/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(cons(X1,X2)) -> cons(active(X1),X2) 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(filter(X1,X2,X3)) -> filter(active(X1),X2,X3) active(filter(X1,X2,X3)) -> filter(X1,active(X2),X3) active(filter(X1,X2,X3)) -> filter(X1,X2,active(X3)) active(nats(N)) -> mark(cons(N,nats(s(N)))) active(nats(X)) -> nats(active(X)) active(s(X)) -> s(active(X)) 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(sieve(X)) -> sieve(active(X)) active(zprimes) -> mark(sieve(nats(s(s(0))))) cons(mark(X1),X2) -> mark(cons(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) filter(mark(X1),X2,X3) -> mark(filter(X1,X2,X3)) filter(ok(X1),ok(X2),ok(X3)) -> ok(filter(X1,X2,X3)) filter(X1,mark(X2),X3) -> mark(filter(X1,X2,X3)) filter(X1,X2,mark(X3)) -> mark(filter(X1,X2,X3)) nats(mark(X)) -> mark(nats(X)) nats(ok(X)) -> ok(nats(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(filter(X1,X2,X3)) -> filter(proper(X1),proper(X2),proper(X3)) proper(nats(X)) -> nats(proper(X)) proper(s(X)) -> s(proper(X)) proper(sieve(X)) -> sieve(proper(X)) proper(0) -> ok(0) proper(zprimes) -> ok(zprimes) s(mark(X)) -> mark(s(X)) s(ok(X)) -> ok(s(X)) sieve(mark(X)) -> mark(sieve(X)) sieve(ok(X)) -> ok(sieve(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) ) Problem 1: Dependency Pairs Processor: -> Pairs: ACTIVE(cons(X1,X2)) -> ACTIVE(X1) ACTIVE(cons(X1,X2)) -> CONS(active(X1),X2) 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),0,M)) -> CONS(0,filter(Y,M,M)) ACTIVE(filter(cons(X,Y),0,M)) -> FILTER(Y,M,M) ACTIVE(filter(X1,X2,X3)) -> ACTIVE(X1) ACTIVE(filter(X1,X2,X3)) -> ACTIVE(X2) ACTIVE(filter(X1,X2,X3)) -> ACTIVE(X3) ACTIVE(filter(X1,X2,X3)) -> FILTER(active(X1),X2,X3) ACTIVE(filter(X1,X2,X3)) -> FILTER(X1,active(X2),X3) ACTIVE(filter(X1,X2,X3)) -> FILTER(X1,X2,active(X3)) ACTIVE(nats(N)) -> CONS(N,nats(s(N))) ACTIVE(nats(N)) -> NATS(s(N)) ACTIVE(nats(N)) -> S(N) ACTIVE(nats(X)) -> ACTIVE(X) ACTIVE(nats(X)) -> NATS(active(X)) ACTIVE(s(X)) -> ACTIVE(X) ACTIVE(s(X)) -> S(active(X)) 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))) -> SIEVE(filter(Y,N,N)) ACTIVE(sieve(cons(0,Y))) -> CONS(0,sieve(Y)) ACTIVE(sieve(cons(0,Y))) -> SIEVE(Y) ACTIVE(sieve(X)) -> ACTIVE(X) ACTIVE(sieve(X)) -> SIEVE(active(X)) CONS(mark(X1),X2) -> CONS(X1,X2) CONS(ok(X1),ok(X2)) -> CONS(X1,X2) FILTER(mark(X1),X2,X3) -> FILTER(X1,X2,X3) FILTER(ok(X1),ok(X2),ok(X3)) -> FILTER(X1,X2,X3) FILTER(X1,mark(X2),X3) -> FILTER(X1,X2,X3) FILTER(X1,X2,mark(X3)) -> FILTER(X1,X2,X3) NATS(mark(X)) -> NATS(X) NATS(ok(X)) -> NATS(X) PROPER(cons(X1,X2)) -> CONS(proper(X1),proper(X2)) PROPER(cons(X1,X2)) -> PROPER(X1) PROPER(cons(X1,X2)) -> PROPER(X2) PROPER(filter(X1,X2,X3)) -> FILTER(proper(X1),proper(X2),proper(X3)) PROPER(filter(X1,X2,X3)) -> PROPER(X1) PROPER(filter(X1,X2,X3)) -> PROPER(X2) PROPER(filter(X1,X2,X3)) -> PROPER(X3) PROPER(nats(X)) -> NATS(proper(X)) PROPER(nats(X)) -> PROPER(X) PROPER(s(X)) -> PROPER(X) PROPER(s(X)) -> S(proper(X)) PROPER(sieve(X)) -> PROPER(X) PROPER(sieve(X)) -> SIEVE(proper(X)) S(mark(X)) -> S(X) S(ok(X)) -> S(X) SIEVE(mark(X)) -> SIEVE(X) SIEVE(ok(X)) -> SIEVE(X) TOP(mark(X)) -> PROPER(X) TOP(mark(X)) -> TOP(proper(X)) TOP(ok(X)) -> ACTIVE(X) TOP(ok(X)) -> TOP(active(X)) -> Rules: active(cons(X1,X2)) -> cons(active(X1),X2) 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(filter(X1,X2,X3)) -> filter(active(X1),X2,X3) active(filter(X1,X2,X3)) -> filter(X1,active(X2),X3) active(filter(X1,X2,X3)) -> filter(X1,X2,active(X3)) active(nats(N)) -> mark(cons(N,nats(s(N)))) active(nats(X)) -> nats(active(X)) active(s(X)) -> s(active(X)) 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(sieve(X)) -> sieve(active(X)) active(zprimes) -> mark(sieve(nats(s(s(0))))) cons(mark(X1),X2) -> mark(cons(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) filter(mark(X1),X2,X3) -> mark(filter(X1,X2,X3)) filter(ok(X1),ok(X2),ok(X3)) -> ok(filter(X1,X2,X3)) filter(X1,mark(X2),X3) -> mark(filter(X1,X2,X3)) filter(X1,X2,mark(X3)) -> mark(filter(X1,X2,X3)) nats(mark(X)) -> mark(nats(X)) nats(ok(X)) -> ok(nats(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(filter(X1,X2,X3)) -> filter(proper(X1),proper(X2),proper(X3)) proper(nats(X)) -> nats(proper(X)) proper(s(X)) -> s(proper(X)) proper(sieve(X)) -> sieve(proper(X)) proper(0) -> ok(0) proper(zprimes) -> ok(zprimes) s(mark(X)) -> mark(s(X)) s(ok(X)) -> ok(s(X)) sieve(mark(X)) -> mark(sieve(X)) sieve(ok(X)) -> ok(sieve(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Problem 1: SCC Processor: -> Pairs: ACTIVE(cons(X1,X2)) -> ACTIVE(X1) ACTIVE(cons(X1,X2)) -> CONS(active(X1),X2) 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),0,M)) -> CONS(0,filter(Y,M,M)) ACTIVE(filter(cons(X,Y),0,M)) -> FILTER(Y,M,M) ACTIVE(filter(X1,X2,X3)) -> ACTIVE(X1) ACTIVE(filter(X1,X2,X3)) -> ACTIVE(X2) ACTIVE(filter(X1,X2,X3)) -> ACTIVE(X3) ACTIVE(filter(X1,X2,X3)) -> FILTER(active(X1),X2,X3) ACTIVE(filter(X1,X2,X3)) -> FILTER(X1,active(X2),X3) ACTIVE(filter(X1,X2,X3)) -> FILTER(X1,X2,active(X3)) ACTIVE(nats(N)) -> CONS(N,nats(s(N))) ACTIVE(nats(N)) -> NATS(s(N)) ACTIVE(nats(N)) -> S(N) ACTIVE(nats(X)) -> ACTIVE(X) ACTIVE(nats(X)) -> NATS(active(X)) ACTIVE(s(X)) -> ACTIVE(X) ACTIVE(s(X)) -> S(active(X)) 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))) -> SIEVE(filter(Y,N,N)) ACTIVE(sieve(cons(0,Y))) -> CONS(0,sieve(Y)) ACTIVE(sieve(cons(0,Y))) -> SIEVE(Y) ACTIVE(sieve(X)) -> ACTIVE(X) ACTIVE(sieve(X)) -> SIEVE(active(X)) CONS(mark(X1),X2) -> CONS(X1,X2) CONS(ok(X1),ok(X2)) -> CONS(X1,X2) FILTER(mark(X1),X2,X3) -> FILTER(X1,X2,X3) FILTER(ok(X1),ok(X2),ok(X3)) -> FILTER(X1,X2,X3) FILTER(X1,mark(X2),X3) -> FILTER(X1,X2,X3) FILTER(X1,X2,mark(X3)) -> FILTER(X1,X2,X3) NATS(mark(X)) -> NATS(X) NATS(ok(X)) -> NATS(X) PROPER(cons(X1,X2)) -> CONS(proper(X1),proper(X2)) PROPER(cons(X1,X2)) -> PROPER(X1) PROPER(cons(X1,X2)) -> PROPER(X2) PROPER(filter(X1,X2,X3)) -> FILTER(proper(X1),proper(X2),proper(X3)) PROPER(filter(X1,X2,X3)) -> PROPER(X1) PROPER(filter(X1,X2,X3)) -> PROPER(X2) PROPER(filter(X1,X2,X3)) -> PROPER(X3) PROPER(nats(X)) -> NATS(proper(X)) PROPER(nats(X)) -> PROPER(X) PROPER(s(X)) -> PROPER(X) PROPER(s(X)) -> S(proper(X)) PROPER(sieve(X)) -> PROPER(X) PROPER(sieve(X)) -> SIEVE(proper(X)) S(mark(X)) -> S(X) S(ok(X)) -> S(X) SIEVE(mark(X)) -> SIEVE(X) SIEVE(ok(X)) -> SIEVE(X) TOP(mark(X)) -> PROPER(X) TOP(mark(X)) -> TOP(proper(X)) TOP(ok(X)) -> ACTIVE(X) TOP(ok(X)) -> TOP(active(X)) -> Rules: active(cons(X1,X2)) -> cons(active(X1),X2) 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(filter(X1,X2,X3)) -> filter(active(X1),X2,X3) active(filter(X1,X2,X3)) -> filter(X1,active(X2),X3) active(filter(X1,X2,X3)) -> filter(X1,X2,active(X3)) active(nats(N)) -> mark(cons(N,nats(s(N)))) active(nats(X)) -> nats(active(X)) active(s(X)) -> s(active(X)) 