0.00/0.52 YES 0.00/0.52 0.00/0.52 Problem 1: 0.00/0.52 0.00/0.52 (VAR v_NonEmpty:S M:S N:S X:S X1:S X2:S X3:S Y:S) 0.00/0.52 (RULES 0.00/0.52 active(filter(cons(X:S,Y:S),s(N:S),M:S)) -> mark(cons(X:S,filter(Y:S,N:S,M:S))) 0.00/0.52 active(filter(cons(X:S,Y:S),0,M:S)) -> mark(cons(0,filter(Y:S,M:S,M:S))) 0.00/0.52 active(nats(N:S)) -> mark(cons(N:S,nats(s(N:S)))) 0.00/0.52 active(sieve(cons(s(N:S),Y:S))) -> mark(cons(s(N:S),sieve(filter(Y:S,N:S,N:S)))) 0.00/0.52 active(sieve(cons(0,Y:S))) -> mark(cons(0,sieve(Y:S))) 0.00/0.52 active(zprimes) -> mark(sieve(nats(s(s(0))))) 0.00/0.52 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.52 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.52 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 0.00/0.52 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 0.00/0.52 filter(active(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(mark(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,active(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,mark(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,X2:S,active(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,X2:S,mark(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) 0.00/0.52 mark(filter(X1:S,X2:S,X3:S)) -> active(filter(mark(X1:S),mark(X2:S),mark(X3:S))) 0.00/0.52 mark(nats(X:S)) -> active(nats(mark(X:S))) 0.00/0.52 mark(s(X:S)) -> active(s(mark(X:S))) 0.00/0.52 mark(sieve(X:S)) -> active(sieve(mark(X:S))) 0.00/0.52 mark(0) -> active(0) 0.00/0.52 mark(zprimes) -> active(zprimes) 0.00/0.52 nats(active(X:S)) -> nats(X:S) 0.00/0.52 nats(mark(X:S)) -> nats(X:S) 0.00/0.52 s(active(X:S)) -> s(X:S) 0.00/0.52 s(mark(X:S)) -> s(X:S) 0.00/0.52 sieve(active(X:S)) -> sieve(X:S) 0.00/0.52 sieve(mark(X:S)) -> sieve(X:S) 0.00/0.52 ) 0.00/0.52 (STRATEGY INNERMOST) 0.00/0.52 0.00/0.52 Problem 1: 0.00/0.52 0.00/0.52 Dependency Pairs Processor: 0.00/0.52 -> Pairs: 0.00/0.52 ACTIVE(filter(cons(X:S,Y:S),s(N:S),M:S)) -> CONS(X:S,filter(Y:S,N:S,M:S)) 0.00/0.52 ACTIVE(filter(cons(X:S,Y:S),s(N:S),M:S)) -> FILTER(Y:S,N:S,M:S) 0.00/0.52 ACTIVE(filter(cons(X:S,Y:S),s(N:S),M:S)) -> MARK(cons(X:S,filter(Y:S,N:S,M:S))) 0.00/0.52 ACTIVE(filter(cons(X:S,Y:S),0,M:S)) -> CONS(0,filter(Y:S,M:S,M:S)) 0.00/0.52 ACTIVE(filter(cons(X:S,Y:S),0,M:S)) -> FILTER(Y:S,M:S,M:S) 0.00/0.52 ACTIVE(filter(cons(X:S,Y:S),0,M:S)) -> MARK(cons(0,filter(Y:S,M:S,M:S))) 0.00/0.52 ACTIVE(nats(N:S)) -> CONS(N:S,nats(s(N:S))) 0.00/0.52 ACTIVE(nats(N:S)) -> MARK(cons(N:S,nats(s(N:S)))) 0.00/0.52 ACTIVE(nats(N:S)) -> NATS(s(N:S)) 0.00/0.52 ACTIVE(nats(N:S)) -> S(N:S) 0.00/0.52 ACTIVE(sieve(cons(s(N:S),Y:S))) -> CONS(s(N:S),sieve(filter(Y:S,N:S,N:S))) 0.00/0.52 ACTIVE(sieve(cons(s(N:S),Y:S))) -> FILTER(Y:S,N:S,N:S) 0.00/0.52 ACTIVE(sieve(cons(s(N:S),Y:S))) -> MARK(cons(s(N:S),sieve(filter(Y:S,N:S,N:S)))) 0.00/0.52 ACTIVE(sieve(cons(s(N:S),Y:S))) -> SIEVE(filter(Y:S,N:S,N:S)) 0.00/0.52 ACTIVE(sieve(cons(0,Y:S))) -> CONS(0,sieve(Y:S)) 0.00/0.52 ACTIVE(sieve(cons(0,Y:S))) -> MARK(cons(0,sieve(Y:S))) 0.00/0.52 ACTIVE(sieve(cons(0,Y:S))) -> SIEVE(Y:S) 0.00/0.52 ACTIVE(zprimes) -> MARK(sieve(nats(s(s(0))))) 0.00/0.52 CONS(active(X1:S),X2:S) -> CONS(X1:S,X2:S) 0.00/0.52 CONS(mark(X1:S),X2:S) -> CONS(X1:S,X2:S) 0.00/0.52 CONS(X1:S,active(X2:S)) -> CONS(X1:S,X2:S) 0.00/0.52 CONS(X1:S,mark(X2:S)) -> CONS(X1:S,X2:S) 0.00/0.52 FILTER(active(X1:S),X2:S,X3:S) -> FILTER(X1:S,X2:S,X3:S) 0.00/0.52 FILTER(mark(X1:S),X2:S,X3:S) -> FILTER(X1:S,X2:S,X3:S) 0.00/0.52 FILTER(X1:S,active(X2:S),X3:S) -> FILTER(X1:S,X2:S,X3:S) 0.00/0.52 FILTER(X1:S,mark(X2:S),X3:S) -> FILTER(X1:S,X2:S,X3:S) 0.00/0.52 FILTER(X1:S,X2:S,active(X3:S)) -> FILTER(X1:S,X2:S,X3:S) 0.00/0.52 FILTER(X1:S,X2:S,mark(X3:S)) -> FILTER(X1:S,X2:S,X3:S) 0.00/0.52 MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) 0.00/0.52 MARK(cons(X1:S,X2:S)) -> CONS(mark(X1:S),X2:S) 0.00/0.52 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.52 MARK(filter(X1:S,X2:S,X3:S)) -> ACTIVE(filter(mark(X1:S),mark(X2:S),mark(X3:S))) 0.00/0.52 MARK(filter(X1:S,X2:S,X3:S)) -> FILTER(mark(X1:S),mark(X2:S),mark(X3:S)) 0.00/0.52 MARK(filter(X1:S,X2:S,X3:S)) -> MARK(X1:S) 0.00/0.52 MARK(filter(X1:S,X2:S,X3:S)) -> MARK(X2:S) 0.00/0.52 MARK(filter(X1:S,X2:S,X3:S)) -> MARK(X3:S) 0.00/0.52 MARK(nats(X:S)) -> ACTIVE(nats(mark(X:S))) 0.00/0.52 MARK(nats(X:S)) -> MARK(X:S) 0.00/0.52 MARK(nats(X:S)) -> NATS(mark(X:S)) 0.00/0.52 MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) 0.00/0.52 MARK(s(X:S)) -> MARK(X:S) 0.00/0.52 MARK(s(X:S)) -> S(mark(X:S)) 0.00/0.52 MARK(sieve(X:S)) -> ACTIVE(sieve(mark(X:S))) 0.00/0.52 MARK(sieve(X:S)) -> MARK(X:S) 0.00/0.52 MARK(sieve(X:S)) -> SIEVE(mark(X:S)) 0.00/0.52 MARK(zprimes) -> ACTIVE(zprimes) 0.00/0.52 NATS(active(X:S)) -> NATS(X:S) 0.00/0.52 NATS(mark(X:S)) -> NATS(X:S) 0.00/0.52 S(active(X:S)) -> S(X:S) 0.00/0.52 S(mark(X:S)) -> S(X:S) 0.00/0.52 SIEVE(active(X:S)) -> SIEVE(X:S) 0.00/0.52 SIEVE(mark(X:S)) -> SIEVE(X:S) 0.00/0.52 -> Rules: 0.00/0.52 active(filter(cons(X:S,Y:S),s(N:S),M:S)) -> mark(cons(X:S,filter(Y:S,N:S,M:S))) 0.00/0.52 active(filter(cons(X:S,Y:S),0,M:S)) -> mark(cons(0,filter(Y:S,M:S,M:S))) 0.00/0.52 active(nats(N:S)) -> mark(cons(N:S,nats(s(N:S)))) 0.00/0.52 active(sieve(cons(s(N:S),Y:S))) -> mark(cons(s(N:S),sieve(filter(Y:S,N:S,N:S)))) 0.00/0.52 active(sieve(cons(0,Y:S))) -> mark(cons(0,sieve(Y:S))) 0.00/0.52 active(zprimes) -> mark(sieve(nats(s(s(0))))) 0.00/0.52 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.52 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.52 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 0.00/0.52 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 0.00/0.52 filter(active(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(mark(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,active(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,mark(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,X2:S,active(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,X2:S,mark(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) 0.00/0.52 mark(filter(X1:S,X2:S,X3:S)) -> active(filter(mark(X1:S),mark(X2:S),mark(X3:S))) 0.00/0.52 mark(nats(X:S)) -> active(nats(mark(X:S))) 0.00/0.52 mark(s(X:S)) -> active(s(mark(X:S))) 0.00/0.52 mark(sieve(X:S)) -> active(sieve(mark(X:S))) 0.00/0.52 mark(0) -> active(0) 0.00/0.52 mark(zprimes) -> active(zprimes) 0.00/0.52 nats(active(X:S)) -> nats(X:S) 0.00/0.52 nats(mark(X:S)) -> nats(X:S) 0.00/0.52 s(active(X:S)) -> s(X:S) 0.00/0.52 s(mark(X:S)) -> s(X:S) 0.00/0.52 sieve(active(X:S)) -> sieve(X:S) 0.00/0.52 sieve(mark(X:S)) -> sieve(X:S) 0.00/0.52 0.00/0.52 Problem 1: 0.00/0.52 0.00/0.52 SCC Processor: 0.00/0.52 -> Pairs: 0.00/0.52 ACTIVE(filter(cons(X:S,Y:S),s(N:S),M:S)) -> CONS(X:S,filter(Y:S,N:S,M:S)) 0.00/0.52 ACTIVE(filter(cons(X:S,Y:S),s(N:S),M:S)) -> FILTER(Y:S,N:S,M:S) 0.00/0.52 ACTIVE(filter(cons(X:S,Y:S),s(N:S),M:S)) -> MARK(cons(X:S,filter(Y:S,N:S,M:S))) 0.00/0.52 ACTIVE(filter(cons(X:S,Y:S),0,M:S)) -> CONS(0,filter(Y:S,M:S,M:S)) 0.00/0.52 ACTIVE(filter(cons(X:S,Y:S),0,M:S)) -> FILTER(Y:S,M:S,M:S) 0.00/0.52 ACTIVE(filter(cons(X:S,Y:S),0,M:S)) -> MARK(cons(0,filter(Y:S,M:S,M:S))) 0.00/0.52 ACTIVE(nats(N:S)) -> CONS(N:S,nats(s(N:S))) 0.00/0.52 ACTIVE(nats(N:S)) -> MARK(cons(N:S,nats(s(N:S)))) 0.00/0.52 ACTIVE(nats(N:S)) -> NATS(s(N:S)) 0.00/0.52 ACTIVE(nats(N:S)) -> S(N:S) 0.00/0.52 ACTIVE(sieve(cons(s(N:S),Y:S))) -> CONS(s(N:S),sieve(filter(Y:S,N:S,N:S))) 0.00/0.52 ACTIVE(sieve(cons(s(N:S),Y:S))) -> FILTER(Y:S,N:S,N:S) 0.00/0.52 ACTIVE(sieve(cons(s(N:S),Y:S))) -> MARK(cons(s(N:S),sieve(filter(Y:S,N:S,N:S)))) 0.00/0.52 ACTIVE(sieve(cons(s(N:S),Y:S))) -> SIEVE(filter(Y:S,N:S,N:S)) 0.00/0.52 ACTIVE(sieve(cons(0,Y:S))) -> CONS(0,sieve(Y:S)) 0.00/0.52 ACTIVE(sieve(cons(0,Y:S))) -> MARK(cons(0,sieve(Y:S))) 0.00/0.52 ACTIVE(sieve(cons(0,Y:S))) -> SIEVE(Y:S) 0.00/0.52 ACTIVE(zprimes) -> MARK(sieve(nats(s(s(0))))) 0.00/0.52 CONS(active(X1:S),X2:S) -> CONS(X1:S,X2:S) 0.00/0.52 CONS(mark(X1:S),X2:S) -> CONS(X1:S,X2:S) 0.00/0.52 CONS(X1:S,active(X2:S)) -> CONS(X1:S,X2:S) 0.00/0.52 CONS(X1:S,mark(X2:S)) -> CONS(X1:S,X2:S) 0.00/0.52 FILTER(active(X1:S),X2:S,X3:S) -> FILTER(X1:S,X2:S,X3:S) 0.00/0.52 FILTER(mark(X1:S),X2:S,X3:S) -> FILTER(X1:S,X2:S,X3:S) 0.00/0.52 FILTER(X1:S,active(X2:S),X3:S) -> FILTER(X1:S,X2:S,X3:S) 0.00/0.52 FILTER(X1:S,mark(X2:S),X3:S) -> FILTER(X1:S,X2:S,X3:S) 0.00/0.52 FILTER(X1:S,X2:S,active(X3:S)) -> FILTER(X1:S,X2:S,X3:S) 0.00/0.52 FILTER(X1:S,X2:S,mark(X3:S)) -> FILTER(X1:S,X2:S,X3:S) 0.00/0.52 MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) 0.00/0.52 MARK(cons(X1:S,X2:S)) -> CONS(mark(X1:S),X2:S) 0.00/0.52 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.52 MARK(filter(X1:S,X2:S,X3:S)) -> ACTIVE(filter(mark(X1:S),mark(X2:S),mark(X3:S))) 0.00/0.52 MARK(filter(X1:S,X2:S,X3:S)) -> FILTER(mark(X1:S),mark(X2:S),mark(X3:S)) 0.00/0.52 MARK(filter(X1:S,X2:S,X3:S)) -> MARK(X1:S) 0.00/0.52 MARK(filter(X1:S,X2:S,X3:S)) -> MARK(X2:S) 0.00/0.52 MARK(filter(X1:S,X2:S,X3:S)) -> MARK(X3:S) 0.00/0.52 MARK(nats(X:S)) -> ACTIVE(nats(mark(X:S))) 0.00/0.52 MARK(nats(X:S)) -> MARK(X:S) 0.00/0.52 MARK(nats(X:S)) -> NATS(mark(X:S)) 0.00/0.52 MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) 0.00/0.52 MARK(s(X:S)) -> MARK(X:S) 0.00/0.52 MARK(s(X:S)) -> S(mark(X:S)) 0.00/0.52 MARK(sieve(X:S)) -> ACTIVE(sieve(mark(X:S))) 0.00/0.52 MARK(sieve(X:S)) -> MARK(X:S) 0.00/0.52 MARK(sieve(X:S)) -> SIEVE(mark(X:S)) 0.00/0.52 MARK(zprimes) -> ACTIVE(zprimes) 0.00/0.52 NATS(active(X:S)) -> NATS(X:S) 0.00/0.52 NATS(mark(X:S)) -> NATS(X:S) 0.00/0.52 S(active(X:S)) -> S(X:S) 0.00/0.52 S(mark(X:S)) -> S(X:S) 0.00/0.52 SIEVE(active(X:S)) -> SIEVE(X:S) 0.00/0.52 SIEVE(mark(X:S)) -> SIEVE(X:S) 0.00/0.52 -> Rules: 0.00/0.52 active(filter(cons(X:S,Y:S),s(N:S),M:S)) -> mark(cons(X:S,filter(Y:S,N:S,M:S))) 0.00/0.52 active(filter(cons(X:S,Y:S),0,M:S)) -> mark(cons(0,filter(Y:S,M:S,M:S))) 0.00/0.52 active(nats(N:S)) -> mark(cons(N:S,nats(s(N:S)))) 0.00/0.52 active(sieve(cons(s(N:S),Y:S))) -> mark(cons(s(N:S),sieve(filter(Y:S,N:S,N:S)))) 0.00/0.52 active(sieve(cons(0,Y:S))) -> mark(cons(0,sieve(Y:S))) 0.00/0.52 active(zprimes) -> mark(sieve(nats(s(s(0))))) 0.00/0.52 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.52 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.52 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 0.00/0.52 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 0.00/0.52 filter(active(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(mark(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,active(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,mark(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,X2:S,active(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,X2:S,mark(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) 0.00/0.52 mark(filter(X1:S,X2:S,X3:S)) -> active(filter(mark(X1:S),mark(X2:S),mark(X3:S))) 0.00/0.52 mark(nats(X:S)) -> active(nats(mark(X:S))) 0.00/0.52 mark(s(X:S)) -> active(s(mark(X:S))) 0.00/0.52 mark(sieve(X:S)) -> active(sieve(mark(X:S))) 0.00/0.52 mark(0) -> active(0) 0.00/0.52 mark(zprimes) -> active(zprimes) 0.00/0.52 nats(active(X:S)) -> nats(X:S) 0.00/0.52 nats(mark(X:S)) -> nats(X:S) 0.00/0.52 s(active(X:S)) -> s(X:S) 0.00/0.52 s(mark(X:S)) -> s(X:S) 0.00/0.52 sieve(active(X:S)) -> sieve(X:S) 0.00/0.52 sieve(mark(X:S)) -> sieve(X:S) 0.00/0.52 ->Strongly Connected Components: 0.00/0.52 ->->Cycle: 0.00/0.52 ->->-> Pairs: 0.00/0.52 SIEVE(active(X:S)) -> SIEVE(X:S) 0.00/0.52 SIEVE(mark(X:S)) -> SIEVE(X:S) 0.00/0.52 ->->-> Rules: 0.00/0.52 active(filter(cons(X:S,Y:S),s(N:S),M:S)) -> mark(cons(X:S,filter(Y:S,N:S,M:S))) 0.00/0.52 active(filter(cons(X:S,Y:S),0,M:S)) -> mark(cons(0,filter(Y:S,M:S,M:S))) 0.00/0.52 active(nats(N:S)) -> mark(cons(N:S,nats(s(N:S)))) 0.00/0.52 active(sieve(cons(s(N:S),Y:S))) -> mark(cons(s(N:S),sieve(filter(Y:S,N:S,N:S)))) 0.00/0.52 active(sieve(cons(0,Y:S))) -> mark(cons(0,sieve(Y:S))) 0.00/0.52 active(zprimes) -> mark(sieve(nats(s(s(0))))) 0.00/0.52 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.52 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.52 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 0.00/0.52 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 0.00/0.52 filter(active(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(mark(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,active(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,mark(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,X2:S,active(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,X2:S,mark(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) 0.00/0.52 mark(filter(X1:S,X2:S,X3:S)) -> active(filter(mark(X1:S),mark(X2:S),mark(X3:S))) 0.00/0.52 mark(nats(X:S)) -> active(nats(mark(X:S))) 0.00/0.52 mark(s(X:S)) -> active(s(mark(X:S))) 0.00/0.52 mark(sieve(X:S)) -> active(sieve(mark(X:S))) 0.00/0.52 mark(0) -> active(0) 0.00/0.52 mark(zprimes) -> active(zprimes) 0.00/0.52 nats(active(X:S)) -> nats(X:S) 0.00/0.52 nats(mark(X:S)) -> nats(X:S) 0.00/0.52 s(active(X:S)) -> s(X:S) 0.00/0.52 s(mark(X:S)) -> s(X:S) 0.00/0.52 sieve(active(X:S)) -> sieve(X:S) 0.00/0.52 sieve(mark(X:S)) -> sieve(X:S) 0.00/0.52 ->->Cycle: 0.00/0.52 ->->-> Pairs: 0.00/0.52 S(active(X:S)) -> S(X:S) 0.00/0.52 S(mark(X:S)) -> S(X:S) 0.00/0.52 ->->-> Rules: 0.00/0.52 active(filter(cons(X:S,Y:S),s(N:S),M:S)) -> mark(cons(X:S,filter(Y:S,N:S,M:S))) 0.00/0.52 active(filter(cons(X:S,Y:S),0,M:S)) -> mark(cons(0,filter(Y:S,M:S,M:S))) 0.00/0.52 active(nats(N:S)) -> mark(cons(N:S,nats(s(N:S)))) 0.00/0.52 active(sieve(cons(s(N:S),Y:S))) -> mark(cons(s(N:S),sieve(filter(Y:S,N:S,N:S)))) 0.00/0.52 active(sieve(cons(0,Y:S))) -> mark(cons(0,sieve(Y:S))) 0.00/0.52 active(zprimes) -> mark(sieve(nats(s(s(0))))) 0.00/0.52 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.52 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.52 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 0.00/0.52 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 0.00/0.52 filter(active(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(mark(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,active(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,mark(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,X2:S,active(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,X2:S,mark(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) 0.00/0.52 mark(filter(X1:S,X2:S,X3:S)) -> active(filter(mark(X1:S),mark(X2:S),mark(X3:S))) 0.00/0.52 mark(nats(X:S)) -> active(nats(mark(X:S))) 0.00/0.52 mark(s(X:S)) -> active(s(mark(X:S))) 0.00/0.52 mark(sieve(X:S)) -> active(sieve(mark(X:S))) 0.00/0.52 mark(0) -> active(0) 0.00/0.52 mark(zprimes) -> active(zprimes) 0.00/0.52 nats(active(X:S)) -> nats(X:S) 0.00/0.52 nats(mark(X:S)) -> nats(X:S) 0.00/0.52 s(active(X:S)) -> s(X:S) 0.00/0.52 s(mark(X:S)) -> s(X:S) 0.00/0.52 sieve(active(X:S)) -> sieve(X:S) 0.00/0.52 sieve(mark(X:S)) -> sieve(X:S) 0.00/0.52 ->->Cycle: 0.00/0.52 ->->-> Pairs: 0.00/0.52 NATS(active(X:S)) -> NATS(X:S) 0.00/0.52 NATS(mark(X:S)) -> NATS(X:S) 0.00/0.52 ->->-> Rules: 0.00/0.52 active(filter(cons(X:S,Y:S),s(N:S),M:S)) -> mark(cons(X:S,filter(Y:S,N:S,M:S))) 0.00/0.52 active(filter(cons(X:S,Y:S),0,M:S)) -> mark(cons(0,filter(Y:S,M:S,M:S))) 0.00/0.52 active(nats(N:S)) -> mark(cons(N:S,nats(s(N:S)))) 0.00/0.52 active(sieve(cons(s(N:S),Y:S))) -> mark(cons(s(N:S),sieve(filter(Y:S,N:S,N:S)))) 0.00/0.52 active(sieve(cons(0,Y:S))) -> mark(cons(0,sieve(Y:S))) 0.00/0.52 active(zprimes) -> mark(sieve(nats(s(s(0))))) 0.00/0.52 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.52 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.52 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 0.00/0.52 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 0.00/0.52 filter(active(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(mark(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,active(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,mark(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,X2:S,active(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,X2:S,mark(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) 0.00/0.52 mark(filter(X1:S,X2:S,X3:S)) -> active(filter(mark(X1:S),mark(X2:S),mark(X3:S))) 0.00/0.52 mark(nats(X:S)) -> active(nats(mark(X:S))) 0.00/0.52 mark(s(X:S)) -> active(s(mark(X:S))) 0.00/0.52 mark(sieve(X:S)) -> active(sieve(mark(X:S))) 0.00/0.52 mark(0) -> active(0) 0.00/0.52 mark(zprimes) -> active(zprimes) 0.00/0.52 nats(active(X:S)) -> nats(X:S) 0.00/0.52 nats(mark(X:S)) -> nats(X:S) 0.00/0.52 s(active(X:S)) -> s(X:S) 0.00/0.52 s(mark(X:S)) -> s(X:S) 0.00/0.52 sieve(active(X:S)) -> sieve(X:S) 0.00/0.52 sieve(mark(X:S)) -> sieve(X:S) 0.00/0.52 ->->Cycle: 0.00/0.52 ->->-> Pairs: 0.00/0.52 FILTER(active(X1:S),X2:S,X3:S) -> FILTER(X1:S,X2:S,X3:S) 0.00/0.52 FILTER(mark(X1:S),X2:S,X3:S) -> FILTER(X1:S,X2:S,X3:S) 0.00/0.52 FILTER(X1:S,active(X2:S),X3:S) -> FILTER(X1:S,X2:S,X3:S) 0.00/0.52 FILTER(X1:S,mark(X2:S),X3:S) -> FILTER(X1:S,X2:S,X3:S) 0.00/0.52 FILTER(X1:S,X2:S,active(X3:S)) -> FILTER(X1:S,X2:S,X3:S) 0.00/0.52 FILTER(X1:S,X2:S,mark(X3:S)) -> FILTER(X1:S,X2:S,X3:S) 0.00/0.52 ->->-> Rules: 0.00/0.52 active(filter(cons(X:S,Y:S),s(N:S),M:S)) -> mark(cons(X:S,filter(Y:S,N:S,M:S))) 0.00/0.52 active(filter(cons(X:S,Y:S),0,M:S)) -> mark(cons(0,filter(Y:S,M:S,M:S))) 0.00/0.52 active(nats(N:S)) -> mark(cons(N:S,nats(s(N:S)))) 0.00/0.52 active(sieve(cons(s(N:S),Y:S))) -> mark(cons(s(N:S),sieve(filter(Y:S,N:S,N:S)))) 0.00/0.52 active(sieve(cons(0,Y:S))) -> mark(cons(0,sieve(Y:S))) 0.00/0.52 active(zprimes) -> mark(sieve(nats(s(s(0))))) 0.00/0.52 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.52 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.52 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 0.00/0.52 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 0.00/0.52 filter(active(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(mark(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,active(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,mark(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,X2:S,active(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,X2:S,mark(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) 0.00/0.52 mark(filter(X1:S,X2:S,X3:S)) -> active(filter(mark(X1:S),mark(X2:S),mark(X3:S))) 0.00/0.52 mark(nats(X:S)) -> active(nats(mark(X:S))) 0.00/0.52 mark(s(X:S)) -> active(s(mark(X:S))) 0.00/0.52 mark(sieve(X:S)) -> active(sieve(mark(X:S))) 0.00/0.52 mark(0) -> active(0) 0.00/0.52 mark(zprimes) -> active(zprimes) 0.00/0.52 nats(active(X:S)) -> nats(X:S) 0.00/0.52 nats(mark(X:S)) -> nats(X:S) 0.00/0.52 s(active(X:S)) -> s(X:S) 0.00/0.52 s(mark(X:S)) -> s(X:S) 0.00/0.52 sieve(active(X:S)) -> sieve(X:S) 0.00/0.52 sieve(mark(X:S)) -> sieve(X:S) 0.00/0.52 ->->Cycle: 0.00/0.52 ->->-> Pairs: 0.00/0.52 CONS(active(X1:S),X2:S) -> CONS(X1:S,X2:S) 0.00/0.52 CONS(mark(X1:S),X2:S) -> CONS(X1:S,X2:S) 0.00/0.52 CONS(X1:S,active(X2:S)) -> CONS(X1:S,X2:S) 0.00/0.52 CONS(X1:S,mark(X2:S)) -> CONS(X1:S,X2:S) 0.00/0.52 ->->-> Rules: 0.00/0.52 active(filter(cons(X:S,Y:S),s(N:S),M:S)) -> mark(cons(X:S,filter(Y:S,N:S,M:S))) 0.00/0.52 active(filter(cons(X:S,Y:S),0,M:S)) -> mark(cons(0,filter(Y:S,M:S,M:S))) 0.00/0.52 active(nats(N:S)) -> mark(cons(N:S,nats(s(N:S)))) 0.00/0.52 active(sieve(cons(s(N:S),Y:S))) -> mark(cons(s(N:S),sieve(filter(Y:S,N:S,N:S)))) 0.00/0.52 active(sieve(cons(0,Y:S))) -> mark(cons(0,sieve(Y:S))) 0.00/0.52 active(zprimes) -> mark(sieve(nats(s(s(0))))) 0.00/0.52 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.52 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.52 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 0.00/0.52 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 0.00/0.52 filter(active(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(mark(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,active(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,mark(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,X2:S,active(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,X2:S,mark(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) 0.00/0.52 mark(filter(X1:S,X2:S,X3:S)) -> active(filter(mark(X1:S),mark(X2:S),mark(X3:S))) 0.00/0.52 mark(nats(X:S)) -> active(nats(mark(X:S))) 0.00/0.52 mark(s(X:S)) -> active(s(mark(X:S))) 0.00/0.52 mark(sieve(X:S)) -> active(sieve(mark(X:S))) 0.00/0.52 mark(0) -> active(0) 0.00/0.52 mark(zprimes) -> active(zprimes) 0.00/0.52 nats(active(X:S)) -> nats(X:S) 0.00/0.52 nats(mark(X:S)) -> nats(X:S) 0.00/0.52 s(active(X:S)) -> s(X:S) 0.00/0.52 s(mark(X:S)) -> s(X:S) 0.00/0.52 sieve(active(X:S)) -> sieve(X:S) 0.00/0.52 sieve(mark(X:S)) -> sieve(X:S) 0.00/0.52 ->->Cycle: 0.00/0.52 ->->-> Pairs: 0.00/0.52 ACTIVE(filter(cons(X:S,Y:S),s(N:S),M:S)) -> MARK(cons(X:S,filter(Y:S,N:S,M:S))) 0.00/0.52 ACTIVE(filter(cons(X:S,Y:S),0,M:S)) -> MARK(cons(0,filter(Y:S,M:S,M:S))) 0.00/0.52 ACTIVE(nats(N:S)) -> MARK(cons(N:S,nats(s(N:S)))) 0.00/0.52 ACTIVE(sieve(cons(s(N:S),Y:S))) -> MARK(cons(s(N:S),sieve(filter(Y:S,N:S,N:S)))) 0.00/0.52 ACTIVE(sieve(cons(0,Y:S))) -> MARK(cons(0,sieve(Y:S))) 0.00/0.52 ACTIVE(zprimes) -> MARK(sieve(nats(s(s(0))))) 0.00/0.52 MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) 0.00/0.52 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.52 MARK(filter(X1:S,X2:S,X3:S)) -> ACTIVE(filter(mark(X1:S),mark(X2:S),mark(X3:S))) 0.00/0.52 MARK(filter(X1:S,X2:S,X3:S)) -> MARK(X1:S) 0.00/0.52 MARK(filter(X1:S,X2:S,X3:S)) -> MARK(X2:S) 0.00/0.52 MARK(filter(X1:S,X2:S,X3:S)) -> MARK(X3:S) 0.00/0.52 MARK(nats(X:S)) -> ACTIVE(nats(mark(X:S))) 0.00/0.52 MARK(nats(X:S)) -> MARK(X:S) 0.00/0.52 MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) 0.00/0.52 MARK(s(X:S)) -> MARK(X:S) 0.00/0.52 MARK(sieve(X:S)) -> ACTIVE(sieve(mark(X:S))) 0.00/0.52 MARK(sieve(X:S)) -> MARK(X:S) 0.00/0.52 MARK(zprimes) -> ACTIVE(zprimes) 0.00/0.52 ->->-> Rules: 0.00/0.52 active(filter(cons(X:S,Y:S),s(N:S),M:S)) -> mark(cons(X:S,filter(Y:S,N:S,M:S))) 0.00/0.52 active(filter(cons(X:S,Y:S),0,M:S)) -> mark(cons(0,filter(Y:S,M:S,M:S))) 0.00/0.52 active(nats(N:S)) -> mark(cons(N:S,nats(s(N:S)))) 0.00/0.52 active(sieve(cons(s(N:S),Y:S))) -> mark(cons(s(N:S),sieve(filter(Y:S,N:S,N:S)))) 0.00/0.52 active(sieve(cons(0,Y:S))) -> mark(cons(0,sieve(Y:S))) 0.00/0.52 active(zprimes) -> mark(sieve(nats(s(s(0))))) 0.00/0.52 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.52 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.52 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 0.00/0.52 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 0.00/0.52 filter(active(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(mark(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,active(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,mark(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,X2:S,active(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,X2:S,mark(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) 0.00/0.52 mark(filter(X1:S,X2:S,X3:S)) -> active(filter(mark(X1:S),mark(X2:S),mark(X3:S))) 0.00/0.52 mark(nats(X:S)) -> active(nats(mark(X:S))) 0.00/0.52 mark(s(X:S)) -> active(s(mark(X:S))) 0.00/0.52 mark(sieve(X:S)) -> active(sieve(mark(X:S))) 0.00/0.52 mark(0) -> active(0) 0.00/0.52 mark(zprimes) -> active(zprimes) 0.00/0.52 nats(active(X:S)) -> nats(X:S) 0.00/0.52 nats(mark(X:S)) -> nats(X:S) 0.00/0.52 s(active(X:S)) -> s(X:S) 0.00/0.52 s(mark(X:S)) -> s(X:S) 0.00/0.52 sieve(active(X:S)) -> sieve(X:S) 0.00/0.52 sieve(mark(X:S)) -> sieve(X:S) 0.00/0.52 0.00/0.52 0.00/0.52 The problem is decomposed in 6 subproblems. 0.00/0.52 0.00/0.52 Problem 1.1: 0.00/0.52 0.00/0.52 Subterm Processor: 0.00/0.52 -> Pairs: 0.00/0.52 SIEVE(active(X:S)) -> SIEVE(X:S) 0.00/0.52 SIEVE(mark(X:S)) -> SIEVE(X:S) 0.00/0.52 -> Rules: 0.00/0.52 active(filter(cons(X:S,Y:S),s(N:S),M:S)) -> mark(cons(X:S,filter(Y:S,N:S,M:S))) 0.00/0.52 active(filter(cons(X:S,Y:S),0,M:S)) -> mark(cons(0,filter(Y:S,M:S,M:S))) 0.00/0.52 active(nats(N:S)) -> mark(cons(N:S,nats(s(N:S)))) 0.00/0.52 active(sieve(cons(s(N:S),Y:S))) -> mark(cons(s(N:S),sieve(filter(Y:S,N:S,N:S)))) 0.00/0.52 active(sieve(cons(0,Y:S))) -> mark(cons(0,sieve(Y:S))) 0.00/0.52 active(zprimes) -> mark(sieve(nats(s(s(0))))) 0.00/0.52 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.52 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.52 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 0.00/0.52 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 0.00/0.52 filter(active(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(mark(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,active(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,mark(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,X2:S,active(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,X2:S,mark(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) 0.00/0.52 mark(filter(X1:S,X2:S,X3:S)) -> active(filter(mark(X1:S),mark(X2:S),mark(X3:S))) 0.00/0.52 mark(nats(X:S)) -> active(nats(mark(X:S))) 0.00/0.52 mark(s(X:S)) -> active(s(mark(X:S))) 0.00/0.52 mark(sieve(X:S)) -> active(sieve(mark(X:S))) 0.00/0.52 mark(0) -> active(0) 0.00/0.52 mark(zprimes) -> active(zprimes) 0.00/0.52 nats(active(X:S)) -> nats(X:S) 0.00/0.52 nats(mark(X:S)) -> nats(X:S) 0.00/0.52 s(active(X:S)) -> s(X:S) 0.00/0.52 s(mark(X:S)) -> s(X:S) 0.00/0.52 sieve(active(X:S)) -> sieve(X:S) 0.00/0.52 sieve(mark(X:S)) -> sieve(X:S) 0.00/0.52 ->Projection: 0.00/0.52 pi(SIEVE) = 1 0.00/0.52 0.00/0.52 Problem 1.1: 0.00/0.52 0.00/0.52 SCC Processor: 0.00/0.52 -> Pairs: 0.00/0.52 Empty 0.00/0.52 -> Rules: 0.00/0.52 active(filter(cons(X:S,Y:S),s(N:S),M:S)) -> mark(cons(X:S,filter(Y:S,N:S,M:S))) 0.00/0.52 active(filter(cons(X:S,Y:S),0,M:S)) -> mark(cons(0,filter(Y:S,M:S,M:S))) 0.00/0.52 active(nats(N:S)) -> mark(cons(N:S,nats(s(N:S)))) 0.00/0.52 active(sieve(cons(s(N:S),Y:S))) -> mark(cons(s(N:S),sieve(filter(Y:S,N:S,N:S)))) 0.00/0.52 active(sieve(cons(0,Y:S))) -> mark(cons(0,sieve(Y:S))) 0.00/0.52 active(zprimes) -> mark(sieve(nats(s(s(0))))) 0.00/0.52 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.52 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.52 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 0.00/0.52 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 0.00/0.52 filter(active(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(mark(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,active(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,mark(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,X2:S,active(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,X2:S,mark(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) 0.00/0.52 mark(filter(X1:S,X2:S,X3:S)) -> active(filter(mark(X1:S),mark(X2:S),mark(X3:S))) 0.00/0.52 mark(nats(X:S)) -> active(nats(mark(X:S))) 0.00/0.52 mark(s(X:S)) -> active(s(mark(X:S))) 0.00/0.52 mark(sieve(X:S)) -> active(sieve(mark(X:S))) 0.00/0.52 mark(0) -> active(0) 0.00/0.52 mark(zprimes) -> active(zprimes) 0.00/0.52 nats(active(X:S)) -> nats(X:S) 0.00/0.52 nats(mark(X:S)) -> nats(X:S) 0.00/0.52 s(active(X:S)) -> s(X:S) 0.00/0.52 s(mark(X:S)) -> s(X:S) 0.00/0.52 sieve(active(X:S)) -> sieve(X:S) 0.00/0.52 sieve(mark(X:S)) -> sieve(X:S) 0.00/0.52 ->Strongly Connected Components: 0.00/0.52 There is no strongly connected component 0.00/0.52 0.00/0.52 The problem is finite. 0.00/0.52 0.00/0.52 Problem 1.2: 0.00/0.52 0.00/0.52 Subterm Processor: 0.00/0.52 -> Pairs: 0.00/0.52 S(active(X:S)) -> S(X:S) 0.00/0.52 S(mark(X:S)) -> S(X:S) 0.00/0.52 -> Rules: 0.00/0.52 active(filter(cons(X:S,Y:S),s(N:S),M:S)) -> mark(cons(X:S,filter(Y:S,N:S,M:S))) 0.00/0.52 active(filter(cons(X:S,Y:S),0,M:S)) -> mark(cons(0,filter(Y:S,M:S,M:S))) 0.00/0.52 active(nats(N:S)) -> mark(cons(N:S,nats(s(N:S)))) 0.00/0.52 active(sieve(cons(s(N:S),Y:S))) -> mark(cons(s(N:S),sieve(filter(Y:S,N:S,N:S)))) 0.00/0.52 active(sieve(cons(0,Y:S))) -> mark(cons(0,sieve(Y:S))) 0.00/0.52 active(zprimes) -> mark(sieve(nats(s(s(0))))) 0.00/0.52 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.52 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.52 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 0.00/0.52 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 0.00/0.52 filter(active(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(mark(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,active(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,mark(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,X2:S,active(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,X2:S,mark(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) 0.00/0.52 mark(filter(X1:S,X2:S,X3:S)) -> active(filter(mark(X1:S),mark(X2:S),mark(X3:S))) 0.00/0.52 mark(nats(X:S)) -> active(nats(mark(X:S))) 0.00/0.52 mark(s(X:S)) -> active(s(mark(X:S))) 0.00/0.52 mark(sieve(X:S)) -> active(sieve(mark(X:S))) 0.00/0.52 mark(0) -> active(0) 0.00/0.52 mark(zprimes) -> active(zprimes) 0.00/0.52 nats(active(X:S)) -> nats(X:S) 0.00/0.52 nats(mark(X:S)) -> nats(X:S) 0.00/0.52 s(active(X:S)) -> s(X:S) 0.00/0.52 s(mark(X:S)) -> s(X:S) 0.00/0.52 sieve(active(X:S)) -> sieve(X:S) 0.00/0.52 sieve(mark(X:S)) -> sieve(X:S) 0.00/0.52 ->Projection: 0.00/0.52 pi(S) = 1 0.00/0.52 0.00/0.52 Problem 1.2: 0.00/0.52 0.00/0.52 SCC Processor: 0.00/0.52 -> Pairs: 0.00/0.52 Empty 0.00/0.52 -> Rules: 0.00/0.52 active(filter(cons(X:S,Y:S),s(N:S),M:S)) -> mark(cons(X:S,filter(Y:S,N:S,M:S))) 0.00/0.52 active(filter(cons(X:S,Y:S),0,M:S)) -> mark(cons(0,filter(Y:S,M:S,M:S))) 0.00/0.52 active(nats(N:S)) -> mark(cons(N:S,nats(s(N:S)))) 0.00/0.52 active(sieve(cons(s(N:S),Y:S))) -> mark(cons(s(N:S),sieve(filter(Y:S,N:S,N:S)))) 0.00/0.52 active(sieve(cons(0,Y:S))) -> mark(cons(0,sieve(Y:S))) 0.00/0.52 active(zprimes) -> mark(sieve(nats(s(s(0))))) 0.00/0.52 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.52 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.52 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 0.00/0.52 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 0.00/0.52 filter(active(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(mark(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,active(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,mark(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,X2:S,active(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,X2:S,mark(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) 0.00/0.52 mark(filter(X1:S,X2:S,X3:S)) -> active(filter(mark(X1:S),mark(X2:S),mark(X3:S))) 0.00/0.52 mark(nats(X:S)) -> active(nats(mark(X:S))) 0.00/0.52 mark(s(X:S)) -> active(s(mark(X:S))) 0.00/0.52 mark(sieve(X:S)) -> active(sieve(mark(X:S))) 0.00/0.52 mark(0) -> active(0) 0.00/0.52 mark(zprimes) -> active(zprimes) 0.00/0.52 nats(active(X:S)) -> nats(X:S) 0.00/0.52 nats(mark(X:S)) -> nats(X:S) 0.00/0.52 s(active(X:S)) -> s(X:S) 0.00/0.52 s(mark(X:S)) -> s(X:S) 0.00/0.52 sieve(active(X:S)) -> sieve(X:S) 0.00/0.52 sieve(mark(X:S)) -> sieve(X:S) 0.00/0.52 ->Strongly Connected Components: 0.00/0.52 There is no strongly connected component 0.00/0.52 0.00/0.52 The problem is finite. 