/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),O(n^1))
* Step 1: Sum.  WORST_CASE(Omega(n^1),O(n^1))
    + Considered Problem:
        - Strict TRS:
            goal(xs,ys) -> revapp(xs,ys)
            revapp(Cons(x,xs),rest) -> revapp(xs,Cons(x,rest))
            revapp(Nil(),rest) -> rest
        - Signature:
            {goal/2,revapp/2} / {Cons/2,Nil/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {goal,revapp} and constructors {Cons,Nil}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:1: Sum.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            goal(xs,ys) -> revapp(xs,ys)
            revapp(Cons(x,xs),rest) -> revapp(xs,Cons(x,rest))
            revapp(Nil(),rest) -> rest
        - Signature:
            {goal/2,revapp/2} / {Cons/2,Nil/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {goal,revapp} and constructors {Cons,Nil}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
*** Step 1.a:1.a:1: Ara.  MAYBE
    + Considered Problem:
        - Strict TRS:
            goal(xs,ys) -> revapp(xs,ys)
            revapp(Cons(x,xs),rest) -> revapp(xs,Cons(x,rest))
            revapp(Nil(),rest) -> rest
        - Signature:
            {goal/2,revapp/2} / {Cons/2,Nil/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {goal,revapp} and constructors {Cons,Nil}
    + Applied Processor:
        Ara {minDegree = 1, maxDegree = 3, araTimeout = 15, araRuleShifting = Just 1, isBestCase = True, mkCompletelyDefined = False, verboseOutput = False}
    + Details:
        Signatures used:
        ----------------
          F (TrsFun "Cons") :: ["A"(0) x "A"(1)] -(1)-> "A"(1)
          F (TrsFun "Cons") :: ["A"(0) x "A"(0)] -(0)-> "A"(0)
          F (TrsFun "Nil") :: [] -(0)-> "A"(1)
          F (TrsFun "goal") :: ["A"(1) x "A"(0)] -(1)-> "A"(0)
          F (TrsFun "main") :: ["A"(1) x "A"(0)] -(1)-> "A"(0)
          F (TrsFun "revapp") :: ["A"(1) 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,ys) -> revapp(xs,ys)
          revapp(Cons(x,xs),rest) -> revapp(xs,Cons(x,rest))
          revapp(Nil(),rest) -> rest
          main(x1,x2) -> revapp(x1,x2)
        2. Weak:
          
        

*** Step 1.a:1.b:1: DecreasingLoops.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            goal(xs,ys) -> revapp(xs,ys)
            revapp(Cons(x,xs),rest) -> revapp(xs,Cons(x,rest))
            revapp(Nil(),rest) -> rest
        - Signature:
            {goal/2,revapp/2} / {Cons/2,Nil/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {goal,revapp} and constructors {Cons,Nil}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          revapp(y,z){y -> Cons(x,y)} =
            revapp(Cons(x,y),z) ->^+ revapp(y,Cons(x,z))
              = C[revapp(y,Cons(x,z)) = revapp(y,z){z -> Cons(x,z)}]

** Step 1.b:1: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            goal(xs,ys) -> revapp(xs,ys)
            revapp(Cons(x,xs),rest) -> revapp(xs,Cons(x,rest))
            revapp(Nil(),rest) -> rest
        - Signature:
            {goal/2,revapp/2} / {Cons/2,Nil/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {goal,revapp} and constructors {Cons,Nil}
    + 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:
          none
        
        Following symbols are considered usable:
          {goal,revapp}
        TcT has computed the following interpretation:
            p(Cons) = [1] x2 + [0]         
             p(Nil) = [0]                  
            p(goal) = [1] x1 + [8] x2 + [2]
          p(revapp) = [8] x2 + [0]         
        
        Following rules are strictly oriented:
        goal(xs,ys) = [1] xs + [8] ys + [2]
                    > [8] ys + [0]         
                    = revapp(xs,ys)        
        
        
        Following rules are (at-least) weakly oriented:
        revapp(Cons(x,xs),rest) =  [8] rest + [0]         
                                >= [8] rest + [0]         
                                =  revapp(xs,Cons(x,rest))
        
             revapp(Nil(),rest) =  [8] rest + [0]         
                                >= [1] rest + [0]         
                                =  rest                   
        
** Step 1.b:2: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            revapp(Cons(x,xs),rest) -> revapp(xs,Cons(x,rest))
            revapp(Nil(),rest) -> rest
        - Weak TRS:
            goal(xs,ys) -> revapp(xs,ys)
        - Signature:
            {goal/2,revapp/2} / {Cons/2,Nil/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {goal,revapp} and constructors {Cons,Nil}
    + 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:
          none
        
        Following symbols are considered usable:
          {goal,revapp}
        TcT has computed the following interpretation:
            p(Cons) = [0]                  
             p(Nil) = [0]                  
            p(goal) = [1] x1 + [8] x2 + [4]
          p(revapp) = [2] x2 + [4]         
        
        Following rules are strictly oriented:
        revapp(Nil(),rest) = [2] rest + [4]
                           > [1] rest + [0]
                           = rest          
        
        
        Following rules are (at-least) weakly oriented:
                    goal(xs,ys) =  [1] xs + [8] ys + [4]  
                                >= [2] ys + [4]           
                                =  revapp(xs,ys)          
        
        revapp(Cons(x,xs),rest) =  [2] rest + [4]         
                                >= [4]                    
                                =  revapp(xs,Cons(x,rest))
        
** Step 1.b:3: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            revapp(Cons(x,xs),rest) -> revapp(xs,Cons(x,rest))
        - Weak TRS:
            goal(xs,ys) -> revapp(xs,ys)
            revapp(Nil(),rest) -> rest
        - Signature:
            {goal/2,revapp/2} / {Cons/2,Nil/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {goal,revapp} and constructors {Cons,Nil}
    + 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:
          none
        
        Following symbols are considered usable:
          {goal,revapp}
        TcT has computed the following interpretation:
            p(Cons) = [1] x2 + [2]          
             p(Nil) = [0]                   
            p(goal) = [13] x1 + [8] x2 + [8]
          p(revapp) = [12] x1 + [8] x2 + [0]
        
        Following rules are strictly oriented:
        revapp(Cons(x,xs),rest) = [8] rest + [12] xs + [24]
                                > [8] rest + [12] xs + [16]
                                = revapp(xs,Cons(x,rest))  
        
        
        Following rules are (at-least) weakly oriented:
               goal(xs,ys) =  [13] xs + [8] ys + [8]
                           >= [12] xs + [8] ys + [0]
                           =  revapp(xs,ys)         
        
        revapp(Nil(),rest) =  [8] rest + [0]        
                           >= [1] rest + [0]        
                           =  rest                  
        
** Step 1.b:4: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            goal(xs,ys) -> revapp(xs,ys)
            revapp(Cons(x,xs),rest) -> revapp(xs,Cons(x,rest))
            revapp(Nil(),rest) -> rest
        - Signature:
            {goal/2,revapp/2} / {Cons/2,Nil/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {goal,revapp} and constructors {Cons,Nil}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

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