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(sieve(X)) -> sieve(active(X)) active(zprimes) -> mark(sieve(nats(s(s(0))))) cons(mark(X1),X2) -> mark(cons(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) filter(mark(X1),X2,X3) -> mark(filter(X1,X2,X3)) filter(ok(X1),ok(X2),ok(X3)) -> ok(filter(X1,X2,X3)) filter(X1,mark(X2),X3) -> mark(filter(X1,X2,X3)) filter(X1,X2,mark(X3)) -> mark(filter(X1,X2,X3)) nats(mark(X)) -> mark(nats(X)) nats(ok(X)) -> ok(nats(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(filter(X1,X2,X3)) -> filter(proper(X1),proper(X2),proper(X3)) proper(nats(X)) -> nats(proper(X)) proper(s(X)) -> s(proper(X)) proper(sieve(X)) -> sieve(proper(X)) proper(0) -> ok(0) proper(zprimes) -> ok(zprimes) s(mark(X)) -> mark(s(X)) s(ok(X)) -> ok(s(X)) sieve(mark(X)) -> mark(sieve(X)) sieve(ok(X)) -> ok(sieve(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: SIEVE(mark(X)) -> SIEVE(X) SIEVE(ok(X)) -> SIEVE(X) ->->-> Rules: active(cons(X1,X2)) -> cons(active(X1),X2) 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(filter(X1,X2,X3)) -> filter(active(X1),X2,X3) active(filter(X1,X2,X3)) -> filter(X1,active(X2),X3) active(filter(X1,X2,X3)) -> filter(X1,X2,active(X3)) active(nats(N)) -> mark(cons(N,nats(s(N)))) active(nats(X)) -> nats(active(X)) active(s(X)) -> s(active(X)) 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(sieve(X)) -> sieve(active(X)) active(zprimes) -> mark(sieve(nats(s(s(0))))) cons(mark(X1),X2) -> mark(cons(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) filter(mark(X1),X2,X3) -> mark(filter(X1,X2,X3)) filter(ok(X1),ok(X2),ok(X3)) -> ok(filter(X1,X2,X3)) filter(X1,mark(X2),X3) -> mark(filter(X1,X2,X3)) filter(X1,X2,mark(X3)) -> mark(filter(X1,X2,X3)) nats(mark(X)) -> mark(nats(X)) nats(ok(X)) -> ok(nats(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(filter(X1,X2,X3)) -> filter(proper(X1),proper(X2),proper(X3)) proper(nats(X)) -> nats(proper(X)) proper(s(X)) -> s(proper(X)) proper(sieve(X)) -> sieve(proper(X)) proper(0) -> ok(0) proper(zprimes) -> ok(zprimes) s(mark(X)) -> mark(s(X)) s(ok(X)) -> ok(s(X)) sieve(mark(X)) -> mark(sieve(X)) sieve(ok(X)) -> ok(sieve(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) ->->Cycle: ->->-> Pairs: S(mark(X)) -> S(X) S(ok(X)) -> S(X) ->->-> Rules: active(cons(X1,X2)) -> cons(active(X1),X2) 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(filter(X1,X2,X3)) -> filter(active(X1),X2,X3) active(filter(X1,X2,X3)) -> filter(X1,active(X2),X3) active(filter(X1,X2,X3)) -> filter(X1,X2,active(X3)) active(nats(N)) -> mark(cons(N,nats(s(N)))) active(nats(X)) -> nats(active(X)) active(s(X)) -> s(active(X)) 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(sieve(X)) -> sieve(active(X)) active(zprimes) -> mark(sieve(nats(s(s(0))))) cons(mark(X1),X2) -> mark(cons(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) filter(mark(X1),X2,X3) -> mark(filter(X1,X2,X3)) filter(ok(X1),ok(X2),ok(X3)) -> ok(filter(X1,X2,X3)) filter(X1,mark(X2),X3) -> mark(filter(X1,X2,X3)) filter(X1,X2,mark(X3)) -> mark(filter(X1,X2,X3)) nats(mark(X)) -> mark(nats(X)) nats(ok(X)) -> ok(nats(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(filter(X1,X2,X3)) -> filter(proper(X1),proper(X2),proper(X3)) proper(nats(X)) -> nats(proper(X)) proper(s(X)) -> s(proper(X)) proper(sieve(X)) -> sieve(proper(X)) proper(0) -> ok(0) proper(zprimes) -> ok(zprimes) s(mark(X)) -> mark(s(X)) s(ok(X)) -> ok(s(X)) sieve(mark(X)) -> mark(sieve(X)) sieve(ok(X)) -> ok(sieve(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) ->->Cycle: ->->-> Pairs: NATS(mark(X)) -> NATS(X) NATS(ok(X)) -> NATS(X) ->->-> Rules: active(cons(X1,X2)) -> cons(active(X1),X2) 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(filter(X1,X2,X3)) -> filter(active(X1),X2,X3) active(filter(X1,X2,X3)) -> filter(X1,active(X2),X3) active(filter(X1,X2,X3)) -> filter(X1,X2,active(X3)) active(nats(N)) -> mark(cons(N,nats(s(N)))) active(nats(X)) -> nats(active(X)) active(s(X)) -> s(active(X)) 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(sieve(X)) -> sieve(active(X)) active(zprimes) -> mark(sieve(nats(s(s(0))))) cons(mark(X1),X2) -> mark(cons(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) filter(mark(X1),X2,X3) -> mark(filter(X1,X2,X3)) filter(ok(X1),ok(X2),ok(X3)) -> ok(filter(X1,X2,X3)) filter(X1,mark(X2),X3) -> mark(filter(X1,X2,X3)) filter(X1,X2,mark(X3)) -> mark(filter(X1,X2,X3)) nats(mark(X)) -> mark(nats(X)) nats(ok(X)) -> ok(nats(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(filter(X1,X2,X3)) -> filter(proper(X1),proper(X2),proper(X3)) proper(nats(X)) -> nats(proper(X)) proper(s(X)) -> s(proper(X)) proper(sieve(X)) -> sieve(proper(X)) proper(0) -> ok(0) proper(zprimes) -> ok(zprimes) s(mark(X)) -> mark(s(X)) s(ok(X)) -> ok(s(X)) sieve(mark(X)) -> mark(sieve(X)) sieve(ok(X)) -> ok(sieve(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) ->->Cycle: ->->-> Pairs: FILTER(mark(X1),X2,X3) -> FILTER(X1,X2,X3) FILTER(ok(X1),ok(X2),ok(X3)) -> FILTER(X1,X2,X3) FILTER(X1,mark(X2),X3) -> FILTER(X1,X2,X3) FILTER(X1,X2,mark(X3)) -> FILTER(X1,X2,X3) ->->-> Rules: active(cons(X1,X2)) -> cons(active(X1),X2) 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(filter(X1,X2,X3)) -> filter(active(X1),X2,X3) active(filter(X1,X2,X3)) -> filter(X1,active(X2),X3) active(filter(X1,X2,X3)) -> filter(X1,X2,active(X3)) active(nats(N)) -> mark(cons(N,nats(s(N)))) active(nats(X)) -> nats(active(X)) active(s(X)) -> s(active(X)) 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(sieve(X)) -> sieve(active(X)) active(zprimes) -> mark(sieve(nats(s(s(0))))) cons(mark(X1),X2) -> mark(cons(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) filter(mark(X1),X2,X3) -> mark(filter(X1,X2,X3)) filter(ok(X1),ok(X2),ok(X3)) -> ok(filter(X1,X2,X3)) filter(X1,mark(X2),X3) -> mark(filter(X1,X2,X3)) filter(X1,X2,mark(X3)) -> mark(filter(X1,X2,X3)) nats(mark(X)) -> mark(nats(X)) nats(ok(X)) -> ok(nats(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(filter(X1,X2,X3)) -> filter(proper(X1),proper(X2),proper(X3)) proper(nats(X)) -> nats(proper(X)) proper(s(X)) -> s(proper(X)) proper(sieve(X)) -> sieve(proper(X)) proper(0) -> ok(0) proper(zprimes) -> ok(zprimes) s(mark(X)) -> mark(s(X)) s(ok(X)) -> ok(s(X)) sieve(mark(X)) -> mark(sieve(X)) sieve(ok(X)) -> ok(sieve(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) ->->Cycle: ->->-> Pairs: CONS(mark(X1),X2) -> CONS(X1,X2) CONS(ok(X1),ok(X2)) -> CONS(X1,X2) ->->-> Rules: active(cons(X1,X2)) -> cons(active(X1),X2) 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(filter(X1,X2,X3)) -> filter(active(X1),X2,X3) active(filter(X1,X2,X3)) -> filter(X1,active(X2),X3) active(filter(X1,X2,X3)) -> filter(X1,X2,active(X3)) active(nats(N)) -> mark(cons(N,nats(s(N)))) active(nats(X)) -> nats(active(X)) active(s(X)) -> s(active(X)) 