVE(X2:S) 2.05/2.15 ACTIVE(zip(X1:S,X2:S)) -> ACTIVE(X1:S) 2.05/2.15 ACTIVE(zip(X1:S,X2:S)) -> ACTIVE(X2:S) 2.05/2.15 ->->-> Rules: 2.05/2.15 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 2.05/2.15 active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) 2.05/2.15 active(incr(X:S)) -> incr(active(X:S)) 2.05/2.15 active(pair(X1:S,X2:S)) -> pair(active(X1:S),X2:S) 2.05/2.15 active(pair(X1:S,X2:S)) -> pair(X1:S,active(X2:S)) 2.05/2.15 active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) 2.05/2.15 active(repItems(nil)) -> mark(nil) 2.05/2.15 active(repItems(X:S)) -> repItems(active(X:S)) 2.05/2.15 active(s(X:S)) -> s(active(X:S)) 2.05/2.15 active(tail(cons(X:S,XS:S))) -> mark(XS:S) 2.05/2.15 active(tail(X:S)) -> tail(active(X:S)) 2.05/2.15 active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) 2.05/2.15 active(take(0,XS:S)) -> mark(nil) 2.05/2.15 active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S) 2.05/2.15 active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S)) 2.05/2.15 active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) 2.05/2.15 active(zip(nil,XS:S)) -> mark(nil) 2.05/2.15 active(zip(X:S,nil)) -> mark(nil) 2.05/2.15 active(zip(X1:S,X2:S)) -> zip(active(X1:S),X2:S) 2.05/2.15 active(zip(X1:S,X2:S)) -> zip(X1:S,active(X2:S)) 2.05/2.15 active(oddNs) -> mark(incr(pairNs)) 2.05/2.15 active(pairNs) -> mark(cons(0,incr(oddNs))) 2.05/2.15 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 2.05/2.15 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 2.05/2.15 incr(mark(X:S)) -> mark(incr(X:S)) 2.05/2.15 incr(ok(X:S)) -> ok(incr(X:S)) 2.05/2.15 pair(mark(X1:S),X2:S) -> mark(pair(X1:S,X2:S)) 2.05/2.15 pair(ok(X1:S),ok(X2:S)) -> ok(pair(X1:S,X2:S)) 2.05/2.15 pair(X1:S,mark(X2:S)) -> mark(pair(X1:S,X2:S)) 2.05/2.15 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(incr(X:S)) -> incr(proper(X:S)) 2.05/2.15 proper(pair(X1:S,X2:S)) -> pair(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(repItems(X:S)) -> repItems(proper(X:S)) 2.05/2.15 proper(s(X:S)) -> s(proper(X:S)) 2.05/2.15 proper(tail(X:S)) -> tail(proper(X:S)) 2.05/2.15 proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(zip(X1:S,X2:S)) -> zip(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(0) -> ok(0) 2.05/2.15 proper(nil) -> ok(nil) 2.05/2.15 proper(oddNs) -> ok(oddNs) 2.05/2.15 proper(pairNs) -> ok(pairNs) 2.05/2.15 repItems(mark(X:S)) -> mark(repItems(X:S)) 2.05/2.15 repItems(ok(X:S)) -> ok(repItems(X:S)) 2.05/2.15 s(mark(X:S)) -> mark(s(X:S)) 2.05/2.15 s(ok(X:S)) -> ok(s(X:S)) 2.05/2.15 tail(mark(X:S)) -> mark(tail(X:S)) 2.05/2.15 tail(ok(X:S)) -> ok(tail(X:S)) 2.05/2.15 take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S)) 2.05/2.15 take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S)) 2.05/2.15 take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S)) 2.05/2.15 top(mark(X:S)) -> top(proper(X:S)) 2.05/2.15 top(ok(X:S)) -> top(active(X:S)) 2.05/2.15 zip(mark(X1:S),X2:S) -> mark(zip(X1:S,X2:S)) 2.05/2.15 zip(ok(X1:S),ok(X2:S)) -> ok(zip(X1:S,X2:S)) 2.05/2.15 zip(X1:S,mark(X2:S)) -> mark(zip(X1:S,X2:S)) 2.05/2.15 ->->Cycle: 2.05/2.15 ->->-> Pairs: 2.05/2.15 TOP(mark(X:S)) -> TOP(proper(X:S)) 2.05/2.15 TOP(ok(X:S)) -> TOP(active(X:S)) 2.05/2.15 ->->-> Rules: 2.05/2.15 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 2.05/2.15 active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) 2.05/2.15 active(incr(X:S)) -> incr(active(X:S)) 2.05/2.15 active(pair(X1:S,X2:S)) -> pair(active(X1:S),X2:S) 2.05/2.15 active(pair(X1:S,X2:S)) -> pair(X1:S,active(X2:S)) 2.05/2.15 active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) 2.05/2.15 active(repItems(nil)) -> mark(nil) 2.05/2.15 active(repItems(X:S)) -> repItems(active(X:S)) 2.05/2.15 active(s(X:S)) -> s(active(X:S)) 2.05/2.15 active(tail(cons(X:S,XS:S))) -> mark(XS:S) 2.05/2.15 active(tail(X:S)) -> tail(active(X:S)) 2.05/2.15 active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) 2.05/2.15 active(take(0,XS:S)) -> mark(nil) 2.05/2.15 active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S) 2.05/2.15 active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S)) 2.05/2.15 active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) 2.05/2.15 active(zip(nil,XS:S)) -> mark(nil) 2.05/2.15 active(zip(X:S,nil)) -> mark(nil) 2.05/2.15 active(zip(X1:S,X2:S)) -> zip(active(X1:S),X2:S) 2.05/2.15 active(zip(X1:S,X2:S)) -> zip(X1:S,active(X2:S)) 2.05/2.15 active(oddNs) -> mark(incr(pairNs)) 2.05/2.15 active(pairNs) -> mark(cons(0,incr(oddNs))) 2.05/2.15 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 2.05/2.15 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 2.05/2.15 incr(mark(X:S)) -> mark(incr(X:S)) 2.05/2.15 incr(ok(X:S)) -> ok(incr(X:S)) 2.05/2.15 pair(mark(X1:S),X2:S) -> mark(pair(X1:S,X2:S)) 2.05/2.15 pair(ok(X1:S),ok(X2:S)) -> ok(pair(X1:S,X2:S)) 2.05/2.15 pair(X1:S,mark(X2:S)) -> mark(pair(X1:S,X2:S)) 2.05/2.15 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(incr(X:S)) -> incr(proper(X:S)) 2.05/2.15 proper(pair(X1:S,X2:S)) -> pair(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(repItems(X:S)) -> repItems(proper(X:S)) 2.05/2.15 proper(s(X:S)) -> s(proper(X:S)) 2.05/2.15 proper(tail(X:S)) -> tail(proper(X:S)) 2.05/2.15 proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(zip(X1:S,X2:S)) -> zip(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(0) -> ok(0) 2.05/2.15 proper(nil) -> ok(nil) 2.05/2.15 proper(oddNs) -> ok(oddNs) 2.05/2.15 proper(pairNs) -> ok(pairNs) 2.05/2.15 repItems(mark(X:S)) -> mark(repItems(X:S)) 2.05/2.15 repItems(ok(X:S)) -> ok(repItems(X:S)) 2.05/2.15 s(mark(X:S)) -> mark(s(X:S)) 2.05/2.15 s(ok(X:S)) -> ok(s(X:S)) 2.05/2.15 tail(mark(X:S)) -> mark(tail(X:S)) 2.05/2.15 tail(ok(X:S)) -> ok(tail(X:S)) 2.05/2.15 take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S)) 2.05/2.15 take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S)) 2.05/2.15 take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S)) 2.05/2.15 top(mark(X:S)) -> top(proper(X:S)) 2.05/2.15 top(ok(X:S)) -> top(active(X:S)) 2.05/2.15 zip(mark(X1:S),X2:S) -> mark(zip(X1:S,X2:S)) 2.05/2.15 zip(ok(X1:S),ok(X2:S)) -> ok(zip(X1:S,X2:S)) 2.05/2.15 zip(X1:S,mark(X2:S)) -> mark(zip(X1:S,X2:S)) 2.05/2.15 2.05/2.15 2.05/2.15 The problem is decomposed in 11 subproblems. 2.05/2.15 2.05/2.15 Problem 1.1: 2.05/2.15 2.05/2.15 Subterm Processor: 2.05/2.15 -> Pairs: 2.05/2.15 ZIP(mark(X1:S),X2:S) -> ZIP(X1:S,X2:S) 2.05/2.15 ZIP(ok(X1:S),ok(X2:S)) -> ZIP(X1:S,X2:S) 2.05/2.15 ZIP(X1:S,mark(X2:S)) -> ZIP(X1:S,X2:S) 2.05/2.15 -> Rules: 2.05/2.15 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 2.05/2.15 active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) 2.05/2.15 active(incr(X:S)) -> incr(active(X:S)) 2.05/2.15 active(pair(X1:S,X2:S)) -> pair(active(X1:S),X2:S) 2.05/2.15 active(pair(X1:S,X2:S)) -> pair(X1:S,active(X2:S)) 2.05/2.15 active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) 2.05/2.15 active(repItems(nil)) -> mark(nil) 2.05/2.15 active(repItems(X:S)) -> repItems(active(X:S)) 2.05/2.15 active(s(X:S)) -> s(active(X:S)) 2.05/2.15 active(tail(cons(X:S,XS:S))) -> mark(XS:S) 2.05/2.15 active(tail(X:S)) -> tail(active(X:S)) 2.05/2.15 active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) 2.05/2.15 active(take(0,XS:S)) -> mark(nil) 2.05/2.15 active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S) 2.05/2.15 active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S)) 2.05/2.15 active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) 2.05/2.15 active(zip(nil,XS:S)) -> mark(nil) 2.05/2.15 active(zip(X:S,nil)) -> mark(nil) 2.05/2.15 active(zip(X1:S,X2:S)) -> zip(active(X1:S),X2:S) 2.05/2.15 active(zip(X1:S,X2:S)) -> zip(X1:S,active(X2:S)) 2.05/2.15 active(oddNs) -> mark(incr(pairNs)) 2.05/2.15 active(pairNs) -> mark(cons(0,incr(oddNs))) 2.05/2.15 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 2.05/2.15 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 2.05/2.15 incr(mark(X:S)) -> mark(incr(X:S)) 2.05/2.15 incr(ok(X:S)) -> ok(incr(X:S)) 2.05/2.15 pair(mark(X1:S),X2:S) -> mark(pair(X1:S,X2:S)) 2.05/2.15 pair(ok(X1:S),ok(X2:S)) -> ok(pair(X1:S,X2:S)) 2.05/2.15 pair(X1:S,mark(X2:S)) -> mark(pair(X1:S,X2:S)) 2.05/2.15 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(incr(X:S)) -> incr(proper(X:S)) 2.05/2.15 proper(pair(X1:S,X2:S)) -> pair(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(repItems(X:S)) -> repItems(proper(X:S)) 2.05/2.15 proper(s(X:S)) -> s(proper(X:S)) 2.05/2.15 proper(tail(X:S)) -> tail(proper(X:S)) 2.05/2.15 proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(zip(X1:S,X2:S)) -> zip(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(0) -> ok(0) 2.05/2.15 proper(nil) -> ok(nil) 2.05/2.15 proper(oddNs) -> ok(oddNs) 2.05/2.15 proper(pairNs) -> ok(pairNs) 2.05/2.15 repItems(mark(X:S)) -> mark(repItems(X:S)) 2.05/2.15 repItems(ok(X:S)) -> ok(repItems(X:S)) 2.05/2.15 s(mark(X:S)) -> mark(s(X:S)) 2.05/2.15 s(ok(X:S)) -> ok(s(X:S)) 2.05/2.15 tail(mark(X:S)) -> mark(tail(X:S)) 2.05/2.15 tail(ok(X:S)) -> ok(tail(X:S)) 2.05/2.15 take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S)) 2.05/2.15 take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S)) 2.05/2.15 take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S)) 2.05/2.15 top(mark(X:S)) -> top(proper(X:S)) 2.05/2.15 top(ok(X:S)) -> top(active(X:S)) 2.05/2.15 zip(mark(X1:S),X2:S) -> mark(zip(X1:S,X2:S)) 2.05/2.15 zip(ok(X1:S),ok(X2:S)) -> ok(zip(X1:S,X2:S)) 2.05/2.15 zip(X1:S,mark(X2:S)) -> mark(zip(X1:S,X2:S)) 2.05/2.15 ->Projection: 2.05/2.15 pi(ZIP) = 1 2.05/2.15 2.05/2.15 Problem 1.1: 2.05/2.15 2.05/2.15 SCC Processor: 2.05/2.15 -> Pairs: 2.05/2.15 ZIP(X1:S,mark(X2:S)) -> ZIP(X1:S,X2:S) 2.05/2.15 -> Rules: 2.05/2.15 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 2.05/2.15 active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) 2.05/2.15 active(incr(X:S)) -> incr(active(X:S)) 2.05/2.15 active(pair(X1:S,X2:S)) -> pair(active(X1:S),X2:S) 2.05/2.15 active(pair(X1:S,X2:S)) -> pair(X1:S,active(X2:S)) 2.05/2.15 active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) 2.05/2.15 active(repItems(nil)) -> mark(nil) 2.05/2.15 active(repItems(X:S)) -> repItems(active(X:S)) 2.05/2.15 active(s(X:S)) -> s(active(X:S)) 2.05/2.15 active(tail(cons(X:S,XS:S))) -> mark(XS:S) 2.05/2.15 active(tail(X:S)) -> tail(active(X:S)) 2.05/2.15 active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) 2.05/2.15 active(take(0,XS:S)) -> mark(nil) 2.05/2.15 active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S) 2.05/2.15 active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S)) 2.05/2.15 active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) 2.05/2.15 active(zip(nil,XS:S)) -> mark(nil) 2.05/2.15 active(zip(X:S,nil)) -> mark(nil) 2.05/2.15 active(zip(X1:S,X2:S)) -> zip(active(X1:S),X2:S) 2.05/2.15 active(zip(X1:S,X2:S)) -> zip(X1:S,active(X2:S)) 2.05/2.15 active(oddNs) -> mark(incr(pairNs)) 2.05/2.15 active(pairNs) -> mark(cons(0,incr(oddNs))) 2.05/2.15 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 2.05/2.15 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 2.05/2.15 incr(mark(X:S)) -> mark(incr(X:S)) 2.05/2.15 incr(ok(X:S)) -> ok(incr(X:S)) 2.05/2.15 pair(mark(X1:S),X2:S) -> mark(pair(X1:S,X2:S)) 2.05/2.15 pair(ok(X1:S),ok(X2:S)) -> ok(pair(X1:S,X2:S)) 2.05/2.15 pair(X1:S,mark(X2:S)) -> mark(pair(X1:S,X2:S)) 2.05/2.15 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(incr(X:S)) -> incr(proper(X:S)) 2.05/2.15 proper(pair(X1:S,X2:S)) -> pair(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(repItems(X:S)) -> repItems(proper(X:S)) 2.05/2.15 proper(s(X:S)) -> s(proper(X:S)) 2.05/2.15 proper(tail(X:S)) -> tail(proper(X:S)) 2.05/2.15 proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(zip(X1:S,X2:S)) -> zip(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(0) -> ok(0) 2.05/2.15 proper(nil) -> ok(nil) 2.05/2.15 proper(oddNs) -> ok(oddNs) 2.05/2.15 proper(pairNs) -> ok(pairNs) 2.05/2.15 repItems(mark(X:S)) -> mark(repItems(X:S)) 2.05/2.15 repItems(ok(X:S)) -> ok(repItems(X:S)) 2.05/2.15 s(mark(X:S)) -> mark(s(X:S)) 2.05/2.15 s(ok(X:S)) -> ok(s(X:S)) 2.05/2.15 tail(mark(X:S)) -> mark(tail(X:S)) 2.05/2.15 tail(ok(X:S)) -> ok(tail(X:S)) 2.05/2.15 take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S)) 2.05/2.15 take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S)) 2.05/2.15 take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S)) 2.05/2.15 top(mark(X:S)) -> top(proper(X:S)) 2.05/2.15 top(ok(X:S)) -> top(active(X:S)) 2.05/2.15 zip(mark(X1:S),X2:S) -> mark(zip(X1:S,X2:S)) 2.05/2.15 zip(ok(X1:S),ok(X2:S)) -> ok(zip(X1:S,X2:S)) 2.05/2.15 zip(X1:S,mark(X2:S)) -> mark(zip(X1:S,X2:S)) 2.05/2.15 ->Strongly Connected Components: 2.05/2.15 ->->Cycle: 2.05/2.15 ->->-> Pairs: 2.05/2.15 ZIP(X1:S,mark(X2:S)) -> ZIP(X1:S,X2:S) 2.05/2.15 ->->-> Rules: 2.05/2.15 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 2.05/2.15 active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) 2.05/2.15 active(incr(X:S)) -> incr(active(X:S)) 2.05/2.15 active(pair(X1:S,X2:S)) -> pair(active(X1:S),X2:S) 2.05/2.15 active(pair(X1:S,X2:S)) -> pair(X1:S,active(X2:S)) 2.05/2.15 active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) 2.05/2.15 active(repItems(nil)) -> mark(nil) 2.05/2.15 active(repItems(X:S)) -> repItems(active(X:S)) 2.05/2.15 active(s(X:S)) -> s(active(X:S)) 2.05/2.15 active(tail(cons(X:S,XS:S))) -> mark(XS:S) 2.05/2.15 active(tail(X:S)) -> tail(active(X:S)) 2.05/2.15 active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) 2.05/2.15 active(take(0,XS:S)) -> mark(nil) 2.05/2.15 active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S) 2.05/2.15 active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S)) 2.05/2.15 active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) 2.05/2.15 active(zip(nil,XS:S)) -> mark(nil) 2.05/2.15 active(zip(X:S,nil)) -> mark(nil) 2.05/2.15 active(zip(X1:S,X2:S)) -> zip(active(X1:S),X2:S) 2.05/2.15 active(zip(X1:S,X2:S)) -> zip(X1:S,active(X2:S)) 2.05/2.15 active(oddNs) -> mark(incr(pairNs)) 2.05/2.15 active(pairNs) -> mark(cons(0,incr(oddNs))) 2.05/2.15 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 2.05/2.15 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 2.05/2.15 incr(mark(X:S)) -> mark(incr(X:S)) 2.05/2.15 incr(ok(X:S)) -> ok(incr(X:S)) 2.05/2.15 pair(mark(X1:S),X2:S) -> mark(pair(X1:S,X2:S)) 2.05/2.15 pair(ok(X1:S),ok(X2:S)) -> ok(pair(X1:S,X2:S)) 2.05/2.15 pair(X1:S,mark(X2:S)) -> mark(pair(X1:S,X2:S)) 2.05/2.15 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(incr(X:S)) -> incr(proper(X:S)) 2.05/2.15 proper(pair(X1:S,X2:S)) -> pair(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(repItems(X:S)) -> repItems(proper(X:S)) 2.05/2.15 proper(s(X:S)) -> s(proper(X:S)) 2.05/2.15 proper(tail(X:S)) -> tail(proper(X:S)) 2.05/2.15 proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(zip(X1:S,X2:S)) -> zip(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(0) -> ok(0) 2.05/2.15 proper(nil) -> ok(nil) 2.05/2.15 proper(oddNs) -> ok(oddNs) 2.05/2.15 proper(pairNs) -> ok(pairNs) 2.05/2.15 repItems(mark(X:S)) -> mark(repItems(X:S)) 2.05/2.15 repItems(ok(X:S)) -> ok(repItems(X:S)) 2.05/2.15 s(mark(X:S)) -> mark(s(X:S)) 2.05/2.15 s(ok(X:S)) -> ok(s(X:S)) 2.05/2.15 tail(mark(X:S)) -> mark(tail(X:S)) 2.05/2.15 tail(ok(X:S)) -> ok(tail(X:S)) 2.05/2.15 take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S)) 