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


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WORST_CASE(?,O(1))
* Step 1: Sum.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict TRS:
            add(0(),X) -> X
            add(s(),Y) -> s()
            dbl(0()) -> 0()
            dbl(s()) -> s()
            first(0(),X) -> nil()
            first(s(),cons(Y)) -> cons(Y)
            sqr(0()) -> 0()
            sqr(s()) -> s()
            terms(N) -> cons(recip(sqr(N)))
        - Signature:
            {add/2,dbl/1,first/2,sqr/1,terms/1} / {0/0,cons/1,nil/0,recip/1,s/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add,dbl,first,sqr,terms} and constructors {0,cons,nil
            ,recip,s}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:1: Sum.  MAYBE
    + Considered Problem:
        - Strict TRS:
            add(0(),X) -> X
            add(s(),Y) -> s()
            dbl(0()) -> 0()
            dbl(s()) -> s()
            first(0(),X) -> nil()
            first(s(),cons(Y)) -> cons(Y)
            sqr(0()) -> 0()
            sqr(s()) -> s()
            terms(N) -> cons(recip(sqr(N)))
        - Signature:
            {add/2,dbl/1,first/2,sqr/1,terms/1} / {0/0,cons/1,nil/0,recip/1,s/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add,dbl,first,sqr,terms} and constructors {0,cons,nil
            ,recip,s}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:2: Ara.  MAYBE
    + Considered Problem:
        - Strict TRS:
            add(0(),X) -> X
            add(s(),Y) -> s()
            dbl(0()) -> 0()
            dbl(s()) -> s()
            first(0(),X) -> nil()
            first(s(),cons(Y)) -> cons(Y)
            sqr(0()) -> 0()
            sqr(s()) -> s()
            terms(N) -> cons(recip(sqr(N)))
        - Signature:
            {add/2,dbl/1,first/2,sqr/1,terms/1} / {0/0,cons/1,nil/0,recip/1,s/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add,dbl,first,sqr,terms} and constructors {0,cons,nil
            ,recip,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 "add") :: ["A"(0, 0, 0) x "A"(0, 0, 0)] -(1)-> "A"(0, 0, 0)
          F (TrsFun "cons") :: ["A"(0, 0, 0)] -(0)-> "A"(0, 0, 0)
          F (TrsFun "dbl") :: ["A"(0, 0, 0)] -(1)-> "A"(0, 0, 0)
          F (TrsFun "first") :: ["A"(0, 0, 0) x "A"(0, 0, 0)] -(1)-> "A"(0, 0, 0)
          F (TrsFun "main") :: ["A"(0, 0, 1)] -(1)-> "A"(0, 0, 0)
          F (TrsFun "nil") :: [] -(0)-> "A"(0, 0, 0)
          F (TrsFun "recip") :: ["A"(0, 0, 0)] -(0)-> "A"(0, 0, 0)
          F (TrsFun "s") :: [] -(0)-> "A"(0, 0, 0)
          F (TrsFun "s") :: [] -(0)-> "A"(0, 0, 1)
          F (TrsFun "sqr") :: ["A"(0, 0, 1)] -(1)-> "A"(0, 0, 0)
          F (TrsFun "terms") :: ["A"(0, 0, 1)] -(1)-> "A"(0, 0, 0)
        
        
        Cost-free Signatures used:
        --------------------------
        
        
        
        Base Constructor Signatures used:
        ---------------------------------
        
        
        
        Following Still Strict Rules were Typed as:
        -------------------------------------------
        1. Strict:
          add(0(),X) -> X
          add(s(),Y) -> s()
          dbl(0()) -> 0()
          dbl(s()) -> s()
          first(0(),X) -> nil()
          first(s(),cons(Y)) -> cons(Y)
          sqr(0()) -> 0()
          sqr(s()) -> s()
          terms(N) -> cons(recip(sqr(N)))
          main(x1) -> terms(x1)
        2. Weak:
          
        

** Step 1.b:1: DependencyPairs.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict TRS:
            add(0(),X) -> X
            add(s(),Y) -> s()
            dbl(0()) -> 0()
            dbl(s()) -> s()
            first(0(),X) -> nil()
            first(s(),cons(Y)) -> cons(Y)
            sqr(0()) -> 0()
            sqr(s()) -> s()
            terms(N) -> cons(recip(sqr(N)))
        - Signature:
            {add/2,dbl/1,first/2,sqr/1,terms/1} / {0/0,cons/1,nil/0,recip/1,s/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add,dbl,first,sqr,terms} and constructors {0,cons,nil
            ,recip,s}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          add#(0(),X) -> c_1()
          add#(s(),Y) -> c_2()
          dbl#(0()) -> c_3()
          dbl#(s()) -> c_4()
          first#(0(),X) -> c_5()
          first#(s(),cons(Y)) -> c_6()
          sqr#(0()) -> c_7()
          sqr#(s()) -> c_8()
          terms#(N) -> c_9(sqr#(N))
        Weak DPs
          