0.00/0.52 0.00/0.52 Problem 1.3: 0.00/0.52 0.00/0.52 Subterm Processor: 0.00/0.52 -> Pairs: 0.00/0.52 NATS(active(X:S)) -> NATS(X:S) 0.00/0.52 NATS(mark(X:S)) -> NATS(X:S) 0.00/0.52 -> Rules: 0.00/0.52 active(filter(cons(X:S,Y:S),s(N:S),M:S)) -> mark(cons(X:S,filter(Y:S,N:S,M:S))) 0.00/0.52 active(filter(cons(X:S,Y:S),0,M:S)) -> mark(cons(0,filter(Y:S,M:S,M:S))) 0.00/0.52 active(nats(N:S)) -> mark(cons(N:S,nats(s(N:S)))) 0.00/0.52 active(sieve(cons(s(N:S),Y:S))) -> mark(cons(s(N:S),sieve(filter(Y:S,N:S,N:S)))) 0.00/0.52 active(sieve(cons(0,Y:S))) -> mark(cons(0,sieve(Y:S))) 0.00/0.52 active(zprimes) -> mark(sieve(nats(s(s(0))))) 0.00/0.52 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.52 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.52 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 0.00/0.52 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 0.00/0.52 filter(active(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(mark(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,active(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,mark(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,X2:S,active(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,X2:S,mark(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) 0.00/0.52 mark(filter(X1:S,X2:S,X3:S)) -> active(filter(mark(X1:S),mark(X2:S),mark(X3:S))) 0.00/0.52 mark(nats(X:S)) -> active(nats(mark(X:S))) 0.00/0.52 mark(s(X:S)) -> active(s(mark(X:S))) 0.00/0.52 mark(sieve(X:S)) -> active(sieve(mark(X:S))) 0.00/0.52 mark(0) -> active(0) 0.00/0.52 mark(zprimes) -> active(zprimes) 0.00/0.52 nats(active(X:S)) -> nats(X:S) 0.00/0.52 nats(mark(X:S)) -> nats(X:S) 0.00/0.52 s(active(X:S)) -> s(X:S) 0.00/0.52 s(mark(X:S)) -> s(X:S) 0.00/0.52 sieve(active(X:S)) -> sieve(X:S) 0.00/0.52 sieve(mark(X:S)) -> sieve(X:S) 0.00/0.52 ->Projection: 0.00/0.52 pi(NATS) = 1 0.00/0.52 0.00/0.52 Problem 1.3: 0.00/0.52 0.00/0.52 SCC Processor: 0.00/0.52 -> Pairs: 0.00/0.52 Empty 0.00/0.52 -> Rules: 0.00/0.52 active(filter(cons(X:S,Y:S),s(N:S),M:S)) -> mark(cons(X:S,filter(Y:S,N:S,M:S))) 0.00/0.52 active(filter(cons(X:S,Y:S),0,M:S)) -> mark(cons(0,filter(Y:S,M:S,M:S))) 0.00/0.52 active(nats(N:S)) -> mark(cons(N:S,nats(s(N:S)))) 0.00/0.52 active(sieve(cons(s(N:S),Y:S))) -> mark(cons(s(N:S),sieve(filter(Y:S,N:S,N:S)))) 0.00/0.52 active(sieve(cons(0,Y:S))) -> mark(cons(0,sieve(Y:S))) 0.00/0.52 active(zprimes) -> mark(sieve(nats(s(s(0))))) 0.00/0.52 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.52 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.52 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 0.00/0.52 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 0.00/0.52 filter(active(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(mark(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,active(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,mark(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,X2:S,active(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,X2:S,mark(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) 0.00/0.52 mark(filter(X1:S,X2:S,X3:S)) -> active(filter(mark(X1:S),mark(X2:S),mark(X3:S))) 0.00/0.52 mark(nats(X:S)) -> active(nats(mark(X:S))) 0.00/0.52 mark(s(X:S)) -> active(s(mark(X:S))) 0.00/0.52 mark(sieve(X:S)) -> active(sieve(mark(X:S))) 0.00/0.52 mark(0) -> active(0) 0.00/0.52 mark(zprimes) -> active(zprimes) 0.00/0.52 nats(active(X:S)) -> nats(X:S) 0.00/0.52 nats(mark(X:S)) -> nats(X:S) 0.00/0.52 s(active(X:S)) -> s(X:S) 0.00/0.52 s(mark(X:S)) -> s(X:S) 0.00/0.52 sieve(active(X:S)) -> sieve(X:S) 0.00/0.52 sieve(mark(X:S)) -> sieve(X:S) 0.00/0.52 ->Strongly Connected Components: 0.00/0.52 There is no strongly connected component 0.00/0.52 0.00/0.52 The problem is finite. 0.00/0.52 0.00/0.52 Problem 1.4: 0.00/0.52 0.00/0.52 Subterm Processor: 0.00/0.52 -> Pairs: 0.00/0.52 FILTER(active(X1:S),X2:S,X3:S) -> FILTER(X1:S,X2:S,X3:S) 0.00/0.52 FILTER(mark(X1:S),X2:S,X3:S) -> FILTER(X1:S,X2:S,X3:S) 0.00/0.52 FILTER(X1:S,active(X2:S),X3:S) -> FILTER(X1:S,X2:S,X3:S) 0.00/0.52 FILTER(X1:S,mark(X2:S),X3:S) -> FILTER(X1:S,X2:S,X3:S) 0.00/0.52 FILTER(X1:S,X2:S,active(X3:S)) -> FILTER(X1:S,X2:S,X3:S) 0.00/0.52 FILTER(X1:S,X2:S,mark(X3:S)) -> FILTER(X1:S,X2:S,X3:S) 0.00/0.52 -> Rules: 0.00/0.52 active(filter(cons(X:S,Y:S),s(N:S),M:S)) -> mark(cons(X:S,filter(Y:S,N:S,M:S))) 0.00/0.52 active(filter(cons(X:S,Y:S),0,M:S)) -> mark(cons(0,filter(Y:S,M:S,M:S))) 0.00/0.52 active(nats(N:S)) -> mark(cons(N:S,nats(s(N:S)))) 0.00/0.52 active(sieve(cons(s(N:S),Y:S))) -> mark(cons(s(N:S),sieve(filter(Y:S,N:S,N:S)))) 0.00/0.52 active(sieve(cons(0,Y:S))) -> mark(cons(0,sieve(Y:S))) 0.00/0.52 active(zprimes) -> mark(sieve(nats(s(s(0))))) 0.00/0.52 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.52 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.52 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 0.00/0.52 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 0.00/0.52 filter(active(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(mark(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,active(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,mark(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,X2:S,active(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,X2:S,mark(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) 0.00/0.52 mark(filter(X1:S,X2:S,X3:S)) -> active(filter(mark(X1:S),mark(X2:S),mark(X3:S))) 0.00/0.52 mark(nats(X:S)) -> active(nats(mark(X:S))) 0.00/0.52 mark(s(X:S)) -> active(s(mark(X:S))) 0.00/0.52 mark(sieve(X:S)) -> active(sieve(mark(X:S))) 0.00/0.52 mark(0) -> active(0) 0.00/0.52 mark(zprimes) -> active(zprimes) 0.00/0.52 nats(active(X:S)) -> nats(X:S) 0.00/0.52 nats(mark(X:S)) -> nats(X:S) 0.00/0.52 s(active(X:S)) -> s(X:S) 0.00/0.52 s(mark(X:S)) -> s(X:S) 0.00/0.52 sieve(active(X:S)) -> sieve(X:S) 0.00/0.52 sieve(mark(X:S)) -> sieve(X:S) 0.00/0.52 ->Projection: 0.00/0.52 pi(FILTER) = 1 0.00/0.52 0.00/0.52 Problem 1.4: 0.00/0.52 0.00/0.52 SCC Processor: 0.00/0.52 -> Pairs: 0.00/0.52 FILTER(X1:S,active(X2:S),X3:S) -> FILTER(X1:S,X2:S,X3:S) 0.00/0.52 FILTER(X1:S,mark(X2:S),X3:S) -> FILTER(X1:S,X2:S,X3:S) 0.00/0.52 FILTER(X1:S,X2:S,active(X3:S)) -> FILTER(X1:S,X2:S,X3:S) 0.00/0.52 FILTER(X1:S,X2:S,mark(X3:S)) -> FILTER(X1:S,X2:S,X3:S) 0.00/0.52 -> Rules: 0.00/0.52 active(filter(cons(X:S,Y:S),s(N:S),M:S)) -> mark(cons(X:S,filter(Y:S,N:S,M:S))) 0.00/0.52 active(filter(cons(X:S,Y:S),0,M:S)) -> mark(cons(0,filter(Y:S,M:S,M:S))) 0.00/0.52 active(nats(N:S)) -> mark(cons(N:S,nats(s(N:S)))) 0.00/0.52 active(sieve(cons(s(N:S),Y:S))) -> mark(cons(s(N:S),sieve(filter(Y:S,N:S,N:S)))) 0.00/0.52 active(sieve(cons(0,Y:S))) -> mark(cons(0,sieve(Y:S))) 0.00/0.52 active(zprimes) -> mark(sieve(nats(s(s(0))))) 0.00/0.52 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.52 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.52 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 0.00/0.52 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 0.00/0.52 filter(active(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(mark(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,active(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,mark(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,X2:S,active(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,X2:S,mark(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) 0.00/0.52 mark(filter(X1:S,X2:S,X3:S)) -> active(filter(mark(X1:S),mark(X2:S),mark(X3:S))) 0.00/0.52 mark(nats(X:S)) -> active(nats(mark(X:S))) 0.00/0.52 mark(s(X:S)) -> active(s(mark(X:S))) 0.00/0.52 mark(sieve(X:S)) -> active(sieve(mark(X:S))) 0.00/0.52 mark(0) -> active(0) 0.00/0.52 mark(zprimes) -> active(zprimes) 0.00/0.52 nats(active(X:S)) -> nats(X:S) 0.00/0.52 nats(mark(X:S)) -> nats(X:S) 0.00/0.52 s(active(X:S)) -> s(X:S) 0.00/0.52 s(mark(X:S)) -> s(X:S) 0.00/0.52 sieve(active(X:S)) -> sieve(X:S) 0.00/0.52 sieve(mark(X:S)) -> sieve(X:S) 0.00/0.52 ->Strongly Connected Components: 0.00/0.52 ->->Cycle: 0.00/0.52 ->->-> Pairs: 0.00/0.52 FILTER(X1:S,active(X2:S),X3:S) -> FILTER(X1:S,X2:S,X3:S) 0.00/0.52 FILTER(X1:S,mark(X2:S),X3:S) -> FILTER(X1:S,X2:S,X3:S) 0.00/0.52 FILTER(X1:S,X2:S,active(X3:S)) -> FILTER(X1:S,X2:S,X3:S) 0.00/0.52 FILTER(X1:S,X2:S,mark(X3:S)) -> FILTER(X1:S,X2:S,X3:S) 0.00/0.52 ->->-> Rules: 0.00/0.52 active(filter(cons(X:S,Y:S),s(N:S),M:S)) -> mark(cons(X:S,filter(Y:S,N:S,M:S))) 0.00/0.52 active(filter(cons(X:S,Y:S),0,M:S)) -> mark(cons(0,filter(Y:S,M:S,M:S))) 0.00/0.52 active(nats(N:S)) -> mark(cons(N:S,nats(s(N:S)))) 0.00/0.52 active(sieve(cons(s(N:S),Y:S))) -> mark(cons(s(N:S),sieve(filter(Y:S,N:S,N:S)))) 0.00/0.52 active(sieve(cons(0,Y:S))) -> mark(cons(0,sieve(Y:S))) 0.00/0.52 active(zprimes) -> mark(sieve(nats(s(s(0))))) 0.00/0.52 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.52 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.52 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 0.00/0.52 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 0.00/0.52 filter(active(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(mark(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,active(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,mark(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,X2:S,active(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,X2:S,mark(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) 0.00/0.52 mark(filter(X1:S,X2:S,X3:S)) -> active(filter(mark(X1:S),mark(X2:S),mark(X3:S))) 0.00/0.52 mark(nats(X:S)) -> active(nats(mark(X:S))) 0.00/0.52 mark(s(X:S)) -> active(s(mark(X:S))) 0.00/0.52 mark(sieve(X:S)) -> active(sieve(mark(X:S))) 0.00/0.52 mark(0) -> active(0) 0.00/0.52 mark(zprimes) -> active(zprimes) 0.00/0.52 nats(active(X:S)) -> nats(X:S) 0.00/0.52 nats(mark(X:S)) -> nats(X:S) 0.00/0.52 s(active(X:S)) -> s(X:S) 0.00/0.52 s(mark(X:S)) -> s(X:S) 0.00/0.52 sieve(active(X:S)) -> sieve(X:S) 0.00/0.52 sieve(mark(X:S)) -> sieve(X:S) 0.00/0.52 0.00/0.52 Problem 1.4: 0.00/0.52 0.00/0.52 Subterm Processor: 0.00/0.52 -> Pairs: 0.00/0.52 FILTER(X1:S,active(X2:S),X3:S) -> FILTER(X1:S,X2:S,X3:S) 0.00/0.52 FILTER(X1:S,mark(X2:S),X3:S) -> FILTER(X1:S,X2:S,X3:S) 0.00/0.52 FILTER(X1:S,X2:S,active(X3:S)) -> FILTER(X1:S,X2:S,X3:S) 0.00/0.52 FILTER(X1:S,X2:S,mark(X3:S)) -> FILTER(X1:S,X2:S,X3:S) 0.00/0.52 -> Rules: 0.00/0.52 active(filter(cons(X:S,Y:S),s(N:S),M:S)) -> mark(cons(X:S,filter(Y:S,N:S,M:S))) 0.00/0.52 active(filter(cons(X:S,Y:S),0,M:S)) -> mark(cons(0,filter(Y:S,M:S,M:S))) 0.00/0.52 active(nats(N:S)) -> mark(cons(N:S,nats(s(N:S)))) 0.00/0.52 active(sieve(cons(s(N:S),Y:S))) -> mark(cons(s(N:S),sieve(filter(Y:S,N:S,N:S)))) 0.00/0.52 active(sieve(cons(0,Y:S))) -> mark(cons(0,sieve(Y:S))) 0.00/0.52 active(zprimes) -> mark(sieve(nats(s(s(0))))) 0.00/0.52 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.52 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.52 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 0.00/0.52 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 0.00/0.52 filter(active(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(mark(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,active(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,mark(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,X2:S,active(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,X2:S,mark(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) 0.00/0.52 mark(filter(X1:S,X2:S,X3:S)) -> active(filter(mark(X1:S),mark(X2:S),mark(X3:S))) 0.00/0.52 mark(nats(X:S)) -> active(nats(mark(X:S))) 0.00/0.52 mark(s(X:S)) -> active(s(mark(X:S))) 0.00/0.52 mark(sieve(X:S)) -> active(sieve(mark(X:S))) 0.00/0.52 mark(0) -> active(0) 0.00/0.52 mark(zprimes) -> active(zprimes) 0.00/0.52 nats(active(X:S)) -> nats(X:S) 0.00/0.52 nats(mark(X:S)) -> nats(X:S) 0.00/0.52 s(active(X:S)) -> s(X:S) 0.00/0.52 s(mark(X:S)) -> s(X:S) 0.00/0.52 sieve(active(X:S)) -> sieve(X:S) 0.00/0.52 sieve(mark(X:S)) -> sieve(X:S) 0.00/0.52 ->Projection: 0.00/0.52 pi(FILTER) = 2 0.00/0.52 0.00/0.52 Problem 1.4: 0.00/0.52 0.00/0.52 SCC Processor: 0.00/0.52 -> Pairs: 0.00/0.52 FILTER(X1:S,X2:S,active(X3:S)) -> FILTER(X1:S,X2:S,X3:S) 0.00/0.52 FILTER(X1:S,X2:S,mark(X3:S)) -> FILTER(X1:S,X2:S,X3:S) 0.00/0.52 -> Rules: 0.00/0.52 active(filter(cons(X:S,Y:S),s(N:S),M:S)) -> mark(cons(X:S,filter(Y:S,N:S,M:S))) 0.00/0.52 active(filter(cons(X:S,Y:S),0,M:S)) -> mark(cons(0,filter(Y:S,M:S,M:S))) 0.00/0.52 active(nats(N:S)) -> mark(cons(N:S,nats(s(N:S)))) 0.00/0.52 active(sieve(cons(s(N:S),Y:S))) -> mark(cons(s(N:S),sieve(filter(Y:S,N:S,N:S)))) 0.00/0.52 active(sieve(cons(0,Y:S))) -> mark(cons(0,sieve(Y:S))) 0.00/0.52 active(zprimes) -> mark(sieve(nats(s(s(0))))) 0.00/0.52 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.52 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.52 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 0.00/0.52 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 0.00/0.52 filter(active(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(mark(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,active(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,mark(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,X2:S,active(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,X2:S,mark(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) 0.00/0.52 mark(filter(X1:S,X2:S,X3:S)) -> active(filter(mark(X1:S),mark(X2:S),mark(X3:S))) 0.00/0.52 mark(nats(X:S)) -> active(nats(mark(X:S))) 0.00/0.52 mark(s(X:S)) -> active(s(mark(X:S))) 0.00/0.52 mark(sieve(X:S)) -> active(sieve(mark(X:S))) 0.00/0.52 mark(0) -> active(0) 0.00/0.52 mark(zprimes) -> active(zprimes) 0.00/0.52 nats(active(X:S)) -> nats(X:S) 0.00/0.52 nats(mark(X:S)) -> nats(X:S) 0.00/0.52 s(active(X:S)) -> s(X:S) 0.00/0.52 s(mark(X:S)) -> s(X:S) 0.00/0.52 sieve(active(X:S)) -> sieve(X:S) 0.00/0.52 sieve(mark(X:S)) -> sieve(X:S) 0.00/0.52 ->Strongly Connected Components: 0.00/0.52 ->->Cycle: 0.00/0.52 ->->-> Pairs: 0.00/0.52 FILTER(X1:S,X2:S,active(X3:S)) -> FILTER(X1:S,X2:S,X3:S) 0.00/0.52 FILTER(X1:S,X2:S,mark(X3:S)) -> FILTER(X1:S,X2:S,X3:S) 0.00/0.52 ->->-> Rules: 0.00/0.52 active(filter(cons(X:S,Y:S),s(N:S),M:S)) -> mark(cons(X:S,filter(Y:S,N:S,M:S))) 0.00/0.52 active(filter(cons(X:S,Y:S),0,M:S)) -> mark(cons(0,filter(Y:S,M:S,M:S))) 0.00/0.52 active(nats(N:S)) -> mark(cons(N:S,nats(s(N:S)))) 0.00/0.52 active(sieve(cons(s(N:S),Y:S))) -> mark(cons(s(N:S),sieve(filter(Y:S,N:S,N:S)))) 0.00/0.52 active(sieve(cons(0,Y:S))) -> mark(cons(0,sieve(Y:S))) 0.00/0.52 active(zprimes) -> mark(sieve(nats(s(s(0))))) 0.00/0.52 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.52 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.52 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 0.00/0.52 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 0.00/0.52 filter(active(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(mark(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,active(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,mark(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,X2:S,active(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,X2:S,mark(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) 0.00/0.52 mark(filter(X1:S,X2:S,X3:S)) -> active(filter(mark(X1:S),mark(X2:S),mark(X3:S))) 0.00/0.52 mark(nats(X:S)) -> active(nats(mark(X:S))) 0.00/0.52 mark(s(X:S)) -> active(s(mark(X:S))) 0.00/0.52 mark(sieve(X:S)) -> active(sieve(mark(X:S))) 0.00/0.52 mark(0) -> active(0) 0.00/0.52 mark(zprimes) -> active(zprimes) 0.00/0.52 nats(active(X:S)) -> nats(X:S) 0.00/0.52 nats(mark(X:S)) -> nats(X:S) 0.00/0.52 s(active(X:S)) -> s(X:S) 0.00/0.52 s(mark(X:S)) -> s(X:S) 0.00/0.52 sieve(active(X:S)) -> sieve(X:S) 0.00/0.52 sieve(mark(X:S)) -> sieve(X:S) 0.00/0.52 0.00/0.52 Problem 1.4: 0.00/0.52 0.00/0.52 Subterm Processor: 0.00/0.52 -> Pairs: 0.00/0.52 FILTER(X1:S,X2:S,active(X3:S)) -> FILTER(X1:S,X2:S,X3:S) 0.00/0.52 FILTER(X1:S,X2:S,mark(X3:S)) -> FILTER(X1:S,X2:S,X3:S) 0.00/0.52 -> Rules: 0.00/0.52 active(filter(cons(X:S,Y:S),s(N:S),M:S)) -> mark(cons(X:S,filter(Y:S,N:S,M:S))) 0.00/0.52 active(filter(cons(X:S,Y:S),0,M:S)) -> mark(cons(0,filter(Y:S,M:S,M:S))) 0.00/0.52 active(nats(N:S)) -> mark(cons(N:S,nats(s(N:S)))) 0.00/0.52 active(sieve(cons(s(N:S),Y:S))) -> mark(cons(s(N:S),sieve(filter(Y:S,N:S,N:S)))) 0.00/0.52 