/export/starexec/sandbox2/solver/bin/starexec_run_default /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- YES Problem 1: (VAR M N X X1 X2 X3 Y) (RULES active(filter(cons(X,Y),s(N),M)) -> mark(cons(X,filter(Y,N,M))) active(filter(cons(X,Y),0,M)) -> mark(cons(0,filter(Y,M,M))) active(nats(N)) -> mark(cons(N,nats(s(N)))) active(sieve(cons(s(N),Y))) -> mark(cons(s(N),sieve(filter(Y,N,N)))) active(sieve(cons(0,Y))) -> mark(cons(0,sieve(Y))) active(zprimes) -> mark(sieve(nats(s(s(0))))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) filter(active(X1),X2,X3) -> filter(X1,X2,X3) filter(mark(X1),X2,X3) -> filter(X1,X2,X3) filter(X1,active(X2),X3) -> filter(X1,X2,X3) filter(X1,mark(X2),X3) -> filter(X1,X2,X3) filter(X1,X2,active(X3)) -> filter(X1,X2,X3) filter(X1,X2,mark(X3)) -> filter(X1,X2,X3) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(filter(X1,X2,X3)) -> active(filter(mark(X1),mark(X2),mark(X3))) mark(nats(X)) -> active(nats(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sieve(X)) -> active(sieve(mark(X))) mark(0) -> active(0) mark(zprimes) -> active(zprimes) nats(active(X)) -> nats(X) nats(mark(X)) -> nats(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) sieve(active(X)) -> sieve(X) sieve(mark(X)) -> sieve(X) ) Problem 1: Dependency Pairs Processor: -> Pairs: ACTIVE(filter(cons(X,Y),s(N),M)) -> CONS(X,filter(Y,N,M)) ACTIVE(filter(cons(X,Y),s(N),M)) -> FILTER(Y,N,M) ACTIVE(filter(cons(X,Y),s(N),M)) -> MARK(cons(X,filter(Y,N,M))) ACTIVE(filter(cons(X,Y),0,M)) -> CONS(0,filter(Y,M,M)) ACTIVE(filter(cons(X,Y),0,M)) -> FILTER(Y,M,M) ACTIVE(filter(cons(X,Y),0,M)) -> MARK(cons(0,filter(Y,M,M))) ACTIVE(nats(N)) -> CONS(N,nats(s(N))) ACTIVE(nats(N)) -> MARK(cons(N,nats(s(N)))) ACTIVE(nats(N)) -> NATS(s(N)) ACTIVE(nats(N)) -> S(N) ACTIVE(sieve(cons(s(N),Y))) -> CONS(s(N),sieve(filter(Y,N,N))) ACTIVE(sieve(cons(s(N),Y))) -> FILTER(Y,N,N) ACTIVE(sieve(cons(s(N),Y))) -> MARK(cons(s(N),sieve(filter(Y,N,N)))) ACTIVE(sieve(cons(s(N),Y))) -> SIEVE(filter(Y,N,N)) ACTIVE(sieve(cons(0,Y))) -> CONS(0,sieve(Y)) ACTIVE(sieve(cons(0,Y))) -> MARK(cons(0,sieve(Y))) ACTIVE(sieve(cons(0,Y))) -> SIEVE(Y) ACTIVE(zprimes) -> MARK(sieve(nats(s(s(0))))) CONS(active(X1),X2) -> CONS(X1,X2) CONS(mark(X1),X2) -> CONS(X1,X2) CONS(X1,active(X2)) -> CONS(X1,X2) CONS(X1,mark(X2)) -> CONS(X1,X2) FILTER(active(X1),X2,X3) -> FILTER(X1,X2,X3) FILTER(mark(X1),X2,X3) -> FILTER(X1,X2,X3) FILTER(X1,active(X2),X3) -> FILTER(X1,X2,X3) FILTER(X1,mark(X2),X3) -> FILTER(X1,X2,X3) FILTER(X1,X2,active(X3)) -> FILTER(X1,X2,X3) FILTER(X1,X2,mark(X3)) -> FILTER(X1,X2,X3) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> CONS(mark(X1),X2) MARK(cons(X1,X2)) -> MARK(X1) MARK(filter(X1,X2,X3)) -> ACTIVE(filter(mark(X1),mark(X2),mark(X3))) MARK(filter(X1,X2,X3)) -> FILTER(mark(X1),mark(X2),mark(X3)) MARK(filter(X1,X2,X3)) -> MARK(X1) MARK(filter(X1,X2,X3)) -> MARK(X2) MARK(filter(X1,X2,X3)) -> MARK(X3) MARK(nats(X)) -> ACTIVE(nats(mark(X))) MARK(nats(X)) -> MARK(X) MARK(nats(X)) -> NATS(mark(X)) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(s(X)) -> S(mark(X)) MARK(sieve(X)) -> ACTIVE(sieve(mark(X))) MARK(sieve(X)) -> MARK(X) MARK(sieve(X)) -> SIEVE(mark(X)) MARK(zprimes) -> ACTIVE(zprimes) NATS(active(X)) -> NATS(X) NATS(mark(X)) -> NATS(X) S(active(X)) -> S(X) S(mark(X)) -> S(X) SIEVE(active(X)) -> SIEVE(X) SIEVE(mark(X)) -> SIEVE(X) -> Rules: active(filter(cons(X,Y),s(N),M)) -> mark(cons(X,filter(Y,N,M))) active(filter(cons(X,Y),0,M)) -> mark(cons(0,filter(Y,M,M))) active(nats(N)) -> mark(cons(N,nats(s(N)))) active(sieve(cons(s(N),Y))) -> mark(cons(s(N),sieve(filter(Y,N,N)))) active(sieve(cons(0,Y))) -> mark(cons(0,sieve(Y))) active(zprimes) -> mark(sieve(nats(s(s(0))))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) filter(active(X1),X2,X3) -> filter(X1,X2,X3) filter(mark(X1),X2,X3) -> filter(X1,X2,X3) filter(X1,active(X2),X3) -> filter(X1,X2,X3) filter(X1,mark(X2),X3) -> filter(X1,X2,X3) filter(X1,X2,active(X3)) -> filter(X1,X2,X3) filter(X1,X2,mark(X3)) -> filter(X1,X2,X3) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(filter(X1,X2,X3)) -> active(filter(mark(X1),mark(X2),mark(X3))) mark(nats(X)) -> active(nats(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sieve(X)) -> active(sieve(mark(X))) mark(0) -> active(0) mark(zprimes) -> active(zprimes) nats(active(X)) -> nats(X) nats(mark(X)) -> nats(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) sieve(active(X)) -> sieve(X) sieve(mark(X)) -> sieve(X) Problem 1: SCC Processor: -> Pairs: ACTIVE(filter(cons(X,Y),s(N),M)) -> CONS(X,filter(Y,N,M)) ACTIVE(filter(cons(X,Y),s(N),M)) -> FILTER(Y,N,M) ACTIVE(filter(cons(X,Y),s(N),M)) -> MARK(cons(X,filter(Y,N,M))) ACTIVE(filter(cons(X,Y),0,M)) -> CONS(0,filter(Y,M,M)) ACTIVE(filter(cons(X,Y),0,M)) -> FILTER(Y,M,M) ACTIVE(filter(cons(X,Y),0,M)) -> MARK(cons(0,filter(Y,M,M))) ACTIVE(nats(N)) -> CONS(N,nats(s(N))) ACTIVE(nats(N)) -> MARK(cons(N,nats(s(N)))) ACTIVE(nats(N)) -> NATS(s(N)) ACTIVE(nats(N)) -> S(N) ACTIVE(sieve(cons(s(N),Y))) -> CONS(s(N),sieve(filter(Y,N,N))) ACTIVE(sieve(cons(s(N),Y))) -> FILTER(Y,N,N) ACTIVE(sieve(cons(s(N),Y))) -> MARK(cons(s(N),sieve(filter(Y,N,N)))) ACTIVE(sieve(cons(s(N),Y))) -> SIEVE(filter(Y,N,N)) ACTIVE(sieve(cons(0,Y))) -> CONS(0,sieve(Y)) ACTIVE(sieve(cons(0,Y))) -> MARK(cons(0,sieve(Y))) ACTIVE(sieve(cons(0,Y))) -> SIEVE(Y) ACTIVE(zprimes) -> MARK(sieve(nats(s(s(0))))) CONS(active(X1),X2) -> CONS(X1,X2) CONS(mark(X1),X2) -> CONS(X1,X2) CONS(X1,active(X2)) -> CONS(X1,X2) CONS(X1,mark(X2)) -> CONS(X1,X2) FILTER(active(X1),X2,X3) -> FILTER(X1,X2,X3) FILTER(mark(X1),X2,X3) -> FILTER(X1,X2,X3) FILTER(X1,active(X2),X3) -> FILTER(X1,X2,X3) FILTER(X1,mark(X2),X3) -> FILTER(X1,X2,X3) FILTER(X1,X2,active(X3)) -> FILTER(X1,X2,X3) FILTER(X1,X2,mark(X3)) -> FILTER(X1,X2,X3) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> CONS(mark(X1),X2) MARK(cons(X1,X2)) -> MARK(X1) MARK(filter(X1,X2,X3)) -> ACTIVE(filter(mark(X1),mark(X2),mark(X3))) MARK(filter(X1,X2,X3)) -> FILTER(mark(X1),mark(X2),mark(X3)) MARK(filter(X1,X2,X3)) -> MARK(X1) MARK(filter(X1,X2,X3)) -> MARK(X2) MARK(filter(X1,X2,X3)) -> MARK(X3) MARK(nats(X)) -> ACTIVE(nats(mark(X))) MARK(nats(X)) -> MARK(X) MARK(nats(X)) -> NATS(mark(X)) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(s(X)) -> S(mark(X)) MARK(sieve(X)) -> ACTIVE(sieve(mark(X))) MARK(sieve(X)) -> MARK(X) MARK(sieve(X)) -> SIEVE(mark(X)) MARK(zprimes) -> ACTIVE(zprimes) NATS(active(X)) -> NATS(X) NATS(mark(X)) -> NATS(X) S(active(X)) -> S(X) S(mark(X)) -> S(X) SIEVE(active(X)) -> SIEVE(X) SIEVE(mark(X)) -> SIEVE(X) -> Rules: active(filter(cons(X,Y),s(N),M)) -> mark(cons(X,filter(Y,N,M))) active(filter(cons(X,Y),0,M)) -> mark(cons(0,filter(Y,M,M))) active(nats(N)) -> mark(cons(N,nats(s(N)))) active(sieve(cons(s(N),Y))) -> mark(cons(s(N),sieve(filter(Y,N,N)))) active(sieve(cons(0,Y))) -> mark(cons(0,sieve(Y))) active(zprimes) -> mark(sieve(nats(s(s(0))))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) filter(active(X1),X2,X3) -> filter(X1,X2,X3) filter(mark(X1),X2,X3) -> filter(X1,X2,X3) filter(X1,active(X2),X3) -> filter(X1,X2,X3) filter(X1,mark(X2),X3) -> filter(X1,X2,X3) filter(X1,X2,active(X3)) -> filter(X1,X2,X3) filter(X1,X2,mark(X3)) -> filter(X1,X2,X3) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(filter(X1,X2,X3)) -> active(filter(mark(X1),mark(X2),mark(X3))) mark(nats(X)) -> active(nats(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sieve(X)) -> active(sieve(mark(X))) mark(0) -> active(0) mark(zprimes) -> active(zprimes) nats(active(X)) -> nats(X) nats(mark(X)) -> nats(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) sieve(active(X)) -> sieve(X) sieve(mark(X)) -> sieve(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: SIEVE(active(X)) -> SIEVE(X) SIEVE(mark(X)) -> SIEVE(X) ->->-> Rules: active(filter(cons(X,Y),s(N),M)) -> mark(cons(X,filter(Y,N,M))) active(filter(cons(X,Y),0,M)) -> mark(cons(0,filter(Y,M,M))) active(nats(N)) -> mark(cons(N,nats(s(N)))) active(sieve(cons(s(N),Y))) -> mark(cons(s(N),sieve(filter(Y,N,N)))) active(sieve(cons(0,Y))) -> mark(cons(0,sieve(Y))) active(zprimes) -> mark(sieve(nats(s(s(0))))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) filter(active(X1),X2,X3) -> filter(X1,X2,X3) filter(mark(X1),X2,X3) -> filter(X1,X2,X3) filter(X1,active(X2),X3) -> filter(X1,X2,X3) filter(X1,mark(X2),X3) -> filter(X1,X2,X3) filter(X1,X2,active(X3)) -> filter(X1,X2,X3) filter(X1,X2,mark(X3)) -> filter(X1,X2,X3) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(filter(X1,X2,X3)) -> active(filter(mark(X1),mark(X2),mark(X3))) mark(nats(X)) -> active(nats(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sieve(X)) -> active(sieve(mark(X))) mark(0) -> active(0) mark(zprimes) -> active(zprimes) nats(active(X)) -> nats(X) nats(mark(X)) -> nats(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) sieve(active(X)) -> sieve(X) sieve(mark(X)) -> sieve(X) ->->Cycle: ->->-> Pairs: S(active(X)) -> S(X) S(mark(X)) -> S(X) ->->-> Rules: active(filter(cons(X,Y),s(N),M)) -> mark(cons(X,filter(Y,N,M))) active(filter(cons(X,Y),0,M)) -> mark(cons(0,filter(Y,M,M))) active(nats(N)) -> mark(cons(N,nats(s(N)))) active(sieve(cons(s(N),Y))) -> mark(cons(s(N),sieve(filter(Y,N,N)))) active(sieve(cons(0,Y))) -> mark(cons(0,sieve(Y))) active(zprimes) -> mark(sieve(nats(s(s(0))))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) filter(active(X1),X2,X3) -> filter(X1,X2,X3) filter(mark(X1),X2,X3) -> filter(X1,X2,X3) filter(X1,active(X2),X3) -> filter(X1,X2,X3) filter(X1,mark(X2),X3) -> filter(X1,X2,X3) filter(X1,X2,active(X3)) -> filter(X1,X2,X3) filter(X1,X2,mark(X3)) -> filter(X1,X2,X3) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(filter(X1,X2,X3)) -> active(filter(mark(X1),mark(X2),mark(X3))) mark(nats(X)) -> active(nats(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sieve(X)) -> active(sieve(mark(X))) mark(0) -> active(0) mark(zprimes) -> active(zprimes) nats(active(X)) -> nats(X) nats(mark(X)) -> nats(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) sieve(active(X)) -> sieve(X) sieve(mark(X)) -> sieve(X) ->->Cycle: ->->-> Pairs: NATS(active(X)) -> NATS(X) NATS(mark(X)) -> NATS(X) ->->-> Rules: active(filter(cons(X,Y),s(N),M)) -> mark(cons(X,filter(Y,N,M))) active(filter(cons(X,Y),0,M)) -> mark(cons(0,filter(Y,M,M))) active(nats(N)) -> mark(cons(N,nats(s(N)))) active(sieve(cons(s(N),Y))) -> mark(cons(s(N),sieve(filter(Y,N,N)))) active(sieve(cons(0,Y))) -> mark(cons(0,sieve(Y))) active(zprimes) -> mark(sieve(nats(s(s(0))))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) filter(active(X1),X2,X3) -> filter(X1,X2,X3) filter(mark(X1),X2,X3) -> filter(X1,X2,X3) filter(X1,active(X2),X3) -> filter(X1,X2,X3) filter(X1,mark(X2),X3) -> filter(X1,X2,X3) filter(X1,X2,active(X3)) -> filter(X1,X2,X3) filter(X1,X2,mark(X3)) -> filter(X1,X2,X3) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(filter(X1,X2,X3)) -> active(filter(mark(X1),mark(X2),mark(X3))) mark(nats(X)) -> active(nats(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sieve(X)) -> active(sieve(mark(X))) mark(0) -> active(0) mark(zprimes) -> active(zprimes) nats(active(X)) -> nats(X) nats(mark(X)) -> nats(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) sieve(active(X)) -> sieve(X) sieve(mark(X)) -> sieve(X) ->->Cycle: ->->-> Pairs: FILTER(active(X1),X2,X3) -> FILTER(X1,X2,X3) FILTER(mark(X1),X2,X3) -> FILTER(X1,X2,X3) FILTER(X1,active(X2),X3) -> FILTER(X1,X2,X3) FILTER(X1,mark(X2),X3) -> FILTER(X1,X2,X3) FILTER(X1,X2,active(X3)) -> FILTER(X1,X2,X3) FILTER(X1,X2,mark(X3)) -> FILTER(X1,X2,X3) ->->-> Rules: active(filter(cons(X,Y),s(N),M)) -> mark(cons(X,filter(Y,N,M))) active(filter(cons(X,Y),0,M)) -> mark(cons(0,filter(Y,M,M))) active(nats(N)) -> mark(cons(N,nats(s(N)))) active(sieve(cons(s(N),Y))) -> mark(cons(s(N),sieve(filter(Y,N,N)))) active(sieve(cons(0,Y))) -> mark(cons(0,sieve(Y))) active(zprimes) -> mark(sieve(nats(s(s(0))))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) filter(active(X1),X2,X3) -> filter(X1,X2,X3) filter(mark(X1),X2,X3) -> filter(X1,X2,X3) filter(X1,active(X2),X3) -> filter(X1,X2,X3) filter(X1,mark(X2),X3) -> filter(X1,X2,X3) filter(X1,X2,active(X3)) -> filter(X1,X2,X3) filter(X1,X2,mark(X3)) -> filter(X1,X2,X3) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(filter(X1,X2,X3)) -> active(filter(mark(X1),mark(X2),mark(X3))) mark(nats(X)) -> active(nats(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sieve(X)) -> active(sieve(mark(X))) mark(0) -> active(0) mark(zprimes) -> active(zprimes) nats(active(X)) -> nats(X) nats(mark(X)) -> nats(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) sieve(active(X)) -> sieve(X) sieve(mark(X)) -> sieve(X) ->->Cycle: ->->-> Pairs: CONS(active(X1),X2) -> CONS(X1,X2) CONS(mark(X1),X2) -> CONS(X1,X2) CONS(X1,active(X2)) -> CONS(X1,X2) CONS(X1,mark(X2)) -> CONS(X1,X2) ->->-> Rules: active(filter(cons(X,Y),s(N),M)) -> mark(cons(X,filter(Y,N,M))) active(filter(cons(X,Y),0,M)) -> mark(cons(0,filter(Y,M,M))) active(nats(N)) -> mark(cons(N,nats(s(N)))) active(sieve(cons(s(N),Y))) -> mark(cons(s(N),sieve(filter(Y,N,N)))) active(sieve(cons(0,Y))) -> mark(cons(0,sieve(Y))) active(zprimes) -> mark(sieve(nats(s(s(0))))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) filter(active(X1),X2,X3) -> filter(X1,X2,X3) filter(mark(X1),X2,X3) -> filter(X1,X2,X3) filter(X1,active(X2),X3) -> filter(X1,X2,X3) filter(X1,mark(X2),X3) -> filter(X1,X2,X3) filter(X1,X2,active(X3)) -> filter(X1,X2,X3) filter(X1,X2,mark(X3)) -> filter(X1,X2,X3) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(filter(X1,X2,X3)) -> active(filter(mark(X1),mark(X2),mark(X3))) mark(nats(X)) -> active(nats(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sieve(X)) -> active(sieve(mark(X))) mark(0) -> active(0) mark(zprimes) -> active(zprimes) nats(active(X)) -> nats(X) nats(mark(X)) -> nats(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) sieve(active(X)) -> sieve(X) sieve(mark(X)) -> sieve(X) ->->Cycle: ->->-> Pairs: ACTIVE(filter(cons(X,Y),s(N),M)) -> MARK(cons(X,filter(Y,N,M))) ACTIVE(filter(cons(X,Y),0,M)) -> MARK(cons(0,filter(Y,M,M))) ACTIVE(nats(N)) -> MARK(cons(N,nats(s(N)))) ACTIVE(sieve(cons(s(N),Y))) -> MARK(cons(s(N),sieve(filter(Y,N,N)))) ACTIVE(sieve(cons(0,Y))) -> MARK(cons(0,sieve(Y))) ACTIVE(zprimes) -> MARK(sieve(nats(s(s(0))))) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> MARK(X1) MARK(filter(X1,X2,X3)) -> ACTIVE(filter(mark(X1),mark(X2),mark(X3))) MARK(filter(X1,X2,X3)) -> MARK(X1) MARK(filter(X1,X2,X3)) -> MARK(X2) MARK(filter(X1,X2,X3)) -> MARK(X3) MARK(nats(X)) -> ACTIVE(nats(mark(X))) MARK(nats(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(sieve(X)) -> ACTIVE(sieve(mark(X))) MARK(sieve(X)) -> MARK(X) MARK(zprimes) -> ACTIVE(zprimes) ->->-> Rules: active(filter(cons(X,Y),s(N),M)) -> mark(cons(X,filter(Y,N,M))) active(filter(cons(X,Y),0,M)) -> mark(cons(0,filter(Y,M,M))) active(nats(N)) -> mark(cons(N,nats(s(N)))) active(sieve(cons(s(N),Y))) -> mark(cons(s(N),sieve(filter(Y,N,N)))) active(sieve(cons(0,Y))) -> mark(cons(0,sieve(Y))) active(zprimes) -> mark(sieve(nats(s(s(0))))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) filter(active(X1),X2,X3) -> filter(X1,X2,X3) filter(mark(X1),X2,X3) -> filter(X1,X2,X3) filter(X1,active(X2),X3) -> filter(X1,X2,X3) filter(X1,mark(X2),X3) -> filter(X1,X2,X3) filter(X1,X2,active(X3)) -> filter(X1,X2,X3) filter(X1,X2,mark(X3)) -> filter(X1,X2,X3) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(filter(X1,X2,X3)) -> active(filter(mark(X1),mark(X2),mark(X3))) mark(nats(X)) -> active(nats(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sieve(X)) -> active(sieve(mark(X))) mark(0) -> active(0) mark(zprimes) -> active(zprimes) nats(active(X)) -> nats(X) nats(mark(X)) -> nats(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) sieve(active(X)) -> sieve(X) sieve(mark(X)) -> sieve(X) The problem is decomposed in 6 subproblems. Problem 1.1: Subterm Processor: -> Pairs: SIEVE(active(X)) -> SIEVE(X) SIEVE(mark(X)) -> SIEVE(X) -> Rules: active(filter(cons(X,Y),s(N),M)) -> mark(cons(X,filter(Y,N,M))) active(filter(cons(X,Y),0,M)) -> mark(cons(0,filter(Y,M,M))) active(nats(N)) -> mark(cons(N,nats(s(N)))) active(sieve(cons(s(N),Y))) -> mark(cons(s(N),sieve(filter(Y,N,N)))) active(sieve(cons(0,Y))) -> mark(cons(0,sieve(Y))) active(zprimes) -> mark(sieve(nats(s(s(0))))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) filter(active(X1),X2,X3) -> filter(X1,X2,X3) filter(mark(X1),X2,X3) -> filter(X1,X2,X3) filter(X1,active(X2),X3) -> filter(X1,X2,X3) filter(X1,mark(X2),X3) -> filter(X1,X2,X3) filter(X1,X2,active(X3)) -> filter(X1,X2,X3) filter(X1,X2,mark(X3)) -> filter(X1,X2,X3) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(filter(X1,X2,X3)) -> active(filter(mark(X1),mark(X2),mark(X3))) mark(nats(X)) -> active(nats(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sieve(X)) -> active(sieve(mark(X))) mark(0) -> active(0) mark(zprimes) -> active(zprimes) nats(active(X)) -> nats(X) nats(mark(X)) -> nats(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) sieve(active(X)) -> sieve(X) sieve(mark(X)) -> sieve(X) ->Projection: pi(SIEVE) = 1 Problem 1.1: SCC Processor: -> Pairs: Empty -> Rules: active(filter(cons(X,Y),s(N),M)) -> mark(cons(X,filter(Y,N,M))) active(filter(cons(X,Y),0,M)) -> mark(cons(0,filter(Y,M,M))) active(nats(N)) -> mark(cons(N,nats(s(N)))) active(sieve(cons(s(N),Y))) -> mark(cons(s(N),sieve(filter(Y,N,N)))) active(sieve(cons(0,Y))) -> mark(cons(0,sieve(Y))) active(zprimes) -> mark(sieve(nats(s(s(0))))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) filter(active(X1),X2,X3) -> filter(X1,X2,X3) filter(mark(X1),X2,X3) -> filter(X1,X2,X3) filter(X1,active(X2),X3) -> filter(X1,X2,X3) filter(X1,mark(X2),X3) -> filter(X1,X2,X3) filter(X1,X2,active(X3)) -> filter(X1,X2,X3) filter(X1,X2,mark(X3)) -> filter(X1,X2,X3) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(filter(X1,X2,X3)) -> active(filter(mark(X1),mark(X2),mark(X3))) mark(nats(X)) -> active(nats(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sieve(X)) -> active(sieve(mark(X))) mark(0) -> active(0) mark(zprimes) -> active(zprimes) nats(active(X)) -> nats(X) nats(mark(X)) -> nats(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) sieve(active(X)) -> sieve(X) sieve(mark(X)) -> sieve(X) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.2: Subterm Processor: -> Pairs: S(active(X)) -> S(X) S(mark(X)) -> S(X) -> Rules: active(filter(cons(X,Y),s(N),M)) -> mark(cons(X,filter(Y,N,M))) active(filter(cons(X,Y),0,M)) -> mark(cons(0,filter(Y,M,M))) active(nats(N)) -> mark(cons(N,nats(s(N)))) active(sieve(cons(s(N),Y))) -> mark(cons(s(N),sieve(filter(Y,N,N)))) active(sieve(cons(0,Y))) -> mark(cons(0,sieve(Y))) active(zprimes) -> mark(sieve(nats(s(s(0))))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) filter(active(X1),X2,X3) -> filter(X1,X2,X3) filter(mark(X1),X2,X3) -> filter(X1,X2,X3) filter(X1,active(X2),X3) -> filter(X1,X2,X3) filter(X1,mark(X2),X3) -> filter(X1,X2,X3) filter(X1,X2,active(X3)) -> filter(X1,X2,X3) filter(X1,X2,mark(X3)) -> filter(X1,X2,X3) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(filter(X1,X2,X3)) -> active(filter(mark(X1),mark(X2),mark(X3))) mark(nats(X)) -> active(nats(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sieve(X)) -> active(sieve(mark(X))) mark(0) -> active(0) mark(zprimes) -> active(zprimes) nats(active(X)) -> nats(X) nats(mark(X)) -> nats(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) sieve(active(X)) -> sieve(X) sieve(mark(X)) -> sieve(X) ->Projection: pi(S) = 1 Problem 1.2: SCC Processor: -> Pairs: Empty -> Rules: active(filter(cons(X,Y),s(N),M)) -> mark(cons(X,filter(Y,N,M))) active(filter(cons(X,Y),0,M)) -> mark(cons(0,filter(Y,M,M))) active(nats(N)) -> mark(cons(N,nats(s(N)))) active(sieve(cons(s(N),Y))) -> mark(cons(s(N),sieve(filter(Y,N,N)))) active(sieve(cons(0,Y))) -> mark(cons(0,sieve(Y))) active(zprimes) -> mark(sieve(nats(s(s(0))))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) filter(active(X1),X2,X3) -> filter(X1,X2,X3) filter(mark(X1),X2,X3) -> filter(X1,X2,X3) filter(X1,active(X2),X3) -> filter(X1,X2,X3) filter(X1,mark(X2),X3) -> filter(X1,X2,X3) filter(X1,X2,active(X3)) -> filter(X1,X2,X3) filter(X1,X2,mark(X3)) -> filter(X1,X2,X3) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(filter(X1,X2,X3)) -> active(filter(mark(X1),mark(X2),mark(X3))) mark(nats(X)) -> active(nats(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sieve(X)) -> active(sieve(mark(X))) mark(0) -> active(0) mark(zprimes) -> active(zprimes) nats(active(X)) -> nats(X) nats(mark(X)) -> nats(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) sieve(active(X)) -> sieve(X) sieve(mark(X)) -> sieve(X) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.3: