/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(n^1))
* Step 1: Sum.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            goal(xs) -> ordered(xs)
            notEmpty(Cons(x,xs)) -> True()
            notEmpty(Nil()) -> False()
            ordered(Cons(x,Nil())) -> True()
            ordered(Cons(x',Cons(x,xs))) -> ordered[Ite](<(x',x),Cons(x',Cons(x,xs)))
            ordered(Nil()) -> True()
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            ordered[Ite](False(),xs) -> False()
            ordered[Ite](True(),Cons(x,xs)) -> ordered(xs)
        - Signature:
            {</2,goal/1,notEmpty/1,ordered/1,ordered[Ite]/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<,goal,notEmpty,ordered,ordered[Ite]} and constructors {0
            ,Cons,False,Nil,S,True}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
* Step 2: Ara.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            goal(xs) -> ordered(xs)
            notEmpty(Cons(x,xs)) -> True()
            notEmpty(Nil()) -> False()
            ordered(Cons(x,Nil())) -> True()
            ordered(Cons(x',Cons(x,xs))) -> ordered[Ite](<(x',x),Cons(x',Cons(x,xs)))
            ordered(Nil()) -> True()
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            ordered[Ite](False(),xs) -> False()
            ordered[Ite](True(),Cons(x,xs)) -> ordered(xs)
        - Signature:
            {</2,goal/1,notEmpty/1,ordered/1,ordered[Ite]/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<,goal,notEmpty,ordered,ordered[Ite]} and constructors {0
            ,Cons,False,Nil,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"(0)
          F (TrsFun "<") :: ["A"(0) x "A"(0)] -(0)-> "A"(0)
          F (TrsFun "Cons") :: ["A"(0) x "A"(0)] -(0)-> "A"(0)
          F (TrsFun "False") :: [] -(0)-> "A"(0)
          F (TrsFun "Nil") :: [] -(0)-> "A"(0)
          F (TrsFun "S") :: ["A"(0)] -(0)-> "A"(0)
          F (TrsFun "True") :: [] -(0)-> "A"(0)
          F (TrsFun "goal") :: ["A"(0)] -(2)-> "A"(0)
          F (TrsFun "notEmpty") :: ["A"(0)] -(1)-> "A"(0)
          F (TrsFun "ordered") :: ["A"(0)] -(1)-> "A"(0)
          F (TrsFun "ordered[Ite]") :: ["A"(0) x "A"(0)] -(1)-> "A"(0)
        
        
        Cost-free Signatures used:
        --------------------------
        
        
        
        Base Constructor Signatures used:
        ---------------------------------
        
        
        
        Following Still Strict Rules were Typed as:
        -------------------------------------------
        1. Strict:
          goal(xs) -> ordered(xs)
          notEmpty(Cons(x,xs)) -> True()
          notEmpty(Nil()) -> False()
          ordered(Cons(x,Nil())) -> True()
          ordered(Nil()) -> True()
        2. Weak:
          ordered(Cons(x',Cons(x,xs))) -> ordered[Ite](<(x',x),Cons(x',Cons(x,xs)))
        
* Step 3: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            ordered(Cons(x',Cons(x,xs))) -> ordered[Ite](<(x',x),Cons(x',Cons(x,xs)))
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            goal(xs) -> ordered(xs)
            notEmpty(Cons(x,xs)) -> True()
            notEmpty(Nil()) -> False()
            ordered(Cons(x,Nil())) -> True()
            ordered(Nil()) -> True()
            ordered[Ite](False(),xs) -> False()
            ordered[Ite](True(),Cons(x,xs)) -> ordered(xs)
        - Signature:
            {</2,goal/1,notEmpty/1,ordered/1,ordered[Ite]/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<,goal,notEmpty,ordered,ordered[Ite]} and constructors {0
            ,Cons,False,Nil,S,True}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
    + Details:
        We apply a matrix interpretation of kind constructor based matrix interpretation:
        The following argument positions are considered usable:
          uargs(ordered[Ite]) = {1}
        
        Following symbols are considered usable:
          {<,goal,notEmpty,ordered,ordered[Ite]}
        TcT has computed the following interpretation:
                     p(0) = [4]                  
                     p(<) = [0]                  
                  p(Cons) = [1] x2 + [2]         
                 p(False) = [0]                  
                   p(Nil) = [2]                  
                     p(S) = [0]                  
                  p(True) = [0]                  
                  p(goal) = [4] x1 + [11]        
              p(notEmpty) = [2] x1 + [2]         
               p(ordered) = [4] x1 + [8]         
          p(ordered[Ite]) = [8] x1 + [4] x2 + [0]
        
        Following rules are strictly oriented:
        ordered(Cons(x',Cons(x,xs))) = [4] xs + [24]                            
                                     > [4] xs + [16]                            
                                     = ordered[Ite](<(x',x),Cons(x',Cons(x,xs)))
        
        
        Following rules are (at-least) weakly oriented:
                               <(x,0()) =  [0]          
                                        >= [0]          
                                        =  False()      
        
                            <(0(),S(y)) =  [0]          
                                        >= [0]          
                                        =  True()       
        
                           <(S(x),S(y)) =  [0]          
                                        >= [0]          
                                        =  <(x,y)       
        
                               goal(xs) =  [4] xs + [11]
                                        >= [4] xs + [8] 
                                        =  ordered(xs)  
        
                   notEmpty(Cons(x,xs)) =  [2] xs + [6] 
                                        >= [0]          
                                        =  True()       
        
                        notEmpty(Nil()) =  [6]          
                                        >= [0]          
                                        =  False()      
        
                 ordered(Cons(x,Nil())) =  [24]         
                                        >= [0]          
                                        =  True()       
        
                         ordered(Nil()) =  [16]         
                                        >= [0]          
                                        =  True()       
        
               ordered[Ite](False(),xs) =  [4] xs + [0] 
                                        >= [0]          
                                        =  False()      
        
        ordered[Ite](True(),Cons(x,xs)) =  [4] xs + [8] 
                                        >= [4] xs + [8] 
                                        =  ordered(xs)  
        
* Step 4: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            goal(xs) -> ordered(xs)
            notEmpty(Cons(x,xs)) -> True()
            notEmpty(Nil()) -> False()
            ordered(Cons(x,Nil())) -> True()
            ordered(Cons(x',Cons(x,xs))) -> ordered[Ite](<(x',x),Cons(x',Cons(x,xs)))
            ordered(Nil()) -> True()
            ordered[Ite](False(),xs) -> False()
            ordered[Ite](True(),Cons(x,xs)) -> ordered(xs)
        - Signature:
            {</2,goal/1,notEmpty/1,ordered/1,ordered[Ite]/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<,goal,notEmpty,ordered,ordered[Ite]} and constructors {0
            ,Cons,False,Nil,S,True}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))