/export/starexec/sandbox2/solver/bin/starexec_run_tct_dci /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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WORST_CASE(?,O(n^2))
* Step 1: NaturalMI.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
            -(x,0()) -> x
            -(0(),y) -> 0()
            -(s(x),s(y)) -> -(x,y)
        - Signature:
            {+/2,-/2} / {0/0,s/1}
        - Obligation:
            innermost derivational complexity wrt. signature {+,-,0,s}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = NoUArgs, urules = NoURules, selector = Just any strict-rules}
    + Details:
        We apply a matrix interpretation of kind triangular matrix interpretation:
        
        Following symbols are considered usable:
          all
        TcT has computed the following interpretation:
          p(+) = [1] x1 + [1] x2 + [11]
          p(-) = [1] x1 + [1] x2 + [4] 
          p(0) = [13]                  
          p(s) = [1] x1 + [4]          
        
        Following rules are strictly oriented:
            +(0(),y) = [1] y + [24]        
                     > [1] y + [0]         
                     = y                   
        
            -(x,0()) = [1] x + [17]        
                     > [1] x + [0]         
                     = x                   
        
            -(0(),y) = [1] y + [17]        
                     > [13]                
                     = 0()                 
        
        -(s(x),s(y)) = [1] x + [1] y + [12]
                     > [1] x + [1] y + [4] 
                     = -(x,y)              
        
        
        Following rules are (at-least) weakly oriented:
        +(s(x),y) =  [1] x + [1] y + [15]
                  >= [1] x + [1] y + [15]
                  =  s(+(x,y))           
        
* Step 2: NaturalMI.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            +(s(x),y) -> s(+(x,y))
        - Weak TRS:
            +(0(),y) -> y
            -(x,0()) -> x
            -(0(),y) -> 0()
            -(s(x),s(y)) -> -(x,y)
        - Signature:
            {+/2,-/2} / {0/0,s/1}
        - Obligation:
            innermost derivational complexity wrt. signature {+,-,0,s}
    + Applied Processor:
        NaturalMI {miDimension = 2, miDegree = 2, miKind = Algebraic, uargs = NoUArgs, urules = NoURules, selector = Just any strict-rules}
    + Details:
        We apply a matrix interpretation of kind triangular matrix interpretation:
        
        Following symbols are considered usable:
          all
        TcT has computed the following interpretation:
          p(+) = [1 1] x1 + [1 0] x2 + [0]
                 [0 1]      [0 1]      [0]
          p(-) = [1 0] x1 + [1 0] x2 + [2]
                 [0 1]      [0 0]      [0]
          p(0) = [2]                      
                 [4]                      
          p(s) = [1 0] x1 + [4]           
                 [0 1]      [4]           
        
        Following rules are strictly oriented:
        +(s(x),y) = [1 1] x + [1 0] y + [8]
                    [0 1]     [0 1]     [4]
                  > [1 1] x + [1 0] y + [4]
                    [0 1]     [0 1]     [4]
                  = s(+(x,y))              
        
        
        Following rules are (at-least) weakly oriented:
            +(0(),y) =  [1 0] y + [6]           
                        [0 1]     [4]           
                     >= [1 0] y + [0]           
                        [0 1]     [0]           
                     =  y                       
        
            -(x,0()) =  [1 0] x + [4]           
                        [0 1]     [0]           
                     >= [1 0] x + [0]           
                        [0 1]     [0]           
                     =  x                       
        
            -(0(),y) =  [1 0] y + [4]           
                        [0 0]     [4]           
                     >= [2]                     
                        [4]                     
                     =  0()                     
        
        -(s(x),s(y)) =  [1 0] x + [1 0] y + [10]
                        [0 1]     [0 0]     [4] 
                     >= [1 0] x + [1 0] y + [2] 
                        [0 1]     [0 0]     [0] 
                     =  -(x,y)                  
        
* Step 3: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
            -(x,0()) -> x
            -(0(),y) -> 0()
            -(s(x),s(y)) -> -(x,y)
        - Signature:
            {+/2,-/2} / {0/0,s/1}
        - Obligation:
            innermost derivational complexity wrt. signature {+,-,0,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^2))