/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^2))
* Step 1: Sum.  WORST_CASE(Omega(n^1),O(n^2))
    + Considered Problem:
        - Strict TRS:
            f(0()) -> true()
            f(1()) -> false()
            f(s(x)) -> f(x)
            g(x,c(y)) -> g(x,g(s(c(y)),y))
            g(s(x),s(y)) -> if(f(x),s(x),s(y))
            if(false(),x,y) -> y
            if(true(),x,y) -> x
        - Signature:
            {f/1,g/2,if/3} / {0/0,1/0,c/1,false/0,s/1,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f,g,if} and constructors {0,1,c,false,s,true}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:1: Sum.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            f(0()) -> true()
            f(1()) -> false()
            f(s(x)) -> f(x)
            g(x,c(y)) -> g(x,g(s(c(y)),y))
            g(s(x),s(y)) -> if(f(x),s(x),s(y))
            if(false(),x,y) -> y
            if(true(),x,y) -> x
        - Signature:
            {f/1,g/2,if/3} / {0/0,1/0,c/1,false/0,s/1,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f,g,if} and constructors {0,1,c,false,s,true}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
*** Step 1.a:1.a:1: Ara.  MAYBE
    + Considered Problem:
        - Strict TRS:
            f(0()) -> true()
            f(1()) -> false()
            f(s(x)) -> f(x)
            g(x,c(y)) -> g(x,g(s(c(y)),y))
            g(s(x),s(y)) -> if(f(x),s(x),s(y))
            if(false(),x,y) -> y
            if(true(),x,y) -> x
        - Signature:
            {f/1,g/2,if/3} / {0/0,1/0,c/1,false/0,s/1,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f,g,if} and constructors {0,1,c,false,s,true}
    + 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 "1") :: [] -(0)-> "A"(0, 0, 0)
          F (TrsFun "c") :: ["A"(0, 0, 0)] -(0)-> "A"(0, 0, 0)
          F (TrsFun "f") :: ["A"(0, 0, 0)] -(1)-> "A"(0, 0, 0)
          F (TrsFun "false") :: [] -(0)-> "A"(0, 0, 1)
          F (TrsFun "false") :: [] -(0)-> "A"(0, 0, 0)
          F (TrsFun "g") :: ["A"(0, 0, 0) x "A"(0, 0, 0)] -(1)-> "A"(0, 0, 0)
          F (TrsFun "if") :: ["A"(0, 0, 1) x "A"(0, 0, 0) x "A"(0, 0, 0)] -(1)-> "A"(0, 0, 0)
          F (TrsFun "main") :: ["A"(0, 0, 1) x "A"(0, 0, 0) x "A"(0, 0, 0)] -(1)-> "A"(0, 0, 0)
          F (TrsFun "s") :: ["A"(0, 0, 0)] -(0)-> "A"(0, 0, 0)
          F (TrsFun "true") :: [] -(0)-> "A"(0, 0, 1)
          F (TrsFun "true") :: [] -(0)-> "A"(0, 0, 0)
        
        
        Cost-free Signatures used:
        --------------------------
        
        
        
        Base Constructor Signatures used:
        ---------------------------------
        
        
        
        Following Still Strict Rules were Typed as:
        -------------------------------------------
        1. Strict:
          f(0()) -> true()
          f(1()) -> false()
          f(s(x)) -> f(x)
          g(x,c(y)) -> g(x,g(s(c(y)),y))
          g(s(x),s(y)) -> if(f(x),s(x),s(y))
          if(false(),x,y) -> y
          if(true(),x,y) -> x
          main(x1,x2,x3) -> if(x1,x2,x3)
        2. Weak:
          
        

*** Step 1.a:1.b:1: DecreasingLoops.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            f(0()) -> true()
            f(1()) -> false()
            f(s(x)) -> f(x)
            g(x,c(y)) -> g(x,g(s(c(y)),y))
            g(s(x),s(y)) -> if(f(x),s(x),s(y))
            if(false(),x,y) -> y
            if(true(),x,y) -> x
        - Signature:
            {f/1,g/2,if/3} / {0/0,1/0,c/1,false/0,s/1,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f,g,if} and constructors {0,1,c,false,s,true}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          f(x){x -> s(x)} =
            f(s(x)) ->^+ f(x)
              = C[f(x) = f(x){}]

