/export/starexec/sandbox/solver/bin/starexec_run_tct_dc /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/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) -> +(x,s(y))
            +(s(x),y) -> s(+(x,y))
        - Signature:
            {+/2} / {0/0,s/1}
        - Obligation:
             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 + [8]
          p(0) = [0]                  
          p(s) = [1] x1 + [0]         
        
        Following rules are strictly oriented:
        +(0(),y) = [1] y + [8]
                 > [1] y + [0]
                 = y          
        
        
        Following rules are (at-least) weakly oriented:
        +(s(x),y) =  [1] x + [1] y + [8]
                  >= [1] x + [1] y + [8]
                  =  +(x,s(y))          
        
        +(s(x),y) =  [1] x + [1] y + [8]
                  >= [1] x + [1] y + [8]
                  =  s(+(x,y))          
        
* Step 2: NaturalMI.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            +(s(x),y) -> +(x,s(y))
            +(s(x),y) -> s(+(x,y))
        - Weak TRS:
            +(0(),y) -> y
        - Signature:
            {+/2} / {0/0,s/1}
        - Obligation:
             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 7] x1 + [1 6] x2 + [2]
                 [0 1]      [0 1]      [2]
          p(0) = [6]                      
                 [1]                      
          p(s) = [1 0] x1 + [0]           
                 [0 1]      [1]           
        
        Following rules are strictly oriented:
        +(s(x),y) = [1 7] x + [1 6] y + [9]
                    [0 1]     [0 1]     [3]
                  > [1 7] x + [1 6] y + [8]
                    [0 1]     [0 1]     [3]
                  = +(x,s(y))              
        
        +(s(x),y) = [1 7] x + [1 6] y + [9]
                    [0 1]     [0 1]     [3]
                  > [1 7] x + [1 6] y + [2]
                    [0 1]     [0 1]     [3]
                  = s(+(x,y))              
        
        
        Following rules are (at-least) weakly oriented:
        +(0(),y) =  [1 6] y + [15]
                    [0 1]     [3] 
                 >= [1 0] y + [0] 
                    [0 1]     [0] 
                 =  y             
        
* Step 3: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),y) -> +(x,s(y))
            +(s(x),y) -> s(+(x,y))
        - Signature:
            {+/2} / {0/0,s/1}
        - Obligation:
             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))