2.05/2.15 take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S)) 2.05/2.15 take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S)) 2.05/2.15 top(mark(X:S)) -> top(proper(X:S)) 2.05/2.15 top(ok(X:S)) -> top(active(X:S)) 2.05/2.15 zip(mark(X1:S),X2:S) -> mark(zip(X1:S,X2:S)) 2.05/2.15 zip(ok(X1:S),ok(X2:S)) -> ok(zip(X1:S,X2:S)) 2.05/2.15 zip(X1:S,mark(X2:S)) -> mark(zip(X1:S,X2:S)) 2.05/2.15 2.05/2.15 Problem 1.1: 2.05/2.15 2.05/2.15 Subterm Processor: 2.05/2.15 -> Pairs: 2.05/2.15 ZIP(X1:S,mark(X2:S)) -> ZIP(X1:S,X2:S) 2.05/2.15 -> Rules: 2.05/2.15 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 2.05/2.15 active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) 2.05/2.15 active(incr(X:S)) -> incr(active(X:S)) 2.05/2.15 active(pair(X1:S,X2:S)) -> pair(active(X1:S),X2:S) 2.05/2.15 active(pair(X1:S,X2:S)) -> pair(X1:S,active(X2:S)) 2.05/2.15 active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) 2.05/2.15 active(repItems(nil)) -> mark(nil) 2.05/2.15 active(repItems(X:S)) -> repItems(active(X:S)) 2.05/2.15 active(s(X:S)) -> s(active(X:S)) 2.05/2.15 active(tail(cons(X:S,XS:S))) -> mark(XS:S) 2.05/2.15 active(tail(X:S)) -> tail(active(X:S)) 2.05/2.15 active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) 2.05/2.15 active(take(0,XS:S)) -> mark(nil) 2.05/2.15 active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S) 2.05/2.15 active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S)) 2.05/2.15 active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) 2.05/2.15 active(zip(nil,XS:S)) -> mark(nil) 2.05/2.15 active(zip(X:S,nil)) -> mark(nil) 2.05/2.15 active(zip(X1:S,X2:S)) -> zip(active(X1:S),X2:S) 2.05/2.15 active(zip(X1:S,X2:S)) -> zip(X1:S,active(X2:S)) 2.05/2.15 active(oddNs) -> mark(incr(pairNs)) 2.05/2.15 active(pairNs) -> mark(cons(0,incr(oddNs))) 2.05/2.15 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 2.05/2.15 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 2.05/2.15 incr(mark(X:S)) -> mark(incr(X:S)) 2.05/2.15 incr(ok(X:S)) -> ok(incr(X:S)) 2.05/2.15 pair(mark(X1:S),X2:S) -> mark(pair(X1:S,X2:S)) 2.05/2.15 pair(ok(X1:S),ok(X2:S)) -> ok(pair(X1:S,X2:S)) 2.05/2.15 pair(X1:S,mark(X2:S)) -> mark(pair(X1:S,X2:S)) 2.05/2.15 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(incr(X:S)) -> incr(proper(X:S)) 2.05/2.15 proper(pair(X1:S,X2:S)) -> pair(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(repItems(X:S)) -> repItems(proper(X:S)) 2.05/2.15 proper(s(X:S)) -> s(proper(X:S)) 2.05/2.15 proper(tail(X:S)) -> tail(proper(X:S)) 2.05/2.15 proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(zip(X1:S,X2:S)) -> zip(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(0) -> ok(0) 2.05/2.15 proper(nil) -> ok(nil) 2.05/2.15 proper(oddNs) -> ok(oddNs) 2.05/2.15 proper(pairNs) -> ok(pairNs) 2.05/2.15 repItems(mark(X:S)) -> mark(repItems(X:S)) 2.05/2.15 repItems(ok(X:S)) -> ok(repItems(X:S)) 2.05/2.15 s(mark(X:S)) -> mark(s(X:S)) 2.05/2.15 s(ok(X:S)) -> ok(s(X:S)) 2.05/2.15 tail(mark(X:S)) -> mark(tail(X:S)) 2.05/2.15 tail(ok(X:S)) -> ok(tail(X:S)) 2.05/2.15 take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S)) 2.05/2.15 take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S)) 2.05/2.15 take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S)) 2.05/2.15 top(mark(X:S)) -> top(proper(X:S)) 2.05/2.15 top(ok(X:S)) -> top(active(X:S)) 2.05/2.15 zip(mark(X1:S),X2:S) -> mark(zip(X1:S,X2:S)) 2.05/2.15 zip(ok(X1:S),ok(X2:S)) -> ok(zip(X1:S,X2:S)) 2.05/2.15 zip(X1:S,mark(X2:S)) -> mark(zip(X1:S,X2:S)) 2.05/2.15 ->Projection: 2.05/2.15 pi(ZIP) = 2 2.05/2.15 2.05/2.15 Problem 1.1: 2.05/2.15 2.05/2.15 SCC Processor: 2.05/2.15 -> Pairs: 2.05/2.15 Empty 2.05/2.15 -> Rules: 2.05/2.15 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 2.05/2.15 active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) 2.05/2.15 active(incr(X:S)) -> incr(active(X:S)) 2.05/2.15 active(pair(X1:S,X2:S)) -> pair(active(X1:S),X2:S) 2.05/2.15 active(pair(X1:S,X2:S)) -> pair(X1:S,active(X2:S)) 2.05/2.15 active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) 2.05/2.15 active(repItems(nil)) -> mark(nil) 2.05/2.15 active(repItems(X:S)) -> repItems(active(X:S)) 2.05/2.15 active(s(X:S)) -> s(active(X:S)) 2.05/2.15 active(tail(cons(X:S,XS:S))) -> mark(XS:S) 2.05/2.15 active(tail(X:S)) -> tail(active(X:S)) 2.05/2.15 active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) 2.05/2.15 active(take(0,XS:S)) -> mark(nil) 2.05/2.15 active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S) 2.05/2.15 active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S)) 2.05/2.15 active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) 2.05/2.15 active(zip(nil,XS:S)) -> mark(nil) 2.05/2.15 active(zip(X:S,nil)) -> mark(nil) 2.05/2.15 active(zip(X1:S,X2:S)) -> zip(active(X1:S),X2:S) 2.05/2.15 active(zip(X1:S,X2:S)) -> zip(X1:S,active(X2:S)) 2.05/2.15 active(oddNs) -> mark(incr(pairNs)) 2.05/2.15 active(pairNs) -> mark(cons(0,incr(oddNs))) 2.05/2.15 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 2.05/2.15 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 2.05/2.15 incr(mark(X:S)) -> mark(incr(X:S)) 2.05/2.15 incr(ok(X:S)) -> ok(incr(X:S)) 2.05/2.15 pair(mark(X1:S),X2:S) -> mark(pair(X1:S,X2:S)) 2.05/2.15 pair(ok(X1:S),ok(X2:S)) -> ok(pair(X1:S,X2:S)) 2.05/2.15 pair(X1:S,mark(X2:S)) -> mark(pair(X1:S,X2:S)) 2.05/2.15 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(incr(X:S)) -> incr(proper(X:S)) 2.05/2.15 proper(pair(X1:S,X2:S)) -> pair(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(repItems(X:S)) -> repItems(proper(X:S)) 2.05/2.15 proper(s(X:S)) -> s(proper(X:S)) 2.05/2.15 proper(tail(X:S)) -> tail(proper(X:S)) 2.05/2.15 proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(zip(X1:S,X2:S)) -> zip(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(0) -> ok(0) 2.05/2.15 proper(nil) -> ok(nil) 2.05/2.15 proper(oddNs) -> ok(oddNs) 2.05/2.15 proper(pairNs) -> ok(pairNs) 2.05/2.15 repItems(mark(X:S)) -> mark(repItems(X:S)) 2.05/2.15 repItems(ok(X:S)) -> ok(repItems(X:S)) 2.05/2.15 s(mark(X:S)) -> mark(s(X:S)) 2.05/2.15 s(ok(X:S)) -> ok(s(X:S)) 2.05/2.15 tail(mark(X:S)) -> mark(tail(X:S)) 2.05/2.15 tail(ok(X:S)) -> ok(tail(X:S)) 2.05/2.15 take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S)) 2.05/2.15 take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S)) 2.05/2.15 take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S)) 2.05/2.15 top(mark(X:S)) -> top(proper(X:S)) 2.05/2.15 top(ok(X:S)) -> top(active(X:S)) 2.05/2.15 zip(mark(X1:S),X2:S) -> mark(zip(X1:S,X2:S)) 2.05/2.15 zip(ok(X1:S),ok(X2:S)) -> ok(zip(X1:S,X2:S)) 2.05/2.15 zip(X1:S,mark(X2:S)) -> mark(zip(X1:S,X2:S)) 2.05/2.15 ->Strongly Connected Components: 2.05/2.15 There is no strongly connected component 2.05/2.15 2.05/2.15 The problem is finite. 2.05/2.15 2.05/2.15 Problem 1.2: 2.05/2.15 2.05/2.15 Subterm Processor: 2.05/2.15 -> Pairs: 2.05/2.15 TAKE(mark(X1:S),X2:S) -> TAKE(X1:S,X2:S) 2.05/2.15 TAKE(ok(X1:S),ok(X2:S)) -> TAKE(X1:S,X2:S) 2.05/2.15 TAKE(X1:S,mark(X2:S)) -> TAKE(X1:S,X2:S) 2.05/2.15 -> Rules: 2.05/2.15 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 2.05/2.15 active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) 2.05/2.15 active(incr(X:S)) -> incr(active(X:S)) 2.05/2.15 active(pair(X1:S,X2:S)) -> pair(active(X1:S),X2:S) 2.05/2.15 active(pair(X1:S,X2:S)) -> pair(X1:S,active(X2:S)) 2.05/2.15 active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) 2.05/2.15 active(repItems(nil)) -> mark(nil) 2.05/2.15 active(repItems(X:S)) -> repItems(active(X:S)) 2.05/2.15 active(s(X:S)) -> s(active(X:S)) 2.05/2.15 active(tail(cons(X:S,XS:S))) -> mark(XS:S) 2.05/2.15 active(tail(X:S)) -> tail(active(X:S)) 2.05/2.15 active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) 2.05/2.15 active(take(0,XS:S)) -> mark(nil) 2.05/2.15 active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S) 2.05/2.15 active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S)) 2.05/2.15 active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) 2.05/2.15 active(zip(nil,XS:S)) -> mark(nil) 2.05/2.15 active(zip(X:S,nil)) -> mark(nil) 2.05/2.15 active(zip(X1:S,X2:S)) -> zip(active(X1:S),X2:S) 2.05/2.15 active(zip(X1:S,X2:S)) -> zip(X1:S,active(X2:S)) 2.05/2.15 active(oddNs) -> mark(incr(pairNs)) 2.05/2.15 active(pairNs) -> mark(cons(0,incr(oddNs))) 2.05/2.15 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 2.05/2.15 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 2.05/2.15 incr(mark(X:S)) -> mark(incr(X:S)) 2.05/2.15 incr(ok(X:S)) -> ok(incr(X:S)) 2.05/2.15 pair(mark(X1:S),X2:S) -> mark(pair(X1:S,X2:S)) 2.05/2.15 pair(ok(X1:S),ok(X2:S)) -> ok(pair(X1:S,X2:S)) 2.05/2.15 pair(X1:S,mark(X2:S)) -> mark(pair(X1:S,X2:S)) 2.05/2.15 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(incr(X:S)) -> incr(proper(X:S)) 2.05/2.15 proper(pair(X1:S,X2:S)) -> pair(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(repItems(X:S)) -> repItems(proper(X:S)) 2.05/2.15 proper(s(X:S)) -> s(proper(X:S)) 2.05/2.15 proper(tail(X:S)) -> tail(proper(X:S)) 2.05/2.15 proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(zip(X1:S,X2:S)) -> zip(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(0) -> ok(0) 2.05/2.15 proper(nil) -> ok(nil) 2.05/2.15 proper(oddNs) -> ok(oddNs) 2.05/2.15 proper(pairNs) -> ok(pairNs) 2.05/2.15 repItems(mark(X:S)) -> mark(repItems(X:S)) 2.05/2.15 repItems(ok(X:S)) -> ok(repItems(X:S)) 2.05/2.15 s(mark(X:S)) -> mark(s(X:S)) 2.05/2.15 s(ok(X:S)) -> ok(s(X:S)) 2.05/2.15 tail(mark(X:S)) -> mark(tail(X:S)) 2.05/2.15 tail(ok(X:S)) -> ok(tail(X:S)) 2.05/2.15 take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S)) 2.05/2.15 take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S)) 2.05/2.15 take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S)) 2.05/2.15 top(mark(X:S)) -> top(proper(X:S)) 2.05/2.15 top(ok(X:S)) -> top(active(X:S)) 2.05/2.15 zip(mark(X1:S),X2:S) -> mark(zip(X1:S,X2:S)) 2.05/2.15 zip(ok(X1:S),ok(X2:S)) -> ok(zip(X1:S,X2:S)) 2.05/2.15 zip(X1:S,mark(X2:S)) -> mark(zip(X1:S,X2:S)) 2.05/2.15 ->Projection: 2.05/2.15 pi(TAKE) = 1 2.05/2.15 2.05/2.15 Problem 1.2: 2.05/2.15 2.05/2.15 SCC Processor: 2.05/2.15 -> Pairs: 2.05/2.15 TAKE(X1:S,mark(X2:S)) -> TAKE(X1:S,X2:S) 2.05/2.15 -> Rules: 2.05/2.15 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 2.05/2.15 active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) 2.05/2.15 active(incr(X:S)) -> incr(active(X:S)) 2.05/2.15 active(pair(X1:S,X2:S)) -> pair(active(X1:S),X2:S) 2.05/2.15 active(pair(X1:S,X2:S)) -> pair(X1:S,active(X2:S)) 2.05/2.15 active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) 2.05/2.15 active(repItems(nil)) -> mark(nil) 2.05/2.15 active(repItems(X:S)) -> repItems(active(X:S)) 2.05/2.15 active(s(X:S)) -> s(active(X:S)) 2.05/2.15 active(tail(cons(X:S,XS:S))) -> mark(XS:S) 2.05/2.15 active(tail(X:S)) -> tail(active(X:S)) 2.05/2.15 active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) 2.05/2.15 active(take(0,XS:S)) -> mark(nil) 2.05/2.15 active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S) 2.05/2.15 active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S)) 2.05/2.15 active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) 2.05/2.15 active(zip(nil,XS:S)) -> mark(nil) 2.05/2.15 active(zip(X:S,nil)) -> mark(nil) 2.05/2.15 active(zip(X1:S,X2:S)) -> zip(active(X1:S),X2:S) 2.05/2.15 active(zip(X1:S,X2:S)) -> zip(X1:S,active(X2:S)) 2.05/2.15 active(oddNs) -> mark(incr(pairNs)) 2.05/2.15 active(pairNs) -> mark(cons(0,incr(oddNs))) 2.05/2.15 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 2.05/2.15 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 2.05/2.15 incr(mark(X:S)) -> mark(incr(X:S)) 2.05/2.15 incr(ok(X:S)) -> ok(incr(X:S)) 2.05/2.15 pair(mark(X1:S),X2:S) -> mark(pair(X1:S,X2:S)) 2.05/2.15 pair(ok(X1:S),ok(X2:S)) -> ok(pair(X1:S,X2:S)) 2.05/2.15 pair(X1:S,mark(X2:S)) -> mark(pair(X1:S,X2:S)) 2.05/2.15 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(incr(X:S)) -> incr(proper(X:S)) 2.05/2.15 proper(pair(X1:S,X2:S)) -> pair(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(repItems(X:S)) -> repItems(proper(X:S)) 2.05/2.15 proper(s(X:S)) -> s(proper(X:S)) 2.05/2.15 proper(tail(X:S)) -> tail(proper(X:S)) 2.05/2.15 proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(zip(X1:S,X2:S)) -> zip(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(0) -> ok(0) 2.05/2.15 proper(nil) -> ok(nil) 2.05/2.15 proper(oddNs) -> ok(oddNs) 2.05/2.15 proper(pairNs) -> ok(pairNs) 2.05/2.15 repItems(mark(X:S)) -> mark(repItems(X:S)) 2.05/2.15 repItems(ok(X:S)) -> ok(repItems(X:S)) 2.05/2.15 s(mark(X:S)) -> mark(s(X:S)) 2.05/2.15 s(ok(X:S)) -> ok(s(X:S)) 2.05/2.15 tail(mark(X:S)) -> mark(tail(X:S)) 2.05/2.15 tail(ok(X:S)) -> ok(tail(X:S)) 2.05/2.15 take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S)) 2.05/2.15 take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S)) 2.05/2.15 take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S)) 2.05/2.15 top(mark(X:S)) -> top(proper(X:S)) 2.05/2.15 top(ok(X:S)) -> top(active(X:S)) 2.05/2.15 zip(mark(X1:S),X2:S) -> mark(zip(X1:S,X2:S)) 2.05/2.15 zip(ok(X1:S),ok(X2:S)) -> ok(zip(X1:S,X2:S)) 2.05/2.15 zip(X1:S,mark(X2:S)) -> mark(zip(X1:S,X2:S)) 2.05/2.15 ->Strongly Connected Components: 2.05/2.15 ->->Cycle: 2.05/2.15 ->->-> Pairs: 2.05/2.15 TAKE(X1:S,mark(X2:S)) -> TAKE(X1:S,X2:S) 2.05/2.15 ->->-> Rules: 2.05/2.15 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 2.05/2.15 active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) 2.05/2.15 active(incr(X:S)) -> incr(active(X:S)) 2.05/2.15 active(pair(X1:S,X2:S)) -> pair(active(X1:S),X2:S) 2.05/2.15 active(pair(X1:S,X2:S)) -> pair(X1:S,active(X2:S)) 2.05/2.15 active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) 2.05/2.15 active(repItems(nil)) -> mark(nil) 2.05/2.15 active(repItems(X:S)) -> repItems(active(X:S)) 2.05/2.15 active(s(X:S)) -> s(active(X:S)) 2.05/2.15 active(tail(cons(X:S,XS:S))) -> mark(XS:S) 2.05/2.15 active(tail(X:S)) -> tail(active(X:S)) 2.05/2.15 active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) 2.05/2.15 active(take(0,XS:S)) -> mark(nil) 2.05/2.15 active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S) 2.05/2.15 active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S)) 2.05/2.15 active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) 2.05/2.15 active(zip(nil,XS:S)) -> mark(nil) 2.05/2.15 active(zip(X:S,nil)) -> mark(nil) 2.05/2.15 active(zip(X1:S,X2:S)) -> zip(active(X1:S),X2:S) 2.05/2.15 active(zip(X1:S,X2:S)) -> zip(X1:S,active(X2:S)) 2.05/2.15 active(oddNs) -> mark(incr(pairNs)) 2.05/2.15 active(pairNs) -> mark(cons(0,incr(oddNs))) 2.05/2.15 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 2.05/2.15 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 2.05/2.15 incr(mark(X:S)) -> mark(incr(X:S)) 2.05/2.15 incr(ok(X:S)) -> ok(incr(X:S)) 2.05/2.15 pair(mark(X1:S),X2:S) -> mark(pair(X1:S,X2:S)) 2.05/2.15 pair(ok(X1:S),ok(X2:S)) -> ok(pair(X1:S,X2:S)) 2.05/2.15 pair(X1:S,mark(X2:S)) -> mark(pair(X1:S,X2:S)) 2.05/2.15 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(incr(X:S)) -> incr(proper(X:S)) 2.05/2.15 proper(pair(X1:S,X2:S)) -> pair(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(repItems(X:S)) -> repItems(proper(X:S)) 2.05/2.15 proper(s(X:S)) -> s(proper(X:S)) 2.05/2.15 proper(tail(X:S)) -> tail(proper(X:S)) 2.05/2.15 proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(zip(X1:S,X2:S)) -> zip(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(0) -> ok(0) 2.05/2.15 proper(nil) -> ok(nil) 2.05/2.15 proper(oddNs) -> ok(oddNs) 2.05/2.15 proper(pairNs) -> ok(pairNs) 2.05/2.15 repItems(mark(X:S)) -> mark(repItems(X:S)) 2.05/2.15 repItems(ok(X:S)) -> ok(repItems(X:S)) 2.05/2.15 s(mark(X:S)) -> mark(s(X:S)) 2.05/2.15 s(ok(X:S)) -> ok(s(X:S)) 2.05/2.15 tail(mark(X:S)) -> mark(tail(X:S)) 2.05/2.15 tail(ok(X:S)) -> ok(tail(X:S)) 2.05/2.15 take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S)) 2.05/2.15 take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S)) 2.05/2.15 take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S)) 2.05/2.15 top(mark(X:S)) -> top(proper(X:S)) 2.05/2.15 top(ok(X:S)) -> top(active(X:S)) 2.05/2.15 zip(mark(X1:S),X2:S) -> mark(zip(X1:S,X2:S)) 2.05/2.15 zip(ok(X1:S),ok(X2:S)) -> ok(zip(X1:S,X2:S)) 2.05/2.15 zip(X1:S,mark(X2:S)) -> mark(zip(X1:S,X2:S)) 2.05/2.15 2.05/2.15 Problem 1.2: 2.05/2.15 2.05/2.15 Subterm Processor: 2.05/2.15 -> Pairs: 2.05/2.15 TAKE(X1:S,mark(X2:S)) -> TAKE(X1:S,X2:S) 2.05/2.15 -> Rules: 2.05/2.15 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 2.05/2.15 active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) 2.05/2.15 active(incr(X:S)) -> incr(active(X:S)) 2.05/2.15 active(pair(X1:S,X2:S)) -> pair(active(X1:S),X2:S) 2.05/2.15 active(pair(X1:S,X2:S)) -> pair(X1:S,active(X2:S)) 2.05/2.15 active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) 2.05/2.15 active(repItems(nil)) -> mark(nil) 2.05/2.15 active(repItems(X:S)) -> repItems(active(X:S)) 2.05/2.15 active(s(X:S)) -> s(active(X:S)) 2.05/2.15 active(tail(cons(X:S,XS:S))) -> mark(XS:S) 2.05/2.15 active(tail(X:S)) -> tail(active(X:S)) 2.05/2.15 active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) 2.05/2.15 active(take(0,XS:S)) -> mark(nil) 2.05/2.15 active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S) 2.05/2.15 active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S)) 2.05/2.15 active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) 2.05/2.15 active(zip(nil,XS:S)) -> mark(nil) 2.05/2.15 active(zip(X:S,nil)) -> mark(nil) 2.05/2.15 active(zip(X1:S,X2:S)) -> zip(active(X1:S),X2:S) 2.05/2.15 active(zip(X1:S,X2:S)) -> zip(X1:S,active(X2:S)) 2.05/2.15 active(oddNs) -> mark(incr(pairNs)) 2.05/2.15 active(pairNs) -> mark(cons(0,incr(oddNs))) 2.05/2.15 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 2.05/2.15 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 2.05/2.15 incr(mark(X:S)) -> mark(incr(X:S)) 2.05/2.15 incr(ok(X:S)) -> ok(incr(X:S)) 2.05/2.15 pair(mark(X1:S),X2:S) -> mark(pair(X1:S,X2:S)) 2.05/2.15 pair(ok(X1:S),ok(X2:S)) -> ok(pair(X1:S,X2:S)) 2.05/2.15 pair(X1:S,mark(X2:S)) -> mark(pair(X1:S,X2:S)) 2.05/2.15 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(incr(X:S)) -> incr(proper(X:S)) 2.05/2.15 proper(pair(X1:S,X2:S)) -> pair(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(repItems(X:S)) -> repItems(proper(X:S)) 2.05/2.15 proper(s(X:S)) -> s(proper(X:S)) 2.05/2.15 proper(tail(X:S)) -> tail(proper(X:S)) 2.05/2.15 proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(zip(X1:S,X2:S)) -> zip(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(0) -> ok(0) 2.05/2.15 proper(nil) -> ok(nil) 2.05/2.15 proper(oddNs) -> ok(oddNs) 2.05/2.15 proper(pairNs) -> ok(pairNs) 2.05/2.15 repItems(mark(X:S)) -> mark(repItems(X:S)) 2.05/2.15 repItems(ok(X:S)) -> ok(repItems(X:S)) 2.05/2.15 s(mark(X:S)) -> mark(s(X:S)) 2.05/2.15 s(ok(X:S)) -> ok(s(X:S)) 2.05/2.15 tail(mark(X:S)) -> mark(tail(X:S)) 2.05/2.15 tail(ok(X:S)) -> ok(tail(X:S)) 2.05/2.15 take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S)) 2.05/2.15 take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S)) 2.05/2.15 take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S)) 2.05/2.15 top(mark(X:S)) -> top(proper(X:S)) 2.05/2.15 top(ok(X:S)) -> top(active(X:S)) 2.05/2.15 zip(mark(X1:S),X2:S) -> mark(zip(X1:S,X2:S)) 2.05/2.15 zip(ok(X1:S),ok(X2:S)) -> ok(zip(X1:S,X2:S)) 2.05/2.15 zip(X1:S,mark(X2:S)) -> mark(zip(X1:S,X2:S)) 2.05/2.15 ->Projection: 2.05/2.15 pi(TAKE) = 2 2.05/2.15 2.05/2.15 Problem 1.2: 2.05/2.15 2.05/2.15 SCC Processor: 2.05/2.15 -> Pairs: 2.05/2.15 Empty 2.05/2.15 -> Rules: 2.05/2.15 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 2.05/2.15 active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) 2.05/2.15 active(incr(X:S)) -> incr(active(X:S)) 2.05/2.15 active(pair(X1:S,X2:S)) -> pair(active(X1:S),X2:S) 2.05/2.15 active(pair(X1:S,X2:S)) -> pair(X1:S,active(X2:S)) 2.05/2.15 active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) 2.05/2.15 active(repItems(nil)) -> mark(nil) 2.05/2.15 active(repItems(X:S)) -> repItems(active(X:S)) 2.05/2.15 active(s(X:S)) -> s(active(X:S)) 2.05/2.15 active(tail(cons(X:S,XS:S))) -> mark(XS:S) 2.05/2.15 active(tail(X:S)) -> tail(active(X:S)) 2.05/2.15 active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) 2.05/2.15 active(take(0,XS:S)) -> mark(nil) 2.05/2.15 active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S) 2.05/2.15 active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S)) 2.05/2.15 active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) 2.05/2.15 active(zip(nil,XS:S)) -> mark(nil) 2.05/2.15 active(zip(X:S,nil)) -> mark(nil) 2.05/2.15 active(zip(X1:S,X2:S)) -> zip(active(X1:S),X2:S) 2.05/2.15 active(zip(X1:S,X2:S)) -> zip(X1:S,active(X2:S)) 2.05/2.15 active(oddNs) -> mark(incr(pairNs)) 2.05/2.15 active(pairNs) -> mark(cons(0,incr(oddNs))) 2.05/2.15 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 2.05/2.15 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 2.05/2.15 incr(mark(X:S)) -> mark(incr(X:S)) 2.05/2.15 incr(ok(X:S)) -> ok(incr(X:S)) 2.05/2.15 pair(mark(X1:S),X2:S) -> mark(pair(X1:S,X2:S)) 2.05/2.15 pair(ok(X1:S),ok(X2:S)) -> ok(pair(X1:S,X2:S)) 2.05/2.15 pair(X1:S,mark(X2:S)) -> mark(pair(X1:S,X2:S)) 2.05/2.15 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(incr(X:S)) -> incr(proper(X:S)) 2.05/2.15 proper(pair(X1:S,X2:S)) -> pair(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(repItems(X:S)) -> repItems(proper(X:S)) 2.05/2.15 proper(s(X:S)) -> s(proper(X:S)) 2.05/2.15 proper(tail(X:S)) -> tail(proper(X:S)) 2.05/2.15 proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(zip(X1:S,X2:S)) -> zip(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(0) -> ok(0) 2.05/2.15 proper(nil) -> ok(nil) 2.05/2.15 proper(oddNs) -> ok(oddNs) 2.05/2.15 proper(pairNs) -> ok(pairNs) 2.05/2.15 repItems(mark(X:S)) -> mark(repItems(X:S)) 2.05/2.15 repItems(ok(X:S)) -> ok(repItems(X:S)) 2.05/2.15 s(mark(X:S)) -> mark(s(X:S)) 2.05/2.15 s(ok(X:S)) -> ok(s(X:S)) 2.05/2.15 tail(mark(X:S)) -> mark(tail(X:S)) 2.05/2.15 tail(ok(X:S)) -> ok(tail(X:S)) 2.05/2.15 take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S)) 2.05/2.15 take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S)) 2.05/2.15 take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S)) 2.05/2.15 top(mark(X:S)) -> top(proper(X:S)) 2.05/2.15 top(ok(X:S)) -> top(active(X:S)) 2.05/2.15 zip(mark(X1:S),X2:S) -> mark(zip(X1:S,X2:S)) 2.05/2.15 zip(ok(X1:S),ok(X2:S)) -> ok(zip(X1:S,X2:S)) 2.05/2.15 zip(X1:S,mark(X2:S)) -> mark(zip(X1:S,X2:S)) 2.05/2.15 ->Strongly Connected Components: 2.05/2.15 There is no strongly connected component 2.05/2.15 2.05/2.15 The problem is finite. 2.05/2.15 2.05/2.15 Problem 1.3: 2.05/2.15 2.05/2.15 Subterm Processor: 2.05/2.15 -> Pairs: 2.05/2.15 TAIL(mark(X:S)) -> TAIL(X:S) 2.05/2.15 TAIL(ok(X:S)) -> TAIL(X:S) 2.05/2.15 -> Rules: 2.05/2.15 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 2.05/2.15 active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) 2.05/2.15 active(incr(X:S)) -> incr(active(X:S)) 2.05/2.15 active(pair(X1:S,X2:S)) -> pair(active(X1:S),X2:S) 2.05/2.15 active(pair(X1:S,X2:S)) -> pair(X1:S,active(X2:S)) 2.05/2.15 active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) 2.05/2.15 active(repItems(nil)) -> mark(nil) 2.05/2.15 active(repItems(X:S)) -> repItems(active(X:S)) 2.05/2.15 active(s(X:S)) -> s(active(X:S)) 2.05/2.15 active(tail(cons(X:S,XS:S))) -> mark(XS:S) 2.05/2.15 active(tail(X:S)) -> tail(active(X:S)) 2.05/2.15 active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) 2.05/2.15 active(take(0,XS:S)) -> mark(nil) 2.05/2.15 active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S) 2.05/2.15 active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S)) 2.05/2.15 active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) 2.05/2.15 active(zip(nil,XS:S)) -> mark(nil) 2.05/2.15 active(zip(X:S,nil)) -> mark(nil) 2.05/2.15 active(zip(X1:S,X2:S)) -> zip(active(X1:S),X2:S) 2.05/2.15 active(zip(X1:S,X2:S)) -> zip(X1:S,active(X2:S)) 2.05/2.15 active(oddNs) -> mark(incr(pairNs)) 2.05/2.15 active(pairNs) -> mark(cons(0,incr(oddNs))) 2.05/2.15 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 2.05/2.15 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 2.05/2.15 incr(mark(X:S)) -> mark(incr(X:S)) 2.05/2.15 incr(ok(X:S)) -> ok(incr(X:S)) 2.05/2.15 pair(mark(X1:S),X2:S) -> mark(pair(X1:S,X2:S)) 2.05/2.15 pair(ok(X1:S),ok(X2:S)) -> ok(pair(X1:S,X2:S)) 2.05/2.15 pair(X1:S,mark(X2:S)) -> mark(pair(X1:S,X2:S)) 2.05/2.15 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(incr(X:S)) -> incr(proper(X:S)) 2.05/2.15 proper(pair(X1:S,X2:S)) -> pair(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(repItems(X:S)) -> repItems(proper(X:S)) 2.05/2.15 proper(s(X:S)) -> s(proper(X:S)) 2.05/2.15 proper(tail(X:S)) -> tail(proper(X:S)) 2.05/2.15 proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(zip(X1:S,X2:S)) -> zip(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(0) -> ok(0) 2.05/2.15 proper(nil) -> ok(nil) 2.05/2.15 proper(oddNs) -> ok(oddNs) 2.05/2.15 proper(pairNs) -> ok(pairNs) 2.05/2.15 repItems(mark(X:S)) -> mark(repItems(X:S)) 2.05/2.15 repItems(ok(X:S)) -> ok(repItems(X:S)) 2.05/2.15 s(mark(X:S)) -> mark(s(X:S)) 2.05/2.15 s(ok(X:S)) -> ok(s(X:S)) 2.05/2.15 tail(mark(X:S)) -> mark(tail(X:S)) 2.05/2.15 tail(ok(X:S)) -> ok(tail(X:S)) 2.05/2.15 take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S)) 2.05/2.15 take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S)) 2.05/2.15 take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S)) 2.05/2.15 top(mark(X:S)) -> top(proper(X:S)) 2.05/2.15 top(ok(X:S)) -> top(active(X:S)) 2.05/2.15 zip(mark(X1:S),X2:S) -> mark(zip(X1:S,X2:S)) 2.05/2.15 zip(ok(X1:S),ok(X2:S)) -> ok(zip(X1:S,X2:S)) 2.05/2.15 zip(X1:S,mark(X2:S)) -> mark(zip(X1:S,X2:S)) 2.05/2.15 ->Projection: 2.05/2.15 pi(TAIL) = 1 2.05/2.15 2.05/2.15 Problem 1.3: 2.05/2.15 2.05/2.15 SCC Processor: 2.05/2.15 -> Pairs: 2.05/2.15 Empty 2.05/2.15 -> Rules: 2.05/2.15 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 2.05/2.15 active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) 2.05/2.15 active(incr(X:S)) -> incr(active(X:S)) 2.05/2.15 active(pair(X1:S,X2:S)) -> pair(active(X1:S),X2:S) 2.05/2.15 active(pair(X1:S,X2:S)) -> pair(X1:S,active(X2:S)) 2.05/2.15 active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) 2.05/2.15 active(repItems(nil)) -> mark(nil) 2.05/2.15 active(repItems(X:S)) -> repItems(active(X:S)) 2.05/2.15 active(s(X:S)) -> s(active(X:S)) 2.05/2.15 active(tail(cons(X:S,XS:S))) -> mark(XS:S) 2.05/2.15 active(tail(X:S)) -> tail(active(X:S)) 2.05/2.15 active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) 2.05/2.15 active(take(0,XS:S)) -> mark(nil) 2.05/2.15 active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S) 2.05/2.15 active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S)) 2.05/2.15 active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) 2.05/2.15 active(zip(nil,XS:S)) -> mark(nil) 2.05/2.15 active(zip(X:S,nil)) -> mark(nil) 2.05/2.15 active(zip(X1:S,X2:S)) -> zip(active(X1:S),X2:S) 2.05/2.15 active(zip(X1:S,X2:S)) -> zip(X1:S,active(X2:S)) 2.05/2.15 active(oddNs) -> mark(incr(pairNs)) 2.05/2.15 active(pairNs) -> mark(cons(0,incr(oddNs))) 2.05/2.15 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 2.05/2.15 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 2.05/2.15 incr(mark(X:S)) -> mark(incr(X:S)) 2.05/2.15 incr(ok(X:S)) -> ok(incr(X:S)) 2.05/2.15 pair(mark(X1:S),X2:S) -> mark(pair(X1:S,X2:S)) 2.05/2.15 pair(ok(X1:S),ok(X2:S)) -> ok(pair(X1:S,X2:S)) 2.05/2.15 pair(X1:S,mark(X2:S)) -> mark(pair(X1:S,X2:S)) 2.05/2.15 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(incr(X:S)) -> incr(proper(X:S)) 2.05/2.15 proper(pair(X1:S,X2:S)) -> pair(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(repItems(X:S)) -> repItems(proper(X:S)) 2.05/2.15 proper(s(X:S)) -> s(proper(X:S)) 2.05/2.15 proper(tail(X:S)) -> tail(proper(X:S)) 2.05/2.15 proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(zip(X1:S,X2:S)) -> zip(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(0) -> ok(0) 2.05/2.15 proper(nil) -> ok(nil) 2.05/2.15 proper(oddNs) -> ok(oddNs) 2.05/2.15 proper(pairNs) -> ok(pairNs) 2.05/2.15 repItems(mark(X:S)) -> mark(repItems(X:S)) 2.05/2.15 repItems(ok(X:S)) -> ok(repItems(X:S)) 2.05/2.15 s(mark(X:S)) -> mark(s(X:S)) 2.05/2.15 s(ok(X:S)) -> ok(s(X:S)) 2.05/2.15 tail(mark(X:S)) -> mark(tail(X:S)) 2.05/2.15 tail(ok(X:S)) -> ok(tail(X:S)) 2.05/2.15 take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S)) 2.05/2.15 take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S)) 2.05/2.15 take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S)) 2.05/2.15 top(mark(X:S)) -> top(proper(X:S)) 2.05/2.15 top(ok(X:S)) -> top(active(X:S)) 2.05/2.15 zip(mark(X1:S),X2:S) -> mark(zip(X1:S,X2:S)) 2.05/2.15 zip(ok(X1:S),ok(X2:S)) -> ok(zip(X1:S,X2:S)) 2.05/2.15 zip(X1:S,mark(X2:S)) -> mark(zip(X1:S,X2:S)) 2.05/2.15 ->Strongly Connected Components: 2.05/2.15 There is no strongly connected component 2.05/2.15 2.05/2.15 The problem is finite. 