/export/starexec/sandbox2/solver/bin/starexec_run_tct_rc /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:
            +(x,0()) -> x
            +(x,s(y)) -> s(+(x,y))
            +(s(x),y) -> s(+(x,y))
            double(x) -> +(x,x)
            double(0()) -> 0()
            double(s(x)) -> s(s(double(x)))
        - Signature:
            {+/2,double/1} / {0/0,s/1}
        - Obligation:
             runtime complexity wrt. defined symbols {+,double} 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:
            +(x,0()) -> x
            +(x,s(y)) -> s(+(x,y))
            +(s(x),y) -> s(+(x,y))
            double(x) -> +(x,x)
            double(0()) -> 0()
            double(s(x)) -> s(s(double(x)))
        - Signature:
            {+/2,double/1} / {0/0,s/1}
        - Obligation:
             runtime complexity wrt. defined symbols {+,double} 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:
            +(x,0()) -> x
            +(x,s(y)) -> s(+(x,y))
            +(s(x),y) -> s(+(x,y))
            double(x) -> +(x,x)
            double(0()) -> 0()
            double(s(x)) -> s(s(double(x)))
        - Signature:
            {+/2,double/1} / {0/0,s/1}
        - Obligation:
             runtime complexity wrt. defined symbols {+,double} and constructors {0,s}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          +(x,y){y -> s(y)} =
            +(x,s(y)) ->^+ s(+(x,y))
              = C[+(x,y) = +(x,y){}]

** Step 1.b:1: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            +(x,0()) -> x
            +(x,s(y)) -> s(+(x,y))
            +(s(x),y) -> s(+(x,y))
            double(x) -> +(x,x)
            double(0()) -> 0()
            double(s(x)) -> s(s(double(x)))
        - Signature:
            {+/2,double/1} / {0/0,s/1}
        - Obligation:
             runtime complexity wrt. defined symbols {+,double} and constructors {0,s}
    + 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(s) = {1}
        
        Following symbols are considered usable:
          all
        TcT has computed the following interpretation:
               p(+) = [2] x1 + [1] x2 + [0]
               p(0) = [4]                  
          p(double) = [3] x1 + [3]         
               p(s) = [1] x1 + [8]         
        
        Following rules are strictly oriented:
            +(x,0()) = [2] x + [4]         
                     > [1] x + [0]         
                     = x                   
        
           +(s(x),y) = [2] x + [1] y + [16]
                     > [2] x + [1] y + [8] 
                     = s(+(x,y))           
        
           double(x) = [3] x + [3]         
                     > [3] x + [0]         
                     = +(x,x)              
        
         double(0()) = [15]                
                     > [4]                 
                     = 0()                 
        
        double(s(x)) = [3] x + [27]        
                     > [3] x + [19]        
                     = s(s(double(x)))     
        
        
        Following rules are (at-least) weakly oriented:
        +(x,s(y)) =  [2] x + [1] y + [8]
                  >= [2] x + [1] y + [8]
                  =  s(+(x,y))          
        
** Step 1.b:2: NaturalPI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            +(x,s(y)) -> s(+(x,y))
        - Weak TRS:
            +(x,0()) -> x
            +(s(x),y) -> s(+(x,y))
            double(x) -> +(x,x)
            double(0()) -> 0()
            double(s(x)) -> s(s(double(x)))
        - Signature:
            {+/2,double/1} / {0/0,s/1}
        - Obligation:
             runtime complexity wrt. defined symbols {+,double} and constructors {0,s}
    + Applied Processor:
        NaturalPI {shape = Linear, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules}
    + Details:
        We apply a polynomial interpretation of kind constructor-based(linear):
        The following argument positions are considered usable:
          uargs(s) = {1}
        
        Following symbols are considered usable:
          all
        TcT has computed the following interpretation:
               p(+) = 4 + x1 + 5*x2
               p(0) = 1            
          p(double) = 4 + 6*x1     
               p(s) = 4 + x1       
        
        Following rules are strictly oriented:
        +(x,s(y)) = 24 + x + 5*y
                  > 8 + x + 5*y 
                  = s(+(x,y))   
        
        
        Following rules are (at-least) weakly oriented:
            +(x,0()) =  9 + x          
                     >= x              
                     =  x              
        
           +(s(x),y) =  8 + x + 5*y    
                     >= 8 + x + 5*y    
                     =  s(+(x,y))      
        
           double(x) =  4 + 6*x        
                     >= 4 + 6*x        
                     =  +(x,x)         
        
         double(0()) =  10             
                     >= 1              
                     =  0()            
        
        double(s(x)) =  28 + 6*x       
                     >= 12 + 6*x       
                     =  s(s(double(x)))
        
** Step 1.b:3: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            +(x,0()) -> x
            +(x,s(y)) -> s(+(x,y))
            +(s(x),y) -> s(+(x,y))
            double(x) -> +(x,x)
            double(0()) -> 0()
            double(s(x)) -> s(s(double(x)))
        - Signature:
            {+/2,double/1} / {0/0,s/1}
        - Obligation:
             runtime complexity wrt. defined symbols {+,double} and constructors {0,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

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