/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(Omega(n^1),O(n^1))
* Step 1: Sum.  WORST_CASE(Omega(n^1),O(n^1))
    + Considered Problem:
        - Strict TRS:
            bits(0()) -> 0()
            bits(s(x)) -> s(bits(half(s(x))))
            half(0()) -> 0()
            half(s(0())) -> 0()
            half(s(s(x))) -> s(half(x))
        - Signature:
            {bits/1,half/1} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {bits,half} and constructors {0,s}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:1: Sum.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            bits(0()) -> 0()
            bits(s(x)) -> s(bits(half(s(x))))
            half(0()) -> 0()
            half(s(0())) -> 0()
            half(s(s(x))) -> s(half(x))
        - Signature:
            {bits/1,half/1} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {bits,half} and constructors {0,s}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:2: DecreasingLoops.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            bits(0()) -> 0()
            bits(s(x)) -> s(bits(half(s(x))))
            half(0()) -> 0()
            half(s(0())) -> 0()
            half(s(s(x))) -> s(half(x))
        - Signature:
            {bits/1,half/1} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {bits,half} and constructors {0,s}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          half(x){x -> s(s(x))} =
            half(s(s(x))) ->^+ s(half(x))
              = C[half(x) = half(x){}]

** Step 1.b:1: Ara.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            bits(0()) -> 0()
            bits(s(x)) -> s(bits(half(s(x))))
            half(0()) -> 0()
            half(s(0())) -> 0()
            half(s(s(x))) -> s(half(x))
        - Signature:
            {bits/1,half/1} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {bits,half} and constructors {0,s}
    + Applied Processor:
        Ara {minDegree = 1, maxDegree = 1, araTimeout = 8, araRuleShifting = Just 1, isBestCase = False, mkCompletelyDefined = False, verboseOutput = False}
    + Details:
        Signatures used:
        ----------------
          F (TrsFun "0") :: [] -(0)-> "A"(5)
          F (TrsFun "0") :: [] -(0)-> "A"(3)
          F (TrsFun "0") :: [] -(0)-> "A"(0)
          F (TrsFun "bits") :: ["A"(5)] -(1)-> "A"(0)
          F (TrsFun "half") :: ["A"(3)] -(1)-> "A"(5)
          F (TrsFun "s") :: ["A"(5)] -(5)-> "A"(5)
          F (TrsFun "s") :: ["A"(3)] -(3)-> "A"(3)
          F (TrsFun "s") :: ["A"(0)] -(0)-> "A"(0)
        
        
        Cost-free Signatures used:
        --------------------------
        
        
        
        Base Constructor Signatures used:
        ---------------------------------
        
        
        
        Following Still Strict Rules were Typed as:
        -------------------------------------------
        1. Strict:
          bits(0()) -> 0()
          bits(s(x)) -> s(bits(half(s(x))))
          half(0()) -> 0()
          half(s(0())) -> 0()
          half(s(s(x))) -> s(half(x))
        2. Weak:
          
        

WORST_CASE(Omega(n^1),O(n^1))