2.05/2.15 2.05/2.15 Problem 1.4: 2.05/2.15 2.05/2.15 Subterm Processor: 2.05/2.15 -> Pairs: 2.05/2.15 S(mark(X:S)) -> S(X:S) 2.05/2.15 S(ok(X:S)) -> S(X:S) 2.05/2.15 -> Rules: 2.05/2.15 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 2.05/2.15 active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) 2.05/2.15 active(incr(X:S)) -> incr(active(X:S)) 2.05/2.15 active(pair(X1:S,X2:S)) -> pair(active(X1:S),X2:S) 2.05/2.15 active(pair(X1:S,X2:S)) -> pair(X1:S,active(X2:S)) 2.05/2.15 active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) 2.05/2.15 active(repItems(nil)) -> mark(nil) 2.05/2.15 active(repItems(X:S)) -> repItems(active(X:S)) 2.05/2.15 active(s(X:S)) -> s(active(X:S)) 2.05/2.15 active(tail(cons(X:S,XS:S))) -> mark(XS:S) 2.05/2.15 active(tail(X:S)) -> tail(active(X:S)) 2.05/2.15 active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) 2.05/2.15 active(take(0,XS:S)) -> mark(nil) 2.05/2.15 active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S) 2.05/2.15 active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S)) 2.05/2.15 active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) 2.05/2.15 active(zip(nil,XS:S)) -> mark(nil) 2.05/2.15 active(zip(X:S,nil)) -> mark(nil) 2.05/2.15 active(zip(X1:S,X2:S)) -> zip(active(X1:S),X2:S) 2.05/2.15 active(zip(X1:S,X2:S)) -> zip(X1:S,active(X2:S)) 2.05/2.15 active(oddNs) -> mark(incr(pairNs)) 2.05/2.15 active(pairNs) -> mark(cons(0,incr(oddNs))) 2.05/2.15 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 2.05/2.15 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 2.05/2.15 incr(mark(X:S)) -> mark(incr(X:S)) 2.05/2.15 incr(ok(X:S)) -> ok(incr(X:S)) 2.05/2.15 pair(mark(X1:S),X2:S) -> mark(pair(X1:S,X2:S)) 2.05/2.15 pair(ok(X1:S),ok(X2:S)) -> ok(pair(X1:S,X2:S)) 2.05/2.15 pair(X1:S,mark(X2:S)) -> mark(pair(X1:S,X2:S)) 2.05/2.15 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(incr(X:S)) -> incr(proper(X:S)) 2.05/2.15 proper(pair(X1:S,X2:S)) -> pair(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(repItems(X:S)) -> repItems(proper(X:S)) 2.05/2.15 proper(s(X:S)) -> s(proper(X:S)) 2.05/2.15 proper(tail(X:S)) -> tail(proper(X:S)) 2.05/2.15 proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(zip(X1:S,X2:S)) -> zip(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(0) -> ok(0) 2.05/2.15 proper(nil) -> ok(nil) 2.05/2.15 proper(oddNs) -> ok(oddNs) 2.05/2.15 proper(pairNs) -> ok(pairNs) 2.05/2.15 repItems(mark(X:S)) -> mark(repItems(X:S)) 2.05/2.15 repItems(ok(X:S)) -> ok(repItems(X:S)) 2.05/2.15 s(mark(X:S)) -> mark(s(X:S)) 2.05/2.15 s(ok(X:S)) -> ok(s(X:S)) 2.05/2.15 tail(mark(X:S)) -> mark(tail(X:S)) 2.05/2.15 tail(ok(X:S)) -> ok(tail(X:S)) 2.05/2.15 take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S)) 2.05/2.15 take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S)) 2.05/2.15 take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S)) 2.05/2.15 top(mark(X:S)) -> top(proper(X:S)) 2.05/2.15 top(ok(X:S)) -> top(active(X:S)) 2.05/2.15 zip(mark(X1:S),X2:S) -> mark(zip(X1:S,X2:S)) 2.05/2.15 zip(ok(X1:S),ok(X2:S)) -> ok(zip(X1:S,X2:S)) 2.05/2.15 zip(X1:S,mark(X2:S)) -> mark(zip(X1:S,X2:S)) 2.05/2.15 ->Projection: 2.05/2.15 pi(S) = 1 2.05/2.15 2.05/2.15 Problem 1.4: 2.05/2.15 2.05/2.15 SCC Processor: 2.05/2.15 -> Pairs: 2.05/2.15 Empty 2.05/2.15 -> Rules: 2.05/2.15 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 2.05/2.15 active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) 2.05/2.15 active(incr(X:S)) -> incr(active(X:S)) 2.05/2.15 active(pair(X1:S,X2:S)) -> pair(active(X1:S),X2:S) 2.05/2.15 active(pair(X1:S,X2:S)) -> pair(X1:S,active(X2:S)) 2.05/2.15 active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) 2.05/2.15 active(repItems(nil)) -> mark(nil) 2.05/2.15 active(repItems(X:S)) -> repItems(active(X:S)) 2.05/2.15 active(s(X:S)) -> s(active(X:S)) 2.05/2.15 active(tail(cons(X:S,XS:S))) -> mark(XS:S) 2.05/2.15 active(tail(X:S)) -> tail(active(X:S)) 2.05/2.15 active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) 2.05/2.15 active(take(0,XS:S)) -> mark(nil) 2.05/2.15 active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S) 2.05/2.15 active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S)) 2.05/2.15 active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) 2.05/2.15 active(zip(nil,XS:S)) -> mark(nil) 2.05/2.15 active(zip(X:S,nil)) -> mark(nil) 2.05/2.15 active(zip(X1:S,X2:S)) -> zip(active(X1:S),X2:S) 2.05/2.15 active(zip(X1:S,X2:S)) -> zip(X1:S,active(X2:S)) 2.05/2.15 active(oddNs) -> mark(incr(pairNs)) 2.05/2.15 active(pairNs) -> mark(cons(0,incr(oddNs))) 2.05/2.15 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 2.05/2.15 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 2.05/2.15 incr(mark(X:S)) -> mark(incr(X:S)) 2.05/2.15 incr(ok(X:S)) -> ok(incr(X:S)) 2.05/2.15 pair(mark(X1:S),X2:S) -> mark(pair(X1:S,X2:S)) 2.05/2.15 pair(ok(X1:S),ok(X2:S)) -> ok(pair(X1:S,X2:S)) 2.05/2.15 pair(X1:S,mark(X2:S)) -> mark(pair(X1:S,X2:S)) 2.05/2.15 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(incr(X:S)) -> incr(proper(X:S)) 2.05/2.15 proper(pair(X1:S,X2:S)) -> pair(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(repItems(X:S)) -> repItems(proper(X:S)) 2.05/2.15 proper(s(X:S)) -> s(proper(X:S)) 2.05/2.15 proper(tail(X:S)) -> tail(proper(X:S)) 2.05/2.15 proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(zip(X1:S,X2:S)) -> zip(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(0) -> ok(0) 2.05/2.15 proper(nil) -> ok(nil) 2.05/2.15 proper(oddNs) -> ok(oddNs) 2.05/2.15 proper(pairNs) -> ok(pairNs) 2.05/2.15 repItems(mark(X:S)) -> mark(repItems(X:S)) 2.05/2.15 repItems(ok(X:S)) -> ok(repItems(X:S)) 2.05/2.15 s(mark(X:S)) -> mark(s(X:S)) 2.05/2.15 s(ok(X:S)) -> ok(s(X:S)) 2.05/2.15 tail(mark(X:S)) -> mark(tail(X:S)) 2.05/2.15 tail(ok(X:S)) -> ok(tail(X:S)) 2.05/2.15 take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S)) 2.05/2.15 take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S)) 2.05/2.15 take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S)) 2.05/2.15 top(mark(X:S)) -> top(proper(X:S)) 2.05/2.15 top(ok(X:S)) -> top(active(X:S)) 2.05/2.15 zip(mark(X1:S),X2:S) -> mark(zip(X1:S,X2:S)) 2.05/2.15 zip(ok(X1:S),ok(X2:S)) -> ok(zip(X1:S,X2:S)) 2.05/2.15 zip(X1:S,mark(X2:S)) -> mark(zip(X1:S,X2:S)) 2.05/2.15 ->Strongly Connected Components: 2.05/2.15 There is no strongly connected component 2.05/2.15 2.05/2.15 The problem is finite. 2.05/2.15 2.05/2.15 Problem 1.5: 2.05/2.15 2.05/2.15 Subterm Processor: 2.05/2.15 -> Pairs: 2.05/2.15 REPITEMS(mark(X:S)) -> REPITEMS(X:S) 2.05/2.15 REPITEMS(ok(X:S)) -> REPITEMS(X:S) 2.05/2.15 -> Rules: 2.05/2.15 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 2.05/2.15 active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) 2.05/2.15 active(incr(X:S)) -> incr(active(X:S)) 2.05/2.15 active(pair(X1:S,X2:S)) -> pair(active(X1:S),X2:S) 2.05/2.15 active(pair(X1:S,X2:S)) -> pair(X1:S,active(X2:S)) 2.05/2.15 active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) 2.05/2.15 active(repItems(nil)) -> mark(nil) 2.05/2.15 active(repItems(X:S)) -> repItems(active(X:S)) 2.05/2.15 active(s(X:S)) -> s(active(X:S)) 2.05/2.15 active(tail(cons(X:S,XS:S))) -> mark(XS:S) 2.05/2.15 active(tail(X:S)) -> tail(active(X:S)) 2.05/2.15 active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) 2.05/2.15 active(take(0,XS:S)) -> mark(nil) 2.05/2.15 active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S) 2.05/2.15 active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S)) 2.05/2.15 active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) 2.05/2.15 active(zip(nil,XS:S)) -> mark(nil) 2.05/2.15 active(zip(X:S,nil)) -> mark(nil) 2.05/2.15 active(zip(X1:S,X2:S)) -> zip(active(X1:S),X2:S) 2.05/2.15 active(zip(X1:S,X2:S)) -> zip(X1:S,active(X2:S)) 2.05/2.15 active(oddNs) -> mark(incr(pairNs)) 2.05/2.15 active(pairNs) -> mark(cons(0,incr(oddNs))) 2.05/2.15 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 2.05/2.15 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 2.05/2.15 incr(mark(X:S)) -> mark(incr(X:S)) 2.05/2.15 incr(ok(X:S)) -> ok(incr(X:S)) 2.05/2.15 pair(mark(X1:S),X2:S) -> mark(pair(X1:S,X2:S)) 2.05/2.15 pair(ok(X1:S),ok(X2:S)) -> ok(pair(X1:S,X2:S)) 2.05/2.15 pair(X1:S,mark(X2:S)) -> mark(pair(X1:S,X2:S)) 2.05/2.15 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(incr(X:S)) -> incr(proper(X:S)) 2.05/2.15 proper(pair(X1:S,X2:S)) -> pair(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(repItems(X:S)) -> repItems(proper(X:S)) 2.05/2.15 proper(s(X:S)) -> s(proper(X:S)) 2.05/2.15 proper(tail(X:S)) -> tail(proper(X:S)) 2.05/2.15 proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(zip(X1:S,X2:S)) -> zip(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(0) -> ok(0) 2.05/2.15 proper(nil) -> ok(nil) 2.05/2.15 proper(oddNs) -> ok(oddNs) 2.05/2.15 proper(pairNs) -> ok(pairNs) 2.05/2.15 repItems(mark(X:S)) -> mark(repItems(X:S)) 2.05/2.15 repItems(ok(X:S)) -> ok(repItems(X:S)) 2.05/2.15 s(mark(X:S)) -> mark(s(X:S)) 2.05/2.15 s(ok(X:S)) -> ok(s(X:S)) 2.05/2.15 tail(mark(X:S)) -> mark(tail(X:S)) 2.05/2.15 tail(ok(X:S)) -> ok(tail(X:S)) 2.05/2.15 take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S)) 2.05/2.15 take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S)) 2.05/2.15 take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S)) 2.05/2.15 top(mark(X:S)) -> top(proper(X:S)) 2.05/2.15 top(ok(X:S)) -> top(active(X:S)) 2.05/2.15 zip(mark(X1:S),X2:S) -> mark(zip(X1:S,X2:S)) 2.05/2.15 zip(ok(X1:S),ok(X2:S)) -> ok(zip(X1:S,X2:S)) 2.05/2.15 zip(X1:S,mark(X2:S)) -> mark(zip(X1:S,X2:S)) 2.05/2.15 ->Projection: 2.05/2.15 pi(REPITEMS) = 1 2.05/2.15 2.05/2.15 Problem 1.5: 2.05/2.15 2.05/2.15 SCC Processor: 2.05/2.15 -> Pairs: 2.05/2.15 Empty 2.05/2.15 -> Rules: 2.05/2.15 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 2.05/2.15 active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) 2.05/2.15 active(incr(X:S)) -> incr(active(X:S)) 2.05/2.15 active(pair(X1:S,X2:S)) -> pair(active(X1:S),X2:S) 2.05/2.15 active(pair(X1:S,X2:S)) -> pair(X1:S,active(X2:S)) 2.05/2.15 active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) 2.05/2.15 active(repItems(nil)) -> mark(nil) 2.05/2.15 active(repItems(X:S)) -> repItems(active(X:S)) 2.05/2.15 active(s(X:S)) -> s(active(X:S)) 2.05/2.15 active(tail(cons(X:S,XS:S))) -> mark(XS:S) 2.05/2.15 active(tail(X:S)) -> tail(active(X:S)) 2.05/2.15 active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) 2.05/2.15 active(take(0,XS:S)) -> mark(nil) 2.05/2.15 active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S) 2.05/2.15 active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S)) 2.05/2.15 active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) 2.05/2.15 active(zip(nil,XS:S)) -> mark(nil) 2.05/2.15 active(zip(X:S,nil)) -> mark(nil) 2.05/2.15 active(zip(X1:S,X2:S)) -> zip(active(X1:S),X2:S) 2.05/2.15 active(zip(X1:S,X2:S)) -> zip(X1:S,active(X2:S)) 2.05/2.15 active(oddNs) -> mark(incr(pairNs)) 2.05/2.15 active(pairNs) -> mark(cons(0,incr(oddNs))) 2.05/2.15 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 2.05/2.15 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 2.05/2.15 incr(mark(X:S)) -> mark(incr(X:S)) 2.05/2.15 incr(ok(X:S)) -> ok(incr(X:S)) 2.05/2.15 pair(mark(X1:S),X2:S) -> mark(pair(X1:S,X2:S)) 2.05/2.15 pair(ok(X1:S),ok(X2:S)) -> ok(pair(X1:S,X2:S)) 2.05/2.15 pair(X1:S,mark(X2:S)) -> mark(pair(X1:S,X2:S)) 2.05/2.15 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(incr(X:S)) -> incr(proper(X:S)) 2.05/2.15 proper(pair(X1:S,X2:S)) -> pair(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(repItems(X:S)) -> repItems(proper(X:S)) 2.05/2.15 proper(s(X:S)) -> s(proper(X:S)) 2.05/2.15 proper(tail(X:S)) -> tail(proper(X:S)) 2.05/2.15 proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(zip(X1:S,X2:S)) -> zip(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(0) -> ok(0) 2.05/2.15 proper(nil) -> ok(nil) 2.05/2.15 proper(oddNs) -> ok(oddNs) 2.05/2.15 proper(pairNs) -> ok(pairNs) 2.05/2.15 repItems(mark(X:S)) -> mark(repItems(X:S)) 2.05/2.15 repItems(ok(X:S)) -> ok(repItems(X:S)) 2.05/2.15 s(mark(X:S)) -> mark(s(X:S)) 2.05/2.15 s(ok(X:S)) -> ok(s(X:S)) 2.05/2.15 tail(mark(X:S)) -> mark(tail(X:S)) 2.05/2.15 tail(ok(X:S)) -> ok(tail(X:S)) 2.05/2.15 take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S)) 2.05/2.15 take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S)) 2.05/2.15 take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S)) 2.05/2.15 top(mark(X:S)) -> top(proper(X:S)) 2.05/2.15 top(ok(X:S)) -> top(active(X:S)) 2.05/2.15 zip(mark(X1:S),X2:S) -> mark(zip(X1:S,X2:S)) 2.05/2.15 zip(ok(X1:S),ok(X2:S)) -> ok(zip(X1:S,X2:S)) 2.05/2.15 zip(X1:S,mark(X2:S)) -> mark(zip(X1:S,X2:S)) 2.05/2.15 ->Strongly Connected Components: 2.05/2.15 There is no strongly connected component 2.05/2.15 2.05/2.15 The problem is finite. 2.05/2.15 2.05/2.15 Problem 1.6: 2.05/2.15 2.05/2.15 Subterm Processor: 2.05/2.15 -> Pairs: 2.05/2.15 PAIR(mark(X1:S),X2:S) -> PAIR(X1:S,X2:S) 2.05/2.15 PAIR(ok(X1:S),ok(X2:S)) -> PAIR(X1:S,X2:S) 2.05/2.15 PAIR(X1:S,mark(X2:S)) -> PAIR(X1:S,X2:S) 2.05/2.15 -> Rules: 2.05/2.15 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 2.05/2.15 active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) 2.05/2.15 active(incr(X:S)) -> incr(active(X:S)) 2.05/2.15 active(pair(X1:S,X2:S)) -> pair(active(X1:S),X2:S) 2.05/2.15 active(pair(X1:S,X2:S)) -> pair(X1:S,active(X2:S)) 2.05/2.15 active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) 2.05/2.15 active(repItems(nil)) -> mark(nil) 2.05/2.15 active(repItems(X:S)) -> repItems(active(X:S)) 2.05/2.15 active(s(X:S)) -> s(active(X:S)) 2.05/2.15 active(tail(cons(X:S,XS:S))) -> mark(XS:S) 2.05/2.15 active(tail(X:S)) -> tail(active(X:S)) 2.05/2.15 active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) 2.05/2.15 active(take(0,XS:S)) -> mark(nil) 2.05/2.15 active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S) 2.05/2.15 active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S)) 2.05/2.15 active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) 2.05/2.15 active(zip(nil,XS:S)) -> mark(nil) 2.05/2.15 active(zip(X:S,nil)) -> mark(nil) 2.05/2.15 active(zip(X1:S,X2:S)) -> zip(active(X1:S),X2:S) 2.05/2.15 active(zip(X1:S,X2:S)) -> zip(X1:S,active(X2:S)) 2.05/2.15 active(oddNs) -> mark(incr(pairNs)) 2.05/2.15 active(pairNs) -> mark(cons(0,incr(oddNs))) 2.05/2.15 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 2.05/2.15 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 2.05/2.15 incr(mark(X:S)) -> mark(incr(X:S)) 2.05/2.15 incr(ok(X:S)) -> ok(incr(X:S)) 2.05/2.15 pair(mark(X1:S),X2:S) -> mark(pair(X1:S,X2:S)) 2.05/2.15 pair(ok(X1:S),ok(X2:S)) -> ok(pair(X1:S,X2:S)) 2.05/2.15 pair(X1:S,mark(X2:S)) -> mark(pair(X1:S,X2:S)) 2.05/2.15 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(incr(X:S)) -> incr(proper(X:S)) 2.05/2.15 proper(pair(X1:S,X2:S)) -> pair(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(repItems(X:S)) -> repItems(proper(X:S)) 2.05/2.15 proper(s(X:S)) -> s(proper(X:S)) 2.05/2.15 proper(tail(X:S)) -> tail(proper(X:S)) 2.05/2.15 proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(zip(X1:S,X2:S)) -> zip(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(0) -> ok(0) 2.05/2.15 proper(nil) -> ok(nil) 2.05/2.15 proper(oddNs) -> ok(oddNs) 2.05/2.15 proper(pairNs) -> ok(pairNs) 2.05/2.15 repItems(mark(X:S)) -> mark(repItems(X:S)) 2.05/2.15 repItems(ok(X:S)) -> ok(repItems(X:S)) 2.05/2.15 s(mark(X:S)) -> mark(s(X:S)) 2.05/2.15 s(ok(X:S)) -> ok(s(X:S)) 2.05/2.15 tail(mark(X:S)) -> mark(tail(X:S)) 2.05/2.15 tail(ok(X:S)) -> ok(tail(X:S)) 2.05/2.15 take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S)) 2.05/2.15 take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S)) 2.05/2.15 take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S)) 2.05/2.15 top(mark(X:S)) -> top(proper(X:S)) 2.05/2.15 top(ok(X:S)) -> top(active(X:S)) 2.05/2.15 zip(mark(X1:S),X2:S) -> mark(zip(X1:S,X2:S)) 2.05/2.15 zip(ok(X1:S),ok(X2:S)) -> ok(zip(X1:S,X2:S)) 2.05/2.15 zip(X1:S,mark(X2:S)) -> mark(zip(X1:S,X2:S)) 2.05/2.15 ->Projection: 2.05/2.15 pi(PAIR) = 1 2.05/2.15 2.05/2.15 Problem 1.6: 2.05/2.15 2.05/2.15 SCC Processor: 2.05/2.15 -> Pairs: 2.05/2.15 PAIR(X1:S,mark(X2:S)) -> PAIR(X1:S,X2:S) 2.05/2.15 -> Rules: 2.05/2.15 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 2.05/2.15 active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) 2.05/2.15 active(incr(X:S)) -> incr(active(X:S)) 2.05/2.15 active(pair(X1:S,X2:S)) -> pair(active(X1:S),X2:S) 2.05/2.15 active(pair(X1:S,X2:S)) -> pair(X1:S,active(X2:S)) 2.05/2.15 active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) 2.05/2.15 active(repItems(nil)) -> mark(nil) 2.05/2.15 active(repItems(X:S)) -> repItems(active(X:S)) 2.05/2.15 active(s(X:S)) -> s(active(X:S)) 2.05/2.15 active(tail(cons(X:S,XS:S))) -> mark(XS:S) 2.05/2.15 active(tail(X:S)) -> tail(active(X:S)) 2.05/2.15 active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) 2.05/2.15 active(take(0,XS:S)) -> mark(nil) 2.05/2.15 active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S) 