active(sieve(cons(0,Y:S))) -> mark(cons(0,sieve(Y:S))) 0.00/0.52 active(zprimes) -> mark(sieve(nats(s(s(0))))) 0.00/0.52 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.52 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.52 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 0.00/0.52 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 0.00/0.52 filter(active(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(mark(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,active(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,mark(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,X2:S,active(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,X2:S,mark(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) 0.00/0.52 mark(filter(X1:S,X2:S,X3:S)) -> active(filter(mark(X1:S),mark(X2:S),mark(X3:S))) 0.00/0.52 mark(nats(X:S)) -> active(nats(mark(X:S))) 0.00/0.52 mark(s(X:S)) -> active(s(mark(X:S))) 0.00/0.52 mark(sieve(X:S)) -> active(sieve(mark(X:S))) 0.00/0.52 mark(0) -> active(0) 0.00/0.52 mark(zprimes) -> active(zprimes) 0.00/0.52 nats(active(X:S)) -> nats(X:S) 0.00/0.52 nats(mark(X:S)) -> nats(X:S) 0.00/0.52 s(active(X:S)) -> s(X:S) 0.00/0.52 s(mark(X:S)) -> s(X:S) 0.00/0.52 sieve(active(X:S)) -> sieve(X:S) 0.00/0.52 sieve(mark(X:S)) -> sieve(X:S) 0.00/0.52 ->Projection: 0.00/0.52 pi(FILTER) = 3 0.00/0.52 0.00/0.52 Problem 1.4: 0.00/0.52 0.00/0.52 SCC Processor: 0.00/0.52 -> Pairs: 0.00/0.52 Empty 0.00/0.52 -> Rules: 0.00/0.52 active(filter(cons(X:S,Y:S),s(N:S),M:S)) -> mark(cons(X:S,filter(Y:S,N:S,M:S))) 0.00/0.52 active(filter(cons(X:S,Y:S),0,M:S)) -> mark(cons(0,filter(Y:S,M:S,M:S))) 0.00/0.52 active(nats(N:S)) -> mark(cons(N:S,nats(s(N:S)))) 0.00/0.52 active(sieve(cons(s(N:S),Y:S))) -> mark(cons(s(N:S),sieve(filter(Y:S,N:S,N:S)))) 0.00/0.52 active(sieve(cons(0,Y:S))) -> mark(cons(0,sieve(Y:S))) 0.00/0.52 active(zprimes) -> mark(sieve(nats(s(s(0))))) 0.00/0.52 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.52 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.52 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 0.00/0.52 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 0.00/0.52 filter(active(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(mark(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,active(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,mark(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,X2:S,active(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,X2:S,mark(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) 0.00/0.52 mark(filter(X1:S,X2:S,X3:S)) -> active(filter(mark(X1:S),mark(X2:S),mark(X3:S))) 0.00/0.52 mark(nats(X:S)) -> active(nats(mark(X:S))) 0.00/0.52 mark(s(X:S)) -> active(s(mark(X:S))) 0.00/0.52 mark(sieve(X:S)) -> active(sieve(mark(X:S))) 0.00/0.52 mark(0) -> active(0) 0.00/0.52 mark(zprimes) -> active(zprimes) 0.00/0.52 nats(active(X:S)) -> nats(X:S) 0.00/0.52 nats(mark(X:S)) -> nats(X:S) 0.00/0.52 s(active(X:S)) -> s(X:S) 0.00/0.52 s(mark(X:S)) -> s(X:S) 0.00/0.52 sieve(active(X:S)) -> sieve(X:S) 0.00/0.52 sieve(mark(X:S)) -> sieve(X:S) 0.00/0.52 ->Strongly Connected Components: 0.00/0.52 There is no strongly connected component 0.00/0.52 0.00/0.52 The problem is finite. 0.00/0.52 0.00/0.52 Problem 1.5: 0.00/0.52 0.00/0.52 Subterm Processor: 0.00/0.52 -> Pairs: 0.00/0.52 CONS(active(X1:S),X2:S) -> CONS(X1:S,X2:S) 0.00/0.52 CONS(mark(X1:S),X2:S) -> CONS(X1:S,X2:S) 0.00/0.52 CONS(X1:S,active(X2:S)) -> CONS(X1:S,X2:S) 0.00/0.52 CONS(X1:S,mark(X2:S)) -> CONS(X1:S,X2:S) 0.00/0.52 -> Rules: 0.00/0.52 active(filter(cons(X:S,Y:S),s(N:S),M:S)) -> mark(cons(X:S,filter(Y:S,N:S,M:S))) 0.00/0.52 active(filter(cons(X:S,Y:S),0,M:S)) -> mark(cons(0,filter(Y:S,M:S,M:S))) 0.00/0.52 active(nats(N:S)) -> mark(cons(N:S,nats(s(N:S)))) 0.00/0.52 active(sieve(cons(s(N:S),Y:S))) -> mark(cons(s(N:S),sieve(filter(Y:S,N:S,N:S)))) 0.00/0.52 active(sieve(cons(0,Y:S))) -> mark(cons(0,sieve(Y:S))) 0.00/0.52 active(zprimes) -> mark(sieve(nats(s(s(0))))) 0.00/0.52 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.52 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.52 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 0.00/0.52 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 0.00/0.52 filter(active(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(mark(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,active(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,mark(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,X2:S,active(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,X2:S,mark(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) 0.00/0.52 mark(filter(X1:S,X2:S,X3:S)) -> active(filter(mark(X1:S),mark(X2:S),mark(X3:S))) 0.00/0.52 mark(nats(X:S)) -> active(nats(mark(X:S))) 0.00/0.52 mark(s(X:S)) -> active(s(mark(X:S))) 0.00/0.52 mark(sieve(X:S)) -> active(sieve(mark(X:S))) 0.00/0.52 mark(0) -> active(0) 0.00/0.52 mark(zprimes) -> active(zprimes) 0.00/0.52 nats(active(X:S)) -> nats(X:S) 0.00/0.52 nats(mark(X:S)) -> nats(X:S) 0.00/0.52 s(active(X:S)) -> s(X:S) 0.00/0.52 s(mark(X:S)) -> s(X:S) 0.00/0.52 sieve(active(X:S)) -> sieve(X:S) 0.00/0.52 sieve(mark(X:S)) -> sieve(X:S) 0.00/0.52 ->Projection: 0.00/0.52 pi(CONS) = 1 0.00/0.52 0.00/0.52 Problem 1.5: 0.00/0.52 0.00/0.52 SCC Processor: 0.00/0.52 -> Pairs: 0.00/0.52 CONS(X1:S,active(X2:S)) -> CONS(X1:S,X2:S) 0.00/0.52 CONS(X1:S,mark(X2:S)) -> CONS(X1:S,X2:S) 0.00/0.52 -> Rules: 0.00/0.52 active(filter(cons(X:S,Y:S),s(N:S),M:S)) -> mark(cons(X:S,filter(Y:S,N:S,M:S))) 0.00/0.52 active(filter(cons(X:S,Y:S),0,M:S)) -> mark(cons(0,filter(Y:S,M:S,M:S))) 0.00/0.52 active(nats(N:S)) -> mark(cons(N:S,nats(s(N:S)))) 0.00/0.52 active(sieve(cons(s(N:S),Y:S))) -> mark(cons(s(N:S),sieve(filter(Y:S,N:S,N:S)))) 0.00/0.52 active(sieve(cons(0,Y:S))) -> mark(cons(0,sieve(Y:S))) 0.00/0.52 active(zprimes) -> mark(sieve(nats(s(s(0))))) 0.00/0.52 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.52 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.52 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 0.00/0.52 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 0.00/0.52 filter(active(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(mark(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,active(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,mark(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,X2:S,active(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,X2:S,mark(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) 0.00/0.52 mark(filter(X1:S,X2:S,X3:S)) -> active(filter(mark(X1:S),mark(X2:S),mark(X3:S))) 0.00/0.52 mark(nats(X:S)) -> active(nats(mark(X:S))) 0.00/0.52 mark(s(X:S)) -> active(s(mark(X:S))) 0.00/0.52 mark(sieve(X:S)) -> active(sieve(mark(X:S))) 0.00/0.52 mark(0) -> active(0) 0.00/0.52 mark(zprimes) -> active(zprimes) 0.00/0.52 nats(active(X:S)) -> nats(X:S) 0.00/0.52 nats(mark(X:S)) -> nats(X:S) 0.00/0.52 s(active(X:S)) -> s(X:S) 0.00/0.52 s(mark(X:S)) -> s(X:S) 0.00/0.52 sieve(active(X:S)) -> sieve(X:S) 0.00/0.52 sieve(mark(X:S)) -> sieve(X:S) 0.00/0.52 ->Strongly Connected Components: 0.00/0.52 ->->Cycle: 0.00/0.52 ->->-> Pairs: 0.00/0.52 CONS(X1:S,active(X2:S)) -> CONS(X1:S,X2:S) 0.00/0.52 CONS(X1:S,mark(X2:S)) -> CONS(X1:S,X2:S) 0.00/0.52 ->->-> Rules: 0.00/0.52 active(filter(cons(X:S,Y:S),s(N:S),M:S)) -> mark(cons(X:S,filter(Y:S,N:S,M:S))) 0.00/0.52 active(filter(cons(X:S,Y:S),0,M:S)) -> mark(cons(0,filter(Y:S,M:S,M:S))) 0.00/0.52 active(nats(N:S)) -> mark(cons(N:S,nats(s(N:S)))) 0.00/0.52 active(sieve(cons(s(N:S),Y:S))) -> mark(cons(s(N:S),sieve(filter(Y:S,N:S,N:S)))) 0.00/0.52 active(sieve(cons(0,Y:S))) -> mark(cons(0,sieve(Y:S))) 0.00/0.52 active(zprimes) -> mark(sieve(nats(s(s(0))))) 0.00/0.52 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.52 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.52 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 0.00/0.52 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 0.00/0.52 filter(active(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(mark(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,active(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,mark(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,X2:S,active(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,X2:S,mark(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) 0.00/0.52 mark(filter(X1:S,X2:S,X3:S)) -> active(filter(mark(X1:S),mark(X2:S),mark(X3:S))) 0.00/0.52 mark(nats(X:S)) -> active(nats(mark(X:S))) 0.00/0.52 mark(s(X:S)) -> active(s(mark(X:S))) 0.00/0.52 mark(sieve(X:S)) -> active(sieve(mark(X:S))) 0.00/0.52 mark(0) -> active(0) 0.00/0.52 mark(zprimes) -> active(zprimes) 0.00/0.52 nats(active(X:S)) -> nats(X:S) 0.00/0.52 nats(mark(X:S)) -> nats(X:S) 0.00/0.52 s(active(X:S)) -> s(X:S) 0.00/0.52 s(mark(X:S)) -> s(X:S) 0.00/0.52 sieve(active(X:S)) -> sieve(X:S) 0.00/0.52 sieve(mark(X:S)) -> sieve(X:S) 0.00/0.52 0.00/0.52 Problem 1.5: 0.00/0.52 0.00/0.52 Subterm Processor: 0.00/0.52 -> Pairs: 0.00/0.52 CONS(X1:S,active(X2:S)) -> CONS(X1:S,X2:S) 0.00/0.52 CONS(X1:S,mark(X2:S)) -> CONS(X1:S,X2:S) 0.00/0.52 -> Rules: 0.00/0.52 active(filter(cons(X:S,Y:S),s(N:S),M:S)) -> mark(cons(X:S,filter(Y:S,N:S,M:S))) 0.00/0.52 active(filter(cons(X:S,Y:S),0,M:S)) -> mark(cons(0,filter(Y:S,M:S,M:S))) 0.00/0.52 active(nats(N:S)) -> mark(cons(N:S,nats(s(N:S)))) 0.00/0.52 active(sieve(cons(s(N:S),Y:S))) -> mark(cons(s(N:S),sieve(filter(Y:S,N:S,N:S)))) 0.00/0.52 active(sieve(cons(0,Y:S))) -> mark(cons(0,sieve(Y:S))) 0.00/0.52 active(zprimes) -> mark(sieve(nats(s(s(0))))) 0.00/0.52 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.52 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.52 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 0.00/0.52 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 0.00/0.52 filter(active(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(mark(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,active(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,mark(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,X2:S,active(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,X2:S,mark(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) 0.00/0.52 mark(filter(X1:S,X2:S,X3:S)) -> active(filter(mark(X1:S),mark(X2:S),mark(X3:S))) 0.00/0.52 mark(nats(X:S)) -> active(nats(mark(X:S))) 0.00/0.52 mark(s(X:S)) -> active(s(mark(X:S))) 0.00/0.52 mark(sieve(X:S)) -> active(sieve(mark(X:S))) 0.00/0.52 mark(0) -> active(0) 0.00/0.52 mark(zprimes) -> active(zprimes) 0.00/0.52 nats(active(X:S)) -> nats(X:S) 0.00/0.52 nats(mark(X:S)) -> nats(X:S) 0.00/0.52 s(active(X:S)) -> s(X:S) 0.00/0.52 s(mark(X:S)) -> s(X:S) 0.00/0.52 sieve(active(X:S)) -> sieve(X:S) 0.00/0.52 sieve(mark(X:S)) -> sieve(X:S) 0.00/0.52 ->Projection: 0.00/0.52 pi(CONS) = 2 0.00/0.52 0.00/0.52 Problem 1.5: 0.00/0.52 0.00/0.52 SCC Processor: 0.00/0.52 -> Pairs: 0.00/0.52 Empty 0.00/0.52 -> Rules: 0.00/0.52 active(filter(cons(X:S,Y:S),s(N:S),M:S)) -> mark(cons(X:S,filter(Y:S,N:S,M:S))) 0.00/0.52 active(filter(cons(X:S,Y:S),0,M:S)) -> mark(cons(0,filter(Y:S,M:S,M:S))) 0.00/0.52 active(nats(N:S)) -> mark(cons(N:S,nats(s(N:S)))) 0.00/0.52 active(sieve(cons(s(N:S),Y:S))) -> mark(cons(s(N:S),sieve(filter(Y:S,N:S,N:S)))) 0.00/0.52 active(sieve(cons(0,Y:S))) -> mark(cons(0,sieve(Y:S))) 0.00/0.52 active(zprimes) -> mark(sieve(nats(s(s(0))))) 0.00/0.52 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.52 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.52 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 0.00/0.52 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 0.00/0.52 filter(active(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(mark(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,active(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,mark(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,X2:S,active(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,X2:S,mark(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) 0.00/0.52 mark(filter(X1:S,X2:S,X3:S)) -> active(filter(mark(X1:S),mark(X2:S),mark(X3:S))) 0.00/0.52 mark(nats(X:S)) -> active(nats(mark(X:S))) 0.00/0.52 mark(s(X:S)) -> active(s(mark(X:S))) 0.00/0.52 mark(sieve(X:S)) -> active(sieve(mark(X:S))) 0.00/0.52 mark(0) -> active(0) 0.00/0.52 mark(zprimes) -> active(zprimes) 0.00/0.52 nats(active(X:S)) -> nats(X:S) 0.00/0.52 nats(mark(X:S)) -> nats(X:S) 0.00/0.52 s(active(X:S)) -> s(X:S) 0.00/0.52 s(mark(X:S)) -> s(X:S) 0.00/0.52 sieve(active(X:S)) -> sieve(X:S) 0.00/0.52 sieve(mark(X:S)) -> sieve(X:S) 0.00/0.52 ->Strongly Connected Components: 0.00/0.52 There is no strongly connected component 0.00/0.52 0.00/0.52 The problem is finite. 0.00/0.52 0.00/0.52 Problem 1.6: 0.00/0.52 0.00/0.52 Reduction Pairs Processor: 0.00/0.52 -> Pairs: 0.00/0.52 ACTIVE(filter(cons(X:S,Y:S),s(N:S),M:S)) -> MARK(cons(X:S,filter(Y:S,N:S,M:S))) 0.00/0.52 ACTIVE(filter(cons(X:S,Y:S),0,M:S)) -> MARK(cons(0,filter(Y:S,M:S,M:S))) 0.00/0.52 ACTIVE(nats(N:S)) -> MARK(cons(N:S,nats(s(N:S)))) 0.00/0.52 ACTIVE(sieve(cons(s(N:S),Y:S))) -> MARK(cons(s(N:S),sieve(filter(Y:S,N:S,N:S)))) 0.00/0.52 ACTIVE(sieve(cons(0,Y:S))) -> MARK(cons(0,sieve(Y:S))) 0.00/0.52 ACTIVE(zprimes) -> MARK(sieve(nats(s(s(0))))) 0.00/0.52 MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) 0.00/0.52 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.52 MARK(filter(X1:S,X2:S,X3:S)) -> ACTIVE(filter(mark(X1:S),mark(X2:S),mark(X3:S))) 0.00/0.52 MARK(filter(X1:S,X2:S,X3:S)) -> MARK(X1:S) 0.00/0.52 MARK(filter(X1:S,X2:S,X3:S)) -> MARK(X2:S) 0.00/0.52 MARK(filter(X1:S,X2:S,X3:S)) -> MARK(X3:S) 0.00/0.52 MARK(nats(X:S)) -> ACTIVE(nats(mark(X:S))) 0.00/0.52 MARK(nats(X:S)) -> MARK(X:S) 0.00/0.52 MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) 0.00/0.52 MARK(s(X:S)) -> MARK(X:S) 0.00/0.52 MARK(sieve(X:S)) -> ACTIVE(sieve(mark(X:S))) 0.00/0.52 MARK(sieve(X:S)) -> MARK(X:S) 0.00/0.52 MARK(zprimes) -> ACTIVE(zprimes) 0.00/0.52 -> Rules: 0.00/0.52 active(filter(cons(X:S,Y:S),s(N:S),M:S)) -> mark(cons(X:S,filter(Y:S,N:S,M:S))) 0.00/0.52 active(filter(cons(X:S,Y:S),0,M:S)) -> mark(cons(0,filter(Y:S,M:S,M:S))) 0.00/0.52 active(nats(N:S)) -> mark(cons(N:S,nats(s(N:S)))) 0.00/0.52 active(sieve(cons(s(N:S),Y:S))) -> mark(cons(s(N:S),sieve(filter(Y:S,N:S,N:S)))) 0.00/0.52 active(sieve(cons(0,Y:S))) -> mark(cons(0,sieve(Y:S))) 0.00/0.52 active(zprimes) -> mark(sieve(nats(s(s(0))))) 0.00/0.52 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.52 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.52 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 0.00/0.52 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 0.00/0.52 filter(active(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(mark(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,active(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,mark(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,X2:S,active(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,X2:S,mark(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) 0.00/0.52 mark(filter(X1:S,X2:S,X3:S)) -> active(filter(mark(X1:S),mark(X2:S),mark(X3:S))) 0.00/0.52 mark(nats(X:S)) -> active(nats(mark(X:S))) 0.00/0.52 mark(s(X:S)) -> active(s(mark(X:S))) 0.00/0.52 mark(sieve(X:S)) -> active(sieve(mark(X:S))) 0.00/0.52 mark(0) -> active(0) 0.00/0.52 mark(zprimes) -> active(zprimes) 0.00/0.52 nats(active(X:S)) -> nats(X:S) 0.00/0.52 nats(mark(X:S)) -> nats(X:S) 0.00/0.52 s(active(X:S)) -> s(X:S) 0.00/0.52 s(mark(X:S)) -> s(X:S) 0.00/0.52 sieve(active(X:S)) -> sieve(X:S) 0.00/0.52 sieve(mark(X:S)) -> sieve(X:S) 0.00/0.52 -> Usable rules: 0.00/0.52 active(filter(cons(X:S,Y:S),s(N:S),M:S)) -> mark(cons(X:S,filter(Y:S,N:S,M:S))) 0.00/0.52 active(filter(cons(X:S,Y:S),0,M:S)) -> mark(cons(0,filter(Y:S,M:S,M:S))) 0.00/0.52 active(nats(N:S)) -> mark(cons(N:S,nats(s(N:S)))) 0.00/0.52 active(sieve(cons(s(N:S),Y:S))) -> mark(cons(s(N:S),sieve(filter(Y:S,N:S,N:S)))) 0.00/0.52 active(sieve(cons(0,Y:S))) -> mark(cons(0,sieve(Y:S))) 0.00/0.52 active(zprimes) -> mark(sieve(nats(s(s(0))))) 0.00/0.52 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.52 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.52 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 0.00/0.52 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 0.00/0.52 filter(active(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(mark(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,active(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,mark(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,X2:S,active(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,X2:S,mark(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) 0.00/0.52 mark(filter(X1:S,X2:S,X3:S)) -> active(filter(mark(X1:S),mark(X2:S),mark(X3:S))) 0.00/0.52 mark(nats(X:S)) -> active(nats(mark(X:S))) 0.00/0.52 mark(s(X:S)) -> active(s(mark(X:S))) 0.00/0.52 mark(sieve(X:S)) -> active(sieve(mark(X:S))) 0.00/0.52 mark(0) -> active(0) 0.00/0.52 mark(zprimes) -> active(zprimes) 0.00/0.52 nats(active(X:S)) -> nats(X:S) 0.00/0.52 nats(mark(X:S)) -> nats(X:S) 0.00/0.52 s(active(X:S)) -> s(X:S) 0.00/0.52 s(mark(X:S)) -> s(X:S) 0.00/0.52 sieve(active(X:S)) -> sieve(X:S) 0.00/0.52 sieve(mark(X:S)) -> sieve(X:S) 0.00/0.52 ->Interpretation type: 0.00/0.52 Linear 0.00/0.52 ->Coefficients: 0.00/0.52 Natural Numbers 0.00/0.52 ->Dimension: 0.00/0.52 1 0.00/0.52 ->Bound: 0.00/0.52 2 0.00/0.52 ->Interpretation: 0.00/0.52 0.00/0.52 [active](X) = X 0.00/0.52 [cons](X1,X2) = X1 0.00/0.52 [filter](X1,X2,X3) = 2.X1 + X2 + 2.X3 0.00/0.52 [mark](X) = X 0.00/0.52 [nats](X) = X 0.00/0.52 [s](X) = X + 1 0.00/0.52 [sieve](X) = X 0.00/0.52 [0] = 0 0.00/0.52 [fSNonEmpty] = 0 0.00/0.52 [zprimes] = 2 0.00/0.52 [ACTIVE](X) = 2.X + 2 0.00/0.52 [CONS](X1,X2) = 0 0.00/0.52 [FILTER](X1,X2,X3) = 0 0.00/0.52 [MARK](X) = 2.X + 2 0.00/0.52 [NATS](X) = 0 0.00/0.52 [S](X) = 0 0.00/0.52 [SIEVE](X) = 0 0.00/0.52 0.00/0.52 Problem 1.6: 0.00/0.52 0.00/0.52 SCC Processor: 0.00/0.52 -> Pairs: 0.00/0.52 ACTIVE(filter(cons(X:S,Y:S),0,M:S)) -> MARK(cons(0,filter(Y:S,M:S,M:S))) 0.00/0.52 ACTIVE(nats(N:S)) -> MARK(cons(N:S,nats(s(N:S)))) 0.00/0.52 ACTIVE(sieve(cons(s(N:S),Y:S))) -> MARK(cons(s(N:S),sieve(filter(Y:S,N:S,N:S)))) 0.00/0.52 ACTIVE(sieve(cons(0,Y:S))) -> MARK(cons(0,sieve(Y:S))) 0.00/0.52 ACTIVE(zprimes) -> MARK(sieve(nats(s(s(0))))) 0.00/0.52 MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) 0.00/0.52 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.52 MARK(filter(X1:S,X2:S,X3:S)) -> ACTIVE(filter(mark(X1:S),mark(X2:S),mark(X3:S))) 0.00/0.52 MARK(filter(X1:S,X2:S,X3:S)) -> MARK(X1:S) 0.00/0.52 MARK(filter(X1:S,X2:S,X3:S)) -> MARK(X2:S) 0.00/0.52 MARK(filter(X1:S,X2:S,X3:S)) -> MARK(X3:S) 0.00/0.52 MARK(nats(X:S)) -> ACTIVE(nats(mark(X:S))) 0.00/0.52 MARK(nats(X:S)) -> MARK(X:S) 0.00/0.52 MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) 0.00/0.52 MARK(s(X:S)) -> MARK(X:S) 0.00/0.52 MARK(sieve(X:S)) -> ACTIVE(sieve(mark(X:S))) 0.00/0.52 MARK(sieve(X:S)) -> MARK(X:S) 0.00/0.52 MARK(zprimes) -> ACTIVE(zprimes) 0.00/0.52 -> Rules: 0.00/0.52 active(filter(cons(X:S,Y:S),s(N:S),M:S)) -> mark(cons(X:S,filter(Y:S,N:S,M:S))) 0.00/0.52 active(filter(cons(X:S,Y:S),0,M:S)) -> mark(cons(0,filter(Y:S,M:S,M:S))) 0.00/0.52 active(nats(N:S)) -> mark(cons(N:S,nats(s(N:S)))) 0.00/0.52 active(sieve(cons(s(N:S),Y:S))) -> mark(cons(s(N:S),sieve(filter(Y:S,N:S,N:S)))) 0.00/0.52 active(sieve(cons(0,Y:S))) -> mark(cons(0,sieve(Y:S))) 0.00/0.52 active(zprimes) -> mark(sieve(nats(s(s(0))))) 0.00/0.52 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.52 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.52 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 0.00/0.52 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 0.00/0.52 filter(active(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(mark(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,active(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,mark(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,X2:S,active(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,X2:S,mark(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) 0.00/0.52 mark(filter(X1:S,X2:S,X3:S)) -> active(filter(mark(X1:S),mark(X2:S),mark(X3:S))) 0.00/0.52 mark(nats(X:S)) -> active(nats(mark(X:S))) 0.00/0.52 mark(s(X:S)) -> active(s(mark(X:S))) 0.00/0.52 mark(sieve(X:S)) -> active(sieve(mark(X:S))) 0.00/0.52 mark(0) -> active(0) 0.00/0.52 mark(zprimes) -> active(zprimes) 0.00/0.52 nats(active(X:S)) -> nats(X:S) 0.00/0.52 nats(mark(X:S)) -> nats(X:S) 0.00/0.52 s(active(X:S)) -> s(X:S) 0.00/0.52 s(mark(X:S)) -> s(X:S) 0.00/0.52 sieve(active(X:S)) -> sieve(X:S) 0.00/0.52 sieve(mark(X:S)) -> sieve(X:S) 0.00/0.52 ->Strongly Connected Components: 0.00/0.52 ->->Cycle: 0.00/0.52 ->->-> Pairs: 0.00/0.52 ACTIVE(filter(cons(X:S,Y:S),0,M:S)) -> MARK(cons(0,filter(Y:S,M:S,M:S))) 0.00/0.52 ACTIVE(nats(N:S)) -> MARK(cons(N:S,nats(s(N:S)))) 0.00/0.52 ACTIVE(sieve(cons(s(N:S),Y:S))) -> MARK(cons(s(N:S),sieve(filter(Y:S,N:S,N:S)))) 0.00/0.52 ACTIVE(sieve(cons(0,Y:S))) -> MARK(cons(0,sieve(Y:S))) 0.00/0.52 ACTIVE(zprimes) -> MARK(sieve(nats(s(s(0))))) 0.00/0.52 MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) 0.00/0.52 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.52 MARK(filter(X1:S,X2:S,X3:S)) -> ACTIVE(filter(mark(X1:S),mark(X2:S),mark(X3:S))) 0.00/0.52 MARK(filter(X1:S,X2:S,X3:S)) -> MARK(X1:S) 0.00/0.52 MARK(filter(X1:S,X2:S,X3:S)) -> MARK(X2:S) 0.00/0.52 MARK(filter(X1:S,X2:S,X3:S)) -> MARK(X3:S) 0.00/0.52 MARK(nats(X:S)) -> ACTIVE(nats(mark(X:S))) 0.00/0.52 MARK(nats(X:S)) -> MARK(X:S) 0.00/0.52 MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) 0.00/0.52 MARK(s(X:S)) -> MARK(X:S) 0.00/0.52 MARK(sieve(X:S)) -> ACTIVE(sieve(mark(X:S))) 0.00/0.52 MARK(sieve(X:S)) -> MARK(X:S) 0.00/0.52 MARK(zprimes) -> ACTIVE(zprimes) 0.00/0.52 ->->-> Rules: 0.00/0.52 active(filter(cons(X:S,Y:S),s(N:S),M:S)) -> mark(cons(X:S,filter(Y:S,N:S,M:S))) 0.00/0.52 active(filter(cons(X:S,Y:S),0,M:S)) -> mark(cons(0,filter(Y:S,M:S,M:S))) 0.00/0.52 active(nats(N:S)) -> mark(cons(N:S,nats(s(N:S)))) 0.00/0.52 active(sieve(cons(s(N:S),Y:S))) -> mark(cons(s(N:S),sieve(filter(Y:S,N:S,N:S)))) 0.00/0.52 active(sieve(cons(0,Y:S))) -> mark(cons(0,sieve(Y:S))) 0.00/0.52 active(zprimes) -> mark(sieve(nats(s(s(0))))) 0.00/0.52 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.52 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.52 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 0.00/0.52 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 0.00/0.52 filter(active(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(mark(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,active(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,mark(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,X2:S,active(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,X2:S,mark(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) 0.00/0.52 mark(filter(X1:S,X2:S,X3:S)) -> active(filter(mark(X1:S),mark(X2:S),mark(X3:S))) 0.00/0.52 mark(nats(X:S)) -> active(nats(mark(X:S))) 0.00/0.52 mark(s(X:S)) -> active(s(mark(X:S))) 0.00/0.52 mark(sieve(X:S)) -> active(sieve(mark(X:S))) 0.00/0.52 mark(0) -> active(0) 0.00/0.52 mark(zprimes) -> active(zprimes) 0.00/0.52 nats(active(X:S)) -> nats(X:S) 0.00/0.52 nats(mark(X:S)) -> nats(X:S) 0.00/0.52 s(active(X:S)) -> s(X:S) 0.00/0.52 s(mark(X:S)) -> s(X:S) 0.00/0.52 sieve(active(X:S)) -> sieve(X:S) 0.00/0.52 sieve(mark(X:S)) -> sieve(X:S) 0.00/0.52 0.00/0.52 Problem 1.6: 0.00/0.52 0.00/0.52 Reduction Pairs Processor: 0.00/0.52 -> Pairs: 0.00/0.52 ACTIVE(filter(cons(X:S,Y:S),0,M:S)) -> MARK(cons(0,filter(Y:S,M:S,M:S))) 0.00/0.52 ACTIVE(nats(N:S)) -> MARK(cons(N:S,nats(s(N:S)))) 0.00/0.52 ACTIVE(sieve(cons(s(N:S),Y:S))) -> MARK(cons(s(N:S),sieve(filter(Y:S,N:S,N:S)))) 0.00/0.52 ACTIVE(sieve(cons(0,Y:S))) -> MARK(cons(0,sieve(Y:S))) 0.00/0.52 ACTIVE(zprimes) -> MARK(sieve(nats(s(s(0))))) 0.00/0.52 MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) 0.00/0.52 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.52 MARK(filter(X1:S,X2:S,X3:S)) -> ACTIVE(filter(mark(X1:S),mark(X2:S),mark(X3:S))) 0.00/0.52 MARK(filter(X1:S,X2:S,X3:S)) -> MARK(X1:S) 0.00/0.52 MARK(filter(X1:S,X2:S,X3:S)) -> MARK(X2:S) 0.00/0.52 MARK(filter(X1:S,X2:S,X3:S)) -> MARK(X3:S) 0.00/0.52 MARK(nats(X:S)) -> ACTIVE(nats(mark(X:S))) 0.00/0.52 MARK(nats(X:S)) -> MARK(X:S) 0.00/0.52 MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) 0.00/0.52 MARK(s(X:S)) -> MARK(X:S) 0.00/0.52 MARK(sieve(X:S)) -> ACTIVE(sieve(mark(X:S))) 0.00/0.52 MARK(sieve(X:S)) -> MARK(X:S) 0.00/0.52 MARK(zprimes) -> ACTIVE(zprimes) 0.00/0.52 -> Rules: 0.00/0.52 active(filter(cons(X:S,Y:S),s(N:S),M:S)) -> mark(cons(X:S,filter(Y:S,N:S,M:S))) 0.00/0.52 active(filter(cons(X:S,Y:S),0,M:S)) -> mark(cons(0,filter(Y:S,M:S,M:S))) 0.00/0.52 active(nats(N:S)) -> mark(cons(N:S,nats(s(N:S)))) 0.00/0.52 active(sieve(cons(s(N:S),Y:S))) -> mark(cons(s(N:S),sieve(filter(Y:S,N:S,N:S)))) 0.00/0.52 active(sieve(cons(0,Y:S))) -> mark(cons(0,sieve(Y:S))) 0.00/0.52 active(zprimes) -> mark(sieve(nats(s(s(0))))) 0.00/0.52 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.52 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.52 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 0.00/0.52 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 0.00/0.52 filter(active(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(mark(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,active(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,mark(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,X2:S,active(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,X2:S,mark(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) 0.00/0.52 mark(filter(X1:S,X2:S,X3:S)) -> active(filter(mark(X1:S),mark(X2:S),mark(X3:S))) 0.00/0.52 mark(nats(X:S)) -> active(nats(mark(X:S))) 0.00/0.52 mark(s(X:S)) -> active(s(mark(X:S))) 0.00/0.52 mark(sieve(X:S)) -> active(sieve(mark(X:S))) 0.00/0.52 mark(0) -> active(0) 0.00/0.52 mark(zprimes) -> active(zprimes) 0.00/0.52 nats(active(X:S)) -> nats(X:S) 0.00/0.52 nats(mark(X:S)) -> nats(X:S) 0.00/0.52 s(active(X:S)) -> s(X:S) 0.00/0.52 s(mark(X:S)) -> s(X:S) 0.00/0.52 sieve(active(X:S)) -> sieve(X:S) 0.00/0.52 sieve(mark(X:S)) -> sieve(X:S) 0.00/0.52 -> Usable rules: 0.00/0.52 active(filter(cons(X:S,Y:S),s(N:S),M:S)) -> mark(cons(X:S,filter(Y:S,N:S,M:S))) 0.00/0.52 active(filter(cons(X:S,Y:S),0,M:S)) -> mark(cons(0,filter(Y:S,M:S,M:S))) 0.00/0.52 active(nats(N:S)) -> mark(cons(N:S,nats(s(N:S)))) 0.00/0.52 active(sieve(cons(s(N:S),Y:S))) -> mark(cons(s(N:S),sieve(filter(Y:S,N:S,N:S)))) 0.00/0.52 active(sieve(cons(0,Y:S))) -> mark(cons(0,sieve(Y:S))) 0.00/0.52 active(zprimes) -> mark(sieve(nats(s(s(0))))) 0.00/0.52 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.52 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.52 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 0.00/0.52 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 0.00/0.52 filter(active(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(mark(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,active(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,mark(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,X2:S,active(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,X2:S,mark(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) 0.00/0.52 mark(filter(X1:S,X2:S,X3:S)) -> active(filter(mark(X1:S),mark(X2:S),mark(X3:S))) 0.00/0.52 mark(nats(X:S)) -> active(nats(mark(X:S))) 0.00/0.52 mark(s(X:S)) -> active(s(mark(X:S))) 0.00/0.52 mark(sieve(X:S)) -> active(sieve(mark(X:S))) 0.00/0.52 mark(0) -> active(0) 0.00/0.52 mark(zprimes) -> active(zprimes) 0.00/0.52 nats(active(X:S)) -> nats(X:S) 0.00/0.52 nats(mark(X:S)) -> nats(X:S) 0.00/0.52 s(active(X:S)) -> s(X:S) 0.00/0.52 s(mark(X:S)) -> s(X:S) 0.00/0.52 sieve(active(X:S)) -> sieve(X:S) 0.00/0.52 sieve(mark(X:S)) -> sieve(X:S) 0.00/0.52 ->Interpretation type: 0.00/0.52 Linear 0.00/0.52 ->Coefficients: 0.00/0.52 Natural Numbers 0.00/0.52 ->Dimension: 0.00/0.52 1 0.00/0.52 ->Bound: 0.00/0.52 2 0.00/0.52 ->Interpretation: 0.00/0.52 0.00/0.52 [active](X) = X 0.00/0.52 [cons](X1,X2) = 2.X1 0.00/0.52 [filter](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 0.00/0.52 [mark](X) = X 0.00/0.52 [nats](X) = 2.X 0.00/0.52 [s](X) = X 0.00/0.52 [sieve](X) = 2.X 0.00/0.52 [0] = 0 0.00/0.52 [fSNonEmpty] = 0 0.00/0.52 [zprimes] = 0 0.00/0.52 [ACTIVE](X) = 2.X + 1 0.00/0.52 [CONS](X1,X2) = 0 0.00/0.52 [FILTER](X1,X2,X3) = 0 0.00/0.52 [MARK](X) = 2.X + 1 0.00/0.52 [NATS](X) = 0 0.00/0.52 [S](X) = 0 0.00/0.52 [SIEVE](X) = 0 0.00/0.52 0.00/0.52 Problem 1.6: 0.00/0.52 0.00/0.52 SCC Processor: 0.00/0.52 -> Pairs: 0.00/0.52 ACTIVE(nats(N:S)) -> MARK(cons(N:S,nats(s(N:S)))) 0.00/0.52 ACTIVE(sieve(cons(s(N:S),Y:S))) -> MARK(cons(s(N:S),sieve(filter(Y:S,N:S,N:S)))) 0.00/0.52 ACTIVE(sieve(cons(0,Y:S))) -> MARK(cons(0,sieve(Y:S))) 0.00/0.52 ACTIVE(zprimes) -> MARK(sieve(nats(s(s(0))))) 0.00/0.52 MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) 0.00/0.52 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.52 MARK(filter(X1:S,X2:S,X3:S)) -> ACTIVE(filter(mark(X1:S),mark(X2:S),mark(X3:S))) 0.00/0.52 MARK(filter(X1:S,X2:S,X3:S)) -> MARK(X1:S) 0.00/0.52 MARK(filter(X1:S,X2:S,X3:S)) -> MARK(X2:S) 0.00/0.52 MARK(filter(X1:S,X2:S,X3:S)) -> MARK(X3:S) 0.00/0.52 MARK(nats(X:S)) -> ACTIVE(nats(mark(X:S))) 0.00/0.52 MARK(nats(X:S)) -> MARK(X:S) 0.00/0.52 MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) 0.00/0.52 MARK(s(X:S)) -> MARK(X:S) 0.00/0.52 MARK(sieve(X:S)) -> ACTIVE(sieve(mark(X:S))) 0.00/0.52 MARK(sieve(X:S)) -> MARK(X:S) 0.00/0.52 MARK(zprimes) -> ACTIVE(zprimes) 0.00/0.52 -> Rules: 0.00/0.52 active(filter(cons(X:S,Y:S),s(N:S),M:S)) -> mark(cons(X:S,filter(Y:S,N:S,M:S))) 0.00/0.52 active(filter(cons(X:S,Y:S),0,M:S)) -> mark(cons(0,filter(Y:S,M:S,M:S))) 0.00/0.52 active(nats(N:S)) -> mark(cons(N:S,nats(s(N:S)))) 0.00/0.52 active(sieve(cons(s(N:S),Y:S))) -> mark(cons(s(N:S),sieve(filter(Y:S,N:S,N:S)))) 0.00/0.52 active(sieve(cons(0,Y:S))) -> mark(cons(0,sieve(Y:S))) 0.00/0.52 active(zprimes) -> mark(sieve(nats(s(s(0))))) 0.00/0.52 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.52 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.52 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 0.00/0.52 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 0.00/0.52 filter(active(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(mark(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,active(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,mark(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,X2:S,active(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 filter(X1:S,X2:S,mark(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.52 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) 0.00/0.52 mark(filter(X1:S,X2:S,X3:S)) -> active(filter(mark(X1:S),mark(X2:S),mark(X3:S))) 0.00/0.52 mark(nats(X:S)) -> active(nats(mark(X:S))) 0.00/0.52 mark(s(X:S)) -> active(s(mark(X:S))) 0.00/0.52 mark(sieve(X:S)) -> active(sieve(mark(X:S))) 0.00/0.52 mark(0) -> active(0) 0.00/0.52 mark(zprimes) -> active(zprimes) 0.00/0.52 nats(active(X:S)) -> nats(X:S) 0.00/0.52 nats(mark(X:S)) -> nats(X:S) 0.00/0.53 s(active(X:S)) -> s(X:S) 0.00/0.53 s(mark(X:S)) -> s(X:S) 0.00/0.53 sieve(active(X:S)) -> sieve(X:S) 0.00/0.53 sieve(mark(X:S)) -> sieve(X:S) 0.00/0.53 ->Strongly Connected Components: 0.00/0.53 ->->Cycle: 0.00/0.53 ->->-> Pairs: 0.00/0.53 ACTIVE(nats(N:S)) -> MARK(cons(N:S,nats(s(N:S)))) 0.00/0.53 