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(sieve(X)) -> sieve(active(X)) active(zprimes) -> mark(sieve(nats(s(s(0))))) cons(mark(X1),X2) -> mark(cons(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) filter(mark(X1),X2,X3) -> mark(filter(X1,X2,X3)) filter(ok(X1),ok(X2),ok(X3)) -> ok(filter(X1,X2,X3)) filter(X1,mark(X2),X3) -> mark(filter(X1,X2,X3)) filter(X1,X2,mark(X3)) -> mark(filter(X1,X2,X3)) nats(mark(X)) -> mark(nats(X)) nats(ok(X)) -> ok(nats(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(filter(X1,X2,X3)) -> filter(proper(X1),proper(X2),proper(X3)) proper(nats(X)) -> nats(proper(X)) proper(s(X)) -> s(proper(X)) proper(sieve(X)) -> sieve(proper(X)) proper(0) -> ok(0) proper(zprimes) -> ok(zprimes) s(mark(X)) -> mark(s(X)) s(ok(X)) -> ok(s(X)) sieve(mark(X)) -> mark(sieve(X)) sieve(ok(X)) -> ok(sieve(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) ->->Cycle: ->->-> Pairs: PROPER(cons(X1,X2)) -> PROPER(X1) PROPER(cons(X1,X2)) -> PROPER(X2) PROPER(filter(X1,X2,X3)) -> PROPER(X1) PROPER(filter(X1,X2,X3)) -> PROPER(X2) PROPER(filter(X1,X2,X3)) -> PROPER(X3) PROPER(nats(X)) -> PROPER(X) PROPER(s(X)) -> PROPER(X) PROPER(sieve(X)) -> PROPER(X) ->->-> Rules: active(cons(X1,X2)) -> cons(active(X1),X2) 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(filter(X1,X2,X3)) -> filter(active(X1),X2,X3) active(filter(X1,X2,X3)) -> filter(X1,active(X2),X3) active(filter(X1,X2,X3)) -> filter(X1,X2,active(X3)) active(nats(N)) -> mark(cons(N,nats(s(N)))) active(nats(X)) -> nats(active(X)) active(s(X)) -> s(active(X)) 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(sieve(X)) -> sieve(active(X)) active(zprimes) -> mark(sieve(nats(s(s(0))))) cons(mark(X1),X2) -> mark(cons(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) filter(mark(X1),X2,X3) -> mark(filter(X1,X2,X3)) filter(ok(X1),ok(X2),ok(X3)) -> ok(filter(X1,X2,X3)) filter(X1,mark(X2),X3) -> mark(filter(X1,X2,X3)) filter(X1,X2,mark(X3)) -> mark(filter(X1,X2,X3)) nats(mark(X)) -> mark(nats(X)) nats(ok(X)) -> ok(nats(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(filter(X1,X2,X3)) -> filter(proper(X1),proper(X2),proper(X3)) proper(nats(X)) -> nats(proper(X)) proper(s(X)) -> s(proper(X)) proper(sieve(X)) -> sieve(proper(X)) proper(0) -> ok(0) proper(zprimes) -> ok(zprimes) s(mark(X)) -> mark(s(X)) s(ok(X)) -> ok(s(X)) sieve(mark(X)) -> mark(sieve(X)) sieve(ok(X)) -> ok(sieve(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) ->->Cycle: ->->-> Pairs: ACTIVE(cons(X1,X2)) -> ACTIVE(X1) ACTIVE(filter(X1,X2,X3)) -> ACTIVE(X1) ACTIVE(filter(X1,X2,X3)) -> ACTIVE(X2) ACTIVE(filter(X1,X2,X3)) -> ACTIVE(X3) ACTIVE(nats(X)) -> ACTIVE(X) ACTIVE(s(X)) -> ACTIVE(X) ACTIVE(sieve(X)) -> ACTIVE(X) ->->-> Rules: active(cons(X1,X2)) -> cons(active(X1),X2) 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(filter(X1,X2,X3)) -> filter(active(X1),X2,X3) active(filter(X1,X2,X3)) -> filter(X1,active(X2),X3) active(filter(X1,X2,X3)) -> filter(X1,X2,active(X3)) active(nats(N)) -> mark(cons(N,nats(s(N)))) active(nats(X)) -> nats(active(X)) active(s(X)) -> s(active(X)) 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(sieve(X)) -> sieve(active(X)) active(zprimes) -> mark(sieve(nats(s(s(0))))) cons(mark(X1),X2) -> mark(cons(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) filter(mark(X1),X2,X3) -> mark(filter(X1,X2,X3)) filter(ok(X1),ok(X2),ok(X3)) -> ok(filter(X1,X2,X3)) filter(X1,mark(X2),X3) -> mark(filter(X1,X2,X3)) filter(X1,X2,mark(X3)) -> mark(filter(X1,X2,X3)) nats(mark(X)) -> mark(nats(X)) nats(ok(X)) -> ok(nats(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(filter(X1,X2,X3)) -> filter(proper(X1),proper(X2),proper(X3)) proper(nats(X)) -> nats(proper(X)) proper(s(X)) -> s(proper(X)) proper(sieve(X)) -> sieve(proper(X)) proper(0) -> ok(0) proper(zprimes) -> ok(zprimes) s(mark(X)) -> mark(s(X)) s(ok(X)) -> ok(s(X)) sieve(mark(X)) -> mark(sieve(X)) sieve(ok(X)) -> ok(sieve(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) ->->Cycle: ->->-> Pairs: TOP(mark(X)) -> TOP(proper(X)) TOP(ok(X)) -> TOP(active(X)) ->->-> Rules: active(cons(X1,X2)) -> cons(active(X1),X2) 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(filter(X1,X2,X3)) -> filter(active(X1),X2,X3) active(filter(X1,X2,X3)) -> filter(X1,active(X2),X3) active(filter(X1,X2,X3)) -> filter(X1,X2,active(X3)) active(nats(N)) -> mark(cons(N,nats(s(N)))) active(nats(X)) -> nats(active(X)) active(s(X)) -> s(active(X)) 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(sieve(X)) -> sieve(active(X)) active(zprimes) -> mark(sieve(nats(s(s(0))))) cons(mark(X1),X2) -> mark(cons(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) filter(mark(X1),X2,X3) -> mark(filter(X1,X2,X3)) filter(ok(X1),ok(X2),ok(X3)) -> ok(filter(X1,X2,X3)) filter(X1,mark(X2),X3) -> mark(filter(X1,X2,X3)) filter(X1,X2,mark(X3)) -> mark(filter(X1,X2,X3)) nats(mark(X)) -> mark(nats(X)) nats(ok(X)) -> ok(nats(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(filter(X1,X2,X3)) -> filter(proper(X1),proper(X2),proper(X3)) proper(nats(X)) -> nats(proper(X)) proper(s(X)) -> s(proper(X)) proper(sieve(X)) -> sieve(proper(X)) proper(0) -> ok(0) proper(zprimes) -> ok(zprimes) s(mark(X)) -> mark(s(X)) s(ok(X)) -> ok(s(X)) sieve(mark(X)) -> mark(sieve(X)) sieve(ok(X)) -> ok(sieve(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) The problem is decomposed in 8 subproblems. Problem 1.1: Subterm Processor: -> Pairs: SIEVE(mark(X)) -> SIEVE(X) SIEVE(ok(X)) -> SIEVE(X) -> Rules: active(cons(X1,X2)) -> cons(active(X1),X2) 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(filter(X1,X2,X3)) -> filter(active(X1),X2,X3) active(filter(X1,X2,X3)) -> filter(X1,active(X2),X3) active(filter(X1,X2,X3)) -> filter(X1,X2,active(X3)) active(nats(N)) -> mark(cons(N,nats(s(N)))) active(nats(X)) -> nats(active(X)) active(s(X)) -> s(active(X)) 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(sieve(X)) -> sieve(active(X)) active(zprimes) -> mark(sieve(nats(s(s(0))))) cons(mark(X1),X2) -> mark(cons(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) filter(mark(X1),X2,X3) -> mark(filter(X1,X2,X3)) filter(ok(X1),ok(X2),ok(X3)) -> ok(filter(X1,X2,X3)) filter(X1,mark(X2),X3) -> mark(filter(X1,X2,X3)) filter(X1,X2,mark(X3)) -> mark(filter(X1,X2,X3)) nats(mark(X)) -> mark(nats(X)) nats(ok(X)) -> ok(nats(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(filter(X1,X2,X3)) -> filter(proper(X1),proper(X2),proper(X3)) proper(nats(X)) -> nats(proper(X)) proper(s(X)) -> s(proper(X)) proper(sieve(X)) -> sieve(proper(X)) proper(0) -> ok(0) proper(zprimes) -> ok(zprimes) s(mark(X)) -> mark(s(X)) s(ok(X)) -> ok(s(X)) sieve(mark(X)) -> mark(sieve(X)) sieve(ok(X)) -> ok(sieve(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) ->Projection: pi(SIEVE) = 1 Problem 1.1: SCC Processor: -> Pairs: Empty -> Rules: active(cons(X1,X2)) -> cons(active(X1),X2) 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(filter(X1,X2,X3)) -> filter(active(X1),X2,X3) active(filter(X1,X2,X3)) -> filter(X1,active(X2),X3) active(filter(X1,X2,X3)) -> filter(X1,X2,active(X3)) active(nats(N)) -> mark(cons(N,nats(s(N)))) active(nats(X)) -> nats(active(X)) active(s(X)) -> s(active(X)) 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(sieve(X)) -> sieve(active(X)) active(zprimes) -> mark(sieve(nats(s(s(0))))) cons(mark(X1),X2) -> mark(cons(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) filter(mark(X1),X2,X3) -> mark(filter(X1,X2,X3)) filter(ok(X1),ok(X2),ok(X3)) -> ok(filter(X1,X2,X3)) filter(X1,mark(X2),X3) -> mark(filter(X1,X2,X3)) filter(X1,X2,mark(X3)) -> mark(filter(X1,X2,X3)) nats(mark(X)) -> mark(nats(X)) nats(ok(X)) -> ok(nats(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(filter(X1,X2,X3)) -> filter(proper(X1),proper(X2),proper(X3)) proper(nats(X)) -> nats(proper(X)) proper(s(X)) -> s(proper(X)) proper(sieve(X)) -> sieve(proper(X)) proper(0) -> ok(0) proper(zprimes) -> ok(zprimes) s(mark(X)) -> mark(s(X)) s(ok(X)) -> ok(s(X)) sieve(mark(X)) -> mark(sieve(X)) sieve(ok(X)) -> ok(sieve(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.2: Subterm Processor: -> Pairs: S(mark(X)) -> S(X) S(ok(X)) -> S(X) -> Rules: active(cons(X1,X2)) -> cons(active(X1),X2) 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(filter(X1,X2,X3)) -> filter(active(X1),X2,X3) active(filter(X1,X2,X3)) -> filter(X1,active(X2),X3) active(filter(X1,X2,X3)) -> filter(X1,X2,active(X3)) active(nats(N)) -> mark(cons(N,nats(s(N)))) active(nats(X)) -> nats(active(X)) active(s(X)) -> s(active(X)) 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(sieve(X)) -> sieve(active(X)) active(zprimes) -> mark(sieve(nats(s(s(0))))) cons(mark(X1),X2) -> mark(cons(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) filter(mark(X1),X2,X3) -> mark(filter(X1,X2,X3)) filter(ok(X1),ok(X2),ok(X3)) -> ok(filter(X1,X2,X3)) filter(X1,mark(X2),X3) -> mark(filter(X1,X2,X3)) filter(X1,X2,mark(X3)) -> mark(filter(X1,X2,X3)) nats(mark(X)) -> mark(nats(X)) nats(ok(X)) -> ok(nats(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(filter(X1,X2,X3)) -> filter(proper(X1),proper(X2),proper(X3)) proper(nats(X)) -> nats(proper(X)) proper(s(X)) -> s(proper(X)) proper(sieve(X)) -> sieve(proper(X)) proper(0) -> ok(0) proper(zprimes) -> ok(zprimes) s(mark(X)) -> mark(s(X)) s(ok(X)) -> ok(s(X)) sieve(mark(X)) -> mark(sieve(X)) sieve(ok(X)) -> ok(sieve(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) ->Projection: pi(S) = 1 Problem 1.2: SCC Processor: -> Pairs: Empty -> Rules: active(cons(X1,X2)) -> cons(active(X1),X2) 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(filter(X1,X2,X3)) -> filter(active(X1),X2,X3) active(filter(X1,X2,X3)) -> filter(X1,active(X2),X3) active(filter(X1,X2,X3)) -> filter(X1,X2,active(X3)) active(nats(N)) -> mark(cons(N,nats(s(N)))) active(nats(X)) -> nats(active(X)) active(s(X)) -> s(active(X)) 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(sieve(X)) -> sieve(active(X)) active(zprimes) -> mark(sieve(nats(s(s(0))))) cons(mark(X1),X2) -> mark(cons(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) filter(mark(X1),X2,X3) -> mark(filter(X1,X2,X3)) filter(ok(X1),ok(X2),ok(X3)) -> ok(filter(X1,X2,X3)) filter(X1,mark(X2),X3) -> mark(filter(X1,X2,X3)) filter(X1,X2,mark(X3)) -> mark(filter(X1,X2,X3)) nats(mark(X)) -> mark(nats(X)) nats(ok(X)) -> ok(nats(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(filter(X1,X2,X3)) -> filter(proper(X1),proper(X2),proper(X3)) proper(nats(X)) -> nats(proper(X)) proper(s(X)) -> s(proper(X)) proper(sieve(X)) -> sieve(proper(X)) proper(0) -> ok(0) proper(zprimes) -> ok(zprimes) s(mark(X)) -> mark(s(X)) s(ok(X)) -> ok(s(X)) sieve(mark(X)) -> mark(sieve(X)) sieve(ok(X)) -> ok(sieve(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.3: Subterm Processor: -> Pairs: NATS(mark(X)) -> NATS(X) NATS(ok(X)) -> NATS(X) -> Rules: active(cons(X1,X2)) -> cons(active(X1),X2) 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(filter(X1,X2,X3)) -> filter(active(X1),X2,X3) active(filter(X1,X2,X3)) -> filter(X1,active(X2),X3) active(filter(X1,X2,X3)) -> filter(X1,X2,active(X3)) active(nats(N)) -> mark(cons(N,nats(s(N)))) active(nats(X)) -> nats(active(X)) active(s(X)) -> s(active(X)) 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(sieve(X)) -> sieve(active(X)) active(zprimes) -> mark(sieve(nats(s(s(0))))) cons(mark(X1),X2) -> mark(cons(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) filter(mark(X1),X2,X3) -> mark(filter(X1,X2,X3)) filter(ok(X1),ok(X2),ok(X3)) -> ok(filter(X1,X2,X3)) filter(X1,mark(X2),X3) -> mark(filter(X1,X2,X3)) filter(X1,X2,mark(X3)) -> mark(filter(X1,X2,X3)) nats(mark(X)) -> mark(nats(X)) nats(ok(X)) -> ok(nats(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(filter(X1,X2,X3)) -> filter(proper(X1),proper(X2),proper(X3)) proper(nats(X)) -> nats(proper(X)) proper(s(X)) -> s(proper(X)) proper(sieve(X)) -> sieve(proper(X)) proper(0) -> ok(0) proper(zprimes) -> ok(zprimes) s(mark(X)) -> mark(s(X)) s(ok(X)) -> ok(s(X)) sieve(mark(X)) -> mark(sieve(X)) sieve(ok(X)) -> ok(sieve(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) ->Projection: pi(NATS) = 1 Problem 1.3: SCC Processor: -> Pairs: Empty -> Rules: active(cons(X1,X2)) -> cons(active(X1),X2) 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(filter(X1,X2,X3)) -> filter(active(X1),X2,X3) active(filter(X1,X2,X3)) -> filter(X1,active(X2),X3) active(filter(X1,X2,X3)) -> filter(X1,X2,active(X3)) active(nats(N)) -> mark(cons(N,nats(s(N)))) active(nats(X)) -> nats(active(X)) active(s(X)) -> s(active(X)) 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(sieve(X)) -> sieve(active(X)) active(zprimes) -> mark(sieve(nats(s(s(0))))) cons(mark(X1),X2) -> mark(cons(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) filter(mark(X1),X2,X3) -> mark(filter(X1,X2,X3)) filter(ok(X1),ok(X2),ok(X3)) -> ok(filter(X1,X2,X3)) filter(X1,mark(X2),X3) -> mark(filter(X1,X2,X3)) filter(X1,X2,mark(X3)) -> mark(filter(X1,X2,X3)) nats(mark(X)) -> mark(nats(X)) nats(ok(X)) -> ok(nats(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(filter(X1,X2,X3)) -> filter(proper(X1),proper(X2),proper(X3)) proper(nats(X)) -> nats(proper(X)) proper(s(X)) -> s(proper(X)) proper(sieve(X)) -> sieve(proper(X)) proper(0) -> ok(0) proper(zprimes) -> ok(zprimes) s(mark(X)) -> mark(s(X)) s(ok(X)) -> ok(s(X)) sieve(mark(X)) -> mark(sieve(X)) sieve(ok(X)) -> ok(sieve(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.4: Subterm Processor: -> Pairs: FILTER(mark(X1),X2,X3) -> FILTER(X1,X2,X3) FILTER(ok(X1),ok(X2),ok(X3)) -> FILTER(X1,X2,X3) FILTER(X1,mark(X2),X3) -> FILTER(X1,X2,X3) FILTER(X1,X2,mark(X3)) -> FILTER(X1,X2,X3) -> Rules: active(cons(X1,X2)) -> cons(active(X1),X2) 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(filter(X1,X2,X3)) -> filter(active(X1),X2,X3) active(filter(X1,X2,X3)) -> filter(X1,active(X2),X3) active(filter(X1,X2,X3)) -> filter(X1,X2,active(X3)) active(nats(N)) -> mark(cons(N,nats(s(N)))) active(nats(X)) -> nats(active(X)) active(s(X)) -> s(active(X)) 