/export/starexec/sandbox/solver/bin/starexec_run_default /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- YES Problem 1: (VAR X X1 X2) (RULES active(f(s(0))) -> mark(f(p(s(0)))) active(f(0)) -> mark(cons(0,f(s(0)))) active(p(s(X))) -> mark(X) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) f(active(X)) -> f(X) f(mark(X)) -> f(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(f(X)) -> active(f(mark(X))) mark(p(X)) -> active(p(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) p(active(X)) -> p(X) p(mark(X)) -> p(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) ) 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(X))) -> MARK(X) CONS(active(X1),X2) -> CONS(X1,X2) CONS(mark(X1),X2) -> CONS(X1,X2) CONS(X1,active(X2)) -> CONS(X1,X2) CONS(X1,mark(X2)) -> CONS(X1,X2) F(active(X)) -> F(X) F(mark(X)) -> F(X) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> CONS(mark(X1),X2) MARK(cons(X1,X2)) -> MARK(X1) MARK(f(X)) -> ACTIVE(f(mark(X))) MARK(f(X)) -> F(mark(X)) MARK(f(X)) -> MARK(X) MARK(p(X)) -> ACTIVE(p(mark(X))) MARK(p(X)) -> MARK(X) MARK(p(X)) -> P(mark(X)) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(s(X)) -> S(mark(X)) P(active(X)) -> P(X) P(mark(X)) -> P(X) S(active(X)) -> S(X) S(mark(X)) -> S(X) -> Rules: active(f(s(0))) -> mark(f(p(s(0)))) active(f(0)) -> mark(cons(0,f(s(0)))) active(p(s(X))) -> mark(X) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) f(active(X)) -> f(X) f(mark(X)) -> f(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(f(X)) -> active(f(mark(X))) mark(p(X)) -> active(p(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) p(active(X)) -> p(X) p(mark(X)) -> p(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) 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(X))) -> MARK(X) CONS(active(X1),X2) -> CONS(X1,X2) CONS(mark(X1),X2) -> CONS(X1,X2) CONS(X1,active(X2)) -> CONS(X1,X2) CONS(X1,mark(X2)) -> CONS(X1,X2) F(active(X)) -> F(X) F(mark(X)) -> F(X) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> CONS(mark(X1),X2) MARK(cons(X1,X2)) -> MARK(X1) MARK(f(X)) -> ACTIVE(f(mark(X))) MARK(f(X)) -> F(mark(X)) MARK(f(X)) -> MARK(X) MARK(p(X)) -> ACTIVE(p(mark(X))) MARK(p(X)) -> MARK(X) MARK(p(X)) -> P(mark(X)) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(s(X)) -> S(mark(X)) P(active(X)) -> P(X) P(mark(X)) -> P(X) S(active(X)) -> S(X) S(mark(X)) -> S(X) -> Rules: active(f(s(0))) -> mark(f(p(s(0)))) active(f(0)) -> mark(cons(0,f(s(0)))) active(p(s(X))) -> mark(X) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) f(active(X)) -> f(X) f(mark(X)) -> f(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(f(X)) -> active(f(mark(X))) mark(p(X)) -> active(p(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) p(active(X)) -> p(X) p(mark(X)) -> p(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: S(active(X)) -> S(X) S(mark(X)) -> S(X) ->->-> Rules: active(f(s(0))) -> mark(f(p(s(0)))) active(f(0)) -> mark(cons(0,f(s(0)))) active(p(s(X))) -> mark(X) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) f(active(X)) -> f(X) f(mark(X)) -> f(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(f(X)) -> active(f(mark(X))) mark(p(X)) -> active(p(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) p(active(X)) -> p(X) p(mark(X)) -> p(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->->Cycle: ->->-> Pairs: P(active(X)) -> P(X) P(mark(X)) -> P(X) ->->-> Rules: active(f(s(0))) -> mark(f(p(s(0)))) active(f(0)) -> mark(cons(0,f(s(0)))) active(p(s(X))) -> mark(X) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) f(active(X)) -> f(X) f(mark(X)) -> f(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(f(X)) -> active(f(mark(X))) mark(p(X)) -> active(p(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) p(active(X)) -> p(X) p(mark(X)) -> p(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->->Cycle: ->->-> Pairs: F(active(X)) -> F(X) F(mark(X)) -> F(X) ->->-> Rules: active(f(s(0))) -> mark(f(p(s(0)))) active(f(0)) -> mark(cons(0,f(s(0)))) active(p(s(X))) -> mark(X) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) f(active(X)) -> f(X) f(mark(X)) -> f(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(f(X)) -> active(f(mark(X))) mark(p(X)) -> active(p(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) p(active(X)) -> p(X) p(mark(X)) -> p(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->->Cycle: ->->-> Pairs: CONS(active(X1),X2) -> CONS(X1,X2) CONS(mark(X1),X2) -> CONS(X1,X2) CONS(X1,active(X2)) -> CONS(X1,X2) CONS(X1,mark(X2)) -> CONS(X1,X2) ->->-> Rules: active(f(s(0))) -> mark(f(p(s(0)))) active(f(0)) -> mark(cons(0,f(s(0)))) active(p(s(X))) -> mark(X) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) f(active(X)) -> f(X) f(mark(X)) -> f(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(f(X)) -> active(f(mark(X))) mark(p(X)) -> active(p(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) p(active(X)) -> p(X) p(mark(X)) -> p(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->->Cycle: ->->-> Pairs: ACTIVE(f(s(0))) -> MARK(f(p(s(0)))) ACTIVE(f(0)) -> MARK(cons(0,f(s(0)))) ACTIVE(p(s(X))) -> MARK(X) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> MARK(X1) MARK(f(X)) -> ACTIVE(f(mark(X))) MARK(f(X)) -> MARK(X) MARK(p(X)) -> ACTIVE(p(mark(X))) MARK(p(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) ->->-> Rules: active(f(s(0))) -> mark(f(p(s(0)))) active(f(0)) -> mark(cons(0,f(s(0)))) active(p(s(X))) -> mark(X) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) f(active(X)) -> f(X) f(mark(X)) -> f(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(f(X)) -> active(f(mark(X))) mark(p(X)) -> active(p(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) p(active(X)) -> p(X) p(mark(X)) -> p(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) The problem is decomposed in 5 subproblems. Problem 1.1: Subterm Processor: -> Pairs: S(active(X)) -> S(X) S(mark(X)) -> S(X) -> Rules: active(f(s(0))) -> mark(f(p(s(0)))) active(f(0)) -> mark(cons(0,f(s(0)))) active(p(s(X))) -> mark(X) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) f(active(X)) -> f(X) f(mark(X)) -> f(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(f(X)) -> active(f(mark(X))) mark(p(X)) -> active(p(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) p(active(X)) -> p(X) p(mark(X)) -> p(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->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(X))) -> mark(X) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) f(active(X)) -> f(X) f(mark(X)) -> f(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(f(X)) -> active(f(mark(X))) mark(p(X)) -> active(p(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) p(active(X)) -> p(X) p(mark(X)) -> p(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.2: Subterm Processor: -> Pairs: P(active(X)) -> P(X) P(mark(X)) -> P(X) -> Rules: active(f(s(0))) -> mark(f(p(s(0)))) active(f(0)) -> mark(cons(0,f(s(0)))) active(p(s(X))) -> mark(X) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) f(active(X)) -> f(X) f(mark(X)) -> f(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(f(X)) -> active(f(mark(X))) mark(p(X)) -> active(p(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) p(active(X)) -> p(X) p(mark(X)) -> p(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->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(X))) -> mark(X) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) f(active(X)) -> f(X) f(mark(X)) -> f(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(f(X)) -> active(f(mark(X))) mark(p(X)) -> active(p(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) p(active(X)) -> p(X) p(mark(X)) -> p(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.3: Subterm Processor: -> Pairs: F(active(X)) -> F(X) F(mark(X)) -> F(X) -> Rules: active(f(s(0))) -> mark(f(p(s(0)))) active(f(0)) -> mark(cons(0,f(s(0)))) active(p(s(X))) -> mark(X) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) f(active(X)) -> f(X) f(mark(X)) -> f(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(f(X)) -> active(f(mark(X))) mark(p(X)) -> active(p(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) p(active(X)) -> p(X) p(mark(X)) -> p(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->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(X))) -> mark(X) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) f(active(X)) -> f(X) f(mark(X)) -> f(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(f(X)) -> active(f(mark(X))) mark(p(X)) -> active(p(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) p(active(X)) -> p(X) p(mark(X)) -> p(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.4: Subterm Processor: -> Pairs: CONS(active(X1),X2) -> CONS(X1,X2) CONS(mark(X1),X2) -> CONS(X1,X2) CONS(X1,active(X2)) -> CONS(X1,X2) CONS(X1,mark(X2)) -> CONS(X1,X2) -> Rules: active(f(s(0))) -> mark(f(p(s(0)))) active(f(0)) -> mark(cons(0,f(s(0)))) active(p(s(X))) -> mark(X) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) f(active(X)) -> f(X) f(mark(X)) -> f(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(f(X)) -> active(f(mark(X))) mark(p(X)) -> active(p(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) p(active(X)) -> p(X) p(mark(X)) -> p(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(CONS) = 1 Problem 1.4: SCC Processor: -> Pairs: CONS(X1,active(X2)) -> CONS(X1,X2) CONS(X1,mark(X2)) -> CONS(X1,X2) -> Rules: active(f(s(0))) -> mark(f(p(s(0)))) active(f(0)) -> mark(cons(0,f(s(0)))) active(p(s(X))) -> mark(X) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) f(active(X)) -> f(X) f(mark(X)) -> f(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(f(X)) -> active(f(mark(X))) mark(p(X)) -> active(p(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) p(active(X)) -> p(X) p(mark(X)) -> p(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: CONS(X1,active(X2)) -> CONS(X1,X2) CONS(X1,mark(X2)) -> CONS(X1,X2) ->->-> Rules: active(f(s(0))) -> mark(f(p(s(0)))) active(f(0)) -> mark(cons(0,f(s(0)))) active(p(s(X))) -> mark(X) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) f(active(X)) -> f(X) f(mark(X)) -> f(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(f(X)) -> active(f(mark(X))) mark(p(X)) -> active(p(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) p(active(X)) -> p(X) p(mark(X)) -> p(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.4: Subterm Processor: -> Pairs: CONS(X1,active(X2)) -> CONS(X1,X2) CONS(X1,mark(X2)) -> CONS(X1,X2) -> Rules: active(f(s(0))) -> mark(f(p(s(0)))) active(f(0)) -> mark(cons(0,f(s(0)))) active(p(s(X))) -> mark(X) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) f(active(X)) -> f(X) f(mark(X)) -> f(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(f(X)) -> active(f(mark(X))) mark(p(X)) -> active(p(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) p(active(X)) -> p(X) p(mark(X)) -> p(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->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(X))) -> mark(X) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) f(active(X)) -> f(X) f(mark(X)) -> f(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(f(X)) -> active(f(mark(X))) mark(p(X)) -> active(p(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) p(active(X)) -> p(X) p(mark(X)) -> p(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.5: Reduction Pair Processor: -> Pairs: ACTIVE(f(s(0))) -> MARK(f(p(s(0)))) ACTIVE(f(0)) -> MARK(cons(0,f(s(0)))) ACTIVE(p(s(X))) -> MARK(X) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> MARK(X1) MARK(f(X)) -> ACTIVE(f(mark(X))) MARK(f(X)) -> MARK(X) MARK(p(X)) -> ACTIVE(p(mark(X))) MARK(p(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: active(f(s(0))) -> mark(f(p(s(0)))) active(f(0)) -> mark(cons(0,f(s(0)))) active(p(s(X))) -> mark(X) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) f(active(X)) -> f(X) f(mark(X)) -> f(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(f(X)) -> active(f(mark(X))) mark(p(X)) -> active(p(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) p(active(X)) -> p(X) p(mark(X)) -> p(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: active(f(s(0))) -> mark(f(p(s(0)))) active(f(0)) -> mark(cons(0,f(s(0)))) active(p(s(X))) -> mark(X) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) f(active(X)) -> f(X) f(mark(X)) -> f(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(f(X)) -> active(f(mark(X))) mark(p(X)) -> active(p(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) p(active(X)) -> p(X) p(mark(X)) -> p(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->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] = 2 [ACTIVE](X) = 2.X + 1 [MARK](X) = 2.X + 1 Problem 1.5: SCC Processor: -> Pairs: ACTIVE(f(s(0))) -> MARK(f(p(s(0)))) ACTIVE(p(s(X))) -> MARK(X) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> MARK(X1) MARK(f(X)) -> ACTIVE(f(mark(X))) MARK(f(X)) -> MARK(X) MARK(p(X)) -> ACTIVE(p(mark(X))) MARK(p(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: active(f(s(0))) -> mark(f(p(s(0)))) active(f(0)) -> mark(cons(0,f(s(0)))) active(p(s(X))) -> mark(X) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) f(active(X)) -> f(X) f(mark(X)) -> f(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(f(X)) -> active(f(mark(X))) mark(p(X)) -> active(p(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) p(active(X)) -> p(X) p(mark(X)) -> p(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(f(s(0))) -> MARK(f(p(s(0)))) ACTIVE(p(s(X))) -> MARK(X) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> MARK(X1) MARK(f(X)) -> ACTIVE(f(mark(X))) MARK(f(X)) -> MARK(X) MARK(p(X)) -> ACTIVE(p(mark(X))) MARK(p(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) ->->-> Rules: active(f(s(0))) -> mark(f(p(s(0)))) active(f(0)) -> mark(cons(0,f(s(0)))) active(p(s(X))) -> mark(X) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) f(active(X)) -> f(X) f(mark(X)) -> f(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(f(X)) -> active(f(mark(X))) mark(p(X)) -> active(p(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) p(active(X)) -> p(X) p(mark(X)) -> p(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.5: Reduction Pair Processor: -> Pairs: ACTIVE(f(s(0))) -> MARK(f(p(s(0)))) ACTIVE(p(s(X))) -> MARK(X) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> MARK(X1) MARK(f(X)) -> ACTIVE(f(mark(X))) MARK(f(X)) -> MARK(X) MARK(p(X)) -> ACTIVE(p(mark(X))) MARK(p(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: active(f(s(0))) -> mark(f(p(s(0)))) active(f(0)) -> mark(cons(0,f(s(0)))) active(p(s(X))) -> mark(X) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) f(active(X)) -> f(X) f(mark(X)) -> f(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(f(X)) -> active(f(mark(X))) mark(p(X)) -> active(p(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) p(active(X)) -> p(X) p(mark(X)) -> p(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: active(f(s(0))) -> mark(f(p(s(0)))) active(f(0)) -> mark(cons(0,f(s(0)))) active(p(s(X))) -> mark(X) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) f(active(X)) -> f(X) f(mark(X)) -> f(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(f(X)) -> active(f(mark(X))) mark(p(X)) -> active(p(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) p(active(X)) -> p(X) p(mark(X)) -> p(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = X [cons](X1,X2) = X1 + 2 [f](X) = X + 2 [mark](X) = X [p](X) = X [s](X) = X + 2 [0] = 2 [ACTIVE](X) = 2.X + 1 [MARK](X) = 2.X + 1 Problem 1.5: SCC Processor: -> Pairs: ACTIVE(f(s(0))) -> MARK(f(p(s(0)))) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> MARK(X1) MARK(f(X)) -> ACTIVE(f(mark(X))) MARK(f(X)) -> MARK(X) MARK(p(X)) -> ACTIVE(p(mark(X))) MARK(p(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: active(f(s(0))) -> mark(f(p(s(0)))) active(f(0)) -> mark(cons(0,f(s(0)))) active(p(s(X))) -> mark(X) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) f(active(X)) -> f(X) f(mark(X)) -> f(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(f(X)) -> active(f(mark(X))) mark(p(X)) -> active(p(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) p(active(X)) -> p(X) p(mark(X)) -> p(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(f(s(0))) -> MARK(f(p(s(0)))) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> MARK(X1) MARK(f(X)) -> ACTIVE(f(mark(X))) MARK(f(X)) -> MARK(X) MARK(p(X)) -> ACTIVE(p(mark(X))) MARK(p(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) ->->-> Rules: active(f(s(0))) -> mark(f(p(s(0)))) active(f(0)) -> mark(cons(0,f(s(0)))) active(p(s(X))) -> mark(X) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) f(active(X)) -> f(X) f(mark(X)) -> f(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(f(X)) -> active(f(mark(X))) mark(p(X)) -> active(p(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) p(active(X)) -> p(X) p(mark(X)) -> p(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.5: Reduction Pair Processor: -> Pairs: ACTIVE(f(s(0))) -> MARK(f(p(s(0)))) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> MARK(X1) MARK(f(X)) -> ACTIVE(f(mark(X))) MARK(f(X)) -> MARK(X) MARK(p(X)) -> ACTIVE(p(mark(X))) MARK(p(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: active(f(s(0))) -> mark(f(p(s(0)))) active(f(0)) -> mark(cons(0,f(s(0)))) active(p(s(X))) -> mark(X) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) f(active(X)) -> f(X) f(mark(X)) -> f(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(f(X)) -> active(f(mark(X))) mark(p(X)) -> active(p(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) p(active(X)) -> p(X) p(mark(X)) -> p(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: active(f(s(0))) -> mark(f(p(s(0)))) active(f(0)) -> mark(cons(0,f(s(0)))) active(p(s(X))) -> mark(X) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) f(active(X)) -> f(X) f(mark(X)) -> f(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(f(X)) -> active(f(mark(X))) mark(p(X)) -> active(p(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) p(active(X)) -> p(X) p(mark(X)) -> p(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = 2.X [cons](X1,X2) = 0 [f](X) = 1 [mark](X) = 2 [p](X) = 1 [s](X) = 0 [0] = 0 [ACTIVE](X) = 2.X [MARK](X) = 2 Problem 1.5: SCC Processor: -> Pairs: ACTIVE(f(s(0))) -> MARK(f(p(s(0)))) MARK(cons(X1,X2)) -> MARK(X1) MARK(f(X)) -> ACTIVE(f(mark(X))) MARK(f(X)) -> MARK(X) MARK(p(X)) -> ACTIVE(p(mark(X))) MARK(p(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: active(f(s(0))) -> mark(f(p(s(0)))) active(f(0)) -> mark(cons(0,f(s(0)))) active(p(s(X))) -> mark(X) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) f(active(X)) -> f(X) f(mark(X)) -> f(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(f(X)) -> active(f(mark(X))) mark(p(X)) -> active(p(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) p(active(X)) -> p(X) p(mark(X)) -> p(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(f(s(0))) -> MARK(f(p(s(0)))) MARK(cons(X1,X2)) -> MARK(X1) MARK(f(X)) -> ACTIVE(f(mark(X))) MARK(f(X)) -> MARK(X) MARK(p(X)) -> ACTIVE(p(mark(X))) MARK(p(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) ->->-> Rules: active(f(s(0))) -> mark(f(p(s(0)))) active(f(0)) -> mark(cons(0,f(s(0)))) active(p(s(X))) -> mark(X) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) f(active(X)) -> f(X) f(mark(X)) -> f(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(f(X)) -> active(f(mark(X))) mark(p(X)) -> active(p(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) p(active(X)) -> p(X) p(mark(X)) -> p(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.5: Reduction Pair Processor: -> Pairs: ACTIVE(f(s(0))) -> MARK(f(p(s(0)))) MARK(cons(X1,X2)) -> MARK(X1) MARK(f(X)) -> ACTIVE(f(mark(X))) MARK(f(X)) -> MARK(X) MARK(p(X)) -> ACTIVE(p(mark(X))) MARK(p(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: active(f(s(0))) -> mark(f(p(s(0)))) active(f(0)) -> mark(cons(0,f(s(0)))) active(p(s(X))) -> mark(X) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) f(active(X)) -> f(X) f(mark(X)) -> f(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(f(X)) -> active(f(mark(X))) mark(p(X)) -> active(p(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) p(active(X)) -> p(X) p(mark(X)) -> p(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: active(f(s(0))) -> mark(f(p(s(0)))) active(f(0)) -> mark(cons(0,f(s(0)))) active(p(s(X))) -> mark(X) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) f(active(X)) -> f(X) f(mark(X)) -> f(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(f(X)) -> active(f(mark(X))) mark(p(X)) -> active(p(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) p(active(X)) -> p(X) p(mark(X)) -> p(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = X [cons](X1,X2) = 2.X1 + 2 [f](X) = X + 2 [mark](X) = X [p](X) = X [s](X) = 2.X + 2 [0] = 0 [ACTIVE](X) = 2.X + 2 [MARK](X) = 2.X + 2 Problem 1.5: SCC Processor: -> Pairs: ACTIVE(f(s(0))) -> MARK(f(p(s(0)))) MARK(f(X)) -> ACTIVE(f(mark(X))) MARK(f(X)) -> MARK(X) MARK(p(X)) -> ACTIVE(p(mark(X))) MARK(p(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: active(f(s(0))) -> mark(f(p(s(0)))) active(f(0)) -> mark(cons(0,f(s(0)))) active(p(s(X))) -> mark(X) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) f(active(X)) -> f(X) f(mark(X)) -> f(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(f(X)) -> active(f(mark(X))) mark(p(X)) -> active(p(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) p(active(X)) -> p(X) p(mark(X)) -> p(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(f(s(0))) -> MARK(f(p(s(0)))) MARK(f(X)) -> ACTIVE(f(mark(X))) MARK(f(X)) -> MARK(X) MARK(p(X)) -> ACTIVE(p(mark(X))) MARK(p(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) ->->-> Rules: active(f(s(0))) -> mark(f(p(s(0)))) active(f(0)) -> mark(cons(0,f(s(0)))) active(p(s(X))) -> mark(X) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) f(active(X)) -> f(X) f(mark(X)) -> f(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(f(X)) -> active(f(mark(X))) mark(p(X)) -> active(p(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) p(active(X)) -> p(X) p(mark(X)) -> p(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.5: Reduction Pair Processor: -> Pairs: ACTIVE(f(s(0))) -> MARK(f(p(s(0)))) MARK(f(X)) -> ACTIVE(f(mark(X))) MARK(f(X)) -> MARK(X) MARK(p(X)) -> ACTIVE(p(mark(X))) MARK(p(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: active(f(s(0))) -> mark(f(p(s(0)))) active(f(0)) -> mark(cons(0,f(s(0)))) active(p(s(X))) -> mark(X) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) f(active(X)) -> f(X) f(mark(X)) -> f(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(f(X)) -> active(f(mark(X))) mark(p(X)) -> active(p(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) p(active(X)) -> p(X) p(mark(X)) -> p(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: active(f(s(0))) -> mark(f(p(s(0)))) active(f(0)) -> mark(cons(0,f(s(0)))) active(p(s(X))) -> mark(X) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) f(active(X)) -> f(X) f(mark(X)) -> f(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(f(X)) -> active(f(mark(X))) mark(p(X)) -> active(p(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) p(active(X)) -> p(X) p(mark(X)) -> p(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = X [cons](X1,X2) = X2 [f](X) = 2.X + 1 [mark](X) = X [p](X) = X [s](X) = X [0] = 2 [ACTIVE](X) = 2.X + 1 [MARK](X) = 2.X + 1 Problem 1.5: SCC Processor: -> Pairs: ACTIVE(f(s(0))) -> MARK(f(p(s(0)))) MARK(f(X)) -> ACTIVE(f(mark(X))) MARK(p(X)) -> ACTIVE(p(mark(X))) MARK(p(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: active(f(s(0))) -> mark(f(p(s(0)))) active(f(0)) -> mark(cons(0,f(s(0)))) active(p(s(X))) -> mark(X) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) f(active(X)) -> f(X) f(mark(X)) -> f(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(f(X)) -> active(f(mark(X))) mark(p(X)) -> active(p(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) p(active(X)) -> p(X) p(mark(X)) -> p(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(f(s(0))) -> MARK(f(p(s(0)))) MARK(f(X)) -> ACTIVE(f(mark(X))) ->->-> Rules: active(f(s(0))) -> mark(f(p(s(0)))) active(f(0)) -> mark(cons(0,f(s(0)))) active(p(s(X))) -> mark(X) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) f(active(X)) -> f(X) f(mark(X)) -> f(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(f(X)) -> active(f(mark(X))) mark(p(X)) -> active(p(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) p(active(X)) -> p(X) p(mark(X)) -> p(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->->Cycle: ->->-> Pairs: MARK(p(X)) -> MARK(X) MARK(s(X)) -> MARK(X) ->->-> Rules: active(f(s(0))) -> mark(f(p(s(0)))) active(f(0)) -> mark(cons(0,f(s(0)))) active(p(s(X))) -> mark(X) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) f(active(X)) -> f(X) f(mark(X)) -> f(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(f(X)) -> active(f(mark(X))) mark(p(X)) -> active(p(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) p(active(X)) -> p(X) p(mark(X)) -> p(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) The problem is decomposed in 2 subproblems. Problem 1.5.1: Reduction Pair Processor: -> Pairs: ACTIVE(f(s(0))) -> MARK(f(p(s(0)))) MARK(f(X)) -> ACTIVE(f(mark(X))) -> Rules: active(f(s(0))) -> mark(f(p(s(0)))) active(f(0)) -> mark(cons(0,f(s(0)))) active(p(s(X))) -> mark(X) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) f(active(X)) -> f(X) f(mark(X)) -> f(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(f(X)) -> active(f(mark(X))) mark(p(X)) -> active(p(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) p(active(X)) -> p(X) p(mark(X)) -> p(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: active(f(s(0))) -> mark(f(p(s(0)))) active(f(0)) -> mark(cons(0,f(s(0)))) active(p(s(X))) -> mark(X) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) f(active(X)) -> f(X) f(mark(X)) -> f(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(f(X)) -> active(f(mark(X))) mark(p(X)) -> active(p(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) p(active(X)) -> p(X) p(mark(X)) -> p(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Interpretation type: Linear ->Coefficients: All rationals ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = X [cons](X1,X2) = X1 [f](X) = 2.X + 2 [mark](X) = X [p](X) = 1/2.X [s](X) = 2.X + 1 [0] = 1 [ACTIVE](X) = 1/2.X + 1 [MARK](X) = 1/2.X + 1 Problem 1.5.1: SCC Processor: -> Pairs: MARK(f(X)) -> ACTIVE(f(mark(X))) -> Rules: active(f(s(0))) -> mark(f(p(s(0)))) active(f(0)) -> mark(cons(0,f(s(0)))) active(p(s(X))) -> mark(X) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) f(active(X)) -> f(X) f(mark(X)) -> f(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(f(X)) -> active(f(mark(X))) mark(p(X)) -> active(p(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) p(active(X)) -> p(X) p(mark(X)) -> p(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.5.2: Subterm Processor: -> Pairs: MARK(p(X)) -> MARK(X) MARK(s(X)) -> MARK(X) -> Rules: active(f(s(0))) -> mark(f(p(s(0)))) active(f(0)) -> mark(cons(0,f(s(0)))) active(p(s(X))) -> mark(X) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) f(active(X)) -> f(X) f(mark(X)) -> f(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(f(X)) -> active(f(mark(X))) mark(p(X)) -> active(p(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) p(active(X)) -> p(X) p(mark(X)) -> p(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->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(X))) -> mark(X) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) f(active(X)) -> f(X) f(mark(X)) -> f(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(f(X)) -> active(f(mark(X))) mark(p(X)) -> active(p(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) p(active(X)) -> p(X) p(mark(X)) -> p(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: There is no strongly connected component The problem is finite.