** Step 1.b:1: Ara.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            f(0()) -> true()
            f(1()) -> false()
            f(s(x)) -> f(x)
            g(x,c(y)) -> g(x,g(s(c(y)),y))
            g(s(x),s(y)) -> if(f(x),s(x),s(y))
            if(false(),x,y) -> y
            if(true(),x,y) -> x
        - Signature:
            {f/1,g/2,if/3} / {0/0,1/0,c/1,false/0,s/1,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f,g,if} and constructors {0,1,c,false,s,true}
    + Applied Processor:
        Ara {minDegree = 1, maxDegree = 2, araTimeout = 3, araRuleShifting = Nothing, isBestCase = False, mkCompletelyDefined = False, verboseOutput = False}
    + Details:
        Signatures used:
        ----------------
          F (TrsFun "0") :: [] -(0)-> "A"(3, 0)
          F (TrsFun "1") :: [] -(0)-> "A"(3, 0)
          F (TrsFun "c") :: ["A"(12, 8)] -(8)-> "A"(4, 8)
          F (TrsFun "c") :: ["A"(7, 0)] -(0)-> "A"(7, 0)
          F (TrsFun "f") :: ["A"(3, 0)] -(1)-> "A"(3, 2)
          F (TrsFun "false") :: [] -(0)-> "A"(3, 0)
          F (TrsFun "false") :: [] -(0)-> "A"(5, 4)
          F (TrsFun "g") :: ["A"(7, 1) x "A"(4, 8)] -(0)-> "A"(4, 8)
          F (TrsFun "if") :: ["A"(3, 0) x "A"(4, 8) x "A"(4, 10)] -(1)-> "A"(4, 8)
          F (TrsFun "s") :: ["A"(3, 0)] -(3)-> "A"(3, 0)
          F (TrsFun "s") :: ["A"(7, 0)] -(7)-> "A"(7, 1)
          F (TrsFun "s") :: ["A"(4, 0)] -(4)-> "A"(4, 8)
          F (TrsFun "s") :: ["A"(7, 0)] -(7)-> "A"(7, 9)
          F (TrsFun "s") :: ["A"(4, 0)] -(4)-> "A"(4, 10)
          F (TrsFun "true") :: [] -(0)-> "A"(3, 0)
          F (TrsFun "true") :: [] -(0)-> "A"(4, 4)
        
        
        Cost-free Signatures used:
        --------------------------
        
        
        
        Base Constructor Signatures used:
        ---------------------------------
          "F (TrsFun \"0\")_A" :: [] -(0)-> "A"(1, 0)
          "F (TrsFun \"0\")_A" :: [] -(0)-> "A"(0, 1)
          "F (TrsFun \"1\")_A" :: [] -(0)-> "A"(1, 0)
          "F (TrsFun \"1\")_A" :: [] -(0)-> "A"(0, 1)
          "F (TrsFun \"c\")_A" :: ["A"(1, 0)] -(0)-> "A"(1, 0)
          "F (TrsFun \"c\")_A" :: ["A"(1, 1)] -(1)-> "A"(0, 1)
          "F (TrsFun \"false\")_A" :: [] -(0)-> "A"(1, 0)
          "F (TrsFun \"false\")_A" :: [] -(0)-> "A"(0, 1)
          "F (TrsFun \"s\")_A" :: ["A"(1, 0)] -(1)-> "A"(1, 0)
          "F (TrsFun \"s\")_A" :: ["A"(0, 0)] -(0)-> "A"(0, 1)
          "F (TrsFun \"true\")_A" :: [] -(0)-> "A"(1, 0)
          "F (TrsFun \"true\")_A" :: [] -(0)-> "A"(0, 1)
        
        

WORST_CASE(Omega(n^1),O(n^2))