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(sieve(X)) -> sieve(active(X)) active(zprimes) -> mark(sieve(nats(s(s(0))))) cons(mark(X1),X2) -> mark(cons(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) filter(mark(X1),X2,X3) -> mark(filter(X1,X2,X3)) filter(ok(X1),ok(X2),ok(X3)) -> ok(filter(X1,X2,X3)) filter(X1,mark(X2),X3) -> mark(filter(X1,X2,X3)) filter(X1,X2,mark(X3)) -> mark(filter(X1,X2,X3)) nats(mark(X)) -> mark(nats(X)) nats(ok(X)) -> ok(nats(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(filter(X1,X2,X3)) -> filter(proper(X1),proper(X2),proper(X3)) proper(nats(X)) -> nats(proper(X)) proper(s(X)) -> s(proper(X)) proper(sieve(X)) -> sieve(proper(X)) proper(0) -> ok(0) proper(zprimes) -> ok(zprimes) s(mark(X)) -> mark(s(X)) s(ok(X)) -> ok(s(X)) sieve(mark(X)) -> mark(sieve(X)) sieve(ok(X)) -> ok(sieve(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) ->Projection: pi(FILTER) = 1 Problem 1.4: SCC Processor: -> Pairs: FILTER(X1,mark(X2),X3) -> FILTER(X1,X2,X3) FILTER(X1,X2,mark(X3)) -> FILTER(X1,X2,X3) -> Rules: active(cons(X1,X2)) -> cons(active(X1),X2) 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(filter(X1,X2,X3)) -> filter(active(X1),X2,X3) active(filter(X1,X2,X3)) -> filter(X1,active(X2),X3) active(filter(X1,X2,X3)) -> filter(X1,X2,active(X3)) active(nats(N)) -> mark(cons(N,nats(s(N)))) active(nats(X)) -> nats(active(X)) active(s(X)) -> s(active(X)) 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(sieve(X)) -> sieve(active(X)) active(zprimes) -> mark(sieve(nats(s(s(0))))) cons(mark(X1),X2) -> mark(cons(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) filter(mark(X1),X2,X3) -> mark(filter(X1,X2,X3)) filter(ok(X1),ok(X2),ok(X3)) -> ok(filter(X1,X2,X3)) filter(X1,mark(X2),X3) -> mark(filter(X1,X2,X3)) filter(X1,X2,mark(X3)) -> mark(filter(X1,X2,X3)) nats(mark(X)) -> mark(nats(X)) nats(ok(X)) -> ok(nats(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(filter(X1,X2,X3)) -> filter(proper(X1),proper(X2),proper(X3)) proper(nats(X)) -> nats(proper(X)) proper(s(X)) -> s(proper(X)) proper(sieve(X)) -> sieve(proper(X)) proper(0) -> ok(0) proper(zprimes) -> ok(zprimes) s(mark(X)) -> mark(s(X)) s(ok(X)) -> ok(s(X)) sieve(mark(X)) -> mark(sieve(X)) sieve(ok(X)) -> ok(sieve(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: FILTER(X1,mark(X2),X3) -> FILTER(X1,X2,X3) FILTER(X1,X2,mark(X3)) -> FILTER(X1,X2,X3) ->->-> Rules: active(cons(X1,X2)) -> cons(active(X1),X2) 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(filter(X1,X2,X3)) -> filter(active(X1),X2,X3) active(filter(X1,X2,X3)) -> filter(X1,active(X2),X3) active(filter(X1,X2,X3)) -> filter(X1,X2,active(X3)) active(nats(N)) -> mark(cons(N,nats(s(N)))) active(nats(X)) -> nats(active(X)) active(s(X)) -> s(active(X)) 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(sieve(X)) -> sieve(active(X)) active(zprimes) -> mark(sieve(nats(s(s(0))))) cons(mark(X1),X2) -> mark(cons(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) filter(mark(X1),X2,X3) -> mark(filter(X1,X2,X3)) filter(ok(X1),ok(X2),ok(X3)) -> ok(filter(X1,X2,X3)) filter(X1,mark(X2),X3) -> mark(filter(X1,X2,X3)) filter(X1,X2,mark(X3)) -> mark(filter(X1,X2,X3)) nats(mark(X)) -> mark(nats(X)) nats(ok(X)) -> ok(nats(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(filter(X1,X2,X3)) -> filter(proper(X1),proper(X2),proper(X3)) proper(nats(X)) -> nats(proper(X)) proper(s(X)) -> s(proper(X)) proper(sieve(X)) -> sieve(proper(X)) proper(0) -> ok(0) proper(zprimes) -> ok(zprimes) s(mark(X)) -> mark(s(X)) s(ok(X)) -> ok(s(X)) sieve(mark(X)) -> mark(sieve(X)) sieve(ok(X)) -> ok(sieve(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Problem 1.4: Subterm Processor: -> Pairs: FILTER(X1,mark(X2),X3) -> FILTER(X1,X2,X3) FILTER(X1,X2,mark(X3)) -> FILTER(X1,X2,X3) -> Rules: active(cons(X1,X2)) -> cons(active(X1),X2) 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(filter(X1,X2,X3)) -> filter(active(X1),X2,X3) active(filter(X1,X2,X3)) -> filter(X1,active(X2),X3) active(filter(X1,X2,X3)) -> filter(X1,X2,active(X3)) active(nats(N)) -> mark(cons(N,nats(s(N)))) active(nats(X)) -> nats(active(X)) active(s(X)) -> s(active(X)) 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(sieve(X)) -> sieve(active(X)) active(zprimes) -> mark(sieve(nats(s(s(0))))) cons(mark(X1),X2) -> mark(cons(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) filter(mark(X1),X2,X3) -> mark(filter(X1,X2,X3)) filter(ok(X1),ok(X2),ok(X3)) -> ok(filter(X1,X2,X3)) filter(X1,mark(X2),X3) -> mark(filter(X1,X2,X3)) filter(X1,X2,mark(X3)) -> mark(filter(X1,X2,X3)) nats(mark(X)) -> mark(nats(X)) nats(ok(X)) -> ok(nats(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(filter(X1,X2,X3)) -> filter(proper(X1),proper(X2),proper(X3)) proper(nats(X)) -> nats(proper(X)) proper(s(X)) -> s(proper(X)) proper(sieve(X)) -> sieve(proper(X)) proper(0) -> ok(0) proper(zprimes) -> ok(zprimes) s(mark(X)) -> mark(s(X)) s(ok(X)) -> ok(s(X)) sieve(mark(X)) -> mark(sieve(X)) sieve(ok(X)) -> ok(sieve(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) ->Projection: pi(FILTER) = 2 Problem 1.4: SCC Processor: -> Pairs: FILTER(X1,X2,mark(X3)) -> FILTER(X1,X2,X3) -> Rules: active(cons(X1,X2)) -> cons(active(X1),X2) 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(filter(X1,X2,X3)) -> filter(active(X1),X2,X3) active(filter(X1,X2,X3)) -> filter(X1,active(X2),X3) active(filter(X1,X2,X3)) -> filter(X1,X2,active(X3)) active(nats(N)) -> mark(cons(N,nats(s(N)))) active(nats(X)) -> nats(active(X)) active(s(X)) -> s(active(X)) 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(sieve(X)) -> sieve(active(X)) active(zprimes) -> mark(sieve(nats(s(s(0))))) cons(mark(X1),X2) -> mark(cons(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) filter(mark(X1),X2,X3) -> mark(filter(X1,X2,X3)) filter(ok(X1),ok(X2),ok(X3)) -> ok(filter(X1,X2,X3)) filter(X1,mark(X2),X3) -> mark(filter(X1,X2,X3)) filter(X1,X2,mark(X3)) -> mark(filter(X1,X2,X3)) nats(mark(X)) -> mark(nats(X)) nats(ok(X)) -> ok(nats(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(filter(X1,X2,X3)) -> filter(proper(X1),proper(X2),proper(X3)) proper(nats(X)) -> nats(proper(X)) proper(s(X)) -> s(proper(X)) proper(sieve(X)) -> sieve(proper(X)) proper(0) -> ok(0) proper(zprimes) -> ok(zprimes) s(mark(X)) -> mark(s(X)) s(ok(X)) -> ok(s(X)) sieve(mark(X)) -> mark(sieve(X)) sieve(ok(X)) -> ok(sieve(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: FILTER(X1,X2,mark(X3)) -> FILTER(X1,X2,X3) ->->-> Rules: active(cons(X1,X2)) -> cons(active(X1),X2) 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(filter(X1,X2,X3)) -> filter(active(X1),X2,X3) active(filter(X1,X2,X3)) -> filter(X1,active(X2),X3) active(filter(X1,X2,X3)) -> filter(X1,X2,active(X3)) active(nats(N)) -> mark(cons(N,nats(s(N)))) active(nats(X)) -> nats(active(X)) active(s(X)) -> s(active(X)) 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(sieve(X)) -> sieve(active(X)) active(zprimes) -> mark(sieve(nats(s(s(0))))) cons(mark(X1),X2) -> mark(cons(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) filter(mark(X1),X2,X3) -> mark(filter(X1,X2,X3)) filter(ok(X1),ok(X2),ok(X3)) -> ok(filter(X1,X2,X3)) filter(X1,mark(X2),X3) -> mark(filter(X1,X2,X3)) filter(X1,X2,mark(X3)) -> mark(filter(X1,X2,X3)) nats(mark(X)) -> mark(nats(X)) nats(ok(X)) -> ok(nats(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(filter(X1,X2,X3)) -> filter(proper(X1),proper(X2),proper(X3)) proper(nats(X)) -> nats(proper(X)) proper(s(X)) -> s(proper(X)) proper(sieve(X)) -> sieve(proper(X)) proper(0) -> ok(0) proper(zprimes) -> ok(zprimes) s(mark(X)) -> mark(s(X)) s(ok(X)) -> ok(s(X)) sieve(mark(X)) -> mark(sieve(X)) sieve(ok(X)) -> ok(sieve(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Problem 