2.05/2.15 active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S)) 2.05/2.15 active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) 2.05/2.15 active(zip(nil,XS:S)) -> mark(nil) 2.05/2.15 active(zip(X:S,nil)) -> mark(nil) 2.05/2.15 active(zip(X1:S,X2:S)) -> zip(active(X1:S),X2:S) 2.05/2.15 active(zip(X1:S,X2:S)) -> zip(X1:S,active(X2:S)) 2.05/2.15 active(oddNs) -> mark(incr(pairNs)) 2.05/2.15 active(pairNs) -> mark(cons(0,incr(oddNs))) 2.05/2.15 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 2.05/2.15 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 2.05/2.15 incr(mark(X:S)) -> mark(incr(X:S)) 2.05/2.15 incr(ok(X:S)) -> ok(incr(X:S)) 2.05/2.15 pair(mark(X1:S),X2:S) -> mark(pair(X1:S,X2:S)) 2.05/2.15 pair(ok(X1:S),ok(X2:S)) -> ok(pair(X1:S,X2:S)) 2.05/2.15 pair(X1:S,mark(X2:S)) -> mark(pair(X1:S,X2:S)) 2.05/2.15 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(incr(X:S)) -> incr(proper(X:S)) 2.05/2.15 proper(pair(X1:S,X2:S)) -> pair(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(repItems(X:S)) -> repItems(proper(X:S)) 2.05/2.15 proper(s(X:S)) -> s(proper(X:S)) 2.05/2.15 proper(tail(X:S)) -> tail(proper(X:S)) 2.05/2.15 proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(zip(X1:S,X2:S)) -> zip(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(0) -> ok(0) 2.05/2.15 proper(nil) -> ok(nil) 2.05/2.15 proper(oddNs) -> ok(oddNs) 2.05/2.15 proper(pairNs) -> ok(pairNs) 2.05/2.15 repItems(mark(X:S)) -> mark(repItems(X:S)) 2.05/2.15 repItems(ok(X:S)) -> ok(repItems(X:S)) 2.05/2.15 s(mark(X:S)) -> mark(s(X:S)) 2.05/2.15 s(ok(X:S)) -> ok(s(X:S)) 2.05/2.15 tail(mark(X:S)) -> mark(tail(X:S)) 2.05/2.15 tail(ok(X:S)) -> ok(tail(X:S)) 2.05/2.15 take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S)) 2.05/2.15 take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S)) 2.05/2.15 take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S)) 2.05/2.15 top(mark(X:S)) -> top(proper(X:S)) 2.05/2.15 top(ok(X:S)) -> top(active(X:S)) 2.05/2.15 zip(mark(X1:S),X2:S) -> mark(zip(X1:S,X2:S)) 2.05/2.15 zip(ok(X1:S),ok(X2:S)) -> ok(zip(X1:S,X2:S)) 2.05/2.15 zip(X1:S,mark(X2:S)) -> mark(zip(X1:S,X2:S)) 2.05/2.15 ->Strongly Connected Components: 2.05/2.15 ->->Cycle: 2.05/2.15 ->->-> Pairs: 2.05/2.15 PAIR(X1:S,mark(X2:S)) -> PAIR(X1:S,X2:S) 2.05/2.15 ->->-> Rules: 2.05/2.15 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 2.05/2.15 active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) 2.05/2.15 active(incr(X:S)) -> incr(active(X:S)) 2.05/2.15 active(pair(X1:S,X2:S)) -> pair(active(X1:S),X2:S) 2.05/2.15 active(pair(X1:S,X2:S)) -> pair(X1:S,active(X2:S)) 2.05/2.15 active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) 2.05/2.15 active(repItems(nil)) -> mark(nil) 2.05/2.15 active(repItems(X:S)) -> repItems(active(X:S)) 2.05/2.15 active(s(X:S)) -> s(active(X:S)) 2.05/2.15 active(tail(cons(X:S,XS:S))) -> mark(XS:S) 2.05/2.15 active(tail(X:S)) -> tail(active(X:S)) 2.05/2.15 active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) 2.05/2.15 active(take(0,XS:S)) -> mark(nil) 2.05/2.15 active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S) 2.05/2.15 active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S)) 2.05/2.15 active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) 2.05/2.15 active(zip(nil,XS:S)) -> mark(nil) 2.05/2.15 active(zip(X:S,nil)) -> mark(nil) 2.05/2.15 active(zip(X1:S,X2:S)) -> zip(active(X1:S),X2:S) 2.05/2.15 active(zip(X1:S,X2:S)) -> zip(X1:S,active(X2:S)) 2.05/2.15 active(oddNs) -> mark(incr(pairNs)) 2.05/2.15 active(pairNs) -> mark(cons(0,incr(oddNs))) 2.05/2.15 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 2.05/2.15 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 2.05/2.15 incr(mark(X:S)) -> mark(incr(X:S)) 2.05/2.15 incr(ok(X:S)) -> ok(incr(X:S)) 2.05/2.15 pair(mark(X1:S),X2:S) -> mark(pair(X1:S,X2:S)) 2.05/2.15 pair(ok(X1:S),ok(X2:S)) -> ok(pair(X1:S,X2:S)) 2.05/2.15 pair(X1:S,mark(X2:S)) -> mark(pair(X1:S,X2:S)) 2.05/2.15 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(incr(X:S)) -> incr(proper(X:S)) 2.05/2.15 proper(pair(X1:S,X2:S)) -> pair(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(repItems(X:S)) -> repItems(proper(X:S)) 2.05/2.15 proper(s(X:S)) -> s(proper(X:S)) 2.05/2.15 proper(tail(X:S)) -> tail(proper(X:S)) 2.05/2.15 proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(zip(X1:S,X2:S)) -> zip(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(0) -> ok(0) 2.05/2.15 proper(nil) -> ok(nil) 2.05/2.15 proper(oddNs) -> ok(oddNs) 2.05/2.15 proper(pairNs) -> ok(pairNs) 2.05/2.15 repItems(mark(X:S)) -> mark(repItems(X:S)) 2.05/2.15 repItems(ok(X:S)) -> ok(repItems(X:S)) 2.05/2.15 s(mark(X:S)) -> mark(s(X:S)) 2.05/2.15 s(ok(X:S)) -> ok(s(X:S)) 2.05/2.15 tail(mark(X:S)) -> mark(tail(X:S)) 2.05/2.15 tail(ok(X:S)) -> ok(tail(X:S)) 2.05/2.15 take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S)) 2.05/2.15 take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S)) 2.05/2.15 take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S)) 2.05/2.15 top(mark(X:S)) -> top(proper(X:S)) 2.05/2.15 top(ok(X:S)) -> top(active(X:S)) 2.05/2.15 zip(mark(X1:S),X2:S) -> mark(zip(X1:S,X2:S)) 2.05/2.15 zip(ok(X1:S),ok(X2:S)) -> ok(zip(X1:S,X2:S)) 2.05/2.15 zip(X1:S,mark(X2:S)) -> mark(zip(X1:S,X2:S)) 2.05/2.15 2.05/2.15 Problem 1.6: 2.05/2.15 2.05/2.15 Subterm Processor: 2.05/2.15 -> Pairs: 2.05/2.15 PAIR(X1:S,mark(X2:S)) -> PAIR(X1:S,X2:S) 2.05/2.15 -> Rules: 2.05/2.15 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 2.05/2.15 active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) 2.05/2.15 active(incr(X:S)) -> incr(active(X:S)) 2.05/2.15 active(pair(X1:S,X2:S)) -> pair(active(X1:S),X2:S) 2.05/2.15 active(pair(X1:S,X2:S)) -> pair(X1:S,active(X2:S)) 2.05/2.15 active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) 2.05/2.15 active(repItems(nil)) -> mark(nil) 2.05/2.15 active(repItems(X:S)) -> repItems(active(X:S)) 2.05/2.15 active(s(X:S)) -> s(active(X:S)) 2.05/2.15 active(tail(cons(X:S,XS:S))) -> mark(XS:S) 2.05/2.15 active(tail(X:S)) -> tail(active(X:S)) 2.05/2.15 active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) 2.05/2.15 active(take(0,XS:S)) -> mark(nil) 2.05/2.15 active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S) 2.05/2.15 active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S)) 2.05/2.15 active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) 2.05/2.15 active(zip(nil,XS:S)) -> mark(nil) 2.05/2.15 active(zip(X:S,nil)) -> mark(nil) 2.05/2.15 active(zip(X1:S,X2:S)) -> zip(active(X1:S),X2:S) 2.05/2.15 active(zip(X1:S,X2:S)) -> zip(X1:S,active(X2:S)) 2.05/2.15 active(oddNs) -> mark(incr(pairNs)) 2.05/2.15 active(pairNs) -> mark(cons(0,incr(oddNs))) 2.05/2.15 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 2.05/2.15 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 2.05/2.15 incr(mark(X:S)) -> mark(incr(X:S)) 2.05/2.15 incr(ok(X:S)) -> ok(incr(X:S)) 2.05/2.15 pair(mark(X1:S),X2:S) -> mark(pair(X1:S,X2:S)) 2.05/2.15 pair(ok(X1:S),ok(X2:S)) -> ok(pair(X1:S,X2:S)) 2.05/2.15 pair(X1:S,mark(X2:S)) -> mark(pair(X1:S,X2:S)) 2.05/2.15 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(incr(X:S)) -> incr(proper(X:S)) 2.05/2.15 proper(pair(X1:S,X2:S)) -> pair(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(repItems(X:S)) -> repItems(proper(X:S)) 2.05/2.15 proper(s(X:S)) -> s(proper(X:S)) 2.05/2.15 proper(tail(X:S)) -> tail(proper(X:S)) 2.05/2.15 proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(zip(X1:S,X2:S)) -> zip(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(0) -> ok(0) 2.05/2.15 proper(nil) -> ok(nil) 2.05/2.15 proper(oddNs) -> ok(oddNs) 2.05/2.15 proper(pairNs) -> ok(pairNs) 2.05/2.15 repItems(mark(X:S)) -> mark(repItems(X:S)) 2.05/2.15 repItems(ok(X:S)) -> ok(repItems(X:S)) 2.05/2.15 s(mark(X:S)) -> mark(s(X:S)) 2.05/2.15 s(ok(X:S)) -> ok(s(X:S)) 2.05/2.15 tail(mark(X:S)) -> mark(tail(X:S)) 2.05/2.15 tail(ok(X:S)) -> ok(tail(X:S)) 2.05/2.15 take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S)) 2.05/2.15 take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S)) 2.05/2.15 take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S)) 2.05/2.15 top(mark(X:S)) -> top(proper(X:S)) 2.05/2.15 top(ok(X:S)) -> top(active(X:S)) 2.05/2.15 zip(mark(X1:S),X2:S) -> mark(zip(X1:S,X2:S)) 2.05/2.15 zip(ok(X1:S),ok(X2:S)) -> ok(zip(X1:S,X2:S)) 2.05/2.15 zip(X1:S,mark(X2:S)) -> mark(zip(X1:S,X2:S)) 2.05/2.15 ->Projection: 2.05/2.15 pi(PAIR) = 2 2.05/2.15 2.05/2.15 Problem 1.6: 2.05/2.15 2.05/2.15 SCC Processor: 2.05/2.15 -> Pairs: 2.05/2.15 Empty 2.05/2.15 -> Rules: 2.05/2.15 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 2.05/2.15 active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) 2.05/2.15 active(incr(X:S)) -> incr(active(X:S)) 2.05/2.15 active(pair(X1:S,X2:S)) -> pair(active(X1:S),X2:S) 2.05/2.15 active(pair(X1:S,X2:S)) -> pair(X1:S,active(X2:S)) 2.05/2.15 active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) 2.05/2.15 active(repItems(nil)) -> mark(nil) 2.05/2.15 active(repItems(X:S)) -> repItems(active(X:S)) 2.05/2.15 active(s(X:S)) -> s(active(X:S)) 2.05/2.15 active(tail(cons(X:S,XS:S))) -> mark(XS:S) 2.05/2.15 active(tail(X:S)) -> tail(active(X:S)) 2.05/2.15 active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) 2.05/2.15 active(take(0,XS:S)) -> mark(nil) 2.05/2.15 active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S) 2.05/2.15 active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S)) 2.05/2.15 active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) 2.05/2.15 active(zip(nil,XS:S)) -> mark(nil) 2.05/2.15 active(zip(X:S,nil)) -> mark(nil) 2.05/2.15 active(zip(X1:S,X2:S)) -> zip(active(X1:S),X2:S) 2.05/2.15 active(zip(X1:S,X2:S)) -> zip(X1:S,active(X2:S)) 2.05/2.15 active(oddNs) -> mark(incr(pairNs)) 2.05/2.15 active(pairNs) -> mark(cons(0,incr(oddNs))) 2.05/2.15 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 2.05/2.15 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 2.05/2.15 incr(mark(X:S)) -> mark(incr(X:S)) 2.05/2.15 incr(ok(X:S)) -> ok(incr(X:S)) 2.05/2.15 pair(mark(X1:S),X2:S) -> mark(pair(X1:S,X2:S)) 2.05/2.15 pair(ok(X1:S),ok(X2:S)) -> ok(pair(X1:S,X2:S)) 2.05/2.15 pair(X1:S,mark(X2:S)) -> mark(pair(X1:S,X2:S)) 2.05/2.15 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(incr(X:S)) -> incr(proper(X:S)) 2.05/2.15 proper(pair(X1:S,X2:S)) -> pair(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(repItems(X:S)) -> repItems(proper(X:S)) 2.05/2.15 proper(s(X:S)) -> s(proper(X:S)) 2.05/2.15 proper(tail(X:S)) -> tail(proper(X:S)) 2.05/2.15 proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(zip(X1:S,X2:S)) -> zip(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(0) -> ok(0) 2.05/2.15 proper(nil) -> ok(nil) 2.05/2.15 proper(oddNs) -> ok(oddNs) 2.05/2.15 proper(pairNs) -> ok(pairNs) 2.05/2.15 repItems(mark(X:S)) -> mark(repItems(X:S)) 2.05/2.15 repItems(ok(X:S)) -> ok(repItems(X:S)) 2.05/2.15 s(mark(X:S)) -> mark(s(X:S)) 2.05/2.15 s(ok(X:S)) -> ok(s(X:S)) 2.05/2.15 tail(mark(X:S)) -> mark(tail(X:S)) 2.05/2.15 tail(ok(X:S)) -> ok(tail(X:S)) 2.05/2.15 take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S)) 2.05/2.15 take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S)) 2.05/2.15 take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S)) 2.05/2.15 top(mark(X:S)) -> top(proper(X:S)) 2.05/2.15 top(ok(X:S)) -> top(active(X:S)) 2.05/2.15 zip(mark(X1:S),X2:S) -> mark(zip(X1:S,X2:S)) 2.05/2.15 zip(ok(X1:S),ok(X2:S)) -> ok(zip(X1:S,X2:S)) 2.05/2.15 zip(X1:S,mark(X2:S)) -> mark(zip(X1:S,X2:S)) 2.05/2.15 ->Strongly Connected Components: 2.05/2.15 There is no strongly connected component 2.05/2.15 2.05/2.15 The problem is finite. 2.05/2.15 2.05/2.15 Problem 1.7: 2.05/2.15 2.05/2.15 Subterm Processor: 2.05/2.15 -> Pairs: 2.05/2.15 INCR(mark(X:S)) -> INCR(X:S) 2.05/2.15 INCR(ok(X:S)) -> INCR(X:S) 2.05/2.15 -> Rules: 2.05/2.15 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 2.05/2.15 active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) 2.05/2.15 active(incr(X:S)) -> incr(active(X:S)) 2.05/2.15 active(pair(X1:S,X2:S)) -> pair(active(X1:S),X2:S) 2.05/2.15 active(pair(X1:S,X2:S)) -> pair(X1:S,active(X2:S)) 2.05/2.15 active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) 2.05/2.15 active(repItems(nil)) -> mark(nil) 2.05/2.15 active(repItems(X:S)) -> repItems(active(X:S)) 2.05/2.15 active(s(X:S)) -> s(active(X:S)) 2.05/2.15 active(tail(cons(X:S,XS:S))) -> mark(XS:S) 2.05/2.15 active(tail(X:S)) -> tail(active(X:S)) 2.05/2.15 active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) 2.05/2.15 active(take(0,XS:S)) -> mark(nil) 2.05/2.15 active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S) 2.05/2.15 active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S)) 2.05/2.15 active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) 2.05/2.15 active(zip(nil,XS:S)) -> mark(nil) 2.05/2.15 active(zip(X:S,nil)) -> mark(nil) 2.05/2.15 active(zip(X1:S,X2:S)) -> zip(active(X1:S),X2:S) 2.05/2.15 active(zip(X1:S,X2:S)) -> zip(X1:S,active(X2:S)) 2.05/2.15 active(oddNs) -> mark(incr(pairNs)) 2.05/2.15 active(pairNs) -> mark(cons(0,incr(oddNs))) 2.05/2.15 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 2.05/2.15 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 2.05/2.15 incr(mark(X:S)) -> mark(incr(X:S)) 2.05/2.15 incr(ok(X:S)) -> ok(incr(X:S)) 2.05/2.15 pair(mark(X1:S),X2:S) -> mark(pair(X1:S,X2:S)) 2.05/2.15 pair(ok(X1:S),ok(X2:S)) -> ok(pair(X1:S,X2:S)) 2.05/2.15 pair(X1:S,mark(X2:S)) -> mark(pair(X1:S,X2:S)) 2.05/2.15 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(incr(X:S)) -> incr(proper(X:S)) 2.05/2.15 proper(pair(X1:S,X2:S)) -> pair(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(repItems(X:S)) -> repItems(proper(X:S)) 2.05/2.15 proper(s(X:S)) -> s(proper(X:S)) 2.05/2.15 proper(tail(X:S)) -> tail(proper(X:S)) 2.05/2.15 proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(zip(X1:S,X2:S)) -> zip(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(0) -> ok(0) 2.05/2.15 proper(nil) -> ok(nil) 2.05/2.15 proper(oddNs) -> ok(oddNs) 2.05/2.15 proper(pairNs) -> ok(pairNs) 2.05/2.15 repItems(mark(X:S)) -> mark(repItems(X:S)) 2.05/2.15 repItems(ok(X:S)) -> ok(repItems(X:S)) 2.05/2.15 s(mark(X:S)) -> mark(s(X:S)) 2.05/2.15 s(ok(X:S)) -> ok(s(X:S)) 2.05/2.15 tail(mark(X:S)) -> mark(tail(X:S)) 2.05/2.15 tail(ok(X:S)) -> ok(tail(X:S)) 2.05/2.15 take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S)) 2.05/2.15 take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S)) 2.05/2.15 take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S)) 2.05/2.15 top(mark(X:S)) -> top(proper(X:S)) 2.05/2.15 top(ok(X:S)) -> top(active(X:S)) 2.05/2.15 zip(mark(X1:S),X2:S) -> mark(zip(X1:S,X2:S)) 2.05/2.15 zip(ok(X1:S),ok(X2:S)) -> ok(zip(X1:S,X2:S)) 2.05/2.15 zip(X1:S,mark(X2:S)) -> mark(zip(X1:S,X2:S)) 2.05/2.15 ->Projection: 2.05/2.15 pi(INCR) = 1 2.05/2.15 2.05/2.15 Problem 1.7: 2.05/2.15 2.05/2.15 SCC Processor: 2.05/2.15 -> Pairs: 2.05/2.15 Empty 2.05/2.15 -> Rules: 2.05/2.15 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 2.05/2.15 active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) 2.05/2.15 active(incr(X:S)) -> incr(active(X:S)) 2.05/2.15 active(pair(X1:S,X2:S)) -> pair(active(X1:S),X2:S) 2.05/2.15 active(pair(X1:S,X2:S)) -> pair(X1:S,active(X2:S)) 2.05/2.15 active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) 