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