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