YES Problem 1: (VAR v_NonEmpty:S X:S X1:S X2:S) (RULES active(f(s(0))) -> mark(f(p(s(0)))) active(f(0)) -> mark(cons(0,f(s(0)))) active(p(s(0))) -> mark(0) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) f(active(X:S)) -> f(X:S) f(mark(X:S)) -> f(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(f(X:S)) -> active(f(mark(X:S))) mark(p(X:S)) -> active(p(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) p(active(X:S)) -> p(X:S) p(mark(X:S)) -> p(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ) (STRATEGY INNERMOST) Problem 1: Dependency Pairs Processor: -> Pairs: ACTIVE(f(s(0))) -> MARK(f(p(s(0)))) ACTIVE(f(0)) -> MARK(cons(0,f(s(0)))) ACTIVE(p(s(0))) -> MARK(0) CONS(active(X1:S),X2:S) -> CONS(X1:S,X2:S) CONS(mark(X1:S),X2:S) -> CONS(X1:S,X2:S) CONS(X1:S,active(X2:S)) -> CONS(X1:S,X2:S) CONS(X1:S,mark(X2:S)) -> CONS(X1:S,X2:S) F(active(X:S)) -> F(X:S) F(mark(X:S)) -> F(X:S) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> CONS(mark(X1:S),X2:S) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(f(X:S)) -> ACTIVE(f(mark(X:S))) MARK(f(X:S)) -> F(mark(X:S)) MARK(f(X:S)) -> MARK(X:S) MARK(p(X:S)) -> ACTIVE(p(mark(X:S))) MARK(p(X:S)) -> MARK(X:S) MARK(p(X:S)) -> P(mark(X:S)) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(s(X:S)) -> S(mark(X:S)) P(active(X:S)) -> P(X:S) P(mark(X:S)) -> P(X:S) S(active(X:S)) -> S(X:S) S(mark(X:S)) -> S(X:S) -> Rules: active(f(s(0))) -> mark(f(p(s(0)))) active(f(0)) -> mark(cons(0,f(s(0)))) active(p(s(0))) -> mark(0) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) f(active(X:S)) -> f(X:S) f(mark(X:S)) -> f(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(f(X:S)) -> active(f(mark(X:S))) mark(p(X:S)) -> active(p(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) p(active(X:S)) -> p(X:S) p(mark(X:S)) -> p(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1: SCC Processor: -> Pairs: ACTIVE(f(s(0))) -> MARK(f(p(s(0)))) ACTIVE(f(0)) -> MARK(cons(0,f(s(0)))) ACTIVE(p(s(0))) -> MARK(0) CONS(active(X1:S),X2:S) -> CONS(X1:S,X2:S) CONS(mark(X1:S),X2:S) -> CONS(X1:S,X2:S) CONS(X1:S,active(X2:S)) -> CONS(X1:S,X2:S) CONS(X1:S,mark(X2:S)) -> CONS(X1:S,X2:S) F(active(X:S)) -> F(X:S) F(mark(X:S)) -> F(X:S) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> CONS(mark(X1:S),X2:S) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(f(X:S)) -> ACTIVE(f(mark(X:S))) MARK(f(X:S)) -> F(mark(X:S)) MARK(f(X:S)) -> MARK(X:S) MARK(p(X:S)) -> ACTIVE(p(mark(X:S))) MARK(p(X:S)) -> MARK(X:S) MARK(p(X:S)) -> P(mark(X:S)) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(s(X:S)) -> S(mark(X:S)) P(active(X:S)) -> P(X:S) P(mark(X:S)) -> P(X:S) S(active(X:S)) -> S(X:S) S(mark(X:S)) -> S(X:S) -> Rules: active(f(s(0))) -> mark(f(p(s(0)))) active(f(0)) -> mark(cons(0,f(s(0)))) active(p(s(0))) -> mark(0) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) f(active(X:S)) -> f(X:S) f(mark(X:S)) -> f(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(f(X:S)) -> active(f(mark(X:S))) mark(p(X:S)) -> active(p(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) p(active(X:S)) -> p(X:S) p(mark(X:S)) -> p(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: S(active(X:S)) -> S(X:S) S(mark(X:S)) -> S(X:S) ->->-> Rules: active(f(s(0))) -> mark(f(p(s(0)))) active(f(0)) -> mark(cons(0,f(s(0)))) active(p(s(0))) -> mark(0) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) f(active(X:S)) -> f(X:S) f(mark(X:S)) -> f(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(f(X:S)) -> active(f(mark(X:S))) mark(p(X:S)) -> active(p(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) p(active(X:S)) -> p(X:S) p(mark(X:S)) -> p(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->->Cycle: ->->-> Pairs: P(active(X:S)) -> P(X:S) P(mark(X:S)) -> P(X:S) ->->-> Rules: active(f(s(0))) -> mark(f(p(s(0)))) active(f(0)) -> mark(cons(0,f(s(0)))) active(p(s(0))) -> mark(0) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) f(active(X:S)) -> f(X:S) f(mark(X:S)) -> f(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(f(X:S)) -> active(f(mark(X:S))) mark(p(X:S)) -> active(p(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) p(active(X:S)) -> p(X:S) p(mark(X:S)) -> p(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->->Cycle: ->->-> Pairs: F(active(X:S)) -> F(X:S) F(mark(X:S)) -> F(X:S) ->->-> Rules: active(f(s(0))) -> mark(f(p(s(0)))) active(f(0)) -> mark(cons(0,f(s(0)))) active(p(s(0))) -> mark(0) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) f(active(X:S)) -> f(X:S) f(mark(X:S)) -> f(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(f(X:S)) -> active(f(mark(X:S))) mark(p(X:S)) -> active(p(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) p(active(X:S)) -> p(X:S) p(mark(X:S)) -> p(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->->Cycle: ->->-> Pairs: CONS(active(X1:S),X2:S) -> CONS(X1:S,X2:S) CONS(mark(X1:S),X2:S) -> CONS(X1:S,X2:S) CONS(X1:S,active(X2:S)) -> CONS(X1:S,X2:S) CONS(X1:S,mark(X2:S)) -> CONS(X1:S,X2:S) ->->-> Rules: active(f(s(0))) -> mark(f(p(s(0)))) active(f(0)) -> mark(cons(0,f(s(0)))) active(p(s(0))) -> mark(0) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) f(active(X:S)) -> f(X:S) f(mark(X:S)) -> f(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(f(X:S)) -> active(f(mark(X:S))) mark(p(X:S)) -> active(p(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) p(active(X:S)) -> p(X:S) p(mark(X:S)) -> p(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->->Cycle: ->->-> Pairs: ACTIVE(f(s(0))) -> MARK(f(p(s(0)))) ACTIVE(f(0)) -> MARK(cons(0,f(s(0)))) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(f(X:S)) -> ACTIVE(f(mark(X:S))) MARK(f(X:S)) -> MARK(X:S) MARK(p(X:S)) -> ACTIVE(p(mark(X:S))) MARK(p(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) ->->-> Rules: active(f(s(0))) -> mark(f(p(s(0)))) active(f(0)) -> mark(cons(0,f(s(0)))) active(p(s(0))) -> mark(0) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) f(active(X:S)) -> f(X:S) f(mark(X:S)) -> f(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(f(X:S)) -> active(f(mark(X:S))) mark(p(X:S)) -> active(p(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) p(active(X:S)) -> p(X:S) p(mark(X:S)) -> p(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) The problem is decomposed in 5 subproblems. Problem 1.1: Subterm Processor: -> Pairs: S(active(X:S)) -> S(X:S) S(mark(X:S)) -> S(X:S) -> Rules: active(f(s(0))) -> mark(f(p(s(0)))) active(f(0)) -> mark(cons(0,f(s(0)))) active(p(s(0))) -> mark(0) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) f(active(X:S)) -> f(X:S) f(mark(X:S)) -> f(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(f(X:S)) -> active(f(mark(X:S))) mark(p(X:S)) -> active(p(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) p(active(X:S)) -> p(X:S) p(mark(X:S)) -> p(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(S) = 1 Problem 1.1: SCC Processor: -> Pairs: Empty -> Rules: active(f(s(0))) -> mark(f(p(s(0)))) active(f(0)) -> mark(cons(0,f(s(0)))) active(p(s(0))) -> mark(0) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) f(active(X:S)) -> f(X:S) f(mark(X:S)) -> f(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(f(X:S)) -> active(f(mark(X:S))) mark(p(X:S)) -> active(p(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) p(active(X:S)) -> p(X:S) p(mark(X:S)) -> p(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.2: Subterm Processor: -> Pairs: P(active(X:S)) -> P(X:S) P(mark(X:S)) -> P(X:S) -> Rules: active(f(s(0))) -> mark(f(p(s(0)))) active(f(0)) -> mark(cons(0,f(s(0)))) active(p(s(0))) -> mark(0) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) f(active(X:S)) -> f(X:S) f(mark(X:S)) -> f(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(f(X:S)) -> active(f(mark(X:S))) mark(p(X:S)) -> active(p(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) p(active(X:S)) -> p(X:S) p(mark(X:S)) -> p(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(P) = 