ACTIVE(sieve(cons(s(N:S),Y:S))) -> MARK(cons(s(N:S),sieve(filter(Y:S,N:S,N:S)))) 0.00/0.53 ACTIVE(sieve(cons(0,Y:S))) -> MARK(cons(0,sieve(Y:S))) 0.00/0.53 ACTIVE(zprimes) -> MARK(sieve(nats(s(s(0))))) 0.00/0.53 MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) 0.00/0.53 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.53 MARK(filter(X1:S,X2:S,X3:S)) -> ACTIVE(filter(mark(X1:S),mark(X2:S),mark(X3:S))) 0.00/0.53 MARK(filter(X1:S,X2:S,X3:S)) -> MARK(X1:S) 0.00/0.53 MARK(filter(X1:S,X2:S,X3:S)) -> MARK(X2:S) 0.00/0.53 MARK(filter(X1:S,X2:S,X3:S)) -> MARK(X3:S) 0.00/0.53 MARK(nats(X:S)) -> ACTIVE(nats(mark(X:S))) 0.00/0.53 MARK(nats(X:S)) -> MARK(X:S) 0.00/0.53 MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) 0.00/0.53 MARK(s(X:S)) -> MARK(X:S) 0.00/0.53 MARK(sieve(X:S)) -> ACTIVE(sieve(mark(X:S))) 0.00/0.53 MARK(sieve(X:S)) -> MARK(X:S) 0.00/0.53 MARK(zprimes) -> ACTIVE(zprimes) 0.00/0.53 ->->-> Rules: 0.00/0.53 active(filter(cons(X:S,Y:S),s(N:S),M:S)) -> mark(cons(X:S,filter(Y:S,N:S,M:S))) 0.00/0.53 active(filter(cons(X:S,Y:S),0,M:S)) -> mark(cons(0,filter(Y:S,M:S,M:S))) 0.00/0.53 active(nats(N:S)) -> mark(cons(N:S,nats(s(N:S)))) 0.00/0.53 active(sieve(cons(s(N:S),Y:S))) -> mark(cons(s(N:S),sieve(filter(Y:S,N:S,N:S)))) 0.00/0.53 active(sieve(cons(0,Y:S))) -> mark(cons(0,sieve(Y:S))) 0.00/0.53 active(zprimes) -> mark(sieve(nats(s(s(0))))) 0.00/0.53 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.53 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.53 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 0.00/0.53 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 0.00/0.53 filter(active(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 filter(mark(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 filter(X1:S,active(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 filter(X1:S,mark(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 filter(X1:S,X2:S,active(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 filter(X1:S,X2:S,mark(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) 0.00/0.53 mark(filter(X1:S,X2:S,X3:S)) -> active(filter(mark(X1:S),mark(X2:S),mark(X3:S))) 0.00/0.53 mark(nats(X:S)) -> active(nats(mark(X:S))) 0.00/0.53 mark(s(X:S)) -> active(s(mark(X:S))) 0.00/0.53 mark(sieve(X:S)) -> active(sieve(mark(X:S))) 0.00/0.53 mark(0) -> active(0) 0.00/0.53 mark(zprimes) -> active(zprimes) 0.00/0.53 nats(active(X:S)) -> nats(X:S) 0.00/0.53 nats(mark(X:S)) -> nats(X:S) 0.00/0.53 s(active(X:S)) -> s(X:S) 0.00/0.53 s(mark(X:S)) -> s(X:S) 0.00/0.53 sieve(active(X:S)) -> sieve(X:S) 0.00/0.53 sieve(mark(X:S)) -> sieve(X:S) 0.00/0.53 0.00/0.53 Problem 1.6: 0.00/0.53 0.00/0.53 Reduction Pairs Processor: 0.00/0.53 -> Pairs: 0.00/0.53 ACTIVE(nats(N:S)) -> MARK(cons(N:S,nats(s(N:S)))) 0.00/0.53 ACTIVE(sieve(cons(s(N:S),Y:S))) -> MARK(cons(s(N:S),sieve(filter(Y:S,N:S,N:S)))) 0.00/0.53 ACTIVE(sieve(cons(0,Y:S))) -> MARK(cons(0,sieve(Y:S))) 0.00/0.53 ACTIVE(zprimes) -> MARK(sieve(nats(s(s(0))))) 0.00/0.53 MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) 0.00/0.53 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.53 MARK(filter(X1:S,X2:S,X3:S)) -> ACTIVE(filter(mark(X1:S),mark(X2:S),mark(X3:S))) 0.00/0.53 MARK(filter(X1:S,X2:S,X3:S)) -> MARK(X1:S) 0.00/0.53 MARK(filter(X1:S,X2:S,X3:S)) -> MARK(X2:S) 0.00/0.53 MARK(filter(X1:S,X2:S,X3:S)) -> MARK(X3:S) 0.00/0.53 MARK(nats(X:S)) -> ACTIVE(nats(mark(X:S))) 0.00/0.53 MARK(nats(X:S)) -> MARK(X:S) 0.00/0.53 MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) 0.00/0.53 MARK(s(X:S)) -> MARK(X:S) 0.00/0.53 MARK(sieve(X:S)) -> ACTIVE(sieve(mark(X:S))) 0.00/0.53 MARK(sieve(X:S)) -> MARK(X:S) 0.00/0.53 MARK(zprimes) -> ACTIVE(zprimes) 0.00/0.53 -> Rules: 0.00/0.53 active(filter(cons(X:S,Y:S),s(N:S),M:S)) -> mark(cons(X:S,filter(Y:S,N:S,M:S))) 0.00/0.53 active(filter(cons(X:S,Y:S),0,M:S)) -> mark(cons(0,filter(Y:S,M:S,M:S))) 0.00/0.53 active(nats(N:S)) -> mark(cons(N:S,nats(s(N:S)))) 0.00/0.53 active(sieve(cons(s(N:S),Y:S))) -> mark(cons(s(N:S),sieve(filter(Y:S,N:S,N:S)))) 0.00/0.53 active(sieve(cons(0,Y:S))) -> mark(cons(0,sieve(Y:S))) 0.00/0.53 active(zprimes) -> mark(sieve(nats(s(s(0))))) 0.00/0.53 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.53 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.53 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 0.00/0.53 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 0.00/0.53 filter(active(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 filter(mark(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 filter(X1:S,active(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 filter(X1:S,mark(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 filter(X1:S,X2:S,active(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 filter(X1:S,X2:S,mark(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) 0.00/0.53 mark(filter(X1:S,X2:S,X3:S)) -> active(filter(mark(X1:S),mark(X2:S),mark(X3:S))) 0.00/0.53 mark(nats(X:S)) -> active(nats(mark(X:S))) 0.00/0.53 mark(s(X:S)) -> active(s(mark(X:S))) 0.00/0.53 mark(sieve(X:S)) -> active(sieve(mark(X:S))) 0.00/0.53 mark(0) -> active(0) 0.00/0.53 mark(zprimes) -> active(zprimes) 0.00/0.53 nats(active(X:S)) -> nats(X:S) 0.00/0.53 nats(mark(X:S)) -> nats(X:S) 0.00/0.53 s(active(X:S)) -> s(X:S) 0.00/0.53 s(mark(X:S)) -> s(X:S) 0.00/0.53 sieve(active(X:S)) -> sieve(X:S) 0.00/0.53 sieve(mark(X:S)) -> sieve(X:S) 0.00/0.53 -> Usable rules: 0.00/0.53 active(filter(cons(X:S,Y:S),s(N:S),M:S)) -> mark(cons(X:S,filter(Y:S,N:S,M:S))) 0.00/0.53 active(filter(cons(X:S,Y:S),0,M:S)) -> mark(cons(0,filter(Y:S,M:S,M:S))) 0.00/0.53 active(nats(N:S)) -> mark(cons(N:S,nats(s(N:S)))) 0.00/0.53 active(sieve(cons(s(N:S),Y:S))) -> mark(cons(s(N:S),sieve(filter(Y:S,N:S,N:S)))) 0.00/0.53 active(sieve(cons(0,Y:S))) -> mark(cons(0,sieve(Y:S))) 0.00/0.53 active(zprimes) -> mark(sieve(nats(s(s(0))))) 0.00/0.53 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.53 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.53 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 0.00/0.53 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 0.00/0.53 filter(active(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 filter(mark(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 filter(X1:S,active(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 filter(X1:S,mark(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 filter(X1:S,X2:S,active(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 filter(X1:S,X2:S,mark(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) 0.00/0.53 mark(filter(X1:S,X2:S,X3:S)) -> active(filter(mark(X1:S),mark(X2:S),mark(X3:S))) 0.00/0.53 mark(nats(X:S)) -> active(nats(mark(X:S))) 0.00/0.53 mark(s(X:S)) -> active(s(mark(X:S))) 0.00/0.53 mark(sieve(X:S)) -> active(sieve(mark(X:S))) 0.00/0.53 mark(0) -> active(0) 0.00/0.53 mark(zprimes) -> active(zprimes) 0.00/0.53 nats(active(X:S)) -> nats(X:S) 0.00/0.53 nats(mark(X:S)) -> nats(X:S) 0.00/0.53 s(active(X:S)) -> s(X:S) 0.00/0.53 s(mark(X:S)) -> s(X:S) 0.00/0.53 sieve(active(X:S)) -> sieve(X:S) 0.00/0.53 sieve(mark(X:S)) -> sieve(X:S) 0.00/0.53 ->Interpretation type: 0.00/0.53 Linear 0.00/0.53 ->Coefficients: 0.00/0.53 Natural Numbers 0.00/0.53 ->Dimension: 0.00/0.53 1 0.00/0.53 ->Bound: 0.00/0.53 2 0.00/0.53 ->Interpretation: 0.00/0.53 0.00/0.53 [active](X) = X 0.00/0.53 [cons](X1,X2) = 2.X1 + 1 0.00/0.53 [filter](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 0.00/0.53 [mark](X) = X 0.00/0.53 [nats](X) = 2.X + 2 0.00/0.53 [s](X) = 2.X 0.00/0.53 [sieve](X) = X 0.00/0.53 [0] = 0 0.00/0.53 [fSNonEmpty] = 0 0.00/0.53 [zprimes] = 2 0.00/0.53 [ACTIVE](X) = 2.X 0.00/0.53 [CONS](X1,X2) = 0 0.00/0.53 [FILTER](X1,X2,X3) = 0 0.00/0.53 [MARK](X) = 2.X 0.00/0.53 [NATS](X) = 0 0.00/0.53 [S](X) = 0 0.00/0.53 [SIEVE](X) = 0 0.00/0.53 0.00/0.53 Problem 1.6: 0.00/0.53 0.00/0.53 SCC Processor: 0.00/0.53 -> Pairs: 0.00/0.53 ACTIVE(sieve(cons(s(N:S),Y:S))) -> MARK(cons(s(N:S),sieve(filter(Y:S,N:S,N:S)))) 0.00/0.53 ACTIVE(sieve(cons(0,Y:S))) -> MARK(cons(0,sieve(Y:S))) 0.00/0.53 ACTIVE(zprimes) -> MARK(sieve(nats(s(s(0))))) 0.00/0.53 MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) 0.00/0.53 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.53 MARK(filter(X1:S,X2:S,X3:S)) -> ACTIVE(filter(mark(X1:S),mark(X2:S),mark(X3:S))) 0.00/0.53 MARK(filter(X1:S,X2:S,X3:S)) -> MARK(X1:S) 0.00/0.53 MARK(filter(X1:S,X2:S,X3:S)) -> MARK(X2:S) 0.00/0.53 MARK(filter(X1:S,X2:S,X3:S)) -> MARK(X3:S) 0.00/0.53 MARK(nats(X:S)) -> ACTIVE(nats(mark(X:S))) 0.00/0.53 MARK(nats(X:S)) -> MARK(X:S) 0.00/0.53 MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) 0.00/0.53 MARK(s(X:S)) -> MARK(X:S) 0.00/0.53 MARK(sieve(X:S)) -> ACTIVE(sieve(mark(X:S))) 0.00/0.53 MARK(sieve(X:S)) -> MARK(X:S) 0.00/0.53 MARK(zprimes) -> ACTIVE(zprimes) 0.00/0.53 -> Rules: 0.00/0.53 active(filter(cons(X:S,Y:S),s(N:S),M:S)) -> mark(cons(X:S,filter(Y:S,N:S,M:S))) 0.00/0.53 active(filter(cons(X:S,Y:S),0,M:S)) -> mark(cons(0,filter(Y:S,M:S,M:S))) 0.00/0.53 active(nats(N:S)) -> mark(cons(N:S,nats(s(N:S)))) 0.00/0.53 active(sieve(cons(s(N:S),Y:S))) -> mark(cons(s(N:S),sieve(filter(Y:S,N:S,N:S)))) 0.00/0.53 active(sieve(cons(0,Y:S))) -> mark(cons(0,sieve(Y:S))) 0.00/0.53 active(zprimes) -> mark(sieve(nats(s(s(0))))) 0.00/0.53 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.53 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.53 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 0.00/0.53 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 0.00/0.53 filter(active(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 filter(mark(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 filter(X1:S,active(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 filter(X1:S,mark(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 filter(X1:S,X2:S,active(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 filter(X1:S,X2:S,mark(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) 0.00/0.53 mark(filter(X1:S,X2:S,X3:S)) -> active(filter(mark(X1:S),mark(X2:S),mark(X3:S))) 0.00/0.53 mark(nats(X:S)) -> active(nats(mark(X:S))) 0.00/0.53 mark(s(X:S)) -> active(s(mark(X:S))) 0.00/0.53 mark(sieve(X:S)) -> active(sieve(mark(X:S))) 0.00/0.53 mark(0) -> active(0) 0.00/0.53 mark(zprimes) -> active(zprimes) 0.00/0.53 nats(active(X:S)) -> nats(X:S) 0.00/0.53 nats(mark(X:S)) -> nats(X:S) 0.00/0.53 s(active(X:S)) -> s(X:S) 0.00/0.53 s(mark(X:S)) -> s(X:S) 0.00/0.53 sieve(active(X:S)) -> sieve(X:S) 0.00/0.53 sieve(mark(X:S)) -> sieve(X:S) 0.00/0.53 ->Strongly Connected Components: 0.00/0.53 ->->Cycle: 0.00/0.53 ->->-> Pairs: 0.00/0.53 ACTIVE(sieve(cons(s(N:S),Y:S))) -> MARK(cons(s(N:S),sieve(filter(Y:S,N:S,N:S)))) 0.00/0.53 ACTIVE(sieve(cons(0,Y:S))) -> MARK(cons(0,sieve(Y:S))) 0.00/0.53 ACTIVE(zprimes) -> MARK(sieve(nats(s(s(0))))) 0.00/0.53 MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) 0.00/0.53 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.53 MARK(filter(X1:S,X2:S,X3:S)) -> ACTIVE(filter(mark(X1:S),mark(X2:S),mark(X3:S))) 0.00/0.53 MARK(filter(X1:S,X2:S,X3:S)) -> MARK(X1:S) 0.00/0.53 MARK(filter(X1:S,X2:S,X3:S)) -> MARK(X2:S) 0.00/0.53 MARK(filter(X1:S,X2:S,X3:S)) -> MARK(X3:S) 0.00/0.53 MARK(nats(X:S)) -> ACTIVE(nats(mark(X:S))) 0.00/0.53 MARK(nats(X:S)) -> MARK(X:S) 0.00/0.53 MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) 0.00/0.53 MARK(s(X:S)) -> MARK(X:S) 0.00/0.53 MARK(sieve(X:S)) -> ACTIVE(sieve(mark(X:S))) 0.00/0.53 MARK(sieve(X:S)) -> MARK(X:S) 0.00/0.53 MARK(zprimes) -> ACTIVE(zprimes) 0.00/0.53 ->->-> Rules: 0.00/0.53 active(filter(cons(X:S,Y:S),s(N:S),M:S)) -> mark(cons(X:S,filter(Y:S,N:S,M:S))) 0.00/0.53 active(filter(cons(X:S,Y:S),0,M:S)) -> mark(cons(0,filter(Y:S,M:S,M:S))) 0.00/0.53 active(nats(N:S)) -> mark(cons(N:S,nats(s(N:S)))) 0.00/0.53 active(sieve(cons(s(N:S),Y:S))) -> mark(cons(s(N:S),sieve(filter(Y:S,N:S,N:S)))) 0.00/0.53 active(sieve(cons(0,Y:S))) -> mark(cons(0,sieve(Y:S))) 0.00/0.53 active(zprimes) -> mark(sieve(nats(s(s(0))))) 0.00/0.53 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.53 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.53 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 0.00/0.53 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 0.00/0.53 filter(active(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 filter(mark(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 filter(X1:S,active(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 filter(X1:S,mark(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 filter(X1:S,X2:S,active(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 filter(X1:S,X2:S,mark(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) 0.00/0.53 mark(filter(X1:S,X2:S,X3:S)) -> active(filter(mark(X1:S),mark(X2:S),mark(X3:S))) 0.00/0.53 mark(nats(X:S)) -> active(nats(mark(X:S))) 0.00/0.53 mark(s(X:S)) -> active(s(mark(X:S))) 0.00/0.53 mark(sieve(X:S)) -> active(sieve(mark(X:S))) 0.00/0.53 mark(0) -> active(0) 0.00/0.53 mark(zprimes) -> active(zprimes) 0.00/0.53 nats(active(X:S)) -> nats(X:S) 0.00/0.53 nats(mark(X:S)) -> nats(X:S) 0.00/0.53 s(active(X:S)) -> s(X:S) 0.00/0.53 s(mark(X:S)) -> s(X:S) 0.00/0.53 sieve(active(X:S)) -> sieve(X:S) 0.00/0.53 sieve(mark(X:S)) -> sieve(X:S) 0.00/0.53 0.00/0.53 Problem 1.6: 0.00/0.53 0.00/0.53 Reduction Pairs Processor: 0.00/0.53 -> Pairs: 0.00/0.53 ACTIVE(sieve(cons(s(N:S),Y:S))) -> MARK(cons(s(N:S),sieve(filter(Y:S,N:S,N:S)))) 0.00/0.53 ACTIVE(sieve(cons(0,Y:S))) -> MARK(cons(0,sieve(Y:S))) 0.00/0.53 ACTIVE(zprimes) -> MARK(sieve(nats(s(s(0))))) 0.00/0.53 MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) 0.00/0.53 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.53 MARK(filter(X1:S,X2:S,X3:S)) -> ACTIVE(filter(mark(X1:S),mark(X2:S),mark(X3:S))) 0.00/0.53 MARK(filter(X1:S,X2:S,X3:S)) -> MARK(X1:S) 0.00/0.53 MARK(filter(X1:S,X2:S,X3:S)) -> MARK(X2:S) 0.00/0.53 MARK(filter(X1:S,X2:S,X3:S)) -> MARK(X3:S) 0.00/0.53 MARK(nats(X:S)) -> ACTIVE(nats(mark(X:S))) 0.00/0.53 MARK(nats(X:S)) -> MARK(X:S) 0.00/0.53 MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) 0.00/0.53 MARK(s(X:S)) -> MARK(X:S) 0.00/0.53 MARK(sieve(X:S)) -> ACTIVE(sieve(mark(X:S))) 0.00/0.53 MARK(sieve(X:S)) -> MARK(X:S) 0.00/0.53 MARK(zprimes) -> ACTIVE(zprimes) 0.00/0.53 -> Rules: 0.00/0.53 active(filter(cons(X:S,Y:S),s(N:S),M:S)) -> mark(cons(X:S,filter(Y:S,N:S,M:S))) 0.00/0.53 active(filter(cons(X:S,Y:S),0,M:S)) -> mark(cons(0,filter(Y:S,M:S,M:S))) 0.00/0.53 active(nats(N:S)) -> mark(cons(N:S,nats(s(N:S)))) 0.00/0.53 active(sieve(cons(s(N:S),Y:S))) -> mark(cons(s(N:S),sieve(filter(Y:S,N:S,N:S)))) 0.00/0.53 active(sieve(cons(0,Y:S))) -> mark(cons(0,sieve(Y:S))) 0.00/0.53 active(zprimes) -> mark(sieve(nats(s(s(0))))) 0.00/0.53 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.53 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.53 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 0.00/0.53 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 0.00/0.53 filter(active(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 filter(mark(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 filter(X1:S,active(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 filter(X1:S,mark(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 filter(X1:S,X2:S,active(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 filter(X1:S,X2:S,mark(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) 0.00/0.53 mark(filter(X1:S,X2:S,X3:S)) -> active(filter(mark(X1:S),mark(X2:S),mark(X3:S))) 0.00/0.53 mark(nats(X:S)) -> active(nats(mark(X:S))) 0.00/0.53 mark(s(X:S)) -> active(s(mark(X:S))) 0.00/0.53 mark(sieve(X:S)) -> active(sieve(mark(X:S))) 0.00/0.53 mark(0) -> active(0) 0.00/0.53 mark(zprimes) -> active(zprimes) 0.00/0.53 nats(active(X:S)) -> nats(X:S) 0.00/0.53 nats(mark(X:S)) -> nats(X:S) 0.00/0.53 s(active(X:S)) -> s(X:S) 0.00/0.53 