1.4: Subterm Processor: -> Pairs: FILTER(X1,X2,mark(X3)) -> FILTER(X1,X2,X3) -> Rules: active(cons(X1,X2)) -> cons(active(X1),X2) 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(filter(X1,X2,X3)) -> filter(active(X1),X2,X3) active(filter(X1,X2,X3)) -> filter(X1,active(X2),X3) active(filter(X1,X2,X3)) -> filter(X1,X2,active(X3)) active(nats(N)) -> mark(cons(N,nats(s(N)))) active(nats(X)) -> nats(active(X)) active(s(X)) -> s(active(X)) 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(sieve(X)) -> sieve(active(X)) active(zprimes) -> mark(sieve(nats(s(s(0))))) cons(mark(X1),X2) -> mark(cons(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) filter(mark(X1),X2,X3) -> mark(filter(X1,X2,X3)) filter(ok(X1),ok(X2),ok(X3)) -> ok(filter(X1,X2,X3)) filter(X1,mark(X2),X3) -> mark(filter(X1,X2,X3)) filter(X1,X2,mark(X3)) -> mark(filter(X1,X2,X3)) nats(mark(X)) -> mark(nats(X)) nats(ok(X)) -> ok(nats(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(filter(X1,X2,X3)) -> filter(proper(X1),proper(X2),proper(X3)) proper(nats(X)) -> nats(proper(X)) proper(s(X)) -> s(proper(X)) proper(sieve(X)) -> sieve(proper(X)) proper(0) -> ok(0) proper(zprimes) -> ok(zprimes) s(mark(X)) -> mark(s(X)) s(ok(X)) -> ok(s(X)) sieve(mark(X)) -> mark(sieve(X)) sieve(ok(X)) -> ok(sieve(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) ->Projection: pi(FILTER) = 3 Problem 1.4: SCC Processor: -> Pairs: Empty -> Rules: active(cons(X1,X2)) -> cons(active(X1),X2) 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(filter(X1,X2,X3)) -> filter(active(X1),X2,X3) active(filter(X1,X2,X3)) -> filter(X1,active(X2),X3) active(filter(X1,X2,X3)) -> filter(X1,X2,active(X3)) active(nats(N)) -> mark(cons(N,nats(s(N)))) active(nats(X)) -> nats(active(X)) active(s(X)) -> s(active(X)) 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(sieve(X)) -> sieve(active(X)) active(zprimes) -> mark(sieve(nats(s(s(0))))) cons(mark(X1),X2) -> mark(cons(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) filter(mark(X1),X2,X3) -> mark(filter(X1,X2,X3)) filter(ok(X1),ok(X2),ok(X3)) -> ok(filter(X1,X2,X3)) filter(X1,mark(X2),X3) -> mark(filter(X1,X2,X3)) filter(X1,X2,mark(X3)) -> mark(filter(X1,X2,X3)) nats(mark(X)) -> mark(nats(X)) nats(ok(X)) -> ok(nats(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(filter(X1,X2,X3)) -> filter(proper(X1),proper(X2),proper(X3)) proper(nats(X)) -> nats(proper(X)) proper(s(X)) -> s(proper(X)) proper(sieve(X)) -> sieve(proper(X)) proper(0) -> ok(0) proper(zprimes) -> ok(zprimes) s(mark(X)) -> mark(s(X)) s(ok(X)) -> ok(s(X)) sieve(mark(X)) -> mark(sieve(X)) sieve(ok(X)) -> ok(sieve(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.5: Subterm Processor: -> Pairs: CONS(mark(X1),X2) -> CONS(X1,X2) CONS(ok(X1),ok(X2)) -> CONS(X1,X2) -> Rules: active(cons(X1,X2)) -> cons(active(X1),X2) 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(filter(X1,X2,X3)) -> filter(active(X1),X2,X3) active(filter(X1,X2,X3)) -> filter(X1,active(X2),X3) active(filter(X1,X2,X3)) -> filter(X1,X2,active(X3)) active(nats(N)) -> mark(cons(N,nats(s(N)))) active(nats(X)) -> nats(active(X)) active(s(X)) -> s(active(X)) 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(sieve(X)) -> sieve(active(X)) active(zprimes) -> mark(sieve(nats(s(s(0))))) cons(mark(X1),X2) -> mark(cons(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) filter(mark(X1),X2,X3) -> mark(filter(X1,X2,X3)) filter(ok(X1),ok(X2),ok(X3)) -> ok(filter(X1,X2,X3)) filter(X1,mark(X2),X3) -> mark(filter(X1,X2,X3)) filter(X1,X2,mark(X3)) -> mark(filter(X1,X2,X3)) nats(mark(X)) -> mark(nats(X)) nats(ok(X)) -> ok(nats(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(filter(X1,X2,X3)) -> filter(proper(X1),proper(X2),proper(X3)) proper(nats(X)) -> nats(proper(X)) proper(s(X)) -> s(proper(X)) proper(sieve(X)) -> sieve(proper(X)) proper(0) -> ok(0) proper(zprimes) -> ok(zprimes) s(mark(X)) -> mark(s(X)) s(ok(X)) -> ok(s(X)) sieve(mark(X)) -> mark(sieve(X)) sieve(ok(X)) -> ok(sieve(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) ->Projection: pi(CONS) = 1 Problem 1.5: SCC Processor: -> Pairs: Empty -> Rules: active(cons(X1,X2)) -> cons(active(X1),X2) 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(filter(X1,X2,X3)) -> filter(active(X1),X2,X3) active(filter(X1,X2,X3)) -> filter(X1,active(X2),X3) active(filter(X1,X2,X3)) -> filter(X1,X2,active(X3)) active(nats(N)) -> mark(cons(N,nats(s(N)))) active(nats(X)) -> nats(active(X)) active(s(X)) -> s(active(X)) 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(sieve(X)) -> sieve(active(X)) active(zprimes) -> mark(sieve(nats(s(s(0))))) cons(mark(X1),X2) -> mark(cons(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) filter(mark(X1),X2,X3) -> mark(filter(X1,X2,X3)) filter(ok(X1),ok(X2),ok(X3)) -> ok(filter(X1,X2,X3)) filter(X1,mark(X2),X3) -> mark(filter(X1,X2,X3)) filter(X1,X2,mark(X3)) -> mark(filter(X1,X2,X3)) nats(mark(X)) -> mark(nats(X)) nats(ok(X)) -> ok(nats(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(filter(X1,X2,X3)) -> filter(proper(X1),proper(X2),proper(X3)) proper(nats(X)) -> nats(proper(X)) proper(s(X)) -> s(proper(X)) proper(sieve(X)) -> sieve(proper(X)) proper(0) -> ok(0) proper(zprimes) -> ok(zprimes) s(mark(X)) -> mark(s(X)) s(ok(X)) -> ok(s(X)) sieve(mark(X)) -> mark(sieve(X)) sieve(ok(X)) -> ok(sieve(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.6: Subterm Processor: -> Pairs: PROPER(cons(X1,X2)) -> PROPER(X1) PROPER(cons(X1,X2)) -> PROPER(X2) PROPER(filter(X1,X2,X3)) -> PROPER(X1) PROPER(filter(X1,X2,X3)) -> PROPER(X2) PROPER(filter(X1,X2,X3)) -> PROPER(X3) PROPER(nats(X)) -> PROPER(X) PROPER(s(X)) -> PROPER(X) PROPER(sieve(X)) -> PROPER(X) -> Rules: active(cons(X1,X2)) -> cons(active(X1),X2) 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(filter(X1,X2,X3)) -> filter(active(X1),X2,X3) active(filter(X1,X2,X3)) -> filter(X1,active(X2),X3) active(filter(X1,X2,X3)) -> filter(X1,X2,active(X3)) active(nats(N)) -> mark(cons(N,nats(s(N)))) active(nats(X)) -> nats(active(X)) active(s(X)) -> s(active(X)) 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(sieve(X)) -> sieve(active(X)) active(zprimes) -> mark(sieve(nats(s(s(0))))) cons(mark(X1),X2) -> mark(cons(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) filter(mark(X1),X2,X3) -> mark(filter(X1,X2,X3)) filter(ok(X1),ok(X2),ok(X3)) -> ok(filter(X1,X2,X3)) filter(X1,mark(X2),X3) -> mark(filter(X1,X2,X3)) filter(X1,X2,mark(X3)) -> mark(filter(X1,X2,X3)) nats(mark(X)) -> mark(nats(X)) nats(ok(X)) -> ok(nats(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(filter(X1,X2,X3)) -> filter(proper(X1),proper(X2),proper(X3)) proper(nats(X)) -> nats(proper(X)) proper(s(X)) -> s(proper(X)) proper(sieve(X)) -> sieve(proper(X)) proper(0) -> ok(0) proper(zprimes) -> ok(zprimes) s(mark(X)) -> mark(s(X)) s(ok(X)) -> ok(s(X)) sieve(mark(X)) -> mark(sieve(X)) sieve(ok(X)) -> ok(sieve(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) ->Projection: pi(PROPER) = 1 Problem 1.6: SCC Processor: -> Pairs: Empty -> Rules: active(cons(X1,X2)) -> cons(active(X1),X2) 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(filter(X1,X2,X3)) -> filter(active(X1),X2,X3) active(filter(X1,X2,X3)) -> filter(X1,active(X2),X3) active(filter(X1,X2,X3)) -> filter(X1,X2,active(X3)) active(nats(N)) -> mark(cons(N,nats(s(N)))) active(nats(X)) -> nats(active(X)) active(s(X)) -> s(active(X)) 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(sieve(X)) -> sieve(active(X)) active(zprimes) -> mark(sieve(nats(s(s(0))))) cons(mark(X1),X2) -> mark(cons(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) filter(mark(X1),X2,X3) -> mark(filter(X1,X2,X3)) filter(ok(X1),ok(X2),ok(X3)) -> ok(filter(X1,X2,X3)) filter(X1,mark(X2),X3) -> mark(filter(X1,X2,X3)) filter(X1,X2,mark(X3)) -> mark(filter(X1,X2,X3)) nats(mark(X)) -> mark(nats(X)) nats(ok(X)) -> ok(nats(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(filter(X1,X2,X3)) -> filter(proper(X1),proper(X2),proper(X3)) proper(nats(X)) -> nats(proper(X)) proper(s(X)) -> s(proper(X)) proper(sieve(X)) -> sieve(proper(X)) proper(0) -> ok(0) proper(zprimes) -> ok(zprimes) s(mark(X)) -> mark(s(X)) s(ok(X)) -> ok(s(X)) sieve(mark(X)) -> mark(sieve(X)) sieve(ok(X)) -> ok(sieve(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.7: Subterm Processor: -> Pairs: ACTIVE(cons(X1,X2)) -> ACTIVE(X1) ACTIVE(filter(X1,X2,X3)) -> ACTIVE(X1) ACTIVE(filter(X1,X2,X3)) -> ACTIVE(X2) ACTIVE(filter(X1,X2,X3)) -> ACTIVE(X3) ACTIVE(nats(X)) -> ACTIVE(X) ACTIVE(s(X)) -> ACTIVE(X) ACTIVE(sieve(X)) -> ACTIVE(X) -> Rules: active(cons(X1,X2)) -> cons(active(X1),X2) 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(filter(X1,X2,X3)) -> filter(active(X1),X2,X3) active(filter(X1,X2,X3)) -> filter(X1,active(X2),X3) active(filter(X1,X2,X3)) -> filter(X1,X2,active(X3)) active(nats(N)) -> mark(cons(N,nats(s(N)))) active(nats(X)) -> nats(active(X)) active(s(X)) -> s(active(X)) 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(sieve(X)) -> sieve(active(X)) active(zprimes) -> mark(sieve(nats(s(s(0))))) cons(mark(X1),X2) -> mark(cons(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) filter(mark(X1),X2,X3) -> mark(filter(X1,X2,X3)) filter(ok(X1),ok(X2),ok(X3)) -> ok(filter(X1,X2,X3)) filter(X1,mark(X2),X3) -> mark(filter(X1,X2,X3)) filter(X1,X2,mark(X3)) -> mark(filter(X1,X2,X3)) nats(mark(X)) -> mark(nats(X)) nats(ok(X)) -> ok(nats(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(filter(X1,X2,X3)) -> filter(proper(X1),proper(X2),proper(X3)) proper(nats(X)) -> nats(proper(X)) proper(s(X)) -> s(proper(X)) proper(sieve(X)) -> sieve(proper(X)) proper(0) -> ok(0) proper(zprimes) -> ok(zprimes) s(mark(X)) -> mark(s(X)) s(ok(X)) -> ok(s(X)) sieve(mark(X)) -> mark(sieve(X)) sieve(ok(X)) -> ok(sieve(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) ->Projection: pi(ACTIVE) = 1 Problem 1.7: SCC Processor: -> Pairs: Empty -> Rules: active(cons(X1,X2)) -> cons(active(X1),X2) 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(filter(X1,X2,X3)) -> filter(active(X1),X2,X3) active(filter(X1,X2,X3)) -> filter(X1,active(X2),X3) active(filter(X1,X2,X3)) -> filter(X1,X2,active(X3)) active(nats(N)) -> mark(cons(N,nats(s(N)))) active(nats(X)) -> nats(active(X)) active(s(X)) -> s(active(X)) 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(sieve(X)) -> sieve(active(X)) active(zprimes) -> mark(sieve(nats(s(s(0))))) cons(mark(X1),X2) -> mark(cons(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) filter(mark(X1),X2,X3) -> mark(filter(X1,X2,X3)) filter(ok(X1),ok(X2),ok(X3)) -> ok(filter(X1,X2,X3)) filter(X1,mark(X2),X3) -> mark(filter(X1,X2,X3)) filter(X1,X2,mark(X3)) -> mark(filter(X1,X2,X3)) nats(mark(X)) -> mark(nats(X)) nats(ok(X)) -> ok(nats(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(filter(X1,X2,X3)) -> filter(proper(X1),proper(X2),proper(X3)) proper(nats(X)) -> nats(proper(X)) proper(s(X)) -> s(proper(X)) proper(sieve(X)) -> sieve(proper(X)) proper(0) -> ok(0) proper(zprimes) -> ok(zprimes) s(mark(X)) -> mark(s(X)) s(ok(X)) -> ok(s(X)) sieve(mark(X)) -> mark(sieve(X)) sieve(ok(X)) -> ok(sieve(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.8: Reduction Pair Processor: -> Pairs: TOP(mark(X)) -> TOP(proper(X)) TOP(ok(X)) -> TOP(active(X)) -> Rules: active(cons(X1,X2)) -> cons(active(X1),X2) 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(filter(X1,X2,X3)) -> filter(active(X1),X2,X3) active(filter(X1,X2,X3)) -> filter(X1,active(X2),X3) active(filter(X1,X2,X3)) -> filter(X1,X2,active(X3)) active(nats(N)) -> mark(cons(N,nats(s(N)))) active(nats(X)) -> nats(active(X)) active(s(X)) -> s(active(X)) 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(sieve(X)) -> sieve(active(X)) active(zprimes) -> mark(sieve(nats(s(s(0))))) cons(mark(X1),X2) -> mark(cons(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) filter(mark(X1),X2,X3) -> mark(filter(X1,X2,X3)) filter(ok(X1),ok(X2),ok(X3)) -> ok(filter(X1,X2,X3)) filter(X1,mark(X2),X3) -> mark(filter(X1,X2,X3)) filter(X1,X2,mark(X3)) -> mark(filter(X1,X2,X3)) nats(mark(X)) -> mark(nats(X)) nats(ok(X)) -> ok(nats(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(filter(X1,X2,X3)) -> filter(proper(X1),proper(X2),proper(X3)) proper(nats(X)) -> nats(proper(X)) proper(s(X)) -> s(proper(X)) proper(sieve(X)) -> sieve(proper(X)) proper(0) -> ok(0) proper(zprimes) -> ok(zprimes) s(mark(X)) -> mark(s(X)) s(ok(X)) -> ok(s(X)) sieve(mark(X)) -> mark(sieve(X)) sieve(ok(X)) -> ok(sieve(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) -> Usable rules: active(cons(X1,X2)) -> cons(active(X1),X2) 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(filter(X1,X2,X3)) -> filter(active(X1),X2,X3) active(filter(X1,X2,X3)) -> filter(X1,active(X2),X3) active(filter(X1,X2,X3)) -> filter(X1,X2,active(X3)) active(nats(N)) -> mark(cons(N,nats(s(N)))) active(nats(X)) -> nats(active(X)) active(s(X)) -> s(active(X)) 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(sieve(X)) -> sieve(active(X)) active(zprimes) -> mark(sieve(nats(s(s(0))))) cons(mark(X1),X2) -> mark(cons(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) filter(mark(X1),X2,X3) -> mark(filter(X1,X2,X3)) filter(ok(X1),ok(X2),ok(X3)) -> ok(filter(X1,X2,X3)) filter(X1,mark(X2),X3) -> mark(filter(X1,X2,X3)) filter(X1,X2,mark(X3)) -> mark(filter(X1,X2,X3)) nats(mark(X)) -> mark(nats(X)) nats(ok(X)) -> ok(nats(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(filter(X1,X2,X3)) -> filter(proper(X1),proper(X2),proper(X3)) proper(nats(X)) -> nats(proper(X)) proper(s(X)) -> s(proper(X)) proper(sieve(X)) -> sieve(proper(X)) proper(0) -> ok(0) proper(zprimes) -> ok(zprimes) s(mark(X)) -> mark(s(X)) s(ok(X)) -> ok(s(X)) sieve(mark(X)) -> mark(sieve(X)) sieve(ok(X)) -> ok(sieve(X)) ->Interpretation