2.05/2.15 active(repItems(nil)) -> mark(nil) 2.05/2.15 active(repItems(X:S)) -> repItems(active(X:S)) 2.05/2.15 active(s(X:S)) -> s(active(X:S)) 2.05/2.15 active(tail(cons(X:S,XS:S))) -> mark(XS:S) 2.05/2.15 active(tail(X:S)) -> tail(active(X:S)) 2.05/2.15 active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) 2.05/2.15 active(take(0,XS:S)) -> mark(nil) 2.05/2.15 active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S) 2.05/2.15 active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S)) 2.05/2.15 active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) 2.05/2.15 active(zip(nil,XS:S)) -> mark(nil) 2.05/2.15 active(zip(X:S,nil)) -> mark(nil) 2.05/2.15 active(zip(X1:S,X2:S)) -> zip(active(X1:S),X2:S) 2.05/2.15 active(zip(X1:S,X2:S)) -> zip(X1:S,active(X2:S)) 2.05/2.15 active(oddNs) -> mark(incr(pairNs)) 2.05/2.15 active(pairNs) -> mark(cons(0,incr(oddNs))) 2.05/2.15 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 2.05/2.15 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 2.05/2.15 incr(mark(X:S)) -> mark(incr(X:S)) 2.05/2.15 incr(ok(X:S)) -> ok(incr(X:S)) 2.05/2.15 pair(mark(X1:S),X2:S) -> mark(pair(X1:S,X2:S)) 2.05/2.15 pair(ok(X1:S),ok(X2:S)) -> ok(pair(X1:S,X2:S)) 2.05/2.15 pair(X1:S,mark(X2:S)) -> mark(pair(X1:S,X2:S)) 2.05/2.15 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(incr(X:S)) -> incr(proper(X:S)) 2.05/2.15 proper(pair(X1:S,X2:S)) -> pair(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(repItems(X:S)) -> repItems(proper(X:S)) 2.05/2.15 proper(s(X:S)) -> s(proper(X:S)) 2.05/2.15 proper(tail(X:S)) -> tail(proper(X:S)) 2.05/2.15 proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(zip(X1:S,X2:S)) -> zip(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(0) -> ok(0) 2.05/2.15 proper(nil) -> ok(nil) 2.05/2.15 proper(oddNs) -> ok(oddNs) 2.05/2.15 proper(pairNs) -> ok(pairNs) 2.05/2.15 repItems(mark(X:S)) -> mark(repItems(X:S)) 2.05/2.15 repItems(ok(X:S)) -> ok(repItems(X:S)) 2.05/2.15 s(mark(X:S)) -> mark(s(X:S)) 2.05/2.15 s(ok(X:S)) -> ok(s(X:S)) 2.05/2.15 tail(mark(X:S)) -> mark(tail(X:S)) 2.05/2.15 tail(ok(X:S)) -> ok(tail(X:S)) 2.05/2.15 take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S)) 2.05/2.15 take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S)) 2.05/2.15 take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S)) 2.05/2.15 top(mark(X:S)) -> top(proper(X:S)) 2.05/2.15 top(ok(X:S)) -> top(active(X:S)) 2.05/2.15 zip(mark(X1:S),X2:S) -> mark(zip(X1:S,X2:S)) 2.05/2.15 zip(ok(X1:S),ok(X2:S)) -> ok(zip(X1:S,X2:S)) 2.05/2.15 zip(X1:S,mark(X2:S)) -> mark(zip(X1:S,X2:S)) 2.05/2.15 ->Strongly Connected Components: 2.05/2.15 There is no strongly connected component 2.05/2.15 2.05/2.15 The problem is finite. 2.05/2.15 2.05/2.15 Problem 1.8: 2.05/2.15 2.05/2.15 Subterm Processor: 2.05/2.15 -> Pairs: 2.05/2.15 CONS(mark(X1:S),X2:S) -> CONS(X1:S,X2:S) 2.05/2.15 CONS(ok(X1:S),ok(X2:S)) -> CONS(X1:S,X2:S) 2.05/2.15 -> Rules: 2.05/2.15 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 2.05/2.15 active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) 2.05/2.15 active(incr(X:S)) -> incr(active(X:S)) 2.05/2.15 active(pair(X1:S,X2:S)) -> pair(active(X1:S),X2:S) 2.05/2.15 active(pair(X1:S,X2:S)) -> pair(X1:S,active(X2:S)) 2.05/2.15 active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) 2.05/2.15 active(repItems(nil)) -> mark(nil) 2.05/2.15 active(repItems(X:S)) -> repItems(active(X:S)) 2.05/2.15 active(s(X:S)) -> s(active(X:S)) 2.05/2.15 active(tail(cons(X:S,XS:S))) -> mark(XS:S) 2.05/2.15 active(tail(X:S)) -> tail(active(X:S)) 2.05/2.15 active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) 2.05/2.15 active(take(0,XS:S)) -> mark(nil) 2.05/2.15 active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S) 2.05/2.15 active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S)) 2.05/2.15 active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) 2.05/2.15 active(zip(nil,XS:S)) -> mark(nil) 2.05/2.15 active(zip(X:S,nil)) -> mark(nil) 2.05/2.15 active(zip(X1:S,X2:S)) -> zip(active(X1:S),X2:S) 2.05/2.15 active(zip(X1:S,X2:S)) -> zip(X1:S,active(X2:S)) 2.05/2.15 active(oddNs) -> mark(incr(pairNs)) 2.05/2.15 active(pairNs) -> mark(cons(0,incr(oddNs))) 2.05/2.15 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 2.05/2.15 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 2.05/2.15 incr(mark(X:S)) -> mark(incr(X:S)) 2.05/2.15 incr(ok(X:S)) -> ok(incr(X:S)) 2.05/2.15 pair(mark(X1:S),X2:S) -> mark(pair(X1:S,X2:S)) 2.05/2.15 pair(ok(X1:S),ok(X2:S)) -> ok(pair(X1:S,X2:S)) 2.05/2.15 pair(X1:S,mark(X2:S)) -> mark(pair(X1:S,X2:S)) 2.05/2.15 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(incr(X:S)) -> incr(proper(X:S)) 2.05/2.15 proper(pair(X1:S,X2:S)) -> pair(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(repItems(X:S)) -> repItems(proper(X:S)) 2.05/2.15 proper(s(X:S)) -> s(proper(X:S)) 2.05/2.15 proper(tail(X:S)) -> tail(proper(X:S)) 2.05/2.15 proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(zip(X1:S,X2:S)) -> zip(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(0) -> ok(0) 2.05/2.15 proper(nil) -> ok(nil) 2.05/2.15 proper(oddNs) -> ok(oddNs) 2.05/2.15 proper(pairNs) -> ok(pairNs) 2.05/2.15 repItems(mark(X:S)) -> mark(repItems(X:S)) 2.05/2.15 repItems(ok(X:S)) -> ok(repItems(X:S)) 2.05/2.15 s(mark(X:S)) -> mark(s(X:S)) 2.05/2.15 s(ok(X:S)) -> ok(s(X:S)) 2.05/2.15 tail(mark(X:S)) -> mark(tail(X:S)) 2.05/2.15 tail(ok(X:S)) -> ok(tail(X:S)) 2.05/2.15 take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S)) 2.05/2.15 take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S)) 2.05/2.15 take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S)) 2.05/2.15 top(mark(X:S)) -> top(proper(X:S)) 2.05/2.15 top(ok(X:S)) -> top(active(X:S)) 2.05/2.15 zip(mark(X1:S),X2:S) -> mark(zip(X1:S,X2:S)) 2.05/2.15 zip(ok(X1:S),ok(X2:S)) -> ok(zip(X1:S,X2:S)) 2.05/2.15 zip(X1:S,mark(X2:S)) -> mark(zip(X1:S,X2:S)) 2.05/2.15 ->Projection: 2.05/2.15 pi(CONS) = 1 2.05/2.15 2.05/2.15 Problem 1.8: 2.05/2.15 2.05/2.15 SCC Processor: 2.05/2.15 -> Pairs: 2.05/2.15 Empty 2.05/2.15 -> Rules: 2.05/2.15 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 2.05/2.15 active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) 2.05/2.15 active(incr(X:S)) -> incr(active(X:S)) 2.05/2.15 active(pair(X1:S,X2:S)) -> pair(active(X1:S),X2:S) 2.05/2.15 active(pair(X1:S,X2:S)) -> pair(X1:S,active(X2:S)) 2.05/2.15 active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) 2.05/2.15 active(repItems(nil)) -> mark(nil) 2.05/2.15 active(repItems(X:S)) -> repItems(active(X:S)) 2.05/2.15 active(s(X:S)) -> s(active(X:S)) 2.05/2.15 active(tail(cons(X:S,XS:S))) -> mark(XS:S) 2.05/2.15 active(tail(X:S)) -> tail(active(X:S)) 2.05/2.15 active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) 2.05/2.15 active(take(0,XS:S)) -> mark(nil) 2.05/2.15 active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S) 2.05/2.15 active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S)) 2.05/2.15 active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) 2.05/2.15 active(zip(nil,XS:S)) -> mark(nil) 2.05/2.15 active(zip(X:S,nil)) -> mark(nil) 2.05/2.15 active(zip(X1:S,X2:S)) -> zip(active(X1:S),X2:S) 2.05/2.15 active(zip(X1:S,X2:S)) -> zip(X1:S,active(X2:S)) 2.05/2.15 active(oddNs) -> mark(incr(pairNs)) 2.05/2.15 active(pairNs) -> mark(cons(0,incr(oddNs))) 2.05/2.15 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 2.05/2.15 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 2.05/2.15 incr(mark(X:S)) -> mark(incr(X:S)) 2.05/2.15 incr(ok(X:S)) -> ok(incr(X:S)) 2.05/2.15 pair(mark(X1:S),X2:S) -> mark(pair(X1:S,X2:S)) 2.05/2.15 pair(ok(X1:S),ok(X2:S)) -> ok(pair(X1:S,X2:S)) 2.05/2.15 pair(X1:S,mark(X2:S)) -> mark(pair(X1:S,X2:S)) 2.05/2.15 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(incr(X:S)) -> incr(proper(X:S)) 2.05/2.15 proper(pair(X1:S,X2:S)) -> pair(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(repItems(X:S)) -> repItems(proper(X:S)) 2.05/2.15 proper(s(X:S)) -> s(proper(X:S)) 2.05/2.15 proper(tail(X:S)) -> tail(proper(X:S)) 2.05/2.15 proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(zip(X1:S,X2:S)) -> zip(proper(X1:S),proper(X2:S)) 2.05/2.15 proper(0) -> ok(0) 2.05/2.15 proper(nil) -> ok(nil) 2.05/2.15 proper(oddNs) -> ok(oddNs) 2.05/2.15 proper(pairNs) -> ok(pairNs) 2.05/2.15 repItems(mark(X:S)) -> mark(repItems(X:S)) 2.05/2.15 repItems(ok(X:S)) -> ok(repItems(X:S)) 2.05/2.15 s(mark(X:S)) -> mark(s(X:S)) 2.05/2.15 s(ok(X:S)) -> ok(s(X:S)) 2.05/2.15 tail(mark(X:S)) -> mark(tail(X:S)) 2.05/2.15 tail(ok(X:S)) -> ok(tail(X:S)) 2.05/2.15 take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S)) 2.05/2.15 take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S)) 2.05/2.15 take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S)) 2.05/2.15 top(mark(X:S)) -> top(proper(X:S)) 2.05/2.15 top(ok(X:S)) -> top(active(X:S)) 2.05/2.15 zip(mark(X1:S),X2:S) -> mark(zip(X1:S,X2:S)) 2.05/2.15 zip(ok(X1:S),ok(X2:S)) -> ok(zip(X1:S,X2:S)) 2.05/2.15 zip(X1:S,mark(X2:S)) -> mark(zip(X1:S,X2:S)) 2.05/2.15 ->Strongly Connected Components: 2.05/2.15 There is no strongly connected component 2.05/2.15 2.05/2.15 The problem is finite. 2.05/2.15 2.05/2.15 Problem 1.9: 2.05/2.15 2.05/2.15 Subterm Processor: 2.05/2.15 -> Pairs: 2.05/2.15 PROPER(cons(X1:S,X2:S)) -> PROPER(X1:S) 2.05/2.15 PROPER(cons(X1:S,X2:S)) -> PROPER(X2:S) 2.05/2.16 PROPER(incr(X:S)) -> PROPER(X:S) 2.05/2.16 PROPER(pair(X1:S,X2:S)) -> PROPER(X1:S) 2.05/2.16 PROPER(pair(X1:S,X2:S)) -> PROPER(X2:S) 2.05/2.16 PROPER(repItems(X:S)) -> PROPER(X:S) 2.05/2.16 PROPER(s(X:S)) -> PROPER(X:S) 2.05/2.16 PROPER(tail(X:S)) -> PROPER(X:S) 2.05/2.16 PROPER(take(X1:S,X2:S)) -> PROPER(X1:S) 2.05/2.16 PROPER(take(X1:S,X2:S)) -> PROPER(X2:S) 2.05/2.16 PROPER(zip(X1:S,X2:S)) -> PROPER(X1:S) 2.05/2.16 PROPER(zip(X1:S,X2:S)) -> PROPER(X2:S) 2.05/2.16 -> Rules: 2.05/2.16 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 2.05/2.16 active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) 2.05/2.16 active(incr(X:S)) -> incr(active(X:S)) 2.05/2.16 active(pair(X1:S,X2:S)) -> pair(active(X1:S),X2:S) 2.05/2.16 active(pair(X1:S,X2:S)) -> pair(X1:S,active(X2:S)) 2.05/2.16 active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) 2.05/2.16 active(repItems(nil)) -> mark(nil) 2.05/2.16 active(repItems(X:S)) -> repItems(active(X:S)) 2.05/2.16 active(s(X:S)) -> s(active(X:S)) 2.05/2.16 active(tail(cons(X:S,XS:S))) -> mark(XS:S) 2.05/2.16 active(tail(X:S)) -> tail(active(X:S)) 2.05/2.16 active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) 2.05/2.16 active(take(0,XS:S)) -> mark(nil) 2.05/2.16 active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S) 2.05/2.16 active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S)) 2.05/2.16 active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) 2.05/2.16 active(zip(nil,XS:S)) -> mark(nil) 2.05/2.16 active(zip(X:S,nil)) -> mark(nil) 2.05/2.16 active(zip(X1:S,X2:S)) -> zip(active(X1:S),X2:S) 2.05/2.16 active(zip(X1:S,X2:S)) -> zip(X1:S,active(X2:S)) 2.05/2.16 active(oddNs) -> mark(incr(pairNs)) 2.05/2.16 active(pairNs) -> mark(cons(0,incr(oddNs))) 2.05/2.16 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 2.05/2.16 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 2.05/2.16 incr(mark(X:S)) -> mark(incr(X:S)) 2.05/2.16 incr(ok(X:S)) -> ok(incr(X:S)) 2.05/2.16 pair(mark(X1:S),X2:S) -> mark(pair(X1:S,X2:S)) 2.05/2.16 pair(ok(X1:S),ok(X2:S)) -> ok(pair(X1:S,X2:S)) 2.05/2.16 pair(X1:S,mark(X2:S)) -> mark(pair(X1:S,X2:S)) 2.05/2.16 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 2.05/2.16 proper(incr(X:S)) -> incr(proper(X:S)) 2.05/2.16 proper(pair(X1:S,X2:S)) -> pair(proper(X1:S),proper(X2:S)) 2.05/2.16 proper(repItems(X:S)) -> repItems(proper(X:S)) 2.05/2.16 proper(s(X:S)) -> s(proper(X:S)) 2.05/2.16 proper(tail(X:S)) -> tail(proper(X:S)) 2.05/2.16 proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2:S)) 2.05/2.16 proper(zip(X1:S,X2:S)) -> zip(proper(X1:S),proper(X2:S)) 2.05/2.16 proper(0) -> ok(0) 2.05/2.16 proper(nil) -> ok(nil) 2.05/2.16 proper(oddNs) -> ok(oddNs) 2.05/2.16 proper(pairNs) -> ok(pairNs) 2.05/2.16 repItems(mark(X:S)) -> mark(repItems(X:S)) 2.05/2.16 repItems(ok(X:S)) -> ok(repItems(X:S)) 2.05/2.16 s(mark(X:S)) -> mark(s(X:S)) 2.05/2.16 s(ok(X:S)) -> ok(s(X:S)) 2.05/2.16 tail(mark(X:S)) -> mark(tail(X:S)) 2.05/2.16 tail(ok(X:S)) -> ok(tail(X:S)) 2.05/2.16 take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S)) 2.05/2.16 take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S)) 2.05/2.16 take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S)) 2.05/2.16 top(mark(X:S)) -> top(proper(X:S)) 2.05/2.16 top(ok(X:S)) -> top(active(X:S)) 2.05/2.16 zip(mark(X1:S),X2:S) -> mark(zip(X1:S,X2:S)) 2.05/2.16 zip(ok(X1:S),ok(X2:S)) -> ok(zip(X1:S,X2:S)) 2.05/2.16 zip(X1:S,mark(X2:S)) -> mark(zip(X1:S,X2:S)) 2.05/2.16 ->Projection: 2.05/2.16 pi(PROPER) = 1 2.05/2.16 2.05/2.16 Problem 1.9: 2.05/2.16 2.05/2.16 SCC Processor: 2.05/2.16 -> Pairs: 2.05/2.16 Empty 2.05/2.16 -> Rules: 2.05/2.16 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 2.05/2.16 active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) 2.05/2.16 active(incr(X:S)) -> incr(active(X:S)) 2.05/2.16 active(pair(X1:S,X2:S)) -> pair(active(X1:S),X2:S) 2.05/2.16 active(pair(X1:S,X2:S)) -> pair(X1:S,active(X2:S)) 2.05/2.16 active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) 2.05/2.16 active(repItems(nil)) -> mark(nil) 2.05/2.16 active(repItems(X:S)) -> repItems(active(X:S)) 2.05/2.16 active(s(X:S)) -> s(active(X:S)) 2.05/2.16 active(tail(cons(X:S,XS:S))) -> mark(XS:S) 2.05/2.16 active(tail(X:S)) -> tail(active(X:S)) 2.05/2.16 active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) 2.05/2.16 active(take(0,XS:S)) -> mark(nil) 2.05/2.16 active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S) 2.05/2.16 active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S)) 2.05/2.16 active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) 2.05/2.16 active(zip(nil,XS:S)) -> mark(nil) 2.05/2.16 active(zip(X:S,nil)) -> mark(nil) 2.05/2.16 active(zip(X1:S,X2:S)) -> zip(active(X1:S),X2:S) 2.05/2.16 active(zip(X1:S,X2:S)) -> zip(X1:S,active(X2:S)) 2.05/2.16 active(oddNs) -> mark(incr(pairNs)) 2.05/2.16 active(pairNs) -> mark(cons(0,incr(oddNs))) 2.05/2.16 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 2.05/2.16 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 2.05/2.16 incr(mark(X:S)) -> mark(incr(X:S)) 2.05/2.16 incr(ok(X:S)) -> ok(incr(X:S)) 2.05/2.16 pair(mark(X1:S),X2:S) -> mark(pair(X1:S,X2:S)) 2.05/2.16 pair(ok(X1:S),ok(X2:S)) -> ok(pair(X1:S,X2:S)) 2.05/2.16 pair(X1:S,mark(X2:S)) -> mark(pair(X1:S,X2:S)) 2.05/2.16 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 2.05/2.16 proper(incr(X:S)) -> incr(proper(X:S)) 2.05/2.16 proper(pair(X1:S,X2:S)) -> pair(proper(X1:S),proper(X2:S)) 2.05/2.16 proper(repItems(X:S)) -> repItems(proper(X:S)) 2.05/2.16 proper(s(X:S)) -> s(proper(X:S)) 2.05/2.16 proper(tail(X:S)) -> tail(proper(X:S)) 2.05/2.16 proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2:S)) 2.05/2.16 proper(zip(X1:S,X2:S)) -> zip(proper(X1:S),proper(X2:S)) 2.05/2.16 proper(0) -> ok(0) 2.05/2.16 proper(nil) -> ok(nil) 2.05/2.16 proper(oddNs) -> ok(oddNs) 2.05/2.16 proper(pairNs) -> ok(pairNs) 2.05/2.16 repItems(mark(X:S)) -> mark(repItems(X:S)) 2.05/2.16 repItems(ok(X:S)) -> ok(repItems(X:S)) 2.05/2.16 s(mark(X:S)) -> mark(s(X:S)) 2.05/2.16 s(ok(X:S)) -> ok(s(X:S)) 2.05/2.16 tail(mark(X:S)) -> mark(tail(X:S)) 2.05/2.16 tail(ok(X:S)) -> ok(tail(X:S)) 2.05/2.16 take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S)) 2.05/2.16 take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S)) 2.05/2.16 take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S)) 2.05/2.16 top(mark(X:S)) -> top(proper(X:S)) 2.05/2.16 top(ok(X:S)) -> top(active(X:S)) 2.05/2.16 zip(mark(X1:S),X2:S) -> mark(zip(X1:S,X2:S)) 2.05/2.16 zip(ok(X1:S),ok(X2:S)) -> ok(zip(X1:S,X2:S)) 2.05/2.16 zip(X1:S,mark(X2:S)) -> mark(zip(X1:S,X2:S)) 2.05/2.16 ->Strongly Connected Components: 2.05/2.16 There is no strongly connected component 2.05/2.16 2.05/2.16 The problem is finite. 