1 Problem 1.2: SCC Processor: -> Pairs: Empty -> Rules: active(f(s(0))) -> mark(f(p(s(0)))) active(f(0)) -> mark(cons(0,f(s(0)))) active(p(s(0))) -> mark(0) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) f(active(X:S)) -> f(X:S) f(mark(X:S)) -> f(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(f(X:S)) -> active(f(mark(X:S))) mark(p(X:S)) -> active(p(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) p(active(X:S)) -> p(X:S) p(mark(X:S)) -> p(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.3: Subterm Processor: -> Pairs: F(active(X:S)) -> F(X:S) F(mark(X:S)) -> F(X:S) -> Rules: active(f(s(0))) -> mark(f(p(s(0)))) active(f(0)) -> mark(cons(0,f(s(0)))) active(p(s(0))) -> mark(0) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) f(active(X:S)) -> f(X:S) f(mark(X:S)) -> f(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(f(X:S)) -> active(f(mark(X:S))) mark(p(X:S)) -> active(p(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) p(active(X:S)) -> p(X:S) p(mark(X:S)) -> p(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(F) = 1 Problem 1.3: SCC Processor: -> Pairs: Empty -> Rules: active(f(s(0))) -> mark(f(p(s(0)))) active(f(0)) -> mark(cons(0,f(s(0)))) active(p(s(0))) -> mark(0) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) f(active(X:S)) -> f(X:S) f(mark(X:S)) -> f(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(f(X:S)) -> active(f(mark(X:S))) mark(p(X:S)) -> active(p(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) p(active(X:S)) -> p(X:S) p(mark(X:S)) -> p(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.4: Subterm Processor: -> Pairs: CONS(active(X1:S),X2:S) -> CONS(X1:S,X2:S) CONS(mark(X1:S),X2:S) -> CONS(X1:S,X2:S) CONS(X1:S,active(X2:S)) -> CONS(X1:S,X2:S) CONS(X1:S,mark(X2:S)) -> CONS(X1:S,X2:S) -> Rules: active(f(s(0))) -> mark(f(p(s(0)))) active(f(0)) -> mark(cons(0,f(s(0)))) active(p(s(0))) -> mark(0) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) f(active(X:S)) -> f(X:S) f(mark(X:S)) -> f(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(f(X:S)) -> active(f(mark(X:S))) mark(p(X:S)) -> active(p(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) p(active(X:S)) -> p(X:S) p(mark(X:S)) -> p(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(CONS) = 1 Problem 1.4: SCC Processor: -> Pairs: CONS(X1:S,active(X2:S)) -> CONS(X1:S,X2:S) CONS(X1:S,mark(X2:S)) -> CONS(X1:S,X2:S) -> Rules: active(f(s(0))) -> mark(f(p(s(0)))) active(f(0)) -> mark(cons(0,f(s(0)))) active(p(s(0))) -> mark(0) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) f(active(X:S)) -> f(X:S) f(mark(X:S)) -> f(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(f(X:S)) -> active(f(mark(X:S))) mark(p(X:S)) -> active(p(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) p(active(X:S)) -> p(X:S) p(mark(X:S)) -> p(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: CONS(X1:S,active(X2:S)) -> CONS(X1:S,X2:S) CONS(X1:S,mark(X2:S)) -> CONS(X1:S,X2:S) ->->-> Rules: active(f(s(0))) -> mark(f(p(s(0)))) active(f(0)) -> mark(cons(0,f(s(0)))) active(p(s(0))) -> mark(0) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) f(active(X:S)) -> f(X:S) f(mark(X:S)) -> f(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(f(X:S)) -> active(f(mark(X:S))) mark(p(X:S)) -> active(p(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) p(active(X:S)) -> p(X:S) p(mark(X:S)) -> p(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.4: Subterm Processor: -> Pairs: CONS(X1:S,active(X2:S)) -> CONS(X1:S,X2:S) CONS(X1:S,mark(X2:S)) -> CONS(X1:S,X2:S) -> Rules: active(f(s(0))) -> mark(f(p(s(0)))) active(f(0)) -> mark(cons(0,f(s(0)))) active(p(s(0))) -> mark(0) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) f(active(X:S)) -> f(X:S) f(mark(X:S)) -> f(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(f(X:S)) -> active(f(mark(X:S))) mark(p(X:S)) -> active(p(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) p(active(X:S)) -> p(X:S) p(mark(X:S)) -> p(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(CONS) = 2 Problem 1.4: SCC Processor: -> Pairs: Empty -> Rules: active(f(s(0))) -> mark(f(p(s(0)))) active(f(0)) -> mark(cons(0,f(s(0)))) active(p(s(0))) -> mark(0) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) f(active(X:S)) -> f(X:S) f(mark(X:S)) -> f(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(f(X:S)) -> active(f(mark(X:S))) mark(p(X:S)) -> active(p(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) p(active(X:S)) -> p(X:S) p(mark(X:S)) -> p(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.5: Reduction Pairs Processor: -> Pairs: ACTIVE(f(s(0))) -> MARK(f(p(s(0)))) ACTIVE(f(0)) -> MARK(cons(0,f(s(0)))) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(f(X:S)) -> ACTIVE(f(mark(X:S))) MARK(f(X:S)) -> MARK(X:S) MARK(p(X:S)) -> ACTIVE(p(mark(X:S))) MARK(p(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: active(f(s(0))) -> mark(f(p(s(0)))) active(f(0)) -> mark(cons(0,f(s(0)))) active(p(s(0))) -> mark(0) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) f(active(X:S)) -> f(X:S) f(mark(X:S)) -> f(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(f(X:S)) -> active(f(mark(X:S))) mark(p(X:S)) -> active(p(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) p(active(X:S)) -> p(X:S) p(mark(X:S)) -> p(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: active(f(s(0))) -> mark(f(p(s(0)))) active(f(0)) -> mark(cons(0,f(s(0)))) active(p(s(0))) -> mark(0) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) f(active(X:S)) -> f(X:S) f(mark(X:S)) -> f(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(f(X:S)) -> active(f(mark(X:S))) mark(p(X:S)) -> active(p(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) p(active(X:S)) -> p(X:S) p(mark(X:S)) -> p(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = X [cons](X1,X2) = 2.X1 + 1 [f](X) = 2.X + 2 [mark](X) = X [p](X) = X [s](X) = 2.X + 2 [0] = 0 [fSNonEmpty] = 0 [ACTIVE](X) = 2.X + 2 [CONS](X1,X2) = 0 [F](X) = 0 [MARK](X) = 2.X + 2 [P](X) = 0 [S](X) = 0 Problem 1.5: SCC Processor: -> Pairs: ACTIVE(f(s(0))) -> MARK(f(p(s(0)))) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(f(X:S)) -> ACTIVE(f(mark(X:S))) MARK(f(X:S)) -> MARK(X:S) MARK(p(X:S)) -> ACTIVE(p(mark(X:S))) MARK(p(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: active(f(s(0))) -> mark(f(p(s(0)))) active(f(0)) -> mark(cons(0,f(s(0)))) active(p(s(0))) -> mark(0) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) f(active(X:S)) -> f(X:S) f(mark(X:S)) -> f(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(f(X:S)) -> active(f(mark(X:S))) mark(p(X:S)) -> active(p(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) p(active(X:S)) -> p(X:S) p(mark(X:S)) -> p(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(f(s(0))) -> MARK(f(p(s(0)))) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(f(X:S)) -> ACTIVE(f(mark(X:S))) MARK(f(X:S)) -> MARK(X:S) MARK(p(X:S)) -> ACTIVE(p(mark(X:S))) MARK(p(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) ->->-> Rules: active(f(s(0))) -> mark(f(p(s(0)))) active(f(0)) -> mark(cons(0,f(s(0)))) active(p(s(0))) -> mark(0) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) f(active(X:S)) -> f(X:S) f(mark(X:S)) -> f(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(f(X:S)) -> active(f(mark(X:S))) mark(p(X:S)) -> active(p(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) p(active(X:S)) -> p(X:S) p(mark(X:S)) -> p(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.5: Reduction Pairs Processor: -> Pairs: ACTIVE(f(s(0))) -> MARK(f(p(s(0)))) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(f(X:S)) -> ACTIVE(f(mark(X:S))) MARK(f(X:S)) -> MARK(X:S) MARK(p(X:S)) -> ACTIVE(p(mark(X:S))) MARK(p(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: active(f(s(0))) -> mark(f(p(s(0)))) active(f(0)) -> mark(cons(0,f(s(0)))) active(p(s(0))) -> mark(0) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) f(active(X:S)) -> f(X:S) f(mark(X:S)) -> f(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(f(X:S)) -> active(f(mark(X:S))) mark(p(X:S)) -> active(p(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) p(active(X:S)) -> p(X:S) p(mark(X:S)) -> p(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: active(f(s(0))) -> mark(f(p(s(0)))) active(f(0)) -> mark(cons(0,f(s(0)))) active(p(s(0))) -> mark(0) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) f(active(X:S)) -> f(X:S) f(mark(X:S)) -> f(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(f(X:S)) -> active(f(mark(X:S))) mark(p(X:S)) -> active(p(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) p(active(X:S)) -> p(X:S) p(mark(X:S)) -> p(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = 2 [cons](X1,X2) = 0 [f](X) = 1 [mark](X) = 2 [p](X) = 0 [s](X) = 1 [0] = 0 [fSNonEmpty] = 0 [ACTIVE](X) = 2.X [CONS](X1,X2) = 0 [F](X) = 0 [MARK](X) = 2 [P](X) = 0 [S](X) = 0 Problem 1.5: SCC Processor: -> Pairs: ACTIVE(f(s(0))) -> MARK(f(p(s(0)))) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(f(X:S)) -> ACTIVE(f(mark(X:S))) MARK(f(X:S)) -> MARK(X:S) MARK(p(X:S)) -> ACTIVE(p(mark(X:S))) MARK(p(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: active(f(s(0))) -> mark(f(p(s(0)))) active(f(0)) -> mark(cons(0,f(s(0)))) active(p(s(0))) -> mark(0) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) f(active(X:S)) -> f(X:S) f(mark(X:S)) -> f(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(f(X:S)) -> active(f(mark(X:S))) mark(p(X:S)) -> active(p(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) p(active(X:S)) -> p(X:S) p(mark(X:S)) -> p(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(f(s(0))) -> MARK(f(p(s(0)))) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(f(X:S)) -> ACTIVE(f(mark(X:S))) MARK(f(X:S)) -> MARK(X:S) MARK(p(X:S)) -> ACTIVE(p(mark(X:S))) MARK(p(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) ->->-> Rules: active(f(s(0))) -> mark(f(p(s(0)))) active(f(0)) -> mark(cons(0,f(s(0)))) active(p(s(0))) -> mark(0) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) f(active(X:S)) -> f(X:S) f(mark(X:S)) -> f(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(f(X:S)) -> active(f(mark(X:S))) mark(p(X:S)) -> active(p(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) p(active(X:S)) -> p(X:S) p(mark(X:S)) -> p(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.5: Reduction Pairs Processor: -> Pairs: ACTIVE(f(s(0))) -> MARK(f(p(s(0)))) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(f(X:S)) -> ACTIVE(f(mark(X:S))) MARK(f(X:S)) -> MARK(X:S) MARK(p(X:S)) -> ACTIVE(p(mark(X:S))) MARK(p(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: active(f(s(0))) -> mark(f(p(s(0)))) active(f(0)) -> mark(cons(0,f(s(0)))) active(p(s(0))) -> mark(0) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) f(active(X:S)) -> f(X:S) f(mark(X:S)) -> f(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(f(X:S)) -> active(f(mark(X:S))) mark(p(X:S)) -> active(p(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) p(active(X:S)) -> p(X:S) p(mark(X:S)) -> p(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: active(f(s(0))) -> mark(f(p(s(0)))) active(f(0)) -> mark(cons(0,f(s(0)))) active(p(s(0))) -> mark(0) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) f(active(X:S)) -> f(X:S) f(mark(X:S)) -> f(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(f(X:S)) -> active(f(mark(X:S))) mark(p(X:S)) -> active(p(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) p(active(X:S)) -> p(X:S) p(mark(X:S)) -> p(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = X [cons](X1,X2) = X1 + 1 [f](X) = 2.X + 2 [mark](X) = X [p](X) = X [s](X) = X + 1 [0] = 1 [fSNonEmpty] = 0 [ACTIVE](X) = X + 1 [CONS](X1,X2) = 0 [F](X) = 0 [MARK](X) = X + 1 [P](X) = 0 [S](X) = 0 Problem 1.5: SCC Processor: -> Pairs: ACTIVE(f(s(0))) -> MARK(f(p(s(0)))) MARK(f(X:S)) -> ACTIVE(f(mark(X:S))) MARK(f(X:S)) -> MARK(X:S) MARK(p(X:S)) -> ACTIVE(p(mark(X:S))) MARK(p(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: active(f(s(0))) -> mark(f(p(s(0)))) active(f(0)) -> mark(cons(0,f(s(0)))) active(p(s(0))) -> mark(0) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) f(active(X:S)) -> f(X:S) f(mark(X:S)) -> f(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(f(X:S)) -> active(f(mark(X:S))) mark(p(X:S)) -> active(p(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) p(active(X:S)) -> p(X:S) p(mark(X:S)) -> p(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(f(s(0))) -> MARK(f(p(s(0)))) MARK(f(X:S)) -> ACTIVE(f(mark(X:S))) MARK(f(X:S)) -> MARK(X:S) MARK(p(X:S)) -> ACTIVE(p(mark(X:S))) MARK(p(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) ->->-> Rules: active(f(s(0))) -> mark(f(p(s(0)))) active(f(0)) -> mark(cons(0,f(s(0)))) active(p(s(0))) -> mark(0) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) f(active(X:S)) -> f(X:S) f(mark(X:S)) -> f(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(f(X:S)) -> active(f(mark(X:S))) mark(p(X:S)) -> active(p(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) p(active(X:S)) -> p(X:S) p(mark(X:S)) -> p(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.5: Reduction Pairs Processor: -> Pairs: ACTIVE(f(s(0))) -> MARK(f(p(s(0)))) MARK(f(X:S)) -> ACTIVE(f(mark(X:S))) MARK(f(X:S)) -> MARK(X:S) MARK(p(X:S)) -> ACTIVE(p(mark(X:S))) MARK(p(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: active(f(s(0))) -> mark(f(p(s(0)))) active(f(0)) -> mark(cons(0,f(s(0)))) active(p(s(0))) -> mark(0) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) f(active(X:S)) -> f(X:S) f(mark(X:S)) -> f(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(f(X:S)) -> active(f(mark(X:S))) mark(p(X:S)) -> active(p(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) p(active(X:S)) -> p(X:S) p(mark(X:S)) -> p(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: active(f(s(0))) -> mark(f(p(s(0)))) active(f(0)) -> mark(cons(0,f(s(0)))) active(p(s(0))) -> mark(0) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) f(active(X:S)) -> f(X:S) f(mark(X:S)) -> f(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(f(X:S)) -> active(f(mark(X:S))) mark(p(X:S)) -> active(p(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) p(active(X:S)) -> p(X:S) p(mark(X:S)) -> p(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = X [cons](X1,X2) = 2.X1 + 2 [f](X) = 2.X + 2 [mark](X) = X [p](X) = X [s](X) = 2.X + 2 [0] = 1 [fSNonEmpty] = 0 [ACTIVE](X) = 2.X + 2 [CONS](X1,X2) = 0 [F](X) = 0 [MARK](X) = 2.X + 2 [P](X) = 0 [S](X) = 0 Problem 1.5: SCC Processor: -> Pairs: ACTIVE(f(s(0))) -> MARK(f(p(s(0)))) MARK(f(X:S)) -> ACTIVE(f(mark(X:S))) MARK(p(X:S)) -> ACTIVE(p(mark(X:S))) MARK(p(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: active(f(s(0))) -> mark(f(p(s(0)))) active(f(0)) -> mark(cons(0,f(s(0)))) active(p(s(0))) -> mark(0) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) f(active(X:S)) -> f(X:S) f(mark(X:S)) -> f(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(f(X:S)) -> active(f(mark(X:S))) mark(p(X:S)) -> active(p(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) p(active(X:S)) -> p(X:S) p(mark(X:S)) -> p(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(f(s(0))) -> MARK(f(p(s(0)))) MARK(f(X:S)) -> ACTIVE(f(mark(X:S))) ->->-> Rules: active(f(s(0))) -> mark(f(p(s(0)))) active(f(0)) -> mark(cons(0,f(s(0)))) active(p(s(0))) -> mark(0) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) f(active(X:S)) -> f(X:S) f(mark(X:S)) -> f(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(f(X:S)) -> active(f(mark(X:S))) mark(p(X:S)) -> active(p(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) p(active(X:S)) -> p(X:S) p(mark(X:S)) -> p(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->->Cycle: ->->-> Pairs: MARK(p(X:S)) -> MARK(X:S) MARK(s(X:S)) -> MARK(X:S) ->->-> Rules: active(f(s(0))) -> mark(f(p(s(0)))) active(f(0)) -> mark(cons(0,f(s(0)))) active(p(s(0))) -> mark(0) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) f(active(X:S)) -> f(X:S) f(mark(X:S)) -> f(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(f(X:S)) -> active(f(mark(X:S))) mark(p(X:S)) -> active(p(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) p(active(X:S)) -> p(X:S) p(mark(X:S)) -> p(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) The problem is decomposed in 2 subproblems. Problem 1.5.1: Reduction Pairs Processor: -> Pairs: ACTIVE(f(s(0))) -> MARK(f(p(s(0)))) MARK(f(X:S)) -> ACTIVE(f(mark(X:S))) -> Rules: active(f(s(0))) -> mark(f(p(s(0)))) active(f(0)) -> mark(cons(0,f(s(0)))) active(p(s(0))) -> mark(0) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) f(active(X:S)) -> f(X:S) f(mark(X:S)) -> f(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(f(X:S)) -> active(f(mark(X:S))) mark(p(X:S)) -> active(p(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) p(active(X:S)) -> p(X:S) p(mark(X:S)) -> p(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: active(f(s(0))) -> mark(f(p(s(0)))) active(f(0)) -> mark(cons(0,f(s(0)))) active(p(s(0))) -> mark(0) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) f(active(X:S)) -> f(X:S) f(mark(X:S)) -> f(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(f(X:S)) -> active(f(mark(X:S))) mark(p(X:S)) -> active(p(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) p(active(X:S)) -> p(X:S) p(mark(X:S)) -> p(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = X [cons](X1,X2) = 2.X1 [f](X) = X + 2 [mark](X) = X [p](X) = 1 [s](X) = 2 [0] = 0 [fSNonEmpty] = 0 [ACTIVE](X) = 2.X + 2 [CONS](X1,X2) = 0 [F](X) = 0 [MARK](X) = 2.X + 2 [P](X) = 0 [S](X) = 0 Problem 1.5.1: SCC Processor: -> Pairs: MARK(f(X:S)) -> ACTIVE(f(mark(X:S))) -> Rules: active(f(s(0))) -> mark(f(p(s(0)))) active(f(0)) -> mark(cons(0,f(s(0)))) active(p(s(0))) -> mark(0) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) f(active(X:S)) -> f(X:S) f(mark(X:S)) -> f(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(f(X:S)) -> active(f(mark(X:S))) mark(p(X:S)) -> active(p(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) p(active(X:S)) -> p(X:S) p(mark(X:S)) -> p(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.5.2: Subterm Processor: -> Pairs: MARK(p(X:S)) -> MARK(X:S) MARK(s(X:S)) -> MARK(X:S) -> Rules: active(f(s(0))) -> mark(f(p(s(0)))) active(f(0)) -> mark(cons(0,f(s(0)))) active(p(s(0))) -> mark(0) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) f(active(X:S)) -> f(X:S) f(mark(X:S)) -> f(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(f(X:S)) -> active(f(mark(X:S))) mark(p(X:S)) -> active(p(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) p(active(X:S)) -> p(X:S) p(mark(X:S)) -> p(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(MARK) = 1 Problem 1.5.2: SCC Processor: -> Pairs: Empty -> Rules: active(f(s(0))) -> mark(f(p(s(0)))) active(f(0)) -> mark(cons(0,f(s(0)))) active(p(s(0))) -> mark(0) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) f(active(X:S)) -> f(X:S) f(mark(X:S)) -> f(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(f(X:S)) -> active(f(mark(X:S))) mark(p(X:S)) -> active(p(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) p(active(X:S)) -> p(X:S) p(mark(X:S)) -> p(X:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite.