/export/starexec/sandbox2/solver/bin/starexec_run_tct_rci /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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WORST_CASE(?,O(1))
* Step 1: Sum.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict TRS:
            filter(cons(X),0(),M) -> cons(0())
            filter(cons(X),s(N),M) -> cons(X)
            nats(N) -> cons(N)
            sieve(cons(0())) -> cons(0())
            sieve(cons(s(N))) -> cons(s(N))
            zprimes() -> sieve(nats(s(s(0()))))
        - Signature:
            {filter/3,nats/1,sieve/1,zprimes/0} / {0/0,cons/1,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {filter,nats,sieve,zprimes} and constructors {0,cons,s}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:1: Sum.  MAYBE
    + Considered Problem:
        - Strict TRS:
            filter(cons(X),0(),M) -> cons(0())
            filter(cons(X),s(N),M) -> cons(X)
            nats(N) -> cons(N)
            sieve(cons(0())) -> cons(0())
            sieve(cons(s(N))) -> cons(s(N))
            zprimes() -> sieve(nats(s(s(0()))))
        - Signature:
            {filter/3,nats/1,sieve/1,zprimes/0} / {0/0,cons/1,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {filter,nats,sieve,zprimes} and constructors {0,cons,s}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:2: Ara.  MAYBE
    + Considered Problem:
        - Strict TRS:
            filter(cons(X),0(),M) -> cons(0())
            filter(cons(X),s(N),M) -> cons(X)
            nats(N) -> cons(N)
            sieve(cons(0())) -> cons(0())
            sieve(cons(s(N))) -> cons(s(N))
            zprimes() -> sieve(nats(s(s(0()))))
        - Signature:
            {filter/3,nats/1,sieve/1,zprimes/0} / {0/0,cons/1,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {filter,nats,sieve,zprimes} and constructors {0,cons,s}
    + Applied Processor:
        Ara {minDegree = 1, maxDegree = 3, araTimeout = 15, araRuleShifting = Just 1, isBestCase = True, mkCompletelyDefined = False, verboseOutput = False}
    + Details:
        Signatures used:
        ----------------
          F (TrsFun "0") :: [] -(0)-> "A"(0, 0, 0)
          F (TrsFun "0") :: [] -(0)-> "A"(0, 0, 1)
          F (TrsFun "cons") :: ["A"(0, 0, 0)] -(0)-> "A"(0, 0, 0)
          F (TrsFun "cons") :: ["A"(1, 2, 1)] -(0)-> "A"(0, 1, 1)
          F (TrsFun "filter") :: ["A"(0, 0, 0) x "A"(0, 0, 0) x "A"(0, 0, 0)] -(1)-> "A"(0, 1, 1)
          F (TrsFun "main") :: [] -(1)-> "A"(0, 0, 0)
          F (TrsFun "nats") :: ["A"(0, 0, 0)] -(1)-> "A"(0, 0, 0)
          F (TrsFun "s") :: ["A"(0, 0, 0)] -(0)-> "A"(0, 0, 0)
          F (TrsFun "sieve") :: ["A"(0, 0, 0)] -(1)-> "A"(0, 0, 0)
          F (TrsFun "zprimes") :: [] -(1)-> "A"(0, 0, 0)
        
        
        Cost-free Signatures used:
        --------------------------
        
        
        
        Base Constructor Signatures used:
        ---------------------------------
        
        
        
        Following Still Strict Rules were Typed as:
        -------------------------------------------
        1. Strict:
          filter(cons(X),0(),M) -> cons(0())
          filter(cons(X),s(N),M) -> cons(X)
          nats(N) -> cons(N)
          sieve(cons(0())) -> cons(0())
          sieve(cons(s(N))) -> cons(s(N))
          zprimes() -> sieve(nats(s(s(0()))))
          main() -> zprimes()
        2. Weak:
          
        

** Step 1.b:1: DependencyPairs.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict TRS:
            filter(cons(X),0(),M) -> cons(0())
            filter(cons(X),s(N),M) -> cons(X)
            nats(N) -> cons(N)
            sieve(cons(0())) -> cons(0())
            sieve(cons(s(N))) -> cons(s(N))
            zprimes() -> sieve(nats(s(s(0()))))
        - Signature:
            {filter/3,nats/1,sieve/1,zprimes/0} / {0/0,cons/1,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {filter,nats,sieve,zprimes} and constructors {0,cons,s}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          filter#(cons(X),0(),M) -> c_1()
          filter#(cons(X),s(N),M) -> c_2()
          nats#(N) -> c_3()
          sieve#(cons(0())) -> c_4()
          sieve#(cons(s(N))) -> c_5()
          zprimes#() -> c_6(sieve#(nats(s(s(0())))),nats#(s(s(0()))))
        Weak DPs
          
        
        and mark the set of starting terms.
** Step 1.b:2: PredecessorEstimation.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            filter#(cons(X),0(),M) -> c_1()
            filter#(cons(X),s(N),M) -> c_2()
            nats#(N) -> c_3()
            sieve#(cons(0())) -> c_4()
            sieve#(cons(s(N))) -> c_5()
            zprimes#() -> c_6(sieve#(nats(s(s(0())))),nats#(s(s(0()))))
        - Weak TRS:
            filter(cons(X),0(),M) -> cons(0())
            filter(cons(X),s(N),M) -> cons(X)
            nats(N) -> cons(N)
            sieve(cons(0())) -> cons(0())
            sieve(cons(s(N))) -> cons(s(N))
            zprimes() -> sieve(nats(s(s(0()))))
        - Signature:
            {filter/3,nats/1,sieve/1,zprimes/0,filter#/3,nats#/1,sieve#/1,zprimes#/0} / {0/0,cons/1,s/1,c_1/0,c_2/0
            ,c_3/0,c_4/0,c_5/0,c_6/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {filter#,nats#,sieve#,zprimes#} and constructors {0,cons
            ,s}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,2,3,4,5}
        by application of
          Pre({1,2,3,4,5}) = {6}.
        Here rules are labelled as follows:
          1: filter#(cons(X),0(),M) -> c_1()
          2: filter#(cons(X),s(N),M) -> c_2()
          3: nats#(N) -> c_3()
          4: sieve#(cons(0())) -> c_4()
          5: sieve#(cons(s(N))) -> c_5()
          6: zprimes#() -> c_6(sieve#(nats(s(s(0())))),nats#(s(s(0()))))
** Step 1.b:3: PredecessorEstimation.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            zprimes#() -> c_6(sieve#(nats(s(s(0())))),nats#(s(s(0()))))
        - Weak DPs:
            filter#(cons(X),0(),M) -> c_1()
            filter#(cons(X),s(N),M) -> c_2()
            nats#(N) -> c_3()
            sieve#(cons(0())) -> c_4()
            sieve#(cons(s(N))) -> c_5()
        - Weak TRS:
            filter(cons(X),0(),M) -> cons(0())
            filter(cons(X),s(N),M) -> cons(X)
            nats(N) -> cons(N)
            sieve(cons(0())) -> cons(0())
            sieve(cons(s(N))) -> cons(s(N))
            zprimes() -> sieve(nats(s(s(0()))))
        - Signature:
            {filter/3,nats/1,sieve/1,zprimes/0,filter#/3,nats#/1,sieve#/1,zprimes#/0} / {0/0,cons/1,s/1,c_1/0,c_2/0
            ,c_3/0,c_4/0,c_5/0,c_6/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {filter#,nats#,sieve#,zprimes#} and constructors {0,cons
            ,s}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1}
        by application of
          Pre({1}) = {}.
        Here rules are labelled as follows:
          1: zprimes#() -> c_6(sieve#(nats(s(s(0())))),nats#(s(s(0()))))
          2: filter#(cons(X),0(),M) -> c_1()
          3: filter#(cons(X),s(N),M) -> c_2()
          4: nats#(N) -> c_3()
          5: sieve#(cons(0())) -> c_4()
          6: sieve#(cons(s(N))) -> c_5()
** Step 1.b:4: RemoveWeakSuffixes.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            filter#(cons(X),0(),M) -> c_1()
            filter#(cons(X),s(N),M) -> c_2()
            nats#(N) -> c_3()
            sieve#(cons(0())) -> c_4()
            sieve#(cons(s(N))) -> c_5()
            zprimes#() -> c_6(sieve#(nats(s(s(0())))),nats#(s(s(0()))))
        - Weak TRS:
            filter(cons(X),0(),M) -> cons(0())
            filter(cons(X),s(N),M) -> cons(X)
            nats(N) -> cons(N)
            sieve(cons(0())) -> cons(0())
            sieve(cons(s(N))) -> cons(s(N))
            zprimes() -> sieve(nats(s(s(0()))))
        - Signature:
            {filter/3,nats/1,sieve/1,zprimes/0,filter#/3,nats#/1,sieve#/1,zprimes#/0} / {0/0,cons/1,s/1,c_1/0,c_2/0
            ,c_3/0,c_4/0,c_5/0,c_6/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {filter#,nats#,sieve#,zprimes#} and constructors {0,cons
            ,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:filter#(cons(X),0(),M) -> c_1()
             
          
          2:W:filter#(cons(X),s(N),M) -> c_2()
             
          
          3:W:nats#(N) -> c_3()
             
          
          4:W:sieve#(cons(0())) -> c_4()
             
          
          5:W:sieve#(cons(s(N))) -> c_5()
             
          
          6:W:zprimes#() -> c_6(sieve#(nats(s(s(0())))),nats#(s(s(0()))))
             -->_1 sieve#(cons(s(N))) -> c_5():5
             -->_1 sieve#(cons(0())) -> c_4():4
             -->_2 nats#(N) -> c_3():3
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          6: zprimes#() -> c_6(sieve#(nats(s(s(0())))),nats#(s(s(0()))))
          5: sieve#(cons(s(N))) -> c_5()
          4: sieve#(cons(0())) -> c_4()
          3: nats#(N) -> c_3()
          2: filter#(cons(X),s(N),M) -> c_2()
          1: filter#(cons(X),0(),M) -> c_1()
** Step 1.b:5: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            filter(cons(X),0(),M) -> cons(0())
            filter(cons(X),s(N),M) -> cons(X)
            nats(N) -> cons(N)
            sieve(cons(0())) -> cons(0())
            sieve(cons(s(N))) -> cons(s(N))
            zprimes() -> sieve(nats(s(s(0()))))
        - Signature:
            {filter/3,nats/1,sieve/1,zprimes/0,filter#/3,nats#/1,sieve#/1,zprimes#/0} / {0/0,cons/1,s/1,c_1/0,c_2/0
            ,c_3/0,c_4/0,c_5/0,c_6/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {filter#,nats#,sieve#,zprimes#} and constructors {0,cons
            ,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(1))