type: Linear ->Coefficients: All rationals ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = X [cons](X1,X2) = X1 [filter](X1,X2,X3) = 2.X1 + X2 + 2.X3 + 2 [nats](X) = X + 1/2 [proper](X) = X [s](X) = X [sieve](X) = 2.X + 1/2 [0] = 0 [mark](X) = X + 1/2 [ok](X) = X [zprimes] = 2 [TOP](X) = 2.X Problem 1.8: SCC Processor: -> Pairs: TOP(ok(X)) -> TOP(active(X)) -> Rules: active(cons(X1,X2)) -> cons(active(X1),X2) 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(filter(X1,X2,X3)) -> filter(active(X1),X2,X3) active(filter(X1,X2,X3)) -> filter(X1,active(X2),X3) active(filter(X1,X2,X3)) -> filter(X1,X2,active(X3)) active(nats(N)) -> mark(cons(N,nats(s(N)))) active(nats(X)) -> nats(active(X)) active(s(X)) -> s(active(X)) 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(sieve(X)) -> sieve(active(X)) active(zprimes) -> mark(sieve(nats(s(s(0))))) cons(mark(X1),X2) -> mark(cons(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) filter(mark(X1),X2,X3) -> mark(filter(X1,X2,X3)) filter(ok(X1),ok(X2),ok(X3)) -> ok(filter(X1,X2,X3)) filter(X1,mark(X2),X3) -> mark(filter(X1,X2,X3)) filter(X1,X2,mark(X3)) -> mark(filter(X1,X2,X3)) nats(mark(X)) -> mark(nats(X)) nats(ok(X)) -> ok(nats(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(filter(X1,X2,X3)) -> filter(proper(X1),proper(X2),proper(X3)) proper(nats(X)) -> nats(proper(X)) proper(s(X)) -> s(proper(X)) proper(sieve(X)) -> sieve(proper(X)) proper(0) -> ok(0) proper(zprimes) -> ok(zprimes) s(mark(X)) -> mark(s(X)) s(ok(X)) -> ok(s(X)) sieve(mark(X)) -> mark(sieve(X)) sieve(ok(X)) -> ok(sieve(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: TOP(ok(X)) -> TOP(active(X)) ->->-> Rules: active(cons(X1,X2)) -> cons(active(X1),X2) 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(filter(X1,X2,X3)) -> filter(active(X1),X2,X3) active(filter(X1,X2,X3)) -> filter(X1,active(X2),X3) active(filter(X1,X2,X3)) -> filter(X1,X2,active(X3)) active(nats(N)) -> mark(cons(N,nats(s(N)))) active(nats(X)) -> nats(active(X)) active(s(X)) -> s(active(X)) 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(sieve(X)) -> sieve(active(X)) active(zprimes) -> mark(sieve(nats(s(s(0))))) cons(mark(X1),X2) -> mark(cons(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) filter(mark(X1),X2,X3) -> mark(filter(X1,X2,X3)) filter(ok(X1),ok(X2),ok(X3)) -> ok(filter(X1,X2,X3)) filter(X1,mark(X2),X3) -> mark(filter(X1,X2,X3)) filter(X1,X2,mark(X3)) -> mark(filter(X1,X2,X3)) nats(mark(X)) -> mark(nats(X)) nats(ok(X)) -> ok(nats(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(filter(X1,X2,X3)) -> filter(proper(X1),proper(X2),proper(X3)) proper(nats(X)) -> nats(proper(X)) proper(s(X)) -> s(proper(X)) proper(sieve(X)) -> sieve(proper(X)) proper(0) -> ok(0) proper(zprimes) -> ok(zprimes) s(mark(X)) -> mark(s(X)) s(ok(X)) -> ok(s(X)) sieve(mark(X)) -> mark(sieve(X)) sieve(ok(X)) -> ok(sieve(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Problem 1.8: Reduction Pair Processor: -> Pairs: TOP(ok(X)) -> TOP(active(X)) -> Rules: active(cons(X1,X2)) -> cons(active(X1),X2) 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(filter(X1,X2,X3)) -> filter(active(X1),X2,X3) active(filter(X1,X2,X3)) -> filter(X1,active(X2),X3) active(filter(X1,X2,X3)) -> filter(X1,X2,active(X3)) active(nats(N)) -> mark(cons(N,nats(s(N)))) active(nats(X)) -> nats(active(X)) active(s(X)) -> s(active(X)) 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(sieve(X)) -> sieve(active(X)) active(zprimes) -> mark(sieve(nats(s(s(0))))) cons(mark(X1),X2) -> mark(cons(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) filter(mark(X1),X2,X3) -> mark(filter(X1,X2,X3)) filter(ok(X1),ok(X2),ok(X3)) -> ok(filter(X1,X2,X3)) filter(X1,mark(X2),X3) -> mark(filter(X1,X2,X3)) filter(X1,X2,mark(X3)) -> mark(filter(X1,X2,X3)) nats(mark(X)) -> mark(nats(X)) nats(ok(X)) -> ok(nats(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(filter(X1,X2,X3)) -> filter(proper(X1),proper(X2),proper(X3)) proper(nats(X)) -> nats(proper(X)) proper(s(X)) -> s(proper(X)) proper(sieve(X)) -> sieve(proper(X)) proper(0) -> ok(0) proper(zprimes) -> ok(zprimes) s(mark(X)) -> mark(s(X)) s(ok(X)) -> ok(s(X)) sieve(mark(X)) -> mark(sieve(X)) sieve(ok(X)) -> ok(sieve(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) -> Usable rules: active(cons(X1,X2)) -> cons(active(X1),X2) 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(filter(X1,X2,X3)) -> filter(active(X1),X2,X3) active(filter(X1,X2,X3)) -> filter(X1,active(X2),X3) active(filter(X1,X2,X3)) -> filter(X1,X2,active(X3)) active(nats(N)) -> mark(cons(N,nats(s(N)))) active(nats(X)) -> nats(active(X)) active(s(X)) -> s(active(X)) 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(sieve(X)) -> sieve(active(X)) active(zprimes) -> mark(sieve(nats(s(s(0))))) cons(mark(X1),X2) -> mark(cons(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) filter(mark(X1),X2,X3) -> mark(filter(X1,X2,X3)) filter(ok(X1),ok(X2),ok(X3)) -> ok(filter(X1,X2,X3)) filter(X1,mark(X2),X3) -> mark(filter(X1,X2,X3)) filter(X1,X2,mark(X3)) -> mark(filter(X1,X2,X3)) nats(mark(X)) -> mark(nats(X)) nats(ok(X)) -> ok(nats(X)) s(mark(X)) -> mark(s(X)) s(ok(X)) -> ok(s(X)) sieve(mark(X)) -> mark(sieve(X)) sieve(ok(X)) -> ok(sieve(X)) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = 2.X [cons](X1,X2) = 2.X1 + 2.X2 + 1 [filter](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 [nats](X) = X [s](X) = 2.X [sieve](X) = 2.X + 1 [0] = 2 [mark](X) = 0 [ok](X) = 2.X + 2 [zprimes] = 2 [TOP](X) = 2.X Problem 1.8: SCC Processor: -> Pairs: Empty -> Rules: active(cons(X1,X2)) -> cons(active(X1),X2) 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(filter(X1,X2,X3)) -> filter(active(X1),X2,X3) active(filter(X1,X2,X3)) -> filter(X1,active(X2),X3) active(filter(X1,X2,X3)) -> filter(X1,X2,active(X3)) active(nats(N)) -> mark(cons(N,nats(s(N)))) active(nats(X)) -> nats(active(X)) active(s(X)) -> s(active(X)) 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(sieve(X)) -> sieve(active(X)) active(zprimes) -> mark(sieve(nats(s(s(0))))) cons(mark(X1),X2) -> mark(cons(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) filter(mark(X1),X2,X3) -> mark(filter(X1,X2,X3)) filter(ok(X1),ok(X2),ok(X3)) -> ok(filter(X1,X2,X3)) filter(X1,mark(X2),X3) -> mark(filter(X1,X2,X3)) filter(X1,X2,mark(X3)) -> mark(filter(X1,X2,X3)) nats(mark(X)) -> mark(nats(X)) nats(ok(X)) -> ok(nats(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(filter(X1,X2,X3)) -> filter(proper(X1),proper(X2),proper(X3)) proper(nats(X)) -> nats(proper(X)) proper(s(X)) -> s(proper(X)) proper(sieve(X)) -> sieve(proper(X)) proper(0) -> ok(0) proper(zprimes) -> ok(zprimes) s(mark(X)) -> mark(s(X)) s(ok(X)) -> ok(s(X)) sieve(mark(X)) -> mark(sieve(X)) sieve(ok(X)) -> ok(sieve(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) ->Strongly Connected Components: There is no strongly connected component The problem is finite.