2.05/2.16 2.05/2.16 Problem 1.10: 2.05/2.16 2.05/2.16 Subterm Processor: 2.05/2.16 -> Pairs: 2.05/2.16 ACTIVE(cons(X1:S,X2:S)) -> ACTIVE(X1:S) 2.05/2.16 ACTIVE(incr(X:S)) -> ACTIVE(X:S) 2.05/2.16 ACTIVE(pair(X1:S,X2:S)) -> ACTIVE(X1:S) 2.05/2.16 ACTIVE(pair(X1:S,X2:S)) -> ACTIVE(X2:S) 2.05/2.16 ACTIVE(repItems(X:S)) -> ACTIVE(X:S) 2.05/2.16 ACTIVE(s(X:S)) -> ACTIVE(X:S) 2.05/2.16 ACTIVE(tail(X:S)) -> ACTIVE(X:S) 2.05/2.16 ACTIVE(take(X1:S,X2:S)) -> ACTIVE(X1:S) 2.05/2.16 ACTIVE(take(X1:S,X2:S)) -> ACTIVE(X2:S) 2.05/2.16 ACTIVE(zip(X1:S,X2:S)) -> ACTIVE(X1:S) 2.05/2.16 ACTIVE(zip(X1:S,X2:S)) -> ACTIVE(X2:S) 2.05/2.16 -> Rules: 2.05/2.16 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 2.05/2.16 active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) 2.05/2.16 active(incr(X:S)) -> incr(active(X:S)) 2.05/2.16 active(pair(X1:S,X2:S)) -> pair(active(X1:S),X2:S) 2.05/2.16 active(pair(X1:S,X2:S)) -> pair(X1:S,active(X2:S)) 2.05/2.16 active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) 2.05/2.16 active(repItems(nil)) -> mark(nil) 2.05/2.16 active(repItems(X:S)) -> repItems(active(X:S)) 2.05/2.16 active(s(X:S)) -> s(active(X:S)) 2.05/2.16 active(tail(cons(X:S,XS:S))) -> mark(XS:S) 2.05/2.16 active(tail(X:S)) -> tail(active(X:S)) 2.05/2.16 active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) 2.05/2.16 active(take(0,XS:S)) -> mark(nil) 2.05/2.16 active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S) 2.05/2.16 active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S)) 2.05/2.16 active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) 2.05/2.16 active(zip(nil,XS:S)) -> mark(nil) 2.05/2.16 active(zip(X:S,nil)) -> mark(nil) 2.05/2.16 active(zip(X1:S,X2:S)) -> zip(active(X1:S),X2:S) 2.05/2.16 active(zip(X1:S,X2:S)) -> zip(X1:S,active(X2:S)) 2.05/2.16 active(oddNs) -> mark(incr(pairNs)) 2.05/2.16 active(pairNs) -> mark(cons(0,incr(oddNs))) 2.05/2.16 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 2.05/2.16 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 2.05/2.16 incr(mark(X:S)) -> mark(incr(X:S)) 2.05/2.16 incr(ok(X:S)) -> ok(incr(X:S)) 2.05/2.16 pair(mark(X1:S),X2:S) -> mark(pair(X1:S,X2:S)) 2.05/2.16 pair(ok(X1:S),ok(X2:S)) -> ok(pair(X1:S,X2:S)) 2.05/2.16 pair(X1:S,mark(X2:S)) -> mark(pair(X1:S,X2:S)) 2.05/2.16 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 2.05/2.16 proper(incr(X:S)) -> incr(proper(X:S)) 2.05/2.16 proper(pair(X1:S,X2:S)) -> pair(proper(X1:S),proper(X2:S)) 2.05/2.16 proper(repItems(X:S)) -> repItems(proper(X:S)) 2.05/2.16 proper(s(X:S)) -> s(proper(X:S)) 2.05/2.16 proper(tail(X:S)) -> tail(proper(X:S)) 2.05/2.16 proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2:S)) 2.05/2.16 proper(zip(X1:S,X2:S)) -> zip(proper(X1:S),proper(X2:S)) 2.05/2.16 proper(0) -> ok(0) 2.05/2.16 proper(nil) -> ok(nil) 2.05/2.16 proper(oddNs) -> ok(oddNs) 2.05/2.16 proper(pairNs) -> ok(pairNs) 2.05/2.16 repItems(mark(X:S)) -> mark(repItems(X:S)) 2.05/2.16 repItems(ok(X:S)) -> ok(repItems(X:S)) 2.05/2.16 s(mark(X:S)) -> mark(s(X:S)) 2.05/2.16 s(ok(X:S)) -> ok(s(X:S)) 2.05/2.16 tail(mark(X:S)) -> mark(tail(X:S)) 2.05/2.16 tail(ok(X:S)) -> ok(tail(X:S)) 2.05/2.16 take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S)) 2.05/2.16 take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S)) 2.05/2.16 take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S)) 2.05/2.16 top(mark(X:S)) -> top(proper(X:S)) 2.05/2.16 top(ok(X:S)) -> top(active(X:S)) 2.05/2.16 zip(mark(X1:S),X2:S) -> mark(zip(X1:S,X2:S)) 2.05/2.16 zip(ok(X1:S),ok(X2:S)) -> ok(zip(X1:S,X2:S)) 2.05/2.16 zip(X1:S,mark(X2:S)) -> mark(zip(X1:S,X2:S)) 2.05/2.16 ->Projection: 2.05/2.16 pi(ACTIVE) = 1 2.05/2.16 2.05/2.16 Problem 1.10: 2.05/2.16 2.05/2.16 SCC Processor: 2.05/2.16 -> Pairs: 2.05/2.16 Empty 2.05/2.16 -> Rules: 2.05/2.16 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 2.05/2.16 active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) 2.05/2.16 active(incr(X:S)) -> incr(active(X:S)) 2.05/2.16 active(pair(X1:S,X2:S)) -> pair(active(X1:S),X2:S) 2.05/2.16 active(pair(X1:S,X2:S)) -> pair(X1:S,active(X2:S)) 2.05/2.16 active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) 2.05/2.16 active(repItems(nil)) -> mark(nil) 2.05/2.16 active(repItems(X:S)) -> repItems(active(X:S)) 2.05/2.16 active(s(X:S)) -> s(active(X:S)) 2.05/2.16 active(tail(cons(X:S,XS:S))) -> mark(XS:S) 2.05/2.16 active(tail(X:S)) -> tail(active(X:S)) 2.05/2.16 active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) 2.05/2.16 active(take(0,XS:S)) -> mark(nil) 2.05/2.16 active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S) 2.05/2.16 active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S)) 2.05/2.16 active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) 2.05/2.16 active(zip(nil,XS:S)) -> mark(nil) 2.05/2.16 active(zip(X:S,nil)) -> mark(nil) 2.05/2.16 active(zip(X1:S,X2:S)) -> zip(active(X1:S),X2:S) 2.05/2.16 active(zip(X1:S,X2:S)) -> zip(X1:S,active(X2:S)) 2.05/2.16 active(oddNs) -> mark(incr(pairNs)) 2.05/2.16 active(pairNs) -> mark(cons(0,incr(oddNs))) 2.05/2.16 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 2.05/2.16 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 2.05/2.16 incr(mark(X:S)) -> mark(incr(X:S)) 2.05/2.16 incr(ok(X:S)) -> ok(incr(X:S)) 2.05/2.16 pair(mark(X1:S),X2:S) -> mark(pair(X1:S,X2:S)) 2.05/2.16 pair(ok(X1:S),ok(X2:S)) -> ok(pair(X1:S,X2:S)) 2.05/2.16 pair(X1:S,mark(X2:S)) -> mark(pair(X1:S,X2:S)) 2.05/2.16 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 2.05/2.16 proper(incr(X:S)) -> incr(proper(X:S)) 2.05/2.16 proper(pair(X1:S,X2:S)) -> pair(proper(X1:S),proper(X2:S)) 2.05/2.16 proper(repItems(X:S)) -> repItems(proper(X:S)) 2.05/2.16 proper(s(X:S)) -> s(proper(X:S)) 2.05/2.16 proper(tail(X:S)) -> tail(proper(X:S)) 2.05/2.16 proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2:S)) 2.05/2.16 proper(zip(X1:S,X2:S)) -> zip(proper(X1:S),proper(X2:S)) 2.05/2.16 proper(0) -> ok(0) 2.05/2.16 proper(nil) -> ok(nil) 2.05/2.16 proper(oddNs) -> ok(oddNs) 2.05/2.16 proper(pairNs) -> ok(pairNs) 2.05/2.16 repItems(mark(X:S)) -> mark(repItems(X:S)) 2.05/2.16 repItems(ok(X:S)) -> ok(repItems(X:S)) 2.05/2.16 s(mark(X:S)) -> mark(s(X:S)) 2.05/2.16 s(ok(X:S)) -> ok(s(X:S)) 2.05/2.16 tail(mark(X:S)) -> mark(tail(X:S)) 2.05/2.16 tail(ok(X:S)) -> ok(tail(X:S)) 2.05/2.16 take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S)) 2.05/2.16 take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S)) 2.05/2.16 take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S)) 2.05/2.16 top(mark(X:S)) -> top(proper(X:S)) 2.05/2.16 top(ok(X:S)) -> top(active(X:S)) 2.05/2.16 zip(mark(X1:S),X2:S) -> mark(zip(X1:S,X2:S)) 2.05/2.16 zip(ok(X1:S),ok(X2:S)) -> ok(zip(X1:S,X2:S)) 2.05/2.16 zip(X1:S,mark(X2:S)) -> mark(zip(X1:S,X2:S)) 2.05/2.16 ->Strongly Connected Components: 2.05/2.16 There is no strongly connected component 2.05/2.16 2.05/2.16 The problem is finite. 2.05/2.16 2.05/2.16 Problem 1.11: 2.05/2.16 2.05/2.16 Reduction Pairs Processor: 2.05/2.16 -> Pairs: 2.05/2.16 TOP(mark(X:S)) -> TOP(proper(X:S)) 2.05/2.16 TOP(ok(X:S)) -> TOP(active(X:S)) 2.05/2.16 -> Rules: 2.05/2.16 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 2.05/2.16 active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) 2.05/2.16 active(incr(X:S)) -> incr(active(X:S)) 2.05/2.16 active(pair(X1:S,X2:S)) -> pair(active(X1:S),X2:S) 2.05/2.16 active(pair(X1:S,X2:S)) -> pair(X1:S,active(X2:S)) 2.05/2.16 active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) 2.05/2.16 active(repItems(nil)) -> mark(nil) 2.05/2.16 active(repItems(X:S)) -> repItems(active(X:S)) 2.05/2.16 active(s(X:S)) -> s(active(X:S)) 2.05/2.16 active(tail(cons(X:S,XS:S))) -> mark(XS:S) 2.05/2.16 active(tail(X:S)) -> tail(active(X:S)) 2.05/2.16 active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) 2.05/2.16 active(take(0,XS:S)) -> mark(nil) 2.05/2.16 active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S) 2.05/2.16 active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S)) 2.05/2.16 active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) 2.05/2.16 active(zip(nil,XS:S)) -> mark(nil) 2.05/2.16 active(zip(X:S,nil)) -> mark(nil) 2.05/2.16 active(zip(X1:S,X2:S)) -> zip(active(X1:S),X2:S) 2.05/2.16 active(zip(X1:S,X2:S)) -> zip(X1:S,active(X2:S)) 2.05/2.16 active(oddNs) -> mark(incr(pairNs)) 2.05/2.16 active(pairNs) -> mark(cons(0,incr(oddNs))) 2.05/2.16 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 2.05/2.16 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 2.05/2.16 incr(mark(X:S)) -> mark(incr(X:S)) 2.05/2.16 incr(ok(X:S)) -> ok(incr(X:S)) 2.05/2.16 pair(mark(X1:S),X2:S) -> mark(pair(X1:S,X2:S)) 2.05/2.16 pair(ok(X1:S),ok(X2:S)) -> ok(pair(X1:S,X2:S)) 2.05/2.16 pair(X1:S,mark(X2:S)) -> mark(pair(X1:S,X2:S)) 2.05/2.16 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 2.05/2.16 proper(incr(X:S)) -> incr(proper(X:S)) 2.05/2.16 proper(pair(X1:S,X2:S)) -> pair(proper(X1:S),proper(X2:S)) 2.05/2.16 proper(repItems(X:S)) -> repItems(proper(X:S)) 2.05/2.16 proper(s(X:S)) -> s(proper(X:S)) 2.05/2.16 proper(tail(X:S)) -> tail(proper(X:S)) 2.05/2.16 proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2:S)) 2.05/2.16 proper(zip(X1:S,X2:S)) -> zip(proper(X1:S),proper(X2:S)) 2.05/2.16 proper(0) -> ok(0) 2.05/2.16 proper(nil) -> ok(nil) 2.05/2.16 proper(oddNs) -> ok(oddNs) 2.05/2.16 proper(pairNs) -> ok(pairNs) 2.05/2.16 repItems(mark(X:S)) -> mark(repItems(X:S)) 2.05/2.16 repItems(ok(X:S)) -> ok(repItems(X:S)) 2.05/2.16 s(mark(X:S)) -> mark(s(X:S)) 2.05/2.16 s(ok(X:S)) -> ok(s(X:S)) 2.05/2.16 tail(mark(X:S)) -> mark(tail(X:S)) 2.05/2.16 tail(ok(X:S)) -> ok(tail(X:S)) 2.05/2.16 take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S)) 2.05/2.16 take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S)) 2.05/2.16 take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S)) 2.05/2.16 top(mark(X:S)) -> top(proper(X:S)) 2.05/2.16 top(ok(X:S)) -> top(active(X:S)) 2.05/2.16 zip(mark(X1:S),X2:S) -> mark(zip(X1:S,X2:S)) 2.05/2.16 zip(ok(X1:S),ok(X2:S)) -> ok(zip(X1:S,X2:S)) 2.05/2.16 zip(X1:S,mark(X2:S)) -> mark(zip(X1:S,X2:S)) 2.05/2.16 -> Usable rules: 2.05/2.16 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 2.05/2.16 active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) 2.05/2.16 active(incr(X:S)) -> incr(active(X:S)) 2.05/2.16 active(pair(X1:S,X2:S)) -> pair(active(X1:S),X2:S) 2.05/2.16 active(pair(X1:S,X2:S)) -> pair(X1:S,active(X2:S)) 2.05/2.16 active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) 2.05/2.16 active(repItems(nil)) -> mark(nil) 2.05/2.16 active(repItems(X:S)) -> repItems(active(X:S)) 2.05/2.16 active(s(X:S)) -> s(active(X:S)) 2.05/2.16 active(tail(cons(X:S,XS:S))) -> mark(XS:S) 2.05/2.16 active(tail(X:S)) -> tail(active(X:S)) 2.05/2.16 active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) 2.05/2.16 active(take(0,XS:S)) -> mark(nil) 2.05/2.16 active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S) 2.05/2.16 active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S)) 2.05/2.16 active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) 2.05/2.16 active(zip(nil,XS:S)) -> mark(nil) 2.05/2.16 active(zip(X:S,nil)) -> mark(nil) 2.05/2.16 active(zip(X1:S,X2:S)) -> zip(active(X1:S),X2:S) 2.05/2.16 active(zip(X1:S,X2:S)) -> zip(X1:S,active(X2:S)) 2.05/2.16 active(oddNs) -> mark(incr(pairNs)) 2.05/2.16 active(pairNs) -> mark(cons(0,incr(oddNs))) 2.05/2.16 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 2.05/2.16 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 2.05/2.16 incr(mark(X:S)) -> mark(incr(X:S)) 2.05/2.16 incr(ok(X:S)) -> ok(incr(X:S)) 2.05/2.16 pair(mark(X1:S),X2:S) -> mark(pair(X1:S,X2:S)) 2.05/2.16 pair(ok(X1:S),ok(X2:S)) -> ok(pair(X1:S,X2:S)) 2.05/2.16 pair(X1:S,mark(X2:S)) -> mark(pair(X1:S,X2:S)) 2.05/2.16 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 2.05/2.16 proper(incr(X:S)) -> incr(proper(X:S)) 2.05/2.16 proper(pair(X1:S,X2:S)) -> pair(proper(X1:S),proper(X2:S)) 2.05/2.16 proper(repItems(X:S)) -> repItems(proper(X:S)) 2.05/2.16 proper(s(X:S)) -> s(proper(X:S)) 2.05/2.16 proper(tail(X:S)) -> tail(proper(X:S)) 2.05/2.16 proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2:S)) 2.05/2.16 proper(zip(X1:S,X2:S)) -> zip(proper(X1:S),proper(X2:S)) 2.05/2.16 proper(0) -> ok(0) 2.05/2.16 proper(nil) -> ok(nil) 2.05/2.16 proper(oddNs) -> ok(oddNs) 2.05/2.16 proper(pairNs) -> ok(pairNs) 2.05/2.16 repItems(mark(X:S)) -> mark(repItems(X:S)) 2.05/2.16 repItems(ok(X:S)) -> ok(repItems(X:S)) 2.05/2.16 s(mark(X:S)) -> mark(s(X:S)) 2.05/2.16 s(ok(X:S)) -> ok(s(X:S)) 2.05/2.16 tail(mark(X:S)) -> mark(tail(X:S)) 2.05/2.16 tail(ok(X:S)) -> ok(tail(X:S)) 2.05/2.16 take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S)) 2.05/2.16 take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S)) 2.05/2.16 take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S)) 2.05/2.16 zip(mark(X1:S),X2:S) -> mark(zip(X1:S,X2:S)) 2.05/2.16 zip(ok(X1:S),ok(X2:S)) -> ok(zip(X1:S,X2:S)) 2.05/2.16 zip(X1:S,mark(X2:S)) -> mark(zip(X1:S,X2:S)) 2.05/2.16 ->Interpretation type: 2.05/2.16 Linear 2.05/2.16 ->Coefficients: 2.05/2.16 All rationals 2.05/2.16 ->Dimension: 2.05/2.16 1 2.05/2.16 ->Bound: 2.05/2.16 4 2.05/2.16 ->Interpretation: 2.05/2.16 2.05/2.16 [active](X) = X 2.05/2.16 [cons](X1,X2) = 3/2.X1 + 1/4.X2 2.05/2.16 [incr](X) = 4/3.X + 1/2 2.05/2.16 [pair](X1,X2) = X1 + 4/3.X2 2.05/2.16 [proper](X) = X 2.05/2.16 [repItems](X) = 3.X + 1 2.05/2.16 [s](X) = 4/3.X 2.05/2.16 [tail](X) = 4.X + 2/3 2.05/2.16 [take](X1,X2) = 4.X1 + 3/2.X2 + 4 2.05/2.16 [top](X) = 0 2.05/2.16 [zip](X1,X2) = X1 + 4/3.X2 + 4/3 2.05/2.16 [0] = 0 2.05/2.16 [fSNonEmpty] = 0 2.05/2.16 [mark](X) = X + 1/3 2.05/2.16 [nil] = 3 2.05/2.16 [oddNs] = 4 2.05/2.16 [ok](X) = X 2.05/2.16 [pairNs] = 2 2.05/2.16 [ACTIVE](X) = 0 2.05/2.16 [CONS](X1,X2) = 0 2.05/2.16 [INCR](X) = 0 2.05/2.16 [PAIR](X1,X2) = 0 2.05/2.16 [PROPER](X) = 0 2.05/2.16 [REPITEMS](X) = 0 2.05/2.16 [S](X) = 0 2.05/2.16 [TAIL](X) = 0 2.05/2.16 [TAKE](X1,X2) = 0 2.05/2.16 [TOP](X) = 1/3.X 2.05/2.16 [ZIP](X1,X2) = 0 2.05/2.16 2.05/2.16 Problem 1.11: 2.05/2.16 2.05/2.16 SCC Processor: 2.05/2.16 -> Pairs: 2.05/2.16 TOP(ok(X:S)) -> TOP(active(X:S)) 2.05/2.16 -> Rules: 2.05/2.16 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 2.05/2.16 active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) 2.05/2.16 active(incr(X:S)) -> incr(active(X:S)) 2.05/2.16 active(pair(X1:S,X2:S)) -> pair(active(X1:S),X2:S) 2.05/2.16 active(pair(X1:S,X2:S)) -> pair(X1:S,active(X2:S)) 2.05/2.16 active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) 2.05/2.16 active(repItems(nil)) -> mark(nil) 2.05/2.16 active(repItems(X:S)) -> repItems(active(X:S)) 2.05/2.16 