/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(Omega(n^1),?)
* Step 1: Sum.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            add(0(),X) -> X
            add(s(X),Y) -> s(add(X,Y))
            dbl(0()) -> 0()
            dbl(s(X)) -> s(s(dbl(X)))
            first(0(),X) -> nil()
            first(s(X),cons(Y,Z)) -> cons(Y,first(X,Z))
            sqr(0()) -> 0()
            sqr(s(X)) -> s(add(sqr(X),dbl(X)))
            terms(N) -> cons(recip(sqr(N)),terms(s(N)))
        - Signature:
            {add/2,dbl/1,first/2,sqr/1,terms/1} / {0/0,cons/2,nil/0,recip/1,s/1}
        - 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 2: Sum.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            add(0(),X) -> X
            add(s(X),Y) -> s(add(X,Y))
            dbl(0()) -> 0()
            dbl(s(X)) -> s(s(dbl(X)))
            first(0(),X) -> nil()
            first(s(X),cons(Y,Z)) -> cons(Y,first(X,Z))
            sqr(0()) -> 0()
            sqr(s(X)) -> s(add(sqr(X),dbl(X)))
            terms(N) -> cons(recip(sqr(N)),terms(s(N)))
        - Signature:
            {add/2,dbl/1,first/2,sqr/1,terms/1} / {0/0,cons/2,nil/0,recip/1,s/1}
        - 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 2.a:1: Ara.  MAYBE
    + Considered Problem:
        - Strict TRS:
            add(0(),X) -> X
            add(s(X),Y) -> s(add(X,Y))
            dbl(0()) -> 0()
            dbl(s(X)) -> s(s(dbl(X)))
            first(0(),X) -> nil()
            first(s(X),cons(Y,Z)) -> cons(Y,first(X,Z))
            sqr(0()) -> 0()
            sqr(s(X)) -> s(add(sqr(X),dbl(X)))
            terms(N) -> cons(recip(sqr(N)),terms(s(N)))
        - Signature:
            {add/2,dbl/1,first/2,sqr/1,terms/1} / {0/0,cons/2,nil/0,recip/1,s/1}
        - 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 "add") :: ["A"(0, 0, 0) x "A"(0, 0, 0)] -(1)-> "A"(0, 0, 0)
          F (TrsFun "cons") :: ["A"(0, 0, 0) x "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") :: ["A"(0, 0, 0)] -(0)-> "A"(0, 0, 0)
          F (TrsFun "s") :: ["A"(1, 2, 1)] -(0)-> "A"(0, 1, 1)
          F (TrsFun "sqr") :: ["A"(0, 0, 0)] -(1)-> "A"(0, 0, 0)
          F (TrsFun "terms") :: ["A"(0, 1, 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(X),Y) -> s(add(X,Y))
          dbl(0()) -> 0()
          dbl(s(X)) -> s(s(dbl(X)))
          first(0(),X) -> nil()
          first(s(X),cons(Y,Z)) -> cons(Y,first(X,Z))
          sqr(0()) -> 0()
          sqr(s(X)) -> s(add(sqr(X),dbl(X)))
          terms(N) -> cons(recip(sqr(N)),terms(s(N)))
          main(x1) -> terms(x1)
        2. Weak:
          
        

** Step 2.b:1: DecreasingLoops.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            add(0(),X) -> X
            add(s(X),Y) -> s(add(X,Y))
            dbl(0()) -> 0()
            dbl(s(X)) -> s(s(dbl(X)))
            first(0(),X) -> nil()
            first(s(X),cons(Y,Z)) -> cons(Y,first(X,Z))
            sqr(0()) -> 0()
            sqr(s(X)) -> s(add(sqr(X),dbl(X)))
            terms(N) -> cons(recip(sqr(N)),terms(s(N)))
        - Signature:
            {add/2,dbl/1,first/2,sqr/1,terms/1} / {0/0,cons/2,nil/0,recip/1,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add,dbl,first,sqr,terms} and constructors {0,cons,nil
            ,recip,s}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          add(x,y){x -> s(x)} =
            add(s(x),y) ->^+ s(add(x,y))
              = C[add(x,y) = add(x,y){}]

WORST_CASE(Omega(n^1),?)