Subterm Processor: -> Pairs: NATS(active(X)) -> NATS(X) NATS(mark(X)) -> NATS(X) -> Rules: active(filter(cons(X,Y),s(N),M)) -> mark(cons(X,filter(Y,N,M))) active(filter(cons(X,Y),0,M)) -> mark(cons(0,filter(Y,M,M))) active(nats(N)) -> mark(cons(N,nats(s(N)))) active(sieve(cons(s(N),Y))) -> mark(cons(s(N),sieve(filter(Y,N,N)))) active(sieve(cons(0,Y))) -> mark(cons(0,sieve(Y))) active(zprimes) -> mark(sieve(nats(s(s(0))))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) filter(active(X1),X2,X3) -> filter(X1,X2,X3) filter(mark(X1),X2,X3) -> filter(X1,X2,X3) filter(X1,active(X2),X3) -> filter(X1,X2,X3) filter(X1,mark(X2),X3) -> filter(X1,X2,X3) filter(X1,X2,active(X3)) -> filter(X1,X2,X3) filter(X1,X2,mark(X3)) -> filter(X1,X2,X3) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(filter(X1,X2,X3)) -> active(filter(mark(X1),mark(X2),mark(X3))) mark(nats(X)) -> active(nats(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sieve(X)) -> active(sieve(mark(X))) mark(0) -> active(0) mark(zprimes) -> active(zprimes) nats(active(X)) -> nats(X) nats(mark(X)) -> nats(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) sieve(active(X)) -> sieve(X) sieve(mark(X)) -> sieve(X) ->Projection: pi(NATS) = 1 Problem 1.3: SCC Processor: -> Pairs: Empty -> Rules: active(filter(cons(X,Y),s(N),M)) -> mark(cons(X,filter(Y,N,M))) active(filter(cons(X,Y),0,M)) -> mark(cons(0,filter(Y,M,M))) active(nats(N)) -> mark(cons(N,nats(s(N)))) active(sieve(cons(s(N),Y))) -> mark(cons(s(N),sieve(filter(Y,N,N)))) active(sieve(cons(0,Y))) -> mark(cons(0,sieve(Y))) active(zprimes) -> mark(sieve(nats(s(s(0))))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) filter(active(X1),X2,X3) -> filter(X1,X2,X3) filter(mark(X1),X2,X3) -> filter(X1,X2,X3) filter(X1,active(X2),X3) -> filter(X1,X2,X3) filter(X1,mark(X2),X3) -> filter(X1,X2,X3) filter(X1,X2,active(X3)) -> filter(X1,X2,X3) filter(X1,X2,mark(X3)) -> filter(X1,X2,X3) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(filter(X1,X2,X3)) -> active(filter(mark(X1),mark(X2),mark(X3))) mark(nats(X)) -> active(nats(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sieve(X)) -> active(sieve(mark(X))) mark(0) -> active(0) mark(zprimes) -> active(zprimes) nats(active(X)) -> nats(X) nats(mark(X)) -> nats(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) sieve(active(X)) -> sieve(X) sieve(mark(X)) -> sieve(X) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.4: Subterm Processor: -> Pairs: FILTER(active(X1),X2,X3) -> FILTER(X1,X2,X3) FILTER(mark(X1),X2,X3) -> FILTER(X1,X2,X3) FILTER(X1,active(X2),X3) -> FILTER(X1,X2,X3) FILTER(X1,mark(X2),X3) -> FILTER(X1,X2,X3) FILTER(X1,X2,active(X3)) -> FILTER(X1,X2,X3) FILTER(X1,X2,mark(X3)) -> FILTER(X1,X2,X3) -> Rules: active(filter(cons(X,Y),s(N),M)) -> mark(cons(X,filter(Y,N,M))) active(filter(cons(X,Y),0,M)) -> mark(cons(0,filter(Y,M,M))) active(nats(N)) -> mark(cons(N,nats(s(N)))) active(sieve(cons(s(N),Y))) -> mark(cons(s(N),sieve(filter(Y,N,N)))) active(sieve(cons(0,Y))) -> mark(cons(0,sieve(Y))) active(zprimes) -> mark(sieve(nats(s(s(0))))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) filter(active(X1),X2,X3) -> filter(X1,X2,X3) filter(mark(X1),X2,X3) -> filter(X1,X2,X3) filter(X1,active(X2),X3) -> filter(X1,X2,X3) filter(X1,mark(X2),X3) -> filter(X1,X2,X3) filter(X1,X2,active(X3)) -> filter(X1,X2,X3) filter(X1,X2,mark(X3)) -> filter(X1,X2,X3) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(filter(X1,X2,X3)) -> active(filter(mark(X1),mark(X2),mark(X3))) mark(nats(X)) -> active(nats(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sieve(X)) -> active(sieve(mark(X))) mark(0) -> active(0) mark(zprimes) -> active(zprimes) nats(active(X)) -> nats(X) nats(mark(X)) -> nats(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) sieve(active(X)) -> sieve(X) sieve(mark(X)) -> sieve(X) ->Projection: pi(FILTER) = 1 Problem 1.4: SCC Processor: -> Pairs: FILTER(X1,active(X2),X3) -> FILTER(X1,X2,X3) FILTER(X1,mark(X2),X3) -> FILTER(X1,X2,X3) FILTER(X1,X2,active(X3)) -> FILTER(X1,X2,X3) FILTER(X1,X2,mark(X3)) -> FILTER(X1,X2,X3) -> Rules: active(filter(cons(X,Y),s(N),M)) -> mark(cons(X,filter(Y,N,M))) active(filter(cons(X,Y),0,M)) -> mark(cons(0,filter(Y,M,M))) active(nats(N)) -> mark(cons(N,nats(s(N)))) active(sieve(cons(s(N),Y))) -> mark(cons(s(N),sieve(filter(Y,N,N)))) active(sieve(cons(0,Y))) -> mark(cons(0,sieve(Y))) active(zprimes) -> mark(sieve(nats(s(s(0))))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) filter(active(X1),X2,X3) -> filter(X1,X2,X3) filter(mark(X1),X2,X3) -> filter(X1,X2,X3) filter(X1,active(X2),X3) -> filter(X1,X2,X3) filter(X1,mark(X2),X3) -> filter(X1,X2,X3) filter(X1,X2,active(X3)) -> filter(X1,X2,X3) filter(X1,X2,mark(X3)) -> filter(X1,X2,X3) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(filter(X1,X2,X3)) -> active(filter(mark(X1),mark(X2),mark(X3))) mark(nats(X)) -> active(nats(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sieve(X)) -> active(sieve(mark(X))) mark(0) -> active(0) mark(zprimes) -> active(zprimes) nats(active(X)) -> nats(X) nats(mark(X)) -> nats(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) sieve(active(X)) -> sieve(X) sieve(mark(X)) -> sieve(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: FILTER(X1,active(X2),X3) -> FILTER(X1,X2,X3) FILTER(X1,mark(X2),X3) -> FILTER(X1,X2,X3) FILTER(X1,X2,active(X3)) -> FILTER(X1,X2,X3) FILTER(X1,X2,mark(X3)) -> FILTER(X1,X2,X3) ->->-> Rules: active(filter(cons(X,Y),s(N),M)) -> mark(cons(X,filter(Y,N,M))) active(filter(cons(X,Y),0,M)) -> mark(cons(0,filter(Y,M,M))) active(nats(N)) -> mark(cons(N,nats(s(N)))) active(sieve(cons(s(N),Y))) -> mark(cons(s(N),sieve(filter(Y,N,N)))) active(sieve(cons(0,Y))) -> mark(cons(0,sieve(Y))) active(zprimes) -> mark(sieve(nats(s(s(0))))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) filter(active(X1),X2,X3) -> filter(X1,X2,X3) filter(mark(X1),X2,X3) -> filter(X1,X2,X3) filter(X1,active(X2),X3) -> filter(X1,X2,X3) filter(X1,mark(X2),X3) -> filter(X1,X2,X3) filter(X1,X2,active(X3)) -> filter(X1,X2,X3) filter(X1,X2,mark(X3)) -> filter(X1,X2,X3) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(filter(X1,X2,X3)) -> active(filter(mark(X1),mark(X2),mark(X3))) mark(nats(X)) -> active(nats(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sieve(X)) -> active(sieve(mark(X))) mark(0) -> active(0) mark(zprimes) -> active(zprimes) nats(active(X)) -> nats(X) nats(mark(X)) -> nats(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) sieve(active(X)) -> sieve(X) sieve(mark(X)) -> sieve(X) Problem 1.4: Subterm Processor: -> Pairs: FILTER(X1,active(X2),X3) -> FILTER(X1,X2,X3) FILTER(X1,mark(X2),X3) -> FILTER(X1,X2,X3) FILTER(X1,X2,active(X3)) -> FILTER(X1,X2,X3) FILTER(X1,X2,mark(X3)) -> FILTER(X1,X2,X3) -> Rules: active(filter(cons(X,Y),s(N),M)) -> mark(cons(X,filter(Y,N,M))) active(filter(cons(X,Y),0,M)) -> mark(cons(0,filter(Y,M,M))) active(nats(N)) -> mark(cons(N,nats(s(N)))) active(sieve(cons(s(N),Y))) -> mark(cons(s(N),sieve(filter(Y,N,N)))) active(sieve(cons(0,Y))) -> mark(cons(0,sieve(Y))) active(zprimes) -> mark(sieve(nats(s(s(0))))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) filter(active(X1),X2,X3) -> filter(X1,X2,X3) filter(mark(X1),X2,X3) -> filter(X1,X2,X3) filter(X1,active(X2),X3) -> filter(X1,X2,X3) filter(X1,mark(X2),X3) -> filter(X1,X2,X3) filter(X1,X2,active(X3)) -> filter(X1,X2,X3) filter(X1,X2,mark(X3)) -> filter(X1,X2,X3) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(filter(X1,X2,X3)) -> active(filter(mark(X1),mark(X2),mark(X3))) mark(nats(X)) -> active(nats(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sieve(X)) -> active(sieve(mark(X))) mark(0) -> active(0) mark(zprimes) -> active(zprimes) nats(active(X)) -> nats(X) nats(mark(X)) -> nats(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) sieve(active(X)) -> sieve(X) sieve(mark(X)) -> sieve(X) ->Projection: pi(FILTER) = 2 Problem 1.4: SCC Processor: -> Pairs: FILTER(X1,X2,active(X3)) -> FILTER(X1,X2,X3) FILTER(X1,X2,mark(X3)) -> FILTER(X1,X2,X3) -> Rules: active(filter(cons(X,Y),s(N),M)) -> mark(cons(X,filter(Y,N,M))) active(filter(cons(X,Y),0,M)) -> mark(cons(0,filter(Y,M,M))) active(nats(N)) -> mark(cons(N,nats(s(N)))) active(sieve(cons(s(N),Y))) -> mark(cons(s(N),sieve(filter(Y,N,N)))) active(sieve(cons(0,Y))) -> mark(cons(0,sieve(Y))) active(zprimes) -> mark(sieve(nats(s(s(0))))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) filter(active(X1),X2,X3) -> filter(X1,X2,X3) filter(mark(X1),X2,X3) -> filter(X1,X2,X3) filter(X1,active(X2),X3) -> filter(X1,X2,X3) filter(X1,mark(X2),X3) -> filter(X1,X2,X3) filter(X1,X2,active(X3)) -> filter(X1,X2,X3) filter(X1,X2,mark(X3)) -> filter(X1,X2,X3) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(filter(X1,X2,X3)) -> active(filter(mark(X1),mark(X2),mark(X3))) mark(nats(X)) -> active(nats(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sieve(X)) -> active(sieve(mark(X))) mark(0) -> active(0) mark(zprimes) -> active(zprimes) nats(active(X)) -> nats(X) nats(mark(X)) -> nats(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) sieve(active(X)) -> sieve(X) sieve(mark(X)) -> sieve(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: FILTER(X1,X2,active(X3)) -> FILTER(X1,X2,X3) FILTER(X1,X2,mark(X3)) -> FILTER(X1,X2,X3) ->->-> Rules: active(filter(cons(X,Y),s(N),M)) -> mark(cons(X,filter(Y,N,M))) active(filter(cons(X,Y),0,M)) -> mark(cons(0,filter(Y,M,M))) active(nats(N)) -> mark(cons(N,nats(s(N)))) active(sieve(cons(s(N),Y))) -> mark(cons(s(N),sieve(filter(Y,N,N)))) active(sieve(cons(0,Y))) -> mark(cons(0,sieve(Y))) active(zprimes) -> mark(sieve(nats(s(s(0))))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) filter(active(X1),X2,X3) -> filter(X1,X2,X3) filter(mark(X1),X2,X3) -> filter(X1,X2,X3) filter(X1,active(X2),X3) -> filter(X1,X2,X3) filter(X1,mark(X2),X3) -> filter(X1,X2,X3) filter(X1,X2,active(X3)) -> filter(X1,X2,X3) filter(X1,X2,mark(X3)) -> filter(X1,X2,X3) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(filter(X1,X2,X3)) -> active(filter(mark(X1),mark(X2),mark(X3))) mark(nats(X)) -> active(nats(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sieve(X)) -> active(sieve(mark(X))) mark(0) -> active(0) mark(zprimes) -> active(zprimes) nats(active(X)) -> nats(X) nats(mark(X)) -> nats(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) sieve(active(X)) -> sieve(X) sieve(mark(X)) -> sieve(X) Problem 1.4: Subterm Processor: -> Pairs: FILTER(X1,X2,active(X3)) -> FILTER(X1,X2,X3) FILTER(X1,X2,mark(X3)) -> FILTER(X1,X2,X3) -> Rules: active(filter(cons(X,Y),s(N),M)) -> mark(cons(X,filter(Y,N,M))) active(filter(cons(X,Y),0,M)) -> mark(cons(0,filter(Y,M,M))) active(nats(N)) -> mark(cons(N,nats(s(N)))) active(sieve(cons(s(N),Y))) -> mark(cons(s(N),sieve(filter(Y,N,N)))) active(sieve(cons(0,Y))) -> mark(cons(0,sieve(Y))) active(zprimes) -> mark(sieve(nats(s(s(0))))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) filter(active(X1),X2,X3) -> filter(X1,X2,X3) filter(mark(X1),X2,X3) -> filter(X1,X2,X3) filter(X1,active(X2),X3) -> filter(X1,X2,X3) filter(X1,mark(X2),X3) -> filter(X1,X2,X3) filter(X1,X2,active(X3)) -> filter(X1,X2,X3) filter(X1,X2,mark(X3)) -> filter(X1,X2,X3) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(filter(X1,X2,X3)) -> active(filter(mark(X1),mark(X2),mark(X3))) mark(nats(X)) -> active(nats(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sieve(X)) -> active(sieve(mark(X))) mark(0) -> active(0) mark(zprimes) -> active(zprimes) nats(active(X)) -> nats(X) nats(mark(X)) -> nats(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) sieve(active(X)) -> sieve(X) sieve(mark(X)) -> sieve(X) ->Projection: pi(FILTER) = 3 Problem 1.4: SCC Processor: -> Pairs: Empty -> Rules: active(filter(cons(X,Y),s(N),M)) -> mark(cons(X,filter(Y,N,M))) active(filter(cons(X,Y),0,M)) -> mark(cons(0,filter(Y,M,M))) active(nats(N)) -> mark(cons(N,nats(s(N)))) active(sieve(cons(s(N),Y))) -> mark(cons(s(N),sieve(filter(Y,N,N)))) active(sieve(cons(0,Y))) -> mark(cons(0,sieve(Y))) active(zprimes) -> mark(sieve(nats(s(s(0))))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) filter(active(X1),X2,X3) -> filter(X1,X2,X3) filter(mark(X1),X2,X3) -> filter(X1,X2,X3) filter(X1,active(X2),X3) -> filter(X1,X2,X3) filter(X1,mark(X2),X3) -> filter(X1,X2,X3) filter(X1,X2,active(X3)) -> filter(X1,X2,X3) filter(X1,X2,mark(X3)) -> filter(X1,X2,X3) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(filter(X1,X2,X3)) -> active(filter(mark(X1),mark(X2),mark(X3))) mark(nats(X)) -> active(nats(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sieve(X)) -> active(sieve(mark(X))) mark(0) -> active(0) mark(zprimes) -> active(zprimes) nats(active(X)) -> nats(X) nats(mark(X)) -> nats(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) sieve(active(X)) -> sieve(X) sieve(mark(X)) -> sieve(X) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.5: Subterm Processor: -> Pairs: CONS(active(X1),X2) -> CONS(X1,X2) CONS(mark(X1),X2) -> CONS(X1,X2) CONS(X1,active(X2)) -> CONS(X1,X2) CONS(X1,mark(X2)) -> CONS(X1,X2) -> Rules: active(filter(cons(X,Y),s(N),M)) -> mark(cons(X,filter(Y,N,M))) active(filter(cons(X,Y),0,M)) -> mark(cons(0,filter(Y,M,M))) active(nats(N)) -> mark(cons(N,nats(s(N)))) active(sieve(cons(s(N),Y))) -> mark(cons(s(N),sieve(filter(Y,N,N)))) active(sieve(cons(0,Y))) -> mark(cons(0,sieve(Y))) active(zprimes) -> mark(sieve(nats(s(s(0))))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) filter(active(X1),X2,X3) -> filter(X1,X2,X3) filter(mark(X1),X2,X3) -> filter(X1,X2,X3) filter(X1,active(X2),X3) -> filter(X1,X2,X3) filter(X1,mark(X2),X3) -> filter(X1,X2,X3) filter(X1,X2,active(X3)) -> filter(X1,X2,X3) filter(X1,X2,mark(X3)) -> filter(X1,X2,X3) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(filter(X1,X2,X3)) -> active(filter(mark(X1),mark(X2),mark(X3))) mark(nats(X)) -> active(nats(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sieve(X)) -> active(sieve(mark(X))) mark(0) -> active(0) mark(zprimes) -> active(zprimes) nats(active(X)) -> nats(X) nats(mark(X)) -> nats(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) sieve(active(X)) -> sieve(X) sieve(mark(X)) -> sieve(X) ->Projection: pi(CONS) = 1 Problem 1.5: SCC Processor: -> Pairs: CONS(X1,active(X2)) -> CONS(X1,X2) CONS(X1,mark(X2)) -> CONS(X1,X2) -> Rules: active(filter(cons(X,Y),s(N),M)) -> mark(cons(X,filter(Y,N,M))) active(filter(cons(X,Y),0,M)) -> mark(cons(0,filter(Y,M,M))) active(nats(N)) -> mark(cons(N,nats(s(N)))) active(sieve(cons(s(N),Y))) -> mark(cons(s(N),sieve(filter(Y,N,N)))) active(sieve(cons(0,Y))) -> mark(cons(0,sieve(Y))) active(zprimes) -> mark(sieve(nats(s(s(0))))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) filter(active(X1),X2,X3) -> filter(X1,X2,X3) filter(mark(X1),X2,X3) -> filter(X1,X2,X3) filter(X1,active(X2),X3) -> filter(X1,X2,X3) filter(X1,mark(X2),X3) -> filter(X1,X2,X3) filter(X1,X2,active(X3)) -> filter(X1,X2,X3) filter(X1,X2,mark(X3)) -> filter(X1,X2,X3) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(filter(X1,X2,X3)) -> active(filter(mark(X1),mark(X2),mark(X3))) mark(nats(X)) -> active(nats(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sieve(X)) -> active(sieve(mark(X))) mark(0) -> active(0) mark(zprimes) -> active(zprimes) nats(active(X)) -> nats(X) nats(mark(X)) -> nats(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) sieve(active(X)) -> sieve(X) sieve(mark(X)) -> sieve(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: CONS(X1,active(X2)) -> CONS(X1,X2) CONS(X1,mark(X2)) -> CONS(X1,X2) ->->-> Rules: active(filter(cons(X,Y),s(N),M)) -> mark(cons(X,filter(Y,N,M))) active(filter(cons(X,Y),0,M)) -> mark(cons(0,filter(Y,M,M))) active(nats(N)) -> mark(cons(N,nats(s(N)))) active(sieve(cons(s(N),Y))) -> mark(cons(s(N),sieve(filter(Y,N,N)))) active(sieve(cons(0,Y))) -> mark(cons(0,sieve(Y))) active(zprimes) -> mark(sieve(nats(s(s(0))))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) filter(active(X1),X2,X3) -> filter(X1,X2,X3) filter(mark(X1),X2,X3) -> filter(X1,X2,X3) filter(X1,active(X2),X3) -> filter(X1,X2,X3) filter(X1,mark(X2),X3) -> filter(X1,X2,X3) filter(X1,X2,active(X3)) -> filter(X1,X2,X3) filter(X1,X2,mark(X3)) -> filter(X1,X2,X3) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(filter(X1,X2,X3)) -> active(filter(mark(X1),mark(X2),mark(X3))) mark(nats(X)) -> active(nats(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sieve(X)) -> active(sieve(mark(X))) mark(0) -> active(0) mark(zprimes) -> active(zprimes) nats(active(X)) -> nats(X) nats(mark(X)) -> nats(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) sieve(active(X)) -> sieve(X) sieve(mark(X)) -> sieve(X) Problem 1.5: Subterm Processor: -> Pairs: CONS(X1,active(X2)) -> CONS(X1,X2) CONS(X1,mark(X2)) -> CONS(X1,X2) -> Rules: active(filter(cons(X,Y),s(N),M)) -> mark(cons(X,filter(Y,N,M))) active(filter(cons(X,Y),0,M)) -> mark(cons(0,filter(Y,M,M))) active(nats(N)) -> mark(cons(N,nats(s(N)))) active(sieve(cons(s(N),Y))) -> mark(cons(s(N),sieve(filter(Y,N,N)))) active(sieve(cons(0,Y))) -> mark(cons(0,sieve(Y))) active(zprimes) -> mark(sieve(nats(s(s(0))))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) filter(active(X1),X2,X3) -> filter(X1,X2,X3) filter(mark(X1),X2,X3) -> filter(X1,X2,X3) filter(X1,active(X2),X3) -> filter(X1,X2,X3) filter(X1,mark(X2),X3) -> filter(X1,X2,X3) filter(X1,X2,active(X3)) -> filter(X1,X2,X3) filter(X1,X2,mark(X3)) -> filter(X1,X2,X3) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(filter(X1,X2,X3)) -> active(filter(mark(X1),mark(X2),mark(X3))) mark(nats(X)) -> active(nats(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sieve(X)) -> active(sieve(mark(X))) mark(0) -> active(0) mark(zprimes) -> active(zprimes) nats(active(X)) -> nats(X) nats(mark(X)) -> nats(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) sieve(active(X)) -> sieve(X) sieve(mark(X)) -> sieve(X) ->Projection: pi(CONS) = 2 Problem 1.5: SCC Processor: -> Pairs: Empty -> Rules: active(filter(cons(X,Y),s(N),M)) -> mark(cons(X,filter(Y,N,M))) active(filter(cons(X,Y),0,M)) -> mark(cons(0,filter(Y,M,M))) active(nats(N)) -> mark(cons(N,nats(s(N)))) active(sieve(cons(s(N),Y))) -> mark(cons(s(N),sieve(filter(Y,N,N)))) active(sieve(cons(0,Y))) -> mark(cons(0,sieve(Y))) active(zprimes) -> mark(sieve(nats(s(s(0))))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) filter(active(X1),X2,X3) -> filter(X1,X2,X3) filter(mark(X1),X2,X3) -> filter(X1,X2,X3) filter(X1,active(X2),X3) -> filter(X1,X2,X3) filter(X1,mark(X2),X3) -> filter(X1,X2,X3) filter(X1,X2,active(X3)) -> filter(X1,X2,X3) filter(X1,X2,mark(X3)) -> filter(X1,X2,X3) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(filter(X1,X2,X3)) -> active(filter(mark(X1),mark(X2),mark(X3))) mark(nats(X)) -> active(nats(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sieve(X)) -> active(sieve(mark(X))) mark(0) -> active(0) mark(zprimes) -> active(zprimes) nats(active(X)) -> nats(X) nats(mark(X)) -> nats(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) sieve(active(X)) -> sieve(X) sieve(mark(X)) -> sieve(X) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.6: Reduction Pair Processor: -> Pairs: ACTIVE(filter(cons(X,Y),s(N),M)) -> MARK(cons(X,filter(Y,N,M))) ACTIVE(filter(cons(X,Y),0,M)) -> MARK(cons(0,filter(Y,M,M))) ACTIVE(nats(N)) -> MARK(cons(N,nats(s(N)))) ACTIVE(sieve(cons(s(N),Y))) -> MARK(cons(s(N),sieve(filter(Y,N,N)))) ACTIVE(sieve(cons(0,Y))) -> MARK(cons(0,sieve(Y))) ACTIVE(zprimes) -> MARK(sieve(nats(s(s(0))))) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> MARK(X1) MARK(filter(X1,X2,X3)) -> ACTIVE(filter(mark(X1),mark(X2),mark(X3))) MARK(filter(X1,X2,X3)) -> MARK(X1) MARK(filter(X1,X2,X3)) -> MARK(X2) MARK(filter(X1,X2,X3)) -> MARK(X3) MARK(nats(X)) -> ACTIVE(nats(mark(X))) MARK(nats(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(sieve(X)) -> ACTIVE(sieve(mark(X))) MARK(sieve(X)) -> MARK(X) MARK(zprimes) -> ACTIVE(zprimes) -> Rules: active(filter(cons(X,Y),s(N),M)) -> mark(cons(X,filter(Y,N,M))) active(filter(cons(X,Y),0,M)) -> mark(cons(0,filter(Y,M,M))) active(nats(N)) -> mark(cons(N,nats(s(N)))) active(sieve(cons(s(N),Y))) -> mark(cons(s(N),sieve(filter(Y,N,N)))) active(sieve(cons(0,Y))) -> mark(cons(0,sieve(Y))) active(zprimes) -> mark(sieve(nats(s(s(0))))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) filter(active(X1),X2,X3) -> filter(X1,X2,X3) filter(mark(X1),X2,X3) -> filter(X1,X2,X3) filter(X1,active(X2),X3) -> filter(X1,X2,X3) filter(X1,mark(X2),X3) -> filter(X1,X2,X3) filter(X1,X2,active(X3)) -> filter(X1,X2,X3) filter(X1,X2,mark(X3)) -> filter(X1,X2,X3) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(filter(X1,X2,X3)) -> active(filter(mark(X1),mark(X2),mark(X3))) mark(nats(X)) -> active(nats(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sieve(X)) -> active(sieve(mark(X))) mark(0) -> active(0) mark(zprimes) -> active(zprimes) nats(active(X)) -> nats(X) nats(mark(X)) -> nats(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) sieve(active(X)) -> sieve(X) sieve(mark(X)) -> sieve(X) -> Usable rules: active(filter(cons(X,Y),s(N),M)) -> mark(cons(X,filter(Y,N,M))) active(filter(cons(X,Y),0,M)) -> mark(cons(0,filter(Y,M,M))) active(nats(N)) -> mark(cons(N,nats(s(N)))) active(sieve(cons(s(N),Y))) -> mark(cons(s(N),sieve(filter(Y,N,N)))) active(sieve(cons(0,Y))) -> mark(cons(0,sieve(Y))) active(zprimes) -> mark(sieve(nats(s(s(0))))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) filter(active(X1),X2,X3) -> filter(X1,X2,X3) filter(mark(X1),X2,X3) -> filter(X1,X2,X3) filter(X1,active(X2),X3) -> filter(X1,X2,X3) filter(X1,mark(X2),X3) -> filter(X1,X2,X3) filter(X1,X2,active(X3)) -> filter(X1,X2,X3) filter(X1,X2,mark(X3)) -> filter(X1,X2,X3) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(filter(X1,X2,X3)) -> active(filter(mark(X1),mark(X2),mark(X3))) mark(nats(X)) -> active(nats(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sieve(X)) -> active(sieve(mark(X))) mark(0) -> active(0) mark(zprimes) -> active(zprimes) nats(active(X)) -> nats(X) nats(mark(X)) -> nats(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) sieve(active(X)) -> sieve(X) sieve(mark(X)) -> sieve(X) ->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 + X2 + 2.X3 + 2 [mark](X) = X [nats](X) = 2.X + 1 [s](X) = 2.X [sieve](X) = 2.X [0] = 0 [zprimes] = 2 [ACTIVE](X) = 2.X + 2 [MARK](X) = 2.X + 2 Problem 1.6: SCC Processor: -> Pairs: ACTIVE(filter(cons(X,Y),0,M)) -> MARK(cons(0,filter(Y,M,M))) ACTIVE(nats(N)) -> MARK(cons(N,nats(s(N)))) ACTIVE(sieve(cons(s(N),Y))) -> MARK(cons(s(N),sieve(filter(Y,N,N)))) ACTIVE(sieve(cons(0,Y))) -> MARK(cons(0,sieve(Y))) ACTIVE(zprimes) -> MARK(sieve(nats(s(s(0))))) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> MARK(X1) MARK(filter(X1,X2,X3)) -> ACTIVE(filter(mark(X1),mark(X2),mark(X3))) MARK(filter(X1,X2,X3)) -> MARK(X1) MARK(filter(X1,X2,X3)) -> MARK(X2) MARK(filter(X1,X2,X3)) -> MARK(X3) MARK(nats(X)) -> ACTIVE(nats(mark(X))) MARK(nats(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(sieve(X)) -> ACTIVE(sieve(mark(X))) MARK(sieve(X)) -> MARK(X) MARK(zprimes) -> ACTIVE(zprimes) -> Rules: active(filter(cons(X,Y),s(N),M)) -> mark(cons(X,filter(Y,N,M))) active(filter(cons(X,Y),0,M)) -> mark(cons(0,filter(Y,M,M))) active(nats(N)) -> mark(cons(N,nats(s(N)))) active(sieve(cons(s(N),Y))) -> mark(cons(s(N),sieve(filter(Y,N,N)))) active(sieve(cons(0,Y))) -> mark(cons(0,sieve(Y))) active(zprimes) -> mark(sieve(nats(s(s(0))))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) filter(active(X1),X2,X3) -> filter(X1,X2,X3) filter(mark(X1),X2,X3) -> filter(X1,X2,X3) filter(X1,active(X2),X3) -> filter(X1,X2,X3) filter(X1,mark(X2),X3) -> filter(X1,X2,X3) filter(X1,X2,active(X3)) -> filter(X1,X2,X3) filter(X1,X2,mark(X3)) -> filter(X1,X2,X3) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(filter(X1,X2,X3)) -> active(filter(mark(X1),mark(X2),mark(X3))) mark(nats(X)) -> active(nats(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sieve(X)) -> active(sieve(mark(X))) mark(0) -> active(0) mark(zprimes) -> active(zprimes) nats(active(X)) -> nats(X) nats(mark(X)) -> nats(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) sieve(active(X)) -> sieve(X) sieve(mark(X)) -> sieve(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(filter(cons(X,Y),0,M)) -> MARK(cons(0,filter(Y,M,M))) ACTIVE(nats(N)) -> MARK(cons(N,nats(s(N)))) ACTIVE(sieve(cons(s(N),Y))) -> MARK(cons(s(N),sieve(filter(Y,N,N)))) ACTIVE(sieve(cons(0,Y))) -> MARK(cons(0,sieve(Y))) ACTIVE(zprimes) -> MARK(sieve(nats(s(s(0))))) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> MARK(X1) MARK(filter(X1,X2,X3)) -> ACTIVE(filter(mark(X1),mark(X2),mark(X3))) MARK(filter(X1,X2,X3)) -> MARK(X1) MARK(filter(X1,X2,X3)) -> MARK(X2) MARK(filter(X1,X2,X3)) -> MARK(X3) MARK(nats(X)) -> ACTIVE(nats(mark(X))) MARK(nats(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(sieve(X)) -> ACTIVE(sieve(mark(X))) MARK(sieve(X)) -> MARK(X) MARK(zprimes) -> ACTIVE(zprimes) ->->-> Rules: active(filter(cons(X,Y),s(N),M)) -> mark(cons(X,filter(Y,N,M))) active(filter(cons(X,Y),0,M)) -> mark(cons(0,filter(Y,M,M))) active(nats(N)) -> mark(cons(N,nats(s(N)))) active(sieve(cons(s(N),Y))) -> mark(cons(s(N),sieve(filter(Y,N,N)))) active(sieve(cons(0,Y))) -> mark(cons(0,sieve(Y))) active(zprimes) -> mark(sieve(nats(s(s(0))))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) filter(active(X1),X2,X3) -> filter(X1,X2,X3) filter(mark(X1),X2,X3) -> filter(X1,X2,X3) filter(X1,active(X2),X3) -> filter(X1,X2,X3) filter(X1,mark(X2),X3) -> filter(X1,X2,X3) filter(X1,X2,active(X3)) -> filter(X1,X2,X3) filter(X1,X2,mark(X3)) -> filter(X1,X2,X3) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(filter(X1,X2,X3)) -> active(filter(mark(X1),mark(X2),mark(X3))) mark(nats(X)) -> active(nats(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sieve(X)) -> active(sieve(mark(X))) mark(0) -> active(0) mark(zprimes) -> active(zprimes) nats(active(X)) -> nats(X) nats(mark(X)) -> nats(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) sieve(active(X)) -> sieve(X) sieve(mark(X)) -> sieve(X) Problem 1.6: Reduction Pair Processor: -> Pairs: ACTIVE(filter(cons(X,Y),0,M)) -> MARK(cons(0,filter(Y,M,M))) ACTIVE(nats(N)) -> MARK(cons(N,nats(s(N)))) ACTIVE(sieve(cons(s(N),Y))) -> MARK(cons(s(N),sieve(filter(Y,N,N)))) ACTIVE(sieve(cons(0,Y))) -> MARK(cons(0,sieve(Y))) ACTIVE(zprimes) -> MARK(sieve(nats(s(s(0))))) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> MARK(X1) MARK(filter(X1,X2,X3)) -> ACTIVE(filter(mark(X1),mark(X2),mark(X3))) MARK(filter(X1,X2,X3)) -> MARK(X1) MARK(filter(X1,X2,X3)) -> MARK(X2) MARK(filter(X1,X2,X3)) -> MARK(X3) MARK(nats(X)) -> ACTIVE(nats(mark(X))) MARK(nats(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(sieve(X)) -> ACTIVE(sieve(mark(X))) MARK(sieve(X)) -> MARK(X) MARK(zprimes) -> ACTIVE(zprimes) -> Rules: active(filter(cons(X,Y),s(N),M)) -> mark(cons(X,filter(Y,N,M))) active(filter(cons(X,Y),0,M)) -> mark(cons(0,filter(Y,M,M))) active(nats(N)) -> mark(cons(N,nats(s(N)))) active(sieve(cons(s(N),Y))) -> mark(cons(s(N),sieve(filter(Y,N,N)))) active(sieve(cons(0,Y))) -> mark(cons(0,sieve(Y))) active(zprimes) -> mark(sieve(nats(s(s(0))))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) filter(active(X1),X2,X3) -> filter(X1,X2,X3) filter(mark(X1),X2,X3) -> filter(X1,X2,X3) filter(X1,active(X2),X3) -> filter(X1,X2,X3) filter(X1,mark(X2),X3) -> filter(X1,X2,X3) filter(X1,X2,active(X3)) -> filter(X1,X2,X3) filter(X1,X2,mark(X3)) -> filter(X1,X2,X3) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(filter(X1,X2,X3)) -> active(filter(mark(X1),mark(X2),mark(X3))) mark(nats(X)) -> active(nats(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sieve(X)) -> active(sieve(mark(X))) mark(0) -> active(0) mark(zprimes) -> active(zprimes) nats(active(X)) -> nats(X) nats(mark(X)) -> nats(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) sieve(active(X)) -> sieve(X) sieve(mark(X)) -> sieve(X) -> Usable rules: active(filter(cons(X,Y),s(N),M)) -> mark(cons(X,filter(Y,N,M))) active(filter(cons(X,Y),0,M)) -> mark(cons(0,filter(Y,M,M))) active(nats(N)) -> mark(cons(N,nats(s(N)))) active(sieve(cons(s(N),Y))) -> mark(cons(s(N),sieve(filter(Y,N,N)))) active(sieve(cons(0,Y))) -> mark(cons(0,sieve(Y))) active(zprimes) -> mark(sieve(nats(s(s(0))))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) filter(active(X1),X2,X3) -> filter(X1,X2,X3) filter(mark(X1),X2,X3) -> filter(X1,X2,X3) filter(X1,active(X2),X3) -> filter(X1,X2,X3) filter(X1,mark(X2),X3) -> filter(X1,X2,X3) filter(X1,X2,active(X3)) -> filter(X1,X2,X3) filter(X1,X2,mark(X3)) -> filter(X1,X2,X3) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(filter(X1,X2,X3)) -> active(filter(mark(X1),mark(X2),mark(X3))) mark(nats(X)) -> active(nats(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sieve(X)) -> active(sieve(mark(X))) mark(0) -> active(0) mark(zprimes) -> active(zprimes) nats(active(X)) -> nats(X) nats(mark(X)) -> nats(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) sieve(active(X)) -> sieve(X) sieve(mark(X)) -> sieve(X) ->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 + X2 + X3 + 2 [mark](X) = X [nats](X) = 2.X + 1 [s](X) = 2.X [sieve](X) = 2.X [0] = 0 [zprimes] = 2 [ACTIVE](X) = 2.X + 2 [MARK](X) = 2.X + 2 Problem 1.6: SCC Processor: -> Pairs: ACTIVE(nats(N)) -> MARK(cons(N,nats(s(N)))) ACTIVE(sieve(cons(s(N),Y))) -> MARK(cons(s(N),sieve(filter(Y,N,N)))) ACTIVE(sieve(cons(0,Y))) -> MARK(cons(0,sieve(Y))) ACTIVE(zprimes) -> MARK(sieve(nats(s(s(0))))) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> MARK(X1) MARK(filter(X1,X2,X3)) -> ACTIVE(filter(mark(X1),mark(X2),mark(X3))) MARK(filter(X1,X2,X3)) -> MARK(X1) MARK(filter(X1,X2,X3)) -> MARK(X2) MARK(filter(X1,X2,X3)) -> MARK(X3) MARK(nats(X)) -> ACTIVE(nats(mark(X))) MARK(nats(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(sieve(X)) -> ACTIVE(sieve(mark(X))) MARK(sieve(X)) -> MARK(X) MARK(zprimes) -> ACTIVE(zprimes) -> Rules: active(filter(cons(X,Y),s(N),M)) -> mark(cons(X,filter(Y,N,M))) active(filter(cons(X,Y),0,M)) -> mark(cons(0,filter(Y,M,M))) active(nats(N)) -> mark(cons(N,nats(s(N)))) active(sieve(cons(s(N),Y))) -> mark(cons(s(N),sieve(filter(Y,N,N)))) active(sieve(cons(0,Y))) -> mark(cons(0,sieve(Y))) active(zprimes) -> mark(sieve(nats(s(s(0))))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) filter(active(X1),X2,X3) -> filter(X1,X2,X3) filter(mark(X1),X2,X3) -> filter(X1,X2,X3) filter(X1,active(X2),X3) -> filter(X1,X2,X3) filter(X1,mark(X2),X3) -> filter(X1,X2,X3) filter(X1,X2,active(X3)) -> filter(X1,X2,X3) filter(X1,X2,mark(X3)) -> filter(X1,X2,X3) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(filter(X1,X2,X3)) -> active(filter(mark(X1),mark(X2),mark(X3))) mark(nats(X)) -> active(nats(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sieve(X)) -> active(sieve(mark(X))) mark(0) -> active(0) mark(zprimes) -> active(zprimes) nats(active(X)) -> nats(X) nats(mark(X)) -> nats(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) sieve(active(X)) -> sieve(X) sieve(mark(X)) -> sieve(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(nats(N)) -> MARK(cons(N,nats(s(N)))) ACTIVE(sieve(cons(s(N),Y))) -> MARK(cons(s(N),sieve(filter(Y,N,N)))) ACTIVE(sieve(cons(0,Y))) -> MARK(cons(0,sieve(Y))) ACTIVE(zprimes) -> MARK(sieve(nats(s(s(0))))) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> MARK(X1) MARK(filter(X1,X2,X3)) -> ACTIVE(filter(mark(X1),mark(X2),mark(X3))) MARK(filter(X1,X2,X3)) -> MARK(X1) MARK(filter(X1,X2,X3)) -> MARK(X2) MARK(filter(X1,X2,X3)) -> MARK(X3) MARK(nats(X)) -> ACTIVE(nats(mark(X))) MARK(nats(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(sieve(X)) -> ACTIVE(sieve(mark(X))) MARK(sieve(X)) -> MARK(X) MARK(zprimes) -> ACTIVE(zprimes) ->->-> Rules: active(filter(cons(X,Y),s(N),M)) -> mark(cons(X,filter(Y,N,M))) active(filter(cons(X,Y),0,M)) -> mark(cons(0,filter(Y,M,M))) active(nats(N)) -> mark(cons(N,nats(s(N)))) active(sieve(cons(s(N),Y))) -> mark(cons(s(N),sieve(filter(Y,N,N)))) active(sieve(cons(0,Y))) -> mark(cons(0,sieve(Y))) active(zprimes) -> mark(sieve(nats(s(s(0))))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) filter(active(X1),X2,X3) -> filter(X1,X2,X3) filter(mark(X1),X2,X3) -> filter(X1,X2,X3) filter(X1,active(X2),X3) -> filter(X1,X2,X3) filter(X1,mark(X2),X3) -> filter(X1,X2,X3) filter(X1,X2,active(X3)) -> filter(X1,X2,X3) filter(X1,X2,mark(X3)) -> filter(X1,X2,X3) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(filter(X1,X2,X3)) -> active(filter(mark(X1),mark(X2),mark(X3))) mark(nats(X)) -> active(nats(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sieve(X)) -> active(sieve(mark(X))) mark(0) -> active(0) mark(zprimes) -> active(zprimes) nats(active(X)) -> nats(X) nats(mark(X)) -> nats(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) sieve(active(X)) -> sieve(X) sieve(mark(X)) -> sieve(X) Problem 1.6: Reduction Pair Processor: -> Pairs: ACTIVE(nats(N)) -> MARK(cons(N,nats(s(N)))) ACTIVE(sieve(cons(s(N),Y))) -> MARK(cons(s(N),sieve(filter(Y,N,N)))) ACTIVE(sieve(cons(0,Y))) -> MARK(cons(0,sieve(Y))) ACTIVE(zprimes) -> MARK(sieve(nats(s(s(0))))) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> MARK(X1) MARK(filter(X1,X2,X3)) -> ACTIVE(filter(mark(X1),mark(X2),mark(X3))) MARK(filter(X1,X2,X3)) -> MARK(X1) MARK(filter(X1,X2,X3)) -> MARK(X2) MARK(filter(X1,X2,X3)) -> MARK(X3) MARK(nats(X)) -> ACTIVE(nats(mark(X))) MARK(nats(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(sieve(X)) -> ACTIVE(sieve(mark(X))) MARK(sieve(X)) -> MARK(X) MARK(zprimes) -> ACTIVE(zprimes) -> Rules: active(filter(cons(X,Y),s(N),M)) -> mark(cons(X,filter(Y,N,M))) active(filter(cons(X,Y),0,M)) -> mark(cons(0,filter(Y,M,M))) active(nats(N)) -> mark(cons(N,nats(s(N)))) active(sieve(cons(s(N),Y))) -> mark(cons(s(N),sieve(filter(Y,N,N)))) active(sieve(cons(0,Y))) -> mark(cons(0,sieve(Y))) active(zprimes) -> mark(sieve(nats(s(s(0))))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) filter(active(X1),X2,X3) -> filter(X1,X2,X3) filter(mark(X1),X2,X3) -> filter(X1,X2,X3) filter(X1,active(X2),X3) -> filter(X1,X2,X3) filter(X1,mark(X2),X3) -> filter(X1,X2,X3) filter(X1,X2,active(X3)) -> filter(X1,X2,X3) filter(X1,X2,mark(X3)) -> filter(X1,X2,X3) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(filter(X1,X2,X3)) -> active(filter(mark(X1),mark(X2),mark(X3))) mark(nats(X)) -> active(nats(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sieve(X)) -> active(sieve(mark(X))) mark(0) -> active(0) mark(zprimes) -> active(zprimes) nats(active(X)) -> nats(X) nats(mark(X)) -> nats(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) sieve(active(X)) -> sieve(X) sieve(mark(X)) -> sieve(X) -> Usable rules: active(filter(cons(X,Y),s(N),M)) -> mark(cons(X,filter(Y,N,M))) active(filter(cons(X,Y),0,M)) -> mark(cons(0,filter(Y,M,M))) active(nats(N)) -> mark(cons(N,nats(s(N)))) active(sieve(cons(s(N),Y))) -> mark(cons(s(N),sieve(filter(Y,N,N)))) active(sieve(cons(0,Y))) -> mark(cons(0,sieve(Y))) active(zprimes) -> mark(sieve(nats(s(s(0))))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) filter(active(X1),X2,X3) -> filter(X1,X2,X3) filter(mark(X1),X2,X3) -> filter(X1,X2,X3) filter(X1,active(X2),X3) -> filter(X1,X2,X3) filter(X1,mark(X2),X3) -> filter(X1,X2,X3) filter(X1,X2,active(X3)) -> filter(X1,X2,X3) filter(X1,X2,mark(X3)) -> filter(X1,X2,X3) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(filter(X1,X2,X3)) -> active(filter(mark(X1),mark(X2),mark(X3))) mark(nats(X)) -> active(nats(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sieve(X)) -> active(sieve(mark(X))) mark(0) -> active(0) mark(zprimes) -> active(zprimes) nats(active(X)) -> nats(X) nats(mark(X)) -> nats(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) sieve(active(X)) -> sieve(X) sieve(mark(X)) -> sieve(X) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = X [cons](X1,X2) = 2.X1 [filter](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 1 [mark](X) = X [nats](X) = 2.X + 2 [s](X) = 2.X [sieve](X) = X [0] = 0 [zprimes] = 2 [ACTIVE](X) = 2.X [MARK](X) = 2.X Problem 1.6: SCC Processor: -> Pairs: ACTIVE(sieve(cons(s(N),Y))) -> MARK(cons(s(N),sieve(filter(Y,N,N)))) ACTIVE(sieve(cons(0,Y))) -> MARK(cons(0,sieve(Y))) ACTIVE(zprimes) -> MARK(sieve(nats(s(s(0))))) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> MARK(X1) MARK(filter(X1,X2,X3)) -> ACTIVE(filter(mark(X1),mark(X2),mark(X3))) MARK(filter(X1,X2,X3)) -> MARK(X1) MARK(filter(X1,X2,X3)) -> MARK(X2) MARK(filter(X1,X2,X3)) -> MARK(X3) MARK(nats(X)) -> ACTIVE(nats(mark(X))) MARK(nats(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(sieve(X)) -> ACTIVE(sieve(mark(X))) MARK(sieve(X)) -> MARK(X) MARK(zprimes) -> ACTIVE(zprimes) -> Rules: active(filter(cons(X,Y),s(N),M)) -> mark(cons(X,filter(Y,N,M))) active(filter(cons(X,Y),0,M)) -> mark(cons(0,filter(Y,M,M))) active(nats(N)) -> mark(cons(N,nats(s(N)))) active(sieve(cons(s(N),Y))) -> mark(cons(s(N),sieve(filter(Y,N,N)))) active(sieve(cons(0,Y))) -> mark(cons(0,sieve(Y))) active(zprimes) -> mark(sieve(nats(s(s(0))))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) filter(active(X1),X2,X3) -> filter(X1,X2,X3) filter(mark(X1),X2,X3) -> filter(X1,X2,X3) filter(X1,active(X2),X3) -> filter(X1,X2,X3) filter(X1,mark(X2),X3) -> filter(X1,X2,X3) filter(X1,X2,active(X3)) -> filter(X1,X2,X3) filter(X1,X2,mark(X3)) -> filter(X1,X2,X3) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(filter(X1,X2,X3)) -> active(filter(mark(X1),mark(X2),mark(X3))) mark(nats(X)) -> active(nats(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sieve(X)) -> active(sieve(mark(X))) mark(0) -> active(0) mark(zprimes) -> active(zprimes) nats(active(X)) -> nats(X) nats(mark(X)) -> nats(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) sieve(active(X)) -> sieve(X) sieve(mark(X)) -> sieve(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(sieve(cons(s(N),Y))) -> MARK(cons(s(N),sieve(filter(Y,N,N)))) ACTIVE(sieve(cons(0,Y))) -> MARK(cons(0,sieve(Y))) ACTIVE(zprimes) -> MARK(sieve(nats(s(s(0))))) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> MARK(X1) MARK(filter(X1,X2,X3)) -> ACTIVE(filter(mark(X1),mark(X2),mark(X3))) MARK(filter(X1,X2,X3)) -> MARK(X1) MARK(filter(X1,X2,X3)) -> MARK(X2) MARK(filter(X1,X2,X3)) -> MARK(X3) MARK(nats(X)) -> ACTIVE(nats(mark(X))) MARK(nats(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(sieve(X)) -> ACTIVE(sieve(mark(X))) MARK(sieve(X)) -> MARK(X) MARK(zprimes) -> ACTIVE(zprimes) ->->-> Rules: active(filter(cons(X,Y),s(N),M)) -> mark(cons(X,filter(Y,N,M))) active(filter(cons(X,Y),0,M)) -> mark(cons(0,filter(Y,M,M))) active(nats(N)) -> mark(cons(N,nats(s(N)))) active(sieve(cons(s(N),Y))) -> mark(cons(s(N),sieve(filter(Y,N,N)))) active(sieve(cons(0,Y))) -> mark(cons(0,sieve(Y))) active(zprimes) -> mark(sieve(nats(s(s(0))))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) filter(active(X1),X2,X3) -> filter(X1,X2,X3) filter(mark(X1),X2,X3) -> filter(X1,X2,X3) filter(X1,active(X2),X3) -> filter(X1,X2,X3) filter(X1,mark(X2),X3) -> filter(X1,X2,X3) filter(X1,X2,active(X3)) -> filter(X1,X2,X3) filter(X1,X2,mark(X3)) -> filter(X1,X2,X3) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(filter(X1,X2,X3)) -> active(filter(mark(X1),mark(X2),mark(X3))) mark(nats(X)) -> active(nats(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sieve(X)) -> active(sieve(mark(X))) mark(0) -> active(0) mark(zprimes) -> active(zprimes) nats(active(X)) -> nats(X) nats(mark(X)) -> nats(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) sieve(active(X)) -> sieve(X) sieve(mark(X)) -> sieve(X) Problem 1.6: Reduction Pair Processor: -> Pairs: ACTIVE(sieve(cons(s(N),Y))) -> MARK(cons(s(N),sieve(filter(Y,N,N)))) ACTIVE(sieve(cons(0,Y))) -> MARK(cons(0,sieve(Y))) ACTIVE(zprimes) -> MARK(sieve(nats(s(s(0))))) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> MARK(X1) MARK(filter(X1,X2,X3)) -> ACTIVE(filter(mark(X1),mark(X2),mark(X3))) MARK(filter(X1,X2,X3)) -> MARK(X1) MARK(filter(X1,X2,X3)) -> MARK(X2) MARK(filter(X1,X2,X3)) -> MARK(X3) MARK(nats(X)) -> ACTIVE(nats(mark(X))) MARK(nats(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(sieve(X)) -> ACTIVE(sieve(mark(X))) MARK(sieve(X)) -> MARK(X) MARK(zprimes) -> ACTIVE(zprimes) -> Rules: active(filter(cons(X,Y),s(N),M)) -> mark(cons(X,filter(Y,N,M))) active(filter(cons(X,Y),0,M)) -> mark(cons(0,filter(Y,M,M))) active(nats(N)) -> mark(cons(N,nats(s(N)))) active(sieve(cons(s(N),Y))) -> mark(cons(s(N),sieve(filter(Y,N,N)))) active(sieve(cons(0,Y))) -> mark(cons(0,sieve(Y))) active(zprimes) -> mark(sieve(nats(s(s(0))))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) filter(active(X1),X2,X3) -> filter(X1,X2,X3) filter(mark(X1),X2,X3) -> filter(X1,X2,X3) filter(X1,active(X2),X3) -> filter(X1,X2,X3) filter(X1,mark(X2),X3) -> filter(X1,X2,X3) filter(X1,X2,active(X3)) -> filter(X1,X2,X3) filter(X1,X2,mark(X3)) -> filter(X1,X2,X3) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(filter(X1,X2,X3)) -> active(filter(mark(X1),mark(X2),mark(X3))) mark(nats(X)) -> active(nats(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sieve(X)) -> active(sieve(mark(X))) mark(0) -> active(0) mark(zprimes) -> active(zprimes) nats(active(X)) -> nats(X) nats(mark(X)) -> nats(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) sieve(active(X)) -> sieve(X) sieve(mark(X)) -> sieve(X) -> Usable rules: active(filter(cons(X,Y),s(N),M)) -> mark(cons(X,filter(Y,N,M))) active(filter(cons(X,Y),0,M)) -> mark(cons(0,filter(Y,M,M))) active(nats(N)) -> mark(cons(N,nats(s(N)))) active(sieve(cons(s(N),Y))) -> mark(cons(s(N),sieve(filter(Y,N,N)))) active(sieve(cons(0,Y))) -> mark(cons(0,sieve(Y))) active(zprimes) -> mark(sieve(nats(s(s(0))))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) filter(active(X1),X2,X3) -> filter(X1,X2,X3) filter(mark(X1),X2,X3) -> filter(X1,X2,X3) filter(X1,active(X2),X3) -> filter(X1,X2,X3) filter(X1,mark(X2),X3) -> filter(X1,X2,X3) filter(X1,X2,active(X3)) -> filter(X1,X2,X3) filter(X1,X2,mark(X3)) -> filter(X1,X2,X3) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(filter(X1,X2,X3)) -> active(filter(mark(X1),mark(X2),mark(X3))) mark(nats(X)) -> active(nats(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sieve(X)) -> active(sieve(mark(X))) mark(0) -> active(0) mark(zprimes) -> active(zprimes) nats(active(X)) -> nats(X) nats(mark(X)) -> nats(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) sieve(active(X)) -> sieve(X) sieve(mark(X)) -> sieve(X) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = X [cons](X1,X2) = 2.X1 + 1 [filter](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 [mark](X) = X [nats](X) = 2.X + 1 [s](X) = 2.X [sieve](X) = 2.X [0] = 0 [zprimes] = 2 [ACTIVE](X) = 2.X [MARK](X) = 2.X Problem 1.6: SCC Processor: -> Pairs: ACTIVE(sieve(cons(0,Y))) -> MARK(cons(0,sieve(Y))) ACTIVE(zprimes) -> MARK(sieve(nats(s(s(0))))) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> MARK(X1) MARK(filter(X1,X2,X3)) -> ACTIVE(filter(mark(X1),mark(X2),mark(X3))) MARK(filter(X1,X2,X3)) -> MARK(X1) MARK(filter(X1,X2,X3)) -> MARK(X2) MARK(filter(X1,X2,X3)) -> MARK(X3) MARK(nats(X)) -> ACTIVE(nats(mark(X))) MARK(nats(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(sieve(X)) -> ACTIVE(sieve(mark(X))) MARK(sieve(X)) -> MARK(X) MARK(zprimes) -> ACTIVE(zprimes) -> Rules: active(filter(cons(X,Y),s(N),M)) -> mark(cons(X,filter(Y,N,M))) active(filter(cons(X,Y),0,M)) -> mark(cons(0,filter(Y,M,M))) active(nats(N)) -> mark(cons(N,nats(s(N)))) active(sieve(cons(s(N),Y))) -> mark(cons(s(N),sieve(filter(Y,N,N)))) active(sieve(cons(0,Y))) -> mark(cons(0,sieve(Y))) active(zprimes) -> mark(sieve(nats(s(s(0))))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) filter(active(X1),X2,X3) -> filter(X1,X2,X3) filter(mark(X1),X2,X3) -> filter(X1,X2,X3) filter(X1,active(X2),X3) -> filter(X1,X2,X3) filter(X1,mark(X2),X3) -> filter(X1,X2,X3) filter(X1,X2,active(X3)) -> filter(X1,X2,X3) filter(X1,X2,mark(X3)) -> filter(X1,X2,X3) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(filter(X1,X2,X3)) -> active(filter(mark(X1),mark(X2),mark(X3))) mark(nats(X)) -> active(nats(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sieve(X)) -> active(sieve(mark(X))) mark(0) -> active(0) mark(zprimes) -> active(zprimes) nats(active(X)) -> nats(X) nats(mark(X)) -> nats(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) sieve(active(X)) -> sieve(X) sieve(mark(X)) -> sieve(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(sieve(cons(0,Y))) -> MARK(cons(0,sieve(Y))) ACTIVE(zprimes) -> MARK(sieve(nats(s(s(0))))) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> MARK(X1) MARK(filter(X1,X2,X3)) -> ACTIVE(filter(mark(X1),mark(X2),mark(X3))) MARK(filter(X1,X2,X3)) -> MARK(X1) MARK(filter(X1,X2,X3)) -> MARK(X2) MARK(filter(X1,X2,X3)) -> MARK(X3) MARK(nats(X)) -> ACTIVE(nats(mark(X))) MARK(nats(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(sieve(X)) -> ACTIVE(sieve(mark(X))) MARK(sieve(X)) -> MARK(X) MARK(zprimes) -> ACTIVE(zprimes) ->->-> Rules: active(filter(cons(X,Y),s(N),M)) -> mark(cons(X,filter(Y,N,M))) active(filter(cons(X,Y),0,M)) -> mark(cons(0,filter(Y,M,M))) active(nats(N)) -> mark(cons(N,nats(s(N)))) active(sieve(cons(s(N),Y))) -> mark(cons(s(N),sieve(filter(Y,N,N)))) active(sieve(cons(0,Y))) -> mark(cons(0,sieve(Y))) active(zprimes) -> mark(sieve(nats(s(s(0))))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) filter(active(X1),X2,X3) -> filter(X1,X2,X3) filter(mark(X1),X2,X3) -> filter(X1,X2,X3) filter(X1,active(X2),X3) -> filter(X1,X2,X3) filter(X1,mark(X2),X3) -> filter(X1,X2,X3) filter(X1,X2,active(X3)) -> filter(X1,X2,X3) filter(X1,X2,mark(X3)) -> filter(X1,X2,X3) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(filter(X1,X2,X3)) -> active(filter(mark(X1),mark(X2),mark(X3))) mark(nats(X)) -> active(nats(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sieve(X)) -> active(sieve(mark(X))) mark(0) -> active(0) mark(zprimes) -> active(zprimes) nats(active(X)) -> nats(X) nats(mark(X)) -> nats(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) sieve(active(X)) -> sieve(X) sieve(mark(X)) -> sieve(X) Problem 1.6: Reduction Pair Processor: -> Pairs: ACTIVE(sieve(cons(0,Y))) -> MARK(cons(0,sieve(Y))) ACTIVE(zprimes) -> MARK(sieve(nats(s(s(0))))) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> MARK(X1) MARK(filter(X1,X2,X3)) -> ACTIVE(filter(mark(X1),mark(X2),mark(X3))) MARK(filter(X1,X2,X3)) -> MARK(X1) MARK(filter(X1,X2,X3)) -> MARK(X2) MARK(filter(X1,X2,X3)) -> MARK(X3) MARK(nats(X)) -> ACTIVE(nats(mark(X))) MARK(nats(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(sieve(X)) -> ACTIVE(sieve(mark(X))) MARK(sieve(X)) -> MARK(X) MARK(zprimes) -> ACTIVE(zprimes) -> Rules: active(filter(cons(X,Y),s(N),M)) -> mark(cons(X,filter(Y,N,M))) active(filter(cons(X,Y),0,M)) -> mark(cons(0,filter(Y,M,M))) active(nats(N)) -> mark(cons(N,nats(s(N)))) active(sieve(cons(s(N),Y))) -> mark(cons(s(N),sieve(filter(Y,N,N)))) active(sieve(cons(0,Y))) -> mark(cons(0,sieve(Y))) active(zprimes) -> mark(sieve(nats(s(s(0))))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) filter(active(X1),X2,X3) -> filter(X1,X2,X3) filter(mark(X1),X2,X3) -> filter(X1,X2,X3) filter(X1,active(X2),X3) -> filter(X1,X2,X3) filter(X1,mark(X2),X3) -> filter(X1,X2,X3) filter(X1,X2,active(X3)) -> filter(X1,X2,X3) filter(X1,X2,mark(X3)) -> filter(X1,X2,X3) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(filter(X1,X2,X3)) -> active(filter(mark(X1),mark(X2),mark(X3))) mark(nats(X)) -> active(nats(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sieve(X)) -> active(sieve(mark(X))) mark(0) -> active(0) mark(zprimes) -> active(zprimes) nats(active(X)) -> nats(X) nats(mark(X)) -> nats(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) sieve(active(X)) -> sieve(X) sieve(mark(X)) -> sieve(X) -> Usable rules: active(filter(cons(X,Y),s(N),M)) -> mark(cons(X,filter(Y,N,M))) active(filter(cons(X,Y),0,M)) -> mark(cons(0,filter(Y,M,M))) active(nats(N)) -> mark(cons(N,nats(s(N)))) active(sieve(cons(s(N),Y))) -> mark(cons(s(N),sieve(filter(Y,N,N)))) active(sieve(cons(0,Y))) -> mark(cons(0,sieve(Y))) active(zprimes) -> mark(sieve(nats(s(s(0))))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) filter(active(X1),X2,X3) -> filter(X1,X2,X3) filter(mark(X1),X2,X3) -> filter(X1,X2,X3) filter(X1,active(X2),X3) -> filter(X1,X2,X3) filter(X1,mark(X2),X3) -> filter(X1,X2,X3) filter(X1,X2,active(X3)) -> filter(X1,X2,X3) filter(X1,X2,mark(X3)) -> filter(X1,X2,X3) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(filter(X1,X2,X3)) -> active(filter(mark(X1),mark(X2),mark(X3))) mark(nats(X)) -> active(nats(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sieve(X)) -> active(sieve(mark(X))) mark(0) -> active(0) mark(zprimes) -> active(zprimes) nats(active(X)) -> nats(X) nats(mark(X)) -> nats(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) sieve(active(X)) -> sieve(X) sieve(mark(X)) -> sieve(X) ->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) = 2.X [sieve](X) = 2.X + 2 [0] = 0 [zprimes] = 2 [ACTIVE](X) = X + 2 [MARK](X) = X + 2 Problem 1.6: SCC Processor: -> Pairs: ACTIVE(zprimes) -> MARK(sieve(nats(s(s(0))))) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> MARK(X1) MARK(filter(X1,X2,X3)) -> ACTIVE(filter(mark(X1),mark(X2),mark(X3))) MARK(filter(X1,X2,X3)) -> MARK(X1) MARK(filter(X1,X2,X3)) -> MARK(X2) MARK(filter(X1,X2,X3)) -> MARK(X3) MARK(nats(X)) -> ACTIVE(nats(mark(X))) MARK(nats(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(sieve(X)) -> ACTIVE(sieve(mark(X))) MARK(sieve(X)) -> MARK(X) MARK(zprimes) -> ACTIVE(zprimes) -> Rules: active(filter(cons(X,Y),s(N),M)) -> mark(cons(X,filter(Y,N,M))) active(filter(cons(X,Y),0,M)) -> mark(cons(0,filter(Y,M,M))) active(nats(N)) -> mark(cons(N,nats(s(N)))) active(sieve(cons(s(N),Y))) -> mark(cons(s(N),sieve(filter(Y,N,N)))) active(sieve(cons(0,Y))) -> mark(cons(0,sieve(Y))) active(zprimes) -> mark(sieve(nats(s(s(0))))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) filter(active(X1),X2,X3) -> filter(X1,X2,X3) filter(mark(X1),X2,X3) -> filter(X1,X2,X3) filter(X1,active(X2),X3) -> filter(X1,X2,X3) filter(X1,mark(X2),X3) -> filter(X1,X2,X3) filter(X1,X2,active(X3)) -> filter(X1,X2,X3) filter(X1,X2,mark(X3)) -> filter(X1,X2,X3) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(filter(X1,X2,X3)) -> active(filter(mark(X1),mark(X2),mark(X3))) mark(nats(X)) -> active(nats(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sieve(X)) -> active(sieve(mark(X))) mark(0) -> active(0) mark(zprimes) -> active(zprimes) nats(active(X)) -> nats(X) nats(mark(X)) -> nats(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) sieve(active(X)) -> sieve(X) sieve(mark(X)) -> sieve(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(zprimes) -> MARK(sieve(nats(s(s(0))))) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> MARK(X1) MARK(filter(X1,X2,X3)) -> ACTIVE(filter(mark(X1),mark(X2),mark(X3))) MARK(filter(X1,X2,X3)) -> MARK(X1) MARK(filter(X1,X2,X3)) -> MARK(X2) MARK(filter(X1,X2,X3)) -> MARK(X3) MARK(nats(X)) -> ACTIVE(nats(mark(X))) MARK(nats(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(sieve(X)) -> ACTIVE(sieve(mark(X))) MARK(sieve(X)) -> MARK(X) MARK(zprimes) -> ACTIVE(zprimes) ->->-> Rules: active(filter(cons(X,Y),s(N),M)) -> mark(cons(X,filter(Y,N,M))) active(filter(cons(X,Y),0,M)) -> mark(cons(0,filter(Y,M,M))) active(nats(N)) -> mark(cons(N,nats(s(N)))) active(sieve(cons(s(N),Y))) -> mark(cons(s(N),sieve(filter(Y,N,N)))) active(sieve(cons(0,Y))) -> mark(cons(0,sieve(Y))) active(zprimes) -> mark(sieve(nats(s(s(0))))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) filter(active(X1),X2,X3) -> filter(X1,X2,X3) filter(mark(X1),X2,X3) -> filter(X1,X2,X3) filter(X1,active(X2),X3) -> filter(X1,X2,X3) filter(X1,mark(X2),X3) -> filter(X1,X2,X3) filter(X1,X2,active(X3)) -> filter(X1,X2,X3) filter(X1,X2,mark(X3)) -> filter(X1,X2,X3) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(filter(X1,X2,X3)) -> active(filter(mark(X1),mark(X2),mark(X3))) mark(nats(X)) -> active(nats(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sieve(X)) -> active(sieve(mark(X))) mark(0) -> active(0) mark(zprimes) -> active(zprimes) nats(active(X)) -> nats(X) nats(mark(X)) -> nats(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) sieve(active(X)) -> sieve(X) sieve(mark(X)) -> sieve(X) Problem 1.6: Reduction Pair Processor: -> Pairs: ACTIVE(zprimes) -> MARK(sieve(nats(s(s(0))))) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> MARK(X1) MARK(filter(X1,X2,X3)) -> ACTIVE(filter(mark(X1),mark(X2),mark(X3))) MARK(filter(X1,X2,X3)) -> MARK(X1) MARK(filter(X1,X2,X3)) -> MARK(X2) MARK(filter(X1,X2,X3)) -> MARK(X3) MARK(nats(X)) -> ACTIVE(nats(mark(X))) MARK(nats(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(sieve(X)) -> ACTIVE(sieve(mark(X))) MARK(sieve(X)) -> MARK(X) MARK(zprimes) -> ACTIVE(zprimes) -> Rules: active(filter(cons(X,Y),s(N),M)) -> mark(cons(X,filter(Y,N,M))) active(filter(cons(X,Y),0,M)) -> mark(cons(0,filter(Y,M,M))) active(nats(N)) -> mark(cons(N,nats(s(N)))) active(sieve(cons(s(N),Y))) -> mark(cons(s(N),sieve(filter(Y,N,N)))) active(sieve(cons(0,Y))) -> mark(cons(0,sieve(Y))) active(zprimes) -> mark(sieve(nats(s(s(0))))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) filter(active(X1),X2,X3) -> filter(X1,X2,X3) filter(mark(X1),X2,X3) -> filter(X1,X2,X3) filter(X1,active(X2),X3) -> filter(X1,X2,X3) filter(X1,mark(X2),X3) -> filter(X1,X2,X3) filter(X1,X2,active(X3)) -> filter(X1,X2,X3) filter(X1,X2,mark(X3)) -> filter(X1,X2,X3) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(filter(X1,X2,X3)) -> active(filter(mark(X1),mark(X2),mark(X3))) mark(nats(X)) -> active(nats(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sieve(X)) -> active(sieve(mark(X))) mark(0) -> active(0) mark(zprimes) -> active(zprimes) nats(active(X)) -> nats(X) nats(mark(X)) -> nats(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) sieve(active(X)) -> sieve(X) sieve(mark(X)) -> sieve(X) -> Usable rules: active(filter(cons(X,Y),s(N),M)) -> mark(cons(X,filter(Y,N,M))) active(filter(cons(X,Y),0,M)) -> mark(cons(0,filter(Y,M,M))) active(nats(N)) -> mark(cons(N,nats(s(N)))) active(sieve(cons(s(N),Y))) -> mark(cons(s(N),sieve(filter(Y,N,N)))) active(sieve(cons(0,Y))) -> mark(cons(0,sieve(Y))) active(zprimes) -> mark(sieve(nats(s(s(0))))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) filter(active(X1),X2,X3) -> filter(X1,X2,X3) filter(mark(X1),X2,X3) -> filter(X1,X2,X3) filter(X1,active(X2),X3) -> filter(X1,X2,X3) filter(X1,mark(X2),X3) -> filter(X1,X2,X3) filter(X1,X2,active(X3)) -> filter(X1,X2,X3) filter(X1,X2,mark(X3)) -> filter(X1,X2,X3) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(filter(X1,X2,X3)) -> active(filter(mark(X1),mark(X2),mark(X3))) mark(nats(X)) -> active(nats(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sieve(X)) -> active(sieve(mark(X))) mark(0) -> active(0) mark(zprimes) -> active(zprimes) nats(active(X)) -> nats(X) nats(mark(X)) -> nats(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) sieve(active(X)) -> sieve(X) sieve(mark(X)) -> sieve(X) ->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 + 2 [mark](X) = 2.X [nats](X) = 2.X [s](X) = 2.X [sieve](X) = 2.X [0] = 0 [zprimes] = 1 [ACTIVE](X) = X [MARK](X) = 2.X Problem 1.6: SCC Processor: -> Pairs: MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> MARK(X1) MARK(filter(X1,X2,X3)) -> ACTIVE(filter(mark(X1),mark(X2),mark(X3))) MARK(filter(X1,X2,X3)) -> MARK(X1) MARK(filter(X1,X2,X3)) -> MARK(X2) MARK(filter(X1,X2,X3)) -> MARK(X3) MARK(nats(X)) -> ACTIVE(nats(mark(X))) MARK(nats(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(sieve(X)) -> ACTIVE(sieve(mark(X))) MARK(sieve(X)) -> MARK(X) MARK(zprimes) -> ACTIVE(zprimes) -> Rules: active(filter(cons(X,Y),s(N),M)) -> mark(cons(X,filter(Y,N,M))) active(filter(cons(X,Y),0,M)) -> mark(cons(0,filter(Y,M,M))) active(nats(N)) -> mark(cons(N,nats(s(N)))) active(sieve(cons(s(N),Y))) -> mark(cons(s(N),sieve(filter(Y,N,N)))) active(sieve(cons(0,Y))) -> mark(cons(0,sieve(Y))) active(zprimes) -> mark(sieve(nats(s(s(0))))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) filter(active(X1),X2,X3) -> filter(X1,X2,X3) filter(mark(X1),X2,X3) -> filter(X1,X2,X3) filter(X1,active(X2),X3) -> filter(X1,X2,X3) filter(X1,mark(X2),X3) -> filter(X1,X2,X3) filter(X1,X2,active(X3)) -> filter(X1,X2,X3) filter(X1,X2,mark(X3)) -> filter(X1,X2,X3) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(filter(X1,X2,X3)) -> active(filter(mark(X1),mark(X2),mark(X3))) mark(nats(X)) -> active(nats(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sieve(X)) -> active(sieve(mark(X))) mark(0) -> active(0) mark(zprimes) -> active(zprimes) nats(active(X)) -> nats(X) nats(mark(X)) -> nats(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) sieve(active(X)) -> sieve(X) sieve(mark(X)) -> sieve(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: MARK(cons(X1,X2)) -> MARK(X1) MARK(filter(X1,X2,X3)) -> MARK(X1) MARK(filter(X1,X2,X3)) -> MARK(X2) MARK(filter(X1,X2,X3)) -> MARK(X3) MARK(nats(X)) -> MARK(X) MARK(s(X)) -> MARK(X) MARK(sieve(X)) -> MARK(X) ->->-> Rules: active(filter(cons(X,Y),s(N),M)) -> mark(cons(X,filter(Y,N,M))) active(filter(cons(X,Y),0,M)) -> mark(cons(0,filter(Y,M,M))) active(nats(N)) -> mark(cons(N,nats(s(N)))) active(sieve(cons(s(N),Y))) -> mark(cons(s(N),sieve(filter(Y,N,N)))) active(sieve(cons(0,Y))) -> mark(cons(0,sieve(Y))) active(zprimes) -> mark(sieve(nats(s(s(0))))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) filter(active(X1),X2,X3) -> filter(X1,X2,X3) filter(mark(X1),X2,X3) -> filter(X1,X2,X3) filter(X1,active(X2),X3) -> filter(X1,X2,X3) filter(X1,mark(X2),X3) -> filter(X1,X2,X3) filter(X1,X2,active(X3)) -> filter(X1,X2,X3) filter(X1,X2,mark(X3)) -> filter(X1,X2,X3) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(filter(X1,X2,X3)) -> active(filter(mark(X1),mark(X2),mark(X3))) mark(nats(X)) -> active(nats(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sieve(X)) -> active(sieve(mark(X))) mark(0) -> active(0) mark(zprimes) -> active(zprimes) nats(active(X)) -> nats(X) nats(mark(X)) -> nats(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) sieve(active(X)) -> sieve(X) sieve(mark(X)) -> sieve(X) Problem 1.6: Subterm Processor: -> Pairs: MARK(cons(X1,X2)) -> MARK(X1) MARK(filter(X1,X2,X3)) -> MARK(X1) MARK(filter(X1,X2,X3)) -> MARK(X2) MARK(filter(X1,X2,X3)) -> MARK(X3) MARK(nats(X)) -> MARK(X) MARK(s(X)) -> MARK(X) MARK(sieve(X)) -> MARK(X) -> Rules: active(filter(cons(X,Y),s(N),M)) -> mark(cons(X,filter(Y,N,M))) active(filter(cons(X,Y),0,M)) -> mark(cons(0,filter(Y,M,M))) active(nats(N)) -> mark(cons(N,nats(s(N)))) active(sieve(cons(s(N),Y))) -> mark(cons(s(N),sieve(filter(Y,N,N)))) active(sieve(cons(0,Y))) -> mark(cons(0,sieve(Y))) active(zprimes) -> mark(sieve(nats(s(s(0))))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) filter(active(X1),X2,X3) -> filter(X1,X2,X3) filter(mark(X1),X2,X3) -> filter(X1,X2,X3) filter(X1,active(X2),X3) -> filter(X1,X2,X3) filter(X1,mark(X2),X3) -> filter(X1,X2,X3) filter(X1,X2,active(X3)) -> filter(X1,X2,X3) filter(X1,X2,mark(X3)) -> filter(X1,X2,X3) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(filter(X1,X2,X3)) -> active(filter(mark(X1),mark(X2),mark(X3))) mark(nats(X)) -> active(nats(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sieve(X)) -> active(sieve(mark(X))) mark(0) -> active(0) mark(zprimes) -> active(zprimes) nats(active(X)) -> nats(X) nats(mark(X)) -> nats(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) sieve(active(X)) -> sieve(X) sieve(mark(X)) -> sieve(X) ->Projection: pi(MARK) = 1 Problem 1.6: SCC Processor: -> Pairs: Empty -> Rules: active(filter(cons(X,Y),s(N),M)) -> mark(cons(X,filter(Y,N,M))) active(filter(cons(X,Y),0,M)) -> mark(cons(0,filter(Y,M,M))) active(nats(N)) -> mark(cons(N,nats(s(N)))) active(sieve(cons(s(N),Y))) -> mark(cons(s(N),sieve(filter(Y,N,N)))) active(sieve(cons(0,Y))) -> mark(cons(0,sieve(Y))) active(zprimes) -> mark(sieve(nats(s(s(0))))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) filter(active(X1),X2,X3) -> filter(X1,X2,X3) filter(mark(X1),X2,X3) -> filter(X1,X2,X3) filter(X1,active(X2),X3) -> filter(X1,X2,X3) filter(X1,mark(X2),X3) -> filter(X1,X2,X3) filter(X1,X2,active(X3)) -> filter(X1,X2,X3) filter(X1,X2,mark(X3)) -> filter(X1,X2,X3) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(filter(X1,X2,X3)) -> active(filter(mark(X1),mark(X2),mark(X3))) mark(nats(X)) -> active(nats(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sieve(X)) -> active(sieve(mark(X))) mark(0) -> active(0) mark(zprimes) -> active(zprimes) nats(active(X)) -> nats(X) nats(mark(X)) -> nats(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) sieve(active(X)) -> sieve(X) sieve(mark(X)) -> sieve(X) ->Strongly Connected Components: There is no strongly connected component The problem is finite.