s(mark(X:S)) -> s(X:S) 0.00/0.53 sieve(active(X:S)) -> sieve(X:S) 0.00/0.53 sieve(mark(X:S)) -> sieve(X:S) 0.00/0.53 -> Usable rules: 0.00/0.53 active(filter(cons(X:S,Y:S),s(N:S),M:S)) -> mark(cons(X:S,filter(Y:S,N:S,M:S))) 0.00/0.53 active(filter(cons(X:S,Y:S),0,M:S)) -> mark(cons(0,filter(Y:S,M:S,M:S))) 0.00/0.53 active(nats(N:S)) -> mark(cons(N:S,nats(s(N:S)))) 0.00/0.53 active(sieve(cons(s(N:S),Y:S))) -> mark(cons(s(N:S),sieve(filter(Y:S,N:S,N:S)))) 0.00/0.53 active(sieve(cons(0,Y:S))) -> mark(cons(0,sieve(Y:S))) 0.00/0.53 active(zprimes) -> mark(sieve(nats(s(s(0))))) 0.00/0.53 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.53 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.53 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 0.00/0.53 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 0.00/0.53 filter(active(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 filter(mark(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 filter(X1:S,active(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 filter(X1:S,mark(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 filter(X1:S,X2:S,active(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 filter(X1:S,X2:S,mark(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) 0.00/0.53 mark(filter(X1:S,X2:S,X3:S)) -> active(filter(mark(X1:S),mark(X2:S),mark(X3:S))) 0.00/0.53 mark(nats(X:S)) -> active(nats(mark(X:S))) 0.00/0.53 mark(s(X:S)) -> active(s(mark(X:S))) 0.00/0.53 mark(sieve(X:S)) -> active(sieve(mark(X:S))) 0.00/0.53 mark(0) -> active(0) 0.00/0.53 mark(zprimes) -> active(zprimes) 0.00/0.53 nats(active(X:S)) -> nats(X:S) 0.00/0.53 nats(mark(X:S)) -> nats(X:S) 0.00/0.53 s(active(X:S)) -> s(X:S) 0.00/0.53 s(mark(X:S)) -> s(X:S) 0.00/0.53 sieve(active(X:S)) -> sieve(X:S) 0.00/0.53 sieve(mark(X:S)) -> sieve(X:S) 0.00/0.53 ->Interpretation type: 0.00/0.53 Linear 0.00/0.53 ->Coefficients: 0.00/0.53 Natural Numbers 0.00/0.53 ->Dimension: 0.00/0.53 1 0.00/0.53 ->Bound: 0.00/0.53 2 0.00/0.53 ->Interpretation: 0.00/0.53 0.00/0.53 [active](X) = X 0.00/0.53 [cons](X1,X2) = X1 0.00/0.53 [filter](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 0.00/0.53 [mark](X) = 2.X 0.00/0.53 [nats](X) = 2.X 0.00/0.53 [s](X) = 2.X 0.00/0.53 [sieve](X) = 2.X + 1 0.00/0.53 [0] = 0 0.00/0.53 [fSNonEmpty] = 0 0.00/0.53 [zprimes] = 2 0.00/0.53 [ACTIVE](X) = X + 2 0.00/0.53 [CONS](X1,X2) = 0 0.00/0.53 [FILTER](X1,X2,X3) = 0 0.00/0.53 [MARK](X) = 2.X + 2 0.00/0.53 [NATS](X) = 0 0.00/0.53 [S](X) = 0 0.00/0.53 [SIEVE](X) = 0 0.00/0.53 0.00/0.53 Problem 1.6: 0.00/0.53 0.00/0.53 SCC Processor: 0.00/0.53 -> Pairs: 0.00/0.53 ACTIVE(sieve(cons(0,Y:S))) -> MARK(cons(0,sieve(Y:S))) 0.00/0.53 ACTIVE(zprimes) -> MARK(sieve(nats(s(s(0))))) 0.00/0.53 MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) 0.00/0.53 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.53 MARK(filter(X1:S,X2:S,X3:S)) -> ACTIVE(filter(mark(X1:S),mark(X2:S),mark(X3:S))) 0.00/0.53 MARK(filter(X1:S,X2:S,X3:S)) -> MARK(X1:S) 0.00/0.53 MARK(filter(X1:S,X2:S,X3:S)) -> MARK(X2:S) 0.00/0.53 MARK(filter(X1:S,X2:S,X3:S)) -> MARK(X3:S) 0.00/0.53 MARK(nats(X:S)) -> ACTIVE(nats(mark(X:S))) 0.00/0.53 MARK(nats(X:S)) -> MARK(X:S) 0.00/0.53 MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) 0.00/0.53 MARK(s(X:S)) -> MARK(X:S) 0.00/0.53 MARK(sieve(X:S)) -> ACTIVE(sieve(mark(X:S))) 0.00/0.53 MARK(sieve(X:S)) -> MARK(X:S) 0.00/0.53 MARK(zprimes) -> ACTIVE(zprimes) 0.00/0.53 -> Rules: 0.00/0.53 active(filter(cons(X:S,Y:S),s(N:S),M:S)) -> mark(cons(X:S,filter(Y:S,N:S,M:S))) 0.00/0.53 active(filter(cons(X:S,Y:S),0,M:S)) -> mark(cons(0,filter(Y:S,M:S,M:S))) 0.00/0.53 active(nats(N:S)) -> mark(cons(N:S,nats(s(N:S)))) 0.00/0.53 active(sieve(cons(s(N:S),Y:S))) -> mark(cons(s(N:S),sieve(filter(Y:S,N:S,N:S)))) 0.00/0.53 active(sieve(cons(0,Y:S))) -> mark(cons(0,sieve(Y:S))) 0.00/0.53 active(zprimes) -> mark(sieve(nats(s(s(0))))) 0.00/0.53 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.53 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.53 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 0.00/0.53 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 0.00/0.53 filter(active(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 filter(mark(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 filter(X1:S,active(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 filter(X1:S,mark(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 filter(X1:S,X2:S,active(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 filter(X1:S,X2:S,mark(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) 0.00/0.53 mark(filter(X1:S,X2:S,X3:S)) -> active(filter(mark(X1:S),mark(X2:S),mark(X3:S))) 0.00/0.53 mark(nats(X:S)) -> active(nats(mark(X:S))) 0.00/0.53 mark(s(X:S)) -> active(s(mark(X:S))) 0.00/0.53 mark(sieve(X:S)) -> active(sieve(mark(X:S))) 0.00/0.53 mark(0) -> active(0) 0.00/0.53 mark(zprimes) -> active(zprimes) 0.00/0.53 nats(active(X:S)) -> nats(X:S) 0.00/0.53 nats(mark(X:S)) -> nats(X:S) 0.00/0.53 s(active(X:S)) -> s(X:S) 0.00/0.53 s(mark(X:S)) -> s(X:S) 0.00/0.53 sieve(active(X:S)) -> sieve(X:S) 0.00/0.53 sieve(mark(X:S)) -> sieve(X:S) 0.00/0.53 ->Strongly Connected Components: 0.00/0.53 ->->Cycle: 0.00/0.53 ->->-> Pairs: 0.00/0.53 ACTIVE(sieve(cons(0,Y:S))) -> MARK(cons(0,sieve(Y:S))) 0.00/0.53 ACTIVE(zprimes) -> MARK(sieve(nats(s(s(0))))) 0.00/0.53 MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) 0.00/0.53 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.53 MARK(filter(X1:S,X2:S,X3:S)) -> ACTIVE(filter(mark(X1:S),mark(X2:S),mark(X3:S))) 0.00/0.53 MARK(filter(X1:S,X2:S,X3:S)) -> MARK(X1:S) 0.00/0.53 MARK(filter(X1:S,X2:S,X3:S)) -> MARK(X2:S) 0.00/0.53 MARK(filter(X1:S,X2:S,X3:S)) -> MARK(X3:S) 0.00/0.53 MARK(nats(X:S)) -> ACTIVE(nats(mark(X:S))) 0.00/0.53 MARK(nats(X:S)) -> MARK(X:S) 0.00/0.53 MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) 0.00/0.53 MARK(s(X:S)) -> MARK(X:S) 0.00/0.53 MARK(sieve(X:S)) -> ACTIVE(sieve(mark(X:S))) 0.00/0.53 MARK(sieve(X:S)) -> MARK(X:S) 0.00/0.53 MARK(zprimes) -> ACTIVE(zprimes) 0.00/0.53 ->->-> Rules: 0.00/0.53 active(filter(cons(X:S,Y:S),s(N:S),M:S)) -> mark(cons(X:S,filter(Y:S,N:S,M:S))) 0.00/0.53 active(filter(cons(X:S,Y:S),0,M:S)) -> mark(cons(0,filter(Y:S,M:S,M:S))) 0.00/0.53 active(nats(N:S)) -> mark(cons(N:S,nats(s(N:S)))) 0.00/0.53 active(sieve(cons(s(N:S),Y:S))) -> mark(cons(s(N:S),sieve(filter(Y:S,N:S,N:S)))) 0.00/0.53 active(sieve(cons(0,Y:S))) -> mark(cons(0,sieve(Y:S))) 0.00/0.53 active(zprimes) -> mark(sieve(nats(s(s(0))))) 0.00/0.53 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.53 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.53 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 0.00/0.53 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 0.00/0.53 filter(active(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 filter(mark(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 filter(X1:S,active(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 filter(X1:S,mark(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 filter(X1:S,X2:S,active(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 filter(X1:S,X2:S,mark(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) 0.00/0.53 mark(filter(X1:S,X2:S,X3:S)) -> active(filter(mark(X1:S),mark(X2:S),mark(X3:S))) 0.00/0.53 mark(nats(X:S)) -> active(nats(mark(X:S))) 0.00/0.53 mark(s(X:S)) -> active(s(mark(X:S))) 0.00/0.53 mark(sieve(X:S)) -> active(sieve(mark(X:S))) 0.00/0.53 mark(0) -> active(0) 0.00/0.53 mark(zprimes) -> active(zprimes) 0.00/0.53 nats(active(X:S)) -> nats(X:S) 0.00/0.53 nats(mark(X:S)) -> nats(X:S) 0.00/0.53 s(active(X:S)) -> s(X:S) 0.00/0.53 s(mark(X:S)) -> s(X:S) 0.00/0.53 sieve(active(X:S)) -> sieve(X:S) 0.00/0.53 sieve(mark(X:S)) -> sieve(X:S) 0.00/0.53 0.00/0.53 Problem 1.6: 0.00/0.53 0.00/0.53 Reduction Pairs Processor: 0.00/0.53 -> Pairs: 0.00/0.53 ACTIVE(sieve(cons(0,Y:S))) -> MARK(cons(0,sieve(Y:S))) 0.00/0.53 ACTIVE(zprimes) -> MARK(sieve(nats(s(s(0))))) 0.00/0.53 MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) 0.00/0.53 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.53 MARK(filter(X1:S,X2:S,X3:S)) -> ACTIVE(filter(mark(X1:S),mark(X2:S),mark(X3:S))) 0.00/0.53 MARK(filter(X1:S,X2:S,X3:S)) -> MARK(X1:S) 0.00/0.53 MARK(filter(X1:S,X2:S,X3:S)) -> MARK(X2:S) 0.00/0.53 MARK(filter(X1:S,X2:S,X3:S)) -> MARK(X3:S) 0.00/0.53 MARK(nats(X:S)) -> ACTIVE(nats(mark(X:S))) 0.00/0.53 MARK(nats(X:S)) -> MARK(X:S) 0.00/0.53 MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) 0.00/0.53 MARK(s(X:S)) -> MARK(X:S) 0.00/0.53 MARK(sieve(X:S)) -> ACTIVE(sieve(mark(X:S))) 0.00/0.53 MARK(sieve(X:S)) -> MARK(X:S) 0.00/0.53 MARK(zprimes) -> ACTIVE(zprimes) 0.00/0.53 -> Rules: 0.00/0.53 active(filter(cons(X:S,Y:S),s(N:S),M:S)) -> mark(cons(X:S,filter(Y:S,N:S,M:S))) 0.00/0.53 active(filter(cons(X:S,Y:S),0,M:S)) -> mark(cons(0,filter(Y:S,M:S,M:S))) 0.00/0.53 active(nats(N:S)) -> mark(cons(N:S,nats(s(N:S)))) 0.00/0.53 active(sieve(cons(s(N:S),Y:S))) -> mark(cons(s(N:S),sieve(filter(Y:S,N:S,N:S)))) 0.00/0.53 active(sieve(cons(0,Y:S))) -> mark(cons(0,sieve(Y:S))) 0.00/0.53 active(zprimes) -> mark(sieve(nats(s(s(0))))) 0.00/0.53 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.53 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.53 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 0.00/0.53 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 0.00/0.53 filter(active(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 filter(mark(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 filter(X1:S,active(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 filter(X1:S,mark(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 filter(X1:S,X2:S,active(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 filter(X1:S,X2:S,mark(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) 0.00/0.53 mark(filter(X1:S,X2:S,X3:S)) -> active(filter(mark(X1:S),mark(X2:S),mark(X3:S))) 0.00/0.53 mark(nats(X:S)) -> active(nats(mark(X:S))) 0.00/0.53 mark(s(X:S)) -> active(s(mark(X:S))) 0.00/0.53 mark(sieve(X:S)) -> active(sieve(mark(X:S))) 0.00/0.53 mark(0) -> active(0) 0.00/0.53 mark(zprimes) -> active(zprimes) 0.00/0.53 nats(active(X:S)) -> nats(X:S) 0.00/0.53 nats(mark(X:S)) -> nats(X:S) 0.00/0.53 s(active(X:S)) -> s(X:S) 0.00/0.53 s(mark(X:S)) -> s(X:S) 0.00/0.53 sieve(active(X:S)) -> sieve(X:S) 0.00/0.53 sieve(mark(X:S)) -> sieve(X:S) 0.00/0.53 -> Usable rules: 0.00/0.53 active(filter(cons(X:S,Y:S),s(N:S),M:S)) -> mark(cons(X:S,filter(Y:S,N:S,M:S))) 0.00/0.53 active(filter(cons(X:S,Y:S),0,M:S)) -> mark(cons(0,filter(Y:S,M:S,M:S))) 0.00/0.53 active(nats(N:S)) -> mark(cons(N:S,nats(s(N:S)))) 0.00/0.53 active(sieve(cons(s(N:S),Y:S))) -> mark(cons(s(N:S),sieve(filter(Y:S,N:S,N:S)))) 0.00/0.53 active(sieve(cons(0,Y:S))) -> mark(cons(0,sieve(Y:S))) 0.00/0.53 active(zprimes) -> mark(sieve(nats(s(s(0))))) 0.00/0.53 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.53 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.53 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 0.00/0.53 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 0.00/0.53 filter(active(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 filter(mark(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 filter(X1:S,active(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 filter(X1:S,mark(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 filter(X1:S,X2:S,active(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 filter(X1:S,X2:S,mark(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) 0.00/0.53 mark(filter(X1:S,X2:S,X3:S)) -> active(filter(mark(X1:S),mark(X2:S),mark(X3:S))) 0.00/0.53 mark(nats(X:S)) -> active(nats(mark(X:S))) 0.00/0.53 mark(s(X:S)) -> active(s(mark(X:S))) 0.00/0.53 mark(sieve(X:S)) -> active(sieve(mark(X:S))) 0.00/0.53 mark(0) -> active(0) 0.00/0.53 mark(zprimes) -> active(zprimes) 0.00/0.53 nats(active(X:S)) -> nats(X:S) 0.00/0.53 nats(mark(X:S)) -> nats(X:S) 0.00/0.53 s(active(X:S)) -> s(X:S) 0.00/0.53 s(mark(X:S)) -> s(X:S) 0.00/0.53 sieve(active(X:S)) -> sieve(X:S) 0.00/0.53 sieve(mark(X:S)) -> sieve(X:S) 0.00/0.53 ->Interpretation type: 0.00/0.53 Linear 0.00/0.53 ->Coefficients: 0.00/0.53 Natural Numbers 0.00/0.53 ->Dimension: 0.00/0.53 1 0.00/0.53 ->Bound: 0.00/0.53 2 0.00/0.53 ->Interpretation: 0.00/0.53 0.00/0.53 [active](X) = X 0.00/0.53 [cons](X1,X2) = X1 0.00/0.53 [filter](X1,X2,X3) = 2.X1 + X2 + 2.X3 0.00/0.53 [mark](X) = X 0.00/0.53 [nats](X) = X 0.00/0.53 [s](X) = X 0.00/0.53 [sieve](X) = X + 1 0.00/0.53 [0] = 1 0.00/0.53 [fSNonEmpty] = 0 0.00/0.53 [zprimes] = 2 0.00/0.53 [ACTIVE](X) = 2.X + 2 0.00/0.53 [CONS](X1,X2) = 0 0.00/0.53 [FILTER](X1,X2,X3) = 0 0.00/0.53 [MARK](X) = 2.X + 2 0.00/0.53 [NATS](X) = 0 0.00/0.53 [S](X) = 0 0.00/0.53 [SIEVE](X) = 0 0.00/0.53 0.00/0.53 Problem 1.6: 0.00/0.53 0.00/0.53 SCC Processor: 0.00/0.53 -> Pairs: 0.00/0.53 ACTIVE(zprimes) -> MARK(sieve(nats(s(s(0))))) 0.00/0.53 MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) 0.00/0.53 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.53 MARK(filter(X1:S,X2:S,X3:S)) -> ACTIVE(filter(mark(X1:S),mark(X2:S),mark(X3:S))) 0.00/0.53 MARK(filter(X1:S,X2:S,X3:S)) -> MARK(X1:S) 0.00/0.53 MARK(filter(X1:S,X2:S,X3:S)) -> MARK(X2:S) 0.00/0.53 MARK(filter(X1:S,X2:S,X3:S)) -> MARK(X3:S) 0.00/0.53 MARK(nats(X:S)) -> ACTIVE(nats(mark(X:S))) 0.00/0.53 MARK(nats(X:S)) -> MARK(X:S) 0.00/0.53 MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) 0.00/0.53 MARK(s(X:S)) -> MARK(X:S) 0.00/0.53 MARK(sieve(X:S)) -> ACTIVE(sieve(mark(X:S))) 0.00/0.53 MARK(sieve(X:S)) -> MARK(X:S) 0.00/0.53 MARK(zprimes) -> ACTIVE(zprimes) 0.00/0.53 -> Rules: 0.00/0.53 active(filter(cons(X:S,Y:S),s(N:S),M:S)) -> mark(cons(X:S,filter(Y:S,N:S,M:S))) 0.00/0.53 active(filter(cons(X:S,Y:S),0,M:S)) -> mark(cons(0,filter(Y:S,M:S,M:S))) 0.00/0.53 active(nats(N:S)) -> mark(cons(N:S,nats(s(N:S)))) 0.00/0.53 active(sieve(cons(s(N:S),Y:S))) -> mark(cons(s(N:S),sieve(filter(Y:S,N:S,N:S)))) 0.00/0.53 active(sieve(cons(0,Y:S))) -> mark(cons(0,sieve(Y:S))) 0.00/0.53 active(zprimes) -> mark(sieve(nats(s(s(0))))) 0.00/0.53 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.53 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.53 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 0.00/0.53 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 0.00/0.53 filter(active(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 filter(mark(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 filter(X1:S,active(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 filter(X1:S,mark(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 filter(X1:S,X2:S,active(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 filter(X1:S,X2:S,mark(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) 0.00/0.53 mark(filter(X1:S,X2:S,X3:S)) -> active(filter(mark(X1:S),mark(X2:S),mark(X3:S))) 0.00/0.53 mark(nats(X:S)) -> active(nats(mark(X:S))) 0.00/0.53 mark(s(X:S)) -> active(s(mark(X:S))) 0.00/0.53 mark(sieve(X:S)) -> active(sieve(mark(X:S))) 0.00/0.53 mark(0) -> active(0) 0.00/0.53 mark(zprimes) -> active(zprimes) 0.00/0.53 nats(active(X:S)) -> nats(X:S) 0.00/0.53 nats(mark(X:S)) -> nats(X:S) 0.00/0.53 s(active(X:S)) -> s(X:S) 0.00/0.53 s(mark(X:S)) -> s(X:S) 0.00/0.53 sieve(active(X:S)) -> sieve(X:S) 0.00/0.53 sieve(mark(X:S)) -> sieve(X:S) 0.00/0.53 ->Strongly Connected Components: 0.00/0.53 ->->Cycle: 0.00/0.53 ->->-> Pairs: 0.00/0.53 ACTIVE(zprimes) -> MARK(sieve(nats(s(s(0))))) 0.00/0.53 MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) 0.00/0.53 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.53 MARK(filter(X1:S,X2:S,X3:S)) -> ACTIVE(filter(mark(X1:S),mark(X2:S),mark(X3:S))) 0.00/0.53 MARK(filter(X1:S,X2:S,X3:S)) -> MARK(X1:S) 0.00/0.53 MARK(filter(X1:S,X2:S,X3:S)) -> MARK(X2:S) 0.00/0.53 MARK(filter(X1:S,X2:S,X3:S)) -> MARK(X3:S) 0.00/0.53 MARK(nats(X:S)) -> ACTIVE(nats(mark(X:S))) 0.00/0.53 MARK(nats(X:S)) -> MARK(X:S) 0.00/0.53 MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) 0.00/0.53 MARK(s(X:S)) -> MARK(X:S) 0.00/0.53 MARK(sieve(X:S)) -> ACTIVE(sieve(mark(X:S))) 0.00/0.53 MARK(sieve(X:S)) -> MARK(X:S) 0.00/0.53 MARK(zprimes) -> ACTIVE(zprimes) 0.00/0.53 ->->-> Rules: 0.00/0.53 active(filter(cons(X:S,Y:S),s(N:S),M:S)) -> mark(cons(X:S,filter(Y:S,N:S,M:S))) 0.00/0.53 active(filter(cons(X:S,Y:S),0,M:S)) -> mark(cons(0,filter(Y:S,M:S,M:S))) 0.00/0.53 active(nats(N:S)) -> mark(cons(N:S,nats(s(N:S)))) 0.00/0.53 active(sieve(cons(s(N:S),Y:S))) -> mark(cons(s(N:S),sieve(filter(Y:S,N:S,N:S)))) 0.00/0.53 active(sieve(cons(0,Y:S))) -> mark(cons(0,sieve(Y:S))) 0.00/0.53 active(zprimes) -> mark(sieve(nats(s(s(0))))) 0.00/0.53 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.53 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.53 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 0.00/0.53 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 0.00/0.53 filter(active(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 filter(mark(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 filter(X1:S,active(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 filter(X1:S,mark(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 filter(X1:S,X2:S,active(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 filter(X1:S,X2:S,mark(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) 0.00/0.53 mark(filter(X1:S,X2:S,X3:S)) -> active(filter(mark(X1:S),mark(X2:S),mark(X3:S))) 0.00/0.53 mark(nats(X:S)) -> active(nats(mark(X:S))) 0.00/0.53 mark(s(X:S)) -> active(s(mark(X:S))) 0.00/0.53 mark(sieve(X:S)) -> active(sieve(mark(X:S))) 0.00/0.53 mark(0) -> active(0) 0.00/0.53 mark(zprimes) -> active(zprimes) 0.00/0.53 nats(active(X:S)) -> nats(X:S) 0.00/0.53 nats(mark(X:S)) -> nats(X:S) 0.00/0.53 s(active(X:S)) -> s(X:S) 0.00/0.53 s(mark(X:S)) -> s(X:S) 0.00/0.53 sieve(active(X:S)) -> sieve(X:S) 0.00/0.53 sieve(mark(X:S)) -> sieve(X:S) 0.00/0.53 0.00/0.53 Problem 1.6: 0.00/0.53 0.00/0.53 Reduction Pairs Processor: 0.00/0.53 -> Pairs: 0.00/0.53 ACTIVE(zprimes) -> MARK(sieve(nats(s(s(0))))) 0.00/0.53 MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) 0.00/0.53 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.53 MARK(filter(X1:S,X2:S,X3:S)) -> ACTIVE(filter(mark(X1:S),mark(X2:S),mark(X3:S))) 0.00/0.53 MARK(filter(X1:S,X2:S,X3:S)) -> MARK(X1:S) 0.00/0.53 MARK(filter(X1:S,X2:S,X3:S)) -> MARK(X2:S) 0.00/0.53 MARK(filter(X1:S,X2:S,X3:S)) -> MARK(X3:S) 0.00/0.53 MARK(nats(X:S)) -> ACTIVE(nats(mark(X:S))) 0.00/0.53 MARK(nats(X:S)) -> MARK(X:S) 0.00/0.53 MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) 0.00/0.53 MARK(s(X:S)) -> MARK(X:S) 0.00/0.53 MARK(sieve(X:S)) -> ACTIVE(sieve(mark(X:S))) 0.00/0.53 MARK(sieve(X:S)) -> MARK(X:S) 0.00/0.53 MARK(zprimes) -> ACTIVE(zprimes) 0.00/0.53 -> Rules: 0.00/0.53 active(filter(cons(X:S,Y:S),s(N:S),M:S)) -> mark(cons(X:S,filter(Y:S,N:S,M:S))) 0.00/0.53 active(filter(cons(X:S,Y:S),0,M:S)) -> mark(cons(0,filter(Y:S,M:S,M:S))) 0.00/0.53 active(nats(N:S)) -> mark(cons(N:S,nats(s(N:S)))) 0.00/0.53 active(sieve(cons(s(N:S),Y:S))) -> mark(cons(s(N:S),sieve(filter(Y:S,N:S,N:S)))) 0.00/0.53 active(sieve(cons(0,Y:S))) -> mark(cons(0,sieve(Y:S))) 0.00/0.53 active(zprimes) -> mark(sieve(nats(s(s(0))))) 0.00/0.53 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.53 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.53 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 0.00/0.53 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 0.00/0.53 filter(active(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 filter(mark(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 filter(X1:S,active(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 filter(X1:S,mark(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 filter(X1:S,X2:S,active(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 filter(X1:S,X2:S,mark(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) 0.00/0.53 mark(filter(X1:S,X2:S,X3:S)) -> active(filter(mark(X1:S),mark(X2:S),mark(X3:S))) 0.00/0.53 mark(nats(X:S)) -> active(nats(mark(X:S))) 0.00/0.53 mark(s(X:S)) -> active(s(mark(X:S))) 0.00/0.53 mark(sieve(X:S)) -> active(sieve(mark(X:S))) 0.00/0.53 mark(0) -> active(0) 0.00/0.53 mark(zprimes) -> active(zprimes) 0.00/0.53 nats(active(X:S)) -> nats(X:S) 0.00/0.53 nats(mark(X:S)) -> nats(X:S) 0.00/0.53 s(active(X:S)) -> s(X:S) 0.00/0.53 s(mark(X:S)) -> s(X:S) 0.00/0.53 sieve(active(X:S)) -> sieve(X:S) 0.00/0.53 sieve(mark(X:S)) -> sieve(X:S) 0.00/0.53 -> Usable rules: 0.00/0.53 active(filter(cons(X:S,Y:S),s(N:S),M:S)) -> mark(cons(X:S,filter(Y:S,N:S,M:S))) 0.00/0.53 active(filter(cons(X:S,Y:S),0,M:S)) -> mark(cons(0,filter(Y:S,M:S,M:S))) 0.00/0.53 active(nats(N:S)) -> mark(cons(N:S,nats(s(N:S)))) 0.00/0.53 active(sieve(cons(s(N:S),Y:S))) -> mark(cons(s(N:S),sieve(filter(Y:S,N:S,N:S)))) 0.00/0.53 active(sieve(cons(0,Y:S))) -> mark(cons(0,sieve(Y:S))) 0.00/0.53 active(zprimes) -> mark(sieve(nats(s(s(0))))) 0.00/0.53 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.53 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.53 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 0.00/0.53 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 0.00/0.53 filter(active(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 filter(mark(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 filter(X1:S,active(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 filter(X1:S,mark(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 filter(X1:S,X2:S,active(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 filter(X1:S,X2:S,mark(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) 0.00/0.53 mark(filter(X1:S,X2:S,X3:S)) -> active(filter(mark(X1:S),mark(X2:S),mark(X3:S))) 0.00/0.53 mark(nats(X:S)) -> active(nats(mark(X:S))) 0.00/0.53 mark(s(X:S)) -> active(s(mark(X:S))) 0.00/0.53 mark(sieve(X:S)) -> active(sieve(mark(X:S))) 0.00/0.53 mark(0) -> active(0) 0.00/0.53 mark(zprimes) -> active(zprimes) 0.00/0.53 nats(active(X:S)) -> nats(X:S) 0.00/0.53 nats(mark(X:S)) -> nats(X:S) 0.00/0.53 s(active(X:S)) -> s(X:S) 0.00/0.53 s(mark(X:S)) -> s(X:S) 0.00/0.53 sieve(active(X:S)) -> sieve(X:S) 0.00/0.53 sieve(mark(X:S)) -> sieve(X:S) 0.00/0.53 ->Interpretation type: 0.00/0.53 Linear 0.00/0.53 ->Coefficients: 0.00/0.53 Natural Numbers 0.00/0.53 ->Dimension: 0.00/0.53 1 0.00/0.53 ->Bound: 0.00/0.53 2 0.00/0.53 ->Interpretation: 0.00/0.53 0.00/0.53 [active](X) = X 0.00/0.53 [cons](X1,X2) = X1 0.00/0.53 [filter](X1,X2,X3) = 2.X1 + 2.X2 + X3 + 1 0.00/0.53 [mark](X) = 2.X 0.00/0.53 [nats](X) = 2.X 0.00/0.53 [s](X) = 2.X 0.00/0.53 [sieve](X) = 2.X 0.00/0.53 [0] = 0 0.00/0.53 [fSNonEmpty] = 0 0.00/0.53 [zprimes] = 2 0.00/0.53 [ACTIVE](X) = X 0.00/0.53 [CONS](X1,X2) = 0 0.00/0.53 [FILTER](X1,X2,X3) = 0 0.00/0.53 [MARK](X) = 2.X 0.00/0.53 [NATS](X) = 0 0.00/0.53 [S](X) = 0 0.00/0.53 [SIEVE](X) = 0 0.00/0.53 0.00/0.53 Problem 1.6: 0.00/0.53 0.00/0.53 SCC Processor: 0.00/0.53 -> Pairs: 0.00/0.53 MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) 0.00/0.53 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.53 MARK(filter(X1:S,X2:S,X3:S)) -> ACTIVE(filter(mark(X1:S),mark(X2:S),mark(X3:S))) 0.00/0.53 MARK(filter(X1:S,X2:S,X3:S)) -> MARK(X1:S) 0.00/0.53 MARK(filter(X1:S,X2:S,X3:S)) -> MARK(X2:S) 0.00/0.53 MARK(filter(X1:S,X2:S,X3:S)) -> MARK(X3:S) 0.00/0.53 MARK(nats(X:S)) -> ACTIVE(nats(mark(X:S))) 0.00/0.53 MARK(nats(X:S)) -> MARK(X:S) 0.00/0.53 MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) 0.00/0.53 MARK(s(X:S)) -> MARK(X:S) 0.00/0.53 MARK(sieve(X:S)) -> ACTIVE(sieve(mark(X:S))) 0.00/0.53 MARK(sieve(X:S)) -> MARK(X:S) 0.00/0.53 MARK(zprimes) -> ACTIVE(zprimes) 0.00/0.53 -> Rules: 0.00/0.53 active(filter(cons(X:S,Y:S),s(N:S),M:S)) -> mark(cons(X:S,filter(Y:S,N:S,M:S))) 0.00/0.53 active(filter(cons(X:S,Y:S),0,M:S)) -> mark(cons(0,filter(Y:S,M:S,M:S))) 0.00/0.53 active(nats(N:S)) -> mark(cons(N:S,nats(s(N:S)))) 0.00/0.53 active(sieve(cons(s(N:S),Y:S))) -> mark(cons(s(N:S),sieve(filter(Y:S,N:S,N:S)))) 0.00/0.53 active(sieve(cons(0,Y:S))) -> mark(cons(0,sieve(Y:S))) 0.00/0.53 active(zprimes) -> mark(sieve(nats(s(s(0))))) 0.00/0.53 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.53 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.53 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 0.00/0.53 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 0.00/0.53 filter(active(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 filter(mark(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 filter(X1:S,active(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 filter(X1:S,mark(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 filter(X1:S,X2:S,active(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 filter(X1:S,X2:S,mark(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) 0.00/0.53 mark(filter(X1:S,X2:S,X3:S)) -> active(filter(mark(X1:S),mark(X2:S),mark(X3:S))) 0.00/0.53 mark(nats(X:S)) -> active(nats(mark(X:S))) 0.00/0.53 mark(s(X:S)) -> active(s(mark(X:S))) 0.00/0.53 mark(sieve(X:S)) -> active(sieve(mark(X:S))) 0.00/0.53 mark(0) -> active(0) 0.00/0.53 mark(zprimes) -> active(zprimes) 0.00/0.53 nats(active(X:S)) -> nats(X:S) 0.00/0.53 nats(mark(X:S)) -> nats(X:S) 0.00/0.53 s(active(X:S)) -> s(X:S) 0.00/0.53 s(mark(X:S)) -> s(X:S) 0.00/0.53 sieve(active(X:S)) -> sieve(X:S) 0.00/0.53 sieve(mark(X:S)) -> sieve(X:S) 0.00/0.53 ->Strongly Connected Components: 0.00/0.53 ->->Cycle: 0.00/0.53 ->->-> Pairs: 0.00/0.53 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.53 MARK(filter(X1:S,X2:S,X3:S)) -> MARK(X1:S) 0.00/0.53 MARK(filter(X1:S,X2:S,X3:S)) -> MARK(X2:S) 0.00/0.53 MARK(filter(X1:S,X2:S,X3:S)) -> MARK(X3:S) 0.00/0.53 MARK(nats(X:S)) -> MARK(X:S) 0.00/0.53 MARK(s(X:S)) -> MARK(X:S) 0.00/0.53 MARK(sieve(X:S)) -> MARK(X:S) 0.00/0.53 ->->-> Rules: 0.00/0.53 active(filter(cons(X:S,Y:S),s(N:S),M:S)) -> mark(cons(X:S,filter(Y:S,N:S,M:S))) 0.00/0.53 active(filter(cons(X:S,Y:S),0,M:S)) -> mark(cons(0,filter(Y:S,M:S,M:S))) 0.00/0.53 active(nats(N:S)) -> mark(cons(N:S,nats(s(N:S)))) 0.00/0.53 active(sieve(cons(s(N:S),Y:S))) -> mark(cons(s(N:S),sieve(filter(Y:S,N:S,N:S)))) 0.00/0.53 active(sieve(cons(0,Y:S))) -> mark(cons(0,sieve(Y:S))) 0.00/0.53 active(zprimes) -> mark(sieve(nats(s(s(0))))) 0.00/0.53 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.53 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.53 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 0.00/0.53 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 0.00/0.53 filter(active(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 filter(mark(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 filter(X1:S,active(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 filter(X1:S,mark(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 filter(X1:S,X2:S,active(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 filter(X1:S,X2:S,mark(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) 0.00/0.53 mark(filter(X1:S,X2:S,X3:S)) -> active(filter(mark(X1:S),mark(X2:S),mark(X3:S))) 0.00/0.53 mark(nats(X:S)) -> active(nats(mark(X:S))) 0.00/0.53 mark(s(X:S)) -> active(s(mark(X:S))) 0.00/0.53 mark(sieve(X:S)) -> active(sieve(mark(X:S))) 0.00/0.53 mark(0) -> active(0) 0.00/0.53 mark(zprimes) -> active(zprimes) 0.00/0.53 nats(active(X:S)) -> nats(X:S) 0.00/0.53 nats(mark(X:S)) -> nats(X:S) 0.00/0.53 s(active(X:S)) -> s(X:S) 0.00/0.53 s(mark(X:S)) -> s(X:S) 0.00/0.53 sieve(active(X:S)) -> sieve(X:S) 0.00/0.53 sieve(mark(X:S)) -> sieve(X:S) 0.00/0.53 0.00/0.53 Problem 1.6: 0.00/0.53 0.00/0.53 Subterm Processor: 0.00/0.53 -> Pairs: 0.00/0.53 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.53 MARK(filter(X1:S,X2:S,X3:S)) -> MARK(X1:S) 0.00/0.53 MARK(filter(X1:S,X2:S,X3:S)) -> MARK(X2:S) 0.00/0.53 MARK(filter(X1:S,X2:S,X3:S)) -> MARK(X3:S) 0.00/0.53 MARK(nats(X:S)) -> MARK(X:S) 0.00/0.53 MARK(s(X:S)) -> MARK(X:S) 0.00/0.53 MARK(sieve(X:S)) -> MARK(X:S) 0.00/0.53 -> Rules: 0.00/0.53 active(filter(cons(X:S,Y:S),s(N:S),M:S)) -> mark(cons(X:S,filter(Y:S,N:S,M:S))) 0.00/0.53 active(filter(cons(X:S,Y:S),0,M:S)) -> mark(cons(0,filter(Y:S,M:S,M:S))) 0.00/0.53 active(nats(N:S)) -> mark(cons(N:S,nats(s(N:S)))) 0.00/0.53 active(sieve(cons(s(N:S),Y:S))) -> mark(cons(s(N:S),sieve(filter(Y:S,N:S,N:S)))) 0.00/0.53 active(sieve(cons(0,Y:S))) -> mark(cons(0,sieve(Y:S))) 0.00/0.53 active(zprimes) -> mark(sieve(nats(s(s(0))))) 0.00/0.53 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.53 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.53 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 0.00/0.53 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 0.00/0.53 filter(active(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 filter(mark(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 filter(X1:S,active(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 filter(X1:S,mark(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 filter(X1:S,X2:S,active(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 filter(X1:S,X2:S,mark(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) 0.00/0.53 mark(filter(X1:S,X2:S,X3:S)) -> active(filter(mark(X1:S),mark(X2:S),mark(X3:S))) 0.00/0.53 mark(nats(X:S)) -> active(nats(mark(X:S))) 0.00/0.53 mark(s(X:S)) -> active(s(mark(X:S))) 0.00/0.53 mark(sieve(X:S)) -> active(sieve(mark(X:S))) 0.00/0.53 mark(0) -> active(0) 0.00/0.53 mark(zprimes) -> active(zprimes) 0.00/0.53 nats(active(X:S)) -> nats(X:S) 0.00/0.53 nats(mark(X:S)) -> nats(X:S) 0.00/0.53 s(active(X:S)) -> s(X:S) 0.00/0.53 s(mark(X:S)) -> s(X:S) 0.00/0.53 sieve(active(X:S)) -> sieve(X:S) 0.00/0.53 sieve(mark(X:S)) -> sieve(X:S) 0.00/0.53 ->Projection: 0.00/0.53 pi(MARK) = 1 0.00/0.53 0.00/0.53 Problem 1.6: 0.00/0.53 0.00/0.53 SCC Processor: 0.00/0.53 -> Pairs: 0.00/0.53 Empty 0.00/0.53 -> Rules: 0.00/0.53 active(filter(cons(X:S,Y:S),s(N:S),M:S)) -> mark(cons(X:S,filter(Y:S,N:S,M:S))) 0.00/0.53 active(filter(cons(X:S,Y:S),0,M:S)) -> mark(cons(0,filter(Y:S,M:S,M:S))) 0.00/0.53 active(nats(N:S)) -> mark(cons(N:S,nats(s(N:S)))) 0.00/0.53 active(sieve(cons(s(N:S),Y:S))) -> mark(cons(s(N:S),sieve(filter(Y:S,N:S,N:S)))) 0.00/0.53 active(sieve(cons(0,Y:S))) -> mark(cons(0,sieve(Y:S))) 0.00/0.53 active(zprimes) -> mark(sieve(nats(s(s(0))))) 0.00/0.53 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.53 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.53 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 0.00/0.53 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 0.00/0.53 filter(active(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 filter(mark(X1:S),X2:S,X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 filter(X1:S,active(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 filter(X1:S,mark(X2:S),X3:S) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 filter(X1:S,X2:S,active(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 filter(X1:S,X2:S,mark(X3:S)) -> filter(X1:S,X2:S,X3:S) 0.00/0.53 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) 0.00/0.53 mark(filter(X1:S,X2:S,X3:S)) -> active(filter(mark(X1:S),mark(X2:S),mark(X3:S))) 0.00/0.53 mark(nats(X:S)) -> active(nats(mark(X:S))) 0.00/0.53 mark(s(X:S)) -> active(s(mark(X:S))) 0.00/0.53 mark(sieve(X:S)) -> active(sieve(mark(X:S))) 0.00/0.53 mark(0) -> active(0) 0.00/0.53 mark(zprimes) -> active(zprimes) 0.00/0.53 nats(active(X:S)) -> nats(X:S) 0.00/0.53 nats(mark(X:S)) -> nats(X:S) 0.00/0.53 s(active(X:S)) -> s(X:S) 0.00/0.53 s(mark(X:S)) -> s(X:S) 0.00/0.53 sieve(active(X:S)) -> sieve(X:S) 0.00/0.53 sieve(mark(X:S)) -> sieve(X:S) 0.00/0.53 ->Strongly Connected Components: 0.00/0.53 There is no strongly connected component 0.00/0.53 0.00/0.53 The problem is finite. 0.00/0.53 EOF