        
        and mark the set of starting terms.
** Step 1.b:2: PredecessorEstimation.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            add#(0(),X) -> c_1()
            add#(s(),Y) -> c_2()
            dbl#(0()) -> c_3()
            dbl#(s()) -> c_4()
            first#(0(),X) -> c_5()
            first#(s(),cons(Y)) -> c_6()
            sqr#(0()) -> c_7()
            sqr#(s()) -> c_8()
            terms#(N) -> c_9(sqr#(N))
        - Weak TRS:
            add(0(),X) -> X
            add(s(),Y) -> s()
            dbl(0()) -> 0()
            dbl(s()) -> s()
            first(0(),X) -> nil()
            first(s(),cons(Y)) -> cons(Y)
            sqr(0()) -> 0()
            sqr(s()) -> s()
            terms(N) -> cons(recip(sqr(N)))
        - Signature:
            {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/0
            ,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
            ,nil,recip,s}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,2,3,4,5,6,7,8}
        by application of
          Pre({1,2,3,4,5,6,7,8}) = {9}.
        Here rules are labelled as follows:
          1: add#(0(),X) -> c_1()
          2: add#(s(),Y) -> c_2()
          3: dbl#(0()) -> c_3()
          4: dbl#(s()) -> c_4()
          5: first#(0(),X) -> c_5()
          6: first#(s(),cons(Y)) -> c_6()
          7: sqr#(0()) -> c_7()
          8: sqr#(s()) -> c_8()
          9: terms#(N) -> c_9(sqr#(N))
** Step 1.b:3: PredecessorEstimation.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            terms#(N) -> c_9(sqr#(N))
        - Weak DPs:
            add#(0(),X) -> c_1()
            add#(s(),Y) -> c_2()
            dbl#(0()) -> c_3()
            dbl#(s()) -> c_4()
            first#(0(),X) -> c_5()
            first#(s(),cons(Y)) -> c_6()
            sqr#(0()) -> c_7()
            sqr#(s()) -> c_8()
        - Weak TRS:
            add(0(),X) -> X
            add(s(),Y) -> s()
            dbl(0()) -> 0()
            dbl(s()) -> s()
            first(0(),X) -> nil()
            first(s(),cons(Y)) -> cons(Y)
            sqr(0()) -> 0()
            sqr(s()) -> s()
            terms(N) -> cons(recip(sqr(N)))
        - Signature:
            {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/0
            ,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
            ,nil,recip,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: terms#(N) -> c_9(sqr#(N))
          2: add#(0(),X) -> c_1()
          3: add#(s(),Y) -> c_2()
          4: dbl#(0()) -> c_3()
          5: dbl#(s()) -> c_4()
          6: first#(0(),X) -> c_5()
          7: first#(s(),cons(Y)) -> c_6()
          8: sqr#(0()) -> c_7()
          9: sqr#(s()) -> c_8()
** Step 1.b:4: RemoveWeakSuffixes.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            add#(0(),X) -> c_1()
            add#(s(),Y) -> c_2()
            dbl#(0()) -> c_3()
            dbl#(s()) -> c_4()
            first#(0(),X) -> c_5()
            first#(s(),cons(Y)) -> c_6()
            sqr#(0()) -> c_7()
            sqr#(s()) -> c_8()
            terms#(N) -> c_9(sqr#(N))
        - Weak TRS:
            add(0(),X) -> X
            add(s(),Y) -> s()
            dbl(0()) -> 0()
            dbl(s()) -> s()
            first(0(),X) -> nil()
            first(s(),cons(Y)) -> cons(Y)
            sqr(0()) -> 0()
            sqr(s()) -> s()
            terms(N) -> cons(recip(sqr(N)))
        - Signature:
            {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/0
            ,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
            ,nil,recip,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:add#(0(),X) -> c_1()
             
          
          2:W:add#(s(),Y) -> c_2()
             
          
          3:W:dbl#(0()) -> c_3()
             
          
          4:W:dbl#(s()) -> c_4()
             
          
          5:W:first#(0(),X) -> c_5()
             
          
          6:W:first#(s(),cons(Y)) -> c_6()
             
          
          7:W:sqr#(0()) -> c_7()
             
          
          8:W:sqr#(s()) -> c_8()
             
          
          9:W:terms#(N) -> c_9(sqr#(N))
             -->_1 sqr#(s()) -> c_8():8
             -->_1 sqr#(0()) -> c_7():7
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          9: terms#(N) -> c_9(sqr#(N))
          8: sqr#(s()) -> c_8()
          7: sqr#(0()) -> c_7()
          6: first#(s(),cons(Y)) -> c_6()
          5: first#(0(),X) -> c_5()
          4: dbl#(s()) -> c_4()
          3: dbl#(0()) -> c_3()
          2: add#(s(),Y) -> c_2()
          1: add#(0(),X) -> c_1()
** Step 1.b:5: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            add(0(),X) -> X
            add(s(),Y) -> s()
            dbl(0()) -> 0()
            dbl(s()) -> s()
            first(0(),X) -> nil()
            first(s(),cons(Y)) -> cons(Y)
            sqr(0()) -> 0()
            sqr(s()) -> s()
            terms(N) -> cons(recip(sqr(N)))
        - Signature:
            {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/0
            ,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
            ,nil,recip,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(1))