active(s(X:S)) -> s(active(X:S)) 2.05/2.16 active(tail(cons(X:S,XS:S))) -> mark(XS:S) 2.05/2.16 active(tail(X:S)) -> tail(active(X:S)) 2.05/2.16 active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) 2.05/2.16 active(take(0,XS:S)) -> mark(nil) 2.05/2.16 active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S) 2.05/2.16 active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S)) 2.05/2.16 active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) 2.05/2.16 active(zip(nil,XS:S)) -> mark(nil) 2.05/2.16 active(zip(X:S,nil)) -> mark(nil) 2.05/2.16 active(zip(X1:S,X2:S)) -> zip(active(X1:S),X2:S) 2.05/2.16 active(zip(X1:S,X2:S)) -> zip(X1:S,active(X2:S)) 2.05/2.16 active(oddNs) -> mark(incr(pairNs)) 2.05/2.16 active(pairNs) -> mark(cons(0,incr(oddNs))) 2.05/2.16 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 2.05/2.16 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 2.05/2.16 incr(mark(X:S)) -> mark(incr(X:S)) 2.05/2.16 incr(ok(X:S)) -> ok(incr(X:S)) 2.05/2.16 pair(mark(X1:S),X2:S) -> mark(pair(X1:S,X2:S)) 2.05/2.16 pair(ok(X1:S),ok(X2:S)) -> ok(pair(X1:S,X2:S)) 2.05/2.16 pair(X1:S,mark(X2:S)) -> mark(pair(X1:S,X2:S)) 2.05/2.16 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 2.05/2.16 proper(incr(X:S)) -> incr(proper(X:S)) 2.05/2.16 proper(pair(X1:S,X2:S)) -> pair(proper(X1:S),proper(X2:S)) 2.05/2.16 proper(repItems(X:S)) -> repItems(proper(X:S)) 2.05/2.16 proper(s(X:S)) -> s(proper(X:S)) 2.05/2.16 proper(tail(X:S)) -> tail(proper(X:S)) 2.05/2.16 proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2:S)) 2.05/2.16 proper(zip(X1:S,X2:S)) -> zip(proper(X1:S),proper(X2:S)) 2.05/2.16 proper(0) -> ok(0) 2.05/2.16 proper(nil) -> ok(nil) 2.05/2.16 proper(oddNs) -> ok(oddNs) 2.05/2.16 proper(pairNs) -> ok(pairNs) 2.05/2.16 repItems(mark(X:S)) -> mark(repItems(X:S)) 2.05/2.16 repItems(ok(X:S)) -> ok(repItems(X:S)) 2.05/2.16 s(mark(X:S)) -> mark(s(X:S)) 2.05/2.16 s(ok(X:S)) -> ok(s(X:S)) 2.05/2.16 tail(mark(X:S)) -> mark(tail(X:S)) 2.05/2.16 tail(ok(X:S)) -> ok(tail(X:S)) 2.05/2.16 take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S)) 2.05/2.16 take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S)) 2.05/2.16 take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S)) 2.05/2.16 top(mark(X:S)) -> top(proper(X:S)) 2.05/2.16 top(ok(X:S)) -> top(active(X:S)) 2.05/2.16 zip(mark(X1:S),X2:S) -> mark(zip(X1:S,X2:S)) 2.05/2.16 zip(ok(X1:S),ok(X2:S)) -> ok(zip(X1:S,X2:S)) 2.05/2.16 zip(X1:S,mark(X2:S)) -> mark(zip(X1:S,X2:S)) 2.05/2.16 ->Strongly Connected Components: 2.05/2.16 ->->Cycle: 2.05/2.16 ->->-> Pairs: 2.05/2.16 TOP(ok(X:S)) -> TOP(active(X:S)) 2.05/2.16 ->->-> Rules: 2.05/2.16 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 2.05/2.16 active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) 2.05/2.16 active(incr(X:S)) -> incr(active(X:S)) 2.05/2.16 active(pair(X1:S,X2:S)) -> pair(active(X1:S),X2:S) 2.05/2.16 active(pair(X1:S,X2:S)) -> pair(X1:S,active(X2:S)) 2.05/2.16 active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) 2.05/2.16 active(repItems(nil)) -> mark(nil) 2.05/2.16 active(repItems(X:S)) -> repItems(active(X:S)) 2.05/2.16 active(s(X:S)) -> s(active(X:S)) 2.05/2.16 active(tail(cons(X:S,XS:S))) -> mark(XS:S) 2.05/2.16 active(tail(X:S)) -> tail(active(X:S)) 2.05/2.16 active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) 2.05/2.16 active(take(0,XS:S)) -> mark(nil) 2.05/2.16 active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S) 2.05/2.16 active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S)) 2.05/2.16 active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) 2.05/2.16 active(zip(nil,XS:S)) -> mark(nil) 2.05/2.16 active(zip(X:S,nil)) -> mark(nil) 2.05/2.16 active(zip(X1:S,X2:S)) -> zip(active(X1:S),X2:S) 2.05/2.16 active(zip(X1:S,X2:S)) -> zip(X1:S,active(X2:S)) 2.05/2.16 active(oddNs) -> mark(incr(pairNs)) 2.05/2.16 active(pairNs) -> mark(cons(0,incr(oddNs))) 2.05/2.16 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 2.05/2.16 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 2.05/2.16 incr(mark(X:S)) -> mark(incr(X:S)) 2.05/2.16 incr(ok(X:S)) -> ok(incr(X:S)) 2.05/2.16 pair(mark(X1:S),X2:S) -> mark(pair(X1:S,X2:S)) 2.05/2.16 pair(ok(X1:S),ok(X2:S)) -> ok(pair(X1:S,X2:S)) 2.05/2.16 pair(X1:S,mark(X2:S)) -> mark(pair(X1:S,X2:S)) 2.05/2.16 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 2.05/2.16 proper(incr(X:S)) -> incr(proper(X:S)) 2.05/2.16 proper(pair(X1:S,X2:S)) -> pair(proper(X1:S),proper(X2:S)) 2.05/2.16 proper(repItems(X:S)) -> repItems(proper(X:S)) 2.05/2.16 proper(s(X:S)) -> s(proper(X:S)) 2.05/2.16 proper(tail(X:S)) -> tail(proper(X:S)) 2.05/2.16 proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2:S)) 2.05/2.16 proper(zip(X1:S,X2:S)) -> zip(proper(X1:S),proper(X2:S)) 2.05/2.16 proper(0) -> ok(0) 2.05/2.16 proper(nil) -> ok(nil) 2.05/2.16 proper(oddNs) -> ok(oddNs) 2.05/2.16 proper(pairNs) -> ok(pairNs) 2.05/2.16 repItems(mark(X:S)) -> mark(repItems(X:S)) 2.05/2.16 repItems(ok(X:S)) -> ok(repItems(X:S)) 2.05/2.16 s(mark(X:S)) -> mark(s(X:S)) 2.05/2.16 s(ok(X:S)) -> ok(s(X:S)) 2.05/2.16 tail(mark(X:S)) -> mark(tail(X:S)) 2.05/2.16 tail(ok(X:S)) -> ok(tail(X:S)) 2.05/2.16 take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S)) 2.05/2.16 take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S)) 2.05/2.16 take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S)) 2.05/2.16 top(mark(X:S)) -> top(proper(X:S)) 2.05/2.16 top(ok(X:S)) -> top(active(X:S)) 2.05/2.16 zip(mark(X1:S),X2:S) -> mark(zip(X1:S,X2:S)) 2.05/2.16 zip(ok(X1:S),ok(X2:S)) -> ok(zip(X1:S,X2:S)) 2.05/2.16 zip(X1:S,mark(X2:S)) -> mark(zip(X1:S,X2:S)) 2.05/2.16 2.05/2.16 Problem 1.11: 2.05/2.16 2.05/2.16 Reduction Pairs Processor: 2.05/2.16 -> Pairs: 2.05/2.16 TOP(ok(X:S)) -> TOP(active(X:S)) 2.05/2.16 -> Rules: 2.05/2.16 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 2.05/2.16 active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) 2.05/2.16 active(incr(X:S)) -> incr(active(X:S)) 2.05/2.16 active(pair(X1:S,X2:S)) -> pair(active(X1:S),X2:S) 2.05/2.16 active(pair(X1:S,X2:S)) -> pair(X1:S,active(X2:S)) 2.05/2.16 active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) 2.05/2.16 active(repItems(nil)) -> mark(nil) 2.05/2.16 active(repItems(X:S)) -> repItems(active(X:S)) 2.05/2.16 active(s(X:S)) -> s(active(X:S)) 2.05/2.16 active(tail(cons(X:S,XS:S))) -> mark(XS:S) 2.05/2.16 active(tail(X:S)) -> tail(active(X:S)) 2.05/2.16 active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) 2.05/2.16 active(take(0,XS:S)) -> mark(nil) 2.05/2.16 active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S) 2.05/2.16 active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S)) 2.05/2.16 active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) 2.05/2.16 active(zip(nil,XS:S)) -> mark(nil) 2.05/2.16 active(zip(X:S,nil)) -> mark(nil) 2.05/2.16 active(zip(X1:S,X2:S)) -> zip(active(X1:S),X2:S) 2.05/2.16 active(zip(X1:S,X2:S)) -> zip(X1:S,active(X2:S)) 2.05/2.16 active(oddNs) -> mark(incr(pairNs)) 2.05/2.16 active(pairNs) -> mark(cons(0,incr(oddNs))) 2.05/2.16 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 2.05/2.16 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 2.05/2.16 incr(mark(X:S)) -> mark(incr(X:S)) 2.05/2.16 incr(ok(X:S)) -> ok(incr(X:S)) 2.05/2.16 pair(mark(X1:S),X2:S) -> mark(pair(X1:S,X2:S)) 2.05/2.16 pair(ok(X1:S),ok(X2:S)) -> ok(pair(X1:S,X2:S)) 2.05/2.16 pair(X1:S,mark(X2:S)) -> mark(pair(X1:S,X2:S)) 2.05/2.16 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 2.05/2.16 proper(incr(X:S)) -> incr(proper(X:S)) 2.05/2.16 proper(pair(X1:S,X2:S)) -> pair(proper(X1:S),proper(X2:S)) 2.05/2.16 proper(repItems(X:S)) -> repItems(proper(X:S)) 2.05/2.16 proper(s(X:S)) -> s(proper(X:S)) 2.05/2.16 proper(tail(X:S)) -> tail(proper(X:S)) 2.05/2.16 proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2:S)) 2.05/2.16 proper(zip(X1:S,X2:S)) -> zip(proper(X1:S),proper(X2:S)) 2.05/2.16 proper(0) -> ok(0) 2.05/2.16 proper(nil) -> ok(nil) 2.05/2.16 proper(oddNs) -> ok(oddNs) 2.05/2.16 proper(pairNs) -> ok(pairNs) 2.05/2.16 repItems(mark(X:S)) -> mark(repItems(X:S)) 2.05/2.16 repItems(ok(X:S)) -> ok(repItems(X:S)) 2.05/2.16 s(mark(X:S)) -> mark(s(X:S)) 2.05/2.16 s(ok(X:S)) -> ok(s(X:S)) 2.05/2.16 tail(mark(X:S)) -> mark(tail(X:S)) 2.05/2.16 tail(ok(X:S)) -> ok(tail(X:S)) 2.05/2.16 take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S)) 2.05/2.16 take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S)) 2.05/2.16 take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S)) 2.05/2.16 top(mark(X:S)) -> top(proper(X:S)) 2.05/2.16 top(ok(X:S)) -> top(active(X:S)) 2.05/2.16 zip(mark(X1:S),X2:S) -> mark(zip(X1:S,X2:S)) 2.05/2.16 zip(ok(X1:S),ok(X2:S)) -> ok(zip(X1:S,X2:S)) 2.05/2.16 zip(X1:S,mark(X2:S)) -> mark(zip(X1:S,X2:S)) 2.05/2.16 -> Usable rules: 2.05/2.16 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 2.05/2.16 active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) 2.05/2.16 active(incr(X:S)) -> incr(active(X:S)) 2.05/2.16 active(pair(X1:S,X2:S)) -> pair(active(X1:S),X2:S) 2.05/2.16 active(pair(X1:S,X2:S)) -> pair(X1:S,active(X2:S)) 2.05/2.16 active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) 2.05/2.16 active(repItems(nil)) -> mark(nil) 2.05/2.16 active(repItems(X:S)) -> repItems(active(X:S)) 2.05/2.16 active(s(X:S)) -> s(active(X:S)) 2.05/2.16 active(tail(cons(X:S,XS:S))) -> mark(XS:S) 2.05/2.16 active(tail(X:S)) -> tail(active(X:S)) 2.05/2.16 active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) 2.05/2.16 active(take(0,XS:S)) -> mark(nil) 2.05/2.16 active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S) 2.05/2.16 active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S)) 2.05/2.16 active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) 2.05/2.16 active(zip(nil,XS:S)) -> mark(nil) 2.05/2.16 active(zip(X:S,nil)) -> mark(nil) 2.05/2.16 active(zip(X1:S,X2:S)) -> zip(active(X1:S),X2:S) 2.05/2.16 active(zip(X1:S,X2:S)) -> zip(X1:S,active(X2:S)) 2.05/2.16 active(oddNs) -> mark(incr(pairNs)) 2.05/2.16 active(pairNs) -> mark(cons(0,incr(oddNs))) 2.05/2.16 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 2.05/2.16 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 2.05/2.16 incr(mark(X:S)) -> mark(incr(X:S)) 2.05/2.16 incr(ok(X:S)) -> ok(incr(X:S)) 2.05/2.16 pair(mark(X1:S),X2:S) -> mark(pair(X1:S,X2:S)) 2.05/2.16 pair(ok(X1:S),ok(X2:S)) -> ok(pair(X1:S,X2:S)) 2.05/2.16 pair(X1:S,mark(X2:S)) -> mark(pair(X1:S,X2:S)) 2.05/2.16 repItems(mark(X:S)) -> mark(repItems(X:S)) 2.05/2.16 repItems(ok(X:S)) -> ok(repItems(X:S)) 2.05/2.16 s(mark(X:S)) -> mark(s(X:S)) 2.05/2.16 s(ok(X:S)) -> ok(s(X:S)) 2.05/2.16 tail(mark(X:S)) -> mark(tail(X:S)) 2.05/2.16 tail(ok(X:S)) -> ok(tail(X:S)) 2.05/2.16 take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S)) 2.05/2.16 take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S)) 2.05/2.16 take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S)) 2.05/2.16 zip(mark(X1:S),X2:S) -> mark(zip(X1:S,X2:S)) 2.05/2.16 zip(ok(X1:S),ok(X2:S)) -> ok(zip(X1:S,X2:S)) 2.05/2.16 zip(X1:S,mark(X2:S)) -> mark(zip(X1:S,X2:S)) 2.05/2.16 ->Interpretation type: 2.05/2.16 Linear 2.05/2.16 ->Coefficients: 2.05/2.16 Natural Numbers 2.05/2.16 ->Dimension: 2.05/2.16 1 2.05/2.16 ->Bound: 2.05/2.16 2 2.05/2.16 ->Interpretation: 2.05/2.16 2.05/2.16 [active](X) = 2.X 2.05/2.16 [cons](X1,X2) = 2.X1 2.05/2.16 [incr](X) = 2.X + 2 2.05/2.16 [pair](X1,X2) = 2.X1 + 2.X2 + 2 2.05/2.16 [proper](X) = 0 2.05/2.16 [repItems](X) = 2.X + 1 2.05/2.16 [s](X) = 2.X 2.05/2.16 [tail](X) = 2.X + 2 2.05/2.16 [take](X1,X2) = 2.X1 + 2.X2 + 2 2.05/2.16 [top](X) = 0 2.05/2.16 [zip](X1,X2) = 2.X1 + 2.X2 + 1 2.05/2.16 [0] = 2 2.05/2.16 [fSNonEmpty] = 0 2.05/2.16 [mark](X) = 1 2.05/2.16 [nil] = 2 2.05/2.16 [oddNs] = 1 2.05/2.16 [ok](X) = 2.X + 2 2.05/2.16 [pairNs] = 2 2.05/2.16 [ACTIVE](X) = 0 2.05/2.16 [CONS](X1,X2) = 0 2.05/2.16 [INCR](X) = 0 2.05/2.16 [PAIR](X1,X2) = 0 2.05/2.16 [PROPER](X) = 0 2.05/2.16 [REPITEMS](X) = 0 2.05/2.16 [S](X) = 0 2.05/2.16 [TAIL](X) = 0 2.05/2.16 [TAKE](X1,X2) = 0 2.05/2.16 [TOP](X) = 2.X 2.05/2.16 [ZIP](X1,X2) = 0 2.05/2.16 2.05/2.16 Problem 1.11: 2.05/2.16 2.05/2.16 SCC Processor: 2.05/2.16 -> Pairs: 2.05/2.16 Empty 2.05/2.16 -> Rules: 2.05/2.16 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 2.05/2.16 active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) 2.05/2.16 active(incr(X:S)) -> incr(active(X:S)) 2.05/2.16 active(pair(X1:S,X2:S)) -> pair(active(X1:S),X2:S) 2.05/2.16 active(pair(X1:S,X2:S)) -> pair(X1:S,active(X2:S)) 2.05/2.16 active(repItems(cons(X:S,XS:S))) -> mark(cons(X:S,cons(X:S,repItems(XS:S)))) 2.05/2.16 active(repItems(nil)) -> mark(nil) 2.05/2.16 active(repItems(X:S)) -> repItems(active(X:S)) 2.05/2.16 active(s(X:S)) -> s(active(X:S)) 2.05/2.16 active(tail(cons(X:S,XS:S))) -> mark(XS:S) 2.05/2.16 active(tail(X:S)) -> tail(active(X:S)) 2.05/2.16 active(take(s(N:S),cons(X:S,XS:S))) -> mark(cons(X:S,take(N:S,XS:S))) 2.05/2.16 active(take(0,XS:S)) -> mark(nil) 2.05/2.16 active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S) 2.05/2.16 active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S)) 2.05/2.16 active(zip(cons(X:S,XS:S),cons(Y:S,YS:S))) -> mark(cons(pair(X:S,Y:S),zip(XS:S,YS:S))) 2.05/2.16 active(zip(nil,XS:S)) -> mark(nil) 2.05/2.16 active(zip(X:S,nil)) -> mark(nil) 2.05/2.16 active(zip(X1:S,X2:S)) -> zip(active(X1:S),X2:S) 2.05/2.16 active(zip(X1:S,X2:S)) -> zip(X1:S,active(X2:S)) 2.05/2.16 active(oddNs) -> mark(incr(pairNs)) 2.05/2.16 active(pairNs) -> mark(cons(0,incr(oddNs))) 2.05/2.16 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 2.05/2.16 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 2.05/2.16 incr(mark(X:S)) -> mark(incr(X:S)) 2.05/2.16 incr(ok(X:S)) -> ok(incr(X:S)) 2.05/2.16 pair(mark(X1:S),X2:S) -> mark(pair(X1:S,X2:S)) 2.05/2.16 pair(ok(X1:S),ok(X2:S)) -> ok(pair(X1:S,X2:S)) 2.05/2.16 pair(X1:S,mark(X2:S)) -> mark(pair(X1:S,X2:S)) 2.05/2.16 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 2.05/2.16 proper(incr(X:S)) -> incr(proper(X:S)) 2.05/2.16 proper(pair(X1:S,X2:S)) -> pair(proper(X1:S),proper(X2:S)) 2.05/2.16 proper(repItems(X:S)) -> repItems(proper(X:S)) 2.05/2.16 proper(s(X:S)) -> s(proper(X:S)) 2.05/2.16 proper(tail(X:S)) -> tail(proper(X:S)) 2.05/2.16 proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2:S)) 2.05/2.16 proper(zip(X1:S,X2:S)) -> zip(proper(X1:S),proper(X2:S)) 2.05/2.16 proper(0) -> ok(0) 2.05/2.16 proper(nil) -> ok(nil) 2.05/2.16 proper(oddNs) -> ok(oddNs) 2.05/2.16 proper(pairNs) -> ok(pairNs) 2.05/2.16 repItems(mark(X:S)) -> mark(repItems(X:S)) 2.05/2.16 repItems(ok(X:S)) -> ok(repItems(X:S)) 2.05/2.16 s(mark(X:S)) -> mark(s(X:S)) 2.05/2.16 s(ok(X:S)) -> ok(s(X:S)) 2.05/2.16 tail(mark(X:S)) -> mark(tail(X:S)) 2.05/2.16 tail(ok(X:S)) -> ok(tail(X:S)) 2.05/2.16 take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S)) 2.05/2.16 take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S)) 2.05/2.16 take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S)) 2.05/2.16 top(mark(X:S)) -> top(proper(X:S)) 2.05/2.16 top(ok(X:S)) -> top(active(X:S)) 2.05/2.16 zip(mark(X1:S),X2:S) -> mark(zip(X1:S,X2:S)) 2.05/2.16 zip(ok(X1:S),ok(X2:S)) -> ok(zip(X1:S,X2:S)) 2.05/2.16 zip(X1:S,mark(X2:S)) -> mark(zip(X1:S,X2:S)) 2.05/2.16 ->Strongly Connected Components: 2.05/2.16 There is no strongly connected component 2.05/2.16 2.05/2.16 The problem is finite. 2.05/2.16 EOF