/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:
            divp(x,y) -> =(rem(x,y),0())
            prime(0()) -> false()
            prime(s(0())) -> false()
            prime(s(s(x))) -> prime1(s(s(x)),s(x))
            prime1(x,0()) -> false()
            prime1(x,s(0())) -> true()
            prime1(x,s(s(y))) -> and(not(divp(s(s(y)),x)),prime1(x,s(y)))
        - Signature:
            {divp/2,prime/1,prime1/2} / {0/0,=/2,and/2,false/0,not/1,rem/2,s/1,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {divp,prime,prime1} and constructors {0,=,and,false,not
            ,rem,s,true}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:1: Sum.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            divp(x,y) -> =(rem(x,y),0())
            prime(0()) -> false()
            prime(s(0())) -> false()
            prime(s(s(x))) -> prime1(s(s(x)),s(x))
            prime1(x,0()) -> false()
            prime1(x,s(0())) -> true()
            prime1(x,s(s(y))) -> and(not(divp(s(s(y)),x)),prime1(x,s(y)))
        - Signature:
            {divp/2,prime/1,prime1/2} / {0/0,=/2,and/2,false/0,not/1,rem/2,s/1,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {divp,prime,prime1} and constructors {0,=,and,false,not
            ,rem,s,true}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:2: DecreasingLoops.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            divp(x,y) -> =(rem(x,y),0())
            prime(0()) -> false()
            prime(s(0())) -> false()
            prime(s(s(x))) -> prime1(s(s(x)),s(x))
            prime1(x,0()) -> false()
            prime1(x,s(0())) -> true()
            prime1(x,s(s(y))) -> and(not(divp(s(s(y)),x)),prime1(x,s(y)))
        - Signature:
            {divp/2,prime/1,prime1/2} / {0/0,=/2,and/2,false/0,not/1,rem/2,s/1,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {divp,prime,prime1} and constructors {0,=,and,false,not
            ,rem,s,true}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          prime1(x,s(y)){y -> s(y)} =
            prime1(x,s(s(y))) ->^+ and(not(divp(s(s(y)),x)),prime1(x,s(y)))
              = C[prime1(x,s(y)) = prime1(x,s(y)){}]

** Step 1.b:1: Ara.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            divp(x,y) -> =(rem(x,y),0())
            prime(0()) -> false()
            prime(s(0())) -> false()
            prime(s(s(x))) -> prime1(s(s(x)),s(x))
            prime1(x,0()) -> false()
            prime1(x,s(0())) -> true()
            prime1(x,s(s(y))) -> and(not(divp(s(s(y)),x)),prime1(x,s(y)))
        - Signature:
            {divp/2,prime/1,prime1/2} / {0/0,=/2,and/2,false/0,not/1,rem/2,s/1,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {divp,prime,prime1} and constructors {0,=,and,false,not
            ,rem,s,true}
    + 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"(2)
          F (TrsFun "0") :: [] -(0)-> "A"(0)
          F (TrsFun "=") :: ["A"(0) x "A"(0)] -(0)-> "A"(0)
          F (TrsFun "and") :: ["A"(0) x "A"(0)] -(0)-> "A"(0)
          F (TrsFun "divp") :: ["A"(0) x "A"(0)] -(1)-> "A"(0)
          F (TrsFun "false") :: [] -(0)-> "A"(0)
          F (TrsFun "not") :: ["A"(0)] -(0)-> "A"(0)
          F (TrsFun "prime") :: ["A"(2)] -(1)-> "A"(0)
          F (TrsFun "prime1") :: ["A"(0) x "A"(2)] -(1)-> "A"(0)
          F (TrsFun "rem") :: ["A"(0) x "A"(0)] -(0)-> "A"(0)
          F (TrsFun "s") :: ["A"(2)] -(2)-> "A"(2)
          F (TrsFun "s") :: ["A"(0)] -(0)-> "A"(0)
          F (TrsFun "true") :: [] -(0)-> "A"(0)
        
        
        Cost-free Signatures used:
        --------------------------
        
        
        
        Base Constructor Signatures used:
        ---------------------------------
        
        
        
        Following Still Strict Rules were Typed as:
        -------------------------------------------
        1. Strict:
          divp(x,y) -> =(rem(x,y),0())
          prime(0()) -> false()
          prime(s(0())) -> false()
          prime(s(s(x))) -> prime1(s(s(x)),s(x))
          prime1(x,0()) -> false()
          prime1(x,s(0())) -> true()
          prime1(x,s(s(y))) -> and(not(divp(s(s(y)),x)),prime1(x,s(y)))
        2. Weak:
          
        

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